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1.1669219.pdf	Spin dynamics in ultrathin film structures with a network of misfit dislocations G. Woltersdorf, B. Heinrich, J. Woltersdorf, and R. Scholz Citation: Journal of Applied Physics 95, 7007 (2004); doi: 10.1063/1.1669219 View online: http://dx.doi.org/10.1063/1.1669219 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/95/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spin transport in Au films: An investigation by spin pumping J. Appl. Phys. 111, 07C512 (2012); 10.1063/1.3676026 Spin dynamics and magnetic anisotropies at the Fe/GaAs(001) interface J. Appl. Phys. 109, 07D337 (2011); 10.1063/1.3556786 Dynamics studies in magnetic bilayer Fe ∕ Au ∕ Fe ( 001 ) structures using network analyzer measurements J. Appl. Phys. 99, 08F303 (2006); 10.1063/1.2173221 Structural and magnetic properties of NiMnSb/InGaAs/InP(001) J. Appl. Phys. 97, 073906 (2005); 10.1063/1.1873036 Using ferromagnetic resonance to measure magnetic moments of ultrathin films (abstract) J. Appl. Phys. 81, 4475 (1997); 10.1063/1.364982 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.11.242.100 On: Wed, 24 Dec 2014 00:49:45Spin dynamics in ultrathin ﬁlm structures with a network of misﬁt dislocations G. Woltersdorfa)and B. Heinrich Simon Fraser University, 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada J. Woltersdorf and R. Scholz Max Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany ~Presented on 8 January 2004 ! Using ferromagnetic resonance ~FMR !and transmission electron microscopy we studied the structural and magnetic properties of lattice mismatched magnetic ultrathin multilayers of thesystem Au/ Fe/Au/Pd/Fe~001!prepared on GaAs ~001!. We observed a correlation between the periodic lattice irregularities due to the misﬁt accommodation processes and the resulting magneticproperties of the multilayer system: In samples with a network of misﬁt dislocations the FMRmeasurements have shown that a signiﬁcant part of the damping is extrinsic and caused by twomagnon scattering.The angular dependence of the FMR linewidth reﬂects the in-plane symmetry ofthe dislocation arrangement. © 2004 American Institute of Physics. @DOI: 10.1063/1.1669219 # I. SAMPLES AND MICROSTRUCTURE ANALYSIS The metallic multilayer ﬁlms studied in this article consist of Fe, Pd, and Au layers and are grown bymolecular beam epitaxy on GaAs ~001!, see details in Ref. 1. The following Fe ultrathin ﬁlms ~shown in bold ! in multilayer samples were studied:20Au/40Fe/40Au/nPd/16Fe/GaAs(001) and 20Au/40Fe/40Au/nPd/ @Fe/Pd #5/16Fe/GaAs(001), the inte- gers and nare in monolayers ~MLs!.@Fe/Pd #5is aL10su- perlattice with ﬁve repetitions. The Fe ~001!mesh is closely matched to Au ~001!(20.5% mismatch !by rotating 45° in the plane ( @100#Fei@110#Au). However Pd has a large lattice mismatch of 4.4% with respect to Fe and 4.9% with respecttoAu, and therefore samples with a sufﬁcient thickness of Pdare affected by the relaxation of lattice strain. The formationof misﬁt dislocations in those samples was evident during thegrowth by reﬂection high energy electron diffraction fanoutstreak patterns on the Au ~001!cap and spacer layers. 2 Aself-assembled network of misﬁt dislocation half loops was observed using transmission electron microscopy byplan view orientation of the layer system 90Au/9Pd/ 16Fe/ GaAs ~001!@cf. Fig. 1 ~a!#. The observed orientation and den- sity of the dislocation arrangement resembles well the misﬁtdislocation networks observed by Woltersdorf 3and Wolters- dorf and Pippel4in epitaxially grownAu/Pd bicrystals of the corresponding thicknesses: During the growth of the ﬁrst Pdmonolayers on Au ~001!substrates complete misﬁt disloca- tions are generated and form a rectangular network located inthe Pd/Au interface. After reaching a critical thickness of 4ML the process of gliding of substrate dislocations can nolonger produce a sufﬁciently high density of dislocations tocompensate the misﬁt; thus an additional generation of dis-location half loops 5started at the top Pd layer and extended to the interface.The corresponding interference of moire ´pat-terns and dislocation contrast phenomena treated in Ref. 4 are also recognizable in Fig. 1 ~a!. The generation mecha- nisms of interface dislocations and their efﬁciency for misﬁtcompensation is outlined in Ref. 6. a!Author to whom correspondence should be addressed; electronic mail: gwolters@sfu.ca FIG. 1.aPlan viewTEM image of the 90Au/9Pd/ 16Fe/GaAs ~001!sample exposing the misﬁt dislocation network. The upper part shows the corre-sponding diffraction pattern. The fourfold symmetry of defects is evident inthe presence of reciprocal sheets. The mean separation between dislocation lines was ;15 nm corresponding to a Fourier component of ;1 310 6cm21. The arrow is along the @110#Aucorresponding to @100#Fe.b Half width half maximum linewidth for the top 40Felayer in the 20Au/40Fe/40Au/4Pd/ @1Fe/1Pd #5/16Fe/GaAs(001) structure at 73 ~!!and 24~d!GHz as function of the in-plane angle wMof the magnetization M with respect to @100#Fe. The Gilbert damping contribution is indicated by the dotted lines. The discontinuities for the 24 GHz measurements arecaused by spin pumping around accidental crossovers of the resonance ﬁelds~see Ref. 7 !.JOURNAL OF APPLIED PHYSICS VOLUME 95, NUMBER 11 1 JUNE 2004 7007 0021-8979/2004/95(11)/7007/3/$22.00 © 2004 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.11.242.100 On: Wed, 24 Dec 2014 00:49:45II. SPIN DYNAMICS IN LATTICE STRAINED STRUCTURES Magnetic relaxation was investigated using ferromag- netic resonance ~FMR !. The FMR experiments were carried out with 14, 18, 24, 36, and 73 GHz systems. In this article itwill be shown that the effective magnetic damping isstrongly enhanced in samples with a self-assembled networkof misﬁt dislocations, and that the enhancement in the FMRlinewidth, DH, can be described by two magnon scattering. Finally we will show that the measured dependence of twomagnon scattering on the microwave frequency and theangle wMof the magnetization with respect to the crystalline axis allow one to identify the two dimensional Fourier com-ponents of magnetic defects. The magnetic anisotropies ofthe top 40Fe~001!layer, in 20Au/ 40Fe/40Au/9Pd/16Fe/ GaAs ~001!, and 20Au/ 40Fe/40Au/4Pd/ @Fe/Pd #5/ 16Fe/GaAs(001) structures are similar to those in nearly lat-tice matched and dislocation free 20Au/ 40Fe/40Au/16Fe/ GaAs ~001!structures. 1The main quantitative difference be- tween the samples with a thick Pd layer, NPd>9 ML, and those with NPd,5 ML was in magnetic damping, where NPd represents the total number of atomic Pd layers in the struc- ture. The top layer ~40Fe!in magnetic double layers with NPd,5 has shown simple Gilbert damping aenhanced only by spin pumping.1,7awas determined from the linear fre- quency dependence of DH. The FMR linewidth in 20Au/ 40Fe/40Au/9Pd/16Fe/GaAs ~001!and 20Au/ 40Fe/40Au/ 4Pd/@Fe/Pd #5/Fe/GaAs(001) samples with NPd>9, was very different. In these samples the FMR linewidth, DH, was strongly dependent on the angle wM@see Figs. 1 ~b!and 2~a!#.DHshows a distinct fourfold symmetry. The minima inDHare along the magnetic hard axis ^110&Fe, and the maxima in DHare along the easy axes ^100&Feat all micro- wave frequencies. The frequency dependence of the FMRlinewidth DHalong the ^100&Feand^110&Feaxes is shown in Fig. 2 ~b!. For the magnetization along the in-plane ^110&Fe directions, the FMR linewidths at 36 and 73 GHz were found to be very close to those caused by the Gilbert damping in20Au/40Fe/40Au/16Fe/GaAs ~001!. The results are different for the FMR measurements with the saturation magnetizationalong the ^100&Fedirections. First, the FMR linewidths are larger than those along the ^110&Fedirections. Second, the microwave frequency dependence is not described by asimple linear dependence as expected for Gilbert damping.In fact the nonlinear frequency dependence @see Fig. 2 ~b!# resembles recent calculations by Arias and Mills’s 8of two magnon scattering in ultrathin ﬁlms.Asimilar frequency de-pendence of DHwas found recently by Twisselmann and McMichael 9for Permalloy ﬁlms grown on NiO and Linder et al.10on Fe4V4superlattices. Obviously, the anisotropic contribution to the FMR line- width is not intrinsic.Asimilar FMR line broadening behav-ior was observed in the following Fe ﬁlms ~in bold !in strain relieved crystalline structures: 20Au/ 40Fe/40Pd/16Fe/ GaAs ~001!, 20Au/20Fe/40Pd/16Fe/GaAs ~001!, 200Pd/30Fe/ GaAs ~001!, and 90Au/9Pd/ 16Fe/GaAs ~001!. This indicates that the extrinsic damping does not depend on the Fe layerthickness and its location inside the structure, and thereforeoriginates in the interior of the Fe ﬁlm. This implies that thedislocation glide along $111%Auplanes propagates across the whole multilayer. III. TWO MAGNON SCATTERING In FMR the uniform mode ( q;0) can be scattered by magnetic inhomogeneities into nonuniform modes ( qÞ0 magnons !. Two magnon scattering has been used to describe extrinsic damping in ferrites11,12and metallic ﬁlms.13The two magnon scattering matrix is proportional to componentsof the Fourier transform A(q)5 drDU(r)e2iqrof magnetic inhomogeneities, where U(r) stands symbolically for a local magnetic energy. The magnon momentum is not conservedin two magnon scattering due to the loss of translationalinvariance, but the energy is. In ultrathin ﬁlms the qvectors are conﬁned to the ﬁlm plane and the magnon dispersionrelation can be found in Ref. 8. The degenerate modes aregiven by crossovers of the magnon manifold with the energyof the homogeneous mode. The direction of magnons is usu-ally determined by the angle cbetween the magnon vector q and the saturation magnetization. The value of cdetermines the magnitude q0of the degenerate magnon. The value of q0 decreases with an increasing angle c. No degenerate modes are available for angles clarger than cmax5arcsin @H/(H 14pMeff)#1 2, where His the applied ﬁeld at FMR, and 4pMeffis the effective demagnetizing ﬁeld perpendicular to FIG. 2. aTypical FMR spectra measured at 24 GHz on a 20Au/40Fe/40Au/4Pd/ @1Fe/1Pd #5/16Fe/GaAs(001) sample. The left spec- tra were taken with the magnetization Min the plane: Mi@110#Fe~solid line ! andMi@100#Fe~dotted line !. The right spectrum ~dashed line !corresponds to the perpendicular conﬁguration ( Mi@001#Fe). Note that the FMR line- widths in the in-plane conﬁguration are anisotropic, and the narrowest line ismeasured in the perpendicular conﬁguration. bFrequency dependence of the FMR linewidth, DH, for the top 40Felayer in the 20Au/40Fe/40Au/4Pd/ @1Fe/1Pd #5/16Fe/GaAs(001) structure along the ^100&Fe~!!and^110&Fe~j!axes, respectively. The purpose of the solid spline ﬁt is to guide the reader’s eye. The dashed line shows the frequencydependence of the intrinsic FMR linewidth ~Gilbert damping !of the 40Fe~001!layer. The Gilbert damping in a double layer with well separated resonance ﬁelds includes the contribution by spin pumping ~Ref. 7 !.7008 J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004 Woltersdorf et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.11.242.100 On: Wed, 24 Dec 2014 00:49:45the ﬁlm surface. When the magnetic moment is inclined with respect to the ﬁlm surface at angles larger than p/4 no de- generate modes are available8and two magnon scattering disappears. In fact, we used this condition to test the appli-cability of two magnon scattering for the interpretation ofextrinsic damping. We would like to emphasize that allsamples in this article satisﬁed this condition @see Fig. 2 ~a!#. This identiﬁes the extrinsic damping as two magnon scatter-ing. The two magnon scattering process is conﬁned to degen- erate magnons which are restricted to lobes, q 0(c), around the direction of the magnetic moment. The lobes resemble inshape the inﬁnity symbol ~‘!with its center at the origin of the reciprocal space. The effectiveness of two magnon scat-tering as a function of the angle of the magnetization withrespect to the crystallographic axis can be tested by evaluat-ing a simpliﬁed expression for the relaxation parameter R.R is the imaginary part of the denominator of the in-plane rfsusceptibility. 8Using the above concept of Fourier compo- nents of inhomogeneities one can write R~wM!;EI~q!d~v2vq!dq3 52E 2cmaxcmaxI~q0,c,wM! ]v ]q~q0,c!q0dc, ~1! whereI(q);A(q)A(q). The expression q0/]v/]qde- scribes a weighting parameter along the two magnon scatter-ing lobe. For a given microwave frequency this factor isnearly independent of c, and therefore the whole lobe con- tributes to Rwith an equal weight. It is also interesting to note that the magnon group velocity ]v/]q(q0,c) in Eq. ~1! is proportional to the strength of dipolar and exchange ﬁeldsand represents the dipole exchange narrowing of the twomagnon scattering mechanism. The maximum magnon momentum q 0in two magnon scattering is small, just of ;33105cm21at 73 GHz and only ;53104cm21at 14 GHz. Two magnon scattering probes mostly the area around the origin of the reciprocalspace. IV. DISCUSSION The angular and microwave frequency dependence of the FMR linewidth allows one to identify the main featuresofI(q, wM). The scattering matrix originates from inhomo- geneous magnetic energy. This leads automatically to an ex-plicit dependence of I(q, wM) on the angle wMof the mag- netization with respect to the defect axes ~in our case ^100&Fe). The dislocations are the source of the magnetic defects, but the magnetic inhomogeneities can manifestthemselves on a different length scale due to the exchangeinteraction and magnetoelastic effects. The angular depen-dence ofI(q, wM) has to satisfy the symmetry of the defects. In our case it is determined by the fourfold symmetry of thedefect lines ( $111%Auglide planes !. Each of the mutually perpendicular sets of linear defects generates a spatially ﬂuc-tuating uniaxial anisotropy ﬁeld. This ﬁeld changes its sign when the magnetization is half way ( wMi^110&Fe) between parallel and perpendicular orientations with respect to thedefects. A similar argument was recently used by Lindneret al. 10to explain the absence of two magnon scattering along the ^110&directions on Fe/V superlattices. Therefore the following ansatz: I(q,wM)5Q(q)cos2(2wM) is appro- priate to interpret the FMR linewidth. Q(q) is the Fourier transform of the magnetic defect distribution satisfying thefourfold symmetry of the misﬁt dislocation network.The fre-quency dependent deviations of the linewidth from sinu-soidal cos 2(2wM) behavior can be accounted for by the func- tional form of Q(uqu,w), where the angle w5wM1cis measured with respect to the crystallographic axis. Angular dependent extrinsic damping created by a rect- angular network of defects appears to be a common phenom-enon. It was observed in our previous studies using the meta-stable bcc Ni/Fe ~001!bilayers grown on Ag ~001! substrates, 14and Fe ~001!ﬁlms grown on bcc Cu ~001!.15In the Ni/Fe bilayers bcc Ni went through a major structuralchange going towards the stable fcc phase of Ni ~001!, result- ing in a network of rectangular lattice defects. The angulardependence of the FMR linewidth indicated that the defectlines were oriented along the ^100&axes of Fe ~001!. The bcc Cu~001!layer went through a lattice transformation after the thickness of the Cu layer was larger than 10 ML. Again astrong anisotropy in DHwas observed for the Fe ~001!ﬁlms grown on the lattice transformed Cu ~001!substrates. The angular dependence indicated that the defect lines in Fe ~001! were along the ^100&crystallographic directions. We ob- served this type of two magnon scattering also in half metal-lic NiMnSb ~001!ﬁlms 16which were affected by two sets of rectangular lattice defects along the ^100&and^110&direc- tions. Consequently, the two magnon scattering was aniso-tropic, but did not disappear in any direction. 1R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 ~2001!. 2G. Woltersdorf and B. Heinrich ~unpublished !. 3J. Woltersdorf, Appl. Surf. Sci. 11Õ12,4 9 5 ~1982!. 4J. Woltersdorf and E. Pippel, Thin Solid Films 116,7 7~1984!. 5D. Bacon and A. Cocker, Philos. Mag. 12, 195 ~1965!. 6J. van der Merwe, J. Woltersdorf, and W. Jesser, Mater. Sci. Eng. 81,1 ~1986!. 7B. Heinrich,Y.Tserkovnyak, G.Woltersdorf,A. Brataas, R. Urban, and G. Bauer, Phys. Rev. Lett. 90, 187601 ~2003!. 8R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 ~1999!. 9D. Twisselmann and R. McMichael, J. Appl. Phys. 93, 6903 ~2003!. 10J. Lindner, L. Lenz, K. Kosubek, K. Baberschke, D. Spoddig, R. Meck- enstock, J. Pelzl, Z. Frait, and D. Mills, Phys. Rev. B 68, 060102 ~R! ~2003!. 11M. Sparks, Ferromagnetic Relaxation Theory ~Mc Graw–Hill, New York, 1966!. 12M. J. Hurben, D. R. Franklin, and C. E. Patton, J. Appl. Phys. 81,7 4 5 8 ~1997!. 13C. E. Patton, C. H. Wilts, and F. B. Humphrey, J. Appl. Phys. 38, 1358 ~1967!. 14B. Heinrich, S. Purcell, J. Dutcher, K. Urquhart, J. Cochran, andA.Arrott, Phys. Rev. 64, 5334 ~1988!. 15Z. Celinski and B. Heinrich, J. Appl. Phys. 70, 5935 ~1991!. 16B. Heinrich, G. Woltersdorf, R. Urban, E. Rozenberg, G. Schmidt, P. Bach, and L. Molenkamp, J. Appl. Phys. 95, 7462 ~2004!, these proceed- ings.7009 J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004 Woltersdorf et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.11.242.100 On: Wed, 24 Dec 2014 00:49:45
1.4953229.pdf	Spin Seebeck effect in a weak ferromagnet Juan David Arboleda, , Oscar Arnache Olmos , Myriam Haydee Aguirre , Rafael Ramos , Alberto Anadon , and Manuel Ricardo Ibarra Citation: Appl. Phys. Lett. 108, 232401 (2016); doi: 10.1063/1.4953229 View online: http://dx.doi.org/10.1063/1.4953229 View Table of Contents: http://aip.scitation.org/toc/apl/108/23 Published by the American Institute of Physics Spin Seebeck effect in a weak ferromagnet Juan David Arboleda,1,a)Oscar Arnache Olmos,1Myriam Haydee Aguirre,2,3,4 Rafael Ramos,5,6Alberto Anadon,2,3and Manuel Ricardo Ibarra2,3,4 1Instituto de F /C19ısica, Universidad de Antioquia, A.A. 1226, Medell /C19ın, Colombia 2Instituto de Nanociencia de Arag /C19on, Universidad de Zaragoza, E-50018 Zaragoza, Spain 3Departamento de F /C19ısica de la Materia Condensada, Universidad de Zaragoza, E-50009 Zaragoza, Spain 4Laboratorio de Microscop /C19ıas Avanzadas, Universidad de Zaragoza, E-50018 Zaragoza, Spain 5WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 6Spin Quantum Rectiﬁcation Project, ERATO, Japan Science and Technology Agency, Sendai 980-8577, Japan (Received 9 March 2016; accepted 23 May 2016; published online 6 June 2016) We report the observation of room temperature spin Seebeck effect (SSE) in a weak ferromagnetic normal spinel Zinc Ferrite (ZFO). Despite the weak ferromagnetic behavior, the measurements ofthe SSE in ZFO show a thermoelectric voltage response comparable with the reported values for other ferromagnetic materials. Our results suggest that SSE might possibly originate from the surface magnetization of the ZFO. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4953229 ] Thermospin effects study the correlations between heat, charge, and spin currents. Nowadays, these phenomena havegained great attention with the emerging ﬁeld of spin calori- tronics. 1,2The spin Seebeck effect (SSE) is one of the most relevant effects of this rapidly growing ﬁeld. In this effect, aspin current is driven through a ferromagnet/normal metal (FM/NM) bilayer structure due to an applied temperature gradient, allowing a powerful alternative for spin currentgeneration using heat currents, which broadens the prospects in the construction of new technological devices based on this principle. 3 Since its discovery in 2008 by Uchida et al.i naF M metal,4the SSE has been observed in a wide variety of mag- netic materials, including semiconductors5and insulators,6 in thin ﬁlms and/or bulk FMs.7–10It is therefore considered as an universal effect among magnetic ordered materials. Up until now, the theoretical models that have been developedare mainly based on the interaction between the localized spin in FM (magnons) and the itinerant spin in the NM at the FM/NM interface. 11–14However, new experimental ﬁndings continue to challenge the SSE understanding, such as the ob- servation of SSE in paramagnetic15and antiferromag- netic16,17materials. Ohnuma et al.18studied theoretically the SSE in ferrimagnets and AFMs concluding that SSE must vanish in AFMs while persists in compensated ferrimagnets even when the saturation magnetization almost vanishes.The detection of the SSE in the latter was an open experi- mental question until today. 19,20 In this paper, we report the observation of the SSE in a weak ferromagnetic material, ZFO (ZnFe 2O4).21Our results show that, despite the negligible inversion degree and very small magnetization, it is possible to generate a spin currentusing a weak FM material. The realization of SSE in spinel ferrites, such as ZFO, could represent a signiﬁcant advantage since these materials combine low cost, mechanical andmagnetic properties, more versatile than any other magnetic material. 22 Bulk ZFO has a normal spinel structure in which all Zn2þions ﬁll tetrahedral sites ( A); hence, the Fe3þions are forced to occupy all of the octahedral sites ( B). It has antifer- romagnetic properties below the N /C19eel temperature of about 10 K and presents the paramagnetic behavior at room tem- perature.23,24The cation distribution of the two interstitial sites of the structure plays a crucial role in the magneticordering. In general, ZFO can be represented by the follow-ing formula: [Zn 1/C0dFed]A[Zn dFe2/C0d]BO4, where dis the inversion parameter. Depending of d, there are uncompen- sated spins in the two sublattices, and ZFO could exhibit aweak ferromagnetic behavior for small values of d, ferrimag- netic order, or even superparamagnetism if the inversioncomes from grain size reduction. 24In spinel ferrites as in ZFO, magnetization monotonically decreases with increas- ing temperature to room temperature25unlike the more com- plex garnet ferrites which has a compensation temperaturewhere the magnetization vanishes. Here, our results do notcontradict the ﬁndings of Geprags et al. in Ref. 20. We have synthesized polycrystalline ZFO by conven- tional solid state reaction method. High purity oxide pow-ders, ZnO (Merk 99%) and a-Fe 2O3(Merk 99.9%), were used as raw materials. The resulting powders were calcinedin air at 1150 /C14C for 12 h, pressed into rectangular pellets under about 5 tons, and sintered at 1300/C14C for 24 h. The width ( Lx), length ( Ly), and thickness ( Lz) of the pellets were 7 mm, 2 mm, and 0.5 mm, respectively. The X-Ray diffrac-tion (XRD) pattern in Fig. 1conﬁrms a single ZFO phase, after sinterization, without any trace of contamination under the detection limit. The corresponding spinel structure hasa cell parameter of 8 :437 ˚A. A ð8:060:5Þnm Pt ﬁlm was deposited at room temperature on the top surface as a spin toelectric-voltage convertor. Before the deposition, the surfacewas carefully polished using diamond abrasive paper ofdown to one micron. This surface allows a ﬂat depositionfree from defects at the interface, as shown in Fig. 2(a).I n a)juan.arboledaj@udea.edu.co 0003-6951/2016/108(23)/232401/4/$30.00 Published by AIP Publishing. 108, 232401-1APPLIED PHYSICS LETTERS 108, 232401 (2016) Figs. 2(b)–2(d) , we can see the crystal analysis by scanning transmission electron microscopy with a high annular angu-lar dark ﬁeld (STEM-HAADF) detector, showing an image of a sample prepared for the SSE measurement. The grain size was in the order of several microns, and the grains pres-ent coherent boundaries (see Fig. 2(b)). Two typical grain orientations are displayed in Figs. 2(c)and2(d), [1–10] and [-112], respectively. The high resolution STEM-HAADFimages show perfect agreement with the simulated structure. The energy dispersive X ray (EDX) measurements show the stoichiometric composition ZnFe 2O4even through the grain boundaries. In addition, the resistivity of the sample was 0.72 M Xcm. The SSE measurements were made in the longitudinal conﬁguration. The sample was placed between two AlN plates with thermal grease to ensure proper thermal contact. A resistive heater was connected to the top plate, while the bottom plate was in direct contact with cryostat providingthe heat sink. A temperature difference was applied in the z direction ( DT). Temperatures between the top and bottom of the sample are stabilized at 300 K þDTand 300 K, respec- tively, which were monitored using two T-type thermocou- ples. The electrical contacts were made using Al wires with 25lm diameter. A magnetic ﬁeld up to 8 kOe was applied along the xdirection. Then, the SSE voltage ( V) in the Pt ﬁlm, produced via the inverse spin Hall effect (ISHE), 26was recorded with a Keithley 2182A nanovoltmeter in the y direction. This conﬁguration is usually used with insulating samples where anomalous Nernst effect (ANE)27,28is not present. We further check the absence of ANE in the sample, and proximity ANE (PANE) in Pt by performing the perpen- dicularly magnetized conﬁguration.29In this conﬁguration, the thermal gradient is applied in the sample plane (x direc- tion), the magnetic ﬁeld is applied parallel to the surface normal (z direction), and the voltage is sensed in the y direc- tion. Under these experimental conditions, the ANE in Pt bythe proximity effects can still be detected while the SSE is forbidden by the sample geometry ( J SkH). To conﬁrm the type of magnetic order of the samples, the magnetic measurements were performed in a Vibrating Sample Magnetometer (VSM), and M €ossbauer spectra (MS) were recorded at room temperature in the standard transmis- sion geometry, using a Co57/Rh source. Fig. 3(a) shows the magnetization as a function of magnetic ﬁeld after subtract- ing the paramagnetic contribution. The results display an hysteretic behaviour with a coercive ﬁelds less than 25 Oe (inset Fig. 3(a)). We made the magnetic measurements by applying the magnetic ﬁeld in plane (IP) as well as in the out of plane (OP) direction without detecting any magnetic anisotropy, which is in agreement with the polycrystalline nature of the samples. We have also veriﬁed that the bulk magnetic properties are unaffected by the Pt ﬁlm deposition. The weak magnetization of about 1.2 emu/g suggests a par- tially inverted spinel structure where a very small fraction of Fe3þoccupies Asites producing a nonzero magnetization. The inversion parameter dshould be less than 4% since it could not be detected by XRD or by MS as in previous stud- ies.30Therefore, our samples exhibit weak ferromagnetism instead of the ideal predicted paramagnetism. Typical SSE samples as magnetite or YIG has bulk magnetization between 92 and 100 emu/g (Ref. 31) and 27 emu/g,32,33 respectively. The room temperature57M€ossbauer spectra (Fig. 3(b)) were taken before and after the sintered process. Both consist of only one doublet, with an isomer shift value of 0.36 mm/s and a quadrupole splitting of 0.32 mm/s, corre- sponding to the characteristic Fe3þcharge state in a direct spinel structure. This is in agreement with earlier reports.34FIG. 1. XRD of ZFO sample after the sintered process. FIG. 2. (a) Typical image in STEM-HAADF for a FIB lamella preparation showing the Pt/ZFO interface, (b) STEM-HAADF image of the bulk samplewith a grain boundary, (c) high resolution STEM-HAADF image in [1-10] zone axis, and (d) [-112] zone axis.232401-2 Arboleda et al. Appl. Phys. Lett. 108, 232401 (2016) The absence of at least a small magnetic component conﬁrms the very small inversion degree of the ZFO sample. A schematic illustration of the SSE measurement setup is shown in Fig. 4(a). In this setup, the magnetic ﬁeld de- pendence of the SSE was obtained at room temperature, as shown in Fig. 4(b). When the magnetic ﬁeld is swept along thexaxis, the sign of Vsignal is clearly reversed following the magnetization behavior. The inset shows the observed linear dependence of DV¼ðVð8 kOeÞ/C0Vð/C08 kOeÞÞ=2a safunction of DT. The above results are in agreement with the expected SSE behavior. Both ANE and PANE measurements show negligible voltages and no dependence with the mag- netic ﬁeld (see Figs. 4(c) and4(d)), therefore ensuring that the SSE signal is not contaminated by ANE or PANE. As shown in Refs. 29and35, the SSE coefﬁcients were esti- mated, after antisymmetrization, as Szy¼Vð8 kOeÞðLz=LyÞ. This way we obtain a SSE response coefﬁcient of about Szy¼28 nV/K which is similar to that reported for NiFe 2O4 (NFO)36(30 nV/K); slightly lower than the value reported for magnetite35(74 nV/K); an order of magnitude lower than the reported value for the archetype YIG3(521 nV/K); but far from the highest value reported so far in magnetic multi- layers of Fe 3O4/Pt by Ramos et al.37(1786 nV/K). The SSE coefﬁcient measured in two different systems, slabs (ZFO and YIG) and thin ﬁlms (NFO and Fe 3O4), revealed the rele- vance of the quality of the Pt interface.38,39However, it is worth to highlight the signiﬁcantly large signal appearing on this sample despite its weak ferromagnetism. The observed results of SSE in ZFO/Pt can be accounted by the dynamics of the magnetization Min the FM, when a temperature gradient is applied in the presence of a magnetic ﬁeld H0, described by the Landau Lifshitz Gilbert (LLG) equation in the linear response theory of the SSE, written as follows:12 @tM¼cH0þh ðÞ /C0Jsd /C22hs/C20/C21 /C2Mþa MsM/C2@tM; where cand aare the gyromagnetic ratio and the Gilbert damping constant, respectively. Jsdis the interface s-d exchange coupling between FM and the NM with itinerant spin density s. The thermal ﬂuctuations are taken into account through the noise ﬁeld h. In the case of weak ferro- magnetism where the saturation magnetization Msis quite small, the damping term becomes stronger in the LLG equa-tion. Some studies have shown a strong enhancement of the spin wave damping in the weak ferromagnetic regime. 40,41 As the magnon propagation lengths scale as 1 =a,42only magnons that are close enough to the interface contribute to the spin current thermally pumped to the NM. This suggests that the mechanism of the SSE in the polycrystalline ZFOFIG. 3. (a) M-H curves in plane (IP) and out of plane (OP) after subtracting the paramagnetic contribution. Inset: Enlargement of a ﬁeld region wherethe coercive ﬁeld is observed. (b) Room temperature 57M€ossbauer spectrum of ZFO after the sintered process. FIG. 4. (a) Schematic illustration of the SSE at room temperature. When applying a temperature gradient rTalong the zdirection in the sample, a spin current Js, polarized in the direction of the magnetic ﬁeld H(xaxis), is pumped from the ZFO to the Pt layer. An electric ﬁeld appears in the ydirection via the ISHE, allowing electrical detection of the SSE by measuring the electric voltage V. (b) SSE response with applied magnetic ﬁeld. (Inset) DT dependence of SSE volt- ageDV. (c) and (d) Schematic illustration and results of the PANE and ANE measurements, respectively.232401-3 Arboleda et al. Appl. Phys. Lett. 108, 232401 (2016) pellet is originated from the surface magnetization which dif- fers in general from the bulk one. This could explain the con- siderable value of the SSE voltage response in ZFO despitethe negligible saturation magnetization. Other factors may affect the SSE signal as: Surface roughness is related directly with the spin mixing conductance; 43the nature of the struc- tures and chemical composition at the atomic level near the interface that could also play important roles on the magnetic anisotropy and magnetic moments unbalance at the interface. Systematic measurements in ZFO thin ﬁlms could clarify the origin of this signal. In summary, we have observed the unexpected presence of the SSE in a weak ferromagnetic system, consisting of a sintered polycrystalline ZFO. The SSE measurement is freefrom artifacts from the thermomagnetic effects from itinerant magnetism, such as ANE or PANE. Despite that saturation magnetization is insigniﬁcant, and the spin wave damping increases considerably in this regime, we report a signiﬁcant Seebeck coefﬁcient of about 28 nV/K that was initially unex-pected. Our results suggest that SSE might possibly originate from the surface magnetization of the ZFO. However, further experiments are needed to elucidate the origin of the SSE inthese weak ferromagnetic ordered materials, for instance, the SSE measurements in the ZFO thin ﬁlms with varying thick- ness might help to clarify the origin of the observed effect. The authors acknowledge Professor P. Algarabel, Dr. I. Lucas, and Professor L. Morell /C19on for enlighted discussion. This work was supported by Solid State Group (GES) at the University of Antioquia in the framework of Sustainability Strategy 2014–2015; Colombian Science, technology andinnovation department (COLCIENCIAS, PhD student grant, conv. 567); Municipality of Medellin through SAPIENCIA agency (EnlazaMundos program, conv. 2014); J.D.A. is thankful to CODI-UdeA by ﬁnancial backing. We also thank the Spanish Ministry of Science (through and MAT2011-27553-C02, including FEDER funding); the Arag /C19on Regional Government (Project No. E26); and Thermo- Spintronic Marie Curie CIG (Grant Agreement No. 304043)-EU. Project No. PRI-PIBJP-2011-0794. 1G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012). 2S. R. Boona, R. C. Myers, and J. P. Heremans, Energy Environ. Sci. 7, 885 (2014). 3K. Uchida, M. Ishida, T. Kikkawa, A. Kirihara, T. Murakami, and E.Saitoh, J. Phys.: Condens. Matter. 26, 343202 (2014). 4K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. 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1.1767613.pdf	Deterministic and finite temperature micromagnetics of nanoscale structures: A simulation study Pierre E. Roy and Peter Svedlindh Citation: Journal of Applied Physics 96, 2901 (2004); doi: 10.1063/1.1767613 View online: http://dx.doi.org/10.1063/1.1767613 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/96/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Exchange-dominated eigenmodes in sub-100nm permalloy dots: A micromagnetic study at finite temperature J. Appl. Phys. 115, 17D119 (2014); 10.1063/1.4862844 Micromagnetic simulation of spin transfer torque switching by nanosecond current pulses J. Appl. Phys. 99, 08B907 (2006); 10.1063/1.2170047 Micromagnetic simulations of hysteresis loops in ferromagnetic Reuleaux’s triangles J. Appl. Phys. 97, 10E318 (2005); 10.1063/1.1858113 Study of in situ magnetization reversal processes for nanoscale Co rings using off-axis electron holography J. Appl. Phys. 97, 054305 (2005); 10.1063/1.1855393 Micromagnetic simulations of thermally activated magnetization reversal of nanoscale magnets J. Appl. Phys. 87, 4792 (2000); 10.1063/1.373161 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 13:59:20Deterministic and ﬁnite temperature micromagnetics of nanoscale structures: A simulation study Pierre E. Roya)and Peter Svedlindh Department of Engineering Sciences, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden (Received 22 December 2003; accepted 5 May 2004 ) Zero and ﬁnite temperature micromagnetic studies have been performed for two nanoscale structures of different geometries by means of numerical integration of the deterministic andstochasticLandau-Lifshitz-Gilbertequationsofmotion.Theresultsindicatethatnotonlydothermalﬂuctuations cause a decrease of the coercivity and the time scales involved in switching, but theycan also alter the magnetization reversal path. In the case of thermally induced changes in theswitching path it is found that with sufﬁcient thermal energy the particle can form other states priorto switching than in the deterministic model. This leads to the identiﬁcation of two switchingregimes in the structures considered, whereby switching from one of the states signiﬁcantlydecreases the coercivity. Furthermore, a study of the time scales involved and the transient magneticconﬁgurations appearing during fast switching was performed. © 2004 American Institute of Physics.[DOI: 10.1063/1.1767613 ] I. INTRODUCTION The use of nanoscale structures in today’s magnetic tech- nology, such as the bits in magnetic storage, requires a thor-ough understanding of the behavior of the magnetizationprocesses in these structures. Detailed experimental studiesof fast spatiotemporal magnetization dynamics have recentlybecome possible through the use of picosecond time scalescanning Kerr microscopy. 1,2Provided that there are reliable magnetic models available, these kinds of investigations areideal for numerical experiments, since a time resolution onthe picosecond scale is required. This paper will make use of micromagnetic modeling, which is a well established method for understanding themagnetization dynamics in conﬁned magnetic structures. 3,4 There are various modeling techniques out of which this pa- per is concerned with the method of direct numerical inte-gration of the Landau-Lifshitz-Gilbert equations of motion.This gives a deterministic (zero temperature )description of the spin dynamics and the static conﬁgurations involved.However, as structures shrink in size, temperature effectswill be more pronounced, something which needs to be takeninto consideration. 5,6 In this paper we present results on both deterministic and ﬁnite temperature simulations of the magnetization processesin nanoscale structures. It is found that apart from a reduc-tion of coercivity, thermal ﬂuctuations alter the magnetiza-tion reversal path. Two such paths have been identiﬁed. Thisgives rise to two switching regimes, where switching throughone of these states signiﬁcantly lowers the coercivity. Someresults on the time scales involved in switching as well as astudy of the transient magnetic states appearing (the reversal path)during fast switching are presented.II. FINITE TEMPERATURE MICROMAGNETIC TECHNIQUE The micromagnetic model consists in discretizing the material into cubic cells whose magnetizations Msrdare rep- resented by classical vectors, all having a constant magnitude corresponding to the spontaneous magnetization Ms.At each sitei, the time evolution of a magnetization vector is gov- erned by the Landau-Lifshitz-Gilbert equations of motion.4 In reduced units the equations of motion are s1+a2ddmi dt=−mi3hieff−afmi3smi3hieffdg, s1d where ais the damping parameter, mi=Mi/Ms,t=g0Mst (where g0is the gyromagnetic ratio ), andhieff=Hieff/Ms.Ata sitei, the direction that the spin relaxes towards is deter- mined by the local effective ﬁeld Hieff. The ﬁrst term on the right hand side of Eq. (1)describes the precessional motion of the vector around the effective ﬁeld direction, whereas thesecond term imposes a damping of the precession, tending toalign the vector along the effective ﬁeld direction. The effective ﬁeld can be derived from the total energy densityE totas follows: Hieff=−]Etot ]Mi. s2d The local effective ﬁeld at any site iis the result of a super- position of all the interaction contributions, Hieff=Happl+Hiex+HiD+HiA+Hifl, s3d whereHapplstems from the applied ﬁeld, Hiexis due to ex- change interactions, HiDrepresents the demagnetizing ﬁeld (dipole-dipole interactions ),HAis the magnetocrystalline an- isotropy ﬁeld and Hiflis a ﬂuctuating ﬁeld due to thermal agitation. The exchange ﬁeld is given by4 a)Electronic mail: pierre.roy@angstrom.uu.seJOURNAL OF APPLIED PHYSICS VOLUME 96, NUMBER 5 1 SEPTEMBER 2004 0021-8979/2004/96 (5)/2901/8/$22.00 © 2004 American Institute of Physics 2901 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 13:59:20Hiex=2A D2o j=nnMj, s4d where Dis the discretization cell lattice constant, Ais the exchange constant, and the summation is carried out over allnearest neighboring snndcells of cell i. The demagnetizing ﬁeld at point iresulting from the dipole interactions with sitesjcan be expressed as H iD=−o jDˆsri−rjd·Mj. s5d HereDˆis the demagnetizing tensor whose elements are func- tions of the difference coordinates between cells. The ana-lytical expressions used were taken from Newell et al. 7and a straightforward recipe for the computation of Dˆis listed in the Appendix. Since the thermal ﬂuctuations result from in-teractions with many microscopic degrees of freedom withidentical statistical properties we can use the central limittheorem that states that in such a case, the distribution func-tion will approach a normal distribution. In effect, thermalﬂuctuations are represented by Gaussian random numbers.The standard deviation is given by 8 s=˛2kBaT g0MSVs1+a2dDt, s6d wherekBis Boltzmann’s constant, Tthe temperature, Vthe discretization cell volume, and Dtthe integration time step. The Gaussian random numbers for the thermal ﬁeld weregenerated from the Box-Muller algorithm. 9When integrating the stochastic Landau-Lifshitz-Gilbert equations care mustbe taken in choosing a numerical scheme that will convergeto the correct solutions. Based on the discussions in Ref. 10,the Heun scheme was chosen for our purposes. III. RESULTS AND DISCUSSION The material parameters used in the simulations are dis- played in Table I. Here K1is the magnetocrystalline aniso- tropy constant and all the other parameters are the same aspreviously deﬁned. Structural elements considered were ofdimensions 100 32535 and 100 35035 nm. An important aspect of the stochastic simulations is the observation timet obsused at each ﬁeld during a hysteresis simulation. This is far from a trivial matter because in the deterministic case,one can use a stability criterion in order to reach equilibrium(not used here )and this is not applicable in the stochastic case (since the spins will always be subject to random pulse ﬁelds ). With a particular application or experimental methodin mind, one can instead tune t obsto imitate the experimental conditions. An example of the effect of tobson the coercive ﬁeld is displayed in Fig. 1. A. 100ˆ50ˆ5 nm particle In this section, results pertaining to the 100 350 35 nm particle are presented and discussed. Hysteresis loops, remanent states, switching paths, and switching timescales have been investigated. 1. Hysteresis loops Figure 2 shows calculated hysteresis loops for the 10035035 nm particle at various temperatures. The ap- plied ﬁeld was ramped from + Msto −Msin 0.33 ms using a tobsof’0.53 ns. The integration time step Dtwas set to <0.05 ps sDt=0.01 d. As can be seen, there is a drastic change in coercivity between 50 and 100 K. This was found to be the result of different switching processes controllingthe magnetization reversal at 50 and 100 K, respectively (see the following section ). 2. Remanent states and switching paths Figure 3 displays the computed remanent states at three different temperatures as well as transient magnetic states forH appl=−30 kA/m. Throughout this paper, three denomina-TABLE I. Material parameters used in the simulations. Ms 860 kA/m K1 0 A 13310−12J/m a 0.02 g0 1.7631011T−1s−1 D 5310−9m FIG. 1. Coercivity as a function of observation time per ﬁeld point (tobs)for the 100 32535 nm particle. The connecting lines are just guides for the eye. FIG. 2. Hysteresis loops for the 100 35035 nm particle at various temperatures.2902 J. Appl. Phys., Vol. 96, No. 5, 1 September 2004 P. E. Roy and P. Svedlindh [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 13:59:20tions describing certain magnetic conﬁgurations will be used; ﬂower,S, andCstates.Aschematic of such conﬁgurations is also shown in Fig. 3. Detailed descriptions of these states canbe found in Ref. 11. Looking at the remanent states, wenotice for the deterministic and the 50 K computations, aﬂower state in both cases, where the spins at the short edgesof the particles are slightly tilted. This is the most stableremanent conﬁguration at zero temperature for this particlesize, as also stated by the authors of Ref. 11. The remanentstate at 100 K displays a somewhat disﬁgured ﬂower con-ﬁguration. Looking at the spins along the short ends of theparticle, there appears to be a tendency towards the Sstate. At H appl=−30 kA/m, Cstates are formed as part of the path to switching for both the deterministic and the 50 Ksimulations. However, at 100 K and Sstate has formed and started to rotate. The CandSstates are almost degenerate, but with the Cstate slightly lower in energy due to the higher degree of ﬂux closure. 4We interpret the ﬁnding of the Sstate at 100 K as a result of sufﬁcient amount of thermal energyenabling the energetically higher Sstate to form. Further- more, the switching path from a Cstate is very different from that of an Sstate and the switching ﬁeld is signiﬁcantly reduced switching from an Sstate. 4This means that we can divide the switching mechanisms into two regimes; oneswitching through a Cstate and another through an Sstate. Plotting the coercive ﬁeld as a function of temperature, theseswitching regimes become more visible. Such a plot isshown in Fig. 4 where we see a jump in the coercive ﬁeld ata certain temperature.This is due to the change in the switch-ing mechanism, as discussed above. 3. Switching time scales and fast switching magnetization reversal paths In this section time scales involved in switching from remanent states as well as the corresponding switching pathsare discussed. Simulations where the particle starting in itsremanent state (those in Fig. 3 )and then subjected to a con- stant applied ﬁeld of −60 kA/m were performed. The corre-sponding time evolution of the magnetization componentsduring switching were recorded at three different tempera-tures. The time evolutions of the magnetization componentsare shown in Fig. 5. Comparing the deterministic simulation to the stochastic ones, one sees that the switching time isgreatly overestimated by the deterministic model. In order toinvestigate the magnetization reversal path during theswitching shown in Fig. 5, snapshots at various points in FIG. 3. Left column: Remanent states. Middle column: Transient states for Happl=−30 kA/m. Right column: Schematic representations of the ﬂower, C, andSstates. FIG. 4. Coercive ﬁeld as a function of temperature for the 100 350 35 nm particle. FIG. 5. Time evolution of the magnetization components for the 100 350 35 nm particle at temperatures 0, 50, and 100 K, applying a constant ﬁeld of −60 kA/m to the remanent states shown in Fig. 3.J. Appl. Phys., Vol. 96, No. 5, 1 September 2004 P. E. Roy and P. Svedlindh 2903 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 13:59:20time of the magnetic conﬁgurations were recorded. Figure 6 visualizes the reversal paths that the particle is undertakingin Fig. 5. The ﬁrst column (from the left )corresponds to the time evolution of the magnetization components in Fig. 5 (a), the middle column to Fig. 5 (b), and the rightmost column to Fig. 5 (c). T=0. In the deterministic simulations the particle passes through a Cstate st=1.13 ns d, which is turned upwards in- stead of downwards (as in Fig. 3 ). This has no signiﬁcance, since there are four energetically equivalent Cstates. 11This is followed by the formation of a domain wall (separating two regions where the spins are aligned in opposite direc-tions ),a tt=1.21 ns. This wall sweeps to the right while the spins to the left of it are gradually rotating towards the ap-plied ﬁeld direction. Finally, at the point of wall annihilation,a curled magnetic structure is formed in the upper right cor-ner and continues to move downwards along the right shortend of the particle, turning the last of the spins into thedirection of the applied ﬁeld. When this has occurred there iswhat could be called precessional ringing under a constantapplied ﬁeld until the system stabilizes. This is manifested inFig. 5 as the oscillations present for some time after switch-ing and is due to the weak damping of the system s a =0.02 d. This phenomenon is observed for the other tempera- tures as well. T=50K. Here, the process appears a little different. A domain wall is nucleated st=0.13 ns d, separating two regions where the spins are aligned in perpendicular directions. As time increases, this wall sweeps through to the left, while thespins behind it rotate towards the applied ﬁeld directions0.17–0.22 ns d. At the event of wall annihilation a curled magnetization distribution or vortex is formed in the upper left corner of the particle st=0.25 ns d. The vortex then be- haves as at T=0, moving downwards along the left short end of the particle and rotating the last of the spins into the ap-plied ﬁeld direction.T=100K. At this temperature, the reversal process is entirely different. Here, reversal is preceded by the formationof anSstate st=0.053 ns dand the magnetization rotates al- most coherently; there is at short time scales a phase lag of the rotations performed by the spins in the central portion ofthe particle comparing to the spins close to the edges. Thetilted spins at the short edges of the particle ﬁrst rotate intothe short axis direction after which the spins in the centralpart coherently rotate into the same directions0.079–0.12 ns d. Then, the opposite order of rotation takes place; the central portion begins to rotate towards the applied ﬁeld direction while the edge spins lag behind s0.13 ns d. B. 100ˆ25ˆ5 nm particle Here, results pertaining to the 100 32535 nm particle are presented. The order in which these results are discussedis the same as that of the 100 35035 nm particle. 1. Hysteresis loops Figure 7 displays calculated hysteresis loops for the 10032535 nm particle at various temperatures. The ap- plied ﬁeld was ramped from + Msto −Msin 0.41 ms.tobsand Dtwere set exactly as before. The coercive ﬁeld for this particle is much higher than that of the 100 35035 nm par- ticle. This is due to the high aspect ratio producing a largeshape anisotropy, thus making it more difﬁcult for the spinsto deviate from the long symmetry axis of the particle. Thereis not the jump in coercive ﬁeld at a certain temperature asfound for the 100 35035 nm particle. Nevertheless, two different reversal mechanisms have been identiﬁed as pre-sented below and the difference in coercive ﬁeld is signiﬁ-cant. 2. Remanent states and switching paths Figure 8 displays remanent states and transient conﬁgu- rations for Happl=−90 kA/m for the 100 32535 nm par- ticle. The remanent states correspond to ﬂower states, al-though not as apparent as for the 100 35035 nm particle. Also here, two switching mechanisms are identiﬁed; that viaaCstate and that through an Sstate. This is seen in Fig. 8 (right column )when applying a ﬁeld of −90 kA/m. As be- FIG. 6. Snapshots of transient magnetic conﬁgurations appearing during switching, applying a constant ﬁeld of −60 kA/m to the remanent statesshown in Fig. 3 at T=0, 50 and 100 K. FIG. 7. Hysteresis loops for the 100 32535 nm particle at various tem- peratures. The inset shows a blow up of a region to make the difference incoercive ﬁeld more visible to the eye. In the inset, the lines represent thetemperatures 0,..., 100 K from left to right.2904 J. Appl. Phys., Vol. 96, No. 5, 1 September 2004 P. E. Roy and P. Svedlindh [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 13:59:20fore, we explain the formation of the Sstate as a conse- quence of the thermal energy promoting its formation. Thetwo different switching regimes (that via a Cstate and that via anSstate)are more clearly seen in Fig. 9, where the coercive ﬁeld is plotted as a function of temperature. Figure9 indicates the two different switching regimes more clearlyin that it indicates two linear regions; that between T=0 to somewhere between 20 and 50 K and that between 50 K andabove. There are not enough data points to resolve exactly atwhat temperature this shift in switching regime occurs, butthe indication is nevertheless clearly seen. 3. Switching time scales and fast switching magnetization reversal paths Following the same line of investigation as for the 10035035 nm particle, simulations in which the remanent states were subjected to a constant ﬁeld of −100 kA/m wereperformed and the time evolution of the magnetization com-ponents during switching recorded at three different tempera-tures. As before, snapshots of transient magnetic conﬁgura-tions during these switchings have been taken in order toinvestigate the reversal paths.The time evolution of the mag-netization components are shown in Fig. 10. Again, the de-terministic representation greatly overestimates the switchingtime of the particle and the same situation with precessional ringing is present. Figures 11–13 show snapshots at various points in time of the magnetic conﬁgurations. The same typeof correspondence between the snapshots and Fig. 10 applyhere, i.e., Fig. 11 corresponds to Fig. 10 (a), Fig. 12 to Fig. 10(b), and Fig. 13 to Fig. 10 (c). T=0. From Fig. 11 one notices that even though the switching in the simulated hysteresis curve occurs via a C state, when a large ﬁeld is applied directly to the remanentstate, the switching path is different, thus indicating the sen-sitivity of the spin dynamics with respect to ﬁeld history.Here, a domain wall is formed, which travels to the lefts2.0–2.10 ns d. The resulting reversal path appears quite complicated with a series of intermediate spin states, includ- ing buckling and curling of the element magnetizationss2.10–2.42 ns d. The reason for this elaborate path is yet un- clear, but it is reasonable to assume that it is due to the comparably high aspect ratio (and also due to the weak FIG. 8. Left: Remanent states at 0, 50, and 100 K. Right: Transient states at Happl=−90kA/m. FIG. 9. Coercive ﬁeld as a function of temperature for the 100 325 35 nm particle. The connecting lines between points are guidelines to the eye. FIG. 10. Time evolution of the magnetization components for the10032535 nm particle at temperatures 0, 50, and 100 K, applying a con- stant ﬁeld of −100 kA/m to the respective states shown in Fig. 8.J. Appl. Phys., Vol. 96, No. 5, 1 September 2004 P. E. Roy and P. Svedlindh 2905 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 13:59:20damping ), making the particle follow elaborate paths in order to minimize effects due to magnetic charge, inﬂicting an en-ergy penalty whenever the directions of the element magne-tizations at the edges deviate from the long symmetry axis. T=50K. Figure 12 again shows a deviation from the path taken during the hysteresis simulation, where switchingoccurred via an Sstate. Instead in the case of direct applica- tion of a large negative ﬁeld to the remanent state at 50 K,the particle chooses to form a Cstate and proceeds from there. At t=0.29 ns, a domain wall is formed. Again the switching path is very elaborate with the formation of across-tie wall (see Ref. 12 for a description of cross ties ). This is visible at 0.37 ns. What follows is the annihilation ofthe cross-tie wall st=0.39 ns dand a series of bucklings until saturation ﬁnally occurs.T=100K. At this temperature (Fig. 13 ), the particle forms an Sstate, just as at −90 kA/m on the hysteresis curve. Here, the usual switching from an Sstate, like that for the 100 35035 nm particle at 100 K, proceeds up to t =0.18 ns. The rest of the path is remarkably elaborate. At t =0.21 ns a column of spins aligns along the vertical directionand then start to curl while moving to the right st =0.23–0.26 ns d.At 0.30 ns there is an indication of a vortex nucleation. This vortex then proceeds to fully nucleate at the right short end of the particle s0.34–0.37 ns d, while leaving the remaining spins into the applied ﬁeld direction. In the end, this vortex will be annihilated at the right short end. IV. CONCLUDING REMARKS In summary, we have performed deterministic and ﬁnite temperature micromagnetic simulations on nanoscale struc-tures. Studies concerning the temperature dependence of thecoercive ﬁeld, switching time scales, and the magnetizationreversal mechanisms havebeen performed. The ﬁndings sug-gest that not only do thermal ﬂuctuations cause a generalreduction in the coercivity and speed up switching times, butcan also alter the path of magnetization reversal. Two whatwe denote as switching regimes have been observed; switch-ing via a Cstate and that through an Sstate. The result is a signiﬁcant change in coercivity depending on which switch-ing regime is dominating. Furthermore, introducing a com-parably large shape anisotropy into these low damped nanos-cale structures seems to severely complicate the reversalpath, where several complex transient states have been ob-served. The found effects of thermal ﬂuctuations only stresseven more, the necessity to include temperature in micro-magnetic modeling of nanoscale magnets. FIG. 11. Snapshots of transient magnetic conﬁgurations appearing during switching from the remanent state, Happl=−100 kA/m and T=0. FIG. 12. Snapshots of transient magnetic conﬁgurations appearing during switching from the remanent state, Happl=−100 kA/m and T=50K. FIG. 13. Snapshots of transient magnetic conﬁgurations appearing during switching from the remanent state, Happl=−100 kA/m and T=100K.2906 J. Appl. Phys., Vol. 96, No. 5, 1 September 2004 P. E. Roy and P. Svedlindh [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 13:59:20ACKNOWLEDGMENTS This work was supported by SSF (Swedish Foundation for Strategic Research ). The authors are also grateful to José Luis García-Palacios for fruitful discussions. APPENDIX: EXPRESSIONS FOR THE DEMAGNETIZING FIELD In this appendix the expressions used for the computa- tion of the demagnetizing ﬁelds are listed. The completederivations can be found in Ref. 7. If the magnetization isdiscretely represented by the distribution hM ijat points hrij, then the magnetostatic ﬁeld at point ridue to all points rjis given by [Eq.(16)in Ref. 7 ] Hi=−o jDˆsri−rjd·Mj, sA1d where the demagnetizing tensor Dˆis a function of the differ- ence coordinates ri−rj. The demagnetizing tensor has nine elements da,bsa,b=x,y,zd), but due to symmetry proper- ties,dxy=dyx,dxz=dzx, anddyz=dzy. Further, all diagonal ele- ments can by evaluated from dxxby permutations of the vari- ablesX,Y,ZandDx,Dy,Dz, whereX,Y,Zare the Cartesian components of ri−rjandDx,Dy,Dzare the discretization cell dimensions. The same applies for the cross elements,which all are of the same form as d xy. In tensor form, Dˆ=C1dxxdxydxz dxydyydyz dxzdyzdzz2, sA2d where the relations between the elements are dxx=dxxsX,Y,Z,Dx,Dy,Dzd, dyy=dxxsY,Z,X,Dy,Dz,Dxd, dzz=dxxsZ,X,Y,Dz,Dx,Dyd, dxy=dxysX,Y,Z,Dx,Dy,Dzd, dxz=dxysZ,X,Y,Dz,Dx,Dyd, dyz=dxysY,Z,X,Dy,Dz,Dxd, sA3d andsC=1/4 pDxDyDzd. Having listed the elemental relation- ships, we now write out the explicit expression for element dxx, dxxsX,Y,Z,Dx,Dy,Dzd=2FsX,Y,Zd−FsX+Dx,Y,Zd −FsX−Dx,Y,Zd, sA4d where FsX,Y,Zd=F1sX,Y+Dy,Z+Dzd−F1sX,Y,Z+Dzd −F1sX,Y+Dy,Zd+F1sX,Y,Zds A5d andF1sX,Y,Zd=F2sX,Y,Zd−F2sX,Y−Dy,Zd −F2sX,Y,Z−Dzd+F2sX,Y−Dy,Z−Dzd. sA6d In the last equation F2=fsX,Y,Zd−fsX,0,Zd−fsX,Y,0d+fsX,0,0d,sA7d where the function fcan be evaluated according to fsx,y,zd=sy/2dsz2−x2dfSy ˛x2+z2D+sz/2dsy2−x2d 3fSz ˛x2+y2D−xyztan−1Syz xRD +s1/6ds2x2−y2−z2dR, sA8d with fsxd;sinh−1sxd;lnsx+˛1+x2dandR=˛x2+y2+z2. Having stated the expression for the dxxelement,dyyanddzz can be computed similarily according to Eq. (A3). We now turn to the expressions for dxy, dxysX,Y,Z,Dx,Dy,Dzd=GsX,Y,Zd−GsX−Dx,Y,Zd −GsX,Y+Dy,Zd +GsX−Dx,Y+Dy,Zd, sA9d where GsX,Y,Zd=G1sX,Y,Zd−G1sX,Y−Dy,Zd−G1sX,Y,Z −Dzd+G1sX,Y−Dy,Z−Dzds A10d and G1sX,Y,Zd=G2sX+Dx,Y,Z+Dzd−G2sX+Dx,Y,Zd −G2sX,Y,Z+Dzd+G2sX,Y,Zd. sA11d In the last equation G2sX,Y,Zd=gsX,Y,Zd−gsX,Y,0d, sA12d where gsx,y,zd=sxyzdsinh−1Sz ˛x2+y2D+sy/6ds3z2−y2d 3sinh−1Sx ˛y2+z2D+sx/6ds3z2−x2d 3sinh−1Sy ˛x2+z2D−sz3/6dtan−1Sxy zRD −szy2/2dtan−1Sxz yRD−szx2/2dtan−1Syz xRD −xyR/3. sA13d The rest of the off-diagonal elements are obtained according to Eq. (A3). 1W. K. Hiebert et al., J. Appl. Phys. 92, 392 (2002 ). 2B. C. Choi et al., Phys. Rev. Lett. 86, 728 (2000 ). 3J. Fidler and T. Schreﬂ, J. Phys. D 33, R135 (2000 ). 4B. Hillebrands and K. Ounadjela, Spin Dynamics in Conﬁned Magnetic Structures I , Topics in Applied Physics Vol. 83 (Springer, Berlin, 2002 ).J. Appl. Phys., Vol. 96, No. 5, 1 September 2004 P. E. Roy and P. Svedlindh 2907 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 13:59:205J. Deak, J. Appl. Phys. 93, 6814 (2003 ). 6G. Brown et al., Phys. Rev. B 64, 134422 (2001 ). 7A. Newell et al., J. Geophys. Res. 98, 9551 (1993 ). 8J-G. Zhu, J. Appl. Phys. 91, 7273 (2002 ). 9W. H. Press et al.,Numerical Recipes , 2nd ed. (Cambridge UniversityPress, New York, 1992 ). 10J. García-Palacios and F. Lázaro, Phys. Rev. B 58, 14937 (1998 ). 11W. Rave et al., IEEE Trans. Magn. 36, 3886 (2000 ). 12A. Hubert and R. Shafer, Magnetic Domains (Springer, Berlin, 1998 ).2908 J. Appl. Phys., Vol. 96, No. 5, 1 September 2004 P. E. Roy and P. Svedlindh [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 13:59:20
1.5023159.pdf	Band-pass Fabry-Pèrot magnetic tunnel junctions Abhishek Sharma , Ashwin. A. Tulapurkar , and Bhaskaran Muralidharan Citation: Appl. Phys. Lett. 112, 192404 (2018); doi: 10.1063/1.5023159 View online: https://doi.org/10.1063/1.5023159 View Table of Contents: http://aip.scitation.org/toc/apl/112/19 Published by the American Institute of Physics Articles you may be interested in Room temperature ferromagnetism in transition metal-doped black phosphorous Applied Physics Letters 112, 192105 (2018); 10.1063/1.5022540 Enhanced interfacial Dzyaloshinskii-Moriya interaction and isolated skyrmions in the inversion-symmetry-broken Ru/Co/W/Ru films Applied Physics Letters 112, 192406 (2018); 10.1063/1.5029857 Spin-orbit torques in high-resistivity-W/CoFeB/MgO Applied Physics Letters 112, 192408 (2018); 10.1063/1.5027855 Dual-mode ferromagnetic resonance in an FeCoB/Ru/FeCoB synthetic antiferromagnet with uniaxial anisotropy Applied Physics Letters 112, 192401 (2018); 10.1063/1.5018809 Mimicking graphene physics with a plane hexagonal wire mesh Applied Physics Letters 112, 191601 (2018); 10.1063/1.5026355 Spin-orbit torque and spin pumping in YIG/Pt with interfacial insertion layers Applied Physics Letters 112, 182406 (2018); 10.1063/1.5025623Band-pass Fabry-Pe `rot magnetic tunnel junctions Abhishek Sharma, Ashwin. A. Tulapurkar, and Bhaskaran Muralidharan Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India (Received 22 January 2018; accepted 26 April 2018; published online 9 May 2018) We propose a high-performance magnetic tunnel junction by making electronic analogs of optical phenomena such as anti-reﬂections and Fabry-Pe `rot resonances. The devices we propose feature anti-reﬂection enabled superlattice heterostructures sandwiched between the ﬁxed and the free fer- romagnets of the magnetic tunnel junction structure. Our predictions are based on non-equilibriumGreen’s function spin transport formalism coupled self-consistently with the Landau-Lifshitz- Gilbert-Slonczewski equation. Owing to the physics of bandpass spin ﬁltering in the bandpass Fabry-Pe `rot magnetic tunnel junction device, we demonstrate an ultra-high boost in the tunnel magneto-resistance ( /C255/C210 4%) and nearly 1200% suppression of spin transfer torque switching bias in comparison to a traditional trilayer magnetic tunnel junction device. The proof of concepts presented here can lead to next-generation spintronic device design harvesting the rich physics ofsuperlattice heterostructures and exploiting spintronic analogs of optical phenomena. Published by AIP Publishing. https://doi.org/10.1063/1.5023159 Spintronics involves the manipulation of the intrinsic spin along with the charge of electrons and has emerged as an active area of research with direct engineering applica- tions for next-generation logic and memory devices. A hall- mark device that leads the development of the technology is the trilayer magnetic tunnel junction (MTJ), which consists of two ferromagnets (FMs) separated by an insulator such as MgO. 1,2The MTJ structure has attracted a lot of attention due to the possibility of engineering a large tunnel magneto- resistance (TMR /C25200%)3and the current driven magnetiza- tion switching via the spin-transfer torque (STT) effect.4–7 Trilayer MTJs ﬁnd their potential applications in magnetic ﬁeld sensors,8,9STT-magnetic random access memory10 devices, and spin torque nano-oscillators (STNOs).11,12The MTJ performance for the aforesaid applications relies onlarge device TMR and low switching bias. 9,12,13There have been consistent efforts in terms of material development14–16 and the device structure designs17–19to enhance the TMR and STT in magnetic tunnel junctions. When it comes to device structures, the double barrier MTJ has been exten- sively explored both theoretically and experimentally to achieve better TMR and switching characteristics.19,20 Owing to the physics of resonant tunneling, the double bar- rier structure has been predicted to provide a high TMR (/C252500%)9,12and nearly 44% lower switching bias19in comparison with the trilayer MTJ device. Superlattice (SL) structures [Fig. 1(a)] consisting of periodic stacks of two dissimilar materials with layer thick- nesses of a few nanometers have been explored extensively in the ﬁeld of photonics, electronics, and thermoelec- tronics.21,22In the area of spintronics, few studies18,23have explored SL structures made of alternate layers of an insula- tor and normal metal (NM) sandwiched between the two FMs as a route to enhance the TMR. As the principal motif of this work, we propose struc- tures that manifest spin selective band-pass transmission spectra as a possible route to achieve superior performance MTJ devices that possess large TMR as well as lowswitching bias. The energy band proﬁles of possible device structures that can be identiﬁed with such a band pass trans- mission spectrum are sketched in Figs. 1(b),1(c), and 1(d) and are termed as band pass—Fabry-Pe `rot magnetic tunnel junction (BP-FPMTJ) I, II, and III, respectively. The struc-tures when sandwiched between two ferromagnets (FMs) can be used to achieve a spin selective band-pass transmis- sion proﬁle. 24–26The structure BP-FPMTJ-I [also identiﬁed as the anti-reﬂective Fabry-Pe `rot magnetic tunnel junction (AR-FPMTJ)] is a regular SL structure terminated by twoanti-reﬂective regions (ARRs) and sandwiched between the ﬁxed and free FMs 24[Fig. 1(b)]. The BP-FPMTJ-I structures can be realized either by an appropriate non-magnetic metal sandwiched between the MgO barriers or via a heterostruc- ture of MgO and a stoichiometrically substituted MgO (Mg x Zn1-xO), whose bandgap and workfunction can be tuned.27 The BP-FPMTJ-II [Fig. 1(c)] is a SL structure having a Gaussian variation in the barrier heights.25Such a structure can be realized via a stoichiometrically substituted MgO (Mg xZn1-xO) whose barrier height can be tuned by chang- ing the Zn mole fraction. The well regime in the BP-FPMTJ- II structure can be realized either via a non-magnetic metal or a lattice matched ZnO.28The BP-FPMTJ-III [Fig. 1(d)] structure is based on a Gaussian distribution of the widths of the MgO barriers in a typical SL structure.26This can be realized either by an appropriate non-magnetic metal sand- wiched between the MgO barriers or via a heterostructure of MgO and stoichiometrically substituted MgO (Mg xZn1-xO) whose band offsets can be tailored.27 To establish the proof of our concept, we present here a detailed analysis of BP-FPMTJ-I or AR-FPMTJ that incor- porates electronic analogs of optical phenomena such asanti-reﬂection coatings (ARCs) and Fabry-Pe `rot resonances. We demonstrate that owing to the bandpass spin-ﬁltering physics of the BP-FPMTJ structure, the proposed AR- FPMTJ device exhibits large non-trivial spin current proﬁles along with an ultra-high tunnel magnetoresistance, leadingto an enhanced switching performance. 0003-6951/2018/112(19)/192404/5/$30.00 Published by AIP Publishing. 112, 192404-1APPLIED PHYSICS LETTERS 112, 192404 (2018) We show in Fig. 2(a)the device schematic of a typical tri- layer MTJ. Device schematics for both the FPMTJ and theAR-FPMTJ structures are depicted in Fig. 2(b) and Fig. 2(c), respectively. We show in Figs. 1(a)and1(b) the band proﬁle schematics of the FPMTJ and the AR-FPMTJ, respectively.The anti-reﬂective (AR) region is a quantum well and a barrier structure, whose well width is the same as that of the SL well and barrier width is half of the SL barrier width, as depicted in Fig.2(d). The AR in a SL structure is analogous to an optical ARC that exploits the wave nature of the electrons. The elec-tronic AR region is designed to get a perfect transmission at a particular energy, simultaneously enhancing the transmission in the entire miniband. We have employed non-equilibriumGreen’s function (NEGF) 29spin transport formalism coupledwith the Landau-Lifshitz-Gilbert-Slonczewski (LLGS)4equa- tion to describe magnetization dynamics of the free FM to substantiate our designs. The details of the calculations are presented in supplementary material Sec. I. In our simulations, we use CoFeB as the FM with its Fermi energy Ef¼2.25 eV and exchange splitting D¼2.15 eV. The effective masses of MgO, the normal metal (NM), and the FM are mOX¼0.18me,mNM¼0.9me, and mFM¼0.8me, respectively,30with mebeing the free electron mass. The barrier height of the CoFeB-MgO interface is UB¼0.76 eV above the Fermi energy.30,31The conduction band offset of the NM and from the FM band edge isU BW¼0.5 eV. We have used a barrier width of 1.2 nm cho- sen such that half of the barrier width is 0.6 nm which is the minimum amount of MgO that can be deposited reliably.32 The quantum well has a width of 3.5 A ˚which is very well within the current fabrication capabilities.33,34It must be noted that resonant effects in metallic quantum wells are lowtemperature phenomena that have been observed experimen- tally in double barrier resonant structures with ferromagnetic contacts. 20 In the results that follow, the parameters chosen for the magnetization dynamics are a¼0.01, the saturation magneti- zation, MS¼1100 emu/cc, c¼17.6 MHz/Oe, uni-axial anisotropy, Ku2¼2.42/C2104erg/cc along the ^x-axis, and the demagnetization ﬁeld of 4 pMsalong the ^z-axis of the free FM.30The cross-sectional area of all the devices considered is 70/C2160 nm2with the thickness of the free FM layer taken to be 2 nm. The critical spin current required to switch the free FM as described by the above parameters is aroundI sc/C250.52 mA.35 Spin-dependent tunneling in spintronic devices results in different amounts of charge currents ﬂowing in the paral-lel conﬁguration (PC) and the anti-parallel conﬁguration (APC) of the FMs at a given applied bias. Figure 3(a)shows the current-voltage (I-V) characteristics of a trilayer MTJdevice in the PC and APC. Spin dependent charge ﬂow is quantiﬁed by the tunnel magnetoresistance (TMR), deﬁned asTMR¼ðR AP/C0RPÞ=ðRPÞ, where RPandRAPare the resis- tances in the PC and the APC, respectively. The TMR varia- tion with the voltage for a trilayer device is shown in Fig. 3(b). The spin current is a rate of ﬂow of angular momentum that can act as a torque on the magnetization of the free FM. The spin current can be resolved into two components, namely, the Slonczewski term ( ISk) and the ﬁeld-like term (IS?), depending on effects of different magnitudes of the spin currents on the magnetization dynamics of the free FM. We show in Fig. 3(c)the variation of the Slonczewski term36 (ISk) of the spin current with bias voltage. The Slonczewski term can act either as a damping term or as an anti-damping term in the magnetization dynamics of the free FM, regu-lated by the direction of the charge current. When the Slonczewski term acts as an anti-damping term in the mag- netization dynamics, it can destabilize the magnetization ofthe free FM and can result in the switching of the free FM magnetization direction. Figure 3(d) shows the variation of the ﬁeld-like term 36(IS?) of the spin current with voltage bias. The ﬁeld-like term of the spin current acts like an effec- tive magnetic ﬁeld in the magnetization dynamics and can switch the free FM. The non-vanishing part of the ﬁeld-like FIG. 1. Equilibrium energy band proﬁle along the ^zdirection: (a) An FPMTJ device. (b) A BP-FPMTJ-I device (also identiﬁed as AR-FPMTJ). The shaded regime is the anti-reﬂective region (ARR) the details of which have been given in supplementary material Sec. II. (c) Gaussian barrier height and (d) Gaussian barrier width distributed BP-FPMTJ-(II) and (III), respectively. FIG. 2. Device schematics: (a) A trilayer magnetic tunnel junction (MTJ) device having a MgO barrier separating ﬁxed and free FM layers, (b) a FPMTJ with 4-barriers or 3-quantum wells having alternating layers of the MgO (red) barrier and normal metal (green) well sandwiched between the free and the ﬁxed FM layers, and (c) the AR-FPMTJ device comprising asuperlattice heterostructure along with anti-reﬂection regions sandwiched between the free and the ﬁxed FM layers.192404-2 Sharma, Tulapurkar, and Muralidharan Appl. Phys. Lett. 112, 192404 (2018)term at zero-bias is a dissipationless spin current and repre- sents the exchange coupling between the FMs due to the tun-nel barrier. 4The nature of the exchange coupling is determined by the relative positioning of the conduction bands in the FM layers and the insulator. In an MgO based trilayer device sandwiched between CoFeB FM layers, the exchange coupling is of anti-ferromagnetic nature. We show in Fig. 4(a) the I-V characteristics of the FPMTJ with 4-barrier/3-quantum well structure in the PC and APC. The I-V characteristics depict a considerabledifference between the PC and APC, which results in an ultra-high TMR as shown in Fig. 4(b). The TMR shows a roll-off with voltage bias and is attributed to the voltagedependent potential proﬁle across the superlattice structure. 30 Figure 4(c) shows the variation of the Slonczewski term ISk of the spin current with voltage bias. The Slonczewski term increases, acquires the maximum value of ISk/C250:1 mA, and then starts to fall with bias due to the off-resonance conduc-tion. The largest value of I Sk/C250:1 mA in the FPMTJ is nearly ﬁve times smaller than the critical spin currentrequired for magnetization switching in the free FM via thespin transfer torque (STT) effect. 11While the FPMTJ has an ultra-high TMR, smaller spin current positions the FPMTJ asan unfavorable choice for STT switching. Although FPMTJcan be designed to provide a large spin current by having anallowed band of the transmission spectrum within the energyrange between DandE f, the device design yields a very low TMR value.37TheIS?(ﬁeld-like term) variation with voltage bias is shown in Fig. 4(d), and it can be inferred from Fig. 4(d) that the ﬁeld-like term here is negligible to induce any signiﬁcant magnetization dynamics of the free FM. We now plot the I-V characteristics for the AR-FPMTJ with a 4-barrier/3-quantum-well structure in Fig. 5(a) in the PC and the APC. The AR-FPMTJ shows a signiﬁcant asym-metry in the current conduction in both the PC and the APCwhich manifests as an ultra-high TMR across the structure.Figure 5(b) shows the TMR variation for AR-FPMTJ with voltage bias, which is seen to have the same order of magni-tude as the TMR of the FPMTJ near zero bias. An ultra-highTMR in the FPMTJ and AR-FPMTJ is ascribed to physics ofspin selective ﬁltering described in supplementary material Sec. IV. We show in Fig. 5(c) the variation of the Slonczewski term I Skof the spin current with the voltage bias. The Slonczewski term ISkin the AR-FPMTJ shows a FIG. 3. Trilayer MTJ device characteristics: (a) I-V characteristics in the PC and the APC, (b) TMR variation with bias voltage, (c) variation of ISk (Slonczewski term), and (d) variation of IS?(ﬁeld-like term) with applied voltage in the perpendicular conﬁguration of the free and ﬁxed FMs. FIG. 4. FPMTJ device characteristics: (a) I-V characteristics in the PC and the APC, (b) TMR variation with applied voltage, (c) variation of ISk (Slonczewski term), and (d) variation of IS?(ﬁeld-like term) with applied voltage in the perpendicular conﬁguration of the free and ﬁxed FMs. FIG. 5. AR-FPMTJ device characteristics: (a) I-V characteristics in the PC and the APC, (b) TMR variation with bias voltage, (c) variation in ISk (Slonczewski term), and (d) variation in IS?(ﬁeld-like term) with applied voltage in the perpendicular conﬁguration of the free and ﬁxed FMs.192404-3 Sharma, Tulapurkar, and Muralidharan Appl. Phys. Lett. 112, 192404 (2018)nearly symmetric behavior around zero bias, which may enable a near symmetric switching bias in this device. It canbe seen clearly from Figs. 5(c),4(c), and 3(c) that the AR- FPMTJ provides a large spin current in comparison to theFPMTJ and the trilayer MTJ due to the physics of selectiveband-pass spin ﬁltering. We have also rationalized theenhance STT in the AR-FPMTJ structure via the analysis ofthe Slonczewski spin current transmission described in sup- plementary material Sec. IV. We show in Fig. 5(d) theI S? (ﬁeld-like term) variation with the voltage bias. The ﬁeld- like term in the AR-FPMTJ is small and has been neglectedto evaluate switching biases (see supplementary material Sec. I). We show in Fig. 6the temporal variation in the ^xcom- ponent of the magnetization vector of the free FM layer dueto the spin transfer torque at a voltage bias slightly higherthan the critical switching voltage. It can be inferred fromFig.6(a) that APC to PC switching (red) for a trilayer MTJ device is induced by the Slonczewski term which signals anunstable oscillation in the magnetization dynamics beforeswitching. The magnetization switching from PC to APC ina trilayer device is difﬁcult to achieve through theSlonczewski term due to the asymmetry in negative bias andhence can be facilitated by ﬁeld-like terms. The magnetiza-tion switching from the PC to APC (blue) is attributed to the ﬁeld-like term as shown in Fig. 6(a)due to its temporal vari- ation during switching. The AR-FPMTJ device shows nearlysymmetric variation in the Slonczewski term with the biasaround zero bias. The symmetric Slonczewski term and asmall ﬁeld-like term in the AR-FPMTJ facilitate the APC toPC and PC to APC switching via the Slonczewski term itselfas shown in Fig. 6(b). A different switching voltage bias is required to switch from APC to PC and PC to APC due tothe angular dependence of the Slonczewski term in the AR-FPMTJ device. The superlattice structure is identiﬁed by the number of alternate quantum barriers and wells. The number of peaksin the transmission spectrum of a superlattice is either equalto the number of quantum wells or one less than the numberof barriers in the SL structure (see supplementary material Sec. II). We show in Fig. 7(a) the TMR variation with the number of barriers in the superlattice of the AR-FPMTJ device. The TMR increases with an increase in the number of barriers as the transmission spectrum transitions fromunity to nearly zero value with the increase in the number ofbarriers (see supplementary material Sec. II). The TMReventually saturates with the number of barriers as the transi- tion in its transmission spectrum approaches a step function. Figure 7(b) shows that the critical switching bias increases with an increase in the number of barriers. In the AR-FPMTJstructure, an increase in the number of barriers increases theﬂuctuation in the band-pass spectra of transmission, whichreduces the band-pass area under the transmission spectra tocontribute to spin and charge ﬂow. This increases the critical bias voltage requirement for magnetization switching due to spin transfer torque. It can be seen from Fig. 7(b) that the critical switching voltage strength for APC to PC switchingis lower than that for PC to APC due to the angular depen-dence of the Slonczewski term in the AR-FPMTJ device. We can also infer from the above discussion that there is nearly a decrease of 1200% and 1300% in the switching bias fromAPC to PC and PC to APC, respectively, in the AR-FPMTJdevice in comparison to the traditional trilayer MTJ device. We show in Fig. 8the effect of quantum states of the AR-FPMTJ structure on the TMR and Slonczewski spin cur- rent. The variation in the width of the quantum wells in theAR-FPMTJ structure changes the position of the transmis-sion spectrum with respect to the Fermi level and manifestsas a periodic variation in the TMR as a function of the well width as seen in Fig. 8(a). Figure 8(b) shows the variation of the Slonczewski spin current as a function of the well width.Due to the quantum states of the structure, the spin currentalso shows a periodic variation with the quantum well width.It can be inferred from Fig. 8that the width of the quantum well at which either the largest TMR or the highest Slonczewski current is observed does not converge to singu- lar points. But still, in the design landscape of the well width, FIG. 6. Spin transfer torque induced magnetization switching proﬁles of the free FM (a) in the trilayer MTJ device and (b) the AR-FPMTJ device with a 3-quantum well structure. FIG. 7. (a) The variation of TMR and (b) critical switching voltage ( VC) for the AR-FPMTJ device as a function of the number of barriers in the super-lattice structure. FIG. 8. (a) The TMR and (b) Slonczewski spin current as a function of quan-tum well width for the 3-barrier AR-FPMTJ device under an applied voltage of 20 mV.192404-4 Sharma, Tulapurkar, and Muralidharan Appl. Phys. Lett. 112, 192404 (2018)there are many possibilities which facilitate the AR-FPMTJ device design with a boosted TMR and low switching bias. We have proposed a fresh route for high-performance spin-transfer torque devices by tapping the band-pass trans-mission proﬁle of an AR-FPMTJ structure sandwiched between the two FM layers. We showed that the physics of spin selective band-pass ﬁltering enabled through the ARregion translates to an ultra-high TMR with ultra-low switch- ing bias. We have estimated that the AR-FPMTJ device caters to a TMR ( /C255/C210 4%) and nearly to a 1200% lower- ing of the switching bias in comparison to a typical trilayer MTJ device. We believe that our idea of using band-passtransmission engineering will open up further theoretical and experimental endeavors in the spintronics ﬁeld. Speciﬁcally, it would be interesting to investigate the BP-FPMTJ struc-tures to provide enhanced thermal spin-transfer torque 38by engineering “box-car” spin selective transmission proﬁles.39 The idea of bandpass spin-ﬁltering can also be extended to similar device structures for “multilevel spin transfer torque devices.”40 Seesupplementary material for details about calcula- tions, anti-reﬂective region design, Slonczewski spin current transmission, and physics of spin ﬁltering. A.S. would like to acknowledge Smarika Kulshrestha for her suggestions on the initial draft of this work. A.S. would also like to acknowledge Pankaj Priyadarshee and Swarndeep Mukherjee for introducing him to the ﬁeld ofsuperlattices. This work was in part supported by the IIT Bombay SEED grant. 1W. H. Butler, X.-G. Zhang, T. C. Schulthess, and J. M. MacLaren, Phys. Rev. B 63, 054416 (2001). 2S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Samant, and S.-H. Yang, Nat. Mater. 3, 862 (2004). 3D. D. Djayaprawira, K. Tsunekawa, M. Nagai, H. Maehara, S. Yamagata, N. Watanabe, S. Yuasa, Y. Suzuki, and K. Ando, Appl. Phys. Lett. 86, 092502 (2005). 4J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 5L. Berger, Phys. Rev. B 54, 9353 (1996). 6S. Assefa, J. Nowak, J. Z. Sun, E. O’Sullivan, S. Kanakasabapathy, W. J. Gallagher, Y. Nagamine, K. Tsunekawa, D. D. Djayaprawira, and N. Watanabe, J. Appl. Phys. 102, 063901 (2007). 7P. Khalili Amiri, Z. M. Zeng, J. Langer, H. 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1.4792214.pdf	Material selection considerations for coaxial, ferrimagnetic-based nonlinear transmission lines J.-W. B. Bragg, J. C. Dickens, and A. A. Neuber Citation: Journal of Applied Physics 113, 064904 (2013); doi: 10.1063/1.4792214 View online: http://dx.doi.org/10.1063/1.4792214 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/113/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Microwave absorption in the frustrated ferrimagnet Cu 2 O Se O 3 Low Temp. Phys. 36, 176 (2010); 10.1063/1.3319505 Volumetric negative-refractive-index metamaterials based upon the shunt-node transmission-line configuration J. Appl. Phys. 102, 094903 (2007); 10.1063/1.2803924 Frequency response analysis of a 1:1 transmission line based transformer Rev. Sci. Instrum. 73, 3652 (2002); 10.1063/1.1505109 Analytical solitons in nonlinear transmission lines loaded with heterostructure barrier varactors J. Appl. Phys. 90, 2595 (2001); 10.1063/1.1388863 Simulations of tapered Goubau line for coupling microwave signals generated by resonant laser-assisted field emission J. Vac. Sci. Technol. B 18, 1009 (2000); 10.1116/1.591345 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 86.148.26.146 On: Wed, 07 May 2014 11:45:34Material selection considerations for coaxial, ferrimagnetic-based nonlinear transmission lines J.-W. B. Bragg, J. C. Dickens, and A. A. Neuber Center for Pulsed Power and Power Electronics, Department of Electrical & Computer Engineering, Texas Tech University, Lubbock, Texas 79409, USA (Received 20 December 2012; accepted 30 January 2013; published online 12 February 2013) The growing need for solid-state high power microwave sources has renewed interest in nonlinear transmission lines (NLTLs). This article focuses speciﬁcally on ferrimagnetic-based NLTLs in acoaxial geometry. Achieved peak powers exceed 30 MW at 30 kV incident voltage with rf power reaching 4.8 MW peak and pulse lengths ranging from 1–5 ns. The presented NLTL operates in S- band with the capability to tune the center frequency of oscillation over the entire 2–4 GHz bandand bandwidths of approximately 30%, placing the NLTL into the ultra-wideband–mesoband category of microwave sources. Several nonlinear materials were tested and the relationship between NLTL performance and material parameters is discussed. In particular, the importance ofthe material’s ferromagnetic resonance linewidth and its relationship to microwave generation is highlighted. For a speciﬁc nonlinear material, it is shown that an optimum relation between incident pulse magnitude and static bias magnitude exists. By varying the nonlinear material’s biasmagnetic ﬁeld, active delay control was demonstrated. VC2013 American Institute of Physics . [http://dx.doi.org/10.1063/1.4792214 ] I. INTRODUCTION The idea of nonlinear transmission lines is not new and traditionally NLTLs, especially coaxial, ferrimagnetic-based, aided slower switches with the ability to provide sub- nanosecond risetime outputs with several nanosecond rise-time inputs. 1,2Recently, NLTLs have gained more attention as possible solutions to ﬁll the need for high power, compact, and solid-state microwave sources. The NLTL is not limitedto coaxial, ferrite-based systems, but encompasses several geometries and modes of operation. A NLTL can be realized in stripline, parallel-plate, microstrip, and lumped elementgeometries in addition to a coaxial geometry. 3–6All geome- tries rely on semiconductors, nonlinear dielectrics, and/or ferrimagnetics, but produce microwaves through differentmeans, i.e., damped gyromagnetic precession, soliton forma- tion, and synchronous wave operation. Frequencies scale from hundreds of MHz up to hundreds of GHz, but comewith an inverse relation between rf power (single cycle peak power) and frequency. 7The coaxial, ferrimagnetic NLTL (from here forth, simply called NLTL) operates through theproduction of a shockwave followed by damped gyromag- netic precession. Two of the main determining factors of successful microwave generation are the magnetic loss tan-gent and internal magnetic ﬁeld of the sample; therefore, this article details the principles of operation and effects of mate- rial properties and magnetic ﬁelds (both incident pulse andstatic bias) on the overall microwave performance of NLTLs. II. BACKGROUND Several mechanisms contribute to pulse sharpening down to sub-nanosecond risetimes in the nonlinear transmis- sion lines (NLTL). The dominant mechanism of pulse sharp-ening is through generating a shockwave which occurs through saturation of the material’s nonlinear permeability,energy dissipation at the traveling pulse front, and spin re- versal in the magnetic material. 1,8As the incident pulse tra- verses the NLTL, the nonlinear permeability undergoessaturation and the degree of saturation depends on the mag- nitude of the pulse. Consequently, the crest of the pulse front travels through a permeability of lower value than the baseof the pulse front. The inverse relation between phase veloc- ity and permeability allows the pulse front crest to “catch- up” to the base. Concurrently, the material is non-ideal, andtherefore has associated losses. These losses result in energy dissipation across the pulse front, which aid in pulse steepen- ing. With the application of an axial, static, biasing magneticﬁeld, the magnetic moments begin to align in the same direc- tion. The azimuthal magnetic ﬁeld from the incident pulse switches the moments from their initial position into a newposition establishing a spin reversal region which leads to magnetic moment switching aiding in the process of pulse sharpening. Microwave generation occurs through damped gyro- magnetic precession and can be accurately described by the Landau-Lifshitz-Gilbert equation (LLG). The normalizedrepresentation of the LLG is presented in Eq. (1). The LLG is a differential equation describing the magnetization dy- namics taking place in the ferrite. @m @t¼/C0cm/C2hef fþam/C2@m @t: (1) The ﬁrst term on the right hand side of Eq. (1)represents the precessional motion of the magnetic moment, m, around an effective magnetic ﬁeld, heff, while the second term is the relaxation term, governing the speed at which the moment 0021-8979/2013/113(6)/064904/4/$30.00 VC2013 American Institute of Physics 113, 064904-1JOURNAL OF APPLIED PHYSICS 113, 064904 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 86.148.26.146 On: Wed, 07 May 2014 11:45:34aligns or relaxes in the direction of the effective ﬁeld. They gyromagnetic factor is represented by cand the damping constant by a. The effective ﬁeld consists of the exchange ﬁelds, anisotropy ﬁelds, demagnetizing ﬁelds, and external ﬁelds,9but due to the size and nature of the sample only demagnetizing ﬁelds and external ﬁelds are consideredthroughout design. The external ﬁelds consist of the axial biasing ﬁeld and incident azimuthal ﬁeld. Here, the axial biasing ﬁeld is produced with a solenoid while the azimuthalﬁeld is due to the incident high voltage pulse. The biasing ﬁeld initially aligns the magnetic moments in the axial direc- tion while the azimuthal ﬁeld pulls the moments away andswitches them into the direction of the effective ﬁeld. III. EXPERIMENTAL SETUP A high voltage dc power supply is utilized to charge a 2.5 nF capacitor bank to a user deﬁned level, typically in the range of 20–40 kV, see Figure 1. Upon purging the pressurized cavity, the spark gap is over-volted and a damped Resistor- Inductor-Capacitor (RLC) signal propagates through a com- mercially available coaxial cable acting as a delay line beforearriving at the NLTL input. After traversing the NLTL, the sig- nal is terminated into a 50 Xresistive load. The delay lines before and after the NLTL provide temporal isolation for diag-nostic purposes in case of reﬂections between the source, load, and NLTL due to the dynamic impedance of the NLTL. Two high speed capacitive voltage probes are used as the maindiagnostic tools, capturing the incident and output waveforms. The NLTL consists of an aluminum or brass coaxial structure with nickel-zinc (NiZn) ferrites snugly ﬁt along theinner conductor. Yttrium iron garnet (YIG), magnesium-zinc (MgZn), manganese-zinc (MnZn), and lithium ferrite have also been used in NLTL technologies, yet presently the mostsuccess has come from various compositions of NiZn. The high voltage levels and small space between conductors necessitates the use of an electrical insulator. Pressurized sul-fur hexaﬂuoride (SF 6at 620 kPa) acts as the dielectric me- dium. Refer to Fig. 7 of Ref. 12for a cross-sectional view of the constructed NLTL. A secondary dc power supply pro-vides the necessary current through a solenoid wrapped around the outer conductor of the NLTL to produce the axi- ally directed, magnetic biasing ﬁeld. The overall system impedance is designed for 50 X, assuming saturated permeability for the ferrite. Typically, the solenoid induced bias saturates the material, but if unsat-urated, the incident azimuthal pulse quickly saturates the ma-terial. The line impedance is varied by varying the outer conductor of the NLTL as the ferrites are generally more dif- ﬁcult to fabricate and brass tubes are available in severalsizes. The electric and magnetic ﬁeld are highly dependent on the inner and outer conductor diameters as well as the fer- rite sizes. Consequently, care must be taken in order to notexceed the voltage breakdown threshold of the ferrite or dielectric insulator as well as aiming to maintain a consistent magnetic ﬁeld throughout the ferrimagnetic material. For agiven driving current, the magnetic ﬁeld has an inverse rela- tion to the samples radius and thus the frequency of opera- tion and bandwidth are affected by the ferrite’s size. In the design stages, the frequency of operation is deter- mined through traditional magnetization dynamics techni- ques, speciﬁcally the Smit and Beljers approximation forcalculating the ferromagnetic resonance (FMR) frequency. The method utilizes the externally applied magnetic ﬁelds and demagnetizing ﬁelds calculated from the sample’sdimensions. A MATLAB program takes an FFT of the output signal and an order of 10% accuracy has been achieved uti- lizing the traditional FMR techniques. IV. RESULTS Various ferrite material parameters signiﬁcantly impact the efﬁciency of microwave generation. Two primaries are the magnetic and electric loss factors. While the dielectricloss value is typically stated in the data sheet of commercially available ferrites, the magnetic loss is generally unknown and needs to be measured in a separate experiment. FMR techni-ques have been utilized to measure the precession linewidth of several tested materials, and the FMR cavity resonance technique 10was employed here. The ferrimagnetic material is sufﬁciently thinned, shaped into an ellipse, and placed in an X-band cavity resonator. The thin, elliptical shape allows the use of known demagnetizing ﬁelds and will not perturbthe ﬁelds in the resonator. A 9.53 GHz source is used to excite the cavity resonator and a dc magnetic ﬁeld is placed around the ferrite sample in order to saturate the material.The magnitude of the ﬁeld is then swept, and the absorbed microwave power into the ferrite is recorded at various dc magnetic ﬁelds. When the magnetic ﬁeld is such that reso-nance in the material is achieved, the power absorption peaks. The fundamental equation relating resonance and magnetic ﬁeld is x¼cH 0, where xis the resonance frequency, cis the gyromagnetic ratio, and H0is the internal magnetic ﬁeld. The FMR linewidth of the ferrite is recorded as the full width-half max (FWHM) of the absorbed power peak. This linewidth isdirectly related to the phenomenological damping factor found in LLG, Ref. 1. The waveforms listed in Figure 2por- tray FMR spectra measured by Metamagnetics, Inc., for sixmaterials with applied dc ﬁeld, measured in kilo-Oersted, on the x-axis and the derivative of the absorbed power, measured in arbitrary units, listed on the y-axis. To complement the FMR spectra are four measured wave- forms from the NLTL output found in Figure 3. These wave- forms represent a commercially available NiZn sample andthree custom NiZn samples developed by Metamagnetics, Inc., narrow linewidth ( <280 Oe) material have produced FIG. 1. Block diagram of the experimental setup. Capacitive voltage probes are used as diagnostics and are found at the input and output of the NLTL.064904-2 Bragg, Dickens, and Neuber J. Appl. Phys. 113, 064904 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 86.148.26.146 On: Wed, 07 May 2014 11:45:34microwave oscillations with narrower linewidths producing higher rf power and a longer duration of precession. A MnZnsample shows very narrow linewidth, but has resistivity 5–8 orders of magnitude lower than the NiZn samples. Due to the high electric loss, this MnZn sample proved to be toolossy and did not produce microwaves. In addition to low magnetic and dielectric loss tangents, the ferrite must have moderate values of initial relative permeability and satura-tion magnetization. The most successful ferrites have relativepermeabilities in the upper hundreds and saturation magnet- izations around 3500 G. In addition to speciﬁc material parameters, the opera- tional performance of the NLTL highly depends on the inci- dent pulse and static biasing ﬁeld magnitudes. Typical trends include increasing frequency with increasing incident pulseamplitude and decreasing frequency with increasing bias ﬁeld. 11,12This can be attributed to the total amount of azi- muthal ﬁeld seen by the ferrite. Since the NLTL geometry iscoaxial, the acting propagation mode is transverse electric and magnetic (TEM) and thus an azimuthally-directed magnetic ﬁeld is primary. Hence, the azimuthal component of the rotat-ing magnetic moment couples to the TEM mode progressing down the line. Due to this coupling, it is evident that as the incident ﬁeld increases, the projection of the rotating momentonto the azimuthal axis increases. In contrast, if the bias ﬁeld increases, the azimuthal contribution decreases. Frequency versus bias ﬁeld plots at varying incident voltage magnitudescan be found in Figure 4. Expectedly, there exists optimal incident amplitude - bias combinations to achieve maximum rf output power, see Fig-ure5. At low bias ﬁeld strengths, the majority of the magnetic moments are not aligned in the axial direction. Consequently, upon application of the azimuthal ﬁeld, there exists a state ofincoherent switching and precession between the moments, resulting in low rf power generation. In contrast, if the magni- tude of the bias ﬁeld is too large, the azimuthal ﬁeld is tooweak to signiﬁcantly move the magnetic moments. Therefore, the azimuthal inﬂuence of the rotating magnetic moments is decreased. Interestingly, the electrical delay of the system can be controlled with the NLTL. This is achieved through control- ling the initial permeability of the ferrite with the biasingﬁeld. As the magnetic bias is increased to higher strengths, the material begins to saturate and the permeability decreases. Thus, the bias can effectively control the initialpermeability seen by the incoming pulse front and therefore control the phase velocity of the wave. By altering the bias magnitude, length of bias, and ferrite, the NLTL can beactively tuned for speciﬁc delay times. This provides an FIG. 2. FMR spectra of ﬁve different materials. The waveforms represent the derivative of absorbed power. The oscillatory nature prior to the main linewidth peak arises from spin-wave excitation. FIG. 3. NLTL Output waveforms for four different materials. The colors ofeach waveform correspond to the colors represented in Figure 2. The materi- als are NiZn-1, MX5, MX7, and MX8 (moving left to right). Each material has a different linewidth (ﬁrst—260 Oe, second—280 Oe, third—130 Oe, and fourth—120 Oe). FIG. 4. Center frequency versus magnetic bias ﬁeld for incident voltages20 kV (black- /H11623), 25 kV (red- /H17034), and 30 kV (blue- D).064904-3 Bragg, Dickens, and Neuber J. Appl. Phys. 113, 064904 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 86.148.26.146 On: Wed, 07 May 2014 11:45:34exciting opportunity for NLTL integration into phased array systems. The delay can be altered through the use of one bias (controlling output power, frequency, and delay), the use of multiple biases on a single line (one for power/frequency,one for delay), or the use of multiple NLTLs and multiple biases. If using multiple biases for each NLTL, the bias located at the output portion of the NLTL determines theoutput power and frequency, while the electrical delay can be altered by a bias located on the front end of the NLTL. 11 At 20 kV, the total electrical delay was demonstrated to vary between 20.5 ns down to 9 ns. Figure 6contains line plots of the changing electrical delay versus bias at various incidentpulse magnitudes. The ﬁgure inset shows the dramatic delay change between unbiased and fully biased (presently 42 kA/ m) lines. The dotted line represents the delay of full satura-tion which all lines asymptotically trend toward. V. CONCLUSIONS Nonlinear transmission lines have proven to be potential alternatives to traditional, vacuum-based high power micro- wave sources. Traditional magnetization dynamics techni-ques to predict frequency and measure microwave losses can be applied to NLTL design for determining operational fre- quency and expected material performance. Additionally,through altering the static bias, frequency tuning and system phasing can be achieved. Two material parameters have been determined with experimental results to verify the im-portance of magnetic and electric loss for microwave genera- tion. This article shows the importance of careful design to achieve large magnetic ﬁelds in the ferrite, tailored magneticﬁeld differential for bandwidth control, and choice of mate- rial for optimal microwave generation. ACKNOWLEDGMENTS This work was supported by the U.S. Ofﬁce of Naval Research (ONR). 1I. Katayev, Electromagnetic Shock Waves (Iliffe, 1966). 2J. Dolan, “Simulation of shock waves in ferrite-loaded coaxial transmis- sion lines with axial bias,” J. Phys. D: Appl. Phys. 32, 1826 (1999). 3I. Romanchenko, V. Rostov, V. Gubanov, A. Stepchenko, A. Gunin, and I. Kurkan, “Repetitive sub-gigawatt rf source based on gyromagnetic nonlin- ear transmission line,” Rev. Sci. Instrum. 83, 074705 (2012). 4J. Darling and P. Smith, “High power pulsed rf generation from nonlinear lumped element transmission lines (NLETLs),” University of Oxford, Technical Presentation, given at the University of New Mexico, 2008. 5N. Seddon, C. Spikings, and J. Dolan, “RF pulse formation in nonlineartransmission lines,” in IEEE 34th International Conference on Plasma Science , Albuquerque, NM, (2007). 6H. Shi, C. W. Domier, and N. C. Luhmann, “A monolithic nonlinear trans- mission line system for the experimental study of lattice solitons,” J. Appl. Phys. 78, 2558 (1995). 7M. Jamshidifar, G. Spickermann, H. Schafer, and P. Haring Bolivar, “200- GHz bandwidth on wafer characterization of CMOS nonlinear transmis- sion line using electro-optic sampling,” Microwave Opt. Technol. Lett. 54(8), 1858 (2012). 8M. Weiner and L. Silber, “Pulse sharpening effects in ferrites,” IEEE Trans. Magn. 17(4), 1472 (1981). 9I. Mayergoyz, G. Bertotti, and C. Serpico, Nonlinear Magnetization Dy- namics in Nanosystems (Elsevier, 2008). 10A. Geiler, private communication (2011). 11J. -W. B. Bragg, J. C. Dickens, and A. A. Neuber, “Nonlinear transmission line performance under various magnetic bias environments,” in Directed Energy Professional Society Annual Directed Energy Symposium ,L a Jolla, CA, (2011). 12J.-W. Bragg, W. W. Sullivan III, D. Mauch, J. C. Dickens, and A. A. Neuber, “Compact pulsed power system realized through integrated SiC photoconductive semiconductor switch and gyromagnetic nonlinear trans- mission line,” Rev. of Sci. Instr. (submitted). FIG. 5. NLTL outputs for 20 kV incident pulse magnitude and varying bias.The waveforms are arranged such that the bias ﬁeld is increasing from left to right. For ease of viewing, the waveforms are time-shifted by 3 ns relative to each other. FIG. 6. The main ﬁgure contains the NLTL electrical delay versus biasing ﬁeld at 20 kV (black- /H11623), 25 kV (red- /H17034), and 30 kV (blue- D). The gray line represents the calculated electrical length of the line when fully saturated. The inset ﬁgure contains waveforms for unbiased (0 kA/m, black, no oscilla- tions) and fully biased (42 kA/m, red, oscillations present) outputs.064904-4 Bragg, Dickens, and Neuber J. Appl. Phys. 113, 064904 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 86.148.26.146 On: Wed, 07 May 2014 11:45:34
5.0013363.pdf	Appl. Phys. Lett. 117, 062403 (2020); https://doi.org/10.1063/5.0013363 117, 062403 © 2020 Author(s).Current-induced torques in black phosphorus/permalloy bilayers due to crystal symmetry Cite as: Appl. Phys. Lett. 117, 062403 (2020); https://doi.org/10.1063/5.0013363 Submitted: 11 May 2020 . Accepted: 01 August 2020 . Published Online: 13 August 2020 Wenxing Lv , Jialin Cai , Zhilin Li , Weiming Lv , Yan Shao , Shangkun Li , Baoshun Zhang , Yukai Chang , Zhongyuan Liu , and Zhongming Zeng ARTICLES YOU MAY BE INTERESTED IN Magnon-drag thermoelectric transport with skyrmion structure Applied Physics Letters 117, 062404 (2020); https://doi.org/10.1063/5.0017272 Cherenkov-type three-dimensional breakdown behavior of the Bloch-point domain wall motion in the cylindrical nanowire Applied Physics Letters 117, 062402 (2020); https://doi.org/10.1063/5.0013002 Two-dimensional BP/ β-AsP van der Waals heterostructures as promising photocatalyst for water splitting Applied Physics Letters 117, 063901 (2020); https://doi.org/10.1063/5.0014867Current-induced torques in black phosphorus/ permalloy bilayers due to crystal symmetry Cite as: Appl. Phys. Lett. 117, 062403 (2020); doi: 10.1063/5.0013363 Submitted: 11 May 2020 .Accepted: 1 August 2020 . Published Online: 13 August 2020 Wenxing Lv,1,2Jialin Cai,2 Zhilin Li,3Weiming Lv,2YanShao,1,2Shangkun Li,2Baoshun Zhang,1,2 Yukai Chang,4 Zhongyuan Liu,4and Zhongming Zeng1,2,a) AFFILIATIONS 1School of Nano Technology and Nano Bionics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China 2Key Laboratory of Multifunctional Nanomaterials and Smart Systems, Suzhou Institute of Nano-Tech and Nano-Bionics, CAS, Suzhou, Jiangsu 215123, People’s Republic of China 3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, CAS, Beijing 100190, People’s Republic of China 4State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, China a)Author to whom correspondence should be addressed: zmzeng2012@sinano.ac.cn ABSTRACT Current-induced spin-torques in two-dimensional (2D) heterostructures have attracted extensive attention due to their importance in under- standing the underlying fundamental physics and developing low-power dissipation nanoelectronics. Here, the Permalloy/black phosphorus(BP) bilayer devices are fabricated, and spin-torque ferromagnetic resonance (ST-FMR) measurements are utilized to investigate thespin-torque effect in the heterostructure. An obvious out-of-plane antidamping torque is observed, which could be associated with the bro-ken mirror symmetry of BP. These results show the possibility of manipulating magnetization by semiconductor ﬁeld-effect devices based on 2D materials and provide a clear avenue for engineering spintronic devices based on 2D materials. Published under license by AIP Publishing. https://doi.org/10.1063/5.0013363 Current-induced torques produced by materials with strong spi- n–orbit coupling (SOC) interactions provide a promising approach for effective manipulation of nano-magnetic devices. 1The SOC interac- tion shows up in crystalline structures that possess structure inversion asymmetry (the Rashba type2) or bulk inversion asymmetry (the Dresselhaus type3). Therefore, spin–orbit torques (SOTs) to date are usually observed in conventional heavy-metal/ferromagnet bilayers4–7 and topological-insulator/ferromagnet thin-ﬁlm bilayers,8where the heavy-metal or topological-insulators are employed as spin-source materials that generate current-induced spin–orbit torques (SOTs) through spin Hall, Rashba–Edelstein, topological spin-momentumlocking, or other spin–orbit effects. 13,14Recently, two-dimensional (2D) materials, such as transition metal dichalcogenides (TMDs),9–12 have been acknowledged as intriguing spin-source materials due tostrong SOC interaction, surface states, and reduced crystal symmetry. For example, strong anisotropic spin–orbit interaction, 15large proximity-induced spin lifetime anisotropy,16and room-temperature spin hall effect17have been reported in WS 2/graphene, MoSe 2/gra- phene, and MoS 2/graphene heterostructure, respectively. A large SOT was observed in the TMD(MoS 2and WSe 2)/CoFeB bilayer.18Moreinterestingly, out-of-plane SOT19,20related to crystal symmetry and ﬁeld-free current-induced magnetization switching21in response to an in-plane current were demonstrated for the WTe 2/Permalloy hetero- structure, which is a highly efﬁcient strategy for the switching of nano- scaled ferromagnets for memory and logic devices. Apart from WTe 2, black phosphorus (BP) also displays low crystal symmetry due to its puckered surface along the x-direction in each monolayer owing to sp3hybridization, which enables a series of new functionalities for future nanodevices due to the anisotropic electric, thermal, optical, and spintronic characteristics.22–30Very recently, Popovic ´et al.29have theoretically reported an anisotropic Rashba effect in monolayer BP. Jung et al.30have experimentally reported that BP material could be a promising candidate for pseu-dospintronics due to its bipolar pseudospin polarization greater than 95% at room temperature. Avsar et al. 31have investigated the spin transport in ultrathin BP and indicated the tunability of spin trans-port by the electric ﬁeld effect. These investigations suggest huge potential for BP in future versatile spintronic applications as it, meanwhile, shows outstanding semiconductor properties, 32–35 such as ultrahigh electric conductivities. However, until now, the Appl. Phys. Lett. 117, 062403 (2020); doi: 10.1063/5.0013363 117, 062403-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplspin–orbit torques originating from the structural asymmetry of BP remain undiscovered. In this work, we fabricated a Permalloy (Py)/BP bilayer device and investigated the spin-torque effect using spin-torque ferromag-netic resonance (ST-FMR) measurements. The measurements show obvious spin–orbit torques, which may originate from the broken mir- ror symmetry in the BP. Our results could be helpful for comprehen-sively understanding the spintronic properties of 2D/magnetic ﬁlmheterostructures. The Py/BP bilayer device was fabricated as follows: ﬁrst, few- layer BP was mechanically exfoliated from a bulk BP crystal onto thepolydimethylsiloxane (PDMS) template and then transferred onto a280 nm SiO 2substrate using a common dry transfer method. The exfoliation procedure was performed inside a nitrogen-ﬁlled glovebox with H 2O and O 2concentrations <0.5 ppm. After the transition, the samples were immediately spin-coated with MMA and PMMA for preventing oxidation. Then, a stripe of 20 nm Py (Ni 80Fe20) was depos- ited on the BP layer by using electron beam lithography combinedwith electron beam evaporation (EBE). Finally, Ti(10 nm)/Au(60 nm)contacts in the shape of a coplanar waveguide were deﬁned using pho- tolithography and deposited using EBE. The BP ultrathin layer was characterized by Raman spectroscopy with a laser radiation of 532 nm and a power of 10 lW. The surface morphology and thickness of the sample were obtained using an atomic force microscope (AFM, Multimode 8, Veeco Instruments, I n c . ,U S A ) .T h es t a t i cm a g n e t i cp r o p e r t i e so fP yw e r ec h a r a c t e r i z e dusing a vibration sample magnetometer (VSM, Lakeshore, USA). TheST-FMR technique in which microwave signals ( I RF) produced by a generator were applied within the sample plane through a bias tee was used to measure the strength of torques.9In the ST-FMR measure- ment, current-induced torques caused the magnetization of Py to pro- cess, creating a device resistance oscillation due to the anisotropic magnetoresistance (AMR) of the ferromagnetic layer. The oscillatedresistance mixes with RF current and further generated a mixing volt-ageV mixacross the sample, which was recorded by the lock-in ampli- ﬁer. All the measurements were performed at room temperature. BP has a puckered lattice in a monolayer and stacked its mono- layer along the z-axis ,a ss h o w ni n Fig. 1(a) . Each monolayer is com- posed of two parallel planes, and every BP atom has three nearest neighbors. Figure 1(b) shows that the Raman spectra of BP, A1 g (359.7 cm/C01),B2g(434.4 cm/C01), and A2 g(461.3 cm/C01)m o d e sa r e observed, which is consistent with the previous studies.35,36The mag- netization hysteresis loop for the 20 nm Py ﬁlm is depicted in Fig. 1(c) applying an external ﬁeld in the plane. The rectangular shape of theloop indicates an in-plane easy axis of Py. Figure 2(a) illustrates an optical image of a ﬁnished device, including contact GSG pads and circuit of the ST-FMR measurement.An in-plane magnetic ﬁeld Hat an angle uwith respect to I RF was applied, and the magnitude of this ﬁeld was sufﬁcient to go through the ferromagnetic resonance condition, as shown in Fig. 2(b) . Figure 3(a) shows the ST-FMR spectra for the Py/BP device in the frequency range of 3.0 GHz to 7.0 GHz at an incident power of10 dBm. The amplitude decreases as the frequency increases. The mix- ing voltage V mixc a nb ee x p r e s s e db yaL o r e n t z i a nf u n c t i o nc o n s i s t i n g of a symmetric and an antisymmetric Lorentzian component asfollows: 19 Vmix¼VSDH2 4H/C0H0 ðÞ2þDH2þVA4DHH/C0H0 ðÞ 4H/C0H0 ðÞ2þDH2;(1) where DHis the linewidth, H0is the resonant ﬁeld, His the applied magnetic ﬁeld, VSis the symmetric Lorentzian amplitude, and VAis the antisymmetric Lorentzian amplitude. As deﬁned in Eq. (1),t h e amplitudes VSandVAare related to the two components of torque as follows:5 VS¼/C0IRF 2dR du/C18/C191 ac2H0þl0Meff ðÞsk; (2) VA¼/C0IRF 2dR du/C18/C19 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þl0Meff=H0p ac2H0þl0Meff ðÞs?; (3) where skis an in-plane torque, s?is an out-of-plane torque, and Ris the device resistance. By ﬁtting the measured Vmixcurve, the parame- ters of DH,VS,VA,a n d H0can be obtained, further analyzing the spin-torque effect. Figure 3(b) shows the resonant fdependence of external mag- netic ﬁeld H.S i m i l a rt ot h ep u r eP yd e v i c e ,t h er e s o n a n t fof the Py/BP device increases as Hincreases and the curves can be well ﬁtted using the Kittel formula,5 FIG. 1. (a) Crystal structure of BP . (b) Raman spectra of BP. (c) Hysteresis loop of the Py ﬁlm applying an external ﬁeld in the plane. FIG. 2. (a) Optical images of the sample geometry including contact pads and schematic of the ST-FMR measurement circuit. (b) Schematic of the bilayer Py/BP sample geometry. FIG. 3. (a) ST-FMR spectra at a series of frequencies from 3.0 GHz to 7.0 GHz with a ﬁxed power of 10 dBm and u¼45/C14. (b) FMR resonant frequency as a func- tion of the applied magnetic ﬁeld for pure Py and Py/BP bilayers, respectively. The solid lines represent the theoretical ﬁtting using the Kittle equation. (c) The linewidthDHextracted from the ﬁtting of the ST-FMR signal vs the resonant frequency f. The solid lines are the linear ﬁttings.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 062403 (2020); doi: 10.1063/5.0013363 117, 062403-2 Published under license by AIP Publishingf¼c 2pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ H0H0þl0Meff ðÞp ; (4) where cis the gyromagnetic ratio, l0is the permittivity in vacuum, andMeffis the effective saturation magnetization of Py. For our Py/BP device, we have l0Meff/C250.8 T. The frequency fdependences of the resonance linewidth DHare demonstrated in Fig. 3(c) .DHincreases as the frequency increases for both Py and Py/BP devices, but DH increases more quickly for the Py/BP device. This parameter is gener- ally determined by intrinsic and extrinsic origins, which is given by37,38 DH¼DH0þ2pa c/C18/C19 f; (5) where DH0is the extrinsic contribution and the second term is the intrinsic contribution to the linewidth. The extrinsic contribution, such as inhomogeneous broadening, is frequency-independent, while the second term is linearly proportional to the frequency fand associ- ated with the Gilbert damping coefﬁcient a.T h e avalues are approxi- mately 0.011 for the Py/BP device and 0.01 for the pure Py device. This suggests that the BP ultralayer has an obvious inﬂuence on the properties of Py. Next, we focus on the spin-torque effect in the Py/BP hetero- structures. For comparison, we performed the ST-FMR measure- ments for Py and Py/BP devices at two given angles, similar to that in the previous study.19The results are plotted in Fig. 4 . It is clearly seen that for the pure Py device [see Fig. 4(a) ], the ST-FMR traces show a nearly identical line shape at u¼/C045/C14and 135/C14after multi- plying u¼/C045/C14by/C01. Note that the dominated symmetric feature in the pure Py layer is related to the thickness of Py.39However, the result of the Py/BP device carried out in the same experiment dis- plays the difference between Vmix(135/C14) and /C0Vmix(/C045/C14) cases. This indicates that there may be extra torques affecting the twofold rotational symmetry of the line shape, similar to the previous report in the WTe 2/Py heterostructure.19,20 To get much insight into this, we conducted the full angular (u¼/C0180/C14to 180/C14)-dependent measurement of ST-FMR signal Vmixwhere the applied magnetic ﬁeld is used to rotate the magnetiza- tion within the Py ﬁlm at a ﬁxed frequency of 3.5 GHz and a power of 10 dBm. By ﬁtting the measured ST-FMR spectra using Eq. (1),t h e symmetric components VSand antisymmetric components VAas a function of the in-plane magnetic ﬁeld angle uare obtained, as shown inFigs. 5(a) and5(b), respectively. In a simple heavy-metal/ferromag- net bilayer with a twofold rotation symmetry, the current-inducedtorque amplitude follows a cos( u) dependence and the AMR in Py has ac o s2(u) angular dependence, which enters Vmixas d R/du/C25 sin(2u). As a consequence, these two contributions yield the same angular dependence for both symmetric and antisymmetric compo-nents, that is, V S¼Scos(u)sin(2 u)a n d VA¼Acos(u)sin(2 u). For the Py/BP device, the VSbehavior was well ﬁtted by this angular dependence, revealing that the spin–orbit torques are present in thePy/BP bilayer since the symmetric component implies an in-plane tor- que and it cannot be produced by an Oersted ﬁeld, as shown in Fig. 5(a). However, the antisymmetric component V Ais different from cos(u)sin(2 u), as demonstrated in Fig. 5(b) . The absolute values of sig- nal amplitudes clearly lack u!u/C0180/C14symmetry, which can be well ﬁtted by adding a term of sin(2 u), VA¼AcosðuÞsinð2uÞþBsinð2uÞ; (6) where A and B are constants independent of the magnetic ﬁeld angle. As we remove the term of sin(2 u) contributing from the angular dependence of the AMR by using Eq. (3),i ti sc l e a r l ys e e nt h a tan e w term torque sBcorresponding to an out-of-plane torque is observed, which is independent of the in-plane magnetization orientation, i.e., iteven can be in ^m. Therefore, this new term torque s Bis found to be a damping-like torque. As the current-induced spin-torques generated by the spin Hall effect, the Rashba–Edelstein effect, or the Oersted ﬁeld strongly rely on magnetization orientation, it is reasonable to infer that the out-of- plane antidamping torque sBis not due to neither the spin Hall effect, the Rashba–Edelstein effect, nor the Oersted ﬁeld. This observation isdifferent from the results of MoS 2/Py bilayers11in which no additional angular-independent spin-torque was found. However, Ralph et al. recently demonstrated a similar out-of-plane antidamping torque in the WTe 2/Py bilayer19,20and an in-plane angular-independent torque in NbSe 2/Py bilayers,10where both extra angular-independent spin- torques were attributed to the broken mirror symmetry of the 2Dmaterial surface. The reduced symmetry of WTe 2is determined by its intrinsic crystal structure characteristics as its surface twofold rota-tional symmetry is broken along the a-axis, while the low symmetry state of the NbSe 2surface may be due to a uniaxial strain caused by external parameters. For our Py/BP device, since BP has a puckeredsurface along the x-direction, leading to a twofold asymmetry along the x and y direction, it is reasonable to suspect that this s Bmay be pri- marily arising from the broken lateral mirror symmetry of the BP sur-face as its spin–orbit coupling is relatively weak. In conclusion, we have experimentally investigated the current- induced spin-torques in Py/BP bilayers. An out-of-plane antidamping FIG. 4. (a) and (b) ST-FMR resonances for pure Py(20 nm) samples and Py(20 nm)/BP(8.8 nm), respectively, with the magnetization oriented at /C045/C14and 135/C14. FIG. 5. (a) Symmetric and (b) antisymmetric ST-FMR resonance components for the Py/BP bilayer as a function of in-plane magnetic ﬁeld angle, at a ﬁxed frequency of 3.5 GHz and a ﬁxed power of 10 dBm.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 062403 (2020); doi: 10.1063/5.0013363 117, 062403-3 Published under license by AIP Publishingtorque is generated in the Py/BP bilayer interface because the BP sur- face has broken mirror symmetry. This study provides an efﬁcient strategy to manipulate magnetization with perpendicular anisotropy and demonstrates a possibility for manipulating magnetization com-patible with semiconductor ﬁeld-effect devices based on 2D materials,which is beneﬁcial for designing and optimizing spintronic devices in the future. This work was supported by the National Natural Science Foundation of China (Nos. 51732010, 51761145025, 11974379, and 51802341) and the China Postdoctoral Science Foundation (Nos. 2020M671592, 2019M661967, and 2019TQ0223). DATA AVAILABILITY The data that support the ﬁndings of this study are available within this article. REFERENCES 1A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012). 2E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960). 3G. Dresselhaus, Phys. Rev. 100, 580 (1955). 4K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008). 5L. Q. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). 6S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. D. Beach, Nat. Mater. 12, 611 (2013). 7X. Wang, J. Tang, X. X. Xia, C. L. He, J. W. Zhang, Y. Z. Liu, C. H. Wan, C. Fang, C. Y. Guo, W. L. Yang, Y. Guang, X. M. Zhang, H. J. Xu, J. W. Wei, M. Z. Liao, X. B. Lu, J. F. Feng, X. X. 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1.4986962.pdf	Effective empirical corrections for basis set superposition error in the def2-SVPD basis: gCP and DFT-C Jonathon Witte , Jeffrey B. Neaton , and Martin Head-Gordon Citation: The Journal of Chemical Physics 146, 234105 (2017); doi: 10.1063/1.4986962 View online: http://dx.doi.org/10.1063/1.4986962 View Table of Contents: http://aip.scitation.org/toc/jcp/146/23 Published by the American Institute of PhysicsTHE JOURNAL OF CHEMICAL PHYSICS 146, 234105 (2017) Eﬀective empirical corrections for basis set superposition error in the def2-SVPD basis: gCP and DFT-C Jonathon Witte,1,2Jeﬀrey B. Neaton,2,3,4and Martin Head-Gordon1,5,a) 1Department of Chemistry, University of California, Berkeley, California 94720, USA 2Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 3Department of Physics, University of California, Berkeley, California 94720, USA 4Kavli Energy Nanosciences Institute at Berkeley, Berkeley, California 94720, USA 5Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Received 31 March 2017; accepted 7 June 2017; published online 20 June 2017) With the aim of mitigating the basis set error in density functional theory (DFT) calculations employing local basis sets, we herein develop two empirical corrections for basis set superposition error (BSSE) in the def2-SVPD basis, a basis which—when stripped of BSSE—is capable of providing near- complete-basis DFT results for non-covalent interactions. Speciﬁcally, we adapt the existing pairwise geometrical counterpoise (gCP) approach to the def2-SVPD basis, and we develop a beyond-pairwise approach, DFT-C, which we parameterize across a small set of intermolecular interactions. Both gCP and DFT-C are evaluated against the traditional Boys-Bernardi counterpoise correction across a set of 3402 non-covalent binding energies and isomerization energies. We ﬁnd that the DFT-C method represents a signiﬁcant improvement over gCP, particularly for non-covalently-interacting molecular clusters. Moreover, DFT-C is transferable among density functionals and can be combined with exist- ing functionals—such as B97M-V—to recover large-basis results at a fraction of the cost. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4986962] I. INTRODUCTION In an electronic structure calculation, two forms of basis set errors arise when local basis sets are employed: basis set superposition error (BSSE), which is a consequence of incon- sistent treatment of a larger supersystem and its constituent subsystems,1–3and intrinsic basis set incompleteness error, the category to which we relegate all remaining basis set errors once BSSE has been removed.4Intrinsic incompleteness error arises from the fact that the Schr ¨odinger equation is being solved in just a fraction of the full Hilbert space, and no system- atic means of removal—short of simply increasing the number of basis functions—has yet been discovered, though adaptive- basis approaches have shown some promise.5–10Basis set superposition error, on the other hand, has a long history within the electronic structure community.11–20In the case of dis- tinct non-covalently interacting units, BSSE can be removed by performing fragment calculations within the basis of the full system, i.e., via the counterpoise correction (CP) ﬁrst introduced by Boys and Bernardi.2 The standard counterpoise correction has two principal shortcomings. First, it requires a partitioning of the full system into a number of fragments, Nfragments ; for some systems, such as those with simple bimolecular interactions, this partition- ing is straightforward, but for many interesting systems—such as those involving substantial intramolecular interactions— it is not. Second, although in principal a good approxima- tion to counterpoise-corrected results may be obtained with a)Electronic mail: mhg@cchem.berkeley.eduminimal extra effort via standard energy decomposition analy- ses,21,22in practice the CP correction often ends up being quite computationally demanding, whereas an uncorrected binding energy requires only one calculation in the full supersystem basis, a counterpoise-corrected one requires Nfragments +1 such calculations. The issues of partitioning and the inability of the CP scheme to address intramolecular BSSE were ﬁrst addressed by Galano and Alvarez-Idaboy with an atom-by-atom counter- poise correction;23Jensen later generalized this into the atomic counterpoise (ACP- n) approach.24In the ACP- nscheme, BSSE is estimated as a sum of atomic BSSEs, where each atomic BSSE is calculated by considering basis functions up to nbonded atoms away. This approach has shown some promise in addressing intramolecular BSSE, though it suffers from the same partitioning problem as CP when ambiguous bonding patterns are involved—e.g., in transition states and hydrogen- bonded systems—and the computational complexity of the method is unchanged. More recently, there have been attempts to develop empir- ical models for BSSE, as such approaches can potentially address both the partitioning and complexity issues. The ﬁrst such model was proposed six years ago by Faver and Merz,25,26who constructed the so-called “proximity func- tions” for molecular fragments from atomic pairs. Since the targets for this method are large biomolecules, the parame- ters are trained on a variety of proteinogenic systems. To date, this is the only empirical correction for BSSE developed for correlated wavefunction-based methods. The chief shortfall of the approach lies in its limited transferability; the param- eters for modeling typical nonpolar, van der Waals-driven 0021-9606/2017/146(23)/234105/10/ $30.00 146 , 234105-1 Published by AIP Publishing. 234105-2 Witte, Neaton, and Head-Gordon J. Chem. Phys. 146 , 234105 (2017) interactions are signiﬁcantly different than those used for modeling hydrogen bonding. Kruse and Grimme more recently introduced the so-called geometrical counterpoise (gCP) scheme,27which was later combined with the DFT-D3(BJ)28,29dispersion correction and either an explicit—in the form of an additional short-range term—or implicit—in the form of a modiﬁed basis set— correction for basis set incompleteness to form the HF-3c, PBEh-3c, and HSE-3c methods.30–32The gCP scheme loosely resembles the proximity function approach of Faver and Merz, in as much as both methods are strictly pairwise atomic corrections. Unlike the proximity function-based correction, however, gCP has gained considerable traction within the elec- tronic structure community,33–35largely due to its low-cost, satisfactory transferability, and ease of use. The gCP approach is utilized in conjunction with very small basis sets—on the order of 6-31G*—and is capable of recovering most of the BSSE in typical systems. Within this work, we adapt the gCP empirical correction for BSSE to the def2-SVPD basis. We focus exclusively on the def2-SVPD basis set36,37due to its good balance of expense and performance; def2-SVPD has low intrinsic incomplete- ness error relative to other comparable-sized bases,38and hence seems to us to be a particularly promising basis set for BSSE correction schemes. In addition, we develop an alter- native beyond-pairwise empirical correction for BSSE within density functional theory: DFT-C. The many-body nature of the method accounts for the overcounting concomitant with any pairwise approach and allows DFT-C to treat both large and small systems in a consistent manner. Whereas gCP is developed for use with exceptionally small basis sets, with the aim of providing semi-quantitative results, we demon- strate that DFT-C can recover near-basis-set-limit results at a fraction of the cost, particularly in the case of non-covalent interactions. II. THEORY AND METHODS A. gCP Here, we will brieﬂy summarize the geometrical counter- poise (gCP) correction for BSSE; for further details, see the original study by Kruse and Grimme.27At the core of gCP lies a function describing the decay of BSSE on atom Adue to the presence of basis functions on atom Ba distance rABaway, which we denote fgCP AB(rAB). This term is given by fgCP AB(rAB)=cABexp r AB (1) and includes a multiplicative constant, cAB, as well as a univer- sal decay parameter and exponent . The contributions of all atom-ghost pairs are summed up to yield the gCP correction for BSSE, EgCP=X AcAX B,AfgCP AB(rAB), (2) where cAare atom-dependent parameters and is an overall scaling parameter. In practice, EgCPis just added to the total electronic energy for a given system. The gCP approach is strictly pairwise additive with respect to nuclear centers.B. Parameterization of gCP Equations (1) and (2) contain several parameters: mul- tiplicative constants cAB, linear coefﬁcients cA, decay fac- torsand, and an overall scaling factor . The pairwise multiplicative constants, cAB, are calculated as cAB=1q SABNvirt B, (3) where Nvirt Bis the number of virtual orbitals on atom B—given byNvirt B=Nbasis functions B1 2Nelectrons B—and SABis a measure of the Slater overlap between atoms AandB. The overlap term is described in detail in the original study;27here, we will simply note that it involves an additional linear parameter, . The atomic linear coefﬁcients, cA, are calculated within the gCP approach as “missing energy” terms, i.e., cAis calcu- lated as the difference in restricted open-shell Hartree-Fock39 energy between atom Ain a target basis (here def2-SVPD) and a large basis, in the presence of an external electric ﬁeld to pop- ulate higher angular momentum functions. We have utilized aug-pc-4 as the large basis.40–42 The remaining parameters—three nonlinear ( ,, and) and one linear ( )—are obtained by minimizing the error in predicted gCP BSSE relative to Boys-Bernardi BSSE with the B3LYP43–46density functional across the S66 8 dataset of intermolecular interactions,47,48within the def2-SVPD basis. As in the original work, the most compressed geometries are weighted for this optimization by a factor of 0.5 in order to emphasize equilibrium and long-distance structures. The optimized set of parameters is provided in the supplementary material; this set of parameters allows the exist- ing gCP approach to be utilized with the def2-SVPD basis set for DFT. Brieﬂy, we mention one particularly interesting aspect of the optimized parameters: the optimal value of — the parameter controlling atomic overlap in the gCP model—in the def2-SVPD basis is 0.000 01, which suggests that for this particular basis set, the gCP expression can be simpliﬁed with- out degrading performance by simply removing the overlap term. We have veriﬁed that this is in fact true; we present in the supplementary material a simpler formulation of gCP for def2-SVPD. C. DFT-C In addition to re-parameterizing the gCP method for use with the def2-SVPD basis, we also present a more com- plex, though physically motivated, geometry-based empirical approximation for BSSE, which will henceforth be referred to as DFT-C. This model is in many ways similar to gCP.27At its core lies a term describing the decay of BSSE on atom A due to the presence of basis functions on atom Ba distance rABaway, which we denote fDFT-C AB(rAB). This term is given by fDFT-C AB(rAB)=cABexp ABr2 AB+ABrAB (4) and includes a multiplicative constant, cAB, a Gaussian decay parameter,AB, and an exponential decay parameter, AB. We expect the decay of BSSE to mirror that of the electron density; the exponential term accounts for the standard decay expres- sion,49and the Gaussian term reﬂects the nature of the basis234105-3 Witte, Neaton, and Head-Gordon J. Chem. Phys. 146 , 234105 (2017) functions employed. The DFT-C approach includes both an exponential term and a Gaussian term, with pair-dependent decay factors; these differences set it apart from the gCP core term given in Eq. (1). In DFT-C, we damp this atomic contribution to BSSE, much as the contribution of gCP is damped in PBEh-3c to potentially address short- rABissues that can arise in thermo- chemical problems.31We employ the same form of damping function as PBEh-3c,31 d(rAB)=1 1 +k1,ABrAB=r0,ABk2,AB, (5) where r0,ABis the sum of the van der Waals radii of atoms AandB, and k1,ABandk2,ABare parameters that control the precise shape of the damping function. Whereas Grimme et al.31setk1= 4 and k2= 6 for all pairs of atoms Aand Bby inspection, we compute them systematically for each atom pair based on the sums of covalent and van der Waals radii such that d(rcov,AB)=0.05 and d(r0,AB) = 0.95. Doing so yields k1,AB= 19 and k2,AB=5.8889logr0,AB=rcov,AB1. Moreover, we propose damping to a ﬁnite value, rather than zero, to more accurately reﬂect the actual short-range behavior of BSSE; after all, BSSE does not simply vanish in the covalent bonding distance regime. Thus, rather than simply multiply- ing the contribution from Eq. (4) by the damping function in Eq. (5), we deﬁne a damped contribution, gDFT-C AB(rAB), as gDFT-C AB(rAB)=d(rAB)fDFT-C AB(rAB) +(1d(rAB)) fDFT-C AB(rcov,AB). (6) At long range, this term reduces to fDFT-C AB(rAB), while at short range, it reduces to a pair-dependent constant, fDFT-C AB(rcov,AB). Whereas the gCP correction is strictly pairwise, we incor- porate a many-body component into DFT-C. We do so in the following physically motivated though ad hoc way, by simply modifying each pairwise contribution by an additional term, hAB(fA,B,:::g), which is given by hAB(fA,B,:::g)=26666641 +X C,A,BNvirt C Nvirt Bterfc (rAC,rAB) terfc (rBC,rAB)37777751 , (7) where Nvirt Bis the number of virtual orbitals on atom B— given by Nvirt B=Nbasis functions B1 2Nelectrons Bas in gCP, with Nelectrons Bbeing the number of electrons on neutral atomic BandNbasis functions Bcorresponding to the number of basis functions centered at atom B—distances are in atomic units, and terfc( x,y) is the attenuator deﬁned by Dutoi and Head- Gordon,50 terfc (x,y)=11 2erf(x+y)+ erf (xy). (8) This additional correction, hAB(fA,B,:::g), addresses the nonzero overlap between the Hilbert space of atom Band the Hilbert spaces of all atoms C,A,B. As more and more atoms are added in the vicinity of atoms AandB, the con- tribution of the ghost functions centered at Bto the atomic FIG. 1. Visualization of how adding a third atom Cimpacts the contribution of basis functions centered at Bto the BSSE on atom A, as per hAB(fA,B,Cg). When atom Cis sufﬁciently far away from AandB(lighter areas), the model reduces to a pairwise approach. In this example, CandBare assumed to have the same number of virtual orbitals, and AandBare located at ( 1.5 a.u., 1.5 a.u.) and (1.5 a.u., 1.5 a.u.), respectively. BSSE of Ashould decrease; eventually, once the space is sat- urated, adding additional atoms (i.e., ghost functions) does not change the BSSE of atom A. This phenomenon is not captured by a strictly pairwise approach. The many-body cor- rection we employ is visualized for a planar 3-atom system in Fig. 1. The ﬁnal form of the DFT-C correction for BSSE is given by EDFT-C=X AcAX B,AgDFT-C AB(rAB)hAB(fA,B,:::g), (9) whereis an overall scaling coefﬁcient, cAis a linear coefﬁ- cient that modiﬁes the contributions of ghost functions on all atoms Bto the BSSE on A, and the damped pairwise contribu- tion, gDFT-C AB(rAB), and many-body correction, hAB(fA,B,:::g) are deﬁned in Eqs. (6) and (7), respectively. With the excep- tion of the many-body term, this expression for the DFT-C energy is mathematically similar to that for gCP—cf. Eqs. (9) and (2). D. Parameterization of DFT-C As can be seen from Eq. (9), DFT-C has a large number of parameters. For each unique set of atom Aand ghost functions centered at atom B, there are exponential and Gaussian decay parameters, ABandAB, and there is a multiplicative constant, cAB. These parameters are obtained by generating BSSE curves for neutral atomic pairs ABusing a form of local spin-density approximation (LSDA), SPW92,51–54in the def2-SVPD basis. For each unique atom Aand corresponding ghost atom B, we perform a least squares ﬁt on a log BSSE curve generated over the rangercov,AB, 5rcov,ABin units of 0.1 a0. To avoid overem- phasizing the long-distance regime—where the atomic BSSE is nearly zero, and hence the logarithm of the BSSE is very large in magnitude—we weight each point by the inverse of the logarithm of the BSSE at each distance. The viability of this approach is demonstrated in Fig. 2 for the neon component of neon-argon BSSE. The DFT-C method does a reasonable job of capturing BSSE throughout the entire distance regime, yielding an root-mean-square error (RMSE) of 0.002 kcal/mol.234105-4 Witte, Neaton, and Head-Gordon J. Chem. Phys. 146 , 234105 (2017) FIG. 2. Dependence of actual and predicted neon atom SPW92/def2-SVPD BSSEs on distance to argon ghost functions. Note that gCP RMSE for this system is an order of magni- tude larger; the lack of a many-body term in gCP necessitates the systematic underprediction of pairwise atomic BSSEs. For pairs ABwhere the Gaussian decay parameter ABoptimizes to a negative value, we set AB=0 and re-optimize, so as to avoid divergence in the large- rABlimit. We have parameterized all 1296 combinations of the ﬁrst 36 elements of the periodic table in this manner; the result- ingcAB,AB, andABare tabulated in the supplementary material. In the cases of manganese, iron, and cobalt, we have taken averages of the BSSEs for the two competing spin states. For elements heavier than krypton, we propose using the parameters from 4th-row analogues, as is done in gCP. The linear coefﬁcients, cA, in Eq. (9) are all unity, with the exception of those for hydrogen, carbon, nitrogen, and oxygen, which are ﬁt via least-squares regression of DFT-C predicted BSSEs to actual BSSEs at the SPW92/def2-SVPD level across the S66 dataset of intermolecular interactions.47The overall scaling parameter, , is by deﬁnition unity for LSDA and is allowed to vary for different density functionals. We have optimizedfor several generalized gradient approximations (GGAs) and meta-GGAs, again by minimizing the root-mean- square error (RMSE) across BSSEs in S66, using the pairwise parameters ( cAB,AB, andAB) and linear coefﬁcients ( cA) obtained at the LSDA level. For GGA functionals, the opti- mal value of is approximately 0.9, while for meta-GGA functionals, it is slightly lower, near 0.85. We thus proposeusing=1 for LSDA, =0.9 for GGAs, and =0.85 for meta-GGAs. Ultimately, almost all of the parameters associated with the DFT-C method are obtained from toy systems—neutral atom-ghost pairs—at the LSDA level. Four linear coefﬁcients are trained on S66 BSSEs, also at the LSDA level, and for non- LSDA density functionals, we allow for one scaling parameter, which is trained on S66 BSSEs. An implementation of this method within the Python programming language is provided in the supplementary material. In practice, the DFT-C cor- rection is applied in the same manner as gCP: the term from Eq. (9) is simply added to the total electronic energy for a given system. E. Datasets and computational details To assess the performance of the gCP and DFT-C methods, we employ a subset of the comprehensive database assembled by Mardirossian and Head-Gordon.55The subset we utilize contains 3402 data points distributed over 48 distinct datasets. These smaller constituent datasets are classiﬁed according to ﬁve distinct datatypes: NCED (easy non-covalent inter- actions of dimers), NCEC (easy non-covalent interactions of clusters), NCD (difﬁcult non-covalent interactions of dimers), IE (easy isomerization energies), and RG10 (binding curves of rare gas dimers). Unlike “easy” interactions, “difﬁcult” interactions are characterized by strong correlation or self- interaction error. A summary of datatypes may be found in Table I. In addition to the version of LSDA on which DFT-C is parameterized—SPW9251–54—we consider in this study three GGA and three meta-GGA functionals. At the GGA level, we examine a pure functional, PBE;102a global hybrid, B3LYP;43–46—the functional with which gCP is parameterized—and a range-separated hybrid, !B97X-V .103 At the meta-GGA level, we test a pure functional, B97M-V;104 a global hybrid, M06-2X;105and a range-separated hybrid, !B97M-V .55 All density functional calculations are performed in the def2-SVPD basis.36,37A ﬁne Lebedev integration grid of 99 radial shells—each with 590 angular points—is used to compute semi-local components of exchange and correlation, while non-local correlation in the VV10-containing func- tionals is calculated with the coarser SG-1 grid.106All cal- culations are performed within a development version of Q-Chem 4.4.107 TABLE I. Summary of datatypes. For more details, see Ref. 55. Datatype No. Constituent datasets References NCED 1744 S66, A24, DS14, HB15, HSG, NBC10, S22, X40, A21 12, BzDC215, HW30, NC15, S66 8, 47, 48, and 56–77 3B-69-DIM, AlkBind12, CO2Nitrogen16, HB49, Ionic43 NCEC 243 H2O6Bind8, HW6Cl, HW6F, FmH2O10, Shields38, SW49Bind345, SW49Bind6, 71 and 78–87 WATER27, H2O20Bind4, 3B-69-TRIM, CE20, H2O20Bind10 NCD 91 TA13, XB18, Bauza30, CT20, XB51 88–92 IE 755 AlkIsomer11, Butanediol65, ACONF, CYCONF, Pentane14, SW49Rel345, SW49Rel6, 79, 81–85, and 93–100 H2O16Rel5, H2O20Rel10, H2O20Rel4, Melatonin52, YMPJ519 RG10 569 RG10 101234105-5 Witte, Neaton, and Head-Gordon J. Chem. Phys. 146 , 234105 (2017) III. RESULTS AND DISCUSSION In this study, we have developed two geometry-based empirical corrections for BSSE in the def2-SVPD basis: gCP and DFT-C. This particular basis was chosen based on its low intrinsic basis set incompleteness error; BSSE-corrected results obtained within this basis are quite near the basis set limit. This is illustrated in Fig. 3, wherein root-mean- square errors (RMSEs) for B97M-V with (CP) and with- out (noCP) counterpoise correction against B97M-V/def2- QZVPPD across the various non-covalent datatypes of Table I are shown. Within the def2-SVP basis, even when BSSE is removed (i.e., the CP SVP speciﬁcation in Fig. 3), the remain- ing basis set incompleteness error is quite large—signiﬁcantly larger than method errors for typical density functionals. This indicates that the def2-SVP basis is not suitable for a high- accuracy BSSE correction scheme; its utility would ultimately be contingent on signiﬁcant cancellation of method and basis set errors. On the other hand, intrinsic incompleteness error in the def2-SVPD basis is quite small, and so a BSSE correc- tion scheme developed in this basis can, in principle, allow for quantitative reproduction of large-basis results. In addition to developing the DFT-C method, we have also parameterized the existing gCP scheme within the def2-SVPD basis for comparison. The ﬁrst of these assess- ments is shown in Fig. 4, wherein we have plotted for the three non-covalent datatypes from Table I normalized FIG. 3. Root-mean-square errors of B97M-V with (CP) and without (noCP) the Boys-Bernardi correction for BSSE in two small basis sets relative to B97M-V in the def2-QZVPPD basis, near the basis set limit. SVP and SVPD correspond to def2-SVP and def2-SVPD, respectively. Methods in the chart are ordered from lowest overall RMSE at the top to highest overall RMSE at the bottom. A table of values is provided below the chart to facilitate quantitative comparison. FIG. 4. Normalized root-mean-square errors (NRMSEs) of gCP and DFT-C predicted BSSEs versus Boys and Bernardi BSSEs at the LSDA level of DFT in the def2-SVPD basis. The datatypes NCED, NCD, and NCEC are deﬁned in Table I. The normalized root-mean-square error is obtained by dividing the RMSE by the mean reference value in the dataset, as described in the text. Direct use of LSDA/def2-SVPD without any correction would result in 100% NRMSE. root-mean-square errors (NRMSEs) for DFT-C and gCP pre- dicted BSSEs at the LSDA level of DFT. The normalized RMSE is simply the RMSE divided by the mean of the refer- ence data, and hence provides a measure of relative error. Its use facilitates comparison between NCED and NCEC since the energy scales of those two datatypes differ by more than an order of magnitude. Within Fig. 4, it is evident that both gCP and DFT-C repro- duce Boys-Bernardi BSSEs at the LSDA level reasonably well; either correction is a substantial improvement over no cor- rection. The performance of DFT-C on molecular dimers is particularly promising, as is its consistency across the various datatypes: the lowest DFT-C NRMSE in SPW92 is 25%, for NCED, and the highest is 33%, for NCD. On the other hand, the performance of gCP is quite variable; the method boasts an exceptionally low NRMSE of 19% across NCEC, but a sig- niﬁcantly worse NRMSE of 56% for NCD. Neither correction can be considered as a quantitative replacement for the full counterpoise correction. This same sort of comparison is made for three popu- lar GGA functionals in Fig. 5. Therein, NRMSEs for DFT-C and gCP BSSEs versus actual BSSEs obtained with a pure functional (PBE), a global hybrid (B3LYP), and a range- separated hybrid with non-local correlation ( !B97X-V) may be found. It is clear that for all three density functionals, both DFT-C and gCP are quite consistent with regard to their per- formances across the various datatypes. Moreover, comparing with Fig. 4, this consistency extends across the LSDA-GGA gap for DFT-C, which bodes well for its transferability. This same level of consistency is not seen for gCP, how- ever, whereas gCP reproduces LSDA cluster BSSEs with unparalleled accuracy, the method is not nearly as good for clusters at the GGA level: the gCP NRMSE across NCEC in!B97X-V is more than double that in SPW92. This is a234105-6 Witte, Neaton, and Head-Gordon J. Chem. Phys. 146 , 234105 (2017) FIG. 5. Normalized root-mean-square errors (NRMSEs) of gCP and DFT-C predicted BSSEs versus Boys and Bernardi BSSEs for three GGA density functionals in the def2-SVPD basis. For further details, see Fig. 4. consequence of the fact that gCP tends to overestimate BSSE in molecular clusters, and BSSEs obtained at the LSDA level are on average larger than those at the GGA level. The excep- tional performance of gCP on SPW92 cluster BSSEs may thus be understood to be largely a consequence of the offsetting of these two phenomena. It is also evident from Fig. 5 that at the GGA level, DFT-C affords signiﬁcant gains over gCP regardless of datatype or density functional. This is quite promising, as DFT-C is param- eterized almost entirely at the LSDA level of theory, with only the overall scaling parameter changing from =1 to=0.9. On the other hand, gCP is parameterized at the GGA level, speciﬁcally with B3LYP. It is still true that use of gCP is signiﬁcantly better than no correction at all. In Fig. 6, we further assess the transferability of the gCP and DFT-C BSSE correction schemes across three distinct meta-GGAs: a pure meta-GGA B97M-V , a global hybrid M06- 2X, and a range-separated hybrid !B97M-V . Again, we see that across the three meta-GGA functionals, the relative perfor- mances of gCP and DFT-C are similar: for all three functionals, gCP exhibits NRMSEs of around 35% for NCED, 50% for NCD, and 60% for NCEC; the corresponding NRMSEs for DFT-C are 25%, 35%, and 20%. Similarly, we see the same sort of consistency for the DFT-C approach at the meta-GGA level as was seen at the GGA and LSDA levels (cf. Figures 5 and 4). On the other hand, gCP is slightly worse at describing molecu- lar clusters at the meta-GGA level than it was at the GGA level. Again, this can be traced back to the facts that gCP systemati- cally overpredicts BSSE in molecular clusters and meta -GGA BSSEs tend to be even lower than their GGA counterparts. This overcorrection by gCP can in turn be attributed to its strictly pairwise nature; owing to the inclusion of a many-bodycorrection, the DFT-C approach does not suffer from this over- counting issue. Note that both gCP and DFT-C can be applied here without modiﬁcation even to the Minnesota family of den- sity functionals—which are renowned for their non-intuitive and slow convergence of BSSE108,109—since the def2-SVPD basis set is too small to capture the unphysical behavior of some of the inhomogeneity correction factors. This same transfer- ability would not be expected in larger, e.g., triple-zeta, basis sets. Across the seven density functionals examined, the aver- age NRMSE of the DFT-C approach across NCED is 30%, compared to the 42% of gCP; this corresponds to an improve- ment of more than 25%. For the NCD datatype, the gCP average NRMSE is 59%, compared to 38%—an improve- ment of 35%. Across the NCEC set of molecular clusters, we see a 46% improvement for DFT-C over gCP: a reduc- tion in average NRMSE from 52% to 28%. It is clear that for a wide variety of systems, across a diverse set of den- sity functionals, in the def2-SVPD basis, the DFT-C method is satisfactorily transferable and represents a signiﬁcant improve- ment over gCP for the reproduction of Boys-Bernardi BSSEs. The remaining DFT-C error of course represents the remaining gap to perfect reproduction of the Boys-Bernardi counterpoise correction. Thus far, with the exception of the basis set com- parison in Fig. 3, all errors have been expressed relative to “exact” BSSEs. Although such metrics are relevant for this particular work, since the DFT-C and gCP methods are designed and trained to reproduce BSSEs, they are not of the same broad interest as, say, errors relative to high- level electronic structure methods. In Fig. 7, we show root- mean-square errors (RMSEs) across the ﬁve datatypes from FIG. 6. Normalized root-mean-square errors (NRMSEs) of gCP and DFT-C predicted BSSEs versus Boys and Bernardi BSSEs for three meta-GGA density functionals in the def2-SVPD basis. For further details, see Fig. 4.234105-7 Witte, Neaton, and Head-Gordon J. Chem. Phys. 146 , 234105 (2017) FIG. 7. Root-mean-square errors of B97M-V versus high-level reference val- ues at ﬁve levels of theory: uncorrected in the def2-SVPD basis (noCP); counterpoise-corrected in def2-SVPD (CP); with the geometrical counter- poise correction in def2-SVPD (gCP); with the correction introduced in this work in the def2-SVPD basis (DFT-C); and near the complete-basis set limit (CBS), in def2-QZVPPD. Methods in the chart are ordered from lowest over- all RMSE at the top to highest overall RMSE at the bottom. A table of values is provided below the chart to facilitate quantitative comparison. Table I for the B97M-V functional relative to high-level (gen- erally CCSD(T)/CBS) results. The noCP and CP designa- tions correspond to uncorrected and counterpoise-corrected B97M-V/def2-SVPD, respectively, and CBS corresponds to B97M-V/def2-QZVPPD—effectively B97M-V at the basis set limit. DFT-C and gCP refer to B97M-V/def2-SVPD with the corresponding approximation for BSSE included. From Fig. 7, it is immediately evident that any sort of BSSE correction is preferable to no correction. By correct- ing using the standard Boys-Bernardi approach, we are able to eliminate 90% of basis set error for NCED, 71% for NCD, 97% for NCEC, and even improve upon CBS results for RG10. Unfortunately, the standard counterpoise correction cannot be applied for the vast majority of isomerization energies—it can only be applied for relative energies, such as relative binding energies—and so the CP and noCP results are almost identi- cal for IE. On the other hand, both gCP and DFT-C offer solid improvements over noCP for every datatype examined, includ- ing isomerization energies, for which we are able to eliminate roughly 60% of basis set error. Errors across the individual datasets comprising each aggregate datatype are provided in Fig. 8. From Fig. 8, it is apparent that there exist datasets in NCED for which gCP outperforms DFT-C; likewise, DFT-C outperforms gCP on a subset of IE. Nevertheless, for B97M- V/def2-SVPD, the DFT-C approach generally offers mod- est improvements over gCP for molecular dimers (NCED, NCD, and RG10), a signiﬁcant improvement for molecular FIG. 8. Root-mean-square errors of B97M-V/def2-SVPD versus “exact” ref- erence values with no correction (noCP), the standard counterpoise correction (CP), the geometrical counterpoise correction (gCP), and the treatment intro- duced here (DFT-C). All RMSEs are in units of kcal/mol. Each row is color- coded for ease of reading, with darker cells corresponding to lower RMSEs. From top to bottom, the blocks correspond to the NCED, NCD, NCEC, IE, and RG10 datatypes. Note for SW49 and most of IE, the standard counterpoise correction is not possible, and so for these datasets the noCP and CP methods are identical. clusters (NCEC), and is slightly inferior for isomerization energies (IE). The DFT-C method outperforms the Boys- Bernardi counterpoise correction across the full dataset, with an overall RMSE of 0.56 kcal/mol compared to a CP RMSE of 0.63 RMSE; the large improvement it affords for NCEC and IE offset the small losses on NCED and NCD. As such, DFT-C is a viable alternative to the traditional counterpoise correction in the def2-SVPD basis set, yielding similar results to CP with effectively no increase in cost over noCP. To further illustrate the power of the DFT-C BSSE- correction scheme, in Fig. 9, we show RMSEs across the234105-8 Witte, Neaton, and Head-Gordon J. Chem. Phys. 146 , 234105 (2017) FIG. 9. Root-mean-square errors in kcal/mol of several pure meta-GGA den- sity functionals relative to high-level reference values. B97M-V-C corresponds to B97M-V with the DFT-C correction. Results for the additional density func- tionals are taken from a previous study.111SVPD corresponds to def2-SVPD and QZVPPD corresponds to def2-QZVPPD. Each datatype category is color- coded, with the darkest color corresponding to the lowest RMSE within that category. four aggregate datatypes for B97M-V with (B97M-V-C) and without (B97M-V) the DFT-C correction for BSSE in the def2-SVPD basis, as well as for four popular pure meta-GGA density functionals—B97M-V ,104MS2-D3(op),110,111M06- L,112and TM113—near the CBS limit, in the def2-QZVPPD basis. From Fig. 9, it is clear that although B97M-V/def2- SVPD is not competitive with standard meta-GGAs at the basis set limit, B97M-V-C/def2-SVPD certainly is—despite requiring a small fraction of the computational effort. IV. DISCUSSION AND CONCLUSIONS In this study, we have introduced a physically motivated empirical correction for basis set superposition error within the def2-SVPD basis set: DFT-C. This correction differs from the existing gCP approach—which we have also re-parameterized for use in the def2-SVPD basis—in two critical areas. First, whereas the linear coefﬁcients within gCP include all manifes- tations of basis set incompleteness error, the DFT-C approach is constructed exclusively from basis set superposition errors. Second, although gCP is a strictly pairwise correction, in DFT-C, each pairwise contribution is reduced by a many- body term to ameliorate the overcounting concomitant with the non-orthogonality of the Hilbert spaces of nearby atoms. We have evaluated both gCP and DFT-C across a diverse dataset containing 3402 non-covalent interactions and isomerization energies. This new method, DFT-C, yields signiﬁcantly more accu- rate BSSEs than gCP for a wide variety of interaction motifs. Moreover, the correction is transferable. DFT-C exhibits roughly the same relative performances across the various non-covalent datatypes regardless of the particular density functional with which it is paired: for non-covalently inter- acting dimers, DFT-C offers a modest improvement over gCP; in the case of molecular clusters—particularly when ameta-GGA functional is employed—the improvement is more pronounced, which is likely attributable to the many-body nature of the method. Whereas gCP has been developed as a general purpose tool that can be relatively easily adapted to any basis set, the DFT-C approach is much more complicated and specialized; tabulating the many pairwise coefﬁcients and decay param- eters is a nontrivial task. In this particular work, we have introduced a correction for def2-SVPD, a double-zeta basis set that has disproportionately low intrinsic basis set incomplete- ness error for how few basis functions it contains.38We are also exploring the possibility of extending this method to triple-zeta basis sets in order to truly push the basis set limit; such may be the focus of work to come. Additionally, we are exploring the impact of the DFT-C correction on thermochemical energies and equilibrium geometries. The gCP correction is an integral component of the small- basis functional PBEh-3c. The DFT-C correction could be incorporated in a similar fashion into a composite small-basis method by allowing some subset of the linear parameters, cA— or simply the overall scaling parameter —to vary. Even with- out modiﬁcation, however, the method is immensely powerful; we have demonstrated that it can be paired with an existing functional, B97M-V , to yield def2-SVPD results on par with def2-QZVPPD results for other state-of-the-art pure meta- GGA density functionals. DFT-C should prove immensely useful for recovering large-basis results for many energetic properties with small-basis effort—the correction scales with the number of atoms, not the number of basis functions, after all, and is essentially free on the scale of an electronic structure calculation—and it can be paired without modiﬁcation with any density functional. This could allow us to obtain high- quality results for large systems which are currently out of the domain of quantitative electronic structure theory. SUPPLEMENTARY MATERIAL See supplementary material for the simple Python imple- mentation of the DFT-C method, as well as parameterizations for both DFT-C and gCP within the def2-SVPD basis and several additional tables and ﬁgures. ACKNOWLEDGMENTS This research was supported by the U.S. Department of Energy, Ofﬁce of Basic Energy Sciences, Division of Chem- ical Sciences, Geosciences and Biosciences under Award No. DE-FG02-12ER16362. This work was also supported by the Director, Ofﬁce of Science, Ofﬁce of Basic Energy Sci- ences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, and a subcontract from MURI Grant No. W911NF-14-1-0359. J.W. would like to thank Narbe Mardirossian for many helpful conversations, as well as for aid in compiling and verifying the datasets. 1H. B. Jansen and P. Ros, Chem. Phys. Lett. 3, 140 (1969). 2S. Boys and F. Bernardi, Mol. Phys. 19, 553 (1970). 3B. Liu and A. D. McLean, J. Chem. Phys. 59, 4557 (1973). 4B. Brauer, M. K. 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1.4729897.pdf	Switching time of a single spin in linearly varying field Yasutaro Uesaka, Yoshio Suzuki, Osamu Kitakami, Yoshinobu Nakatani, Nobuo Hayashi, and Hiroshi Fukushima Citation: Journal of Applied Physics 111, 123907 (2012); doi: 10.1063/1.4729897 View online: http://dx.doi.org/10.1063/1.4729897 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/111/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Atomic level micromagnetic model of recording media switching at elevated temperatures Appl. Phys. Lett. 98, 192508 (2011); 10.1063/1.3589968 Effect of the anisotropy distribution on the coercive field and switching field distribution of bit patterned media J. Appl. Phys. 106, 103913 (2009); 10.1063/1.3260240 A study of hard:soft layer ratios and angular switching in exchange coupled media J. Appl. 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Downloaded to ] IP: 128.82.252.58 On: Mon, 22 Dec 2014 12:07:01Switching time of a single spin in linearly varying field Y asutaro Uesaka,1Y oshio Suzuki,1Osamu Kitakami,2Y oshinobu Nakatani,3 Nobuo Hayashi,4and Hiroshi Fukushima5 1College of Engineering, Nihon University, Koriyama, Fukushima 963-8642, Japan 2IMRAM, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan 3University of Electro-communications, Chofu, Tokyo 182-8585, Japan 4Independent, Kichijyoji, 2-2-22, Kichijyojikita, Musashinoshi, Tokyo 180-0001, Japan 5Independent, 3-73, Honda, Midoriki, Chiba 266-0005, Japan (Received 27 February 2012; accepted 21 May 2012; published online 20 June 2012) We studied the switching time of a single spin in a ﬁeld varying linearly in time using a micromagnetics simulation based on the Landau-Lifshitz-Gilbert equation. The applied ﬁeld largerthan the switching ﬁeld or coercivity is not enough for a spin to switch but some duration of time is also necessary. We found that the value of C 1deﬁned by C1¼ÐðH/C0H1Þdtwas constant when the rate of change in the ﬁeld was larger than 10 /C2cHk2, where cis the gyromagnetic ratio with g value¼2,His the applied ﬁeld, H1is a constant, and Hkis the anisotropy ﬁeld of the spin. The integration is taken from the time the spin begins switching to the switching time. The equation is a generalized form of the equation, C0¼ðH/C0H0Þssw, in a constant ﬁeld H. Here, C0andH0are constants, and sswis the switching time. We found that C1in the region dH=dt>10/C2cH2 KandC0 in the region H/C29HKare the same, but that H1does not coincide with H0. We found that the head ﬁeld rise time has a very small effect on the switching ﬁeld and time of recording media. VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4729897 ] I. INTRODUCTION Some duration of time, which we call the switching time, is necessary for a spin to switch even when a very largeﬁeld is applied. Several researchers have studied switching times of magnetic materials (single spin, particle, and thin ﬁlm) in constant or short-pulse ﬁelds. 1–6In constant ﬁeld, the following equation for the switching times, ssw, of a sin- gle spin2,3or a particle4,5is known to hold exactly at the temperature T¼0 when the angle nbetween the easy direc- tion and the applied ﬁeld His zero or when the applied ﬁeld is much larger than the switching ﬁeld in the case n=0/C14. ssw¼C0 H/C0H0: (1) Here, C0is a constant and H0is the anisotropy ﬁeld Hkwhen n¼0.3However, H0does not coincide with the switching ﬁeld when n=0 because Eq. (1)does not hold at n=0 when the applied ﬁeld is near the switching ﬁeld. Igarashi et al.6studied the switching time of a thin ﬁlm in short-pulse ﬁelds applied in-plane, and the easy axis of each grain in the ﬁlm was randomly distributed in-plane.They proposed the equation C 1¼ðssw s0ðH/C0H1Þdt; (2) where sswis the switching time and s0is the time when the magnetization begins switching. Equation (2)is a general- ized form of Eq. (1). That is, C1corresponds to C0andH1 does to H0. In this paper, we study the conditions in which Eq.(2)holds in a ﬁeld linearly varying in time and the rela- tions between C0and C1and between H0and H1. Theswitching time of particles in a magnetic recording medium will be able to be estimated from C1,dH/dt , and the maxi- mum head ﬁeld, and this will make it possible to estimatethe maximum transfer rate. II. CALCULATION METHOD We calculated the magnetization direction of a single spin using the Landau-Lifshitz-Gilbert equation. The effect of temperature was not taken into account. The magnetizationwas initially oriented to the easy direction ( zdirection). We deﬁned the initial and ﬁnal switching times ( s swiandsswf)a s the times when the zcomponent of the spin, Mz, becomes zero initially and ﬁnally, respectively, in constant or linearly vary- ing ﬁelds. ( Mzbecomes zero many times when the applied ﬁeld angle nis large and Gilbert’s damping constant ais small.) The initial or ﬁnal switching ﬁelds ( HswiandHswf)i n linearly varying ﬁelds were deﬁned as sswi/C2dH/dt and sswf/C2dH/dt , respectively, where His the applied ﬁeld and tis the time. We derived C1andH1by ﬁtting several dH/dt (rate of change in ﬁeld) and switching ﬁeld ( Hsw¼dH/dt /C2ssw) data with Eq. (2). We also derived C0andH0by ﬁtting several applied ﬁeld Hand switching time sswdata with Eq. (1). III. RESULTS AND DISCUSSION We ﬁrst report the results in a constant ﬁeld. Figure 1 shows the inverse of the switching time, 1/ ssw, plotted against the constant applied ﬁeld for spins with three Hkval- ues (0.1 kOe, 1 kOe, and 10 kOe) and Gilbert’s damping constant a¼0.01 for n¼0.001/C14and 45/C14. In the n¼45/C14 case, there are two switching ﬁelds, the initial switching ﬁeld (Hswi) and the ﬁnal switching ﬁeld ( Hswf), although only one switching ﬁeld is observed for n¼0.001/C14. We draw three 0021-8979/2012/111(12)/123907/5/$30.00 VC2012 American Institute of Physics 111, 123907-1JOURNAL OF APPLIED PHYSICS 111, 123907 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.82.252.58 On: Mon, 22 Dec 2014 12:07:01conclusions from the ﬁgure. The ﬁrst is that Eq. (1)holds for both n¼0.001/C14and 45/C14for an applied ﬁeld H/C29HK. The second is that the switching times sswwith different Hkin the same applied ﬁeld Hseem to coincide with each other when the applied ﬁeld is very large, but they are not exactly the same. The second conclusion is easily understood fromEq.(1)considering that H 0is similar to Hk(shown later) and H/C29H0. The third is that the deviation of the switching time from the one expected from Eq. (1)near the switching ﬁeld atn¼0.001/C14is much larger than that at n¼45/C14. We ﬁnd that the switching time of a spin in a constant ﬁeld can be estimated from the switching time of anotherspin with different H kin a different constant ﬁeld. Table I shows the relation between the switching times of three Hk (0.1 kOe, 1 kOe, 10 kOe) single spins. The switching time sswof the spin with Hk¼0.1 kOe under the ﬁeld His exactly 10 times as long as the switching time of the spin with Hk¼1 kOe under 10 H, and it is exactly 100 times as long as the switching time of the spin with Hk¼10 kOe under 100H. This suggests that H/Hkis a suitable scaled constant ﬁeld. We now give the results in a linearly varying ﬁeld. Figure 2shows the effect of the rate of change in the ﬁeld, dH/dt , on the switching time, ssw, and the switching ﬁeld, Hsw ðssw/C2dH=dtÞ, obtained in a ﬁeld linearly varying in time for n¼0.001/C14,a¼0.01, and three Hk(0.1, 1, 10 kOe) cases. Figure 3shows the same relationship for n¼45/C14,a¼0.01 andHk¼0.1 kOe. In both cases, ssw/1=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ dH=dtp andHsw/ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ dH=dtp for large dH/dt ,a n d ssw/1=ðdH=dtÞandHsware almost constant for small dH/dt . The relations between ssw anddH/dt and between HswanddH/dt for large dH/dt can be explained from Eq. (2)using the relations H¼(dH/dt )/C1t,Hsw /C29H1andssw/C29s0:C1¼ðssw s0ðH/C0HÞ1dt/C25ðssw s0ðdH=dtÞ/C1tdt/C251 2ðdH=dtÞ/C1s2 sw ¼1 2ðdH=dtÞH2 sw: (3) The switching ﬁeld Hswapproaches the Stoner-Wohlfarth switching ﬁeld with decreasing dH/dt . Therefore, Hswis almost constant for small dH/dt . We see from Fig. 2that the switching time ssw(and switching ﬁeld Hsw) of the spins with different Hkalmost coincide with each other when dH/ dtis large, but they are not exactly the same. This is similar to what we observe in a constant ﬁeld with a large applied ﬁeld (Fig. 1). This result can be understood from Eqs. (2) and(3)considering that H1is similar to Hkandssw/C29s0. We found that the switching time of a spin in a linearly varying ﬁeld can be estimated from the switching time of another spin with different Hkin a different linearly varying ﬁeld. The relations between Hkof a spin, rate of change in ﬁeld dH/dt , and switching time (switching ﬁeld) are shown in Table II. The switching time sswand switching ﬁeld Hsw of the spin with Hk¼0.1 kOe at dH/dt exactly coincides with 10 sswand (1/10) Hswof the spin with Hk¼1 kOe at 100dH/dt and coincides with 100 sswand (1/100) Hswof the spin with Hk¼10 kOe at 10 000 dH/dt . We understand from these results that ( dH/dt )/(cHk2) is a non-dimensional nor- malized rate of change in a ﬁeld linearly varying in time.Here, cis the gyromagnetic ratio with g value ¼2.FIG. 1. Effect of constant applied ﬁeld on inverse of switching time. n ¼0.001/C14and 45/C14,a¼0.01, Hk¼0.1, 1, and 10 kOe. The letter “i” means the initial switching and “f” the ﬁnal switching. TABLE I. Applied ﬁeld and corresponding switching time for three Hksin- gle spins in constant ﬁelds. Hk(kOe) Switching time Applied field 10/C01ssw H 100ssw/10 10 H 101ssw/100 100 HFIG. 2. Dependence of switching ﬁeld ( Hsw) and time ( ssw) on rate of change in ﬁeld ( dH/dt ) for three Hk(0.1 kOe, 1 kOe, 10 kOe) spins with a¼0.01, n¼0.001/C14. FIG. 3. Dependence of Hswand sswon rate of change in ﬁeld ( dH/dt )a t Hk¼0.1 kOe, a¼0.01, n¼45/C14.123907-2 Uesaka et al. J. Appl. Phys. 111, 123907 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.82.252.58 On: Mon, 22 Dec 2014 12:07:01Figure 4shows the dependence of C1ondH/dt for a¼0.01 and for n¼0.001/C14and 45/C14. The dependence of HswondH/dt is also shown for comparison. The horizontal axis in the ﬁgure is the normalized rate of change in ﬁeld, (dH/dt )/(cHk2). We obtained the results in Fig. 4using the least squares method applied to the following equation derived from Eq. (2): dH dt¼ðHsw/C0HstÞðHsw/C02H1þHstÞ=2C1: (4) Here, Hst¼dH=dt/C1s0. Figure 4reveals that C1is constant at large dH/dt , and it is not constant at small dH/dt . This is expected from the change in Hswwith dH/dt . Figures 5and6show the angular dependence of C1of spins with several avalues (0.01–1) for large dH/dt (constant C1) for initial and ﬁnal switching, respectively ( C1iandC1f mean C1for initial and ﬁnal switching, respectively). These ﬁgures also show the same dependence of C0in constant ﬁelds ( C0iand C0fmean C0in initial and ﬁnal switching, respectively). We obtained the C0values for large H(H/C29 HK). We see that C0andC1coincide with each other in all cases. Figures 7and8show the angular dependence of H1and H0at initial and ﬁnal switching (using the scaled values H1/ HkandH0/Hk) for a¼0.01, 0.1, and 1. Figure 7is for initial switching and Fig. 8for ﬁnal switching. The switching ﬁelds, Hsw, in constant ﬁeld for a¼0.01, 0.1, and 1 are also shown in both ﬁgures. A switching ﬁeld Hswfora¼0.1 is a little larger than that for 0.01 with the same nand the Hswfor a¼1 is much larger than both. We obtained the values of H1 andH0for large dH/dt and for high ﬁelds, respectively. We see that H1,H0, and Hsware similar and in the same order ofmagnitude as in Hkand that H1andH0are usually smaller than Hsw. We also see that H1andH0do not coincide with each other in contrast to the case of C1andC0. This can be understood as follows. When a linearly varying ﬁeld is applied, the spin will begin switching when the ﬁeldbecomes larger than the switching ﬁeld in a constant ﬁeld 6 (the Stoner-Wohlfarth switching ﬁeld or a smaller ﬁeld7,8). We see from Fig. 1that Eq. (1)does not hold when the applied ﬁeld is near the switching ﬁeld in a constant ﬁeld. That is, in a linearly varying ﬁeld, the switching time should include the effect of switching near the switching ﬁeld in aconstant ﬁeld where Eq. (1)does not hold. On the other hand, H 0was derived at constant high ﬁeld H/C29Hk. There- fore, we can understand that H1andH0do not necessarily coincide with each other. The difference between H0and H1is large especially when n¼0.001/C14. This may be because the deviation of the switching time from the prediction of Eq. (1)atn¼0.001/C14 is much larger than that at the other angles when the ﬁeld is near the switching ﬁeld in a constant ﬁeld (see Fig. 1forn ¼0.001/C14andn¼45/C14).TABLE II. Rate of change in ﬁeld and corresponding switching time and ﬁeld for three Hksingle spins in ﬁelds linearly varying in time. Hk(kOe) Switching time (switching field) Rate of change in field 10/C01ssw(Hsw) dH/dt 100ssw/10 (10 Hsw)1 02dH/dt 101ssw/100 (100 Hsw)1 04dH/dt FIG. 4. Dependence of C1andHswondH/dt .n¼0.001/C14,4 5/C14,a¼0.01.FIG. 5. Angular dependence of constants C1andC0at initial switching with several a. The superscript i means initial switching. FIG. 6. Angular dependence of constants C1andC0at ﬁnal switching for several avalues. The superscript f means ﬁnal switching.123907-3 Uesaka et al. J. Appl. Phys. 111, 123907 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.82.252.58 On: Mon, 22 Dec 2014 12:07:01As mentioned above, Hswis almost constant for small dH/dt andC1is almost constant in for large dH/dt . This indi- cates that there are two types of switching, static switchingand dynamic switching; static switching occurs in small dH/dt region and dynamic switching occurs in large dH/dt region. Figure 9shows the normalized maximum dH/dt values in the static switching region of constant H sw(with a deviation of less than 20%) for the initial switching, and Fig. 10shows the same values for the ﬁnal switching. Figure 11shows the nor- malized minimum dH/dt in the dynamic switching region of constant C1(the deviation is less than 20%) for the initial switching, and Fig. 12shows the same values for the ﬁnal switching. We see from Figs. 11and12thatC1is almost con- stant for dH/dt >10/C2cHk2at any applied ﬁeld angle. The maximum (Figs. 9and10) or the minimum (Figs. 11and12) dH/dt values at n¼0.001/C14are smaller than those at the other angles. This means that a dH/dt region for constant Hswat n¼0.001/C14is narrower than at the other angles and that a dH/ dtregion for constant C1atn¼0.001/C14is wider than at the other angles. This may be because the magnetic torque at n/C2115/C14is much larger than at n¼0.001/C14.T h a ti s ,l a r g et o r - que at n/C2115/C14brings about switching near the Stoner- Wohlfarth switching ﬁeld even at large dH/dt , which causes aFIG. 8. Dependence of H0/Hkfand H1/Hkfon applied ﬁeld angle at ﬁnal switching for three avalues. The superscript f means ﬁnal switching. The switching ﬁelds Hswin constant ﬁeld for three avalues are also shown. FIG. 9. Effect of Gilbert’s damping constant aon the normalized maximum dH/dt for constant Hsw(20%) for initial switching.FIG. 10. Effect of Gilbert’s damping constant aon the normalized maxi- mum dH/dt for constant Hsw(20%) for ﬁnal switching. FIG. 7. Dependence of H0/HkiandH1/Hkion applied ﬁeld angle at initial switching for three avalues. The superscript i means initial switching. The switching ﬁelds Hswin constant ﬁeld for three avalues are also shown. FIG. 11. Effect of Gilbert’s damping constant aon the normalized mini- mum dH/dt for constant (20%) C1for initial switching.123907-4 Uesaka et al. J. Appl. Phys. 111, 123907 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.82.252.58 On: Mon, 22 Dec 2014 12:07:01large dH/dt region of constant Hswatn/C2115/C14. A larger maxi- mum dH/dt value for constant Hswmay increase the minimum dH/dt value for constant C1. In Figs. 9–12, the black line shows a typical ( dH/dt )/cHk2 value for a recording head and medium. We obtained the value by assuming a head speed of 10 m/s, head ﬁeld rise time of 2/C2102Oe/nm, and medium Hkof 10 kOe. The value is smaller than the minimum ( dH/dt )/cHk2value for constant C1 at any nand a, and it is also smaller the maximum ( dH/dt )/ cHk2value for constant Hswatn¼30/C14and a/C210.004. Several research groups obtained avalue of CoCrPt magneticrecording media using several experimental methods.9–12 The avalues range from 0.004 to 0.3. Considering the above results and that a head ﬁeld is tilted typically by 30/C14from the easy direction of a recording medium, we conclude that the head ﬁeld rise time has a very small effect on switching ﬁeld and switching time. Static switching occurs in the ( dH/dt )/(cHk2) region smaller than the maximum ( dH/dt )/(cHk2) in Figs. 9and10. Dynamic switching occurs in the ( dH/dt )/(cHk2) region larger than the minimum ( dH/dt )/(cHk2) in Figs. 11and12 and Eq. (2)holds in this region. 1R. Kikuchi, J. Appl. Phys. 27(11), 1352 (1956). 2E. M. Gyorgy, J. Appl. Phys. 28(9), 1011–1015 (1957). 3H. Fukushima, Y. Uesaka, Y. Nakatani, and N. Hayashi, IEEE Trans. Magn. 38(5), 2394 (2002). 4Y. Uesaka, H. Endo, T. Takahashi, Y. Nakatani, N. Hayashi, and H. Fukushima, Phys. Status Solidi A 189(3), 1023 (2002). 5Y. Uesaka, H. Endo, Y. Nakatani, N. Hayashi, and. H. Fukushima, IEEE Trans. Magn. 42(7), 1892 (2006). 6M. Igarashi, F. Akagi, A. Nakamura, H. Ikekame, H. Takano, and K. Yoshida, IEEE Trans. Magn. 36(1), 154 (2000). 7H. Fukushima, Y. Uesaka, Y. Nakatani, and N. Hayashi, J. Magn. Magn. Mater. 290–291 , 526 (2005). 8M. d’Aquino, D. Suess, T. Schreﬂ, C. Serpico, and J. Fidler, J. Magn. Magn. Mater. 290–291 , 506 (2005). 9N. Inaba, Y. Uesaka, A. Nakamura, M. Futamoto, Y. Sugita, and S. Nar- ishige, IEEE Trans. Magn. 33(5), 2989 (1997). 10I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann, J. Stohr, G. Ju, and D. Weller, Nature 428, 831 (2004). 11N. Mo, J. Hohlfeld, M. ul Islam, C. S. Brown, E. Girt, P. Krivosik, W. Tong, A. Rebei, and C. E. Patton, Appl. Phys. Lett. 92, 022506 (2008). 12S. Mizukami, D. Watanabe, T. Kubota, X. Zhang, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, Appl. Phys. Exp. 3, 123001 (2010).FIG. 12. Effect of Gilbert’s damping constant aon the normalized mini- mum dH/dt for constant (20%) C 1for ﬁnal switching.123907-5 Uesaka et al. J. Appl. Phys. 111, 123907 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.82.252.58 On: Mon, 22 Dec 2014 12:07:01
1.2710737.pdf	Micromagnetic modal analysis of spin-transfer-driven ferromagnetic resonance of individual nanomagnets L. Torres, G. Finocchio, L. Lopez-Diaz, E. Martinez, M. Carpentieri, G. Consolo, and B. Azzerboni Citation: Journal of Applied Physics 101, 09A502 (2007); doi: 10.1063/1.2710737 View online: http://dx.doi.org/10.1063/1.2710737 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/101/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Micromagnetic calculation of dynamic susceptibility in ferromagnetic nanorings J. Appl. Phys. 105, 083908 (2009); 10.1063/1.3108537 Micromagnetic computation of interface conductance of spin-transfer driven ferromagnetic resonance in nanopillar spin valves J. Appl. Phys. 105, 07D112 (2009); 10.1063/1.3058623 Broadband ferromagnetic resonance measurements of a micromagnetic disk array using a meander-line technique J. Appl. Phys. 104, 063920 (2008); 10.1063/1.2982429 Micromagnetic study of full widths at half maximum in spin-transfer-driven self-oscillations of individual nanomagnets J. Appl. Phys. 103, 07B107 (2008); 10.1063/1.2832882 Coupling of spin-transfer torque to microwave magnetic field: A micromagnetic modal analysis J. Appl. Phys. 101, 053914 (2007); 10.1063/1.2435812 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.24.51.181 On: Sat, 29 Nov 2014 14:31:19Micromagnetic modal analysis of spin-transfer-driven ferromagnetic resonance of individual nanomagnets L. T orresa/H20850 Departamento de Fisica Aplicada, University of Salamanca, Plaza de la Merced, E-37008, Salamanca, Spain G. Finocchio Departamento di Fisica della Materia, University of Messina, Contrada da di Dio, I-98100, Messina, Italy L. Lopez-Diaz Departamento de Fisica Aplicada, University of Salamanca, Plaza de la Merced, E-37008, Salamanca,Spain E. Martinez Departamento de Ingeniería Electromecánica, University of Burgos, Plaza Misael Bañuelos, E-09001Burgos, Spain M. Carpentieri, G. Consolo, and B. Azzerboni Departamento di Fisica della Materia, University of Messina, Contrada da di Dio, I-98100, Messina, Italy /H20849Presented on 9 January 2007; received 26 October 2006; accepted 4 December 2006; published online 3 April 2007 /H20850 In a recent investigation Sankey et al. /H20851Phys. Rev. Lett. 96, 227601 /H208492006 /H20850/H20852demonstrated a technique for measuring spin-transfer-driven ferromagnetic resonance in individual ellipsoidal PyCunanomagnets as small as 30 /H1100390/H110035.5 nm 3. In the present work, these experiments are analyzed by means of full micromagnetic modeling ﬁnding quantitative agreement and enlightening the spatialdistribution of the normal modes found in the experiment. The magnetic parameter set used in thecomputations is obtained by ﬁtting static magnetoresistance measurements. The temperature effectis also included together with all the nonuniform contributions to the effective ﬁeld as themagnetostatic coupling and the Ampere ﬁeld. The polarization function of Slonczewski /H20851J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850/H20852is used including its spatial and angular dependences. Experimental spin-transfer-driven ferromagnetic resonance spectra are reproduced using the same currents as inthe experiment. The use of full micromagnetic modeling allows us to further investigate the spatialdependence of the modes. The dependence of the normal mode frequency on the dc and the externalﬁeld together with a comparison to the normal modes induced by a microwave current is alsoaddressed. © 2007 American Institute of Physics ./H20851DOI: 10.1063/1.2710737 /H20852 Full micromagnetic modeling /H20849FMM /H20850is used in this work in order to analyze in detail the recent work by Sankeyet al. about a technique for measuring ferromagnetic reso- nance /H20849FMR /H20850in individual nanomagnets. 1It will be shown in the present FMM study how this question can be discussed from a more ample point of view, considering the spin-transfer torque as an excitation source for the magnetic nor-mal modes of the system and the magnetoresistance mea-surement as an ad hoc detection method of the mentioned modes. The technique described in Ref. 1provides either the spectral density of spin-transfer dc-driven oscillations or theFMR spectra /H20849V mix/Irf2/H20850induced by an oscillating current with a frequency in the microwave range. These innovative mea- surements are performed in individual ellipsoidal PyCu na-nomagnets as small as 30 /H1100390/H110035.5 nm 3, avoiding in this way the possible coupling effects of measurements in nano-magnet arrays. 2In our FMM computations, it is possible to access to the detailed magnetization spatial conﬁguration ofthe nanomagnet at each time instant. Consequently, we canaccess the normal modes observing directly the magnetiza- tion conﬁguration. Our goal will be to compare these modeswith the experimental measurements trying to identify thespectrum peaks. In this way a deeper understanding of themagnetization dynamics is gained and further trends in suchprocesses can be proposed. The nanopillar under study consists of a pinned layer /H20849PL/H20850of Permalloy /H20849Py/H20850, 20 nm thickness, and a free layer /H20849FL/H20850of Py 65Cu35alloy, 5.5 nm thick. The two magnetic lay- ers are separated by a 12 nm copper spacer and the pillar hasan elliptical section of 90 /H1100330 nm 2. The results presented here have been obtained by means of a FMM two-dimensional /H208492D/H20850computation of the FL. Details on the implementation of FMM can be found in former works. 3 Brieﬂy, a ﬁnite difference scheme is used; the FL is dis-cretized in prismatic cells of 2.5 /H110032.5/H110035.5 nm 3. The Landau-Lifshitz-Gilbert equation is solved including in theeffective ﬁeld the magnetostatic coupling from the PL, theAmpere classical ﬁeld from the electric current, and all thestandard micromagnetic terms. The spin-transfer torque isconsidered by means of the Slonzewski’s term 4including both the angular and the spatial dependence of the polariza-a/H20850Author to whom correspondence should be addressed; FAX: 34 923 294584; electronic mail: luis@usal.esJOURNAL OF APPLIED PHYSICS 101, 09A502 /H208492007 /H20850 0021-8979/2007/101 /H208499/H20850/09A502/3/$23.00 © 2007 American Institute of Physics 101 , 09A502-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.24.51.181 On: Sat, 29 Nov 2014 14:31:19tion function for each computational cell.5Thermal activa- tion is also taken into account by means of a stochastic ﬁeld6 and the actual temperature of the sample for each current iscalculated supposing a thermal bath of 10 K and using theempirical expressions of Ref. 7. The time step used is 32.5 fs and the simulations are carried out for 20 ns, providing aspectral frequency resolution of ±0.05 GHz. Larger simula-tions, lower time steps, and different cell sizes have beentested, showing no signiﬁcant differences either in time orfrequency domain results. The magnetic parameters used for the FL /H20849M s=27.85 /H11003104A/m, and exchange constant A=1.0/H1100310−11J/m /H20850 were obtained by ﬁtting, in a FMM three-dimensional /H208493D/H20850 computation of the whole nanopillar, the experimental varia-tion of the magnetoresistance with the magnetic ﬁeld, as re-ported in Ref. 1. The nonuniform magnetostatic coupling and the different initial magnetization conﬁgurations of bothmagnetic layers at each applied magnetic ﬁeld are also cal-culated by means of 3D simulations. The external ﬁeld isapplied perpendicularly to the plane. The dynamics of the FLis computed in 2D using a damping /H9251=0.04 /H20849Ref. 1/H20850and a polarization factor of 0.3 for calculating the value of theSlonczewski’s polarization function. 4 In Fig. 1the spectral density of dc-driven precessional normal modes obtained from the fast fourier transform /H20849FFT /H20850 ofMxis shown. In the insets of the ﬁgure, density plots of the spatial distribution of the power at the frequencies of thetwo main peaks are also shown. These plots have been ob-tained following the so-called micromagnetic spectral map-ping technique 8/H20849MSMT /H20850which allows to observe which parts of the sample are oscillating with more strength at thefrequency analyzed. In this way the oscillation mode is visu-alized and assessments about its nature can be formulated.As shown in Fig. 1the main mode is the uniform one; the slight differences shown in the density plot are probably dueto the nonuniform magnetostatic coupling ﬁeld from thepinned layer which affects the effective ﬁeld and accordinglythe oscillation frequency. The second peak is the “1,0” modewhere one-half spatial wavelength is detected along the x direction /H20849long axis of the ellipsoid /H20850while no clear variation appears along the ydirection. 8Comparison with the experi- mental dc-driven modes of Ref. 1yields good agreement. It is to be noted that the currents applied are exactly the sameas the experiment and the power spectrum intensity of Figs.1/H20849a/H20850and1/H20849b/H20850is divided by 8 and 2 so as in the experimental results, the full dependent polarization function of Sloncze-wski is used for each computational cell and no free ﬁttingparameters are used. These results conﬁrm the assertions ofSankey et al. about modes A 0andA1found experimentally,1 which are now clearly identiﬁed as the uniform and 1,0 modes, respectively. In the experiment, for the highest cur-rent /H20849645 /H9262A/H20850of Fig. 1, just one large peak is detected, which shifts clearly to higher frequencies. This behavior was attributed to a mode hopping from A0toA1.1This is not the case in our FMM. A large peak is also detected /H20849spectral power is divided by 8 /H20850but 1,0 /H20849A1/H20850mode is also found al- though with lower intensity /H20851see inset of Fig. 1/H20849a/H20850/H20852.I no u r opinion, when increasing the current, the dynamics is closerto the magnetization switching; perturbations to the normalmodes begin to be present, leading to more frequencies in the FFT and the consequent broadening of the main mode peakshape. 9This broadening is evident in Fig. 1/H20849a/H20850and also the low frequency deformation of the peak shape is announcingthe proximity of the nonuniform switching which is obtainedfor currents around 700 /H9262A. Regarding the FMR modes obtained by means of an ac with a frequency in the microwave range, the spectra ob- FIG. 1. /H20849Color online /H20850Spectral density of dc-driven oscillations obtained from the FFT of Mxwith/H92620H=420 mT and currents Idc=645 /H20849a/H20850,5 8 5 /H20849b/H20850, 505 /H20849c/H20850, 445 /H20849d/H20850, and 305 /H20849e/H20850/H9262A/H20849Iac=0/H20850.09A502-2 T orres et al. J. Appl. Phys. 101 , 09A502 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.24.51.181 On: Sat, 29 Nov 2014 14:31:19tained by FMM also reveal the same normal modes observed in spin-transfer dc-driven experiments. In Fig. 2/H20849a/H20850theMx FFT spectrum is presented; a sinusoidal ac of 300 /H9262Aa ta frequency of 5 GHz and the same ﬁeld used in Fig. 1 /H20849420 mT /H20850have been applied; no dc is present in this case. The uniform, 1,0, and 2,0 modes are clearly observed which could be identiﬁed in principle with A0,A1, and A2modes found in the experiment. The experimental frequencies of A0 and A1modes are close to the ones found in our FMM. However, the frequency of the 2,0 mode seems signiﬁcantlyhigher than the experimental A2mode of Ref. 1. A plausible explanation is that in the experiment and due to the imper-fections of the samples a hybrid mode /H20849A 2/H20850is excited at lower frequencies than the normal 2,0 mode which should be identiﬁed with some of the Cmodes also found experimen- tally at higher frequencies.1In Fig. 2/H20849b/H20850the ﬁeld dependence of the frequency of the FMR normal modes induced by theac of 5 GHz is depicted. This dependence is very similar tothe one reported in Ref. 1conﬁrming the identiﬁcation of the normal modes. As summary, the following can be achieved. /H20849i/H20850 FMM modeling of spin transfer driven is presented. The modes revealed in the experiments are identiﬁed. /H20849ii/H20850No free ﬁtting parameters are needed since the full dependences of the polarization function and the non-uniform contributions of magnetostatic coupling fromthe PL and classical Ampere ﬁeld are used. /H20849iii/H20850The same normal modes are excited either by dc or ac, conﬁrming that at the values of the currents andﬁelds studied in our FMM the physics is the same: themagnetic system is excited and the normal modesarise. If larger excitations /H20849either current or ﬁeld /H20850are used, chaotic or switching behaviors could beinduced. This work was partially supported by the Spanish Gov- ernment under Project No. MAT2005–04287. 1J. C. Sankey et al. , Phys. Rev. Lett. 96, 227601 /H208492006 /H20850. 2R. D. Cowburn et al. , Phys. Rev. Lett. 83,1 0 4 2 /H208491999 /H20850. 3L. Torres et al. , J. Magn. Magn. Mater. 286,3 8 1 /H208492005 /H20850. 4J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 5M. Carpentieri et al. , J. Magn. Magn. Mater. /H20849in press /H20850. 6L. Lopez-Diaz et al. , Phys. Rev. B 65, 224406 /H208492002 /H20850. 7I. N. Krivorotov et al. , Phys. Rev. Lett. 93, 166603 /H208492004 /H20850. 8R. D. McMichael et al. , J. Appl. Phys. 97, 10J901 /H208492005 /H20850. 9D. V. Berkov et al. , Phys. Rev. B 72, 094401 /H208492005 /H20850. FIG. 2. /H20849Color online /H20850/H20849a/H20850Spectral density of FMR ac-driven oscillations obtained from the FFT of Mxwith/H92620H=420 mT, Iac=300/H9262A, and f =5 GHz /H20849Idc=0/H20850./H20849b/H20850Field dependence of the modes, Iac=300/H9262A and f =5 GHz /H20849Idc=0/H20850.09A502-3 T orres et al. J. Appl. Phys. 101 , 09A502 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.24.51.181 On: Sat, 29 Nov 2014 14:31:19
1.4795115.pdf	Vortex annihilation in magnetic disks with different degrees of asymmetry Chao-Hsien Huang, Kuo-Ming Wu, Jong-Ching Wu, and Lance Horng Citation: J. Appl. Phys. 113, 103905 (2013); doi: 10.1063/1.4795115 View online: http://dx.doi.org/10.1063/1.4795115 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v113/i10 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 08 Apr 2013 to 129.252.86.83. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsVortex annihilation in magnetic disks with different degrees of asymmetry Chao-Hsien Huang, Kuo-Ming Wu, Jong-Ching Wu, and Lance Hornga) Department of Physics, National Changhua University of Education, Changhua 50007, Taiwan (Received 4 December 2012; accepted 25 February 2013; published online 13 March 2013) We investigate the inﬂuence of one-side-ﬂat asymmetric degrees on vortex annihilation behavior in different chirality, clockwise or counterclockwise. The vortex annihilation ﬁelds are found to depend not only on the vortex chirality but also strongly on the degrees of asymmetry. The sequence of vortex annihilation from the ﬂat to the round edges is observed in low asymmetric disks,and interestingly, the sequence is reversed in high asy mmetric disks. Fast and non-contact vortex chirality detection can be realized in high asymmetric disks b y analyzing hysteresis loop of focused magneto-optic Kerr effect on vortex annihilation. The experimen tal results agree well w ith the micromagnetic simulations. VC2013 American Institute of Physics .[http://dx.doi.org/10.1063/1.4795115 ] I. INTRODUCTION Patterned magnetic domain structures have revealed many useful properties that can be applied to advanced mag- netic sensors and logic memories.1,2It is well known that a single domain magnetic layer builds up a distribution of strayﬁeld around itself. 3One of the main challenges in manufac- turing high-density magnetic-based devices is that the inter- action between adjacent elements may suppress anticipantmagnetization switching. A single magnetic vortex structure, stabilized in a ferromagnetic disk with diameter less than a micrometer, has high potential for operating in a unit cell forfuture storage media. Because the magnetic vortex at the re- manent state forms a magnetic ﬂux closure and induces no (or little) stray ﬁeld outside of itself, 4,5a more close layout can be achieved. The chirality of vortex, which is the rota- tional direction of magnetic moments separated into clock- wise (CW) or counterclockwise (CCW), determines thetrajectory of the vortex core motion, which is caused by the in-plane external ﬁeld. When the magnetic ﬁeld is applied, the vortex core moves towards the dot perimeter to increasethe net magnetization component along the direction of the applied ﬁeld. 6,7However, the appearance of CW or CCW vortices cannot be deliberately controlled by applying an in-plane magnetic ﬁeld, 8because the in-plane shape of the ani- sotropic energy is isotropic. To achieve ideal single vortex elements (SVEs), it is important to be able to control the vor-tex chirality reliably. The introduced asymmetry controls the chirality of the vortex. In recent years, some research groups have made asymmetric disks with small, chipped areas toachieve control over the vortex state appearing in a magnetic disk with an in-plane magnetic ﬁeld. 9–11Theoretical calcula- tion predicts that the vortex state still exists in nano-sizeddots with diameters smaller than 100 nm. 12In a micro- or nano-sized magnetic disk with a ﬂat edge on one side, the vortex will nucleate from the ﬂat edge. This is becausethe demagnetizing ﬁeld at the ﬂat edge is larger than that at the round edge when the external ﬁeld is applied parallel tothe ﬂat edge. Some researchers investigating vortex nuclea- tion, displacement, and annihilation adopt magnetic forcemicroscopy, 7,13Lorentz microscopy,7,14and magnetoresist- ance.13,15,16However, few researchers17–19have studied the effect on vortex nucleation and annihilation of varying thegeometric asymmetry and changing the chipped area. The magnetic domains in one-side-ﬂat magnetic disks with an external ﬁeld parallel to the ﬂat edge nucleate into a CW or CCW vortex determined by the original saturation direction. The nucleation ﬁelds of CW and CCW vortices arethe same because the one-side-ﬂat disk has mirror symmetry for the external ﬁeld reversal. However, the annihilation ﬁelds of CW and CCW vortices are different because thevortex cores move towards opposite directions, while a mag- netic ﬁeld is applied. The inﬂuence of geometric asymmetry on vortex motion has been studied in our previous report. 16 Dumas et al. reported on chirality control by manipulating the size and/or thickness of asymmetric Co dots.19A differ- ent and opposite chirality control mechanism through thenucleation and coalescence of double vortices were investi- gated. In this letter, our main goal is to investigate the anni- hilation ﬁelds for CW and CCW vortices in magnetic diskswith different degrees of asymmetry. Interestingly, a reversal of the CW and CCW vortex annihilation sequence was observed in low and high asymmetry disks. II. EXPERIMENTAL Asymmetric Ni 80Fe20(Permalloy, Py) disks, 800 nm in diameter and 30 nm thick are prepared by electron-beam li-thography and lift-off technique. Disks with an excised angle of 30 /C14represent a low asymmetry case and those with an excised angle of 75/C14represent a high asymmetry case, where the excised angle ( h) is half of the central angle correspond- ing to the excised arc, as shown in Fig. 1(a). Scanning elec- tron microscopy images for low and high asymmetry caseswith excised angles of 30 /C14and 75/C14are shown in Figs. 1(b) and1(c), respectively. Py disks are arranged in a square lat- tice with a period of 1.6 lm and a 100 /C2100lm2area. Focused magneto-optic Kerr effect (focused MOKE) in a longitudinal conﬁguration is adopted to characterize thea)Author to whom correspondence should be addressed. Electronic mail: phlhorng@cc.ncue.edu.tw. Tel.: 886-4-723-2105. Fax: 886-4-721-1153. 0021-8979/2013/113(10)/103905/5/$30.00 VC2013 American Institute of Physics 113, 103905-1JOURNAL OF APPLIED PHYSICS 113, 103905 (2013) Downloaded 08 Apr 2013 to 129.252.86.83. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionshysteresis loop at room temperature with an external ﬁeld parallel to the ﬂat edge. An S-polarized laser beam of632.8 nm passes through a diverging lens and a converging lens and is focused to a spot of 50 lm diameter, striking the Py array with an incidence angle of 45 /C14. Note that the focused MOKE hysteresis loop represents a superposition of the signal from all identical elements in the array. Micromagnetic domain structures in the Py disk of 800 nmin diameter, 30 nm in thickness, and with a 75 /C14excised angle, are calculated by object oriented micromagnetic framework ( OOMMF ) software. III. RESULTS AND DISCUSSION In this paper, we describe a technique of analyzing the minor hysteresis loops for determining vortex chirality in submicro-scaled disks by focused MOKE. After forcing the magnetic moments to saturation with different direction, asstate A and B in Fig. 2(a), release them to remanent state. The disk is in the state of A (B) originally then forms a CW/ CCW vortex, as shown in Fig. 2(b). The chirality of vortex is known to be controlled by the asymmetry in a disk. Under a small external ﬁeld of H as shown in Fig. 2(c), the two vor- texes with different chirality are deformed by forcing theircores to move to the opposite directions perpendicular to the applied ﬁeld direction. It should be noted that at a remanence state, the magnetizations in an asymmetric disk nucleatefrom the ﬂat edge easier than from the round edge, and simi- larly a vortex core pushed to the ﬂat edge annihilates easier than that to the opposite round edge. Therefore, in the caseof Fig. 2(c), the annihilation ﬁeld (H an) of CCW vortex is smaller than the H anof CW vortex, while the ﬁeld isincreasing to the right side. While the external ﬁeld increases to the H anof CCW, the disk in CCW vortex state saturates, but a CW vortex structure can still survive at that ﬁeld, as shown in Fig. 2(d). Here, the saturation is considered as the state where the most domains are approximately aligned par-allel to external ﬁeld. As mentioned above, the core moving toward the ﬂat edge annihilates earlier than that away from the ﬂat edge.Therefore, the CW and CCW vortices will present different trajectories on the hysteresis loops near the annihilation ﬁeld. For the case of a disk at remanent state, in which the vortexchirality is unknown, one could measure the minor hysteresis loop with external ﬁeld from zero to the ﬁeld, which is in the middle of the two annihilation ﬁelds for different chirality.As shown in Fig. 2(d), if the vortex is in CW conﬁguration originally then no annihilation signal will be observed. On the other hand, if the vortex is in CCW then an annihilationsignal will be observed. It is important that the detected disk by this technique, its chirality hold still when it is released back to the remanent state. In one word, all detected vortexstates will not be changed after the reading process. So, the apparent ﬂat edge in SVE not only achieves the control of vortex chirality but also separates the annihilation ﬁeld ofCW/ CCW vortex states. In the low asymmetry case, the full loop is measured at an initial saturation ﬁeld of þ800 Oe, which then sweeps down to an opposite saturation ﬁeld of /C0800 Oe, before ﬁnally sweeping back up to þ800 Oe again, as shown in Fig.3. Here, the saturation state is deﬁned as a state where the magnetic moments are aligned approximately parallel, and where a small angle of departure from the means of thetotal magnetization direction is acceptable. Moreover, the direction of positive magnetic ﬁeld is deﬁned as the right side and the direction of negative magnetic ﬁeld as the leftside. The other loops in Fig. 3(a)are the minor reverse loops. For example, the L-100 reverse loop begins at the same ini- tial saturation ﬁeld of þ800 Oe, then sweeps down to /C0100 Oe, and subsequently returns to þ800 Oe. The reverse loops are obtained in a similar way but for different reversing ﬁelds. Because the front parts of all these loops are coinci-dent, only the rear loops are displayed for clarity. From the L150, L100 and L0 reverse loops in Fig. 3(a), about 50%, FIG. 1. Schematic illustrations of (a) one-side-ﬂat disk with excised angle h, adopted to represent the degree of asymmetry. Scanning electron micros- copy (SEM) images of 800 nm Py disk for: (b) low asymmetry case with 30/C14 excised angle; and (c) high asymmetry case with 75/C14excised angle, respectively. FIG. 2. Schematic illustrations of magnetic moments in disk with eminent asymmetry at: (a) longitudinal positive/negative saturation state, (b) rema- nent state in CW/CCW conﬁguration, (c) a small right direction external ﬁeld, and (d) annihilation ﬁeld of CCW vortex.103905-2 Huang et al. J. Appl. Phys. 113, 103905 (2013) Downloaded 08 Apr 2013 to 129.252.86.83. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions90%, and 100% of disks nucleating into CCW vortex are observed, respectively, which is in accordance with the prin-ciple of asymmetry control on vortex chirality, as mentioned above. Upon 0 Oe, the reverse loop presents complete disks nucleation of vortex. The distribution of vortex nucleationﬁeld (H n) or vortex annihilation ﬁeld (H an) is induced by the existence of a difference between the disks caused by the fabricating process. It is observed in Fig. 3(a) that the CW vortex annihilation occurs at about þ400 Oe in the ascending branch of the full loop, and that the CCW vortex annihilation occurs at about þ510 Oe in the ascending branch of the L0 and L-100 reverse loops. In the ascending branch, the CCW vortex annihilation ﬁeld from the ﬂat edge is þ520 Oe (H an for CCW), which is larger than the CW vortex annihilation ﬁeld from the round edge of þ400 Oe (H anfor CW). Similar experimental results are observed in another report using low asymmetry magnetic disks.11Fig.3(b)shows the full, L-300, L-400, L-500, and L-600 Oe reverse loops; some reserve loops are presented as only the rear part of the loop for clarity. From the reverse loops, it is observed that about 0%,10%, 90%, and 100% disks achieve negative direction satu- ration and enter gradual transition saturation on hysteresis trajectories between the H anof CCW and CW vortices, as shown by the circled area in Fig. 3(b), due to the proportion of CW to CCW vortex changing.In the high asymmetry case, the full and minor reverse hysteresis loops are obtained in a similar way to the low asymmetry case and for clarity, some reverse loops are dis-played as only the rear curves, as shown in Fig. 4. From the L0, L-25 and L-50 reverse loops in Fig. 4(a), it is observed that about 40%, 80%, and 95% of disks nucleate into CCWvortex, respectively. The full loop presents a gentle annihila- tion trajectory at about þ500 Oe, and other minor reverse loops show steep vortex annihilation at about þ285 Oe, as shown in Fig. 4(a). Here, the CCW vortex annihilation ﬁeld from the ﬂat edge is þ285 Oe (H anfor CCW), which is much smaller than the CW vortex annihilation ﬁeld from the roundedge of þ500 Oe (H anfor CW). Fig. 4(b)exhibits the full, L- 450, L-500, L-600, and L-700 reverse loops; some reverse loops are presented as only the rear part of loop for clarity. Itis observed in Fig. 4(b) that partial disks achieve negative saturation in L-450 and L-500 reverse loops and enter a mixed situation, where some disks are in CCW and some inCW conﬁgurations at remanent state, as shown in the lower insets of Fig. 4(b). The two loops show annihilation of both CW and CCW vortices in the circled area of Fig. 4(b). Upon L-600 and L-700 Oe reverse loops, all disks arrive at nega- tive saturation ﬁeld and the curves overlap with the full hys- teresis loop. Comparing the low asymmetry case with the high asym- metry case, a reversal of annihilation sequence of the CCW FIG. 3. Full hysteresis loop of low asymmetry Py (30/C14excised angle) disk array with (a) reverse loops returned at external ﬁelds of 150, 100, 0 and/C0100 Oe, and (b) reverse loops returned at external ﬁelds of /C0300,/C0400, /C0500, and /C0600 Oe. The insets schematically represent the vortex types in different loops at remanent state. FIG. 4. Full hysteresis loop of high asymmetry Py (75/C14excised angle) disk array with (a) reverse loops returned at external ﬁeld of 0, /C025,/C050,/C0100, and/C0200 Oe, and (b) reverse loops returned at external ﬁeld of /C0450, /C0500,/C0600, and /C0700 Oe. The insets schematically represent the vortex types in different loops at remanent state.103905-3 Huang et al. J. Appl. Phys. 113, 103905 (2013) Downloaded 08 Apr 2013 to 129.252.86.83. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsand CW vortices is observed. That is, H anfor the CCW vor- tex in the low asymmetry case is larger than that for the CW vortex, (in Fig. 3(a)), but H anfor the CCW vortex in the high asymmetry case is smaller than that for the CW vortex (in Fig.4(a)). In the high asymmetry case, the vortex core mov- ing towards the ﬂat edge annihilates earlier than that movingaway from the ﬂat edge, because the long ﬂat edge forces the domains into a straight alignment instead of forming a rota- tional conﬁguration, and the large cut area depresses the vor-tex motion. With an increasing external ﬁeld, the local magnetostatic energy rises quickly when a vortex core is pushed towards the ﬂat edge. As mentioned above, the largecut area and edge suppress a vortex core moving towards the cut side, which leads to a large decrease of vortex annihila- tion ﬁeld from the ﬂat edge. Therefore, the CW and CCWvortices exhibit different annihilation sequences and differ- ent trajectories on hysteresis loops in low and high asymme- try disks. This reverse sequence of vortex annihilation in thehigh asymmetry case offers a way to detect directly the vor- tex chirality by analyzing the hysteresis loop. For example, as in the case of Fig. 4(a), once the hysteresis loop is meas- ured from the remanent state to a positive ﬁeld a little larger than H anof CCW, the curve will present vortex annihilation behavior if the detected disk is originally in CCW conﬁgura-tion at the remanent state and will show no vortex annihila- tion behavior if the disk is originally in CW conﬁguration. It should be noted that after detecting vortex chirality in highasymmetry disks, the annihilated vortex originally in CCW returns to CCW vortex at the remanent state, and the disk in CW conﬁguration is abiding still in CW conﬁguration. Inother words, all detected vortex states are preserved at the re- manent state after detection. Numerical micromagnetic simulations based on the Landau-Lifshitz-Gilbert equation (LLG equation) are solved using OOMMF software. Here, the cell size, the saturation magnetization Ms, the exchange constant, and the dampingparameter are 5 nm, 800 kA/m, 1 /C210 /C011J/m, and 0.5, respectively. Fig. 5(a) shows the hysteresis loops simulated for a Py disk with a diameter of 800 nm, a thickness of30 nm, and an excised angle of 75 /C14. The full loop was calcu- lated with the external ﬁeld decreasing from þ5000 to /C05000 Oe, and then increasing back to þ5000 Oe. The L- 150 reverse loop is for the magnetic ﬁeld decreasing from þ5000 to /C0150 Oe, and increasing back to þ5000 Oe. In the descending branch, the vortex nucleation occurs at /C0150 Oe, which is identical for both full and reverse loops. In the ascending branch, the vortex annihilation for CCW in reverse loop is solved as þ275 Oe (H anfor CCW), and that for CW in full loop is þ512.5 Oe (H anfor CW). These results are in good agreement with the experimental data of þ285 Oe for the CCW vortex and þ500 Oe for the CW vor- tex. Moreover, the simulated hysteresis loop trajectory shows the same tendency with the focused MOKE record at the CCW vortex annihilation, which arrives at saturationsteeply, and at the CW vortex annihilation, which arrives at saturation gently. To further understand the domain struc- tures in the asymmetric disk, the simulated magnetizationdistributions for full and reverse loops at ﬁve different external ﬁelds (as the positions (i) to (v) shown in Fig. 5(a)during ascending branch) were selected. Fig. 5(b) exhibits the disk magnetic structures at (i) þ125 Oe, (ii) þ260 Oe, (iii)þ300 Oe, (iv) þ500 Oe, and (v) þ575 Oe, respectively. From Fig. 5(b) ((i) and (ii)), it is observed that the vortices are in CW and CCW conﬁgurations for full and /C0150 Oe minor reverse loops. At the ﬁeld between þ275 and þ512.5 Oe (the H anof CCW and CW), the disk originally in CCW vortex is saturated, but the disk in CW vortex is still preserved, as shown in Fig. 5(b) ((iii) and (iv)). Finally, in Fig.5(b) (v), the two disks both arrive at saturation state at large external ﬁeld. It should be noted that a reversal of the CW and CCW vortex annihilation sequence was observed in low and high asymmetry disks. Dumas et al. reported on a different and opposite chirality control mechanism through the nucleationand coalescence of double vortices by manipulating the size and/or thickness of asymmetric Co dots. 19Comparing these results, it is similar that the analysis of the annihilation ﬁledbehavior along major and half hysteresis loops is a useful and reproducible technique to determine vortex chirality. IV. CONCLUSION In conclusion, we studied the inﬂuence of the degree of asymmetry on vortex annihilation with different chirality. The annihilation ﬁeld is found to depend not only on the FIG. 5. (a) Numerical simulated hysteresis loop for Py disk with diameter of 800 nm, thickness of 30 nm, and excised angle of 75/C14. The full loop is calcu- lated with external ﬁeld from þ5000 to /C05000 Oe and back to þ5000 Oe. The/C0150 Oe loop is from þ5000 to /C0150 Oe and back to þ5000 Oe. For easy comparison with Fig. 3, the hysteresis loops only show the range from þ800 to /C0800 Oe. (b) Simulated magnetization distributions of the disk in full and reverse loops during ascending branch at external ﬁeld of: (i) þ125 Oe, (ii) þ260 Oe, (iii) þ300 Oe, (iv) þ500 Oe, and (v) þ575 Oe, respectively.103905-4 Huang et al. J. Appl. Phys. 113, 103905 (2013) Downloaded 08 Apr 2013 to 129.252.86.83. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsvortex chirality but also on the degree of asymmetry in the magnetic disk. H anfrom the ﬂat edge is larger than that from the round edge in low asymmetric magnetic disks. Thesequence of vortex annihilatio ni sr e v e r s e di nh i g ha s y m m e t r i c magnetic disks because the magnetostatic energy increases quickly near the long ﬂat region. Numerical simulations and ex- perimental results indicate simila r trajectories of hysteresis loop and annihilation ﬁelds for diffe rent chirality vortices. In addi- tion, the sequence of vortex annihilation and the apparently dif-ferent hysteresis trajectorie si nh i g ha s y m m e t r i cd i s k sh a v e high potential for fast vortex chirality detection. ACKNOWLEDGMENTS This work was supported by the National Science Council, Taiwan, under Grant No. NSC 99-2112- M-018- 003-MY3. 1J. G. Zhu, Y. F. Zheng, and G. A. Prinz, J. Appl. Phys. 87, 6668 (2000). 2E. Saitoh, M. Kawabata, K. Harii, H. Miyajima, and T. Yamaoak, J. Appl. Phys. 95, 1986 (2004). 3T. Aign, P. Meyer, S. Lemerle, J. P. Jamet, J. Ferre, V. Mathet, C. Chappert, J. Gierak, C. Vieu, F. Rousseaux, H. Launois, and H. Bernas, Phys. Rev. Lett. 81, 5656 (1998). 4R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E. Welland, and D. M. Tricker, Phys. Rev. Lett. 83, 1042 (1999).5T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, Science 289, 930 (2000). 6J. Raabe, R. Pulwey, R. Sattler, T. Schweinbock, J. Zweck, and D. Weiss,J. Appl. Phys. 88, 4437 (2000). 7M. Natali, I. L. Prejbeanu, A. Lebib, L. D. Buda, K. Qunadjela, and Y. Chen, Phys. Rev. Lett. 88, 157203 (2002). 8M. Schneider, H. Hoffmann, and J. Zweck, Appl. Phys. Lett. 77, 2909 (2000). 9M. Schneider, H. Hoffmann, and J. Zweck, Appl. Phys. Lett. 79, 3113 (2001). 10R. Nakatani, T. Yoshida, Y. Endo, Y. Kawamura, M. Yamamoto, T.Takenaga, S. Aya, T. Kuroiwa, S. Beysen, and H. Kobayashi, J. Magn. Magn. Mater. 286, 31 (2005). 11T. Kimura, Y. Otani, H. Masaki, T. Ishida, R. Antos, and J. Shibata, Appl. Phys. Lett. 90, 132501 (2007). 12H. Hoffmann and F. Steinbauer, J. Appl. Phys. 92, 5463 (2002). 13C. A. Ross, F. J. Castano, D. Morecroft, W. Jung, H. I. Smith, T. A. Moore, T. J. Hayward, J. A. C. Bland, T. J. Bromwich, and A. K. Petford- Long, J. Appl. Phys. 99, 08S501 (2006). 14T. J. Bromwich, A. K. Petford-Long, F. J. Castano, and C. A. Ross, J. Appl. Phys. 99, 08H304 (2006). 15C. C. Wang, A. O. Adeyeye, and Y. H. Wu, J. Appl. Phys. 97, 10J902 (2005). 16K. M. Wu, J. F. Wang, Y. H. Wu, C. M. Lee, J. C. Wu, and L. Horng,J. Appl. Phys. 103, 07F314 (2008). 17N. M. Vargas, S. Allende, B. Leighton, J. Escrig, J. Mej /C19ıa-L/C19opez, D. Altbir, and I. K. Schuller, J. Appl. Phys. 109, 073907 (2011). 18R. K. Dumas, T. Gredig, C.-P. Li, I. K. Schuller, and K. Liu, Phys. Rev. B 80, 014416 (2009). 19R. K. Dumas, D. A. Gilbert, N. Eibagi, and K. Liu, Phys. Rev. B 83, R060415 (2011).103905-5 Huang et al. J. Appl. Phys. 113, 103905 (2013) Downloaded 08 Apr 2013 to 129.252.86.83. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
1.466877.pdf	Theory of activated rate processes in the weak and intermediate friction cases: New analytical results for one and many degrees of freedom A. I. Shushin Citation: The Journal of Chemical Physics 100, 7331 (1994); doi: 10.1063/1.466877 View online: http://dx.doi.org/10.1063/1.466877 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/100/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Transition rate prefactors for systems of many degrees of freedom J. Chem. Phys. 124, 164102 (2006); 10.1063/1.2188943 Erratum: Theory of activated rate processes in the weak and intermediate friction cases: New analytical results for one and many degrees of freedom [J. Chem. Phys. 100, 7331 (1994)] J. Chem. Phys. 101, 8266 (1994); 10.1063/1.468199 Effective Feynman propagators and Schrödinger equations for processes coupled to many degrees of freedom J. Chem. Phys. 96, 5952 (1992); 10.1063/1.462662 Activated barrier crossing for many degrees of freedom: Corrections to the low friction Kramers result J. Chem. Phys. 86, 2444 (1987); 10.1063/1.452095 The Response of a Resonant System with One Degree of Freedom as a Limiting Case Am. J. Phys. 39, 577 (1971); 10.1119/1.1986220 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.107 On: Sun, 23 Nov 2014 07:03:15Theory of activated rate processes in the weak and intermediate friction cases: New analytical results for one and many degrees of freedom A. I. Shush in Institute of Chemical Physics, Academy of Sciences, GSP-1, 117977, Kosygin str. 4, Moscow, Russia (Received 7 October 1993; accepted 30 December 1993) Simple analytical expressions for the reaction rate of activated rate processes are derived in the weak/intermediate friction limit for one and many degrees of freedom and for finite microcanonical reaction rates. The expressions are obtained by analytical solution of the steady-state integral master equations (in energy variables). The microcanonical reaction rate is taken to be independent of energy (higher than the activation energy). Irreversible transitions from one state and reversible transitions between many states are discussed in detail. A simple interpolation formula for the reaction rate is derived which describes a turnover from the weak friction regime to a strong friction one. The formula takes into account an interplay between activation and reaction at energies close to the activation energy. When applied to unimolecular gas phase reactions this interpolation formula bridges between the weak and strong collision limits. The formulas obtained are generalized to multidimensional activated rate processes. I. INTRODUCTION The theory of activated rate processes has been devel oped very intensively in the last few years. The interest in this problem is inspired by many applications in physics and chemistry. A comprehensive review of recent works in this field can be found in Ref. I. This article deals with activated rate processes in the weak and intermediate friction limits in which the problem is known to reduce to an integral master equation (ME) in en ergy (action) variables. Essential progress in treating the problem within the framework of the conventional Kramers theory was made in Refs. 2 and 3 where the Wiener-Hopf method was proposed for solution of the MEs. The method made it possible to reasonably describe the change-over from the weak damping to a strong one. Later the method was modified to treat the process as activation along the unstable mode of the Kramers problem near the top of the barrier.4 However. the modified method being rather accurate ap peared to be somewhat complicated for applications. It is known I that in the weak and intermediate friction limits any assumptions on properties of the friction force fluctuations manifest themselves in the mathematical form of the kernel in the ME. For example. the Gaussian approach for fluctuations is known to lead to the Gaussian kernels.2,3 There are also some other popular types of kernels such as the exponential one.S,6 The Wiener-Hopf method enables us to obtain the universal solutions of the MEs for any type of kernels and thus for any type of the friction force fluctua tions. Surprisingly. so far it has been applied only to the processes in which activation is induced by the Gaussian friction forces. In all the above-mentioned works the Wiener-Hopf method has been applied to the one-dimensional (lD) MEs. However. sometimes more complicated multidimensional MEs are required to properly describe molecular reactions in which many molecular degrees of freedom are involved. For example. in the case of strong ro/vibrational interaction the unimolecular gas phase reactions are often analyzed via 2D MEs.7-12 No simple analytical expressions for the reaction rates in this case have been proposed so far. The only ex ample of the analytically solvable multidimensional MEs is the 2D exponential model proposed by Troe8,9 but the solu tion found is too complicated to be useful. The main goal of this paper is to demonstrate that a number of well-known problems of the theory of activated rate processes in the weak and intermediate friction limits (which are treated in the ME approach) can be solved in a unified way by the Wiener-Hopf method. Moreover, this method enables us to generalize some known solutions. In particular, analytical expression for the reaction rate in the case of a finite (and independent of energy) microcanonical reaction rate is derived. This expression is used to obtain a simple interpolation formula providing a physically reason able description of an interplay of activation and reaction at energies close to the activation energy and bridging the un derdamped and overdamped regimes. Simple analytical ex pressions are also derived for the rate of reversible transi tions between several states and for the multidimensional reaction rate are also derived. II. GENERAL FORMULATION OF THE PROBLEM In the weak friction limit the activated rate processes are described by a ME for the population density in the multidi mensional space of energy (or, more correctly, action) vari ables E={EJ (j= 1, ... ,m)1,S,6 p= -v(1-g)p-kp, (1) where v is a collision frequency and k(E) is a microcanonical reaction rate. The integral operator g is defined by gp= f dE' g(E-E')p(E'). (2) The kernel g(E-E') is hereafter called the transition prob ability. In accordance with definition (2) we assume that g(E,E') depends only on the difference E-E'. This assump tion is not very restrictive although, in reality, some devia- J. Chem. Phys. 100 (10), 15 May 1994 0021-9606/94/100(10)17331/9/$6.00 © 1994 American Institute of Physics 7331 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.107 On: Sun, 23 Nov 2014 07:03:157332 A. I. Shushin: Theory of activated rate processes tion from E-E' dependence is possible, e.g., due to the ef fect of the degeneracy N(E) which is incorporated in g(E,E'). The deviation is, in general, rather weak because the reaction kinetics is determined mainly by the narrow re gion of energy IEI-Ea-T near the activation energy Ea where N(E) is usually smooth. In addition, the above mentioned assumption is valid for the exponential N(E) de pendence. In this case the effect of N(E) reduces to a renor malization of the temperature. To take into account this effect we introduce different effective temperatures T· for different coordinates E j • J The kernel geE) satisfies the normalization condition f dE g(E)= 1 (3) and the detailed balance relation g(E-E')= geE' -E)exp[ -F·(E-E')], (4) where F={lIT) is the vector of inverse temperatures in the E space. The analytical form of geE) depends on the activation mechanism. Very often, however, in the absence of any in formation on g (E) except for the average energy transfer per collision AE= f dE Eg(E) (5) and the correlation matrix S Sjk=Skj= (1I2)AE jAEk =(1I2)f dE(Ej-AEj)(Ek-AEk)G(E). (6) one has to restrict oneself to simple physically reasonable models for geE). The most popular are (1) The Gaussian modell,6 in which geE) = (4'7T det S) -112 exp[ -(E-SF)T( 4S) -I(E-SF)]. (7) (2) The exponential model (for the diagonal matrix Sik=SiOik) in which g(E)~A exp( -F·El2-j. ~JiEjl/2 ) (8) with 'Pj = ~FJ+4/Sj and A = IIfI/Sj'Pj)' For the high activation energy Ea~ 1I1FI the transition rate W is expressed in terms of the solution of the steady state ME1,5,6 v(l-g)p+kp=O with the boundary condition at IEI~Ea p(E)-exp( -F·E). Namely, W=(l/Z)f dE k(E)p(E), where (9) (10) (ll) z= ( dE peE) J {i} (12) is the quasiequilibrium partition function of the initial state {i}. In this work we apply the Wiener-Hopf method to an analytical solution of the steady state ME9 with the stepwise reaction rate k(E)= KO[b·(E-Ea)], (13) where ~x) is the Heaviside step function and b is the unit vector perpendicular to the reaction surface. III. ONE-DIMENSIONAL PROBLEM A. Escaping from the well In the weak and intermediate friction limits escaping from aID well is described by the I D ME V[P(E)- f dE' g(E-E')p(E') 1 +k(E)p(E) =0. (14) In accordance with Eq. (13) the rate k(E) is taken in the form (15) To solve Eq. (14) let us introduce the Fourier transforms R±(q)=Ff~oo dE e(iq+1I2)FE p(E)O[±(E-Ea)]. (16) The transition rate is then given by W=(K,Z)foo dE p(E)=KR+(il2). Ea (17) It is easily seen that R ± (q) satisfy the Wiener-Hopf equationl3 (18) in which and I-G Q=----I-G· v/(v+ K) G(q)= f:oodE g(E)e(iq+1I2)FE. (19) (20) The boundary condition (10) shows that R_(q) has a pole 1 i R_(q)=-Zq+il2 at Iq+il21~1. (21) The functions R+(q) and R_(q) are analytical in the upper and lower half-planes of complex q, respectively. Solution of Eq. (18) by the Wiener-Hopf method givesl3 . Q_( -i12) R_(q) = (-I) Q_(q)(q+ i12) (22) and J. Chem. Phys., Vol. 100, No. 10, 15 May 1994 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.107 On: Sun, 23 Nov 2014 07:03:15A. I. Shushin: Theory of activated rate processes 7333 (23) where [ I foo In Q(A) ] Q:t(q)=exp ±-2' dA ~ --'0 . 7T1 _00 I\. q+l (24) Substituting Eq. (24) into Eq. (18) one obtains the formula [ I foo dA ] W= Wsc exp 27T -00 1I4+A2 In Q(A) (25) in which f k(E) K Wsc=(vIZ) dE v+k(E) e-FE=v V+K e-FEa (26) is the reaction rate in the strong collision limit (S -+00) and Q(A) is given by Eq. (19). Formula (25) represents the first example of the general analytical expression for the rate of activated processes with the finite microcanonical transition rate k(E) derived within the ME approach. This formula allows us to take into ac count the interference of activation and reaction at E> Ea for any kernel g(E). Now let us consider the particular models of g(E) men tioned earlier. B. Exponential kernels The exponential model for g(E) is rather popular in the theory of gas phase unimolecular reactions.5,6 It is easily seen that for any general ID m-exponential model with m g(E) = e -FEf2 ~ gje -'I)EI/2 j= I (27) the problem reduces to solving algebraic equations. Indeed, since the Fourier transform of g(E) is given by (28) one can obtain the integral in Eq. (25) analytically and get finally m ( ) 2 1I2+q_j w=wscil 112+ . ' j= I q+} (29) where q _ j and q + j are the positive roots of the algebraic equations I-GUq)=O K and 1 +--GUq)=O, v (30) respectively. Note, that the lowest root q -I can be found explicitly: q -I = 112, because p=:exp( -FE) is the solution of Eq. (14) at E<Ea. 1. Monoexponentlal model In the simplest monoexponential model (m =: 1)8 g(E) = (1 /S<p )exp( -<pIE1/2 -FEf2), (31) where <p = ,jF2+4/S and S is defined by Eq. (8). In accor dance with Eq. (29) (1 )-2 W=Wsc 2' +q+l , (32a) 2 (1I2-q+l) =v(2F1<p) 1I2+q+1 exp(-FE a), (32b) where q+1 = (l/2),jl+(l+vIK) I(F2S) I. The expres sions (32a) and (32b) can also be represented as W/W -( v) ~ sc =FILlEI 1 +-, 1-W/Wsc K (33) where LlE=fdE Eg(E)=-FS< O. At vlK~l formula (33) reduces to that derived earlier.8 For finite vi K it describes W dependence on the collision frequency v and correctly reproduces the behavior of W both in the strong and in the weak collision limits thus providing an interpolation formula for any values of v and S. Formulas (32a), (32b), and (33) can be obtained by an alternative method based on the fact that the integral equa tion (14) with the monoexponential kernel (31) is equivalent to the differential equationl2,14 for d20' (F2S k(E») S dE2 -4 + v+k(E) 0'=0, p(E) O'(E) v+k(E) (34) (35) Simple solution of Eq. (34) for the stepwise dependence k(E) [Eq. (15)] results in the expressions (32a), (32b), and (33). For any general dependence k(E) 8(E -Ea) the solution is obtained by matching (at E=Ea) the solution O'_(E) for E<Ea with O'+(E) for E>Ea. Substitution of this solution into the general expression shows that the rate W is given by Eq. (32b) with I 1 dO'l q+l='F O'(E)dE E=E . a (36) It is worth noting that Eq. (34) can be solved analytically for some dependencies k(E) interesting for applications: (1) for k(E)-(E- Ea) the solution is expressed in terms of the confluent hypergeometric functions, 15 (2) for k(E) =ko exp[A(E-Ea)]+kl and k(E)=ko tanh[A(E-E a)] + k I the solution is expressed in terms the hypergeometric function 2FI(a,b,c,z).15 Here we restrict ourselves only to these brief comments on the new methods of analytical solution of the integral MEs with monoexponential kernels. Any detailed investiga tion of the behavior of W for the above-mentioned depen dencies k(E) should be done for particular systems by com parison with experimental data and, if possible, using numerically calculated k(E). J. Chern. Phys., Vol. 100, No. 10, 15 May 1994 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.107 On: Sun, 23 Nov 2014 07:03:157334 A. I. Shushin: Theory of activated rate processes 2. Biexponential model Rather simple analytical solution of the problem can also be obtained in the biexponential model16 geE) = (gle-rpdEl12+ g2e-'I'2IEI/2)e-FEI2, (37) for which Eqs. (32) and (33) reduce to simple quadratic equations. In this model [ 1I4+(q-l +q-2)/2+q-lq-2 ] (38) W=Wsc 1I4+(q+l+q+2)/2+q+lq+2 . The simple combinations of roots in Eq. (42) can be obtained analytically (39) and q:!:.l +q:!:.2= ~Xl +X2+Y:!:.1 +Y:!:.2+2Y:!:., (40) where xj=('Pif2F)2, Y_j=gj'P/F2, and Y+j=gj'P/(l+K)F. In the important limit of high reactivity (K-+OO) formulas (39) and (40) are simplified since b+j=O and thus q+j = .J;;. The expression for W in this limit was obtained in Ref. 16 by an alternative method which, however, results in a rather inconvenient analytical representation of this expres sion. C. Turnover formula The method developed in Sec. II A enables us to derive a simple bridging formula which correctly describes the be havior of the rate for low-to-high frictions. The problem of bridging the low and high friction limits has been considered in a number of papers.I-4 Different approximations have been proposed to describe the intermediate friction region. Some of them treat this problem empirically using simple kinetic arguments (for references see Ref. 1). A more rigor ous approach is proposed in Ref. 4 and is based on the ap proximation of linear coupling of the reaction coordinate with the oscillator bath. This approach describes the behavior of the rate constant rather accurately. Unfortunately, the ob tained expression for W appeared to be cumbersome and inconvenient for applications. The approach we will develop in this section is not quite rigorous but is based on clear physical arguments and per mits derivation of a simple interpolation formula useful for applications. To obtain this formula let us consider the escaping of a particle from the potential well U(x) (see Fig. 1) in the model of activation induced by a Gaussian fluctuating force f(t). It is known that the motion of this particle is described by the generalized Langevin equation 1 (t dU mX-J/ t' f(t-t')i+ dx = f(t), (41) where m is the particle mass and f(t)=(2IF)(j(t)f(t') is the generalized friction induced by the fluctuating force. When the rate is determined by the dynamics of passing over the barrier then according to the Grote-Hynes theoryl7 in the strong friction limit u(x) x FIG.!. Schematic picture of the potential welJ U(x). (42) where V=nl27T is a frequency of the potential well near the bottom and, in fact, is a frequency of collisions with the barrier. The parameter p < 1 is the positive root of the equa tion 1 p= p+ Y(P)IWb (43) in which Wb is the frequency of the barrier near the top and "y(p) is the Laplace transform of the friction f(t) Y(P)=Jo'" dt f(t)e-wbPt• (44) In the low and intermediate friction cases the process of activation into the transition region near the top becomes important. The effect of of activation in this region can be properly described within the proposed model. First, note that the parameter p in Eq. (43) can be interpreted as a re action probability per collision with the barrier at energies E>Ea (it is seen from Eq. (43) that p<I). Second, the pro posed model treats the collision frequency v as the rate of transitions between energy levels. This means that the reac tion probability at each energy level is given by p = KI( v+ K) and thus Klv=pl(l-p). Substitution of Eq. (45) into Eq. (25) gives W= vp exp[ 21 7T J~", 11:~}" 2 In Q(}..) ]exP( -FEa), where V=nt27T and I-G(}") Q(}..)= l-(l-p)G(}") with G(}") =exp[ -SF2( 114+}" 2)]. (45) (46) (47) J. Chern. Phys., Vol. 100, No. 10, 15 May 1994 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.107 On: Sun, 23 Nov 2014 07:03:15A. I. Shush in: Theory of activated rate processes 7335 The expression for the parameter S in terms of the charac teristics of the dynamical motion in the potential U(x) and the fluctuations of l(t) is given in Refs. 1-4. It is easily seen that formula (46) reproduces correctly both the weak (p "'" I) and the strong (p ~ 1) friction limits. Unlike the majority of interpolation expressions proposed earlier' Eq. (46) takes into account an interplay between ac tivation and reaction at E~Ea in a physically reasonable way. Calculation shows that for the conventional Kramers problem expression (46) predicts a turnover behavior of W which is close to the predictions of other well known inter polation formulas:'-3 The deviation does not exceed 10%- 20%. It should be noted, however, that this expression makes it possible to describe quite reasonably the peculiarities of the above-mentioned interplay of excitation and reaction at E> Ea which seems to become rather important in the mod els with coordinate-dependent friction'8,19 and with poten tials U(x) highly anharmonic near the top of the barrier. D. Transitions between two wells The proposed model can easily be generalized to de scribe transitions between two wells. Let p,,2(E) denote the population densities in the two states I and 2, respectively. The rate of reversible transitions is obtained by solution of the system of MEs v, (I -8, )p, + K, 8(E-Ea)P' -K2[ 8(E-Ea)P2] = 0, (48) v2(l -82)P2 + K28(E-Ea)P2 -K ,[ 8(E-Ea)p,] =0. In accordance with relation (4S) we take (49) The first terms in Eqs. (48) represent activation transitions while the second and third terms describe the forward and backward reactions, respectively. The forward reaction flux is modeled by the microcanonical rates K',2 similarly to the single well transitions discussed earlier. As for the backward reaction, we consider here two models of this process. Model I. In this model we treat the backward reactions in the same way as the forward ones, i.e., taking Kj= Kj' j=I,2. Model 2. To describe more realistically the backward reaction flux let us take into account that the direction of velocities of the particles which pass over the barrier, say, from weIl I to well 2 is opposite to that required to pass back so that the particles have to spend one period in well 2 before passing back. This means that at the moment of backward reaction the particles will have an additional spread of en ergy g2(E-E') due to activation process in the well 2. Similar spread g, (E -E') will appear in well 1. This model implies the following definition of the back ward reaction terms: K,p,=K,82[8(E-E a)pd and K2P2 = K28, [8(E-Ea)P2]' In both models the ME (S2) can be solved by the Wiener-Hopf method. To calculate the transition rate W2, = (1IZ,/,lO dE[kiPI (E) -k2P2(E)]= (1IZI)R+(il2), lEa (SO) we need to obtain the function p= KIP, -K2P2 or its Fourier transform R= K,R, -K2R2 defined by Eq. (16). It is easily seen that R(q) satisfies the Wiener-Hopf equation -R_=[I+I~P (~, +~J]R+' in modell, (Sla) - R -= [ I + I ~ P (~, + ~ 2 -I ) ] R +, in model 2, (Sib) in which Equations (Sla) and (Sib) should be solved with the bound ary condition corresponding to the initial population of one of the wells, say, the well I: R + I (q) and R + 2( q) are ana lytical in the upper half-plane of q, R _ 2( q) is analytical in the lower half-plane of q and R _ , (q) has a pole of type (21) in the lower half-plane. Solutions of Eqs. (Sla) and (SIb) with this boundary condition yields in both models U,U2 W2,=v,p -u;; exp(-FE'a), (S3) where Uj are defined by the relation [ I roo dz ] Uj=exp 27T Jo 114+>..2 In[Nj(>..)] , (j=O,1,2), (S4) in which N,,2 are given by Eq. (S2). As for No(>") it is dif ferent in the two models considered {(l-P)N,N 2+P(N,+N 2), in model I No= (l-p)N,N 2+p(l-G,G 2), in model 2' (55) At first sight, the two models give different formulas for W2, due to the difference in No, but in reality the expressions are close to each other and lead to the same results in all physically important limits. For example, in the case of high reactivity at E> Ea when p=l _{2-G,(>..)+G 2(>..), in model 1 No(>")-I-G,(>")G2(>"), in model 2 . (56) It is easily seen that in the weak friction limit (S I 2F2~ 1) when G/>..)=1-SjF2(>..2+1I4) one' has No(>") ""'(S, + S2)F2(>.. 2+ 114) in both models and thus W'2=V,(S,S2)F2/(SI+S2) in agreement with earlier investigations.' In another limit S, ~S2 one can set Gt(>..)= I and No'" 1 - G , (>..), and get for both model 1 and model 2 W 2' = WI' where WI is the escaping rate from well 1 defined by Eq. (46). Similarly, in the strong friction limit corresponding to the low reactivity p ~ 1 both models give the same result W2,=v,p. J. Chern. Phys., Vol. 100, No. 10, 15 May 1994 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.107 On: Sun, 23 Nov 2014 07:03:157336 A. I. Shushin: Theory of activated rate processes Thus both models provide reasonable interpolation for mulas for the rate of transitions between two wells. The for mulas are as easy as that proposed in Ref. 2 and 3 but take into account the interference of excitation and reaction at energies E close to Ea. IV. MULTIDIMENSIONAL PROCESSES In the multidimensional case there is a large variety of possible analytical forms of the kernel g(E). It is impossible to derive the general formula for the reaction rate W without any assumptions on the analytical properties of g(E) even in the model of the stepwise microcanonical rate k(E). Here we restrict ourselves to two rather general and flexible models for the transition operator § [recall that g(E)=(EI§IO)] men tioned in Sec. II. It will be shown that in both cases the multidimensional problem can be reduced to the lD one and then solved by the Wiener-Hopf method. A. Covariant kernels Let us consider kernels of the type §=Go(i) with i=VS(V+F), (57) where Go(x) satisfies the conditions Go(O) = I and Go(x-+ -00) = O. The positive definite "diffusion" operator S [defined in Eq. (6)] is, in general, nondiagonal in E coor dinates. It is easily seen that the corresponding kernel g(E-E/)=(EI §IE/) satisfies the normalization and detailed balance relations [Eqs. (3) and (4)]. Notice, that for the Gaussian transition probability geE) = (47T det S) -112 exp[ -(E-SF)T( 4S) -I(E-SF)] =(Elexp(i)IO). (58) The lD exponential transition probability (34) can also be expressed in terms of i: g (E) = (11 S 'P ) Xexp( -lEI 'P12 -FEI2) = (EI( 1-i) -110). Unfortunately, this formula is incorrect in the multidimensional case. To reduce the problem to a lD one let us make a linear transformation to new coordinates (59) where c=SI12b/(bTSb)112 and b is the unit vector normal to the reaction surface [see Eq. (13)]. Transformation (59) is a product of (1) the nonunitary transformation X=S-I12E which results in spherically symmetric "diffusion" matrix S x and (2) the rotation from X coordinates to those perpendicu lar (u) and parallel (v) to the reaction plane determined in X coordinates by c·(X-Xo)=O. The coordinate u is called hereafter the reaction coordinate. In the coordinates (u,v) Eq. (57) reduces to v[I-Go(iu+iv)]p+K8(u-uo)P=0 (60) in which ij=Vj(Vj+F)(j=u,v) (61) with Fu=(Fx .c)=(FTSb)/(bTSb)112 and Fv=F-c·F u' It is evident that the solution p(u,v) satisfying the boundary con dition (10) can be represented by p(u, v) = a(u )exp( -v·F v), (62) where a(u) obeys a ID equation v[ 1-Go(iu)]a+ K8(u -uo)a= O. (63) Solution of Eq. (63) by the Wiener-Hopf method yields [ 1 J'" dh. W = W sc exp 2 7T _ 00 114 + h. 2 (64) In Eq. (64) 1 J k(E) Wsc= v z dE v+ k(E) exp( -E·F) (65) is the reaction rate in the strong collision limit and Q(h.) is given by Eq. (19) with G(h.)=Go(_A(~+h.2», A=(bTSF)2/(bTSb). (66) It is important to note that deviation of the reaction rate from W sc (corresponding to A-+oo) is characterized by only two parameters Klv and A. The compact and simple formulas (64)-(66) represent the final result of the theory for the kernels of type (57). B. Noncovariant kernels Here we consider other types of kernels for which the multidimensional problem can be reduced to a lD one. They are called noncovariant because the corresponding operators §=GO(il,·.·,i n), ik=SkV k(V k+ Fk), (67) are represented in terms of components ik of the covariant Smoluchowsky operator. Definition (67) implies that S is diagonal in E coordinates: Sjk=SjOjk' Note, that the 2D separable exponential kernel g(EI,E2) =gl(EI)g2(E2) -exp{ -~ ['PdEd+'P2IE21+(EIFI+E2F2)]} -(EIE21(1-il)-I(1-i 2)-1100) (68) proposed in Refs. 8 and 9 is a simple example of the kernels (67). The method of reduction to a ID problem is similar to that discussed in the previous Sec. IV A. After the transfor mation (59) using the unsatz (62) one obtains [1-Go(AI , ... ,An)]a+ K8(u -uo)a= O. (69) Here the operators A k k Ak=Ck(CkV u-FJ2-Fv)(V u+ Ful2) (70) in which ,JS;bk Ck= (ISjbJ) 112 ' k r;:,- -Fv= "Sk(Fk-bkF) Fk=C (IS ob2)1/2p u k J J ' (71) are the projections of the vectors c, Fu, and Fv , respectively, on the axis Ek and P=(bTSF)/(bTSb). It is easily seen that if Go depends on iu = IAk then Eq. (69) reduces to Eq. (63). J. Chem. Phys., Vol. 100, No. 10, 15 May 1994 .. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.107 On: Sun, 23 Nov 2014 07:03:15A. I. Shushin: Theory of activated rate processes 7337 I\. '\. {Il f'" \. . ~ ¢.>. \. 0 tb.. ~ \ I' '<; 0<c> \. V ~ "" '\. V ~ , ~. \. 0-:> \. \. FIG. 2. Picture of the reaction (shaded) and the molecular (empty) regions. The straight line separating these two regions is the reaction line, b is the vector normal to the reaction line and F is the vector of inverse temperatures. Solution of Eq. (69) by the developed method again gives expression (64) but with G(q) = Go[A I (q ) •...• An(q )], (72) where . A/q)= _(F~)2(q-il2)(q+il2+igj) and gj = Fb/ F~. Function (72) is complex in contrast to function (66) corresponding to the covariant kernels, however, simple analysis shows that any observables expressed in terms of the inverse Fourier transform of G(q) in Eq. (72) are real values. C. Two-dimensional model To illustrate the general results obtained in this section let us consider the simple 2D model which is used for inves tigating the effect of the ro/vibrational coupling in the uni molecular reactions.8•9 Two coordinates Eland E2 corre spond to the rotational and the vibrational energies of the molecule. In the most simple version the model assumes un correlated activation along both coordinates Eland E2 which implies separability of the transltlOn probability: g(E)=lljg/E). It is evident that this relation will be satiAs fied if one takes the covariant kernel in the form g = exp( L) X[Go(x) =exp(x)]. where i is given by Eq. (57) with diag onal S: S;j=Sj8ij. In this case g~Ej) are Gaussian functions of type (58) with the widths 2 "Sj' Figure 2 shows schematic picture of the reaction and the molecular regions in E space. In the 2D case the reaction surface reduces to a reaction line. The ro/vibrational cou pling manifests itself in the orientation of this line. For the sake of simplicity and brevity we shaH consider only the high reactivity limit (K!v~ I) typical for gas phase processes. According to formula (64) in this limit W/Wsc=l(.~) =exp( 21'TT J~oo In{J-exp[ -Ll(1I4+A2)]} dA ) Xl/4+A2 • (73) where A=(SlbIFl+S2b2F2)2/(Slbi+S2b~). In Eq. (73) 1(Ll) is a monotonously increasing function which changes from 0 to 1 as A increases from 0 to infinity. The analytical properties of this function are discussed in detail in Ref. 2. Expression (73) shows the following characteristic prop erties of the reaction rate dependence on the parameters of the model: (1) In the case of "isotropic" activation: S I = S 2 = S. one gets the expected result Ll = S (F· b)2 which means that the 2D reaction rate is determined by the ID flux along the reactive coordinate (parallel to b). Obviously. "an isotropic" activation (S) S2) complicates this simple pic ture. (2) In the case of a highly anisotropic activation when. say. S) ~ S 2 one has Ll =S ,Fi. i.e .• the main contribution to the reaction rate comes from the reaction flux along axis 1. (3) If the reaction coordinate is parallel to the axes Ej X(bi= 8i). then the reaction rate is determined only by the reaction flux along this axis and in agreement with this state ment one gets from Eq. (73) Ll=SjFJ. As has already been pointed out. the remarkable feature of the covariant models is that the reaction rate dependence on all parameters of the model reduces to the dependence on the only parameter A (recall that we assume K!v~ 1). The great advantages of this property becomes evident in the 2D model considered. Now let us briefly discuss the noncovariant 2D exponen tial model with the kernel (68). It is shown in the previous Sec. IV B that in this model the reaction rate is also given by Eq. (73) in which. however. the term exp[ -Ll(l/4+A2)] in the integrand should be replaced by the function G(A)=(1 -A,(A)-A2(A)+A,(A)A2(A»-1 [A/A) are defined in Eq. (72)]. It is easily seen that this function is similar to that appeared in the ID exponential model [see Eq. (28)]. Therefore one can apply the method developed in Sec. III B to reduce the problem of calculating the rate W to solving the algebraic equations (30) which is of order of 4 for the model considered. We shall not present here the complicated final expression for the reaction rate. It is clear that it depends not only on Ll but also on a number of other parameters. Simple analysis shows that the exponential 2D model predicts the limiting properties of the reaction rate similar to those men tioned in the discussion of the covariant (Gaussian) model. Concluding of the discussion of the exponential 2D model it is worth noting that the roots A;:j of the above mentioned fourth order equations are real and thus W is also real (as it should be) in accordance with the foregoing gen eral statement. Complex A;:j would mean oscillatory behav ior of geE) and peE). Such a behavior is not inherent in activated rate processes. in general, and in the model consid ered, in particular. J. Chern. Phys., Vol. 100, No. 10, 15 May 1994 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.107 On: Sun, 23 Nov 2014 07:03:157338 A. I. Shushin: Theory of activated rate processes V. SUMMARY Here we will summarize and briefly discuss the most important results of this paper. (1) The 1D master equation (14) with an arbitrary tran sition probability gee) is solved analytically for a finite mi crocanonical rate k(E) = K()(E-Ea)' This solution allows to investigate analytically an interplay of activation and reac tion at energies E> Ea. In particular, it enables us to derive a new simple interpolation formula which accurately de scribes a turnover between two different types of behavior of the transition rate corresponding to the weak and the strong friction limits. Predictions of this formula for the conven tional Kramers problem appear to deviate only weakly from those of other known interpolation formulas. I However, manifestation of a finite microcanonical rate which models the effect of friction in the region near the top of the barrier is expected to be stronger when the strengths of the friction within the well and near the top of the barrier are different, i.e., for the coordinate dependent friction, and when the po tential is strongly anharmonic near the top. Recently the effects of the coordinate dependence of friction have been studied in a number of articles (see Refs. 18 and 19 and references therein). Some discrepancies be tween numerically calculated transition rates and those ob tained by known analytical expressions have been found. The model proposed in this work can help to reduce the discrepancies by more rigorously taking into account an in terplay of activation and reaction for E> Ea. (2) The method developed enables us to generalize the considerations to multi state activated rate processes. In this article we have discussed transitions between two states, however, the method can be applied to any number of states. In general, the problem reduces to diagonalization of some matrices of linear systems of the Wiener-Hopf equations. (3) The multidimensional activated rate processes are shown to be described by the iD ME along the reaction coordinate perpendicular to the reaction surface. The simple expression (64) for the transition rate is obtained for the co variant kernels of type (57). According to this expression deviation of W from Wsc is governed b¥ the only parameter (except for Klv) Ll=(bTSF?/(bTSb)=J~/Et where J b = bTSF is the component of the drift flux perpendicular to the reaction surface induced by the constant "force" F and E~ is a mean square of energy transfer perpendicular to the reaction surface (E b = 2, jE i j) in the absence of the drift. Expression (64) describes both the correlated and the uncorrelated activation along different coordinates Ej• The uncorrelated activation implies separability of the transition probability: g(E)=IIjg/E j). The general covari ant kernel (57) corresponds to the (Gaussian) separable tran sition pr?babilit)' and thus to the uncorrelated activation if g =exp(L) and S is diagonal in E coordinates. Unfortunately, the non-Gaussian separable transition probabilities corre spond to the noncovariant kernels as it follows from the con siderations of Sec. IV B. In this case the expression for W is not as compact as that for the covariant kernels and the ratio W/W sc depends not only on Ll but also on a number of other parameters. Nevertheless, in some models the expression is still rather simple as has been demonstrated in Sec. IV C for the model with the 2D exponential kernel (68). In addition, it is worth noting that predictions of any separable noncovari ant models and the corresponding nonseparable covariant models do not differ essentially. The term "corresponding" means that in the covariant model S should be taken diagonal in E coordinates and the dependence of both kernels on each particular operator ij (i.e., for ik=o with k j) must be the same. For example, the noncovariant exponential kernel (68) can be reasonably we~ approximated by the covariant g(EI,E2)=(EIE21(1-LI-L2)-IIOO) and thus (making use of the results of Sec. IV B) the corresponding exponen tial 2D model can be reduced to the monoexponential i D one (Sec. III B 2). The covariant kernels with nondiagonal S describe cor related activation along different coordinates Ej• These ker nels are of special interest for investigating effects of in tramolecular energy transfer on unimolecular reactions. This energy transfer gives rise to negative nondiagonal elements: Sij<O U* j). In general, according to formula (64) with Ll defined by Eq. (66) W increases with the increase of the intramolecular energy transfer rate. However, for some ori entations of b and F this formula predicts a negative sign of the effect of the energy transfer, i.e., the decrease of W. This result can be easily obtained in the simple 2D model with b=(1,O). Such a prediction may be correct in some cases (although detailed analysis by comparison with numerical calculations is required) but when applying formula (64) we should keep in mind that in some regions of the parameters of the model it is inapplicable. The formula is certainly in valid when bTSF is negative (and thus definitely Sij<O) be cause in this case the effective "potential" V(u)=Fuu along the reaction coordinate u is repulsive and thus the steady state approximation used in deriving formula (64) is incor rect. This case corresponds to very fast intramolecular energy transfer which, in reality, can be incorporated in advance by changing the orientation of the reaction surface. (4) The formulas presented in this article are derived in the assumption that the reaction surface is a plane, however, they can easily be generalized to the smooth curved reaction surfaces by making use of a local plane approximation for the surfaces. This approximation implies the (energy) coor dinate dependence of the normal vector band Ll. In such a case the reaction rate W cannot be represented in form (64), i.e., as a product of Wsc and the function independent of E coordinates [in the 2D model considered in Sec. IV C this is the function [(Ll) given by Eq. (73)]. Since the parameter Ll depends on E this function is also E dependent and thus it should be inserted in the integral over E in the expression for W sc [see Eq. (65)]. This is the onl)' modification required. The smoothly changing k(E) and SeE) can be treated in a similar way. VI. CONCLUSION It is well known that the kinetics of activated rate pro cesses in the 'weak and intermediate friction limit is de scribed by MEs in energy (or action) variables. The idea of this work is to demonstrate that the Wiener-Hopf method provides a unified way of general solution of different MEs J. Chern. Phys., Vol. 100, No. 10, 15 May 1994 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.107 On: Sun, 23 Nov 2014 07:03:15A. I. Shushin: Theory of activated rate processes 7339 which occur in the gas and condensed phase reaction-rate theories. This method makes it possible to generalize the well known formulae for the reaction rate to the cases of a finite microcanonical reaction rate, multidimensional activa tion, etc. The considerations performed in this work shows that the Wiener-Hopf approach can help us to get very simple and general analytical solutions of the MEs and thus to get deeper insight into the kinetics of activation processes and the interplay of activation and reaction in the reaction re gions. ACKNOWLEDGMENT This work has been partially supported by a grant (N 93-03-5205) of the Russian Fond of Fundamental Investiga tions. I P. Hanggi, P. Talkner, and M. Berkovec, Rev. Mod. Phys. 62, 251 (1990). 2V. I. Mel'nikov and S. V. Meshkov, J. Chern. Phys. 85, 1018 (1986). 3V. I. Mel'nikov, Phys. Rep. 201, 1 (1991). 4E. Pollak, H. Grabert, and P. Hanggi, J. Chern. Phys. 91, 4073 (1989). sH. Hippler and J. Troe, in Bimolecular Collisions, edited by G. E. Baggott and M. N. Ashfold (Royal Society of Chemistry, London, 1989), p. 209. 6I. Oref and D. C. Tardy, Chern. Rev. 90, 1407 (1990). 7J. C. Keck, J. Chern. Phys. 46, 4211 (1967). 8 J. Troe, J. Chern. Phys. 66,4745 (1977). 91. Troe, Z. Phys. Chern. Neue Fo1ge 154, 73 (1987). lOS. C. Smith and R. G. Gilbert, Int. J. Chern. Kinet. 20, 307 (1988); 20, 979 (1988). II M. Berkovec and B. J. Berne, J. Chern. Phys. 84, 4327 (1986); J. Phys. Chern. 89, 3994 (1985). 12S. H. Robertson, A. I. Shushin, and D. M. Wardlaw, J. Chern. Phys. 98, 8673 (1993). 13p. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw Hill, New York, 1953), Vol. 1. 14 A. I. Shushin, Chern. Phys. 144,201 (1990). 15 Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stigan (National Bureau of Standards, Washington, D.C., 1964). 16 J. Troe, 1. Chern. Phys. 97, 288 (1992). 17R. F. Grote and J. T. Hynes, J. Chern. Phys. 73, 2715 (1980). 181. B. Straus and G. A. Voth, J. Chern. Phys. 96, 5460 (1992). 19 J. B. Straus, J. M. Liorente, and G. A. Voth, J. Chern. Phys. 98, 4082 (1993). J. Chern. Phys., Vol. 100, No. 10, 15 May 1994 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.107 On: Sun, 23 Nov 2014 07:03:15
1.1832144.pdf	Taylor–Couetteﬂowwith an imposed magnetic ﬁeld — linear and nonlinear results WolfgangDobler† Kiepenheuer-Institutfür Sonnenphysik, Schöneckstr.6, D-79104 Freiburg,Germany Newaddress: Departmentof Physics and Astronomy,Universityof Calgary,2500 UniversityDrive NW,Calgary AB T2N 1N4, Canada Abstract. Using numerical simulations we investigate the (in)-stability and saturation behaviour of moderately compressible, cylindrical Taylor–Couette ﬂow in the presence of a uniform axial magneticﬁeld.ForRayleigh-stableconﬁgurations,weﬁndmagneticallyinducedTaylorvorticesas predicted by linear theory, with both axisymmetric and non-axisymmetric solutions, depending on theHartmann number. Theﬂowshowsclearindicationsofthemagneto-rotationalinstabilitywhichiswell-knownfrom numericalsimulationsinaccretiondiscgeometry.Inthesaturatedstate,thestructureoftheﬂowand themagnetic ﬁeld can be verydifferentfrom the linear phase of the instability. 1. TAYLOR–COUETTEFLOW Taylor–Couetteﬂow—theviscousﬂowbetweentworotatingcoaxialcylindersis—one of the most intensively studied ﬂows in hydrodynamics [for a comprehensive overview, see1].Foraﬂuidofconstantdynamicviscosity,theNavier–Stokesequationhasasimple andhighly symmetric solution, the so-called (cylindrical)Couette ﬂow, ω(r)≡uϕ(r) r=Ω2R2 2−Ω1R2 1 R2 2−R2 1−(Ω2−Ω1)R2 1R2 2 R2 2−R2 11 r2, (1) whereR1andR2denote the radius of the inner and outer cylinder, Ω1andΩ2are the corresponding angular velocities, and rdenotes cylindrical radius. The other two velocitycomponents vanishfor Couette ﬂow, ur=uz=0. (2) In the limit of vanishing viscosity, the solution (1), (2) is unstable whenever the speciﬁc angular momentum l1≡Ω1R2 1,l2≡Ω2R2 2at the two cylinders satisﬁes the Rayleighcriterion sgnΩ1(l2−l1)<0, (3) (wheresgnΩ1denotes the sign of Ω1) and is stable otherwise. For real ﬂuids, viscosity candamp the instability,and it will only occur if the Taylornumber Ta≡4Ω2 1(R2−R1)4 ν2(4) 142exceedsa certain threshold ( νdenotesthe kinematic viscosity of the ﬂuid). If the ﬂuid is electrically well conducting, the presence of a magnetic ﬁeld can change the stability properties completely [2–5]. The strength of the magnetic ﬁeld is characterizedby a newdimensionless parameter,the Hartmann number Ha≡B(R2−R1)√ µ0ρνη=vA(R2−R1)√ νη, (5) whereµ0denotes vacuum permeability, ρandηdensity and magnetic diffusivity of the ﬂuid, and vA≡B/√ µ0ρis the Alfvén speed. For an ideal ﬂuid ( ν=η=0), the change in stability properties is particularly drastic. For strong magnetic ﬁelds or thin gapsR2−R1, the magnetic ﬁeld rather has a stabilizing function [3, 6]. But in the limit of weak magnetic ﬁelds, the Couette ﬂow becomes unstable provided that the angular velocitiesof the twocylinderssatisfy the condition sgnΩ1(Ω2−Ω1)<0, (6) whichismuchweakerthantheRayleighcriterion(3).Thischangecanbeinterpretedin terms of the dominant mechanism of angular momentum transport: In the nonmagnetic case advection is the only radial transport mechanism, and thus the gradient of speciﬁc angular momentum l(i.e. the deviation from the state l=constwhere no angular mo- mentumwouldbetransported)determinesthestabilityoftheﬂow.Themagnetictension force,ontheotherhand,triestosynchronizeangularvelocity ω,sointhemagneticcase the direction of angular momentum transport is determined by the gradient dω/dr. Evenas \|B\| →0,thefastestgrowthrateisoforder \|Ω1\|,i.e.remainsﬁnite.However, the wave number corresponding to that fastest growing eigenmode scales like k∼ \|Ω1\|/vA, and for very weak magnetic ﬁelds, dissipative effects will eventually destroy the instability [see also 7] Thismagnetorotational instability (MRI), i.e. the destabilizing effect of magnetic ﬁelds on rotating shear ﬂows, is thought to be the main mechanism for rendering accre- tion discs turbulent, and thus viscous. While nonlinear instabilities have been proposed to explain turbulent accretion discs as well [8] and may be relevant for very cool discs, theMRIcertainlyplaysa centralroleinthetheoryof accretiondiscs,andmanynumer- icalsimulationshaveconﬁrmedthatitisindeedveryefﬁcientintransformingalaminar accretion disc into a turbulentone [see e.g. 5, 9–11]. While even weak magnetic ﬁelds are enough for the MRI to maintain an accretion disc in a turbulent state, the magnetic ﬁelds must eventually be maintained against Ohmicdecay.Fromdynamotheory,weknowthatmagneticﬁeldgenerationisanatural consequence of the turbulent, three-dimensional nature of the rotating accretion ﬂow. This gives rise to a very elegant scenario for magnetized accretion discs, in which turbulenceandmagneticﬁeldmaintaineachothersymbiotically.Thisscenariohasbeen veriﬁedin a number of numerical experiments[9, 10, 12]. The importance of the MRI for accretion discs and possibly also for galactic discs [13,14]isoneofthemotivationsforbuildingmagneticTaylor–Couettelaboratoryexper- iments, a topic that will be discussed at length in other chapters of this book. One chal- lenge for experiments is the low electrical conductivity of liquid metals, which makes Ohmicdissipationamuchmoreprominenteffectthaninastrophysicalobjects.Thelow 143conductivity also makes it difﬁcult to numerically model laboratory experiments, since the low magnetic Prandtl number ( Pm≡ν/η∼10−5for liquid sodium and similar or lower for other liquid metals) leads to vastly different scales for the ﬂow and the mag- neticﬁeld.Webelieveneverthelessthatmodelswith Pmoforderunitycanteachusalot aboutmagneticTaylor–Couetteﬂow.Thesevaluesmakeitfeasibletoconductparameter studies, and more expensive calculations with lower Pmcan be targeted at particularly interesting parameter regimes once these have been identiﬁed. Also, highly turbulent mediaare often modelled with a turbulentmagnetic Prandtl number close to unity. In the context of dissipative magnetic Taylor–Couette ﬂow, the MRI will manifest itself in a modiﬁed (lowered) threshold for the formation of Taylor-like vortices, and in thefactthatsuchvorticesformforratios Ω2/Ω1wheretheCouetteproﬁle(1),(2)would bestable in the absence of magnetic ﬁelds. Previous studies have focused on the onset of dynamo action in both the linear [15] and nonlinear case [16]. A related ﬂow, even easier capable of dynamo action, is the so-called helical Couette ﬂow, where the cylinders also move in the axial direction. The resulting velocity ﬁeld gives rise to the well-known screw dynamo, which is well- investigatedboth theoretically [17–21] and numerically [22–26]. In the present paper we take a different approach and consider cylindrical Taylor– Couette ﬂow in the presence of an axial, uniform magnetic ﬁeld B0ez. In this system, magnetic induction due to the imposed ﬁeld and intrinsic dynamo action cannot easily bedisentangled(ifatall),butstilltheresultingﬂowcanhavepropertiesthatwouldmake ita dynamo in the absence of the externalﬁeld. Linear stability analysis of this conﬁguration has shown that the imposed magnetic ﬁeld indeed gives rise to the MRI as discussed above [27], and for certain parameters, non-axisymmetricTaylorvorticesarethepreferredmodes[28].Theseresultshavebeen conﬁrmedin the limits of verylowand veryhigh magnetic Prandtl number [29]. 2. OUR MODEL 2.1. Equations We consider the ﬂow between two concentric cylinders as described in Sec. 1. Our numerical code uses cylindrical coordinates (r,ϕ,z)and solves the compressible MHD equations for (logarithmic) density lnρ, ﬂuid velocity u, and magnetic vector potential A, Dlnρ Dt=−divu, (7) Du Dt=−1 ρgradp+j×B ρ+1 ρdiv(2ρνS), (8) ∂A ∂t=u×curlA−/angbracketleft(u×B)·er/angbracketrightϕ,zer+ηΔA, (9) whereD/Dt≡∂/∂t+u·graddenotes the advective time derivative, B=curlAis the magnetic ﬂux density, µ0j=curlBthe current density, and Sik≡[∂iuk+∂kui− 144TABLE 1. Parameters and properties of the different runs. Other parameters are R1=0.5, R2=1,Lz=1,ν=η=7×10−4. Linear modes are characterized by their axial and azimuthal wave numbers k,m; longitudinal wave numbers kare listed in units of 2π/Lz.γdenotes the growthrate dln\|\|uz\|\|/dtof the mode. RunΩ1Ω2B0l2/l1HaLinear structureγSaturated structure 1a2.0 0 .5 0 .00 1 .0 0 .0k=2,m=0 0 .13k=2,m=0 1b2.0 0 .5 0 .02 1 .0 14 .3k=3,m=0 0 .48k=3,m=0 1c2.0 0 .5 0 .05 1 .0 35 .7k=3,m=0 0 .72k=3,m=0 1d2.0 0 .5 0 .10 1 .0 71 .4k=2, “wavy” 0.64k=1, wavy 1e2.0 0 .5 0 .20 1 .0 142 .9k=1,m=0 0 .42k=1, wavy 1f2.0 0 .5 0 .50 1 .0 357 .1k=1,m=0 −0.03— 1g2.0 0 .5 1 .00 1 .0 714 .3k=1,m=0 −0.03— 2a2.0 0 .667 0 .10 1 .33 71 .4k=2,m=1/“wavy” 0.56k=2, wavy 3a2.0 1 .0 0 .05 2 .0 35 .7k=3,m=0 ≈0.36 3b2.0 1 .0 0 .10 2 .0 71 .4k=2,m=1 0 .36k=2,m=0 3c2.0 1 .0 0 .20 2 .0 142 .9k=1, “wavy” ≈0.26k=1, wavy (2/3)δikdivu]/2isthetracelessrate-of-straintensor.Thesecondtermontheright-hand- sideoftheinductionequation(9)doesnotcontributetothemagneticﬁeldandispresent forpurelynumericalreasons.ToevolveEqs.(7)–(9),weuse6th-orderﬁnitedifferences inspace and 3rd-order Runge–Kuttatime-stepping scheme. While the code solves the compressible MHD equations (and uses an isothermal equation of state), we think that our results are only moderately inﬂuenced by the compressibility of the ﬂuid (but see Sec. 2.3.3 below). For reasons of efﬁciency, we haveusedaMachnumberoforderunityandthecorrespondingdensitycontrastisabout ρ2/ρ1≈1.5. In other simulations [25], we had found that for a Mach number of about 0.3weakly compressible and incompressible results are almost identical. Ourinitialmagneticﬁeldispurelyverticalanduniform, B=B0ez.Theinitialvelocity is the Couette proﬁle (1), (2), superimposed with white noise at verylowamplitude. Theverticalboundaryconditionsareperiodic,whileradiallywehaveno-slip,impen- etrable conditions for the velocity and perfectly conducting conditions for the magnetic ﬁeld. We note that these magnetic boundary conditions do not allow the total magnetic ﬂux between the cylinders to change and thus our magnetic ﬁeld has no chance of de- caying. 2.2. Parameters The inner and outer radius are chosen as R1=0.5,R2=1, while the full height of the (periodic) cylinders is Lz=1. In all runs presented here, viscosity and magnetic permeability are equal, i.e. the magnetic Prandtl number is Pm=1. Table 1 lists other parameters of the individual runs, together with some of the properties of the ﬂow and magnetic ﬁeld. 1450.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8zFIGURE1. Horizontalsectionsofvelocity u(left)andresidualmagneticﬁeld Bres≡B−B0ez(right)for thelinearphaseofRun1b.Arrowsindicatethetangentialcomponents uϕ,uz,whilecoloursrepresentthe radial component with bright (dark) colours representing a component towards (away from) the viewer, i.e. positive (negative) Br. [As an exception, the sign has been reversed for uϕ, so bright colour means positiveuϕhere to avoid excessive dark colours.] Both velocity and magnetic ﬁeld are axisymmetric in thisrun. 2.3. Results 2.3.1. Geometry As Table 1 shows, varying the Hartmann number changes the structure of the linear modes considerably. For weak magnetic ﬁelds (Runs 1b, 1c), the preferred mode has three nodes in the vertical direction, i.e. the vertical size of the cylinder accommodates six Taylor half-cells. This conﬁguration is shown in Fig. 1. With increasing Hartmann numberHa, the vertical wave number gets larger, as the magnetic ﬁeld is able to synchronize velocity over a larger vertical distance. For both weak and strong magnetic ﬁeld,the vorticesare axisymmetric (azimuthal wavenumber m=0). However,forRun1dwithitsmoderateHartmannnumberofabout 70,thelinearstage shows a “wavy” mode (see below), and the same holds for Run 2a which has the same Hartmann number. The velocity for the latter case is shown in Figure 2, which shows thefullvelocityandtheresidualmagneticﬁeld Bres≡B−B0ez,onacylindricalsurface, while Fig. 3 shows the same in a vertical section. One can clearly see the vertical wave numberk=2k1,wherek1≡2π/Lzisthelowestnon-vanishingwavenumbercompatible withthe verticalsize Lzofthe cylinder. The azimuthal structure is a superposition of different wave numbers with at least m=0,±1prominently present. Note that here during the linear phase these modes evolve independently and must thus have very similar growth rates to coexist for a long time. We note that, from Taylor–Couette experiments, the “wavy mode” is known [1], where nodal surfaces of uz(or similar diagnostics) are not planar, but oscillate in ϕ. 1460 1 2 3 4 5 6 ϕ0.00.20.40.60.8z 0 1 2 3 4 5 6 ϕ0.00.20.40.60.8z 0 1 2 3 4 5 6 ϕ0.00.20.40.60.8z 0 1 2 3 4 5 6 ϕ0.00.20.40.60.8zFIGURE 2. Velocity u(top) and residual magnetic ﬁeld Bres(bottom) on a cylindrical shell r=0.75 forthelinearstageofRun2a.RepresentationisasinFig.1,inparticularbrightcoloursrepresentpositive radialcomponents ur,Br. 0.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8z FIGURE3. As in Fig. 1, butfor the linear phase of Run 2a. The simplest wavy mode can be described as a superposition of modes (k,m=0)and (k,m=±1)withappropriatephasefactors.However,evenifsomecombinationoflinear modes looks likea wavymode, this can only be a coincidence, since the relative phases oflinearmodesarearbitrary,andevenevolveintime.Realwavymodesareanonlinear phenomenon. 1470 1 2 3 4 5 6 ϕ0.00.20.40.60.8z 0 1 2 3 4 5 6 ϕ0.00.20.40.60.8z 0 1 2 3 4 5 6 ϕ0.00.20.40.60.8z 0 1 2 3 4 5 6 ϕ0.00.20.40.60.8zFIGURE4. Same as Fig. 2, butfor the saturated phase of Run 2a. Aftersometime,theexponentialgrowthofthekinematicphaseslowsdownandeven- tually a saturated state is reached, in which magnetic and kinetic energy are stationary or vary by a moderate percentage around some average value. This saturated regime can look quite different from the kinematic phase as is shown in Figs. 4 and 5. The az- imuthal structure is obviously no longer described by the ﬁrst three azimuthal modes \|m\|=0,±1,butinvolveshigherharmonicsaswell(Fig.4).Thispatternvisuallyresem- bles the hydrodynamical “wavy mode” of Taylor-Couette ﬂow, although we ﬁnd here a less symmetric and more structured geometry compared to the simplest manifestations ofthe hydrodynamicalwavymode. Not all Runs maintain their geometric structure in the nonlinear regime. As can be seen in Table 1, some Runs (1e and 3a) switch from axisymmetric to non-axisymmetric behaviour when saturating. On the other hand, Run 3a switches from a clear m=1 mode during the linear phase to an axisymmetric saturated state. These ﬁndings clearly demonstrate that it can be misleading to extrapolate linear results to the nonlinear regime. 2.3.2. Velocityproﬁle In Fig. 6, we have plotted uϕas a function of radius for the saturated phase of Run 2a, with three different representations for the azimuthal velocity component: We compare the radial proﬁles of angular velocity ω≡uϕ/r, azimuthal velocity uϕ, and speciﬁc angular momentum l≡ruϕ. While the boundary values for all three curves are determined by R1,Ω1,R2, andΩ2, the proﬁles between the boundaries reﬂect the physics of angular momentum transport. If angular momentum was transported mainly duetoradialadvection,theproﬁle l(r)wouldberoughlyconstant(likeentropyismostly 1480.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8zFIGURE5. Same as in Fig. 3, butfor the saturated phase of Run 2a. 0.50.60.70.80.91.0 r1.01.52.0ω 0.50.60.70.80.91.0 r0.60.70.80.91.01.1uϕ 0.50.60.70.80.91.0 r0.40.50.60.7lz FIGURE 6. Scatter plot showing different representations of the azimuthal velocity component as function of rfor the saturated phase of Run 2a. Left: Angular velocity ω. Middle: Azimuthal velocity uϕ. Right: Speciﬁc angular momentum l. constantinthermalconvectionlayers);thisisclearlynotthecaseinFig.6.Itisratherthe angular velocity ω(r)that has been synchronized by the ﬂow, which is a clear indicator for the magnetic tension force being the dominant mechanism of angular momentum transport. 2.3.3. A potential compressibilityeffect For Runs 1a–g, the speciﬁc angular momenta on the two cylinders are equal, l1=l2, thusweexpectthehydrodynamicCouetteﬂowtobemarginallystableintheabsenceof viscosity according to Rayleigh’s criterion. If viscosity is present, it is natural to expect that the system becomes stable because perturbations will now be damped, even if that damping is small. To our surprise, we found however that Run 1a, where the magnetic 149ﬁeldiszero,developsTaylorvorticesjustlikethemagneticcases,albeitthegrowthrate is lower. This is hard to understand, since lis not only equal on the two cylinders, but accordingto Eq. (1), we have l(r) =l1=const (10) everywhere. Thus, advection will not have any effect on the distribution of angular momentum,and there is no obviousother mechanism that could transport it at all. However,compressibilitycanmakeadifference.Sincewehaveset kinematic viscos- ityν=const,ratherthanassumingconstant dynamic viscosity,the r-dependentequilib- rium solution is not exactly the Couette proﬁle (1). But any deviation from the Couette proﬁlewillintroduceagradient dl/dr/negationslash=0whichtakesonbothsigns(weﬁndthat lhas aminimumnear r=0.7andthus dl/dr<0nearR1anddl/dr>0closerto R2).Ifvis- cosityislowenough,thepartwith dl/dr<0willbeunstableanddriveTaylorvortices. We thus believe that the ﬂow we ﬁnd in Run 1a is due to compressibility effects, which causeviscous angular momentum transport. 2.3.4. Helicity and alpha effect In the context of mean-ﬁeld theory [30], a crucial parameter describing the magnetic ﬁeld generation properties of many dynamo systems is the α-effect [31]. In their work on linear properties of magnetic Taylor-Couette ﬂow, Rüdiger & Zhang [27] discussed thepossibilityofan α-effectinthattypeofﬂow.Forinﬁnitelylongcylindersandinthe linearregime,theﬂowisstrictlyperiodicin zandthusthe αeffectoscillatesaroundzero along that direction. Noting that such a system has no net αeffect, the authors seem to concludethat it is not suitable as a mean-ﬁeld dynamo. We note however that it is too restrictive to judge the dynamo properties solely by the net sign of the α-effect. In fact, most cosmic dynamos have almost exactly vanishing αnet≡Rα(x)dV, because αis antisymmetric with respect to their equatorial or symmetry plane. Nevertheless the α-effect in these objects is able to generate all kinds of cosmic magnetic ﬁelds. It is not a priori clear that a periodic array of cells of alternating kinetic helicity cannot be an interesting dynamo system in its own. InFigs.7and8,weshowthedistributionofdifferentquantitiesrelatedtothe αeffect. In the quasi-linear approximation, the αeffectis givenby [32, 33] α=τ 3µ −u/prime·curlu/prime®+1 ρB/prime·j/prime®¶ , (11) whereτis the turbulent turnover time and u/prime≡u−/angbracketleftu/angbracketright, etc. For our geometry, the averages /angbracketleft·/angbracketrightare conveniently taken over azimuth ϕ. In the ﬁgures we show kinetic helicityHkin≡/angbracketleftu/prime·curlu/prime/angbracketright, the current helicity Hcur≡/angbracketleftB/prime·j/prime/angbracketright, their combination (11), and the verticalcomponent of the “turbulent”electromotiveforce, Ez≡u/primerB/primeϕ−u/primeϕB/primer. 1500.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8zFIGURE 7. Dynamo properties for the linear phase of Run 2a. (a) Kinetic helicity Hkin. (b) Current helicityHcur.(c) Alpha effectaccording to Eq. (11). (d) Verticalcomponent Ezof ﬂuctuating EMF. 0.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8z 0.50.60.70.80.91.0 r0.00.20.40.60.8z FIGURE 8. As in Fig. 7, butfor the saturated stage of Run 2a. 2.3.5. Connection to classical MRI Finally,inFig.9weshowiso-surfacesofangularvelocity ωforboth,theearlynonlin- ear and the saturated phase. These surfaces approximately also trace the magnetic ﬁeld lines (which are predominantly vertical). The structure of the iso-surfaces is very remi- niscent of the so-called “channel ﬂow” observed in MRI simulations of accretion discs [34], but with a “wavy” ϕ-dependence superimposed. This once more exempliﬁes how the MRI in accretion discs and Taylor vortices become one and the same phenomenon inmagnetic Taylor–Couetteﬂow. 151FIGURE9. Channel ﬂowin the moderately nonlinear case (left) and the saturated case (right). 3. CONCLUSIONS We have presented a set of numerical simulations of Taylor–Couette ﬂow with an axial magnetic ﬁeld, and see many indications that for not too low Hartmann numbers the Taylorvorticesturn into a manifestation of the MRI. WehavenotyetcarriedoutcalculationsformagneticPrandtlnumberslessthanunity, and it would be very interesting to see whether the different geometries of linear and nonlinearevolutionscan be found for these more “realistic” parameters as well. In any case the fact that we can have non-axisymmetric linear modes developing into axisymmetric saturated ﬂows and vice versa should be a clear warning to refrain from extrapolatinglinear results REFERENCES 1. Koschmieder, E. L., Bénard cells and Taylor vortices , Cambridge Univ. Press, 1993, ISBN 0-521- 40204-2. 2. Velikhov,E. P., Sov.Phys. JETP ,36, 1398ff(1959). 3. Chandrasekhar,S., Hydrodynamicand HydrodynamicStability ,Clarendon, Oxford, 1961. 4. Balbus,S. A., and Hawley,J. F., Astroph.J. ,376,214–222 (1991). 5. Hawley,J. F.,and Balbus,S. A., Astroph.J. ,376,223–233 (1991). 6. Donnelly,R. J., and Ozima, M., Phys.Rev.Lett. ,4,497–498 (1960). 7. Fleming, T.P.,Stone, J. M., and Hawley,J. F., Astroph.J. ,530, 464–477 (2000). 8. Richard, D., and Zahn, J.-P., Astron.Astrophys. ,347,734–738 (1999). 9. Brandenburg,A.,Nordlund,Å.,Stein,R.F.,andTorkelsson,U., Astrophys.J. ,446,741–754(1995). 10. Ziegler,U., and Rüdiger,G., Astron.Astrophys. ,356, 1141–1148 (2000). 11. Hawley,J. 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1.4808102.pdf	Irradiation-induced tailoring of the magnetism of CoFeB/MgO ultrathin films T. Devolder, I. Barisic, S. Eimer, K. Garcia, J.-P. Adam, B. Ockert, and D. Ravelosona Citation: Journal of Applied Physics 113, 203912 (2013); doi: 10.1063/1.4808102 View online: http://dx.doi.org/10.1063/1.4808102 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/113/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Amorphous FeCoSiB for exchange bias coupled and decoupled magnetoelectric multilayer systems: Real- structure and magnetic properties J. Appl. Phys. 116, 134302 (2014); 10.1063/1.4896662 Interfacial perpendicular magnetic anisotropy in CoFeB/MgO structure with various underlayers J. Appl. Phys. 115, 17C724 (2014); 10.1063/1.4864047 Precessional magnetization induced spin current from CoFeB into Ta Appl. Phys. Lett. 103, 252409 (2013); 10.1063/1.4853195 Damping of CoxFe80−xB20 ultrathin films with perpendicular magnetic anisotropy Appl. Phys. Lett. 102, 022407 (2013); 10.1063/1.4775684 In-situ characterization of rapid crystallization of amorphous CoFeB electrodes in CoFeB/MgO/CoFeB junctions during thermal annealing Appl. Phys. Lett. 95, 242501 (2009); 10.1063/1.3273397 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 163.118.172.206 On: Fri, 03 Jul 2015 11:02:53Irradiation-induced tailoring of the magnetism of CoFeB/MgO ultrathin films T. Devolder,1,2,a)I. Barisic,1,2S. Eimer,1,2K. Garcia,1,2J.-P . Adam,1,2B. Ockert,3 and D. Ravelosona1,2,4 1Institut d’Electronique Fondamentale, CNRS, UMR 8622, Orsay, France 2Univ. Paris-Sud, 91405 Orsay, France 3Singulus Technology AG, Hanauer Landstrasse 103, 63796 Kahl am Main, Germany 4Siltene Technologies, 86 rue de Paris, 9140 Orsay, France (Received 4 April 2013; accepted 14 May 2013; published online 30 May 2013) We study perpendicularly magnetized Ta/CoFeB/MgO ﬁlms and investigate whether their irradiation with light ions can improve their properties by inducing a different crystallization dynamics. We report the magnetization, anisotropy, g-factor, and damping dependence upon irradiation ﬂuence and discuss their evolutions with collisional mixing simulations and its expectedconsequence on magnetic properties. We show that after a short irradiation at 100 /C14C, the anisotropy increases close to the value obtained by conventional high temperature annealing. Higher irradiation-induced increase of anisotropy can be obtained but with a detrimental effect onthe damping that can be understood from spin-orbit contributions. VC2013 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4808102 ] I. INTRODUCTION Magnetic tunnel junctions with ultrathin CoFeB free layers in contact with the MgO oxide can exhibit both a Perpendicular Magnetic Anisotropy1(PMA) and a reason- ably low damping,2which makes them one of the most promising systems for the free layer of the next generation of magnetic random access memories.3In such materials, the giant tunnel magnetoresistance (TMR) requires a nearly per-fect crystalline state and a ﬂat interface, while the high inter- face anisotropy requires an oxygen stoichiometry close to nominal. The degree of CoFeB crystallization, 4,5the grain size,6and the oxidation state7near the interface are thus the essential features affecting their performance. Classically, roughness is minimized by ﬁrst depositing Ta-based smoothing layers and then depositing CoFe in the amorphous state through the inclusion of a large content of boron. So far the crystallization is obtained by a subsequentthermal annealing 8at typically 300/C14C, sometimes preceded by an in situ infrared annealing.6The Ta/CoFeB interface remains reasonably sharp because the solubility of Fe in Tais very low 9and that of Co in Ta is only a few percent.10The CoFeB/MgO interface is free of galvanic corrosion since Fe is more noble than Mg; experimentally, this interface getssharper upon annealing 4,11since O atoms initially incorpo- rated in the transition metal due to the deposition conditions migrate to the MgO. Unfortunately, the dynamics of oxygenmigration (ruling the PMA) and of CoFeB crystallization (ruling the TMR) are different so that ﬁnding annealing con- ditions that both optimize the PMA and the TMR is deli-cate. 8This is why PMA magnetic tunnel junctions have always TMR values substantially lower than their in-plane magnetized counterparts for which PMA is not needed. The isotropic diffusion of boron during annealing is also critical. Being a metalloid, it incorporates both in the MgO and in Ta.12,13While the incorporation of B in Ta isdesirable, the presence12of B in the MgO barrier is not: in addition to its detrimental effect on PMA, it creates conduct- ing spots inside the MgO which degrade the TMR and lead to early dielectric breakdown.14 Alternative material science strategies to induce CoFeB crystallization deserve attention, with the objective of avoid- ing boron incorporation in MgO, minimizing the interfaceroughness, and getting the correct in-depth oxygen concentra- tion proﬁle. One elegant approach to efﬁciently control the structural and magnetic properties of thin ﬁlms is to use lightion irradiation. 15,16The low interaction cross section together with the low energy transfer lead to short range atomic dis- placements and pairwise exchange of atomic positions. Thisprocess allows a very precise control of magnetic properties through atomic short range order modiﬁcations. For instance, 17,18combining irradiation with heating, a signiﬁcant reduction of the L0 1ordering temperature (300/C14C instead of 670/C14C) was observed in the case of FePt and FePd alloys. The purpose of this work is to determine whether light ion (Heþ) irradiation at moderate temperatures is operative in inducing crystallisation of amorphous Ta-CoFeB-MgO ultrathin ﬁlms. We follow two main arguments. The ﬁrst oneis that light ions knocking on heavier atoms transfer much more energy to B atoms than to the heavier species (Mg, Fe, Ta). In addition, we will see that the energies needed to dis-place O, Mg, and Ta atoms are substantially larger than the one of B. Our second guiding idea is that momentum conser- vation in primary He-B collisions ends in B knocked-onatoms being preferentially sent to Ta than towards MgO. The possible drawback is that a moderate ion mixing may occur. 19We shall mitigate the mixing rate by using a mild annealing during the irradiation to beneﬁt20from the nega- tive heat of mixing of Fe and Ta. II. SAMPLES AND SETUP Our samples are 1 nm thick CoFeB layers of composi- tions substrate/Ta (5 nm)/Co 20Fe60B20(1 nm)/MgO (2 nm)/Taa)Electronic mail: thibaut.devolder@u-psud.fr 0021-8979/2013/113(20)/203912/4/$30.00 VC2013 AIP Publishing LLC 113, 203912-1JOURNAL OF APPLIED PHYSICS 113, 203912 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 163.118.172.206 On: Fri, 03 Jul 2015 11:02:53(5 nm). They were grown in a Singulus Timaris deposition machine by sputtering on oxidized silicon. The CoFeB layers are initially amorphous. Two different post growth treatmentswere applied to induce the crystallization. The ﬁrst one is con- ventional annealing at 300 /C14C during 2 h, which is known to induce crystallization of CoFeB. The second one is using15 keV He þion irradiation performed at 100/C14C after a 1-h stay at this temperature and using a current density of 5lA=cm2. The working temperature of 100/C14C was chosen with the sole criterion of being much smaller than that of con- ventional annealing; however, qualitatively similar results were obtained when ion irradiation was performed at roomtemperature. The total duration of irradiation was in the 1 min range, i.e., much shorter than the conventional annealing time. Irradiation was performed for variable ﬂuences up to5/C210 19ions=m2; a ﬂuence of 1020ions=m2rendered the sample not magnetic at room temperature. The magnetization of the ﬁlms was studied using alternative gradient ﬁeld mag-netometry (AGFM). The consistency of the magnetization and the Kerr signal was checked with polar magneto-optical Kerr effect (PMOKE). The high frequency properties were studied by broad- band (0.1–70 GHz) Vector Network Analyzer FerroMagnetic Resonance (VNA-FMR 21) in the open-circuit total reﬂection conﬁguration,22using a 2.4 T ﬁeld applied perpendicular to the sample surface. Data analysis was conducted using meth- ods similar to those previously validated in Ref. 2to extract Gilbert damping a, Land /C19e factor g, and the effective uniaxial anisotropy Hef f kdeﬁned as the magneto-crystalline anisot- ropy ﬁeld minus the magnetization, i.e., Hk1/C0MS(see Ref. 2). The permeability levels in VNA-FMR spectra were checked to be consistent with the evolutions of the magnet- ization extracted from AGFM. III. RESULTS Fig. 1displays the ﬁeld dependence of the FMR fre- quencies of some of our ﬁlms after either of the two post- treatments. These curves are used to extract the effectiveanisotropy ﬁelds and the Land /C19e spectroscopic splitting fac- torsgof all samples, as listed in Figures 3(a)and3(b). The ﬁeld dependence of the FMR linewidth (Fig. 2) has been used to extract the Gilbert damping (Fig. 3(d)). As well known,4,23,24we conﬁrm a substantial increase of the effec- tive anisotropy Hef f kupon annealing and a slight increase of g. More interestingly, irradiation also induces an increase of Hef f k, up to a value slightly above that obtained by conven- tional annealing. This evolution of Hef f khappens in two steps. In the low ﬂuence regime (i.e., F<2/C21019ions=m2) the increase of Hef f kis linear with irradiation ﬂuence. The Land /C19e factor increases slightly in a correlated manner, while the damping (0.01) does not vary within the precision of our measurements. The magnetization decreases a bit, but thisonly partly accounts for the increase of the effective anisot- ropy: the magneto-crystalline anisotropy is strengthened dur- ing this evolution. These evolutions are consistent with theevolutions observed generally during the CoFeB crystalliza- tion 2with an unaltered abrupt CoFeB/MgO interface. The material evolutions are different at higher irradia- tion ﬂuences (i.e., F>2/C21019ions=m2). The magnetiza- tion reduces substantially, and the magneto-crystalline anisotropy also decreases. Overall, the effective anisotropyincreases and then stabilizes. This irradiation induced increase of H ef f kis predominantly due to a decrease of the magnetization. The Land /C19e factor and the damping both undergo substantial increase. These evolutions are consistent with an important mixing at the CoFe/Ta interface and a reduction of the abruptness of the CoFe/MgO interface.Indeed, CoFeTa alloys have lower magnetizations than CoFe (Refs. 25and26) such that interface mixing will naturally decrease the layer’s magnetization. Also, it is known 27that Ta dopants increase the damping of transition metal alloys. For instance, the Fe 90Ta10alloys have FMR linewidths typi- cally 5 times greater than iron.25Finally, the Land /C19e factor in- crement ( g– 2) is a measurement of the orbital momentum contribution to the magnetization. In transition metals, the orbital momentum is quenched by the electron itinerancy, FIG. 1. Field dependence of the ferromagnetic resonance frequencies for 1 nm thick Ta/CoFeB/MgO layers that have undergone either an annealing of 300/C14C or an annealing of 100/C14C with Heþion irradiation. Inset: PMOKE hysteresis loop for a ﬂuence of 1019ions=m2. The coercivity is 0.35 mT.FIG. 2. Example of a permeability spectrum recorded on 1 nm of Co20Fe60B20in an irradiated state for F¼1:5/C21019ions=m2at an applied ﬁeld of 0.86 T (red dots) perpendicular to the sample. The ﬁt (bold black line) is done with an effective linewidth parameter Dx=ð2xFMRÞ¼0:022 that includes an inhomogeneous broadening contribution.203912-2 Devolder et al. J. Appl. Phys. 113, 203912 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 163.118.172.206 On: Fri, 03 Jul 2015 11:02:53but it is partly restored by the spin-orbit coupling. Therefore, the spin-orbit coupling of Ta, which is 4 times larger28than that of Fe, is expected to increase gto levels unaccessible in CoFe alloys,29in qualitative agreement with our data. Let us now simulate the material’s evolution during our irradiation process to shed light onto the evolution of itsmagnetic properties. IV. DISCUSSION In ion-atom collisions, the transferred energy Thas a maximum value Tmaxthat depends on the ion and atom masses (see Table I). Because B atoms are the lightest target atoms, they can receive the highest recoiling energy. A sec-ond important fact is that the velocity of 15 keV helium ions is small enough for the ion-atom collisions to be screened Coulomb interactions, 30leading to cross sections varying as 1=T2: most collisions transfer a small energy, leading to short range displacements. The last important fact is thatwhen knocked-on, atoms do not leave their position unless T is greater than the displacement threshold energy Edwhich is material dependent and anisotropic. For instance, thedirection-averaged displacement energy of B and Fe in Fe 75B25are both equal to 22 63 eV,31while that of Mg and O in MgO are 55 eV.32Considering the different Tmax, the 1=T2cross-section, and the Edvalues, one can anticipate that the number of displacement per B atom will substantially exceed that of the other atomic species in our sample. Theseexpectations can be confronted to simulations of composi- tion proﬁles as obtained by ballistic mixing simulations. The modeling of the effect of post-growth irradiation is based on SRIM (Stopping and Ranges of Ions in Matter) cal- culations, 33with the material parameters of Table I. Changes in the concentration proﬁles were calculated by assuminglinear mixing (i.e., assuming that the probability of an al- ready displaced atom to be moved once again is negligible). In that limit, changes in the composition proﬁles can beobtained from ( r/C0v)FVfor homoatoms, and rFVfrom het- eroatoms, where randvare the recoil and vacancy distribu- tions per unit depth and per incident ion, Fis the ﬂuence, andVthe typical volume of an atom in the layer of interest. We have taken V/C251:16/C210 /C029m3for the CoFeB layer, assuming interstitial position for all B atoms. The changes inthe composition proﬁle are displayed in Figure 4, which shows that the concentration of foreign atoms near the inter- faces stay below p¼8% for a ﬂuence F¼10 19ions/m2. This justiﬁes a posteriori the linear mixing assumption until p2/C210:1, which corresponds to 4 /C21019ions/m2. The main outcomes of the simulations are the atoms’ displacements, which are relevant for the onset ofTABLE I. Parameters used for the collisional mixing simulation. Layer Layer density (g/cm3) Ed(eV) Tmax=E MgO 3.58 55 [Ref. 32] Mg: 68%; O: 64% CoFeB 8.2 22 [Ref. 31] B: 79%; Fe: 25% Ta 16.6 33 [Ref. 34] Ta: 8.5% FIG. 3. Evolution of the magnetic properties with annealing at 300/C14C (circles) or with irradiation for a Ta/Co 20Fe60B20(1 nm)/MgO ﬁlm which is amorphous in its initial state (empty rectangles). The lines are guide to theeye in all panels. Panel (a), squares: effective anisotropy ﬁeld H k1/C0MS; tri- angles: magneto-crystalline anisotropy ﬁelds Hk1. Panel (b): spectroscopic splitting factor g. Panel (c), black squares: magnetization obtained from the moments measured in AGFM and assuming an invariant CoFeB thickness of 1 nm; green triangles: Kerr rotation at magnetization saturation. Panel (d): Gilbert damping factor. FIG. 4. Changes in the concentration proﬁles as simulated assuming linear collisional mixing for a ﬂuence of F¼1019ions=m2. Negative values indi- cate a deﬁcit of material.203912-3 Devolder et al. J. Appl. Phys. 113, 203912 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 163.118.172.206 On: Fri, 03 Jul 2015 11:02:53crystallization, and the mixing rates, which are relevant for the high ﬂuence material evolutions. One can ﬁrst notice that irradiation induces a deﬁcit of material in CoFeB layer nearMgO/CoFeB and CoFeB/Ta interfaces. This is less marked at the CoFeB/Ta since Ta atoms are 54% bigger than Fe atoms. This is expected to favor the crystallization preferen-tially from the MgO/CoFeB interface, in agreement with the increase of magneto-crystalline anisotropy seen at low ﬂuences. One can also see that some Fe or Co atoms are incorpo- rated into Ta-rich regions. If only those atoms became para- magnetic, irradiation would lead to a reduction of the totalspin moment of 1% every 10 19ions=m2. This is an order of magnitude smaller than the measurements (Fig. 3(c)) and comes from the fact that in bulk TaFe alloys, the ferromag-netism is lost 26when Fe concentrations fall below 65%. A magnetic dead layer is thus likely to be formed within the CoFeB near its interface with Ta, as some ﬂuence thatdepends on the degree of initial intermixing. In addition, the incorporation of high spin-orbit paramagnetic Ta atoms in CoFe is probably the main factor increasing a. One can get an estimate of this effect by using the tables of Ref. 27that gather the increase of damping by doping Ta in permalloy. Assuming permalloy’s conclusion is valid for CoFe; onewould expect typically 0.001 extra damping for each 10 19ions=m2. This number is compatible with our ﬁndings at ﬂuences below 1 :5/C21019ions=m2, but the damping degra- dation seems much more dramatic at higher ﬂuences. This last evolution is not quantitatively understood. V. CONCLUSION In conclusion, we have studied the effect of 15 keV Heþ irradiation onto the properties of Ta/CoFeB/MgO layers withperpendicular anisotropy. We have reported the evolutions of the magnetization, the anisotropy, the Land /C19e factor, and the Gilbert damping. We have modeled the material evolu- tion by collisional mixing simulations. At low ﬂuences, there is an increase of the magneto-crystalline anisotropy, whilethe damping, the g-factor, and the magnetization are almost unaffected, consistent with an irradiation-induced crystalli- zation of the CoFeB layer with the correct oxygen composi-tion proﬁle within the stack. 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1.2402032.pdf	Switching times of a single-domain particle in a field inclined off the easy axis Hiroshi Fukushima, Yasutaro Uesaka, Yoshinobu Nakatani, and Nobuo Hayashi Citation: Journal of Applied Physics 101, 013901 (2007); doi: 10.1063/1.2402032 View online: http://dx.doi.org/10.1063/1.2402032 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/101/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of a dc bias field on the dynamic hysteresis of single-domain ferromagnetic particles J. Appl. Phys. 107, 073914 (2010); 10.1063/1.3359722 Reversal of single-domain magnetic nanoparticles induced by pulsed magnetic fields J. Appl. Phys. 103, 07F502 (2008); 10.1063/1.2829594 Reversal time of the magnetization of single-domain ferromagnetic particles with cubic anisotropy in the presence of a uniform magnetic field J. Appl. Phys. 101, 093909 (2007); 10.1063/1.2728783 Fast switching in a single-domain particle under sub-Stoner–Wohlfarth switching fields Appl. Phys. Lett. 81, 4008 (2002); 10.1063/1.1522132 Integral relaxation time of single-domain ferromagnetic particles (abstract) J. Appl. Phys. 81, 4744 (1997); 10.1063/1.365449 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.187.254.4 On: Fri, 21 Nov 2014 18:42:12Switching times of a single-domain particle in a ﬁeld inclined off the easy axis Hiroshi Fukushimaa/H20850 3-73 Honda, Midori-ku, Chiba-shi, 266-0005 Japan Yasutaro Uesaka Nihon University, Kohriyama-shi, 963-8642 Japan Yoshinobu Nakatani The University of Electro-Communications, Chofu-shi, 182-8585 Japan Nobuo Hayashi 2-2-22 Kichijyoujikita, Musashino-shi, 180-0001 Japan /H20849Received 16 May 2006; accepted 9 October 2006; published online 3 January 2007 /H20850 By solving Brown’s Fokker-Planck equation numerically with the spherical harmonics, the magnetization reversal of a single-domain particle in a ﬁeld at an oblique angle up to 45° to the easyaxis is investigated. Different from the usual deﬁnition, the switching time is deﬁned as the timewhen the averaged zcomponent of the magnetization reaches 90% of its ﬁnal value. The switching times of the particle under various conditions are calculated. When the oblique angle of the ﬁeld is30°−45° and its magnitude is larger than the Stoner-Wohlfarth limit /H20849H sw/H20850, the switching time is dependent slightly on the oblique angle and magnitude of the ﬁeld, and the temperature. For the oblique angle of 5°, the switching time depends largely on the magnitude of the ﬁeld and thetemperature. When the magnitude of the ﬁeld is less than H sw, the switching time is dependent largely on the oblique angle and the temperature. Effects of the damping constant are also studied.©2007 American Institute of Physics ./H20851DOI: 10.1063/1.2402032 /H20852 I. INTRODUCTION Recently the magnetization reversal of a single-domain particle has been an interesting ﬁeld associated with isolatedferromagnetic nanostructures, such as patterned recordingmedia. Dependence of the switching time of the particle onan applied ﬁeld has been studied by using the Landau-Lifshitz-Gilbert equation in disregard of temperature, theLangevin equation with random thermal ﬁeld, and Brown’sFokker-Planck equation with thermal term. 1–5Brown’s Fokker-Planck equation describes time-dependent behaviorsof the magnetization of a single-domain particle with thermaldiffusion through the stochastic method. Since this equationcannot be solved analytically, it was solved by using a ﬁnitedifference method in the polar coordinate with a truncatedFourier series expansion in the azimuthal coordinate. 3–5In this method, however, an oblique angle of the applied ﬁeld tothe easy axis was limited to less than a few degrees. In thispaper, in order to extend this limitation to a large angle, atruncated expansion in the spherical harmonics is employed.Switching times of the particle with uniaxial anisotropy sub-jected to a ﬁeld applied at an oblique angle up to 45° to theeasy axis are investigated under various conditions. II. METHOD The coordinate system used in this paper is shown in Fig. 1. Initially angles of the magnetization direction distrib- ute according to a Boltzmann distribution for an applied ﬁelddirecting to the + zdirection. Then the ﬁeld is reversed at an oblique angle /H9264to the − zdirection in the x-zplane. With this model the energy density Vof the particle is written as V=Kusin2/H9258−MsH/H20849− cos/H9264cos/H9258+ sin/H9264sin/H9258cos/H9278/H20850, /H208491/H20850 where Ku/H20849=MsHk/2/H20850is the anisotropy constant, Msthe mag- nitude of the magnetization, and Hthe magnitude of the applied ﬁeld. To investigate the magnetization reversal of theparticle with the above model, Brown’s Fokker-Planck equa-tion is employed. This equation can be expressed as a/H20850Electronic mail: hcb00125@nifty.com FIG. 1. Deﬁnition of the coordinate system used in this paper. Uniaxial easy axis is along the zaxis. Initially a ﬁeld is applied to the + zdirection, and then it is reversed at an oblique angle /H9264to the − zdirection in the x-zplane.JOURNAL OF APPLIED PHYSICS 101, 013901 /H208492007 /H20850 0021-8979/2007/101 /H208491/H20850/013901/7/$23.00 © 2007 American Institute of Physics 101 , 013901-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.187.254.4 On: Fri, 21 Nov 2014 18:42:12/H11509W /H11509/H9270=/H11612/H20875/H11612/H20873V MsHk/H20874·W/H20876−1 /H9251/H20875/H11612/H20873V MsHk/H20874/H11003/H11612W/H20876 +kBT MsvHk/H116122W, /H208492/H20850 with /H11612/H11013/H20873/H11509 /H11509/H9258,1 sin/H9258/H11509 /H11509/H9278/H20874 /H208493/H20850 and /H116122/H110131 sin/H9258/H11509 /H11509/H9258/H20873sin/H9258/H11509 /H11509/H9258/H20874+1 sin2/H9258/H115092 /H11509/H92782, where Wis the probability density of the magnetization vec- torMon the unit sphere, /H9270=/H9251/H9253Hkt//H208491+/H92512/H20850normalized time, ttime,/H9251the damping constant, /H9253the gyromagnetic constant, Hkthe anisotropy ﬁeld, kBthe Boltzmann constant, vthe volume of the particle, and Tthe temperature.1The ﬁrst term in Eq. /H208492/H20850is a dissipative term, the second a precessional one, and the third a thermal diffusive one. Function Win Eq. /H208492/H20850can be expanded by using a truncated spherical harmon- ics as follows: W/H20849/H9258,/H9278,/H9270/H20850=/H20858 n=0N /H20858 m=−nn an,m/H20849/H9270/H20850Yn,m/H20849/H9258,/H9278/H20850, /H208494/H20850 where Nis the cutoff wave number, an,m/H20849/H9270/H20850is the coefﬁcient and is generally complex number, and Yn,m/H20849/H9258,/H9278/H20850is the spherical harmonics with the degree nand order m.I ti s deﬁned in this paper as Yn,m/H20849/H9258,/H9278/H20850=Pnm/H20849cos/H9258/H20850exp /H20849im/H9278/H20850, /H208495/H20850 where Pnm/H20849/H9258,/H9278/H20850is the associated Legendre function with the degree nand order m. Here a Fokker-Planck operator LFPis deﬁned as the operator equivalent to the right-hand side ofEq. /H208492/H20850. With Eq. /H208494/H20850and this operator, Eq. /H208492/H20850can be ex- pressed as /H11509W /H11509/H9270=/H20858 n=0N /H20858 m=−nn/H11509an,m /H11509/H9270Yn,m=LFPW=/H20858 n=0N /H20858 m=−nn an,mLFPYn,m. /H208496/H20850 In addition, LFPYn,mcan be expanded in terms of Yn+p,m+qas follows:6 LFPYn,m=/H20858 p=−22 /H20858 q=−rr cn,m,n+p,m+qYn+p,m+q, /H208497/H20850 where r=1 for /H20841p/H20841/H113491 and r=0 for /H20841p/H20841=2. Combining Eqs. /H208496/H20850and /H208497/H20850yields the following equa- tion as /H20858 n=0N /H20858 m=−nn/H11509an,m /H11509/H9270Yn,m =/H20858 n=0N /H20858 m=−nn an,m/H20858 p=−22 /H20858 q=−rr cn,m,n+p,m+qYn+p,m+q. /H208498/H20850 To obtain the coefﬁcient for Yn,mon the right-hand side ofEq. /H208498/H20850, the indices of Yn+p,m+qneed to be shifted to Yn,m. Correspondingly the indices of an,mandcn,m,n+p,m+qare also shifted as follows: /H20858 n=0N /H20858 m=−nn/H11509an,m /H11509/H9270Yn,m =/H20858 n=0N /H20858 m=−nn/H20875/H20858 p=−22 /H20858 q=−rr an−p,m−qcn−p,m−q,n,m/H20876Yn,m, /H208499/H20850 where n−p/H113500 and /H20841m−q/H20841/H11349n−p. Multiplying both sides of Eq. /H208499/H20850byYn,mand integrating them, a set of simultaneous differential equations for an,mcan be obtained on the basis of the orthogonality of the spherical harmonics. That is, /H11509an,m /H11509/H9270=/H20858 p=−22 /H20858 q=−rr cn−p,m−q,n,man−p,m−q. /H2084910/H20850 The formulas for cn−p,m−q,n,mcan be obtained through very complicated manipulations of formulas for the spherical har-monics as follows: 6 cn−2,m,n,m=/H20849n+1/H20850/H20849n−m−1/H20850/H20849n−m/H20850/ /H20851/H208492n−3/H20850/H208492n−1/H20850/H20852, cn−1,m−1,n,m=hx/H20849n+1/H20850//H208512/H208492n−1/H20850/H20852, cn−1,m,n,m=hz/H20849n+1/H20850/H20849n−m/H20850//H208492n−1/H20850 −im/H20849n−m/H20850//H20851/H9251/H208492n−1/H20850/H20852, cn−1,m+1,n,m=−hx/H20849n+1/H20850/H20849n−m−1/H20850/H20849n−m/H20850//H208512/H208492n−1/H20850/H20852, cn,m−1,n,m=ihx//H208492/H9251/H20850, cn,m,n,m=/H20851n/H20849n+1/H20850−3m2/H20852//H20851/H208492n−1/H20850/H208492n+3/H20850/H20852 −imhz//H9251−n/H20849n+1/H20850kBT//H20849MsvHk/H20850, cn,m+1,n,m=ihx/H20849n+m+1/H20850/H20849n−m/H20850//H208492/H9251/H20850, cn+1,m−1,n,m=hxn//H208512/H208492n+3/H20850/H20852, cn+1,m,n,m=−hzn/H20849n+m+1/H20850//H208492n+3/H20850 −im/H20849n+m+1/H20850//H20851/H9251/H208492n+3/H20850/H20852, cn+1,m+1,n,m=−hxn/H20849n+m+2/H20850/H20849n+m+1/H20850//H208512/H208492n+3/H20850/H20852, cn+2,m,n,m=−n/H20849n+m+2/H20850/H20849n+m+1/H20850//H20851/H208492n+5/H20850/H208492n+3/H20850/H20852, /H2084911/H20850 where hx=Hsin/H9264/Hk,hz=−Hcos/H9264/Hkand i=/H20881−1. The number of the coefﬁcients an,misN/H20849N+2/H20850+1. With Eq. /H208494/H20850and formulas of the spherical harmonics, the integral of Wover the unit sphere is obtained as /H20885 02/H9266/H20885 0/H9266 Wsin/H9258d/H9258d/H9278=4/H9266a0,0. /H2084912/H20850013901-2 Fukushima et al. J. Appl. Phys. 101 , 013901 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.187.254.4 On: Fri, 21 Nov 2014 18:42:12In addition, a0,0keeps the initial value throughout following time integration, because /H11509a0,0//H11509/H9270=0 from Eq. /H2084911/H20850. There- fore when 4 /H9266a0,0=1 is set initially, the condition that the integral of Wover the unit sphere must be 1 is always ful- ﬁlled. Furthermore the average of Mz/Ms/H20849=cos/H9258/H20850over the unit sphere is given as a simple formula /H20885 02/H9266/H20885 0/H9266 Wcos/H9258sin/H9258d/H9258d/H9278=4 3/H9266a1,0. /H2084913/H20850 Hereafter the averaged MzandMz/Msare denoted as /H20855Mz/H20856 and /H20855Mz/Ms/H20856, respectively. Then, Eq. /H2084910/H20850is discretized with the Crank-Nicholson scheme to make a set of linear equations for an,m. Thus the time evolution of an,mcan be calculated from the initial state to a speciﬁed state by solving the equations with a /H20849LU /H20850- decomposition method /H20849decomposition of a matrix A into a product of a lower triangular matrix L and an upper triangu-lar matrix /H20850. In this paper, material parameters are chosen to be typi- cal for memory media as follows: M s=3/H11003105A/m /H20849300 emu/cm3/H20850,v=10−24m3, and Hk=7.96/H11003105A/m /H20849104Oe/H20850. It is to be noted that a change in Tcan be assumed to be, for example, a change in Ms, because the values of T, Ms, and vare applied together only to the coefﬁcient kBT//H20849MsvHk/H20850of the third term on the right-hand side of Eq. /H208492/H20850. Parameters for the time integration are following. The time step is 10−13s. The cutoff wave number Nis required to be 60 in order that switching times can be calculated withinerrors of 1%. The switching time is usually deﬁned as the time that M z reaches zero. When, however, the ﬁeld is applied at a large oblique angle, dozens of degrees, to the easy axis, this deﬁ-nition is not appropriate, because /H20855M z/H20856oscillates changing its sign during early period of switching process. For example, Fig.2shows the time evolution of /H20855Mz/Ms/H20856during switching process for H/Hsw=1.2 and 2.0 with /H9264=45°, /H9251=0.01, and T=350 K, where Hswstands for the Stoner-Wohlfarth limit. Therefore, in this paper, the switching time is deﬁned as thetime that /H20855M z/H20856reaches 90% of its ﬁnal value correspondingto the energy minimum depending on the applied ﬁeld. With this deﬁnition, switching times under various conditions arecalculated. III. RESULTS AND DISCUSSION Figures 3/H20849a/H20850and3/H20849b/H20850show the switching time versus the applied ﬁeld for different values of the damping constant /H9251 and the oblique angle /H9264atT=350 K: /H20849a/H20850/H9251=0.01 and /H20849b/H208500.1. For/H9264=5° in Fig. 3/H20849a/H20850, an increase in the switching time in the range H/Hsw=1.0−1.4 is due to an increase in the dura- tion of the oscillation of /H20855Mz/Ms/H20856. For/H9264/H1135015° and both /H9251’s, switching times are slightly dependent on the magnitude and oblique angle of the applied ﬁeld. In Fig. 2, although the two curves differ in the early period, they approach nearly thesame value in the late period. Similar phenomena occur inthe case of /H9251=0.1, though frequency of the oscillation is less than that in Fig. 2. Figures 4/H20849a/H20850–4/H20849c/H20850present the switching time versus the applied ﬁeld for different values of the oblique angle /H9264and the temperature Twith/H9251=0.01: /H20849a/H20850/H9264=45°, /H20849b/H2085015°, and /H20849c/H20850 5°. For /H9264=45° and 15°, the switching time increases slightly with increasing temperature. For /H9264=5°, however, this tem- perature dependence becomes large, especially in the regionofh/H20849=H/H sw/H20850=1.0−1.4. To elucidate this phenomenon, the time evolution of /H20855Mz/Ms/H20856under these conditions is shown in Fig. 5. In this ﬁgure /H20855Mz/Ms/H20856’s for H/Hsw=1.4 at T=50 and 350 K decrease more slowly with oscillation than those FIG. 2. Time evolution of /H20855Mz/Ms/H20856forH/Hsw=1.2 and 2.0 with the oblique angle /H9264=45°, /H9251=0.01, and T=350 K. Here Hswstands for the Stoner- Wohlfarth limit. FIG. 3. Switching time versus applied ﬁeld for different values of /H9251and/H9264at T=350 K: /H20849a/H20850/H9251=0.01 and /H20849b/H208500.1.013901-3 Fukushima et al. J. Appl. Phys. 101 , 013901 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.187.254.4 On: Fri, 21 Nov 2014 18:42:12forH/Hsw=1.0 in the early period. As regards inﬂuence of the temperature, /H20855Mz/Ms/H20856forH/Hsw=1.4 at T=350 K de- creases more quickly than that at T=50 K in the early period, but it decreases more slowly in the late period. Therefore theswitching time at T=350 K is larger than that at T=50 K. A similar phenomenon takes place for H/H sw=1.0. Figures 6/H20849a/H20850–6/H20849c/H20850display the normalized switching time /H9253/H9251Hkts//H208491+/H92512/H20850/H20849ts: switching time /H20850versus the applied ﬁeld for different values of /H9264and/H9251atT=350 K: /H20849a/H20850/H9264=45°, /H20849b/H20850 15°, and /H20849c/H208505°. When H/Hsw/H113501.2, the damping constant almost does not affect the normalized switching time for ev-ery /H9264. For H/Hsw=1.0 and /H9264=5° and 15°, the normalized switching time for /H9251=0.1 is larger than those for /H9251=0.03 and FIG. 4. Switching time versus applied ﬁeld for different values of /H9264andT with/H9251=0.01: /H20849a/H20850/H9264=45°, /H20849b/H2085015°, and /H20849c/H208505°. FIG. 5. Dependence of time evolution of /H20855Mz/Ms/H20856on temperature and ap- plied ﬁeld for T=350 K, 50 K and h=1.0, 1.4 with /H9264=5° and /H9251=0.01, where hdenotes H/Hsw. FIG. 6. Normalized switching time /H9253/H9251Hkts//H208491+/H92512/H20850versus applied ﬁeld for different values of /H9264and/H9251atT=350 K: /H20849a/H20850/H9264=45°, /H20849b/H2085015°, and /H20849c/H208505°. Here Hkstands for the anisotropy ﬁeld, and tsthe switching time.013901-4 Fukushima et al. J. Appl. Phys. 101 , 013901 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.187.254.4 On: Fri, 21 Nov 2014 18:42:120.01. This can be explained by considering the time evolu- tion of /H20855Mz/Ms/H20856with normalized time for /H9251=0.01, 0.03, and 0.1 with /H9264=5°, H/Hsw=1, and T=350 K, shown in Fig. 7, and the following discussion. In the early period in Fig. 7, /H20855Mz/Ms/H20856for/H9251=0.1 decreases more slowly than those for the other /H9251’s. In connection with this phenomenon, Figs. 8/H20849a/H20850–8/H20849c/H20850show samples of the locus of Min terms of Mxand Mycalculated from the Langevin equation on the contourmap of the potential energy, while Mz/H110220 and elapsed nor- malized time is within a speciﬁed maximum value /H9270max, with the same /H9264and Tas those in Fig. 7;/H20849a/H20850H/Hsw=1.0, /H9251 =0.01, /H20849b/H208501.0, 0.1, and /H20849c/H208501.2, 0.1, respectively. They are marked with solid circles at temporal intervals of a normal-ized time step /H9004 /H9270. Thus the speed of motion of Mcan be estimated from the spatial interval of the adjacent marks di-vided by /H9004 /H9270. The spatial intervals near the start point in Fig. 8/H20849a/H20850are longer than those in Fig. 8/H20849b/H20850, and/H9004/H9270in Fig. 8/H20849a/H20850is shorter than that in Fig. 8/H20849b/H20850. Therefore the moving speed of the magnetization in Fig. 8/H20849a/H20850is faster than that in Fig. 8/H20849b/H20850. As the path of the locus in Fig. 8/H20849b/H20850elongates to the right- hand side compared with that in Fig. 8/H20849c/H20850, inﬂuence of this difference of the speed becomes larger in Fig. 8/H20849b/H20850than in Fig. 8/H20849c/H20850. Therefore in the case of H/Hsw=1, the normalized switching time for /H9251=0.1 becomes larger than those for /H9251 =0.01 and 0.03. For the case of the applied ﬁeld below the Stoner- Wohlfarth limit, Fig. 9presents the switching time versus the oblique angle for different values of Twith H/Hsw=0.9 and /H9251=0.1. In this ﬁgure the temperature dependence of the switching time decreases with increasing oblique angle. Asregards this temperature dependence, the time evolution of/H20855M z/Ms/H20856with/H9251=0.1 for different values of /H9264andTis pre- sented in Fig. 10. For /H9264=15°, /H20855Mz/Ms/H20856’s decrease with FIG. 7. Time evolution of /H20855Mz/Ms/H20856with normalized time for different val- ues of /H9251with/H9264=5°, H/Hsw=1, and T=350 K. FIG. 8. Samples of the locus of Min terms of MxandMycalculated from the Langevin equation on the contour map of potential energy with /H9264=5° andT=350 K under various conditions. The loci and the contour maps are plotted while Mz/H110220 and elapsed normalized time is within /H9270maxbelow. The parameters H/Hsw,/H9251,/H9004/H9270, and/H9270maxare /H20849a/H208501.0, 0.01, 0.01, and 0.5, /H20849b/H208501.0, 0.1, 0.04, and 3.0, and /H20849c/H208501.2, 0.1, 0.04, and 1.96, respectively. FIG. 9. Switching time versus oblique angle for different values of Twith H/Hsw=0.9 and /H9251=0.1. FIG. 10. Time evolution of /H20855Mz/Ms/H20856for different values of /H9264andTwith H/Hsw=0.9 and /H9251=0.1.013901-5 Fukushima et al. J. Appl. Phys. 101 , 013901 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.187.254.4 On: Fri, 21 Nov 2014 18:42:12nearly the same slopes while they are larger than 0, but after they reach 0, they decay with different slopes depending ontemperature. In association with inﬂuence of the oblique angle pre- sented in Fig. 10, Figs. 11/H20849a/H20850and11/H20849b/H20850show the loci of M forT=350 K on the contour map of potential: /H20849a/H20850 /H9264=15° and /H20849b/H2085045°. In both ﬁgures, while Mz/H110220,Mmoves along the contour to the periphery without precessional rotation. AfterM z=0, it rotates with precession /H20849open circles /H20850. Therefore Mz decays rapidly while it is approaching 0, then it decreases slowly with rotation. As Hxfor/H9264=45° is larger than that for /H9264=15°, the moving speed for /H9264=45° is faster than that for /H9264=15°, and Mzdecays more rapidly. Concerning the damping constant dependence, Fig. 12 displays the normalized switching time /H9253/H9251Hkts//H208491+/H92512/H20850ver- sus the oblique angle for different values of /H9251with H/Hsw =0.9 and T=350 K. In this ﬁgure, the curves for /H9251=0.01 and0.03 are similar, but that for /H9251=0.1 differs from them. This difference can be explained almost with the difference of themoving speed of the magnetization vector with respect tonormalized time, as Figs. 13and14present. Figure 13shows the time evolution of /H20855M z/Ms/H20856with normalized time for different values of /H9251where H/Hsw =0.9, /H9264=15°, and T=350 K. In this ﬁgure, /H20855Mz/Ms/H20856for/H9251 =0.1 decays more slowly than the others. As an explanation for this effect of the damping constant, the loci of Mwith /H9264=15°, and H/Hsw=0.9 are plotted in Figs. 14/H20849a/H20850and14/H20849b/H20850: /H20849a/H20850/H9251=0.01 and /H20849b/H208500.1. The moving speed in Fig. 14/H20849b/H20850is approximately half of that in Fig. 14/H20849a/H20850. Therefore /H20855Mz/Ms/H20856 in Fig. 14/H20849b/H20850decays more slowly than that in Fig. 14/H20849a/H20850. Generally when His larger than Hswand the oblique angle is over 15°, the damping constant affect slightly thenormalized switching time. But when His smaller than H sw and the oblique angle is below 30°, the normalized switching time depends on the damping constant. FIG. 11. Samples of the locus of Mwith H/Hsw=0.9, /H9251=0.1, and T =350 K for different values of /H9264:/H20849a/H20850/H9264=15° and /H20849b/H2085045°. The loci are marked with solid circles for Mz/H110220 and with open circles for Mz/H110210 at temporal intervals of 0.04 in normalized time. Dotted lines present the contour map ofpotential for M z/H110220. FIG. 12. Normalized switching time /H9253/H9251Hkts//H208491+/H92512/H20850versus oblique angle for different values of /H9251with H/Hsw=0.9 and T=350 K. FIG. 13. Time evolution of /H20855Mz/Ms/H20856with normalized time for different values of /H9251with/H9264=15°, H/Hsw=0.9, and T=350 K. FIG. 14. Samples of the locus of Mwith/H9264=15°, H/Hsw=0.9, and T =350 K, plotted through the same procedure as in Fig. 8for different values of parameters: /H20849a/H20850/H9251=0.01, /H9004/H9270=0.01, and /H9270max=0.18, and /H20849b/H208500.1, 0.04, and 2, respectively.013901-6 Fukushima et al. J. Appl. Phys. 101 , 013901 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.187.254.4 On: Fri, 21 Nov 2014 18:42:12IV. CONCLUSION Switching times of the particle in a ﬁeld at an oblique angle up to 45° to the easy axis were investigated under various conditions by solving Brown’s Fokker-Planck equa-tion with the spherical harmonics. The switching times at anoblique angle of 30°–45° are only slightly dependent on theapplied ﬁeld larger than the Stoner-Wohlfarth limit. In thecase of the ﬁeld smaller than this limit the magnetizationreversal takes place in a different way from the former case.1W. F. Brown, Jr., IEEE Trans. Magn. 15, 1196 /H208491979 /H20850. 2Y . Uesaka, H. Endo, T. Takahashi, Y . Nakatani, N. Hayashi, and H. Fuku- shima, Phys. Status Solidi A 189, 1023 /H208492002 /H20850. 3H. Fukushima, Y . Uesaka, Y . Nakatani, and N. Hayashi, J. Magn. Magn. Mater. 242–245 , 1002 /H208492002 /H20850. 4H. Fukushima, Y . Uesaka, Y . Nakatani, and N. Hayashi, IEEE Trans. Magn. 38,2 3 9 4 /H208492002 /H20850. 5H. Fukushima, Y . Uesaka, Y . Nakatani, and N. Hayashi, IEEE Trans. Magn. 39,2 5 1 9 /H208492003 /H20850. 6L. J. Geoghegan, W. T. Coffey, and B. Mulligan, “Differential Recurrence Relations for Non-axially Symmetric Rotational Fokker-Planck Equa-tions,” Advances in Chemical Physics /H20849Wiley, New Jersey, 1997 /H20850,V o l . 100, pp. 475–641.013901-7 Fukushima et al. J. Appl. Phys. 101 , 013901 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.187.254.4 On: Fri, 21 Nov 2014 18:42:12
1.3643046.pdf	Spin-transfer-torque reversal in perpendicular anisotropy spin valves with composite free layers I. Yulaev, M. V. Lubarda, S. Mangin, V. Lomakin, and Eric E. Fullerton Citation: Appl. Phys. Lett. 99, 132502 (2011); doi: 10.1063/1.3643046 View online: http://dx.doi.org/10.1063/1.3643046 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v99/i13 Published by the American Institute of Physics. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 30 Apr 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsSpin-transfer-torque reversal in perpendicular anisotropy spin valves with composite free layers I. Yulaev,1,2M. V. Lubarda,1,2S. Mangin,3V . Lomakin,1,2and Eric E. Fullerton1,2,a) 1Center for Magnetic Recording Research, University of California, San Diego, La Jolla, California 92093, USA 2Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, California 92093, USA 3Institut Jean Lamour, CNRS, Nancy Universite ´, UPV Metz, Vandoeuvre les Nancy, France (Received 6 June 2011; accepted 25 August 2011; published online 26 September 2011) We describe modeling of spin-transfer-torque (STT) driven reversal in nanopillars with strong out-of-plane magnetic anisotropy where the free layer is a magnetically hard-soft composite structure. By adjusting the exchange coupling between the hard and soft layers, we observed reduced current amplitude and pulse durations required to reverse the magnetization compared to a homogeneous free layer of comparable thermal stability. The reduction in critical current comes from the increased STT efﬁciency acting on the soft layer. As such, the switching current isrelatively insensitive to the damping parameter of the magnetic hard layer. These properties make composite free layers promising candidates for STT-based magnetic memories. VC2011 American Institute of Physics . [doi: 10.1063/1.3643046 ] Spin-transfer-torque (STT) based magnetic random access memories (MRAM) are a promising technologies for implementing non-volatile storage in commercial integratedcircuits. 1One of the present challenges in implementing STT-MRAM is the reduction in critical current ( IC), the cur- rent required to change the magnetization of the free mem-ory element while maintaining a sufﬁcient thermal stability for non-volatile applications. A reduction in I Cis essential for reducing power dissipation and current. STT devices exhibiting perpendicular anisotropy pro- vide a pathway to low critical current and high thermal sta- bility.2In the perpendicular geometry, the critical current for the onset of spin-torque reversal for a macrospin is given by IC0¼2e /C22h/C18/C192a gðhÞpEB (1) assuming zero temperature and no applied ﬁelds. Here, EBis the energy barrier for reversal, ais the Gilbert damping con- stant of the free layer, g(h) is the angular dependence of spin torque transfer efﬁciency, and pis the spin polarization of the current. As EBis set by the thermal stability requirements of the device (typically EB>50 k BT), further reductions require decreasing aand increasing g(h)pas in recent dem- onstrations of STT switching of perpendicular anisotropy CoFeB/MgO/CoFeB tunnel junctions.3For current pulses of ﬁnite time ( s), modeling and experiments of fast time switch- ing of perpendicular anisotropy nanopillars show that s/C01¼AðIC/C0IC0Þ; (2) where ICis the current required to reverse the magnetization and the parameter Agoverns the switching rate.4,5 In this letter, we describe STT-driven switching of exchange coupled magnetically hard/soft b i-layer as the free layer. Suchcomposite structures have been extensively studied for their efﬁ- ciency for magnetic ﬁeld switching, particularly for magnetic recording6–9and for microwave assisted magnetic recording.10 The model used in the present calculation is schematically depicted in Fig. 1. The reference layer is ﬁxed in the calcula- tions. The free layer has a relativ ely soft layer #1 that interacts with the reference layer via the S TT interactions. The soft layer is ferromagnetically exchange coupled ( Jex¼0.2-5 ergs/cm2)t o the magnetically harder layer # 2 having relatively higher magneto-crystalline anisotropy.11,12For the examples discussed in this paper, the soft layer has i dentical parameters as the hard layer, except with magneto-crystalline anisotropy KUreduced by a factor of /C247 relative to the hard layer. We further alter the damping parameters of hard and soft layers for selected calculations. We run time-domain simulations of both current and ﬁeld induced switching dynamics for the composite free layer bysolving the Landau-Lifshitz-Gilbert (LLG) equation modiﬁedto include spin-torque term. We assumed that there is no spintorque interaction between layers #1 and #2 such that the spin-torque interaction from the reference layer only affects layer #1 of the composite layer. We have performed bothmacrospin (where each layer of the composite is treated as amacrospin) and micromagnetic simulations and have veriﬁedthat the results are self-consistent. The micromagnetic simula- tions used the LLG Micromagnetics Simulator where the magnetic layers were discretized into 4 /C24n m 2squares. For both simulations, we assume the symmetric Slonczewskiapproximation for g(h)¼ q ASSþBSScosðhÞ,w h e r e ASSandBSSare functions of polarization Pthat approximate a metallic spacer.13,14We also performed macrospin calculations using g(h)¼g0which more closely represents a tunnel junction. An example time trace showing reversal of the compos- ite free layer is shown in Fig. 1forJex¼1 erg/cm2and I¼433 lA. As a result of the ﬁnite exchange between the hard and soft layers, the softer layer (dashed line) responds more strongly to the STT interaction as compared to the harda)Author to whom correspondence should be addressed: Electronic mail: efullerton@ucsd.edu. 0003-6951/2011/99(13)/132502/3/$30.00 VC2011 American Institute of Physics 99, 132502-1APPLIED PHYSICS LETTERS 99, 132502 (2011) Downloaded 30 Apr 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionslayer (solid line) leading to larger precession angles prior to reversal. The soft layer initializes reversal as seen most clearly in the M zscan. This behavior is similar to that observed in microwave assisted magnetic recording simula-tions for composite structures. 10To quantify the efﬁciency of this approach for spin-torque reversal, we compare the de- pendence of EB, the coercive ﬁeld HC, and the critical current for reversal ( IC) on the interlayer exchange coupling (Fig. 2). Each quantity is normalized to the values for large Jexwhere the free layer can be considered a single macrospin. We calculate EBfor the composite system using an approach described in Ref. 15.EBis only weakly dependent on the exchange coupling (Fig. 2(a)) decreasing less than 10% down to Jex¼0.4 ergs/cm2.W ea l s os e ead e c r e a s ei n HCof t h ef r e el a y e r( F i g . 2(a)) with decreasing exchange coupling from the strong-coupling limit until a minimum is reached at approximately 2.5 ergs/cm2and then increases quickly with fur- ther decrease of the coupling. Th is behavior, in agreement with previous experimental data16and theoretical modeling,17,18 results from incoherent reversal of the composite structure. A similar but enhanced behavior is obtained for STT switch- ing. We assume square write pulses of duration ( s) and deter- mine the minimum current for reversal. Results for s¼20 ns ands¼2n sa r es h o w ni nF i g s . 2(a)and2(b), respectively. The soft layer is initiated with a /C240.6/C14tilt in magnetization away from the easy axis. There are strong reductions in ICwith reduced coupling strength that depend on both the functional form of g(h)a n d s.F o rt h ec o n s t a n t g(h), the parallel-to-antipar- allel (P-to-AP) and AP-to-P swit ching are equivalent, and we see a 50% and 40% reduction of the cri tical current relative the strong coupling limit for 20 and 2 ns, respectively. For the Slonc-zewski form of g(h), there is a strong diff erence between P-to- AP and AP-to-P switching, particularly at longer pulse times where the composite structure is more effective in reducing theP-to-AP current for low coupling strengths. As the P-to-AP switching is the least efﬁcient, i.e., requiring greater current to induce switching of the magneti zation, this reduces the asymme- try when comparing I Cfor P-to-AP versus AP-to-P switching. This difference in AP-to-P and P-to-AP switching is also seen in micromagnetic calculations (Fig. 3) where wecalculate ICvs.sforJex¼1 ergs/cm2compared to the refer- ence model where Jex¼10 ergs/cm2. In the simulation, we use an AP reference layer to minimize the dipolar interactionof the reference and free layers. 12The results in Fig. 3are plotted as 1/ svs. I C. The simulated results generally follow the expectations of Eq. (2)and only deviate somewhat for the shortest pulse durations. For switching from AP to P, we observe only a small change of IC0(the x-axis intercept) but an increase of a factor 2 in the Aparameter for the composite structure. This results in signiﬁcantly reduced current at ﬁ- nite pulse widths reaching a 50% reduction in the critical current at 1 ns. For P-to-AP switching, we see a roughlytwo-fold decrease in I C0for the coupled bi-layer in addition to modest increase in A. Again, we see a roughly 50% decrease in the switching current at 1 ns. For both orienta-tions, we observe that the critical current is more strongly reduced compared to the reduction in E B. While the general trends observed in Fig. 3are maintained for different Jex (and also observed in macrospin calculations), the detailed behavior of IC0andAdepend on the interlayer coupling. The general trends shown in Figs. 1–3can be understood from the angular dependence of the spin-torque interaction. The soft layer moves out of the P or AP orientation relative to the reference layer much faster than the hard layer (Fig.1). As the soft free layer moves, there is an increase in the spin-torque N stwhich depends on g(h)sin(h).14For the sym- metric Sloncewski form of g(h), there is a stronger increase inNstwith angle for AP-to-P switching. Only small angles are needed for the soft layer to effectively pull the hard layer and the optimum coupling is relatively strong (Fig. 1(a)). FIG. 1. (Color online) STT switching results of a composite free layer struc- tures (inset) that is made up of two ferromagnetic layers that are ferromag- netically exchange coupled. The lower anisotropy layer 1 interacts with thereference layer via the STT interactions. The time traces show the three components of magnetization for layers 1 (dashed) and 2 (solid) during re- versal where M zis normal to the layers. FIG. 2. (Color online) Switching ﬁeld ( HC), switching current ( IC), and energy barrier ( EB) results for a composite free layer as a function of ferro- magnetic exchange coupling ( Jex) in the composite free layer. All quantities are normalized to the values for large Jex. The values for HCandEBare given in (a). ICvalues are given both AP-to-P and P-to-AP switching for the symmetric Slonczewski (SS) approximation for g(h) and assuming a ﬂat g(h)¼g0. The values for a 20-ns current pulses are given in (a) and 2-ns pulses in (b).132502-2 Yulaev et al. Appl. Phys. Lett. 99, 132502 (2011) Downloaded 30 Apr 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsConversely, for P-to-AP switching, the increase in Nstis weaker such that larger deviations of the soft layer areneeded and the optimum coupling is weaker so that the com- posite layer is more effective. The case for a constant g(h)i s in between these limits. An additional beneﬁt may arise using a composite bilayer when ais unequal in the soft and hard layers. The damping of the soft layer plays a greater role in determining the effectivedamping of the structure and hence the switching current. Using the macrospin model, we simulate the switching behav- ior of the composite bilayer for different combinations ofdamping parameter values (Fig. 4). For these calculations, we use a relatively large s¼100 ns such that I Cis dominated by IC0. If we increase the damping in both layers, ICis propor- tional to the damping as expected from Eq. (1).I n c r e a s i n ge i - ther the damping parameter of the soft layer ( asoft)o rt h eh a r d layer ( ahard) while holding the other layer ﬁxed at 0.01 also yields a linear increase in IC. However, the slope is shallower when increasing the damping in the harder layer. For example shown in Fig. 4, a 10-fold increase in asoftyields a 7-fold increase in ICwhere a 10-fold increase in ahardyields only a4.50-fold increase in IC. This property may have important practical applications as many recent experiments have shown aandKUto be positively correlated.3,19–24 In conclusion, we described mo deling of spin-transfer-tor- que reversal in nanopillars with strong out-of-plane magnetic anisotropy where the free layer i s a magnetically hard-soft com- posite structure. We ﬁnd for modest coupling between the hardand soft layers that there is a reduction in I C0a n da ni n c r e a s ei n Awhich results in improved switc hing efﬁciency without a cor- responding reduction in E B. The reduction in critical current comes from the increased STT efﬁciency acting on the soft layer combined with incoherent re versal of the composite struc- ture. As such, the switching current is relatively insensitive to the damping parameter of the ma gnetic hard layer. These prop- erties make these structures pr omising candidates for spin- transfer-torque based magnetic memories. We would like to thank A. Kent for helpful discussions. This work is supported by NSF Award No. DMR-1008654 and by theFriends contract of the French National Research Agency (ANR). 1J. A. Katine and E. E. Fullerton, J. Magn. Magn. Mater. 320, 1217 (2008). 2S. Mangin, Y. Henry, D. Ravelosona, J. A. Katine, and E. E. Fullerton, Appl. Phys. Lett. 94, 012502 (2009). 3S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nature Mater. 9, 721 (2010). 4D. Bedau, H. Liu, J. J. Bouzaglou, A. D. Kent, J. Z. Sun, J. A. Katine, E. E. Fullerton, and S. Mangin, Appl. Phys. Lett. 96, 022514 (2010). 5D. Bedau, H. Liu, J. Z. Sun, J. A. Katine, E. E. Fullerton, S. Mangin, and A. D. Kent, Appl. Phys. Lett. 97, 262502 (2010). 6N. F. Supper, D. T. Margulies, A. Moser, A. Berger, H. Do, and E. E. Fullerton, IEEE Trans. Magn. 41, 3238 (2005). 7E. E. Fullerton, H. V. Do, D. T. Margulies, and N. Supper, U.S. patent 7,425,377 (Sept. 16, 2008). 8R. H. Victora and X. Shen, IEEE Trans. Magn. 41, 2828 (2005). 9D. Suess, T. Schreﬂ, S. Fahler, M. Kirschner, G. Hrkac, F. Dorfbauer, and J. Fidler, Appl. Phys. Lett. 87, 12504 (2005). 10S. Li, B. Livshitz, H. N. Bertram, M. Schabes, T. Schreﬂ, E. E. Fullerton, and V. Lomakin, Appl. Phys. Lett. 94, 202509 (2009). 11Antiferromagnetically (AF)-coupled free layers are less efﬁcient that ferro- magntically coupled free layers. In the AF-coupled case, the preferred pre-cession direction for the harder layer (#2) is counterclockwise to the direction of the anisotropy ﬁeld, whereas the softer layer (#1) wishes to precess counterclockwise to exchange ﬁeld coming from layer #2 (which is opposite to the direction of the hard layer anisotropy ﬁeld for AF cou- pling). Thus, the two layers wish to precess with opposite cyclicity, but the exchange coupling opposes this dynamic suppressing reversal consistent with experimental observations (Ref. 12) 12I. Tudosa, J. A. Katine, S. Mangin, and E. E. Fullerton, Appl. Phys. Lett. 96, 212504 (2010). 13J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 14J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 72, 014446 (2005). 15W .J .C h e n ,S .F .Z h a n g ,a n dH .N .B e r t r a m , J. Appl. Phys. 71, 5579 (1992). 16T. Hauet, E. Dobisz, S. Florez, J. Park, B. Lengsﬁeld, B. D. Terris, and O. Hellwig, Appl. Phys. Lett. 95, 262504 (2009). 17H. N. Bertram and B. Lengsﬁeld, IEEE Trans. Magn. 43, 2145 (2007). 18T. P. Nolan, B. F. Valcu, and H. J. Richter, IEEE Trans. Magn. 47,6 3( 2 0 1 1 ) . 19A. Barman, S. Wang, O. Hellwig, A. Berger, E. E. Fullerton, and H. Schmidt, J. Appl. Phys. 101, 09D102 (2007). 20J. M. Beaujour, D. Ravelosona, I. Tudosa, E. E. Fullerton, and A. D. Kent, Phys. Rev. B 80, 180415 (2009). 21E. P. Sajitha, J. Walowski, D. Watanabe, S. Mizukami, F. Wu, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, IEEE Trans. Magn. 46, 2056 (2010). 22S. Mizukami, E. P. Sajitha, D. Watanabe, F. Wu, T. Miyazaki, H. Naga- numa, M. Oogane, and Y. Ando, Appl. Phys. Lett. 96, 152502 (2010). 23N. Fujita, N. Inaba, F. Kirino, S. Igarashi, K. Koike, and H. Kato, J. Magn. Magn. Mater. 320, 3019 (2008). 24S. Pal, B. Rana, O. Hellwig, T. Thomson, and A. Barman, Appl. Phys. Lett. 98, 082501 (2011). FIG. 3. (Color online) Micromagnetic results for switching critical currents of a composite free layer for varying current pulse durations ( s) assuming a symmetric Slonczewski approximation for g(h). Results are shown both for a Jex¼1 ergs/cm2and a reference calculations where Jex¼10 ergs/cm2. FIG. 4. (Color online) Changes in the critical current as a function of damp- ing parameter in the hard, soft, or both layers. The critical current is normal- ized to value for asoft¼ahard¼0.1.132502-3 Yulaev et al. Appl. Phys. Lett. 99, 132502 (2011) Downloaded 30 Apr 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions
1.1709138.pdf	Wall Streaming in Ferromagnetic Thin Films K. U. Stein and E. Feldtkeller Citation: J. Appl. Phys. 38, 4401 (1967); doi: 10.1063/1.1709138 View online: http://dx.doi.org/10.1063/1.1709138 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v38/i11 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 18 Apr 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsJOURNAL OF APPLIED PHYSICS VOLUME 3&, NUMBER 11 OCTOBER 1967 Wall Streaming in Ferromagnetic Thin Films K. U. STEIN AND E. FELDTKELLER Forschungslaboratorium der Siemens AG, Munchen, Germany (Received 24 May 1967) In nickel-iron films thicker than about 100 nm (1000 A), fast-rising field pulses along the hard anisotropy axis lead to a new kind of wall displacement called "wall streaming," which does not require any field component parallel to the wall necessary for all other wall motion processes. The walls are displaced with a very regular wall step width per pulse which strongly depends on the strength of the pulse field and on the pulse rise and fall times, but not on the pUlse duration. The direction of wall motion reverses if the pulse polarity is reversed. During the application of a sequence of uniform pulses, the direction of wall motion may also reverse, beginning at one end of the wall, with a reversing line (interpreted as being a Neelline moving along the wall. The phenomenon is explained as a consequence of the gyromagnetic behavior of the magnetization in the Bloch walls. A detailed theory is presented, in which the intrinsic spin damping and the film inhomogeneity (characterizable by the coercivity) are to be taken into account. Wall streaming may contribute to a destruction of the information stored in magnetic films thicker than 100 nm (e.g., in plated-wire memories) as soon as fast-rising (t,;S 20 nsec) pulses are applied. INTRODUCTION Two different kinds of wall motion have been reported in the literature on ferromagnetic thin films: Barkhausen jumps and wall creeping. Barkhausen jumps are effected by the magnetic field component oriented parallel to the wall, and are explained simply by the force of this field acting upon the wall. Creeping is effected by changes of a field component perpendic ular to the wall during the existence of a field compo nent parallel to the wall, and may be explained by changes of both the wall angle and the wall structure. Reviews on detailed observations and possible explana tions of both effects have been given by several authors.I-5 Common to both phenomena is the fact that a field component parallel to the wall is necessary to determine the sense of motion. In both cases, the adjacent domain favored by the field grows at the expense of the other one. A third kind of wall motion observable in ferro magnetic thin films is reported here. It is called "wall streaming" because of its appearence in visual observa tion. This kind of wall motion occurs even if no compo nent parallel to the wall exists, i.e., no adjacent domain is favored by a field. It is shown that streaming may be explained as due to the gyro magnetic behavior of the wall spins. Wall streaming was detected by Stein in 1964, both by observing the voltage induced in a sense line during repeated rotation of the magnetization, and by mag neto-optic observation of the domain walls. It was not published immediately because a vivid means of presentation was being looked for. This problem was solved by a motion picture on the wall streaming and on the two other kinds of wall motion, photographed by Hillebrand6 in our laboratory. This motion picture and a preliminary report on wall streaming, and its possible explanation have been presented.3 A techni cally improved version of it has been prepared.7 Possibly some unexplained observations reported by Edwards8 may now be understood also in terms of wall streaming. EXPERIMENTAL TECHNIQUES Zero-magnetostriction nickel-iron films of different thicknesses have been prepared by vacuum deposition on glass surfaces in a homogeneous magnetic field. All observations reported here have been carried out with the Kerr magneto-optic effect, in the homogeneous part of the field of either the stripline described formerly9 containing a window, or on a strip conductor (5 mm wide, 35 ~m thick) similar to that described by Middelhoek,lO with a return conductor relatively far from the strip conductor. The films have been coated with ZnS for increasing the Kerr rotationY·12 A special alignment of the photographic objective lens proposed recentlyl3 for getting high contrast and sharp images (taking into account the angle of incidence of the light beam to the sample) was used for the observations and for taking the improved version of our motion picture, on which some of the drawings presented here are based. Short-rise-time field pulses are generated by the aid of a cable-discharge pulse generator with a mercury relay. The pulse-field strength could be varied between o and 5 A/cm, and the pulse duration (measured be- 6 H. Hillebrand (unpublished). 7 E. Feldtkeller and K. U. Stein, Encyclopaedia Cinemato graphica (Institut fUr den Wissenschaftlichen Film Gottingen I A. Greene, K. D. Leaver, and M. Prutton, J. App\. Phys. 35, 1967, 16-mm silent film. " 812 (1964). 8 J. G. Edwards, Nature 201,359 (1964). 2 T. H. Beeforth, Internat. Control 1, 375 (1965). 9 K. U. Stein, Z. Angew. Physik, 20, 36 (1965). 3 E. Feldtkeller, in Struktur und Eigenschaften Magnetischer 10 S. Middelhoek, IBM J. Res. Develop. 10,351 (1966). Festkorper Magnetismus (Deutscher Verlag fur Grundstoffin- 1l J. Kranz and W. Drechsel, Z. Physik 150, 632 (1958). dustrie, Leipzig, 1966), p. 215. 12 For a review on Kerr observation techniques, see J. Kranz and • W. K!lyser, IEEE Trans. ~ag. 3, 141 (1967). A. Hubert, Z. Angew. Physik, 15, 220 (1963). 6 S. Mlddelhoek and D. Wild, IBM J. Res. Develop. 11, 93 13 E. Feldtkeller and K. U. Stein Z. Angew. Physik 23 100 (1967). (1967)." 4401 Downloaded 18 Apr 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions4402 K. U. STEIN AND E. FELDTKELLER 0.6r-------------------,.1 mm WALL POS/TiQ,/ AFTER f -0 0.4 .::::: 80 ~ :=180 y~ ~ ,EASY AXIS, PULSES OO~~-;====~~~~~~--~~ 0It Q8 12 16 mm 2.0 x- FIG. 1. Location of a Bloch wall in a IS0-nm thick nickel-iron film (H. = 1.2 A/cm, HK=2.4 A/cm) after application of different numbers of pulses along the hard anisotropy axis. Pulse-field strength H=O.3H K. No field along the easy axis is applied. The figure is drawn after different single frames of a motion picture. tween 5.0% amplitude points) between 2 nsec and 1000 nsee. The field rise and fall times (defined by the 10% and 90% values) have been varied, from 0.5 to 20 nsee, and from 1 to 40 nsec, respectively, by the use of pulse-forming low-pass filters at the output of the generator or at the end of the discharge cable, re spectively. Therefore, the rise time tr was always smaller than the fall time t,. The pulse repetition rate was usually 100 pps. Lower rates and single pulses were also used. The anisotropic nickel-iron films were adjusted within the stripline such that domain splitting was observed if large field pulses (H> HK) were applied. The pulse field was calibrated in terms of the anisot ropy field HK of the films by determining the generator voltage necessary for causing domain-splitting in the films. Besides this, a calculated field calibration and the measure~ent of the anisotropy field HK with the Kobelev method14 have been used to check the normal ized field calibration. The influence of the earth's magnetic field has been eliminated by a compensating field, and by the repeti tion of every quantitative measurement with reversed fields. QUALITATIVE EXPERIMENTAL RESULTS If unidirectional fast-rising field pulses along the hard axis, with an amplitude H =O.4HK, e.g., are applied to a nickel-iron thin film previously saturated along an easy direction, the combination of the pulse field and the film's demagnetizing field causes creeping. However, creeping increases the size of reversed domains and the demagnetizing field is thereby diminished. If the field combination finally decreases below the creep threshold, no further wall motion is observed. This behavior has to be expected and is observed for all films not thicker than about 100 nm (1000 A) film thickness. For films thicker than the thickness threshold of about 100 nm in a pulse field applied along the hard axis, creeping also occurs after saturation, as was expected; in the demagnetized state, however, the walls do not stop moving. Instead, they keep moving to and fro as long 14 V. V. Kobelev, Fiz. Met. Metalloved. 13, 467 (1962) [Eng!. trans!': Phys. Met. Metallography 13, 146 (1963)]. as field pulses are applied, such that the film always gets demagnetized again. Since this process appears to the observer similar to plants in streaming water or flags streaming in the wind, we have chosen the name "streaming." This phenomenon is surprising at first sight, since thete seems to be no force at all acting upon the wall. (The adjacent domains are magnetized along the easy directions, with equal magnetostatic energies as to the pulse field). Four processes mainly contribute to it: (1) wall-displacement directed parallel or anti parallel to the pulse field (d. Fig. 1), (2) creation or annihilation of domain nuclei, which are increased or decreased by process (1) at the edges of the film, . (3) migration of reversal lines which separate wall sections moving parallel and antiparallel to the pulse field, along the wall (d. Fig. 1), and (4) creation or annihilation of reversal lines in a wall at the film edge, or of a line pair anywhere within the wall. Which of these processes dominate depends upon the pulse-field strength. In low fields only (1) and (2) are observed, while in higher fields, processes (3) and (4) become increasingly important in addition to (1), as is shown in the paragraph on quantitative experimental results. Because the wall streaming looks random at first glance, a kind of creeping, caused by thermal fluctu ating fields and the pulse field, was supposed at first. However, the further qualitative results listed below already show that the wall streaming cannot be an effect of a statistically fluctuating field. (a) The step width per pulse of a distinct wall remains constant over a large number of pulses (d. Figs. 1 and 2). (b) The step width per pulse of a line reversing the direction of wall motion also remains constant over a large number of pulses (d. Fig. 1). (c) If the pulse sequence is interrupted for an arbi trary time interval and continued afterwards, wall '" z 0.6 g I- i!l o!o w Q. N ~ 02 a 0 200 I,()Q 600 800 1000 1200 1400 NUMBER OF PULSES FIG. 2. Dependence of the position y (along the hard anisotropy axis) of a Bloch wall for a fixed x value on the number of pulses applied along the hard axis. The pulse field strength was H=O.3HK• At the symbols w the influence of a neighboring wall and at N the influence of a nucleus at neighboring x values is detectable. Same film as in Fig. 1. Downloaded 18 Apr 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsWALL STREAMING IN FERROMAGNETIC FILMS 4403 streaming continues with the same directions and step widths for the wall and reversing-line motion as those observed before the interruption. (d) If the pulse polarity is reversed, the directions of the wall motion and of the reversing-line motion likewise reverse. This indicates that the direction is not controlled by any unknown (e.g., stray) field compo nent along the easy axis. (e) The direction of reversing-line motion is the same as the direction of the magnetization of that domain which lies on the side of the wall in the di rection of the pulse field. This relation between the direction of reversing-line motion L, domain magneti zations M, and pulse field H is illustrated in Fig. 3. (f) The wall streaming is strongly dependent upon the pulse-field rise and fall times. Only with rise or fall times in the nanosecond region, is streaming observed. (g) A remarkable dependence upon the pulse dura tion has not been found for pulse durations (measured between 50% amplitude points, fall time 35 nsec) longer than 25 nsec. (h) If the streaming is interrupted as in C and a field of, e.g., 0.7 HK is applied along the hard axis during the interruption, observation C is no longer valid. (The same holds if the wall is caused to make a large Barkhausen jump by applying a large enough field along the easy axis during the interruption.) If the pulse sequence is continued afterwards, the wall either starts moving with an arbitrary direction, or seems to be irresolutely trying to move in different directions in different wall sections, and a number of pulses is needed until the entire wall again moves in the same direction. This change from alternating motion to uniform motion always begins at one end of the wall and the wall gets thereby inclined. QUANTITATIVE EXPERIMENTAL RESULTS From the four above-mentioned processes contribut ing to the streaming, the creation and annihilation processes (2) and (4) are regarded to be less funda mental than the displacement processes (1) and (3), because they seem to be essentially the effects of edge and wall stray fields. Therefore, only the wall displace ment and the reversal-line displacement have been investigated quantitatively. The wall and reversal-line step widths per pulse are well suited for separately characterizing these processes quantitatively. For getting reproducible quantitative results, it is necessary to observe an isolated wall in a nearly ~iH ~IH ~iH ~!H ~ ~~ ~ ~ FIG. 3. Connection between the direction L of motion of a line reversing the wall-motion direction, the magnetization directions M in the adjacent domains, and the sign of a fast-rising slowly falling field pulse H applied along the hard anisotropy axis. (The reversing lines are interpreted to be Neellines separating differ ently magnetized Bloch wall sections.) ILl ~ 06 a.1Il1 Q: ILl a. :z: 0.4 ... c ~ a. ILl ... Vl 0.2 ...J ~ 010 20 50 100 200 500 ns 1000 PULSE DURATION FIG.4. Influence of the pulse duration on the wall step width per pulse in a 200-nm thiek Ni-Fe film. Pulse field H=0.23HK, rise time 0.5 nsee, fall time 35 nsee, HK=0.7 A/em, H,=2.5 A/em. demagnetized state of the film, because the demagnet izing field of the film and stray fields of nearby walls may influence the step widths. Without special pre cautions, the film will leave again and again the demagnetized state. In the measurements reported here, the film is therefore held near a demagnetized state by reversing the pulse-field polarity as soon as a significant departure from the demagnetized state is observed. If these conditions are not observed, a large irreproducibility of the results is obtained. The experimental results reported in Figs. 4-8 show the influence of the pulse parameters on the wall step width per pulse in a 2oo-nm thick nickel-iron film with Hc=0.7 A/cm and HK=2.5 A/cm. Figure 4 shows that the wall step width per pulse does not depend upon the pulse duration even if the pulse duration is varied by several orders of magnitude. Figures 5 and 6 show the dependence of the wall . step width on the pulse field amplitude for three different pulse rise times and for three different pulse fall times, respectively. As may be seen, the largest wall steps result from the pulses with the shortest rise time and the largest fall time. In Fig. 7, the loss of information regarding the former wall displacement direction by the application of a large dc field during an interruption of the pulse sequence [mentioned above in paragraph (h) ] is demonstrated quantitatively. The lowest field disturb ing this information is 0.57 HK in the case shown. As may be seen from Fig. 5 a distinct threshold pulse field exists for every pulse rise time. The depend ence of this streaming threshold field on the pulse rise time is shown in Fig. 8. The field disturbing the wall displacement sign (Fig. 7) is also indicated in Fig. 8. Only in the pulse-field-vs-pulse-rise time region bounded by these curves as drawn in Fig. 8 may the wall stream ing be observed. Since near the high-field limit in Fig. 8 the direction of wall displacement is frequently reversed, and since for measuring the wall step width, a uniform wall motion over a distance of-about 70 ~m:was neces sary, with pulse rise times larger than 17 nsec no step Downloaded 18 Apr 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions4404 K. U. STEIN AND E. FELDTKELLER lil 1.0 sum a. '" w a. I >- Q 05 :;: a. w tn -" ~ FIG. 5. Wall-step width per pulse vs the normalized pulse field H/H" for slow-falling pulses (t,?35 nsec) with differ ent rise times, for the same film as in Fig. 4. width could be measured. The threshold field for a rise time of 20 nsec indicated in Fig. 8 could therefore be only roughly estimated. No streaming at all could be resolved for pulse rise times essentially larger than 20 nsec. 'Figure 9 shows the influence of the film thickness on the wall streaming. A film containing sections with different film thicknesses has been used in order to make sure that differences in the film inhomogeneity, for instance, have no effect upon the result. As may be seen, there is no detectable influence of the film thick ness on the wall step width as long as the film is thicker than a certain threshold near 100 nm. No streaming at all has been detected in thinner films. The step width per pulse of the reversing lines moving along the walls has been determined by evaluat ing our motion picture frames. Figure 10 shows the dependence of the reversing line step width as well as of the wall step width per pulse on the pulse field for the nickel-iron film used for Figs. 1 and 2. There is a field region where the walls, but not any reversing lines, are displaced (i.e., only the processes 1 and 2 take place) . QUALITATIVE THEORETICAL DISCUSSION The observations reported in the previous para graphs, especially the existence of a film-thickness treshold and the significance of the pulse rise time, suggest an explanation on the basis of the gyromagnetic behavior of the Bloch wall spins. Gyromagnetic Behavior of Bloch Wall Spins In nickel-iron films more than about 100 nm thick, the Bloch wall type is more favorable than the Neel lil 1.0,--------tr----, ~ J.,Im ffi a. I b 0.5 ~ a. w >(/) -" ~ __ ~~~L-~_~_~. 00 0.5 NORMALIZED PULSE FIELD FIG. 6. Wall step width per pulse vs the normalized pulse field H /HK, for fast-rising (tr= 0.5 nsec) pulses with different pulse fall times, for the same film as in Fig. 4. +0.51tl=r~"..."",,,~::;..,~...;..;, ..... ~ .. ~m ~ o~--~~--~----~--~--~ 'J:' <I -0.5 \-------\----+----4......;------1 o 0.2 0.4 0.8 FIELD H APPLIED BETWEEN 1 AND 2 FIG. 7. Influence of a temporarily applied field 11 along the hard axis on the continuity of the wall streaming. The wall step width per pulse, AY2, measured with 0.23-HK pulses (tr=0.5 nsec, t,=35 nsec) after applying and removing the field H, is drawn positive (negative) if its sign is equal (opposite) to the sign of the wall step AYI observed before applying the field H. Same film as in Fig. 4. wall type for 1800 walls. This has been shown both experimentally and theoretically.I5-18 In the center of a 1800 Bloch wall the magnetization is oriented perpen dicular to the film plane as shown schematically in Fig. 11. If a pulse field along the hard axis is applied, the magnetization of the domains initially precesses around the field axis, thereby leaving the film plane, then it precesses around the demagnetizing field and finally reaches a stable direction within the film plane with the help of the intrinsic damping of the precession. This fast coherent rotation and the significance of 1.0,---------------, '3 0.8 w iL DISPLACEMENT REVERSAL POSSIBLE FOR EVERY PULSE ~ Q6F---,N.",E",E",-L ...!W.!::A~LL:.....!.!TR!!:A~N:='SI~TI~O!:'..N_---, -" ~ a. '" w N 0.4 ~ 02 ::;: '" o STREAMING z 00~-7--~1~0--*15-~20~~2=5-ns~30 PULSE RISE TIME FIG. 8. Range in the pulse field-vs-pulse rise time diagram with in which the wall streaming occurs in a 200-nm thick Ni-Fe film for slowly falling pulses. At the low-field boundary the wall step width is zero, d. Fig. 5. At the high-field boundary a Bloch-to N eel wall transition occurs. Because the Bloch wall polarity deter mines the direction of streaming, the direction mav be reversed after every pulse with a higher field strength (d. Fig. 7). 15 S. Methfessel, S. Middelhoek, and H. Thomas, IBM J. Res. Develop. 4,96 (1960); J. App!. Phys. 31,')46 (1960). 16 H. Thomas (private communication). 17 S. Middelhoek, Dissertation Amsterdam 1961, J. App!. Phvs. 34, 1054 (1963). . 18 E. Feldtkeller and E. Fuchs,'Z. Angew. Physik 18, 1 (1964). Downloaded 18 Apr 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsWALL STREAMING IN FERROMAGNETIC FILMS 440S FILM THICKNESS SYMBOL 90 nm A 125 nm 180 nm • ~ 1.0 220 nm 2IJm 265 nm D 310 nm + a: ~ e 0 0 0 0 :I: § a f-e8 aD .;. + ~ 0.5 +e e o .)" e a.. ~ + '" UJ Iii B + + ::J ;; ~ 0 0 0.1 Q.2 0.3 NORMALIZED Pl1.SE FIELD FIG. 9. Wall step width per pulse vs the normalized pulse field H/Hx, for fast-rising slowly falling field pulses (4=0.5 nsec 1,=35 nsec) applied to Ni-Fe films of different film thicknesses. The wall streaming is only observed in films where 1800 Bloch walls are stable. the damping within it has been discussed in many papers.19-21 Within a wall the magnetization also precesses around the field axis, but no additional demagnetizing field perpendicular to the film plane is thereby estab lished, because there are locations where the magnet ization component normal to the film plane is increased, and others where it is decreased. The precessional rotation is indicated in Fig. 11 by thin arrows. As may be seen from the figure, the effect of this precession is to displace the wall center. Our qualitative observa tions (a), (c), (d), and (g) may be well understood on this basis. A quantitative theoretical estimation of the wall step width on this basis will be given in a separate paragraph. ~ :5 0.. 30r------------:"'::---, ffi tr= 0.5 ns ifi a.. lJm ai", "'", tf; 35 ns 5 ~~ ~ 20 :11r fu ffi~ ~ >'" ~ 1i!f5 z ~~ ::J 10 -' ~ ~ z Vi a: i!;! I '" ______________ WALL STEP wtDTH a: 00~--~~~~--~O~4----M~~ NORMALIZED PULSE FIELD FIG. 10. Reversing-line (Neel line) step width per pulse vs the normalized pulse field H/Hx for the film presented in Fig.1. 1,=0.5 nsec, 1,=35 nsec. The wall step width in the same film for the same rise and fall times is shown for comparison. 19 R. Kikuchi, J. Appl. Phys. 27,1352 (1956). 20 D. O. Smith, J. Appl. Phys. 29, 264 (1958). 21 E. Feldtkeller, Z. Angew_ Physik 12, 257 (1960). ~ <,:'l </yC!Y ~ 9.~ 'l' -EASY AXIS- FIG. 11. Orientation of the local magnetization vectors (thick white arrows) in a Bloch wall, and their precessional rotation (thin black arrows) around a field applied perpendicular to the wall plane. N eel Line Motion A 1800 domain wall in nickel-iron films thicker than about 100 nm may consist, like in bulk material, of Bloch wall sections separated by Neellines.18,22,23 The Bloch wall sections are alternatingly magnetized in their centers along one of the two directions normal to the film. The structure of a Neel line is drawn sche matically in Fig. 12(a) _ Because the central mag netization of a Neelline lies in the film plane, its stray field causes a characteristic curvature of the local magnetization direction in its neighborhood, containing a b d / initial , , final -' FIG_ 12. Schematic diagram of a Neelline separating differently magnetized Bloch wall sections. To avoid magnetic space charges the Neelline carries a cross wall (a), that must be reoriented if a field is applied perpendicular to the wall (b). If the magnetiza tion in the domains rotates considerably faster than the cross wall is reoriented, the field of space charges (c) may cause a dis placement of the Neelline(d). 22 H. J. Williams and M. Goertz, J. App!. Phys. 23, 316 (1952). 23 Formerly these lines were sometimes called Bloch lines like those separating different Neel wall sections. Downloaded 18 Apr 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions4406 K. U. STEIN AND E. FELDTKELLER x-FIG. 13. Spherical coordinates used in the theoretical treat- ~~ m~t of the wall be ~<,; havlOr. ~ty a short cross-wall analogous to the well-known cross ties in the cross-tie walls in thinner films. If a field along the hard axis is applied, the mag netization in the domains rotates toward the field direction. In order to remain free of magnetic poles, the cross walls have to rearrange themselves to a new orientation, Fig. 12 (b). Because this reorientation corresponds to a wall motion of the cross-walls, the cross-walls may not be able to follow the rotation of the magnetization in the domains immediately and may thereby get magnetically charged, Fig. 12(c). The field of these charges may displace the Neel line, thereby making the cross-wall reorientation easier, since now the mean cross-wall step necessary for a correct reorientation is smaller, Fig. 12 (d). This process is to be expected only if fast-rising pulses are applied, because during a slow field change, the cross walls have time enough to reorient without displacing the Neellines. Therefore, the Neel-line dis placement will be larger during the fast rises than during the slow falls of our pulses. The expected sign of the net displacement in this case is exactly that observed and agrees with the rule indicated in Fig. 3. Thus, our qualitative observations (b), (c), (d), and ( e) maybe explained if the reversing lines are assumed to be N eel lines. For every film thickness, a certain wall angle, and hence a certain field along the hard axis, exists beyond which only Neel walls are stable.s,17,24--26 If this field strength is exceeded, the walls will therefore completely lose their former Bloch-wall polarity and the location of any Neel lines. Thus, our observation (h) may be understood. It may be expected that the step width per pulse of the Neel lines strongly increases if the pulse-field strength approaches this critical field, and that the sign of wall displacement is random for every pulse (yielding no visible wall streaming) if the pulse field exceeds the Bloch-to-Neel wall transition field (Fig. 8). The field of about 0.5 HK reported by Middelhoek and WildS to be necessary for a Bloch-to Neel wall transition near 200-nm film thickness agrees well with the threshold value of 0.57 HK indicated in Fig.7. Our observation (f) concerning the dependence on the 24 S. Middelhoek, J. AppJ. Phys. 34, 1054 (1963), 25 E. ]. Torok, A, L. Olson, and H. N, Oredson, ]. AppJ. Phys. 36, 1394 (1965). 2<1 E. Feldtkeller, in Basic Problems in Thin Film Physics, Pro ceedings of the International Symposium Clausthal GOttingen 1965 (Vandenhoeck & Ruprecht, G6ttingen, 1966), p. 451. rise time may be understood on the basis of the quanti tative theory presented in the next paragraph. QUANTITATIVE THEORY OF THE WALL STEP WIDTH In this paragraph the local magnetization direction will be described by the spherical coordinates rp, t/!, where t/!(y, t) indicates the angle between the local magnetization and the wall plane, and cp(y, t) indicates the angle between the projection of the magnetization into the wall plane and the film plane (i.e., rp =0, 7r in the domains and rp=7r/2 in the wall center), Fig. 13. The positional coordinate y runs normal to the wall plane, and an index m indicates the instantaneous wall center. For a quantitative discussion of the wall streaming, the Landau-Lifshitz phenomenological equation, de scribing the gyromagnetical behavior of the magnet ization vector, has to be applied to every element of the wall. An exact calculation for the whole Bloch wall is, however, too complicated to be done with a reason able effort, since even an equilibrium computation of a not-180° wall in uniaxial films is still missing, and only rough estimations are available.I 6,17,24 On the other hand, a calculation by regarding only the wall center at any moment has proven to yield sufficient results without presuming very arbitrary simplifications that would be necessary for a calculation of the whole wall, and is thus more valuable for our problem. The only simplification needed is, that the equations rp=7r/2, Ot/Ijay=y/ =0, and for y=ym, ( 1) shall be valid in the wall center, i.e., that these three wall center definitions define an identical wall center Ym(t) during the process.27 This assumption may well be approximately valid, and is much less arbitrary than assuming a symmetry of the whole wall during the process, or even a special wall structure. The Gilbert modification of the Landau-Lifshitz equation has been used in its most general form (see Refs. 20, 21) : aM/at = (-Y/JLoM)[M xF]+(a/M)[M x (aM/at)], (2) where F is a generalized force with the components F",= -fiE/at/! and F",= -(l/cost/!)flE/flrp, and 'Y= -2.2X10 7 sec1(A/cm)-1 is the gyromagnetic ratio, and a is the phenomenological damping constant. Since our configuration contains variations only in one '}{T For the case of a field applied parallel to a Bloch or Neel wall, the same assumption is sufficient for deriving the 20w-wall mobility dv/ dH = a I 'Y I /20t8w in an analogous manner, and to show that this formula is valid independently of the special wall structures presumed in previous calculations28,119 if a wall thickness defini tion [Eq. (10) of the present paper] based on the spherical angle element, is used. Downloaded 18 Apr 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsWALL STREAMING IN FERROMAGNETIC FILMS 4407 dimension Y (i.e., no dependence on x and z is con tained), the variational derivatives are oE/o1/;=aE/a..{t-d(aE/a1/;')/dy, and oE/ocp=aE/acp-d(aE/acp')/dy. (3) By using the variational instead of the partial deriva tives of the free energy E(cp, 1/;, cp', 1/;'), not only the magnetostatic and anisotropy contributions, but also exchange contributions to the local torque, are regarded. As has been shown,21 Eq. (2) and its components may be written in an explicit form: a..{t/at=[ -'Y/,uoM(1-a2)] X [(1/cos1/;) (oE/ocp) -a (oE/Oif;) ] and acp/at= ['Y / ,uoM (1-a2) cos1/;] X [(oE/o1/;) + (a/cos1/;)(oE/ocp)]. (4) In the following, a homogeneous film, i.e., a film with wall coercivity Hc=O, will be regarded at first (Eqs. 5-12), and a correction necessary for regarding a real film, will be made afterwards. Ideal Film For a homogeneous film, our assumptions (1) lead to the following equations for the instantaneous wall center: (5) because cp=7r/2 is a symmetry plane for the anisotropy and for all fields, (which is equivalent to the usual a = 20w/O'm where only a one-dimensional rotation is allowed), the velocity of the wall center is (11) If the wall thickness a is assumed to depend only on 1/;m (and not, e.g., onfm), Eq. (11) may be integrated with the result (12) This means, in an ideal film, that the wall step does not depend upon the pulse rise time nor on the speed of rotation, and is perfectly cancelled by the back rotation at the end of the field pulse. Thus, this micromagnetic calculation, valid for ideal homogeneous films, is not sufficient for discussing the wall streaming. As for understanding the Barkhausen jumps and the wall creeping, a consideration of the film inhomogeneities is necessary in addition. Real film In wall-mobility calculations,28 the film inhomoge neities have usually been regarded by taking into account a local force acting on the wall, or by postu lating a local effective field He(Y) corresponding to this force and oriented parallel to the wall. This yields (1/cosif;m) (oE/oCP)m= -}LoMHe (13) for the wall center, instead of Eqs. (5) and (6). Equa tion (9) has then to be replaced by (acp/at)m= (fm-I'Y I H.)/a cos1/;m, (14) [d(aE/acp') /dY]m=O, because aE/acp' =2Acp' cos2if;, and (6) (where a2«1 has been assumed), and Eq. (11) by (a..{t / at) m = d1/;m/ dt=fm' (7) This means that and or by combining these equations: This statement is equivalent to the statement derived formerly21 that the rotation proceeds under a constant angle arctan a against the constant-energy lines, which is a consequence of Eq. (2) and of the fact that I [MxaM/at]1 M I = I aM/at I for a rotation. If the wall thickness a is defined in spherical coordi nates by a = [(cpyco-f-o;> -CPY~OJ)/ (acp/aY)m cos1/;m] = (7r /cp'm cos1/;m), (10) (15) In the known theoretical treatments28,29 of the wall mobility, the simple, though not very realistic assump tion, He(Y) =Ho with the effective field always opposing the wall motion, has been used. The experimental results on the wall mobility may be fitted by the result of this assumption within the experimental error. The same assumption may be used here. The wall displacement in an inhomogeneous film resulting from this simple assumption is ~ym= (7ra)-ljfb ad1/;m=F L'Y I Holtbadt for fmZO, fa 7ra ta (16) where the integration intervals are the intervals within which I fm I > I 'Y I Ho is valid. In order to discuss this result, the dependence of 1/;m on t must be regarded. After a field change, 1/;m will 28 J. K. Galt, Phys. Rev. 85, 664 (1952). 29 K. U. Stein, Dissertation, Stuttgart 1965. Downloaded 18 Apr 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions4408 K. U. STEIN AND E. FELDTKELLER approach a new equilibrium valueY;me(H). Torok, et al.25 have shown that dY;me/dh is larger than one, but in the order of unity, as long as the wall has not become a Neel wall. h=H/HK is the normalized field along the hard axis. The time needed for the change in Y;m will be larger than the rise or fall time of the field. Therefore, I -.1m I will be of the sa1lle order or smaller than I1h/tr or I1h/tf, respectively, if I1h is the normalized pulse-field amplitude. Because the pulse rise and the pulse fall may con tribute to the resulting step width per pulse, the follow ing cases may be regarded: Slow-rising, slow-falling pulses: If I1h/ tr« I 'Y I Ho and I1h/t,« I 'Y I Ho, I -.1m I will be always smaller than I 'Y I Ho and the wall will not be displaced at all. Fast-rising, slow-falling pulses: If I1h/tr» I 'Y I Ho and I1h/t,« I 'Y I Ho, only the pulse rise will lead to a wall displacement. We may then assume for the limits in Eq. (16) Y;b-Y;a~(dY;me/dh)l1h and tb-ta~T+tr, where T is the switching time contributing to the wall dis placement in the case of tr=O. After introducing an averaged wall thickness (a) = J adY;m/ J dY;m~ J adt/ J dt, the resulting wall step width per pulse is I1Ym = (a )/Tra) [ (dt/;me/ dh )l1h-I 'Y I Ho( T+tr)]. (17) There is a good qualitative agreement between Eq. (17) and our experimental observations presented in Fig. 5. From this figure, the values T=5 nsec, Hill (dY'me/dh )=0.6 A/cm, and (a )/a=8j.1m may be deduced for establishing a quantitative agreement. These values are quite reasonable for the following reasons: The coercivity of this film is He=0.7 A/cm which should be equal to Ho. The switching time T contains the duration of rotation in the adjacent domains which has been found to lie between 1.5 and 2 nsec for H <HK, and the unknown duration of the Y;m change within the wall in the wall stray field created by the'rotation in the domains. An experimental value of a Bloch wall thickness is a= (0.15±0.1) j.lm deter mined by comparing the width of black and white wall images in Lorentz micrographs taken by Fuchs from a nickel-iron film 0.12 j.lm thick.30 Damping constants of a= 0.01, for a O.l-j.lm thick film and a~0,08 for a 0.27-j.lm thick film with He/ HK= 0.25 have been deter mined from the speed of coherent rotation.3l Further more, the a/a value may be compared with wall-mobility 30 C. E. Patton and F, B, Humphrey, l Appl. Phys. 38, 1998 (1967) . 31 K_ U. Stein, Z_ Angew, Physik 18, 528 (1965). measurements, since the mobility is m= I 'Y I a/Tra for a 1800 wall. The a/a values resulting from mobility measurements by different authors lo,32,33 lie between 6 j.lm and 21 j.lm for 200-nm thick films. As may be seen, the value derived from our wall-step measurements lies well between these values. All experimental wall thickness values do not, how ever, agree with the theoretical values computed with the exchange constant derived from spin-wave reso nance experiments. The reason for this disagreement is not yet known. According to the mobility measurements men tioned,1O,33 the mobility, and hence, a/a increases slightly with increasing film thickness. On the other hand, (dY;me! dh) decreases slightly with increasing film thick ness.25 Thus it may be understood why the product (dY;me! dh )a/ a and hence, the step width for fast-rising pulses does not depend remarkably on the film thick ness, as has been shown in Fig. 9. Fast-rising fast-falling pulses: If l1k/tr and iJ.h/t,» I 'Y I Ho, the pulse rise and fall will contribute, and the resulting wall step width per pulse will be the difference of Eq. (15) and the corresponding expression for the pulse fall. If the integration of Eq. (16) would lead to Eq. (17) with the same average wall thickness and with the same contributing intervals for the rise and fall of a pulse, the resulting step width should be predictable from the step widths observed with slowly falling pulses with the corresponding rise times. This predic tion is indicated by dotted lines in Fig. 6. The reason for the disagreement between prediction and observa tion is not yet clear. A more careful integration of Eq. (16) might be necessary for calculating the small differences between the displacements due to the rise and fall of these pulses. CONCLUSION It must be expected that wall streaming reduces the disturbed field in magnetic-film memories with films thicker than 100 nm (e.g., plated wires) as soon as fast-rising (t. ;520 nsec) pulses are applied, and that the disturb field depends on the pulse rise and fall time in a manner corresponding to the observations reported in this paper. 32 E. N. Il'icheva and I. S. Kolotov, Izv. Akad. Nauk SSSR, Ser. Fiz. 29, 552 (1965) .[Engl. transl.: Bull. Acad. Sci. USSR, Phys. Ser. 29, 559 (1965)]. 33 C. E. Patton and F. B. Humphrey, l Appl. Phys. 37, 4269 (1966) . Downloaded 18 Apr 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
1.3109243.pdf	Magnetization reversal in enclosed composite pattern media structure Chi-Keong Goh, Zhi-min Yuan, and Bo Liu Citation: Journal of Applied Physics 105, 083920 (2009); doi: 10.1063/1.3109243 View online: http://dx.doi.org/10.1063/1.3109243 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/105/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Microwave assisted magnetization reversal in composite media Appl. Phys. Lett. 94, 202509 (2009); 10.1063/1.3133354 Microwave-assisted magnetization reversal and multilevel recording in composite media J. Appl. Phys. 105, 07B909 (2009); 10.1063/1.3076140 Effects of laminated soft layer on magnetization reversal of exchange coupled composite media J. Appl. Phys. 105, 07B729 (2009); 10.1063/1.3075557 High frequency switching in bit-patterned media: A method to overcome synchronization issue Appl. Phys. Lett. 92, 012510 (2008); 10.1063/1.2831692 Thermally induced magnetization reversal in antiferromagnetically coupled media J. Appl. Phys. 93, 7405 (2003); 10.1063/1.1558237 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 120.117.138.77 On: Mon, 22 Dec 2014 01:34:15Magnetization reversal in enclosed composite pattern media structure Chi-Keong Goh,a/H20850Zhi-min Yuan, and Bo Liu Data Storage Institute, (A *STAR) Agency for Science, Technology, and Research, 5 Engineering Drive 1, Singapore 117608, Singapore /H20849Received 4 December 2008; accepted 26 February 2009; published online 27 April 2009 /H20850 The bit patterned media use one single domain magnetic island to record a bit. It can effectively push the superparamagnetic limit to higher recording densities. In this paper, we present andinvestigate a composite patterned media structure comprising of soft layer enclosed magneticislands to signiﬁcantly improve the writing capability of conventional writer. Systematicmicromagnetic simulation studies reveal that the proposed structure has a different domain wallpropagation mechanism and has less stringent requirement on the exchange-coupling strength ascompared to conventional composite structures. © 2009 American Institute of Physics . /H20851DOI: 10.1063/1.3109243 /H20852 I. INTRODUCTION The signal-to-noise ratio in present magnetic recording media is determined by the number of grains in each data bit.In order to increase areal density, the size of these grainsmust be reduced to maintain a minimum number of grainsper bit. However, thermal magnetic instability, which ariseswhen grain size is too small, can lead to potential loss ofinformation. Thermal stability can be improved by matchingthe reduction in grain dimension with a corresponding in-crease in the magnetic anisotropy. This places a tremendouschallenge on write ﬁeld requirement. Current head ﬁeld islimited by the highest magnetic moment available. The alternative way to circumvent thermal instability and achieve recording densities beyond 1 Tbit /in 2is to use bit patterned media /H20849BPM /H20850.1In BPM, the grains within each magnetic island are strongly coupled and, hence, behave likea large single magnetic domain. As one island records onebit, the areal density is increased signiﬁcantly. This workexamines the effect of exchange-coupling on patterned me-dia and presents an enclosed composite patterned media/H20849ECPM /H20850structure. All analyses presented in this paper are based on micromagnetic simulations. II. COMPOSITE PATTERNED MEDIA „CPM … The straightforward approach of extending exchange- coupling to BPM is to adopt the conventional two layer hard-soft structure 2,3as illustrated in Fig. 1/H20849a/H20850. This structure is denoted as CPM in this paper. However, the requirement of athick soft layer for such a composite structure poses morefabrication challenges and large head keeper spacing, whichreduce the head writing capability. It also introduces issuessuch as large demagnetization ﬁelds 4and strong dependence on the exchange-coupling strength. Here, we present and in-vestigate an alternative structure, which is illustrated in Fig.1/H20849b/H20850. Instead of the two layer structure adopted in CPM, the proposed structure has the hard layer magnetically coupledto the soft layer that covers it from the top and sides. Thebasic motivation is to exploit the potentially strongerexchange-coupling effect due to a larger surface of interac- tion between the hard and soft magnetic layers. In this paper, magnetic reversal is simulated using mi- cromagnetic modeling where the gyromagnetic motion ofmagnetization is governed by the reduced Landau–Lifshitz–Gilbert equation. The effective ﬁeld includes the exchangeﬁeld, the anisotropy ﬁeld that is oriented in the /H11006zdirection, the demagnetization ﬁeld, and the external write ﬁeld. Theangle between the easy axis of hard region and external writeﬁeld is 5°. The reduced damping constant is set as 0.1. In all simulations, the CPM structure is modeled by an array of 1 nm cubic cells. Each hard magnetic island isformed by 16 cubic cells and has a dimension of 4 /H110034 /H110033n m 3. The dimension of the soft layer in CPM is set as 4/H110034/H110036n m3. In ECPM, the hard magnetic island is sur- rounded by a layer of cubic cells. The soft layers for bothCPM and ECPM have the same physical and magnetic vol-ume. Unless otherwise stated, the average saturation magne-tization is maintained at 300 emu/cc. The K uof the soft and hard layer is 100 erg/cc and 20 Merg/cc, respectively.Exchange-coupling constant of 9 Merg/cm is used, which isfeasible for fabrication. III. RESULTS AND ANALYSIS The comparison between magnetic behavior of the con- ventional patterned media, CPM, and ECPM is shown in Fig.2. As expected, both CPM have a much smaller coercivity as compared to the conventional patterned media. This reduc-tion in coercivity is due to the exchange ﬁeld contribution tothe effective ﬁeld. However, it is observed that CPM has twoswitching states. This is a phenomenon that is associatedwith insufﬁcient exchange-coupling between the soft andhard regions. In the case of ECPM, there is a single switch-ing region and the reduction in coercivity is much more sig-niﬁcant. Note that similar magnetic switching behavior hasbeen discovered due to the presence of cavities in magneticgrains. 5 Figure 3shows the coercivity as a function of ﬁeld angle for both CPM and ECPM. There is a signiﬁcant reduction inthe coercivity of CPM as the ﬁeld angle is increased. For a/H20850Electronic mail: meitikor@yahoo.com.sg.JOURNAL OF APPLIED PHYSICS 105, 083920 /H208492009 /H20850 0021-8979/2009/105 /H208498/H20850/083920/3/$25.00 © 2009 American Institute of Physics 105 , 083920-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 120.117.138.77 On: Mon, 22 Dec 2014 01:34:15example, the coercivity at 10° is almost 0.1 HKlower than the coercivity at 5°. On the other hand, the coercivity of ECPMremains relatively constant. This is an important characterbecause ﬁeld angle-induced coercivity variation is a sourceof recording noise.Figure 4shows the switching time of CPM and ECPM as a function of the switching ﬁeld. Switching time is deﬁnedas the time required for the magnetization to cross the originand to reach a magnitude of 0.9 M shere. The switching ﬁeld is incremented in steps of 0.05 HKfrom 0 Oe, and no points are plotted in cases where no switching is achieved. ECPMstarts to switch from a ﬁeld of 0.05 H Konward, and the switching time reduces by a factor of three from 0.12 to 0.04ns when the switching ﬁeld is increased to 0.2 H K. As ex- pected from Fig. 2, a signiﬁcant higher minimum switching ﬁeld of 0.3 HKis observed for CPM. Switching times are slightly lower for CPM at 0.95 HKand 1 HK, but there is no advantage gained in writability at such high ﬁelds. In all simulations, both CPM and ECPM have the same magnetic volume ratio and exchange-coupling strength. Theresults have clearly indicated that ECPM has a lowerexchange-coupling requirement. This can be attributed to thehigher exchange ﬁeld contribution to the effective ﬁeld whenthe interface area between the soft and hard regions is in-creased. The dependence of coercivity on the area of hard-soft layer interface is shown in Fig. 5, and coercivity is clearly an inverse function of interface area. This is becausethe increase in soft layer volume will also increase the Zee-man energy during reversal. 6 Another important consequence of the ECPM structure is the way domain wall propagates into the hard magneticregion. Figure 6illustrates the process of domain wall propa- gation from the soft region to the hard region for both CPMand ECPM. While the domain wall penetrates from the softSoft magnetic layerHard magnetic layer Soft magnetic layer Hard magnetic layer (a) (b) FIG. 1. /H20849Color online /H20850The conventional CPM is illustrated in /H20849a/H20850, while the proposed ECPM structure is shown in /H20849b/H20850. −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1−0.500.51 Applied field ( Ha/Hk)Magnetization ( M/Ms) FIG. 2. /H20849Color online /H20850Hysteresis loop of patterned media /H20849solid /H20850, CPM /H20849dashed /H20850, and the ECPM /H20849dash-dotted /H20850.5 10 15 20 25 3000.050.10.150.20.250.30.350.4Switching field ( Hs/Hk) Applied field an gle (de gree) FIG. 3. /H20849Color online /H20850Coercivity as a function of ﬁeld angle for CPM /H20849/H17040/H20850 and ECPM /H20849/H17034/H20850. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.020.040.060.080.10.12 Normalized A pplied FieldSwitching Time (ns) FIG. 4. /H20849Color online /H20850Switching time as a function of ﬁeld strength for CPM /H20849/H17034/H20850and ECPM /H20849/H17005/H20850.083920-2 Goh, Yuan, and Liu J. Appl. Phys. 105 , 083920 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 120.117.138.77 On: Mon, 22 Dec 2014 01:34:15region into the hard region in one direction in CPM, the domain wall in the case of ECPM penetrates into the hardregion from three different directions. Therefore, this mecha-nism of domain wall penetration provides for a more effec-tive assisted switching. Recording simulations are performed fo r a 6 bit array of conventional patterned media and ECPM. The same mediaparameters are assumed as before, and the writing ﬁelds forthe patterned media and ECPM are set as 1 H Kand 0.5 HK, respectively. Interactions between the magnetic islands dueto the overlap in the covering soft layer are modeled by a 2nm thick soft layer between the ECPM islands as shown in Fig. 1/H20849b/H20850. The bit pattern of /H208531,1,/H110021,1,/H110021,/H110021/H20854is written and the readback signal is plotted in Fig. 7. Transitions between similar patterns are less distinctive in ECPM. The differencein readback signal for the two media structure is due to themagnetization of the soft layer between the magnetic islandsin ECPM. IV. CONCLUSION A CPM structure is proposed to improve switching ﬁeld reduction for high density recording in patterned media. Weobserve that lower switching ﬁelds can be achieved with thenew structure, and it has lower exchange-coupling strengthrequirement as compared to the conventional stack compos-ite structure. In addition, the ECPM structure can use lesshead keeper spacing than CPM. 1E. Chunsheng, D. Smith, J. Wolfe, D. Weller, S. Khizroev, and D. Litvi- nov, J. Appl. Phys. 98, 024505 /H208492005 /H20850. 2R. H. Victora and X. Shen, IEEE Trans. Magn. 41, 537 /H208492005 /H20850. 3K.-Z. Gao, J. Fernandez de Castro, and H. N. Bertram, IEEE Trans. Magn. 41, 4236 /H208492005 /H20850. 4Y . Isowaki, T. Maeda, and A. Kikitsu, Dig. Perpendicular Mag. Rec. Conf., 214 /H208492007 /H20850. 5J.-G. Zhu, H. Yuan, S. Park, T. Nuhfer, and D. E. Laughlin, IEEE Trans. Magn. 45,9 1 1 /H208492009 /H20850. 6K.-Z. Gao and J. Fernandez de Castro, J. Appl. Phys. 99, 08K503 /H208492006 /H20850.0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.040.060.080.10.120.140.160.18Switching Field ( Hs/Hk) Normalized surface area between soft and hard la yer FIG. 5. /H20849Color online /H20850Coercivity as a function of interface area for ECPM. /CID1CPM ECPM FIG. 6. /H20849Color online /H20850Illustration of domain wall propagation in CPM and ECPM.0 20 40 60 80 100 120−1−0.500.51 Down Track (nm )Normalized Readback Signal FIG. 7. /H20849Color online /H20850Readback signal for the bit pattern of /H208531,1,/H110021,1, /H110021,/H110021/H20854for conventional patterned media /H20849--/H20850and ECPM /H20849–/H20850.083920-3 Goh, Yuan, and Liu J. Appl. Phys. 105 , 083920 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 120.117.138.77 On: Mon, 22 Dec 2014 01:34:15
1.5094763.pdf	Force on the lips of a trumpet player N. Giordano Citation: The Journal of the Acoustical Society of America 145, 1521 (2019); doi: 10.1121/1.5094763 View online: https://doi.org/10.1121/1.5094763 View Table of Contents: https://asa.scitation.org/toc/jas/145/3 Published by the Acoustical Society of America ARTICLES YOU MAY BE INTERESTED IN Extended source models for wind turbine noise propagation The Journal of the Acoustical Society of America 145, 1363 (2019); https://doi.org/10.1121/1.5093307 Mice can learn phonetic categories The Journal of the Acoustical Society of America 145, 1168 (2019); https://doi.org/10.1121/1.5091776 Volumetric reconstruction of acoustic energy flows in a reverberation room The Journal of the Acoustical Society of America 145, EL203 (2019); https://doi.org/10.1121/1.5092820 The percept of reverberation is not affected by visual room impression in virtual environments The Journal of the Acoustical Society of America 145, EL229 (2019); https://doi.org/10.1121/1.5093642 Numerical investigation on turbulent oscillatory flow through a jet pump The Journal of the Acoustical Society of America 145, 1417 (2019); https://doi.org/10.1121/1.5094346 Task-dependence of articulator synergies The Journal of the Acoustical Society of America 145, 1504 (2019); https://doi.org/10.1121/1.5093538Force on the lips of a trumpet player N.Giordanoa) Department of Physics, Auburn University, Auburn, Alabama 36849, USA (Received 17 July 2018; revised 25 February 2019; accepted 3 March 2019; published online 25 March 2019) When modeling a brass instrument such as the trumpet, an estimate of the pressure at the player’s lips is essential, since the resulting force drives the oscillations of the lips which are needed to pro- duce a musical tone. In most work to date, the calculation of the force on the lips has relied on val- ues of the pressure derived from the Bernoulli equation, even though that relation assumes steadyﬂow in contrast to the pulsating ﬂow caused by vibrations of the lips. This paper uses a quantitative application of the Navier-Stokes equations to calculate the ﬂow through a model of vibrating lips attached to a toy model of the trumpet. The results are used to explore when the Bernoulli equationcan and cannot be used. The Bernoulli equation is found to fail badly during signiﬁcant portions of each oscillation cycle of the lips. The reasons for this breakdown are elucidated. VC2019 Acoustical Society of America .https://doi.org/10.1121/1.5094763 [TRM] Pages: 1521–1528 I. INTRODUCTION Modeling wind instruments such as the trumpet is now possible using the fundamental equations of ﬂuid dynamics,the Navier-Stokes (NS) equations. While results for instru-ment geometries that are based accurately on real instru-ments are not yet available, results for a trumpet-likeinstrument were recently reported by the present author. 1 Those simulations included a simpliﬁed dynamical model ofthe lips, and yielded detailed results for how air ﬂowsthrough the mouthpiece of a trumpet and the lips of theplayer while a note is being played. Essentially all previous discussions of the pressure in the mouthpiece and lip region of the trumpet (e.g., Refs.2–11) make use of the Bernoulli equation. Those works, including those focused on modeling, have yielded many useful insights by using the Bernoulli equation to estimate the pressure between the lips, which is a major contributor tothe force that drives the lips and locks their motion into syn-chrony with the resonant modes of the instrument. However,the classic Bernoulli equation is derived using assumptionsthat are not strictly correct in the mouthpiece of a windinstrument. A primary purpose of the present paper is toevaluate the accuracy of the using the Bernoulli equation inthis case so as to understand when it can and cannot be usedreliably. The starting point for the analysis in the present paper is a ﬁrst principles calculation, based on the NS equations, ofthe ﬂow through the lips of a trumpet player along with thepressure in the vicinity of the lips and the associated forcethat drives the lip motion. The Bernoulli equation is anapproximation that can be derived from the NS equationswith several assumptions, including the assumptions that the effects of viscosity and the acceleration of the ﬂuid can both be ignored. In order to test what one might term the“Bernoulli approximation” we will use our NS results toestimate the size and importance of the contributions of vis- cosity and ﬂuid acceleration, and thereby obtain an estimatefor the force on the lips subject to the approximations inher- ent in the Bernoulli approximation. The lip force estimated using the Bernoulli equation to relate the ﬂow velocity to thepressure is then compared to the force found using the fullsolution of the NS equations. We will see that this Bernoulli result for the force can, for the simple instrument geometry we have studied, differ from the actual force by as much asan order of magnitude or even more. Besides revealing short-comings in previous treatments of this class of instruments, our results may also help in the development of improved approaches to estimating the lip force without the need toresort to a full, ﬁrst-principles calculation with the NS equa-tions. In a way, one goal of our work is to use the NS results to gain an understanding of how nonsteady and viscous ﬂow effects contribute to the forces on the lips of a brass player. II. THE MODEL A direct numerical simulation was used to solve the compressible NS equations for the ﬂow of air through a three-dimensional model of the trumpet with an explicitﬁnite different algorithm using the ﬂuid properties (viscosity,etc.) of air at room temperature. The numerical algorithm and other details of the calculation are described elsewhere. 1 All of the results shown and analyzed in this paper were obtained with the conical bore model referred to as model #3in Ref. 1(see also Figs. 1and2of that paper). The lowest resonant mode of the instrument was near 1000 Hz, and all of the results reported here were obtained with the naturalfrequency of the lips equal to this frequency. As discussed inRef. 1, the instrument geometry studied here is smaller than a real trumpet, so we should not expect quantitative agree- ment for quantities such as the blowing pressure. The Adachi-Sato lip model was used to describe the lip motion in which the channel between two ﬂexible lips is rectangular (and three dimensional). 5With that model the lips are able to swing along the direction of net ﬂow whilea)Electronic mail: njg0003@auburn.edu J. Acoust. Soc. Am. 145(3), March 2019 VC2019 Acoustical Society of America 1521 0001-4966/2019/145(3)/1521/8/$30.00 simultaneously stretching or compressing in the perpendicu- lar direction thereby narrowing or widening the channel between the lips. In the classiﬁcation of lip models this istermed a one mass-two degree of freedom model. The swinging-stretching motion of the lips during an oscillation cycle is shown schematically in Fig. 1. The region upstream from the lips (the mouth) is much larger in volume than the lip channel and serves to buffer the ﬂow and pressure. For our particular model and simulation conditions, the variationof the pressure well upstream in the mouth is typically about 15%, which is comparable to that found experimentally. 12 The magnitude of the blowing pressure agrees with those experiments to within about a factor of 2, which also seems acceptable given the different model geometry. It is also worth noting that the Adachi-Sato lip model is just one of afamily of lip models, with other models containing addi- tional degrees of freedom, additional (distributed) masses, and more realistic shapes including rounded edges. 13It seems unlikely that the incorporation of more realistic lip models would change any of our qualitative conclusions or even have much effect quantitatively, although vena con-tracta effects near the sharp corners of the Adachi-Sato lip models, which do not play a signiﬁcant role in our analysis, would possibly be affected somewhat.While the lip motion is driven by the pressure associated with the ﬂow, the lips also experience several other forces. 1,5 These include a harmonic restoring force that tends to pull each lip back to its undisplaced position and to itsunstretched-uncompressed dimensions, a damping forcewith a magnitude proportional to the lip velocity, a nonlinear contact force that is signiﬁcant when the two lips touch, and a force that prevents the lips from compressing excessivelyalong the transverse direction (to account for the inevitablenonlinearities when the lip motion is very large). Figures 2and 3show the results of a NS simulation; these results are similar to the behavior presented previ- ously, 1showing maps of the ﬂow speed (Fig. 2) and air den- sity (Fig. 3) on a plane that cuts through the instrument and lips. Results are shown at six different instances during oneperiod of the lip oscillations. The colors indicate the magni- tude of the ﬂow speed or the value of the density relative to the background density. In part (a) of Figs. 2and3the lip channel is open and the lips have swung well forward alongthe ﬂow direction, similar to that shown schematically inpart 1 of Fig. 1. In part (b) of Figs. 2and3the lips have swung even further to the right (downstream), with each lip compressed even more so that the width of the lip channel is at its maximum, as in part 2 of Fig. 1. In part (c) of Figs. 2 and3the lips have begun to swing back and the channel has narrowed (compare with part 3 of Fig. 1), while in (d) the channel is even narrower. Note also that in (d) the lips have swung very slightly to the left, upstream against the ﬂow; such behavior is found when the mouth pressure is largeresulting in a large the amplitude for the lip motion. In (e)the lips are at their closest approach and the ﬂow is nearlyblocked (compare with part 4 of Fig. 1). In (f) the lips have swung forward and the channel has opened slightly, and air is just beginning to ﬂow into the channel. The lips then move to the position in (a) and the cycle repeats. The behavior seen in Figs. 2and3is also in good quali- tative agreement with experiments (e.g., Ref. 12) and model- ing (e.g., Ref. 4). As noted above, our instrument geometry is not identical to that of a real instrument, but we will arguethat certain key aspects of our results apply in general. III. ANALYSIS STRATEGY The simulations that yielded Figs. 2and3gave quantita- tive results for the ﬂow velocity and density throughout the mouth, lip channel, and in the bore downstream from the lips, as well as the net force on the lips along both the paral-lel and perpendicular directions, all as a function of time. Togain insight from those results and understand how the pres-sure between the lips varies as the lips oscillate, it is useful to recall the Bernoulli equation P 1þ1 2qu2 1¼P2þ1 2qu2 2; (1) where qis the ﬂuid density, the subscripts refer to the pres- sure Pand ﬂow speed uat two different locations in the ﬂuid, and we assume that gravitational forces can beneglected. There are several ways to derive Eq. (1); in ele- mentary texts it is commonly derived using work-energyFIG. 1. Schematic motion of the lips during one oscillation cycle. The lips are the black parallelograms and are attached along one edge to rigid por-tions of the mouth (shown as shaded rectangles above and below the lips). The net air ﬂow between the lips is from left to right, parallel to the xdirec- tion, with approximate streamlines shown. The lips swing back and forth along the direction of net ﬂow in response to the pressures upstream (to the left, in the mouth) and downstream (to the right, in the mouthpiece). At that same time, the lips stretch and compress in the perpendicular direction ( y)i n response to the pressure in the channel between the lips. This motion resultsin the narrowing and widening of the channel between lips; that is, as each lip compresses and becomes “thinner” the channel becomes wider, while when each lip stretches they become “thicker” leading to a narrower ﬂow channel. In (1) the lips have swung to the right, the channel between the lips is open, and the lips are somewhat compressed, as compared to their undis- placed dimensions. As the oscillation cycle progresses (2) the lips have swung forward (to the right, downstream) even more and have compressedfurther making the channel wider. A short time later in (3) the lips have begun to swing back and are less compressed and the channel has narrowed. In (4) the lips they have swung back even more and have stretched further, coming together so that the lip channel is nearly closed; the ﬂow is now greatly reduced or even stopped. The lips then move to the positions shown in (1) and the cycle repeats. 1522 J. Acoust. Soc. Am. 145(3), March 2019 N. GiordanoFIG. 2. Results from the NS simula- tions for the air speed near the lips at aseries of times in one oscillation cycle. The arrows indicate the order of the images in time, starting in frame (a), then (b)–(f), and then returning to frame (a); compare with the schematic lip conﬁgurations in Fig. 1. The color indicates the air speed with blue indi-cating a low speed and red a large speed. The maximum speed was about 200 m/s, and occurred in the center of the channel when the lips were nearly closed. The lips are 2 mm long along the ﬂow direction, and are outlined in white. These images were approxi-mately equally spaced in time (with about 20% of the oscillation period between each image), except for images (a) and (f) which were more closely spaced in time and chosen to illustrate changes in the ﬂow when the lip channel was narrow and rapidlychanging. FIG. 3. Results from the NS simula- tions for the density near the lips at a series of times in one oscillation cycle.The arrows indicate the order of the images in time, starting in frame (a), then (b)–(f), and then returning to frame (a). These images were recorded as the same times as the corresponding images in Fig. 2. The color indicates the air density and pressure relative totheir background values, with blue indicating a low density and pressure, and red a high density and pressure. Quantitative values of these quantities will be given below. The lips are 2 mm long along the ﬂow direction, and are outlined in white. The software used toconstruct these color maps attempts to interpolate in regions were the density changes abruptly and this causes the lip edges to sometimes be displaced an amount equal to the NS grid spacing. J. Acoust. Soc. Am. 145(3), March 2019 N. Giordano 1523arguments as a portion of the ﬂuid moves from point 1 to point 2, with the assumptions of steady ﬂow, an incompress-ible ﬂuid, and negligible viscosity. The Bernoulli equation can also be derived from the NS equation that describes changes in the momentum with time. Denoting the compo- nents of the velocity along x,y, and z(see Fig. 1)b y u,v, andw, respectively, the kinematic viscosity by /C23, and taking thexdirection to be the direction of net ﬂow (the horizontal direction in the plane of Fig. 2), the NS equation for the time variation of ufor the air ﬂow can be written as @u @tþu@u @xþv@u @yþw@u @zþc2 q@q @x/C0/C23r2u¼0; (2) where cis the speed of sound. The Bernoulli equation can be derived by integration of Eq.(2)with the assumptions that the ﬂow is along xso that v¼w¼0, that the ﬂow is steady so that udoes not vary with time, and that viscous forces are negligible. With these assumptions, only the second and ﬁfth terms on the left-hand side in Eq. (2)are nonzero. This approach to deriving the Bernoulli equation allows one to estimate the effects of non- steady ﬂow and the importance of the viscosity as theseeffects give rise to the ﬁrst and last terms on the left hand side of Eq. (2). In our analysis we will also need the relation between the density and pressure, which is DP¼c 2Dq; (3) where DPandDqare the variations of the pressure and den- sity relative to their background values of P0(1 atm) and q0 (the density of air). This relation assumes that viscous and other losses are small, which should be a good approximation. Using Eqs. (1)and(2)along with the simulation results for the velocity and density as functions of time throughout the lip region, we can test the accuracy of the Bernoulli equation as applied to this ﬂow problem. Our NS solution includes the effects of nonsteady ﬂow and viscosity while the Bernoulliequation does not, so we can test if, when, and to what degree these effects make a signiﬁcant contribution. Speciﬁcally, we will use results for the ﬂow velocity obtained from the NS cal- culation together with the Bernoulli equation to obtain the “Bernoulli prediction” for the pressure. This Bernoulli pressure will then be compared to the actual pressure found in the NS calculation. If there are signiﬁcant differences, we then exam-ine the magnitudes of the acceleration and viscous terms in Eq. (2)to gain insight into their importance. We will ﬁnd that at certain times during the lip oscilla- tion cycle the Bernoulli equation works well, but that at other times, when the lip channel is narrow, the predictionsof the Bernoulli equation fail signiﬁcantly. We then also use the Bernoulli pressure to calculate the force on the lips and compare with the lip force obtained from the NS calculation. IV. QUANTITATIVE ANALYSIS AND RESULTS We ﬁrst consider a case in which the lip channel is wide open, Fig. 2(b). Figure 4shows results at this moment in thelip oscillation cycle for the pressure as a function of position along the axis that passes along the center of the lip channel, from left to right in Fig. 4(parallel to the xaxis in Fig. 1). In Fig. 4the pressure is calculated in two different ways. The solid curve is the value of the pressure derived from the solu-tion of the NS equations for the density together with Eq.(3); we will refer to this as the actual pressure, P actual. The solid symbols labeled as PBernoulli in Fig. 4are values of the pressure P2derived from the Bernoulli equation [Eq. (1)] with location 1 deep inside the mouth (far to left and beyond the range of Fig. 3) where the pressure is approximately independent of location. In evaluating Eq. (1)we obtain P1 from the actual pressure in the mouth and use the ﬂow speed calculated from the NS equations on the horizontal axis thatruns along the center of the lip channel in Fig. 2(note that the variation of the density on this axis is also included, buthas only a small effect). It is seen that P actual andPBernoulli in Fig. 4agree quite well throughout the entire lip region; at the instant considered in this case the lip channel begins atabout x¼22 mm and ends at x¼24 mm. Similarly good agreement between the Bernoulli pressure and the actualpressure was found whenever the lip channel was wide openas in Figs. 2(b) and2(c). This suggests that the terms associ- ated with the viscosity and with the acceleration in Eq. (2) make negligible contributions under these conditions. Hence, in the case of a fairly open lip channel it is quite accurate, quantitatively, to use the Bernoulli equationtogether with the ﬂow velocity to estimate the pressure in thelip channel. This “success” of the Bernoulli equation, asdemonstrated in Fig. 4is interesting for another reason. Since the relation between the ﬂow velocity and pressure aredescribed by the Bernoulli equation, one might think that FIG. 4. Variation of the actual pressure and the pressure calculated with the Bernoulli equation on the axis parallel to the net ﬂow velocity running along the center of the lip channel. These results were obtained at the time corre- sponding to the images in part (b) in Figs. 2and3. In the evaluation of the Bernoulli equation, point #1 in Eq. (1)was deep inside the mouth region at x¼10 mm which was to the left of the region shown in Figs. 2and3. The values of the Bernoulli pressure shown as the solid symbols were then obtained as P2in Eq. (1). Note that these results were obtained long after the lip oscillation and the variations of the pressure and ﬂow velocity had reached steady state. (Steady state was typically reached after about 20 peri- ods of the fundamental frequency while these results were obtained after about 50 periods.) The vertical dashed lines show the location of the start and end of the lip channel. 1524 J. Acoust. Soc. Am. 145(3), March 2019 N. Giordanothis situation is analogous to stationary, frictionless ﬂow in a simple wide channel. However, in that case we would expectthe ﬂow velocity be constant along the channel which means that, according to the Bernoulli equation, the pressure should also be constant. That is deﬁnitely not the case in Fig. 4, since the pressure decreases signiﬁcantly along the channel. The explanation of this behavior seems to be that there is some jet formation as the air enters the channel causing thevelocity to increase as one moves through the channel. At the same time, the air farthest along the channel entered ear- lier in time when the channel had a different width (i.e., abreakdown of the assumption of stationary ﬂow) also caus- ing a variation in the velocity. The key insight here is that even though the Bernoulli equation seems to correctly relatethe ﬂow velocity and pressure, the assumptions underlying the Bernoulli approximation may still be breaking down. This may have important implications for modeling. Results for the pressure for a case in which the lip chan- nel was narrow are shown in Fig. 5. These results for P actual and PBernoulli were obtained at the instant corresponding to image (d) in Figs. 2and3. The Bernoulli pressure now devi- ates quite signiﬁcantly from the actual pressure. The leading edge of the lips in this case was at x¼20.9 mm, while the Bernoulli pressure begins to deviate strongly from Pactual at x¼20.0 mm, which is well before the start of the lip channel. Intuitively, it is natural to suspect that the breakdown of the Bernoulli equation seen in Fig. 5is due to the effect of viscous forces when the lip channel is very narrow. The vis-cous forces are largest in this case, due to the no-slip ( u¼0) boundary conditions for the ﬂow at the edges of the lip chan- nel. The importance of viscous forces is conﬁrmed by exam-ining the spatial variation of the viscous term in the NSequation, Eq. (2). We consider the situation in which the ﬂow is along the xdirection (horizontal in Fig. 6) so that the velocity components v¼w¼0, in which case this term becomes /C23(@ 2u/@y2). The variation of this term is shown in Fig. 6as a function of position along the axis that runs through the lip channel for the case of the open channel inFig. 4, and the narrow channel in Fig. 5. For the narrow channel (the open symbols in Fig. 6) the magnitude of the viscous term becomes large even before the channel entranceatx/C2520.9 mm and then decreases in magnitude on leaving the channel. It is interesting that the viscous term alsoincreases in magnitude inside the channel even when thechannel is quite open (the closed symbols in Fig. 6), but is still somewhat smaller than found for the nearly closed channel. The key points shown in Fig. 6are that the viscous term is small and slowly varying outside and upstreamfrom the lip channel as compared to inside the channel, andthat this term is much larger inside the channel when it isnarrow as compared to when the channel is wide. Note thatdownstream from the lip channel we certainly expect the viscous term to be nonzero, as the jet emerging from the lips breaks up. It is also interesting to compare the acceleration term in the NS equation, Eq. (2),@u/@t, for the cases of wide open and nearly closed channels. Results for this term are shownin Fig. 7, where we show the variation of @u/@t, the ﬁrst term on the left-hand side on Eq. (2), along the lip axis for the instants corresponding to Figs. 4and5, with the open symbols again corresponding to the nearly closed channel and the closed symbols corresponding to the wide open channel. Physically, this term is the acceleration of the ﬂuidat a particular location, and is thus large when the assump-tion of steady ﬂow breaks down. For the open channel theacceleration term is small throughout the channel, andincreases to the right of the lip channel as the air jet breaks up [Fig. 2(d)]. For the nearly closed channel the acceleration FIG. 5. Variation of the actual pressure Pactual shown as the solid curve and the pressure P2¼PBernoulli calculated with the Bernoulli equation on the axis parallel to the net ﬂow velocity running along the center of the lip chan- nel. These results were obtained at the time corresponding to part (d) inFigs. 2and3, when the lip channel was relatively narrow and the ﬂow was still signiﬁcant. Note that these results were obtained long after the lip oscil- lation and the variations of the pressure and ﬂow velocity had reached steady state. (Steady state was typically reached after about 20 periods of the fundamental frequency while these results were obtained after about 50 periods.) The vertical dashed lines show the location of the start and end of the lip channel.FIG. 6. Viscous term in the NS equation Eq. (2),/C23(@2u/@y2), as a function of position along the axis passing through the center of the lip channel when the channel is wide open corresponding to Fig. 4(solid symbols) and when the channel is nearly closed corresponding to Fig. 5(open symbols). The start and end of the lip channel for these two cases are shown in Figs. 4and 5, and were slightly different in the two cases because of the swinging motion of the lips. J. Acoust. Soc. Am. 145(3), March 2019 N. Giordano 1525term increases rapidly in magnitude inside the channel as the ﬂuid slows rapidly due to the opposing viscous forces. Asfound with the viscous term, the acceleration term is seen to be small and slowly varying outside and upstream from the lip channel as compared to inside the channel. This term isalso much larger inside the narrow channel as compared to when the channel is wide. Note that downstream from the lip channel we also expect the acceleration term to be nonzero,as the jet must slow as it emerges from the lips. Finally, a comparison with Fig. 6shows that for the narrow channel the acceleration term is larger in magnitude than the viscousterm by roughly an order of magnitude The results in Figs. 2and3show that the effects of ﬂuid acceleration and viscosity both become largest when the lipchannel is narrow and changing most rapidly in width. The units are the same in the two ﬁgures allowing a direct com- parison between the two effects, and it is seen that the accel-eration effect is generally larger. However, we caution that in the analysis in Figs. 4and 5the effects are integrated from deep in the mouth, so a simple comparison of the mag-nitudes at one or a few positions may not the best way to compare. The behavior seen in Figs. 6and 7is in qualitative accord with expectations. One certainly expects the assump- tions of steady ﬂow and negligible viscosity to break down at some point as the lip channel becomes narrow. However,the truly important question with regards to the accurate modeling of the trumpet concerns how much this breakdown affects the calculation of quantities such as the lip motion. Inparticular, how does the lip force calculated using the Bernoulli equation to estimate the pressure near and between the lips compare with the actual force on the lips? Here, to be precise, we use the term “actual force” to mean the force on the lips as found using the NS calculation. The most straightforward way to calculate the actual force on a lip isto compute the product of the pressure at the lip surface andthe lip area, and integrate this product over the surface of the lip. The actual force will have a component perpendicular to the direction of net ﬂow which will act to expand or shrink the width of the lip channel, and a component parallel to the direction of net ﬂow which will cause the lips to swing back and forth along the ﬂow direction. It was explained in Ref. 1 that we used the immersed boundary algorithm to implement the motion of the lips through the NS grid. It turns out that this algorithm yields the actual lip force directly from the algorithm as the force required to keep the lips moving con-sistently with the ﬂow velocity at the lip surfaces. 14,15We can thus calculate the actual force on the lips in two ways: (1) Using the immersed boundary algorithm and (2) by inte- grating the product of the lip area and the pressure obtained from the NS calculation. We found that these two ways to ﬁnd the lip force gave the same result (to within the uncer- tainties), thus providing a consistency check on our NS cal- culations. The solid curve in Fig. 8shows the actual force on one of the lips in the direction perpendicular to the net ﬂow; that is, along the ydirection in Fig. 1. (The actual force on the other lip is equal in magnitude but opposite in sign and is not shown here.) The dotted curve in Fig. 8was calculated using the Bernoulli pressure to estimate the perpendicular lip force. Speciﬁcally, the dotted curve is PBernoulli /C2Aintegrated over the surface of the lip inside the lip channel; we will refer to this quantity as the “Bernoulli force.” (Note that the contri- bution to the Bernoulli force from surfaces outside the lip channel is very small and does not change any of our conclu- sions below.) It is clear that the Bernoulli force differs greatly from the actual force during about a quarter of each oscillation cycle. Indeed, at certain times during each cycle, such as near t¼51.9, 52.9, and 53.8 ms the Bernoulli force is about a factor of 5 larger than the maximum value of the actual force at any point in each cycle, while the actual force during these times is very close to zero. At these times, the Bernoulli force is qualitatively incorrect and would certainly not be reliable for calculations of the lip motion. It is worth noting that the times at which the Bernoulli force is FIG. 7. Acceleration term in the NS equation, @u/@t, as a function of posi- tion along the axis passing through the center of the lip channel when the channel is wide open corresponding to Fig. 4(solid symbols) and when the channel is nearly closed corresponding to Fig. 5(open symbols). The start and end of the lip channel for these two cases are shown in Figs. 4and5, and were slightly different in the two cases because of the swinging motion of the lips.FIG. 8. Force on one of the lips in the direction perpendicular to the direc- tion of mean ﬂow. Solid curve: Perpendicular force obtained directly from the NS simulation using the immersed boundary method. Dotted curve: Estimate for the perpendicular force calculated using the Bernoulli equation to calculate the pressure in the lip channel. 1526 J. Acoust. Soc. Am. 145(3), March 2019 N. Giordanoqualitatively incorrect are the times during each oscillation cycle at which the lip channel is narrow, as in the case con- sidered in Fig. 5. At other times the Bernoulli force agrees reasonably well with the actual force, for example, near t /C2552.3 and 53.2 ms, and these are times at which the lip channel is fairly open as in the case in Fig. 4. (Results for the lip width are given below in Fig. 10.) It is also interesting to consider the force on the lips in the direction parallel to the direction of net ﬂow, that is, alongthexdirection in Fig. 1. The dotted curve in Fig. 9shows the parallel force on one lip, while the solid curve in this ﬁgureshows the perpendicular force for comparison (this is theresult shown as the solid curve in Fig. 8). These are the actual forces in the parallel and perpendicular directions. It is per-haps surprising that the parallel force is much larger than theperpendicular force during much of each oscillation cycle.However, this is readily understood when one realizes thatduring much of each cycle the upstream surface of the lip isdirectly exposed to the incoming ﬂow, which essentiallypushes the lip forward. This parallel force is important since it drives the swinging motion of the lip; indeed, this swinging motion is seen to be quite important in Figs. 2and3. Our results conﬁrm that it is necessary to include the par- allel force and the parallel lip motion as well, which is cer-tainly not surprising given that this lip model allows the lipsto swing along the direction of net ﬂow, and is in accord withthe results of Adachi and Sato 4and other previous modeling. In addition to being large in magnitude, the parallel force isseen to vary considerably during each oscillation cycle. Some of this variation in the parallel force is due to the changes in the orientations of the upstream and downstream lip surfacesas is evident in Fig. 3, and that orientation has been included in some modeling of the lip motion (e.g., Adachi and Sato 4). However, this component of the lip force also depends on thepressure on the upstream and downstream lip edges. To thebest of our knowledge the few modeling studies that havetreated the parallel lip motion (including the work of Adachiand Sato) have all assumed that the pressure on the upstreamedge of the lip is constant and equal to the mouth pressure farupstream. The images in Fig. 3suggest that the pressure on the upstream surfaces of the lips varies substantially during each oscillation cycle. This is conﬁrmed in Fig. 10which shows the pressure quantitatively as a function of time at sev-eral locations. The pressure deep in the mouth varies onlyabout 15% during each oscillation cycle, an amount similar tothat found experimentally. 12In contrast, the pressure right at the entrance to the lips, which we will term Pentrance ,v a r i e s quite substantially with time, approaching the mouth pressureonly when the lip channel is nearly closed. This large varia-tion of P entrance is important for our discussion since the paral- lel force on the upstream edge of a lip is equal to the product ofPentrance and the lip area integrated over the upstream sur- face of the lip (with allowance for the slope of the lip). Theparallel force on the downstream edge of a lip is given by asimilar relation involving the pressure at the lip exit.However, the pressure at the lip exit is smaller and variesmuch less than P entrance , so the parallel lip force and its time dependence is dominated by the force on the upstream edge.Our results thus show that it is important to include the varia-tion of P entrance with time. That variation has not been included in previous discussions of the lip motion, but can bereadily accounted for in NS based calculations. V. CONCLUSIONS The goal of this paper has been to report a careful analy- sis of the ﬂow velocity, density, and lip force obtained froma direct numerical simulation of a lip reed instrument usingFIG. 10. Top: Variation of the width of the lip channel width with time. Bottom: Pressure as a function of time derived from the NS calculation atvarious locations on the axis parallel to the direction of net ﬂow (the xdirec- tion in Fig. 1) that passes along the center of the lip channel. The pressure in the mouth shown here was recorded far upstream from the lips; the mouth pressure is essentially independent of position if one is more than about 1 mm upstream from the lip channel entrance. The other curves show the pressure at the entrance to the lip channel, the center of this channel, and at the channel exit.FIG. 9. Actual force on one of the lips in the direction parallel to the direc- tion of mean ﬂow (dotted curve) and perpendicular to the direction of mean ﬂow (solid curve). The perpendicular force is also shown as the black curve in Fig. 8. Note that the vertical scale here is different than in Fig. 8. The results shown here for both the perpendicular and parallel forces were obtained using the immersed boundary method algorithm. J. Acoust. Soc. Am. 145(3), March 2019 N. Giordano 1527the NS equations. All previous simulations of such instru- ments have used approximate treatments of the ﬂow alongwith the Bernoulli equation to relate the ﬂow properties to the pressure and then to the force on the lips. Estimates of the lip force are essential for calculating the associated lipmotion and oscillations. Our analysis shows clearly the lim- its of the Bernoulli equation for estimating the pressure near and between the lips. When the lip channel is fairly open,the Bernoulli equation seems to work well, but this is only during a portion of the oscillation cycle. During much of the oscillation cycle, when the lip channel is closed or nearlyclosed, the Bernoulli equation leads to a prediction for lip force that differs drastically in magnitude and time depen- dence from the actual lip force. In addition, we ﬁnd that theparallel force on the lips, that is, the force in the direction of net ﬂow, also varies quite signiﬁcantly during each lip oscil- lation cycle, and drives strong lip motion in this direction. Aquantitative description of this parallel motion must include the variation of the upstream pressure with time, a variation which has not been described in nearly all previous model-ing, and is very challenging to access experimentally. Our results show that accurate modeling of lip reed instruments such as the trumpet with direct solutions of theNS equations is now feasible using available supercom- puters. At the same time, our results indicate that if one wants to build new simpler models that avoid the NS equa-tions in favor of Bernoulli-like approximations, one must include the effects of ﬂuid acceleration and viscous forces in the ﬂow calculations. ACKNOWLEDGMENTS The author thanks D. A. Maurer for a useful discussion. This work was supported by the U.S. National Science Foundation Grant No. PHY1513273. The computations werecarried out at the Rosen Center for High Performance Computing at Purdue University and with computationalfacilities supported by the Ofﬁce of Information Technology at Auburn University. 1N. Giordano, “Physical modeling of a conical lip reed instrument,” J. Acoust. Soc. Am. 143, 38–50 (2018). 2S. J. Elliott and J. M. Bowsher, “Regeneration in brass instruments,” J. Sound Vib. 83, 181–217 (1982). 3J. Saneyoshi, H. Teramura, and S. Yoshikawa, “Feedback oscillation in reed woodwind and brasswind instruments,” Acta Acust. Acust. 62, 194–210 (1987). 4S. Adachi and M. Sato, “Time-domain simulation of sound production in the brass instrument,” J. Acoust. Soc. Am. 97, 3850–3861 (1995). 5S. Adachi and M. Sato, “Trumpet sound simulation using a two- dimensional lip vibration model,” J. Acoust. Soc. Am. 99, 1200–1209 (1996). 6J. S. Cullen, J. Gilbert, and D. M. Campbell, “Brass instruments: Linearstability analysis and experiments with an artiﬁcial mouth,” Acta Acust. Acust. 86, 704–724 (2000). 7C. E. Vilain, X. Pelorson, A. Hirschberg, L. Le Marrec, W. Op’t Root, and J. Willems, “Contribution to the physical modeling of the Lips. Inﬂuenceof the mechanical boundary conditions,” Acta Acust. Acust. 89, 882–887 (2003). 8M. Campbell, “Brass instruments as we know them today,” Acta Acust.Acust. 90, 600–610 (2004). 9I. Lopez, A. Hirschberg, A. Van Hirtum, N. Ruty, and X. Pelorson, “Physical modeling of buzzing artiﬁcial lips: The effect of acoustical feedback,” Acta Acust. Acust. 92, 1047–1059 (2006). 10S. Bromage, M. Campbell, and J. Gilbert, “Open areas of vibrating lips in trombone playing,” Acta Acust. Acust. 96, 603–613 (2010). 11R. Tournemenne, J-F. Petiot, and J. Gilbert, “The capacity for simulation by physical modeling to elicit perceptual differences between trumpetsounds,” Acta Acust. Acust. 102, 1072–1081 (2016). 12H. Boutin, N. Fletcher, J. Smith, and J. Wolfe, “Relationships between pressure, ﬂow, lip motion, and upstream and downstream impedances for the trombone,” J. Acoust. Soc. Am. 137, 1195–1209 (2015). 13M. D €ollinger and M. Kaltenbacher, “Preface: Recent advances in under- standing the human phonation process,” Acta Acust. Acust. 102, 195–208 (2016). 14R. Mittal and G. Iaccarino, “Immersed boundary methods,” Annu. Rev. Fluid Mech. 37, 239–261 (2005). 15J. Pederzani and H. Haj-Hariri, “A numerical method for the analysis of ﬂexible bodies in unsteady viscous ﬂows,” Int. J. Numer. Methods. in Eng. 68, 1096–1112 (2006). 1528 J. Acoust. Soc. Am. 145(3), March 2019 N. Giordano
1.5045629.pdf	Selective activation of an isolated magnetic skyrmion in a ferromagnet with microwave electric fields Akihito Takeuchi , and Masahito Mochizuki Citation: Appl. Phys. Lett. 113, 072404 (2018); doi: 10.1063/1.5045629 View online: https://doi.org/10.1063/1.5045629 View Table of Contents: http://aip.scitation.org/toc/apl/113/7 Published by the American Institute of Physics Articles you may be interested in Dual-mode ferromagnetic resonance in an FeCoB/Ru/FeCoB synthetic antiferromagnet with uniaxial anisotropy Applied Physics Letters 112, 192401 (2018); 10.1063/1.5018809 Tuning Slonczewski-like torque and Dzyaloshinskii–Moriya interaction by inserting a Pt spacer layer in Ta/ CoFeB/MgO structures Applied Physics Letters 112, 232402 (2018); 10.1063/1.5026423 180°-phase shift of magnetoelastic waves observed by phase-resolved spin-wave tomography Applied Physics Letters 112, 232403 (2018); 10.1063/1.5030342 Direct detection of spin Nernst effect in platinum Applied Physics Letters 112, 162401 (2018); 10.1063/1.5021731 Layer-selective microwave-assisted magnetization switching in a dot of double antiferromagnetically coupled (AFC) layers Applied Physics Letters 112, 162404 (2018); 10.1063/1.5027127 A fully electric field driven scalable magnetoelectric switching element Applied Physics Letters 112, 182401 (2018); 10.1063/1.5023003Selective activation of an isolated magnetic skyrmion in a ferromagnet with microwave electric fields Akihito Takeuchi1,a)and Masahito Mochizuki2,3,b) 1Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa 252-5258, Japan 2Department of Applied Physics, Waseda University, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan 3PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan (Received 22 June 2018; accepted 31 July 2018; published online 15 August 2018) We theoretically reveal that pure eigenmodes of an isolated magnetic skyrmion embedded in a ferromagnetic environment can be selectively activated using microwave electric ﬁelds without exciting gigantic ferromagnetic resonances, in contrast to conventional methods using microwavemagnetic ﬁelds. We also demonstrate that this selective activation of a skyrmion can efﬁciently drive its translational motion in a ferromagnetic nanotrack under application of an external magnetic ﬁeld inclined from the normal direction. We ﬁnd that a mode with combined breathingand rotational oscillations induces much faster skyrmion propagation than the breathing mode studied in a previous work [Wang et al. , Phys. Rev. B 92, 020403(R) (2015)]. Published by AIP Publishing. https://doi.org/10.1063/1.5045629 A skyrmion crystal, a hexagonally crystalized state of magnetic skyrmions, 1–6has characteristic resonance modes at microwave frequencies,7–10which give rise to intriguing physi- cal phenomena11such as microwave directional dichroism,12–15 spin-voltage induction,16,17and spin-current generation.18 When a static magnetic ﬁeld Hexis applied perpendicular to a thin-plate specimen of the skyrmion-hosting magnet, severaltypes of spin-wave modes emerge depending on the microwave polarization. 7A microwave magnetic ﬁeld Hxnormal to the skyrmion plane ( Hx ?) activates the so-called breathing mode in which all the skyrmions constituting the skyrmion crystal uni- formly expand and shrink in an oscillatory manner. On the other hand, the Hxﬁeld within the skyrmion plane ( Hx k) acti- vates two types of rotation modes with opposite rotational senses, in which cores of all the skyrmions circulate uniformly in counterclockwise and clockwise ways. In addition to the crystallized form, skyrmions can appear as individual defects in a ferromagnetic state; such skyrmions are also expected to have peculiar collectivemodes. 19Isolated skyrmions conﬁned in a nano-ferromagnet are potentially useful for applications20to memory devices,21 magnonics devices,22–24spin-torque oscillators,25,26and microwave sensing devices.27As such, it is necessary to clar- ify the microwave-active eigenmodes of a single skyrmion in a ferromagnetic specimen. In addition, it is important to establish a way to manipu- late isolated skyrmions using microwaves for their device application. As the microwave ﬁeld Hx ?cannot activate pre- cessions of the magnetizations when the microwave ﬁeld is applied parallel to them, we can activate pure breathing-type skyrmion oscillations with Hx ?under a perpendicular Hex ﬁeld without exciting the background ferromagnetic state. However, once the Hexﬁeld is inclined, the microwave mag- netic ﬁeld Hx ?excites huge ferromagnetic resonances, whichdrown out the weaker skyrmion resonances. Moreover, the microwave energy is absorbed by the sample when exciting the gigantic ferromagnetic resonances, which would inevita- bly result in high energy consumption and considerable tem- perature rise. Therefore, a technique to selectively activate an isolated skyrmion is urgently required. In this letter, we ﬁrst theoretically show that the eigenmo- des of an isolated skyrmion embedded in a ferromagnetic envi- ronment can be selectively activated with a microwave electricﬁeld E xvia oscillatory modulation of the Dzyaloshinskii- Moriya interaction (DMI) without activating ferromagnetic resonances. We then demonstrate that translational motion of the skyrmion can be driven through activating its resonance modes using this electric technique in an inclined Hexﬁeld. The latter part of the research was motivated by the recent the- oretical work by Wang et al. that demonstrated skyrmion prop- agation via activating the breathing mode with a microwave Hx ?ﬁeld.28Our study reveals that skyrmion motion can be driven not only by the breathing mode but also by other Ex- active modes. Furthermore, we ﬁnd that a mode with com- bined clockwise and breathing oscillations can induce much faster propagation of the skyrmion than the previously studiedbreathing mode. Our ﬁndings pave a way toward efﬁcient manipulation of isolated skyrmions in nano-devices via the application of microwaves. We consider a magnetic bilayer system composed of a ferromagnetic layer and a heavy-metal layer with strongspin-orbit interactions [see Fig. 1(a)], where the spatial inversion symmetry is broken at their interface, and thereby the DMI is active. An inclined magnetic ﬁeld H ex¼ðHx;0;HzÞ with Hx¼Hztanhis applied, where his the inclination angle [see Fig. 1(b)]. For 0/C14<h/C2090/C14, the Hexﬁeld is inclined toward the positive xdirection. The DMI favors a rotating alignment of the magnetizations, which results in the formation of a Neel-type skyrmion. The skyrmion has acircular symmetry under a perpendicular H exﬁeld ( h¼0/C14), but has disproportionate weight in distributions of thea)Electronic mail: akihito@phys.aoyama.ac.jp b)Electronic mail: masa_mochizuki@waseda.jp 0003-6951/2018/113(7)/072404/5/$30.00 Published by AIP Publishing. 113, 072404-1APPLIED PHYSICS LETTERS 113, 072404 (2018) magnetizations and scalar spin chiralities [see Figs. 1(c)and 1(d)]. To describe the magnetism in this magnetic bilayer system, we employ a classical Heisenberg model on a square lattice with magnetization vectors miwhose norm mis unity.29,30The Hamiltonian contains the ferromagnetic- exchange interaction, the Zeeman coupling to the magnetic ﬁelds, and the interfacial DMI H¼/C0 JX hi;jimi/C1mj/C0HexþHðtÞ ½/C138 /C1X imi þDðtÞX iðmi/C2miþ^xÞ/C1^y/C0ðmi/C2miþ^yÞ/C1^x/C2/C3:(1) Here, HðtÞ¼ð 0;0;HzðtÞÞandEðtÞ¼ð 0;0;EzðtÞÞrepresent time-dependent magnetic and electric ﬁelds applied perpen-dicular to the sample plane, respectively. We neglect magnetic anisotropies because they do not alter the results qualitatively although stability of the skyrmions and resonant frequenciesof the eigenmodes may be slightly changed. The strength of the interfacial DMI can be tuned by applying a gate electric ﬁeld normal to the plane via varying the extent of the spatial inversion asymmetry. 31,32The DMI coefﬁcient DðtÞ¼D0 þDDðtÞhas two components, namely, a steady component D0and a EðtÞ-dependent component DDðtÞ¼jEzðtÞwith j being the coupling constant. We take J¼1 for the energy units and take D0=J¼0.27. For the inclined magnetic ﬁeld, we take Hz¼0.057 with hbeing a variable. The unit conver- sions when J¼1 meV are summarized in Table I. We simulate the magnetization dynamics by numeri- cally solving the Landau-Lifshitz-Gilbert equation using the fourth-order Runge-Kutta method. The equation is given by dmi dt¼/C0cmmi/C2Heff iþaG mmi/C2dmi dt: (2) Here, aG(¼0.04) and cmare the Gilbert-damping constant and the gyrotropic ratio, respectively. The effective ﬁeld Heff iis calculated as Heff i¼/C0 ð 1=cm/C22hÞ@H=@mi.We ﬁrst calculate the dynamical electromagnetic and magnetic susceptibilities vemandvmm vemðxÞ¼ﬃﬃﬃﬃﬃl0 /C150rDMx z Epulse;vmmðxÞ¼DMx z l0Hpulse: (3) After applying a short pulse HzðtÞorEzðtÞwith duration of Dt¼1 in the units of J=/C22h, we trace time proﬁles of the net magnetization MzðtÞ¼ð 1=NÞP imziðtÞandDMzðtÞ¼MzðtÞ /C0Mzð0Þand obtain the Fourier transform DMx z. Dividing this quantity by an amplitude of the pulse HpulseorEpulse,w e obtain these susceptibilities. The calculations are performedusing a system of N¼160/C2160 sites with periodic bound- ary conditions. Figure 2(a) displays the imaginary parts of the electro- magnetic susceptibilities Im v emðxÞfor several values of h, which describe the response of the magnetizations to the Exﬁeld. When h¼0/C14, the spectrum exhibits a single peak corresponding to the breathing mode activated by the oscil-lating DMI under the application of an AC electric ﬁeld.When the H exﬁeld is inclined with h6¼0/C14, two additional modes emerge, one with a higher and the other with a lower frequency than the breathing mode. The intensities of theadditional modes increase, whereas the intensity of theoriginal breathing mode is increasingly suppressed as h increases. The imaginary parts of the magnetic susceptibilities Imv mmðxÞin Fig. 2(b) describe the response of the magnet- izations to the Hxﬁeld. We ﬁnd that only a breathing mode (m-mode 2) appears when the Hexﬁeld is perpendicular (h¼0/C14); however, its intensity decreases as hincreases. Remarkably, a large ferromagnetic resonance from the sur-rounding ferromagnetic magnetizations emerges under an inclined H exﬁeld, whereas it is totally silent under the per- pendicular Hexﬁeld. We also ﬁnd an additional mode (m- mode 1) at lower frequencies. In reality, the skyrmion has another Hx-active mode (m- mode 3) at higher frequencies, but it is hidden behind the gigan- tic ferromagnetic resonance and thus cannot be seen in thespectra of Im v mmðxÞ. We can see this weak response of the skyrmion to the Hxﬁeld by focusing on the vector spin chiral- ities, si¼P cmi/C2miþc(c¼^x;^y). The calculated imaginary parts of the dynamical susceptibilities Im vmc aðxÞfor the a-com- ponent of the vector spin chira lity Sa¼ð1=NÞP isai (a¼x;y)a r es h o w ni nF i g . 2(c). We ﬁnd that they coincide with the spectra of Im vemðxÞin Fig. 2(a). These results indicate that the magnetic method using Hxcannot selectively activate the eigenmodes of an isolated skyrmion in the ferromagneticspecimen; however, the results show that the electrical methodusing E xcan achieve this. This electrical technique is antici- pated to play a crucial role for developing future skyrmion- based devices. FIG. 1. (a) Schematics of a magnetic bilayer system hosting skyrmions sta- bilized by the interfacial Dzyaloshinskii-Moriya interaction. (b) External magnetic ﬁeld Hex, where his an inclination angle from the normal direc- tion. (c) and (d) Color maps of the normal component of magnetizations mz (c) and scalar spin chiralities cs(d) of a skyrmion under a perpendicular Hex ﬁeld. In-plane components of the magnetizations ( mx,my) are displayed by arrows. (e) and (f) Those under an inclined Hexﬁeld with h¼30/C14.TABLE I. Unit conversion table when J¼1 meV. Exchange int. J¼1 1 meV Time t¼1 0.66 ps Frequency f¼x=2px ¼0:01 2.41 GHz Magnetic ﬁeld H¼1 8.64 T072404-2 A. Takeuchi and M. Mochizuki Appl. Phys. Lett. 113, 072404 (2018)Based on the susceptibility data, we ﬁnd that an isolated skyrmion in a ferromagnetic specimen has three low-lyingeigenmodes. In Fig. 3, we show simulation results of snap- shots for h¼30 /C14. It is found that all of these modes have the breathing component, i.e., they show oscillatory expansionand shrinkage. Among these three modes, the higher- frequency mode ( x¼0.0664) can be regarded as a pure breathing mode, whereas the other two modes show distinctbehaviors. The lower-frequency mode ( x¼0:0453) is accompanied by a clockwise rotation of skyrmion in an ellip- tical orbit oriented horizontally against the inclined directionofH exas shown in Fig. 3(a). The moderate-frequency mode (x¼0:0513) is also accompanied by the clockwise rotation of the skyrmion in an elliptical orbit, but its trajectory is ori-ented vertically against the inclined direction of H exas shown in Fig. 3(b). The higher-lying pure breathing mode atx¼0.0664 does not show any rotational component [Fig. 3(c)]. The three Ex-active modes are referred to as e-modes 1, 2, and, 3 [see Figs. 2(a)and3]. Next, we numerically investigate the translational motion of a skyrmion driven by the electrically activated res-onance modes under an inclined H exﬁeld [see Fig. 4(a)]. Here, the inclination angle of Hexis ﬁxed at h¼30/C14. The amplitude of the AC electric ﬁeld EzðtÞ¼Ex zsinxtis ﬁxed atjEx z¼0:05D0¼0:0135. A recent experiment for Ta/ FeCoB/TaO xreported a huge E-ﬁeld-induced variation of the interfacial DMI that reaches 140% when the applied volt-age is 10 V. This observation supports the experimental fea- sibility of the 5% modulation assumed here. Figure 4(b) shows simulated snapshots of the skyrmion motion when theE xﬁeld with x¼0.0513 is applied, which indeed displays propagation in the negative xdirection. The trajectories of the propagating skyrmion during a time period from t¼0t ot¼5000 are shown in Fig. 4(c)for three different Ex-active modes at x¼0.0453, 0.0513, and 0.0664. They were obtained by tracing the center-of-masscoordinate ( j x,jy) of the topological-charge distribution jc¼X i¼ðix;iyÞiccsðix;iyÞ/C30X i¼ðix;iyÞcsðix;iyÞ; (4) with cs¼1 8pmi/C1ðmiþ^x/C2miþ^yÞþmi/C1ðmi/C0^x/C2mi/C0^yÞ/C2/C3:(5) We ﬁnd that the direction and velocity of the motion vary depending on the skyrmion resonance mode. For all the modes, the skyrmion moves mainly in the negative xdirec- tion. However, the trajectories for e-modes 2 and 3 are slightly slanted toward the negative ydirection; meanwhile, the trajectory for e-mode 1 is perfectly parallel to the xaxis. FIG. 2. Imaginary parts of (a) the electromagnetic susceptibility Im vemðxÞ, (b) the magnetic susceptibility Im vmmðxÞ, and (c) the chirality susceptibility Imvmc aðxÞfor several values of h. Here, an inclined magnetic ﬁeld Hex¼ðHztanh;0;HzÞwith Hz¼0.057 is applied. Three Ex-active modes are labeled as e- modes 1–3 in (a), whereas the three Hx-active modes are labeled as m-modes 1–3 in (b) and (c). The extremely intense mode around x/C240:05–0:06 in (b) is the ferromagnetic resonance (FMR). FIG. 3. Simulated snapshots of the magnetization dynamics for three Ex- active eigenmodes (e-modes 1–3) of an isolated skyrmion in the ferromag- netic specimen under an inclined Hexﬁeld, where Hz¼0.057 and h¼30/C14.072404-3 A. Takeuchi and M. Mochizuki Appl. Phys. Lett. 113, 072404 (2018)Interestingly, despite the slanted trajectory for e-mode 2, its traveling distance along the xaxis is identical to that of the trajectory for e-mode 1, which is directed perfectly along the xaxis. It can also be seen that the directions of the skyrmion movement for e-modes 2 and 3 are the same, even though their traveling distances are different. The traveling distancefor e-mode 3 is much shorter than that for e-mode 2 because of the smaller intensity of the latter mode, as can be seen in the inset of Fig. 4(c). In Figs. 4(d)and4(e), we plot the velocities v¼ðv x;vyÞ of the skyrmion for three different Ex-active modes (see the right vertical axes) as functions of the strength of the time- dependent DMI, i.e., jEx z. The values are calculated from the simulated displacements of the skyrmion in the xandy directions for a time period from t¼2000 to t¼10 000 (see the left vertical axes) by assuming J¼1 meV and a¼5A˚ with abeing the lattice constant. It can be seen that the velocities are proportional to the square of jEx z, and they are on the order of 10/C01m/s. In fact, the traveling speed of the skyrmions achieved using this technique turns out to be rela- tively slow compared with the speed achieved in techniques based on electric-current injection.33–40However, the presenttechnique has an advantage: it is free from the Joule heating, and thus the energy consumption and the temperature risecould be signiﬁcantly suppressed. We ﬁnally study the skyrmion motion driven by AC mag- netic ﬁelds H xunder an inclined Hexﬁeld with Hz¼0.057 andh¼30/C14.T h e Hxis applied perpendicular to the skyrmion plane, which is given by Hx zsinxtwith Hx z¼0:05Hz ¼0:00285. The simulated trajectories and velocities v ¼ðvx;vyÞare shown in Figs. 4(f)and4(g), respectively. We ﬁnd that the trajectories are again oriented almost in the nega-tive xdirection; however, for the ferromagnetic resonance mode with x¼0.0664, the trajectory is slanted toward the positive ydirection, which contrasts with the case of the E x-active mode. Noticeably, the higher-frequency mode with x¼0.0664 has a much faster propagation of the skyrmion than the other two modes. However, the usage of this mode isnot energetically efﬁcient because this mode is not an eigen-mode of the isolated skyrmion but a very intense resonance of the vast ferromagnetic magnetizations, which necessarily leads to large energy consumption and considerable rise ofdevice temperatures. In summary, we have theoretically found that resonance modes of an isolated skyrmion in a ferromagnet can be acti- vated by application of AC electric ﬁelds through oscillatory variation of the interfacial DMI. The advantage of thismethod compared with conventional methods using AC magnetic ﬁelds is that we can selectively excite skyrmions without activating gigantic ferromagnetic resonances; thisresults in a signiﬁcant suppression of both energy consump- tion and temperature rise. Our result revealed that among the three E x-active modes, the mode with combined clockwise and breathing oscillations induces much faster skyrmionpropagation than the previously studied breathing mode. Our ﬁndings will pave a way toward the efﬁcient manipulation of isolated skyrmions and thus will be useful for futureskyrmion-based devices. This work was supported by JSPS KAKENHI (Grant No. 17H02924), Waseda University Grant for Special Research Projects (Project Nos. 2017S-101 and 2018K-257), and JST PRESTO (Grant No. JPMJPR132A). 1A. N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP 68, 101 (1989). 2A. Bogdanov and A. Hubert, J. Magn. Magn. Mater. 138, 255 (1994). 3S. M €uhlbauer, B. Binz, F. Jonietz, C. Pﬂeiderer, A. Rosch, A. Neubauer, R. Georgii, and P. B €oni,Science 323, 915 (2009). 4X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901 (2010). 5N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8, 899 (2013). 6S. Seki and M. Mochizuki, Skyrmions in Magnetic Materials , Springer Briefs in Physics (Springer, 2016). 7M. Mochizuki, Phys. Rev. Lett. 108, 017601 (2012). 8O. Petrova and O. Tchernyshyov, Phys. Rev. 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Ch /C19erif, A. Stashkevich, S. Auffret, O. Boulle, G. Gaudin, M. Chshiev, C. Baraduc, and H. B /C19ea, preprint arXiv:1804.09955 . 33F. Jonietz, S. M €uhlbauer, C. Pﬂeiderer, A. Neubauer, W. M €unzer, A. Bauer, T. Adams, R. Georgii, P. B €oni, R. A. Duine, K. Everschor, M. Garst, and A. Rosch, Science 330, 1648 (2010). 34X. Z. Yu, N. Kanazawa, W. Z. Zhang, T. Nagai, T. Hara, K. Kimoto, Y. Matsui, Y. Onose, and Y. Tokura, Nat. Commun. 3, 988 (2012). 35J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Commun. 4, 1463 (2013). 36J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Nanotechnol. 8, 742 (2013). 37F. B €uttner, C. Moutaﬁs, M. Schneider, B. Kr €u g e r ,C .M .G €unther, J. Geilhufe, C. v. Korff Schmising, J. Mohanty, B. Pfau, S. S c h a f f e r t ,A .B i s i g ,M .F o e r s t e r ,T .S c h u l z ,C .A .F .V a z ,J .H . Franken, H. J. M. Swagten, M. Kl €aui, and S. Eisebitt, Nat. Phys. 11, 225 (2015). 38S. Woo, K. Litzius, B. Kr €u g e r ,M . - Y .I m ,L .C a r e t t a ,K .R i c h t e r ,M .M a n n ,A . Krone, R. M. Reeve, M. Weigand, P. Agrawal, I. Lemesh, M.-A. Mawass, P. Fischer, M. Kl €aui, and G. S. D. Beach, Nat. Mater. 15, 501 (2016). 39G. Yu, P. Upadhyaya, Q. Shao, H. Wu, G. Yin, X. Li, C. He, W. Jiang, X. Han, P. K. Amiri, and K. L. Wang, Nano Lett. 17, 261 (2017). 40W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. B. Jungﬂeisch, J. E. Pearson, X. Cheng, O. Heinonen, K. L. Wang, Y. Zhou, A. Hoffmann, and S. G. E. te Velthuis, Nat. Phys. 13, 162 (2017).072404-5 A. Takeuchi and M. Mochizuki Appl. Phys. Lett. 113, 072404 (2018)
1.3549438.pdf	Optimum design consideration for interferometric spin wave logic operations Y. Nakashima, K. Nagai, T. Tanaka, and K. Matsuyama Citation: Journal of Applied Physics 109, 07D318 (2011); doi: 10.1063/1.3549438 View online: http://dx.doi.org/10.1063/1.3549438 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/109/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Hybrid yttrium iron garnet-ferromagnet structures for spin-wave devices J. Appl. Phys. 117, 17E101 (2015); 10.1063/1.4906209 Design of a spin-wave majority gate employing mode selection Appl. Phys. Lett. 105, 152410 (2014); 10.1063/1.4898042 Non-stationary excitation of two localized spin-wave modes in a nano-contact spin torque oscillator J. Appl. Phys. 114, 153906 (2013); 10.1063/1.4825065 Modulation of propagation characteristics of spin waves induced by perpendicular electric currents Appl. Phys. Lett. 95, 142508 (2009); 10.1063/1.3243687 Magnetic soliton-based logic with fan-out and crossover functions Appl. Phys. Lett. 85, 2367 (2004); 10.1063/1.1794850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 163.118.172.206 On: Thu, 02 Jul 2015 16:12:10Optimum design consideration for interferometric spin wave logic operations Y . Nakashima, K. Nagai, T. Tanaka,a)and K. Matsuyama Kyushu University, Department of Electronics, Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan (Presented 16 November 2010; received 24 September 2010; accepted 9 November 2010; published online 25 March 2011) In the present study, the operational modes and the structural design are optimized to realize potential performance in the interferometric spin wave logic circuits. Successive functional operations, such as generation, propagation, and inductive detection of spin wave packets are numerically studied by using micromagnetic simulations. The logic input is coded with the phaseof pulsed microwave currents applied through the generators. Among the various kinds of the investigated spin wave (SW) modes, the backward volume mode exhibits superior performance. Various structural and operational parameters, including the pulsed microwave frequency and theﬁlm thickness of the magnetic strip, were optimized by taking the inductive output voltage ( V out)a s a quantitative criterion. The several orders of difference obtained in the Voutfor the different logic inputs demonstrates the successful exclusive-not-OR operation. VC2011 American Institute of Physics . [doi: 10.1063/1.3549438 ] I. INTRODUCTION Magnetostatic spin waves (MSW) have recently attracted practical interest as novel information carriers in spintronicsdevices. Spin wave interferometers and current controlling of the phase were studied with Mach-Zehnder-type device structures. 1–3Another type of magnetic logic device utilizing the interference between the spin wave packets (SWPs) has also been proposed.4,5Deep understanding of the dynamic properties of the SWP and optimum design consideration arekey issues to realize the spin based interferometric devices. In the present study, fundamental device operations of nucleation, propagation, mutual interference and inductivedetection of SWPs are numerically studied by using micro- magnetic simulations. The device performances are also compared for the three different spin wave modes of magne-tostatic surface wave (MSSW), magnetostatic forward vol- ume wave (MSFVW), and magnetostatic backward volume wave (MSBVW). II. NUMERICAL SIMULATIONS Cooperative spin dynamics were investigated by solving the Landau-Lifshitz-G ilbert equation with a ﬁnite differential method. The accuracy of the numerical simulation was determined from comparison with the previously reported ex-perimental results. 6The experimentally observed spin wave packet group velocity of 13.1 km/s reasonably agrees with the simulation result of 12.8 km/s, assuming the same materialparameters and device structures. In the following simulations standard material parameters for a ferromagnetic material are assumed; the saturation magne tization, exchange stiffnessconstant, and Gilbert damping constant are assumed to be 680 emu/cm 3,3 . 0/C210/C07erg/cm and 0.01, respectively. The longi- tudinal bias ﬁeld Hb, x, the transverse bias ﬁeld Hb, y,a n dt h ep e r - pendicular anisotropy ﬁeld Hk, za r ea s s u m e dt od e ﬁ n et h e precession axis for the MSBVW, MSSW and MSFVW modes,respectively. The values of H b, x,Hb, y,a n d Hk, zare chosen so that the ferromagnetic resonance frequency of the uniform mode ( k¼0) becomes identical (7.9 GHz) for the three conﬁgurations. The schematic ﬁgure of designed device structure is shown in Fig. 1. A ferromagnetic strip with 82 mm length and 5 mm width was assumed as a spin wave guide. Hair-pin shape conductors (SWG1, SWG2) with 0.3 mm width and 0.2mm gap are used for the generation of the SWPs. The SWPs are nucleated with nonuniform Oersted ﬁelds induced from the one cycle application of pulsed microwave current applied through the generators. The initial phase angle uof the pulsed microwave is used as the bit information. The data “1” and “0” are coded by u¼0 and prad. The interfero- metric operation is veriﬁed from the inductive output voltageV outcalculated from the time differential of the whole mag- netic ﬂux Uinside the spin wave detector with 0.2 mm gap width, where the SWPs emitted from the two generators aresuperposed. The frequency of the pulsed microwave and the FIG. 1. Schematic ﬁgure of the interferometric spin wave logic device, con- structed with the spin wave guide of a ferromagnetic strip, spin wave genera- tors (SWG1, SWG2) and the spin wave detector (SWD).a)Author to whom correspondence should be addressed: Electronic mail:t-tanaka@ed.kyushu-u.ac.jp. 0021-8979/2011/109(7)/07D318/3/$30.00 VC2011 American Institute of Physics 109, 07D318-1JOURNAL OF APPLIED PHYSICS 109, 07D318 (2011) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 163.118.172.206 On: Thu, 02 Jul 2015 16:12:10ﬁlm thickness of the SW guide are optimized for three differ- ent spin wave modes so that the Voutfor a single SWP emit- ted with the same conductor current amplitude becomesmaximum. The optimized values of f g, opt and doptfor the three spin wave modes are 8.9 GHz and 100 nm for MSSW, 7.9 GHz and 130 nm for MSFVW, and 6.9 GHz and 900 nmfor MSBVW, respectively. III. RESULTS AND DISCUSSION The SWP logic operations performed for the MSBVW mode are shown in Fig. 2, where the waveforms of the Vout are plotted for various logic inputs. The notation “N” means no input. The Voutfor the logic input “11” is almost twice that for the single SWP (“1N”), which demonstrates the superposi- tion of the two SWPs with the same u. Several orders of smaller Voutfor the logic input “01” represents the cancella- tion of the two SWPs with the relative phase difference of p rad. In contrast, the values of Voutfor the MSSW mode are markedly different depending on the propagation direction of SWP, as shown in Figs. 3(a)and3(b). Resultantly, the cancel- lation for the out of phase SWP was imperfect, as shown inFig.3(c). The results originated from the nonreciprocal prop- erties of the SWP emission in the MSSW mode. 6 Figure 4shows the deterioration of the logic perform- ance due to the dislocation of the detector position. Since the relative phase angle at the detector depends on the propaga- tion distance of the SWP, the maximum output voltageV out, max for the logic input “11” decreases with the increase of the dislocation distance txand vice versa for “01”. Although the structural dislocation results in the degradedinterferometric operation, the t xof 0.1 mm still exhibits apractical Vout, max difference of 113 mV/mm2for the input “11” and 47 mV/mm2for “01”. The ratio of Vout, max for the input “11” and “01” is plot- ted in Fig. 5as a function of tx.The results for the MSBVW and MSFVW modes are compared in the ﬁgure. Superior logic operation is realized in the MSBVW, where the ratio of Vout, max for the two logic inputs reaches 115 dB at tx¼0mm. This excellent performance is due to the almost perfect can- cellation for the logic input “01”, where the Vout, max is only 0.21 nV/ mm2. A soliton like behavior of the colliding SWPs is demon- strated in Fig. 6. In the simulation, the detector position is artiﬁcially dislocated at tx¼/C02mm so that the SWPs from both sides of the generators can be individually detected with a time delay. Fig. 6(a) presents the reference Voutfor the single SWP. The ﬁrst and second Voutsignals shown in Fig. 6(b) come from the SWPs generated at the left- and right- hand side generators. No distinguishable difference is observed for the Voutin Figs. 6(a) and6(b), noted as SWG2 in the ﬁgure. The results indicate that the collision of the SWPs does not affect the following propagation. Figure 7presents an interaction of the propagating SWP with the current induced pulsed magnetic ﬁeld. An additional control conductor with the same design as the generator is located at the mid-to-mid 1 mm separation from the genera- tor. The simulations are performed for the pulsed microwave control currents with opposite polarities noted as u¼0,pin the ﬁgure, and various delay times Td. As shown in the ﬁg- ure, the control ﬁelds enlarge or suppress the Vout, max , depending on the polarity and the values of Td. The FIG. 2. Wave forms of inductively detected output voltage in MSBVW mode for different logic inputs; (a) “11”, (b) “01”, and (c) “1N”. The logic data “1” and “0” are coded by the phase angle u¼0 and pof the pulsed microwave current for the spin wave packet generation. The notation “N” means no input. FIG. 3. Wave forms of inductively detected output voltage in MSSW mode for different logic inputs; (a) “0N”, (b) “N1”, and (c) “01”. FIG. 4. Maximum output voltage as a function of the dislocation distance tx of the detector. The dislocation affects the relative phase difference at the detector, which modulates the interferometric output. FIG. 5. Ratio of maximum output voltage for logic inputs of “11” and “10”versus the dislocation distance of detector; (a) MSBVW, (b) MSFVW.07D318-2 Nakashima et al. J. Appl. Phys. 109, 07D318 (2011) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 163.118.172.206 On: Thu, 02 Jul 2015 16:12:10considerable modulation of the output signal amplitude with the control ﬁelds demonstrates the possibility of the ampliﬁ-cation and the attenuation of the SWP in the logic circuit. IV. CONCLUSION Interferometric logic operations utilizing spin wave packets were numerically studied. Fundamental operations of nucleation, propagation, interference, and inductive detec-tion of spin wave packets are numerically studied by using micromagnetic simulations. The operation modes and the structural design were optimized to realize a practical per-formance. The logic inputs were coded by the relative phase difference of the pulsed microwave currents applied through the generators. Among the various investigated spin wavemodes, the backward volume mode exhibited superior performance. The numerically predicted 115 dB difference in the output voltage for the logic inputs of “11” and “01” demonstrates the successful logic operation. 1Y. K. Fetisov and C. E. Patton, IEEE Trans. Magn. 35, 1024 (1999). 2M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Appl. Phys. Lett. 87, 13501 (2005). 3T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps, and M. P. Kostylev, Appl. Phys. Lett. 92, 022505 (2008). 4A. Khitun and K. L. Wang, Superlattices Microstruct .38, 184 (2005). 5A. Khitun, M. Bao, and K. L. Wang, IEEE Trans. Magn .44, 2141 (2008). 6K. Sekiguchi, K. Yamada, S. M. Seo, K. J. Lee, D. Chiba, K. Kobayashi, and T. Ono, Appl. Phys. Lett. 97, 022508 (2010). FIG. 6. Wave forms of inductively detected output voltage; (a) single spin wave packet, (b) two conﬂicting spin wave packets. Almost the same wave forms, noted as SWG2, suggest that the conﬂict causes no distinguishableinﬂuence on the spin wave packets propagation. FIG. 7. Control the propagating spin wave packets with pulsed microwavecurrent with opposite polarity ( u¼0,p) and various delay times T d. The current induced magnetic ﬁelds enhance or attenuate the precession of the spin wave, depending on the relative phase difference.07D318-3 Nakashima et al. J. Appl. Phys. 109, 07D318 (2011) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 163.118.172.206 On: Thu, 02 Jul 2015 16:12:10
1.3243985.pdf	Nonlinear motion of coupled magnetic vortices in ferromagnetic/nonmagnetic/ferromagnetic trilayer Su-Hyeong Jun, Je-Ho Shim, Suhk-Kun Oh, Seong-Cho Yu, Dong-Hyun Kim et al. Citation: Appl. Phys. Lett. 95, 142509 (2009); doi: 10.1063/1.3243985 View online: http://dx.doi.org/10.1063/1.3243985 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v95/i14 Published by the American Institute of Physics. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 29 Mar 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsNonlinear motion of coupled magnetic vortices in ferromagnetic/ nonmagnetic/ferromagnetic trilayer Su-Hyeong Jun,1Je-Ho Shim,1Suhk-Kun Oh,1Seong-Cho Yu,1Dong-Hyun Kim,1,a/H20850 Brooke Mesler,2and Peter Fischer3 1Department of Physics, Chungbuk National University, Cheongju 361-763, South Korea 2Lawrence Berkeley National Laboratory, Center for X-ray Optics, Berkeley, California 94720, USA and Applied Science and Technology Graduate Group, UC Berkeley, Berkeley, California 94720, USA 3Lawrence Berkeley National Laboratory, Center for X-ray Optics, Berkeley, California 94720, USA /H20849Received 20 May 2009; accepted 15 September 2009; published online 9 October 2009 /H20850 We have investigated a coupled motion of two parallel vortex cores in ferromagnetic/nonmagnetic/ ferromagnetic trilayer cylinders by means of micromagnetic simulation. Dynamic motion of twovortices with parallel and antiparallel relative chiralities of curling spins around the vortex coreshave been examined after excitation by 1 ns pulsed external ﬁeld, revealing a nontrivial coupledvortices motion. © 2009 American Institute of Physics ./H20851doi:10.1063/1.3243985 /H20852 Recently, spin dynamics in conﬁned geometry has at- tracted growing interest due to potential application of spin-tronics and magnetic storage in ferromagnetic nanoelements.It is well known that a magnetic vortex structure is formed invarious conﬁned geometries due to minimization of magne-tostatic and exchange energies. Numerous studies have beendevoted to understand the vortex structure 1and dynamics.2–5 In particular, the understanding of magnetic vortex dynamics becomes more essential in the realization of nanometer scalespintronic devices and memory devices. 6,7We know that a magnetic vortex is driven to move under an external mag-netic ﬁeld 2,8or excited to move by a spin transfer torque.6 The motion of the vortex is known to be gyrotropic with circular trajectories in excited states.2,6,8 Interaction between magnetic vortices was recently in- vestigated for trilayer ﬁlms with two ferromagnetic layersdisplaced by a nonmagnetic spacer layer, where the simu-lated motion of vortices was predicted to be similar to adamped simple harmonic oscillation with a weak couplingbetween two vortices. 9An experimental study has been re- ported on coupled vortex dynamics in multilayer structure,10 where a depth-resolved gyrotropic motion of vortex intrilayer with a relative 180° phase shift due to an interlayercoupling has been observed. However, no detailed study hasbeen addressed to a strong interaction behavior between vor-tex cores in a multilayer system, while the interaction be-tween vortex cores in a multilayer system becomes strongeras the system size becomes smaller down to nanometerscales. In this work, we have carried out a micromagneticsimulation study of coupled motion of vortices in the trilayerwith systematic control of vortex chirality at the top andbottom layers as well as with variation in nonmagneticspacer layer thickness. We have carried out micromagnetic simulations using the object oriented micromagnetic framework /H20849 OOMMF /H20850/H20849Ref. 11/H20850based on the Landau–Lifshitz–Gilbert equation to inves- tigate the coupled motion of vortices in a cylindricalferromagnetic/nonmagnetic/ferromagnetic trilayer ﬁlm. Inour simulation, material parameters of the ferromagnetic lay-ers are chosen to be those of Permalloy with the exchange stiffness coefﬁcient of 13 /H1100310 −12J/m and the saturation magnetization Msof 8.0/H11003105A/m. The thickness of each Permalloy layer has been chosen to be 5 nm and the radius ofthe cylinder is set to be 250 nm. Spacer layer thickness isvaried from 0 to 20 nm. The cell size of the micromagneticsimulation is 5 /H110035/H110035n m 3and the damping constant is 0.03. Polarities of two vortices have been chosen to be thesame with an upward direction. The dynamic motion of twovortices with parallel and antiparallel relative chiralities hasbeen examined under a pulsed external ﬁeld. The duration ofthe pulse ﬁeld is 1 ns with a rising time of 0.1 ns and afalling time of 0.1 ns. A pulse with a strength of 3.14 mT hasbeen applied in the plane to excite vortex motion. Time-dependent vortex core positions are determined by process-ing and analyzing simulated images. a/H20850Author to whom correspondence should be addressed. Electronic mail: donghyun@cbnu.ac.kr. FIG. 1. /H20849Color online /H20850Time-resolved trajectories of vortices cores for dif- ferent spacer layer thickness /H20849d=5, 10, and 20 nm /H20850for parallel and antipar- allel chiralites. Single layer case for d=0 nm is shown on the top ﬁgure.APPLIED PHYSICS LETTERS 95, 142509 /H208492009 /H20850 0003-6951/2009/95 /H2084914/H20850/142509/3/$25.00 © 2009 American Institute of Physics 95, 142509-1 Downloaded 29 Mar 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsAs demonstrated in Fig. 1, the direction of initial motion of two vortex cores is same in the cases of parallel relativechiralities, whereas the direction becomes opposite in thecases of antiparallel relative chiralities. Although the initialmotion depends on the handedness of vortex structure, thegyrotropic motion only depends on the vortex core polarityso that two vortex cores exhibit the same sense of a counter-clockwise rotation. In the case with a spacer layer thicknessd=20 nm, overall gyrotropic motions becomes similar irre- spective of the relative chirality. However, in the case with athinner spacer layer /H20849d=5 nm /H20850, a faster decaying behavior of the gyrotropic vortex core motion is vividly observed for an antiparallel relative chiralities as in the ﬁgure, which impliesthat there is a signiﬁcant coupling phenomenon between twovortex cores of the top and bottom layers. To investigate further details of coupling between two vortex cores, time-dependent vortex core positions of the topand bottom layers were analyzed, as shown in Fig. 2. Posi- tion of the vortex core along the yaxis is presented together with an average radial position and a lateral distance betweentwo vortex cores with respect to time. The lateral distance isdeﬁned to be a relative distance between the two core posi-tions projected onto the x-yplane. The average radial dis- tance is deﬁned as an average radial distance of the two corepositions projected onto the x-yplane from the center of the dot. The position of the core in all cases exhibits a dampedoscillatory behavior. In the case of antiparallel relativechiralities, oscillatory behavior decays faster as the spacerlayer becomes thinner. It becomes more evident by notingthat an average radial distance and a lateral distance betweentwo cores reach zero fastest when the spacer layer thicknessis thinnest /H20849d=5 nm /H20850. This can be explained by taking into account the fact that a mutual interaction of two cores on the bottom and top layers is attractive. Since the cores have thesame polarities /H20849p=+1 /H20850, they prefer to have shortest lateral distance between them due to the ﬂux closure of the core magnetization. In the case of parallel chiralities, a faster decay of an oscillatory behavior does not exist. There is a signiﬁcant at-tractive magnetic force between the two cores, however itdoes not contribute to the faster decay of the oscillation sincethe two cores are quite close together right from the motionstart. If there is no coupling between the two cores, lateral FIG. 2. /H20849Color online /H20850Vortex core position projected on the yaxis with respect to the time for different spacer layer thickness /H20849d=5, 10, 15, and 20 nm /H20850, together with average radius of two cores from the center and lateral distance between two cores are plotted. The inset ﬁgures show the core position an d the average radius of the top and bottom vortices projected on the xaxis.142509-2 Jun et al. Appl. Phys. Lett. 95, 142509 /H208492009 /H20850 Downloaded 29 Mar 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsdistance between them will be zero all the time. However, even when the spacer layer is thickest /H20849d=20 nm /H20850, the lat- eral distance is not zero all the time but oscillating although the amount of oscillation of the lateral distance is relativelysmall. Very interestingly, when d=5 nm, the coupled oscil- lation behavior survives longest as the average radial dis-tance accordingly survives longest. Moreover, it should benoted that oscillatory behavior of the two cores becomes cha-otic after about two cycles of gyrotropic motion. During thechaotic coupled motion, the lateral distance between the twocores becomes even greater than the average radial distance.Considering the interaction between the two cores is alwaysattractive, it seems that a strong non-linear interaction playsa key role in the chaotic motion regime. Coupled oscillatory behavior with variation of the spacer layer thickness has been ﬁtted, as demonstrated in the Fig. 3. Vortex core position projected onto the yaxis is ﬁtted at the time when the pulse ﬁeld is switched off /H20849t=1 ns /H20850. For com- parison, a simple gyrotropic motion of the single layer vortex has been ﬁtted with a damped simple harmonic oscillationmotion as y=y 0exp /H20849−/H9252t/H20850sin/H20849/H92750t/H20850, where y0=−41.7 nm, /H9252 =6.9/H11003108s−1, and /H92750=1.1/H11003109rad /s/H20850. In the case of thicker spacer layer /H20849d=20 nm /H20850, very weak coupling re- sulted in almost independent gyrotropic motions of two vor- tices irrespective of relative chiralities. Thus, the position ofeach vortex core is still ﬁt with a damped simple harmonicoscillatory motion, as demonstrated in the bottom of Fig. 3.Chaotic motion of the vortex core due to the signiﬁcant attractive interaction for the case of thinner spacer layer /H20849d =5 nm /H20850is explainable based on a simple model. We assume that each vortex core with M zcomponent as a magnetic dipole m/H6023vortex and estimate dipole-dipole interaction energy to be /H20851/H92620//H208494/H9266r3/H20850/H20852/H208513/H20849m/H6023vortex1 ·u/H602312/H20850/H20849m/H6023vortex2 ·u/H602312/H20850 −m/H6023vortex1 ·m/H6023vortex2 /H20852, where /H92620is the permeability, ris the in- terdistance between two dipoles, and u/H602312is a unit vector along the direction of the relative displacement of two di-poles. The lateral attractive force is then the derivative of theenergy with respect to s, the lateral displacement between two dipoles, where r= /H20881s2+d2for the nonmagnetic spacer layer thickness d. The force f/H20849s/H20850is found to have a form of f/H20849s/H20850=/H20851A·s/H20849−1+B·s2/H20850/H20852//H20851/H208491+C·s2/H208507/2/H20852with ﬁtting parameters A,B, and C. The attractive force is used as a weak coupling force in the simultaneous differential equations of twoweakly coupled damped simple harmonic oscillators. By nu-merically solving the simultaneous differential equation withproper selection of ﬁtting parameters, we reproduced thechaotic motion of coupled vortices, as demonstrated in Fig. 3 /H20849A=1.5/H1100310 19,B=1.0/H110031016, and C=4.0/H110031016/H20850. The cha- otic behavior after few initial oscillations is qualitatively re-produced for the trilayer with 5 nm spacer thickness in thecases of parallel chiralities and the faster oscillation of vortexcore with a shorter lateral distances between two cores areobserved as well in the cases of antiparallel chiralities. This work was supported by the Korea Research Foun- dation Grant funded by the Korean Government /H20849Grant No. KRF-2007-331-C00097 /H20850. B.M. acknowledges ﬁnancial sup- port from the NSF Extreme Ultraviolet Engineering Re-search Center. P.F. acknowledges ﬁnancial support by theDirector, Ofﬁce of Science, Ofﬁce of Basic Energy Sciences,Materials Sciences and Engineering Division, of the U.S.Department of Energy. 1M. Bode, O. Pietzsch, A. Kubetzka, W. Wulfhekel, D. McGrouther, S. McVitie, and J. N. Chapman, Phys. Rev. Lett. 100, 029703 /H208492008 /H20850. 2S. B. Choe, Y. Acremann, A. Scholl, A. Bauer, A. Doran, J. Stohr, and H. A. Padmore, Science 304, 420 /H208492004 /H20850. 3K. Y. Guslienko, B. A. Ivanov, V. Novosad, Y. Otani, H. Shima, and K. Fukamichi, J. Appl. Phys. 91, 8037 /H208492002 /H20850. 4K. Y. Guslienko, Appl. Phys. Lett. 89, 022510 /H208492006 /H20850. 5K.-S. Lee, K. Y. Guslienko, J.-Y. Lee, and S.-K. Kim, Phys. Rev. B 76, 174410 /H208492007 /H20850. 6K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A. Thiaville, and T. Ono, Nature Mater. 6, 270 /H208492007 /H20850. 7H.-G. Piao, D. Djuhana, S.-K. Oh, S.-C. Yu, and D.-H. Kim, Appl. Phys. Lett. 94, 052501 /H208492009 /H20850. 8B. Van Waeyenberge, A. Puzic, H. Stoll, K. W. Chou, T. Tyliszczak, R. Hertel, M. Fähnle, H. Brückl, K. Rott, G. Reiss, I. Neudecker, D. Weiss,C. H. Back, and G. Schütz, Nature /H20849London /H20850444,4 6 1 /H208492006 /H20850. 9K. Y. Guslienko, K. S. Buchanan, S. D. Bader, and V. Novosad, Appl. Phys. Lett. 86, 223112 /H208492005 /H20850. 10K. W. Chou, A. Puzic, H. Stoll, G. Schütz, B. Van Waeyenberge, T. Tyl- iszczak, K. Rott, and G. Reiss, J. App. Phys. 99, 08F305 /H208492006 /H20850. 11M. J. Donahue and D. G. Porter, OOMMF User’s Guide, http:// math.nist.gov/oommf /H208492002 /H20850. FIG. 3. /H20849Color online /H20850/H20849a/H20850Fitted motion of the vortex motion of the single layer /H20849d=0 nm /H20850with a simple damped harmonic oscillation. /H20849b/H20850Fitted mo- tion of coupled vortices cores for d=5 nm in case of the parallel/antiparallel chiralities. /H20849c/H20850Fitted motion of coupled vortices cores for d=20 nm in case of the parallel/antiparallel chiralities.142509-3 Jun et al. Appl. Phys. Lett. 95, 142509 /H208492009 /H20850 Downloaded 29 Mar 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions
1.2740588.pdf	Size dependent damping in picosecond dynamics of single nanomagnets A. Barman, S. Wang, J. Maas, A. R. Hawkins, S. Kwon, J. Bokor, A. Liddle, and H. Schmidt Citation: Applied Physics Letters 90, 202504 (2007); doi: 10.1063/1.2740588 View online: http://dx.doi.org/10.1063/1.2740588 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/90/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Femtosecond laser excitation of multiple spin waves and composition dependence of Gilbert damping in full- Heusler Co2Fe1−xMnxAl films Appl. Phys. Lett. 103, 232406 (2013); 10.1063/1.4838256 Detecting single nanomagnet dynamics beyond the diffraction limit in varying magnetostatic environments Appl. Phys. Lett. 98, 052502 (2011); 10.1063/1.3549302 Spin dynamics and damping in nanomagnets measured directly by frequency-resolved magneto-optic Kerr effecta) J. Appl. Phys. 102, 103909 (2007); 10.1063/1.2812541 Precessional dynamics in microarrays of nanomagnets J. Appl. Phys. 97, 10A706 (2005); 10.1063/1.1849057 Arrays of nanoscale magnetic dots: Fabrication by x-ray interference lithography and characterization Appl. Phys. Lett. 85, 4989 (2004); 10.1063/1.1821649 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Tue, 25 Nov 2014 14:04:44Size dependent damping in picosecond dynamics of single nanomagnets A. Barmana/H20850and S. Wang School of Engineering, University of California, Santa Cruz, 1156 High Street, Santa Cruz, California 95064 J. Maas and A. R. Hawkins Department of Electrical and Computer Engineering, Brigham Young University, 459 Clyde Building,Provo, Utah 84604 S. Kwon,b/H20850J. Bokor, and A. Liddle Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720 H. Schmidtc/H20850 School of Engineering, University of California, Santa Cruz, 1156 High Street, Santa Cruz, California 95064 /H20849Received 7 February 2007; accepted 24 April 2007; published online 17 May 2007 /H20850 The authors use time-resolved cavity-enhanced magneto-optical Kerr spectroscopy to study the damping of magnetization precession in individual cylindrical nickel nanomagnets. A wide range ofshapes /H20849diameters of 5 /H9262m–125 nm and aspect ratio: 0.03–1.2 /H20850is investigated. They observe a pronounced difference in damping between the micro- and nanomagnets. Microscale magnets showlarge damping at low bias ﬁelds, whereas nanomagnets exhibit bias ﬁeld-independent damping. Thisbehavior is explained by the interaction of in-plane and out-of-plane precession modes in microscalemagnets that results in additional dissipative channels. The small and robust damping values on thenanoscale are promising for implementation of controlled precessional switching schemes innanomagnetic devices. © 2007 American Institute of Physics ./H20851DOI: 10.1063/1.2740588 /H20852 Nanomagnets will form the building blocks for magnetic data storage and future spintronic devices. Beyond data stor-age they have the potential to open additional avenues formedical diagnostics such as medical resonance imaging anddynamic methods for cancer treatments. 1Future technology requires faster magnetic switching in magnetic nanostruc-tures and large-angle precessional switching has the potential to increase the operation speed to picoseconds. Two veryimportant dynamic parameters are the precession frequencyand damping. Microscale elements show complicated behav-ior as a result of inhomogeneous internal magnetic ﬁelds. 2–5 Understanding the ultrafast dynamics of single nanomagnets is a subject of intense interest. Precessional dynamics fromnanomagnet ensembles has been explored, 6albeit at the ex- pense of dipolar broadening and dynamic dephasing effectsfrom the neighboring elements. This problem may be over-come by combining the increased spatial sensitivity of cavityenhancement of magneto-optical Kerr effect 7,8/H20849CE-MOKE /H20850 with the femtosecond time resolution offered by ultrafast la-sers. We have recently reported the picosecond dynamics ofsingle Ni nanomagnets using such an all-optical time-resolved CE-MOKE technique. 9Here we report the damping of the precessional motion of single Ni nanomagnets as afunction of size /H20849aspect ratio /H20850and bias magnetic ﬁelds. Ni magnets with 150 nm thickness and diameters ranging be-tween 5 /H9262m and 125 nm as shown in Fig. 1/H20849a/H20850are studied to cover a large range of aspect ratio between 0.03 and 1.2. Thisallows us to study damping across the transition from multi-domain microscale to single-domain nanoscale magnets.The Ni magnets, spaced by 5 /H9262m for optical and mag- netostatic isolation, were fabricated by electron beamlithography. 8One batch of the sample was coated with a 70 nm silicon nitride /H20849SiN/H20850layer for improving the spatial sensitivity by enhancing the Kerr rotation as well as by re-ducing the background noise. 8The magnetic force micro- scope images in Fig. 1/H20849a/H20850show the remanent magnetic states of the Ni elements, which conﬁrm multidomain states forlarger magnets down to 250 nm and a single domain for the 125 nm magnet. The experimental setup is based on a two-color optical pump-probe technique described in detailelsewhere. 9The sample was optically pumped10by 15 mW linearly polarized 400 nm laser pulses of about 100 fs pulse-width, which induces precession in the sample. 2 mW lin- a/H20850Also at: Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India. b/H20850Present address: School of Electrical Engineering, Seoul National Univer- sity, Seoul 155-744, South Korea. c/H20850Electronic mail: hschmidt@soe.ucsc.edu FIG. 1. /H20849Color online /H20850/H20849a/H20850Magnetic force microscope images showing the domain structures of the magnetic elements. /H20849b/H20850Time-resolved magneto- optical Kerr rotation /H20849raw data /H20850with the time scale broken between 10 and 15 ps to show the three regions of interests clearly. /H20849c/H20850Fast Fourier trans- forms of the time-resolved data after double exponential backgroundsubtraction.APPLIED PHYSICS LETTERS 90, 202504 /H208492007 /H20850 0003-6951/2007/90 /H2084920/H20850/202504/3/$23.00 © 2007 American Institute of Physics 90, 202504-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Tue, 25 Nov 2014 14:04:44early polarized laser pulses of 800 nm wavelength were time delayed with respect to the pump beam to probe the dynam-ics by detecting polar magneto-optical Kerr rotation. A per-pendicular static magnetic ﬁeld up to 3.73 kOe was appliedto bias the samples during the experiment. The two-colorscheme and spectral ﬁltering before probe detection com-bined with the CE-MOKE provided the required sensitivityto probe single nanomagnet dynamics down to 125 nm. Figure 1/H20849b/H20850shows the time-resolved dynamics of mag- nets between 5 /H9262m and 125 nm at a bias ﬁeld of 1.68 kOe. The time scale is broken between 10 and 15 ps to show threedifferent regimes of interest clearly. Within the initial 10 psthe fast demagnetization and the quick recovery are shown.The precession is observed after 10 ps on top of a slow ex-ponential decay, which is subtracted from these signals andthe corresponding fast Fourier transform spectra are shownin Fig. 1/H20849c/H20850. Multiple frequency modes are observed for al- most all samples where the highest frequency mode is iden-tiﬁed as the uniform precession mode. The uniform preces-sion mode shows a clear size dependence which has beendescribed in detail elsewhere. 9The lower frequency mode does not show any clear size dependence, indicating differentorigins for magnets of varying size as discussed below. To extract the damping of the uniform precession mode we have processed the time-resolved data by a fast Fouriertransform /H20849FFT /H20850high-pass ﬁltering to eliminate low fre- quency modes. The FFT spectra also do not show any peakabove the uniform precession mode. The ﬁltered time-resolved data possess a single uniform precession frequencyand are ﬁtted through a least square ﬁtting routine with adamped sine function of the form M/H20849t/H20850=M/H208490/H20850e −t//H9270sin/H20849/H9275t−/H9278/H20850, /H208491/H20850 where /H9270is the decay time of the precession deﬁned as /H9270 =1//H9275/H9251eff,/H9275is the angular frequency of the uniform preces- sion mode given by /H9275=/H9253Heff, and/H9278is the initial phase of oscillation. /H9251effis the “effective” damping coefﬁcient as op- posed to the intrinsic Gilbert damping,11/H9253is the gyromag- netic ratio, and Heffis the effective magnetic ﬁeld which includes the external bias ﬁeld, demagnetizing ﬁeld, volumeand surface anisotropy ﬁelds, and an offset ﬁeld as discussedin Ref. 9Using the best ﬁt values of /H9275/H20849same as obtained from the FFT power spectra of time-resolved data /H20850and/H9270 from Eq. /H208491/H20850/H9251effis extracted. Since Eq. /H208491/H20850is valid only in the limit of small damping /H20849/H9251/H112701/H20850, for a further conﬁrmation we have also extracted the damping coefﬁcient /H20849/H9251eff/H20850from numerical solution of Landau-Lifshitz-Gilbert equation of motion12under macrospin model /H20849not shown /H20850and obtained similar results. Figure 2/H20849a/H20850shows the FFT ﬁltered experi- mental time-resolved dynamics and the best ﬁt curves withEq. /H208491/H20850for magnets of varying diameters at a bias ﬁeld of 1.68 kOe and Fig. 2/H20849b/H20850shows the extracted /H9251effas a function of magnet diameter. Large qualitative and quantitative differ-ences in /H9251effbetween the micro- and nanoscale are found. For magnets with diameters between 5 and 3 /H9262m/H9251eff reaches a maximum value of nearly 0.17. At 2 /H9262m,/H9251effre- duces sharply followed by a gradual decrease down to500 nm. At 500 nm /H9251effsettles down at around 0.04, compa- rable to the reported damping coefﬁcient 0.05 of continuousNi thin ﬁlms measured by all-optical method.10The nearly fourfold increase in /H9251efffor magnets /H110222/H9262m diameter may originate from extrinsic sources such as multimagnonscattering,11spin wave propagation for large angleprecession,13spin pumping process,14interfacial effects,15 and dephasing of incoherent spin waves.16This size depen- dence of /H9251effis also evident in a bias ﬁeld series. Figure 2/H20849c/H20850 shows the time-resolved experimental and ﬁtted data for a400 nm element and Fig. 2/H20849d/H20850shows the bias ﬁeld depen- dence of /H9251effversus magnet diameter. For magnets /H110222/H9262ma strong bias ﬁeld dependence of /H9251effis observed, while for magnets with diameter /H110211/H9262m, no bias ﬁeld dependence is observed. For magnets with intermediate diameters weakbias ﬁeld dependence is observed. Even for magnets/H110222 /H9262m,/H9251effapproaches smaller values at larger bias ﬁelds, nearly the value obtained for the nanoscale magnets. Biasﬁeld /H20849frequency /H20850dependent damping has been observed in previous studies from ferromagnetic resonance linewidth17,18 measurements and an increase in damping at reduced bias ﬁeld /H20849frequency /H20850typically has been ascribed to inhomoge- neous line broadening caused by dispersion in the anisotropyﬁeld. In order to understand the size and bias ﬁeld-dependent damping behavior we investigate the frequency spectra ofthe experimental results carefully. In all spectra we have ob-served lower frequency modes of signiﬁcant amplitudes withfrequencies around 1–2 GHz. Figures 3/H20849a/H20850and 3/H20849b/H20850show the bias ﬁeld-dependent FFT power spectra for 5 /H9262m and 400 nm magnets, representing regions 1 and 2 in Figs. 2/H20849b/H20850 and2/H20849d/H20850, respectively. The gray solid lines show Gaussian ﬁts to peak 1 and peak 2. For the 5 /H9262m magnet the two resonant modes have large splitting at stronger bias ﬁelds.With reduction of the bias ﬁeld, the uniform precession modefrequency decreases while the low frequency mode in-creases. Consequently, a signiﬁcant overlap of the two modes FIG. 2. /H20849a/H20850Experimental /H20849open circles /H20850and ﬁtted /H20849gray lines /H20850time-resolved data from magnetic dots of varying diameter at an external bias field=1.68 kOe. /H20849b/H20850The extracted effective damping coefﬁcient /H20849 /H9251eff/H20850as a func- tion of magnet diameter. The hatched rectangle shows the transition regionfrom a high to low /H9251eff./H20849c/H20850Experimental /H20849open circles /H20850and simulated /H20849gray lines /H20850time-resolved data from a magnetic dot of 400 nm diameter at varying external bias ﬁelds. /H20849d/H20850The extracted /H9251efffor magnets with varying diameter as a function of the external bias ﬁeld. A high-pass FFT ﬁltering was appliedto the experimental time-resolved data for ﬁtting with a single damped sinefunction.202504-2 Barman et al. Appl. Phys. Lett. 90, 202504 /H208492007 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Tue, 25 Nov 2014 14:04:44is observed from intermediate bias ﬁelds, the amount of which increases with further reduction in the bias ﬁeld,shown by the hatched region in Fig. 3/H20849c/H20850. It is the dephasing of these two modes that opens another dissipative channeland causes the enhancement of the effective damping of theuniform precession mode for samples with diameter /H333561 /H9262m. In comparison, the FFT power spectra for the 400 nm samplealso show two frequency modes but they remain well split/H20851Fig. 3/H20849d/H20850/H20852for the whole bias ﬁeld range down to 1.25 kOe. The weaker bias ﬁeld dependence of the uniform precessionmode for nanoscale magnets due to their very large aniso-tropy is the main reason for this. The low frequency modefor samples /H110211 /H9262m is most likely associated with the trans- lation of the vortex core that has been predicted by staticmicromagnetic simulation.9For magnets with diameter /H333561/H9262m/H20849where static micromagnetic simulation predicts multiple domains and signiﬁcant in-plane magnetization9/H20850, the low frequency mode is most likely associated with theprecession of an in-plane magnetization component aboutH in-plane /H20849in-plane anisotropy ﬁeld plus small in-plane com- ponent of the bias ﬁeld due to the slight tilt of the sampleplane with the bias ﬁeld direction /H20850. The frequency of the in-plane precession mode may be expressed as19 fin-plane =g/H9262B h/H20881/H20849Hin-plane /H20850/H20849Hin-plane +4/H9266Min-plane /H20850, /H208492/H20850 where gis the Lande gfactor, /H9262Bis the Bohr magneton, his Planck’s constant, and Min-plane is the in-plane component of the magnetization parallel to Hin-plane . With the reduction of the out-of-plane bias ﬁeld, Min-plane must increase in magni- tude and frequency of mode 2 increases accordingly. Simul-taneously, the frequency of mode 1 decreases with the reduc-tion of bias ﬁeld, which results in the increased overlap of the two modes for magnets /H333561 /H9262m. Other possible explana- tions for the observed damping behavior can be ruled outimmediately including spin pumping 14and interface effects15 /H20849due to the use of a single layer material /H20850and multimagnon scattering11/H20849due to unsaturated magnetic states of samples /H20850. High-frequency spin waves may originate from nonuniformexcitation along the thickness due to much shorter skin depth/H20849/H1101110 nm /H20850of the pump laser /H20849perpendicular standing spin waves /H20850and ﬁnite magnet size /H20849forward volume magneto- static modes /H20850, but are not observed in the spectra due to both smaller amplitudes and small separation from the dominantuniform mode. 9 In summary, we have studied the damping of preces- sional motion from single nanomagnets while eliminatingextrinsic ensemble effects such as dynamic dephasing 9that further complicate the dynamics. The observed damping ofthe uniform precession mode in microscale magnets has alarge extrinsic contribution due to dephasing with a lowerfrequency in-plane precessional mode. On the other hand,the observation of ﬁeld-independent damping for nanoscalemagnets close to the thin ﬁlm value ensures the reliability ofcoherent control of precessional switching by a straightfor-ward pulse shaping scheme 20in nanomagnets. The authors thank B. Hillebrands and T. J. Silva for fruitful discussions, the National Science Foundation /H20849Grant No. ECS-0245425 /H20850, and Ofﬁce of Science, Ofﬁce of Basic Energy Sciences, of the U.S. Department of Energy /H20849Con- tract No. DE-AC02-05CH11231 /H20850for ﬁnancial support. 1G. Reiss and A. Hütten, Nat. Mater. 4,7 2 5 /H208492005 /H20850. 2Y . Acremann, C. H. Back, M. Buess, O. Portmann, A. Vaterlaus, D. Pescia, and H. Melchior, Science 290, 492 /H208492000 /H20850. 3J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, and P. A. Crowell, Phys. Rev. Lett. 89, 277201 /H208492002 /H20850. 4J. Jorzick, S. O. Demokritov, B. Hillebrands, M. Bailleul, C. Fermon, K. Y . Guslienko, A. N. Slavin, D. V . Berkov, and N. L. Gorn, Phys. Rev. Lett. 88, 047204 /H208492002 /H20850. 5M. Belov, Z. Liu, R. D. Sydora, and M. R. Freeman, Phys. Rev. B 69, 094414 /H208492004 /H20850. 6V . V . Kruglyak, A. Barman, R. J. Hicken, J. F. Childress, and J. A. Katine, Phys. Rev. B 71, 220409 /H20849R/H20850/H208492005 /H20850. 7A. V . Sokolov, Optical Properties of Metals /H20849Blackie, London, 1967 /H20850, p. 311. 8N. Qureshi, S. Wang, M. Lowther, A. R. Hawkins, S. Kwon, B. Hartle-neck, and H. Schmidt, Nano Lett. 5, 1413 /H208492005 /H20850. 9A. Barman, S. Wang, J. D. Maas, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, Nano Lett. 6,2 9 3 9 /H208492006 /H20850. 10M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W. J. M. de Jonge, and B. Koopmans, Phys. Rev. Lett. 88, 227201 /H208492002 /H20850. 11D. L. Mills and S. M. Rezende, Spin Dynamics in Conﬁned Magnetic Structures /H20849Springer, Heidelberg, 2003 /H20850, V ol. II, p. 27. 12L. D. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8,1 5 3 /H208491935 /H20850;T .L . Gilbert, Phys. Rev. 100, 1243 /H208491955 /H20850. 13T. J. Silva, M. R. Pufall, and P. Kabos, J. Appl. Phys. 91, 1066 /H208492002 /H20850. 14G. Woltersdorf, M. Buess, B. Heinrich, and C. H. Back, Phys. Rev. Lett. 95, 037401 /H208492005 /H20850. 15R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 /H208492001 /H20850. 16A. Barman, V . V . Kruglyak, R. J. Hicken, J. M. Rowe, A. Kundrotaite, J. Scott, and M. Rahman, Phys. Rev. B 69, 174426 /H208492004 /H20850. 17P. Wolf, J. Appl. Phys. 32,S 9 5 /H208491961 /H20850. 18Z. Celinski and B. Heinrich, J. Appl. Phys. 70, 5937 /H208491991 /H20850. 19A. H. Morrish, The Physical Principles of Magnetism , 1st ed. /H20849Wiley- IEEE, New York, 2001 /H20850, p. 545. 20Th. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. Bar, and Th. Rasing, Nature /H20849London /H20850418, 509 /H208492002 /H20850. FIG. 3. FFT power spectrum of the time-resolved data /H20849ﬁlled dots /H20850for the /H20849a/H208505/H9262m and /H20849b/H20850400 nm magnets with two ﬁtted /H20849Gaussian /H20850peaks as gray lines. The extracted frequencies /H20849points /H20850and the width of the peaks /H20849error bars /H20850are plotted as a function of the external bias ﬁeld for the /H20849c/H208505/H9262m and /H20849d/H20850400 nm magnets. The hatched region in /H20849c/H20850shows the overlap between the two modes.202504-3 Barman et al. Appl. Phys. Lett. 90, 202504 /H208492007 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Tue, 25 Nov 2014 14:04:44
1.1858784.pdf	Numerical integration of Landau–Lifshitz–Gilbert equation based on the midpoint rule M. d’Aquino, C. Serpico, G. Miano, I. D. Mayergoyz, and G. Bertotti Citation: J. Appl. Phys. 97, 10E319 (2005); doi: 10.1063/1.1858784 View online: http://dx.doi.org/10.1063/1.1858784 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v97/i10 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 02 Jan 2013 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsNumerical integration of Landau–Lifshitz–Gilbert equation based on the midpoint rule M. d’Aquino,a!C. Serpico, and G. Miano Department of Electrical Engineering, University of Napoli Federico II, Napoli, Italy I. D. Mayergoyz Department of Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742 G. Bertotti Istituto Elettrotecnico Nazionale, Galileo Ferraris, Strada delle Cacce, 91 I-10135, Torino, Italy sPresented on 8 November 2004; published online 9 May 2005 d The midpoint rule time discretization technique is applied to Landau–Lifshitz–Gilbert sLLG d equation. The technique is unconditionally stable and second-order accurate. It has the importantproperty of preserving the conservation of magnetization amplitude of LLG dynamics. In addition,for typical forms of the micromagnetic free energy, the midpoint rule preserves the main energybalance properties of LLG dynamics. In fact, it preserves LLG Lyapunov structure and, in the caseof zero damping, the system free energy. All the above preservation properties are fulﬁlledunconditionally, namely, regardless of the choice of the time step. The proposed technique is thentested on the standard micromagnetic problem No. 4. In the numerical computations, themagnetostatic ﬁeld is computed by the fast Fourier transform method, and the nonlinear system ofequations connected to the implicit time-stepping algorithm is solved by special and reasonably fastquasi-Newton technique. © 2005 American Institute of Physics .fDOI: 10.1063/1.1858784 g Numerical integration of the Landau–Lifshitz–Gilbert sLLG dequation has been widely used in micromagnetics for the analysis of dynamical magnetization processes. In moststudies, the time discretization is obtained by using “off-the-shelf” algorithms such as Euler, linear multistep se.g., Ad- ams, Crank–Nicholson, etc. d, and Runge–Kutta methods. 1 These standard techniques usually corrupt intrinsic geometri- cal properties of LLG time evolution and may lead to inac-curate results especially when long-term behaviors of micro-magnetic systems have to be investigated. In this respect, itis important to develop numerical schemes that have moreappropriate geometrical properties ssee, e.g., Refs. 2–5 d. In this paper, the simplicit dmidpoint rule is used for the numerical time integration of the LLG equation.This methodleads to an unconditionally stable, second-order accuratescheme, which has very important geometric preservationproperties. 6 We start our discussion with a brief review of the LLG equation and its relevant properties. The equation can bewritten in the following normalized form: ]m ]t=−m3Sheff−a]m ]tD, s1d wheremst,rd=M/Mssumu=1d,Mis the magnetization vec- tor ﬁeld, Msis the saturation magnetization, ais the dimen- sionless Gilbert damping constant, and the time is measuredin units of su guMsd−1sgis the gyromagnetic ratio d.The vector ﬁeldmst,rdis nonzero for rPV, where Vis the region occupied by the magnetic body. The normalized effectiveﬁeldheff=Heff/Mscan be deﬁned by the variational deriva- tive of the micromagnetic free-energy functional Gsmd, i.e., heff=−dG/dm.7The effective ﬁeld is typically constituted by the sum of four terms: the applied ﬁeld hastd, the exchange ﬁeldhex=2A/sm0Ms2d„2msAis the exchange constant d, the anisotropy ﬁeld han=f2K1/sm0Ms2dgeansean·mdsK1is the uniaxial anisotropy constant and eanis the easy axis unit vector d, and the magnetostatic ﬁeld hm, which can be expressed by the usual Coulomb convolution integralh m=−„reV„r8f1/s4pur−r8udg·mst,r8ddVr8. The magnetiza- tionmst,rdis also assumed to satisfy the Neumann condition ]m/]n=0 at the body surface. The ﬁrst fundamental property of LLG dynamics is the time preservation of magnetization magnitude, umst,rdu=umst0,rdu"rPV, s2d which can be easily derived from Eq. s1dby dot multiplying both sides of the equation by m. The second fundamental property can be derived, in the case of constant applied ﬁeld,by scalar multiplying both sides of the equation by sh eff −a]m/]tdand using the fact that heff=−dG/dm. This leads immediately to the following energy balance equation: d dtGstd=−E VaU]m ]tU2 dV, s3d which has very important implications. First, we notice that, for constant applied ﬁeld, the LLG dynamics has a Lyapunovstructure, namely, the free energy is always a decreasingfunction of time. This property is very important because itguarantees that the system tends toward minima of free en-ergy si.e., metastable equilibrium points d. Second, for a=0,adElectronic mail: mdaquino@unina.itJOURNAL OF APPLIED PHYSICS 97, 10E319 s2005 d 0021-8979/2005/97 ~10!/10E319/3/$22.50 © 2005 American Institute of Physics 97, 10E319-1 Downloaded 02 Jan 2013 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsthe free energy is preserved, and the LLG equation takes the Hamiltonian form ]m/]t=−m3sdG/dmd. Although the LLG dynamics is always dissipative, it is interesting to con- sider this special case since in most experimental situationsthe dissipative effect are quite small stypically a!1d.I n other terms, the LLG dynamics, on relatively short timescale, is a perturbation of the conservative precessional dy-namics. We want now to investigate the preservation properties of midpoint rule when it is applied to the LLG equation. Tothis end, let us assume that the magnetic body has been sub-divided in Ncells or ﬁnite elements. We denote the magne- tization vector associated with the k-th cell or node by m kstdPR3, and the collection of all vectors mkstdby the vectormstdPR3N. Analogous notations are used for haand heff. Let us notice that the mathematical form of the effective ﬁeld isheffsm,td=−Cm+hastd, where Cis a linear integro- differential operator. Usual spatial discretization techniques se.g., ﬁnite elements and ﬁnite difference dgenerally preserve this structure of the effective ﬁeld, and the discretized ver-sion ofh effis typically given by heffsm,td=−]G ]m=−C·m+hastd, s4d whereGsmd=s1/2dmT·C·m−haT·mis the discretized free energy and Ci sa3N33Nsymmetric matrix. Using this notation, the spatially semidiscretized LLG equation can bewritten as follows: d dtm=−Lsmd·Fheffsm,td−ad dtmG, s5d where Lsmd=diag fLsm1d,..., LsmNdgis a block-diagonal matrix with blocks Ls·dPR333such that Lsvd·w=vˆw. Equation s5dcan be numerically integrated by using the im- plicit midpoint rule which leads to the following implicittime-stepping algorithm: mn+1−mn Dt=−LSmn+1+mn 2D· ·FheffSmn+1+mn 2,tn+Dt 2D−amn+1−mn DtG, s6d which, for the generic k-th cell, can be written as mkn+1−mkn Dt=−Smkn+1+mkn 2D 3Fheff,kSmn+1+mn 2,tn+Dt 2D−amkn+1−mkn DtG.s7d Let us study the relevant properties of the midpoint dis- cretized LLG equation. First, by dot multiplying both sidesof Eq. s7dbym kn+1+mkn, it can be easily veriﬁed that umkn+1u =umknu, i.e., at each cell the magnitude of the vector magne- tization remains constant. Thus, the midpoint rule preservesexactly the LLG property s2d. Next, let us assume constant applied ﬁeld si.e., that heffdoes not depend on tdand let us multiply both sides of Eq. s6dbyfheffssmn+1+mnd/2d −asmn+1−mnd/Dtg. By using the symmetry of the matrix Cand the antisymmetry of the 3 33 blocks of the matrix Lone can readily derive the following equation: Gsmn+1d−Gsmnd Dt=−aUmn+1−mn DtU2 . s8d Notice that the proof of this equation is crucially connected with the fact that the free energy Gsmdis given by the sum of a quadratic form and a linear form in m. Equation s8dhas very important consequences. First, independently from thetime step, the discretized energy Gsmndis decreasing. Sec- ond, for a=0, the energy is exactly preserved regardless of the time step.These two properties conﬁrm the unconditionalstability of the midpoint rule, but more importantly they in-dicate that, the midpoint rule will tend to correctly reproducethe most important part in the LLG dynamics, i.e., the pre-cessional magnetization motion. The properties we have just discussed are strongly re- lated to the implicit nature of midpoint rule.As consequenceof this implicit nature, we have to solve Eq. s6dfor the un- known mn+1at each time step which amounts to solve a system of 3 Nnonlinear equations in the 3 Nunknowns mn+1. The solution of this system of equations can be obtained byusing Newton–Raphson sNRdalgorithm for the equation F nsmn+1d=0, where Fnsyd=FI−aLSy+mn 2DG·sy−mnd −DtfnSy+mn 2D, s9d andfnsmd=−Lsmd·heffsm,tn+Dt/2d. The main difﬁculty in applying NR method is that the Jacobian JnsydofFnsydis a full matrix, due to the long-range character of magnetostatic interactions. The inversion of the matrices Jnsydat each NR iteration would lead to an exceedingly high computational cost. In this respect, as it is usual in solving ﬁeld problemswith implicit time stepping, we have used a quasi-Newtonmethod by considering a reasonable approximation of theJ nsyd. We have considered the approximated sparse Jacobian J˜nsyd, obtained by neglecting in Jnsydall the terms related to magnetostatic interactions. The inversion of the sparse ma- trixJ˜nsydcan be then achieved by using fast iterative solvers. In particular, since the matrices J˜nsydto be inverted are non- symmetric, we have opted for the generalized minimal re- sidual sGMRES dmethod.8 Up to this point, the considerations we have made about the properties and the implementation of midpoint rule wererather independent from the spatial discretization techniqueused. In the following, in order to test the method, we havechosen to perform the spatial discretization by using the ﬁ-nite difference method.The magnetic body is subdivided intoa collection of rectangular prisms with edges parallel to thecoordinate axes. The magnetization is uniform within eachcell. The exchange ﬁeld is computed by means of a ssecond- order accurate in space dseven-point ﬁnite difference Laplac- ian. The magnetostatic ﬁeld is written as a discrete convolu-tion by using analytical formula proposed in Ref. 9. The10E319-2 d’Aquino et al. J. Appl. Phys. 97, 10E319 ~2005 ! Downloaded 02 Jan 2013 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsdiscrete convolution is then computed by means of three- dimensional fast Fourier transform using the zero-paddingalgorithm. We apply the above numerical technique to solve the m-mag standard problem No. 4 ssee Ref. 1 d. This problem concerns the study of magnetization reversal dynamics in athin-ﬁlm subject to a constant external ﬁeld, applied almostantiparallel to the initial magnetization. The geometry of themedium is sketched in Fig. 1. The material parameters areA=1.3 310 −11J/m,Ms=8.0 3105A/m, and K1=0J/m3. The dimension of the cells are 3.125 nm 33.125 nm 33 nm. The total number of cells is N=6400. The external ﬁeld is applied at an angle of 190° off the xaxis, with x-y components such that m0Mshax=−35.5 mT, m0Mshay= −6.3 mT, and magnitude m0Msha=36 mT. In the following, we report the comparison between the numerical solution obtained by using the proposed imple-mentation of the midpoint rule and the solutions submittedby other researchers to the m-mag website.1The time step of the midpoint algorithm is constant and has been chosen suchthats gMsd−1Dt=2.5 ps. We observe that the time steps used in the algorithms developed by other authors1are consider- ably smaller sless than 0.2 ps d. In Fig. 2 the plots of kmyls k·l means spatial average das a function of time are reported. In Fig. 3 the plot of magnetization vector ﬁeld, when the ﬁrstzero crossing of kmxloccurs, is reported. Numerical simula- tionsofthesameproblemwereperformedwithasmallercell edge s2.5 nm d, showing that the results do not depend on the mesh size. Finally, we notice that the numerical implementation of the midpoint rule fulﬁlls the preservation properties dis-cussed above only within certain accuracy. This is a naturalconsequence of the fact that we solve the time-steppingequationF nsmn+1d=0 by an iterative procedure within a cer- tain numerical tolerance. It is then important to verify a pos- teriori the accuracy in the preservation of magnetizationmagnitude and energy balance properties. To this end, wehave veriﬁed the uniformity of the magnetization vector ﬁeldby computing, at each time step, the average and the qua-dratic deviation of the values um ku, withk=1,...,N:mav =sok=1Numkud/N,sm2=ok=1Nsmav−umkud2/N. We have veriﬁed thatumav−1u,10−16andsm2,10−30. To check also the accu- racy of energy balance property preservation we have com-puted the sequence saccording to a procedure proposed in Ref. 10 d aˆn=−hfGsmn+1d−Gsmndg/Dtj/usmn+1−mnd/Dtu2 and we have veriﬁed that the relative deviation ean=uaˆn −au/ais always less than 10−7. This work is partially supported by the Italian MIUR- FIRB under Contract No. RBAU01B2T8 and by “Pro-gramma Scambi Internazionali, University di Napoli Fe-derico II.” 1m-mag group website, http://www.ctcms.nist.gov/ãrdm/mumag.org.html 2C. Serpico, I. D. Mayergoyz, and G. Bertotti, J. Appl. Phys. 89,6 9 9 1 s2001 d. 3P. S. Krishnaprasad and X. Tan, Physica B 306,1 9 5 s2001 d. 4D. Lewis and N. Nigam, J. Comput. Appl. Math. 151, 141 s2003 d. 5C. J. Budd and M. D. Piggott, Geometric Integration and Its Applications , http://www.maths.bath.ac.uk/ ˜cjb/s2001 d. 6M. A. Austin and P. S. Krishnaprasad, J. Comput. Phys. 107,1 0 5 s1993 d. 7A. Aharoni, Introduction to the Theory of Ferromagnetism sOxford Uni- versity Press, New York, 2001 d. 8Y. Saad and M. H. Schultz, SIAM sSoc. Ind. Appl. Math. dJ. Sci. Stat. Comput. 7, 856 s1986 d. 9M. E. Schabes and A. Aharoni, IEEE Trans. Magn. 23, 3882 s1987 d. 10G. Albuquerque, J. Miltat, and A. Thiaville, J. Appl. Phys. 89, 6719 s2001 d. FIG. 1. Thin-ﬁlm geometry for m-mag standard problem No. 4. FIG. 2. Plots of kmyl=kMyl/Msvs time. The external ﬁeld is applied 190° off thexaxis. FIG. 3. Snapshot of magnetization vector ﬁeld when the ﬁrst zero crossing ofkmxloccurs. The external ﬁeld is applied at an angle of 190° off the x axis.10E319-3 d’Aquino et al. J. Appl. Phys. 97, 10E319 ~2005 ! Downloaded 02 Jan 2013 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
5.0023242.pdf	Appl. Phys. Lett. 117, 122412 (2020); https://doi.org/10.1063/5.0023242 117, 122412 © 2020 Author(s).Magnon-mediated spin currents in Tm3Fe5O12/Pt with perpendicular magnetic anisotropy Cite as: Appl. Phys. Lett. 117, 122412 (2020); https://doi.org/10.1063/5.0023242 Submitted: 27 July 2020 . Accepted: 10 September 2020 . Published Online: 24 September 2020 G. L. S. Vilela , J. E. Abrao , E. Santos , Y. Yao , J. B. S. Mendes , R. L. Rodríguez-Suárez , S. M. Rezende , W. Han, A. Azevedo , and J. S. Moodera ARTICLES YOU MAY BE INTERESTED IN Robust spin–orbit torques in ferromagnetic multilayers with weak bulk spin Hall effect Applied Physics Letters 117, 122401 (2020); https://doi.org/10.1063/5.0011399 Spin current generation and detection in uniaxial antiferromagnetic insulators Applied Physics Letters 117, 100501 (2020); https://doi.org/10.1063/5.0022391 Strong interface-induced spin-charge conversion in YIG/Cr heterostructures Applied Physics Letters 117, 112402 (2020); https://doi.org/10.1063/5.0017745Magnon-mediated spin currents in Tm 3Fe5O12/Pt with perpendicular magnetic anisotropy Cite as: Appl. Phys. Lett. 117, 122412 (2020); doi: 10.1063/5.0023242 Submitted: 27 July 2020 .Accepted: 10 September 2020 . Published Online: 24 September 2020 .Corrected: 26 October 2020 G. L. S. Vilela,1,2,a) J. E.Abrao,3 E.Santos,3 Y.Yao,4,5J. B. S. Mendes,6 R. L. Rodr /C19ıguez-Su /C19arez,7 S. M. Rezende,3W.Han,4,5 A.Azevedo,3 and J. S. Moodera1,8 AFFILIATIONS 1Plasma Science and Fusion Center, and Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2F/C19ısica de Materiais, Escola Polit /C19ecnica de Pernambuco, Universidade de Pernambuco, Recife, Pernambuco 50720-001, Brazil 3Departamento de F /C19ısica, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil 4International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China 5Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 6Departamento de F /C19ısica, Universidade Federal de Vic ¸osa, Vic ¸osa, Minas Gerais 36570-900, Brazil 7Facultad de F /C19ısica, Pontiﬁcia Universidad Cat /C19olica de Chile, Casilla 306, Santiago, Chile 8Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA a)Author to whom correspondence should be addressed: gilvania.vilela@upe.br ABSTRACT The control of pure spin currents carried by magnons in magnetic insulator (MI) garnet ﬁlms with a robust perpendicular magnetic anisotropy (PMA) is of great interest to spintronic technology as they can be used to carry, transport, and process information. Garnet ﬁlmswith PMA have labyrinth domain magnetic structures that enrich the magnetization dynamics and could be employed in more efﬁcientwave-based logic and memory computing devices. In MI/non-magnetic (NM) bilayers, where NM is a normal metal providing a strong spin–orbit coupling, the PMA beneﬁts the spin–orbit torque-driven magnetization switching by lowering the needed current and rendering the process faster, crucial for developing magnetic random-access memories. In this work, we investigated the magnetic anisotropies inthulium iron garnet (TIG) ﬁlms with PMA via ferromagnetic resonance measurements, followed by the excitation and detection of magnon-mediated pure spin currents in TIG/Pt driven by microwaves and heat currents. TIG ﬁlms presented a Gilbert damping constant of a/C250:01, with resonance ﬁelds above 3.5 kOe and half linewidths broader than 60 Oe, at 300 K and 9.5 GHz. The spin-to-charge current conversion through TIG/Pt was observed as a microvoltage generated at the edges of the Pt ﬁlm. The obtained spin Seebeck coefﬁcient was0.54lV/K, also conﬁrming the high interfacial spin transparency. Published under license by AIP Publishing. https://doi.org/10.1063/5.0023242 Spin-dependent phenomena in systems composed of layers of magnetic insulators (MIs) and non-magnetic heavy metals (NMs)with strong spin–orbit coupling have been extensively explored ininsulator-based spintronics. 1–6Among the MI materials, YIG (Y3Fe5O12) is widely employed in devices for generation and transmis- sion of pure spin currents. The main reason is its very low magneticdamping with the Gilbert parameter on the order of 10 /C05and its large spin decay length, which permits spin waves to travel distances of sev-eral orders of centimeters inside it before they vanish. 7–9When com- bined with heavy metals such as Pt, Pd, Ta, or W, many intriguingspin-current related phenomena emerge, such as the spin pumpingeffect (SPE), 10–14spin Seebeck effect (SSE),7,15–18spin Hall effect (SHE),19–21and spin–orbit torque (SOT).22–25The origin of these effects relies mainly on the spin diffusion length and the quantum-mechanical exchange and spin–orbit interactions at the interface andinside the heavy metal. 26All these effects turn the MI/NM bilayer into a fascinating playground for exploring spin–orbit driven phenomenaat interfaces. 27–30 Well investigated for many years, intrinsic YIG(111) ﬁlms on GGG(111) (GGG ¼Gd3Ga5O12) exhibit in-plane anisotropy. To obtain YIG single-crystal ﬁlms with perpendicular magnetic anisot- ropy (PMA), it is necessary to grow them on top of a different Appl. Phys. Lett. 117, 122412 (2020); doi: 10.1063/5.0023242 117, 122412-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplsubstrate or partially substitute yttrium ions with rare-earth ions, to cause strain-induced anisotropy.31–33Even so, it is well-known that magnetic ﬁlms with PMA play an important role in spintronic tech- nology. The PMA enhances the spin-switching efﬁciency, which reduces the current density for observing the spin–orbit torque (SOT) effect, and it is useful for developing SOT-based magnetoresistive ran- dom access memory (SOT-MRAM).34–36Besides that, PMA increases the information density in hard disk drives and magnetoresistive random access memories,37–39and it is crucial for breaking the time- reversal symmetry in topological insulators (TIs) aiming toward quantized anomalous Hall states in MI/TI.40–42 Recently, thin ﬁlms of another rare-earth iron garnet, TIG (Tm 3Fe5O12), have caught the attention of researchers due to their large negative magnetostriction constant, which favors an out-of-plane easy axis.4,43,44TIG is a ferrimagnetic insulator with a critical tempera- ture of 549 K, a crystal structure similar to YIG, and a Gilbert dampingparameter on the order of a/C2410 /C02:4,45Investigations of spin trans- port effects have been reported in TIG/Pt45,46and TIG/TI,42,47where the TIG was fabricated by the pulsed laser deposition (PLD) technique. The results showed a strong spin mixing conductance at the interface of these materials that made it possible to observe spin Hall magneto- resistance, spin Seebeck, and spin–orbit torque effects. In this paper, we ﬁrst present a study of the magnetocrystalline and uniaxial anisotropies, as well as the magnetic damping of sput- tered epitaxial TIG thin ﬁlms using the ferromagnetic resonance (FMR) technique. For obtaining the cubic and uniaxial anisotropy ﬁelds, we analyzed the dependence of the FMR spectra on the ﬁlm thickness and the orientation of the dc applied magnetic ﬁeld at room temperature and 9.5 GHz. Then, we swept the microwave frequency for getting their magnetic damping at different temperatures. Subsequently, we focused this investigation on the excitation of magnon-mediated pure spin currents in TIG/Pt via the spin pumping and spin Seebeck mechanisms for different orientations of the dc applied magnetic ﬁeld at room temperature. Pure spin currents trans- port spin angular momentum without carrying charge currents. They are free of Joule heating and could lead to spin-wave based devicesthat are energetically more efﬁcient. Employing the inverse spin Hall effect (ISHE), 12we observed the spin-to-charge conversion of these currents inside the Pt ﬁlm, which was detected as a developed microvoltage. TIG ﬁlms with thicknesses ranging from 15 to 60 nm were depos- ited by rf sputtering from a commercial target with the same nominal composition of Tm 3Fe5O12and a purity of 99.9%. The deposition pro- cess was performed at room temperature, at a pure argon working pressure of 2.8 mTorr and a deposition rate of 1.4 nm/min. To improve the crystallinity and the magnetic ordering, the ﬁlms were post-growth annealed for 8 h at 800/C14C in a quartz tube in ﬂowing oxy- gen. After the thermal treatment, the ﬁlms yielded a magnetization sat- uration of 100 emu/cm3, and an RMS roughness below 0.1 nm was conﬁrmed using a superconducting quantum interference device (SQUID) and high-resolution x-ray diffraction measurements, as detailed in our recent article.44Moreover, the out-of-plane hysteresis loops showed curved shapes, which might be related to labyrinth domain structures very common in garnet ﬁlms with PMA.48The next step of sample preparation consisted of an ex situ deposition of a 4 nm-thick Pt ﬁlm over the post-annealed TIG ﬁlms using the dc sput- tering technique. Platinum ﬁlms were grown under an Ar gas pressureof 3.0 mTorr, at room temperature, and a deposition rate of 10 nm/ min. The Pt ﬁlms were not patterned. Ferromagnetic resonance (FMR) is a well-established technique for the study of basic magnetic properties such as saturation magneti-zation, anisotropy energies, and magnetic relaxation mechanisms. Furthermore, FMR has been central to the investigation of microwave- driven spin-pumping phenomena in FM/NM bilayers. 11,12,49First, we used a homemade FMR spectrometer running at a ﬁxed frequency of 9.5 GHz, at room temperature, where the samples were placed in themiddle of the back wall of a rectangular microwave cavity operating in the TE 102mode with a Q factor of 2500. Field scan spectra of the deriv- ative of the absorption power ( dP=dHÞwere acquired by modulating t h ed ca p p l i e dﬁ e l d ~H0with a small sinusoidal ﬁeld ~hat 100 kHz and using lock-in ampliﬁer detection. The resonance ﬁeld HRwas obtained as a function of the polar and azimuthal angles ( hH;/HÞof the applied magnetic ﬁeld ~H, as illustrated in Fig. 1(d) ,w h e r e ~H¼~H0þ~hand h/C28H0. The FMR spectra for TIG(t) ﬁlms are shown in Figs. 1(a)–1(c) for thicknesses t ¼15, 30, and 60 nm, respectively. The spectra were measured for Happlied along three different polar angles: hH¼0/C14 (blue), hHﬃ45/C14(green), and hH¼90/C14( r e d ) .T h ec o m p l e t ed e p e n - dence of HR, for each sample, as a function of the polar angle (0/C14/C20hH/C2090/C14)i ss h o w ni n Figs. 1(e)–1(g) . For all samples, HRwas minimum for hH¼0/C14, conﬁrming that the perpendicular anisotropy ﬁeld was strong enough to overcome the demagnetization ﬁeld. While the ﬁlms with t ¼15 nm and 30 nm exhibited the maximum value of HRforhH¼90/C14(in-plane), the sample with t ¼60 nm showed a maximum HRathH/C2460/C14.T oe x p l a i nt h eb e h a v i o ro f HRas a func- tion of the out-of-plane angle hH, it is necessary to normalize the FMR data to compare with the theory described as follows. The most relevant contributions to the free magnetic energy den- sity/C15for GGG(111)/TIG(111) ﬁlms are /C15¼/C15Zþ/C15CAþ/C15Dþ/C15U; (1) where /C15Zis the Zeeman energy density, /C15CAis the cubic anisotropy energy density for (111)-oriented thin ﬁlms, /C15Dis the demagnetization energy density, and /C15Uis the uniaxial energy density. Taking into con- sideration the reference frame shown in Fig. 1(d) , each energy density term can be written as:50 /C15Z¼/C0MSHsinhsinhHcos//C0/H ðÞ þcoshcoshH ðÞ ; (2) /C15CA¼K1=12 3/C06cos2hþ7cos4hþ4ﬃﬃ ﬃ 2p coshsin3/sin3h/C0/C1 ;(3) /C15Dþ/C15U¼2p~M/C1^e3/C0/C12/C0K? 2~M/C1^e3=MS/C0/C12 /C0K? 4~M/C1^e3=MS/C0/C14;(4) where hand/are the polar and azimuthal angles of the magnetization vector ~M,MSis the saturation magnetization, K1is the ﬁrst order cubic anisotropy constant, and K? 2andK? 4are the ﬁrst and second order uniaxial anisotropy constants. The uniaxial anisotropy terms come from two sources: growth-induced and stress-induced anisot-ropy. The relation between the resonance ﬁeld and the excitation fre- quency xcan be obtained from: 51,52 x=cðÞ2¼1 M2sin2h/C15hh/C15///C0/C15h/ðÞ2hi ; (5) where cis the gyromagnetic ratio. The subscripts indicate partial derivatives with respect to the coordinates, /C15hh¼@2/C15=@h2jh0;/0,Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122412 (2020); doi: 10.1063/5.0023242 117, 122412-2 Published under license by AIP Publishing/C15//¼@2/C15=@/2jh0;/0,a n d /C15h/¼@2/C15=@h@/jh0;/0,w h e r e h0and/0 are the equilibrium angles of the magnetization determined by the energy density minimum conditions, @/C15=@hjh0;/0¼0a n d @/C15=@/jh0;/0¼0. The best ﬁts to the data obtained using Eq. (5)are shown in Figs. 1(e)–1(g) by the solid red lines. The main physical parameters extracted from the ﬁts, including the effective magnetiza-tion 4 pM eff, are summarized in Table I . Here, 4 pMeff¼4pM /C02K? 2=MS, where the second term is the out-of-plane uniaxial anisot- ropy ﬁeld HU2¼2K? 2=M,a l s on a m e d H?. It is important to notice that the large negative values of HU2w e r es u f ﬁ c i e n t l ys t r o n gt os a t u - rate the magnetization along the direction perpendicular to the TIG ﬁlm’s plane, thus overcoming the shape anisotropy. We used the satu- ration magnetization as the nominal value of MS¼140:0G.A st h ethickness of the TIG ﬁlm increased, the magnitude of the perpendicu- lar magnetic anisotropy ﬁeld, HU2, decreased due to the relaxation of the induced growth stresses as expected. To obtain the Gilbert damping parameter ðaÞof the TIG thin ﬁlms, we used the coplanar waveguide technique in the variable tem-perature insert of a physical property measurement system (PPMS). Avector network analyzer measured the amplitude of the forward com- plex transmission coefﬁcients ( S 21) as a function of the in-plane mag- netic ﬁeld for different microwave frequencies ðfÞand temperatures (T).Figure 2(a) shows the FMR spectra ðS21vsHÞfor TIG(30 nm) corresponding to frequencies ranging from 2 GHz to 14 GHz at 300 K, with a microwave power of 0 dBm, after normalization by background subtraction. Fitting each FMR spectrum using the Lorentz function,we were able to extract the half linewidth DHfor each frequency, as shown in Fig. 2(b) .T h e n , awas estimated based on the linear approxi- mation DH¼DH 0þ4pa=cðÞ f,w h e r e DH0reﬂects the contribution of magnetic inhomogeneities, the linear frequency part is caused by the intrinsic Gilbert damping mechanism, and cis the gyromagnetic ratio.53The same analysis was performed for lower temperature data, and it was extended to TIG(60 nm). Due to the weak magnetization ofthe thinnest TIG (15 nm), the coplanar waveguide setup was not able to detect its FMR signals. Figure 2(c) shows the Gilbert damping dependence with T. At 300 K, a¼0:015 for TIG(60 nm), which is in agreement with the values reported in the literature, 4,45and it increases by 130% as Tgoes down to 150 K.54 Next, this work focused on the generation of pure spin currents carried by spin waves in TIG at room T, followed by their propagation FIG. 1. FMR absorption derivative spectra vs ﬁeld scan H for (a) TIG(15 nm), (b) TIG(30 nm), and (c) TIG(60 nm), at room T and 9.5 GHz. The half linewidths ( DH) for TIG(15 nm) with Happlied along hH¼0/C14;50/C14, and 90/C14are 112 Oe, 74 Oe, and 72 Oe, respectively. For TIG(30 nm), DHvalues are 82 Oe, 72 Oe, and 65 Oe for hH¼0/C14;50/C14, and 90/C14, respectively. For TIG(60 nm), DHvalues are 72 Oe, 75 Oe, and 61 Oe for hH¼0/C14;45/C14, and 90/C14, respectively. These values were extracted from the ﬁts using the Lorentz function. (d) Illustration of the FMR experiment where the magnetization ( M) under an applied magnetic ﬁeld (H) is driven by a microwave. (e)–(g) show the dependence of the resonance ﬁeld HRwithhHfor different thicknesses of TIG. The red solid lines are theoretical ﬁts obtained for the FMR condition. Magnetization curves are given in Ref. 44. TABLE I. Physical parameters extracted from the theoretical ﬁts of the FMR response of the TIG thin ﬁlms with thickness t, performed at room Tand 9.5 GHz. 4pMeffis the effective magnetization, H 1Cis the cubic anisotropy ﬁeld, and H U2and HU4are the ﬁrst and second order uniaxial anisotropy ﬁelds, respectively. H U2is the out-of-plane uniaxial anisotropy ﬁeld, also named H?. TIG film’s thickness t 15 nm 30 nm 60 nm 4pMeff(G) /C0979 /C0799 /C0383 H1C¼2K1=MS(Oe) 31 26 /C0111 HU2¼4pMeff/C04pMS(Oe) /C02739 /C02559 /C02143 HU4¼2K? 4=MS(Oe) 311 168 432Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122412 (2020); doi: 10.1063/5.0023242 117, 122412-3 Published under license by AIP Publishingthrough the interface between TIG and Pt and their spin-to-charge conversion inside the Pt ﬁlm. Initially, we explored the FMR-driven spin-pumping effect in TIG(60 nm)/Pt(4 nm), where the coherentmagnetization precession of the TIG injected a pure spin current J s into the Pt layer, which was converted as a transverse charge current Jcby means of the inverse spin Hall effect, expressed as ~Jc¼hSHr^/C2~Js/C0/C1 ,w h e r e hSHis the spin Hall angle and r^is the spin polarization.55As the FMR was excited using a homemade spectrome- ter at 9.5 GHz, a spin pumping voltage ( VSP) was detected between the two silver painted electrodes placed on the edges of the Pt ﬁlm, as illus- trated in Fig. 3(a) . It is important to note that when the magnetization vector was perpendicular to the sample’s plane, no V SPwas detected. The sample TIG(60 nm)/Pt(4 nm) had dimensions of 3 /C24m m2and a resistance between the silver electrodes of 48 Xat zero ﬁeld. VSP showed a peak value of 0 :85lVin the resonance magnetic ﬁeld for anincident power of 185 mW and an in-plane dc magnetic ﬁeld (hH¼90/C14)a ss h o w ni n Fig. 3(b) . The signal reversed when the ﬁeld direction went through a 180/C14rotation. The dependence of VSPon the microwave incident power was linear, as shown in Fig. 3(c) , whereas the spin pumping charge current ( ISP¼VSP=R) had the dependence of VSP/sinhH,a ss h o w ni n Fig. 3(d) , for a ﬁxed micro- wave power of 100 mW. The ratio between the microwave-driven volt- age and the microwave power was 4 lV/W. We also excited pure spin currents via the spin Seebeck effect (SSE) in TIG(60 nm)/Pt(4 nm) at room T. The SSE emerges from the interplay between the spin and heat currents, and it has the potential to harvest and reduce power consumption in spintronic devices.16,18 When a magnetic material is subjected to a temperature gradient, a spin current is thermally driven into the adjacent non-magnetic (NM) layer by means of the spin-exchange interaction. The spin accumula- tion in the NM layer can be detected by measuring a transversal charge current due to the ISHE. To observe the SSE in our samples, the uncovered GGG surface was placed over a copper plate, acting as athermal bath at room T, while the sample’s top was in thermal contact with a 2 /C22mm 2commercial Peltier module through a thermal paste, as illustrated in Fig. 4(a) . The Peltier module was responsible for creating a controllable temperature gradient across the sample. On the other hand, the temperature difference ( DT) between the bottom and top of the sample was measured using a differential thermocouple. The ISHE voltage due to the SSE ( VSSE) was detected between the two silver painted electrodes placed on the edges of the Pt ﬁlm. The behavior of VSSEby sweeping the dc applied magnetic ﬁeld (H), while DT,hH,a n d /Hwere kept ﬁxed, was investigated. Fixing /H¼0/C14and varying the magnetic ﬁeld from out-of-plane ( hH¼0/C14Þ to in-plane along the x-direction ( hH¼90/C14Þ,VSSEwent from zero to its maximum value of 5.5 lVforDT¼20K,a ss h o w ni n Fig. 4(b) . Around zero ﬁeld, no matter the value of hH, the TIG’s ﬁlm magneti- zation tended to rely along its out-of-plane easy axis, which zeroes VSSE. For in-plane ﬁelds ( hH¼90/C14)w i t h DT¼12K,VSSEwas maxi- mum when /H¼0/C14, and it was zero for /H¼90/C14.T h er e a s o nt h a t VSSEwent to zero for /H¼90/C14may be attributed to the generated charge ﬂow along the x-direction, while the silver electrodes were placed along the y-direction, thus not enabling the current detection [seeFig. 4(c) ]. The analysis of the spin Seebeck amplitude DVSSEvs hH,/H,a n dDTshowed a sine, cosine, and linear dependence, respec- tively, as can be seen in Figs. 4(d) and4(e),w h e r et h er e ds o l i dl i n e s are theoretical ﬁts. The Spin Seebeck coefﬁcient (SSC) extracted from the linear ﬁt of DVSSEvsDTwas 0.54 lV/K. FIG. 2. (a) Ferromagnetic resonance spectra vs in-plane applied ﬁeld Hfor a 30 nm-thick TIG ﬁlm at frequencies ranging from 2 GHz to 14 GHz and a temperature of 300 K, after normalization by background subtraction. (b) Half linewidth DHvs frequency for TIG(30 nm) at 300 K. The Gilbert damping parameter awas extracted from the linear ﬁtting of the data. (c) Damping avs temperature Tfor TIG ﬁlms with thicknesses of 30 nm and 60 nm. FIG. 3. Spin pumping voltage (V SP) excited by a FMR microwave of 9.5 GHz, at room T, in TIG(60 nm)/Pt(4 nm). (a) Illustration of the spin pumping setup. (b) In- plane ﬁeld scan of V SPfor different microwave powers. (c) Linear dependence of the maximum V SPwith the microwave power. (d) hHscan of the charge current (ISP) generated by means of the inverse spin Hall effect in the Pt ﬁlm. (e) In-plane ﬁeld scan of the FMR absorption derivative spectrum for 5 mW.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122412 (2020); doi: 10.1063/5.0023242 117, 122412-4 Published under license by AIP PublishingIn conclusion, we used the FMR technique to probe the magnetic anisotropies and the Gilbert damping parameter of the sputtered TIGthin ﬁlms with perpendicular magnetic anisotropy. The results showedhigher resonance ﬁelds ( >3.5 kOe) and broader linewidths ( >60 Oe) when comparing with YIG ﬁlms at room T. Thinner TIG ﬁlms (t¼15 nm and 30 nm) presented a well-deﬁned PMA; on the other hand, the easy axis of the thicker TIG ﬁlm (60 nm) showed a deviationof 30 /C14from normal to the ﬁlm plane. By numerically adjusting the F M Rﬁ e l dd e p e n d e n c ew i t ht h ep o l a ra n g l e ,w ee x t r a c t e dt h ee f f e c t i v e magnetization, the cubic (H 1C), and the out-of-plane uniaxial anisot- ropy (H U2¼H?) ﬁelds for the three TIG ﬁlms. The thinnest ﬁlm pre- sented the highest intensity for H ?as expected, even so H ?was strong enough to overcome the shape anisotropy and gave place to a perpendicular magnetic anisotropy in all the three thickness of TIGﬁlms. The Gilbert damping parameters ðaÞfor TIG(30 nm) and TIG(60 nm) ﬁlms were estimated to be /C2510 /C02, by analyzing a set of FMR spectra using the coplanar waveguide technique at various microwave frequencies and temperatures. As Twent down to 150 K, the damping increased monotonically 130%. Furthermore, spin waves (magnons) were excited in the TIG(60 nm)/Pt(4 nm) heterostructure through the spin pumping and spin Seebeck effects, at room Tand 9.5 GHz. The generated pure spin currents carried by the magnons were converted into charge currentsonce they reached the Pt ﬁlm by means of the inverse spin Hall effect. The charge currents were detected as a microvoltage measured at the edges of the Pt ﬁlm, and they showed sine and cosine dependenceswith the polar and azimuthal angles, respectively, of the dc appliedmagnetic ﬁeld. This voltage was linearly dependent on the microwave power for the SPE and on the temperature gradient for the SSE. Theseresults conﬁrmed a good spin-mixing conductance in the interface TIG/Pt and an efﬁcient conversion of pure spin currents into chargecurrents inside the Pt ﬁlm, which is crucial for the employment of TIGﬁlms with a robust PMA in the development of magnon-based spin-tronic devices for computing technologies. This research was supported in the USA by the Army Research Ofﬁce (Nos. ARO W911NF-19-2-0041 and W911NF-20-2-0061),NSF (No. DMR 1700137), and ONR (No. N00014-16-1-2657), inBrazil by CAPES (No. Gilvania Vilela/POS-DOC-88881.120327/2016-01), FACEPE (Nos. APQ-0565-1.05/14 and APQ-0707-1.05/14), CNPq, UPE (No. PFA/PROGRAD/UPE 04/2017), andFAPEMIG-Rede de Pesquisa em Materiais 2D and Rede deNanomagnetismo, in Chile by Fondo Nacional de DesarrolloCient /C19ıﬁco y Tecnol /C19ogico (FONDECYT) No. 1170723, and in China by the National Natural Science Foundation of China (No.11974025). DATA AVAILABILITY The data that support the ﬁndings of this study are available from the corresponding author upon reasonable request. REFERENCES 1P. Pirro, T. Br €acher, A. V. Chumak, B. L €agel, C. Dubs, O. Surzhenko, P. G€ornert, B. Leven, and B. Hillebrands, Appl. Phys. Lett. 104, 012402 (2014). 2A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015). 3L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Nat. 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1.5007324.pdf	Effect of external magnetic field on locking range of spintronic feedback nano oscillator Hanuman Singh , K. Konishi , A. Bose , S. Bhuktare , S. Miwa , A. Fukushima , K. Yakushiji , S. Yuasa , H. Kubota , Y. Suzuki , and A. A. Tulapurkar Citation: AIP Advances 8, 056010 (2018); View online: https://doi.org/10.1063/1.5007324 View Table of Contents: http://aip.scitation.org/toc/adv/8/5 Published by the American Institute of Physics Articles you may be interested in Conversion of spin current into charge current at room temperature: Inverse spin-Hall effect Applied Physics Letters 88, 182509 (2006); 10.1063/1.2199473 Spin transfer torque nano-oscillators based on synthetic ferrimagnets: Influence of the exchange bias field and interlayer exchange coupling Journal of Applied Physics 121, 013903 (2017); 10.1063/1.4973525 Efficient micromagnetic modelling of spin-transfer torque and spin-orbit torque AIP Advances 8, 056008 (2017); 10.1063/1.5006561 Antidamping spin-orbit torques in epitaxial-Py(100)/ β-Ta Applied Physics Letters 111, 232407 (2017); 10.1063/1.5007202 Nanoconstriction-based spin-Hall nano-oscillator Applied Physics Letters 105, 172410 (2014); 10.1063/1.4901027 Opposite signs of voltage-induced perpendicular magnetic anisotropy change in CoFeB\|MgO junctions with different underlayers Applied Physics Letters 103, 082410 (2013); 10.1063/1.4819199AIP ADV ANCES 8, 056010 (2018) Eﬀect of external magnetic ﬁeld on locking range of spintronic feedback nano oscillator Hanuman Singh,1,aK. Konishi,2A. Bose,1S. Bhuktare,1S. Miwa,2 A. Fukushima,3K. Yakushiji,3S. Yuasa,3H. Kubota,3Y. Suzuki,2 and A. A. Tulapurkar1 1Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India 2Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan 3National Institute of Advanced Industrial Science and Technology (AIST), Spintronics Research Center, Ibaraki 305-8568, Japan (Presented 9 November 2017; received 2 October 2017; accepted 26 October 2017; published online 13 December 2017) In this work we have studied the effect of external applied magnetic ﬁeld on the locking range of spintronic feedback nano oscillator. Injection locking of spintronic feedback nano oscillator at integer and fractional multiple of its auto oscillation fre- quency was demonstrated recently. Here we show that the locking range increases with increasing external magnetic ﬁeld. We also show synchronization of spintronic feedback nano oscillator at integer (n=1,2,3) multiples of auto oscillation frequency and side band peaks at higher external magnetic ﬁeld values. We have veriﬁed experimental results with macro-spin simulation using similar conditions as used for the experimental study. © 2017 Author(s). All article content, except where oth- erwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5007324 The spin transfer-torque (STT)1–7effect has been used to realize nano-scale microwave oscilla- tors (STNO). The low output power and large linewidth of STNO are considered as biggest challenges for practical applications of STNO. Recently our group demonstrated a new type of spintronic oscillator based on the magnetic ﬁeld feedback (SFNO). SFNO8,9shows very high quality factor (Q= frequency/Line width). The working of SFNO is based on the tunneling magneto-resistance (TMR) effect and can work without STT. SFNO comprises a magnetic tunnel junction (MTJ) nano- pillar and a co-planer wave guide (CPW) insulated from it. Such a system can amplify rf signals passing through the waveguide when dc current ﬂows through the MTJ. The rf signal through the waveguide excites the magnetization of the free layer via the Oersted magnetic ﬁeld, which is con- verted into rf voltage by the dc current via the TMR effect. For dc current exceeding a threshold value, the input signal can be ampliﬁed. Such an amplifying system can work as oscillator if positive feedback is provided from the output to the input side. A distinctive feature of the SFNO is the presence of side band peaks along with the main peak in the power spectral density (PSD), which arises due to the delayed feedback. A large output power out can be obtained by mutual synchronization of oscillators. A ﬁrst step in such a study is the injection locking of a single oscillator to a microwave source. These studies are also important to understand the non-linear dynamics of auto-oscillator. Many extensive theoretical and experimental studies have been carried out on synchronization of STNO. In previous studies many research groups have shown injection locking of STNOs10–23to external rf current and microwave magnetic ﬁeld. In the case of SFNO, the oscillator itself is based on feedback of rf magnetic ﬁeld, which is created by the oscillation of the free layer. In addition to that, we need to add external rf aCorresponding Author: Hanuman Singh Email: hanumanbana20@gmail.com 2158-3226/2018/8(5)/056010/5 8, 056010-1 ©Author(s) 2017 056010-2 Singh et al. AIP Advances 8, 056010 (2018) magnetic ﬁeld for injection locking experiments. Recently our group carried out injection locking study of SFNO to microwave magnetic ﬁeld.24It was demonstrated that SFNO can be injection locked at integer as well as fractional multiple of the free running auto-oscillation frequency.24It was also shown that the SFNO can be injection locked on its side band peaks.24The concept of SFNO is not limited to feedback from Oersted magnetic ﬁeld but it can be realized with different technique of feedback like inverse spin Hall effect25,26and Rashba interfacial coupling.27,28We can use the effect of SFNO concept with STT29to improvement in output power and quality factor. In this study we shown the synchronization of SFNO to external microwave magnetic ﬁeld at integer multiples (n=1,2,3) and side band peaks of its auto-oscillation frequency with higher external magnetic ﬁeld perpendicular to easy axis. In particular, we have investigated the dependence of n=1 locking range on the external magnetic ﬁeld. The SFNO studied here comprises of MTJ nano pillar of size 300 x 500 nm2in elliptical shape. The MTJ nano pillar were fabricated using e-beam lithography and Ar ion milling from multilayer stack of the following structure: Ta(5nm)/Cu(20nm)/Ta(5nm)/Cu(20nm)/Ta(3nm)/Ru(5nm)/IrMn (7nm)/CoFe(3nm)/Ru(0.8nm)/CoFeB(3nm)/CoFe(0.4nm)/MgO(0.9nm)/CoFeB(3nm)/Ta(5nm)/Cu (30nm)/Ta(5nm)/Ru(5nm) on thermally grown SiO 2(500 nm). A 1 m wide co-planar wave guide (CPW) was fabricated on top of MTJ. The 100 nm thick SiO 2layer is used to isolate MTJ top contact and CPW. The orientation of fabricated CPW is adjusted such that current passing through it creates magnetic ﬁeld along x axis as shown in ﬁg. 1(a). The pinned layer magnetization is pinned along x-axis. Which is also the easy axis of the free layer. To generate auto oscillation in free layer of MTJ we applied external magnetic ﬁeld perpendicular to easy axis and a constant bias current through bias-tee network. The rf output from the MTJ is divided into two parts using a power splitter: one part is measured using network analyzer and second part of output signal is ampliﬁed and fed into CPW. The realization of feedback without ampliﬁer is also possible if we have high TMR device and very thin CPW as discussed in Ref. 9. The synchronization of SFNO to external rf magnetic ﬁeld is realized by adding rf current from rf signal generator to the feedback line through one port of direc- tional coupler as shown in ﬁg. 1(a). The measurement to study effect of external magnetic ﬁeld on the locking range of SFNO is carried out by changing external magnetic ﬁeld applied perpendicular to easy axis. All the measurement was carried out at room temperature. The SFNO power spectral density (PSD) in log scale is shown in ﬁg. 1(b) when we applied external magnetic ﬁeld ( Hext) of 100 Oe along y-axis and bias current of -2.3 and 27 dB gain of ampliﬁer. We have shown the variation of auto-oscillation frequency of SFNO with applied external FIG. 1. (a) Show the schematic for experimental set-up used to study the synchronization of spintronic feedback nano oscillator (SFNO). The SFNO consist of MTJ and an electrically insulated coplanar wave guide on top. The DC current is used to bias the MTJ through bias-tee. An oscillating voltage is generated through TMR effect of MTJ due to oscillation of free layer and passage of dc current. The generated output signal is collected through bias-tees and split into two parts using power splitter. One part of output signal is ampliﬁed and used as feedback signal to CPW through directional coupler and second part of signal is directly observed through spectrum analyzer. The one port of directional coupler is used to add the external rf signal in feedback line. Fig. 1(b) shows Power spectral density of the free running oscillator (i.e. oscillator with feedback but without external locking magnetic ﬁeld) in log scale, with applied magnetic ﬁeld of 100 Oe along y-axis, bias current of -2.3mA and 27 dB gain of ampliﬁer. The ﬁrst inset of ﬁg. 1(b) shows the variation of free running oscillator frequency with external ﬁeld (hext), the second inset shows zoomed in spectrum of main peak in linear scale. The main peak shows a high Q factor of 4800 and the third inset shows the TMR of the device.056010-3 Singh et al. AIP Advances 8, 056010 (2018) magnetic ﬁeld ( Hext) in the inset (i) of ﬁg. 1(b). The SFNO have high quality factor (Q = frequency (f)/linewidth ( f)) of 4800 as shown in inset (ii) of ﬁg. 1(b) with same bias and external magnetic ﬁeld. The observed TMR value of the MTJ device used for experiment is 56% as shown in inset (iii) of ﬁg. 1(b). The PSD spectrum in log scale shows that the main peak is accompanied by side peaks with separation of around 120 MHz which is corresponding to delay of 8 ns in feedback circuit. These separation of side peak can be changed by changing delay as shown in Ref. 9. To demonstrate the synchronization of SFNO to rf magnetic ﬁeld, we passed rf current through the CPW via a directional coupler as shown in ﬁg. 1(a). The rf current creates rf magnetic ﬁeld ( he) along x axis. Fig. 2(a) shows the 2D color plot of PSD as a function of external frequency ( fe) obtained athe=5.5Oe with applied external magnetic ﬁeld of Hext=100 Oe. This ﬁgure clearly shows the synchronization phenomena observed at integer (n=1,2,3) multiples of the auto-oscillation frequency, f0of SFNO . Fig. 2(b) shows PSD plots obtained for three different values of feclose to f0,2f0, and 3f0(i.e. for n=1,2,3 phase locking) along with the PSD of free running SFNO. We have studied the effect of external dc magnetic ﬁeld ( Hext) on the locking range of SFNO. We measured locking range for n=1 for different values of Hext. We deﬁne locking range as frequency range over which oscillator frequency matches with the external frequency. It was found that the locking range increases with Hextas shown in the ﬁg. 2(c). We further explored the effect of injection locking on the side band peaks. The result shown in ﬁg. 3(a–c) are obtained at h e=5.5Oe with applied external magnetic ﬁeld Hext=100 Oe along y axis. The 2D color plots in ﬁg. 3(a–c) shows synchronization of side band peaks and main peaks and its effect on the other peaks. Panel (a) shows that when left side peak is locked the main peak and right side peak are suppressed. Panel (b) shows that when center peak is locked the left side peak and right side peak are suppressed. Similarly, panel (c) shows that when right side peak is locked the main peak and left side peak are suppressed. Similar results were obtained in Ref. 24 for lower values of dc magnetic ﬁeld. These results show that SFNO can be injection locked on the side band peaks, which effectively increases the locking range. As the position of side band peaks can be controlled by the feedback delay (Ref. 9) this provides a useful technique for locking oscillators with large difference in their free running frequencies. We carried out macro-spin simulation of the injection locking of feedback oscillator to support the experimental results. The LLG equation was modiﬁed to include the effect of magnetic ﬁeld feedback (Refs. 8, 9, and 24) as: ˙ˆm= ˆm(¯Heff+¯hr+¯hfb+¯he) +( ˆm˙ˆm) FIG. 2. (a) 2D plot of power spectral density (PSD) as a function of external frequency (f e) applied at integer (n=1, 2, 3) of the auto-oscillation frequency (f 0) of the free running SFNO at h e=5.5Oe. (b) PSD obtained for three values of f eclose to f 0, 2f0, and 3f 0(i.e. for n=1,2,3 phase locking) respectively with the free running PSD of SFNO shown for comparison. (c) shows locking range for n=1 multiple as a function of external dc magnetic ﬁeld (H ext) applied along y-axis. Rf magnetic ﬁeld of he=5.5 Oe was applied.056010-4 Singh et al. AIP Advances 8, 056010 (2018) FIG. 3. (a-c) 2D color plot shows locking of side band peaks and main peaks and its effect on main peak and side band peaks when locked as a function of f eat h e=5.5Oe with applied 100 Oe magnetic ﬁeld. (d) shows Simulation results at T=300K: PSD plot obtained for three values of f eclose to f 0(i.e. for n=1 phase locking) with the free running PSD of SFNO at external ﬁeld H extof 100 Oe along Y axis and -35 Oe ﬁeld along X axis and external rf ﬁeld h e=2 Oe. In above equation ˆ mdenotes unit vector along magnetization, is the gyromagnetic ratio, Heffis the effective magnetic ﬁeld comprising the external ﬁeld ( Hext) and the anisotropy ﬁeld ( Hani), the random magnetic ﬁeld ( hr), feedback ﬁeld (hfb)and external rf magnetic ﬁeld (he).is the Gilbert damping constant. The random magnetic ﬁeld satisﬁes the following statistical properties:8 hr,i(t)=0,D hr,i(t)hr,j(s)E =2Dij(ts),D=kBT=( 0MSV) where kBis the Boltzmann constant , Tis temperature ,0is magnetic permeability , MSis saturation magnetization, and Vis volume. Dis taken as the strength of the thermal ﬂuctuations. The feedback ﬁeld is given by,8hfb(t)=Idcˆx[R(tt)R0]=[2w(R0+RT)] , where Idc,w,R0andRTdenotes the dc current, the width of the feedback line, the average resistance of MTJ, and the termination resistance (50 ) respectively. tandR(t-t)also denotes the feedback delay and the resistance of MTJ at time t-trespectively. The following parameters are used in the simulation: =0.01, =2.21X105A/ms, T=300 K, MS=1000 emu/cc, V= (500 nm X 300 nm X 3nm). The anisotropy magnetic ﬁeld ( Hani) is given by: Hani=H==mx+H?mz, where H==andH?denote the in-plane and out-of-plane anisotropy ﬁelds. Positive values of H==andH?imply that x-axis is the easy axis and z-axis is out-of-plane hard axis. We have used H===50 Oe and H?=104Oe. We assumed feedback strip width as 1 m and ampliﬁcation gain of 20 dB. The parameters used for simulations, are similar to the experimental conditions. The simulation results for synchronization of SFNO at n=1 are shown in ﬁg. 3(d). This ﬁgure shows PSD plot obtained for three values of f eclose to f 0(i.e. for n=1 phase locking) with the free running PSD of SFNO for he=2 Oe with Hext=100 Oe along Y axis and 35 Oe ﬁeld along X axis. As can be seen from the TMR data shown in in the inset ii of ﬁg. 1(b), the center of TMR loop is shifted by about -35 Oe. To account for this, 35 Oe ﬁeld along x-axis was used in the simulation. The simulations results shown in ﬁg. 3(d) show the n=1 locking phenomena. The red curve in ﬁg. 3(d) is the psd of the SFNO oscillator without any locking signal. When external rf magnetic ﬁeld with frequency close to the free running frequency is applied, the oscillation frequency follows the applied rf frequency. This can be seen from the dark blue and light blue curves in ﬁg. 3(d). These simulation results match with the experimental data as shown in ﬁg. 2(b). The simulations were also carried out for different values of external magnetic ﬁelds along y-axis. It was also found that the locking range increases with increasing magnetic ﬁeld in agreement with the experimental results. In summary, we have studied the injection locking of SFNO for different external dc magnetic ﬁelds. We showed the injection locking at integer multiples of the free running frequency as well as injection locking at the side band peak positions. We found that the locking range increases with increasing external dc magnetic ﬁeld. The experimental results are supported by maco-spin LLG simulations.056010-5 Singh et al. AIP Advances 8, 056010 (2018) ACKNOWLEDGMENTS We are thankful to the Centre of Excellence in Nanoelectronics (CEN) at the IIT-Bombay Nanofabrication facility (IITBNF) and Ministry of Electronics and Information Technology (Meity), Government of India for its support. 1J. C. Slonczewski, “Current-driven excitation of magnetic multilayers,” J. Magn. Magn. Mater. 159, L1–L7 (1996). 2S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, “Microwave oscillations of a nanomagnet driven by a spin-polarized current,” Nature (London) 425, 380 (2003). 3S. Sharma, B. Muralidharan, and A. 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1.5129954.pdf	AIP Advances 9, 125130 (2019); https://doi.org/10.1063/1.5129954 9, 125130 © 2019 Author(s).Micromagnetic simulations of first-order reversal curves in nanowire arrays using MuMax3 Cite as: AIP Advances 9, 125130 (2019); https://doi.org/10.1063/1.5129954 Submitted: 02 October 2019 . Accepted: 03 November 2019 . Published Online: 23 December 2019 R. G. Eimerl , K. S. Muster , and R. Heindl COLLECTIONS Paper published as part of the special topic on 64th Annual Conference on Magnetism and Magnetic Materials Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials. AIP Advances ARTICLE scitation.org/journal/adv Micromagnetic simulations of first-order reversal curves in nanowire arrays using MuMax3 Cite as: AIP Advances 9, 125130 (2019); doi: 10.1063/1.5129954 Presented: 7 November 2019 •Submitted: 2 October 2019 • Accepted: 3 November 2019 •Published Online: 23 December 2019 R. G. Eimerl,a)K. S. Muster,a)and R. Heindlb) AFFILIATIONS Department of Physics and Astronomy, San Jose State University, San Jose, California 95112, USA Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials. a)These authors contributed equally to this work. b)Electronic mail: ranko.heindl@sjsu.edu. ABSTRACT We perform simulations of magnetic reversal in a 3 ×3 array of nanowires using MuMax3 micromagnetic simulation program. We record a series of first-order reversal curves (FORCs) that form distinct branches of ascending minor curves depending on the ini- tial magnetization state. We calculate the FORC distribution, which shows 9 positive primary peaks, representing single reversals of the 9 simulated nanowires. The primary peaks form an interaction field distribution (IFD), a common feature in experimental FORC distri- butions due to demagnetizing interactions. The FORC distribution also contains positive and negative secondary peaks due to differing magnetization during reversal. We demonstrate the use of MuMax3 simulations to relate FORC distribution features to visualized magnetic configurations. ©2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5129954 .,s I. INTRODUCTION Magnetic nanowire arrays are promising nanostructures for applications in high-density magnetic recording,1,2microwave devices,3,4spintronic devices,5,6and sensors.7,8The magnetic rever- sal fields of these arrays can be mapped by the first-order reversal curve (FORC) distribution.9However, large interaction fields can reshape the FORC distribution of arrays and influence the over- all switching behavior.10,11Better understanding of nanowire array switching is needed to mitigate or utilize interaction effects in future devices. Previous simulations of nanowire arrays have modeled grids of point-dipoles,12grids of point-elements,13–15or collections of hys- teretic elements,11which exhibit FORC distribution features sim- ilar to those observed experimentally. These models allow simu- lation of arrays composed of many nanowires but at the cost of approximating or neglecting intra-wire interactions and inhomoge- neous states. Our goal has been to simulate the reversal processes in nanowire arrays and to map changes in magnetization to features in the FORC diagram. We have used micromagnetic simulations of 3-D nanowires to investigate how the internal magnetization rever- sals, including mixed or inhomogeneous states, of the nanowires inthe system appear in FORC distributions. Since full micromagnetic simulations are more computationally intensive than simulations of simplified models, we are limited to small arrays. In this study, we have simulated 3 ×3 and 4 ×4 nanowire arrays. Here, we report results for the 3 ×3 array, noting that the 4 ×4 array exhibits similar results. II. MICROMAGNETIC SIMULATION Simulation has been performed with the open-source program MuMax3. MuMax3 simulates magnetization dynamics in nano- and microscale ferromagnetic structures.16The magnetization at all points in the model is solved simultaneously and over time from the Landau–Lifshitz–Gilbert equation.17The program computes finite- difference solutions in CUDA-enabled GPUs.16Prior to simulation, we have tested and verified MuMax3 by matching the published results for the Standard Problems.18 The 3 ×3 array of nanowires is composed of identical cylin- ders with diameter ( D= 80 nm) and length ( L= 1μm). The chosen length is in the regime for which modeling nanowires as point- dipoles may be limited.19The wires are arranged in a 2-D square AIP Advances 9, 125130 (2019); doi: 10.1063/1.5129954 9, 125130-1 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv grid with equal spacing between nanowire centers ( r= 200 nm). Every nanowire has the same exchange constant ( Aex= 1.3×10−11 J m−1), saturation magnetization ( Ms= 800 kA m−1), and damp- ing ( α= 0.008). Therefore, we have not introduced differences in reversal behavior of the nanowires themselves, which would oth- erwise alter the FORC distribution.15,20The system is discretized into 4 nm ×4 nm ×4 nm cells, width smaller than the exchange length ( lex=√ 2Aex/μ0M2s≤5.69 nm). Thermal effects are not accounted for. After the model is constructed, and before solving the Landau– Lifshitz–Gilbert equation, the demagnetization kernel is calculated for each finite-difference cell in the simulation. MuMax3 does this by assuming a uniform magnetization for each cell, directly cal- culating the demagnetization field, solving for the demagnetizing field kernel, then convolving with the magnetization of a particu- lar state.16When determining the kernel, MuMax3 allows for peri- odic boundary conditions that extend the system. This significantly reduces run-time due to the lack of zero-padding required for the Fourier transforms.16This also prevents interactions being weaker for nanowires along the edges or corners of the simulation win- dow for better representation of an infinite array.15Our simula- tions repeat the system four times on each side of the simulation window. III. FORC SIMULATION The simulated external magnetic field is varied similarly to the applied field in experimental FORC measurements. First, a field sufficient to saturate the system is applied ( Hs= 119 kA m−1). The descending major curve is simulated while the 3-D magneti- zation and initial external field ( HR) are recorded. Unlike experi- ment, where saturation and the descending sweep are performed between each FORC, we initialize each FORC using these saved configurations to reduce total run time by 50%. A particular FORC, or ascending minor curve, is then mea- sured by recording the normalized magnetic moment ( m) as the external field ( H) is increased to positive saturation at Hs. The net moment and the external magnetic field are recorded as two 3-D vectors, although only the components parallel to the nanowire axis are analyzed. This process is repeated for 60 values of HR between + Hsand−Hs. This provides a data set describing the mag- netic moment as a function of the reversal field and the external field [ m(HR,H)], known as the FORC family (Fig. 1). Since each FORC is an independent simulation from known initial states, mul- tiple FORCs can be simulated in parallel to further reduce total run time. The low Hportion of some curves forms a branch, a visible ver- tical gap indicating a greater mthan that of the curve labeled “Low” at the same H. The branches demonstrate hysteresis, a dependence ofmon the initial state at HRas well as on Hcommonly seen in FORC measurement. Above a particular value of Hand below a value of HR, the FORCs merge with Low. We refer to this thresh- old ( H,HR) as the “merge point.” There are 9 distinct hysteretic branches, labeled “A” to “I,” which we have verified on an ( H,HR) plot after subtracting the Low curve for improved contrast. Each branch is composed of several overlapping FORCs (Fig. 1 inset). The curves have similar slope at Hvalues between merge points. FIG. 1 . Simulated FORC family of the 3 ×3 nanowire array with color-coded branch features. The inset shows branch D, a set of 7 overlapping FORCs with different starting fields separated from the ascending major curve labeled “Low.” Nine such branches are observed and labeled alphabetically. States imaged in Fig. 3 and Fig. 4 are labeled for reference in Section V (squares). The sloped component of mis dependent on Hbut independent of HR(discussed below). At fields above the merge point, some FORCs exhibit tem- porary separation from the Low curve. The states along different FORCs but at the same Hhave nearly equal mexcept near the merge points. IV. FORC ANALYSIS The analysis method of the simulated FORC data is identical to that of experimental FORC data. The Preisach model describes mag- netic reversal as the collective behavior of elementary hysteretic ele- ments called “hysterons.”21The FORC density, representing change inmdue to a single hysteron reversal, is calculated as the mixed second derivative of FORC data,9 ρ(H,HR)=−1 2∂2m(H,HR) ∂H∂HR. (1) This density calculated over the ( H,HR) half-plane ( H>HR) is the FORC distribution.9The FORC distribution is often displayed as a contour or density plot in the basis of the interaction field [Hu= (H+HR)/2] and the coercive field [ Hc= (H−HR)/2]. These new fields represent the center (interaction) and the hysteresis width (coercivity) of each hysteron.20 The resulting FORC distribution [ ρ(Hc,Hu) calculated using Eq. (1)] is shown in Fig. 2. The hysteretic branch features of Fig. 1 can be described by their merge points, labeled by branch in Fig. 2. Each merge point has a corresponding positive primary peak in the calculated FORC distribution. Positive and negative secondary peaks are seen at similar Hbut lower HR. Dobrot ˘a and Stancu have observed comparable FORC dis- tributions of a simulated array of 1600 nanowires, simulated as point-elements. Individual nanowires reverse at different field values between FORCs, creating multiple positive and negative contribu- tions across the ( H,HR) plane with fine field steps.15Dobrot ˘a and AIP Advances 9, 125130 (2019); doi: 10.1063/1.5129954 9, 125130-2 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv FIG. 2 . The calculated FORC distribution of the simulated array (color) and the merge points of the 9 observed branched features in Fig. 1 representing a net moment contribution. Stancu verify the net contribution of a single nanowire is a unit of m, identified by a single positive primary contribution at highest HR.15 We observe similar notable contributions to m(merge points) from 9 primary peaks, possibly one for each nanowire. The observed hysterons form a ridge parallel to the Huaxis. This feature is comparable to an intrinsic field distribution (IFD) in experimental nanowire array FORC distributions. The IFD has been attributed to geometry-dependent demagnetizing interactions between nanowires in the array.19,22In the moving Preisach model, hysterons are shifted from their intrinsic positions on the Hcaxis by the mean interaction field ( Hint).10,11This model has been usedto interpret experimental results.11,19The intrinsic field that causes hysterons to reverse is Hc=H+mH int.10Linear fit of the merge point fields versus moments provide intrinsic hysteron coerciv- ity (Hc= 1.18 ±0.07 kAm−1) and maximum mean interaction field ( Hint=−111.3 ±0.1 kAm−1), or effective array axial demag- netizing field ( Naxial =−Hint/Ms= 0.1391 ±0.0001) with small root mean square error (RMSE = 0.4 kAm−1). The small error in the linear trend suggests primary reversals occur with simi- lar coercivity and an interaction field that is uniform between elements. We do not observe peaks forming a clear coercive field distri- bution (CFD), which often appears in experimental FORC distri- butions.22From the simulation study by Dobtrot ˘a and Stancu, this can be attributed to our number of nanowires being small com- pared to the number of field steps.15The individual contributions produce finer features in the FORC distribution instead of statistical distributions.15 As mentioned, the constant-slope component of min Fig. 1 is independent of HR. As expected from Eq. (1), the sloped component does not have a corresponding positive density in the FORC distri- bution, which only shows hysteretic or irreversible change in mag- netic moment.9Instead, the sloped component represents a large reversible change in magnetic moment. V. VISUALIZATION 3-D visualization allows us to deduce the mechanisms respon- sible for the three observed FORC features: hysteretic branches (pri- mary peaks at merge points), variations between FORCs (secondary peaks at equal Hand lower HR), and non-hysteretic slope (all Hbut not visible in the FORC distribution). Fig. 3 shows the 3-D visual- ization comparing the magnetization of FORCs starting in branches D and Low near the merge point D. From Fig. 2, FORCs from ini- tialization to merge point D should differ from the Low curve by the contribution of peak D. In images L(i) to L(iv) of Fig. 3, nanowire β2 evolves from a mixed state to saturated up. Nanowire γ2 appears nearly saturated in L(i) but relaxes to a mixed state in L(iv), so we consider this FIG. 3 . 3-D visualization and comparison of a FORC from the D branch [D(i)-D(iv)] with the Low curve [L(i)-L(iv)] at Hnear that of the D merge point in Fig. 1 and reported in Table I. The images have varying number of nanowires sat- urated up (blue) and down (red), demonstrating hysteresis. The labeling convention of nanowires is overlaid in D(i); for example, the center nanowire is referred to as “ β2.” AIP Advances 9, 125130 (2019); doi: 10.1063/1.5129954 9, 125130-3 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv TABLE I . Normalized full and relative moments associated with the two FORCs shown in Fig. 3. H(kA m−1) HR(kA m−1) −11.966 −7.977 −3.988 0.000 −119.655 −0.1253 −0.0880 −0.0530 −0.0125 −31.908 −0.1062 −0.0836 −0.0532 −0.0184 Relative m 0.0191 0.0044 −0.0002 −0.0059 nanowire to be mixed in L(i) to L(iv). Nanowires α3,β1, and γ1 are saturated in the initial state. The number of nanowires fully sat- urated up along branch Low changes from 3 to 4. From D(ii) to D(iii), nanowire β2 evolves from saturated down to a mixed state. The number of nanowires saturated up in branch D remains con- stant at 4. Therefore, merge point D and its primary peak represent a net increase in the number of saturated nanowires by one, showing agreement with the study by Dobrot ˘a and Stancu despite simulating fewer nanowires.15 Images of L(ii) to L(iv) and D(ii) to D(iv) show states at Hnear that of the merge point D. The magnetizations in wires β2 and γ2 are visibly different between images L(iv) and D(iv). Despite this, the net moment values of these states are nearly equal as seen in Table I and the Fig. 1 inset. Dobrot ˘a and Stancu observe differences between FORCs at the same Hdue to the coercivity of the particular reversed nanowire, resulting in secondary features in the FORC distribu- tion.15Although we simulated identical nanowires, the nanowirereversal field may be influenced by different inhomogeneous states before reversal. Domain growth in mixed-state nanowires is gradual, seen between each image of both branches in Fig. 3. Domain growth even occurs at fields between the merge points or observed primary peaks. Thus, the resulting maccounts for the sloped component of the FORCs which, as discussed earlier, does not result in FORC peaks. The imaged states show primary peak D only describes the late stages of reversal (saturation) in a single nanowire, β2 in the case of the Low curve. This is a newly described detail enabled by micromagnetic simulation. 3-D magnetization of states along the Low curve is shown in Fig. 4. States have been chosen such that the number of nanowires saturated up increases by 1 in each consecutive image. The imaged measurements are shown as points in Fig. 2. Most of these images are states at Hbetween those of the 9 primary peaks, support- ing the hypothesis that secondary peaks and a primary peak FIG. 4 . 3-D visualizations of distinct states along the ascending major curve, Low, showing sequential nanowire rever- sals. The fields of each state are recorded in Table II and labeled in Fig. 1. TABLE II . External fields at which distinct states can be seen in the Low curve, shown as 3-D visualizations in Fig. 4. State L0 L1 L2 L3 L4 H(kA m−1) −99.713 −73.389 −50.255 −20.740 −3.191 L5 L6 L7 L8 L9 11.966 43.874 59.030 94.130 102.106 AIP Advances 9, 125130 (2019); doi: 10.1063/1.5129954 9, 125130-4 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv collectively describe multiple reversals of an individual nanowire between FORCs.15Discrepancies in Hmay be accounted for by con- tributions of secondary peaks, related to variations in switching field between FORCs. VI. CONCLUSION Micromagnetic simulation allows combining 3-D visualization with standard FORC analysis. The FORC distribution shows 9 pri- mary peaks, which are also seen as the family of FORCs forming separate branches. From visualization, these peaks are found to rep- resent saturation of a single additional nanowire. Gradual domain growth in mixed-state nanowires is not represented in the FORC dis- tribution. The peaks occupy different Hupositions, forming an IFD, attributed to demagnetizing interactions between elements in the array.12Our simulation suggests that identical nanowires in a region of an array would reverse one-by-one at different external fields. This verifies common explanations for the IFD in FORC distributions of nanowire arrays in experiment12,22and simulation.12,15 REFERENCES 1C. Ross, “Patterned magnetic recording media,” Annu. Rev. of Mat. Res 31, 203–235 (2001). 2S. Bochmann, A. Fernandez-Pacheco, M. Ma ˇckovi ´c, A. Neff, K. R. Siefer- mann, E. Spiecker, R. P. Cowburn, and J. Bachmann, “Systematic tuning of seg- mented magnetic nanowires into three-dimensional arrays of ‘bits’,” RSC Adv. 7, 37627–37635 (2017). 3A. Saib, M. Darques, L. Piraux, D. Vanhoenacker-Janvier, and I. 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1.5090455.pdf	J. Appl. Phys. 125, 223903 (2019); https://doi.org/10.1063/1.5090455 125, 223903 © 2019 Author(s).Spintronic terahertz-frequency nonlinear emitter based on the canted antiferromagnet-platinum bilayers Cite as: J. Appl. Phys. 125, 223903 (2019); https://doi.org/10.1063/1.5090455 Submitted: 28 January 2019 . Accepted: 27 May 2019 . Published Online: 12 June 2019 P. Stremoukhov , A. Safin , M. Logunov , S. Nikitov , and A. Kirilyuk COLLECTIONS Note: This paper is part of the Special Topic section “Advances in Terahertz Solid-State Physics and Devices” published in J. Appl. Phys. 125(15) (2019). This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN 3D nanoprinting via focused electron beams Journal of Applied Physics 125, 210901 (2019); https://doi.org/10.1063/1.5092372 Carbon dots for energy conversion applications Journal of Applied Physics 125, 220903 (2019); https://doi.org/10.1063/1.5094032 Observation of longitudinal spin-Seebeck effect in magnetic insulators Applied Physics Letters 97, 172505 (2010); https://doi.org/10.1063/1.3507386Spintronic terahertz-frequency nonlinear emitter based on the canted antiferromagnet-platinum bilayers Cite as: J. Appl. Phys. 125, 223903 (2019); doi: 10.1063/1.5090455 View Online Export Citation CrossMar k Submitted: 28 January 2019 · Accepted: 27 May 2019 · Published Online: 12 June 2019 P. Stremoukhov,1,2,a) A. Sa ﬁn,3,4 M. Logunov,3S. Nikitov,2,3and A. Kirilyuk1 AFFILIATIONS 1FELIX Laboratory, Radboud University, Toernooiveld 7c, 6525 ED Nijmegen, The Netherlands 2Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141700, Russia 3Kotel ’nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, 125009 Moscow, Russia 4National Research University “Moscow Power Engineering Institute, ”111250 Moscow, Russia Note: This paper is part of the Special Topic section “Advances in Terahertz Solid-State Physics and Devices ”published in J. Appl. Phys. 125(15) (2019). a)Electronic mail: pavel.stremoukhov@ru.nl ABSTRACT In this paper, we propose an approximate nonlinear theory of a spintronic terahertz-frequency emitter based on canted antiferromagnet- platinum bilayers. We present a model accounting for the excitation of nonlinear oscillations of the Néel vector in an antiferromagnet using terahertz pulses of an electromagnetic ﬁeld. We determine that, with increasing amplitude of the pumping pulse, the spin system ’s response increases nonlinearly in the fundamental quasiantiferromagnetic mode. We demonstrate control of the Néel vector trajectory by changingthe terahertz pulse peak amplitude and frequency and determine the bands of nonlinear excitation using Fourier spectra. Finally, wedevelop an averaging method which gives the envelope function of an oscillating output electromagnetic ﬁeld. The nonlinear dynamics of the antiferromagnet-based emitters discussed here is of importance in terahertz-frequency spintronic technologies. Published under license by AIP Publishing. https://doi.org/10.1063/1.5090455 I. INTRODUCTION Antiferromagnetic spintronics is an emerging ﬁeld of magne- tism where several fundamental discoveries have been made in thepast few years, 1–3resulting in novel technological ideas.4,5The key goal of antiferromagnetic spintronics is to demonstrate devicesthat enable information processing and storage on the terahertz- frequency scale. The latest advances have thus made it possible to observe and control low-energy excitations on picosecond andfemtosecond timescales. 1,6,7 Until recently, antiferromagnets were considered as theoretically interesting but still without any practical applications. Nevertheless, the dynamics of spin order in antiferromagnets were shown to beintrinsically ultrafast 8–10(see the overview in Ref. 11for more details). These results unlock a multitude of known and newly identi ﬁed unique features of antiferromagnets relevant for applications in spin- tronics.11Experimentally, terahertz excitation of antiferromagneticresonance was shown for NiO.2,12Moreover, it has recently been pro- posed and demonstrated that when driving a macroscopic electrical current through certain antiferromagnetic crystals (e.g., CuMnAs,Mn 2Au), a ﬁeldlike Néel spin –orbit torque emerges that allows for a reversible switching of antiferromagnetic moments.10,13Metallic CuMnAs and insulating NiO mentioned above are examples of the simplest two-spin-sublattice collinear antiferromagnets. Recently, it was reported that terahertz-frequency emission can be realized in nanosized structures composed of heavy metal (HM) and ferromagnetic (FM) thin ﬁlms upon excitation of the latter by short laser pulses.14–16The transient currents are generated via the inverse spin Hall e ﬀect (ISHE) on the spin current injected into the HMﬁlm from the demagnetized FM ﬁlm. Because the AFM and HM layers have di ﬀerent transport properties, a net current along thez-axis is launched. In addition, the product of the density, band-velocity, and lifetime of spin-up (majority) electrons, whichJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 125, 223903 (2019); doi: 10.1063/1.5090455 125, 223903-1 Published under license by AIP Publishing.is signi ﬁcantly higher than that of the spin-down (minority) electrons, is strongly spin-polarized.17On entering the HM layer, spin –orbit coupling de ﬂects the spin-up and spin-down electrons in opposite directions.18This ISHE converts the longitudinal spin current density into a transverse charge current density. The detec-tion of the net spin current ﬂowing into the HM can be achieved electrically via the ISHE, as was demonstrated before. 18Such gen- eration of a transverse electric current by a spin current injectedinto a paramagnetic metal can also be employed to detect theeﬀect of spin pumping, 19resulting from the excitation of ferro- magnetic or antiferromagnetic resonances.20 The FM material in such experiments is usually magnetized by an applied magnetic ﬁeld, which also sets the frequency of the exci- tation (ferromagnetic resonance). One of the possible ways todevelop magnetic-based terahertz emitters without applied magneticﬁelds is to use AFM materials with strong internal magnetic ﬁelds (originating from, e.g., exchange magnetic ﬁeld between sublattices). A detailed theoretical study of the electric ﬁeld arising due to the ISHE in a nonmagnetic metal resulting from the spin current froman antiferromagnet is presented in Refs. 21and 22. We use the results obtained in these papers to calculate the output electric ﬁeld generated by the AFM emitter. While the terahertz-driven nonlinear spin response of a thin AFM ﬁlm was discovered in Ref. 3, the understanding of the nonlinear ultrafast processes in antiferromag-nets is still in its infancy. In this work, we propose and theoretically analyze a terahertz- frequency nonlinear emitter based on the canted AFM (takinghematite as an example) and HM (such as platinum). The article isorganized as follows. In Sec. II, we consider a physical structure of the AFM-based terahertz-frequency nonlinear emitter. Then, we present (Sec. III) a model of excitation of nonlinear spin oscillations of the Néel vector in an antiferromagnet under the action of tera-hertz pulses delivered by an electromagnetic ﬁeld. We demonstrate the control of the Néel vector trajectory by changing the terahertzpulse peak amplitude and frequency. In Sec. IV, we develop the averaging method, which gives the envelope function of an oscillat- ing output electromagnetic ﬁeld. Finally, Sec. Vis devoted to numerical simulations used to determine the bands of nonlinearexcitations using Fourier spectra. Because of the ISHE, we ﬁnd that increasing the pumping pulse amplitude causes the spin system ’s response amplitude to increase nonlinearly in the fundamental qua- siantiferromagnetic mode. II. PHYSICAL STRUCTURE Figure 1 schematically shows a bilayered AFM-based terahertz emitter consisting of a Pt/ α-Fe 2O3under photoexcitation. A tera- hertz laser pulse pumps the AFM resonance mode of α-Fe2O3and, via spin pumping, drives a spin current into the HM layer. TheISHE transforms the spin current into a picosecond pulse of trans-verse charge current. As a result, an electromagnetic wave with near-terahertz frequency (depending speci ﬁcally on the pulse ’s tem- poral pro ﬁle) is produced by the transient charge current. This tera- hertz pulse is radiated out of the ﬁlm plane, with its polarization being set by the direction of the electric current. Here, we do not take into account any externally applied constant magnetic ﬁeld, which could reorient the magnetic sublattices in the AFM. Since anexternal ﬁeld is not applied to the sample, the two-magnon modes in the AFM are degenerate. 23 We consider hematite α-Fe 2O3as a prototypical example of an AFM. The bulk Dzyaloshinskii –Moriya interaction (DMI)24,25 inside the AFM layer leads to the canting of the magnetization M1andM2within the AFM sublattices, thus creating a small net magnetization. Once excited by terahertz laser pulse, the magne-tization of each sublattice M 1andM2exhibits periodical preces- sionlike dynamics. The directions of the magnetization vectors and the anisotropy axes, with respect to the sample geometry, are shown in Fig. 2 . Figure 3(a) shows the typical pulse with pumping frequency Ω=2π¼0:265 THz and pulse amplitude μ0/C1Hmax¼0:1T .W es e t FIG. 2. Schematic representation of the rotating sublattice-speci ﬁc magnetiza- tion M1and M2in an AFM. The presence of the easy plane magnetic anisot- ropy in the AFM layer leads to a variable in time rotation speed of the AFM sublattice magnetizations with oscillation frequency. The magnetization of theAFM sublattices are canted by a small angle θ 0due to the DMI. FIG. 1. Schematic view of the terahertz-frequency AFM-based emitter. A terahertz pulse pumps an AFM/HM heterostructure of thickness dand generates nonequilib- rium pure spin current injected into the HM layer. The spin current is converted into a transient charge current due to the inverse spin Hall effect (ISHE) in the HMlayer. This charge current generates the outgoing terahertz pulse.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 125, 223903 (2019); doi: 10.1063/1.5090455 125, 223903-2 Published under license by AIP Publishing.the shape of the magnetic ﬁeld pulse HinðtÞ, which acts as a com- ponent of the e ﬀective magnetic ﬁeld, using a Gaussian function with the terahertz ﬁlling and linear polarization along the ex-axis, HinðtÞ¼Hmax/C1exp/C0t/C0t0 τ/C16/C17 2/C20/C21 /C1cosðΩðt/C0t0ÞÞex, where t0is the pulse envelope delay time and τis the width of the pulse. Here, the terahertz pump pulse has a wide spectrum in thefrequency domain, with a tuneable central frequency. The terahertz emitter is the oscillation system with an oscillation frequency, which is deﬁned by the easy plane anisotropy ﬁeld and exchange ﬁeld. III. MATHEMATICAL MODEL We consider the spin dynamics in an AFM excited by tera- hertz pulses using the sigma-model widely used in the theoryof antiferromagnetism. 26,27For this purpose, using the Landau – Lifshitz –Gilbert (LLG) equations of motion to describe the magne- tization of sublattices M1andM2, we resort to the dynamic vari- ablel¼ðM1/C0M2Þ=2Ms(Néel vector), where Msis the saturation magnetization. We take in to account the fact that the total magne- tization m¼ðM1þM2Þ=2Msis a small value and m/C28l(see Fig. 2 ). We express the dynamics of the vector mthrough thevector land its time derivative in the following form: Hexm¼Heff/C0lðHeff/C1lÞþ1 γ/C16@l @t/C2l/C17 , (1) where γis the gyromagnetic ratio, Hexis the exchange ﬁeld between the sublattices, and Heffis the e ﬀective magnetic ﬁeld, which takes into account the anisotropy ﬁeldHan, DMI ﬁeldHDMI, and the magnetic ﬁeld of the terahertz pulse HinðtÞ. Here, we do not take into account any constant applied magnetic ﬁeld. The last term in (1)describes the dynamic part of the total magnetization m.28 We parameterize the vector lðtÞin terms of polar θðtÞand azi- muthal wðtÞangles in the spherical coordinate system, lz¼cosðθÞ,lx¼sinðθÞcosðwÞ, and ly¼sinðθÞsinðwÞ: From the experimental data (see, e.g., Refs. 26and 29), it is known that vector lis oriented almost at a constant angle θ¼θ0 with respect to the ez-axis, and so the dynamics can be described using just the azimuthal angle wðtÞ. By varying the Lagrangian L½l/C138¼/C22h 2γHex@l @t/C18/C192 /C0WaðlÞ/C0/C22h HexHeff/C1l/C2@l @t/C20/C21 /C18/C19 FIG. 3. (a) The pro ﬁle of the pump pulse with central frequency Ω=2π¼0:265 THz, time offset t0¼5 ps, and pulse width τ¼1 ps. (b) Numerical solution of Eq. (2), describing the response, which is normalized to the maximum value (dashed line corresponds to the envelope function). (c) Spectra of the pump teraher tz signal and output response.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 125, 223903 (2019); doi: 10.1063/1.5090455 125, 223903-3 Published under license by AIP Publishing.over the angle wðtÞwith θ¼const, we obtain the second-order equation for wðtÞ,26 d2w dt2þαωexdw dtþω2 0 2sinð2wÞþωDMI/C1γ/C1HinðtÞcosðwÞ ¼γdHinðtÞ dt: (2) Here α/C2510/C04is the Gilbert damping constant, /C22his the reduced Planck constant, ωex¼γHex(where the exchange ﬁeld is Hex¼9 MOe), ωDMI¼γHDMI (where the DMI ﬁeld is HDMI¼22 kOe), and ω0¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃωaωexpis the quasiantiferromagnetic resonance frequency (where ωa¼γHais the frequency related with the anisotropy ﬁeldHa¼200 Oe). The assumed parameter values correspond to those of hematite.5,30The action of the optical tera- hertz pump pulse in our model is considered in Eq. (2)as a time- varying magnetic ﬁeld, which drives the induced inertial dynamics. The ISHE electric ﬁeld in the HM is calculated using the ana- lytical expression,31 Eout¼θSHg"#eλρ 2πdPttanh/C16dPt 2λ/C17dw dt¼κ/C1dw dt, (3) where g"#¼6:9/C21014cm/C02is the spin-mixing conductance at the Pt-AFM interface, θSH/C250:1 rad is the spin Hall angle in Pt, eis the modulus of the electron charge, λ¼7:3 nm is the spin- diﬀusion length in the Pt layer, ρ¼4:8/C210/C07Ωm is the electrical resistivity of Pt, and dPt¼20 nm is the thickness of the Pt layer.4,5 For the chosen parameters, κ/C251:35/C210/C09V/m (rad/s)/C01. The presence of the DMI, which is represented by the 4th term in Eq. (2), leads to inertial dynamics of the Néel vector in the AFM. In the inertial mechanism, during the action of the drivingforce, the orientation of the Néel vector is hardly changed, but it is enough to overcome the potential barrier afterward. 9On the onehand, the presence of DMI is essential for the nonlinear depen- dence of the oscillations on the excitation frequency. On the other hand, in its absence, the forced dynamics of the AFM is deter-mined only by the gyroscopic mechanism γdHinðtÞ dt, which is too weak to give rise to oscillations on its own. IV. AVERAGED EQUATIONS Figure 3(b) shows the result of solving Eqs. (2)and (3)in terms of the outgoing electric ﬁeld Eoutnormalized to the maximum value Eoutobtained by the numerical integration. Since the shape of the envelope function of the output electric ﬁeld could be recorded using pump-and-probe experiments, we present herethe method of its theoretical determination. Equation (2)corre- sponds to the equation of motion of a driven pendulum and can be examined using standard methods of the theory of nonlinear oscil- lations. We ﬁnd the solution wðtÞin the form of a quasiharmonic response at a frequency of a forced oscillation, wðtÞ¼β0þΩ/C1tþβ1ðtÞsinðΩtÞ, (4) where the amplitude β1ðtÞis slowly varying with time (accounted for by the envelope function) and β0is a certain constant phase. The solution (4)represents oscillations of the lycomponent the Néel vector. Upon substituting Eq. (4)into Eq. (2)and decomposing the nonlinear term in Eq. (2)using a Fourier series, we obtain the fol- lowing nonlinear equations characterizing β0andβ1: 2Ωdβ1 dtþαωexΩβ1/C0ω2 0J1ðβ1ÞðJ0ðβ1ÞþJ2ðβ1ÞÞsinð2β0Þ /C0ωDMIωmaxJ1ðβ1ÞfðtÞcosðΩt0Þ ¼ωmaxh ΩfðtÞsinðΩt0Þþf0ðtÞcosðΩt0Þi , (5a) FIG. 4. 2D color plot of the terahertz temporal traces at different applied magnetic ﬁeld maximum values from 0 to 1 T (a) and excitation frequencies from 0 to 1 THz (b).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 125, 223903 (2019); doi: 10.1063/1.5090455 125, 223903-4 Published under license by AIP Publishing.αωexΩþω2 0J2 1ðβ1Þ 2sinð2β0Þ¼0, (5b) where ωmax¼γHmaxJnðβ1Þis the nth order Bessel function, fðtÞ¼ exp½/C0/C0t/C0t0 τ/C12/C138is the envelope of a probing pulse, and f0ðtÞ¼dfðtÞ=dt. The result of solving the averaged equations (5a) and(5b) in terms of the output electric ﬁeld is shown in Fig. 3(b) in comparison with the solution of the initial equation (2). Thus, the averaged equation characterizing the envelope is an adequate approximation of the original model (2).Figure 3(c) shows the spectra of the pump terahertz signal and output response. V. SIMULATION RESULTS In response to the application of a constant magnetic ﬁeld to the structure, the magnetic moments are displaced slightlyfrom their equilibrium states. In turn, the spins undergo dampedoscillations toward their initial state. The amplitude of these nonlinear oscillations strongly depends on the amplitude and frequency of excitation [see Figs. 4(a) and 4(b)]. Increasing the amplitude of the constant magnetic ﬁeld results in the oscilla- tions being observed more clearly, and after the amplitude of themagnetic ﬁeld reaches 0.6 T, the sample ’s response becomes similar to the waveform of the excitation. Changes in the excita- tion frequency on the other hand a ﬀect the response di ﬀerently: t h er e s p o n s ep u l s eh a se x a c t l yt h es a m ef r e q u e n c ya st h ee x c i t a -tion pulse. The amplitude of the AFM precession also depends on both the amplitude of the constant magnetic ﬁeld and the exact frequency of the terahertz excitation [see Figs. 5(a) and 5(b)]. An increase of the excitation frequency results in a nonlinearbehavior of the amplitude of AFM precession such that har- monics are observed in the outgoing wave. This growth has a peak point after which the amplitude starts to slowly decrease.The amplitude of the fundamental frequency also increases nonlinearly with an increase of the amplitude of the input tera- hertz pulse. Figure 6 shows the dependency of the total obtained spec- trum as a function of the frequency of external excitation forthe amplitude of the constant magnetic ﬁeldH max¼0:15 T. It is clearly observable that, depending on the excitation fre- quency, a harmonic could be excited in the sample supplemen-tary to the AFM resonance mode. The frequency of thisadditional harmonic is exactly the same as the frequency of theexternal electromagnetic wave and is not the eigenfrequency of the magnetic system. FIG. 5. Dependencies of the normalized response fundamental frequency spectrum peak on the (a) excitation frequency and (b) applied magnetic ﬁeld. FIG. 6. The dependence of the spectrum of the excited AFM precession on the central frequency of excitation. Applied magnetic ﬁeld is Hmax¼0:15 T .Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 125, 223903 (2019); doi: 10.1063/1.5090455 125, 223903-5 Published under license by AIP Publishing.VI. CONCLUSION In this work, we have theoretically demonstrated that a bilayered AFM-based heterostructure can be used as a nonlinear terahertz-frequency emitter, where the amplitude of the output excitation on the fundamental frequency varies nonlinearly with the amplitude of the pump pulse. We demonstrate control of the Néel vector trajectoryin the AFM by tuning the peak amplitude and frequency of the tera-hertz pulse. We determine the bands of nonlinear excitation in awide range of applied magnetic pulse amplitude using Fourier spectra. The proposed mathematical model and averaging method can be used for the description of nonlinear dynamics for a wideclass of antiferromagnets and ferrimagnets. 32The obtained results could become crucial for the development of terahertz-frequencynanosized emitters and electromagnetic oscillators. SUPPLEMENTARY MATERIAL See the supplementary material for more detailed information on the two-parameter dependency of the resonance spectrum on both the excitation frequency and the constant magnetic ﬁeld. ACKNOWLEDGMENTS Financial support from the Government of the Russian Federation (Agreement No. 074-02-2018-286) within the laboratory“Terahertz spintronics ”of the Moscow Institute of Physics and Technology (State University) and the Russian Foundation for BasicResearch (Project Nos. 18-57-76001, 18-37-20048, 18-29-27020, 18-29-27018, and 18-07-00485) is acknowledged. We gratefully acknowledge the Nederlandse Organisatie voor WetenschappelijkOnderzoek (NWO-I) for their ﬁnancial contribution, including the support of the FELIX Laboratory. REFERENCES 1P. N ěmec, M. Fiebig, T. Kampfrath, and A. V. Kimel, “Antiferromagnetic opto- spintronics, ”Nat. Phys. 14, 1 (2018). 2T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. Mährlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. 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Kimel, “Macrospin dynamics in antiferromagnets trig- gered by sub-20 femtosecond injection of nanomagnons, ”Nat. Commun. 7, 10645 (2016). 13K. Olejník, V. Schuler, X. Martí, V. Novák, Z. Ka šp a r ,P .W a d l e y ,R .P .C a m p i o n , K. W. Edmonds, B. L. Gallagher, J. Garcés et al. “Antiferromagnetic cumnas multi-level memory cell with microelectronic compatibility, ”Nat. Commun. 8, 15434 (2017). 14T. Seifert, S. Jaiswal, U. Martens, J. H a n n e g a n ,L .B r a u n ,P .M a l d o n a d o , F .F r e i m u t h ,A .K r o n e n b e r g ,J .H e n r i z i ,I .R a d u et al. “Eﬃcient metallic spin- tronic emitters of ultrabroadband terahertz radiation, ”Nat. Photonics. 10, 483 (2016). 15G. Torosyan, S. Keller, L. Scheuer, R. Beigang, and E. T. Papaioannou, “Optimized spintronic terahertz emitters based on epitaxial grown Fe/Pt layer structures, ”Sci. Rep. 8, 1311 (2018). 16D. Yang, J. Liang, C. Zhou, L. Sun, R. Zheng, S. Luo, Y. Wu, and J. 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Münzenberg, “Perspective: Ultrafast magnetism and THz spintronics, ”J. Appl. Phys. 120, 140901 (2016). 30A. H. Morrish, Canted Antiferromagnetism: Hematite (World Scienti ﬁc, 1994). 31H. Nakayama, K. Ando, K. Harii, T. Yoshino, R. Takahashi, Y. Kajiwara, K.-I. Uchida, Y. Fujikawa, and E. Saitoh, “Geometry dependence on inverse spin Hall e ﬀect induced by spin pumping in Ni 81 Fe 19/Pt ﬁlms, ”Phys. Rev. B 85, 144408 (2012). 32R. Mikhaylovskiy, E. Hendry, A. S ecchi, J. H. Mentink, M. Eckstein, A .W u ,R .P i s a r e v ,V .K r u g l y a k ,M .K a t s n e l s o n ,T .R a s i n g et al. “Ultrafast optical modi ﬁcation of exchange interactions in iron oxides, ”Nat. Commun. 6, 8190 (2015).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 125, 223903 (2019); doi: 10.1063/1.5090455 125, 223903-6 Published under license by AIP Publishing.
1.1555374.pdf	Semiclassical theory of spin transport in magnetic multilayers R. Urban, B. Heinrich, and G. Woltersdorf Citation: J. Appl. Phys. 93, 8280 (2003); doi: 10.1063/1.1555374 View online: http://dx.doi.org/10.1063/1.1555374 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v93/i10 Published by the American Institute of Physics. Related Articles Influence of microstructure and interfacial strain on the magnetic properties of epitaxial Mn3O4/La0.7Sr0.3MnO3 layered-composite thin films J. Appl. Phys. 112, 083910 (2012) Parallel-leaky capacitance equivalent circuit model for MgO magnetic tunnel junctions Appl. Phys. Lett. 101, 162404 (2012) Half-metallic ferromagnetism in (CrN)1/(GaN)1 (001) and (VN)1/(InN)1 (001) superlattices J. Appl. Phys. 112, 083703 (2012) Helical domain walls in constricted cylindrical NiFe nanowires Appl. Phys. Lett. 101, 152406 (2012) Assembled Fe3O4 nanoparticles on graphene for enhanced electromagnetic wave losses Appl. Phys. Lett. 101, 153108 (2012) Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 18 Oct 2012 to 136.159.235.223. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsSpin Dynamics and Relaxation II: Multilayers and Thin Films Robert McMichael, Chariman Semiclassical theory of spin transport in magnetic multilayers R. Urban,a)B. Heinrich, and G. Woltersdorf Simon Fraser University, 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada ~Presented on 15 November 2002 ! A semiclassical model of the spin momentum transfer in ferromagnetic ﬁlm ~FM!/normal metal ~NM!structures is presented. It is based on the Landau–Lifshitz equation of motion and the exchange interaction in FM, and on the spin diffusion equation in NM. The internal magnetic ﬁeldis treated by employing Maxwell’s equations. A precessing magnetization in FM creates a spincurrent which is described by spin pumping proposed by Tserkovnyak et al.The back ﬂow of spins from NM into FM is assumed to be proportional to the spin accumulation in NM as proposed bySilsbeeet al.These theoretical calculations are tested against the experimental results obtained by different groups.Agood agreement was found for Py/Cu samples, but spin pumping is signiﬁcantlyenhanced in Py/Pt systems. © 2003 American Institute of Physics. @DOI: 10.1063/1.1555374 # In our recent ferromagnetic resonance ~FMR !studies 1–4 it was shown that the transfer of the spin momentum across ferromagnetic ~FM!/normal metal ~NM!interfaces can result in nonlocal interface Gilbert damping G 8. The generation of spin momentum in magnetic ultrathin ﬁlms was theoreticallydescribed by Tserkovnyak et al. 5and the effect was called ‘‘spin pumping.’’ The presence of a second magnetic layercreates a spin sink. 3,4,6,7The combination of spin pump and spin sink in the ballistic limit leads to an additional interfaceGilbert damping. In this article we extend the spin pump andspin sink mechanisms to the nonballistic electron transportwhich includes a full treatment of the Landau–Lifshitz ~LL! equation of motion in FM and diffusion equation in NM andMaxwell’s equations accounting for a ﬁnite penetration ofthe rf ﬁelds. The coordinate system was chosen in such a way that the sample normal is parallel to the zaxis. The external dc ﬁeld, H, lies in the sample plane and is parallel to the yaxis, and the internal electromagnetic rf ﬁelds are h5(h,0,0),e 5(0,e,0). The LLequations of motion in FM and NM layers can be written as 1 g]MF ]t52~MF3HeffF!1G0 g2Ms2SMF3]MF ]tD, ~1! 1 g]MN ]t52~MN3HeffN!1D g„2dMN2dMN gtsf, ~2! where gis the absolute value of the electron gyromagnetic ratio,Msis the saturation magnetization of FM, G0is the intrinsic Gilbert damping, Dis the diffusion constant in NM (D5vF2tel/6,vFis the Fermi velocity and telis the electron momentum relaxation time !,tsfis the spin–ﬂip relaxationtime, and dMN5MN2xPhis the excess magnetization in NM, where xPis the Pauli susceptibility. The effective ﬁeld HeffFis derived from the total Gibbs free energy which con- tains the external ﬁelds, magnetocrystaline anisotropies, and exchange interaction.8The effective ﬁeld HeffNin NM con- tains only the external dc, internal ﬁeld H, and the demag- netizing ﬁeld perpendicular to the sample plane. Equations~1!and~2!were solved in a small angle approximation using M F5~mxF,Ms,mzF!, ~3! dMN5~mxN2xPh,xPH,mzN!, ~4! whereHis the external applied magnetic ﬁeld. The time and spatial variations of the rf components were assumed to be;exp(i vt2kz), wherekis the propagation wave vector and vis the rf angular frequency. Maxwell’s equations in Gauss- ian units neglecting the displacement current for this geom-etry are 24 pk02imx1~k22ik02!h50, ~5! e h5kc 4ps,bz50, ~6! where sis the appropriate conductivity, cis the velocity of light in free space, and k025(4pisv)/c2. The skin depth d 5c/A(2psv). Equations ~1!,~2!,~5!, and ~6!provide the secular equa- tion fork. In both cases, FM and NM, the secular equation results in a cubic equation in k2which leads to six kwave numbers with corresponding six waves of propagation. Therf magnetization and electromagnetic ﬁeld components aregiven by a linear superposition of six waves.The coefﬁcientsare evaluated by matching the boundary conditions at theﬁlm interfaces. a!Electronic mail: rurban@sfu.caJOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 10 15 MAY 2003 8280 0021-8979/2003/93(10)/8280/3/$20.00 © 2003 American Institute of Physics Downloaded 18 Oct 2012 to 136.159.235.223. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsWe assume no direct exchange interaction between the FM and NM layers. The coupling between FM and NM iscaused by spin currents across the FM/NM interface. Weconsider three contributions to the net spin ﬂow: I FM!NM5\g"# 4pMs2SMF3]MF ]tD, ~7! INM!FM5vFtNM 4gdMN, ~8! Idiff5Dk gdMN. ~9! IFM!NMis described by the spin pumping model proposed byTserkovnyak et al.5,9Parameter g"#represents is the num- ber of conducting channels per unit area9which is directly related the interface mixing conductance G"#by g"#5h e2G"#, ~10! whereeis the electron charge, and his Planck’s constant. G"#were evaluated for different interfaces by ﬁrst principle band calculations by Xia et al.10 INM!FMwas proposed by Silsbee et al.11,12from a simple kinematic argument. tNMis the transmission coefﬁ- cient for conduction electrons from NM into FM. Tserk- ovnyaket al.used forINM!FMa similar term ( Isbackin their notation !. The transmission coefﬁcient tNMcan be deter- mined by direct comparison of INM!FMandIsback~Ref. 9 !and is found to be tNM5pg"# kF2, ~11! wherekFis the Fermi wave vector. Note, that the coefﬁcient in Eq. ~7!andtNMare proportional to the number of conduct- ing channels, which reduces the number of free ﬁtting pa- rameters. Since g"#’kF2/4p,13the transmission coefﬁcient is ’0.25. Idiffis present only in NM. It represents the ﬂux of non- equilibrium spins away from the FM/NM interface into theNM bulk. dMrelaxes back to equilibrium with the rate of 1/tsf. At each interface there are two electromagnetic bound- ary conditions ~continuity of hande!. In addition, the fol- lowing four boundary conditions satisfy the magnetic andspin ﬂow requirements at the FM/NM and NM/FM inter-faces. FM: S22A Msk2Ks MsDMz1IFM!NM~x!5INM!FM~x!, ~12!2A MskMx1IFM!NM~z!5INM!FM~z!, whereAis the strength of the bulk exchange coupling and Ks is the interface perpendicular uniaxial anisotropy ( Es5 2Kscos2(u)@erg/cm2#), see Ref. 8. The term in the roundbracket arises from the interface torques generated by the exchange coupling and the interface perpendicular uniaxialanisotropy. NM: I FM!NM5INM!FM1Idiff. ~13! The calculations were carried out for symmetric driving. This means that the rf components of hat both outer surfaces are equal. It is interesting to explore the following aspects of the above theory: ~A!The strength of g"#:g"#can be found in Ref. 10 and ranges between 1 and 2 31015cm22. In the limit of tNM !0 there is no backﬂow of the spin momentum from NM into FM. This corresponds to a ‘‘perfect spin sink’’and givesthe maximum effect regardless of d NM~thickness of NM !,D, andtsf. ~B!FMR linewidth, DHvsdFM: Figure 1 ~a!shows the total FMR linewidth as a function of the FM layer thicknessd FM. The dotted line does not include spin pumping ( g"# 50). In this case, there are two regions: ~i!FordFM ,300Å DHis dominated by the intrinsic damping G0of a single layer; ~ii!fordFM.500Å the additional broadening arises from eddy currents.The solid line includes spin pump-ing (g "#5131015cm22). Amazingly, the additional broad- eningalwaysscales like 1/ dFM. FordFM.500Å the addi- tional interface damping is negligible ( DHwith and without g"#are within 1 Oe !. ~C!GvsdNM, inﬂuence of lsf: In Fig. 1 ~b!the solid lines represent calculated total Gilbert damping G assuminga perfect mirror at the back side of NM @d(dM)/dz50#. For dNM!lsf5vFAteltsf/6 the rf magnetization accumulates in NM and the spin current INM!FMcompensates the spin pumping current IFM!NMresulting in zero interface damping (G5G0!. WhendNMbecomes comparable to lsfthe spin currentINM!FMis not sufﬁcient to compensate IFM!NMre- sulting in increased Gilbert damping. This increase eventu-ally saturates and its ﬁnal value depends on the ratio of tel/tsf. The dashed line in Fig. 1 ~b!represents a perfect spin sink at the back side of NM ( dM50). Note, that in this case FIG. 1. ~a!Total FMR linewidth DHas a function of dFMatf510GHz. Calculations were carried out for FM using permalloy ~Py, Ni80Fe20, 4pMs510.7kOe, G050.73108s21,r515mVcm!and NM using Cu ~r51mVcm,tel52.5310214s,D595cm2/s). The dashed line corre- sponds to a single Py layer and therefore DHis caused by the bulk proper- ties. The solid line shows the linewidth which includes spin pumping ( g"# 5131015cm22), assuming a perfect sink at the FM/NM interface ( tNM 50).~b!DHas a function of the NM thickness for two different values of tsf. The solid lines correspond to a perfect mirror at the back side of NM @d(dM)/dz50#. The dashed lines correspond to a perfect spin sink at the back side of NM ( dM50).8281 J. Appl. Phys., Vol. 93, No. 10, Part s2&3 ,1 5M a y 2003 Urban, Heinrich, and Woltersdorf Downloaded 18 Oct 2012 to 136.159.235.223. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsfordNM!lsfone obtains a perfect spin sink ~equivalent to tNM50) and for a large Cu thickness there is no difference between the Cu/ perfect-sink and Cu/ perfect-mirror. ~D!Inﬂuence of the skin depth d: It is interesting to discuss the limit when the skin depth dbecomes comparable or even less then lsf. Figure 2 simulates the effect of de- creasing temperature. The solid line corresponds to Py/Cu atroom temperature ~RT!, and the dashed line corresponds to a cryogenic temperature ~CT!with the resistivity ratio equal to 10. In this calculation the ratio between telandtsfwas assumed to be temperature independent. The spin diffusionlengths for RT and CT are 0.1 and 1 mm, respectively. The corresponding skin depths are 0.5 and 0.2 mm, respectively. For RT the ratio R5d/lsf’5 while for CT R’1/6. Note, that in both cases the additional linewidth increases whend NMbecomes comparable to lsf, and saturates for dNM !lsf. In the remainder of this article some recent experimental results will be discussed. Mizukami et al.14and Invarsson et al.15investigated the FMR linewidth in Py ﬁlms which were surrounded by NM layers. In both cases they observedan interface damping. Their studies were carried out at dif-ferent microwave frequencies. The strength of the interfacedamping in the same type of samples ~Pt/Py/Pt !scaled with the microwave frequency. It is, therefore, appropriate to in-terpret their results using the spin pumping theory as outlinedabove. The strength of the interface damping in the Py ﬁlmssurrounded by Pt and Pd is surprisingly high. Even for thecase when these layers act as perfect sinks ( t NM50) one needs to use g"#52.5 and 1.4 31015cm22for Pt and Pd, respectively. The number of transversal channels for lightelectrons ~m */mel’2!of the sixth band16is the same for Pt and Pd and leads to g"#50.731015cm22.This number can be perhaps enhanced by a factor of 2 by including the heavyholes. Therefore, one can expect g "#to be in a range be- tween 0.7 to 1.4 31015cm22. This is clearly at variance with the experimental value for Pt (2.5 31015cm22). Apos- sible explanation is being offered by the Stoner enhancementfactor which enhances the strength of spin pumping, see Si-manek and Heinrich. 17 Recently we studied the increase of the Gilbert damping in GaAs/16Fe/10Pd/20Au ~001!, where integers represent the number of atomic layers. This sample was prepared by mo-lecular beam epitaxy ~MBE !where atomic intermixing be-tween Fe and Pd is kept at its minimum. The additional Gilbert damping at f524 GHz was found to be 0.3 310 8s21. This value is small compared to the increase in G(1.73108s21) that was measured by Mizukami et al.14 for the same FM thickness. In interpretation of our data we have to invoke a ﬁnite spin diffusion length. The requiredvalue is l sf570 Å. However, that needs tsf5telwhich we ﬁnd unrealistic. The mean free path exceeds signiﬁcantly thePd thinkness; therefore, we are in the ballistic limit whereour theory does not apply. In the ballistic limit it is morereasonable to interpret the measured data by determining thefraction of I FM!NMwhich was absorbed in Pd. In Mizuka- mi’s experiment everything is absorbed, in our measure-ments only 20% is lost in Pd. In separate experiments Mizukami et al. 18studied the Gilbert damping as a function of dCu~from 10 nm to over 1 mm!in glass/Cu(5 nm !/Py/Cu(dCu) and glass/ Cu(5 nm !/Py(3 nm !/Cu(dCu!/Pt samples. Their results are similar to those shown in Fig. 1 ~b!fortsf5200tel(lsf 50.2mm). Notice, that Cu on its own is a poor spin sink even fordCu@lsf. In glass/Cu/Py/Cu( dCu!/Pt structures one was able to explore the role of the Pt layer when separatedfrom Py by a variable thickness of Cu. The experimentalresults were possible to explain by assuming that the Cu/Ptinterface acted as a perfect spin sink and therefore the in-crease in the Gilbert damping can be explained by the maxi-mum strength of spin pumping in Cu. The authors thankY. Tserkovnyak, E. Simanek, and J. F. Cochran for valuable discussions. Financial support from theNatural Sciences and Engineering Research Council ofCanada ~NSERC !and Canadian Institute for Advanced Re- search ~CIAR !is gratefully acknowledged. G.W. thanks the German Academic Exchange Service ~DAAD !for a gener- ous scholarship. 1R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 ~2001!. 2B. Heinrich, R. Urban, and G. Woltersdorf, J.Appl. Phys. 91, 7523 ~2002!. 3B. Heinrich, R. Urban, and G. Woltersdorf, IEEE Trans. Magn. 38, 2496 ~2002!. 4B. Beinrich, G. Woltersdorf, R. Urban, and E. Simanek, J.Appl. Phys. ~to be published !. 5Y. Tserkovnyak, A. Brataas, and G. Bauer, Phys. Rev. Lett. 88, 117601 ~2002!. 6M. Stiles and A. Zangwill Phys. Rev. B 66, 014407 ~2002!. 7B. Heinrich, G. Woltersdorf, R. Urban and E. Simanek, J. Magn. Magn. Mater. ~in press !. 8B. Heinrich and J. F. Cochran, Adv. Phys. 42,5 2 3 ~1993! 9Y. Tserkovnyak, A. Brataas, and G. Bauer, e-print cond-mat/0208091. 10K. Xia, J. Kelly, G. B.A. Brataas, and I.Turek, Phys. Rev. B 65, R220401 ~2002!. 11R. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19,4 3 8 2 ~1979!. 12P. Sparks and R. Silsbee, Phys. Rev. B 35, 5198 ~1987!. 13A. Brataas, Y. Tserkovnyak, G. Bauer, and B. Halperin e-print cond-mat/0205028. 14S. Mizukami,Y.Ando, andT. Miyazaki, Jpn. J.Appl. Phys., Part 1 40,5 8 0 ~2001!. 15S. Ingvarsson, L. Ritchie, X. Liu, G. Xiao, J. Slonczewski, P. Trouilloud, and R. Koch, e-print cond-mat/02008207. 16C. Lehmann, S. Sinning, P. Zahn, H. Wonn, and I. Mertig, Fermi surface database ~1996-1998 !, URL http://www.physik.tu-dresden.de/ ;fermisur/. 17E. Simanek and B. Heinrich, e-print cond-mat/0207471. 18S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. Magn. Mater. 239,4 2 ~2002!. FIG. 2. DHfor Py ~20 Å!covered by Cu( dNM) for RT ~solid line !and a cryogenic temperature ~dashed line !with the resistivity ratio equal to 10. Calculations were carried out at f510GHz. For RT tel52.5310214s and e5tsf/tel5100.ewas assumed to be temperature independent.8282 J. Appl. Phys., Vol. 93, No. 10, Part s2&3 ,1 5M a y 2003 Urban, Heinrich, and Woltersdorf Downloaded 18 Oct 2012 to 136.159.235.223. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
5.0013402.pdf	J. Appl. Phys. 128, 033907 (2020); https://doi.org/10.1063/5.0013402 128, 033907 © 2020 Author(s).Skyrmion-based spin-torque nano-oscillator in synthetic antiferromagnetic nanodisks Cite as: J. Appl. Phys. 128, 033907 (2020); https://doi.org/10.1063/5.0013402 Submitted: 11 May 2020 . Accepted: 02 July 2020 . Published Online: 20 July 2020 Sai Zhou , Cuixiu Zheng , Xing Chen , and Yaowen Liu Skyrmion-based spin-torque nano-oscillator in synthetic antiferromagnetic nanodisks Cite as: J. Appl. Phys. 128, 033907 (2020); doi: 10.1063/5.0013402 View Online Export Citation CrossMar k Submitted: 11 May 2020 · Accepted: 2 July 2020 · Published Online: 20 July 2020 Sai Zhou,1Cuixiu Zheng,1Xing Chen,2and Yaowen Liu1,a) AFFILIATIONS 1Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, People ’s Republic of China 2Department of Optical Science and Engineering, Shanghai Ultra-Precision Optical Engineering Center, Fudan University, Shanghai 200433, People ’s Republic of China a)Author to whom correspondence should be addressed: yaowen@tongji.edu.cn ABSTRACT The skyrmion-based spin-torque nano-oscillator is a potential next-generation nano microwave signal generator. In this paper, the self- sustained oscillation dynamics of magnetic skyrmions are investigated in a nanodisk with synthetic antiferromagnetic (SAF) multilayerstructure, in which the skyrmion Hall effect can be effectively suppressed. An analytical model based on the Thiele equation is developed todescribe the dynamics of a pair of skyrmions formed in the SAF nanodisks. Combining the analytical solutions with the micromagneticsimulations, we demonstrate that circular rotations with opposite directions for a skyrmion pair could be suppressed by increasing the antiferromagnetic (AF) coupling in a nanopillar with dual spin polarizers. However, a stable circular rotation can be achieved in a nanopillar with a single spin polarizer, in which one skyrmion plays as a master whose rotation is driven by spin torque, while the other skyrmion is aslaver whose motion is dragged by the AF coupling between the two free layers. Moreover, we found that the effective mass factor in theSAF structure rather than the gyrotropic torque plays the dominant role in the circular rotation of skyrmions. The rotation orbit radius andfrequency gradually increase with the decrease of damping factor and increase of applied current strength. Published under license by AIP Publishing. https://doi.org/10.1063/5.0013402 I. INTRODUCTION Magnetic skyrmions, 1–4topologically stable spin texture found in chiral magnets with Dzyaloshinskii –Moriya interaction (DMI),5,6 can be used for the development of different information devices such as racetrack memory,7–9spin logics,10and spin-torque nano-oscillators (STNOs).11–17The skyrmion motion is driven by spin transfer torque (STT)18,19or spin –orbit torque (SOT).20,21 Recently, very small-sized skyrmions with a few nanometers dimension have been reported in ultrathin ferromagnetic (FM)22or antiferromagnetic (AF)23thin films in the presence of interfacial DMI induced by the proximity to an adjacent heavy metal (HM) layer (such as Pt or Ta)24with strong spin –orbit coupling (SOC).8,25,26Moreover, it was discovered that the interfacial DMI induced Néel-type skyrmions can be stabilized at roomtemperature. 22,25–29In contrast with vortex-based STNOs,30,31it has been suggested that the skyrmion-based STNOs could reach a higher and steadier maximum working frequency and broader tenability.15–17Several methods have been proposed to increase thefrequency of the skyrmion-based oscillator by using skyrmion array,11enhanced perpendicular magnetic anisotropy edge,13,17 modified profile of DMI,15and antiferromagnetic skyrmions.16 However, an unexpected feature has been reported that the skyrmion cannot move in a straight line along the driving current direction due to the Magnus force. This behavior is referred to the skyrmion Hall effect.32–36Recent studies show that the skyrmion Hall effect can be effectively suppressed for paired skyrmions with opposite spin moment direction formed in the top and bottom layers of a synthetic antiferromagnetic (SAF) nanowire.37–39The SAF nanowire is composed of two FM layers with one non- magnetic (NM) spacer (such as Ru),40in which the Ruderman – Kittel –Kasuya –Yosida (RKKY) exchange coupling plays an impor- tant role.41On the other hand, the STNOs with dual free layers (FLs) have been suggested to have the advantages of large output signals and frequency because the angle between the magnetiza- tions of the dual free layers changes twice as fast as the angle between the magnetizations of the free and reference layers.42,43Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 033907 (2020); doi: 10.1063/5.0013402 128, 033907-1 Published under license by AIP Publishing.In this paper, the characteristic of a pair of skyrmions with opposite topological numbers will be modeled in the SAF free layers. We developed a simple analytical expression for this type ofSTNO by using the Thiele approach. 44The analytical results are supported by our micromagnetic simulations, showing that thecurrent-driven skyrmion pair motion can be suppressed by the increase of interlayer AF coupling of the SAF structure when the two skyrmions are driven by two spin polarizers. Instead, astable circular motion of the skyrmion pair can be achieved whenthe STT driving force acts on the bottom skyrmion only. Therefore,the top skyrmion is dragged by the strong AF coupling effect. II. THEORETICAL MODEL Here, we consider a single skyrmion formed in the free layer of spin valves as shown in Figs. 1(a) and 1(b), respectively, in which the two skyrmions have different topological numbers ofQ=−1 and Q= +1 and move in a clockwise (CW) or counter-clockwise (CCW) circular motions, respectively, where Q¼ 1 4πÐdr2m@m @x/C2@m @y/C16/C17 . The heavy metal (HM) layer (Ta or Pt) adjacent to the free layer is used to generate strong SOC and inter- facial DMI to stabilize the skyrmions [in order to generate positiveDMI to the free layer, in this study we suppose HM = Pt in Fig. 1(a) and HM = Ta in Fig. 1(b) ]. The fixed spin polarizer layer has a vortex-like magnetization configuration, and the polarizationvector m pis spatially dependent as a function of coordinated (x,y) and can be written as mp= (cos Φ, sinΦ, 0), with Φ= arctan (y/x)+w.In this study, the angle wis set to 0°. When the currentdirection or mpis reversed, the two skyrmions will move back to their equilibrium position (i.e., the center of disks) due to the reversed direction of Fgyro. Generally, for a rigid skyrmion (without any or with very slight deformation), its motion driven by a spin polarized currentcan be described by a generalized Thiele equation, 44,45 G/C2vþD$ /C1vþFJþFedge/C0MFMa¼0, (1) where v¼dX/dtis the velocity of skyrmion and X¼(X,Y) is the position of skyrmion center. The first term in Eq. (1)is the gyrotropic (or Magnus) force38andG¼4πQezis the gyromagnetic coupling vector. The second term describes the dissipative process and D$ is the dissipative tensor with the elements of Dxx¼Dyy¼D¼αÐ dr2@xm@xm, where αis the Gilbert damping coefficient. The third term represents the current-induced drivingforce F J,15,16which can be decomposed into two orthogonal directions along the radial direction erand tangential direction eτ of the skyrmion motion: FJ¼FJrerþFJτeτand FJi¼/C0aJÐdr2 [(m/C2mp)/C1@im], where i = (r, τ).aJis the STT strength defined as aJ¼/C22hγJg(θ,P)/(eμ0MSd), where ħ,γ,P,e, and μ0are the Planck constant, gyromagnetic ratio, spin polarization, electron charge, and vacuum permeability, respectively. MSanddare the saturation magnetization and thickness of the free layer. g(θ,P) is a scalar function that depends on the spin polarization Pand angle θ between the magnetization vector of the free layer and that ofthe polarizer layer. 18,46,47The fourth term in Eq. (1)describes the FIG. 1. Schematic models of skyrmion-based nano-oscillators. A single skyrmion with topological number of Q=−1 (a) and Q= + 1 (b) formed in the ferromagnetic (FM) free layer. The spin polarizer layer has a magnetic vortex configuration. The schematic of the forces exerted on the current-driven skyrmion dynamic s is also illustrated. The spin torque FSTTand damping torque Fdamp lead to tangential forces, where the sign of FSTTdepends on the current and mpdirections. The FSTTwith different direc- tions in (a) and (b) drive the skyrmions rotate in clockwise and counter-clockwise directions, respectively. The gyrotropic force Fgyroand restoring force Fedge give rise to radial forces. (c) Schematic illustration of a skyrmion pair formed in a SAF free layer of nanopillar. The snapshots show the motion trajectories of th e current-driven sky- rmion pair as a function of AF coupling strength. Here, the trajectories are recorded within the first 2 ns.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 033907 (2020); doi: 10.1063/5.0013402 128, 033907-2 Published under license by AIP Publishing.repulsive (or restoring) force originated from the boundary of the sample, Fedge=−▿U, where Uis the potential energy. As the sky- rmion moves closer to the boundary of sample, the repulsive forcewill increase due to the distribution of the magnetic charges at theside face of the cylinder. 48This edge-induced force acts as a centripetal force to sustain the circular motion, as illustrated in Figs. 1(a) and1(b). We would like to mention that without driving source the equilibrium position of skyrmions will be in the centerof nanodisk due to the boundary effect. After the current isapplied, the skyrmion will slowly departure from the center andfinally reach a persistent circular precession orbit. This process usually will take ∼20 ns or even more time. 11In this study, without loss of generality, an effective method for saving computation timeis done as follows: 15the initial position of the skyrmions is initially set at a position out of the disk center, and then the system hasbeen relaxed for 1 ns before the driving current or field is applied. The last term in Eq. (1)describes the acceleration process of sky- rmion motion, where M FMis the effective mass of skyrmion and a is the acceleration.45Generally, this acceleration process is very fast (within several picoseconds) for the current-driven skyrmionmotion. The acceleration term will vanish once the skyrmion reach its steady velocity (i.e., a τ= 0).23 Theoretically, the Thiele equation of Eq. (1)for a rigid sky- rmion at aτ= 0 can be decomposed in the radial and tangential directions,15,16 vτG/C2eτ/C0MFMarerþFedgeer¼0, (2a) DvτeτþFJτeτ¼0, (2b) where vτandvrare the tangential and radial velocities, respectively. For a circular motion, we have vr= 0. In this case, the contribution of the effective mass of skyrmion is usually very small in the FMmonolayer and can be ignored; 15–17therefore, the repulsive force Fedgetogether with the remanent gyrotropic force Fgyroin Eq. (2a) will provide the centripetal force for circular motion. In this study, we will consider a heterostructure with symmet- ric dual free layers by combining these two spin valves, as shown inFig. 1(c) , in which the two free layers are antiferromagnetically coupled to each other by Ru layer via the RKKY interlayer interac-tion. 41,49The strong AF coupling results in the two FM layers (top FL and bottom FL) having the opposite magnetization configurations and forming a so-called SAF free layer. The initialmagnetization configuration of the top FL is assumed to pointupward and the bottom FL is pointing downward. Here, dual spinpolarizers are used, 42,50and both of them have the vortex-like magnetization configuration.17We consider a couple of Néel-type skyrmions with opposite polarity initially generated in the SAF freelayers, see Fig. 1(c) . The two skyrmions in the top and bottom FLs have the different topological numbers of Q T=–1 and QB= +1, respectively. In this case, the magnetization dynamics of the SAF free layer is described by the Landau –Lifshitz –Gilbert (LLG)equation including the STT terms,51 @mk @t¼/C0γmk/C2Hk effþαmk/C2@mk @tþak J(mk/C2mk p/C2mk) /C0bk Jmk/C2mk p, (3) where m(=M/MS) is the unit magnetization vector. The superscript k(=T or B) is used to indicate the top and bottom free layer, respectively. In this study, we suppose the STT from spin polarizeronly act on the neighboring magnetic layer. 42,43,52This is based on the fact that the spin polarization could be strongly reduced by the Ru layer within the SAF.53The third term in Eq. (3)is the Slonzewski damping-like STT18and the fourth is the field-like STT.54Here, aJ¼/C22hγJg(θ,P)/(eμ0MSd) and bJ¼βaJwith g(θ,P)¼PΛ2/[(Λ2þ1)þ(Λ2/C01)m/C1mp],47,55where β= 0.15 is the strength ratio of the field-like torque, Λ= 1 is an asymmetry factor of spin torque, and mpis the spin polarization direction. Heff is the effective field derived from the free energy density of the system with respect to the local magnetization. The energy densityE, includes the intralayer exchange, anisotropy, demagnetization, DMI, and interlayer RKKY interaction energy, given by E¼A(∇m) 2/C0K(n/C1m)2/C0μ0MS 2m/C1Hd þDi[mz(∇/C1m)/C0(m/C1∇)mz]þEAF, (4) where A,K, and Diare the exchange, magnetic anisotropy, and interfacial DMI constants, respectively. nis the unit vector of anisotropy easy axis. Hdis the demagnetizing field. EAFis the AF interlayer coupling energy and reads as56 EAF¼/C0JAF(1/C0mT/C1mB), (5) where JAFis the AF coupling strength. Following the Thiele approach, the motion of the skyrmion pair at the top and bottom FM layers of SAF structure is governedby 57 GT/C2vTþD$ /C1vTþFT JþFT edgeþFT AF/C0MT SAFaT¼0, (6a) GB/C2vBþD$ /C1vBþFB JþFB edgeþFB AF/C0MB SAFaB¼0, (6b) where FAFis the driving force originated from interlayer AF cou- pling effect and M SAFis the effective mass of the skyrmion pair, MSAF¼μ0MSDl2/(2αγA),23,58,59where lis the lattice constant. In the SAF structures, the Fedgeis hard to calculate from the energy profile of U. But instead, it can be indirectly estimated from the balance with FJ.16 The dynamics of the skyrmion pair strongly depend on the strength of AF coupling in the SAF multilayer structure. For very strong interlayer AF coupling, the skyrmion pair could be tightlybound and the two skyrmions at the top and bottom free layersmove together with the same velocity vand acceleration a. In this case, the total interlayer AF forces can be canceled out because of F T AFþFB AF¼0, and the SAF free layer system behaves as a singleJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 033907 (2020); doi: 10.1063/5.0013402 128, 033907-3 Published under license by AIP Publishing.FM layer. Thus, the two Thiele equations of (6a) and(6b) could be rewritten as a single equation, Gtotal/C2vþ2D$ /C1vþFtotal JþFtotal edge/C0MSAFa¼0: (7) Here, Gtotal¼4πQtotalezwith Qtotal¼QTþQB¼/C01þ1¼0. Therefore, the Magnus force is also canceled, implying that the sky- rmion Hall effect can be strongly suppressed in this case. Consequently, the radial and the tangential component of sky-rmion pair motion can be derived by solving Eq. (7)as follows: /C0M SAFarerþFtotal edgeer¼0, (8a) 2DvτeτþFtotal Jτeτ¼0: (8b) It is noticeable that the effective mass effect becomes domi- nant in Eq. (8a). This feature is completely different from the case of the single free layer given in Eq. (2a). The tangential velocity of the skyrmion pair can be easily derived by solving Eq. (8b) as follows:16 vτ¼aJπ2RSK 2D, (9) where RSK¼A/K/C21/C0Diπ/2ﬃﬃﬃﬃﬃﬃﬃ AKp /C0/C1 /C2/C3 1/2is the skyrmion radius.60Equation (9)indicates that the velocity of the skyrmion pair is proportional to the STT strength (i.e., current) and is inversely proportional to the damping factor D. III. SIMULATION RESULTS In order to verify the above analytical result, micromagnetic simulations have been performed with the OOMMF publiccode. 56The typical parameters of Pt/Co multilayers with perpen- dicular magnetic anisotropy (PMA) are used for the SAF struc- tures:8,61MS= 580 kA m−1,K= 800 kJ m−3,A= 15 pJ/m, Di= 1.5 – 4.0 mJ m−2, and α= 0.1. In our simulations, we consider the SAF structure sample having a circular shape with a diameter of100 nm. Both the top and bottom free layers have the thickness of1 nm. The discrete cell is 1 × 1 × 1 nm 3, which is smaller than the Néel exchange length λN/C19eel¼[2A/(μ0M2 S)]1/2¼8:42 nm as well as the Bloch exchange length λBloch¼(A/K)1/2¼4:33 nm. The fixed spin polarizing layer has a vortex-like magnetic configuration. Figure 1(c) shows the typical motion trajectories of a sky- rmion pair as a function of different strengths of AF coupling, excited by the top and bottom dual spin polarizers with applied current density of J=5×1 010A/m2. For a weak coupling of JAF= 1.5 × 10−7J/m2, we can see that the two skyrmion process in an opposite direction independently, showing the top skyrmionmoves in the clockwise circular motion while the bottom skyrmion moves in the counterclockwise circular motion. Such a motion behavior is as same as that of the individual free layer case with agiven single spin polarizer [see Figs. 1(a) and 1(b)]. However, we can see that the independence circular motion tendency of the sky- rmion pair will be gradually suppressed with the increase of AF interaction strength between the two free layers [ Fig. 1(c) ]. Weattribute this to the strong AF interaction between the two sky- rmions blocking their opposite motion. In the extreme case, when the AF coupling increases above J AF= 1.5 × 10−4J/m2, the skyrmion pair can be completely imprisoned in the center of the nanodisk(i.e., the equilibrium position). In order to achieve a stable circular motion of the skyrmions driven by current in the nanopillar with a SAF free layer, we have modified our model by replacing the top spin polarizing layer witha heavy metal layer, as shown in Fig. 2(a) . In this case, the top heavy metal layer will generate DMI effect to act on the top FL.The STT from the bottom polarizer only act on its neighboring bottom free layer so that the skyrmion at the bottom FL (namely, master skyrmion) will be driven by STT into a CCW rotation. Thetop FL is not subjected to the STT but the strong AF coupling FIG. 2. (a) Schematic illustration of a skyrmion pair-based STNO, containing a SAF free layer and a single spin polarizer. The top heavy metal generates a strong interface DMI to the top free layer. Current-driven circular oscillation of skyrmion pair is achieved with the same precession directions in the top andbottom layers of SAF , where the motion trajectories are recorded in the first0.16 ns. (b) The size dependence of skyrmion pair on the DMI strength for three different configurations: both the top and bottom free layers have the same DMI strength (black curve); the bottom free layer has no DMI (blue curve); and thebottom free layer has a fixed strength of DMI of 1.5 mJ/m 2(red curve).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 033907 (2020); doi: 10.1063/5.0013402 128, 033907-4 Published under license by AIP Publishing.between the top FL and bottom FL will drag the top skyrmion (slaver) to follow the CCW motion. For that, the stability of the skyrmion pair has been investigated in such a SAF structure withdifferent DMI configurations. Figure 2(b) shows the size depen- dence of the skyrmion pair on the DMI strength, characterized bythe stabilized radius R SK, where the AF coupling between the two FM layers of SAF is fixed at JAF= 1.5 × 10−3J/m2.35In these simu- lations, we have checked three types of DMI configurations: Thebottom free layer has a changing or fixed or no DMI effect. Theresults indicate that with the increase of the DMI, the skyrmion pair can be stabilized in all these three types of configurations but the threshold ( D c) of DMI to sustain a stable skyrmion pair is dif- ferent: Dc= 2.5, 3.4, or 5.2 mJ/m2, respectively. From these results, what is especially interesting is that for the top layer having a DMIlarger than 5.2 mJ/m 2, a skyrmion with opposite number of Q=+ 1 can be stabilized in the bottom free layer through the AF coupling between the SAF layers, even though the bottom layer has no DMIeffect (red curve). This result implies that the strong AF couplingcan sustain a stable skyrmion pair in this SAF structure withoutneed of the bottom HM layer. The right panel of Fig. 2(a) shows the motion trajectory of the skyrmion pair under the action of STT applied in the bottom freelayer. 62At the same time, the spin polarization rate in the lower NM is far more than that in the upper NM.63In this simulation, we suppose the top free layer has the DMI of 4.0 mJ/m28,27,64and the bottom free layer has the DMI of 1.5 mJ/m2.64We can see that the bottom master skyrmion ( Q= +1) is driven into a CCW FIG. 3. (a) Displacement evolutions of the FM single skyrmion (blue curve) and SAF skyrmion pair (olive curve). The applied current is J=5×1 010A/m2. (b) Time evolution of displacement ( X, Y ) for the single skyrmion and skyrmion pair. FIG. 4. (a) Time evolution of displacement ( X,Y) for the skyrmion pair driven by a current density of 6 × 1010A/m2. (b) The corresponding frequency spectrum calculated by fast Fourier transform.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 033907 (2020); doi: 10.1063/5.0013402 128, 033907-5 Published under license by AIP Publishing.circular motion oscillation excited by the STT from the bottom spin polarizer; and then due to the strong AF coupling, the slaver skyrmion at the top free layer with opposite number of Q=−1i s also dragged into the CCW circular motion. To characterize the feature difference between the single sky- rmion and skyrmion pair, we show the time and space evolutions of skyrmion centers for two same lateral size of samples in Fig. 3 , where the single skyrmion is taken from the model of Fig. 1(b) and the skyrmion pair is taken from the model of Fig. 2(b) with DT= 4.0 mJ/m2andDB= 1.5 mJ/m2, respectively. The dimension of two samples is 100 nm in diameter. The same current strength of J=5×1 010A/m2is applied. We can clearly see that in both cases the skyrmions can oscillate steadily and rotate in the CCW circularmotion. However, the amplitudes of precession orbit show big dif-ferences although the same current is applied with the relativelylarger precession radius of r c= 34.53 nm for the FM single sky- rmion case and smaller radius of rc= 4.60 nm for the SAF sky- rmion pair, see Fig. 3(a) . The time evaluations of the skyrmion precession in Fig. 3(b) indicate that the SAF skyrmion pair has a higher oscillation frequency. The reduction of orbit radius impliesthat the skyrmion pair is far away from the sample boundary, and therefore the repulsive force F edgealmost goes to zero in the SAF structure. We know that in the single skyrmion case, the repulsiveforce of Fedgeis used to balance the gyrotropic force of Fgyrothat is associated to the skyrmion Hall effect [ Fig. 1(b) ]. Conversely, the skyrmion Hall effect has been effectively suppressed in the sky-rmion pair formed in the SAF structure. This leads to the effectivemass M SAFplays the dominate role for the circular precession motion of the skyrmion pairs. The result agrees well with the theory as discussed in Eqs. (7)and (8). The reduction of precession radius of skyrmion pair opens the potential increase of oscillation frequency with the applied currentdensity in such a STNO with the SAF free layer. In order to obtaina steady periodic oscillation, we have run a 50-ns simulation and show the result in Fig. 4 . The current density is fixed at 6 × 10 10A/ m2. The calculated precession frequency of the skyrmion pair is 1.02 GHz, see Fig. 4(b) . Finally, we summarize the features of skyrmion pair-based STNO in Fig. 5 by showing the dependence of oscillation fre- quency, tangential velocity, and trajectory radius as functions of damping constant α(left column) and applied current density (right column). With the decrease of damping α, the oscillation fre- quency gradually increases. In particular, for α= 0.02, the preces- sion frequency can reach 6.4 GHz. For the damping αlower than 0.02, the skyrmion pair will move out of the edge of the sample. Accordingly, the motion velocity vτincreases with the decrease of α FIG. 5. The dependence of frequency f,velocity vτ, and orbit radius rcof skyrmion pair motion on the damping α (left column, J= 3.3 × 1011A/m2) and applied current density J(right column, α= 0.1). The brown curves show the analytical results based on Eq. (9). The inset plots in the right column show the wide range of current Jfrom 0 to 25 × 1011A/m2.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 033907 (2020); doi: 10.1063/5.0013402 128, 033907-6 Published under license by AIP Publishing.[Fig. 5(b) ]. For comparison, the theoretical curves derived from Eq.(9)are also shown in Fig. 5(b) by the solid curves. Very good agreements have been achieved between the theory and simulations.The dependence of the motion velocity vs the applied currentalso displays the good agreements between the theory and simula-tions, showing a good linear increase with the increased current. The inset shows the current Jchanging at large scale up to 25 × 10 11A/m2. We can see that the motion velocity vτalmost line- arly increases until J=2 0×1 011A/m2. Consider that the trajectory radius of skyrmions depends on the increasing velocity and inertialforce, and, therefore, the radius of skyrmion motion r cincreases with the decreasing α[Fig. 5(c) ]. The frequency of skyrmion pair motion can be tuned by current strength, as shown in Fig. 5(d) . The frequency fincreases from 1.0 to 2.5 GHz when the applied current density Jchanges from 0.6 × 1011to 3.3 × 1011A/m2. Accordingly, both the motion velocity and the trajectory radius of the skyrmion pair increase with the current strength J[Figs. 5(e) and5(f)]. IV. CONCLUSION In summary, we have presented a study on the skyrmion- based STNO by using SAF free layers to host two skyrmionshaving opposite topological numbers. The well-established Thieleapproach has been extended to describe this type of skyrmion pair.The validity of the analytical solution was supported by micromag- netic simulations based on the LLG equation. Compared with the single skyrmion formed in a ferromagnetic layer, the rotation orbitradius and oscillation frequency gradually increase with thedamping factor as well as the increase of current. All these findingsopen new insights into the understanding of the application of sky- rmions in spintronics. ACKNOWLEDGMENTS This work was supported by the National Basic Research Program of China (No. 2018YFB0407600) and the National Natural Science Foundation of China (NNSFC) (Grant Nos. 11774260 and 51971161). DATA AVAILABILITY The data that support the plots within the paper are available within the article. 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1.4737017.pdf	Network analyzer measurements of spin transfer torques in magnetic tunnel junctions Lin Xue, Chen Wang, Yong-Tao Cui, J. A. Katine, R. A. Buhrman et al. Citation: Appl. Phys. Lett. 101, 022417 (2012); doi: 10.1063/1.4737017 View online: http://dx.doi.org/10.1063/1.4737017 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v101/i2 Published by the American Institute of Physics. Related Articles Manipulating spin injection into organic materials through interface engineering Appl. Phys. Lett. 101, 022416 (2012) Low magnetisation alloys for in-plane spin transfer torque devices J. Appl. Phys. 111, 113904 (2012) Room temperature single GaN nanowire spin valves with FeCo/MgO tunnel contacts Appl. Phys. Lett. 100, 182407 (2012) Observation of anomalous Hall effect in Cu-Py-crossed structure with in-plane magnetization J. Appl. Phys. 111, 07D307 (2012) Micro-fabrication process for small transport devices of layered manganite J. Appl. Phys. 111, 07E129 (2012) Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 16 Jul 2012 to 128.95.104.109. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsNetwork analyzer measurements of spin transfer torques in magnetic tunnel junctions Lin Xue,1Chen Wang,1Y ong-Tao Cui,1J. A. Katine,2R. A. Buhrman,1and D. C. Ralph1,3 1Cornell University, Ithaca, New York 14853, USA 2Hitachi Global Storage Technologies, San Jose, California 95135, USA 3Kavli Institute at Cornell, Ithaca, New York 14853, USA (Received 27 March 2012; accepted 23 June 2012; published online 13 July 2012) We demonstrate a simple network-analyzer technique to make quantitative measurements of the bias dependence of spin torque in a magnetic tunnel junction. We apply a microwave current toexert an oscillating spin torque near the ferromagnetic resonance frequency of the tunnel junction’s free layer. This produces an oscillating resistance that, together with an applied direct current, generates a microwave signal that we measure with the network analyzer. An analysis of theresonant response yields the strength and direction of the spin torque at non-zero bias. We compare to measurements of the spin torque vector by time-domain spin-torque ferromagnetic resonance. VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4737017 ] Spin transfer torque provides the possibility of efﬁ- ciently manipulating the magnetic moment in a nanoscale magnetic device using applied current.1–3Understanding the strength of the spin torque, and particularly its bias depend-ence, is important for applications that include spin torque magnetic random access memory and frequency-tunable microwave oscillators. 4Several different techniques have been developed to measure the bias dependence of the spin torque vector in magnetic tunnel junctions (MTJs), with results that in some cases are inconsistent with each other.These include measurements of the bias dependence of the magnetic precession frequency and linewidth, 5–10DC-volt- age-detected spin torque ferromagnetic resonance (ST-FMR), 11–13ﬁts to the statistics of magnetic switching as a function of current and magnetic ﬁeld,14,15analyses of the current dependence of magnetic astroids and switching phasediagrams, 16,17and time-domain detection of ST-FMR.18Of these, in the high bias regime that is relevant for applica- tions, we believe that the time-domain ST-FMR technique isthe most accurate and trustworthy, since it measures directly the amplitude and phase of small-angle magnetic precession in response to an oscillating spin torque and therefore is leastsusceptible to artifacts associated with heating, spatially non- uniform magnetic dynamics, and changes in the DC resist- ance in response to spin torque. 13,18However, time-domain ST-FMR requires expensive, specialized equipment (i.e., a high-bandwidth oscilloscope and multiple pulse generators). Here we show that it is possible to use a simple network ana-lyzer measurement to determine the bias dependence of the spin torque vector, by studying the resonant response of a magnetic tunnel junction subject to both DC and microwavecurrents. We ﬁnd excellent agreement with time-domain ST- FMR measurements 18made on the same devices. The MgO-based MTJs that we study came from the same batches measured in Refs. 18and19, with resistance- area products for the tunnel barriers equal to RA¼1.5Xlm2 and 1.0 Xlm2. We will present data for one sample with RA¼1.5Xlm2, a resistance of 272 Xin the parallel state, and a tunneling magnetoresistance (TMR) of 91%, but we found similar behavior in three other samples. The device onwhich we will focus has the layer structure (in nm): bottom electrode, IrMn pinned synthetic antiferromagnet (SAF) [IrMn(6.1)/CoFe(1.8)/Ru/CoFeB(2.0)], tunnel barrier [MgO x], magnetic free layer [CoFe(0.5)/CoFeB(3.4)], and cappinglayer [Ru(6.0)/Ta(3.0)/Ru(4.0)]. Both the pinned layer and the free layer were patterned into a circular cross section with a nominal 90 nm diameter. All the measurements weredone at room temperature. We conﬁrmed that the device properties did not degrade during the process of measure- ment 20by checking that the device resistance and TMR remained unchanged. We will use a sign convention that positive values of current correspond to electron ﬂow from the free layer to the reference layer (giving spin torque favor-ing antiparallel alignment). We performed measurements with a commercial net- work analyzer (Agilent 8722ES, 50 MHz–40 GHz) using thecircuit in Fig. 1. We measured the microwave response in a reﬂection geometry, using a bias tee to allow simultaneous application of a DC bias to the MTJ. Before routing thereﬂected microwave signal to the network analyzer, we ampliﬁed it using a 15-dB ampliﬁer in combination with a directional coupler. The microwave gain of the ampliﬁer andtransmission losses in other circuit components were cali- brated by standard methods. Figure 2shows an example of the real and imaginary parts of the reﬂected signal as a func-tion of frequency, in the frequency range exhibiting spin- torque-driven magnetic resonance. These data correspond to a DC current of /C00.4 mA and an applied magnetic ﬁeld H¼200 Oe oriented 70 /C14from the exchange bias of the SAF reference layer, so that the initial offset angle of the two IdcMgOVin Direc/g415onal Coupler VrefNetwork Analyzer+15 dB FIG. 1. The network analyzer circuit used in the measurement. 0003-6951/2012/101(2)/022417/4/$30.00 VC2012 American Institute of Physics 101, 022417-1APPLIED PHYSICS LETTERS 101, 022417 (2012) Downloaded 16 Jul 2012 to 128.95.104.109. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsmagnetic layers is approximately h¼61/C14. The microwave excitation signal Vinthat we applied to the sample had an amplitude always less than 22 mV. Within the model dis- cussed below this results in magnetic precession angles <3/C14, and we veriﬁed that the output signals scaled linearly with Vinas expected in the linear-response regime. To interpret these data, and to use them to measure the strength of the spin transfer torque, we analyze the reﬂected microwave signal Vrefwithin a macrospin model of the mag- netic dynamics, combining the Landau-Lifshitz-Gilbert-Slonczewski equation of motion for a magnetic tunnel junc- tion subject to an oscillating spin torque together with appro- priate microwave circuit equations (see Ref. 19for details). The resulting (complex-valued) reﬂection coefﬁcient corre-sponding to the resonant magnetic response can be written S 11/C17Vref Vin¼R0/C0ð50XÞ R0þð50XÞþð50XÞ R0þð50XÞIDCvðxÞ;(1) where vðxÞ/C17DRðxÞ=Vin¼/C0@R @h/C12/C12/C12/C12/C12 IR0 R0þð50XÞc MSVol1 x/C0xm/C0iri@sjj @V/C12/C12/C12/C12/C12 hþcNxMeff xm@s? @V/C12/C12/C12/C12/C12 h"# ; (2) and the resonance frequency and current-dependent resonant linewidth are xm/C25cMeffﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ NxNy/C01 MeffMSVol@s? @h/C12/C12/C12/C12/C12 Vþð50XÞ R0þð50XÞIDC@R @h/C12/C12/C12/C12/C12 I@s? @V/C12/C12/C12/C12/C12 h ! "#vuut; (3) r/C25acMeffðNxþNyÞ 2 /C0c MsVol@sjj @h/C12/C12/C12/C12/C12 Vþ1 2ð50XÞ R0þð50XÞIDC@R @h/C12/C12/C12/C12/C12 I@sjj @V/C12/C12/C12/C12/C12 h ! : (4) Here R0is the differential resistance of the MTJ, DRðxÞ is the oscillating part of the DC resistance, his the angle between the magnetizations of the two electrodes of the MTJ,ais the Gilbert damping parameter, M SVolis the total mag- netic moment of the free layer, sjjðV;hÞands?ðV;hÞare the “in-plane” and “perpendicular” components of the spin tor-que, Vis the voltage across the MTJ including both DC and high-frequency terms, c¼2l B=/C22his the absolute value of the gyromagnetic ratio, Nx¼4pþH=Meff,Ny/C25H=Meff,4pMeff is the strength of the easy-plane anisotropy ﬁeld, and His the component of applied magnetic ﬁeld along the precession axis. When both in-plane and perpendicular components oftorque are present, both the real and imaginary parts of the resonant signal consist of a sum of frequency-symmetric andantisymmetric Lorentzian curves. Both torque components can therefore be extracted by ﬁtting the symmetric and anti- symmetric parts of either the real or imaginary response. The solid lines in Figs. 2(a) and2(b) show an example of the good agreement we ﬁnd when ﬁtting Eq. (1)to our res- onance measurements. We observe two resonances in each panel in Fig. 2, one with large amplitude near 5.9 GHz and a second with smaller amplitude near 7.5 GHz. We performseparate ﬁts to the real and imaginary curves, employing four free parameters for each resonance in a ﬁt: the center frequency of the resonance, the amplitude of the frequency-symmetric and antisymmetric Lorentzians, and the linewidth (taken to be the same for both the symmetric and antisym- metric components). We allow for a small nonzero constantslope in the non-resonant background signals (dashed lines in Fig. 2) that may be associated with an imperfect capaci- tance calibration. The dependences on HandI DCfor the real part of the resonances are shown in Figs. 3(a) and3(b). As in Fig. 2(a), the spectra contain one primary dip in Re( S11) together with a smaller side resonance at a higher frequency. The primary resonance shifts with Has expected from the Kittel formula(a) (b) 4 5 6 7 89 1 0 11 Frequency (GHz)0.7300.7320.7340.7360.7380.740Re(S11) 4 5 6 7 89 1 0 11 Frequency (GHz)−0.008−0.006−0.004−0.0020.0000.0020.004Im(S11) (a) (b)FIG. 2. The measured (a) real part and (b) imaginary part of the reﬂection signal (S 11) for IDC¼/C00.4 mA and a magnetic ﬁeld H¼200 Oe applied 70/C14from the exchange bias direction, giving h¼61/C14. The solid lines are a ﬁt to Eq. (1). The dashed lines are the nonresonant back- grounds used in the ﬁts.022417-2 Xue et al. Appl. Phys. Lett. 101, 022417 (2012) Downloaded 16 Jul 2012 to 128.95.104.109. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionswhile the secondary signal shifts more slowly and decreases in amplitude with increasing ﬁeld strength. We suspect that the secondary peak may involve coupled motion of the magneticlayers in the synthetic antiferromagnet polarizing layer. To avoid having this mode interfere with our measurements of spin torque, we select values of magnetic ﬁeld and magneticﬁeld angle such that the secondary mode has small amplitude and maximum separation in frequency from the primary mode. These are the same selection criteria used in Ref. 18. B a s e do nE q s . (1)and(2), for any value of bias we can determine the spin transfer “torkances” 21@sjj=@Vjhand @s?=@Vjhfrom ﬁts to the frequency-symmetric and antisym- metric parts of the primary resonance in either Re( S11)o r Im(S11). In calculating the torkan ces from the resonant ampli- tudes, we use the following parameters: MSVol¼1.8 /C210/C014emu (615%),184pMeff¼1361k O ed e t e r m i n e df r o m high-ﬁeld measurements of th e resonance frequency, and a¼0.01660.001 determined by measuring the resonance line- width at positive and negative biases and interpolating to zero bias. In Fig. 3(c), we plot the bias dependence of the resulting torkances as found by the network analyzer technique. We nor-malize the results by sin hsince the spin torque of a MTJ is pre- dicted to have this angular dependence. 21We note that the torkance values determined by independent ﬁts to the real andimaginary parts of the resonance agree, as is required in order that our analysis procedure be self-consistent. Figure 3(c) also shows a comparison to measurements on the same sampleusing the time-domain ST-FMR technique introduced in Ref. 18, whereby the magnetic precession driven by a resonant spin torque is detected by a fast oscilloscope. We ﬁnd excellentagreement between the two types of measurements. The in- plane component of the torkance, @s jj=@Vjh,m e a s u r e db yt h e two techniques agrees in magnitude near zero bias with thesame moderate dependence on bias, with no adjustment of pa- rameters for either technique. The perpendicular component @s ?=@Vjhdisplays the same approxim ately linear bias depend-ence at low bias. In Fig. 3(d), we plot the full bias dependent torques skðVÞands?ðVÞ, obtained by numerical integration of the torkances. Neither the network-analyzer ST-FMR technique nor the time-domain ST-FMR technique can be used at V¼0, because a non-zero DC bias is required to generate the oscil-latory voltage signal that is measured (see Fig. 3(c)). (For measurements near zero bias, DC-voltage-detected ST-FMR can provide accurate torque measurements without artifactsin the mixing signal. 11–13) The time-domain ST-FMR tech- nique allows measurements to higher biases, because it is naturally implemented using short bias pulses that are lesslikely to produce dielectric breakdown in the tunnel barrier, compared to the constant DC biases used in our network ana- lyzer technique. However, in the bias range shown in Figs.3(c) and3(d), the network analyzer method provides a more convenient approach in that it does not require specialized, expensive equipment, while it yields a sensitivity compara-ble to time-domain ST-FMR. In summary, we demonstrate that it is possible to use a simple network-analyzer technique to measure the strengthand direction of the spin transfer torque vector as a function of bias in magnetic tunnel junctions. This technique provides roughly similar sensitivity as the time-domain ST-FMRmethod, 18making it useful as a simple and rapid means for characterizing spin-torque devices. Cornell acknowledges support from ARO, NSF (DMR- 1010768), ONR, and the NSF/NSEC program through theCornell Center for Nanoscale Systems. We also acknowl- edge NSF support through use of the Cornell Nanofabrica- tion Facility/NNIN and the Cornell Center for MaterialsResearch facilities (DMR-1120296). 1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 2L. Berger, Phys. Rev. B 54, 9353 (1996). 3D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008).FIG. 3. (a) Measured frequency dependence of the real part of S11for several values of magnetic ﬁeld applied 70/C14from the exchange bias direction, with IDC¼/C00.4 mA. The curves are offset by 0.01 vertically. (b) Measured frequency depend- ence of the real part of S11for several values of DC current, with H¼200 Oe applied 70/C14from the exchange bias direction. The curves are offset vertically by 0.01. (c) Bias dependence of the in-plane and perpendicular components of the torkance @s=@Vjhdetermined by ﬁtting to the fre- quency dependence of Re( S11) (red circles) and Im(S11) (blue diamonds) at different values of the DC bias. These data correspond to H¼200 Oe applied 70/C14from the exchange bias direction, giv- ingh¼61/C14. For comparison, we also show in gray the results on the same device from time-domain ST-FMR measurements (triangles: for H¼250 Oe applied 95/C14from the exchange bias direction giv- ingh¼85/C14; squares: H¼200 Oe applied at 68/C14 giving h¼64/C14). (d) Integrated in-plane and per- pendicular components of the spin torque vector determined by integrating the network-analyzer data in (c), with representative error bars.022417-3 Xue et al. Appl. Phys. Lett. 101, 022417 (2012) Downloaded 16 Jul 2012 to 128.95.104.109. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions4J. A. Katine and E. E. Fullerton, J. Magn. Magn. Mater. 320, 1217 (2008). 5S. Petit, C. Baraduc, C. Thirion, U. Ebels, Y. Liu, M. Li, P. Wang, and B. Dieny, Phys. Rev. Lett. 98, 077203 (2007). 6S. Petit, N. de Mestier, C. Baraduc, C. Thirion, Y. Liu, M. Li, P. Wang, and B. Dieny, Phys. Rev. B 78, 184420 (2008). 7A. M. Deac, A. Fukushima, H. Kubota, H. Maehara, Y. Suzuki, S. Yuasa, Y. Nagamine, K. Tsunekawa, D. D. Djayaprawira, and N. Watanabe, Nat. Phys. 4, 803 (2008). 8M. H. Jung, S. Park, C.-Y. You, and S. Yuasa, Phys. Rev. B 81, 134419 (2010). 9O. G. Heinonen, S. W. Stokes, and J. Y. Yi, Phys. Rev. Lett. 105, 066602 (2010). 10P. K. Muduli, O. G. Heinonen, and J. Akerman, Phys. Rev. B 83, 184410 (2011). 11J. C. Sankey, Y.-T. Cui, J. Z. Sun, J. C. Slonczewski, R. A. Buhrman, andD. C. Ralph, Nat. Phys. 4, 67 (2008). 12H. Kubota, A. Fukushima, K. Yakushiji, T. Nagahama, S. Yuasa, K. Ando, H. Maehara, Y. Nagamine, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and Y. Suzuki, Nat. Phys. 4, 37 (2008).13C. Wang, Y.-T. Cui, J. Z. Sun, J. A. Katine, R. A. Buhrman, and D. C. Ralph, Phys. Rev. B 79, 224416 (2009). 14Z. Li, S. Zhang, Z. Diao, Y. Ding, X. Tang, D. M. Apalkov, Z. Yang, K. Kawabata, and Y. Huai, Phys. Rev. Lett. 100, 246602 (2008). 15S.-C. Oh, S.-Y. Park, A. Manchon, M. Chshiev, J.-H. Han, H.-W. Lee, J.-E. Lee, K.-T. Nam, Y. Jo, Y.-C. Kong, B. Dieny, and K.-J. Lee, Nat. Phys. 5, 898 (2009). 16T. Devolder, J.-V. Kim, C. Chappert, J. Hayakawa, K. Ito, H. Takahashi, S. Ikeda, and H. Ohno, J. Appl. Phys. 105, 113924 (2009). 17S.-Y. Park, Y. Jo, and K.-J. Lee, Phys. Rev. B 84, 214417 (2011). 18C. Wang, Y.-T. Cui, J. A. Katine, R. A. Buhrman, and D. C. Ralph, Nat. Phys. 7, 496 (2011). 19L. Xue, C. Wang, Y.-T. Cui, J. A. Katine, R. A. Buhrman, and D. C. Ralph, Appl. Phys. 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1.4817281.pdf	Dependence of spin torque diode voltage on applied field direction Tomohiro Taniguchi and Hiroshi Imamura Citation: J. Appl. Phys. 114, 053903 (2013); doi: 10.1063/1.4817281 View online: http://dx.doi.org/10.1063/1.4817281 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v114/i5 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 13 Aug 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsDependence of spin torque diode voltage on applied field direction Tomohiro Taniguchi and Hiroshi Imamura Spintronics Research Center, AIST, 1-1-1 Umezono, Tsukuba 305-8568, Japan (Received 18 June 2013; accepted 16 July 2013; published online 2 August 2013) The optimum condition of an applied ﬁeld direction to maximize spin torque diode voltage was theoretically derived for a magnetic tunnel junction with a perpendicularly magnetized free layer and an in-plane magnetized pinned layer. We found that the diode voltage for a relatively small applied ﬁeld is maximized when the projection of the applied ﬁeld to the ﬁlm-plane is parallel oranti-parallel to the magnetization of the pinned l ayer. However, by increasing the applied ﬁeld magnitude, the optimum applied ﬁeld direction shift s from the parallel or anti-parallel direction. These analytical predictions were conﬁrmed by numerical simulations. VC2013 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4817281 ] I. INTRODUCTION Magnetization dynamics induced by spin torque in nano-structured ferromagnets1–4have provided interesting phenomena, such as magnetization switching and oscillation. Many spintronics devices utilizing the spin torque have been proposed, such as a magnetic random access memory(MRAM) based on magnetic tunnel junctions (MTJs) and a microwave oscillator. 5,6The spin torque diode effect7–21is also an important phenomenon, in which an alternating cur-rent applied to an MTJ is rectiﬁed by synchronizing the reso- nant oscillation of tunnel magnetoresistance (TMR) 22,23by spin torque with the alternating current. The spin torquediode effect has been used to quantitatively evaluate the strength of the spin torque. 8,9 The spin torque diode effect is applicable to a magnetic sensor application, where a small magnetic ﬁeld from a fer- romagnetic or paramagnetic particle modulates the reso- nance condition of the spin torque diode.19In such a sensor application, a large diode voltage is required to enhance sen- sitivity, deﬁned as the ratio between the input power and the diode voltage.17It should also be noted that the direction of the applied ﬁeld, which is proportional to the spin of the par- ticle, points in an arbitrary direction. Thus, it is important to clarify the relation between the spin torque diode voltageand the applied ﬁeld direction, and to maximize the spin tor- que diode voltage. In this paper, we derive the optimum condition of the applied ﬁeld direction to maximize the spin torque diode volt- age of an MTJ with a perpendicularly magnetized free layer and an in-plane magnetized pinned layer. This type of MTJwas recently developed in experiments, 14,24,25and is consid- ered an ideal candidate for spin torque diode application because of its narrow linewidth and high diode voltage. Weﬁrst derived the general formula of the spin torque diode volt- age, and then, applied the formula to the system under consid- eration. The main result is Eq. (40), which represents the applied ﬁeld direction at the maximized diode voltage. The diode voltage for a relatively small applied ﬁeld is maximized when the projection of the applied ﬁeld to the ﬁlm-plane is par-allel or anti-parallel to the magnetization of the pinned layer. However, the optimum applied ﬁeld direction shifts from theparallel or anti-parallel direction by increasing the applied ﬁeld magnitude. These results are conﬁrmed numerically. The paper is organized as follows. In Sec. II, we derive the analytical solution to the linearized Landau-Lifshitz- Gilbert (LLG) equation of the free layer. In Sec. III,t h eg e n - eral formula of the spin torque diode voltage and its depend- ence on the magnetization alignment are discussed. Section IVis the main section in this paper, where we derive the opti- mum condition for the applied ﬁeld direction in an MTJ with a perpendicularly magnetized free layer and an in-plane mag- netized pinned layer. Section Vis devoted to the conclusions. II. SOLUTION TO THE LINEARIZED LLG EQUATION In this section, we solve the linearized LLG equation for an arbitrary magnetization alignment. The system we consider is schematically shown in Fig. 1, where the MTJ consists of free and pinned layers separated by a thin nonmagnetic spacer. Thex,y,a n d zaxes are parallel to the uniaxial anisotropy axes of the free layer. The unit vectors pointing in the direction ofthe magnetizations of the free and the pinned layers are denoted as mandp¼ðsinh pcosup;sinhpsinup;coshpÞ; respectively, where the zenith and the azimuth angles of themagnetization of the pinned layer are denoted as h pandup; FIG. 1. Schematic view of an MTJ. The unit vectors pointing to the magnet- ization directions of the free and the pinned layers are denoted as mandp, respectively. The positive current is deﬁned as the electron ﬂow from the pinned to the free layer. 0021-8979/2013/114(5)/053903/7/$30.00 VC2013 AIP Publishing LLC 114, 053903-1JOURNAL OF APPLIED PHYSICS 114, 053903 (2013) Downloaded 13 Aug 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsrespectively. We assume that the magnetization dynamics in the presence of the spin torque are well described by the mac- rospin LLG Eqs. (1)–(4)26–28 dm dt¼/C0cm/C2H/C0caJm/C2ðp/C2mÞ þcbJp/C2mþam/C2dm dt; (1) where candaare the gyromagnetic ratio and the Gilbert damping constant, respectively. The magnetic ﬁeld His deﬁned as the derivative of the energy density Ewith respect to the magnetization, i.e., H¼/C01 M@E @m; (2) where Mis the saturation magnetization. The energy density Eis given by E¼/C0MH appl½sinhHsinhcosðu/C0uHÞþcoshHcosh/C138 þX ‘¼x;y;z2pM2~N‘m2 ‘;(3) where Happl;hH;anduHin the ﬁrst term are the magnitude, the zenith angle, and the azimuth angle of the applied ﬁeld, respectively. The second term of Eq. (3)describes the uniax- ial anisotropy energy. The coefﬁcient ~N‘(‘¼x;y;z)i s deﬁned as 4 pM~N‘¼4pMN ‘/C0HK‘;where 4 pMN ‘andHK‘ are the shape anisotropy ﬁeld (demagnetization ﬁeld) and the crystalline anisotropy ﬁeld along the ‘-axis, respectively. The demagnetization coefﬁcients satisfy NxþNyþNz¼1: The two components of the spin torque in Eq. (1), the Slonczewski torque and the ﬁeld like torque, are denoted as aJandbJ, respectively, whose explicit forms are given by aJ¼/C22hgI 2eMV(4) andbJ¼baJ:Here, Iis the current and Vis the volume of the free layer, respectively. The positive current corresponds tothe electron ﬂow from the free to the pinned layer. We assume that both the direct (dc) and alternating (ac) currents are applied to the MTJ, i.e., I¼I dcþIacðtÞ:Thus, aJandbJare decomposed into the dc and the ac parts as aJ¼aJðdcÞþ aJðacÞandbJ¼bJðdcÞþbJðacÞ;respectively. The magnitude of the direct current is on the order of 0.1–1.0 mA, while that ofthe alternating current is 0.1 mA. 7,8As shown below, the pres- ent formula is valid for jaJðdcÞj<ajHj;where the Gilbert damping constant is on the order of 10/C02.29The ratio of the Slonczewski torque to the ﬁeld like torque, b, is on the order of 0.1 for MTJs.7–9The factor gcharacterizes the spin polar- ization of the current, and is given by1–3 g¼g 1þkm/C1p: (5) The dimensionless parameters, gandk, characterize the magnitude of the spin polarization and the dependence of the spin torque on magnetization alignment, respectively. Although the relation among g,k, and the material parame- ters depends on the theoretical models, the form of Eq. (5)isapplicable to spin torque in both MTJs and giant- magnetoresistive system.30For example, in the ballistic transport theory in MTJ, gis proportional to the spin polar- ization of the density of state of the free layer and k¼g2.1,6,31Below, we set k¼0 for simplicity. The spin torque diode voltage and its optimum condition with ﬁnite k are discussed in the Appendix. The solution to the LLG equation is derived in an XYZ- coordinate in which the Z-axis is parallel to the steady state of the magnetization of the free layer. We denote mat the steady state as mð0Þ;where the condition d mð0Þ=dt¼0can be expressed in terms of the zenith and the azimuth angles,ðh;uÞ;as 32,33 Happl½sinhHcoshcosðu/C0uHÞ/C0coshHsinh/C138 /C04pMð~Nxcos2uþ~Nysin2u/C0~NzÞsinhcosh /C0aJðdcÞsinhpsinðu/C0upÞ þbJðdcÞ½sinhpcoshcosðu/C0upÞ/C0coshpsinh/C138¼0;(6) HapplsinhHsinðu/C0uHÞ/C04pMð~Nx/C0~NyÞsinhsinucosu /C0aJðdcÞ½sinhpcoshcosðu/C0upÞ/C0coshpsinh/C138 /C0bJðdcÞsinhpsinðu/C0upÞ¼0: (7) In the absence of the direct current, ðh;uÞsatisfying Eqs. (6) and(7)correspond to the equilibrium state, i.e., the mini- mum state of the energy density E. The transformation from thexyz-coordinate to the XYZ-coordinate is performed by multiplying the following rotation matrix to Eq. (1): R¼cosh0/C0sinh 01 0 sinh0 cos h0 @1 Acosu sinu0 /C0sinucosu0 00 10 @1 A:(8) For example, the components of pin the XYZ-coordinate can be expressed as pX pY pZ0 @1 A¼coshsinhpcosðu/C0upÞ/C0sinhcoshp /C0sinhpsinðu/C0upÞ sinhsinhpcosðu/C0upÞþcoshcoshp0 @1 A:(9) The alternating current exerts a small amplitude oscilla- tion of the magnetization around the Z-axis. Then, the LLG equation can be linearized by assuming mZ’1 and jmXj;jmYj/C281;and is given by 1 cd dtmX mY/C18/C19 þ/C0H YXþaHXHY/C0aHXY /C0H X/C0aHYXHXYþaHY/C18/C19mX mY/C18/C19 ¼/C0aJðacÞpXþbJðacÞpY /C0aJðacÞpY/C0bJðacÞpX ! ; (10) where we use the approximation that 1 þa2’1.29The com- ponents of Hare deﬁned as HX¼HXþbJðdcÞpZ; (11) HY¼HYþbJðdcÞpZ; (12) HXY¼HXY/C0aJðdcÞpZ; (13)053903-2 T. Taniguchi and H. Imamura J. Appl. Phys. 114, 053903 (2013) Downloaded 13 Aug 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsHYX¼HYXþaJðdcÞpZ: (14) Here, HX¼HZZ/C0HXX;HY¼HZZ/C0HYY:The ﬁeld Hij (i;j¼X;Y;Z) are the i-components of the magnetic ﬁeld in the XYZ-coordinate proportional to mj,H¼ðHXXmXþ HXYmY;HYXmXþHYYmY;HZZþHZXmXþHZYmYÞ;where the explicit forms of Hijare given by HXX¼/C04pM½ð~Nxcos2uþ~Nysin2uÞcos2hþ~Nzsin2h/C138; (15) HXY¼HYX¼4pMð~Nx/C0~NyÞcoshsinucosu; (16) HYY¼/C04pMð~Nxsin2uþ~Nycos2uÞ; (17) HZX¼/C04pMð~Nxcos2uþ~Nysin2u/C0~NzÞsinhcosh;(18) HZY¼/C04pMð~Ny/C0~NxÞsinhsinucosu; (19) HZZ¼Happl½sinhHsinhcosðu/C0uHÞþcoshHcosh/C138 /C04pM½ð~Nxcos2uþ~Nysin2uÞsin2hþ~Nzcos2h/C138: (20) By assuming that the alternating current is given by Iacsinð2pftÞ;the solutions to ðmX;mYÞin Eq. (10) are, respec- tively, given by34 mX’Im~cðifþ~cHXYÞpX/C0~c2HYpY f2/C0f2 res/C0ifDfe2pift"# ~aJðacÞ /C0Im~cðifþ~cHXYÞpYþ~c2HYpX f2/C0f2 res/C0ifDfe2pift"# ~bJðacÞ;(21) mY’Im~cðif/C0~cHYXÞpY/C0~c2HXpX f2/C0f2 res/C0ifDfe2pift"# ~aJðacÞ /C0Im/C0~cðif/C0~cHYXÞpXþ~c2HXpY f2/C0f2 res/C0ifDfe2pift"# ~bJðacÞ;(22) where ~c¼c=ð2pÞ:~aJðacÞand ~bJðacÞare deﬁned as aJðacÞ ¼~aJðacÞsinð2pftÞand bJðacÞ¼~bJðacÞsinð2pftÞ;respectively. The resonance frequency fresand the linewidth Dfare, respectively, given by fres¼c 2pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HXHY/C0H XYHYXp ; (23) Df¼c 2p½aðHXþH YÞþH XY/C0H YX/C138: (24) In the absence of the direct current, Eq. (23) is the ferromag- netic resonance (FMR) frequency, fFMR¼cﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HXHY/C0H2 XYp =ð2pÞ:Since bis on the order of 0.1,7–9andaJðdcÞis on the order of a small parameter a,abJðdcÞin Eq. (24) is negligible. Thus,Dfcan be approximated to Df’c 2p½aðHXþHYÞ/C02aJðdcÞpZ/C138: (25) III. SPIN TORQUE DIODE VOLTAGE The magnetoresistance of an MTJ is given by R ¼RPþðDR=2Þð1/C0m/C1pÞ;where DR¼RAP/C0RPis thedifference in the resistances between the parallel ( RP) and the anti-parallel ( RAP) alignments of the magnetizations. The spin torque diode voltage is given by Vdc¼T/C01ÐT 0IðtÞRðtÞdt; where T¼1=fis the period of the alternating current. By using Eqs. (21) and(22), the explicit form of the spin torque diode voltage is given by Vdc¼DRIac 4Re/C0½ifðp2 Xþp2 YÞþ~cHa/C138~c~aJðacÞþ~c2Hb~bJðacÞ f2/C0f2 res/C0ifDf"# ¼DRIac 4½LðfÞþAðfÞ/C138; (26) whereHaandHbare, respectively, given by Ha¼H XYp2 X/C0H YXp2Yþð H X/C0H YÞpXpY; (27) Hb¼H Yp2 XþH Xp2Yþð H XYþH YXÞpXpY: (28) The Lorentzian and the anti-Lorentzian parts, LðfÞand AðfÞare, respectively, given by LðfÞ¼f2Df~c~aJðacÞð1/C0p2 ZÞ ðf2/C0f2 resÞ2þðfDfÞ2; (29) AðfÞ¼/C0ðf2/C0f2 resÞ~c2ðHa~aJðacÞ/C0Hb~bJðacÞÞ ðf2/C0f2 resÞ2þðfDfÞ2: (30) As shown, the Lorentzian part depends on the Slonczewski torque only while the anti-Lorentzian part depends on boththe Slonczewski torque and the ﬁeld like torque, in general. The peak of the spin torque diode voltage appears around the resonance frequency, f res;where the Lorentzian part shows a peak while the anti-Lorentzian part is zero. At f¼fres;the spin torque diode voltage is VdcðfresÞ¼DRIac 4~aJðacÞsin2w aðHXþHYÞ/C02aJðdcÞcosw; (31) where w¼cos/C01pZ¼cos/C01mð0Þ/C1pin the relative angle between the magnetizations of the free and the pinned layers. Equation (31) is maximized when the relative angle of the magnetizations is given by wopt¼cos/C01Ic Idc7ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ic Idc/C18/C192 /C01s2 43 5; (32) where the double sign “ 7” means the upper ( /C0)f o r Idc=Ic>0 and the lower ( þ)f o r Idc=Ic<0:The critical current of the spin torque induced magnetization dynamics in the case of mð0Þkp;Ic,i sg i v e nb y Ic¼2aeMV /C22hgHXþHY 2/C18/C19 : (33) Since wis a real number, the following condition should be satisﬁed: Ic Idc/C12/C12/C12/C12/C12/C12/C12/C12>1: (34)053903-3 T. Taniguchi and H. Imamura J. Appl. Phys. 114, 053903 (2013) Downloaded 13 Aug 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsThis condition means that the linear approximation cannot be applied to the LLG equation when the spin torque over- comes the damping. The maximized spin torque diode volt-age is given by V opt dcðfresÞ¼DRI2 ac 4IdcIc Idc7ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ic Idc/C18/C192 /C01s2 43 5: (35) Equations (32) and(35) are the main results in this section, and can be regarded as generalizations of the result in Ref. 32. We emphasize that the optimum condition, Eq. (32),d e p e n d s on not only the material (sample) parameters and the applied ﬁeld but also the magnitude and direction of the direct current. It should be noted that Eq. (32) is 90/C14forIdc¼0;and shifts from this orthogonal alignment for a ﬁnite Idc: Equation (32) is the optimum condition of the magnet- ization alignment to maximize the spin torque diode voltage.However, in experiments, the direction of the applied ﬁeld is more easily controlled, than the magnetization alignment, because the direction of the magnetization of the pinnedlayer is ﬁxed by the exchange bias from an anti- ferromagnetic layer. In Sec. IVby using Eq. (32), we derive the analytical formula of the optimum condition of theapplied ﬁeld direction to maximize the spin torque diode voltage in an MTJ with a perpendicularly magnetized free layer and an in-plane magnetized pinned layer. IV. OPTIMUM CONDITION OF APPLIED FIELD DIRECTION In an MTJ with a perpendicularly magnetized free layer and an in-plane magnetized pinned layer, ðhp;upÞin Fig. 1 areð90/C14;0/C14). The free layer has uniaxial anisotropy along the easy axis which is normal to the ﬁlm plane, and has a circu- lar cross section. The components of the anisotropy ﬁeld are4pM~N x¼4pM~Ny¼0 and 4 pM~Nz¼/C0HKþ4pM;where thez-axis is parallel to the easy axis. Since we are interested in the perpendicularly magnetized free layer, the anisotropyﬁeld H Kshould be larger than the demagnetization ﬁeld 4pM:The x-axis is parallel to the magnetization of the pinned layer. We assume that the magnetic ﬁeld is appliedtilted from the z-axis with the angle h Hð0<hH<pÞ:In the following, we investigate the optimum direction of the applied ﬁeld in the ﬁlm-plane, uH;to maximize the diode voltage. We assume that the steady state ðh;uÞis determined by Eqs. (6)and (7)by neglecting the spin torque term, i.e., ðh;uÞcorresponds to the equilibrium state, because the spin torque term is on the order of a small parameter a. The equi- librium state of the free layer satisﬁes Happlsinðh/C0hHÞþð HK/C04pMÞsinhcosh¼0( 3 6 ) andu¼uH:Then, the critical current Ic¼2aeMV /C22hgHapplcosðh/C0hHÞþH?cos2hþcos2h 2/C20/C21 ;(37) is independent of uH:Equation (32)can be expressed assinhcosu¼Ic Idc7ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ic Idc/C18/C192 /C01s : (38) As mentioned above, uon the left-hand side of Eq. (38) can be replaced by uH:When the condition 1 sinhIc Idc7ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ic Idc/C18/C192 /C01s2 43 5/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12/C12<1 (39) is satisﬁed, the diode voltage is maximized at the ﬁeld direction u H¼cos/C011 sinhIc Idc7ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ic Idc/C18/C192 /C01s2 43 58 < :9 = ;: (40) Equation (40) is the main result in this paper. For a relatively small applied ﬁeld magnitude, the equilibrium state is close to the easy axis, i.e., h’sinh/C281;and Eq. (39) is not satis- ﬁed. Then, the diode voltage is maximized at uH¼0o r p, depending on the direction of the current. However, by increasing the applied ﬁeld magnitude, the magnetizationtilts from the easy axis, and Eq. (39) is satisﬁed. The opti- mum direction of the applied magnetic ﬁeld then shifts from u H¼0;ptouHgiven by Eq. (40). The physical meaning of the condition (39) is as fol- lows. As mentioned in the paragraph below Eq. (35), the spin torque diode voltage is maximized near the orthogonalalignment of the magnetization. When the magnitude of the applied ﬁeld is small, this condition is approximately satis- ﬁed. Then, the magnetization should oscillate in the xz-plane to obtain a large oscillation amplitude of TMR because p points to the x-direction. Thus, Eq. (40) is 0 or p. However, the magnetization moves to the xy-plane for a relatively large applied ﬁeld. To keep the relative angle of the magnetiza- tions close to Eq. (32), the magnetization of the free layer should shift from the x-axis by changing the ﬁeld direction. Thus, the optimum ﬁeld direction shifts from u H¼0;p; according to Eq. (40). The reason why the analytical solution of the optimum applied ﬁeld direction, Eq. (40), can be obtained in this sys- tem is that, because of the axial symmetry, uin Eq. (38) can be replaced by uH:In the general system, both sides of Eq. (32) depend on the applied ﬁeld direction ðhH;uHÞthrough Eqs. (6)and(7). Consequently, an analytical expression of the optimum ﬁeld direction cannot be obtained. Let us quantitatively estimate the optimum direction, uH:Figure 2shows the dependence of the spin torque diode voltage, VdcðfresÞ;on the applied ﬁeld direction, uH;for sev- eral values of HapplandIdc:The values of the parameters25 are M¼1313 emu/c.c., HK¼17:9k O e , hH¼60/C14;V ¼p/C250/C250/C22n m3,c¼17:32 MHz/Oe, a¼0:005;g ¼0:33;b¼0:1;Iac¼0:1 mA, and DR¼100X, respec- tively. The values of Happland Idcare (a) ðHapplðkOeÞ; IdcðmAÞ Þ¼ð 1:0;0:2Þ;(b)ð1:0;/C00:2Þ;(c)ð5:0;0:2Þ;and (d) ð5:0;/C00:2Þ;respectively, where the value of Idcis chosen to observe the shift of the optimum uHfrom 0 or pto a certain angle in a typical range of Happlin experiments.25The053903-4 T. Taniguchi and H. Imamura J. Appl. Phys. 114, 053903 (2013) Downloaded 13 Aug 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionscurrent magnitude (0.2 mA) is also a typical value used in experiments (for example, Ref. 8). The steady state of the magnetization of the free layer is h¼26:3/C14forHappl¼1:0 kOe. In this case, Eq. (39)is not satisﬁed, and thus, the diode voltage is maximized at uH¼0 for Idc=Ic>0 and at uH ¼pforIdc=Ic<0;as shown in Figs. 2(a) and2(b), respec- tively. On the other hand, the steady state is given by h¼52:0/C14forHappl¼5:0 kOe. The condition, Eq. (39), is satis- ﬁed, and the optimum direction of the applied ﬁeld is given byuH¼63:7/C14and 296 :3/C14forIdc¼0:2 mA and 73 :9/C14and 106:1/C14for/C00.2 mA, respectively. The maximized voltage is estimated to be 272 lV while the diode voltages at uH¼0 andpforIdc>0 are estimated to be 146 and 73 lV, respec- tively. Since the relative angle between the magnetizationsdecreases as the applied ﬁeld magnitude increases, the maxi- mized voltage for H appl¼5:0 kOe is smaller than that for Happl¼1:0 kOe. We perform numerical simulations35to conﬁrm the above analytical results. Figures 3(a) and 3(b) show the dependences of the spin torque diode voltage at uH ¼uopt H;0;andpon the frequency of the alternating current with Idc¼0:2 mA and /C00.2 mA, respectively. The magni- tude of the applied magnetic ﬁeld is Happl¼5:0 kOe. A sharp peak of the diode voltage appears near the FMR fre- quency, fFMR’13:8 GHz. The magnitudes of the diode volt- age at f¼fresagree well with the results shown in Fig. 2, demonstrating the validity of the above analytical formula. V. CONCLUSIONS In conclusion, we derive the optimum condition of the applied ﬁeld direction to maximize the diode voltage of an MTJ with a perpendicularly magnetized free layer and an in- plane magnetized pinned layer, which was recently devel-oped in experiments. For a relatively small applied ﬁeld, the diode voltage is maximized when the projection of the applied ﬁeld to the ﬁlm-plane is parallel or anti-parallel tothe magnetization of the pinned layer. However, the voltage is maximized at a certain direction shifted from the parallel or anti-parallel direction by increasing the applied ﬁeld mag-nitude. These results are conﬁrmed by numerically solving the Landau-Lifshitz-Gilbert equation.FIG. 2. The dependence of the diode voltage at the resonance, VdcðfresÞ;on the applied ﬁeld direction, uH:The values of the applied ﬁeld and the current are(a)ðH applðkOeÞ;IdcðmAÞÞ ¼ ð 1:0;0:2Þ; (b)ð1:0;/C00:2Þ;(c)ð5:0;0:2Þ;and (d) ð5:0;/C00:2Þ;respectively. FIG. 3. The dependences of the spin torque diode voltage at uH¼uopt H (black, solid),0 (red, dotted), and p(blue, dashed) on the frequency of the applied alternating current. The values of the direct current are (a) Idc¼0:2 mA and (b) /C00.2 mA, respectively.053903-5 T. Taniguchi and H. Imamura J. Appl. Phys. 114, 053903 (2013) Downloaded 13 Aug 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsACKNOWLEDGMENTS The authors would like to acknowledge H. Kubota, H. Maehara, A. Emura, T. Yorozu, H. Arai, S. Yuasa, K. Ando, and S. Miwa for the valuable discussions they had with us. This work was supported by JSPS KAKENHI Number23226001. APPENDIX: SPIN TORQUE DIODE VOLTAGE AND ITS MAXIMIZED CONDITION WITH FINITE k In this appendix, the spin torque diode voltage with ﬁ- nitekin Eq. (5)is derived. First, let us brieﬂy describe the importance of k, which arises from the dependence of the tunneling probability on the magnetization alignment.10Since the magnitude of kis small, for simplicity, we assume kis zero in some cases.35 However, when the magnetization alignment of the free and the pinned layers in equilibrium is orthogonal ( mð0Þ?p), a ﬁ- nitekplays a key role in the spin torque induced magnetiza- tion dynamics. For example, when kis neglected, the critical current for the magnetization dynamics, Eq. (A7) shown below, diverges for mð0Þ?p(i.e., pZ¼0). This is because the work done by spin torque is zero at this alignment, and thus, the spin torque cannot overcome the damping.However, if k6¼0;the critical current remains ﬁnite because the work done by spin torque is also ﬁnite, and can be larger than the energy dissipation due to the damping. 36 Now we calculate the diode voltage with ﬁnite k. Let us redeﬁne aJandbJas aJ/C17/C22hgI 2eMVð1þkpZÞ(A1) andbJ¼baJ:Also, we introduce Kas K/C17k 1þkpZ: (A2) Then, instead of Eqs. (11)–(14), we redeﬁne HX;HY;HXY; andHYXas HX¼HXþbJðdcÞpZþaJðdcÞKpXpYþbJðdcÞKp2 X;(A3) HY¼HYþbJðdcÞpZ/C0aJðdcÞKpXpYþbJðdcÞKp2 Y; (A4) HXY¼HXY/C0aJðdcÞpZ/C0aJðdcÞKp2 Y/C0bJðdcÞKpXpY;(A5) HYX¼HYXþaJðdcÞpZþaJðdcÞKp2 X/C0bJðdcÞKpXpY:(A6) By using these H, the resonance frequency, the linewidth, andHare redeﬁned according to Eqs. (23),(24),(27), and (28), respectively. Then, the diode voltage is given by Eqs. (26),(29), and (30). The critical current for the magnetiza- tion dynamics is given by Ic¼2aeMV /C22hgð1/C0abÞ½pZþKð1/C0p2 ZÞ/C138HXþHY 2/C18/C19 :(A7)It is difﬁcult for an arbitrary k(/C01<k<1) to derive the optimum condition. However, the optimum condition for jkj/C281 can be derived as follows. In this case, the diode voltage at f¼fresis given by Eq. (31)in which aJis replaced by Eq. (A1). Then, the diode voltage is maximized when the relative angle of mð0Þandpis given by wopt¼cos/C01ðIc=IdcÞ7ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðIc=IdcÞ2/C0½1/C0kðIc=IdcÞ/C1382q 1/C0kðIc=IdcÞ8 < :9 = ;; (A8) where Icis deﬁned by Eq. (33). Equation (A8) is identical to Eq.(32)in the limit of k!0:The maximized voltage at f¼ fresis given by Vopt dc¼DRI2 ac 4IdcðIc=IdcÞ7ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðIc=IdcÞ2/C0½1/C0kðIc=IdcÞ/C1382q ½1/C0kðIc=IdcÞ/C13828 < :9 = ;: (A9) The spin torque diode effect is useful for estimating the value of kexperimentally. For example, let us consider the spin torque diode effect of MTJ with the perpendicularly magnetized free layer and the in-plane magnetized pinned layer discussed in Sec. IV. The direct current is assumed to be zero. By ﬁxing the magnitude ( Happl) and the tilted angle (hH) of the applied ﬁeld, the resonance frequencies and the linewidths at a certain uHandp/C0uHare identical. Then, the ratio of the diode voltages at uHandp/C0uHis given by Vdcðf¼fres;uHÞ Vdcðf¼fres;p/C0uHÞ¼1/C0ksinh 1þksinh; (A10) where the factor 1 7ksinhappears from ~aJðacÞ/1=ð1þ kpZÞin the numerator of Eq. (31). Since the value of his determined by Eq. (36), the value of kcan be estimated by this ratio. This method of estimating kis applicable to gen- eral system if there are at least two equilibrium states with identical resonance frequencies and linewidths and differentrelative angles with the magnetization of the pinned layer. 1J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989). 2J. C. Slonczewski, J. Magn. Magn. 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1.2940734.pdf	Spin dynamics triggered by subterahertz magnetic field pulses Zhao Wang, Matthäus Pietz, Jakob Walowski, Arno Förster, Mihail I. Lepsa et al. Citation: J. Appl. Phys. 103, 123905 (2008); doi: 10.1063/1.2940734 View online: http://dx.doi.org/10.1063/1.2940734 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v103/i12 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 13 Apr 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsSpin dynamics triggered by subterahertz magnetic ﬁeld pulses Zhao Wang,1Matthäus Pietz,1Jakob Walowski,1Arno Förster,2Mihail I. Lepsa,3and Markus Münzenberg1,a/H20850 1IV . Physikalisches Institut, Georg-August-Universität, Göttingen 37077, Germany 2Fachhochschule Aachen, Jülich 52428, Germany 3Institute für Bio- und Nanosysteme (IBN-1), Forschungszentrum Jülich GmbH, Jülich 52425, Germany /H20849Received 18 November 2007; accepted 14 April 2008; published online 18 June 2008 /H20850 Current pulses of up to 20 A and as short as 3 ps are generated by a low-temperature-grown GaAs photoconductive switch and guided through a coplanar waveguide, resulting in a 0.6 T subterahertzmagnetic ﬁeld pulse. The pulse length is directly calibrated using photocurrent autocorrelation.Magnetic excitations in Fe microstructures are studied by time-resolved Kerr spectroscopy. Anultrafast response time /H20849within less than 10 ps of the magnetization /H20850to the subterahertz electromagnetic ﬁeld pulse is shown. © 2008 American Institute of Physics . /H20851DOI: 10.1063/1.2940734 /H20852 I. INTRODUCTION There are different ways to drive magnetization dynam- ics to the limits: In the time domain, the most prominent arefemtosecond all-optical excitations 1–3and ﬁeld pulse excitations.4–7In the ﬁrst case the time scales are extremely short /H20849approximately picoseconds /H20850,8but the direction of the excitation cannot be controlled. In general, the ultrafast per-turbation by the femtosecond laser pulse generates a broadspectrum of excitations from high energy /H20849high k-vector /H20850to low energy modes of the coherent precession /H20849k=0/H20850. 9 Øersted ﬁeld pulses are generally limited to ﬁeld strength /H20849approximately a few milliteslas /H20850and temporal res- olution /H20849/H1102230 ps /H20850since they are restricted to the capabilities of high frequency electronics. The record is held by an alter- native cost-intensive approach: The generation of a magneticﬁeld pulse by relativistic electron bunches. At the StanfordLinear Accelerator Center /H20849SLAC /H20850, the magnetic ﬁeld yields up to more than 5 T in amplitude and less than a picosecondin pulse length. 10,11From the load of the ultrafast and strong ﬁeld pulses, a fracture of the magnetization is observable.Tudosa et al. 11therefore postulate a limit for the fastest switching of a recording media determined by the magneti-zation breakup and driven by the intrinsic nonlinearity of theLandau–Lifshitz–Gilbert equation. Random thermal ﬂuctua-tions, always present in the magnetic system, are ampliﬁedby the driving ﬁeld pulse. However, to make an electronicdevice spin ultrafast, a ﬁeld of about 10 T is needed in orderto switch the magnetization within a picosecond. In the following, we shall present an on-chip geometry approach, which uses optical switches, as a source of pico-second and high-power current pulses to drive the magneti-zation dynamics toward a similar value range. The transientmagnetic ﬁeld is generated by a photoconductive /H20849or Auston /H20850 switch 12and the magnetization dynamics are probed with a delay by a probe pulse via the magneto-optic Kerr effect/H20849MOKE /H20850, as shown in Fig. 1/H20849b/H20850. The method, namedmagneto-optic sampling, has been intensively developed by M. R. Freeman over the last few years /H208514,13/H20852and allows the observation of the magnetization transient directly in time.Because of the ultrashort carrier lifetime, low-temperature-grown GaAs /H20849LT-GaAs /H20850is of special interest for applications up to terahertz bandwidths and is widely used. 14Here we connect both techniques, magneto-optic sampling and tera-hertz pulse generation, to establish a SLAC on-chip. Theprocess is as follows: First the terahertz-current pulse is char-acterized by a photocurrent autocorrelation technique. Thenthe magnetic response of an Fe stripe to the subterahertz ﬁeldpulse, experimentally determined by magneto-optic sam-pling, is given. II. EXPERIMENT In the following, the preparation details and dimensions of the on-chip devices are given. The photoconductiveswitches are prepared by optical lithography on a 1 /H9262m thick LT-GaAs ﬁlm grown by molecular beam epitaxy on asemi-insulating GaAs wafer at 200 °C and annealed at a/H20850Author to whom correspondence should be addressed. Electronic mail: mmuenze@gwdg.de. FIG. 1. /H20849Color online /H20850/H20849a/H20850Scanning electron microscope image of the opti- cal switch with schematic representation of the photocurrent autocorrelationexperiment to determine the current pulse characteristics of the LT-GaAsphoto switch. /H20849b/H20850Optical microscope image of the optical switch area in- cluding the patterned magnetic Fe structures at both sides: an array of twomicron-sized structures to the left and an Fe stripe pattern to the right. Ontop, the schematic representation of the experiment to monitor the magneti-zation dynamics is given.JOURNAL OF APPLIED PHYSICS 103, 123905 /H208492008 /H20850 0021-8979/2008/103 /H2084912/H20850/123905/4/$23.00 © 2008 American Institute of Physics 103, 123905-1 Downloaded 13 Apr 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions600 °C for 10 min inside the chamber in As-rich conditions.15Characterization of the photocarrier lifetimes by time-resolved reﬂectivity measurements reveal two domi-nating relaxation times of the carriers of 70 and 140 fs, re-spectively. In the next step, by using optical lithography, a22.5 /H9262m wide center conductive strip /H208495 nm Ti/ 30 nm Al /H20850 with a gap of 3 /H9262m is evaporated onto the LT-GaAs sub- strate. Figure 1/H20849a/H20850shows a scanning electron microscope im- age of the metal-semiconductor-metal /H20849MSM /H20850gap and its dimensions. In addition the experimental geometry is givenschematically for the photocurrent autocorrelation experi-ment. Two pulses delayed by a time /H9270illuminate the 3 /H9262m MSM gap and the photocurrent is determined. The electricalpulses are generated by the femtosecond laser illumination ofthe MSM gap and then transmitted through the coplanarwaveguide /H20851Fig. 1/H20849b/H20850/H20852. When passing the coplanar wave- guide, an ultrafast magnetic ﬁeld pulse is generated with adominating in-plane component in the middle of the centerconductor. To complete the magneto-optic sampling device, the magnetic structures are patterned directly on top of the tera-hertz waveguide close to the MSM gap /H20849on-chip geometry /H20850 as seen in the optical microscope image shown in Fig. 1/H20849b/H20850: A 5 nm MgO/ 30 nm Fe ﬁlm is evaporated on the centerconductor and structured using electron-beam lithography bya lift-off process. The Fe structure similar to the one dis-cussed in Sec. III B can be seen in Fig. 1/H20849b/H20850,2 0 /H9262m /H11003100/H9262m in size, close to the MSM gap. On top of the microscopy image, a schematic drawing of the magneto-optic sampling experiment is given. While the ﬁrst pulse stillilluminates the MSM gap, the second pulse is reﬂected at themagnetic structure. Via the MOKE, the magnetization dy-namics are determined at a delay time /H9270using a double modulation technique.16Since the skin depth for an Fe ﬁlm is 3.5 nm at 1 THz frequency only, a considerable currentﬂow through the Fe ﬁlm itself has to be avoided by theinsertion of a thin insulating 5 nm MgO layer. The lasersystem used for the carrier excitation is a Ti:sapphire oscil-lator with a RegA ampliﬁer that generates 60 fs pulses/H20849/H110111 /H9262J/H20850with a central wavelength of 800 nm and a repeti- tion rate of frep=250 kHz. III. EXPERIMENTAL RESULTS A. Current pulse characteristics The advantage of the photocurrent autocorrelation tech- nique presented here is that as opposed to other techniques/H20849e.g., picosecond electro-optic 17or photoconductive sam- pling using a dual photoconductor circuit /H2085018the same sample geometry as for the magneto-optic sampling can be used tocharacterize the electric pulse length. Only a single photo-conductor is needed for the photocurrent autocorrelationmeasurement. A prerequisite is that the photocurrent in-creases nonlinearly with the rise of laser power as seen inFig. 2/H20849a/H20850: At a constant voltage, the photocurrent saturates for high ﬂuence. Because of the high defect density of theLT-GaAs ﬁlm, the MSM contact has Ohmic-likecharacteristics. 15,19It has been shown in Jacobsen et al.20that from the photocurrent autocorrelation experiments, the time-dependent carrier density can be extracted. Therefore the photocurrent autocorrelation curve can be analyzed using anexponential decay function where the time constants are re-lated to carrier relaxation times. In the following we allowtwo relaxation times /H20849 /H9270eland/H9270geom /H20850to describe the experi- mental data; then the photocurrent as a function of the delaytime /H9270between the laser pulses is given by I/H20849/H9270/H20850=I0−Iele−/H20841/H9270/H20841//H9270el−Igeome−/H20841/H9270/H20841//H9270geom, /H208491/H20850 where I0is the maximum photocurrent /H20851Fig.2/H20849b/H20850/H20852. Parameter sets Iel,/H9270eland Igeom,/H9270geomcharacterize the electrical pulse decay. It is found that the ﬁrst relaxation time of /H9270el =1–1.5 ps is related to the carrier recombination time. Theratio of the current amplitudes is about I el:Igeom/H110221.5:1. For a ﬁnger-switch geometry where the gap region is curved inorder to increase the optically active area, the second, slowerdecay /H20849 /H9270geom=5–25 ps, dependent on the alignment /H20850can be suppressed. Therefore from the geometry dependence weconclude that antenna effects of the metallization interactingwith the femtosecond-light pulse are responsible for the sec-ond, slower contribution. 21The average pulse length for the 3/H9262m gap switch geometry extracted from the photocurrent autocorrelation experiments is therefore estimated to be /H9270¯ =3/H110061 ps from the autocorrelation experiment. This value is in good agreement with the results from photoconductivesampling experiments using a second photoconductive FIG. 2. /H20849Color online /H20850/H20849a/H20850Current vs voltage characteristics of the photo- switch structure /H208493/H9262m gap /H20850under illumination varying the laser power from 1 to 17 mW, showing the nonlinearity of the photocurrent with theillumination power. /H20849b/H20850Photocurrent autocorrelation /H208493 /H9262m gap structure, 3 V gap voltage /H20850. The solid line shows the analysis using a double exponential decay of the photocurrent toward zero delay /H9270between the two laser pulses illuminating the gap.123905-2 Wang et al. J. Appl. Phys. 103, 123905 /H208492008 /H20850 Downloaded 13 Apr 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsswitch as it was determined earlier.18An estimate of the maximum current is given by Imax=Iav−Idark frep/H9270¯. It can be easily seen that a high resistance of the nonil- luminated switch is needed to suppress the dark current. Theaverage current is up to 16 /H9262A for an 80 V bias voltage and 6 mW average laser power /H20849250 kHz repetition rate /H20850and results in a current amplitude of Imax=20/H110068 A. Assuming a homogeneous current distribution throughout the coplanarwaveguide /H20849the skin depth of the center conductor materials at 1 THz is about 100 nm /H20850, the numerical calculations of the magnetic ﬁeld distribution above the center conductor resultin a homogeneous ﬁeld component /H20849B y/H20850parallel to the sur- face of the center conductor of Bmax=0.6/H110062 T. Compared with prior, standard approaches which commonly use Austonswitches or electrical pulsers for magneto-optic sampling andsynchrotron-based experiments /H20849and which are becoming in- creasingly important as a novel tool to image magnetizationdynamics /H20850, this is a signiﬁcant increase in magnetic ﬁeld strength. The out-of-plane ﬁeld component /H20849B z/H20850has a strong contribution at the edges of the conductor, with opposite sign, but it is zero at the center and will be neglected in thefollowing. B. Magnetization dynamics For the magneto-optic sampling experiments as depicted in Fig. 1/H20849b/H20850, the experimental geometry and a schematic dia- gram of the sample in the on-chip geometry are shown. Anexternal magnetic static ﬁeld of 0.03 T is applied along the20 /H9262m/H11003100/H9262m 30 nm Fe ﬁlm structure to saturate the ﬁlm along that direction. The Fe ﬁlm senses the magneticnear ﬁeld of the terahertz pulse propagating through the cen-ter conductor that is directed perpendicular to the static mag-netic ﬁeld. The evolution of the time-resolved Kerr rotation/H9004 /H9258Kerr/H20849/H9270/H20850is shown in Fig. 3./H9004/H9258Kerr/H20849/H9270/H20850is probed for 0 /H20849ref- erence /H20850, 28 and 60 V voltage applied to the gap. This voltage corresponds to about 0.20 /H110068 and 0.40 /H1100616 T. As a refer- ence a Gaussian function with 3 ps width at half maximum isshown. A steep rising edge of the differential Kerr signal/H9004 /H9258Kerr/H20849/H9270/H20850well below 10 ps is found. This response time does not depend signiﬁcantly on the ﬁeld pulse strength. Only the amplitude is about doubled as a result of doubling the ﬁeldstrength. For the reference experiment with zero voltage ap-plied across the photoconductive switch, the observed differ-ential Kerr signal /H9004 /H9258Kerr/H20849/H9270/H20850is zero and thus excludes a direct demagnetization by the laser pump pulse. Micromagnetic simulations using OOMMF /H20849Ref. 22/H20850represented by the dashed lines are overlaid on the experimental data. For themicromagnetic simulations, an Fe structure of 5 /H1100315 /H9262m2in size and 30 nm in thickness and a cell size of dcell/H1134950 nm were used. The results were tested for convergence forsmaller cell sizes. Surprisingly we ﬁnd a good agreementwith the experimental data without adjusting any parameters/H20849dashed lines in Fig. 3/H20850. In the inset of Fig. 3, the Kerr signal shown for the time scale of up to 500 ps /H20849gap voltage of 28 V/H20850reveals a critically damped oscillation. The high dampingfound in response to the terahertz pulse indicates an increase in the apparent damping. This may be interpreted as a signa-ture of the broad spectrum of spin-wave excitations leadingto a strong decay of the signal in total. Increasing the ﬁeldpulse strength can lead to increased damping, as shown inprevious cases; the activation of additional damping channelsis actually a strongly debated ﬁeld 23–26to which we hope to contribute through determining the results of driving the ﬁeldpulse strength even higher. Details of this increase have yetto be veriﬁed in further experiments. The major aim in fur-ther experiments will be to realize a full 180° switching ofthe magnetization of the Fe ﬁlm within one magnetic ul-trashort pulse. This will be possible in future photoconduc-tive switch devices approaching 10 T ﬁeld amplitude. IV. CONCLUSIONS Applying high voltages up to 80 V and an average laser power of 10 mW, the devices are driven to the limit of theirstability in the present design. Also low probe beam intensi-ties limit the sensitivity of the Kerr signal detection. How-ever, we have shown that it is possible to generate 0.6 /H110062T , 3/H110061 ps long magnetic ﬁeld pulses and to study the magne- tization dynamics excited by a subterahertz electromagneticﬁeld pulse on a chip. The response time of the magneticsignal is found to be within the order of 10 ps, as expectedfrom micromagnetic calculations. An improved switch de-sign using a ﬁnger-switch structure with a larger gap areawill stabilize the photoconductive switch and allow pulsestrengths of a few teslas /H20849similar to the SLAC experiments, but without using a linear accelerator or synchrotron /H20850in an FIG. 3. /H20849Color online /H20850Magnetic response of a 30 nm thick Fe stripe pattern on the center conductor for the short time scale for different voltages, 0/H20849reference /H20850, 28, and 60 V, applied to the photoconductive switch. Overlaid on the data, the results of the micromagnetic simulation are shown /H20849dashed line /H20850. In the inset, the signal for 28 V is shown on a larger time scale. As a reference a Gaussian function /H208493p s /H20850is plotted to indicate the ﬁeld pulse /H20849dotted line /H20850.123905-3 Wang et al. J. Appl. Phys. 103, 123905 /H208492008 /H20850 Downloaded 13 Apr 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionson-chip experiment with comparatively simple laboratory environment in the future. We expect to study similar effectsto these, driving the terahertz radiation emission in ultrafastdemagnetization experiments 27using devices with pulse-rise times below the picosecond range in the future. ACKNOWLEDGMENTS Support by the Deutsche Forschungsgemeinschaft within the priority program SPP 1133 is gratefully acknowledged. 1E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Phys. Rev. Lett. 76, 4250 /H208491996 /H20850. 2M. van Kampen, C. 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McMichael, 52nd Annual Conference on Magnetism and MagneticMaterials, Tampa, FL, 6 November 2007 /H20849unpublished /H20850. 24H. Suhl, J. Phys. Chem. Solids 1, 209 /H208491957 /H20850. 25M. L. Schneider, Th. Gerrits, A. B. Kos, and T. J. Silva J. Appl. Phys. 102, 053910 /H208492007 /H20850; Th. Gerrits, P. Krivosik, M. L. Schneider, C. E. Patton, and T. J. Silva, Phys. Rev. Lett. 98, 207602 /H208492007 /H20850. 26G. Müller, M. Münzenberg, G.-X. Miao, and A. Gupta, Phys. Rev. B 77, 020412 /H20849R/H20850/H208492008 /H20850. 27E. Beaurepaire, G. M. Turner, S. M. Harrel, M. C. Beard, J.-Y. Bigot, and C. A. Schmuttenmaer, Appl. Phys. Lett. 84, 3465 /H208492004 /H20850; S. M. Harrel, J. M. Schleicher, E. Beaurepaire, J.-Y. Bigot, and C. A. Schmuttenmaer,Proc. SPIE 5929 , 592910 /H208492005 /H20850.123905-4 Wang et al. J. Appl. Phys. 103, 123905 /H208492008 /H20850 Downloaded 13 Apr 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. 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1.4972672.pdf	Nonlinear analysis and control of an aircraft in the neighbourhood of deep stall Sébastien Kolb , Laurent Hétru , Thierry M. Faure , and Olivier Montagnier Citation: AIP Conference Proceedings 1798 , 020080 (2017); doi: 10.1063/1.4972672 View online: http://dx.doi.org/10.1063/1.4972672 View Table of Contents: http://aip.scitation.org/toc/apc/1798/1 Published by the American Institute of Physics Articles you may be interested in Quasi-periodic dynamics of a high angle of attack aircraft AIP Conference Proceedings 1798 , 020131 (2017); 10.1063/1.4972723Nonlinear Analysis and Control of an Aircraft in the Neighbourhood of Deep Stall S´ebastien Kolb1,a), Laurent H ´etru1,b), Thierry M. Faure1,c)and Olivier Montagnier1,d) 1CReA (French Air Force Research Centre) BA 701, 13661 Salon Air, France a)Corresponding author: sebastien.kolb@defense.gouv.fr b)laurent.hetru@defense.gouv.fr c)thierry.faure@defense.gouv.fr d)olivier.montagnier@defense.gouv.fr Abstract. When an aircraft is locked in a stable equilibrium at high angle-of-attack, we have to do with the so-called deep stall which is a very dangerous situation. Airplanes with T-tail are mainly concerned with this phenomenon since the wake of the main wing ﬂows over the horizontal tail and renders it ine ective but other aircrafts such as ﬁghters can also be a ected. First the phase portrait and bifurcation diagram are determined and characterized (with three equilibria in a deep stall prone conﬁguration). It allows to diagnose the conﬁgurations of aircrafts susceptible to deep stall and also to point out the di erent types of time evolutions. Several techniques are used in order to determine the basin of attraction of the stable equilibrium at high angle-of-attack. They are based on the calculation of the stable manifold of the saddle-point equilibrium at medium angle-of-attack. Then several ways are explored in order to try to recover from deep stall. They exploits static features (such as curves of pitching moment versus angle-of-attack for full pitch down and full pitch up elevators) or dynamic aspects (excitation of the eigenmodes and improvement of the aerodynamic e ciency of the tail). Finally, some properties of a deep stall prone aircraft are pointed out and some control tools are also implemented. We try also to apply this mathematical results in a concrete situation by taking into account the captors speciﬁcities or by estimating the relevant variables thanks to other available information. INTRODUCTION Deep stall occurs when an aircraft is at high angle-of-attack and moreover the horizontal tail which creates the pitch- ing moment is ine ective mainly due to the main wing wake which degrades its aerodynamics. This study deals with the global aircraft behaviour. After modelling the ﬂight dynamics, the core of the analysis focuses itself on the dy- namic features such as the characteristics of the eigenmodes, the phase portrait, recovery procedures but some more classical static aspects are also observed such as the multiple longitudinal equilibria (of the pitching moment) and the bifurcation diagram. MODELLING A classical (barycentric) model of ﬂight dynamics is taken in this study. Nevertheless the aerodynamics must take into account the e ects of deep stall. This is the case for example for the Learjet aircraft model for which data like wind tunnel tests are published in [1] and [2]. ICNP AA 2016 World Congress AIP Conf. Proc. 1798, 020080-1–020080-7; doi: 10.1063/1.4972672 Published by AIP Publishing. 978-0-7354-1464-8/$30.00020080-1As far as the pitching moment is concerned, it is divided here into two parts. A static part Cm static(; e) depending on angle-of-attack and elevator e. This part is the most important in the sense that it determines the propensity to deep stall. From the practical point of view, an engineer tries to verify that for full pitch up or full pitch down command, no deep stall appears. Another dynamic part renders mostly the damping e ect of the tail. Its dependance towards the pitch rate qis linearized. In deep stall, the aerodynamic derivative Cm qis lower in absolute value since the tail is less e cient. Its mathematical form is identiﬁed as a function of :Cm q(). All in all the aerodynamic coe cient of the pitching moment is Cm(;q;e)=Cm static(; e)+Cm q()cW 2Vq (1) with the chord cWand the speed V. The other aerodynamic features linked to stall is a lower lift (also an unsteady aerodynamics, not modeled here) and a huge drag which implies amongst other a negative ﬂight-path angle. Indeed the aircraft thrust is no more sucient to compensate the drag to maintain a level ﬂight in such a way that the aircraft ﬂies down. After modelling the ﬂight dynamics and aerodynamics, it is possible to analyze the characteristics of a deep stall prone aircraft. SHORT PERIOD MODE A ﬁrst indicator is linked to the characteristics of the short period mode which is a longitudinal (oscillatory) mode and which involves angle-of-attack and pitch rate qand of low time period. Indeed the classical linearization of the aircraft model (equations of lift and pitching moment) gives the following formula for the pulsation !spmand the damping spm. !2 spm =V2 ecWSW 2IYY"cW Ve SWVe 2mCz+T mVe! Cm q+Cm# (2) 2spm!spm =SWVe 2mCzc2 WSWVe 2IYYCm q+T mVecose (3) with the mass m,IYYthe moment of inertia about the y-axis,the air density, SWthe main wing surface and the engine thrust T. In stall, the -derivative of the lift coe cient Czis lower than in normal operational ﬂight. But in deep stall, the (normalized) q-derivative of the pitching moment coe cient Cm qis also smaller, since its main contribution comes from the horizontal tail which is under the wake of the main wing and is thus far less e ective. As an overall consequence, the damping of the short period mode is smaller at high angle-of-attack than at low angle-of-attack and is far smaller in deep stall. This is a good point to remark in order to develop a sense of danger. Besides since the damping of the phugoid mode (exchange of altidude /ﬂight-path angle and speed V) depends on the lift-to-drag ratio which is very degraded due to the huge drag, generally in stall it is also low. But this remark cannot be used as a discriminant indicator. Moreover as the short period mode is far quicker than the phugoid mode, it will often be assumed that both modes can be decoupled. Thus we will often isolate the behaviour of the short period mode and of the variables (;q). PITCHING MOMENT Amongst others the study of the (static) pitching moment allows to know the longitudinal equilibria since it corresponds to angles-of-attack and elevator position efor which the pitching moment is zero Cm static(; e)=0 020080-2(for the static study, the pitch rate q=0). Moreover when the curve of the pitching moment in function of the angle-of-attack decreases (derivative@Cm @<0), the longitudinal equilibrium is statically stable and when the curve increases (derivative@Cm @>0), the longitudinal equilibrium is statically unstable [3]. FIGURE 1. Pitching moment coe cient The ﬁgure 1 is typical from a deep stall prone aircraft in the sense that apart from the classical stable equi- librium at low angle-of-attack and the a ne (decreasing) pitching moment Cm, there are two more equilibria. The equilibrium at medium angle-of-attack is unstable and the one at high angle-of-attack is stable. This last one is linked to the deep stall since it is a stable equilibrium at high angle-of-attack and thus it is a very dangerous situation. After dealing with the modelling and the static characteristics of deep stall, we will next consider the dynamic aspects. The nonlinear analysis focuses on the classical diagrams of phase portrait (time simulations), bifurcation diagram (curve of equilibria) and then tries to exploit them so as to conclude about the inﬂuence of some parameters. TYPICAL PHASE PORTRAIT Performing time simulations of the aircraft ﬂight dynamics is the most direct way to study its behaviour since it shows the equilibria, the type and the duration of the involved movements. In the ﬁgure 2, the phase portrait shows the three mentioned equilibria. The aircraft can converge to the stable equilibria at low or high angle-of-attack and is repelled from the unstable equilibrium at medium angle-of-attack. This last equilibrium is a so-called saddle point since the Jacobian matrix has two real eigenvalues, one positive and one negative. FIGURE 2. Phase portrait 020080-3Furthermore it can be noted that the stable manifold of the saddle point is a frontier for the basin of attraction of the equilibrium at high angle-of-attack [4]. This statement comes from the theorem that for su cient regular planar systems, the trajectories cannot cut themselves [5]. BIFURCATION DIAGRAM The classical bifurcation diagram of the aircraft can also be drawn with a matlab toolbox like matcont [6]. The ﬁgure 3 represents the angle-of-attack at equilibrium in function of the elevator position eand is quite interesting. FIGURE 3. Bifurcation diagram On the one side, there is a range of medium elevator angles for which there are three equilibria. The equilibria at lowest and highest angles-of-attack are stable. They are mostly oscillatory but can be aperiodic stable (the so-called short period mode can be destroyed before becoming unstable at the bifurcation point. Indeed the pair of complex conjugate eigenvalues becomes a pair of negative reals ﬁrst before one eigenvalue becomes ﬁnally positive). As far as the equilibrium at medium angle-of-attack is concerned, it is a saddle point with one real negative eigenvalue and another one real positive. On the other side, for high (positive and negative) elevator angles, there is one unique oscillatory stable equilibrium. The bifurcation points are saddle-nodes since they correspond to a real eigenvalue being zero at this critical parameter. Moreover near a bifurcation point, a jump can occur after a little deplacement of the elevator. This sudden event may be dangerous since the pilot does not foresee it and is then locked at high angle-of-attack. Indeed it is not so easy to recover as a phenomenon of hysteresis is visible on the bifurca- tion diagram (a larger deplacement of the elevator is required so as to recover) and leave the branch of stable equilibria. After presenting the classical diagrams of dynamical systems linked to this deep stall issue, we will next try to use these elements so as to draw conclusions and to give practical advices to the pilot when ﬂying in the neighbourhood of such a phenomenon. COMPARISON OF BASINS OF ATTRACTION The comparison of the sizes of the basins of attraction allows to get an insight about the susceptibility to deep stall. Here the e ects of the ﬂight control is assessed and especially concerning the pitch damper. A simple model is here computed with a low and high Cm q(normalized q-derivative of the pitching moment) in absolute value. With an activated pitch damper, the q-derivative of the pitching moment is higher (in absolute value), that is to say when there is some pitch rate, the elevator gives rise to a higher pitching moment. In the ﬁgure 4, it is visible that the activation of the pitch damper produces a larger basin of attraction. As an advice, the pilot does have to switch othe pitch damper when reaching a deep stall region. It is thus easier to ﬂy back towards the equilibrium at low angle-of-attack. 020080-4FIGURE 4. Basins of attraction for di erent Cm q(with or without pitch damper) For instance, a passive observation of the aircraft near deep stall was done. Next we will try to become active by ﬁnding ways to recover or by adapting the avionics so as to be able to keep on ﬂying with a good level of information in this situation. RECOVERY Several recovery procedures are evaluated. Indeed after predicting the deep stall, it is necessary to react adequately if possible. First a static recovery is performed with a classical pitch down command in order to make the airplane recover and the wing aerodynamics to be restored with higher speed. Next a dynamic method based on an oscillating elevator control is made. Once the Learjet aircraft is stabilized at high AOA, a pitch down command is applied. Depending on the moment this action is applied, the aircraft succeeds in recovering from deep stall or not. The ﬂaps are here down and the center-of-gravity is at 25% of the chord. FIGURE 5. Pitch down maneuver for the Learjet aircraft recovery One question concerns the moment for which a pitch down command is applied. It seems better to do it when the angle-of-attack decreases (with a negative pitch rate moreover) and not when the angle-of-attack increases. 020080-5The study performed in the analytical section shows that an abnormally low damping of the short period mode can alert the pilot in advance of a forecoming deep stall. This indication allows to react far in advance and thus to take an appropriate decision as long as it is still possible to do something. Besides for the F 16 aircraft, a recovery procedure is described in the NASA technical paper [7]. A pitching down moment is created with the speedbrakes and the short period mode is excited so as to create a resonance by an in-phase oscillating action of the pilot. FIGURE 6. Dynamic recovery for the F 16 aircraft In the ﬁgure 6, with an aft-centered aircraft (37 :5% of the chord), the described procedure allows well to recover from the deep stall angle-of-attack. The study is based for instance hion numerical calculations and on theoretical works of modelling and analysis. It is mainly performed with the scientiﬁc software matlab . But in order to use concretely these knowledges, some practical aspects must be taken into account. ADAPTED A VIONICS All this analytical study assumes a good knowledge of the state variables. But the measure of the high angles-of- attack may give problems since it is outside of the range of validity of the usual probes or because some disturbances (vortices) appearing in these conditions may disturb its measure. As a consequence, it is necessary to estimate it with other ways. FIGURE 7. -vane employed usually to measure the angle-of-attack. The ﬁrst mean consists in choosing probes with higher range of validity like a 5-hole probe (ﬁgure 8) instead of the classical -vane (ﬁgure 7). The di erence between the pressures allows to determine the angle-of-attack and the sideslip[8]. 020080-6CP;=P4P3 Pt;ind(P1+P2+P3+P4)(4) CP;=P2P1 Pt;ind(P1+P2+P3+P4)(5) FIGURE 8. 5-hole probes. The second mean relies on exploiting other informations so as to build estimations of the angle-of-attack or of the variables describing the trajectory. Amongst others GPS, gyroscopes, accelerometers may help estimating such variables even if the AoA probes are out of use. Algorithms such as Kalman ﬁltering or Madgwick method [9] may be implemented in order to use eciently these captors and to estimate the attitudes. For example, an abnormally huge negative ﬂight-path angle indicates a dangerous situation even if no alert rings elsewhere because the classical probes are not working exceptionally. CONCLUSION This study focuses on the deep stall phenomenon. After modelling the aircraft behavior in these conditions, the ﬂight dynamics was analyzed. The typical phase portrait and bifurcation diagram were drawn. Besides an abnormal damping of the short period mode was pointed out in this situation. At the end, some proposals are made in such a way that the pilot recovers from deep stall or that the avionics keeps on working at high angle-of-attack. REFERENCES [1] R. Stengel, 2014, available at http://www.princeton.edu/˜stengel/FDcodeB.html. [2] P. Soderman and T. Aiken, “Full scale wind tunnel tests of a small unpowered jet aircraft with a t-tail,” Technical Note TN D-6573 (NASA, 1971). [3] B. Etkin, Dynamics of Atmospheric Flight (Dover Publications Inc, 2000). [4] Z. G. Goman, M.G. and A. V . Khramtsovsky, Progress in Aerospace Sciences 33, 539–586 (1997). [5] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, 2002). [6] W. Dhooge A., Govaerts and Y . Kuznetsov, ACM TOMS 29. [7] L. Nguyen, M. Ogburn, W. Gilbert, K. Kibler, P. Brown, and P. Deal, “Simulator study of stall/post-stall characteristics of a ﬁghter airplane with relaxed longitudinal static stability,” Technical Paper 1538 (NASA, 1979). [8] T. Dudzinski and L. Krause, “Flow direction measurement with ﬁxed-position probes,” Tech. Rep. TM X- 1904 (NASA, 1969). [9] S. O. Madgwick, “An ecient orientation ﬁlter for inertial and inertial/magnetic sensor arrays,” Tech. Rep. (University of. Bristol, 2010). [10] R. Montgomery and M. Moul, Journal of Aircraft 3(1966). [11] R. Taylor and E. Ray, “Deep stall aerodynamic characteristics of t-tail aircraft,” in NASA conference on aircraft operating problems, SP-83 (1965). [12] R. Taylor and E. Ray, “A systematic study of the factor contributing to post-stall longitudinal stability of t-tail transport conﬁgurations,” in AIAA conference on aircraft design and technology meeting (1965). 020080-7
1.2450645.pdf	Microwave assisted switching in a Ni 81 Fe 19 ellipsoid H. T. Nembach, P. Martín Pimentel, S. J. Hermsdoerfer, B. Leven, B. Hillebrands, and S. O. Demokritov Citation: Applied Physics Letters 90, 062503 (2007); doi: 10.1063/1.2450645 View online: http://dx.doi.org/10.1063/1.2450645 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/90/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Complex pulsed field magnetization behavior and Walker breakdown in a NiFe thin-film J. Appl. Phys. 108, 073926 (2010); 10.1063/1.3490233 Field-dependent ultrafast dynamics and mechanism of magnetization reversal across ferrimagnetic compensation points in GdFeCo amorphous alloy films J. Appl. Phys. 108, 023902 (2010); 10.1063/1.3462429 Kerr microscopy studies of microwave assisted switching J. Appl. Phys. 102, 063913 (2007); 10.1063/1.2783997 Dependence of spatial coherence of coherent suppression of magnetization precession upon aspect ratio in Ni 81 Fe 19 microdots J. Appl. Phys. 97, 10A710 (2005); 10.1063/1.1850834 Mesofrequency switching dynamics in epitaxial CoFe and Fe thin films on GaAs(001) J. Appl. Phys. 97, 053903 (2005); 10.1063/1.1854205 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39 On: Sat, 20 Dec 2014 09:40:04Microwave assisted switching in a Ni 81Fe19ellipsoid H. T. Nembach, P . Martín Pimentel,a/H20850S. J. Hermsdoerfer, B. Leven, and B. Hillebrands Fachbereich Physik and Forschungsschwerpunkt MINAS, Technische Universität Kaiserslautern, Erwin-Schrödinger-Str. 56, 67663 Kaiserslautern, Germany S. O. Demokritov Institut für Angewandte Physik, Westfälische Wilhelms-Universität Münster, Corrensstr. 2-4,48149 Münster, Germany /H20849Received 11 September 2006; accepted 4 January 2007; published online 5 February 2007 /H20850 The authors demonstrate the stimulation of the magnetization switching process of a Ni 81Fe19 ellipsoid, which is dominated by domain nucleation and propagation, by applying a transverse microwave ﬁeld. The study of the quasistatic switching behavior under the inﬂuence of a microwaveﬁeld was performed using longitudinal magneto-optic Kerr effect magnetometry. A strong reductionof the coercive ﬁeld for microwave frequencies between 500 and 900 MHz has been observed,which can be attributed to two different mechanisms: microwave stimulated enhancement of domainnucleation and microwave stimulated growth of the reversed domain. The authors prove that heatingis not the origin of the reduction of the coercive ﬁeld. © 2007 American Institute of Physics . /H20851DOI: 10.1063/1.2450645 /H20852 Fundamental studies of the switching behavior of thin ﬁlm elements in the high-frequency regime are of basic in-terest for magnetic data processing. One important aspect inthis framework is the modiﬁcation of the switching ﬁeld. Apromising approach to realize this requirement is microwaveassisted switching, as has been demonstrated by Thirion et al. 1for single cobalt nanoparticles. A microwave ﬁeld is also expected to reduce the switching ﬁeld if the switching pro-cess is dominated by domain nucleation and growth. 2The underlying mechanism can be described as follows: the ap-plied microwave ﬁeld induces large angle oscillations of themagnetization of the element if the resonance condition isfulﬁlled. This reduces the effective energy barrier for domainnucleation, which cannot be overcome by thermal ﬂuctua-tions alone. 3–6Thus, the magnetization switching can be as- sisted by the microwave ﬁeld. We study the quasistatic switching behavior of a 160 /H1100380/H9262m2Ni81Fe19ellipsoid under the inﬂuence of a trans- versal microwave ﬁeld characterized by magneto-optic Kerreffect magnetometry in longitudinal geometry. A picoseconddiode laser /H20849/H9261=407 nm /H20850with a pulse width of 83 ps is em- ployed. The sample is placed on top of a 160 /H9262m wide mi- crowave antenna. The long axis of the ellipsoid is aligned parallel to the quasistatic magnetic ﬁeld, which is generatedby external coils. The microwave ﬁeld is oriented transver-sally, i.e., perpendicular to the quasistatic ﬁeld in the plane ofthe element. To generate the microwave ﬁeld, an IFR 2032signal generator with a frequency range of 10 kHz–5.4 GHzin combination with an Aldetec microwave ampliﬁer APL-0520P433 is used. The required synchronization of the mi-crowave signal with the laser has been realized by triggeringthe pulsed laser with a DG535 Stanford Research Systemsdelay generator. This delay generator is triggered by the ref-erence signal of the signal generator /H2084910 MHz /H20850, which sends a trigger signal with programmable delay to the pulsed laser. The ellipsoid was produced by a combination of UV-photolithography and molecular beam epitaxy on a 100 /H9262mthick glass substrate. The 10 nm thick Ni 81Fe19layer is capped by a 2 nm Al protection layer. An uniaxial anisotropywith the easy axis oriented along the long axis of the ellip-soid was induced by applying a magnetic ﬁeld during thegrowth process. To characterize the basic magnetic properties of the sample, time-domain ferromagnetic resonance experimentswere carried out. 7For a ﬁxed microwave ﬁeld chosen in the range of 0.5–2.0 GHz, the oscillation of the magnetizationwas measured for different magnetic ﬁelds. The amplitude ofthe oscillations was determined for each frequency as a func-tion of the applied ﬁeld. To determine the resonance ﬁeld forfrequencies higher than 1.1 GHz, the obtained data were ﬁt-ted using a Lorentzian proﬁle. 8For lower frequencies of the microwave ﬁeld, the magnetic state of the ellipsoid becomesunstable and therefore no reasonable resonance ﬁeld can bedetermined. The linewidth for 1.5 GHz is 12.9 Oe, whichcorresponds to a Landau Lifshitz Gilbert /H20849LLG /H20850damping parameter of 0.012. To determine the induced uniaxial aniso-tropy, the Kittel equation 9,10 /H9275=/H9253/H20881/H20849Hstat+Huni+Ms/H20849Ny−Nx/H20850/H20850 /H11003/H20881/H20849Hstat+Huni+Ms/H20849Nz−Nx/H20850/H20850 /H20849 1/H20850 has been employed. Msmarks the saturation magnetization, Hunithe uniaxial anisotropy ﬁeld, and Hstatthe static mag- netic ﬁeld. With the demagnetizing factors Nx,Ny, and Nz taking the values of 1.022 /H1100310−3, 2.410 /H1100310−3, and 4 /H9266−Nx −Ny=12.562, respectively,114/H9266Ms=10 800 G, and for the case where Hstat,Huni/H112704/H9266Msholds, Eq. /H208491/H20850is rewritten as /H9275=/H9253/H20881/H20849Hstat+Huni+Ms/H20849Ny−Nx/H20850/H20850/H208494/H9266−Nx/H20850Ms. /H208492/H20850 Thus, the induced uniaxial anisotropy can be determined from a plot of the linear ﬁt of the squared frequency as afunction of the static ﬁeld. The uniaxial anisotropy ﬁeld, ob-tained from the intersection with the xaxis, is H uni=7.7 Oe. Furthermore, we obtain /H9253=0.0198 ns−1Oe−1, as deduced from the slope yielding g=2.25. This result is higher than the standard value for Ni 81Fe19, but similarly high values for Ni81Fe19elements have been occasionally reported.12 a/H20850Electronic mail: pimentel@physik.uni-kl.deAPPLIED PHYSICS LETTERS 90, 062503 /H208492007 /H20850 0003-6951/2007/90 /H208496/H20850/062503/3/$23.00 © 2007 American Institute of Physics 90, 062503-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39 On: Sat, 20 Dec 2014 09:40:04The study of the switching behavior of the Ni 81Fe19el- lipsoid was carried out by measuring hysteresis curves byapplying a transversal microwave ﬁeld, i.e., the ﬁeld is in theplane perpendicular to the quasistatic magnetic ﬁeld /H20849range between −60 and 60 Oe /H20850. The frequency of the microwave ﬁeld was varied in the range from 500 MHz to 2.0 GHz insteps of 100 MHz and the microwave power was increasedfrom 3.2 mW to 3.2 W for each frequency. In Fig. 1hysteresis loops for a ﬁxed frequency of 500 MHz and different microwave power values are shown.To highlight the observed hysteresis loops, only the ﬁeldwindow ±20 Oe is shown. Three different cases of magneti-zation reversal are plotted exemplarily, without microwaveﬁeld /H20849/H17040/H20850, for a microwave power of 3.2 mW /H20849/L50098/H20850, and for a microwave power of 3.2 W /H20849/H17034/H20850. The curve for the higher microwave power /H20849/H17034/H20850shows a dramatic reduction of the coercivity. In the other two cases /H20849/H17040and/L50098/H20850, the curves exhibit the behavior of a standard hysteresis loop in the easyaxis conﬁguration. In these two cases, with low microwave power applied and without microwave, the hysteresis loopsare rectangular with coercive ﬁelds of±3 Oe. For the case ofhigh microwave power, not only the coercive ﬁeld but alsothe shape of the hysteresis curve has been changed drasti-cally reﬂecting a modiﬁcation of the underlying magnetiza-tion reversal mechanism, which occurs between −5 and5 Oe. This modiﬁcation in shape can be understood as fol-lows: the amplitude of the oscillating ﬁeld generated in theantenna is approximately 9.8 Oe for the maximum outputpower of 3.2 W, which is clearly larger than the coerciveﬁeld observed without /H20849/H17040/H20850and with low microwave power applied /H20849/L50098/H20850. Thus, the effective applied ﬁeld is no longer parallel to the easy axis, and the reversal process is thendominated by the combined static and microwave ﬁelds. As a consequence the magnetization component along the easyaxis of the element, which corresponds to the longitudinalgeometry, is reduced due to the increased angle of the mag-netization precession; a modiﬁcation of the shape of the hys-teresis curve occurs /H20849Fig.1/H20850. Figure 2shows a map summarizing the full data set. The coercive ﬁeld is plotted using a multicolor code as a functionof the frequency /H20849xaxis /H20850and the power /H20849yaxis /H20850of the ap- plied microwave ﬁeld. As can be seen from Fig. 2, the coer- civity is strongly reduced for microwave frequencies be-tween 500 and 900 MHz. This reduction of the coercive ﬁeld of the ellipsoid under the inﬂuence of high-power, transversally applied microwaveﬁeld can be attributed to two dominating mechanisms. Theﬁrst one is the enhancement of domain nucleation by a mi-crowave ﬁeld. 13,14To nucleate a reversed domain an energy barrier has to be overcome. The microwave-induced preces-sion of the magnetization lowers the effective height of theenergy barrier and allows the system to overcome this barrierat smaller reversed ﬁelds, i.e., supports the magnetizationreversal processes. In our experiments the resonance condi-tion for a reversed magnetic ﬁeld of −5 Oe /H20849starting ﬁeld of the magnetization reversal, see Fig. 1/H20850is fulﬁlled for a fre- quency of 670 MHz. The observed strong reduction of thecoercive ﬁeld between 500 and 900 MHz /H20849see Fig. 2/H20850is due to the fact that domains are favorably nucleated in areas withreduced internal ﬁelds, i.e., lower resonance frequency. Thesecond mechanism is characterized by an enhanced growth FIG. 1. /H20849Color online /H20850Magneto-optic kerr effect hysteresis curves of a Ni81Fe19ellipsoid under the inﬂuence of different microwave ﬁelds. Open squares correspond to a hysteresis curve measured without an applied mi-crowave ﬁeld, while closed circles and open circles correspond to a micro-wave frequency of 500 MHz and powers of 3.2 mW and 3.2 W, respec-tively. The three hysteresis curves clearly demonstrate the reduction of thecoercive ﬁeld under the inﬂuence of a microwave ﬁeld with a power of3.2 W. FIG. 2. /H20849Color online /H20850Magnitude of the coercive ﬁeld is shown for appliedmicrowave ﬁelds in the range of0.5–2 GHz and a power between3.2 mW and 3.2 W /H20849logarithmic scale /H20850. A strong reduction of the coer- cive ﬁeld can clearly be observed inthe range between 500 and 900 MHz.062503-2 Nembach et al. Appl. Phys. Lett. 90, 062503 /H208492007 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39 On: Sat, 20 Dec 2014 09:40:04of the reversed domain. This is due to the fundamental prin- ciple that physical systems favor the state with the lowestGibbs free energy.15,16The more microwave power is ab- sorbed, the higher the entropy of the system becomes and thelower the Gibbs free energy is. Therefore, a reversed domain,which is in resonance with the applied microwave ﬁeld,grows accordingly. These two effects cause the reduction ofthe coercivity and thus enable the microwave assistedswitching process. To exclude that the switching process is heat assisted due to absorption of the microwave power, 17measurements with variable duty cycles were performed. To determine how themicrowave ﬁeld affects the magnetization of the sample, mi-crowave pulses with a length of 230 ns and repetition ratesof 23.7, 185, and 714 kHz were applied. The microwavefrequency was 600 MHz. No modiﬁcation of the coerciveﬁeld was observed for the three different repetition rates. Asa result no effects of heating could be identiﬁed in the ex-periment. In conclusion, we demonstrate that the switching process of micron size magnetic thin ﬁlm elements, which is domi-nated by domain nucleation and propagation, can be assistedby applying a transversal microwave ﬁeld. High-power mi-crowave ﬁelds cause a drastic modiﬁcation of the reversalmechanism, revealing a distinct reduction of the coerciveﬁelds. Thus, this effect provides less power consuming mag-netization reversal processes, which could deﬁne a promisingimprovement for magnetic data storage and processing de-vices. The authors thank the Nano+Bio Center of the Univer- sity of Technology of Kaiserslautern for technical support,Andreas Beck for sample deposition and Patrizio Candelorofor thorough discussion. Financial support by the EuropeanCommission within the EU-RTN ULTRASWITCH /H20849HPRN- CT-2002-00318 /H20850is gratefully acknowledged. Furthermore, the work and results reported in this publication were ob-tained with the research funding from the European Commu-nity under the Sixth Framework Programme Contract No.510993: MAGLOG. The views expressed are solely those ofthe authors, and the other Contractors and/or the EuropeanCommunity cannot be held liable for any use that may bemade of the information contained herein. 1C. Thirion, W. Wernsdorfer, D. Mailly, Nat. Mater. 2,5 2 4 /H208492003 /H20850. 2A. Krasyuk, F. Wegelin, S. A. Nepijko, H. J. Elmers, G. Schoenhense, M. Bolte, and C. M. Schneider, Phys. Rev. Lett. 95, 207201 /H208492005 /H20850. 3Y. C. Chang, C. C. Chang, W. Z. Hsieh, H. M. Lee, and J. C. Wu, IEEE Trans. Magn. 41, 959 /H208492005 /H20850. 4W. Scholz, D. Suess, T. Schreﬂ, and J. Fidler, J. Appl. Phys. 91, 7047 /H208492002 /H20850. 5A. Hubert and R. Schäfer, Magnetic Domains: The Analysis of Magnetic Structures /H20849Springer, New York, 2000 /H20850, p. 462. 6W. K. Hiebert, L. Lagae, and J. De Boeck, Phys. Rev. B 68, 020402 /H20849R/H20850 /H208492003 /H20850. 7F. R. Morgenthaler, J. Appl. Phys. 31,S 9 5 /H208491960 /H20850. 8Spin Dynamics in Conﬁned Magnetic Structure II , Topics in Applied Phys- ics, edited by B. Hillebrands and K. Ounadjela /H20849Springer, Berlin, 2003 /H20850,p . 31. 9C. Kittel, Phys. Rev. 71, 270 /H208491947 /H20850. 10C. Kittel, Phys. Rev. 73, 155 /H208491948 /H20850. 11M. Hanson, C. Johansson, B. Nilsson, P. Isberg, and R. Wäppling, J. Appl. Phys. 85,2 7 9 3 /H208491999 /H20850. 12T. J. Silva, C. S. Lee, T. M. Crawford, C. T. Rogers, J. Appl. Phys. 85, 7849 /H208491999 /H20850. 13E. Schloemann, IEEE Trans. Magn. 11,1 0 5 1 /H208491975 /H20850. 14E. Schloemann, J. J. Green, and U. Milano, J. Appl. Phys. 31, s386 /H208491960 /H20850. 15P. J. Thompson, and R. Street, J. Phys. D 29, 2779 /H208491996 /H20850. 16R. C. Smith, M. J. Dapino, and S. Seelecke, J. Appl. Phys. 93,4 5 8 /H208492003 /H20850. 17A. Yamaguchi, S. Nasu, H. Tanigawa, T. Ono, K. Miyake, K. Mibu, and T. Shinjo, Appl. Phys. Lett. 86, 012511 /H208492005 /H20850.062503-3 Nembach et al. Appl. Phys. Lett. 90, 062503 /H208492007 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39 On: Sat, 20 Dec 2014 09:40:04
1.366144.pdf	Theory of microwave propagation in dielectric/magnetic film multilayer structures R. E. Camley and D. L. Mills Citation: Journal of Applied Physics 82, 3058 (1997); doi: 10.1063/1.366144 View online: http://dx.doi.org/10.1063/1.366144 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/82/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quasi magnetic isotropy and microwave performance of FeCoB multilayer laminated by uniaxial anisotropic layers J. Appl. Phys. 115, 17A310 (2014); 10.1063/1.4863257 Quantifying thickness-dependent charge mediated magnetoelectric coupling in magnetic/dielectric thin film heterostructures Appl. Phys. Lett. 103, 232906 (2013); 10.1063/1.4839276 Magnetic states of multilayer Fe ∕ Cr structures with ultrathin iron layers Low Temp. Phys. 36, 808 (2010); 10.1063/1.3493419 Magnetic and microwave properties of CoFe ∕ PtMn ∕ CoFe multilayer films J. Appl. Phys. 99, 08C901 (2006); 10.1063/1.2163843 Investigation of interlayer coupling in [ Fe/Cr ] n magnetic multilayer structures by the ferromagnetic resonance method (Review) Low Temp. Phys. 28, 581 (2002); 10.1063/1.1511701 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 16:28:56Theory of microwave propagation in dielectric/magnetic ﬁlm multilayer structures R. E. Camley Department of Physics, University of Colorado, Colorado Springs, Colorado 80933 D. L. Millsa) Department of Physics and Astronomy, University of California, Irvine, California 92697 ~Received 17 March 1997; accepted for publication 4 June 1997 ! We explore the theory of microwave propagation in dielectric ﬁlms, on which thin metallic ferromagnetic ﬁlms have been deposited. Our aim is to study coupling between the microwaveelectromagnetic ﬁelds, and spin excitations in the ferromagnetic ﬁlms. We present quantitativestudies of attenuation provided by coupling to spin excitations, for various model structuresincluding superlattices. We ﬁnd strong attenuation of the microwaves, for frequencies near theferromagnetic resonance frequency of Fe. Modest magnetic ﬁelds place this resonance above 20GHz, and allow its frequency to be tuned. We note a transmission minimum occurs near thefrequency g(H014pMs), which is in the 70 GHz range for external magnetic ﬁelds H0of a few kilograms. We explore the dependence of these phenomena on ﬁlm thicknesses, and argue that suchstructureswillmovesuitablyforhighfrequencymicrowavedevices. © 1997AmericanInstituteof Physics. @S0021-8979 ~97!08717-3 # I. INTRODUCTION During the past decade, there has been impressive progress in the growth of very high quality thin metallicﬁlms, and multilayer structures such as superlattices formedfrom such ﬁlms. Multilayers can be synthesized from diverseconstituents, and growth by either sputtering techniques ormolecular beam epitaxy ~MBE !provide samples with inter- faces of very high quality. 1 There has been particularly strong interest in structures which contain ﬁlms of metallic ferromagnets such as Fe, Co,or Ni and their alloys. Phenomena such as giant magnetore-sistance ~GMR ! 2and spin dependent tunneling make such structures suitable for various applications, such as magneticsensors, or elements in high density memory devices. Forthis reason, there has been a very high level of activity inrecent years, devoted to the synthesis and characterization ofnew multilayer structures. While the remarks above have in mind metallic ﬁlms, and multilayers formed from them, it is the case that veryhigh quality metallic ﬁlms may be grown on semiconductorsas well, by methods such as MBE. There is a good latticematch between Fe, and the ~100!surface of GaAs. Also, high quality Fe ﬁlms may be grown on ZnSe. Progress in this areahas been summarized in a review article by Prinz. 4 Such semiconductor/ferromagnetic ﬁlm combinations offer new device possibilities. The semiconductor, viewedhere as simply a dielectric ﬁlm, may support the propagationof a microwave signal, or perhaps also an optical beam. Inaddition, the magnetic ﬁlm possesses collective excitationsreferred to as spin waves. These are magnetic analogs of thesound waves in elastic media. Optical beams or microwavesmay couple to the spin waves in the magnetic ﬁlm, sincetheir electric and magnetic ﬁelds penetrate the metal ﬁlm byvirtue of its ﬁnite skin depth. One may envision possibledevice applications, made possible through use of the spin waves as a means of modifying the propagation characteris-tics of the electromagnetic wave supported by the dielectricﬁlm. For many years, garnet ﬁlms have been used as the basis for microwave and integrated optics devices. A recent ex-ample is the development of the magneto-optic Bragg cell. 5 In the garnet ﬁlms, the maximum spin wave frequencieswhich may be realized for device applications are in therange of 10 GHz, or slightly above. Large external magneticﬁelds must be applied to exceed this frequency range, andthese are difﬁcult to realize in device geometries. The use of ﬁlms such as Fe offer the possibility, at least in principle, of operating at much higher frequencies. Thereason is as follows. When spin motions are excited in aferromagnet, the spin precession frequency, and hence that ofthe spin wave or collective excitation, is the Larmor fre-quency of the spin in the externally applied magnetic ﬁeldH 0, supplemented by an internal ﬁeld generated from the ferromagnetically aligned spin array. A measure of thestrength of this internal ﬁeld is 4 pMs, withMsthe satura- tion magnetization of the ferromagnet. In the garnets, 4 pMs is roughly 2 kG, whereas in ferromagnetic Fe, this internalﬁeld is 21 kG at room temperature. In the absence of anisot-ropy, the ferromagnetic resonance frequency V FMof thin ferromagnetic ﬁlms is given by g@H0(H014pMs)#1/2, where gis the gyromagnetic ratio. Application o fa2k G ﬁeld to a Fe ﬁlm provides a resonance frequency a bit above20 GHz, while the same ﬁeld applied to a garnet ﬁlm gives aresonance frequency of roughly 8 GHz. The obvious disadvantage of utilizing Fe and other me- tallic ferromagnets in devices is the Ohmic dissipation nec-essarily introduced into the structure. For this reason,dielectric/ferromagnetic multilayers are most attractive, sincethe electromagnetic energy is stored mainly in the dielectriccomponents, where the electrical conductivity is extremely a!Electronic mail: dlmills@uci.edu 3058 J. Appl. Phys. 82(6), 15 September 1997 0021-8979/97/82(6)/3058/10/$10.00 © 1997 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 16:28:56low. There is then the question of achieving strong coupling to the spin waves. This paper is devoted to a theoreticalstudy of this question, for several model structures, and formicrowave propagation in the 20 GHz frequency range. The possibility of utilizing Fe ﬁlms deposited on GaAs ~100!as the basis for a notch ﬁlter was considered some years ago by Schlo ¨mannet al. 7The GaAs ﬁlm serves as a dielectric waveguide, and as mentioned above coupling tospins in the Fe ﬁlm is achieved through the skin effect. Mi-crowaves are absorbed as they propagate down the structure,in a frequency band centered around V FM, with width con- trolled by the ferromagnetic resonance linewidth. Schlo ¨mann and his co-workers presented both theoretical studies of thisstructure, and data in the 10 GHz frequency range onsamples. The calculations presented here are in very goodaccord with his, when we examine the structures exploredearlier. This paper is organized as follows. In Sec. II, we sum- marize our theoretical approach, with emphasis on physicalconsiderations that enter importantly. In Sec. III, we presentresults of our studies of microwave attenuation in variousmodel structures, and in Sec. IV we summarize our principalconclusions. II. ANALYSIS Two examples of the model structures explored here are illustrated in Fig. 1. In Fig. 1 ~a!, we have a system which is patterned after that used previously by Schlo ¨mann and his colleagues. We have a dielectric ﬁlm of thickness D, with metallic ferromagnetic ﬁlms of thickness ddeposited on both the top and bottom surface. The only difference between thisconﬁguration, and that explored in Ref. 7 is that these au-thors had only one magnetic ﬁlm, and not two. We shall also explore other forms of multilayer structure, such as the su-perlattice illustrated in Fig. 1 ~b!. We suppose a magnetic ﬁeld is applied in the plane of the magnetic ﬁlm, as illustrated in Fig. 1. The microwavespropagate parallel to the zdirection, along which the mag- netic ﬁeld is directed. All quantities thus exhibit the spatialvariation exp( ikz). IfVis the frequency of the disturbance, both the real and imaginary part of the wave vector kare determined from an implicit dispersion relation described be-low for the various structures of interest. If we consider a single isolated ferromagnetic ﬁlm, and examine the spin waves which propagate in this geometry,the conﬁguration is such that one realizes entities referred toin the literature as ‘‘backward volume waves.’’ These havestanding wave character in the direction perpendicular to theﬁlm surfaces, and they propagate down the ﬁlm, with groupvelocity that is antiparallel to the phase velocity. It will beapparent that our interest will center on very thin ferromag-netic ﬁlms, for which kd!1. In this limit, the backward vol- ume waves have frequency V BVW(k) described by a simple dispersion relation. If VFMis the ferromagnetic resonance frequency of the thin ﬁlm discussed in Sec. I, we have VBVW~k!25VFM222pg2H0Mskd, ~1! whereMsis the saturation magnetization and gthe gyromag- netic ratio. Recall that VFM5g@H0~H014pMs!#1/2. ~2! Whenkd!1, the modes of interest to which the microwaves couple all lie very close in frequency to VFM. We will also be interested in structures formed from thin dielectric ﬁlms, so we have kD!1 as well. The microwave mode of interest is then the lowest frequency TM mode ofthe structure. For this mode the magnetic ﬁeld His parallel to thexdirection. When kD!1, thisHﬁeld is almost spa- tially uniform throughout the dielectric ﬁlm. Since tangentialcomponents of Hare conserved across the dielectric/metal interface, the full Hﬁeld penetrates into the metal ﬁlms, to excite the spins. For the TM mode, the dominant componentof electric ﬁeld is parallel to the ydirection, as illustrated in Fig. 1 ~a!. There is a small longitudinal ( z) component of electric ﬁeld as well. The structure illustrated will supportthe TM mode just discussed, for all frequencies down to zerofrequency. There are higher order TM modes, which only exist for frequencies above a cutoff frequency the order of ( c/ Ae) 3(np/D), with ethe dielectric constant of the dielectric ﬁlm. For a GaAs ﬁlm with thickness in the 300 mm range ~e>12!, the cutoff frequency of the n51 mode is roughly 150 GHz, well above the frequency range of interest here.The structure also supports TE modes, in which the electricﬁeld is parallel to the xdirection. The TE modes all have a cutoff frequency in the range just estimated, and thus will notpropagate. We now turn to our analysis. First we explore the very simple case where the metal ﬁlms are not ferromagnetic.This will allow us to assess the inﬂuence of Ohmic dissipa-tion on microwave propagation down the structure. This is a FIG. 1. Two examples of the structures explored in the present paper. In ~a!, we have a dielectric ﬁlm of thickness D, with a ferromagnetic ﬁlm of thickness ddeposited on both the top and the bottom. We consider micro- waves launched down the structure, which propagate in the zdirection. In ~b!we have a superlattice formed from ferromagnetic ﬁlms ~shaded !, and dielectric ﬁlms. Again the microwaves propagate in the zdirection. 3059 J. Appl. Phys., Vol. 82, No. 6, 15 September 1997 R. E. Camley and D. L. Mills [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 16:28:56concern for any device which incorporates metallic overlay- ers. This discussion is straightforward, and will enable us toestablish the notation and approach. We remark that in thissection, we just obtain the implicit dispersion relations forour various models. We present results based on their solu-tion in Sec. III. A. Microwave attenuation in a dielectric waveguide cladded with metal ﬁlms 1. The structure depicted in Fig. 1(a) Here we explore the particular structure illustrated in Fig. 1 ~a!, wherein a dielectric waveguide has deposited on both its top and bottom surface a thin ﬁlm of conductingmaterial, with air outside each metal ﬁlm. If these ﬁlms aremade from a material ~such as Fe !which may oxidize, quite commonly one adds an overlayer of a noble metal such asAg or Au, whose role is to suppress oxidation. We shallconsider the inﬂuence of such overlayers in the next subsec-tion. We shall see, when the results in Sec. III are presented,that the presence of such overlayers strongly inﬂuences thenature of the Ohmic damping present in such systems. The dielectric waveguide occupies the regime 0 ,y ,D, while the lower metal ﬁlm lies in the region 2d,y ,0, and the upper metal ﬁlm D,y,D1d. It is straightforward to synthesize ﬁelds within the di- electric waveguide from Maxwell’s equations applied to theTM mode of interest. We write the electric ﬁeld Eand mag- netic ﬁeld Hin the form E5E 'HyˆcosFQSy21 2DDG 2iQ kzˆsinFQSy21 2DDGJeikze2iVt~3a! and H52eV ckxˆE'cosFQSy21 2DDGeikze2iVt. ~3b! We note that, in accord with our earlier discussion, His an even function about the midplane of the structure. Here, eis the dielectric constant of the waveguide, assumed real for thenumerical calculations reported below. Then cis the velocity of light, and Qalong with the propagation constant kare related through requiring each Cartesian component of theﬁelds to satisfy the wave equation. One has Q 25eV2 c22k2. ~4! Given the frequency V, our aim is to solve for the propa- gation constant k. We shall do this through an implicit dis- persion relation derived below. We must ﬁnd the electromagnetic ﬁelds within the metal ﬁlms, and then match them to the ﬁelds in Eqs. ~3!through the appropriate boundary conditions. With microwave fre-quencies in mind, we neglect the displacement current termin Maxwell’s equation, since its inﬂuence in metals is quitenegligible at such frequencies. The structure in Fig. 1 ~a!has reﬂection symmetry through the plane y5D/2. Thus, wemay conﬁne our attention only to one of the two metal ﬁlms, which we take to be that between y52dandy50. The most general forms for the electric and magnetic ﬁelds in themetal is then E5 FE'~1!Syˆ2k kzˆDeiky1E'~2!Syˆ1k kzˆDe2ikyGeikz2iVt ~5a! and H52xˆck2 Vk~E'~1!eiky1E'~2!e2iky!eikze2iVt. ~5b! In these expressions: k51 d0~11i!. ~6! Here d0is the classical skin depth of the metal: d05c ~2ps0V!1/2~7! with s0its conductivity. Relations between the three amplitudes E',E'(1), and E'(2)follow upon applying the electromagnetic boundary conditions at y50. Conservation of tangential components ofEprovide us with sinS1 2QDDE'5ik Q~E'~1!2E'~2!!, ~8a! while conservation of tangential Hyields cosS1 2QDDE'5c2k2 eV2~E'~1!1E'~2!!. ~8b! No new information follows from the requirement that nor- mal components of Dbe conserved. We must now consider the region below y52d, which we assume is occupied by air with dielectric constant unity.The ﬁelds in this region will be evanescent in character, forwaves conﬁned to and guided by the structure. For y,2d, we have ﬁelds which we write as E5E '~,!Syˆ1ia0 kzˆDea0~y1d!eikze2iVt~9a! and H52V ckE'~,!xˆea0~y1d!eikze2iVt~9b! where the wave equation in vacuum gives a05Sk22V2 c2D1/2 . ~10! For all the modes we consider, the imaginary part of k, namelyk2, will be very small compared to its real part. Waves are bound or guided by the structure when k1, the real part of k, is larger than V/c. We always choose the square root in Eq. ~10!so that 3060 J. Appl. Phys., Vol. 82, No. 6, 15 September 1997 R. E. Camley and D. L. Mills [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 16:28:56Re~a0!.0. ~11! Once again we require tangential EandHbe conserved, but now at the interface y52d. Conservation of tangential E gives E'~1!e2ikd2E'~2!eikd52ia0 kE'~,!~12a! while conservation of tangential Hrequires E'~1!eikd1E'~2!eikd5V2 c2k2E'~,!. ~12b! In Eqs. ~8!and~12!, we have four homogeneous equa- tions in the four ﬁeld amplitudes E',E'(1),E'(2), and E'(,). Once the frequency Vis chosen, these equations ad- mit nonzero solutions only for one, or perhaps a discrete setof propagation constants k. It is a straightforward matter to derive an implicit equation from which the allowed propaga-tion constants may be obtained. This has the form cot S1 2QDD5ikQ ek02~12zei2kD! ~11zei2kd!, ~13! where z5a0k1ik02 a0k2ik02~14a! and we have deﬁned k05V c. ~14b! In Sec. III, we shall discuss numerical solutions of Eq. ~13!for structures of interest, and we shall also obtain ap- proximate analytic solutions applicable to particular regimes. 2.Theinﬂuenceofmetalliccapsonthestructurein Fig. 1(a) As noted above, the expression in Eq. ~13!provides us with the implicit dispersion relation of microwaves whichpropagate down the structure illustrated in Fig. 1 ~a!, where it is assumed that we have air outside the two metallic ﬁlms. Inpractice, most particularly if the thin ﬁlms are a metal suchas Fe, a noble metal overlayer will be deposited over the Feﬁlms to prevent oxidation. In what follows, we assume such overlayers are present, and furthermore that they are sufﬁciently thick that they mayeach be supposed to be of inﬁnite thickness. Let the conduc-tivity of the overlayer material be s1, and its skin depth be d1. It is straightforward to modify the discussion given in the previous subsection to describe this case. When this isdone, the implicit dispersion relation has precisely the formgiven in Eq. ~13!, except the quantity zis replaced by z 1, where z15Sk2k1 k1k1D ~15! andk151 d1~11i!. ~16! Again we present calculations which address this structure in Sec. III. B. Microwave attenuation in a dielectric waveguide cladded with ferromagnetic metal ﬁlms 1. The structure depicted in Fig. 1(a) We now turn to the case where the metal ﬁlms in Fig. 1~a!are not only metallic, as in the discussion above, but ferromagnetic as well. Here we discuss the situation illus-trated in Fig. 1 ~a!where the ferromagnetic ﬁlms are un- capped, with air above. In the next subsection, we discuss theextension to the case where each ferromagnetic ﬁlm is cov-ered by a thick metallic ﬁlm. The electromagnetic ﬁelds within the dielectric wave- guide, and those in the air outside the ferromagnetic ﬁlm aredescribed as in Sec. ~II A!. Thus, we do not display their form here, but we will use the expressions given in the pre-vious section. We do discuss the inﬂuence of the ferromag-netism on the ﬁelds within the two ferromagnetic ﬁlms. Aswe proceed, we will invoke an approximation described be-low which we believe to be quite accurate, for the systemsstudied here. The two principal Maxwell equations we explore are ¹3E5ik 0B ~17a! and ¹3H54ps0 cE ~17b! where once again we ignore the displacement current contri- bution to Eq. ~17b!, since in the frequency regime of interest, its inﬂuence is quite negligible. Recall that k05V/c, from Eq.~14b!. The conditions ¹E5¹B50 are appended to Eqs.~17!. To proceed, we require a constitutive relation between B andH. This takes the form, for a ferromagnet oriented such as that in Fig. 1: Bx5m1Hx1im2Hy, ~18a! By52im2Hx1m1Hy, ~18b! and Bz5Hz. ~18c! Expressions for the frequency dependent magnetic response functions are derived standardly from the Landau–Lifshitzequations. 8Letgbe the gyromagnetic ratio, and deﬁne VM 5gMSandVH5gH0. One then ﬁnds8 m15114pVM~VH2iGV! ~VH2iGV!22V2~19a! and m254pVMV ~VH2iGV!22V2. ~19b! 3061 J. Appl. Phys., Vol. 82, No. 6, 15 September 1997 R. E. Camley and D. L. Mills [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 16:28:56The dimensionless parameter Gin Eqs. ~19!has its origin in dissipation in the spin system.9Its value controls the ferro- magnetic resonance linewidth DH, deﬁned by Heinrich and Cochran as8 DH51.16SV gDG. ~20! We proceed by combining Eqs. ~18!with~17!, and seek- ing eigensolutions with the form exp( 6iky)exp(ikz)exp (2iVt). There are two eigensolutions, neither of which has the character of a pure TM mode, or a pure TE mode. Thegyrotropic character of the response of the ferromagnet, withorigin in m2, produces normal modes in which all the Car- tesian components of the ﬁeld are nonzero. It follows fromthis that the electromagnetic ﬁelds within the dielectricwaveguide are also no longer of pure TE or TM character,but are mixtures of the TE and TM mode. If, in the metal ﬁlm, we take the mathematical ~but un- physical !limit m2!0, one of the two modes reduces to a TM mode, and one reduces to a TE mode. For ease of dis-cussion, when m2Þ0, we refer to one of the exact modes as the TM mode, and the second the TE mode, labeling each bytheir behavior in the limit m2!0. This nomenclature is ap- propriate, for reasons we shall appreciate below. Consider the electric and magnetic ﬁelds associated with the TM mode of the ferromagnetic ﬁlm. These have the form E~6!5E'~6!Sib ~mv21!xˆ1yˆ7k˜ kzˆDe6ik˜yeikze2Vt~21a! B~6!5E'~6!F2Sk˜2 kk0Dxˆ2i 3k k0Sb mv21Dyˆ6ik˜ k0Sb mv21DzˆGe6ik˜yeikze2iVt ~21b! and H~6!5E'~6!F2Sk˜2 kk0mvD~xˆ1ibyˆ!1k k01 mvSib mv21D 3S2ibxˆ1yˆ7k˜ kmvzˆDGe6ik˜yeikzeiVt. ~21c! In these expressions: mv5m122m22 m1~22a! is often referred to as the Voigt permeability: b5m2 m1~22b! and k˜5~mv!1/2 d0~11i!. ~22c! Some general comments on the structure of these expres- sions are in order. First note, as remarked above, that when m2and consequently bare nonzero, as mentioned earlier, theﬁelds do not have pure TM character. As a consequence, the mode as a whole is no longer a TM mode, by virtue of thegyrotropic response of the ferromagnet. We argue below,however, that under circumstances of interest to us, devia-tions from pure TM character are very small. The microwave skin depth is affected strongly by the magnetic response of the ﬁlm, as one sees from Eq. ~22c!. The effective skin depth is deff5d0 ~mv!1/2. ~23! If, in the interest of simplicity, we set the damping constant Gto zero, then mv5VB22V2 VFM22V2~24! where VFMis the ferromagnetic resonance frequency of the ﬁlm discussed in Sec. I @VFM25VH(VH14pVM)#, and VB5VH14pVM. AsVapproaches the ferromagnetic resonance fre- quency, mvincreases dramatically, and the skin depth de- creases by a large amount. This will have important conse-quences for the calculations presented in Sec. III. This is anunfortunate situation, because the reduced skin depth ‘‘cutsoff’’ coupling between the microwave ﬁeld and the spins,precisely when it is most desired, on resonance. Notice that mvhas a zero, and thus the metallic ﬁlms ‘‘open up’’ near the frequency VB, which for Fe is in the 70 GHz range, w h e na2k g ﬁeld is present. We shall explore consequences of this as well. We next consider the order of magnitude of the various parameters that enter Eqs. ~21!, with the 20 GHz frequency range in mind. We have k05V/c>4c m21. One expects k >k0Ae;8–10cm21, for a typical semiconducting wave- guide. The parameter Qwill be in the same range, since Q21k25k02efor these propagating modes. However, kandk˜are very much larger indeed than the three parameters just described. For Fe at 20 GHz, the skindepth d0off resonance is very close to 1024cm, or 1 mm. Hence, k'104cm21,@k0,Q,o rk. The argument given above suggests that near ferromagnetic resonance, k˜is in fact much larger than k. If one examines Eq. ~21a!with the above numerical es- timates in mind, one sees that the zˆcomponent of the electric ﬁeld~the component parallel to the biasing ﬁeld H0!is larger than thexˆandyˆcomponents by roughly three or four orders of magnitude. For the magnetic ﬁeld H, whose tangential components are conserved across the interface, the xˆcompo- nent~parallel to the interface !and theyˆcomponent ~normal to the interface !are larger than the zˆcomponent by three or four orders of magnitude. It is the presence of the xˆcomponent of E, and the zˆ component of Hwhich are responsible for ‘‘mixing in’’ ﬁelds of the TE character in the dielectric waveguide. Wehave just seen that these two components are three to fourorders of magnitude smaller than the dominant componentsofEandH, which can be matched appropriately to ﬁelds of TM character in the dielectric. 3062 J. Appl. Phys., Vol. 82, No. 6, 15 September 1997 R. E. Camley and D. L. Mills [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 16:28:56With these remarks in mind, we shall proceed by ap- proximating the ﬁelds in the dielectric ﬁlm by ﬁelds of pureTM character, as given in Eq. ~3!. We match them to ﬁelds in the metal ﬁlm, which are linear combinations of the ﬁeldsE (6), andH(6)given in Eqs. ~21a!and~21c!. When we match the ﬁelds, we require only that tangential componentsofEandHbe conserved across the boundary, and ignore the very small quantitative errors introduced by requiring conti-nuity of the other small components. Once this approximation is accepted, the implicit disper- sion relation may be derived by a discussion that follows thatgiven in the previous section. We thus simply quote theresult: cot S1 2QDD5ik˜Q ek02mvS12z˜ei2k˜d 11z˜ei2k˜dD ~25! where z˜5a0k˜1ik02mv a0k˜2ik02mv. ~26! 2.Theinﬂuenceofmetalliccapsonthestructurein Fig. 1(a) We handle this with the approximation described in the previous subsection, in regard to the ﬁelds within the ferro-magnetic ﬁlms, presently capped by a thick conducting ﬁlm.The thick conducting ﬁlms are treated as in Sec. II A 1. Thederivation is straightforward, and the effective dispersion re-lation has form identical to Eq. ~25!, with the factor z ˜re- placed by z˜5Sk˜2mvk1 k˜1mvk1D, ~27! with k1deﬁned by Eq. ~16!. C. Microwave propagation in the superlattice structure depicted in Fig. 1(b) As one sees from the ﬁgure, one has a superlattice struc- ture whose basic unit cell consists of a dielectric ﬁlm ofthickness D, and a ferromagnetic ﬁlm of thickness d. The unit cells are stacked together as indicated in the ﬁgure. Weshall assume here that we have an inﬁnite number of unitcells, so the structure ﬁlls the entire space from y52`to y51`. It should be remarked that our interest will be in the case where the dielectric ﬁlm thickness Dis rather small. Thus, a practical sample will consist of many unit cells. We shall treat the ﬁelds within the scheme described in Sec. II B, where we regard the mode as very well approxi-mated by one TM character. We match tangential compo-nents of EandHin the dielectric ﬁlm to the very large tangential components of EandHin the ferromagnetic ﬁlm. For the superlattice which consists of the inﬁnite stack of unit cells, from the perspective of any individual dielectricﬁlm, the structure has reﬂection symmetry through the mid-plane of the ﬁlm. Thus, within each dielectric ﬁlm, the elec-tromagnetic ﬁeld may be taken to have a form identical tothat described by Eqs. ~3!. These apply to the particular ﬁlmlocated between y50 andy5D, and through the appropri- ate translation describe the remaining dielectric ﬁlms. If wesit in one of the ferromagnetic ﬁlms, the structure also hasreﬂection symmetry through the midpoint of the ferromag-netic ﬁlm. A consequence is that the Cartesian componentsof the electric and magnetic ﬁeld have well-deﬁned parity inthese ﬁlms. Within the ferromagnetic ﬁlm centered between y50 andy52d, for the tangential components of EandH we have E t52ik˜ kE~M!zˆsinFk˜Sy11 2dDG ~28a! and Ht52k˜2 kk0mvE~M!xˆcosFk˜Sy11 2dDG. ~28b! Since these ﬁeld forms are repeated throughout the structure unchanged in shape, we obtain the implicit dispersion rela-tion by tangential components of EandHto be conserved across the interface at y50. A short calculation gives us cotS1 2QDD52k˜Q ek02mvcotS1 2k˜dD. ~29! In the next section, we present both analytical results in special limits, and numerical results for the various structuresconsidered in this section. III. RESULTS AND DISCUSSION Next we turn to a discussion of the results of a series of calculations based on the implicit dispersion relations ob-tained in Sec. II. It should be noted that all calculations areperformed for a frequency of 20 GHz. We have in minddielectric waveguide thicknesses in the range of a few tens toa few hundred microns. The skin depth of Fe is very close toone micron at 20 GHz, so Fe ﬁlm thicknesses will be at mosta few microns. As we shall see, in fact, a few hundred ang-stroms of Fe will allow one to achieve optimum coupling. The ﬁrst question we address is the inﬂuence of the metal ﬁlms on the microwave propagation length, in the ab-sence of magnetism. That is, we inquire the extent to whichthe presence of the metal ﬁlms leads to attenuation over andabove that provided by losses in the dielectric ﬁlm itself. If the two metal ﬁlms are very thick compared to the skin depth, then one may derive a very simple analytic for-mula for the attenuation length. One uses Eq. ~13!, and takes the limitd!`, where exp( i2 kd)!0. We recall from Sec. II that under the conditions of interest ( k/k0);104. The right- hand side of Eq. ~13!is then very large compared to unity, and we are in the limit QD!1. Thus, cot(1/2 QD) is well approximated by 2/ QD, and thus Q2>2(2iek02/kD), then recall that k5~ek022Q2!1/2>e1/2k0S11i kDD. ~30! We write k5k11ik2, and note that k052p/l0, where l0is the free space wavelength of the radiation ﬁeld, at the wave-length of interest. We then have a very simple result: 3063 J. Appl. Phys., Vol. 82, No. 6, 15 September 1997 R. E. Camley and D. L. Mills [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 16:28:56k25pe1/2 l0d0 D,d@d0. ~31! From Eq. ~13!, we may calculate k2, for the case where the metal ﬁlms have ﬁnite thickness. We have done this nu-merically. For the structure illustrated in Fig. 1 ~a!, where we have a dielectric waveguide with a metal ﬁlm deposited ontop and on bottom, the results are surprising. As the thick-ness of the metal ﬁlms is decreased, the Ohmic dissipationincreases rather than decreases, as one might expect intu-itively. Of course, as d!0, as one sees from Eq. ~13!,Qand consequently kare purely real, as they must be for the loss- less dielectric assumed here. But, as just remarked, initiallythe attenuation rises, as ddecreases. We illustrate this in the curve labeled A ~see Fig. 3 !. The calculations assume the dielectric waveguide has a thicknessof 100 mm, with a ~real!dielectric constant of 12, a value typical for the common semiconductors. The metal ﬁlmshave the conductivity of Fe at room temperature. From curve A in Fig. 2, we see the conductivity damping depends weakly on ﬁlm thickness when d.2 mm, as ex- pected from the value of the skin depth mentioned above. Asthe ﬁlm thicknesses drop below 1 mm or so, quite surpris- ingly once again, we see a dramatic increase. If one exam-ines the electric ﬁeld within the metal ﬁlm, the dominantcomponent of the electric ﬁeld is the longitudinal component~zcomponent !. The combined effect of the boundary condi- tions at the air/metal interface, and metal/dielectric interfacesis to cause the strength of the zcomponent of electric ﬁeld to increase as 1/ dwhen kd!1. In the end, k2increases as 1/das well, as a consequence of this ﬁeld enhancement. One may extract the behavior just described from Eq. ~13!, and derive as well an estimate of the ﬁlm thickness below which k2will eventually fall to zero as d!0. When kd!1, we make the replacement exp(2 ikd)>112ikdto ﬁndcotS1 2QDD5kQ ek02@k021~a0k1ik02!kd# @a0k1i~a0k1ik02!kd#. ~32! In both the numerator and the denominator of this expres- sion, ( a0k1ik02) may be replaced by a0k, and in the de- nominator, the term in kdmay be dropped altogether. When this is done, Eq. ~32!may be written cotS1 2QDD5Q ea0S112ia0d k02d02D. ~33! The metal overlayers continue to assert their presence so long as 2 a0d/k02d02is large compared to unity. Recall that k052p/l0, with l0the free space wavelength of the radia- tion. The wave vector kis always close in value to e1/2k0,a s one sees from the example in Eq. ~30!. Hence the metal ﬁlms control the behavior of the structure so long as d.p ~e21!1/2d02 l0[dc. ~34! For Fe, as noted, d0>1024cm, and at 20 GHz, d0/l0 ;1024. Hence, the conducting overlayer has a very strong inﬂuence on the propagation characteristics until the cover-age is down to the atomic monolayer level! We are remindedof a study of ultrathin Ag ﬁlms on GaAs some years ago,which demonstrated that monolayer quantities of Ag com-pletely screened electric ﬁelds generated by atomic motionsin the GaAs from the outside world. 10 In the regime dc!d!d0, we may ignore the factor of unity on the right-hand side of Eq. ~33!, and we still have QD!1. Upon proceeding as in the derivation of Eq. ~31!,w e have k2>pe1/2 l0d02 Dd~dc!d!d0!. ~35! As discussed above, the attenuation rate increases inversely with the metal ﬁlm thickness. This expression provides agood account of curve A in Fig. 2, when d! d0. If, as is commonly done to prevent oxidation, the Fe ﬁlms are covered with a noble metal ﬁlm, the behavior of k2 differs qualitatively from the case just described. Curve B in Fig. 2 are calculations for a dielectric ﬁlm 100 mm in thick- ness, where now the Fe ﬁlms of thickness dare covered with very thick Ag ﬁlms. We now see that as the Fe ﬁlmsare made progressively thinner, the attenuation decreasessubstantially. These calculations show that capping the Fe ﬁlms with thick metallic overlayers plays a most important role in lim-iting the conductivity damping, as the Fe ﬁlms are madethinner. For reasons discussed below, Fe ﬁlms in the 300–500 Å range will be proven to be of primary interest. In theabsence of capping, the conductivity damping would be se-vere, to the point where the metal coated dielectric wave-guide would be of limited usefulness. We now turn our attention to the coupling between the microwave ﬁelds, and spin motions in ferromagnetic ﬁlmslocated on the top and bottom of the dielectric waveguide.We shall conﬁne our attention to the case where thin ferro- FIG. 2. The microwave attenuation at 20 GHz, as a function of Fe ﬁlm thickness, for two cases. Case ~A!is a lossless dielectric ﬁlm ~e512!with a thickness of 100 mm, and air outside the Fe ﬁlms on the top and bottom of the dielectric. Case ~B!is the same dielectric ﬁlm, but now deposited on each Fe ﬁlm are overlayers of Ag, whose thickness is large compared to theskin depth in Ag. 3064 J. Appl. Phys., Vol. 82, No. 6, 15 September 1997 R. E. Camley and D. L. Mills [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 16:28:56magnetic ﬁlms of thickness dare placed on the waveguide, and these are each capped by very thick metallic overlayerswhich we take to be silver. In Fig. 3, we show the frequency dependence of the attenuation introduced by coupling between the microwaves,and the ferromagnetic ﬁlms. These calculations explore vari-ous Fe ﬁlm thicknesses, for a dielectric waveguide whosethickness is 100 mm. An external ﬁeld of 1.85 kG renders the ferromagnetic resonance frequency to be 20 GHz. One might believe that to achieve maximal coupling, the Fe ﬁlms must have a thickness of a few microns, since theskin depth d0is the order of 1 mm. We have seen, however, that near resonance, the effective skin depth is very muchsmaller. Right on resonance, for the parameters we have used in these calculations, 9Amv>35, so in fact the skin depth is reduced to only about 300 Å. Thus, maximal coupling isachieved even with very thin Fe ﬁlms. We illustrate this in Fig. 2, where we display the attenu- ation introduced by coupling to the ferromagnetic resonanceresponse of the spin system. We see almost no differencebetween the peak attenuation produced by a ﬁlm 0.1 mm ~1000 Å !in thickness, and that produced by a ﬁlm 0.05 mm ~500 Å !in thickness. It is not until the Fe ﬁlm thickness drops well below 300 Å that one begins to see a falloff in thepeak attenuation. This is illustrated by the curve labeled0.0125 mm~125 Å !in Fig. 3. We remark that conclusions very similar to these are evident in the calculations displayedin the paper by Schlo ¨mann and co-workers. 7Indeed, our cal- culations are in very good accord with theirs in all regards, ifone realizes we have two ferromagnetic ﬁlms deposited onthe dielectric waveguide, while they have only a single ﬁlm.Our calculated peak attenuation rates are thus quite close, as expected, to twice theirs. One virtue of the strong dependence of the skin depth on frequency is that it reduces the sensitivity of the peak attenu-ation to the linewidth of the ferromagnetic resonance line ofthe ferromagnetic ﬁlm. In our treatment, this is controlled bythe parameter Gwhich enters Eqs. ~19!. In general, on reso- nance, the absorption rate is inversely proportional to G. Here, the peak scales as G 21/2, so the peak absorption is somewhat less sensitive to linewidth than one might expect.If one increases the linewidth of the ferromagnet, the ampli-tude of the spin response on resonance is of course, reduced.However, the skin depth is larger at resonance; this allowsthe microwave ﬁeld to sample more spins than before topartially compensate for the loss in amplitude of the spinresponse. The peak attenuation realized in the geometry employed in Fig. 3 is affected sensitively by the thickness of the di-electric waveguide. As D, the thickness of the waveguide decreases, the peak attenuation increases dramatically as il-lustrated in Fig. 4. Note, that if the dielectric waveguidethickness is decreased from 100 to 50 mm, then the calcu- lated peak attenuation rate increases to a value in excess of75 dB/cm. If strong coupling between the microwave ﬁeldsand spins in the ferromagnetic ﬁlms is highly desirable, quiteclearly one should fabricate samples from the thinnest pos-sible dielectric waveguide. The results presented suggest that to obtain strong cou- pling, one wishes to make the dielectric waveguide very thin,as just discussed. However, in practice, of course, there is alower limit to the thickness that may be utilized. In the par-ticular example of a GaAs based structure explored here, it isour understanding that a structure based on the use of a 50 mm thick GaAs ﬁlm would be quite fragile. These remarks suggest one should utilize a superlattice structure such as that illustrated in Fig. 1 ~b!. The notion is FIG. 3. The frequency dependent attenuation, for a dielectric waveguide ~e512!of thickness D5100mm, upon which two Fe ﬁlms of thickness d have been deposited. Each Fe ﬁlm is assumed capped by a thick layer of Ag, as illustrated in the inset. We assume an external ﬁeld H051.85 kG is present, which gives a ferromagnetic resonance frequency very near 20 GHzin the Fe ﬁlms. FIG. 4. For the structure studied in Fig. 3, we plot the peak attenuation as a function of the thickness Dof the dielectric waveguide. We have taken the thickness of the Fe ﬁlm to be 500 Å. 3065 J. Appl. Phys., Vol. 82, No. 6, 15 September 1997 R. E. Camley and D. L. Mills [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 16:28:56that one can create a macroscopic sample by stacking to- gether many unit cells, as depicted in Fig. 1 ~b!. In Fig. 5, we show calculations for such a superlattice, in which the dielectric ﬁlms have a thickness of 1 mm, and the intervening Fe ﬁlms have a thickness of only 100 Å. Theattenuation at the peak is now quite enormous, in excess of300 dB/cm. Thus, by fabricating such a structure, one canrealize very strong coupling between microwaves and spinexcitations. Off resonance, in the presence of the metal ﬁlms,the attenuation remains modest. In the 15 GHz range, forinstance, one realizes 0.3 dB/cm, with a similar value at 25GHz. In Sec. II, it was noted that the Voigt susceptibility mv has a zero in the near vicinity of the frequency VB5VH 14pVM, which is near 70 GHz, for Fe exposed to an ex- ternal magnetic ﬁeld in the 2 kG range. In this frequencyregime, the skin depth in the ferromagnetic ﬁlm opens up,and becomes very large, as one sees from Eq. ~23!. In ferro- magnetic resonance studies of thin ﬁlms, there is a transitionresonance, discovered some years ago by Heinrich andMescharyakov. 11This feature is referred to as an antireso- nance of the ﬁlm. If one prepares a superlattice such as that displayed in Fig. 1 ~b!that is metal rich, then at frequencies removed from the antiresonance, the conductivity damping is very strong.However, near V B, in the antiresonance region, the structure opens up and transmits. We illustrate this in Fig. 6, where weshow the frequency dependence of the transmissivity of astructure fabricated from dielectric ﬁlms 1 mm thick, with Fe ﬁlms 3 mm thick interspersed between them. We see the dramatic attenuation minimum near 70 GHz. At the mini-mum, the attenuation falls to 4 dB/cm; the depth of the mini-mum is controlled by the damping parameter G. This struc- ture may be appropriate for use as a tunable band pass ﬁlter.Since V B5VH14pVMthe frequency of the dip may be tuned by varying the externally applied magnetic ﬁeld. In the limit that both constituents in the superlattice are very thin, a simple analytic expression for the propagationconstant kfollows from Eq. ~29!.I fQD!1, and also k˜d !1, each cotangent may be replaced by its small argument limit. This yieldsQ252d Dek02mv, ~36! from which one ﬁnds the simple result k25ek0S11d DmvD. ~37! An expression equivalent to this emerges from the effective medium theory of magnetic superlattices.12 IV. SUMMARY AND CONCLUSIONS The calculations presented in Sec. III have elucidated a number of features of the microwave propagation character-istics of structures such as those depicted in Fig. 1. Our prin-cipal conclusions may be summarized as follows: ~a!Capping of the Fe ﬁlms by a nonmagnetic conductor does more than simply prevent oxidation of the Fe ﬁlm. Itcontrols the dependence of the off-resonance conductivitydamping on Fe ﬁlm thickness, when the Fe ﬁlms becomeconsiderably thinner than the off-resonant skin depth. With-out the capping, as the Fe ﬁlms become very thin, one real-izes a very strong conductivity damping off resonance, asillustrated by curve A in Fig. 2. If the ﬁlms are capped by athick conducting layer, then the off-resonance conductivitydamping remains quite small, for all Fe ﬁlm thicknesses con-sidered, for the structures explored here. ~b!One would think that to achieve maximal coupling of microwaves to spins, one needs Fe ﬁlms a few micronsthick. That is, they should be thicker than the nominal skindepth, so the microwave ﬁeld comes into contact with asmany spins as possible. This is not the case. The fact that theapparent skin depth decreases dramatically on resonance al-lows one to achieve maximal coupling with rather thin ~500 Å!Fe ﬁlms, as illustrated in Fig. 3. FIG. 5. The attenuation as a function of frequency near the ferromagnetic resonance frequency, for a superlattice structure such as that depicted in Fig.1~b!. The thickness of the dielectric ﬁlm ~ e512!is 1mm, and that of the Fe ﬁlm is 100 Å. FIG. 6. The frequency variation of the propagation length near the Fe ﬁlm antiresonance, at the frequency VH14pVM. The dielectric ﬁlms ~e512! have a thickness of 1 mm, and the Fe ﬁlms a thickness of 3 mm. 3066 J. Appl. Phys., Vol. 82, No. 6, 15 September 1997 R. E. Camley and D. L. Mills [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 16:28:56~c!One can increase absorption on resonance by using the thinnest possible dielectric ﬁlm. The range of 50 mm seems interesting, if such a thin ﬁlm structure can be fabri-cated. ~d!The use of multilayer or superlattice structures seems of great interest, if one seeks strong coupling betweenmicrowaves and the spins in the structure, for the followingreasons: ~i!One can achieve very large attenuation on resonance, as illustrated in Fig. 5, by making such a structure with ratherthin~order of 1 mm!dielectric ﬁlms. ~ii!There is a dramatic attenuation dip at high frequen- cies, due to the ‘‘opening up’’ of the skin depth at antireso-nance. This is illustrated in Fig. 6. The development ofmetal-rich structures will present very large attenuation awayfrom the antiresonance region, with attenuation dips here asillustrated. ~iii!Both effects ~i!and~ii!just described can be achieved in samples made with rather low quality Fe ﬁlms.For~i!, the attenuation maxima are very high, so relatively low quality ﬁlms with only modest linewidths can providestrong coupling to spins, if such ﬁlms are incorporated into asuperlattice. For ~ii!, the overall shape of the attenuation dip is controlled by the real part of the Voigt susceptibility,which is not so sensitive to linewidth, save quite near thezero in the real part which drives the phenomenon. Theseconsiderations suggest sputtered samples should prove quiteadequate, for the superlattice structures. It is our hope that the calculations presented here pro- vide an orientation of the inﬂuence of sample geometry andmicrostructure, for combinations of semiconductor and fer- romagnetic metal ﬁlms which may prove useful for high fre-quency microwave devices. ACKNOWLEDGMENTS One of us ~D.L.M. !appreciates numerous conversations with Professor C. S. Tsai, and with Professor H. Hopster.This research was supported by the Army Research Ofﬁce,under Grant No. CS0013132. 1For a discussion of the growth of high quality ultrathin ﬁlms and the means of characterizing them see: Ultrathin Magnetic Structures , edited by J. A. C. Bland and B. Heinrich ~Springer, Heidelberg, 1994 !, Vol. 1, Chap. 5, p. 177. 2M. N. Babich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P.Eitenne, G. Creuzet, A. Frederick, and J. Chazelas, Phys. Rev. Lett. 61, 2472 ~1988!. 3For a discussion of the application of spin dependent tunneling to mag- netic memory devices see Z. Wang and Y. Nakamura, J. Magn. Magn.Mater.159, 233 ~1996!. 4G. A. Prinz, Ultrathin Magnetic Structures , edited by J. A. C. Bland and B. Heinrich ~Springer, Heidelberg, 1994 !, Vol. II, p. 1. 5C. S. Tsai, IEEE Trans. Magn. 32, 4118 ~1996!. 6C. Kittel, Phys. Rev. 71, 270 ~1947!;ibid.73, 155 ~1948!. 7E. Schlo¨mann, R. Tutison, J. Weissman, H. J. Van Hook, and T. Vatimos, J. Appl. Phys. 63, 3140 ~1988!. 8B. Heinrich and J. F. Cochran, Adv. Phys. 42, 523 ~1993!. 9In the conventional form of the Landau–Lifshitz equations, one encoun- ters the Gilbert damping constant G~Ref. 8 !. We have G5G/gMs [G/VM. For Fe at room temperature, G50.83108s21. 10L. H. Dubois, G. P. Swartz, R. E. Camley, and D. L. Mills, Phys. Rev. B 29, 3208 ~1984!. 11B. Heinrich and V. F. Mescharyakov, Sov. Phys. JETP 32, 232 ~1971!. 12N. S. Almeida and D. L. Mills, Phys. Rev. B 38, 6698 ~1988!;ibid.39, 12 339 ~1989!. 3067 J. Appl. Phys., Vol. 82, No. 6, 15 September 1997 R. E. Camley and D. L. 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1.4863377.pdf	Conversion of pure spin current to charge current in amorphous bismuth H. Emoto, Y. Ando, E. Shikoh, Y. Fuseya, T. Shinjo, and M. Shiraishi Citation: Journal of Applied Physics 115, 17C507 (2014); doi: 10.1063/1.4863377 View online: http://dx.doi.org/10.1063/1.4863377 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A proposal for time-dependent pure-spin-current generators Appl. Phys. Lett. 101, 213109 (2012); 10.1063/1.4764557 Photoexcited charge current for the presence of pure spin current Appl. Phys. Lett. 96, 262108 (2010); 10.1063/1.3455887 Tunable pure spin currents in a triple-quantum-dot ring Appl. Phys. Lett. 92, 042104 (2008); 10.1063/1.2838310 Conversion of spin current into charge current at room temperature: Inverse spin-Hall effect Appl. Phys. Lett. 88, 182509 (2006); 10.1063/1.2199473 Pure SpaceChargeLimited Electron Current in Silicon J. Appl. Phys. 37, 2412 (1966); 10.1063/1.1708829 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 62.178.183.194 On: Thu, 15 May 2014 21:09:22Conversion of pure spin current to charge current in amorphous bismuth H. Emoto,1Y . Ando,1E. Shikoh,2Y . Fuseya,3T. Shinjo,1and M. Shiraishi1,a) 1Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan 2Graduate School of Engineering, Osaka City University, Osaka 558-8585, Japan 3Department of Applied Physics and Chemistry, The University of Electro-Communications, Tokyo 182-8585, Japan (Presented 5 November 2013; received 22 September 2013; accepted 28 October 2013; published online 30 January 2014) Spin Hall angle and spin diffusion length in amorphous bismuth (Bi) are investigated by using conversion of a pure spin current to a charge current in a spin pumping technique. In Bi/Ni 80Fe20/Si(100) sample, a clear direct current (DC) electromotive force due to the inverse spin Hall effect of the Bi layer is observed at room temperature under a ferromagnetic resonance condition of the Ni 80Fe20layer. From the Bi thickness dependence of the DC electromotive force, the spin Hall angle and the spin diffusion length of the amorphous Bi ﬁlm are estimated to be 0.02and 8 nm, respectively. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4863377 ] Bismuth (Bi) has been extensively studied from the early days of solid state physics. Some important physical phenom- ena in solid state physics such as diamagnetism, the Nernst-Ettingshausen effect, the Shubnikov-de Haas effect, and the de H a a s - v a nA l p h e ne f f e c th a v eb e e nﬁ r s td i s c o v e r e di nB i . 1–4 Such noteworthy discoveries by using Bi are attributed to its unique physical properties, e.g., a small effective mass of the Dirac electrons, a strong spin-orbit coupling, and so on. In a spintronics ﬁeld, Bi attracts much attention,5–7a n di np a r t i c u - lar, its large spin-orbit coupling is quite important because conversion between a charge current and a pure spin current is a main subject for future spintronic devices.8–10Recently, a large spin Hall angle, hSHE, and a relationship between dia- magnetism and spin Hall conductivity in Bi were theoretically predicted.11–13An experimental investigation of the hSHEin Bi was also performed in a ferromagnetic metal (FM)/Bi bilayer systems7by means of the spin pumping technique.14–22In this method, the hSHEwas evaluated with a direct current (DC) electromotive force (EMF) in Bi because the inverse spin Hall effect (ISHE) generated the DC EMF from the spin current.18 However, anomalous Bi-thickness dependence of the EMF, which cannot be explained within the framework of the con- ventional spin pumping theory, was obtained. Although this curious behavior is currently explained as a result of contribu-tion of an intermixing layer between Bi and Ni 80Fe20(Py) layers, the estimated hSHEin the intermixing layer was consid- erably large and opposite in sign to that of Bi layer. Since thissituation impedes accurate evaluation of h SHEin Bi, a precisely controlled FM/Bi sample is strongly desired. In this study, we demonstrate spin injection into amorphous Bi by using spinpumping technique, and estimate the h SHEand the spin diffu- sion length, kBi, in amorphous Bi, where no such anomalous behavior was observed. The Bi-thickness dependence of EMFis clearly reproduced by theoretical ﬁtting function of the con- ventional spin pumping theory. Figure 1(a) shows a schematic illustration of a sample used in this study. A 16 nm-thick ferromagnetic Py layer wasformed at room temperature (RT) on a low-doped Si(100) substrate (carrier concentration at RT is /C2410 13cm/C03)b ye l e c - tron beam evaporation. After deposition of the Py layer, a Bilayer was formed at RT by using resistance heating evapora- tion. In order to estimate the spin diffusion length in Bi, thick- ness of the Bi layer, d Bi, was varied from 5 to 250 nm. The crystal structure and surface roughness of the Bi layer were investigated by means of X-ray diffraction (XRD) and atomic force microscopy (AFM), respectively. Conductivities of thePy,r Py, and Bi, rBi, were measured by means of the standard four probe method using Bi/Py Hall bar samples, where the thickness of the Py layer, dPy,w a s5 n ma n d dBi,w a sv a r i e d from 50 to 150 nm. Since the conductance of the sample is expressed as ðdPyrPyþdBirBiÞwHall/lHall,w h e r e wHallandlHall are the width and the length of the Hall bar, rPyandrBiwere evaluated independently. In the spin pumping measurements, the Bi/Py/Si(100) sample was placed in a nodal position of a TE011cavity of an electron spin resonance (ESR) system (JEOL FA-200), where the alternating electric and magnetic ﬁeld components were a minimum and a maximum, respec- tively (the microwave frequency, f, is 9.12 GHz). An external static magnetic ﬁeld was applied at an angle, hH,a si l l u s t r a t e d in Fig. 1(a). In the ferromagnetic resonance (FMR) condition of the Py layer, spin angular momentum is transferred fromthe Py layer to the Bi layer, resulting in generation of a spin current propagated along the normal direction of the ﬁlm plane of the Bi layer. Then, the spin current is converted intoa charge current due to the ISHE of Bi. Figures 1(c)–1(e) show the XRD h-2hpatterns of (c) Si(100), (d) Bi ( d Bi¼100 nm)/Si(100), and (e) Bi ( dBi¼100 nm)/Py/Si(100) samples. The (003), (006), and (009) peaks of Bi ﬁlm are observed in the Bi/Si(100) sample as shown in Fig.1(d), suggesting that highly orientated Bi(100) was grown on the Si(100). The AFM observation in Fig. 1(b) reveals that the Bi layer has a textured structure and the surface roughness is considerably large (Root-mean-square, RMS,roughness ¼12 nm), which is consistent with previous stud- ies. 23,24However, in order to realize quantitative investigation of the spin current conversion properties by means of the spinpumping technique, a smooth Bi/Py interface is necessarya)Author to whom correspondence should be addressed. Electronic mail: shiraishi@ee.es.osaka-u.ac.jp. 0021-8979/2014/115(17)/17C507/3/$30.00 VC2014 AIP Publishing LLC 115, 17C507-1JOURNAL OF APPLIED PHYSICS 115, 17C507 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 62.178.183.194 On: Thu, 15 May 2014 21:09:22because FMR does not uniformly occur when the interface is rough.25Thus, the Bi/Si(100) sample shown in Fig. 1(d)is not suitable in this study. By contrast, since very small RMS rough- ness (0.4 nm) was realized at the surface of the Py layer depos- ited on Si(100), efﬁcient spin injection by the spin pumping isexpected in the Bi/Py/Si(100) sample. As can be seen in Fig.1(e), the XRD pattern of the Bi/Py/Si(100) sample shows no apparent peaks except for the peaks of Si (100), indicatingthat the Bi layer is almost amorphous. r Biwas also different between the Bi/Si(100) sample ðrBi¼1:5/C2105X/C01m/C01Þand Bi/Py/Si(100) samples ðrBi¼0:94/C2105X/C01m/C01Þand tem- perature dependence of rBiof the Bi/Py/Si(100) samples exhib- ited semiconductor behavior. These results also suggest that the Bi layer of the Bi/Py/Si(100) samples is amorphous.26 Figure 2(a) shows the FMR spectra, i.e., d I(H)/dHas a function of H-H FMR, of the Bi/Py/Si(100) and the Py/Si(100) sample at hH¼0/C14,w h e r e I,H,a n d HFMRare the microwave absorption intensity, the extern al magnetic ﬁeld, and FMR ﬁeld, respectively. The HFMRis 90 mT for both samples. As can be seen in the inset of Fig. 2(a), the line width, W,o ft h eF M Rs p e c - tra is obviously enhanced by attaching the Bi layer (WBi/Py¼2.7 mT, WPy¼2.4 mT). The Wis expressed as W¼(2x=ﬃﬃﬃ 3p c)a,w h e r e a,x,a n d care the Gilbert damping constant, the angular frequency of magnetization precession, and the gyromagnetic ratio, respectively.19Thus, increment of Wis due to enhancement of the Gilbert damping constant of the Pylayer, suggesting that the spin angular momentum was pumped from the Py layer to the Bi layer. Figure 2(b) shows the DC EMF, V EMF, observed at RT. A clear signal was observed around the HFMR(the red open circles). Since the EMF of the anomalous Hall effect (AHE) in the Py layer also contributes the obtained VEMF, the obtained VEMF-Hsignals were analyzed using a deconvolution ﬁtting function as follows:18 VEMFðHÞ¼ VISHEC2 ðH/C0HFMRÞ2þC2 þVAHE/C02CðH/C0HFMRÞ ðH/C0HFMRÞ2þC2; (1) where U,VISHE,a n d VAHEare the damping constant in this deﬁnition, the amplitude of the ISHE, and that of the AHE,respectively. As shown in Fig. 2(b), a theoretical ﬁt using Eq.(1)(the black solid line) reproduces good agreement with the experimental results, and jVISHEjandjVAHEjare estimated to be 35.9 and 14.1 lV, respectively. The microwave power, PMW, dependence of jVISHEjshown in Fig. 2(c) reveals that jVISHEjis proportional to the microwave power. This result corresponds to the theory of the spin pumping. Figure 2(d) shows the VISHEas a function of the hH. The polarity reversal of the VISHEwas observed when the hHwas changed from 0/C14 to 180/C14, and in addition, no VISHE signal was observed at hH¼90/C14. In the ISHE theory, the charge current, Jc,i s expressed as18 Jc¼hSHE2e /C22h/C18/C19 Js/C2r; (2) where Js,r,e, and /C22hare the spin current, the spin polarized vector, the elementary charge, and the Dirac constant,respectively. Therefore, the experimental result of the h Hde- pendence of the VISHEin Fig. 2(d)is consistent with the theo- retical one. These results indicate that the VISHE in this study is attributed to the ISHE in the Bi layer. Hereafter, we focus on the dBidependence of the VISHE. Taking into account the spin relaxation and diffusion in theBi layer, the spin current density along the ydirection can be written as 27 JsðyÞ¼sinhðdBi/C0yÞ=kBi ½/C138 sinhðdBi=kBiÞJ0 s; (3) where J0 sis the spin-current density at the Bi/Py interface (y¼0), and expressed as19 J0 s¼g"# rc2h2/C22h4pMscﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð4pMsÞ2c2þ4x2q/C20/C21 8pa2½ð4pMsÞ2c2þ4x2/C138; (4) FIG. 1. (a) A schematic illustration of the Bi/Py structure for spin pumping. MandHdenote the magnetization and the external static magnetic ﬁeld, respectively. (b) The surface proﬁle of the Bi layer deposited on Si(100). XRD patterns of (c) Si(100) substrate, (d) Bi ( dBi¼100 nm)/Si(100) sample, and (e) Bi ( dBi¼100 nm)/Py/Si(100) sample. FIG. 2. (a) FMR spectra, d I(H)/dHof the Bi/Py/Si(100) and Py/Si(100) samples measured at RT. The microwave excitation power is 200 mW. (b) Hdependence of DC output voltage in the Bi/Ni 80Fe20/ Si(100) sample at RT. The red circles show experimental data and the blue and green solid lines show theoretical ﬁtting of VISHE andVAHE, respectively. The black solid line shows theoretical ﬁtting using Eq. (1). (c) Microwave power, PMW, dependence of \| VISHE\|. The solid line shows a linear ﬁt to the data. Inset shows VEMFversus H-H FMRunder various PMW. (d)hHdependence of VISHE. Inset shows VEMFversus H-H FMRathH¼180/C14.17C507-2 Emoto et al. J. Appl. Phys. 115, 17C507 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 62.178.183.194 On: Thu, 15 May 2014 21:09:22where g"# r,h, and Msare the real part of the mixing conduct- ance, the amplitude of an alternating magnetic ﬁeld, and the saturation magnetization of the Py layer, respectively. Thereal part of the mixing conductance is given by 17 g"# r¼4pMsdPy glBðaPy=Bi/C0aPyÞ; (5) where dPy,g,a n d lBare the thickness of the Py layer, the Lande g-factor, and the Bohr magneton, respectively. Here, for a soft-ferromagnetic ﬁlm such as Py, the reso- nance condition isx c/C0/C12¼ðHFMRþ4pMsÞHFMR ;athH¼0/C14 andx c/C0/C12¼ðHFMRþ4pMsÞ2;athH¼90/C14.1From these equa- tions with HFMR¼90 mT at hH¼0/C14andHFMR¼1298mT at hH¼90/C14,Msand care estimated to be 79mT and 1.8/C21011T/C01s/C01, respectively. From Eqs. (2)and(3),t h e average of the charge current density along the xdirection can be written hJcðxÞi ¼ hSHE2e /C22h/C18/C19dBi kBitanhdBi 2kBi/C18/C19 J0 s: (6) Taking into account the equivalent circuit as shown in Fig. 3(b), the VISHEis expressed as VISHE¼whSHEkBitanhðdBi=2kBiÞ dBirBiþdPyrPy2e /C22h/C18/C19 J0 s; (7) where wis the length between two electrodes. Here, w¼2 mm, h¼0.077 mT, rBi¼0.94/C2105X/C01m/C01, and rPy¼1.3/C2106X/C01m/C01. As can be seen in Fig. 3(a), the VISHE is increased with increasing the dBiup to around 50 nm, and then moderately decreased, which is obviously different from that of the previous study.7A theoretical ﬁt using Eq. (7)(the black solid line) reproduces good agree- ment with the experimental results, indicating that our sample precisely controlled Bi/Py bilayer system. From the ﬁtting, the hSHEand the kBiin the Bi layer at RT are esti- mated to be 0.02 for the hSHEand 8 nm for the kBi, respec- tively. This hSHEis comparable with those of the other heavy metals.19,28Although a DC EMF with a Lorentzian shape in aV-H curve caused by self-induced ISHE in the Py layer isreported,29this spurious effect does not show such a typical dBidependence. Therefore, the dBidependence of the VISHE strongly manifests successful spin injection from the Py layer into the Bi layer and reliable estimation of the hSHEand thekBiin the Bi layer. It is also noted that the anomalous behavior appeared in the previous study7was not observed. ThekBiof this study is considerably smaller than that of the previous study.7This difference can be explained as a result of a crystal structure. Since Bi layer of this study isamorphous, there are a lot of scattering centers, which reduce the electron mobility. As a result, it is expected that the spin diffusion length of amorphous Bi is shorter than that of crys-tal Bi. In summary, the h SHEand the kBiin amorphous Bi were investigated by using the spin pumping technique. Under theferromagnetic resonance condition, the V ISHE due to the ISHE of the amorphous Bi layer was observed at RT. The spin Hall angle and the spin diffusion length of the amor-phous Bi ﬁlm were estimated to be 0.02 for the h SHEand 8 nm for the kBi, respectively. This research was supported in part by a Grant-in-Aid for Scientiﬁc Research from the MEXT, Japan, by the Adaptable & Seamless Technology Transfer Program through Target-driven R&D from JST, and by the TorayScience Foundation. 1C. Kittel, Introduction to Solid State Physics (Wiley, New York, 2005). 2A. V. Ettingshausen and W. Nernst, Ann. Phys. 265, 343 (1886). 3L. Schubnikov and W. J. de Haas, Commun. Phys. Lab. Univ. Leiden 207d , 35 (1930). 4W. J. de Haas and P. M. van Alphen, Commun. Phys. Lab. Univ. Leiden 212a , 3 (1930). 5L. Wu et al.,Hyperﬁne Interact. 69, 509 (1992). 6J. Fan and J. Eom, Appl. Phys Lett. 92, 142101 (2008). 7D. Hou et al.,Appl. Phys. Lett. 101, 042403 (2012). 8M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 1790 (1985). 9F. J. Jedema, A. T. Filip, and B. J. van Wees, Nature 410, 345 (2001). 10S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006). 11Y. Fuseya, M. Ogata, and H. Fukuyama, Phys. Rev. Lett. 102, 066601 (2009). 12Y. Fuseya, M. Ogata, and H. 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Phys. 109, 103913 (2011). 29A. Tsukahara, Y. Kitamura, E. Shikoh, Y. Ando, T. Shinjo, and M. Shiraishi, e-print arXiv:1301.3580 . FIG. 3. (a) Bi-thickness dependence of the VISHE. The solid line shows a ﬁt- ting curve using Eq. (7). (b) The equivalent circuit model of the Bi/ Py/Si(100) sample.17C507-3 Emoto et al. J. Appl. Phys. 115, 17C507 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 62.178.183.194 On: Thu, 15 May 2014 21:09:22
1.4893949.pdf	Different temperature scaling of strain-induced magneto-crystalline anisotropy and Gilbert damping in Co2FeAl film epitaxied on GaAs H. C. Yuan, S. H. Nie, T. P. Ma, Z. Zhang, Z. Zheng, Z. H. Chen, Y. Z. Wu, J. H. Zhao, H. B. Zhao, and L. Y. Chen Citation: Applied Physics Letters 105, 072413 (2014); doi: 10.1063/1.4893949 View online: http://dx.doi.org/10.1063/1.4893949 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/105/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The thickness-dependent dynamic magnetic property of Co2FeAl films grown by molecular beam epitaxy Appl. Phys. Lett. 105, 172406 (2014); 10.1063/1.4900792 Bending strain-tunable magnetic anisotropy in Co2FeAl Heusler thin film on Kapton® Appl. Phys. Lett. 105, 062409 (2014); 10.1063/1.4893157 Femtosecond laser excitation of multiple spin waves and composition dependence of Gilbert damping in full- Heusler Co2Fe1−xMnxAl films Appl. Phys. Lett. 103, 232406 (2013); 10.1063/1.4838256 Magnetic and Gilbert damping properties of L 21-Co2FeAl film grown by molecular beam epitaxy Appl. Phys. Lett. 103, 152402 (2013); 10.1063/1.4824654 Relationship between Gilbert damping and magneto-crystalline anisotropy in a Ti-buffered Co/Ni multilayer system Appl. Phys. Lett. 103, 022406 (2013); 10.1063/1.4813542 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.149.200.5 On: Sun, 30 Nov 2014 18:55:12Different temperature scaling of strain-induced magneto-crystalline anisotropy and Gilbert damping in Co 2FeAl film epitaxied on GaAs H. C. Yuan,1S. H. Nie,2T. P. Ma,3Z. Zhang,1Z. Zheng,1Z. H. Chen,3Y . Z. Wu,3 J. H. Zhao,2,a)H. B. Zhao,1,b)and L. Y . Chen1 1Shanghai Ultra-precision Optical Manufacturing Engineering Research Center, and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China 2State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China 3Department of Physics, State Key Laboratory of Surface Physics and Advanced Materials Laboratory, Fudan University, Shanghai 200433, China (Received 28 June 2014; accepted 13 August 2014; published online 21 August 2014) The temperature dependence of the Gilbert damping and magnetic anisotropy are investigated in L21Co2FeAl ﬁlms epitaxially grown on GaAs (001) substrate by the time resolved magneto- optical Kerr effect. We found that the in-plane biaxial anisotropy increases by more than 90% with the temperature decreasing from 300 K to 80 K, which is mainly due to the strong variation of themagneto-elastic coefﬁcients. In contrast, the intrinsic Gilbert damping rises only about 10%, which is mainly attributed to the reduction of the electron phonon scattering rate, independent of the strain-induced spin-orbit coupling energy. VC2014 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4893949 ] The Co-based full-Heusler alloys such as Co 2FeAl (CFA) may ﬁnd important applications in magnetic random access memory (MRAM) devices employing the spin transfertorque (STT) effect. 1,2These alloys normally exhibit very small Gilbert damping aas a result of their half metallic na- ture,3so their implementation in MRAM has the advantage of low current threshold required for the magnetization switching. When epitaxially grown as a single crystalline thin ﬁlm on the GaAs (001) or MgO (001) substrate, the Co-basedHeusler alloy often displays a pronounced lattice strain- induced magneto-crystalline anisotropy (MCA) with its strength comparable or even larger than that of the intrinsicbulk MCA. 4–6Considering that the strain-dependent MCA, i.e., magneto-elastic anisotropy (MEA), originates from the alteration of spin orbit coupling (SOC) due to the change ofthe distance between the magnetic atoms, and the intrinsic a is strongly correlated with the SOC, the possible impacts of the strain and the resultant MEA on aneed to be examined before designing the appropriate STT-based structures. According to the torque correlation model proposed by Kambersk /C19y, 7the intrinsic ain ferromagnetic (FM) medium has a quadratic scaling with SOC strength nin the tempera- ture regime where inter-band transitions dominate. On the other hand, the MCA is thought to arise from the second-order energy correction of SOC in the perturbation treatment and it exhibits the same scaling with nasa, 8,9thus a linear relationship between aand MCA is expected if other impor- tant factors affecting the two are kept unchanged. This rela- tionship was recently conﬁrmed in L10FePd 1–xPtxalloys where only nis widely tuned by changing x.10Besides n, other parameters including the spin-polarized band width, and spin and orbital moments, involved in the spin-orbitinteraction, will also affect both aand MCA.8,11These pa- rameters may change upon the lattice distortion under stress, leading to the alteration of aand the formation of the MEA. To investigate the correlation between aand MEA in the FM thin ﬁlms, an intuitive approach is to modify the strain by altering the ﬁlm thickness or substrate property. However, theﬁlm thickness and substrate alterations may modify not only the strain but also the crystalline order, and the other interface related characteristics, all of which would have differentimpacts on the magnetic anisotropy as well as the damping behavior. For example, the enhancement of interface- or defect-induced two-magnon scattering 12,13and spin pumping effect14,15in a thinner ﬁlm will lead to an increase of awith- out involvement of SOC. The thinner ﬁlm, however, typically shows a decreased in-plane biaxial anisotropy due to thereduced crystalline order, 16but increased uniaxial anisotropy as a result of the enhanced interface bonding effect.17Because of these complexities, both linear and nonlinear dependenceofaon the magnetic anisotropy were observed, 16,18–21and the intrinsic impact of MEA energy on aremains unclear. Here, we investigate the temperature dependence of MCA and ain a compressive strained L21CFA ﬁlm grown on GaAs (001) substrate, without inﬂuencing the chemical composition, crystalline order, and interface structure. Verysurprisingly, we found that the in-plane biaxial and uniaxial magnetic anisotropies increase by more than 90% and 40%, respectively, when decreasing the temperature from 300 K to80 K. Such large anisotropy alteration mainly arises from the strong variation of the magneto-elastic energy with tempera- ture. In contrast, aonly rises about 10% in the same tempera- ture range. These results thus indicate that aand MEA may evolve independently of each other. The sample consists of a 10-nm-thick CFA ﬁlm epitax- ially grown on the GaAs (001) substrate. X-ray diffraction measurements reveal the L2 1structure and a lattice constanta)E-mail: jhzhao@red.semi.ac.cn b)E-mail: hbzhao@fudan.edu.cn 0003-6951/2014/105(7)/072413/5/$30.00 VC2014 AIP Publishing LLC 105, 072413-1APPLIED PHYSICS LETTERS 105, 072413 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.149.200.5 On: Sun, 30 Nov 2014 18:55:12of 0.568 nm of the CFA ﬁlm, and thus, a compressive strain of/C241% is expected within the ﬁlm plane. In order to deter- mine aand magnetic anisotropy, time resolved magneto- optical Kerr effect (TRMOKE) measurements wereperformed using a Ti:sapphire regenerative ampliﬁer laser with /C24120 fs pulses at the central wavelength of 800 nm and 1 KHz repetition rate. The pump pulses with energy densityof/C242 mJ/cm 2instantaneously modulate the magnetic anisot- ropy, and trigger the uniform magnetization precession which is monitored by the time delayed probe beam in the crossed polarizer geometry. We applied the external mag- netic ﬁeld along the in-plane [100] and [1–10] directions toacquire the ﬁeld dependence of precession frequencies which is used to determine the magnetic anisotropy. To suppress the contribution of the dephasing effect to a, the ﬁeld was applied up to 1.2 T, much larger than the anisotropy ﬁelds. Figure 1shows the time evolution of the polar magnet- ization component after pump laser excitation at ﬁeld of1.2 T applied along the [100] direction for different tempera- tures. The major feature of the magnetization dynamics man- ifests as a slowly decayed oscillatory signal corresponding tothe weakly damped uniform magnetization precession. 22The oscillations can be well ﬁtted by a damped sine function hk¼Ae/C0Ctsinð2pftþuÞ, with precession amplitude A, time delay t, decay rate C, and precession frequency f. The ﬁeld dependence of ffor different temperatures is summarized in Fig.2. We note that fincreases with decreasing temperatures at low ﬁelds for both [100] and [1–10] directions, indicating larger anisotropy ﬁelds at lower temperatures. From the hys- teresis loops measured for these two ﬁeld directions,6we can see that the CFA ﬁlm exhibits both in-plane biaxial and uni- axial anisotropies, with both easy axes along the [110] axis. To quantitatively determine the magnetic anisotropy values, we ﬁtted the ﬁeld dependence of ffor both ﬁeld directions with the Kittel equation for the uniform precession 2pf¼cf½Hcosðd–/ÞþH1/C138½Hcosðd–/ÞþH2/C138g1=2;(1) with H1¼4pMsþ2Kv/Ms–2Kusin2//MsþKjj(2–sin2(2/))/ Ms,a n d H2¼2Kucos (2 /)/Msþ2Kjjcos (4 /)/Ms,w h e r e cis the gyromagnetic ratio, dand/represent the angles of Handin-plane equilibrium Mwith respect to the CFA [110] axis, Msis the saturated magnetization which increases from 1100 to 1131 emu/cm3with temperature decreasing from 300 to 80 K, and Kjj,Ku,a n d Kv, denote the in-plane biaxial, uniaxial, and out-of-plane uniaxial magnetic anisotropies, respectively. We obtain from the best ﬁtting Kjj¼14 kJ/m3,Ku¼34.5 kJ/ m3,a n d Kv¼/C059.5 kJ/m3at 300 K. Both in-plane anisotropy values are much larger than that of the strain free bulk materi- als or very thick ﬁlms.6,23–25The stronger Kumay come from the interface bonding effect as for other FM metals grown on GaAs,26on the other hand, shear strain e12may also contrib- ute to the uniaxial anisotropy energy as Eu¼2B2u1u2e12,27 where B2is a second order magneto-elastic coefﬁcient, and u1,a n d u2denote the cosine of the angle of the magnetization with respect to the in-plane [110], and [1 /C010] directions, respectively. In contrast, the enhanced four-fold anisotropy can only originate from the strain effect related to the fourth- order magneto-elastic coefﬁcients ( C1,C2), with the aniso- tropic energy E4¼(C2e33/C02C1e11)u12u22for biaxial strain e11¼e22ande33. Such in-plane biaxial MEA is also observed in CFA thin ﬁlms grown on MgO (001) substrate, and it scalesFIG. 1. Transient Kerr rotation (color dots) with external ﬁeld of 1.2 T applied along the [100] axis of Co 2FeAl for different temperatures. The solid lines are ﬁtted curves using the damped sine function.FIG. 2. Magnetic ﬁeld dependence of precession frequency at different tem-perature, with the ﬁeld direction along (a) [1 /C010] and (b) [100] axes. Solid lines represent the ﬁtted results. The measurements were performed with ﬁeld-decreasing from the saturation ﬁeld to nearly zero.072413-2 Yuan et al. Appl. Phys. Lett. 105, 072413 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.149.200.5 On: Sun, 30 Nov 2014 18:55:12linearly with the strain magnitude.5The negative value of Kv indicates an easy out-of-plane anisotropy in this thin ﬁlm, and this perpendicular anisotropy can originate from the biaxial strain induced energy Ev¼2B1(e33/C0e11), where B1is another second order magneto-elastic coefﬁcient. The best ﬁtting of the ﬁeld dependence of fat different temperatures in Fig. 2reveals that Kjj,Ku, and Kvincrease by /C2495%, /C2445%, and /C2440%, with the temperature decreasing from 300 K to 80 K, respectively, as shown in Fig. 3.28Since Msonly changes by less than 3% in the range of 80–300 K,6 the enhancement of K jjshould be much larger than that of the intrinsic four-fold bulk anisotropy. Thus, we may assignthis anisotropy change mainly to the variation of the MEA. However, the lattice mismatch between CFA and GaAs is estimated to decrease about 5% based on the lattice changeof Co 2FeGe29and other FM metals30in this temperature range. Therefore, the signiﬁcant enhancement of Kjjmust arise from the large variation of the C1andC2constants with temperature. In addition to the strain, the interface bonding may also contribute to Ku, but this contribution may have lit- tle change with the temperature; therefore, we speculate thatthe variation of MEA contribution to K uis larger than 45%. In comparison with Ku,Kvdisplays a smaller variation with temperature, and this indicates that B1has weaker tempera- ture dependence than B2does. As a summary of the above discussions, we may attrib- ute the temperature inﬂuence on the magnetic anisotropiesmainly to the temperature dependence of C 1,C2,B1, and B2. Although we are unaware of any report about the tempera- ture dependence of these magneto-elastic coefﬁcients inCFA, the parameter B 2in Fe was reported to increase about 45% with the temperature decreasing from 300 K to 80 K,and the change ratio of B1is smaller,31similar to the case of the CFA ﬁlm. Our results further indicate that the fourth order magneto-elastic coefﬁcient may have larger variation with temperature than the second order one. Several theoreti-cal works pointed out that the pronounced change of the magneto-elastic coefﬁcients can only have its origin from the perturbing inﬂuences of SOC. 32,33Although such SOC perturbation may dramatically change the MEA energy, its impact on the Gilbert damping ahas not been identiﬁed previously. We now turn to the discussion of the temperature de- pendence of a. We obtain the effective damping aefffrom the ﬁtted precession decay rate Cat different ﬁelds, using aeff¼2C=½cð2Hcos ðd/C0/ÞþH1þH2Þ/C138: (2) Thus, obtained damping exhibits a maximum at the applied ﬁeld strength close to the anisotropy ﬁeld values, as shown in Fig. 4(a). This ﬁeld dependence of aeffis mainly due to the dephasing effect as a result of the inhomogeneous anisot- ropy distribution.34The dephasing can be suppressed at applied ﬁelds much stronger than the anisotropy ﬁelds,35 thus we can see that aeffremains almost constant for H>7900 Oe, and this value can be regarded as the intrinsic a. To improve the accuracy for determining the intrinsic a, we average the values of aeffforH>7900 Oe and plot them FIG. 3. Temperature dependence of the in-plane biaxial (a), uniaxial (b), and out-of-plane uniaxial (c) magnetic anisotropy constants.FIG. 4. (a) Field dependence of the effective Gilbert damping parameter aeff at different temperatures. (b) Temperature dependence of the intrinsic a averaged from the values of aeffat H >7900 Oe in the inset of (a).072413-3 Yuan et al. Appl. Phys. Lett. 105, 072413 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.149.200.5 On: Sun, 30 Nov 2014 18:55:12as a function of the temperature in Fig. 4(b).36Thus obtained aat 300 K has the value of /C240.0044, much smaller than that obtained with the ﬁeld applied perpendicular to the ﬁlm,37 where the dephasing effect may not be completely sup- pressed since the applied ﬁeld is not strong enough to satu- rate the magnetization. It can be seen from Fig. 4(b) that the intrinsic ashows a slight increase of /C2410% as the tempera- ture is lowered from 300 K to 80 K. The increase of the intrinsic awith lowering temperature was also observed in other FM metals,38and this enhance- ment was thought to mainly arise from the reduction of the electron phonon scattering rate 1/ s.39The magnetization pre- cessions generate via SOC electron hole pairs which exist for lifetime sbefore relaxing by lattice scattering. Because the amount of the spin angular momentum dissipation to thelattice depends on how far from equilibrium the system gets, the damping increases with longer sand they scale linearly with each other. The ﬁrst principle calculations performedby Gilmore and co-workers conﬁrmed this tendency as a result of the variation of 1/ sother than the temperature ﬂuc- tuations of SOC and density of states in Fe, Co, and Ni at thelow temperature regime where intra-band transitions domi- nate. 40We therefore also attribute the reduction of 1/ swith decreasing temperature in CFA as the dominant factor to theenhancement of the intrinsic a. From the above discussions, we may conclude that the large variation of strain induced spin-orbit coupling energyhas negligible inﬂuences on the intrinsic ain CFA. Since both MCA and ahave the quadratic scaling with SOC strength n, we may exclude the signiﬁcant variation of nin the temperature range of 80–300 K. Nevertheless, it was found from the ﬁrst-principles calculations that the magneto- elastic coefﬁcient may be very sensitive to the details of theband structure near the Fermi surface. 41In other words, when there is a small change of the energy band in CFA as a result of the temperature variation, it will have a great impacton the strain-induced MCA. In contrast, such small band structure modulation associated with the slightly modiﬁed spin-polarized band width, and spin and orbit moments mayhave little impact on the intrinsic Gilbert damping a. In summary, we reveal distinctly different change ratios of the MCA and awith temperature within 80–300 K in the L2 1CFA ﬁlm. The temperature dependence of magneto- elastic coupling coefﬁcients and electron scattering time are the dominant factors for the variations of the MCA and a, respectively, while the SOC strength in this temperature re- gime has negligible variation. The CFA ﬁlm may keep its small Gilbert damping value nearly unchanged in spite of thelarge increasing of the strain-induced magneto-crystalline energy, which is ideal for magnetic storage applications based on STT strategy. This work was supported by the National Natural Science Foundation of China with Grants Nos. 61222407 and 51371052, NCET (No. 11-0119), and MOST of China with Grants Nos. 2013CB922303 and 2011CB921801. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1–L7 (1996). 2D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008).3I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Phys. Rev. B 66, 174429 (2002). 4M. S. Gabor, T. Petrisor, Jr., C. Tiusan, M. Hehn, and T. Petrisor, Phys. Rev. B 84, 134413 (2011). 5M. Belmeguenai, H. Tuzcuoglu, M. S. Gabor, T. Petrisor, Jr., C. Tiusan, D. Berling, F. Zighem, T. Chauveau, S. M. Cherif, and P. Moch, Phys. Rev. B 87, 184431 (2013). 6K. K. Meng, S. L. Wang, P. F. Xu, L. Chen, W. S. Yan, and J. H. Zhao, Appl. Phys. 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1.4709727.pdf	Electromagnetic and absorption properties of urchinlike Ni composites at microwave frequencies T. Liu, P. H. Zhou, J. L. Xie, and L. J. Deng Citation: Journal of Applied Physics 111, 093905 (2012); doi: 10.1063/1.4709727 View online: http://dx.doi.org/10.1063/1.4709727 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/111/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electromagnetic properties of Fe-Si-Al/BaTiO3/Nd2Fe14B particulate composites at microwave frequencies J. Appl. Phys. 115, 17C722 (2014); 10.1063/1.4865218 Influence of Ag on the microstructure and magnetic properties of perpendicular exchange coupled composite L10-[FePt-Ag]/[Co/Ni]N films Appl. Phys. Lett. 102, 252410 (2013); 10.1063/1.4812836 Microwave absorption properties of the hierarchically branched Ni nanowire composites J. Appl. Phys. 105, 053911 (2009); 10.1063/1.3081649 Analyses on double resonance behavior in microwave magnetic permeability of multiwalled carbon nanotube composites containing Ni catalyst Appl. Phys. Lett. 92, 042507 (2008); 10.1063/1.2839382 Electromagnetic and microwave absorption properties of ( Co 2 + – Si 4 + ) substituted barium hexaferrites and its polymer composite J. Appl. Phys. 101, 074105 (2007); 10.1063/1.2716379 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.111.164.128 On: Wed, 19 Aug 2015 11:36:27Electromagnetic and absorption properties of urchinlike Ni composites at microwave frequencies T. Liu, P . H. Zhou, J. L. Xie, and L. J. Deng State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China (Received 23 November 2011; accepted 29 March 2012; published online 2 May 2012) In this paper, nearly monodispersed urchinlike Ni powders were synthesized by a simple hydrogen-thermal reduction method. Electromagnetic and absorption characteristics were theninvestigated at 0.5–18 GHz. The permeability spectra present four resonance peaks over the whole frequency range. The resonance absorption property was discussed by ﬁtting the permeability spectrum using the well-known Landau-Lifshitz-Gilbert equation and Maxwell-Garnett mixing rule.Correspondingly, the magnetic loss of the ﬁrst band observed is attributed to the natural resonance, while the other three bands are considered to originate from non-uniform exchange resonance in the permeability spectra. The maximum reﬂection loss can reach /C043 dB at about 10 GHz with 2 mm in absorber thickness. VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4709727 ] I. INTRODUCTION Nowadays, with the rapid advancements in wireless communication, the high density of electromagnetic radia- tions in our surroundings is becoming a serious problem. The use of electromagnetic absorbers that absorb electro-magnetic radiations can handle this problem up to a certain extent. Among all absorbers, magnetically soft metallic materials may be potential candidates for microwave absorp-tion materials because their permeability remains high in the gigahertz range due to high saturation magnetization. The effect of hierarchical architecture on the electromagneticproperties of microwave absorber candidates is important for absorption design. Although attentions have been paid to the static magnetic characteristics of these assembledstructures, 1,2the effects of the superstructures, especially hierarchical ones, on the microwave electromagnetic proper- ties are rarely reported. The authors have studied the micro-wave electromagnetic properties of dendritic Co composites previously. 3In this paper, urchinlike Ni powders have been prepared by a simple hydrothermal reduction method. Alarge number of nanocones radiating from the surface may exhibit signiﬁcant impacts on polarization relaxation. The purpose of this work is to explore the inﬂuence of urchinlikemicrostructure on the microwave electromagnetic and absorption properties of Ni. II. EXPERIMENT AND CHARACTERIZATION A. Preparation of urchinlike Ni microstructures The urchinlike Ni sample was synthesized via a facile hydrothermal reduction route, as described in previouswork. 4In brief, all the reagents were of analytical grade and used without further puriﬁcation. First, 0.476 g NiCl 2/C16H2O, 1.95 g glycine, and 1.6 g NaOH were dissolved into 30 mlH 2O under sonication. Then, 5 ml of 80% N 2H4aqueous so- lution was added dropwise into the mixture under simultane- ous violently stirring for 10 min. The ﬁnal mixture wastransferred into a teﬂon-lined stainless steel autoclave,sealed, and held at 160 /C14C for 24 h. Finally, the products were collected by centrifugal settling and washed with abso- lute alcohol and deionized water, and dried in vacuum fur- nace at 55/C14C for 2 h. B. Characterization Morphology and structure of the samples were charac- terized using a ﬁeld emission scanning electron microscope (FE-SEM, JEOL JSM-7600 F). X-ray diffraction (XRD) pat-terns of the samples were recorded by SHIMADZU XRD- 7000 x-ray diffractometer with Cu K aat voltage of 40 kV and a current of 30 mA. Magnetic hysteresis at room temper-ature of the sample was measured in magnetic ﬁelds between /C010 and 10 kO e, using Lakeshore 7300 vibrating sample magnetometer (VSM). The urchinlike Ni/parafﬁn compositesample was prepared by randomly and homogeneously dis- persing the 16.83 vol. % urchinlike Ni powders in parafﬁn and was then pressed into toroidal shape with an inner diam-eter of 3.04 mm and outer diameter of 7 mm for microwave measurements. The complex permittivity and complex per- meability were calculated from the S-parameters measuredwith an Agilent 8720ET vector network analyzer with a transverse EM mode in a frequency range of 0.5–18 GHz. III. RESULTS AND DISCUSSION A. Structural characterization Figures 1(a)and1(b) indicate that the samples consist of urchinlike spheres with diameter of 0.5–2 lm. Such spheres are in fact built from nanocones with diameters of 50–150 nm at the root and lengths of 100–500 nm from the root to the tip. All XRD peaks in Fig. 1(c) can be clearly indexed as Ni with face centered cubic (fcc) structure. The average crystal size of urchinlike Ni particles is about 29.4 nm, according to the Sherrer formula at the main XRDpeaks (2 h¼44.5 /C14). It can be seen in Fig. 1(d) that the value of the saturation magnetization ( Ms) of the urchinlike Ni is 54.1 emu/g, close to bulk Ni (55 emu/g) at room temperature. 0021-8979/2012/111(9)/093905/5/$30.00 VC2012 American Institute of Physics 111, 093905-1JOURNAL OF APPLIED PHYSICS 111, 093905 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.111.164.128 On: Wed, 19 Aug 2015 11:36:27B. Complex permittivity Figure 2(a) shows the frequency dependence of the real part ( e0) and imaginary part ( e00) of the relative complex per- mittivity in the 0.5–18 GHz range. It demonstrates that e0of the sample shows variations ( e0¼15:561:5) in the whole frequency range. Meanwhile, e00is almost constant ( e00¼2) in the 0.5–6 GHz range but has a remarkable increase over6 GHz. It goes without saying that the speciﬁc surface area of urchinlike structure increase with developing nanocones, so a large area of interfaces is formed between urchinlike Niﬁllers and parafﬁn matrix. Hence, the interface polarizability is one of the important factors contributing to the obvious frequency-dependent dielectric response, 5similar to the den- dritic Co structure.3 C. Complex permeability The real part ( l0) and the imaginary part ( l00) of the rela- tive permeability are plotted in Fig. 2(b). It reveals that the values of l0decrease from 1.8 to 0.8 gradually with increasing frequency. As to the l00-fspectrum, clearly, four resonance FIG. 1. FE-SEM images of the as- synthesized urchinlike Ni at low magni- ﬁcation (a) and high magniﬁcation (b). (c) The XRD pattern of urchinlike Ni. (d) Magnetization hysteresis loops meas- ured at room temperature. FIG. 2. Frequency dependences of effective permittivity (a) and permeability (b) of the composites with 16.83 vol. % of urchinlike Ni particles.093905-2 Liu et al. J. Appl. Phys. 111, 093905 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.111.164.128 On: Wed, 19 Aug 2015 11:36:27absorption peaks are observed, which are denoted as P1, P2, P3, and P4, respectively. Actually, m ulti-resonance is still a subject of controversy. The most accepted one may be relevant to thepresent observations of the non-uniform exchange resonance mode, 6which is attributed to the small size effect of nanocrys- talline structure, surface effe ct, and spin wave excitations.7–9 The eddy current loss contribution to the imaginary part perme- ability is related to thickness (d) and the electric conductivity (r) of the composites: l00ðl0Þ/C02f/C01¼2pl0d2r,10where l0is the permeability of vacuum. The calculated evolution l00ðl0Þ/C02f/C01with frequency is plotted in Fig. 3.I ft h e observed magnetic loss only results from eddy current loss,the value l 00ðl0Þ/C02f/C01should be constant. As we can see, however, the value of l00ðl0Þ/C02f/C01for the sample varies drastically in the whole frequency range, meaning that theeddy current loss is suppressed in the measured frequency range. As far as measure method is concerned, generally speaking, dimensional resonance on coaxial line related tothe thickness of sample in microwave test is one of the main reasons of measurement inaccuracy. We calculate possible resonance frequencies associated with the dimensional reso-nance. Figure 4demonstrates the relationship between the wavelength of electromagnetic wave entering the sample andfrequency in the 0.5–18 GHz range. If the thickness of sam- ple equals to an integer multiple of half the wavelength, the dimensional resonance can be excited, namely d¼ k 2nðn¼1;2;3…Þ; (1) where dandkdenote the thickness of sample and the wave- length of electromagnetic wave entering the sample, respec- tively. According to Fig. 4, the wavelength minimum is 4.72 mm. If the physics origin of the peaks is the dimensionalresonances, dis at least 2.36 mm. However, the thickness of real sample is 1.71 mm; hence we can exclude the dimen- sional resonances and the peaks most probably are ferromag-netic origin. Moreover, multiple resonance behaviors also have been observed in the composites with Co microﬂowers ﬁllers due to the natural and exchange resonances in Ref. 11. D. Study on magnetic loss mechanism To further understand our experimental results, the reso- nance spectrum will be ﬁtted as a linear superposition of the overlapped peaks P1, P2, P3, and P4. In order to simplify thecalculation method of intrinsic permeability, the urchinlike Ni could be regarded as isotropic particulates. Hence, the intrinsic permeability of urchinlike Ni homogeneously dis-persed in parafﬁn matrix can be obtained 12 li¼1þ2 3xmðx0þiaxÞ ðx0þiaxÞ2/C0x2; (2) where xm¼cMs,x0¼cHe.candHedenote the gyromag- netic factor and effective anisotropy ﬁeld, respectively. Then the resonance peaks P1, P2, P3, and P4 could be separatedfrom each other by ﬁtting Landau-Lifshitz-Gilbert (LLG) equation as 13,14 li¼1þX4 i¼1Ii2 3xmðx0þiaxÞ ðx0þiaxÞ2/C0x2"# ; (3) where Iiis the intensity of the peak. FIG. 3. The l00ðl0Þ/C02f/C01values for mixture sample as a function of frequency. FIG. 4. Frequency dependence of the wavelength of electromagnetic wave entering the urchinlike Ni composites. FIG. 5. The experimental curves and ﬁtting curves of the imaginary partand real part of the effective permeability.093905-3 Liu et al. J. Appl. Phys. 111, 093905 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.111.164.128 On: Wed, 19 Aug 2015 11:36:27For the composites consisting of urchinlike Ni randomly and uniformly distributed in parafﬁn matrix, the effectivepermeability is isotropic and given as 15 leff¼lmþflmli/C0lm lmþ1/C0f fc/C18/C1913ðl i/C0lmÞ; (4) where liandlmare the permeability of inclusion and ma- trix; fandfcare the volume fraction of urchinlike Ni and percolation threshold, respectively. LLG equation (Eq. (3)) and modiﬁed Maxwell-Garnett mixing rule (Eq. (4))a r e used to calculate the intrinsic permeability and effective permeability, respectively. Figure 5demonstrates the ex- perimental curves and calculated curves of effective perme- ability. First, the four resonance peaks in the l00-fcurve over 0.5–18 GHz are ﬁtted. Then the l0-fcurve is calculated using the obtained ﬁtting param eters. All the ﬁtting parame- ters are listed in Table I. The magnetocrystalline anisotropy ﬁeld H0¼4jK1j= ð3l0MsÞ, the anisotropic coefﬁcient ( K1) for the fcc-type bulk nickel is about /C05/C2103J/m3,16soH0is about 140 Oe, which is small compared to Heof P1 band. To check the mechanism of P1, the Kittel equation17is adopted to calcu- late the natural resonance frequency ( fr) and the shape anis- tropy could be ignored. When using H0instead of Heand taking the damping factor ( a) into account, the frequency (fmax) at which the l00maximum appears is given by 2pfr¼cH0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þa2p ;fmax¼frﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þa2p : (5)Thus, the calculated resonance frequency should be around several hundreds of megahertz. Because of the existence of nanocones, the anisotropy energy will be signiﬁcantlyincreased due to the enhanced surface anisotropy, according to a simple model ( K eff¼Kvþ6Ks=d).18KvandKsrefer to the volume and surface contributions to anisotropies, respec-tively. In addition, H eis markedly inﬂuenced by the follow- ing factors: lattice defects, interior and magnetic exchange coupling, and so on.18,19Consequently, the actual value of fr could be obviously higher. Toneguzzo et al.20have sug- gested that the natural resonance occurs at a lower frequency than the exchange resonance. In this case, consequently, theﬁrst resonance band P1 can be ascribed to the natural reso- nance. For exchange model, the exchange resonance fre- quencies ( f ex) are calculated by6 2pfex r¼Heff¼HeþCl2 kn R2Ms; (6) where Cis the exchange constant (C ¼2/C210/C07erg/cm);21 R is the crystal size; lknare the roots of the differential spherical Bessel function ( lkn¼l10,l14, and l15for P2, P3, and P4, respectively). According to fmax¼fex=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þa2p , the calculated values of fmax are 4.32 GHz, 6.01 GHz, and 7.84 GHz, which accord with the ﬁtted fmaxvalues of P2 (4.08 GHz), P3 (5.59 GHz), and P4 (7.68 GHz), respectively. Therefore, the exchange resonance mode is proven valid inexplaining the resonance mechanism of P2, P3, and P4. E. Electromagnetic wave absorption properties The reﬂection loss (RL) curves are calculated from the relative permeability and permittivity measured at the givenfrequency and thickness of an absorber according to the transmission theory. Figure 6shows the relationship between the RL and the frequency for the urchinlike Ni dispersed inparafﬁn matrix in the 0.5–18 GHz range. The frequency for RL exceeding /C010 dB is in the 5–16 GHz range with a thick- ness from 1.5 to 2.5 mm, while the frequency for RL exceed-ing/C020 dB is in the 9.5–10.5 GHz and 12.5–14 GHz with a thickness 2 mm and 1.5 mm, respectively. FIG. 6. Frequency dependency of the microwave RL of the urchinlike Ni dispersed in parafﬁn with various thickness. FIG. 7. Frequency dependence of the dielectric- and magnetic-loss factorsof urchinlike Ni dispersed in parafﬁn.TABLE I. Fitting parameters for permeability dispersion spectra. Peak Heff(Oe) Ia f c P1 628.32 0.90 0.74 P2 1507.96 0.34 0.26 0.20P3 2010.62 0.16 0.12P4 2764.60 0.16 0.12093905-4 Liu et al. J. Appl. Phys. 111, 093905 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.111.164.128 On: Wed, 19 Aug 2015 11:36:27Compared to the dielectric-loss, the magnetic-loss factor exhibits higher values at low (0.5–8 GHz) and lower value at high (13–18 GHz), respectively, as shown in Fig. 7. In par- ticular, it can be seen that the optimal RL reach /C043 dB at about 10 GHz (Fig. 6), perfectly consistent with the fre- quency corresponding to the same value of dielectric lossfactor and magnetic loss factor. IV. CONCLUSIONS In summary, urchinlike Ni powders have been prepared by a facile hydrothermal reduction method. The microwavecharacterization of urchinlike Ni/parafﬁn composites is ana- lyzed in the frequency range of 0.5–18 GHz and four reso- nance behaviors are detected. The combination of themodiﬁed LLG equation and Maxwell-Garnett mixing rule is used to predict the effective permeability spectrum of com- posites. The agreement of the ﬁtted and calculated resultsreveals the coexistence of natural and exchange resonances. The excellent microwave-absorption properties are a conse- quence of a proper electromagnetic match and strong multi-resonance peaks. ACKNOWLEDGMENTS This work has been supported by “the Fundamental Research Funds for the Central Universities” and the NSFC(Grant Nos. 61001026 and 51025208).1Z. G. An, J. J. Zhang, and S. L. Pan, Mater. Chem. Phys. 123, 795 (2010). 2H. Li, Z. Jin, H. Y. Song, and S. Liao, J. Magn. Magn. Mater. 322,3 0 (2010). 3T. Liu, P. H. Zhou, J. L. Xie, and L. J. Deng, J. Appl. Phys. 110, 033918 (2011). 4Y. Wang, Q. S. Zhu, and H. G. Zhang, Mater. Res. Bull. 42, 1450 (2007). 5C. P. Smyth, Dielectric Behavior and Structure (McGraw-Hill, New York, 1955), pp. 52–74. 6A. Aharoni, J. Appl. Phys. 69, 7762 (1991). 7Q. Zhang, C. F. Li, Y. N. Chen, Z. Han, H. Wang, Z. J. Wang, D. Y. Geng, W. Liu, and Z. D. Zhang, Appl. Phys. Lett. 97, 133115 (2010). 8P. H. Zhou, L. Zhang, and L. J. Deng, Appl. Phys. Lett. 93, 112510 (2010). 9L. J. Deng, P. H. Zhou, J. L. Xie, and L. Zhang, J. Appl. Phys. 101, 103916 (2007). 10M. Z. Wu, Y. D. Zhang, S. Hui, T. D. Xiao, S. H. Ge, W. A. Hines, J. I.Budnick, and G. W. Taylor, Appl. Phys. Lett. 80, 4404 (2002). 11Z. Ma, Q. F. Liu, J. Yuan, Z. K. Wang, C. T. Cao, and J. B. Wang, Phys. Status Solidi B 249, 575 (2012). 12S. B. Liao, Ferromagnetic Physics (3) (Science, Beijing, 2000 ). 13F. Ma, Y. Qin, and Y. Z. Li, Appl. Phys. Lett. 96, 202507 (2010). 14F. S. Wen, H. B. Yi, L. Qiao, H. Zheng, D. Zhou, and F. S. Li, Appl. Phys. Lett. 92, 042507 (2008). 15K. N. Rozanov, A. V. Osipov, D. A. Petrov, S. N. Starostenko, and E. P. Yelsukov, J. Magn. Magn. Mater. 321, 738 (2009). 16L. L. Diandra and D. R. Reuben, Chem. Mater. 8, 1770 (1996). 17C. Kittel, Phys. Rev. 73, 155 (1948). 18F. Bødker, S. Mørup, and S. Linderoth, Phys. Rev. Lett. 72, 282 (1994). 19X. G. Liu, D. Y. Geng, H. Meng, B. Li, Q. Zhang, D. J. Kang, and Z. D. Zhang, J. Phys. D: Appl. Phys. 41, 175001 (2008). 20P. Toneguzzo, G. Viau, O. Acher, F. Fievet-Vincent, and F. Fievet, Adv. Mater. (Weinheim, Ger.) 10, 1032 (1998). 21P. A. Voltatas, D. I. Fotiadis, and C. V. Massalas, J. Magn. Magn. Mater. 217, L1 (2000).093905-5 Liu et al. J. Appl. Phys. 111, 093905 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.111.164.128 On: Wed, 19 Aug 2015 11:36:27
1.1453358.pdf	Recording physics of perpendicular recording with single layered medium and ring head K. J. Lee , Y. S. Kim , E. S. Kim , Y. H. Im , K. M. Lee , J. W. Kim , B. K. Lee , H. S. Oh , K. Y. Kim , N. Y. Park , and T. D. Lee Citation: Journal of Applied Physics 91, 8700 (2002); View online: https://doi.org/10.1063/1.1453358 View Table of Contents: http://aip.scitation.org/toc/jap/91/10 Published by the American Institute of PhysicsRecording physics of perpendicular recording with single layered medium and ring head K. J. Lee,a)Y. S. Kim, E. S. Kim, Y. H. Im, K. M. Lee, J. W. Kim, B. K. Lee, H. S. Oh, K. Y. Kim, and N. Y. Park Storage Laboratory, Samsung Advanced Institute of Technology, P.O. Box 111, Suwon, Korea T. D. Lee Department of Materials Science and Engineering, KAIST, 373-1, Taejon, Korea In this article, the recording physics of a ring head with a single layered perpendicular medium ~Ring/SL !is studied and the results are compared with a single pole head with a double layered perpendicular medium ~SPT/DL !by using various simulations and experiments. The Ring/SL has much lower effective medium coercivity than the SPT/DL due to the substantial longitudinal ﬁeldcomponent of ring head and the incoherent rotation mode of medium magnetizations. Furthermore,switching time of the Ring/SL is estimated as only 10% ;30% of the SPT/DL. In the Ring/SL, the head ﬁeld gradient of 40 Oe/nm is enough for maximizing SNR.The Ring/SLshows very low noisecharacteristics especially at high linear density. The signal output of the Ring/SLis smaller than theSPT/DL, but it is large enough to be detected. Therefore, it can be concluded that the combinationof single layered perpendicular medium and ring head is highly promising for ultrahigh densityrecording. © 2002 American Institute of Physics. @DOI: 10.1063/1.1453358 # Perpendicular magnetic recording is a promising candi- date beyond longitudinal magnetic recording because it hasbetter thermal stability at high linear density and can be usedwith a smaller writing ﬁeld than in longitudinal type. 1,2There are two possible combinations in perpendicular recordingtechnology. One is a combination of a single pole head and adouble layered medium ~SPT/DL !and the other is a combi- nation of a ring head and a single layered medium ~Ring/SL !. So far the Ring/SL has been considered as an inadequateapproach to perpendicular recording because of its smallerhead ﬁeld, output signal, and ﬁeld gradient compared withthose of the SPT/DL. However, the SPT/DL has a lot oftechnical problems, such as large medium noise, spike noise,antenna effect, etc. We have reported 1000 kFCI recording with carrier-to- noise ratio ~CNR!of 25 dB by using the Ring/SL, where we could observe very small signal reduction with increasinglinear density around the 1,000 kFCI. 3Therefore, we strongly believe that the possibility of the Ring/SL for ultra-high density recording should be seriously reconsidered. In this work, the recording physics of the Ring/SL are studied by using various simulations and experiments. Wefocus especially on three well-known weak points of theRing/SL: small head ﬁeld, slant head ﬁeld gradient, andsmall output signal. Micromagnetic simulations have been performed to in- vestigate effective medium coercivity and magnetizationswitching time in the Ring/SL and the SPT/DL. A recordinglayer is modeled by 10 310 tetragonal shaped grains.To con- sider the incoherent rotation mode properly, each grain issubdivided into 4 34325 cubic cells, each with a volume of 2n m 3. Easy axes are tilted 1° off the ﬁlm thickness direction. The adopted magnetic parameters are as follows: saturationmagnetization Ms5250 emu/cm3, damping constant a 50.05, exchange constant within grains Aex51 31026erg/cm, zero exchange constant across grain bound- aries. Uniaxial anisotropy Kuis varied from 1 3106to 4 3106erg/cm3. The Lindholm head ﬁeld is used as the ring head ﬁeld. We assume that the SPT head produces only a perpendicularﬁeld and a four times larger maximum ﬁeld magnitude thanthe ring head due to the soft underlayer. The writing processis simulated by applying a pulsed type head ﬁeld on therecording layer with remanent state. Effective coercivity isdeﬁned as the smallest head ﬁeld magnitude to reverse me-dium magnetizations at the medium center. Switching time isdeﬁned as the elapsed time of the magnetization reversal.The time evolution of the magnetization is obtained by inte-grating the Landau–Lifshitz–Gilbert equation. The newly developed 2D Preisach model 4is used to study the effect of the head ﬁeld gradient on read/write char-acteristics. Precise description and validity of this modelwere well described in Ref. 4, in which the calculated wave-form showed excellent coincidence with the measuring one. For experimental measurements, a single layered me- dium ~CoCrX/Ti/Pt !and a double layered medium ~CoCrX/ Ti/NiFe !are prepared. Coercivity and squareness of the CoCrX are 3000 Oe and 0.7, respectively. Read/write char-acteristics are investigated by using Guzik RWA2585. In theGuzik test, a ring head with track width of 0.45 mm and write gap of 0.15 mm is used for writing. Flying height and head-medium velocity are 20 nm and 10 m/s, respectively.Amerged magnetoresistive ~MR!head with shield-to-shield spacing of 0.11 mm and read track width of 0.35 mm is used to measure readback signal and noise. Magnetization reversal in the medium of the Ring/SL system shows very oscillatory behavior, while that in theSPT/DL shows small oscillations and monotonic decay @Fig. a!Electronic mail: leekj@sait.samsung.co.krJOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 10 15 MAY 2002 8700 0021-8979/2002/91(10)/8700/3/$19.00 © 2002 American Institute of Physics1~a!#. This difference arises from the longitudinal ﬁeld com- ponent and the rotation mode. To ﬁnd the rotation mode, weuse one grain’s exchange energy normalized by 180° wallenergy @E W54Agrain(AexKu)1/2, whereAgrainis the area of the grain’s basal plane #as an index because it can measure degree of nonuniformity of magnetizations during rotation.5 Magnetizations of the medium in the Ring/SL follow an in-coherent mode while those in the SPT/DL follow a coherentmode @Fig. 1 ~b!#. In the Ring/SL, the magnetization near the top surface of a grain is reversed ﬁrst and then the rest isreversed @inset of Fig. 1 ~a!#. The incoherent mode results from nonuniform distribution of perpendicular and longitu-dinal components of the ring head along the medium’s thick-ness direction @inset of Fig. 1 ~b!#. Switching time of medium magnetization tin the Ring/SL is less than 0.5 ns and is only 10%–30% of theSPT/DL @Fig. 2 ~a!#. This indicates the Ring/SL is more at- tractive for high data rate recording than the SPT/DL. Effec-tive medium coercivity, H C,effof the Ring/SL is about 0.4 Hk while that of the SPT/DL is almost same as Hk@inset of Fig. 2~a!#, where the Hkis the anisotropy ﬁeld of the medium. A distinguishing feature of the ring head is the substantial lon-gitudinal component. The longitudinal components exertlarge torque on spins, and therefore the tandHC,effof the Ring/SL are signiﬁcantly reduced. Another origin of the smaller tandHC,effin the Ring/SL is the incoherent rotation mode. We made an approximateestimation to ﬁnd the contribution of the incoherent mode totheH C,effreduction of the Ring/SL. If an external ﬁeld is applied to a single domain particle with tilting angle of uh and the rotation mode is coherent, coercivity of the particlecan be calculated from H c/Hk5sinucosuand cos( uh2u) 50 where uis angle between magnetization and easy axis.6 In our case the average angle between the easy axis and the ring head ﬁeld is 40° @5tan21(0.678/0.555) #, since relative values of thickness-averaged perpendicular and longitudinalﬁeld components are 0.678 and 0.555, respectively @inset of Fig. 1 ~b!#. Consequently, the coercivity of the single domain particle is 0.49 H k. There is 0.09 Hkdifference between the 0.4Hk~HC,effof the Ring/SL !and 0.49Hk, which is the con- tribution of the incoherent mode to the HC,effreduction of the Ring/SL. It should be noted that our simulations have beenperformed at zero temperature. If a thermal effect is consid- ered, the tandHC,effof the Ring/SL are further lowered due to the incoherent mode in which the thermal activation vol-ume must be smaller than the grain volume. The minimum magnetomotive force to switch magneti- zations with varying medium anisotropy is shown in Fig.2~b!. TheK uvalues in square boxes indicate the top limit of the medium anisotropy that can be recorded when BSof write head is 1.6 T. The writing ability of the Ring/SL isabout 62% ~52.18/3.55 !of the SPT/DLwhen head-medium spacing ~HMS !is 40 nm. When the HMS is reduced to 20 nm, the writing ability of the Ring/SL is 83% ~52.94/3.55 ! of the SPT/DL. By using the modiﬁed Preisach model, the writing pro- cess is simulated with a varying head ﬁeld gradient at thetrailing pole of a ring head from 9 to 100 Oe/nm.As the ﬁeldgradient becomes larger than 40 Oe/nm, the pulse width at50% threshold ~PW 50) of the Ring/SL converges @Fig. 3 ~a!#. We have reported that the perpendicular head ﬁeld strengthand its gradient could be improved by trimming the top poleedge of a conventional ring head. 7A ﬁnite element method ~FEM!calculation shows that the perpendicular ﬁeld gradi- ent of a ring head increases signiﬁcantly with trimmed thick-ness@inset of Fig. 3 ~a!#. When the total thickness of the top pole is 3.0 mm the ﬁeld gradient of 40 Oe/nm can be ob- tained by 2.75 mm trimming. For the experimental Guzik test of the Ring/SL, two dif- ferent ring heads are prepared. One is a conventional ringhead without trimming and the other is a modiﬁed ring headtrimmed by 2.5 mm using focused ion beam etching.To com- pare the Guzik test with simulation, ﬁeld gradients of the twoheads are calculated by FEM: the ﬁeld gradient of the con-ventional ring head is 9 Oe/nm and that of the 2.5 mm trim- ming head is 25 Oe/nm. Figure 3 ~b!and its inset show the experimental and simulation results, respectively. Both track-averaged ampli-tude and noise voltage ( V noise) increase with ﬁeld gradient. This means that the main noise of the Ring/SL is not thetransition noise. If it is the transition noise, the noise mustdecrease with increasing ﬁeld gradient because the jitter sizeis inversely proportional to the ﬁeld gradient in a transitionnoise dominant system such as the SPT/DL. 8 FIG. 1. Switching dynamics of magnetizations in Ring/SL and SPT/DL: ~a! magnetization decay with time evolution and ~b!variations of exchange energy in a grain with time evolution. Inset of ~a!showsMdistribution in a grain during switching when ^MZ&is zero. Inset of ~b!shows variations of normalized perpendicular and longitudinal components of the ring headalong medium thickness at position of maximum perpendicular ﬁeld. FIG. 2. Switching time and effective coercivity of the Ring/SLand SPT/DL:~a!switching time with varying anisotropy ﬁeld of medium and ~b!mini- mum magnetomotive force for full switching of magnetizations versus me-dium anisotropy. Inset of ~a!shows the effective coercivity versus anisot- ropy ﬁeld of medium.8701 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Lee et al.Interestingly, Vnoiseof the Ring/SL decreases with in- creasing linear density. This noise reduction indicates thedc-erase noise is the main noise of the Ring/SL.The dc-erasenoise is dependent only on magnetization ﬂuctuation innerbits. It can be reduced by magnetostatic interactions betweenneighboring bits, especially at high linear density. Figure 4 shows the effect of ﬁeld gradient on SNR in the Ring/SL. We can obtain higher SNR with the trimmed head.A steep ﬁeld gradient is more effective to enhance SNR at ahigher linear density. However, the dependency of SNR onﬁeld gradient is almost saturated for all linear densities whenthe ﬁeld gradient is larger than 40 Oe/nm. The calculated MR sensitivity function 4of the SLshows smaller amplitude and narrower width at the read gap centerthan the DL ~inset of Fig. 5 !. Although it depends on ﬂying height, magnetic property of the soft underlayer, and MRhead geometry, the DLshows 50% larger amplitude and 25%broader width than the SL in our calculation. This indicatesthat the SL shows a smaller signal output but better resolu- tion than the DL. It should be noted that the base line of thesensitivity function for the DL is not zero. In the DL, mag-netization positioned far from the MR sensor can affect read-ing signal through the high permeable soft underlayer.This isthe origin of low frequency noise of the DL. By Fouriertransformation of the MR sensitivity functions, we can di-rectly compare the noise spectra between the SL and the DL~Fig. 5 !. Both calculation and experiment show that the DL has larger low frequency noise. Since the experimental noisespectrum is measured after 50 kFCI recording, it should in-clude medium noise.Therefore, the discrepancy between cal-culation and experiment of the DL indicate that the DL haslarger medium noise than the SL. Therefore, it can be said that both signal and noise of the Ring/SL are intrinsically smaller than those of the SPT/DL.Furthermore, the SLshows better reading resolution. Smallersignal output in the Ring/SLis not a crucial demerit. Even inthe Ring/SL, signal amplitude is sufﬁciently large to be de-tected since the Mrtof perpendicular medium is larger than that of longitudinal one. ACKNOWLEDGMENTS The authors would like to thank Dr. K. Ouchi ofAIT for assisting the media preparation, and the advanced group ofReadRite Corp. for supporting head fabrication. 1M. Futamoto, Y. Honda, Y. Hirayama, K. Itoh, H. Ide, and Y. Maruyama, IEEE Trans. Magn. 32, 3789 ~1996!. 2N. Honda, and K. Ouchi, IEEE Trans. Magn. 33, 3097 ~1997!. 3K. M. Lee, J. W. Kim, K. J. Lee, B. K. Lee, and N.Y. Park, Digests of the International Conference on Materials for Advanced Technologies ~IC- MAT 2001 !, Singapore, 1–6 July 2001, E4-02. 4K. J. Lee, Y. H. Im, Y. S. Kim, K. M. Lee, J. W. Kim, N. Y. Park, G. S. Park, and T. D. Lee, J. Magn. Magn. Mater. 235, 398 ~2001!. 5D. Suess, T. Schreﬂ, and J. Fidler, IEEE Trans. Magn. 37,1 6 6 4 ~2001!. 6B. D. Cullity, Introduction to Magnetic Materials ~Addison-Wesley, Read- ing, MA, 1972 !, Chap. 9. 7E. S. Kim,Y. H. Im,Y. S. Kim, K. J. Lee, K. M. Lee, and N.Y. Park, IEEE Trans. Magn. 37, 1382 ~2001!. 8K. Miura, H. Muraoka, and Y. Nakamura, IEEE Trans. Magn. 37, 1926 ~2001!. FIG. 3. Effects of the ﬁeld gradient on ~a!PW50~simulation !, and ~b!output signal and noise voltage ~GUZIK test !in the Ring/SL. Inset of ~a!shows variations of the ﬁeld gradient with varying trimmed thickness ~FEM calcu- lation !where total thickness of top pole is 3 mm. Inset of ~b!shows simu- lation results for same condition as Fig. 3 ~b!. FIG. 4. Effect of the ﬁeld gradient on SNR in the Ring/SL: ~a!GUZIK test and~b!simulation. The vertical axis of ~b!is normalized by SNR at a ﬁeld gradient of 9 Oe/nm. FIG. 5. Noise spectra of perpendicular media ~GUZIK test and calculation !. Inset shows the calculated MR sensitivity functions. Signal peaks ~50 kFCI ! are removed from the measured spectra for clarity.8702 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Leeet al.
1.4985119.pdf	Static and dynamic magnetic properties of FeMn/Pt multilayers Ziyan Luo , Yumeng Yang , Yanjun Xu , Mengzhen Zhang , Baoxi Xu , Jingsheng Chen , and Yihong Wu Citation: Journal of Applied Physics 121, 223901 (2017); doi: 10.1063/1.4985119 View online: http://dx.doi.org/10.1063/1.4985119 View Table of Contents: http://aip.scitation.org/toc/jap/121/22 Published by the American Institute of Physics Articles you may be interested in Perpendicular magnetic anisotropy of TiN buffered Co 2FeAl/MgO bilayers Journal of Applied Physics 121, 223902 (2017); 10.1063/1.4984891 Tuning static and dynamic properties of FeGa/NiFe heterostructures Applied Physics Letters 110, 242403 (2017); 10.1063/1.4984298 Making the Dzyaloshinskii-Moriya interaction visible Applied Physics Letters 110, 242402 (2017); 10.1063/1.4985649 Tuning the magnetoresistance symmetry of Pt on magnetic insulators with temperature and magnetic doping Applied Physics Letters 110, 222402 (2017); 10.1063/1.4984221 Temperature dependence of the anisotropy field of L1 0 FePt near the Curie temperature Journal of Applied Physics 121, 213902 (2017); 10.1063/1.4984911 Electric field induced broadening of magnetic resonance line in ferrite/piezoelectric bilayer Journal of Applied Physics 121, 224103 (2017); 10.1063/1.4985069Static and dynamic magnetic properties of FeMn/Pt multilayers Ziyan Luo,1Yumeng Y ang,1Y anjun Xu,1,2Mengzhen Zhang,3Baoxi Xu,2Jingsheng Chen,3 and Yihong Wu1,a) 1Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117583, Singapore 2Data Storage Institute, ASTAR (Agency for Science, Technology and Research), 2 Fusionopolis Way, 08-01 Innovis, Singapore 138634, Singapore 3Department of Materials Science and Engineering, National University of Singapore, Singapore 117575 (Received 26 February 2017; accepted 25 May 2017; published online 8 June 2017) Recently, we have demonstrated the presence of spin-orbit torque in FeMn/Pt multilayers which, in combination with the anisotropy ﬁeld, is able to rotate its magnetization consecutively from 0/C14 to 360/C14without any external ﬁeld. Here, we report on an investigation of the static and dynamic magnetic properties of FeMn/Pt multilayers using the combined techniques of magnetometry, fer- romagnetic resonance, inverse spin Hall effect, and spin Hall magnetoresistance measurements. The FeMn/Pt multilayer was found to exhibit ferromagnetic properties, and its temperature depen-dence of saturation magnetization can be ﬁtted well using a phenomenological model by including a ﬁnite distribution in Curie temperature due to subtle thickness variations across the multilayer samples. The non-uniformity in static magnetic properties is also manifested in the ferromagneticresonance spectra, which typically exhibit a broad resonance peak. A damping parameter of around 0.106 is derived from the frequency dependence of ferromagnetic resonance linewidth, which is comparable to the reported values for other types of Pt-based multilayers. Clear inversespin Hall signals and spin Hall magnetoresistance have been observed in all samples below the Curie temperature, which corroborate the strong spin-orbit torque effect observed previously. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4985119 ] I. INTRODUCTION Multilayer structures consisting of ultrathin nonmag- netic (NM) layers, particularly, Pt and Pd, and ferromagnetic(FM) layers such as Co and Fe, have been of both fundamen- tal and technological interest since the late 1980s. 1When the thicknesses of both NM and FM layers are controlled within a certain range, typically less than 1.5 nm, the multilayer as a whole exhibits ferromagnetic properties with dominantlyperpendicular magnetic anisotropy (PMA). Some of these multilayer ﬁlms have already been applied in magneto-optic recording 2and more recently also in magnetic tunnel junc- tions as part of the reference layer.3,4Stimulated by earlier work on proximity effect at the FeMn and Pt interface,5we have recently carried out a systematic study of FeMn/Pt mul- tilayers.6,7Despite the fact that FeMn is an antiferromagnet (AFM), FeMn/Pt multilayers with ultrathin FeMn and Pt layers ( <1 nm) were found to exhibit global FM ordering with in-plane magnetic anisotropy. A large ﬁeld-like spin-orbit torque (SOT) was found to be present in the multilayer when a charge current ﬂows through it. 7Quantiﬁcation of the SOT strength was carried out by varying the thicknesses of both FeMn and Pt systematically, and the results corrobo- rate the spin Hall effect (SHE) scenario, i.e., spin current isgenerated and absorbed by the multilayer, thereby generating the SOT. We have further demonstrated that the SOT is able to rotate the magnetization of FeMn/Pt multilayers by 360 /C14 without any external ﬁeld. These results demonstrate clearlythe potential of FeMn/Pt multilayers in memory and sensor applications. In order to gain further insights into the SOT genera- tion mechanism in FeMn/Pt multilayers, in this paper, we report on ferromagnetic resonance (FMR), inverse spinHall effect (ISHE), and spin Hall magnetoresistance (SMR)studies of multilayer samples, which exhibit a clear SOTeffect. Before proceeding to dynamics studies, the staticmagnetic properties of the multilayers were characterizedusing magnetometry at variable temperatures. Specialemphasis was placed on the understanding of the tempera-ture dependence of the saturation magnetization. From theﬁtting of the experimental dat a using different models, it is found that the multilayers exhibit the characteristic of three-dimensional Heisenbe rg universality class with a ﬁnite Curie temperature distribution. This correlates wellwith the large linewidth of resonance peaks observed inboth FMR and ISHE. A large damping parameter ( /C240.106) is derived from the frequency dependence of the FMR,which is comparable to the values reported previously forother types of Pt-based multi layers. The observation of both ISHE and SMR suggests the presence of spin currentgeneration/absorption processes, corroborating the strong SOT effect observed previously. The role of asymmetric FeMn/Pt and Pt/FeMn interfaces in generating the SOT isdiscussed for samples with relatively thick FeMn and Ptlayers, whereas for samples with ultrathin Pt as well as co-sputtered samples, extrins ic SHE/ISHE may play a more important role. a)Author to whom correspondence should be addressed: elewuyh@nus.edu.sg 0021-8979/2017/121(22)/223901/10/$30.00 Published by AIP Publishing. 121, 223901-1JOURNAL OF APPLIED PHYSICS 121, 223901 (2017) II. EXPERIMENTAL [FeMn( tFeMn)/Pt(tPt)]nmultilayers (here ndenotes the repeating period) with different FeMn and Pt thicknessest FeMn andtPtwere prepared on SiO 2/Si substrates by magne- tron sputtering with a base pressure of 2 /C210/C08Torr and a working pressure of 3 /C210/C03Torr, respectively. The nomi- nal composition of Fe:Mn is 50:50. The structural propertiesof the multilayers were characterized using both X-ray dif- fraction (XRD) and X-ray reﬂectivity (XRR) analysis. Magnetic measurements were carried out using a QuantumDesign vibrating sample magnetometer (VSM) with the sam-ples cut into a size of 2 mm /C22 mm. The FMR measure- ments were performed at room temperature via a coplanarwaveguide (CPW), designed to have an impedance of 50 X within a broad frequency range up to 20 GHz. The wave- guide, 5 mm long, has a signal line of 150 lm and a signal to ground line spacing of 20 lm. The two signal lines of the CPW were connected to a Vector Network Analyzer (VNA)via high-frequency probes. The FMR spectra were obtainedby placing a 2 mm /C22 mm sample directly on the CPW with sample surface facing down and taking readings of theS 21signal while sweeping a DC magnetic ﬁeld in the signal line direction. For ISHE measurements, the sampleswere patterned into Hall bars with a lateral dimension of2000 lm/C2120lm by combined techniques of photolithog- raphy, sputtering deposition, and lift-off. Following the Hallbar fabrication, a 100 nm SiO 2insulating layer was deposited to isolate electrical conduction between the waveguide and the multilayer with the contacts to the Hall bar uncovered for subsequent electrical measurements. The last step was todeposit a 150– lm wide and 200-nm thick Cu coplanar wave- guide and four 500 lm/C2500lm contact pads. The same Hall bar was used to measure the SMR, which was obtainedby rotating the samples under a constant ﬁeld of 30 kOe inthexy,yx, and zxplanes, respectively. III. RESULTS AND DISCUSSION A. Structural properties The as-deposited multilayers were characterized using both high-angle XRD and small-angle XRR. Figure 1shows the XRD pattern of Pt(1)/[FeMn(0.6)/Pt(0.4)] 10, covering the range of bulk fcc Pt (111) peak at 39.8/C14and bulk fcc FeMn (111) peak at 43.5/C14, using the Cu K aradiation ( k¼1.541 A ˚). Here, the number and symbols inside the parentheses denotethe thickness of individual FeMn and Pt layers in nm.I n order to prevent oxidation, all the samples, unless otherwise stated, were all covered by a 1 nm Pt capping layer. As canbe seen from the ﬁgure, the diffraction pattern is dominatedby the main peak at 40.5 /C14– 40.6/C14, which falls between the bulk Pt (111) and FeMn (111) peaks. This suggests that themultilayer is (111) textured and its lattice spacing is the aver- age of those of Pt and FeMn, though it is more dominantly of Pt characteristic. The FeMn (111) peak is almost at the samelevel of the baseline, which is presumably caused by thecombined effect of ultrathin thickness, interface mixing, andsmall scattering cross sections of Fe and Mn as compared toPt. Similar phenomena have also been observed in Co/Ptmultilayers, in which the peak position is near that of Pt and increases with increase in the Co thickness. 8–10The small- angle XRR was measured with an incident angle in the rangeof 0 /C14–1 0/C14with a step of 0.02/C14. Figure 2shows the XRR of a multilayer with a structure: Pt(1)/[FeMn(0.6)/Pt(0.6)] 30 and another co-sputtered sample, i.e., Pt and FeMn were deposited simultaneously using the same deposition time and power. The n¼1 Bragg maximum corresponding to a period of 1.06 nm (about 20% smaller than the nominal values) isclearly observed in the spectrum of the multilayer sample (red solid line). In contrast, only thickness induced fringes are observed in the spectrum of the co-sputtered sample(blue dotted line). The result demonstrates that the multilayer has a well-deﬁned periodicity. B. Magnetic properties All the multilayers with tFeMn<0.8 nm and tPt<1n m exhibit ferromagnetic properties with in-plane magnetic anisotropy. The Curie temperature ( TC) varies from 250 K to 380 K, depending on both the total and individual layer thicknesses. Figure 3(a) shows the hysteresis loop of Pt(1)/ [FeMn(0.6)/Pt(0.3)] 10at 50 K and 300 K, respectively. The coercivity at 50 K is around 240 Oe, but it decreases rapidly to about 1 Oe at 300 K. Such kind of behavior is typical of samples exhibiting FM properties above room temperature(RT). Figure 3(b) shows the saturation magnetization as a function of the temperature ( M-T) with the FeMn layer thick- ness ( t FeMn) ﬁxed at 0.6 nm and Pt layer thickness ( tPt) rang- ing from 0.1 nm to 0.8 nm. As it was found that a minimum repeating period of 3–4 is required for most samples toFIG. 1. X-ray diffraction pattern of Pt(1)/[FeMn(0.6)/Pt(0.4)] 10. Dotted lines indicate the (111) peak position of Pt and FeMn, respectively. FIG. 2. XRR patterns of Pt(1)/[FeMn(0.6)/Pt(0.6)] 30multilayer sample (red solid line) and co-sputtered sample (blue dotted line) deposited under the same condition.223901-2 Luo et al. J. Appl. Phys. 121, 223901 (2017)exhibit ferromagnetic properties above RT, we ﬁxed the rep- etition period for all the samples at 10. Although the polar-ized Pt also contributes to the measured magnetic moment, itis difﬁcult to quantify it for samples with different thicknesscombinations and at different temperatures. Therefore, asan approximation, we take only the overall FeMn volumeinto consideration when calculating the saturation magneti- zation. As shown in the ﬁgure, the M sat low-temperature increases with increasing tPt, although the sample with tPt¼0.1 nm has a signiﬁcantly smaller magnetization. An opposite trend is observed for TCwhich decreases with tPt, saturating at about 300 K when the adjacent FeMn layers arecompletely separated magnetically by the Pt layer. Bothtrends are in qualitative agreement with ﬁndings reported forCo/Pt multilayers, 11which can be accounted for by the prox- imity effect at Pt/FeMn interfaces. Pt is known to be justunder the Stoner limit that can be readily polarized when it isin direct contact with ferromagnetic materials. In the present case, although FeMn is an AFM in bulk phase, it shall behave like a “superpara-AFM” when it is ultrathin, i.e.,t FeMn<1 nm. This can be inferred from exchange bias stud- ies in FeMn-based AFM/FM bilayers, which have revealedthat a minimum thickness of 4–5 nm is required for FeMn toestablish a measurable exchange bias to the FM at RT. 12 Despite its superpara-AFM nature, when it forms a multi-layer with Pt, the mutual interaction at their interfaces pro-motes FM order in both layers, which eventually extendsthroughout the multilayer when both layers are ultrathin.Therefore, as long as Pt is thin enough to allow complete polarization by the adjacent FeMn layers, the average mag- netic moment at low temperature will increase with the Ptthickness. On the other hand, the decrease of T Cat increasing Pt thickness is presumably due to the weakening of exchangecoupling throughout the multilayer caused by the incompletepolarization of the Pt layers at central regions. The anomalyatt Pt¼0.1 nm can be readily understood by taking into account the effect of interface roughness. At this thickness,Pt is probably partially discontinuous, resulting in the directcoupling of neighboring FeMn layers at certain locations andthereby reduces the saturation magnetization and T C. In order to gain more insights on the magnetic properties of the multilayers, we examine the M-T curves using differ- ent models. The temperature-dependence of magnetizationfor a ferromagnet at low-temperature can be calculated from the number of thermally excited magnons—quanta of spin-wave. Associated with each magnon is a magnetic momentgl B, and therefore, the total moment of magnon is given by N¼glBX k1 exp/C22hxk kBT/C18/C19 /C01; (1) where gis the electron g-factor, lBis the Bohr magneton, /C22h is the reduced Planck’s constant, xkis the magnon fre- quency, and kBis the Boltzmann’s constant. Under the long wavelength limit, the magnon dispersion relation may, in general, be written as /C22hxk¼Dkn, where Dis the spin-wave stiffness, and n¼2 for a ferromagnet, and n¼1 for an AFM. Substituting the dispersion relation into Eq. (1), one obtains N¼4pglB 2pðÞ3ð1 0k2dk exp Dkn=kBT ðÞ /C01 ¼1 2p2nglBf3 n/C18/C19 C3 n/C18/C19 kBT D/C18/C193=n ; (2) where fis the Riemann zeta function and Cis the Gamma function. Equation (2)can be used to calculate the tempera- ture dependence of magnetization in FM or stagger orderparameter in AFM. Since the FeMn/Pt multilayers exhibit ferromagnetic properties despite the fact that bulk FeMn is an AFM, in what follows we only focus on FM. By substitut-ingn¼2 into Eq. (2), we obtain the Bloch T 3/2law, i.e., MðTÞ¼Mð0Þð1/C0B3=2T3=2Þ; (3) where B3/2is a constant proportional to D/C03/2. Although the Bloch T3/2law can satisfactorily explain the M-T dependence at low temperature, it fails at high temperature because of the neglect of magnon-magnon interactions and deviation of the dispersion relation from /C22hxk¼Dk2at large k. For a Heisenberg ferromagnet, the high-temperature effect may beincluded in M(T) by introducing a temperature-dependent D, namely, DðTÞ¼Dð0Þð1/C0B 5=2T5=2Þ, where B5/2is a con- stant.13As a result, the M(T) in a wide temperature range can be modelled by MTðÞ¼M0ðÞ 1/C0B3=2T 1/C0B5=2T5=2 !3=20 @1 A:(4) When B5/2is small, M(T) can be approximated as MTðÞ¼M0ðÞ 1/C0B3=2T3=2/C03 2B3=2B5=2T4/C18/C19 : (5) Although Eq. (5)improves the ﬁtting at a higher temperature as compared to the Bloch T3/2law, it is still unable to repro- duce the M-T curve in the entire temperature range, and the deviation from experimental value tends to increase near TC due to the critical behavior of ferromagnet. In order to improve the ﬁtting near TCby taking into account the critical behavior, we invoke the semi-empiricalFIG. 3. (a) Hysteresis loop of Pt(1)/[FeMn(0.6)/Pt(0.3)] 10measured at 50 K (square) and 300 K (circle), respectively. (b) Saturation magnetization as afunction of temperature. The legend ( t 1,t2) denotes a multilayer with a FeMn thickness of t1and Pt thickness of t2. The number of the period for all sam- ples is ﬁxed at 10.223901-3 Luo et al. J. Appl. Phys. 121, 223901 (2017)model developed by Kuz’min,14which turned out to be very successful in ﬁtting the M-T curves of many different types of magnetic materials. According to this model, the temperature- dependent magnetization of a ferromagnet is given by MTðÞ¼M0ðÞ1/C0sT TC/C18/C193=2 /C01/C0sðÞT TC/C18/C195=2"#b ;(6) where M(0) is the magnetization at zero temperature, TCis the Curie temperature, sis the so-called “shape parameter” with a value in the range of 0–2.5, and bis the critical expo- nent whose value is determined by the universality classof the material: 0.125 for two-dimensional Ising, 0.325 forthree-dimensional (3D) Ising, 0.346 for 3D XY, 0.365for 3D Heisenberg, and 0.5 for mean-ﬁeld theory. 15,16On the other hand, for surface magnetism, bis in the range of 0.75–0.89.17,18The shape parameter sis determined by the dependence of exchange interaction, including its sign, oninteratomic distance in 3D Heisenberg magnets. 19This may have implications for multilayer samples as lattice distortionand strain are unavoidable at the interfaces due to large lat- tice match between and FeMn and Pt. The M-T dependence shown in Fig. 3(b) can be ﬁtted reasonably well using Eq. (6)with b¼1.01/C242.55 and s¼–0.85/C24–0.45, except that the ﬁtted magnetization drops to zero more quickly as compared to the experimental data. The large bvalues seem to suggest that the M-T of FeMn/Pt multilayers follows the surface scaling behavior. However,a careful examination of the results suggests that this maynot be the case because we found that bdecreases as t Pt increases. An opposite trend would have been observed if it was due to surface mechanism because a thick Pt layer would help enhance the 2D nature of ferromagnetism at the interfaces. This prompted us to consider other possible fac-tors that may affect the shape of M-T of the multilayers in a more prominent way as compared to the case of a uniform3D ferromagnet. The one that came to our attention is thehigh sensitivity of T Cto the Pt thickness as manifested in the M-T curves in Fig. 3(b); this may lead to a ﬁnite distribution ofTCthroughout the multilayer due to thickness variation induced by the interface roughness. When this happens, themagnetization may drop more slowly near T C, as experimen- tally observed. To this end, we modiﬁed Eq. (6)by including a normal distribution of TC, which leads to MTðÞ¼M0ðÞð1 01/C0sT TC/C18/C193=2 /C01/C0sðÞT TC/C18/C195=2"#b /C21ﬃﬃﬃﬃﬃﬃ 2pp DTCexp/C0TC/C0TC0 ðÞ2 2DT2 C"# dTC; (7) where TC0is the mean value of TCandDTCis its standard deviation. As shown in Fig. 4(a), all the M-T curves can be ﬁtted very well using Eq. (7)with a ﬁxed bvalue of 0.365, especially near the TCregion. Note that b¼0.365 is the criti- cal exponent for 3D Heisenberg ferromagnet. For the sake of clarity, all the curves in Fig. 4(a) except for the one fortPt¼0.1 nm are shifted vertically. In the ﬁgure, the symbols are the experimental data, and solid-lines are the ﬁttingresults. The ﬁtting values for M(0) ,T C0, andDTC, and sas a function of Pt thickness are shown in Figs. 4(b)–4(d) , respec- tively. Except for the sample with smallest tPt, the trends of M(0)/C0tPtandTC/C0tPtare opposite to each other, i.e., the former increases whereas the latter decreases with tPt. Both are a manifestation of the fact that the global FM ordering inFeMn/Pt multilayers originates from the proximity effect atFeMn/Pt interfaces, as discussed above. It is interesting tonote that DT Calso increases when tPtdecreases, and impor- tantly, the range of DTCfor samples with tPt¼0.1–0.8 nm corresponds to the range of average TCof all samples with tPtranging from 0.1 nm to 0.8 nm. These results are consis- tent with the TCﬂuctuation scenario, i.e., a larger ﬂuctuation inTCis expected in samples with smaller tPtdue to interface roughness, and its range should be corresponding to thedifference in average T Cwhen tPtvaries from 0.1 to 0.8 nm or less. Another important result derived from curve ﬁtting isthet Pt-dependence of the shape parameter s. According to Kuz’min et al. , for 3D Heisenberg magnets, sis determined by the dependence of exchange interaction on interatomicdistance. 19It is generally positive with a small s(<0.4) corresponding to metallic FMs with long-range ferromag-netic ordering and high T C, whereas a large s(>0.8) is indic- ative of competing exchange interactions and the resultant material typically has a low TC. As shown in Fig. 4(d),sis small and positive for samples with tPt¼0.6 nm and 0.8 nm, but it turns negative for smaller tPt. When sis negative, theT3/2term of the base of Eq. (6)becomes positive, or in other words, it contributes positively to M(T) when the temperature increases. This is counterintuitive for 3DHeisenberg ferromagnet. It suggests that, in addition to iso- tropic exchange coupling, interfacial Dzyaloshinskii-Moriya interaction (DMI) may play a role, particularly in samplesFIG. 4. (a) Experimental M-T curves (open symbols) and ﬁtted results (solid lines). The experimental data are the same as those shown in Fig. 3(b), but are shifted for clarity (except for the tPt¼0.1 nm sample). (b) M 0, (c) T C0 (triangle) and DTC(square), and (d) s, as a function of tPtobtained from the ﬁttings.223901-4 Luo et al. J. Appl. Phys. 121, 223901 (2017)with smaller tPt. As DMI favors non-collinear alignment of spins, a weakening of DMI at moderately elevated tempera-ture may give a relative boost of isotropic exchange cou- pling, thereby resulting in a positive contribution to the magnetic moment at intermediate temperature range. Thismay explain why sis negative, though further studies are required to quantify the effect of DMI on the temperature dependence of magnetization in these multilayers. C. FMR measurements and damping constant Magnetic damping plays a key role in the magnetization dynamics of magnetic materials, which can be treated phe-nomenologically by including a damping term a~M/C2ðd~M=dtÞ in the Landau-Lifshitz-Gilbert (LLG) equation. Here, ais the Gilbert damping constant, which characterizes the strength ofdamping. It is commonly assumed that the origin of Gilbert damping is spin-orbit coupling (SOC), same as that of mag- netic anisotropy. Since SOC is also the origin of spin-orbit tor-que, naturally it would be of interest to measure the damping constant of FeMn/Pt multilayers and correlate it with SOT or ISHE. The effective damping constant, including both intrin-sic and extrinsic contributions, can be deduced from the FMR line width as a function of the resonance frequency. Figure 5(a)shows the FMR spectra of a Pt(1)/[FeMn(0.6)/Pt(0.5)] 80 multilayer extracted by VNA at different frequencies ranging from 2 GHz to 4 GHz with a sweeping DC magnetic ﬁeld. Compared with a homogeneous FM layer, the FMR peak israther broad. This is presumably caused by the variation in T C and Msthroughout the multilayer as discussed earlier. Nevertheless, the average resonance ﬁelds at different fre-quencies can still be described by the Kittel equation 20 2pf¼l0cﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HFMRðHFMRþMsÞp ; (8) where fis the frequency, cis the effective gyromagnetic ratio, Msis the saturation magnetization, HFMRis the reso- nance ﬁeld, and lois the vacuum permeability. The FMR spectra near the resonance region can be roughly ﬁtted by the superposition of a symmetric and an antisymmetric peak.As an example, Fig. 5(b)shows the ﬁtting result at f¼3 GHz for Pt(1)/[FeMn(0.6)/Pt(0.5)] 80. The full width at half maxi- mum (FWHM) of the symmetric peak with Lorentz shape isplotted in Fig. 5(c) (empty square) as a function of fre- quency. The solid-line is the linear ﬁtting to the relation 21 DHfðÞ¼4p l0c/C18/C19 afþDH0; (9) where ais the effective damping parameter and DH0is zero- frequency linewidth caused by magnetic inhomogeneity of the sample. The large DH0value is consistent with the distri- bution of TCdiscussed in Sec. III B. From the linear ﬁtting, we obtained an effective damping parameter of 0.106 for this speciﬁc sample, which is approximately one order of magni- tude larger than that of permalloy at the same thickness,22 but is comparable to that of Pt/Co multilayers.23,24This afﬁrms our previous argument of the twofold role of Pt in Pt/ FeMn multilayers,7i.e., it promotes global FM ordering via proximity effects at Pt/FeMn interfaces and at the same time it functions simultaneously as both a spin current generator and an absorber. It is postulated that both the proximityeffect and spin-current absorption contribute to the enhance- ment of a, 25,26although it is difﬁcult to determine which fac- tor is dominant. When Pt is magnetized, it will be an FM with large SOC which will lead to large damping. FeMn is known to have a small SOC. However, being sandwiched byPt in the multilayer structure, the precession of its magneti- zation under ferromagnetic resonance will pump spin current into the neighboring Pt layers, which again will lead to the enhancement of damping. Although a large damping con- stant is undesirable for applications which require the use ofspin torque transferred from other layers to switch its magne- tization, it can be effectively exploited for SOT-based appli- cations, i.e., to generate SOT internally by a charge current. This is exactly what we have reported in our earlier work, in which we have demonstrated that it is possible to switch themagnetization of FeMn/Pt multilayers by SOT without any external ﬁeld. 7It is worth pointing out that the damping parameter extracted above may be overestimated considering the fact that sample inhomogeneity may also contribute to the large FM linewidth. D. Inverse spin Hall effect In the aforementioned FMR measurements, we attribute the enhancement of apartially to the absorption of spin cur- rent by the Pt layers. As we will discuss shortly in the SMR experiments, for multilayers with relatively thick Pt and FeMn, we may treat them as consisting of alternating FMand HM layers. However, if the Pt and FeMn layers are ultra- thin, it is more appropriate to treat the multilayer equiva- lently as a single FM layer. We consider the multilayer case ﬁrst. If we focus on a speciﬁc FeMn layer inside the multi- layer structure, there are two interfaces with the adjacent Ptlayers. To differentiate these two interfaces, we call Pt/FeMn the upper interface and FeMn/Pt the lower interface. These two interfaces are not necessarily to be identical due to the large lattice mismatch between Pt and FeMn. 27Although theFIG. 5. (a) FMR spectra of Pt(1)/[FeMn(0.6)/Pt(0.5)] 80at ﬁxed frequency ranging from 2 GHz to 4 GHz. (b) Data (square symbol) and ﬁtting (line) for FMR signal at f¼3 GHz. (c) Full width at half maximum of the resonance peak (triangle symbol) are plotted against the frequency. The solid line is a linear ﬁt to the data.223901-5 Luo et al. J. Appl. Phys. 121, 223901 (2017)FeMn/Pt multilayer behaves like a single phase FM, the magnetic moment is presumably mainly from the FeMn layer. Under the FMR condition, the precession of magneti-zation in the FeMn layer pumps spin current into the adja- cent Pt layers, which is subsequently absorbed either completely or partially depending on the Pt layer thickness.This leads to the enhancement of damping constant as dis- cussed above. If the two interfaces are symmetrical, there should not be a net spin current following inside the multi-layer after we take into account the contributions of all the individual layers. However, if the two interfaces are asym- metrical and have different spin-mixing conductance, a netspin current will be generated due to broken inversion symmetry. When this happens, a transverse electromotive force (EMF) will be generated due to ISHE, which can bedetected as a voltage signal under open circuit condition. In this context, we have measured the voltage across the two side-contacts of the sample simultaneously with the FMRmeasurements. Figure 6(a) shows the measurement geometry, where h m is the rfdriving ﬁeld, and His the external ﬁeld. The mea- surement was ﬁrst performed on multilayer sample Pt(1)/ [FeMn(0.6)/Pt(0.5)] 50with n¼50. The peak position of the ISHE signal in Fig. 6(b) and FMR spectrum in Fig. 6(c) show good correspondence with each other, suggesting that the ISHE signal might be directly related to the FMR absorp- tion. Following that, we carried out the same measurementson Pt(1)/[FeMn(0.6)/Pt( t Pt)]10samples with tPtranging from 0.1 nm to 0.8 nm, respectively. Although the FMR signal of the sample with n¼10 was too weak to be detected due to small absorption, the voltage could still be detected in sam- ples with relatively large Msat RT with tPt¼0.2–0.5 nm; however, we could not detect any voltage signal for sampleswith tPt¼0.1 nm, 0.6 nm, and 0.8 nm due to the small Msat RT. As an example, Fig. 6(d) shows the measured voltage for Pt(1)/[FeMn(0.6)/Pt(0.4)] 10as a function of external magnetic ﬁeld at ﬁxed frequency of 3 GHz. As can be seen, the peak contains both symmetric and antisymmetric compo- nents with respect to the resonance ﬁeld and its polaritychanges when the ﬁeld reverses. Although the transversevoltage can be readily detected under FMR, the analysis of the signal is not straightforward because, in addition to ISHE, it also contains contributions due to non-ISHE relatedeffects such as spin rectiﬁcation effect (SRE) and anomalousNernst effect (ANE). The ANE is caused by the temperature gradient due to microwave heating, and as reported in sev- eral studies, is generally smaller than the SRE effect. 28,29 The SRE signal contains both anisotropic magnetoresistance (AMR) and anomalous Hall effect (AHE) contributions and exhibits complex symmetry and sign dependence on the applied external ﬁeld, H. Based on previous FMR studies in different measurement geometries,30,31there are mainly three contributions to the measured voltage signal in the pre-sent case: (i) symmetric component due to the ISHE, (ii) symmetric component due to AHE, and (iii) antisymmetric component due to AMR. Based on this, we ﬁrst decomposethe obtained voltage signal into the symmetric and antisym-metric components. Figure 6(e) shows the symmetric and antisymmetric voltage components of the sample Pt(1)/ [FeMn(0.6)/Pt(0.4)] 10. In this speciﬁc case, the peak value of the symmetric component is around 0.97 lV. Based on its symmetry and polarity, the symmetric component should contain both ISHE and AHE contributions. As our experi- mental setup does not allow us to perform an accurate angle-dependent measurement, here we estimate the magnitude ofAHE signal using known parameters. Following Chen et al. , 32the Lorentzian contribution of AHE is approximately given by VAHE;L¼Irf;sDRAHEhmcosh0cosU 2a2HFMRþMs ðÞ; (10) where Irf;s¼Irf;0Rwg=Rswith Irf;0the magnitude of rfdriv- ing current and Rwg,Rsthe resistance of coplanar waveguide and sample, respectively, DRAHEis the anomalous Hall resis- tance, Msis the saturation magnetization, HFMRis the reso- nant magnetic ﬁeld, hmis the rfmagnetic ﬁeld along the x direction, h0is the angle between the direction of external magnetic ﬁeld and coplanar waveguide, and Uis the phase ofrfﬁeld with respect to rfdriving current. In the present case, Irf;s/C250:2360:03 mA (calculated from the microwave power assuming maximum delivery efﬁciency), RAHE /C251:0660:11X(from static measurement), Ms/C25262:462:8 emu/cm3,HFMR/C25548:769:2 Oe, hm/C2536:864:7 Oe (cal- culated from rfcurrent), h0/C250/C14, and a/C250:10660:01. Based on these parameters, we obtain VAHE;L/C25ð1:7960:8Þ /C210/C07V, which is around one order of magnitude smaller than the measured symmetric voltage component. Since the phase difference between the rfﬁeld and rfcurrent is unknown, we assume U¼0 in the calculation, which might have led to a slight overestimation of the AHE signal.Based on the discussion made earlier, we believe that the FIG. 6. (a) Measurement geometry of ISHE and FMR. (b) ISHE and (c) FMR spectra for Pt(1)/[FeMn(0.6)/Pt(0.5)] 50measured at 3.0 GHz. (d) Voltage signal as a function of positive (circle) and negative (square) mag- netic ﬁeld for Pt(1)/[FeMn(0.6)/Pt(0.4)] 10at 3 GHz. (e) Decomposition of measured voltage signal for Pt(1)/[FeMn(0.6)/Pt(0.4)] 10at 3 GHz into sym- metric and antisymmetric components. Symbols are raw data as shown in (d). Dashed dotted and dashed lines show the symmetric and antisymmetric components, respectively. The solid-line shows the combined ﬁtting results.223901-6 Luo et al. J. Appl. Phys. 121, 223901 (2017)symmetric component of the measured voltage signal is mainly from the ISHE. Before we end this section, it is worth pointing out that the above discussion based on asymmetry in upper and lower interfaces may not apply to multilayers with ultrathin FeMn and Pt as the interfaces are not well deﬁned. This poses a question as to whether the ISHE signal can still be detected in these kinds of samples. As we will discuss in the SMR section, we believe that in this case, we still can detect the ISHE due to extrinsic spin Hall and inverse spin Hall effect. E. SMR measurement Both FMR and ISHE measurements conﬁrm that spin current generation and absorption occur simultaneously in the multilayer. This is exactly the ingredient for generating both SOT and SMR, which themselves are complementary pro- cesses of each other.33To conﬁrm this, we have performed SMR measurements for the same batch of samples used for the ISHE measurements. Figure 7(a)shows the geometry of the SMR, or angle-dependent magnetoresistance (MR) meas- urements, which were carried out with an applied ﬁeld of 30 kOe rotating in the zy,zx, and xyplanes, respectively. All the multilayer samples exhibit clear SMR signal. As a typical example, Fig. 7(b) shows the angle-dependent MR of Pt(1)/ [FeMn(0.6)/Pt(0.3)] 10. From the angle-dependence, we can see that only AMR is observed when the ﬁeld is rotated in the zxplane, whereas the signal obtained in the zyplane is domi- nantly from SMR. When the ﬁeld is rotated in the xyplane, both AMR and SMR are detected. Recently, Manchon devel- oped a model for SMR in AFM/HM bilayer,34which applies to the collinear AFM with well-deﬁned Neel order n¼m* 1 /C0m* 2, where m* 1,m* 2are the unit vector of the two spin sublat- tices, respectively. According to this model, the SMR of AFM/HM bilayers is given byDR Rxx¼kNrN dNrNþdAFrAFh2 SH1/C0cosh/C01dN kN/C18/C192 /C2ckgk/C0c?g? ðÞ 1þckgktanh/C01dN kN/C18/C19 1þc?g?tanh/C01dN kN/C18/C19 ; (11) with gk;?¼1þðrk;?rAF k;?=kAF k;?ÞtanhðdAF=kAF k;?Þ, ck;? ¼ðkAF k;?rN=kNrAF k;?Þtanh/C01ðdAF=kAF k;?Þ,kAF k¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ DAF ksAF sfq , and kAF ?¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ DAF ?ð1=sAF sfþ1=sAF uÞq . Here, the subscript kð ? Þ refers to the conﬁguration when the spin polarization aligns parallel (transverse) to the Neel order parameter, hSHis the spin Hall angle, DAFis the electron diffusion coefﬁcient in the AFM, sAF sfis the conventional isotropic spin relaxation time, sAF uis the spin dephasing time that relaxes only the spin component that is transverse to the Neel order parameter, ris the interfacial resistivity, kN,rN(rAF), and dN(dAF)a r es p i n diffusion length, conductivity, and thickness of the HM (AFM) layer, respectively. As shown in Fig. 7(c),b yv a r y i n gt h eP t thickness systematically, we found that the thickness- dependence of SMR of FeMn(3)/Pt( tPt) bilayers can be ﬁtted well using the following parameters: kN¼1.0560.05 nm, kFeMn k¼4.160.1 nm, kFeMn ?¼1.7060.07 nm, hSH¼0.28 60.03, sFeMn sf¼(4.2560.25)/C210/C014s,sFeMn u¼(7.7560.25) /C210/C015s,rN¼4.0/C2106S/m, rFeMn k¼1.0/C2106S/m, and rFeMn ?¼1.5/C2106S/m. The inset of Fig. 7(c)shows the region with small Pt thickness in log-scale, together with the experi- mental SMR of Pt(1)/[FeMn(0.6)/Pt( tPt)]10multilayers. As can be seen, the experimental SMR values for multilayers are sig- niﬁcantly larger than the simulat ed results for bilayers, particu- larly at very small Pt thickne sses. The difference becomes smaller when the Pt thickness increases. This suggests that when Pt is thick, the multilayer can be considered as compris- ing of magnetically decoupled bilayers, and therefore, the SMR ratio should be the same for both types of samples. However, when tPtis very small, the multilayer behaves more like a “single phase” FM; this is the reason why the SMR is different from that of bilayers with small tPt. The observation of large SMR in the multilayers suggests that there is spin cur- rent generation/absorption process taking place inside the mul- tilayer, presumably due to either intrinsic (for samples withthick Pt) or extrinsic SHE/ISHE (for samples with ultrathin Pt) or the combination of both. This is also the reason why a large SOT was observed in these structures. To shed some light on the origin of SMR, particularly, in structures with ultrathin Pt layers, we have also fabricated and measured the SMR of co-sputtered samples. At the same nominal thickness and composition, the co-sputtered sample is more resistive than its multilayer counterpart, consistentwith its more disordered structure. Despite the structural dif- ference, SMR of similar magnitude as that of multilayers was also observed in the co-sputtered samples. Figure 8(a) shows the angle-dependent MR of a co-sputtered FeMn:Pt sample with an overall nominal thickness of t FeMn¼6n m FIG. 7. (a) Geometry of angle-dependent MR measurement. (b) Angle- dependent MR of Pt(1)/[FeMn(0.6)/Pt(0.3)] 10. (c) Data (squares) and ﬁtting (line) of SMR ratio as a function of tPtfor FeMn(3)/Pt( tPt) bilayers. Inset shows the calculated SMR (line) for FeMn(0.6)/Pt( tPt) bilayers at small Pt thickness as well as experimentally obtained SMR ratio (triangles) for Pt(1)/ [FeMn(0.6)/Pt( tPt)]10multilayers.223901-7 Luo et al. J. Appl. Phys. 121, 223901 (2017)and tPt¼3 nm (calculated from the deposition power and duration). Both the AMR and SMR components are present in the angle-dependent MR. In Figs. 8(b) and8(c), we show the normalized AMR and SMR curve for both the co-sputtered and Pt(1)/[FeMn(0.6)/Pt(0.3)] 10multilayer sample. The nominal thickness and composition are the same for the two samples, and both samples are capped with a 1 nm Pt.We have conﬁrmed that the SMR for Pt(1)/[FeMn(0.6)/Pt(0.3)] 10and [FeMn(0.6)/Pt(0.3)] 10is almost the same, and therefore, the 1 nm Pt capping layer is not responsible for the SMR observed in both cases. Although further studies arerequired to elucidate the SMR mechanism in both co- sputtered and multilayer samples with ultrathin layers, the observed SMR can be qualitatively explained using the drift-diffusion model by taking into account both the precession and dephasing of SHE-generated spin inside a single FM with large spin-orbit coupling. The dynamics of the SHE-generated spin accumulation ^Sis governed by the coupled equations 35 r2^S¼1 k2 u^S/C2^mþ1 k2 h^m/C2^S/C2^mðÞ þ1 k2 S^S; (12a) ^JS i¼/C0Dðr^SiþhSH^ei/C2rnÞ; (12b) where ^Sis the non-equilibrium spin density generated by SHE, ^mis direction of the local magnetization, ku,kh, and kSare spin precession, dephasing, and spin-ﬂip diffusionlength, respectively, ^JS iis the ith component of spin current with polarization in ^Sdirection, Dis the diffusion coefﬁ- cient, nis the charge density, hSHis the spin Hall angle, and ^eiis a unit vector. The angle-dependence of MR (or simply SMR) appears due to the additional electromotive force gen- erated by ^JS ivia ISHE. Eq. (12a) can be better understood by considering the special cases: (i) ku,kh/C29kS, (ii) kh/C29kS, ku, and (iii) kS/C29kh,ku. In case (i), the ﬁrst two terms at the right-hand-side of Eq. (12a) , which leads to the spin dif- fusion equation for a non-magnetic metal, can be ignored. In this case, there will be no SMR-like angle-dependent MR unless it is in contact with a ferromagnetic layer. In case (ii), the 2nd term, which leads to r2^S¼1 k2 u^S/C2^mþ1 k2 S^S, can be ignored. This is similar to the case of Hanle MR (HMR) in HM except that the spin precession in HMR is caused by an external ﬁeld,36whereas in the present case, it is caused by the exchange ﬁeld of the FM itself. In the last case, both spin precession and dephasing terms have to be taken into account on equal footing. To estimate the inﬂuence of thesetwo terms on the spin density, we consider two special cases which are related to the transverse and vertical MR, i.e .,(i) ^m¼ð0;1;0Þand (ii) ^m¼ð0;0;1Þ. In the thin ﬁlm geometry, we are mainly concerned about the spin accumulation on the top and bottom surfaces which have a spin polarization dom-inantly in the y-direction. In this case, when ^m¼ð0;1;0Þ, both the precession and dephasing terms can be ignored. Under this condition, spin accumulation occurs on both sur-faces, resulting in a diffusion spin current reﬂecting back to the sample. This will lead a smaller resistivity due to ISHE effect. On the other hand, when ^m¼ð0;0;1Þ, the dephasing and diffusion term can be combined, leading to r 2^S¼1 k2 u^S/C2^mþ1 k2^S, where1 k2¼1 k2 hþ1 k2 S. This expression is similar to the case of HMR except that the spin diffusion length is replaced by an equivalent diffusion length. We can letkS¼kFeMn k¼4:1 nm and kh¼ku¼kFeMn ?¼1:7n m , and then k¼1:57 nm. Since this equation is similar to the case of HMR, we can use the solution given in the supple- mentary material of V /C19elez et al.36to estimate the SMR-like resistance change due to the ﬁrst term, which is given by DR Rxx/C253h2 SH 4k ku/C18/C192k d; (13) where dis the sample thickness, DRis the change in longi- tudinal resistance and Rxxis the longitudinal resistance at zero ﬁeld. Using hSH¼0:1,d¼10 nm, ku¼1:7n m , a n d k¼1:57 nm, we obtain an MR ratioDR Rxx¼0:1%, which is on the same order of magnitude of SMR observed experi- mentally. Although the exact value depends on the parame-ters used, we believe it does explain the salient features of the MR response observed in both the co-sputter and multi- layer samples with ultrathin Pt and FeMn layer. However, when the Pt layer is sufﬁciently thick, the bilayer model seems to be more appropriate as manifested in the agree-ment between experiment and theoretical model shown in Fig.7.FIG. 8. (a) Angle-dependent MR of a co-sputtered sample; (b) AMR and (c) SMR of co-sputtered and multilayer samples with same nominal composi- tion and thicknesses.223901-8 Luo et al. J. Appl. Phys. 121, 223901 (2017)F. Discussion In this study, we investigated the static and dynamic properties of [FeMn/Pt] nmultilayers by the combined techni- ques of magnetometry, FMR, ISHE, and SMR, and found agood correlation in the results obtained by the different tech- niques. First, the FMR and ISHE signals can only be detected in samples with sufﬁciently large M sat room tem- perature, which typically happens in samples with a largerepetition period, and magnetic inhomogeneity due tothickness-sensitive T cvariation is well reﬂected in the broad peak appeared in the FMR and ISHE spectra. Second, theFMR peak positions correspond well with those of ISHE.Third, SMR with a magnitude comparable to that of FeMn/Pt bilayer was observed, supporting the presence of largeSOT. All these results in combination with the fact that the multilayer behaves like a 3D Heisenberg ferromagnet and exhibits a large SOT seem to suggest that there is a brokeninversion symmetry (BIS) inside the multilayers. The mostlikely origin of the BIS in the multilayer is the crystallineasymmetry of the FeMn/Pt and Pt/FeMn interface caused bythe different atomic size. According to Liu et al. , 27the atom radii of Pt and FeMn are 0.139 nm and 0.127 nm, respec-tively. When depositing Pt on fcc (111) textured FeMn layer,the crystal direction and atom packing will have to change inorder to accommodate the large Pt atoms as the (111) plane is already close-packed. On the other hand, the situation will be different when smaller Fe and Mn atoms are deposited onfcc (111) textured Pt layer. This will lead to local inversionasymmetry in the multilayer. A similar phenomenon has alsobeen reported for Co/Pt 37,38and Co/Pd39multilayers. This explains why a large SOT is generated when a charge currentis applied to the multilayer, as we demonstrated previously.However, the observation of SMR in co-sputtered sampleswith a magnitude comparable to the multilayer suggests theobserved phenomena can also be explained by the simulta- neous actions of extrinsic SHE and ISHE, particularly in multilayers with ultrathin FeMn and Pt. Further studies arerequired to evaluate the relative contribution of intrinsic andextrinsic SHE and ISHE in FeMn/Pt multilayers with differ-ent thickness combinations. IV. CONCLUSIONS The static and dynamic magnetic properties of FeMn/Pt multilayers have been studied using combined techniques ofmagnetometry, FMR, ISHE, and SMR. Despite the fact that FeMn is an AFM in the bulk phase, FeMn/Pt multilayers with ultrathin FeMn ( t FeMn<0.8 nm) and Pt ( tPt<1.0 nm) layers exhibit ferromagnetic properties with in-plane magneticanisotropy. The temperature dependence of saturation magne-tization can be ﬁtted well using a phenomenological modeldeveloped for 3D Heisenberg magnet by including a ﬁnitedistribution in T C. The latter is attributed to the high sensitiv- ity of magnetic properties to subtle changes in the individuallayer thicknesses. The ﬁnite distribution of T Ccorrelates well with the broad absorption peaks observed in the FMR spectra. A large damping parameter ( /C240.106) is derived from the frequency dependence of FMR linewidth, which is compara-ble to the values reported for Co/Pt multilayers. Clear ISHEsignals and SMR have been observed in all samples below the Curie temperature, which corroborate the strong SOTeffect observed previously. The latter is attributed to the crys-talline asymmetry between the top FeMn/Pt and bottom Pt/FeMn interfaces when the Pt layer is relatively thick.However, for samples with ultrathin Pt, extrinsic SHE/ISHEmay play a more important role in the phenomena observed. ACKNOWLEDGMENTS Y.W. would like to acknowledge the support by the Singapore National Research Foundation, Prime Minister’sOfﬁce, under its Competitive Research Programme (GrantNo. NRF-CRP10-2012-03), and the Ministry of Education,Singapore, under its Tier 2 Grant (Grant No. MOE2013-T2-2-096). Y.W. and J.C. are members of the SingaporeSpintronics Consortium (SG-SPIN). 1P. Carcia, J. Appl. Phys. 63, 5066 (1988). 2Y. Ochiai, S. Hashimoto, and K. Aso, IEEE Trans. Magn. 25,3 7 5 5 (1989). 3J.-H. Park, C. Park, T. Jeong, M. T. Moneck, N. T. Nufer, and J.-G. Zhu,J. Appl. Phys. 103, 07A917 (2008). 4H. Sato, S. Ikeda, S. Fukami, H. Honjo, S. Ishikawa, M. Yamanouchi, K. Mizunuma, F. Matsukura, and H. Ohno, Jpn. J. Appl. Phys., Part 1 53, 04EM02 (2014). 5Y. Liu, C. Jin, Y. Fu, J. Teng, M. Li, Z. Liu, and G. Yu, J. Phys. D: Appl. Phys. 41, 205006 (2008). 6Y. Yang, Y. Xu, X. Zhang, Y. Wang, S. Zhang, R.-W. Li, M. S. Mirshekarloo, K. Yao, and Y. Wu, Phys. Rev. B 93, 094402 (2016). 7Y. Xu, Y. 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1.3640230.pdf	Dynamic and temperature effects in microwave assisted switching: Evidence of chaotic macrospin dynamics Dorin Cimpoesu and Alexandru Stancu Citation: Applied Physics Letters 99, 132503 (2011); doi: 10.1063/1.3640230 View online: http://dx.doi.org/10.1063/1.3640230 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/99/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electrical switching dynamics and broadband microwave characteristics of VO2 radio frequency devices J. Appl. Phys. 113, 184501 (2013); 10.1063/1.4803688 Microwave switching of graphene field effect transistor at and far from the Dirac point Appl. Phys. Lett. 96, 103105 (2010); 10.1063/1.3358124 Effect of dipole interaction on microwave assisted magnetization switching J. Appl. Phys. 107, 033904 (2010); 10.1063/1.3298929 Media damping constant and performance characteristics in microwave assisted magnetic recording with circular ac field J. Appl. Phys. 105, 07B902 (2009); 10.1063/1.3067839 Frequency modulation effect on microwave assisted magnetization switching Appl. Phys. Lett. 93, 142501 (2008); 10.1063/1.2996573 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.193.164.203 On: Mon, 22 Dec 2014 16:03:23Dynamic and temperature effects in microwave assisted switching: Evidence of chaotic macrospin dynamics Dorin Cimpoesua)and Alexandru Stancu Department of Physics, Alexandru Ioan Cuza University of Iasi, Iasi 700506, Romania (Received 9 June 2011; accepted 28 August 2011; published online 26 September 2011) Microwave assisted switching (MAS) is a method that can be used in magnetic recording in order to reduce the writing ﬁeld. In order to have a robust method, the factors inﬂuencing MAS have tobe systematically analyzed. In this paper we use the stochastic Landau-Lifsitz-Gilbert simulations to examine MAS in terms of microwave amplitude and frequency, damping, and the parameters describing the pulse ﬁeld. Also, we discuss about the troubling aspect of numerical induced chaos. VC2011 American Institute of Physics . [doi: 10.1063/1.3640230 ] The recording industry requires technologies able to simultaneously increase the recording density and the speed ofread/write processes. In order to thermally stabilize the mag- netic moments of small ferromagnetic particles, the anisotropy should be rather high, i.e., the writing magnetic ﬁeld requiredincreases as well. A recently proposed method of decreasing the write ﬁeld is the microwave assisted switching (MAS), 1 which consists in applying a microwave with a radio fre- quency magnetic ﬁeld perpendicular to the magnetization easy axis (EA) simultaneously with a pulse ﬁeld along EA. Due to the ac ﬁeld, the magnetization precession will increase its pre-cessional angle and the switching can take place within dura- tion of the pulsed reversing ﬁeld, which is smaller that what the switching ﬁeld would be in the absence of the ac ﬁeld. Theexperimental investigation reported in Ref. 2demonstrated the efﬁciency of this method. Subsequently, the method has been studied both for isolated (e.g., Refs. 3–5) and coupled thin ﬁlms. 6,7The microwave induced magnetization reversal is fundamentally different from that of a static ﬁeld, because a static ﬁeld is not an energy source while a microwave can be.8 Most often, the magnetization dynamics is described by Landau-Lifshitz-Gilbert (LLG) equation.9This imply a highly nonlinear dynamic system that may exhibit very complex fea-tures, like nonlinear resonance and chaos. 10In Ref. 11,t h e expression “strong chaos” is used to express the abundance of phenomena in magnetic resonance. Only under simplifyingconditions the problem has exact solutions. For uniaxial ani- sotropy and circularly polarized microwave ﬁeld, chaos is pre- cluded and the symmetry permits to derive all admissibledynamical regimes. 10Nonlinear resonance and foldover insta- bilities can be used to achieve magnetization switching below Stoner-Wohlfarth (SW) limit. For elliptically or linearlypolarized microwave ﬁelds, the LLG exhibits different forms of instabilities and chaotic behaviors, i.e., it has a complex and irregular pattern and a sensitivity to initial conditions.“Chaos” is a tricky thing to deﬁne, and even textbooks devoted to chaos do not really deﬁne the term. In fact, it is much easier to list properties that a system described as“chaotic” has, rather than to give a precise deﬁnition of chaos. A simple way of describing chaos is by sensitive dependence on initial conditions and by evolution that appears to be quiterandom. The chaos driven by the ac ﬁeld is fundamentallydifferent from the chaos due to spatially nonuniform magnet- ization dynamics, which is not an intrinsic property of theac ﬁeld driven oscillator. All these statements are for zero Kelvin approximation. Regarding the effect of microwave polarization on switching, linear polarization seems to bemore efﬁcient than circular polarization. 12In Ref. 13, the reducing the coercivity is related to the onset of chaotic mag- netization dynamics. Okamoto et al.14have shown that depending on the frequency and amplitude of the ac ﬁeld MAS can be stable or unstable. For the unstable switching, the switching ﬁeld is large and its distribution is very broad.Ref. 15explores thermal effects in MAS based on optimal re- versal path and logarithmic susceptibility concepts. Numeri- cal integration of LLG in this case can be extremely sensitiveto the integration time step, and even to the numerical algo- rithms used, as a consequence of repeated ampliﬁcation of truncation and rounding errors. Instabilities and chaos can beeasily excited by very small perturbations, on the order of mentioned errors, giving rise to numerical chaos. 16For exam- ple, in Ref. 17is reported that “The magnetization dynamics and the magnitude of the DC bias ﬁeld required to switch the media depends on the DC ﬁeld rise time and time step used in the computation.” These are not regarding programming mis-takes, but unavoidable errors in computing, and no computed chaotic solutions independent of integration time step can be found. On the other hand, a solution sensitive to initial condi-tions is not necessarily sensitive to time step. In this letter, we consider a single domain magnetic par- ticle, ellipsoid shaped with saturation magnetization M s ¼10.8/C2106/4pA/m (permalloy) and no intrinsic anisot- ropy. The ellipsoid’s principal axes are along x,y,and z: 100 nm, 50 nm, and 5 nm, leading to in-plane uniaxial shapeanisotropy ﬁeld l 0Hsh,1¼l0(Ny–Nx)Ms¼55.5 mT, and to Kittel resonance frequency with no applied ﬁeld f0¼6.36 GHz.18Magnetic ﬁelds presented throughout the paper are normalized by Hsh,1and ac ﬁeld frequency facbyf0. First, we have used LLG and we have considered two schemes for the applied ﬁelds [see Figs. 1(a) and1(b)]. In the ﬁrst scheme the pulse is applied after a time long enough to surpass the initial transient state characteristic to forced oscillations. However, as the amplitude of the ac ﬁeld is increased beyondthe threshold for linear excitation, outstanding strong nonlin- ear processes appear which can give rise to numerical insta- bilities, as seen in Figs. 1(c)and1(d). Time evolution shows a)Electronic mail: cdorin@uaic.ro. 0003-6951/2011/99(13)/132503/3/$30.00 VC2011 American Institute of Physics 99, 132503-1APPLIED PHYSICS LETTERS 99, 132503 (2011) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.193.164.203 On: Mon, 22 Dec 2014 16:03:23the divergence of the computed results for two time steps Dt. A systematic decrease of Dtdoes not lead to a convergent pattern, rather irregularly ﬂuctuating results are noted due to instabilities. Initially, the trajectories almost coincide (orbits around EA and out-of-plane orbits) and then they diverge.Similar behaviors were noted with different numerical algo- rithms and different error tolerance parameters. The jumps between the two type of orbits are completely random andviolate the LLG equation. This is an intrinsic stochastic pro- cess driven by a deterministic external perturbation. Irregular and unpredictable motion is not desired in practical applica-tions. The main goal of our letter is to determine the parame- ters of MAS method to achieve a fast and stable switching. Our starting point is the observation that computers intro-duce noise due to numerical errors just as the real system introduces noise in a physical experiment. And accordingly, we have to use a speciﬁc method for stochastic processes,namely we have to use the stochastic LLG (SLLG) equa- tion 19instead of its deterministic counterpart. In this way, the switching mechanism becomes statistical in nature andthe variable states are not described by unique values, but rather by probability distributions. We note that various models are used in the literature to account for thermaleffects (e.g., Refs. 20–23). We numerically integrate the SLLG equation using an implicit midpoint technique, for two values of temperature T, so that K sh,1V/kBT¼2500 and 75. A set of 1000 stochastic realizations has been performed. Thus there is no time step dependence of the statistical val- ues, and we can determine when a reliable switch takesplace. Starting from the negative saturation, for a given ac ﬁeld frequency f acand amplitude hac, there is a minimal bias ﬁeld amplitude, referred to as reversal ﬁeld, required for themagnetization xcomponent to reach the opposite level of nM s. In Figs. 2and3n¼1, but no signiﬁcant differences have been obtained for n¼0.95. For a smaller ﬁeld, either there are oscillations without reversal or the switch has a chaotic character. At low frequencies, the ac ﬁeld assists the switch like a dc ﬁeld (SW limit). At ac frequencies higherthan f0, the magnetization cannot follow the ac ﬁeld and switch can be expected for ﬁelds above SW ﬁeld. For inter- mediate frequencies, MAS at ﬁelds below SW ﬁeld can beobserved. Reduction in the reversal ﬁeld depends on the details of the bias ﬁeld (sweep rate t Hand pulse width TH). Generally, increasing the bias ﬁeld duration, and conse-quently increasing the microwave duration, the reversal ﬁeld decreases ﬁrst because the reversal occurs when the gained energy from the ac ﬁeld exceeds the energy barrier, but then FIG. 1. (Color online) Time dependence of the applied magnetic ﬁelds: a bias pulse reversing ﬁeld along x(easy) axis together with a continuous applied ac excitation along yaxis (a) and a pulsed ac excitation (b), respectively. Sinusoi- dal time dependence for the pulse rise/fall are assumed. The waiting time is long enough to reach the equilibrium. (c) and (d) Chaotic time evolution of magnetization when only a continuous ac excitation is applied, for a¼0.01, fac¼0.75 and hac¼0.2:f0Dt¼5/C210/C03(line) and f0Dt¼10/C03(dash). FIG. 2. (Color online) Contour plot of the reversal ﬁeld as function of ac ﬁeld frequency and amplitude, with a continuous applied ac excitation. In the hatched regions, the switch is driven only by the ac excitation. The levelcurves for 0.2, 0.4, 0.6, and 0.8 are also presented. In the white regions, referred to as nonswitching regions, no switches have been obtained for h pulse,max /C202. The vertical grayscale bar gives the required dc ﬁeld along EA to reverse the magnetic moment, if a dc ﬁeld with the value indicated in the vertical axis is applied along yaxis (SW limit).132503-2 D. Cimpoesu and A. Stancu Appl. Phys. Lett. 99, 132503 (2011) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.193.164.203 On: Mon, 22 Dec 2014 16:03:23approaches a lower limit, due to the equilibrium between the gained energy and the damping. For a continuous applied microwave excitation (Fig. 2), there is a region (hatched region) where the switch is driven only by the ac ﬁeld, and astable reversal cannot be obtained. This region is enlarged by the temperature and reduced by the increase of damping a. Fora¼0.01 and short pulses ( T H¼0 and 0.5 ns), the switch- ing diagrams have layerlike structures, and depend on the pulse lenght and the sweep rate (because both pulse rise/fall time and energy change rate depend on tH). Increasing tH generally increases the reversal ﬁeld. A more stable switch- ing is obtained by increasing a, [see Fig. 2(b)]. As tH increases, the reversal ﬁeld is decreasing for a¼0.05 because the out-of-plane orbits are hardly activated. Temper-ature increases the reversal ﬁeld mainly for short pulses and small damping. For longer pulses, the temperature affects mainly the reversal at high frequencies. For a pulsed micro- wave excitation (Fig. 3), switching and nonswitching areas alternate with increasing frequency for a¼0.01. As the tem- perature increases, the layer structure fades away and a unin- terrupted nonswitching region arises. These nonswitchingregions are similar with those obtained for a continuous microwave excitation, but are shifted toward higher frequen- cies. The nonswitching area can be reduced by turning off the microwave ﬁeld before the beginning of the descending part of h pulse. The reversal ﬁeld reduction through MAS is obtained at frequencies lower than f0due to the large angle gyration of magnetization (i.e., nonlinear response). In summary, we have shown that numerical integration of LLG equation, which most often is used to describe the magnetization dynamics in MAS method, can give rise to numerical chaos. We managed this problem using the sto-chastic LLG equation. We have determined the reversal ﬁeld, and we have identiﬁed switching and nonswitching regions in the switching diagrams. This study can offeranswers to different types of technological requirements involving an optimization, like the parameters describing the applied ﬁelds or the magnetic material. This work was supported by Romanian Grant PNII-RP3 No. 9/1.07.2009. 1N. D. Rizzo and B. N. Engel, U.S. patent 6,351,409 B1 (26 February 2002). 2C. Thirion, W. Wernsdorfer, and D. Mailly, Nature Mater. 2, 524 (2003). 3T. Moriyama, R. Cao, J. Q. Xiao, J. Lu, X. R. Wang, Q. Wen, and H. W. Zhang, Appl. Phys. Lett. 90, 152503 (2007). 4H. T. Nembach, H. Bauer, J. M. Shaw, M. L. Schneider, and T. J. Silva, Appl. Phys. Lett. 95, 062506 (2009). 5Z. Wang, K. Sun, W. Tong, M. Wu, M. Liu, and N. X. Sun, Phys. Rev. B 81, 064402 (2010). 6S. S. Cherepov, V. Korenivski, and D. C. Worledge, IEEE Trans. Magn. 46, 2112 (2010). 7S. S. Cherepov, B. C. Koop, Yu. I. Dzhezherya, D. C. Worledge, and V. Korenivski, Phys. Rev. Lett. 107, 077202 (2011). 8Z. Z. Sun and X. R. Wang, Phys. Rev. B 74, 132401 (2006). 9T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 10G. Bertotti, I. Mayergoyz, and C. Serpico, Nonlinear Magnetization Dy- namics in Nanosystems (Elsevier, Amsterdam, 2009). 11P. E. Wigen, Nonlinear Phenomena and Chaos in Magnetic Materials (World Scientiﬁc, Singapore, 1994), Chap. 6. 12S. Okamoto, N. Kikuchi, and O. Kitakami, Appl. Phys. Lett. 93, 102506 (2008). 13M. d’Aquino, C. Serpico, G. Bertotti, I. D. Mayergoyz, and R. Bonin,IEEE Trans. Magn. 45, 3950 (2009). 14S. Okamoto, M. Igarashi, N. Kikuchi, and O. Kitakami, J. Appl. Phys. 107, 123914 (2010). 15X. Wang and P. Ryan, J. Appl. Phys. 108, 083913 (2010). 16M. J. Ablowitz, C. Schober, and B. M. Herbst, Phys. Rev. Lett. 71, 2683 (1993). 17S. Batra and W. Scholz, IEEE Trans. Magn. 45, 889 (2009). 18C. Kittel, Phys. Rev. 73, 155 (1948). 19W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963). 20Z. Li and S. Zhang, Phys. Rev. B 69, 134416 (2004). 21N. Kazantseva, D. Hinzke, U. Nowak, R. W. Chantrell, U. Atxitia, and O. Chubykalo-Fesenko, Phys. Rev. B 77, 184428 (2008). 22X. K. Pu and B. L. Guo, Sci. China Math. 53, 3115 (2010). 23I. Mayergoyz, G. Bertotti, and C. Serpico, J. Appl. Phys. 109, 07D312 (2011). FIG. 3. (Color online) Similar with Fig. 2but with a pulsed ac excitation.132503-3 D. Cimpoesu and A. Stancu Appl. Phys. Lett. 99, 132503 (2011) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.193.164.203 On: Mon, 22 Dec 2014 16:03:23
1.3515928.pdf	Atomistic spin model simulation of magnetic reversal modes near the Curie point J. Barker, R. F. L. Evans, R. W. Chantrell, D. Hinzke, and U. Nowak Citation: Appl. Phys. Lett. 97, 192504 (2010); doi: 10.1063/1.3515928 View online: http://dx.doi.org/10.1063/1.3515928 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v97/i19 Published by the AIP Publishing LLC. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 06 Aug 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsAtomistic spin model simulation of magnetic reversal modes near the Curie point J. Barker,1,a/H20850R. F . L. Evans,1R. W. Chantrell,1D. Hinzke,2and U. Nowak2 1Department of Physics, The University of York, York YO10 5DD, United Kingdom 2Fachbereich Physik, Universität Konstanz, Universitätsstrasse 10, D-78464 Konstanz, Germany /H20849Received 23 September 2010; accepted 25 October 2010; published online 10 November 2010 /H20850 The so-called linear reversal mode is demonstrated in spin model simulations of the high anisotropy material L1 0FePt. Reversal of the magnetization is found to readily occur in the linear regime despite an energy barrier /H20849KV /kBT/H20850that would conventionally ensure stability on this timescale. The timescale for the reversal is also established with a comparison to the Landau–Lifshitz–Bloch equation showing good agreement. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3515928 /H20852 In order for the current increase in magnetic storage den- sity to continue, one must overcome the so-called magneticrecording trilemma; namely, that smaller grains are requiredfor higher data densities and to ensure their thermal stability,materials with a high anisotropy are required. The highercoercive ﬁeld that this produces also becomes a limiting fac-tor as the maximum ﬁeld produced by the recording head isconstrained by the saturation magnetization of the pole. Oneproposed solution to the trilemma is the use of heat assistedmagnetic recording /H20849HAMR /H20850, which utilizes the temperature dependence of the anisotropy to enable writing of materialswith a high coercivity. For the highest anisotropy media, thiswill require heating to the Curie temperature /H20849T C/H20850of the material. Close to TC, longitudinal ﬂuctuations in the magne- tization can have a signiﬁcant impact on the expected energybarriers and therefore the relaxation time of the magnetiza-tion. These effects become especially important when at-tempting to minimize the time to reverse the magnetizationstate of the media that will be important at higher storagedensities. Recently, the existence of a so-called linear reversal mode was predicted 1from the Landau–Lifshitz–Bloch /H20849LLB /H20850 equation.2During linear reversal, the magnetization does not coherently rotate, but instead linearly reduces along the easy axis, reappearing in the opposite sense in the same manner.This reversal mode is found in materials with a very highanisotropy and only occurs close to T C/H20849although at tempera- tures less than TC/H20850, where this reduction of magnetization becomes more energetically favorable than coherent rotation.Analytic work by Kazantseva et al.1with the LLB equation has suggested that linear reversal occurs on a much fastertimescale than coherent rotation. The linear reversal mechanism seems to be a contribu- tory factor in the optomagnetic reversal phenomenon ob-served by Stanciu et al. 3These experiments circularly used a polarized laser light to demonstrate magnetization reversalusing 100 fs pulses in the absence of an externally appliedﬁeld. This is a timescale that is much shorter than expectedfor magnetization reversal by precession, but one accessibleto the linear reversal mechanism according to the model ofKazantseva et al. 1Using a model based on the LLB equa- tion, Vahaplar et al.4showed that reversal on a subpicosec- ond timescale is possible via the linear reversal mechanismand experiments supported the predicted criticality of the onset of the linear reversal mechanism, which occurs at atemperature determined by the ratio of the longitudinal andtransverse susceptibilities. 1Thus linear reversal seems to be central to the optomagnetic reversal mechanism. Consequently, linear reversal is an important mecha- nism, justifying detailed investigation of its physical basis. Inthis letter, we use atomistic scale dynamic simulations todemonstrate the existence of linear and elliptical reversalmodes. We also show that reversal readily occurs whereconventionally, a Stoner–Wohlfarth type barrier /H20849KV /k BT/H20850 would ensure thermal stability on a long timescale. Finally, we make a direct comparison between the reversal times inthe atomistic spin simulation and the values calculated usingthe LLB equation. The model Hamiltonian uses the Heisenberg form of ex- change for moments well localized to atomic sites. It is im-portant in this work to have such microscopic detail so thattemperatures close to the Curie point and through the phasetransition can be reproduced in the model. In this paper wemodel the high anisotropy material L1 0ordered FePt. This material is known to have a very large uniaxial magneticanisotropy of K/H1101510 8erg /cc, making it a good candidate for next generation hard drive devices.5The model is parameter- ized with ab initio data for the exchange interaction and anisotropy as found by Mryasov et al.6The large anisotropy in L1 0FePt arises due to the two-ion exchange that exists between the alternating layers of Fe and Pt. Mryasov et al. showed that the moment induced in the Pt ions has a direc-tion and magnitude that is linearly dependent on the ex-change ﬁeld from the surrounding Fe. This allows the Ptspins to be combined onto the Fe lattice sites. The result is aHamiltonian that only contains Fe spins, S i, but has a long range exchange that is mediated by the Pt sites.6Equation /H208491/H20850 gives the Hamiltonian, where J˜ijis the effective exchange, di/H208490/H20850is the single ion anisotropy energy and dij/H208492/H20850is the two-ion anisotropy energy, His the applied ﬁeld, and /H9262˜=/H9262Fe+/H9262Pt H=−J˜ij/H20858 i/HS11005jSi·Sj−di/H208490/H20850/H20858 i/H20849Siz/H208502−dij/H208492/H20850/H20858 i/HS11005jSizSjz −/H9262˜H·/H20858 iSi. /H208491/H20850 Due to the dependence of the anisotropy on the ordering ofa/H20850Electronic mail: jb544@york.ac.uk.APPLIED PHYSICS LETTERS 97, 192504 /H208492010 /H20850 0003-6951/2010/97 /H2084919/H20850/192504/3/$30.00 © 2010 American Institute of Physics 97, 192504-1 Downloaded 06 Aug 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsthe FePt, perfect L1 0phase is assumed by the model, as is a 1:1 stoichiometry. The ab initio characterization in this model produces a ferromagnetic state with a Curie temperature of TC /H11015700 K for a bulk /H20849periodic /H20850system that is just slightly lower than the experimentally observed value of TC =750 K.7For granular /H20849open /H20850systems, the atomic sites are characterized using the same long ranged exchange andtwo-ion anisotropic energies as for the bulk, but the totalexchange on many sites is reduced due to the absence ofinteracting neighboring spin sites at the surface. This ap-proximation is used in the absence of detailed experimentalorab initio characterization of FePt surfaces. It is noted that because the large anisotropy depends on the long range two-ion anisotropy, this will also be reduced at the surface. The dynamics of each atomistic moment is described by the Landau–Lifshitz–Gilbert /H20849LLG /H20850equation, /H11509Si /H11509t=−/H9253 /H208491+/H92612/H20850/H9262sSi/H11003/H20853Hi/H20849t/H20850+/H9261/H20851Si/H11003Hi/H20849t/H20850/H20852/H20854. /H208492/H20850 The effective ﬁeld on each spin is given by Hi=−/H11509H//H11509Si +/H9256i, where /H9256iare three independent stochastic processes that satisfy the conditions /H20855/H9256i/H20856=0 ; /H208493/H20850 /H20855/H9256i/H20849t/H20850/H9256j/H20849t/H11032/H20850/H20856=/H9254ij/H9254/H20849t−t/H11032/H208502/H9261kBT/H9262s//H9253, where /H9251is the Gilbert damping parameter; kBis the Boltz- mann constant; and Tis the temperature of the thermal heat bath. In ultrafast laser experiments to which this work iscomparable, it would be the conduction electron heat baththat this represents. In such experiments, the phonon andelectron heat baths are in a highly nonequlibrium state, thusthe electron heat bath temperature can be considerably higherthan that of the phonons. 8,9/H9262sis the atomic moment that is 3.23 /H9262Bfor the localized Fe+Pt combined moments; /H9253is the gyromagnetic ratio given the value 1.76 /H110031011rad s−1T−1. The LLG equation was integrated using the Heun method with a time step of 0.5 fs. For the fastprocesses being investigated in this paper and the high tem-poral resolution used, the validity of white noise isquestionable. 10Recent attempts have been made to include colored noise into the Langevin equation of motion;11how- ever the approach of Atxitia et al. requires at least two un- known parameters, the heat bath correlation time and thebath coupling strength. Therefore we will use white noisethermal processes. A reversal path for the system was calculated by allow- ing the system to evolve at thermal equilibrium for a longperiod of time in zero ﬁeld. Comparing the mean values ofthe longitudinal magnetization, m zand transverse magnetiza- tion, mt, as the magnetization moves through the conﬁgura- tion space, the mean reversal path was obtained. This calcu-lation was performed for both “up” and “down” initialconﬁgurations so that a range of motion can be establishedfor systems that do not undergo thermal reversal within thetimescale of the simulations. The results in Fig. 1are from an ensemble of periodic systems with a total combined integration time of 2 ns. It canbe seen that as the system approaches T C, the ellipticity of the mean reversal path increases. Very close to TC, the be- havior changes to a linear mode where mtremains very smallfor all values of mz. The small remanent mtis a ﬁnite size effect. A calculation of the Stoner–Wohlfarth type barrier/H9004E=KV /k BTgives /H9004E/H11015100 for 620 K and /H9004E/H1101530 for 670 K. For both of these values, reversal would be veryunlikely within the 2 ns of total simulation time, yet within atemperature change of 50 K the system goes from beingthermally stable to superparamagnetic, suggesting a dramaticreduction of the energy barrier associated with the onset ofthe linear reversal mode. The reversal time is also signiﬁcant, as it governs the fundamental speed of magnetic phenomena. We have there- fore calculated the magnetization reversal time t 01for com- parison with the analytic solution of the LLB equation by Kazantseva et al.1for reversal times. The time t01is deﬁned as the time taken to for the system to change from a state ofm z=1 to mz=0 with a reversing ﬁeld applied along the easy axis. The results are the average of many simulations to es-tablish a mean reversal time. We now compare the atomisticresults with analytic LLB calculations. We note that inreference 1the LLB equation has been parameterized for a system size of 6 nm. To allow a direct comparison with theanalytical results we have carried out atomistic calculationsfor a system size of 6 nm to ensure that the susceptibilitiesand equilibrium magnetization had the same temperaturevariation as those used in the LLB model. Due to ﬁnite sizeeffects, which are especially pronounced in FePt due to thelong range nature of the exchange, this system has a smallerT Cof 600 K.12 Figure 2shows a very good agreement between the ato- mistic simulation results and the analytic solution of the LLBequation. 1Importantly, the reversal time changes by an order of magnitude across TCin the linear reversal regime. This effect is signiﬁcant for HAMR, as it shows that reversal ispossible in the magnetically hardest materials, but heating close to T Cwill be necessary. To illustrate this effect more explicitly, we have simu- lated the reversal probability o fa6n m grain under a 10 ps heat pulse and 1 T ﬁeld, as shown in Fig. 3. The reversal probability is zero at low temperatures, consistent with the710 K670 K620 K mtmz 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.100.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 FIG. 1. /H20849Color online /H20850Mean reversal path of a periodic system /H20849TC /H11015700 K /H20850showing the equilibrium magnetization vector for given tempera- tures. Dashed lines are guides representing a circular reversal path for eachtemperature.192504-2 Barker et al. Appl. Phys. Lett. 97, 192504 /H208492010 /H20850 Downloaded 06 Aug 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionslarge energy barriers noted earlier. At a critical temperature of/H11011640 K, the reversal probability increases rapidly. Note that this temperature is consistent with the estimate of/H11011642 K for the critical temperature T /H11569for the onset of linear reversal. T/H11569is determined1by the condition /H9273˜/H20648//H9273˜/H11036=1 /2, where /H9273˜/H20648,/H9273˜/H11036are the longitudinal and transverse suscepti- bilities respectively. We note that the maximum reversalprobability reached is less than unity. This reﬂects the ther-mal equilibrium probability, which is determined bytanh /H20849 /H9262H/kT/H20850where /H9262is the total spin moment of the nano- particle. The sharp transition at the critical temperature fa- vorably compares with Fig. 2and the results of Kazantseva et al. Importantly, the criticality of the reversal mechanism isdemonstrated in agreement with the experimental data of Va- haplar et al.4 In summary, we have demonstrated the so-called linear reversal mode within atomistic spin model simulations. Thishas shown that in the linear regime, reversal occurs within atimescale much shorter than the expected relaxation time forthe conventional Stoner–Wohlfarth barrier /H20849KV /k BT/H20850. The onset of this linear regime also appears to be very critical with thermal stability and reversal being separated by a rela-tively small change in temperature. The atomistic spin simulations performed here support the analytic solution of the LLB equation by Kazantsevaet al. with respect to reversal times. Again these results con- ﬁrm the apparent criticality of the onset of the linear reversalmode. Very close to T Cthe reversal time of the system changes by at least an order of magnitude. These results alsodemonstrate that for temperatures and ﬁelds achievable inthe nonequilibrium regime of ultrafast laser experiments, re-versal is possible on a subpicosecond timescale, which isconsistent with the optomagnetic reversal experiments ofStanciu et al. 3 J.B. is grateful to the EPSRC Contract No. EP/ P505178/1 for provision of a PhD studentship. D.H. ac-knowledges support by the Deutsche Forschungsgemein-schaft through Grant No. SFB 767. Financial support of theEU FP7 program /H20851Grant No. NMP3-SL-2008-214469 /H20849Ultra- Magnetron /H20850and Grant No. 214810 /H20849FANTOMAS /H20850/H20852is also gratefully acknowledged. 1N. Kazantseva, D. Hinzke, R. W. Chantrell, and U. Nowak, Europhys. Lett. 86, 27006 /H208492009 /H20850. 2D. A. Garanin, Phys. Rev. B 55, 3050 /H208491997 /H20850. 3C. D. Stanciu, F. Hansteen, A. V . Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and Th. Rasing, Phys. Rev. Lett. 99, 047601 /H208492007 /H20850. 4K. Vahaplar, A. M. Kalashnikova, A. V . Kimel, D. Hinzke, U. Nowak, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, and Th. Rasing, Phys. Rev. Lett. 103, 117201 /H208492009 /H20850. 5D. Weller, A. Moser, L. Folks, M. E. Best, W. Lee, M. F. Toney, M. Schwickert, J.-U. Thiele, and M. F. Doerner, IEEE Trans. Magn. 36,1 0 /H208492000 /H20850. 6O. N. Mryasov, U. Nowak, K. Y . Guslienko, and R. W. Chantrell, Euro- phys. Lett. 69, 805 /H208492005 /H20850. 7S. Okamoto, N. Kikuchi, O. Kitakami, T. Miyazaki, and Y . Shimada, Phys. Rev. B 66, 024413 /H208492002 /H20850. 8P. B. Allen, Phys. Rev. Lett. 59, 1460 /H208491987 /H20850. 9N. Kazantseva, U. Nowak, R. W. Chantrell, J. Hohlfeld, and A. Rebei, Europhys. Lett. 81, 27004 /H208492008 /H20850. 10W. F. Brown, Phys. Rev. 130, 1677 /H208491963 /H20850. 11U. Atxitia, O. Chubykalo-Fesenko, R. W. Chantrell, U. Nowak, and A. Rebei, Phys. Rev. Lett. 102, 057203 /H208492009 /H20850. 12U. Nowak, O. N. Mryasov, R. Wieser, K. Guslienko, and R. W. Chantrell, Phys. Rev. B 72, 172410 /H208492005 /H20850.10 T1T T[K]t1 0[ps] 1200 1100 1000 900 800 700 600100 10 1 0.1 FIG. 2. /H20849Color online /H20850A comparison of the characteristic reversal time t01as a function of temperature, through TCi na6n mc u b eo fF e P t /H20849TC=660 K for this small ﬁnite size system /H20850. Reversing ﬁelds /H20849an applied ﬁeld along the z-axis opposing the magnetization /H20850of 1 and 10 T are compared. Atomistic spin simulations are represented by points and the solid lines are the analyticsolution of the LLB equation /H20849see Ref. 1/H20850. /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc T[ K ]Reversa lProbability 740 720 700 680 660 640 6201 0.8 0.60.4 0.2 0 FIG. 3. /H20849Color online /H20850The reversal probability o fa6n mF e P tg rain in a 1 T reversing ﬁeld after the application of a 10 ps square heat pulse. Theshaded area represents reversal /H20849m z/H110210/H20850.192504-3 Barker et al. Appl. Phys. Lett. 97, 192504 /H208492010 /H20850 Downloaded 06 Aug 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions
1.1623491.pdf	A multilevel-based dynamic approach for subgrid-scale modeling in large-eddy simulation M. Terracola) Ofﬁce National d’Etudes et de Recherches Ae ´rospatiales, 29 av. de la Division Leclerc, BP 72, 92322 Cha ˆtillon cedex, France P. Sagaut Ofﬁce National d’Etudes et de Recherches Ae ´rospatiales, 29 av. de la Division Leclerc, BP 72, 92322 Cha ˆtillon cedex, France and Laboratoire de Mode ´lisation en Me ´canique, Universite ´Pierre et Marie Curie, 4 place Jussieu, BP 162, 75252 Paris cedex 5, France ~Received 27 November 2002; accepted 9 September 2003; published 21 October 2003 ! In this paper we present a new dynamic methodology to compute the value of the numerical coefﬁcient present in numbers of subgrid models, by mean of a multilevel approach. It is based onthe assumption of a power law for the spectral density of kinetic energy in the range of the highestresolved wave numbers. It is shown that this assumption also allows us to deﬁne an equivalent lawfor the subgrid dissipation, and to obtain a reliable estimation for it through the introduction of athree-level ﬂow decomposition. The model coefﬁcient is then simply tuned dynamically during thesimulation to ensure the proper amount of subgrid dissipation. This new dynamic procedure hasbeen assessed here in inviscid homogeneous isotropic turbulence and plane channel ﬂowsimulations ~with skin-friction Reynolds numbers up to 2000 !.©2003 American Institute of Physics. @DOI: 10.1063/1.1623491 # I. INTRODUCTION In large-eddy simulation, only the largest scales of mo- tion are resolved. They are deﬁned through the use of a low-pass ﬁltering operator G, associated with a cut-off wave numberk c. However, the presence of unresolved ~subgrid ! scales must be taken into account by the use of a subgridmodel. Despite the considerable effort devoted to the devel-opment of subgrid closures ~see Ref. 1 for a review !, actual models are still restricted in practice to a limited range ofapplications. Indeed, the subgrid-viscosity models widelyused in actual simulations have been developed in the re-stricted framework of isotropic and homogeneous turbu-lence, and thus appear not able to account correctly for thepresence of inhomogeneous subgrid scales. Scale similarityand deconvolution closures appear well-suited to account forcomplex phenomena, since no particular form of the subgridterms is assumed. They exhibit a high degree of correlationwith the real subgrid terms in a priori tests, 2but appear generally underdissipative in practical simulations and re-quire an additive regularization. Some examples are the ad-dition of an eddy-viscosity term as in the mixed model pro-posed by Zang et al., 3or a relaxation term in the set of ﬁltered equations as in the deconvolution approach of Stolzet al. 4,5Domaradzki et al.6have also found it necessary to include a secondary regularization step in the original formof the velocity estimation model 7to account for high Rey- nolds numbers. It thus appears that actual models still sufferfrom a general unadapted level of subgrid dissipation. An alternative is the incorporation of a coefﬁcient in the model,which should ensure a good level of dissipation, as it wasproposed for scale-similarity models by Liu et al., 2Cook,8or Maurer and Fey,9for instance, or equivalently a modiﬁcation of the constant present in eddy-viscosity closures. Germanoet al. 10have introduced a way of modifying the coefﬁcient of a given subgrid model by the mean of a dynamic approach,originally applied to the Smagorinsky model. 11It has then been extended to the case of linear combination models byseveral authors. 3,12–14This approach relies on mathematical considerations, and more particularily on the Germano’sidentity. While this treatment leads to generally satisfactorybehavior, it can produce some numerical instabilities due tolarge variations of the coefﬁcient, or an antidissipative be-havior of the model. Thus, some stabilization techniquessuch as space averaging, clipping, or more complexapproaches 15–17have to be introduced. In the present study, a new dynamic procedure, based on physical considerations is proposed. It is based on the as-sumption of a power law for the spectral density of energy inthe highest resolved wave numbers. The use of a three-levelﬁeld decomposition allows us to estimate dynamically theexpected slope of the spectrum, which is not arbitrarily im-posed, thus allowing the method to account for disequilib-rium effects.The three-level decomposition then allows us toget an evaluation of the proper amount of subgrid dissipationand adapt the model in consequence. The organization of the paper is as follows: in the ﬁrst part, the general framework of a multilevel decomposition of a!Author to whom correspondence should be addressed. Telephone: ~133!1.46.73.42.89; fax: ~133!1.46.73.41.66; electronic mail: Marc.Terracol@onera.frPHYSICS OF FLUIDS VOLUME 15, NUMBER 12 DECEMBER 2003 3671 1070-6631/2003/15(12)/3671/12/$20.00 © 2003 American Institute of Physicsthe solution, developed in previous studies18–20is recalled since it is one of the bases of the present approach. Then, theset of ﬁltered governing equations and basic subgrid closuresare detailed.The third part of the paper is then devoted to thedescription of the multilevel dynamic procedure itself. Someapplications are then presented in the fourth section. Thecases that have been considered include ~i!homogeneous and isotropic turbulence in the inviscid limit which is themost representative case of the ability of the model to pro-duce the proper amount of subgrid dissipation; and ~ii!a plane channel ﬂow conﬁguration that allows us to assess thedynamic procedure in the case of more practical ﬂows withboundary conditions such as walls. Some rather high valuesof the skin-friction Reynolds numbers have been considered~up to Re t52000). Finally, some conclusions are drawn in the last part of the paper. II. MULTILEVEL DECOMPOSITION We ﬁrst recall the framework of a multilevel decompo- sition of any variable fof the ﬂow by the use of Ndifferent ﬁltering levels. Each level is deﬁned by mean of a family of low-pass ﬁlters $Gn%,nP@1,N#that are characterized by their cutoff length scales Dn, associated with the cutoff wave numbers kn5p/Dnin spectral space. The ﬁltering operation is then formally deﬁned as the convolution product with the ﬁlter kernel Gn: Gnf~x,t!5E VGn~x2j!f~j,t!dj, ~1! wherexPV,R3is the space coordinates vector, tPR1is time, and f:V3R1!Rrepresents any ﬂow variable. Hereafter, the case Dn11.Dnwill be considered, or, equivalently, kn11,kn. The ﬁltered variables at the ﬁnest level of resolution are deﬁned as f¯(1)5G1f. At levelnP@2,N#, the ﬁltered variables are then recur- sively deﬁned as f¯~n!5GnGn21ﬂG2G1f5G1n~f!, ~2! with, for any mP@1,n#:Gmn()5GnGn21ﬂGm11Gm (). That is to say that level 1 corresponds to the ﬁnest rep- resentation of the solution, while levels with increasing val-ues ofncorrespond to coarser and coarser representations. This multilevel formalism also allows us to introduce a multilevel decomposition of any ﬂow variable fas f5f¯~n!1( l51n21 dfl1f9, ~3! where f¯(n)5G1n(f),dfl5f¯(l)2f¯(l11), and f95f2f¯(1) 5df0. In the multilevel decomposition ~3!,f¯(n)corresponds to the resolved scales at the nth level of resolution. The details dflcorrespond to the scales resolved at the level l, which are unresolved at the level l11, and, ﬁnally, f9corresponds to the ﬁnest level unresolved scales. Figure 1 illustrates de-composition ~3!in spectral space, in the particular case of sharp cut-off primary ﬁlters Gn. Remark that for N51, the classical LES decomposition is recovered. In the compressible case, density-weighted ﬁltering is used. In that case, density-weighted ﬁltered variables at levelnare deﬁned as f˜~n!5G1n~rf! G1n~r!5Gnrf~n21! Gnr¯~n21!. ~4! III. GOVERNING EQUATIONS A. Filtered Navier–Stokes equations We consider the compressible Navier–Stokes equations under the following compact form: ]V ]t1N~V!50, ~5! whereV5(r,rUT,rE)T,U5(u1,u2,u3)T, and N~V!5S"~rU! "~rU^U!1p2"s "~rE1p!U2"~s:U!1"QD, ~6! wherepis the pressure, ris the density, Uis the velocity vector, and rEis the total energy. Classical expressions are used for the viscous stress tensor sand viscous heat ﬂux vectorQ, i.e., s522mSd, ~7! Q52kT, ~8! where the exponentddenotes the deviatoric part of a tensor, Tis temperature, and Sis the rate-of-strain tensor: S51 2"U1~"U!T. ~9! The temperature is linked to the pressure by the perfect gas state law, and Sutherland’s law is used to compute the vis-cosity mas a nonlinear function of T. Finally, the thermal conductibility coefﬁcient kis linked to viscosity through the FIG. 1. Multilevel decomposition ~sharp cut-off ﬁlters !.3672 Phys. Fluids, Vol. 15, No. 12, December 2003 M. Terracol and P. Sagautuse of a Prandtl number assumption ~Pr50.7 in this study !as k5Cpm/Pr, where Cpis the isopressure heat coefﬁcient. The ﬁltered equations at any level nP@1,N#are then simply obtained by applying the ﬁltering operator G1nto Eq. ~5!. Assuming classically the commutation of the ﬁltering operation with time derivative, the ﬁltered equations at levelnare ]Vˆ~n! ]t1N~Vˆ~n!!52T~n!~10! where T(n)is the subgrid term, deﬁned as T~n!5N~V!~n!2N~Vˆ~n!!. ~11! In these equations, Vˆ(n)has been substituted to V¯(n)because of density-weighted ﬁltering. Indeed, the ﬁltered variables arer¯(n),u˜i(n),p¯(n)from which the vector of resolved vari- ablesVˆ(n)5(r¯(n),r¯(n)u˜i(n),rEd(n))Tis computed, with rEd(n) the resolved energy at level n: rEd~n!5p¯~n! g2111 2r¯~n!~u˜i~n!!2. ~12! Commutation errors between space derivatives and ﬁlters are included in T(n). However, if the ﬁlters used commute with space derivatives, the only remaining term in T(n)comes from the ~nonlinear !convective term. Indeed, Vreman showed in a detailed study21that subgrid quantities resulting from the nonlinearity of the viscous terms are negligible infront of those coming from the convective terms. These classical hypothesis will be made in the following, thus leading at each ﬁltering level to the following expres-sion for the ~uncomputable !subgrid terms: T ~n!5S0 "t~n! "~t~n!:U˜~n!!1"q~n!D, ~13! with the following expressions for the subgrid stress tensor t(n)and subgrid heat ﬂux vector q(n)at leveln: t~n!5r¯~n!~U^Ug ~n!2U˜~n!^U˜~n!!, ~14! q~n!5r¯~n!Cv~UTg~n!2U˜~n!T˜~n!!, ~15! whereCvis the isovolume heat coefﬁcient. B. Subgrid closure To close the system of ﬁltered equations ~10!, a param- etrization is needed for the two subgrid terms t(n)andq(n). Several subgrid closures have been developed and can be found in litterature ~see Ref. 1 for a review !, ranging from simple eddy-viscosity closures to more recentdeconvolution-like ones ~Stolzet al. 4,5and Domaradzki et al.6,7!. In the following, a ‘‘generic’’ expression will be considered for the subgrid terms, depending on the resolvedquantities at level n, under the form t~n!5C3Mt~r¯~n!,U˜~n!,Dn!, ~16! q~n!5C3Mq~r¯~n!,T˜~n!,U˜~n!,Dn!. ~17!The parameter Caccounts for the fact that most of the sub- grid models include a numerical constant in their expression.This coefﬁcient is generally calibrated by considering theparticular case of an isotropic homogeneous turbulence, orby comparison with experiments. For instance the two mod-els considered in this study are the classical Smagorinsky 11 closure and the scale-similarity closure of Liu et al.,2which provide at level one the following expressions for Mt: ~i!Smagorinsky closure, Mt52r¯~1!~D1!2uS˜~1!uS˜~1!; ~18! ~ii!Liuet al.scale-similarity closure: Mt5G2r¯~1!U˜~1!^U˜~1!2r¯~2!U˜~2!^U˜~2!, ~19! where ﬁltering level two is used as a test level. IV. MULTILEVEL DYNAMIC PROCEDURE First, a power-law is assumed for the energy spectrum in the range of the highest resolved wave numbers, i.e., E~k!5E0ka. ~20! Such a scaling law was proposed by many authors to modify the original 25/3 scaling by Kolmogorov. It is worth noting that both E0andacan be Reynolds-number-dependent, as suggested by Barenblatt.22Most of these modiﬁcations can be recast as follows: E~k!5CKe2/3k25/3~kL!z, ~21! whereCKis the Kolmogorov constant, ethe mean viscous dissipation, La length scale, and za real parameter, leading toa5z25/3. Under this assumption, it can be shown that the expression of the mean subgrid dissipation at a given wavenumberk n, obeys also to a power-law, i.e., ^e~kn!&52^t~n!:S˜~n!&5e0kng, ~22! whereE0,e0are some functions of z~or equivalently a!and the brackets denote ensemble averaging. In the present study,only some averages over the entire computational domainwill be used. Considering an eddy-viscosity-type parametrization of the form t(n)522nsgs(kn)S˜(n)for the subgrid-stress tensor, the following expression is obtained for the mean subgriddissipation at the wave number k5k n: ^e~kn!&5^2nsgs~kn!S˜~n!:S˜~n!&, ~23! where nsgsis a subgrid viscosity, and S˜(n)is the resolved rate-of-strain tensor at level n. A simple dimensional analysis1gives an expression for ^nsgs(kn)&as a function of knand^e(kn)&: ^nsgs~kn!&5n0^e~kn!&1/3kn24/3, ~24! where n0is a constant. Under the assumption that ^2nsgs(kn)S˜(n):S˜(n)& .^2nsgs(kn)&^S˜(n):S˜(n)&, the following analytical expression is obtained for the mean subgrid dissipation at level n:3673 Phys. Fluids, Vol. 15, No. 12, December 2003 A multilevel-based dynamic approach^e~kn!&.2n0^e~kn!&1/3kn24/3E 0knk2E~k!dk 52E0n0 31a^e~kn!&1/3kn31a24/3, ~25! which ﬁnally yields to a power-law form for ^e(kn)&: ^e~kn!&5S2E0n0 31aD3/2 kng5S2CKe2/3Lzn0 31aD3/2 kng, with g53a15 253 2z. ~26! Remark that for an equilibrium turbulence, for which the subgrid dissipation is constant along the spectrum ~g5z50!, a classical Kolmogorov energy spectrum is recovered with a525/3. The multilevel formalism presented in Sec. II is then considered, with N53. The primary level is the one at which the computation is performed, and at which a subgrid closureis required to close the ﬁltered Navier–Stokes equations,while levels 2 and 3 are secondary ﬁltering levels that will beused for the dynamic procedure ~often referred to as ‘‘test’’ levels !. For the following developments, the ratio R n,n11 5kn/kn115Dn11/Dnis introduced. The mean subgrid dissipation at level n,^e(kn)&will be now referred to as e(n). From ~22!and~26!,w eg e t e~n! e~n11!5Rn,n11g. ~27! From this expression, the value of gis then simply given by g5log~e~2!/e~3!! log~R2,3!. ~28! As stated in Sec. III, a subgrid term appears in the ﬁltered momentum equations at each level n51, 2, 3, and more particularily the subgrid stress tensor t(n). The aim of the present approach is to propose a reliable closure for the sub-grid terms of the ﬁnest resolution level, i.e., t(1). As stated in Sec. IIIB3, a ‘‘generic’’ parametrization is adopted for thisterm, under the form t~1!5C3Mt~r¯~1!,U˜~1!,D¯~1!!, ~29! where the global parameter Cis introduced to ensure the proper amount of subgrid dissipation. For R1,2.2 andR2,3 .1, some reliable approximations of the subgrid stress ten- sors at level two and three can be obtained by the followingexpressions: t~2!5G2~r¯~1!U˜~1!^U˜~1!!2r¯~2!U˜~2!^U˜~2!, ~30! t~3!5G23~r¯~1!U˜~1!^U˜~1!!2r¯~3!U˜~3!^U˜~3!. ~31! Indeed, following Domaradzki et al.23and Kerr et al.,24the main part of the subgrid energy transfer at a given level isdue to local interactions with wave numbers lower than twicethe cut-off wave number. This property is widely used indeconvolution-like approaches, 4,5,7,25based on a reconstruc- tion of scales two times smaller than the resolved ones thatare then used to compute the subgrid terms. Thus, an estimation of the mean subgrid dissipations e(2) ande(3)can be obtained with the previous expressions for t(2)andt(3)by e~n!52^t~n!:S˜~n!&,n52,3. ~32! The corresponding value of gis then obtained by relation ~28!. Since t(1)5C3Mt, the subgrid dissipation at level one is given by e(1)5C3e8, where the quantity e8 52^Mt:S˜(1)&is computable, and the parameter Cremains to be evaluated. The correct amount of dissipation at levelone is given by setting n51 in relation ~27!: e~1!5R1,2ge~2!. The value of Cis then ﬁnally given by C5R1,2ge~2! e8. ~33! Remarks: ~i!ForR1,25R2,3, a simple relation is directly obtained: C5~e~2!/e8!3~e~2!/e~3!!. ~34! ~ii!In the particular case of a Kolmogorov spectrum, a525/3, and g5z50, the expression of Cbecomes simply C5e(2)/e8to ensure a constant dissipation along the spectrum. V. APPLICATIONS The numerical scheme used in this study is a second- order accurate nondissipative cell-centered ﬁnite-volumescheme. The skew-symmetric form of the convective ﬂuxeshas been retained to reduce the aliasing errors, 26coupled with a staggered formulation of the viscous ones. Time inte-gration is performed with a classical explicit low-storagethird-order accurate Runge–Kutta scheme, with a CFL num-ber value of 0.95 to neglect time-ﬁltering effects. The ability of the proposed method to estimate the proper level of subgrid dissipation is ﬁrst analyzed by con-sidering the simple case of an homogeneous and isotropicturbulence. Indeed, this case is one of the most representativeof the dissipative behavior of a subgrid model, since theenergy spectrum in k 25/3expected at sufﬁciently high Rey- nolds numbers is not well reproduced with over- or under-dissipative simulations. A. Homogeneous isotropic turbulence The case considered here deals with a fully turbulent homogeneous isotropic turbulence. All the simulations aredone in the limit of an inﬁnite Reynolds number such that theonly dissipation of energy is due to the subgrid model used.The computational domain is a cubic box of side 2 p, with periodic boundary conditions in the three space directions,and 64 3uniformly distributed meshpoints.3674 Phys. Fluids, Vol. 15, No. 12, December 2003 M. Terracol and P. SagautThe initial ﬂow is a random ﬁeld, the spectral energy distribution of which satisﬁes the following law: E~k,t50!;k4exp~22k2/k02!. ~35! The mode k0corresponding to the initial integral scale is set here tok052. The initial ﬁeld is solenoidal, while the turbu- lent Mach number Mt5urms/c, withcthe sound celerity is set to 0.2. These two characteristics ensure a quasi-incompressible ﬂow all along the simulations. They are car-ried out from t50t ot510, where tis the time, nondimen- sionalized by L 0/urms, withL0the initial integral scale, and urmsthe rms velocity. Despite its apparent simplicity, this case is critical in the sense that the results are very dependent on the mean subgriddissipation level provided by the subgrid model. Indeed, inthe absence of molecular viscosity, the establishment of ahigh level of turbulence at t.6 implies the use of a sufﬁ- ciently dissipative subgrid model to prevent any blow-up ofthe simulation due to energy accumulation at the cut-off. Itthus appears as a good ﬁrst test case to see the potentiality ofthe new dynamic procedure to give the proper amount ofsubgrid dissipation. The procedure has been applied here tothe determination of the coefﬁcient C sused in the classical Smagorinsky closure. The expression for t(1)is thus here given by Eq. ~29!, where the ‘‘generic’’term Mtis given by the Smagorinsky model @see Eq. ~18!#. We then propose to compute dynamically in time the value of the Smagorinskycoefﬁcient C s5C1/2by the use of the new multilevel dy- namic procedure. One simulation using the Smagorinsky closure, together with the dynamic determination of Csby the proposed mul- tilevel method, has thus been carried out ~case S-NEW !. The ﬁlters used here to deﬁne the different ﬁltering levels are thethree-point discrete ﬁlters proposed by Sagaut andGrohens, 27withR1,25R2,352, applied successively in the three space directions. These discrete ﬁlters are equivalent tothe second order to Gaussian ﬁlters. As a comparison, twoother simulations have been carried out: one with the stan-dard version of the Smagorinsky model, 11where the theoret- ical value of Cs50,18 has been retained ~case S-018 !, and one using the dynamic Smagorinsky model of Germanoet al. 10~case S-DYN !. It is to be noted that numerical simu- lations with no subgrid model blew up rapidly, and couldthus not be performed. This point highlights the strong effectof the subgrid dissipation in this test case. Figure 2 illustrates for each case considered here the temporal evolution of the coefﬁcient C s. For the two simulations using a dynamic coefﬁcient, one notes that its value grows during the transitional phase, toreach a quasiconstant value from t.5.7 to the end of the simulation. Indeed, during this phase, the ﬂow has reached afully turbulent self-similar state. The two mean values of C s given by the two dynamic methods during the self-similar phase differ slightly ~0.178 for the S-DYN case, and 0.188 for the S-NEW case !. The slightly higher value obtained in the S-NEW case remains in very good agreement with thetheoretical value of 0.18, however, not reaching the value of0.2 suggested by Deardorff. 28 Figure 3 presents the resolved kinetic energy spectra ob-tained at t510. One can note for each case a large inertial zone in perfect agreement with the theoretical k25/3slope. Figures 4 and 5 show, respectively, the temporal evolu- tions of kinetic energy and enstrophy V5^uÙUu&~spatial integration over the computational domain !. First, we note that the kinetic energy decay is faster during the transitionalphase for case S-018. This is generally attributed to a tooimportant value of C sduring this phase in which the ﬂow is the place of many anisotropic events. The too strong inten-sity of subgrid dissipation in this last case also results in adrop on the amplitude of the enstrophy peak, which alsoappears later. During the self-similar phase, all the simula-tions exhibit a kinetic energy decay in t 2b, with here b51.97, however greater than the decay rate of 1.38 given by the eddy-damped quasi-normal Markovian ~EDQNM ! theory or by spectral DNS.29 FIG. 2. Temporal evolution of the Smagorinsky coefﬁcient. : S-NEW; : S-018; : S-DYN. FIG. 3. Kinetic energy spectrum at t510. : S-NEW; : S-018; : S-DYN.3675 Phys. Fluids, Vol. 15, No. 12, December 2003 A multilevel-based dynamic approachFinally, one can note slightly different positions of the enstrophy peak in each case. It is obtained at t54.12 in case S-018,t53.93 in case S-DYN, and t53.78 in case S-NEW. This last position is the closest from the one predicted by theEDQNM theory t.5.9/V(0) 1/2, which is equal to 3.74 here. Globally, one can thus note a very good behavior of the simulation S-NEW, which gives results appreciably equiva-lent to those obtained in case S-DYN, generally cited as areference. As a ﬁrst result, it shows that the new dynamicprocedure is efﬁcient, and that, in particular, the subgrid dis-sipation estimations provided at levels n52 andn53 are sufﬁciently accurate to give a reliable estimation of theproper subgrid dissipation at level n51. Some additional tests have thus been carried out to as- sess the dynamic method in the context of more practicalﬂows such as wall-bounded ﬂows. Some channel ﬂow com- putations are reported in the next section, for several valuesof the Reynolds number. B. Plane channel ﬂow The dynamic multilevel procedure is applied here to the well-known plane channel ﬂow conﬁguration. The nominalMach number value is M50.5, and several values of the Reynolds number Re mbased on the channel width and the mean bulk values have been considered. This time, the scale-similarity model of Liu et al.has been retained as subgrid closure, combined with the pro-posed dynamic approach. The subgrid stress tensor is thusexpressed as C3M t, with Mt5G2r¯~1!U˜~1!^U˜~1!2r¯~2!U˜~2!^U˜~2!. ~36! This model, which shows a very high correlation with exact subgrid terms during a prioritests, is generally not able to give the correct amount of subgrid dissipation with C51. Thus, it is proposed here to combine the good structuralproperties of the scale similarity model with some goodproperties in terms of subgrid dissipation by means of theproposed dynamic approach. The different simulation casesare referred to as cases A, B, C, D, which correspond totargeted skin-friction Reynolds number values of 180, 590,1050, and 2000, respectively. The domain sizes are 2 p 34p/332 for case A, 2 p3p32 for case B, and 2.5 p 3p/232 for cases C and D, in the respective x~streamwise !, y~spanwise !, andz~wall–normal !directions. The character- istics of the computational grids used in each case are sum-marized in Table I. It should be noted that the grid resolu-tions are relatively coarse for the second-order accuratenumerical scheme used in this study. First of all, some simulations without a model ~LoDNS ! have been performed for each case. For casesAand B, simu-lations with a classical plane-averaged dynamic Smagorin-sky model 10~SDYN !have also been carried out. For the four cases A, B, C, and D, simulations using the new closure~NEW !have been performed. Again, the secondary ﬁltering levels are obtained through the use of the discrete three-pointﬁlter proposed in Ref. 27, with R 1,25R2,352. For all the computations using the new dynamic procedure, a simpleaverage of the coefﬁcient Cover the entire computational domain has been considered. Table II summarizes all thesimulations that have been done, and displays the skin-friction parameters obtained in each case, which agree wellwith the targeted values, except the ‘‘SDYN’’ computations,which tend to underestimate these parameters. This can beexplained by the purely dissipative behavior of the model inthese cases. FIG. 4. Kinetic energy decay. : S-NEW; : S-018; : S-DYN. FIG. 5. Temporal evolution of enstrophy. : S-NEW; : S-018; : S-DYN.TABLE I. Computational parameters. Case Rem Nx3Ny3Nz Dx1Dy1Dz1 A 5600 22 362364 51 12 1–10 B2 1 8 5 04 0 392364 92 20 1–30 C4 2 1 4 08 2 382364 100 20 1.25–50 D 85000 156 3156380 100 20 1.5–803676 Phys. Fluids, Vol. 15, No. 12, December 2003 M. Terracol and P. SagautFigures 6–9 present the mean streamwise velocity pro- ﬁles obtained in each case, in wall units. For the casesAandB, the results are compared to the DNS results of Moseret al. 30It is observed that the use of the new scale-similarity model improves the results in comparison with other simu-lations. All the simulations fail to give the correct slope ofthe velocity proﬁle in the logarithmic region, but this is aknown feature of LES performed with second-order accurateschemes ~see the numerical studies of Kravchenko and Moin 26and Shah and Ferziger,25for instance !. For the high Reynolds cases C and D, the agreement with the theoreticalwall law U 152.5logz115.5 is very satisfactory, even with the coarse grid resolution used here. Again, it is observedthat the simulations performed with the new dynamic proce-dure improve the results, in comparison with the simulationswithout a model. Figures 10–13 present, in wall units, the rms velocity ﬂuctuation proﬁles. For the cases A and B, it is seen that allthe LES performed tend to overestimate the peak value ofthe streamwise component, and to underestimate the valuesof the spanwise and wall–normal components. This is also aparticularity of second-order schemes. However, it is strikingthat the results obtained with the new dynamic scale-similarity closure are closer to DNS results than the otherones, in particular, for the amplitude and position of the FIG. 6. Mean streamwise velocity proﬁle for case A. : ‘‘LoDNS’’; : ‘‘SDYN’’. : ‘‘NEW’’; symbols: DNS, Moser et al. ~Ref. 30 !;: wall law. FIG. 7. Mean streamwise velocity proﬁle for case B. Same key as Fig. 6. FIG. 8. Mean streamwise velocity proﬁle for case C. Same key as Fig. 6. FIG. 9. Mean streamwise velocity proﬁle for case D. Same key as Fig. 6.TABLE II. Simulations parameters and skin-friction values. Case SGS Model Ret(%error) ut3102 A-LoDNS No model 178 ~21.1! 6.15 A-SDYN Dyn. Smag. 172 ~24.4! 5.90 A-NEW New model 183 ~11.6! 6.30 B-LoDNS No model 590 ~0.0! 5.16 B-SDYN Dyn. Smag. 570 ~23.3! 5.00 B-NEW New model 607 ~12.8! 5.32 C-LoDNS No model 1035 ~21.4! 4.67 C-NEW New model 1072 ~12.2! 4.85 D-LoDNS No model 1934 ~23.3! 4.35 D-NEW New model 1980 ~21.0! 4.453677 Phys. Fluids, Vol. 15, No. 12, December 2003 A multilevel-based dynamic approachstreamwise component peak. Cases C and D, despite the fact that they are performed using coarse grids, yield satisfactoryresults, which are comparable to those obtained by Domar-adzki and Loh 7at Re t.1000 with the subgrid-scale estima- tion model and a high-order pseudospectral numericalscheme. The simulations performed without model lead,however, to a strong overestimation of the peak values. Thiscan be explained by the lack of dissipation in the productionzone of the ﬂow. Table III presents, for each simulation performed with the new scale-similarity model, the mean computed values~time and domain averages !ofC, g, and a, given, respec- tively, by relations ~33!,~28!, and ~26!. It is observed that a 25/3 slope is not recovered, leading to a nonzero value of z. Another interesting point is that zis not constant, and exhib- its a Reynolds number dependence. In order to check Baren-blatt’s hypothesis 22of a dependence of the form z~Re!5z8/ln~Re!,z8is displayed as a function of the friction Reynolds number in Fig. 14. It is seen that the present simu-lations suggest that an asymptotic value z8.21.4 is valid for high Reynolds numbers. However, further investigations atvery high Reynolds numbers are required to conclude on thispoint. Several other authors 30,31have raised the issue of a Rey- nolds number dependence in channel ﬂow simulations. In-deed, the momentum equations, written in wall coordinatesdo not exhibit a Reynolds number dependence. 31However, as observed in both DNS calculations and experiments, theresults generally exhibit such a dependence for the range oflow and moderate Reynolds numbers that were studied. In-deed, the authors observe an increase of the level of turbulent FIG. 10. The rms velocity ﬂuctuations for caseA. : ‘‘LoDNS’’; : ‘‘SDYN’’. : ‘‘NEW’’; symbols: DNS, Moser et al. ~Ref. 30 !. FIG. 11. The rms velocity ﬂuctuations for case B. Same key as Fig. 10. FIG. 12. The rms velocity ﬂuctuations for case C. Same key as Fig. 10. FIG. 13. The rms velocity ﬂuctuations for case D. Same key as Fig. 10. TABLE III. Mean values of C,g, and a. Case C ga A-NEW 0.55 20.775 22.18 B-NEW 0.73 20.385 21.92 C-NEW 0.77 20.310 21.87 D-NEW 0.79 20.280 21.853678 Phys. Fluids, Vol. 15, No. 12, December 2003 M. Terracol and P. Sagautﬂuctuations as the Reynolds number increases, and thus raise the issue of some low-Reynolds number effects that wouldnaturally disappear when increasing the Reynolds number.Fischeret al. 31indicate that the Reynolds dependence close to the wall originates from the behavior of a sink term in thedissipation rate equation that is Reynolds number dependentin the limit of two-component two-dimensional turbulenceclose to the wall. The simulations performed in this study, up to a signiﬁ- cant value of the Reynolds number have thus also been usedto conﬁrm the observations from these authors, and, in par-ticular, the existence of a universal behavior of the ﬂowwhen increasing sufﬁciently the Reynolds number. Figure 15displays, in wall coordinates, the rms streamwise velocityﬂuctuations proﬁle obtained in each case with the proposedapproach. This ﬁgure clearly shows that the two ‘‘high-Reynolds’’ cases C and D lead to very similar results, thusclaiming for some universal properties of the turbulent ﬂowclose to the wall at high Reynolds numbers. Figure 16 shows, as in Ref. 31, the limiting value of U rms8/Uat the wall obtained in the present simulations, together with previousDNS and experimental results. It appears, as expected, thatthe results obtained in our high-Reynolds simulations lead to an asymptotic behavior. The asymptotic value of U rms8/Uat the wall is of about 0.435 in the limit of inﬁnite Reynoldsnumber, slightly higher than the one suggested by Fischeret al. 31This, however, conﬁrms the trend of a universal be- havior at high Reynolds numbers observed by Moser et al.30 and Fischer et al.31in their moderate Reynolds number chan- nel ﬂows analysis. Another point, which was investigated by Moser et al.30 in their channel ﬂow DNS is the behavior of the streamwise velocity proﬁle in the overlap region between inner and outerscalings in wall-bounded turbulence. As in Ref. 30, Fig. 17shows for each Reynolds case the values of the two coefﬁ- FIG. 14. The evolution of zwith the Reynolds number. FIG. 15. Streamwise rms velocity ﬂuctuations. : Case A-NEW; : Case B-NEW; : Case C-NEW; : Case D-NEW. FIG. 16. Limiting behavior of limz!0Urms8/Uat different Reynolds numbers. Experimental ~Ref. 30 !and DNS ~Refs. 33–40 !results. FIG. 17. Proﬁles of gandb.: Case A-NEW; : Case B-NEW; : Case C-NEW; : Case D-NEW.3679 Phys. Fluids, Vol. 15, No. 12, December 2003 A multilevel-based dynamic approachFIG. 18. Instantaneous 1-D energy spectra : Volume average; :z1.12;:ka slope. FIG. 19. Mean subgrid dissipation proﬁles. :^e&; :^e1&; :^e2&.3680 Phys. Fluids, Vol. 15, No. 12, December 2003 M. Terracol and P. Sagautcients g5z1(dU1/dz1) and b5(z1/U1)(dU1/dz1). If the proﬁle of U1obeys a log law, gshould be a constant, while bshould be constant if U1obeys a power law. The tendencies observed on these curves are very similar to thosereported by Moser et al.for the two moderate Reynolds number cases A and B, which do not show any real plateauof gorbaway from the wall. For the two high-Reynolds cases C and D, it seems that greaches a nearly constant value, although slowly decreasing, but with a less importantslope than b. This indicates that the ﬂow behavior is slightly more consistent with a log law than with a power law. Thevalues of 1/ gobtained in our simulations in the log layer are 0.415 for case C and 0.313 for case D. This last value islower than the theoretical one of 0.4, and can be attributed tothe coarse resolution used in this last case. Although somemore accurate simulations and/or some higher Reynoldsnumbers cases would be needed to conﬁrm these results, thegeneral trend for the streamwise velocity proﬁle observedhere is a log law. Figure 18 shows some instantaneous monodimensional streamwise energy spectra. Plane-averaged ( z 1.12) and ensemble- ~volume !-averaged spectra are plotted, and com- pared to the slope ka, where ais taken fromTable III.Avery good agreement is observed between the spectra and the‘‘analytical’’ a-slope, with the value of aobtained from Eqs. ~26!and~28!, for both spectra.An exception is the low Rey- nolds number case A. Indeed, in this case, only a very short a-slope is obtained because of viscous effects. Moreover, a is estimated at k25k1/2, with the hypothesis that E(k) obeys a law in kafor all the wave numbers greater than k2. This hypothesis appears valid for higher Reynolds numbers, but isnot true in case A associated with a rather low turbulencelevel. Finally, Fig. 19 displays the mean plane-averaged sub-grid dissipation ~ e!proﬁles obtained with the new scale- similarity model, together with its forward @e15max(e,0)# and backward @e25inf(e,0)#contributions. For all the com- putational cases, the proﬁles of eexhibit a strong maximum atz1512, which is very satisfactory.This ﬁgure also reveals that the model accounts for backscatter, which becomes moreand more important as the Reynolds number increases ~it represents up to 35% of the global dissipation for case D !, with a peak value at z 1.20 consistent with the observations of Horiuti.32 VI. CONCLUSIONS A new dynamic procedure has been proposed and as- sessed in the present paper. It is based on an estimation of thelevel of subgrid dissipation that must be provided by thesubgrid model, by means of the introduction of two additiveﬁltering levels of the solution. This procedure was ﬁrstproven to be very accurate and efﬁcient by numerical testsperformed in the case of an isotropic and homogeneous tur-bulence, in the inviscid limit, with results that are at leastcomparable to—and even better than—those obtained withthe classical dynamic Smagorinsky model. Then, the newdynamic procedure has been assessed when combined with ascale-similarity closure, in the case of wall-bounded ﬂows.The plane channel ﬂow simulations that have been per-formed show a rather high improvement of the quality of the results in comparison with simulations performed with theclassical dynamic Smagorinsky model or without model.This can be related to the use of a scale-similarity modelallowing to reproduce backscatter effects, while the dynamicprocedure allows us to provide the proper amount of meansubgrid dissipation. In these simulations, the robustness ofthe method has also been assessed by considering somerather high values of the skin-friction Reynolds number. 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1.3650706.pdf	Currentinduced coupled domain wall motions in a twonanowire system I. Purnama, M. Chandra Sekhar, S. Goolaup, and W. S. Lew Citation: Applied Physics Letters 99, 152501 (2011); doi: 10.1063/1.3650706 View online: http://dx.doi.org/10.1063/1.3650706 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/99/15?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Current-induced motion of a transverse magnetic domain wall in the presence of spin Hall effect Appl. Phys. Lett. 101, 022405 (2012); 10.1063/1.4733674 Current-induced domain wall motion in a multilayered nanowire for achieving high density bit J. Appl. Phys. 111, 07D314 (2012); 10.1063/1.3679760 Current-induced domain wall motion in permalloy nanowires with a rectangular cross-section J. Appl. Phys. 110, 093913 (2011); 10.1063/1.3658219 Field- and current-induced domain-wall motion in permalloy nanowires with magnetic soft spots Appl. Phys. Lett. 98, 202501 (2011); 10.1063/1.3590267 Spin-current-induced magnetization reversal in magnetic nanowires with constrictions J. Appl. Phys. 97, 10C705 (2005); 10.1063/1.1851434 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.193.242.21 On: Thu, 27 Nov 2014 10:55:20Current-induced coupled domain wall motions in a two-nanowire system I. Purnama,1M. Chandra Sekhar,1S. Goolaup,1,2and W. S. Lew1,a) 1School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore 2Department of Electrical and Electronic Engineering, University of Mauritius, Reduit, Mauritius (Received 23 May 2011; accepted 17 September 2011; published online 10 October 2011) In two closely spaced nanowires system, where domain walls exist in both of the nanowires, applying spin-polarized current to any of the nanowire will induce domain wall motions in the adjacent nanowire. The zero-current domain wall motion is accommodated by magnetostatic interaction between the domain walls. As the current density is increased, chirality ﬂipping isobserved in the adjacent nanowire where no current is applied. When current is applied to both nanowires, the coupled domain wall undergoes oscillatory motion. Coupling breaking is observed at a critical current density which varies in a non-linear manner with respect to the interwirespacing. VC2011 American Institute of Physics . [doi: 10.1063/1.3650706 ] There has been increasing interest to understand the motion of domain walls (DWs) driven by spin-polarized cur-rent, particularly for developing the next generation data storage 1and logic devices.2For data storage, magnetic domains inside a nanowire are used as the data bits, withDWs separating each of them. Spin-polarized current is then used to move the DWs along the nanowire. Under applied current, DWs will move along the same direction irrespec-tive to the DW type, contrary to the case where magnetic ﬁeld was used; 3the length of the magnetic domains is then expected to remain constant. The operating speed of the datastorage depends on how fast the DWs can be moved within the nanowires, while the density is determined by how close the nanowires can be placed to each other. Many effortshave been spent to understand the motion of DWs inside a single nanowire. For instance, it was found that adjusting the rise time of the applied pulse current will amplify the motionof a DW. 4,5It has also been shown that when the applied cur- rent density is higher than a critical value, a transverse DW undergoes a chirality ﬂipping,6,7the phenomenon is known as Walker breakdown. High data density design implies that the nanowires will be placed very close to each other. Mag- netostatic interaction between the DWs from adjacent nano-wires then becomes important. It has been shown that the interaction can act as a pinning mechanism. 8,9To overcome the pinning, external magnetic ﬁeld has to be applied to thesystem. However, no report has been made on how the mag- netostatic interaction will affect the motions of the DWs within the nanowires that are being driven by spin-polarizedcurrent. In this paper, by using micromagnetic simulation, we show how the magnetostatic interaction affects the motion of DWs in two nanowires system. The Walker break-down limit of such system is found to be shifted to higher current density. Applying current to both of the nanowires with each in different direction results in an oscillatorymotion of the two DWs. The interaction between the two DWs can then be modelled as two bodies with ﬁnite masses that are connected by a spring.We consider Ni 80Fe20nanowires with width of 100 nm and thickness of 10 nm. At these dimensions, transverse DWsare the only stable conﬁgurations. 10The distance between the two wires was set to 100 nm. The object oriented micromag- netic framework code (OOMMF) extended by incorporatingthe spin transfer torque term 11to the Landau Lifshitz Gilbert (LLG) equation for the DW motion was used. The materials parameters are chosen for permalloy. The damping coefﬁcient(a)i sﬁ x e dt o0 . 0 0 5a n dt h en o na d i a b a t i cc o n s t a n t bhas been chosen as 0.04 in our simulations. The unit cell size for all simulations was set to be 5 nm /C25n m/C25n m . We have studied the interaction between two types of transverse DWs: head-to-head (HH) and tail-to-tail (TT) in two adjacent nanowires. The system is relaxed at zero ﬁeldand zero current. The two DWs are attracted to each other via their stray magnetic ﬁeld, reaching an equilibrium posi- tion where the two DWs are aligned along each other asshown in the inset of Fig. 1(a). The interaction can be pic- tured as two magnetic charges with different polarities being attracted to each other. 12In this stable conﬁguration, the total energy of the system is minimized. Spin-polarized current is then applied to move the nano- wire with the TT DW. Our results show that as the TT DWmoves, the HH DW in the adjacent nanowire also moves in the same direction. Similar phenomenon is observed when spin-polarized current is applied only to the wire with theHH DW. Both cases reveal that coupling between the two DWs is strong enough to induce DW motion within nano- wires where spin-polarized current is not applied. The twoDWs system can be considered as a coupled domain wall system (CDWS). Shown in Fig. 1(a)is the displacement of the CDWS as a function of time for various current densities. For current densities J/C20J awhere Ja¼2.755/C21012A/m2, the CDWS moves with a constant speed along the nanowire. The magni-tudes of the speed are 326.96 m/s for J¼2.120/C210 12A/m2 and 407.82 m/s for J¼2.755/C21012A/m2. The speed of the CDWS is increasing linearly with respect to the current den-sity value. The DWs also retain their shapes as they propa- gate along the nanowire. Here in Fig. 1(a), we show the displacement of the CDWS as a function of time fora)Author to whom correspondence should be addressed. Electronic mail: wensiang@ntu.edu.sg. 0003-6951/2011/99(15)/152501/3/$30.00 VC2011 American Institute of Physics 99, 152501-1APPLIED PHYSICS LETTERS 99, 152501 (2011) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.193.242.21 On: Thu, 27 Nov 2014 10:55:20J¼3.179/C21012A/m2andJ¼3.391/C21012A/m2. The aver- age speeds are 163.21 m/s and 118.73 m/s, respectively. Increasing the current density beyond Jaresults in a drastic drop of the average velocity. Thus, Jais the Walker break- down current density limit of the CDWS. It is higher than the Walker breakdown limit of a single nanowire which in our simulation was found to be Jb¼1.696/C21012A/m2. Shown in Fig. 1(b) is the transverse component of the mag- netization of a CDWS and a single DW as a function of time. The applied current density is J¼3.179/C21012A/m2, which is well above Walker breakdown current density limit for both cases. The maximas and the minimas of the graph represent the times where transverse DWs are observed.The increase and decrease of the magnetization along the y- direction represent the chirality ﬂipping of the DWs. Chiral- ity ﬂipping in CDWS is observed to occur in both of thenanowires, even though current is only applied to one of the nanowires. The timeframe where a DW retains its transverse shape in CDWS is found to be extended compared to the sin-gle nanowire case. To understand the characteristics of the coupling in the CDWS, spin polarized current is applied to the TT DW alongþxdirection and to the HH DW along –xdirection. The applied current causes the two DWs to move in the opposite direction, while the magnetostatic coupling tries to bring thetwo DWs together. The resultant motion of the DWs is due to the competition between the two forces. Shown in Fig. 2(a)is the separation between the two DWs along the horizontaldirection as a function of simulation time. The distance between the two DWs increases and decreases until it reaches a certain equilibrium position. At any time, the velocities ofthe two DWs are equal in magnitude but opposite in direc- tion. The ﬁnal separation ( x f) between the two DWs increases linearly with respect to the current density as shown inFig.2(b). According to the one-dimensional model, the force exerted by spin-polarized current ( Fs) on a DW is a linear function of current density;13xfincreases linearly as Fsis increased linearly. In equilibrium, the force from the spin-cur- rent is equal to the force from the coupling, thus both forces are linear functions of xf. The behaviour of the coupling force is similar to the behaviour of a spring. The CDWS can be modelled as two masses connected by a spring. The spring constant of the CDWS gives the information of the coupling strength and also can be used in determining the motion of the two DWs under various applied current density. To obtain the spring constant, we look at how theenergy of the system evolves. The total energy of the system is a sum of its demagnetization energy and exchange energy. The demagnetization energy represents how the stray mag-netic ﬁeld affects the magnetization while the exchange energy represents the shape of the DWs. In this case where current is applied to both of the nanowires, the two DWsretain their transverse shapes as the magnitude of the applied current density is below the Walker breakdown current FIG. 1. (Color online) (a) The displacement of the CDWS as a function of time for various current density values. Inset is the remanent state of theCDWS. (b) The normalized transverse component of the magnetization as a function of time for a CDWS and a single nanowire. The applied current density is J¼3.179/C210 12A/m2. FIG. 2. (Color online) (a) Separation between the two DWs as a function of simulation time. Inset shows the directions of the applied current on both nanowires. The magnitude of the applied current is equal at anytime. (b) Theﬁnal separation between the two DWs as a function of current density. (c) The demagnetization energy of the system as a function of the ﬁnal separa- tion between the DWs in the equilibrium states. FIG. 3. (Color online) (a) The period of the oscillation and the spring con-stant of the CDWS as a function of interwire spacing. (b) The mass of the DWs in CDWS as a function of interwire spacing.152501-2 Purnama et al. Appl. Phys. Lett. 99, 152501 (2011) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.193.242.21 On: Thu, 27 Nov 2014 10:55:20density limit. The exchange energy of the system thus remains constant; the evolution of the total energy of the sys- tem comes mainly from the evolution of the demagnetization energy. Fitting the demagnetization energy as a function ofthe equilibrium positions to a quadratic function will give us the spring constant of the system as shown in Fig. 2(c). Shown in Fig. 3(a)are the calculated spring constant and the oscillation period of the CDWS as a function of the interwire spacing. The spring constant is found to be decreasing as the distance between the nanowires is increased. This shows thatthe coupling between the two DWs is weaker for higher interwire spacing. The oscillation period increases as the dis- tance is increased. The mass of the coupled domain wall canbe found by using m¼ kT2 4p2. The mass shown in Fig. 3(b)is of the order 10/C024kg which is in a good agreement with the values reported before.14,15 Coupling between the two DWs is not observed when the current density is increased beyond a certain critical value. The coupling is broken and the two DWs move irre-spective of each other. Shown in Fig. 4(a)is the critical cur- rent density as a function of interwire spacing. The critical current density decreases in a non linear manner with respectto the distance between the wires, which shows that the cou- pling is weaker for larger interwire spacing. To understand the coupling breaking process, we con- sider the change in the internal structure of the DWs. When a transverse DW is driven into motion, the internal structure of the DW changes; part of the magnetization of the DW willstart to point to the z axis. The direction that the magnetiza- tion faces, whether it is in the þz or the –z direction, is deter- mined by the chirality of the DW and the direction of themotion. Fig. 4(b) shows the normalized values of the mag- netization of the two DWs along the z axis. Here, we can see that for current below the critical value, the magnetostaticinteraction induces a periodic change in the out-of-plane magnetization component of the DW. Beyond t /C251.5 ns,where the two DWs start to move closer to each other again, the out-of-plane component of the magnetization now points to theþz direction. However, for current above the critical value, the magnetization of the system after t /C251.5 ns keeps on building up to the –z direction. The different behavior of the system below and above the critical current density can, therefore, be explained as the two DWs being unable toreverse the direction of their out-of-plane magnetization component when current above the critical value is applied. The non-linear change of the magnetization in the early stage of the simulation (t <1.5 ns) is due to the non-linearity of the stray magnetic ﬁeld. In conclusion, we have shown how current-driven DW motion is affected when the DW is coupled to adjacent DWs of opposite polarity. The coupled DW within the adjacentnanowire is induced to move in the same direction as the cur- rent-driven DW. In the CDWS, the Walker breakdown is shifted to higher current density limit. It is interesting to seethat the chirality ﬂipping is observed on both nanowires, even though spin-polarized current is only applied to one of the nanowires. Coupling two DWs or more can also be an al-ternative method to move DWs with only applying spin- polarized current to speciﬁc wires. When current is applied to both nanowires in opposite direction, the two DWsundergo a damped oscillation motion, revealing the spring- like nature of the magnetostatic coupling. Increasing the cur- rent density in this manner results in the breaking of themagnetostatic coupling, the critical current density varies with the interwire spacing in a non-linear manner. This work was supported in part by the ASTAR SERC grant (082 101 0015) and the NRF-CRP program (Multifunc-tional Spintronic Materials and Devices). 1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 2D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn, Science 309, 1688 (2005). 3A. Kunz, Appl. Phys. Lett. 94, 132502 (2009). 4H. H. Langner, L. Bocklage, B. Kruger, T. Matsuyama, and G. Meier, Appl. Phys. Lett. 97, 242503 (2010). 5L. Bocklage, B. Kruger, T. Matsuyama, M. Bolte, U. Merkt, D. Pfann- kuche, and G. Meier, Phys. Rev. Lett. 103, 197204 (2009). 6A. Vanhaverbeke, A. Bischof, and R. Allenspach, Phys. Rev. Lett. 101, 107202 (2008). 7M. Hayashi, L. Thomas, C. Rettner, R. Moriya, and S. S. P. Parkin, Nat. Phys. 3, 21 (2007). 8T. J. Hayward, M. T. Bryan, P. W. Fry, P. M. Fundi, M. R. J. Gibbs, M.-Y. Im, P. Fischer, and D. A. Allwood, Appl. Phys. Lett. 96, 052502 (2010). 9M. D. Mascaro, C. Nam, and C. A. Ross, Appl. Phys. Lett. 96, 162501 (2010). 10Y. Nakatani, A. Thiaville, and J. Miltat, J. Magn. Magn. Mater. 290–291 , 750 (2005). 11See supplementary material at http://dx.doi.org/10.1063/1.3650706 for LLG parameter and code used in simulation. 12L. O’Brien, D. Petit, H. T. Zeng, E. R. Lewis, J. Sampaio, A. V. Jausovec,D. E. Read, and R. P. Cowburn, Phys. Rev. Lett. 103, 077206 (2009). 13A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69, 990 (2005). 14M. Jamali, K.-J. Lee, and H. Yang, Appl. Phys. Lett. 98, 092501 (2011). 15E. Saitoh, H. Miyajima, T. Yamaoka, and G. Tatara, Nature 432, 203 (2004). FIG. 4. (Color online) (a) The critical current density as a function of inter- wire spacing in CDWS. Inset is the direction of the applied current in bothof the nanowires. (b) The normalized values of the magnetization of the sys- tem pointing along the zaxis as a function of simulation time for various applied current density. The interwire spacing here is 100 nm with critical current J ¼1.908/C210 12A/m2.152501-3 Purnama et al. Appl. Phys. Lett. 99, 152501 (2011) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.193.242.21 On: Thu, 27 Nov 2014 10:55:20
1.1611871.pdf	Raman band shape analysis of a low temperature molten salt Ary O. Cavalcante and Mauro C. C. Ribeiro Citation: The Journal of Chemical Physics 119, 8567 (2003); doi: 10.1063/1.1611871 View online: http://dx.doi.org/10.1063/1.1611871 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/119/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Analysis and prediction of absorption band shapes, fluorescence band shapes, resonance Raman intensities, and excitation profiles using the time-dependent theory of electronic spectroscopy J. Chem. Phys. 127, 164319 (2007); 10.1063/1.2770706 Response to “Comment on ‘Ion association dynamics in aqueous solutions of sulfate salts as studied by Raman band shape analysis’” [J. Chem. Phys.124, 247101 (2006)] J. Chem. Phys. 124, 247102 (2006); 10.1063/1.2205862 Ion association dynamics in aqueous solutions of sulfate salts as studied by Raman band shape analysis J. Chem. Phys. 123, 034508 (2005); 10.1063/1.1931660 Inertial solvent dynamics and the analysis of spectral line shapes: Temperature-dependent absorption spectrum of β-carotene in nonpolar solvent J. Chem. Phys. 120, 4344 (2004); 10.1063/1.1644534 Molecular-dynamics simulations of solvent effects on the C–H stretching Raman bands of cyclohexane- d 11 in supercritical CO 2 and liquid solvents J. Chem. Phys. 110, 1687 (1999); 10.1063/1.477816 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Sun, 30 Nov 2014 08:51:25Raman band shape analysis of a low temperature molten salt Ary O. Cavalcante and Mauro C. C. Ribeiroa) Laborato´rio de Espectroscopia Molecular, Instituto de Quı ´mica, Universidade de Sa ˜o Paulo, C.P. 26077, 05513-970, Sa ˜o Paulo, SP, Brazil ~Received 2 June 2003; accepted 29 July 2003 ! The salt tetra ~n-butyl !ammonium croconate, @(n-C4H9)4N#2C5O54H2O,~TBCR !, is a very viscous glassforming liquid which undergoes a glass transition at room-temperature. Raman band shape analysis of the totally symmetric ring breathing mode of the croconate dianion, C 5O522, was performed by Fourier analysis. The vibrational time correlation functions obtained from theisotropic Raman spectra were modelled with well-known models for vibrational dephasing. Thetime correlation functions of pureTBCR and ofTBCR in acetonitrile solutions were compared withprevious results for the simple salt Li 2C5O5in aqueous solution. It has been found remarkable changes of the dynamic parameters characterizing the vibrational dephasing of C 5O522in these different environments. Discontinuous temperature dependence of the dephasing parameters wasobserved at the glass transition temperature of pure TBCR. In glassy TBCR, however, commonmodels for vibrational dephasing are not strictly valid because the Raman bands display clearasymmetric shapes. The experimental data in glassy TBCR were also reproduced with a model thatconsiders the second and the third order terms in the cumulant expansion of the vibrationalcorrelation function. © 2003 American Institute of Physics. @DOI: 10.1063/1.1611871 # I. INTRODUCTION Throughout the last four decades, the dynamics of vibra- tional and reorientational relaxation in liquids have been ex-tensively investigated by Fourier analysis of some appropri-ate bands in the Raman spectra. 1By Fourier transforming a chosen Raman band, in principle one can obtain the vibra-tional and the reorientational time correlation functions. Inthe particular case of ~high temperature !molten salts, most of these Raman spectroscopy investigations concerned alkali salts of simple anions, for instance, NO 32, SCN2,C O322.2 Due to the high symmetry of such simple anions, their Ra- man spectra have few nonoverlapping bands, what promptthem as good candidates for a detailed Raman band shapeanalysis. On the other hand, much more complex species arepresent in the ionic systems which are liquids at room-temperature, that is, the so-called ionic liquids. 3Typical cat- ions in ionic liquids include alkylammonium, imidazolium,and pyridinium derivatives, and typical anions include tet-raﬂuoroborate, hexaﬂuorophospate, triﬂuoroacetate, and bis-~triﬂuoromethylsulfonyl !imide. Due to their technological relevance, for instance, solvents for several chemical processor electrolytes for electrodeposition, an extensive literature on ionic liquids is now available and many different systemshave been synthesised. Although ionic liquids have beencharacterized by Raman spectroscopy, 4,5in the authors knowledge no previous Raman band shape analysis has beenundertaken in such complex systems in order to reveal theirmicroscopic dynamics. Of course, transport coefﬁcients ofionic liquids, such as viscosity and ionic conductivity, havebeen extensively investigated. However, the microscopicstructure and dynamics of ionic liquids are less understood, for which NMR spectroscopy has been one of the most im-portant experimental tool. 6,7Very recently, neutron scattering investigations8,9and molecular dynamics simulations10,11 have been undertaken in order to reveal the structure anddynamics of ionic liquids in a microscopic level. The croconate anion, C 5O522~see the inset in Fig. 1 !,i s one member of the oxocarbon dianions, which are planar species with general formula C nOn22(n53,4,5,6). Oxocar- bon ions are well known species in Organic Chemistrysynthesis. 12Recently,13it was shown that a hydrated salt with a low melting point based on the croconate dianion isobtained by replacing a simple alkali cation by a much morelarge cation, namely, tetra ~n-butyl !ammonium croconate ~TBCR !, @(n-C4H9)4N#2C5O54H2O. Pure TBCR is a pale yellow very viscous liquid at room temperature that hardlycrystallize. In fact, differential scanning calorimetry ofTBCR indicated a glass transition temperature at T g ’120.0°C.13TBCR has been characterized as a fragile glassforming liquid, that is, a glassforming liquid whose vis-cosity, h, at temperature close to Tgincreases in a steeper behavior than anArrhenius dependence.14@SiO2is the arche- typical strong glassformer, so that a linear dependence isobtained in an Arrhenius plot of the viscosity, log( h)3T21]. As one would expect in a system made of a tetraalkylammo- nium cation with long alkyl chains, the viscosity of TBCR isvery high, h’103cP at 30.0°C. For comparison purposes, ionic liquids with viscosity three magnitude orders smallerare obtained in the well-known mixtures 1-ethyl-3-methyl-imidazolium chloride/AlCl 3(Tgof such mixtures can be smaller than 290.0°C at some molar fractions of AlCl 3).3 We showed that the oxocarbon ions are good probes for a detailed Fourier analysis of the Raman bands in order toobtain the corresponding time correlation functions. Such ana!Author to whom correspondence should be addressed. Electronic mail: mccribei@quim.iq.usp.brJOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 16 22 OCTOBER 2003 8567 0021-9606/2003/119(16)/8567/10/$20.00 © 2003 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Sun, 30 Nov 2014 08:51:25investigation has been undertaken for the croconate and the squarate, C 4O422, species in Li 2C5O5and Li 2C4O4aqueous solutions.15It has been found similar bandwidths in both the polarized and the depolarized Raman spectra of the oxocar-bon ions, pointing to very hindered reorientation of thesespecies in water. The analysis of the vibrational time corre-lation functions of the oxocarbon ions in aqueous solutionindicated fast modulation of the forces experienced by theoscillators. The physical picture emerging from these Ramaninvestigations is that the oxocarbon anions perform oscilla-tory and librational motions in relatively long-lived cagesformed by neighboring water molecules. Fast modulation ofthe short-range forces experienced by the oxocarbon ion insuch a cage ensures the fast modulation regime of vibrationaldephasing, and simultaneously the slow reorientation of theoxocarbon ion as a whole. This picture of the microscopic dynamics has been re- cently corroborated by a molecular dynamics simulation of C 4O422in aqueous solution.16The computer simulations in- dicated that three water molecules are hydrogen bonded to each oxygen atom of the C 4O422anion, with a relatively long residence time ~more than 20.0 ps !. The ﬁrst hydration shell around C 4O422includes a whole of 18 water molecules, as additional three water molecules are founded above andother three below the oxocarbon plane. These additional sixwater molecules are not directly hydrogen bonded to the an-ion, and they are replaced by other molecules of the bulkmuch more frequently ~residence time ’2.0 ps !. Proper to the high quality of the Raman spectra of oxo- carbon ions, TBCR is very much appropriate for a Ramaninvestigation of the microscopic dynamics in a prototype ofan ionic system with low melting point. Low-frequency ~5– 200 cm 21!Raman spectra of TBCR across the glass- transition has been already reported.13As TBCR is cooled down to the glass-transition temperature, one observes asharp decrease of the relaxational contribution close to zerowave number, and an increase of the vibrational contribution~the so-called boson peak !at’20.0 cm 21. Whereas Raman spectra in such a low-frequency range probes directly theintermolecular dynamics of the liquid, this information is implicit in the Raman band shape of the high-frequency in-tramolecular modes. The purpose of the present paper is tofurther elucidate the microscopic dynamics in TBCR by aRaman band shape analysis of the intramolecular modes ofthe croconate anion. In addition, whereas the solubility ofsimple alkali oxocarbon salts is appreciable only in water,TBCR is also soluble in organic solvents. Interestingly, weare in position of comparing the microscopic dynamics of C 5O522in pure TBCR at different temperatures, TBCR in acetonitrile solutions, and the previous results15for Li2C5O5 aqueous solution. It will be shown that the parameters char-acterizing the vibrational dephasing of the ring-breathing mode of the C 5O522ion change by a signiﬁcant amount in these different environments. The paper is organized as follows: Sec. II presents ex- perimental details on the synthesis of TBCR, and on the dataacquisition and treatment. Section III presents the results anddiscussion in three subsections. An overview of the Ramanspectra of TBCR is given in Sec. IIIA, in which we dis-cussed the observed frequency shifts and the issue of slowreorientational relaxation of the croconate dianion. Vibra-tional dephasing of the croconate dianion in different envi-ronments at room temperature is discussed in Sec. IIIB. Theeffect of the glass transition on the vibrational dephasing ofthe croconate dianion in pure TBCR is discussed in Sec.IIIC. Concluding remarks are given in Sec. IV. II. EXPERIMENT The synthesis of TBCR has been reported previously.13 Brieﬂy, TBCR is obtained by the reaction in methanol solu-tion between @(n-C4H9)4N#Cl and the simple salt Ag 2C5O5, followed by separating the AgCl precipitate and evaporatingthe solution. Raman spectra have been recorded with aU-1000 Jobin-Yvon double monochromator spectrometer ﬁt-ted with a photomultiplier tube. The spectra were excitedwith the 647.1 nm line of a Kr 1laser ~Coherent model 400 !, with ’150 mW of output power. The spectral resolution, Dvsp, was kept at 1.0 cm21. This spectral resolution is ac- ceptable as it is rather small in comparison with the typicalfull width at half height ~FWHH !of the Raman bands ~see Table I !. In fact, we found that the band shapes suffered of no artifacts due to the effect of instrumental slit proﬁle byrecording several spectra with different spectral resolution.Spectra of pure TBCR were recorded from a high tempera-ture state ~320 K !down to the glassy state at 230 K, the temperature control being achieved by using the Optistat DN cryostat of Oxford Instruments. Spectra at low temperatureshave been obtained by stepwise cooling from room tempera-ture, and then followed by ’1 h period for thermal equili- bration at the target temperature. The more appropriate normal mode for a Raman band shape analysis is the totally symmetric ring breathing mode of the C 5O522anion, n2(a18), at ’625 cm21.15This band is free of overlapping bands at the high frequency side, al-though the low frequency side is overlapped by the ring bending mode, n11(e28), at ’550 cm21. In such a situation, one normally would assume that the band shape is symmetric FIG. 1. Polarized Raman spectra, IVV(v), of Li2C5O5aqueous solution ~thin line !, pure TBCR ~bold line !, and TBCR in a dilute acetonitrile solu- tion~dashed line !at room temperature. The inset shows the structure of the croconate dianion, C5O522.8568 J. Chem. Phys., Vol. 119, No. 16, 22 October 2003 A. O. Cavalcante and M. C. C. Ribeiro This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Sun, 30 Nov 2014 08:51:25and would use only the high frequency side for a quantitative analysis, that is, the high frequency side of the band wouldbe reﬂected on the low frequency side. However, it will beshown in the following that a clear asymmetric shape devel-ops on the n2band as TBCR is cooled down to the glassy state. Thus, we used instead an alternative procedure inwhich the n11and the n8bands ~see Fig. 1 !are ﬁtted by a mixed Gaussian–Lorentzian function and subtracted fromthe experimental spectrum in order to obtain the proﬁle ofthe ring breathing mode n2. The spectra were acquired each 0.2 cm21and recorded for typically vmax5100.0cm21from each side of the center of the n2band. Such a value of vmax is a satisfactory one as it implies that the time resolution ofthe resulting correlation function is Dt5(2c vmax)21 50.17ps,17wherecis the speed of the light. This is a rea- sonable resolution in light of the smooth decay of the presenttime correlation functions ~see Fig. 3 !. Furthermore, the spectral resolution of D vsp51.0cm21implies that the re- sulting correlation function would be reliable up to tmax 5(2cDvsp)21516.7ps,17which is larger than the typical re- laxation of the correlation functions. The spectra were recorded in the so-called VV–VH experiment,1in which one records both the polarized and the depolarized Raman scattering intensities, where the polariza-tion of the scattered light is either parallel or perpendicular tothe polarization of the incident light, respectively. Polariza-tion discrimination of the scattered light was achieved byusing a polarizer and a polarization scrambler placed beforethe entrance slit of the spectrometer. The polarized, I VV(v) ~vertical–vertical or i!, and the depolarized, IVH(v) ~vertical–horizontal or ’!, spectra were recorded in the usual 90° geometry, and the isotropic, Iiso(v), and the anisotropic, Ianiso(v), spectra were obtained,1,2 Iiso~v!5IVV~v!24/3IVH~v!, ~1! Ianiso~v!5IVH~v!. ~2! By Fourier transforming Iiso(v) andIaniso(v), one ob- tains the corresponding time correlation functions, Ciso(t) andCaniso(t). The experimental Ciso(t) andCaniso(t) are re- lated to the vibrational, Cv(t), and the reorientational, Cr(t), time correlation functions,1,2 Ciso~t!5Cv~t!, ~3!Caniso~t!5Cv~t!Cr~t!. ~4! In the latter, the usual separability of Caniso(t) as a simple product was assumed, so that the pure reorientational func-tion could be obtained by the ratio between the experimentaltime correlation functions, C r~t!5Caniso~t!/Ciso~t!. ~5! III. RESULTS AND DISCUSSION A. Overview of the Raman spectra As a ﬁrst insight on the interactions which the C 5O522 anion experiences in different environments, Fig. 1 shows the polarized Raman spectra of pure TBCR; TBCR in a di-lute acetonitrile solution ~0.07 M !, together with the previous result 15of a saturated Li 2C5O5aqueous solution ~0.40 M !. One can see signiﬁcant frequency shifts and changes in theFWHH of the bands, and these values for the n2mode are collected in Table I. Interestingly, in aqueous solution, wherethere are strong hydrogen bonds with the neighboring watermolecules, the n2mode is shifted to highwave numbers. This is not a common behavior. One usually expects theelongation of the bond, the decrease of the force constant,and the concomitant low frequency shift of the vibrationalfrequency, upon formation of the hydrogen bond. Unusual high frequency shifts due to hydrogen bonding is an issue which has been recently addressed by ab initio Quantum Chemistry calculations, for instance, C–H bonds inchloroform and ﬂuoroform. 18In the present case of the C5O522anion, the high frequency shift in aqueous solution is being observed in the species which is the acceptor of thehydrogen bond. A similar ﬁnding has been analyzed byDinur 19in the case of HCN–HF complexes, in which the CN stretching frequency increases in comparison to the mono-mer at the gas phase. The reasons for a contraction of thebond length, and a concomitant increase of the vibrationalfrequency, upon formation of the hydrogen bond is not asunderstood as the usual lengthening of the bond length andthe decrease of vibrational frequency. In the usual case,simple electrostatic arguments apply, whereas the unusualTABLE I. Best ﬁt parameters of the Kubo’s model @Eq.~6!#and the Rothschild et al. ~Ref. 29 !model @Eq.~8!# for the vibrational dephasing of the ring breathing mode n2(a18) of the croconate anion, C5O522, at different environments at room temperature. n ~cm21!FWHH ~cm21!tv ~ps!a^Dv(0)2& ~cm22!btc ~ps!chdV ~cm21!cg ~ps21!c Li2C5O5a qe636.5 14.5 0.73 428 0.09 0.35 90.0 25.0 Pure TBCR 625.5 11.5 1.14 98 0.33 0.61 45.0 17.0TBCR in CH 3CN 0.50 M 622.5 8.5 1.54 42 0.60 0.73 9.0 5.90.37 M 621.5 9.0 1.44 60 0.40 0.58 11.6 8.00.07 M 620.0 7.5 1.73 45 0.43 0.54 15.0 6.2 atv5Cv(t)dt. bEquation ~7!. The estimated uncertainty is 610 cm22. cThe estimated uncertainty is 10%.dh5^Dv(0)2&1/2tc. eReference 15.8569 J. Chem. Phys., Vol. 119, No. 16, 22 October 2003 Raman band shape of a molten salt This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Sun, 30 Nov 2014 08:51:25high frequency shift has been considered as a manifestation of the inherent covalent nature of the hydrogen bond.18 Gilli and Gilli20proposed that stronger and shorter hy- drogen bonds occur when the covalent contribution in-creases. X-ray diffraction of Na 2C4O43H2O and Na 2C5O5 3H2O crystals indicated that the distance between the oxy- gen atom of the oxocarbon ion and the hydrogen atom ofwater molecules is in the 1.8–1.9 Å range. 21In several hy- drated alkali salts of the C 5O522anion, it has been shown that the distance between the oxygen atoms of the oxocarbon andthe water molecules is in the 2.6–2.9 Å range. 22Similar distances have been also obtained in an x-ray investigation ofthe Li 2C5O52H2O crystal.23Molecular dynamics simula- tion of the C 4O422species in aqueous solution indicated that the average distances between the oxygen atom of the oxo-carbon ion and the hydrogen and the oxygen atoms of hy-drogen bonded water molecules are 1.6 and 2.6 Å,respectively. 16All of these values would correspond to rela- tively strong hydrogen bonds in the Gilli and Gilli’sclassiﬁcation. 20Thus, the lower vibrational frequency ob- served in pure TBCR and in TBCR in acetonitrile solutions,in comparison to the Li 2C5O5aqueous solution, corroborates the picture of strong interactions between the oxocarbon ionand the water molecules in the latter. These interactions arestrong enough for compressing the electron density on theoxocarbon ring, so that the bonds are shortened and the vi-brational frequency increases in aqueous solution. The effect on the electron density of the oxocarbon ring by the environment would be also responsible for thechanges on the relative intensities of the Raman bands. InFig. 1, the spectra have been arbitrarily normalized by theintensity of the n11band, so that the relative intensity of the n2band display a clear dependence on the environment in which the oxocarbon ion is immersed. Of course, the lack inthis ﬁgure of some absolute reference for the intensities pre-cludes saying which normal mode had its intensity increased.Nevertheless, there is an obvious dependence of the relativeintensities which could be the result of compressing effectsof the environment on the electron density of the oxocarbon ring. Consistently, the electronic absorption band of C 5O522 ~at’360 nm in aqueous solution !displays a signiﬁcant sol- vatochromic shift on going from aqueous to acetonitrile so-lution ~’10.0 nm redshift !. 24As one would expect that such strong interactions between the C 5O522anion and the solvent molecules should be diminished in acetonitrile solution, theobserved vibrational frequency in the latter is close to pureTBCR. On the other hand, the ﬁnding of similar vibrationalfrequency in pure TBCR and in its acetronitrile solution alsosuggests that no signiﬁcant short-range interactions between the C 5O522anion and the tetra ~n-butyl !ammonium cation take place in pure TBCR. In fact, the Raman spectra in thefrequency range corresponding to the modes of the tetra ~n- butyl!ammonium cation is very much the same ~not shown here!both in pure TBCR and in the simple salt tetra ~n- butyl!ammonium chloride in a saturated aqueous solution. Anisotropic Raman bands I aniso(v) are usually broader than isotropic ones Iiso(v) since the time correlation function Caniso(t) decays faster than Ciso(t) because the former probes both the vibrational and the reorientational relaxation,whereas the latter probes only the vibrational relaxation @see Eqs.~3!and~4!#. We found15that the bandwidths of both the Iiso(v) and the Ianiso(v) of the n2mode of C 5O522in aque- ous solution are almost the same, which was an indication of very hindered reorientations of C 5O522in water. In other words, if both the Caniso(t) and the Ciso(t) decay in a com- parable rate, their ratio Cr(t)@see Eq. ~5!#is almost constant at 1.0 in the accessible time range. A similar ﬁnding is seen in Fig. 2, which shows Iiso(v) andIaniso(v) of the n2mode of C 5O522in pure TBCR at room-temperature and in a 0.07 M acetonitrile solution. Inboth the cases, the bandwidths of I iso(v) andIaniso(v) are similar, indicating that the reorientational dynamics of the C5O522anion is also hindered in these environments. The slow reorientational relaxation of C 5O522found previously in aqueous solution persists in spite of the absence of stronghydrogen bond interactions with the solvent, and also byheating pure TBCR up to 320 K ~not shown in Fig. 2 !.I ti s remarkable that reorientational relaxation of C 5O522is very hindered both in pure TBCR and in a dilute acetonitrile so-lution, despite of the huge difference in viscosity of thesessystems. Thus, the reorientational relaxation time, tr,o f C5O522does not scale with the viscosity, which is contrary to ﬁndings in simpler systems, for instance, molten alkali ni-trates, in which tr}hholds according to the Stokes– Einstein–Debye relation.25Therefore, we are led to the physical picture that such slow reorientational relaxation of C5O522is due to the particular size and shape of the oxocar- bon species itself. In fact, it has been suggested from system-atic comparison between anions with different charge, sizeand symmetry, that the microscopic reorientational dynamicsis very much dependent on the shape of the ion. 26For in- stance, it has been suggested that the elongated shape of theSCN 2accounts for its relatively slow reorientational relax- ation in water ( tr’10.0ps),27in comparison with the more symmetric CN2ion (tr’0.9ps).28Taking SO422and CO322 as common examples of dianions, the faster reorientational relaxation of SO422(tr’4.2ps) than CO322(tr’6.2ps) in FIG. 2. Isotropic ~white symbols !and anisotropic ~black symbols !Raman bands of pure TBCR ~circles !and TBCR in a dilute acetonitrile solution ~triangles !of the ring breathing mode n2(a18) of the croconate anion, C5O522, at room temperature. The spectra have been normalized by their maximum intensity and frequency shifted so that the center of the band islocated at zero wave number.8570 J. Chem. Phys., Vol. 119, No. 16, 22 October 2003 A. O. Cavalcante and M. C. C. Ribeiro This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Sun, 30 Nov 2014 08:51:25water is understood by the spherical symmetry of the former.26Therefore, we do not assign the slow reorientation of C5O522as the result of strong interactions with the neigh- boring molecules, as the same ﬁnding was obtained for thesespecies in very different environments. Instead, we assignedits slow reorientation to the special unusual planar symmetry of the C 5O522dianion. Finally, it is worth mentioning the detailed analysis of the reorientational dynamics of oxocar- bon species revealed by recent MD simulations of the C 4O422 dianion in aqueous solution.16It has been found that the reorientational dynamics of the parallel and the perpendicu-lar components to the main symmetry axis do not scale withthe corresponding moments of inertia. For comparison pur-poses, the calculated reorientational relaxation time for the C 4O422species in water is ca. 40.0 ps,16which is in fact much larger than the vibrational relaxation times discussedbelow ~see Table I !. B. Vibrational dephasing of the croconate anion at room temperature By assuming that vibrational dephasing is the main mechanism of relaxation acting on the isotropic Raman com-ponent, the experimental vibrational time correlation func-tions were ﬁtted by the well known Kubo’s model, 1,2 Cv~t!5exp$2^Dv~0!2&tc@tc~exp~2t/tc!21!1t#%, ~6! where ^Dv(0)2&is the average amplitude of the vibrational frequency ﬂuctuations, and tcis the relaxation time of the correlation function of such ﬂuctuations, x(t) 5^Dv(t)Dv(0)&, which is assumed to be a simple expo- nential in the Kubo’s model, x(t)5exp(2t/tc). The experi- mentalCv(t) were ﬁtted by using Eq. ~6!with only tcas an adjustable parameter, since ^Dv(0)2&is the second moment of the isotropic component, which was evaluated by the in-tegral on the experimental spectrum, 1,2 ^Dv~0!2&52‘‘~v2v0!2Iiso~v!dv *2‘‘Iiso~v!dv, ~7! where v0is the center of the band. Figure 3 shows the experimental Cv(t) of the ring breathing mode n2of C5O522in different environments, to- gether with the best ﬁt Cv(t) according to the Kubo’s model. The inset in Fig. 3 also shows the best ﬁt Cv(t) by using the Rothschild et al.29model which will be discussed below.The corresponding parameters of the Kubo’s model are shown inTable I, which also gives the vibrational relaxation time, tv, evaluated by the time integral of the experimental Cv(t). The relaxation time tvwould be directly related to the full width at half height of the isotropic Raman band, tv 5(pcFWHH)21, at the fast modulation limit, in which Cv(t) would be a simple exponential function and Iiso(v) would be a Lorentzian function. The criteria of fast and slowmodulation regime is the parameter h5^Dv(0)2&1/2tc shown in Table I, respectively, h!1 and h@1.1,2 We ﬁrst discuss the average amplitude of vibrational fre- quency ﬂuctuations ^Dv(0)2&of the n2mode of C 5O522in the different environments. It is clear from Table I the largedecrease in ^Dv(0)2&on going from the aqueous solution to pure TBCR. This indicates a more homogeneous environ-ment experienced by the probe oscillator in TBCR thanin water. It has been suggested that the size and shape ofanions play a crucial role on the static distribution ofwater molecules around the anion, what would correlatewith the magnitude of the average amplitude ^Dv(0)2&.30 For comparison purposes, ^Dv(0)2&544cm22(SO422), 97cm22(CO322), and 650cm22(CH3CO22), in aqueous solution.30It is also interesting to compare ^Dv(0)2&in pure TBCR at room temperature ~98 cm22!with other simple molten salts, for instance, LiNO 3and RbNO 3at 600 K, for which ^Dv(0)2&of the totally symmetric stretching mode n1(a18) of the NO32anion is 420 and 130 cm22, respectively.2 The relative small value of ^Dv(0)2&in pure TBCR is also consistent with the relative small FWHH ~11.5 cm21!in comparison to the FWHH values of the n1stretching modes in Li2CO3at 1193 K and LiNO 3at 623 K, respectively, 36.7 and 25.6 cm21.31 A further signiﬁcant reduction in ^Dv(0)2&, indicating an even more homogeneous environment experienced by theprobe oscillator, is observed on going from pure TBCR toTBCR in acetonitrile solutions. This ﬁnding indicates thatthe probe oscillator is less perturbed in acetonitrile solutionthan in aqueous solution or in TBCR. A key concept here isthe effectiveness of the interactions between the probe oscil-lator and the environment. When the neighborhood around a given C 5O522anion is made of hydrogen-bonded water mol- ecules, slight displacements or reorientations of the neigh-boring molecules in the ﬁrst solvation shell imply major per-turbation and frequency ﬂuctuation of the probe oscillator. It should be noted that the C 5O522anion in TBCR is sur- rounded by 11.0 charged species plus four water molecules per@(n2C4H9)4N#2C5O5unity. In acetonitrile solution, the nearest neighbor shell around the C 5O522anion is not well deﬁned, but ^Dv(0)2&is relatively small because the inter- action between the C 5O522species and the acetonitrile mol- FIG. 3. Vibrational time correlation function, Cv(t), of the n2mode of the C5O522anion in Li2C5O5aqueous solution ~circles !, pure TBCR ~up tri- angles !, 0.37 M acetonitrile solution of TBCR ~squares !, and 0.07 M aceto- nitrile solution of TBCR ~down triangles !, at room temperature. The full lines on the experimental data is the best ﬁt Cv(t) according to Kubo’s model @Eq.~6!#. The inset shows the same data in a log scale together with the best ﬁt according Eq. ~8!~dashed line !.8571 J. Chem. Phys., Vol. 119, No. 16, 22 October 2003 Raman band shape of a molten salt This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Sun, 30 Nov 2014 08:51:25ecules is not too strong and, therefore, not too effective in causing vibrational dephasing. On the point of view of theprobe oscillator, it experiences a more homogeneous envi-ronment in acetonitrile solution. Interestingly, ^Dv(0)2&does not vary smoothly with the concentration in acetonitrile solution, instead a maximum in ^Dv(0)2&is observed at 0.37 M, which is a solution ’1:1 volume/volume proportion. Following the maximum in ^Dv(0)2&, we found the largest FWHH and the smallest tv at this concentration. This ﬁnding is analogous to previousones in several mixtures of simple molecular liquids, for in-stance, CHCl 3/CS2,32CH3CN/CCl 4,33CH3I/CDCl 3,34and CH2I2/CCl4.35The maximum in ^Dv(0)2&, the maximum in FWHH, and the minimum in tv, at the 0.37 M acetonitrile solution of TBCR is assigned to composition ﬂuctuationswhich are maximized at this concentration. In other words, inthe limit that a given species is either the major or the minorcomponent, it will experience a more homogeneous environ-ment, whereas wider distribution in the environment devel-ops when the number of the molecules of each species iscomparable. Furthermore, Fig. 4 shows that the Raman band of the n2mode of C 5O522is symmetric in aqueous solution, in pure TBCR, and in dilute acetonitrile solution, whereas itis clearly asymmetric in the 0.37 M acetonitrile solution. Infact, it is usually observed that asymmetric band shapes de-velop when local concentration ﬂuctuations play a role onthe broadening mechanisms of the Raman bands. 35Proposed models for broadening mechanism in liquid mixtures assignsthe asymmetry in the band shapes to a weighted superposi-tion of symmetric band shapes due to different localenvironments. 35 It is clear from Table I, that the relaxation time of vibra- tional frequency ﬂuctuation tcof the n2mode of C 5O522is larger in pure TBCR than in aqueous solution. tcof different anions in water is also correlated with the size and shape of the ion, for instance, tc50.42ps ~SO422), 0.37ps ~CO322), and 0.12ps ~CH3CO22).30In case of the oxocarbon anions, it has been suggested15that the small tcin aqueous solution is due to the fast modulation of short-range forces acting on theprobe oscillator proper to the fast oscillatory and librational dynamics of the anions in cages formed by neighboring wa-ters.The absence of these relatively rigid cages in the case ofpure TBCR is reﬂected on the slower modulation of theforces on the probe oscillators. tcfurther increases by dilut- ing TBCR in acetonitrile, and again a non smooth concen-tration dependence is seen as the smallest tcin acetonitrile solution is observed at 0.37 M. It is remarkable the differ-ence of tcin water and in dilute acetonitrile solution, that is, the forces acting on the probe oscillator are much moreslowly ﬂuctuating in acetonitrile than in aqueous solutions. The small FWHH in pure TBCR and in acetonitrile so- lution, in comparison with the aqueous solution, implyslowly decaying C v(t) and larger vibrational relaxation times tv. The limits of slow or fast modulation regime is more appropriately characterized by the hparameter shown in Table I. One sees that the fastest modulation regime of the n2mode of C 5O522occurs in aqueous solution. For compari- son purpose, hvalues for other simple anions in water are 0.52(SO422), 0.69(CO322), 0.57(CH 3CO22), and 0.50(NO32).30The modulation regime of C 5O522in water seems to be only slower than CN2(h50.17), the latter be- ing assigned to the fast reorientational motion of the CN2 which also results in fast ﬂuctuations of the vibrational fre-quency ( tc50.04ps in CN2).30In summary, the systematic comparison of the FWHH, tv,^Dv(0)2&,tc, and h, in pure TBCR, in TBCR in acetonitrile solution, and in Li 2C5O5 aqueous solution, indicates that the oxocarbon species in themolten salt TBCR experiences a relatively homogeneousslowly ﬂuctuating environment. The above discussion relies on applying the Kubo’s model for the vibrational dephasing of the ring breathing n2 mode of C 5O522. Different models have been proposed in which the assumption of a single exponential decay for thetime correlation function of the vibrational frequency ﬂuc-tuations, x(t)5^Dv(t)Dv(0)&, is replaced by other func- tional form. From the picture outlined above, a temptingmodel is the one proposed by Rothschild et al. 29in which x(t) is assumed to be a damped-oscillatory function with average frequency Vand damping g, resulting in the follow- ing vibrational correlation function: lnCv~t!52^Dv~0!2&V2$gt1V2@~g22V2! 3~exp21/2gtcosv8t21!1g~g223V2! 3~2v8!21exp21/2gtsinv8t#%, ~8! where v85@V22(4g2)21#1/2. This model has been used al- ready in the previous study on Li 2C5O5aqueous solution,15 and it is instructive to ﬁnd how the parameters change in theenvironments of pure TBCR and in acetonitrile solution, in particular the librational frequency Vof the C 5O522ion as a whole. The inset in Fig. 3 shows the best ﬁt Cv(t) according to Eq. ~8!, which seems to be even a better model than the Kubo’s model due to the better agreement with the experi-mentaldataatlongtime.TableIgivesthecorrespondingbestﬁt parameters of applying Eq. ~8!to the experimental C v(t) of the n2mode of C 5O522. The most important conclusion drawn from Table I is that the average frequency Vin pure TBCR is much smaller than in aqueous solution, and it is yet FIG. 4. Isotropic Raman bands of the n2mode of the croconate anion, C5O522, at room temperature in Li2C5O5aqueous solution ~dotted line !, pureTBCR ~bold line !,TBCR in a 0.37 M acetonitrile solution ~circles !, and TBCR in a 0.07 M acetonitrile solution ~dashed line !. The spectra have been normalized by their maximum intensity and frequency shifted so that thecenter of the band is located at zero wave number.8572 J. Chem. Phys., Vol. 119, No. 16, 22 October 2003 A. O. Cavalcante and M. C. C. Ribeiro This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Sun, 30 Nov 2014 08:51:25further reduced in acetonitrile solution. The relative small librational frequency Vand damping gin pure TBCR and in acetonitrile solution fully corroborates the picture of a muchless rigid, slowing ﬂuctuating microenvironment, experi-enced by the oxocarbon ion in TBCR than in aqueous solu-tion. It is worth mentioning that the previous low-frequencyRaman study of pure TBCR indicated the presence of abroad band centered at ’35 cm 21at room temperature.13 This band was assigned to the librational motion of the oxo- carbon ring, an interpretation which nicely agrees with thepresent result of V545 cm 21in pureTBCR obtained here by the band shape analysis of the high frequency n2mode of C5O522. This ﬁnding is also in line with very recent results of Kerr-effect spectroscopy of ionic liquids based on deriva-tives of imidazolium salts, 36in which the librational motion of the imidazolium ring was observed in a similar frequencyrange ~30–100 cm 21!at room temperature. Finally, it is interesting to compare the present results with a recent Raman band shape analysis of the molecularglassforming liquid phenyl salicylate ~salol!in the pure liq- uid, in a dilute solution in CCl 4, and under conﬁnement in nanoporous silica glasses.37The analysis of the vibrational dephasing of both the C–H stretching and the C–H bendingmodes indicated that tvincreases, whereas ^Dv(0)2&and FWHH decrease, on going from pure salol to the dilute so- lution, in line with the present ﬁndings for C 5O522. In con- trast with the present results, however, tcfor the C–H bend- ing mode is smaller in the dilute solution than in pure salol.Perhaps more interestingly is the ﬁnding that by spatiallyconﬁning salol, the dephasing parameters change in a direc- tion which is similar to the present case of C 5O522on going from pure TBCR to the aqueous solution, namely, FWHHand ^Dv(0)2&increase, whereas tcandtvdecrease. Thus, the rigid cage made by the neighboring water molecules in the case of the C 5O522aqueous solution plays a similar role observed by conﬁning salol. Actually, the magnitude of the observed changes on the dephasing parameters in C 5O522is much larger than the results for salol,37indicating that the oxocarbon species are excellent probes for revealing the dy-namical nature of the environment in which they are im-mersed. C. Glass transition signatures on the vibrational dephasing Both of the models given by Eqs. ~6!and~8!were ap- plied to ﬁt the experimental Cv(t) of the n2mode of C 5O522 in pure TBCR as the system was cooled down to the glassy state. Figure 5 shows the best ﬁt parameters as a function ofthe temperature across the glass-transition. It is clear thatthere are discontinuous changes of the dephasing parameters atT g, indicating that vibrational dephasing of the C 5O522 moiety is also an excellent indicative of the glass transition. The local environment experienced by the probe oscillatorbecomes increasingly homogeneous in the glass, as ^Dv(0)2&decrease below Tg. Conversely, tcincreases in the glassy state, which should be assigned to the slowingdown of the ﬂuctuating forces experienced by the probe os-cillator. The slowing down of the microscopic dynamics inglassy TBCR is also manifested in the tvand the hparam-eters, as both of them increase below Tg. Concerning the usage of the Rothschild et al.29model for ﬁtting the experi- mental data, one sees the smaller damping gand smaller librational frequency VbelowTg. The frequency range which Vspreads from the high temperature state down to the low temperature one ~’45–25 cm21!nicely matches the same frequency range of the broad band previously observedby the analysis of the low frequency Raman spectra as afunction of the temperature in pure TBCR. 13 It is interesting to compare the temperature dependence of the dephasing parameters shown in Fig. 5 with the recentresults of Kirillov and Yannopoulos 38for the glassformers As2O3and 2BiCl 3–KCl. In line with the present results, the authors of Ref. 38 also found that ^Dv(0)2&decreased and tvincreased below Tgin these glassformers. In contrast with our results, however, tcdecreased below Tgfor both As 2O3 and 2BiCl 3–KCl. This unexpected ﬁnding has been consid- ered as an indication that one should actually consider theeffectiveness of the interactions which cause the vibrationaldephasing. Complex structural arrangements and the en-hancement of orientational order of the environment at lowtemperature would result in more successful interactionswith the probe oscillators and lower tcbelowTgin As2O3 and 2BiCl 3–KCl.38In the present case of pure TBCR, we found conversely that tcincreased below Tg, which sug- gests that this is due here to the simple slowing down of thedynamics of the ﬂuctuating forces around the probe oscilla-tor at low temperature. Figure 6 shows the isotropic Raman spectra of the n2 mode of C 5O522in pure TBCR at room temperature ~dashed line!and below Tg~circles !. Both the spectra are superim- posed in the high frequency side, so that the ﬁgure makesclear that the band shape below T gis asymmetric. In both of the models for vibrational dephasing used in the previous FIG. 5. Temperature dependence of the vibrational dephasing parameters of then2mode of the C5O522anion in pure TBCR. The vertical dotted lines indicate the glass-transition temperature.8573 J. Chem. Phys., Vol. 119, No. 16, 22 October 2003 Raman band shape of a molten salt This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Sun, 30 Nov 2014 08:51:25section, it was assumed that the Raman bands are symmetric. This in turn arises by expanding the vibrational correlationfunction, C v~t!5KexpHiE 0t dt8Dv~t8!JL, ~9! onto a cumulant expansion truncated at the second order term. This truncation of the cumulant expansion is justiﬁedby assuming that the probability distribution of D v(t)i s Gaussian, which comes from the central limit theorem.1 Thus, asymmetric band shapes have been considered as anindication of the fail of the central limit theorem. Asymmet-ric bandshape has been also observed in the infrared spectraof the 1030 cm 21mode of glassy quinoline at 115 K, for which the best ﬁt ^Dv(0)2&andtcparameters were 16.3 cm22and 3.0 ps, respectively.1Non-Gaussian distribution of Dv(t), and therefore asymmetric bandshapes, would be also realized in a physical situation in which the probe oscillatorexperiences relatively few interactions sites, for instance,HCl in a dilute CCl 4solution39and CO chemisorbed on platinum dispersed on Al 2O3.40It should be noted that these asymmetric band shapes arise from a different source of theones observed in the 0.37 M acetonitrile solution ofTBCR atroom temperature ~see Fig. 4 !. The latter is due to composi- tion ﬂuctuation and a less homogeneous environment expe-rienced by the probe oscillator as indicated by a correspond-ing increase in ^Dv(0)2&~see Table I !. In contrast, ^Dv(0)2&decreases in pure TBCR below Tg~Fig. 5 !reveal- ing that the oscillator is probing a more homogeneous envi-ronment in the glassy state. Therefore, we are lead to thepicture that the asymmetry in the band shapes due to thefailing of the central limit theorem arises from fewer, al-though more homogeneous, interaction sites experienced by the C 5O522anion in glassy TBCR. The asymmetric band shapes in glassy TBCR imply that the models given by Eqs. ~6!and~8!are strictly no longer valid. More generally, higher order terms in the cumulantexpansion of Eq. ~9!should be considered, 1lnCv~t!521 2E 0tE 0t dt1dt2^Dv~t1!Dv~t2!&c 2i 6E 0tE 0tE 0t dt1dt2dt3 3^Dv~t1!Dv~t2!Dv~t3!&c1ﬂ, ~10! where ^ﬂ&cstands for a cumulant average. In the models outlined in the previous section, only the ﬁrst term of theabove expression is considered and some functional form for ^Dv(t1)Dv(t2)&is assumed, for instance, a single exponen- tial in the Kubo’s model. If some simple assumption is alsoproposed for ^Dv(t1)Dv(t2)Dv(t3)&, asymmetric band shapes could be modelled. We follow here the proposal ofRothschild and Yao, 40namely, the exponential ansatz for ^Dv(t1)Dv(t2)&with a relaxation time tcas the Kubo’s model, and a double exponential ansatz for ^Dv(t1)Dv(t2)Dv(t3)&with two relaxation times t1and t2. This model result in the following vibrational correlation function:40 Cv~t!5exp$2^Dv~0!2&fs~t!2i^Dv~0!3&fa~t!%,~11! wherefs(t) is deﬁned by reference to the previous Kubo’s model, Eq. ~6!, andfa(t) accounts for the complex Cv(t) and therefore asymmetric band shape, fa~t!5t1t2t2t13 12~t1/t2!~e2t/t121! 2t23 12~t2/t1!~e2t/t221!. ~12! ^Dv(0)3&is given by the experimental third moment of the isotropic Raman spectra, which of course would be zero fora symmetric band shape. In order to reduce the number ofadjustable parameters, it was further assumed that tc5t1,s o that only two relaxation times were left to be varied. By Fourier transforming Eq. ~11!one obtains the corre- sponding spectra, and Fig. 6 shows the ability of the model to reproduce the Raman band of the n2mode of C 5O522in pure TBCR at 230 K. The temperature dependence of theresulting dephasing parameters are shown in Fig. 7. It isclear from Fig. 6 that a good agreement between the modeland the experimental data is achieved. Although lacking amore clear physical meaning for the relaxation times in-volved in the model, it is interesting to note in Fig. 7 thediscontinuous variation at T gand the increase of the relax- ation times below Tg. It is well known that the microscopic dynamics of relaxation of many different properties in super-cooled glassforming liquids spans two very different timescales, but both of the t1and the t2parameters in the present case of vibrational dephasing in C 5O522are similar and small in comparison with the very long time scale of structuralrelaxation. Therefore, we are led to a physical picture of amore local short-range contribution to the frequency modu-lation in TBCR even in the vitreous state. FIG. 6. Isotropic Raman bands of the n2mode of the C5O522anion in pure TBCR at 300 K ~dashed line !and at 230 K ~circles !.The bold full line is the best ﬁt according to Eq. ~11!. The spectra have been normalized by their maximum intensity and frequency shifted so that the center of the band islocated at zero wave number.8574 J. Chem. Phys., Vol. 119, No. 16, 22 October 2003 A. O. Cavalcante and M. C. C. Ribeiro This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Sun, 30 Nov 2014 08:51:25IV. CONCLUSIONS The extensive comparison of the Raman band shape of the totally symmetric ring breathing mode n2at’625 cm21 of the croconate dianion in pure TBCR, in acetonitrile solu- tion, and the previous results15for an aqueous solution of the simple salt Li 2C5O5, lead us to the following conclusions on the microscopic dynamics of the C 5O522anion. The reorien- tational motion of C 5O522is very hindered despite of the very different microenvironments in these systems. The rela- tively slow reorientational relaxation of C 5O522is thus as- signed to its unusual shape in comparison with more com-mon anions. The analysis of the vibrational dephasing of the n2mode indicates a more homogeneous environment expe- rienced by the C 5O522anion in pure TBCR than in aqueous solution. The relaxation of the ﬂuctuation of the vibrationalfrequency of the n2mode is signiﬁcantly slower in pure TBCR than in aqueous solution. On going from liquidTBCRto glassy TBCR, we found a slowing down of this relaxationdynamics, and the regime of the vibrational dephasing goesto a slower modulation one. The regime of modulation foundhere in TBCR even at high temperature is slower than theprevious ﬁnding in aqueous solution. This corroborates theproposition that fast rattling dynamics of the oxocarbon spe-cies inside relatively long lived cages of water molecules isthe main contribution to the ﬂuctuation of short-range repul- sive forces experienced by the probe oscillator of C 5O522in aqueous solution. In TBCR, the microenvironment around a given C 5O522anion changes relatively slowly, so that a faster regime is obtained upon dilution ofTBCR in acetonitrile, butit is still slower than in aqueous solution. Finally, in glassyTBCR, the usual expressions for modelling vibrational cor-relation functions are strictly not valid, because the Ramanbands are clearly asymmetric. It was shown that the isotropicRaman spectra of pure TBCR below T gcould be modelled by using a previously proposed model40which considers the cumulant expansion of the vibrational correlation function upto the third order term. TBCR has been used here as a pro-totype system of molten salts with low melting point. Itwould be interesting to ﬁnd whether the present conclusionswould be also valid for the usual technologically relevantionic liquids mentioned in the Introduction, as long as onecould record Raman spectra for these systems with such ahigh quality as the case of TBCR. ACKNOWLEDGMENTS The authors acknowledge Dr. Luiz F. C. de Oliveira and Dr. Munir S. Skaf for helpful discussions. The authors areindebted to FAPESP and CNPq for ﬁnancial support. 1W. G. Rothschild, Dynamics of Molecular Liquids ~Wiley, New York, 1984!. 2S. A. Kirillov, J. Mol. Liq. 76,3 5~1998!. 3T. Welton, Chem. Rev. 99, 2071 ~1999!. 4R. J. Gale, B. Gilbert, and R. A. Osteryoung, Inorg. 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Perrot, Faraday Discuss. 66, 216 ~1978!. 29W. G. Rothschild, J. S. Jacob, J. Bessiere, and J.V. Geisse, J. Chem. Phys. 79, 3002 ~1983!. 30M. Perrot and W. G. Rothschild, J. Mol. Struct. 80, 367 ~1982!. 31S. Okazaki, M. Matsumoto, and I. Okada, Mol. Phys. 79,6 1 1 ~1993!. 32A. F. Bondarev and A. I. Mardaeva, Opt. Spectrosc. 35,1 6 7 ~1973!. 33A. Morresi, P. Sassi, M. Ombelli, R. S. Cataliotti, and G. Paliani, J. Raman Spectrosc. 31, 577 ~2000!. FIG. 7. Temperature dependence of the vibrational dephasing parameters of Eq.~11!for the n2mode of the C5O522anion in pure TBCR. The vertical dotted lines indicate the glass-transition temperature.8575 J. Chem. Phys., Vol. 119, No. 16, 22 October 2003 Raman band shape of a molten salt This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Sun, 30 Nov 2014 08:51:2534E. W. Knapp and S. F. Fischer, J. Chem. 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5.0011873.pdf	J. Appl. Phys. 128, 013903 (2020); https://doi.org/10.1063/5.0011873 128, 013903 © 2020 Author(s).Underlayer effect on the soft magnetic, high frequency, and magnetostrictive properties of FeGa thin films Cite as: J. Appl. Phys. 128, 013903 (2020); https://doi.org/10.1063/5.0011873 Submitted: 27 April 2020 . Accepted: 15 June 2020 . Published Online: 02 July 2020 Adrian Acosta , Kevin Fitzell , Joseph D. Schneider , Cunzheng Dong , Zhi Yao , Ryan Sheil , Yuanxun Ethan Wang , Gregory P. Carman , Nian X. Sun , and Jane P. Chang ARTICLES YOU MAY BE INTERESTED IN Enhancing the soft magnetic properties of FeGa with a non-magnetic underlayer for microwave applications Applied Physics Letters 116, 222404 (2020); https://doi.org/10.1063/5.0007603Underlayer effect on the soft magnetic, high frequency, and magnetostrictive properties of FeGa thin films Cite as: J. Appl. Phys. 128, 013903 (2020); doi: 10.1063/5.0011873 View Online Export Citation CrossMar k Submitted: 27 April 2020 · Accepted: 15 June 2020 · Published Online: 2 July 2020 Adrian Acosta,1 Kevin Fitzell,1 Joseph D. Schneider,2 Cunzheng Dong,3 Zhi Yao,4 Ryan Sheil,1 Yuanxun Ethan Wang,4Gregory P. Carman,2Nian X. Sun,3 and Jane P. Chang1,a) AFFILIATIONS 1Department of Chemical and Biomolecular Engineering, University of California, Los Angeles, California 90095, USA 2Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, California 90095, USA 3Department of Electrical and Computer Engineering, Northeastern University, Boston, Massachusetts 01225, USA 4Department of Electrical and Computer Engineering, University of California, Los Angeles, California 90095, USA a)Author to whom correspondence should be addressed: jpchang@ucla.edu ABSTRACT The soft magnetic, microstructural, and magnetostrictive properties of Fe 81Ga19(FeGa) film sputter deposited onto 2.5-nm Ta, Cu, and Ni80Fe20(NiFe) underlayers were investigated. The films deposited with an underlayer showed increased in-plane uniaxial anisotropy and a decrease in in-plane coercivity. The smallest coercivity was observed in FeGa deposited with a NiFe underlayer at 15 Oe, compared to 84 Oefor films deposited directly on Si. In addition, an effective Gilbert damping coefficient ( α eff) as low as 0.044 was achieved for a 100-nm FeGa film with a NiFe underlayer. The coercivity and αeffwere shown to decrease further as a function of FeGa film thickness. The FeGa films were also able to retain or increase their saturation magnetostriction when deposited on an underlayer. This enhancement is attributable to the impact of the underlayer to promote an increased (110) film texture and smaller grain size, which is correlated to the lattice match of theunderlayer of the sputtered FeGa film. Among the underlayers studied, NiFe promoted the best enhancement in the soft magnetic propertiesfor FeGa thin films, making it an attractive material for both strain-mediated magnetoelectric and microwave device applications. Published under license by AIP Publishing. https://doi.org/10.1063/5.0011873 I. INTRODUCTION Recent research has shown great potential for the voltage control of magnetism at the nanoscale in magnetoelectric (ME) or multiferroic materials and heterostructures. 1,2This is motivated by the promise of next-generation electrical and electronic deviceswith a lower energy cost. Many applications that have received sig-nificant attention in recent years, exploiting ME coupling, includerandom access memory, spintronics, mechanically actuated anten-nas, and RF or microwave devices. 3–12While there are many mate- rials with either large magnetostriction or soft magnetic properties,the current bottleneck in achieving high efficiency and ME cou-pling in these devices is synthesizing ferromagnetic materials that exhibit simultaneously a large magnetostriction and soft magnetic properties in order to achieve a high magnetoelastic coupling. 13 Furthermore, to achieve fast switching and low loss operations inME spintronic and microwave applications, it is necessary to have a low Gilbert damping coefficient.13–16This functional property often requires materials engineering to realize as the large spin-lattice coupling that is typically responsible for high magnetostric- tion also results in high magnetic hysteresis and large Gilbert damping coefficients. 17 FexGa1−xalloys have been of interest due to the high magneto- striction observed for bulk and polycrystalline alloys which makesthem promising for integration in strain-mediated ME devices. 18–22 However, one of the barriers for high frequency applications of FeGa thin films has been their large ferromagnetic resonance(FMR) linewidths ( ∼620–700 Oe at X-band). 23,24For sputtered FeGa thin films, it has been well documented that the structure,magnetic softness, and magnetostrictive properties can be heavily influenced by their deposition parameters. 25–27Indeed, more recent works have shown that the fabrication of high quality epitaxialJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 013903 (2020); doi: 10.1063/5.0011873 128, 013903-1 Published under license by AIP Publishing.films can be used to achieve greatly reduced linewidths ( ∼80–220 at X-band).28,29Similarly, the addition of small interstitial atoms (e.g., C, B, and N) to FeGa thin films have been explored and foundto promote excellent soft magnetic properties by reducing their grainsize and diminishing their magnetocrystalline anisotropy. 20,30,31 One approach that has been previously explored to enhance the soft magnetic properties of FeCo alloys has been to use an underlayer material between the sputtered film and substrate.32–41 The resulting enhancement in the soft magnetic properties was attributed to the refined grain size and impact of the stress at theinterface on the magnetoelastic energy. 32,39,42,43Similarly, for FeGa thin films, a recent study showed that a non-magnetic underlayer (Cu) can be used to improve its soft magnetic and high frequencycharacteristics. 44In this work, the underlayer effect in FeGa with Ta (a non-magnetic material with a lattice constant different thanthat of Cu) and NiFe (a magnetic material with a lattice constant similar to that of Cu) was explored and compared to that of Cu. The choice of these materials is primarily to assess the effect of thelattice match at the interface with the FeGa film. II. EXPERIMENTAL DETAILS The films in this study were grown via DC magnetron sputter deposition using a ULVAC JSP 8000 sputter system with a basepressure of 2 × 10 −7Torr at room temperature with 4-in. targets. Si (100) substrates were used for all of the depositions without theremoval of the native oxide. The FeGa films were sputtered using a target with 80/20 Fe/Ga composition at 200 W DC bias power and an Ar pressure of 0.5 mTorr; the Ta, Cu, and Ni 80Fe20(NiFe) underlayers were sputtered at 100 W DC bias power and an Arpressure of 0.5 mTorr in the same chamber. SEM imaging was usedto confirm the thickness of the films. The resulting composition (81% Fe and 19% Ga) of the sputtered FeGa films was determined via x-ray photoelectron spectroscopy (XPS) with a monochromatedAl K αsource. The structural characterization of the films was determined via x-ray diffraction (XRD) using a Panalytical X ’Pert Pro x-ray powder diffractometer with a Cu K αsource and fit with the Fityk software package. 45Atomic Force Microscopy (AFM) imaging of the surface microstructure was performed using aBruker Dimension Icon Scanning Probe Microscope with a BrukerRTESPA-300 AFM tip with an 8-nm nominal tip radius. The room temperature magnetic hysteresis curves of the mul- tilayers were measured via superconducting quantum interferencedevice (SQUID) magnetometry using a Quantum Design MPMS3.The high frequency magnetic linewidth was measured using ashort-circuited strip line connected to a vector network analyzer (VNA) with details described elsewhere. 46For these measurements, the samples were placed facing the strip line and a large saturatingmagnetic field was first applied parallel to the strip line to establisha baseline for the measurement. The reflection coefficient (S 11)w a s then measured as a function of bias magnetic field (0 –600 Oe) and frequency (100 MHz to 6 GHz). The magnetostrictive properties were performed by depositing FeGa, with and without an under-layer, on thin Si cantilever substrates (100 μm thickness) and utiliz- ing an MTI-2000 fiber-optic sensor to detect the deflection of the cantilever tip due to changes in the stress of the FeGa thin films. Details are described elsewhere. 30III. RESULTS AND DISCUSSION In this study, 100-nm FeGa thin films were deposited either directly onto Si substrates or with a thin 2.5-nm Ta, Cu, or NiFe underlayer. Figure 1 shows the in-plane magnetic hysteresis loops for the 100-nm FeGa films deposited on different underlayermaterials normalized to the saturation magnetization. All of thefilms exhibited strong in-plane magnetic anisotropy. The FeGa film deposited directly onto a Si substrate, without an underlayer, showed a coercivity of 84 Oe. The coercivity of FeGa was reducedto 54 Oe when deposited onto a 2.5-nm Ta underlayer and furtherdecreased to 17 and 15 Oe when deposited on 2.5-nm Cu and NiFeunderlayers, respectively. These results follow a similar trend to that previously observed for Fe 65Co35films where a Ta underlayer resulted in a modest decrease in easy-axis coercivity while Cu andNiFe underlayers promoted a larger decrease. 32In addition, the FeGa films deposited with both Cu and NiFe underlayers displayed an enhanced uniaxial anisotropy, as observed from the increase in remnant magnetization (M r) as summarized in Table I . FIG. 1. Normalized in-plane magnetic hysteresis loops of 100-nm FeGa sput- tered on a Si substrate with different underlayer materials. TABLE I. Summary of in-plane coercivity, normalized remnant magnetization (Mr/Ms), effective Gilbert damping coefficient ( αeff), relative change in (110) peak intensity ( ΔI110), and relative change in film strain ( Δϵ) for 100-nm FeGa grown on different underlayer materials on a Si substrate . UnderlayerIn-plane coercivity (Oe) M r/MsαeffΔI110 (%)Δϵ (%) None 83 0.83 0.206 …… 2.5-nm Ta 54 0.84 0.118 0 −0.06 2.5-nm Cu 17 0.97 0.053 30 −0.28 2.5-nm NiFe 15 0.92 0.044 29 −0.21Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 013903 (2020); doi: 10.1063/5.0011873 128, 013903-2 Published under license by AIP Publishing.The high frequency characteristics of FeGa films deposited with different underlayers were studied using broadband FMR spectroscopy. Figure 2 shows the S 11absorption as a function of the magnetic bias field (0 –600 Oe) at a fixed frequency of 6 GHz. These are the cross sections of the entire FMR spectra collectedfor the frequency range of 100 MHz –6 GHz (see Fig. S1 in the supplemental material ). For a 100-nm FeGa film deposited without an underlayer, the FMR spectra are characterized by a very lowpeak absorption ( ∼0.3%) and very broad FMR linewidth (>600 Oe at 6 GHz) that extends beyond the maximum magnetic fieldapplied. For an FeGa film deposited with a Ta underlayer, a small enhancement in the FMR linewidth ( ∼465 Oe at 6 GHz) can be observed. In contrast, the FeGa films deposited on a Cu or a NiFeunderlayer were characterized by a dramatic enhancement in theFMR response with linewidths decreasing to as low as ∼178 Oe and ∼160 Oe at 6 GHz, respectively. The effective Gilbert damping coefficient, α eff, is calculated and reported in Table I by fitting the FMR linewidth of the absorp- tion as a function of frequency for the entire FMR spectra inFig. S1 in the supplemental material to the following equation: ΔH¼2αeffω γþΔH0, where ωis the frequency, γis the gyromag- netic ratio ( ≈2.8 MHz/Oe), and ΔH0is the frequency-independent linewidth broadening. The FeGa films deposited with Cu and NiFeunderlayers show a significant decrease ( ∼75%–78%) in their effec- tive Gilbert damping coefficient compared to an FeGa film without an underlayer. The enhanced soft magnetic properties of the FeGa films grown on the Cu and NiFe underlayers must originate from theimpact of the underlayer on its microstructure. The structural char-acterization of the FeGa films grown on different underlayers wasfirst investigated with XRD. All of the FeGa films showed primarily a bcc (110) diffraction as the strongest diffraction line. Figure 3 shows the spectra highlighting the bcc (110) diffraction for a100-nm FeGa film without an underlayer compared to those sput-tered on Ta, Cu, and NiFe underlayers. The films deposited ontoCu and NiFe underlayers, which show the largest enhancement in their soft magnetic properties also displayed the largest shift of the (110) diffraction line position which is caused by a relative change instrain compared to FeGa deposited directly onto a Si substrate.Compared to FeGa deposited directly on a Si substrate, this shift inpeak position represents a relative increase of 0.28% and 0.21% com- pressive film strain for the FeGa films on Cu and NiFe underlayers, respectively. This relative change in strain was calculated from theXRD data using Braggs law, d¼λ/2dsinθ,w h e r eac h a n g ei nt h e relative strain between the two samples causes a shift, Δd,i nt h e lattice constant: Δε¼Δd/d 1¼(d2/C0d1)/d1¼sinθ2/sinθ1/C01. The FeGa films deposited on both the Cu and NiFe under- layers showed an increase ( ∼30%) in the intensity of their (110) diffraction peak compared to Ta or no underlayer, indicating anincreased (110) polycrystalline texture. This is consistent with pre-vious studies where a Cu buffer layer encourages a (110) crystalline texture along the growth direction for FeGa films. 47This enhance- ment can be attributed to the close lattice match of the FeGa (110)(d= 2.06 Å) film texture to the underlying Cu (111) ( d=2.09 Å) and NiFe (111) ( d=2.05 Å) film texture which is highlighted in Fig. S2 in the supplemental material . In contrast to Cu and NiFe, FIG. 2. S11absorption spectra as a function of magnetic bias field at 6 GHz for 100-nm FeGa films sputtered on a 2.5-nm underlayer of different materials (Ta,Cu, and NiFe). FIG. 3. (Left) XRD spectra of the main bcc (110) FeGa peak when grown on different underlayer materials. Solid lines are the best Voigt fit of the data in circles. Vertical dashed lines are used to highlight the shift in the (110) peakacross samples. (Right) AFM imaging of the same corresponding samples.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 013903 (2020); doi: 10.1063/5.0011873 128, 013903-3 Published under license by AIP Publishing.Ta exhibits a preferential β-(002) diffraction at 33.7° ( d= 2.66) that has a large lattice mismatch with FeGa. AFM imaging ( Fig. 3 ) was used to probe the differences in the microstructure of the FeGa films that can appear in their surfacemorphology when grown on the different underlayers. The surfaceroughness remained in the range of 1.1 –1.4 nm for all of the FeGa films. More interestingly, the magnetically softer FeGa films depos- ited on Cu or NiFe underlayers exhibited a smaller and moreuniform grain width distribution (29 ± 7 nm and 29 ± 6 nm, respec-tively) than the magnetically harder FeGa samples depositeddirectly on Si or with a Ta underlayer (46 ± 23 nm and 39 ± 14 nm, respectively). The properties discussed thus far for the FeGa films deposited on the different underlayers are summarized in Table I .T h i ss e r v e st o highlight the correlation and change in microstructure with the staticand dynamic magnetic properties. Note that the calculated change in the bcc (110) peak intensity ( ΔI 110) and the change in film strain (Δϵ) are reported relative to the FeGa films without an underlayer. In order to study the impact of the thickness of FeGa on its soft magnetic properties with an underlayer, varying thicknesses ofFeGa films were deposited using Cu as the underlayer. The ratio- nale for Cu as the underlayer material is to decouple the effect of two magnetic phases present if NiFe were used (e.g., exchangeeffects across the interface which becomes a greater fraction of themagnetic volume at smaller FeGa thickness). For the same reason, all of the samples were also capped with a 2.5-nm Cu layer toreduce the oxidation of the FeGa layer that becomes a more signifi- cant fraction of the total volume at smaller thicknesses. The normalized in-plane magnetic hysteresis for these samples is shown in Fig. 4 . The saturation magnetization before normalization (not pictured) decreases linearly with the film thick-ness. A clear dependence of the coercivity on the FeGa thickness can be observed, where the coercivity decreases from ∼17 Oe for a 100-nm film down to ∼12 Oe for a 10-nm film. In addition, from the corresponding XRD spectra it can be seen that there is anincrease in the linewidth of the bcc (110) FeGa diffraction peak asthe thickness decreases (0.55° for a 100-nm film to ∼1.3° for a 10-nm film). This is indicative of a trend toward smaller grain size as the film thickness decreases. The value of α efffor the FeGa films on a 2.5-nm Cu under- layer as a function of thickness was determined based on the FMRspectra in Fig. S3 in the supplemental material . It was found that α effdecreases from 0.053 to 0.004 for a 100-nm film compared to a 10-nm film. This trend, along with the decrease in coercivity, issummarized in Fig. 5 . These trends are consistent with the previous studies on FeGa films where coercivity and α effincrease with film thickness due to an increase in film roughness and inhomogeneity.21,48 In order to obtain the magnetostriction measurements for the FeGa films, a perpendicular AC magnetic field is applied along theshort axis of the silicon cantilever, while initially a constant 100 Oe bias field is applied in the long axis in order to saturate the magne- tization and assess the full magnetostriction during the measure-ment. The magnetic field induced stress, b, is calculated from the deflection at the cantilever tip using the following relation: 49 b¼/C0 dt2 sEs/3tfl2(1þvs), where dis the deflection, tsand tfare the substrate and film thicknesses (100 μm and 100 nm, respectively), FIG. 4. (Left) In-plane magnetic hysteresis loops of varying thicknesses of FeGa sputtered on Si with a 2.5-nm Cu underlayer. (Right) XRD spectra of the main bcc (110) FeGa peak for the same corresponding samples. Solid lines are the best Voigt fit of the XRD data in circles. All samples were capped with2.5-nm Cu to reduce the oxidation of the FeGa films. FIG. 5. Trend in in-plane coercivity and effective Gilbert damping coefficient (αeff) for thicknesses of 100, 25, and 10 nm of FeGa sputtered on Si with a 2.5 nm Cu underlayer.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 013903 (2020); doi: 10.1063/5.0011873 128, 013903-4 Published under license by AIP Publishing.lis the distance between the clamping edge and the probe location (27 mm), and Esandνsare Young ’s modulus and Poisson ratio of the Si substrate [169 GPa and 0.069, respectively, along the [110]in-plane direction for a Si (100) substrate 50]. For thin films, the magnetostrictive stress is the more relevant parameter to describe the magnetostrictive effects because the in-plane strain is prevented by the substrate clamping and thus one can measure only the stress; this also avoids the need to measurethe elastic properties of thin films which can be difficult andcannot necessarily be assumed to be the same as the bulk.However, for comparison with other literature on magnetostrictive thin films, magnetostriction in terms of strain can be calculated from the relation: λ¼/C0 2 31þνf Ef/C16/C17 /C2b, where Efand vfare Young ’s modulus and Poisson ratio of the FeGa film which are approxi- mated from the relation thatEf 1þνf/C16/C17 ¼50 GPa.51 From the data in Fig. 6 , the FeGa film deposited without an underlayer reached a saturation magnetostriction of 99 ppm. The FeGa film grown on the Cu underlayer largely maintained the same magnetostriction, displaying a saturation magnetostriction of95 ppm. Interestingly, the FeGa film grown on the NiFe underlayershowed a 27% increase in the saturation stress, reaching 125 ppm.While the literature values of magnetostriction reported for FeGa thin films can vary significantly across an order of magnitude, which may be due to differences in deposition parameters andmeasurement techniques, 31,52,53the importance of the results here is to highlight that the enhancement in the soft magnetic propertiesof the FeGa films can be achieved without a trade-off in magnetostriction.IV. CONCLUSIONS In summary, the effect of 2.5-nm Ta, Cu, and NiFe under- layers on the soft magnetic and microstructural properties of FeGathin films was compared. It was found that up to an 82% decrease in coercivity and ∼78% decrease in effective Gilbert damping coef- ficient can be achieved with the optimal NiFe underlayer material.Both Cu and NiFe, which have a good lattice match to the FeGafilms, influence the microstructure of the FeGa films by promotingan increased (110) polycrystalline texture, smaller grain size, and an increase in compressive film strain. Additionally, the films were able to maintain their high magnetostriction with an underlayer,making it an excellent material for application in both microwaveand strain-mediated magnetoelectric devices. SUPPLEMENTARY MATERIAL See the supplementary material for the full FMR spectra of the FeGa films with different underlayers and the complete XRDspectra of the FeGa, Ta, Cu, and NiFe films. ACKNOWLEDGMENTS We acknowledge the use of the fabrication facility at the Integrated Systems Nanofabrication Cleanroom (ISNC), the Nano and Pico Characterization Lab, and the Molecular Instrumentation Center (MIC) at the California NanoSystems Institute (CNSI) atUCLA. This work was also supported by the NSF NanosystemsEngineering Research Center for Translational Applications of Nanoscale Multiferroic Systems (TANMS) under the Cooperative Agreement Award (No. EEC-1160504). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1C. A. F. Vaz, J. Phys. Condens. Matter 24(33), 333201 (2012). 2Y. Cheng, B. Peng, Z. Hu, Z. Zhou, and M. Liu, Phys. Lett. A 382(41), 3018 (2018). 3T. Nan, H. Lin, Y. Gao, A. Matyushov, G. Yu, H. Chen, N. Sun, S. Wei, Z. Wang, M. Li, X. Wang, A. Belkessam, R. Guo, B. Chen, J. Zhou, Z. Qian,Y. Hui, M. Rinaldi, M. E. McConney, B. M. Howe, Z. Hu, J. G. Jones, G. J. 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1.1851426.pdf	1 ∕ f -type noise in a biased current perpendicular to the plane spin valve: A numerical study A. Rebei, L. Berger, R. Chantrell, and M. Covington Citation: Journal of Applied Physics 97, 10E306 (2005); doi: 10.1063/1.1851426 View online: http://dx.doi.org/10.1063/1.1851426 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Suppression of spin torque noise in current perpendicular to the plane spin-valves by addition of Dy cap layers Appl. Phys. Lett. 93, 103506 (2008); 10.1063/1.2978958 Dual current-perpendicular-to-plane giant magnetoresistive sensors for magnetic recording heads with reduced sensitivity to spin-torque-induced noise J. Appl. Phys. 99, 08S305 (2006); 10.1063/1.2165930 Spin transfer excited regular and chaotic spin waves in current perpendicular to plane spin valves J. Appl. Phys. 95, 6630 (2004); 10.1063/1.1689171 Spin momentum transfer in current perpendicular to the plane spin valves Appl. Phys. Lett. 84, 3103 (2004); 10.1063/1.1707227 High-frequency noise measurements in spin-valve devices J. Vac. Sci. Technol. A 21, 1167 (2003); 10.1116/1.1582458 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sat, 20 Dec 2014 23:44:291/f-type noise in a biased current perpendicular to the plane spin valve: A numerical study A. Rebei,a!L. Berger, R. Chantrell, and M. Covington Seagate Research Center, Pittsburgh, Pennsylvania 15222 sPresented on 8 November 2004; published online 6 May 2005 d We add the spin momentum-transfer torque to the stochastic Landau–Lifshitz equation and use it to study the noise spectrum as a function of the current and easy axis ﬁeld for conﬁgurations close toequilibrium. The current perpendicular to the plane structure is biased by a constant ﬁeldperpendicular to the polarization axis of the pinned layer. We show that this structure can exhibitlarge 1/f-type noise for frequencies in the microwave regime. This 1/ fnoise is not due to the spin torque. The spin torque can only change the amplitude of the noise. © 2005 American Institute of Physics.fDOI: 10.1063/1.1851426 g It is very well established that a current in a spin valve can be used to control the dynamics of the magnetization inone of the layers in a current perpendicular to the planesCPPddevice by transferring angular momentum from other layers. 1–4In magnetic recording, this additional property may not be desirable for diverse reasons. It has recently beenfound that CPP structures can show unwanted behavior 5 which may be detrimental to a giant magnetoresistancesGMR dsignal. These latter measurements show that in the presence of a current, a sizeable amount of noise is detectedin the microwave regime and below for certain conﬁgura-tions of the CPP valve. This behavior was attributed to spinmomentum transfer since the magnitude of noise showeddependency on the sign of the current. Theoretical studies 6,7 also showed that spin momentum transfer can induce randomswitching in a spin valve. In this paper we study numerically the effect of current in these structures when they are biased by an in-plane ﬁeldwhich has a large component perpendicular to the magneti-zation of the thick layer. This conﬁguration has been studiedexperimentally in Ref. 5. Here we model a similar structureusing the stochastic Landau–Lifshitz–Gilbert sLLG dequation with and without the spin momentum term. The use of LLGwith the spin torque for nonhomogeneous magnetic systemsis not well justiﬁed. A satisfactory theory that takes into ac-count nonuniformities does not exist yet. The CPP structure has two layers, one is very thick and pinned while the other one is thin and free to precess underthe inﬂuence of current or an external ﬁeld. This structure isimportant because of its potential use in recording devices.Hence it is important to study its stability under various con-ditions. It has been observed in Ref. 5 that current in a biased spin valve can give rise to 1/ f-type noise even at relatively high frequencies, i.e., in the megahertz range for a devicewith a ferromagnetic resonance sFMR dfrequency of the or- der of 10 GHz. Frequencies in this range may interfere withdesigns of recording heads and hence the need to understandany potential source for this noise. Since the current is per-pendicular to the plane, it is natural to ask if spin momentum has a role to play in the generation of this noise. For ex-ample, Ref. 7 considers this scenario and attributes the noiseto spin momentum transfer. However, this scenario does notexplain the trends measured in Ref. 5. The numerical simulation uses the LLG equation with a random white Gaussian ﬁeld hthat simulates thermal ﬂuc- tuations at T=100 °C.The damping constant ais taken to be 0.020–0.025. The use of this damping and noise term is welljustiﬁed only at temperatures below the Curie temperatureand for situations where the dynamic is adiabatic and closeto equilibrium. 8All these conditions are approximately met in our experiment. The mesh size has been varied between232n m 2and 15 315 nm2; however, all the results pre- sented here were carried out at 10 310 nm2. Below the 232-nm2mesh, deviations in the transfer curve of resistance versus current become apparent at zero temperature. Thesedeviations were mostly attributed to the spin transfer torquerather than the exchange. At ﬁnite temperature, these devia-tions are more pronounced and hence we avoided any com-parable mesh sizes in the simulations. For the exchange term,the continuum approximation has been avoided and no at-tempt has been made to self-consistently renormalize the ex-change with the mesh size. It should be pointed out that thespin torque is also expected to depend on the exchange con-stant, the mesh size, and the thickness of the ﬁlm. However,since the polarization parameter pis unknown, we will ig- nore these difﬁculties. In the presence of spin torque, theLLG equation takes the following form: dm dt=−gm3FHeff+am3Sdm dtD+hG+apI dm 3smp3md, wheremis the normalized magnetization of the free layer with thickness dandmpis the local direction of the magne- tization in the ‘pinned’ layer. The current Iis taken positive when it ﬂows from the pinned layer to the free layer. pis the polarization coefﬁcient which is taken to be 0.5 in this studyandais a geometrical factor.The random ﬁeld hsatisﬁes the usual simplest correlation functions which are assumed inde- adElectronic mail: arebei@mailaps.orgJOURNAL OF APPLIED PHYSICS 97, 10E306 s2005 d 0021-8979/2005/97 ~10!/10E306/3/$22.50 © 2005 American Institute of Physics 97, 10E306-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sat, 20 Dec 2014 23:44:29pendent of the current and the ellipticity of the precession khistdhjst8dl=2akBTdijdst−t8d. In the actual experiment, the structure of the spin valve is somewhat more complicated than a simple pinned layerand a free layer. However, the noise measurements were car-ried out in a region where only the free layer is sensitive tothe external bias ﬁeld. Hence in the following, we assume apinned layer and apply the bias ﬁeld to the free layer. Themagnetization of the pinned layer is kept ﬁxed in plane alongthe easy axis which is assumed to have a uniaxial anisotropyof 50 Oe. The bias ﬁeld has a constant hard axis componentof 300 Oe but a variable easy axis component. This lattercomponent will be swept between −200 and 200 Oe whichhappens to be the region where most of the 1/ f-type noise is detected in the simulation as well as in the experiment. Itshould be stressed that in all the measurements the current isbelieved to be below the critical current for switching. Thisis an important difference with previous works where noisehas been detected in the switching process. The single-particle studies do not show the noise which was observed in the experiments and hence it was soon re-alized that the inhomogenous conﬁguration of the magneti-zation should play a fundamental role in the generation ofthis noise. The single-particle simulation shows smooth R -Hcurves; however, because of the hard axis bias ﬁeld the micromagnetic simulations show curves which are notsmooth, i.e., they have discontinuous slope. The discontinu-ity in the slopes is dependent on the magnitude of the currentbut not on its direction. Hence it is clear that the oersted ﬁeldplays a role in the structure of these transfer curves. Themeasured GMR in the original experiment is believed to be ameasure of the xcomponent of the magnetization in the free layer. Hence, any detected noise is related to ﬂuctuations inthis component C xxsvd=Edtkmxst+tdmxstdlexpf−ivtg.In the simulations, we can measure the noise in any component. We found that the noise is largest in the xcom- ponent for easy axis ﬁelds around zero but it is largest in the ycomponent for large easy axis ﬁelds; this agrees with the experiment. In Figs. 1 and 2, we show a real time trace of themagnetization components m xandmyvery close to one of the discontinuities in the slope of the R-Hcurve. We see in this case that the ﬂuctuation is largest for the xcomponent sMs=1440 emu/cc d. They are negligible for the out-of plane component, the zcomponent snot shown d. Figure 1 clearly shows that the magnetization is switching between two statesthat differ only in the easy axis component. The switchingseems to be thermally activated and not due to spin momen-tum transfer. This is the main point of the numerical simula- FIG. 1. The magnetization component parallel to the polarization axis, i.e., the easy axis sorxaxisdas a function of time. hx=90Oe,hy=300 Oe, and I=5 mA. FIG. 2. The magnetization component along the direction perpendicular to the polarization axis of the current. hx=90Oe,hy=300 Oe, and I=5 mA. FIG. 3. The Cphase: This phase is stable in the absence of thermal ﬂuc- tuations. The horizontal arrows pointing to the right are those for the mag-netization of the pinned layer. This conﬁguartion is for h x=90Oe, hy =300 Oe, and I=5 mA.10E306-2 Rebei et al. J. Appl. Phys. 97, 10E306 ~2005 ! [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sat, 20 Dec 2014 23:44:29tions. The spin torque, however, tends to make one of the states more or less stable than the other.To further check thispoint, we show in Figs. 3 and 4 two stable conﬁgurations ofthe magnetization which are very close in energy. In fact, weﬁnd that the difference in energy, uE 90−E110u<2kBT,i so ft h e order of the thermal ﬂuctuations in the system. Finally, inFigs. 5 and 6 we show that the turning off the spin momen-tum torque in the LLG equation does not affect the 1/ f-type noise which is another indication that the origin of this noiseis mainly due to the balance between the bias ﬁeld and theoersted ﬁeld. Similar simulations without a hard axis ﬁeld donot show this excessive noise. Depending on the relativedirections of the oersted ﬁeld and the bias ﬁeld, the spinmomentum torque can amplify or slightly suppress the noise in these conﬁgurations. Figures 5 and 6 show one conﬁgura-tion where the spin momentum transfer acts to stabilize the C phase. This shows that a detailed knowledge of the phasediagram of the CPP system is required when it comes to thestudy of noise. The simulations show, as in the experiment,that the higher the current, the larger the easy axis ﬁeld atwhich we observe the most noise. This can be easily under-stood from Fig. 1 and the curling of the magnetization due tothe oersted ﬁeld. In fact, at much higher currents, e.g., forI.40 mA, the oersted ﬁeld dominates and the magnetization assumes a vortex conﬁguration. This latter conﬁguration hasno 1/fnoise. The simulations also show that the state with sh x=90 Oe, I=5mA dis about twice noisier than the state with shx=90 Oe,I=−5mA d. This relative value is, however, much smaller than those measured in the experiment. It issuspected that the ﬁeld from the leads is also a contributingfactor to the noise. In conclusion, we have shown that in spin valves with bias ﬁelds almost perpendicular to the polarization axis, themagnetization becomes highly nonuniform in the presence ofthe ﬁeld from the current. Because of thermal ﬂuctuations,the conﬁguration of the magnetization may show transitionsbetween two conﬁgurations. These transitions are the originof the 1/ f-type noise and not the spin momentum transfer. We thank G. J. Parker for his LLG solver. 1J. Slonczewski, J. Magn. Magn. Mater. 159,L 1 s1996 d. 2L. Berger, Phys. Rev. B 54, 9353 s1996 d. 3M. Tsoiet al., Phys. Rev. Lett. 80, 4281 s1998 d. 4J. Katine et al., Phys. Rev. Lett. 84, 4212 s2000 d. 5M. Covington et al., Phys. Rev. B 69, 184406 s2004 d. 6Z. Li and S. Zhang, Phys. Rev. B 68, 024404 s2003 d. 7J.-G. Zhu, Ninth Joint MMM-Intermag Conference 2004, Anaheim, Cali- fornia, USA sunpublished d. 8A. Rebei and M. Simionato sunpublished d. FIG. 4. The Sphase with slightly higher bias ﬁeld than for the Cphase. hx=110 Oe, hy=300 Oe, and I=5 mA. FIG. 5. The power spectrum density sPSDdin theMxcomponent with the spin torque set to zero in the LLG equation. FIG. 6. The PSD in the Mxcomponent using the LLG equation with a spin torque term. Same scale as in Fig. 6.10E306-3 Rebei et al. J. Appl. Phys. 97, 10E306 ~2005 ! [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sat, 20 Dec 2014 23:44:29
1.3369582.pdf	The critical current density in composite free layer structures for spin transfer torque switching Chun-Yeol You Citation: Journal of Applied Physics 107, 073911 (2010); doi: 10.1063/1.3369582 View online: http://dx.doi.org/10.1063/1.3369582 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/107/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of interlayer exchange coupling parameter on switching time and critical current density in composite free layer J. Appl. Phys. 115, 17D111 (2014); 10.1063/1.4861215 Reduced spin transfer torque switching current density with non-collinear polarizer layer magnetization in magnetic multilayer systems Appl. Phys. Lett. 100, 252413 (2012); 10.1063/1.4730376 Reduction of switching current density in perpendicular magnetic tunnel junctions by tuning the anisotropy of the CoFeB free layer J. Appl. 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Downloaded to ] IP: 131.91.169.193 On: Tue, 11 Aug 2015 03:58:42The critical current density in composite free layer structures for spin transfer torque switching Chun-Yeol Youa/H20850 Department of Physics, Inha University, Incheon 402-751, Republic of Korea /H20849Received 10 November 2009; accepted 23 February 2010; published online 12 April 2010 /H20850 The critical current density for spin transfer torque switching with composite free layer structures is investigated using the macrospin Landau–Lifshitz–Gilbert equation. We consider a magnetictunneling junction with a rigid ﬁxed layer, and a composite free layer consisting of two coupledferromagnetic layers in which the coupling is parallel or antiparallel. The dependence of criticalcurrent density on thickness, coupling sign and strength, spin torque efﬁciency, and magnetizationof the composite free layer is explored. We determine that reduction in the critical current densitycan be achieved only through careful design of the composite free layer structures. We show thedetailed spin dynamics of the composite free layer when the reduction in the critical current densityis accomplished. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3369582 /H20852 I. INTRODUCTION The spin transfer torque /H20849STT /H20850in magnetic tunneling junction /H20849MTJ /H20850nanopillar geometry1,2is attracting many re- searchers due to its rich physics3–5and technical importance.6The magnetization direction of the free layer is switched at the critical current density Jcin MTJ structures due to the STT. Reducing Jcin order to realize STT- magnetoresistive random access memory /H20849MRAM /H20850is a sig- niﬁcant challenge because a higher Jcrequires a larger tran- sistor size and causes severe Joule heating.7–10Jcis proportional to the thickness of the free layer. Therefore, thethinner the free layer, the smaller the value of J cthat can be achieved. However, when the volume of the free layer issmall, thermal activation energy becomes an important issue.Since the thermal activation energy is proportional to thevolume of the free layer and J cis proportional to its thick- ness, there is a trade-off relationship between them. To main-tain the thermal activation energy while reducing J c, com- posite free layers /H20849CFLs /H20850consisting of two ferromagnetic layers with various types of coupling, including syntheticferrimagnetic free layer structures, have been proposed andtested. 11–17It is clear that CFLs have a better thermal activa- tion energy than single layers due to their larger volume.However, surprisingly, there is no systematic theoretical ap-proach to the determination of the J cvalue for CFL struc- tures, while determination of the Jcvalue of single free lay- ers has been examined in detail.18In this study, we consider theJcvalue for various CFL structures by employing the macrospin Landau–Lifshitz–Gilbert /H20849LLG /H20850equation includ- ing the STT.19We found that Jcstrongly depends not only on the detailed structure of the CFL /H20849thickness and magnetiza- tion /H20850, but also on the strength and type of the interlayer ex- change coupling. II. MACROSPIN LLG EQUATIONS We considered MTJ stacks consisting of ﬁxed ferromag- netic /H20849FFix/H20850, insulator /H20849I/H20850, ﬁrst ferromagnetic, /H20849F1/H20850, nonmag-netic /H20849NM /H20850, and second ferromagnetic /H20849F2/H20850layers, as shown in Fig. 1. The thicknesses of the F 1and F 2layers are d1and d2, respectively. We assumed that the F Fixlayer is rigid and that its magnetization direction P/H6023=/H208491,0,0 /H20850. A positive cur- rent means that the electrons ﬂow from F Fixto F 1.F1prefers a parallel conﬁguration with F Fix. Initially, the magnetization of the F 1layer /H20849M1/H20850is parallel to the − xdirection, while the magnetization of F 2/H20849M2/H20850is aligned in the + x/H20849−x/H20850direction for antiparallel /H20849parallel /H20850coupling. The LLG equations with the STT term for the F 1and F 2layers are dM/H60231 dt=−/H9253/H20849M/H60231/H11003H/H6023eff1/H20850+/H92511 M1/H20873M/H60231/H11003dM/H60231 dt/H20874+ STT 1, /H208491/H20850 a/H20850Electronic mail: cyyou@inha.ac.kr.Acurrent > 0 z y NMF2A xy d2 FFixIF1 d1 electron flow FIG. 1. /H20849Color online /H20850Schematic diagram of the layered structure. The CFL consists of two ferromagnetic layers /H20849F1and F2/H20850separated by a NM layer with a rigid ﬁxed layer /H20849FFix/H20850. The direction of the positive current is deﬁned as being from the free layer to ﬁxed layer. The thickness of F1and F2ared1 andd2, respectively.JOURNAL OF APPLIED PHYSICS 107, 073911 /H208492010 /H20850 0021-8979/2010/107 /H208497/H20850/073911/4/$30.00 © 2010 American Institute of Physics 107 , 073911-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.91.169.193 On: Tue, 11 Aug 2015 03:58:42STT 1=−/H9253a1J M1/H20851M/H60231/H11003/H20849M/H60231/H11003P/H6023/H20850/H20852−/H9253/H20849b0J+b1J2/H20850/H20849M/H60231 /H11003P/H6023/H20850−/H9253a2,1/H20849−J/H20850 M1M2/H20851M/H60231/H11003/H20849M/H60231/H11003M/H60232/H20850/H20852 −/H9253b2,1 M2/H20849M/H60231/H11003M/H60232/H20850, /H208492/H20850 dM/H60232 dt=−/H9253/H20849M/H60232/H11003H/H6023eff2/H20850+/H92512 M2/H20873M/H60232/H11003dM/H60232 dt/H20874+ STT 2, /H208493/H20850 and STT 2=−/H9253a2,2J M1M2/H20851M/H60232/H11003/H20849M/H60232/H11003M/H60231/H20850/H20852−/H9253b2,2 M1/H20849M/H60232/H11003M/H60231/H20850. /H208494/H20850 Here, H/H6023eff1,2are the effective ﬁelds, /H92511,2are the Gilbert damping parameters of F 1and F 2, respectively, and /H9253is the gyromagnetic ratio. The STT 1terms are the torques acting on F1due the STT. a1is the so-called Slonczewski term from FFix, deﬁned by a1=/H9257p/H6036/2e/H92620M1d1, where /H9257p,e, and/H92620are the spin torque efﬁciency of F Fix, the electron charge, and the permeability, respectively. b0andb1are the linear and qua- dratic ﬁeldlike terms, respectively. a2,1is the Slonczewski torque acting on F 1due to F 2, where a2,1=/H92572/H6036/2e/H92620M1d1 and/H92572is the spin torque efﬁciency of F 2.b2,1andb2,2are other ﬁeldlike terms between the F 1and F 2layers, which can also be referred to as the interlayer exchange coupling en-ergy E EXwith the relation b2,/H208491,2 /H20850=EEX//H92620M1,2d1,2.20,21We considered the NM layer to be a metallic layer and b2,/H208491,2 /H20850to be independent of J. A negative /H20849positive /H20850b2,/H208491,2 /H20850represents antiparallel /H20849parallel /H20850exchange coupling. STT 2is the torque acting on F 2due to F 1. Here, a2,2=/H92571/H6036/2e/H92620M2d2, where /H92571 is the spin torque efﬁciency of F 1. More details can be found in the literature.19 III. CALCULATION RESULTS Only the spin dynamics of the CFL F 1/H20849d1/H20850/NM /F2/H20849d2/H20850 structures were investigated because we assumed a rigid ﬁxed layer. For simplicity, we ignored the anisotropy ﬁelds/H20849except for the shape anisotropy /H20850because it is dominant in STT-MRAM applications. We set H ext=0,/H92511=/H92512=0.01, and /H9257p=0.7 through all our studies. The dimensions of F 1/H20849F2/H20850 are 100 /H1100350/H11003d1/H20849d2/H20850nm3and the corresponding demagne- tization factors were evaluated22and used in our calculations to include shape anisotropy. The ﬁeldlike terms /H20849b0andb1/H20850 between F Fixand F 1were ignored. The effects of b0andb1 were not small for a large value of Jbut their contributions around Jcwere not signiﬁcant. They will affect some details of the dynamics but not the overall trends. The magnitudes of a2,1anda2,2should be discussed. /H92571,2 can be reduced from their individual bulk values due to the low thickness of F 1,2. Therefore, we varied /H92571,2in our calcu- lations, but assumed that /H92571=/H92572=/H9257for simplicity. In this study, we ignored the angular dependence of the Sloncze-wski term and considered only the switching current densityfrom the parallel to the antiparallel cases.First, we considered the antiparallel exchange coupling /H20849E EX/H110210/H20850. Figures 2/H20849a/H20850–2/H20849d/H20850show Jcford1=1.5 nm and M1=M2=1.1/H11003106A/m with various values of d2/H20849from 1.0 to 2.0 nm /H20850. The meaning of the Greek numbers will be ex- plained subsequently. For comparison, the values of Jcfor single layers with ds=1.5 to 4.0 nm are also indicated in the ﬁgures using red open symbols. For weak coupling /H20849EEX= −0.05 mJ /m2/H20850,Jcdecreases /H20849increases /H20850with increasing /H9257 when d1/H11022d2,/H20849d1/H11021d2/H20850. However, its dependence on /H9257is weak for strong EEX, especially for low values of d2. When EEXis strong, F 1and F 2are already coupled together tightly and the main dynamics is governed by b2, regardless of /H9257. From Fig. 2, we speculate that a lower value of d2and weaker EEXare preferable to obtain a smaller value of Jc. Only in some special cases, marked by the blue circles inFigs. 2/H20849a/H20850and2/H20849b/H20850, is the value of J csmaller than that of the corresponding single layers /H20849ds=d1+d2/H20850. Therefore, the tai- loring of b2will be important, and this can be achieved by controlling the thickness and composition of the NMlayer. 23,24We note that the dependence of EEXis the opposite of recent experimental observation.13 Figures 3/H20849a/H20850–3/H20849d/H20850show the Jcvalues for d2=1.5 nm and M1=M2=1.1/H11003106A/m with various values of d2/H20849from 1.0 to 2.0 nm /H20850. From Figs. 2and3, we ﬁnd that Jcincreased /H20849decreased /H20850as/H9257increased when d1/H11021d2,/H20849/H11022d2/H20850. The detailed physics of the dependence of Jcon/H9257is not yet clear and will require additional analytic study. When EEXis strong, the dependence of Jcon/H9257is very weak for d1/H11022d2. We note that the reduction in Jcfor small /H9257was achieved only with small EEX, as marked by the blue circles in Figs. 3/H20849a/H20850and3/H20849b/H20850. This will be discussed in more detail subsequently.151.8 1.8(b)d2(nm) = 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 4.0 4.0 3.5dsm2)EEX= -0.05 mJ/m2(a) EEX= -0.1 mJ/m2 3.5 0.91.21.5 1.21.5 212.4 2.8(d) (c)2.53.0 2.0Jc(1011A/m d1=1.5 nm I2.53.0 EEX= -0.5 mJ/m2EEX=- 1 . 0m J / m2 0.0 0.3 0.61.21.51.82.1 0.0 0.3 0.61.62.02.4 II3.03.54.0Jc(1011A/m2) /CID753.54.0 /CID75/CID75 FIG. 2. /H20849Color online /H20850Jcas a function of /H9257ford1=1.5 nm with various values of d2.EEX=−0.05, −0.01, −0.5, and −1.0 mJ /m2for /H20849a/H20850–/H20849d/H20850, respec- tively. The Jcvalues of the single layers are also depicted. The meaning of the Greek numbers is explained in Fig. 6.073911-2 Chun-Yeol You J. Appl. Phys. 107 , 073911 /H208492010 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.91.169.193 On: Tue, 11 Aug 2015 03:58:42Next, we investigated the case in which EEX/H110220. The results are plotted in Figs. 4/H20849a/H20850and4/H20849b/H20850forEEX=0.05 and 1.0 mJ /m2, respectively, with various values of d1and d2. For strong EEX, the values of Jcare almost the same as those of the corresponding single layer cases, as shown in Fig.4/H20849b/H20850. However, in the weak coupling case when d 1=1.0 and d2=1.5 nm, a noticeable reduction in Jcis achieved, as marked by the blue circle in Fig. 4/H20849a/H20850. These results are con- sistent with recent observations and interpretations by Yen et al.15and Lee et al.17 Finally, we simulated JcforM1/HS11005M2with selected val- ues of d1and d2for only weak antiparallel and parallel /H20849EEX=−0.05,0.05 mJ /m2/H20850couplings, which showed the most interesting results of the previous calculations. The sample structures of each alphabet are summarized in Table I and the results are plotted in Figs. 5/H20849a/H20850and5/H20849b/H20850. In the antiparallel coupling case, sample D shows a reduction in Jc /H20849=6.8/H110031010A/m2/H20850. The most pronounced reduction was found for parallel coupling in sample G /H20849Jc=5.2 /H110031010A/m2/H20850. At this stage, it is not yet clear what the mechanism governing the reduction in Jcin our study is, due to the lack of an analytic formalism for the Jcvalue of CFLs. IV. DISCUSSION Consider the physical origins of the reduction in Jc. Since the spin dynamics of CFLs depend on many param-eters and there is no analytic expression for J c, it is impos- sible to present clear and solid physics for the reductionmechanism. We explored the spin dynamics in greater detailfor cases I through VI /H20849the Greek numbers are found in Figs. 2–5/H20850, and the results are depicted in Figs. 6/H20849a/H20850–6/H20849f/H20850, respec- tively. The normalized M xandMyfor F 1and F 2as a function of time are shown in this ﬁgure. First, Figs. 6/H20849a/H20850and6/H20849b/H20850 correspond to the cases of the smallest Jcin Figs. 2/H20849a/H20850and 2/H20849d/H20850with EEX=−0.05 and −0.1 mJ /m2/H20849I and II /H20850, respec- tively. In case I, where two spins precess with a phase dif-ference of /H9266with a single frequency at the beginning /H20849t /H110213n s /H20850, after t/H110223 ns their precession with multifrequen- cies is shown and the phase differences become complicated. In case II, where strong interlayer exchange coupling is ob-served and which has a larger J c, the two layers move to- gether with a single frequency, as shown in Fig. 6/H20849b/H20850. Next, we compare Figs. 6/H20849c/H20850and 6/H20849d/H20850, which show the detailed spin dynamics of cases III and IV in Fig. 4/H20849a/H20850, respectively. The only difference between cases III and IV is d1; however, both Jcand the detailed spin dynamics are quite different. In case III, the two spins precess with different phases withmultifrequencies as in case I, whereas case IV showsstrongly coupled switching. Due to the different phase of thespin dynamics, each spin precession assists another spin mo-tion, as previously claimed by Yen et al. , 15for weak ferro- magnetic coupling layers. Figures 6/H20849e/H20850and6/H20849f/H20850, correspond- ing to cases V and VI /H20849Fig. 5/H20850, respectively, support our hypothesis. In the case of either parallel or antiparallel inter-layer exchange coupling, when one spin starts to precessmore easily and interacts with the other of a different phase,the reduction in J cis accomplished. We have found that multifrequency precession modes and different phases are the key requirements for loweringtheJ cvalue. Accordingly, we present one possible scenariod1(nm) = 1.0 1.1 1.2 1.3 1.4 15 16 17 18 19 20 1.21.41.61.8 1.21.41.61.8 (a)EEX= -0.05 mJ/m21.5 1.6 1.7 1.8 1.9 2.0Jc(1011A/m2)ds 3.5 3.0 2.5(b)EEX= -0.1 mJ/m2 3.5 3.0 2.5 1.01.0 141.61.82.0 2.02.4Jd2=1.5 nm2.0 (c)EEX= -0.5 mJ/m2011A/m2)4.0 3.5 30(d)EEX=- 1 . 0m J / m2 4.0 3 0.0 0.3 0.61.21.4 0.0 0.3 0.61.21.6Jc(1 /CID753.0 2.5 /CID753.5 3.0 FIG. 3. /H20849Color online /H20850Jcas a function of /H9257ford2=1.5 nm with various values of d1.EEX=−0.05, −0.1, −0.5, and −1.0 mJ /m2for /H20849a/H20850–/H20849d/H20850, respec- tively. The Jcvalues of the single layers are also depicted. 182.0 ds(a)EEX= 0.05 mJ/m2d1;d2= 1.0; 1.5 1.5; 1.0 1.5; 1.5 1.5; 2.0 2.0; 1.5 Single Layer 4.0(b)EEX=1 . 0m J / m2 0.60.81.01.21.41.61.8 IIIsJc(1011A/m2)3.5 3.0 2.5 2.0 1.5IV 0.0 0.3 0.6 0.0 0.3 0.6 0.9/CID75/CID75 FIG. 4. /H20849Color online /H20850Jcas a function of /H9257for various values of d1andd2. /H20849a/H20850EEX=0.05 and /H20849b/H208501.0 mJ /m2are considered for parallel coupling.10(b)EEX= -0.05 mJ/m2(b)EEX=0 . 0 5m J / m2 0.81.01.2 0.40.60.81.0 A C E B D FJc(1011A/m2) VI G H IV 0.0 0.3 0.6 0.0 0.3 0.6 /CID75/CID75 FIG. 5. /H20849Color online /H20850Jcas a function of /H9257for selected sample structures with different magnetizations and thicknesses for EEX=−0.05 and /H20849b/H20850 +0.05 mJ /m2. The sample structures are shown in Table I.073911-3 Chun-Yeol You J. Appl. Phys. 107 , 073911 /H208492010 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.91.169.193 On: Tue, 11 Aug 2015 03:58:42for the reduction mechanism. When the spin-polarized cur- rent is injected, the lowest energy spin wave will be excitedeasily in F 1or F 2. Since there is coupling between F 1and F 2, another layer spin wave will be excited more easily than inthe case without coupling. Since the characteristics of thespin wave of each layer are different but they are weaklycoupled, they can assist each other. Therefore, we conjecturethat the spin motion of the different frequencies and phasesand their mutual assistance are important to achieving thereduction in J c. V. CONCLUSION We investigated the Jcvalues of CFL structures with various thicknesses, coupling strengths and signs, and mag-netizations by means of macrospin LLG. We found that thereduction in J cis achieved only within a very narrow set of conditions and is not substantial. From our study, we con-clude that the beneﬁt of CFL structures is not signiﬁcant,even though they allow for some gain in thermal activationenergy. Furthermore, in general, the J cvalue of a CFL isgreater than that of the corresponding single layer. Only carefully designed CFL structures have smaller Jcvalues with better thermal activation energy. Further studies are re-quired to clarify the detailed mechanism underlying the re-duction in J c. ACKNOWLEDGMENTS This work was supported by a 2010 Inha University re- search grant. The author thanks Professor K.-J. Lee and Dr.B.-C. Min for their helpful discussions. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2L. Berger, J. Appl. Phys. 49, 2156 /H208491978 /H20850;Phys. Rev. B 54, 9353 /H208491996 /H20850. 3E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285,8 6 7 /H208491999 /H20850. 4S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoe- lkopf, R. A. Buhrman, and D. C. Ralph, Nature /H20849London /H20850425,3 8 0 /H208492003 /H20850. 5Y . Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 /H208492005 /H20850. 6R. Beach, T. Min, C. Horng, Q. Chen, P. Sherman, S. Le, S. Young, K. Yang, H. Yu, X. Lu, W. Kula, T. Zhong, R. Xiao, A. Zhong, G. Liu, J.Kan, J. Yuan, J. Chen, R. Tong, J. Chien, T. Torng, D. Tang, P. Wang, M.Chen, S. Assefa, M. Qazi, J. DeBrosse, M. Gaidis, S. Kanakasabapathy, Y .Lu, J. Nowak, E. O’Sullivan, T. Mafﬁtt, J.Z. Sun, and W.J. Gallagher,Tech. Dig. - Int. Electron Devices Meet 2008 , 305. 7C.-Y . You, S.-S. Ha, and H.-W. Lee, J. Magn. Magn. Mater. 321, 3589 /H208492009 /H20850. 8H. Kubota, A. Fukushima, K. Yakushiji, S. Yakata, S. Yuasa, K. Ando, M. Ogane, Y . Ando, and T. Miyazaki, J. Appl. Phys. 105, 07D117 /H208492009 /H20850. 9Y . Jiang, T. Nozak, S. Abe, T. Ochiai, A. Hirohata, N. Tezuka, and K. Inomata, Nature Mater. 3, 361 /H208492004 /H20850. 10J. C. Lee, C.-Y . You, S.-B. Choe, K.-J. Lee, and K.-H. Shin, J. Appl. Phys. 101, 09J102 /H208492007 /H20850. 11T. Ochiai, Y . Jiang, A. Hirohata, N. Tezuka, S. Sugimoto, and K. Inomata, Appl. Phys. Lett. 86, 242506 /H208492005 /H20850. 12J. Hayakawa, S. Ikeda, Y . M. Lee, R. Sasaki, T. Meguro, F. Matsukura, H. Takahashi, and H. Ohno, Jpn. J. Appl. Phys., Part 2 45, L1057 /H208492006 /H20850. 13J. Hayakawa, S. Ikeda, K. Miura, M. Yamanouchi, Y . Min Lee, R. Sasaki, M. Ichimura, K. Ito, T. Kawahara, R. Takemura, T. Meguro, F. Matsukura,H. Takahashi, H. Matsuoka, and H. Ohno, IEEE Trans. Magn. 44, 1962 /H208492008 /H20850. 14M. Ichimura, T. Hamada, H. Imamura, S. Takahashi, and S. Maekawa, J. Appl. Phys. 105, 07D120 /H208492009 /H20850. 15C.-T. Yen, W.-C. Chen, D.-Y . Wang, Y .-J. Lee, C.-T. Shen, S.-Y . Yang, C.-H. Tsai, C.-C. Hung, K.-H. Shen, T.-J. Tsai, and M.-J. Kao, Appl. Phys. Lett. 93, 092504 /H208492008 /H20850. 16X. Yao, R. Malmhall, R. Ranjan, and K.-P. Wang, IEEE Trans. Magn. 44, 2496 /H208492008 /H20850. 17K. Lee, W.-C. Chen, X. Zhu, X. Li, and S.-H. Kang, J. Appl. Phys. 106, 024513 /H208492009 /H20850. 18J. Z. Sun, Phys. Rev. B 62, 570 /H208492000 /H20850. 19C.-Y . You, Curr. Appl. Phys. 10, 952 /H208492010 /H20850. 20I. Theodonis, N. Kioussis, A. Kalitsov, M. Chshiev, and W. H. Butler, Phys. Rev. Lett. 97, 237205 /H208492006 /H20850. 21D. M. Edwards, F. Federici, J. Mathon, and A. Umerski, Phys. Rev. B 71, 054407 /H208492005 /H20850. 22J. A. Osborn, Phys. Rev. 67, 351 /H208491945 /H20850. 23C.-Y . You, C. H. Sowers, A. Inomata, J. S. Jiang, S. D. Bader, and D. D. Koelling, J. Appl. Phys. 85, 5889 /H208491999 /H20850. 24C.-Y . You and S. D. Bader, J. Appl. Phys. 92, 3886 /H208492002 /H20850.TABLE I. Magnetizations and thicknesses of CFL samples. ABCD E F GHI /H20851M1,M2/H20852 /H20849106A/m/H20850 0.86, 1.1 0.86, 1.1 1.1, 0.86 1.1, 0.86 1.1, 1.1 1.1, 1.1 0.86, 1.1 1.1, 0.86 1.1, 1.1 /H20851d1,d2/H20852 /H20849nm /H20850 1.0, 1.5 1.5, 1.0 1.0, 1.5 1.5, 1.0 1.0, 1.5 1.5, 1.0 1.0, 1.5 1.0, 1.5 1.0, 1.5 0.51.0 (a) I Jc= 1.06x1011A/m2(b) II Jc=1.76x1011A/m2 -1.0-0.50.0 0.51.0M1x/Ms M2x/Ms M1y/Ms M2y/MsMx/Ms (c) III Jc=7 . 2 x 1 010A/m2(d) IVJ c=1.84x1011A/m2 -1.0-0.50.0 0.51.0Mx/Ms (e) V Jc=6 . 8 x 1 010A/m2(f) VIJ c=5 . 2 x 1 010A/m2 0.0 2.0 4.0 6.0 8.0 10.0-1.0-0.50.0 0.0 2.0 4.0 6.0 8.0 10.0Mx/Ms t(ns) t(ns) FIG. 6. /H20849Color online /H20850Normalized MxandMyfor F1and F2as a function of time. /H20849a/H20850–/H20849f/H20850correspond to cases I–VI in Figs. 2–5, respectively.073911-4 Chun-Yeol You J. Appl. Phys. 107 , 073911 /H208492010 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.91.169.193 On: Tue, 11 Aug 2015 03:58:42
1.5143447.pdf	J. Appl. Phys. 127, 085101 (2020); https://doi.org/10.1063/1.5143447 127, 085101 © 2020 Author(s).Determining absolute Seebeck coefficients from relative thermopower measurements of thin films and nanostructures Cite as: J. Appl. Phys. 127, 085101 (2020); https://doi.org/10.1063/1.5143447 Submitted: 23 December 2019 . Accepted: 08 February 2020 . Published Online: 24 February 2020 S. J. Mason , A. Hojem , D. J. Wesenberg , A. D. Avery , and B. L. Zink COLLECTIONS This paper was selected as an Editor’s Pick Determining absolute Seebeck coefficients from relative thermopower measurements of thin films and nanostructures Cite as: J. Appl. Phys. 127, 085101 (2020); doi: 10.1063/1.5143447 View Online Export Citation CrossMar k Submitted: 23 December 2019 · Accepted: 8 February 2020 · Published Online: 24 February 2020 S. J. Mason,a)A. Hojem,b)D. J. Wesenberg,c)A. D. Avery,d) and B. L. Zinke) AFFILIATIONS Department of Physics and Astronomy, University of Denver, Denver, CO 80208, USA a)Current address: Broadcom Ltd. Fort Collins, CO 80525, USA. b)Current address: Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA. c)Current address: Lam Research, Portland, OR 97062, USA. d)Current address: Department of Physics, Metropolitan State University of Denver, Denver, CO 80204, USA. e)Author to whom correspondence should be addressed: barry.zink@du.edu ABSTRACT Measurements of thermoelectric effects such as the Seebeck effect, the generation of electric field in response to an applied thermal gradient, are important for a range of thin films and nanostructures used in nanoscale devices subject to heating. In many cases, a clear understand-ing of the fundamental physics of these devices requires knowledge of the intrinsic thermoelectric properties of the material, rather than theso-called “relative ”quantity that comes directly from measurements and always includes contributions from the voltage leads. However, for a thin film or nanostructure, determining the absolute Seebeck coefficient, α abs, is challenging. Here, we first overview the challenges for measuring αabsand then present an approach for determining αabsfor thin films from relative measurements made with a micromachined thermal isolation platform at temperatures between 77 and 350 K. This relies on a relatively simple theoretical description based on theMott relation for a thin film sample as a function of thickness. We demonstrate this technique for a range of metal thin films, which showthatα absalmost never matches expectations from tabulated bulk values, and that for some metals (most notably gold) even the sign of αabs can be reversed. We also comment on the role of phonon and magnon drag for some metal films. Published under license by AIP Publishing. https://doi.org/10.1063/1.5143447 I. INTRODUCTION The Seebeck effect, the generation of an electric field by the application of a thermal gradient, has been a topic of study for more than 100 years and is still very actively explored for both novel mate-rials in bulk form 1–4and a wide range of mesoscopic to nanoscale materials.5–7The Seebeck effect is a consequence of the thermody- namics of charge carriers, which flow in metallic and semiconduct- ing samples in response to an applied thermal gradient. In an often used but overly simple description, the thermal gradient generates atemperature difference across the sample that generates a thermo-electric voltage. Scaling this voltage by the temperature gives aSeebeck coefficient, or thermopower, α¼ΔV=ΔT. This and related thermoelectric effects have had a recent impact in fields as diverse as energy harvesting, 8,9spintronics,10–12and 2D materials.13,14Measurements of the Seebeck effect, especially when the sample to be studied has any dimension &100 nm, are often chal- lenging since the experimentalist must control and measure bothelectric fields and thermal gradients. The issue of controllingthermal gradients on nanoscale samples ranging from magnetictunnel junctions 15and other micro- and nanofabricated magnetic,16–20van der Waals21,22or other electronic materials,23–28 or various nanowire systems29–37is a serious one for many very active fields and has been recently addressed by severalauthors. 38–44Here, we focus on another issue, that of the contribu- tion of the measurement leads to the overall thermopower of thinfilm and nanoscale samples. This lead contribution is an unavoid-able consequence of all Seebeck measurements. The researchcommunity focusing on bulk materials typically uses carefullyJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 085101 (2020); doi: 10.1063/1.5143447 127, 085101-1 Published under license by AIP Publishing.constructed measurement circuits with leads made from materials with painstakingly calibrated absolute thermopower. This allows simple subtraction of the lead contribution. However, thin filmsamples and nanostructures are very commonly measured withmicro- and nano-machined devices where the leads themselves areformed from thin films. In this case, such calibration is simply impossible, since the defects, grain boundaries, strains, and surfaces that are always present in thin films and nanostructures, in additionto classical size effects, change the Seebeck coefficient, oftendramatically. 28,37,45–51The only exception to this is if a supercon- ducting lead material can be used below its Tc, a situation that is often difficult or impossible to achieve in practical measurements. As a result, it is common in Seebeck and related measure- ments of thin films and nanostructures, when the contributions ofthe leads are recognized at all, to either report the value thatincludes the lead, typically referred to as a relative thermopower, α rel, or to subtract literature values tabulated for bulk materials. Both approaches can lead to significant uncertainties that canundermine explorations of basic materials physics and understand-ing of devices using or experiencing thermal gradients. When a thermal gradient ~∇Tis applied to a sample in open- circuit conditions, charges flow from hot to cold until a steady-state electric field ~Ebuilds up to counter this flow. In the most general case, this field is given by ~E¼~α~∇T, (1) where ~αis the thermoelectric tensor that relates the direction of thermal gradients to the resulting electric field. For many materials,it is reasonable to assume this tensor is diagonal, so that the thermally-generated electric field is parallel to the applied thermal gradient, and these diagonal elements are the Seebeck coefficientsof the material. Note that anisotropies in the material forming thesample can in theory make the Seebeck coefficient direction- dependent and also introduce off-diagonal entries in ~α. 52In applied magnetic fields, Lorentz forces impart transverse momen-tum to charge carriers, introducing the off-diagonal ordinaryNernst effect. 53In magnetic systems, spin –orbit coupling intro- duces well-known off-diagonal terms in applied field or when a sample is magnetized, leading to anomalous Nernst and Ettingshausen effects,54–56and the Righi-Leduc effect.57,58 When a real sample is heated, the Seebeck effect generates electric fields everywhere in the measurement circuit where athermal gradient exists on a material with a finite Seebeck coeffi- cient. As shown in Fig. 1 , this potentially complicated situation is typically dramatically simplified to consider only three relevanttemperatures, that of the heated end after the sample has reachedsteady-state, T H, the colder end of the sample, TS, and the tempera- ture of the meter measuring the voltage, TM. If the thermal gradi- ents generated on the sample and each of the leads is constant not only in magnitude, but also in direction, and assuming its directionis aligned with the path of charge flow established by the voltageleads, then the electric field on the sample is given by E x¼αxx@T @x Es¼αsTH/C0TS ‘s, (2)where ‘sis the length of the sample along the direction of the uniform thermal gradient TH/C0TS ðÞ =‘sand we use αsfor the Seebeck coefficient of the sample, which is the absolute Seebeck coefficient that describes the physical generation of field by athermal gradient on the sample itself. With the assumption ofconstant thermal gradient always parallel to the charge flow direc-tion for the leads that connect the sample to the meter, we have similar expressions for the electric field generated on the two leads, E L,H¼αleadTH/C0TM ‘L, (3) EL,S¼αleadTS/C0TM ‘L: (4) The voltage measured at the meter is defined by the path integral V¼ðþ /C0~E/C1d~l, (5) which then has three contributions, V¼VL,HþVsþVL,S, (6) FIG. 1. Simple schematics of measurement of the Seebeck effect, drawn for both electron-like (a) and hole-like (b) charge carriers. The electric fields that result from the thermal gradient shown with maroon vectors are indicated byblack vectors, and the voltmeter is indicated by the dotted line.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 085101 (2020); doi: 10.1063/1.5143447 127, 085101-2 Published under license by AIP Publishing.where VL,H¼/C0αleadTH/C0TM ‘L‘L, (7) Vs¼αsTH/C0TS ‘s‘s, (8) VL,S¼αleadTS/C0TM ‘L‘L, (9) where the negative sign of VL,Hcomes from the dot product of the anti-parallel ~Eand d~lalong that lead. The voltage then is V¼(αs/C0αlead)(TH/C0TS): (10) Thus, the typical ratio of the thermovoltage to the temperature dif- ference gives αs/C0αlead¼αrel¼V TH/C0TS, (11) where αrel;αs/C0αlead: Some authors have explored measurement techniques for nanoscale samples that attempt to remove the lead contributionsand thus allow estimates of the absolute Seebeck coefficient from measurements on micro- and nanopatterned systems, but these attempts are rare and often challenging. For example, one approachfocuses on nanomagnetic systems and uses the dependence of thesample ’s Seebeck coefficient on magnetic field to separate magnetic contributions from non-magnetic contributions. 59We have also explored such a technique, but this is not only limited to magnetic systems with measurable change in αin response to applied field, it also cannot separate any purely electronic contributions to theSeebeck effect of the sample from those from the leads, which are often large contributions even to ferromagnetic metals. A second approach, focused on correct identification of the magneto-Seebeckratio for domain walls in permalloy nanowires, relies as we will onthe Mott relation. 20This should include magnetic field indepen- dent contributions to the sample thermopower, but it is fairly nar- rowly focused on the application to domain-wall effects in ferromagnetic nanowires. Recently, another group used thin film Ptas the lead to measure a bulk Au wire that could be assumed tofollow tabulated values of α abs, which is a clever hybrid of bulk and film thermal techniques that should allow reliable αabsmeasure- ments as long as the bulk wire does not modify the thermal gradi- ents of the Pt thin films near their junctions, and if thedeformation of the bulk Au wire where it is bonded to the Pt thinfilm does not introduce defects that modify its absolute thermo-power. 28A more general approach to extract the absolute Seebeck coefficient from thermoelectric measurements using thin film com- ponents remains an open challenge. In this paper, we present a potential solution to this challenge based on sequential measurements of a single micromachined structure as a series of gold thin films are added. The analysis relies only on the Mott equation for diffusion thermopower, which weuse to model the changes in the sample as the thickness grows. The extrapolation of the Mott equation gives the thickness-independent lead contribution, which we determine as a function of temperaturefrom 80 to 350 K. We use this approach to extract absolute Seebeckcoefficients for a range of metallic films, which reveal strong contri-butions from vacancies and defects that in some cases drive large deviations from bulk values of α abs. II. EXPERIMENT As shown in Fig. 2 , our work on thermal and thermoelectric properties of thin films and nanostructures uses micromachinedthermal isolation platforms, where suspended silicon-nitride (Si-N)membranes with lithographically patterned heaters and thermome- ters are used to control the thermal gradient on the thin film sample. 60–64The ability to separately heat the thermal isolation structure introduces the intermediate temperature of the bulk sub-strate of the device, T 0, but such additional temperatures cannot change the essential mathematics of Eqs. (2)–(11), and the lead contribution to the measured thermopower always contributes, remaining as given by Eq. (11). The scanning electron micrograph clarifies the contributions to the voltage generated across the thinfilm sample in response to heating with the electrically separate thermometer patterned on the left Si-N island. For illustration, we have chosen to show ~Eas generated by leads with positive α abs, though this changes depending on the material used and the tem-perature. For the experiments described below we used two types ofthermal platforms, both formed from 500 nm thick Si-N mem- branes and with the same ratio of sample width and length and the same relative sizes of patterned heaters and thermometers. Onetype uses 50 nm thick Pt leads with a 10 nm thick Cr adhesion-promoting underlayer and has a total length of the sample platformof/difference2 mm. The Cr and Pt layers were grown by e-beam evaporation in high vacuum at typical deposition rate of 0 :1n m =s for Cr and 0:2n m =s for Pt. The second type uses two different materials for FIG. 2. Tilted-angle scanning electron micrograph of an example Si-N thermal isolation platform with false color shading to indicate heating. The maroon arrows indicate the direction of ~rT, and the silver arrows the direction of the Seebeck effect-generated ~E, drawn here with the assumption of a positive αabs. The dashed line indicates the measurement path for V.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 085101 (2020); doi: 10.1063/1.5143447 127, 085101-3 Published under license by AIP Publishing.leads, with a pair formed from 20 nm thick Au and a second pair from 200 nm thick Mo. Au was e-beam evaporated in high vacuum at a/C211n m =s deposition rate, and the Mo sputtered after reaching a base pressure of /difference5/C210/C08Torr or better. The Cr/Pt and Au leads are patterned via photoresist lift-off, and the Mo is etchedusing a commercially-available aluminum etchant. Samples are deposited on the thermal isolation platforms by a variety of deposi- tion techniques either through shadow masks after the Si-N struc-tures are released using anisotropic Si wet etching, or in some casesbefore release of the Si-N by deep-trench plasma etching. Samplefilms are deposited on the bridge through shadow masks aligned to allow contact to the voltage leads or lithographically patterned before the Si-N release. We show data below for a range of filmsincluding those thermally evaporated in /difference10 /C06Torr (Au and AuPd), e-beam evaporated after reaching a base pressure of/difference10 /C08Torr or better (Cu, Co, Fe) or sputtered (Ni, Fe). For measurements of thermopower, the thermal isolation platforms are clamped to radiation-shielded gold-plated samplestages and connections to the platform heaters, thermometers,and sample leads are wire-bonded to custom circuit boards. Thesample stage is bolted to the cold finger of a sample-in-vacuum cryostat, which allows the platform base temperature to be moni- tored and regulated over the range from 77 to 350 K for the exper-iments presented here. The platform and substrate resistivethermometers are measured using an ac resistance bridge, and the power applied to the sample heater and monitored using a dc sourcemeter. The thermovoltage is measured using a digital mul-timeter or nanovoltmeter. For the resistivity measurementsrequired for the lead estimation, we flow current to the sampleusing a precision current source or sourcemeter using a separate set of leads, eliminating contributions from contact and lead resis- tance. All equipment is computer controlled and data recordedusing custom software. Two examples of a typical thermopower measurement are shown in Fig. 3 . As indicated in the simple thermal diagram showninFig. 3(a) , applying power to a heater on the left Si-N island raises this structure to temperature T H, while heat flow through the sample and supporting Si-N bridge (with combined thermal con-ductance K BþKSi/C0N) raises the right island to TS. The temperature difference ΔT¼TH/C0TSis set by the balance of heating power applied and the thermal conductances, but in our measurements it is always measured using calibrated lithographically-patterned resis- tive thermometers. As shown in Figs. 3(b) –3(d) for three choices of T0, we measure Vfor a series of ΔTand determine αrelfrom the slope of a linear fit. This removes the additional thermovoltagecontribution from the leads running from the substrate of the thermal platform to the voltmeter outside of the cryostat, which gives a ΔT-independent offset. These plots already clearly show the change in the sign of α relfor the 20 nm thick evaporated Au film as a function of T.Figure 3(e) shows the resulting αrelas a function of the average temperature of the bridge, Tav. All subsequent plots of αuse the average bridge temperature as well, though we simplify the notation as T. III. ANALYSIS: THE MOTT EQUATION Our approach to estimating the lead contribution to αrelis based on adding a sample film to the thermal isolation platform with a thermopower that is dominated by a single thickness-dependent contribution. We then use a single set of leads tomeasure a sample as a function of thickness by growing additionalsample layers on the same thermal platform. This requires a sample where the thickness dependence of both α relandρcan be easily resolved by the relevant measurements. Furthermore, sincefor experiments to date we cannot measure a film without breakingvacuum and exposing the surface to oxidation, we must use a mate- rial that does not oxidize and therefore dramatically alter the thick- ness dependence of ρandα s. We have found that an evaporated Au thin film meets these assumptions reasonably well, though withlimitations that are discussed below. FIG. 3. Clockwise from top left. (a) Schematic of the thermal isolation platform with temperatures TH,TS, and T0, and thermal conductances KB,KS, and KLindicated. (b)–(d) Thermovoltage VvsTH/C0TSfor three selected T0for Au and AuPd sample films. The lines show linear fits used to determine αrel, given by the slope. (e) Resulting αrelvs average bridge temperature Tavfor the Au and AuPd films.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 085101 (2020); doi: 10.1063/1.5143447 127, 085101-4 Published under license by AIP Publishing.To analyze the film contribution, one must use a theoretical model of the thickness dependence of the sample film, ideally based only on measurable properties of the film itself. The simplesttheory appropriate for thermopower of metals is the Mott relationthat relates expected diffusion thermopower to fundamental con-stants and the resistivity of the film, α¼/C0 π2k2 BT 3e1 ρ@ρ @E/C20/C21 E¼EF, (12) where kBis Boltzmann ’s constant, eis the fundamental charge on the electron, Tis the temperature, ρis the sample electrical resistiv- ity, and @ρ=@Eis the energy derivative of the resistivity taken here at the Fermi energy, EF. This expression itself assumes not only conduction through isotropic s-like bands, but also via carriers that obey the Wiedemann-Franz law. Note that the thickness depen-dence of a sample enters this expression most strongly through thethickness dependence of ρ, since the energy derivative is driven mostly by the type of the dominant carrier scattering, which it is reasonable to assume does not change for a given material as afunction of the thickness of the sample. With this assumption, and explicitly writing the dependence on sample film thickness t, we can write α rel(t)¼αfilm(t)/C0αlead (13) ¼/C0π2k2 BT 3e1 ρ(t)@ρ @E/C20/C21 E¼EF/C0αlead: (14) Here, we observe that a plot of αrelvs 1=ρ(t) for a set of measure- ments made across sample thickness tat a given Tshould have a y-intercept that gives the t-independent αleadvalue at that T. We demonstrate this for one thermal isolation platform in Fig. 4 . Here, we measured αrelandρ(t) after deposition of 9 subse- quent Au films on a single thermal isolation platform. The total thickness of the film ranges from /difference20 nm to more than 300 nm. These films were thermally evaporated in high vacuum at relativelylow 0 :1/C00:2n m =s evaporation rates. Under these growth condi- tions we expect a relatively high level of defects and imperfectionsin the polycrystalline films that form on the Si-N platform. Figure 4 presents α relvs 1=ρfor 11 representative temperatures from 78 to 318 K, with a linear fit to each set of data shown withthe red line. For each Tthe data to the right of the plot represents thicker films. This plot shows that the tvariation indeed changes both ρandα relso that the extrapolation of these linear fits gives a smoothly varying curve. Note also that the fit is quite strongly affected for most Tby the thinner samples. As discussed else- where,65,66these thinner films obey the Wiedemann-Franz law more closely than the thicker films. We also considered use of more complicated models of the thickness-dependence of thin film thermopower such as extensions of the Fuchs –Sondheimer67or Mayadas –Shatzkes68–70models. However, we found that our experimental data for evaporated Aufilms never fit these models well, and that the implementation wasmuch more cumbersome than the simple approach based only on the Mott equation. If a different film system can be employed thatresults in ρ(t) and α(t) that follow one of these models more closely, it is likely that replacing the Mott equation with the rele-vant expression for thickness-dependent thin film thermopowerwould allow more accurate extraction of the lead thermopower. Figure 5 shows the results of the lead estimation technique for the two different types of thermal isolation platform. Figure 5(a) compares α absfor the Cr/Pt leads to bulk values.71Note that since both lead and film thermopowers that together generate αrelrepre- sent absolute thermopower for each, we label these with the “abs” subscript. Any thermopower not specifically labeled with the “rel” subscript is an absolute thermopower. The blue data points inFig. 5(a) are taken from the y-intercepts of the linear fits in Fig. 4 , where the error bars represent statistical uncertainties from the fit taking into account estimated measurement error on α rel. Three red data points use a fitting algorithm that also considers estimatederrors on 1 =ρ. These points agree well with the simpler fits. The estimated α absshows strong qualitative agreement with the bulk Pt values, reproducing the sign change that is typically viewed as a positive low Tphonon drag component added to a typical negative diffusive thermopower.47Indeed above 175 K, the estimated thin film Cr/Pt values agree quantitatively with bulk within estimatedstatistical uncertainties. Below this temperature, the reduced α abs suggests a reduction of phonon drag compared to the bulk system. Such a reduction in αabsfor Pt in the phonon drag regime was also recently seen by another group28using the bulk Au reference tech- nique mentioned above, though their high Tvalues are also signifi- cantly reduced from that of bulk Pt. However, differences from one set of Pt films to another made in a different chamber from differ- ent source material are far from surprising. FIG. 4. Plot of αrelvs 1=ρfor a series of Au films deposited on a thermal isola- tion platform with Cr/Pt leads, plotted for several T0with values as indicated. The solid red lines show linear fits to each T’s data, and according to Eq. (14) the negative value of the yintercept gives the sample film thickness indepen- dent value of αleadfor each T. The change from negative lead αto positive with increasing temperature is clear from this plot.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 085101 (2020); doi: 10.1063/1.5143447 127, 085101-5 Published under license by AIP Publishing.Figure 5(b) compares αabsfor Au and Mo leads formed on a single thermal isolation platform determined using the same esti- mation approach outlined for the Cr/Pt leads. These αabsare com- pared vs Tto bulk Au72and Mo.73Again, error bars are estimated based on the statistical values from the fitting procedure, but as isthe case for Cr/Pt, these statistical errors are likely smaller than sys- tematic deviations that come from deviations from the assumptions of the lead extraction technique that are difficult to estimate. Here,the e-beam evaporated 20 nm thick Au lead has the same sign asbulk Au, but it is reduced from the bulk values significantly, as isoften seen for thin films. The thicker sputtered Mo film, which had been exposed to air for an extended period at the time of this mea- surement and could have been oxidized, has a similar low tempera-ture upturn as seen in bulk, but is otherwise larger and alwaysnegative where bulk Mo has a positive sign at most T. Again, con- sidering the additional defects and impurities in thin films, these deviations are not unreasonable.IV. RESULTS AND DISCUSSION: ESTIMATED ABSOLUTE SEEBECK COEFFICIENTS Having extracted reasonable α absfor the lead materials we used in our thermal isolation platforms, Eq. (13) indicates that determining the absolute thermopower of the sample film αfilm requires only addition of αleadtoαrelat each T. We now present these estimated αabsfor a range of thin metal films. Figure 6 shows αabsvsTfor Cu, Co, Ni, and Au-Pd thin films, again compared to bulk values. Here the Cu, Co, and Ni were measured with Mo leads, and the Au-Pd measured with Pt leads. Also note that for Ni, Co, and Au-Pd, the Seebeck coefficient is relatively large such thatthe uncertainties introduced from the lead subtraction proceduremost likely do not cause large errors in α abs. In addition to bulk comparisons, for the Ni sample we are able to estimate the diffu- sion thermopower since we previously used magnetic field depen- dence to estimate the energy derivative of ρ, which is normally difficult to determine experimentally.75This allows calculation of the Mott thermopower using only the measured ρand the FIG. 5. Estimated lead αabsfor two thermal platforms vs T. (a)αabsdetermined for Cr/Pt leads vs Tcompared to bulk Pt.71(b)αabsfor Au and Mo leads vs T compared to bulk Au72and bulk Mo.73 FIG. 6. αabsvsTfor (a) thin film Cu compared to bulk, and (b) thin film Co, Ni, and Au-Pd compared to bulk, and in the case of Ni also to the diffusion thermo- power contribution estimated from the Mott equation. Au-Pd even in bulk shows αabsthat varies significantly with composition.74Here, we show two composi- tions that roughly span the range of values at these T.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 085101 (2020); doi: 10.1063/1.5143447 127, 085101-6 Published under license by AIP Publishing.estimated @ρ=@E¼3:8/C210/C07Ω/C0m=eV. The result is given by the dashed-dotted green line labeled “Diffusion. ”This agrees rea- sonably well with the measured values for Ni. This suggests thatour film, as has long been discussed for bulk Ni, 76does not have significant contributions from magnon drag thermopower. The lack of magnon drag in Ni is often contrasted with the case for Fe, where the different nature of electronic scattering allows lower Gilbert damping while keeping a strong enoughelectron-magnon scattering to introduce the drag effect. 76,77Our measurements of αsupport this picture for thin films of Fe as well,62,78though in our earlier publications we presented only rela- tive thermopower values. We show the estimated αabsvalues vs T for three Fe films compared to the bulk data in Fig. 7 .E a c ho f these films were measured using Mo leads. αabsis qualitatively similar for all three films, with each showing positive thermopowerwith a broad peak. The increase in α absfrom 65 nm and 75 nm evaporated films (grown at relatively low rates /difference0:1/C00:2n m =s) to the 50 nm sputtered film shows the same trend as electronic con-ductivity, which is highest for the sputtered film in this case.Though magnon drag should be expected to be less sensitive todefect, surface, and disorder scattering than phonon drag, such a trend could indicate additional scattering of magnons by defects. All films are within the range of bulk Fe with modest level ofimpurities. Finally, we examine α absfor the evaporated Au films we used for the extraction of the Cr/Pt lead contributions as shown in Figs. 4 and5.Figure 8 compares estimated αabsfor these thermally evaporated Au films grown on a thermal isolation platform withCr/Pt leads. We show a subset of the films for clarity, the valuesselected span the range of α absfor the entire range of the films from 20 to 380 nm Cr =Pt leads. We compare these to both bulkvalues measured for Au wires,72and older values measured for unannealed Au thin films.79At the lowest measured T, the thickest films match expectations of bulk Au well, while thinner films withsmaller average grain size and more grain-boundary scattering ofelectrons show reduced values but with the positive sign expected for bulk Au. As temperature increases however, α absreduces and becomes negative near 150 K. Negative contributions to thermo-power of Au can be driven by both impurities 80–82and vacancies and/or size effects.45,46Absolute Seebeck coefficients of thin films of Au have been measured in a handful of older studies that usually measured αrelagainst thicker wire leads of the same material, and these reveal a wide range of values, including negative αabs, in par- ticular, for films measured without additional thermal anneal-ing, 48,79as is the case for our measurements. The maroon dashed line in Fig. 8 shows the report from Angus and Dalgliesh for 30–400 nm thick unannealed films. These match our data for thinner films extremely well. However as is clear from Fig. 5(b) and also seen in αabsestimated for the Au films used to extract the Mo and Au lead thermopower (not shown but with values that cross αabs¼0 but vary ,1μV=K at all T),αabsfor Au thin films depends strongly on the growth conditions and the nature of scat-tering in the film and should not be assumed to follow bulk valueswithout more careful investigation. Despite what is perhaps better-than-expected agreement of α abswith bulk values (in the case of the Cr/Pt films) or with older values measured for unannealed films (for some thermally evapo-rated Au films), we emphasize that there are still limitations in thisapproach to determining α abs, and the values must always be con- sidered only an estimate. As one example of the possible systematic errors in the assumptions underlying this approach, consider that if FIG. 7. αabsvsTfor three Fe films compared to values of various bulk Fe samples.76The broad, positive peak that is relatively insensitive to sample quality is evidence of magnon drag in Fe. FIG. 8. αabsfor a thermally evaporated Au film on a platform with Cr/Pt leads. The film was built up in nine subsequent depositions, here we present a subset of these for clarity. Total film thickness is indicated in the legend. The film values are compared to bulk Au wires72and to older measurements of unan- nealed Au films.79Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 085101 (2020); doi: 10.1063/1.5143447 127, 085101-7 Published under license by AIP Publishing.a thin film sample has a contribution to αthat is independent of thickness as the sample film is built up, application of Eq. (14) will treat this as an additional lead contribution. This erroneous leadthermopower then would affect all α absvalues for films measured with those leads. Phonon drag contributions could be an exampleof a non-diffusive thermopower that would not be captured by Eq.(14), though the defects and grain boundaries always expected in a film are usually assumed to strongly limit phonon drag, and inthe case of Au phonon drag contributions should be limited toquite low temperatures due to the low Debye temperature. V. CONCLUSIONS In summary, we first clarified the challenges in measuring the Seebeck coefficient for thin films and nanostructures and thetypical assumptions, including the direction and value of thermalgradients and the introduction of lead contributions to the Seebeckeffect. We then described a relatively simple route for estimating lead contributions from relative measurements using a microma- chined thermal isolation platform, and demonstrated this usingsequential measurements of gold thin films which we assume haveα abs(t) that is described by the Mott equation. This allowed deter- mination of the lead contribution for several types of leads used on the platforms. We then removed these contributions and showedthatα absfor a range of metal thin films almost always deviates sub- stantially from bulk values. ACKNOWLEDGMENTS We thank L. O ’Brien and D. Phelan for helpful discussions, J. Nogan and the IL staff at CINT for guidance and training in fab-rication techniques, and G. Hilton, J. Beall, and D. Schmidt for dis-cussions and support in nanofabrication at NIST, and gratefullyacknowledge support from the National Science Foundation (NSF) (Nos. DMR-1410247, DMR-1709646, and EECS-1610904). 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1.2670151.pdf	Noncollinear magnetism in Permalloy Markus Eisenbach, G. Malcolm Stocks, and Don M. Nicholson Citation: Journal of Applied Physics 101, 09G503 (2007); doi: 10.1063/1.2670151 View online: http://dx.doi.org/10.1063/1.2670151 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/101/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic structure variation in manganese-oxide clusters J. Chem. Phys. 136, 134315 (2012); 10.1063/1.3698279 Magnetization reversal in individual micrometer-sized polycrystalline Permalloy rings J. Appl. Phys. 97, 063910 (2005); 10.1063/1.1858055 Magnetic structure of iron inclusions in copper J. Appl. Phys. 95, 6684 (2004); 10.1063/1.1687253 Magnetic structures of CrPt 3 by first-principles calculations J. Appl. Phys. 93, 7151 (2003); 10.1063/1.1558609 Effects of Ta on the magnetic structure of permalloy J. Appl. Phys. 89, 6886 (2001); 10.1063/1.1356032 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.143.1.222 On: Fri, 12 Dec 2014 16:12:56Noncollinear magnetism in Permalloy Markus Eisenbacha/H20850and G. Malcolm Stocks Material Sciences and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 Don M. Nicholson Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 /H20849Presented on 10 January 2007; received 31 October 2006; accepted 7 November 2006; published online 21 March 2007 /H20850 Permalloy is an important material in a wide variety of magnetic systems, most notably in giant-magnetoresistive read heads. However, despite this great interest, its properties are not fullyunderstood. For an in depth analysis of important physical properties as, e.g., electric transport ormagnetic anisotropy, a detailed understanding of the distribution of magnetic moments on an atomiclevel is necessary. Using our ﬁrst principles locally self-consistent multiple scattering method, wecalculate the magnetic ground state structure for a large supercell model of Permalloy. Our codeallows us to solve both the usual nonrelativistic Schrödinger equation as well as the fully relativisticDirac equation and to ﬁnd the magnitude and direction of the magnetic moments at each atomic site.While the nonrelativistic calculation yields a collinear ground state in accordance with previouscalculations, we ﬁnd the ground state for the fully relativistic calculation to be slightly noncollinear.We also investigate the inﬂuence of variations in the iron concentration on the distribution ofmagnetic moments. © 2007 American Institute of Physics ./H20851DOI: 10.1063/1.2670151 /H20852 I. INTRODUCTION Magnetism in iron-nickel alloys has attracted consider- able interest in the past and both the iron rich and the nickelrich alloys are of technological importance. While the mag-netic ground state of Invar has been extensively studied, 1less effort has been spent on Permalloy. In the case of Invar it isgenerally agreed that local magnetic order is essential forunderstanding the Invar effect. 2Consequently, numerous theoretic studies have been performed. In recent years differ-ent groups have reported highly noncollinear magneticground states for Invar and its connection to the volume,thereby providing a very likely explanation of the Invar ef-fect. While Permalloy seems to be a good ferromagnetic ma- terial macroscopically with technological importance in highpermeability applications, its microscopic magnetic order hasreceived less attention. In most calculation its magnetic orderis assumed to be collinear, 3which would be compatible with the average cubic symmetry. Yet locally, this average sym-metry will be broken by a random arrangement of Fe and Nineighbors. As Sandratskii has conjectured, 4this should, in general, lead to a noncollinear magnetic order, since the col-linear order is no longer supported by the symmetry of thesystem if special relativistic effects, spin-orbit coupling inparticular, are included in the Hamiltonian. Collinear calculations of electron transport in Permalloy indicate that the resistance in the majority channel is verylow and that it exceeds that of the minority channel by sev-eral orders of magnitude. The low measured electrical resis-tance of Permalloy and the large increase in resistance whenPermalloy layers are arranged antiFerromagnetically in spinvalves are consistent with this picture. However, the pictureis not complete as indicated by the large discrepancy be- tween the calculated and measured resistivities. While thevalue calculated by various researchers ranges from essen-tially zero 5to 2/H9262/H9024cm,3the measured value is 4 /H9262/H9024cm. The amount of spin ﬂip scattering induced by noncollinearlocal moments affects not only the bulk resistance but alsothe contribution to transport behavior from interfaces anddefects. A quantitative understanding of the noncollinearmagnetic structure of Permalloy is central to understandingand optimizing many giant-magnetoresistive devices. In the present report we will investigate this emergence of noncollinear magnetism in Fe 0.2Ni0.8Permalloy and its dependence on volume and concentration. II. METHOD We employ the ﬁrst principles framework of density functional theory in the local spin density approximation. Tosolve the Kohn-Sham equations arising in the above context,we use the multiple scattering formalism of Korringa, Kohn,and Rostoker /H20849KKR /H20850. Since our interest here lies in calculat- ing properties related to magnetism beyond the reach of anordinary Schrödinger equation, we have to take into accounteffects due to relativistic electron behavior, especially thecoupling between electron spins and their orbital motion. Wedo this by utilizing relativistic spin density functional theoryas formulated by many different authors. As usually done inthese calculations, we neglect the coupling between orbitalcurrents and the vector potential. This leads to solving aKohn-Sham-Dirac equation of the form /H20851−i/H6036c /H9251·/H11633+/H9252mc2+V/H20849r;n,m/H20850+/H9252/H9268·B/H20849r;n,m/H20850−E/H20852/H9274 =0 , where VandBare the local density approximation /H20849LDA /H20850a/H20850Electronic mail: eisenbachm@ornl.govJOURNAL OF APPLIED PHYSICS 101, 09G503 /H208492007 /H20850 0021-8979/2007/101 /H208499/H20850/09G503/3/$23.00 © 2007 American Institute of Physics 101 , 09G503-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.143.1.222 On: Fri, 12 Dec 2014 16:12:56functionals of the charge n/H20849r/H20850and magnetization m/H20849r/H20850den- sities. The details of the KKR method for calculating the Green’s function and the total ground state energyE/H20851n/H20849r/H20850,m/H20849r/H20850/H20852are described elsewhere. 6,7Our relativistic lo- cally self-consistent multiple scattering /H20849Rel-LSMS /H20850method allows the possibility of noncollinear magnetism.8 The new orientation eˆiof the magnetic moment for each site is determined by eˆi=/H20885 /H9024idrmi/H20849r/H20850/H20882/H20879/H20885 /H9024idrmi/H20849r/H20850/H20879. As the LDA self-consistent ﬁeld /H20849SCF /H20850iterations proceed, eˆi will rotate in order to minimize the total energy. Thus, we can ﬁnd a noncollinear magnetic structure in which the ori-entations of the local moments vary from site to site. Sincean arbitrary arrangement is not a density functional theory/H20849DFT /H20850ground state, we will have to deal with a constrained general state as presented by Wang and co-workers. 9,10In the constrained local moment /H20849CLM /H20850model the local spin den- sity approximation /H20849LSDA /H20850equations are solved subject to a constraint /H20885 /H9024imi/H20849r/H20850/H11003eidr=0 /H208491/H20850 that ensures that the local magnetizations lay along the di- rections prescribed by /H20853ei/H20854. The result is that in order to maintain the speciﬁc orientational conﬁguration, a localtransverse constraining ﬁeld must be applied at each site. Theconstraining ﬁeld is obtained from the condition /H9254Econ/H20851/H20853ei/H20854,/H20853Bicon/H20854/H20852 /H9254ei=0 /H208492/H20850 applied to all sites and where Eis the generalized energy functional in the presence of the constraining ﬁeld. To makeuse of the CLM model to ﬁnd magnetic ground state con-ﬁgurations, we note that the internal effective ﬁeld that ro-tates the spins is just the opposite of the constraining ﬁeld,i.e.,B ieff=−Bicon. Using these effective ﬁelds, we can evolvethe moment directions using a Landau-Lifshitz-Gilbert equa- tion, where the damping constant can be adjusted to ensurerapid convergence to the ground state /H20849or, at least, the nearest local minimum /H20850. III. MAGNETIC GROUND STATE OF PERMALLOY Using the method describe above, we investigate the magnetic ground state of Permalloy /H20849Fe0.2Ni0.8/H20850. We set up randomly generated supercell instances of Permalloy by placing Fe and Ni atoms on a fcc lattice to correspond withthe desired concentrations. In the present discussion we in-vestigate different fcc supercells of 108 atomic sites each at alattice constant of a=6.7 a 0. The two instances contain 84 Ni and 24 Fe atoms or 88 Ni and 20 Fe atoms, respectively,corresponding to iron concentrations of 22.2% and 18.5%. Self-consistent calculations using the scalar-relativistic version of LSMS ﬁnd a collinear ferromagnetic ground statein agreement with previous results. 3Employing a fully rela- tivistic version of our code, i.e., solving the Dirac equationdescribed above as opposed to a Schrödinger equation, im-mediately leads to nonvanishing constraining ﬁelds for thecollinear ordering. By iterating the procedure described above, we can ﬁnd the ground state conﬁguration for the relativistic case. Herewe ﬁnd that the moment directions on both the iron and thenickel sites deviate from the mean magnetization direction.To illustrate this we have plotted the x-yprojection of the Fe moments in Fig. 1. The deviation from a collinear arrangement of the mo- ments turns out to be very small. We have calculated thedeviation from the mean orientation direction and plotted theresults for the two concentrations we have investigated inFig.2. While all the angles are small, it is obvious that the two different atomic species exhibit different behaviors. TheFe moments show a much narrower distribution than the Nimoments in both cases. On the other hand, the change in FIG. 1. Distribution of xandycom- ponents of the Fe magnetization direc-tions in Fe 0.22Ni0.78.09G503-2 Eisenbach, Nicholson, and Stocks J. Appl. Phys. 101 , 09G503 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.143.1.222 On: Fri, 12 Dec 2014 16:12:56concentrations appears to have a much more signiﬁcant in- ﬂuence on the Fe magnetic moment directions than at the Nisites. IV. CONCLUSIONS As we have shown in this report, random order in the composition of a material will usually lead to some degree ofdisorder in the magnetic state. While this should not be a reason for surprise, since any disorder would break the sym-metries in the system that would require collinear order, 4the actual degree of magnetic disorder will depend on the spe-ciﬁcs of the system under consideration. While the results presented here are sufﬁcient to estab- lish the existence of variations in the magnetization directioninside Permalloy, further calculations for larger supercellsare under way to obtain better quantitative results for themoment distributions both as function of Fe/Ni ratio andpressure. Furthermore, it will be necessary to evaluate theinﬂuence of the magnetic state found here on the electricconductivity of Permalloy. ACKNOWLEDGMENTS This research was supported in part by an appointment /H20849M.E. /H20850to the Postgraduate Research Program at the Oak Ridge National Laboratory administered by the Oak RidgeInstitute for Science and Education. This was sponsored byDOE-OS, BES-DMSE, and OASCR-MICS under ContractNo. DE-AC05-00OR22725 with UT-Battelle LLC. The cal-culations presented in this paper were performed at the Cen-ter for Computational Sciences /H20849CCS /H20850at ORNL and at the National Energy Research Scientiﬁc Computing Center/H20849NERSC /H20850. 1Y. Wang, G. M. Stocks, D. M. Nicholson, W. A. Shelton, V. P. Antropov, and B. N. Harmon, J. Appl. Phys. 81, 3873 /H208491997 /H20850. 2M. van Schilfgaarde, I. A. Abrikosov, and B. Johansson, Nature /H20849London /H20850 400,4 6 /H208491999 /H20850. 3D. M. Nicholson, W. H. Butler, W. A. Shelton, Y. Wang, X.-G. Zhang, and G. M. Stocks, J. Appl. Phys. 81, 4023 /H208491997 /H20850. 4L. M. Sandratskii, Phys. Rev. B 64, 134402 /H208492001 /H20850. 5I. Mertig, R. Zeller, and P. H. Dederichs, Mater. Res. Soc. Symp. Proc. 253, 249 /H208491993 /H20850. 6Yang Wang, G. M. Stocks, W. A. Shelton, D. M. C. Nicholson, Z. Szotek, and W. M. Temmerman, Phys. Rev. Lett. 75, 2867 /H208491995 /H20850. 7M. Eisenbach, B. L. Györffy, G. M. Stocks, and B. Újfalussy, Phys. Rev. B65, 144424 /H208492002 /H20850. 8G. M. Stocks, Y. Wang, D. M. C. Nicholson, W. A. Shelton, Z. Szotek, W. M. Temmerman, B. N. Harmon, and V. P. Antropov, Mater. Res. Soc.Symp. Proc. 408,1 5 7 /H208491996 /H20850. 9G. M. Stocks, B. Újfalussy, X. Wang, D. M. C. Nicholson, W. A. Shelton, Y. Wang, A. Canning, and B. L. Györffy, Philos. Mag. B 78,6 6 5 /H208491998 /H20850. 10B. Újfalussy, X. Wang, D. M. C. Nicholson, W. A. Shelton, G. M. Stocks, Y. Wang, and B. L. Györffy, J. Appl. Phys. 85, 4824 /H208491999 /H20850. FIG. 2. /H20849Color online /H20850A histogram showing the distribution of the angles between the average moment direction and the actual direction on the Feand Ni sites. The horizontal axis represents the angle in degrees from themean direction and the vertical axis indicates the number of atoms. The topﬁgure shows the distribution for Fe 0.18Ni0.82and the bottom one for Fe0.22Ni0.78.09G503-3 Eisenbach, Nicholson, and Stocks J. Appl. Phys. 101 , 09G503 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.143.1.222 On: Fri, 12 Dec 2014 16:12:56
1.4854956.pdf	A numerical approach to incorporate intrinsic material defects in micromagnetic simulations J. Leliaert, B. Van de Wiele, A. Vansteenkiste, L. Laurson, G. Durin, L. Dupré, and B. Van Waeyenberge Citation: Journal of Applied Physics 115, 17D102 (2014); doi: 10.1063/1.4854956 View online: http://dx.doi.org/10.1063/1.4854956 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The influence of individual lattice defects on the domain structure in magnetic antidot lattices J. Appl. Phys. 113, 103907 (2013); 10.1063/1.4795147 A micromagnetic study of the reversal mechanism in permalloy antidot arrays J. Appl. Phys. 111, 053915 (2012); 10.1063/1.3689846 Micromagnetic studies of domain structures and switching properties in a magnetoresistive random access memory cell J. Appl. Phys. 97, 10E310 (2005); 10.1063/1.1852193 Micromagnetic simulation studies of ferromagnetic part spheres J. Appl. Phys. 97, 10E305 (2005); 10.1063/1.1850073 Micromagnetic configurations and switching mechanism in Pac-man-shaped submicron Ni 80 Fe 20 magnets J. Appl. Phys. 97, 073904 (2005); 10.1063/1.1874297 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.100.58.76 On: Wed, 08 Oct 2014 13:55:58A numerical approach to incorporate intrinsic material defects in micromagnetic simulations J. Leliaert,1,2,a)B. Van de Wiele,1A. Vansteenkiste,2L. Laurson,3G. Durin,4,5L. Dupr /C19e,1 and B. Van Waeyenberge2 1Department of Electrical Energy, Systems and Automation, Ghent University, Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium 2Department of Solid State Science, Ghent University, Krijgslaan 281/S1, 9000 Gent, Belgium 3COMP Centre of Excellence, Department of Applied Physics, Aalto University, PO Box 14100, Aalto 00076, Finland 4Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135 Torino, Italy 5ISI Foundation, Via Alassio 11/c, 10126 Torino, Italy (Presented 5 November 2013; received 23 September 2013; accepted 6 October 2013; published online 6 January 2014) Spintronics devices like racetrack memory rely on the controlled movement of domain walls in magnetic nanowires. The effects of distributed disorder on this movement have not yet been studied extensively. Defects give rise to a pinning potential that can be characterized in terms of a depth and an interaction range. We investigate how the effects of defects can be realisticallyintroduced in micromagnetic simulations by comparing the properties of the pinning potential to experimental results in the literature. W e show that the full 3-dimensional simulations can be replaced by equivalent 2-dimensional ones and propose two approaches to include defects. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4854956 ] I. INTRODUCTION Many future spintronics devices are based on the con- trolled movement of domain walls in magnetic nanowires.1,2 Consequently, it is important to have a complete understand- ing of the dynamics governing this motion. Domain wall mo- bility is extensively described in the literature for perfect nanowires3,4or nanowires with edge roughness.5However, real wires always contain a large number of intrinsic material defects.6Recently, it has been observed that this distributed disorder throughout the whole wire has an important effecton the domain wall mobility, 7,8e.g., domain walls can get pinned to trapping sites. Several experiments have been con- ducted to characterize the nature of these trapping sites andquantify their properties. 9–11It is found that defects give rise to potential wells in the micromagnetic energy landscape. These potentials can be characterized in terms of their depthand interaction range. In this contribution, we numerically investigate the properties of defects implemented in different ways and propose a method to realistically include the inﬂu-ence of intrinsic defects in 2-dimensional (2D) numerical simulations. Polycrystalline magnetic materials are built up out of grains with possibly varying lattice orientations and imperfect grain boundaries. Despite some controversy, 6,11there are indi- cations that mainly the grains inﬂuence the magnetic proc-esses in Permalloy (Py) because the measured trapping site density is correlated with the grain density. 6Experiments are able to characterize the resulting pinning potential. The depthof the well is found to be 1–5 eV, 9–11while its interactionrange is of the same size as the vortex core diameter used to probe the defect, i.e., approximately 20 nm.9,11 In numerical simulations, the energy of the system is ac- cessible which makes measuring the properties of the poten-tial well a much less challenging task than for experiments, where only macroscopic quantities are measureable. This advantage is used to perform a systematic study of differentpossibilities to include trapping sites in numerical simula- tions. Because of the suspected link, we investigate two dif- ferent implementations that are reminiscent of the grains. II. METHODS To determine the properties of the potential well in micromagnetic simulations, we simulate a disk (diameter:750 nm, thickness: 10 nm) in which a defect is introduced in the central region, see Fig. 1(a). A magnetic vortex is inserted 200 nm from the center. From that point, the vortexrelaxes, following a spiralling trajectory towards the disk center. During this slow relaxation (over 400 ns), the energy of the system is probed, see Fig. 1. The depth of the potential well is extracted from the difference between the energy with and without defect. The interaction range is measured from the center of the defect and is determined by the radiusover which the potential is deeper than 10% of its maximum, as shown in Fig. 1. One way to simulate a grain is to focus on its physical size: not every grain has the same thickness. We perform 3- dimensional (3D) simulations in which we simulate the grain as a region with a different thickness. A disk is simulated inwhich the thickness of the center region is reduced by 2.5, 5, or 7.5 nm, corresponding with 1, 2, or 3 ﬁnite difference (FD) cells. We also investigated if we can replace these simula-tions by performing an equivalent and faster 2D simulation in a)Electronic mail: jonathan.leliaert@ugent.be. 0021-8979/2014/115(17)/17D102/3/$30.00 VC2014 AIP Publishing LLC 115, 17D102-1JOURNAL OF APPLIED PHYSICS 115, 17D102 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.100.58.76 On: Wed, 08 Oct 2014 13:55:58which the saturation magnetization is changed. This method has been used before8to investigate the effects of disorder on domain wall motion. A second way to simulate a grain is to introduce a grain boundary by deﬁning a region with a reduced exchange stiffness constant at the boundaries. All micromagnetic simulations were performed using the GPU-based micromagnetic software package MuMax3.12 Typical material parameters for Py are used: saturation mag- netization 860 /C2103A/m, Gilbert damping parameter a¼0:02, and exchange stiffness constant 13 /C210/C012J/m. In the 2D simulations, the disks are discretized using cells of3:125/C23:125/C210 nm 3. In the 3D simulations, the thick- ness of the disk is further discretized using cells with a thick- ness of 2.5 nm. Simulations were performed for defectregions of different sizes of 1 /C21u pt o4 /C24 FD cells with a reduction in the saturation magnetization/exchange stiffness constant at the boundaries ranging from 10% to 100%. In thesimulations in which the saturation magnetization is reduced, the exchange length at the boundary of the defect region is reset to its original value. III. RESULTS AND DISCUSSION A. Grain thickness The grain thickness reduction was simulated by remov- ing FD cells from the top layer. The results are shown asgreen points in Fig. 2(a). It is observed that the depth of the potential well rises as a function of reduction in thickness, and is larger for larger defect regions. An effort is made to investigate if these 3D simulations can be reduced to equivalent 2D simulations in which defects are simulated as regions with a reduced saturation magnetiza-tion. The depth of the resulting potential well is linearly de- pendent on the reduction in saturation magnetization and is larger for larger defects, see Fig. 2(a). For sizes larger than 1/C21 FD cells, a jump is observed for defects with the satura- tion magnetization set to 0. This jump is caused by thedisappearance of the vortex core in the defect. To make the 2D simulations equivalent to the 3D ones, it is not sufﬁcient to reduce the saturation magnetization the same amount asthe reduction in thickness. There are two different approaches possible to make the simulations equivalent. A ﬁrst approach is to include regions with larger sizes in the 2D simulations,e.g., in Fig. 2(a), it can be seen that the 2 /C22 FD cell sized defects in the 3D simulations lie on the same curve as the 3 /C2 3 FD cell sized defects in the 2D simulations. A secondapproach is to reduce the saturation magnetization more than the corresponding reduction in thickness. To estimate the size of this reduction Fig. 2(a)can be used as a guide. The interaction range is weakly dependent on the thick- ness reduction and seems to be dependent on the size of the defect. However, this dependency arises mainly because theinteraction range is measured from the center of the defect. If the size of the defect is deducted from the interaction range, it is found that the resulting distance is almost constant andequal to the vortex core diameter. This observation is sup- ported by Refs. 10and11, where it is stated that the measured energy is convolved with the energy proﬁle of the vortexcore, resulting in an interaction range of approximately the same size as the diameter of the vortex core. The interaction FIG. 1. The magnetic energy of a vortex in a disk with (full line) and without (dotted line) a defect, implemented as a region in the center of size 10/C210 nm with the exchange stiffness constant reduced by 70% at the boundaries. Without the defect the energy landscape has a parabolic shape. The defect causes an additional potential well to this, for which the depth and interaction range are shown. Inset (a) depicts the initial magnetization in the disk and the trajectory the vortex core follows while it relaxes into the defect. Inset (b) depicts the energy of the system. FIG. 2. The depth (dotted blue lines) and interaction range (full red lines) ofthe potential well originating from simulated material defects of differentsizes. Defects are simulated in two different ways. (a) First, the saturation magnetization within a region is reduced. The green points show the depth of the potential wells in the 3-dimensional simulations, where the reduction in saturation magnetization is equal to the reduction in the thickness of the defect region. (b) Second, the exchange stiffness constant is reduced at the boundaries of a region.17D102-2 Leliaert et al. J. Appl. Phys. 115, 17D102 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.100.58.76 On: Wed, 08 Oct 2014 13:55:58ranges for the 3D simulations are not shown and are approxi- mately 20 nm, which is a factor two larger than in the 2D sim- ulations. This can be explained by the fact that the vortexcore is larger in 3D simulations. B. Grain boundary In the simulations with defects implemented as regions with a reduced exchange stiffness constant at the boundaries,it is found that the depth of the potential well slowly rises as a function of the reduction in the exchange stiffness constant. It is observed that larger defects give rise to deeper potentialwells. See Fig. 2(b). The interaction range is weakly dependent on the reduc- tion in the exchange stiffness constant and rises for largerdefects. This dependency is just as in the previous case caused by the method, i.e., the interaction range is measured from the centre of the defect region and not from the edge. Based on these results, the following methods to realisti- cally include defects in micromagnetic simulations are pro- posed. First, defects can be included as regions with a size ofapproximately 10 by 10 nm (similar to the ﬁlm thickness) with their saturation magnetization reduced by 50%. Alternatively, defects can be deﬁned as regions with theexchange constant reduced by 70% at the boundaries. The potential well caused by such defects is shown in Fig. 1and has a depth close to the average of the experimentally meas-ured values. For both approaches, the interaction range is approximately the same as the vortex core diameter. IV. CONCLUSION In conclusion, we investigated two different ways to include material defects in micromagnetic simulations, both based on the crystal structure in Permalloy. One way to sim- ulate a grain is deﬁning a region with reduced thickness in3D simulations, or equivalently, an appropriately chosen defect size or reduction in saturation magnetization in 2D simulations. A second way consists in reducing the exchangestiffness constant at the boundaries of the defect region. The interaction range and depth of the potential well are deter- mined by the size of the defect and the reduction of themicromagnetic parameter. Based on this characterization, we propose two ways to realistically include the inﬂuence of defects in micromagnetic simulations. ACKNOWLEDGMENTS This work had been supported by the Flanders Research Foundation (B.V.d.W. and A.V.), the Academy of Finlandthrough a Postdoctoral Researcher’s Project (L.L., Project No. 139132), an Academy Research Fellowship (L.L., Project No. 268302), and Progetto Premiale MIUR-INRIM“Nanotecnologie per la metrologia elettromagnetica” and MIUR-PRIN2010-11 Project2010ECA8P3 “DyNanoMag” (G.D.). 1S. E. 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1.4967798.pdf	Effect of perpendicular magnetic field on bubble-like magnetic solitons driven by spin- polarized current with Dzyaloshinskii–Moriya interaction Chengkun Song , Chendong Jin , Senfu Zhang , Shujun Chen , Jianbo Wang , and Qingfang Liu, Citation: J. Appl. Phys. 120, 183901 (2016); doi: 10.1063/1.4967798 View online: http://dx.doi.org/10.1063/1.4967798 View Table of Contents: http://aip.scitation.org/toc/jap/120/18 Published by the American Institute of Physics Effect of perpendicular magnetic field on bubble-like magnetic solitons driven by spin-polarized current with Dzyaloshinskii–Moriya interaction Chengkun Song,1Chendong Jin,1Senfu Zhang,1Shujun Chen,1Jianbo Wang,1,2 and Qingfang Liu1,a) 1Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, People’s Republic of China 2Key Laboratory for Special Function Materials and Structural Design of the Ministry of Education, Lanzhou University, Lanzhou 730000, People’s Republic of China (Received 29 July 2016; accepted 29 October 2016; published online 11 November 2016) The topological properties of bubble-like magnetic solitons can be modiﬁed by interfacial Dzyaloshinskii-Moriya interaction (DMI). In this paper, the dynamic responses of bubble-like magnetic solitons nucleated in the free layer of the spin-torque nano-oscillators (STNOs) areinvestigated in the presence of DMI and the perpendicular magnetic ﬁeld by using micromagnetic simulations. We observed that the oscillation frequency of bubble-like magnetic solitons can be manipulated by the perpendicular magnetic ﬁeld. Moreover, the magnetic structures keep stable insmall DMI. With an increase in the DMI strength, rich kinds of bubble-like magnetic solitons appear at different spin-polarized current and perpendicular magnetic ﬁeld. These results provide a further understanding of bubble-like magnetic solitons structures and direct applications in STNOs.Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4967798 ] I. INTRODUCTION In recent years, the spin-wave excitations in magnetic nanostructures have attracted a considerable attention. Magnetic solitons are self-localized spin-wave excitationsthat exist in dissipative magnetic materials. 1–5They behave like particles with both static and dynamic forms. In particu-lar, static form includes skyrmions, 6–10vortices,11magnetic bubbles,12domain walls (DW),13etc. Dynamic form includes droplets,1,5spin wave bullets,14,15dynamical skyrmions,16 vortex-pairs,17etc. A magnetic bubble is a circular bi-domain state,12,18which can be described as two concentric domains (up and down) separated by an in-plane domain wall (DW).Magnetic solitons can be locally nucleated in a disk-shapedferromagnetic layer with strong perpendicular magneto-crystalline anisotropy (PMA) in nanoscale oscillators andmanipulated by the spin-polarized current based on spintransfer torque (STT). STT provides the exchange betweenangular momentum from the spin-polarized electron andmagnet, which is an effective way to manipulate magneticsolitons, such as sustaining a stable rotation of vortices andskyrmions in spin-torque nano-oscillators (STNOs) 16,19,20 and driving skyrmions and domain walls in nanowires, which can be used in the racetrack memory.21–23Recently, the dynamics of bubble-like magnetic solitons have been investi-gated theoretically 4and experimentally.24 In the presence of Dzyaloshinskii-Moriya interaction (DMI), bubble-like magnetic solitons can be nucleated asdifferent magnetic structures in STNOs, such as magneticskyrmions 25and droplets.16In our previous work, three different bubble-like magnetic solitons (pseudonormalmagnetic droplet, singular magnetic droplet, and dynamicalskyrmion) are nucleated in the presence of DMI. 26Due to the different topological charges, these bubble-like magnetic solitons have different dynamic responses as the current den-sity varies. However, the effects of the perpendicular mag- netic ﬁeld on the nucleation and dynamics of bubble-like magnetic solitons are not clear in the presence of differentDMI. In this paper, we report the dynamic responses of bubble-like magnetic solitons with the combined action ofperpendicular magnetic ﬁeld and current in the presence of different DMI. Using micromagnetic simulations, bubble- like magnetic solitons are nucleated in nanocontact STNOsbased on a spin-valve with a free layer presented PMA.Then, we investigated the dynamic properties of bubble-like magnetic solitons in the presence of the perpendicular magnetic ﬁeld. In the presence of small DMI, the oscillationfrequency increases with an increase in the perpendicular magnetic ﬁeld while the magnetic structure of the pseudo- normal magnetic droplet keeps stable. Further increasing theDMI strength, the perpendicular magnetic ﬁeld affects both oscillation frequency and the structure of bubble-like mag- netic solitons. II. MICROMAGNETIC SIMULATION DETAILS As shown in Fig. 1, we consider a nanocontact STNO based on the spin-valve structure with a soft thin ferromag- netic layer presented PMA (free layer) and a thick hard layer(ﬁxed layer), which are separated by a nonferromagneticlayer (space layer). The ﬁxed layer is assumed to be magne- tized along the zdirection. There are two electrodes at the top and bottom of the spin-valve, where randRrepresent the radius of electrodes and nanodisk with the value of 20 nm and 125 nm, respectively, and the thickness of the free layer is 2 nm. The spin-polarized current ﬂows through point a)Author to whom correspondence should be addressed. Electronic mail: liuqf@lzu.edu.cn. Tel.: þ86-0931-8914171. Fax: þ86-0931-8914160. 0021-8979/2016/120(18)/183901/6/$30.00 Published by AIP Publishing. 120, 183901-1JOURNAL OF APPLIED PHYSICS 120, 183901 (2016) contact electrodes perpendicularly and locally, and then drives the reversal of local magnetization to generate mag- netic solitons. Micromagnetic simulations are performed with the GPU- accelerated open-source simulation software MuMax3.27The dynamics of magnetization in the free layer is governed bythe Landau-Lifshitz-Gilbert (LLG) equation, including the STT term 28 dm dt¼/C0cm/C2Heffþam/C2dm dt/C18/C19 /C0aJm/C2mP/C2m ðÞ ;(1) where the unit magnetization vector of the free layer is m¼M=MS;MSis the saturation magnetization, cis the gyromagnetic ratio, and ais the Gilbert damping factor. The third term in Eq. (1)describes the STT; mPis the unit mag- netization vector of the ﬁxed layer, the factor aJðhÞ ¼/C22hcgðhÞJ=ð2jejMStÞ, where Jis the current density, eandt are electron charge and the thickness of the free layer, andgðhÞis a scalar function depending on the spin polarization P and the angle hbetween mandm P. Here, the effective mag- netic ﬁeld Heff¼/C0 ð 1=l0MSÞ@E=@Mis derived from the free energy of this system. The free energy includes the con- tributions from the exchange interaction energy, the magne- tostatic energy, the magneto-crystalline anisotropy energy,the Zeeman energy, and the DMI energy, and the energy induced by the Oersted ﬁeld is also considered. DMI energy can be written as 25 EDM¼dðð Dm x@mz @x/C0mz@mx @x/C18/C19/C20 þmx@mz @x/C0mz@mx @x/C18/C19 /C21 d2r; (2) where Dis the strength of DMI. The parameters of the free layer used in our simulations are as follows:29saturation magnetization is MS¼6.5/C2105A/m, exchange constant is A¼1.3/C210/C011J/m, PMA constant is Ku¼3.3/C2105J/m3, andPandaare set as 0.01 and 0.4, respectively. The free layer was divided into cells with 2 /C22/C22n m3. In our simu- lations, the temperature is ignored, and the stray ﬁeldproduced by ﬁxed layer is introduced as an additional contri- bution to the effective magnetic ﬁeld.III. RESULTS AND DISCUSSIONS A bubble-like magnetic soliton can be nucleated in the free layer with PMA of a STNO by injecting a large enoughspin-polarized current through the nanocontact electrode.When D¼0.1 mJ/m 2, a pseudonormal magnetic droplet is nucleated in the free layer at zero perpendicular magnetic ﬁeldunder the spin-polarized current J a¼12/C21011A/m2,a s shown in the top row of Fig. 2(a)(Multimedia view), where the dashed circle represents a point contact area. The black arrows in Fig. 2(a)represent the in-plane magnetization com- ponents of the DW, and the red and blue regions are the out-of-plane magnetization components with opposite orientation,where the red region represents the magnetization along thez-axis and the blue region is along the z-axis in the opposite direction. When the pseudonormal magnetic droplet is nucle- ated, the local magnetizations of blue region rotate around a point contact region under the current. The down row ofFig. 2(a) is the associated topological density distribution, which shows that the pseudonormal magnetic droplet is com-posed of two pairs of bubble/antibubble, 30and the topological density is positive in the bubble area and negative in the anti-bubble area. It is possible to characterize bubble-like magnetic solitons on the basis of their topology, like the skyrmion num- berN¼ 1 4pÐnðx;yÞdxdy,w h e r e nx;yðÞ ¼m/C1@m @x/C2@m @y/C16/C17 is the topological charge density, which indicates how the unit magnetization vector mvaries at every point of the ﬁlm.25 According to this deﬁnition, the skyrmion number of the pseu- donormal magnetic droplet is N¼0. When applying a perpen- dicular magnetic ﬁeld, the size of the pseudonormal magneticdroplet can be modulated, as shown in Figs. 2(b) and2(c). Fig.2(b)gives the magnetization conﬁguration (top row) and the associated topological density distribution (down row) ofthe pseudonormal magnetic droplet by applying a current J a ¼12/C21011A/m2at perpendicular magnetic ﬁeld /C030 mT along the z-axis in the opposite direction. This indicates that the size of the pseudonormal magnetic droplet is larger thanthat shown in Fig. 2(a).F i g . 2(c)shows that the pseudonormal magnetic droplet size is smaller than that in a zero magneticﬁeld when a magnetic ﬁeld of 30 mT is applied. Then, withthe aim of investigating the dynamic response of the pseudo- normal magnetic droplet in the presence of perpendicular magnetic ﬁeld, a systematic study in different current and ﬁeldis performed. The phase diagram of the pseudonormal magnetic drop- let oscillation frequency as a function of the current and per-pendicular magnetic ﬁeld is summarized in Fig. 3(a), where the magnetic ﬁeld varies from /C050 mT to 100 mT, and the current density varies from J a¼10/C21011A/m2to Ja¼20/C21011A/m2. It describes the ﬁeld- and current- dependent tunability of the oscillation frequency of the pseu- donormal magnetic droplet, which is obtained from the fast Fourier transform (FFT) of in-plane magnetization compo-nent m x. The result indicates that the oscillation frequency is linearly increasing with the increasing perpendicular mag-netic ﬁeld and does not depend much on current. The dashedlines in Fig. 3(a)represent the oscillation frequency in three different perpendicular magnetic ﬁelds 0 mT, /C030 mT, and FIG. 1. The schematic illustration of a nanocontact STNO based spin-valve structure.183901-2 Song et al. J. Appl. Phys. 120, 183901 (2016) 30 mT, as shown in Fig. 3(b). Under a zero magnetic ﬁeld, the oscillation frequency decreases slowly with increasing current. When applying a perpendicular ﬁeld along the z-axis, the oscillation frequency increases obviously when compared with the case of 0 mT, and decreases obviouslywith an perpendicular ﬁeld along the z-axis in the opposite direction. In the presence of a perpendicular magnetic ﬁeld,the oscillation frequency still decreases slowly at differentcurrents, while the trend is more obvious than the case of azero ﬁeld. In detail, the oscillation frequency increases withthe decrease of the size of the pseudonormal magnetic drop-let at the magnetic ﬁeld along the z-axis, and it decreases with an increase in the size of the pseudonormal magneticdroplet under a magnetic ﬁeld along the z-axis in the oppo- site direction. In particular, the magnetization structure remains stable with varying current and the perpendicularmagnetic ﬁeld. When D¼0.5 mJ/m 2, a dynamical skyrmion is nucle- ated under the current density Ja¼12/C21011A/m2at a zeroﬁeld. Then, we focus our attention on the effect of the per- pendicular magnetic ﬁeld on dynamical skyrmion. Fig. 4(a) shows the inﬂuence of the perpendicular magnetic ﬁeld onthe oscillation frequency and the magnetic structure of thedynamical skyrmion. The magnetization conﬁguration ofthe dynamical skyrmion (left) and corresponding topological density distribution (right) are shown in Fig. 4(b) (Multimedia view), which exhibits a strong breathing modeunder the large enough spin-polarized current. 12This breath- ing mode leads to a variation in magnetization componentm z, and the oscillation frequency can be obtained from FFT ofmz. The dynamical skyrmion is a topologically protected particle with the skyrmion number N¼/C01, which is the same as the static skyrmion, while the dynamic responses ofthe dynamical skyrmion are different from the static sky-rmion under the spin-polarized current and magnetic ﬁeld. Itis noteworthy that the perpendicular magnetic ﬁeld can affectboth the oscillation frequency and magnetic structure for the dynamical skyrmion. The inset ﬁgure in Fig. 4(a) shows FIG. 2. Magnetization conﬁgurations of the pseudonormal magnetic droplet (top row) and corresponding topological den- sity distributions (down row) at D¼0.1 mJ/m2in the presence of the magnetic ﬁeld at (a) zero ﬁeld, (b) /C030 mT, and (c) 30 mT. The dashed circle represents a nanocontact area. Red and blue colors(top row) indicate positive and negative magnetization component along the z-axis. (Multimedia view) [URL: http:// dx.doi.org/10.1063/1.4967798.1 ] FIG. 3. (a) The phase diagram of pseudonormal magnetic droplet frequency as a function of current density and perpendicular ﬁeld at D¼0.1 mJ/m2, and (b) three dashed lines indicate the inﬂuence of current on oscillation frequency at 0 mT, /C030 mT, and 30 mT.183901-3 Song et al. J. Appl. Phys. 120, 183901 (2016) different kinds of nucleated magnetic structures as the mag- netic ﬁeld varying from /C050 mT to 50 mT. When applying magnetic ﬁeld along the z-axis in the opposite direction, the nucleated magnetic structures are still dynamical skyrmionwhen the magnetic ﬁeld varies from 0 mT to /C020 mT, while the oscillation frequency decreases with an increase in thebreathing amplitude. When the magnetic ﬁeld increases from/C020 mT to /C050 mT, a new kind of magnetic soliton is nucle- ated. In-plane magnetization components in the DW of this soliton are antisymmetric, and the in-plane magnetization components of the both ends orientate inside. It is a topologi-cal magnetic soliton, and the skyrmion number of this objectisN¼1. To distinguish with the dynamical skyrmion and static skyrmion, we called this cigar-shaped skyrmion, asshown in the left column of Fig. 4(c) (Multimedia view). The corresponding topological density distribution is pre-sented in the right column of Fig. 4(c). Under a spin- polarized current, the cigar-shaped skyrmion rotates as a whole around the point contact area in a counterclockwisemode. For this kind of bubble-like magnetic soliton, the spa-tial averages of magnetization component m xand myare zero all the time, and mzis non-zero with the value of 0.84 all the time, so the oscillation frequency from FFT of mx,my, andmzis zero. To get the oscillation frequency, we choose a zoom that both ends of the cigar-shaped skyrmion go through periodically and calculate this frequency from FFT.The results show that the frequency is smaller than the oscil-lation frequency of dynamical skyrmion, and increases whenapplying a ﬁeld along the z-axis, while decreases with a ﬁeld applied along the z-axis in the opposite direction. It is clear that the shape of cigar-shaped skyrmion in a positive ﬁeld is ﬂatter than that in a negative ﬁeld for the scale-up of the blue region in a negative ﬁeld. When increasing the magneticﬁeld beyond /C050 mT, a FM (AP) state with downward orien- tation appears. Then with magnetic ﬁeld along the z-axis from 0 mT to 30 mT applied, the cigar-shaped skyrmion is nucleated. While the magnetic ﬁeld increases beyond 30 mT,the dynamical skyrmion transforms to the pseudonormalmagnetic droplet, and the oscillation frequency increases lin-early with an increase in the magnetic ﬁeld. This phenome-non is similar to the effect of perpendicular magnetic ﬁeldon the pseudonormal magnetic droplet in the case of D¼0.1 mJ/m 2. When the perpendicular magnetic ﬁeld is greater than 550 mT, the free layer keeps a FM (P) state with upwardmagnetization. In order to investigate the nucleation states of different kinds of magnetic structures in the presence of the perpen-dicular magnetic ﬁeld, further analysis of the nucleation inthe presence of different DMI is shown in Fig. 5. The nucle- ation phase diagram indicates the nucleated steady magneticstructures when current and ﬁeld are considered together,where the perpendicular magnetic ﬁeld l 0Hvaries from /C060 mT to 100 mT, and the current density Javaries from 6/C21011A/m2to 20 /C21011A/m2. As shown in Fig. 5(d), ﬁve different magnetic structures of bubble-like magnetic solitons can be identiﬁed: a pseudonormal magnetic droplet,a singular magnetic droplet, a cigar-shaped skyrmion, adynamical skyrmion, and a skyrmion. There also exists a FM(AP) state and an unstable state. Fig. 5(a)(Multimedia view) exhibits the nucleation phase diagram in the case of D¼0.3 mJ/m 2, the singular magnetic droplet (vertical purple ellipse) nucleated at zero ﬁeld. With increasing current, dynamicalskyrmions (green ﬁlled circle) are nucleated. When applyingthe perpendicular magnetic ﬁeld along the z-axis, pseudonor- mal magnetic droplets (red circle) are nucleated when the magnetic ﬁelds are greater than 20 mT. Cigar-shaped FIG. 4. (a) The effect of the perpendic- ular magnetic ﬁeld on the frequency of dynamical skyrmion at D¼0.5 mJ/m2 when Ja¼12/C21011A/m2, the inset shows bubble-like magnetic solitons nucleated in the low ﬁeld region. (b) and (c) are magnetization conﬁgurations and associated topological density distributions of dynamical skyrmion and cigar-shaped skyrmion. The cigar- shaped skyrmion rotates in a counter-clockwise direction under the current. (Multimedia view) [URL: http:// dx.doi.org/10.1063/1.4967798.2 ][ U R L : http://dx.doi.org/10.1063/1.4967798.3 ]183901-4 Song et al. J. Appl. Phys. 120, 183901 (2016) skyrmions (slanted blue ellipse) are generated at /C040 mT when the current density varies from 12 /C21011A/m2to 20/C21011A/m2. In a small ﬁeld and a high current density, the dynamical skyrmions are stable. When a perpendicular magnetic ﬁeld of /C060 mT is applied, the free layer keeps the FM (AP) state with downward orientation of magnetization conﬁguration (blue ﬁlled square). When D¼0.5 mJ/m2, more dynamical skyrmions are nucleated with a large varia- tion of nucleation current at zero ﬁeld than the case of D¼0.3 mJ/m2, and the singular magnetic droplets disappear with more cigar shaped skyrmions nucleated, as shown in Fig. 5(b). There are also more unstable states (black ﬁlled square), which cannot be nucleated in any kind of bubble- like magnetic solitons all the time. FM (AP) states with downward orientation appear at /C060 mT, which is the same as that in D¼0.3 mJ/m2. When further increasing the DMI strength, dynamical skyrmion nucleated in a large scope at different current and ﬁeld, and pseudonormal magnetic drop- let states reduce. Fig. 5(c) shows the nucleation phase dia- gram when D¼0.8 mJ/m2. Due to its topological property, dynamical skyrmion is stable over a large current and ﬁeld in the presence of large DMI relatively. In particular, sky- rmions (red ﬁlled circle) are nucleated at /C040 mT current density Javaried from 6 /C21011A/m2to 16 /C21011A/m2. When D¼0.8 mJ/m2, the unstable state occurs at a low cur- rent and low ﬁeld; however, the unstable state occurs over a large current and ﬁeld range in the case of D¼0.5 mJ/m2, which indicates that the stability of the nucleation ofmagnetic solitons can be enhanced with a large DMI. When the perpendicular magnetic ﬁeld varies, the pseudonormalmagnetic droplet appears in a large enough magnetic ﬁeldalong the z-axis, and the free layer keeps FM states in a spe- ciﬁc perpendicular magnetic ﬁeld along the z-axis in the opposite direction. In the presence of the strong current andﬁeld, the pseudonormal magnetic droplet disappears and isreplaced by other magnetic structures with a relatively large DMI. By varying the spin-polarized current and perpendicu- lar ﬁeld in the presence of DMI, different kinds of magneticsolitons appear. IV. CONCLUSIONS In summary, we have investigated the nucleation and dynamical responses of bubble-like magnetic solitons withthe combined action of the perpendicular magnetic ﬁeld andcurrent in the presence of different DMI. In a small DMIwith D¼0.1 mJ/m 2, the perpendicular magnetic ﬁeld affects the size of pseudonormal magnetic droplet while the magne-tization structure keeps stable. The oscillation frequency ofpseudonormal magnetic droplet increases rapidly with anincrease in the perpendicular magnetic ﬁeld along the z-axis. Furthermore, in a large DMI, both magnetization structureand frequency are affected by the perpendicular magneticﬁeld, and rich kinds of magnetic solitons are nucleated. With an increase in the DMI strength, the dynamical skyrmion occupies most as a stable magnetic structure. These results FIG. 5. The nucleation phase diagrams at different current and ﬁeld when (a) D¼0.3 mJ/m2, (b) D¼0.5 mJ/m2and (c)D¼0.8 mJ/m2. Schematic represen- tations of different magnetic structures are shown in (d), a red circle represents pseudonormal magnetic droplet; a vertical purple ellipse represents the singular magnetic droplet; a slanted blue ellipse represents cigar-shaped skyrmion; a green ﬁlled circle repre-sents dynamical skyrmion; a red ﬁlled circle represents static skyrmion; a blue ﬁlled square represents the FM (AP) state with downward orientation, and unstable state is represented by a black ﬁlled square. (Multimedia view) [URL: http://dx.doi.org/10.1063/1.4967798.4 ]183901-5 Song et al. J. Appl. Phys. 120, 183901 (2016) provide a new understanding of the effect of the perpendicu- lar magnetic ﬁeld on the dynamics of bubble-like magneticsolitons in the presence of DMI, and contribute to a manipu-lation of bubble-like magnetic solitons in STNOs and othermagnetic devices. 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5.0031957.pdf	AIP Advances 11, 015205 (2021); https://doi.org/10.1063/5.0031957 11, 015205 © 2021 Author(s).Nonlinear dynamics of magnetization evolution in orthogonal spin torque devices: Phases and classification Cite as: AIP Advances 11, 015205 (2021); https://doi.org/10.1063/5.0031957 Submitted: 07 October 2020 . Accepted: 11 December 2020 . Published Online: 05 January 2021 Yuan Hui , Zheng Yang , and Hao Yu COLLECTIONS Paper published as part of the special topic on Chemical Physics , Energy , Fluids and Plasmas , Materials Science and Mathematical Physics ARTICLES YOU MAY BE INTERESTED IN Precession coupled spin current in spin torque driven magnetic tunnel junctions AIP Advances 11, 015006 (2021); https://doi.org/10.1063/9.0000020 Bias-field-free high frequency microwave emission of spin-transfer nano-oscillator with magnetizations all in-plane Applied Physics Letters 118, 012405 (2021); https://doi.org/10.1063/5.0031507 Field-free and sub-ns magnetization switching of magnetic tunnel junctions by combining spin-transfer torque and spin–orbit torque Applied Physics Letters 118, 092406 (2021); https://doi.org/10.1063/5.0039061AIP Advances ARTICLE scitation.org/journal/adv Nonlinear dynamics of magnetization evolution in orthogonal spin torque devices: Phases and classification Cite as: AIP Advances 11, 015205 (2021); doi: 10.1063/5.0031957 Submitted: 7 October 2020 •Accepted: 11 December 2020 • Published Online: 5 January 2021 Yuan Hui, Zheng Yang, and Hao Yua) AFFILIATIONS Department of Physics, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road, Suzhou 215123, People’s Republic of China a)Author to whom correspondence should be addressed: hao.yu@xjtlu.edu.cn ABSTRACT The magnetization evolution of the free layer in an orthogonal spin torque device is studied based on a macrospin model. The trajectory of the magnetization vector under various conditions has shown rich nonlinear properties. The phase diagram is obtained in the parameter spaces of current density and the polarization distribution (the ratio of polarization of in-plane to out-of-plane layers), where two critical currents and three phases are found. These dynamic phases can be classified according to their nonlinear behaviors, which are different in terms of limit cycles and limit points. The classification is meaningful to design ultra-fast spin torque devices under different dynamic conditions toward various applications, such as in memory and oscillators. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0031957 I. INTRODUCTION Spin-transfer torque (STT) discovered by Slonczewski and Berger1,2is an effect that conductive electrons carrying angular momentum reorient the local spins, which enables the manipula- tion of magnetization by a spin-polarized current flowing through a multi-layered junction. Current-induced STT expands the writ- ing technique for data storage, such as the magnetic random-access memory (MRAM).3,4It is also promising for microwave oscillators owing to the ultrafast precessional oscillation of the magnetic free layer.5An STT nanopillar magnetic tunnel junction (MTJ) con- sists of a free layer and one or two reference layer(s) where the magnetization of layers can be either in-plane (IP) or out-of-plane (OOP). The combination of orthogonal or OOP polarization layer with IP reference layers, first proposed by Kent,6has been demon- strated to be able to achieve more efficient layer magnetization.7 In such orthogonal spin torque (OST) devices, the maximum spin- transfer torque from the beginning of the current pulse causes faster reversal of layer magnetization and less switching energy is cost, and therefore, it could be seen that the polarization comprising both an IP reference polarizer and an OOP polarizer optimizes the original STT sandwich model and realizes a more ideal writing technique.The evolution dynamics of magnetization in OST devices have been studied by previous researchers using analytical calculation and numerical simulation on various models.8–13They have shown that (i) the free layer magnetization can be ten times faster in OST than that of an IP-only device;8(ii) steady precession can be excited by a current and the precession frequency depends on the strength of the orthogonal polarizer;9(iii) time for complete reversal, namely, from parallel (P) to anti-parallel (AP) or from AP to P, can be shortened by adjusting the pulse width. A ratio ( r) of the in-plane (IP) reference to the perpendicular polarizer was defined to analyze and compare the influence of two reference layers. Pinna et al. developed a theoretical model11,12in energy space to study the STT/OST magnetization dynamics considering ther- mal effects. In their model, the parameter ωindicating the ratio of spin polarization efficiency was derived analytically to OOP pre- cession. Two critical currents ICand IOOP as the function of ω were built,13which separate the energy spaces into three regimes. They also showed that the shape of pulse can determine the final magnetization state. The current-field state diagram14for an OST device has been presented experimentally illustrating the range of different states. Low temperature OST memory element has been studied15 over a wide range of parameter space, which is the first clear AIP Advances 11, 015205 (2021); doi: 10.1063/5.0031957 11, 015205-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv experimental evidence of magnetization precession in an OST device where the dynamics of magnetization are dominated by the OOP polarization. Previous research studies have demonstrated that a macrospin model is effective in reflecting the intrinsic property of the magneti- zation dynamics of STT or OST devices. However, the dynamics of such spin oscillators are very complicated with rich physics in terms of their nonlinearity. In a conventional STT device (only with an in- plane reference layer), the transition to chaotic dynamics has been revealed in a study16on the Landau–Lifshitz–Gilbert–Slonczewski (LLG) equation, showing a series of period doubling bifurca- tions. The nonlinear phenomena suggest that we could classify the dynamic system in terms of its nonlinearity, such as limit cycle/limit point formed by the evolution trajectory of the magnetization vec- tor. In fact, in this macrospin OST model, we discovered that the dynamic process of magnetization evolves with horizontal equi- libria bifurcation under particular conditions of current density and polarization distribution (the ratio of IP layer to OOP layer), which offers a reference to different applications. For each phase, a specific range of current density and polarization distribution are provided. In this paper, we demonstrate the dynamic magnetization pro- cess in OST by numerical simulations and discuss the results from the perspective of nonlinearity in which both the current density ( J) and polarization ratio ( r) determine the magnetization states. The following contents will be orderly introducing the model, results along with the analysis of the relationship between parameters, and final conclusion with related applications. II. MODEL We adopted a macrospin model from Ref. 10, which was initially constructed according to the Landau–Lifshitz– Gilbert–Slonczewski equation,1,2,17 d⃗m dt=−γ⃗m×⃗He f f+αγ⃗m×(⃗m×⃗He f f)−τ∥⃗m×(⃗m×⃗ny) +τ/⊙◇⊞⃗m×(⃗m×⃗nz), τ∥=γa∥;τ/⊙◇⊞=γa/⊙◇⊞,⃗Heff=−Hdmz⃗nz+HKmy⃗ny,(1) where⃗m=⃗Ms Msis the normalized magnetization and ⃗Msis the satu- ration magnetization of the free layer, ⃗nyand⃗nzare the unit vector along yand z, respectively, γ=γ0/(1+α2), whereγ0is the gyro- magnetic ratio and αis the Gilbert damping parameter, and ⃗He f f is the effective field defined as ⃗Heff=−Hdmz⃗nz+HKmy⃗ny, where Hd is the demagnetizing field and HKis the uniaxial shape anisotropy field. The two coefficients τ∥andτ/⊙◇⊞are torques due to the parallel and perpendicular reference layers, respectively: τ∥(τ/⊙◇⊞)=̵hη∥(η/⊙◇⊞) 2eμ0MsdzJ, where eis the electron charge, μ0is the permeability of vacuum,̵his the Planck constant, dzis the thickness of the free layer, Jis the cur- rent density, and η∥(η/⊙◇⊞)is the current polarization of the parallel and perpendicular reference layers. The external current is provided as pulse, which is downward along the z axis, as shown in Fig. 1(b). The free layer made of iron18is of length dy=100 nm, width dx =50 nm, and thickness dz=5 nm, and its easy-axis is along y. The free layer is only affected by the spin-transfer torque generated by FIG. 1. Schematic of a cell of sandwich spin valve device: (a) a conventional STT nanopillar with in-plane polarizing magnetization, consisting of an analyzer and a free layer (from the top to the bottom); (b) an OST nanopillar with an out-of-plane polarizing reference layer in the bottom. The easy-axis of magnetization of the free layer is along the y direction. two polarizers through current pulse. No external field is added. The detailed definition and value of parameters are defined in Table I as the Appendix. The polarization distribution factor ris defined as the ratio of the spin torque amplitude of the in-plane analyzer to the perpendic- ular polarizer.10When r=0, the perpendicular polarizer dominates the polarization of the free layer; meanwhile, r=∞means the major influence is from the in-plane analyzer. The study is mainly focused on the dynamic evolution of the magnetization vector of the free layer. Numerical simulation is done for each fixed randJby apply- ing external current pulse with width 1 ns (on 1 ns and then off 1 ns), which is a periodic rectangular wave. III. RESULTS The spin torque is maximum at the beginning owing to the orthogonal polarization reference layer, and therefore, tilting or switching occurs very fast after a while. As the result, Fig. 2 shows the transition as an example taken r=1 when the current den- sity increases [(a) J=3×1010A/m2, (b) J=3×1011A/m2, and (c) J=4×1011A/m2]. The left-hand side lists 3D diagrams of the nor- malized magnetization vector for each case and is attached on the right-hand side with corresponding oscillation of my, namely, the magnetization along the easy axis of the free layer. Transi- tion appears as the current density increases. Specifically, Phase 1 [Fig. 2(a)] represents the condition when the current density ( J= 3×1010A/m2) is below the critical value that is insufficient for com- plete switching. Phase 2 [Fig. 2(b)] shows the stage when the current density is enough for magnetization ( J=3×1011A/m2) where the free layer could achieve a reversal of my, namely, P to AP. Within the first half period 1 ns, myoscillates fast with a stable precessional frequency.9,12There are two equilibrium states for Phase 2, one limit circle in the first half period and one limit point in the second half period. The stability of these equilibria changes when the current density increases. It is different from Phase 1 where the two states are both limit points. This would be discussed in the following con- tents. For the last stage, Phase 3 [Fig. 2(c)], when the current density is large enough ( J=4×1011A/m2), the middle oscillation converges as the limit circle transforms into another limit point. AIP Advances 11, 015205 (2021); doi: 10.1063/5.0031957 11, 015205-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 2. Each phase of the magnetization process with r=1 and (a) J=3×1010A/m2, (b) J=3×1011A/m2, and (c) J=4×1011A/m2, respectively. The diagram on the left: 3D magnetization vector (normalized mx,my, and mz) evolution; right: my, the magnetization along the easy axis responds to the current pulse in two periods. Nonlinear behaviors can be found from the left diagrams as (a) two limit points with different mx; (b) one limit point and one limit circle; and (c) two limit points along z. Because of the contribution of the perpendicular polarizer, the free layer magnetization vector oscillates before reaching the final state. Therefore, the oscillation of the free layer has different forms in each situation, known from Fig. 2. Particularly, the appearance of well-defined frequency in Phase 2 is related to the system equilibria asymptotically approaching to the limit cycle.In addition, each phase counters a range of current density and the limiting case causes bifurcation of the system.16,19When ris fixed, the main contribution of magnetization from the in-plane ana- lyzer or perpendicular polarizer is determined. Therefore, the critical current density to bifurcation could be obtained for each r. In our simulation, the fixed current width δ=1 ns promises the following AIP Advances 11, 015205 (2021); doi: 10.1063/5.0031957 11, 015205-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv results, which are three different phases. However, by changing the width of the current pulse, the corresponding critical current den- sity for transition could also be altered.10The shape of pulse also determines the final state of magnetization.13The following expla- nation and the phase diagram (Fig. 3) are about the relationship of the current density and polarization distribution. The diagram of dynamic phases (Fig. 3) shows the boundary between phases, which indicate that there are two critical currents. It is noted that in Refs. 12 and 13, Pinna et al. have developed an ana- lytical model based on the energy orbit approach. In their model, a fixed point in energy space corresponds to a limit cycle of the stable precessional state of magnetization dynamics. One feature of their model is that there are two critical currents ICandIOOP; both are the function of spin torque efficiency ω, which is defined as tan ω≡ηpol/ηref, whereηpolandηrefare the spin torque of the perpen- dicular polarizer and reference layer, respectively. The feature of two critical currents is the same as our simulation, so we compare our results to their analytical ones and find that ηpol/ηrefis just propor- tional to rdefined in our work, which derives tan ω∝1/r.In their work, IC∝1/cosωandIOOP∝1/sinω, which means that the two critical currents have different monotonicity. However, in our sim- ulation, the two critical currents both monotonically increase with respect to r. This difference may be due to the distinction of mod- els and the complexity of nonlinear dynamics systems, where the dynamic behaviors are sensitive to parameters including r,J, and even the shape and width of current pulse. As mentioned in previous work,10the frequency of oscilla- tion in mydecreases as rbecomes larger. This is because for a certain value of current density, the value of rdetermines how much torque can be transferred from the in-plane analyzer and perpendicular polarizer. Hence, when rincreases, the horizontal polarization dominates, which weakens the influence of perpendic- ular polarization, so that there is less oscillation. Since our study shows that for each condition with respect to different r,Jvaries FIG. 3. Phase diagram shows the critical current density for each case with differ- entr. For rin the range of (0, 10), the system transforms through all three phases. Asrkeeps increasing, the available range of current density enabling the system to reach Phase 2 expands, where well-defined frequency appears. FIG. 4. Frequency with corresponding Phase 2 current density for each r. The fit lines illustrate the linear relationship between the current density and oscillation frequency in the y plane. as well in case to cause the complete polarization of the free layer, we restricted situation to only Phase 2, where the stable frequency would appear to study the further result in frequency and relation- ship between Jand frequency f. Precessional oscillation frequency fof Phase 2 is shown in Fig. 4 for different cases of rwith respect to J. It indicates that fis a linear function of J, and when rbecomes larger, the corresponding range of current density for Phase 2 expands. The result is consistent with that in Ref. 9 where frequency is linearly proportional to current and the spin torque of the OOP polarizer and therefore fshould be inversely proportional to r. Figure 4 clearly shows that for fixed J, when rincreases, fdecreases. IV. SUMMARY In conclusion, our study is based on the model of spin-transfer- torque MTJ with an additional perpendicular polarizer, namely, an OST device. The results show that due to the different current den- sity and the spin torque polarization ratio of IP to OOP, the dynamic features of magnetization evolution trajectories vary in terms of non- linearity, which are summarized as follows: (1) two limit points for Phase 1 where the current density is below the critical value and no switching occurs; (2) one limit point and one limit cycle for Phase 2 with current density greater than the critical point, where OOP precessional oscillation occurs within one pulse; (3) two limit points for Phase 3 with even larger current, the system switches between two limit points. The category of magnetization evolution of the free layer according to the nonlinear dynamics can help us understand the phase diagram of dynamic behaviors of OST spin valves, and it also offers a reference to design MTJ devices for various application purposes, such as in ultra-fast microwave oscillators or MRAM. One can adjust the parameters of devices, namely, randJ, to make their value fall in the appropriate phase and category. ACKNOWLEDGMENTS This research was supported by the Key Programme Special Fund (Grant No. KSF-E-22) the and Research Enhancement Fund (Grant No. REF17-1-7) of Xi’an Jiaotong-Liverpool University. AIP Advances 11, 015205 (2021); doi: 10.1063/5.0031957 11, 015205-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv TABLE I. Parameters for simulation. Parameter (variables) Definition Expression (value/range) ⃗m The normalized magnetization vector ⃗m=[⃗mx,⃗my,⃗mz]T γ0Gyromagnetic ratio 8.861 ×106rad/(s T) α Gilbert damping parameter 0.02 γγ0 1+α2 8.86×106rad/(s T) η∥(η/⊙◇⊞) Current polarization layer for in-plane (perpendicular) η∥=0.3×r/√ 1+r2η/⊙◇⊞=η∥/r a∥(a/⊙◇⊞) Spin polarizing amplitude for in-plane (perpendicular) polarizer̵hη∥(η/⊙◇⊞) 2eμ0MsdzJ Hd The demagnetizing field 1.2 ×106A/m Ms Saturation of magnetization of free layer 1.2 ×106A/m HK The uniaxial shape anisotropy field 4 ×104A/m ⃗ny unit vector along the in-plane y axis [0, 1, 0]T ⃗nz unit vector along the perpendicular z axis [0, 0, 1]T Ra∥ a/⊙◇⊞=η∥ η/⊙◇⊞[0,∞) J Density of current pulse [0,∞)A/m2 APPENDIX: SIMULATION PARAMETERS Table I shows the parameters for simuation. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 2L. Berger, Phys. Rev. B 54, 9353 (1996). 3A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012). 4A. D. Kent and D. C. Worledge, Nat. Nanotechnol. 10, 187 (2015). 5S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature 425, 380 (2003). 6A. D. Kent, B. Özyilmaz, and E. del Barco, Appl. Phys. Lett. 84, 3897 (2004).7O. J. Lee, V. S. Pribiag, P. M. Braganca, P. G. Gowtham, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 95, 012506 (2009). 8Z. Hou, Z. Zhang, J. Zhang, and Y. Liu, Appl. Phys. Lett. 99, 222509 (2011). 9H. Zhang, Z. Hou, J. Zhang, Z. Zhang, and Y. Liu, Appl. Phys. Lett. 100, 142409 (2012). 10A. Mejdoubi, B. Lacoste, G. Prenat, and B. Dieny, Appl. Phys. Lett. 102, 152413 (2013). 11D. Pinna, A. D. Kent, and D. L. Stein, Phys. Rev. B 88, 104405 (2013). 12D. Pinna, D. L. Stein, and A. D. Kent, Phys. Rev. B 90, 174405 (2014). 13D. Pinna, C. A. Ryan, T. Ohki, and A. D. Kent, Phys. Rev. B 93, 184412 (2016). 14L. Ye, G. Wolf, D. Pinna, G. D. Chaves-O’Flynn, and A. D. Kent, J. Appl. Phys. 117, 193902 (2015). 15G. E. Rowlands, C. A. Ryan, L. Ye, L. Rehm, D. Pinna, A. D. Kent, and T. A. Ohki, Sci. Rep. 9, 803 (2019). 16Z. Yang, S. Zhang, and Y. C. Li, Phys. Rev. Lett. 99, 134101 (2007). 17C. Abert, Eur. Phys. J. B 92, 120 (2019). 18S. Yuasa, K. Hono, G. Hu, and D. C. Worledge, MRS Bull. 43, 352 (2018). 19M. Lakshmanan, Philos. Trans. R. Soc. A 369, 1280 (2011). AIP Advances 11, 015205 (2021); doi: 10.1063/5.0031957 11, 015205-5 © Author(s) 2021
5.0014879.pdf	Appl. Phys. Lett. 117, 062405 (2020); https://doi.org/10.1063/5.0014879 117, 062405 © 2020 Author(s).Regulating the anomalous Hall and Nernst effects in Heusler-based trilayers Cite as: Appl. Phys. Lett. 117, 062405 (2020); https://doi.org/10.1063/5.0014879 Submitted: 22 May 2020 . Accepted: 01 August 2020 . Published Online: 14 August 2020 Junfeng Hu , Tane Butler , Marco A. Cabero Z. , Hanchen Wang , Bohang Wei , Sa Tu , Chenyang Guo , Caihua Wan , Xiufeng Han , Song Liu , Weisheng Zhao , Jean-Philippe Ansermet , Simon Granville , and Haiming Yu ARTICLES YOU MAY BE INTERESTED IN Magnon-drag thermoelectric transport with skyrmion structure Applied Physics Letters 117, 062404 (2020); https://doi.org/10.1063/5.0017272 Current-induced torques in black phosphorus/permalloy bilayers due to crystal symmetry Applied Physics Letters 117, 062403 (2020); https://doi.org/10.1063/5.0013363 Magnetic domain wall curvature induced by wire edge pinning Applied Physics Letters 117, 062406 (2020); https://doi.org/10.1063/5.0010798Regulating the anomalous Hall and Nernst effects in Heusler-based trilayers Cite as: Appl. Phys. Lett. 117, 062405 (2020); doi: 10.1063/5.0014879 Submitted: 22 May 2020 .Accepted: 1 August 2020 . Published Online: 14 August 2020 Junfeng Hu,1,2 Tane Butler,3 Marco A. Cabero Z.,1,4Hanchen Wang,1 Bohang Wei,1SaTu,1Chenyang Guo,5 Caihua Wan,5Xiufeng Han,5 Song Liu,4Weisheng Zhao,1 Jean-Philippe Ansermet,2 Simon Granville,3,6,a) and Haiming Yu1,b) AFFILIATIONS 1Fert Beijing Institute, School of Microelectronics, Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100191, China 2Institute of Physics, Station 3, Ecole Polytechnique F /C19ed/C19erale de Lausanne, 1015 Lausanne-EPFL, Switzerland 3Robinson Research Institute, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand 4Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China 5Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese Academy of Sciences, ChineseAcademy of Sciences, Beijing 100190, China 6The MacDiarmid Institute for Advanced Materials and Nanotechnology, Wellington 6140, New Zealand a)Electronic mail: simon.granville@vuw.ac.nz b)Electronic mail: haiming.yu@buaa.edu.cn ABSTRACT Anomalous Hall and anomalous Nernst properties of thin MgO/Co 2Fe0:4Mn 0:6Si/Pd stacks with perpendicular magnetic anisotropy (PMA) revealed the presence of the magnetic proximity effect (MPE) in the Pd layer. The MPE is evidenced by nanometer range thickness-dependenttransport measurements. A three-layer model that combines bulk and interface contributions accounts for our experimental data and providesquantitative estimates for the contributions to the total anomalous Nernst voltage of the ferromagnet Heusler [ þ0:97lV/(K nm)] and the proximity-magnetized Pd layers [ /C00:17lV/(K nm)]. The anomalous Nernst effect (ANE) reverses its sign by tuning the thickness of the Heusler layer, which is useful for designing ANE thermopiles. Published under license by AIP Publishing. https://doi.org/10.1063/5.0014879 Spin caloritronics 1,2is a branch of spintronics focused on the interplay between spin and thermoelectric effects. Harvesting energy in thin-ﬁlm heterostructures stimulates the ﬁeld of spin caloritronics. The anomalous Nernst effect (ANE) occurs in a ferromagnetic (FM) mate- rial when a longitudinal gradient of temperature establishes an electric potential transverse to the spontaneous magnetization of the ferromag- net and the temperature gradient.3So far, it has been investigated in normal ferromagnetic (FM) metals,4–8Heusler thin ﬁlms,9–11Fe3O4 single crystals,12rare-earth alloys,13[Pt/Co] nmultilayered structures,14 ferromagnetic semiconductors,15ferromagnetic semimetals,16–18Weyl/ Dirac semimetals,19and antiferromagnetic materials.20,21 Half-metallic Heusler compounds22,23could be ideal materi- als to study the ANE.24The observation of a large spin polariza- tion in Co 2MnSi has stimulated the study of these kinds of materials.25,26I no u rp r e v i o u sw o r k ,27we found that increasingthe thickness of Pd from 2.64 nm to 4.62 nm in MgO/ Co2Fe0:4Mn 0:6Si (CFMS)/Pd stacks enhanced the anomalous Nernst coefﬁcient. This showed the important role of the non- magnetic (NM) layer in ferromagnetic/nonmagnetic (FM/NM) heterostructures for tuning the spin-dependent transport proper- ties and, in particular, the thermoelectric effects. A phenomenonthat can inﬂuence the spin transport properties in such FM/NM heterostructures is the magnetic proximity effect (MPE), 28,29 which results in magnetic moments present in the layers of theNM at the interface with FM. Several groups have pointed out that it is important to consider the possible presence of the MPE inthese heterostructures, with magnetotransport effects observed up to the spin-diffusion length of the NM. 30–35The interplay mechanisms between ferromagnetic and normal metal layers in ANE heterostructures are yet to be revealed. Appl. Phys. Lett. 117, 062405 (2020); doi: 10.1063/5.0014879 117, 062405-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplHere, we studied the anomalous Hall effect (AHE) and ANE of magnetic Heusler compounds in the MgO/CFMS/Pd stack [ Fig. 1(a) ], which shows the perpendicular magnetic anisotropy (PMA) at certain CFMS and Pd thicknesses (Fig. S1, supplementary material ). Samples were grown on Si(100) wafers at room temperature by magnetronsputtering in a Kurt J Lesker CMS-18 UHV system with a base pres- sure of 2 /C210 /C08Torr. All samples were post-growth annealed in situ for 1 h at 300/C14C. The MgO layer was grown by RF sputtering, whereas both CFMS and Pd layers were grown by DC sputtering, in Ar pres-s u r e so f3m T o r r ,5m T o r r ,a n d8m T o r r ,r e s p e c t i v e l y .T h ed e v i c e sconsist of stacks prepared on 10 /C210 mm Si/SiO 2substrates in the sequence MgO (1.60)/CFMS ( tH)/Pd (2.50) and MgO (1.60)/CFMS (1.64)/Pd ( tPd), where the number in parentheses is the nominal layer thickness in nm with an error bar of 0.05 nm for each thickness. Thec o m p o s i t i o no ft h eH e u s l e rﬁ l mw a sv e r i ﬁ e dt ob eC o 2Fe0:4Mn 0:6Si by energy dispersive x-ray analysis in a scanning electron microscope (SEM), and more details can be found in Ref. 36. The AHE and ANE were measured in the same probe holder, and more details for themeasurement setup can be found in the supplementary material .A l l samples were patterned into a Hall bar structure (9 mm /C22m m )w i t h six Cr (3 nm)/Au (100 nm) electrodes. All measurements were per- formed at room temperature. The AHE is usually proportional to the out-of-plane magnetiza- tion; therefore, measurements of the AHE [ Fig. 1(a) ] can be used to probe PMA in thin ﬁlms. 36–38The AHE in Fig. 1(b) conﬁrms the PMA for CFMS ﬁlm thicknesses tH¼1.25 nm, in which the coercive ﬁeld is around 3.5 mT. While the AHE is usually proportional to the out-of-plane mag- netization, its sign and magnitude also depend on other properties ofthe material, including the band structure, Berry curvature, scattering mechanisms (side-jump effect or skew-scattering), and spin-polarization. 39The anomalous Hall resistance RAHEis deﬁned here as the data for positive saturated magnetic ﬁelds (Fig. S2, supplementary material ). For the CFMS ﬁlm in MgO/CFMS/Pd stacks, we observe a thickness-dependent AHE, as shown in Fig. 1(c) .T h e r ei sas i g n change of RAHE between tH¼2.00 and 2.50 nm. Interestingly, the AHE also changes the sign when the Pd thickness is varied, as shown inFig. 1(d) . We attribute the slight mismatch between the Hall resis- tance values from the two sets of samples to the CFMS thickness in the MgO (1.60)/CFMS (1.64)/Pd ( tPd) stack being a bit thicker than the nominal 1.64 nm, which is further conﬁrmed by the results of the ANE as shown later. A sign change of the AHE in CFMS thin ﬁlms was previously reported by Schneider et al. in samples with varying ratios of Fe:Mn composition, which was related to changes in the band structure and spin-dependent bandgap.40However, in our multilayers, the CFMS composition is ﬁxed. Xuet al.41found that the opposite signs of Hall resistivity of Fe and Gd single-layers can induce the AHE sign change of the Fe/Gd bilayers. Here, we can argue that the Pd layer has the opposite sign of the anomalous Hall coefﬁcient to the CFMS layer, and competition between the AHE from these two layers gives rise to the sign change. In other words, the trilayer structure could be simpliﬁed as a three- layer model: the magnetic Heusler layer, the magnetized Pd layer, and the nonmagnetic Pd layer. The origin of the magnetized Pd layer could be attributed to the MPE, which magnetizes the part of the Pd closest to the interface with the magnetic Heusler layer. It needs to be noted that the interface contributions may also need to be taken into account.42 In order to gain further insight into the respective contributions of layers and the interface in our structures, we measured their ther- moelectric properties. The Nernst voltage was measured as a function of the magnetic ﬁeld [ Fig. 2(d) ]. The ANE in samples with a ﬁxed Pd thickness of 2.50 nm and a CFMS thickness varying from 1.25 nm to 3.00 nm is shown in Fig. 2(a) . The different CFMS thicknesses affect not only the amplitude of the ANE but also its sign. There is also an apparent change in the magnetic properties, shown by a shape change of the ANE signals [ Fig. 2(a) ]. In particu- lar, the sample with tH¼1.75 nm does not have the sharp change of voltage with the magnetic ﬁeld as shown for tH¼1.25 nm. The coercive ﬁeld is increased from 1.5 mT for tH¼1.25 nm up to 5.0 mT for tH¼1.75 nm. Moreover, when tH¼2.00 nm, the hyster- esis in the ANE has disappeared. Upon increasing tHto 2.50 nm and 3.00 nm, the shape of the ANE returns back to the form it has when there is perpendicular magnetization. In contrast, when the Pd layer thickness is varied [ Fig. 2(b) ], the PMA is present for all of tPdand the ANE does not change sign. The anomalous Nernst volt- age induced by the rTcan be described by Vxy¼wNrT,w h e r e N is the anomalous Nernst coefﬁcient and wis the sample width. With a ﬁxed in-plane temperature gradient rTof 5.2 K/cm, we extract the anomalous Nernst coefﬁcients and plot them vs thick- ness in Figs. 2(c) and2(e). The Seebeck coefﬁcients also show very strong thickness dependence for both CFMS and Pd layers (Fig. S3, supplementary material ). Similar to the AHE, the thickness dependence of the ANE obtained by varying both CFMS and Pd thicknesses indicates that the sign change of the ANE signal could also be attributed to the competi- tion of the bulk contribution of CFMS and Pd layers. Interestingly, a simpliﬁed equation [Eq. (1)] ﬁts our experimental data well. However, FIG. 1. (a) Schematics of the Hall effect measurement and the trilayer stack. (b) Hall resistance of MgO (1.60)/CFMS (1.25)/Pd (2.50 nm). (c) Anomalous Hall resis-tance of MgO (1.60)/CFMS ( t H)/Pd (2.50 nm). (d) Anomalous Hall resistance of MgO (1.60)/CFMS (1.64)/Pd ( tPd). The dashed line marks the position of zero anomalous Hall resistance. All the measurements were performed with a 50 lAD C applied along the x-axis. The error bars for Hall resistances are extracted from the variation of the average signal at the saturated ﬁeld.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 062405 (2020); doi: 10.1063/5.0014879 117, 062405-2 Published under license by AIP Publishingit should be noted that the actual relationship should be more complex than this simple equation, especially for samples in the thicker range,41,42 N¼NSþtHNH BþtPdNPd B; (1) where NSis the interface ANE contribution and NH Band NPd Bare the bulk ANE contribution of CFMS and magnetized Pd layers, respectively. By ﬁtting our experimental data, we get the averaged interface contribution to be /C01.80lV/K, the CFMS bulk contribu- tion to be þ0:97lV/(K nm), and the Pd bulk contribution to be /C00:17lV/(K nm). These results conﬁrm our hypothesis about the opposite contributions of CFMS and Pd layers. Although Pd is a paramagnetic material by nature, it plays an important role at the interface with the CFMS layer. At tPd¼2.50 nm, the magnitude of the contribution of Pd together with the CFMS/Pdinterface contribution is larger than that of the contribution of CFMS (t H<2.50 nm). The total ANE signal carries the sign of the Pd layer (negative). However, by increasing the thickness of CFMS (tH>2.50 nm), the bulk contribution of CFMS is dominant and the ANE signal will now carry the sign of the bulk contribution of the CFMS layer (positive). This is evidence for the importance of both the Pd and Heusler layers for the ANE in this system. The bulk contribu- tion of our CFMS thin ﬁlms would no longer hold in thicker ﬁlms as has been observed in conventional ferromagnetic materials, Fe, Co, Ni, and Py.43The same can be argued concerning the Pd ﬁlms. Extensive literature9,30,33,42–44has been written about the enhancement of the ANE and its separation from the spin Seebeck effect (SSE) in different structures and combinations of materials. Theinsertion of additional metallic layers like Cu has been used to avoid the MPE and separate the ANE from the SSE. Such an insertion is notfeasible here since the sample would lose PMA if a Cu layer was added in between. 36The large interface contribution to the anomalous Nernst coefﬁcient is quite interesting and needs to be further under-stood. Gabor et al. 45proposed that the interdiffusion at the Heusler/ MgO interface could cause the increase in the Gilbert damping coefﬁ- cient and also form strong PMA. Since CFMS is a ferromagnet with high spin polarization and Pd has a strong spin–orbit interaction, oneneeds to note there are many possible mechanisms, which could alsocontribute to the observed signal, especially for the interface compo- nent, such as the spin Hall effect (SHE), Rashba effect, and interfacial Dzyaloshinskii–Moriya interaction (iDMI). To gain further insight regarding the relationship between the ANE and AHE, a comparison of these two effects is needed. In Fig. 3 , we normalized both the anomalous Hall resistance and anomalous Nernst coefﬁcients by the value it has for the thinnest layer in the cor-responding series of samples, e.g., the normalization for the varying C F M St h i c k n e s ss e r i e si sb yt h e i rv a l u e sf o rt h e t H¼1.25 nm sample. The normalized data are plotted in Fig. 3(a) for CFMS thickness dependence and Fig. 3(b) for the Pd thickness dependence. It is clear from Fig. 3 that the AHE and ANE both show strong dependences on the thicknesses of both the CFMS and Pd layers, though the AHE has a larger relative change than the ANE. For example, when the CFMSthickness changed from 1.25 nm to 3.00 nm, the anomalous Hall resis-tance changes more sharply than the anomalous Nernst coefﬁcient. However, an identical sign reverse occurs in the range of 2.00 to FIG. 2. (a) Nernst voltage measured as a function of magnetic ﬁeld in MgO(1.60)/CFMS( tH)/Pd(2.50) stacks. (b) Nernst voltage as a function of magnetic ﬁeld in MgO (1.60)/ CFMS (1.64)/Pd ( tPd) stacks. (d) Schematics of the ANE measurement setup. (c) and (e) are the extracted anomalous Nernst coefﬁcients with thickness changing of CFME and Pd, respectively. The arrows in (a) and (b) represent the ﬁeld sweep direction. The red curves in (c) and (e) are the ﬁtting results. The error bars fo r Nernst coefﬁcients are extracted from the variation of the average signal at the saturated ﬁeld.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 062405 (2020); doi: 10.1063/5.0014879 117, 062405-3 Published under license by AIP Publishing2.50 nm. Although a different behavior is observed when the thickness of Pd changes, sharp changes with sign reversal are observed in anom-alous Hall resistance, but no sign reversal for the anomalous Nernstcoefﬁcient. These phenomena could be interpreted by the differentinterface and bulk contributions for the AHE and ANE. The Heusler-based PMA system could be employed to design nanostructured thermoelectric devices for energy harvesting with asmall magnetic ﬁeld or even operating in zero ﬁeld in the remanentstate where the stack is still magnetized out-of-plane. Also, a large volt-age can be expected by increasing the separation between the voltagecontacts as illustrated in Fig. 4(a) . Our work furthers the possibility to harness magnetothermoelectric effects in thin-ﬁlm heterostructureswith PMA, by controlling the magnitude and sign of the anomalousNernst coefﬁcient through changing the thicknesses of the multilayers.One example of a simple device that could be produced from suchPMA stacks is the ANE thermopile, 20,46which is made by connecting a series of Heusler stacks with opposite signs of the ANE. In the ANEthermopile, the total voltage arises from the difference of the ANE vol- tages in the individual layers connected in series, and so for layers with opposite sign, the total voltage is increased [ Fig. 4(b) ]. For example, we have shown here that changing the CFMS thickness from 1.25 nm to3.0 nm can change the anomalous Nernst coefﬁcient from /C01:3lV/K toþ0:4lV/K; thus, a total coefﬁcient up to 1.7 lV/K could be avail- able in a single two-layer cell at zero ﬁeld. In conclusion, we have characterized the MgO/Co 2Fe0:4Mn 0:6Si/ Pd trilayers with PMA by measuring anomalous Hall and Nernsteffects as a function of CFMS and Pd layer thicknesses. Very similarthickness-dependent behaviors are observed in these two transversetransport measurements. We ﬁnd a contribution to the anomalousHall and Nernst effects from both CFMS and Pd layers. We attributethe effect in Pd to the MPE. By studying two sets of samples, onewhere the CFMS thickness is varied and the other where the Pd thickness is varied, we are able to identify an interfacial contribution. Tuning the thickness of the layers can change the total anomalousNernst (Hall) effect from negative ( /C01:3lV/K) to positive (þ0:4lV/K), depending on which layer has the dominant contribu- tion. A three-layer model accounts for our experimental ﬁndings,which provide evidence for the existence of the magnetic proximity effect in the Pd layer. See the supplementary material for the MOKE signal and Hall resistance measured as a function of magnetic ﬁeld and Seebeck coefﬁ-cient in MgO(1.60)/CFMS( t H)/Pd(2.50) and MgO(1.60)/CFMS(1.64)/ Pd(tPd)s t a c k s . AUTHORS’ CONTRIBUTIONS J.H., T.B., and M.A.C.Z. contributed equally to this work. The authors thank Albert Fert for helpful discussion. This work was supported by the National Natural Science Foundation of China under Grant Nos. 11674020 and U1801661, by the Sino-Swiss Science and Technology Cooperation (SSSTC, Grant No. EG01–122016 for J.H.), by the Program of Introducing Talents of Discipline to Universities in China “111 Program” No. B16001, by the National Key Research and Development Program of China(254), and by the National Key Research and Development Program of China MOST, Grants No. 2017YFA0206200. 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1.4894512.pdf	Tuning microwave magnetic properties of FeCoN thin films by controlling dc deposition power Y. P. Wu, Yong Yang, Z. H. Yang, Fusheng Ma, B. Y. Zong, and Jun Ding Citation: Journal of Applied Physics 116, 093905 (2014); doi: 10.1063/1.4894512 View online: http://dx.doi.org/10.1063/1.4894512 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Tuning of magnetization dynamics in sputtered CoFeB thin film by gas pressure J. Appl. Phys. 111, 07A304 (2012); 10.1063/1.3670605 Influence of the magnetic field annealing on the extrinsic damping of FeCoB soft magnetic films J. Appl. Phys. 108, 073902 (2010); 10.1063/1.3489954 High-frequency permeability spectra of FeCoSiN / Al 2 O 3 laminated films: Tuning of damping by magnetic couplings dependent on the thickness of each ferromagnetic layer J. Appl. Phys. 105, 043902 (2009); 10.1063/1.3078112 Soft magnetism of Fe–Co–N thin films with a Permalloy underlayer J. Appl. Phys. 92, 1477 (2002); 10.1063/1.1491017 Soft magnetic properties of as-deposited Fe–Hf–C–N and Fe–Hf–N nanocrystalline thin films J. Appl. Phys. 83, 6652 (1998); 10.1063/1.367526 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.216.129.208 On: Sun, 07 Dec 2014 03:01:48Tuning microwave magnetic properties of FeCoN thin films by controlling dc deposition power Y. P. Wu ,1Yong Y ang,2Z. H. Y ang,1Fusheng Ma,1B. Y. Zong,1and Jun Ding2 1Temasek Laboratories, National University of Singapore, 5A Engineering Drive 1, Singapore 117411 2Department of Materials Science and Engineering, National University of Singapore, Singapore 119260 (Received 13 July 2014; accepted 21 August 2014; published online 3 September 2014) In this work, we deposited FeCoN thin ﬁlms by reactive dc magnetron sputtering under various deposition powers. Composition, microstructu re, static magnetic properties, and microwave permeability of as-sputtered ﬁlms were examined. The permeability spectra were theoretically ana- lyzed based on LLG equation. When high deposition power was applied, l0 0improved signiﬁcantly due to the increased M sand decreased H k. On the other hand, the damping coefﬁcient kincreased with the power, which resulted in the widen perme ability spectra. The physical origin of the inﬂuences should be related to the change in the ﬁlm composit ion and microstructure, which have immediate impact on static magnetic properties and damping effect of the ﬁlm. VC2014 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4894512 ] I. INTRODUCTION With expanded applications in electronic, computer, and telecommunication industries, ferromagnetic thin ﬁlms haveattracted intensive interest due to high microwave permeabil- ity in recent decades. Theoretically, in order to obtain mag- netic thin ﬁlms with attractive high-frequency properties,high saturation magnetization M s, low coercivity H c, in- plane anisotropy, and controllable anisotropy ﬁeld H kare preconditions.1–5Msis an intrinsic parameter controlled by the material composition. FeCo-based ﬁlm is one of the most competitive candidates because of high M sof FeCo alloy6 (2.45 T for Fe 65Co357). In comparison, H cand H kare extrin- sic parameters which are determined by microstructure as well as composition. A lot of work has been done to reduce Hcby inserting underlayer3,4,8and controlling deposition conditions,9,10both are effective ways to modify the micro- structure of ﬁlms. In reality, the damping of the magnetic moment preces- sion is also a determining parameter.11Several damping mechanisms in theory have been studied12and the Gilbert damping is the most conventional mechanism. The dampingparameter kin actual materials is greatly dependent on their structure. It has been proved that kcan be effectively modi- ﬁed by different methods, such as doping the ferromagneticmaterials 13,14and making use of exchange-bias effect.15Xu et al. also reported that the damping coefﬁcients in the mag- netization dynamics can be conveniently and effectivelytuned by controlling the sputtering gas pressure. 16The possi- ble physical origin is suggested as the change in the stress of the ﬁlms. In this work, we will show that the dynamic mag-netic properties in FeCoN ﬁlms can be tuned by varying dep- osition power. The high-frequency magnetic properties are investigated by experimental measurement and theoreticalanalysis of the permeability spectra. It was found that the change in the magnetization dynamics is contributed by the variation of the ﬁlm composition and microstructure, whichhave immediate impact on static magnetic properties and damping coefﬁcient of the ﬁlms.II. EXPERIMENTAL DETAILS The FeCoN ﬁlms were deposited on Si (100) substrates by reactive dc magnetron sputtering in a background pres-sure below 5 /C210 /C07Torr. A customized alloy target with Fe50Co50composition was used. An argon and nitrogen (8%) gas mixture was used as the ambient gas which was main-tained at 3.0 mTorr. FeCoN ﬁlms were fabricated by varying the dc source power from 150 W to 1000 W. During deposi- tion, a magnetic ﬁeld of 200 Oe was applied to induce in-plane anisotropy. The ﬁlm thickness was tested by a surface step proﬁlometer and veriﬁed by a transmission electron mi- croscopy (TEM) images. The ﬁlm thickness was controlledby about 150–170 nm by controlling the deposition time. Magnetic properties were measured using a vibrating sample magnetometer (VSM). The microstructure was investigatedusing the TEM and the composition was measured by a X- ray photoelectron spectroscopy (XPS). The permeability fre- quency spectra from 0.5 to 5.5 GHz were characterized by anetwork analyzer (Agilent 5230 A) using a shorted micro- strip transmission-line perturbation ﬁxture. 17 III. RESULTS AND DISCUSSION A. TEM investigation The thickness of FeCoN ﬁlms deposited under various powers was listed in Table I. The deposition rate is estimated by the ﬁlm thickness and deposition time. As shown in Fig. 1, an almost linear relationship between deposition rate and applied source power was observed. The FeCoN depositionrate is about 14 nm/min for 150 W of dc power. When the power increases, the deposition rate is increased gradually. For the ﬁlm deposited under 1000 W, the deposited rate is ashigh as 102 nm/min. It is reasonable because the Sputter yield of Fe and Co increases almost linearly with incident ion energy, 18which has immediate relation with applied power. Fig.2shows typical cross-sectional TEM graphs of the FeCoN ﬁlms grown under various deposition powers. The 0021-8979/2014/116(9)/093905/5/$30.00 VC2014 AIP Publishing LLC 116, 093905-1JOURNAL OF APPLIED PHYSICS 116, 093905 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.216.129.208 On: Sun, 07 Dec 2014 03:01:48images were captured under the dark ﬁeld. The left image is for the ﬁlm deposited with 150 W of power. Separated small nano grains are evenly distributed and embedded in an amor-phous matrix. No obvious texture is observed. It is noted that the contrast between each grains sometimes appeared mot- tled due to the impact of magnetic ﬁlms on the movement ofelectrons under extra-high magniﬁcations. However, with the deposition power increased to 250 W and above, the ﬁlms exhibit completely different microstructures. As shown inthe right image of Fig. 2, pronounced and well-deﬁned co- lumnar crystal grains are formed. The grains have an average diameter of 10 nm, and all of them extended across the entire thickness of the ﬁlms. The reason for the dramatic change in the microstructure remains unknown. Nevertheless, it maybe related with the formation of different nitrides since the incorporation of N content varies with the deposition power which will be mentioned in XPS results.B. Static magnetic properties of the films As shown in Fig. 3, remarkably improved soft magnetic properties and formation of in-plane anisotropy of FeCoNﬁlms are observed by changing the deposition power. The measurement was conducted along two directions, parallel and perpendicular to the magnetic aligning ﬁeld during ﬁlmdeposition. Fig. 3(a) shows the hysteresis loops for the ﬁlm deposited under 150 W of source power. The ﬁlm possesses almost same shape of hysteresis loops along two perpendicu-lar directions, which is identiﬁed as in-plane isotropy. However, when the applied power is higher than 150 W, the deposited FeCoN ﬁlms exhibit excellent in-plane anisotropy,which is indicated by the noticeable discrepancy of the hys- teresis loops along different directions. Typical hysteresis loops are shown in Fig. 3(b). It is obvious that well-deﬁnedTABLE I. The parameters of thickness, static magnetic properties, and microwave permeability for the ﬁlms deposited under various dc powers. Hk for the ﬁlm deposited with 150 W of power is based on theoretical estima- tion and given in brackets. Power (W) 150 250 350 500 750 1000 Thickness (nm) 164 150 172 158 157 154M s(T) 1.22 1.51 1.64 1.78 1.87 1.91 Hk(Oe) (65) 126 85 53 45 43 l0 0 79 148 270 420 537 510 l00 max 911 1388 2093 2564 2885 1885 fr(GHz) 2.5 3.7 3.3 2.9 2.7 2.5 fR(GHz) 2.5 3.7 3.3 2.9 2.7 2.5 kA 0.42 0.91 0.99 0.99 0.96 0.97 FIG. 1. The dependence of ﬁlm deposition rate on applied dc source power. The power is changed from 150 W to 1000 W. FIG. 2. Typical cross-sectional TEM images in the FeCoN ﬁlms deposited with power of (a) 150 W and (b) 250 W and above. FIG. 3. The typical hysteresis loops forthe ﬁlms deposited under different dep- osition powers. (a) 150 W of deposi- tion power and (b) 500 W of deposition power. The measurements are parallel (easy axis) and perpendicu-lar (hard axis) to the aligning magnetic ﬁeld during deposition.093905-2 Wu et al. J. Appl. Phys. 116, 093905 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.216.129.208 On: Sun, 07 Dec 2014 03:01:48in-plane anisotropic ﬁlms are produced under high dc power. The loop measured along the easy axis is close to a narrow rectangle that is an indication of small demagnetization on the ﬁlm. The anisotropy ﬁeld H kis estimated by the extrapo- lation of the hard axis loop and listed in Table I. The saturation magnetization M s, easy axis coercivity Hce, and hard axis coercivity H chof the FeCoN ﬁlms are plotted in Fig. 4as a function of applied source power. With the increase in deposition power from 150 W to 1000 W, M s is found to gradually increase from 1.2 T to 1.9 T. The varia- tion in M sis believed to relate with the concentration of N in the ﬁlms. The XPS spectra provide the evidence of different incorporation of N in the ﬁlms with changed source power.Fe 2p, Co 2p, and N 1s XPS scans were carried out for the ﬁlms deposited with various dc powers. The typical XPS spectra are shown in Fig. 5. The concentration of N isestimated as 9.2%, 6.9%, and 5.6% for the ﬁlms deposited with 150 W, 500 W, and 1000 W dc power, respectively. That means, with the increase in the deposition power, theincorporation of N in the ﬁlms is reduced gradually. According to the XPS results, Fe and Co regions reveal that both metal and metal nitride peaks are present. Fe and Coatoms can both have þ3 and/or þ2 states. However, XPS spectra indicate that most of the Fe and Co atoms are in þ3 valence state. It has been reported that FeN or FeCoN thinﬁlms can form several nitrides with different structures and properties, which depend on the incorporated nitrogen con- centration. 19,20Except for the phases of a-Fe8N and a00- Fe16N2, the saturation magnetization of the other ferromag- netic phases is generally lower than that of the a-Fe, which has been proved by many studies.21–23Therefore, the M sis changed with the concentration of nitrogen in the ﬁlms. Signiﬁcant reduction in both H ceand H chis observed when the deposition power changed from 150 W to 250 W.For the FeCoN ﬁlm deposited under 150 W, the coercivities along the easy H ceand hard H chdirections are as high as 39 Oe and 54 Oe, respectively. When the dc power increasesto 250 W, H cedramatically drops to 1.5 Oe and H chis 10 Oe. With further increase in deposition power, H ceand H chhave the minimum values of 0.8 Oe and 1.5 Oe, respectively, fordeposition power of 750 W. For the ﬁlm deposited under 1000 W, H ceslightly goes up to 2.1 Oe and H chis up to 5.7 Oe. As mentioned above, the deposition rate increaseslinearly with the deposition power. Giving the condition of high deposition rate, more initial nucleation centers are formed. This results in ﬁne-grained and smooth depositswhich become continuous at small thickness. 24Therefore, the ﬁlms deposited under high power tend to possess small grain size, which leads to low coercivity.25The slightFIG. 4. Saturation magnetization and coercivity of FeCoN thin ﬁlms depos- ited at varied power which changes from 150 W to 1000 W. FIG. 5. Typical XPS spectra of FeCoN ﬁlms in the (a) survey, (b) Fe 2p, (c) Co 2p, and (d) N 1s.093905-3 Wu et al. J. Appl. Phys. 116, 093905 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.216.129.208 On: Sun, 07 Dec 2014 03:01:48increase in H cfor the ﬁlm deposited at 1000 W should be related with the increased defects and stress due to high ener- getic bombarding particles. C. Permeability spectra and damping analysis The hard-axis magnetic permeability, both real and imaginary parts, of the FeCo ﬁlms was measured using ahome-developed microstrip ﬁxture in the frequency range of 0.5–5.5 GHz. The permeability spectra of the ﬁlms deposited under various sputtering powers are presented in Fig. 6. All ﬁlms exhibit a strong ferromagnetic resonance within the measured frequency range. The static initial permeability l 0 0 is deﬁned as the real part at 0.5 GHz and the ferromagnetic resonance frequency frcorresponds to the maximum imagi- nary permeability l00 max. The values of measured high- frequency parameters, fr,l0 0, and l00 max, for the ﬁlms depos- ited with various powers are listed in Table I. Signiﬁcant change in both l0 0andl00 maxwas observed when the different power was applied. With the increase in depositing powerfrom 150 W to 1000 W, l 0 0increases gradually from 79 to 444, respectively, while l00 maxreached the maximum of 2344 at the power of 750 W. In comparison, slight variation in fris observed with the depositing power. frreached the maximum of 3.7 GHz when the power is 250 W and then decreased with the further decrease or increase in the power. Based on the LLG equation,11the resonance spectra can be expressed as26 lfðÞ¼v01þikf fR/C18/C19 1þikf fR/C18/C192 /C0f fR/C18/C192þ1; (1) where v0¼l0/C01 is the static susceptibility, fRis the intrin- sic resonance frequency,27andkis the damping coefﬁcient.The curve-ﬁtting results are also shown in Fig. 6and repre- sented by lines. The dependence of kon the applied power is plotted in Fig. 7.kincreases from 0.036 to 0.14 when the deposition power changed from 150 W to 1000 W, respec- tively. It indicates that more signiﬁcant damping effect is produced with the increase in the applied power. The damp-ing coefﬁcient kin actual materials is greatly dependent on their microstructures. In ideal single crystal samples, kis the result of the interaction between the electron-spin system,conduction electrons, and lattice. However, for polycrystal- line materials, kis contributed by the inhomogeneity of the magnetic properties as well, such as a spread in directions ofthe anisotropy ﬁeld 28or nonuniformity of magnetization due to domain walls.29It has been proved that the deposition rate has an almost linear dependence on the deposition power.Giving the condition of high deposition rate, more initial nucleation centers are formed and some atoms may not have enough time to transport to low energy state. In this way, thedefect points and inhomogeneity of the ﬁlm may increase, which directly contribute to the increased damping coefﬁ- cient k. The most obvious dependence of the spectra on damping effect is the full-width at half-maximum (FWHM) Dfof the imaginary permeability. As plotted in Fig. 7,Df becomes broader with the increase in deposition power,which is in the same trend as the damping coefﬁcient k. As listed in Table I, the values of the intrinsic resonance frequency f Rare same as those of the measured resonance frequency fr. Based on Eq. (1), the relationship between fr andfRcan be obtained as fr¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1þk2s fR: (2) Hence, the discrepancy between frandfRis determined by the damping coefﬁcient k. For the studied thin ﬁlms, the damping coefﬁcient kis quite small (the largest value is only 0.14), so the values of frandfRare close to each other. In addition, the resonance frequency can be theoretically given by fr¼cﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4pMs/C2Hkp ; (3) FIG. 6. The permeability spectra in the hard axis for FeCoN ﬁlms deposited under various dc source powers. The measurement is done with a home- developed microstrip ﬁxture in the frequency range of 0.5–5.5 GHz. FIG. 7. The dependence of kandDf on the deposition power which varies from 150 W to 1000 W.093905-4 Wu et al. J. Appl. Phys. 116, 093905 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.216.129.208 On: Sun, 07 Dec 2014 03:01:48where cis the gyromagnetic ratio ( /C253 MHz/Oe for FeCo- based alloy30) and 4 pMsis the saturation magnetization of the ﬁlm which can be obtained from VSM results. Hence, forthe ﬁlm deposited with 150 W of power, H kcan be theoreti- cally estimated as about 65 Oe and shown in Table Iin brackets. On the other hand, the high-frequency performance of the ﬁlm can be evaluated by Acher’s constant31 kA¼l0 0f2 r c4pMs ðÞ2: (4) kAis calculated based on the measured static and dynamic magnetic results and listed in Table I. For the ﬁlms deposited under the deposition power of 250 W, kAis close to 1. This means that these ﬁlms have excellent in-plane magnetic ani- sotropy which is induced by the applied magnetic ﬁeld dur-ing deposition. The in-plane anisotropy results in the good alignment of all magnetic moments along the parallel direc- tion, which is called the easy axis of the anisotropic ﬁlms.Hence, the impressive high permeability along the hard axis (perpendicular to the easy axis) is expected and has been veriﬁed by the measurement. In comparison, k Ais only 0.42 for the ﬁlm deposited under 150 W, which is in good agree- ment with the VSM results. Furthermore, it indicates that most of the magnetic moments are controlled within the ﬁlmplane despite in-plane isotropy. IV. CONCLUSIONS In this work, we show dynamic magnetic properties in as-sputtered FeCoN thin ﬁlms are effectively tuned by changing the deposition power. With the increase in deposi- tion power from 150 W to 1000 W, l0 0increases signiﬁcantly from 79 to 444, while l00 maxreached a maximum of 2344 at the power of 750 W. In comparison, slight change was observed for the resonance frequency fr. The curve-ﬁtting results show that with the increase in the deposition power, the damping coefﬁcient increases, which results in the wid- ening permeability spectra ( Df). The variation of magnetic dynamics is closely related to the static magnetic properties and damping effect, which are controlled by ﬁlm microstruc- ture and composition.1L. Landau and E. Lifschitz, Phys. Z. Sowjetunion 8, 153 (1935). 2O. Acher, P. M. Jacquart, J. M. Fontaine, P. Baclet, and G. Perrin, IEEE Trans. Magn. 30, 4533 (1994). 3H. S. Jung, W. D. Doyle, J. E. Wittig, J. F. Al-Sharab, and J. Bentley, Appl. Phys. Lett. 81, 2415 (2002). 4C. L. Platt, A. E. Berkowitz, D. J. Smith, and M. R. McCartney, J. Appl. Phys. 88, 2058 (2000). 5J. Shim, J. Kim, S. H. Han, H. J. Kim, K. H. Kim, and M. Yamaguchi, J. Magn. Magn. Mater. 290–291 , 205 (2005). 6R. S. Sundar and S. C. Deevi, Int. Mater. Rev. 50, 157 (2005). 7G. Y. Chin and J. H. Wernick, Ferro Magnetic Material (North-Holland Publishing Company, 1980), Vol. 2, pp. 55–188. 8Y. P. Wu, G.-C. Han, and L. B. Kong, J. Magn. Magn. Mater. 322, 3223 (2010). 9J. Yu, C. Chang, D. Karns, G. Ju, Y. Kubota, and W. Eppler, J. Appl. Phys. 91, 8357 (2001). 10M. K. Minor, T. M. Crawford, T. J. Klemmer, Y. Pend, and D. E. Laughlin, J. Appl. Phys. 91, 8453 (2002). 11T. L. Gilbert, Phys. Rev. 100, 1243 (1955). 12L. Kraus, Z. Frait, and J. Schneider, Phys. Status Solidi A 64, 449 (1981). 13J. O. Rantschler, R. D. McMiael, A. Castillo, A. J. Shapiro, W. F. Egelhoff, B. B. Maranville, D. Pulugytha, A. P. Chen, and L. M. Conners,J. Appl. Phys. 101, 033911 (2007). 14J. Fassbender and J. Mccord, Appl. Phys. Lett. 88, 252501 (2006). 15J. McCord, R. Kaltofen, T. Gemming, R. Huhne, and L. Schultz, Phys. Rev. B 75, 134418 (2007). 16F. Xu, N. N. Phouc, X. Zhang, Y. Ma, X. Chen, and C. K. Ong, J. Appl. Phys. 104, 093903 (2008). 17V. Bekker, K. Seemann, and H. Leiste, J. Magn. Magn. Mater. 270, 327 (2004). 18N. Matsunami, Y. Yamamura, Y. Itikawa, N. Itoh, Y. Kazumata, S.Miyagawa, K. Morita, R. Shimizu, and H. Tawasa, At. Data Nucl. Data Tables 31, 1 (1984). 19V. Hari Babu, J. Rajeswari, S. Venkatesh, and G. Markandeyulu, J. Magn. Magn. Mater. 339, 1 (2013). 20P. Schaaf, Prog. Mater. Sci. 47, 1 (2002). 21S. Iwastsubo and M. Naoe, Vacuum 66, 251 (2002). 22J. M. D. Coey and P. A. I. Smith, J. Magn. Magn. Mater. 200, 405 (1999). 23N. Takahashi, Y. Toda, and T. Nakamura, Mater. Lett. 42, 380 (2000). 24W. Kiyotaka, K. Makoto, and A. Hideaki, Thin Film Materials Technology (William Andrew Publishing, 2003), p. 37. 25G. Herzer, in Handbook of Magnetism and Advanced Magnetic Materials: Novel Materials , edited by H. Kronmuller and S. Parkin (John Wiley & Sons, 2007), Vol. 4, p. 1882. 26C. Kittle, J. Phys. Radium 12, 332 (1951). 27A. N. Lagarkov, K. N. Rozanov, N. A. Simonov, and S. N. Starostenko, Handbook of Advanced Magnetic Materials , (Tsinghua University Press, Springer, Beijing, 2005), Vol. 4, p. 414. 28K. Nakanishi, O. Shimizu, and S. Yoshida, IEEE Trans. J. Magn. Jpn. 8, 340 (1993). 29T. Taffary, D. Autisser, F. Boust, and H. Pascard, IEEE Trans. Magn. 34, 1384 (1998). 30O. Kohmoto, J. Phys. D: Appl. Phys. 30, 546 (1997). 31O. Acher and S. Dubourg, Phys. Rev. B 77, 104440 (2008).093905-5 Wu et al. J. Appl. Phys. 116, 093905 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.216.129.208 On: Sun, 07 Dec 2014 03:01:48
1.3081638.pdf	Current-driven ferromagnetic resonance in magnetic trilayers with a tilted spin polarizer Peng-Bin He, Zai-Dong Li, An-Lian Pan, Qing-Lin Zhang, Qiang Wan, Ri-Xing Wang, Yan-Guo Wang, Wu-Ming Liu, and Bing-Suo Zou Citation: Journal of Applied Physics 105, 043908 (2009); doi: 10.1063/1.3081638 View online: http://dx.doi.org/10.1063/1.3081638 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/105/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spin-torque-driven ferromagnetic resonance in point contacts J. Appl. Phys. 109, 07C912 (2011); 10.1063/1.3553942 Optimal control of magnetization dynamics in ferromagnetic heterostructures by spin-polarized currents J. Appl. Phys. 108, 103717 (2010); 10.1063/1.3514070 Measurement of spin current using spin relaxation modulation induced by spin injection J. Appl. Phys. 105, 07C913 (2009); 10.1063/1.3063081 Exchange biased spin polarizer with an embedded nano-oxide layer for a substantially lower switching current density Appl. Phys. Lett. 89, 094103 (2006); 10.1063/1.2337532 Spin-polarized current-driven switching in permalloy nanostructures J. Appl. Phys. 97, 10E302 (2005); 10.1063/1.1847292 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.135.12.127 On: Thu, 02 Oct 2014 10:47:16Current-driven ferromagnetic resonance in magnetic trilayers with a tilted spin polarizer Peng-Bin He,1,a/H20850Zai-Dong Li,2An-Lian Pan,1Qing-Lin Zhang,1Qiang Wan,1 Ri-Xing Wang,1Yan-Guo Wang,1Wu-Ming Liu,3and Bing-Suo Zou1 1Micro-Nano Technologies Research Center, College of Physics and Microelectronics Science, Hunan University, Changsha 410082, China 2Department of Applied Physics, Hebei University of Technology, Tianjin 300401, China 3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China /H20849Received 8 August 2008; accepted 12 January 2009; published online 27 February 2009 /H20850 We theoretically investigate the current-excited and adjusted ferromagnetic resonance in magnetic trilayers with a tilted spin polarizer. The current- and frequency-swept resonant spectra are obtainedby the linearization method. We ﬁnd that the precessional frequency, the equilibrium position, theenergy pumping and damping, and the resonant linewidth and location can be adjusted by changingthe current and the magnetization in the pinned layer. By optimizing the current density and thedirection of the pinned magnetization, the energy pumping will be more efﬁcient. © 2009 American Institute of Physics ./H20851DOI: 10.1063/1.3081638 /H20852 I. INTRODUCTION In a magnet, the total magnetic moment precesses around the direction of the static magnetic ﬁeld. When thefrequency of a transverse microwave ﬁeld coincides with theprecessional frequency, there is a maximal energy absorptionfrom the microwave ﬁeld. This phenomenon is named asferromagnetic resonance /H20849FMR /H20850, the inevitability of which was initially indicated by Landau and Lifshitz after bringingforward their famous equation. With the development ofultrahigh-frequency technique, FMR was ﬁrst observed byGrifﬁths. 1Since then, there have been large numbers of work on FMR in different materials such as ultrathin magneticﬁlms. 2Recently, the FMR technique was used to investigate the properties of magnetic multilayers, which have compre-hensive application in data storage and information process-ing. Many magnetic information of ultrathin ﬁlms can beobtained by FMR technique 3–15such as magnetization,3mag- netic anisotropy,3–6Landé gfactor,3,5spin-pump effect on relaxation mechanism,7–11interlayer exchange coupling,12–14 spin diffusion,15etc. Besides being a probe for measure- ments, FMR can be used to pump spin-polarized electrons into normal metal16,17and generate dc voltage.18,19Further- more, a method of FMR-assisted switching was proposed todecrease the switching ﬁeld. 20–22 Traditionally, FMR is excited by an rf magnetic ﬁeld. Recently, a new experimental technique based on spin-torque-driven FMR was developed in magnetic tunneljunctions 23and spin valves.24–26In these experiments, the FMR signals were excited by the rf current and detected bythe dc voltage output and the resonance was realized byfrequency-swept 23–25or ﬁeld-swept methods.26This new technique enables FMR studies on the individual sub-100-nm magnets. Many important parameters, such as mag-netic anisotropy and damping, can be determined. Subse-quently, the experimental spin-torque-driven FMR spectra were reproduced by micromagnetic modeling. 27In theory, a microscopic theory on the spin-torque-driven FMR indicatedthat the output dc voltage results from rectiﬁcation of boththe applied rf current and the spin current emitted by the precessing magnetization. 28The resonance of this dc voltage was suggested to detect the vibrational modes excited by thespin current. 29Before the experiments, the dc spin-current effect on the rf-ﬁeld driven FMR and the possibility of theFMR excited by an ac current were investigated by Xi et al. 30 Heretofore, most studies were concentrated on the mag- netic multilayers with parallel or perpendicular anisotropy.Magnetic ﬁlms with tilted anisotropy are also attractive forpotential application. A method has been proposed to obtaintilted magnetic anisotropy in TbFe thin ﬁlms. 31The effects of oblique anisotropy on FMR modes have been studied insingle and coupled layers. 32Furthermore, magnetic thin-ﬁlm media with tilted anisotropy has potential application in re-cording with the advantages of unprecedented storage densi-ties, thermal stability, and fast switching speeds. 33–36It is also intriguing to investigate the magnetic behavior of mul-tilayers with tilted anisotropy. Recently, the current-drivenmagnetization dynamics, such as the microwave signalgeneration 37and the current effect on the ﬁeld-swept FMR,38 was studied in the magnetic multilayers with the tiltedpinned magnetization. The tilted anisotropy in the free orpinned magnetic layers provides an alternate choice to con-trol the magnetization dynamics in multilayers and minimizethe switching current. Investigation of FMR in the tiltedmagnetic multilayers is helpful to understand and control themagnetic damping which is important for optimizing themagnetic conﬁguration. Inspired by recent experiments on the spin-torque-driven FMR and the possible realization of the tilted anisotropy inmultilayers, we present a theoretical study on current-drivenand adjusted FMR in the magnetic trilayers with tilted aniso- a/H20850Electronic mail: pbhe1026@yahoo.com.cn.JOURNAL OF APPLIED PHYSICS 105, 043908 /H208492009 /H20850 0021-8979/2009/105 /H208494/H20850/043908/7/$25.00 © 2009 American Institute of Physics 105 , 043908-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.135.12.127 On: Thu, 02 Oct 2014 10:47:16tropy. Starting from the Landau-Lifshitz-Gilbert- Slonczewski /H20849LLGS /H20850equation,39,40we investigate the linear response of the magnetization to an ac current. Then, the dccurrent-adjusted and ac current-frequency-adjusted FMRspectra are obtained and the dependence of the FMR prop-erties on the current and the direction of the pinned magne-tization are discussed in detail. II. LINEARIZATION We consider magnetic trilayers consisted of two ferro- magnetic metallic layers with a sandwiched normal metallicspacer. The magnetization in the thick layer is pinned. Weassume that different directions of the pinned magnetizationcan be achieved by magnetic ﬁeld or by fabrication. Themagnetization in the thin layer is free and the easy axis isdeﬁned along the xdirection. The polar and azimuth angles of the magnetization in the pinned layer are denoted by /H9258p and/H9278p, respectively. With electrical current ﬂowing perpen- dicularly through the trilayer, spin polarization takes place inthe pinned layer and the magnetization in the free layer un-dergoes a spin-transfer torque generated by the current. Thistorque arises from the exchange coupling between the spinsof the conductive electrons and the local magnetization. 39,40 Phenomenologically, the magnetic dynamics of the free layer can be described by the LLGS equation,39 dM dt=/H9253M/H11003/H11509E /H11509M+/H9251 MsM/H11003dM dt+/H9253aJ MsM /H11003/H20849M/H11003mp/H20850, /H208491/H20850 where /H9253is the gyromagnetic ratio, /H9251is the Gilbert damping constant, Mis the magnetization vector of the free layer, Ms is the saturation magnetization, and mp =/H20849sin/H9258pcos/H9278p,sin/H9258psin/H9278p,cos/H9258p/H20850is the unit vector of the pinned magnetization. The spin-torque parameter aJis proportional to the current density and dependent on the ma-terials and the angle between the pinned and free magnetiza- tion. In general, a J=/H6036 2eg MsdJ, with g=1 //H20851−4+ /H208493 +M·mp/Ms/H20850/H208491+P/H208503//H208494P3/2/H20850/H20852, where dis the ﬁlm thickness of the free layer, Pis the spin polarization of the incident current, and the current density Jtakes positive value when the current ﬂows from the ﬁxed layer to the free one. Thecurrent includes dc component and ac one, i.e., J=J dc +Jacei/H9275t. In our consideration, the energy density consists of anisotropic energy and demagnetization energy, i.e., E =K/H208491−sin2/H9258cos2/H9278/H20850+2/H9266Ms2cos2/H9258. In the typical experi- ment, FMR is realized by a swept magnetic ﬁeld and a nor- mal rf magnetic ﬁeld with small amplitude. In our model,they are replaced by a dc current and an ac current, respec-tively. FMR can be realized by adjusting the dc current den-sity or the ac current frequency. Minimizing the energy density, we ﬁnd that /H9258=/H9266/2o r 3/H9266/2 and /H9278=0 or /H9266at equilibrium without applying current. Assuming the initial condition /H9258=/H9266/2 and /H9278=0 and intro- ducing the parameter /H9274=/H9266/2−/H9258, we can transform Eq. /H208491/H20850 into a pair of coupled differential equations,41−/H208491+/H92512/H20850d/H9274 dt=/H9251/H9253/H208732/H9266Ms+HK 2cos2/H9278/H20874sin 2/H9274 −/H9253HK 2cos/H9274sin 2/H9278+/H9253aJ/H20853/H20851−/H20849mpxcos/H9278 +mpysin/H9278/H20850sin/H9274+mpzcos/H9274/H20852 −/H9251/H20849mpxsin/H9278−mpycos/H9278/H20850/H20854, /H208492/H20850 /H208491+/H92512/H20850d/H9278 dt=−/H9253/H208494/H9266Ms+HKcos2/H9278/H20850sin/H9274 −/H9251/H9253HK 2sin 2/H9278+/H9253aJ/H20853/H9251/H20851−/H20849mpxcos/H9278 +mpysin/H9278/H20850tan/H9274+mpz/H20852 +/H20849mpxsin/H9278−mpycos/H9278/H20850sec/H9274/H20854, /H208493/H20850 where HK=2K/Ms, representing the anisotropy ﬁeld. Owing to the small amplitude of the rf current, the de- viation of magnetization from the equilibrium position isvery slight. It is sound to take a linearization approximation.In the usual FMR experiment of bulk magnet, the trajectoryof magnetization forms a cone about the direction of effec-tive static ﬁeld. Nevertheless, in an ultrathin ﬁlm, the verylarge demagnetization ﬁeld forces this cone into a very ﬂatellipse. 41So it is reasonable to assume that the rotation of magnetization can be conﬁned in the ﬁlm plane and can bedescribed by a one-dimensional equation. Under the dc spin-transfer torque, the magnetization in the free layer reaches anew equilibrium location /H9278=/H9278eq. Applying an ac current, a small deviation /H9254/H9278from this equilibrium location comes into being and /H9278=/H9278eq+/H9254/H9278. Performing a tedious derivation /H20849see the Appendix /H20850, a forced oscillation equation about /H9254/H9278is ob- tained from Eqs. /H208492/H20850and /H208493/H20850, d2/H9254/H9278 dt2+2/H9252d/H9254/H9278 dt+/H927502/H9254/H9278=fei/H9275t. /H208494/H20850 This equation describes a forced vibration with the damping constant /H9252, the nature angular frequency /H92750, and the periodic driving force fei/H9275t. The dynamical equilibrium location /H9278eqis determined by /H20849AD+A/H11032D/H11032/H20850/H20849A2+C/H11032D/H11032/H20850=0. The parameters above are given in the Appendix. By solving Eq. /H208494/H20850,w eg e t the amplitude of deviation /H9254/H9278/H20849/H9254/H9278=/H9004/H9278ei/H9275t/H20850, /H9004/H9278=/H9004/H9278/H11032+i/H9004/H9278/H11033, /H208495/H20850 where /H9004/H9278/H11032=1 /H9003/H20851/H20849/H927502−/H92752/H20850fR+2/H9252/H9275fI/H20852, /H9004/H9278/H11033=1 /H9003/H20851/H20849/H927502−/H92752/H20850fI−2/H9252/H9275fR/H20852, /H208496/H20850 with/H9003=/H20849/H927502−/H92752/H208502+4/H92522/H92752, and fRand fIare the real and imaginary parts of f, respectively. On the basis of Eqs. /H208495/H20850 and /H208496/H20850, we will discuss the FMR by adjusting the dc current and the frequency of ac current in Secs. IV and V.043908-2 He et al. J. Appl. Phys. 105 , 043908 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.135.12.127 On: Thu, 02 Oct 2014 10:47:16III. CURRENT-ADJUSTED FERROMAGNETIC RESONANCE In this section, we ﬁx the frequency of ac current and investigate the linear response of the free magnetization un-der the varying dc current. In the following discussions, wetake CoFeB as an illustrative example. 25The values of re- lated parameters are the gyromagnetic ratio /H9253=1.75 /H11003107Oe−1s−1, the saturate magnetization Ms=1178 G, the anisotropic ﬁeld HK=850 Oe, the Gilbert damping param- eter/H9251=0.014, the spin polarization P=0.5, the thickness of free layer d=3.5 nm, and the frequency of ac current /H9275 =65 GHz. The spin-torque parameter aJhas the dimension of magnetic ﬁeld. With the current density J=0.1 A //H9262m2,aJ varies from 181 to 1299 Oe by changing the orientation of the pinned magnetization. aJtakes the maximal value in the antiparallel conﬁguration in which the pinned magnetizationis antiparallel to the easy axis of the free layer, while it takesminimum value in the parallel one. The calculations reveal that the equilibrium value /H9278eq and the precessional frequency /H92750are dependent on the di- rection of the pinned magnetization and the dc current. Forexample, with the pinned magnetization in the x−zplane /H20849x/H110220/H20850, /H92750increases slightly with increasing current density in the parallel conﬁguration. When /H9258pdeviates from 90°, /H92750 decreases with increasing current density more and more rap- idly. It is well known that the precessional frequency is de-termined by the torques exerted on the magnetization. In thepresent of current, apart from the static torques generated bythe effective ﬁeld, there is a spin-transfer torque produced bythe spin-polarized current. This dynamical torque providesan effect magnetic ﬁeld a J/Ms/H20849M/H11003mp/H20850, which is weakest in the parallel conﬁguration while strongest in the perpendicu- lar one. So, its inﬂuence on the precessional frequency isvery little in the parallel conﬁguration. The spin-polarizedcurrent has the similar effect on the equilibrium location /H9278eq. In the parallel conﬁguration, /H9278eqis unchanged by varying the current. With /H9258pbeing an acute angle, /H9278eqdecreases with increasing current density while increases in the case of ob-tuse angle. In the perpendicular conﬁguration, the change of /H9278eqby applying the current has the most effect. The current has two roles on the energy change in the magnetic system: whether pumping energy into the free layeror dissipating energy from it depends on the directions of thecurrent and the pinned magnetization. With the spin torque inthe direction of the Gilbert damping torque, the energy ispumped into the magnet. Otherwise, with the spin torque inthe opposite direction, the energy is dissipated from the mag-net. In the parallel conﬁguration, the magnet draws energyfrom the positive current, whereas loses energy for the nega-tive current. In the antiparallel conﬁguration, negative cur-rent pumps energy thoroughly and positive current dissipatesenergy thoroughly. With the pinned magnetization deviatingfrom the easy axis of the free layer, the case is more com-plicated. When the precessional angle is smaller than theangle included between the precessional axis and the pinnedmagnetization, energy pumping occurs in half of the preces-sional circle and dissipation occurs in another half circle. Inour consideration, the small amplitude of ac current implies asmall-angle precession. Thus, the ac spin torque may be a source of precession or may serve as a damping source.These arguments can be manifested by the effects of thecurrent density and the direction of pinned magnetization onthe effective damping constant /H9252. By adjusting /H9252from posi- tive value to negative one, the force of friction turns intodriven force and the pumping energy exceeds the dissipativeenergy. In terms of a detail analysis, we ﬁnd that /H9252decreases with the positive current and approaches zero in a certaincurrent density with the pinned magnetization parallel to theeasy axis of the free layer. This indicates that the current-related energy pumping is enhanced by increasing currentdensity and may exceed the intrinsic damping. However, inthe antiparallel conﬁguration, /H9252becomes negative when the negative current density is greater than 0.009 A //H9262m2.I n Fig. 1, we show the dependence of /H9252on the direction of pinned magnetization. With the negative current, /H9252/H110210 in the inner of contour line that /H9252=0, as shown in Fig. 1/H20849a/H20850./H9252takes the minimal value in antiparallel conﬁguration. Thus, in thecase of negative current and antiparallel conﬁguration, thepumping energy from the current has the most value. Withthe positive current, /H9252/H110210 in the outer of contour line that /H9252=0, as shown in Fig. 1/H20849b/H20850./H9252takes the maximal value in the antiparallel conﬁguration. Contrary to the case of negativecurrent, the dissipative energy from the magnet is most in theantiparallel conﬁguration.0 180 360090180 −1e+010−5e+009 0(a) φp(deg)θp(deg ) 0 180 360090180 00 5e+0091e+010(b) φp(deg)θp(deg ) FIG. 1. /H20849Color online /H20850The contour lines of the damping constant /H9252as a function of both azimuth and polar angles of the pinned magnetization with/H20849a/H20850J dc=−0.08 A //H9262m2and /H20849b/H208500.08 A //H9262m2, respectively. The unit of /H9252is hertz.043908-3 He et al. J. Appl. Phys. 105 , 043908 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.135.12.127 On: Thu, 02 Oct 2014 10:47:16Instead of static magnetic ﬁeld, FMR can be realized by adjusting the dc current density. With the pinned magnetiza-tion taking different orientations, the resonant curves showdiverse line shapes, resonant locations, and peak heights. InFig. 2, we show the resonant curves with M pin the x−z plane /H20849x/H110220/H20850. With the pinned magnetization parallel to the ﬁlm, the resonance is weakest and the resonant dc current density is highest; whereas, with the pinned magnetizationperpendicular to the ﬁlm, the case is on the contrary. Theinset in Fig. 2/H20849a/H20850indicates that the resonant current density is maximal in the parallel conﬁguration and decreases with thepinned magnetization sloping from the ﬁlm plane. When thepinned magnetization is slanted out of the ﬁlm plane, thepeak height increases rapidly. When /H9258p=72° and 108°, the resonant amplitude of /H9004/H9278reaches the maximal value, as shown in the inset of Fig. 2/H20849b/H20850. In resonant state, it is easy to reverse the magnetization because of the most deviationfrom the equilibrium location. Therefore, by combining thedc and ac spin torques, it is possible to decrease the criticalcurrent density in the current-induced magnetization rever-sal. With the pinned magnetization deﬂected from the ﬁlmplane slightly, we can obtain a relatively low dc current den- sity and a relatively high resonant peak.IV. FREQUENCY-ADJUSTED FERROMAGNETIC RESONANCE Since the frequency of ac current can be adjusted in a wide range, several recent experiments23–25have actualized FMR measurements by adjusting the frequency of a small-amplitude radio-frequency ac current. In these experiments,the pinned magnetization is parallel to the easy axis of thefree layer and a static magnetic ﬁeld is applied. In the fol-lowing, we will investigate the frequency-adjusted FMR inany directions of pinned magnetization without magneticﬁeld. Figure 3shows the FMR curves in a certain direction of pinned magnetization and different dc current densities. Withthe dc current density increasing, the FMR peak turns blunterfor the negative current while turns sharper for the positiveone. The inset of Fig. 3/H20849a/H20850indicates that the resonant line- width approaches zero when J dc=0.075 A //H9262m2. In this situ- ation, the pumping energy may compensate the damping en-ergy. As shown in the inset of Fig. 3/H20849b/H20850, the resonant frequency takes the maximum when J dc=0.024 A //H9262m2. The dependence of resonant frequency on the current density isqualitatively coincident with the experimental result of Ref.−0.1 −0.05 0 0.05 0.1 0.1 3−0.08−0.06−0.04−0.020(a) Jdc(A/µm2)∆φ′(rad) −0.1 −0.05 0 0.05 0.1 0.1 3−0.06−0.04−0.0200.020.04(b) Jdc(A/µm2)∆φ′′(rad)0 90 18000.040.08 θp(deg)Jr(A/µm2) 0 90 180−0.100.1 θp(deg)∆φ′ ex(rad)0°75° 0°75°90°90° FIG. 2. /H20849Color online /H20850/H20849a/H20850The real part and /H20849b/H20850imaginary parts of /H9004/H9278as a function of dc current density for different directions of the pinned magne-tization. In both ﬁgures, /H9278p=0° and /H9258ptakes values from 0° to 90° in 15° steps. The insets of /H20849a/H20850and /H20849b/H20850show the dependences of the resonant loca- tionJrand the resonant amplitude /H9004/H9278ex/H11032on the polar angle /H9258p, respectively.50 55 60 65 70−0.15−0.1−0.0500.050.1(a) ω(GHz)∆φ′(rad) 50 55 60 65 70−0.2−0.15−0.1−0.0500.05(b) ω(GHz )∆φ′′(rad)−0.1 0 0.1024 Jdc(A/µm2)∆ω (GHz) −0.1 0 0.1506070 Jdc(A/µm2)ωr(GHz) 0.06A/ µm2−0.06A/ µm2−0.06A/ µm2 0.06A/ µm2 FIG. 3. /H20849Color online /H20850Frequency-adjusted FMR curves for different dc current densities in the case that /H9258p=30° and /H9278p=60°. The dependences of the real and imaginary parts of /H9004/H9278on the frequency are shown in /H20849a/H20850and /H20849b/H20850, respectively. In both ﬁgures, Jdctakes the values from −0.06 A //H9262m2to 0.06 A //H9262m2in 0.03 A //H9262m2steps. The insets of /H20849a/H20850and /H20849b/H20850show the resonant linewidth /H9004/H9275and the resonant frequency /H9275ras a function of cur- rent density, respectively.043908-4 He et al. J. Appl. Phys. 105 , 043908 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.135.12.127 On: Thu, 02 Oct 2014 10:47:1642 /H20849Fig. 2/H20850. The resonant frequency is maximal in a small positive current density and decreases with increasing cur-rent density /H20849red shift /H20850in both positive and negative direc- tions. Figure 4shows the FMR curves in a certain current and different directions of the pinned magnetization. In the x−z plane /H20849x/H110220/H20850, the more the pinned magnetization slants out of the ﬁlm plane, the sharper the FMR peak becomes. In the perpendicular conﬁguration, the peak is highest /H20849not shown in the ﬁgure and about 1.066 8 rad /H20850. The inset of Fig. 4/H20849a/H20850 shows that the resonant frequency is maximal in the parallelconﬁguration and minimal in the perpendicular one. Asshown in the inset of Fig. 4/H20849b/H20850, the resonant amplitude is lowest in the parallel conﬁguration and highest in the perpen-dicular one. The susceptibility of the free magnetization to the ac current can be understood by the behavior of /H9004 /H9278. It is con- venient to deﬁne the susceptibility of /H9254/H9278to the ac current Jacei/H9275t, /H9273=/H9254/H9278 Jacei/H9275t=1 Jac/H20849/H9004/H9278/H11032+i/H9004/H9278/H11033/H20850. /H208497/H20850 Under the ac spin-transfer torque, the change in the free magnetization can be divided into two parts: one is in phasewith the ac current and the other has a 90° lag in phase. The parameters /H9273/H11032/H20849=1 Jac/H9004/H9278/H11032/H20850and/H9273/H11033/H20849=1 Jac/H9004/H9278/H11033/H20850represent the ratio of in-phase part and out-of-phase part of /H9254/H9278to the ac current, respectively. Now, we illustrate the relation between /H9004/H9278and the mag- netic energy. From the expression of energy density and Eqs./H208492/H20850and /H208493/H20850and taking a linear approximation, we obtain the power density, P=/H9008/H20851/H9011 1/H20849/H9004/H9278/H110322+/H9004/H9278/H110332/H20850+/H90112/H9004/H9278/H11032+/H90113/H9004/H9278/H11033+/H9018/H20852, /H208498/H20850 where /H9008=Kcos2/H9278eq+2/H9266Ms2,/H90111=/H9253B/H11032/H20849AB+A/H11032B/H11032/H20850/A2 −A/H11032//H20849/H9253A2/H20850/H92752,/H90112=/H9253aac/H20849ABE /H11032−AB/H11032E−2A/H11032B/H11032E/H11032/H20850/A2,/H90113 =aac/H20849AE−2A/H11032E/H11032/H20850/A2/H9275, and /H9018=/H9253/H20851/H20849AE−A/H11032E/H11032/H20850E/H11032aac2−/H20849AD +A/H11032D/H11032/H20850/H20852/A2. In Fig. 5, we show the dependences of power density at resonance on the dc current density and the direc- tion of the pinned magnetization in two special cases. Figure5/H20849a/H20850indicates that the power density has an extremum at J dc=0.075 A //H9262m2. This is consistent with the result in the inset of Fig. 3/H20849a/H20850that the linewidth approaches zero with Jdc=0.075 A //H9262m2. From Fig. 5/H20849b/H20850, we ﬁnd that the power density has two maxima at /H9258p=12° and 168°. This coincides with the consequence in Fig. 1that the damping constant /H9252is zero in the case that /H9278p=0 and /H9258p=12° and 168°. In the parallel conﬁguration, the extremely small value of thepower density indicates that the inﬂuence of current on thefree layer is very feeble. From these two particular examples,we can infer that with suitable dc current density and orien-tation of the pinned magnetization the most pumping powercan be obtained. V. CONCLUSION We have investigated the current-excited and adjusted FMR in the magnetic trilayers with a tilted pinned magneti-zation. By the linearization method, the current- andfrequency-adjusted FMR spectra are obtained. The preces-sional frequency, the damping constant, and the equilibriumposition of the free magnetization can be adjusted by the dccurrent density and the direction of the pinned magnetiza-tion. In some regions deﬁned by the dc current and thepinned magnetization, the damping constant is negative andthe energy pumping is more efﬁcient. The resonant linewidthor amplitude and the resonant location are dependent on thedc current and the direction of the pinned magnetization. Inthe end, we connect the power density with the FMR spectra50 55 60 65 70−0.1−0.0500.050.1(a) ω(GHz)∆φ′(rad) 50 55 60 65 70−0.0500.050.10.150.20.25(b) ω(GHz )∆φ′′(rad)0 90 180−101 θp(deg)∆φ′′ ex(rad)0 90 180506070 θp(deg)ωr(GHz) 15°15°90° 90° FIG. 4. /H20849Color online /H20850Frequency-adjusted FMR curves for different direc- tions of the pinned magnetization in the case that Jdc=0.1 A //H9262m2.T h e dependences of the real and imaginary parts of /H9004/H9278on the frequency are shown in /H20849a/H20850and /H20849b/H20850, respectively. In both ﬁgures, /H9278p=0° and /H9258ptakes values from 15° to 90° in 15° steps. The insets of /H20849a/H20850and /H20849b/H20850show the resonant frequency /H9275rand the resonant amplitude /H9004/H9278ex/H11033as a function of /H9258p, respectively.−0.1 −0.05 0 0.05 0.1100102104106108(a) Jdc(A/µm2)P (W/cm3) 0 90 18010−4010−3010−2010−101001010(b) θp(deg)P (W/cm3) FIG. 5. /H20849Color online /H20850/H20849a/H20850The dependence of the power density Pon the dc current density Jdcin the case that /H9258p=30° and /H9278p=60°. /H20849b/H20850The dependence ofPon/H9258pfor/H9278p=0 and Jdc=0.08 A //H9262m2.043908-5 He et al. J. Appl. Phys. 105 , 043908 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.135.12.127 On: Thu, 02 Oct 2014 10:47:16and discuss the inﬂuence of the dc current and the pinned magnetization on the power. With the dc current density andthe pinned magnetization taking certain values, the powerdensity may reach an extremum. By selecting the currentdensity and the direction of the pinned magnetization, wecan acquire the most energy pumping from the current to thefree layer. This is useful for improving the efﬁciency ofcurrent-driven microwave oscillation and current-inducedmagnetization reversal. ACKNOWLEDGMENTS This work was supported by the NSF of China under Grants Nos. 10747128 and 10804028 and the 985 project ofHunan University. Z.D.L. was supported by NSF of Chinaunder Grant No. 10874038, by NSF of Hebei Province underGrant Nos. A2007000006. APPENDIX: LINEARIZATION OF THE EQUATION OF MOTION IN THE SPHERICAL COORDINATES Taking derivative with respect to time in the two sides of Eq. /H208492/H20850,w eg e t /H208491+/H92512/H20850d2/H9278 dt2=−/H9253/H20875/H20849HKcos/H9274cos2/H9278+4/H9266Mscos/H9274/H20850d/H9274 dt +HK/H20849/H9251cos 2/H9278− sin/H9274sin 2/H9278/H20850d/H9278 dt/H20876 +/H9253aJ/H20875/H20849−/H9251/H90211sec/H9274+/H90212tan/H9274/H20850sec/H9274d/H9274 dt +/H20849/H90211sec/H9274+/H9251/H90212tan/H9274/H20850d/H9278 dt/H20876/H9253daJ dt /H11003/H20851−/H9251/H90211tan/H9274+/H90212sec/H9274+/H9251mpz/H20852, /H208499/H20850 where /H90211=mpxcos/H9278+mpysin/H9278, /H90212=mpxsin/H9278−mpycos/H9278. In the following calculations, we insert /H9278=/H9278eq+/H9254/H9278into Eq. /H208499/H20850and expand every term in Eqs. /H208492/H20850,/H208493/H20850, and /H208499/H20850up to linear terms in /H9254/H9278,/H9274, and Jac. Then, we obtain three linear equations about termsd/H9274 dt,d/H9254/H9278 dt, andd2/H9254/H9278 dt2, respectively. Re- moving the variables /H9274andd/H9274 dtin the equation withd2/H9254/H9278 dt2by use of the other two equations, we get Eq. /H208494/H20850and the damp- ing constant /H9252, the nature angular frequency /H92750, and the driving force fare given as follows: /H9252=/H9253 2/H208491+/H92512/H20850/H20875A/H11032+B/H11032+C/H11032 A2/H20849AD+2A/H11032D/H11032/H20850/H20876, /H2084910/H20850 /H92750=/H9253 1+/H92512/H20875AB+A/H11032B/H11032−C A/H20849AD+A/H11032D/H11032/H20850+C/H11032 A/H20849BD /H11032 +B/H11032D/H20850+2A/H11032B/H11032C/H11032D/H11032 A2/H208761/2 , /H2084911/H20850f=/H9253aac 1+/H92512/H20849fR+ifI/H20850, /H2084912/H20850 where fR=/H9253 1+/H92512/H20875A/H11032E/H11032−AE+C/H11032 A/H20849DE /H11032−D/H11032E/H20850−2A/H11032C/H11032D/H11032E/H11032 A2 +AD+A/H11032D/H11032 A/H20849/H9251Z2+mpzGE /H11032/H20850/H20876, fI=/H9275/H208731+D/H11032 AmpzG/H20874E/H11032. The parameters in the above equations are written as A=/H208494/H9266Ms+HKcos2/H9278eq/H20850+adc/H20851/H9251Z2+mpz/H20849Z1 +/H9251mpz/H20850G/H20852, A/H11032=/H9251/H208494/H9266Ms+HKcos2/H9278eq/H20850−adc/H20851Z2−mpz/H20849/H9251Z1 −mpz/H20850G/H20852, B=HKcos 2/H9278eq+adc/H20851/H9251Z2+Z1/H20849/H9251Z1−mpz/H20850G/H20852, B/H11032=/H9251HKcos 2/H9278eq−adc/H20851Z2+Z1/H20849Z1+/H9251mpz/H20850G/H20852, C=HKsin 2/H9278eq+adc/H20851/H9251Z1−Z2/H20849/H9251Z1+mpz/H20850G −2Z1mpz/H20849Z1+/H9251mpz/H20850G2/H20852, C/H11032=adc/H20851Z1+Z2/H20849Z1+3/H9251mpz/H20850G+2mpz2/H20849Z1+/H9251mpz/H20850G2/H20852, D=HK 2sin 2/H9278eq+adcE,D/H11032=HK 2sin 2/H9278eq−adcE/H11032, E=/H9251Z1−mpz,E/H11032=Z1+/H9251mpz, where Z1=mpxsin/H9278eq−mpycos/H9278eq, Z2=mpxcos/H9278eq +mpysin/H9278eq,adc/H20849ac/H20850=/H208512/H6036//H20849eM st/H20850/H20852/H20851P3/2G//H208491+P/H208503/H20852Jdc/H20849ac/H20850, and G=/H208491+P/H208503//H20851−16P3/2+/H208491+P/H208503/H208493+Z2/H20850/H20852. 1J. 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1.1522421.pdf	REVIEWS OF ACOUSTICAL PATENTS Lloyd Rice 11222 Flatiron Drive, Lafayette, Colorado 80026 The purpose of these acoustical patent reviews is to provide enough information for a Journal reader to decide whether to seek more information from the patent itself.Any opinions expressed here are those of reviewers as individuals and are not legal opinions. Printed copies of United States Patents may be ordered at $3.00 each from the Commissioner of Patents and Trademarks, Washington, DC 20231. Patents are available via the Internet at http://www.uspto.gov. Reviewers for this issue: GEORGE L. AUGSPURGER, Perception, Incorporated, Box 39536, Los Angeles, California 90039 MARK KAHRS, Department of Electrical Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261 DAVID PREVES, Micro-Tech Hearing Instruments, 3500 Holly Lane No., Suite 10, Plymouth, Minnesota 55447 DANIEL R. RAICHEL, 2727 Moore Lane, Fort Collins, Colorado 80526 KEVIN P. SHEPHERD, Mail Stop 463, NASA Langley Research Center, Hampton, Virginia 23681 WILLIAM THOMPSON, JR., Pennsylvania State University, University Park, Pennsylvania 16802 ROBERT C. WAAG, Department of Electrical and Computer Engineering, Univ. of Rochester, Rochester, New York 14627 6,386,755 43.28.Vd ACOUSTIC PYROMETER Dean E. Draxton et al., assignors to Combustion Specialists, Incorporated 14 May 2002 Class 374 Õ117; ﬁled 5 January 2000 In a coal-ﬁred boiler, pendant tubes 38may occupy a space more than 50 ft across. Slag accumulates on the tubes and must be removed periodi-cally using blasts of steam. However, the cleaning process introduces addi-tional corrosion and should be performed no more often than necessary. Forvarious technical reasons discussed in the patent, accurate measurements of gas temperature can facilitate more efﬁcient operation of the boiler and alsodetermine when cleaning is required. The invention includes a generator ofpulsed acoustic signals 55and one or more receivers 60.Asignal processing system ﬁlters out background noise and then calculates temperature as afunction of transit time.—GLA5,822,272 43.30.Gv CONCENTRIC FLUID ACOUSTIC TRANSPONDER Donald E. Ream, Jr., assignor to the United States of America as represented by the Secretary of the Navy 13 October 1998 Class 367 Õ2; ﬁled 13 August 1997 A passive acoustic transponder consists of two concentric thin-walled spherical shells. The region interior to the inner sphere and the annularregion between them are each ﬁlled with different refracting ﬂuids. Throughthe choice of these ﬂuids, the sphere sizes, the shell materials, and wallthicknesses, it is possible to realize transponders with a wide variety offrequency responses and target strengths.—WT 5,877,460 43.30.Jx DEVICE FOR TALKING UNDERWATER Ritchie C. Stachowski, Moraga, California 2 March 1999 Class 181 Õ127; ﬁled 16 September 1997 A more or less horn-shaped structure made of a rigid plastic has a smaller open end with an elastomeric mouth ﬁtting for forming a water-tightseal around a user’s mouth while the larger end is sealed with a thin dia-phragm, which could be the same material as the rest of the horn. The sidesof the horn’s body contain one-way blow valves for releasing the user’sexhausted air in the form of small bubbles. When one speaks, while under-water, the thin diaphragm is set into vibrations which reradiate the soundsinto the water. The small released air bubbles allegedly do not radiate muchinterfering noise.—WT 6,366,534 43.30.Lz UNDERWATER HIGH ENERGY ACOUSTIC COMMUNICATIONS DEVICE Robert Woodall and Felipe Garcia, assignors to the United States of America as represented by the Secretary of the Navy 2 April 2002 Class 367 Õ145; ﬁled 2 April 2001 Two metal spherical shells are held in concentric spaced-apart posi- tions by a large number of radially oriented springs. The space interior toboth spheres may be ﬁlled with pressurized gas or liquid. The inner surfaceof the outer sphere supports many explosive devices consisting of a squib, aSOUNDINGS 2507 J. Acoust. Soc. Am. 112(6), December 2002 0001-4966/2002/112(6)/2507/20/$19.00 © 2002 Acoustical Society of America Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:40radially oriented tube, and a projectile such as a small metal sphere. Firing signals from a micro-controller detonate selected sets of the squibs in pre-programmed sequences thereby launching the projectiles to strike the innersphere. The resulting vibrations of this shell subsequently radiate high-level,broadband, acoustic signals into the surrounding acoustic medium. An aux-iliary hydrophone on the assembly allows a remote source to communicate with the micro-controller, thereby varying the ﬁring signals to produce acoded acoustic signal.—WT 6,396,770 43.30.Nb STEERABLE THERMOACOUSTIC ARRAY Charles A. Carey et al., assignors to BAE Systems Information and Electronic Systems Integration Incorporated 28 May 2002 Class 367 Õ141; ﬁled 28 June 1982 To provide communication between an airplane and a submerged sub- marine ~or some other submerged hydrophone system !, a modulated laser beam ~or particle beam !is radiated from the airplane towards the surface of the ocean and moved across that surface at a speed equal to the speed ofsound in water divided by the sine of the incident angle.The beam of energyincrementally heats the water causing thermal expansion or explosive va-porization, either of which effects create a sound wave in the water.—WT 6,377,514 43.30.Tg ACOUSTIC LENS-BASED SWIMMER’S SONAR Thomas E. Linnenbrink et al., assignors to Q-Dot, Incorporated 23 April 2002 Class 367 Õ11; ﬁled 6 April 2000 A hand-held diver’s ultrasonic imaging system consists of a two- dimensional grid array of identical acoustical video converter elements.Each of these elements consists of a set of polymethylpentene acousticlenses, a multi-element focal plane transducer array fashioned from 1–3composite piezoceramic, and associated electronics to drive a VGA displaymounted in the diver’s mask in C-scan format. The transducer array is usedin both transmit and receive modes. The whole sonar unit, with the excep-tion of the battery pack strapped to the swimmer, is housed in a cylindricalcan 6.7 in. in diameter by 15 in. long.—WT 6,377,515 43.30.Vh SYNCHRONIZED SONAR Robert W. Healey, assignor to Brunswick Corporation 23 April 2002 Class 367 Õ88; ﬁled 4 August 2000 An electronic control system is described which can interconnect a number of closely spaced identical sonars, such as ﬁsh-ﬁnders or depth-ﬁnders, and allow them to be energized simultaneously so that all of theunits are listening for the echo at the same time. This will presumablyreduce cross-unit interference. Alternatively, the many sonars are electroni-cally interconnected and energized in a predetermined temporal manner,designed so that again the sonars do not interfere with each other.—WT6,349,791 43.30.Wi SUBMARINE BOW DOME ACOUSTIC SENSOR ASSEMBLY Daniel M. Glenning and Bruce E. Sandman, assignors to the United States of America as represented by the Secretary of theNavy 26 February 2002 Class 181 Õ140; ﬁled 3 April 2000 A submarine bow dome acoustic sensor comprises an acoustically transparent outer hull 10and an inner pressure hull 12that deﬁne a free- ﬂooded compartment 14. Within this compartment, an acoustic panel 16, which may be planar ~as shown !or hemispherical, with an optically reﬂect- ing surface 20, is mounted on acoustically isolating supports 18. The panel is fashioned of a relatively stiff plastic or aluminum. A laser scanner 22~or possibly a number of such scanners !, also mounted in the free-ﬂooded com- partment 14, casts a laser beam 24onto the surface 20. This beam ~or beams !can be moved rapidly over portions of surface 20. Sensor26, which may be mounted on each scanner housing as indicated, receives the reﬂec-tions of the laser beam 24from panel surface 20. Doppler shifts of these reﬂected light waves, because of vibrations of panel 16caused by a noise generating or reﬂecting object in the surrounding acoustic medium, provideinformation for the calculation of the location and speed of that object. Anarray of acoustic sources 28permits an active mode of operation of this vibrometer-sensor system.—WT 6,370,084 43.30.Xm ACOUSTIC VECTOR SENSOR Benjamin A. Cray, assignor to the United States of America as represented by the Secretary of the Navy 9 April 2002 Class 367 Õ141; ﬁled 25 July 2001 An acoustic vector sensor is realized by encasing a commercially available tri-axial accelerometer within a sphere of syntactic foam of sufﬁ-cient size to render the whole structure neutrally buoyant. This, in turn, issurrounded by a thin spherical shell of viscoelastic rubber which is acous-tically transparent yet isolates the accelerometer from structure-borne soundthat may enter through any attachment points.—WTSOUNDINGS 2508 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:405,822,271 43.30.Yj SUBMARINE PORTABLE VERY LOW FREQUENCY ACOUSTIC AUGMENTATION SYSTEM Richard M. Ead and Robert L. Pendleton, assignors to the United States of America as represented by the Secretary of the Navy 13 October 1998 Class 367 Õ1; ﬁled 1 April 1998 To augment the acoustic signature of a submarine or to duplicate the signatures of other submarines for training purposes, a mobile target very-low-frequency projector is suitably mounted within a dedicated torpedoshell which is then positioned within the ﬂooded torpedo tube of any sub-marine. Control and power are provided from the submarine to the device.The device operates wholly within the torpedo tube and does not need to belaunched.—WT 5,859,812 43.30.Yj SELF POWERED UNDERWATER ACOUSTIC ARRAY Michael J. Sullivan et al., assignors to the United States of America as represented by the Secretary of the Navy 12 January 1999 Class 367 Õ130; ﬁled 14 October 1997 A housing, to which a line array of sensors is attached, contains a shrouded impeller and a shielded electric generator. As the assembly istowed through the water, enough electric power is generated locally to sup-ply the needs of the sensor electronics without having to plumb electricpower down from the tow ship. This signiﬁcantly reduces the size of thetowing cable and electromagnetic interference problems associated withhigh power levels on long cables. Data transmission from the sensors backto the tow ship is via a ﬁber optic cable in the tow cable.—WT 5,878,000 43.30.Yj ISOLATED SENSING DEVICE HAVING AN ISOLATION HOUSING Neil J. Dubois, assignor to the United States of America as represented by the Secretary of the Navy 2 March 1999 Class 367 Õ188; ﬁled 1 October 1997 A‘‘windscreen’’to isolate a hydrophone from ﬂow noise consists of a more or less cylindrically shaped two-part housing made of an acousticallytransparent material such as PVC. The base part of the housing is perma-nently attached to the host structure, e.g., a naval vessel. The cap portion ofthe housing features a number of holes to allow for free-ﬂooding of theinterior and for gas bubbles to escape, as well as containing a number ofresilient elements, such as rubber bands, to suspend and vibration isolate thehydrophone within that free-ﬂooded but sealed cavity.—WT 6,370,085 43.30.Yj EXTENDABLE HULL-MOUNTED SONAR SYSTEM Jonathan Finkle et al., assignors to the United States of America as represented by the Secretary of the Navy 9 April 2002 Class 367 Õ173; ﬁled 3 August 2001 Asystem is described to deploy one or more arrays of sonar transduc- ers~and other sensors !away from the hull of a submarine while it is in motion thereby increasing their effective aperture. This is accomplished viaa set of support arms that extend radially outward from an attachment pointat the bow of the hull much like the ribs of an umbrella. These arms are positioned in an approximately equispaced circumferential arrangementaround the hull.The transducers, either sources or receivers, can be mountedon these arms, or extend between the arms, or be located between the armsand the hull on auxiliary supports, or there could be towed line arraysattached to the ends of the arms. This system of arms and transducer arraysfolds into longitudinal grooves on the hull during high-speed transit of thesubmarine to reduce self-noise.—WT 6,377,516 43.30.Yj ULTRASONIC TRANSDUCER WITH LOW CAVITATION John Whiteside and Craig Mehan, assignors to Garmin Corporation 23 April 2002 Class 367 Õ173; ﬁled 8 December 2000 An ultrasonic sonar transducer, such as used with a depth-ﬁnder or ﬁsh-ﬁnder system, is described. The sonar includes a conventional trans-ducer element 12in housing 14designed for mounting on a ship’s hull. The bottom face 34of the housing is curved such that no major portion of it is parallel to the active face of the transducer element 12. The front end of the housing is raised relative to the rear, creating a positive angle of attackrelative to the direction of water ﬂow. This produces a pressure gradientover the bottom face which promotes laminar ﬂow, thereby reducing thenoise associated with turbulence and cavitation.—WT 6,404,701 43.30.Yj ENCAPSULATED VOLUMETRIC ACOUSTIC ARRAY IN THE SHAPE OF A TOWEDBODY Thomas R. Stottlemyer, assignor to the United States of America as represented by the Secretary of the Navy 11 June 2002 Class 367 Õ20; ﬁled 16 July 2001 The central, cylindrical portion of a towed underwater body is envi- sioned to contain a set of electroacoustic transducers encapsulated in a solidcasting of polyurethane which both prevents water intrusion into the trans-ducer elements and allegedly increases the cavitation threshold, thereby al-lowing the array of transducers to be driven to a greater acousticintensity.—WTSOUNDINGS 2509 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:406,400,645 43.30.Yj SONOBUOY APPARATUS Bruce W. Travor, assignor to the United States of America as represented by the Secretary of the Navy 4 June 2002 Class 367 Õ4; ﬁled 11 October 2001 A sonobuoy apparatus, sized to ﬁt into a standard canister, includes a set of telescopic arms 34which elongate as the weighted canister 20and weighted acoustic projector unit 40both fall downward. Hinge arrange- ments36at the upper ends of each of the telescopic arms cause the arms to rotate to a near-horizontal orientation after the canister falls away. These deployed arms support hydrophones ~not shown !at various positions along the arms and also along tension lines stretched between the arms at theirouter ends. Surface ﬂotation unit 50houses the transmitter/receiver equip- ment and antenna 51while28damps vertical motion of the assembly.—WT 5,884,650 43.35.Ei SUPPRESSING CAVITATION IN A HYDRAULIC COMPONENT Anthony A. Ruffa, assignor to the United States of America as represented by the Secretary of the Navy 23 March 1999 Class 137 Õ13; ﬁled 26 February 1997 It is suggested that the cavitation threshold in some ﬂow region can be raised through the use of an acoustic transducer that radiates an acousticﬁeld into the ﬂow region thereby increasing the ambient pressure. There is no discussion of what happens during the half of the cycle in which thatradiated acoustic pressure ﬁeld subtracts from the ambient pressure.—WT 6,396,484 43.35.Pt ADAPTIVE FREQUENCY TOUCHSCREEN CONTROLLER USING INTERMEDIATE-FREQUENCY SIGNAL PROCESSING Robert Adler et al., assignors to Elo Touchsystems, Incorporated 28 May 2002 Class 345 Õ177; ﬁled 29 September 1999 This controller is intended for an acoustic touchscreen in which acous- tic or ultrasonic waves are generated and directionally propagated across thetouchscreen surface utilizing the phenomena of surface acoustic waves. Thecontroller can either utilize look-up tables to achieve the desired outputfrequency or it can use a multi-step process in which it ﬁrst determines thefrequency requirements of the touchscreen and then adjusts the burst fre-quency characteristics, the receiver circuit center frequency, or both, in ac-cordance with the touchscreen requirements. In one embodiment, the adap-tive controller compensates for global mismatch errors through a digitalmultiplier that modiﬁes the output of a crystal reference oscillator. In an-other embodiment, a digital signal processor provides corrections based onstored values that compensate for both global and local signal variations.—DRR 5,900,533 43.35.Sx SYSTEM AND METHOD FOR ISOTOPE RATIO ANALYSIS AND GAS DETECTIONBY PHOTOACOUSTICS Mau-Song Chou, assignor to TRW Incorporated 4 May 1999 Class 73Õ24.01; ﬁled 3 August 1995 The system includes a tunable laser that is directed into a sample at energy levels sufﬁcient to generate detectable acoustic emissions. A micro-phone detects these emissions for processing and analysis.—WT 6,404,536 43.35.Sx POLARIZATION INDEPENDENT TUNABLE ACOUSTO-OPTICAL FILTER AND THEMETHOD OF THE SAME Eric Gung-Hwa Lean et al., assignors to Industrial Technology Research Institute 11 June 2002 Class 359 Õ308; ﬁled in Taiwan, Province of China 30 December 2000 In this ﬁlter, input light is diffracted into two light beams, one affected by acoustic waves and the other not.Apolarization beam displacer/combineris employed to separate the input light beam into two orthogonal beams.Several acousto-optical polarized rotators are used to rotate the polarizationof a particular light wavelength by 90 degrees. The two beams are properlycombined to form orthogonal beams.—DRR 6,391,020 43.35.Ty PHOTODISRUPTIVE LASER NUCLEATION AND ULTRASONICALLY-DRIVEN CAVITATIONOF TISSUES AND MATERIALS Ron Kurtz et al., assignors to The Regents of the University of Michigan 21 May 2002 Class 606 Õ2; ﬁled 6 October 1999 This apparatus creates a cavitation nucleus in a target material by focusing optical radiation, in the form of a short pulse laser beam, at aSOUNDINGS 2510 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:40portion of the material and then causing mechanical disruption in another portion of the materials adjacent to the cavitation nucleus by subjecting the cavitation nucleus to ultrasound waves.—DRR 6,392,540 43.35.Ty NON-AUDITORY SOUND DETECTION SYSTEM Mary E. Brown, Rochester, New York 21 May 2002 Class 340 Õ540; ﬁled 4 May 2000 This is a device for converting sound into nonauditory signals to alert a user to the presence of a predetermined sound. The system consists of amain control unit and a transceiver. The transceiver is used to transmit anactivation signal upon receiving the predetermined sound. A remote assem-bly contains an attachment device which can be worn by the user and cangenerate a nonauditory signal ~such as a ﬂash of light !.—DRR 6,390,979 43.35.Yb NONINVASIVE TRANSCRANIAL DOPPLER ULTRASOUND COMPUTERIZEDMENTAL PERFORMANCE TESTING SYSTEM Philip Chidi Njemanze, Owerri IMO, Nigeria 21 May 2002 Class 600 Õ438; ﬁled 24 August 2001 The device purports to determine the mental performance capacity of a human subject for performing a given task by measuring the subject’s base-line blood ﬂow velocity in cerebral arteries using a transcranial Dopplerultrasound instrument. Two probes are placed on the temples and the later-ality index for both arteries is calculated. A computer is used to presentmental tasks on a monitor while simultaneously monitoring in real-time themean blood ﬂow velocity during each stage of the task. The acquired data isprocessed to yield mental performance indices that may be relayed via cel-lular telephony to a remote computer or mission control.—DRR6,390,982 43.35.Yb ULTRASONIC GUIDANCE OF TARGET STRUCTURESFORMEDICALPROCEDURES Frank M. Bova, Gainesville, Florida et al. 21 May 2002 Class 600 Õ443; ﬁled 21 July 2000 The system described in this patent combines an ultrasound probe with both passive and active infrared tracking systems to provide a real timeimage display of the entire region of interest. This is done without probemovement. Real time tracking of the target region permits physiological gating and probe placement during image acquisition so that all externaldisplacements introduced by the probe can be monitored during the time oftreatment. The system may be used in the surgical arena for image guidanceduring radiation therapy and surgery.—DRR 6,390,983 43.35.Yb METHOD AND APPARATUS FOR AUTOMATIC MUTING OF DOPPLER NOISEINDUCED BY ULTRASOUND PROBE MOTION Larry Y. L. Mo and Dean W. Brouwer, assignors to GE Medical Systems Global Technology Company, LLC 21 May 2002 Class 600 Õ453; ﬁled 7 September 2000 The device monitors the blood vessel wall signal input to a spectral Doppler processor to check for clutter induced by probe motion. The clutteris typically of higher frequency than that due to normal vessel wall motion.Threshold logic is applied to check for energy within a frequency bandgreater than the normal wall signal frequencies. If signiﬁcant energy abovesome ‘‘rattle’’ threshold is detected for a predeﬁned time interval, the Dop-pler audio is automatically muted.This can be effected at one or more pointswithin the Doppler audio signal path in a conventional scanner. If the rat-tling clutter is no longer detected, the Doppler audio is reactivated orramped up smoothly.—DRR 6,398,732 43.35.Yb ACOUSTIC BORDER DETECTION USING POWER MODULATION George A. Brock-Fisher and David M. Prater, assignors to Koninklijke Philips Electronics, N.V. 4 June 2002 Class 600 Õ443; ﬁled 11 February 2000 The described method entails the control of an ultrasound system to identify a boundary between a tissue region and a blood-ﬁlled region thatlies within a region of interest ~ROI!. A contrast agent is initially adminis- tered to the ROI and then ultrasound beams are transmitted at differentpower levels into the ROI. Signal returns from the beams are processed todetermine a phase difference. It is claimed that under certain circumstancesa phase change in echo returns occurs at the boundary between tissue andSOUNDINGS 2511 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:40blood-containing contrast agent. Detection of the phase change provides for precise identiﬁcation of the boundary on the basis of the time interval inwhich the phase change is detected.—DRR 6,398,736 43.35.Yb PARAMETRIC IMAGING ULTRASOUND CATHETER James B. Seward, assignor to Mayo Foundation for Medical Education and Research 4 June 2002 Class 600 Õ466; ﬁled 20 October 1999 The subject device provides parametric images of a surrounding in- sonated environment. Parametric imaging is deﬁned as the imaging of quan-tiﬁable ‘‘parameters’’ for visible two-, three-, four-, or nonvisible, higher-dimensional, temporal physiological events. Visible motion is a fourth-dimensional event and includes surrogate features of cardiac muscularcontraction, wall motion, valve leaﬂet motion, etc. Nonvisible motion is ahigher-dimensional event that encompasses slow nonvisible occurrences~e.g., remodeling, transformation, aging, healing, etc. !or fast nonvisible events ~i.e., heat, electricity, strain, compliance, perfusion, etc. !. An ultra- sound catheter with parametric imaging capability can obtain dynamic digi-tal or digitized information from the surrounding environment and displayinformation features or quanta as static or dynamic geometric ﬁgures fromwhich discrete or gross quantiﬁable information can be obtained.—DRR 5,756,898 43.35.Zc PASSIVE ACOUSTIC METHOD OF MEASURING THE EFFECTIVE INTERNALDIAMETER OF A PIPE CONTAINING FLOWINGFLUIDS Victor Diatschenko et al., assignors to Texaco Incorporated 26 May 1998 Class 73Õ592; ﬁled 27 June 1994 A change in the internal diameter of a pipe, whether it is decreased because of the accumulation of nonﬂowing material or increased because ofcorrosion or erosion, is detected by observing a shift in the characteristicfrequency of the pipe. The measuring system is entirely passive in that itrequires only that vibrations of the pipe be excited by the noise generated bythe ﬂow within the pipe. Furthermore, because the diameter measurement isnot based on acoustic signal transit time, the present system is not dependenton the assumption of uniform ﬂow conditions within the pipe.—WT 6,371,095 43.35.Zc ULTRASOUND WHISTLES FOR INTERNAL COMBUSTION ENGINE Walter E. Sacarto, Denver, Colorado 16 April 2002 Class 123 Õ590; ﬁled 21 August 2000 Ultrasonic whistles are proposed to improve the mixing of air and fuel prior to ignition in an internal combustion engine. They may be placed in acylinder head, around a valve stem, or in a carburetor. Various whistle de-signs are described.—KPS 6,402,769 43.38.Ar TORSIONAL ULTRASOUND HANDPIECE Mikhail Boukhny, assignor to Alcon Universal Limited 11 June 2002 Class 606 Õ169; ﬁled 21 January 2000 This handpiece design features a set of piezoelectric elements con- structed of segments capable of both longitudinal and torsional motion. Anappropriate ultrasound driver drives the set of elements at their respectiveresonant frequencies to produce longitudinal and torsional oscillations.—DRR6,332,029 43.38.Bs ACOUSTIC DEVICE Henry Firouz Azima et al., assignors to New Transducers Limited 18 December 2001 Class 381 Õ152; ﬁled in the United Kingdom 2 September 1995 This is another in a long line of recent NXT patents. This particular issue delineates 48 pages of examples of vibratory panels in ceilings, easels,pianos, vending machines, and more. The patent does not discuss any of themore substantive issues such as shaker placement, panel construction, etc.These are discussed in British Patent 235008 ~or European Patent 1068770 !.—MK 6,399,870 43.38.Bs MUSICAL INSTRUMENTS INCORPORATING LOUDSPEAKERS Henry Azima et al., assignors to New Transducers Limited 4 June 2002 Class 84Õ744; ﬁled in the United Kingdom 2 Septem- ber 1995 This is another NXT patent. Place a vibration panel on the back of an electronic musical instrument. Evidently, they forgot to include this appli-cation in their application compendium ~United States Patent 6,332,029, reviewed above !.—MK 5,898,642 43.38.Fx SONAR ANTENNA Jean-Marie Wagner, assignor to Etat Francais represente par le Delegue General pour l’Armement 27 April 1999 Class 367 Õ158; ﬁled in France 28 September 1995 Normally a planar array of Tonpilz-type sonar transducers is realized by ﬁrst making individual transducers and then bonding or otherwise attach-ing them in a grid arrangement to an acoustically transparent elastomericmaterial layer which constitutes the acoustic window. Here it is proposed toﬁrst bond to the acoustic window layer a continuous layer of material of thesize and shape that represents the ensemble of head masses of the entirearray. The head mass layer has a set of predrilled and tapped holes corre-sponding to the positions of the stress bolts of the set ofTonpilz transducers.Then the head mass layer is cut into a series of orthogonal grooves ~groove depths equal to head mass layer thickness and spacings equal to head massdimensions !to produce a grid arrangement of individual head masses. The remainder of each of the Tonpilz transducers is then assembled onto theseseparate but spatially arranged head masses in standard fashion.—WT 6,386,041 43.38.Fx STEP COUNTING DEVICE INCORPORATING VIBRATION DETECTINGMECHANISM David Yang, Taipei, Taiwan, Province of China 14 May 2002 Class 73Õ651; ﬁled 1 February 2000 Those who choose walking for excercise often use a pedometer to keep track of the distance covered, or more accurately the number of stepstaken. The invention is a small, self-contained step-counting device that canbe attached to shoes or clothing. A piezoelectric transducer detects vibra-tions which are then analyzed, counted, and displayed.—GLASOUNDINGS 2512 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:406,362,726 43.38.Ja SOUNDER DEVICE WHICH DEFLECTS SOUND AWAY FROM A HOUSING Kieron Chapman, assignor to Fulleon Limited 26 March 2002 Class 340 Õ384.7; ﬁled in the European Patent Ofﬁce 27 February 1997 An acoustical alarm device comprises a base member 1for mounting against a ceiling 2~the ﬁgure is drawn inverted !that supports a sounder plate1awhich is generally of concave shape. The transducer 7and its housing4are displaced from the sounder plate by a series of spacers 9 creating gap 8. Sound radiated by 7is reﬂected off the sounder plate 1a outwardly through gap 8. Central channel 3permits the routing of electrical cables to auxiliary detectors and/or lights indicated schematically by dashedlines11.T a g12indicates an electronics module mounted within housing 4. An alleged advantage of this design is that both the transducer module andthe electronics module can be encapsulated simultaneously as opposed toprior art designs that required separate encapsulations.—WT 6,377,696 43.38.Ja LOUDSPEAKER SYSTEMS Stuart Michael Nevill, assignor to B & W Loudspeakers Limited 23April 2002 Class 381 Õ345; ﬁled in the United Kingdom 2 May 1997 The intent, as in some transmission line loudspeaker designs, is to completely absorb rear radiation from speaker 28without introducing acous- tic resonances or excessive cone damping. Those readers familiar with 1970s speaker systems will ﬁnd that the patent document—text, illustra-tions, and claims—almost perfectly describes the Webb transmission linesystem featured in ‘‘Audio Amateur Loudspeaker Projects, 1970–1979.’’—GLA6,381,334 43.38.Ja SERIES-CONFIGURED CROSSOVER NETWORK FOR ELECTRO-ACOUSTICLOUDSPEAKERS Eric Alexander, South Ogden, Utah 30 April 2002 Class 381 Õ99; ﬁled 23 February 1999 With resistive loads, series and parallel frequency dividing networks perform equally well. With reactive loudspeaker loads, the series networkhas the annoying property of altering the high-frequency ﬁlter in response tothe woofer’s changing impedance. Prior art also includes at least one speaker-level constant-voltage network in whicha relatively highimpedancetweeter was bridged across the inductor of a second-order low-pass section.The patent describes a number of conﬁgurations that seem to make use of alittle of each.—GLA 6,381,337 43.38.Ja SOUND REPRODUCTION DEVICE OR MICROPHONE Marc Adam Greenberg, assignor to Floating Sounds Limited 30 April 2002 Class 381 Õ345; ﬁled in the United Kingdom 9 De- cember 1995 From time to time inventors come up with the idea of pumping air in andoutofaballoontoreproducesound.Inthiscase,however,thesurfaceof an inﬂatable balloon is mechanically driven at two or more points. Theconﬁguration can also be used as a microphone.—GLA 6,384,550 43.38.Ja SPEAKER AND DRIVE DEVICE THEREFOR Hideaki Miyakawa et al., assignors to Canon Kabushiki Kaisha 7 May 2002 Class 318 Õ116; ﬁled in Japan 6 September 1994 One might guess that this circuit is intended to drive a digital loud- speaker. Not quite. It drives a supersonic vibration wave motor, which inSOUNDINGS 2513 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:40turn drives a loudspeaker cone. The patent teaches us that since such a wave motor has relatively high mass and substantial friction, ‘‘...no resonancephenomenon takes place.’’ Moreover, since the wave motor generates noback emf, it follows that ‘‘...no group delay phenomenon takes place.’’Theconcept of minimum-phase response seems to have eluded the fourinventors.—GLA 6,385,324 43.38.Ja BROADBAND LOUDSPEAKER Karl Heinz Ko ¨ppen, assignor to Sorus Audio AG 7 May 2002 Class 381 Õ336; ﬁled in Germany 17 March 1997 It is certainly not unusual to see loudspeakers mounted on convex spherical surfaces. The novel feature of this design is 45-degree rear deﬂec- tion plane 9. We are told that reﬂected waves must pass several times through internal damping material and, as a result, axial standing wavescannot develop.—GLA 6,389,144 43.38.Ja SOUND FIELD EQUALIZING APPARATUS FOR SPEAKER SYSTEM Deog Jin Lee, assignor to LG Electronics Incorporated 14 May 2002 Class 381 Õ340; ﬁled in the Republic of Korea 29 July 1997 In the 1940s, Altec-Lansing patented a loudspeaker design in which a little multicell horn was mounted in front of a loudspeaker. This invention eliminates the cells.—GLA 6,389,146 43.38.Ja ACOUSTICALLY ASYMMETRIC BANDPASS LOUDSPEAKER WITH MULTIPLEACOUSTIC FILTERS James J. Croft III, assignor to American Technology Corporation 14 May 2002 Class 381 Õ345; ﬁled 17 February 2000 In 1994, the bandpass conﬁguration shown was thoroughly analyzed and documented by JBL in an unpublished research project. However, thepatent includes one variant in which sealed chamber 21is quite large and uncontrolled, such as the trunk cavity of an automobile. Another variant includes an internal acoustic notch ﬁlter. Neither of these speciﬁc geometrieswas anticipated by the JBL study.—GLA 6,389,140 43.38.Kb CERAMIC PIEZOELECTRIC TYPE MICROPHONE Jose Wei, Hsin Tien City, Taipei, Taiwan, Province of China 14 May 2002 Class 381 Õ173; ﬁled 30 November 1999 This patent describes a contact ~throat !microphone in which high- density foam is used to conduct mechanical vibrations from the user’s skinto a piezoelectric transducer. With proper selection of the foam material andits thickness, the patent asserts that improved high-frequency ﬁdelity can beachieved and background noise suppressed.—GLASOUNDINGS 2514 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:406,363,156 43.38.Lc INTEGRATED COMMUNICATION SYSTEM FOR A VEHICLE Timothy S. Roddy, assignor to Lear Automotive Dearborn, Incorporated 26 March 2002 Class 381 Õ86; ﬁled 18 November 1998 Microphones are distributed within the passenger compartment of a van so that passengers in the rear can communicate with those in the front.Furthermore, each passenger is also able to operate an on-board cell phoneand control the vehicle’s sound system.Adigital signal processor is used toreduce unwanted microphone signals and feedback.—KPS 6,377,862 43.38.Md METHOD FOR PROCESSING AND REPRODUCING AUDIO SIGNAL Hidetoshi Naruki and Shoji Ueno, assignors to Victor Company of Japan, Limited 23 April 2002 Class 700 Õ94; ﬁled in Japan 19 February 1997 ADVD has sufﬁcient space to consider the addition of other informa- tion about the stored tracks, speciﬁcally data concerning playback param-eters such as equalization and reverb.—MK 6,378,010 43.38.Md SYSTEM AND METHOD FOR PROCESSING COMPRESSED AUDIO DATA David Burks, assignor to Hewlett-Packard Company 23 April 2002 Class 710 Õ68; ﬁled 10 August 1999 Filed in 1999, this patent proposes a very simple computer architecture for a DSP system that compresses audio data on a CD-ROM. Even in 1989,this would have been starkly obvious.—MK 6,392,133 43.38.Md AUTOMATIC SOUNDTRACK GENERATOR Alain Georges, assignor to dBtech SARL 21 May 2002 Class 84Õ609; ﬁled 17 October 2000 This conceptual patent uses two pages and three claims to propose mixing an external audio track with an existing video track. Owing to thebrevity, nothing close to real is described.—MK 6,392,576 43.38.Md MULTIPLIERLESS INTERPOLATOR FOR A DELTA-SIGMA DIGITAL TO ANALOGCONVERTER Gerald Wilson and Robert S. Green, assignors to Sonic Innovations, Incorporated 21 May 2002 Class 341 Õ143; ﬁled 21 August 2001 Delta sigma converters use oversampling to achieve high SNR. How- ever, there is a tradeoff between the oversampling rate and the processingspeed required in the DSP circuitry. The inventors propose using interpola-tion and decimation combined with a zero-order hold and a lattice ﬁlterdesign. The lattice ﬁlter implementation is noteworthy for using shifts andadds to avoid multiplication. The patent writing is clear and concise.—MK6,393,401 43.38.Md PICTURE DISPLAY DEVICE WITH ASSOCIATED AUDIO MESSAGE Alan R. Loudermilk and Wayne D. Jung, assignors to LJ Laboratories, L.L.C. 21 May 2002 Class 704 Õ272; ﬁled 6 December 2001 In 2002, a ‘‘talking picture’’ has a different interpretation from the 1930s. Here, the inventors propose adding a sound storage chip that can beactivated by a switch on the picture frame. They also propose other systemsthat show the same sense of originality.—MK 6,385,320 43.38.Vk SURROUND SIGNAL PROCESSING APPARATUS AND METHOD Tae-Hyun Lee, assignor to Daewoo Electronics Company, Limited 7 May 2002 Class 381 Õ17; ﬁled in the Republic of Korea 19 De- cember 1997 Several earlier patents describe methods of producing virtual surround sound sources from a single pair of loudspeakers. Using head-related trans-fer functions, the circuitry shown is intended to create two rear virtual sound images in addition to the two real front sound images. At the same time,reverberation is added to simulate a more spacious sonic environment.—GLA 6,360,844 43.50.Gf AIRCRAFT ENGINE ACOUSTIC LINER AND METHOD OF MAKING THE SAME William H. Hogeboom and Gerald W. Bielak, assignors to The Boeing Company 26 March 2002 Class 181 Õ213; ﬁled 2 April 2001 An acoustic liner for use in the nacelle of an aircraft engine consists of alternating sections of absorptive liner and low-resistance liner. The latterserves to scatter low-mode-order noise into higher modes that are morereadily absorbed.The low-resistance liner is composed of a perforated sheet,having percent open area of at least 15%, a honeycomb layer, and an im-pervious backing sheet. Methods of manufacture to achieve varying cavitydepths in the honeycomb layer are described, thus enabling absorption overa broad frequency range.Applications to circular inlets as well as to splittersin the aft engine duct are described.—KPS 6,364,054 43.50.Gf HIGH PERFORMANCE MUFFLER John Bubulka et al., assignors to Midas International Corporation 2 April 2002 Class 181 Õ264; ﬁled 27 January 2000 This patent describes a mufﬂer intended for use with high-performance automobiles that creates a ‘‘deep throaty high performance sound.’’ Thecross section is rectangular with the width being two and one-half timesSOUNDINGS 2515 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:40greater than the height. Arranged along the length are deﬂector plates, such as24. These plates are perforated and form a series of chambers and a sinuous path for the exhaust gases.—KPS 6,367,580 43.50.Gf SOUND ADJUSTABLE TAIL PIPE STRUCTURE Ming-Tien Chang, assignor to Liang Fei Industry Company, Limited 9 April 2002 Class 181 Õ241; ﬁled 11 July 2000 A mufﬂer tail pipe is adjustable, so that the same mufﬂer can be used on a range of different cars and engines. The alignment of 23and24can be altered by means of the arrangement using the oblong slot 11, thus adjusting the airﬂow through the tail pipe.—KPS 6,404,152 43.50.Gf MOTOR CONTROL DEVICE Takashi Kobayashi et al., assignors to Hitachi, Limited 11 June 2002 Class 318 Õ254; ﬁled in Japan 29 May 1998 The device is essentially a feed-back system that reduces undesired sound caused by ring oscillations in the radial direction of the stator insidean electric motor. The electromagnetic force drift is measured to providecorrective coefﬁcients to the activation current.—DRR 6,394,655 43.50.Ki METHOD AND APPARATUS FOR INFLUENCING BACKGROUND NOISE OFMACHINES HAVING ROTATING PARTS Ju¨rgen Schnur and Silvia Tomaschko, assignors to DaimlerChrysler AG 28 May 2002 Class 384 Õ247; ﬁled in Germany 13 June 1998 This method of reducing the perceptible vibrations of rotating parts entails the monitoring of mounting conditions between a shaft bearing and arotating part. By varying the mounting conditions, the transfer function isvaried, particularly for vibrations between the rotating part and the shaft bearing or a component in contact with the shaft bearing. A radial adjust-ment unit containing piezoelectric elements varies the mounting conditions.The transfer function of the vibrations is thereby varied.—DRR 6,360,607 43.50.Lj SOUND DETECTOR DEVICE Francois Charette et al., assignors to Ford Global Technologies, Incorporated 26 March 2002 Class 73Õ587; ﬁled 9 August 1999 A device aimed at detecting and locating squeaks and rattles in an automobile consists of a pair of microphones 50connected to headphones 68. A mechanism is provided which allows the operator to vary the separa- tion distance between the microphones. This capability, along with the se- lection of center frequency and bandwidth, allows the operator to efﬁcientlylocate the source of sounds in a vehicle.—KPS 6,363,984 43.50.Lj TIRE TREAD PITCH SEQUENCING FOR REDUCED NOISE Christopher D. Morgan, assignor to Kumho & Company, Incorporated 2 April 2002 Class 152 Õ209.2; ﬁled 25 October 1999 Atread design is proposed in which repeated pitches of three different lengths are arranged around the circumference of a tire in order to reducerolling tire noise. One hundred different pitch sequences are deﬁned, all ofwhich claim reduced noise.—KPS 6,399,868 43.55.Lb SOUND EFFECT GENERATOR AND AUDIO SYSTEM Makoto Yamato and Tony Williams, assignors to Roland Corporation 4 June 2002 Class 84Õ701; ﬁled 28 September 2000 Given a two-channel ~stereo !input, the question is how to produce a ﬁve-channel output. The Roland unit uses an unspeciﬁed multichannel re-SOUNDINGS 2516 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:40verb12followed by a low-pass ﬁlter to create ‘‘presence’’ for the center ~surround !channel. While low frequencies do add ‘‘presence,’’ they do not make a soundﬁeld or even close ~contrary to claims of the patent !.—MK 6,386,037 43.58.Gn VOID DETECTOR FOR BURIED PIPELINES AND CONDUITS USING ACOUSTICRESONANCE William F. Kepler and Fred A. Travers, assignors to the United States ofAmerica as represented by the Secretary of the Interior 14 May 2002 Class 73Õ579;ﬁ l e d6J u n e2 0 0 1 This void detection device is a little robotic car that can move through buried conduit. Power and telemetry signals pass through umbilical cord 28. An acoustic exciter 12repeatedly taps the wall of the conduit and the re- sulting acoustic waves are detected by sensor 18. If a hidden void Vis encountered, the acoustic signal changes. External circuitry analyzes thefrequency response signals in comparison with a baseline response obtainedfrom a known good area of conduit.—GLA 6,403,944 43.58.Kr SYSTEM FOR MEASURINGABIOLOGICAL PARAMETER BY MEANS OF PHOTOACOUSTICINTERACTION Hugh Alexander MacKenzie and John Matthew Lindberg, assignors to Abbott Laboratories 11 June 2002 Class 250 Õ214.1; ﬁled in the United Kingdom 7 March 1997 This is a system intended to measure a biological parameter such as blood glucose. The system operates by directing laser pulses from a light guide into soft tissue, such as the tip of a ﬁnger, thereby producing a pho- toacoustic interaction. The resulting acoustic signal is detected by a trans-ducer and analyzed to provide the desired parametric reading.—DRR 6,390,014 43.58.Wc ACOUSTIC SIGNALING DEVICE FOR CULINARY-USE VESSELS, IN PARTICULAR FORKETTLES Tiziano Ghidini, assignor to Frabosk Casalinghi, S.P.A. 21 May 2002 Class 116 Õ150; ﬁled in the European Patent Ofﬁce 24 September 1999 True tea aﬁcianados know that black teas demand 100°C water. There- fore, many tea kettles feature whistles to alert the brewer that the water has reached boiling. This patent claims that whistles are subject to clogging dueto calcium deposits and therefore proposes a steam driven ‘‘clanger.’’—MKSOUNDINGS 2517 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:406,394,874 43.58.Wc APPARATUS AND METHOD OF USE FOR SOUND-GENERATING FINGER PUPPET Takao Kubo and Todd Miller Lustgarten, assignors to Hasbro, Incorporated 28 May 2002 Class 446 Õ327; ﬁled 4 February 2000 Sound generator chips can show up anywhere, including ﬁnger pup- pets. The ﬁnger can close a switch that activates a battery powered micro- processor 24that outputs to a ~very!small speaker 25. When will infrared communication for all ﬁve ﬁngers be patented?—MK 6,394,875 43.58.Wc BICYCLE MOUNTED NOISE-MAKING DEVICE Terry Smith, Tustin, California 28 May 2002 Class 446 Õ404; ﬁled 31 March 2000 Using a clothespin to attach a playing card to a bicycle’s rear forks doesn’t make the bike sound like a motorcycle. So, if you add a pipe and ahorn, you can imitate various motorcycle sounds. The inventors claim ‘‘Theﬂexible contact is designed to be easily replaceable, even by a child.’’—MK 6,400,275 43.58.Wc AUDITORY CUES FOR NOTIFICATION OF DEVICE ACTIVITY Michael C. Albers, assignor to Sun Microsystems, Incorporated 4 June 2002 Class 340 Õ635; ﬁled 23 June 1999 Many devices, like toasters, lack necessary and sufﬁcient displays. So, when attaching such devices to a computer network, why not use audibletonestoindicatetheconnectionstatus?Thisobviousconceptisthetotalsumof this patent.—MK 6,402,580 43.58.Wc NEARLY HEADLESS NICK NOISEMAKER CANDY TOY Thomas J. Coleman, Abingdon, Virginia et al. 11 June 2002 Class 446 Õ72; ﬁled 11 April 2001 Harry Potter fans know Nearly Headless Nick. The inventors know they can reuse their earlier patent ~United States Patent 5,855,500 !to create a rattle with a skeleton head.—MK6,337,999 43.60.Àc OVERSAMPLED DIFFERENTIAL CLIPPER Robert A. Orban, assignor to Orban, Incorporated 8 January 2002 Class 700 Õ94; ﬁled 18 December 1998 Clipping or compression of audio waveforms is a necessity in any system with a ﬁxed dynamic range. However, clipping introduces new har-monics that can alias down to baseband. The inventor, who has been work-ing on audio processors for many years, proposes to oversample and then clip. Further, the ‘‘clippings’’ are ﬁltered and then downsampled, ‘‘pro-cessed’’ ~e.g., ﬁltered !, and subtracted from the delayed input. The patent writing is succinct and clear.—MK 6,402,782 43.64.Yp ARTIFICIAL EAR AND AUDITORY CANAL SYSTEM AND MEANS OF MANUFACTURINGTHE SAME Alastair Sibbald and George Derek Warner, assignors to Central Research Laboratories, Limited 11 June 2002 Class 623 Õ10; ﬁled in the United Kingdom 15 May 1997 This device is a laminated artiﬁcial pinna having a concha, fossa, and auditory canal. The auditory canal is constructed and arranged with respectto the concha so the center of the entrance of the auditory canal is 15 to 20 mm from the rear wall of the concha and 9 to 15 mm from the concha ﬂoor and the alignment of the turning point of the entrance of the auditory canalis substantially horizontal.—DRRSOUNDINGS 2518 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:406,390,971 43.66.Ts METHOD AND APPARATUS FOR A PROGRAMMABLE IMPLANTABLE HEARING AID Theodore P. Adams et al., assignors to St. Croix Medical, Incorporated 21 May 2002 Class 600 Õ25; ﬁled 4 February 2000 A completely implanted middle ear hearing aid is adjusted by the physician or wearer remotely via infrared, ultrasonic, or rf wireless means.The programmer-transmitter sends encoded acoustic signals to the implantthat may be in the form of pulse code modulation telemetry. Post-ﬁttingadjustments that can be made by the wearer during use without the need forsurgery include volume control, frequency response, and device on-off.—DAP 6,393,130 43.66.Ts DEFORMABLE, MULTI-MATERIAL HEARING AID HOUSING Paul R. Stonikas and Robert S. Yoest, assignors to Beltone Electronics Corporation 21 May 2002 Class 381 Õ322; ﬁled 16 July 1999 A manufacturing method is described for a compliant hearing aid in which the multi-material housing deforms in response to ear canal shapechanges as the hearing aid wearer moves his or her jaw. Temporary defor-mation of the soft shell is also used during manufacture to slide componentsthrough constricted channels in the internal cavity of the housing. After-ward, remaining spaces inside the shell are ﬁlled in with a curablematerial.—DAP 6,402,682 43.66.Ts HEARING AID Patrik Johansson, assignor to Nobel Biocare AB 11 June 2002 Class 600 Õ25; ﬁled in Sweden 11 April 1997 A detachable electronics module is mounted externally in the mastoid bone where it can be accessed for battery replacement and servicing. Am-pliﬁed sound is conveyed from the module through the skin into the middleear cavity via a surgically implanted tube. The result is that the naturalmovement of the eardrum caused by sounds from the outside is enhanced byampliﬁed sounds impinging on the middle ear side of the eardrum. Theadvantage of this approach is ampliﬁcation while leaving the ear canal open, which eliminates occlusion problems inherent with conventional air conduc-tion hearing aids.—DAP 6,404,895 43.66.Ts METHOD FOR FEEDBACK RECOGNITION IN A HEARING AID AND A HEARING AIDOPERATING ACCORDING TO THE METHOD Tom Weidner, assignor to Siemens Audiologische Technik GmbH 11 June 2002 Class 381 Õ318; ﬁled in Germany 4 February 1999 Acoustic feedback is recognized by monitoring the hearing aid output signal level while attenuating a frequency band in the signal transmissionpath between the hearing aid receiver and microphone in which the feedbackcould occur. When feedback is present in a monitored frequency band, thesignal level is reduced by the attenuation more than would be expected without feedback. While detecting whether feedback is present, the micro-phone output can be switched between no attenuation and attenuation in aparticular frequency band, causing the duration of the attenuation to bevaried so that this scheme can be implemented continuously in multiplefrequency bands in which feedback is likely to occur.—DAP 6,366,883 43.72.Ja CONCATENATION OF SPEECH SEGMENTS BY USE OF A SPEECH SYNTHESIZER Nick Campbell and Andrew Hunt, assignors to ATR Interpreting Telecommunications 2 April 2002 Class 704 Õ260; ﬁled in Japan 15 May 1996 This speech synthesizer stores multiple versions of each phoneme for synthesis use. As the training speech is analyzed, multiple allophones ofeach phoneme are stored, along with feature information to be used forindexing and a weighting value based on the number of similar units alreadystored and the degree of similarity of the new unit to the stored units. Duringsynthesis, a feature search locates the stored phoneme most suitable for usein constructing the output utterance.—DLR 6,366,884 43.72.Ja METHOD AND APPARATUS FOR IMPROVED DURATION MODELING OF PHONEMES Jerome R. Bellegarda and Kim Silverman, assignors to Apple Computer, Incorporated 2 April 2002 Class 704 Õ266; ﬁled 8 November 1999 This speech synthesis system uses a sum-of-products model to scale and average the durations of phonemes extracted from the training speechdata. Minimum and maximum durations of the training phonemes and thenumber and position of the phonemes in the utterance form the basis of aSOUNDINGS 2519 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:40nonexponential, root sinusoidal transformation used to model phoneme du- rations in the synthesis output.—DLR 6,366,887 43.72.Ja SIGNAL TRANSFORMATION FOR AURAL CLASSIFICATION William J. Zehner and R. Lee Thompson, assignors to the United States of America as represented by the Secretary of the Navy 2 April 2002 Class 704 Õ278; ﬁled 12 January 1998 Certain auditory tasks, such as listening to sonar echoes, depend on the ability of the human listener to classify or categorize the signal in somemeaningful way. This patent proposes a method of altering the signal so asto make it sound more speechlike and so easier for the perceptual system tomake quick judgments of similarity or signal class. Certain temporal andspectral patterns are detected and their amplitudes and spectra are mapped toproduce more speechlike amplitudes, spectra, and redundancy patterns.—DLR 6,366,885 43.72.Lc SPEECH DRIVEN LIP SYNTHESIS USING VISEME BASED HIDDEN MARKOV MODELS Sankar Basu et al., assignors to International Business Machines Corporation 2 April 2002 Class 704 Õ270; ﬁled 27 August 1999 Video and audio speech data is analyzed to train either a hidden Mar- kov model or a neural network in order to generate video frames with lipmotions synchronized to a supplied audio sound track. During the trainingphase, audio and video data streams are simultaneously processed to deter-mine a suitable phoneme classiﬁcation. Video frames are selected directlyaccording to cepstral feature similarities. There is a brief mention withoutfurther elaboration of a video smoothing process which would occur duringthe synthesis reconstruction.—DLR 6,363,348 43.72.Ne USER MODEL-IMPROVEMENT-DATA- DRIVEN SELECTION AND UPDATE OFUSER-ORIENTED RECOGNITION MODEL OF AGIVEN TYPE FOR WORD RECOGNITIONAT NETWORK SERVER Stefan Besling and Eric Thelen, assignors to U.S. Philips Corporation 26 March 2002 Class 704 Õ270.1; ﬁled in the European Patent Ofﬁce 20 October 1997 Pattern recognition is enabled for a wide range of subjects and for many clients by selecting a recognition model from several recognitionmodels of the same type using an adaptation proﬁle to cover commonly usedsequences and speciﬁc areas of interest. Selection is further enhanced byusing the model improvement data provided by acoustic training with a fewsentences from the user. Only one basic model, for a given type, and anumber of much smaller adaptation models need be stored. Cited advantagesinclude not having to store a speciﬁc model for each user, reusing recogni-tion models for many users, and a reduced amount of training required foreach user.—DAP6,370,504 43.72.Ne SPEECH RECOGNITION ON MPEG ÕAUDIO ENCODED FILES Gregory L. Zick and Lawrence Yapp, assignors to University of Washington 9 April 2002 Class 704 Õ251; ﬁled 22 May 1998 For use in automatic video indexing applications, a technique is de- scribed capable of recognizing continuously spoken words in compressedMPEG/audio. Decompression of the MPEG/audio ﬁle and creation of an intermediate ﬁle are not required because training and feature recognitionare performed based on the extracted subbands of the ﬁles using a hiddenMarkov model as a speech recognizer.—DAP 6,374,219 43.72.Ne SYSTEM FOR USING SILENCE IN SPEECH RECOGNITION Li Jiang, assignor to Microsoft Corporation 16 April 2002 Class 704 Õ255; ﬁled 20 February 1998 To date, speech recognition systems have treated silence as a special word in the lexicon. However, in an isolated speech recognition system,taking into account transitions from silence to other words and the reverse iscomputationally intensive. In this patent, a feature extraction module ﬁrstdivides words into codewords representing phonemes. Possible words are provided as a preﬁx tree including several phoneme branches connected atnodes ~e.g., ‘‘Orange’’ in the ﬁgure !. Several phoneme branches are brack- eted by at least one input silence branch and at least one output silencebranch. Optionally, several silence branches are provided in the preﬁx treethat represent context-dependent silence periods.—DAPSOUNDINGS 2520 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:406,374,222 43.72.Ne METHOD OF MEMORY MANAGEMENT IN SPEECH RECOGNITION Yu-Hung Kao, assignor to Texas Instruments Incorporated 16 April 2002 Class 704 Õ256; ﬁled 16 July 1999 Speech recognition involves expanding a search tree according to the size of the vocabulary, frequently resulting in very large storage require-ments. To reduce the search space and to speed up the search involved incomparing the input speech to speech models, slots with bad scores areremoved from the storage space while expanding the search tree. Thesememory spaces are later replaced with slots that have better scores and aremore likely to match the input speech. There are three levels of hiddenMarkov model ~HMM !utilization when a speech frame enters: all the pos- sible words in the sentence HMM are expanded by expanding their indi-vidual phone sequence in the lexicon HMM which requires, in turn, expand-ing their phonetic HMMs that contain acoustic observations.—DAP 6,403,870 43.75.Bc APPARATUS AND METHOD FOR CREATING MELODY INCORPORATING PLURALMOTIFS Eiichrio Aoki, assignor to Yahama Corporation 11 June 2002 Class 84Õ609; ﬁled in Japan 18 July 2000 Automatic composition has been attempted by various musicians since the creation of the digital computer. As usual with Yamaha patents, manycritical details such as motive variation are not detailed ~and are barely mentioned !. The inventor seems totally unaware of the large body of work on using generative grammars for analysis and composition ~e.g., Lerdahl and Jackendoff !.—MK 6,392,137 43.75.Gh POLYPHONIC GUITAR PICKUP FOR SENSING STRING VIBRATIONS IN TWO MUTUALLYPERPENDICULAR PLANES Osman K. Isvan, assignor to Gibson Guitar Corporation21 May 2002 Class 84Õ726; ﬁled 27 April 2000 Since Fender’s original patent in 1961, electric guitar manufacturers use magnetic pickups for ferromagnetic strings. It is well known that vibrat-ing strings exhibit both horizontal and vertical modes, so more than 20 years ago separate transducers for each mode were proposed. Here, the use of twosensors25and27is proposed. These are fed to an RMS scaler and mixer.—MK6,380,468 43.75.Hi DRUM HAVING SHELL CONSISTING OF MORE THAN ONE KIND OF VIBRATORYELEMENT ARRANGED IN PARALLEL WITHRESPECT TO SKIN Fumihiro Shigenaga, assignor to Yamaha Corporation 30 April 2002 Class 84Õ411 R; ﬁled in Japan 30 September 1999 Drum materials affect the timbre of the hit as a factor of the material properties. So if the drum body is replaced with a composite, drummers areunhappy. The inventor proposes ~in broken English: ‘‘The difference in propagation sheep resulted in sound quality’’ !insertion of metal rods or bars in the shell to increase propagation speed.—MK 6,376,759 43.75.Mn ELECTRONIC KEYBOARD INSTRUMENT Satoshi Suzuki, assignor to Yamaha Corporation 23 April 2002 Class 84Õ615; ﬁled in Japan 24 March 1999 Akeyboard instrument has limited expressiveness—the action is ﬁxed, the pitches ﬁxed, etc. How can this be made more ﬂexible so that keyboardperformers can use their training and technique to control a synthesizer?The answer presented here depends on ~1!more pedals with pressure and veloc- ity sensors and ~2!a complex ﬁnite state machine that guides the instrument modes.—MK 6,380,469 43.75.Mn KEYBOARD MUSICAL INSTRUMENT EQUIPPED WITH KEY ACTUATORS ACCURATELYCONTROLLING KEY MOTION Haruki Uehara, assignor to Yamaha Corporation 30 April 2002 Class 84Õ439; ﬁled in Japan 21 June 2000 Yet another Disklavier™ patent, this time it’s devoted just to the key mechanism. As shown, the solenoids 240,241, operated by controller 230,SOUNDINGS 2521 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:40can actuate key 112, which will control action 120and hammer 130. The controller can also sense key motions via sensors 210.—MK 6,392,136 43.75.Mn MUSICAL TONE GENERATION STRUCTURE OF ELECTRONIC MUSICALINSTRUMENT Masao Kondo et al., assignors to Yamaha Corporation 21 May 2002 Class 84Õ718; ﬁled in Japan 19 June 2000 In a computer-controlled piano like the Disklavier™, the strings can be damped so that no sound is audible. If so, then the electronics can generate the sound, but now the problem is how to make it ‘‘originate’’ from thepiano. The solution is to use a set of loudspeakers in the lid. Uprights neednot apply.—MK 6,403,872 43.75.Mn KEYBOARD MUSICAL INSTRUMENT FAITHFULLY REPRODUCING ORIGINALPERFORMANCE WITHOUT COMPLICATED TUNINGAND MUSIC DATA GENERATING SYSTEMINCORPORATED THEREIN Shigeru Muramatsu et al., assignors to Yamaha Corporation 11 June 2002 Class 84Õ724; ﬁled in Japan 16 December 1999 This is yet another Disklavier™ patent, this time focusing on the hammer/string interface. Eleven different embodiments are disclosed.—MK6,380,470 43.75.St TRAINING SYSTEM FOR MUSIC PERFORMANCE, KEYBOARD MUSICALINSTRUMENT EQUIPPED THEREWITH ANDTRAINING KEYBOARD Yuji Fujiwara et al., assignors to Yamaha Corporation 30 April 2002 Class 84Õ470 R; ﬁled in Japan 13 April 1999 The principle behind this invention is simply this: a MIDI stream with on/off events can light up LEDs on the tops of the keys. Among the em-bodiments is a ten-key keyboard reminiscent of Englebart’s chord-set. TheEnglish in the patent is almost unbearable, e.g., ‘‘The reason why the be-ginners feel hard is that optical indicators are too many to quickly searchthem for the radiation.’’—MK 6,392,132 43.75.St MUSICAL SCORE DISPLAY FOR MUSICAL PERFORMANCE APPARATUS Haruki Uehara, assignor to Yamaha Corporation 21 May 2002 Class 84Õ477 R; ﬁled in Japan 21 June 2000 Essentially, this is an automatic page turner for a piano player. Using an automatic speech recognition system, the performer speaks a commandand the display changes accordingly. However, good human page turners aresilent and use automatic nod recognition.—MK 6,390,923 43.75.Wx MUSIC PLAYING GAME APPARATUS, PERFORMANCE GUIDING IMAGE DISPLAYMETHOD, AND READABLE STORAGE MEDIUMSTORINGPERFORMANCEGUIDINGIMAGEFORMING PROGRAM Kensuke Yoshitomi et al., assignors to Konami Corporation 21 May 2002 Class 463 Õ43; ﬁled in Japan 1 November 1999 Continuing the tradition of arcade contest of man versus machine, this patent compares human performance on an artiﬁcial drum or guitar againsta known template. This is reminiscent of United States Patent 6,342,665@reviewed in J. Acoust. Soc. Am. 112~3!, 803 ~2002!#.—MK 6,392,135 43.75.Wx MUSICAL SOUND MODIFICATION APPARATUS AND METHOD Toru Kitayama, assignor to Yamaha Corporation 21 May 2002 Class 84Õ622; ﬁled in Japan 7 July 1999 This could have been an interesting and educational patent. But it lacks speciﬁcity and the translation of Japanese to English is poor including‘‘truck’’ for ‘‘track,’’ ‘‘memorized’’ for ‘‘stored,’’ and so forth. The patentconcerns the reuse of acoustic data by a MIDI stream digital instrument.TheSOUNDINGS 2522 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:40acoustic data is analyzed into amplitude and frequency, with the frequency mistakenly called ‘‘timbre.’’The signal is also analyzed for attack, sustain,decay, and release by unknown and undisclosed methods.After being stored,these parameters can be recombined to make new sounds. If your only toolis a MIDI hammer, then you see all instruments as MIDI nails.—MK 6,403,871 43.75.Wx TONE GENERATION METHOD BASED ON COMBINATION OF WAVE PARTS ANDTONE-GENERATING-DATA RECORDING METHODAND APPARATUS Masahiro Shimizu and Hideo Suzuki, assignors to Yamaha Corporation 11 June 2002 Class 84Õ622; ﬁled in Japan 27 September 1999 Essentially, this patents the user interface for a waveform editor. Each waveform has the following time domain representations: waveform, pitch,amplitude, ‘‘spectral template,’’and ‘‘time template.’’These names are com-pletely arbitrary since Yamaha says nothing about how these ‘‘templates’’are created from acoustic sounds. Further, they say nothing about how toresynthesize notes from this representation. Perhaps this will become clearin a later patent.—MK6,390,995 43.80.Gx METHOD FOR USING ACOUSTIC SHOCK WAVES IN THE TREATMENT OF MEDICALCONDITIONS John A. Ogden and John F. Warlick, assignors to Healthtronics Surgical Services, Incorporated 21 May 2002 Class 601 Õ2; ﬁled 25 June 1999 The idea of this method, intended for medical treatment of a variety of pathological conditions associated with osseous or musculoskeletal environ-ments, is to apply a sufﬁcient number of acoustic shock waves to the site ofa pathological condition to generate micro-disruptions, nonosseous tissuestimulation, increased vascularization, and circulation and induction ofgrowth factors to induce or accelerate a body’s healing processes andresponses.—DRR 6,402,965 43.80.Gx SHIP BALLAST WATER ULTRASONIC TREATMENT Patrick K. Sullivan et al., assignors to Oceanit Laboratories, Incorporated 11 June 2002 Class 210 Õ748; ﬁled 13 July 2000 Aship’s ballast water is pumped through intake and outtake manifolds which are internally lined with piezoelectric material to create long continu-ous electroacoustic transducers within these pipes. The high-intensity soundﬁelds created by these transducers can potentially destroy micro-organisms,algae, diatoms, veligers, ﬁsh larvae, and plankton, thus preventing theintroduction of nonindigenous species into new and unwelcomeenvironments.—WT 6,396,402 43.80.Ka METHOD FOR DETECTING, RECORDING AND DETERRING THE TAPPING ANDEXCAVATING ACTIVITIES OF WOODPECKERS Robert Paul Berger and Alexander Leslie McIlraith, assignors to Myrica Systems Incorporated 28 May 2002 Class 340 Õ573.2; ﬁled 12 March 2001 The device, which consists of a housing with mounting ﬂanges for direct attachment to a utility pole or the like, can receive vibrations resultingfrom woodpeckers’activities.Atransducer attached to the mounting wall ofthe housing converts the vibrations into signals. A circuit compares thevibrations with a long-term average and emits an output in response todetection above a threshold. The outputs are counted and, if the numberwithin a predetermined time exceeds a preset minimum, a sound transmitteris actuated to emit a deterrent sound. A memory contains various deterrentsounds, including those usually made by predators, to discourage the wood-peckers. The power source consists of a solar cell charging a battery.—DRR 6,379,304 43.80.Qf ULTRASOUND SCAN CONVERSION WITH SPATIAL DITHERING Jeffrey M. Gilbert et al., assignors to TeraTech Corporation 30 April 2002 Class 600 Õ447; ﬁled 23 November 1999 This scan conversion accepts lines of ultrasonic b-scan echoes in a polar format and produces data in a Cartesion format by using either soft-ware or hardware in a computer that is connected to an ultrasonic scan head.Ultrasonic echo, positional, and other data are sent from the scan head to thecomputer. The conversion uses spatial dithering to approximate pixel valuesthat fall between two input data points.—RCWSOUNDINGS 2523 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:406,398,734 43.80.Qf ULTRASONIC SENSORS FOR MONITORING THE CONDITION OF FLOWTHROUGH A CARDIAC VALVE George E. Cimochowski and George W. Keilman, assignors to VascuSense, Incorporated 4 June 2002 Class 600 Õ454; ﬁled 12 August 1999 A parameter indicative of the condition of a cardiac valve is estab- lished by monitoring blood ﬂow and/or velocity in a vessel that is coupled tothe cardiac valve or in a chamber adjacent to an artiﬁcial heart valve. One ormore ultrasonic transducers are supplied either in a cuff or a wall arrange-ment deployed about a cardiac vessel to monitor the parameter with respectto a natural or artiﬁcial valve. Transient or Doppler measurements are madeusing an appropriate number of transducers to determine either blood volu-metric ﬂow or velocity. Various implantable electronic circuits enable atransducer to be driven and to receive an ultrasonic signal indicative of thestatus of the blood ﬂow and thus, the condition of the heart valve. A radiofrequency coil coupled to an external power supply and monitoring consoleconveys power to the ultrasonic transducers and receives the blood ﬂow datasignals.—DRR 6,405,069 43.80.Qf TIME-RESOLVED OPTOACOUSTIC METHOD AND SYSTEM FOR NONINVASIVEMONITORING OF GLUCOSE AlexanderA. Oraevsky and AlexanderA. Karabutov, assignors to Board of Regents, the University of Texas System 11 June 2002 Class 600 Õ407; ﬁled 6 October 1999 Awideband optoacoustic transducer measures the spatial in-depth pro- ﬁle of optically-induced acoustic pressure transients in tissues in order to determine the laser-induced proﬁle of absorbed optical distribution. Thistype of technique can be applied to monitor glucose concentration in varioushuman or nonhuman tissues, cell cultures, solutions, or emulsions.—DRR6,379,306 43.80.Qf ULTRASOUND COLOR FLOW DISPLAY OPTIMIZATION BY ADJUSTING DYNAMICRANGE INCLUDING REMOTE SERVICES OVER ANETWORK Michael J. Washburn et al., assignors to General Electric Company 30 April 2002 Class 600 Õ454; ﬁled 27 December 1999 Values corresponding to color ﬂow signals from an ultrasonic Doppler imaging instrument are stored in a memory. A dynamic range compressionscheme based on an analysis of the signals in the memory is used to deter-mine a second set of values that are stored and used for display locally aswell as at a remote facility via communication over a network.—RCW 6,394,955 43.80.Sh DEVICE ATTACHABLE TO A THERAPEUTIC HEAD FOR ADJUSTABLY HOLDINGAN ULTRASOUND TRANSDUCER, ANDTHERAPEUTIC HEAD IN COMBINATION WITHSUCH A DEVICE Lucas Perlitz, assignor to Siemens Aktiengesellschaft 28 May 2002 Class 600 Õ439; ﬁled in Germany 1 February 1999 The device provides adjustability in holding an ultrasonic transducer and it can be attached at a therapeutic head that emits acoustic waves con-verging into a focus. The device has at least one element that can be swiv-eled about an axis. The ultrasonic transducer is at least indirectly attachableat the element such that its acoustic axis, as well as the swivel axis when thedevice is attached, proceed substantially through the focus of the therapeutichead.—DRR 6,394,956 43.80.Sh RF ABLATION AND ULTRASOUND CATHETER FOR CROSSING CHRONIC TOTALOCCLUSIONS Chandru V. Chandrasekaran et al., assignors to Scimed Life Systems, Incorporated 28 May 2002 Class 600 Õ439; ﬁled 29 February 2000 This catheter combines an ultrasound transducer and a rf ablation elec- trode. The ultrasound transducer transmits into and receives echos from ablood vessel. The echo signals are processed and used to produce an imageof the tissue surrounding the catheter. A driveshaft rotates the transducer toSOUNDINGS 2524 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:40yield a 360-degree view of the vessel wall.At the distal end of the driveshaft is an electrode that is coupled to a rf generator delivering rf energy forablating occluding material inside the vessel.—DRR 6,398,753 43.80.Sh ULTRASOUND ENHANCEMENT OF PERCUTANEOUS DRUG ABSORPTION David H. McDaniel, Virginia Beach, Virginia 4 June 2002 Class 604 Õ22; ﬁled 9 October 1998 This is a system for enhancing and improving the transcutaneous de- livery of topical chemicals or drugs. A disposable container contains a sub-stantially sterile unit dose of an active agent adapted for a single use in amedical instrument. The unit dose is formulated to enhance transport of theactive agent through mammalian skin when the active agent is applied toskin that is exposed to light and/or ultrasound.—DRR 6,384,516 43.80.Vj HEX PACKED TWO DIMENSIONAL ULTRASONIC TRANSDUCER ARRAYS John Douglas Fraser, assignor to ATL Ultrasound, Incorporated 7 May 2002 Class 310 Õ334; ﬁled 21 January 2000 These transducer arrays are comprised of elements closely packed in a hexagonal conﬁguration such as shown in the ﬁgure.—RCW 6,390,981 43.80.Vj ULTRASONIC SPATIAL COMPOUNDING WITH CURVED ARRAY SCANHEADS James R. Jago, assignor to Koninklijke Philips Electronics N.V. 21 May 2002 Class 600 Õ443; ﬁled 23 May 2000 Spatial compounding is accomplished by beam steering that depends on both the curvature of the array and electronic phasing in ways advanta-geous for beamforming and image registration coefﬁcients, uniformity in sampling, reduction of speckle, and achievement of a large compoundedarea.—RCW 6,390,984 43.80.Vj METHOD AND APPARATUS FOR LOCKING SAMPLE VOLUME ONTO MOVINGVESSEL IN PULSED DOPPLER ULTRASOUNDIMAGING Lihong Pan et al., assignors to GE Medical Systems Global Technology Company, LLC 21 May 2002 Class 600 Õ453; ﬁled 14 September 2000 A gate selecting a volume for Doppler analysis is locked onto the selected vessel by using pattern matching in images from successive framesprocessed in the space domain or the Fourier domain to determine howmuch a vessel in the image has translated and rotated from one frame to thenext.—RCW 6,394,967 43.80.Vj METHOD AND APPARATUS FOR DISPLAYING LUNG SOUNDS AND PERFORMINGDIAGNOSIS BASED ON LUNG SOUNDANALYSIS Raymond L. H. Murphy, Wellesley, Massachusetts 28 May 2002 Class 600 Õ586; ﬁled 30 October 2000 This lung sound diagnostic system contains a plurality of transducers that may be placed at various sites around the patient’s chest. The micro-phones are coupled to signal processing circuitry and A/D converters thatsupply digitized data to a computer system. A program in the computercollects and organizes the data and formats the data into a combinatorial display that can be shown on a monitor screen or printed out. The systemmay also include application programs for detecting and classifying abnor-mal sounds. An analysis program can compare selected criteria correspond-ing to the detected abnormal sounds with predeﬁned thresholds in order toprovide a likely diagnosis.—DRRSOUNDINGS 2525 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:406,396,931 43.80.Vj ELECTRONIC STETHOSCOPE WITH DIAGNOSTIC CAPABILITY Cicero H. Malilay, Los Angeles, California 28 May 2002 Class 381 Õ67; ﬁled 8 March 1999 This stethoscope is a self-contained, hand-held, electronic unit that includes a built-in chestpiece, speaker, and visual monitor. It includes amemory containing prerecorded heart and lung sounds along with a briefdescription of the malady producing the sounds so that a technician maycompare the actual sounds with the prerecorded sounds and obtain a sug-gested diagnosis on the monitor.—DRR 6,398,731 43.80.Vj METHOD FOR RECORDING ULTRASOUND IMAGES OF MOVING OBJECTS Bernard M. Mumm et al., assignors to Tomtec Imaging Systems GmbH 4 June 2002 Class 600 Õ437; ﬁled in Germany 25 July 1997 Images of a moving object are acquired with a moving transducer. Based on the amount of object movement, images are either not acquired ornot processed. Images not processed are omitted when the images are as-sembled and displayed.—RCW6,398,733 43.80.Vj MEDICAL ULTRASONIC IMAGING SYSTEM WITH ADAPTIVE MULTI-DIMENSIONAL BACK-END MAPPING Constantine Simopoulos et al., assignors to Acuson Corporation 4 June 2002 Class 600 Õ443; ﬁled 24 April 2000 This back-end mapping uses beamformed signals after detection. The processing consists of logarithmic compression followed by a mappingbased on the system noise level and a target display value of soft-tissueechoes. The mapping includes gain and dynamic range.—RCW 6,398,735 43.80.Vj DETECTING A RELATIVE LEVEL OF AN ULTRASOUNDIMAGINGCONTRASTAGENT David W. Clark, assignor to Koninklijke Philips Electronics N.V. 4 June 2002 Class 600 Õ458; ﬁled 7 March 2000 The relative level of an ultrasound contrast agent is detected by deter- mining a level from ultrasonic echoes produced in a region with contrastagent present, determining a level of echoes from the region after destruc-tion of the contrast agent, and forming the ratio of the two levels.—RCWSOUNDINGS 2526 J. Acoust. Soc. Am., Vol. 112, No. 6, December 2002 Reviews of Acoustical Patents Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 158.42.28.33 On: Tue, 02 Dec 2014 12:51:40
5.0051875.pdf	J. Appl. Phys. 129, 210905 (2021); https://doi.org/10.1063/5.0051875 129, 210905 © 2021 Author(s).Lab-on-a-chip based mechanical actuators and sensors for single-cell and organoid culture studies Cite as: J. Appl. Phys. 129, 210905 (2021); https://doi.org/10.1063/5.0051875 Submitted: 28 March 2021 . Accepted: 10 May 2021 . Published Online: 02 June 2021 Jaan Männik , Tetsuhiko F. Teshima , Bernhard Wolfrum , and Da Yang COLLECTIONS This paper was selected as Featured This paper was selected as Scilight ARTICLES YOU MAY BE INTERESTED IN Magnetism in curved geometries Journal of Applied Physics 129, 210902 (2021); https://doi.org/10.1063/5.0054025 Acoustic nonreciprocity Journal of Applied Physics 129, 210903 (2021); https://doi.org/10.1063/5.0050775 Special optical performance from single upconverting micro/nanoparticles Journal of Applied Physics 129, 210901 (2021); https://doi.org/10.1063/5.0052876Lab-on-a-chip based mechanical actuators and sensors for single-cell and organoid culture studies Cite as: J. Appl. Phys. 129, 210905 (2021); doi: 10.1063/5.0051875 View Online Export Citation CrossMar k Submitted: 28 March 2021 · Accepted: 10 May 2021 · Published Online: 2 June 2021 Jaan Männik,1,a) Tetsuhiko F. Teshima,2,3 Bernhard Wolfrum,2,3 and Da Yang1 AFFILIATIONS 1Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA 2Neuroelectronics, Department of Electrical and Computer Engineering, Technical University of Munich, 85748 Garching, Germany 3Medical and Health Informatics Laboratories, NTT Research Incorporated, Sunnyvale, California 94085, USA a)Author to whom correspondence should be addressed: jmannik@utk.edu ABSTRACT All living cells constantly experience and respond to mechanical stresses. The molecular networks that activate in cells in response to mechanical stimuli are yet not well-understood. Our limited knowledge stems partially from the lack of available tools that are capable ofexerting controlled mechanical stress to individual cells and at the same time observing their responses at subcellular to molecular resolu- tion. Several tools such as rheology setups, micropipetes, and magnetic tweezers have been used in the past. While allowing to quantify short-time viscoelastic responses, these setups are not suitable for long-term observations of cells and most of them have low throughput. Inthis Perspective, we discuss lab-on-a-chip platforms that have the potential to overcome these limitations. Our focus is on devices that applyshear, compressive, tensile, and confinement derived stresses to single cells and organoid cultures. We compare different design strategiesfor these devices and highlight their advantages, drawbacks, and future potential. While the majority of these devices are used for funda- mental research, some of them have potential applications in medical diagnostics and these applications are also discussed. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0051875 I. INTRODUCTION All living organisms interact with their environment. How these interactions alter the growth and the behavior of the organ- ism is a key question in biology. Frequently, the interactions between the organism and its surrounding environment can be analyzed in terms of continuum mechanics by coarse-graining the underlying molecular interactions. However, the mechanicalresponse of a biological system to an external force is by far more complex than the one encountered in non-living systems. Cells as continuum mechanical objects show responses to mechanical stress that are highly non-linear and history-dependent. Moreover, beyond mechanical responses cells surmount different biochemicalresponse mechanisms, referred to as mechanotransduction, that unfold over different time scales. Understanding the latter is an area of active research in the fields of cancer biology, immunology, developmental biology, and microbiology, among others. Studiesinvolving mechanical properties and responses of biological systems are increasingly considered a distinct branch of biology byitself, referred to as mechanobiology. 1In addition to providing a fundamental understanding of cellular behavior, the goal for manymechanobiological studies is to understand how diseases alter the mechanical properties of cells and tissues, and how an altered mechanical environment can trigger pathological processes. ThisPerspective discusses some of the emerging tools used in mecha-nobiology with a particular emphasis on new trends. Taking thebroad scope of the field, we will limit our coverage to devices and techniques that are used to study individual cells and organoid cul- tures. The latter mimics the responses of individual organs ortissues and consists typically of only a few types of cultivated cellsin much smaller numbers than present in an actual organism.Before settling on device aspects, we will first outline key concepts that have been addressed in mechanobiology research. We will then discuss the state-of-the-art tools in mechanobiology. Our mainJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210905 (2021); doi: 10.1063/5.0051875 129, 210905-1 Published under an exclusive license by AIP Publishingfocus will be on lab-on-a-chip approaches and we will highlight the benefits and limitations of such devices. We would like to empha- size that this article is not intended to be a review but as aPerspective that represents the authors ’own experience and inter- pretation of a diverse and extended body of literature. As a conse-quence, we focus on a limited number of techniques and applications without covering all developments in the field. II. MECHANOBIOLOGY AND MECHANICAL SIGNALING IN CELLS In broadest terms, the cellular responses to mechanical forces can be grouped as passive or active ( Fig. 1 ). The passive responses arise from short-time scale mechanical interactions occurring at atime scale less than about a second and are not accompanied bythe expenditure of chemical or electrochemical energy in the form of ATP, GTP, or membrane potential. During these interactions, cells behave largely as a viscoelastic material. This regime is themost commonly studied in biophysics experiments. 2However, cel- lular level responses unfold typically on longer time scales. The elastic part of the viscoelastic response involves typically the cell envelope, underlying cytoskeletal network, and internalorganelles. For the most part, these layered shell-like structureshave small bending rigidity but high areal stretch modulus. The compressibility of the liquid interior of a cell is low and corre- sponds approximately to the compressibility of water. As a result,cells deform after the application of step-like compression to themwith no change in their volume and small changes in their surfacearea at short time scales ( Fig. 1 ). If the cell survives such deforma- tion without rupture of its outer membrane layer(s), then the flow of water and solutes across the cell envelope becomes the dominantresponse. The flow lowers the stresses in the outer membranelayer(s). The flow occurs predominantly via different membranechannels. Some of these channels activate as a response to stress in the lipid bilayer membrane(s) surrounding the cells. This activation is highly non-linear. For small stresses, channels are closed butthey open above the threshold via a conformational change in thechannel proteins. In bacteria, these so-called mechanosensitivechannels have a size cutoff but otherwise, they are not selective to types of molecules that can pass through. 3The largest mechanosen- sitive channels in bacteria (MscL) have pore openings of about3 nm allowing passage of small proteins, ∼9 kDa in size. 4In cells of most known higher organisms (eukaryotes), mechanosensitive channels typically are selective for the type of ions/molecules that can pass. Commonly, these channels allow passage of specific ionssuch as K +,N a+,o rC a2+and are referred to as stretch-gated ion channels or mechanosensitive ion channels.5The opening of ion channels in response to increased tension in the membrane leads to changes in the membrane potential. A change in membrane poten- tial expends energy accumulated by metabolic processes of the cell,which categorizes this response as an active response ( Fig. 1 ). Stretch-activated ion channels are responsible for the initial depola-rization or hyperpolarization from a mechanical stimulus and are involved in the sensing of touch and hearing. Opening of these channels as a response to a mechanical stimulus can trigger propa-gation of action potential in excitable cells. Responses that occur on a time scale longer than about 1 s (rough order of magnitude estimate) are all driven by active, energy-consuming processes. The already mentioned passive pro- cesses are also present at these time scales but one can considerthem to result from the active responses to initial step-like pertur-bation. Among the long-time active responses, two different time regimes can be furthermore distinguished ( Fig. 1 ). On the time scale of seconds to minutes, the response is driven by protein – protein interactions. In eukaryotic cells, this leads to reorganizationof the actomyosin cortex and tubulin networks via processes thatconsume energy in the form of ATP and GTP, respectively. 6Part of cytoskeletal re-arrangements is driven by the influx of extracellular Ca+2via the opening of stretch-gated ion channels.7These cytoskel- etal re-arrangements lead to cell morphogenesis8and in some cell types drive cellular motility.9While the dynamical behavior of actin polymers and myosin motors as single entities is relatively well understood, their collective behavior in networks is much less clear. Understanding this behavior constitutes an exciting newdirection in soft matter physics known as active gel studies. 10 FIG. 1. Dominant cellular responses to applied step-like mechanical compres- sion of a cell at different time scales. The associated time scales represent approximate rankings and rough order of magnitude estimates. Differentresponses have significant temporal overlaps.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210905 (2021); doi: 10.1063/5.0051875 129, 210905-2 Published under an exclusive license by AIP PublishingFrom a minute to hour time scale, the cells respond to mechanical stimuli by re-arranging their transcriptional networks. Mechanical forces acting via transcriptional networks are thoughtto play a key role in the differentiation of stem cells, 11,12controlled cell death,13and activating tumor growth,14among others.15 Similar to eukaryotic cells, bacteria are sensing stresses in the cellenvelope and they activate specific transcription factors in responseto these stresses. 16In all domains of life, these system-level responses are yet poorly understood. The lack of understanding is related to the absence of suitable systems that are able to actuate these responses in a controlled manner. The shape of most bacteria, archaea, fungi, and plant cells is determined by their cell wall. The energy-driven response to exter- nal mechanical forces also involves remodeling of the meshworkmaking up the cell wall in these organisms. These responses unfold on a time scale at which the cells double their mass and cell wall area. The remodeling of the cell wall appears to occur as a directconsequence of stresses in the cell wall. It has been postulated thatthe stress acts on enzymes that synthesize the cell wall; in particularto lytic enzymes that cut covalent bonds in this macromolecularstructure before new polymer strands can be inserted. 17These decades-old predictions yet wait to be verified in experiments.III.“CONVENTIONAL ”ACTUATORS AND SENSORS FOR PROBING SHORT TIME-SCALE MECHANICAL RESPONSES A diverse set of tools has been developed over the years to probe the mechanical properties of cells and new techniques are emerging. The main focus of this Perspective lies on actuators andsensors that can be implemented in lab-on-a-chip devices. To putthese devices into Perspective, we will first discuss some of themost promising techniques to measure cellular forces and viscoelas- tic properties in general ( Table I ). We refer to these techniques as “conventional methods. ”Nevertheless, it should be stated that these techniques are also state-of-an-art and several of them haveemerged quite recently. Our discussion of this very broad materialis inevitably brief. In-depth reviews can be found in the references added to Table I . The techniques listed in Table I have been divided into three groups. The first group comprises techniques where a solid probecomes into direct contact with cells. These techniques includeAFM, parallel and rotating plate rheology setups, and micropipetes. The first three techniques allow determining complex elastic moduli of cells considering them as homogenous soft material. 18 However, the moduli determined by these techniques differ by TABLE I. Comparison of “conventional ”methods for mechanobiology. Viscoelastic property signifies the quantity typically extracted from experiments although other rheological functions can also be calculated. Force range for all techniques except FRET probes shows the force that the probe is capable of exerting but for FRET is the measurement range. Dynamic time scale indicates the typical time scale of the relaxation process that is probed. Technique Main viscoelastic property Force rangeDynamic time scaleLinear dimension of the probed regionMeas. through-put (cell/h) Reference AFM Young modulus Y=E(ω→0)10 pN –10μNm s –s1 0 n m –5μm5 –10 2and96 Parallel plate rheologyElastic modulus E(ω)1 0 p N –10μNm s –s Whole cell 5 –10 2 Rotating plate rheologyShear modulus G(ω) Limited by the adhesive properties of cellsms–s Whole cell <106(5–6h needed to prepare cells)2and97 MicropipeteaspirationStretch modulus of envelope, strength of cell-to-cell and cell-to-substrate attachments0.1 nN –1μNm s –min >1 μm5 –10 20 Magnetic tweezers Shear modulus G(ω) 0.1 –100 pN ms –min 1 –10μm 1000 2,21, 98and99 Particle tracking rheologyDiffusion coefficient →G(ω) N/A ms –h 0.01 –1μm1 0 –100 2and22 FRET probes Force, stress 1 –100 pN ms –min 1 nm to whole cell 10 –100 24 Brillouin scattering Spatial distribution of longitudinal modulus M(x,y)N/A ns >250 nm 10 –100 25and26 Optical stretchers Creep function J(t) <100 pN ms –s Whole cell 100 2and27 Laser ablation Creep function J(t) N/A ms –s >250 nm 5 –10 29Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210905 (2021); doi: 10.1063/5.0051875 129, 210905-3 Published under an exclusive license by AIP Publishingseveral orders of magnitude even when the same cell line in almost identical culturing conditions is used.18Part of the variation in the moduli can be explained by differences in the contact area betweenthe cell and the probe, loading rate, and stress level although arange of other factors could also play a role. The mechanicalresponse in the above three techniques arises from the mechanical properties of both the envelope and the internal structures of the cell. In contrast, response to micropipete aspiration arises predomi-nantly from the viscoelastic properties of cell membranes. 2,19,20 This technique can also be used to estimate the attachment strength of cells to a substrate and other cells. The second group includes techniques that track small parti- cles or beads either attached to the cell surface or freely diffusing inthe cytosol ( Table I ). These techniques include magnetic tweez- ers 2,21and particle tracking rheology.2,22We exclude here optical tweezers, which make use of beads attached to the cell surface, as this technique has been rarely used due to its rather limited force range and phototoxic effects. Extraction of complex moduli usingmagnetic tweezers and particle tracking rheology is less direct com-pared to the techniques in the first group in Table I . For magnetic tweezer experiments, the determination of shear moduli has relied on finite element modeling of a cell as an isotropic medium. 2,23In particle tracking rheology, the complex shear modulus is deter-mined from the mean square displacement of particles making useof the fluctuation-dissipation theorem. 18,19As an advantage, both techniques allow more cells to be studied in parallel compared to the techniques in the first group.18 The last group in Table I comprises techniques that use light to directly probe the mechanical properties of cells. Our listincludes FRET-based force sensors, Brillouin scattering microscopy, fiber-based optical stretchers, and laser ablation. As these techni- ques have emerged more recently and can be used in conjunctionwith lab-on-a-chip platforms, we will next also discuss briefly theconcepts of these techniques. In a FRET-based force sensor, a donor –acceptor pair of fluores- cent molecules is covalently attached to the opposite ends of a force- transducer molecule, which acts as a linear or non-linear spring. 24 The transducer molecule can be an unstructured peptide chain, aDNA hairpin, or a receptor –ligand pair that unbinds when the force exceeds a threshold value. In the latter two cases, the transducer acts as an ON –OFF switch that responds to a force exceeding a threshold value. Nevertheless, a continuous force response near the thresholdcan still be found when the signals from a larger number of molecu-lar sensors are averaged. The drawback of FRET and its extensions is the cumbersome labeling technique. Also, one has to be careful to ascertain that labeling itself and/or the applied light intensities donot alter cellular behavior. In Brillouin microscopy, no labeling of cells is required. In this technique, light inelastically scatters from the acoustic phonons. 25,26 Frequency shifts and linewidths of Brillouin Stokes and anti-Stokes peaks are measured as a function of spatial coordinates. Based onthese quantities, the real and imaginary parts of the longitudinalelastic modulus, M, can be calculated. However, this calculation requires knowledge of the local density and refractive index. Moreover, the longitudinal modulus is distinct from the elastic modulus ( E). As of now, Brillouin scattering measurements allow distinguishing a denser elastic environment from a less dense viscousenvironment within the cell. A further drawback is that Brillouin scattering is very weak; typically only one in 10 12of incident photons scatters.25As a result, measurements use high light intensity over long integration times causing phototoxic effects to cells. Optical fields can also be used for the mechanical actuation of cells. A widespread technique, referred to as the optical stretcher,27,28uses two aligned optical fibers that are separated by a small gap. Cells are trapped into this gap by scattering forces,which arise at the cell surface due to the difference in the refractiveindex. The same forces also stretch the cell. The setup typicallymeasures the creep function of the cell. 18It is less cytotoxic than the conventional optical tweezer because the light is not focused and thus the power density within the cells is lower. The techniquehas some potential as a tool for medical diagnostics, but itsthroughput is small compared to microfluidic approaches as will bediscussed later. The last technique on our list is laser ablation. In this largely qualitative technique, high-energy laser pulses are used to obliteratecytoskeletal elements, intracellular junctions, and whole cells inmulti-cellular cultures. 29The surroundings of the ablated structures usually undergo damped recoil. Creep function can be extracted from these measurements, but its magnitude remains unknown because the stress in the structure before and after ablation is notdirectly measured. Nevertheless, the viscoelastic properties of thestructures can be inferred from the speed and magnitude of recoil. These inferences typically rely on the mechanical modeling of a cell or multicellular network. Its usefulness in mapping out strains incells and cellular networks remains limited by its destructiveone-time readout. After a cell or network is probed, it is question-able if it could be measured again because the extent of the damage from the laser pulse remains unknown. IV. USING LAB-ON-A-CHIP DEVICES AS MECHANICAL ACTUATORS While the “conventional ”techniques discussed in Sec. IIIhave made great strides in understanding cells as soft matter systems, they are not well applicable to address questions on how cells and tissuesrespond to mechanical stimuli on longer, biologically relevant timescales for most active responses (cf. Fig. 1 ). For such studies, mea- surement setups are needed where actuation can be applied over hours to days and cellular responses can be observed in real-time. For meaningful interpretation of such cellular responses, cells needto be in a well-designed microenvironment during the experiment,which closely mimics their native environment. Otherwise, the cellu-lar response to a foreign microenvironment rather than to an intended controlled mechanical perturbation is studied. A critical requirement for the reproducibility of such studies is also that thenumber of cells or organoid cultures is sufficiently large. It is wellknown that the physical properties of individual cells have large var-iations within the cell population. 30Probing small numbers of them can lead to misinterpretation of results. Lab-on-a-chip-based devices offer possibilities to overcome the above limitations. First and foremost, lab-on-a-chip approachesallow the growth of cells in a steady and well-defined environment during the measurement. The techniques based on particle tracking and optical readout (groups 2 and 3 in Table I , respectively) can beJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210905 (2021); doi: 10.1063/5.0051875 129, 210905-4 Published under an exclusive license by AIP Publishingcarried out on lab-on-a-chip platforms to provide a cellular envi- ronment that better resembles the native one. Such transfer is not easily applicable to AFM/cantilevers and parallel plate rheologysetups but devices with similar functionalities can be constructedon the lab-on-a-chip platform as will be discussed later. Integrating“conventional ”techniques with lab-on-a-chip platforms thus enables mechanical stimulation and monitoring of cells over much longer periods. Possibilities of lab-on-a-chip platforms for mechanobiology go beyond mere cell culture improvement. Frequently, lab-on-a-chipplatforms enable the expansion of the measurement throughput from single cells to hundreds and thousands of cells in parallel. Lab-on-a-chip platforms also enable monitoring chemical and elec-trical cues using on-chip sensor arrays. 31–33It is feasible to combine mechanical and (electro)chemical sensing on the sameplatform in the future. In the following, we discuss the advantages and disadvantages of different device concepts that have been used to mechanicallystimulate and probe cells. We do not dwell on different fabricationaspects of these devices but point the reader to excellent discus-sions on the topic. 34,35 A. Confinement induced forces Cell growth, shape, division, and differentiation are all expected to be affected by mechanical forces. In particular, embry-onic, stem, and primary tumor cells are imposed to confinement and mechanical stress. Many pathogens penetrate host tissue by processes where a mechanical pushing force is important. 36The growth of single-celled organisms in the interior of colonies alsorequires overcoming mechanical stresses. It has been suggested thatmechanical confinement can be the main growth-limiting factor forE. coli colonies on agar plates instead of nutrient availability. 30 In all these cases, the force is internally generated by the cells. Forces arise as the cell pushes against surrounding cells and theextracellular environment that opposes its growth. The mechanical work performed by cells in such situations is frequently coupled to enzymatic activities that can remodel the extracellular environment.However, most microfluidic devices, especially those which arebased on inorganic materials, are sufficiently inert so that the lattertype of remodeling activity has a negligible effect. A fundamental question related to cell physiology is how much force a cell can generate by its growth before stalling. Thequestion about stall force has been extensively studied in individualmotor proteins in vitro conditions using optical and magnetic tweezers. However, the “conventional ”tools are not suitable to answer this question in cells because of the much higher forces involved. Using microfluidics, the stall force can be readily mea-sured using simple circular-shaped microfluidic chambers made ofsoft polydimethylsiloxane (PDMS), which is deformable by individ-ual cells [ Fig. 2(a) ]. It has been found that the fission yeast (and some other fungal) cells are capable of exerting up to about 10 μN forces to their surrounding environment. 37,36So far, it is not clear how these findings translate to the other types of cells. It can beexpected that the above approach will be more extensively used to map out growth-generated forces in different single-celled organ- isms. Making use of the same setup and fluorescent reporters, onewill also be able to better understand how forces affect biochemical pathways involved in cell growth. Microfluidic channels can also be used to study how cells respond to mechanical deformations that squeeze them in thedirection perpendicular to their main symmetry axes, which fre- quently determines their direction of movement. The migration of squeezed cells is relevant for metastatic cancer cells and immune FIG. 2. Confinement induced forces in microchannels and chambers. (a) Time-lapse images of growing fission yeast cell in a circular-shaped micro-chamber for 3 h. The deformable chamber is fabricated using soft PDMS with Young ’s modulus of 0.16 MPa. Scale bar, 10 μm. [Reproduced with permission from Minc et al., Curr. Biol. 19, 1096 –1101 (2009). Copyright 2009 Elsevier Ltd.] (b) Time-lapse images of an MDA-MB-231 breast cancer cell during migrationthrough a microfluidic constriction. The bright region corresponds to the cell nucleus. Scale bar, 5 μm. [Reproduced with permission from Denais et al. , Science 352, 353 –358 (2016). Copyright 2016 AAAS.] (c) Left: E. coli growing in slit-like channels with widths about half the size of the typical diameter of thecells (0.8 μm). Confined cells transform into large irregular shapes. Right: regular unconfined E. coli cell for comparison. [Reproduced from Männik et al ., Proc. Natl. Acad. Sci. U.S.A. 106, 14861 –14866 (2009). Copyright 2009 National Academy of Sciences.] (d) Schematics of fabrication of self-foldablemicrotubes. By removing the sacrificial layer, a strained bilayer is released and bent, resulting in a microtubular structure. (e) Encapsulation and arrangement of yeast cells in a self-foldable microtube. [Reproduced with permission fromMei et al ., Adv. Mater. 20, 4085 –4090 (2008). Copyright 2008 John Wiley and Sons, Ltd.]Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210905 (2021); doi: 10.1063/5.0051875 129, 210905-5 Published under an exclusive license by AIP Publishingcells. Many of these studies have made use of slit-like channels and pores. It has been found that the dimension of the nucleus signifi- cantly limits the migration of cancer cells. However, the cell cansqueeze through pores smaller than the linear dimension of thenucleus in its relaxed state [ Fig. 2(b) ]. In doing so, cell ’s nuclear envelope can rupture, which could result in genomic instability. 38 The latter refers to the fragmentation of DNA and its subsequentpartial degradation. Interestingly, the cells are usually able to repairthe damage to the nuclear envelope while the damage to the DNAcan be irreversible. The stiffness of the nuclear envelope has shownto be also a limiting factor for the migration of other cell types through confined spaces. 39,40The next anticipated frontier for these studies would be to combine cell migration studies with single-cellsequencing to quantitatively assess the damage to the genome.Furthermore, these studies can be combined with single-cell prote-omic analysis to understand how mechanical stress stimulates the excretion of enzymes and small molecules that are capable of remodeling extracellular matrix and cell-to-cell junctions. The migration of bacteria in narrow slit-like channels has also revealed unexpected behavior. It was found that E. coli in slit-likechannels with widths about half their unperturbed diameter trans- formed to very wide and irregularly shaped cells 41,42[Fig. 2(c) ]. Interestingly, most cellular processes including DNA replicationand mass growth only showed a minor perturbance in response tothe change in cell shape and size. 42On the other hand, the widen- ing of bacteria under uniaxial stress indicates that the synthesis of the E. coli cell wall is strongly influenced by the stresses in it. Similar conclusions were also drawn from different types of experi-ments in microfluidics chips. 43,44In the latter, E. coli cells grew out of short side channels into the main channel. A fluid flow in themain channel exerted stress effectively bending the cells in the flow direction. Both types of measurements indicate that regions of the cell wall, which are exposed to tensile stresses and strains, favor theaddition of new cell wall material. While these experiments estab-lished the phenomenological relationship between cell wall growthand mechanical stress, the underlying molecular mechanisms remain to be understood. For that end, one could tag enzymes involved in the cell wall synthesis with fluorescent fusions and/oruse recently developed fluorescent precursors for cell wall 45in imaging of cells that grow in microfluidic channels. In the above studies, cells have been strongly confined by one spatial direction. Studies are mostly making use of uniaxial instead of biaxial confinement because of the difficulty in fabricating arange of channels of different heights on a single device using con-ventional soft lithography methods. Some workarounds for this limitation have been demonstrated recently but these are applicable for large channels. 46Promising alternative to conventional methods is to use self-foldable tubes.47–49Their circular geometry mimics better in vivo constraints than the rectangular geometry of conven- tional microchannels. Self-foldable tubes form when a thin film with the tube mate- rial is released from a sacrificial layer by etching [ Fig. 2(d) ]. The radius of the self-foldable tube is controlled by the lattice mismatchelastic properties of two layers that form the wall of the tube. Awide range of tube materials can be used including inorganic mate- rials, cell-friendly polymers, 47or graphene.50Typically, the self- foldable tubes are not part of a fluidic circuitry, as usually encoun-tered in lab-on-a-chip devices. Nevertheless, different tubes can bearranged into a network using micropipetes. 47The cells can enter these channels on their own without any external force applied49 [Fig. 2(e) ] or encapsulated during the folding process.47As for 2D cases, the propagation of cells in the channels depends sensitivelyon the channel diameter. Below a certain diameter, which for HeLacells is about 8 μm, disruptions to cell divisions and genome insta- bility occur. 51,52The advantage of self-foldable tubes over conven- tional microfabricated channels is the flexibility to use a range ofdifferent materials as the channel walls. This will enable future sys-tematic studies on how cells respond to tight contact with differentsurface materials. Moreover, electrical, 50magnetic,53or optical sensing54capabilities in microchannels of self-foldable tubes have been demonstrated recently. These sensing modalities can be usedfor real-time measurements of encapsulated cells. 55A potential advantage of self-foldable tubes over conventional microfabricatedchannels is also a simpler determination of forces exerted by the cells on the channel walls. Thanks to the simple geometry of the tube, the average stress acting on the cells can be analytically calcu-lated from the deformation of the tube. 48,49For conventional FIG. 3. Fluidic and contact shear forces in microchannels. (a) T op: schematics for shear flow deformability cytometry where cells are flown from wider channelsto narrower ones. Bottom: in narrower channels, cells deform from spherical(not shown) to bullet-like shapes. [Reproduced with permission from Mietke et al ., Biophys. J. 109, 2023 –2036 (2015) Copyright 2017 Author(s), licensed under a Creative Commons Attribution (CC BY-NC-ND) license.] (b) T op: sche-matics for extensional flow deformability cytometry. Bottom: the extensional flowstretches the cell to prolate spheroid shape. [Reproduced from Gossett et al. , Proc. Natl. Acad. Sci. U.S.A. 109, 7630 –7635 (2012). Copyright 2012 National Academy of Sciences.] (c) Schematics of a constricting channel fabricatedwithin a silicon cantilever. The presence of the cell is detected by the change ofresonance frequency of the cantilever. [Reproduced from Byun et al., Proc. Natl. Acad. Sci. U.S.A. 110, 7580 –7585 (2013) Copyright 2013 National Academy of Sciences.]Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210905 (2021); doi: 10.1063/5.0051875 129, 210905-6 Published under an exclusive license by AIP Publishingmicrofluidic channels, such calculations are much more compli- cated and prone to uncertainties. B. Fluidic and solid shear forces in microchannels Cells in blood vessels, such as red and white blood cells and endothelial cells, experience shear flows and stresses.56Deformability of these cells is important for their proper function.57The deforma- tion of the cells in shear flow can help to distinguish diseased cells from normal cells in medical diagnostics. This so-called mechanicalphenotyping can be also used for other cell types, in particular, forthe detection of metastatic cancer cells. 58,59The underlying group of techniques has been referred to as deformability cytometry.60Note that the term deformability has a rather loose definition in the bio- medical field and is mostly used as a relative measure when compar-ing one cell type to another. In some measurements, it may refer tothe change in the aspect ratio of cells while in others, deformability is characterized by the time of how fast a cell can pass a constriction. Several microfluidic approaches for implementing deformabil- ity cytometry have emerged. In one class of devices, the cells arepushed through a constricting channel where one of the lateraldimensions is smaller than the relevant dimension of the cell. 61 This setup is similar to the one discussed in Sec. IV A but with the difference that the driving force to push cells through the constric-tion is created by advection and not by the cells themselves. In other types of devices, cells are not in contact with channel walls and shear arises solely from the flow profile [ Figs. 3(a) and3(b)]. Here, two approaches have been used more broadly. In the first approach, referred to as a shear flow deformability cytome-try, the fluid with the cell suspension passes from wider to anarrower channel [ Fig. 3(a) ]. 62Shear stress and pressure in the channel deform from a spherical cell into a bullet-like shape. This deformation is imaged by a high frame-rate camera and deform- ability is calculated based on a segmented image of the cell in realtime [ Fig. 3(a) , bottom]. In the second approach, cells are trapped and stretched by an extensional flow that forms in a four-way junc- tion of fluidic channels [ Fig. 3(b) ]. 63,64To achieve stretching, high Reynolds number ( Re.50) flows are required. In this approach too, cell deformation is determined from images acquired by ahigh-speed camera. The method has shown very promising resultsallowing to screen cancer cells at rates of 2000 cells/s and to distin- guish normal and malignant leukocytes with about 90% accuracy. 63 At the same time, it appears to be less sensitive in detecting pertur- bations of cell stiffness compared to shear flow deformabilitycytometry. 60Whether any of the above-mentioned approaches will be able to compete in their reliability with the existing detection methods for cancer markers in body fluids remains to be seen. Their current use in cell cytometry applications is also hindered bythe need for expensive high-resolution microscopes and high-speedcameras. The need for a high-speed camera and a microscope is cir- cumvented in an approach where a microfluidic channel with a constriction is fabricated within a cantilever. 60,65In this measure- ment scheme, the readout is obtained from the resonant frequencyshift of the cantilever. Deformability is characterized by the time the cells spend in the constricting region where it is in contact with channel walls [ Fig. 3(c) ]. The technique also allows the readout ofthe cell ’s buoyant mass. This precision measurement adds a high- quality feature, which in addition to deformability helps in distin- guishing diseased cells from normal ones. Adding further modali-ties to this setup would be potentially useful. For example, onecould perform impedance spectroscopy in this setup. The potentialfor higher-dimensional datasets and at the same time simpler readout may give the cantilever-based setup an advantage in medical diagnostics compared to the purely fluidic based deform-ability measurements. C. Cell stretchers and organs-on-a-chip devices Many types of animal cells, including myocytes, lung, and epi- thelial cells, experience tensile mechanical forces in their native environment. Studying these cells in a static environment is notguaranteed to represent their native responses to different stimuli/insults such as drugs and pathogens. These recognitions have led to the development of tools where cells are stretched via a flexible sub- strate to which these cells are attached. 66,67Many macro-scale systems exist that implement this concept; some of them beingcommercially available. These systems differ primarily by the actua-tion mechanism, which includes for example stepper motors, as well as electromagnetic and piezoelectric actuators. 66,67Different approaches allow different strain rates and magnitudes to beapplied. Here, we focus on a pneumatic actuation mechanism thatis compatible with the lab-on-a-chip platform. In our opinion, ithas the highest potential to yield transformative biological insights and to develop a broadly applicable screening platform for biomed- ical research. One of the pioneering works in this direction was byHuh et al. 68who developed a so-called lung-on-a-chip device that allowed periodic stretching of lung cells. Further work in this fieldhas produced a heart-on-a-chip, 69gut-on-a-chip,70and kidney-on-a-chip platforms.71The common design element for all these platforms is a stretchable microporous substrate made ofPDMS, which is cyclically stretched by applying vacuum to the sidechannels [ Fig. 4(a) ]. The stretchable porous substrate separates two parallel channels where different fluids or gases can be present. The substrate can also be stretched by fabricating a pressure channelunderneath it and applying overpressure to this channel so that thesubstrate bulges [ Fig. 4(b) ]. The first of the two designs is more dif- ficult to fabricate but allows the flowing of different liquids/gases on either side of the membrane. Also, different cell types can be grown on the two sides of the membrane. For example, in alung-on-a-chip platform, lung epithelial cells have been cultivatedon one side of the membrane and endothelial cells on the otherside. Epithelial cells are exposed to airflow in the top channel while endothelial cells are exposed to the flow of cell culture medium in the bottom channel as depicted in Fig. 4(a) . In some measure- ments, neutrophils (type of white blood cell) were also present inthe bottom channel. It was found that mechanical stimulationincreased recruitment of neutrophils to the epithelium via the endothelial layer upon exposure of the former to silica nanoparti- cles or to E. coli bacteria. In current studies, organ-on-a-chip devices have been used to study drug efficacies, toxicity, and interactions of tissue with patho- gens. 52Organ-on-a-chip devices hold great promise in reducing the number of in vivo studies with animal models. Using cultivatedJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210905 (2021); doi: 10.1063/5.0051875 129, 210905-7 Published under an exclusive license by AIP Publishinghuman cells, which closely mimic cells in a tissue, could provide a more representative model than a living animal for certain research questions and be ethically less problematic. Such a platform wouldalso allow a more detailed observation of individual cells as manyorgan-on-a-chip platforms are compatible with high-resolutionoptical imaging. Finally, these platforms facilitate the integration of on-chip electrical sensors that can detect metabolites and signaling molecules from cells in real time 72although this integration is yet in the early stages of development. A challenge for manyorgan-on-a-chip platforms is the difficulty to stably preserve cellcultures over one month. 52The latter is still a relatively short time in terms of (human) development. The factors that affect the viabil- ity of cells in lab-on-chip devices are not completely understood.They involve degradation of adhesive layers (typically fibronectinor collagen) and absorption of small hydrophobic molecules from culture medium into PDMS, 73which has been the dominant mate- rial for stretchable lab-on-a-chip devices. It has also been arguedthat PDMS has some cytotoxic effects. 74The toxicity has been assigned to PDMS oligomers that are not cross-linked and candiffuse out from the PDMS matrix. Finding easily processable bio- compatible alternatives to PDMS, such as stretchable hydrogels, 75is therefore desirable. Despite these shortcomings, we expect that incoming years organ-on-a-chip platforms will find more and moreadoption by pharmacological companies and that this adoptionwill help to speed up the lengthy and costly drug-screening process. The platforms will also allow relevant in basic research and help to elucidate how individual cells in a tissue respond to mechanicalforces. D. Compressive stresses Lab-on-a-chip devices allow the application of compressive stresses to groups or individual cells, as well as sub-cellular regions.A promising approach of applying compressive stresses to cells makes use of PDMS-based pneumatic microvalves. These devices were originally designed to turn on and off flow in microfluidicchannels. 76The microvalve consists of two perpendicular fluidic lines separated by a thin (tens of micrometers) elastomer mem- brane [ Figs. 5(a) and5(b)]. The application of hydrostatic pressure to one of the fluidic channels (control line) leads to the expansionof this channel and deflection of the membrane between the fluidiclines. If a cell is placed underneath the valve, a compressive force is applied to the cell [ Fig. 5(c) ]. Stresses of several hundreds of kPa can be applied to cells this way. Early attempts to use this approach employed valves where the fluid line had either a concave (semi-circular) or rectangular crosssection. A wide fluidic line (100 –200μm) with a concave cross section can be fully closed by the pressure in the control line. In principle, such valves can be used to study very large cells, such asneurons. However, this approach is not practical for smaller cells,including bacteria, because the possible range of uniaxial stress that can be applied is very limited. For higher applied stresses, the elas- tomer ceiling curves effectively embedding the cells. 77 Consequently, the cells are cut off from the media supply andbecome dehydrated as water and possibly some small molecules arepressed out. 78Both are highly undesired outcomes for most studies, although there are also useful applications.78The embed- ding of cells can be avoided in valves where the flow line has a rec-tangular cross section. However, this induces in addition to thecompressive force also a lateral force that pushes cells away fromthe center of the valve [ Fig. 5(d) ]. 79To overcome this shortcoming, a different design of the valve has emerged, featuring a protrusion in the ceiling of the flow line at the center of the valve [ Figs. 6(a) and6(b)].80–83This design, referred to as the microanvil, has been used to study both eukaryotic and prokaryotic cells. By growingaxons across the microvalve area, it was found that at loads higher than about 95 kPa, axons were instantaneously transected. 80 However, nearly half of these axons were able to regrow within about a 10 h period in the absence of exogenous stimulating factors[Fig. 6(c) ]. A similar approach was also used to study damage to vascular tissue. 83The approach holds much promise to better understand how different tissues respond to mechanical trauma at FIG. 4. Designs of pneumatic cell stretchers integrated into lab-on-a-chip platforms. (a) Schematics for a lung-on-a-chip. Lung epithelial and endotheli al cells are cultivated on different sides of a stretchable membrane. Vacuum suction is applied to side chambers to stretch the membrane. [Reproduced with permission from Hu het al. , Science 328, 1662 –1668 (2010). Copyright 2010 AAAS.] (b) Schematics of a cell stretching device in which overpressure is applied to a pneumatic channel underneath the c ells. [Reproduced with permission from Kamble et al. , Lab Chip 16, 3193 –3203 (2016). Copyright 2016 the Royal Society of Chemistry.]Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210905 (2021); doi: 10.1063/5.0051875 129, 210905-8 Published under an exclusive license by AIP Publishingthe single-cell level. It will also help to elucidate the molecular pathways involved in the immediate response and repair pathways. To study bacterial cells, the dimensions of the valve were scaled down about 20 times reaching lateral dimensions of about4×8 μm 2and a height of about 1 μm[Fig. 6(b) ]. Using this device, filamentous E. coli with a length of about 8 –10μm have been studied. Among other findings, the measurements have given new insights into how fast cytosolic water and small molecules can leave E. coli cells. The initial slow speed measurements showed loss of about 30% of the cytosolic volume upon compression in the 1-minrange. 81A more recent high-speed measurement revealed that the outflow of water occurs in a mere 50 ms [ Figs. 6(d) and 6(e)]. Interestingly, the outflow of water was reversible and also the influx occurred on a fast time scale ( ≈100 ms) [ Fig. 6(e) ]. The rapid outflow of water can be explained by the opening of mechanosensi-tive channels when there is a sudden increase in cytosolic pressureand cell membrane tension but these channels should be closed during the restoration of cell volume when there is no overpressure. It remains to be determined what mechanism allows cells to refilltheir volume rapidly. In future work, the dimensions of the valveshould be scaled down to accommodate regularly sized E. coli and other bacterial cells. Also, deformation of the anvil portion could be used to determine the actual stress applied to the cells.E. On-chip electromagnetic tweezers As mentioned earlier (cf. Table I ), experiments with magnetic tweezers are typically carried out at low sample numbers. Animplementation of electromagnetic actuator techniques within alab-on-a-chip device promises to open up the possibility for auto-mation and high-throughput experiments. 84Prominent methods to apply on-chip tweezers rely on magnetophoretic and dielectropho- retic particle actuation.85,86While the required magnetic fields to generate magnetophoresis are conventionally produced using exter-nal coils, the suitable structures can also be directly incorporatedinto the chip design, e.g., making use of a crossbar layout. 87 Driving specific loop-like current patterns through the crossbararray allows generating magnetic fields at specific locations of thechip in a time-dependent manner. The generated fields apply aforce on an induced (or a permanent) dipole and can be used to actuate multiple particles in different locations of the chip surface (Fig. 7 ). Since most biological cells show only negligible response to FIG. 6. Microanvil valve actuators. (a) A conceptual design of the microanvil. A protrusion (anvil) is fabricated in the ceiling of the valve that contacts the cell and compresses it. (b) A phase-contrast image of the microanvil and trappedE. coli cell. The scale bar is 2 μm. [Panels (a) and (b) are reproduced with per- mission from Yang et al ., Mol. Microbiol. 113, 1022 –1037 (2020). Copyright 2020 John Wiley and Sons, Ltd.) (c) Transection of the axon by a microanvil. 0 min: an axon (green) before transection, 20 min: the same axon right aftertransection. The later images show the regrowth of this axon. [Reproduced withpermission from Hosmane et al. , Lab Chip 11, 3888 –3895 (2011). Copyright 2011 the Royal Society of Chemistry.] (d) Fluorescence images of E. coli cells under rapid compression (left) and release (right) of the microvalve.Fluorescence originates from the HupA-mCherry label that stains bacterial DNA(the nucleoids). (e) Fluorescence intensity from the center of the same cell as a function of time. Panels (d) and (e) are authors ’unpublished results. FIG. 5. Pneumatic microvalves for cell squeezing experiments. (a) A basic design of the valve. Pneumatic valve forms at the region where flow and control lines overlap. (b) A photograph of a finished microfluidic chip showing the flow(blue) and the control (red) lines. (c) If a cell is placed in the valving region andexternal pressure is applied to the control line, then the cell will be squeezed. (d) Calculated profiles for the ceilings of the flow lines at pressures of 0, 2, and 4 bars in the control line. Due to a convex shape, the ceiling of a half-closedvalve exerts a lateral force (blue arrows) that tends to displace cells from thecenter of the channel. [Reproduced with permission from Yang et al ., J. Vac. Sci. T echnol., B 33, 06F202 (2015). Copyright 2015 AIP Publishing LLC.]Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210905 (2021); doi: 10.1063/5.0051875 129, 210905-9 Published under an exclusive license by AIP Publishingmoderate magnetic fields themselves, magnetic tags need to be attached to them.88,89While the generation of chip-based magneto- phoretic forces is intrinsically limited to 2D using the crossbararchitecture, it is possible to additionally generate dielectrophoreticforces with the same chip design. To this end, the DC sequences for the generation of the magnetic field are overlaid with an AC sequence, generating an inhomogeneous electrical field. Thisapproach allows a precise actuation in three dimensions using asingle chip without any moving parts or external components[Fig. 7(d) ]. 90Yet, care has to be taken to avoid the generation of excessive temperatures at the chip surface caused by resistive heating from the applied currents. Such temperature changes mightinterfere with the response of cells to mechanical stimuli. Futureapplications of on-chip electromagnetic tweezers for high-throughput mechanobiological cell experiments will thus critically depend on advanced chip designs, which allow the application of significant forces and minimize heat dissipation. This could enablethe precise and localized generation of tensile and compressiveforces to adherent cell networks in a parallel configuration usingarrays of individually trapped and actuated particles. V. CONCLUSIONS Cells actively respond to mechanical cues in their environ- ment. Understanding these responses at the molecular and cellular level is currently limited because of a lack of available tools to stim-ulate individual cells and cell networks in a controlled manner.Lab-on-a-chip platforms provide unique capabilities to study responses in conditions that closely match the cellular environment over extended periods. Parallelization that is inherent tomicrofabrication technology enables to drastically increase the mea- surement throughput. Despite these promises, many challenges remain that limit the use of such devices. An important issue is thecomplexity of fabrication and device handling. Devices that allowthe application of forces to individual cells in a controlled mannertypically require multilayer processes and the capability to align dif- ferent layers together. This is usually done manually and as such the process is time-consuming and prone to a high failure rate.Multi-layered devices can also be fabricated using 3D printing. 34 Although more reliable, 3D printing is also a serial process and asof now, its resolution is not sufficient to fabricate devices for smaller cell types such as bacteria. Material compatibility issues with long-term cell cultures remain also a concern. Making furtheruse of hydrogels and biopolymers instead of traditional PDMScould alleviate these issues. Recently, intensive efforts have beendirected at 3D printing of diverse materials for the fabrications of microfluidic devices, 91–95which could potentially be used in studies related to mechanotransduction. Further improvement is alsoneeded in determining the exact force and stress magnitudes thatmicrofluidic actuators apply to cells. As of now, most in situ force- sensing in lab-on-a-chip devices relies on rather complicated mechanical or fluid dynamic simulations. Furthermore, in many measurements force and stress magnitudes are not known and thedevice is just used to stimulate the cells. Miniaturized pressure andforce sensors that are integrated into microfluidic circuits could sig- nificantly improve the accuracy of the force readout. Different fluo- rescent probes represent a promising approach in this direction asmost lab-on-a-chip devices are used in conjunction with fluores-cent microscopy setups. The development of new integrated sensorarrays could also facilitate adding electrical and electrochemical detectors to the microfluidic chips that are capable of monitoring signaling molecules, cell metabolites, and secreted enzymes in a cellculture medium. These chemical signals form an integral part ofcellular response to a mechanical stimulus. Finally, to investigateorganoids, the lab-on-a-chip platform needs to extend from mostly 2D cell cultures to more realistic 3D systems. Including blood vessels within these cultures would be highly desired. Indeed, engi-neering cells so that they partly fabricate the microfluidic device ontheir own is a logical next step in the development of lab-on-a-chipplatforms. ACKNOWLEDGMENTS The authors thank Jaana Männik for useful discussions. A part of this research was conducted at the Center for NanophaseMaterials Sciences, which is sponsored at Oak Ridge NationalLaboratory by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy. This work has been supported by the US-Israel BSF Research Grant (No. 2017004)(J.M.), and the National Institutes of Health Award (No.R01GM127413) (J.M.). DATA AVAILABILITY The data that support the findings of this study are available within the article. FIG. 7. On-chip electromagnetic tweezers implemented by a crossbar array architecture. (a) Concept of attractive magnetic and repulsive dielectrophoretic forces that can be generated simultaneously by driving DC and AC currents,respectively, through the crossbar array. (b) Microscopic image of a single mag-netic particle captured at the center of a crossbar array. (c) and (d) Defined application of forces resulting in 3D actuation of a particle along defined trajecto- ries above the chip surface (c: microscopic top view at the levitation plane over-laid with the particle trajectory and d: 3D plot of the particle trajectory.The corresponding time is indicated by the color code). Scale bars in (b) and (c) are 15 μm. [Reproduced with permission from Rinklin et al ., Lab Chip 16, 4749 –4758 (2016). Copyright 2016 the Royal Society of Chemistry.]Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 210905 (2021); doi: 10.1063/5.0051875 129, 210905-10 Published under an exclusive license by AIP PublishingREFERENCES 1K. A. Jansen, D. M. Donato, H. 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5.0038778.pdf	Appl. Phys. Lett. 118, 182401 (2021); https://doi.org/10.1063/5.0038778 118, 182401 © 2021 Author(s).Two oscillation states in free/hard bilayered nano-pillars Cite as: Appl. Phys. Lett. 118, 182401 (2021); https://doi.org/10.1063/5.0038778 Submitted: 25 November 2020 . Accepted: 09 April 2021 . Published Online: 03 May 2021 X. Yuan , Z. Lu , Z. Zhang , M. Cheng , J. Liu , D. Wang , and R Xiong ARTICLES YOU MAY BE INTERESTED IN Materials, physics, and devices of spin–orbit torque effect Applied Physics Letters 118, 180401 (2021); https://doi.org/10.1063/5.0054652 Spin-orbit torques: Materials, physics, and devices Applied Physics Letters 118, 120502 (2021); https://doi.org/10.1063/5.0039147 Strain modulated quantum spin Hall effect in monolayer NiB Applied Physics Letters 118, 183101 (2021); https://doi.org/10.1063/5.0048423Two oscillation states in free/hard bilayered nano-pillars Cite as: Appl. Phys. Lett. 118, 182401 (2021); doi: 10.1063/5.0038778 Submitted: 25 November 2020 .Accepted: 9 April 2021 . Published Online: 3 May 2021 X.Yuan,1,2 Z.Lu,1,2,a) Z.Zhang,3M.Cheng,3J.Liu,1,2D.Wang,4and R Xiong3,a) AFFILIATIONS 1The State Key Laboratory of Refractories and Metallurgy, Wuhan University of Science and Technology, Wuhan 430081, China 2School of Materials and Metallurgy, Wuhan University of Science and Technology, Wuhan 430081, China 3Key Laboratory of Artiﬁcial Micro- and Nano-Structures of Ministry of Education, School of Physics and Technology, Wuhan University, Wuhan 430072, China 4College of Science, Wuhan University of Science and Technology, Wuhan 430065, China a)Authors to whom correspondence should be addressed: zludavid@live.com andxiongrui@whu.edu.cn ABSTRACT The magnetization oscillation driven by spin transfer torque (STT) in a nano-pillar composed of a large in-plane anisotropy ﬁxed layer and a soft free layer is studied. It is found that instead of frequency continuously changing with current as most nano-oscillators do, this kind of nano-oscillator can only oscillate in two stable states with speciﬁc frequencies. In each state, the frequency is almost invariant with current density. The oscillation state could be easily manipulated by the magnetization state of the free layer or an applied pulse magnetic ﬁeld as theworking current density is lower than a critical value ( J c). The critical current density and the frequency difference of the two states can be tuned by the saturation magnetization ( Ms) of the two layers and the anisotropy constant Kof ﬁxed layer. Phase-locked oscillation is obtained in a two-nanopillar system, suggesting that it may be possible to amplify the oscillation signal by building an array of this kind of nanopillar system. This kind of STT-based nano-oscillator may have various applications in the ﬁeld of spintronics. Published under license by AIP Publishing. https://doi.org/10.1063/5.0038778 Since the discovery of the spin transfer torque (STT) effect by Slonczewski1and Berger,2tremendous and continuous interest has been aroused in the study of current-driven spin dynamics in conﬁnedmagnetic structures due to the basic physics involved and the prospectfor technological applications. 3–7In a conﬁned magnetic structure such as a sandwich nano-pillar8or a nano-wire9–11as a proper direct current is passed through, the transfer of spin angular momentumfrom conduction electron to local spins drives a persistent precessionof local spins in the free layer or an oscillation of a pinned domain wall in a nano-wire. The magnetization oscillations in magnetic nano- structures due to the current-driven spin precession or domain wall(DW) oscillation open a new method for the generation of microwaveoscillations. The magnetic nano-structures can be utilized to buildnano-scale oscillators with the tunable frequency in the microwaverange, which has important applications in various areas such as wire-less communication 12a n dr e a dh e a d .13So far, two types of current- driven magnetic nano-oscillators have been most studied: one is theso-called spin torque nano-oscillator (STNO) based on a magnetic sand-wich structure. 14The other is the domain wall-based nano-oscillator (DWNO),6,9based on a magnetic nano-wire with pinning notches. It was reported that a ﬁxed DW spanning along a ferromagneticnano-pillar can be driven to rotate by a direct current.3,15In composite magnetic nano-structures composed of two magnetic materials withdifferent anisotropy directions, a DW is naturally formed and pinnedat the interface of two materials. Recently, interesting magnetizationoscillation behaviors induced by the rotation of the interface DW orspins were found in this kind of composite nano-structure, which isdependent on the geometry of the nano-structure and the relativedirection of anisotropies. In a composite nano-wire composed of asmall perpendicular anisotropy (PMA) region and a large in-plane anisotropy (IMA) region, when a current enters from PMA part into IMA part, transverse domain walls will be periodically emitted fromthe interface due to the rotation of the interface spins and propagatealong the nano-wire, leading to an oscillation of magnetization. 16In bilayered nano-pillars with a PMA ﬁxed layer and soft free layer, theDW that is naturally formed and ﬁxed at the interface will rotate sus-tainability as a proper current applied. Depending on the geometry ofthe free layer, the rotation of the DW will either lead to a precession ofa virtual vortex domain 17or induce a periodic motion of a transverse DW in the free layer.18Either behavior will also lead to a stable oscillation of the magnetization of the free layer. The magnetizationoscillation of composite nano-structures occurs without the aid of an Appl. Phys. Lett. 118, 182401 (2021); doi: 10.1063/5.0038778 118, 182401-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplexternal ﬁeld. Considering that they are easily fabricated and various interesting spin dynamic behaviors are found in them, such a kind ofcomposite nano-structure is of great interest for study. In a PMA/IMA composite nano-pillar, the naturally formed DW spans along the thickness, which is almost in the same direction of the stray ﬁeld of the hard layer. Therefore, as a current is applied, the DW rotates around the z-axis. The magnetization oscillation is induced by this kind of DW rotation. In an IMA (ﬁxed)/IMA (soft) composite nano-pillar, the stray ﬁeld of the ﬁxed layer is perpendicular to the DW spanning direction. The magnetization oscillation in this kind ofnano-pillar may show different features. In this study, the spin dynamics in a bilayered nano-pillar with a s o f tl a y e ra n da ni n - p l a n em a g n e t i z e dﬁ x e dl a y e ri ss t u d i e d .I ti sf o u n d that the magnetization of the free layer can oscillate in two states depending on the current density. In each state, the frequency is quitestable and almost independent of current density. There is a critical current ( J c) density at which the frequency is subject to a sudden change. When the current density is lower than Jc, oscillation states can be manipulated by the magnetizing direction of the free layer or a pulse magnetic ﬁeld along the z-axis. All simulations in this paper are performed using the OOMMF code19with the contribution of the STT effect being considered. The system considered in our study is a bilayered nano-pillar system com-posed of two magnetic layers with different easy axis directions. In this kind of bilayered system, although material is discontinuous along the thickness, the magnetization direction changes continuously evenaround the interface due to the exchange interaction. In this case, spin torque forms proposed by Zhang et al. are applicable to describe the magnetization dynamics. Therefore, to include the STT terms, the class, spinTEvolve, is applied. The magnetization dynamics of the sys-tem is, thus, described by the Landau—Lifshitz—Gilbert (LLG) equa-tion with two spin transfer terms, 20 ~dm dt¼/C0 j cj~Heff/C2~mþa~m/C2~dm dtþu/C1~m /C2~m/C2@~m @z/C18/C19 þb/C1u/C1~m/C2@~m @z; (1) with~mbeing the unit magnetization, cthe gyromagnetic constant, ~Heffthe effective ﬁeld including the contributions of the exchange ﬁeld, anisotropy ﬁeld, demagnetizing ﬁeld, and Zeeman ﬁeld, athe Gilbert damping constant, and bthe coefﬁcient of the nonadiabatic term. Here, uis deﬁned as u¼JPglB 2eM s,w i t h Jbeing the current density, Pthe polarization rate, gthe Land /C19ef a c t o r , lBthe Bohr magneton, e the electron charge, and Msthe saturation magnetization. In this study, the nano-pillars are composed of two materials: Permalloy and CoPtCr alloy are used. The length ( l), width ( w), and thickness (t) of each layer are 32 nm (along the y-axis), 20 nm (along thex-axis), and 32 nm (along the z-axis), respectively. The Permalloy layer is the free layer, while CoPtCr with strong magnetocrystallineanisotropy is used as the ﬁxed layer. The inset of Fig. 1(a) shows the FIG. 1. (a) Relaxed initial state of the nano-pillar; the inset shows the model of the bilayered nano-pillar. The direction of the current Jisþz. (b) The dependences of the oscil- lation frequency and amplitude on current density. The inset shows the angle between the rotation axis and the xy plane ( h) evaluated at different Jvalues. (c) The time dependences of three components of magnetic moments (m x,m y, and m z) when J¼10.4/C21011A/m2. (d) The time dependences of three components of magnetic moments (mx,m y, and m z) when J¼13.1/C21011A/m2.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 182401 (2021); doi: 10.1063/5.0038778 118, 182401-2 Published under license by AIP Publishingmodel of the nano-pillar. The parameters of Permalloy21are as fol- lows: a saturation magnetization of Ms-free 8.0/C2105A/m and an anisotropy constant of K 0. The saturation magnetization ( Ms-ﬁxed) and anisotropy constant K of CoPtCr22are set to 6.8 /C2105A/m and 4.0/C2105J/m3, respectively. For both layers, the magnetic stiffness constant A is ﬁxed at 1.0 /C210/C011J/m, the Gilbert damping constant a is set to 0.04,23and the non-adiabatic constant bis ﬁxed at 0.04. The unit cell size is set as 2 nm /C22n m/C22 nm, and the spin polarization is ﬁxed at 0.4. These cell dimensions are smaller than the exchange length lex(lex¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2A=l0M2 sp )24of the materials used in this study. In our study, a relaxed initial state of the nano-pillar as shown in Fig. 1(a) is used except for a special mention. To obtain the relaxed initial state, the magnetizations of the ﬁxed layers are set to theþx-direction ﬁrst, and then the whole system is allowed to relax. When a proper current is applied, the spins of the free layer begin to precess, leading to an oscillation of the magnetization of the nano- pillar with a constant frequency and amplitude. To investigate the tuning effect of the magnitude of applied cur- rent, the magnetic dynamics of the nano-pillar at different currentdensities ( J) are simulated. Figure 1(b) shows the dependences of the oscillation frequency and amplitude on current density. The oscillation amplitude is deﬁned as mymax/C0mymin 2,w i t h mymax/mymin being the maximum/minimum value of the ycomponent of magnetization. The current density ranges from 5.53 /C21011A/m2to 13.83 /C21011A/m2. Below 5.53 /C21011A/m2, oscillation cannot be established, while over 13.83 /C21011A/m2, the oscillation becomes unstable. Two important features are observed: (1) there is a critical current density ( Jc)o f 11.75 /C21011A/m2, under which the frequency and amplitude are sub- ject to a sudden change. (2) On both sides of Jc,t h ef r e q u e n c ya n d amplitude do not show a signiﬁcant dependence on current density.Usually, in a conﬁned nano-structure, the oscillation frequency andamplitude highly depend on applied current due to the change in theSTT. 3In our soft/hard bilayer, the nano-pillar shows completely differ- ent behaviors. It seems that there exist two stable oscillation states, each having its own frequency and amplitude. We may name the state with lower frequency a low-frequency state (L state), and the one withhigher frequency a high-frequency state (H state). Before the currentdensity reaches the critical value for state switching, the frequency andamplitude are almost unvaried in a fairly larger range of current densi- ties. To explore the reasons for two states with different frequency, the spin conﬁgurations during one oscillation period in either state arestudies (the results are not shown). It is found that in the L state, thespins rotate around an out-of-plane axis, while in the H state, the spinsrotates around the x- a x i s .T h et i m ed e p e n d e n c e so ft h r e ec o m p o n e n t s of magnetic moments ( m x,my,a n d mz)s h o w ni n Figs. 1(c) and1(d) can clearly illustrate the different precession behaviors of the two states. In the L state ( J¼10.4/C21011A/m2), the mean values and oscil- lation amplitudes of mx,my,a n d mzare different, suggesting that the spins rotate around an out-of-plane axis. However, in the H state(J¼13.1/C210 11A/m2), the mean values of myandmzare almost equal to 0, and the oscillation amplitudes of myandmzare close, indicating that the rotation axis is almost along the x-axis. Based on the mean values and amplitudes of mx,my,a n d mz, the angle between the rota- tion axis and the xyplane ( h)i se v a l u a t e du n d e rd i f f e r e n t Jvalues [shown in the inset of Fig. 1(b) ]. When the Jvalue is lower than 11.75 /C21011A/m2,t h e hvalue decreases slowly from 50/C14to 45/C14at beginning and becomes almost unvaried as Jis higher than8/C21011A/m2. A sudden drop of hfrom 45/C14to 0/C14occurs at J¼11.75 /C21011A/m2, beyond which hdoes not change much with the increase in current density. Based on the results shown above, the spins in the free layer can perform precession around two axes—one is 45–50/C14from the xyplane and the other is in the xyplane (close to the x-axis). The oscillation fre- quency is determined by the precession frequency of the spins. Whenthe spins precess around the x-axis, the effective ﬁeld is larger because the stray ﬁeld of the ﬁxed layer is along the x-direction. As a result, the precession frequency of the spins is higher and the oscillation of the magnetic moment is in the H state. On either side of the critical cur- rent, the change in current density only slightly changes the effective ﬁeld. Since the precession frequency is mainly determined by the pre- cession torque M/C2H eff,at i n yc h a n g ei n Heffwill only lead to a tiny change in frequency. It is noted that this kind of two-state oscillation only occurs in nano-pillars with the free layer having the similar value of length and thickness. If the thickness and length are signiﬁcantly different, only one stable oscillation state can be obtained. We ﬁnd that when t¼40 nm and l¼32 nm, only the L state can be observed, while at t¼32 nm and l¼40 nm, only the H state can be found. This phenom- enon is understandable because the extra anisotropy ﬁeld induced by the dimension difference changes the direction of the effective ﬁeld and restrains the occurrence of one oscillation state. From above, we learn that in the studied nano-pillar, two differ- ent oscillation states can occur at different current densities due to the change in the effective ﬁeld direction. As a DC density is applied along thez- a x i s ,t h es p i n sw o u l db ec o m p r e s s e dt o w a r dt h e xyplane due to the STT effect.14The change in the spin direction will lead to a slight change in the effective ﬁeld due to the small change in the demagnetiz- ing ﬁeld and exchange ﬁeld. As a result, the precession axis gradually tilts toward xyas the current density increases. In this period, the oscil- lation is in the L state. When the precession axis reaches some critical angle, a further increase in the current density will make the precession axis sudden jump to the xyplane. The oscillation, thus, enters the H state. Since the direction of the precession axis depends on the spin directions, it may be possible to manipulate the oscillation state by set- ting the initial magnetization direction of the free layer. To prove this, the magnetization of the free layer is ﬁrst set to the z-,y-o rx-direction. For comparison, an unmagnetized free layer is also considered. A dc current with a density of 10.4 /C21011A/m2or 13.1 /C21011A/m2is then applied. The former value is lower, and the latter one is higher than Jc, as shown in Fig. 1(b) .Figure 2 exhibits the dependences of oscillation frequency and amplitude on the initial magnetization direction at dif- ferent current densities. When J<Jc, the oscillation state strongly depends on the initial state: when the free layer is unmagnetized or magnetized to the z-direction, the magnetization oscillates in the L state; as it is magnetized in advance to the x-o r y-direction, H state oscillation occurs. If J>Jc, no matter what the initial state is, only H state oscillation happens. These results suggest that L and H states are two stable oscillation states as J<Jc. The selection between the states can be easily achieved by manipulating the magnetization direction of t h ef r e el a y e r . At proper current density, the oscillation state depends on the effective ﬁeld direction, which can be manipulated by the spin direc- tions in the free layer. The spin directions can be instantaneously changed by an applied pulse magnetic ﬁeld. Therefore, it is possible toApplied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 182401 (2021); doi: 10.1063/5.0038778 118, 182401-3 Published under license by AIP Publishingdynamically switch between two states using a pulse magnetic ﬁeld. To achieve this, a pulse ﬁeld with a magnitude of 8 mT and a duration of4 n si sa p p l i e da l o n gt h e þz-o r– z-direction during the oscillation. Figure 3 shows the time dependences of oscillation frequency and amplitude. There are gray regions. These regions are the transitionperiod during which a pulse ﬁeld applied, and the frequency andamplitude change with time. In this study, the current density is kept at 10.4 /C210 11A/m2. Obviously, after each transition period, the oscil- lation switches from the state to the other state. A pulse ﬁeld along the–z-direction can change the oscillation from the L state to the H state, while, a þz pulse ﬁeld can switch the oscillation from the H back to L state. Therefore, by applying a proper pulse magnetic ﬁeld, it is veryconvenient to accomplish the conversion between two oscillationstates. Based on the above study, this kind of bilayered nano-pillar can oscillate in two stable states. In each state, the frequency and amplitude will not vary signiﬁcantly with the change in current density. Therefore, the oscillation frequency will have good tolerance for thecurrent deviation. Moreover, when working at a current density lower than J c, this kind of nano-pillar can output two different stablefrequencies, which can be manipulated by the initial magnetization or a proper pulse magnetic ﬁeld. Therefore, this kind of nano-pillar may have important applications in building nano-oscillators with two con- stant frequencies. For application consideration, low working currentdensity is desirable. Since the working current density is supposed tobe lower than J c, to achieve low working current, we need to diminish Jc. The nano-pillar can output two kinds of frequency—one is low fre- quency and the other is high frequency. It may also be important toenlarge the difference between the frequencies ( Df) for better applica- tion. For the purpose of lowering working current density and enhanc-ing the frequency difference, the dependence of J candDfon the magnetic properties of the free layer and ﬁxed layer is studied and shown in Fig. 4 . It is found that for the ﬁxed layer, high K and low Ms-ﬁxed are beneﬁcial for obtaining low Jcand high Df;a st ot h ef r e e layer, Jcincreases while Dfdecreases fast with the increase in Ms-free due to the strong dependence of STT on Ms-free. Therefore, to reach low Jcand high Df, the ﬁxed layer needs to use material with high FIG. 2. The dependences of oscillation frequency and amplitude on the initial mag- netization direction at different current densities. FIG. 3. A pulse ﬁeld with a magnitude of 8 mT and a duration of 4 ns is applied along the þz or –z direction during the oscillation. The regions are transition period during which a pulse ﬁeld applied, and the frequency and amplitude change withtime. FIG. 4. (a) The dependence of JcandDfon the Ms of the free layer. (b) The dependence of JcandDfon the Msof the ﬁxed layer. (c) The dependence of Jc andDfonKof the ﬁxed layer.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 182401 (2021); doi: 10.1063/5.0038778 118, 182401-4 Published under license by AIP Publishinganisotropy but low saturation magnetization, while material with low saturation magnetization should be applied to the free layer. One example can illustrate how signiﬁcantly JcandDfcan change by the optimization of the materials for the free layer and ﬁxed layer. If theﬁxed layer with K ¼6/C210 5J/m3and Ms-ﬁxed¼5.0/C2105A/m and free layer with Ms-free¼7.0/C2105A/m, JcandDfare 5.75 /C21011A/m2 and 2.67 GHz, respectively, which are in strong contrast to 11.75 A/m2 and 0.6 GHz for the nano-pillar composed of permalloy and CoPtCr. Although only effects of the material parameters are considered in ourstudy, J candDfcan also be tuned by the dimensions of the layers. Lower Jcand larger Dfmay be achievable by the additional geometry optimization. For nano-oscillator applications, the strength of the output signal is an important issue to be considered. An effective way forenlarging the signal is to build an array of nano-pillars whose oscil-lations are phase-locked. To investigate the feasibility of phase- locked oscillation among nano-pillars, two nano-pillars separated by different space are studied. It is found that when they are sepa-rated by 44 nm, the oscillations of them are coupled and synchro-nous oscillation can be achieved (results are not shown). Theoscillation amplitude of the two nano-pillar system is about 1.5 times of a single nano-pillar. Moreover, compared with its single nano-pillar counterpart, the two nano-pillar system has larger Df but lower J cdue to the inter-pillar coupling. This ﬁnding suggests that it is possible to achieve phase-locked oscillation in an array ofnano-pillars and greatly enhance the output signal. In summary, the spin dynamic behavior of a nano-pillar com- posed of an IMA ﬁxed layer and a soft free layer is studied. Two oscil-lation states with different frequencies are observed in this kind ofnano-pillar depending on the current density: below J c,t h em a g n e t i z a - tion oscillates in a low-frequency state; otherwise, it oscillates in a high-frequency state. The frequency of each state is quite stable andalmost independent of current density. J cand the frequency difference between states can be tuned by Msof the free layer and Msand K of the ﬁxed layer. When working at a current density just below Jc,t h e oscillation state can be manipulated by the magnetized direction of the free layer and a pulse magnetic ﬁeld along the z-axis. We ﬁnd that the oscillation mode could be changed by adjusting the initial magnetiza-tion states when the current density is located at the lower frequency.Particularly, low and high frequency states could be switched at will through a pulse magnetic ﬁeld. Msof two layers and K of the ﬁxed layer all have effects on J cand oscillation frequency. In a two-nanopillar system, phase-lock oscillation can be obtained, suggesting it may bepossible to enhance the oscillation amplitude by building an array ofthis kind of nanopillars. This kind of nanopillar can be utilized to fabri- cate STT-based nano-oscillators for different applications.The authors would like to acknowledge the ﬁnancial support from the National Natural Science Foundation of China (Nos.51871170 and 11774270). DATA AVAILABILITY The data that support the ﬁndings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 2L. Berger, Phys. Rev. B 54, 9353 (1996). 3M. Franchin, T. Fischbacher, G. Bordignon, P. de Groot, and H. Fangohr, Phys. Rev. B 78, 054447 (2008). 4R. A. van Mourik, T. Phung, S. S. P. Parkin, and B. Koopmans, Phys. Rev. B 93, 014435 (2016). 5Z. Yu, Y. Zhang, Z. Zhang, M. Cheng, Z. Lu, X. Yang, J. Shi, and R. Xiong, Nanotechnology 29, 175404 (2018). 6M. Cheng, X. Yuan, S. Li, C. Chen, Z. Zhang, Z. Yu, Y. Liu, Z. Lu, and R. Xiong, Nanotechnology 31, 235201 (2020). 7R. Li, Z. Yu, Z. Zhang, Y. Shao, X. Wang, G. Finocchio, Z. Lu, R. 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5.0040874.pdf	Appl. Phys. Lett. 118, 122401 (2021); https://doi.org/10.1063/5.0040874 118, 122401 © 2021 Author(s).Spin–torque dynamics for noise reduction in vortex-based sensors Cite as: Appl. Phys. Lett. 118, 122401 (2021); https://doi.org/10.1063/5.0040874 Submitted: 16 December 2020 . Accepted: 06 March 2021 . Published Online: 22 March 2021 Mafalda Jotta Garcia , Julien Moulin , Steffen Wittrock , Sumito Tsunegi , Kay Yakushiji , Akio Fukushima , Hitoshi Kubota , Shinji Yuasa , Ursula Ebels , Myriam Pannetier-Lecoeur , Claude Fermon , Romain Lebrun , Paolo Bortolotti , Aurélie Solignac , and Vincent Cros COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Spin-orbit torques: Materials, physics, and devices Applied Physics Letters 118, 120502 (2021); https://doi.org/10.1063/5.0039147 Impact of oxygen on band structure at the Ni/GaN interface revealed by hard X-ray photoelectron spectroscopy Applied Physics Letters 118, 121603 (2021); https://doi.org/10.1063/5.0033165 Tuning the linear field range of tunnel magnetoresistive sensor with MgO capping in perpendicular pinned double-interface CoFeB/MgO structure Applied Physics Letters 118, 122402 (2021); https://doi.org/10.1063/5.0041170Spin–torque dynamics for noise reduction in vortex-based sensors Cite as: Appl. Phys. Lett. 118, 122401 (2021); doi: 10.1063/5.0040874 Submitted: 16 December 2020 .Accepted: 6 March 2021 . Published Online: 22 March 2021 Mafalda Jotta Garcia,1,a) Julien Moulin,2Steffen Wittrock,1 Sumito Tsunegi,3 KayYakushiji,3 Akio Fukushima,3 Hitoshi Kubota,3 Shinji Yuasa,3 Ursula Ebels,4Myriam Pannetier-Lecoeur,2 Claude Fermon,2Romain Lebrun,1 Paolo Bortolotti,1Aur/C19elieSolignac,2and Vincent Cros1 AFFILIATIONS 1Unit /C19e Mixte de Physique, CNRS, Thales, Universit /C19e Paris-Saclay, 91767 Palaiseau, France 2SPEC, CEA, CNRS, Universit /C19e Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France 3National Institute of Advanced Industrial Science and Technology, Research Center for Emerging Computing Technologies, Tsukuba, Ibaraki 305-8568, Japan 4Univ. Grenoble Alpes, CEA/IRIG, CNRS, GINP, SPINTEC, 38054 Grenoble, France a)Author to whom correspondence should be addressed: mafalda.jotta@cnrs-thales.fr ABSTRACT The performance of magnetoresistive sensors is today mainly limited by their 1/f low-frequency noise. Here, we study this noise component in vortex-based TMR sensors. We compare the noise level in different magnetization conﬁgurations of the device, i.e., vortex state or uniform par- allel or antiparallel states. We ﬁnd that the vortex state is at least an order of magnitude noisier than the uniform states. Nevertheless, by activat- ing the spin-transfer-induced dynamics of the vortex conﬁguration, we observe a reduction of the 1/f noise, close to the values measured in theAP state, as the vortex core has a lower probability of pinning into defect sites. Additionally, by driving the dynamics of the vortex core by anon-resonant rf ﬁeld or current, we demonstrate that the 1/f noise can be further decreased. The ability to reduce the 1/f low-frequency noisein vortex-based devices by leveraging their spin-transfer dynamics thus enhances their applicability in the magnetic sensors’ landscape. Published under license by AIP Publishing. https://doi.org/10.1063/5.0040874 Magnetoresistive ﬁeld sensors have a wide range of uses, such as in biomedical applications, 1the automotive industry,2robotics,3and smart city technologies like power-grid monitoring4or navigation.5 Figures of merit like detectivity, sensitivity, and spatial resolution areused to evaluate the performance of such sensors. 6,7At low- frequencies, the 1/f noise component is dominant and is, in fact,responsible for limiting the device’s detectivity and, consequently, itsperformance. 8,9Tackling this limitation brings about an active research effort to reduce this noise component.10,11 Vortex-based devices, in which the free layer exhibits a vortex magnetization distribution in its equilibrium state, are promising mag- netic ﬁeld sensors due to their large linear detection range12and the fact that they show practically no hysteresis in this range. In addition,these devices are often considered as model systems for the study ofmagnetization dynamics. In this study, we focus on the investigation of the 1/f noise in a particular type of magnetic sensor based on a vor- tex magnetic conﬁguration integrated in a magnetic tunnel junction(MTJ) spin torque nano-oscillator (STNO). STNOs present very goodrf characteristics for future radio frequency (rf) devices andapplications, 13such as rf generation,14,15detection,16,17or neuromor- phic computing.18,19While the use of vortex-based STNOs for applica- tions such as these referred here has been largely studied, they arenewcomers in the magnetic sensor’s landscape. H e r e ,w es t u d yt h e1 / fl o w - f r e q u e n c yn o i s ei nv o r t e x - b a s e d STNOs, in the ﬁrst instance, to assess their performance as magneticﬁeld sensors. While there have been studies regarding the noise prop-erties in the low offset frequency regime in the dynamical modes ofthese devices 15,20(pertaining to the emitted rf signal), their 1/f low- frequency noise, related to the resistance ﬂuctuations, is largelyunstudied. Ultimately, we provide some solutions relying on theSTNO’s functionalities to decrease the devices’ 1/f noise as a means toimprove their performance as sensors. The studied magnetic tunnel junction (MTJ) stack is composed of (Si/SiO 2) substrate/buffer layer/synthetic antiferromagnet (SAF)/ MgO (1)/FeB (6)/MgO (1)/capping layers (thickness in nanometers).The pinned SAF layer is a PtMn (15)/CoFe 29(2.5)/Ru (0.86)/CoFeB (1.6)/CoFe 30(2.5) multilayer. The free layer with a magnetic vortex as the ground state is the FeB layer, with a diameter of 350 nm. The Appl. Phys. Lett. 118, 122401 (2021); doi: 10.1063/5.0040874 118, 122401-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldesign of the studied MTJ stack optimizes the characteristics that are central to the operation of STNOs. Speciﬁcally, the use of FeB for the free layer has been shown to improve the sample’s rf characteristics, namely, its emission power and Q factor.21Additionally, the use of MgO for the capping layer leads to a decrease in the sample’s Gilbert damping. The sample has a tunneling magnetoresistance (TMR) of 85% and an average resistance R 0¼60 Ohm. An inductive line sits 300 nm above the magnetic tunnel junction. InFig. 1(a) ,w ed e s c r i b et h em e a s u r e m e n tc i r c u i tt h a th a sb e e n designed to allow the simultaneous study of the magnetization dynam-ics, in the hundreds of MHz range, and the low-frequency noise of thedevice. The high-frequency component of the circuit consists of aspectrum analyzer and an rf power source. The low-frequency noisemeasurements were performed by biasing the STNO through a bal-anced Wheatstone bridge using a dc current source. The role of theWheatstone bridge is to allow the precise measurement of small volt-age ﬂuctuations of the device. The output signal is pre-ampliﬁed by anINA103 ampliﬁer, followed by a second ampliﬁcation and ﬁlteringchain, reaching a total gain of about 10 3. The output temporal signal is then acquired by a 16-bit acquisition card. A Fast Fourier Transform(FFT) is performed on the measured signal in order to obtain the noisespectral density, S V.Noise spectral density (NSD) curves typically have a low-frequency 1 =fcomponent, thermal white noise, and lorentzian random telegraph noise (RTN).22,23Each noise curve was obtained from averaging over 20 acquisitions, and its analysis is done by ﬁtting the different noise components in the range between 1 and 5000 Hz.Here, we are interested in the 1/f noise component.The Hooge parameter, a, is a commonly used phenomenological parameter 24used to compare the 1 =fnormalized noise level of differ- ent devices with the same RA product. This parameter is extractedfrom the ﬁtting of the experimental 1/f noise spectral density compo- nent, S1=f V, using the following equation: S1=f V¼aV2 Af; (1) where Vis the average voltage of the device during each measurement, Athe device’s surface area, and fthe frequency. The STNO device is placed between the two poles of an electro- magnet. We position it at an angle such that the applied ﬁeld has both in-plane and out-of-plane components, respectively, HIPand HOOP. The vortex magnetization distribution in the studied STNOs is charac- terized by two parameters, its polarity (P), which is the direction of the vortex’ core magnetization, and its chirality (C), which is the sense ofthe rotation of the magnetization in the vortex’ body. Vortex-based STNOs present four possible polarity/chirality magnetic conﬁgura- tions. The polarity ( 6P) of the vortex can be set through the applica- tion of a large out-of-plane magnetic ﬁeld, around 6700 mT. In our experiments, the sign of the out-of-plane ﬁeld determines the vortexpolarity. The vortex chirality ( 6C) can be set through the injection of a large dc current, around 65 mA. The direction of the ortho-radial Oersted ﬁeld generated by the current, which itself depends on the current sign, determines the chirality. When an in-plane magnetic ﬁeld is applied, the vortex core is dis- placed from the disk’s center perpendicularly to the applied ﬁeld. 25For FIG. 1. (a) Schematic of the measurement setup. (b) Evolution of the STNO device’s resistance with the applied ﬁeld in-plane component, for a –2.0 mA bias curr ent, in the negative chirality conﬁguration of the vortex. [(c) and (d)] Noise level at low frequency (1 Hz–5 kHz), represented by the Hooge Parameter, as a functi on of the applied ﬁeld in- plane component, swept from the anti-parallel (AP) state to the parallel state (P) (in orange) and vice versa (in blue), for the (c) positive and (d) neg ative chirality.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 122401 (2021); doi: 10.1063/5.0040874 118, 122401-2 Published under license by AIP Publishinga large enough HIP, the vortex core reaches the MTJ’s edge (also called annihilation ﬁeld, HA) and the disk’s magnetization becomes uniform, aligning itself with the applied ﬁeld. In the case where the uniform freelayer’s magnetization follows the same direction as the ﬁxed reference layer (SAF)—parallel (P) state—the STNO resistance is the lowest due to the magnetoresistance effect. 26,27Inversely, when the free and ﬁxed layers’ magnetization directions oppose each other, the device is in its highest resistance conﬁguration—anti-parallel (AP) state. By decreas-ingH IP, the magnetic vortex is recovered at the nucleation ﬁeld, HN. A sc a nb es e e ni n Fig. 1(b) , when the device is in the AP (P) state and the applied magnetic ﬁeld is decreased (increased), there is renuclea-tion of the vortex core in the -C þP (-C-P) conﬁguration. In order to compare the low-frequency noise in the different magnetic states, the Hooge parameter is determined experimentally,for the positive ( I dc¼þ2:0m A )a n dn e g a t i v e( Idc¼/C02:0 mA) chi- rality conﬁgurations of the vortex, at different magnetic ﬁeld values between the P and AP states, passing through the vortex state, and vice versa [see Figs. 1(c) and1(d)]. Note that for such Idcvalues, the STNOs are still in the so-called subcritical regime, meaning that thespin-transfer torques are small enough not to generate a sustained dynamical state of the vortex core. For each magnetic conﬁguration, we calculate an average Hooge parameter value. This average value is determined from all the ﬁtted 1/f NSD slopes that were measured in a certain device conﬁguration. The magnetic conﬁguration for eachmeasurement is determined by the resistance of the device [see Fig. 1(b) ]. We ﬁnd that, for a positive chirality, the vortex state has an average noise level a V¼2:1/C210/C010lm2at least one order of magni- tude greater than the parallel and anti-parallel states, with Hooge parameters of aP¼1:4/C210/C011lm2and aAP¼3:2/C210/C011lm2, respectively [see Fig. 1(c) ]. It is to be noticed that there is a large dis- persion of the measured Hooge parameters in the vortex state, of 2:6/C210/C010lm2, which is much larger than the error bars, contrary to what is measured in the saturated cases, where the dispersion is 0:2/C210/C011lm2and 0 :4/C210/C011lm2, in the P and AP states, respec- tively. This dispersion is most probably associated with the fact that in the displacement of the vortex core perpendicular to the applied mag- netic ﬁeld lines,25asHIPchanges, the vortex core moves between pin- ning sites and/or material grains.14,28The interaction of the vortex core with such defects exhibits a Mexican-hat shaped potential, repul- sive at short-range and attractive at long-range.29This interaction deviates the vortex core from its trajectory, causing it to become pinned close to the defects. When these are present, there is an increase in the measured low-frequency noise as it gets pinned. We ﬁnd that the Hooge parameter in the AP state is threefold that of the P state. This difference is well explained by the electrical 1 =fnoise dependence on the number of open conduction channels in the tunneling barrier, which is higher in the parallel state.30This is indeed a classical behavior in TMR-based sen- sors.31Hence, in the case of a pure electrical origin, it could be expected that in the vortex state, the noise would be limited between the parallel and the anti-parallel states’ noise levels. Given that we ﬁnd a noiseamplitude much larger than that of the AP one, we elaborate that the magnetic noise component is behind the increase in ain the vortex state, when compared to the saturated states, where the magnetic noise is min-imized. 22Interestingly, we ﬁnd that the measured 1/f noise is indepen- dent of the vortex chirality and polarity conﬁguration, given that the Hooge parameter has comparable values in the different conﬁgurations [seeFigs. 1(c) and1(d)]. After having characterized the 1/f noise in thevortex conﬁguration, we propose in the following some strategies to reduce the low-frequency noise of the vortex states close to the valuesobtained in the uniform states. A ﬁrst approach is based on the use of a dc current injected into the STNO device that generates a spin-transfer torque that acts uponthe layer’s magnetization. For I dc<0, the induced spin-transfer torque acts as an extra-damping term, and as such, no self-sustained preces-sion of the vortex core occurs. In these measurements, the appliedmagnetic ﬁeld is fully out-of-plane, l 0HOOP¼170 mT. As can be observed in Fig. 2 , we ﬁrst observe a reduction of the Hooge parameter forIdcbetween /C01m Aa n d /C03mA,r e a c h i n g a¼1:4/C210/C010lm2,a typical value for the vortex state (see Fig. 1 ). ForIdc>0, we ﬁrst see that up to Idc¼3 mA, the Hooge parame- ter remains in the range of what is obtained at zero current. For Idc between 3 and 5 mA, we ﬁnd that the Hooge parameter gradually decreases. Then, for a large enough current Idc>Icritof 6 mA, the STNO enters the self-sustained oscillation regime, as can be seen bythe increase in the oscillations’ power in the inset of Fig. 2 . While the system’s sustained dynamics occur in the radio frequency range, inthe case of the studied device around 240 MHz, we study here howthey inﬂuence the low-frequency noise of the device. In this regime,we determine a decrease and stabilization of the Hooge parametervalue, with the device achieving a¼3:6/C210 /C011lm2. Moreover, we also ﬁnd a clear decrease in the dispersion of the measured values,reducing to 1 :2/C210 /C011lm2(seeFig. 2 ). The precessional movement of the vortex core, in the self-sustained regime, makes it less sensibleto material defects of the free layer, therefore decreasing the measuredlow-frequency noise. This noise reduction is in the magnetic compo-nent of the 1/f low-frequency noise. We ﬁnd that the vortex magnetization dynamics strongly inﬂu- ence the low-frequency noise of the device. There is a reduction of the1/f noise of the STNO, while still exhibiting a vortex magnetizationdistribution at the free layer. The self-sustained oscillations of the vor-tex do not signiﬁcantly alter the sensor’s large linear detection range,thus keeping its advantage. The measured Hooge parameter in thisregime is comparable to that measured in the AP state in the sub-critical regime (see the red dotted line in Fig. 2 ). Another approach to improve the 1/f noise amplitude is by rely- i n go nt h ei n j e c t i o no fa nr fs i g n a li n t ot h eS T N O .I nf a c t ,t h e r ea r e FIG. 2. Evolution of the Hooge parameter with the applied bias current. The inset shows the oscillation power of the rf emission due to the emerging vortex dynamics.The red dotted line represents the Hooge parameter measured in the AP state.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 122401 (2021); doi: 10.1063/5.0040874 118, 122401-3 Published under license by AIP Publishingtwo possibilities to generate rf torques acting on the vortex core dynamics, either by using an rf ﬁeld generated in an rf line close to the device or by using an rf current directly injected into the device. Boththese options are tested hereafter. We study the inﬂuence of an alternating magnetic ﬁeld acting on the vortex magnetization. The injection of an oscillating current into theinductive line generates an oscillating in-plane magnetic ﬁeld at the free layer. In this case, the device is operating in the self-sustained regime, forI dc¼8.0 mA and l0HOOP¼170 mT (see Fig. 2 ). For an injection power of 1 mW, the rf current amplitude in the inductive line is Irf¼6 . 9 m A .F o ra nS T N Oi nt h es e l f - s u s t a i n e do p e r a t i o nr e g i m e ,t h e measured noise without any rf ﬁeld is slightly higher than the noise measured in the anti-parallel state, as presented in Fig. 2 for large posi- tive current. In Fig. 3(a) , we observe that by applying an rf ﬁeld with a frequency in the range of 200–280 MHz, the noise level at the studied operation conditions (I dc¼8:0m Aa n d l0HOOP¼170 mT) is reduced from 5 :4/C210/C011lm2to a third of this value. The Hooge parameter value measured without an applied rf ﬁeld in these operation conditions is represented by a red dotted line in Fig. 3 . The average Hooge parame- ter obtained in this case is aHrf¼1:8/C210/C011lm2. We purposefully chose to sweep a frequency range, which includes the STNO resonance frequency, 243 MHz. We ﬁnd that the achieved noise reduction is simi-lar whether the signal is off resonance or in resonance. Increasing the intensity of the rf ﬁeld, the decrease in the 1/f noise level is more pronounced. As the vortex core movement is faster, withless probability of pinning, we ﬁnd a decrease in the measured Hooge parameter, as shown in Fig. 3(b) . This noise reduction is limited by the noise level of the parallel state, which is the minimum achievable noiselevel of the device, represented by a green dotted line in Fig. 3 .W i t h this strategy, the device’s detectivity can be improved – a fundamental factor for magnetic ﬁeld sensors. A second approach investigated here to drive the dynamics of the vortex system is by directly injecting an rf current, I rf,i n t ot h e STNO, while keeping Idc<Icrit, so that the STNO remains in the sub- critical (damped) regime. Note that this second series of measure- ments has been performed on a different STNO from the same waferbut having comparable operation and noise properties. When a 3.2 nW rf current is injected we measure a Irf¼1:8/C210/C010lm2,w h i l e without an rf current we have aV¼3:0/C210/C010lm2.A l t h o u g ht h e r e is a reduction of the noise level, the Hooge parameter is still an orderof magnitude larger than aPdue to the absence of self-sustained oscil- lations of the vortex core. We observe this decrease for Irfwith fre- quencies close to the nano-oscillator resonance frequency—around 290 MHz—but also below it, down to 500 kHz, which is the lower fre- quency limit of the instruments used in the experimental work. We ﬁnd that the noise reduction derived from the rf driven vortex core motion is a non-resonant effect (see Fig. 4 ). Compared to the situation where an rf ﬁeld is applied, much lower rf powers are necessary for the same relative reduction of the noise level, with a few nW being supplied in this case vs slightly below 0.1 mW in the previous case. This is due to the increased efﬁciency of the rf current in driving the vortex core motion ( Fig. 4 ). In summary, we analyze the 1/f low-frequency noise in vortex- based spin-torque nano-oscillators by determining the Hooge parame- ter,a, in different conditions. First, we ﬁnd that in the uniform states, theaof the studied device is comparable to that of typical state-of-the- art TMR sensors, while in the vortex state, it is over one order of mag- nitude larger. This is due to the increased probability of pinning of the vortex core into defects or inhomogeneities of the free layer. Second, we determine that the dynamics of the vortex core strongly inﬂuence the noise level of the device. In the self-sustained oscillations’ regime, the noise decreases to a level close to that of the AP state. Furthermore, we present a strategy for reducing the 1/f low- frequency magnetic noise, through the application of an in-plane rf ﬁeld or injection of an rf current. By using this approach while the device is operating in the self-sustained regime, we are capable of further decreas- ing the measured noise level to values close to the minimum attainable. As such, we can have a vortex-based STNO with relevant noise properties, comparable to those of state-of-the-art TMR ﬁeld sensors. At the same time, we proﬁt from the speciﬁc advantages of vortex-based STNOs for sensing applications: large linear detection range and high spatial resolu-tion. This noise reduction technique based on the spin-torque dynamics of the vortex can have an impact on the sensors’ industry, which may proﬁt from the advantages of the vortex conﬁguration. This work was supported by the French ANR projects “SPINNET” ANR-18-CE24-0012 and “CARAMEL” ANR-18-CE42-0001. DATA AVAILABILITY The data that support the ﬁndings of this study are available from the corresponding author upon reasonable request. FIG. 3. (a) Hooge parameter as a function of the frequency of the applied oscillating ﬁeld, with ﬁxed Prf¼1 mW. The red line indicates the value measured in the absence of this ﬁeld. (b) Hooge parameter as a function of the power amplitude ofthe ﬁeld at a ﬁxed frequency, f¼f res¼243:1 MHz. The green dotted lines repre- sent the Hooge parameter measured in the P state. FIG. 4. Hooge parameter as a function of the injected rf current frequency, for the operating conditions: l0HIP¼400 mT, Idc¼1.0 mA, and Prf¼3:2n W .Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 122401 (2021); doi: 10.1063/5.0040874 118, 122401-4 Published under license by AIP PublishingREFERENCES 1S. Cardoso, D. Leitao, T. Dias, J. Valadeiro, M. 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1.4792434.pdf	THE JOURNAL OF CHEMICAL PHYSICS 138, 084102 (2013) Useful lower limits to polarization contributions to intermolecular interactions using a minimal basis of localized orthogonal orbitals: Theory and analysis of the water dimer R. Julian Azar,a)Paul Richard Horn,b)Eric Jon Sundstrom,c)and Martin Head-Gordond) Department of Chemistry, University of California Berkeley, Berkeley, California 94720, USA and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Received 23 November 2012; accepted 3 February 2013; published online 22 February 2013) The problem of describing the energy-lowering associated with polarization of interacting molecules is considered in the overlapping regime for self-consistent ﬁeld wavefunctions. The existing ap-proach of solving for absolutely localized molecular orbital (ALMO) coefﬁcients that are block- diagonal in the fragments is shown based on formal grounds and practical calculations to often overestimate the strength of polarization effects. A new approach using a minimal basis of polar-ized orthogonal local MOs (polMOs) is developed as an alternative. The polMO basis is minimal in the sense that one polarization function is provided for each unpolarized orbital that is occu- pied; such an approach is exact in second-order perturbation theory. Based on formal grounds andpractical calculations, the polMO approach is shown to underestimate the strength of polarization effects. In contrast to the ALMO method, however, the polMO approach yields results that are very stable to improvements in the underlying AO basis expansion. Combining the ALMO and polMOapproaches allows an estimate of the range of energy-lowering due to polarization. Extensive nu- merical calculations on the water dimer using a large range of basis sets with Hartree-Fock the- ory and a variety of different density functionals illustrate the key considerations. Results are alsopresented for the polarization-dominated Na +CH 4complex. Implications for energy decomposi- tion analysis of intermolecular interactions are discussed. © 2013 American Institute of Physics . [http://dx.doi.org/10.1063/1.4792434 ] I. INTRODUCTION There is no question that interest in intermolecular inter- actions with an eye toward elucidating the interplay of forces underlying weak potentials has grown in recent years. With it, so has the number of so-called energy decomposition analysis (EDA) schemes, designed to resolve a quantum mechanical (QM) interaction energy into physically based components.In addition to direct use for insight or interpretive purposes, EDAs can serve as high-level QM tools in applications rang- ing from guiding drug functionalization 1,2to designing force ﬁelds for molecular mechanics (MM) simulations.3 The physical contributions that give rise to weak interac- tions between distant molecules whose densities do not over-lap have long been well-characterized. 4At a given separation, the magnitude of interactions can be directly evaluated from properties of the individual (isolated) molecules. They include(i) long-range permanent electrostatic interactions coupling the monopole, dipole, quadrupole, and higher-order moments of the isolated species; (ii) additional induced electrostatic in- teractions, which arise from distortions of the charge densities due to electric ﬁelds emanating from nearby molecules. For a given ﬁeld, induction is determined by static molecular polar- izabilities, e.g., dipole, quadrupole, etc.; and (iii) weaker dis- a)julianazar2323@berkeley.edu. b)prhorn@berkeley.edu. c)esundstr@berkeley.edu. d)mhg@cchem.berkeley.edu.persive forces, or van der Waals interactions, resulting from instantaneous multipole interactions, of strength governed toleading-order by the C 6coefﬁcients of the molecules. When the interacting molecules overlap, additional in- teractions arise. In qualitative terms, these effects are well-known and are usually described in molecular orbital (MO) language. 5Speciﬁcally, they include (iv) Pauli repulsions that distort the density due to the overlap between occupied levelson neighboring molecules, and (v) attractive donor-acceptor (dative) interactions that arise when there is sufﬁcient overlap between occupied and empty levels of neighboring molecules,leading to partial charge transfer. In quantitative terms, there is no unique prescription for partitioning the observable bind- ing energy in the overlapping regime. For example, in MO terms, when molecular neighbors overlap, there are many ways to infer occupied and empty orbitals of each molecule,which affects the relative values of induction and charge trans- fer. The task of an EDA is to provide a well-deﬁned procedure for calculating each contribution. Exploring different deﬁni-tions of these components and resolving differences between different EDAs (provided they are physically defensible) is a basis for deepening our understanding of intermolecularinteractions. While summarizing the full range of available EDAs is a task for a detailed review, it is useful to identify someof the most widely used methods, and to distinguish those that decompose a given level of calculation (e.g., density functional theory (DFT)) from those which also aim to 0021-9606/2013/138(8)/084102/14/$30.00 © 2013 American Institute of Physics 138 , 084102-1 084102-2 Azar et al. J. Chem. Phys. 138 , 084102 (2013) provide a method for efﬁciently calculating the interactions. Considering ﬁrst the constructive approaches to intermolec- ular interactions, symmetry-adapted perturbation theory(SAPT) 6,7is a many-body generalization of Heitler-London polarization theory that treats the inter-monomer coupling as the ﬂuctuation potential. SAPT has become popular, partic-ularly with the development of inexpensive DFT approaches for computing previously demanding terms. 8,9Direct use of the many-body expansion10to separate pairwise, three-body, and higher terms is another strategy in approaches such as the fragment MO method.11,12Finally, it is important to note that results on the form of intermolecular interactions from decomposition methods such as those discussed below have been incorporated into efﬁcient computational approachessuch as the effective fragment potential (EFP) method, 13,14 a step toward even more highly simpliﬁed methods such as polarizable MM potentials.15,16 Regarding decompositions, the pioneering variational Kitaura-Morokuma (KM) EDA17partitions the binding en- ergy into (including, but not limited to) geometric distor-tion, electrostatic, polarization, and charge-transfer compo- nents. The related Ziegler-Rauk procedure was developed essentially at the same time. 18,19The natural orbital EDA (NEDA)20scheme is used as part of the widely employed natural bond orbital analysis.5,21,22A markedly different “density-based” EDA23has been proposed recently, which constrains the electrostatic density to remain identical to the superposed density while the electrostatic interaction en- ergy is determined variationally. Many other EDAs have provided useful modiﬁcations and improvements to the ba- sic KM framework.24–28One class of improvements is the use of block-localized,29,30or, equivalently, absolutely local- ized MOs (ALMOs)31,32to variationally describe polariza- tion. The ALMOs are determined by solving nonorthogonal,locally-projected SCF equations for molecular interactions (SCF MI) as ﬁrst proposed by Stoll, 33and later recast in dif- ferent ways,34,35and then efﬁciently implemented.36 The ALMO EDA provides a self-consistent determina- tion of intramolecular polarization as the energy-lowering upon solving the SCF-MI equations with the MO coefﬁcientmatrix constrained to be block-diagonal in each of the clus- ter molecules. The molecule-blocking prevents intermolecu- lar charge transfer while simultaneously allowing for relax-ation of each MO in the ﬁeld of all other electrons and nuclei. This natural separation of charge transfer and self-consistent polarization is a merit of the approach. Formally, the bindingenergy in the ALMO EDA is the sum of four terms: (i) ge- ometric distortion (gd), deﬁned as the energy required to de- form an isolated molecule’s internal coordinates to those con- sistent with the cluster geometry, and evaluated as the energy difference between the complex in its equilibrium geometryand the sum of its elements’, each taken at its vacuum mini- mum, /Delta1E gd=EAB\|complex −EA\|min−EB\|min, (1) (ii) frozen orbital (frz) interactions, accounting for both per- manent electrostatic contributions and Pauli repulsions, corre- sponding to bringing inﬁnitely separated, distorted molecules together, and operationally determined from the energy asso-ciated with the supermolecular density matrix formed from the converged MO matrices of the isolated molecules, each in its complex geometry, /Delta1Efrz=EAB{Pfrz(CA,CB)}\|complex −EA(CA)\|complex −EB(CB)\|complex , (2) (iii) polarization (pol), deﬁned as the relaxation of fragment ALMOs in the ﬁeld of all other ALMOs, but with the block- diagonal constraint in place, /Delta1Epol=EAB(Ppol)−EAB(Pfrz), (3) and (iv) charge transfer (ct), stabilization due to intermolecu- lar relaxation of ALMOs to the canonical orbitals, /Delta1Ect=EAB(Pcan.)−EAB(Ppol). (4) Taken together, these contributions sum to the full binding energy, /Delta1ESCF bind, /Delta1ESCF bind=/Delta1Egd+/Delta1Efrz+/Delta1Epol+/Delta1Ect. (5) Though the ALMO EDA in its current form gives a rea- sonable decomposition and has enjoyed much recent success in application37–42and extension to explicit correlation,43we acknowledge here that the polarization term has no well-deﬁned basis set limit because there is a point of over- completeness of the underlying basis beyond which relaxation of the ALMO constraint can no longer improve the fragment-localized orbitals. In other words, there is enough variational freedom near the basis-set limit in the constrained orbitals to completely describe their delocalized counterparts, thus inva-liding the physical insight of the orbital constraint and ren- dering polarization and charge-transfer no longer separable. While this may seem like a purely formal objection, it has thepractical implication that one cannot converge the polariza- tion and charge-transfer components of the ALMO EDA to a well-deﬁned basis set limit. While reasonable stability hasalready been demonstrated in the aug-cc-pVXZ, X =D,T,Q sequence for the water dimer, 44it is worthwhile to emphasize that this is at best metastability. This paper focuses on exploring several aspects of the deﬁnition and stability of the polarization and charge-transfercontributions to intermolecular interaction energies. First, we present a proposal for the deﬁnition of polarization that is de- signed to yield stable contributions across a wide range ofbasis set sizes by removing near-linear dependencies between the virtual spaces describing polarization on different frag- ments. This is accomplished by deﬁning small numbers of po-larization functions for each fragment based on singular value decomposition (SVD) of the ﬁrst-order singles amplitudes as- sociated with the frozen MOs, which are then orthogonalizedamongst themselves and relocalized. SVD has been useful in deﬁning the most important orbitals in applications ranging from analyzing excited states, 45to donor-acceptor orbitals in EDA,32to analyzing electron correlation effects46and MP2.47 Using the resulting minimal polarization basis, we retainthe general structure and terms of the ALMO approach, no- tably the feature of self-consistent polarization, emphasizing that the added beneﬁt of orthogonal MOs allows for trivial084102-3 Azar et al. J. Chem. Phys. 138 , 084102 (2013) extension of the method beyond a mean-ﬁeld treatment. This procedure is described in detail in Sec. II. The second main aspect of the paper consists of numer- ical results that compare the new approach to polarization against the existing fragment-blocked SCF-MI method as a function of basis set size and composition, energy functional,and geometry, for the model system of the water dimer. These comparisons are undertaken in Sec. III. It is interesting to assess the dependence of calculated polarization and chargetransfer contributions for different sequences of basis sets: cc-pVXZ, aug-cc-pVXZ and d-aug-cc-pVXZ, as well as to compare the results obtained at the mean-ﬁeld Hartree-Fock level against the components calculated with various density functionals. Additionally, since the difﬁculty in disentanglingpolarization and charge transfer arises directly from the de- gree of intermolecular overlap, it is interesting to assess the separation dependence of the differences in results betweenthe new approach and polarization evaluated by the SCF-MI procedure. Some results are also given for the Na +CH 4com- plex, where polarization effects are dominant. We summarizeour main conclusions in Sec. IV. II. THEORY The Einstein summation convention of Refs. 48and49 applies where a co- or contravariant index pair occurs, except for fragment labels. OAandVArefer to the number of occu- pied and virtual spin-orbitals on molecule A. NAis the number of AO functions centered on A. Fis the number of fragments. The indices i,j,k,l, . . . denote MOs spanning the occupied subspace; a,b,c,d, . . . are mean virtual MOs; p,q,r,s,... a r e any spin-orbitals; and μ,ν,σ,λ, . . . are AO basis functions. We ﬁrst discuss the behavior of the SCF-MI eigenvec- tors as the basis approaches completeness, and then detail a procedure for determining the optimal fragment-tagged vari- ational subspaces to obtain polarized molecular states in the supermolecular ﬁeld, taken as solutions of a set of constrained equations. A. Basis set superposition error (BSSE) and the drawbacks of SCF-MI as a basis for EDA The term taken as intramolecular polarization in the ALMO scheme has no basis set limit. It is instructive to ex- amine the general BSSE problem50to understand why. Con- sider computing the binding energy /Delta1Eof the molecular com- plex X◦Ywithin the subtractive supermolecule approach, /Delta1E=E(X◦Y)−E(X)−E(Y). Self-consistent diagonaliza- tion of the Hamiltonian operator in the full AO basis will yield a set of orthonormal eigenfunctions, each with an associated eigenvalue equation: ˆf\|φpX/angbracketright=εpX\|φpX/angbracketright. (6) While each MO formally has amplitudes on all fragments, the fact that this is a complex means that, in general, the MOs can be fragment-localized by standard methods such as Boys51or Edmiston-Ruedenberg localization.52It is the fact that even after localization ˆf\|φpX/angbracketrightcan be resolved into projections onto the fragment basis and its orthogonal complement that givesrise to the BSSE that pockmarks such calculations: ˆf\|φpX/angbracketright= ˆfˆ1\|φpX/angbracketright= ˆfˆPX\|φpX/angbracketright+ ˆf(ˆ1−ˆPX)\|φpX/angbracketright,(7) where ˆPX=\|ωXμ/angbracketrightS−1 Xμν/angbracketleftωXν\|. Speciﬁcally, the second term on the right-hand side of Eq. (7)allows for variational opti- mization of \|φpX/angbracketrightvia access to functions that are not centered on that fragment. The consequence is systematic overestima- tion of binding energies due to inﬂation of the E(X◦Y)t e r m . Of course, at the complete basis set limit, the second term approaches zero, and the basis functions centered on fragment X span a sufﬁcient space to describe X’s eigenvectors withoutany borrowing. Away from this limit, many methods to coun- teract BSSE have been developed, of which the most popu- lar is probably the counterpoise method, 53where the energy- lowering due to borrowing of extra-fragment functions is ex- plicitly subtracted from the supermolecular result E(X◦Y). Other approaches include forcible elimination of the BSSEterm of Eq. (7)from the Roothaan equations, 54,55but at the expense of the Hermiticity of the matrix representation of the Hamiltonian operator. Another strategy is that employed by the SCF-MI ap- proach, detailed above, which constrains the MO coefﬁcient vectors {CpX}to be block-diagonal (absolutely localized) in the interacting fragments, \|φpX/angbracketright=\|ωμX/angbracketrightCμX •pX. (8) By performing variational optimization with fragment- blocking of the MO coefﬁcients, BSSE is prohibited by con-struction: ALMOs tagged to a given fragment cannot employ basis functions from other fragments. The use of the SCF-MI procedure within an EDA for describing the energy-lowering due to polarization relies on the physically intuitive assumption that fragment-blocking the MO coefﬁcient matrix also prohibits charge transfer from agiven fragment to any other. Since an ALMO tagged to a given fragment cannot contain contributions from AOs on other fragments, dative interactions should be prohibited.Thus polarization is any energy-lowering where the trace of the on-fragment density matrix operator is preserved, or in other words, where electrons are not shuttled between molecules. Charge transfer can then be associated with any remaining energy-lowering that is achieved when the ALMOconstraint is lifted. However, separating the energy-lowering associated with polarization and charge transfer based on the SCF MI con-straint of Eq. (8)has deﬁciencies. It can only be used with one-particle basis sets that are atom- or fragment-tagged, and thus is natural with AO basis sets, but cannot be used di-rectly with a plane wave expansion. Furthermore, even with AO basis sets, at or near the complete basis set limit, a given fragment-tagged MO will already be described optimally andwill not beneﬁt from any projection onto the basis functions of a neighboring fragment. Alternatively put, the second term of Eq.(7)will be reduced to zero (as will the associated charge- transfer term, /Delta1E ct), and it becomes obvious that the magni- tude of the EDA components is basis set-dependent. Thus the success of the ALMO EDA in practice depends upon using a basis set that is not too small (inaccurate total interaction energies), but also not too large (as the ct contribution will084102-4 Azar et al. J. Chem. Phys. 138 , 084102 (2013) progressively be damped away). In practice, the aug-cc-pVTZ basis has appeared to be a reasonable compromise. B. General construction Our goal is to obtain a small set of linearly-independent functions that are still local (fragment-ascribable) which maybe used to described the energy-lowering due to polarization in a way that is stable with respect to basis set extension. Should these functions be non-orthogonal (like AOs), or orthogonal (like MOs)? While either is possible, we shall employ the orthogonal choice here because it ensures zerooverlap between the Hilbert spaces associated with different fragments. Furthermore, orthogonalization will generate a ba- sis of fragment-tagged linearly-independent orbitals whoseshapes and extents are parametrized by allcenters, not just a subset, which is appropriate to properly respect antisymme- try between electrons on different fragments. 56For instance, when two ﬂuorine atoms approach each other to form F 2, the electrons occupying the σ∗ uorbital avoid collapsing into the nuclei by maintaining orthogonality to the mostly unper-turbed core 1 sstates, a property that neither of the individ- ual atomic 2 p zstates exhibited with respect to the core of the other nucleus before the bond was formed. In the same way,the spatial extent and nodal structure of orthogonalized func- tions centered on one molecule in the ﬁeld of another should reﬂect these intermolecular exchange interactions. While or-thogonalization will produce delocalization tails extending to other fragments, fragment identity can be maintained via the transformations that we shall detail below. The functions we shall employ are eigenvectors of the intramolecular response density, whose number will be equal to the number of elec-trons on each fragment. The starting point for treating intra-fragment polariza- tion is the result of calculations on individual fragments inthe basis of their own AOs, and without any consideration of neighboring molecules, which is the so-called “frozen” orbital calculation. The eigenvectors of the Hamiltonian op-erator in the frozen orbital representation are (i) orthogonal within a fragment, satisfying {p∪q}∈X\|g X pq=δpq, where gX pqis the matrix of MO overlaps belonging to Xand (ii) strongly orthogonal , whereby all functions lying in the fragment’s virtual space are orthogonal to functions lyingin its occupied space, {i∪a}X/vextendsingle/vextendsingleg X ia=0. These properties are the direct consequence of solving the SCF equations for each fragment independently, guaranteeing a block-diagonal Hamiltonian matrix and idempotent one-particle frozen den- sity. However, any inter -fragmental MO pair is neither orthog- onalwithin a subspace, nor is it orthogonal between sub- spaces. That is, {(pεX )∪(qεY )}/vextendsingle/vextendsinglegpq/negationslash=0. As we would like to unambiguously determine non- overlapping occupied and virtual subspaces, we construct a new set of “projected” virtual orbitals, {\|φ/prime a/angbracketright}, strongly orthog- onal to the global set of occupied orbitals, and related to theold set { \|φ a/angbracketright}b y \|φ/prime a/angbracketright= ˆQ\|φa/angbracketright=(ˆ1−ˆP)\|φa/angbracketright=(ˆ1−\|φi/angbracketrightgij/angbracketleftφj\|)\|φa/angbracketright,(9)and the projection matrix Qin the AO representation is given by Qμ •ν=δμ •ν−Cμ •igijC†σ j•Sσν, (10) where Sis the AO overlap matrix and Cis the frozen coef- ﬁcient matrix. The transformation C/primeμ •p=Qμ •νCν •psmoothly guarantees strong orthogonality, gia=0, as ˆQ→ˆ1i nt h e non-overlapping limit, preserving the original spaces. We seek next to orthonormalize the orbitals within each subspace separately, noting that the frozen density is invariant to such transformations. Having already orthogonalized the subspaces, this is sufﬁcient to enforce /angbracketleftφp\|φq/angbracketright=δpqfor all p and q. Generally, we want to transform the nonorthogonal set by ˜\|φqY/angbracketright=/summationdisplay Z\|φpZ/angbracketrightXpZ •qY, (11) where Xis the orthogonalizer that takes the non-orthogonal set {\|φp/angbracketright} to the orthogonal set {˜\|φq/angbracketright}. (to keep the notation uncluttered, we have dropped the “/prime” that denoted the frozen, projected set.) Schemes rooted in symmetric orthogonalization repre- sent a least-squares minimization of the Hilbert-space dis- tance between a function of the nonorthogonal set { \|φp/angbracketright} and the corresponding function of the orthonormal set {˜\|φp/angbracketright}. Most generally,57the sum Zto be minimized is the difference in the vectors pre- and post-orthogonalization, Z=/summationdisplay pwp/integraldisplay/vextendsingle/vextendsingle/vextendsingle/vextendsingle˜\|φp/angbracketright−\|φp/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2dτ, (12) where { \|φp/angbracketright} is the non-orthogonal vector set, {˜\|φp/angbracketright}is the orthogonal set, {wp}is a set of weighting scalars, and dτrep- resents inﬁnitesimal Hilbert space. If each non-orthogonal MO contributes equivalently in the construction of the orthogonal spin-orbital (i.e., the ma- trix of weighting scalars is chosen as Wp=1) then Eq. (12) is minimized by choosing Xp •q=(g)−1 2pq≡gp •q, and we have arrived at the familiar Löwdin (symmetric) prescription for orthogonalization.58We transform the occupied space accord- ing to ˜\|φiA/angbracketright=/summationdisplay B\|φjB/angbracketrightgjB •iA. (13) The “absolute” locality in the AO basis afforded by the ALMO scheme is sacriﬁced at this point since the freshly orthogonalized functions span the entire occupied space, butthey are still imputable to parent fragments because of the relative compactness of the occupied subspace and the least- squares connection, Eq. (12). The orthonormal occupied set is subsequently tightened by the Boys’ localization scheme 51 which, again, leaves the frozen density invariant. The story is more bleak for the virtual functions since they are more delocalized to begin with, and the problem is only exacerbated by the necessary inclusion of diffuse AOs for applications of interest. Symmetric orthogonalization of this subspace will treat the basis too democratically, mix- ing on equal footing a relatively tight MO on one fragment084102-5 Azar et al. J. Chem. Phys. 138 , 084102 (2013) with some diffuse MO centered far away, for instance. Conse- quently, evenly mixed virtual MOs become hardly imputable to a speciﬁc molecule. The crux of the problem is thus thecareful delineation of a space belonging to each fragment which a least-squares minimal orthogonalization will not ap- preciably distort. More speciﬁcally, we want to develop a partitioning of the Hilbert space Hinto a minimal valence space (relevant for intramolecular polarization) spanned by the set Mand a “low-impact” space spanned by the more diffuse, Rydberg- like functions R,H=M⊕R, where M=/circleplusdisplay AMAandR=/circleplusdisplay ARA,with (14) MA=VA⊕OA, (15) and where OAandVAare the minimal occupied and virtual spans centered on molecule A and all subspaces are orthog- onal,VA⊥VB⊥RB. In the subsection below, we develop an approach to obtain the minimal valence space, M, from a perturbation theory of intramolecular polarization in the su- permolecular ﬁeld. Once Mis available, the functions that span it can be orthogonalized via essentially the same schemedescribed above for the frozen occupied space. C. A minimal basis for polarization The Fock matrix built from the frozen density will necessarily contain non-zero occupied-virtual coupling ele-ments f iX •aYthat arise in response to the perturbation of each isolated fragment by the supermolecular environment. A sub- space partitioning for the purpose of variationally determin-ing molecular eigenstates in the supermolecular ﬁeld can be guided by this fact. Speciﬁcally, we can use perturbation the- ory to examine the on-fragment polarization response, and ex-tract a small set of polarization functions (not exceeding the number of electrons on the fragment) that can exactly repre- sent it. Those functions can be used as a basis for a variationaltreatment of polarization after orthogonalization. Perturbation theory for either DFT or Hartree-Fock is conveniently cast in terms of the one-particle density ma-trix,P, and the Fock matrix, F. Given a frozen density, P (0), we evaluate the Fock matrix as F=F(P(0)). Since we are interested only in the intramolecular polarization, we shall consider perturbation theory for a single fragment in the orthonormal space of its frozen occupied orbitals, and its(orthormalized) projected frozen virtuals, deﬁned in Eq. (9). In other words, each fragment now has its own perturbation problem, and we neglect the interfragment coupling on thegrounds that it is charge-transfer-related. The fragment Fock operator is partitioned into zeroth- order pieces (the OO and VV blocks), and a ﬁrst-order per-turbation (the OV and VO blocks) due solely to the presence of the supermolecular environment: Thus F=(F OO+FVV)(0)+(FOV+FVO)(1). (16) The problem of minimizing the energy for a perturbed Fock matrix in an orthogonalized basis may be expressed in sev- eral equivalent ways: (i) block diagonalization to zero thecoupling between the occupied and virtual blocks in the new basis, (ii) ﬁnding a valid one-particle density matrix that com- mutes with the Fock matrix, FP=PF, or (iii) solving the fol- lowing set of quadratic equations:59,60 FVO+FVVXVO−XVOFOO−XVOFOVXVO=0VO, (17) and then evaluating the energy-lowering (relative to the un- perturbed problem) as δE=Tr(FOVXVO). (18) This last form is convenient for doing perturbation theory with the partitioning given in Eq. (16). To zeroth order, it is immediately clear from Eq. (17) thatX(0) VO=0VO. First-order perturbation theory applied to Eq. (17) is straightforward as- suming that F(0) VVandF(0) OOare initially diagonalized such that F(0) ab=ε(0) aδabandF(0) ij=ε(0) iδij. The resulting ﬁrst-order per- turbed amplitudes, X(1) VO,a r eg i v e na s X(1) ai=−F(1) ai/slashbig/parenleftbig ε(0) a−ε(0) i/parenrightbig (19) with a corresponding second-order energy-lowering obtained by substituting into Eq. (18). Note that these are exactly of the form obtained when doing Hartree-Fock perturbation theory in the space of single substitutions, but are equally valid for Kohn-Sham DFT. A solution of the form of Eq. (19) can be constructed in- dividually for each fragment, say A, describing the ﬁrst-order polarization effects on that fragment due to the presence ofthe other components of the complex. On an individual frag- ment, there are O AVAcoupling parameters within X(1) VO,s o the ﬁrst-order correction to \|ηi/angbracketrightwill contain VAcomponents in the orthogonal complement: \|η(1) iA/angbracketright=\|ηaA/angbracketrightXa(1) •iA. (20) Since typically OA/lessmuchVA, it is desirable to condense the in- formation encoded in this sum, by ﬁnding a minimal virtualbasis sufﬁcient to describe the ﬁrst-order wavefunction. The ﬁrst-order result (on a fragment) can be exactly recaptured in this way by performing a singular value (SVD) decomposi-tion of X (1) VO. The SVD is deﬁned as LVVX(1) VOR† OO=x(1) VO. (21) Here x(1) VOis a rectangular matrix with only OAnon-zero en- tries lying along the diagonal; these are the singular values. The left eigenvectors, LVV, describe transformations of the original virtual functions into a reduced set of essential virtu- als whose number is no greater than OA. \|γa/primeA/angbracketright=\|ηa/angbracketrightLaA •a/primeA,a/prime=1,...,O A. (22) All other virtual orbitals have corresponding singular values of zero, and retaining only the virtuals above, we are guar-anteed to recover the energy-lowering of Eq. (18) but in a rank-reduced polarization basis. This is our deﬁnition of the minimal virtual space, V A, on fragment A. This transformation gives an intuitive bond-antibond pic- ture of polarization whereby relaxations through ﬁrst-order in084102-6 Azar et al. J. Chem. Phys. 138 , 084102 (2013) perturbation theory on a given fragment can be exactly ex- pressed via a minimal polarization basis no larger than the occupied space. The presence of polarization is a direct con-sequence of violating Brillouin’s theorem, ( F VO=0), in the intermolecular environment, as the inhomogeneous term of Eq.(17) isFVO. The null space of the SVD spanned by the vectors of Lwith vanishing singular values will naturally in- clude diffuse molecular states especially as the AO basis set is extended. As the polarization energy of Eq. (18) is con- vergent with the basis set provided the perturbation theory is well-behaved, so too will the SVD and minimal polarization orbitals. The minimal set of virtual functions for the complex can now be deﬁned as the union over fragments of the minimalvirtual set, {V A}. However, these functions will not, in gen- eral, be orthogonal between fragments, and so we orthogo- nalize the minimal set of virtual vectors amongst themselvesvia Eq. (13), then Boys-localize across the orthonormalized minimal virtual space, paralleling the procedure used for the occupied orbitals to complete the speciﬁcation of the func-tions spanning the minimal basis set for polarization, M.T h e null space spanning Ris discarded in the minimal scheme because its vectors are extraneous to a description of polar-ization at the level of perturbation theory, as follows from the SVD, Eq. (21). However, re-introducing Rafter polarization is necessary to guarantee recovery of the full SCF energy anddelocalized eigenfunctions. We note that orthogonalization will reintroduce BSSE into the polarization term, but this is not before the varia- tional subspaces are determined as the span of the minimal eigenset of the ﬁrst-order orbital response. It is thus assumedthat mutual frozen interactions in the approach induce defor- mations ﬁrst among a molecule’s own electronic distributions, followed by inter-fragment distortions in the interest of or-thogonality. We also note that, though we make no explicit reference to the underlying AO basis in the equations de- termining each molecule’s variational space, each set of re-sponse amplitudes {X aA(1) •iA}remains ultimately parametrized by fragment-allotted AO functions originating from the frozen set of ALMOs. D. Orthogonal SCF for molecular interactions and associated EDA We solve the problem of computing the energy-lowering due to intramolecular polarization in a manner similar to the original SCF-MI approach. The polarization energy is taken as the energy-lowering on self-consistently solving subspace-projected fragment-labeled SCF equations constrained to con- serve the number of electrons on a fragment, e.g., the trace of the fragment density projector in the fragmental basis remains constant. The charge-transfer stabilization is subsequently de- termined as the difference between the full SCF energy andthe energy of the polarized wavefunction as in the ALMO scheme: /Delta1E bind=/Delta1ESCF bind=/Delta1E gd+/Delta1E frz+/Delta1E pol+/Delta1E ct. (23)The above constraint is realized by demanding that the self-consistently polarized set { \|ψpX/angbracketright} – which corresponds to/Delta1E polfrom the energy of the super-system computed with the frozen density (given in Eq. (2)) – simultaneously be de- scribed by OV mixings strictly among the vectors of the sub-system X, \|ψ pX/angbracketright=\|γqX/angbracketrightUqX •pX, (24) and satisfy the variational eigenvalue equation ˆf\|ψpX/angbracketright=\|ψpX/angbracketrightεpX, (25) where ˆfis the standard mean-ﬁeld or DFT Hamiltonian, and \|ψpX/angbracketrightis what we term a polarized orthogonal local molec- ular orbital (polMO) eigenfunction labeled pof the super- molecular Hamiltonian matrix projected into the variational space spanned by fragment X. Developing the working equa- tions, we resolve the identity into properly idempotent pro- jectors onto the individual fragment subspaces, ˆ1=/summationtextF XˆRX =/summationtext X(ˆPX+ˆQX), with ˆPXˆPY=ˆPXδXYand ˆPXˆQY=0,and insert above, F/summationdisplay Y/parenleftbig \|γqY/angbracketright/angbracketleftγqY •\|/parenrightbigˆfF/summationdisplay Z/parenleftbig \|γrZ/angbracketright/angbracketleftγrZ •\|/parenrightbig \|ψpX/angbracketrightεpX=\|ψpX/angbracketrightεpX. (26) Left-multiplying by /angbracketleftψsW•\|and expanding \|ψtX/angbracketrightin the minimal basis respecting the local constraint of Eq. (24), we arrive at UsX† •qXfqX •rXUrX •pX=δs pεpX. (27) Thus, we have Fprojected sets of SCF equations for the po- larized eigenvectors and eigenvalues, UX† •XfX •XUX •X=εX. (28) Solving these projected equations is equivalent to block- diagonalizing the Hamiltonian matrix in the minimal polar-ization basis. All remaining orbital mixings (either between fragments in the minimal space, M, or coupling to any mem- ber of the Rydberg space, R) account for the remaining energy-lowering necessary to approach the full SCF calcula- tion. Within the minimal polarization space, the utility of ini-tially neglecting interfragment mixings, U X •Y, is that it serves to cleanly separate intra- and intermolecular effects. The motivation for neglecting the Rydberg space is that it is not as-sociated with intramolecular polarization to leading-order in perturbation theory. We expect the locally-projected polMO wavefunction and energy to approach exactness in the limitthat the fragments make negligible use of charge-transfer ro- tations to relax their orbitals, for instance, in the case of very weakly interacting systems, or for systems near dissociation.We emphasize that the polarized wavefunction is an exact eigenfunction of ˆF (0)with energy complete through second- order perturbation theory, as discussed in Subsection II C. Once the polMOs are obtained self-consistently for the polarization energy, the vectors spanning Rare re-introduced and the full Hamiltonian matrix is diagonalized to self- consistency. The energy-lowering is due to charge-transfer delocalizations connecting to the observable binding energy.What follows is a sketch of the polMO-based EDA:084102-7 Azar et al. J. Chem. Phys. 138 , 084102 (2013) 1. Perform Findependent self-consistent HF calculations to obtain {CX •p}. 2. Build PandF(P) and compute /Delta1Efrzby Eq. (2). 3. Project off the occupied space from virtual space follow- ing Eq. (9). 4. Symmetrically orthogonalize across the occupied space following Eq. (13), then Boys-localize. 5. Semi-canonicalize the occupied and virtual subspaces by fragment to make the denominator of Eq. (19) diagonal. 6. Construct and SVD XVOby Eq. (21), then transform to the natural polarization basis M, discarding R. 7. Symmetrically orthogonalize across the minimal virtual set, then Boys-localize. 8. Solve the locally-projected Eq. (28) self-consistently, obtaining /Delta1Epol. 9. Re-introduce Rand semi-canonicalize across full occu- pied and virtual subspaces. 10. Relax the constraint of Eq. (24) to obtain full-space eigenvectors and /Delta1Ect. III. RESULTS AND DISCUSSION It will, of course, be essential to inspect the results of the EDA when applied to a wide variety of molecular complexes, but for the present purpose of uncovering trends particular tothe decomposition methods, we limit our scope to the widely studied water dimer interaction potential. The C s-symmetry global minimum places the molecular dipoles at an apprecia-ble offset presumably to enhance the p(O)→σ(OH)i n - teraction, hinting at a delicate balance between dative and electrostatic interactions. So there is no question that a satis-factory description of the water dimer interaction is difﬁcult, and the question of what elements are important is not with- out considerable controversy. 44,61We performed all compu- tations within a development version of Q-Chem.62AO ba- sis set parameters for all 5z63and doubly augmented basis sets64were obtained from the EMSL Basis Set Exchange with h-angular-momentum functions removed. The C s-symmetry CCSD(T)/cc-pvqz-level optimized water dimer minimum was taken from the S22 set.65The data were not corrected for BSSE. A. Stability with respect to basis set extensions It is desirable for any EDA scheme that, in the same way that the binding energy is convergent with respect to basis set extension, its resolved components likewise converge onsome limiting value. If this were not the case, there would be no reason to take the components of the EDA in one AO basis versus another as superior. In general, there is good rea-son to prefer larger- to smaller-basis results (if the former are feasible) simply because increasing the variational degrees of freedom available to the wavefunction leads to a descriptioncloser to the complete basis set limit. For well-posed meth- ods, this corresponds to a more precise description of the intermolecular interactions themselves. Within Hartree-Fock theory, we compare the behavior of EDA terms in the min- imal polMO approach and the existing ALMO scheme withTABLE I. HF minimal-basis polMO and ALMO EDA components of the interaction between equilibrium water dimer in kJ/mol. Basis frz pol(polMO) pol(ALMO) ct(polMO) ct(ALMO) Bind dz 9.88 2.22 3.34 12.11 11.00 24.21 tz 8.30 2.54 4.03 7.57 6.08 18.41qz 6.82 2.57 4.49 7.05 5.13 16.44 5z 5.72 2.79 4.84 6.89 4.84 15.40 aug-dz 6.32 2.79 4.62 6.86 5.03 15.96aug-tz 5.61 2.93 5.60 6.62 3.95 15.16 aug-qz 5.29 2.91 5.75 6.93 4.09 15.13 aug-5z 5.21 2.87 6.03 6.94 3.78 15.01d-aug-dz 6.07 2.83 4.94 7.30 5.19 16.20 d-aug-tz 5.37 2.96 6.05 6.98 3.89 15.31 d-aug-qz 5.24 3.06 6.82 6.91 3.15 15.21 respect to enhancements of the AO basis in Table I.W ev i - sualize the p(O) andσ(OH) guess orbitals and their mutu- ally polarized and delocalized versions in Fig. 1. The data are grouped according to basis diffusivity. Augmented basis sets add a single diffuse shell of each angular momentum to ev-ery atom. For instance, in the case of aug-cc-pVDZ, an ex- tra set of low- ζs- and p-type functions on hydrogen, and s-,p-, and d-type functions on oxygen. Doubly augmented sets add a second yet more diffuse shell of each angular mo- mentum. A negative sign in front of a contribution meansit will destabilize the complex. The HF limit for binding is estimated at 15.19 kJ/mol from a CBS extrapolation within the doubly augmented series according to a ﬁtted equationB(L)=B(CBS)+Xe −AL, where Lis the quantum number of the highest-angular momentum function in the set, e.g., L(dz) =2,L(tz)=3, and L(qz)=4. The magnitude of the favorable frozen contribution (la- beled frz in Table I) decreases as the basis set is extended, converging most slowly in the non-augmented set and com-prising the biggest contribution to the net change in bind- ing energy beyond the double-zeta level in that series. The frozen interactions include permanent electrostatics, which in σ∗(OH ) : [guess] [pol] [del] p(O):[guess ] [pol] [del] FIG. 1. σ(OH)a n d p(O) guess, polMO, and delocalized orbital pair set plotted at a contour value of 0.12. Guess and polarized orbitals have mostly local amplitudes in spite of orthogonality.084102-8 Azar et al. J. Chem. Phys. 138 , 084102 (2013) TABLE II. Behavior of polarization in non-equilibrium H-bonds depends strongly on the AO basis when overlaps are large. The “ −” sign reads as repulsive. contracted 20% frz pol(ALMO) ct(ALMO) pol(polMO) ct(polMO) dz −0.96 4.91 15.51 2.78 17.64 tz −3.49 6.43 10.12 3.89 12.67 qz −5.60 7.18 9.18 4.21 12.15 5z −7.22 8.09 8.68 4.68 12.09 aug-dz −5.38 7.59 8.18 5.25 10.52 aug-tz −7.24 9.06 7.58 4.39 12.25 aug-qz −7.80 9.47 7.58 4.60 12.45 aug-5z −7.78 9.94 6.96 4.57 12.34 d-aug-dz −5.59 7.74 8.42 4.84 11.32 d-aug-tz −7.43 9.81 7.11 4.45 12.47 d-aug-qz −7.80 10.71 6.41 4.94 12.19 the water dimer primarily reﬂect favorable dipole-dipole in- teractions and unfavorable exchange repulsions. When the lo-cal orbitals have access to increasingly diffuse functions to de- scribe their spatial extents, exchange repulsions are felt more strongly and must distort their densities to respect Pauli ex- clusion. We ﬁnd it quite interesting that the frozen contri- bution converges so slowly with respect to basis set size, asit is still changing slightly when the total binding energy is apparently converged. It follows that if the frozen contribu- tion is unfavorable, e.g., when the molecules are squeezed to-gether, a diffuse basis should only increase its magnitude until convergence. Though the sum of charge transfer and polarization is stable, the individual ALMO quantities themselves are man- ifestly not, with unacceptably large ranges of ∼3.5 and ∼8 kJ/mol for, respectively, polarization and charge transfer. Even if we exclude the small double-zeta results (which suf- fer walloping BSSE due to incompleteness), and all results obtained without augmented basis sets, the range remains∼1.2 kJ/mol for polarization and ∼0.8 kJ/mol for charge transfer, while the range in the total binding energy is only ∼0.1 kJ/mol. The deﬁciency is especially palpable in the doubly augmented trend where the largest number of near- linear dependencies exist. These results are consistent withthe fact that the ALMO polarization contribution will steadily increase as the basis approaches completeness. The effect be- comes larger when the inter-fragment H-bond length is con-tracted by about 20%, as given in Table II.I nt h eA L M O scheme, polarization continues to increase as the lone-pair on the donor oxygen atom’s freedom to inﬁltrate the core regionof the other oxygen increases with the size of the variational space allotted to the donor. By contrast, of course, the instabil- ity of the individual ALMO terms diminishes when the bondlength is protracted by about 13% (given in Table III). This is because the molecules overlap only weakly, and therefore the extent to which basis functions on one water moleculecan mimic charge transfer to the other molecule is greatly re- duced. The total interaction energy is similar ( ∼9k J / m o l )a t both displacements. Turning to the behavior of the polMO treatment of po- larization, it is signiﬁcantly more stable than the ALMO po-TABLE III. polMO polar ization is slightly smaller than ALMO polarization at intermediate separation. protracted 13% frz pol(ALMO) ct(ALMO) pol(polMO) ct(polMO) dz 8.58 0.32 5.57 0.25 5.64 tz 8.74 0.45 2.68 0.34 2.78qz 8.71 0.51 1.30 0.38 1.43 5z 8.64 0.56 0.48 0.41 0.63 aug-dz 8.55 0.53 0.70 0.38 0.85aug-tz 8.51 0.59 0.46 0.44 0.61 aug-qz 8.52 0.63 0.28 0.43 0.48 aug-5z 8.51 0.66 0.21 0.43 0.44d-aug-dz 8.62 0.59 0.71 0.38 0.93 d-aug-tz 8.49 0.64 0.48 0.43 0.70 d-aug-qz 8.50 0.67 0.30 0.42 0.56 larization for the water dimer at its equilibrium separation, as shown in Table I. The reduction in the spread of results as the basis set improves is about a factor of four across all basis setsconsidered. However, if we again exclude the small double- zeta basis, and the non-augmented calculations (which con- verge slowly), the resulting range in the polMO polarization is less than 0.2 kJ/mol, a roughly sixfold reduction over the spread in the corresponding ALMO polarization results. This0.2 kJ/mol range is very comparable to the range in the total binding energies across the same selection of basis sets. The converged value of polMO polarization is ∼3 kJ/mol which gives a roughly 35%:20%:45% frz:pol:ct decomposition of the interaction energy in these essentially CBS-limit HF cal- culations. If the EDA components are normalized to the binding energy and plotted against the basis diffusivity (in the same order as above), then the slope will be a measure of basisset dependence (which we hope will approach zero if the binding energy is converged). Any intersections will suggest a fundamental change in (the assessment of) the characterof the interaction. We plot the components in Fig. 2and note the minimal-basis polarization and charge transfer sta- bilize quickly and never cross, while the ALMO polarization crosses the frozen and charge-transfer contributions well af- ter the binding energy is converged (by the aug-tz level byTable I), though the polarization is quasi-stable in the singly augmented trend where it is most likely to be used. The impact of the Boys orbital localization steps on the stability of the polarization term is assessed in Table IV.B o y s localization of the occupied and virtual spaces serves to atten- uate the real-space extent of the individual subspace spannedby each fragment’s orbitals while, of course, leaving the full span intact. The consequence of this is a considerable im- provement in the stability of the method. The dependenceis increasingly noticeable in the doubly diffuse trend since a larger spatial extent allows the converged polMOs a degree of artiﬁcial charge-transfer energy-lowering that the ALMOsenjoy, albeit less dramatic. Both the Boys localization pro- cedure and that the subspaces associated with different frag- ments have no overlap serves to attenuate the contributions to polarization associated with the polMO description while still providing the variational ﬂexibility in the full space of084102-9 Azar et al. J. Chem. Phys. 138 , 084102 (2013) FIG. 2. The character of the C swater dimer interaction is basis-set dependent in the SCF MI scheme, but stable in the minimal-basis scheme. virtual functions associated with second-order perturbation theory. B. DFT decomposition quantities and exchange effects A post-mean-ﬁeld treatment of intermolecular interac- tions is vital for any serious application of the EDA, and DFT represents a parsimonious ﬁrst thrust in this direction. When TABLE IV. polMO polarization contributions in the basis set extension de- pend slightly on the localization, increasing gently with diffusivity. basis polMO(Boys) polMO(w/o Boys) ALMO dz 2.22 2.19 2.78 tz 2.54 2.66 3.69 qz 2.57 2.64 4.495z 2.79 2.77 4.97 aug-dz 2.79 2.98 4.15 aug-tz 2.93 3.33 6.19aug-qz 2.91 3.07 6.63 aug-5z 2.87 3.40 7.11 d-aug-dz 2.83 3.24 4.40d-aug-tz 2.96 3.74 6.93 d-aug-qz 3.06 3.62 7.81 FIG. 3. Component magnitudes in the aug-dz basis scale with e.e. in accor- dance with the form of the decomposition term described. The 20% e.e. point corresponds to optimized B3LYP. the exchange contribution is adjusted and inter-electronic cor- relations are included, the results of the equilibrium decom-position differ from those at the HF level (Table V). Larger polarization and charge transfer effects tend to result from smaller HOMO-LUMO gaps (standard functionals tend tounderestimate the gap, 66,67while HF overestimates it). Of course electron correlation effects generally strengthen inter- molecular interactions, so the HF values should not be re- garded as true. Frozen interactions are sensitive to the dipole moment, which is overestimated at the HF level. Thus densityfunctionals may typically exhibit less favorable frozen inter- actions than HF, but based on this criterion, the resulting value should be more reliable. The frozen interactions are also sensitive to the treatment of exchange, but it is difﬁcult to guess the effect of func- tional approximations on this term. To test the dependenceof all EDA terms on the composition of a density functional more carefully, we vary the amount of exact exchange (e.e.) in the three-parameter B3LYP exchange-correlation potentialexplicitly, keeping Slater exchange at a constant 8% and ad- justing %B88 exchange to allow for the desired %HF ex- change. Consistent with the general considerations alreadygiven, the results of Fig. 3suggest roughly linear behavior of frz and shallow inverse dependence of ct with e.e., and weak-to-zero e.e.-dependence of pol in either scheme. Since the B3LYP functional energy is linear in the HF exchange pa- rameter, we can only expect the total energy to scale linearlywith e.e. ap r i o r i , as observed. That frz and ct clearly depend on e.e. in a way consistent with their deﬁnitions demonstrates correspondence between the terms of our decompositions anda totally independent metric describing exchange forces. In other words, these terms appear well-suited to describe the physical phenomena for which they were designed. We note that dispersion is not considered explicitly in the energy decomposition, and even if a dispersion-corrected functional is employed, the correction will formally be spreadout between the decomposition terms, none of which is alone adequate to entirely capture this force. We might, however, expect the leading effect of dispersion to be contained in thefrozen interactions, with smaller, density-dependent correc- tions contained in the polarization and charge-transfer terms. If the dispersion is not density-dependent at all, such as for “-D” functionals, then it will be entirely contained in the frozen term, As an illustration, we decompose the essentially084102-10 Azar et al. J. Chem. Phys. 138 , 084102 (2013) TABLE V. KS-DFT polMO augmented-series components (kJ/mol) in the basis and exact exchange (e.e.). Charge transfer decreases with increasing e.e., while frozen interactions increase. contribution/basis frz pol ct bind XC/%e.e. dz tz qz dz tz qz dz tz qz dz tz qz B3LYP/20685.92 5.80 5.53 2.49 2.73 2.85 11.02 10.38 10.60 19.43 18.91 18.99 M06L/0695.57 6.45 4.96 2.41 2.60 2.77 10.81 10.90 12.37 18.78 19.95 20.09 M06/27707.56 7.16 5.67 2.59 2.72 2.74 10.05 10.24 11.51 20.21 20.11 19.92 M06-2X/547010.59 10.50 9.87 2.58 2.95 2.90 8.61 8.11 8.59 21.77 21.56 21.36 PBE/0713.22 3.11 3.30 2.50 2.86 2.98 12.54 11.66 11.92 11.82 11.41 11.60 PBE0/25728.48 7.92 7.60 2.55 2.71 2.82 10.48 10.15 10.40 21.50 20.78 20.82 quantitative interaction energy for water dimer furnished by the range-separated ωB97X-D73hybrid functional in Table VI.T h e ωB97X-D binding energy approaches the CCSD(T)/CBS extrapolation of 20.8kJ/mol.65With density- independent dispersion, it is no surprise that the frozen in-teraction is some ∼2kJ/mol larger in magnitude than at the B3LYP level. Comparing the most accurate decompositions atωB97X-D and B3LYP, the “-D” augmentation of the frozen interaction appears to be the principle contribution to the binding energy difference between them. C. Breaking the hydrogen bond of the water dimer If one goal of the EDA is that it be a true quantum- mechanical basis for force ﬁeld parameters in molecular me- chanics simulations, the EDA components should be well-behaved across the potential energy surface, each weighted in accordance with the true intermolecular force it designates and decaying to zero at the dissociation limit. Thus, whenthe two molecules of a dimer are squeezed together along some interaction coordinate, Pauli and electrostatic repulsions will begin to trump all other forces. Conversely long-rangeelectrostatic forces should exert their effects long before the wavefunction assumes the equilibrium supermolecular con- ﬁguration that will be determined on polarization and chargetransfer. Polarization gives a sense in which permanent poles are deformed in the supermolecular ﬁeld and should decay classically as the inverse of some (induced) multipole orderin the interaction coordinate R, while charge transfer is con- TABLE VI. ωB97X-D-level decomposition components in kJ/mol. Most of the dispersion is captured in frozen electrostatics. basis frz pol(polMO) pol(ALMO) ct(polMO) ct(ALMO) bind dz 11.22 2.19 2.78 19.60 19.01 33.01 tz 10.94 2.32 3.69 12.56 11.20 25.83 qz 9.51 2.57 4.49 10.94 9.07 23.025z 8.48 2.75 4.97 10.40 8.18 21.63 aug-dz 8.79 2.53 4.15 10.24 8.62 21.56 aug-tz 8.16 2.73 6.19 10.11 6.65 20.99aug-qz 7.52 2.77 6.63 10.03 6.66 20.81 aug-5z 7.44 2.77 7.11 10.43 6.08 20.64 d-aug-dz 8.61 2.53 4.40 10.11 8.68 21.24d-aug-tz 7.99 2.72 6.93 10.36 6.15 21.08 d-aug-qz 7.46 2.90 7.81 10.62 5.65 20.98tingent on intermolecular overlaps and dies off exponentially, and so at the very least these components must cross. Be-cause the PES is sampled a great deal in the course of ther- mal ﬂuctuations, an accurate description of the interaction potential along the entire weak-bond-breaking coordinate isindispensable. We plot the potential energy across the H-bond-breaking coordinate of the water dimer in Fig. 4, showing the contri- bution of the polMO terms for B3LYP calculations on the left-hand side, and a log-log plot showing the R-dependence FIG. 4. (a) polMO binding components of B3LYP/aug-cc-pvdz C swater dimer traversing its H-bond-breaking coordinate have the correct limiting be-havior and a complicated binding interaction; (b) log( /Delta1E)-log(r) plot of the frozen and polarization contributions indicate scaling consistent with appro- priate classical inverse square-power.084102-11 Azar et al. J. Chem. Phys. 138 , 084102 (2013) of the non-ct terms on the right-hand side. From Fig. 4(b), we observe the appropriate distance-dependence of all elec- trostatic terms in the long-range limit (at a separation greaterthan 1.5 R eqwhere this asymptotic analysis becomes valid). Frozen interactions are dominated by dipole-dipole interac- tions, while the polarization terms are dominated by dipole-induced dipole contributions. It is only in the strictly non- overlapping regime that we should expect slopes of exactly three and six for frozen and polarization contributions re-spectively. Inclusion of quantum mechanical exchange in the Hamiltonian will be responsible for slight deviations away from the correct curve. The effective power law behavior for decay of the polar- ization via the ALMO and polMO treatments are particularlyinteresting. Both ALMO and polMO should be exact (within the chosen basis) in the non-overlapping regime (as the in- teraction becomes weak enough that the perturbative modelfor minimal virtual functions in the polMO method becomes increasingly suitable). At small separations, we have argued that a part of the ALMO polarization is in fact attributableto charge transfer, and this error will increase as the basis set is improved. Since this error diminishes with the overlap of the fragments, its presence should cause the ALMO powerlaw for decay of the polarization contribution to exceed six. The data show a power law with exponent 6.1 for the ALMO model. On the other hand, the polMO model, with polariza- tion described in a minimal orthogonal space, will likely underestimate the polarization contribution in the strongly overlapping regime while approaching exactness in the non- overlapping limit. This will result in a power law exponentof less than six (5.9 for the polMO model). We conclude that the data shown in Fig. 4(b) is consistent with polMO polariza- tion being a lower-limit estimate of the true polarization in theoverlapping regime, while the ALMO polarization should be regarded as an upper limit to polarization in the overlapping regime. Armed with the potential surface, we can read off the story of the gas-phase water dimer interaction from the right in Fig. 4(a): Two approaching water molecules in the ap- propriate orientation ﬁrst see each other’s vacuum dipoles at a separation greater than 3 R eq. As they approach closer along the axis that becomes a hydrogen bond, dipole polar-ization occurs along the H-bonding axis (in Fig. 1, the polMO p(O) donor has changed its orientation w.r.t. the symmetry axis of its vacuum analogue to better respond to the acceptorσ(OH) in its ﬁeld). As separation of the two water molecules is further decreased, the dative interaction, which decreases the equilibrium H-bond length for the dimer well within the frozen minimum by some 0.5Å begins to rapidly increase. At the equilibrium geometry, all three types of contributions(frozen, polarization, and charge-transfer) are important. D. The Na+CH4monopole-induced-dipole polarization The case of the water dimer illustrates what we believe is the most common paradigm in complex intermolecular in- teractions: there is a rich distance-dependent admixture of permanent (frozen), polarization, and charge transfer inter-TABLE VII. B3LYP decomposition components of the polarization- dominated Na+CH 4interaction in kJ/mol. Basis frz pol(ALMO) pol(polMO) ct(ALMO) ct(polMO) Bind dz 0.72 20.62 15.30 12.75 18.07 34.09 tz 0.74 26.68 19.08 5.17 12.79 32.59qz 0.82 29.54 24.74 3.34 8.13 33.70 5z 0.62 30.99 25.48 0.76 7.89 33.99 aug-dz −0.18 30.74 21.87 1.77 10.64 32.33 aug-tz 0.54 31.58 19.02 1.19 13.75 33.31 aug-qz 0.64 32.08 27.70 2.63 6.96 35.34 aug-5z a0.53 32.23 25.84 2.19 8.45 34.83 aNa was treated at the 5z level. actions. It is also useful to brieﬂy examine an interaction in which charge transfer effects are expected on chemicalgrounds to be negligible, while polarization effects are very important. Such a case will be a potentially difﬁcult challenge for the polMO approach, because it generally will underes-timate polarization, and therefore overestimate ct. In a case where ct is negligible, such a result would be spurious. A speciﬁc system that is anticipated to have negligible ct is the Na +CH 4interaction, which one may intuitively think of as a problem of describing a Stark-shifted methane molecule. Thepositive charge resides on Na +, and its occupied orbitals are very deep in energy, and therefore donation into σ* orbitals on methane is blocked. For ct in the other direction, methaneis a poor donor, and Na +does not have low-lying afﬁnity or- bitals, so ct is expected to be very small. The polMO and ALMO decomposition terms for the Na+CH 4interaction were obtained from a 6-311++G**/B3LYP-optimized geometry and are given in Table VII for a wide range of basis sets (without counter- poise correction). In both schemes, polarization is the chief contribution to binding with a disparity between the decom- positions decreasing steeply on inclusion of higher-angularmomentum functions to ∼5 and∼6 kJ/mol for 5z and aug-5z calculations, respectively. If it is accepted based on the arguments above that there is no “real” charge-transfer in this interaction, then it follows that ALMO polarization should be stable and an adequate estimate of the true polarization.The numerical results support this contention, as the ALMO polarization is converging smoothly in both basis set se- quences, and ct is very small. ALMO-based ct is roughly 7%of the polarization value in the largest basis set reported in Table VII. The polMO results improve signiﬁcantly in the larger ba- sis sets, in the sense that polMO polarization generally in- creases in magnitude as the basis set is improved. The gap between ALMO ct and polMO ct diminishes ranges from 4 to12 kJ/mol depending on the basis set, with somewhat smaller differences for the larger basis sets. Relative to the size of the interaction (35 kJ/mol with this density functional), thegap between ALMO and polMO ct is only about 20% in the larger basis sets, so both treatments give a qualitatively similar picture of a polarization-dominated interaction. These considerations bolster the contention that polMO polarization represents a lower-limit estimate, while ALMO polarization084102-12 Azar et al. J. Chem. Phys. 138 , 084102 (2013) represents an upper limit. This issue will be discussed further in Sec. III E . E. Bracketing intrinsic polarization effects We understand the polMO minimal-basis polarization as a lower bound to true polarization since we neglect the vec- tors inRduring polarization, as well as enforce orthogonality between the variational spaces associated with different frag- ments. While neglect of Rhas zero error at the second-order perturbation level, it must lead to an underestimate relativeto the energy-lowering evaluated with a fragment-based par- titioning of the orbital space that includes all functions. The magnitude of this difference will depend strongly on the ge-ometry, since the error tends to zero when perturbation the- ory amplitudes approximate the space of the true polarized wavefunction well (which will be the case when polarizationis small). Thus in the polMO scheme, charge-transfer effects are overestimated, or, from a physical standpoint, they contain some contaminating polarization contributions. By contrast, as we discussed in detail in Sec. II A,u s e of ALMO polarization will tend to be an overestimate, be- cause the non-orthogonal one-particle Hilbert spaces of dif- ferent fragments have an intersection, and that intersection in- creases as the basis set is improved. Thus ALMO polarizationis contaminated with some energy-lowering that is in fact re- lated to charge transfer. We therefore bracket true polarization as lying between the upper-bound ALMO polarization and thelower-bound minimal-basis polMO polarization. How large or small is the difference between the upper and lower bounds for the water dimer at the basic Hartree-Fock level of theory, at equilibrium? The tightest upper bound for polarization by the ALMO approach comes from the smallest value of polarization when the sum of polar-ization and charge transfer energy is converged. Referring back to Table I, the smallest upper limit might be the cc- pVQZ pol(ALMO), at ∼4.6 kJ/mol. If we take the lower- estimate pol(polMO) quantity as ∼3.0 kJ/mol, then true po- larization is bounded within a 1.6 kJ/mol range (3.0, 4.6). We would obtain an essentially identical result with the ωB97X- D functional, as can be seen from Table VI. For HF at the compressed geometry shown in Table II, the corresponding bracket for true polarization is (4.9,8.1) kJ/mol, which is nec- essarily wider because of the increased overlap between the fragments. At the stretched geometry shown in Table III,t h e estimated bracket for true polarization is narrower, at (0.4,0.6) kJ/mol. Finally, for comparison, we can infer a bracket for polarization of roughly (26,30) kJ/mol for the polarization-dominated Na +CH 4interaction using the data from Table VII. What are the implications for the overestimation of po- larization effects in the ALMO EDA as it is commonly em-ployed? Let us consider the water dimer at the equilibrium geometry again, and assume that a standard application of the ALMO EDA employs the aug-cc-pVTZ basis. In that case,the calculated ALMO polarization is ∼5.6 kJ/mol, while our bracket for true polarization is (3.0, 4.6) kJ/mol. We there- fore conclude that polarization is overestimated by at least 1 kJ/mol, and not more than 2.6 kJ/mol in the ALMO EDA/aug- cc-pVTZ method. Thus the true polarization is no less than20% smaller than the ALMO value, and it could be as much as 46% smaller. Since the errors depend on the identity of the basis it is likely that they are quite systematic, so trendsin the ALMO polarization estimate are likely to be reliable. Nonetheless, it is clear that improved procedures for calcu- lating polarization are important for future work. The polMOmethod is one such candidate. IV. CONCLUSIONS AND OUTLOOK In energy decomposition analysis of intermolecular in- teractions, one important issue is disentangling the separate contributions associated with intramolecular polarization in the ﬁeld of neighboring molecules and intermolecular charge transfer (dative bonding) between molecules. This issue is challenging because such a separation in all likelihood cannotbe uniquely deﬁned in the important regime where the molec- ular partners overlap signiﬁcantly. We have studied some as- pects of this issue with the aim of attempting to understandstrengths and weaknesses of existing EDAs, and introduce a new and complementary approach. Our main results and con- clusions are as follows: 1. We have demonstrated that fragment-blocking the molecular orbital coefﬁcient matrix as employed in theALMO EDA 31,32and the related BLW-EDA29,30overes- timates the energy-lowering due to polarization effects in intermolecular interactions. In essence, this arises be-cause the one-particle Hilbert spaces of different frag- ments are allowed to have non-zero intersection, and the extent of the intersection increases with the size ofthe basis set. Therefore in the ALMO EDA, the energy- lowering due to polarization becomes contaminated with charge-transfer effects as one improves the basis set. 2. We have developed a new method that uses fragment- blocked variations to obtain a minimal basis of polar- ized orthogonal local MOs (polMOs) describing stabi-lization due to polarization. Only one polMO is pro- vided per occupied MO of the isolated fragments by SVD of the ﬁrst-order polarization response on each fragment followed by symmetric orthogonalization and relocalization. 3. The polMO approach will underestimate polarization because strict orthogonality is maintained between vari- ational subspaces that describe polarization on differ-ent fragments, and a large fraction of the virtual orbitals is discarded. Therefore, taken together, the ALMO and polMO estimates of polarization are expected to bracketthe true value. 4. Numerical tests of the ALMO and polMO polarization energies have been carried out on the water dimer us-ing a large sequence of cc-pVXZ, aug-cc-pVXZ and d-aug-cc-pVXZ (X =D,T,Q,5) basis sets. The polMO scheme is stable with respect to basis set extensionseven in the strongly overlapping regime. By contrast, the ALMO polarization contribution is not stable with respect to basis set extensions. Analysis of the power law decay of ALMO and polMO polarization as a func- tion of intermolecular distance is consistent with ALMO084102-13 Azar et al. J. Chem. Phys. 138 , 084102 (2013) overestimating and polMO underestimating polariza- tion. Results were also calculated for the Na+CH 4in- teraction, which is dominated by polarization in both theALMO and polMO approaches. 5. Within the Hartree-Fock method, for the water dimer at the equilibrium geometry, the estimated range withinwhich the true polarization energy-lowering lies is (3.0, 4.6) kJ/mol. 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1.3056407.pdf	Amplitude-phase coupling in a spin-torque nano-oscillator Kiwamu Kudo, Tazumi Nagasawa, Rie Sato, and Koichi Mizushima Citation: J. Appl. Phys. 105, 07D105 (2009); doi: 10.1063/1.3056407 View online: http://dx.doi.org/10.1063/1.3056407 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v105/i7 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 05 Jun 2013 to 128.226.37.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsAmplitude-phase coupling in a spin-torque nano-oscillator Kiwamu Kudo,a/H20850T azumi Nagasawa, Rie Sato, and Koichi Mizushima Corporate Research and Development Center, Toshiba Corporation, Kawasaki 212-8582, Japan /H20849Presented 12 November 2008; received 17 September 2008; accepted 15 October 2008; published online 3 February 2009 /H20850 The spin-torque nano-oscillator in the presence of thermal ﬂuctuation is described by the normal form of the Hopf bifurcation with an additive white noise. By the application of the reductionmethod, the amplitude-phase coupling factor, which has a signiﬁcant effect on the power spectrumof the spin-torque nano-oscillator, is calculated from the Landau–Lifshitz–Gilbert–Slonczewskiequation with the nonlinear Gilbert damping. The amplitude-phase coupling factor exhibits a largevariation depending on an in-plane anisotropy under the practical external ﬁelds. © 2009 American Institute of Physics ./H20851DOI: 10.1063/1.3056407 /H20852 When a direct current Iﬂows into a magnetoresistive /H20849MR /H20850device, a stationary magnetic state becomes unstable and a steady magnetic oscillation is excited by the spin-transfer torque. The oscillation is expected to be applicableto a nanoscale microwave source, i.e., the spin-torque nano-oscillator /H20849STNO /H20850. 1,2According to the theory based on the spin-wave Hamiltonian formalism,3–6the frequency nonlin- earity plays a key role to determine the behavior of the os-cillator. It has been shown that the strong frequency nonlin-earity leads to signiﬁcant effects on the power spectrum ofSTNO in the presence of thermal ﬂuctuation: a linewidthenhancement 5and non-Lorentzian lineshapes.6In this paper, the important nonlinearity is examined. From the Landau–Lifshitz–Gilbert–Slonczewski /H20849LLGS /H20850equation as the model of STNO, we explicitly calculate the magnitude of the quan-tity corresponding to the normalized frequency nonlinearityN//H9003 eff/H20851see, e.g., Eq. /H208494/H20850in Ref. 6/H20852of the spin-wave ap- proach. In particular, we take account of the in-plane aniso-tropy of a magnetic ﬁlm which has been neglected in theearly studies, 3–6ﬁnding the large effect of the anisotropy on the nonlinearity. We describe STNO by a generic oscillator model. It is known that small-amplitude oscillations near the Hopf bifur-cation point are generally governed by a simple evolutionequation for a complex variable W/H20849t/H20850known as the Stuart- Landau /H20849SL/H20850equation. 7The SL equation is derived as a nor- mal form of the supercritical Hopf bifurcation from the gen-eral system of ordinary differential equations. Accordingly,the LLGS equation similarly reduces to the SL equation inthe case where the Hopf bifurcation, which represents a gen-eration of magnetic oscillations in STNO, occurs. The reduc- tion of the LLGS equation can be executed by the reductiveperturbation method based on the center-manifold theorem.At ﬁnite temperature, there exists inevitable thermal magne-tization ﬂuctuation in STNO. 8,9We include the thermal effect into the magnetization dynamics by just adding white noiseterm to the SL equation, i.e., STNO in the presence of ther-mal ﬂuctuation is described by the “noisy” Hopf normalform as follows:dW ˜ dt˜=i/H9024˜W˜+/H208491+i/H9254/H20850/H20849p−/H20841W˜/H208412/H20850W˜+/H9257˜/H20849t˜/H20850, /H208491/H20850 where W˜is the normalized complex variable representing the amplitude and phase of a magnetization vector M/H20851see Eq. /H208497/H20850below /H20852. In Eq. /H208491/H20850,/H9024˜represents a fundamental frequency, t˜is a normalized dimensionless time, and /H9257˜/H20849t˜/H20850is the zero- mean, white Gaussian noise with the only nonvanishing sec- ond moment given by /H20855/H9257˜/H20849t˜/H20850/H9257˜¯/H20849t˜/H11032/H20850/H20856=4/H9254/H20849t˜−t˜/H11032/H20850.pis the bifur- cation parameter. An oscillation is generated when p becomes positive. In the context of STNO, p/H11008/H20849I−Ic/H20850where Icis the threshold current. The parameter /H9254quantiﬁes the coupling between the amplitude and phase ﬂuctuations and iscalled the amplitude-phase coupling factor .I ti s /H9254that we calculate numerically in this paper and that corresponds tothe normalized frequency nonlinearity N//H9003 effof the spin- wave approach. The amplitude-phase coupling factor /H9254af- fects the power spectrum of an oscillator and leads to a line-width enhancement and non-Lorentzian lineshapes. 10,11Due to its effect, the factor /H9254is also called the linewidth enhance- ment factor .12Equation /H208491/H20850is often used as the simplest model of a noisy auto-oscillator in many ﬁelds, for example,electrical engineering, chemical reactions, optics, biology,and so on. 10,13Therefore, we can easily compare STNO with conventional oscillators and clarify its features. The amplitude-phase coupling factor /H9254is obtained in the procedure of the reduction of the LLGS equation. In thefollowing, we ﬁrst explain the LLGS equation. Then, follow-ing Kuramoto’s monograph, 7we consider an instability of a steady solution and execute the reduction of the LLGS equa-tion. The magnetic energy density of the free layer of STNO is assumed to have the form E=−M·H ext−Ku Ms2/H20849M·xˆ/H208502+1 24/H9266M·N·M, /H208492/H20850 where Msis the saturation magnetization, Hext=Hxxˆ+Hyyˆ +Hzzˆis an external ﬁeld, Kuis an uniaxial anisotropy along thexdirection, and Nis the demagnetizing tensor; N =diag /H20849Nx,Ny,Nz/H20850. Using the spherical coordinate system /H20849seea/H20850Electronic mail: kiwamu.kudo@toshiba.co.jp.JOURNAL OF APPLIED PHYSICS 105, 07D105 /H208492009 /H20850 0021-8979/2009/105 /H208497/H20850/07D105/3/$25.00 © 2009 American Institute of Physics 105 , 07D105-1 Downloaded 05 Jun 2013 to 128.226.37.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsFig.1/H20850, we describe the magnetization dynamics of STNO by the LLGS equation /H20877cos/H9274/H9278˙=−/H9251/H20849/H9264/H20850/H9274˙−F1/H20849/H9278,/H9274,/H9275J/H20850 /H9274˙=/H9251/H20849/H9264/H20850cos/H9274/H9278˙+F2/H20849/H9278,/H9274,/H9275J/H20850,/H20878 /H208493/H20850 where F1/H20849/H9278,/H9274,/H9275J/H20850/H11013/H20849/H9253/Ms/H20850/H11509E//H11509/H9274−a/H20849/H9278,/H9275J/H20850 and F2/H20849/H9278,/H9274,/H9275J/H20850/H11013/H20851/H9253//H20849Mscos/H9274/H20850/H20852/H11509E//H11509/H9278+b/H20849/H9278,/H9274,/H9275J/H20850./H9253is the gyromagnetic ratio. The second terms of Firesult from the Slonczewski term TJ=/H20849/H9253aJ/Ms/H20850M/H11003/H20849M/H11003p/H20850in which aJis proportional to the current density Jthrough the free layer.14 Therefore, a/H20849/H9278,/H9275J/H20850/H11013/H9275Jcos/H9274psin/H20849/H9278−/H9278p/H20850and b/H20849/H9278,/H9274,/H9275J/H20850 /H11013/H9275J/H20851cos/H9274psin/H9274cos/H20849/H9278−/H9278p/H20850−sin/H9274pcos/H9274/H20852, where /H9275J=/H9253aJ. /H9251/H20849/H9264/H20850terms of Eq. /H208493/H20850are the generalized Gilbert damping terms proposed by Tiberkevich and Slavin.15We take into account only the ﬁrst nontrivial term of the Taylor seriesexpansion for /H9251/H20849/H9264/H20850by the magnetization change rate /H9264 /H11013/H20849/H11509m//H11509t/H208502//H20849/H92534/H9266Ms/H208502;/H9251/H20849/H9264/H20850=/H9251G/H208491+q1/H9264/H20850. According to Ref. 15, the nonlinear LLGS model with q1=3 gives a good agreement with the experimental results of Refs. 1and16. An instability of a steady solution of Eq. /H208493/H20850is consid- ered. A steady solution /H20849/H92780/H20849/H9275J/H20850,/H92740/H20849/H9275J/H20850/H20850is derived from Fi/H20849/H92780,/H92740,/H9275J/H20850=0. Shifting the variables as u1/H11013/H9278−/H92780and u2/H11013/H9274−/H92740, we have the Taylor series of Eq. /H208493/H20850as follows: u˙=Lu+N2uu+N3uuu+¯, /H208494/H20850 where u=/H20849u1,u2/H20850T. Here, the diadic and triadic notations7 have been used. The stability of a steady solution is deter- mined by the eigenvalues of the linear coefﬁcient matrix L: /H9261/H11006=/H9003/H11006 /H20849/H90032−det L/H208501/2./H9003is deﬁned as /H9003=/H9003/H20849/H9275J/H20850/H11013/H208491/2/H20850trL and plays the role as a control parameter since it depends on /H9275J. We conﬁne ourselves to the case where the Hopf bifur- cation occurs. Then, /H9261/H11006is a pair of complex-conjugate ei- genvalues. The point, /H9003=0, is the Hopf bifurcation point; while a steady solution remains stable for /H9003/H110210, it becomes unstable for /H9003/H110220. The bifurcation point corresponds to the threshold /H9275Jcwhich is determined by tr L=0 and Fi/H20849/H92780,/H92740,/H9275Jc/H20850=0. Near the bifurcation point, we divide L into the two parts: L=L0+/H9003L1, where L0is the critical part and/H9003L1is the remaining part. Corresponding to L,/H9261+is also divided into the two parts; /H9261+=/H92610+/H9003/H92611. Although L1and/H92611 generally depend on /H9003further, we neglect their dependence and evaluate them by the values at /H9003=0. Accordingly, /H92610 =i/H92750and/H92611=1−1 2i/H92750/H20879d d/H9003detL/H20879 /H9003=0, /H208495/H20850 where /H92750/H11013/H20881detL0. The right and left eigenvectors of L0 corresponding to the eigenvalue /H92610are denoted as UandU, respectively. These are normalized as UU=UU¯=1 where U¯means a complex conjugate of U. Let us apply the reduction method to Eq. /H208494/H20850. The SL equation for a complex amplitude W/H20849t/H20850, W˙=/H9003/H92611W−g/H20841W/H208412W /H208496/H20850 and the neutral solution for the magnetization dynamics, /H20873/H9278 /H9274/H20874=/H20873/H92780 /H92740/H20874+W/H20849t/H20850ei/H92750tU+W¯/H20849t/H20850e−i/H92750tU¯/H208497/H20850 are obtained within the lowest order approximation.7Under the approximation, only the Taylor expansion coefﬁcients upto the third order are needed. The complex constant gin Eq. /H208496/H20850is given by g/H11013 /H92631+i/H92632=−3 /H20849U,N3U¯UU /H20850+4/H20849U,N2UV 0/H20850 +2/H20849U,N2U¯V+/H20850, /H208498/H20850 where V0=L0−1N2UU¯and V+=/H20849L0−2i/H92750/H20850−1N2UU. The amplitude-phase coupling factor /H9254is obtained from the com- plex constant gand is given by /H9254=/H92632//H92631. /H208499/H20850 In this way, the factor /H9254for STNO can be calculated numeri- cally from the parameters of the LLGS equation. The noisy Hopf normal form given by Eq. /H208491/H20850is derived when we add the noise term f/H20849t/H20850with /H20855f/H20849t/H20850f¯/H20849t/H11032/H20850/H20856=4D/H92532/H9254/H20849t −t/H11032/H20850to the SL Eq. /H208496/H20850.f/H20849t/H20850has the dimension of a magnetic ﬁeld. The components in Eq. /H208491/H20850are deﬁned as W˜/H20849t/H20850=/H20849D/H92532//H92631/H20850−1 /4W/H20849t/H20850ei/H20849/H92750+/H9003/H9254−/H9003Im/H92611/H20850t,t˜/H11013/H20881D/H92532/H92631t,p /H11013/H9003//H20881D/H92532/H92631, and/H9024˜/H11013/H92750//H20881D/H92532/H92631. Therefore, we can make the most of many well-known properties of Eq. /H208491/H20850/H20849Refs. 10 and11/H20850to examine the behavior of STNO. It is known, for example, that the spectrum linewidth /H9004/H9275FWHM far above the threshold /H20849p/H333560/H20850is increased by a factor of /H208491+/H92542/H20850.10In the context of STNO, when /H9003/H333560, the linewidth can be ex- pressed as /H9004/H9275FWHM =/H9004/H9275res/H11003kBT Eosci/H110031 2/H208491+/H92542/H20850, /H2084910/H20850 which corresponds to Eq. /H2084911/H20850in Ref. 5. Here, kBTis the thermal energy. /H9004/H9275resis the linewidth at thermal equilibrium /H20849/H9275J=0/H20850given by /H9004/H9275res=2/H9003eq, where /H9003eq/H11013−/H9003/H20849/H9275J=0/H20850. More- over, Eosciis the magnetization oscillating energy and can be written as Eosci/H112292U†/H20851/H11509/H20849/H11509u1E,/H11509u2E/H20850//H11509/H20849u1,u2/H20850/H20852u=0UPWVfree =1 2/H20849/H9003eqkBT/D/H92532/H20850PWwhen it is assumed that Eosci/H11229kBTnear thermal equilibrium /H20849energy equipartition /H20850. Here, Vfreeis the volume of the free layer and PWis the total power of W/H20849t/H20850 given by PW=/H20881D/H92532//H92631/H20853p+2 /F/H20849p/H20850/H20854with F/H20849p/H20850/H11013/H20881/H9266ep2/4/H208511 +erf /H20849p/2/H20850/H20852. From the expression of Eq. /H2084910/H20850, it is found that the MR device in STNO itself is nothing but a resonator on the analogy of electrical circuits. The other one of well-φHH M xHxyyz ψz uK pI FIG. 1. The spherical coordinate system /H20849/H9278,/H9274/H20850for the direction of the free layer magnetization m=M/Msof STNO. pdenotes the direction of the pinned layer magnetization; p=/H20849cos/H9274pcos/H9278p,cos/H9274psin/H9278p,sin/H9274p/H20850.07D105-2 Kudo et al. J. Appl. Phys. 105 , 07D105 /H208492009 /H20850 Downloaded 05 Jun 2013 to 128.226.37.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsknown properties of Eq. /H208491/H20850is that the amplitude-phase cou- pling factor distorts the power spectrum to non-Lorentzianlineshapes especially near the threshold /H20849see, e.g., Fig. 5 of Ref. 11/H20850. The degree of the lineshape distortion is determined by the magnitude of /H9254andp, corresponding to the calcula- tion in Ref. 6. We comment on the validity of Eq. /H208491/H20850for large-amplitude oscillations. In Fig. 2, the theoretical ﬁtting curves based on the model Eq. /H208491/H20850are compared with the experimental data of Ref. 16and give a good agreement with them up to I/H112295.6 mA /H20849p/H112298.2/H20850beyond the threshold current Ic=4.8 mA /H20849p=0/H20850estimated by the ﬁtting.17Therefore, al- though the derivation of Eq. /H208491/H20850is based on a perturbation expansion around the bifurcation point, it is considered to bevalid for rather large-amplitude oscillations with p/H1101110. We brieﬂy mention the oscillating frequency /H9275osci. From Eqs. /H208491/H20850and /H208497/H20850, the oscillating frequency of a free layer magnetization far above threshold is written as /H9275osci=/H92750 −/H9003/H9254+/H9003Im/H92611. Although the results of the calculation for Im/H92611from Eq. /H208495/H20850are not shown here, we have found that this quantity has a small value with Im /H92611/H11011/H9251Gfor wide range of parameters of the LLGS equation. Accordingly, /H9275osciis approximately given by /H9275osci/H11229/H92750−/H9003/H9254. Since /H9003/H11008 /H20849I −Ic/H20850, while the frequency /H9275oscidecreases as the current I/H20849/H11022Ic/H20850increases when /H9254/H110220/H20849redshift /H20850,/H9275osciincreases when /H9254/H110210/H20849blue shift /H20850in accordance with the spin-wave models.3–6 As illustrated above, the amplitude-phase coupling fac- tor/H9254plays a key role to determine the behavior of an oscil- lator. Therefore, the features of STNO can be found out bythe calculation of /H9254. Some calculation examples of /H9254are shown in Fig. 3.I ti s considered the case where a free layer is an in-plane mag-netic ﬁlm with an in-plane external ﬁeld applied along the x direction, H ext=Hxˆ. It is assumed that N=diag /H208490,0,1 /H20850,/H9251G =0.02, and /H20849/H9278p,/H9274p/H20850=/H208490,0 /H20850. In Fig. 3/H20849a/H20850, the dependence of /H9254 on the nonlinearity of the damping q1is shown. It is found that/H9254monotonically decreases for q1and the variation of /H9254 is very large. This result suggests that a nonlinear damping signiﬁcantly changes the LLG dynamics.15In Fig. 3/H20849b/H20850, thedependence of /H9254on an external magnetic ﬁeld Hfor various values of an uniaxial anisotropy ﬁeld Hk/H20849=2Ku/Ms/H20850is shown. The nonlinearity of the damping is taken as q1=3.15 In the practical external ﬁeld region, /H9254is very sensitive to an uniaxial anisotropy ﬁeld and varies largely. Therefore, whenthe dynamics of STNO is considered, it is necessary to takethe effect of a uniaxial anisotropy ﬁeld into account seri-ously. This is the main result of the present paper. In summary, we have considered the dynamics of STNO by reducing the LLGS equation to a generic oscillator modeland calculated explicitly the amplitude-phase coupling factorwhich is the key factor for the power spectrum. Theamplitude-phase coupling factor /H9254is very sensitive to mag- netic ﬁelds, in-plane anisotropy, and the nonlinearity ofdamping. The large variation of /H9254is the remarkable feature of STNO in comparison with conventional oscillators. Thecalculation way for /H9254shown is applicable for an arbitrary magnetization conﬁguration and may be useful for ﬁnding astable STNO with small /H9004 /H9275FWHM /H20851Eq. /H2084910/H20850/H20852, which is pref- erable for applications. 1S. I. Kiselev et al. ,Nature /H20849London /H20850425, 380 /H208492003 /H20850. 2W. H. Rippard et al. ,Phys. Rev. Lett. 92, 027201 /H208492004 /H20850. 3A. N. Slavin and P. Kabos, IEEE Trans. Magn. 41, 1264 /H208492005 /H20850. 4V. Tiberkevich, A. N. Slavin, and J.-V. Kim, Appl. Phys. Lett. 91, 192506 /H208492007 /H20850. 5J.-V. Kim, V. Tiberkevich, and A. N. Slavin, Phys. Rev. Lett. 100, 017207 /H208492008 /H20850. 6J.-V. Kim et al. ,Phys. Rev. Lett. 100, 167201 /H208492008 /H20850. 7Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence /H20849Springer- Verlag, Berlin, 1984 /H20850, Chap. 2. 8J.-V. Kim, Phys. Rev. B 73, 174412 /H208492006 /H20850. 9K. Mizushima, K. Kudo, and R. Sato, J. Appl. Phys. 101, 113903 /H208492007 /H20850. 10H. Risken, Fokker-Planck Equation /H20849Springer-Verlag, Berlin, 1989 /H20850, Chap. 12. 11J. P. Gleeson and F. O’Doherty, SIAM J. Appl. Math. 66, 1669 /H208492006 /H20850. 12C. H. Henry, IEEE J. Quantum Electron. QE-18 , 259 /H208491982 /H20850. 13H. Haken, Advanced Synergetics /H20849Springer-Verlag, New York, 1993 /H20850. 14J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 15V. Tiberkevich and A. Slavin, Phys. Rev. B 75, 014440 /H208492007 /H20850. 16Q. Mistral et al. ,Appl. Phys. Lett. 88, 192507 /H208492006 /H20850. 17The dimensionless power in Fig. 2/H20849a/H20850is given by P/R0I2=a/H20853p+2 /F/H20849p/H20850/H20854 with a/H112292.3063 /H1100310−9and p/H1122910.202 /H20849I−Ic/H20850. To obtain the linewidth in Fig. 2/H20849b/H20850, we have used the parameters of /H20881D/H92532/H92631/2/H9266=11.24 MHz and /H9254=0.5, and have solved the eigenvalue problem of the Fokker–Planck equation corresponding to Eq. /H208491/H20850as done in Ref. 10or6.FIG. 2. /H20849Color online /H20850/H20849a/H20850Power Pdivided by R0I2with R0=13.6 /H9024and /H20849b/H20850 linewidth /H20851full width at half maximum /H20849FWHM /H20850/H20852of the signal of STNO as a function of applied current I. Dots are experimental data at T=150 K taken from Ref. 16. Red lines are theoretical ﬁtting curves based on the model of Eq. /H208491/H20850.FIG. 3. /H20849Color online /H20850/H20849a/H20850Dependence of /H9254on the nonlinearity of the damp- ingq1for various values of an external magnetic ﬁeld H. An uniaxial an- isotropy ﬁeld is taken as Hk/4/H9266Ms=0.04. /H20849b/H20850Dependence of /H9254on an exter- nal magnetic ﬁeld Hfor various values of an uniaxial anisotropy ﬁeld Hk.07D105-3 Kudo et al. J. Appl. Phys. 105 , 07D105 /H208492009 /H20850 Downloaded 05 Jun 2013 to 128.226.37.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
1.47901.pdf	Single component plasma bibliography Compiled by J. Fajans Citation: AIP Conference Proceedings 331, 271 (1995); doi: 10.1063/1.47901 View online: http://dx.doi.org/10.1063/1.47901 View Table of Contents: http://aip.scitation.org/toc/apc/331/1 Published by the American Institute of PhysicsSingle Component Plasma Bibliography Compiled by J. Fajans Papers in the bibliography are categorized as follows: A: Equilibrium and stability. B: Transport and Kinetic Effects. C: Waves. D: 2-d fluid effects. E: Cyclotron motion effects. F: High density plasmas and energetic plasmas. G: Correlations and microplasmas. H: Ion plasmas. I: Antimatter plasmas. J: Non-neutral plasmas and atomic physics. K: Exotic traps or particles. L: Numerical techniques. M: Applications. N: Other. O: Experimental P: Theoretical Entries were submitted by the workshop participants and are limited to published papers and thesis. This bibliography can be obtained by emailing a request to fajans~physics.berkeley.edu. The bibliography is available in LaTeX format and as a BiBTeX database. Cross-Reference A: Equilibrium and stability: avin:91, bhat:92, bhat:92a, bhat:92b, boge:70, bolh93, boll:94a, brew:88, brow:86, chan:90, chen:90, chen:90a, chen:91a, chen:94a, chen:94b, chu:93, chu:93a, clar:76, daug:67, daug:69, davi:69, davi:70, davi:71, davi:72, davi:72a, davi:73, davi:74, davi:75, davi:77, davi:79, davi:82, davi:84, davi:84a, davi:84b, davi:84c, davi:85, davi:85a, davi:85b, davi:86, davi:86a, davi:87, davi:88, davi:88a, davi:90, davi:90a, davi:91, davi:93, davi:94, davi:94a, degr:77, dris:76, dris:85, dris:86b, dris:86c, dris:89, dris:92, dris:94, dubi:86, dubi:86b, dubi:88, dubi:90, dubi:91, dubi:92, dubi:93a, dubi:93b, dubi:94, fang93a, fine:89, gabr:92, glis:94, hein:91, ho11:93, huan:93, itan:82, itan:88, iwat:94, kerv:86, kerv:89a, kerv:91, kerv:94, khir:93, kwon:83, kwon:92, lars:86, lurid:93, hmd:93a, maim:82, mahn:84, malta:92, mite:93, miti:93a, miti:93b, mora:88, murp:92, nott:93b, onei:80b, onei:80c, onei:81, onei:88, onei:94, onei:94a, pass:89, peur:90, peur:92d, pras:79, pras:81, pras:86, pras:87a, pras:88, pras:89, 9 1995 American Institute of Physics 271 272 Single Component Plasma Bibliography raiz:921, robe:88, smit:89, smit:90, smit:90a, smit:91, smit:92, smit:92a, smit:92b, spen:92, spen:93a, tan:94, tots:81a, tots:88, tots:88a, tots:93, turn:87, turn:90, turn:91, turn:92, turn:93, turn:93a, turn:93b, turn:94, turn:94a, turn:94b, uhm:78, uhm:80, uhm:80a, uhm:82, uhm:83, weim:94, zave:92 B: Transport and Kinetic Effects: avin:92, avin:92a, beck:90, beck:92, boll:91, brig:70, brow:86, chan:90, chen:90, chen:90a, chen:91a, chen:93, chen:93a, chen:94, chen:94a, chen:94b, corn:93, craw:85, craw:87, croo:94, davi:69, davi:70, davi:71, davi:72, davi:72a, davi:73, davi:74, davi:75, davi:77, davi:79, davi:82, davi:85, davi:85a, davi:85b, davi:86, davi:86a, davi:88, davi:90, davi:90a, davi:91, davi:93, davi:94, davi:94a, degr:77, degr:77a, degr:80, doug:78, dris:82, dris:83, dris:85, dris:86a, dris:86b, dris:86c, dris:92, dubi:86a, dubi:88a, dubi:94, eggh87a, faja:93, gabr:92, glin:91, glin:91a, 'glin:92, hart:91, hjor:86, hjor:87, hjor:88, holl:93, huan:93, huan:94, hyat:87, hyat:88, kape:73, kein:84, kerv:89a, kriv:93, lamb:83, levy:69, malta:82, malm:92, miti:94, mood:92, murp:90, onei:80a, onei:88, onei:90, onei:94, onel:85a, peti:87, peur:90, peur:92d, peur:93e, rama:93, rasb:93, robe:88, spen:90, spen:92, spen:93a, spen:93b, tots:80, turn:93a, turn:93b, uhm:78, uhm:80, uhm:80a, uhm:82, uhm:83, wine:85a C: Waves: bo11:92, bolh93, boll:94a, boll:94b, brig:70, chart:90, chen:90, chen:90a, chen:91a, craw:85, craw:86, craw:87, croo:94, davi:69, davi:70, davi:73, davi:74, davi:77, davi:82, davi:84, davi:84a, davi:84b, davi:84c, davi:85, davi:85a, davi:85b, davi:86, davi:86a, davi:87, davi:88, davi:88a, davi:90, davi:90a, davi:91, davi:93, degr:77, degr:77a, degr:80, dimo:81, dris:85, dris:86b, dris:90b, dris:92, dris:94, dubi:86a, dubi:91, dubi:91a, dubi:93, dubi:93a, eggl:87, eggl:87a, faja:93, fine:88, fine:89, goul:91, goul:92, grea:94b, hein:91, huan:93, kape:73, kein:81, kein:84, kerv:89b, lamb:83, levy:65, levy:69, realm:82, malta:92, mitc:93, miti:93b, miti:94, mood:92, mora:88, nott:93a, nott:93b, onei:80a, onei:94, peti:87, peur:92d, peur:93b, peur:93c, peur:93e, pi11:94, prad:93, pras:81, pras:83, pras:84, pras:86, pras:87a, pras:88, pras:89, rama:93, robe:88, rose:87, rose:90, smit:90, smit:92a, smit:92b, spen:90, spen:93b, tink:94, tink:95, tots:80a, tots:81a, tots:81b, tots:92a, turn:94b, uhm:78, uhm:80, uhm:80a, uhm:82, uhm:83, weim:94, whit:82, wine:87 D: 2-d fluid effects: brig:70, davi:74, davi:90, dris:90b, dris:92, dris:94, dubi:90, fine:89, hart:91, huan:94, kadt:94, kerv:89b, kerv:89c, kerv:91, malta:92, miti:93a, miti:94, onei:94, peur:92d, peur:93a, peur:93b, peur:93c, peur:93f, pilh94, pras:86, robe:88, rose:87, rose:90, smit:89, smit:90, smit:90a, smit:91, smit:92, smit:92a, smit:92b, spen:90, spen:92, spen:93b, tots:75, tots:76, tots:78 E: Cyclotron motion effects: avin:92, beck:92, beu:91, boll:92, boll:93, boll:94a, bolh94b, chen:91, chen:93, chen:94a, davi:90, glin:92, gors:93, gouh91, goul:92, hans:89a, hans:89b, hein:91, hjor:86, hjor:87, hyat:87, jeff:83, kerv:85, kerv:89a, kerv:91, lauk:86, onei:90, peur:94a, pras:85, pras:87, robe:88, tots:93, xian:93 F: High density plasmas and energetic plasmas: boll:92, bolh93, boll:94a, boll:94b, davi:74, davi:90, hein:91, reich:89, robe:88 C: Correlations and microplasmas: alex:84, bo11:84, boll:90, bolh94a, boll:94b, chen:93a, chen:94, davi:74, davi:90, dubi:86, dubi:86b, dubi:88, dubi:89, dubi:90, dubi:9Oa, dubi:91, dubi:92, dubi:93b, dubi:94, gilb:88, hang:91:, hang:94, hass:90, itan:82, itan:88, krau:94, lars:86, J. Fajans 273 malm:84, niel:94, onei:80b, onei:88, onei:94, rafa:91, rahm:86a, rahm:88, raiz:92, raiz:921, rama:93, robe:88, schi:85, schi:86, schi:88, schi:88a, schi:89, schi:91, schi:93, schi:93a, schi:93b, schi:94, schi:94a, schu:88, tots:75, tots:76, tots:78, tots:79, tots:80, tots:80a, tots:81, tots:81a, tots:81b, tots:82, tots:84, tots:84a, tots:86, tots:87, tots:88, tots:88a, tots:92, tots:92a, whit:82, wine:87 H: Ion plasmas: barh86, barn:93, boll'84, boll:85, boll:90, bolh91, bolh92, boll:93, boll:94a, boll:94b, brew:88, brow:86, daug:68, davi:74, davi:90, dimo:81, dris:85, dris:86b, dubi:88, dubi:90, dubi:92, dubi:93b, gilb:88, hang:91:, hang:94, hass:90, hein:91, itan:88, kerv:85, kerv:89a, kerv:91, krau:94, lars:86, mich:89, mora:88, nieh94, pras:87a, pras:88, pras:89, tara:91, rahm:86a, rahm:88, raiz:92, raiz:921, robe:88, rose:87, rose:90, schi:85, schi:86, schi:88, schi:88a, schi:89, schi:91, schi:93, schi:93a, schi:93b, schi:94, schi:94a, schu:88, tan:94, tots:84a, tots:88, tots:88a, tots:92a, tots:93, whit:82, wine:85a, wine:87, wine:90 I: Antimatter plasmas: cowa:91, cowa:93, davi:74, davi:90, fang93a, gabr:92, glis:94, grea:94a, hang:91:, hang:94, hass:90, iwat:93, iwat:94, krau:94, kwon:83, murp:90, murp:91, murp:92, niel:94, onei:80b, rafa:91, rahm:86a, rahm:88, robe:88~ schi:85, schi:86, schi:88, schi:88a, schi:89, schi:91, schi:93, schi:93a, schi:93b, schi:94, schi:94a, surk:86a, surk:86b, surk:86c, surk:88, surk:89, surk:90, surk:93, tang:92, tink:94, turn:87, wine:93, wyso:88 J: Non-neutral plasmas and atomic physics: barn:93, boll:84, bolh85, boll:90, bolh91, boll:93, boll:94a, boll:94b, brew:88, davi:74, davi:90, fang93a, fang93b, fang94, gilb:88, glis:94, itan:82, itan:88, iwat:93, iwat:94, kerv:85, kwon:83, kwon:89a, kwon:89b, kwon:90, kwon:92, kwon:93a, kwon:93b, lars:86, murp:90, murp:91, pass:89, raiz:92, raiz:921, robe:88, schu:88, surk:88, surk:90, surk:93, tan:94, tang:92, weim:94, whit:82, wine:85, wine:85a, wine:87, wine:90 K: Exotic traps or particles: avin:91, avin:92, barl:86, bhat:92a, bhat:92b, brow:86, chen:93a, chen:94, clar:76, daug:67, daug:68, daug:69, davi:74, davi:90, gabr:92, glin:91, gors:92, hang:91:, hang:94, hass:90, khir:93, krau:94, mich:89, mora:88, niel:94, onei:94a, prad:93, pras:87a, pras:88, pras:89, rafa:91, rahm:86a, rahm:88, robe:88, schi:85, schi:86, schi:88, schi:88a, schi:89, schi:91, schi:93, schi:93a, schi:93b, schi:94, schi:94a, schu:88, tots:93, turn:90, turn:91, turn:92, turn:93, turn:93a, turn:93b, turn:94, turn:94b, wang:89, yin:92, zave:92 L: Numerical techniques: davi:74, davi:90, fang93b, robe:88, spen:93a, spen:93b, tots:84a, tots:92 M: Applications: chan:90, chen:91a, daug:68, davi:72, davi:74, davi:82, davi:84a, davi:84b, davi:85a, davi:85b, davi:88, davi:88a, davi:90, davi:93, glin:91, peti:87, robe:88, schu:88, turn:92, turn:93a, turn:93b, turn:94b, uhm:80, uhm:82, uhm:83 N: Other: cowa:91, cowa:93, davi:74, davi:90, eggh92, mich:89, robe:88 O: Experimental: beck:90, beck:92, brow:86, chu:93, clar:76, cowa:91, cowa:93, daug:67, daug:68, daug:69, davi:74, davi:90, degr:77, degr:77a, degr:80, dimo:81, dris:83, dris:85, 274 Single Component Plasma Bibliography P: dris:86a, dris:86b, dris:88, dris:90, dris:90a, dris:90b, dris:92, dris:94, dubi:86a, eggl:84, eggl:87, eggl:87a, eggh92, fine:88, fine:89, fine:91, gabr:92, glis:94, goul:91, goul:92, grea:94a, grea:94b, huan:93, huan:94, hyat:87, hyat:88, iwat:93, iwat:94, kadt:94, malm:75, malta:80, realm:82, malm:84, realm:88, malm:92, mitc:93, miti:93a, miti:93b, miti:94, mood:92, murp:90, murp:91, murp:92, nott:92, nott:93a, nott:93b, nott:94, onei:90, onei:94, pass:89, peur:92d, peur:93a, peur:93b, peur:93c, peur:93e, peur:93f, pill:94, robe:88, rose:87, rose:90, schu:88, surk:86a, surk:86b, surk:86c, surk:88, surk:89, surk:90, surk:93, tang:92, tink:95, whit:82, wyso:88 Theoretical. avin:92a, barn:93, bhat:92b, boge:70, bolh92, brow:86, chen:93, chen:93a, chen:94, chu:93, chu:93a, corn:93, craw:85, craw:86, craw:87, croo:94, davi:69, davi:70, davi:71, davi:73, davi:74, davi:90, doug:78, dris:76, dris:82, dris:86c, dris:89, dubi:86, dubi:86a, dubi:86b, dubi:88, dubi:88a, dubi:89, dubi:90a, dubi:91, dubi:91a, dubi:92, dubi:93, dubi:93a, dubi:93b, dubi:94, faja:93, fine:92, gabr:92, glin:91, glin:91a, glin:92, hart:91, hjor:86, hjor:87, hjor:88, itan:82, kein:84, levy:65, levy:69, malm:77, nott:92, nott:93a, onei:79, onei:80, onei:80a, onei:80b, onei:80c, onei:81, onei:83, onei:85, onei:87, onei:88, onei:90, onei:92, onei:94, onei:94a, onel:85a, peur:90, peur:93f, pras:79, pras:81, pras:83, pras:84, pras:86, rahm:86a, raiz:921, schu:88, smit:89, smit:90, smit:90a, smit:91, smit:92, smit:92a, smit:92b, tan:94, tots:75, tots:76, tots:78, tots:79, tots:80, tots:80a, tots:81, tots:81a, tots:81b, tots:82, tots:84, tots:84a, tots:86, tots:87, tots:88, tots:88a, tots:92, tots:92a, tots:93, turn:87, turn:90, turn:91, turn:92, turn:93a, turn:93b, turn:94, turn:94a, turn:94b, weim:94 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5.0044042.pdf	Appl. Phys. Lett. 118, 072406 (2021); https://doi.org/10.1063/5.0044042 118, 072406 © 2021 Author(s).Rotated read head design for high-density heat-assisted shingled magnetic recording Cite as: Appl. Phys. Lett. 118, 072406 (2021); https://doi.org/10.1063/5.0044042 Submitted: 15 January 2021 . Accepted: 29 January 2021 . Published Online: 17 February 2021 Wei-Heng Hsu , and R. H. 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Victora1,2,a) AFFILIATIONS 1Center for Micromagnetics and Information Technologies, University of Minnesota, Minneapolis, Minnesota 55455, USA 2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA a)Author to whom correspondence should be addressed: victora@umn.edu ABSTRACT In heat-assisted shingled magnetic recording, recorded tracks are erased on one side. The transitions are no longer symmetric relative to the track center, especially when the transitions are highly curved as a result of the temperature proﬁle generated by the near-ﬁeld transducer. To optimally utilize these asymmetrically curved transitions, the read head is rotated to match the curvature. For a single rotated head, a morethan 10% improvement in user density is achieved compared to that of a single non-rotated head. We found that the optimal rotation anglegenerally follows the transition shape. With an array of two rotated heads, a track pitch of 15 nm, and a minimum bit length of 6.0 nm, theuser areal density reaches 6.2 Tb/in 2, more than 30% above previous projections for recording on granular media. Published under license by AIP Publishing. https://doi.org/10.1063/5.0044042 New data are created and stored every second; the total global data storage may exceed 175 zettabytes by 2025.1To accommodate this stored data growth, hard disk drives remain solid candidates for data centers because they are less costly than solid-state drives. Heat- assisted magnetic recording (HAMR)2has been introduced as the suc- cessor to contemporary perpendicular magnetic recording to extend areal density growth. In conventional HAMR, tracks are written in random order. At high recording density with closely spaced tracks, adjacent track era- sure (ATE) occurs and a given track can experience two-sided erasure from writing to the neighboring tracks.3,4Maximum user density (UD) is achieved by optimizing the track pitch (TP), and so the ATE does not distort the data on the previously written track.5In heat- assisted shingled magnetic recording (HSMR), tracks are written adja- cently in sequence; each newly written track overlaps the ﬁxed side of the previously written track. In HSMR, only one-sided erasure occurs. Ideally, HSMR yields a higher UD than conventional HAMR with the same TP. However, two factors limit further improvement in HSMR’s areal density. First, only the track edge remains after ATE; the edge’s recording quality is worse than that of the track center. Also, the highly curved transitions in HSMR are asymmetric relative to the track center and are inevitable as long as the temperature proﬁle generated by thenear-ﬁeld transducer remains elliptical in the media plane. 6,7These asymmetrically curved transitions lead to signal loss in readback and limit the recording density.In this paper, we report on a study of HSMR through micromag- netic simulations. We found that by rotating the read head to compen- sate for asymmetrically curved transitions, we improved the UD morethan 10% over that achievable with a non-rotated head. The relation- ship between the optimal rotation angle and the TP is explored. Finally, we show that the UD could go beyond 6.2 Tb/in 2by combin- ing the rotated read head with multiple sensor magnetic recording (MSMR). Magnetization dynamics were modeled with the Landau–Lifshitz–Gilbert equation using 1.5 nm cubic renormalized cells.8The recording medium in this work was an ac-erased exchange- coupled composite (ECC) structure with a superparamagnetic writelayer. 5,9The ECC media consisted of a 4.5 nm-thick superparamag- netic write layer and a 9-nm-thick storage layer. The recording grains were modeled as Voronoi cells with an average grain pitch of 4.8 nm, astandard deviation of 18%, and a 1 nm non-magnetic grain boundary. The storage layer was L1 0FePt. A 2% Curie temperature variation among the grains was included. The Curie temperatures of the write layer and the storage layer were 900 K and 700 K, respectively. The magnetic properties of the renormalized cell were temperature depen-dent, 8and their values at 300 K can be found in Table I . The temperature proﬁle was approximated as a 2D Gaussian function in space with a peak temperature of 850 K and a full width at half maximum (FWHM) of 30 nm.8The head velocity was 20 m/s, and the applied writing ﬁeld was 8 kOe with a canting angle of 22.5/C14. Appl. Phys. Lett. 118, 072406 (2021); doi: 10.1063/5.0044042 118, 072406-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplTo simulate HSMR, three adjacent tracks were written on the media sequentially. The data sequence of the middle track was a ﬁxed 31-bit pseudorandom binary sequence (PRBS) generated by the polynomial x5þx3þ1, while the sequences on the remaining two tracks were arbitrary 31-bit PRBSs. The minimum bit length was 6 nm. Sixty-four recording simulations were performed on different media realizations. The noise-free magnetoresistive (MR) read head for readback had a shield-to-shield spacing (SSS) of 11 nm, a square-shaped 4 nm-thick MR element, and a 6 nm magnetic ﬂy height. The readback signals were obtained through cross correlation of the read head sensitiv- ity10,11and the magnetization as prescribed by the reciprocity principle.12 The signal-to-noise ratio (SNR), the bit error rate (BER), and the UD were calculated from the readback signals.5,13For the BER calcula- tion, 16 readback signals (496 bits) were used to train the equalizer function and 48 readback signals (1488 bits) were used in a Viterbi detector to calculate the BER. The Shannon channel capacity Cwas deﬁned as C¼1þBER/C1log2ðBERÞþð 1/C0BERÞ/C1log2ð1/C0BERÞ, and the UD was calculated as C=ðBL/C1TPÞ.Figure 1(a) shows the average recording pattern of HSMR over 64 simulations and the relative conﬁguration of the read head. Onlythe second track is visible after averaging. The read head was offset along the cross-track direction by Dnm to the track center and was rotated counterclockwise by the angle h.Figures 1(b)–1(e) show the relations between the track pitch and the track width, SNR, BER, and UD; Fig. 1(f) displays SNR as a function of track width for various reader widths (RW) from 12 nm to 18 nm. The ﬁgures include resultsfor both HAMR and HMSR. Here, we consider typical readback: the head was offset to the location that gave the maximum SNR without rotation ( h¼0 /C14). From the integral of the absolute value of the average magnetization along the down-track direction after writing, the track width (TW) can be deﬁned as the full width at 50% of the maximum magnetization value. For a complete track, the TW was 24.3 nm for the heat proﬁle and media design used here. The TW is narrower than the TP indicating the occurrence of the ATE. The HAMR TW decaysfaster than the HSMR TW due to the former’s two-sided erasure. The resulting wider track width in HSMR yields better SNR, BER, and UD compared to HAMR. The FWHMs of reader sensitivity in the cross-track direction for 12 nm, 15 nm, and 18 nm heads are 16.1 nm, 19.0 nm, and 21.9 nm,respectively. As the track width decreases, the wider heads start to pick up undesired signals outside the track and show worse recording met- rics than the narrowest head. It can be seen in Fig. 1(e) that the 18 nm head has no gain in UD for HSMR since the decrease in channel capacity Coffsets the gain from increasing TP. This shows that the UD can achieve 5 Tb/in 2with a TP of 15 nm and a 12 nm head in HSMR. Beyond that point, ATE dominates and UD drops. One would expect HAMR to perform better than HSMR for a given track width since HSMR uses a curved track edge. Surprisingly,TABLE I. Magnetic properties at 300 K for recording simulations. Parameters Write layer Storage layer Saturation magnetization (emu/cm3) 550 942 Uniaxial anisotropy (erg/cm3)0 :7/C21074:4/C2107 Exchange stiffness (erg/cm) 1 :4/C210/C061:1/C210/C06 Gilbert damping 0.02 0.02 FIG. 1. (a) Sample of the average recording pattern of HSMR over 64 simulations. The top track and the bottom track are imperceptible due to averaging. (b) Trac k width, (c) SNR, (d) BER, and (e) UD as a function of track pitch and (f) SNR as a function of track width for HSMR (solid curve) and HAMR (dashed curve) with various rea der widths at h¼0/C14. The black dashed line in (b) indicates the track width without ATE.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 072406 (2021); doi: 10.1063/5.0044042 118, 072406-2 Published under license by AIP Publishingthe SNRs in HSMR are comparable to the SNRs in HAMR [ Fig. 1(f) ], and they become slightly better when the TP is smaller. This may bebecause HAMR experiences two times erasure from adjacent trackwrites (ATWs), and the recorded quality of the center track isdegraded. Moreover, there will be more than one ATW in a real HAMR drive, whereas there is only one ATW in a real HSMR drive. The same argument can be applied to heat-assisted interlaced mag-netic recording (HIMR); 14the underneath track may experience many ATWs resulting in bad recording quality. To achieve a higher UD, the asymmetrically curved transitions in HSMR need to be addressed. One approach is to rotate the read headto match the transition curvature. Figure 2 shows the SNR, BER, and UD vs the rotation angle hfor a 12 nm head and an 18 nm head. At small h, all recording metrics are considerably improved, indicating that the rotated head starts to match the curve transition. If we furtherrotate the head, the SNR shows a monotonic increase, while BERreaches its minimum and UD is at its maximum at a certain h,w h i c h is deﬁned as the optimal h(h opt). The hoptvalues are estimated from the quadratic polynomial ﬁts of the UD curves. It can be seen that a large SNR does not necessarily imply a low BER or high UD. The larger effective SSS from the rotation improves the SNR because morelow-frequency signals or long-bit signals can pass and strengthen thesignal, but a larger SSS also leads to a resolution loss in high-frequencysignals or short-bit signals, and thus, BER grows and UD drops whenhexceeds h opt.F o ra1 2 n mh e a d , hoptis larger when the TP becomes narrower, which indicates that the transitions are more asymmetric [Fig. 2(c) ]. For an 18 nm head, hoptvalues are smaller than those of a 12 nm head and the difference in hoptbetween different TPs is smaller [Fig. 2(f) ]. This is because the 18 nm head tends to average out the transition and, thus, is less sensitive to the transition shape.To understand how hoptchanges with TP and RW, we focus on a single transition. The inset in Fig. 3(a) shows the average of transitions from a single tone signal. The transition is ﬁtted with quadratic poly-nomial as a function of cross-track position (red line). Depending onthe position of the read head, the corresponding transition angle with respect to the track center can be extracted by taking the arctangent of the tangent to the transition. We compare the extracted angle with theh optobtained from Fig. 2 and show them according to the read head position with respect to different TP in Fig. 3(a) . The read head is closer to the track center with a wider TP and vice versa. It is clear that the rotation of the head generally follows the shape of the transition. The difference between different read head widths may originate fromthe ﬁnite width of the reader sensitivity along down-track and cross-track directions. A wide read head tends to rotate less, while a narrowhead rotates more to capture the asymmetry. Figure 3(b) shows the UDs of rotated and non-rotated heads. By simply rotating the head, the UD demonstrates a 5.9% enhancement; it reaches a maximum of5.4 Tb/in 2from 5.1 Tb/in2a ta1 5n mT Pa n da3 0/C14hoptwith a 12 nm head. For a 15 nm head, a 14% improvement can be seen (from 4.4Tb/in 2to 5.0 Tb/in2) .T h ei n c r e a s ei sm o r ep r o n o u n c e dw h e nt h eh e a d is wide (18 nm) and the TP is narrow (12 nm) where a more than 20% enhancement is achieved. Again, the maximum UD for an 18 nm head does not occur at a 15 nm TP due to the excess noise outside thetrack. The previously published work, 15which adopted a 90/C14rotated head design, utilized a side shield in the cross-track direction andaggressive oversampling to achieve its gain, while in this work, the improvement comes from matching the physical shape of the transi- tion. This rotated head design is not compatible with HIMR since thetransitions are symmetric as found in conventional HAMR, suggestingthat HSMR has potential for higher UD over HIMR with proper RW = 18 nmRW = 12 nm RW = 12 nm RW = 12 nm RW = 18 nm RW = 18 nm(a) (d)(b) (e)(c) (f) FIG. 2. SNR, BER, and UD in HSMR as a function of hat various TPs with (a)–(c) a 12 nm head and (d)–(f) an 18 nm head where the read head is offset to the track center. The gray solid lines in (c) and (f) are quadratic polynomial ﬁts.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 072406 (2021); doi: 10.1063/5.0044042 118, 072406-3 Published under license by AIP Publishingoptimization. It should be noted that the rotated read head is used to address the asymmetrically curved transitions. For a PRBS with a lon-ger bit length, the number of transitions per unit length is fewer, and soh optmay not follow the angle derived from the transition shape. We note that in our conﬁguration, unlike the previous work with its longer bit length,16inter-granular exchange coupling ( Aex;inter)d o e s not improve the recording. By introducing up to 10% of intra-grainexchange ( A ex;intra) in the write layer, the maximum UD drops from 5.4 to 4.6 Tb/in2(Fig. 4 ). One reason for this is that the media used here are thick enough (13.5 nm) to reduce DC noise; another reason is that the minimum bit length used here is 6 nm, which is very close tothe grain pitch, while A ex;interfavors a longer bit. To further extend the feasibility of the rotated head, we combine it with MSMR, which is proven to have UD gains.17We consider a two-reader MSMR, where the ﬁnal signal ( Stot) is a linear combinationof the signal from head 1 ( S1) and the signal from head 2 ( S2), Stot¼3/C2S1/C02/C2S2. From the fabrication point of view, the rota- tion angle for both heads is set to be the same. Figures 5(a) and5(b) show the UD as a function of the head 1 offset ( D1) and the head 2 off- set (D2) for 12 nm heads and 15 nm heads where the track center is at /C06 nm and TP is 15 nm. We can see that the maximum UD is 5.9 Tb/ in2for the 12 nm head and is 5.4 Tb/in2for the 15 nm head. Both occur at 37.5/C14h,a/C09n mD1value, and a /C07.5 nm D2value. The hopt value is different from the single head hopt,w h i c hi s3 0/C14. The dual read heads show a 15.7% and 22.7% improvement over a single non- rotated 12 nm head and 15 nm head, respectively. Finally, if we allowboth heads to rotate independently, 6.2 Tb/in 2of UD is achieved for 12 nm heads when head 1 is rotated 7.5/C14,D1is/C01.5 nm, head 2 is rotated 37.5/C14,a n dD2is 1.5 nm [ Fig. 5(c) ]. Including additional heads FIG. 3. (a)hoptvs head position in the cross-track direction with various RWs. The top axis is the corresponding track pitch. The dashed line is the hderived from the shape of the transition. The inset shows the transition and the ﬁtted line. (b) Comparison ofUD between the rotated head and the non-rotated head for various RWs.FIG. 4. SNR, BER, and UD as a function of Aex;inter =Aex;intra at TP ¼15 nm and hopt¼30/C14for a 12 nm head. FIG. 5. UD as a function of the head 1 offset ( D1) and the head 2 offset ( D2) for (a) 12 nm heads and (b) 15 nm heads at h¼37.5/C14and TP ¼15 nm. (c) UD as a func- tion of head 2 h(h2) and the head 2 offset ( D2) for 12 nm heads where h1¼7.5/C14 andD1¼/C0 1.5 nm.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 072406 (2021); doi: 10.1063/5.0044042 118, 072406-4 Published under license by AIP Publishingand utilizing advanced signal processing techniques will likely push the UD well beyond 6.2 Tb/in2. It is more than 30% greater than a pre- vious projection for UD5and more than 50% greater than our predic- tion for non-shingled recording displayed in Fig. 1(e) . In this work, we demonstrated that UD can be increased by more than 10% by simply rotating the read head. The gain in UD originates from the improved ability of the head to match the asymmetrically curved transitions through its optimal angle, which roughly followsthe shape of the transitions. By combining the rotated read head withMSMR and advanced signal processing techniques, the UD canpotentially be extended beyond 6.2 Tb/in 2. The reduced ATW and the compatibility with the rotated head make HSMR suitable for very high-density applications. This work was supported by the Advance Storage Research Consortium (ASRC). The authors would like to thank Dr. Y. Wang at Shanghai Jiao Tong University for the help in the BERcalculations. The authors would also like to thank Dr. N. Natekarand Dr. Z. Liu for useful discussions. DATA AVAILABILITY The data that support the ﬁndings of this study are available from the corresponding author upon reasonable request.REFERENCES 1D. R.-J. G.-J. Rydning, IDC Report: The Digitization of the World From Edge to Core (International Data Corporation, Framingham, 2018). 2M. H. Kryder, E. C. Gage, T. W. McDaniel, W. A. Challener, R. E. Rottmayer, G. Ju, Y.-T. Hsia, and M. F. Erden, Proc. IEEE 96, 1810 (2008). 3S. Kalarickal, A. Tsoukatos, S. Hernandez, C. Hardie, and E. Gage, IEEE Trans. Magn. 55, 1 (2019). 4N. A. Natekar and R. Victora, IEEE Trans. Magn. (published online, 2020). 5Z. Liu, Y. Jiao, and R. Victora, Appl. Phys. Lett. 108, 232402 (2016). 6Y. Qin, H. Li, and J.-G. Zhu, IEEE Trans. Magn. 53, 3001604 (2017). 7R. H. Victora and A. Ghoreyshi, IEEE Trans. Magn. 55, 1 (2019). 8R. H. Victora and P.-W. Huang, IEEE Trans. Magn. 49, 751 (2013). 9Z. Liu and R. H. Victora, IEEE Trans. Magn. 52, 3201104 (2016). 10R. H. Victora, W. Peng, J. Xue, and J. Judy, J. Magn. Magn. Mater. 235, 305 (2001). 11Y. Dong and R. Victora, IEEE Trans. Magn. 45, 3714 (2009). 12H. N. Bertram, Theory of Magnetic Recording (Cambridge University Press, 1994). 13Y. Jiao, Y. Wang, and R. Victora, IEEE Trans. Magn. 51, 1 (2015). 14S. Granz, W. Zhu, E. C. S. Seng, U. H. Kan, C. Rea, G. Ju, J.-U. Thiele, T. Rausch, and E. C. Gage, IEEE Trans. Magn. 54, 1 (2018). 15Y. Wang, M. Erden, and R. H. Victora, IEEE Magn. Lett. 3, 4500304 (2012). 16N. Natekar, W. Tipcharoen, and R. H. Victora, J. Magn. Magn. Mater. 486, 165253 (2019). 17C. Rea, P. Krivosik, V. Venugopal, M. F. Erden, S. Stokes, P. Subedi, M. Cordle,M. Benakli, H. Zhou, D. Karns et al. ,IEEE Trans. Magn. 53, 3001506 (2017).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 072406 (2021); doi: 10.1063/5.0044042 118, 072406-5 Published under license by AIP Publishing
1.4864046.pdf	Tuning of the spin pumping in yttrium iron garnet/Au bilayer system by fast thermal treatment Lichuan Jin, Dainan Zhang, Huaiwu Zhang, Qinghui Yang, Xiaoli Tang, Zhiyong Zhong, and John Q. Xiao Citation: Journal of Applied Physics 115, 17C511 (2014); doi: 10.1063/1.4864046 View online: http://dx.doi.org/10.1063/1.4864046 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic and high frequency properties of nanogranular CoFe-yttrium-doped zirconia films J. Appl. Phys. 115, 17A337 (2014); 10.1063/1.4866391 Electric field tuning of domain magnetic resonances in yttrium iron garnet films Appl. Phys. Lett. 102, 222407 (2013); 10.1063/1.4809580 Highly flexible poly (vinyldine fluoride)/bismuth iron oxide multiferroic polymer nanocomposites AIP Conf. Proc. 1447, 1309 (2012); 10.1063/1.4710494 Enhanced spin pumping at yttrium iron garnet/Au interfaces Appl. Phys. Lett. 100, 092403 (2012); 10.1063/1.3690918 Structural effects on the magnetic character of yttrium–iron–garnet nanoparticles dispersed in glass composites J. Appl. Phys. 93, 7199 (2003); 10.1063/1.1555900 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 37.191.221.36 On: Tue, 13 May 2014 05:58:17Tuning of the spin pumping in yttrium iron garnet/Au bilayer system by fast thermal treatment Lichuan Jin,1,a)Dainan Zhang,2Huaiwu Zhang,1,a)Qinghui Y ang,1Xiaoli Tang,1 Zhiyong Zhong,1and John Q. Xiao3 1State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology, Chengdu 610054, People’s Republic of China 2Department of Electrical and Computer Engineering, University of Delaware, Newark, Delaware 19716, USA 3Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA (Presented 5 November 2013; received 23 September 2013; accepted 4 November 2013; published online 12 February 2014) In this Letter, we investigated the inﬂuence of the fast thermal treatment on the spin pumping in ferromagnetic insulator yttrium iron garnet (YIG)/normal metal Au bilayer system. The YIG/Au bilayer thin ﬁlms were treated by fast annealing process with different temperatures from 0 to 800/C14C. The spin pumping was studied using ferromagnetic resonance. The surface evolution wasinvestigated using a high resolution scanning microscopy and an atomic force microscopy. A strong thermal related spin pumping in YIG/Au bilayer system has been revealed. It was found that the spin pumping process can be enhanced by using fast thermal treatment due to the thermal modiﬁcations ofthe Au surface. The effective spin-mixing conductance of the fast thermal treated YIG/Au bilayer has been obtained. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4864046 ] Magnetic insulator (MI)/normal metal (NM) bilayer structure has been attracted intensive attentions in many fun- damental as well as applied spintronics areas.1–4Pure spin current generated in magnetic insulator is beneﬁcial for spin-tronic operations with ultra low energy consuming, which is the most promising candidate in next generation electronic devices. By using spin-orbit interaction in MI/NM bilayer,spin current generated from both spin Hall effect and spin pumping process can be obtained. Spin Hall effect is a DC electrical current induced pure spin current process. 5–7 While in the latter approach, GHz regime microwave pho- tons are used to resonantly excite magnetization dynamics in a ferromagnet and thus drive a spin current into an adjacentnormal metal. 8–12Besides, a thermal gradient can also gener- ate a spin current, which is named as spin Seebeck effect (SSE).13 Very recently, Heinrich et al. studied the spin pumping at ferromagnetic insulator yttrium ion garnet (YIG)/normal metal (Au) interface.14–16An enhanced spin pumping efﬁ- ciency was obtained with YIG surface modiﬁcation. In this work, we report a tunable spin pumping efﬁciency in yttrium iron garnet/Au bilayer system by using a fast thermal treat-ment. It is revealed that the spin pumping process can be sig- niﬁcantly inﬂuenced by changing the surface morphology of the Au layer. Moreover, with the temperature of fast thermaltreatment above 700 /C14C, typical gold nano-particles (NPs) gradually show up. It will totally kill the spin pumping with inducing a surface anisotropy at YIG/Au interface. High quality single crystal YIG (Y 3Fe5O12) thin ﬁlms with thickness 100 nm were deposited on (111) GGG (Gd 3Ga5O12) single crystal substrates by using pulsed laser deposition (PLD) method. Top Au thin ﬁlms with thickness15 nm were deposited by using high vacuum thermal evapo- ration method. Series of YIG/Au bilayers were then treated with 60 s light assisted annealing from 400 to 800/C14Ci na vacuum oven. The spin pumping properties were studied bycoplanar waveguide (CPW) vector-network-analyzer ferro- magnetic resonance (VNA-FMR) spectrometer with an in-plane conﬁguration. Surface morphology was investigatedusing high resolution scanning electron microscopy (HR-SEM) and an atomic force microscopy (AFM). The chemical composition was determined by energy dispersivex-ray spectroscopy (EDS). The crystal structures were char- acterized by x-ray diffraction (XRD) with a Cu Kasource. Figures 1(a)–1(d)show the typical FMR absorption lines for bare YIG (blue line) and YIG/Au bilayers (red line) treated at different fast annealing temperatures, measured at a ﬁxed microwave frequency 9 GHz. Inset ﬁgures are atomicforce microscopy scan images of the YIG/Au bilayers treated at different temperatures (the scan area is 5 /C25lm). It can be clearly seen, with covering Au thin ﬁlms, the linewidth ofYIG/Au bilayers has been pronounced broadened, which is contributed by the spin pumping. 17Without thermal treat- ment, as shown in Fig. 1(a), the linewidth for bare YIG was obtained as 6.5 Oe, while the linewidth was enhanced as 9.7 Oe for YIG/Au bilayer. Figures 1(b) and1(c)show typi- cal FMR curves obtained in thermal treated YIG/Au thinﬁlms. A signiﬁcant improvement of FMR linewidth with the increase of thermal temperature was presented. The line- width was enhanced to 14.2 Oe and 16.7 Oe for 400 /C14C and 600/C14C thermal treatments, respectively. As shown in inset AFM graphs of Figs. 1(b) and1(c), the surface morphology of the YIG/Au thin ﬁlms also changes remarkably. Smallraindrop shaped Au covering ﬁlms have been formed due to the fast heat treatment. More interestingly, as presented in Fig. 1(d), the spin pumping induced linewidth broadening has been killed with 800 /C14C thermal treatment. Meanwhile, aa)Authors to whom correspondence should be addressed. Electronic mail: lichuanj@udel.edu and hwzhang@uestc.edu.cn. 0021-8979/2014/115(17)/17C511/3/$30.00 VC2014 AIP Publishing LLC 115, 17C511-1JOURNAL OF APPLIED PHYSICS 115, 17C511 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 37.191.221.36 On: Tue, 13 May 2014 05:58:17FMR frequency ( fr) shifting was observed. Top covering Au ﬁlm has been formed into Au nano-particles, as shown in theinsetted AFM graph of Fig. 1(d). It is proved that the surface morphology of the normal metal layer plays an important role in MI/NM spin pumping process. Figures 2(a)–2(d) show the obtained SEM pictures at different annealing temperatures. It can be seen that 15 nm Au ﬁlm covers well without any thermal treatment, as shownin Fig. 2(a). And there is no Au clusters or grains on top of the YIG surface. However, with proper annealing tempera- ture, the 15 nm Au ﬁlm became reunion into small clustersdue to the thermal stress, as shown in Fig. 2(b). By further increasing the annealing temperature up to 700 /C14C, the Au clusters would gradually grow into Au NPs, while the Auﬁlm covering area (yellow mask) is reduced, as shown in Fig.2(c).I nF i g . 2(d), Au NPs are formed on top of the YIG surface at 800 /C14C. EDS results convinced there is no chemi- cal property change in YIG layer with such a fast thermal treatment process.The FMR linewidth ( DH) as a function of frequency at different Tais presented in Fig. 3(a), the measured frequency is from 9 to 12 GHz. The Gilbert damping parameter can be extracted from the frequency dependence of the ﬁeld-swept linewidth DHðxÞ¼2ax=ﬃﬃ ﬃ 3p cþDH0.DH0is the zero- frequency intercept and it is usually considered to be an ex- trinsic inhomogeneous contribution to the linewidth.18–21An important parameter for spin pumping is the real part of thespin-mixing conductance ( g "# ef f). The real part of the spin-mixing conductance is proportional to the ﬂux of angular momentum in the form of spin current which ﬂow throughthe YIG/Au interface. The spin pumping enhanced damping for the ferromagnetic YIG layer is predicted as 17,22–24 FIG. 1. Typical FMR absorption lines for bare YIG (100 nm) and YIG (100 nm)/Au (15 nm) bilayer treated at different temperatures (all samples are measur ed at 9 GHz). Inset ﬁgures are atomic force microscopy scan images of the YIG/Au bilayers treated at different temperatures (the scan area is 5 /C25lm). FIG. 2. The surface evolution of YIG (100 nm)/Au (15 nm) bilayers treated at different temperatures. (a) 0/C14C, (b) 400/C14C, (c) 700/C14C, and (d) 800/C14C. FIG. 3. (a) The FMR linewidth DHas functions of frequency for YIG/Au bilayers at different Ta. (b) The extracted effective spin mixing conductance g"# ef fas a function of Ta.17C511-2 Jin et al. J. Appl. Phys. 115, 17C511 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 37.191.221.36 On: Tue, 13 May 2014 05:58:17Da¼glBg"# ef f 4pMs1 t; (1) where g"# ef fis the real part of the effective spin mixing con- ductance, tis the thickness of the ferromagnetic YIG layer, andlBis the Bohr magneton. The additional Gilbert damping constant can be calculated with Da¼aYIG =Au-aBareYIG ,25 where aYIG =Auis the damping constant of YIG/Au bilayer. The extracted g"# ef fas a function of Tawas plotted in Fig. 3(b). The experimental obtained g"# ef fin YIG (100 nm)/Au (15 nm) without any thermal treatment is 1.15 /C21014cm/C02, which is the same order as the previous report.14,15Surprisingly, a dra- matic g"# ef fenhancement can be obtained in proper tempera- ture thermal treated YIG/Au bilayers. After 400/C14C annealing, a signiﬁcant enhanced g"# ef fabout 3.18 /C21014cm/C02has been obtained. Moreover, g"# ef ffor 500/C14C and 600/C14C thermal treat- ment are 2.71 /C21014cm/C02and 2.4 /C21014cm/C02, respectively. These values are double or three times comparing with the non-treated YIG/Au sample. For 700/C14C annealed YIG/Au ﬁlm, the value of g"# ef fsharply reduces to 0.61 /C21014cm/C02 due to a partial coverage YIG surface. The results indicate that a fast thermal annealing process can inﬂuence the efﬁ-ciency of spin current injection from YIG to Au. One possible reason is normal metal surface roughness can contribute to the interfacial spin scattering. As mentioned above, a shortthermal annealing process can relieve the inner stress of the evaporation Au thin ﬁlm with forming Au clusters. It means a modiﬁcation of normal metal ﬁlm morphology can effectivelychange the spin pumping efﬁciency in MI/NM system. The well known Kittle formula x=cðÞ 2¼HrHrð þ4pMef fÞdescribes the frequency-dependence of resonanceﬁelds,26where Hris the in-plane resonance ﬁeld, cis the gyromagnetic ratio. And, 4 pMef fis the effective saturation magnetization deﬁned as8,26,27 4pMef f¼4pMsþð2Ks=MsÞt/C01 FM; (2) where Ksis the surface/interface anisotropy. 4 pMsis the mag- netization of the YIG ﬁlm, which was obtained as 1.74 kG from the VSM measurements. So, the values of Kshave been ﬁtted for YIG/Au bilayers at different Ta,a ss h o w ni nF i g . 4(b). The derived surface anisotropy constant Ksis quite small (about 0.024 erg/cm2) for low temperature ( Ta/C20600/C14C) annealed samples, but dramatically increases from 0.024 erg/cm2up to 0.092 erg/cm2asTaincreases up to 700/C14C. These results indi- cate that Au NPs pinned in YIG surface can enhance the surface/interface anisotropy. Moreover, the surface anisotropyvalue is inﬂuenced with T a. This phenomenon may offer valua- ble choice in the design of spin transfer torque related devices and spin current injection related devices. Financial support from the National Basic Research Program of China under Grant No. 2012CB933104, the Foundation for Innovative Research Groups of the NationalNatural Science Fund of China under Grant No. 61021061, National Natural Science Foundation of China (Grant Nos. 61271037, 61021061, and 60801027), and Internationalcooperation Project Nos. 2012DFR10730, 2013HH0003, and 111 Project No. B13042. 1S. S.-L. Zhang and S. F. Zhang, Phys. Rev. Lett. 109, 096603 (2012). 2D. Qu et al.,Phys. Rev. Lett. 110, 067206 (2013). 3J. Xiao and G. E. W. Bauer, Phys. Rev. Lett. 108, 217204 (2012). 4Y. Kajiwara et al.,Nature 464(7286), 262 (2010). 5J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 6B. F. Miao et al.,Phys. Rev. Lett. 111, 066602 (2013). 7M. C. Beeler et al.,Nature 498, 201–204 (2013) 8R. Urban et al.,P h y s .R e v .L e t t . 87, 217204 (2001). 9Y. Tserkovnyak et al.,Phys. Rev. Lett. 88, 117601 (2002). 10B. Heinrich et al.,Phys. Rev. Lett. 90, 187601 (2003). 11M. V. Costache et al.,Phys. Rev. Lett. 97, 216603 (2006). 12F. D. Czeschka et al.,Phys. Rev. Lett. 107, 046601 (2011). 13K. Uchida et al.,Nature 455, 778 (2008). 14B. Heinrich et al.,Phys. Rev. Lett. 107, 066604 (2011). 15C. Burrowes et al.,Appl. Phys. Lett. 100, 092403 (2012). 16E. Montoya et al.,J. Appl. Phys. 111, 07C512 (2012). 17S. M. Rezende et al.,Appl. Phys. Lett. 102, 012402 (2013). 18See, for example, D. L. Mills and S. M. Rezende, in Spin Dynamics in Conﬁned Magnetic Structures II , edited by B. Hillebrands and K. Ounadjela (Springer, Heidelberg, 2002), pp. 27–58. 19R. D. McMichael et al.,Phys. Rev. Lett. 90, 227601 (2003). 20B. Heinrich et al.,J. Appl. Phys. 57, 3690 (1985). 21Z. Celinski and B. Heinrich, J. Appl. Phys. 70, 5935 (1991). 22Y. Tserkovnyak et al.,Rev. Mod. Phys. 77, 1375 (2005). 23J. M. Shaw et al.,Phys. Rev. B 85, 054412 (2012). 24O. Mosendz et al.,Phys. Rev. B 82, 214403 (2010). 25T. Yoshino et al.,J. Phys.: Conf. Ser. 266, 012115 (2011). 26J.-M. L. Beaujour et al.,Phys. Rev. B 74, 214405 (2006). 27Y. C. Chen et al.,J. Appl. Phys. 101, 09C104 (2007). FIG. 4. (a) FMR resonance ﬁeld ( Hr) as a function of annealing temperature (Ta) in YIG/Au bilayers. (b) Extracted surface anisotropy energy constant (Ks) as a function of Ta.17C511-3 Jin et al. J. Appl. Phys. 115, 17C511 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 37.191.221.36 On: Tue, 13 May 2014 05:58:17
1.5047066.pdf	Reuse of AIP Publishing content is subject to the terms at: <a href="https://publishing.aip.org/authors/rights-and-permissions">https://publishing.aip.org/authors/rights- and-permissions</a>. Downloaded to: 5.101.222.216 on 07 November 2018, At: 02:41Ferromagnetic resonance manipulation by electric fields in Ni 81Fe19/ Bi3.15Nd0.85Ti2.99Mn0.01O12 multiferroic heterostructures Rongxin Xiong , Wanli Zhang , Bin Fang , Gang Li , Zheng Li , Zhongming Zeng , and Minghua Tang Citation: Appl. Phys. Lett. 113, 172407 (2018); doi: 10.1063/1.5047066 View online: https://doi.org/10.1063/1.5047066 View Table of Contents: http://aip.scitation.org/toc/apl/113/17 Published by the American Institute of Physics Articles you may be interested in Tunneling anomalous Hall effect in a ferroelectric tunnel junction Applied Physics Letters 113, 172405 (2018); 10.1063/1.5051629 Study of spin-orbit torque induced magnetization switching in synthetic antiferromagnet with ultrathin Ta spacer layer Applied Physics Letters 113, 162402 (2018); 10.1063/1.5045850 Enhanced magnon spin transport in NiFe 2O4 thin films on a lattice-matched substrate Applied Physics Letters 113, 162403 (2018); 10.1063/1.5049749 Thickness dependence of ferrimagnetic compensation in amorphous rare-earth transition-metal thin films Applied Physics Letters 113, 172404 (2018); 10.1063/1.5050626 An effect of capping-layer material on interfacial anisotropy and thermal stability factor of MgO/CoFeB/Ta/CoFeB/ MgO/capping-layer structure Applied Physics Letters 113, 172401 (2018); 10.1063/1.5050486 Temperature invariable magnetization in Co-Al-Fe alloys by a martensitic transformation Applied Physics Letters 113, 172402 (2018); 10.1063/1.5055350Ferromagnetic resonance manipulation by electric fields in Ni81Fe19/Bi3.15Nd0.85Ti2.99Mn0.01O12multiferroic heterostructures Rongxin Xiong,1,2Wanli Zhang,3,4BinFang,1,a)Gang Li,4Zheng Li,4Zhongming Zeng,1 and Minghua Tang4,a) 1Key Laboratory of Nanodevices and Applications, Suzhou Institute of Nano-tech and Nano-bionics, Chinese Academy of Sciences, Ruoshui Road 398, Suzhou 215123, China 2School of Physics and Optoelectronics, Xiangtan University, Xiangtan, Hunan 411105, China 3School of Electronic Information Engineering, Yangtze Normal University, Chongqing, Sichuan 408100, China 4Key Laboratory of Key Film Materials and Application for Equipments (Hunan Province), School of Material Sciences and Engineering, Xiangtan University, Xiangtan, Hunan 411105, China (Received 4 July 2018; accepted 14 October 2018; published online 26 October 2018) We investigated electric-ﬁeld modulation of ferromagnetic resonance (FMR) in Ni 81Fe19(NiFe)/ Bi3.15Nd0.85Ti2.99Mn 0.01O12(BNTM) heterostructures at room temperature. BNTM thin ﬁlms were deposited on a Pt (111)/Ti/SiO 2/Si (100) substrate by the sol-gel method. The strain effect is pro- duced by the electric ﬁeld applied to the BNTM layer, which results in the FMR spectrum shift bytuning of the magnetic anisotropy of the NiFe microstrip devices. A strain-induced magnetic anisotropy change of 332 fJ/Vm is obtained by analyzing the experimental FMR spectra. We dis- cussed an inﬂuence on spin orbit torques by applying an electric ﬁeld to a ferroelectric (FE) layervia coupling to polarization with FMR experiments evidencing. The torque ratios s a/sbincreased at ﬁrst and then declined from the positive to negative electric ﬁeld. As the value of the applied elec- tric ﬁeld changes from 129 kV/cm to 0 kV/cm, the variation of the torque ratios sa/sb(the ﬁeld-like torque saand damping-like torque sb) is about 0.07. Our results reported in this work demonstrate a route to realize a large magneto-electric coupling effect at room temperature and provide some insights into possible applications of the ferromagnetic/FE device. Published by AIP Publishing. https://doi.org/10.1063/1.5047066 The magnetoelectric coupling effect has drawn exten- sive attention since it can either induce changes in polariza-tion and order parameters using the magnetic ﬁeld or enable the control of magnetism using the electric ﬁeld. 1,2The latter offers a promising approach for energy-efﬁcient manipula-tion in a range of applications, including nonvolatile mem- ory, microwave devices, and magnetoelectric recording. 2To date, various material systems have been demonstrated toexhibit the magnetoelectric coupling effect at a certain con- dition. A few single-phase materials that consist of distinct ferroelectric (FE), ferromagnetic, and ferroelastic phases have shown the magnetoelectric coupling effect under extreme conditions. 3Among them, multiferroic heterostruc- tures are of great interest due to their relatively strong mag- netoelectric coupling effect at room temperature in comparison to the single-phase materials.4,5In multiferroic heterostructures, the magnetoelectric coupling effect may arise from several mechanisms, such as exchange-bias medi- ated, charge-mediated, and strain-mediated.6–13Previous studies demonstrated that a strong magnetoelectric coupling effect existed in ferromagnetic/piezoelectric substrate heter-ostructures, while a high operation bias voltage (at the level of several hundred voltages, e.g., 400 V in Refs. 10–13)i s required. From a practical application point of view, it ishighly desirable to achieve a strong magnetoelectric cou- pling effect with a low operation bias voltage/electric ﬁeld.Recent works have shown that ferromagnetic/ferroelec- tric (FM/FE) bilayers need only a small operation bias volt- age to be applied to the ferroelectric layer. 14–16Many research studies demonstrated the magnetoelectric coupling effect manipulation of various magnetic properties in ferro- magnetic/ferroelectric (FM/FE) heterostructures such asCurie temperature, spin polarization, magnetic ordering, magnetic anisotropy, magnetotransport measurements (pla- nar and anomalous Hall effect, magnetoresistance), and fer-romagnetic resonance (FMR). 17–24Meanwhile, an approach which holds promise for energy-efﬁcient manipulation of nonvolatile magnetic logic and memory technologies iscurrent-induced torques generated by materials with strong spin–orbit (S–O). 25For example, in metallic bilayer sys- tems such as ferromagnet (FM)/heavy metal (HM) multi-layer systems and magnetically doped topological insulator heterostructures, spin orbit torques that originate from the spin Hall effect or Rashba effect can be used for magnetiza-tion reversal. 26–29For a coupled FM/FE bilayer, a spiral magnetic ordering at the FM/FE interface that builds up in FM is within a nanometer range determined by spin diffu- sion length. This non-collinear ordering is coupled to theFE polarization, 30which entails a spin orbital coupling.31 By combining nature of the ferroelectric layer and its inter-face coupling with the FM layer, we expect thus an inﬂu-ence on spin orbit torques by applying a voltage to the FE layer via coupling to polarization. Nevertheless, the study of spin-orbit torques in such FM/FE bilayers is largelylacking. a)Authors to whom correspondence should be addressed: bfang2013@sinano. ac.cn and mhtang@xtu.edu.cn 0003-6951/2018/113(17)/172407/5/$30.00 Published by AIP Publishing. 113, 172407-1APPLIED PHYSICS LETTERS 113, 172407 (2018) In this work, we investigated the electric-ﬁeld effect of magnetic anisotropy and spin orbit torques in the ferromag-netic Ni 81Fe19(NiFe)/ferroelectric Bi 3.15Nd0.85Ti2.99Mn0.01O12 (BNTM) bilayer by using the FMR technique. Until now, thework on the electric-ﬁeld modulation of magnetism at roomtemperature in NiFe/BNTM structures using the ferromagnetic resonance (FMR) is still lacking. With the application of an electric ﬁeld, a magnetic anisotropy modulation of 332 fJ/Vmwas observed. Furthermore, we found that the spin orbit torque can be also modulated by the electric ﬁeld. BNTM ﬁlms were prepared by a sol-gel method on Pt/Ti/ SiO 2/Si(001) substrates as follows. The BNTM precursor solu- tion was prepared by mixing deﬁned molar ratios of bismuth nitrate Bi(NO 3)3/C15H2O, neodymium nitrate Nd(NO 3)3/C16H2O, titanium isopropoxide Ti(OC 3H7)4, and manganese acetate Mn(CH 3COO) 2/C14H2O. Glacial acetic acid was used as the sol- vent, and the resulting solution was diluted by 2-methoxyethanolto adjust the viscosity. Crystallographic orientations (pre- dominantly along [117] and [200] orientation) of BNTM thin ﬁlms were controlled by spinning and heating rates. The pre-cursor solution was aged for 3–7 days before its use for spin coating. The BNTM precursor solution was spin coated on Pt/Ti/SiO 2/Si substrates, followed by a drying process at 180/C14C for 5 min. The as-deposited ﬁlms were pyrolyzed at 400/C14C for 5 min in air and annealed at 700/C14C for 5 min under an O 2pressure of 1.5 atm to produce a layered perov- skite phase. The rapid thermal annealing (RTP) method was used for the annealing process at a ramping rate of 15/C14C/C1s/C01. Crystalline phases of the prepared ﬁlms were analyzed by x-ray diffraction (XRD). Figure 1(a)displays the X-ray diffrac- tion results, revealing a highly [117]-preferred growth of BNTM. The thickness of the BNTM ﬁlm is estimated to beabout 387 nm according to the cross-sectional scanning elec- tron microscopy (SEM) image as shown in Fig. 1(b).T oc h e c k the electrical properties of BNTM ﬁlms, Pt top electrodeswith a diameter of 200 lm were deposited on BNTM ﬁlms using DC sputtering. Typical polarization-electric ﬁeld ( P–E) hysteresis loops under various voltages at 1 kHz are shown inFig.1(c), indicating the good ferroelectric nature of the ﬁlms. Figure 1(d) shows that the grains have 180 /C14phase and high amplitude under an electric ﬁeld of 200 kV/cm.To fabricate FM/FE bilayers, magnetically soft alloy NiFe was used as a ferromagnetic material owing to its lowcoercivity, low magnetocrystalline anisotropy, low magneto-striction, and low damping ( a). 32,33A 15 nm thick layer of NiFe was deposited on a BNTM layer by using electron beam evaporation. Afterwards, the NiFe micro-strips with a 13lm width and a 65 lm length were deﬁned using UV pho- tolithography and Ar ion-beam etching. The FMR measure-ment set-up is shown in Fig. 2. All measurements presented in this work were carried out at room temperature. The mag-netic ﬁeld Hwas applied in the direction of 45 /C14with respect to the NiFe bar. The electric ﬁeld applied on the BNTM is supplied by a voltage source meter (Keithley 2400). Amicrowave signal of 15 dBm produced by a signal generator(N5183B) is applied to the bar through a bias tee. When theRF current is passed through the uniformly magnetized bar,there are magnetization dynamics. This magnetization pre- cession induces a time-dependent resistance change of the bar owing to anisotropy magneto-resistance (AMR), and theresulting rectiﬁed d.c. voltage across the bar was recorded bya nano-voltmeter. Figure 3(a) shows FMR spectra for different values of magnetic ﬁeld magnitudes without external electric ﬁeld applied to the BNTM layer. The FMR spectra can be well ﬁt-ted by the sum of anti-symmetric and symmetric Lorentzians V dc¼VADHðHext/C0HrÞ ðHext/C0HrÞ2þDH2þVsDH2 ðHext/C0HrÞ2þDH2 þV0; (1) where ﬁtting parameters are regulated by anti-symmetric and symmetric voltages ( VAandVS), resonant ﬁeld Hr, magnetic ﬁeld Hext, linewidth DHof the spectrum, and V0is the FIG. 1. (a) XRD patterns of the BNTM thin ﬁlm. (b) SEM cross-sectional image of the BNTM thin ﬁlm. (c) P–Ehysteresis loops at 1 kHz. (d) Piezoelectric hysteresis loops. FIG. 2. Sample structure of the device and schematic of the measurement set-up. Hlies in the ﬁlm plane. FIG. 3. (a) FMR spectra for different magnetic ﬁelds with zero external elec- tric ﬁeld applied to the BNTM layer with hH¼45/C14. The solid lines represent theoretical ﬁtting using Eq. (1). (b) FMR frequency as a function of resonant magnetic ﬁeld at zero external electric ﬁeld of the BNTM layer. The solid lines represent theoretical ﬁtting using Eq. (3).172407-2 Xiong et al. Appl. Phys. Lett. 113, 172407 (2018)constant voltage offset.34,35The resonant frequency fvs. res- onant ﬁeld curve is well ﬁtted with the Kittel formula, as will be discussed later, as shown in Fig. 3(b). In order to investigate the electric-ﬁeld-modulation of magnetic anisotropy through the strain-mediated interaction,we characterized FMR spectra in the presence of electric ﬁelds. Figure 4(a) shows the representative electric-ﬁeld- controlled FMR spectra as a function of magnetic ﬁelds. TheFMR ﬁeld is shifted from 325 Oe to 299 Oe with a gap of26 Oe as the value of the applied electric ﬁeld from 0 kV/cm to 129 kV/cm as shown in Fig. 4(a). There are several possi- ble causes for the electric-ﬁeld-induced shift of the reso-nance ﬁeld. One is the charge-induced magnetic anisotropychange, which was found in a thin magnetic ﬁlm (generally around 1 nm) with a ferroelectric material. 36The other one is the strain-induced magnetic anisotropy change by the appli-cation of piezo-strain to the magnetic layer. Moreover, it hasbeen reported that the anisotropy change in strain-induced magneto-electric (ME) coupling is independent of the mag- netic ﬁlm thickness. 37Zhou et al. had demonstrated that CoFe (1.2 nm)/BSTO heterostructures might have a co-existence of strain and charge mediated magnetoelectric cou- plings and that in CoFe (50 nm)/BSTO heterostructures, there was only strain-mediated magnetoelectric coupling dueto the non-existence of charge-mediated magnetoelectriccoupling in relatively thick magnetic ﬁlms ( /C291n m ) . 5 Since the resonance ﬁeld change was caused by the strain effect, we can quantify the strain-induced surfaceanisotropy change DK s(V) using the energy equation. The total energy of a NiFe ﬁlm can be represented as5 Etotal¼/C01 2l0MsþKuþKsþDKsðVÞ tN; (2) where Etotalis the total energy of the NiFe thin ﬁlm, l0is the permeability of the free space, l0MS(¼1.1 T) is the satura- tion magnetization of the NiFe thin ﬁlm,33Kuis the bulk anisotropy, KSis the surface anisotropy between NiFe and BNTM layers, and tNis the magnetic ﬁlm thickness. Similar modulation of Hrhas also been observed in other material systems. The resonance ﬁeld Hrcan be deter- mined by the Kittel equation5 f¼c 2pl0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðHrþHkÞðHrþHkþMeffÞp ; (3) where fis the resonance frequency, c/2p¼28 GHz/T is the gyromagnetic ratio, Hk¼2Ku/l0Msis the bulk anisotropyﬁeld, and Meffis the effective saturation magnetization incor- porating out-of-plane magnetic anisotropy37 Meff¼Ms/C02ðKsþDKsðVÞÞ l0tNMs: (4) Figure 4(b) shows the FMR ﬁeld as a function of the applied electric ﬁeld to the BNTM layer. The FMR ﬁeld curve as a function of electric ﬁeld shared a similar shape to the pie- zoelectric hysteresis loop of the BNTM layer as shown in Fig. 1(d), which has a sudden jump at the electric ﬁeld of around 630 kV/cm. This link between the Hr-Eand piezoelectric hysteresis loops demonstrates the control of the magnetism by the electric ﬁeld via the strain mechanism. The strain- mediated interaction was also observed in CoFe 2O4(CFO)/ Bi3.15Nd0.85Ti3O12(BNT) bilayer thin ﬁlms on a conventional Pt(111)/Ti/SiO 2/Si(100) substrate. This fact shows that the ori- entation of the BNT layer has a strain-mediated interfacial effect that can substantially affect the magnetoelectric cou- pling behavior of the bilayer structures.38 At a ﬁxed frequency f¼5 GHz, the change in resonance ﬁeldDHrinduced by the strain effect can be solved as37 DHr¼DKsðVÞ l0tNMs: (5) Under a constant applied electric ﬁeld of þ129 kV/cm or/C0129 kV/cm, the change in perpendicular surface anisot- ropy is estimated to be 4.5 lJ/m2for NiFe/BNTM nanowire devices with tN¼15 nm. This corresponds to a giant electric ﬁeld effect on magnetic anisotropy of 332 fJ/Vm, which is little higher than the value of 263 fJ/Vm for NiFe/PLZT mul- tiferroic thin ﬁlm heterostructures.37It is much higher than the experimental results of 100 fJ/Vm for the CoFeB/MgO interface inserted with a Mg layer of 0.1–0.3 nm.39 In order to further understand the origin of VSandVA,w e performed a comprehensive full angular ( u)-dependent mea- surement of spin torque (ST)-FMR signal Vdc. The ST-FMR measurement was conducted at a series of in-plane magnetic- ﬁeld angles ufrom 0/C14to 360/C14at a ﬁxed frequency of 5 GHz and a power of 15 dBm. The symmetric ST-FMR resonance components VSand antisymmetric components VAas a func- tion of the in-plane magnetic-ﬁeld angle were obtained, as shown in Figs. 5(a)and5(b), respectively. The angle depen- dence of VSandVAcan be ﬁtted by the following equation: V¼V0cosðuÞsinð2uÞ; (6) FIG. 4. (a) The electric-ﬁeld controlled FMR spectra as a function of magnetic ﬁeld. (b) The FMR ﬁeld as a function of applied electric ﬁeld to the BNTM layer.172407-3 Xiong et al. Appl. Phys. Lett. 113, 172407 (2018)where Vstands for VSorVA,V0is a coefﬁcient, and uis in- plane ﬁeld angle.40As shown in Figs. 5(a) and5(b), the well-ﬁtted cos( u)sin(2 u) angular dependence behavior for both VSandVAconﬁrms the two contributions from interfa- cial spin orbit coupling and AMR rectiﬁcation on the FMRresonance signal. 40 As have been reported recently, FE polarization which stems actually from the FE surface leads to the spin spiralin a FM; with a FE, a non-collinear spin density ~s(~r,t ) develops in the FM interface on the scale of the spin diffusion length k m.30In the system, the spin orbit torque (SOT) has two components: ﬁeld-like torques of the forms a~m/C2~r(FL) with different directions and damping-like torques of the form sa~m/C2~r/C2~m(DL, or Slonczewski- like), where ~mand ~rare the direction and vectors of ferro- magnet’s magnetization and spin of the current, respec- tively.41The magnetization dynamics driven by the spin- orbital interaction (SOI) follows the generalized Landau-Lifshitz-Gilbert equation d~m=dt¼/C0c~m/C2~H effþa~m/C2d~m=dt þsa~m/C2~rþsb~m/C2ð~r/C2~mÞ; (7) where cis the gyromagnetic ratio, ~Heffis the total effective ﬁeld including the external ﬁelds ~Hex, anisotropy ﬁelds ~Ha, and the Oersted ﬁeld ~Hoested generated by the current, ais the damping, ~ris the unit vector that is in-plane and orthogo- nal to the electric current, and saandsbcorrespond to the ﬁeld-like and damping-like torque induced from the spin- orbital interaction (SOI), respectively.42The ﬁeld-like torque (sa) and damping-like torque ( sb) amplitudes deﬁned in the FMR spectrum contribute to the symmetric and antisymmet-ric parts of the V dclineshape, respectively.25We can deter- minate the torque as a function of applied magnetic ﬁeld to a sum of symmetric and antisymmetric Lorentzians [Eq. (1)]. The amplitudes of the Lorentzians are related to the twocomponents of torque byV s¼/C0IRF 2dR d//C18/C191 aGcð2Hrþl0MeffÞsa; (8) VA¼/C0IRF 2dR d//C18/C19 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þl0Meff=Hrp aGcð2Hrþl0MeffÞsb: (9) The torque ratio sa/sbcan be obtained from Eqs. (8)and (9)as follows: sa sb¼VA VSﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 1þl0Meff=Hrp ; (10) where Ris the device resistance, /is the angular orientation of the magnetization relative to the direction of applied cur- rent in the sample, d R/d/is due to the AMR in the Py, Hris the resonance ﬁeld, l0Meffis the out-of-plane demagnetiza- tion ﬁeld, IRFis the microwave current in the device, aGis the Gilbert damping coefﬁcient, and cis the gyromagnetic ratio.25,42We estimate Mefffrom the frequency dependence ofHrusing the Kittel formula [Eq. (3)]. The Gilbert damping aGis estimated from the frequency dependence of the line- width via DH¼2pfaG=cþDH0; (11) where DH0is the inhomogeneous broadening.25In our devi- ces,l0Meff¼1.07 T and aG¼0.013, as determined by the ST-FMR resonance frequency and linewidth, respectively. Figure 5(c) shows the dependences of sa/sbon the applied electric ﬁeld of BNTM layer. As expected, we observeappreciable dependence of s a/sbas a function of the applied electric ﬁeld. The torque ratios sa/sbincreased at ﬁrst and then declined from the positive to negative electric ﬁeld. As the value of the electric ﬁeld changes from 129 kV/cm to0 kV/cm, the ﬁeld-like torque decreases; otherwise, thedamping-like torque increases, and the variation of the tor-que ratios s a/sbis about 0.07. The observation we discussed has important implications for future applications of com- posite FM/FE systems to spintronic applications. In summary, we experimentally demonstrated electric- ﬁeld modulations of FMR at room temperature in NiFe/BNTM heterostructures. Our results indicate that the electri- cal control of FMR spectra depends sensitively on electric ﬁelds. An experimental strain-mediated magnetic anisotropyof 332 fJ/Vm has been obtained. We discussed an inﬂuenceon spin orbit torques by applying an electric ﬁeld to the FElayer via coupling to polarization with FMR experiments evidencing. The torque ratios s a/sbincreased at ﬁrst and then declined from the positive to negative electric ﬁeld. As thevalue of the applied electric ﬁeld changes from 129 kV/cm to0 kV/cm, the variation of the torque ratios s a/sb(the ﬁeld- like torque saand damping-like torque sb) is about 0.07. Our results could be important for future NiFe/BNTM (FM/FE) based spintronic device applications. The authors would like to thank the ﬁnancial support from the National Natural Science Foundation of China (NSFC)under Grant Nos. 51761145025, 51472210, 11474311 and 11804370. This work was also supported by the executive programme of scientiﬁc and technological cooperation betweenItaly and China for the years 2016–2018 (Code Nos. FIG. 5. (a) Symmetric ST-FMR resonance components VSas a function of the in-plane magnetic-ﬁeld angle u. (b) Antisymmetric ST-FMR resonance components VAas a function of the in-plane magnetic-ﬁeld angle u. (c) Spin orbit torques ratio (ﬁeld-like torque sa: damping-like torque sb) as a function of applied electric ﬁeld to BNTM layer.172407-4 Xiong et al. Appl. Phys. Lett. 113, 172407 (2018)CN16GR09 and 2016YFE0104100). This work was also supported in part by the National Postdoctoral Program forInnovative Talents (No. BX201700275). The authors thankSteven Louis from Oakland University for his help and usefuldiscussions. 1N. A. Pertsev, H. Kohlstedt, and B. 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1.3151859.pdf	Variation of magnetization reversal in pseudo-spin-valve elliptical rings C. Yu, T. W. Chiang, Y. S. Chen, K. W. Cheng, D. C. Chen et al. Citation: Appl. Phys. Lett. 94, 233103 (2009); doi: 10.1063/1.3151859 View online: http://dx.doi.org/10.1063/1.3151859 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v94/i23 Published by the American Institute of Physics. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 01 Apr 2013 to 171.67.34.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsVariation of magnetization reversal in pseudo-spin-valve elliptical rings C. Yu,1T . W. Chiang,1,2Y . S. Chen,1,2K. W. Cheng,1D. C. Chen,1S. F . Lee,1,a/H20850Y . Liou,1 J. H. Hsu,2and Y . D. Yao1 1Institute of Physics, Academia Sinica, Taipei, Taiwan 115, Republic of China 2Department of Physics, National Taiwan University, Taipei, Taiwan 106, Republic of China /H20849Received 26 December 2008; accepted 15 May 2009; published online 8 June 2009 /H20850 We studied nanoscale elliptical ring shaped NiFe/Cu/NiFe trilayer pseudo-spin-valve structures. The magnetization reversal processes showed simultaneous-reversal single-step transition or double-steptransition involving ﬂux closure states. For various aspect ratios /H20849short axis to long axis /H20850and linewidths, transition between single-step and double-step magnetization reversals was measured toform a phase diagram. When the linewidth was reduced, edge roughness became important.Simulations of the magnetization reversal behavior agreed qualitatively with our results. © 2009 American Institute of Physics ./H20851DOI: 10.1063/1.3151859 /H20852 Study on the ring shaped micromagnet has steadily re- ceived growing interest. 1–7The spin valve multilayer ellipti- cal ring /H20849ER /H20850was introduced8after the proposal from Zhu et al.6Besides the similar magnetization reversal behavior to the ring shape, i.e., the bidomain and the vortex states, theER has the advantage that the domain wall location is easilycontrolled by the shape anisotropy. 9To read and write the signals into the rings or ERs, various ways have beenproposed. 10–14The magnetization conﬁguration can be deter- mined by measuring the electron spin chemical potential in alateral nonlocal spin-valve structure 10,14or by analyzing the characteristic response of two-dimensional electron gas lyingbeneath the ring. 11When the magnetoresistance /H20849MR /H20850is measured, the spin valve multilayer can enhance the MRratio from the anisotropic MR to the giant MR /H20849GMR /H20850 effect. 15Hayward et al.12placed asymmetric electrical con- tacts to read and applied off axis external ﬁelds to write thevortex circulations in pseudo-spin-valve ERs. When the ringsare put into an array, interaction between units 16,17and the aspect ratio of an ER become important issues. The aspectratio of an ER was usually taken as a simple ratio, e.g., 2:1 12 or 8:5.13Here, we present a systematic study of the magne- tization reversal for different aspect ratio pseudo-spin-valveERs. The sample fabrication procedure can be found elsewhere. 14The long axis of the spin valve ER samples, Ni80Fe20/H2084940 nm /H20850/Cu/H2084910 nm /H20850/Ni80Fe20/H2084910 nm /H20850, were changed from 3.2 to 2.0 /H9262m but the circumferences were kept around 6.3 /H9262m. The samples’ names, short axis lengths, and their aspect ratios are given in Table I. The linewidths of the samples were varied from 60 to 160 nm because thedomain wall has simple head-to-head structure. Below thethin limit of 60 nm, edge roughness could affect the magne-tization reversal behavior. The scanning electron microscopy/H20849SEM /H20850pictures of samples with linewidth 100 nm, short axis 0.2 /H20849with electrical contacts /H20850, 0.3, 0.8, and 2.0 /H9262m/H20849ell02, ell03, ell08, and ell20 /H20850are shown in Fig. 1. MR was mea- sured by the ac lock-in technique with 10 /H9262A at room tem- perature. The electrical contacts were located on the longaxis to reduce the inﬂuence of contact geometry on theMR. 18The GMR ratio is deﬁned as /H20853/H20851R/H20849H/H20850−R/H20849Hs/H20850/H20852/R/H20849Hs/H20850/H20854 /H11003100%, where Hsis the saturation ﬁeld. Figure 2shows the GMR loops of the samples ell02 and ell04 with 100 nmlinewidth. The MR loops of ell00, ell02, and ell03 havesingle-step shape. The rest of the ERs all have double-stepshape MR curves, which are similar to the results of Ref 8. These results are listed in Table Itogether with the switching ﬁelds H c1,Hc2,Hc3, and the GMR ratios. In the range of linewidth we chose, edge roughness is important and we didnot observe any systematic change in the switching ﬁelds.The three stable resistance states are labeled as P1, P2, andP3 in Fig. 2. A review on the switching behavior phase diagram of a single layer ring was reported in Ref. 3. For similar outer diameter and linewidth, ﬁlms 15 nm and thicker showedtypical double-step reversal with stable vortex state. As thethickness became thinner, the ﬁeld range for which the stablevortex state existed became smaller. For ﬁlms thinner than15 nm, the rings showed single-step reversal with a smallswitching ﬁeld. This is the result of competition between theZeeman energy and the exchange plus the demagnetizationenergies. A vortex state has higher Zeeman energy than thebidomain state but eliminates the exchange energy and de-magnetization energy of two domain walls. For thicker ringﬁlms, the tradeoff favors the formation of vortex states.When ﬁlms are thin, the domain wall energy is small and asingle-step switching is favorable. Our samples have 10 nmof Cu spacer layers. No exchange coupling between the lay-ers needs to be considered. The effect of dipolar interactionis considered in the simulation described below. The magnetization reversal process of our ERs was in- vestigated by the magnetic force microscopy /H20849MFM /H20850with real-time applied ﬁeld. From the MFM images /H20849not shown here /H20850, the 10 nm top layer showed a forward bidomain state changed into a reverse bidomain state after the switchingﬁeld. The vortex state of the top layer was missing when the a/H20850Electronic mail: leesf@phys.sinica.edu.tw. FIG. 1. SEM pictures of selected samples /H20849a/H20850ell02 /H20849with I/Vcontact leads /H20850, /H20849b/H20850ell03, /H20849c/H20850ell08, and /H20849d/H20850ell20 /H20849ring shape /H20850.APPLIED PHYSICS LETTERS 94, 233103 /H208492009 /H20850 0003-6951/2009/94 /H2084923/H20850/233103/3/$25.00 © 2009 American Institute of Physics 94, 233103-1 Downloaded 01 Apr 2013 to 171.67.34.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsﬁeld was reversed after saturation for all samples, as reported in Ref. 8. The bar shape /H20849ell00 /H20850sample showed the GMR effect of two ferromagnetic layers. For samples ell02 and ell03, theswitching behaviors of the thin top layer and the thick bot-tom layer are shown in Fig. 2/H20849a/H20850. For the other samples, the switching behaviors are shown in Fig. 2/H20849b/H20850. In Table Iwe list the distinct shapes of the MR curves, which suggested thetransition of the thick layer from single-step /H20849ell03 /H20850to double-step /H20849ell04 /H20850as the aspect ratio increased. The vortex state cannot survive in ERs when the aspect ratio was smallerthan about 0.13. Thus, we had a transition around 0.097–0.13. When the linewidths of our samples were increased,this transition aspect ratio remained the same, as shown inFig. 3. The solid circles and triangles are single-step and double-step experimental results, respectively. When thelinewidths were decreased, single-step transition became fa-vorable such that the 70 and 60 nm wide ell04 samplesshowed double-step and single-step MR curves, respectively.Our phase diagram suggests that when the ER having vortexconﬁguration is favorable, the aspect ratio should be 0.13 orlarger. When the linewidth is reduced to less than 70 nm,larger aspect ratio is needed. Simulations of the ER magnetization were made by three-dimensional micromagnet simulation OOMMF .19The magnetizations are determined by solving the Landau–Lifschitz–Gilbert /H20849LLG /H20850equation. Linewidth from 25 to 200 nm and more speciﬁc aspect ratios near the transition weresimulated than the experiments. The parameters used wereexchange constant 3 /H1100310 11J/m, saturation moment 8.6 /H11003105A/m, no anisotropy, damping parameters 0.5, and meshes size 10 nm cube. The thin top layer from simulationshowed single-step switching for all linewidths, agreed withthe results of MFM. The simulation results of MH curves and the magnetization are indicated schematically in Figs.2/H20849a/H20850and2/H20849b/H20850. In Fig. 3, we showed hollow stars and reversetriangles as single-step and double-step simulation results, respectively. The results showed good agreements with ourexperimental results. The /H20851M/H20849H/H20850/M S/H208522curves of the simula- tion hysteresis loops show qualitatively the same shapes with our measured MR curves.20The dipolar interaction between the top and bottom layers plays important role for the switch-ing ﬁelds behaviors. In our simulation, a 10 nm thick singleﬁlm of ell02 sample has a switching ﬁeld of 960 Oe. In thetrilayer sample, the switching ﬁeld of the 10 nm layer isreduced to 360 Oe due to the dipolar ﬁeld from the 40 nmthick layer. Vortex states for the thin layer were observed in minor loop studies both experimentally and in simulation. MR mi-nor loop results on ell04 are presented in Fig. 4/H20849a/H20850. The data showed that when the ﬁeld was reversed at the P2 plateau, aP4 state was observed in a narrow ﬁeld range, where the MRwas zero. This indicated that the two layers were both invortex state and parallel to each other. In some othersamples, a similar behavior was observed but with a P5 state/H20849not shown here /H20850, where the MR was equal to the P1 state, an antiparallel vortex state conﬁguration. Simulation showedsimilar results as presented in Fig. 4/H20849b/H20850for sample ell04. The blue solid line showed half of the full hysteresis loop whensaturation ﬁeld was applied. When the ﬁeld was reversedbefore the jump to saturation, the domain wall in the thinlayer moved stochastically through one of the two arcs. 21 The magnetization switched back to the positive direction intwo steps, as the red dash line in Fig. 4/H20849b/H20850showed. The detailed conﬁguration depended on the crystalline defects,roughness, etc. In simulations, antiparallel vortex states areenergetically favorable. From application point of view, theTABLE I. Sample list with the short axis, short axis to long axis aspect ratios, the shape of measured MR curves /H20851single /H20849S/H20850or double /H20849D/H20850steps /H20852, the measured switching ﬁelds, and the MR percentages at the plateaus for one series of samples. The circumferences of all samples are 6.3 /H9262m. The linewidth of the listed samples is 100 nm. For other sets of samples the linewidth is varied between 60 and 160 nm. Sample ell00 ell02 ell03 ell04 ell05 ell06 ell08 ell10 ell12 ell20 Short axis /H20849/H9262m/H20850 0 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 2.0 Aspect ratio /H20849b/a/H20850 0 0.064 0.097 0.130 0.165 0.200 0.275 0.358 0.449 1 Shape of MR curves S S S D DDDDD D Switching ﬁelds Hc1/Hc2/Hc3/H20849Oe/H20850 30/258/N/A 34/273/N/A 37/196/N/A 20/123/259 27/45/197 35/68/304 28/60/322 26/58/380 18/44/316 10/176/366 MR at P1/P2 /H20849%/H20850 1.43/N/A 0.92/N/A 0.98/N/A 0.86/0.43 0.66/0.35 0.84/0.49 0.75/0.48 1.06/0.55 0.68/0.36 0.85/0.52 FIG. 2. /H20849Color online /H20850Measured MR loop in symbols /H20849left and bottom axes /H20850 and simulated M-Hcurve in red dash line /H20849right and top axes /H20850of/H20849a/H20850ell02, and /H20849b/H20850ell04. P1, P2, and P3 states have magnetization conﬁgurations as shown. Thin and thick arrows represent top and bottom layers, respectively. FIG. 3. /H20849Color online /H20850Magnetization switching phase diagram of the thick elliptical layer as functions of linewidth and aspect ratio. The solid circlesand triangles are experimental results for single-step and double-step tran-sition, respectively; hollow stars and reverse triangles are single-step anddouble-step simulation results, respectively. The line is a guide for the eye.233103-2 Yu et al. Appl. Phys. Lett. 94, 233103 /H208492009 /H20850 Downloaded 01 Apr 2013 to 171.67.34.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsoccurrence of the transient P4 and P5 states could be avoided by fully saturating the sample. Magnetization reversals can occur by magnetic ﬁelds or currents in nanostructures due to the spin transfer torqueeffect.22Recently, the nanoring shaped magnetic tunnel junc- tions /H20849MTJs /H20850had been designed in magnetic random access memory devices.23Magnetization switching induced by both currents and magnetic ﬁelds was measured and simulatedwell by the LLG equation in a nanoring MTJ.22,23In psudo- spin-valve NiFe/Cu/Co trilayer rings measured with current-in-plane, with a biasing ﬁeld between 32 and 37 Oe for their6 nm thick, 370 nm wide, 5 /H9262m in diameter NiFe layer, a current density in the order of 107A/cm2induced domain wall motion in the direction of electron ﬂow.24Thus, the domain wall in the bidomain state can be moved to form aclockwise of counterclockwise vortex state by the appliedcurrent. For larger ﬁelds, domain wall motion could be in-duced but the moving direction was determined by the bias-ing ﬁeld. These results are applicable to our trilayer ERs sothat the above mentioned P4 and P5 states could be wellcontrolled. In summary, ER shape is an alternative structure for magnetic information storage due to its elimination of theenergetic vortex core of a disk and its better control of thedomain wall locations compared to the circular ring. Weshowed the limits to the aspect ratio between the single-stepand double-step magnetization reversal with various line-widths. In pseudo spin valves, the parallel and antiparallelvortex domain states between the top and bottom layers are unstable. Formation and control of these states is a topicwhich needs more study from the application point of view. Financial support of the National Science Council and the Academia Sinica of Taiwan, Republic of China is ac-knowledged. 1J. Rothman, M. Kläui, L. Lopez-Diaz, C. A. F. Vaz, A. Bleloch, J. A. C. Bland, Z. Cui, and R. Speaks, Phys. Rev. Lett. 86, 1098 /H208492001 /H20850. 2T. Uhlig and J. Zweck, Phys. Rev. Lett. 93, 047203 /H208492004 /H20850. 3M. Kläui, C. A. F. Vaz, L. J. Heyderman, U. Rüdiger, and J. A. C. Bland, J. Magn. Magn. Mater. 290-291 ,6 1 /H208492005 /H20850. 4I. Neudecker, M. Kläui, K. Perzlmaier, D. Backes, L. J. Heyderman, C. A. F. Vaz, J. A. C. Bland, U. Rüdiger, and C. H. Back, Phys. Rev. Lett. 96, 057207 /H208492006 /H20850. 5M. Eltschka, M. Kläui, U. Rüdiger, T. Kasama, L. Cervera-Gontard, R. E. Dunin-Borkowski, F. Luo, L. J. Heyderman, C.-J. Jia, L.-D. Sun, andC.-H. Yan, Appl. Phys. Lett. 92, 222508 /H208492008 /H20850. 6J. G. Zhu, Y. Zheng, and G. A. Prinz, J. Appl. Phys. 87, 6668 /H208492000 /H20850. 7F. Q. Zhu, G. W. Chern, O. Tchernyshyov, X. C. Zhu, J. G. Zhu, and C. L. Chien, Phys. Rev. Lett. 96, 027205 /H208492006 /H20850. 8F. J. Castaño, D. Morecroft, W. Jung, and C. A. Ross, Phys. Rev. Lett. 95, 137201 /H208492005 /H20850. 9F. J. Castaño, C. A. Ross, and A. Eilez, J. Phys. D 36, 2031 /H208492003 /H20850. 10T. Kimura, Y. Otani, and J. Hamrle, Appl. Phys. Lett. 87, 172506 /H208492005 /H20850. 11M. Hara, J. Shibata, T. Kimura, and Y. Otani, Appl. Phys. Lett. 88, 082501 /H208492006 /H20850. 12T. J. Hayward, J. Llandro, R. B. Balsod, J. A. C. Bland, F. J. Castaño, D. Morecroft, and C. A. Ross, Appl. Phys. Lett. 89, 112510 /H208492006 /H20850. 13W. Jung, F. J. Castaño, and C. A. Ross, Phys. Rev. Lett. 97, 247209 /H208492006 /H20850. 14D. C. Chen, Y. D. Yao, J. K. Wu, C. Yu, and S. F. Lee, J. Appl. Phys. 103, 07F312 /H208492008 /H20850. 15W. Jung, F. J. Castaño, and C. A. Ross, Appl. Phys. Lett. 91, 152508 /H208492007 /H20850. 16T. Miyawaki, K. Toyoda, M. Kohda, A. Fujita, and J. Nitta, Appl. Phys. Lett. 89, 122508 /H208492006 /H20850. 17L. J. Chang, C. Yu, T. W. Chiang, K. W. Cheng, W. T. Chiu, S. F. Lee, Y. Liou, and Y. D. Yao, J. Appl. Phys. 103, 07C514 /H208492008 /H20850. 18D. Morecroft, F. J. Castaño, W. Jung, J. Feuchtwanger, and C. A. Ross, Appl. Phys. Lett. 88, 172508 /H208492006 /H20850. 19A three-dimensional code on http://math.nist.gov/oommf. 20J.-E. Wegrowe, D. Kelly, A. Franck, S. E. Gilbert, and J.-Ph. Ansermet, Phys. Rev. Lett. 82, 3681 /H208491999 /H20850. 21T. J. Hayward, T. A. Moore, D. H. Y. Tse, J. A. C. Bland, F. J. Castano, and C. A. Ross, Phys. Rev. B 72, 184430 /H208492005 /H20850. 22H. X. Wei, J. X. He, Z. C. Wen, X. F. Han, and S. Zhang, Phys. Rev. B 77, 134432 /H208492008 /H20850. 23X. F. Han, Z. C. Wen, and H. X. Wei, J. Appl. Phys. 103, 07E933 /H208492008 /H20850. 24C. Nam, B. G. Ng, F. J. Castano, M. D. Mascaro, and C. A. Ross, Appl. Phys. Lett. 94, 082501 /H208492009 /H20850. FIG. 4. /H20849Color online /H20850/H20849a/H20850MR minor loop of ell04. The solid line is the s a m ea si nF i g . 2/H20849b/H20850. When the ﬁeld was reversed at the P2 state at the negative ﬁeld, a transient state was found in a narrow ﬁeld region on thepositive side, indicated by the dash line. /H20849b/H20850Simulation results of magneti- zation of upper half of full loop /H20849solid line /H20850and minor loop /H20849dash line /H20850of ell04. When the ﬁeld was reversed at the P2 state, a transient state P5 wasfound in a narrow ﬁeld region, as in the MR measurements. Magnetizationconﬁgurations are shown as thin and thick arrows for top and bottom layers,respectively.233103-3 Yu et al. Appl. Phys. Lett. 94, 233103 /H208492009 /H20850 Downloaded 01 Apr 2013 to 171.67.34.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions
1.3536657.pdf	Landau–Lifshitz magnetization dynamics driven by a random jump-noise process (invited) I. Mayergoyz, , G. Bertotti , and C. Serpico Citation: Journal of Applied Physics 109, 07D312 (2011); doi: 10.1063/1.3536657 View online: http://dx.doi.org/10.1063/1.3536657 View Table of Contents: http://aip.scitation.org/toc/jap/109/7 Published by the American Institute of PhysicsLandau–Lifshitz magnetization dynamics driven by a random jump-noise process (invited) I. Mayergoyz,1,a)G. Bertotti,2and C. Serpico3 1Department of Electrical and Computer Engineering, UMIACS and AppEl Center, University of Maryland College Park, College Park, Maryland 20742, USA 2Instituto Nazionale di Ricerca Metrologica (INRiM), Torino, Italy 3Dipartimento di Ingegneria Elettrica Universita `di Napoli “Federico II”, Napoli, Italy (Presented 16 November 2010; received 22 September 2010; accepted 29 October 2010; published online 23 March 2011) In the paper, a jump-noise process is introduced in magnetization dynamics equations in order to account for random thermal effects. It is demonstrat ed that in the case of small noise, Landau–Lifshitz and Gilbert damping terms emerge as average effects caused by the jump-noise process. This approachleads to simple formulas for the damping constant in terms of the scattering rate of the jump-noise process. These formulas also reveal the dependence of the damping constant on magnetization. The analysis of random switching of magnetization cau sed by the jump-noise process is presented. It is shown that the switching rate at very low temperat ures may appreciably deviate from the predictions of thermal activation theory, which is consistent w ith experimental observations of low temperature switchings and is usually attributed to the phenomenon of “macroscopic tunneling” of magnetization. VC2011 American Institute of Physics . [doi: 10.1063/1.3536657 ] I. INTRODUCTION Random thermal effects on magnetization dynamics have been studied for many years. This continuous interest in stochastic magnetization dynamics has been motivated bynumerous scientiﬁc and technological applications, which range from the study of thermally activated magnetization switching in magnetic data storage devices to the analysis ofpower spectral density of spin-torque nano-oscillators in the area of spintronics. Traditionally, the random thermal effects on magnetization dynamics have been modeled by introduc-ing two distinct and somewhat disjointed terms: 1–4(1) the deterministic Landau–Lifshitz or Gilbert damping term in magnetization dynamics equations and (2) the white-noisetorque term in the same equations. The reason behind this two-fold approach is that the white-noise process alone can- not fully and adequately describe the random thermal effectsbecause its expected value is zero. The common physical origin of these two terms has long been recognized and it is somewhat accounted for by imposing ﬂuctuation-dissipationrelations on them. These relations are justiﬁable under close- to-equilibrium conditions but questionable when dealing with far-from-equilibrium magnetization dynamics. It isclearly preferable and beneﬁcial to describe random thermal effects by a single random process and to extract the damp- ing term as its expected value. It is demonstrated in this paper that this can be accom- plished by modeling random thermal effects by a single jump-noise (Poisson-type) process. This stochastic jump pro-cess may also reﬂect random discontinuous magnetization transitions occurring on the fundamental quantum mechani-cal level. It is shown in this paper that in the case of a small jump-noise process, the Landau–Lifshitz 5and Gilbert6 damping terms can be derived as average effects caused by the jump-noise process. Furthermore, the damping constants in these terms can be directly related to the scattering rate of the jump-noise process. The latter is clearly consistent withthe physical origin of damping and may serve as a bridge for connecting damping with fundamental scattering processes. The derived formulas for damping constants reveal theirexplicit dependence on magnetization through magnetic free energy. Stochastic magnetization dynamics is studied in this paper on two equivalent levels: (1) stochastic dynamicsequations and (2) transition probability density which satis- ﬁes the partial differential equations with nonlocal Boltz- mann-type “collision integral” terms. The second level isvery convenient for the study of random magnetization switching caused by the jump-noise process. It is demon- strated that at very low temperatures the rate of such switch-ings may appreciably deviate from the predictions of thermal activation theory. This is consistent with experimental obser- vations of low temperature switchings and is usually attrib-uted to the phenomenon of “macroscopic tunneling” of magnetization. 7,8 II. STOCHASTIC DYNAMICS EQUATION To start the discussion, consider the following magnet- ization dynamics equation: dM dt¼/C0cM/C2Heff ðÞ þ TrðtÞ: (1) Here Mis the magnetization, cis the gyromagnetic ratio, andHeffis the effective magnetic ﬁeld, while TrðtÞis thea)Author to whom correspondence should be addressed. Electronic mail: isaak@eng.umd.edu. 0021-8979/2011/109(7)/07D312/6/$30.00 VC2011 American Institute of Physics 109, 07D312-1JOURNAL OF APPLIED PHYSICS 109, 07D312 (2011)jump-noise process which accounts for random thermal effects. The process TrðtÞcan be deﬁned as follows: TrðtÞ¼X imidðt/C0tiÞ; (2) where miare random jumps of magnetization occurring at random times ti: Equations (1)and(2)imply that the stochastic magnetiza- tion dynamics consists of continuous magnetization precessions randomly interrupted by random jumps in magnet-ization (see Figure 1). To fully describe the process T rðtÞ,s t a - tistics of miandtimust be deﬁned. Furthermore, the random jump process must be deﬁned in such a way that the dynamicsdescribed by Eq. (1)occurs on the sphere R, jMðtÞj ¼ M s¼const ; (3) where Msis the spontaneous magnetization. The latter con- straint is due to the strong local exchange interaction, which prevails over all other interactions at the smallest spatialscale compatible with the continuous media description. To fully specify the jump process (2), the transition probability rate SðM i;Miþ1Þis introduced, where Mi¼Mðt/C0 iÞandMiþ1¼Mðtþ iÞ¼Miþmiare magnetiza- tions immediately before and after a jump at t¼ti, respec- tively. To satisfy the fundamental constraint (3), the function SðMi;Miþ1Þis required to be deﬁned on the sphere R.B y using the function SðMi;Miþ1Þ, the following formula describes the random timing of magnetization jumps: Probðtiþ1/C0ti>sÞ¼exp /C0ðtiþs tikMðtÞ½/C138 dt/C26/C27 ; (4) where kMðtÞ½/C138 is the scattering rate of the jump process, which is given by the formula kMðtÞ½/C138 ¼þ RSMðtÞ;M0½/C138 dR0; (5) with the integration being performed over all M0such that Mjj¼Ms. It is clear that kMðtÞ½/C138 dthas the physical meaning of the probability that a magnetization jump will occurduring the time interval ðt;tþdtÞ. Assuming that a jump event occurs at some time ti, the probability density function vðmijMiÞof magnetization jump is speciﬁed by the formula vðmijMiÞ¼SMi;Miþmi ðÞ kðMiÞ: (6) It is apparent from formula (5)that the probability density function vðmijMiÞsatisﬁes the normalization condition þ RvmijMi ðÞ dRmi¼1; (7) where the integration is performed over all misuch that jMiþmij¼ Ms: Formulas (3)–(6)completely deﬁne the random jump process (2), provided that the transition probability rate SðMi;Miþ1Þis known. The physically reasonable expressions for the transition rate SðMi;Miþ1Þare discussed later in the paper. It is worthwhile to point out that the stochastic magnet- ization dynamics equations deﬁned by formulas (1)–(6)are (in many ways) similar to the semiclassical transport equa-tions used in semiconductor physics. 9 Next, we shall discuss how the damping terms for mag- netization dynamics can be extracted as average effectscaused by the jump process T rðtÞ. To this end, we shall write the process (2)in the form TrðtÞ¼ETrðtÞ½/C138 þ Tð0Þ rðtÞ; (8) where the symbol EðÞdenotes the expected value, while Tð0Þ rðtÞhas the meaning of ﬂuctuations. It can be shown10that ETrðtÞ½/C138 ¼ kMðtÞ½/C138 EmðtÞ½/C138 : (9) It will be assumed in the sequel that the process TrðtÞis small in the sense that only small jumps mðtÞhave non-neg- ligible probability to occur. Under this assumption, it can beshown that MðtÞ/C1EmðtÞ½/C138 ’ 0: (10) Indeed, from formula (3)and the relation jMðtÞj ¼ j Mðt /C0Þj ¼ j Mðt/C0Þþmj¼Ms; (11) we ﬁnd that 2MðtÞþm ½/C138 /C1 m¼0: (12) Since the process TrðtÞis assumed to be small, it can be con- cluded that for jumps with non-negligible probability the fol- lowing inequality holds: jmj/C282Ms: (13) From the last inequality and formula (12), we ﬁnd MðtÞ/C1m/C250; (14) which implies formula (10). It follows from formulas (9)and (10) that FIG. 1. Jump-noise magnetization dynamics on the sphere.07D312-2 Mayergoyz, Bertotti, and Serpico J. Appl. Phys. 109, 07D312 (2011)MðtÞ/C1ETrðtÞ½/C138 ’ 0: (15) This means that the expected value ETrðtÞ½/C138 is in the plane normal to MðtÞ. By choosing in this plane the basis vectors M/C2HeffandM/C2M/C2Heff ðÞ ; (16) we ﬁnd ETrðtÞ½/C138 /C25 /C0 c0 LM/C2Heff ðÞ /C0 aLM/C2M/C2Heff ðÞ ½/C138 :(17) By substituting formula (17) into formula (8)and then into Eq.(1), we arrive at the following equation: dM dt¼/C0 ~cLM/C2Heff ðÞ /C0 aLM/C2M/C2Heff ðÞ þ Tð0Þ rðtÞ;(18) where ~c¼cþc0 L. The last equation is the randomly per- turbed [by ﬂuctuations Tð0Þ rðtÞ] Landau–Lifshitz equation, which justiﬁes the use of subscript Lforcanda. It is also apparent from formula (17) that average (deterministic) action caused by random thermal effects as described by the jump process (2)results in the Landau–Lifshitz damping and slight correction of the gyromagnetic ratio in the precession term. It is easy to see that if the basis vectors M/C2HeffandM/C2dM dt(19) are chosen in the plane normal to MðtÞinstead of the basis vectors (16), then we can write ETrðtÞ½/C138 ’ /C0 c0 GM/C2Heff ðÞ /C0 aGM/C2dM dt: (20) By substituting formula (20) into formula (8)and then into dynamics Eq. (1), we arrive at the following equation: dM dt¼/C0 ~cGM/C2Heff ðÞ /C0 aGM/C2M/C2Heff ðÞ þ Tð0Þ rðtÞ;(21) where ~cG¼cþc0 G: The last equation is the randomly perturbed Landau– Lifshitz–Gilbert equation (hence the use of subscript G). It is clear from the presented discussion that the choice of different basis vectors in the plane normal to MðtÞleads to different (but mathematically equivalent) forms of the magnetizationdynamics equation. It is also clear that damping constant aas well as gyromagnetic ratio correction ccan be found by eval- uating the expected value ET rðtÞ½/C138 of the jump process (2) and then decomposing this expected value in terms of appro- priate basis vectors in the plane normal to MðtÞ. To evaluate ETrðtÞ½/C138 , the expressions for SðMi;Miþ1Þare needed. III. EQUATION FOR TRANSITION PROBABILITY DENSITY It turns out that the physically meaningful expressions forSðMi;Miþ1Þcan be found by studying the stochastic magnetization dynamics deﬁned by Eq. (1)on the level of transition probability density wðM;t;M0;t0Þ. For the sake of notational simplicity, the “backward variables” M0andt0 will be suppressed (omitted) in the sequel. It can be shown(see, for instance Ref. 11) that wðM;tÞis the solution of the following integral partial differential equation: @w @t¼/C0cdivRM/C2r Rg ðÞ w ½/C138 þ ^CðwÞ; (22) where ^CðwÞis the Boltzmann-type “collision integral” given by the formula ^CðwÞ¼þ RSðM0;MÞwðM0;tÞ/C0SðM;M0ÞwðM;tÞ ½/C138 dR0(23) andgis the magnetic free energy related to the effective ﬁeld Heffby the expression Heff¼/C0 r Rg: (24) It is evident that Eq. (22) contains the collision integral term instead of a “diffusion” term which corresponds to the white noise process. It is also clear that the collision inte-gral represents the net probability “ﬂow” due to the scatter- ings from Mto all M 0on the sphere Rand from all M0on RtoM. Equation (22) is very convenient for the derivation of constraints on SðM;M0Þ, which follow from the consistency of this equation with thermodynamics. At thermal equilib-rium the following relations are satisﬁed: wðM;tÞ¼w 0ðMÞ¼Ae/C0gðMÞ kT; (25) @w0ðMÞ @t¼0: (26) Furthermore, it can be proved that divRM/C2r Rg ðÞ w0 ½/C138 ¼ 0: (27) Indeed, divR½ðM/C2r RgÞw0ðgðMÞÞ/C138 ¼ w0½gðMÞ/C138divRðM/C2r RgÞ þM/C2r Rg ðÞ /C1dw0 dgrRgðMÞ; (28) and divRM/C2r Rg ðÞ ¼ 0; (29) M/C2r Rg ðÞ /C1 r Rg¼0: (30) By using formulas (25),(26), and (27) in Eq. (22), we obtain þ RSðM;M0Þw0ðMÞ/C0SðM0;MÞw0ðMÞ ½/C138 dR0¼0:(31) It is clear that the last equation is valid if the “detailed bal- ance” condition SðM;M0Þw0ðMÞ¼SðM0;MÞw0ðM0Þ (32) is fulﬁlled. This condition is quite natural from the physical point of view and expresses the pr obability balance in the back- and-forth scattering between any pair of MandM0.F r o mt h e07D312-3 Mayergoyz, Bertotti, and Serpico J. Appl. Phys. 109, 07D312 (2011)mathematical point of view, th e detailed balance condition (32) can be used for the symmetrization of the kernel of the collision integral (23) when the ratio wðM;tÞ=w0ðMÞis treated as an unknown function. This, in turn, can be used to prove the uniqueness of the equilibrium distribution as well as to establish the following generalized H-th eorem for magnetization dynam- ics driven by the jump-noise process, d dtþ Rw0ðMÞFwðM;tÞ w0ðMÞ/C20/C21 dR<0( 3 3 ) for any function FðxÞwith the property that its derivative F0ðxÞis a monotonically increasing function. In the particular case of the function Fðw=w0Þ¼ð w=w0Þlnðw=w0Þ, the last formula is reduced to d dtþ RwðM;tÞlnwðM;tÞ w0ðMÞ/C20/C21 dR<0: (34) It is clear that these theorems are similar to the celebrated Boltzmann H-theorem in s tatistical mechanics,12and they reveal the existence of a class of Lyapunov functionals that aremonotonically decreased with tim e during the magnetization dynamics. These issues are not treated in this paper because they will be discussed elsewhere. 13Instead, we shall proceed to the derivation of the physically reasonable expression for the transition rate SðM;M0Þ. To this end, we substitute formula (25) into the detailed balance condition (32) a n dt h e nd i v i d e both sides by Aexp/C0gðMÞþgðM0Þ 2kThi . As a result, we obtain /ðM;M0Þ¼SðM;M0ÞexpgðM0Þ/C0gðMÞ 2kT/C20/C21 ¼SðM0;MÞexpgðMÞ/C0gðM0Þ 2kT/C20/C21 : (35) It is clear from the last formula that the function /ðM;M0Þis symmetric: /ðM;M0Þ¼/ðM0;MÞ: (36) It is also clear from formula (35) that SðM;M0Þ¼/ðM;M0ÞexpgðMÞ/C0gðM0Þ 2kT/C20/C21 : (37) In general, function /ðM;M0Þis expected to be found through some identiﬁcation procedure based on experimental data.However, it is natural to assume on physical grounds that /ðM;M 0Þ¼/jM/C0M0j ðÞ (38) and it is narrow peaked at M¼M0. Then, by using the formula /ðxÞ¼eln/ðxÞ;x¼jM/C0M0j ðÞ ; (39) andtheﬁrstthreetermsintheTaylorexpansionforln /ðxÞ,weﬁnd ln/ðxÞ’ln/ð0Þ/C0j/00ð0Þj 2/ð0Þx2: (40) By using formulas (38),(39) and(40), the expression (37) can be transformed as follows:SðM;M0Þ¼Bexp /C0jM/C0M0j2 2r2 ! /C2expgðMÞ/C0gðM0Þ 2kT/C20/C21 ;(41) where r2¼/ð0Þ=j/00ð0Þj: It must be remarked that the expression for Sidentical to formula (41) was postulated in Ref. 14and used in the study of nucleation rate. Here, we shall use formula (35) for the derivation of expressions for the damping constant aand the scattering rate kMðtÞ½/C138 . To this end, by using formulas (6), (9), and (41), as well as the smallness of r2, we ﬁnd ETrðtÞ½/C138 ¼ Bð mexp/C0jM/C0M0j2 2r2 !" /C2expgðMÞ/C0gðM0Þ 2kT/C18/C19 /C21 dR: (42) By taking into account that M0/C0M¼mandgðMÞ/C0gðM0Þ ’/C0m/C1rRg, we end up with the following Gaussian-type integral: ETrðtÞ½/C138 ’ Bð mexp/C0jmj2 2r2þm/C1rRg 2kT !"# dR:(43) Due to the smallness of r2, the integration in formula (43) [and in (42)] is performed only for small min the plane tan- gential to the sphere R. By evaluating this integral, we ﬁnd ETrðtÞ½/C138 ’ /C0pr4 kTBrRgexp1 2rjrRgj2 2kT !22 43 5: (44) In a similar way, by using formula (41) in Eq. (5), we derive kðMÞ’2pr2Bexp1 2rjrRgj2 2kT !22 43 5: (45) By substituting formula (45) into(44) and taking into account that rRg¼1 M2 sM/C2M/C2Heff ðÞ ; (46) as well as relation (17), we obtain aL’kðMÞr2 2kTM2 s: (47) The last two formulas clearly reveal the dependence of aLon the properties of the jump-noise process as well as magnetiza-tionM, and can be used, for instance, to estimate the range of variation of a Lduring the magnetization switching dynamics. In the presented calculations, EðTrÞhas only a component along the vector M/C2M/C2Heff ðÞ and, consequently, c0 Lin for- mula (17) is equal to zero. This occurs because function /in Eq.(37) has been chosen to be isotropic, i.e., it depends only07D312-4 Mayergoyz, Bertotti, and Serpico J. Appl. Phys. 109, 07D312 (2011)onm¼jM/C0M0j[see formulas (38) and(41)]. It is expected that when /is anisotropic, c0 Lmay be different from zero. This means that random thermal effects may result in some correc-tions of the gyromagnetic ratio c. These corrections (if they exist) are experimentally observable and may provide infor- mation concerning the nature of the function /ðM;M 0Þ. In the presented calculations, the ﬁrst-order approxima- tion for gðMÞ/C0gðM0Þhas been used. More accurate results can be obtained in the axially symmetric case by using thesecond-order approximation for gðMÞ/C0gðM 0Þand evaluat- ing the resulting Gaussian-type integrals. The ﬁnal formulas are presented below: aL¼kðhÞr2 2kTM2 sQ2; (48) kðhÞ¼2pr2B Qexp1 2r 2kTQ@g @h/C18/C192"# ; (49) Q¼1/C0r2 2kT@2g @h2: (50) In the above formulas, his the angle between Mand the symmetry axis, while Qplays the role of the correction fac- tor which accounts for the second-order approximation in the expansion of gðMÞ/C0gðM0Þ. It is apparent from formula (50) that for sufﬁciently small r2the factor Qis close to 1 and formulas (48) and (49) are reduced to formulas (47) and (45), respectively. IV. RANDOM SWITCHING Next, we consider the problem of random switching of magnetization in uniaxial nanoparticles with only two possi-ble equilibrium (minimum energy) states located in two energy wells D 1andD2such that D1þD2¼R. More com- plicated energy landscapes can be treated in a similar way. Itis known that noise-driven switchings occur on a slow time scale. As a result, an equilibrium distribution of magnetiza- tion may be achieved in a well before thermal switching isrealized. This justiﬁes the following Kramers-Brown quasi- local equilibrium approximation for the transition probability density function: 2,15 wðM;tÞ’X2 i¼1PiðtÞwoiðMÞ: (51) In the last formula, PiðtÞis the probability of M2Diand w0iðMÞ¼wiðMÞ Ziexp/C0gðMÞ/C0gi kT/C20/C21 ; (52) where wiðMÞ¼1f o r M2DiandwiðMÞ¼0f o r M62Di.gi are energy minima, while Ziare normalization constants such thatð Diw0iðMÞdR¼1: (53) This normalization is consistent with the deﬁnition PiðtÞ¼ð DiwðM;tÞdR: (54)We shall next use the Kramers–Brown approximation (51) for the transformation of Eq. (22) into the Master equation. To start this transformation, we integrate both sides of (22) over Dkðk¼1;2Þ: ð Dk@w @tdR¼/C0cð DkdivRM/C2r Rg ðÞ w ½/C138 dR þð Dk^CðwÞdR: ð55Þ It is clear thatð Dk@w @tdR¼dPk dt: (56) By using the divergence theorem on the sphere R, we ﬁnd ð DkdivRM/C2rRg ðÞ w ½/C138 dR¼þ LwM/C2rRg ðÞ /C1 mdl¼0:(57) This is because on the common boundary LofD1andD2the magnetic free energy gis constant and, consequently, M/C2r Rg is orthogonal to the vector mof unit normal to LonR. Finally, the last term in Eq. (55)can be transformed as follows: ð Dk^CðwÞdR¼X2 i¼1ð Dkð DiSðM0;MÞwðM0;tÞdR0/C20/C21 dR /C0X2 i¼1ð DkwðM;tÞð DiSðM;M0ÞdR0/C20/C21 dR: (58) By substituting formula (51) into(58) and then into Eq. (55) and taking into account relations (56) and(57), we arrive at the following Master equation: dPk dt¼X2 i¼1kkiPi/C0PkX2 i¼1kik; (59) where kki¼ð Dkð DiSðM0;MÞw0iðM0ÞdR0/C20/C21 dR: (60) By using formulas (41) and(52), we ﬁnd kki¼B Zið Dkð Die/C0jM/C0M0j2 2r2e/C0gðMÞþgðM0Þ/C02gi 2kT dR0/C20/C21 dR:(61) The Master equation (59)–(61) is very convenient for the study of random switching. To demonstrate this, consider theinitial stage of random switching from the energy well D 1 into the energy well D2. During this initial stage, P1’1 and P2’0: (62) Consequently, from Master equation (59) we ﬁnd dP2 dt’k21: (63)07D312-5 Mayergoyz, Bertotti, and Serpico J. Appl. Phys. 109, 07D312 (2011)It is clear from formulas (61) and(63) that there are two dis- tinct regimes that are controlled by r2andT. For sufﬁciently large Tand small r2, the integrand in (61) is strongly peaked in the narrow region near the boundary Lbetween D1andD2. In this narrow region, the second factor of the integrand in (61) is close to exp /C0ðgmax/C0giÞ=kT ½/C138 . This leads to the classi- cal Arrhenius Law for the switching rate, and the temperature dependence for this switching rate is typical for thermally acti- vated switching phenomena. Another distinct case is when atvery low temperatures the second factor in the integrand of (61) dominates and it is strongly peaked for MandM 0being around respective energy minima. This “peaking” is especiallypronounced when g 2<g1, which may be realized in the pres- ence of external bias magnetic ﬁeld. It is apparent that this will result in a different temperature dependence of the switch-ing rate at very low temperatures. This is consistent with ex- perimental observations of low temperature magnetization switching, and it is usually attributed to the phenomena ofmacroscopic tunneling of magnetization. It is also predictable that for intermediate values of Tthe crossover between the above distinct regimes may occur 8,16which may be used for the identiﬁcation of r2[or function /ðM;M0Þin(37)]. V. CONCLUSION It is proposed to use a jump-noise (Poisson-type) pro- cess rather than a white-noise process to account for ran-dom thermal effects on magnetization dynamics. It is shown that in the case of small jump noise, Landau–Lifshitz and Gilbert damping terms emerge as average effects pro-duced by the jump-noise process. Simple formulas for damping constants in terms of the scattering rate of the jump-noise process are derived. These formulas reveal thedependence of the damping constants on magnetization. The analysis of random magnetization switchings caused by the jump-noise process is outlined. It is demonstrated thatthe switching rate at very low temperatures may appreciably deviate from the predictions of thermal activation theory. This fact is consistent with experimental observations andis usually attributed to the “macroscopic tunneling” magnet- ization phenomena. ACKNOWLEDGMENTS This research has been supported by NSF and by ONR. 1W. F. Brown, Micromagnetics (Krieger, New York, 1963). 2W. F. Brown, Phys. Rev. 130, 1677 (1963). 3D. R. Fredkin, Physica B 306, 26 (2001). 4G. Bertotti, I. Mayergoyz, and C. Serpico, Nonlinear Magnetization Dynamics in Nanosystems (Elsevier, Oxford, 2009). 5L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935). 6T. L. Gilbert, Phys. Rev. 100, 1243, 1955 [Abstract only; full report, Armor Research Foundation Project No. A059, Supplementary Report,May 1, 1956] (unpublished). 7E. M. Chudnovsky and L. Gunther, Phys. Rev. B 37, 9455 (1988). 8L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, and B. Barbara, Nature 383, 145 (1996). 9C. Korman and I. Mayergoyz, Phys. Rev. B 54, 17620 (1996). 10D. Kannan, An Introduction to Stochastic Processes (North-Holland, New York, 1979). 11I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations (Springer Verlag, New York, 1972). 12R. Kubo, Statistical Mechanics (North-Holland, New York, 1990). 13I. Mayergoyz, G. Bertotti, and C. Serpico, Generalized H-theorems for magnetization dynamics driven by a jump-noise process, J. Appl. Phys. (submitted). 14J. S. Langer, Phys. Rev. Lett. 21, 973 (1968). 15H. A. Kramers, Physica (Amsterdam) 7, 284 (1940). 16W. Wernsdorfer, E. B. Orozco, K. Hasselbach, A. Benoit, D. Maily, O. Kubo, H. Nakano, and B. Barbara, Phys. Rev. Lett. 79, 4014 (1997).07D312-6 Mayergoyz, Bertotti, and Serpico J. Appl. Phys. 109, 07D312 (2011)
1.2172893.pdf	10GHz bandstop microstrip filter using excitation of magnetostatic surface wave in a patterned Ni78Fe22 ferromagnetic film Marina Vroubel, Yan Zhuang, Behzad Rejaei, and Joachim N. Burghartz Citation: J. Appl. Phys. 99, 08P506 (2006); doi: 10.1063/1.2172893 View online: http://dx.doi.org/10.1063/1.2172893 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v99/i8 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 04 Dec 2012 to 132.210.244.226. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions10 GHz bandstop microstrip ﬁlter using excitation of magnetostatic surface wave in a patterned Ni 78Fe22ferromagnetic ﬁlm Marina Vroubel,a/H20850Yan Zhuang, Behzad Rejaei, and Joachim N. Burghartz Laboratory of High Frequency Technology and Components, Delft University of Technology, 2600 GA Delft, The Netherlands /H20849Presented on 3 November 2005; published online 24 April 2006 /H20850 Various microstrips with a ferromagnetic core were designed and fabricated on a silicon substrate. The core was formed by a 0.5- /H9262m-thick Ni 78Fe22ﬁlm, patterned into rectangular prisms. Measurement results for attenuation constant versus frequency show a peak value of /H1101150 dB/cm around 10 GHz. Electromagnetic simulations show that the attenuation observed is due to theenergy exchange between the quasi-TEM mode of the microstrip and magnetostatic surface modesexcited in the direction perpendicular to the signal line. © 2006 American Institute of Physics . /H20851DOI: 10.1063/1.2172893 /H20852 INTRODUCTION Microstrip transmission lines with hybrid magnetic/ dielectric cores have been extensively explored for micro-wave applications. 1–6In particular, it was demonstrated for a microstrip with yttrium iron garnet magnetic core, saturatedperpendicular to the plane of the strip line, that transmissioncharacteristics of the line show strong coupling phenomenabetween quasi-TEM and magnetostatic surface wave/H20849MSSW /H20850modes. 2On the other hand, for a microstrip with a tangentially /H20849in-plane /H20850magnetized magnetic core, the effect of the MSSW mode has been ignored; even the possibletracks of these modes can be seen from experimental data. 1,7 In this paper, the attenuation of a quasi-TEM propagatingmode due to coupling with MSSWs will be demonstrated fora fully monolithically integrated microstip with a thin-ﬁlmNi 78Fe22ferromagnetic /H20849FM /H20850core with an in-plane magneti- zation. EXPERIMENT Microstrips with signal lines of different lengths /H20849L=1, 2, and 4 mm /H20850and widths /H20849W=20, 30, and 50 /H9262m/H20850were de- signed and fabricated on a silicon substrate /H20849Fig. 1 /H20850. The core of the microstrips was formed by a 0.5- /H9262m-thick Ni 78Fe22 layer, sputtered in the absence of an external dc magneticﬁeld. Wet etching was then used to pattern the FM ﬁlm intotwo groups of rectangular prisms with widths /H20849W FM/H20850of 100/H9262m/H20849sample A /H20850and 200 /H9262m/H20849sample B /H20850. In each case the prisms had the same length as the signal line. SiO 2insu- lation layers 1 /H9262m thick separate the FM core from the sig- nal and ground lines. B-Hloop measurements showed a shape-induced easy axis orientated along the long side of theFM prism with effective shape-induced anisotropy ﬁelds H eff of 60 and 40 Oe for samples A and B, respectively, and a saturation magnetization 4 /H9266Mof 1.2 T for both samples. This yields the ferromagnetic resonance /H20849FMR /H20850frequencies of 2.4 GHz /H20849sample A /H20850and 1.9 GHz /H20849sample B /H20850according to Refs. 1 and 8:fFMR=/H9253 2/H9266/H20881Heff/H20849Heff+4/H9266M/H20850. /H208491/H20850 The high-frequency properties of the microstrip lines were extracted from S-parameter measurements performed on a HP-8510 network analyzer. The microstrips were measuredusing ground-signal-ground /H20849G-S-G /H20850rf probes in a two-port conﬁguration. Through-reﬂect line /H20849TRL /H20850calibration was performed, and open dummy structures were used to mea-sure and deembed parasitic capacitances appearing due to themeasurement patches. RESULTS AND DISCUSSION Figures 2 /H20849a/H20850and 2 /H20849b/H20850show the measured attenuation constant /H20849real part of the propagation constant /H20850as a function of frequency for 2-mm-long microstrips with signal lines andmagnetic cores of different widths. For a narrow FM corewith a high shape-induced anisotropy /H20851sample A, Fig. 2 /H20849a/H20850/H20852, the attenuation peak can be divided in two components: alow-frequency component with a maximum around 5 GHz/H20849peak LF /H20850and a high-frequency component with a maximum at 10 GHz /H20849peak HF /H20850/H20851Fig. 2 /H20849a/H20850/H20852. The low-frequency peak of attenuation constant versus frequency can be associated withFMR of a ferromagnetic core. 1–8The observed frequency shift with respect to the expected thin-ﬁlm FMR frequency/H208492.4 GHz /H20850is partly due to the combination of a high mag- netic damping constant and a small width of a signal line, 9as well as the inﬂuence of a high-frequency component /H20849peak HF/H20850. a/H20850FAX: /H1100131 15 262 3271; electronic mail: m.vroubel@ewi.tudelft.nl FIG. 1. Microstrip line with a FM core.JOURNAL OF APPLIED PHYSICS 99, 08P506 /H208492006 /H20850 0021-8979/2006/99 /H208498/H20850/08P506/3/$23.00 © 2006 American Institute of Physics 99, 08P506-1 Downloaded 04 Dec 2012 to 132.210.244.226. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsFor sample B, the attenuation constant versus frequency demonstrates no low-frequency FMR component but only ahigh-frequency contribution close to 10 GHz /H20851Fig. 2 /H20849b/H20850/H20852. The absence of clear tracks of the FMR in microstrips with wideFM cores /H20849sample B /H20850can be attributed to the low value of a shape-induced anisotropy and high nonuniformity of both dcand ac magnetic ﬁelds and magnetizations. As a result, de-magnetizing ﬁelds, effective anisotropy ﬁelds, and, conse-quently, FMR frequency become local characteristics, de-pending on the position inside the FM core. The same isvalid for sample A resulting in a broadening of FMR, but thiseffect is more pronounced for sample B due to a smallereffective anisotropy. This leads not only to a lower FMRfrequency but also to a lower amplitude of the FMR attenu-ation peak according to our own simulations as well as thoseof Ref. 8. The position of the high-frequency maximum /H20849peak HF /H20850 in attenuation constant versus frequency depends on thewidth of the microstrip. For sample A /H20851Fig. 2 /H20849a/H20850/H20852this depen- dence is partly masked by the presence of the FMR peak. Forsample B, where the effect of FMR is washed out, the fre-quency of peak HF shifts from 8 to 10 GHz, when the widthof the signal line changes from 50 to 20 /H9262m. Measurements carried out on 1-mm- and 4-mm-long microstrip lines led tothe same values for the attenuation constant and, therefore,will not be shown here. While the nature of the low-frequency attenuation peak is clear, the source of the high-frequency peak should beexplained. The microstrip devices were simulated with HFSS 9.21 /H20849Ansoft /H20850software package providing a three-dimensional/H208493D/H20850full-wave analysis. Ferromagnetic core was modeled as aB-Hnonlinear material with FM characteristics determined by the saturation magnetization 4 /H9266M=1.2 T, a dc bias ﬁeld Hdc=1 Oe which—in our case—models the dc demagnetiz- ing ﬁeld /H20849should not be confused with the dc effective aniso- tropy ﬁeld Heff/H20850, and the FMR linewidth /H9004H=4/H9266/H9251f//H9253which accounts for the FM losses /H20849/H9251is the Gilbert damping con- stant and /H9253is the gyromagnetic constant /H20850. This approach is equivalent to the description of the magnetic material based on a permeability tensor /H20849e.g., Ref. 8 /H20850. Though the nonuni- formity of the dc magnetization and demagnetizing ﬁeldswas neglected, the nonuniformity of the ac ﬁelds was natu-rally included from 3D boundary conditions. The calculated attenuation constant versus frequency for signal lines of different width is given in Fig. 3 /H20849a/H20850. The small disparity between the measured and calculated results ap-pears in the region of FMR frequency and, in our opinion,comes partly from additional magnetic losses, possibly asso-ciated with magnetic pinning due to the wet etching on theedges of FM core 10and partly from the neglected nonunifor- mity of dc demagnetizing ﬁelds and magnetization. The dis-parity can be decreased by increasing the value of dissipationparameter, as it is shown for comparison in Fig. 3 /H20849a/H20850. The amplitudes and frequencies of the high-frequency attenuationpeak are in a very good agreement with the measured data/H20851Figs. 2 /H20849a/H20850and 3 /H20849a/H20850/H20852. To explain the nature of the high-frequency peak, we FIG. 2. /H20849a/H20850Measured attenuation constant vs frequency for FM core with the width of 100 /H9262m/H20849sample A /H20850./H20849b/H20850Measured attenuation constant vs frequency for FM core with the width of 200 /H9262m/H20849sample B /H20850. The numbers correspond to the width of a signal line in /H9262m. All microstrip lines are 2 mm long. FIG. 3. /H20849a/H20850Calculated attenuation constant vs frequency for FM core with the width of 100 /H9262m/H20849sample A /H20850.4/H9266M=1.2 T, Hdc=1 Oe, and the conduc- tivity of the FM core /H9268=6.4/H11003106S/m. The numbers correspond to the width of a signal line in /H9262m. For solid lines, a dissipation parameter /H9251 =0.005 is used. For a dashed line, a dissipation parameter /H9251=0.025 and W=50/H9262m were used. /H20849b/H20850Calculated attenuation constant vs frequency for FM core with the width of 100 /H9262m. 4/H9266M=1.2 T, Hdc=1 Oe, /H9251=0.005, and /H9268=6.4/H11003102S/m. The numbers correspond to the width of a signal line in /H9262m. All microstrip lines are 2 mm long.08P506-2 Vroubel et al. J. Appl. Phys. 99, 08P506 /H208492006 /H20850 Downloaded 04 Dec 2012 to 132.210.244.226. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsshould notice that it does not appear in the parallel plate model, when in-plane propagation perpendicular the signalline is neglected. 1,8Hence, the high-frequency attenuation is caused by an excitation of in-plane waves perpendicular tothe signal line. In the frequency range above the FMR fre-quency, only MSSWs can exist in the FM core. 10–14In the magnetostatic limit, the dispersion relation for these wavescan be written as 13 /H208491+/H9260/H208502+/H20849/H9260/H9251+1/H20850/H20875/H9260/H9251− tanh/H20873tD tFMktFM/H20874/H20876+/H208491+/H9260/H208502 /H11003/H208751 + tanh/H20873tD tFMktFM/H20874/H20876coth /H20849ktFM/H20850=0 , /H208492/H20850 with/H9260=/H9275H/H9275M//H20849/H9275H2−/H92752/H20850,/H9260/H9251=/H9275/H9275 M//H20849/H9275H2−/H92752/H20850,/H9275M=/H92534/H9266M, and/H9275H=/H9253Heff. Here /H9275=2/H9266f, and tDand tFMdenote the thickness of the dielectric and ferromagnetic layers, respec-tively. A quasi-TEM mode propagating along the signal linecouples to MSSW modes with a corresponding wave vectorkwhich is of the order of /H9266/W/H20849Wis the width of the signal line /H20850. This, of course, is a very coarse approximation, but gives a qualitative explanation of the effect of the strip widthon the frequency of the attenuation maximum. Figure 3 /H20849b/H20850 gives the HFSS results for attenuation constant versus fre- quency for two signal lines of different widths /H2084920 and 50/H9262m/H20850. The set of parameters used was the same as in Fig. 3/H20849a/H20850, except for the conductivity of the FM core, which was reduced from 6.4 /H11003106to 6.4/H11003102S/m in order to clearly see the nondamped magnetostatic wave effects. When con-ductivity of FM ﬁlm decreases, the high-frequency peaksplits into harmonics appearing due to the ﬁnite width of FMcore.CONCLUSION An additional peak for attenuation constant versus fre- quency is observed at frequency far above FMR in micros-trips with NiFe FM cores. Electromagnetic /H20849EM /H20850simulations are in a good agreement with experiments and lead to theconclusion that the attenuation of a signal at frequencyaround 10 GHz is due to the energy exchange between thequasi-TEM mode of the microstrip and magnetostatic surfacemodes excited in the direction perpendicular to the magneti-zation. 1V. S. Liau, T. Wong, W. Stacey, S. Ali, and E. Schloemann, IEEE MTT-S Int. Microwave Symp. Dig. 3,9 5 7 /H208491991 /H20850. 2M. Tsutsumi and K. Okubo, IEEE Trans. Magn. 28, 3297 /H208491992 /H20850. 3E. Saluhun, P. Queffelec, G. Tanne, A.-L. Adenot, and O. Acher, J. Appl. Phys. 91, 5449 /H208492002 /H20850. 4S. Ikeda, T. Sato, A. Ohshiro, K. Yamasawa, and T. Sakuma, IEEE Trans. Magn. 37,2 9 0 3 /H208492001 /H20850. 5N. Cramer, D. Lucic, R. E. Camley, and Z. Celinski, J. Appl. Phys. 87, 6911 /H208492000 /H20850. 6Y. Zhuang, B. Rejaei, E. Boellaard, M. Vroubel, and J. N. Burghartz, IEEE Microw. Wirel. Compon. Lett. 12, 473 /H208492002 /H20850. 7E. Saluhun, G. Tanne, P. Queffelec, M. LeFloc’h, A.-L. Adenot, and O. Acher, Microwave Opt. Technol. Lett. 30,2 7 2 /H208492001 /H20850. 8R. J. Astalos and R. E. Camley, J. Appl. Phys. 83, 3744 /H208491998 /H20850. 9M. Vroubel, Y. Zhuang, B. Rejaei, and J. N. Burghartz, Trans. Magn. Soc. Jpn. 2, 371 /H208492002 /H20850. 10T. W. O’keeffe and R. W. Patterson, J. Appl. Phys. 9, 4886 /H208491978 /H20850. 11R. W. Damon and J. R. Eshbash, J. Phys. Chem. Solids 19,3 0 8 /H208491961 /H20850. 12A. Ganguly and D. C. Webb, IEEE Trans. Microwave Theory Tech. 23, 998 /H208491975 /H20850. 13T. Yukawa, J. Yamada, K. Abe, and J. Ikenoue, Jpn. J. Appl. Phys. 12, 2187 /H208491977 /H20850. 14M. Bailleul, D. Olligs, C. Fermon, and S. O. Demokritov, Europhys. Lett. 56,7 4 1 /H208492001 /H20850.08P506-3 Vroubel et al. J. Appl. Phys. 99, 08P506 /H208492006 /H20850 Downloaded 04 Dec 2012 to 132.210.244.226. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
5.0004837.pdf	J. Chem. Phys. 152, 214108 (2020); https://doi.org/10.1063/5.0004837 152, 214108 © 2020 Author(s).Coupled-cluster techniques for computational chemistry: The CFOUR program package Cite as: J. Chem. Phys. 152, 214108 (2020); https://doi.org/10.1063/5.0004837 Submitted: 14 February 2020 . Accepted: 12 April 2020 . Published Online: 03 June 2020 Devin A. Matthews , Lan Cheng , Michael E. Harding , Filippo Lipparini , Stella Stopkowicz , Thomas-C. Jagau , Péter G. Szalay , Jürgen Gauss , and John F. Stanton COLLECTIONS Paper published as part of the special topic on Electronic Structure Software Note: This article is part of the JCP Special Topic on Electronic Structure Software. ARTICLES YOU MAY BE INTERESTED IN Recent developments in the general atomic and molecular electronic structure system The Journal of Chemical Physics 152, 154102 (2020); https://doi.org/10.1063/5.0005188 PSI4 1.4: Open-source software for high-throughput quantum chemistry The Journal of Chemical Physics 152, 184108 (2020); https://doi.org/10.1063/5.0006002 TURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed- matter simulations The Journal of Chemical Physics 152, 184107 (2020); https://doi.org/10.1063/5.0004635The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Coupled-cluster techniques for computational chemistry: The CFOUR program package Cite as: J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 Submitted: 14 February 2020 •Accepted: 12 April 2020 • Published Online: 3 June 2020 Devin A. Matthews,1,a) Lan Cheng,2,b) Michael E. Harding,3,c) Filippo Lipparini,4,d) Stella Stopkowicz,5,e) Thomas-C. Jagau,6,f) Péter G. Szalay,7,g) Jürgen Gauss,5,h) and John F. Stanton8,i) AFFILIATIONS 1Department of Chemistry, Southern Methodist University, Dallas, Texas 75275, USA 2Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, USA 3Institut für Physikalische Chemie, Karlsruher Institut für Technologie (KIT), Kaiserstr. 12, D-76131 Karlsruhe, Germany 4Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Via G. Moruzzi 13, I-56124 Pisa, Italy 5Department Chemie, Johannes Gutenberg-Universität Mainz, Duesbergweg 10-14, D-55128 Mainz, Germany 6Department of Chemistry, University of Munich (LMU), Butenandtstr. 5-13, D-81377 Munich, Germany 7ELTE Eötvös Loránd University, Institute of Chemistry, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary 8Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, Florida 32611, USA Note: This article is part of the JCP Special Topic on Electronic Structure Software. a)Electronic mail: damatthews@smu.edu b)Electronic mail: lcheng24@jhu.edu c)Electronic mail: michael.harding@kit.edu d)Electronic mail: filippo.lipparini@unipi.it e)Electronic mail: stella.stopkowicz@uni-mainz.de f)Electronic mail: th.jagau@lmu.de g)Electronic mail: szalay@chem.elte.hu h)Author to whom correspondence should be addressed: gauss@uni-mainz.de i)Electronic mail: johnstanton@ufl.edu ABSTRACT An up-to-date overview of the CFOUR program system is given. After providing a brief outline of the evolution of the program since its inception in 1989, a comprehensive presentation is given of its well-known capabilities for high-level coupled-cluster theory and its application to molecular properties. Subsequent to this generally well-known background information, much of the remaining content focuses on lesser- known capabilities of CFOUR , most of which have become available to the public only recently or will become available in the near future. Each of these new features is illustrated by a representative example, with additional discussion targeted to educating users as to classes of applications that are now enabled by these capabilities. Finally, some speculation about future directions is given, and the mode of distribution and support for CFOUR are outlined. Published under license by AIP Publishing. https://doi.org/10.1063/5.0004837 .,s I. INTRODUCTION The origin of the CFOUR (Coupled-Cluster techniques for Com- putational Chemistry) program package1is deeply connected with the story of several young scientists crossing paths at an early stage of their careers in Rodney J. Bartlett’s group at the QuantumTheory Project at the University of Florida in Gainesville, near the dawn of the 1990s. After attending the inaugural Molecular Quantum Mechanics (MQM) meeting in honor of John A. Pople in Athens, GA, in October 1989, John F. Stanton was inspired by the rapid development around the world in high-accuracy quan- tum chemical methods and especially by the rapid progress that J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp was being made in their application to interesting and “real” chem- ical problems. Educated in the Bartlett group, he had been fully convinced of the power of high-level many-body methods and was determined to develop a new set of programs to bring these approaches to bear on meaningful chemical applications. Upon his return to Gainesville, Stanton started a project that has now lasted more than three decades, which has led to what is now known as CFOUR . By the end of 1989, he had written interfaces to the self- consistent field (SCF) and integral packages used in the Bartlett group—the ACES (Advanced Concepts in Electronic Structure) pro- gram system.2In 1990, Jürgen Gauss arrived in Gainesville for a postdoc in the Bartlett group, which fueled the development of the project. Together, Stanton and Gauss wrote many-body perturba- tion theory (MBPT)3and coupled-cluster (CC)4codes—the latter through CC with singles and doubles (CCSD)—which included ana- lytic gradients5as well as the exploitation of molecular point-group symmetry ( D2hand subgroups).6 John D. Watts, another postdoc in the Bartlett group at that time, contributed code for triple excitations, and Walter J. Laud- erdale, a graduate student, wrote a new SCF and integral transforma- tion program. Together with atomic orbital (AO) integrals coming from the MOLECULE package7of Jan Almlöf (one- and two-electron integrals; the code had recently been extensively modified for perfor- mance on vector processors by Peter R. Taylor), the VPROPS package8 (dipole and other one-electron property integrals that can trace their lineage back to the POLYATOM package9), and integral derivatives com- ing from the ABACUS package10of Trygve Helgaker et al. , the main core of what was to become CFOUR had already emerged. Apart from AO integral and integral derivative evaluation, all other codes were completely new; nothing associated with Hamiltonian construc- tion, MBPT and CC energy and density evaluation was taken from another source; indeed, even input parsing and general processing of output (vibrational frequencies, for example) was written from scratch. With this nucleus, a number of chemical applications11–13 were done at the dawn of the 1990s, and a first paper14describ- ing the code—called ACES II at that time—was published in 1992. Following the move of the main developers, Stanton to Austin, TX, and Gauss to Karlsruhe (Germany) and later Mainz (Germany), the main development centers of ACES II migrated from the original Gainesville location, taking their exposure to many-body methods with them. This eventually resulted in a bifurcation of ACES II , from which the Mainz–Austin–Budapest (MAB) version originated—the Budapest center (Hungary) involving Péter G. Szalay as another main author. In Gainesville, this was followed by a complete rewrite of the overall package devised to target emergent parallel comput- ers. This is known now as the ACES III package.15Finally, in 2008, the Mainz–Austin–Budapest version of ACES II , by now containing many new features and enhanced computational sophistication, was renamed as CFOUR .16 Since its beginnings, CFOUR has specialized in high-accuracy quantum chemical methods, targeting applications in the field of thermodynamic, spectroscopic, and kinetic phenomena of small- to medium-sized molecular systems. While some of its nearly 30- year-old primordial core remains in the current version, much has also changed since its inception. Incremental algorithmic improvements have been made to the existing capabilities, and new methodologies have been continuously added to the package by developers throughout the world. Some of thecapabilities included today (together with their first appearance in CFOUR ) are nuclear magnetic resonance (NMR) chemical shifts ranging from second-order MBPT through CCSD(T) (1990s),17–23 equation-of-motion coupled cluster (EOM-CC) methods for elec- tronically excited and ionized states,24–29analytic second deriva- tives for MBPT and CC through CCSDT (1990s);23,30–33auto- mated evaluation of anharmonic (quartic) force fields and computa- tion of associated rovibrational spectroscopic constants (1990s),34,35 new open-shell CC methods (1990s),36,37properties associated with high-resolution spectroscopy such as spin-rotation tensors (1990s and 2000s),35,38–41arbitrarily high-order CC gradients and sec- ond derivatives (as interfaced to the MRCC program package42,43 of Mihály Kállay, 2000s),44–47diagonal Born–Oppenheimer correc- tions (2000s),48,49couplings between quasidiabatic states (2010s),50,51 relativistic quantum chemical methods (2010s),52–60multireference CC methods (2010s),61highly efficient code for high-accuracy [post- CCSD(T)] methods (2010s),62and many more. Following the work of the original team and beginning their careers in the groups of the main authors, many more young sci- entists actively contributed to CFOUR . The primary authors of CFOUR now include Lan Cheng, who has contributed extensively with rela- tivistic quantum chemical methods56,58–60for both energy and prop- erty calculations; Devin A. Matthews, who has written a new and very fast coupled-cluster module ( xncc )62for CFOUR and contributed significantly to some of the spectroscopic extensions of CFOUR ;63,64 and Michael E. Harding, who has been in charge of many issues related with code infrastructure, parallelization,65,66and general organization. An accurate characterization of CFOUR is that it is a program system with many capabilities for the highly accurate calculation of parameters that play a role in diverse areas of chemical physics. Largely through methods based on coupled-cluster theory,4one can calculate potential energy surfaces, couplings between electronic states, a vast number of one- and two-electron properties that play a role in various branches of molecular spectroscopy, and relativistic corrections to electronic structure, and generally obtain informa- tion that can be extracted from accurate electronic wavefunctions and their response to external perturbations. Beyond this, there are auxiliary tools that make use of this fundamental information. For example, vibrational perturbation theory (VPT)67can be used to obtain accurate positions for the fundamental vibrational levels of semirigid polyatomic molecules (using the efficiently calculated anharmonic force field); information can be extracted to construct vibronic Hamiltonians in a diabatic representation; extrapolation to the basis set limit can be done in an automated fashion;68and molec- ular structures can be fitted to rotational constants,35both the raw experimental data and the equilibrium constants corrected (by CFOUR calculations) for the effects of vibration–rotation interaction.34,67 The capabilities of CFOUR can be also used in conjunction with the features of other computational chemistry programs (e.g., MRCC ,42,43GIMIC ,69NEWTON-X ,70–72and GECCO73,74) to which CFOUR has been interfaced. While providing powerful tools for the quantum chemical study of small-sized to medium-sized molecules, CFOUR does not have a great deal to offer in the area of large molecules. Devel- opments in CFOUR have focused on many-body treatments of elec- tron correlation, and the methods of density functional theory are completely absent from its repertoire. The coupled-cluster methods J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp available in CFOUR are mainly single-reference methods, meaning that calculations are built upon a single Slater determinant that is usually (but need not be) composed of orbitals associated with the Hartree–Fock self-consistent field (HF-SCF) solution. While some multireference effects can certainly be treated within the framework of equation-of-motion coupled-cluster (EOM-CC) methods75—this area represents a decided strength of CFOUR —more traditional meth- ods based on multiconfigurational zeroth-order wavefunctions are needed to describe phenomena associated with bond-breaking, to construct (semi-)global potential energy surfaces, and even to treat certain classes of transition states. While some of these limita- tions described above have been addressed by implementing mul- tireference variants of CC theory61,76and incorporating a rigor- ous second-order complete-active space SCF (CASSCF) scheme77 inCFOUR , the currently available version of the program exclusively offers single-reference treatments of the correlation problem. The remainder of this paper elaborates on the strengths and capabilities of the CFOUR program system. Section II summarizes the “core features” of CFOUR , specifically its treatments of the non- relativistic electronic Schrödinger equation based on CC and MBPT methods and its capabilities for calculating properties within these approximations using analytic derivative techniques. Many users of CFOUR are likely to be familiar with these capabilities, and Sec. II doc- uments these features with some remarks about the current status of implementations. We continue with a short section (Sec. III) about practical aspects such as input and use of CFOUR . Section IV describes new developments that are present in CFOUR , either in the current version (V2.1) or versions likely to come in the near future. After the discussion of the present state of the CFOUR project, we proceed in Sec. V with some remarks about the general long-term perspec- tive of CFOUR and close by describing the method by which the code is distributed. II. ESTABLISHED FEATURES A. Treatments of electron correlation The available treatments of electron correlation in CFOUR are based on many-body perturbation theory [MBPT, also known as Møller–Plesset (MP) perturbation theory]3,78and coupled-cluster (CC) theory,4,79,80collectively referred to as single-reference meth- ods, as their description of electron correlation starts from a single Slater determinant. CC theory was originally formulated for the quantum-chemical treatment of nuclear matter.81,82After its introduction into elec- tronic structure theory by ˇCížek,83,84it developed to one of the most powerful schemes quantum chemistry nowadays has to offer for the electron-correlation treatment and for high-accuracy computations. The success of CC theory is probably illustrated best by the fact that the CCSD(T) method,85to be described in detail below, often is referred to as the “gold standard” in quantum chemistry. CC theory uses an exponential ansatz for the wavefunction ∣ψ⟩=exp(T)∣0⟩, (1) where \|0 ⟩denotes the reference determinant (often, but not neces- sarily chosen as the HF state), and Tdenotes the cluster operator, which is an excitation operator and consists of the weighted sum of all excitations, T=T1+T2+. . .TN. (2)The sum in Eq. (2) runs up to TNwith Nas the number of elec- trons. T1,T2,. . .denote the weighted sums of single, double, etc., excitations with the unknown parameters given by the weighting coefficients that are usually referred to as amplitudes. The chosen wavefunction ansatz in Eq. (1) has significant advantages over the corresponding linear choice in configuration-interaction (CI) theory, as it ensures size-consistency86/size-extensivity87of the electron-correlation treatment even within a truncated scheme that does not include all excitations. CC theory, therefore, is, by construc- tion, a size extensive approach. Because of the exponential ansatz, the CC wavefunction is typ- ically not determined via the variational principle. Instead, one uses a projection approach in which the CC wavefunction is inserted into the electronic Schrödinger equation; the latter is then multi- plied from the left with exp( −T), and an expression for the energy is obtained by projection onto the reference determinant E=⟨0∣exp(−T)Hexp(T)∣0⟩, (3) and nonlinear equations for the amplitudes are obtained by projec- tion onto the excited determinants 0=⟨ΦP∣exp(−T)Hexp(T)∣0⟩. (4) In Eqs. (3) and (4), Hdenotes the usual molecular Hamiltonian andΦPdenotes a determinant from the manifold of excited deter- minants. The nonlinear amplitude equations [Eq. (4)] consequently need to be solved for all possible ΦP. Without any truncation, CC theory is equivalent to, though more involved than, full configuration interaction (FCI) and hence, in that form, not particularly useful. CC theory demonstrates its advantages only when used with a truncated cluster operator. The usual choices are here T=T2[CC doubles (CCD)],88–90T=T1+T2 [CC singles and doubles (CCSD)],91T=T1+T2+T3[CC sin- gles, doubles, triples (CCSDT)],92,93and T=T1+T2+T3+T4 [CC singles, doubles, triples, quadruples (CCSDTQ)],44,94,95etc. While initially the implementation of CC methods was quite cum- bersome,89–91the use of intermediates together with a rewrite of the equations in terms of matrix-vector products has enabled more straightforward access to CC methods6,95,96and also forms the basis of the CCSD implementations in CFOUR, which is described in detail in Ref. 6. CFOUR also offers the possibility to perform CCSDT92,93,97 as well as CCSDTQ calculations.44,62,94,95In addition, through an interface to the MRCC code,42,43CC computations with arbitrary excitations are possible.44 While CCSD is for many applications not accurate enough and CCSDT with an M8scaling ( Mdenotes here the system size, which is assumed to be proportional to both the number of occupied and virtual orbitals) too expensive, approximate CC methods have been developed in which not only the cluster operator is truncated but (expensive) terms in the CC equations are also neglected. This leads, in the case of triple excitations in a straightforward manner, to the CCSDT- nmethods.98,99The key idea is here to (a) skip the M8 terms and (b) avoid storage of the triples amplitudes. The selection of the terms in the triples equations is then based on perturbation theory and leads to CCSDT-1a,98CCSDT-1b,98CCSDT-2,99and CCSDT-3.99Somewhat related to CCSDT-1b is the CC3 model,100 which has been introduced by the Aarhus group in the context of CC response theory.101All these models (CCSDT- nwith n= 1–3 J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and CC3) scale with M7and do not require storage of triple excita- tion amplitudes. The CC3 model furthermore is part of an alterna- tive hierarchy of CC methods: CC2, CCSD, CC3, CCSDT, CC4, etc., in which CC2102is the simplest choice and a cheap approximation to CCSD with an M5scaling and no need to store double excitation amplitudes. In this context, one should also mention the quadratic CI singles and doubles (QCISD) scheme103by Pople et al. , which was introduced as a simpler alternative to CCSD. However, as there are nowadays no difficulties to implement CCSD, QCISD now plays only a minor role. The CCSDT- nand CC3 models are significantly more efficient in computational terms than the full CCSDT model, but they are, for many applications, still not affordable due to the need to con- sider triple excitations in each iteration. This issue can be amelio- rated by just considering a perturbative correction for triple exci- tations on top of a CCSD computation. Starting with early ideas based on corrections taken from fourth order MBPT or MP the- ory99,104this ultimately led to the development of the (T) correction, which involves the fourth-order correction due to triple excita- tions,85,105though computed with the converged CCSD amplitudes, together with one fifth-order correction, namely, the one that cou- ples singles and triples. Justifications for this choice have been, for example, given in Refs. 106 and 107. Similar ideas as in the case of CCSD(T) for triple excitations can be also pursued for the per- turbative treatment of quadruple excitations, which leads to the CCSDT(Q) approach.108,109More elaborate triple and quadruple corrections110[referred to as Λ-CCSD(T)111andΛ-CCSDT(Q)109] can be obtained by using the solution of the Λequations [Eq. (7)] in addition to those of the amplitude equations for the evaluation of the perturbative corrections.107,109,111,112 Considering the treatment of closed- and open-shell sys- tems, CFOUR offers spin-adapted treatments for closed-shell sys- tems and open-shell treatments based on unrestricted HF (UHF) and restricted open-shell HF (ROHF) reference determinants.12 The UHF-CC treatment is a straightforward spin-orbital based approach, though with spin integration, while ROHF-CC113for- mally classifies as a non-HF CC approach as the occupied-virtual block of the Fock matrix in the spin-orbital basis does not vanish. However, this only requires the trivial inclusion of off-diagonal ele- ments of the Fock matrix in the CC equations within a standard CC treatment, but some thought is required to formulate appro- priate perturbative corrections.114,115The latter are most efficiently implemented using so-called semicanonical orbitals.116CC calcula- tions can also be carried out using the quasi-RHF (QRHF) deter- minant113as reference (here, the orbitals for the reference determi- nant are obtained in an RHF calculation with a different number of electrons). Further options involve Brueckner CC (B-CC)117,118and orbital-optimized CC calculations.119In both cases, the orbitals are determined in the presence of electron correlation, which, though more expensive, sometimes turns out to be more efficient. MBPT can be derived using perturbative techniques together with the Møller–Plesset partitioning120of the electronic Hamilto- nian. Alternatively, expressions for the various orders of MBPT can be obtained through perturbative expansions of the CC energy expression as well of the CC amplitude equations. Second-order MBPT, known as MBPT(2) or MP2, has evolved over the years to the standard scheme for a first (and not particularly accurate) description of electron correlation at low cost (the formal scalingis only of the order of M5) for otherwise rather well behaved sys- tems. Higher-order MBPT schemes (up to sixth order) have also been formulated and implemented86,121–125but are only rarely used. The reasons are the now well established convergence problems of MBPT126,127as well as the availability of the more robust CC methods. Nevertheless, MBPT(3) (equivalent to MP3) and MBPT(4) (equivalent to MP4) are accessible through CFOUR . MBPT(5) and MBPT(6) are only available in specialized codes,123–125while even higher order MBPT corrections so far can only be extracted from a perturbative dissection of FCI.128,129 MBPT is rather straightforward to formulate for restricted and unrestricted HF (RHF and UHF) reference functions. However, after some experimentation,130–132a satisfactory formulation of MBPT for restricted open-shell HF (ROHF) reference functions has been sug- gested.116,133,134The perturbed Hamiltonian contains here also the virtual-occupied blocks of the Fock matrices in a spin-orbital formu- lation, and a non-iterative treatment is possible when semicanonical orbitals are used.116 Table I summarizes the CC and MBPT/MP methods that are available in the current public version (V2.1) of CFOUR together with information about the possible choices for the reference determinants. B. Analytic derivatives for the computation of molecular properties A particular strength of CFOUR is its ability to provide analytic derivatives of the energy and thus easy access to molecular prop- erties for most of the implemented quantum-chemical methods. Analytic derivative techniques136,137play an important role for the computation of molecular geometries, as only analytically evaluated forces render geometry optimizations routinely doable. CFOUR offers geometrical derivatives5,32,45,114,138–141for most of the implemented CC and MBPT methods and thus allows the routine determina- tion of equilibrium geometries [preferably via the Broyden-Fletcher- Goldfarb-Shanno (BFGS) scheme142] but also of transition-state geometries using methods based on eigenvector following.143 In CC theory, analytic gradients have been formulated144,145 and implemented144rather late. The main reason is the non- variational character of the standard CC approaches. Straightfor- ward differentiation of the CC energy expression [Eq. (3)], with respect to a perturbation x, thus leads to an expression that involves the derivatives of the cluster operator dE dx=⟨0∣exp(−T)dH dxexp(T)∣0⟩+⟨0∣[exp(−T)Hexp(T),dT dx]∣0⟩. (5) Evaluation of gradients based on this expression would offer lit- tle advantage over a finite-difference approach. However, based on the interchange theorem of perturbation theory,146the deriva- tive expression can be reformulated such that the derivatives of the cluster operator Tare no longer needed. This has been shown by Adamowicz, Laidig, and Bartlett,147thereby introducing the perturbation-independent Λequations, and used by Scheiner et al.144for their implementation of analytic closed-shell CCSD gradients. A modern formulation of CC derivative theory is based on the Lagrangian formalism introduced by Helgaker and Jørgensen.148–150In order to cope here with the non-variational J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . CC and MBPT/MP methods available in the CFOUR program package.aA single x designates that only energy evaluations are possible, while xx indicates that both energies and gradients can be calculated, and xxx indicates that analytic second derivatives are available. Method RHF UHF ROHF Remarks HF xxx xxx xxx MBPT(2)/MP2 xxx xxx xx MBPT(3)/MP3 xxx xxx xx SDQ-MBPT(4)/SDQ-MP4 x x x MBPT(4)/MP4 xxx xxx x CCD xxx xxx CCSD xxx xxx x Also Brueckner, orbital-optimized CCSD, QRHF-CCSD CCSDT xxx x x CCSDTQ xx CCSDT- n,n= 1a, 1b, 2, 3 xxx x CCSDTQ- n,n= 1a, 1b, 3 xx CC2 xxx xxx Inefficient code, M6scaling CC3 xxx x CC4 xx CCSD + T(CCSD) x x CCSD(T) xxx xxx xx Also Brueckner, orbital-optimized CCSD, QRHF-CCSD Λ-CCSD(T) xx x CCSD + TQ(CCSD) x CCSD + TQ∗(CCSD) x CCSDT + Q(CCSDT) x CCSDT(Q) xx CCSDT(Q)/A x Differs from CCSDT(Q) for closed-shell non-HF reference CCSDT(Q)/B x Differs from CCSDT(Q) for closed-shell non-HF reference Λ-CCSDT(Q) x CCSD(T- n),n= 2, 3, 4, 5bx CCSD(TQ- n),n= 2, 3, 4cx CCSDT(Q- n),n= 2, 3, 4, 5, 6cx LCCDdx x LCCSDex x CISD x x x QCISD xxx xxx QCISD(T) xxx xxx aAdditional methods, in particular, open-shell variants of higher-order coupled cluster methods, including in many cases gradients and analytic second derivatives, are available through the interface to the MRCC program [see the MRCC manual (www.mrcc.hu) for a complete list]. bSee Ref. 107. cSee Ref. 135. dLCCD stands for linearized CCD. eLCCSD stands for linearized CCSD. character of standard CC theory, a Lagrangian Lis introduced, which consists of the CC energy augmented by the CC equations (as the so-called constraints) premultiplied with Lagrange multipliers L=⟨0∣(1 +Λ)exp(−T)Hexp(T)∣0⟩. (6) In this equation, a compact notation is used in which the Lagrange multipliers are subsumed into the Λoperator, a de-excitation oper- ator that gathers all of them. At this point, it should be mentioned that this CC energy functional was actually first suggested by Arpo- nen151in order to cast CC theory in a variational framework. The Lagrangian is then made stationary. Stationarity with respect tothe amplitudes in the Λoperator recovers the CC amplitude equa- tions, while stationarity with respect to the amplitudes in the cluster operator leads to the linear equations for the amplitudes of the Λ operator ⟨0∣(1 +Λ)(exp(−T)Hexp(T)−E)∣ΦP⟩=0. (7) Due to the stationarity of L, differentiation with respect to a pertur- bation xyields, for the derivative, dE dx=∂L ∂x=⟨0∣(1 +Λ)exp(−T)dH dxexp(T)∣0⟩, (8) which forms the basis of CC gradient theory. J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The discussion so far has ignored orbital relaxation. The con- sideration of this effect requires coupled-perturbed HF theory152,153 but is, in CC gradient theory, treated using the Z-vector approach by Handy and Schaefer.154CFOUR is able to handle orbital relaxation for RHF and UHF reference functions5and also in the case of ROHF and some classes of QRHF reference determinants.138,139 All analytic gradient implementations in CFOUR (see Table I for methods marked “xx” or “xxx”) make use of a density-based for- mulation of the first derivatives155,156such that, in the final step, the perturbation-independent quantities, i.e., the one- and two-particle density matrices as well as some intermediates, are contracted with the derivatives of the one- and two-electron AO integrals without any need to store the latter. Analytic second derivatives have been formulated and imple- mented within CC theory.23,30–32,46,157,158CFOUR offers here a range of options with all implementations based on the so-called asymmetric formulation33,159that results from a straightforward differentiation of the gradient expression given in Eq. (8) with respect to a second perturbation y. This means that the first derivatives of TandΛneed to be computed but, at no point, are these quantities required for two different perturbations at the same time. Geometrical analytic second derivatives allow the computa- tion of quadratic force constants (and thus harmonic vibrational frequencies) and, via numerical differentiation,34,160of cubic and semidiagonal quartic force constants (and thus in the framework of second order vibrational perturbation theory (VPT2) anharmonic corrections to vibrational frequencies, i.e., the computation of fun- damental frequencies as well as the frequencies of overtone and combination bands).67CFOUR offers the corresponding capabilities and renders such computations doable on a routine basis. The cor- responding computations can furthermore be easily performed in a parallel manner. We also note that CFOUR offers capabilities to per- form such calculations on the basis of numerical differentiation of analytically evaluated forces as well. Table I also summarizes the CC and MBPT/MP methods for which analytic second derivatives are available in CFOUR (methods marked with “xxx”). Note that so far no analytic second derivatives are available for schemes based on a ROHF reference function. Analytic differentiation schemes are particularly useful for the computation of the corresponding geometrical derivatives. How- ever, analytic derivatives also provide access to a range of other properties. To be mentioned here in the context of first deriva- tives are first-order properties such as dipole moments, quadrupole moments, and nuclear electric-field gradients. There is an additional point to be discussed here, namely, whether these first-order properties are computed with or without orbital relaxation effects included. CFOUR offers both options, and it has been argued161that CC theory takes (via single excitations) care of orbital relaxation effects162in an adequate manner. The issue of orbital relaxation is also of relevance when dealing with frequency-dependent properties in the framework of CC response theory.163The consideration of orbital relaxation can lead here to artificial poles and is therefore avoided. CFOUR offers, based on the existing analytic second derivative technol- ogy, access to frequency-dependent polarizabilities at the CCSD,164 CC3,163and CCSDT level.165In addition, using analytic third deriva- tives, frequency-dependent hyperpolarizabilities can be evaluated at the same levels of theory.166–168Further analytic third derivativesinclude Raman intensities computed as gradients of the frequency- dependent polarizability at the CCSD level169and Verdet constants computed as quadratic response functions at the CCSD and CCSDT levels of theory.170 Concerning the computation of magnetic properties, i.e., nuclear magnetic shielding tensors and magnetizabilities, care has to be taken with respect to the gauge-origin problem. As amply demon- strated in the literature, the use of gauge-including atomic orbitals (GIAOs,171–174also known as London orbitals175) is here an ade- quate choice, and they are hence used by default in CFOUR .CFOUR offers unique capabilities to compute magnetic properties at various CC levels with high accuracy for both nuclear magnetic shielding constants20–23,46as well as magnetizabilities.176The implementation of shielding constants at the MP2 level17,18in CFOUR was the first presented in the literature, but by now this option is also offered by other quantum chemical program packages177–180together with advancements that facilitate large-scale calculations. The capabili- ties of CFOUR concerning magnetic properties also allow the com- putation of closely related properties such as nuclear spin-rotation and rotational gtensors181via the use of so-called rotational Lon- don orbitals.38In the context of NMR properties, we also note that the second derivative capabilities of CFOUR allow the computation of indirect spin–spin coupling constants at CCSD,182CC3,183and CCSDT and higher CC levels (both via the MRCC program). To be noted here is that (a) these calculations must be performed in an orbital-unrelaxed manner182and (b) CFOUR allows the computation of all four contributions to the indirect spin–spin coupling constants [i.e., Fermi-contact, spin–dipole, paramagnetic spin–orbit (SO), and diamagnetic spin–orbit terms].184,185 To conclude this section, we mention that CFOUR also offers the capability to compute vibrational corrections to a range of properties via VPT2.186These corrections turn out to be essential in the case of high-accuracy computations that are compared to experimental values from precise gas-phase measurements. C. Excited state treatments via equation-of-motion/linear response methods Single-reference methods based on MBPT and CC theory are excellent approaches to study the potential energy surfaces asso- ciated with ground electronic states near their equilibrium struc- ture but generally cannot be straightforwardly applied to study excited states. In particular, all such methods are subject to varia- tional collapse (through the reference function \|0 ⟩) or convergence to the lower-lying states with the same (spatial and spin) symme- try. For closed-shell systems, the lowest singlet excited states often have a symmetry different than the ground state (for example, the lowest excited state of formaldehyde has1A2symmetry, while the ground state has1A1symmetry), but such states are described (in zeroth order) by a linear combination of two Slater determi- nants and therefore not amenable to standard MBPT or CC cal- culations. For many radicals, however, excited states are properly described by a single determinant (for example, the excited2Σ state of OH), and the usual toolkit of “ground state” MBPT/CC methods can indeed be employed. The same holds for excited triplet states where a single determinant is often a valid descrip- tion for the high-spin components. However, when one speaks generally of excited states in the context of quantum chemistry, it J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp can be assumed that standard single-determinant methods are not suitable. The major advance in extending CC theory to excited states was identified in an insightful paper by Monkhorst187and has ultimately come to be known as both “equation-of-motion CC” (EOM-CC) theory24,75,188,189and “linear response CC” (LR-CC)190–194theory. Both of these approaches give the same excitation and final state energies (see below) but differ in the way that certain properties are defined (see Subsection II D). It should be noted that the symmetry- adapted-cluster configuration-interaction (SAC-CI) method,195–198 which is similar in spirit to the EOM-CC approach, was developed for excited, ionized, and electron-attached states by Nakatsuji and Hirao in the late 1970s. In EOM-CC methods, the final state energies are obtained by diagonalization of the similarity-transformed Hamiltonian ¯H, ¯H≡exp(−T)Hexp(T), (9) a non-Hermitian operator that is obtained from the usual electronic Hamiltonian using the CC amplitudes in the transformation step. The excited states are described by the wavefunctions ∣ΨEOM-CC⟩=Rexp(T)∣0⟩, (10) ⟨˜ΨEOM-CC∣=⟨0∣Lexp(−T), (11) whereRandLare the right and left eigenvectors of ¯H. The characterization of EOM-CC above applies strictly only to “complete” CC methods such as CCSD, CCSDT, etc., but must be modified somewhat for methods in which certain classes of exci- tation are not treated completely (CC2,102CCSDT-1,92,98and so on). In such a case, the excitation energies are obtained again by diagonalization of a non-symmetric matrix, but one that cannot be written as ¯His designated above. Rather, one differentiates the CC amplitude equations [Eq. (4) of Sec. II A], which leads to the linear equation dT dx=A−1bx, (12) where Ais the “CC Jacobian” that is diagonalized to obtain the exci- tation energies. This perspective on EOM-CC applies equally well to the normal (CCSD, CCSDT, etc.) case, in which A=¯H, and is illu- minating in that one can easily see the correspondence between the eigenvalues of Aand the excitation energies from the point of view of first-order perturbation theory. The first EOM-CC calculations were based on the CCSD approximation and appeared more than 30 years ago,191but the method began to gain momentum with a flurry of activity that took place both in Gainesville and in Aarhus after 1990.24,189,193For excited states that can be characterized as “single excitations”, EOM- CCSD theory gives excitation energies that are usually no more than 0.25 eV in error and tends toward overestimation.199–201Later developments led to EOM-CCSDT202–204and EOM-CCSDTQ,205,206 as well as general arbitrary-order EOM-CC47via the MRCC pack- age.42,43With these methods, excitation energies become systemat- ically more accurate as the cost of calculation grows significantly. As for ground-state methods, the high cost of EOM-CCSDT calcu- lations has driven efforts to find suitable approximations, and this remains an area of important research. Such approximations includegeneralizations of the CCSDT- nmethods mentioned earlier, CC3, which is probably the most popular and perhaps successful such approach,207and a great variety of non-iterative methods. While many such methods have been identified and tested,28,202,208–218a recent non-iterative technique [EOM-CCSD(T)(a)∗]29shows con- siderable promise200,219–221and might be the method of choice for future applications. While sometimes thought of as strictly a means to compute excitation energies, EOM-CC methods can also be used to com- pute states that differ from the ground state in terms of the number of electrons. That is, their domain of application includes “excited states” in which electrons are “excited” to the continuum (ioniza- tion) or electrons are excited from the continuum (electron attach- ment). EOM-CC methods belonging to the former class are called EOMIP-CC27(removal of one electron), EOMDIP-CC222(two elec- trons), etc., while those in the latter class are EOMEA-CC223(attach- ment of one electron), EOMDEA-CC, and so on. EOM-CC meth- ods, in which the number of electrons in the initial and final state are identical, are then called EOMEE-CC (EE standing for excitation energy). CFOUR has extensive capabilities for all the variants men- tioned above [EOMEE-CC, EOM(D)IP-CC, and EOMEA-CC], the state of which is summarized in Table II. It should be noted that the capabilities indicated in Table II are only for efficient implementations of the methods. This is impor- tant because it has been shown224that an EOMEE-CC code can be used to do EOM(D)IP-CC or EOMEA-CC calculations by making use of continuum orbitals; excitation of one electron to this contin- uum orbital is equivalent to EOMIP-CC, excitation from an occu- pied continuum orbital is equivalent to EOMEA-CC, etc. That is, while Table II indicates that, for example, EOMEA-CCSDT is not “available” in CFOUR , such calculations can indeed be done by this means, although the resulting implementation has the same cost as the corresponding EOMEE-CCSDT calculation. CFOUR allows the straightforward use of these continuum orbital techniques, and the capabilities extend to both energy and gradient calculations. In addition to EOMEE-CC methods, CFOUR is also able to per- form calculations using configuration interaction singles225(CIS, also known as the Tamm–Damcoff approximation226,227), the perturbatively corrected CIS(D) method,228and an approximate method known as EOM-CCSD(2).229All of these methods work at the excitation energy level, and both EOMEE-CCSD(2) and EOMIP- CCSD(2) are implemented. Several functionalities are available to direct the program into the desired excited state. The character of the excitation can be spec- ified in terms of dominant orbitals as further explained in Sec. III. Alternatively, one can simply request the lowest excited state(s) of a particular spin and spatial symmetry. It is also possible with CFOUR to compute excited states near a particular target energy. D. Analytic derivatives and molecular properties for excited states While the pioneering work with EOM-CC theory dealt strictly with energy differences (vertical excitation energies, ionization potentials, and electron attachment energies), the central impor- tance of excited states in chemical physics has demanded that the associated potential energy surfaces be characterized computation- ally. Such studies are relevant not only for analysis and predictions J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . EOM-CC methods available in the CFOUR program package for closed-shell reference functions. A single x desig- nates that only energy evaluations are possible, while xx indicates that both energies and gradients can be calculated. The interface to the MRCC program also does allow general CC(n) (n >1) computations of energies and gradients for open- and closed-shell references. Method EOMEE EOMIP EOMEA Remarks CCSD xx xx xx Also open-shell \|0 ⟩and EOMDIP CCSDT xx x Also EOMDIP CCSDTQ x x CCSDT- n,n= 1a, 1b, 2, 3 x CC2 xx x Inefficient code, M6scaling CC3 x CCSD∗ax x CCSD(T)bx CCSD(T)(a)cx x CCSD(T)(a)∗cx CCSDR( n),n= T, 1a, 1b, 3dx CIS xx CIS(D) xx CCSD(2) xx xx aSee Ref. 28. bSee Ref. 214. cSee Ref. 29. dSee Ref. 210. of electronic spectroscopy but also to study photochemical behav- ior and interactions between excited states. Accordingly, analytic derivative techniques similar in the spirit of application to those mentioned in Sec. II B were developed for EOM-CC methods in the early 1990s25–27and were present in the first version of CFOUR . The EOM-CC energy gradient is given by dE dx=⟨0∣L∂¯H ∂xR∣0⟩+⟨0∣Z∂¯H ∂x∣0⟩, (13) and, apart from contractions between the differentiated electronic Hamiltonian and the right- and left-eigenvectors of ¯H(note that a calculation of the excitation energies requires only that one of these eigenvectors be evaluated), involves an additional de-excitation operator Z, which is analogous to the Λoperator in ground-state CC gradient theory. The amplitudes that make up this operator are obtained from solving the linear system ⟨0∣Z∣ΦP⟩=−⟨0∣Ξ∣ΦP⟩[⟨ΦP∣¯H−ECC∣ΦP⟩]−1, (14) where matrix elements of the auxiliary operator Ξare defined by ⟨0∣Ξ∣ΦP⟩≡∑ Q⟨0∣L¯H∣ΦQ⟩⟨ΦQ∣R∣ΦP⟩ (15) withΦQrepresenting a determinant in the space of excitations beyond that defined by the particular truncated CC approach (for example, triply excited determinants in CCSD). As for ground state CC methods, the general gradient for- mula [Eq. (13)] is recast in terms of one- and two-electron density matrices. Contraction of these with the geometric derivatives of the Hamiltonian gives the gradient, while contraction of the densities with other operators again provides other properties. EOM-CCSD and EOM-CC2 gradients are available in CFOUR for all methods(EOMEE, EOMIP, and EOMEA), for both closed-shell and open- shell reference functions, and offer a very efficient means to study potential energy surfaces of the final states. EOMEE-CCSDT gradi- ents for closed-shell references are a very recent addition, and gen- eral EOMEE-CC( n) gradients are available with the MRCC interface. It is a straightforward matter here to calculate properties such as dipole moments, higher multipole moments, Mulliken populations, and so on, using the one-electron density; these properties are all equivalent to those calculated as energy derivatives. In addition to gradients, one-electron transition densities involving only the ground-state Tamplitudes and the LandR vectors25are available. These yield, among other things, transition moments. It is here (and only here) that EOM-CC and CCLR meth- ods provide different results.230–232The transition moments eval- uated in CFOUR calculations—those mentioned here—are not size- intensive, becoming so only in the limit of a full CC (i.e., CCSDTQ for a four-electron system) calculation. In CCLR theory, the transi- tion moments satisfy size-intensivity but involve the cost associated with solving an additional set of linear equations for each excited state considered. III. INPUT AND USE OF CFOUR CFOUR calculations are rather straightforward to perform. After having installed CFOUR (for information concerning the installation ofCFOUR , see the CFOUR website www.cfour.de and Appendix A) and with all executables placed either in the working directory of the cal- culation or in a directory (e.g., ../cfour/bin/ ) that is part of the path, all calculations (unless otherwise advised) are invoked by the command xcfour . This command calls a driver program that, after having analyzed the input file ZMAT (see below), determines the var- ious modules that need to be run and in what order to call them. J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The input for a CFOUR calculation consists of a single file. This file, called ZMAT , consists of several sections as shown in the fol- lowing example. The first three sections are always necessary, while the fourth is optional (dependent on the chosen computational approach). EOMEE-CCSD/cc-pVDZ calculation for water H O 1 R H 1 R 2 A R=0.958 A=104.5 ∗CFOUR(CALC=CCSD,BASIS=PVDZ,EXCITE=EOMEE) %excite∗ 1 1 1 5 0 6 0 1.0 The ZMAT file starts with a mandatory one-line title, which is followed by the geometry information, either in the Z-matrix for- mat (shown here and which is currently mandatory for geometry optimizations) or in Cartesian coordinates. The geometry is fol- lowed by a list of keywords in a sequence of lines that starts with ∗CFOUR . There are roughly 250 active keywords, but virtually all of them take on default values (or are modified by default according to other keywords in the input file). Common keywords to supply, as shown in the example file above, include information about the chosen quantum chemical method ( CALC=CCSD obviously invokes a CCSD calculation), basis set ( BASIS=PVDZ requests the use of the cc-pVDZ basis), and calculation type ( EXCITE =EOMEE requests an EOMEE-CCSD treatment). Additional parameters such as conver- gence thresholds, maximum number of iterations, etc., can also be modified, but they have appropriate default values and do not need to be supplied. The final section (initiated by a % sign) provides addi- tional information. In the example chosen here, this information guides the choice of guess vectors for the EOMEE-CCSD compu- tation with this particular example instructing the EOM-CC pro- gram to start with the HOMO →LUMO guess in the Davidson diagonalization procedure. Basis set information is provided via the file GENBAS , which can be either customized and externally supplied or used from the default location ( ../cfour/basis/ ). The same holds also for infor- mation about effective core potentials (ECPs), which is supplied via the related file ECPDATA . Of course, more elaborate input files can be created, and it is sometimes advantageous or necessary to include additional files (beyond ZMAT ) in the running directory. Examples include here the fileFCMINT (which contains the force constants in Z-matrix internal coordinates), which can be supplied to facilitate geometry optimiza- tion (this permits the force constants in FCMINT to be used as a starting guess for the Hessian as opposed to a naive set of initial parameters). The ZMAT file below,Calculation of LVC parameters for nitrogendioxide O N 1 R O 2 R 1 A R = 1.26 239 A = 116.4431 ∗CFOUR(CALC=CCSD,BASIS=AUG-PVDZ,FROZEN_CORE=ON EXCITE=EOMIP,SCF_EXPSTART=10 CC_MAXCYC=200,LINEQ_MAXCYC=200 FCGRADNEW=0 CHARGE=−1 TRANGRAD=ON,DERIV_LEV=1) %excite∗ 1 1 1 0 10 0 1.0 together with the file FCMFINAL , which, in this example, contains the force constants for the NO 2anion, calculated separately, provides the input to calculate the linear vibronic coupling (LVC) parame- ters in Table IX ( vide infra ) for the Ã2B2state (theκA s,vide infra ). In addition to directing CFOUR to do an EOMIP calculation with the NO 2anion as reference, it specifies the calculation of a gra- dient ( DERIV_LEV =1), that this gradient should be transformed to the normal coordinate representation associated with the force con- stants in FCMFINAL , that the frozen core approximation is to be used, and also some other parameters about the algorithm used for the frozen-core gradient calculation, and specifications for the maximum number of cycles for various equations that are solved. Clearly, it is not possible or appropriate here to give an exhaus- tive list of examples. The point is simply to show a few representative cases and to state that the input is generally quite simple: the ZMAT file and perhaps another file or two, depending on the type of cal- culation. More examples can be found on the CFOUR website (see Appendix A). IV. NEW FEATURES A. Higher-order coupled cluster methods: xncc Highly accurate calculations often require treatment of the cor- relation energy beyond CCSD(T). For example, many common ther- mochemical protocols such as HEAT,233–235Wn,236–238and ANL n239 include not only CCSDT contributions but additional contributions from quadruple excitations [CCSDT(Q) or CCSDTQ] and in some cases even quintuple excitations [CCSDTQ(P) or CCSDTQP]. Such corrections are critical (in combination with corrections for rela- tivistic effects, basis set convergence, etc., described in Secs. IV C, IV F, and IV G) to reaching sub-kJ/mol accuracy, and enabling real- world applications using these methods has long been a design goal ofCFOUR . For many years, CFOUR has supported CCSDT energy calcula- tions for both closed and open-shell references, as well as prop- erties, gradients, and even second derivatives at the closed-shell CCSDT level. Additionally, the CCSDT(Q) method,108which pro- vides a cost-effective and often highly accurate approximation to full J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp CCSDTQ, was originally implemented in a development version of CFOUR . Recently, the full hierarchy of coupled cluster methods has been made accessible via the interface between CFOUR and the MRCC program of Kállay.44 However, in the last several years, we have become interested in writing a new implementation of CCSDT(Q), CCSDTQ, and other higher-order coupled cluster methods, which maximizes effi- ciency and scalability on modern computers, as well as develop- ing new theoretical techniques to facilitate such an implementa- tion. For closed-shell references, we developed a general algebraic and graphical interpretation of the non-orthogonal spin-adaptation approach240,241first pioneered by Kucharski and Bartlett242and later used by one of us (JG) to develop an efficient closed-shell CCSDT code in CFOUR . In order to maximize efficiency, we coupled this math- ematical technique with a storage format and set of implementa- tion techniques designed to minimize data movement (from disk as well as from main memory) and to avoid costly tensor trans- poses.62We also made code quality a major design goal, and we put a large focus on modularity and code reusability, maintainability, and extensibility. Finally, we included explicit OpenMP parallelization to effectively make use of modern multi-core processors. The end product of this work is a new CFOUR module, xncc ,62,240 which implements a full suite of coupled cluster methods for closed- shell molecules through CCSDTQ, including, in most cases, gradi- ents (see Table I for the full list of supported methods). Calcula- tions with xncc can be requested with CC_PROGRAM=NCC , but, in most cases, this is not necessary as xncc is the default program for CCSDT(Q) and CCSDTQ. Sample timings from Ref. 62 are listed in Table III as the number of minutes per iteration (for CCSDT and CCSDTQ) or the time in minutes required for the (Q) correction. The hardware used here was one core of an Intel Xeon E5620 proces- sor with 22 GB of memory allocated to CFOUR . From these results, it is immediately clear that significant speed-ups can be achieved with xncc compared to other programs—while these results use only one core, the multi-core scalability of xncc is also very good with paral- lel efficiencies (achieved parallel speed-up divided by the number of cores used) of ∼50% for eight or more cores. xncc also includes implementations of EOMEE-CC and EOMIP-CC methods through CCSDTQ, with gradients avail- able for EOMEE-CCSD and EOMEE-CCSDT. In addition to full EOMEE-CCSDT, a number of approximate methods are also included: EOMEE-CCSD∗,28,243EOMEE-CCSD(T)(a) and EOMEE- CCSD(T)(a)∗,29EOMEE-CC3,207EOMEE-CCSDT- nand EOMEE- CCSD(T),209,211and EOMEE-CCSDR(T), EOMEE-CCSDR(1a), and TABLE III . Timing of CCSDT(Q) and CCSDTQ calculations in minutes (from Ref. 62) for a representative set of small molecules. Two basis sets are listed for some molecules: in this case, the first basis set refers to the CCSDT(Q) calculation, while the second refers to the CCSDTQ calculation. The time for the CCSDT part (per iteration) and the (Q) correction in CCSDT(Q) are listed separately. CCSDT (Q) CCSDTQ HSOH cc-pVTZ/cc-pVDZ 3.7 85.5 9.3 H2O cc-pVQZ/aug-cc-pVTZ 0.3 5.9 19.7 H2CCCCH 2 cc-pVDZ/DZ 1.2 43.9 35.1 O3 aug-cc-pVDZ 0.2 7.5 99.6 FO 3–cc-pVDZ 0.5 12.3 241.3EOMEE-CCSDR(3).210Corrections to excited state energies, geome- tries, and vibrational frequencies can be rather large; for exam- ple, in a calculation of the geometries and harmonic frequencies of the S1excited state potential energy surface of C 2H2, we found that triples contributions to the harmonic frequencies can be in excess of 100 cm−1, while quadruples corrections can be as large as 35 cm−1.206While the current release includes analytic gradients for EOMEE-CCSDT, transition properties at this level have not yet been implemented but will be included in the next version along with EOMEE-CCSDT natural transition orbitals. Another unique feature of xncc is the use of sub-iteration con- vergence acceleration for the CCSDT, CCSDTQ, and approximate CCSDT (CC3 and CCSDT- n) methods.244For CCSDT and other iterative triples methods, this technique essentially “freezes” the higher-order cluster amplitudes and their contributions to the sin- gles and doubles, while a number of (modified) CCSD iterations are performed. The triples amplitudes are then updated and the cycle repeats. For CCSDTQ, two levels of sub-iteration are possible, and xncc utilizes both of them simultaneously by default. For all meth- ods, but especially for approximate methods such as CC n, CCSDT- n, and CCSDTQ- n, this technique can drastically reduce the number of iterations required for convergence. The current version includes sub-iteration for the amplitude equations, optional direct inversion in the iterative subspace (DIIS) for the triples and/or quadruples amplitudes, and optional amplitude damping that can help in cases where oscillatory behavior is encountered. The next version will extend the sub-iteration technique to linear equations (e.g., the Λ equations) and potentially to EOM-CC as well. The availability of a high-performance yet easily extensible plat- form for higher-order coupled cluster has also allowed us to rapidly implement new coupled cluster-based methods. Perhaps the best example of this is the recent development of bivariational coupled cluster perturbation theory methods CCSD(T- n), CCSD(TQ- n), and CCSDT(Q- n)107,135for which we have implemented up to n= 5, 4, and 6, respectively. These methods, with the exception of the lowest-order correction, scale formally the same as the full method (CCSDT or CCSDTQ), but, by recovering essentially all of the higher-order correlation energy in only a small number of high-scaling steps, a steep reduction in computational cost can be achieved. As an example, errors in total atomization energies for a test set of small molecules are summarized in Table IV with respect to full CCSDTQ.135From these results, we can see that TABLE IV . Total atomization energy errors with respect to CCSDTQ in kJ/mol for various approximate quadruples methods (from Ref. 135). Errors are summarized by MeanSigned Error,MeanAbsolute Error, and MAX imum-amplitude signed error. CCSDT CCSDT(Q) Λ-CCSDT(Q) MSE −3.06 0.55 0.35 MAE 3.06 0.56 0.36 MAX −14.06 4.01 1.92 CCSDT(Q–2) CCSDT(Q–3) CCSDT(Q–4) MSE −0.70 −0.01 −0.15 MAE 0.70 0.08 0.15 MAX −2.58 −0.29 −0.97 J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp CCSDT(Q-3) can reduce errors by approximately one order of magnitude compared to CCSDT(Q) at the expense of one M10step. All the capabilities described here (except where noted) are available in the current version. The next release of xncc will focus on implementing open-shell alternatives for all supported methods, in particular, CCSDT(Q) and CCSDTQ. Additionally, the version of xncc under development has included further perfor- mance improvements due to transpose-free tensor contraction oper- ations from the TBLIS library,245including extension to tensors with explicit point-group symmetry.246We also hope to include scalable distributed-parallel implementations in the next release. B. Quadratically convergent SCF and complete active space SCF methods A rigorous treatment of multireference systems can usually not be achieved by using a single-reference method (see Sec. II C). In order to have not only a method to describe such systems in an unbiased and qualitatively correct way but also a starting point for internally contracted multireference correlated treatments, an implementation of the Complete Active Space–Self-Consistent Field (CASSCF) method247,248has been recently added to CFOUR . In CASSCF, the orbital space is partitioned into the following three groups: (i) internal orbitals that are always doubly occupied, (ii) active orbitals with floating occupation, and (iii) external orbitals that are always empty. The molecular wavefunction is written as the linear combination of all the symmetry allowed Slater determinants that can be formed by varying the occupation of the active orbitals for a given number of active electrons. Both the orbitals and the CI coefficients are then fully optimized. Such a non-linear optimiza- tion problem is typically difficult to converge and ill-conditioned, making the use of advanced numerical strategies mandatory. Many CASSCF algorithms have been developed in the past. The numerical strategies proposed can be grouped into two main classes depending on their convergence properties, namely, first-order methods249–254 and second-order methods.255–261The latter strategy is particularly attractive, because the second-order methods offer rigorous con- vergence and are particularly robust, so that achieving convergence requires little to no case-by-case calibration by the user. The implementation strategy pursued for the CASSCF module ofCFOUR is based on the Norm Extended Optimization (NEO) algo- rithm of Jensen and co-workers.77,258–260The CI operations are han- dled in a direct fashion using a string-based determinant CI formal- ism,262–264and the CI implementation follows the integral-driven, vector implementation by Bendazzoli and Evangelisti.265 A second-order optimization strategy is based on the definition of a quadratic model Qof the energy, obtained by expanding it in Taylor series with respect to the variational parameters xup to the second order around a starting point x0, Q(x)=E(x0)+g†x+1 2x†Gx, (16) where gandGare the energy gradient and Hessian evaluated at the expansion point. The straightforward minimization of the quadratic model corresponds to the Newton–Raphson (NR) method142and prescribes to take a step δNR=−G−1g. (17)The NR method enjoys quadratic convergence if the starting point is close to a local minimum but is known to exhibit erratic behav- ior or even to diverge if, at the starting point, the Hessian is not positive definite. This issue can be solved by defining a trust region, i.e., a maximum stepsize Rtwithin which the quadratic model of the energy is deemed to provide an accurate representation. This con- straint can be imposed by means of a Lagrange multiplier ν. By doing so, one gets, for the step, the following coupled equations: {(G+νI)δ=−g, ∥δ(ν)∥=Rt.(18) The trust-radius Newton method is also known as Levenberg– Marquardt (LM) method.142If the LM method is coupled with an adaptive choice of the trust radius Rt, as proposed by Fletcher,142 depending on the agreement of the quadratic model with the energy, it is possible to prove that, under certain regularity hypotheses of the energy that can be assumed to be satisfied, the procedure always converges to the closest local minimum. The NEO algorithm is an elegant practical implementation of the Fletcher–Levenberg– Marquardt (FLM) strategy, thus enjoying its convergence proper- ties.259The NEO scheme is the default for state-specific CASSCF calculations. The implementation in CFOUR also includes another second-order algorithm, in particular, a simplified version of the one proposed by Meyer, Werner, and Knowles,74,256,261which can be used for state-averaged CASSCF. CASSCF calculations are requested via the CALC=CASSCF keyword and require one to provide, as an additional input, the definition of the orbital spaces. This is done by adding a section to the ZMAT input file that specifies the number of active alpha and beta electrons and the number of active orbitals and then the actual definition of the active space. The latter can be provided in two different ways. The first possibility, invoked with the keyword CAS_INPUT=ORBITALS , is to specify a list of active orbitals (in HF energy order), and the second possibility, invoked with the keyword CAS_INPUT=OCCUPATION , is to specify, for each irreducible representation, the number of internal orbitals and then the number of active orbitals. The following example provides the input for a CASSCF calculation on benzene, in D2hsymmetry, corre- lating the six πelectrons in the six πorbitals, using the first strategy, where the order of the orbitals is obtained from a HF calculation using the cc-pVDZ266basis set: %casscf 3 3 6 17 20 21 22 23 30 The same calculation, using the second input method, is obtained with the following route: %casscf 3 3 6 6 4 5 3 0 0 0 0 0 0 0 0 2 1 2 1 J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Other options that control the CASSCF calculations can be found in the CFOUR online manual (see Appendix A). CASSCF can be used with either non-relativistic or spin-free relativistic Hamilto- nians, which are detailed in Sec. IV C. At the moment, the CASSCF code is still experimental, and it is thus not included in the current public release of CFOUR . The code will be made available with the next release. The quadratically convergent machinery developed for CASSCF can also be employed to deal with a particularly important sub- case, i.e., regular SCF. These equations can be notoriously dif- ficult to converge using a standard SCF algorithm even when Pulay’s DIIS267is used to accelerate convergence, especially for open-shell systems. Furthermore, even for well-behaved systems, it can be difficult to achieve very tight convergence, which is required, for instance, when computing numerical derivatives of post-HF Hessians in anharmonic force field calculations. In all these cases, the user must try and adjust a combination of SCF convergence parameters, such as whether to damp the first iter- ations and what damping parameters to use, how many points to use for the DIIS extrapolation, and when to start it. Tun- ing all these parameters on a system-dependent basis can be very time consuming, especially if one has to perform a large num- ber of calculations for which different parameters need to be used. In such a situation, the robust convergence properties of a second-order scheme are particularly useful. A quadratically con- vergent implementation of restricted and unrestricted HF based on the solution of Eq. (18) is available in the last public release of CFOUR and can be used by adding the SCF_PROG=QCSCF keyword. The cur- rent implementation works in the MO basis and requires to fully assemble and diagonalize the MO rotation Hessian and is, therefore, much more computationally demanding than regular SCF. How- ever, as HF is typically an intermediate step in a correlated calcu- lation, this is, in practice, not an issue for the standard CFOUR user. A new, direct, AO-based implementation that uses the NEO algo- rithm exists and can be accessed by specifying SCF_PROG=DQCSCF . However, this implementation is not mature enough to be released at the moment and will be made available with the next release of the code. The QCSCF program can be considered as an almost black- box SCF code. However, there are a few precautions that the user needs to take. The code performs, at the beginning of the calcula- tion, a few regular SCF iterations that are used in order to get a better starting point for the QC solver and, if a calculation is run with symmetry, to try to guess the correct occupation numbers for each irreducible representation. These are fixed during the QC opti- mization so that QCSCF will converge to a minimum for that given occupation. The user should therefore make sure that the occupation numbers guessed are correct or provide the correct ones in input. A second aspect that should be considered is the general condition- ing of the problem. If a very large basis set is used, linear depen- dence problems can be encountered, as it can be seen by looking at the eigenvalues of the overlap matrix. In such cases, it will not be possible to converge the SCF equations beyond a certain thresh- old due to numerical precision limitations. This issue can be easily detected by looking at the QCSCF iterations. If the residual norm starts oscillating or iterations are stagnating, it means that the best numerical solution that can be achieved for the chosen basis sethas been reached, and the user should either consider the calcula- tion converged or, if not satisfied with the result, remove redundant basis functions. A third aspect concerns UHF cases for which multi- ple SCF solutions with different spin contamination exist. QCSCF is guaranteed to converge to the closest local minimum, which might be different from the one found with regular SCF. In the experience of the authors, QCSCF tends to converge to the solution that is low- est in energy and more spin-contaminated. Whether this solution is acceptable is something that the user needs to check. Neverthe- less, a subsequent post-HF treatment is usually able to remove most of the spin contamination. An interesting aspect of QCSCF is that, when regular SCF converges to an unstable solution, QCSCF usually manages to converge to a stable one, at least within the symmetry of the electronic wavefunction. However, convergence can be diffi- cult, especially if the MO rotation Hessian has several small and close eigenvalues. In order to illustrate the robustness of QCSCF, we propose two examples. As an example of a routine application where very tight convergence is required, we compute the SCF solution for benzene (C–C distance 1.3989 Å and C–H distance 1.0808 Å) with the aug- cc-pVTZ266basis set. This is a standard calculation; however, we require the wavefunction to be converged to 10−11in the root mean square (rms) norm of the MO rotation gradient. Using the default parameters for the calculation and starting from a guess obtained by diagonalizing the core Hamiltonian, QCSCF performs six regu- lar SCF iterations, until the rms variation of the density matrix is smaller than 0.1, and then manages to converge in only four FLM iterations. On the other hand, the regular SCF code easily achieves an intermediate convergence (maximum change of the density matrix smaller than 10−7) but then struggles to further refine the solu- tion, exhibiting an oscillating behavior. The convergence profiles of the two algorithms are reported in Fig. 1. The superlinear con- vergence of QCSCF is particularly apparent, as two convergence profiles can be seen focusing on the green line. The regular SCF iter- ations exhibit a linear convergence profile. As soon as the FLM iter- ations start, the energy error drops very rapidly until convergence is achieved. FIG. 1 . Convergence profile for the regular SCF code and QCSCF for benzene. The converged energy is −230.780 571 677 Eh. J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A second, more challenging example concerns a weakly bonded complex of molecular oxygen and argon (O–O distance 1.25 Å, O–Ar distance 2.1748 Å, O–O–Ar angle 174.21○), the ground state of which is a triplet. For this molecule, described using a UHF ref- erence, the regular SCF code converges with some difficulty to an unstable solution, which has both a UHF–UHF instability that pre- serves the symmetry of the wavefunction and a UHF–UHF insta- bility to a broken-symmetry solution. QCSCF manages to converge to a minimum within the symmetry (about 1 μEhlower in energy than the regular SCF solution), although convergence requires as many as 36 FLM iterations. An instability with respect to a broken symmetry solution is, however, still present. Interestingly, the NEO based code xdqcscf converges effortlessly to a stable solution—no instability is found even with respect to a broken symmetry UHF solution. While the latter result is a fortunate occurrence that can, in general, not be expected, the better convergence properties of the NEO based code can be explained by the fact that the NEO algorithm introduces an augmented Hessian so that the presence of small and close eigenvalues in the original MO rotation Hes- sian has a small effect on the overall convergence of the optimiza- tion. The convergence profile of the three algorithms is reported in Fig. 2. It is interesting to comment on the behavior of QCSCF. The first iterations manage to quickly locate the same solution found by the regular SCF code. However, the iterations are not stopped as QCSCF detects the instability in the form of a negative eigenvalue in the MO rotation Hessian. A large number of iterations are then spent trying to reach the local minimum. As the lowest eigenval- ues of the Hessian are both small and very close, convergence is very slow. On the other hand, the NEO based xdqcscf code does not suffer from this problem and converges smoothly to the global minimum. FIG. 2 . Convergence profile (absolute energies are reported) for the regular SCF code, the default QCSCF code, xqcscf , and the NEO based code, xdqcscf , for Ar⋯O2. The converged energy is −676.391 261 81 Ehfor the regular SCF, which finds a solution unstable both with respect to a UHF solution with the same symmetry and with broken symmetry, −676.391 262 67 Ehforxqcscf , which finds a solution that is unstable with respect to a UHF solution with broken symmetry, and −676.391 300 17 Ehforxdqcscf , which finds a stable solution.C. Relativistic quantum chemical methods Treatment of relativistic effects268,269is indispensable for cal- culations of molecules containing heavy elements and also plays an important role in high-accuracy calculations of molecules that com- prise lighter atoms from the first few rows of the periodic table. The development of relativistic quantum-chemical methods in CFOUR has focused on obtaining relativistic corrections to energies and prop- erties with a CC treatment of electron correlation. Initial efforts on the perturbative treatment of scalar-relativistic effects were focused on the framework of standard (non-relativistic) CC gradient theory and the Breit–Pauli Hamiltonian.270,271First-order scalar-relativistic corrections to energies can be conveniently obtained in a calcula- tion of first-order properties ( PROP=FIRST_ORDER ) and are widely used in well-established protocols for the computation of ther- mochemical parameters.233Calculations of scalar-relativistic cor- rections to geometrical parameters and electrical properties have been enabled by using nonrelativistic analytic CC second-derivative techniques.272Perturbative techniques for treating relativistic effects have been extended to using direct perturbation theory (DPT),52 a four-component formalism that permits a rigorous treatment of two-electron contributions.273–275In the released version of CFOUR , the use of the keyword RELATIVISTIC=DPT2 in geometry opti- mizations and evaluation of first-order properties is a convenient way of obtaining leading relativistic corrections to geometries and first-order electrical properties. Uncontracted basis sets are rec- ommended for DPT calculations, since DPT requires an accurate description for both the non-relativistic and the relativistic wave- functions. DPT corrections to energies have been implemented in CFOUR through fourth order with respect to c−1(DPT4) as ana- lytic second derivatives of non-relativistic energies, including both scalar-relativistic corrections and spin–orbit corrections,53and have been further extended to sixth order for scalar-relativistic correc- tions.276Furthermore, DPT4 corrections to electrical properties can be computed.54The development of DPT has also provided relativis- tic one- and two-electron integrals required for the development of non-perturbative approaches. Subsequent development of relativistic quantum chemical methods within CFOUR has involved a rigorous non-perturbative treatment of scalar-relativistic effects augmented with a perturbative treatment of spin–orbit coupling. In these calculations, the cost of the coupled-cluster steps of a scalar-relativistic calculation is essen- tially identical to that of the corresponding non-relativistic calcu- lation. In contrast, spin-symmetry breaking due to spin–orbit cou- pling leads to substantial computational overhead; a spin–orbit CC calculation requires more than an order of magnitude more com- puting time and storage than a corresponding nonrelativistic or scalar-relativistic calculation.277Meanwhile, the magnitude of the impact of scalar-relativistic effects on properties is usually substan- tially larger than that of spin–orbit effects. Therefore, a natural idea for a cost-effective treatment of relativistic effects at CC levels is to treat the larger but computationally less expensive scalar relativistic effects rigorously and then address spin–orbit effects by means of perturbation theory. In this context, the spin-free exact two-component theory in its one-electron variant (SFX2C-1e)56,278,279is highly recommended for a rigorous treatment of scalar-relativistic effects in routine chemical applications. The SFX2C-1e scheme performs an exact J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp block-diagonalization of the spin-free version of the matrix rep- resentation of the Dirac Hamiltonian to decouple electronic and positronic degrees of freedom and uses the electronic block of the resulting matrix representation of the Hamiltonian together with non-relativistic two-electron integrals in the subsequent many- electron treatment. As scalar-relativistic corrections are dominated by one-electron contributions,278,280the SFX2C-1e scheme is capa- ble of providing an accurate treatment of scalar-relativistic effects on energies and properties. An SFX2C-1e calculation requires only additional manipulation of one-electron Hamiltonian integrals as compared to a non-relativistic calculation and thus essentially has the same computational cost, as mentioned above. The SFX2C-1e energy and analytic gradients56,281are available in the released version of CFOUR . SFX2C-1e calculations of energies and first-order properties and geometry optimizations can conve- niently be carried out. That is, the same input file used for the corresponding non-relativistic calculation needs only an instruction that the SFX2C-1e scheme is to be used ( RELATIVISTIC=X2C1E ), and then, an appropriate basis set (recontracted for the SFX2C-1e scheme) needs to be selected. Table V summarizes the geomet- rical parameters for gold-containing molecules computed at the non-relativistic and SFX2C-1e CCSD(T) levels. These SFX2C-1e CCSD(T) calculations have essentially identical computational cost as the corresponding non-relativistic ones; scalar-relativistic effects are obtained for free. In this demonstration, the availability of ana- lytic gradients and the efficiency of the SFX2C-1e scheme allow a quick prediction for the geometry of an unknown gold-containing species (AuCH 3) with reasonably good accuracy, with one optimiza- tion cycle (one gradient calculation) taking only around 15 min using a single core of an Intel Xeon E5-2698v3@2.30GHz processor and 4 GB memory. More rigorous treatments of scalar-relativistic effects using the spin-free Dirac–Coulomb (SFDC) approach282or SFX2C in its mean-field variant (SFX2C-mf)283have also been implemented in CFOUR . The SFDC approach features a spin sep- aration in the four-component framework and is perhaps the most rigorous treatment of scalar-relativistic effects. SFDC is avail- able in the released version of CFOUR for calculations of energies and first-order electrical properties ( RELATIVISTIC=SFREE ).55The SFX2C-mf scheme recently implemented284in CFOUR performs the TABLE V . Geometrical parameters of AuF, AuCN, and AuCH 3computed at the non- relativistic and SFX2C-1e-CCSD(T) levels (bond lengths in Å and bond angles in degree). 1s electrons of C, N, and F as well as 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, and 4d electrons of gold have been kept frozen in the CC treatment. The ANO basis sets of triple-zeta quality used here have been obtained by recontracting ANO-RCC primitive sets285,286using non-relativistic and SFX2C-1e CCSD atomic densities and can be found at www.cfour.de. Nonrelativistic SFX2C-1e Experiment AuF R(Au–F) 2.094 1.921 1.918 AuCN R(Au–C) 2.151 1.902 1.912 R(C–N) 1.171 1.168 1.159 AuCH 3R(Au–C) 2.207 1.984 . . . R(C–H) 1.088 1.088 . . . ∠(Au–C–H) 107.7 107.3 . . .block-diagonalization at the HF level and will be available in the next released version. Perturbative treatment of spin–orbit effects can be obtained using either the SFDC or the SFX2C-1e scheme as the zeroth-order treatment.287–289For the latter, the corresponding spin–orbit inte- grals are defined as first derivatives of SFX2C-1e Hamiltonian inte- grals, thereby treating spin–orbit integrals in the four-component formulation as the perturbation and using the analytic SFX2C- 1e derivative technique. In this way, scalar-relativistic effects on spin–orbit integrals, which represent the coupling between scalar relativistic effects and spin–orbit coupling, have been taken into account. This greatly extends the applicability of the perturbative treatment of spin–orbit coupling in CFOUR to molecules contain- ing heavy elements. Two-electron spin–orbit contributions can be taken into account using the molecular mean-field (MMF) or the atomic mean-field (AMF) spin–orbit approach.288,290,291The result- ing effective one-electron spin–orbit integrals can be contracted with one-electron transition density matrices to obtain spin–orbit matrix elements between two electronic states. The EOM-CCSD transition density matrices (also needed for the quasidiabatic couplings in Sec. IV E) have been shown to provide accurate spin–orbit param- eters289,292,293and are highly recommended for routine applications. Spin–orbit splittings of representative2Πstates computed using MMF and AMF spin–orbit integrals within the SFX2C-1e scheme at the EOM-CCSD level are summarized in Table VI. The computed splittings compare very well with the experimental values, with the biggest discrepancy being about 4% in the case of TeH. SFX2C-1e EOM-CCSD calculation of spin–orbit coupling matrix elements will be available in the next release of CFOUR . CFOUR has also included options for non-perturbative treat- ment of spin–orbit coupling to obtain benchmark results or for studying heavy elements such as those in the 6p or 7p blocks for which these effects are too large to be handled perturbatively. The released version of CFOUR provides a spin–orbit CCSD(T) scheme for closed-shell systems.294In this scheme, a HF calcu- lation using scalar-relativistic effective core potentials (ECP) is first performed to obtain orbitals. A corresponding ECP spin– orbit term is then included to augment the Fock matrix in sub- sequent CC calculations. Analytic first and second derivatives are available for this scheme in the released version of CFOUR .294–296 Recent developments along this line include EOMEE-, EOMEA-, TABLE VI . Spin–orbit splittings (in cm−1) of2Πradicals calculated at the SFX2C-1e- EOM-CCSD level using uncontracted ANO-RCC basis sets. “MMF” and “AMF” refer to molecular mean field and atomic mean field, respectively. The experimental values are given as compiled in Ref. 289. MMF AMF Expt. OH 135.2 132.7 139 SH 369.5 369.3 377 SeH 1701.2 1700.9 1763 TeH 3675.3 3675.1 3816 FO 195.0 193.8 197 ClO 319.6 316.9 322 BrO 985.2 984.5 975 IO 2126.3 2124.0 2091 J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and EOMIP-CCSD methods.297–299Recently, an X2C300–305AMF approach has been developed for the non-perturbative treatment of spin–orbit coupling.59Based on this approach, coupled-cluster methods [CCSD(T), EOM-CCSD, and EOM-CCSD(T)(a)∗] with spin–orbit coupling included at the orbital level have been imple- mented.60,221,306The focus of these studies is on efficient implemen- tation using atomic orbital based algorithms and rigorous treatment of spin–orbit coupling in X2C. Users requesting more information about these relativistic methods in CFOUR are encouraged to make inquiries on the CFOUR mailing list (see Appendix B). D. Multireference coupled-cluster methods The treatment of quasidegenerate systems with chemical accu- racy is one of the most intriguing problems of electronic-structure theory. Although certain patterns of quasidegeneracy can be treated by means of EOM-CC methods (see Sec. II C) or in terms of gen- eralized single-reference CC methods,307,308all these methods are subject to limitations, in particular, a bias toward the selected ref- erence determinant. The development of genuine multireference CC (MRCC) methods therefore remains an important goal of CC theory. Much effort has been devoted to generalize the CC ansatz to the multireference domain, but this has turned out to be not straightfor- ward: Many MRCC methods have been suggested and successfully applied to actual chemical problems, but a theory as elegant and robust as single-reference CC theory discussed in Sec. II A has yet to emerge. Comprehensive overviews of the field are provided, for example, in Refs. 61 and 309. The development of MRCC theory in CFOUR has concentrated on the method suggested by Mukherjee and co-workers (Mk- MRCC).310,311This is a state-specific MRCC variant relying on the Jeziorski–Monkhorst ansatz,312 ∣Ψα⟩=d ∑ μexp(Tμ)∣Φμ⟩cα μ. (19) The reference determinants Φμdiffer in the occupation of the active orbitals; they form a model space of dimension d, and their weighting coefficients cα μare optimized for a particular tar- get stateα. The cluster operators Tμare specific to reference Φμand can be partitioned into excitation classes in analogy to Eq. (2). So-called internal excitations that map Φμonto another reference determinant Φνneed to be excluded from Tμ. The energy Eαand the coefficients cα μare obtained as the eigenvalue and eigenvector of an effective Hamiltonian, whose elements are Heff μν=⟨Φμ∣exp(−Tν)Hexp(Tν)∣Φν⟩. The amplitude equations take on the form ⟨ΦP(μ)∣exp(−Tμ)Hexp(Tμ)∣Φμ⟩cα μ +∑ ν≠μ⟨ΦP(μ)∣exp(−Tμ)exp(Tν)∣Φμ⟩Heff μνcα ν=0, (20) withΦP(μ) as an excitation manifold specific to reference Φμ. The first term of Eq. (20) can be interpreted as a generalization of Eq. (4), whereas the second term couples the amplitude equations for dif- ferent cluster operators Tμ,Tν. In practice, the cluster operators are usually truncated in analogy to the single-reference case, giving rise to the Mk-MRCCSD,310,311,313Mk-MRCCSDT,314etc., models.Distinct advantages of Mk-MRCC theory include rigorous size- extensivity,311the unbiased treatment of all references Φμin the model space,61and conceptual simplicity, resulting in relatively sim- ple working equations.313However, all truncated MRCC methods based on Eq. (19) are not invariant with respect to rotations among the active orbitals,61,315and it has also been shown that the com- putation of excitation energies and frequency-dependent proper- ties by means of linear-response theory is problematic with Mk- MRCC methods because the pole structure of the linear-response function is flawed.316,317Furthermore, the number of amplitudes to be determined is proportional to the size of the model space. As a consequence, the computational cost scales with system size as dtimes that of the corresponding single-reference model, that is, d⋅M6for Mk-MRCCSD, d⋅M8for Mk-MRCCSDT, and so forth, making Mk-MRCC impractical for large model spaces.61 CFOUR offers efficient Mk-MRCCSD318and Mk-MRCCSDT66 implementations for a model space of two closed-shell determi- nants. An implementation of Mk-MRCC for arbitrary excitation levels and model spaces has been presented elsewhere.319The CFOUR implementation is adequate for biradical species and single-bond breaking and therefore applicable to many multireference cases. In these calculations, orbitals can be taken from either an HF or a two- configurational SCF calculation. The application of Mk-MRCCSDT to larger molecules is greatly facilitated by means of paralleliza- tion, that is, computing the triple amplitudes and their contributions to the singles and doubles residuals in a distributed manner. Mk- MRCCSDT computations using well over 200 basis functions have been carried out with CFOUR .66A non-iterative treatment of triple excitations, termed Mk-MRCCSD(T), has also been implemented into CFOUR for model spaces of two closed-shell determinants.320The treatment of open-shell states is possible at the Mk-MRCCSD level using a model space of two open-shell determinants and orbitals from a low-spin ROHF calculation.321The case of a full model space of two electrons distributed among two orbitals (comprising four reference determinants) can also be treated at the Mk-MRCCSD level. Larger model spaces are required if more than two orbitals are (quasi-)degenerate. Examples include the breaking of double and triple bonds as well as many transition-metal compounds.61Such cases can be treated by means of internally contracted (ic)-MRCC methods73,322–326implemented in the GECCO program73that has been interfaced to CFOUR .74In ic-MRCC theory, a single cluster operator acts on a multideterminantal reference. ic-MRCC methods maintain full orbital invariance and size extensivity, and their computational cost is roughly comparable to that of the corresponding single- reference method.61,73However, the working equations are con- siderably more complicated mandating automated implementation techniques.73 As a unique feature, CFOUR offers efficient implementations of analytic gradients at the Mk-MRCCSD318,327and Mk-MRCCSDT328 levels of theory. The theory is formulated starting from a Lagrangian in analogy to single-reference CC gradient theory (see Sec. II B). The Mk-MRCC gradient can be written as318 dE dx=∑ μ¯cμcμ⟨Φμ∣(1 +Λμ)exp(−Tμ)dH dxexp(Tμ)∣Φμ⟩ (21) J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-15 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VII . C1–C2distances for o-benzyne, C 2–C6distances for m-benzyne, and C1–C4distances for p-benzyne in Å computed at the Mk-MRCCSD, CCSD, and CCSD(T) levels of theory using the cc-pCVTZ basis set. The weights of the reference determinants [see Eq. (19)] are also shown. For further details, see Ref. 321. o-benzyne m-benzyne p-benzyne R [CCSD] 1.2436 . . .a2.7071 R [CCSD(T)] 1.2567 2.0432 2.7183 R [Mk-MRCCSD] 1.2505 2.0141 2.6865 \|c1\|2[Mk-MRCCSD] 0.942 0.921 0.724 \|c2\|2[Mk-MRCCSD] 0.058 0.079 0.276 aCCSD calculations for m-benzyne favor a bicyclic structure without multireference character. withΛμas an analog to the Λoperator from Eqs. (6) and (7) and ¯cμas additional Lagrange multipliers. Equation (21) is evalu- ated based on density matrices; the relevant details are discussed in Ref. 318. Besides enabling geometry optimizations of polyatomic molecules,318,321,327these analytic gradients also provide conve- nient access to harmonic vibrational frequencies through numerical differentiation. To give an example of Mk-MRCC geometry optimizations, Table VII shows selected structural parameters for the ground states of the three isomers of benzyne depicted in Fig. 3. The electronic structure of these biradicals can be understood qualita- tively in terms of two frontier MOs that are a bonding and an antibonding combination of the atomic orbitals hosting the rad- ical electrons. The wavefunctions are dominated by two closed- shell determinants whose weights computed with Mk-MRCCSD are also included in Table VII; this illustrates that the multirefer- ence character increases from the o-isomer to the m-isomer to the p-isomer. Owing to the shape of the frontier MOs, the distance between the two radical centers provides a measure for the influ- ence of the two reference determinants on the molecular equilib- rium structures.318,327Table VII illustrates good agreement between CCSD and Mk-MRCCSD for o-benzyne, whereas larger deviations are observed for the other two isomers with stronger multireference character. FIG. 3 . Optimized structures of the ground states of the three isomers of benzyne computed at the Mk-MRCCSD/cc-pCVTZ level of theory. Taken from Ref. 321.TABLE VIII . Spin–orbit splittings in cm−1calculated at the Mk-MRCCSD/cc-pVQZa level of theory using the spin–orbit mean-field approximation. Experimental data are also given. For further details, see Ref. 329. Molecule Mk-MRCCSD Expt. OH 135.1 139.2 SH 375.2 377.0 SeH 1707.9 1763.3 NCS 360.8 325.3 ag-functions have been omitted. In addition to geometrical derivatives, CFOUR can compute spin– orbit (SO) splittings for2Πstates based on degenerate perturbation theory as a first-order property at the Mk-MRCCSD level of the- ory.329This constitutes an alternative to the computation of these quantities by means of EOM-CC theory (see Table VI) and is also helpful for the theoretical analysis of MRCC models relying on Eq. (19). For such methods, the symmetry properties of the SO operator allow for a decomposition of the SO splitting expression into two terms: a similarity-transformed SO operator times a cou- pling term intimately related to the coupling term from Eq. (20). It has been argued329that SO splittings provide a quality measure for this coupling term. As a numerical example, Table VIII shows SO splittings for the2Πstates of a few diatomic and triatomic molecules. E. Vibronic Hamiltonians and electronic spectroscopy A relatively common application of quantum chemistry is to electronic spectroscopy, the full understanding of which requires knowledge of electronic, vibrational, and (sometimes) rotational energy levels. While many electronic transitions, photoioniza- tion, and electron detachment processes are well-described by the Franck–Condon approximation, this is not always the case. A standard approach for treating these difficult cases—which involve Herzberg–Teller or true non-adiabatic effects—is to con- struct a molecular Hamiltonian in an electronic basis that does not consist of the usual adiabatic states typically obtained in quantum chemical calculations. A convenient framework for such an analysis was devised by Köppel, Domcke, and Cederbaum (KDC),330who applied it long ago with great success to a num- ber of photoelectron spectra in which ionization to the lowest- lying ionic states was inadequately treated by the Franck–Condon picture.331 In such calculations, the molecular Hamiltonian is written in a basis of “quasidiabatic” electronic states that, by construction, vary smoothly and slowly as the nuclei are displaced. This assump- tion motivates the form of the (diagonal) kinetic energy operator but means that the potential energy (the usual electronic Hamilto- nian) is not diagonal. For a two state problem, this model vibronic Hamiltonian takes the form HKDC=T+V=(Ta0 0Tb)+(VaaVab VabVbb), (22) J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-16 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp which is usually projected onto a vibrational basis and then diag- onalized to compute the spectrum and intensities. A particularly simple form is given by the so-called linear vibronic coupling model (LVC), viz., VLVC=⎛ ⎝∑sκa sqs+1 2∑kωkq2 k ∑cλcqc ∑cλcqc Δab+∑sκb sqs+1 2∑kωkq2 k⎞ ⎠, (23) which, in this form, is applicable to the pseudo-Jahn–Teller case, where interaction between two (generally quite proximate) non- degenerate states is important. Treatments of electronic spectra with the KDC model can involve an arbitrarily large number of electronic states (for exam- ple, a proper treatment of the NO 3radical requires at least five states332), and going beyond the LVC is sometimes necessary to obtain qualitative understanding and always necessary for quanti- tative agreement with the measured spectra. Moreover, true Jahn– Teller cases (interaction between degenerate states) can also be treated with largely the same framework. Nevertheless, the very sim- ple non-degenerate two-state LVC model is an appropriate example to explain what tools are available in CFOUR for such calculations. Details for how more elaborate calculations are done can be found elsewhere.333,334 The form of the LVC Hamiltonian above involves a choice of normal coordinates ( q), the gap between the electronic states at the coordinate origin ( Δab, assumed to be positive below), linear diag- onal terms with coefficients κsthat correspond to gradients along totally symmetric coordinates ( qs) on the adiabatic potential energy surface, quadratic force constants for all modes on the diagonal (in the LVC model, these are assumed to be equal to the reference state for which the normal coordinates are calculated), and—critically— an off-diagonal coupling in which modes qcof a certain symmetry (for example, the asymmetric b2NO stretching mode if the two states are the ˜X2A1and Ã2B2states of NO 2) carry quasidiabatic coupling constants λc. Without sacrificing simplicity, a useful exten- sion of the LVC model is to maintain the assumption of linear off- diagonal coupling but to allow the quadratic force constants to relax from those of the reference state, which leads to V=⎛ ⎝∑sκa sqs+1 2∑klga klqkql ∑cλcqc ∑cλcqc Δab+∑sκb sqs+1 2∑klgb klqkql⎞ ⎠. (24) The computation of all parameters begins with the determina- tion of a set of normal coordinates, which usually are those of the same molecule in a different (reference) electronic state, with the absorbing state in the spectroscopic experiment being the most log- ical choice. For example, to study photodetachment of NO− 2, one would choose the anion. To do an LVC calculation, the first and second derivatives of the energies at the origin of the coordinate sys- tem (i.e., the geometry of NO− 2) are evaluated using the derivative techniques in CFOUR and then transformed into the normal coordi- nates. CFOUR contains a module called xquadmodel for effecting this transformation. The quasidiabatic coupling constants ( λc) above areevaluated according to a diabatization scheme based on EOM-CC theory that is described in detail elsewhere,50and their evaluation is based on an algorithm that is quite similar to that for adiabatic EOM- CC gradients. However, transition one- and two-electron densities are used in this case, and there are additional minor modifications necessitated by the different physical situation under consideration. It is important to note that these are not “non-adiabatic couplings” (which are off-diagonal terms in the kinetic energy in the adia- batic basis rather than off-diagonal terms in the potential energy in the quasidiabatic basis) but are intimately related to them, as discussed in Refs. 51, 335, and 336. In any event, once the qua- sidiabatic couplings are calculated, the force constants of the cou- pling modes appearing in the diagonal blocks of the potential are “diabatized” via ga cc′=(fA cc′)adiabatic +2λcλ′ c Δab, (25) gb cc′=(fA cc′)adiabatic−2λcλ′ c Δab, (26) where fcc′are the quadratic force constants on the adiabatic poten- tial surfaces. For coefficients gklwhere qkand qldo not couple the states, these are simply equal to the corresponding adiabatic force constants on the two surfaces. Together with the trivially calculated Δab, all parameters for the Hamiltonian are now avail- able, and the xsim module of CFOUR can then carry out the spectral simulation. It should be emphasized that the crucial coupling of states that characterizes these situations makes special demands on the quantum-chemical method. Approaches appropriate for the param- eterization are many but generally do not include ground-state single determinant MBPT and CC methods. It has been recog- nized that EOM-CC methods are ideally suited for problems of this sort75,337and are recommended for applications. For the exam- ple above (the photodetachment spectrum of NO− 2), EOMIP-CC is the most appropriate method, and the gradients available in CFOUR (together with the quasidiabatic coupling calculation) greatly facilitate the calculations that need to be done to construct the Hamiltonian. Quasidiabatic couplings can currently be routinely evaluated with EOMEE-CCSD only, with the continuum orbital approach recommended for EOMIP-CCSD and EOMEA-CCSD calculations. Documentation about vibronic Hamiltonian construction and diagonalization calculations is spotty, and the process of carrying out these calculations (apart from the simplest LVC treatment) is slightly arduous and tedious. In general, the procedure involves three phases. First, the reference state (which is used to define normal coordi- nates and is usually the absorbing state in the experiment) is charac- terized by means of geometry optimization and second derivative calculations. Then, the first and second derivatives are calculated for the final states and transformed to the reference state normal coordinates. Beyond this, the quasidiabatic couplings are calculated and similarly transformed. For an LVC (or slightly elaborated LVC calculation, as is demonstrated in the following paragraph), these are the three required phases of quantum chemistry calculation. Any investigators who require assistance with such calculations or J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-17 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp intend to explore more elaborate vibronic coupling models with CFOUR are encouraged to seek advice and assistance from the CFOUR mailing list (see Appendix B). This will also permit them to be directed to the tools that have been created by the authors to facilitate this particular type of spectroscopic application and instructed in their use. Parameters and a simulation are shown for the NO− 2photode- tachment spectrum in Table IX and Fig. 4, respectively, where the latter may be compared to the laboratory spectrum. The calculations were done at the frozen-core CCSD/cc-pVDZ level of theory (the anion is treated with simple single-reference CCSD, and the ˜X2A1 and Ã2B2states of NO 2are treated with EOMIP-CCSD), and the simulated spectrum shows indeed the power of the LVC model for capturing the salient qualitative features of electronic spectra. It is an entirely straightforward matter to do this parameterization and spectroscopic calculation with CFOUR ; the entire procedure can easily be done in a few hours of work. Finally, for simpler electronic spectra in which interactions between electronic states can be neglected, CFOUR has a highly efficient Franck–Condon program xfc_squared ,339and clear doc- umentation for running it is available on the CFOUR website (see Appendix A). F. Automatized composite schemes and basis-set extrapolations Additivity schemes and basis-set extrapolation340,341are nowa- days popular tools to minimize both basis-set truncation errors and correlation errors and to provide high-accuracy quantum-chemical results.233,236,342,343While these schemes are easily handled (with a TABLE IX . Parameters of the LVC Hamiltonian describing the photodetachment spectrum of NO− 2obtained at the fc-(EOMIP)-CCSD/cc-pVDZ level of theory. The geometry of the anion is R(N–O) = 1.262 Å, θ= 116.44○, and the anion harmonic frequencies are ω1= 1356.7 cm−1,ω2= 794.6 cm−1, andω3= 1322.7 cm−1. The first two modes have a1symmetry, and the third mode (which couples the two states) hasb2symmetry. All parameters are in cm−1. Parameter κX 1 −2614.4 κX 2 1400.1 κA 1 803.3 κA 2 −2034.1 gX 11 984.2 gX 12 137.5 gX 22 902.2 gX 33 1148.8 gA 11 1500.8 gA 12 −70.7 gA 22 463.3 gA 33 1100.5 Δab 8039.8 λa 3 530.2 aGeometric mean of λAXandλXA(see Ref. 50). FIG. 4 . Simulation of the 266 nm (4.66 eV) photodetachment spectrum of the NO 2 anion using the parameters in Table IX and calculated with the xsim module of CFOUR . The vertical energies have been adjusted by +0.2 eV so as to have the origin (peak at highest eKE) approximately coincide with that in the laboratory measurement of Ref. 338 (inset). This shift accounts for an underestimation of the electron affinity at the EOMIP-CCSD level of theory with the aug-cc-pVDZ basis set. In the simulation, each state in the stick spectrum has been convoluted with Gaussians having a width of 0.05 eV. Note that the experimental spectrum reveals a higher excited state of NO 2(at low electron kinetic energy), which was not included in the simulation. The two-state Hamiltonian was projected onto a vibrational basis comprising 25 functions per mode and diagonalized using 1000 Lanczos recursions. Transition moments for the two ionization processes are assumed to be equal. The inset was reproduced with permission from Weaver et al. , J. Chem. Phys. 90, 2070–2071 (1989). Copyright 1989 AIP Publishing LLC. calculator or a spreadsheet) when focusing on energies, their appli- cation is much more cumbersome in the context of geometry opti- mization or the computation of other properties. CFOUR offers here an automatized scheme,68,344which within a geometry optimization sets up and runs all individual computations that are needed, gathers the result, and computes the total energy and gradient. As input for computations involving basis-set extrapolation as well as composite schemes, CFOUR requires (a) the property to be computed (energy, geometry, or harmonic frequencies), (b) infor- mation concerning the basis sets used in the extrapolation (three basis sets from one of the correlation-consistent hierarchies of basis sets266,345are required for the extrapolation at the HF level;340two sets are needed for the extrapolation at the correlated level341), (c) information about the additional corrections to be applied, i.e., those from CCSDT, CCSDTQ, or all-electron CCSD(T) computations, and (d) keywords for the individual calculations to be performed. Detailed information about the input can be found on the CFOUR website (see Appendix A). It should be pointed out that the computation of equilibrium geometries and harmonic vibrational frequencies in this way pro- vides results that are consistent with the potential energy surface defined by the extrapolated energy. This is accomplished by using, for the gradient or the corresponding second derivatives, expres- sions that are derived from the original extrapolated energy by means of straightforward differentiation.68 J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-18 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Computed and semi-experimental equilibrium structure of cyclic SiS 2. The semi-experimental structure was obtained via a least-squares fit of the geometrical parameters of cyclic SiS 2to the experimentally determined rotational constants of three isotopologues, and the theoretical one (in parentheses) was obtained via composite computations as described in the text. All distances are given in Å, and all angles are given in degrees. For further details, see Ref. 346. To give an example, Fig. 5 compares the equilibrium geometry of cyclic SiS 2obtained at the HF/ ∞Z + CCSD(T)/ ∞Z +ΔT/cc- pVTZ + ΔQ/cc-pVDZ + core/cc-pCV5Z level at which compu- tations using the cc-pVQZ, cc-pV5Z, and cc-pV6Z basis sets are used to estimate the HF limit and computations at the cc-pV5Z and cc-pV6Z level to obtain the basis-set limit for the fc-CCSD(T) correlation energy. Additional contributions involve a correction obtained at the fc-CCSDT level [in comparison to fc-CCSD(T)] computed with the cc-pVTZ set, a correction obtained at the fc- CCSDTQ level (in comparison to fc-CCSDT) evaluated with cc- pVDZ, and a correction for core-correlation effects obtained at the CCSD(T)/cc-pCV5Z level [in comparison to frozen-core CCSD(T)]. The experimental equilibrium geometry346has been obtained from rotational constants determined for three isotopologues of cyclic SiS 2. These rotational constants have been adjusted using vibra- tional corrections computed at the CCSD(T)/cc-pCVTZ level using VPT2.67Concerning harmonic vibrational frequencies, the extrapo- lation scheme yields 1647 cm−1, 3836 cm−1, and 3947 cm−1, which can be compared to the experimental inferred values of 1648.5 cm−1, 3832.2 cm−1, and 3942.5 cm−1.347 Statistical analyses of the performance of these extrapolation schemes can be found in Ref. 68 for equilibrium geometries and in Ref. 35 for rotational constants derived from the computed geome- tries after taking account of vibrational corrections. In passing, we note that the extrapolation scheme can be further augmented by scalar-relativistic corrections computed either at the DPT2 level or using the X2C scheme. G. Analytic calculation of the Diagonal Born–Oppenheimer Correction (DBOC) The Born–Oppenheimer approximation348(BOA) is a funda- mental assumption used in the description of molecules: not only are quantum-chemical calculations mostly based on it but also chemical intuition relies on the notion of potential energy surfaces defined by the BOA. It is a quite good approximation, and as cause for its breakdown typically (near-)degeneracy of coupled electronic states349is mentioned. The first-order correction to the BOA350is, however, not related explicitly to other electronic states;351it comes from the (parametric) dependence of the electronic wavefunctionon the nuclear coordinates, which results in a nonzero expecta- tion value of the nuclear kinetic energy operator over the electronic wavefunction, ΔEDBOC(R)=∫drΨ∗(r;R)TN(R)Ψ(r;R), (27) withΨas the normalized electronic wavefunction obtained within the BOA and TNas the nuclear kinetic energy operator. In Eq. (27), the electronic coordinates are collectively denoted by r, while the nuclear coordinates are represented by R. The integration in Eq. (27) is over electronic coordinates only; thus, the so-called diagonal Born–Oppenheimer correction (DBOC) depends parametrically on the nuclear coordinates and represents a mass-dependent increment to the potential energy surface. Thus, with the DBOC included in the calculation, the adiabatic picture is kept352(the DBOC is some- times also called the adiabatic correction), and the notion of poten- tial energy surfaces is retained, although they now become mass- dependent. The DBOC is numerically small, but the high accuracy reached by electronic structure methods, as also discussed in several parts of this paper, sometimes necessitates its inclusion in the final energy. Since the kinetic energy operator in Eq. (27) includes the sec- ond derivative with respect to nuclear coordinates ( RAi), the key to the computation of the DBOC is the evaluation of the expecta- tion value of the operator ∇2 RAiover the electronic wavefunction.353 Replacing this second derivative by first derivatives of both the right- and left-hand CC wavefunctions, we were able to formulate the DBOC at the general CI level.48However, calculation of the DBOC from the coupled-cluster electronic wavefunction is complicated by the biorthogonal approach with different right- and left-hand wavefunctions,151,187especially by issues associated with normaliza- tion. These problems have been resolved in Ref. 48, and the DBOC expression could be formulated using derivatives of the cluster and Λoperators, the antisymmetric CC derivative density matrix, as well as the one- and two-particle unrelaxed density matrices. Evaluation of the DBOC formulas is possible with gradient and second derivative techniques available in CFOUR and MRCC : the deriva- tive of the amplitudes and the Λparameters can be taken directly from analytic force constant calculations. The same also holds for the calculation of the unperturbed one- and two-particle density matri- ces. Two differences need to be mentioned: (a) for the DBOC, unre- laxed density matrices are required, while the relaxed density matri- ces are used for the force constants; (b) translational invariance, which is exploited in force-constant calculations, cannot be used for the DBOC since derivatives with respect to all nuclear coordinates are required. The latter difference makes a slight increase in com- putational time, while the first one precludes the possibility of doing DBOC and force constant calculations at the same time. The depen- dence of the computational effort on the size of the system is the same as for the underlying CC model, but the loop over the complete set of nuclear coordinates introduces an additional factor of 3 Natoms with Natoms being the number of atoms in the considered molecule. Thus, the calculation of the DBOC is rather expensive compared to a single-point energy evaluation; nevertheless, it can always be rou- tinely performed when harmonic frequencies and zero-point energy corrections to the energy can be calculated analytically. J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-19 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE X . DBOC contribution to atomization energies (in kJ mol−1) of selected small systems calculated by different methods. The raw data are taken from Ref. 49 and obtained with the aug-cc-pCVQZ basis at CCSD(T)/cc-pVQZ geometries. SCF MP1 MP2 CCSD SCF MP1 MP2 CCSD C2H2 0.50 0.39 0.36 0.33 HCN 0.31 0.24 0.20 0.20 CCH 0.13 0.10 0.14 0.10 HCO −0.14 −0.19 −0.21 −0.21 CF 0.03 0.01 0.01 0.01 HF 0.00 0.00 0.00 0.00 CH −0.35 −0.41 −0.42 −0.44 HNO −0.23 −0.31 −0.36 −0.40 CH 2 0.13 0.06 0.06 0.06 HO 2 0.03 −0.01 −0.03 −0.04 CH 3 0.19 0.07 0.08 0.06 N 2 0.08 0.04 0.01 0.03 CN −0.07 −0.07 −0.02 0.06 NH −0.21 −0.24 −0.24 −0.24 CO 0.07 0.03 0.02 0.02 NH 2−0.05 −0.12 −0.12 −0.14 CO 2 0.20 0.16 0.14 0.15 NH 3 0.56 0.44 0.44 0.42 F2 0.02 0.02 0.00 0.01 NO −0.48 −0.39 −0.11 0.02 H2 0.22 0.13 0.09 0.06 O 2 0.05 0.03 0.01 0.01 H2O 0.52 0.45 0.42 0.41 OF −0.02 −0.02 −0.01 0.00 H2O2 0.52 0.44 0.39 0.36 OH 0.04 0.01 0.00 −0.01 Availability of the DBOC for CC (and CI) methods in CFOUR is the same as that of the analytic second derivative, as shown in Table I. The only exceptions are non-iterative methods such as CCSD(T), where, due to the lack of a well-defined wavefunc- tion, the DBOC cannot be expressed in the above formalism. For more details, see Ref. 48. We note that according to numerical tests,48triples contributions are rather small even at the full CCSDT level; therefore, a CCSD(T)-type DBOC would not bring substantial improvement over CCSD. To offer a reduced-cost alternative to CC methods, in Ref. 49, approximations to the above theory within many-body perturbation theory were presented. The first one, termed MP1, uses first-order amplitudes in the formula and its perhaps unusual name reflects the fact that, contrary to the total energy, there is a first-order cor- rection to the DBOC even in the Møller–Plesset partitioning. MP1- level DBOC just requires the evaluation of first-order (MP2) double excitation amplitudes and their contraction with the corresponding DBOC integrals, i.e., no significant additional cost compared to the HF-SCF evaluation of the DBOC is incurred (provided the CPHF equations are solved). The next level is MP2, which requires the knowledge of the first- and second-order single and double excita- tion amplitudes. Higher order formulas have not been worked out since the cost of their evaluation would be similar to CCSD. The calculated DBOC has found most of its application in accu- rate prediction of thermochemical values235,354–356as well as in spec- troscopy.357–364To demonstrate its importance, the DBOC contri- butions to the atomization energies of selected small molecules are given in Table X, as obtained by different methods. Table X shows that the DBOC contribution can be as large as several tenths of a kJ mol−1, therefore non-negligible in certain applications. Indeed, as has been shown, e.g., in Refs. 49 and 355, the DBOC contribu- tion increases with the number hydrogen atoms, and its role can be even more important for larger molecules with many hydrogen atoms. The importance of electron correlation and the accuracy of different methods is represented graphically in Fig. 6. Here, theaverage DBOC contribution to atomization energies with respect to the CCSD value (100%) is presented. One can conclude that (a) the correlation contribution is important, and its size is unpredictable (as shown by the large standard deviation of the SCF values); and (b) both MP1 and MP2 give good estimates with decreasing error bars. H. Core-level spectroscopy Core electron photoelectron and absorption spectra have served as useful tools for probing local chemical environments in molecules and solids.365,366Recent developments of x-ray light sources have also led to a rapid growth in investigations of x-ray FIG. 6 . Average DBOC contribution to atomization energies with respect to the CCSD value (in %). Standard derivations are given as error bars. Data from Table X have been used, NO and OF excluded. J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-20 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp induced ultrafast dynamics.367,368Accurate calculations of core ion- ization and excitation energies and of x-ray absorption spectra are of significant interest and have been a longstanding challenge for quantum chemistry.369Benefiting from the available efficient imple- mentation of EOM-CC methods, CFOUR offers EOM-CC machin- ery ranging from EOM-CCSD (available using the xvcc ,xecc , and xncc modules) to EOM-CCSDT and EOM-CCSDTQ (using the new xncc module) for high-accuracy calculations of core ionization and excitation energies. In order to eliminate spurious coupling between core ionized or excited states and high-lying valence excited states, Cederbaum and collaborators370proposed a generic idea of core– valence separation (CVS). The original formulation of CVS neglects coupling between core and valence orbitals in Hamiltonian inte- grals. An efficient implementation of this scheme has recently been reported by Vidal et al. for the EOM-CCSD method.371We have adopted a variant of CVS suggested for EOM-CCSD by Coriani and Koch372in which CVS is only applied to the EOM vectors, i.e., only excitation operators containing targeted core orbitals are retained in the EOM vectors. EOM-CC methods using this variant of CVS (hereafter referred to as “CVS-EOM-CC” methods) have ini- tially been implemented in CFOUR by using a projector that sets pure valence excitations in the EOM vector to zero in a regular EOM-CC calculation. As shown in Table XI, benchmark studies have demon- strated the systematic convergence of CVS-EOM-CC methods and the high accuracy of computed core ionization energies when triples contributions are taken into account.373 We have also recently explored the use of both perturbative and iterative approximations to full CVS-EOM-CCSDT coupled with efficient techniques for implementing the core–valence sepa- ration for higher-order excitation amplitudes.375Among the best- performing approximations was the CVS-EOM-CCSD∗method, which is a straightforward modification of the original method of Stanton and Gauss.28We have recently implemented these approx- imations in xncc (along with full CVS-EOM-CCSDT and CVS- EOM-CCSDTQ) using an algorithm that explicitly discards triple and quadruple excitation amplitudes with only valence occupied or inactive core indices. When only a constant number of core orbitals are active (in most calculations only one core orbital is active), this implementation leads to reduced scaling of the EOM- CC calculation. Importantly, the scaling is reduced to fully M6for CVS-EOM-CCSD∗. TABLE XI . Maximum absolute deviations (MADs) and standard deviations (SDs) of CVS-EOM-CC373and CVS- ΔCC374results from experimental values for chemical shifts of 21 1s ionization energies of C, N, O, and F in 14 molecules (in eV). SD MAD CVS-EOM-CCSD 0.40 0.94 CVS-EOM-CCSDT 0.20 0.45 CVS-EOM-CCSDTQ 0.10 0.24 CVS-ΔHF 0.70 1.67 CVS-ΔCCSD 0.19 0.53 CVS-ΔCCSD(T) 0.10 0.20Although EOM-CC methods are capable of providing accurate results for core ionization energies, relatively slow convergence of the computed results with respect to the rank of excitation has been observed. This can be attributed to strong relaxation of the wave- function due to the removal of core electron(s). The convergence is expected to be even slower for calculation of double core hole states. An alternative option for computing core ionization energies using ΔCC methods374,376has also been implemented in CFOUR and will be available in the next release. ΔCC methods perform separate HF and CC calculations for the neutral molecule and the core-ionized state. Due to the local nature of core holes, the HF wavefunction of a core-ionized state can usually be obtained using the maximum over- lap method.377The convergence problem of the CC equations for core-ionized states due to coupling to valence continuum states can be handled using a generalization of the CVS scheme.374Favorable accuracy has been obtained for CVS- ΔCC results of core ioniza- tion energies, with CVS- ΔCCSD(T) providing results as accurate as CVS-EOM-CCSDTQ, as shown in Table XI. I. Vibrational perturbation theory and effective Hamiltonians CFOUR allows for the determination of harmonic vibrational fre- quencies for a wide range of quantum-chemical methods. When analytic Hessians are not available, the Hessian may be com- puted numerically by finite differences of gradients and/or single- point energies. Additionally, anharmonic vibrational frequencies and intensities may be obtained by finite differences (preferably of analytical Hessians). The xcubic module calculates anharmonic con- tributions based on second-order vibrational perturbation theory (VPT2).380–384While VPT2, when paired with a sufficient level of electron correlation and basis set completeness, can provide highly accurate frequencies and intensities compared to gas-phase experi- ments,385–390the presence of near-degeneracies in the harmonic fre- quencies can lead to a breakdown in the perturbation theory. Most commonly, VPT2 is affected by Fermi391and Darling–Dennison392 resonances (although the latter is better described as a missing vibra- tional interaction rather than a PT breakdown). xcubic automat- ically attempts to detect cases of Fermi resonance and provides “deperturbed” frequencies and intensities, but a more accurate treat- ment requires the construction and diagonalization of an effective vibrational Hamiltonian as in contact transformation perturbation theory (Van Vleck perturbation theory).393,394 In order to treat these more difficult cases, the xguinea module is provided as a standalone program. xguinea reads the output of an anharmonic calculation, in particular, the files rota, coriolis, dipole[xyz], quadratic, cubic, and quartic . The CFOUR job archive files ( JOBARC andJAINDX ) are used if present to deter- mine symmetry and axis frame information. xguinea offers an inter- active command-line input so that different options and structures of the effective Hamiltonian can be quickly explored. Alternatively, an input file can be fed to xguinea using shell redirection, e.g., xguinea<input . An example input file for treating multiple Fermi resonances in formaldehyde is given below (here, ω5≈ω2+ ω6≈ω3+ω6—the Darling–Dennison coupling between the latter two states is also included). The full xguinea manual is available on the CFOUR website (see Appendix A). J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-21 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp states 3 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 vibration vpt2 diagonalize calc The construction of the effective Hamiltonian requires two steps: first, the diagonal elements are set equal to the deperturbed anharmonic frequencies. These differ from the standard VPT2 fre- quencies by removal of terms with a nearly degenerate energy denominator. Second, the off-diagonal elements are determined by coupling formulas specific to the type of resonance (Fermi or Darling–Dennison) and the relationship between the two states. The Fermi coupling coefficients, also called the Fcoefficients, are simply equal to scaled cubic force constants. The Darling–Dennison or K coefficients395are much more complicated in form and arise from the second-order transformed Hamiltonian. The expressions for the effective Hamiltonian for the formaldehyde example above are given in Ref. 390, Heff=51 2161 3161 ⎛ ⎜⎜ ⎝ν∗ 51√ 8ϕ2561√ 8ϕ356 1√ 8ϕ256ν2+ν6+x∗ 26 K 1√ 8ϕ356 Kν3+ν6+x∗ 36⎞ ⎟⎟ ⎠,(28) K=1 46 ∑ i=1Ki2,i3+1 2K26,36, (29)where an asterisk indicates deperturbation of the frequencies or anharmonicity coefficients xij, and the Kij,klcoefficients are tabu- lated in the literature.63,395,396 The treatment of Darling–Dennison resonances is especially important for accurately calculating the overtone and combination bands of molecules with multiple hydrogen stretching modes. For example, in water, the symmetric and antisymmetric O–H stretch- ing modes interact strongly for 2 νOHand higher. The results from Ref. 63 for the nνOH,n= 1, 2, 3, 4, levels of water computed with CCSD(T)/ANO2397are reproduced in Table XII. Overtone levels ofν3are reproduced extremely well, as are combination and ν1 overtone levels for νOH≤3. In the 4νOHpolyad, additional inter- actions with bending mode overtones nν2begin to affect the sym- metric stretching mode. Effective Hamiltonians for the fixed polyad numbers are easily specified in xguinea , e.g., polyad 2 1 0 0 0 0 1 vibration vpt2 states 1 0 0 0 1 diagonalize calc !set states 1 0 0 0 2 diagonalize calc . . . TABLE XII . Stretching levels of water obtained at the CCSD(T) level of theory with the ANO2 basis set. Italicized level energies correspond to states of b2vibrational symmetry. The VPT2 values are ordered in terms of decreasing ν1quantum numbers (i.e., the 3 νOHlevels are ordered 300, 201, 102, and 003), and the VPT2 + K levels are ordered in terms of those with dominant eigenvector projections along the same zeroth-order levels. νOH 2νOH 3νOH 4νOH VPT2 VPT2 + K VPT2 VPT2 + K VPT2 VPT2 + K VPT2 VPT2 + K Calc.3659 . . . 7231 7201 10 718 10 591 14 119 14 215 3757 . . . 7249 7249 10 656 10 604 13 977 13 804 . . . . . . 7415 7445 10 742 10 869 13 982 13 801 . . . . . . . . . . . . 10 976 11 028 14 136 14 309 . . . . . . . . . . . . . . . . . . 14 439 14 525 Expt.a3657 . . . . . . 7202 . . . 10 600 . . . 13 828 3756 . . . . . . 7250 . . . 10 613 . . . 13 831 . . . . . . . . . 7445 . . . 10 869 . . . 14 221 . . . . . . . . . . . . . . . 11 032 . . . 14 319 . . . . . . . . . . . . . . . . . . . . . 14 538 aReferences 378 and 379. J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-22 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In addition to computing anharmonic frequencies, intensi- ties, and vibrationally averaged dipole moments at the VPT2 level, xguinea can also compute frequencies using fourth-order vibrational perturbation theory (VPT4).398VPT4 calculations in xguinea additionally require the didq, quintic , and sextic files—the latter two are not calculated as part of a standard CFOUR anharmonic calculation, but they may be manually com- puted either by finite differences of fourth-order force fields or by fitting to a local potential energy surface (the “off-diagonal” quartic constants ϕijklwith i≠j≠k≠lare also required). In a future version, we hope to extend xguinea to rotational and ro-vibrational spectroscopies and the calculation of higher- order vibration–rotation interaction and centrifugal distortion constants. V. FUTURE DIRECTIONS The discussion so far has focused on the current status of CFOUR and is limited to features provided via either the current pub- lic version or a version to be released shortly. However, there are many other long-term developments concerning CFOUR either ini- tially underway or in the planning stages, which will extend its capabilities in the future. While most of these ideas are still in the planning stages and not yet appropriate for discussion, a few rep- resentative examples are provided here. Specifically, we will briefly discuss ongoing work on the use of Cholesky decomposition (CD) in order to facilitate computations on large molecules and on the development of methods for treating atoms and molecules in the exotic but astrophysically relevant environment of finite and strong magnetic fields. A. Cholesky decomposition representation of the electron repulsion integrals While the main focus of CFOUR is, by design, on the high-level treatment of small- to medium-sized molecules, extending the appli- cability of rigorous, ab initio methods to larger systems is becoming more and more desirable. The asymptotically rate-determining step of such calculations is the solution of the amplitude equations; how- ever, calculations on medium to large molecules with reasonable but not too large basis sets can often become overwhelming due to the cost of handling the two-electron repulsion integrals (ERIs). Opera- tions such as the full or partial transformation of the ERIs from the AO to the MO basis may often become the limiting step in prac- tice. This is due to two factors. First, it is usually safe to assume that the ERIs do not fit in memory, which is usually true for ERIs in the AO basis and even more so for ERIs in the MO basis. This means that handling the integrals involves slow disk I/O, which can be a serious limiting factor. Second, integrals are computed (and stored) in an order that depends on the shell structure of the basis set and accessed (or re-computed, for integral-direct implementations399–401) in buffers. This makes writing subsequent code with optimal han- dling of memory accesses virtually impossible, as the order in which the integrals are available is system-dependent and, in gen- eral, not optimal for vectorization or use of highly optimized BLAS routines.The ERI matrix is, however, not a full rank one. While there are, in principle, O(M4)nonzero integrals, due to the localization of Gaussian basis functions, many of these will be small or negli- gible.399,400,402This induces sparsity in the ERI matrix that can be exploited by introducing low rank approximations (μν∣ρσ)≈n ∑ KL(μν∣K)SKL(L∣ρσ), (30) where nis the rank of the decomposition and is assumed to be much smaller than the full rank N=M(M+ 1)/2, where Mis the number of basis functions. Popular choices are the so-called resolution of the identity (RI)403–407[or density fitting (DF)] approximation and the Cholesky decomposition (CD)408–415technique. In RI, an auxiliary basis set is introduced in order to approximate four-center integrals with products of three-center ones according to Eq. (30). CD, on the other hand, is, in principle, the exact decomposition of the ERI matrix in the product of a (full rank) lower triangular matrix times its transpose, i.e., (μν∣ρσ)=N ∑ K=1LK μνLK ρσ. (31) However, the decomposition in Eq. (31) can be truncated at n≪Nin a way that allows for both compression, to the point that the resulting Cholesky vectors can often be kept in mem- ory, and a rigorous a priori control of the approximation error. The latter feature is particularly attractive, as the accuracy of a CD-based calculation can be precisely controlled, which is not the case for the RI approximation. On the other hand, RI computa- tions can be performed using the same machinery used to compute the ERIs themselves, with little modifications, and many auxiliary basis sets are available in the literature,406,416–419while CD needs an ad hoc implementation to compute the decomposition itself. The same applies for integral derivatives.420–422We believe that this price is worth paying to retain full control of the precision of the cal- culation. For this reason, CD of the ERIs has been implemented inCFOUR . The long term goal of this development is to offer all the main features of CFOUR in conjunction with a CD representation of the ERIs. CD allows for large computational savings in opera- tions on the integral tensor as it reduces the scaling of AO to MO transformations from M5toM4. However, it does not change the scaling of the correlated treatment, with the exception of scaled- opposite-spin second-order many body perturbation theory (SOS- MP2).423,424Nevertheless, it can make a large difference as a formu- lation based on the CD of the integrals is intrinsically well suited for writing all the operations involving the ERIs as matrix–matrix multiplications, which can be performed with very efficient level 3 BLAS routines. Furthermore, as each Cholesky vector LKcon- tributes to the final quantity independently of the others, it is possi- ble to parallelize CD-based calculations by distributing the Cholesky vectors. At the moment, we are just starting to explore the use of CD inCFOUR .425A particularly promising development is the coupling of CD with quadratically convergent solvers for both SCF and CASSCF. To show an example of the potential benefits of such a technique, J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-23 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp we present here some preliminary results obtained with a serial, CD-based implementation of quadratically convergent SCF. This implementation is part of the experimental xdqcscf code described in Sec. IV B. As an example of the use of CD to extend the applicability of methods implemented in CFOUR , let us consider a medium-sized molecule such as caffeine (C 8H10N4O2). Using Dunning’s corre- lation consistent cc-pVTZ basis set, 560 basis functions are used, which is starting to be borderline for many post-HF applications. The SCF optimization can however still be performed using the AO- based code in xdqcscf . The calculation requires, on a single core, about 2.5 h and is heavily dominated by disk I/O. The same calcu- lation using CD ERIs can be performed in little more than 5 min using a threshold for the CD of 10−4, which is a reasonable choice for most applications. All the timings were obtained on a single core of an Intel Xeon Gold 6140M processor. The main difference is that the CD ERIs can easily fit in memory, avoiding thus slow I/O oper- ations, and that the vast majority of the operations performed are done with highly efficient BLAS-3 routines. It is interesting to note that the same calculation, when performed forcing the use of an out- of-core algorithm and thus reading the Cholesky vectors from disk, requires slightly less than 10 min on the same machine. Therefore, even though the calculation is slower by a factor of two than the same performed with the Cholesky vectors in core, it is still much faster than the traditional one. As a second example, we computed the SCF wavefunction for taxol (C 47H51NO 14), a large molecule for which we employ again Dunning’s cc-pVTZ basis set and the same settings for CD for a total of 1947 basis functions. The SCF optimiza- tion can be performed in 4.5 h on the same cluster node used before. While these are very preliminary results and simple-minded appli- cations, we believe that they offer a convincing argument in favor of using CD as a method to handle larger molecular systems. We have also recently completed an implementation in xncc of a CD-based algorithm for the expensive particle–particle ladder contribution to MP3 and CCSD, which avoids explicit storage of the ⟨ab∥cd⟩integrals. We plan to extend this pilot implementation to additional terms in the CCSD equations that deal with the ⟨ab∥ci⟩ integrals in order to further reduce storage and I/O bottlenecks. B. Reduced-scaling coupled cluster methods While the Cholesky decomposition approach (or RI/DF) can drastically reduce the memory and I/O requirements of both the SCF and correlated calculations, by itself it cannot reduce the scal- ing of post-Hartree–Fock methods, except for SOS-MP2. In order to reduce the scaling of the same-spin (exchange) part of the MP2 energy as well, Hohenstein et al. introduced a further factorization of the ERIs termed the tensor hypercontraction (THC) decomposi- tion,426 (μν∣ρσ)≈∑ RSXR μXR νVRSXS ρXS σ. (32) This factorization, combined with a Laplace quadrature representa- tion of the orbital energy denominators, reduces the scaling of full MP2 to M4and SOS-MP2 to M3. Parrish et al. refined the THC method by assuming that the factor matrices Xtake the form of a real-space collocation of the orbitals over a set of grid points: XR μ=ϕμ(xR).427This reduces the problem of finding the interactionmatrix Vto a linear least squares problem with closed-form solution, VRS=∑ R′S′∑ μνρσ(S−1)RR′XR′ μXR′ ν(μν∣ρσ)XS′ ρXS′ σ(S−1)SS′, (33) SRS=∑ μνXR μXR νXS μXS ν. (34) This procedure scales as M5for exact ERIs but reduces to M4when paired with an additional CD/DF/RI approximation. We have recently used this LS-THC factorization to implement reduced-scaling MP2 and MP3 methods (both scale as M4). In par- ticular, we have found that using a Cholesky decomposition of the real-space metric matrix Sallows for defining “pruned” grids spe- cific to particular classes of transformed MO integrals, e.g., ( ai\|bj) vs (ab\|cd).428The accuracy of the LS-THC-DF-MP2 energy and size of the pruned grids were found to be similar or superior to hand-optimized429or other automatically generated430,431grids. We are now turning to the THC factorization of the double excitation amplitudes432and the efficient implementation of a reduced-scaling THC-CCSD method. C. Atoms and molecules in finite magnetic fields Strong magnetic fields lead, due to the interplay between Coulomb and Lorentz forces, to a fascinating and complex elec- tronic structure.433For example, the lowest triplet state of the hydro- gen molecule (3Σ+ u) becomes bound and even assumes the role of the ground state of the molecule at a sufficiently strong magnetic field by the so-called perpendicular paramagnetic bonding mech- anism even though the formal bond order is zero.434Such strong field strengths are of astrophysical relevance as they can be found on magnetic White Dwarf stars (WDs). Spectra from WDs are, how- ever, very difficult to interpret since the magnetic field strength as well as the composition of the atmosphere are a priori unknown. As the magnetic field changes the electronic spectra completely, accurate quantum-chemical predictions are crucial prerequisites to interpretation. For such predictions, perturbation theory is inade- quate because the field is by no means only a small perturbation to the system and finite-field methods have to be used instead. The predictions face the challenge that due to the structure of the Hamil- tonian for a molecule in a magnetic field, the wavefunction becomes (in general) complex-valued, such that the implementation needs to allow for complex wavefunction parameters, integrals, etc. It is hence the goal to develop high-accuracy methods for the investi- gation of atoms and molecules in strong magnetic fields. Finite- field full-CI implementations exist and have led to the discovery of strongly magnetized WDs with helium atmospheres435and to the above-mentioned bonding mechanism.434However, since finite- field full-CI only allows the study of systems with very few electrons, alternative high-accuracy finite-field methods with lower computa- tional scaling, such as finite-field methods based on coupled-cluster and equation-of-motion coupled-cluster theory, are desirable.436–438 In order to use these methods within CFOUR , a new integral code using gauge-including atomic orbitals based on the McMurchie–Davidson scheme439,440together with an SCF driver is being written and will be interfaced with the QCUMBRE program.441The latter is written in C++ and designed in an object-oriented manner. A hierarchical data-type structure ensures that changes can be made on a low level without J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-24 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp having to modify the existing top-level code. A key feature of QCUM- BREis a black-box contraction routine that allows one to code in a manner that resembles the equations on paper, while efficient com- plex BLAS algorithms such as ZGEMM3M are being used internally to carry out matrix multiplications. ACKNOWLEDGMENTS Those who have contributed to CFOUR extend well beyond the author list of this paper. In particular, J.G., P.G.S., and J.F.S. would like to acknowledge R. J. Bartlett at the University of Florida, in whose research group the three were educated, allowed to flourish as postdoctorals and to develop the bonds that ultimately led to their career-long collaboration. The many others who have contributed to CFOUR have made important developments that have paid benefits to all of us, and the complete list of authors can be found on the CFOUR website (http://www.cfour.de). CFOUR development in Gainesville has been supported by the U.S. National Science Foundation (Grant CHE-1664325). In Mainz, the work on CFOUR has been supported by the Deutsche Forschungs- gemeinschaft, the Fonds der Chemischen Industrie, and the Alexan- der von Humboldt foundation. The CFOUR development in Budapest has been supported by the National Research, Innovation and Devel- opment Fund (NKFIA) of Hungary (Grant No. 124293). In Dallas, the CFOUR development has been supported by a generous start-up grant from SMU, and in Baltimore, the work on CFOUR has been sup- ported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0020317. DATA AVAILABILITY Data available on request from the authors. APPENDIX A: WEBSITE AND ONLINE DOCUMENTATION Already in 2005, at the time of the ACES II Mainz–Austin– Budapest (MAB) version, a wiki-based website was implemented to replace the old latex based manual in order to increase the up-to- dateness and to facilitate documentation of old and new features of the program package. With the renaming to CFOUR, the current wiki-based website www.cfour.de was introduced, which provides detailed information how to obtain, install, and use the CFOUR pro- gram package, what features are available, as well as many illustra- tive examples together with a bibliography, which provides refer- ences for methods, basis sets, and the underlying implementations inCFOUR . APPENDIX B: MAILING LIST Besides the aforementioned online manual (see Appendix A), there is a mailing list available (cfour@lists.uni-mainz.de) to which any CFOUR user may subscribe. This mailing list, which is hosted at the University of Mainz, is meant as a forum for the exchange of experiences between users of the CFOUR program sys- tem. Users may join at any time via the website https://lists.uni- mainz.de/sympa/subscribe/cfour. Note that in order to preventspam, subscription requests are monitored and require that sub- scribers are accepted manually. After having subscribed, one can post questions and comments via email to cfour@lists.uni-mainz.de. A searchable message archive of previous postings to the CFOUR mailing list, which goes back to about 2009, is available at https://lists.uni-mainz.de/sympa/arc/cfour. APPENDIX C: LICENSING AND MODE OF DISTRIBUTION For non-commercial purposes, there is no charge to obtain CFOUR for academic users (individuals, universities, and research institutes). The CFOUR license agreement, which is available from the aforementioned website, has to be signed and sent via regular mail or fax to the indicated address. After reception of the properly signed unmodified CFOUR license agreement, instructions will be provided for downloading CFOUR from a GitLab server hosted by the University of Florida. This portal offers a user interface similar to other popular git-based portals such as GitHub and Bitbucket. From there, users can easily download any released CFOUR version. Bug fixes that fall between versions are dis- tributed through this system as well, and users can either download a new version or receive updates through git version control. REFERENCES 1J. F. Stanton, J. Gauss, L. Cheng, M. E. Harding, D. A. Matthews, and P. G. Szalay, CFOUR, Coupled-Cluster techniques for Computational Chemistry, a quantum- chemical program package, with contributions from A. A. Auer, R. J. Bartlett, U. Benedikt, C. Berger, D. E. Bernholdt, Y. J. Bomble, O. Christiansen, F. Engel, R. Faber, M. Heckert, O. Heun, M. Hilgenberg, C. Huber, T.-C. Jagau, D. Jonsson, J. Jusélius, T. Kirsch, K. Klein, W. J. Lauderdale, F. Lipparini, T. Metzroth, L. A. Mück, D. P. O’Neill, D. R. Price, E. Prochnow, C. Puzzarini, K. Ruud, F. Schiff- mann, W. Schwalbach, C. Simmons, S. Stopkowicz, A. Tajti, J. Vázquez, F. Wang, J. D. Watts and the integral packages MOLECULE (J. Almlöf and P. R. Taylor), PROPS (P. R. Taylor), ABACUS (T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, and J. Olsen), and ECP routines by A. V. Mitin and C. van Wüllen. For the current version, see http://www.cfour.de. 2G. D. Purvis III and R. J. Bartlett, “ACES, a program to perform MBPT and CC calculations,” in Quantum Theory Project (University of Florida, Gainesville, FL, 1977). 3R. J. Bartlett, “Many-body perturbation theory and coupled cluster theory for electron correlation in molecules,” Annu. Rev. Phys. Chem. 32, 359–401 (1981). 4I. Shavitt and R. J. Bartlett, Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory (Cambridge University Press, 2009). 5J. Gauss, J. F. Stanton, and R. J. Bartlett, “Coupled-cluster open-shell analytic gradients—Implementation of the direct product decomposition approach in energy gradient calculations,” J. Chem. Phys. 95, 2623–2638 (1991). 6J. F. Stanton, J. Gauss, J. D. Watts, and R. J. Bartlett, “A direct product decom- position approach for symmetry exploitation in many-body methods. I. Energy calculation,” J. Chem. Phys. 94, 4334–4345 (1991). 7J. Almlöf, “The MOLECULE integral program,” Technical Report No. 74-09 (University of Stockholm Institute of Physics, 1974). 8P. R. Taylor, “VPROPS: A program for the evaluation of one-electron property integrals over Gaussians.” 9J. W. Moskowitz and L. C. Snyder, “POLYATOM: A general computer program forab initio calculations,” in Methods of Electronic Structure Theory , Modern The- oretical Chemistry Vol. 3, edited by H. F. Schaefer III (Springer, Boston, 1977), pp. 387–411. 10T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, J. Olsen, and P. R. Taylor, ABACUS: A Gaussian integral and integral derivative program. J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-25 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 11J. D. Watts, J. F. Stanton, J. Gauss, and R. J. Bartlett, “A coupled-cluster study of the ground state of C+ 3,” J. Chem. Phys. 94, 4320–4327 (1991). 12J. F. Stanton, J. Gauss, and R. J. Bartlett, “Potential nonrigidity of the NO 3 radical,” J. Chem. Phys. 94, 4084–4087 (1991). 13J. F. Stanton, J. Gauss, R. J. Bartlett, T. Helgaker, P. Jørgensen, H. J. Aa. Jensen, and P. R. Taylor, “Interconversion of diborane(4) isomers,” J. Chem. Phys. 97, 1211–1216 (1992). 14J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, and R. J. Bartlett, “The ACES II program system,” Int. J. Quantum Chem. 44, 879–894 (1992). 15V. Lotrich, N. Flocke, M. Ponton, A. D. Yau, A. Perera, E. Deumens, and R. J. Bartlett, “Parallel implementation of electronic structure energy, gradient and Hessian calculations,” J. Chem. Phys. 128, 194104 (2008). 16J. Gauss, J. F. Stanton, M. E. Harding, and P. G. Szalay, “Coupled cluster tech- niques for computational chemistry,” in Invited Lecture at the 8th WATOCM- meeting in Sydney, Australia, 2008. 17J. Gauss, “Calculation of NMR chemical shifts at second-order many-body per- turbation theory using gauge-including atomic orbitals,” Chem. Phys. Lett. 191, 614–620 (1992). 18J. Gauss, “Effects of electron correlation in the calculation of nuclear magnetic resonance chemical shifts,” J. Chem. Phys. 99, 3629–3643 (1993). 19J. Gauss, “GIAO-MBPT(3) and GIAO-SDQ-MBPT(4) calculations of nuclear magnetic shielding constants,” Chem. Phys. Lett. 229, 198–203 (1994). 20J. Gauss and J. F. Stanton, “Gauge-invariant calculation of nuclear magnetic shielding constants at the coupled–cluster singles and doubles level,” J. Chem. Phys. 102, 251–253 (1995). 21J. Gauss and J. F. Stanton, “Coupled-cluster calculations of nuclear magnetic resonance chemical shifts,” J. Chem. Phys. 103, 3561–3578 (1995). 22J. Gauss and J. F. Stanton, “Perturbative treatment of triple excitations in coupled-cluster calculations of nuclear magnetic shielding constants,” J. Chem. Phys. 104, 2574–2583 (1996). 23J. Gauss, “Analytic second derivatives for the full coupled-cluster singles, dou- bles, and triples model: Nuclear magnetic shielding constants for BH, HF, CO, N 2, N2O, and O 3,” J. Chem. Phys. 116, 4473–4776 (2002). 24J. F. Stanton and R. J. Bartlett, “The equation of motion coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transi- tion probabilities, and excited state properties,” J. Chem. Phys. 98, 7029–7039 (1993). 25J. F. Stanton, “Many-body methods for excited state potential energy surfaces. I. General theory of energy gradients for the equation-of-motion coupled-cluster method,” J. Chem. Phys. 99, 8840–8847 (1993). 26J. F. Stanton and J. Gauss, “Analytic energy gradients for the equation- of-motion coupled-cluster method: Implementation and application to the HCN/HNC system,” J. Chem. Phys. 100, 4695–4698 (1994). 27J. F. Stanton and J. Gauss, “Analytic energy derivatives for ionized states described by the equation-of-motion coupled cluster method,” J. Chem. Phys. 101, 8938–8944 (1994). 28J. F. Stanton and J. Gauss, “A simple correction to final state energies of doublet radicals described by equation-of-motion coupled cluster theory in the singles and doubles approximation,” Theor. Chem. Acc. 93, 303–313 (1996). 29D. A. Matthews and J. F. Stanton, “A new approach to approximate equation-of- motion coupled cluster with triple excitations,” J. Chem. Phys. 145, 124102 (2016). 30J. Gauss and J. F. Stanton, “Analytic CCSD(T) second derivatives,” Chem. Phys. Lett.276, 70–77 (1997). 31P. G. Szalay, J. Gauss, and J. F. Stanton, “Analytic UHF-CCSD(T) second deriva- tives: Implementation and application to the calculation of the vibration-rotation interaction constants of NCO and NCS,” Theor. Chem. Acc. 100, 5–11 (1998). 32J. Gauss and J. F. Stanton, “Analytic first and second derivatives for the CCSDT- n (n = 1 – 3) models: A first step towards the efficient calculation of CCSDT properties,” Phys. Chem. Chem. Phys. 2, 2047–2059 (2000). 33J. F. Stanton and J. Gauss, “Analytic second derivatives in high-order many- body perturbation and coupled-cluster theories: Computational considerations and applications,” Int. Rev. Phys. Chem. 19, 61–96 (2000). 34J. F. Stanton, C. L. Lopreore, and J. Gauss, “The equilibrium structure and fun- damental vibrational frequencies of dioxirane,” J. Chem. Phys. 108, 7190–7196 (1998).35C. Puzzarini, J. F. Stanton, and J. Gauss, “Quantum-chemical calculation of spectroscopic parameters for rotational spectroscopy,” Int. Rev. Phys. Chem. 29, 273–367 (2010). 36P. G. Szalay and J. Gauss, “Spin-restricted open-shell coupled-cluster theory,” J. Chem. Phys. 107, 9028–9038 (1997). 37M. Heckert, O. Heun, J. Gauss, and P. G. Szalay, “Towards a spin-adapted coupled-cluster theory for high-spin open-shell states,” J. Chem. Phys. 124, 124105 (2006). 38J. Gauss, K. Ruud, and T. Helgaker, “Perturbation-dependent atomic orbitals for the calculation of spin-rotation constants and rotational g tensors,” J. Chem. Phys. 105, 2804–2812 (1996). 39J. Gauss and D. Sundholm, “Coupled-cluster calculations of spin-rotation constants,” Mol. Phys. 91, 449–458 (1997). 40J. Gauss, M. Kállay, and F. Neese, “Calculation of electronic g-tensors using coupled cluster theory,” J. Phys. Chem. A 113, 111541 (2009). 41G. Tarczay, P. G. Szalay, and J. Gauss, “First-principles calculation of electron spin-rotation tensors,” J. Phys. Chem. A 114, 9246–9252 (2010). 42M. Kállay, P. R. Nagy, Z. Rolik, D. Mester, G. Samu, J. Csontos, J. Csóka, B. P. Szabó, L. Gyevi-Nagy, I. Ladjánszki, L. Szegedy, B. Ladóczki, K. Petrov, M. Farkas, P. D. Mezei, and B. Hégely, MRCC, a quantum chemical program. See also Z. Rolik, L. Szegedy, I. Ladjánszki, B. Ladóczki, and M. Kállay, J. Chem. Phys. 139, 094105 (2013), as well as: www.mrcc.hu. 43M. Kállay, P. R. Nagy, D. Mester, Z. Rolik, G. Samu, J. Csontos, J. Csóka, P. B. Szabó, L. Gyevi-Nagy, B. Hégely, I. Ladjánszki, L. Szegedy, B. Ladóczki, K. Petrov, M. Farkas, P. D. Mezei, and Á. Ganyecz, “The MRCC program system: Accurate quantum chemistry from water to proteins,” J. Chem. Phys. 152, 074107 (2020). 44M. Kállay and P. R. Surján, “Higher excitations in coupled-cluster theory,” J. Chem. Phys. 115, 2945–2954 (2001). 45M. Kállay, J. Gauss, and P. G. Szalay, “Analytic first derivatives for gen- eral coupled-cluster and configuration interaction models,” J. Chem. Phys. 119, 2991–3004 (2003). 46M. Kállay and J. Gauss, “Analytic second derivatives for general coupled-cluster and configuration-interaction models,” J. Chem. Phys. 120, 6841–6848 (2004). 47M. Kállay and J. Gauss, “Calculation of excited-state properties using general coupled-cluster and configuration-interaction models,” J. Chem. Phys. 121, 9257– 9269 (2004). 48J. Gauss, A. Tajti, M. Kállay, J. F. Stanton, and P. G. Szalay, “Analytic calculation of the diagonal Born–Oppenheimer correction within configuration-interaction and coupled-cluster theory,” J. Chem. Phys. 125, 144111 (2006). 49A. Tajti, P. G. Szalay, and J. Gauss, “Perturbative treatment of the electron- correlation contribution to the diagonal Born–Oppenheimer correction,” J. Chem. Phys. 127, 014102 (2007). 50T. Ichino, J. Gauss, and J. F. Stanton, “Quasidiabatic states described by coupled- cluster theory,” J. Chem. Phys. 130, 174105 (2011). 51A. Tajti and P. G. Szalay, “Analytic evaluation of the nonadiabatic coupling vector between excited states using equation-of-motion coupled-cluster theory,” J. Chem. Phys. 131, 124104 (2009). 52S. Stopkowicz and J. Gauss, “Relativistic corrections to electrical first-order properties using direct perturbation theory,” J. Chem. Phys. 129, 164119 (2008). 53S. Stopkowicz and J. Gauss, “Direct perturbation theory in terms of energy derivatives: Fourth-order relativistic corrections at the Hartree–Fock level,” J. Chem. Phys. 134, 064114 (2011). 54S. Stopkowicz and J. Gauss, “Fourth-order relativistic corrections to electri- cal properties using direct perturbation theory,” J. Chem. Phys. 134, 204106 (2011). 55L. Cheng and J. Gauss, “Analytical evaluation of first-order electrical properties based on the spin-free Dirac–Coulomb Hamiltonian,” J. Chem. Phys. 134, 244112 (2011). 56L. Cheng and J. Gauss, “Analytic energy gradients for the spin-free exact two- component theory using an exact block diagonalization for the one-electron Dirac Hamiltonian,” J. Chem. Phys. 135, 084114 (2011). 57L. Cheng and J. Gauss, “Analytic second derivatives for the spin-free exact two- component theory,” J. Chem. Phys. 135, 244104 (2011). 58L. Cheng, S. Stopkowicz, and J. Gauss, “Analytic energy derivatives in relativistic quantum chemistry,” Int. J. Quantum Chem. 114, 1108–1127 (2014). J. Chem. Phys. 152, 214108 (2020); doi: 10.1063/5.0004837 152, 214108-26 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 59J. Liu and L. Cheng, “An atomic mean-field spin-orbit approach within exact two-component theory for a non-perturbative treatment of spin-orbit coupling,” J. Chem. Phys. 148, 144108 (2018). 60A. Asthana, J. Liu, and L. Cheng, “Exact two-component equation-of-motion coupled-cluster singles and doubles method using atomic mean-field spin-orbit integrals,” J. Chem. Phys. 150, 074102 (2019). 61A. Köhn, M. Hanauer, L. A. Mück, T.-C. Jagau, and J. Gauss, “State-specific multireference coupled-cluster theory,” Wiley Interdiscip. Rev.: Comput. Mol. Sci. 3, 176–197 (2013). 62D. A. Matthews and J. F. Stanton, “Non-orthogonal spin-adaptation of cou- pled cluster methods: A new implementation of methods including quadruple excitations,” J. Chem. Phys. 142, 064108 (2015). 63D. A. Matthews, J. Vazquéz, and J. F. Stanton, “Calculated stretching overtone levels and Darling–Dennison resonances in water: A triumph of simple theoretical approaches,” Mol. Phys. 105, 19–22 (2007). 64D. A. Matthews and J. F. Stanton, “Quantitative analysis of Fermi resonances by harmonic derivatives of perturbation theory corrections,” Mol. Phys. 107, 213–222 (2009). 65M. E. Harding, T. Metzroth, J. Gauss, and A. A. Auer, “Parallel calculation of CCSD and CCSD(T) analytic first and second derivatives,” J. Chem. Theory Comput. 4, 64–74 (2008). 66E. Prochnow, M. E. Harding, and J. Gauss, “Parallel calculation of CCSDT and Mk-MRCCSDT energies,” J. Chem. Theory Comput. 6, 2339–2347 (2010). 67I. M. 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1.4850415.pdf	Control of domain wall motion at vertically etched nanotrench in ferromagnetic nanowires Kulothungasagaran Narayanapillai and Hyunsoo Yang Citation: Applied Physics Letters 103, 252401 (2013); doi: 10.1063/1.4850415 View online: http://dx.doi.org/10.1063/1.4850415 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/103/25?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Application of local transverse fields for domain wall control in ferromagnetic nanowire arrays Appl. Phys. Lett. 101, 192402 (2012); 10.1063/1.4766173 Current-induced domain wall motion in permalloy nanowires with a rectangular cross-section J. Appl. Phys. 110, 093913 (2011); 10.1063/1.3658219 Currentinduced coupled domain wall motions in a twonanowire system Appl. Phys. Lett. 99, 152501 (2011); 10.1063/1.3650706 Field- and current-induced domain-wall motion in permalloy nanowires with magnetic soft spots Appl. Phys. Lett. 98, 202501 (2011); 10.1063/1.3590267 Magnetic imaging of the pinning mechanism of asymmetric transverse domain walls in ferromagnetic nanowires Appl. Phys. Lett. 97, 233102 (2010); 10.1063/1.3523351 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 148.88.244.115 On: Sun, 04 May 2014 17:02:36Control of domain wall motion at vertically etched nanotrench in ferromagnetic nanowires Kulothungasagaran Narayanapillai and Hyunsoo Y anga) Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576 (Received 2 August 2013; accepted 29 November 2013; published online 16 December 2013) We study ﬁeld-induced domain wall motion in permalloy nanowires with vertically etched nanotrench pinning site. Micromagnetic simulations and electrical measurements are employed to characterize the pinning potential at the nanotrench. It is found that the potential proﬁle for a transverse wall signiﬁcantly differs from that of a vortex wall, and there is a correlationbetween the pinning strength and the potential proﬁle. Reliable domain wall pinning and depinning is experimentally observed from a nanotrench in permalloy nanowires. This demonstrates the suitability of the proposed nanotrench pinning sites for domain wall deviceapplications. VC2013 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4850415 ] Domain wall (DW) based devices have been proposed as a promising solution for future high density storage andlogic devices. 1–6Implementing these devices requires pre- cise and reliable control of DWs, which can be achieved through pinning centers. Automotion happens in ideal nano-wires, 7however, in reality, lithographic imperfections in nanowires result in natural pinning sites, which are inher- ently hard to control. Controllable pinning sites can be intro-duced by engineering artiﬁcially structured variations into the geometry along the nanowires, which are known as “notches.” Complex DWs are formed at these notches, andits depinning behavior is very sensitive to the initial state of the DW, its structure, and chirality, as well as the excitation method. 7–10Even the stochastic behavior of the pinning and depinning process can be controlled by varying the notch dimensions.11Static and dynamic pinning strengths for a DW at the pinning sites have also shown deviations.12All these results show that the control of DWs depends predomi- nantly on the notch proﬁle. A few mechanisms are typically used to pin and control a DW in ferromagnetic nanowires of in-plane anisotropy sys- tems. Ion irradiation is an approach in which a portion of the nanowire is implanted with ions to soften the magnetic prop-erties, thereby creating a pinning site. 13,14The common approach is to introduce lateral constrictions along the nano- wire, as a result, giving rise to a notch15or alternatively a lat- eral protrusion known as anti-notch.12However, in nanowires, it is challenging to precisely control the lateral dimensions at the nano-scale due to the limit of modern li-thography techniques. In this Letter, we demonstrate an alternative approach to control the notch proﬁle vertically by removing a rectangularshaped portion of the magnetic material along the nanowire (hereafter referred to as nanotrench). It enables us to utilize the advantage of controlling the vertical etching depth accu-rately down to a few monolayers with ion milling. We study the ﬁeld-induced pinning and depinning of DWs, from the nano-trench pinning site, in the permalloy nanowires.Micromagnetic simulations and electrical measurements areemployed to characterize the potential strength of these pin- ning sites. The pinning strength linearly increases with thedepth of the nanotrench for transverse and vortex DWs. Above a certain length of the nanotrench, the depinning strength begins to saturate. The stochastic nature of DW gen-eration and depinning is also presented in a nanotrench. Micromagnetic simulations of the depinning studies are performed using the object oriented micromagnetic frame-work (OOMMF). Two different dimensions of nanowires are utilized for studying transverse (a width of 100 nm and thick- ness of 10 nm) and vortex (a width of 200 nm and thicknessof 40 nm) DWs. Cell dimensions of 4 /C24/C22n m 3and 4/C24/C25n m3have been used, for the transverse and vortex case, respectively. A saturation magnetization ofM S¼8.6/C2105A/m, exchange constant of A¼13/C210/C012 J/m, and an anisotropy constant of K¼0 are assumed. The simulations were performed at the quasi-static regime andthe Gilbert damping parameter ( a) is set to 0.5 to improve the speed of the simulations. A nanotrench is placed at the center along the nanowire as shown in Fig. 1(a). The DWs are initially located at the right edge of the nanotrench and then released to relax for several nanoseconds to form an energetically favorable andstable structure in each simulation. Examples of a similarly initialized transverse and a vortex DW at the nanotrench are shown in Figs. 1(b) and1(c), respectively. In both cases, the length of the nanotrench (LN) is 240 nm, while the depth (DN) is 6 and 20 nm, respectively, for transverse and vortex cases. The position of the nanotrench is highlighted using adark shade. During the relaxation process, the vortex DW moves outwards of the nanotrench and is stabilized, while the transverse DW moves towards the center of thenanotrench. The effect of varying the nanotrench dimensions on the pinning and depinning of both types of DWs has also beenstudied. The lengths LN and DN of the nanotrench are grad- ually varied. For the case of a transverse wall, when the depth increases from 2 to 6 nm, the depinning ﬁeld strengthincreases almost linearly as shown in Fig. 2(a). This trend remains the same for all different lengths investigated. However, when the length increases from 20 to 240 nm, it is a)Electronic mail: eleyang@nus.edu.sg 0003-6951/2013/103(25)/252401/4/$30.00 VC2013 AIP Publishing LLC 103, 252401-1APPLIED PHYSICS LETTERS 103, 252401 (2013) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 148.88.244.115 On: Sun, 04 May 2014 17:02:36observed that the pinning ﬁeld increases linearly and, subse- quently, saturates as shown in Fig. 2(b). For the case of 6 nm deep nanotrench, the depinning ﬁeld saturates at a length of 100 nm. A similar trend is also seen for the vortex DWs,which have been plotted in Figs. 2(c) and2(d). The depin- ning ﬁeld increases with the depth and length of the nano- trench. However, it is evident that irrespective of the depthof the nanotrench, the depinning strength proﬁle saturates around a length of /C24100 nm for both cases of transverse and vortex DWs. This saturation behavior can be understood bythe energy landscape of the pinning sites as we discuss below. In order to provide a more quantitative understanding of the depinning behavior, the potential landscape for the DW states is calculated by micromagnetic simulations. The geo- metrical variations along the nanowire generate an energylandscape that a DW experiences while traversing through the wire. The change in the potential proﬁle reﬂects the inter- action between the spin structure of DW and the pinningsite. In order to understand the energy landscape of the nano- trench proﬁle, energy terms like demagnetization and exchange energy are taken into consideration. A DW is ini-tially placed on the left side, at a distance of 1.5 lm away from the center of the nanotrench. A constant magnetic ﬁeld is applied along the þx-direction to drive the DW towards the right end of the nanowire, thereby passing through the nanotrench at the center of the nanowire. The absence of the anisotropy energy term makes the energy equation convergetoE Tot¼EDemþEEx, where EDem is the demagnetization energy and EExis the exchange energy. The contributions of each energy term in the system for a transverse DW (LN ¼240 nm; DN ¼6 nm) is plotted inFig.3(a)with respect to the DW position. The energy is nor- malized with respect to the total energy ( ETot). The center of the nanotrench is set to zero on the x-axis. The total energy of the system is locally reduced forming a potential wellaround the center of the nanotrench in case of the transverse DW as shown in Fig. 3(a). The demagnetization energy con- tributes 90%, while the exchange energy is /C2410% of the total energy contribution outside the potential well. The exchange energy increases while entering into the nano- trench area, as a result, providing resistance to the movingDW, indicated by a small peak in E Exat/C00.19lm in Fig. 3(a). The energy landscape of the vortex wall (LN ¼240 nm; DN¼15 nm) are shown in Fig. 3(b). It should be noted that the interactions generated at the nanotrench edge by a larger vortex DW structure will make the energy landscape differ- ent from the transverse DW case. The transverse DW pins atthe center of the pinning site, whereas the vortex DW is repelled away from the center of the nanotrench, but pins at either edges of the nanotrench due to a dual-dip energy pro-ﬁle. This phenomenon is also observed in the conventional constriction type notches, where the vortex DW has to realign its spin structure at the expense of increasing energyterms while passing through the notch. 9,16However, the con- tributions from the demagnetization energy and exchange energy remain around 90% and 10% outside the potentialwell, respectively, which is similar to the transverse DW case. Similar energy contributions have been reported for pinning sites deﬁned by ion implantation. 13 The total energy is plotted in Figs. 3(c)and3(d)for both transverse (DN ¼6 nm) and vortex (DN ¼15 nm) walls for various lengths (LN ¼40 to 240 nm) of nanotrenches in order to achieve a better understanding of the energy FIG. 1. (a) Schematics of a nanowire with a nanotrench. Simulated a trans- verse (b) and vortex (c) DW at the nanotrench. FIG. 2. Dependence of depinning ﬁeld with the depth (a) and the length (b) of nanotrench for a transverse wall. (c) and (d) show the dependence of depinning ﬁeld for vortex walls. FIG. 3. Energy proﬁles with respect to DW position for a transverse (a) and a vortex (b) wall. (c) and (d) show the total energy for transverse and vortexwalls for various lengths of nanotrenches. The drop in the energy proﬁle (DE Tot) is plotted in the insets for respective DW types.252401-2 K. Narayanapillai and H. Y ang Appl. Phys. Lett. 103, 252401 (2013) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 148.88.244.115 On: Sun, 04 May 2014 17:02:36landscape at the pinning site. It can be inferred from both cases that the shape of the energy proﬁle is almost con- served. Furthermore, the depth of the dip in the energy pro- ﬁle increases with increasing the length of the nanotrench.The curvature as well as the depth of the energy landscape determines the pinning strength. The insets of Figs. 3(c)and 3(d) show the drop in the energy proﬁle, DE Totwith respect to the energy at /C00.5lm which follows the same trend as that of the depinning strength discussed in Figs. 2(b) and 2(d) for both types of the DWs. For the vortex DW case, the drop in energy on both sides of the dip (labeled left and right) shows a very similar behavior as shown in the inset ofFig.3(d). The proposed pinning sites are experimentally veriﬁed in permalloy nanowires. Thin ﬁlms with the stack structureof substrate/Ta (3 nm)/Ni 81Fe19(30 nm)/Ta (3 nm)/Ru (2 nm) are deposited in a dc-magnetron sputter tool at a base pres- sure of 1 /C210/C09Torr. Sub-micrometer wires are patterned by electron beam lithography (EBL) followed by Ar ion milling. The measurement contact pads are deﬁned by EBL, and followed by deposition and a lift-off process. The top2 nm of the nanowire was partially etched before depositing the contact pads to provide ohmic contacts between the nanowire and the contact pads. Finally, the nanotrench isdeﬁned using an etch mask designed by EBL followed by Ar ion milling to remove a portion of the nanowire in order to form the required vertical nanotrench. Figure 4(a) shows a scanning electron micrograph with the measurement sche- matics along with the nanotrench highlighted in red color within the nanowire. The width of the nanowire is 650 nmand the length is 12 lm. Anisotropic magnetoresistance (AMR) is a suitable choice for DW detection. It reduces the resistance of thenanowires due to the presence of a DW. The following sequence is employed for DW generation and detection. First, a saturation magnetic ﬁeld, H SAT¼1 kOe, is applied in theþx-direction and reduced to zero. Then, the saturation re- sistance RSATis measured with a dc current of 30 lA applied across A 1B1contacts, which is /C24145.80 X. Second, a shortpulse is applied across A 1A2contacts to generate a DW by utilizing the Oersted ﬁeld generation method17and simulta- neously a constant assist ﬁeld of 30 Oe, HASSIST is applied in the/C0x-direction to push the DW to the nanotrench. Then, the resistance ( RI) across A 1B1contacts is measured again at zero ﬁelds. The difference between two resistance values (¼RSAT/C0RI) is associated with the DW resistance ( RDW). Subsequently, 1 kOe is applied along the /C0x-direction to remove any effects from remanence. The above process is repeated to gain a statistical distribution. The histogram of the DW generation process is shown in Fig. 4(b). The three different DW resistances (RDW¼/C00.16,/C00.13, and /C00.10X) can be explained by the existence of transverse and vortex DWs with different chirality at the nanotrench. The high occurrence of two typesof DWs around /C00.16 and /C00.13Xcould be due to the anti- clockwise and clockwise vortex DWs, while the relatively small occurrence around /C00.10Xcan be attributed to the transverse DW conﬁgurations. 18,19The depinning strength of the pinned DWs at the nanotrench is also investigated. After a DW is pinned at the nanotrench as discussed earlier, themagnetic ﬁeld increases in steps of 2 Oe in the /C0x-direction and a representative depinning proﬁle is shown in Fig. 4(c). As shown in Fig. 4(c), when the DW is removed from the nanotrench and moves out of the A 1B1portion, the resistance reaches the RSATvalue (145.80 X). The depinning strength depends on the DW type and chirality. From the histogramplot shown in Fig. 4(d), we can see that the DW depinning is distributed. This could be understood by the presence of dif- ferent DW types generated during the DW generation pro-cess and stochastic behavior of the depinning process. 20,21 In summary, we have demonstrated DW wall pinning and depinning in the proposed vertical nanotrench site. Themicromagnetic simulations show that the depinning strength can be effectively controlled by the proper selection of nano- trench dimensions. Different shapes of the potential proﬁleare observed for transverse and vortex type DWs. In permal- loy nanowires with nanotrench pinning sites, both types of DWs have been experimentally shown to exist. Reliable pin-ning and depinning behaviors from a vertical nanotrench are observed. Compared to the lateral constrictions, our pro- posed method has a higher precision in deﬁning the dimen-sions of the pinning sites in the sub-nanoscale. This work is partially supported by the Singapore National Research Foundation under CRP Award No. NRF-CRP 4-2008-06. 1M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin, Science 320, 209 (2008). 2D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn, Science 309, 1688 (2005). 3S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 4A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205 (2004). 5D. A. Allwood, G. Xiong, and R. P. Cowburn, Appl. Phys. Lett. 85, 2848 (2004). 6T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito, and T. Shinjo,Science 284, 468 (1999). 7M. Jamali, K. J. Lee, and H. Yang, New J. Phys. 14, 033010 (2012). 8M. Hayashi, L. Thomas, C. Rettner, R. Moriya, X. Jiang, and S. S. P. Parkin, Phys. Rev. Lett. 97, 207205 (2006). FIG. 4. (a) Scanning electron micrograph image of the device with the mea- surement schematics. (b) The histogram plot of the generated DW resist-ance. (c) A typical depinning proﬁle of a DW from a pinning site. (d) The histogram plot of depinning ﬁelds.252401-3 K. Narayanapillai and H. Y ang Appl. Phys. Lett. 103, 252401 (2013) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 148.88.244.115 On: Sun, 04 May 2014 17:02:369L. K. Bogart, D. Atkinson, K. O’Shea, D. McGrouther, and S. McVitie, Phys. Rev. B. 79, 054414 (2009). 10M. Jamali, H. Yang, and K. J. Lee, Appl. Phys. Lett. 96, 242501 (2010). 11M. Y. Im, L. Bocklage, G. Meier, and P. Fischer, J. Phys.: Condens. Matter. 24, 024203 (2012). 12A. Kunz and J. D. Priem, IEEE. Trans. Magn. 46, 1559 (2010). 13A. Vogel, S. Wintz, T. Gerhardt, L. Bocklage, T. Strache, M. Y. Im, P. Fischer, J. Fassbender, J. McCord, and G. Meier, Appl. Phys. Lett. 98, 202501 (2011). 14M. A. Basith, S. McVitie, D. McGrouther, and J. N. Chapman, Appl. Phys. Lett. 100, 232402 (2012).15M. Klaui, C. A. F. Vaz, J. Rothman, J. A. C. Bland, W. Wernsdorfer, G. Faini, and E. Cambril, Phys. Rev. Lett. 90, 097202 (2003). 16M. Klaui, J. Phys.: Condens. Matter. 20, 313001 (2008). 17M. Hayashi, L. Thomas, C. Rettner, R. Moriya, and S. S. P. Parkin, Nat. Phys. 3, 21 (2007). 18F. U. Stein, L. Bocklage, T. Matsuyama, and G. Meier, Appl. Phys. Lett. 100, 192403 (2012). 19M. Munoz and J. L. Prieto, Nat. Commun. 2, 562 (2011). 20G. Meier, M. Bolte, R. Eiselt, B. Kruger, D. H. Kim, and P. Fischer, Phys. Rev. Lett. 98, 187202 (2007). 21X. Jiang, L. Thomas, R. Moriya, M. Hayashi, B. Bergman, C. Rettner, and S. S. P. Parkin, Nat. Commun. 1, 25 (2010).252401-4 K. Narayanapillai and H. Y ang Appl. Phys. Lett. 103, 252401 (2013) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 148.88.244.115 On: Sun, 04 May 2014 17:02:36
1.373388.pdf	Rotationally symmetric solutions of the Landau–Lifshitz and diffusion equations I. D. Mayergoyz, G. Bertotti, and C. Serpico Citation: Journal of Applied Physics 87, 5511 (2000); doi: 10.1063/1.373388 View online: http://dx.doi.org/10.1063/1.373388 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/87/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Analytical solutions of Landau–Lifshitz equation for precessional switching J. Appl. Phys. 93, 6909 (2003); 10.1063/1.1557278 Comparison of analytical solutions of Landau–Lifshitz equation for “damping” and “precessional” switchings J. Appl. Phys. 93, 6811 (2003); 10.1063/1.1557275 Soliton solutions of XXZ lattice Landau–Lifshitz equation J. Math. Phys. 42, 5457 (2001); 10.1063/1.1407839 Some exact nontrivial global solutions with values in unit sphere for two-dimensional Landau–Lifshitzequations J. Math. Phys. 42, 5223 (2001); 10.1063/1.1402955 Coupling between eddy currents and Landau–Lifshitz dynamics J. Appl. Phys. 87, 5529 (2000); 10.1063/1.373394 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.49.251.30 On: Fri, 05 Dec 2014 06:26:51Rotationally symmetric solutions of the Landau–Lifshitz and diffusion equations I. D. Mayergoyz Department of Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742 G. Bertotti IEN Galileo Ferraris, Corso M. d’Azeglio 41, I-10125 Torino, Italy C. Serpico Department of Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742 The problem of isotropic conducting ferromagnetic ﬁlm subject to in-plane circular polarized magnetic ﬁelds is discussed. This problem requires simultaneous solution of diffusion and Landau–Lifshitz equations. It is observed that the mathematical formulation of the problem is invariant withrespect to rotations in the ﬁlm plane. By exploiting this invariance, the rotationally symmetricsolutions of the Landau–Lifshitz equation coupled with the diffusion equation are obtained andexamined. © 2000 American Institute of Physics. @S0021-8979 ~00!32108-9 # The theoretical and experimental study of spin dynamics governed by the Landau–Lifshitz ~LL!equation has been the focus of considerable research for many years. Traditionally,this study has been motivated by the ferromagnetic reso-nance problems. 1–3Recently, the spin dynamics has received renewed attention in the area of magnetic recording4,5due to the increasing rate of data transfer. In write heads of mag-netic recording, spin dynamics is accompanied by eddy cur-rents. For this reason, the solution of the LL equationcoupled with the diffusion equation is required. This problemis also of general theoretical interest because its solution willreveal how the dynamic constitutive properties of ferromag-netic media described by the LL equation affect the diffusionof electromagnetic ﬁelds. In the paper, the problem of isotropic conducting ferro- magnetic ﬁlm subject to a constant perpendicular magneticﬁeld and in-plane circularly polarized ac magnetic ﬁelds isstudied. Mathematically, this problem is described by thefollowing coupled diffusion equation and LL equation in theGilbert form, ]2H’ ]z25m0sD2S]H’ ]t1]M’ ]tD, ~1! ]M ]t52gM3SHeff2a gMs]M ]tD, ~2! Heff’52A m0Ms2D2]2M’ ]z21H’, ~3! Heffz5Haz2Mz12A m0Ms2D2]2Mz ]z2, subject to the boundary conditions, H’~61 2,t!5Hm@excos~vt1u!1eysin~vt1u!#,~4!]M ]z~61 2,t!50, ~5! where Dis the ﬁlm thickness, zis normalized by D,Hais the applied ﬁeld, subscript ‘‘ ’’’ indicates in-plane components, while all other symbols have their usual meaning. Equations ~1!–~3!are strongly nonlinear and this, in general, makes their analytical solution very difﬁcult. How-ever, in the case of the boundary value problem ~1!–~5!the following observation is very instrumental. The mathemati-cal form of the boundary value problem ~1!–~5!isinvariant ~up to inessential values of initial phase u!with respect to rotations of coordinate axes xandyin the ﬁlm plane. This suggests that the time periodic solutions of the boundaryvalue problem ~1!–~5!may exist that are invariant with re- spect to the above rotations as well. The latter means that H ’ andM’are uniformly rotating ~circularly polarized !vectors. By introducing phasors for these vectors and by using moreor less straightforward algebraic transformations, the bound-ary value problem ~1!–~5!for the partial differential equa- tions can be exactlyreduced to the following boundary value problem for the ordinary differential equations: d 2Hˆx dz25jb~Hˆx1Mˆx!, ~6! 2A m0Ms2D2d2Mˆx dz25FSMz,d2Mz d2zDMˆx2Hˆx, ~7! Hˆx~61 2!5Hˆax,dMˆx dz~61 2!50, ~8! where b5vm0sD2, FSMz,d2Mz dz2D5gHeffz2v gMz1jav gMs, ~9!JOURNAL OF APPLIED PHYSICS VOLUME 87, NUMBER 9 1 MAY 2000 5511 0021-8979/2000/87(9)/5511/3/$17.00 © 2000 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.49.251.30 On: Fri, 05 Dec 2014 06:26:51Ms25uMˆxu21Mz2, ~10! andHeffzis given by Eq. ~3!andHˆax5Hmeju. The math- ematically identical boundary value problem can be written forHˆyandMˆy. Theexacttransformation of the boundary value problem ~1!–~5!into the boundary value problem ~6!–~8!can be con- strued as a mathematical proof that the boundary value prob-lem~1!–~5!admits the solutions in the form of uniformly rotating vectors H ’andM’. This also proves the remark- able fact that there is no generation of higher order harmon-ics despite the nonlinear nature of the LL equation. In otherwords, the time harmonic solutions may exist for arbitrarystrong applied in-plane ac magnetic ﬁelds. This is in contrastwith the generally held opinion that sinusoidal solutions tothe LL equation can be only obtained for sufﬁciently smalldriving sinusoidal ﬁelds by using the linearization technique. Ordinary differential Eqs. ~6!and~7!are still strongly nonlinear due to relations ~9!and~10!. However, this difﬁ- culty can be circumvented in the practically important casewhen the ﬁlm thickness D~and/or its conductivity s!is suf- ﬁciently small. In this case, parameter bcan be regarded as a small one and we can use the perturbation technique withrespect to b. According to this technique, we look for the solution of the boundary value problem ~6!–~10!in the form, Mˆx5Mˆx(0)1bMˆx(1)1ﬂ,Hˆx5Hˆx(0)1bHˆx(1)1ﬂ. ~11! By substituting Eq. ~11!into Eqs. ~6!–~10!and equating the terms of the same order of smallness with respect to b,w e end up with the following equations for zero and ﬁrst orderterms, respectively: Hˆ x(0)5Hˆaxfor 21 2,z,1 2, ~12! 2jgHˆaxMz(0)5Fj~v2gHeffz(0)!1av MsMz(0)GMˆx(0),~13! Heffz(0)5Hax2Mz(0),Mz(0)56AMs22uMˆx(0)u2, ~14! and d2Hˆx(1) dz25j~Hˆax1Mˆx(0)!,Hˆx(1)~61 2!50, ~15! 2A m0Ms2D2d2Mˆx(1) dz25Fˆ(0)Mˆx(1)1F(1)Mˆx(0)2Hˆx(1), ~16! dMˆx(1) dz~61 2!50, ~17! where Fˆ(0)5g~Haz2Mz(0)!2v gMz(0) 1jav gMs, ~18!F(1)5ReHMˆx(0)* ~Mz(0)!2FS11g~Haz2Mz(0)!2v gMz(0)DMˆx(1) 22A m0Ms2D2d2Mˆx(1) dz2GJ, ~19! and ‘‘’’ denotes the complex conjugate quantity. Thus, the calculation of the zero order terms requires the uniform mode solution of the nonlinear LL equation writtenin the phasor ~algebraic !form Eqs. ~13!–~14!. The calcula- tion of the ﬁrst order terms requires the solution of linearsecond order ODE’s Eqs. ~15!–~16!. The latter task is sim- pler, and it can be even further simpliﬁed if we are only interested in average ~over ﬁlm thickness !values of Mˆ x(1). These average quantities are measurable and, for this reason,they are of practical interest. By integrating Eq. ~16!over the ﬁlm thickness and by taking into account the boundary con-dition ~17!and formulas ~18!and~19!, we arrive at the fol- lowing linear algebraic equation for the average value Mˆ x(1): Fˆ(0)Mˆx(1)1Mˆx(0) ~Mz(0)!2Re@Mˆx(0)~11Re@Fˆ(0)#!Mˆx(1)#5Hˆx(1), ~20! where, according to Eq. ~15!, the average value of the mag- netic ﬁeld is Hˆx(1)52j(Hˆax1Mˆx(0))/12. Finally, to characterize the losses in the thin ﬁlm we compute the following quantity: x95201/2Re@2jvm0Hˆx~u!Mˆx~u!1suEˆx~u!u2#du m0vuHˆaxu2, ~21! which, in the linear case, is reduced to the imaginary part of the magnetic susceptibility. By substituting the expansions~11!in Eq. ~21!, one ends up with the expansion, x9 5x9(0)1bx9(1), where x9(0)5Re@22jHˆaxMˆx(0)#/uHˆaxu2and x9(1)depends only on Hˆax,Mˆx(0),Mˆx(1), andHˆx(1). To start the calculations, the uniform mode solution of the phasor LL equations ~13!–~14!is ﬁrst obtained. These equations can be reduced to the following quartic equation: Vmz Vmz2V056VA12mz2 a2211mz2, ~22! wheremz5Mz(0)/Ms,V05(gHaz2v)/av,V5gMs/av, a5guHˆaxu/av. Equation ~22!is similar to the equation de- rived in Ref. 6 for the uniform mode solution of LL equation. Depending on the value of uHˆaxuand other parameters, this equation may have two or four real solutions. One of thesesolutions has M zopposite to Hazand is usually of no physi- cal interest. Thus, for sufﬁciently small uHˆaxu, there is only one physically meaningful real solution. This solution re-veals the resonance behavior which is qualitatively similar tothat described by the linear theory. The main difference isthe shift in the resonance frequency with the small increase in uHˆaxu. When uHˆaxuis further increased and reaches a cer-5512 J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 Mayergoyz, Bertotti, and Sepico [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.49.251.30 On: Fri, 05 Dec 2014 06:26:51tain critical value, then two more real solutions of Eq. ~22! appear. This leads to the foldover phenomena of resonancecurves which exhibit well-known bistable behavior usuallyobserved in ferromagnetic thin ﬁlm. 6–10A sample example of nonlinear resonance curves computed for various values ofuHˆaxuby using Eq. ~22!is shown in Fig. 1. By using the described uniform mode solution ~zero or- der terms !and Eq. ~21!, ‘‘eddy current’’ corrections for non- linear resonance curves can be computed. These correctionsare shown in Figs. 2 and 3, for the applied ﬁeld below andabove the critical ﬁeld, respectively. From these ﬁgures it is apparent that ‘‘eddy current’’ corrections exhibit resonancebehavior as well. Finally, it is interesting to know that eddy current losses are ‘‘generated’’ by the ﬁrst order term Hˆ x(1). From Eq. ~15!, we ﬁnd that Hˆx(1)5j(Hˆax1Mˆx(0))(z221 4)/2, which leads to the following expression for the electric ﬁeld Eˆy(z) 5b(Hˆax1Mˆx(0))z/s. This ﬁeld varies linearly with respect tozas in the case of classical eddy current losses. However, the frequency dependence of eddy current losses may signiﬁ-cantly deviate from the classical ‘‘ f 2’’-law. This deviation is caused by the resonance dependence of uMˆx(0)uon frequency. Thus, the conclusion can be reached that eddy current lossesexhibit resonance behavior which is controlled by the ap-plied ~constant in time !perpendicular magnetic ﬁeld H az. This observation suggests that it may be difﬁcult to separateexperimentally the losses due to the eddy currents from thosedue to the phenomenological damping term in the LL equa-tion because these two types of losses exhibit similar reso-nance behavior. This work is supported by the U.S. Department of En- ergy, Engineering Research Program. 1P. W. Anderson and H. Suhl, Phys. Rev. 100,1 7 8 8 ~1955!. 2H. Suhl, Proc. IRE 44, 1270 ~1956!. 3A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves ~Chemical Rubber, Boca Raton, 1996 !. 4N. Smith, IEEE Trans. Magn. 27, 4754 ~1991!. 5G. M. Sandler and H. N. Bertram, J. Appl. Phys. 81, 4513 ~1997!. 6G. V. Skrotskii and Yu. I. Alimov, Sov. Phys. JETP 8, 899 ~1959!. 7M. T. Weiss, Phys. Rev. Lett. 1,2 3 9 ~1958!. 8D. J. Seagle, S. H. Charap, and J. O. Artman, J. Appl. Phys. 57,3 7 0 6 ~1985!. 9A. Prabhakar and D. D. Stancil, J. Appl. Phys. 85, 4859 ~1985!. 10R. D. McMichael and P. E. Wigen, in Nonlinear Phenomena and Chaos in Magnetic Materials , edited by P. E. Wigen ~World Scientiﬁc, Singapore, 1994!, pp. 167–189. FIG. 1. Foldover of resonant curves for the function x9(0)vsvfor increas- ing values of incident rotating ﬁeld Hm. Values of the parameters, Haz 51.1Ms,a50.01,Ms58105. Legend ‘‘—’’ for linear theory, symbols ‘‘’’ forHm5Ms1025, symbols ‘‘ 3’’ forHm55Ms1025, symbols ‘‘ s’’ and ‘‘ L’’ forHm58Ms1025~‘‘s,’’ stable rotating solutions; ‘‘ L,’’ un- stable rotating solutions !. FIG. 2. Resonant curve for x9vsvincluding eddy current losses. Values of the parameters, Haz51.1Ms,a50.08,Ms58105,Hm52.5Ms1024,b 53.61023~at the resonant frequency !. Legend ‘‘ 2’’ for linear theory, symbols ‘‘ 1’’ for zero order nonlinear theory, symbols ‘‘ s’’ for ﬁrst order nonlinear theory. FIG. 3. Resonant curve for x9vsvincluding eddy current losses with foldover. Values of the parameters, Haz51.1Ms,a50.08,Ms58105, Hm58Ms1024,b53.61023~at the resonant frequency !. Legend ‘‘ 2’’ for linear theory, symbols ‘‘ *’’ and ‘‘ 1’’ for zero order nonlinear theory, symbols ‘‘ s’’ and ‘‘ L’’ for ﬁrst order nonlinear theory.5513 J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 Mayergoyz, Bertotti, and Sepico [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.49.251.30 On: Fri, 05 Dec 2014 06:26:51
1.4768958.pdf	Simulation of inhomogeneous magnetoelastic anisotropy in ferroelectric/ferromagnetic nanocomposites Nicolas M. Aimon, Jiexi Liao, and C. A. Ross Citation: Appl. Phys. Lett. 101, 232901 (2012); doi: 10.1063/1.4768958 View online: http://dx.doi.org/10.1063/1.4768958 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v101/i23 Published by the American Institute of Physics. Related Articles Enhanced magneto-impedance in Fe73.5Cu1Nb3Si13.5B9 ribbons from laminating with magnetostrictive terfenol-D alloy plate Appl. Phys. Lett. 101, 251914 (2012) Development of FeNiMoB thin film materials for microfabricated magnetoelastic sensors J. Appl. Phys. 112, 113912 (2012) Induced additional anisotropy influences on magnetostriction of giant magnetostrictive materials J. Appl. Phys. 112, 103908 (2012) Discuss on using Jiles-Atherton theory for charactering magnetic memory effect J. Appl. Phys. 112, 093902 (2012) Microstructure and magnetostriction of melt-spun Fe73Ga27 ribbon Appl. Phys. Lett. 101, 144106 (2012) Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 25 Dec 2012 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsSimulation of inhomogeneous magnetoelastic anisotropy in ferroelectric/ ferromagnetic nanocomposites Nicolas M. Aimon,a)Jiexi Liao, and C. A. Rossb) Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 1 November 2012; accepted 12 November 2012; published online 3 December 2012) The magnetic response of CoFe 2O4/BiFeO 3(CFO/BFO) nanocomposite thin ﬁlms, in which ferromagnetic CFO nanopillars are embedded in a ferroelectric BFO matrix, has been modeled by including the position-dependent magnetoelastic anisotropy of the CFO. A ﬁnite element simulationof the strain state of an arrangement of CFO pillars was performed in which the BFO matrix surrounding one or all of the pillars was subject to a piezoelectric strain. The strain transferred to the CFO pillars was calculated and transformed into a spatially varying magnetoelastic anisotropy in theCFO, and a micromagnetic model was then used to calculate the hysteresis of the pillar, which differed signiﬁcantly from a macrospin model. The position-dependent anisotropy led to a complex reversal process and to a reorientation of the easy axis to the in-plane direction at sufﬁcient appliedelectric ﬁelds. VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4768958 ] Films of perovskite/spinel se lf-assembled vertical magne- toelectric nanocomposites, in which ferrimagnetic pillars (e.g.,CoFe 2O4or NiFe 2O4) are epitaxially embedded in a ferroelec- tric matrix (e.g., BiFeO 3,P b T i O 3or BaTiO 3), can be grown by codeposition of the perovskite and spinel phases onto a singlecrystal substrate. 1–3Coupling between the two order parame- ters, ferrimagnetism and ferroelectricity, is achieved via strain transfer at the vertical interfaces between the two phases.4,5 The electric ﬁeld polarizes the f erroelectric, leading to a piezo- electric strain, which is transferred to the ferrimagnet across the interface, affecting its magnetic anisotropy via magnetoe-lastic effects. The degree of coupling in these two-phase nano- composites is signiﬁcant, especially considering that intrinsic magnetoelectrics have yet to display large magnetoelectriccoefﬁcients at room temperature. 6Moreover, unlike multi- layered composites where the fer roelectric and ferromagnetic phases form in-plane layers and t he strain is limited by substrate clamping, vertical nanocompos ites can support higher strains and, consequently, exhibit highe r magnetoelectric coupling.7 In BiFeO 3/CoFe 2O4(BFO/CFO) composites, the CFO has a strong magnetoelastic anisotropy, so an electric ﬁeld which strains the piezoelectric BFO matrix can reorient the magnetic easy axis of the CFO from out-of-plane to in-plane.8Storage and computation devices relying o n the electric ﬁeld-assisted control of the pillar magnetization state9have since been pro- posed.10Most of the modeling of the reversal mechanism of the magnetic pillars by combined el ectric and magnetic ﬁelds has been based on simple energetic arguments relying on the assumption of homogeneous st rain and magnetization within the pillars. 3D simulations by phase-ﬁeld modeling of the strain-mediated magnetoelectric coupling in other vertical nano- composites (BaTiO 3/CoFe 2O4) have been reported,11–14which showed the strong dependence of the coupling coefﬁcient on the boundary conditions at the interface between the ﬁlm and substrate and at the top free surface. However, these reportsonly considered the applica tion of a global magnetic and/or electric ﬁeld to the whole ﬁlm, and the feasibility of writing themagnetic state of a single pillar b y local ﬁelds without affecting its neighbors has not been studied. In this Letter, we describe a combined model including ﬁnite element analysis for strain calculations and micromag- netic modeling to calculate the magnetic switching of a CFO pillar when the BFO matrix surrounding it has been piezoelec-trically strained. The full strain ﬁeld in the pillar is converted into a magnetic anisotropy ﬁeld that is imported into a micro- magnetic solver, in this case, OOMMF (the NIST Object Ori-ented MicroMagnetic Framework), from which the reversal mechanism can be determined. This approach shows that the pillars reverse inhomogeneously due to the nonuniform strainstate imposed by the matrix, with dramatic consequences for the magnetic behavior of the nanocomposite. Fig. 1(a) shows a scanning electron microscope plan- view image of the typical morphology of BFO/CFO vertical nanocomposite thin ﬁlms grown on an (001) oriented SrTiO 3 (STO) substrate using pulsed laser deposition methods described previously.5The BFO is a single crystal ﬁlm with a cube-on-cube orientation with the substrate, and the CFO pillars grow vertically within the BFO as rectangular prismswith tapered facets at the top surface as well as at the ﬁlm- substrate interface. 2The facet orientations are summarized in the inset of Fig. 1(a). Using ﬁnite element analysis (ADINA 8.7, Automatic Dynamic Incremental Nonlinear Analysis), the strain in a CFO pillar was calculated when the BFO matrix around itundergoes piezoelectric deformation upon application of an electric ﬁeld. Two cases were considered. In the ﬁrst (local) case, only a small region of the BFO phase is strained arounda selected pillar. The strained region of the BFO is shown in Figs. 1(b) and1(c). This corresponds to the application of a local electric ﬁeld from, for example, a scanning probemicroscope tip, or a patterned electrode. In the second (global) case, the entire BFO volume is strained, which cor- responds to the application of an electric ﬁeld from a large- a)Electronic mail: naimo@mit.edu. b)Electronic mail: caross@mit.edu. 0003-6951/2012/101(23)/232901/5/$30.00 VC2012 American Institute of Physics 101, 232901-1APPLIED PHYSICS LETTERS 101, 232901 (2012) Downloaded 25 Dec 2012 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsarea top electrode. The ﬁnite element simulation mesh and dimensions which represent a unit cell of a BFO/CFO ﬁlmgrown on an STO substrate are detailed in Figs. 1(b) and 1(c). The pillars are 60 nm apart center-to-center and occupy a square lattice, and each pillar is 32 nm across its diagonal.Displacements perpendicular to the lateral boundaries of the model unit cell (which was 60 nm /C260 nm in plane and con- tained four quarter pillars, Fig. 1(c)) were prohibited to ensure the reconstruction, by symmetry, of an inﬁnite ﬁlm in the in-plane directions. The bottom surface of the unit cell was ﬁxed (zero displacement) to model the clamping con-straints exerted by the thick substrate. BFO was approximated as a cubic material, like STO and CFO, although it has a perovskite structure which under-goes a tetragonal distortion when grown epitaxially on (100) STO. The elastic constants used for BFO were 15 E¼189 GPa (Young modulus), G ¼68 GPa (shear modulus), and /C23¼0.35 (Poisson ratio); for CFO,16E¼188 GPa, G¼70 GPa, and /C23¼0.33; and for STO,17E¼284 GPa, G¼127 GPa, and /C23¼0.24. The initial epitaxial strains were not included in this work, i.e., all three materials were assumed to be unstrained in zero electric ﬁeld, allowing the effects of piezoelectric strains to be seen. In experimentalsamples, BFO is under an epitaxial strain from the substrate, and the strain in CFO can vary from compressive in the out- of-plane direction to fully relaxed, so there is a magnetoelas-tic anisotropy present in the as-grown nanocomposite even in the absence of an electric ﬁeld. 5 To include piezoelectricity in the ﬁnite element model, a ﬁctitious anisotropic thermal expansion was used to generatestrain in the BFO matrix locally. To model the electrostric- tion constant d 33/C2450 pm :V/C01of BFO,18,19a unidirectional thermal expansion coefﬁcient a33¼5/C210/C04K/C01was intro- duced in the BFO so that raising the temperature of a selected region of the matrix by 1 K in the model corre- sponded to applying a voltage of 1 V across a 100 nm thick ﬁlm. The strain distribution in the pillar was then calculated after equilibrating the structure. To determine the effect of the strain imposed by the BFO matrix on the magnetic behavior of the CFO pillar, the magnetoelastic anisotropy was calculated from the full straintensor and the magnetoelastic coefﬁcients of CFO, then imported into the OOMMF micromagnetic simulator. The micromagnetic simulator uses an Euler energy minimizationscheme to numerically solve the Landau-Lifshitz-Gilbert (LLG) equation, which can be expressed as dM dt¼/C0cM/C2Hef f/C0a MsM/C2ðM/C2Hef fÞ; (1) where Mis the magnetization, ais the damping coefﬁcient, c the gyromagnetic ratio, and Msthe saturation magnetization. Heffis the effective ﬁeld deﬁned as the derivative of the total energy utotwith respect to the magnetization, i.e., Hef f¼/C01 l0dutot dM; (2) utot¼uZeeman þudemagþuexchþuanis; (3) where uZeeman is the Zeeman energy due to an external applied ﬁeld, udemag the energy due to the demagnetizing ﬁelds, uexchthe exchange energy, and uanisthe sum of the magnetocrystalline and magnetoelastic anisotropy energies. The dimensions and shape of the pillars used in the micromagnetic simulation were the same as in the ﬁnite ele- ment model. Unlike the ﬁnite element analysis which uses an irregular mesh size for better accuracy in regions of highstrain gradients, a grid with 4 nm cubic cells was used in OOMMF. This length was chosen to be in the range of the exchange length of CFO, l ex¼ðA=l0M2 sÞ1=2, where the exchange constant Aand the saturation magnetization Ms were, respectively, 11 /C210/C012Jm/C01and 4 /C2105Am/C01. The damping constant awas set to 0.5, which ensured a fast relaxation of the magnetization to its equilibrium state with- out accounting for the high frequency dynamics of the system. The strong cubic magnetocrystalline anisotropy of CFO, withanisotropy constants K 1¼2/C2105Jm/C03and K2¼0Jm/C03 (Refs. 11,13,20) was also included in the simulation. The magnetoelastic anisotropy energy umeis given by ume¼B1ð/C15xxa2 xþ/C15yya2 yþ/C15zza2 zÞ þB2ð/C15xyaxayþ/C15xzaxazþ/C15yzayazÞ; (4) where /C15ijare the strain tensor components, aithe direction cosines of the magnetization, and Bithe magnetoelastic coef- ﬁcients. The latter are calculated using21 FIG. 1. (a) Top view SEM image of a typical BFO/CFO ﬁlm and inset show- ing the morphology of a CFO pillar including its facets. (b) Mesh and loads used in the ﬁnite element model for the local ﬁeld case. (c) Summary of the ﬁnite element model geometry and dimensions. Blue represents the CFO pil- lars and orange the BFO matrix. In (b) and (c), the regions over which theelectric ﬁeld is applied in the local case are indicated.232901-2 Aimon, Liao, and Ross Appl. Phys. Lett. 101, 232901 (2012) Downloaded 25 Dec 2012 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsB1¼/C03 2k100ðC11/C0C12Þand B2¼/C03k111C44;(5) where Cijare the components of the elastic compliance, which are related to the elastic constants E, G, and /C23in a cubic mate- rial, and khklrepresents the magnetostriction coefﬁcients of CFO ( k100¼/C0590/C210/C06andk111¼120/C210/C06).20 The magnetoelastic energy is quadratic in the direction cosines of the magnetization. It is thus characterized by threeorthogonal principal axes in which its matrix representation is diagonal, i.e., u me¼ax ay az0 @1 ATB1/C15xxB2 2/C15xyB2 2/C15xz B2 2/C15xyB1/C15yyB2 2/C15yz B2 2/C15xzB2 2/C15yzB1/C15zz0 BBBB@1 CCCCAax ay az0 @1 A;(6) ¼ax0 ay0 az00 @1 ATKx000 0Ky00 00 Kz00 @1 Aax0 ay0 az00 @1 A; (7) ¼Kx0a2 x0þKy0a2 y0þKz0a2 z0; (8) where ax0;ay0, and az0are the direction cosines of the mag- netization with respect to the set of principal axes, and Kx0;Ky0, and Kz0are the corresponding principal values of the anisotropy. The magnetoelastic anisotropy can thus be interpreted as a sum of three uniaxial anisotropies. The magnitude and orien-tation of these three orthogonal anisotropy terms varies within the magnetic pillar because the strain is position-dependent. To import the magnetoelastic anisotropy into the micromag-netic simulation, a linear interpolation was used to convert the strain ﬁeld from the irregular mesh of the ﬁnite element analy- sis to the cubic grid of the micromagnetic model, and then themagnetoelastic energy was diagonalized to calculate the three principal anisotropy components along principal axes x 0;y0, andz0at every cubic cell of the simulation. Fig.2(a)illustrates the largest diagonal component, /C1533, of the strain tensor calculated for the case of a localized elec- tric ﬁeld applied only to the BFO surrounding one “selected”pillar out of the four pillars in the unit cell as in Fig. 1(d).I n this simulation, a voltage of 1 V is applied across the 100 nm thickness of the selected region of the BFO, which wouldproduce a strain of E:d 33¼5/C210/C04in bulk BFO where E is the electric ﬁeld. A nonuniform strain state can be seen in the BFO region subjected to the electric ﬁeld and in the pillarwhich it encloses. The other regions of the BFO have no electric ﬁeld applied, and the strain decays rapidly in those regions. The strain in the unselected pillars is also small, atmost 1/6 of the maximum strain in the selected pillar. The model shows strain is also developed in the substrate just under the BFO, so the substrate cannot be approximated asfully rigid. The strain transferred to the pillar governs its magnetoelastic anisotropy. The strain /C15 33is smaller than that in the BFO matrix, with a maximum value of 3 :3/C210/C04, and is highly inhomogeneous. The other, smaller strain com- ponents /C1511and /C1522(not shown) are also inhomogeneous throughout the pillar.In the case of an electric ﬁeld applied globally, i.e., throughout the entire volume of the BFO matrix (Fig. 2(b)), the strain is higher in both the BFO and the CFO. The CFO pillars have a maximum strain of /C15max 33¼5/C210/C04, but as in the local strain case of Fig. 2(a), the strain in the CFO pillar is inhomogeneous. In both cases, the strain is lowest at the top and bottom of the CFO pillar and higher at the center. The strong magnetoelastic coefﬁcients of CFO imply thatthese strain inhomogeneities will induce large spatial varia- tions of the magnetic anisotropy. The diagonalization of the magnetoelastic anisotropy into three uniaxial magnetic anisotropies along principal axes x 0;y0, and z0is illustrated in Fig. 2(c) for a cell near a top facet of the strained pillar. In this cell, in contrast withthe overall out-of-plane tensile strain that was established in the bulk of the pillar by coupling from the BFO matrix, there is a small tensile in-plane strain. This leads to an out-of-plane easy axis as shown by the dimple in the total magne- toelastic energy surface, because of the negative magneto- striction of CFO. This is an example; the principal axes andmagnitudes of the anisotropy terms are in general different for each cell of the simulation. We now demonstrate the effect of the magnetoelastic anisotropy on the magnetic reversal process of the nanocom- posite. Fig. 3(a) shows the results for the magnetic ﬁeld- driven switching of a pillar when voltages of 0, 10, or 20 Vwere applied across the surrounding BFO, in the case corre- sponding to Fig. 2(a) (a local electric ﬁeld). The micromag- netic calculation was based on the strain state derived fromthe ﬁnite element model, in which the strain and the magne- toelastic anisotropy are inhomogeneous. At ﬁrst, when the electric ﬁeld is zero and the pillar is unstrained, its magnetic FIG. 2. /C1533in the (a) local case, (b) global case, and (c) decomposition of the full magnetoelastic anisotropy at a cell near the top of the pillar into three uniaxial anisotropies along x0;y0, and z0.232901-3 Aimon, Liao, and Ross Appl. Phys. Lett. 101, 232901 (2012) Downloaded 25 Dec 2012 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsswitching is characterized by a square loop with a high coer- cive ﬁeld of 875 kA/m governed by the strong cubic magne- tocrystalline anisotropy and the uniaxial shape anisotropy ofthe CFO pillar. As the voltage is applied and the out-of-plane tensile strain builds up in the pillar, the coercive ﬁeld decreases as the magnetoelastic anisotropy makes the pillareasier to magnetize in-plane. The loop at 0 V is also characteristic of the reversal of unselected pillars, i.e., those whose matrix was not subject tothe electric ﬁeld and which, therefore, have very little strain. The decrease in switching ﬁeld with applied voltage suggests that a magnetic ﬁeld of, e.g., 700 kA/m could switch theselected pillars (those surrounded by BFO with 10 V applied voltage) without affecting the unselected pillars. The out-of-plane direction becomes a hard axis at 20 V, with a reversible low-ﬁeld region and an out-of-plane satura- tion ﬁeld of 900 kA/m. Considerable structure is evident in the hysteresis loop at 20 V, caused by the differences in mag-netoelastic anisotropy between the top, middle, and bottom of the pillar, which allow these three regions to be magne- tized in different directions and switch at different ﬁelds. The corresponding magnetic states of the pillar are illustratedschematically in the ﬁgure. Fig.3(b)shows a similar calculation for a global electric ﬁeld, where the entire volume of the BFO matrix is piezo-electrically strained by voltages of 5 and 10 V. The magnetic hysteresis loop of the CFO pillar exhibits a similar depend- ence on the applied voltage as the local case of Fig. 3(a). Because the strain is higher for the same voltage applied globally rather than locally, the reorientation of the anisot-ropy from out-of-plane to in-plane occurs at lower voltage. The ﬁne structure in the hysteresis loops is different from Fig.3(a), reﬂecting the differences in the spatial dependence of the strain ﬁeld. A contrast to Fig. 3(a)is given in Fig. 3(c), which shows the reversal of a pillar subject to a uniform strain state. Thisstrain was obtained by averaging the strain over the volume of the pillar. The magnitude of the strain was found by aver- aging the diagonal components of the strain tensor in theoriginal x, y, z axes throughout the pillar (this comes out to /C15 xx¼/C15yy¼/C03:62/C210/C05and /C15zz¼1:35/C210/C04at 1 V, and is correspondingly higher for higher voltages). Off-diagonalterms were ignored for simplicity. The uniform strain state leads to a reorientation in anisotropy similar to that of Fig. 3(a), but the loop lacks the multiple small steps characteristic of the piecewise reversal of the inhomogeneously strained pillar. This illustrates the importance of the inhomogeneity in the strain state in determining the reversal process of thepillars. In conclusion, electric ﬁeld-assisted magnetic switch- ing of CFO pillars in BFO/CFO nanocomposites has beenmodeled in detail by taking the full piezoelectric strain state into account. The different boundary conditions at the top, sides, and bottom of the CFO pillars lead to largeinhomogeneities in magnetoelastic anisotropy, and conse- quently promote a complex incoherent switching process. Prior models 9,10which assume a homogeneous strain in the pillar, or macrospin models based on coherent rotation of the magnetization, cannot capture the details of the mag- netization reversal. The coupled strain and magnetoelasticmodel can provide insight into the behavior and design of devices based on multiferroic nanocomposites, and the diagonalization scheme used to simplify the magnetoelasticanisotropy terms can also be used in a broad range of micromagnetics problems involving inhomogeneous strain ﬁelds. The support of DARPA, NRI, and the NSF is gratefully acknowledged. The authors are grateful for support of the Center for Materials Science and Engineering, an NSF MRSEC. The authors sincerely thank M. Buehler and A. P.Garcia for helpful discussions on ﬁnite element modeling. 1H. Zheng, J. Wang, S. E. Loﬂand, Z. Ma, L. Mohaddes-Ardabili, T. Zhao, L. Salamanca-Riba, S. R. Shinde, S. B. Ogale, F. Bai, D. Viehland, Y. Jia, D. G. Schlom, M. Wuttig, A. Roytburd, and R. Ramesh, Science 303, 661 (2006). 2I. Levin, J. Li, J. Slutsker, and A. L. Roytburd, Adv. Mater. 18, 2044 (2006). FIG. 3. Magnetic hysteresis loops with magnetic ﬁeld along the axis of thepillars, for various applied electric ﬁelds. (a) electric ﬁeld applied locally to the BFO surrounding one pillar; (b) electric ﬁeld applied globally to the BFO; (c) similar to (a) except the strain in the pillar was set to an uniform average value.232901-4 Aimon, Liao, and Ross Appl. Phys. Lett. 101, 232901 (2012) Downloaded 25 Dec 2012 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions3Q. Zhan, R. Yu, S. P. Crane, H. Zheng, C. Kisielowski, and R. Ramesh, Appl. Phys. Lett. 89, 172902 (2006). 4N. Dix, R. Muralidharan, J. Guyonnet, B. Warot-Fonrose, M. Varela, P. Paruch, F. Sanchez, and J. Fontcuberta, Appl. Phys. Lett. 95, 062907 (2009). 5N. M. Aimon, D. H. Kim, H. Kyoon Choi, and C. A. Ross, Appl. Phys. Lett. 100, 092901 (2012). 6L. Martin and R. Ramesh, Acta Mater. 60, 2449 (2012). 7J. Ma, J. Hu, Z. Li, and C.-W. Nan, Adv. Mater. 23, 1062 (2011). 8F. Zavaliche, H. Zheng, L. Mohaddes-Ardabili, S. Yang, Q. Zhan, P. Shafer, E. Reilly, R. Chopdekar, Y. Jia, P. Wright et al.,Nano Lett. 5, 1793 (2005). 9F. Zavaliche, T. Zhao, H. Zheng, F. Straub, M. P. Cruz, P.-L. Yang, D. Hao, and R. Ramesh, Nano Lett. 7, 1586 (2007). 10M. Kabir, M. R. Stan, S. A. Wolf, R. B. Comes, and J. Lu, in Proceedings of the 21st edition of the Great Lakes Symposium on VLSI (ACM, New York, 2011), p. 25. 11J. X. Zhang, Y. L. Li, D. G. Schlom, L. Q. Chen, F. Zavaliche, R. Ramesh,and Q. X. Jia, Appl. Phys. Lett. 90, 052909 (2007). 12X. Lu, B. Wang, Y. Zheng, and E. Ryba, J. Phys. D: Appl. Phys. 42, 015309 (2009).13P. Wu, X. Ma, J. Zhang, and L. Chen, Philos. Mag. 90, 125 (2010). 14H. T. Chen, L. Hong, and A. K. Soh, J. Appl. Phys. 109, 094102 (2011). 15J. X. Zhang, D. G. Schlom, L. Q. Chen, and C. B. Eom, Appl. Phys. Lett. 95, 122904 (2009). 16Z. Li, E. S. Fisher, J. Z. Liu, and M. Nevitt, J. Mater. Sci. 26, 2621 (1991). 17Y. Li, S. Choudhury, J. Haeni, M. Biegalski, A. Vasudevarao, A. Sharan, H. Ma, J. Levy, V. Gopalan, S. Trolier-McKinstry, D. Schlom, Q. Jia, and L. Chen, Phys. Rev. B 73, 184112 (2006). 18J. X. Zhang, B. Xiang, Q. He, J. Seidel, R. J. Zeches, P. Yu, S. Y. Yang, C. H. Wang, Y.-H. Chu, L. W. Martin, A. M. Minor, and R. Ramesh, Nat. Nanotechnol. 6, 98 (2011). 19C. Daumont, W. Ren, I. C. Infante, S. Lisenkov, J. Allibe, C. Carr /C19et/C19ero, S. Fusil, E. Jacquet, T. Bouvet, F. Bouamrane, S. Prosandeev, G. Geneste, B.Dkhil, L. Bellaiche, A. Barth /C19el/C19emy, and M. Bibes, J. Phys. Condens. Mat- ter24, 162202 (2012). 20R. M. Bozorth, E. F. Tilden, and A. J. Williams, Phys. Rev. 99, 1788 (1955). 21R. C. O’Handley, Modern Magnetic Materials: Principles and Applica- tions , 1st ed. (Wiley, New York, 2000), p. 768.232901-5 Aimon, Liao, and Ross Appl. Phys. Lett. 101, 232901 (2012) Downloaded 25 Dec 2012 to 152.14.136.96. 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1.2838332.pdf	Micromagnetic modeling of ferromagnetic resonance assisted switching Werner Scholz and Sharat Batra Citation: Journal of Applied Physics 103, 07F539 (2008); doi: 10.1063/1.2838332 View online: http://dx.doi.org/10.1063/1.2838332 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/103/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Generalized Stoner-Wohlfarth model accurately describing the switching processes in pseudo-single ferromagnetic particles J. Appl. Phys. 114, 223901 (2013); 10.1063/1.4839895 Magnetic force microscopy study of microwave-assisted magnetization reversal in submicron-scale ferromagnetic particles Appl. Phys. Lett. 91, 082510 (2007); 10.1063/1.2775047 Numerical study for ballistic switching of magnetization in single domain particle triggered by a ferromagnetic resonance within a relaxation time limit J. Appl. Phys. 100, 053911 (2006); 10.1063/1.2338128 Micromagnetic investigation of resonance frequencies in ferromagnetic particles J. Appl. Phys. 97, 10E313 (2005); 10.1063/1.1852432 Micromagnetic study of nonlinear effects in soft magnetic materials J. Appl. Phys. 93, 7456 (2003); 10.1063/1.1557362 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Thu, 18 Dec 2014 09:53:29Micromagnetic modeling of ferromagnetic resonance assisted switching Werner Scholza/H20850and Sharat Batra Seagate Technology, 1251 Waterfront Place, Pittsburgh, Pennsylvania 15222, USA /H20849Presented on 7 November 2007; received 12 September 2007; accepted 29 November 2007; published online 14 March 2008 /H20850 We studied the steady state behavior and magnetization switching process of single domain particles subject to ac and dc magnetic ﬁelds using analytical and numerical models based on the Landau–Lifshitz–Gilbert equation. We compared the analytical solutions for circularly polarized ﬁelds witha numerical single spin model and circularly and linearly polarized ac magnetic ﬁelds. It has beenfound, that the initial conditions and the dynamics of the external ﬁelds /H20849ﬁeld ramps and amplitude changes /H20850strongly determine which precession orbit the magnetization converges to, if the magnetization precession is stable, and if the magnetization switches. We also studied the effects ofﬁeld amplitudes, ﬁeld angles, and damping on the switching behavior. The presented results can beapplied to high power ferromagnetic resonance experiments and ferromagnetic resonance assistedmagnetic recording schemes. © 2008 American Institute of Physics ./H20851DOI: 10.1063/1.2838332 /H20852 I. INTRODUCTION To detect the existence /H20849or nonexistence /H20850of a bubble domain for applications in bubble memory devices, Dötschet al. proposed a method based on ferrimagnetic resonance. 1 This led them and other groups to the discovery that the creation of bubbles can be induced/facilitated by the micro-wave ﬁeld. 1,2Artman et al. followed up on these ﬁndings and performed numerical computer simulations to investigate thephenomenon further. 3They describe sudden changes in the magnetization state as “foldover” and present a switching/H20849“ﬂipover” /H20850phase diagram. More recently a “FMR assisted recording” method has been proposed to use these resonanceeffects for magnetic recording on thin ﬁlm media 4as well as discrete memory devices.5Recent experimental work by Thirion and Mailly show switching ﬁeld reductions when theStoner–Wohlfarth astroid of Co nanoparticles is measured ina rf ﬁeld. 6In addition, theory and modeling of microwave assisted switching has been discussed in several papers.7–9 II. METHOD For this study, we used the Landau–Lifshitz–Gilbert /H20849LLG /H20850equation of motion dm dt=−/H9253 1+/H92512/H20849m/H11003H/H20850−/H9251/H9253 /H208491+/H92512/H20850/H20851m/H11003/H20849m/H11003H/H20850/H20852, /H208491/H20850 where mis the normalized magnetization vector M /Msand His the effective ﬁeld H=Hani+Hext/H20849+Hexch+Hdemag /H20850. Since we are only interested in the behavior of spherical single domain particles with uniaxial anisotropy, the ex-change and demagnetizing ﬁelds have been omitted. We useda single domain particle /H20849single spin /H20850model and integrated the LLG equation using a second order Runge–Kutta timeintegration method with ﬁxed time step.The periodic steady state solutions for the LLG equation with magnetocrystalline anisotropy and homogeneous dc andcircularly polarized ac ﬁelds have been derived by Bertotti et al. 7and they can be written in the form of a fourth order equation bn2 1−mz2−/H20849bz+mz/H208502 mz2−/H90242=0 , mz=Mz/Ms,/H9024=/H9251/H9275/H9260eff,/H9260eff=2K1//H20849/H92620Ms2/H20850. bz=/H20849Hbias /Ms−/H9275/H20850//H9260eff,bn=/H20849Hac/Ms/H20850//H9260eff, As a result, for any choice of the parameters Hbias,Hac, /H9275=2/H9266f, and K1, there are always four solutions, even though they can be pairwise degenerate. In addition, ﬁrst order per-turbation theory can be applied to determine the stability ofthe solutions as described in Ref. 7. This leads to the follow- ing stability conditions: • Stable: det A/H110220, tr A/H110210 • Unstable: det A/H110220, tr A/H110220 • Saddle: det A/H110210 with detA= /H9260eff2 1+/H92512/H20851/H92632−/H208491−mz2/H20850/H9263+/H90242mz2/H20852, /H208492/H20850 trA=−2/H9251/H9260eff 1+/H92512/H20873/H9263−1−mz2 2+/H9024mz /H9251/H20874,/H9263=bz mz+1 . III. RESULTS Figure 1shows the magnetization component Mz/H20849paral- lel to the anisotropy axis /H20850as a function of frequency of the circularly polarized ac ﬁeld for Hac=0.02 T, HK=1.1 T, Hext=0, and /H9251=0.02. For the numerical simulation, a fre- quency sweep was performed with frequency steps of /H9004fac =0.03 GHz. The Larmor precession frequency of the magne-a/H20850Electronic mail: werner.scholz@seagate.com.JOURNAL OF APPLIED PHYSICS 103, 07F539 /H208492008 /H20850 0021-8979/2008/103 /H208497/H20850/07F539/3/$23.00 © 2008 American Institute of Physics 103 , 07F539-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Thu, 18 Dec 2014 09:53:29tization in the anisotropy ﬁeld is fani=/H9253HK//H208492/H9266/H20850 =30.8 GHz. Thus, at frequencies above fani, the magnetiza- tion cannot follow the ac ﬁeld and remains close to the an-isotropy axis. As the frequency is reduced, the magnetizationcomes into resonance with the ac ﬁeld and tilts away fromthe anisotropy axis. At a frequency of 18.3 GHz, the preces-sion becomes unstable and the magnetization jumps/H20849foldover /H20850from solution 1 /H20849“analytical 1” /H20850to solution 2 /H20849“analytical 2 /H20850, which happens to be close to M z=1 /H20849cf. Fig. 1/H20850. This critical frequency is conﬁrmed by the stability con- ditions of the analytical model /H20851Eq. /H208492/H20850/H20852. However, for larger frequency steps /H9004fac, the foldover occurred at higher fre- quencies due to the disturbance introduced by the discontinu-ous change in the ac frequency. Subsequent increase of thefrequency keeps the magnetization following solution 2 untilit disappears at 27.8 GHz and the magnetization jumps tosolution 1 again. This behavior is quite remarkable because itshows an open hysteresis loop instead of the typical reso-nance dip. The latter can be observed for signiﬁcantlysmaller ac ﬁelds, for which solution 2 becomes unstable /H20849i.e., the ac ﬁeld is too weak to overcome the intrinsic damping /H20850. The third solution, not plotted in Fig. 1, lies between solu- tions 1 and 2 but it is unstable. Finally, the fourth solution,which is in fact always stable, is close to M z=−1 because the magnetization precesses in the opposite direction than the acﬁeld and, thus, cannot resonate. This chirality effect obvi-ously disappears for linearly polarized ﬁelds. However, dueto the loss of symmetry, there seems to be no analyticalsolution but numerical simulations with linearly polarized acﬁelds showed a qualitatively similar behavior with solution 1being shifted toward larger M zvalues due to the lower power of linearly polarized ﬁelds. In order to induce ferromagnetic resonance assisted switching /H20849e.g., from Mz/H110220t o Mz/H110210/H20850the parameters of the system should be chosen such that all solutions with Mz /H110220 become unstable. In this case, the magnetization would /H20849eventually /H20850switch, independent of initial conditions, ﬁeld ramps, ﬁeld timing issues, thermal perturbations, etc. Figure2shows the switching phase diagram for a single domain particle with H K=1.1 T, Hac=0.1 *HK=0.11 T, and /H9251=0.2 as a function of dc bias ﬁeld and ac frequency. The dc bias ﬁeldis applied along the anisotropy axis antiparallel to the mag-netization while the ac ﬁeld is applied perpendicular to theanisotropy axis. The phase diagram was obtained from 50/H1100350 simulations /H20849/H9004H bias=0.03 T, /H9004f=0.3 GHz /H20850.At low frequencies /H20849compared to the Larmor frequency in the local ﬁeld, i.e., f/H11270/H9253/H20849Hani+Hbias /H20850/2/H9266/H20850, the ac ﬁeld as- sists the switching process like a dc ﬁeld. Thus, switching can be expected if the effective ﬁeld /H20849calculated based on the Stoner–Wohlfarth astroid /H20850exceeds the aniosotropy ﬁeld: Heff=/H20849Hac2/3+Hbias2/3/H208503/2/H11022HK. On the other hand, at ac frequen- cies higher than the Larmor frequency, the magnetization cannot follow the ac ﬁeld and switching can be expected forH bias/H11022HKbecause the ac ﬁeld effectively cancels out. This behavior is indeed observed, as shown in the phase diagramin Fig. 2. However, for intermediate frequencies, true FMR as- sisted switching at dc ﬁelds below the Stoner–Wohlfarth ﬁeldcan be observed. The transition line between switching andnonswitching /H20849for increasing frequency on the left edge of the triangle in Fig. 2/H20850is very sharp and independent of the damping constant and various timing parameters /H20849ﬁeld rise times, dc before/after ac ﬁeld /H20850. The optimum /H20849switching at FIG. 3. /H20849Color online /H20850Switching phase diagram for ac frequency and ac ﬁeld amplitude with color coding of the switching time /H20849Hbias=0.77 T; no delay between ac and bias ﬁeld, 1 ns ﬁeld rise time of ac and bias ﬁeld,switching time measured from the beginning of the ﬁeld ramp /H20850. Conﬁgura- tions which do not switch within 3 ns /H20849or not at all /H20850appear white. FIG. 1. /H20849Color online /H20850Periodic solutions for the magnetization precessing around the anisotropy axis in a circularly polarized ac ﬁeld. For sufﬁcientlyhigh ac ﬁeld amplitudes, an open hysteresis loop /H20849as shown here /H20850is found. FIG. 2. /H20849Color online /H20850Switching phase diagram for a single domain particle with HK=1.1 T, Hac=0.11 T, and /H9251=0.2 as a function of dc bias ﬁeld /H20849x axis /H20850and frequency of the ac ﬁeld /H20849yaxis /H20850. The switching time is color coded /H20849nanosecond /H20850.07F539-2 W. Scholz and S. Batra J. Appl. Phys. 103 , 07F539 /H208492008 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Thu, 18 Dec 2014 09:53:29the lowest dc bias ﬁeld /H20850occurs at about fac=8 GHz and Hbias=0.6 T, which corresponds to a switching ﬁeld reduc- tion of 45% compared to the anisotropy ﬁeld of 1.1 T and areduction of 22% compared to the low frequency switchingﬁeld of 0.77 T. The optimum ac frequency to induce switch-ing is typically between 1 /2 and 2 /3 of the Larmor preces- sion frequency in the local ﬁeld /H20849H ani+Hbias /H20850. However, the transition line at the highest FMR assisted switching fre- quencies /H20849top edge of the triangle /H20850strongly depends on these parameters, especially the damping constant, which reducesthe area of the FMR assisted switching regime with increas-ing damping. For the phase diagram in Fig. 2, the ac and dc ﬁelds had a ﬁeld rise time of 1 ns and the dc ﬁeld wasramped up starting 2 ns after the ac ﬁeld. This, in fact, turnsout to be the most favorable conﬁguration because the slowramp of the dc ﬁeld /H20849while the ac ﬁeld is on /H20850effectively scans the whole range of dc bias ﬁelds and makes it morelikely that the resonance conditions, which induce switching,occur. In addition, the high frequency end of the FMR as-sisted switching triangle is not well deﬁned because there arestill stable unswitched states for the magnetization to go into.Thus, there is no sharp transition between switching andnonswitching areas and it seems very hard to control if themagnetization will switch or not. Figure 3shows the f ac/Hacphase diagram for a dc bias ﬁeld of 0.77 T. At low frequencies, we observe again thatswitching occurs when the effective ﬁeld exceeds the aniso-tropy ﬁeld /H20849atH ac=0.11 T /H20850. As the ac frequency increases, FMR assisted switching occurs and the optimum /H20849switching with the lowest ac ﬁeld /H20850is found at fac=6 GHz and Hac =0.043 T. In another aspect, Fig. 3shows that FMR assisted switching at low frequencies/low ac ﬁelds can be a slowprocess which takes many precessions of the magnetization/H20849and up to 3 ns from the start of the ﬁeld ramp /H20850to absorb the energy from the ac ﬁeld. Figure 4shows the relationship between damping and the ac ﬁeld amplitude required to induce switching. As theintrinsic damping increases, the required ac ﬁeld amplitudeincreases linearly until it saturates at high damping values when the effective ﬁeld reaches the anisotropy ﬁeld. Finally, Fig. 5shows a phase diagram for which the ac and dc bias ﬁelds were kept constant while the angle of theac and dc ﬁelds /H20849with respect to the anisotropy axis /H20850was varied. The smallest switching ﬁeld in conventional /H20849Stoner– Wohlfarth-type /H20850switching occurs at a ﬁeld angle of 45°. It seems reasonable to assume that the optimum ﬁeld angles forFMR assisted switching could be found at 90° with respectto the anisotropy axis for the ac ﬁeld /H20849for symmetry reasons /H20850 and at less than 45° for the dc ﬁeld /H20849to minimize the loss of symmetry of the dc ﬁelds while maintaining some angle as-sist /H20850. Figure 5shows that this is indeed correct since switch- ing with the smallest dc ﬁeld amplitude occurs at an ac ﬁeldangle of about 80° and a dc ﬁeld angle of about 25°. IV. CONCLUSION We studied ferromagnetic resonance assisted switching of single domain particles with numerical simulations. Weidentiﬁed three different regimes: angle assisted Stoner–Wohlfarth-type switching at low frequencies, unassistedStoner–Wohlfarth-type switching at high frequencies andFMR assisted switching at sub-Stoner–Wohlfarth bias ﬁeldsand ac ﬁeld frequencies between approximately 1 /2 and 2 /3 of the Larmor precession frequency in the local ﬁeld. Theamount of the reduction in switching ﬁeld depends on the acfrequency and amplitude, damping, ﬁeld ramps and timing.The lowest switching ﬁelds were found for dc ﬁelds at 25°and ac ﬁelds at 80° from the anisotropy axis. 1H. Dötsch, H. J. Schmitt, and J. Müller, Appl. Phys. Lett. 23,6 3 9 /H208491973 /H20850. 2A. M. Mednikov, S. I. Ol’Khovskii, V. G. Redko, V. I. Rybak, V. P. Sondaevskii, and G. Chirkin, Sov. Phys. Solid State 19, 698 /H208491977 /H20850. 3J. Artman, S. Charap, and D. Seagle, IEEE Trans. Magn. 19, 1814 /H208491983 /H20850. 4J. Zhu, MMM 2005 conference /H20849unpublished /H20850, Paper No. CC-12; J. Zhu, X. Zhu, and Y. Tang, TMRC 2007 conference /H20849unpublished /H20850, Paper No. B6. 5K. Rivkin and J. B. Ketterson, Appl. Phys. Lett. 89, 252507 /H208492006 /H20850. 6C. Thirion and W. W. D. Mailly, Nat. Mater. 2, 524 /H208492003 /H20850. 7G. Bertotti, C. Serpico, and I. Mayergoyz, Phys. Rev. Lett. 86,7 2 4 /H208492001 /H20850. 8H. K. Lee and Z. Yuan, Intermag 2006, San Diego, CA /H20849unpublished /H20850, Paper No. AD-04. 9Z. Z. Sun and X. R. Wang, Phys. Rev. Lett. 97, 077205 /H208492006 /H20850. FIG. 4. /H20849Color online /H20850Phase diagram for ac ﬁeld amplitude vs damping constant at fac=9 GHz and Hbias=0.62 T. The switching time is color coded /H20849nanoseconds /H20850. FIG. 5. /H20849Color online /H20850Phase diagrams for the effect of the ﬁeld angle of ac and dc bias ﬁelds /H20849measured from the anisotropy axis /H20850for dc ﬁeld amplitude of 0.45 T /H20849left /H20850and 0.35 T /H20849right /H20850. The switching time is color coded /H20849nanoseconds /H20850.07F539-3 W. Scholz and S. Batra J. Appl. Phys. 103 , 07F539 /H208492008 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Thu, 18 Dec 2014 09:53:29
1.3514070.pdf	Optimal control of magnetization dynamics in ferromagnetic heterostructures by spin-polarized currents M. Wenin, A. Windisch, and W. Pötz Citation: J. Appl. Phys. 108, 103717 (2010); doi: 10.1063/1.3514070 View online: http://dx.doi.org/10.1063/1.3514070 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v108/i10 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 03 Aug 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsOptimal control of magnetization dynamics in ferromagnetic heterostructures by spin-polarized currents M. Wenin,a/H20850A. Windisch,b/H20850and W. Pötzc/H20850 Institut für Physik, Theory Division Karl Franzens Universität Graz, Universitätsplatz 5, 8010 Graz, Austria /H20849Received 16 September 2010; accepted 8 October 2010; published online 30 November 2010 /H20850 We study the switching-process of the magnetization in a ferromagnetic-normal-metal multilayer system by a spin polarized electrical current via the spin transfer torque. We use a spindrift-diffusion equation /H20849SDDE /H20850and the Landau–Lifshitz–Gilbert equation /H20849LLGE /H20850to capture the coupled dynamics of the spin density and the magnetization dynamic of the heterostructure.Deriving a fully analytic solution of the stationary SDDE we obtain an accurate, robust, and fast self–consistent model for the spin–distribution and spin transfer torque inside generalferromagnetic/normal metal heterostructures. Using optimal control theory we explore the switchingand back-switching process of the analyzer magnetization in a seven-layer system. Starting from aGaussian, we identify a uniﬁed current pulse proﬁle which accomplishes both processes within aspeciﬁed switching time. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3514070 /H20852 I. INTRODUCTION Spin transfer torque in nanoscaled ferromagnetic/ normal-metal /H20849FN/H20850heterostructures has potential application for data storage and manipulation.1–3Apart from the experi- mental studies many theoretical investigations have beenmade since the pioneering work by Slonczewski andBerger. 4–8The problem to describe the physics in FN hetero- structures arises from the need to consider the dynamics ofthe conduction electrons as spin carriers and the dynamics ofthe localized magnetic moments in parallel and in differentregions of the heterostructure. The electron dynamics isfaster by several orders of magnitude than that of the latter. 9 Moreover, the spin dynamics in normal metal regions differssigniﬁcantly from that in ferromagnetic regions: the formeris characterized by fast diffusion and slow spin relaxation,while in the latter the opposite is the case. This time hierar-chies make it difﬁcult to provide a fully numerical solution.A Boltzmann-transport theory for magnetic multilayer sys-tems including the spin was developed by Valet–Fert. 10,11On the next level of approximation a drift-diffusion equationwas applied for mobile spins. 12The dynamics of the local- ized magnetic moments is governed by the Landau–Lifshitz–Gilbert equation /H20849LLGE /H20850, extended by additional spin trans- fer terms. A similar investigation has been performed forsemiconductor/ferromagnetic multilayers assuming ballistictransport, but using nonequilibrium Green’s functions. 13 While spin-torque assisted switching on magnetization is a vividly and thoroughly studied phenomenon, here we offera new and efﬁcient numerical approach for which is based onan entirely analytic solution to the spin drift-diffusion equa-tion /H20849SDDE /H20850, accounting for the three-dimensional spin den- sity vector, as well as self-consistency with the time-dependent magnetization. This allows for an efﬁcient self-consistent analysis of spin-torque phenomena in complexmagnetic heterostructures with piece-wise constant /H20849within each layer /H20850material parameters. Thus, main assumptions of our approach are the validity of the drift-diffusion equation/H20849i.e., classical diffusive transport rather than tunneling /H20850and the presence of magnetic monodomains in polarizers andanalyzers. The paper is organized as follows. In section II we present our model for FN multilayer system. Sections III andIV are devoted to the mathematical description of the mag-netization dynamics /H20849LLGE /H20850and the dynamics of the con- duction electrons /H20849SDDE /H20850, respectively. The exact solution of our SDDE is given, with details deferred to the Appendix.In section V we present numerical results for a symmetricseven-layer system. Optimal current pulse proﬁles to switchthe magnetization in a given time from parallel to antiparallelstate /H20849and in opposite direction /H20850is shown. Our results are compared with our fully numerical simulations to conﬁrmthe validity of our approach. Our results regarding criticalswitching currents versus switching time agree well with ear-lier work by others. 14,15 II. MODEL Our model of the heterostructure assumes three different physical building blocks: /H20849i/H20850the normal-metal leads and spacer layers, /H20849ii/H20850ferromagnetic polarizers, and /H20849iii/H20850ferro- magnetic analyzers. The leads and the spacer layers are cho-sen to be nonmagnetic /H20849N/H20850metals with equal material prop- erties. A lead is assumed to be inﬁnitely thick and serving asa spin bath with vanishing spin polarization. We describe awide ferromagnetic hard polarizer layer /H20849P 1,P2in Fig. 1/H20850as static and homogeneous. A thin ferromagnetic /H20849soft /H20850analyzer layer /H20849region A in Fig. 1/H20850is treated as a ferromagnetic mon- odomain described by a single time-dependent variable, aunit-vector m/H20849t/H20850pointing in the direction of the magnetization. 16,17The conduction spin-electrons are treated as classical magnetic moments moving in an external mag-netic ﬁeld created by localized magnetic moments in the fer-romagnet. The spin density S/H20849x,t/H20850is the dynamical variable a/H20850Electronic mail: markus.wenin@uni-graz.at. b/H20850Electronic mail: 06windis@edu.uni-graz.at. c/H20850Electronic mail: walter.poetz@uni-graz.at.JOURNAL OF APPLIED PHYSICS 108, 103717 /H208492010 /H20850 0021-8979/2010/108 /H2084910/H20850/103717/8/$30.00 © 2010 American Institute of Physics 108, 103717-1 Downloaded 03 Aug 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsto describe the spin distribution.18It is deﬁned for an isolated ferromagnet with magnetization direction mas S=nP/H6036 2m. /H208491/H20850 Here n=n↑+n↓is the free electron number density, where n↑,↓is the particle density with spin up/down, respectively, andP=/H20849n↑−n↓/H20850/ncorresponds to the spin density polariza- tion, extracted from the experiment.19In this work we use for the spin density the dimensionless quantity s=S//H20849n/H6036/2/H20850. For simplicity we do not consider spin-resolved quantities but use mean values instead /H20849diffusion constant, electric conduc- tivity, spin diffusion length etc. /H20850. III. MAGNETIZATION DYNAMICS A. Landau-Lifshitz-Gilbert equation The temporal evolution of the magnetization Mis gov- erned by the LLGE.16,20Using the saturation magnetization Ms, we deﬁne the quantities M=Msm,h=/H9253H, where /H9253is the gyromagnetic ratio and /H20849/H11509m//H11509t/H20850st=1 /Ms/H20849/H11509M //H11509t/H20850stto obtain an equation of motion for the dimensionless magnetization: dm dt=−1 1+/H92512m/H11003h−/H9251 1+/H92512m/H11003/H20849m/H11003h/H20850+/H20873/H11509m /H11509t/H20874 st. /H208492/H20850 Here h=han+hexis the effective ﬁeld containing the aniso- tropy ﬁeld and external ﬁelds measured in units of a fre-quency and /H9251the Gilbert damping constant. With a unit vec- tornwe set for the anisotropy ﬁeld han=/H9275ann/H20849m·n/H20850, /H208493/H20850 where /H9275anis the corresponding frequency. /H20849/H11509M //H11509t/H20850stdenotes the spin-transfer term,4,5,17 /H20873/H11509m /H11509t/H20874 st=/H9264m/H11003/H20849/H9004Is/H11003m/H20850. /H208494/H20850 Here/H9004Is/H11013/H20849Is/H20850in−/H20849Is/H20850outstands for the spin current absorbed inside the domain, whereas /H9264is a constant.21,22Without ex- ternal torque the equilibrium magnetization is either parallel/H20849P/H20850or antiparallel /H20849AP/H20850ton.B. Dipole ﬁeld In this paper we consider the control of the magnetiza- tion by spin currents only. So the only contribution to hex from the outside are the dipole ﬁelds originating from the polarizers. In order to obtain a simple result and a crudeestimate of the order of magnitude of the dipole ﬁelds weconsider a polarizer /H20849here written for P 1in Fig. 1/H20850as a cyl- inder with radius Rand thickness x1which is homogeneous magnetized and compute the ﬁeld at the position xm=/H20849x2 +x3/H20850/2. Evaluation of the general integral for a dipole density23we obtain /H20849/H20853ex,ey,ez/H20854is the canonical basis /H20850 Hd−d=1 4Msez/H20877xm−x1 /H20881R2+/H20849x1−xm/H208502−xm /H20881R2+xm2/H20878. /H208495/H20850 IV. DYNAMICS OF THE CONDUCTION ELECTRONS A detailed derivation of the balance equation for the spin density sj/H20849x,t/H20850is a many particle problem.24We use the phe- nomenological expression for the spin current density,18 jk/H20849x,t/H20850=−/H9262sk/H20849x,t/H20850E/H20849t/H20850−D/H20849x/H20850/H11612/H9254sk/H20849x,t/H20850. /H208496/H20850 jkis the spin current density for electrons with spin- polarization along the k-axis. /H9262is the electron mobility, which we assume as material-independent, and E/H20849t/H20850is the time-dependent electric ﬁeld. D/H20849x/H20850stands for the material- dependent diffusion constant and /H9254sk/H20849x,t/H20850/H11013sk/H20849x,t/H20850 −skeq/H20849x,t/H20850is the nonequilibrium spin density /H20849spin- accumulation /H20850,seq/H20849x,t/H20850is the space- and time-dependent equilibrium spin density. We compute the latter using the SDDE, as explained in the next section. Equation /H208496/H20850is in general valid for ferromagnetic as well as for nonmagneticmaterials. Because of /H11612·E=0 inside the metal, we obtain the spin drift-diffusion equation /H20849SDDE /H20850, 18,25 /H11509s /H11509t=/H20841e/H20841 ms/H11003B+/H20849/H11612D·/H11612/H20850/H9254s+D/H9004/H9254s+/H9262/H20849E·/H11612/H20850s +/H20873/H11509s /H11509t/H20874 sf. /H208497/H20850 /H20841e/H20841is the elementary charge and mthe electron mass. For the spin ﬂip term we make a spin-relaxation-time ansatz, /H20873/H11509s /H11509t/H20874 sf=−/H9254s /H9270/H20849x/H20850, /H208498/H20850 with the space dependent relaxation time /H9270/H20849x/H20850. For simplicity we assume an isotropic /H9270inside each layer. In Eq. /H208497/H20850the magnetic induction is related to the magnetization and anexternal ﬁeld trough B/H20849x,t/H20850= /H92620/H20849H/H20849x,t/H20850+M/H20849x,t/H20850/H20850 /H20849/H92620is the permeability of the vacuum /H20850. A. Spin and charge currents From now on we will consider quasi-one-dimensional systems along the x-axis, such as sketched in Fig. 1. Using Eq. /H208496/H20850we obtain for the spin current Is FIG. 1. /H20849Color online /H20850Geometry of the seven-layer system. The outside layers act as spin-carrier /H20849electron /H20850reservoirs with polarization P=0. The regions P1andP2are the two polarizers, and A is the analyzer layer whose magnetization is to be manipulated.103717-2 Wenin, Windisch, and Pötz J. Appl. Phys. 108, 103717 /H208492010 /H20850 Downloaded 03 Aug 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsIs/H20849x,t/H20850=−A/H20873/H9262E/H20849t/H20850s/H20849x,t/H20850+D/H20849x/H20850/H11509/H9254s /H11509x/H20849x,t/H20850/H20874. /H208499/H20850 Here Ais the cross-section of the sample, and E/H20849t/H20850=E/H20849t/H20850ex. In the drift-diffusion model the charge current density is given by26 j/H20849x,t/H20850=n/H20841e/H20841/H9262E/H20849t/H20850−D/H20849x/H20850/H11509n /H11509x. /H2084910/H20850 We assume homogeneity, /H11509n//H11509x=0 to obtain j/H20849t/H20850=n/H20841e/H20841/H9262E/H20849t/H20850. /H2084911/H20850 A related quantity is the drift-velocity vd, used for numerical computations, deﬁned by vd/H20849t/H20850=−j/H20849t/H20850/n/H20841e/H20841. For electrons j and vdhave opposite sign. vd/H110220,j/H110210 means electrons /H20849spin-carrier /H20850move in the positive x-direction. B. Equilibrium spin density Because we consider an arbitrary movement of the ana- lyzer magnetization vector m/H20849t/H20850we have a time-dependent equilibrium spin density /H20849we neglect spin pumping processes induced by moving magnetization17,27/H20850. For a ﬁxed time tthe equilibrium spin density seq/H20849x,t/H20850is space-dependent, with a ﬁrst order approximation /H20849isolated layers /H20850, seq/H20849x,t/H20850/H20841/H208491/H20850/H11013s˜=/H20902PB/H20849t/H20850 /H20841B/H20849t/H20850/H20841,x/H33528F; 0, x/H33528N./H20903/H2084912/H20850 This expression reﬂects our choice of dimensionless spin density sin Eq. /H208496/H20850and Eq. /H208497/H20850. To obtain the equilibrium spin density in the complete structure we use the generalstationary solution Eq. /H2084914/H20850presented in the next section, in which we use the ﬁrst order expression Eq. /H2084912/H20850fors eq/H20849x,t/H20850. We use the boundary conditions for transparent interfaces in F/N-junctions:28s/H20849x,t/H20850andIs/H20849x,t/H20850are continuous. For E/H20849t/H20850 =0 we obtain in second order seq/H20849x,t/H20850/H20841/H208492/H20850. In the following we omit the subscript /H20841/H208492/H20850. Note that, for noncollinear magnetic layers, all components of seq/H20849x,t/H20850/HS110050 in general. C. Stationary solution of the SDDE for constant parameters For a given layer, we consider Eq. /H208497/H20850for constant cur- rent, constant material parameters and time- and space-independent magnetic ﬁeld. We use s eq=s˜given by Eq. /H2084912/H20850. Setting /H11509s//H11509t=0 in Eq. /H208497/H20850, we have for a one dimensional structure the equation Ds/H11033/H20849x/H20850+/H9262Es/H11032/H20849x/H20850+/H9275s/H20849x/H20850/H11003b1−s/H20849x/H20850−s˜ /H9270=0 . /H2084913/H20850 Here we have deﬁned b1=B//H20841B/H20841, and B=/H20849m//H20841e/H20841/H20850/H9275b1, with /H9275 the Larmor frequency. The general solution of Eq. /H2084913/H20850is a quite lengthy expression, containing 6 integration constants,denoted as c 1...c6. To ﬁnd it we split s/H20849x/H20850into two parts, one part parallel to the magnetic ﬁeld, and the other perpendicu- lar to it,s/H20849x/H20850=s/H20648/H20849x/H20850+s/H11036/H20849x/H20850. /H2084914/H20850 We deﬁne an orthonormal, positive oriented basis /H20853b1,b2,b3/H20854. One ﬁnds for the parallel part /H20849where ld=/H9262E/H9270is the drift length with sign determined by E/H20850, s/H20648/H20849x/H20850=b1/H20877c1exp/H20875−x/H20851ld+/H20881ld2+4/H92612/H20852 2/H92612/H20876+c2exp/H20875 −x/H20851ld−/H20881ld2+4/H92612/H20852 2/H92612/H20876/H20878+s˜. /H2084915/H20850 s/H20648/H20849x/H20850does not depend either on /H20841B/H20849t/H20850/H20841or the saturation mag- netization. The second part is given by s/H11036/H20849x/H20850=b2/H20853c3G4/H20849x/H20850+c4G3/H20849x/H20850+c5G2/H20849x/H20850+c6G1/H20849x/H20850/H20854 +b3/H20853c3G3/H20849x/H20850−c4G4/H20849x/H20850−c5G1/H20849x/H20850+c6G2/H20849x/H20850/H20854. /H2084916/H20850 Here the functions Gi/H20849x/H20850,i=1,...4 are given in Appendix. They depend on the magnetic ﬁeld and the electric current, not indicated here to simplify the notation. Equation /H2084914/H20850 with Eqs. /H2084915/H20850and /H2084916/H20850present the complete solution of Eq. /H2084913/H20850used in our numerical simulations. We make the follow- ing remarks: /H20849i/H20850 The solution of of the SDDE for spin-orientation- dependent material parameters is straightforward. /H20849ii/H20850 Using this solution one can study different boundary conditions when linking layers. /H20849iii/H20850Because the solutions for spin densities parallel and normal /H20849to the magnetic ﬁeld /H20850can be separated, it is immediately possible to reﬁne the model using differ-ent times /H92701and/H92702for spin relaxation and dephasing. D. Validity of the quasistatic solution Here we develop a scheme to estimate the errors from our quasistatic approach. We use the stationary solution fromthe previous section to compute the spin density for a time-dependent current and magnetization vector. In general thisapproximation is valid as long as the variation of j/H20849t/H20850and m/H20849t/H20850is slow compared to the shortest relaxation time /H9270/H20849qua- sistatic time-evolution, QSE /H20850. A more rigorous estimate of the accuracy of the QSE in comparison with the solution ofthe full time-dependent equation is a nontrivial task. This isdue the different relevant processes and time-scales in thedifferent layers. To get a quantitative picture we set s/H20849x,t/H20850 =s qs/H20849x,t/H20850+/H9254sqs/H20849x,t/H20850, where sqs/H20849x,t/H20850denotes the quasistatic solution Eq. /H2084914/H20850and/H9254sqs/H20849x,t/H20850the deviation from the exact solution, denoted as s/H20849x,t/H20850. For /H9254sqs/H20849x,t/H20850we have inside a single layer the equation /H11509/H9254sqs /H11509t=/H20841e/H20841 m/H9254sqs/H11003B+D/H9004/H9254sqs+/H9262/H20849E·/H11612/H20850/H9254sqs−/H9254sqs /H9270 −s˙qs. /H2084917/H20850 The inhomogeneity is deﬁned as103717-3 Wenin, Windisch, and Pötz J. Appl. Phys. 108, 103717 /H208492010 /H20850 Downloaded 03 Aug 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionss˙qsª/H11509sqs /H11509jdj dt+/H20873dm dt·/H11612m/H20874sqs, /H2084918/H20850 and is the source for a nonvanishing /H9254sqs. Let us discuss the spin-relaxation in the ferromagnetic layers. Here the typicalrelaxation time is /H9270/H110151 ps and it is reasonable to neglect, in a ﬁrst approximation, the Larmor, diffusion, and drift terms.The Larmor term is of the order of 1 / /H9275, the diffusion is characterized by a time scale /H9270d=l2/D=/H9270/H20849l//H9261/H208502, where lis a characteristic ﬁnite length /H20849layer thickness /H20850. The drift term goes as /H9270j=l//H20841vd/H20841. For l=3 nm /H20849analyzer thickness as a worst case /H20850this gives /H9270d/H110150.3/H9270and /H9270j/H110150.03 ns for j /H11015108A/cm2. If we integrate Eq. /H2084917/H20850under this assump- tions we ﬁnd as a ﬁrst-order correction /H20849for a ferromagnetic layer /H20850, /H9254sqs/H208491/H20850/H20849x,t/H20850=−/H20885 0t dt/H11032e−/H20849t−t/H11032/H20850//H9270s˙st/H20849x,t/H11032/H20850. /H2084919/H20850 If we consider now a spacer layer /H20849x1...x2in Fig. 1/H20850and compare /H9270with/H9270d=/H9270/H20849l//H9261/H208502using the parameters given in Table Iwe can see that /H9270d/H11270/H9270. Diffusion is dominant in the spacer-layers. It occurs on a time-scale /H9270d/H1101510−3ps. In fact, this quite different time-scales in different layers are the rea-son why an integration of Eq. /H208497/H20850by a discretization- procedure used in usual PDE-toolboxes leads to numerical problems. To obtain an estimate of /H9254sqs/H208491/H20850/H20849x,t/H20850inside the spacer-layer we solve Eq. /H2084917/H20850with boundary-conditions given by Eq. /H2084919/H20850. Numerical results of this strategy to esti- mate the accuracy of the QSE will be given below. V. SEVEN-LAYER SYSTEM Figure 1shows the seven-layer structure, for which we apply the general formalism. We select this system becausesuch structures where used for low-critical currentexperiments. 14,15This allows testing of the present approach. As indicated in the ﬁgure we use two opposite alignedpolarizer-layers, polarizer P 1points in the + zand polarizer P2in the − zdirection. P1deﬁnes the parallel position of the analyzer A, where a small deviation from P1is needed for a nonvanishing initial-spin torque. Both polarizers have thesame material and geometric properties. As a consequencethe dipole ﬁeld Eq. /H208495/H20850vanishes exactly at the position of the analyzer A and the anisotropy ﬁeld Eq. /H208493/H20850produces two energetically equivalent stable positions /H20849degenerate two- level system /H20850. We expect and proof that, if a current pulse j/H20849t/H20850switches the magnetization from P →AP, then − j/H20849t/H20850does the inverse operation, AP →P.A. Numerical strategy All computations are done with the help of Math- ematica . We use the solution Eqs. /H2084914/H20850–/H2084916/H20850to compute the time- and space-dependent spin density inside of each layerfor given direction of Band current j. The solution of the total system requires the determination of all integration con-stants c 1...c36. Whereas the boundary conditions /H20849continuous spin density and spin current density /H20850are formulated analyti- cally, the solution is computed numerically as a function ofmandj. The spin current density and the spin torque in Eq. /H208492/H20850are then calculated self-consistently using Eq. /H208494/H20850. The last step requires the numerical solution of the LLGE, Eq./H208492/H20850. B. Optimized switching procedure We now address the switching of the analyzer magneti- zation /H20849for optimized switching using external magnetic ﬁelds see Ref. 29/H20850. We ﬁrst note that, due to the nonlinearity inmof the LLGE, it is impossible to identify a single current pulse proﬁle which switches both from P →AP and AP →P /H20849initial-state-independent switching /H20850. However, using the symmetry of the structure, one can identify a current pulseproﬁle which, when changing the current direction only, pro-motes both processes. To ﬁnd a simple pulse-shape which performs the desired task it is convenient to use an optimization procedure basedon a suitably deﬁned cost-functional J. 30We set J=/H20648m/H20849tf/H20850−mT/H20648,0/H11349J/H113492, /H2084920/H20850 where mTis the target magnetization and m/H20849tf/H20850is its actual value at the prescribed target time tf. We choose the time- dependent current as j/H20849t;X1,X2,X3/H20850=XAexp/H20877−XB tf2/H20849t−tf/2/H208502/H20878 +/H20858 l=13 Xlsin/H20849l/H9266t/tf/H20850, /H2084921/H20850 with three variational parameters X1,...X3/H20849one can use also more parameters. Global optimization algorithms howeverwork best with a few parameters /H20850. The Gauss-pulse, charac- terized by X A,XB, is selected by hand such that it is sufﬁcient to switch the magnetization from P →AP. It is used as a reference pulse. However, to steer the magnetization in theprescribed time additional current contributions are needed.The additional terms in Eq. /H2084921/H20850are constructed to ensure that at the end points of the control-time interval /H208510,t f/H20852the current vanishes for arbitrary X1,2,3. To ﬁnd the minimum ofTABLE I. Material-parameters used in the simulation. Mat./H9261 /H20849nm/H20850/H9270 /H20849ns/H20850Ms /H20849A/m /H20850/H9275 /H20849GHz /H20850 P Cu 450 0.024 0 0 0 Fe 5 0.001 17 /H11003105230 0.45 Py 5 0.001 8 /H11003105110 0.37103717-4 Wenin, Windisch, and Pötz J. Appl. Phys. 108, 103717 /H208492010 /H20850 Downloaded 03 Aug 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsJ/H20849X1,...X3/H20850a standard line search method or genetic algo- rithm can be used. C. System-Parameters We use the material-parameters typical for a Cu/H11009/Fe15/Cu3/Py2/Cu3/Fe15/Cu/H11009 /H20849in nm /H20850multilayer- system. The relevant material-parameters are listed in TableI. 19,31,32We use a material-independent electrical conductiv- ity and free-electron-density of n=84 /nm3. We obtain a mi- croscopic expression for the coupling constant /H9264given by /H9264 =−/H20841e/H20841n/H6036/2mM sd=−0.969 /d, where d=x3−x2is the analyzer thickness. For the Gilbert damping parameter we set /H9251 =0.01.33The direction of the anisotropy ﬁeld is chosen as n=/H208510,sin /H20849/H9272/H20850,−cos /H20849/H9272/H20850/H20852, with /H9272=0.9/H9266, and its modulus /H9275an =2 GHz. D. Results 1. Switching into constant current For a ﬁrst example we consider the dynamics of the seven-layer system in Fig. 1for a current that we switch onaccording to j/H20849t/H20850=j0/H208491−e−t/T/H20850, with T=0.5 ns. We have inte- grated the LLGE for different values j0, as shown in part /H20849a/H20850 of Fig. 2. Part /H20849b/H20850shows the z-component of the magnetiza- tion as a function of time. In all three cases the magnetiza-tion switches from P →AP, however, the lowest current leads to a switching time of more than 100 ns. These inves-tigations agree well with basic experimental results in theliterature: 14,15the critical current /H20841jc/H20841is of the order /H20841jc/H20841 /H11015106A/cm2, for the parameters chosen here, and depends on the saturation magnetization, Gilbert damping, and aniso-tropy ﬁeld. 16As seen in Fig. 2, switching into a constant spin-polarized current leads to damped oscillations of themagnetization vector. Above the critical current, they resultin a ﬂipping of the magnetization vector into the new /H20849AP/H20850 equilibrium position. For currents /H20841j 0/H20841/H11021/H20841jc/H20841one induces damped oscillations without switching. We should remarkthat the equilibrium-positions of mfor a constant /H20849spin- /H20850 current are no more given by the directions of /H11006n, but there is small deviation due to the spin current, however, not re-solved in Fig. 2. The seven-layer structure with antiparallel polarizer- orientations is crucial for the occurrence of low /H20841j c/H20841. Compu- tations for parallel polarizer orientations /H20849P1/H20648P2/H20850give vastly different critical currents for P →AP and AP →P ﬂips. Fig- ure3reveals the reason for this result. For antiparallel ori- entation of the polarizers the z-component of the spin density shows a large gradient inside the analyzer layer. As a conse-quence large spin currents can be generated compared toparallel oriented polarizers. In fact for a simpliﬁed modelwith vanishing dipole ﬁeld /H20849for sample radius R→/H11009/H20850and parallel polarizers the critical current is /H20841j c/H20841/H11022108A/cm2for this structure. As investigated, switching times for the ana-lyzer magnetization tend to decrease with increasing /H20841j 0/H20841.34 2. Optimal pulse-sequences We now consider the problem of switching of the mag- netization musing an optimized time-dependent electric cur- rent, where we set the switching time to tf=5 ns. The ﬁrst current pulse should switch the magnetization from P →AP. Initial and desired ﬁnal value of the analyzer magnetizationm/H20849t/H20850, respectively, are m/H208490/H20850=nandm/H20849t f/H20850=!mT=−n. /H2084922/H20850 A numerical minimization of Eq. /H2084920/H20850, limiting ourselves to the pulse shape Eq. /H2084921/H20850, gives as a result the ﬁrst pulse shown in Fig. 4. We stopped the computation when the cost10 20 30 40 50t/LParen1ns/RParen10.51.01.52.02.5/Minusj/Cross107/LParen1A/Slash1cm2/RParen1 10 20 30 40 50t/LParen1ns/RParen1 /Minus1.0/Minus0.50.51.0mza/RParen1 b/RParen1 FIG. 2. /H20849Color online /H20850/H20849a/H20850Electric current and /H20849b/H20850z-component of the mag- netization in the analyzer vs time. Associated quantities are plotted in thesame line style. For decreasing current the switching time increases. Theelectric current is plotted as − jaccording to a positive drift velocity. FIG. 3. /H20849Color online /H20850Equilibrium spin density for two different polarizer alignments. The dashed line corresponds to parallel orientations of the two polarizers, the solid line is for antiparallel orientation, as used in low-current spin-torque experiments. The analyzer is at the position m=n, and therefore, small perpendicular components of the spin density are present.103717-5 Wenin, Windisch, and Pötz J. Appl. Phys. 108, 103717 /H208492010 /H20850 Downloaded 03 Aug 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsfunctional was J/H110150.006. This means that the optimal control pulse, rather than relying on intrinsic Gilbert damping, ac-tively drives the magnetization precisely into the target stateAP; likewise for the back ﬂip, see Fig. 4. Note that the pulse shape is chosen such that the current is zero at the boundariesof the time interval. To ensure that the magnetization remainsin the AP state after the ﬁrst ﬂip, a few ns later we apply thesame pulse once more. Only a weak deviation from the equi-librium position in form of a few damped oscillations arevisible demonstrating stability, see Fig. 4. However when weapply the same pulse proﬁle with opposite current direction we switch the magnetization back from AP →P. In addition tom/H20849t/H20850we have plotted in Fig. 5the time-dependent spin density during the ﬁrst current pulse. The rows /H20849a/H20850and /H20849b/H20850, respectively, show the equilibrium spin density and its devia-tion from equilibrium inside the multilayer device. One ob-serves the degree to which the equilibrium spin density de-pends on the time-dependent magnetization m/H20849t/H20850: due to the choice of the magnetization of P 1andP2/H20849as collinear /H20850only FIG. 4. /H20849Color online /H20850/H20849a/H20850Optimized time-dependent electric current for switching the analyzer magnetization from P →AP and vice versa. Three 5 ns pulses with the same shape are applied. The ﬁrst switches from P →AP, the second is used to test stability, and the third switches back AP →P./H20849b/H20850z-component of the analyzer magnetization vector m/H20849t/H20850as a function of time. /H20849c/H20850Plot of the three-dimensional trajectory of m/H20849t/H20850during the ﬁrst /H20849solid /H20850and last pulse /H20849dashed /H20850. FIG. 5. /H20849Color online /H20850/H20849a/H20850The ﬁrst row shows the equilibrium spin density for the switching process P →AP for the ﬁrst pulse in Fig. 4. It depends on m/H20849t/H20850. /H20849b/H20850Nonequilibrium spin density induced by the electrical-current pulse. The computed values for /H9254sx,/H9254sy, however, are at the limit of the accuracy of the QSE.103717-6 Wenin, Windisch, and Pötz J. Appl. Phys. 108, 103717 /H208492010 /H20850 Downloaded 03 Aug 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsthez–component of the spin density shows signiﬁcant devia- tion from equilibrium. The order of magnitude of the devia-tion of x- and y-components is of the order of the error made by the QSE. The nonequilibrium spin density as function oftime is inﬂuenced by the actual position of m/H20849t/H20850and the current j/H20849t/H20850, as well as the magnetization of P 1andP2. 3. Error estimate We have used the results from the previous section to test the numerical validity of the stationary solution as dis-cussed in section IV D. Figure 6shows the estimate for the deviation of the z- component of the spin density in selected parts of the structure. The solid line is for the center of P 1, while the dashed line is for the center of the spacer layer to the left of the analyzer. The ﬁgure shows that /H20851/H9254sqs/H208491/H20850/H20852z, de- pending on position, is of the order of 10−5–10−4, compared with szeq/H110150.1–0.3 /H20849see Fig. 3/H20850. The dominant contribution in Eq. /H2084918/H20850comes from the moving magnetization, whereas the current contribution is negligible. For the other componentswe obtained similar results regarding relative errors. VI. CONCLUSIONS AND OUTLOOK We have presented a self-consistent model for magneti- zation switching by spin-polarized electric current in metallicferromagnetic heterostructures. Our method is founded uponan analytic solution of the stationary spin drift-diffusionequation /H20849SDDE /H20850for each layer using constant material pa- rameters, electric current, and magnetic ﬁeld. Matching lay-ers, using continuity of spin density and spin current densityat the interfaces as boundary conditions, we obtain an ana-lytic solution for the spin density of the entire heterostruc-ture. Making a quasistatic approximation in which the timedependence of the spin density depends on time solely viathe electric current and net magnetic ﬁeld, the time evolutionof the spin density is computed in parallel to the Landau–Lifshitz–Gilbert /H20849LLGE /H20850equation. Both equations couple via the spin torque effect and the time-dependent magnetizationin the SDDE. This method allows for an efﬁcient and robustmathematical description of the coupled carrier spin andmagnetization dynamics in metal/ferromagnet heterostruc-tures. Because the model is based on a completely analyticsolution of the stationary SDDE for given electric current and magnetic ﬁeld for each layer, it is applicable to hetero-structures of high complexity, for example for tilted polariz-ers or structures exposed to external magnetic ﬁelds. 35 We have demonstrated the efﬁciency of this semianalytic approach by investigating a seven-layer system with antipar-allel oriented polarizers, as studied in recent experiments,and computed optimized current pulses to switch the magne-tization from P →AP→P in speciﬁed time of 5 ns. As ex- pected for the system under investigation, the obtained cur-rent densities are in the range of 10 8A/cm2, with a critical current of about 106A/cm2. Using optimal control theory, we identify solutions for current proﬁles which allow forprecise switching in predetermined switching times. We pro-vide and discuss one example. Furthermore, a detailed investigation of the validity of the quasistatic time evolution of the SDDE is given. It con-ﬁrms excellent accuracy for the example of the simulatedseven-layer heterostructure. Several future applications of the presented formalism can be envisioned. A combined variation of material- andgeometric parameters to obtain optimal current pulses withlow critical currents. A description of thermal ﬂuctuationsusing temperature-dependent effective /H20849Langevin- /H20850ﬁelds in the LLGE /H20849via the spin torque in the SDDE /H20850and the search of “thermally robust” current pulses by averaging over manyﬁeld conﬁgurations. ACKNOWLEDGMENT We wish to acknowledge ﬁnancial support of this work by FWF Austria, project number P21289-N16. APPENDIX: STATIONARY SOLUTION OF THE SDDE Here we summarize the remaining analytic expressions for the stationary solution and constant material parametersas presented in section IV C. We use the dimensionless quan-tities /H9260ª/H9275/H9270and/H9267ªld//H9261. Further we deﬁne a=R e /H20851/H208814+/H92672+4i/H9260/H20852 =/H208814+/H92672 8+/H20881/H9260 2/H208751+/H208734+/H92672 4/H9260/H208742/H20876, /H20849A1/H20850 b=I m /H20851/H208814+/H92672+4i/H9260/H20852 =/H20881−4+/H92672 8+/H20881/H9260 2/H208751+/H208734+/H92672 4/H9260/H208742/H20876. /H20849A2/H20850 Using the auxiliary functions, F1/H20849x/H20850=e−/H9267x/2/H9261cos/H20873bx 2/H9261/H20874sinh/H20873ax 2/H9261/H20874, /H20849A3/H20850 F2/H20849x/H20850=e−/H9267x/2/H9261cosh/H20873ax 2/H9261/H20874sin/H20873bx 2/H9261/H20874, /H20849A4/H20850 F3/H20849x/H20850=e−/H9267x/2/H9261cos/H20873bx 2/H9261/H20874cosh/H20873ax 2/H9261/H20874, /H20849A5/H20850 FIG. 6. /H20849Color online /H20850Numerical estimate of the error within the QSE relative to an exact treatment of the SDDE. The inset shows the locations where we compute /H9254sqs/H208491/H20850. Inside the polarizer /H20849solid line /H20850we use Eq. /H2084919/H20850to estimate the deviation from the exact result, whereas inside the spacer-layer,Eq. /H2084917/H20850is integrated numerically.103717-7 Wenin, Windisch, and Pötz J. Appl. Phys. 108, 103717 /H208492010 /H20850 Downloaded 03 Aug 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsF4/H20849x/H20850=e−/H9267x/2/H9261sin/H20873bx 2/H9261/H20874sinh/H20873ax 2/H9261/H20874, /H20849A6/H20850 the four dimensionless functions Gi/H20849x/H20850, entering in Eq. /H2084916/H20850 are, as follows: G1/H20849x/H20850=/H20851−4a3/H9267/H9260+4a/H9267/H9260/H208494+3 b2+/H92672/H20850/H20852F1/H20849x/H20850+/H208514b3/H9267/H9260 +4b/H9267/H9260/H208494−3 a2+/H92672/H20850/H20852F2/H20849x/H20850−4/H9260/H20849a2+b2/H20850/H20849−4 +a2−b2−/H92672/H20850F3/H20849x/H20850−8ab/H20849a2+b2/H20850/H9260F4/H20849x/H20850,/H20849A7/H20850 G2/H20849x/H20850=a/H9267/H20851a4+/H208494+5 b2+/H92672/H20850/H208494−2 a2+b2+/H92672/H20850/H20852F1/H20849x/H20850 +b/H9267/H208515a4+/H208494+b2+/H92672/H208502−2a2/H2084912 + 5 b2 +3/H92672/H20850/H20852F2/H20849x/H20850−/H20849a2−b2/H20850/H20851/H20849a2+b2−/H92672−4/H208502 −4a2b2/H20852F3/H20849x/H20850+4ab/H20849a2−b2/H20850/H9260/H208514−a2+b2 +/H92672/H20852F4/H20849x/H20850, /H20849A8/H20850 G3/H20849x/H20850=/H2085124a2b/H9260−8b/H9260/H208494+b2+/H92672/H20850/H20852F1/H20849x/H20850+/H2085124ab2/H9260 +8a/H9260/H208494−a2+/H92672/H20850/H20852F2/H20849x/H20850, /H20849A9/H20850 G4/H20849x/H20850=/H20851−8a3/H9260+8a/H9260/H208494+3 b2+/H92672/H20850/H20852F1/H20849x/H20850+/H208518b3/H9260 +8b/H9260/H208494−3 a2+/H92672/H20850/H20852F2/H20849x/H20850. /H20849A10 /H20850 The integration of the normal component of Eq. /H2084913/H20850requires the solution of two second order differential equations. It isadvantageous to transform this two second order equationsinto four ﬁrst order equations and solve this system by ma-trix exponentiation. This procedure, after some simpliﬁca-tions, leads to the four functions G i/H20849x/H20850, which build the fun- damental solution. 1S. I. Kiselev, J. Sankey, I. Krirovotov, N. Emley, R. Schoelkopf, R. Bu- hrman, and D. Ralph, Nature /H20849London /H20850425, 380 /H208492003 /H20850. 2H. Dassow, R. Lehndorff, D. Bürgler, M. Buchmeier, P. Grünberg, C. Schneider, and A. van der Hart., IFF Scientiﬁc Report, 2004/2005, http:// www.fz-juelich.de/iff/datapool/iff2/sr2004.pdf . 3W. Pötz, J. Fabian, and U. Hohenester, Modern Aspects of Spin Physics ,Lecture Notes in Physics V ol. 712 /H20849Springer-Verlag, Berlin, 2006 /H20850. 4J. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 5L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 6M. Wilczy ński, J. Barnas, and R. Swirkovicz, Phys. Rev. B 77, 054434 /H208492008 /H20850. 7D. M. 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Phys. 77, 1375 /H208492005 /H20850. 18J. Fabian, Acta Phys. Slov. 57,5 6 5 /H208492007 /H20850. 19M. Ziese and M. J. Thornton, Spin Electronics , Lecture Notes in Physics /H20849Springer-Verlag, Berlin, 2001 /H20850. 20L. D. Landau and E. M. Lifschitz, Lehrbuch der theoretischen Physik, IX, Statistische Physik, Teil 2 /H20849Akademie Verlag, Berlin, 1975 /H20850. 21M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407 /H208492002 /H20850. 22Z. Li and S. Zhang, Phys. Rev. B 70, 024417 /H208492004 /H20850. 23J. D. Jackson, Classical Electrodynamics ,/H20849Wiley, New York, 1999 /H20850. 24C. Heide and P. E. Zilberman, Phys. Rev. B 60, 14756 /H208491999 /H20850. 25I. Žuti ć, J. Fabian, and S. D. Sarma, Phys. Rev. Lett. 88, 066603 /H208492002 /H20850. 26K. Seeger, Semiconductor Physics /H20849Springer-Verlag, Berlin, 1973 /H20850. 27T. Taniguchi and H. Imamura, Phys. Rev. B 76, 092402 /H208492007 /H20850. 28I. Žuti ć, J. Fabian, and S. D. Sarma, Rev. Mod. 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1.1657441.pdf	Rigorous Dynamic Analysis of ExchangeCoupled Films Y. S. Lin and H. Chang Citation: Journal of Applied Physics 40, 604 (1969); doi: 10.1063/1.1657441 View online: http://dx.doi.org/10.1063/1.1657441 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/40/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Aligned and exchange-coupled FePt-based films Appl. Phys. Lett. 99, 172504 (2011); 10.1063/1.3656038 Ultrafast laser-induced magnetization precession dynamics in FePt/CoFe exchange-coupled films Appl. Phys. Lett. 97, 172508 (2010); 10.1063/1.3510473 Effects of the thickness asymmetry of nanostructured exchange-coupled trilayers on their dynamic magnetization switching Appl. Phys. Lett. 93, 152507 (2008); 10.1063/1.3001805 Coupled precession modes in indirect exchange-coupled [ Pt ∕ Co ] – Co thin films J. Appl. Phys. 101, 09D115 (2007); 10.1063/1.2712320 Analysis of ExchangeCoupled Magnetic Thin Films J. Appl. Phys. 41, 2491 (1970); 10.1063/1.1659251 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Thu, 18 Dec 2014 18:06:29604 LEE ET AL. and, therefore, NbSe2 should be metallic. This model is consistent with the observed properties of WSe2 and NbSe2. It is not readily apparent, however, why there should be a change in the sign of the charge carrier in the half-filled band of NbSe2. Cohen et aU! have proposed a band model for the normal state of some superconductors with the (3- tungsten structure to explain the anomalous tempera ture dependence of a variety of properties. This band model could conceivably give rise to a p-to-n transition in any substance with a similar band structure. Their model requires two bands which slightly overlap or almost overlap. The ratio of the density of states (a) in the two bands must be greatly different from one (e.g., as in an s and a d band). If the Fermi level (EF) lies close enough to the region of overlap (or near over lap) so that a change in temperature could shift EF from one band to the other, then this could change the 11 R. W. Cohen, G. D. Cody, and J. J. Halloran, Phys. Rev. Letters 19, 840 (1967). behavior of the charge carriers in the magnetic field from electron like to holelike. Goodenough's model indicates that the two bands in the vicinity of the Fermi level are d bands and would be expected to have approximately equal densities of state. Therefore, it is not clear in this case which bands could be present in order to give a value of a greatly different from 1. Furthermore, since the Debye temperature is not known for NbSe2, it is not possible to make any definite cal culation using this model. It is apparent that at present not enough is known about the band structure of NbSe2. Further studies of the band structure and the Fermi surface are necessary for the complete understanding of the electrical proper ties of NbSe2. ACKNOWLEDGMENTS The authors wish to thank R. Kershaw for growing the single crystals of NbSe2 and R. Adams for analyzing for iodine content in the crystals. JOURNAL OF APPLIED PHYSICS VOLUME 40, NUMBER 2 FEBRUARY 1969 Rigorous Dynamic Analysis of Exchange-Coupled Films Y. S. LIN AND H. CHANG IBM Watson Research Center, Yorktown Heights, New York 10598 (Received 27 July 1967; in final form 16 October 1968) The Landau-Lifshitz equation for distributed film systems, which has been formulated by Chang, Lin, and Priver, is applied to analyze both the quasistatic and dynamic magnetization reversal in exchange coupled films. Each elementary layer of infinitesimal thickness is assumed to reverse magnetization by coherent rotation, subject to both magnetostatic and exchange forces of the other layers. As an example, detailed analysis is made on "dual uniaxial films," which consist of two uniaxial films of different anisotropy fields in intimate contact, and with their easy axes in parallel. Numerical solutions are obtained for mag netization distribution and switching modes, as functions of film parameters and driving fields. Unique in dual uniaxial films, simultaneous switching, sequence-field switching, and separate switching occur in films with strong, moderate, and weak exchange couplings respectively. A convenient method of con structing the critical curves and a comprehensive classification of switching characteristics are given. Finally, a critique is made of Goto's quasistatic analysis based on lumped-constant approximation. I. INTRODUCTION Since the observation in 1964 by the Grenoble groupl that exchange interaction between separate but proxi mate magnetic layers modifies the usual uniaxial switch ing curves, and affects domain configuration, much interest has been stimulated in studying film structures involving exchange coupling. Generally in multilayer exchange-coupled films, the magnetization orientation is nonuniform along the film thickness in quiescent stable states; and/or during switching. Three types of exchange-coupled films have been reported in the literature: (1) structures with two or more uniaxial films of different anisotropy constants either in suffi cient proximity! or in intimate contact;2 (2) structures 1 J. C. Bruyere et at., IEEE Trans. Magnetics MAG-I, 174 (1965) . 2 E. Goto et al., J. Appl. Phys. 36, 2951 (1965). with the orientation of uniaxial easy axis varying along the thickness direction;3-6 and (3) structures with the magnitude of magnetization or exchange constant varying along the thickness direction.7 It should be noted that even in films with homogeneous· property, non uniform field along the thickness direction can be created by sending current through the film,8 thereby bringing exchange force into play. The essential feature of the exchange-coupled films is that the magnetization vector M everywhere is sub jected, not only to the applied field, anisotropy field, 3 w. T. Siegle, J. Appl. Phys. 36, 1116 (1965). 4 N. Hayashi and E. Goto, J. Appl. Phys. 37, 3715 (1966). 5N. Hayashi, JapanJ. Appl. Phys. 5, 1148 (1966). & D. A. Thompson et al., J. Appl. Phys. 37,1274 (1966). 7 W. P. Lee and D. A. Thompson, IEEE Trans. Magnetics MAG-4,520 (1968). 8 S. Methfessel et al., J. Phys. Soc. Japan 17, 607 (1962). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Thu, 18 Dec 2014 18:06:29DYNAMIC ANALYSIS OF COUPLED FILMS 605 and magnetostatic field, but also to the exchange inter action, on account of the nonuniform magnetization distribution. The nonuniform magnetization distribu tion results in a spatial integral of magnetization orien tation for the expression of total free energy. Hence in the study of quasistatic magnetization reversal, which is based on energy minimization principle for determining stable and critical states, calculus of variations6 is em ployed and a second-order nonlinear differential-inte gral equation is obtained. A detailed analytical treat ment appears extremely complex. Even with numerical solution, it is still necessary to confirm the stability of each equilibrium state and to select the correct switch ing path as a critical state is reached. Simplified solu tions such as lumped-constant approximation1,2 and small-angle approximation3-5 often omit essential features of the problem. The purpose of this paper is twofold. One aim is to present the method of rigorous analysis of exchange coupled films in general by using the magnetodynamic Landau-Lifshitz equation for distributed film systems which has been formulated in an earlier paper.9 The treatment is "dynamic," "distributed," and of large angle. It yields both dynamic (such as switching speed) and quasistatic (such as stable and critical states) magnetization reversal characteristics. The unambig uous selection of the correct switching path is auto matic with the transient (dynamic) solution. The second aim is to delineate both dynamic and quasistatic magnetization-reversal characteristics, spe cifically for dual uniaxial films. The structure consists of two uniaxial films of different anisotropy fields in intimate contact with their easy axes in parallel. Various magnetization reversal modes, such as simultaneous switching, sequence-field switching, and separate switching are observed respectively in structures with strong, moderate, and weak coupling between the two films. A convenient method of constructing the critical curves and a comprehensive classification of switching characteristics are described. Finally, the results are compared with those of a preliminary analysis by Goto et al.,2 based on quasistatic lumped-constant approxi mation, by examining the validity of their assumptions, which have led to some misleading conclusions. II. PHYSICAL PRINCIPLE AND MATHEMATICAL MODEL The stable equilibrium state of a magnetization vec tor (or its stable orientation) corresponds to a relative minimum energy. The stable orientation varies with the applied field H. The variation is continuous until the field H reaches a critical value, then the magnetiza tion will change discontinuously to achieve a new minimum-energy orientation. In the quasistatic studies, the stable and critical 9 H. Chang, Y. S. Lin, and A. Priver, J. App!. Phys. 38, 2294 (1967). states are determined by the minimization of the total free energy G with respect to the internal coordinates (such as direction cosines or polar angles) of the mag netiza tion vector, with the external field H as parameter. In the dynamic studies, the equation of motion of magnetization vector M is determined from the Hamilton's principle1o; viz., the path of motion will be such that the time integral of the difference between the kinetic and potential (free) energies is minimum. Under the constraint of constant magnitude ofM, the equation of motion can be derived from the Lagrange formulation by recognizing the analogy between the motions of a magnetization vector and a gyroscope. The equation of motion can be alternatively derived by considering the motion of the magnetization vector subject to various torques resulting in the Landau-Lifshitz, or Gilbert, equations. 9 It has been shown that the Lagrange formu lation yields an equation of motion identical to that depicted by the Landau-Lifshitz (or Gilbert) equation after the latter has been expanded into scalar formY In the present paper, the general magnetodynamic Landau-Lifshitz equation for distributed film ?ystems is applied to analyze the exchange-coupled films. The equation, which has been formulated in an earlier paper,9 expresses the equation of motion for M in the form of two coupled nonlinear differential-integral equations. The treatment is not only "distributed," but also of large angle. The magnetization reversal within each elementary layer of infinitesimal thickness is assumed to follow a simple coherent rotational model. The interaction between layer and layer is through both the magnetostatic coupling among all layers and the exchange coupling between adjacent layers. The equation of motion of M is expressed in terms of the total free energy "density" instead of effective field "in tensi ty." The magnetization orientation is described in terms of two polar angles cp and if;. The azimuthal angle cp extends between the projection of M on film plane (X-Y) and the reference axis (x), while the polar angle (ni2-if;) extends between M and the normal to the film plane (Z). The direction cosines of M along x, y, and z axes are then cosif; cosrjJ, cosif; sinc/>, and sinif;, respectively. The total free energy "density," which consists of anisotropy energy (EK) , exchange energy (EA) , self-magnetostatic energy (EM)' and applied field energy (Eo), can then be expressed as functions of external field (Ho), film parameters, and the two polar angles of the magnetization orientation. The uniaxial anisotropy energy density is given by where K is the uniaxial anisotropy constant, and C/>O is the easy axis orientation with respect to the X axis. 10 Y. S. Lin, Ph.D. thesis, Carnegie Institute of Technology. Pittsburgh, Pa. (1967), p. 64, 11 Ref. 10, Chap. 3. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Thu, 18 Dec 2014 18:06:29606 Y. S. LIN AND H. CHANG The exchange-energy density is EA=A(z)[(aNaz)2+ cos2t/!(acjJ/az)2], where A is the exchange constant. (2) The self-magnetostatic energy density is obtained by considering the film as composed of many elementary layers, each of which is approximated by a flat ellipsoid of thickness dz9: EM = (JJ.o/2) M2(Z) sin2t/!+ (JJ.o/2)(N a/ D)M (z) cost/! coscjJ· f M (z') cost/! coscjJdz' +CJJ.o/2)·(N b/D)·M(z) cost/!sincp.j M(z') cost/!sincjJdz', (3) where Na and Nb are the demagnetizing factors along the easy and hard axis, respectively, for an ellipsoid with thickness D. The applied field-energy density due to an external field Ho=aJI,,+~ is given by: EO=-JJ.OM·H o = -JJ.oMH x cost/! coscjJ-JJ.oMH y cost/! sincjJ. (4) The equation of motion for M (z, t) is described by the following two coupled nonlinear-integro-differential equations (in mkS)9: iJV;/at= (I 'Y 1/JJ.oM cost/!)· {(aE/acp) -(d/dz) [aE/a(acf>/az)]} -(VJJ.oM2). {(aE/at/!) -(d/dz) [aE/a(at/!/az) ]}; (S) acp/at= -(I 'Y 1/JJ.oM cost/!).{(aE/at/!) -(d/dz)[ (a~ )]} -(VJJ.oM2CoS2tJ!){(aE/acp) -(d/dz)[ aE ]}: (6) a 0tJ! az a (acjJ/az) . with the boundary conditions: acjJ/az=at/!/az=o at film outer surfaces, and [A (acjJ/az) ] 1.=.o+=[A (acjJ/az) ] 1_'0- [A (aNaz) ] [z=zo+=[A (aNaz)] 1'='0- at interface z=zo, t7) (8) where E is the total free-energy density equal to i!."K+ EA+EM+Eo, 'Y is the gyromagnetic ratio (= -2.21X IOS/ At/m·sec) , X is the damping constant, and JJ.o is the permeability of free space. As iJV;/at~, and acf>/at~, Eqs. (5) and (6) reduce to Euler equations and are equivalent to Brown's micro magnetic equations.12 The analytical solution of Eqs. (5) and (6) appears extremely complicated and is not considered further in the present paper. The numerical approach, however, is relatively easy and has been discussed in Ref. 9, includ ing considerations such as time-saving subroutine, com putational stability, and determination of initial con ditions. The computer program begins by entering the input parameters such as A, K, D, and the driving-field function. The dynamic magnetization reversal is readily obtained by monitoring the change of M(dcjJ/dt and 12 W. F. Brown, Jr., Muromagnetics (Interscience Publishers, Inc., New York, 1963), p. 48. dt/!/dt) and the magnetization orientation (cp and t/!) for distance incremental throughout the film thickness and also for time incremental. The studies of quasistatic magnetization reversal, however, require an extremely slow-varying field, sO that there will be no dynamic effect (damping). The program is useful in solving for magnetization distribution, critical curves, various switching modes and switching speed. As a specific example, it will be applied to dual uniaxial films in the remaining part of the paper. The structure consists of a hard film (with anisotropy constant K" or anisotropy field Hkh, ex change constant A", and thickness D,,) and a soft film (with K., Hk., A., and D.) in intimate contact with their easy axes in parallel along x axis (see Fig. 1). To facili tate analysis, both soft and hard films are assumed to have the same saturation magnetization (M =8XIOS At/m, or 471'M=1()4 G), and damping constant (X= 6.28X109/sec, or X=SXIOS G/Oe·sec). III. STABLE STATES AND MAGNETIZATION DISTRIBUTION The switching occurs in dual uniaxial exchange coupled films in essentially the same manner as in a single uniaxial film, viz., by gyromagnetic precession. In response to an applied field in the film plane, the magnetization in each elementary layer of infinitesimal thickness begins to precess around the applied field and out of the film plane. The normal component of mag netization then results in a large demagnetizing field [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Thu, 18 Dec 2014 18:06:29DYNAMIC ANALYSIS OF COUPLED FILMS 607 antiparallel to it, which in turn causes the planar component of M to precess around it, and thus rotate in the film plane toward the applied-field direction. On account of the inhomogeneity of magnetic pro perties (specifically Hk), there will be a nonuniform magnetization distribution along the direction normal to film plane. Hence, in addition to the torques due to the anisotropy field and applied field, to which a single domain film is subjected, the magnetization vector in each elementary layer is also subjected to the torque due to the exchange interaction. The stable magnetiza tion distribution is realized as the "total" torque acting on M, and is everywhere zero. A pictorial drawing and a detailed configuration of the magnetizaton orientation of dual uniaxial films in the presence of an external field are depicted in Figs. 1 and 2 respectively. Note that the strong demagnetizing field normal to the film plane limi ts M everywhere to precessing only a few degrees (1/;":;' 2°) away from the film plane during switching and to staying in the film plane (1/;=0°) at stable equilibrium. Hence the stable magnetization distribution can be represented by IjI alone. Figures 2 (a) and 2 ( c) show the magnetization orientation throughout the film under the influence of a hard-direction field (=3H k.). Figures 2 (b) and 2 (d), however, show only the magnetization orientation at both outer surfaces for various hard-direction fields. In one case [Figs. 2(a) and 2(b)], the hard-film thick ness is kept constant [Dh=0.5(A./K.)1/2] while the soft film thickness is varied. In the other case [Figs. 2 (c) and 2 (d)] the soft-film thickness remains unchanged [D. = (A./K.)l/2] while the hard-film thickness is varied. The essential features of the stable magnetization distribution are described below: (1) The magnetization varies its orientation con tinuously along the film thickness [see Figs. 2(a) and 2 (c) J. The gradient of magnetization orientation (dcp/dz) also varies continuously, and conforms to the boundary conditions as well. It changes rapidly but continuously at the interface between two films, but is zero at both outer surfaces to satisfy the boundary conditions stated in Eqs. (7) and (8). (2) The magnetization twist occurs only within a fraction of the soft layer thickness, when it exceeds a critical value. The critical thickness per radian of magnetization twist is given as (A./ K.) 1/2, where A. is the exchange constant, and K. is the anisotropy con stant. It is of interest to note that both the width of a 180° Bloch wall in bulk materials and the critical thick ness for helical magnetization in a half-turn helical anisotropy film6 are on the order of 7r(A/ K) 1/2. For dual uniaxial films with thickness much greater than 7r(A./K).li2, multiple-tum magnetization twist may be induced under rotating-field excitation. (3) Adequate thickness of the soft film does not ensure the occurrence of magnetization twist. At the INITIA~ Z APPLIED FIELD SOFT LAYER As.Ks ",~~L Y HARD AXIS FIG. 1. Schematic drawing of dual uniaxial films and its magnetiza tion distributIon under the excitation of an external field. same time, the hard-film anisotropy energy (K"Dh) must be sufficiently large so that the hard film cannot be switched by the combined torque due to the applied field and exchange field from the soft film [see Fig. 2 (c)]. The hard film can provide an anchoring action for the soft-film magnetization adjacent to the interface. (4) Although the nonuniform magnetization may result in both short-range exchange coupling and long range magnetostatic coupling, in general the former overpowers the latter. For a circular film with a thick ness to diameter ratio less than 10-3, the magnetostatic (demagnetizing and stray) fields have negligible effect on magnetization distribution. With the above qualitative picture in mind, we may proceed to explore the dependence of magnetization distribution on material parameters, geometrical param eters and excitation conditions. Of the material param eters (A, K, and M), K is much more readily con trolled than A and M. For instance, by varying perm alloy composition, or angle of incidence in evaporation, or the use of sequenced field, Hk may be varied from a half to tens of oersteds, a range of more than one hundredfold. On the other hand, the variation of ex change constant and saturation magnetization are confined to a much smaller range.13 We therefore select the thickness ratio D./D" (s=soft, h=hard) and aniso tropy-energy ratio K"D,,/ K.D. as the variable param eters in investigating the magnetization- distribution variation of dual uniaxial films. 11 R. Kimura and H, Nose, J. Phys. Soc. Japan 17, 604 (1962). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Thu, 18 Dec 2014 18:06:29608 Y. S. LIN A_N D H. C HAN G lT/4 ,"/4 OL-~O-----LI ----2~--~--~ ~--~L---~--------~o FIG. 2. Magnetization-orientation distribu tion as a function of film thickness and external field (AA=A.) Kh=lOK,). (a) and (c) Mag netization distribution (q,) vs Z as applied field=3Hk •. (b) and (d) magnetization orienta tion at top and bottom surfaces vs hard direction field. z/(As/Ks)1I2 OL----oL---~----------~ (c) Figure 3, based on the numerical solution of Eqs. (5) and (6), serves both as an index of the areas of our in vestigation, and later on as a summary of the results. Along the line Kh=K., single-film behavior should be observed. The region below this line is physically un realizable, since by definition KhC"K •. When D./ Dh becomes very large (upper region), the behavior of the dual films is predominantly that of the soft film. When KhDh/ K.D. becomes very large (right region), the be havior of the dual films is predominantly that of the hard film. The interesting dual-film behavior therefore should be sought in the region bounded by the above three regions. In general, films with large D. or large KhDh/ K.D. permit large magnetization twist and result in weak exchange coupling between the soft and hard films. Various magnetization reversal modes, such as simul taneous switching, sequential switching, and separate switching, prevail in regions of strong, moderate, and weak couplings respectively. These different switching modes will be discussed in Sec. IV. The range of param eters studied by Goto et at.2 is also indicated in Fig. 3 by assuming the identity between the true soft-film thick ness D. and the effective mean value D. used by Goto et at. Discussion of Goto's analysis and comments on its shortcomings will be given in Sec. V. IV. CRITICAL STATES AND SWITCHING MODES The stable state of a magnetization vector M corre sponds to a relative minimum energy. As the applied field H is increased from zero, the magnetization orien tation, starting from a given quiescent state, will adjust continuously to reach a different relative minimum energy. However, when the applied field is sufficiently different in orientation from the quiescent magnetiza tion direction, and is increased beyond a critical value, 10 (b) ,"/4 10 a discontinuous and irreversible, rather than continuous and reversible, change in magnetization orientation is necessary in order to reach a new relative minimum energy. This phenomenon is called irreversible rota tional switching, and the field is called critical field. The locus of Critical fields in the Hz-Hy coordinate plane is called a critical curve or rotational-threshold curve (H=axHx+ayHy with the x-v plane being the film plane). The curve usually consists of branches corresponding to different quiescent magnetization orientation. In this section we will present the critical curves of dual uniaxial films, obtained from the numeri cal solution of the magnetodynamic equation. The method, which we consider to be universal, and superior GOTO GOTO CASE 2 CASE 3 0.5 REGION A,PHYSICAlLY IMPOSSIBLE B'STRONG COUPLING. SIMULT. SWITCHING " CMOOERATE COUPLING, '\.. ~~9¥t~I~~-FlELD "'-DWEAK COUPLING, D "'-SEPARATE SWITCHING " "'-'\.. '" /Q '\b , 50 100 FIG. 3. Schematic of various switching modes, and magnetiza tion twist as a function of film parameters. A.= Ah, D.= r(A./K.)1f2. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Thu, 18 Dec 2014 18:06:29DYNAMIC ANALYSIS OF COUPLED FILMS 609 to conventional quasistatic method, will be described in detail in Appendix 1. It is well known that for a thin film with uniaxial anisotropy, the critical curve has the shape of an astroid. In single films with higher-order anisotropies (such as biaxial anisotropy), the branches of critical curves may crisscross with each other. However, for field increasing in a given direction, the irreversible switching only occurs once. Even in magnetostatically coupled films, the coupling causes the two films to switch simultaneously, and irreversible switching can only occur once in response to field increasing in a given direction. As we shall see presently, when two films are not too strongly coupled by exchange forces, it is possible to cause irreversible switching in the two films to occur separately at different critical fields. This feature is unique in exchange-coupled films, and cannot be found in single or magnetostatically coupled films undergoing rotational-magnetization reversal. Depending on the strength of exchange coupling, dual uniaxial films ex hibit at least three types of switching modes (see Fig. 3) : (1) Strongly coupled films: When the soft-film thick ness (DB) is below the critical value, or when the aniso- 5 CURVE A -HARD FILM THRESHOLD B -SOFT FILM THRESHOlD C -DUAL FILM THRESHOLD (01 ~1~ ______ ~c=J_H_O~ ____ ~~ ____ ___ He (bl 0 m m+1 n n+1 B (e) ~~----~~~~--~~---m~ n .n+1 l:rr <" ~ otL ______ ~~~-,~--~-··.~.~--_- __ ~~~--_-L_ __ . __ _ m m+1 n n+1 TIME (tiT) REVERSIBLE IRREVERSIBLE SWITCHING SWITCHING FIG. 4. Determination of critical curve for strongly coupled films. (A.""Ah, KA=5K., D.=2D1= (A./K.) 1/2. (a) Critical curves for soft film, hard film, and dual uniaxial films; simul taneous switching. (b) Pulse field waveform; magnitude is in creased by steps of 0.2 Hk. per :pulsej pulse width T= 10 nsec. (c) Easy-axis sense signal dM./dt (relative magnitude). (d) Surfaces magnetization orientationj solid line: top (soft film) surface; dashed line: bottom (hard film) surface. 20 15 \ '" \ CURVE A-HARD FILM THRESHOLD 8-S0FT FILM THRESHOLD C-DUAL FILM THRESHOLD FOR BOTH FILM SWITCHING D-DUAL FILM THRESHOLD FOR SOFT FILM SWITCHING flO \ A '<: ""~. SEPARATE SWITCHING ';.. I Ho 5 (0) (b) Ht __ .c~~~-~- __ r::,,--,/-,-,/-,--(~d)~ m m+1 n n+1 p p+1 SIMULTANEOUS REVERSIBLE SWITCHING TIME (tiT) SEPARATE SWITCHING SIMULTANEOUS IRREVERSIBLE SWITCHING FIG. 5. Determination of critical curve for weakly coupled films [A.""AA, KII.",,20K., D.",,10D h=5(A./K.)1/2]. (a) One-film and two-film irreversible-switching threshold. (b) Separate switching mode occurs in shaded area. Pulse-field pattern; O.2Hka/pulsej T= 10 nsec. (c) dM~/dt vs t. (d) "'top and "'bottom VS t. tropyenergy (KhD,,) for the hard film is not sufficient to anchor the hard-layer magnetization, only small magnetization twist (::;71"/4, see Fig. 3) is permitted. The constituent films, when strongly coupled, will switch simultaneously, and irreversible switching can only occur once in response to field increasing in a given direction, resulting in a single critical curve [Fig. 4(a)]. The magnetization distribution in quiescent state is uniform. From symmetry considerations, there are two such quiescent states. The films exhibit distorted astroidal critical curves elongated in the hard direction and bounded within the astroids of the two constituent films. (2) Weakly coupled films: When the soft film is sufficiently thick, and the hard film has sufficiently large anisotropy energy, large magnetization twist G:: 71", see Fig. 3) is possible. The two films, when weakly coupled, undergo irreversible switching separately at different critical fields in a given field direction, resulting in two separate critical curves [Fig. 5 (a) J. The quiescent magnetization distribution may be either uniform as a consequence of irreversible switching for both films, or helical as a consequence of irreversible switching for the soft film. From symmetry considerations, there are a total of four (two uniform, two helical) quiescent magnetization states. The critical curves are generally astroids elongated along the hard direction and bounded [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Thu, 18 Dec 2014 18:06:29610 Y. S. LIN AND H. CHANG >:I: x :I: "r\ CURVE A -HARD FILM THRESHOLD B -SOFT FILM THRESHOLD C -DUAL FILM THRESHOLD DUE TO UNIPOLAR HX,Hy D -DUAL FILM THRESHOLD \ 4 \ o "A " " "- "-, DUE TO UNIPOLAR Hx , BIPOLAR Hy P(Hx,Hy)', SEQ -FIELD' "- SWITCHING ---___ [ ~ 4 6 Hx/Hks ~--\-~~-----y--o I _______ J (0) L~/----Ll- :::J ___ ,--- (b) o 5 10 15 20 25 I, I \ I , ~-\ o ~~(C) (d) o 5 10 15 20 25 TIME (ns) FIG. 6. Determination of critical curve for moderately coupled films. (Ah= 4.= 1()-6 erg/em, HkA=20 Oe, H/u=3 Oe, Dh= 810 A, D.= 10 000 A). (a) Sequence-field switching mode prevails in shaded areas. Curve C is identical to that generated from a radial field. (b) Sequence-field waveform. (c) dM./dt vs t. (d) "'top and "'bottom VS t. within the two astroids of the two constituent films. As the exchange coupling is reduced, the two critical curves of the dual films approach those of the constituent films. (3) M odemtely coupled films: When the soft-film thickness and hard-film anisotropy energy are in be tween the values for the above two extreme cases, moderate magnetization twist between 71"/4 and 71" will result (see Fig. 3). In this range, both films undergo irreversible switching simultaneously, similar to the strongly coupled case. But the magnitude of the critical field is dependent on the driving field trajectory. For instance, a pulse sequence with overlapping unipolar hard-and easy-direction fields results in critical curve identical to that generated from a radial field. A pulse sequence with unipolar easy-direction and bipolar hard direction field, however, reduces the irreversible switch ing threshold, [see Figs. 6(a) and 6(b)]. Note that the above phenomenon is not due to the dynamic effect. V. A CRITIQUE OF GOTO'S LUMPED-CONSTANT APPROXIMATION The analysis based on the distributed model reveals two major differences from the results of Goto et al.2 (1) Regardless of film parameters, the critical curve always resembles an astroid with various degrees of elongation along the hard direction. The infinite elono-a tion, open loop in the hard direction, or the multi~le bends in the critical curve as predicted by Goto et al. hav~ not been observed in the present analysis (see Section IV). (2) For structures with moderate coupling strength between soft and hard films the threshold field is reduced by the use of sequence' field instead of field applied in one given direction (see Fig. 6). How ever, the sequence field is not absolutely essential in achieving irreversible switching as implied by Goto et at. T.he assumptions and the predicted switching be haVIOrs of the approximate analysis by Goto et at. are summarized below. They will be analyzed to reveal how the discrepancy in the predictions arise. (1) For the soft layer, an average magnetization orientation <\>. znd an effective thickness D. are as sumed. The free-energy function E. can then be sim plified from an integral form iDs E.= [A.(dl/>/dz)2+K. sin21/>-MH z cosl/> o -MHy sin<p]dz (9) to a function of two variables E.=D.{A.[(<\>.-<p,,)/D.]2+K. sin2<\>. -MHz cos<\>.-MH y sin<\>.j, (10) where 1/>" is the magnetization orientation at the inter face (or of the hard layer) . (2) For the hard layer, it is assumed that Kh ap proaches infinity while D" approaches zero. Hence the magnetization is uniform and may be represented by a single angle <Ph. Furthermore, the applied field energy :s assumed to be negligible in comparison with the anisotropy energy. Thus the free energy Eh, which is similar in form to Eq. (9), reduces to (11) The two assumptions above simplify the total free energy from an expression involving the integral of magnetization orientation I/>(z) to a function of two single variables, <\>. and I/>h. Hence only calculus, instead of calculus of variations, is needed to analyze the prob lem. Furthermore, with the omission of the applied field energy in the hard layer, <\>. becomes a single-valued function of I/>h at equilibrium. The energy reduces to a single-valued function of <\>., and by the use of envelope theory, the stable and critical states can be presented in a concise manner in the Hz-Hy plane. Goto et at. then analyzed the rotational-magnet ization reversal as a function of the ratio Ha/H2 (= A./D.K"D h) and described three cases of NDRO behavior: In case 1 (Ha/ H2?:.1), the astroid curve is elongated in the hard direction. The elongation may be infinite. In case 2 (1 > II a/ H2?:. 0.217), the critical curve is open in the hard direction and extends to infinitv along an asymptotic line at a certain angle. The mag- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Thu, 18 Dec 2014 18:06:29DYNAMIC ANALYSIS OF COUPLED FILMS 611 netization can be reversed only with significant easy direction field in addition to hard-direction field. In case 3, (Ha/H2<0.217), each branch of the critical curve degenerates into several sections, and the hard film can be switched irreversibly only by applying a sequence field, i.e., a field in at least two directions in a suitable time sequence. Since the distributed model is not restricted by the . two assumptions above, it is used to examine the validity of each of the assumptions. First, in the pres ence of an applied field, the magnetization rotates rapidly and unsymmetrically near the film interface and levels off at both film surfaces (see Fig. 2). To replace the z-dependent magnetization orientation cf>(z) by an effective mean value cjJ. is valid only when the soft layer is much thicker than the transition layer. Furthermore, since the exchange energy is dependent on the actual magnetization distribution, the replacement of the spatial-dependent magnetization orientation gradient (dc/>/dz) by a discrete expression [see Eqs. (9) and (10)] may result in appreciable error. For example, the pre diction of the soft-film-magnetization-orientation angle increasing monotonically and indefinitely without ro tating the hard-film magnetization is due to the under estimation of exchange energy in the soft film. Second, the variation of magnetization orientation in the hard film is relatively small (see Fig. 2) because the anisotropy energy is predominant in that layer. Hence the omission of exchange energy is reasonable as long as the hard film is relatively thin. However, in the presence of a hard-direction field, the ratio between anisotropy energy (Kh sin2cf>h) and applied field energy (MHo sincf>h) is (Hkh/2Ho) sinc/>h' The omission of applied-field energy is valid only when the applied field is much smaller than the anisotropy field (e.g., Hkh/ Ho'2:.lO). Goto's omission of the applied-field energy results in the prediction of infinite elongation of the critical curve along the hard direction (see Figs. 6 and 7 of Ref. 2) and of perfect nondescructive read against any single-direction applied field (see Fig. 8 of Ref. 2). In fact, the experiments of Goto et at. have already revealed the inadequacy of their analysis. Their experi ment purported to show the elongation of the critical curve for case 1 films, whereas they actually chose film parameters (Hkh=20 Oe, Hk8=3 Oe, Dh=270 A, D.= 10 000 A) belonging to case 2. However, only finite elongation of the critical curve along the hard axis (case 1) instead of infinite elongation along an angular asymptote (case 2) has been observed. The measured hard-direction threshold field (50e) agrees well with the theoretical value computed from the distributed model. In their experiment, which purported to illus trate sequence-field switching, the following film param eters are used: Hkh=20 Oe, Hk.=30e, D,,=81O A, D. = 10 000 A. A bipolar hard-direction field (Hy) is varied, and a unipolar easy direction field (Hz) is maintained with a constant peak amplitude. In con trast to their theoretical predictions that the sequence-field switching prevails over the entire field range, a limited field range has been observed (Hy = 1.5 to 2.8 Oe for Hx=5.5 Oe, and Hy=2.2 to 50e for Hx=4.1 Oe). The film parameters correspond to the case of moderate coupling strength (see Fig. 3), and the observed switch ing behavior also conforms to the prediction of the distributed-model analysis (see Fig. 6). VI. CONCLUSION The magnetodynamic Landau-Lifshitz equation for distributed magnetization has been applied to study the magnetization reversal of exchange-coupled films. In contrast to the conventional dynamic modes, such as Walker mode14 and spin-wave mode,15 which only ap plies to small-angle magnetization motion, the present dynamic-distributed model is a large-angle theory and yields transient as well as steady-state solutions for magnetization reversal. In comparison to the "rigorous" quasistatic treatment, which is based on energy minimization principle, the dynamic-distributed model lends itself easily to the numerical solution of both quasistatic and dynamic-magnetization reversals. Fur thermore, the transient solution automatically selects the correct switching path. Detailed analysis has been focused on dual uniaxial films. In contrast to the results of Goto, based on lumped-constant approximation, the present rigorous analysis has shown that for any coupling strength, only an elongated astroid has been observed. The sequence field switching is found to reduce rotational threshold in moderately coupled films, but it is not the only mode to effect irreversible switching. These results also reconcile the inconsistencies between the analysis and experiments of Goto et al. A new classification of dual uniaxial films according to possible magnetization twist has been proposed. The magnetization twist depends on the ratios of thicknesses as well as anisotropy energies. Distinct features of the various classes include simultaneous switching, reduced threshold for sequential fields, and separate switchings. The switching waveforms and speeds for the various classes have also been examined. In order to focus on some essential features, we have made a number of assumptions in the analysis. However, the analytical formulation is sufficiently general to allow the removal of these assumptions: (i) The analy sis can be extended to other exchange-coupled struc tures. (ii) The films are assumed to be of infmite extent. In practical devices of discrete size, the planar magnetization distribution and consequently the cur rent requirement are influenced by demagnetizing fields.9 (iii) When the analvsis is extended to two di mensions, structures such a~ magnetization ripple and domain walls can be taken into account,u (iv) The 14L. R. Walker, Phys. Rev. lOS, 390 (1957). 15 C. Herring and C. Kittel, Phys. Rev. 81, 869 (1951). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Thu, 18 Dec 2014 18:06:29612 Y. S. LIN AND H. CHANG magnetodynamic equation used is in the Landau Lifshitz form. For large damping the equation can be easily modified into Gilbert's form.!l ACKNOWLEDGMENTS Helpful discussions with G. S. Almasi, L. A. Finzi, J. C. Slonczewski, and D. A. Thompson are acknow ledged. APPENDIX I: CRITICAL CURVE CONSTRUCTION FROM THE 'MAGNETODYNAMIC EQUATION In quasi static magnetization reversal, critical state corresponds to the vanishing of a stable state which is at a relative minimum energy. In the study of a single domain film, the critical state can be defined mathe matically as a point of inflection for the energy-vs magnetization-orientation function. In the study of two or more coupled single-domain films,I6,l7 it is more convenient to define the critical state as the limiting case of stable state as proposed by Chang.!6 A new method of constructing a critical curve utilizing magnetodynamic equations is to be described. The method derives from the recognition that critical state also corresponds to the initiation of irreversible switch ing in rotational magnetization reversal. The mechanics of generating a critical curve from magnetodynamic equations [Eqs. (5) and (6)], which resulted in Figs. 4, 5, and 6, will now be discussed. For a given initial quiescent-magnetization distribution, a reversing field may be applied at various angles. The reversing field at each angle consists of a train of pulses of gradually increasing amplitudes [see Figs. 4(b) and 5(b)]. Each pulse may be assumed to have instantan eous rise and fall times, but a duration T (e.g., 10 nsec) sufficiently long such that the magnetization orientation will reach an equilibrium state before the field is re moved. After each pulse, sufficient time (e.g., 10 nsec) is allowed for an equilibrium state (a final quiescent magnetization distribution) to be reached. Then the final quiescent magnetization distribution is compared to the initial one, [see Figs. 4( d) and 5 (d)]. If they are the same, the reversal was a reversible one, and pulse of larger amplitude is to be applied. If they are different, the reversal was an irreversible one, and the amplitude of the last pulse is a critical value. It is not necessary to examine the magnetization distribution in total to determine the switching mode. For example, in dual uniaxial films, it is sufficient to examine only the mag netization orientations at the top and bottom surfaces. After the critical field or fields for one field orientation have been determined, the magnetization distribution is reset into its original state, and a pulse train is applied I6H. Chang, IBM J. Res. & Develop. 6, 419 (1962). 17 E. Fulcomer et ai., J. Appl. Phys. 37, 4451 (1966). when the field orientation is advanced to another value. When the field orientation has swept through 3600 (usually a small sector of it), a complete critical curve can be constructed. As mentioned earlier in Sec. IV, in structures with several films coupled by exchange forces, there may be more than one critical curve [e.g., Fig. 5(a)] corresponding to one quiescent magnetization state. Hence the pulse train for each field orientation must be carried sufficiently far to detect mUltiple successive switchings [e.g., Fig. 5(b)]. Furthermore, in some structures, the critical curves are dependent on the sequence and polarity of the easy-and hard-direction fields [e.g., Fig. 6(a)]. A complete picture of switching modes can only be obtained by surveying a sufficient range of field trajectories [e.g., Fig. 6 (b) ]. There are several advantages unique to the magneto dynamic method of constructing critical curves: (1) It is applicable to both film structures with one or more single domains as well as distributed magnetization. (21 It may be used to generate both quasi static and dynamic critical curves just by varying the time rate of applied field. For example, for pulses of short duration, insufficient time is allowed to rotate the hard-film magnetization in dual uniaxial films, and hence a more elongated critical curve results ~see Fig. A1). (3) It yields automatically the final state of magnetization distribution. This feature is very useful when there are crisscrossing critical curves (as in single-domain biaxial films) and multiple critical curves (as in weakly coupled dual uniaxial films), and when the switching is dependent on field trajectory (as in moderately coupled dual uniaxial films). (4) Since the method is based on the detection of irreversible switching, it lends itself to ready comparison with experimental results. 5 \ \ 4 \ o \ CURVE A HARD FILM THRESHOLD B SOFT FILM THRESHOLD \ \A \ C\ \. "i} DUAL FILM THRESHOLD " ~ 2 PULSE DURATION ~ =2n5 .............-- FIG. Al. Dynamic and quasistatic thresholds. Applied-field waveforms are identical to that in Figs. 4 and 5. AA= A., KA = SK., D.=5D h= (As/K.)1/'. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Thu, 18 Dec 2014 18:06:29DYNAMIC ANALYSIS OF COUPLED FILMS 613 APPENDIX II: SWITCHING SPEED AND SIGNAL 5 The switching speed of dual uniaxial films undergoing 1800 rotational switching can be estimated by balancing the applied-field energy 2MHo with the increase of the free energy of the system, which includes the normal demagnetizing-field energy H'; /87r, and exchange energy A [(¢top-¢bottom)/D]2, but omits the anisotropy energy (in cgs) : Hi/87r+A[(¢top-¢bottom)/D]2 = 2MHo, (AI) where A is the exchange constant, ¢top-¢bottom is the magnetization twist between the two outer surfaces, and D is the film thickness. Typicallv, A = 10-6 erg/ cm, 47rM = 104 gauss, Ho= 1 Oe, D=3000 X, ¢top-¢bottom = 1 radian, and Eq. Al yields H<P:'-130 Oe. The preces sional angular velocity (w) of M due to this demagnetiz ing field is then w= I 'Y I H<P:'-1.3X109 secI, (A2) where I'Y I =1.76X107 (Oe·sec)-l is the gyromagnetic ratio. The switching time is approximately the half period of the precession: T.=H27r/w)~1.5Xl0-9 sec. For single uniaxial films, however, the exchange term is negligible, and the switching time is only T.~10-9 sec for the same applied field. Computed from the dynamic-distributed model [Eqs. (5) and (6)], the curve of inverse-switching time as a function of hard-direction step field for different values of Kh/ K. is presented in Fig. A2. The switching time t. is 4 3 '0 Q) en C1I 2 Q _II) .... FIG. A2. Inverse switching time as a function of hard-direction field with Kh/K. as parameters. t.=time for net flux parallel to easy axis to decrease by 90% from its initial saturation value. [Ah=A., D.=5D h= (A./K.)1I2]. 4 (a) RESPONSE TO STEP_ FUNCTION X = 5 x 108 (DAMPING CONSTANT) (b) RESPONSE TO STEP FUNCTION X=2x109 FIG. A3. Longitudinal output signal as a function of transverse field waveform and damping constant. Ah= A" Kh= 10K., D,= 5Dh= (A./K.)il2. defined as the time for the net flux parallel to easy direction to decrease by 90% from its initial saturation value. The switching curves have two regions. At low fields, the speed of the dual uniaxial films is reduced in comparison to that of single films. This reduction is due to the expenditure on exchange energy. At high fields, both films switch simultaneously and behave as a single-domain film. Relatively small dependence on the ratio Kh/K. is observed. Finally, it is of interest to present the calculated switching waveform of dual films. Figure A3 depicts the dependence of longitudinal (easy axis) output signal on the waveform of the transverse driving field and the damping constant. Note that a large output signal and oscillation arise with a large step function drive [see Fig. A3(a)]. Large damping reduces oscillation as well as output [see Fig. A3(b)]. The critical dampi'ng con stant is around 109 GjOe-sec. Nevertheless, in practical circuits, the drive field has a rise time which tends to eliminate the oscillation and reduce the output signal [see Fig. A3(c)]. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Thu, 18 Dec 2014 18:06:29
1.5013667.pdf	First-principles study of perpendicular magnetic anisotropy in ferrimagnetic D0 22- Mn3X (X = Ga, Ge) on MgO and SrTiO 3 B. S. Yang , L. N. Jiang , W. Z. Chen , P. Tang , J. Zhang , X.-G. Zhang , Y. Yan , and X. F. Han Citation: Appl. Phys. Lett. 112, 142403 (2018); doi: 10.1063/1.5013667 View online: https://doi.org/10.1063/1.5013667 View Table of Contents: http://aip.scitation.org/toc/apl/112/14 Published by the American Institute of Physics Articles you may be interested in Dynamics of a magnetic skyrmionium driven by spin waves Applied Physics Letters 112, 142404 (2018); 10.1063/1.5026632 Perpendicular magnetic tunnel junctions with Mn-modified ultrathin MnGa layer Applied Physics Letters 112, 062402 (2018); 10.1063/1.5002616 High performance perpendicular magnetic tunnel junction with Co/Ir interfacial anisotropy for embedded and standalone STT-MRAM applications Applied Physics Letters 112, 092402 (2018); 10.1063/1.5018874 Observation of large exchange bias and topological Hall effect in manganese nitride films Applied Physics Letters 112, 132402 (2018); 10.1063/1.5025147 Magnetic phase dependence of the anomalous Hall effect in Mn 3Sn single crystals Applied Physics Letters 112, 132406 (2018); 10.1063/1.5021133 Size dependence of the spin-orbit torque induced magnetic reversal in W/CoFeB/MgO nanostructures Applied Physics Letters 112, 142410 (2018); 10.1063/1.5022824First-principles study of perpendicular magnetic anisotropy in ferrimagnetic D022-Mn 3X( X5Ga, Ge) on MgO and SrTiO 3 B. S. Yang,1,2L. N. Jiang,2W. Z. Chen,2P.Tang,2J.Zhang,3X.-G. Zhang,4Y.Yan,1,a) and X. F . Han2,b) 1Key Laboratory of Physics and Technology for Advanced Batteries (Ministry of Education), Department of Physics, Jilin University, Changchun 130012, People’s Republic of China 2Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 3School of Physics and Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, 430074 Wuhan, China 4Department of Physics and the Quantum Theory Project, University of Florida, Gainesville, Florida 32611, USA (Received 14 November 2017; accepted 18 March 2018; published online 3 April 2018) The magnetic anisotropy energy (MAE) of bulk D022-Mn 3X( X ¼Ga, Ge), Mn 3X/MgO, and Mn 3X/STiO 3(STO) heterostructures is calculated from ﬁrst principles calculations. The main source of the large perpendicular magnetic anisotropy (PMA) of bulk Mn 3X is identiﬁed as Mn atoms in the Mn-Mn layer. In the four heterostructures, the magnetic moment of interfacial Mn atoms was reversed when Mn 3X was epitaxially grown on MgO and STO substrates. More impor- tantly, a large in-plane tensile strain induced by lattice mismatch between Mn 3X and MgO signiﬁ- cantly changes the MAE, explaining the difﬁculty in experiments to obtain PMA in epitaxial Mn 3X/MgO. Furthermore, interface and surface Mn atoms also help to enhance the PMA of Mn 3X/ STO (MgO) heterostructures due to dxyanddz2states changing from occupied states in bulk Mn 3X to unoccupied states in the interface (surface) Mn of the heterostructures. These results suggest that the PMA of manganese compound heterostructures can be produced by decreasing the lattice mis-match with substrates and will guide the search for ultrathin manganese compound ﬁlms with high PMA epitaxially grown on substrates for the application of spintronic devices. Published by AIP Publishing. https://doi.org/10.1063/1.5013667 Perpendicular magnetic tunnel junctions (p-MTJs) have received much attention for high-density nonvolatile memory,high thermal stability, and low critical current in spin-transfer- torque or spin-orbital-torque based spintronic devices. 1–4In a conventional MTJ, the interface between the magnetic elec-trode CoFeB and the MgO barrier contributes to a perpendicu-lar magnetic anisotropy (PMA). 4–10When the magnetic ﬁlms are sufﬁciently thin, the demagnetization ﬁeld can be over- come and the moments become perpendicular to the ﬁlm.11 The interfacial PMA of CoFeB/MgO is about 1.3 erg/cm2and is too weak to overcome thermal ﬂuctuation at room tempera- ture when the thickness of the electrode layer is less than 20 nm.12Moreover, annealing above 300/C14C rapidly decreases PMA in the Ta/CoFeB/MgO system.13,14High thermal stabil- ity is easier to achieve using magnetic electrode materialswith large bulk PMA instead of interfacial PMA. Therefore, people have recently turned to tetragonally distorted Heusler alloys, such as ferrimagnetic D0 22-Mn 3Ga and Mn 3Ge, to look for suitable materials.15,16The tetragonally distorted Mn3Ga and Mn 3Ge have PMA greater than 10 Merg/cm3, which is comparable to the well-known CoPt and FePt ﬁlms.17–20Due to their low saturation magnetization,21–23low Gilbert damping constant,24,25and high spin-polarization rate, these materials are promising candidates for minimizing theswitching current and enlarging switching speed according to the Slonczewski-Berger equation. 2,3,26Experimentally, it was demonstrated that due to different in-plane lattice constants of the substrates, PMA of Mn 3Ga epitaxially grown on the SrTiO 3substrate is greater than that grown on the MgO substrate.27T h ef o r m e rh a sab e t t e rl a t t i c e match with Mn 3Ga. Interestingly, when the MnGa ﬁlm thick- ness is smaller than 1 nm, a well squared perpendicular mag- netization hysteresis can be achieved by inserting a CoGa buffer layer between Mn 3Ga and MgO.28Cr, Ti-Mg-O, TaN, and CsCl-type NiGa buffer layers were also used to obtain large PMA of manganese compounds epitaxially grown on the MgO substrate.11,16,29Both facts, that the SrTiO 3substrate yields a larger PMA and that a buffer layer can boost the PMA of the Mn 3Ga ﬁlm, suggest an important role played by the interfacial strain induced by the substrate. In this paper, we present a ﬁrst-principles study of the magnetic anisotropy energy (MAE) of bulk Mn 3X( X ¼Ga, Ge) and heterostructures epitaxially grown on MgO and SrTiO 3(STO) substrates. The calculation allows us to iden- tify the source of large magnetic anisotropy in the bulk Mn 3X. In addition, the calculations of the MAE of the heter- ostructures as a function of tensile strain are performed to understand the effect of strain on PMA. All calculations were performed within the framework of the density functional theory (DFT) implemented in the Vienna ab-initio simulation package (VASP).30–32The ion-electron interaction was described by the projector-augmented plane wave (PAW) potentials.33The exchange-correlation potential was treated with the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof functional.34Thea)Electronic mail: yanyu@jlu.edu.cn b)Electronic mail: xfhan@iphy.ac.cn 0003-6951/2018/112(14)/142403/4/$30.00 Published by AIP Publishing. 112, 142403-1APPLIED PHYSICS LETTERS 112, 142403 (2018) energy cutoff was 500 eV, and 13 /C213/C27a n d1 7 /C217/C21 Monkhorst-Pack k-point meshes were used in the calculations of bulk Mn 3Xa n dM n 3X/STO (MgO) heterostructures, respec- tively. All atom positions were fully relaxed until the force on each atom was less than 0.001 eV/A ˚and the total energy change was smaller than 10–6eV. As shown in Fig. 1, the unit cell of bulk D022-Mn 3Xc o n - tains eight atoms in total and the Mn atoms in the Mn-Mn l a y e ra n dM n - Xl a y e rw e r el a b e l e da sM n 1and Mn 2,r e s p e c - tively. The optimized lattice constants of Mn 3Ga are a¼3.780 A ˚and c ¼7.106 A ˚, and those of Mn 3Ge are a¼3.740 A ˚and c ¼7.106 A ˚, similar to previous calcula- tions.35The in-plane lattice constants of SrTiO 3(3.905 A ˚)a n d MgO (4.211 A ˚)36were used in Mn 3X/STO and Mn 3X/MgO heterostructures, respectively. A vacuum region of 10 A ˚in the normal direction to the heterostructure was used to avoid the interaction between adjacent periodic layers. The Mn 3Xﬁ l m in the heterostructures contains seven monolayers, and thus, the thickness of the Mn 3X ﬁlm is approximately 1.0 nm. In view of the important role of the interface electronic and mag- netic structures in the MAE of the heterostructures,5,37several possible conﬁgurations with different interface structures werecalculated. Among them, the most stable Mn 3Ga/STO (MgO) heterostructure is shown in Fig. 1.T h em a g n e t i ca n i s o t r o p y energy (MAE) was calculated by the force theorem approach.38,39In this method, we ﬁrst perform the self- consistent calculation without spin-orbital coupling with theoptimized structure. Then, using the obtained charge density in self-consistent calculations as input, the spin-orbital cou- pling was treated as perturbation to calculate the energy for two different orientations of the magnetic moment. At last, the MAE was calculated from the energy difference betweenthe magnetic moment aligning in-plane direction [(100) axis] and the out-of-plane direction [(001) axis]. For bulk Mn 3Ga and Mn 3Ge, the calculated magnetic moment of Mn 1is 2.32 and 1.98 lBand that of Mn 2is 2.87 and 2.90 lB, respectively. Moreover, the magnetic moment of Mn 2is antiparallel to that of Mn 1. As a result, the mag- netic moments of the unit cell for Mn 3Ga and Mn 3Ge are 3.56 and 2.03 lB, respectively. The calculated MAEs of Mn 3Ga and Mn 3Ge are 1.75 and 1.54 meV/cell, and the posi- tive sign indicates that the easy magnetic axis is along the (001) direction. For both materials, the MAE comes mainly from the Mn 1atom, with a negligible contribution from theMn 2atom. The orbital-resolved MAE of the Mn 1atom is shown in Fig. 2, where the matrix element between the dz2 anddyzorbitals of Mn 1provides the largest contribution to the positive MAE in Mn 3Ga. While for Mn 3Ge, the matrix element between dxyanddx2/C0y2,dyzanddx2/C0y2, as well as dyz anddz2orbitals of Mn 1provides large positive values. These positive contributions produce the perpendicular magnetic anisotropy of bulk Mn 3X. In contrast to the opposite alignment of the moments of Mn1and Mn 2a t o m si nb u l kM n 3X, the magnetic moment of the Mn 2atom closest to the interface is parallel to that of the Mn1atom in both heterostructures. The energy differences between the antiparallel conﬁguration and the parallel conﬁg- uration for the magnetic moments at the interface and the sec-ond layer Mn atom are 0.77, 1.38, 0.83, and 0.27 eV for Mn 3Ga/MgO, Mn 3Ga/STO, Mn 3Ge/MgO, and Mn 3Ge/STO, respectively. Furthermore, the calculated MAEs of Mn 3Ga/ MgO, Mn 3Ge/MgO, Mn 3Ga/STO, and Mn 3Ge/STO hetero- structures are 0.028, 0.160, 2.522, and 2.146 meV, respec- tively. It is clear that the MAE of Mn 3X/STO is much higher than that of Mn 3X/MgO. To investigate whether the different in-plane tensile strain induced by the MgO and STO substrate may be the reason for the difference in the MAE of STO- and MgO-based heterostructures, we calculated the MAEs of bulk Mn3X as a function of in-plane lattice constant ain the range of 3.70 A ˚to 4.25 A ˚. The results are shown in Fig. 3, where we also indicate the optimized in-plane lattice constant of Mn 3X, STO, and MgO. When the in-plane lattice constant is smaller FIG. 1. Atomic structure of bulk D022-Mn 3Ga(Ge) (top), Mn 3Ga/STO (mid- dle), and Mn 3Ga/MgO heterostructures (bottom). FIG. 2. Orbital-resolved MAE of the Mn 1atom in bulk (a) Mn 3Ga and (b) Mn 3Ge. FIG. 3. MAE as a function of in-plane lattice constant for Mn 3Ga (red line) and Mn 3Ge (blue line). The inset shows contributions from Mn 1and Mn 2 atoms in bulk Mn 3X.142403-2 Y ang et al. Appl. Phys. Lett. 112, 142403 (2018)than 4.00 A ˚, the MAEs of Mn 3X are nearly constant. In con- trast, when the in-plane lattice constant is larger than 4.00 A ˚, the MAEs decrease rapidly with the increase in the in-plane lat- tice constant. When the in-plane lattice constant of Mn 3Xi s the same as that of STO and MgO, the corresponding MAEs of Mn3Ga are 1.74 and /C00.61 meV/cell, and those of Mn 3Ge are 1.61 and /C00.33 meV/cell, respectively. These results indicate that a large in-plane tensile strain induced by the large lattice mismatch between Mn 3X and MgO can make the easy axis rotate from the out-of-plane orientation to the in-plane orienta- tion. The inset in Fig. 3shows that the variations of the MAEs in Mn 3Ga and Mn 3Ge mainly come from the Mn 1-atom and that of the Mn 2atom contributes a negligible constant value. To elucidate the change in the MAE as a function of in- plane lattice constant, we perform a comparison of the orbital-resolved MAEs of Mn 1atoms in bulk Mn 3Ga at in-plane lattice constants of 3.905 A ˚and 4.221 A ˚, as shown in Figs. 4(a)and 4(b). It is clear that the most important anisotropy change induced by the variation of the in-plane lattice constant is the change of sign in the matrix element between dyzanddz2as well as its symmetric part. Therefore, in the following, we willmainly discuss the change in the matrix element between d yz anddz2. According to the second-order perturbation theory by Wang et al.,38the MAE is expressed as DE¼EðxÞ/C0E ðzÞ¼n2Ro;uðj<ojLzju>j2/C0j<ojLxju>j2Þ=ðEu/C0EoÞ.F r o m this equation, we can deduce that the value of the matrix ele- ment between two different dorbitals is the same as its sym- metric part. On the other hand, when an orbital moves from the unoccupied to occupied state, a positive term in the energy dif- ference disappears and a negative term appears. In Figs. 4(a) and4(b),t h e dyzorbital has no dramatic change around the Fermi level, while the dz2orbital changed from an unoccupied to an occupied state when the in-plane lattice constant increasesfrom 3.905A ˚to 4.211A ˚. This explains the change in the MAE. Similarly, the decrease in MAE of Mn 3Ge as the in-plane lat- tice constant increases can also be explained by analyzing theDOS and the orbital-resolved MAE of Mn 1in Mn 3Ge. In both cases, the in-plane tensile strain causes a decrease in the MAEs of bulk Mn 3X.The layer-resolved MAEs in Fig. 5show that in both MgO- and STO-based heterostructures, the MAE contribu- tions from the interface (ﬁrst layer) and the surface (seventh layer) Mn atoms have positive values. However, in bulkMn 3X, the MAE contributions from Mn atoms have negative values when the in-plane lattice constant is 4.221 A ˚(see the inset in Fig. 3). This difference in the sign of the MAE contri- butions from bulk and from the surface and interface is again examined by plotting orbital resolved MAE in Fig. 4. Comparing Figs. 4(b)to4(c), we see that the contributions to MAE from the matrix element between dxyanddx2/C0y2as well asdyzanddz2orbitals in the two systems show opposite signs. The cause of this sign change is that both dxyanddz2states are occupied in the bulk [Fig. 4(e)], while they become unoccu- pied in the surface [Fig. 4(f)]. Although the dyzanddx2/C0y2 orbitals also undergo changes from the bulk to the surface, they do not obviously change the occupation of orbitals. Except the sign change, the magnitude of both matrix ele-ments for the surface Mn 1atom also increases compared to that for Mn 1in bulk Mn 3Ga. Summing up all contributions, the positive and negative MAEs are obtained for surface Mn 1 in Mn 3Ga/MgO and Mn 1in bulk Mn 3Ga, respectively. Our calculations clearly show that due to the small lat- tice mismatch between Mn 3X and STO, Mn 3X/STO hetero- structures produce large PMA. But for Mn 3X/MgO with the large lattice mismatch between Mn 3X and MgO when the thickness of Mn 3X is small, there is no PMA. Furthermore, interface and surface Mn atoms also contribute to the PMA due to the change of the dxyanddz2states from occupied in bulk Mn 3X to unoccupied on the interface (surface) of Mn 3X/MgO. These results explain why PMA is difﬁcult to produce in Mn 3X/MgO when its thickness is small and also explain the PMA of Mn 3Ga epitaxially grown on the SrTiO 3 substrate. We also calculated the MAE of Mn 3Ga by GGA þU calculations and found that the coulomb correla- tions have some inﬂuence on the value of MAE, which issimilar to the results in Ref. 40. The GGA þU calculations also show that a large in-plane tensile strain induced by the large mismatch between Mn 3Ga and MgO even can make FIG. 4. Orbital-resolved MAE of Mn 1(a) in bulk Mn 3Ga with an in-plane lattice constant of a ¼3.795 A ˚, (b) in bulk Mn 3Ga with an in-plane lattice constant of a¼4.221 A ˚, and (c) on the surface of the Mn 3Ga/MgO heterostructure and (d)–(f) the corresponding density of states projected onto Mn 1. The circles in (d)–(f) indicate the electronic states of dz2anddxynear the Fermi level.142403-3 Y ang et al. Appl. Phys. Lett. 112, 142403 (2018)the easy axis of Mn 3Ga rotate from the out-of-plane orienta- tion to the in-plane orientation, and therefore, we did not include the coulomb correlations in this paper. In summary, the magnetic anisotropy of bulk D022-Mn 3X (X¼Ga, Ge) and Mn 3X/MgO (SrTiO 3) heterostructures is investigated by ﬁrst principles calculations. In heterostruc- tures, the in-plane strain induced by the lattice mismatchbetween the D0 22structure and the substrate has a crucial inﬂuence on the MAE. This is the reason why perpendicular magnetic anisotropy is difﬁcult to produce in Mn 3Xﬁ l m so n MgO. Through layer- and orbital-resolved MAEs combined with DOS analysis, we ﬁnd that the interface (surface) anisot- ropy also plays an important role in these heterostructures.These results suggest that using a buffer layer to release the strain in the interface is important. They also provide further guidance on how to enhance the PMA of ultrathin Mn 3X ﬁlms for the application of spintronic devices. This project was supported by the National Key Research and Development Program of China (Grant No.2017YFA0206200) and the National Natural Science Foundation of China (NSFC, Grant Nos. 11434014, 51620105004, 11674373, and 51701203), and partiallysupported by the Strategic Priority Research Program (B) (Grant No. XDB07030200), the Key Research Program of Frontier Sciences (Grant No. QYZDJ-SSW-SLH016), and theInternational Partnership Program (No.112111KYSB20170090) of the Chinese Academy of Sciences (CAS). We are grateful to the National Supercomputer Center in Tianjin for providing thecomputational facility and the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase) under Grant No. U1501501. 1A. D. Kent, Nat. Mater. 9, 699 (2010). 2L. Berger, Phys. Rev. B 54, 9353 (1996). 3J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 4S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. 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1.4983993.pdf	A prospectus on kinetic heliophysics Gregory G. Howes Citation: Physics of Plasmas 24, 055907 (2017); doi: 10.1063/1.4983993 View online: http://dx.doi.org/10.1063/1.4983993 View Table of Contents: http://aip.scitation.org/toc/php/24/5 Published by the American Institute of Physics Articles you may be interested in Electron holes in phase space: What they are and why they matter Physics of Plasmas 24, 055601 (2017); 10.1063/1.4976854 Announcement: The 2016 Ronald C. Davidson Award for Plasma Physics Physics of Plasmas 24, 055906 (2017); 10.1063/1.4983992 Turbulent transport in 2D collisionless guide field reconnection Physics of Plasmas 24, 022104 (2017); 10.1063/1.4975086 Edge transport bifurcation in plasma resistive interchange turbulence Physics of Plasmas 24, 055905 (2017); 10.1063/1.4983624 Key results from the first plasma operation phase and outlook for future performance in Wendelstein 7-X Physics of Plasmas 24, 055503 (2017); 10.1063/1.4983629A prospectus on kinetic heliophysics Gregory G. Howesa) Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242, USA (Received 14 March 2017; accepted 1 May 2017; published online 23 May 2017) Under the low density and high temperature conditions typical of heliospheric plasmas, the macroscopic evolution of the heliosphere is strongly affected by the kinetic plasma physics governing fundamental microphysical mechanisms. Kinetic turbulence, collisionless magneticreconnection, particle acceleration, and kinetic instabilities are four poorly understood, grand- challenge problems that lie at the new frontier of kinetic heliophysics. The increasing availability of high cadence and high phase-space resolution measurements of particle velocity distributions bycurrent and upcoming spacecraft missions and of massively parallel nonlinear kinetic simulations of weakly collisional heliospheric plasmas provides the opportunity to transform our understanding of these kinetic mechanisms through the full utilization of the information contained in the particlevelocity distributions. Several major considerations for future investigations of kinetic heliophysics are examined. Turbulent dissipation followed by particle heating is highlighted as an inherently two-step process in weakly collisional plasmas, distinct from the more familiar case in ﬂuid theory.Concerted efforts must be made to tackle the big-data challenge of visualizing the high- dimensional (3D-3V) phase space of kinetic plasma theory through physics-based reductions. Furthermore, the development of innovative analysis methods that utilize full velocity-space meas-urements, such as the ﬁeld-particle correlation technique, will enable us to gain deeper insight into these four grand-challenge problems of kinetic heliophysics. A systems approach to tackle the multi-scale problem of heliophysics through a rigorous connection between the kinetic physics atmicroscales and the self-consistent evolution of the heliosphere at macroscales will propel the ﬁeld of kinetic heliophysics into the future. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4983993 ] I. INTRODUCTION Humanity continually strives to understand its environ- ment, not only to ensure its continued survival, but also for the sake of knowledge itself. The heliosphere—the realm of inﬂuence of our Sun, within which the planets of our solarsystem orbit—is our home in the universe. Nuclear fusion within the core of the Sun is the source of energy that ena- bles life to thrive on our planet. The majority of this energyemerges as light, but a small fraction of this energy also drives a supersonic ﬂow of diffuse ionized gas, or plasma, that blows radially outward toward the outer reaches of theheliosphere. Carrying along with it an embedded magnetic ﬁeld, this solar wind varies dramatically in response to con- ditions at the Sun, and is strongly disturbed during periods ofviolent activity on the Sun’s surface. It is the dynamics of this magnetized plasma that governs the interaction of the Sun with the Earth and the other planets of our solar system.Humankind has spent billions of dollars to launch spacecraft to explore our heliosphere in the scientiﬁc endeavor to understand and predict the dynamics of the interplanetaryplasma that affect the Earth and its environment, and here I highlight some issues at the frontier of that effort. The inaugural Ronald C. Davidson Award for Plasma Physics recognized the development of a simple analytical model and supporting numerical simulations of the turbulentcascade and its kinetic dissipation at small scales, 1but to progress further requires kinetic theory. The low density and high temperature conditions of the plasma that ﬁlls the helio-sphere—as well as many more remote astrophysical sys- tems—lead to a mean free path for collisions that is often longer than the length scales relevant to many dynamical processes of interest. Under these weakly collisional condi- tions, the dynamics of the plasma require investigation using the equations of kinetic plasma physics. For example, many space and astrophysical plasmas are found to be turbulent. One of the key impacts of this turbu- lence is the nonlinear transfer of the energy of large-scale electromagnetic ﬁelds and plasma ﬂows down to small scales at which the turbulent energy is ultimately converted to plasma heat, or to some other energization of the plasma ions and electrons. At the largest scales of the astrophysical turbulent cascade in the interstellar medium, 2,3the length scales of the turbulent ﬂuctuations may be much larger than the collisional mean free path, meaning that a ﬂuid descrip- tion of the turbulent dynamics at these scales is generally sufﬁcient. But at all scales of the turbulence in the solar wind,4,5and at the smallest scales of the turbulence in the interstellar medium,6the length scales of the turbulent ﬂuctu- ations are much smaller than the collisional mean free path. Under these conditions, the effect of collisions is negligible on the timescale of the turbulent ﬂuctuations. Not only are collisions insufﬁcient to maintain the Maxwellian particle velocity distributions that motivate the use of a ﬂuida)gregory-howes@uiowa.edu 1070-664X/2017/24(5)/055907/12/$30.00 Published by AIP Publishing. 24, 055907-1PHYSICS OF PLASMAS 24, 055907 (2017) description, but collisionless interactions generally dominate the energy exchange between the ﬂuctuating electromagnetic ﬁelds and the plasma particles.7,8Therefore, the equations of kinetic plasma physics are essential to describe the mecha- nisms responsible for removing energy from the turbulent ﬂuctuations and consequently energizing the plasma particles. For the weakly collisional interplanetary plasma, such “microphysical” kinetic processes govern the heating of theplasma and the energization of particles, and thereby they exert a signiﬁcant inﬂuence on the macroscopic evolution of the heliosphere. Plasma turbulence, magnetic reconnection, particle acceleration, and instabilities are four fundamental plasma processes operating under weakly collisional condi-tions that signiﬁcantly impact the evolution of the helio- sphere. These four grand-challenge topics lie at the frontier of heliophysics. The details of these kinetic plasma processesremain relatively poorly understood, motivating the helio- physics community to pursue a coordinated effort of space- craft observations, numerical simulations, kinetic plasmatheory, and even laboratory experiments to develop a thor- ough understanding, and ultimately a predictive capability, of these processes in kinetic heliophysics. This prospectusexamines important issues in our exploration of the kinetic plasma physics of the heliosphere. A. The transport of mass, momentum, and energy in the heliosphere The key impact of these four fundamental kinetic plasma physics processes—kinetic turbulence, collisionless magnetic reconnection, particle acceleration, and instabil- ities—is their effect on the transport of particles, transfer ofmomentum, and ﬂow of energy throughout the heliosphere. Extreme space weather illustrates concisely how the transport of mass, momentum, and energy by these kinetic plasma physics mechanisms governs conditions within the heliosphere, possibly leading to adverse impacts on the Earthand its near-space environment. Magnetic buoyancy instabil- ities cause the strong magnetic ﬁelds generated by the solar magnetic dynamo to rise out of the turbulently boiling solarconvection zone, emerging through the photosphere and building up strong magnetic ﬁelds in lower solar atmosphere, or corona. Eventually, some type of explosive instability caninitiate vigorous magnetic reconnection, hurling tons of mag- netized plasma out into the heliosphere at thousands of kilo- meters per second, an event known as a coronal mass ejection. Magnetic energy released through the process of reconnection can also accelerate electrons back downtowards the photosphere, often causing a powerful solar ﬂare that enhances x-ray and UV ﬂuxes radiating from the Sun. In addition, as the magnetized cloud of ejected plasma barrelsat supersonic and super-Alfv /C19enic speeds through the slower ambient solar wind, a collisionless shock forms on the lead- ing edge, frequently accelerating protons, electrons, andminor ions to nearly the speed of light, showering the helio- sphere in a solar energetic particle event. These energetic particles stream through the helio- sphere, being scattered by ﬂuctuations in the turbulent inter- planetary magnetic ﬁeld. Because these energetic particlespose a serious hazard to communication and navigation sat- ellites as well as manned spacecraft missions, predictingtheir ﬂuxes in the near-Earth environment is a critical ele-ment of space weather forecasting, requiring an understand-ing of the transport of these particles through the turbulentsolar wind. In addition, the enhanced x-ray and UV ﬂuxesfrom a strong solar ﬂare can boost ionization in the iono-sphere, interfering with or even totally disrupting radio com-munications with satellites and aircraft on polar ﬂight paths. If the coronal mass ejection is directed towards the Earth, its momentum can lead to a severe compression of theEarth’s magnetosphere, altering the system of currents thatmodify the Earth’s magnetic ﬁeld, and triggering a geomag-netic storm. During a geomagnetic storm, the magnetic ﬁeld embedded within the ejected coronal plasma can undergo reconnection with Earth’s protective magnetic ﬁeld, greatlyenhancing the penetration of interplanetary plasma into themagnetosphere, thereby boosting the density of the ring cur-rent caused by the azimuthal (longitudinal) drift of ions andelectrons trapped in Earth’s dipolar magnetic ﬁeld. Duringparticular strong geomagnetic storms, this enhancement ofthe ring current can depress the magnitude of the magneticﬁeld at the Earth’s surface by a few percent, causing intensegeomagnetically induced currents that may damage criticalcomponents of the electrical power grid. As the geomagneticstorm rages, the aurorae at the poles light up, driven eitherby particles streaming down along open ﬁeld lines towardthe ionosphere or by the acceleration of electrons by Alfv /C19en waves which transmit shifts in Earth’s distant magnetospherealong ﬁeld lines down to the Earth. This complicated interplay of the different phenomena that constitute space weather illustrates the fundamentalimportance of turbulence, magnetic reconnection, particleacceleration, and instabilities to the dynamics of the helio-sphere and its impact on Earth and society. It is important toemphasize that most of the processes mentioned aboveremain poorly understood in detail. An overarching aim ofheliophysics is to improve our understanding of these funda-mental processes and their effect on the transport of par-ticles, momentum, and energy, with the ultimate aim todevelop a predictive capability for space weather and itsimpact on our lives. The path forward is through the applica-tion of kinetic plasma physics to the study of heliosphericprocesses, giving birth to the new frontier of kinetic heliophysics . B. A coordinated approach Although spacecraft missions enable in situ measure- ments of the ﬂuctuating electric ﬁeld Eand magnetic ﬁeld B and of the particle velocity distribution functions in thethree-dimensions of velocity space f s(v), many of these fun- damental kinetic processes in heliospheric plasmas remainpoorly understood. One reason is that spacecraft measure-ments suffer the signiﬁcant limitation that we measure infor-mation only at a single point, or at most a few points, inspace. To circumvent this limitation of spacecraft observa-tions, many of these kinetic processes can alternatively beexplored in laboratory experiments under more controlled055907-2 Gregory G. Howes Phys. Plasmas 24, 055907 (2017)conditions and with the ability to make measurements at many points in space, even if it is not possible to achieve the same plasma parameters or scale separations found in space.A further complication in exploring kinetic heliophysics is the inherent high dimensionality of kinetic plasma theory, with its fundamental variables being the particle distributionfunctions for each species sin six-dimensional phase space (3D-3V, three dimensions in physical space and three dimen- sions in velocity space), f s(r,v,t). Theoretical insights from kinetic plasma theory are vital to reduce this six-dimensional phase space to a more tractable, smaller number of essentialdimensions, for either space-based or laboratory investiga- tions. Finally, kinetic numerical simulations provide a criti- cal bridge between the often idealized conditions susceptibleto analytical theory and the more complex, nonlinear evolu- tion of actual space or laboratory plasmas. A closely coordinated approach of analytical theory, numerical simulations, spacecraft measurements, and labora- tory experiments has the greatest potential for transformingour understanding of the kinetic plasma physics that inﬂuen- ces the evolution of the heliosphere. Here, we discuss some important considerations for the next generation of investiga-tions into kinetic heliophysics. II. DAMPING, DISSIPATION, AND HEATING IN WEAKLY COLLISIONAL PLASMAS A subtle but important issue arises in the investigation of the conversion of the electromagnetic energy of ﬁelds andthe kinetic energy of bulk plasma ﬂows into plasma heat by kinetic physical mechanisms in weakly collisional helio- spheric plasmas. That bottom line is that, unlike in the morewell-known case of ﬂuid systems, in weakly collisional plas- mas, the dissipation of turbulent energy into plasma heat is inherently a two-step process. Fluid systems are derived from the strongly collisional, or small mean free path, limit of the Boltzmann equation inkinetic theory. In this limit, frequent microscopic collisions maintain the Maxwellian equilibrium velocity distributions of local thermodynamic equilibrium. A hierarchy of momentequations may be derived in the limit of small mean free path (relative to the characteristic length scales of gradients in the system) by the Chapman-Enskog procedure 9for neu- tral gases, or an analogous procedure for plasma systems.10 Microscopic collisions in the limit of ﬁnite mean free pathgive rise to the diffusion of velocity ﬂuctuations by viscosity and of magnetic ﬁeld ﬂuctuations by resistivity. Because vis- cosity and resistivity are ultimately collisional, the diffusionof the velocity and magnetic ﬁeld ﬂuctuations by these mechanisms is irreversible, dissipating the kinetic and elec- tromagnetic energy of these ﬂuctuations, and consequentlyrealizing thermodynamic plasma heating and the associated increase of the system entropy. This picture of plasma heat- ing, based on the physical intuition derived from the ﬂuidsystem, implies that energy removed from the velocity and electromagnetic ﬁeld ﬂuctuations through viscosity and resistivity is immediately converted into plasma heat. But in weakly collisional plasmas, the removal of energy from the electromagnetic ﬁeld ﬂuctuations and bulkplasma ﬂows is a separate process from the irreversible con- version of that energy into plasma heat. In fact, the energyremoved by kinetic processes may not all be irreversiblyconverted into heat, but rather some energy may be chan-neled instead into nonthermal particle energization, such asthe acceleration of small fraction of particles to high energy,in apparent deﬁance of the ﬁrst law of thermodynamics. These subtleties require a signiﬁcantly different approach to the study of the dissipation of plasma turbulence and theresulting energization of the plasma under the typicallyweakly collisional conditions of heliospheric plasmas. To consider in more detail the dynamics and dissipation of turbulence in weakly collisional heliospheric plasmas, weturn to the Boltzmann equation which governs the evolutionof the six-dimensional velocity distribution function f s(r,v, t) for a plasma species s @fs @tþv/C1rfsþqs msEþv/C2B c/C20/C21 /C1@fs @v¼@fs @t/C18/C19 coll: (1) Combining a Boltzmann equation for each species with Maxwell’s equations forms the closed set of Maxwell-Boltzmann equations that govern the nonlinear kinetic evolu-tion of a plasma. In the inner heliosphere (within 1 AU of thesun), the typical conditions of the interplanetary plasma leadto a collisional mean free path that is of order 1 AU, approxi-mately 10 8km.5In comparison, the largest scale structures of the interplanetary turbulent cascade have a length scale of10 6km. The upshot is that the collisional term on the right- hand side of (1)is subdominant, not signiﬁcantly affecting the turbulent dynamics on the timescale of the turbulentﬂuctuations. Since the collisional term in (1)is insufﬁcient to dimin- ish the turbulent ﬂuctuations in heliospheric plasmas, theremoval of energy from the turbulent electromagnetic ﬁeldand bulk plasma ﬂow ﬂuctuations occurs through interac-tions between the electromagnetic ﬁelds and the chargedplasma particles, 8,11and these interactions are governed by the Lorentz force term, the third term on the left-hand side of(1). The linear collisionless wave-particle interactions—such as Landau damping, 12,13Barnes damping,14and cyclotron damping15—provide familiar examples of such interactions. But it is important to note that a net transfer of energy fromﬁelds to particles, depleting the energy of electromagneticﬂuctuations and boosting the microscopic kinetic energy ofthe particles, can occur under more general circumstancesthat do not require the persistent presence of waves.Fundamentally, when collisions are weak, the only avenue toremove energy from electromagnetic ﬁeld and bulk plasmaﬂow ﬂuctuations is through collisionless interactionsbetween the ﬁelds and the particles, where the electromag-netic forces do net work on the plasma particles. One of the key fundamental distinctions, compared to the viscous and resistive dissipation in a ﬂuid system, is thatthe net energy transfer between ﬁelds and particles by thework of electromagnetic forces is reversible, with no associ-ated increase in the system entropy. In a kinetic system,Boltzmann’s HTheorem shows that the increase of entropy, and therefore irreversible plasma heating, can only be055907-3 Gregory G. Howes Phys. Plasmas 24, 055907 (2017)accomplished through collisions.16,17So, how can one achieve irreversible heating in a plasma of arbitrarily weakcollisionality? To accomplish irreversible heating requires a subsequent process that enhances the effectiveness of even arbitrarily weak collisions, as explained below. When the collisionless interaction between ﬁelds and particles mediates the removal of energy of the electromag-netic ﬁeld ﬂuctuations, that energy is transferred to the par- ticles, appearing as ﬂuctuations in velocity space of the particle distribution functions. The general form of the colli- sion operator involves second-derivatives in velocity, so the rate of change of the distribution function due to collisionstakes the form 17@f=@t/C24/C23v2 t@2f=@v2/C24/C23=ðDv=vtÞ2f.W e must compare the rate of the collisional evolution to the typi- cal frequency xof the turbulent ﬂuctuations, governed by the other terms of the Boltzmann equation. Even if the colli- sional frequency /C23is arbitrarily small, /C23/C28x, the rate of change of the distribution function due to collisions can com- pete with the frequency of turbulent ﬂuctuations if the scaleof the velocity space ﬂuctuations Dvis sufﬁciently small rel- ative to the typical thermal velocity v t,Dv=vt/C24ð/C23=xÞ1=2. Note that these small ﬂuctuations in velocity space typically contribute little to the ﬁrst moment of the distribution func- tions (which yields the bulk plasma ﬂows and current den-sity), so the turbulent ﬂuctuations are insigniﬁcantly affected by these small ﬂuctuations in velocity space. 7 How do the ﬂuctuations generated by collisionless inter- actions reach sufﬁciently small scales in velocity space that collisions can effectively smooth them out? The answerdepends on the associated spatial length scale of the ﬂuctua- tions. One process is the linear phase mixing governed by the ballistic term of the Boltzmann equation [the second term on the left-hand side of (1)]. Note, however, it has been recently suggested that, for length scales large relative to the thermal Larmor radius of particle species s,k ?qs/C281, an anti-phase mixing mechanism in the presence of turbulencemay prevent these ﬂuctuations in velocity space from reach- ing sufﬁciently small velocity scales, Dv=v ts/C24ð/C23=xÞ1=2,t o be thermalized by weak collisions.18,19At length scales smaller than the Larmor radius, k?qs/H114071, a nonlinear phase- mixing mechanism, arising from differential drifts due to theparticle-velocity-dependent Larmor averaging of the electro- magnetic ﬁelds, 20also known as the entropy cascade,17,21–24 may effectively drive velocity-space ﬂuctuations to sufﬁ- ciently small scales to achieve irreversible heating through collisions. The primary message here is that the physical mecha- nisms governing the damping of turbulent ﬂuctuations and the subsequent irreversible heating in weakly collisional heliospheric plasmas has an inherently different nature from dissipation in the strongly collisional, ﬂuid systems with which most people are more familiar. Kinetic plasma physics plays a central role in the process of particle energization,deﬁning a key frontier in kinetic heliophysics. Below, we will highlight how these key differences motivate powerful new approaches to the study of the ﬂow of energy throughout the heliosphere, approaches that fully utilize the measure- ments routinely made by modern spacecraft missions.III. VELOCITY SPACE: THE NEXT FRONTIER Tackling the six-dimensional phase space of kinetic plasma physics presents the new challenge of interpretingnot only the ﬂuctuations in space and time, as necessary inﬂuid theory as well, but also the dynamics in velocity space.By utilizing the full information content of velocity-space measurements, however, we have the tremendous opportu- nity to realize a transformative leap in our understanding ofkinetic heliophysics. Visualization of the high-dimensionaldatasets of modern spacecraft instrumentation and cutting-edge kinetic numerical simulations represents a new, big-data challenge. Physics-driven reduction of the data is essen-tial for interpreting the results of complicated nonlinearkinetic dynamics, and innovative new analysis methodspromise to shed new light on how particles in differentregions of velocity space contribute to the dynamics. Here, Ipresent some thoughts on exploiting velocity space, the next frontier in kinetic heliophysics. A. New insights lurking in velocity space Spacecraft suffer the inherent limitation that measure- ments are made at only a single point in space (or in the caseof multi-spacecraft missions, a few points in space). But, atthat single point in space, ion and electron instruments canmeasure the full three-dimensional distribution of particle velocities. Velocity space is a messy place, especially in the turbulent state typical of heliospheric plasmas, and althoughthe ﬂuctuations in the particle velocity distribution functionsare hard to interpret, they contain a vast store of informationthat has been largely underutilized. Often spacecraft measurements of the velocity distribu- tions are used to compute moments of the distributions,yielding the density, bulk ﬂow velocity, and (possibly aniso-tropic) temperature of the plasma, while more sophisticatedapproaches may compute other dynamic quantities, such asthe heat ﬂux. For example, over the last ﬁfteen years, severalbreakthrough observational investigations have illuminatedthe role of kinetic temperature anisotropy instabilities in reg-ulating the temperature anisotropy of the solar windplasma. 25–28Kinetic plasma theory29provided critical guid- ance in this case, suggesting that the action of these kinetictemperature anisotropy instabilities is most clearly illustrated on a plot of the ðb k;T?=TkÞplane, often called a Brazil plot because the distribution of measurements of the near-Earthsolar wind plasma on this plane resembles the geographicoutline of Brazil. Velocity space, however, contains far more information about the kinetic dynamics of heliospheric plasmas than just these low-order moments. In particular, velocity spaceretains an imprint of the collisionless interactions betweenthe electromagnetic ﬁelds and the plasma particles, so theinvestigation of the morphology of the velocity distributionfunctions can be used to gain insight into the processeswhich govern the plasma evolution. Early measurements from the Helios spacecraft within 1 AU showed proton velocity distributions with a stronglyanisotropic core (having a characteristic temperature perpen-dicular to the local magnetic ﬁeld that is greater than the055907-4 Gregory G. Howes Phys. Plasmas 24, 055907 (2017)temperature parallel to the ﬁeld) and a signiﬁcant ﬁeld- aligned beam.30Subsequent detailed examinations of the equilibrium proton velocity distribution functions measuredin the solar wind have sought evidence for the quasilineardiffusion of proton distribution functions through pitch anglescattering by ion cyclotron waves 31–35and for the develop- ment of a plateau (quasilinear ﬂattening) in the distributionfunction along the ﬁeld-aligned direction through Landaudamping. 35Currently, proton and electron velocity distribu- tion functions measured at unprecedented phase-space reso-lution and cadence by the Magnetospheric Multiscale (MMS ) mission 36are providing a detailed view of the kinetic plasma dynamics associated with collisionless magneticreconnection in the Earth’s magnetosphere. 37 Indeed, searching for evidence of the quasilinear evolu- tion of the mean velocity distribution functions by examiningstructures in velocity space can provide important cluesabout the kinetic evolution of the plasma, but there is actu-ally much more information contained within the ﬂuctua- tions in velocity space. For example, the Morrison Gtransform 38is an integral transform of the perturbations in the velocity distributionfunction for an electrostatic system. This transform enablesthe perturbation to the distribution function to be written as aweighted sum of Case-Van Kampen modes, a continuousspectrum of solutions to the Vlasov equation. 39,40With rea- sonable assumptions, the Morrison Gtransform can be exploited to reconstruct the spatial dependence of the electricﬁeld from measurements of the perturbed distribution func-tion made at just a single location in space. 41This example illustrates the potential for fully exploiting the informationcontained in the ﬂuctuations in velocity space to gain muchdeeper insight into the kinetic plasma dynamics, an approachthat requires detailed guidance from kinetic plasma theory. Transformative progress can be made in kinetic helio- physics by capitalizing on the power of kinetic plasma theoryto devise insightful new analysis techniques that can beapplied to the high cadence and high phase-space resolutionmeasurements of particle velocity distributions enabled bymodern spacecraft instrumentation. One such promising newmethod is the ﬁeld-particle correlation technique, 8,11 described in Sec. III D. Cutting-edge nonlinear kinetic simu- lations of the plasma dynamics provide a valuable tool bothto test these new techniques under realistic plasma condi-tions and to interpret the results of their application to space-craft measurements. Finally, the development of powerfulnew diagnostics for measuring the velocity distribution func-tions in the laboratory will enable complementary experi-ments that test critical aspects of kinetic physical processesin space plasmas. B. Visualizing velocity space A key challenge for fully utilizing velocity distribution measurements is the visualization and analysis of the high-dimensional data arising from the six-dimensional (3D-3V)phase space of kinetic plasma theory. In particular, nonlinearkinetic numerical simulations are able to compute the fullsix-dimensional velocity distribution functions for eachspecies, f s(r,v,t), at each point in time, resulting in a big- data challenge for the analysis of kinetic heliophysics prob-lems. Six-dimensions is more than can easily visualized, so a physics-driven reduction of this high-dimensional data is essential for the interpretation of the complicated nonlinearkinetic dynamics. Even for spacecraft observations, where particle veloci- ties are measured at only a single point in space as a functionof time, visualizing the three-dimensional velocity distribu- tions can be awkward, but theoretical considerations can point to helpful simpliﬁcations. The recent study by Heet al. , 35for example, presents cross-sections through the three-dimensional proton velocity distribution functions measured by the WIND spacecraft. Physical considerations led them to orient these cross-sections relative to the direc- tions of the solar wind ﬂow velocity and the local magnetic ﬁeld, enabling the characteristic structures in the meanvelocity distribution functions to be more easily seen. But rather than taking cross-sections through a three- dimensional velocity space, which effectively discards thebulk of the 3-V information that lies outside of that cross- section, integrating the data over an ignorable coordinate incorporates the full data set, yielding an improved signal-to-noise ratio. Under the strongly magnetized conditions typical of heliospheric plasmas—speciﬁcally meaning that the typi- cal radius of a particle’s Larmor motion about the magneticﬁeld is much smaller than the length scale of spatial gra- dients in the plasma equilibrium, a condition that is almost always well satisﬁed in space plasmas 16—the local magnetic ﬁeld establishes a preferred direction in the plasma. In this case, the helical motion of a charged particle about the mag- netic ﬁeld, caused by the Lorentz force, is most efﬁcientlyexpressed using cylindrical coordinates for velocity space, ðv ?;h;vkÞ. Here, v?is the velocity perpendicular to the local magnetic ﬁeld, his the angle of the particle’s gyromotion about the magnetic ﬁeld, and vkis the particle velocity paral- lel to the local magnetic ﬁeld. If the characteristic frequencies for the evolution of both the equilibrium and the ﬂuctuations are smaller than the cyclotron frequency, x/C28X, then the distribution function turns out to be gyrotropic ,42meaning that it is independent of the gyrophase angle habout the magnetic ﬁeld, fðv?;vkÞ. Therefore, integrating over the gyrophase angle incorporates all of the data in three-dimensional velocity space to yield anoptimal representation in gyrotropic velocity space ,ðv ?;vkÞ. Note that, in the case of spacecraft measurements, the origin of the velocity-space coordinate system should be centeredat the plasma bulk ﬂow velocity. It is worthwhile noting that, even in cases where the fre- quencies of the ﬂuctuations violate the low frequencyapproximation, x/H11407X, and thereby the physics cannot be described using a gyrotropic model, gyrotropic velocity space ðv ?;vkÞmay still be a useful reduction of the three- dimensional velocity space for visualization. For example, in the case of cyclotron damping, the dynamics are inherently not gyrotropic, but the effect on the distribution function is abroadening of the distribution in the plane perpendicular to the local magnetic ﬁeld, 43,44an impact that can be usefully visualized in gyrotropic velocity space ðv?;vkÞ.055907-5 Gregory G. Howes Phys. Plasmas 24, 055907 (2017)In summary, determining the optimal, physics-based reductions of the six-dimensional (3D-3V) phase space of kinetic plasma theory for important heliophysics problemswill enable better utilization of the full information contentof velocity space. Tackling this challenge to visualize efﬁ-ciently the high-dimensional data will enable the heliophy- sics community not only to maximize the scientiﬁc return from the high phase-space resolution plasma measurementsof current and upcoming spacecraft missions, but also togain deeper insight into the underlying kinetic physicalmechanisms governing the evolution of massively parallel, nonlinear kinetic numerical simulations. C. Spectral decomposition of velocity space In a weakly collisional plasma, valuable insight into the ﬂux of free energy in velocity space can be gained by using an appropriate spectral decomposition of the structure of the perturbations to the velocity distribution function. Thekinetic equation for the evolution of ﬂuctuations in velocityspace parallel to the magnetic ﬁeld is simpliﬁed by recastingthe perturbed distribution functions in terms of Hermite pol- ynomials, an approach exploited in early investigations of kinetic plasma physics. 45–50Speciﬁcally, the linear parallel phase mixing due to the ballistic term in the kinetic equationreduces to a coupling between adjacent Hermite moments,and the Lenard-Bernstein collision operator 51takes on a par- ticularly simple form since its eigenfunctions are the Hermite polynomials. Recent studies of the dissipation of weakly collisional plasma turbulence have exploited thisHermite representation of parallel velocity ﬂuctuations todiagnose the ﬂow of energy through velocity space. 18,19,52–57 Likewise, the perpendicular velocity-space structure arising from nonlinear phase mixing17,20–22can be conveniently rep- resented using a Hankel transform,21–23,58,59enabling the ﬂow of energy to smaller scales in perpendicular velocity tobe diagnosed clearly. Further use of these optimal spectraldecompositions of the structure of ﬂuctuations in velocity space will facilitate greater insights into the nature of particle energization in weakly collisional heliospheric plasmas. D. Field-particle correlations To make the most of the velocity-space information pro- vided by modern spacecraft instrumentation and high- performance kinetic numerical simulations, it is essential todevelop innovative analysis methods that enable us to gaindeeper insight into the grand-challenge problems of kineticheliophysics: turbulence, collisionless magnetic reconnec- tion, particle acceleration, and instabilities. The recently developed ﬁeld-particle correlation technique 8,11employs the electromagnetic ﬁeld ﬂuctuations along with ﬂuctuationsin the particle velocity distribution functions to determinethe energy transfer between the ﬁelds and particles. The idea of using correlated ﬁeld and particle measure- ments to explore the kinetic physics of space plasmas hasfound limited application in the aurora 60–66and the Earth’s magnetosphere67,68using wave-particle correlator instru- ments ﬂown on sounding rockets and spacecraft. A detailedreview of these previous efforts is found in Howes, Kleinand Li. 8These early instrumental efforts largely focused on seeking electron phase-space bunching in ﬁnite amplitudeLangmuir waves, in a regime where the electron count ratewas generally signiﬁcantly lower than the frequency of the Langmuir waves. With the modern instrumentation on current ( MMS 37), upcoming ( Solar Probe Plus69,70andSolar Orbiter71), and proposed ( Turbulence Heating ObserveR ,THOR72) space- craft missions, particle velocity distribution function meas-urements can now be made with unprecedented phase-spaceresolution and at cadences sufﬁcient to resolve the frequen- cies of the electromagnetic turbulent ﬂuctuations involved in the damping of the turbulence and resulting energization ofthe particles. With access now to such high quality velocity-space data from spacecraft observations, and with cutting- edge numerical simulations now capable of simulating the full high-dimensional phase-space of kinetic plasma physics,advanced analysis methods based on kinetic plasma theoryhave the potential to break new ground in our understanding of kinetic heliophysics. The ﬁeld-particle correlation technique was developed to exploit these new instrumental and computational capabil- ities to provide a new window on the kinetic mechanisms at play in heliospheric plasmas. The novel aspect of thismethod is that it determines the energy transfer betweenﬁelds and particles as a function of the particle velocity, yielding a velocity-space signature that characterizes the kinetic mechanism responsible for the energy transfer. This technique was primarily developed to diagnose the particle energization in plasma turbulence as energy is removed from the turbulent magnetic ﬁeld and plasma ﬂow ﬂuctuations through collisionless interactions between theﬁelds and particles. The method, however, is simply based on the equations of nonlinear kinetic plasma theory. 8At the most basic level, collisionless magnetic reconnection, parti-cle acceleration, and kinetic instabilities are simply nonlinearkinetic plasma physics phenomena, mediated by interactions between the electromagnetic ﬁelds and particles. Therefore, the ﬁeld-particle correlation approach is a fundamental wayto explore the evolution of these other processes and theirimpact on the plasma environment (often signiﬁcantly inﬂuencing the large-scale, macroscopic evolution of the system). As emphasized earlier in Sec. II, under the weakly colli- sional conditions relevant to most heliospheric plasmas, the collisional term in the Boltzmann Equation (1)cannot be responsible for the damping of the turbulent ﬂuctuations.Instead, the Lorentz force term, the third term on the left- hand side of (1), governs the collisionless interactions that lead to the net transfer of energy from the electromagneticﬁelds to the microscopic kinetic energy of individual plasmaparticles. Therefore, we may drop the collisional term on the right-hand side of (1)to obtain the Vlasov equation for the following analysis. As an example of the application of the ﬁeld-particle correlation technique, we brieﬂy derive here the appropriate ﬁeld-particle correlation for Landau damping in a 3D, elec-tromagnetic plasma. We begin by multiplying the Vlasov055907-6 Gregory G. Howes Phys. Plasmas 24, 055907 (2017)equation by msv2/2 to obtain an expression for the rate of change of the phase-space energy density @wsr;v;tðÞ @t¼/C0v/C1rws/C0qsv2 2E/C1@fs @v/C0qs cv2 2v/C2BðÞ /C1@fs @v; (2) where the energy density in six-dimensional phase space for a particle species sis given by wsðr;v;tÞ¼msv2fsðr;v;tÞ=2. Under appropriate boundary conditions, such as periodic or inﬁnite spatial boundaries, the ﬁrst and third terms on theright-hand side of (2)yield zero net energy transfer upon integration over all phase-space, including both spatial vol-ume and velocity space. Therefore, this fundamental applica-tion of nonlinear kinetic plasma theory shows that it is thesecond term that is responsible for the net energy transfer between ﬁelds and particles in a collisionless plasma. Since Landau damping is mediated by the component of the elec-tric ﬁeld parallel to the local magnetic ﬁeld, E k, the term that is responsible for the energy transfer from ﬁelds to particlesthrough Landau damping has the form 73 /C0qsv2 k 2@fs @vkEk: (3) Note that the v2¼v2 kþv2 ?factor is reduced to v2 khere because the net energy change is zero for the v2 ?contribution when integrated over velocity. But the term in (3)not only governs the physics of the net transfer of energy to the particles through the collisionlessLandau damping of the electromagnetic ﬂuctuations, but alsocontains a signiﬁcant contribution from the undamped oscilla-tory motion in the plasma that yields no net energization of particles. 8To eliminate this contribution of the oscillatory energy transfer, which often has a larger amplitude than thesecular transfer of energy that does yield a net energization ofparticles, we perform a correlation of the two factors in (3) over a suitably chosen correlation interval s C Ekv;t;sðÞ ¼C/C0qsv2 k 2@fsr0;v;t ðÞ @vk;Ekr0;tðÞ ! :(4) This unnormalized correlation gives the phase-space energy transfer rate between species sand the parallel electric ﬁeld, and retains its functional dependence on velocity space. We emphasize here that this method requires measure- ments of fs(v,t) and EkðtÞat only a single-point in space r0. In order to achieve the cancellation of the oscillatory energy transfer, the measurements simply need to span at least 2 pin the phase of the ﬂuctuations.8Essentially, this method is complementary to the approach used in quasilinear theory,where a spatial integration over all volume is used to elimi-nate any oscillatory contribution; here, we integrate overtime, rather than space, to sample the full 2 pphase of the ﬂuctuations. Note however, that in the presence of ﬂuctua-tions with different characteristic frequencies (for example, with dispersive waves that are common in plasma physics, such as kinetic Alfv /C19en waves, or in a plasma exhibiting broadband turbulent ﬂuctuations), the integration over timeachieves only an approximate cancellation of the oscillatory component, rather than the exact cancellation that is achieved in quasilinear theory using integration over all space. It can be shown, through the integration of term (3)over velocity-space, that the net energy transfer rate to a species s is equivalent to j ksEk, the rate of net work done on the par- ticles by the parallel electric ﬁeld,8an approach previously used with spacecraft observations as a direct measure of the plasma heating.37,74,75However, by not integrating over velocity space, the ﬁeld-particle correlation technique pro- vides much more information than just the net rate of energy transfer to the particles—it provides the distribution of thatenergy transfer in velocity-space, denoted here the velocity- space signature , potentially enabling different mechanisms of energy transfer to be distinguished. E. Example: Velocity-space signature of the Landau damping of a kinetic Alfv /C19en wave Here, I present the application of the ﬁeld-particle corre- lation technique to determine the velocity-space signature ofthe particle energization due to the Landau damping of a kinetic Alfv /C19en wave. A useful reduction of the six-dimensional phase-space of kinetic plasma theory for the modeling of the Landau dampingof kinetic Alfv /C19en waves is the gyrokinetic approximation. The derivation of gyrokinetics, a rigorous low-frequency aniso- tropic limit of kinetic plasma theory, 16,17,76–82systematically averages out the particle cyclotron motion, leading to a reduc- tion of the three-dimensional velocity space ðv?;h;vkÞto the two-dimensional gyrotropic velocity space ðv?;vkÞ. This pro- cedure orders out the fast magnetosonic and whistler waves as well as the cyclotron resonances, but retains ﬁnite Larmor radius effects and the collisionless Landau resonance. We employ here the Astrophysical Gyrokinetics code AstroGK83 to perform a nonlinear gyrokinetic simulation of the Landau damping of a single kinetic Alfv /C19en wave. AstroGK evolves the perturbed gyroaveraged distribu- tion function hs(x,y,z,k,e) for each species s, the scalar potential u, the parallel vector potential Ak, and the parallel magnetic ﬁeld perturbation dBkaccording to the gyrokinetic equation and the gyroaveraged Maxwell’s equations.16,80 Velocity space coordinates are k¼v2 ?=v2ande¼v2/2. The domain is a periodic box of size L2 ?/C2Lk, elongated along the straight, uniform mean magnetic ﬁeld B0¼B0^z, where all quantities may be rescaled to any parallel dimension sat- isfying Lk=L?/C291. Uniform Maxwellian equilibria for ions (protons) and electrons are chosen, with the correct mass ratio mi/me¼1836. Spatial dimensions ( x,y) perpendicular to the mean ﬁeld are treated pseudospectrally; an upwind ﬁnite- difference scheme is used in the parallel direction, z. Collisions employ a fully conservative, linearized collisionoperator with energy diffusion and pitch-angle scattering. 84,85 We initialize a single kinetic Alfv /C19en wave with k?qi¼1.3 for plasma parameters bi¼1 and Ti/Te¼1i na simulation domain of size L?¼2pqi/1.3 and Lk¼L?=/C15, where /C15/C281 is the gyrokinetic expansion parameter. The simulation resolution is ( nx,ny,nz,nk,ne,ns)¼(10, 10, 32, 64, 64, 2). The initialization procedure86speciﬁes the initial055907-7 Gregory G. Howes Phys. Plasmas 24, 055907 (2017)perturbed distribution functions and electromagnetic ﬁelds according to the eigenfunction from the linear collisionlessgyrokinetic dispersion relation 16for the kinetic Alfv /C19en wave. The solution for this kinetic Alfv /C19en wave has a linear fre- quency x/xA¼1.237 and collisionless damping rate c/ xA¼–0.0445, yielding a normalized period TxA¼5.079, where xA¼kkvA¼2pvA=Lkis the characteristic angular frequency associated with crossing the parallel domainlength L kat the Alfv /C19en speed vA. The initialization procedure includes a short linear evolution of ﬁve wave periods with enhanced collisionality /C23i¼/C23e¼0.02xAto eliminate any transients in the initial conditions that do not satisfy the properties of the desired kinetic Alfv /C19en wave. After the lin- ear transient elimination, the nonlinear evolution of the sim-ulation begins with /C23 i¼/C23e¼0.002 xA, leading to weakly collisional conditions with /C23s/x/C2410/C03. As the nonlinear simulation evolves, the distribution functions for each species and the electromagnetic ﬁelds are sampled at one spatial point in the simulation domain. We choose a correlation interval sxA¼10.0, which is approxi- mately equal to two periods of the kinetic Alfv /C19en wave. In Figure 1(a), we present the gyrotropic ðvk;v?Þvelocity-space signature CEkð/C0ðqiv2 k=2Þ@fiðvÞ=@vk;EkÞfor the ions, show- ing the localization in velocity space of the energy transfer between the parallel electric ﬁeld and the ions around the phase velocity of the kinetic Alfv /C19en wave, vk=vti¼ x=ðkkvtiÞ¼1:237 (vertical black line). Similar to the case of the Landau damping of electrostatic ﬂuctuations (Langmuir waves) in a 1D-1 V Vlasov-Poisson plasma,8,11the energy gain (red) by ions with v>x=kkand energy loss (blue) by ions with v<x=kkis a signature of the familiar quasilinear ﬂattening of the distribution function in the parallel directionas a result of Landau damping. In Figure 1(b), we plot the corresponding ﬁeld-particle cor- relation C Ekð/C0ðqev2 k=2Þ@feðvÞ=@vk;EkÞfor the electrons using the same correlation interval sxA¼10.0. One can see a similar localization in velocity space of the energy transfer near the res- onant electron velocity vk=vte¼x=ðkkvteÞ¼0:029 (vertical black line), shown in more detail in Figure 1(c)where we have zoomed into the vkrange containing the resonant energy trans- fer. Also apparent in Figure 1(b) are two broader regions of energy transfer at 0 :5/C20jvk=vtej/C202:0. This component of the energy transfer is odd in vk, and therefore cancels upon integra- tion over vk, leading to no net transfer of energy between ﬁelds and particles. This component ar ises from the incomplete can- cellation of the larger-amplitude oscillating energy transfer, both because the correlation interval sis not exactly an integral multiple of the wave period and because the damping of the wave amplitude leads to incomplete cancellation in the second- half of a wave period. Note that performing the ﬁeld-particlecorrelation analysis at other spatial points in the simulation gives qualitatively the same result. A key point to emphasize about the ﬁeld-particle corre- lation technique is that the distribution of the energy transfer in velocity space is expected to depend on the kinetic mecha- nism of energy transfer. Other physical mechanisms—suchas transit-time damping, 14ion cyclotron damping,43,44sto- chastic ion heating,87–93or collisionless magnetic reconnec- tion54,94–105—are expected to yield velocity-space signaturesthat are qualitatively distinct from that of Landau damping. Ongoing work shows that this ﬁeld-particle correlation tech-nique, when an appropriate correlation interval is chosen, still works in the presence of strong, broadband kinetic plasma turbulence. 73In addition, the same technique can be used to explore the transfer of free energy in kinetic instabil- ities from unstable particle velocity distributions to electro- magnetic ﬂuctuations.106FIG. 1. The gyrotropic ( vk;v?Þvelocity-space signature of Landau damping of a kinetic Alfv /C19en wave with k?qi¼1.3,bi¼1, and Ti/Te¼1 using a correla- tion interval skkvA¼10. (a) The correlation CEkð/C0ðqiv2 k=2Þ@fiðvÞ=@vk;EkÞ for ions shows a clear signature at the resonant velocity, vk=vti¼x=ðkkvtiÞ ¼1:237 (vertical black line). (b) The correlation CEkð/C0ðqev2 k=2Þ@feðvÞ= @vk;EkÞfor electrons shows a signature at the resonant velocity vk=vte¼x= ðkkvteÞ¼0:029 (vertical black line). (c) Zooming into the region /C00:5/C20 vk=vte/C200:5 for CEkfor the electrons, detailing the distribution of the energy transfer near the resonant velocity vk=vte¼x=ðkkvteÞ¼0:029.055907-8 Gregory G. Howes Phys. Plasmas 24, 055907 (2017)IV. THE ROAD AHEAD The increasing availability of high cadence and high phase-space resolution measurements of particle velocitydistributions by spacecraft and of gyrokinetic 3D-2 V or fully kinetic 3D-3 V nonlinear simulations of weakly collisional heliospheric plasmas motivates a concerted effort to developnew methods to maximize the scientiﬁc return from thesehigh-dimensional datasets. Plasma turbulence, magneticreconnection, particle acceleration, and instabilities are four fundamental kinetic plasma processes operating under weakly collisional conditions that signiﬁcantly impact theevolution of the heliosphere. The application of kineticplasma physics to the study of these heliospheric processes is primary driver of the new frontier of kinetic heliophysics . Unlike in a traditional (strongly collisional) ﬂuid, the removal of energy from turbulent ﬂuctuations and conver-sion of that energy to plasma heat is a two-step process under the weakly collisional plasma conditions relevant to many heliospheric environme nts. Learning to handle the high-dimensional phase-space of kinetic plasma theory and to exploit the information contained in velocity space holds the potential for transformational progress in our under- standing of kinetic heliophysical processes. The develop-ment of innovative methods based on kinetic plasmaphysics, such as the ﬁeld-particle correlation techniquehighlighted here, will enable us to gain much deeper insight into the dynamics and energetics of the heliosphere, our home in the universe. And, beyond laying the foundation offundamental knowledge needed to construct a predictivecapability for heliophysics phenomena, such as extreme space weather, advances in our understanding of fundamen- tal physics through in situ measurements of heliospheric plasmas may be applied to better comprehend the dynamicsof more remote or extreme astrophysical systems that lie out of reach of direct measurements. Can we go further than some of the new directions dis- cussed here to exploit the information contained in velocityspace of kinetic theory? New capabilities enable fundamen- tal aspects of the kinetic physics of space plasmas to the explored in the laboratory under controlled or reproducibleconditions. The development of improved experimentaldiagnostics to measure the particle velocity distribution func-tions will enable some of the novel kinetic plasma physics methods endorsed here to be applied in a laboratory setting. Can machine learning, coupled with sufﬁcient physicsinsight from kinetic plasma theory, be used to discover pat-terns in the high-dimensional phase space of kinetic plasma theory? And, of course, our urgent need to understand these essentially microphysical processes—turbulence, reconnec-tion, particle acceleration, and instabilities—is motivated bytheir effect on the macroscopic evolution of the heliosphere, in particular, their impact on Earth and society. Using our reﬁned knowledge of these kinetic physical mechanisms, wemay attempt to build next-generation models that coupletheir impact to the global evolution of the heliosphere,enabling us to treat near-Earth space, and other heliospheric environments, as complex systems. Efforts to tackle the multi-scale problem of heliophysics through a rigorousconnection between the kinetic physics at microscales and the self-consistent evolution of the heliosphere at macro-scales will propel the ﬁeld of kinetic heliophysics into thefuture. 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1.4977974.pdf	Antiferromagnetic spin current rectifier Roman Khymyn , Vasil Tiberkevich , and Andrei Slavin Citation: AIP Advances 7, 055931 (2017); doi: 10.1063/1.4977974 View online: http://dx.doi.org/10.1063/1.4977974 View Table of Contents: http://aip.scitation.org/toc/adv/7/5 Published by the American Institute of Physics Articles you may be interested in Comparative determination of Y3Fe5O12/Pt interfacial spin mixing conductance by spin-Hall magnetoresistance and spin pumping AIP Advances 110, 062402062402 (2017); 10.1063/1.4975704 Effect of NiO inserted layer on spin-Hall magnetoresistance in Pt/NiO/YIG heterostructures AIP Advances 109, 032410032410 (2016); 10.1063/1.4959573 Spin transport in antiferromagnetic NiO and magnetoresistance in Y3Fe5O12/NiO/Pt structures AIP Advances 7, 055903055903 (2016); 10.1063/1.4972998 Magneto-Seebeck effect in magnetic tunnel junctions with perpendicular anisotropy AIP Advances 7, 015035015035 (2017); 10.1063/1.4974972 Magnon-photon coupling in antiferromagnets AIP Advances 110, 082403082403 (2017); 10.1063/1.4977083 Spin Seebeck effect in insulating epitaxial γ-Fe2O3 thin films AIP Advances 5, 026103026103 (2017); 10.1063/1.4975618AIP ADV ANCES 7, 055931 (2017) Antiferromagnetic spin current rectiﬁer Roman Khymyn,aVasil Tiberkevich, and Andrei Slavin Department of Physics, Oakland University, Rochester, Michigan 48309, USA (Presented 3 November 2016; received 23 September 2016; accepted 28 November 2016; published online 2 March 2017) It is shown theoretically, that an antiferromagnetic dielectric with bi-axial anisotropy, such as NiO, can be used for the rectiﬁcation of linearly-polarized AC spin cur- rent. The AC spin current excites two evanescent modes in the antiferromagnet, which, in turn, create DC spin current ﬂowing back through the antiferromag- netic surface. Spin diode based on this effect can be used in future spintronic devices as direct detector of spin current in the millimeter- and submillimeter- wave bands. The sensitivity of such a spin diode is comparable to the sensitiv- ity of modern electric Schottky diodes and lies in the range 102-103V/W for 3030 nm2structure. © 2017 Author(s). All article content, except where oth- erwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4977974] Recent advances in understanding of fundamental spin-dependent phenomena in solid state put the development of ﬁrst prototypes of spintronic devices within experimental reach. Such devices will operate using pure spin currents, which are dissipationless and, therefore, can provide much more energy efﬁcient information processing than current electron-based devices. However, realization of practically useful spintronic circuits requires development of spintronic analogs of various nonlinear electronic components, such as spin diodes and spin transistors. One of the widely used elements in the modern electronics is a quadratic detector of AC currents, which is often based on a Schottky diode.1,2The quadratic detector performs the rectiﬁcation of the microwave input signal into the output DC voltage VDC, which is propotional to the power of the input signal PAC. Consequently, one of the main characteristics of such a rectiﬁer is the sensitivity R=VDC=PAC, which typically lies in the range of 102-104V/W for detectors based on Schottky diodes.1Another important characteristic of a diode is the frequency range of operation (typically, from 1 GHz to 1 THz). The closest spintronic analog of a quadratic detector is a diode based on the spin-transfer torque (STT) diode effect in ferromagnetic tunnel junctions.3Although the STT-based diodes can have rather high sensitivity, they can not operate using pure spin currents, and, therefore, are not particularly well-suited as building elements of future spintronic circuits. In addition, the high sensitivity of STT diodes is achieved only in a rather narrow frequency region near the ferromagnetic resonance (FMR) frequency, which may be a serious drawback of such devices. Also, the FMR frequency of magnetic materials is determined, mainly, by the applied bias magnetic ﬁeld and it is practically impossible to increase it above several tens of GHz. One possible way of realization of spintronic detectors with large frequency range, comparable to that of Schottky diodes, is to use antiferromagnetic (AFM) materials as an active medium of the detectors. AFM materials have natural eigen-frequencies of spin excitations lying in the sub-THz to THz frequency range, do not require bias magnetic ﬁeld, and often have rather low intrinsic damping, which makes the antiferromagnets very attractive for use in spintronic devices. Although the investi- gation of spintronic phenomena in antiferromagnets are still at initial stage, recent experimental and theoretical studies showed rather encouraging results. For example, it was demonstrated that a thin layer of nickel oxide (NiO) – AFM dielectric with bi-axial magnetic anisotropy – can efﬁciently trans- fer pure spin currents generated by magnetization precession in an adjacent ferromagnetic layer.4–6 aElectronic mail: khiminr@gmail.com 2158-3226/2017/7(5)/055931/6 7, 055931-1 ©Author(s) 2017 055931-2 Khymyn, Tiberkevich, and Slavin AIP Advances 7, 055931 (2017) Also, the recent theoretical work predicted a possibility of developing a THz-range oscillator based on spin-current-induced precession in AFM dielectrics.7,8 In this work we show theoretically that an AFM dielectric with bi-axial crystallographic anisotropy (e.g., NiO) can be used as a rectiﬁer of linearly-polarized AC spin currents, transforming them into a DC spin voltage. The proposed rectiﬁer demonstrates high sensitivity, comparable to the sensitivity of electrical diodes, in a wide frequency range – from very low frequencies up to the frequency of the AFM resonance ( 200 GHz for NiO). The bi-axial anisotropy of the AFM, which enables the angular momentum exchange between the spin subsystem and the crystallographic lattice of the AFM, plays a crucial role in the rectiﬁcation process, and the DC voltage vanishes in the limit of uniaxial anisotropy. The predicted rectiﬁcation effect does not require use of thin AFM layers and can be observed even in bulk AFM samples, which signiﬁcantly simpliﬁes its experimental observation. We consider a device, schematically shown in Fig. 1(a), which consists of a layer of bi-axial AFM, driven by the input AC spin current jACﬂowing from the adjacent layer of a normal metal (NM). For convenience, we shall use electrical units for both the spin current density (A/m2) and spin voltage (V). Then, the input spin current density jACcan be related to the AC spin voltage VAC in NM as jAC=G"#VAC, where G"#is the spin mixing conductance of the NM/AFM interface.9 Respectively, the power of the input signal can be evaluated as PAC=SjACVAC=Sj2 AC=G"#, where Sis the cross-section area of the device. We assume that the spin-polarization of the input spin current is perpendicular to the easy axis (e3) of the AFM and, thus, jACcan be written as jAC=jAC(e1cos+e2sin)sin!t, (1) where jACis the amplitude of the spin current, e1,2are the intermediate and hard axes of the AFM, andand!are, respectively, the polarization angle and frequency of the input current. We shall demonstrate below, that, under such conditions, the spin dynamics in the AFM generates a DC spin current jDCthat ﬂows back into the NM. The output current is polarized along the easy AFM axis e3and has the form FIG. 1. (a) The scheme of the AC-DC spin current rectiﬁer, (b) The proposed experimental sample for the observation of the spin current rectiﬁcation.055931-3 Khymyn, Tiberkevich, and Slavin AIP Advances 7, 055931 (2017) jDC=e3Bj2 AC, (2) where the efﬁciency Bdepends on the AFM properties, angle and frequency !, but is independent of the properties of the NM/AFM interface. The output current (2) corresponds to the output spin voltage VDC=jDC=G"#and, accordingly, the sensitivity of the spin rectiﬁer can be found as R=VDC=PAC=B=S. (3) Calculation of BandRfor realistic parameters of NiO-based spin rectiﬁer is presented below and is the main result of this paper. Here we shall use the approach and notations used in Ref. 10, where the spin dynamics of an AFM dielectric under the action of an input spin current was studied in detail. The AFM dynamics can be described using the unit-length Neel vector (or vector of antiferromagnetism) l= (M1 M2)/(2Ms), where M1,2are the magnetization vectors of the two AFM sublattices, and Ms=\|M1,2\| is the sublattice saturation magnetization ( Ms'350 kA/m for NiO at room temperature). The dynamics of the vector lis described by the second-order equation11 l @2l @t2c2@2l @y2+ˆ 2l+! ex@l @t! (4) = ~!ex 2eMsl(ljAC)(y) . Here c'38 km/s is the speed of AFM magnons, ˆ 2=diag(!2 1,!2 2, 0) is the matrix of crystallographic anisotropy (in frequency units), !1=2'220 GHz and !2=2'1.1 THz are the frequencies of the AFM resonances,12'6104is the Gilbert damping constant, and !ex'27.5 THz is the exchange frequency.11,13The coordinate yis the direction of spin current propagation and the NM/AFM inter- face is located at y= 0. The direction of the spin current propagation yin the AFM is completely unrelated to the anisotropy axis e1,e2, because the spin and the spatial degrees of freedom are inde- pendent of each other. The term in the right-hand side of (4) describes the inﬂuence of the input spin current, and effectively determines the boundary conditions at the NM/AFM interface. The boundary conditions at the other AFM interface, y=d, are chosen in the form @l=@y=0 , (5) which, physically, means that no spin current ﬂows through the y=dsurface. Here we shall consider the case of relatively low input frequencies, !<! 1. Our numerical analysis shows that in this case the inﬂuence of the Gilbert damping on the AFM dynamics is negligible, and, for simplicity, the damping term will be ignored in the following. Also, we assume that the input spin current is relatively weak and (4) can be solved to the linear order in jAC. Then, the solution of (4) and (5) has the form l(t,y)=e3+[e1l1(y)+e2l2(y)] sin!t, where lj=ajcosh(( dy)=j) cosh( d=j), (6) where a1= ~!ex1 2eMsc2sinjAC, (7) a2= ~!ex2 2eMsc2cosjAC, (8) and j=c=q !2 j!2. (9) In an AFM with bi-axial anisotropy the rates of the spatial decay 1,2of spin excitations along axes e1,2are not equal, and, as a result, the vectors land@l=@yare not parallel to each other. Consequently, the total spin current in the AFM,10 j=4eMsc2 ~!ex @l @yl! (10) acquires a non-zero DC component orthogonal to both land@l=@y, i.e., it becomes polarized along the easy AFM axis e3. This generated DC current has the form (2) with the intrinsic efﬁciency055931-4 Khymyn, Tiberkevich, and Slavin AIP Advances 7, 055931 (2017) B= ~!ex ec2Mssin(2)[2tanh( d=1)1tanh( d=2)]. (11) This expression clearly shows that for 1=2(uniaxial anisotropy) B= 0, and the DC rectiﬁed voltage is absent. Also, as it follows from (11), the output DC current reaches maximum at ==4 – in the case when the input current excites two orthogonal AFM modes (7) with equal amplitudes, i.e., the output current is proportional to the product of the mode amplitudes a1a2. Another important consequence that follows from (11) is that the rectiﬁed voltage does not vanish in the limit of thick AFM layer d!1 . In this case the efﬁciency Bapproaches the limiting value B1= ~!ex ec2Mssin(2)(21). (12) This property is in a striking contrast with the previously observed properties of the spin transfer through AFM dielectrics which requires nm-thick AFM layers, and allows one to observe the spin rectiﬁcation effect even in bulk single-crystal AFM samples. In Fig. 2 we showed the dependence of the spin diode sensitivity Ron the thickness of the AFM layer for NiO spin diode with cross-section area S=3030 nm2at low input frequency (!=2=10 GHz). As one can see, there is a weak maximum of the sensitivity at d10 nm, but the sensitivity remains rather large ( R1=463 V/W) even for thick AFM layers. Fig. 3 shows the dependence of the sensitivity Ron the frequency !of the input spin current for a thick AFM layer. The sensitivity (and, consequently, the output DC spin current) increases with the increase of !and exceeds 1000 V/W for frequencies close to the AFM resonance frequency !1=2=220 GHz. Such a high value of the sensitivity is comparable with the sensitivity of modern FIG. 2. The dependence of the sensitivity Rof NiO-based spin rectiﬁer on the thickness dof the AFM layer at !=2=10 GHz andS=3030 nm2. FIG. 3. The dependence of the sensitivity Rof NiO-based spin rectiﬁer on the frequency !of the input spin current.055931-5 Khymyn, Tiberkevich, and Slavin AIP Advances 7, 055931 (2017) electrical rectiﬁers based on Schottky diodes. The increase of the efﬁciency of the spin rectiﬁer near !!1is explained by the divergence of the decay length 1near this point. The above calculated sensitivity of the proposed spin current rectiﬁer relates to all-spintronic regime of operation, when both input and output signals are pure spin currents and spin voltages. Nowadays, spintronic experiments are usually performed using electrical injection of spin currents by means of the spin Hall effect in heavy metals (typically, Pt) and electrical detection of the output spin currents via the inverse spin Hall effect. The efﬁciency of electrical generation and detection of spin currents depends on the spin Hall angle SHin a normal metal, which is rather low ( SH'0.1 in Pt14). Respectively, the efﬁciency of the proposed AFM spin diode in such schemes will be too low to be interesting for practical applications. Nevertheless, the standard setups employing spin Hall effect can be used for experimental observation and veriﬁcation of the rectiﬁcation effect predicted here. The simplest possible scheme of experimental observation of the AFM spin rectiﬁcation is shown in Fig. 1(b) and consists of a bulk AFM sample with an adjacent Pt strip. When an AC electric current is passed through the Pt line it will generate, via the spin Hall effect, an AC spin current ﬂowing into the AFM sample. The rectiﬁed DC spin current has the orthogonal spin polarization and will induce, via the inverse spin Hall effect, an electric ﬁeld in the direction perpendicular to the Pt line. Thus, in the setup Fig. 1(b) the spin rectiﬁcation effect will lead to the appearance of a ﬁnite DC electric voltage at the opposite sides of the Pt strip, which can be easily measured experimentally. The maximum output voltage will be observable when the Neel vector lis parallel to the electrical current jcand the hard axis of the AFM e2makes an angle ==4 with the interface. To estimate the magnitude of the induced DC voltage, we take the width of the Pt strip (distance between output electrodes) to be equal to L=100m, and the Pt thickness dPt=20 nm. In this case, the AC spin current ﬂowing into AFM can be calculated as15 jAC=jcG"#SHPttanh( dPt=2Pt) , (13) where jcis the density of electric current ﬂowing in Pt, SH=0.1 is the spin-Hall angle, Pt=7.3 nm is the spin diffusion length, =4.8107 m is the Pt resistivity,14andG"#=2.61014 1m2is the spin-mixing conductance of the Pt/NiO interface.16The output electric voltage is related to the output spin current by17 Vc=jDCLSHPttanh( dPt=2Pt)=dPt. (14) Using the above equations, one can estimate the output electric voltage to be Vc=41V at the input AC electric current density jc= 107A/m2, frequency !=2=10 GHz, and polarization angle ==4. Such an output DC voltage should be easily observable experimentally. In conclusion, we showed, that an AFM with biaxial anisotropy can be used as an active element for the AC-DC conversion of spin currents. The sensitivity of such a spin diode is in the range of 102103V/W, which is comparable with the modern electrical Schottky diodes. Both the sign and magnitude of the rectiﬁed DC current are determined by the mutual arrangement of the crystallo- graphic axes of the AFM and the direction of polarization of the input AC spin current. Thus, the maximum efﬁciency of the AC-DC spin conversion is achieved when the AC spin current polarization is perpendicular to the Neel vector of the AFM, and forms the angle of =4 with both axes of magnetic anisotropy. The presence of the bi-axial anisotropy is crucial for the spin rectiﬁcation, since the effect is caused by the angular momentum exchange between the spin sub-system and the crystal lattice of the AFM, and is absent in uniaxial AFM materials. The effect can be easily observed experimentally using the electric injection and detection of spin currents via the spin Hall effects in Pt/NiO bilayers. ACKNOWLEDGMENTS This work was supported in part by Grant No. EFMA-1641989 from the National Science Foundation of the USA, by the contract from the US Army TARDEC, RDECOM, and by the Center for NanoFerroic Devices (CNFD) and the Nanoelectronics Research Initiative (NRI). 1J. L. Hesler and T. W. Crowe, in 2007 Joint 32nd International Conference on Infrared and Millimeter Waves and the 15th International Conference on Terahertz Electronics (IEEE, 2007), pp. 844–845. 2B. Sharma, Metal-semiconductor Schottky barrier junctions and their applications (Springer Science & Business Media, 2013).055931-6 Khymyn, Tiberkevich, and Slavin AIP Advances 7, 055931 (2017) 3O. Prokopenko, G. Melkov, E. Bankowski, T. Meitzler, V . Tiberkevich, and A. Slavin, Applied Physics Letters 99, 032507 (2011). 4H. Wang, C. Du, P. C. Hammel, and F. Yang, Physical review letters 113, 097202 (2014). 5C. Hahn, G. De Loubens, V . V . Naletov, J. B. Youssef, O. Klein, and M. Viret, EPL (Europhysics Letters) 108, 57005 (2014). 6Z. Qiu, J. Li, D. Hou, E. Arenholz, A. T. NDiaye, A. Tan, K.-i. Uchida, K. Sato, S. Okamoto, Y . Tserkovnyak et al. , Nature Communications 7(2016). 7E. V . Gomonay and V . M. Loktev, Low Temp. Phys. 40, 17 (2014). 8R. Cheng, D. Xiao, and A. Brataas, Physical Review Letters 116, 207603 (2016). 9Y . Tserkovnyak, A. Brataas, and G. E. Bauer, Physical Review B 66, 224403 (2002). 10R. Khymyn, I. Lisenkov, V . S. Tiberkevich, A. N. Slavin, and B. A. Ivanov, Phys. Rev. B 93, 224421 (2016). 11T. Satoh, S.-J. Cho, R. Iida, T. Shimura, K. Kuroda, H. Ueda, Y . Ueda, B. A. Ivanov, F. Nori, and M. Fiebig, Physical Review Letters 105, 077402 (2010). 12M. T. Hutchings and E. J. Samuelsen, Physical Review B 6, 3447 (1972). 13A. J. Sievers and M. Tinkham, Physical Review 129, 1566 (1963). 14H. L. Wang, C. H. Du, Y . Pu, R. Adur, P. C. Hammel, and F. Y . Yang, Physical Review Letters 112, 197201 (2014). 15Y .-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. Goennenwein, E. Saitoh, and G. E. Bauer, Physical Review B87, 144411 (2013). 16R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Physical review letters 113, 057601 (2014). 17H. Nakayama, K. Ando, K. Harii, T. Yoshino, R. Takahashi, Y . Kajiwara, K. Uchida, Y . Fujikawa, and E. Saitoh, Physical Review B 85, 144408 (2012).
1.5114789.pdf	J. Chem. Phys. 151, 121102 (2019); https://doi.org/10.1063/1.5114789 151, 121102 © 2019 Author(s).Discovery of blue singlet exciton fission molecules via a high-throughput virtual screening and experimental approach Cite as: J. Chem. Phys. 151, 121102 (2019); https://doi.org/10.1063/1.5114789 Submitted: 11 June 2019 . Accepted: 28 August 2019 . Published Online: 24 September 2019 Collin F. Perkinson , Daniel P. Tabor , Markus Einzinger , Dennis Sheberla , Hendrik Utzat , Ting-An Lin , Daniel N. Congreve , Moungi G. Bawendi , Alán Aspuru-Guzik , and Marc A. Baldo ARTICLES YOU MAY BE INTERESTED IN Morphology independent triplet formation in pentalene films: Singlet fission as the triplet formation mechanism The Journal of Chemical Physics 151, 124701 (2019); https://doi.org/10.1063/1.5097192 Defect energy levels in carbon implanted n-type homoepitaxial GaN Journal of Applied Physics 126, 125301 (2019); https://doi.org/10.1063/1.5109237 Damage evolution in LiNbO 3 due to electronic energy deposition below the threshold for direct amorphous track formation Journal of Applied Physics 126, 125105 (2019); https://doi.org/10.1063/1.5116667The Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp Discovery of blue singlet exciton fission molecules via a high-throughput virtual screening and experimental approach Cite as: J. Chem. Phys. 151, 121102 (2019); doi: 10.1063/1.5114789 Submitted: 11 June 2019 •Accepted: 28 August 2019 • Published Online: 24 September 2019 Collin F. Perkinson,1 Daniel P. Tabor,2,a) Markus Einzinger,3Dennis Sheberla,2,b) Hendrik Utzat,1 Ting-An Lin,3Daniel N. Congreve,4Moungi G. Bawendi,1,c)Alán Aspuru-Guzik,2,5,c) and Marc A. Baldo3,c) AFFILIATIONS 1Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA 3Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 4Rowland Institute at Harvard University, Cambridge, Massachusetts 02142, USA 5Department of Chemistry and Department of Computer Science, University of Toronto, Toronto, Ontario M5S 3H6, Canada Note: This paper is part of the JCP Special Collection on Singlet Fission. a)Current address: Department of Chemistry, Texas A&M University, College Station, Texas 77843, USA. b)Current address: Kebotix, Inc., Cambridge, MA 02139, USA. c)Authors to whom correspondence should be addressed: mgb@mit.edu; alan@aspuru.com; and baldo@mit.edu ABSTRACT Singlet exciton fission is a mechanism that could potentially enable solar cells to surpass the Shockley-Queisser efficiency limit by converting single high-energy photons into two lower-energy triplet excitons with minimal thermalization loss. The ability to make use of singlet exciton fission to enhance solar cell efficiencies has been limited, however, by the sparsity of singlet fission materials with triplet energies above the bandgaps of common semiconductors such as Si and GaAs. Here, we employ a high-throughput virtual screening procedure to discover new organic singlet exciton fission candidate materials with high-energy ( >1.4 eV) triplet excitons. After exploring a search space of 4482 molecules and screening them using time-dependent density functional theory, we identify 88 novel singlet exciton fission candidate materials based on anthracene derivatives. Subsequent purification and characterization of several of these candidates yield two new singlet exciton fission materials: 9,10-dicyanoanthracene (DCA) and 9,10-dichlorooctafluoroanthracene (DCOFA), with triplet energies of 1.54 eV and 1.51 eV, respectively. These materials are readily available and low-cost, making them interesting candidates for exothermic singlet exciton fission sensitization of solar cells. However, formation of triplet excitons in DCA and DCOFA is found to occur via hot singlet exciton fission with excitation energies above ∼3.64 eV, and prominent excimer formation in the solid state will need to be overcome in order to make DCA and DCOFA viable candidates for use in a practical device. ©2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5114789 .,s INTRODUCTION Introduction to singlet exciton fission Singlet exciton fission is a down-conversion process in organic semiconductors that spontaneously converts one spin- singlet electron-hole pair (exciton) into two spin-triplet excitons.1 Each triplet exciton carries approximately half the energy of theinitial singlet exciton. Conventional single-junction solar cells are limited in efficiency to about 34% (the Shockley-Queisser limit), largely due to loss from unabsorbed below-bandgap photons and thermalization of high-energy excitons.2When combined with a lower-bandgap semiconductor, singlet exciton fission materials raise the theoretical efficiency limit of a single-junction solar cell by reducing thermalization of excitons generated by high-energy J. Chem. Phys. 151, 121102 (2019); doi: 10.1063/1.5114789 151, 121102-1 © Author(s) 2019The Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp photons. It has been calculated that the maximum power conversion efficiency of a single-junction photovoltaic device incorporating a layer of materials that can undergo singlet exciton fission is 44.4%.3 On its own, singlet exciton fission yields no advantage to the power efficiency of solar cells because the potential increase in pho- tocurrent is matched by a decrease in the open circuit voltage.4 A benefit can be realized, however, if a singlet fission material is matched with a second material that absorbs low-energy photons. For example, in combination with silicon, a singlet exciton fission material ideally absorbs all photons with energies greater than twice the silicon bandgap.5The resulting excitons are split into two exci- tons at or just above the silicon bandgap and transferred to silicon, where they supplement silicon photocurrent generated from direct absorption of photons with energies between the silicon bandgap and twice the silicon bandgap.6 Singlet exciton fission requires that the energy of the singlet exciton is approximately twice the energy of the triplet exciton. The exchange energy splitting between singlet and triplet excitons scales with the degree of overlap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital.7As a result, most polyacene molecules exhibit singlet-triplet exchange energies of approximately 1.0–1.3 eV.8The ideal singlet exciton energy of a singlet fission material is therefore approximately 2.0–2.6 eV; above this singlet energy, fission is typically exothermic, while below this singlet energy, fission is typically endothermic. Indeed, the best- known blue fission material, anthracene, is only capable of hot fis- sion, an inefficient process where the dissociation of singlet excitons competes with internal conversion and vibrational relaxation.9,10 Modifying anthracene by incorporating chemical side groups can perturb its singlet and triplet energies to make singlet exciton fis- sion thermodynamically feasible directly from the S1singlet state. To date, however, such anthracene derivatives have reported triplet energies around 1.1–1.2 eV, providing minimal enthalpic driving force for sensitization of silicon solar cells.11,12 The bluest, efficient class of singlet exciton fission materials is based on a core of tetracene, with a singlet exciton energy of approximately 2.4 eV. Fission of these excitons yields triplets with energies almost identical in energy to the silicon bandgap.1,13At these energies in the short wavelength infrared, molecules are typ- ically weakly luminescent. Thus, coupling dark triplet excitons to photons for sensitization of photovoltaics requires the use of inor- ganic materials such as PbS nanocrystals, but with minimal ener- getic allowance for their Stokes shift or the spectral width of their photoluminescence.2,14–16 To improve the prospects of radiative coupling of triplet exci- tons to silicon, here we employ high-throughput virtual screening (HTVS) to search for blue exciton fission materials based on an anthracene core. We seek to find viable singlet exciton fission candi- dates with triplet exciton energies in excess of 1.4 eV. The triplet energy threshold of 1.4 eV is chosen because it should allow for exothermic transfer of the triplet excitons to silicon (with a bandgap of about 1.1 eV), even after accounting for uncertainty in calculated triplet energies. Introduction to high-throughput virtual screening High-throughput virtual screening (HTVS) combines quantum chemical calculations and cheminformatics methods to reduce alarge molecular space to a set of promising leads that experimen- tal chemists can then synthesize and characterize.17–26Functional organic molecules are particularly well-suited for high-throughput virtual screening, since the important properties of the materials can be approximated by studying individual molecules in the material, often with relatively computationally inexpensive methods, such as density functional theory (DFT).18,19 RESULTS Library generation In order to limit our parameter space, we focus on known molecules with an anthracene core and a smaller fraction of combi- natorically generated anthracene derivatives, in contrast to the large combinatorial fragment-based libraries that are often employed in high-throughput virtual screening projects for organic mate- rials.11,27–29The chemical space of this study consists of known and commercially available molecules. The libraries are obtained by searching eMolecules and Reaxys databases for molecules with an anthracene substructure. A total of 4482 candidates are examined. Calculation method and benchmark Before running full-scale calculations, we benchmark several methods to predict S1and T1values. In order to test the calcu- lations, we use a dataset of 26 published molecules with experi- mentally determined S1and T1energies. Since a large contribu- tion from a multireference character in the ground state is not expected, standard hybrid DFT methods are employed. The over- all pipeline for calculations is outlined in Fig. 1(a). At the first stage, molecules encoded as Simplified Molecular-Input Line-Entry Sys- tem (SMILES) strings from the generated library are fed into a conformer generator. The conformer generator samples conform- ers using a random distance matrix method, as implemented in RDKit.30The samples generated are optimized using the MMFF94 force field31and duplicates are eliminated. Next, the conformers are optimized using the DFT-B3 method,32which gives improved ground state geometries for pi-conjugated systems compared to the MMFF94 force field, and duplicates are again eliminated. The DFT calculations are performed using the OChem 4.0 package.33The DFT-B3 conformers are optimized at the B3LYP/6-31G(d) level of theory for both singlet and triplet ground state electron config- urations. Finally, excited state calculations are performed on the S0geometry of the optimized conformers with three commonly used functionals: hybrid B3LYP,34range-separated ωB97X-D,35and range-separated ωLC-PBE0,36,37using the 6-31G(d) basis set. For the calibration dataset, B3LYP/6-31G(d) outperforms both of the range-separated functionals with respect to predict- ing S1and T1energies (Figs. S1 and S2). We compare two approaches of calculating T1energies: vertical time-dependent DFT and a method where triplet energies are estimated by calculat- ing the energy difference between the optimized structures at the ground state and the triplet excited state (calculated using open- shell DFT). The latter performs better in predicting T1energies (Fig. S2). With these considerations in mind, we select the pipeline depicted in Fig. 1(a) for calculations of the full candidate material library. J. Chem. Phys. 151, 121102 (2019); doi: 10.1063/1.5114789 151, 121102-2 © Author(s) 2019The Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp FIG. 1 . Calculation pipeline and screening criteria for singlet exciton fission candidates. (a) Computational workflow. (b) Density plot showing the calculated electronic structure of 4482 singlet exciton fission candidate materials resulting from the search from eMolecules. The shaded region indicates the molecules for which S1= 2T1±0.2 eV and T2>2T1, which are the first two criteria applied to screen singlet exciton fission candidates. (c) Distribution of T1adiabatic energies for the molecules in the shaded region of (b). The vertical dashed line is at 1.4 eV, an approximate threshold for triplet energy transfer to silicon, when accounting for uncertainty in the molecular candidate excited state calculations. Computational screening results In this section, we describe our successive criteria for narrowing the library of 4482 molecules to the most promising blue-absorbing singlet exciton fission candidates for exothermic coupling to silicon solar cells. The first two criteria [depicted in Fig. 1(b)] are related to maximizing the energy level compatibility of the S1andT1states for singlet exciton fission. In Fig. S3(a), the distributions of calculated S1andT1energies are jointly plotted. The shaded region indicates the region where the differences between the S1energy and twice the T1energy are within 0.2 eV of each other, which we use as the cutoff for compatibility, given the observation that singlet exciton fission in tetracene is about 0.2 eV uphill.38About 20% of the library satisfies this criterion (929 molecules). The second criterion, which evaluates the difference between the T2andT1energies, is motivated by the desire to avoid molecules for which two T1excitons could upcon- vert into a T2exciton. The shaded region in Fig. 2(b) shows the 541 molecules for which T2>2T1and for which the S1/T1criterion is satisfied. Next, we screen for materials with T1energies above 1.4 eV, reducing the number of eligible candidates to 157 [Fig. 1(c)]. Ofthe 384 molecules that are removed at this stage, the vast majority contain tetracene or anthraquinone substructures. To maximize the probability that the molecules will perform well in an aggregate structure, we seek to minimize the variance in the conformer excited state energies. Screening for molecules where the range of S1and T1adiabatic excitation energies is less than 0.05 eV further reduces the number of eligible candidate materials to 116 (Figs. S4 and S5). As expected from chemical principles, most of the molecules excluded at this stage contain multiple rotatable bonds, often to another aromatic ring. We next screen for molecules with S1energies predominantly in the blue but not in the UV, since such molecules are better suited to the solar spectrum and may be less susceptible to photoinduced chemical degradation. This corresponds to an S0→S1transition between 2.64 eV and 3.26 eV. Of the 116 molecules remaining before this criterion, all absorb in this range (Fig. S6). However, this crite- rion would exclude some molecules if a lower T1cutoff energy (such as 1.2 eV) is employed for compatibility to Si, and it could also be a useful screening criterion for larger libraries. Finally, to narrow the focus of our study to true anthracene derivatives, we filter out the remaining molecules containing more J. Chem. Phys. 151, 121102 (2019); doi: 10.1063/1.5114789 151, 121102-3 © Author(s) 2019The Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp FIG. 2 . Steady-state optical characterization of DCA and DCOFA. Chemical structures of (a) DCA and (b) DCOFA. Absorbance and fluorescence of dilute solutions of (c) DCA in cyclohexane and (d) DCOFA in chloroform. (e) Triplet phosphorescence of a drop-cast film of DCA doped at 2 wt. % in 4BrPS, obtained at 77 K, using out-of-phase optical choppers to filter out prompt singlet emission. (f) Triplet phosphorescence of a drop-cast film of DCOFA doped at 2 wt. % in [PMMA:BP] 3:1at room temperature. Delayed singlet emission resulting from triplet-triplet annihilation is subtracted from the original spectra to obtain these results, as detailed in Fig. S7. The rising edges of the phosphorescence indicate triplet energies of 1.54 eV and 1.51 eV for DCA and DCOFA, respectively. than five fused rings, leaving us with 88 singlet exciton fission candidate materials, approximately 2.0% of our original library. Experimental characterization of singlet exciton fission candidates From the pool of 88 singlet exciton fission candidate mate- rials, two are chosen for experimental characterization: 9,10- dicyanoanthracene (DCA) and 9,10-dichlorooctafluoroanthracene(DCOFA). DCA and DCOFA are selected for this study because their calculated triplet energies are high (1.47 eV and 1.45 eV, respectively), making them promising candidates for exothermic triplet transfer and sensitization of silicon. Additionally, both DCA and DCOFA have known crystal structures reported in the Cam- bridge Crystallographic Data Centre and are readily available from commercial suppliers, having been used as synthetic precursors in prior experimental studies.39–41Furthermore, both materials are reported to have a single conformer, and therefore, issues related J. Chem. Phys. 151, 121102 (2019); doi: 10.1063/1.5114789 151, 121102-4 © Author(s) 2019The Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp to stacking and energy mismatch between conformers should be minimized. DCA and DCOFA are purchased from Sigma-Aldrich and further purified using a sublimation furnace. Their chemical structures are shown in Figs. 2(a) and 2(b). To confirm the calculated singlet energies of DCA and DCOFA, steady-state absorbance and photoluminescence spectra are mea- sured [Figs. 2(c) and 2(d)]. Both materials exhibit absorbance spectra with a clear vibrational structure. From the onset of their bluest emission features, we estimate the S1energies of DCA and DCOFA in solution to be 2.99 eV and 3.10 eV, respectively, compared to the calculated values of 2.98 eV and 2.97 eV. Note that these singlet energies are likely to vary slightly depending on the specific solvent chosen.42 To prepare for measuring the triplet energy of DCA, the mate- rial is dropcast from solution at 2 wt. % in poly(4-bromostyrene) (4BrPS), at a total concentration of ∼20 mg/ml in methoxybenzene. At 2 wt. %, it is not expected that DCA undergoes efficient sin- glet exciton fission. That said, the presence of bromine in the host material causes enhanced intersystem singlet-to-triplet crossing via spin-orbit coupling and the heavy-atom effect.43Moreover, as methoxybenzene dries, 4BrPS forms a rigid polymer matrix, which is expected to reduce nonradiative recombination from coupling to molecular vibronic modes.44The film is encapsulated in a nitrogen glovebox before being transferred to a cryostat and pumped down to ∼1×10−7Torr at a temperature of 77 K. To experimentally measure T1phosphorescence from DCA, we employ an optical gating method using out-of-phase choppers to mechanically filter out prompt emission and capture only delayed emission, which is expected to result from longer-lived triplet state emission [as well as delayed emission following subsequent triplet- triplet (TT) fusion back to the singlet state]. The resulting spectrum (after subtraction of singlet emission, Fig. S7) is shown in Fig. 2(e). To measure DCOFA phosphorescence, we use a room temper- ature phosphorescence method described by Reineke and Baldo.44 DCOFA is doped in a film of PMMA and benzophenone (BP), a well-known triplet sensitizer with an intersystem crossing effi- ciency close to 100% at room temperature45and a triplet energy of 2.96 eV.44The film is excited at 270 nm, where BP absorbs strongly, and the resulting emission spectrum is compared to the spectrum when excited at 385 nm, where DCOFA is the dominant absorber. Subtracting the neat DCOFA singlet exciton spectrum from the spectrum of DCOFA doped in BP yields the phosphorescence spec- trum shown in Fig. 2(f). The triplet energies of DCA and DCOFA are estimated from the onset of their emission to be 1.54 eV and 1.51 eV, respectively (compared to the calculated triplet energies of 1.47 eV and 1.45 eV). The experimental triplet energies of DCA and DCOFA are in close agreement with the values calculated during material screening, as summarized in Table I. To demonstrate that DCA and DCOFA undergo singlet exciton fission, we measure the magnetic field effect (MFE) on photolumi- nescence intensity. The theory of MFEs in organic molecular crystals was developed by Merrifield in 1968.46When a molecule undergoes singlet exciton fission, it forms a triplet-triplet pair state with an overall spin-singlet character. Singlet excitons can only effectively couple to the subpopulation of the nine possible triplet-triplet (TT) pair states with a singlet character. In the absence of a magnetic field, only three of the nine TT states have partial singlet character. As theTABLE I . Comparison of singlet and triplet energies from experiment and theory. Sin- glet and triplet energies calculated with density functional theory (DFT), compared against experimentally measured values. DCA Theory (eV) Experiment (eV) S1 2.98 2.99 T1 1.47 1.54 DCOFA Theory (eV) Experiment (eV) S1 2.97 3.10 T1 1.45 1.51 field increases, the small splitting in triplet energies results in more of the TT states developing the singlet character, thus increasing the rate of singlet exciton fission (and resulting in an initially negative MFE on the singlet PL intensity). As the magnetic field is increased further, the number of TT states with singlet character decreases to two, consequently reducing the rate of singlet exciton fission (and resulting in a positive MFE on the singlet PL intensity at higher mag- netic fields). The typical MFE signature of singlet exciton fission is therefore a negative MFE at low fields, switching to a positive MFE at higher fields, which plateaus as the magnetic field is increased and the number of TT states with singlet character approaches two. Figure 3 shows the MFE on singlet photoluminescence from purified DCA and DCOFA crystals under 340 nm light-emitting diode (LED) excitation. Both DCA and DCOFA exhibit a nonmono- tomic, positive MFE with zero-MFE crossings at ∼0.1 T, characteris- tic of singlet exciton fission. Curiously, however, the MFE on photo- luminescence in DCA and DCOFA is inverted when using excitation wavelengths longer than 340 nm and 365 nm, respectively (Fig. S8). To make sense of these differences, we suggest that the observed MFE shapes reflect changes in the origin of delayed fluorescence for varying pump wavelengths. A positive MFE, as is observed under 340 nm excitation, suggests that delayed fluorescence is due to the annihilation of correlated triplet excitons originally formed by sin- glet exciton fission, as described above. Meanwhile, the inverted and negative MFE observed at longer excitation wavelengths is indicative of delayed fluorescence due to the annihilation of initially uncor- related triplet excitons (possibly formed through weak intersystem crossing). We speculate therefore that, as in neat anthracene,9,10sin- glet exciton fission in DCA and DCOFA is mediated by excited states above S1, while at longer excitation wavelengths, singlet exciton fission is unfavorable. Indeed, the highest-energy emission peaks from DCA and DCOFA are red-shifted by around 100 nm in crystalline powder as compared to those in solution (Fig. S9). Based on this red-shift and their large spectral width, the prompt emission peaks in Fig. S9 likely contain contributions from emissive excimer states of DCA and DCOFA. From the rising edge of the crystalline DCA and DCOFA prompt emission peaks, we estimate the energies of these states to be 2.59 eV and 2.71 eV, respectively, making singlet exciton fission uphill by more than 300 meV in each case. The substantial endother- micity associated with splitting such states into two triplets further suggests that singlet exciton fission in crystalline DCA and DCOFA is mediated by excited states above S1. J. Chem. Phys. 151, 121102 (2019); doi: 10.1063/1.5114789 151, 121102-5 © Author(s) 2019The Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp FIG. 3 . Evidence of singlet exciton fission. Relative change in fluorescence at 340 nm excitation from purified crystals of (a) DCA and (b) DCOFA as a function of external magnetic field. The change in fluorescence is positive for both samples at higher field strengths, with a zero-MFE crossing at 0.05–0.1 T, strongly suggesting that both materials undergo singlet exciton fission. Hot singlet exciton fission has previously been reported in anthracene, tetracene, and several other organic crystalline semi- conductors.8,9,47Hot fission in these materials has generally been attributed to coupling of the TT pair state to vibrationally excited states,48,49with recent studies emphasizing the importance of excited states with charge-transfer (CT) character.50,51Given the excitation wavelength dependence that we observe in the MFE of DCA and DCOFA, it is possible that singlet exciton fission in these mate- rials is mediated by an excited CT state. Rational design of DCA and DCOFA dimers could be used to reduce excimer formation and make singlet exciton fission from the ground excited state more favorable.52 CONCLUSION In this study, we employ high-throughput virtual screening to reduce an initial set of 4482 singlet exciton fission candidate materi- als to a list of 88 candidates with calculated triplet energies in excess of 1.4 eV. The effectiveness of the virtual screening procedure is demonstrated via the discovery of two new singlet exciton fission materials: DCA and DCOFA. These materials are readily available, low-cost, and have high triplet energies ( ∼1.5 eV), making them promising candidates for exothermic singlet exciton fission sensi- tization of solar cells. Practical use of DCA and DCOFA as singlet fission materials is limited, however, by substantial excimer forma- tion in the solid-state, requiring excitation wavelengths of ∼340 nm or lower in order to observe singlet exciton fission. Further tailoring of molecular coupling in DCA and DCOFA, for example through dimerization, may reduce excimer formation and make these mate- rials more practical for use in a device. Integrating intermolecular coupling into virtual screening criteria is expected to result in higher hit rates for screened materials. This study shows that computational screening can lead to molecules that have necessary but not suffi- cient properties for function, highlighting one of the most importantchallenges and opportunities in the field of virtual high-throughput virtual screening. EXPERIMENTAL METHODS Materials The materials used have acronyms as follows. DCA: 9,10- dicyanoanthracene, DCOFA: 9,10-dichlorooctafluoroanthracene, 4BrPS: poly(4-bromostyrene), MB: methoxybenzene, BP: benzophe- none, and PMMA: poly(methyl 2-methylpropenoate). All materials except DCA and DCOFA were used as received without further purification. DCA ( >99.7% sublimed grade) and DCOFA ( >99% sublimed grade) were further purified via sublimation in a tube fur- nace. The experimental work in this manuscript relied on the mate- rial deposited near the center of the tube furnace, since this region had the lowest temperature gradient and was expected to yield the purest material. Further details on material purification are included in the supplementary material. Vendor information: DCA, DCOFA, 4BrPS, MB, chloroform, and cyclohexane: Sigma Aldrich; PMMA: Alfa Aesar. Steady-state absorbance Absorbance spectra in solution were collected using a UV-Vis absorbance spectrometer (Cary 5000, Agilent). Samples were pre- pared by dissolving purified DCA and DCOFA in cyclohexane and chloroform, respectively. All solution absorbance measurements were made using 1 cm path length quartz cuvettes. Solid-state absorbance spectra of purified, crystalline powders of DCA and DCOFA were obtained by diffuse reflection (Cary 5000, Agilent). KBr was used as a reference, and the powders were diluted at 1 wt. % in KBr. The measured reflectance was converted to the Kubelka-Munk parameter, which is proportional to the absorption coefficient. J. Chem. Phys. 151, 121102 (2019); doi: 10.1063/1.5114789 151, 121102-6 © Author(s) 2019The Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp Singlet state photoluminescence Photoluminescence spectra were collected using a spectroflu- orometer (Fluoromax-3, Horiba). To minimize spectral distortion due to reabsorption, samples were diluted so that the optical den- sity at their peak absorbance wavelengths is below 0.03 cm−1. All spectra were collected using an excitation wavelength of 375 nm with entrance and exit slits set at a bandpass of 5 nm and 1 nm, respectively. Triplet state phosphorescence DCA phosphorescence was measured using a thermoelec- trically cooled silicon camera (PRO-EM-HS:512BX3, Princeton Instruments). Samples were prepared using 5 mm ×5 mm quartz substrates (MTI Corp). The substrates were cleaned via sequential sonication in detergent solution (Micro-90), deionized water, and acetone. They were then immersed in boiling isopropanol, dried with a nitrogen spray gun, and transferred to a nitrogen glovebox. Solutions of 4BrPS (20 mg/ml in MB) and DCA (1 mg/ml in MB) were prepared and subsequently combined to form a solution of DCA (2 wt. % in 4BrPS). Films of DCA/4BrPS were formed via drop- casting onto the quartz substrates at 70○C and maintaining at 70○C until dry. Samples were encapsulated in the glovebox using a UV curable epoxy (OG159-2, Epoxy Technology) and a second, 2-side polished quartz substrate (MTI Corp). Samples were then loaded into a helium closed-cycle cryo- stat (Montana), pumped under vacuum to 1 ×10−7Torr, and cooled to 77 K. Phosphorescence spectra were obtained using two chopper wheels operated out-of-phase using phase-locked chop- per controllers (MC2000B, Thorlabs) to time-gate sample exci- tation and emission collection, so as to block prompt emission and collect only the delayed portion of the photoluminescence. A 340 nm LED (M340L4, Thorlabs) was used for excitation, chopped at 270 Hz, defining approximately 2 ms optical gates. Sample emis- sion was collimated, refocused into a monochromator (SP-2300, Princeton Instruments), and subsequently imaged on a thermo- electrically cooled silicon camera (PRO-EM-HS:512BX3, Princeton Instruments). Both singlet photoluminescence and triplet phospho- rescence contributed to the recorded spectra. To remove the con- tribution from singlet emission, the blue portion of the emission spectrum (from 450 to 700 nm) was fit to the 77 K ungated sin- glet emission spectrum, and the singlet PL contribution was sub- tracted from the measured spectrum, yielding the phosphorescence spectrum shown in Fig. 2(c). DCOFA phosphorescence was measured at room temperature according to a method adapted from Reineke et al.44A 20 mg/ml solution of DCOFA doped at 2 wt. % in [PMMA:BP] 3:1was prepared and dropcast from methoxybenzene onto a cleaned 10 mm ×10 mm quartz substrate at 70○C in a nitrogen glovebox. Once dry, the sam- ple was encapsulated using a second quartz substrate and a UV cur- able epoxy. Phosphorescence was detected on a spectrofluorometer (Fluoromax-3, Horiba) by exciting the sample at 270 nm (where BP absorbs) and comparing against spectra obtained at 385 nm exci- tation (where only DCOFA absorbs). To remove the contribution from singlet emission, the blue portion of the emission spectra (from 450 to 700 nm) was fit and the singlet PL contribution was sub- tracted from the measured spectrum, yielding the phosphorescence spectrum shown in Fig. 2(d).Magnetic field effect Measurements of the magnetic field effect on DCA and DCOFA photoluminescence were performed according to the procedure described by Congreve et al.4A monochromatic 340 nm light- emitting diode (M340L4, Thorlabs) was used to excite the samples. Light from the diode was cleaned with a bandpass filter and mechan- ically chopped. While the sample was under illumination, an electro- magnet was switched between positive and zero magnetic fields at a frequency of 33 mHz and a duty cycle of 50%. A 400 nm longpass filter was used to filter the scatter from the LED and ensure that only sample emission was detected, while a 750 nm shortpass filter was used to ensure that any phosphorescence that might contribute to the sample emission at room temperature was excluded. Photolumi- nescence from the sample was detected using a silicon photodetector (818-UV, Newport) connected to a lock-in amplifier (SR830, Stan- ford Research Systems). The magnetic field was monitored using a transverse gaussmeter probe (HMMT-6J04-VF, Lakeshore). The emission intensity and magnetic field were recorded at a frequency of 1 Hz. For each data point in the plot of the magnetic field effect, the change in photoluminescence was calculated using the following steps: First, the photoluminescence was averaged over the full period in which the positive magnetic field was applied. Second, the PL was averaged over the full period in which the magnetic field was zero. The amplitude of the magnetic field effect is then given by the rel- ative change in signal percentage, MFE = 100%∗(PL B−PL0)/PL 0, where PL Band PL 0are the averaged emission intensities with and without applied magnetic field. Powder x-ray diffraction Powder x-ray diffraction PXRD patterns were measured on purified, crystalline powders of DCA and DCOFA using a diffrac- tometer (Advance II, Bruker) equipped with θ/2θBragg-Brentano geometry and Ni-filtered Cu K αradiation (K α1= 1.5406 Å, Kα2= 1.5444 Å, K α1/Kα2= 0.5). The tube voltage and current were set to 40 kV and 40 mA, respectively. Samples were prepared as a thin layer of powder on a zero-background silicon crystal plate. The angle was scanned from 2 θ= 3○–50○in increments of 0.02○using a slit size of 1.0 mm and a scan rate of 1 s/step and resulting PXRD patterns compared to calculated patterns from the data reported in the Cambridge Crystallographic Data Centre. SUPPLEMENTARY MATERIAL See the supplementary material for additional figures related to the materials screening procedure and experimental characteriza- tion. A full list of singlet exciton fission calibration data and candi- date materials considered for this study is included in the associated spreadsheet files uploaded with this manuscript. ACKNOWLEDGMENTS C.F.P. was supported by the Center for Excitonics, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0001088 (MIT). C.F.P. was also supported by the National Science Foundation Graduate Research Fellowship under Grant No. J. Chem. Phys. 151, 121102 (2019); doi: 10.1063/1.5114789 151, 121102-7 © Author(s) 2019The Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp 1122374. M.E. and T.-A.L. were supported by the U.S. Depart- ment of Energy, Office of Basic Energy Sciences (Award No. DE- FG02-07ER46474). H.U. was funded by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sci- ences and Engineering (Award No. DE-FG02-07ER46454). D.N.C. acknowledges the support of the Rowland Fellowship at the Row- land Institute at Harvard University. D.P.T. and A.A.-G were funded by the Innovation Fund Denmark via the Grand Solutions project “ORBATS” (File No. 7045-00018B). D.S. and A.A.-G. were sup- ported by the Harvard Climate Solution Fund. A.A.-G. was addition- ally supported by the Canada 150 Research Chair Program, as well as generous support from Anders Frøseth. The authors would also like to thank Dong-Gwang Ha and Ruomeng Wan for their assistance in collecting the PXRD and solid-state absorbance measurements reported in Figs. S9–S11. The authors declare no competing financial interests. REFERENCES 1M. B. Smith and J. Michl, “Singlet fission,” Chem. Rev. 110, 6891–6936 (2010). 2A. Rao and R. H. 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1.1662989.pdf	Applications of the gyrocoupling vector and dissipation dyadic in the dynamics of magnetic domains A. A. Thiele Citation: Journal of Applied Physics 45, 377 (1974); doi: 10.1063/1.1662989 View online: http://dx.doi.org/10.1063/1.1662989 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/45/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetization dependent vector model and single domain nanostructures J. Appl. Phys. 105, 07D516 (2009); 10.1063/1.3068009 Highly directional receivers using various combinations of scalar, vector, and dyadic sensors J. Acoust. Soc. Am. 114, 2427 (2003); 10.1121/1.4778888 Dynamics of magnetic domains and walls J. Appl. Phys. 69, 4590 (1991); 10.1063/1.348320 DYNAMICS OF MAGNETIC DOMAIN WALLS AIP Conf. Proc. 5, 170 (1972); 10.1063/1.3699416 Dynamics of Magnetic Bubble Domains with an Application to Wall Mobilities J. Appl. Phys. 42, 1977 (1971); 10.1063/1.1660475 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41Applications of the gyrocoupling vector and dissipation dyadic in the dynamics of magnetic domains A. A. Thiele Bell Laboratories. Murray Hill. New Jersey 07974 (Received 9 April 1973; in final form 10 July 1973) This paper extends the theory of magnetic domains with emphasis on recent developments in "hard bubbles". A spin configuration of a planar Bloch wall containing periodic Bloch lines is presented which minimizes the magnetostatic energy to first order in the parameter 21T M.~/ K u for arbitrary period. The form of this solution is found to suggest the form of the dynamic breakdown of this spin configuration. The remainder of the paper consists of applications of the gyrocoupling force and vector, fg = g X v and g = -(M ,.-1yD sin8(V8) X (Vc/», respectively, and the dissipation force and dyadic, F = d . v and d = -a(M ,.-1yl)[(v8)(V8) + sin'8(vc/»(Vc/»]. The use of fg and F produces results with fewer assumptions and with less calculation than with previous methods. The magnitude of g is found to be an invariant local measure of the "hardness" of the domains. Integrating ~ and fa produces a general planar wall response function from which the hard bubble dynamic equation is obtained. It is found that the difference between the hard bubble and normal bubble damping parameters can be accounted for by examination of the hard bubble spin-wave spectrum. An estimate of the velocity required for the production of horizontal Bloch lines is made using ~. This velocity is a substantial fraction of the Walker velocity. The vector g is used as an aid in the visualization of the mechanism by which ion implantation suppresses hard bubbles. From the point of view of both mobility and hard bubble suppression, materials having a large in-plane anisotropy are found to be desirable. I. INTRODUCTION Recent papers1,2 report the observation of anomalous cylindrical magnetic domains which have unusually high collapse fields (hard bubbles) and do not propagate in the direction of the applied field gradient. These pro perties have been ascribed to the inclusion of multiple Bloch-line pairs in the Bloch walls, as have been ob served directly in cobalt films. 3 In subsequent papers, the consequences of particular assumed spin configura tions were examined with respect to determining the static2-s and dynamic6-s properties of these domains. In all cases agreement between theoretical predictions and experimental results was obtained. The present paper considers various static and dy namic properties of magnetiC domains (with emphasis on hard magnetic bubbles) as applications of two quadratic functions of the spatial derivatives of the magnetization, the gyrocoupling vector and dissipation dyadic. The gyrocoupling force corresponds to the gyroscopic term in the magnetic equation of motion. Likewise the dissipative force corresponds to the dissi pative term in the equation of motion. The gyrocoupling and dissipative forces, together with the usual energy derivative force, completely determine the steady-state motion of a domain. (The forces are all computed from the spin distribution.) The magnitude of the gyrocoupling vector emerges as an excellent choice for a general quantitative measure of the "hardness" of bubbles or of domains in general. Before introducing the formal expressions for the gyrocoupling vector, the dissipation dyadic, and the associated forces, it is convenient to describe the assumed static spin configuration of the domains and to discuss the plausibility of this spin configuration as an approximation to the steady-state dynamic spin con figuration of the domain. The description is in terms of right-handed Cartesian (x,y,z) and cylindrical (r, cf>b'Z) coordinate systems. The orientation of the magnetiza- 377 Journal of Applied Physics, Vol. 45, No.1, January 1974 tion vector M (I M I = M.), is described by the polar angle e (e=o is the z-axis direction) and the azimuthal angle cf> (e = i7f, cf> =0 is the x-axis direction). The sym bol cf> will always be used in specifying magnetization orientations, while the symbol cf>b will be used in specifying field positions. The magnetic energy density function of the material in which the domains reside will, in the absence of applied fields or surface demag netizing effects, be assumed to be PE=A[(ve)2 +sin2e(Vcf»2] (1 ) In (1), A is the isotropic exchange constant, Ku is the uniaxial anisotropy constant, and the last term is a local demagnetizing term in the Winter approximation. 9 In the local demagnetizing term, it is assumed that the wall normal lies in the xy plane, its orientation being described by cf>o' At this point it is convenient to define several auxiliary parameters in terms of the parameters which appear in the energy density expression (1). These parameters are the wall energy denSity in the absence of Bloch lines Gwo=4(AK)1/2, the expansion parameter q=K/27TM!, the domain characteristic material length I=G /47fM2 wO s' (2) (3) (4) the isolated Bloch linewidth (an exchange length) de fined so as to agree with the usual wall width definition 110 = 7T(A/27TM!)1 /2 = 7Tl/2ql /2, and the wall width in the absence of Bloch lines Iwo = 7f(A/K)1 /2 = 7Tl/2q. Copyright © 1974 American Institute of Physics (5) (6) 377 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41378 A.A. Thiele: Dynamics of magnetic domains x FIG. 1. Configuration of the cylindrical magnetic domain whose wall contains five pairs of Bloch lines (n = 6). The static spin configuration for a hard bubble may be approximated by cos e =tanh[1T(r -ro)/l)<1»], (7a) <1> = <1>(<1>0) = <1>(<1>0 + 21T/p) , (7b) where ro is the domain radius, lw(<1» is the wall width function, <1>0 specifies a particular point of the domain wall, and p is an integer. The spin configuration (7) is depicted in Fig. 1 for lw=const and <1>=6<1>0' with emphasis on r=ro' For <1> = <1>0± t1T, the spin configura tion (7) is of the form used by DeBonte10 for describing finite wall width effects, except that he added a term to remove the exchange singularity at r = O. Under the approximations which will be used here, it will not be necessary to consider this additional complexity. The spin distribution (7) is similar to those used in Refs. 2 and 4-7 but is somewhat more flexible and therefore allows a more detailed description. In the Appendix, a solution to first order in the ex pansion parameter q-l is given to the Euler equations associated with the minimization of the energy density (1). The form of this solution is cos e = tanh [1Tx/lw (<1»], <1>(Y)=<1>(Y+s). (8a) (8b) This is a planar wall with internal periodic structure of period s. A portion of such a wall is depicted in Fig. 2 with lw=const, <1> = <1>0 + 21TY/S , J. Appl. Phy •. , Vol. 45, No.1, January 1974 (9a) (9b) 378 which is the configuration considered in Refs. 4 and 6. The fundamental assumption in most of what follows is that the lw and <1> functions obtained in the Appendix for the planar wall configuration (8) may be applied to the cylindrical domain (7) using (10) This is equivalent to assuming that the planar wall solu tion may be rolled up to form a cylindrical domain without destroying its relevant properties. Note that the gyrocoupling vector and diSSipation dyadic derivations given in Ref. 8 are independent of this assumption. II. QUALITATIVE DISCUSSION OF PLANAR WALL DYNAMICS Before proceeding with the dynamics of cylindrical domains, it is instructive to consider qualitatively some of the dynamic properties of the infinite planar wall. Propagation of the structure, (8) with (9), to the left in Fig. 2 under the influence of a uniform applied field H~ =H is described for the case I s I = 00 (p = 0) by the steady-state domain-wall solution of Walker. 11 In this solution the Gilbert damping term12 is exactly can celled at all points by the applied field term. When the wall moves in the x direction in Fig. 2, the z-axis angular momentum per unit wall area of the spin sys tem changes at the rate, Idn/dtl =2IM.v/YI, where Y is the gyromagnetic ratio. Consequently, the host lat tice must exert, on the average through anisotropy and demagnetizing effects, the corresponding z-axis torque on the spin system in order to conserve angular mo mentum. The applied field makes no contribution since it has only a z component, there is no dissipative z axis torque, since d<1>/dt=O in the Walker solution. The z-axis torque equation of the Walker solution is inde pendent of e (which is quite remarkable in itself) and differs from the static equation only by the addition of the constant gyroscopic term required by conservation ®M(X~-Q)) @) H .. x FIG. 2. Configuration of the infinite planar Bloch wall contain ing uniformly spaced Bloch lines. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41379 A.A. Thiele: Dynamics of magnetic domains of angular momentum. In (A8) of the Appendix, it is indicated that the static equation is the pendulum equa tion, in which the pendulum orientation angle /; is re placed by 21> and time t is replaced by y. Since in the Walker solution iJ1>/ay==o everywhere, the first term in (A8) is zero. Solving the remaining algebraic ex pression (with the gyroscopic term added) for 1> com pletes the solution of the z-axis torque equation. The Walker solution thus corresponds to a stationary pen dulum having an applied torque which is a function of the wall velocity. The torque (that is, the velocity) is re quired to be sufficiently small so that the pendulum is not carried over the top. When I s I < 00, it is easily shown for I Vx I > 0 in Fig. 2, that the dissipative torque due to d8/dt and the applied field torque again have exactly the same x de pendence as in the I s I == 00 case, so that at some veloc ity (neglecting the y dependence of the wall width) they could cancel exactly. Although the z-axis torque re quired by conservation of angular momentum remains unchanged, the z-axis torque situation is radically changed in other respects. The exchange term [the first term in (A8)] is now generally nonzero; but since exchange is a spin-spin interaction, it may only trans fer torque from one point to another, making no con tribution to the average torque. The demagnetizing torque term does not, in this case, provide the torque required by angular momentum conservation. This may be seen, as in Ref. 7, by conSidering a rotation of 1> at all points by a small angle with respect to the static solution in the direction it is pushed by the gyroscopic torque. (The gyroscopic torque direction is the same at all pOints.) In contrast to the case of the Walker solu tion in which an initial angle could be found such that the demagnetizing torque began to cancel the gyroscopic torque at all points as the 1> variation was increased, the demagnetizing torque in the present case adds to the gyroscopic torque at as many points as it subtracts from it. Although it is possible to conjure up 1> varia tions which do contribute a net demagnetizing torque, they are not solutions to the field equations. Returning to the pendulum analogy, the I s I < 00 case corresponds to adding a constant torque to a pendulum which already has sufficient energy to swing over the top. In the absence of losses, the pendulum accelerates without bound. The direction of the gyroscopic torque produced by negative v" on the configuration shown in Fig. 2 is such that this torque tends to accelerate the configuration in the negative y direction. As shown by Malozemoff and Slonczewski in Ref. 7 and discussed below in terms of the dissipation-gyrocoupling force theorem, the "miss ing" gyroscopiC torque may be provided by dissipative drag if Vy is large with respect to v" (assuming Q, the Gilbert damping parameter, is less than unity and the Bloch lines are isolated). Although such a motion may be stable at low velocities, it seems improbable that the domain wall can transmit all of its z-axis angular momentum to the cystallattice by purely dissipative processes at high velocity. A buildup of excess z-axis angular momentum could result in the generation of additional Bloch lines (acceleration of the pendulum). Such a nucleation process could account for the obser- J. Appl. Phys., Vol. 45, No.1, January 1974 379 vation1•4 that hard bubbles often contain a surprisingly large number of Bloch lines. From the above information, it is clear that there is a qualitative change in the motion of the domain walls when Bloch lines are introduced. In this connection Vella-Coleiro et al.I3 have recently shown that in cer tain materials, the limiting velocity observed by the mobility method of Bobeck et al. 14 can be explained in terms of an initial (at the beginning of the applied field pulse) Walker breakdown followed by a low-mobility turbulent motion. In the pendulum analogy, the turbu lent motion corresponds, very roughly, to the accelera tion of the pendulum. In the case that the Bloch lines are so tightly com pressed that (A7c) applies and there is no dissipation, it is easy to show that there exists an approximate solution to the field equations in which there is an applied field in the positive z direction, Vx is zero, and Vy is negative in Fig. 2. Consider the effect of the rapid application of a z-axis field to the static spin configura tion, (8) with (9). Each spin begins to precess about the z axis at a rate w == I y I H~, bringing the configuration to an equally good (in the Ito == const approximation) static configuration with a new 1>0. The applied field and gyrocoupling terms in the field equations thus cancel at each point, the exchange and anisotropy terms cancel because the motion is a sequence of static solutions, and the resultant precession appears as a motion of the entire spin configuration in the negative y direction in Fig. 2. The velocity-field relation implied by this particular motion, H~ == -21TV/S I yl , (11) is a special case of the infinite wall solution obtained in Refs. 6 ~d 7. This infinite wall solution will, in turn, be shown to be easily obtainable from the gyro coupling dissipation force equation. III. THE GYROCOUPLlNG·DISSIPATION FORCE EQUATION The preceding discussion represents an attempt to provide physical insight into the dynamics of domain walls containing Bloch lines. The gyrocoupling-dissipa tion force equation which was derived in Ref. 8 is now introduced. In the remainder of this paper, lower-case vectors will indicate field quantities while upper case vectors will indicate quantities associated with the en tire domain. Thus, f is a force density and F is a force on a domain, and x is a field point and X is a domain position. The gyrocoupling-dissipation force density equation for domains moving with constant velocity v is (12a) where the f' (for a=g,Q ,r) are ordinary force density vectors (force per unit volume associated with transla tion, as opposed to generalized force densities). The force densities whose sum in (12a) is the total force density, £f, are the gyrocoupling force density F==gXv, (12b) in which the gyrocoupling vector denSity is [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41380 A.A. Thiele: Dynamics of magnetic domains g == (-M/I )'1) sin8 (V8) X (vcp), the Gilbert drag force density is f"==d·v in which the dissipation dyadic is d ==-aM 11)'1 [(V8)(V8) +sin28(Vcp)(vcp)], s (12c) (12d) (12e) and the reversible (energy derivative) force density is r-OPE (V 8) + OPE (vcp) -08 ocp , (12f) in which 0 denotes the variational derivative. Note that the validity of (12), when the magnitude of M, is a func tion of position, has not been investigated at the present time. Consider the matrixLJ ==/h T/h where /h is the 2 x3 matrix 08 oy sin8oq> oy so that d== (-aM,1 1)'1) f), and the exchange energy den sity is pex==ATr(f)). Since the rank of/h is no greater than 2, the rank of j) is no greater than 2. Since LJ is a positive definite form, the eigenvalues of LJ are non negative, and at least one of them is zero. The sum of these eigenvalues is proportional to the exchange ener gy, and it can be shown that their product is proportional to g2. If Igl is zero, thenj) has at most one nonzero eigenvalue, so that in some coordinate system the only nonzero element is (d8 I dX)2 + sin2 8 (dCPI dX)2. Thus 8 and cp are functions of x only (locally). The magnitude of the gyrocoupling vector is thus an invariant measure of the extent to which the magnetic distribution locally depends on two spatial dimensions. [Consider, for example, the case in which the magnetization depends on only one dimension. Then in (12c), the gradients are coplanar and I g I == 0.] The appearance of a measure of the extent to which the magnetic distribution depends on two spatial coordinates in the steady-state dynamics of domains may be qualitatively understood in the following way: When the spin distribution depends on two spatial coor dinates, the gyroscopic terms in the equations of mo tion are able to couple these dependences. In steady-state motion, 8 and cp are functions of (x -X), where x denotes the field point and X==vt (13a) denotes the pOSition of the domain. The gradients in (12f) are x gradients, which account for the unexpected sign. In the steady state, it follows from (13a) that l..==-v.V. dt From (12a) v·ft==O or J. Appl. Phys., Vol. 45, No.1, January 1974 (13b) (14a) (14b) 380 so that using (12b)-(12f) and (13b), the usual dissipation equation °6! == -1~7s [(::) 2 +sin28 (~~)r (15) is reproduced. In (15), 0 denotes only that part of the energy variation which results from variations of 8 and cp. The present paper may thus be considered as a generalization of previous dissipation treatments of domain mobility15-1B to the case of nonzero gyrocoupling vector magnitude. For a well-defined domain moving in steady state, it is possible to define total domain forces as (16a) so that the gyrocoupling-dissipation force density equation (12a) becomes the domain gyrocoupling-dissi pation force equation (16b) In (16a) the integration is over some volume including the domain, the surface bounding the region of integra tion being either outside the magnetic material or in a region in which the magnetization is constant. Since the velocity vector is constant by assumption, the total gyrocoupling and dissipation forces may be written FK==Gxv, (17a) where the domain gyrocoupling vector is defined by (17b) arid F"==D'v, ( 18a) where the domain dissipation dyadic is defined by D== J ddV y (18b) It was shown formally in Ref. 8 that if the domain reversible force term is divided into internal and ex te rnal parts Frln+Frex== r frlndV+J frexdV Jy y , (19a) then (19b) This is obvious physically, since for a domain in steady-state motion, the internal energy is constant by assumption. Note that in order to achieve steady-station motion Frex must be constant, and this requirement generally means that the external driving force density frex must be time dependent. (The drive field configura tion must move along with the domain.) As shown in Ref. 8, the domain gyrocoupling integral may be evaluated in general. Consider, for example, the z component of (17b) G =--' - ----smBdxdydz, M 1 [a8 acp a8 aCP] . ~ 1)'1 6~6y6x ax ay ay ax (20a) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41381 A.A. Thiele: Dynamics of magnetic domains G Mal a(coso, cJ» d d d =- X Y z ~ I yl t:.zAyAx a(x,y) , (20b) Mal G~=-I -I dcosOdcJ>dz. y A~A</JA case (20c) Similar expressions apply for Gy and Gr' In order for the mapping x, y -cos 0, cJ> to be one to one, neither g 1 ~ nor If; must be zero. In an exchange coupled material, the exchange energy prevents If;l from being zero, although g. may be zero. For example, in a plate con taining many bubbles (hard or not) each domain exists in a cell bounded by g" =0 surfaces (the plate normal being along the z axis). In a material containing many bubbles (20c) applies in each cell defined by a g" = 0 surface. The integral (20c) is easily evaluated in those special cases in which either cosO or cJ> is a constant on the bounding surface. The planar wall and hard bubbles are two such special cases which will be evaluated here. There is no corresponding first integral for d; however, since this integral is a quadratic measure of the rough ness of the magnetic distribution, it is relatively insen sitive to the exact details of the distribution o Note that when the domain motion is only approxi mately steady state, the gyrocoupling and dissipation force expressions may be usefully applied as approxi mations. The remaining force in the total domain force equation, the total force due to reversible effects, may be computed by global methods since the total force is the integral of r, which is the ordinary energy deriva tive force density. Thus, for example, the total domain force may be computed from the total domain energy as a function of position16; then as an approximation, the total instantaneous gyro coupling and diSSipation forces may simply be added. This procedure is only a first order approximation since it is difficult (if not impos sible) to define steady-state motion in a nonuniform medium, and in this paper, the gyration and dissipation forces have only been defined for steady-state motion. Additional difficulties may be encountered when M is a function of position (see Ref. 8). Note, finally, th~t in the quasi-steady-state approximation, inertial forces (Wall mass effects) are included as a part of fT. IV. EVALUATION OF THE PLANAR WALL GYROCOUPLING VECTOR AND DISSIPATION DYADIC The gyrocoupling vector for a section of area ~x~y of the infinite planar domain wall depicted in Fig. 2 and described in the Appendix is -M •. 1 (ao acJ> ao acJ» G==-I-I-1 - ----sinOdV "}' "vax ay ay ax -M. 'lA·1AY(j""' ao ) acJ> = ""iYIl~ 0 0 _'" smO ax dx ay dy dz (21) where i" is the unit vector in the z direction and ~cJ> is the change in cJ> corresponding to ~y. Since only G~ is J. Appl. Phys., Vol. 45, No.1, January 1974 381 nonzero, in (21), G is unchanged if the wall plane is rotated about the z axis. In the case of an infinite planar 1800 wall in a plate of thickness h, in which the wall is oriented so that the wall plane contains iii' the plate normal, G. may be evaluated exactly. In this case only G" need be evalu ated, since all velocity and force vectors of interest are in the xy plane. The region of integration is between the g" =0 planes parallel to the wall plane and infinitely far removed from the wall plane in opposite directions. At one plane 0 == 0 and at the other plane 0 = 7r. If the mag netic distribution has no singularities, then the cJ> = const planes are smooth, although they may be dis torted somewhat near the plate surface of the magnetic material by surface demagnetizing effects. 18,19 In this case (20c) becomes (22) so that (21) is exact for strips of constant ~cJ>. In particular, since by symmetry the internal wall struc ture must be periodic, the relation is exact for any strip whose width is multiple of a full period. A result equivalent to (22) is obtained in Ref. 22 using equations from Ref. 20 which are re stricted to K /21TMl » 1 u • • Although the evaluation of the diSSipation dyadic is no more difficult than the evaluation of the wall energy (involving the same elliptic integrals), attention here will be largely restricted to the high and low Block-line density limits (A6) and (A7). In the general case, all z components of D are zero, since 0 and cJ> are not func tions of z and Dxy=Dyx=O, since _ aM. rfJ alw 1'" 2 (7rX) dx -I yl [3 a x sech l w y -"" w =0. (23a) (23b) Neglecting terms in (ao/ay)2 as being higher order in q-1, yields for the nonzero diagonal elements of D -aM fA. (AY 7r 100 (7rX) ( ) Dxx= ~ 10 10 lw _00 sech2 z:: d ;: dydz (24a) and where d(7rx/l) is the differential of 7rx/l . The x and z integrations may be carried out immedi:tely to yield (24c) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41382 A.A. Thiele: Dynamics of magnetic domains D = -2Q1M • ..E....~ziAy <7w(y) d xx 1'1/1 laY' , wO 0 wO (24d) where (24d) is obtained from (24c) using (A4) and D = -2Q1M. ~zlAY lw (ilrp)2 d . yy I rl 0 11 oy y (24e) Since the integral in (24d) is proportional to the average wall energy, Dxx may be obtained from the average wall energy expressions in the Appendix. In the limit of well spaced Bloch-line pairs, the first two terms of (A6d) yield for Dxx Setting lw = lwo in (24e) and using (A6b), and (A6c) yields for D yy (25a) D =-2Q1M'41!.lq-l/2~y~z. (25b) yy I rl s In the case of a wall containing no Bloch-line pairs (s =00), summing the drive force produced by an applied field H and the drag force from (12d) and (25) yields 2M H ~y~z -D v == O. This then reduces to the usual mobility relatix~nx J.Lw = v/H == (I rl /QI)(lwo/11) (Ref. 17). The terms linear in 1/ I s I in (25) describe the drag added by the Bloch-line pairs, Since these terms make equal contributions to Dxx and Dyy' the Bloch-line drag may be said to be isotropic in the xy plane. Evaluating (24) in the limit of highly compressed Bloch lines (A3) and (A7) yields and D == -2Q1M. ~ (OcfJ) 2 ~ ~z. yy Irl 11 oy y (26a) (26b) The form of (26a) is the same as the first (normal) term of (25a), the only change being that the normal wall width is replaced by the actual wall width lw' This follows from the approximation that cfJ is independent of x and that the form of the x dependence of (} in the highly compressed case differs from the normal case only by a change in the constant lw' In the limit of extreme com pression, ts < lwo' 11/lw -dcfJ/dy, so that the entire dissipation dyadic is isotropic, Dxx -Dyy• The results obtained above for the infinite planar wall with straight Bloch lines will now be applied to various special cases, assuming that the results remain apprOximately true locally for finite curved walls and lines. Attention will first be restricted to cases in which the magnetic distribution is independent of z, the Bloch lines lying in the z direction. This restriction carries with it the implicit assumption that demagnetiz ing effects at the surface of the platelet are ignored.la,l9 These effects will be discussed separately later. V. APPLICATION OF THE GYROCOUPLlNG DISSIPATION RESULT TO PLANAR WALL AND COLLAPSING BUBBLE DYNAMICS The results of Sec. IV will now be applied to the cal- J. Appl. Phys., Vol. 45, No.1, January 1974 382 culation of the velocity-field relation for the planar wall structure of the Appendix (Fig 2). The velocity of this spin system induced by the application of a uniform field H is calculated assuming that surface demagnetiz ing effects are neglected and that apart from the uniform translation, the dynamic spin configuration does not differ significantly from the static spin configuration. Under these assumptions, only the applied field makes a net contribution to the reversible energy force, since in the steady state the total exchange energy, the total anisotropy energy, and the total demagnetizing energy remain constant (see Ref. 8). The force equations (per unit area) corresponding to the assumed motion are ~ == Dxx v _ 2M. ~cfJ v -2M H==O (27a) ~y~z ~y~z x I rl ~y y • and ~ == 2M. ~cfJ v +~v =0. (27b) ~y~z I rl ~y x ~y~z y Solving these equations using (25), (26), and ~cfJ/~y ==211/S yields the velocity-drive expressions in the two limiting cases. In the limit of uncompressed, is > lID' Bloch lines (28a) and I vYI ==!!.. QI-lql/2. (28b) Vx 2 The third term inside the parentheses in (28a) is in cluded for completeness, since it results from solving the equations as stated (it results from the x component of the Bloch-line drag). Since this term is one order higher in q-l than the other terms in the parentheses, it should be dropped. From (28b) it can be seen that the Bloch-line velocity will be much greater than the wall velocity in the wall normal direction, since in low-loss bubble domain materials q < 1 and a «1. This result was previously obtained in Refs. 6 and 7. In the limit of com pressed, i I s I < 1,0, Bloch lines (29a) and I~I_ -1!.2..!. -a 2l ' Vx w (29b) where lw is given by (A7c). Note that in the limit a -0, v -0 and v --(S/211) I rlH which is the solution ob- x' y tained in Sec. II (Eq. 11) by a different method. In the case of extreme Bloch-line compression, h < lwo. (29) becomes (30a) and (30b) All other parameters being fixed, Vx and Vv in the ex tremely compressed case are seen to decrease [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41383 A.A. Thiele: Dynamics of magnetic domains -;/ - -X FIG. 3. Configuration of a domain with one Bloch-line pair in which the total rp rotation is zero, the Bloch-line pair being shown at its stable dynamic riding point. linearly with decreasing I" I. Thus, if it is assumed that high-speed Bloch-line motion is unstable with re spect to the generation of more Bloch lines, then the process is found to be self-limiting. If dynamic effects other than the steady-state forces are neglected, (28) and (29) may be applied to the bubble collapse mobility method of Bobeck et al. 7,14 Identifying Vx and v" of Fig. 2 with vr and v ~ of Fig. 1, respectively, setting s = 11d/n where d is the domain diameter and n is the number of Bloch-line pairs, and setting H= 1 oET 411M"roh oro ' (31 ) where ET is the total cylindrical domain energy and h is the plate thickness yields the mobility expressions for this motion. If the highest-order term inq-l is dropped from the low Bloch-line density expression (28), the results obtained directly from the field equa tions by Malozemoff and Slonczewski7 are found to be reproduced. The high Bloch-line density formula, when written in terms of the wall width as in (29), is identical to the expressions obtained in Refs. 6 and 7 for the motion of a planar wall. If differs from Ref. 7 in that instead of the wall width being given by (A7c) they use lw =lwo' VI. APPLICATION OF THE GYROCOUPLlNG· DISSIPATION RESULTS TO PROPAGATING CYLINDRICAL DOMAINS CONTAINING BLOCH LINES In this section the translation with constant velocity v of a cylindrical domain containing Bloch lines parallel to the cylinder axis is considered. The gyrocoupling vectors for equal small areas in the planar wall and the cylindrical domain wall are denoted by r! and gc, re spectively. Similarly the corresponding dissipation J. Appl. Phys., Vol. 45, No.1, January 1974 383 dyadics are denoted by l)I> and I)C. Applying the planar wall results to a cylindrical domain yields gc=r! , (32) (33a) (33b) (33c) By using (21) and (32) with tlz =h, the total gyro coupling vector and gyrocoupling force for a cylindrical domain are and G-411Mshn . -I yl 1. (34a) (34b) since tlcp=211n, where n is the number of times the magnetization rotates about the z axis when the perim eter of the domain is traversed once in the direction of increasing CPb' In normal orthoferrite domains cP "" const, so that n is the number of Bloch-line pairs. In garnet bubbles n = p ± 1, where p is the number of Bloch lines, since in a domain with no Bloch lines cP = CPb ± t11 (see Fig. 3, Where n=O,p=-l). The z component of the total gyrocoupling vector expression (34a) is exact for a cylindrical domain. It does not depend on the apprOximation of applying the planar wall results to the cylindrical domain. This may be seen by carrying out the integration of (20c). The cosO integration is carried out first. This is an approximately radial integration along cP = const z = const lines from a presumed g. = 0, 0"" 11 line at ap proximately the axis of the domain to either a g=O, 0=0 surface at infinity, or a g. = 0, 0"" 0 surface separating this domain from other domains (see Fig. 1). The cP integration is carried out next. If the magnetic distribution is to be nonsingular, then tlcp = 211n, inde pendent of z. Carrying out the z integration in a plate of thickness h yields the z component of (34a). Note that the total gyrocoupling force is independent of either the distribution of the Bloch-line pairs about the circum ference of the domain or of any twisting (z dependence) of the magnetic distribution which may occur at the sur face of the plate. Assuming the Bloch lines to be uniformly distributed and using only the average values of D yields, in general, and Dxx=Dyy, so that the total dissipative force is FOt=-IDxxlv. Using (25) yields in the low Bloch line density aM 111 I D =---' -11d+8nq-l/2 h xx I yl lwo ' (35a) (35b) (35c) (36a) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41384 A.A. Thiele: Dynamics of magnetic domains while using (26) yields in the high Bloch-line density limit D = _ aM. j.!!.. + lw (2n) 21rrdh xx I s I lw rr d I ' (36b) where h is the plate thickness and d is the domain diameter. Since expressions (36a) and (36b) are only approximate, it is not worthwhile to distinguish between nand p as in (34) where the exact integral is available. Since the total internal reversible force is zero for cylindrical domains propagating with constant velocity in a uniform material (see Ref. 8), F' and Fa must be balanced by the externally applied field gradient. The motion is thus completely determined (for a given spin distribution) by F' + Fa + Frex = 0, (37) in which Frex is the externally applied field force. In contrast to the gyrocoupling vector, the dissipation dyadic is affected by a rearrangement of the Bloch lines. In general, any rearrangement of the Bloch lines from a uniform distribution (at constant n and d) pro duces an increase in the diagonal elements of the dis sipation dyadic. (It also produces an increase in the total wall energy so that the uniform distribution is statically stable.) The off-diagonal component, D"y' of the domain dissipation only becomes nonzero if the dissipation distribution about the domain perimeter contains a component in sinCPb cos CPo' In the low Bloch-line density limit, it is quite easy for the Bloch lines to redistribute, since the repulsion potential between them is exponential (A6d). If the do main is set in uniform motion by an applied field gra dient in the low Bloch-line density limit, the Bloch lines, being nearly free to move, will move so as to cluster about those points on the domain perimeter at which the sum of the gyrocoupling and dissipation force vectors associated with the individual Bloch lines, F" + Fa" has no component along the domain perimeter. Note that for q> 1, as is required in cylindrical do main devices, the characteristic Bloch linewidth l,o is smaller than the minimum cylindrical domain diameter15 q1/2lWO=l,o < (8/rr)q1/2l,0=4l"'dm1n, (38) so that cylindrical domains are geometrically capable of supporting many pairs of uncompressed Bloch lines. In Fig. 3 a configuration containing one negative cP ro tation Bloch-line pair is depicted. Since the general cP rotation is positive in the spin configuration of Fig. 3, the total D.cp is zero and the domain will propagate in the direction of the applied force. The single Bloch-line gyrocoupling force F" - _ 4rrM.h. x -I yl l~ v (39a) is obtained from the cross product of the Single Bloch line pair coupling vector [(21) with D.z =h and D.cp = -2rr] and the velocity. The Bloch-line drag force is defined as that part of the drag which is associated with J. Appl. Phys., Vol. 45, No.1, January 1974 384 the Bloch line itself. As noted in Sec. IV, the Bloch-line drag force is Fa, = _ ~~s 8q-1 /2hv. (39b) The angle which specifies the points on the domain perimeter at which F" + Fa' has no component in the plane of the domain wall is given by t A, __ I Fa, I __ ~ -1/2 an'f'o'-IF"I- rraq . (40) In Fig. 3, the spin distribution at r = ro is depicted with the Bloch-line pair shown at the stable solution to (40). The vector F'l + Fa, is shown positioned at both the stable (S) and unstable (U) solutions to (40). Since in cylindrical domains q > 1, the Bloch lines in moving domains in low-loss materials, a« 1, ride on the side of the domain. When coercivity is ignored, as it has been here, the angle at which the Bloch lines ride is independent of the velocity. It should be possible to ob serve this effect in materials that have some small amount of coercivity by propagating the domains at a sufficiently high velocity so that coercivity may be neglected, and then reducing the bias field on the static domain so that the domain runs out into a strip. The orientation of the Bloch lines should then produce a observable effect on the direction in which the domain runs out. (It is assumed here that the presence of the Bloch lines will cause a local increase in the wall mo tion coercivity.) Note that in the general (low Bloch line density) case, the formula for the angle between the velocity and the Bloch-line cluster point remains valid, although the angle between the applied field force and the velocity is no longer zero. The effect of the gyrocoupling force in causing the domains to propagate at an angle to the applied field force is most pronounced in the high Bloch-line density limit. It is in this region that the data reported in Refs. 1 and 21 were taken. In order to compare the theory with experiment, it is convenient (in order to introduce coercivity) to specify the solution to the force equation formed from (34), (35), (36b), and (37) in terms of the components of the applied field gradient required to maintain steady-state motion along the y axis as shown in Fig. 1. For the description of uniform gradients in the applied bias field H~, it is convenient to denote the field difference across the domain as (41a) With this notation, the force on a cylindrical domain produced by a uniform gradient in H~ is (41b) which is the force produced on a dipole oriented along the z axis of strength -rrr~h2M. in a uniform H~ gradient [see Eq. (60) of Ref. 15]. If the component of the applied field gradient perpendicular to the direction of propagation is denoted by H1, then from (34) and (41) D.H 8nv. 1 = dl Y11x' since only the externally applied force and the gyro coupling force have components in this direction. (42) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41385 A.A. Thiele: Dynamics of magnetic domains v FIG. 4. Field gradient vector diagram fora domain of configu ration shown in Fig. 1 when it is moving with velocity v in a coercive medium. Similarly, denoting the component of the applied field gradient necessary to overcome the Gilbert damping induced drag by AH"a and using (35), (36b), and (41) yields AH"a =01 ~;~a f(N}iy, (43a) where 1 +2N2 fiN) = 2N(1 + N2)1 12 , (43b) feN) == 2~ (1 + %N2 + ... ), (43c) (43d) and N_nl _~ nlwO -aq-rr a . (43e) Adding a domain coercivity field gradient AHcd to AH and AB"a as was done by Slonczweski22 produces the vector diagram Fig. 4, in which the total applied field gradient is denoted by AHa and the angle between the applied field direction and the velocity is denoted by t. The domain coercivity AHed is to be considered as a parameter which is adjusted to fit the experiment, the best value probably being considerably greater than the normal value of (8/rr)He, where He is the minimum field necessary to produce wall motion in an infinite planar wall containing no Bloch lines [see Eq. (63) of Ref. 15]. Resolving the vector diagram (Fig. 4) and using (42) and (43) yields and v=Iyla 1 {[(1+0'2f2)A~-A~ ]1/2_OIFAH } 8n 1 +0I2f2 a cd ,. cd cott aly I AHqI +O'f. 8nv (44a) (44b) J. Appl. PhyS., Vol. 45, No.1, January 1974 The result (44) has been obtained independently by Slonczewski and Malozemoff23 by another method. 385 Before comparing this velocity-drive relation with experiment, it is useful to interpret it in terms of the qualitative discussion of Sec. II. When the domain spin configuration specified by (7) with (A7) and depicted in Fig. 1 is moved in the y direction, the local component of the velocity of the Bloch lines in the plane of the wall is (45) If the velocity-field relation (11) which was obtained for a stationary domain with moving Bloch lines in the lossless case is applied, the associated field is H = _ 2nv COS<Pb .on I yl d . (46) This corresponds in <Pb distribution and magnitude to a uniform bias field gradient which may be denoted by (47) The force corresponding to (46) is directed at each point on the wall perimeter in a direction normal to the wall. The component of this force in the direction of the domain motion, the y direction, at each point on the domain is canceled by an equal force of opposite sign at some other point on the domain. The x component of the force corresponding to (46) does not integrate to zero over the domain. It is given by jKn __ 4Mshnlvi 2-/. x -dl yl cos 'l'b· (48) Since (47) accounts for half of the perpendicular field gradient (42), since the x component of the force attributable to the motion of the Bloch lines within the domain wall in the y direction is distributed according to COS2<Pb' and since the gyro coupling force density is independent of <Pb, the following may be concluded: The gyrocoupling force density at the sides of a moving cylindrcial domain is completely accounted for by the moving Bloch-line formula (11). The other half of the total gyrocoupling force must be attributable to the interaction of the normal motion of the wall with the Bloch lines as discussed in Sec. II. The x component of the associated force per unit wall length, obtained by subtracting (48) from the constant force density asso ciated with (42), is fKP--4M.hlvl . 2-/. x -alyl sm'l'b· (49) For a domain in uniform motion this must be balanced by a gradient component (since AH~n + AH~p = AH) 4nlvl. AH1P=dfYj" Ix. (50) The x component of the force density associated with (50) is distributed according to cos2<P&. Since this den sity distribution does not match the force density dis tribution (49), the domain must distort in such a way that the exchange force carries the excess x-direction gyro coupling force from the leading and trailing edges of the domain to the sides where it is compensated by [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41386 180 160 :;; 140 V> Q; 0.'20 E ~ ~ 100 U '3 w 80 > 60 40 20 A.A. Thiele: Dynamics of magnetic domains o~-LL-~~-J __ ~ __ ~ __ -L __ -L __ ~ __ ~ __ ~ o 2 3 4 6 7 8 9 10 FIG. 5. Domain velocity transverse to the drive force, 0, and parallel to the drive force, e, as a function of the drive force for a domain 4.5 j.I in diameter. Theoretical curves are for n = 52, a =0. 012, t.H",,=0.75, andf(N) = 1. O. the applied field gradient force. Such a bunching of the Bloch lines has no effect on the total gyrocoupling force. The bunching will, however, cause a second order increase in the total dissipation which will, in turn, tend to cause the domain to propagate in a direction more nearly parallel to the driving force than if there were no bunching. The component of the applied field gradient required by the Gilbert damping induced drag has been written in terms of the function j{N). Since rrd is the domain circumference the parameter N, (43e) is a measure of the Bloch-line density in terms of the wall width lwo' The value N = 1 corresponds to the Bloch-line pairs being compressed in width from the isolated line pair width of 2l/o=2ql/2lwO to a pair width of 2lwo' Note that this is the value at which the wall width begins con tracting with increasing compression, which was pre viously termed the region of extreme compression. The factors in fiN) may be accounted for in the following way: the N'l factor is due to the increase in the trans verse force when the number of Bloch-line pairs is increased, the 1 +2N2 factor is due to the increase in exchange energy (and therefore of diSSipation density with increasing n), and the (1 +N2)-1/2 factor is due to the decrease in wall width with increasing n. For n:= 0 (as in orthoferrites for example), j{N):= 00, and the motion is parallel to the applied field gradient. The discussion above illustrates the use of the gyro coupling force and diSSipation force as an aid to deter mining the internal structure of a moving domain, as well as in determining the over-all motion of the do main. In Ref. 21 the detailed internal structure of a moving domain containing highly compressed Bloch lines is discussed. In particular, the distortion of the domain necessary to transmit the excess gyrocoupling force from the leading and trailing edges to the sides of the domain is explicitly given. The gyro coupling and J. Appl. Phys., Vol. 45, No.1, January 1974 386 diSSipation force concepts were a useful aid in obtaining the internal structure given in Ref. 21. The data for the motion of a hard bubble in a garnet film presented by Tabor et al. in Refs. 1 and 6 have been discussed theoretically by Slonczewski and Malozemoff in Ref. 23. More extensive velocity-drive data for domains with highly compressed Bloch lines together with microwave linewidth data on an epitaxial film of Y1Gd1 TmlG~.BFe4.2012 were presented in Ref. 21. The velocity-drive data of Ref. 21 are replotted in Figs. 5 and 6. The theoretical curves (44) are plotted for the parameter values a := 0.012, AHcd:= 0.75, and fiN) := 1. 0, with n := 52 for the 4.5 -J-L-diam domain of Fig. 5 and with n:= 57 for the 5.8-J-L-diam domain of Fig. 6. (The data were taken on two different domains. ) The measured domain coercivity (defined as the mini mum pulse amplitude necessary to produce observable domain motion) was 1. 0 Oe for the 4.5 -J-L-diam domain and 0.9 Oe for the 5.8-J-L-diam domain. SiI'lce only the product a j(N) appears in (44), the approximation j(N) := 1. 0 affects only the a value. USing the fitted n values and the measured d values together with the exchange and uniaxial anisotropy constants for this material ,24 A=2.5x10-7 erg/cm and Ku:=8.0 X103 erg/cm3 in (6) and (43e) yield the N values for the domains in Figs. 5 and 6. The values obtained are N = 1 . 3 for the 4.5 -J-L diam domain and N = 1.1 for the 5.8-J-L-diam domain. From (43d) it is seen that the error in ('J produced by assuming j(N):= 1 is less than 10% for either domain. The agreement between the data and the theoretical curves in Figs. 5 and 6 is seen to be quite good. In the figures or (44) it is seen that for drive field gradients only slightly greater than ARcd the velocity is parallel to the applied field force. For very large applied field gradients, the angle between the applied field gradient and the velocity approaches !; L := tan-! (a-lFl). Figures 180 160 ~ 140 V> 8. E 120 u i: 100 ~ ~ 80 60 40 20 o L-~ __ ~ __ -L __ ~ __ ~ __ L-~ __ ~ __ -L __ ~ o 2 3 4 5 6 7 8 9 10 lIHa(Qe) FIG. 6. Domain velocity transverse to the drive force, 0, and parallel to the drive force, e, as a function of the drive force for a domain 5.8 j.I in diameter. Theoretical curves are for n =57, a=0.012, t.Hcd=0.75, andj(N)=1.0. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41387 A.A. Thiele: Dynamics of magnetic domains 5 and 6 clearly show that the drive gradient must pe many times illlCd before the limiting angle ~L is approached. The reason for this is that as the drive increases, ~ increases so that the applied force is not as effective in overcoming the coercive drag as it was when the velocity and field were parallel. Because of the large role played by coercivity at all velocities, the data may be fitted quite well for any a value below some maximum value. The data are almost equally well fitted with a = 0.03, illlcd = 0.70, and the same n values as above. For a values a little greater than 0.03, the fit to the data becomes poor (see Fig. 2 of Ref. 21). The a value determined from a resonance linewidth measuremene5 at 13.15 GHz on a section of the same film used for obtaining the data in the Figs. 5 and 6 is O. 03. The a values extracted from mobility measurements on normal bubbles in this materiaP6 0.2. The discrepancy in a values appears to be resolved by examination of the spin-wave spectrum of a domain wall containing highly compressed Bloch lines. 27 In order to make the problem tractable, in-plane aniso tropy effect are ignored [the last term in (1) is taken to be zero or K/27TM;=00]. In this case, the solution specified by (A7) is exact. The spin-wave spectrum in the body of the domains is normal for a material with uniaxial anisotropy except that the spin waves are partly reflected by the domain wall. The presence of the Bloch lines, on the other hand, greatly modifies the spectrum of the spin waves bound to the domain wall. In the notation of the Appendix the wall-bound spin-wave spectrum is w=2 ~~A {[I + (2~wo) 2r ~ +k!}, Ikyl <2;ll+(2~worl corresponding to an unnormalized mode amplitude (51a) (51b) m == sech( l: x) exp ~y 2!w x + i(kyY + k~z -wt~. (51c) [The eigenvalue equation from which (51) is obtained is derived in exactly the same way as described in Appen dix A of Ref. 28. The spin-wave amplitude may be specified by a single component, since here the ampli tude of the two components is equal (see Ref. 28). The volume spin-wave states will not be specified here since this is quite length process.] The spectrum [(51a) and (51b)] reduces to the normal bound spin-wave spectrum when I s I == 00 (and q2 == 0 in the notation of Ref. 28). For I s I < 00, the w values are seen to be reduced from their I s I = 00 values. (This does not, in this case, correspond necessarily to a decrease in the stiffness of the wall, since there is an increase in the mode volume.) The dominant effect is not the change in the mode frequen cies but instead is the disappearance of the bound modes having a ky magnitude greater than that allowed by (51b). The bound modes above this limit disappear into an initially totally reflecting part of the volume spin-wave spectrum. The effect is greatest (the maximum allowed value of I k y I having its minimum value) when h = lwo' The corresponding N value is N == 1. The effect is thus J. Appl. Phys., Vol. 45, No.1, January 1974 387 nearly maximized in the data in Figs. 5 and 6 in which N=1.3 andN==I.1. The disappearance of the short-wavelength wall-bound spin-wave modes removed them as intermediaries for the dissipation of motion energy. This apparently re duces the a value for wall motion from its usual value to the microwave linewidth value. This confirmation of the importance of the wall-bound spin-wave modes sug gests that there is an optimum value for the "intrinsic" a value for the attainment of high bit rates in garnet film devices. Clearly, when a is decreased from a large value, the domain mobility will increase. If the ('j value is decreased too far, however, the amplitude of the wall-bound spin waves in a moving domain will increase to the point where Bloch lines are produced. The Bloch-line production will then cause the domain dynamics to become erratic (assuming that the material has been ion implanted so that domains containing Bloch lines are not statically stable). Such an effe ct was de scribed in Ref. 13 as dynamic conversion. This type of breakdown is distinguished from the breakdown dis cussed in Sec. II and Ref. 13 in connection with the do main collapse mobility method Bobeck et al. in that the production of Bloch lines from large spin-wave ampli tudes will be an incoherent process. VII. ESTIMATION OF SOME THRESHOLD VELOCITIES FOR THE GENERATION OF BLOCH LINES IN NORMAL BUBBLES The discussion of the effect of the presence of Bloch lines on the dynamic properties of cylindrical domains has thus far been restricted to the case in which the Bloch lines are parallel to the cylinder axis. In this section, both horizontal and generally oriented Bloch lines are considered. The problem of the nucleation or pulling away of initially horizontal (perpendicular to the cylinder axis) Bloch lines from the ends of a cylindrical domain considered here is identical or very similar to that conSidered in Refs. 18 and 29 In these references, it is proposed that the generation of horizontal Bloch lines produces a limiting velocity which is considerably below Walker's limiting velocity, 11 the ratio depending on the plate thickness. The velocity necessary for hori zontal Bloch-line generation will now be estimated here using the gyro coupling force method. The initial static domain configuration to be con sidered is shown in Fig. 7. The magnetic distribution is that of a cylindrical domain in a plate in which the plate normal and the uniaxial easy axis lie along the z axis. The magnetization vector in the body of the magnetic material lies along the positive z axis and within the domain along the negative z axis. Within the domain wall, midway between the plate surfaces, the wall is a Bloch wall, the magnetization lying in the wall plane, and pointing in the direction of increasing cpo The cylin drical shape of the wall produces a gyrocoupling vector of small magnitude pointing along the z axis everywhere within the domain wall. The effect of this gyrocoupling vector is to produce a small gyrocoupling force at right angles to the direction of propagation of the domain when the domain is moved. This effect will be ignored for the present, since the effect to be considered in- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41388 A.A. Thiele: Dynamics of magnetic domains FIG. 7. Magnetization, gyro coupling vector, and gyrocoupling force configurations in a normal moving garnet film domain. volves a larger gyrocoupling density. In Refs. 18, 19, and 29, it is pointed out that in the case of a cylindrical domain with a wall of zero thick ness the radial component of the demagnitizing field at the wall plane has a logarithmic singularity. This singularity, evaluated from the general elliptic integral expression for the radial field,30 is in this case ~=~lnlz..1 h«d (52a) 47TM. 7T h' 4:~s =;lnl~l, d«h, (52b) where z is the distance in the wall plane to the plate surface. By using again the general elliptic integral expressions for the radial field in the special case d = 2h, the initial slope of the radial field in the wall plane at the midpoint between the plate surfaces is I (h/47TM.)(ilH r/ilz) I '" 1, the field reaching IHr I /4M. = 0.5 at z/h"'O.14, and IHrl/47TMs"'1.0 at z/h"'O,03. Of interest here is the total rotation of cp as the plate sur face is approached 0 The recent variational calculation of Schl6mann19 yields a total rotation angle of nearly h over a wide range of parameters, even though he used the trial function cp = CPo sin[ 7T(z/h +~)], which excludes rapid cp rotation near the surfaces. Since the most rapid cp rotation should be concentrated near the plate surface, the static magnetic distribution and gyro coupling vector may be represented approxi mately as in Fig. 7. In the present calculation, it is convenient to assume that the total cp rotation at each plate surface has its largest possible value, LJ..CP = I7T. Using (12b) and (12c) yields a magnitude for the force per unit line length on a generally oriented half Bloch line of I fA'1 I = (7TM.I I y I ) I v I . The directions of the gyro coupling forces produced when the domain is moved in the y direction are shown in Fig. 7. Note that at the upper leading and lower trailing edges of the domain the gyro coupling forces tend to push the end structure toward the plate surface. On the other hand, at the lower leading and upper trailing edges of the domain, the gyro coupling forces tend to pull the end structure into the body of the domain wall. If the gyrocoupling force is strong enough to pull a Bloch line into the body of the domain wall, then the structure which results is as shown in Fig. 8 (assuming J. Appl. Phys., Vol. 45, No.1, January 1974 388 for ease of presentation that this happens only at the upper trailing edge of the domain) The structure at the plate surface is seen to have increased in stability as a result of the production of the Bloch line, since the tendency to generate additional Bloch lines by this method has been eliminated.· It will be shown that the over-all domain velocity large enough to generate a Bloch line in this case is nearly the Walker breakdown velocity,11 and at this over-all domain velocity the Bloch-line velocity which is reached when the structure develops to the point shown in Fig. 8 is very high. For this reason the entire structure is not expected to be ve ry stable. In order to estimate the over-all domain velocity required to pull a Bloch line into the domain wall, it is necessary to introduce some rather crude assumptions about the structure and mechanism of this process. It is assumed that the domain retains the static structure shown in Fig. 7 at the beginning of the process. Since the structures at the ends of the domain have the same energy before and after the production of the Bloch line, it is assumed that the maximum force necesary to create the Bloch line is the force associated with the maximum Bloch-line energy density which can be computed from (1) and (A6). This maximum force per line length is I fell = 47TQ-l/2A/llo' (53) Assuming that the initial twist only starts the creation process, so that the full (LJ..CP = 7T) gyrocoupling force is available to overcome the creation force, yields for the minimum velocity for the creation of Bloch lines vc = (2/7T) I YI27™s(lwo/7T), '" (2/7T)Vw' where (54a) (54b) (54c) is Walker'S limiting velocity, 11 to the extent that lwo '" lw' the dynamic wall width. In the preceding calculation, it has been assumed that in the creation of a Bloch line the restoring forces are linear (energy quadratic). The restoring force propor tionality constant was estimated from the total energy and the characteristic length llo' the Bloch line width. z~ M 9 FIG. 8. Magnetization, gyrocoupling vector, and gyro coupling force configurations in a garnet film domain in which a Bloch line is being pulled off of the upper film surface. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41389 A.A. Thiele: Dynamics of magnetic domains M 9 FIG. 9. Magnetization, gyrocoupling vector, and gyrocoupling force configurations in a garnet film domain in which Bloch line ring is being nucleated. As the Bloch line develops, the restoring force and the gyrocoupling force are thus porportional to some L!t.cp, characterizing the extent of development of the Bloch line. Hagedorn31 has calculated the critical velocity for the production of Bloch lines using the gyrocoupling force and the partial Bloch-line energy expression from the Appendix of Ref. 29. The critical velocity obtained by this method agrees with that obtained in Ref. 29. This critical velocity is lower than Walker's critical velocity by a factor proportional to ql /2, when the plate thickness and domain diameter are proportional to l. The discrepancy results from the fact that the restoring force computed in the Appendix of Ref. 29 is quadratic in L!t.cp (cubic energy). This implausible result is probably due to the fact that the partial Bloch line is not properly joined to the remainder of the domain. This results in too wide a partial Bloch line and too low a partial Bloch-line energy. The error being particu~ larly pronounced at small L!t.cp. The result (54) is somewhat self-contradictory, since in obtaining it the spin system was assumed to maintain a configuration derived from the static spin configura tion and the result approaches the Walker limiting velocity where the static spin configuration is known to be incorrect. The result (54) does, however, provide an approximate lower bound on the velocity required for the generation of horizontal Bloch lines. After the Bloch lines have pulled away as in Fig. 8, the reversible energy force for the creation of a segment of Bloch line evolves into the much smaller force as sociated with the lengthening of the line (the line curva ture force). The dominant forces in this state are thus the gyrocoupling and dissipation forces, so that the approximate Bloch-line velocity is the very high velocity I v, I = hO!-lql /2vc. [This is obtained by setting Vx = Vc and V~=V, in (28b), which is equivalent to (27b) for one Bloch line and Vx driven to vc.1 The gyro coupling force may be used to investigate yet another mechanism for the nucleation of Bloch lines in cylindrical domains. The mechanisms to be considered is depicted in Fig. 9. The initial reversal in the sense of rotation of the magnetization of the Bloch wall is considered to be the result of the interaction of the mo tion of the domain wall with a defect. The minimum diameter of such a reversal for which the boundary of J. Appl. Phys., Vol. 45, No.1, January 1974 389 the reversal may be considered to be a well-formed circular Bloch line is clearly liD. By ignoring line curvature effects, the energy required to nucleate such a structure, computed from the energy per unit line length (A6), is (55) If the gyrocoupling force vector of this Bloch-line ring points inward, the structure will collapse immediately. If, on the other hand, the rotation sense and orientation are as shown in Fig. 9, the gyrocoupling force is out ward. Therefore, at a sufficiently large velocity, the gyrocoupling force will equal or exceed the Bloch-line curvature force and the structure will expand. Com puting the Bloch-line curvature force from the Bloch line energy (A6), and setting this equal to the gyro coupling force from (21), or with L!t.cp = 7T, yields for the critical velocity if = (4/7T)21 YI27TM.<Zwo/7T) (56a) VC = (4/7T)2vw' (56b) where again in (56b) the difference between static and dynamic wall width has been ignored. The result again contradicts the assumption that the dynamic spin con figuration is the same as the static spin configuration, since this time the computed velocity is actually larger than the Walker breakdown velocity. The calculation does, however, show that if the critical velocity is to be much less than Walker's breakdown velocity, then the diameter (and nucleation energy) of the initially nu cleated Bloch-line ring must be proportionally greater than the minimum diameter, l/O" VIII. THE SUPPRESSION OF HARD BUBBLES BY MULTILAYERING AND ION IMPLANTATION The concepts introduced in Secs. I-Vll may be used as an aid to further speculation as to the mechanism by which multilayering32 or ion implantation33 suppress the formation of hard bubbles in garnet films. As a first example, the increasing of the nucleation threshold of horizontal Bloch lines at the ends of the domain by the presence of the added layer is considered. In the case of ion implantation, the ion-implanted layer could act to "short out the radial field" or "feather out the mag netization" or both at the upper end of the domain. If there were at the outset a similar effect at the lower r 1 r ,..----.......... ! "t /" M 9 FIG. 10. Magnetization and gyrocoupling configurations in a type-I double-layer garnet film domain. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41390 A.A. Thiele: Dynamics of magnetic domains /, I I I '1 r I I '----'...---1 I 1 ---1--1 ~ ------- 1<:'" .,----.''1 / / , '...---I' 1 \ t- ' I I "....... I '--... \1...;;-.... I M 9 FIG. 11. Magnetization and gyrocoupling vector configurations showing a domain with a single Bloch-line pair connected to a modified exchange singularity. magnetic-nonmagnetic boundary, then both horizontal Bloch-line nucleating regions would be eliminated. This mechanism can be ruled out, however, since multilayer films achieve similar results to ion-im planted films and since it is clear from Fig. 8 that the radial field must be eliminated at both the upper and lower surfaces in order to suppress horizontal Bloch line nucleation. Subsequent comments will be in terms of the case of a type-I double-layer film as defined in Ref. 32, the generalization to type-II and ion-impanted films being straightforward. The magnetic distribution of double-layer film with a cylindrical domain containing no Bloch lines is de picted in Fig. 10. The rotation of the magnetization vector at the ends of the domain has been ignored, since in the present case it has no effect on the argument to be presented. On the other hand, the magnetization pro ducing the weak gyrocoupling vector within the wall has been shown. The energy density function for the domain wall which forms the lower cap of the domain wall is in the very crude approximation needed here (differences in the properties of the upper and lower films being ignored) which is obtained by adding a local demagnetizing energy term to a uniaxial anisotropy term and an ex change term as in (1). Note that the zero of energy in (1) and (57) are not the same. The most remarkable property of (57) for the present discussion is that the anistropy function is independent of </>, so that the </> distribution depends only on the boundary conditions This symmetry property would clearly be maintained in a more precise treatment of the energy density function. The e distribution is approximately cos e == tanh( nZ/l) , (n/l)2 == (Ku -2nM!)/ A (58a) (58b) as usual as long as (V</»2 makes only a small contribu tion to the exchange energy. When the solution (58) is joined to the usual side wall solution as in Fig. 10, there is found to be no way to avoid an exchange sin gularity with respect to </> somewhere on the lower do- J. Appl. Phys., Vol. 45, No.1, January 1974 390 main wall. In Fig. 10, the exchange singularity is represented by the central point from which the gyro coupling vectors diverge _ The system doubtless finds a way to avoid this singularity by modification of (58) over an area of the wall whose characteristic length is lw' Since the wall width is much smaller than the minimum domain diameter, the area of such a modified singularity is only a small fraction of the total area of the lower domain wall, most of the domain wall being described by (58). The singularity can be eliminated entirely if a Bloch-line pair is included in the side wall so that the total rotation of </> in the side wall is zero, as in Fig. 3. The zero total </> rotation property is a property of the final configuration previously described by Rosencwaig34 in his discussion of the mechanism of hard bubble suppression. Consider now a domain in which the total rotation in the side walls is several times 2n. In this case, there will be several modified exchange singularities in the lower domain wall. Since the Bloch-line energy of the Bloch lines in the lower domain wall is very low (the width being determined only by the boundary conditions and the energy being inversely related to the width), the associated forces are correspondingly small. Be cause of the weakness of these forces, movement of the domain as a whole will allow the gyro coupling forces of the Bloch lines in the side wall to pull these Bloch lines to one side as in Fig. 3. The magnetization and gyrocoupling vector configuration for one Bloch line pair is depicted in Fig. 11. (The gyrocoupling vec tors in Fig. 11 are oppositely directed to those in Fig. 3. ) The Bloch-line energy density is highest in the do main side walls and at the modified exchange singularity (the point from which the Bloch lines diverge in Fig. 11). The Bloch lines In the lower domain wall which connect the exchange singularity to the side wall have the lowest energy, since here the rate of change of </> is the lowest. Although the energy density of these sec tions of the Bloch lines is low, they are oriented so as to tend to pull the modified exchange singularities to the side of the domaino Once at the side wall, the ex change singularity will be pulled up the wall by the Bloch lines in the side wall, and so this Bloch-line pair will be eliminated. A recent series of experi ments35-37 can be interpreted as indicating that in many cases the state reached by the domain is neither the n == 0 or n == ± 1 state but is one in which n is some small number greater than unity. The reason for this is that since the Bloch-line energy is very low in the lower do main wall, it competes with demagnetizing energies which are local to the lower domain wall but are not local in the sense of being included in (58) 0 The situa tion is similar to that in Permalloy thin films and, in deed, domain patterns have been observed in ion-im planted films which are similar to those observed in Permalloy. 36,37 In the Permailoy-like spike domains which are observed, each spike point presumably cor responds to a modified exchange singularity. In a recent experiment, Henry et al. 38 have observed that there is a characteristic temperature TH above which hard bubble generation does not take place in many materials. They find that hard bubbles generated below this temperature remain hard bubbles when the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41391 A.A. Thiele: Dynamics of magnetic domains temperature is raised above T H' This can be inter preted in terms of the previous discussion if it is as sumed that in this case that Bloch-line termination structures once formed do not tend to link up (form exchange singularities). This must be true even when the Bloch lines are pushed together, since in the ex periment the domains were moved. In another experi ment by Wolfe and North39 in which films containing hard bubbles were ion implanted, it was observed that while the demagnetized domain pattern was hardly dis turbed, the hard bubbles were eliminated. In this case, the termination structures were apparently driven hard enough to link up. The over-all picture that emerges from the above discussion is that the end cap allows any g which is generated at high domain velocity to relax out at some subsequent time when the domain velocity is low. This prevents the number of Bloch lines from becoming large, and so eliminates the worst of the difficulties caused by the presence of the Bloch lines. IX. CONCLUSION In cylindrical domain device applications, the existence of domains with nonzero gyrocoupling vectors is undesirable, since the mobility of such domains is generally reduced as compared with domains in which the gyro coupling vector is zero. In materials such as garnet films in which the only effective in-plane aniso tropy comes from demagnetizing effects, the amplitude of the wall-bound spin-wave modes is expected to in crease with decreasing ex. If ex is sufficiently small, this will result in the generation of Bloch lines. There is therefore an optimum value of ex in such materials. In a material such as an orthoferrite in which there is a strong in-plane anisotropy which effectively constrains the magnetization to lie in a Single plane, the gyro coupling vector is zero. The presence of a large in plane anisoptropy will thus effectively eliminate hard bubbles. The decrease in the damping parameter ex apparently produced by the removal of the wall-bound spin-wave modes offers evidence of the importance of these modes in limiting the mobility of domain walls. In a material having a strong in-plane anisotropy, the stiff ness of both the wall-bound and volume spin-wave modes is increased by the in-plane anisotropy. The mobility increase produced by the stiffening of both the wall bound modes and the volume modes should be greater than in the case in which the presence of the Bloch lines removed the wall-bound modes but did not alter the volume spin-wave disperSion relation. DeLeeuw40 has recently reported an increase of over an order of magnitude in the limiting velocity of a garnet film, the velocity increase having been pro duced by the application of an in-plane field. Such a field makes ~cp = rr Bloch lines unstable, generally sup presses the formation of Bloch lines, and raises the frequencies of the wall-bound spin-wave modes. The experimental result may thus be taken as tending to confirm the general picture given in the text. The prescription for both the elimination of the gyrocoupling force and the attainment of high velocity is indicated in J. Appl. Phys., Vol. 45, No.1, January 1974 391 Ref. 27: start with an uniaxial easy-plane material then induce an orthogonal easy axis. ACKNOWLEDGMENTS The author would like to acknowledge R. C. LeCraw and G. P. Vella-Coleiro for making available their unpublished data, R. Wolfe,.W. J. De Bonte , and F. B. Hagedorn for many useful discussions, and, in partic ular, F. B. Hagedorn for his persistance in reading the manuscript through many editions, pointing out many flaws, and making suggestions for their removal. APPENDIX: STATIC DOMAIN WALL CONTAINING BLOCH LINES In this Appendix, a spin configuration is specified which minimizes the energy density expression (1) to first order in q-l. The spin configuration is depicted in Fig. 2. The wall normal is the x axis [<p"=O in (1)), the periodic direction is the y axis, and the solution is in dependent of z. The boundary conditions are 8(x=oo,y) =0, (Ala) 8(x=-oo,y)=rr, (Alb) and 8(x,y) = 8(x,y + s), (A1c) cp(x,y)= cp(x,y+s), (A1d) and the Euler equations corresponding to the energy density (1) may be arranged in the form (P8 [2rr~ (0<P)2J 0x2 -sin8cos8 --r(q +cos2cp)+ ay (Ale) sin8 = _ sin8 a2 cp _ cos 8(~ a cp +~ a CP) . (Alf) ax2 ax ax ay ay The solution is obtained in the following way: First qJ is assumed independent of x and the left-hand side of (Alf) is set equal to zero and solved. This solution is then used in the left-hand side of (Ale) which is set equal to zero and solved. It may then show that the right-hand sides of the equations are of higher order in q-l than the left-hand sides and that the energy is, in fact, minimized to first order in q-l. In order to present the solution, it is necessary to introduce the elliptic function parameter m which is related to the periodicity by q s I /4)(rr/1,o) =ml/2 K(m), (A2) where K(m) is the complete elliptic integral of the first kind (m =k2). The solution to first order in q-l to the Euler equations associated with the energy density (1) is cos8 =tanh(rrx/1w) ' sin8 = sech(1Tx/1w) ' sincp = ± sn(rry/1, 1m), (A3a) (A3b) (A3c) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41392 A.A. Thiele: Dynamics of magnetic domains coscfJ=cn(1Ty/lllm), (A3d) where the upper sign corresponds to ¢ increasing with increasing y, the length parameter in the elliptic func tions sn and cn is (1T/lI)2 = (1/m)(1T/lIO)2 and the wall width is (l:r = ~ [ Ku +21TM! (2cos2¢ + 1 :m) J. In general, the wall energy density is O"w(Y) = O"wo[zwo/lw (y) J. (A3e) (A3f) (A4) The average wall energy density is obtained by inte grating over a period after expanding the square root which occurs. From here on and in the text, O"w will be assumed to denote this average wall energy density which is [( 1 1 ;\ 1/2 O"w = O"wO 1 +(j :mJ +!. (1 +!. 1 _m)-1/2 E(m) -(l-m)K(m) ] (A5) q q m mK(m) , where E(m) is the complete elliptic integral of the se cond kind. The sense in which the spin configuration (A3) mini mizes the energy to lowest order in q-l is a function of s. This may be discussed in terms of the relative mag nitude of the three terms inside the bracket in (A3f). For uncompressed Bloch lines, is> llo, the last two terms are of equal magnitude and are smaller than the first term by q-l. The Winter approximation9 is in this case the lowest-order approximation to the local de magnetizing term in q-l since llo = ql /2lwo' so that the Bloch lines are wider than they are thick by ql/2 and the stray fields are small. As the Bloch lines are com pressed, there is a gradual transition until in the very high compression limit, is« llo, the last term domi nates the first term. The second term, which depends on the Winter approximation, is no longer valid, its actual value being reduced by flux closure effects. How ever, this term is now the smallest of the three terms. The greatest relative error occurs at is ""lwo' In the limit of uncompressed Bloch lines is> llo, 1 -m = 16 exp(- 1T I s I /2l10) , (A6a) sin¢ = ± tanh(1Ty/llo) , (A6b) cos¢ = sech(1Ty/llo) , (Tw=(TWO+\~\T[l +4exp(_1T~l:~)J. T = 16Aq-l /2. (A6c) (A6d) (A6e) Equations (A6b) and (A6c) describe one-half of a Bloch line pair (5 = 00). The othe r half is obtained by changing the sign in (A6b). The density T is the additional Bloch line pair energy per unit length. The interaction energy between the lines is seen to be exponential. In the opposite limiting case in which the Bloch lines are compressed, is < llo, J. Appl. Phys .• Vol. 45. No.1. January 1974 392 (A7a) and ¢=±21TY/S, (A7b) the wall width approaching the constant l~2 = lj + (is)-2. (A7c) Since the wall width is constant, the wall energy may be obtained from (A4). Note that (A7c) has the same form as was obtained in Ref. 4. Note that in (A3) the functions describing the depen dence of both e and ¢ on the spatial coordinates obey the pendulum equation tf2l: + 2 • ,. 0 dt2 Wo Sin!, = , (AS) where Wo is a constant, t (time) is x or y, and l:, the orientation angle, is 2e or 2¢. The infinite period solu tion in e was first applied to domains by Landau, 41 the extension to the finite period case being carried out by Shirobokov . 42 lW.J. Tabor, A.H. Bobeck, G.P. Vella-Coleiro, and A. Rosencwaig, Bell Syst. Tech. J. 51, 1427 (1972). 2A.P. Malozemoff, Appl. Phys. Lett. 21, 149 (1972). 3p.J. Grundy, D.C. Hothersall, and R.S. Tebble, J. Phys. D 4, 174 (1971). 4A. Rosencwaig, W.J. Tabor, and T.J. Nelson, Phys. Rev. Lett. 29, 946 (1972). 5p.J. Grundy, D.C. Hothersall, G.A. Jones, B.K. Middleton, and R. S. Tebble, Phys. Status Solidi A 9, 79 (1972). 6G. P. Vella-Coleiro, A. Rosencwaig, and W. J. Tabor, Phys. Rev. Lett. 29, 949 (1972). 7A. P. Malozemoff and J. C. Slonczewski, Phys. Rev. Lett. 29, 952 (1972). BA.A. Thiele, Phys. Rev. Lett. 30, 230 (1973). 9J.M. Winter, Phys. Rev. 124, 452 (1961). 10W.J. DeBonte, in Proceedings of the Seventeenth Annual Conference on Magnetism and Magnetic Materials, Chicago, editedbyC.D. Graham, Jr. andJ.J. Rhyne (AlP, New York, 1972), p. 140. llL. R. Walker (unpublished); described by J. F. Dillon in A Treatise on Magnetism, edited by G. T. Rado and H. Suhl (Academic, New York, 1963), Vol. lIT, p. 450. 12T. L. Gilbert, Phys. Rev. 100, 1243 (1955). 13G.p. VeIla-Coleiro, F.B. Hagedorn, Y.S. Chen, andS.L. Blank, Appl. Phys. Lett. 22, 324 (1973). l4A. H. Bobeck, I. Danylchuk, J. P. Remeika, L. G. Van Vitert, and E. M. Walters, in Ferrites: Proceedings of the Interna tional Conference, edited by Y. Hoshino, S. !ida, and M. Sugimoto (University of Tokyo Press, Tokyo, 1971), p. 361. 15A.A. Thiele, Bell Syst. Tech. J. 50, 725 (1971). 16A.A. Thiele, A.H. Bobeck, E. Della Torre, and V.F. Gianola, Bell Syst. Tech. J. 50, 711 (1971). 17J.K. GaIt, Phys. Rev. 85, 664 (1952). l8B. E. Argyle, J. C. S!onczewski, and A. F. Mayadas, AlP Conf. Proc. 5, 175 (1972). 19E. SchlOmann, Appl. Phys. Lett. 21, 227 (1972); J. Appl. Phys. 44, 1837 (1973); 44, 1850 (1973). 20J. C. Slonczewski, Int. J. Magn. 2, 85 (1972). 21A.A. Thiele, F.B. Hagedorn, and G.P. Vella-Coleiro, Phys. Rev. B 8, 241 (1973). 22J. C. Slonczewski, Phys. Rev. Lett. 29, 1679 (1972). 23 J . C. Slonczewski and A. P. Malozemoff, in Proceedings of the Eighteenth Annual Coriference on Magnetism and Magnetic Materials, Denver, editedbyC.D. Graham, Jr. andJ.J. Rhyne (AlP, New York, 1973), p. 458. 24F. B. Hagedorn (private communication). 25R. C. LeCraw (private communication). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41393 A.A. Thiele: Dynamics of magnetic domains 26G• P. Vella-Coleiro, D. H. Smith, and L. G. Van Uitert, Appl. Phys. Lett. 21, 36 (1972). 27A.A. Thiele (unpublished). 28A.A. Thiele, Phys. Rev. B 7, 391 (1973). 29J.C. Slonczewski, J. Appl. Phys. 44, 1769 (1973). 30 F. C. Rossol and A. A. Thiele, J. Appl. Phys. 41, 1163 (1970) . 31F.B. Hagedorn (private communication). 32A. H. Bobeck, S. L. Blank, and H.J. Levinstein, Bell Syst. Tech. J. 51, 1431 (1972). 33R. Wolfe and J. C. North, Bell Syst. Tech. J. 51, 1436 (1972) • 34A. Rosencwaig, Bell Syst. Tech. J. 51, 1440 (1972). 35A. W. Anderson (private communication). 36R. Wolfe, J.C. North, W.A. Johnson, R.R. Spiwak, L.J. J. Appl. Phys., Vol. 45, No.1, January 1974 Varnerin, and R. F. Fischer, in Proceedings of the Eigh teenth Annual Conference on Magnetism and Magnetic Ma terials, Denver, editedbyC.D. Graham, Jr. andJ.J. Rhyne (AlP, New York, 1973), p. 339. 393 37R. Wolfe, J. C. North, and Y. P. Lai, Appl. Phys. Lett. 22, 683 (1973). 38R.D. Henry, P.J. Besser, R. G. Warren, and E.C. Whitcomb, lntermag. Conference, Washington, D. C., 1973 (unpublished) • 39R. Wolfe and J. C. North (unpublished). 4oF.H. de Leeuw, Intermag. Conference, Washington, D.C., 1973 (unpublished). 41L. D. Landau, Collected Papers of L. D. Landau (Gordon and Breach, New York, 1965), p. 101. 42Y.A. Shirobokov, Zh. Eksp. Teor. Fiz. 15, 57 (1945). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.101.79.200 On: Thu, 11 Sep 2014 14:26:41
1.2712946.pdf	Influence of the Oersted field in the dynamics of spin-transfer microwave oscillators G. Consolo, B. Azzerboni, G. Finocchio, L. Lopez-Diaz, and L. Torres Citation: Journal of Applied Physics 101, 09C108 (2007); doi: 10.1063/1.2712946 View online: http://dx.doi.org/10.1063/1.2712946 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/101/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Micromagnetic study of phase-locking in spin-transfer nano-oscillators driven by currents and ac fields J. Appl. Phys. 109, 07C914 (2011); 10.1063/1.3559476 Micromagnetics of single and double point contact spin torque oscillators J. Appl. Phys. 105, 083923 (2009); 10.1063/1.3110185 Spin-transfer induced noise in nanoscale magnetoresistive sensors J. Appl. Phys. 101, 073911 (2007); 10.1063/1.2720094 Spin transfer oscillators emitting microwave in zero applied magnetic field J. Appl. Phys. 101, 063916 (2007); 10.1063/1.2713373 Frequency modulation of spin-transfer oscillators Appl. Phys. Lett. 86, 082506 (2005); 10.1063/1.1875762 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Fri, 21 Nov 2014 19:31:46Inﬂuence of the Oersted ﬁeld in the dynamics of spin-transfer microwave oscillators G. Consolo,a/H20850B. Azzerboni, and G. Finocchio Dipartimento di Fisica della Materia e Tecnologie Fisiche Avanzate, University of Messina, Salita Sperone 31, Villa. Sant’ Agata, 98166 Messina, Italy L. Lopez-Diaz and L. T orres Departamento de Fisica Aplicada, University of Salamanca, Plaza de la Merced 37008, Spain /H20849Presented on 10 January 2007; received 31 October 2006; accepted 4 January 2007; published online 3 May 2007 /H20850 This paper focuses on the magnetization dynamics induced by the balance between the physical positive damping and the negative induced by spin-transfer torque in point-contact devices. Weconsider an applied ﬁeld perpendicular to the device plane which both saturates the magnetizationof the free layer and tilts the one of the pinned layer about 30° out-of-the-ﬁlm plane. The inﬂuenceof the nonuniform current-induced magnetic /H20849Oersted or Ampere /H20850ﬁeld on the magnetization dynamics of such oscillators has been taken into account within a micromagnetic framework.Results of micromagnetic calculations show that the Oersted ﬁeld yields spatial asymmetries in themagnetization conﬁguration, which do not introduce any modiﬁcations in the frequency domain.Finally, Slonczewski’s analytical formulation /H20851J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850/H20852about the spatial geometry of the current-excited spin-wave modes has been validated numerically. © 2007 American Institute of Physics ./H20851DOI: 10.1063/1.2712946 /H20852 Since the pioneer work of Slonczewski in 1996, 1it is well-known that the sd-like conduction electrons’ spins can induce a torque on the magnetization of a thin magneticlayer. This torque, justiﬁable by means of angular momen-tum conservation laws, creates a negative damping whichopposes to the positive phenomenological damping. 1,2Ac- cording to the intensity of both applied ﬁeld and current, wecan induce either reversal or precessional processes. Thepossibility to sustain persistent oscillations is discovering alot of interest because of its potential applications as current-controlled microwave oscillators as well as in nanoscale tele-communication systems. It has been also shown 3,4that a multilayer point-contact system, like the one depicted in Fig.1, could be employed to generate microwave steady magne- tization precessions in a thin free layer /H20849FL/H20850. Most of the understanding of the dynamics has been successfullyachieved by using single-domain theories, 5,6although the full comprehension could be carried out by taking into accountall the nonstandard /H20849and nonuniform /H20850contributions of the effective ﬁeld. In that sense, we will focus on the preces-sional dynamics induced by spin-polarized currents in point-contact geometries by using a full micromagnetic frame-work. From the computational point of view, the large lateraldimensions employed in point-contact experiments would re-quire prohibitive memory allocation and times, so that it isnecessary to deal with a computational area smaller than thephysical one introducing some absorbing boundary condi-tions to avoid the noisy contribution arising from the re-ﬂected waves. 7–9In our model, the absorption is guaranteed by introducing an abrupt change of the dissipation proﬁle atthe computational boundaries, checking that the reduced dis-tance from the contact edges does not affect the spin-wave properties. 7,9The spin-valve structure under investigation is composed by a trilayer Co 90Fe10 /H2084920 nm /H20850/H20851pinned layer /H20849PL/H20850/H20852/Cu /H208495n m /H20850/Ni 80Fe20/H208495n m /H20850/H20849FL/H20850, whose lateral dimen- sions, for the computational reasons cited above, have been set as 600 /H11003600 nm2. The point contact area is assumed to be circular in shape, whose diameter is varied between 40and 160 nm. Our micromagnetic approach 10starts from the Landau- Lifshitz-Gilbert-Slonczewski /H20849LLGS /H20850equation, a/H20850Electronic mail: consolo@ingegneria.unime.it FIG. 1. /H20849Color online /H20850Sketch of the point-contact device together with the spatial distribution of the implemented current ﬂow and the related Oersted/H20849Ampere /H20850ﬁeld.JOURNAL OF APPLIED PHYSICS 101, 09C108 /H208492007 /H20850 0021-8979/2007/101 /H208499/H20850/09C108/3/$23.00 © 2007 American Institute of Physics 101 , 09C108-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Fri, 21 Nov 2014 19:31:46dM dt=−/H92530 1+/H92512M/H11003Heff−/H9251 Ms/H92530 1+/H92512M/H11003/H20849M/H11003Heff/H20850 −g/H9262BJ /H208491+/H92512/H20850dMs3Pse/H9255/H20849/H9278,/H9257/H20850M/H11003/H20849M/H11003P/H20850 +g/H9262B/H9251J /H208491+/H92512/H20850dMs2Pse/H9255/H20849/H9278,/H9257/H20850M/H11003P, /H208491/H20850 where M=M/H20849r/H20850andP=P/H20849r/H20850are the spatially dependent magnetization vectors of FL and PL, respectively, /H9251=0.015 is the damping constant, /H9262Bis the Bohr magneton, gis the Landè factor, dis the FL thickness, Jis the current density, e is the absolute value of the electron charge, MS=640 kA/m is the saturation magnetization of the FL /H20849PS=1500 kA/m for the PL /H20850, and /H92530is the gyromagnetic ratio; Heffis the effective ﬁeld which carries the contribution of exchange,anisotropy, external, Oersted, and magnetostatic; the latterincluding also the interaction between PL /H20849whose dynamics has been neglected due to its large saturation magnetizationand thickness /H20850and FL; 7/H9255/H20849/H9278,/H9257/H20850is the spin-torque efﬁciency function,1which depends both on the angle /H9278between the magnetization of the PL and the FL and on the effectiveFermi-level polarization factor /H9257. We introduce a system of Cartesian unit vectors /H20849xˆ,yˆ,zˆ/H20850, where the x-ycoordinates de- ﬁne the device plane. The current density spatial distribution is assumed to be uniform inside the contact area, with asharp cutoff outside the contact. 9Under this approximation, the current ﬂow involves an inner cylinder of the structurewhich yields a spatial distribution of the Oersted ﬁeld alongthex-yplane only, with no out-of-plane component /H20849see Fig. 1/H20850. Considering the experimental results obtained by Tsoi et al.in Ref. 3, where the frequency of the persistent magneti- zation dynamics has been found to be unaffected by the Jouleheating, our simulations were performed neglecting the ran-dom ﬂuctuating ﬁeld representing thermal noise. The com-putational time step used for the simulations is t S=40 fs; tests performed with tS=20 fs gave exactly the same results. The computational region has been discretized in a two-dimensional /H208492D/H20850mesh of 4 /H110034/H110035n m 3prismatic cells in order to integrate the LLGS numerically. Calculations per-formed with a cell size of 2.5 /H110032.5/H110035n m 3do not produce qualitative changes in the spatial conﬁguration of the mag-netization and do not quantitatively modify the frequencyspectrum /H20849the frequency of the main mode varies by less than 3% /H20850. A perpendicular-to-the-plane ﬁeld H ext=0.9 T is applied to the system. Three-dimensional /H208493D/H20850micromag- netic simulations of the whole structure show that this ﬁeld isable to saturate completely the magnetization of the freelayer along the ﬁeld direction, while it cants the magnetiza-tion of the pinned layer 30° out-of-the-ﬁlm plane. A tiltingangle of 3° with respect to the zaxis has been considered in order to better approximate the experimental setup /H20849it is also used to control the in-plane component of the magnetiza-tion /H20850. In that conﬁguration, it was ﬁrst observed experimentally 3and then explained qualitatively by simpli- ﬁed analytical theory5that if the current density exceeds a given threshold, a radial spin-wave generation is conceiv-able. In detail, the excitation is ﬁrst triggered in the contact area, which behaves like a transmitting antenna, whose spinsare in a standing-wave conﬁguration. Outside the contactarea, the outwards spin-wave propagation occurs via the pre-dicted cylindrical symmetry. This is what we observed in ourcalculations, which are in full agreement with the theorygiven by Slonczewski in Ref. 5, where Bessel-type spin waves generated in the contact area propagate outwards ac-cording to Hankel-type functions /H20849see Fig. 2/H20850. Once this fundamental step is performed, we study the inﬂuence of the nonuniform spatial distribution related to thecurrent-induced magnetic /H20849Oersted /H20850ﬁeld on the magnetiza- tion dynamics. Firstly, results of our numerical simulationsshow that the computed frequencies and wavelengths do notchange if the Oersted ﬁeld is removed. This could be due tothe fact that the Oersted ﬁeld does not introduce any contri-bution to the out-of-plane component of the effective ﬁeld. 2 On the other hand, preliminary results of investigations car-ried out by using in-plane applied ﬁelds point out its rel-evance in the magnetization dynamics /H20849both in the magneti- zation trajectory and in the frequency spectrum /H20850, as also reported in Ref. 8. It has been also theoretically shown how a precessing vortex mode state can be generated by the Oersted ﬁeld if themagnetization is kept pinned at the center of the pointcontact. 11This analytical formulation produces a magnetiza- tion conﬁguration with an odd symmetry, which also yields alarge frequency shift. With this in mind, we analyze the inﬂuence of the current-induced magnetic ﬁeld on the spatial conﬁguration ofthe magnetization during the precession. To perform properlythis kind of investigation, it is better to distinguish betweenthe low-amplitude and the high-amplitude oscillation re-gimes. As reported in our previous work, 7it is possible to identify two different behaviors in both time and frequencydomains. 12In detail, when a given current value is exceeded /H20849e.g., for the proposed geometry, these bounds are about I =9, 14, and 40 mA for contact diameters of D=40, 64, and 160 nm, respectively /H20850, the average out-of-plane component FIG. 2. /H20849Color online /H20850Representation of the spin-wave generation and propagation through a point-contact excitation. Spins inside the contact areaprecess keeping ﬁxed their maximum and minimum positions /H20849standing spin waves with Bessel-type proﬁle /H20850, whereas spins outside the contact have their maximum and minimum shiftings in space /H20849propagating spin waves with Hankel-type proﬁle /H20850.09C108-2 Consolo et al. J. Appl. Phys. 101 , 09C108 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Fri, 21 Nov 2014 19:31:46of the magnetization inside the contact is reversed and, at the same time, a change of slope /H20849from positive to negative /H20850in the relationship frequency versus current occurs. In the low-amplitude oscillation regime, results of nu- merical simulations indicate clearly that the magnetizationdynamics is not affected by the inclusion of the Oersted ﬁeldcontribution. In addition, the expected cylindrical symmetryis even guaranteed in the presence of the current-inducedmagnetic ﬁeld. In the high-amplitude oscillation regime, due to the larger intensity of the Oersted ﬁeld, some minor differencescan be appreciated in the spatial domain, but not in the fre-quency spectrum. In this regime, the spatial magnetizationconﬁguration obtained neglecting the Oersted ﬁeld does notpreserve the cylindrical symmetry exhibited in the low-current region /H20851see Fig. 3/H20849a/H20850/H20852because the in-plane propaga- tion of spins occurs with two different velocities /H20849and wave- lengths /H20850. When the Oersted ﬁeld is taken into account, its contribution is to add an asymmetry to the torque whichslightly modify the magnetization pattern described previ-ously /H20851see Fig. 3/H20849b/H20850/H20852. In other words, in the proposed out-of- plane applied ﬁeld conﬁguration because the excited modesdid not exhibit any difference in the frequency domain whenthe Oersted ﬁeld is neglected, we can argue that the maincontribution to the magnetization precession arises from thespin-transfer terms. These conclusions can be enlarged to allthe investigated contact diameters /H20849in a wide range of current values /H20850and also to the different formulations proposed for the spin-torque efﬁciency function /H9255/H20849 /H9278,/H9257/H20850/H20849for a detailed discussion, see Refs. 9and13/H20850. In both cases, we do not obtain any vortex state as well as an odd symmetric conﬁgu-ration. This result could be strongly related to either the ini-tial pinning assumption /H20849which differs from our initial state computed by solving the Brown equation m/H11003H eff=0/H20850or the simpliﬁed model built in Ref. 11. Furthermore, as widely discussed in Ref. 7under the high-amplitude oscillation re- gime, it is perhaps necessary to introduce additional nonlin-earities to the LLGS equation in order to better describe themagnetization dynamics observed experimentally. In summary, the inﬂuence of the Oersted ﬁeld on the dynamics observed in spin-transfer microwave oscillatorsbased on perpendicularly saturated point-contact systems hasbeen carried out. Results of micromagnetic investigationsconﬁrm Slonczewski’s theory about the generation andpropagation of spin waves and the expected even symmetryof the magnetization conﬁguration through the free layer ofthe device. This result holds for the low-amplitude oscilla-tion regime in both spatial and frequency domains, where theOersted ﬁeld brings a negligible contribution. Under thehigh-amplitude oscillation regime, the Oersted ﬁeld intro-duces spatial asymmetries, which do not affect the Fourierspectrum and do not yield the formation of vortexlike states. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2A. N. Slavin and P. Kabos, IEEE Trans. Magn. 41, 1264 /H208492005 /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. 4W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett. 92, 027201 /H208492004 /H20850. 5J. C. Slonczewski, J. Magn. Magn. Mater. 195, L261 /H208491999 /H20850. 6A. N. Slavin and V. Tiberkevich, Phys. Rev. Lett. 95, 237201 /H208492005 /H20850. 7G. Consolo, L. Lopez-Diaz, L. Torres, and B. Azzerboni, Phys. Rev. B /H20849submitted /H20850; see also the digest book of the International Workshop on Spin Transfer, Nancy, France, 2006. 8D. V. Berkov and N. L. Gorn, J. Appl. Phys. 99, 08Q701 /H208492006 /H20850. 9G. Consolo, L. Lopez-Diaz, L. Torres, and B. Azzerboni, IEEE Trans. Magn. /H20849to be published /H20850. 10L. Torres, L. Lopez-Diaz, E. Martinez, M. Carpentieri, and G. Finocchio, J. Magn. Magn. Mater. 286, 381 /H208492005 /H20850. 11M. A. Hoefer and T. J. Silva, e-print cond-mat/0609030. 12The frequency has been calculated by performing the fast Fourier trans- form of the giant magnetoresistance /H20849GMR /H20850signal over the contact area. The temporal window is 30 ns in all simulations. 13M. Carpentieri, L. Torres, B. Azzerboni, G. Finocchio, G. Consolo, and L.Lopez-Diaz, J. Magn. Magn. Mater. /H20849to be published /H20850. FIG. 3. /H20849Color online /H20850Snapshot of the magnetization conﬁguration inside the contact area in the high-amplitude oscillation regime /H20849a/H20850without and /H20849b/H20850 with Oersted ﬁeld. The contact diameter is D=40 nm and the applied cur- rent is I=20 mA. The color map is representative of the xcomponent of the magnetization /H20849blue is negative and red is positive /H20850.09C108-3 Consolo et al. J. Appl. Phys. 101 , 09C108 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Fri, 21 Nov 2014 19:31:46
5.0048825.pdf	Appl. Phys. Lett. 118, 162402 (2021); https://doi.org/10.1063/5.0048825 118, 162402 © 2021 Author(s).Bistable nanomagnet as programmable phase inverter for spin waves Cite as: Appl. Phys. Lett. 118, 162402 (2021); https://doi.org/10.1063/5.0048825 Submitted: 26 February 2021 . Accepted: 03 April 2021 . Published Online: 20 April 2021 Korbinian Baumgaertl , and Dirk Grundler COLLECTIONS Paper published as part of the special topic on Mesoscopic Magnetic Systems: From Fundamental Properties to Devices ARTICLES YOU MAY BE INTERESTED IN Introduction to spin wave computing Journal of Applied Physics 128, 161101 (2020); https://doi.org/10.1063/5.0019328 Compact tunable YIG-based RF resonators Applied Physics Letters 118, 162406 (2021); https://doi.org/10.1063/5.0044993 Spin wave propagation in a ferrimagnetic thin film with perpendicular magnetic anisotropy Applied Physics Letters 117, 232407 (2020); https://doi.org/10.1063/5.0024424Bistable nanomagnet as programmable phase inverter for spin waves Cite as: Appl. Phys. Lett. 118, 162402 (2021); doi: 10.1063/5.0048825 Submitted: 26 February 2021 .Accepted: 3 April 2021 . Published Online: 20 April 2021 .Publisher error corrected: 27 April 2021 Korbinian Baumgaertl1 and Dirk Grundler1,2,a) AFFILIATIONS 1Laboratory of Nanoscale Magnetic Materials and Magnonics, Institute of Materials (IMX), /C19Ecole Polytechnique F /C19ed/C19erale de Lausanne (EPFL), 1015 Lausanne, Switzerland 2Institute of Microengineering (IMT), /C19Ecole Polytechnique F /C19ed/C19erale de Lausanne (EPFL), 1015 Lausanne, Switzerland Note: This paper is part of the APL Special Collection on Mesoscopic Magnetic Systems: From Fundamental Properties to Devices. a)Author to whom correspondence should be addressed: dirk.grundler@epﬂ.ch ABSTRACT To realize spin wave logic gates, programmable phase inverters are essential. We image using phase-resolved Brillouin light scattering microscopy propagating spin waves in a one-dimensional magnonic crystal consisting of dipolarly coupled magnetic nanostripes. We dem- onstrate phase shifts upon a single nanostripe of opposed magnetization. Using micromagnetic simulations, we model our experimental ﬁnd-ing in a wide parameter space of biasﬁelds and wave vectors. We ﬁnd that low-loss phase inversion is achieved, when the internal ﬁeld of theoppositely magnetized nanostripe is tuned such that the latter supports a resonant standing spin wave mode with an odd quantization num-ber at the given frequency. Our results are key for the realization of phase inverters with optimized signal transmission. Published under license by AIP Publishing. https://doi.org/10.1063/5.0048825 Spin wave (SW) computing is promising for future low-power consuming data processing. 1–3A common approach for SW logic gates relies on encoding the logic output in the combined amplitude oftwo (or more) interfering SWs. 4–9By inverting the phase of one of the incoming SWs, the output level can be switched from constructive interference with high amplitude (logic “1”) to destructive interferencewith low amplitude (logic ‘0’). For technological applications, an idealphase inverter should be efﬁciently gateable, introduce little SW atten-uation, and operate at the nanoscale. In initial works, phase inversionwas achieved by exposing SWs to an inhomogeneous magnetic ﬁeldcreated by a current carrying wire. 5–8To obtain higher efﬁciency, volt- age controlled anisotropy9,10and magnetic defects11–13have been explored. In previous works such as Refs. 12and13, phase shifts were detected electrically by propagating spin-wave spectroscopy. Spatiallyresolved data were not provided. The critical dimension for the phaseinversion process and its optimization remained unclear. In this work, we use phase-resolved Brillouin light scattering microscopy ( lBLS) 14–16to spatially resolve SW wavefronts in a 1D MC with a programmable magnetic defect. We evidence that thepreviously reported phase shift 12DHoccurs locally within the individ- ual magnetic defect. In our experiment, its width amounts to 325 nmmuch smaller than the SW wavelength k.S of a r ,e x p e r i m e n t a l l y observed phase shifts were concomitant with a reduction gin thetransmitted SW amplitudes, 12,13hindering the performance of the phase inverter. Using micromagnetic simulations, we show that thereduction in amplitude can be circumvented by tuning the eigenfre-quency of the magnetic defect such that resonant coupling is achieved. Our ﬁndings are promising for the realization of a low-loss nanoscale phase inverter in magnonics. Figure 1(a) shows a scanning electron microscopy (SEM) image of the investigated sample. The 1D MC consisted of dipolarly cou-pled Co 20Fe60B20nanostripes arranged periodically with a period of p¼400 nm. Nanostripes were 325 nm wide, ð1962Þnm thick, and 80lm long. A single stripe in the center of the MC was elongated on both sides by 8 lm to increase its coercivity. By tuning the mag- netic history, we magnetized the short stripes in the þy-direction, while the prolonged stripe was magnetized in the – y-direction [ Fig. 1(b)]. In this state, the prolonged stripe is magnetized in the oppo- site direction compared to the rest of the MC, i.e., the short stripes,and we refer to it as a magnetic defect . On top of the MC, two coplanar waveguides (CPWs) with signal and ground line widths of0.8lm were prepared out of 5 nm thick Ti and 110 nm Au. For phase-resolved spin wave transmission experiments based on all-electrical spectroscopy, both CPWs were used. Spectra were reportedin Ref. 12(sample MC1). In the present study, we go beyond the earlier studies 11–13and exploit focused laser light to investigate Appl. Phys. Lett. 118, 162402 (2021); doi: 10.1063/5.0048825 118, 162402-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplmicroscopic aspects of the phase shifting process with high spatial resolution. We excited SWs by applying a microwave current atCPW1 and used lBLS for detection. Figure 1(c) shows an optical image of MC1 taken using a lBLS camera. We focused a laser with a wavelength of 473 nm and a power of 1 mW to a spot with adiameter of about 350 nm onto the sample surface. The laser spotwas scanned in the þx-direction in 100 nm steps, while SW inten- sity and phase were measured [see the yellow scan path in Fig. 1(c)]. A magnetic ﬁeld l 0HBwas applied in the þy-direction, corre- sponding to the magnetization direction of the short stripes. The 1D MC with a magnetic defect was investigated for several l0HBvalues. Then, the prolonged stripe was magnetized in the þy- direction and measurements were performed on the defect-free MC1using identical instrument settings. The microwave frequency f exused for excitation was adapted for each HBin order to excite an SW with k1¼2p=k¼2:0r a d lm/C01.12Thereby, we excite an SW with k¼3:1lm, which is more than 9 times larger than the width of the magnetic defect. We use micromagnetic simulations using MuMax317 to explore DHandgfor a wide range of bias ﬁelds and wave vectors. We simulated a slice of MC1 in the x-z-plane, while in the y-direction, the periodic boundary condition (PBC) approach18with 1024 repeti- tions in þyand – y-directions was applied, assuming a constant mag- netization of nanostripes along their lengths. We used l0Ms¼1:8T as saturation magnetization,12a¼0:006 as Gilbert damping,19Aex ¼20 pJ m/C01as an exchange constant,20and a grid size of 2 nm /C220 nm /C22n mðDx;Dy;DzÞ. For band structure simulations, a chain of 40 stripes and for SW propagation, a chain of 164 stripes wereconsidered. Following Refs. 21and22, we simulated band structures by excit- ing the MC with a spatially and temporally varying magnetic ﬁeld(given by sinc functions) and subsequent computation of the Fourieramplitudes of the dynamic magnetization components m xðx;tÞand myðx;tÞ. We obtained a good agreement with the measured band structure of MC1,12when the simulated ﬁlm thickness was reduced to d¼10 nm ( supplementary material , Fig. S1). The discrepancy with the nominal value of dmight be due to ﬁlm roughness in the real sam- ple, which reduced the surface pinning23and was not considered in the simulations.For simulating SW transmission through a magnetic defect, SWs were excited locally at x¼0lm by a sinusoidal hrfexploring different frequencies fexat different HBvalues. An individual stripe at xD ¼6lm was magnetized in the – y-direction. The other stripes were magnetized in the þy-direction. We analyzed amplitudes and phases ofmxðx;tÞand mzðx;tÞafter the simulations had run for t0¼10 ns and propagating SWs had reached a steady state. To avoid back reﬂec-tion, an absorbing boundary condition following Ref. 24was applied at the outer edges of the MC. Figure 1(d) displays the SW phase signal measured for MC1 with magnetic defects (green lines) and without (blue lines) at speciﬁcl 0HB. Phase-resolved lBLS allowed us to measure cos ðHðxÞ/C0H0Þ, whereHðxÞis the SW phase at a position xandH0is a reference phase.25H0is a constant for a given frequency fex.F o ra l l l0HB,w e observed sinusoidal waves with a wavelength of k’3:1lm. In the defect-free state (all stripes magnetized in one direction), the sinusoi-dal wave proﬁle was unperturbed at the position x Dof the prolonged stripe. We assume that due to the large aspect ratio of the investigatednanostripes, the demagnetization factor in the y-direction was already close to zero for all nanostripes 26and the additional prolongation had little impact on the nanostripe’s demagnetization ﬁeld. When the pro-longed stripe was oppositely magnetized (magnetic defect state), thephase was clearly modiﬁed at x¼x D.F o r l0HB¼0m T [ t o p r o wi n Fig. 1(d) ], a localized phase jump (dip) was observed at xD.F o r x>xD, the phase proﬁles with and without defects were still in good agreement. We attribute the localized phase jump at xDto an in-plane dynamic coupling of the defect’s magnetization to its neighboringstripes, as suggested in the study by Huber et al. 27Due to magnetic gyrotropy, the sense of spin-precessional motion in the defect is oppo-site to the rest of the MC. Consequently, the in-phase coupling resultsin apphase jump of the dynamic out-of-plane magnetization compo- nent, which is detected by lBLS. For increasing l 0HB[second and third rows in Fig. 1(d) ], the phase proﬁles with and without defects were signiﬁcantly displacedrelative to one another for x>x D. The displacements indicate phase shifts of SWs. Strikingly, the relative displacements were pronounced directly at the defect, i.e., the phase shifts were established on thelength scale of the individual stripe of width w¼325 nm. We quantify FIG. 1. (a) SEM and (b) magnetic force microscopy image of the central region of MC1. The elongated nanostripe was intentionally magnetized in the – y-direction opposed to the magnetization of the short stripes in order to form a magnetic defect. (c) Optical image of MC1 as seen in the BLS microscope. The probing laser spot w as scanned along the yellow dashed line, while SW intensity and phase signal were recorded. Microwave excitation was applied to CPW1. (d) Phase signal measured with (g reen lines) and with- out (blue lines) magnetic defect for different HBvalues (rows). The position xDof the magnetic defect is highlighted.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 162402 (2021); doi: 10.1063/5.0048825 118, 162402-2 Published under license by AIP Publishingphase shifts DHby ﬁtting cosine functions cos ðHðxÞ/C0H0Þfor SWs, which passed the defect (yellow) and SWs in the defect-free state [black dashed lines in Fig. 1(d) ] .T h eb l u el i n ei n Fig. 2 displays the extracted DHas a function of l0HB. In good agreement with Ref. 12, we ﬁnd a monotonous increase in DH with l0HBreaching a phase shift close to p/2 at 15 mT. The experiments of Fig. 1(d) reveal the nanoscale nature of the phase shifting mechanism. The phase shift was concomitant with a reduction in the amplitude of transmitted SWs (details are given in the supplementary material ,F i g .S 2 ) .T h eg r e e n line in Fig. 2 displays the ratio gof measured SW amplitudes with and without defects for x>xD. At 15 mT, the defect reduced the SW amplitude by a factor of 2. InFig. 3 , we present micromagnetic simulations displaying results in the vicinity of the defect for l0HB¼0m Ta n da nS Ww i t h kx/C252r a d lm/C01. The ﬁrst row in Fig. 3 displays a snapshot of mxðx;t0Þandmzðx;t0Þatt0. Due to the ellipticity of the magnetization precession, the amplitude of mzwas small compared to mxand multi- plied by a factor of 10 for better visibility. We recorded mxandmzfor t/C21t0in 10 ps steps during a time span of 2 ns and computed the FFT amplitude (Amp) and phase ( H) at the driving frequency fex. Amp( mx)a n dA m p ( mz) are displayed in the second row of Fig. 3 .T h e SWs decayed exponentially with a decay length of d¼3:15lm, which was in good agreement with d¼2:960:2lm observed in the experi- ment (see Fig. S2). At the defect, no signiﬁcant change in amplitude was visible for l0HB¼0 mT. In the third row, we plot cos ðH/C0H0Þ forHðmxÞandHðmzÞextracted at the center of each stripe. HðmxÞ followed a sinusoidal wave with k/C243lm( b l u ed a s h e dl i n ei n Fig. 3 ) without deviation at the defect. For HðmzÞ,h o w e v e r ,al o c a lp h a s e jump of pis apparent at the defect (orange dashed line in Fig. 3 ), which agrees well with the measurement observation on MC1 for l0HB¼0m T[ c f .ﬁ r s tr o wi n Fig. 1(d) ]. For a quantitative analysis of amplitude and phase changes intro- duced by the defect, we repeated simulations for the defect-free state as a reference. We computed gas the ratio of Amp( mx)v a l u e sw i t h and without defects. Furthermore, we computed the magnitude of thephase shift jDHjbased on the difference in Hðm xÞwith and without defects. Both gandjDHjwere evaluated in the region x¼8–12 lm and then averaged. Bias ﬁelds from 0 to 44 mT were simulated in4 mT steps. We note that in the simulations, the defect was not switched up to 44 mT, while in our experiments, the defect switched at 23 mT. Switching ﬁelds in real stripes with rough edges have alreadybeen reported to be smaller compared to stripes with ideal edges insimulations. 28For each HB, we computed the dispersion relation and extracted frequencies of the ﬁrst miniband at kx¼k1¼2r a d lm/C01 (as used in the experiment) and kx¼4r a d lm/C01(in the middle of the ﬁrst Brillouin zone of the MC). Then SW propagation was simu- lated for the extracted frequencies. In this manner, we evaluated gand jDHjas a function of HBwithout signiﬁcantly varying the wave vector [Figs. 4(a) and4(b)]. For kx/C252r a d lm/C01, we observe a decrease in transmission with HBuntil l0HB¼24 mT, where greaches a mini- mum value of 0.06. In the same ﬁeld regime, we extract an approxi-mately linear increase of jDHjfrom 0 to 0 :88p, which is in good qualitative agreement with our experimental data ( Fig. 2 ). Strikingly, above 24 mT, the simulated transmission coefﬁcient g started to increase with H B. Concomitantly, jDHjfurther increased. Forkx¼2r a d lm/C01,w ef o u n d g¼0:65 at 40 mT. jDHjpeaked at 36 mT, amounting to 0 :95p.F o r kx/C254r a d lm/C01, the maximum in jDHjcoincided with the local maximum in gforl0HB¼36 mT. We found jDHj¼0:92pand an appreciable transmission with g¼0:73, allowing for low-loss phase inversion. In the following, we discuss the origin of the large gfor high HB. Figure 4(c) shows Amp and Hfor SWs with kx/C254r a d lm/C01excited atfex¼11:73 GHz and l0HB¼36 mT. The plotted Hhas been unwrapped, and the slope (black dashed line) representsH/C0H 0¼kxx. The phase evolution of stripes neighboring the defect behaves regularly. The phase shift occurs right at the center of the defect, where Habruptly shifts by about p. At the same position, a node in the SW amplitude is observed. The dynamic magnetizationproﬁle along the width of the defect agrees well with a standing wave with a quantization number of n¼1. 29To identify the eigenfrequen- cies of the magnetic defect, we simulated with MuMax3 thermallyexcited magnons at a ﬁnite temperature of T¼300 K. 30The simula- tion was run over an extended time period of 100 ns, and then the power spectral density SDðfÞofmxðtÞat the position of the defect was computed. Allowed SW eigenfrequencies are apparent as peaks inS DðfÞ.30–32By considering thermal magnons, we are not limited to SW modes compatible with the symmetry of an exciting hrf. Figure 4(d) compares the band structure of the 1D MC and SDðfÞof the magnetic defect for l0HB¼36 mT. Here, the frequencyFIG. 2. Measured phase shift DH (blue line) and attenuation ratio g(green) for SWs at x>xDwith magnetic defect compared to SWs without defect.FIG. 3. Simulated mxand mzfor a propagating SW excited at x¼0lm with kx /C252 rad lm/C01shown for l0HB¼0 mT. The stripe at xD¼6lm (marked in red) was oppositely magnetized. The ﬁrst row shows a snapshot of mxand mzat t0¼10 ns. The second and third row depicts the precessional amplitudes and cosine of the phase. At the defect, a pphase jump of the phase of mzis observed.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 162402 (2021); doi: 10.1063/5.0048825 118, 162402-3 Published under license by AIP Publishingof SWs with kx¼4r a d lm/C01in the ﬁrst miniband ( q¼0) matches well with the frequency of the second allowed state ( n¼1) observed in SDðfÞ. On the contrary, for 24 mT, where the transmission was low, the relevant fexwas between the n¼0a n d n¼1p e a k so f SDðfÞ(sup- plementary material Fig. S3), i.e., inside a forbidden frequency gap. Our ﬁnding suggests that high transmission is obtained when one of the eigenfrequencies of the defect is resonantly tuned to fex. We specu- late that for larger HBnot considered in the simulation, further max- ima in gare achieved every time the frequency of SWs excited in a miniband qof the regular magnetized stripes overlaps with a higher eigenfrequency state n¼qþi(with i2Nand i/C212) at the defect. Based on the dynamic magnetization proﬁles known for laterally standing waves in a nanostripe,29,33,34we anticipate a phase shift of /C24pin case n–qis odd and /C240pin case n–qis even. To conclude, via phase-resolved lBLS, we measured the phase evolution of propagating SWs in a 1D MC consisting of dipolarly cou- pled nanostripes. When a single nanostripe was magnetized in the opposing direction, a local phase jump of the out-of-plane dynamic component was detected. For l0HB>0 mT, phase shifts occurred on the length scale of 325 nm much smaller than kand were concomitant with a reduction in transmission amplitude. Using micromagnetic simulations, we found, however, an increase in transmission, once the bias ﬁeld was sufﬁcient to align the magnon miniband with the eigen- frequency of the second laterally quantized mode in the defect. Due to the resonant coupling of the defect, a high transmission and a phase shift of close to pwere achieved, allowing for a low-loss phase inverter. For future experimental studies, it will be relevant to either increase the switching ﬁeld (e.g., by using a material with an appropriate mag- netocrystalline anisotropy) or reduce the frequency spacing of the minibands. The latter could be realized for 1D MCs prepared from Yttrium iron garnet. Our results pave the way for efﬁcient and low- loss phase inverters in nanomagnonics.See the supplementary material for a comparison of the simu- lated and experimental dispersion relation, SW intensities measuredwith lBLS, and the simulated dispersion relation and S DðfÞat l0HB¼24 mT. We thank funding by SNSF via Grant No. 163016. We thank F. Stellacci, E. Athanasopoulou, and S. Watanabe for supportconcerning MFM. DATA AVAILABILITY The data that support the ﬁndings of this study are openly avail- able in Zendo at https://doi.org/10.5281/zenodo.4680409 ,R e f . 35. REFERENCES 1A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015), review. 2G. Csaba, /C19A. Papp, and W. Porod, Phys. Lett. A 381, 1471 (2017). 3A. Mahmoud, F. Ciubotaru, F. Vanderveken, A. V. Chumak, S. Hamdioui, C. Adelmann, and S. Cotofana, J. Appl. Phys. 128, 161101 (2020). 4R. Hertel, W. Wulfhekel, and J. Kirschner, Phys. Rev. Lett. 93, 257202 (2004). 5M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Appl. Phys. Lett. 87, 153501 (2005). 6K.-S. Lee and S.-K. Kim, J. Appl. Phys. 104, 053909 (2008). 7T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps, and M. P. Kostylev, Appl. Phys. Lett. 92, 022505 (2008). 8O. Rousseau, B. 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(d) Dispersion relation of the 1D MC and the power spectral density SDðfÞof thermally excited magnons at the defect simulated for l0HB¼36 mT. The frequency of SWs with 4 rad lm/C01in the ﬁrst miniband ( q¼0) of the MC matches with the frequency of the second spin wave reso- nance n¼1 of the defect.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 162402 (2021); doi: 10.1063/5.0048825 118, 162402-4 Published under license by AIP Publishing14A. A. Serga, T. Schneider, B. Hillebrands, S. O. Demokritov, and M. P. Kostylev, Appl. Phys. Lett. 89, 063506 (2006). 15V. E. Demidov and S. O. Demokritov, IEEE Trans. Magn. 51, 1 (2015). 16T. Sebastian, K. Schultheiss, B. Obry, B. Hillebrands, and H. Schultheiss, Front. Phys. 3, 35 (2015). 17A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, AIP Adv. 4, 107133 (2014). 18H. Fangohr, G. Bordignon, M. Franchin, A. Knittel, P. A. J. de Groot, and T. Fischbacher, J. Appl. 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B66, 132402 (2002). 30J. Leliaert, J. Mulkers, J. De Clercq, A. Coene, M. Dvornik, and B. Van Waeyenberge, AIP Adv. 7, 125010 (2017). 31N. Smith, J. Appl. Phys. 90, 5768 (2001). 32J. Yoon, C. You, Y. Jo, S. Park, and M. Jung, J. Korean Phys. Soc. 57, 1594 (2010). 33C. Mathieu, J. Jorzick, A. Frank, S. O. Demokritov, A. N. Slavin, B. Hillebrands,B. Bartenlian, C. Chappert, D. Decanini, F. Rousseaux, and E. Cambril, Phys. Rev. Lett. 81, 3968 (1998). 34G. Gubbiotti, S. Tacchi, G. Carlotti, P. Vavassori, N. Singh, S. Goolaup, A. O. Adeyeye, A. Stashkevich, and M. Kostylev, P h y s .R e v .B 72, 224413 (2005). 35K. Baumgaertl and D. Grundler (2021). “Bistable nanomagnet as programablephase inverter for spin waves,” Zenodo https://doi.org/10.5281/zenodo.4680409 .Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 162402 (2021); doi: 10.1063/5.0048825 118, 162402-5 Published under license by AIP Publishing
1.3590017.pdf	Low axial drift stage and temperature controlled liquid cell for z-scan fluorescence correlation spectroscopy in an inverted confocal geometry Edward S. Allgeyer, Sarah M. Sterling, David J. Neivandt, and Michael D. Mason Citation: Review of Scientific Instruments 82, 053708 (2011); doi: 10.1063/1.3590017 View online: http://dx.doi.org/10.1063/1.3590017 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/82/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Characterization of the fluorescence correlation spectroscopy (FCS) standard Rhodamine 6G and calibration of its diffusion coefficient in aqueous solutions J. Chem. Phys. 140, 094201 (2014); 10.1063/1.4867096 Note: Multi-confocal fluorescence correlation spectroscopy in living cells using a complementary metal oxide semiconductor-single photon avalanche diode array Rev. Sci. Instrum. 84, 076105 (2013); 10.1063/1.4816156 Total internal reflection fluorescence microscopy imaging-guided confocal single-molecule fluorescence spectroscopy Rev. Sci. Instrum. 83, 013110 (2012); 10.1063/1.3677334 A versatile dual spot laser scanning confocal microscopy system for advanced fluorescence correlation spectroscopy analysis in living cell Rev. Sci. Instrum. 80, 083702 (2009); 10.1063/1.3205447 Color imaging with a low temperature scanning tunneling microscope Rev. Sci. Instrum. 73, 305 (2002); 10.1063/1.1433946 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sun, 21 Dec 2014 20:23:25REVIEW OF SCIENTIFIC INSTRUMENTS 82, 053708 (2011) Low axial drift stage and temperature controlled liquid cell for z-scan ﬂuorescence correlation spectroscopy in an inverted confocal geometry Edward S. Allgeyer,1Sarah M. Sterling,2,3David J. Neivandt,2,3,4, a) and Michael D. Mason3,4, b) 1Department of Physics and Astronomy, University of Maine, Orono, Maine 04469, USA 2Graduate School of Biomedical Sciences, University of Maine, Orono, Maine 04469, USA 3Department of Chemical and Biological Engineering, University of Maine, Orono, Maine 04469, USA 4Institute for Molecular Biophysics, Orono, Maine 04469, USA (Received 9 March 2011; accepted 20 April 2011; published online 20 May 2011) A recent iteration of ﬂuorescence correlation spectroscopy (FCS), z-scan FCS, has drawn attention for its elegant solution to the problem of quantitative sample positioning when investigating two-dimensional systems while simultaneously providing an excellent method for extractingcalibration-free diffusion coefﬁcients. Unfortunately, the measurement of planar systems using (FCS and) z-scan FCS still requires extremely mechanically stable sample positioning, relative to a microscope objective. As axial sample position serves as the inherent length calibration, instabilities in sample position will affect measured diffusion coefﬁcients. Here, we detail the design and function of a highly stable and mechanically simple inverted microscope stage that includes a temperaturecontrolled liquid cell. The stage and sample cell are ideally suited to planar membrane investigations, but generally amenable to any quantitative microscopy that requires low drift and excellent axial and lateral stability. In the present work we evaluate the performance of our custom stage systemand compare it with the stock microscope stage and typical sample sealing and holding methods. © 2011 American Institute of Physics . [doi: 10.1063/1.3590017 ] I. INTRODUCTION Since its inception in 1972 by Magde et al. ,1ﬂuorescence correlation spectroscopy (FCS) has proven to be an invalu-able research tool allowing for the study of the photophysics of ﬂuorescent dyes 2,3and quantum dots,3–5translational and rotational diffusion,2conformational ﬂuctuations of biomolecules,6and live cells7,8among others. FCS is a technique that applies auto or cross correlation to a recorded ﬂuorescence intensity as a function of time from a system ofinterest. Statistical ﬂuctuations in the local concentration of ﬂuorescent probes results in a ﬂuctuating ﬂuorescent signal. The intensity ﬂuctuations are recorded using a light sensitivedetector and the signal is correlated with itself (in the case of autocorrelation). The resulting correlation curve may be analyzed to yield diffusion information, triplet kinetics, and photophysical properties of the probe molecules. 9 Comprehensive overviews of the technique with the relevant theory have been detailed elsewhere.2,10,11 Since the application of a confocal geometry to FCS in 1993 by Rigler et al. ,12FCS has grown in popularity and become well suited for situations that require elucidation of dynamics at the single molecule level. Due to the extremely small volume probed in the confocal geometry,9FCS is an excellent choice for systems that require minimal pertur- bation by utilizing low probe concentrations.13,14Since the amplitude of a measured autocorrelation curve is proportionalto the inverse of the average concentration in the observation a)Electronic mail: dneivandt@umche.maine.edu. b)Electronic mail: michael.mason@maine.edu.volume,2systems with extremely low labeling densities are not only advantageous since they provide minimal sample perturbation but are preferred as the optimum operating regime. Additionally, FCS is routinely employed in boththree-dimensional (3D)(Ref. 3) and two-dimensional (2D) (Ref. 15) systems and combinations thereof. 13,16All of these factors conspire to make FCS an excellent choice for the study of dynamics and transport in various model cellular membrane systems and live cells. Despite its excellent qualiﬁcations, a long standing prob- lem in the application of confocal FCS to thin (4 or 5 nm) (Ref. 17) planar membranes is axial sample positioning.15,17 Although a planar membrane can easily be positioned to maximize the detected count rate, this axial position does not necessarily correspond to the maximum count rate permolecule 17(the quantity that deﬁnes signal to noise in a FCS measurement18). Simply stated, axial positioning of planar membrane systems is often qualitative. Additionally, sam-ple drift during a FCS acquisition on a planar system of greater than ∼100 nm may introduce artifacts in the mea- sured correlation curve that appear as ﬁctitious (erroneous)slowly diffusing species. 15Since experimental acquisition times are potentially on the order of minutes or tens of min- utes, long enough for signiﬁcant sample drift, using unchar- acterized stock microscope stages may be a poor choice for the researcher interested in quantitative FCS on thin planarsystems. The former of these problems (sample position relative to the focal plane) has been eloquently addressed by Bendaet al. 19with the introduction of the novel technique termed “z-scan” FCS. In this mode the sample position is stepped 0034-6748/2011/82(5)/053708/6/$30.00 © 2011 American Institute of Physics 82, 053708-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sun, 21 Dec 2014 20:23:25053708-2 Allgeyer et al. Rev. Sci. Instrum. 82, 053708 (2011) through the focal volume using, for example, a high resolu- tion piezo scan stage. At each axial position an autocorrela- tion measurement is performed. The correlation curve at eachposition is analyzed with the appropriate model and the dif- fusion time is plotted against sample position relative to the focal plane. The diffusion time and particle number in a 2Dsystem have a parabolic dependence on axial sample position and the resulting diffusion time ( τ D) and particle number (PN) versus sample position may be ﬁt using19 τD=w2 0 4D/parenleftbigg 1+λ2 0/Delta1z2 π2n2w4 0/parenrightbigg , (1) PN=πcw2 0/parenleftbigg 1+λ2 0/Delta1z2 π2n2w4 0/parenrightbigg , (2) where Dis the lateral diffusion coefﬁcient, w0is the radius of the beam waste in the focal plane, λ0is the excitation wave- length, /Delta1zis the sample distance from the focal plane, and n is the refractive index of the surrounding medium. This yields a calibration-free diffusion coefﬁcient, particle number, and beam size. However, application of z-scan FCS is still depen- dent on a stable microscope setup and repeatable sample po- sitioning. FCS on planar systems inherently requires minimalaxial drift, z-scan FCS further compounds the problem by re- quiring not only excellent stability at each measurement posi- tion but also excellent stability for the duration of the samplescan. Because the change in sample position serves as the in- trinsic calibration for z-scan FCS, uncertainty in the sampleposition will inherently affect the measured diffusion coefﬁ- cient. As axial sample scanning is most conveniently accom- plished using a piezo scan stage, the sample cell weight mustalso be considered (we employ here a three-axis piezo scan stage with a maximum recommended vertical load of 200 g). Sample positioning and measurement is further complicatedfor studies of membranes by the necessary incorporation of temperature control equipment and the ability to mount a sam- ple that must be maintained in solution. In order to maintain the physiological relevance of a model membrane system, or the viability of live cells, carefultemperature regulation of the surrounding media or buffer so- lution is required. The ability to control the temperature over a wide range also permits the researcher to perform studiesrelated to cellular stress via heat shock 20and in the case of model membrane systems, controlled temperature changes al- low for more thorough investigation of membrane character-istics, speciﬁcally the changes in phospholipid diffusion and phase transition temperatures. A sample cell and stage setup that meets the above requirements, in an inverted microscopegeometry, would additionally be ideal for long acquisition quantitative imaging. Here we present a simple, robust, liquid sample cell and low axial drift stage designed speciﬁcally for the study of model membrane systems using z-scan FCS on an inverted confocal microscope. The setup addresses the previously mentioned axial drift requirements, the need for temperature control, and the necessity of sample preparation in solution.It is also amenable to traditional sample or laser scanning confocal imaging or imaging utilizing a CCD camera. We detail the sample cell and stage design and experimentallyevaluate the performance of the custom setup. II. INSTRUMENTATION A. Low drift stage and objective holder An Olympus IX71 inverted microscope is used as the base. The factory shipped XY translation stage and objec-tive turret were removed. In place of the original XY stage a 12.7 mm thick 178 by 240 mm block of aluminum is em- ployed. A 32 mm square was milled in the middle of the plateto accommodate our custom objective holder (detailed later) and four M6 clearance holes were drilled allowing the plate to be mounted on the IX71 base in place of the Olympus XY stage. Four holes were drilled and tapped for 1/4-20 screw size allowing for the attachment of a large aperture high perfor-mance two axis translation stage (Newport Corporation, 406) and our objective holder. Schematics of the IX71 base were used to ensure correct dimensioning. Mounted on top of the aluminum plate is the afore men- tioned large aperture Newport course XY stage. This stage has a 57.15 mm diameter aperture to accommodate our ob-jective holder and 13 mm of travel in the X and Y directions allowing the sample to be manually positioned. A three axis piezo scan stage (Mad City Labs, Nano-T115) is mounted, viaa connecting plate, on top of the Newport XY stage. A custom fabricated acrylic sample cell holder that thermally insulates the scan stage from the sample cell sits atop the piezo scanstage. The objective holder consists of an externally threaded 28.757 mm outside diameter stainless steel rod, a brass mounting plate, and a brass lock ring. The brass mounting plate and the lock ring have internal threads mated to the ex-ternal threads of the rod. The stainless steel rod is 88.9 mm long and has 76.2 mm of external 44 threads per inch (tpi) threads (giving a 577 μm axial advance per revolution). The inner diameter of the rod is 0.8 in. with internal Royal Mi- croscope Society threads (20.32 mm diameter and 36 tpi) at one end allowing attachment of a microscope objective. Thebrass mounting plate is attached to the top of the large alu- minum plate while the stainless steel rod and lock ring thread in from beneath. Rotation of the threaded rod, accessed fromunder the aluminum plate, allows the microscope objective to be moved up and down. The lock ring allows the position of the objective to be ﬁxed and the high tpi provides adequateaxial resolution of the objective. Fine tuning of the focus is easily accomplished with the piezo scan stage. A schematic of the stage and objective holder is given in Fig. 1(a). With the setup described the sample cell is ﬁxed to the top of the scan stage and the position of the objective relative to the sample is held constant once the lock ring is tightened. Thus, for a given piezo z-position, the critical feature of a ﬁxed distance between the sample and objective is fulﬁlledwhile the Newport XY translation stage allows for the equally critical feature of course sample positioning. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sun, 21 Dec 2014 20:23:25053708-3 Allgeyer et al. Rev. Sci. Instrum. 82, 053708 (2011) Dichroic MirrorPiezo Scan Stage(a) (b)Course XY Stage Objective HolderLock RingMicroscope Base Top BottomCover glass O-RingExcitation BeamOBJTE Heater/Cooler See (c) Sample Cell See (b) To Liquid Cooler Power ConnectionFront Side TE Heater/CoolerReservoir(c)Acrylic Insulator FIG. 1. (Color online) (a) Stage and objective holder as detailed in the text. (b) Liquid sample cell forming a cover glass o-ring sandwich for planar mem- brane systems. (c) TE heater/cooler sandwiched between circulating liquid reservoir and an aluminum plate that friction-ﬁts on top of the sample cell presented in (b). B. Liquid cell and heater/cooler stage The sample cell is composed of two 300 series stainless steel plates, one with a boss (top), and the other with a matching pocket (bottom) as shown in Fig. 1(b).T h el o w e r portion of the sample cell is a 4.7625 mm thick stainlesssteel bar comprised of a 22 ×22 mm pocket milled in the center, two 4-40 clearance holes for direct mounting to the scan stage, and a 7.9375 mm aperture to accommodate themicroscope objective. The upper portion of the sample cell has a boss that ﬁts inside the pocket of the lower portion. The depth of the pocket and height of the boss have been chosenso two No. 1.5 microscope coverslips may sandwich a PTFE coated o-ring inside the cavity and compress the o-ring by ∼5%. The top portion of the sample cell attaches to the lower portion using four 4-40 stainless steel screws. Stainless steel is employed throughout to allow all parts to be rigorously cleaned. A 3.175 mm hole machined into the side of the lowerplate of the sample cell houses a friction-ﬁt removable brass rod that penetrates to within 1.5 mm of the liquid cavity. A 1.2 mm hole machined longitudinally into the brass rod houses a thermistor (RTD) (TE Technology, MP-2444) mounted with thermal paste. This allows easy removal of 1000 1100 1200 1300 1400 0 50 100 150 200 250 300 3.2 3.5 3.8 4.1 4.4Count Rate (kHz) Power (%) Time (seconds)Intensity Power FIG. 2. (Color online) Effect of the TE heater’s output power on the lipid bilayer’s measured count rate when operating under PID control. Although the TE’s output power only varies from 3.1% to 4.5% the resulting change in the bilayer’s signal is signiﬁcant and clearly trends. the RTD rod unit for cleaning and sample mounting while providing accurate temperature data of the liquid cavity environment. For measurements requiring temperature control a peltier-thermoelectric (TE) heater/cooler (TE Technology, TE-127-1.0-0.8) is sandwiched between an aluminum reser-voir with liquid input and output ports and an aluminum block machined to ﬁt on top of the sample cell as shown in Fig. 1(c). The aluminum reservoir is connected to a circulating liquidcooler (Thermaltake, BigWater 780e ESA) to dissipate heat or cool one side of the TE heater/cooler. The fully assembled sample cell with the TE heater/cooler weighs 196 g just underthe maximum recommended vertical load of the scan stage of 200 g. Proportional-integral-derivative (PID) control (TE Tech- nology, TC-36-25-RS232) was initially used to regulate the temperature of the TE heater/cooler and sample cell. After ju-dicious determination of PID parameters, the PID controller was able to control the sample cell’s temperature to within a tenth of a degree Celsius of the set point, however, doingso required the controller to modulate the TE heater/cooler’s power. It was quickly noted, however, that the cycling of the power to the TE heater/cooler, although providing a relativelyconstant temperature, produced a change in the detected count rate from lipid bilayer samples (detailed later) as may be seen in Fig. 2(the deﬁnite cause of this affect is undetermined). This affect produced gross distortions in measured correlation curves and, subsequently, PID control was eliminated in favor of a constant power heating scheme. The controller was set toa constant power and the resulting steady state temperature was observed and recorded using a custom LABVIEW inter- face across a range of experimentally relevant temperatures. The results were ﬁt using a second order polynomial allow- ing the power needed to achieve a desired temperature to bepredicted. This scheme resulted in average steady state tem- peratures with standard deviations of only a few hundredths of a degree although the time necessary to reach the desiredtemperature was longer than that observed using PID control. III. PERFORMANCE EVALUATION AND DISCUSSION The affect of drift and sample stability on the resultant z-scan FCS diffusion coefﬁcient were assessed by simulat- ing the confocal point spread function (PSF) for our system21 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sun, 21 Dec 2014 20:23:25053708-4 Allgeyer et al. Rev. Sci. Instrum. 82, 053708 (2011) -1500-1000-500 0 500 1000 1500 2000 0 500 1000 1500 2000 2500 3000 3500Fitted Center Position (nm) Time (seconds)(b) Custom Sample Holder Nail Polish Sealed Wax Sealed Nunc Well-400-200 0 200 400 600 800 1000 0 500 1000 1500 2000 2500 3000 3500Fitted Center Position (nm ) Time (seconds)(a) Custom Sample Holder Stock Stage Typical Stock Stage Atypical FIG. 3. (Color online) (a) Axial stability of the custom setup described herein compared with a typical and atypical result from a stock stage and turret. (b) Stability of our sample cell compared with nail polish and wax sealed coverglass and an eight-chambered well. and, subsequently, the 1 /e2radius as a function of axial po- sition (the form of the PSF used did not account for back- aperture over/under ﬁlling). The diffusion time as a functionof axial position was computed using τ D=w2/4D, where w is the 1 /e2radius and Dwas ﬁxed. Assuming a 30 s acqui- sition time per position, τDwas computed for the equivalent of 8 and 100 nm/min of uniform unidirectional sample drift. The resulting simulated z-scan curves were ﬁt with Eq. (1) and the resulting diffusion coefﬁcients were found to deviate from the zero drift case by 30% and 100%, respectively. To assess the affect of sample instability, τDwas computed us- ing a position that deviated from ideal by an amount equal to a normally distributed random number generated at each position with a mean of zero and a standard deviation of 8and 100 nm. The simulated z-scan curves were ﬁt and in both cases the resulting diffusion coefﬁcients were found to vary from the zero drift case by roughly 30%. Although simula-tion does not account for artifacts in the correlation curves due to sample drift during acquisition, the results suggest that sample positioning and stability are critical for reliable z-scanFCS measurements. Evaluation of the sample cell and stage’s stability was carried out by loading the sample cell with two No. 1.5 cov-erslips and a PTFE coated o-ring as would be done for pla- nar membrane investigations but with liquid omitted from the cavity. The sample cell was mounted on the scan stage and the inner surface of the bottom coverslip was brought into focus using a 60X 1.2 N.A. UPlanApo/IR water immersionobjective (Olympus) by monitoring the scattered laser light from the glass/air interface. Excitation and collection fol- lowed a standard confocal geometry using a 50 μmd i a m e - ter ﬁber in place of a pin hole and a ﬁber-coupled avalanche photo diode (APD) as the detector. Once the objective was brought into position, the brass lock ring was secured anda custom LABVIEW program was used to monitor the APD count rate and control the position of the piezo scan stage. The sample cell was scanned through the axial range of the scan stage, the APD count rate at each position was saved, ﬁt to a Gaussian, and the center position of the ﬁtted Gaussianrecorded. The software was then set to scan the sample cell through the initial center position every 5 min. Scanning was performed from 3 μm below to 3 μm above the initial center position, in 500 nm steps, and the APD count rate at each po- sition was recorded along with the time at which each scan began. The ﬁtted center position for each scan along withthe time stamp allowed the position as a function of time to be tracked. Axial scanning of the sample cell using the cus- tom stage setup yielded an average drift rate of (3.47 ±0.22) nm/min (averaged from an equal number of scans with and without TE heater/cooler in place) and a typical result is pre- sented in Fig. 3(a). Considering the requirement of less than 100 nm per acquisition of drift and typical aggregated z-scan FCS acquisition times on the order of 5 to 10 min, this drift rate is well within the criteria for planar FCS measurements although simulations suggest that at this drift rate there will be some affect on the recovered z-scan diffusion coefﬁcient. Similarly, the scan stage and sample cell were mounted on the factory shipped XY stage and the objective was mounted on the original objective turret. As also presentedin Fig. 3(a) scanning the sample cell in the same manner as described above yielded an average drift rate of (7.25 ±1.06) nm/min which is a factor of two larger than our custom setup. Further, Fig. 3(a) also presents occasional me- chanical instabilities beyond the reported average rate were observed suggesting that the stability of the stock stage andturret have the potential to heavily affect measurements on planar systems without operator awareness. Although the exact sources of drift in the conventional and custom system are not explicitly known, vibration and thermal currents have the potential to play some role. How- ever, as the optical setup described is situated on a large vi- bration damping table (TMC, Peabody, Massachusetts) it is unlikely that high frequency vibration plays any major rolein system instability. The laboratory housing this system is climate controlled via forced air but no correlation between heating/cooling from the climate control system has been ob-served, likely because climate control vents are spatially sep- arated from instrumentation. A more likely source of drift in the conventional system, however, is the lack of coupling be-tween the objective turret and the sample position. Although both are supported via the same microscope base, neither is explicitly linked to the other unlike the custom setup em-ployed here. Even though the conventional turret allows easy switching between multiple objectives this functionality intro- duces more moving parts and inevitably more instability. Al-though our setup does not include this capability, it eliminates any possible instabilities due to a rotating turret. Further, axial This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sun, 21 Dec 2014 20:23:25053708-5 Allgeyer et al. Rev. Sci. Instrum. 82, 053708 (2011) 0 0.2 0.4 0.6 0.8 1 0.01 0. 1 1 10 100 1000 10000 100000Normalized Autocorrelation G( τ) τ(ms)Measured 2D Diffusion Model FIG. 4. (Color online) Measured autocorrelation function of a DMPG lipid bilayer using the setup detailed herein. movement of the objective can only be accomplished by rotat- ing the objective holder with an applied force perpendicular to the direction of translation. This means that simply pressingdown on the objective, as gravity does, will not move or adjust the objective position. In fact, it would increase the normal force between the contact surface on the threads of the ob-jective holder and brass mounting plate and subsequently in- crease the frictional force between these surfaces making ax- ial objective movement more difﬁcult. Our custom setup also includes a lock ring explicitly employed for immobilizing the objective, whereas the conventional system does not includeany mechanism for this purpose. Unlike the conventional sys- tem which employs a turret, suspended at a 90 ◦angle from an axially translating arm, our system supports the objectivedirectly underneath eliminating another potential source of in- stability. The same stability veriﬁcation method was also em- ployed to evaluate more typical, or easily available, sample mounting methods. Cover glass was mounted to welled slide glass using wax and separately, nail polish. The welled slideglass was inverted and held on the scan stage using clips typ- ically found on microscope stages for said purpose. Typical results from investigating the stability of wax and nail pol-ish sealed cover glass are shown in Fig. 3(b) along with the stability of our custom setup for comparison. Finally, a No. 1.5 bottom thickness eight-chambered well (Nunc, Lab-Tek II) was tested and the results are also presented in Fig. 3(b). As it can be seen in Fig. 3(b) the wax and nail polish sealed welled slide glass, as well as the eight-chambered well, are relatively unstable when compared with our custom setup and would be a poor choice for z-scan FCS investigations. It hasalso been noted that nail polish is not well suited for use with ﬂuorescence techniques. 22 As it is clearly evidenced in Fig. 3by comparing data presented in parts (a) and (b), although our custom stage setup does provide a factor of two decrease in axial drift, relative to the conventional system, it is clear that the custom sample cellemployed herein plays an even more critical role. For typical z-scan step sizes (200–300 nm) we ﬁnd the sample’s maxi- mum axial speed is 20.4 μm/s (1.22 mm/min) and likely not a cause of instability. Both nail polish and wax seals have been observed to al- low air bubbles into the sealed sample well if left for as little as 12 h. This suggests that the wax and nail polish may form an unreliable seal and allow liquid to evaporate. In equivalent 2 3 4 5 6 7 8 9 10 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 25 30 35 40 45 50 55 60 65 70 75Diffusion Time (ms) Particle Number Distance Δz(μm)Diffusion Time Particle Number FIG. 5. (Color online) Z-Scan of a DMPG bilayer membrane on glass, inner leaﬂet labeled with 0.0005 mol% Rho-PE at 36◦C. The recovered diffusion coefﬁcient, particle number, and beam waste are (9.78 ±0.206) μm2s, (94.93 ±0.665)/ μm2, and (307 ±4.4) nm, respectively. unsealed samples, we observe that as samples dry the loss of liquid due to evaporation causes a visually apparent concave deformation in the cover glass. If the cover glass, which sup-ports the sample, is deforming as a function of time this will clearly affect the absolute position of the sample from scan to scan. This is in contrast to our custom sample cell whichallows samples to be examined promptly prior to the poten- tial formation of air bubbles and without having to wait for nail polish or wax to dry. Bubbles have been observed to formin our custom sample cell but only after a few days. More importantly, our sample holder is physically attached to the scan stage using two 4-40 machine screws. This ensures goodrigidity between the sample cell and scan stage, whereas in the case of conventional microscope stage clips the sample is held by pressure and direct attachment of the sample to thestage is not possible. Evaluation of the setup for FCS and z-scan FCS on planar systems was performed using 1,2-dimyristoyl- sn- glycero-3-phospho-(1 /prime-rac-glycerol) (sodium salt) (DMPG) (Avanti Polar Lipids, 840445) bilayers labeled with 0.0005mol% 1,2-dimyristoyl- sn-glycero-3-phosphoethanolamine- N-(lissamine rhodamine B sulfonyl) (ammonium salt) (Rho-PE) (Avanti Polar Lipids, 810157) in the innerleaﬂet deposited via standard Langmuir-Blodgett/Langmuir Schäfer methods 23–25employing cleaned No. 1.5 cover- slips as the substrate. All parts of the sample cell werecleaned by ﬁrst soaking in a dilute solution of liquid detergent (Decon, Contrad 70), rinsed in 18.2 M /Omega1cm water (Millipore, Milli-Q), soaked in 70% nitric acid, andagain copiously rinsed in 18.2 M /Omega1cm water. Bilayers were mounted in the sample cell in solution ensuring sample ﬁ- delity. The sample cell was mounted on the scan stage and theTE heater/cooler set in place. The desired temperature was achieved as described previously and z-scan FCS performed. FCS and z-scan FCS were carried out using a cus- tom built confocal/FCS microscope. This instrument uses the afore mentioned Olympus IX71 microscope body as the base This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sun, 21 Dec 2014 20:23:25053708-6 Allgeyer et al. Rev. Sci. Instrum. 82, 053708 (2011) (a) (b) 5μm 5μm FIG. 6. Images of drop cast 200 nm ﬂuorescent beads imaged (a) after initial focus and (b) 75 min later. Note that the contrast of image (b) has been ad- justed for ease of comparison as photobleaching over the 75 min period has had an affect. and a 543 nm Helium-Neon (HeNe) laser for excitation. The HeNe’s beam is expanded and recollimated to ﬁll approx- imately 60% of the previously mentioned Olympus objec- tive’s back aperture and a 50 μm diameter silica core ﬁber acts as the pin hole. The ﬁber’s position was optimized for count rate per molecule and the setup’s validity tested with the popular and robust ﬂuorophore Alexa-Fluor 546 (Invitro-gen, A20002). Measurement of the Alexa-Fluor ﬂuorophore in solution (3 nM in 18.2 M /Omega1cm H 2O), excited with ∼3μW at the sample, yielded excellent correlation curves that ﬁt wellto a 3D diffusion model with a non-divergent structure fac- tor of less than six and no need to include triplet kinetics. Figure 4presents an autocorrelation curve from a DMPG bi- layer on glass excited with <2μW at the sample that is well ﬁt by a 2D diffusion model ((1 /N)(1+τ/τ D)−1). Figure 5 presents z-scan FCS results for DMPG on glass at 36◦Ci n - dicating that our setup possesses the required mechanical sta- bility for successful measurements on planar systems. To test the imaging stability of our setup a dilute 3 nM so- lution of 200 nm ﬂuorescent beads (Invitrogen, F8784) were dried on a No. 1.5 cover glass and mounted in the sample cell.The sample cell was mounted on the piezo scan stage and the beads were brought into focus. An EMCCD camera (Andor, Luca) was used to capture an image of the beads every 5 minfor 75 min after the initial focus was achieved. Initial and ﬁ- nal images are shown in Fig. 6. Features in each image were ﬁt to a 2D Gaussian using a custom MATLAB script and the resulting 1 /e2radius of each feature was stored (features with ﬁtted 1/ e2radii larger than would be expected for this imag- ing system and features not well spatial separated were ex-cluded). Over the 75 min period the average ﬁtted 1 /e 2radius increases by (209.4 ±137) nm (or (2.6 ±1) nm/min.) indi- cating the viability of this system for long term imaging. IV. CONCLUSION Mechanical stability and low drift are features critical for any researcher interested in high resolution diffractionlimited quantitative microscopy. We have shown that our custom setup has half the drift of a stock stage and tur- ret. Although the stock stage and turret perform reasonablyin most instances, occasional instability was observed sug- gesting the necessity of a custom setup as detailed in this work. When using techniques that require repeatable sam-ple positioning, such as z-scan FCS, the necessity of a rigid sample cell that can be mounted directly and with minimal drift to the scanning apparatus is critical. Conventional sam- ple sealing methods were found to be inadequate and are not recommended. ACKNOWLEDGMENTS This material is based upon work supported by the Na- tional Science Foundation under Grant No. CHE0722759. The authors thank Daniel Breton and Gilbert Hopler for their invaluable help and advice in the machine shop, Amos Clinefor initial temperature control setup, and Samuel T. Hess and Travis J. Gould for excellent conversations regarding FCS. 1D. Magde, E. Elson, and W. W. Webb, Phys. Rev. Lett. 29, 705 (1972). 2O. Krichevsky and G. Bonnet, Rep. Prog. Phys. 65, 251 (2002). 3J. A. Rochira, M. V . Gudheti, T. J. Gould, R. R. Laughlin, J. L. Nadeau, a n dS .T .H e s s , J. Phys. Chem. C 111, 1695 (2007). 4T. J. Gould, J. Bewersdorf, and S. T. Hess, Z. Phys. Chem. 222, 833 (2008). 5R. F. Heuff, J. L. Swift, and D. T. Cramb, Phys. Chem. Chem. Phys. 9, 1870 (2007). 6G. Bonnet, O. Krichevsky, and A. Libchaber, Proc. Natl. Acad. Sci. U.S.A. 95, 8602 (1998). 7K. Bacia and P. Schwille, Methods 29, 74 (2003). 8S. A. Kim, K. G. Heinze, and P. Schwille, Nat. Methods 4, 963 (2007). 9J. R. Lakowicz, Principles of Fluorescence Spectroscopy , 3rd ed. (Springer, New York, 2006). 10E. Haustein and P. Schwille, Annu. Rev. Biophys. Biomol. Struct. 36, 151 (2007). 11R. Rigler and E. S. Elson, Fluorescence Correlation Spectroscopy ,1 s te d . (Springer, New York, 2001). 12R. Rigler, Ü. Mets, J. Widengren, and P. Kask, Eur. Biophys. J. 22, 169 (1993). 13N. Kahya and P. Schwille, Mol. Membr. Biol. 23, 29 (2006). 14L. Zhang and S. Granick, J. Chem. Phys. 123, 211104 (2005). 15J. Ries and P. Schwille, Phys. Chem. Chem. Phys. 10, 3487 (2008). 16Y . Takakuwa, C.-G. Pack, X.-L. An, S. Manno, E. Ito, and M. Kinjo, Bio- phys. Chem. 82, 149 (1999). 17R. Machá ˘na n dM .H o f , BBA-Biomembranes 1798 , 1377 (2010). 18D. E. Koppel, Phy. Rev. A 10, 1938 (1974). 19A. Benda, M. Bene ˘s, V . Mare ˘cek, A. Lhotský, W. T. Hermens, and M. Hof, Langmuir 19, 4120 (2003). 20A. Jackson, S. Friedman, X. Zhan, K. A. Engleka, R. Forough, and T. Maciag, Proc. Natl. Acad. Sci. U.S.A. 89, 10691 (1992). 21Confocal and Two-Photon Microscopy F oundations, Applications, and Ad- vances , edited by A. Diaspro (Wiley-Liss, New York, 2007). 22G. Callis, Biotech. Histochem. 81, 4 (2006). 23T. Baumgart and A. Offenhäusser, Langmuir 19, 1730 (2003). 24J. Liu and J. C. Conboy, Biophys. J. 89, 2522 (2005). 25J. Y . Wong, J. Majewski, M. Seitz, C. K. Park, J. N. Israelachvili, and G. S. Smith, Biophys. J. 77, 1445 (1999). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sun, 21 Dec 2014 20:23:25
1.1560212.pdf	Numerical and experimental modal analysis of the reed and pipe of a clarineta) Matteo L. Facchinetti Laboratoire de Me ´canique des Solides and Laboratoire d’Hydrodynamique, CNRS-E´cole Polytechnique, 91128 Palaiseau Cedex, France Xavier Boutillonb) Laboratoire d’Acoustique Musicale, CNRS-Universite ´Paris 6-Ministe `re de la Culture, 11 rue de Lourmel, 75015 Paris, France Andrei Constantinescu Laboratoire de Me ´canique des Solides, CNRS-E´cole Polytechnique, 91128 Palaiseau Cedex, France ~Received 12 December 2001; revised 8 August 2002; accepted 24 January 2003 ! Amodal computation of a complete clarinet is presented by the association of ﬁnite-element models of the reed and of part of the pipe with a lumped-element model of the rest of the pipe. In the ﬁrstpart, we compare modal computations of the reed and the air inside the mouthpiece and barrel withmeasurements performed by holographic interferometry. In the second part, the complete clarinet ismodeledbyadjoiningaseriesoflumpedelementsfortheremainingpartofthepipe.Theparametersof the lumped-resonator model are determined from acoustic impedance measurements. Computedeigenmodes of the whole system show that modal patterns of the reed differ signiﬁcantly whetherit is alone or coupled to air. Some modes exhibit mostly reed motion and a small contribution of theacousticpressureinsidethepipe.Resonancefrequenciesmeasuredonaclarinetwiththemouthpiecereplaced by the cylinder of equal volume differ signiﬁcantly from the computed eigenfrequencies ofthe clarinet taking the actual shape of the mouthpiece into account and from those including the~linear !dynamics of the reed. This suggests revisiting the customary quality index based on the alignment of the peaks of the input acoustical impedance curve. © 2003 Acoustical Society of America. @DOI: 10.1121/1.1560212 # PACS numbers: 43.75.Ef @NHF# I. INTRODUCTION The clarinet is usually considered as the association of a linear resonator, the pipe, and a nonlinear excitor, the reed,subject to the air ﬂow from the mouth.Alternatively, one canconsider the air column and the reed as a linear system sub-ject to nonlinear boundary conditions. This is the approachretained in this article where the reed is considered as a lin-ear mechanical system coupled to the pipe and where theinteraction with the player is not treated. Nonlinear phenom-enon such as the interaction between the reed and the jetacross the reed-slit, the contact forces between the reed andthe lay, and the interaction between the reed and the player’slip will be included in a subsequent piece of work as nonlin-ear boundary conditions to the normal modes that are de-scribed here. Humidity of the reed and the player’s lip alsohave a damping role which is not considered in this modalanalysis of a pipe coupled to a ~dry!reed. Acoustical studies of the clarinet have so far representedthe mouthpiece of a wind instrument by its equivalent vol- ume. This study goes beyond this approximation and pre-sents the three-dimensional distribution of the pressure in theupper part of the instrument. Studies of the pipe of the clarinet have traditionally been expressed in the frequency domain and were based on mea-surements or computations of input acoustic impedance.However, numerical simulations of this instrument operate inthe time domain and are usually based on the reﬂection func-tion of the pipe. Recent experimental studies have adoptedthe time domain approach with direct measurements of thisreﬂection function. Abundant literature extensively coversthese subjects: for general presentations, see Refs. 1–4. Studies of the reeds are far less extensive and the me- chanical behavior of cane is still subject to discussion. Thesimplest reed model, a spring, is implicitly used by reed-makers when they rate them by their so-called ‘‘strength,’’which corresponds to the mechanical compliance. Experi-mental studies have proposed values for the compliance ofthe reed. 1,5,6Associated with various models of the pipe and excitation, this model has been used in numerical simulationswhich were successful in describing basic features of thedynamics of clarinet-like system. 7–9Music-oriented algo- rithms have also been proposed in which the values of theparameters describing the excitor and the resonator are ad-justed in order to obtain realistic sounds instead of accuratelydescribing their mechanical behavior. 10,11However, this model is obviously insufﬁcient to describe quality-based cri-teria: otherwise all reeds in a given commercial box ~witha!Part of this work was presented in ‘‘Application of modal analysis and synthesis of reed and pipe to numerical simulations of a clarinet,’’ invitedpaper at the 140th meeting of the ASA, Newport Beach, CA, December2000 @J. Acoust. Soc. Am. 108, 2590 ~A!#,i n‘ ‘ E´tude modale d’une clari- nette,’’ Proceedings of the Colloque National en Calcul de Structures,Giens, France, May 2001, and in ‘‘Modal analysis of a complete clarinet,’’Proceedings of the International Conference on Acoustics, Rome, Italy,September 2001. b!Electronic mail: boutillon@lms.polytechnique.fr; present address: Labora- toire de Mecanique des Solides, E´cole Polytechnique, 91128 Palaiseau Cedex, France. 2874 J. Acoust. Soc. Am. 113(5), May 2003 0001-4966/2003/113(5)/2874/10/$19.00 © 2003 Acoustical Society of America Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Thu, 18 Dec 2014 20:28:59similar strength !would suit a given player, but this is not the case. The next modeling step is the single-degree-of-freedom oscillator. Although some simulation algorithms12have been very successful in producing realistic sounds,13,14this is not sufﬁcient in itself to assert the physical validity of a model.One degree of freedom is not sufﬁcient to account for criteriasuch as reed quality. Stewart and Strong 15and Sommerfeld and Strong16used a reﬁned model of the reed as a nonuni- form beam. In the latter study, the pipe was only slightlysimpliﬁed compared to a real clarinet and the player’s aircolumn ~including the lungs !was also taken into account. There is no fundamental difference between this simulationand those based on a simple oscillator model for the reedsince the interaction with the rest of the system is averagedalong the beam. The beam model is needed if one wants totake into account the curved shape of the mouthpiece onwhich the reed beats during large amplitude motions.Gazengel 17derived a simple oscillator model from a beam equation. In his time-domain simulation, the mass of the os-cillator is recalculated at each time step as a function of theposition of the reed, introducing by this means the nonlinearbehavior of the reed contact. Modeling the reed as a continuous system is the current state of research. Several examples of modal analysis ofclarinet reeds with holographic interferometry have been pre-sented in conferences over the recent years, 18–20but never published. One example of ﬁnite-element modeling based onmeasurements of the mechanical properties of cane has beenreported. 21Another ~unpublished !pioneering study has been done by Pinard and Laine when they were students at theE´cole Polytechnique ~France !. The experimental modal analysis and the ﬁnite-element modeling of isolated reedsthat are presented in the following are a development of thisunpublished work. To our knowledge, no model of the reedas a continuous system in association with the air column hasbeen proposed. The model proposed here is aimed at overcoming sev- eral limitations of previous approaches. Besides giving ameans to review the approximations of the classical model,this new approach is also a ﬁrst step toward numerical simu-lations of the instrument based on modal projection 22,23 rather than on propagation schemes represented by reﬂectionfunctions. The different parts of a clarinet—reed, mouthpiece, bar- rel, upper and lower parts of the pipe, bell—are shown inFig. 1 together with their respective models. Fluid and solidﬁnite-element models ~FEM!for the reed and the beginning of the pipe and a lumped elements model for the main part ofthe pipe are used. The work presented here begins with the modal analysis of the isolated reed. In each subsequent section, another el-ement of the model is added, ﬁnally resulting in a completeinstrument. In addition, the modes of the reed associatedwith the mouthpiece and barrel are compared with the resultsof experimental modal analysis. II. THE REED A. Construction of the numerical model Establishing a ﬁnite element model requires the determi- nation of the geometry of the reed, the choice of a constitu-tive law, the determination of the mechanical parameters, aswell as the appropriate boundary conditions. A series of three reeds have been measured. The thick- ness of each reed was measured with a coordinate measuringmachine ~Mitutoyo EURO-M 574 and Johansson Saphir 7 were used !. Approximately 200 points, arbitrarily chosen on the reed surface, have been measured @Fig. 2 ~a!#. The geo- metrical data for the model are interpolated from the mea-sured values. Interpolation between measured points wasdone by using a fourth-order polynomial, resulting in andgiving the thickness map shown in Fig. 2 ~b!. The reed is assumed to be symmetrical with regard to its longitudinalaxis. The shape of the reed was measured using a high preci- sion optical projector ~Macro Dynascope 5D, by Vision En- gineering with Metronics Quadra-Check 4000 interpolatingsoftware !with the results shown in Fig. 2 ~c!. The precision of the geometrical measurements of the reed can be esti-mated to ’2 mm. Reeds are made out of cane which is considered here as a purely elastic, transversely isotropic, homogeneous mate-rial. Viscosity and plasticity, related to energetic losses, havebeen neglected at this step of the analysis. The homogeneityhypothesis will be analyzed a posteriori in Sec. V. In thecurrent state of knowledge, we have found no other plausibledescription that could be expressed quantitatively. A discussion of losses in cane has been given lately by Marandas et al. 24and Obataya and Norimoto.25The former found out that dry cane is viscoelastic and turns viscoplasticwhen wet. This implies that static tests on wet cane are notappropriate to measure Young’s moduli. Obataya proposedvalues of the quality factor Qof the order of magnitude of FIG. 1. The clarinet: its parts and their respective models ~not to scale !. 2875 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 Facchinetti et al.: Modal analysis of the clarinet Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Thu, 18 Dec 2014 20:28:59100 varying with frequency, relative humidity, and internal state of cane. Since only individual modes of the reed areconsidered here losses can be ignored. They would need tobe taken into account in modeling the actual dynamics of theinstrument. Under these assumptions, ﬁve parameters are needed to describe the material: density rs, longitudinal and transverse Young’s moduli ELandET, transverse to longitudinal shear modulus GLT, and longitudinal-transverse Poisson ratio nLT. The values adopted here are given in Table I. The val- ues for rs,EL, and nLTwere obtained by Pinard and Laine and result from static measurements on a piece of dry canegiven by a reed maker. Obataya and Norimoto give roughlythe same value for the mainYoung’s modulus E Lof dry cane in the frequency that is relevant here ~2–6 kHz !. Their mea- surements show that this value decreases linearly with therelative humidity level ~RH!,E Ldecreasing by around 30% for a variation of 100% in RH. The other parameters werealso obtained by Pinard and Laine. Their work has been pio- neering in several respects. In particular, they were the ﬁrstto match eigenfrequencies and modal patterns of reeds ob-tained by holographic interferometry with those obtainedwith a ﬁnite-element model. As boundary condition, we consider the reed rigidly clamped on the section corresponding to the ligature andhaving a stress-free boundary elsewhere. B. Computed eigenmodes This model has been implemented on a standard PC ~450 MHz, 250 Mbyte RAM, Linux !using linear Love– Kirchoff plate elements in the CAST3Mﬁnite-element code. The ﬁrst modes of a reed are presented in Fig. 3. A classiﬁ-cation of the modes is needed for referencing and an attemptis made here. Since modal patterns with closed modallines have not been encountered, it is intuitively appealing tolabel the modes according to the number of intersectionsbetween the nodal lines and the edges of the reed. Forthe symmetric reed considered here, a mode is labeled LnTm. ‘‘L’’ stands for longitudinal and the ﬁrst index nis the number of intersections of nodal lines with the edge ~s! parallel to the main axis. Such nodal lines include the oneimposed by the boundary condition at the ligature. ‘‘ T’’ stands for transverse and the index mis the number of inter- sections of the nodal lines with the tip edge of the reed.Modes appear in an order which can be expected(L1T0,L1T1,L2T0,L1T2,L2T1), given the larger ﬂex- ibility in the direction transverse to the reed and the thick-ness distribution. The generalized mass of a mode is: m5u TMsu, ~1! whereurepresents the reed displacement for the mode and Mis the mass matrix of the reed. For a unit value of the maximum displacement in each mode, the modal masses are7, 0.35, 0.47, 0.063, and 0.094 mg for the L1T0,L1T1, L2T0,L1T2, andL2T1 modes, respectively. Along with modal patterns, these values establish a comparison betweenmodes. These mass values can also be compared to the orderof magnitude of the real mass of the moving reed. At the tipof the reed, the thickness is about 1/10 mm and the width 13mm. For a density rs5450 kgm23, the mass of a moving portion of the reed of length l~in mm !is (0.59 3l) mg. III. MODAL COMPUTATION OF THE REED ASSOCIATED WITH MOUTHPIECE AND BARREL This section analyzes how the dynamics of the reed is inﬂuenced by air loading and provides a comparison betweenresults given by the model and experiments presented in Sec. FIG. 2. Geometry of the reed, with dimensions in mm: ~a!points actually measured, ~b!interpolated thickness, ~c!estimated contour.TABLE I. Material properties for dry cane used in reeds, as given by Pinard and Laine. Density rs5450kg/m3 Longitudinal Young modulus EL510000 MPa Transverse Young modulus ET5400 MPa Shear modulus GLT51300MPa Poisson ratio nLT50.22 2876 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 Facchinetti et al.: Modal analysis of the clarinet Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Thu, 18 Dec 2014 20:28:59II. The system considered now is composed of the reed, the mouthpiece, and the barrel and is represented using acoupled ﬂuid–solid model. A. Numerical model The full model of reed, mouthpiece, and open barrel is shown in Fig. 4. The internal shape of the mouthpiece ~a Selmer HS *) has been carefully measured by means of the coordinate measuring machine used for the reed. The barrelis considered as a cylindrical bore with a diameter of 15 mm.The air volume inside the mouthpiece and the barrel is mod-eled with linear tetrahaedric and prismatic ﬁnite elements ofcompressible elastic ﬂuid.The acoustic pressure at points of the open air surfaces is considered to be zero. The normal derivative of the acous-tic pressure on the walls of the mouthpiece and the barrel,corresponding to air ﬂow, is also set to zero. The boundarycondition coupling the reed and the mouthpiece involves thestress in the solid and the velocity of the ﬂuid and will begiven explicitly in the following. The eigenvalue problem for a coupled solid–ﬂuid sys- tem is expressed in the continuous formulation by thefollowing: 26 divC„u2v2rsu50, ~2! div1 rfp1v21 c2rfp50, ~3! whereprepresents the acoustic pressure in the ﬂuid. The densities of solid and ﬂuid are rsandrf, respectively. The speed of sound is c, the angular frequency of the motion is v, andCdenotes the elasticity matrix of the solid. The boundary conditions coupling the ﬂuid and the solid parts are s"n52pn, ~4! ]p ]n5rfv2u"n, ~5! wherenrepresents the unit vector normal to the solid surface ands5C„uthe stress tensor. In order to formulate these equations as a standard ei- genvalue problem, a new variable p52(1/v2)pmust be introduced.26The equations and boundary conditions become divC„u2v2rsu50, ~6! div1 rfp21 c2rfp50, ~7! v2p1p50, ~8! s"n52pn, ~9! ]p ]n52rfu"n. ~10! To the preceding equations, we can associate the follow- ing Lagrangian Ldenoting the variational formulation of the problem: FIG. 3. First ﬁve computed modes of an isolated reed. Modes are labeled according to the number of modal lines perpendicular to the main axis ~Ln! and parallel to it ~Tm!. FIG. 4. Reed and volume of air inside the mouthpiece mounted on an open barrel. 2877 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 Facchinetti et al.: Modal analysis of the clarinet Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Thu, 18 Dec 2014 20:28:59L51 2E Vs„uC„udv11 2E Vf1 rfc2p2dv 2v2S1 2E Vsrsu2dv21 2E Vf1 rfc2~p!2dv 2E Vf1 rfc2ppdv2E ]Vps"ndsD, ~11! where VsandVfrepresent the solid and ﬂuid volumes, re- spectively, and ]Vrepresents the boundary between these volumes. Finally, the problem is expressed in its discrete form by the following eigenvalue problem: SFKs00 0Kf0 00 0G2v2FMs0 2N 00 Kf 2NTKfT2MfGDFu p pG5F000G, whereKs~respectively, Kf) andMs~respectively, Mf) are rigidity and mass matrices of the solid ~respectively, ﬂuid ! part of the system and Nis the operator corresponding to the coupling boundary condition ~10!related to the normal vec- torn. Details of the derivation can be found in Ref. 26. B. Experimental modal analysis An experimental modal analysis on reeds by means of holographic interferometry was performed in order to checkthe validity of the numerical model of the reed coupled to air.Recent works have been reported in shortcommunications. 18–20For various reeds mounted on a mouthpiece under dry conditions Pinard and Laine observedone mode corresponding to a longitudinal ﬂexion at around2200 Hz; one family of modes around 3500–3700 Hz, withpatterns varying from reed to reed, some of them being in-dicative of torsion, others being closer to ﬂexion; and one family of modes around 5800–6000 Hz, with more complexpatterns. In measurements presented here, the reed was attached to the mouthpiece exactly as on the real instrument. Since theligature was producing strong light reﬂexions, it wasreplaced with adhesive tape placed slightly further from thetip. A sinusoidally driven loudspeaker was placed closeto the reed to excite its vibration. For determining the reso-nance frequencies a very thin PVDF piezoelectric ﬁlm@poly~vinylideneﬂuoride !, thickness 0.05 mm, mass 0.06 g, of which only a part was actually moving #was glued onto the lower thicker part of the reed, yielding the average de- formation near the ligature. Resonance frequencies were de-termined using the maximum of the piezoelectric signal. Theexperiments were performed under natural humidity. A satu-rated atmosphere would have been preferable but was notpossible with the interferometry equipment. The eigenmodes were visualized by means of laser transmission interferometry. Complete details of the imple-mentation of this classical method are described in Ref. 27.The images in Fig. 5 represent variations of equal normal-displacement of the reed. The resolution of the system is halfthe wavelength of the laser, approximately 0.3 mm. The reed was measured either alone, associated with an open mouthpiece, or with the mouthpiece mounted on anopen barrel. The ﬁrst four measured modes shown in Fig. 5correspond to the barrel conﬁguration ~see Fig. 4 !. They are compared with the corresponding computed modal patterns~see the next section for computation of the eigenmodes !on top. Results of the holographic measurements show that themaximum displacement of the reed is negligible compared tothe distance between the mouthpiece and the reed at thatlevel of excitation. Thus one can be conﬁdent that contactbetween the reed and the lay, which could possibly make thesystem nonlinear, does not occur. FIG. 5. Projection of four eigenmodes on the reed ~see the text for labels !. Top pictures: computed normalized eigenmodes of the association of a reed with mouth-piece and barrel. In this representation, a cyclic grayscale produces fringes of equal differences in normaldisplacement, allowing a comparison with the modalpatterns observed experimentally. Bottom pictures:modal patterns measured by holographic interferometryon one good reed mounted on the mouthpiece attachedto the barrel. The resonance is not very sharp owing todamping, hence the rounded eigenfrequencies.The pho-tographed section of the reed does not have the sameheight between the various experiments and the simu-lations. 2878 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 Facchinetti et al.: Modal analysis of the clarinet Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Thu, 18 Dec 2014 20:28:59C. Results A comparison between computed and measured modes is displayed in Fig. 5 for the situation described by Fig. 4. Inthis comparison with holographic measurements, the ‘‘liga-ture’’of the reed had to be placed slightly beyond its normalposition. This led to a slightly more ﬂexible reed than in thenormal situation. When the reed is coupled to air, one shouldstress that eigenmodes concern the whole system, not just thereed. Strictly speaking, expressions such as ‘‘reed modes’’are inappropriate and refer instead to modes for which en-ergy ismostlylocalized in the reed. Each mode has been labeled using the notation proposed earlier. The ‘‘ R’’ preﬁx indicates that we regard the result just as the projection of the four ﬁrst eigenmodes on the reed subspace. In order to sim-plify the discussion, we have not attempted to label the airconﬁguration. One can notice that the L2T2 pattern did not appear in the isolated reed case. One can also notice that theL1T0 pattern of the reed appears in the two ﬁrst modes of the coupled system. The computed modes appear in the same order as the measured ones with eigenfrequencies deviating by 10%–20% from measured resonance frequencies. The modal pat-terns are globally the same despite the fact that no real reedis symmetric whereas the numerical model has been chosensymmetric. As expected, the modes are mainly localized atthe tip of the reed where it becomes very thin, showing theimportance of a precise measurement of the geometry. Al-though some of the mechanical parameters come themselvesfrom a ﬁt between observation and computation of modes ofan isolated reed, the mixed ﬂuid–solid model can be consid-ered as valid within the range of approximations retainedhere. Real reeds have natural asymmetries due to their geom- etry or to nonuniform mechanical properties. One noticesthat the asymmetry seems stronger for the lowest mode thanfor any other one. D. Sensitivity analysis The sensitivity analysis of the eigenfrequencies to varia- tions in mechanical parameters describing the reed and inacoustical properties of the air is presented in Table II. Theair volume is that of Fig. 4. Parameters are varied by 5%above and below their average values ~i.e., 10% overall !and the corresponding overall variations of eigenfrequencies arereported. The value of the Poisson ratio appears to be irrel-evant. Eigenfrequencies 1190, 2680, and 4010 Hz vary lin- early with the speed of sound.This is also almost the case forthe mode at 5280 Hz. Without looking at the modal patternof air pressure or reed displacement, one can infer that theycorrespond to ‘‘‘air modes,’’with energy mostly localized inthe~short!pipe. Conversely, the mode at 3700 Hz is not inﬂuenced by air characteristics; sensitivity to the shearmodulusG LTindicates that the reed is subject to torsion ~see the second mode of Fig. 3 !and is poorly coupled to the pipe ~Fig. 6 !. To a lesser degree, this is also the case of the mode at 6300 Hz. The mode at 4740 reveals a ( EL/rs)1/2depen- dency of the eigenfrequency. It is mostly a ‘‘reed mode’’involving primarily a longitudinal deformation. The mode at2010 Hz is apparently a mode in which air and reed arestrongly coupled. It is interesting to notice that the transverseYoung’s modulus does not seem to inﬂuence any frequency.The measurement of its precise value is therefore less par-ticularly important. E. Evolution of the eigenfrequencies Another way of examining how the reed is coupled to the acoustic ﬁeld is to follow the evolution of the eigenfre-quencies when the reed is loaded by the air volume ofmouthpiece and barrel. A decrease of the eigenfrequenciesand a dominance of the longitudinal ﬂexion occurs in theeigenmodes ~Fig. 7 !. The frequencies of the ﬁrst two modes of the $reed, mouthpiece, barrel %system are mainly imposed by the reso- nance of the air cavity. In both modes, the reed undergoesmainly longitudinal ﬂexion. The frequency of the torsionmodeL1T1~3257 Hz for the isolated reed !does not vary FIG. 6. Computed eigenmode at 4119 Hz in a mixed solid-air situation: acoustic pressure inside the mouthpiece and barrel.TABLE II. Sensitivity analysis: changes in eigenfrequencies when mechanical characteristics of the reed and acoustical properties of the air vary. Changes are given in % for a 10% variation of each parameter. D510% mean valuesEL 104MPaET 400 MPaGLT 1300 MPanLT 0.22rs 450 kgm23c 340 ms21rf 1.23 kgm23 1190 Hz 0 0 0 0 0 9.8 0 2010 Hz 2.4 0 0 0 22.2 0.7 20.4 2680 Hz 0.1 0 0 0 20.2 9.6 0 3700 Hz 1.5 0 3.1 0 24.6 0 0 4010 Hz 0.2 0 0 0 20.2 9.3 0 4740 Hz 4.9 0 0 0 24.8 2.7 20.1 5280 Hz 0.6 0 0 0 20.9 8.1 20.1 6300 Hz 1.7 0.9 4.9 0 26.4 3.2 0 2879 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 Facchinetti et al.: Modal analysis of the clarinet Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Thu, 18 Dec 2014 20:28:59signiﬁcantly, meaning that this mode is weakly coupled to the air cavity. The same phenomenon can be noticed for themodeL1T2 at 5840 Hz for the isolated reed. One can con- clude from Fig. 7 and from the observation of the ratheruniform pressure in the pipe at these modes ~not shown here ! that this mode also is weakly coupled to the pipe. IV.MODALCOMPUTATIONOFTHEWHOLECLARINET In order to simulate the modal behavior of the complete clarinet, we have associated a ﬁnite-element model of ’10 cm of pipe with lumped elements representing the rest of thepipe and matching its acoustic input impedance. This can bedone since at the outlet of the barrel, the acoustic ﬁeld con-sistsessentiallyofplanewaves.Anexampleofacousticpres-sure in the mouthpiece is represented in Fig. 8. The mode isthat of a complete clarinet and corresponds to the lowestmode at 311 Hz of the medium C ]ﬁngering combined with the opening of the register key ~see the following for the complete list of modes in this conﬁguration !. The length of mouthpiece represented here is 32 mm and corresponds tothe tapered part. One can see that the acoustic waves canalready be considered as plane waves within a very goodapproximtation. The lumped-element oscillators ~shown in generic form in Fig. 1 !are coupled to the ﬁnite-element barrel by meansof a rigid plate of negligible mass, as shown in Fig. 9. The plate and the lumped-element oscillators are supposed tomove only in the longitudinal axis of the instrument. Thelumped elements are placed at the ~virtual !junction between the barrel and the lower part of the clarinet. It is now explained how the numerical values of the lumped elements are calculated on the basis of measure-ments provided by Gibiat 28on several notes of a Noblet B [ clarinet. Results of these measurements are supposed to rep-resent the inputacoustical impedance of the instrument. In order to measure this input impedance, a reference plane wasdeﬁned by Gibiat et al.by replacing the mouthpiece with a portion of cylindrical tube of equal volume. This is the usual‘‘equivalent volume’’ approximation which we discuss lateron. Prior to matching the impedance of the lumped elementsto the measured input acoustical impedance of the pipe, thelatter must therefore be transported from the input plane to-ward the open end of the pipe. The ‘‘transportation distance’’is equal to the length of a cylinder having the volume of themouthpiece and the barrel. 22,29 An oscillator is associated with each measured imped- ance peak. At the angular frequency vthe mechanical im- pedance of each elementary oscillator in Fig. 1 is Zm~v!5iS1 mv2v k1ivrD21 , ~12! wherem, r, kare respectively the mass, damping, and stiff- ness of the lumped elements. In this ‘‘comb-like’’ association, the impedances of the oscillators add. The dual association where the admittancesadd is ‘‘chain-like.’’Each elementary oscillator of Fig. 1 is amass chained with a comb of a damper and a spring, leadingto Eq. ~12!. The parameters m i,ri,kiof each oscillator ~a tooth of the large comb !are identiﬁed by minimizing a cost func- tional Jmeasuring the distance between computed and mea- sured moduli and phase of the impedance: J5auMod~Zcomp!2Mod~Zmeas!u 1buArg~Zcomp!2Arg~Zmeas!u. ~13! The initial values of the parameters for each oscillator are obtained by identifying each single resonance peak andthe ﬁnal values are obtained by running a Nelder–Mead sim-plex search algorithm. A comparison between the measuredand the identiﬁed modulus and phase of the acoustic imped-ance of the lowest F ﬁngering ~E[heard !of the clarinet is presented in Fig. 10. The impedance represented is not theinput acoustical impedance but the impedance of the lowerpart taken at the ~virtual !junction between the barrel and the lower part of the clarinet.Therefore, the peak frequencies arenot the eigenfrequencies of the instrument. The acousticalimpedance represented here is the ratio of the acousticalpressure to the air velocity, normalized by rc. The average modulus on a logarithmic scale would be 1 for an ideal longcylindrical pipe. According to Gibiat, it is less here due tointernal losses, radiation, and presumably the complexity ofthe pipe. FIG. 7. Evolution of the eigenfrequencies ~left scale, in hertz !when the system evolves from the isolated reed ~left!to$reed1mouthpiece %~middle ! and$reed1mouthpiece 1barrel %~right!. Black lines represent ‘‘primary reed’’ modes, dotted lines ‘‘primary air’’ modes, and dash-dot lines,‘‘mixed’’ modes. FIG. 8. Acoustic pressure inside the tip part of the mouthpiece for a 311 Hzmode of the complete clarinet. The acoustic pressure decreases monotoni-cally from the tip to the largest section by 14%. 2880 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 Facchinetti et al.: Modal analysis of the clarinet Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Thu, 18 Dec 2014 20:28:59Two eigenmodes of the complete instrument for differ- ent ﬁngerings are shown in Fig. 9. One eigenmode has noamplitude per se. For each eigenmode in Fig. 9 the ~relative ! amplitude of the motion of the oscillators is represented bythe length of a straight line extending from the plate. Onenotices that the pressure distribution is not uniform in themouthpiece. Examining other similar ﬁgures reveals that themotion of the reed can differ signiﬁcantly from mode tomode of a given note, even if it follows a L1T1 pattern.This means that, although the ﬁrst modes of the isolated reed oc-cur at signiﬁcantly higher frequencies than those consideredhere, a single degree of freedom for the reed is not appropri-ate since it would not account for these differences. Whenthe reed undergoes mostly longitudinal ﬂexion, it is to beexpected that the beam model used by several authors 15,16,30 would give comparable results. For the low F ﬁngering ~sounding one tone lower !, the ﬁrst eigenfrequencies are 166, 464, 743, 1147, 1436, 1620,1950, 2058, and 2201 Hz. They are 373, 1035, 1541, 1687,1893, 1930, and 2309 Hz for the medium G ﬁngering and311, 735, 1213, 1467, 1578, 1865, and 2211 Hz for the highG], played with medium C ]ﬁngering and opening of the register key. These frequencies are represented in Fig. 11 inorder to evaluate their harmonicity. Eigenfrequencies arenormalized by their ratio to the theoretical musical frequencyfor the note under consideration ~respectively, 156, 349, and 740 Hz !, rounded to the nearest integer. For example, a 900 Hz eigenfrequency for note A4 ~440 Hz !would be normal- ized by 2, nearest integer to 900/440. For this high note, theregister key does not eliminate the ﬁrst mode of the instru-ment but the sound will be locked approximately on the sec- ond mode. The lowest mode is very roughly at half the pitchof the note and is therefore normalized by the integer 2. The sets of solid lines in Fig. 11 represent the computed eigenfrequencies listed above of the complete instrument.The sets of dashed lines are resonances of the pipe as ex-tracted from the measurements of the input impedance of thepipe. This set represents the traditional view of the instru- FIG. 9. Modal representation of a complete clarinet: amplitude of the motion of the lumped-element oscillators ~left!, air pressure in the upper part of the pipe ~middle !, and deformation of the reed ~right!. Eigenmode 2 for note treeble F# ~ﬁngering of C# medium plus opening of register key !and eigenmode 8 for note low E [~low F ﬁngering !. FIG. 10. Acoustical impedances ~ratio of the acoustical pressure to the air velocity, normalized by rc) for the low F note of the clarinet. Solid lines: acoustical impedance of the pipe as measured at the closed end of the pipeand transported at the ~virtual !junction between the barrel and the lower part of the clarinet. Dashed lines: impedance of the set of lumped oscillatorsbest matching the impedance of the pipe at the junction. 2881 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 Facchinetti et al.: Modal analysis of the clarinet Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Thu, 18 Dec 2014 20:28:59ment where the volume of the mouthpiece has been replaced by a cylindrical pipe having the same volume and closed atone end. The sets of dotted lines represent computed eigen-frequencies of the air column with a rigid boundary on thereed surface. Instead of the completely closed pipe of thetraditional modeling, one assumes here a slight opening be-tween the reed surface and the lay of the mouthpiece with azero pressure condition. V. DISCUSSION AND PERSPECTIVES A. Alignment of resonances and low-frequency approximation The traditional model of the mouthpiece is that of a cylinder of equivalent volume. Within this approximationthere is no point in measuring the input acoustic impedanceabove a certain limit. This limit can be evaluated by thelength scale at which the mouthpiece geometry differs from acylinder.Taking as an order of magnitude for these geometri-cal differences a length of 1 cm is consistent with a 2.5 kHzfrequency limit beyond which input acoustical impedanceswould begin to differ noticeably. In the approach followed inthis paper, the equivalent volume approximation is aban-doned and the acoustical input impedance of the pipe wouldkeep full utility and validity up to the frequency of the ﬁrsttransverse mode of the pipe ~13.3 kHz for the clarinet !. Thecylinderofequivalentvolumeapproximationforthe mouthpiece is assumed to be correct for low frequencies. Itappears in Fig. 11 that this approximation is not acceptableenough to be used in conjunction with an alignment of peakscriteria. One can see in Fig. 11 that variations in eigenfre-quencies due to the model change are signiﬁcant with regardto the alignment of resonances, even at low frequencies .I n other words, the deviations from alignment in the traditionalview ~equivalent volume approximation !are of the same or- der of magnitude as the frequency shifts due to the presenceof the reed and the prismatic shape of the mouthpiece.B. Coupling of torsion modes to the air The association of reed, mouthpiece, and a short open portion of the pipe is shown in Fig. 6. The modal acousticpressure at an eigenfrequency of 4119 Hz is displayed in Fig.6. In this mode, the reed undergoes torsion in a pattern verysimilar to the L1T1 mode of the isolated reed ~Fig. 3 !. The characteristic distance of this modal deformation is signiﬁ-cantly smaller than half the wavelength in air at that fre-quency ( l’10 cm); the resulting acoustical short-circuit prevents any efﬁcient coupling of the reed to the air in themouthpiece. This explains the fairly uniform acoustic pres-sure for this mode, except very near to the reed. However,there are several reasons why these modes may be importantin the actual playing. First of all, the ﬂow entering the air channel between the reed and the lay is governed by a nonlinear equation. There-fore, antisymmetric reed modes may have an inﬂuence onthe global ﬂow entering the pipe. It has been shown that the antisymmetric reed modes are very weakly coupled to the acoustic ~far!ﬁeld in the clarinet. This is not to say that these modes play no role in the dy-namics. Asymmetries or, better said, unevenness in the geo-metric or constitutive properties of reeds induce asymmetriesof longitudinal reed modes and consequently an asymmetryin the local acoustical ﬁeld. Due to its small relative modalmass, the torsion mode can be easily excited at a frequencydifferent from its resonant frequency and therefore may playa signiﬁcant role in the actual dynamics of the reed. Thecoupling factor would then be the local acoustic ﬁeld. Thismay be an explanation for the player’s experience that fordifferent mouthpieces, the preferred reeds are also different. This modal analysis is performed on a symmetric reed. This is not the case in reality as shown for example by theﬁrst mode in Fig. 5.The so-called torsion modes are likely tobe associated in the ﬂuid domain to a ﬂow different fromzero and therefore couple to the plane waves inside the pipe. C. Symmetry Experimental modal analysis shows that some reeds have strong asymmetries. Makers can be expected to be suc-cessful in controlling the symmetry of the geometry; there-fore, the cause of modal asymmetries lies most probably inthe lack of homogeneity of the cane used for the reed due toits natural character. Pinard, Laine, and Vach 31examined 24 reeds, ranked by two professional players. They observedthat the two reeds ranked as good and very good were sym-metric whereas the poor reed had asymmetrical high modes.Based on limited sampling of reeds and players, no deﬁniteconclusion can be drawn. Intuition would suggest that asym-metry is not a desirable feature for a reed. However, we thinkthat it might not be so. Visualizing the lip motion in brass playing shows that lips do not move symmetrically and that this factor variesfrom player to player. Since brass mouthpieces are symmet-ric, one can conclude that the mechanical properties of lips~possibly coupled to dentition and the mouth cavity !are not symmetric for all brass players. One can hypothesize that thesame is true among clarinet and saxophone players. Another FIG. 11. Normalized eigenfrequencies ~logarithmic scale !of the complete clarinet, pipe with reed ~solid symbols !, of the pipe with a ﬁxed reed ~dash dot!, and normalized resonance frequencies measured on the pipe where the mouthpiece replaced by its equivalent volume ~dashed !. See the text for the deﬁnition of the normalization. Fingerings are low F, medium G, and highG]~medium C ]with register key !corresponding to notes E [3 ,F4 ,a n d G]5. 2882 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 Facchinetti et al.: Modal analysis of the clarinet Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Thu, 18 Dec 2014 20:28:59observation is that different players do not always prefer the same reeds in a given box, even for common musical tasks,style, etc., and the same clarinet and mouthpiece. A goodmatch between a player and a reed could mean that a givenasymmetry in a reed would ﬁt well the natural asymmetry ofa given player and not so well with another one. It has evenbeen observed that a few players use reeds which ﬁt almostnone of their colleagues. It would be interesting to test theseplayers and their preferred reeds with regard to the symmetryhypothesis. VI. CONCLUSION The modal analyses of reeds and of a few notes of the whole clarinet were performed. Results have shown the fol-lowing points. ~1!A numerical model of cane based on the hypothesis of transverse isotropy is suited to describe modal patterns ofreeds. Some of the numerical hypotheses ~homogeneity, symmetry, damping !can be released but this would necessi- tate additional measurements. ~2!When coupled to air, the reed is subject to deforma- tion patterns which are not always those of its own normalmodes. Therefore, the normal modes of isolated reeds cannotbe taken as a source for the acoustic ﬁeld in the mouthpiece.Speciﬁcally, coupling must be taken into account. ~3!Torsion modes of reeds generate a strong but very localized acoustic ﬁeld in the mouthpiece. It remains to beexamined how this would interact with asymmetries in lowermodes through the excitation process. ~4!Acoustic waves are already plane within a very good approximation in the cylindrical part of the mouthpiece.Since ﬁnite-element modeling of air is interesting insofar asthe waves are not plane, the air volume in the barrel and alarge proportion of that in the mouthpiece can be included inthe lumped-element model, reducing signiﬁcantly the com-putational burden. ~5!The shape of the mouthpiece and the dynamics of the reed inﬂuence the alignment of resonances in the same pro-portion as the misalignment derived from the customary ob-servation of the input acoustical impedance. Therefore, theapproximation of the equivalent volume is too coarse to beused when looking at harmonicity of resonances. This study shows the need for input impedance measure- ments at higher frequencies than usually performed. It callsfor simpliﬁed formulations of the acoustic ﬁeld in the mouth-piece. The procedure outlined here could be used to testthese formulations. Finally, the method paves the way fornumerical simulations of the dynamics of the clarinet basedon modal projection and taking into account the whole com-plexity of the reed. ACKNOWLEDGMENTS We express our gratitude to Holger Vach for his decisive help in the experimental part of this study, to Vincent Gibiatfor providing us with the measurements of the acoustic inputimpedances and the associated software, and to Brian Katzfor many language corrections.1C. J. Nederveen, Acoustical Aspects of Woodwind Instruments ~Illinois University Press ~ﬁrst ed. Frits Knuf Pub., Amsterdam !, Dekalb, 1998 ~ﬁrst ed. 1968 !!. 2A. H. Benade, Fundamentals of Musical Acoustics ~Oxford University Press, New York, 1976 !. 3J. Kergomard, ‘‘Elementary considerations on reed-instrument oscilla- tions,’’in Mechanics of Musical Instruments ~Springer, New York, 1995 !. 4D. Campbell, ‘‘Nonlinear dynamics of musical reed and brass wind instru- ments,’’ Contemp. Phys. 40, 415–431 ~1999!. 5J. Gilbert, ‘‘E´tude des instruments a `anche simple’’ ~On simple reed in- struments !, Ph.D. thesis, Universite ´du Maine-Le Mans, 1991. 6X. Boutillon andV. Gibiat, ‘‘Evaluation of the acoustical stiffness of saxo- phone reeds under playing conditions by using the reactive power ap-proach,’’ J. Acoust. Soc. Am. 100, 1178–1189 ~1996!. 7R. Schumacher, ‘‘ Ab initio calculations of the oscillations of a clarinet,’’ Acustica 48,7 3 –8 5 ~1981!. 8M. Mcintyre, R. Schumacher, and J. Woodhouse, ‘‘On the oscillations of musical-instruments,’’ J. Acoust. Soc. Am. 74, 1325–1345 ~1983!. 9C. Maganza, R. Causse, and F. Laloe, ‘‘Bifurcations, period doublings and chaos in clarinet-like systems,’’ Europhys. Lett. 1, 295–302 ~1986!. 10X. Rodet and C. Vergez, ‘‘Nonlinear dynamics in physical models: Simple feedback-loop systems and properties,’’ Comput. Music J. 23,1 8 – 3 4 ~1999!. 11J. Smith, ‘‘Physical modeling using digital wave-guides,’’Comput. Music J.16,7 4– 9 8 ~1992!. 12B. Gazengel, J. Gilbert, and N. Amir, ‘‘Time-domain simulation of single reed wind instrument—From the measured input impedance to the syn-thesis signal—Where are the traps?,’’Acta Acustica 3,4 4 5 –4 7 2 ~1995!. 13E. Ducasse, ‘‘Modelisation et simulation dans le domaine temporel d’instruments a `vent a anche simple en situation de jeu’’ ~Time-domain model and simulation of simple-reed instruments in playing conditions !, Ph.D. thesis, Universite ´du Maine-Le Mans, 2001. 14E. Ducasse, ‘‘Models of musical-instruments for sound synthesis: Appli- cation to woodwind instruments,’’ J. Phys. ~France !51, 837–840 ~1990!. 15S. Stewart and W. Strong, ‘‘Functional-model of a simpliﬁed clarinet,’’ J. Acoust. Soc. Am. 68, 109–120 ~1980!. 16S. Sommerfeldt and W. Strong, ‘‘Simulation of a player clarinet system,’’ J. Acoust. Soc. Am. 83, 1908–1918 ~1988!. 17B. Gazengel and J. Gilbert, ‘‘Numerical simulations in time and frequency domains—Comparative-study, application to single-reed woodwind in-struments,’’ J. Phys. IV 4, 577–580 ~1994!. 18P. Hoekje and G. Roberts, ‘‘Observed vibration patterns of clarinet reeds,’’ J. Acoust. Soc. Am. 99, 2462 ~A!~1996!. 19I. Lindevald and J. Gower, ‘‘Vibrational modes of clarinet reeds,’’ J. Acoust. Soc. Am. 102, 3085 ~A!~1997!. 20B. Richardson ~private communication !. 21D. Casadonte, ‘‘The perfect clarinet reed? Vibrational modes of realistic clarinet reeds,’’ J. Acoust. Soc. Am. 94, 1807 ~A!~1993!. 22M. Facchinetti, ‘‘Etude des vibrations de l’anche de la clarinette’’ and ‘‘Analisi del comportamento dinamico di un clarinetto,’’ EcolePolytechnique-Paris and Politecnico-Milano ~1999!. 23M. Facchinetti, X. Boutillon, and A. Constantinescu, ‘‘Application of modal analysis and synthesis of reed and pipe to numerical simulations ofa clarinet,’’ J. Acoust. Soc. Am. 108, 2590 ~A!~2000!. 24E. Marandas, V. Gibiat, C. Besnainou, and N. Grand, ‘‘Mechanical char- acterization of woodwind reeds,’’ J. Phys. IV 4, 633–636 ~1994!. 25E. Obataya and M. Norimoto, ‘‘Acoustic properties of a reed ~Arundo donax L. !used for the vibrating plate of a clarinet,’’ J. Acoust. Soc. Am. 106, 1106–1110 ~1999!. 26R.-J. Gibert, Vibrations des Structures-Interactions avec les Fluides ~Ey- rolles, Paris, 1988 !. 27K. Menou, B. Audit, X. Boutillon, and H. Vach, ‘‘Holographic study of a vibrating bell:An undergraduate laboratory experiment,’’Am. J. Phys. 66, 380–385 ~1998!. 28V. Gibiat and F. Laloe, ‘‘Acoustical impedance measurements by the 2-microphone-3-calibration ~tmtc!method,’’ J. Acoust. Soc. Am. 88, 2533–2545 ~1990!. 29V. Gibiat, ‘‘Mesures d’impe ´dance acoustique pour la clarinette,’’ propri- etary software and private communication, 1999. 30B. Gazengel, ‘‘Caracte ´risation ... des instruments a `anche simple’’ ~Char- acterization ... of simple reed instruments !, Ph.D. thesis, Universite ´du Maine-Le Mans, 1994. 31F. Pinard, B. Laine, and H. Vach, ’’Musical quality assessment of clarinetreeds using optical holography,’’ J. Acoust. Soc. Am. 113, 1736–1742 ~2003!. 2883 J. Acoust. Soc. Am., Vol. 113, No. 5, May 2003 Facchinetti et al.: Modal analysis of the clarinet Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Thu, 18 Dec 2014 20:28:59
1.1480476.pdf	Coherent suppression of magnetic ringing in microscopic spin valve elements H. W. Schumacher, C. Chappert, P. Crozat, R. C. Sousa, P. P. Freitas, and M. Bauer Citation: Applied Physics Letters 80, 3781 (2002); doi: 10.1063/1.1480476 View online: http://dx.doi.org/10.1063/1.1480476 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/80/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Current-in-plane magnetoresistance of spin valve elliptical rings Appl. Phys. Lett. 91, 152508 (2007); 10.1063/1.2798495 Thermal magnetization fluctuations in CoFe spin-valve devices (invited) J. Appl. Phys. 91, 7454 (2002); 10.1063/1.1452685 Coherently suppressed ringing of the magnetization in microscopic giant magnetoresistive devices J. Appl. Phys. 91, 8043 (2002); 10.1063/1.1450818 Use of a permanent magnet in the synthetic antiferromagnet of a spin-valve J. Appl. Phys. 91, 2176 (2002); 10.1063/1.1433937 AP-pinned spin valve GMR and magnetization J. Appl. Phys. 87, 5723 (2000); 10.1063/1.372501 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.55.98 On: Mon, 01 Dec 2014 19:04:46Coherent suppression of magnetic ringing in microscopic spin valve elements H. W. Schumacher,a)C. Chappert, and P. Crozat Institut d’Electronique Fondamentale, CNRS UMR 8622, Universite ´Paris-Sud, Ba ˆtiment 220, 91405 Orsay, France R. C. Sousa and P. P. Freitas Instituto de Engenharia de Sistemas e Computadores, Rua Alves Redol, 9, 10 Dt., P-1000 Lisbon, Portugal M. Bauer Laboratoire de Physique des Solides, CNRS UMR 8502, Universite ´Paris-Sud, Ba ˆtiment 510, 91405 Orsay, France ~Received 14 August 2001; accepted for publication 24 March 2002 ! We demonstrate the coherent suppression of magnetic precession in microscopic spin valve elements after the decay of ultrashort magnetic ﬁeld pulses. The magnetization dynamics in1 mm34mm wide giant magnetoresistance devices are studied by measuring the magnetotransport response to ultrashort magnetic ﬁeld pulses ~pulse length 0.2–4 ns !. Under the inﬂuence of a static ﬁeld perpendicular to the pulsed ﬁeld, pronounced magnetic precession is observed after the onsetof the pulse as well as upon pulse termination.The precession after the pulse decay ~‘‘ringing’’ !can be effectively suppressed by adapting the effective pulse length to the precession period. © 2002 American Institute of Physics. @DOI: 10.1063/1.1480476 # Precise control of the ultrafast magnetization dynamics in microscopic magnetic structures is a crucial requisite toachieve reliable and fast switching in future magneticmemory devices. One fundamental limitation of the read/write time in such memory structures is due to the weaklydamped precession ~‘‘ringing’’ !of the magnetization in mag- netic thin ﬁlm elements that occurs during and after the ap-plication of short magnetic ﬁeld pulses. 1Time resolved magneto-optic1–3and magneto-transport4,5measurements on various magnetic materials revealed that ringing of the mag-netization can persist up to several ns. One intriguing con-cept by which to overcome the problem of ringing withoutlimiting the switching bandwidth is the so-called coherentsuppression of precession. Here, the ﬁeld pulse parametersare matched to the frequency and phase of the magnetic ex-citation. It has so far been observed in magneto-optical ex-periments on macroscopic thin ﬁlms by varying the length 6 or the shape7of the magnetic ﬁeld pulse. In this letter we study the magnetization dynamics of microscopic spin valve~SV!elements by measuring the time resolved magneto- transport response to ultrashort magnetic ﬁeld pulses. Wedemonstrate that by proper adjustment of the pulse lengthwith respect to the precession period coherent suppression ofringing in microscopic magnetic memory cells is possible.The extension of this technique to precessional switching 8of magnetic memory prototypes should allow a stable, ballistic magnetization reversal9,10on extremely short time scales of the order of half a single precession period. The samples used for our experiments are 1 mm 34mm wide spin valve elements in a buried pulse line conﬁguration.11The 2 mm long and 50 mm wide pulse lineconductor is made of 250 nm thick Al grown on glass. Over a length of about 70 mm at the center the width is decreased down to 5 mm and locally increases the magnetic ﬁeld cre- ated by transient current pulses. The pulse line is covered bya 250 nm thick sputtered Si oxide layer to provide electricalinsulation between the pulse and sense line. The magneticﬁlm consisting of a Ta 65 Å/NiFe 40 Å/MnIr 80 Å/CoFe 43Å/Cu 24 Å/CoFe 20 Å/NiFe 30 Å/T a8ÅS V stack is sputter deposited on top. Microscopic elements are then deﬁned byoptical lithography and ion milling before 250 nm thick Alsense line contacts are added.All conductors are designed as 50Vadapted coplanar waveguides, allowing the transmis- sion of ultrashort voltage pulses and the measurement of thehigh frequency SV response. The two terminal resistance ofthe SV is around 90 V. The giant magnetoresistance ~GMR ! is of the order of 6%. For electric and magnetic characterization the waveguides are contacted by spring loaded microwaveprobes ~40 GHz bandwidth !. Magnetic ﬁeld pulses are cre- ated by injecting voltage pulses ~maximum amplitude of 5 V, pulse length 0.2–4 ns, rise time 60 ps !from a commercial pulse generator through the pulse line. The pulse transmittedis recorded using a 50 GHz bandwidth sampling oscillo-scope. To detect the high frequency GMR response dc cur-rents of 61 mA are applied through the SV via a bias tee. 4,5 The voltage response pulse of the SV is picked up on one side of the sense line by the second oscilloscope channelwhile the other sense line contact is terminated to 50 V. Subtraction of two oscilloscope traces for positive and nega-tivedcbiasallowsonetoseparatetheSV’sresistancechangeDR SVfrom the current independent crosstalk between the crossed lines. By averaging up to 4000 oscilloscope tracesnoise levels down to 20 mV peak to peak can be obtained. Additional static ﬁelds up to 1 kOe in arbitrary in planedirections can be applied via an external revolving ﬁeld coil.a!Author to whom correspondence should be addressed; electronic mail: schumach@ief.u-psud.frAPPLIED PHYSICS LETTERS VOLUME 80, NUMBER 20 20 MAY 2002 3781 0003-6951/2002/80(20)/3781/3/$19.00 © 2002 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.55.98 On: Mon, 01 Dec 2014 19:04:46The amplitude of the magnetic ﬁeld pulse is calibrated by measuring easy axis hysteresis loops as a function of theexternal ﬁeld while applying constant pulse line currents inthe range of 610 mA. The local easy axis ﬁeld created by the current leads to a shift of the measured SV GMR loops.From the ﬁeld shift as a function of the line current a pulseline ﬁeld ratio of ;1.1 Oe/mA is deduced. In Fig. 1 ~a!a 4 ns wide rectangular voltage pulse after transmission through the pulse line is shown. Due to trans-mission and reﬂection losses the initial pulse amplitude isreduced by approximately 5% while the pulse shape is notsigniﬁcantly changed. The measured rise and fall times ~be- tween 20% and 80% !of the transient pulse are 60 and 275 ps, respectively. Similar rise times can be expected for themagnetic ﬁeld pulse. From the static ﬁeld ratio we estimatean easy axis ﬁeld of ;33 Oe for the pulse given. Sketched in the two insets of Fig. 1 ~a!are the geometry ~right inset !of the SV cell ~dark gray !with respect to the pulse line ~PL! ~light gray !and the conﬁguration of the applied ﬁelds ~left inset!during the pulse experiment. The pulse line current I PL ﬂows perpendicular to the easy ~long!axis of the SV. The magnetization of the pinned layer MPis ﬁxed parallel to the easy axis by coupling to the IrMn antiferromagnetic layer.Static external hard axis ﬁelds up to H S5250 Oe are applied to turn the free layer magnetization Mout of the easy axis. The change in GMR under application of a hard axis ﬁeld HS is plotted in the inset of Fig. 2. Due to an increase of the tilting angle between MandMPwith an increase in HSthe SV resistance ﬁrst increases before saturation sets in at about75 Oe ~arrow !. Above this ﬁeld MandM Pare aligned al- most perpendicular to each other, leading to maximum sen-sitivity of the GMR to small tilts of Mduringandafter pulse application. The pulsed ﬁeld H Pis applied parallel to the easy axis, i.e., perpendicular to HS. Neglecting furtheranisotropies one obtains a tilted total ﬁeld Htotduring pulse application given by the sum of vectors HSandHPas sketched in the left inset of Fig. 1 ~a!. The change in SV resistance DRSVas a response to the pulse shown in Fig. 1 ~a!and with HS591 Oe is plotted in Fig. 1 ~b!. During pulse application the GMR decreases, showing a more parallel alignment of MandMP. Immedi- ately after the pulse onset pronounced oscillations of the re-sistance are found @seen at 1 in ~Fig. 1 !#. They are due to damped precession of Mabout the new total ﬁeld H tot sketched in the left inset. Mrelaxes by precession from its former direction parallel to HSto the new direction parallel toHtot.The precession period is Tprec5475 ps corresponding to a precession frequency of fprec51/Tprec52.1 GHz. The decay of the damped precession below the noise level takesapproximately 2 ns. This means that at least ﬁve dampedprecession cycles are needed before Mis aligned along H tot. Such weak damping can be described by a low Gilbertdamping factor ain the Landau–Lifshitz–Gilbert ~LLG! equation.12From comparison of the SV response to numeri- cal solutions of the LLG equation for a Stoner particle10we derive a50.031 for the device given. After decay of the pulse, ringing is again found @seen at 2 in ~Fig. 1 !#. Now,M relaxes from HtottowardsHSby precession about the static ﬁeld ~see the right inset !. However, due to the relatively slower decay of the pulse the oscillations are less pro- nounced than at the pulse onset ~1!. The known angle depen- dence of the giant magnetoresistance13allows one to derive the average tilt of Mduring pulse application.After decay of the initial precession Mis expected to be aligned parallel to Htot. Here, we measure a tilt of 20° in good agreement with the calculated direction of Htotfor the given ﬁelds HS 591 Oe and HP533 Oe. In Fig. 2 the measured precession frequencies fprecdur- ing~open circles !and after pulse application ~closed squares ! are plotted as a function of the static hard axis ﬁeld HS.fprec during pulse application was derived from Fourier transfor- mation of the response during 33 Oe, 4 ns pulses at variousH Swhereasfprecafter the pulse was derived from the ringing after 200 ps long pulses. To calculate the ferromagnetic reso- FIG. 1. Precession in a 1 mm34mm SV element due to the application of a 4 ns rectangular magnetic ﬁeld pulse: ~a!transmitted voltage pulse. Insets: Sample geometry ~right!and ﬁeld conﬁgurations during the pulsed experi- ment. ~b!SV magneto-resistance response to the pulse in ~a!. dc SV current 1 mA, static ﬁeld 91 Oe. Precession occurs ~1!during and ~2!after pulse application. Precession period at ~1!:Tp5475 ps. Insets: Precession con- ﬁgurations at ~1!~left!and~2!. FIG. 2. Measured precession frequencies fprecupon ~open dots !and after ~squares !pulse application as a function of the static hard axis ﬁeld. The lines represent the calculated frequencies fFMRduring ~dashed line !and after ~solid line !pulse application. Inset: SV resistance change vs static hard axis ﬁeld.3782 Appl. Phys. Lett., Vol. 80, No. 20, 20 May 2002 Schumacher et al. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.55.98 On: Mon, 01 Dec 2014 19:04:46nance ~FMR !frequencies fFMRfor the two ﬁeld conﬁgura- tions we use the Kittel formula:14 fFMR5g 2pA@H1~Ny2Nz!4pMS#@H1~Nx2Ny!4pMS, whereHis the applied ﬁeld, Nx,y,zthe demagnetizing fac- tors, 4 pMS510800 Oe the saturation magnetization of per- malloy, and g50.2212 3106m/As the gyromagnetic ratio. Nx,y,zare approximated by Nx,y’t/wx,yfor the in plane components and by Nz512Nx2Nyout of plane with tand wx,ythe total thickness and lateral extensions of the free layer, respectively. Using H5HSandH5uHtotu 5AHS21HP2,15respectively, we derive fFMR(HS) given by the straight line ~after the pulse !and the dashed line ~on the pulse !in Fig. 2. Both curves describe the measured fprecvs HSdependence well. In Fig. 3 three shorter ﬁeld pulses and the corresponding SV responses of the same device are plotted. The pulse du-rations are 230, 470, and 760 ps, respectively ~full width at half maximum !from top to bottom. The curves are offset for clarity. The static ﬁeld is again 91 Oe. The pulse amplitudesare comparable to the amplitude of the pulse in Fig. 1 ~a! leading to similar H totandfprecas those for the 4 ns pulse. For the shortest pulse of 230 ps ~upper trace !as well as for the longest pulse ~760 ps, lower trace !strong ringing is found after decay of the pulse @see Fig. 2 ~b!, upper and lower curves, respectively !. In the case of the 470 ps long pulse, however, practically no ringing is present after pulse termi-nation ~shown by the arrow !. Here, the effective pulse length T pulse5470 ps matches the measured precession period Tprec5475 ps at the given Htot(Tpulse’Tprec). During pulse application Mthus precesses exactly once about Htotbeforethe pulsed ﬁeld is switched off again. As discussed earlier due to the low damping Mhas to pass several precession cycles until parallel alignment with Htotis attained. After a single precession cycle Mis thus still quite well aligned with HS. Only a small tilt between MandHtotis present when the pulse is switched off.As a consequence, almost no relax-ation is needed to reach the ﬁnal equilibrium magnetizationparallel to H Sand the ringing is suppressed. The two other pulses in Fig. 3 ~Tpulse5230 and 760 ps !show very pro- nounced ringing at the HSgiven. Here, the pulse ends after ’0.5 and ’1.6Tprec, respectively. At these times the tilting angles between Mand the ﬁnal precession axis HSare close to their maximum @cf. the left inset in Fig.1 ~b!#leading to an increase in precession amplitude and thus to strong ringingafter pulse decay. Such coherent suppression and ampliﬁca-tion can be observed at various H S, but then the pulse pa- rameters have to be adapted to the ﬁeld dependence of fFMR ~cf. Fig. 2 !to meet the coherence criterion. In conclusion, we have demonstrated the coherent sup- pression of ringing in microscopic magnetic memory de-vices. It was achieved by matching the pulse length to themagnetic precession period of the SV element. This coherentsuppression of ringing after pulse decay presents the ﬁrststep towards stable, ballistic magnetization switching 9,10in future magnetic random access memory cells. One of the authors ~H.W.S. !acknowledges ﬁnancial sup- port by European Union ~EU!Marie Curie Fellowship No. HPMFCT-2000-00540. The work was supported in part bythe EU Training and Mobility of Researchers Program underContract No. ERBFMRX-CT97-0147, and by a NEDO con-tract, Nanopatterned Magnets. 1W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, Phys. Rev. Lett. 79, 1134 ~1997!. 2T. M. Crawford, T. J. Silva, C. W. Teplin, and C. T. Rogers, Appl. Phys. Lett.74, 3386 ~1999!. 3Y.Acreman, C. H. Back, M. Buess, O. Portmann,A. Vaterlaus, D. Pescia, and H. Melchior, Science 290, 492 ~2000!. 4R. H. Koch, J. G. Deak, D. W. Abraham, P. L. Trouilloud, R. A. Altman, Y. Lu, W. J. Gallagher, R. E. Scheuerlein, K. P. Roche, and S. S. P. Parkin,Phys. Rev. Lett. 81, 4512 ~1998!. 5S. E. Russek, S. Kaka, and M. J. Donahue, J.Appl. Phys. 87,7 0 7 0 ~2000!. 6M. Bauer, R. Lopusnik, J. Fassbender, and B. Hillebrands, Appl. Phys. Lett.76, 2758 ~2000!. 7T. M. Crawford, P. Kabos, and T. J. Silva, Appl. Phys. Lett. 76,2 1 1 3 ~2000!. 8C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, D. Weller, E. L. Garwin, and H. C. Siegmann, Science 285, 864 ~1999!. 9J. Miltat, G.Aburquerque, andA. Thiaville, in Spin Dynamics in Conﬁned Magnetic Structures , edited by B. Hillebrands and K. Ounadjela ~Springer, Berlin, 2001 !. 10M. Bauer, J. Fassbender, B. Hillebrands, and R. L. Stamps, Phys. Rev. B 61, 3410 ~2000!. 11R. C. Sousa, V. Soares, F. Silva, J. Bernardo, and P. P. Freitas, J. Appl. Phys.87, 6382 ~2000!. 12T. L. Gilbert, Phys. Rev. 100,1 2 4 3 ~1955!. 13B. Dieny, V. S. Speriosu, S. S. P. Parkin, B.A. Gurney, D. R. Wilhoit, and D. Mauri, Phys. Rev. B 43, 1297 ~1991!. 14C. Kittel, Introduction to Solid States Physics , 4th ed. ~Wiley, New York, 1971!. 15Here, we neglect the tilt of Htotwith respect to HSwhich is feasible for the small value of HPat the static ﬁelds given but, however, leads to a slight underestimation of fFMRfor the pulse. FIG. 3. Suppressed ringing in a 1 mm34mm SV by variation of the pulse length: ~a!transmitted voltage pulses. Pulse lengths are 230, 470, and 760 ps from top to bottom. The graphs are offset for clarity. ~b!Corresponding SV magneto-resistance response to the pulses in ~a!. dc current 1 mA, static ﬁeld 91 Oe. For the 470 ps pulse ~middle curve !ringing after pulse termi- nation is suppressed ~shown by the arrow !.3783 Appl. Phys. Lett., Vol. 80, No. 20, 20 May 2002 Schumacher et al. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.55.98 On: Mon, 01 Dec 2014 19:04:46
5.0031193.pdf	J. Chem. Phys. 153, 234704 (2020); https://doi.org/10.1063/5.0031193 153, 234704 © 2020 Author(s).Surface-enhanced Raman spectroscopic investigation on adsorption kinetic of carbon monoxide at the solid–gas interface Cite as: J. Chem. Phys. 153, 234704 (2020); https://doi.org/10.1063/5.0031193 Submitted: 28 September 2020 . Accepted: 30 November 2020 . Published Online: 21 December 2020 Ming Ge , Minmin Xu , Yaxian Yuan , Qinghua Guo , Renao Gu , and Jianlin Yao COLLECTIONS Paper published as part of the special topic on Spectroscopy and Microscopy of Plasmonic Systems ARTICLES YOU MAY BE INTERESTED IN Probing the deformation of [12]cycloparaphenylene molecular nanohoops adsorbed on metal surfaces by tip-enhanced Raman spectroscopy The Journal of Chemical Physics 153, 244201 (2020); https://doi.org/10.1063/5.0033383 Quantifying the enhancement mechanisms of surface-enhanced Raman scattering using a Raman bond model The Journal of Chemical Physics 153, 224704 (2020); https://doi.org/10.1063/5.0031221 Machine learning with bond information for local structure optimizations in surface science The Journal of Chemical Physics 153, 234116 (2020); https://doi.org/10.1063/5.0033778The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Surface-enhanced Raman spectroscopic investigation on adsorption kinetic of carbon monoxide at the solid–gas interface Cite as: J. Chem. Phys. 153, 234704 (2020); doi: 10.1063/5.0031193 Submitted: 28 September 2020 •Accepted: 30 November 2020 • Published Online: 21 December 2020 Ming Ge,1,2Minmin Xu,1Yaxian Yuan,1,a)Qinghua Guo,1Renao Gu,1and Jianlin Yao1,a) AFFILIATIONS 1College of Chemistry, Chemical Engineering and Materials Science, Soochow University, Suzhou 215123, China 2College of Chemistry and Chemical Engineering, Nantong University, Nantong 226001, China Note: This paper is part of the JCP Special Topic on Spectroscopy and Microscopy of Plasmonic Systems. a)Authors to whom correspondence should be addressed: yuanyaxian@suda.edu.cn and jlyao@suda.edu.cn ABSTRACT A molecular-level understanding of CO adsorption behavior would be greatly beneficial to resolving the problem of CO poisoning in fuel cells and medical science. Herein, an efficient borrowing strategy based on surface enhanced Raman scattering (SERS) has been developed to investigate the adsorption behavior of CO at the gas–solid interface. A composite SERS substrate with high uniformity was fabricated by electrochemical deposition of optimal Pt over-layers onto an Au nanoparticle film. The results indicated that the linearly bonded mode follows the Langmuir adsorption curve (type I), while the multiply bonded did not. It took a longer time for the C–O Mvibration to reach the adsorption equilibrium than that of C–O L. The variation tendency toward the Pt–CO Lfrequency was in opposition to that of C–O L, caused by the chemical and dipole–dipole coupling effects. The increase in dynamic coupling effects of the CO molecules caused a blue shift inνCOand a red shift of the Pt–CO band, while its shielding effect on SERS intensity cannot be ignored. Additionally, higher pressure is more conducive for linear adsorption to achieve saturation. Density functional theory calculations were employed to explore the adsorption mechanisms. It should also be noted that the substrate with good recycling performance greatly expands its practical application value. The present study suggested that the SERS-based borrowing strategy shows sufficient even valuable capacity to investigate gas adsorption kinetics behavior. Published under license by AIP Publishing. https://doi.org/10.1063/5.0031193 .,s I. INTRODUCTION Detection and adsorption investigations of carbon monox- ide (CO) are of great importance since it is the most common adsorbate intermediate on transition metal surfaces. Its diverse applications include pollution controls for the automobile indus- try, Fischer–Tropsch synthesis, polymer electrolyte membrane fuel cells, and so on. Therefore, many techniques have been developed, mainly containing metal oxide semi-conductor sensors,1electro- chemical approaches,2,3gas chromatography,4laser infrared absorp- tion,5and colorimetric sensing.6Among these techniques, optical measurements exhibit a significant advantage. For example, Bly- holder and Allen investigated infrared spectra and a molecularorbital model for CO adsorbed on metals.7Kruppe et al. performed polarization-dependent reflection absorption infrared spectroscopy to study CO adsorption on the surface of a Pd/Cu (111) single-atom alloy, obtaining the molecular level informa- tion on the bonding properties.8However, there exists a signif- icant challenge in trace gas analysis owing to the low molecular density and small cross section. Therefore, a convenient detec- tion technology with high selectivity and sensitivity is still highly desired. Surface enhanced Raman scattering (SERS) has attracted much attention in diverse areas since it was discovered in the 1970s.9–12 The technique can uniquely identify molecules by their vibrational fingerprint and has limits of detection down to the single molecule J. Chem. Phys. 153, 234704 (2020); doi: 10.1063/5.0031193 153, 234704-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp level.13Now, SERS has become one of the most powerful techniques for exploring the structural configuration at the solid–liquid inter- face. However, there is still an essential desire for further investi- gation into gas-phase analyses. It has been reported that SERS was developed for vapor or gas sensing with Ag14,15or Au nanoparti- cles16–19at a lower limit of detection.20Recently, SERS based flow- through gas detection methods have been reported successfully. Chou et al. constructed a miniaturized flow-through system for the detection of vapor from model explosive compounds. It was believed that the detection of the vapor molecules on the surface of the gold coated silicon substrate was dependent on the net physicochemical force.21Khaing Oo et al. developed a novel 3D SERS platform using Au nanoparticle-modified multi-hole capillaries for a rapid and ultrasensitive vapor detection of 4-nitrophenol,22which exhibited more than a sixfold SERS enhancement in comparison with a similar 2D structure. Although some attempts have been made, challenges still exist on gas detection, especially with regard to absorbability and reproducibility. In addition to its promising practical application, CO has the advantage of being both the adsorbate and the SERS reporter, since it is well recognized that CO molecules strongly adsorb on surfaces of most Pt-based catalysts.23The chemical and physical properties of CO were demonstrated through well-documented intramolecular C–O ( νCO) and metal–CO ( νMCO) stretching vibrations. However, SERS enhancement factors of Pt-based surfaces are relatively low when compared with noble metals. Therefore, significant effort has been made to increase SERS sensitivity from the transition metals, for example, through the borrowing SERS activity strategy.24–26In this strategy, the long-range electromagnetic fields from a SERS- active material are employed to enhance the surface Raman signal of molecules adsorbed on the less SERS-active materials, such as Pt, Pd, and Rh.27Weaver and co-workers first used under-potential deposi- tion and a chemical redox replacement method to prepare a pinhole- free Pt-group film coated Au substrate for the detection of selected adsorbates.28–32The SERS signal is remarkably enhanced because Au nanoparticles with well-defined shape and size provide suffi- cient SERS activity toward the ultrathin Pt over-layers. Ren fabri- cated Au@transition-metal nanoparticles to extend the applications of composite nanoparticles by using CO as the probe molecule33and deduced that about two orders for the SERS enhancement were bor- rowed from the Au source. However, these above-mentioned inves- tigations mainly described the CO detection and adsorption modes at liquid–solid interfaces, focusing on the investigation into charge- transfer and electro-oxidation in the electric double layer. As for the solid–gas environments, no electrochemical reactions and the potential tuning cause the poor SERS enhancements, resulting in difficulties to obtain the interfacial configuration of CO by SERS. To the best of our knowledge, few reports have been made on the CO adsorption and oxidation at the gas–solid interface. For exam- ple, the SERS was explored to investigate the CO catalytic oxidation at the gas–solid interface.34Nanba et al. observed the SERS of CO gas adsorbed on an Ag surface at a very low temperature of about 120 K and demonstrated the coverage dependent CO stretching vibrational frequencies.35Renet al. reported the adsorption behav- ior of gas CO on a rough pure Pt surface in a three-phase Raman cell. A simple adsorption configuration was demonstrated for the saturation adsorption of CO.36Theoretical methods such as DFT- AER have also been proven to give the information of inducedsurface reconstruction of Pt during CO adsorption.37However, it is possible to miss some weak adsorption configuration by SERS on the pure Pt surface due to poor enhancement. Moreover, the adsorption configuration and the frequencies of CO are sensitive to the nature of the substrates, the coverage, and so on. Thus, a sub- strate with chemical and SERS effect uniformity is highly desired for the adsorption and kinetics studies on CO at the solid–gas interface. Herein, a novel SERS-based strategy for the adsorption kinet- ics studies of CO at the gas–solid interface has been developed. An Au monolayer nanoparticle film was fabricated at a gas–liquid inter- face with chemical and SERS effect uniformity and transferred to an Indium Tin Oxide (ITO) substrate. Controlled electrochemical deposition of Pt over-layers on the ITO/Au monolayer/substrate served as a SERS-active substrate. The gas flows through the ITO substrate covered with an Au/Pt film, resulting in CO molecules being captured. The intensity and frequency of the CO stretch vibra- tional band were very sensitive to the interaction between the sub- strate and gas molecules and used as the probe to resolve the sur- face configuration of CO. The SERS-based adsorption isotherm of CO at the gas–solid interface, as well as that at the liquid–solid interface, was determined to systematically clarify the adsorption kinetics behavior. Simple CO adsorption modeling approaches from different pressures are also described, which are suitable for the application of CO detection. In addition, density functional the- ory (DFT) findings based on nanoclusters of Pt 5are used as an assistance to explain the experimental adsorption kinetics behavior. Finally, it should be noted that the SERS-active substrates exhib- ited excellent performance for recycling, which allows regenera- tion and even the in situ continuous detection in a convenient way. II. EXPERIMENTAL SECTION A. Materials HAuCl 4⋅4H 2O, H 2PtCl 6⋅6H 2O, and sodium citrate were pur- chased from Shanghai Reagent Co. Ltd. ITO was purchased from CSG Holding Co. Ltd. Polyvinylpyrrolidone (M w= 10 000) was sup- plied by Sigma. Ultra-high purity (99.99%) CO gas was obtained from Shanghai WuGang Gas Co. Ltd. All chemicals used were of analytical reagent grade. Milli-Q water (18.2 MΩ cm) was used throughout the whole experiments. All the electrolyte solutions were bubbled by continuous N 2flow. B. Measurements The electrochemical measurements were performed on a CHI660B electrochemical workstation (Shanghai Chenhua) in a three-electrode system. The ITO substrate with an Au monolayer nanoparticle film acted as the working electrode, and platinum wire and a saturated calomel electrode (SCE) were used as the counter and reference electrodes, respectively. SERS measurements were car- ried out with a microprobe Raman system (HR800 from Horiba Jobin Yvon, France) using a He–Ne laser (632.8 nm). The slit and pinhole were 100 μm and 400 μm, respectively. A portable digi- tal barometer was purchased from AZ Instrument Co. Ltd. with a pressure range of 0 psi–15 psi. J. Chem. Phys. 153, 234704 (2020); doi: 10.1063/5.0031193 153, 234704-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp SCHEME 1 . Schematic diagram of a lab-made chamber for the gas detection. C. Fabrication of highly uniform ITO/Au/Pt substrate The SERS-active Au monolayer nanoparticle film was pre- pared through our previous work.38The detailed preparation pro- cess is described in the supplementary material. SEM and TEM images are presented in Fig. S1. The ITO/Au substrate was then electrochemically polished and reactivated in a 0.1M HCl solu- tion. Constant-current deposition of Pt over-layers was performed in a 1 mM H 2PtCl 6+ 0.1M H 2SO 4solution with a current of 0.1 mA for 15 s, forming thin and SERS-active Pt over-layers. The potential time and cyclic voltammetry curves are shown in Figs. S2 and S3. D. CO adsorption The gas detection was conducted by using a lab-made gas- flow apparatus (Scheme 1). The SERS substrate was fixed at the bottom of the glass chamber with a magnetic sheet stuck in the middle while holding parallel to an optical glass viewport mounted on the detector channel. The gas flow was detected in situ by the digital barometer. The sample chamber could be evacuated to 10−3Torr by a mechanical pump before SERS measurements. The CO gas was delivered to the chamber via a flow control system. An adjustable confocal Raman objective was used to monitor the adsorption behavior. The different pressure is read by the digital barometer. Each spectrum was collected with 60 s under a laser power of 2 mW. All the experiments were performed at room temperature. E. Theoretical calculations All DFT calculations were performed with the Gaussian 09 program package.39Optimization of the complexes and vibrational frequency calculation for the optimized structures were carried out by DFT with the Becke 3-parameter hybrid functional combined with the Lee–Yang–Parr exchange–correlation functional (B3LYP). The 6-31 + G(d) basis set was used for CO, while the LANL2DZ basis set, as well as effective core potentials, was used for Pt atoms. III. RESULTS AND DISCUSSION A. Optimization of borrowing strategy It was well known that the poor SERS activity of a pure Pt surface (about two to four orders of magnitude) results in diffi- culties to detect the surface Raman signal of the adsorbent, partic- ularly for the gas–solid interface. Thus, the borrowing strategy is explored to dramatically enhance the Raman signal of moleculesadsorbed on the Pt surface through the long-distance effect of the SERS effect from the inner Au nanoparticles. For this strategy, two critical factors should be considered, involving the probe adsorp- tion site on the Pt surface rather than the inner Au nanoparticles and the appropriate thickness of the Pt over-layer for the reasonable SERS effect from the Au nanoparticles. Fortunately, the frequen- cies of the probe, such as CO, are sensitive to the substrate, i.e., the significant different frequencies of Pt and Au surfaces. Accord- ingly, it allows to contribute the surface Raman signal to the exact metal surfaces. As for the second factor, the thicker over-layer of Pt resulted in the damping of the SERS effect from the underneath Au nanoparticles, and it is quite difficult to detect the surface Raman signal of the molecules adsorbed onto Pt over-layers. The smaller amount of Pt result in discontinuous Pt over-layers. It produced the pinhole effect and the complexity of elucidation of SERS spectra. Along this line, it is essential to tune the thickness of Pt over-layers to ensure the effective origination of the observed surface Raman signal from transition metal layers. Our previous studies demon- strated that the Au nanoparticle monolayer film exhibited excellent SERS performance, particularly for reproducibility and uniformity. It improved the capability of SERS to probe the molecules that are remarkably sensitive to the composite and structure of the sub- strate. Moreover, the SERS performance holds similar to that of the Au nanoparticle monolayer after the deposition of Pt over-layers. Figure 1(A) presents the SER spectra of CO adsorbed on an Au/Pt film with different electrochemical deposition durations. A broad band of the region from 1800 cm−1to 1900 cm−1was assigned to the CO intramolecular vibration of multiply bonded (C–O M). The high frequency band of ∼2040 cm−1was attributed to the CO intramolecular vibrational mode of linearly bonded (C–O L).40 However, when compared with the C–O Mband, a relatively strong band of C–O Lmeans more CO molecules preferred the linear form. The stretching mode frequency of CO adsorbed on Au (at ∼2120 cm−1) is distinctively different from that on Pt, Pd, and the similar transition metal surfaces (below 2100 cm−1). There- fore, the absence of νCO–Au indicates that the effect of pinholes is neglectable.33In this case, two bands in the νCOregion were away from the characteristic C–O vibrational mode at the Au sur- face (above 2100 cm−1), suggesting that the Pt over-layers are con- tinuous and pinhole free. The observed surface exactly originated from the Pt over-layer rather than from the inner Au nanoparti- cles. With increasing deposition duration time, the band frequency of C–O Lblueshifted steadily from 2030 cm−1to 2040 cm−1[as shown in Fig. 1(A)]. The following two effects should be pointed out: (i) The formation of small Pt particles with higher surface energy occurred after more over-layers were deposited. Li and his co-workers reported that Pt over-layers grew smoothly with an epi- taxial mode in several atomic layers.41The small Pt particles can lead to a higher CO frequency as a result of higher surface energy.42 (ii) Au is 4% large than Pt in lattice constant.43The blue shift can be attributed to the large strain of the lattice mismatch between Au and small Pt particles as well as the strain-induced repulsion between neighboring Pt particles. From Fig. 1(B), the band inten- sity of C–O Lreached the maximum as the deposition time was 15 s. With prolonging deposition time, CO adsorption was initially dependent on the amount of Pt in agreement with the SERS peak increasing, followed by a decrease in the SERS signal after attaching more Pt over-layers. It was mainly due to the dramatic damping of J. Chem. Phys. 153, 234704 (2020); doi: 10.1063/5.0031193 153, 234704-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . SERS spectra of CO adsorbed on Au/Pt films with different deposition times (A), SERS intensity–deposition time profile of band of CO L(B), and the relative intensity ofνCOL–Pt over-layer thickness film profile (C). the SERS effect. Interestingly, the abnormal intensity of the SERS signal was observed as the Pt over-layer deposited by 5 s (about 1 monolayer). It should be associated with the maximum enhance- ment and, thus, contributed to the strongest SERS signal. However, the actual coverage of Pt layers is rather low and results in a smaller amount of Pt (1–2 layers) depositing on the Au film. It results in the quite low coverage of CO; therefore, the number of CO molecules bound to it decreases significantly and it resulted in the SERS sig- nal damping. Moreover, the blue shift toward the frequencies of the CO Lvibrational mode provided the other fact to support the above assumption. The Pt thickness dependence of the SERS intensity of CO was crucial to optimize the borrowing strategy. The thickness was con- trolled by the deposition duration varying from 15 s to 250 s (15 s, 25 s, 50 s, 100 s, and 250 s), approximately associating with 2–46 Pt atomic layers. A detailed estimation of the number of Pt monolay- ers is presented in Fig. S4. As presented in Fig. 1(C), the normalized integrated intensity of C–O Lexponentially decreased with increas- ing Pt over-layer thickness. The strongest signal was about sixfold higher than the case of ∼46 monolayers. It was in good agreement with the mechanism of electromagnetic enhancement (EM), i.e., exponential relationship between the distance and the SERS effect. It was also noted that the bandwidth of CO for 15 s was narrower as against that of other deposition times. This result further indicated that the substrate with a deposition time of 15 s had a more uni- form surface in comparison with other conditions. Therefore, this deposition duration was believed to be the optimal condition for CO detection. All the following experiments were performed at the Au/Pt surface with a Pt deposition duration of 15 s, i.e., about 2.8 Pt over-layers. B. Adsorption kinetics of CO at solid–gas interface The adsorption kinetics is one of the most important issues for predicting and reorganizing the nature of molecular interaction at the solid–gas interface. Figures 2(A) and 2(B) present the adsorption time dependent SER spectra of CO in the high and low frequency regions, respectively. According to the previous electrochemical in situ SERS studies on Au@Pt nanoparticles,44,45the band located at∼480 cm−1is attributed to the metal–adsorbate stretching vibration of linearly bonded CO (Pt–CO L) and the lower band of ∼380 cm−1is assigned to the hollow site adsorption of multiply bonded CO (Pt– CO M). We also detected obvious C–O Land C–O Mbands shortly after the introduction of CO gas. As seen in Figs. 2(A) and 2(B), the intensities and frequencies of the C–O and Pt–CO bands appeared to change with the increasing adsorption time. However, not only the tendency of diverse bands but also their intensities and fre- quencies were quite different. From Fig. 2(A), the intensities of C–O Land C–O Mpeaks gradually increased during the initial period, and they would eventually reach a minor fluctuation. For the clear description, the time dependent SERS intensities of different bands were included in Fig. 3. The intensities of both C–O Land C–O M FIG. 2 . SER spectra of CO adsorption (A) in the high frequency region and (B) in the low frequency region with different adsorption times: (a) no CO adsorption, (b) 1 min, (c) 8 min, (d) 16 min, (e) 24 min, (f) 32 min, (g) 40 min, (h) 48 min, (i) 56 min, and (j) 64 min. J. Chem. Phys. 153, 234704 (2020); doi: 10.1063/5.0031193 153, 234704-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Adsorption curves of CO adsorption for various vibrational bonds. vibrations reached the maximum at ∼20 min and ∼45 min, respec- tively, achieving the balance of dynamic adsorption–desorption. It revealed that a longer duration to reach the adsorption/desorption equilibrium was required for the C–O Mvibrational band, result- ing from the fact that C–O Ltended to directly adsorb on the sur- face, while C–O Mmolecules might adsorb on or in the adjacent Pt atoms of both the surface and the interior. However, the adsorption isotherm of linearly bonded CO Lseemed to follow the Langmuir isotherm models as a result of its configuration and orientation (top- adsorption).46,47It is of special interest that multiply bonded CO seemed consistent with isotherm (IV) with a H4 hysteresis loop.48,49 Hence, it turned out that C–O Mmolecules of the multi-adsorption mode may adsorb into the pores of the layered structure. It should be noted that a slight decrease in SERS signals was observed after adsorption equilibrium. It can be explained by the fact that after reaching the absorption equilibrium, some molecular CO was des- orbed from the substrate in the gas phase and that the Raman signals were screened to a certain extent by strong dipole–dipole effects. Additionally, as indicated in Fig. 2(A), a somewhat significant Raman blue shift of C–O Land C–O Mfrom 2025 cm−1to 2031 cm−1 and 1851 cm−1to 1859 cm−1, respectively, were found in the high frequency region. However, the increase in dynamic dipole–dipole coupling effects of the adsorbed CO molecules can cause a blue shift of the C–O band frequencies, revealing the strong interaction between CO molecules.40,41 In the low frequency region, the appearance of Pt–CO L (∼475 cm−1) and Pt–CO M(∼380 cm−1) bands after CO introduc- tion in Fig. 2(B) confirmed the binding of CO to the Pt surface. The intensities of both metal–CO bands increased with the experimental duration at the initial stage. However, after ∼30 min adsorption, the intensity of Pt–CO Lexhibited a steady fluctuation, while the inten- sity of Pt–CO Mwas still increasing (Fig. 3). Assumedly, it was due to the desorption of top-adsorption CO molecules while CO molecules permeated through the tiny holes of the surface. One can find thatthe band of Pt–CO Lwas well correlated with the C–O Lstretch- ing; nevertheless, the tendency of C–O Land C–O Mband intensi- ties was quite different. With regard to the frequency of metal–CO, the Pt–CO band gradually redshifted with the increasing adsorp- tion time, i.e., from 464 cm−1to 454 cm−1for Pt–CO Land from 380 cm−1to 375 cm−1for Pt–CO M. More correctly, weakening of the Pt–CO bond was due to the chemical effects and the superimpo- sition as a result of the enhanced CO–CO repulsion. Nevertheless, the dipole–dipole coupling effects of Pt–CO is nearly two orders smaller than those of C–O.50However, with the increase in surface coverage, a clear change was observed in the Pt–CO Lband inten- sity together with a red shift of its band frequency, while a com- paratively small change was observed for that of Pt–CO M, suggest- ing that the interaction force of Pt–CO Mwas stronger than that of Pt–CO L. However, as shown in Fig. 4, the changing tendency of the Pt–CO Lfrequency was opposite compared to the frequency of C–O L. It has been claimed that the frequency change of Pt–CO L was mainly caused by chemical effects, while both chemical effects and dipole–dipole coupling effects were believed to be responsible for the frequency change of C–O L.45Thus, it is reasonable to con- clude that strong dipole–dipole coupling effects produced a strong force between neighboring CO molecules, while the electron den- sity of the π∗orbital of C–O bond is decreased due to the declining d-π∗backdonation effect between Pt and CO,40thereby confirm- ing that the strength between Pt and CO Lbecame weaker, forcing CO Lto desorb from the Pt surface. It should be noted that the con- figuration description is limited to C–O Land Pt–CO Lstretching vibrations because the relative broad and weak Pt–CO Mand C–O M bands resulted in difficulties in the illustration. C. Comparison of CO adsorption at gas–solid and liquid–solid interfaces In contrast to CO adsorption at the gas–solid ( g–s) interfaces, the adsorption behavior at the liquid–solid ( l–s) interfaces to sim- ulate the electrochemical environment was also explored by in situ SERS. Figure 5 shows the time dependent SER spectra of CO at the same surface in the CO saturated 0.5M H 2SO 4solution together FIG. 4 . Contrast diagram of the band frequency for Pt–CO Land C–O L. J. Chem. Phys. 153, 234704 (2020); doi: 10.1063/5.0031193 153, 234704-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Time dependent SER spectra (A) of CO at the same surface in the CO sat- urated 0.5M H 2SO4solution: (a) 1 min, (b) 8 min, (c) 16 min, (d) 24 min, (e) 32 min, (f) 40 min, (g) 48 min, and (h) 56 min. The time–band intensity– frequency profiles (B) at the l–s interfaces. with the time–band intensity–frequency profiles at the l–sinter- faces. As seen in Fig. 5(A), the SER spectra of C–O Land C–O M were captured even though their intensities were fairly low, rela- tive to that of the gas–solid interface. Despite the poor signal-to- noise ratio, we could still observe the blue shift phenomenon of the C–O Lband, i.e., from 2031 cm−1to 2037 cm−1. From Fig. 5(B), with increasing immersion time to the CO-saturated solution, the band intensity of C–O Lin the l–sinterface increased steadily up to a maximum at ∼40 min, while at the same time, the intensity has achieved a balance at the g–sinterface. It should be noted that it takes longer time to reach adsorption saturation than for the g–s interface, since the existence of solvent molecules hinder the inter- action between adsorbed CO and the Pt surface. Furthermore, the higher gas velocity in the gas phase than that of the liquid phase was becoming another factor. Concerning the frequency of CO L, the tendency toward the frequency was similar to that of the intensity before maximizing the coverage. However, after reaching adsorp- tion saturation, the tendency toward the frequency and intensity was quite different. As mentioned above, the strong dipole–dipole cou- pling effect has a more shielding impact on Raman intensity, which decreases the intensity and blue shifts the frequency. In addition, a blue shift of ∼5 cm−1–10 cm−1for the C–O Lfrequency in water solution to that in the gas phase was observed. Factors that induce the change of the C–O Lfrequency are originated from the inter- action of hydrogen with the CO molecules36and a slight compres- sion of CO molecules in the presence of co-adsorbed water hydro- gen, which may shift a small fraction of the CO molecules located at near top to the exact top position.37,51With the exception of these two reasons, we hold the opinion that as an electron acceptor, hydrion adsorbs on the Pt surface, which makes the Pt surface pos- sess some positive charge and therefore induces the blue shifts of CO frequencies. D. Pressure effect on adsorption behavior Figure 6 presents the C–O Lband–pressure profile of CO at 1000 mbar, 750 mbar, 500 mbar, and 100 mbar. The time dependent SER spectra of CO adsorbed on Au/Pt surfaces is shown in Fig. S5. It was found that the time for achieving saturated adsorption decreasedwith increasing pressure. Band intensities fluctuated after reaching the maximum coverage due to the adsorption/desorption equilib- rium. The adsorption curves were similar to the Langmuir adsorp- tion isotherm.52The intensity of four spectra exhibits a slightly downward tendency. It is believed that the increase in repulsion among adsorbed CO molecules at high CO surface coverage results in the decrease in the intensity, which is in good agreement with previous literature.53There may also be a few molecules with rel- atively weak force falling off the surface “hotspots.” Neverthe- less, it was reported that there would be a slight red shift in the C–O Lstretching frequency when lowering the CO surface coverage, which can be attributed to the decrease in the dipole–dipole cou- pling interaction in addition to an increase in d-π∗backdonation.40 Although it is difficult to accurately figure out the change of cover- age associated with the spectroscopic properties of CO, we attempt to investigate the relationship through DFT calculation in Sec. III E. Unfortunately, it was also still difficult to quantify the contribution from dipole–dipole coupling effects to the coverage dependent band intensity. FIG. 6 . The C–O Lband–pressure profile of CO at different pressures. J. Chem. Phys. 153, 234704 (2020); doi: 10.1063/5.0031193 153, 234704-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . Models of Pt 5(4,1) clusters and CO adsorbed at Pt for various coverages. E. Comparison of experimental results with DFT calculations The two-layer nanocluster Pt 5(4-1) (representing four Pt atoms in the first layer and one Pt atom in the second layer) was con- structed and CO can be adsorbed atop Pt atoms at various cover- ages (Fig. 7). In order to simulate the Pt surface structure, geometry parameters for Pt clusters are fixed in bulk platinum geometry with a Pt–Pt distance of 2.775 Å. The C–O bond length of CO molecule is calculated as 1.144 Å. The binding energy of CO individually is calculated from −ECO/surface =−Etotal CO/surface +Esurface +Efree CO, (1) where Etotal CO/surface andEsurface represent the total energy of the surface with adsorbed CO and the energy of bare clusters, respectively, and Efree COis the energy of the free gas-phase CO molecule. The CO binding energies on the top site at various coverages were calculated and are reported in Table I. It should be noted that CO adsorption is a spontaneous exothermic reaction on the Pt (100) surface. The binding energies increase with the coverage, and there is a good linear relationship between the binding energies and the cov- erage rate. The larger the binding energy, the more stable it tends to be. It also can be seen that the bond length of Pt–C increases with the coverage, while the bond length of C–O gradually decreases. FIG. 8 . The frequency–adsorption time profiles (A) of the νPt–CO andνCObands from experiment (inset) and the frequency–coverage profiles (B) of calculated results. These results verified the previous experiments, and the increase of CO adsorption led to a decrease in the bonding force between Pt and C atoms and a stronger binding effect between C and O atoms. Figure 8 shows the contrast curves from experimental (A) and calculated (B) results of Pt–CO and C–O vibration frequencies. The spectra of C–O blue shifts more significantly at lower coverage; how- ever, it becomes inconspicuous at higher coverage. Obviously, the variation tendency toward the Pt–C frequency is opposite to the for- mer. When these calculated results are compared with the available experimental results, one can find that the CO adsorption at the top site on the Pt surface has generally reached saturation around 20 min, the coverage is ∼50%–60% at this time, consistent with the full saturation coverage on poly-crystalline Pt electrodes.55The present study suggested that the SERS technique with appropriate attractive metal over-layers provided a significant and possibly even valuable approach to explore the absorptive behavior and kinetics at gas–solid interfaces. TABLE I . Calculated bond length ( d) and vibrational frequencies ( υ) at different coverages. Coverage −ECO/surface (eV) d(Pt–C)ad(C–O) υ(C–O)bυ(Pt–C) υ(C–O) FSFcυ(Pt–C) FSF 0.25 MLa44.48 1.8221 1.1606 2064.8 524.8 1990 505.7 0.5 MLb89.07 1.8415 1.1545 2090.7 513.38 2015 494.7 0.75 MLc133.55 1.8470 1.1541 2100.6 515.79 2024 497.0 1 ML 177.45 1.8496 1.1526 2092.6 510.61 2016 492.0 adis the bond length in angstroms (Å) and υis the vibrational frequency (cm−1). bυ(C–O) is determined with the usual DFT method. cFSF denotes the frequency scaling factors, and υ(C–O) FSFis calculated by fitting the FSFs. The FSF of 6-31 + G(d) is 0.9636.54 J. Chem. Phys. 153, 234704 (2020); doi: 10.1063/5.0031193 153, 234704-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 9 . SERS-based recycling perfor- mance investigation: (a) original, (b) first cycle, (c) second cycle, (d) third cycle, (e) fourth cycle, (f) fifth cycle, and intensity–cycle time profiles (inset). In addition, the obtained calculation can help explain why the C–O frequency blue shifts slightly and its intensity diminished over 40 min. We assumed that some CO molecules have been desorbed from the Pt surface, resulting in the weakened Raman signals. In the meantime, the desorption will lead to an increase of molecular CO in the gas phase. Theoretical calculation shows that the frequency of the isolated CO molecule is at ∼2122 cm−1(after correction with the FSFs) and higher than the combined state. Thus, we believed that there exists an increasing contribution of molecular CO to the Raman scattering. F. Regeneration of substrate It was found that the composite substrate is quite reproducible after electrochemical cleaning. The modified SERS substrate with CO adsorption was handled with an anodic CO oxidation procedure in 0.5M H 2SO 4(Fig. S6). Then, the constant-voltage experiment was performed in 0.5M H 2SO 4at a potential of 1.0 V vs SCE, cor- responding to the more positive oxidation potential for CO. The running time was 10 s to wipe off adsorbed CO by oxidization. Finally, the electrochemically treated substrate is reused for the next CO gas detection. Figure 9 compares the SERS spectra after several regenerations. A total of five cycles (labeled as first, second, third, fourth, fifth, respectively, and original is the initial signal) have been carried on by SERS investigation. It exhibits a slight decrease with the recycling number increasing. It can be estimated that ∼15% of the original substrate signal decreased after five cycles. The results revealed that the detection substrate has an acceptable recycling per- formance. The most intriguing phenomenon observed in the present case was the slight upward signal after first regeneration (Fig. 9). Pos- sible reasons that may cause such a spectral change are as follows: (i) the residual reagents were removed and active adsorption sites of CO increased after electrochemical processes and (ii) the “hotspots” increase due to the formation of appropriate spacing between adja- cent nanoparticles. However, after the regeneration processes were further repeated, the SERS signals declined steadily, since the Auor Pt nanoparticles detached from the substrate by regeneration for several times. After several electrochemical cleaning, it was found that the compactness of the Au film obviously decreased compared with the initial substrate. It should be noted that gap distances among nanoparticles increased accordingly after the electrochemical process due to a few Au nanoparticles detaching from the substrate (see Fig. S7). Consequently, the increase in the gap distance resulted in the damping of the “hotspot” effect, causing the decrease in the SERS intensity. IV. CONCLUSIONS A borrowing strategy has been developed to explore the adsorp- tion behavior of CO at the gas/solid interface. A composite SERS substrate was fabricated by the electrochemical deposition of Pt over-layers onto an Au nanoparticle monolayer film surface. The Au nanoparticle monolayer film assembled at the gas/liquid inter- face and transferred to the ITO electrode exhibited high unifor- mity for the SERS measurements and enhanced the surface Raman signal of the molecules adsorbed onto the Pt over-layers through the long-distance enhancement SERS effect. With the aid of a lab- made gas detection device, the preparation conditions of the sub- strate for CO detection and modulating the transition metal cov- ering layer in order to achieve the optimum enhancement and adsorption capacity were optimized accordingly. The CO adsorp- tion kinetics was deeply studied on the basis of Pt–CO and C–O band analysis due to its sensitivity to the surface configuration. More specifically, the linearly bonded stretching mode follows the Langmuir adsorption curve (type I), while the multiply bonded did not. It took a longer time for the C–O Mvibration to reach the adsorption equilibrium than that of C–O Ldue to the number of tiny pores in the layered structure. Dipole–dipole effects played an important role in CO adsorption, screening the Raman signal to some extent. The variation tendency toward the Pt–CO Lfre- quency was in opposition to that of C–O L. The frequencies of C–O L blue shifted due to the charge transfer and dipole–dipole effects, J. Chem. Phys. 153, 234704 (2020); doi: 10.1063/5.0031193 153, 234704-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp while the latter was not responsible for the Pt–CO Lvibrational mode. In addition, there was a blue shift of ∼5 cm−1–10 cm−1for the C–O Lband in water solution by contrast with the gas phase, owing to the adsorption of hydrion, interaction of hydrogen of water with the CO molecules, and a compression of CO molecules in the existence of co-adsorbed H (from water) as a result of the adsorption configuration changes for CO molecules. DFT calcula- tions were employed to explain the adsorption mechanisms. Bind- ing energy, structures, and vibrational frequencies for the Pt sur- face are studied by considering different adsorption coverages and comparing them with the experimental data. These results clearly demonstrated that the SERS technique shows sufficient sensitivity to investigate the CO adsorption kinetics behavior for understand- ing the nature and the CO adsorption mechanism. This SERS-based method will have an important role in kinetics studies. We empha- size that the detection substrate with good recycling performance greatly expands its practical application value. Further investigations are underway to obtain a deeper understanding of SERS-based gas detection. SUPPLEMENTARY MATERIAL The supplementary material contains one table and one figure. ACKNOWLEDGMENTS This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 21773166 and 21673152), the Natural Science Fundamental Research Project of Jiangsu Col- leges and Universities (Grant No. 18KJA150009), the Priority Aca- demic Program Development of Jiangsu (PAPD), and the Project of Scientific and Technologic Infrastructure of Suzhou (Grant No. SZS201708). DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1G. F. Fine, L. M. Cavanagh, A. Afonja, and R. 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1.2899955.pdf	Magnetic coupling of pinned, asymmetric Co Pt Ru Co Fe trilayers Chengtao Yu, Bryan Javorek, Michael J. Pechan, and S. Maat Citation: Journal of Applied Physics 103, 063914 (2008); doi: 10.1063/1.2899955 View online: http://dx.doi.org/10.1063/1.2899955 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/103/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ferromagnetic resonance studies of surface and bulk spin-wave modes in a Co Fe Pt Mn Co Fe multilayer film J. Appl. Phys. 103, 07B525 (2008); 10.1063/1.2839337 Microstructure and magnetic properties of Co Cr Pt – Si O 2 perpendicular recording media with synthetic nucleation layers J. Appl. Phys. 103, 07F512 (2008); 10.1063/1.2831499 Pulsed inductive measurement of ultrafast magnetization dynamics in interlayer exchange coupled Ni Fe Ru Ni Fe films J. Appl. Phys. 101, 09C101 (2007); 10.1063/1.2693852 Spin valves with synthetic exchange bias Co 70 Ni 10 Pt 20 Ru CoFe J. Appl. Phys. 99, 08R501 (2006); 10.1063/1.2162811 Antiferromagnetically coupled Co Fe B Ru Co Fe B trilayers Appl. Phys. Lett. 85, 2020 (2004); 10.1063/1.1792375 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.35.41 On: Thu, 21 Aug 2014 10:25:48Magnetic coupling of pinned, asymmetric CoPt/Ru/CoFe trilayers Chengtao Yu,1,a/H20850Bryan Javorek,1Michael J. Pechan,1and S. Maat2 1Miami University, Oxford, Ohio 45056, USA 2San Jose Research Center, Hitachi Global Storage Technologies, 3403 Yerba Buena Rd., San Jose, California 95135, USA /H20849Received 6 August 2007; accepted 24 January 2008; published online 26 March 2008 /H20850 Magnetic exchange coupling in pinned, asymmetric CoPt 18/H2084950/H20850/Ru/H20849x/H20850/CoFe 16/H2084938/H20850trilayers with 0/H33355x/H3335525 Å has been investigated with magnetometry and ferromagnetic resonance. We found the parameters associated with coupling /H20849remanence, coerctivity, and resonance position /H20850to be oscillatory as a function of Ru thickness with extrema at x=7 Å /H20851antiparallel /H20849AP/H20850/H20852,1 4 Å /H20851parallel /H20849P/H20850/H20852, and 20 Å /H20849AP/H20850, consistent with observations for Ru spacer material in unpinned, more symmetric systems. Utilizing analysis methods unique to pinned systems with resonance arisingfrom the soft layer only, we were able to extract coupling strengths of 0.55, −0.29, and 0.27 erg /cm 2 at Ru thicknesses of 7, 14, and 20 Å, respectively. Noteworthy in the analysis method is the ability to extract P coupling strength of both signs from magnetization data. The resonance linewidthcorrelates with coupling, where minimum relaxation rates occur at low coupling strengths. Variabletemperature magnetization loops revealed that the exchange coupling monotonically increases withdecreasing temperatures. © 2008 American Institute of Physics ./H20851DOI: 10.1063/1.2899955 /H20852 INTRODUCTION The current perpendicular to the plane /H20849CPP /H20850giant mag- netoresistance /H20849GMR /H20850effect in magnetic multilayers pres- ently is a matter of great interest. This interest is founded inthe low resistance-area /H20849RA/H20850products combined with high GMR ratios observed in CPP spin valves and multilayers 1–3 making it possible to build low resistance nanometer size GMR devices for future high data rate magnetic recordingapplications. Moreover, the spin-torque effect, 4–7and the use of spin-polarized current to switch the magnetization of a magnetic layer in the CPP geometry, is useful to realizenovel magnetoelectronic devices such as spin-torquememory and is also a ﬁeld of new and interesting physics. A generalized model to describe CPP-GMR transport in metallic multilayers was proposed by Fert and co-workers. 8,9 It accounts for the different scattering rates of majority and minority electrons resulting in different spin diffusionlengths. Generally, antiferromagnetic /H20849AF/H20850pinning layers such as PtMn or IrMn do not contribute to the magnetoresis-tance signal, rather they constitute a parasitic resistance thatsigniﬁcantly decreases the overall signal since its magnitudecan be similar or greater than the total resistance of the activepart of the spin-valve structure. Typical resistivities are193 /H9262/H9024cm as deposited and 227 /H9262/H9024cm after annealing for 4 h at 255 °C for PtMn and 150 /H9262/H9024cm as deposited and 162/H9262/H9024cm after annealing for 4 h at 255 °C for IrMn. PtMn needs to be about 150 Å thick in order to becomeantiferromagnetically ordered upon annealing to induce ex-change in the pinned layer; IrMn needs to be about 80 Å toobtain optimum exchange bias. This translates to serial RAproduct resistance values of 34 /H9024 /H9262m2for a PtMn pinned spin valve and 12 m /H9024/H9262m2for a IrMn pinned spin valve due to the antiferromagnet only /H20849excluding possible underlayers /H20850.Recently, it was proposed to utilize CoPt x/H2084916/H33355x /H3335524 at. % /H20850thin layers grown onto Cr underlayers as pin- ning material in CPP spin valves as they exhibit low resis- tivity /H20849/H1101130/H9262/H9024cm/H20850, and about 40 Å thin layers can exhibit high remanence and coercivities of /H110221.5 kOe.10These prop- erties help us to minimize serial resistance and magnetic sta-bility and thus enhance the magnetoresistance. Another im-portant feature is that CoPt pinning layers exhibit strongantiparallel /H20849AP/H20850coupling to a CoFe reference layer through Ru spacer layers, much as has been observed in more sym-metric, unpinned systems. 11This is necessary to minimize magnetostatic coupling to the free layer and to keep the freelayer magnetically soft. Hysteresis loops have been exten-sively used to indicate the strength of AP coupling, but donot as readily yield information about parallel /H20849P/H20850coupling. On the other hand, ferromagnetic resonance /H20849FMR /H20850has proven to be an effective tool in determining both P and APcouplings in systems with resonances arising from both mag-netic layers. 12Here, we demonstrate methods to obtain both AP and P coupling strengths from each of these techniques inpinned, asymmetric CoPt 18/H2084950/H20850/Ru/H20849x/H20850/CoFe 16/H2084936/H20850trilayers /H208490/H33355x/H3335525 Å /H20850with resonance arising from CoFe only. EXPERIMENT A series of Ta /H2084950/H20850/Cr/H2084950/H20850/CoPt 18/H2084950/H20850/ Ru/H20849t/H20850/CoFe 16/H2084938/H20850/Ta/H2084950/H20850ﬁlms was grown by dc magnetron sputtering at room temperature on glass substrates, where the CoPt 18alloy was formed by codeposition from separate Co and Pt targets and the Ru thickness was varied from zero to25 Å. The thicknesses of the CoPt 18and CoFe 16layers were chosen to be 50 and 38 Å, respectively, to achieve equalmagnetization in the two magnetic layers, similar to what isneeded in an antiferromagnetically coupled pinned layerstructure of a spin valve. For comparison, three referencesamples consisting of single magnetic layers, Ta /H2084950/H20850/ a/H20850Electronic mail: yuc@muohio.eduJOURNAL OF APPLIED PHYSICS 103, 063914 /H208492008 /H20850 0021-8979/2008/103 /H208496/H20850/063914/5/$23.00 © 2008 American Institute of Physics 103 , 063914-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.35.41 On: Thu, 21 Aug 2014 10:25:48Cr/H2084950/H20850/CoPt 18/H2084950/H20850/Ta/H2084950/H20850,T a /H2084950/H20850/Cr/H2084950/H20850/CoFe 18/H2084938/H20850/ Ta/H2084950/H20850, and Ta /H2084950/H20850/Cr/H2084950/H20850/Ru/H2084925/H20850/CoFe 16/H2084938/H20850/Ta/H2084950/H20850, were prepared under the same experimental condition. The magnetization loops were measured by vibrating samplemagnetometry /H20849VSM /H20850. FMR spectra were taken at 35 GHz, with the applied ﬁeld P to the ﬁlm plane /H20849in-plane geometry /H20850. RESULTS AND DISCUSSION Figure 1shows the magnetization loops for some repre- sentative Ta /H2084950/H20850/Cr/H2084950/H20850/CoPt 18/H2084950/H20850/Ru/H20849t/H20850/CoFe 16/H2084938/H20850/ Ta/H2084950/H20850samples with different Ru thicknesses. The sample with a Ru thickness of 3 Å exhibits a simple square hyster- esis loop with high remanence, low saturation ﬁled, and rela-tively low coercive ﬁeld /H20849/H11011500 Oe /H20850, characteristic of F cou- pling. When the Ru thickness is increased to 7 Å, the magnetization loop exhibits AF coupling with almost zeroremanence, high saturation ﬁeld, and large coercive ﬁeld/H20849/H110112.8 kOe /H20850. For AF coupled samples, we refer to the coer- cive ﬁeld as the centroid ﬁeld of the hard layer reversal, as indicated in Fig. 1. When the magnetic ﬁeld is decreased after saturating the sample in a high positive magnetic ﬁeld,a broad reversal from 300 to 500 Oe is observed. Since themoments of the hard CoPt and soft CoFe layers are similar, itis difﬁcult to distinguish between the two layers in magneti-zation loops. However, simulations assuming coupling of thesoft CoFe to randomly oriented uniaxial CoPt grains indicatethat the initial reversal is associated with the CoFe layer. Atzero ﬁeld, the CoPt and CoFe layers are AP. As the ﬁeld isdecreased further, a sharper transition from −2500 to−3000 Oe is observed, which is attributed to the harder CoPtlayer. Note that this reversal ﬁeld is much larger than theintrinsic coercivity /H208491136 Oe /H20850of the single CoPt reference ﬁlm, reﬂecting the inﬂuence of the AF coupling to CoFe in retarding its switching. For a Ru thickness of 15 Å, the system exhibits F cou- pling, resulting in a square magnetization loop with rela-tively low saturation and coercive ﬁeld but high remanence.At further increased of Ru thickness of 20 Å, the magnetiza-tion loop exhibits the typical antiferromagnetically coupledloop features, similar to the loop for 7 Å of Ru, but with asmaller saturation and coercive ﬁeld. For a Ru thickness of25 Å, it is not straightforward to determine if the system isantiferromagnetically coupled or decoupled. The loop dis-plays almost full remanence, indicative of parallel alignmentof the CoPt and CoFe layers at zero ﬁeld. The reversal of thesoft layer starts once a small negative ﬁeld is applied, and thereversal of the hard layer follows at a ﬁeld around−1000 to −1500 Oe, which is similar to the reversal ﬁeldrange of the single CoPt ﬁlm. This behavior points to a com-pletely decoupled or weakly ferromagnetically coupledsample. In such a case, the loop is a superposition of the hardand soft layer loops. Another possible scenario is that the AFcoupling is present in the system, but the coupling strength istoo small, and it is not sufﬁcient to overcome the intrinsiccoercivity of the soft layer to reverse its magnetization atzero ﬁeld. In this case, it may show a full remanence thoughthe system is antiferromagnetically coupled. We will revisitthis in the discussion of the FMR data.Figure 2shows the remanence and the coercive ﬁeld as a function of Ru thicknesses. As expected, the remanenceshows oscillatory behavior with Ru thicknesses, and the AF FIG. 1. Hysteresis loops for sample Ta /H2084950/H20850/Cr/H2084950/H20850/ CoPt18/H2084950/H20850/Ru/H20849t/H20850/CoFe16 /H2084938/H20850/Ta/H2084950/H20850with different Ru thicknesses. Note that the coercive ﬁeld for antiferromagnetically coupled sample /H20849Ru thick- ness of 7 Å /H20850is marked by the reversal ﬁeld of the hard CoPt18layer.063914-2 Yu et al. J. Appl. Phys. 103 , 063914 /H208492008 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.35.41 On: Thu, 21 Aug 2014 10:25:48coupling peaks at 7 and 20 Å with a period of about 13 Å. Oscillatory behavior is also observed in the coercive ﬁeldassociated with the CoPt reversal, as well as saturation ﬁeldsand minor loop shifts /H20849not shown /H20850. The oscillatory behavior of these parameters reﬂects the interlayer exchange coupling,however, none alone is a direct measure of the couplingstrength. FMR is known for its ability to directly measure internal ﬁelds in a sample and it is employed herein to access thecoupling ﬁeld. All FMR measurements were taken at a ﬁxedfrequency of 35 GHz while varying an in-plane ﬁeld. Thesamples exhibit no signiﬁcant in-plane anisotropy, so the in-plane ﬁeld is not applied along a particular direction. Figure 3displays the FMR spectra for some representative samples in the series with Ru=7, 15, 20, and 25 Å. Also shown is thespectrum for the Ta /H2084950/H20850/Cr/H2084950/H20850/Ru/H2084925/H20850/CoFe 16/H2084938/H20850/Ta/H2084950/H20850 reference sample, for which the resonance occurs at 6235 Oe. A variation in resonance ﬁeld with Ru thickness isobserved, illustrating the inﬂuence of coupling contributionsto the internal ﬁeld. The dependence of the resonance ﬁeldand linewidth with Ru thickness are shown in Fig. 4/H20849a/H20850. While a clear oscillatory behavior of the resonance ﬁeld isobserved, the linewidth peaks at the Ru thicknesses exhibitminimum resonance ﬁelds. It should be noted that the oneand only resonance observed in this CoPt 18/Ru /CoFe 16mag- netic system arises from the CoFe layer. No resonance isobserved from the CoPt 18, presumably owing to its granular nature and its hard magnetic properties. Although a reso-nance is not observed from the CoPt, that layer does inﬂu-ence the resonance of the CoFe through local coupling ﬁelds.At Ru thicknesses below 5 Å, the FMR signal is very weakand broad, probably due to the existence of strong F couplingthrough pinholes in the Ru layer, in which case the entire trilayer takes on the resonance properties of a single CoPthard layer. A measure of the coupling strength can be obtained from both the magnetization and the FMR data. First, we focus onthe magnetization data. In the single CoPt 18ﬁlm, the coer- cive ﬁeld is determined by the intrinsic properties of theCoPt 18alone, whereas in the CoPt 18/Ru /CoFe 16trilayers, the AF /H20849F/H20850coupling ﬁeld opposes /H20849assists /H20850the applied ﬁeld in FIG. 3. FMR spectra of representative samples /H2084935 GHz /H20850. The amplitude of CoFe single layer sample has been reduced by a factor of 2.5 to show in acomparable scale. FIG. 4. /H20849a/H20850FMR ﬁeld /H20849solid circles /H20850and linewidth /H20849open circles /H20850as a func- tion of Ru thickness; /H20849b/H20850interlayer coupling strength expressed in ﬁeld /H20849left/H20850 and energy density /H20849right /H20850as determined from both resonance ﬁeld /H20849solid squares /H20850and coercive ﬁeld /H20849open squares /H20850./H20849Note: J/H110220 implies AF coupling. /H20850 FIG. 2. Magnetization remanence /H20849solid dots /H20850and CoPt coercive ﬁeld /H20849open circles /H20850as a function of Ru thicknesses.063914-3 Yu et al. J. Appl. Phys. 103 , 063914 /H208492008 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.35.41 On: Thu, 21 Aug 2014 10:25:48switching the CoPt 18. Therefore, a direct measure of the cou- pling ﬁeld /H20849assuming no variation in intrinsic CoPt 18coerciv- ity between samples /H20850is given by Hcoupling =Hc,CoPt,trilayer −Hc,CoPt,monolayer , /H208491/H20850 whose variation with Ru thickness is shown /H20849open squares /H20850 in Fig. 4/H20849b/H20850. It should be noted that this approach /H208491/H20850assumes negligible variation in intrinsic CoPt 18coercivity between samples and /H208492/H20850averages over any variation of the coupling across a sample that might pin domain walls, thus alteringthe rotation mechanism. Before discussing the details of thisvariation, we discuss determination of coupling ﬁelds fromFMR data. Based upon the fact that in the CoPt 18/Ru /CoFe 16 trilayer samples only the CoFe 16layer is contributing to the FMR signal observed, only the areal energy density of thesoft layer is considered and can be written as E=−dMH cos /H9278sin/H9258−2d/H9266M2sin2/H9258−Jexcos/H9278sin/H9258, /H208492/H20850 where the ﬁrst term is the Zeeman energy, the second term is the out-of-plane shape anisotropy energy, and the last is theareal exchange coupling energy term expressed as a unidi-rectional anisotropy energy. Here, Mis the saturation mag- netization of the soft layer, dis the CoFe layer thickness, H is the externally applied magnetic ﬁeld, and /H9278is the angle between the CoFe magnetization MandH. Since the hard layer does not resonate at 35 GHz, its magnetization isaligned with the external ﬁeld direction, /H9258is the angle be- tween Mand the ﬁlm normal /H20849in our case, /H9266/2/H20850andJexis the exchange constant expressed in areal energy density. Bysolving the Landau–Lifschitz equation, the in-plane reso-nance equation in magnetic saturation /H20849MandHaligned with the CoPt magnetization, /H9278=0/H20850can be obtained as follows: /H20873/H9275 /H9253/H208742 =/H20873H+Jex Md/H20874/H20873H+Jex Md+4/H9266M/H20874, /H208493/H20850 where /H9275is the frequency and /H9253the gyromagnetic ratio. For the single CoFe layer sample, the exchange term is zero;therefore, one observes the effect of exchange coupling inthis system as a shift in the resonance position by the amountH ex/H20849=Jex/Md/H20850.13,14The resulting exchange coupling ﬁeld and areal energy density are plotted as a function of Ru thickness in Fig. 4/H20849b/H20850, where one observes good agreement in exchange values obtained from FMR and coercivity. Smalldiscrepancies between results obtained from the two methodsare most likely due to slight variations in intrinsic CoPt co-ercivity between the reference sample and the trilayersamples. As expected, only in the region between 11 and16 Å is a negative coupling ﬁeld or F coupling present. Onthe other hand, AF coupling exists below 11 Å and above16 Å. Noteworthy here is the direct observance of the signand strength of the exchange coupling term. Moreover, agradual approach to zero coupling at larger Ru thickness isobserved, where the more sensitive and accurate FMR tech-nique /H20849owing to its ability to directly measure internal ﬁelds and the magnetically soft CoFe /H20850indicates zero to slight F coupling at Ru thickness of 25 Å. The exchange couplingstrength for Ru=7 Å, which is adapted in the real spin-valve structure, 10is around 0.58 erg /cm2in areal energy density. This coupling strength is comparable to the values found inPtMn exchange biased structures. 15,16 The FMR linewidth reﬂects the dynamics of the magne- tization with narrower resonances indicating longer relax-ation times. In comparing Figs. 4/H20849a/H20850and4/H20849b/H20850, one observes linewidth maxima coincident with peaks in F and AF cou-pling strengths, whereas linewidth minima coincide with Ruthicknesses /H2084911 and 16 Å /H20850for which the system is switching coupling sign—i.e., low coupling strengths. This is can beunderstood in terms of coupling acting as a damper on theprecession of the CoFe layer. Strong coupling of the CoFe tothe magnetically hard /H20849and, as mentioned earlier, nonresonat- ing/H20850CoPt layer causes the CoFe precession to quickly relax FIG. 5. Magnetization loops at different temperatures for sample with Ru thicknesses of /H20849a/H208507 Å and /H20849b/H2085020 Å. /H20849c/H20850Temperature dependence of the corrected coercive ﬁeld, which is an indicator of the coupling strength.063914-4 Yu et al. J. Appl. Phys. 103 , 063914 /H208492008 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.35.41 On: Thu, 21 Aug 2014 10:25:48to the equilibrium orientation resulting in a large FMR line- width. The extreme limit of this occurs in the very thin Ruspacer samples where pinholes likely produce very strongcoupling to the CoPt, broadening the FMR signal to the pointof where it is no longer discernable. By differentiating theresonance equation, 17,18estimates of relaxation rates /H20849/H9004f/H20850 can be obtained from the observed resonance linewidth /H20849/H9004H/H20850 as follows: /H9004f=/H9004H/H9253 /H9266/H208811+/H20873M/H9253 f/H208742 . /H208494/H20850 From this, one deduces relaxation rates of approximately 2.1 GHz /H20849/H9004Hres=300 G /H20850for weakly coupled trilayers to 5.6 GHz /H20849/H9004Hres=800 G /H20850for those most strongly coupled. The corresponding relaxation times range from 0.48 to 0.18 ns. Since the FMR linewidth correlates stronglywith the coupling strength, one led to model the Ru thicknessdependent component in terms of the Landau–Lifshitz orGilbert damping, 19 /H9004H=2/H9251/H9275 /H9253. /H208495/H20850 From the linewidth in excess of the intrinsic CoFe linewidth /H20849approximately 300 G /H20850and the measurement frequency /H2084935 GHz /H20850, one obtains coupling induced damping parameters ranging from 0 to 0.14 in the system investigated. /H20849Note that in the absence of variable frequency measurements, one can-not separate inhomogeneous broadening and the Gilbertdamping contributions to the linewidth of the intrinsic CoFeresonance. /H20850 The temperature dependence of the interlayer coupling has also been investigated. The magnetization loops for theAP and P coupled samples have been measured with variabletemperature VSM. Shown in Figs. 5/H20849a/H20850and5/H20849b/H20850are a series of magnetization loops for the sample with 7 and 20 Å thickRu, respectively, corresponding to the peaks of AF coupling.The coercive ﬁelds increase as the temperature is decreased. The temperature dependence of the coercivity of a single CoPt layer has also been measured and was used accordingto Eq. /H208491/H20850to extract the temperature dependence of the cou- pling strength. The corrected coercivity as a function of tem-peratures for some typical samples is shown in Fig. 5/H20849d/H20850. Coupling strengths for the ferromagnetically coupled sample/H20849t Ru=15 Å /H20850, antiferromagnetically coupled samples /H20849tRu=7 and 20 Å /H20850, and weakly coupled sample /H20849tRu=25 Å /H20850exhibitweak, monotonic temperature dependence. The weak, mono- tonic temperature dependence is the result of two mecha-nisms. One is an intrinsic temperature dependence arisingfrom the change of the Ru Fermi surface with temperatureand the other is a more extrinsic effect driven by the disor-dering of the ferromagnet moments /H20849CoPt and CoFe /H20850with increasing temperature. 11 CONCLUSIONS Magnetic coupling in CoPt /Ru /CoFe trilayers has found to be oscillatory with peaks at x=7 Å /H20849AP/H2085014 Å /H20849P/H20850, and 20 Å /H20849AP/H20850as demonstrated by both FMR and magnetization loops. By comparing the internal ﬁeld of the trilayers withsingle CoFe layer, the coupling strength was extracted to be0.55 /H20849AP/H20850, −0.29 /H20849P/H20850, and 0.27 /H20849AP/H20850erg /cm 2at the coupling peak positions, respectively. The magnetic damping corre-lates with coupling, with minimum relaxation rates occuringat low coupling strengths. The magnitude of the exchangecoupling monotonically increases, but weakly with decreas-ing temperature. 1W. P. Pratt, S. F. Lee, P. Holody, Q. Yang, R. Loloee, J. Bass, and P. A. Schroeder, J. Magn. Magn. Mater. 126, 406 /H208491993 /H20850. 2W. P. Pratt, S. F. Lee, P. Holody, Q. Yang, R. Loloee, J. Bass, and P. A. Schroeder, Phys. Rev. B 51, 3226 /H208491995 /H20850. 3S. D. Steenwyk, S. Y . Hsu, R. Loloee, J. Bass, and W. P. Pratt, J. Magn. Magn. Mater. 170,L 1 /H208491997 /H20850. 4J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 5J. C. Slonczewski, J. Magn. Magn. Mater. 195, L261 /H208491999 /H20850. 6J. A. Katine, F. J. Albert, and R. A. Buhrman, Appl. Phys. Lett. 76,3 5 4 /H208492000 /H20850. 7F. J. Albert, J. A. Katine, R. A. Buhrman, and D. C. Ralph, Appl. Phys. Lett. 77, 3809 /H208492000 /H20850. 8T. Valet and A. Fert, Phys. Rev. B 48, 7099 /H208491993 /H20850. 9A. Fert and S. Lee, Phys. Rev. B 53, 6554 /H208491996 /H20850. 10S. Maat, J. Checkelsky, M. J. Carey, J. A. Katine, and J. R. Childress, J. Appl. Phys. 98, 113907 /H208492005 /H20850. 11S. S. Parkin, Phys. Rev. Lett. 67, 3598 /H208491991 /H20850. 12Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Phys. Rev. B 50,6 0 9 4 /H208491994 /H20850. 13J. A. C. Bland and B. Heinrich, Ultrathin Magnetic Structures II /H20849Springer-Verlag, Berlin, 1994 /H20850. 14Y . Wu, K. Li, J. Qiu, Z. Guo, and G. Han, Appl. Phys. Lett. 80, 4413 /H208492002 /H20850. 15M. Saito, N. Hasegawa, F. Koike, H. Seki, and T. Kuriyama, J. Appl. Phys. 85, 4928 /H208491999 /H20850. 16Y . Sugita, Y . Kawawake, M. Satomi, and H. Sakakima, J. Appl. Phys. 89, 6919 /H208492001 /H20850. 17V . Kambersky and C. Patton, Phys. Rev. B 11, 2668 /H208491975 /H20850. 18S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. Schneider, P. Kabos, T. J. Silva, and J. P. Nibarger, J. Appl. Phys. 99, 093909 /H208492006 /H20850. 19R. Soohoo, Magnetic Thin Films /H20849Harper, New York, 1965 /H20850.063914-5 Yu et al. J. Appl. Phys. 103 , 063914 /H208492008 /H20850 [This article is copyrighted as indicated in the article. 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1.4938390.pdf	Traveling surface spin-wave resonance spectroscopy using surface acoustic waves P. G. Gowtham , T. Moriyama , D. C. Ralph , and R. A. Buhrman Citation: J. Appl. Phys. 118, 233910 (2015); doi: 10.1063/1.4938390 View online: http://dx.doi.org/10.1063/1.4938390 View Table of Contents: http://aip.scitation.org/toc/jap/118/23 Published by the American Institute of Physics Traveling surface spin-wave resonance spectroscopy using surface acoustic waves P. G . Gowtham,1T.Moriyama,2D. C. Ralph,1,3and R. A. Buhrman1 1Cornell University, Ithaca, New York 14853, USA 2Institute for Chemical Research, Kyoto University, Kyoto, Japan 3Kavli Institute at Cornell, Ithaca, New York 14853, USA (Received 14 October 2015; accepted 9 December 2015; published online 21 December 2015) Coherent gigahertz-frequency surface acoustic wa ves (SAWs) traveling on the surface of a piezoelec- tric crystal can, via the magnetoelas tic interaction, resona ntly excite traveling s urface spin waves in an adjacent thin-ﬁlm ferromagnet. These excited su rface spin waves, traveling with a deﬁnite in-plane wave-vector qkenforced by the SAW, can be detected by measuring changes in the electro-acoustical transmission of a SAW delay line. Here, we provi de a demonstration that such measurements consti- tute a precise and quantitative technique for spin-w ave spectroscopy, providing a means to determine both isotropic and anisotropic contributions to th e spin-wave dispersion and damping. We demonstrate the effectiveness of this spectroscopic techniqu e by measuring the spin-wave properties of a Ni thin ﬁlm for a large range of wave vectors, jqkj¼2.5/C2104–8/C2104cm/C01, over which anisotropic dipolar interactions vary from being ne gligible to quite signiﬁcant. VC2015 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4938390 ] I. INTRODUCTION Spin waves in magnetic materials can transport spin in- formation with a high degree of efﬁciency over distances that far exceed the limitations of spin diffusion in metals.1–3 Consequently, control over the generation and propagation of spin waves in micron-scale and nano-scale structures is of interest for next-generation spin-based technologies. Considerable thought and effort has been brought to bear on exploiting systems with novel anisotropic spin-spin interac- tions (e.g., Dzyaloshinskii-Moriya interactions)4–10and on engineering new structures (e.g., magnonic crystals) for use in tailoring the propagation characteristics of spin waves.11–18These systems can possess highly anisotropic spin-wave dispersion and damping—both of which can be used to advantage for guiding and manipulating spin waves in magnetic heterostructures. Developing a spectroscopic technique that is capable of quantitatively measuring such anisotropies (and on length scales that are technologically relevant) is therefore imperative. Recent experiments19–22have shown that surface acous- tic wave (SAW) delay line devices can (via the magnetoelas- tic interaction) be used to launch and detect spin waves in magnetic thin ﬁlms that are coupled to piezoelectric sub- strates. Here we extend these initial results to show that the SAW-based excitation of traveling surface spin waves pro- vides a sensitive spectroscopic technique for making quanti- tative measurements of anisotropic contributions to the spin-wave dispersion and damping. Unlike spin-wave mea- surement techniques which possess no wave-vector selectiv- ity (e.g., anomalous Nernst effect,23and spin-pumping24–26/ inverse spin Hall effect detection schemes1,27–29), a SAW can excite a single traveling surface spin wave mode with a deﬁnite in-plane wave vector qkthat is matched to the wave vector chosen for the SAW. While other acoustical techni- ques (e.g., bulk opto-acoustical techniques) for spin-wavespectroscopy can be used to spectrally select and measure single spin-wave modes, these techniques suffer from thedifﬁculty that quantitative analysis of the spin-wave ampli- tude line-shapes and spin-wave resonance frequencies can be very challenging 30—a difﬁculty that we will show SAW- based traveling surface spin-wave spectroscopy does notshare. The SAW delay line measurement scheme differsfrom other electrical spin-wave spectroscopy techniques (e.g., microstrip delay lines 31,32) in that the SAW imposes an effective pump ﬁeld modulated at wave-vector qkthroughout the entire magnetic ﬁlm that drives the traveling spin waveand therefore provides a fairly direct measure of thedynamic, wave-vector dependent spin-wave susceptibility— whereas microstrip delay line techniques provide a measure of the local spin-wave excitation/non-local propagator (i.e.,Green’s function) for a magnetic ﬁlm. The low velocity of aRayleigh SAW implies that at the GHz frequency scalescharacteristic of spin-wave resonance, the wave-vector q kof the effective pump ﬁeld and traveling spin-wave can span characteristic scales ( qk/C24104–106cm/C01) over which various isotropic and anisotropic spin-spin interactions start tobecome important in determining the propagation of thespin-wave. While operable only at discrete values of jq kjand requiring both spin-wave excitation and detection, unlike wave-vector resolved optical spin-wave measurementschemes such as Brillouin light scattering (BLS), 12,33–36 SAW-driven spin wave spectroscopy can, in principle, be operated at wave-vectors (e.g., jqkj>3/C2105cm/C01), cur- rently inaccessible to BLS, that extend deep into the dipole- exchange regime—as will be discussed more thoroughly inSection V. SAW-based spin-wave spectroscopy allows for quantita- tive studies of the traveling surface spin-wave susceptibility, dispersion, and damping as a function of varying the anglebetween the orientation of the magnetization m 0and ﬁxed 0021-8979/2015/118(23)/233910/9/$30.00 VC2015 AIP Publishing LLC 118, 233910-1JOURNAL OF APPLIED PHYSICS 118, 233910 (2015) qk. Such analysis as a function of jqkjand angle provides a simple means to directly determine anisotropic and wave- vector-dependent contributions (e.g., from anisotropic spin- spin interactions) to the spin-wave dispersion and damping.To demonstrate these capabilities, we perform SAW-drivenspin-wave resonance measurements as a function of appliedﬁeld and in-plane ﬁeld angle for a d¼10 nm thick Ni thin ﬁlm microstrip on a piezoelectric YZ-cut LiNbO 3substrate. We examine a range of larger wave vector, jqkj/C242:5/C2104 to 8/C2104cm/C01, than has been studied previously by SAW experiments. This range is chosen to span from the regionwhere anisotropic dipolar interactions should be negligible to the region where they contribute signiﬁcantly to the spin- wave dispersion. By implementing a quantitative analysis ofabsorption measurements for SAW delay lines, we demon-strate that it is possible to achieve a comprehensive, quantita- tive determination of the wave-vector and angular structure of this dipolar interaction. The same experimental techniqueshould also be more generally applicable to measure otherinteractions that modify spin-wave propagation, e.g., in mag-nonic crystals and magnetic heterostructures designed to have strong Dzyaloshinskii-Moriya interactions. II. ANALYTICAL CALCULATION OF SAW POWER ABSORBED BY EXCITATION OF A SPIN WAVE The spin-wave dynamics of an ultrathin magnetic ﬁlm driven coherently by a SAW traveling with an in-plane wavevector q k, and the resultant SAW power absorption, can be derived within the framework of the Landau-Lifshitz-Gilbert (LLG) equation37 dmrðÞ dt¼/C0cmrðÞ/C2HeffrðÞþCqk;m0 ðÞ mrðÞ/C2dmrðÞ dt; (1) where mðrÞis the local magnetization, HeffðrÞis the local effective ﬁeld exerting a torque on mðrÞ,a n d m0is the equi- librium magnetization in the absence of the RF pump ﬁeld. We allow for the spin-wave damping term Cðqk;m0Þto depend on qkandm0rather than assuming it is a constant, as is sometimes done, but we do employ the assumptionCðq k;m0Þ/C281 so that the damping term can be treated as a perturbation. Eq. (1)is analyzed using both a fgnand a xyz coordinate system (Figure 1). The fgncoordinate system is deﬁned such that qkof the SAW lies along the þgdirection, thefaxis corresponds to the magnetic easy axis of our thin Ni microstrip (which is in-plane), and the naxis lies normal to the ﬁlm plane. The xyzcoordinate system, which is deﬁned such that m0lies along the zaxis, is convenient for deriving the linearized LLG spin-wave dynamics about equilibrium. The various components of the effective ﬁeld relevant to a continuous Ni microstrip are Heff¼HappþHkþH? anisþhRFr;tðÞ þ2Aex Msr2mr;tðÞþhdr;tðÞ: (2) Happis the external ﬁeld applied in the plane of the ﬁlm. #H is deﬁned as the angle Happmakes with respect to the gaxis(see Figure 1). The internal contributions to Heffare the in- plane anisotropy ﬁeld Hk¼Hkmf^f, a perpendicular anisot- ropy ﬁeld H? anis¼2K? Msmn^nthat partially counteracts the out- of-plane demagnetization ﬁeld, the magnetoelastic pump ﬁeld hRFðr;tÞat wave vector qkgenerated by the SAW via the magnetoelastic interaction, the exchange ﬁeld2Aex Msr2m ðr;tÞwhere Aexis the exchange stiffness, and ﬁnally the dipolar ﬁeld hdðr;tÞthat encompasses both the out-of-plane demagnetization energy as well as a ﬂuctuating, spatiallyvarying component generated by the temporal and spatial variation of the magnetization in the spin wave. This non- local dipolar interaction serves to couple various parts ofthe spin-wave together and thus affects the propagation ofthe spin-wave. The non-local ﬂuctuating part of the dipolar interaction has not been included in previous studies analyz- ing SAW-driven spin-wave resonance in magnetic thinﬁlms. 21,22 The in-plane equilibrium magnetic orientation m0is completely determined by HappandHk. We transform Eqs. (1)and(2)into the xyzcoordinate system and linearize the LLG equation about m0. The magnetoelastic pump ﬁeld hRFðr;tÞ¼hqk RFeiðqk/C1r/C0xtÞarising from the traveling SAW derives from the relation hRFðr;tÞ¼ /C01 MsrmfME, where fME is the magnetoelastic part of the magnetic free energy density fME¼/C0Bef feggeiðq/C1r/C0xtÞsin2hcos2uþBshearegneiðq/C1r/C0xtÞ /C2coshsinhsinuþBnnenneiðq/C1r/C0xtÞcos2h: (3) The angle udenotes the in-plane azimuthal angle that m makes with the faxis, his the magnetization polar angle deﬁned with respect to the naxis, and Bef f,Bshear, and Bnn are, respectively, the effective magnetoelastic couplings to the in-plane longitudinal strain amplitude egg, shear strain amplitude egn, and strain amplitude perpendicular to the Ni ﬁlm plane associated with the traveling SAW. Since m0in our measurements is in-plane, we ignore the component of FIG. 1. Illustration of a traveling SAW with wave vector qkgenerating a time-dependent traveling strain wave ﬁeld in a Ni ﬁlm and driving a spin- wave resonance. This ﬁgure deﬁnes the fgncoordinate system, the equilib- rium magnetization m0and angle u0, the xyzcoordinate system used for lin- earizing the LLG equations about equilibrium, and the in-plane appliedmagnetic ﬁeld and ﬁeld angle # H.233910-2 Gowtham et al. J. Appl. Phys. 118, 233910 (2015) fMEthat is sensitive to enn. We also ignore the shear strain egn as the Ni ﬁlm under study is a d¼10 nm thick ﬁlm and is in the regime where d/C28kSAWfor all the SAW bandpasses employed in this experiment. In this thin-ﬁlm regime, thezero-shear-strain boundary condition at the substrate surface (which the SAW must satisfy) implies that the shear strain e gnthrough the ﬁlm is small. Then hqk RFcan be expressed in thexyzcoordinate system as hqk RF¼þ2Bef f Mseggsinu0ðÞcosu0ðÞ^x; (4) where u0is the angle that m0makes with the f(easy) axis, andMsis the saturation magnetization. We solve for the traveling spin-wave amplitude dmqk (averaged over the ﬁlm thickness) in terms of hqk RFby self consistently solving both the LLG equations linearized aboutm 0for the dynamic magnetization proﬁle across the Ni ﬁlm thickness (Eqs. (5)and(6)) as well as the magnetostatic equations (Eqs. (7)and(8)) for the nonlocal dipolar ﬁelds hdðr;tÞthat depend on the instantaneous magnetization pro- ﬁle of the traveling spin-wave /C0ixdmqk xyðÞ¼/C0c/C18 Hkcos2u0ðÞþHappsinu0þ#H ðÞ þ2Aex Ms/C20 jqkj2/C0@2 @y2/C21/C19 dmqk yyðÞþchy dyðÞ þchqk RF;yþixCqk;m0 ðÞ dmqk yyðÞ; (5) /C0ixdmqk yyðÞ¼þc/C18 Hkcos 2 u0ðÞ þHappsinu0þ#H ðÞ þ2Aex Ms/C20 jqkj2/C0@2 @y2/C21/C19 dmqk xyðÞ /C0chx dyðÞ/C0chqk RF;x/C0ixCqk;m0 ðÞ dmqk xyðÞ; (6) r/C1hd¼/C04pr/C1m; (7) r/C2 hd¼0: (8) This enables us to express hdin terms of magnetostatic potential Uashd¼/C0 rUand then solve for the potential U anddmqkðyÞ. The thickness dependence of the ﬂuctuating component dmqkðyÞof the magnetization is non-zero and arises from considering the evanescence of the traveling SAW into the Ni ﬁlm (with decay length of order kSAW)a s well as boundary conditions on the magnetostatic potential U associated with hdðr;tÞat the surfaces of the magnetic ﬁlm. The calculation is adapted from Stamps and Hillebrands38 and further details of the solution can be found there. The relationship between the thickness averaged spin- wave amplitude and the pump ﬁeld is expressed as dmqk ¼/C22v/C1hqk RF, where /C22v¼/C22v0þi/C22v00is the susceptibility tensor. We have restricted ourselves to the condition jqkjd/C281, where dis the ﬁlm thickness, appropriate for the wave- vector range jqkj¼2.5/C2104–8/C2104cm/C01andd¼10 nm thin ﬁlm microstrips employed in this study. The imaginary part of the susceptibility governing the out-of-phase responseof the magnetization to the relevant component of the mag- netoelastic pump ﬁeld is then /C22v00½/C138xx¼xp cCqk;m0 ðÞ !2þxp c/C18/C192 ! xres c/C18/C192 /C0xp c/C18/C192 !2 þxpCqk;m0 ðÞ Wþ!ðÞ c !2; (9) where xp¼cSAWjqkjis the ﬁxed SAW pump frequency, cSAWis the Rayleigh SAW sound speed, and xres¼cﬃﬃﬃﬃﬃﬃﬃﬃ W!p can be identiﬁed as the traveling surface dipole-exchange spin-wave resonance frequency, and the quantities Wand! are W¼Hkcos 2 u0ðÞ þHappsinu0þ#H ðÞ þ2Aex Msjqkj2 þ2pMsjqkjdcos2u0; !¼Hkcos2u0þHappsinu0þ#H ðÞ þ2Aex Msjqkj2 þ4pMs/C02K? Ms/C18/C19 /C04pMsjqkjd 2/C18/C19 : (10) The SAW power absorbed due to the excitation of a traveling surface dipole-exchange spin wave at wave vector qk, using the relation Pabs¼xp 2hqk RF†/C1/C22v00/C1hqk RF,21,39is Pabs¼xp 2/C22v00½/C138xx2Bef f Msegg/C18/C192 sin2u0cos2u0: (11) The angular structure of the absorbed SAW power derives from a combination of the magnetoelastic RF ﬁeld contribu- tionjhqk RFj2¼ð2Bef f MseggÞ2sin2u0cos2u0and the susceptibility ½/C22v00/C138xx. The pump ﬁeld itself depends on the angle that m0 makes with respect to the gaxis (the longitudinal strain axis) and possesses four-fold symmetry in u0with maxima when u0¼ð2nþ1Þp=4 for n2Z. This magnetoelastic pump component has the same form for any spin wave within an in-plane magnetized polycrystalline thin ﬁlm excited by a coherent Rayleigh SAW. It is the ½/C22v00/C138xxcomponent of Pabs that carries both the information about the internal magnetic anisotropy energies present in a speciﬁc system and also, more importantly, the angular and wave-vector dependence of the excited spin wave. An inspection of Eqs. (9)–(11) shows that delay line measurements of Pabsas a function of u0andjqkjcan be used to determine both the isotropic and anisotropic parts of the spin-spin interactions embedded within ½/C22v00/C138xx, given an independent determination of Ms,Hk, andK?. For our continuous Ni ﬁlm, the isotropic component of the spin-wave corrections comes from (1) the exchange interaction and (2) the term 4 pMsðjqkjd=2Þin!(Eq. (10)) whose origin is the dipolar interaction. The anisotropic com- ponent in our Ni ﬁlms is expected to arise solely from the dipolar interaction and is completely encoded in the term 2pMsjqkjdcos2u0within W. We show in the remainder of the paper that SAW based spectroscopy can sensitively and233910-3 Gowtham et al. J. Appl. Phys. 118, 233910 (2015) coherently map out the u0andjqkjdependence arising from the dipolar interaction even in the low MsNi system at mod- erate wave-vectors. III. EXPERIMENT We performed SAW power absorption measurements on Al(10 nm)/AlO x(2 nm)/Ni(10 nm)/Pt(15 nm) microstrips deﬁned in the middle of a SAW delay line. The Al/AlO xunderlayer and Pt overlayer were deposited to enable a separate inverse spin Hall effect (ISHE) quantiﬁcation of the SAW-induced spin-wave excitation (not discussed in this paper). The ﬁlm stack was magnetron sputter-deposited on 0.5-mm-thicksingle-side polished YZ-cut LiNbO 3substrates and patterned via lift-off process into 100 lm/C2500lm microstrips with the long axis orthogonal to the LiNbO 3Z-axis SAW propa- gation direction. The crystal Z-axis thus corresponds to the g axis and the long axis of the wire corresponds to the faxis. The base pressure for our chamber was P0<4/C210/C09Torr and the working Ar gas pressure was kept at 2 mTorr throughout the deposition process. The samples were depos- ited with no applied magnetic ﬁeld. The AlO xlayer was formed by native oxidation in air. Al(100 nm) was depositedand patterned via lift-off for the delay line metallization. Our SAW delay line was designed to have a center frequency f 0¼300 MHz and we use higher order bandpasses at fpump ¼1.48, 2.67, 3.26, 3.86, and 4.45 GHz to excite magnetization dynamics in the Ni. Details about our SAW delay line are pro- vided in Figure 2. The SAW resonant absorption was determined as a function of Happand#Hby measuring the quantity jSloss 21 ðHapp;#HÞj ¼ j S21ðHapp;#HÞj /C0 j S21ðHapp¼3 kOe ;#HÞjat selected bandpasses of the SAW delay line using a vector network analyzer (VNA). At the frequencies employed in our experiment, when Happ¼3 kOe the magnetic system is far from any spin-wave resonance condition. In such case, the loss is determined by changes in the transmission line im-pedance due to mass-loading and capacitive coupling to the magnetic metallic ﬁlm. Figure 3shows a density plot of jS loss 21ðHapp;#HÞjat the various bandpasses of our SAW delay line. The VNA transmission measurements were carried out using an input RF power of þ10 dBm. The ðHapp;#HÞ dependence of the jSloss 21ðHapp;#HÞjas a function of fpumpare consistent with the fact that the SAW is driving a magneticresonance. Calculated density plots of P absðHapp;#HÞas a function of the various fpumpemployed in the experiment are shown in the bottom row of Figure 3. In performing these calculations, we used an in-plane anisotropy Hk/C24380610 Oe with easy axis along the faxis as measured by Anisot ropic Magnetoresistance (AMR) measurements, Ms¼485 emu/cm3as measured by SQUID magnetometry, and K?/C241.13/C2106ergs/cm3as measured by out-of-plane SQUID scans for inputs into PabsðHapp;#HÞof Eq. (11). SQUID and AMR transport char- acterization of the Ni ﬁlms are shown in Figure 4. Both the large Hkand substantial K?in our 10-nm thick Ni ﬁlm are likely due to the magnetoelastic interaction and high aniso- tropic strains arising from depositing the ﬁlm stack on theLiNbO 3substrate. We assumed a value Aex¼8/C210/C07erg/cmfor the exchange stiffness of Ni,40a speed cSAW¼3:488 /C2105cm/s for SAW propagation along the Z-axis of YZ-LiNbO 3,41andd¼10 nm for the Ni ﬁlm thickness. The only remaining undetermined quantity in the formula for PabsðHapp;#HÞ(Eq.(11)) is the spin-wave damping Cðqk;m0Þ. The measured log scale jSloss 21ðHapp;#HÞjwas converted to lin- ear scale and normalized between /C01a n d0 . We ﬁnd good quantitative agreement between this nor- malized linear jSnorm 21ðHapp;#HÞjdata and normalized PabsðHapp;#HÞover the full wave-vector regime studied with a single value for the damping of the SAW excited spinwave, C¼0.14260.008, that is independent of both jq kj and angle. The precision with which we can quantify the spin-wave damping and its angular and wave-vector depend-ence via the SAW power absorption measurements is shown in Figure 5. 42Our extracted value for Cis of the same order as the spin-wave damping values ( C/C240.1) estimated from simulations in previous SAW work on Ni ﬁlms.21These val- ues of the spin-wave damping are signiﬁcantly higher than the typical values for the Gilbert damping in Ni ( a0/C240.048) as measured by uniform mode resonance. We estimate that the additional damping contribution arising from spin-pumping from our 10 nm Ni ﬁlm into the 15 nm Pt overlayer isa SP/C240.006. For this estimation, we have assumed that the real part of the mixing conductance is g"# r¼2/C21015cm/C02 for the Ni jPt interface and kPt s¼1:4 nm for the spin-FIG. 2. (a) Illustration of inter-digitated electrodes used for launching a SAW. The signal and ground ﬁnger pattern is repeated to produce the full IDT electrode used in our study with N¼40 total ﬁngers. We have used electrodes with pitch p¼k0=2¼5.8lm and metallization ratio m¼a=p ¼0.4, where k0is the fundamental band pass center wavelength and ais the IDT ﬁnger width. The wave vector of the SAW launched by the IDT is qSAW/C24qk.46(b) Time-gated S21transmission spectrum for our fundamental f0¼300 MHz SAW delay line where emitter and receiver electrodes are placed at a center-to-center distance of 700 lm. High-order bandpass center frequencies are visible at fpump¼1.48 GHz, 2.08 GHz, 2.67 GHz, 3.26 GHz, 3.86 GHz, and 4.45 GHz. The gate center time for the spectrum was set at /C240.2ls in order to maximize the single-transit signal, and we used a gate span of 0.05 ls. All of our SAW-driven resonance measurements used a time-gate with this set of speciﬁcations.233910-4 Gowtham et al. J. Appl. Phys. 118, 233910 (2015) diffusion length in Pt43and we neglect any enhancement of aSPbeyond the macrospin theory associated with the fact that the excitation is a surface spin-wave44(justiﬁable in the limit d/C28kSAW). The physical mechanisms responsible for this large spin-wave damping are as yet unclear. Wespeculate that small magnetostrictive deformations in the Ni ﬁlm generated by the precessing spin-wave can couple into various elastic modes (i.e., bulk longitudinal and transverse phonons) of the LiNbO 3substrate. This coupling could potentially lead to a large spin-wave damping such as wehave extracted from experiment. IV. DISCUSSION In this section, we show that SAW power absorption measurements can be used to perform traveling surface spin-wave spectroscopy and quantitatively measure the aniso-tropic dipolar spin-spin contributions to the spin-wave dis-persion—even at moderate wave-vectors in the low M sNi system where these contributions are not particularly strong.The structure of the anisotropic dipole interactions, formingthe leading order spin-wave contribution to the dispersion, is embedded within the resonant SAW absorption measure- ments. We ﬁrst show that this is the case by demonstratingthat the power absorption matches well to an analyticaltheory for the traveling surface spin-wave including the mag-netic dipolar interaction. This is ﬁrst done for two ﬁeld scansat different # Hwhere the dipolar interactions are expected to be weak and strong and it is shown that the full dipolartheory quantitatively captures the line-shapes of both scans, whereas the theory excluding dipolar interactions agree with the data only where the dipolar corrections to the spin-wavedispersion are expected to be weak. Then we show that theSAW power absorption can be mapped to and sampled overa large part of ðH app;u0Þspace and agrees quantitatively with the analytical theory including dipolar interactions.Given the quantitative agreement between the power FIG. 3. Simultaneous plots of the measured log scale jSloss 21ðHapp;#HÞjtransmission loss and the normalized Pabscalculation as a function of ﬁeld angle #Hand ﬁeld magnitude Happ. FIG. 4. (a) AMR curves for 0/C14(blue) and 90/C14(red) ﬁeld angle with respect to the SAW propagation direction show that the Ni has a substantial in-plane anisotropy ﬁeld Hkwith 90/C14(i.e., the faxis) being the easy-axis direction. (b) Out-of-plane ﬁeld scan of Al(10)/AlO x(2)/Ni(10)/Pt(15) bilayer on YZ- cut LiNbO 3.233910-5 Gowtham et al. J. Appl. Phys. 118, 233910 (2015) absorption data and the traveling surface dipolar spin-wave theory, we then directly extract the strength of the dipoleinteractions from the power absorption data at the differentq kinvestigated in this study. We ﬁrst plot in Figure 6normalized jSnorm 21ðHapp;#HÞj lineshapes together with the results of calculations forP absðHappÞin which both dipolar and exchange interactions are included or in which the dipolar interactions areneglected (i.e., exchange-only), at q k¼6.9/C2104cm/C01^g (fpump¼3.86 GHz) for two ﬁeld angles #H¼20/C14and #H¼60/C14. The two particular ﬁeld angles were chosen due to the very different range of u0sampled in the two ﬁeld scans. Due to the large in-plane Hk,u0runs through a continuous range of values as Happis swept at some ﬁxed #H(except for sweeps on the easy faxis or along directions very close to f where switching events can occur at low Happ). The #H¼20/C14 ﬁeld scan (i.e., a near hard axis sweep) is such that u0 quickly converges to a large value as Happis increased. For example, u0is/C2460/C14for the absorption maximum at Happ¼940 Oe. The #H¼60/C14ﬁeld scan, on the other hand, is such that u0is small for all Happand the magnetization lies close to the easy axis. The angle u0is small, /C2417/C14, for the absorption maximum at Happ¼460 Oe. The anisotropic dipolar contribution to the lineshape, 2 pMsjqkjdcos2u0inEq.(10), should thus be small for the #H¼20/C14scan and large for the #H¼60/C14scan. Indeed, the two calculations (with and without the dipolar contribution) nearly overlap inthe upper panel of Figure 6for the # H¼20/C14scan and are in good agreement with the jSnorm 21ðHapp;#HÞjdata. There is however a large discrepancy between the data and theexchange-only calculation when # H¼60/C14, where the aniso- tropic dipolar correction should be signiﬁcant, while the PabsðHappÞcalculation including dipolar interactions agrees well with the measured jSnorm 21ðHapp;#HÞjresonance for the #H¼60/C14scan. This agreement is not limited to the ðHapp;u0Þspace subtended by the #H¼20/C14and #H¼60/C14ﬁeld scans. Knowledge of Hk,Happ, and #Hallows for a direct mapping ofjSnorm 21ðHapp;#HÞjtojSnorm 21ðHapp;u0Þjvia the relation Hksinu0cosu0¼Happcosðu0þ#HÞ, where u0is the orien- tation of the magnetization (Figure 1). A plot of jSnorm 21ðHapp;u0Þjatjqkj¼6.9/C2104cm/C01is shown in the top panel of Figure 7.T h ea n g u l a rd e p e n d e n c eo ft h ed i p o l a rc o r - rection in the spin-wave dispersion is reﬂected in the meas-ured SAW absorption for a broad range of u 0/C2410/C14–80/C14.T h i s can be seen by the good agreement between jSnorm 21ðHapp;u0Þj andPabsðHapp;u0Þover this entire set of u0values. The angu- lar range over which SAW absorption due to spin-wave exci- tation can be accessed is only limited by the fact that the magnetoelastic interaction itself becomes vanishingly small as u0approaches 0/C14or 90/C14. The comparison shown in the bot- tom panel of Figure 7between the measured jSnorm 21ðHapp;u0Þj and the isotropic exchange-only calculation shows graphi- cally where and how the anisotropic dipolar correction 2pMsjqkjdcos2u0becomes important as a function of u0.FIG. 5. (a) Spin wave damping measured as a function of #Hat ﬁxed jqkj¼6:9/C2104cm/C01. Scans at other wave vectors show similar angle- independent damping. Error bars indicate the standard error as obtained from least-squares ﬁts to normalized, linear power jSnorm 21ðHapp;#HÞjline- shapes. (b) Spin-wave damping averaged over ﬁeld angle as a function of wave vector shows that spin-wave damping is wave-vector independent.Error bars for damping as a function of jq kjindicate standard error obtained from least squares ﬁts to the series of #Hscans (at ﬁxed jqkj)f o r which there is appreciable signal.FIG. 6. Comparison of normalized jSnorm 21ðHapp;#HÞjtransmission data with the power absorption Pabspredicted by an exchange-only theory and also a full theory including dipolar corrections, for #H¼20/C14and 60/C14at fpump¼3.86 GHz ( jqkj¼6.9/C2104cm/C01). The data normalization have been carried out by converting the S21loss to linear power and rescaling each S21 vsHappcurve to lie between /C01 and 0.233910-6 Gowtham et al. J. Appl. Phys. 118, 233910 (2015) As expected, the inﬂuence of dipolar interactions on the SAW power absorption is seen to be strong at low u0and weak for large u0nearer to 90/C14. ThejSnorm 21ðHapp;u0Þjdata quantitatively match the Pabs ðHapp;u0Þcalculation also not just at jqkj¼6.9/C2104cm/C01 but at all the other qkemployed in the study. We show this by plotting the ﬁelds at which maximum SAW absorption occurs at the different jqkjallowed by our delay line and at a ﬁxed #H¼45/C14(Figure 8). The positions of the absorption maxima cannot be simultaneously ﬁt for all jqkjusing the pure-exchange theory and treating Aex,Ms, and Hkas free pa- rameters or using a uniform-mode resonance theory. Acomparison between the pure-exchange theory and the data shows that the corrections to the dispersion beyond pure- exchange model have the wave-vector dependence corre- sponding to the anisotropic dipolar interaction. We note thatprevious studies 19,21operated in a low wave-vector regime (jqkj¼3/C2103–4/C2104cm/C01) where the leading order wave- vector dependent contributions beyond the uniform-mode theory are small. These studies model their absorption data using a uniform-mode resonance theory—as is appropriateto most of this low wave-vector range in a Ni ﬁlm. Ourexperiment performed at higher wave-vectors clearly meas-ures the impact of the leading order spin-wave contributions to the dispersion (i.e., the dipolar interaction), thus providing a ﬁrst conﬁrmation that the SAW is resonantly exciting atraveling surface spin-wave matched to the SAW’s in-planewave-vector q k. This outstanding quantitative agreement between our measurements and the theory including anisotropic dipolar interactions allows us to use SAW-driven spin-wave spec-troscopy to directly extract the strength of the dipole interac-tions. To do so, we perform a least-squares ﬁt to the jS norm 21ðHapp;u0Þjdataset at various higher jqkjwhere the effects of the dipolar interaction clearly affect the angulardependence of the absorption. We ﬁt the data to the formula(Eqs. (9)–(11)) for P absðHapp;u0Þ, with three free parame- ters: a constant spin-wave damping Cand the coefﬁcients A1 andA2, deﬁned in Eq. (12) W¼Hkcos 2 u0ðÞ þHappsinu0þ#H ðÞ þ2Aex Msjqkj2þA1jqkjcos2u0; !¼Hkcos2u0þHappsinu0þ#H ðÞ þ2Aex Msjqkj2 þ4pMs/C02K? Ms/C18/C19 þA2jqkj: (12) The analytical theory of Eq. (10) predicts A1¼/C0A2¼ 4pMsd 2¼(3.060.1)/C210/C03Oe cm arising from the dipolar spin-wave corrections (with the uncertainty arising from the determination of Ms). The best-ﬁt parameters extracted from the data at different fpumpandjqkjare given in Table I. The coefﬁcients A1andA2as well as the constraint A1 ¼/C0A2have been obtained directly from the jSnorm 21ðHapp;u0Þj measurements at several jqkjand agree within an accuracy of FIG. 8. Comparison of the ﬁeld values where the absorption maximum should occur as a function of pump frequency for #H¼45/C14scans as pre- dicted by the uniform mode resonance theory, an exchange-only theory, and a full surface dipole-exchange theory, and as measured from the SAW absorption.TABLE I. Coefﬁcients for the dipolar interaction A1andA2as well as the spin-wave damping Cextracted from least-squares ﬁtting to jSnorm 21ðHapp;u0Þj data at various value of jqkj. Direct extraction of A1and A2from the jSnorm 21ðHapp;u0Þjdataset at fpump¼2.67 GHz ( jqkj¼4.8/C2104cm/C01)i st r i c k y due to the weaker effect that the dipolar contribution has on the dispersion at this wave-vector and over most of the larger u0at which jSnorm 21ðHapp;u0Þjhas appreciable amplitude at this fpump. fpump (GHz)jqkj (104cm/C01) CA1 (10/C03Oe cm)A2 (10/C03Oe cm) 3.26 5.9 0.143 60.006 3.0 60.2 /C0(3.160.1) 3.86 6.9 0.139 60.007 3.1 60.1 /C0(2.960.2) 4.45 8.0 0.14 60.01 2.9 60.3 /C0(2.860.3) FIG. 7. Plot of the normalized jSnorm 21ðHapp;u0Þjatjqkj¼6.9/C2104cm/C01. The bottom panel has the contours of the dipole-corrected jPabsðHapp;u0Þj and the exchange-only calculation overlaid on top of the jSnorm 21ðHapp;u0Þj data. The contours for each calculation vary in equal steps from Pabs¼/C00.8 (innermost) to Pabs¼/C00.2 (outermost).233910-7 Gowtham et al. J. Appl. Phys. 118, 233910 (2015) a few percent with the expected contributions to the spin wave dispersion arising from the anisotropic dipolar interaction. V. CONCLUSION/OUTLOOK We have quantitatively demonstrated that SAWs can excite a single traveling surface spin-wave mode with an in-plane wave-vector qkmatched to the wave-vector of the SAW. Measurements of the SAW power absorption on such traveling surface spin-wave modes can be used to directly extract precise, quantitative information about the angle and wave-vector dependence of the spin-wave dispersion anddamping. For the Ni thin ﬁlms of the present study we ﬁnd, as expected, that the spin-wave anisotropy is accounted for quan- titatively by the dipolar spin-wave interaction. However, this spectroscopic technique can also be implemented to examine other sources of anisotropy. The same technique can beextended to study spin-wave physics that emerges at higher wave vectors (e.g., to map the angular structure of the Dzyaloshinskii-Moriya exchange interaction) by using piezo-electric substrates with ultra-low SAW speeds, initially devel- oped with the objective of obtaining signal delay lines operating at long delay times. For example, wave vectors as large as jq kj¼7.9/C2105cm/C01at 11 GHz could be achieved using SAWs propagating on a (110)-cut Tl 3TaSe 4substrate, which have a speed of just 8.9 /C2104cm/s.45This large wave vector is accessible by operating a SAW delay line deﬁned by deep UV lithography at a high-order bandpass (e.g., 11th overtone). Anisotropic Dzyaloshinskii-Moriya contributions to the spin-wave dispersion should be as large as several hun- dreds of Oe at this wave vector in a system with high interfa-cial DMI strength (e.g., the Pt jCo system with an interfacial DMI strength of /C240.5 ergs/cm 2).5Measurements of the angu- lar dependence of jSnorm 21ðHapp;u0Þjwould readily allow one to extract the DMI strength. In addition, comparison of two- dimensional jSnorm 21ðHapp;u0Þjplots with jSnorm 12ðHapp;u0Þj plots would allow for a direct measure of the spin-wave non- reciprocity ( qk!/C0 qk) associated with the DMI. A SAW-based spectroscope should also be an excellent tool for carrying out wave-vector and magnetization-angle- resolved studies on 1D and 2D magnonic crystals composedof ultra-thin magnetic elements. In such a case, SAW modiﬁ- cation associated with periodic reﬂections off the magnonic crystal (arising from, e.g., mass loading) should be verysmall, providing SAW based spectroscopy the capability to map spin-wave dynamics throughout the magnonic Brillouin zone. ACKNOWLEDGMENTS We acknowledge G. E. Rowlands for help with Figure 1 of the manuscript. We also thank S. Aradhya and G.Finnochio for suggestions on the manuscript. We are grateful to T. Gosavi and S. Bhave for their encouragement throughout the course of the project. This work was supported in part by the Ofﬁce of Naval Research and the Army Research Ofﬁce. This work made use of the CornellCenter for Materials Research Shared Facilities, which are supported through the NSF MRSEC program (DMR-1120296). This work was performed in part at the Cornell NanoScale Facility, a node of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation (Grant No. ECCS-0335765). 1Y. Kajiwara, K. 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1.2920804.pdf	Theoretical investigation of the relationships between magnetic circular dichroism signals and Gilbert damping coefficient in magnetic films Jie Lu and Peng Yan Citation: Appl. Phys. Lett. 92, 203108 (2008); doi: 10.1063/1.2920804 View online: http://dx.doi.org/10.1063/1.2920804 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v92/i20 Published by the American Institute of Physics. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 11 Jun 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsTheoretical investigation of the relationships between magnetic circular dichroism signals and Gilbert damping coefﬁcient in magnetic ﬁlms Jie Lua/H20850and Peng Yan Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong Special Administrative Region, People’ s Republic of China /H20849Received 28 March 2008; accepted 16 April 2008; published online 22 May 2008 /H20850 Inspired by the traditional ferromagnetic resonance /H20849FMR /H20850approach, the relationships between two kinds of magnetic circular dichroism signals and the Gilbert damping coefﬁcient in magnetic ﬁlmsare theoretically investigated within linear response framework. These results may provideinspirations on potential experimental strategies to remeasure the Gilbert damping coefﬁcient, whichis traditionally obtained from FMR technique. © 2008 American Institute of Physics . /H20851DOI: 10.1063/1.2920804 /H20852 In real magnets, the energy losing process, which is usu- ally called damping, always exists. Its intensity is usuallydescribed by the phenomenological, dimensionless Gilbertdamping coefﬁcient /H9251. The damping mechanism plays cru- cial roles in many micromagnetism problems, such as mag-netization reversal problem in Stoner particles,1domain wall propagating problem in magnetic nanowires,2etc. Tradition- ally, the Gilbert damping coefﬁcient /H9251is mainly obtained from ferromagnetic resonance /H20849FMR /H20850technique.3–5Consid- ering the importance of the damping coefﬁcient in the de-scription of the dynamics of magnets, the investigation ofother alternative techniques which can independently pro-vide /H9251is then meaningful, and can provide references and complements to the results from traditional FMR technique.Among the numerous modern experimental techniques, themagnetic circular dichroism /H20849MCD /H20850spectroscopy 6is a good candidate. MCD signals denotes the difference between theabsorption of right-circular and left-circular propagatingelectromagnetic waves by materials with nonzero magnetiza-tion M. This magneto-optics spectroscopy is a powerful technique in the investigations of electronic structures of ma-terials in modern physics. In this letter, inspired by traditional FMR approach, 7–9 we investigate the connection between several MCD signals and the Gilbert damping coefﬁcient in magnetic ﬁlms withina linear response framework. These relationships may beuseful to construct potential experimental setups to measurethe Gilbert damping coefﬁcient /H9251. A magnetic ﬁlm with magnetization M/H6023is placed in a static magnetic ﬁeld H/H6023, as shown in Fig. 1. The orientations ofM/H6023andH/H6023are described by /H20849/H9258,/H9278/H20850and /H20849/H9258H,/H9278H/H20850, respec- tively. The motion of macrospin M/H6023is described by the Laudau–Lifshitz–Gilbert /H20849LLG /H20850equation, dM/H6023 dt=−/H9253M/H6023/H11003H/H6023eff+/H9251 MsM/H6023/H11003dM/H6023 dt, /H208491/H20850 where /H9253=g/H9262B//H6036is the gyromagnetic ratio, Msis the satura- tion magnetization of the ﬁlm, and /H9251is the dimensionless Gilbert damping coefﬁcient. The effective ﬁeld is obtainedfrom H eff=−/H11612M/H6023F, where Fis the total energy density func- tion of the ﬁlm. In spherical coordinates shown in Fig. 1, Eq. /H208491/H20850becomes10/H208491+/H92512/H20850/H9258˙=−/H9253 Mssin/H9258/H20849/H9251sin/H9258F/H9258+F/H9278/H20850 /H208491+/H92512/H20850/H9278˙=/H9253 Mssin/H9258/H20873F/H9258−/H9251 sin/H9258F/H9278/H20874. /H208492/H20850 The equilibrium of the magnetization M/H6023demands /H9258˙=/H9278˙ =0, which is just equivalent to /H20841F/H9258/H20841/H20849/H92580,/H92780/H20850=/H20841F/H9278/H20841/H20849/H92580,/H92780/H20850=0. This deﬁnes the equilibrium position /H20849/H92580,/H92780/H20850ofM/H6023under H/H6023. Next a small-amplitude microwave h/H6023is projected onto the ﬁlm. The magnetization M/H6023will then take a slight re- sponse motion around the equilibrium position /H20849/H92580,/H92780/H20850, /H9254M/H6023=/H20849/H9254M/H9258/H20850eˆ/H9258+/H20849/H9254M/H9278/H20850eˆ/H9278=Ms/H20849/H9254/H9258eˆ/H9258+ sin/H92580/H9254/H9278eˆ/H9278/H20850/H20849 3/H20850 and the right hand of Eq. /H208492/H20850can be divided into two parts, /H208491+/H92512/H20850d/H9258 /H9253dt=/H20849/H9251h/H9258+h/H9278/H20850−/H9251sin/H9258F/H9258+F/H9278 Mssin/H9258 /H208491+/H92512/H20850sin/H9258d/H9278 /H9253dt=/H20849/H9251h/H9278−h/H9258/H20850+sin/H9258F/H9258−/H9251F/H9278 Mssin/H9258, /H208494/H20850 where h/H9258/H20849h/H9278/H20850is the component of h/H6023along eˆ/H9258/H20849eˆ/H9278/H20850direction at /H20849/H92580,/H92780/H20850point and Fis total energy density function of the ﬁlm excluding the Zeeman part coming from h/H6023. Suppose the frequency of the projected microwave h/H6023is /H9275, then the forced movement /H9254M/H6023ofM/H6023has the same fre- quency /H9254/H9258,/H9254/H9278/H11011ei/H9275t. Within the linear response framework, we expand the total energy density function Fto the second order derivatives and after some simple algebra, Eq. /H208494/H20850ﬁ- nally reads, /H20873/H9254M/H9258 /H9254M/H9278/H20874=1 D/H20875−/H20849Q+i/H9251/H9024/H20850 R−i/H9024 R+i/H9024 −/H20849P+i/H9251/H9024/H20850/H20876/H20873h/H9258 h/H9278/H20874, /H208495/H20850 where P/H11013F/H9258/H9258/Ms2, Q/H11013F/H9278/H9278 //H20849Ms2sin2/H92580/H20850, R /H11013F/H9258/H9278//H20849Ms2sin/H92580/H20850,/H9024/H11013/H9275//H20849/H9253Ms/H20850, and D/H11013/H20851/H208491+/H92512/H20850/H90242 −/H20849PQ−R2/H20850/H20852−i/H9251/H9024/H20849P+Q/H20850. Equation /H208495/H20850indeed provides the dynamic susceptibility of the magnetic ﬁlm under any incident radio-frequency mi- crowave h/H6023, assuming that h/H6023is weak enough so that the mag- netization M/H6023will not depart from its equilibrium positiona/H20850Electronic mail: lujie@ust.hk.APPLIED PHYSICS LETTERS 92, 203108 /H208492008 /H20850 0003-6951/2008/92 /H2084920/H20850/203108/3/$23.00 © 2008 American Institute of Physics 92, 203108-1 Downloaded 11 Jun 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions/H20849/H92580,/H92780/H20850too much. This is the theoretical basement of the traditional FMR approach and also the start point of our following discussions. In order to analyze the MCD signals, certain forms of h/H6023 will be chosen and investigated. Different choice of h/H6023corre- sponds to different strategy of detecting the MCD signals. Strategy 1 , as shown in Fig. 2/H20849a/H20850, a linearly polarized radio-frequency microwave /H20849LP-MW /H20850h/H6023is projected onto the ﬁlm with its polarization direction being normal to the equi- librium position /H20849/H92580,/H92780/H20850ofM/H6023under H/H6023.h/H6023can be viewed as the combination of two equal-module right- and left- circularly polarized microwave /H20849CP-MW /H20850components h/H6023/H11006. Due to the difference between the absorption of h/H6023/H11006by the magnetic ﬁlm, the transmitted microwave is usually nolonger linearly but elliptically polarized. The intensity ratio /H9260 of the transmitted elliptically polarized microwave, which isdeﬁned as the intensity on the long axis over that on the short axis /H20851/H20849h +/H11032+h−/H11032/H208502//H20849h+/H11032−h−/H11032/H208502/H20852, is one kind of the MCD signals. We hereby relate /H9260to the dimensionless Gilbert damping coefﬁcient /H9251of the magnetic ﬁlm. The LP-MW h/H6023can be expressed as h/H6023 =2hei/H9275t/H20849cos/H9257, sin/H9257/H20850+, where /H9257is the angle of the polariza-tion direction respective to eˆ/H9258within /H20849eˆ/H9258,eˆ/H9278/H20850/H20849/H92580,/H92780/H20850plane. The matrix expression is written within the local /H20849eˆ/H9258,eˆ/H9278/H20850/H20849/H92580,/H92780/H20850 coordinate system and the same below. Thus the two anti- chiral CP-MW components h/H6023/H11006ofh/H6023are h/H6023/H11006=hei/H20849/H9275t/H11006/H9257/H20850 /H11003/H208491,/H11007i/H20850+. From the LLG Eq. /H208491/H20850, the energy density change rate of any time-dependent microwave h/H6023/H20849t/H20850that travels through a magnetic sample is10dEh/H20849t/H20850/dt=Re /H20851M/H6023†·h/H6023˙/H20849t/H20850/H20852. Therefore the energy density change rate of h/H6023/H11006can be straightforwardly obtained, dEh/H11006/dt=−/H9261/H11006Eh/H11006/H208496/H20850 with Eh/H11006=h2,/H9261+=2/H9275/H20851−XZ+Y/H9251/H9024/H20849P+Q/H20850/H20852 Z2+/H92512/H90242/H20849P+Q/H208502, /H9261−=2/H9275/H20851/H20849X+2/H9251/H9024/H20850Z+/H20849Y+2/H9024/H20850/H9251/H9024/H20849P+Q/H20850/H20852 Z2+/H92512/H90242/H20849P+Q/H208502, X=Rcos 2/H9257+/H20849P−Q/H20850sin/H9257cos/H9257−/H9251/H9024, Y=Rsin 2/H9257−Pcos2/H9257−Qsin2/H9257−/H9024, Z=/H208491+/H92512/H20850/H90242−/H20849PQ−R2/H20850. Usually /H9261+/HS11005/H9261−, thus the transmitted microwave h/H11032/H6023is no longer linearly but elliptically polarized. Suppose the thick-ness of the ﬁlm is rather small, then one can neglect the Faraday effect, i.e., assuming the traveling speeds of h/H6023/H11006are the same /H20849and equal to the speed of h/H6023/H20850. This assumption results in a universal transmitting time t1. Then from Eq. /H208496/H20850, the energy densities of the transmitted left- and right-circular components are E/H11006/H11032=e−/H9261/H11006t1E0, where E0is the initial energy density of the two components before projecting. Thus onehas /H9260=/H20873/H20881E+/H11032+/H20881E−/H11032 /H20881E+/H11032−/H20881E−/H11032/H208742 = coth2/H20875−/H20849X+/H9251/H9024/H20850Z+/H9251/H90242/H20849P+Q/H20850 Z2+/H92512/H90242/H20849P+Q/H208502·/H9275t1/H20876. /H208497/H20850 To eliminate the unnecessary complexity originated from the arbitrariness of the polarization direction inside the /H20849eˆ/H9258,eˆ/H9278/H20850/H20849/H92580,/H92780/H20850plane, one can just take the resonance condition Z=0 and Eq. /H208497/H20850is eventually reduced to /H9260= coth2/H20875−/H9275t1 /H9251/H20849P+Q/H20850/H20876, /H208498/H20850 where PandQshould take the values at the resonance point. Eq. /H208498/H20850relates the MCD signal /H9260and the Gilbert damp- ing coefﬁcient /H9251in a simple manner. Once we know the precise anisotropy form of the sample /H20849indeed only the form is needed, the values of the anisotropy parameters are notnecessary /H20850, we can extract out /H9251at the FMR point. In strategy 2 , the ﬁlm and static ﬁeld conﬁgurations are the same with strategy 1 but with a different pattern of mi-crowave. Rather than a LP-MW, a right CP-MW is projectedfrom two opposite directions with its polarization vector in the /H20849eˆ /H9258,eˆ/H9278/H20850/H20849/H92580,/H92780/H20850plane, which is shown in Fig. 2/H20849b/H20850. This is FIG. 1. Illustration of the usual experimental setup of magnetic measure- ments. /H9018is a magnetic ﬁlm produced on some substrate, with a nonzero magnetization M/H6023/H20849/H9258,/H9278/H20850.M/H6023reaches a equilibrium position M/H60230/H20849/H92580,/H92780/H20850/H20849de- noted by the thick dash line /H20850under a static external ﬁeld H/H6023/H20849/H9258H,/H9278H/H20850. /H20849eˆr,eˆ/H9258,eˆ/H9278/H20850are the local unit vectors at the equilibrium position. k/H6023is the wave vector of a projected small-amplitude radio-frequency microwave h/H6023/H20849t/H20850.U n - der this microwave, M/H6023will take a weak response motion /H9254M/H6023/H20849t/H20850around M/H60230. FIG. 2. Illustration of the two strategies of detecting the MCD signals. /H20849a/H20850 Strategy 1: a LP-MW is projected. Due to the MCD effect the transmittedmicrowave is no longer linearly but elliptically polarized. Its intensity ratio /H20849h +/H11032+h−/H11032/H208502//H20849h+/H11032−h−/H11032/H208502is one kind of the MCD signals. /H20849b/H20850Strategy 2: a right- circular polarized microwave is projected into the magnetic ﬁlm from two opposite directions with its polarization vector in the /H20849eˆ/H9258,eˆ/H9278/H20850/H20849/H92580,/H92780/H20850plane. The difference between the energy densities of two transmitted CP-MW, /H9273MCD=/H20849EI/H11032−EII/H11032/H20850/E0, is another kind of MCD signals.203108-2 J. Lu and P . Yan Appl. Phys. Lett. 92, 203108 /H208492008 /H20850 Downloaded 11 Jun 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsequivalent to let k/H6023/H20648M/H60230. The difference between the energy densities of two transmitted CP-MWs, /H9273MCD=/H20849EI/H11032−EII/H11032/H20850/E0, is another kind of MCD signals. Under macrospin assump- tion, a right CP-MW projecting from the opposite direction isequivalent to a left one projecting along the original direc-tion. Thus, the MCD signal can also be expressed as, /H9273MCD =/H20849E+/H11032−E−/H11032/H20850/E0. Suppose the thickness of the ﬁlm is small enough, after some algebra similar with those in strategy 1, one can obtain /H9273MCD=−4/H9275t2 /H9251/H20849P+Q/H20850, /H208499/H20850 where t2is the time that the right-circular microwave travels through the magnetic ﬁlm and P,Qshould also be the cor- responding values at the resonance point. Equations /H208498/H20850and /H208499/H20850are the main results of this paper. One can see that the two strategies have the same level ofpracticability. However, strategy 2 has an advantage in thedetection of the polarity of the signal. To illustrate the possible usage of the two strategies, one may consider a simplest example. 3A thin magnetic ﬁlm is produced on some substrate. Its thickness is far less than theother two dimensions, thus the demagnetizing factors areN /H11036=4/H9266,N/H20648/H20849Nx,Ny/H20850=0. As the simplest case, one can assume magnetic isotropy in the ﬁlm plane and only consider the crystalline anisotropy in the normal direction which is char- acterized by the parameter K/H11036. A static magnetic ﬁeld H/H6023is applied out of the ﬁlm plane. Then the total energy densityfunction of the ﬁlm is F=−M/H6023·H/H6023+M s 2/H208494/H9266Meff/H20850cos2/H9258, /H2084910/H20850 where 4 /H9266Meff=4/H9266Ms−2K/H11036/Ms. The equilibrium position /H20849/H92580,/H92780/H20850of the magnetization M/H6023can be obtained through F/H9258=F/H9278=0, i.e., /H208494/H9266Meff/H20850sin 2/H92580=2Hressin/H20849/H92580−/H9258H/H20850;/H92780=/H9278H. /H2084911/H20850 From Eq. /H208495/H20850, P=H1/Ms,Q=H2/Ms,R=0 , /H2084912/H20850 where H1=Hrescos/H20849/H92580−/H9258H/H20850−/H208494/H9266Meff/H20850cos 2/H92580 H2=Hrescos/H20849/H92580−/H9258H/H20850−/H208494/H9266Meff/H20850cos2/H92580. Usually, the Gilbert damping coefﬁcient satisﬁes /H9251/H112701, then the FMR resonance condition Z=0 is reduced to /H9275//H9253=/H20881H1/H11003H2 /H2084913/H20850 and the MCD signals /H9260and/H9273MCDbecome /H9260= coth2/H20875−Ms/H9275t1 /H9251/H20849H1+H2/H20850/H20876;/H9273MCD=−4Ms/H9275t2 /H9251/H20849H1+H2/H20850. /H2084914/H20850 In real experiments, one may ﬁx the microwave frequency /H9275 and propose the following experimental procedure: /H208491/H20850Measure the saturated magnetization Msvia static mag- netic method./H208492/H20850Hresvs/H9258His numerically calculated using Eqs. /H2084911/H20850–/H2084913/H20850, and is ﬁtted to the experimental Hresvs/H9258H curve by adjusting the values of gand 4/H9266Meff. /H208493/H20850For a certain /H9258H, using the value of 4 /H9266Meffobtained in step /H208492/H20850,P,Q/H20849i.e.,H1,H2/H20850are calculated through Eqs. /H2084911/H20850and /H2084912/H20850. /H208494/H20850The traveling time t1/H208492/H20850can be obtained by the ratio of the distance D1/H208492/H20850that the microwaves travel over the corresponding speed v1/H208492/H20850of it. In strategy 1, v1is set to be the speed of the incident LP-MW, while in strategy 2, v2is just the speed of the right CP-MW. /H208495/H20850The MCD signal /H9260or/H9273MCDis measured. /H208496/H20850Put the above parameters into Eq. /H2084914/H20850, the Gilbert damping coefﬁcient /H9251/H20849/H92580/H20850can be extracted out. /H208497/H20850Change the polar angle /H9258Hof the static magnetic ﬁeld H/H6023 /H20851which is equivalent to vary the polar angle /H9258M/H20849=/H92580/H20850of the magnetization M/H6023/H20852, the angle dependence of Gilbert damping coefﬁcient, /H9251vs/H9258Mcan be eventually revealed. Compared with the traditional FMR technique, our pro- posal is less universal because in order to extract /H9251, we must have some knowledge about the form of crystalline aniso-tropy in the sample in advance. This is the main constraint ofour MCD approach as an independent method. On the otherhand, the signiﬁcance of our MCD proposal is that it pro-vides an alternative way to evaluate /H9251of magnets rather than the traditional FMR technique. We can ﬁrst use FMR scan toreveal the anisotropy form of the magnets, and then measurethe damping coefﬁcient /H9251via FMR and MCD techniques, respectively. Comparison of the results from both two tech-niques will increase the accuracy and reliability of /H9251mea- surement. This is the central purpose of our present work. In conclusion, we present the relationship between two kinds of MCD signals and the Gilbert damping coefﬁcient inmagnetic thin ﬁlms. Based on these results, potential experi-mental proposals are suggested to measure the Gilbert damp-ing coefﬁcient, which is traditionally obtained through FMRtechnique. The signiﬁcation and disadvantages of the MCDapproach are discussed. The authors would like to thank Professor X. R. Wang for valuable discussions in this work. This work is supportedby RGC CERG /H20849Grant Nos. 603007 and 603106 /H20850. 1Z. Z. Sun and X. R. Wang, Phys. Rev. B 71, 174430 /H208492005 /H20850;Phys. Rev. Lett. 97, 077205 /H208492006 /H20850;98, 077201 /H208492007 /H20850; 2N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 /H208491974 /H20850. 3S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 /H208492002 /H20850. 4S. J. Yuan, L. Sun, H. Sang, J. Du, and S. M. Zhou, Phys. Rev. B 68, 134443 /H208492003 /H20850. 5M. Díaz de Sihues, C. A. Durante-Rincón, and J. R. Fermin, J. Magn. Magn. Mater. 316,e 4 6 2 /H208492007 /H20850. 6W. Roy Mason, A Pratical Guide to Magnetic Circular Dichroism Spec- troscopy /H20849Wiley, Hoboken, New Jersey, 2007 /H20850. 7C. Kittel, Phys. Rev. 73, 155 /H208491948 /H20850. 8H. Suhl, Phys. Rev. 97, 555 /H208491955 /H20850. 9S. V. Vonsovskii, Ferromagnetic Resonance: The Phenomeonon of Reso- nant Absoption of a High-frequency Magnetic Field in FerromagneticSubstances /H20849Pergamon, Oxford, London, 1966 /H20850. 10Z. Z. Sun and X. R. Wang, Phys. Rev. B 73, 092416 /H208492006 /H20850;74, 132401 /H208492006 /H20850.203108-3 J. Lu and P . Yan Appl. Phys. Lett. 92, 203108 /H208492008 /H20850 Downloaded 11 Jun 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions
1.3540415.pdf	Current driven oscillation and switching in Co/Pd perpendicular giant magnetoresistance multilayer C. H. Sim, S. Y. H. Lua, T. Liew, and J.-G. Zhu Citation: J. Appl. Phys. 109, 07C905 (2011); doi: 10.1063/1.3540415 View online: http://dx.doi.org/10.1063/1.3540415 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v109/i7 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 07 Jun 2013 to 193.1.100.108. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsCurrent driven oscillation and switching in Co/Pd perpendicular giant magnetoresistance multilayer C. H. Sim,1,2,a)S. Y . H. Lua,2T. Liew,2and J.-G. Zhu1 1Data Storage Systems Center, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA 2A*STAR (Agency for Science, Technology and Research), Data Storage Institute, Singapore 117608 (Presented 17 November 2010; received 7 October 2010; accepted 4 November 2010; published online 22 March 2011) Spin torque transfer (STT) induced magnetization oscillation and switching in metallic spin valves with Co/Pd electrodes of perpendicular magnetic anisotropy are demonstrated. The bottom Co/Pdmultilayer, acting as a perpendicular spin-polarizing/reference layer, is relatively thick with a strong perpendicular anisotropy and a perpendicular switching ﬁeld of 8 kOe. An in-plane spin valve is placed on the top for reading back magnetization oscillation of the middle Co layer, whosethickness is varied from 6 to 30 A ˚. When the middle Co layer is thin, current driven magnetization switching is observed. When the middle Co layer is relatively thick, perpendicular spin torque oscillation is clearly observed with oscillating frequency at 4 GHz. STT-included micromagneticmodeling has been performed which predicts the exact observed behavior and illustrates the signiﬁcance of magnetization conﬁguration of the Co layer on determining STT-induced dynamics. VC2011 American Institute of Physics . [doi: 10.1063/1.3540415 ] I. INTRODUCTION The transfer of spin angular momentum between a spin-polarized electric current and the magnetic moment in a multilayer nanostructure, known as spin torque transfer(STT) effect, has unveiled an exciting ﬁeld of studies on current-induced magnetization dynamics. 1,2STT can either induce a magnetic layer to reverse its magnetization direc-tion, or drive steady-state magnetization precession, as pre- dicted theoretically by Slonczewski 3and Berger.4For a spin valve structure of a nonmagnetic metal layer sandwichedbetween two ferromagnetic (FM) layers, these dynamics can be exploited through the giant magnetoresistance (GMR) effect for potential applications in high density nonvolatilemagnetic memory (MRAM) 5–7or as current-tunable micro- wave nano-oscillator (STNO).8,9 The relative strengths and directions of anisotropies of the two FM layers, namely the reference layer (RL) and free layer (FL), play an important role in determining the magnet- ization response of the system in the presence of a spin polar-ized current. Increasing attention has been paid to Co/Pd and Co/Pt multilayer because they exhibit large perpendicular magnetic anisotropy (PMA) which provides high STT efﬁ-ciency. 10–12Recently, the use of a bilayer FL, in which a magnetic layer with PMA is coupled to a spin torque driven magnetic layer adjacent to the spacer/barrier, is shown toprovide a reduction in switching current density through the exchange-spring effect. 13–15On the other hand, Zhu has dem- onstrated steady magnetization precession and zero externalﬁeld operation of a STNO with similar geometry. 16So far, this stack combination has not been fully explored and it remains unclear how the material parameters of the bilayerFL will alter STT magnetization dynamics of the system. Inthis paper, we demonstrate that the variation in thickness of the middle spin torque driven magnetic layer can modify the magnetic potential energy landscape in Co/Pd perpendicular multilayer to induce various STT-related phenomena. II. EXPERIMENTAL PROCEDURES Two series of ﬁlms were deposited using ultrahigh vac- uum direct current (dc) magnetron sputtering onto thermally oxidized Si wafers at base pressures below 7 /C210/C09Torr. The basic stacking structure used is [Co 3 A ˚/Pd 8 A ˚]/C25/ Co 6 A ˚/Cu 20 A ˚/CotA˚/[Pd 8 A ˚/Co 5 A ˚]/C23, where the thickness of the middle Co layer, tCois varied from 6 A ˚ (sample 1) to 30 A ˚(sample 2), as shown in Fig. 1. The RL is a Co/Pd multilayer with high perpendicular magnetic anisot- ropy, and the FL is constructed from a bilayer structure FIG. 1. (Color online) Schematic of pillar device with a reference layer which comprises of Co/Pd multilayer and a free layer made up of a Co layer and Co/Pd multilayer. The Co thickness, tCois varied in the experiment.a)Author to whom correspondence should be addressed. Electronic mail: cheowhin@cmu.edu. 0021-8979/2011/109(7)/07C905/3/$30.00 VC2011 American Institute of Physics 109, 07C905-1JOURNAL OF APPLIED PHYSICS 109, 07C905 (2011) Downloaded 07 Jun 2013 to 193.1.100.108. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsconsisting of a spin torque driven middle Co layer exchange coupled to a Co/Pd multilayer. An in-plane spin valve con- sisting of Co/IrMn is placed on top of the FL to readout mag-netization oscillation. We patterned the ﬁlms into nanopillars of 80 nm lateral dimension by e-beam lithography, followed by Ar ion milling and a lift-off process. The completed devi-ces were annealed at 215 /C14C under an in-plane magnetic ﬁeld of 4 kOe to pin the magnetization direction of the analyzer. The samples were ﬁrst saturated in a large positive per- pendicular ﬁeld to set the RL magnetization. Spin torque switching (STS) characteristics were measured using lock-in d e t e c t i o nw i t ha na cc u r r e n to fr m sa m p l i t u d e1 0 mA superim- posed onto a pulsed or dc current. The microwave spectra were measured by contacting the samples using microcoaxial probes, which were connected to a current source through abias tee, and fed to a spectrum analyzer through a 40 dB gain broadband ampliﬁer. In our convention, positive current is deﬁned for electrons ﬂowing from the bottom to top electrode. III. RESULTS AND DISCUSSION It is generally known that the magnetic anisotropy of a magnetic layer is made up of a combination of the interfacialcontribution which favors perpendicular magnetization, and a volume contribution which favors an in-plane magnetiza- tion. 17,18In particular, for a certain range of Co thickness, a Pd layer can induce strong interfacial perpendicular anisot- ropy which pulls its magnetization out of the ﬁlm plane. Figure 2shows the current-perpendicular-to-plane (CPP) re- sistance as a function of perpendicular magnetic ﬁeld ( R-H) measured for the two samples. We note that the origin of the tilted baseline comes from GMR contribution between thetop in-plane analyzing layer and FL. Sample 1 exhibits a well-deﬁned switching behavior between parallel state (low resistance) and antiparallel state (high resistance), with aswitching ﬁeld of 3.0 and 8.0 kOe for the FL and RL, respec- tively. On the other hand, in sample 2 we observed a gradual convex increase in resistance before the RL switched at6.0 kOe. This suggests that the magnetization of the FL is undergoing a rotation from in-plane toward out of the ﬁlm plane with increasing ﬁeld. Based on these data, we candeduce that the FL in sample 1 has a preferred perpendicular direction of magnetic orientation, whereas the FL magnetiza- tion in sample 2 is estimated to be tilted 3.6 /C14out of plane from the in-plane easy axis, which may be attributed to the exchange coupling at the interface of the in-planemagnetized thick Co layer when it adjoins the top perpendic- ular Co/Pd multilayer. Quasistatic transport measurements are shown in Fig. 3, which compares the variation in differential resistance dV/dI as a function of current Ibetween the two samples. In sample 1 where the Co thickness is thin, the device exhibits clearSTT switching from a high resistance to low resistance state and vice versa, at pulse currents of 1.0 and /C02.0 mA, respec- tively. This magnitude change in resistance is consistentwith the value obtained from R-H curve, implying that spin torque from current has effectively drove the FL magnetiza- tion between the two energy minima. On the other hand, a dc FIG. 2. R-Hcurve with magnetic ﬁeld Happlied in the out-of-ﬁlm plane direction at current I¼2 mA measured for (a) sample 1 with tCo¼6A˚, and (b) sample 2 with tCo¼30 A˚. FIG. 3. (Color online) (a) STS curve measured for sample 1, using a pulsed current with pulse width of 0.5 ms. (b) STS plot for sample 2, using a dc cur- rent sweep with perpendicular ﬁeld applied at 0 and 500 Oe. The inset shows the microwave spectrum obtained under a ﬁeld Hof 500 Oe and current I¼/C03.8 mA.07C905-2 Sim et al. J. Appl. Phys. 109, 07C905 (2011) Downloaded 07 Jun 2013 to 193.1.100.108. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionscurrent sweep was used to measure dV/dI of sample 2. The observed parabolic increase in differential resistance is attributed to joule heating of the device. Under zero appliedﬁeld, the device exhibits a minor dip at I ¼/C02.4 mA. At 500 Oe, the ﬁeld favors the antiparallel alignment and the dip becomes more distinct while shifting to a higher currentatI¼– 4.0 mA. Many previous experiments have associated steady-state spin torque oscillations as peaks or dips in dV/dI versus I scans. 1,2,9,10To conﬁrm this point, we acquired the power spectrum density (PSD) of the signal emitted by sam- ple 2. The PSD plotted in the inset of Fig. 3(b) veriﬁes the appearance of a narrow peak at 4 GHz which correlates withthe onset of the dip in the dV/dI scan. This peak broadens and disappears into the background when the injected current is increased beyond /C04.2 mA. The results indicate that modifying the middle Co thick- ness in the bilayer FL can induce changes in STT magnetiza- tion dynamics. When t Cois thin, interfacial effects are dominant and majority of the spin current gets dissipated in the Co/Pd multilayer, resulting in high effective damping which facilitates reversal at the onset of magnetization insta-bility. Conversely, when t Cois thick, magnetic anisotropy in the FL has a nonzero tilting angle with respect to the perpen- dicular axis under quiescent condition. In this case, it ispossible for spin torque from current to excite magnetization into new dynamic equilibrium positions corresponding to closed magnetization orbits in the absence of an appliedﬁeld. A more detailed examination of the interplay between magnetic anisotropies, interlayer coupling and STT is neces- sary to understand the ensuing dynamics in the FL.The experimental observations are qualitatively in agree- ment with micromagnetic simulation calculated using the Landau–Lifshitz–Gilbert (LLG) equations of motion with theSlonczewski spin-transfer torque term. In the simulation, the top perpendicular layer is assumed to be 6 nm thick with an intrinsic perpendicular anisotropy constant K u¼5/C2106erg/ cm3, saturation magnetization M s¼550 emu/cm3and damp- ing constant a¼0.02; the middle spin torque driven magnetic layer has M s¼1440 emu/cm3,a¼0.01 with its thickness var- ied. We use a spatial discretization cell of 4 /C24/C2thickness nm in size to simulate 40 /C240 nm2devices. The calculated zero-temperature dynamic phase diagram is presented in Fig.4. The dark area labeled 1 corresponds to region with no steady oscillation, while the shaded area corresponds to vary- ing magnitude of oscillation frequency, with the highestoccurring in the area labeled 3. In consistent with our experi- mental data, the plot illustrates that a minimum layer thick- ness of about 0.5 nm is required to stabilize perpendicularoscillation; below this critical thickness, the device will undergo STT switching instead. IV. CONCLUSION In summary, we have demonstrated that the thickness of the middle Co layer in a bilayer FL is a crucial parameter to control the magnetization proﬁle and current induced mag- netization dynamics. When the Co layer is thin, STT switch-ing is obtained. For a larger thickness, a current bias of the proper polarity can excite uniform magnetization precession of the FL. The means to manipulate STT dynamics byadjustment of the magnetic layer thickness could potentially simplify future designs of spin torque driven nanodevices. ACKNOWLEDGMENTS This research is supported in part by the NSF/MRSEC program at Johns Hopkins University and the Data StorageSystems Center at Carnegie Mellon University and its indus- trial sponsors. 1E. B. Myers et al.,Science 285, 867 (1999). 2M. Tsoi et al.,Phys. Rev. Lett. 80, 4281 (1998). 3J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 4L. Berger, Phys. Rev. B 54, 9353 (1996). 5R. Law et al.,J. Appl. Phys. 105, 103911 (2009). 6N. Nishimura et al.,J. Appl. Phys. 91, 5246 (2002). 7T. Kawahara et al.,IEEE J. Solid-State Circuits 43, 109 (2008). 8W. H. Rippard et al.,Phys. Rev. Lett. 92, 027201 (2004). 9S. I. Kiselev et al.,Nature 425, 380 (2003). 10S. Mangin et al.,Nature Mater. 5, 210 (2006). 11J.-H. Park et al.,J. Appl. Phys. 105, 07D129 (2009). 12S. Ikeda et al.,Nature Mater. 9, 721 (2010). 13O. G. Heinoen and D. V. Dimitrov, J. Appl. Phys. 108, 014305 (2010). 14X. Zhu and J.-G. Zhu, IEEE Trans. Magn. 43, 2739 (2006). 15R. Victoria and X. Shen, IEEE Trans. Magn. 41, 537 (2005). 16X. Zhu and J.-G. Zhu, IEEE Trans. Magn. 42, 2670 (2006). 17F. J. A. den Broeder, W. Hoving, and P. J. H. Bloemen, J. Magn. Magn. Mater. 93, 562 (1991). 18P. F. Carcia, A. D. Meinhaldt, and A. Suna, Appl. Phys. Lett. 47, 178 (1985). FIG. 4. (Color online) (a) Simulated phase diagram of a 40 /C240 nm2device as functions of the spin torque driven layer thickness and current density. (b) The microwave spectra correspond to the positions of the white dots denoted in (a).07C905-3 Sim et al. J. Appl. Phys. 109, 07C905 (2011) Downloaded 07 Jun 2013 to 193.1.100.108. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions
1.4928383.pdf	Parametric study of a Schamel equation for low-frequency dust acoustic waves in dusty electronegative plasmas Akbar Sabetkar and Davoud Dorranian Citation: Phys. Plasmas 22, 083705 (2015); doi: 10.1063/1.4928383 View online: http://dx.doi.org/10.1063/1.4928383 View Table of Contents: http://aip.scitation.org/toc/php/22/8 Published by the American Institute of Physics Parametric study of a Schamel equation for low-frequency dust acoustic waves in dusty electronegative plasmas Akbar Sabetkar and Davoud Dorraniana) Laser Lab., Plasma Physics Research Center, Science and Research Branch, Islamic Azad University, Tehran, Iran (Received 8 May 2015; accepted 22 July 2015; published online 13 August 2015) In this paper, our attention is ﬁrst concentrated on obliquely propagating properties of low-frequency (x/C28xcd) “fast” and “slow” dust acoustic waves, in the linear regime, in dusty electronegative plas- mas with Maxwellian electrons, kappa distributed positive ions, negative ions (following the combi- nation of kappa-Schamel distribution), and negatively charged dust particles. So, an explicit expression for dispersion relation is derived by linearizing a set of dust-ﬂuid equations. The resultsshow that wave frequency xin long and short-wavelengths limit is conspicuously affected by physi- cal parameters, namely, positive to negative temperature ion ratio ( b p), trapping parameter of nega- tive ions ( l), magnitude of the magnetic ﬁeld B0(viaxcd), superthermal index ( jn;jp), and positive ion to dust density ratio ( dp). The signature of the penultimate parameter (i.e., jn) on wave frequency reveals that the frequency gap between the modes reduces (escalates) for k<kcr(k>kcr), where kcr is critical wave number. Alternatively, for weakly nonlinear analysis, reductive perturbation theory has been used to construct 1D and 3D Schamel Korteweg-de Vries (S-KdV) equations, whose nonli- nearity coefﬁcient prescribes only compressive soliton for all parameter values of interest. The sur- vey manifests that deviation of ions from Maxwellian behavior leads intrinsic properties of solitarywaves to be evolved in opposite trend. Additionally, at lower proportion of trapped negative ions, solitary wave amplitude mitigates, whilst the trapping parameter has no effect on both spatial width and the linear wave. The results are discussed in the context of the Earth’s mesosphere of dusty elec-tronegative plasma. VC2015 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4928383 ] I. INTRODUCTION Undoubtedly, dust grain is an integral and inseparable ingredient of astrophysical and space environments, the so- called dusty plasmas.1–3Due to acquiring ﬁnite electric potential4,5through charging process,6,7charged dust grains are now responsible for development of the novel types ofelectrostatic collective modes the most notable of which isthe dust acoustic waves (DAWs) 8,9which has a frequency of just a few Hz typically. Strictly speaking, the DAWs are acompressional disturbance which propagates through thedust and directly involves the dynamics of the dust particles.An electronegative plasma is deﬁned as a plasma composedof negative and positive ion species as well as electrons.This plasma can appear as a result of elementary processes,such as dissociative and non-dissociative electron attachmentto neutral particles when an electronegative gas is introducedin an electrical gas discharge or injected from external sour-ces. Also, the existence of negative ions has frequently been detected in space plasma (such as Earth’s mesosphere, 10 upper layers of Earth’s ionosphere,11and F-ring of Saturn12). Furthermore, the importance of negative ion plasmas to theﬁeld of plasma physics is growing up day by day because oftheir wide technical applications in neutral beam sources, 13 plasma processing reactors,14etc. A large number of both theoretical and experimental investigations have also been conﬁrmed that the presence ofnegative ions in dusty plasma plays a pivotal role in modi- fying intrinsic properties of waves.15–17For instance, Adhikary et al.18reported empirically the velocity of rare- factive ion solitary wave in multicomponent plasma withnegative ions is greater than that in the presence of nega-tively charged dust. In another study on the role of negativeions, Misra and Barman 19have found that the effect of the concentration of charged dusts on dispersion relation is toreduce the frequency of dust ion acoustic waves, whereasits effect on amplitude and width of soliton is reverse. Mostrecently, Hussain and his coworker 20have studied the for- mation of shock structures in negative ion and pair-ion non-thermal plasmas. The authors pointed out that the phasespeed of ion acoustic shocks in pair-ion plasma becomeslarger in comparison to multi-ion plasmas for similar valueof negative ion density and strength of the shock waveincreases as the number of superthermal electrons in theplasma system is decreased. Tasnim et al. 21investigated nonlinear propagation of planar and non-planar solitarywaves in a quantum dusty electronegative plasma. It hasbeen also found that the properties of dust ion-acoustic soli-tary waves in non-planar cylindrical or spherical geometrydiffer from those in planar one-dimensional geometry aswell as the effect of positive and negative ion mass ratio onwidth and amplitude of both negative and positive solitarywave is reverse. Furthermore, spacecraft observations have provided evi- dence of the occurrence of abnormal energetic particle pres-ence in the Earth’s magnetosphere 22and solar wind.23,24Onea)Author to whom correspondence should be addressed. Electronic mail: doran@srbiau.ac.ir. Tel.: þ98 21 44869654. Fax: þ98 21 44869640. 1070-664X/2015/22(8)/083705/12/$30.00 VC2015 AIP Publishing LLC 22, 083705-1PHYSICS OF PLASMAS 22, 083705 (2015) of the well-ﬁtted models to describe such kind of energetic particles is kappa distribution.25,26The kappa or generalized Lorentzian distribution, a generalization of the Maxwellian, represents a family of velocity distributions, ranging from an extreme “hard” spectrum associated with j’1:5/C02, to the Maxwell-Boltzmann distribution for j!1 . Recently, the most evident theoretical and simulation features on the superthermality effects have been employed successfully inplasma physics. 27–29For instance, Borhanian and his co- worker30investigated the existence and characteristics of propagation of dust acoustic waves in a superthermal dustyplasma with the help of energy integral equations. They con- clude that with increasing obliqueness the existence range of solitons would be restricted to the lower values of superther-mal index. In another work, Aoutou et al. 31have investigated existence of arbitrary amplitude DAWs by using Sagdeev potential approach. It is shown that due to electron and ionsuperthermality, the present dusty plasma model may support subsonic as well as supersonic electrostatic solitary waves involving cusped potential humps. In another study, Arshadand Mahmood 27calculated the damping rates for electrostatic ion waves in Lorentzian electronegative plasmas. They have found that the Landau damping rate of the ion plasma wave isincreased in Lorentzian plasmas in comparison with Maxwellian pair-ion plasmas. Dorranian and Sabetkar 32 recently proved that for certain magnitude of nonthermal pa- rameter there is a condition for generation of an evanescent dust acoustic solitary wave in a dusty plasma. Another phenomenon that has broadly been viewed in both space and laboratory plasmas is particle trapping. In this case, proportion of particles is restricted to ﬁnite region of phase space where they bounce back and forth. There aresome solid evidences of the existence of trapped particles in the space contexts. 22,33One of forerunners developed a pseudo potential method for the construction of equilibriumsolutions, and also derived a modiﬁed Korteweg-de Vries equation (KdV) equation, often called the Schamel equation, for weakly nonlinear ion acoustic waves which are modiﬁedby the presence of trapped electrons. 34,35Ahmad and his co- workers36investigated the effects of dust polarity and trap- ping of plasma particles on dust acoustic wave within thesmall amplitude regime by using modiﬁed KdV equation as well as pseudo-potential approach; and they concluded that in case of electron trapping (ion trapping), an increase inboth electron and ion trapping parameters will increase the depth of positive (negative) Sagdeev potential, with enhanced amplitude of compressive (rarefactive) soliton. As far as we know, there is no investigation about propa- gation of DAWs in the dusty electronegative plasmas in the presence of the combination of kappa-Schamel distribution.In the present article, we are interested in extending and pro- viding the novel standpoints into previously published work in Ref. 37. For this purpose, we consider the propagation of obliquely dust acoustic solitary waves in an electronegative plasma subjected to external magnetic ﬁeld. At a ﬁrst step, we show the existence of both fast and slow modes and discusshow these modes are inﬂuenced by trapping parameter, super- thermality index, positive ion to dust density ratio, magnetic ﬁeld, and so on. We then proceed by employing a reductiveperturbation technique to derive an evolution equation in the form of one and three dimensional Schamel Korteweg-de Vries (S-KdV) equations, and ﬁnally aforementioned agentsare interpreted on intrinsic features of solitary wave. The manuscript is structured as follows. After introduc- tion, the basic equations governing our plasma model areprovided in Sec. II. Then, Section IIIdeals with linear dust acoustic wave analysis. A weakly nonlinear analysis is car- ried out in Sec. IV. Discussions (Conclusions) are respec- tively given in Sec. V(Sec. VI). II. BASIC MODEL EQUATIONS We are modeling three-dimensional propagation of DAWs in a collisionless and magnetized dusty electronega-tive plasma, which consists of electrons, positive, and nega- tive ions as well as negatively charged warm dusts ( T d6¼0). Where in such kind of plasma, the formation of the wave isdue to the inertia and the pressure contribution from the dust ﬂuid and the restoring force either by electron or ion. The ambient magnetic ﬁeld ~Bð¼B 0^zÞis assumed stationary, pointing along the z-axis. Also, we introduce the direction cosines lzð¼coshÞ, where his the direction of wave propa- gation with respect to ~B. At equilibrium, the charge neutral- ity condition requires that nn0þne0þnd0zd¼np0; (1) where ne0;nn0;np0;andnd0are the unperturbed number den- sities of electrons, negative and positive ions, and dusts, respectively, and zdð>0Þis the number of electrons residing on the dust grain surface at equilibrium. It is clear that basedon Eq. (1), the positive ion density is larger than that of nega- tive ion, so dusts become negatively charged. The nonlinear dynamics a low frequency DAW, whose wave phase speed lies between the ion and electron thermal speeds, viz., t thi/C28x=k/C28tthe, can be described by set of three-dimensional and normalized equations (continuity, mo-mentum, and Poissons equations) as @nd @tþ~r/C1 nd~tdðÞ ¼0; (2a) @~td @t¼~rw/C0~td/C1~r/C0/C1 ~td/C0~td/C2xcd^z ðÞ /C05 3bdn/C01=3 d~rnd; (2b) r2w¼ndþdnnnþdene/C0dpnp: (2c) Charge neutrality condition in the normalized form is dp¼dnþdeþ1, where the quantities dnð¼nn0=zdnd0Þ; dpð¼np0=zdnd0Þ,a n d deð¼ne0=zdnd0Þare deﬁned as density ratios. In the latter equation, we assume that the normalized number densities of kappa distributed positive ions, Schamel-kappa distributed negative ions, and Maxwellian electrons, respectively, are given by the following relations: 38 np’ð1/C0a1pwþa2pw2þOðw3ÞÞ; (3a) nn’ð1þa1nw/C0a2nw3=2þOðw2ÞÞ; (3b) ne¼expðbewÞ; (3c)083705-2 A. Sabetkar and D. Dorranian Phys. Plasmas 22, 083705 (2015) with expansion parameters a1p;a1n;a2p, and a2nas functions of the jp;nð>3=2Þandlare given below a1p¼2jp/C01 ðÞ 2jp/C03 ðÞ;a1n¼bp2jn/C01 ðÞ 2jn/C03 ðÞ;a2p¼4j2 p/C01/C16/C17 22jp/C03 ðÞ2; a2n¼8ﬃﬃﬃﬃﬃﬃﬃﬃ 2=pp 1/C0lðÞ b3=2 p 32jn/C03 ðÞ3=2Cjnþ1 ðÞ Cjn/C01=2 ðÞ: The limit of npand nnin Eq. (3)asjp;n!1 leads to Boltzmann and Schamel distribution, respectively.35,38In the above expressions, we have deﬁned the number densityn jof the j-species ðj¼n;p;e;anddfor negative and positive ions, electrons, and negative dust grains, respectively).These number densities are normalized by their respectiveunperturbed densities. Eq. (2b) includes adiabatic pressure term, i.e., P¼P 0ðnd=nd0Þcwith Pd0¼nd0kBTd, where adia- batic index c¼5=3½¼ ðNþ2Þ=N/C138, and Nis the number of degrees of freedom (in this model N¼3).~tdis the dust-ﬂuid velocity normalized by dust acoustic speed Cd¼xpdkD ½¼ ðzdTp=mdÞ1=2/C138andwis the electrostatic wave potential normalized by Tp=e, where eis the magnitude of the electric charge. ~r¼ð ^x@=@x;^y@=@y;^z@=@zÞwhere x,y, and zare the space coordinates, which are normalized by kD, where kD½¼ ðTp=4pnd0zde2Þ1=2/C138is the dusty plasma Debye radius, the time variable ( t) is normalized by the inverse of the dust-plasma frequency xpd½¼ ð4pnd0z2 de2=mdÞ1=2/C138,mdis the dust grain mass, and xcd½¼B0=ð4pnd0mdÞ1=2/C138is the dust- cyclotron frequency, which is normalized by xpd.lis a pa- rameter determining the population of trapped negative ions,whose magnitude is deﬁned as the ratio of the free negativeions temperature T nfto the trapped negative ions temperature Tnt, i.e., l¼Tnf=Tnt. When l¼1, this corresponds to a Boltzmann distribution which has no trapped electrons, andatl¼0, we have a ﬂat-topped distribution. 38Furthermore, bp¼Tp=Tn;be¼Te=Tp, and bd¼Td=zdTp, where Tjis tem- perature of species jð¼p;n;e;d). III. LINEAR WAVE ANALYSIS Looking for linear solution of dust acoustic waves for small perturbation propagating obliquely in a collisionlessmagnetized dusty electronegative plasma, we linearize Eq. (1) by using ‘¼‘ð0Þþ‘ð1Þ,w h e r e ‘¼ðnd;tdx;tdy;tdz;wÞare physical quantities of plasma and ‘ð0Þ¼ð1;0;0;0;0Þare cor- responding unperturbed parts. The perturbed quantities, ‘ð1Þ, are proportional to exp ½ið~k/C1~r/C0xtÞ/C138,w h e r e ~kandxare the wave number and the frequency. By replacing all derivatives such as ~rand@=@tbyi~kand/C0ixin main equations, respec- tively, the linearized system of Eq. (1)takes the form /C0xnð1Þ dþkxtð1Þ dxþkytð1Þ dyþkztð1Þ dz¼0; (4a) i/C0xt1ðÞ dx/C0kxw1ðÞþ5 3bdkxn1ðÞ d/C18/C19 þxcdt1ðÞ dy¼0; (4b) i/C0xt1ðÞ dy/C0kyw1ðÞþ5 3bdkyn1ðÞ d/C18/C19 /C0xcdt1ðÞ dx¼0; (4c) /C0xt1ðÞ dzþ5 3bdkzn1ðÞ d¼0; (4d) ðk2 xþk2 yþk2 zÞwð1Þþnð1Þ dþdebewð1Þ þdna1nwð1Þþdpa1pwð1Þ¼0: (4e) To linearize Eq. (4e), we have dropped the second and higher terms in the Taylor expansion for densities. From the x, y,a n d z-components of momentum equation Eqs. (4b)–(4d), we derive the following equations for tð1Þ dx;tð1Þ dy,a n d tð1Þ dz: t1ðÞ dx¼/C0kxw1ðÞ xþ5bdkxn1ðÞ d 3x/C0xcdkyw1ðÞ ix2/C0x2 cd/C0/C1 /C0x2 cdkxw1ðÞ xx2/C0x2 cd/C0/C1 /C05bdxcdixky/C0kxxcd ðÞ n1ðÞ d 3xx2/C0x2 cd/C0/C1 ;(5a) t1ðÞ dy¼/C0xkyw1ðÞ x2/C0x2 cd/C0/C1 þxcdkxw1ðÞ ix2/C0x2 cd/C0/C1 þ5bdixky/C0kxxcd ðÞ n1ðÞ d 3ix2/C0x2 cd/C0/C1 ; (5b) t1ðÞ dz¼/C0kzw1ðÞ xþ5bdkzn1ðÞ d 3x: (5c) Substituting the expressions from Eq. (5)into Eq. (4a), the perturbed density of the dusts is expressed as n1ðÞ d¼/C0k2 xþk2 z/C0/C1 x2þx2 cdk2 x x2x2/C0x2 cd/C0/C1 þk2 y x2/C0x2 cd/C0/C1() w1ðÞ 1/C05bdk2 xþk2 z/C0/C1 3x2þ5bdxcdkxixky/C0kxxcd ðÞ 3x2x2/C0x2 cd/C0/C1 /C05bdkyixky/C0kxxcd ðÞ 3ixx2/C0x2 cd/C0/C1() : (6) Next, substituting Eq. (6)into Eq. (4e)we obtain the general dispersion relation x4/C05 3bdþ1 k2þDi;e/C18/C19 k2þx2 cd/C26/C27 x2þk2 zx2cd5 3bdþ1 k2þDi;e/C18/C19 ¼0; (7) which is a quartic equation and has four roots (or two symmetrical roots) given as follows:083705-3 A. Sabetkar and D. Dorranian Phys. Plasmas 22, 083705 (2015) x2 6kðÞ¼1 253b dþ1 k2þDi;e/C18/C19 k2þx2 cd65 3bdþ1 k2þDi;e/C18/C192 k4þx4 cdþ2x2 cd5 3bdþ1 k2þDi;e/C18/C19 k2/C02k2cos2h ðÞ"# 1 28 < :9 = ;;(8a) Di;e¼dnbp2jn/C01 ðÞ 2jn/C03 ðÞþdp2jp/C01 ðÞ 2jp/C03 ðÞþdebe() ; (8b) where k¼ðk2 ?þk2 jjÞ1=2;kjjð¼kzÞand k?ð¼kx;kyÞare the parallel and perpendicular components of the wave vector ~k. By replacing the expression Di¼dnbpð2jn/C01Þ ð2jn/C03Þþdpð2jp/C01Þ ð2jp/C03Þno instead of Di;ein Eq. (8b), one can obtain the general disper- sion relation in which the electron component is depleted (in which kz¼klz). The upper curves (viz., fast mode) show the frequency of the mode corresponding to ðþÞsign, while the lower curves (i.e., slow mode) are for ð/C0Þ sign (see Eq.(8a)). The modiﬁcation of wave frequency is presented in Fig. 1due to plasma parameters. Furthermore, analyzing Eq.(8), we now focus on the two extreme limits as follows: Limiting Case 1—parallel propagation :S u c hk i n do f propagation satisﬁes at kx;ky!0;kz¼kk6¼0. In this case, the transverse velocity components of dust ﬂuid vanish [see Eq.(5)] as well as dusts move parallel to the magnetic ﬁeld ~B. So, the general linear dispersion relation reduces to x2 þkzðÞ¼5 3bdþ1 k2 zþDi;e/C18/C19 k2 z;x2 /C0¼x2 cd; (9) where minus and plus signs refer to slow mode, which is a non-propagating mode, and fast mode, respectively. For cold dust ( bd!0) together with neglecting the concentra- tion of negative ion ( dn!0) in an electron depleted mag- netized dusty plasma this equation is similar to Eq. (13)that has been obtained by Piel and Goree.39Also, we notice that wave dispersion is unaffected by the magnetic ﬁeld, and hence in the long-wavelength (i.e., for k/C28D1=2 i;e) turn into x2 k2z¼5 3bdþ1 Di;e/C16/C17 . Limiting Case 2—perpendicular propagation : In this case kz!0;k2 xþk2 y¼k2 ?6¼0. So, motion of dusts occurs only in ðx/C0yÞplane and ﬁnally by considering the follow- ing assumptions Eq. (8)becomes x2 þk?ðÞ¼5 3bdþ1 k2 ?þDi;e/C18/C19 k2 ?þx2 cd;x2 /C0¼0:(10) As before, slow wave becomes non-propagating mode, while fast waves are dependent on the wave number k?.F o ra n unmagnetized limit, Eq. (10) recovers Eq. (9)as well as by taking into account aforementioned limitations (viz., bd;dn; de!0), in the limit of jp!1 above equation is in a good agreement with those obtained by Shukla and Rahman.40 Furthermore, by removing electron component of plasma we found that the inﬁnitesimal frequency gap is seen to occur ascompared to the presence of ele ctrons in plasma (ﬁgure not shown), which is dispensable. The comparison of wave fre- quency given by Eq. (10)in two limits is delineated in Fig. 2.IV. WEAKLY NONLINEAR ANALYSIS A. Derivation of the Schamel Korteweg-de Vries equation In calculation of Sec. III(i.e., linear waves), we deal ex- plicitly with perturbed terms of ﬁrst-order such as nð0Þ dtð1Þ dx, whereas in weakly nonlinear analysis derivations are basedon perturbed terms of second or higher-order terms like n ð1Þ dtð1Þ dx. In fact, higher-order perturbations play pivotal role in the evolution of solitons when the wave grows in ampli-tude. Therefore, in order to study the dynamic of weakly nonlinear DAWs, we adopt the reductive perturbation tech- nique (RPT) 35of Schamel to derive the S-KdV equation. The stretched coordinates are deﬁned as n¼e1=4ðlxxþlyyþlzz/C0ktÞ;s¼e3=4t; (11) where lx,ly,a n d lzare the direction cosines in x,y,a n d zdirec- tions, respectively, and follows the relation, l2 xþl2 yþl2 z¼1 andedenotes a small parameter characterizing the strength of the nonlinearity as well as kis the unknown nonlinear wave speed, normalized by Cd,t ob ed e t e r m i n e dl a t e r .T h ed y n a m i - cal variables are expanded in the powers of eas follows: nd tdz tdx tdy w0 BBBB@1 CCCCA¼1 0 00 00 BBBB@1 CCCCAþen ð1Þ d tð1Þ dz e1=4tð1Þ dx e1=4tð1Þ dy wð1Þ0 BBBBBBBBB@1 CCCCCCCCCAþe 3=2nð2Þ d tð2Þ dz tð2Þ dx tð2Þ dy wð2Þ0 BBBBBBBBB@1 CCCCCCCCCAþ/C1/C1/C1 :(12) Note that in Eq. (12), the appearance of transverse velocity components ( t dx;y) at a higher order of e(compared to parallel component tdz) comes from anisotropy applied into the system by the magnetic ﬁeld.19By substituting the above expansions ofnd;w;andtdx;y;zin terms of the corresponding perturbed quantities in the basic equation Eq. (1)and making use of Eq. (11), we obtain the lowest order of efor continuity, z-compo- nent of momentum, and Poisson equations as follows: n1ðÞ d t1ðÞ dx w1ðÞ t1ðÞ dz0 BBBBB@1 CCCCCA¼003l2 z 5bdl2 z/C03k2/C0/C1 0 00 0 /C0ly lx /C01 Di;e00 0 k lz00 00 BBBBBBBBBBBB@1 CCCCCCCCCCCCAn 1ðÞ d t1ðÞ dx w1ðÞ t1ðÞ dy0 BBBBBB@1 CCCCCCA: (13)083705-4 A. Sabetkar and D. Dorranian Phys. Plasmas 22, 083705 (2015) By taking into account space plasma parameters with bd>0, the expression 3 l2 z=ð5bdl2 z/C03k2Þis always negative fork2>5bdl2 z=3, whose the accuracy is proved from theterm of kshortly. Comparing nð1Þ dandwð1Þfrom solution of Eq.(13) and using Eq. (8b), we obtain the expression for the normalized nonlinear wave speedFIG. 1. Plot of frequency xof dust acoustic waves against wave number k[appeared in Eq. (8a)] for different values of (a) weak magnetic ﬁeld xcd ð¼0:07;0:1;0:13Þ; (b) strong magnetic ﬁeld xcdð¼1;1:3;1:6Þ; (c) ratio of positive to negative ion temperature bpð¼1;2;3Þ; (d) obliqueness hð¼20/C14;30/C14;40/C14Þ; (e) dust temperature bdð¼0:001;0:01;0:03Þ; (f) superthermal index jpð¼3;5;15Þ; (g) positive ion to dust density ratio dpð¼2:06;2:30;2:50Þ.W eh a v et a k e n here in subplots h¼30/C14;jn¼1:7, and jp¼3; the typical data used here for space plasmas with negatively charged dusts are given by np0/C242:1 /C2106cm/C03;nn0/C24106cm/C03;mp/C245:02/C210/C022g,mn/C244:7/C210/C023g,Te/C24Tp/C24Tn/C24200 K, Td/C240:1e V , B0/C240:5G ,zdnd0/C24106cm/C03,a n d md/C241012mi.083705-5 A. Sabetkar and D. Dorranian Phys. Plasmas 22, 083705 (2015) k¼lzﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 dnbp2jn/C01 ðÞ 2jn/C03 ðÞþdp2jp/C01 ðÞ 2jp/C03 ðÞþdebe ! þ5 3bdvuuuut; (14) where for lz¼1 (viz., for wave propagation parallel to themag- netic ﬁeld), this is the same to that obtained in Eq. (9).I ti si m - portant to mention here that the Schamel trapping parameter l has no effect on k,a n df o r jn;jp!1 (Maxwellian limit), k!/C19kwhere /C19k¼lzﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1=/C19D2 i;eð¼dnbpþdpþdebeÞþ5bd=3q as well as for the same limits in the absence of thermal pressure of dust, above equation is identical to Eq. (14) of Ref. 37.T h e variation of this phase speed kwith different plasma parame- ters is shown in Fig. 3. On the other hand, we can derive the lowest order of e (i.e., e5=4ande3=2) from xandy-components of the momen- tum equation, after some algebraic manipulation, as t1ðÞ dy;x¼6lx;y xcd@w1ðÞ @n75 3bdlx;y xcd@n1ðÞ d @n; t2ðÞ dy;x¼6k xcd@t1ðÞ dx;y @n: (15)In Eq. (15) fortð1Þ dy;xandtð2Þ dy;x, the sign 6is corresponding to xand y-components of momentum equation, whereas the sign7in the last term of tð1Þ dy;xis in contrast to the former statement. Note that using of the perturbative expansions Eq. (12)in Eq. (2), we arrive at np’ð1/C0ea1pwð1Þ/C0e3=2a1pwð2Þþ/C1/C1/C1 Þ ¼ðnð0Þ pþenð1Þ pþe3=2nð2Þ pþ/C1/C1/C1 Þ ; (16a) nn’ð1þea1nwð1Þþe3=2½a1nwð2Þ/C0a2nðwð1ÞÞ3=2/C138þ/C1/C1/C1 Þ ¼ðnð0Þ nþenð1Þ nþe3=2nð2Þ nþ/C1/C1/C1 Þ ; (16b) ne’ð1þebewð1Þþe3=2bewð2Þþ/C1/C1/C1 Þ ¼ðnð0Þ eþenð1Þ eþe3=2nð2Þ eþ/C1/C1/C1 Þ : (16c) Similarly, by employing the same previous procedure, for the coefﬁcients of e7=4from z-component of the momentum equation and continuity equation and of e3=2 from Poisson’s equation, we have the set of following equations: @n1ðÞ d @s/C0k@n2ðÞ d @nþlx@t2ðÞ dx @nþly@t2ðÞ dy @nþlz@t2ðÞ dz @n¼0;(17a) @t1ðÞ dz @s/C0k@t2ðÞ dz @n/C0lz@w2ðÞ @nþ5 3bdlz@n2ðÞ d @n¼0; (17b) n2ðÞ dþdnn2ðÞ nþden2ðÞ e/C0dpn2ðÞ p/C0@2w1ðÞ @n2¼0: (17c) Differentiating Eq. (17c) with respect to n, we obtain @n2ðÞ d @n¼/C0dn@n2ðÞ n @n/C0de@n2ðÞ e @nþdp@n2ðÞ p @nþ@3w1ðÞ @n3:(18) Eliminating the expression @tð2Þ dz=@nfrom Eqs. (17a) and (17b) leads to FIG. 3. Plot of the nonlinear wave speed kof the DAWs [based on Eq. (14)] against (a) temperature ratio bpfor varying lzð¼1;0:9;0:7Þwith jn¼1:7 and jp¼3; (b) both jnandjpfor varying bdð¼0:001;0:01;0:02Þwith lz¼0.8; The other typical data for space plasmas are considered as the same as in Fig. 1. FIG. 2. Comparison of frequency of wave perpendicular propagation between non-Maxwellian (lower surface) [given by (10)] and Maxwellian limit (upper surface). The typical data for space plasmas are considered as the same as in Fig. 1.083705-6 A. Sabetkar and D. Dorranian Phys. Plasmas 22, 083705 (2015) @t1ðÞ dz @sþk lz@n1ðÞ d @sþklx lz@t2ðÞ dx @nþkly lz@t2ðÞ dy @n þ5 3bdlz/C0k2 lz ! @n2ðÞ d @n/C0lz@w2ðÞ @n¼0: (19) Again, combining Eqs. (19) and(18) with the help of (13), we obtain @w1ðÞ @sþk1/C0l2 z/C0/C1 5bdþ3 Di;e/C18/C19 6x2 cd@3w1ðÞ @n3þl2 z 2Di;ek@w2ðÞ @n þl2 z 2D2 i;ek/C0dn@n2ðÞ n @n/C0de@n2ðÞ e @nþdp@n2ðÞ p @nþ@3w1ðÞ @n3() ¼0: (20) Substituting the expressions for nð2Þ e;nð2Þ n, and nð2Þ p,t ob e obtained from Eq. (16) and noting that coefﬁcient of @wð2Þ=@nvanishes after some straightforward simpliﬁca- tions, and eventually we derive the following equation: @W @sþAW1 2@W @nþB@3W @n3¼0; (21) where wð1Þ/C17W. Equation (21) is called a one-dimensional Schamel Korteweg-de Vries (S-KdV)35equation and, more recently, a similar equation has been already derived byWilliams et al. , 38We note that the coefﬁcients AandBcan be expressed explicitly as A¼ﬃﬃﬃ 8p ﬃﬃﬃppl2 z D2 i;ekdn1/C0lðÞ b3=2 p 2jn/C03 ðÞ3=2Cjnþ1 ðÞ Cjn/C01 2/C18/C19 ; (22a)B¼5bdk1/C0l2 z/C0/C1 6x2 cdþ1 2Di;ek1/C0l2 z/C0/C1 x2 cdþl2 z Di;ek() ;(22b) where Di;epreviously introduced in Eq. (8b). By substituting the term Difrom linear regime instead of Di;ein(22),w e obtain the expressions for Aand Bin which the electron component is depleted. This is similar to that reported byAlinejad, 41but our expression includes the superthermal pa- rameter. The Ais the coefﬁcient of nonlinearity, which deter- mines the steepness of the wave and is proportional to the population of trapped negative ions l, while Bis the coefﬁ- cient of dispersion, causing wave broadening in Fourierspace, and the effect of magnetic ﬁeld (via x cd) appears in B. If we consider there is no superthermality (i.e., jn;jp¼1 ), A!l2 zdnð1/C0lÞb3=2 p=ﬃﬃﬃpp/C19D2 i;e/C19k, and by substituting /C19D2 i;eand /C19kinstead of its corresponding expressions, respectively, which are mentioned after Eq. (14), we obtain coefﬁcient of dispersion Bin Maxwellian limit. The nonlinearity and dis- persion coefﬁcients (i.e., AandB) from S-KdV equation (22) are plotted in Fig. 4. In order to seek a stationary solitary waves solution of Eq. (21), we introduce a variable transformation38into moving frame, viz., g¼vðn/C0M0sÞ,w h e r e M0is the constant speed of the wave frame (normalized by Cd). With the help of hyperbolic tangent method42as detailed in the Appendix , together with employing appropriate boundary conditions (viz., W!0; @W=@g!0, and @2W=@g2!0a tg!61), we obtain WðgÞ¼Wmsech4ðg=L0Þ; (23) where Wm¼15M0 8A/C16/C172 andL0¼ﬃﬃﬃﬃﬃﬃ 16B M0q are amplitude (height) and width of the solitary waves with B>0, respectively. It is FIG. 4. Plot of the coefﬁcients of Schamel equations (S-KdV and S-ZK) [represented by Eqs. (22) and(26)] (a) AandA0versus both jnandjpwith ﬁxed values of l¼0:5 and lz¼0.8; (b)BandC0versus both jnandjpat lz¼0.8; (c) Aversus bpfor varying l (¼/C00:3,/C00.2, 0.1); (d) BandD0ver- sus xcd; in subplots (c) and (d): jn¼1:7,lz¼0.8, and jp¼3. The other parameters for space plasmas areconsidered as the same as in Fig. 1.083705-7 A. Sabetkar and D. Dorranian Phys. Plasmas 22, 083705 (2015) evident from height expression of soliton, unlike the ﬁndings of Ref. 43, we are associated with compressive (positive po- larity) solitary waves. We know that the characteristics ofsmall amplitude of compressive solitary waves essentially depend on the form of nonlinearity Aand dispersion Bcoefﬁ- cients in the S-KdV equation. We have, therefore, plottedcompressive solitons (23) for different values of the relevant parameters, as shown in Figs. 5(a)–5(d) . B. Derivation of the Schamel Zakharov-Kuznetsov equation For the sake of completeness, we extend the one- dimensional S-KdV equation to three dimensional case byemploying the RPT in which new stretched coordinates 45are deﬁned as Z¼e1=4ðz/C0kzktÞ;X¼e1=4x;Y¼e1=4y;and s¼e3=4t; (24) where kzk½¼1 Di;eþ5 3bd/C16/C171=2/C138is the wave phase velocity nor- malized by Cd. Now, making use of the above stretched coordinates and Eq. (12) and inserting into Eq. (1)and after some simpliﬁcations, we get a Schamel Zakharov-Kuznetsov (S-ZK) equation as @Wzk @sþA0W1 2 zk@Wzk @ZþC0@3Wzk @Z3 þD0@3Wzk @Z@X2þ@3Wzk @Z@Y2/C18/C19 ¼0;(25) where wð1Þ/C17Wzk, the coefﬁcients of nonlinearity A0, disper- sion C0, and mixed derivative D0are given, respectively, by A0¼2C0ﬃﬃﬃ 8p ﬃﬃﬃppdn1/C0lðÞ b3=2 p 2jn/C03 ðÞ3=2Cjnþ1 ðÞ Cjn/C01 2/C18/C19 ; C0¼5bd/C03k2 zk/C16/C172 18kzk;D0¼C0þk3 zk 2x2 cd ! :(26)By putting lz¼1 into nonlinearity and dispersion coefﬁcients of (1D) S-KdV equation, namely, Eq. (22), we, respectively, reach to the coefﬁcients of A0andC0. The variation of A0, C0, and D0with physical parameters is depicted in Fig. 4. Solitary wave solution of Eq. (25) can be found by using the well-known hyperbolic tangent method42as already dis- cussed in the Appendix . In this method, we transform the space coordinates to a new coordinate, i.e., g¼v0ðlxX þlyYþlzZ/C0M0tÞwhere M0is a constant velocity normal- ized by Cd. Also, the inverse of v0gives the width of the soli- ton. The sum of the squared direction cosines along x,y, and z-axes must always be unity. Finally, using the vanishing boundary condition W!0 and their derivatives up to sec- ond order for g!61, we obtain the steady state solution for S-ZK45equation as WzkðgÞ¼Wmzksech4ðg=LzkÞ; (27) where the amplitude Wmzkand the width Lzk¼1=v0of the soliton are given by Wmzk¼15M0 8A0lz/C16/C172 , and Lzk¼ﬃﬃﬃﬃﬃﬃﬃ 16F0 M0q with F0¼fC0l3 zþD0lzð1/C0l2 zÞg. Figure 6provides how ampli- tude and width of the Schamel’s solitons vary with oblique- ness, trapping, and difference between temperature of two species ions bpð¼Tp=TnÞ. V. NUMERICAL INVESTIGATIONS AND DISCUSSIONS The propagation of the three dimensional DAWs with opposite polarity ions is investigated. The linear and corre- sponding 1D and 3D Schamel equations with the help of the RPT35are derived. In the numerical investigations, we have used parameters that may be representative of space plasma environments (e.g., in the Earth’s mesosphere,10a dusty elec- tronegative plasma region at an altitude of about 95 km) where the respective value of plasma parameters are men- tioned in the caption of Fig. 1. To discuss the effect of rele- vant physical parameters, the dispersion relation Eq. (8), such as (i) strength of external magnetic ﬁeld ðxcdÞ, (ii) positive to negative ion temperature ratio ðbp), (iii) obliqueness ( lz), (iv) dust temperature ( bd), (v) superthermality of positive and FIG. 5. Plot of the S-KdV soliton solution W[represented by Eq. (23)] against gfor different values of (a) lð¼0:3;0:2;/C00:2;/C00:4Þ; (b) xcd ð¼0:07;0:1;0:13;0:2Þ; (c) bdð¼0:001;0:01;0:02;0:03Þ; (d) jnð¼1:6;1:7;1:8;1:9Þ. The other parameters here are taken: jn¼1:7;l¼0:5,lz¼0.8, jp¼3, and M0¼0:06 as well as the remaining parameters for space plasmas are considered as the same as in Fig. 1.083705-8 A. Sabetkar and D. Dorranian Phys. Plasmas 22, 083705 (2015) negative ions ( jn;jpÞ, and (vi) positive ion to dust density ratioðdpÞon the linear properties of DAWs, we have numeri- cally investigated these modes separately in Figs. 1(a)–1(g) . (i) Effect of magnetic ﬁeld: The effect of magnetic ﬁeld on frequency xof fast and slow wave is illustrated in Figs. 1(a) and 1(b). In Fig. 1(a) for the weaker (xcd<1) magnetic ﬁeld strength, the fast mode fre- quency sharply increases as k!0 with increasing xcdand eventually converges at higher values of k. On the other hand, the change of frequency xof slow mode is noticeable for wave numbers gratifyingk/H114070:2. Indeed, the change of frequency gap between two modes at long and short-wavelength follows con- spicuously a reverse trend, which is unobservedbefore in electronegative plasmas. 19For stronger magnetic ﬁeld strength ( xcd>1) in Fig. 1(b), signiﬁ- cant increase in the slow wave frequency starts atgreater values of wave numbers, i.e., k/H114073:5 and con- versely the inﬂuence of x cdon slow wave in the long- wavelength limit, i.e., k!0 is almost negligible. Also, for strong magnetic ﬁeld (via xcd>1), in the absence of dust thermal pressures (i.e., bdtends to zero) there is a large frequency gap as compared tothe presence of b dfor wave number that satisfying k/H1140713, as exhibited in Fig. 1(b), whereas such kind of scenario has not been observed for weak xcd/C281. These interesting features of slow modes have notreported in any pair-ion plasma up to now. (ii) Effect of positive to negative ion temperature ratio: Figure 1(c) shows the impact of temperature ratio b p on fast and slow wave. As bpincreases (i.e., as the discrepancy of temperature between ions enhances),the wave frequency xof fast (slow) mode enervates but in the opposite wave wavelength, respectively, in short (long)-wavelength limits. In other words,increasing the temperature ratio leads to a reduction of the frequency gap between the modes. However, it is obvious from Fig. 1(c)that the effect of b pon slow mode is comparatively weak as k!1 in comparison to its effects on fast mode in the short wavelength limit (i.e., k!1). (iii) Effect of obliqueness on wave propagation: Inﬂuence of obliqueness or propagation angle hon frequency x of fast (upper curves) and slow (lower curves) modesare depicted in Fig. 1(d). This ﬁgure represents that as the direction of wave propagation with respect tomagnetic ﬁeld increases, the wave frequency xof the fast mode increases, where such kind of behavior isappreciable starts at k/H114070:07, while the reverse behavior is acceptable for slow mode, and the growthof the slow wave frequency with increasing the anglehis seen to occur in 0 :1/H11351k/H113510:3, and eventually is saturated. As a result, an enhancement of frequencygap between modes is viewed by the signature of h. Furthermore, the fast wave frequency (upper curves)tends to approach a constant value for k/H114071. (iv) Effect of dust grains temperature: The effectiveness of the thermal pressures of dusts or dust temperatureon dispersion properties of the low-frequency modesare displayed in Fig. 1(e). This ﬁgure clearly eluci- dates that impact of an increase in dust temperatureb don slow mode frequency, thereby increasing of slow wave frequency x, is restricted only to a less tract of wave numbers, i.e., 0 /H11351k/H113510:28, while the same effect (i.e., increasing in fast mode frequency)on fast mode is more noticeable, and the effectstrength (i.e., an increase in degree of separationbetween each subﬁgure on the frequency xof fast mode) covers more tract of wave numbers that satis-ﬁes 0 :15/H11351k/H113511. Also, for slow wave, as k!1, wave frequency is found to remain constant with k. (v) Effect of superthermality of negative and positive ions: Figures 1(f)exhibits the effectiveness of super- thermality of positive ions (represented by j p)o nb o t h upper mode (fast) and lower mode (slow). In contrastto Fig. 1(c), as superthermality of both types of ions reduces (i.e., the value of j n;pis increased), the fre- quency xof both fast and slow wave increase, but the effect of jpon slow mode is negligible. Furthermore, the signature of jnon frequency of both modes is exactly similar to bd(ﬁgure not shown). Overall, the inﬂuence of jnon frequency xof fast and slow mode is stronger than the effect of jpask!1. However, the increase of frequency gap between the modes atshort wavelengths is visible as in Fig. 1(f). (vi) Effect of the positive ion to dust density ratio: Signature of the positive ion to dust density ratio isplotted in Fig. 1(g). Opposite to the effects of positiveFIG. 6. Plot of the amplitude and width of both S-KdV and S-ZK soliton versus (a) obliqueness ðlzÞand trapping parameter ðlÞwith lz¼0.8 and l¼/C00:4; (b) temperature ratio bp ð¼Tp=TnÞwith lz¼0.9 and l¼0:8; in subplots (a) and (b): jn¼1:7, jp¼3, and M0¼0.06. The other pa- rameters for space plasmas are consid- ered as the same as in Fig. 1.083705-9 A. Sabetkar and D. Dorranian Phys. Plasmas 22, 083705 (2015) ion superthermality, as the positive ion to dust density ratio dpescalates (i.e., subsequently increasing the concentration of electron deto satisfy the charge neu- trality condition), the frequency xof fast modes ener- vates in the short-wavelength limit ( k!1). Hence, the frequency gap between the modes is found todecrease, while modiﬁcation of frequency of slow mode is almost dispensable and remains constant for k/H114070:28. This feature appears completely in contrast to ﬁndings of Ref. 19. The features of the fast mode for parallel and perpendic- ular propagations are, respectively, modiﬁed in the form of linear and non-linear by physical parameters. It is found that (not shown in the ﬁgures) the effects of increasing the valuesofb d;jn;jp, andðbpanddpÞare to increase (decrease) the frequency of the fast modes in short-wavelength limit (k!1). Furthermore, Fig. 2presents wave frequency of per- pendicular propagation in both non-Maxwellian and Maxwellian limits. This ﬁgure reveals that wave frequency xof fast mode in Maxwellian limit ( jn;p!1 ; blue surface) is higher as compared to non-Maxwellian one (green sur- face), and it seems that both surfaces overlap each other at short-wavelength (i.e., k/C291). Typical variations of the phase velocity kwith relevant parameters are delineated in Fig. 3. Figure 3(a) shows that for plasma with negatively charged dusts, the value of k tends to decrease with increasing values of the positive to negative ion temperature ratio bp, while the opposite trend is acceptable with increasing of lz(i.e., as angle of propagation increases, subsequently the nonlinear wave speed enervates). Also, it is obvious as jn;jp!1 (Maxwellian limit), the values of kare greater, as evident from Fig. 3(a). According to Fig. 3(b), it is necessary to point out that strong superther- mality (low value of jn;jp) makes the phase speed enhance non-monotonously and then remains practically constant for weak superthermality (i.e., large value). Thus, the impact of superthermality of negative ions jnonkis stronger than pos- itive ion superthermality jp. Moreover, the effect of thermal pressure of dusts on the behavior of phase speed illustrate that an enhancement in the dust temperature bdleads to an increase of k. Typically, the existence of compressive (viz., positive potential) and rarefactive (viz. , negative potential) solitons is reliant highly on the sign of a Schamel nonlinearity coefﬁ- cients ( AandA0). Thus, here, we have depicted the results numerically. Figures 4(a)and4(b)exhibit the variation of non- linearity and dispersion coefﬁcients of S-KdV and S-ZK against superthermality. We point out that for all realistic val- ues of space plasma parameters, we have A;B;A0;C0;D0>0, as physically excepted in order to conﬁrm reality of soliton width ( L0andLzk)i n(23)and(27). So, the present model sup- ports only compressive solitary wave. We see that the nonli-nearity coefﬁcient of Schamel equation illustrates different behaviors against positive and negative ions superthermality j nandjp, and subsequently shows inverse signature on ampli- tude of solitons. In Fig. 4(a),w eo b s e r v et h a t AandA0enhan- ces with increasing jpin value, whilst at stronger superthermality cases (1 :6/H11351jp/H113512:5), these coefﬁcientsprimarily represents a linear response and then saturates for higher values of jp(/H114072:5), and hence the amplitude sup- presses. Contrary to jp, variations of them in terms of jnfol- lows opposite scenario; overall, both AandA0are strongly affected by superthermality. In Fig. 4(b), on the other hand, the dispersion coefﬁcients of the Schamel (i.e., BandC0)a r eo n l y weakly dependent on the jpparameter, enhancing slightly at very low value of jp(/H113512:6). Inversely, signature of jnonBis signiﬁcant. Most importantly, evolution of S-KdV dispersion coefﬁcient ( B) is greater than S-ZK one. Based on numerical results, we have noticed that there is an inﬁnitesimal gapbetween the presence and the absence electrons in this model. The role of negative ions trapping parameter lis also high- lighted in Figure 4(c), one can see that with enervating in val- ues of l(i.e., less ratio of the negative ions are trapped), the values of nonlinearity coefﬁcient Aare enhanced, but with increasing values of b pð¼Tp=TnÞ, evolution A0of S-ZK equa- tion is more in comparison to A; moreover, it is worth mention- ing here that in Maxwellian limit (for jn;p!1 ), the amplitude ( WmandWmzk) of compressive solitary waves are signiﬁcantly impressed as compared to non-Maxwellian case (see Fig. 4(c)). Interestingly, we remark that smaller tempera- ture ratio of positive to negative ions leads nonlinearity Ato grow faster than for bp/H114075. The variation of S-KdV dispersion coefﬁcient Band mixed derivative D0of S-ZK equation with strength magnetic ﬁeld xcdare depicted in Fig. 4(d). With increasing values of xcdð/B0Þ, both of them are mitigated noticeably. In other words, coefﬁcient Bis saturated immedi- ately for small values of xcdð/H114070:4Þas compared to mixed derivative D0. Figure 5introduces the electrostatic potential Wof S- KdV soliton for DAWs which is expressed in Eq. (23). Figure 5(a) demonstrates how the solitary wave solution varies with different proportions of negative ions trapping. Due to the fact that only amplitude of solitons is reliant on l parameter (see nonlinearity coefﬁcient), trapping does not have an effect on the width of soliton, whereby decreased the values of l(smaller proportion of the negative ions are trapped) has the effect of reducing the amplitude Wmsolitary wave signiﬁcantly. In contrast to Fig. 5(a), Figure 5(b)shows that the magnetic ﬁeld has inﬂuence only on the width L0of the soliton, since the effect of B0(viaxcd) is only entered into the dispersion coefﬁcient B. A stronger magnetic ﬁeld leads to reduce the width of compressive DAWs, while theheight remains constant, which is consistent with the result of Ref. 44. In other words, physically, the role of magnetic ﬁeld in system is to attach the components of plasma stifﬂyto the lines of force so that transverse movements of particles are restrained within the ﬂuid element. The dependence of the height W mand width of compressive solitons on the tem- perature dust bdis illustrated in Fig. 5(c). This ﬁgure eluci- dates that thermal energy is vigorous enough to escalate the amplitude and width L0of soliton. Figure 5(d) manifests the characteristic features of the S-KdV soliton with the varia- tion of superthermality of negative ions jn. We see that at higher levels of superthermality (i.e., great in value of jn) amplitude Wmof solitary wave become higher and wave structures become wider, whilst superthermality of positive ions jphas signiﬁcant and opposite effects on the soliton083705-10 A. Sabetkar and D. Dorranian Phys. Plasmas 22, 083705 (2015) proﬁles, which is overt from Fig. 4(a). In the latter, we also have examined the signature of the positive ion to dust den-sity ratio d pon solitary wave graphically but not shown here. The examination reveals that the amplitude Wmof soliton increases signiﬁcantly with increasing dp, whereas its effect on width L0of soliton is reverse. Furthermore, Fig. 6suggests that solitons generated in one dimensional (S-KdV) and three dimensional (S-ZK)include the identical width (both L 0andLzk) and amplitude (WmandWmzk) for ﬁxed values of physical parameters de- spite the fact that its corresponding coefﬁcients are different(see Figs. 4(a)–4(d) ). From Fig. 6(a), we have noticed that the height of the Schamel solitons mitigates for increasingthe obliqueness l z(red-line), whereas an enhancement (miti- gation) of solitary wave width (i.e., L0andLzk) is seen to occur for the range of 0 /H11351lz/H113510:55 (0 :55/H11351lz/H113511), as obvious from the blue-line. Also, a change of lon ampli- tude is overt from the green line. Opposition to the effect ofb don proﬁle soliton, with enhancing in value of temperature ratio of two species ions bpð¼Tp=TnÞ, the intrinsic proper- ties of compressive solitons shrink, as clear from Fig. 6(b), but width of compressive solitary waves (green line) issaturated immediately as compared to its correspondingamplitude ( W mandWmzk). Finally, the effect of remainder parameters on proﬁle S-ZK soliton is identical to that men-tioned in Fig. 5. VI. SUMMARY In this work, we have presented the linear and nonlinear properties of low-frequency dust acoustic solitary wave propa-gating in a dusty electronegative plasma, whose constituents are the Maxwellian electrons, positive, and negative ions. Negative ions were assumed to obey a kappa-Schamel densitydistribution, while positive ions were assumed to be onlysuperthermal particles. In the linear regime, we have obtaineda dispersion relation, leading to two separate modes. Theproperties of these modes are addressed with the effects ofpropagation angle with magnetic ﬁeld ( h), magnetic ﬁeld strength ( x cd), dust temperature ( bd), ion temperature ratio (bp), positive ion to dust density ratio ( dp), and superthermal index of two species ions ( jn;p)( s e eF i g s . 1and2). We have found that a change of wave frequency of fast and slow modesappears at different wavelength for weak magnetic ﬁeld. Also,it is clear that the frequency xof the fast (slow) mode increases (decreases) with increasing obliqueness, andthus obliqueness causes the frequency gap between the modesto escalate. Furthermore, a reverse behavior of change infrequency of both modes is observed by the effect of b pand jn. Similar to the impact of jn, for a wave number larger (smaller) than its critical value kcr, separation between these two modes is enhancing (enervating) with an increase valueofb d. In nonlinear theory, a standard reductive perturbation technique is employed to derive a 1D and 3D Schamel equa-tion, and its corresponding solitary wave solution with thehelp of hyperbolic tangent method. According to Fig. 3,i ti s obvious that the changes of phase velocity kagainst b pand superthermality parameters are in opposite trend, and thesignature of jnis more noticeable on phase velocity. Additionally, the coefﬁcients of the Schamel equations areplotted against relevant physical parameters (see Fig. 4). It is found that variations of nonlinearity coefﬁcients in terms of j n;pfollow conﬂicting behavior. Furthermore, properties of aforementioned parameters on solitary wave are investigated.Most importantly, the effect of landx cdappears on different aspects of soliton proﬁle and follows the same trend (see Figs. 5and6). Our results are practical to elucidate the fea- tures of electrostatic waves in dusty electronegative plasmas,which are commonly exists in the Earth’s mesosphere, 10etc. ACKNOWLEDGMENTS The authors gratefully acknowledge the anonymous referee for useful comments which greatly helped to improve the manuscript. APPENDIX: DERIVATION OF SOLITARY WAVE SOLUTION OF EQ. (21) The general solution of Equation (21) using the hyper- bolic tangent (tanh) method is detailed below. Let Wðn;sÞ ¼WðgÞ, where g¼vðn/C0M0sÞ. So, Eq. (21)becomes /C0M0v@W @gþAvW1 2@W @gþBv3@3W @g3¼0: (A1) LetW1=2¼/. Integrating with respect to the variable g,a n d assuming a solution in which /!0;@/=@g!0;@2/=@g2!0 atg!61), we obtain /C0M0/2þ2A 3/3þ2v2B/@2/ @g2þ@/ @g/C18/C192() ¼0:(A2) Using the transformation y¼tanhðgÞ, and noting that ð@=@gÞ ¼ð1/C0y2Þðd=dyÞand considering a solution /such that: / ¼Panyn¼a0þa1yþa2y2þ/C1/C1/C1 ,E q u a t i o n (A2) becomes /C0M0X anyn/C16/C172 þ2A 3X anyn/C16/C173 þ2v2B /C2X anyn/C16/C17 1/C0y2/C0/C1 d dy1/C0y2/C0/C1 d dyX anyn/C16/C17/C20/C21 /C26 þ1/C0y2/C0/C1 d dyX anyn/C16/C17/C20/C212) ¼0: (A3) Truncating at n¼2 and equating to zero the different coefﬁ- cients of different powers of yfunctions, one can obtain the following set of algebraic equations for a0;a1;a2, and v: y6:20a2 2v2Bþ2A 3a3 2¼0; (A4) y5:24a1a2v2Bþ2a1a2 2A¼0; (A5) y4:ð6a2 1/C032a2 2þ12a0a2Þv2B þð2a2 1a2þ2a0a2 2ÞA/C0a2 2M0¼0; (A6) y3:/C036a2a1þ4a1a0 ðÞ v2B þ2a3 1 3þ4a0a1a2/C18/C19 A/C02a1a2M0¼0; (A7)083705-11 A. Sabetkar and D. Dorranian Phys. Plasmas 22, 083705 (2015) y2:ð12a2 2/C08a2 1/C016a0a2Þv2Bþð2a0a2 1þ2a2 0a2ÞA þð /C0 a2 1/C02a2a0ÞM0¼0: (A8) We assume that A;B>0:Solving for a0;a1;a2, and v,w e ﬁnd a2¼/C030v2B A; (A9) a1¼0; (A10) a0¼1 A5M0 8þ20v2B/C18/C19 ; (A11) v¼M0 16B/C18/C191=2 : (A12) By substituting above coefﬁcients in /¼a0þa1yþa2y2 þ/C1/C1 /C1 and considering W¼/2, our ﬁnal solution is, there- fore, given by W¼fa0þa2tanh2½vðn/C0M0sÞ/C138g2: (A13) 1C. K. Goertz, Rev. Geophys. 27, 271, doi:10.1029/RG027i002p00271 (1989). 2E. C. Whipple, T. G. Northrop, and D. A. Mendis, J. Geophys. Res. 90, 7405, doi:10.1029/JA090iA08p07405 (1985). 3U. de Angelis, V. Formisano, and M. Giordano, J. Plasma Phys. 40, 399 (1988). 4T. G. Northrop, Phys. Scr. 45, 475 (1992). 5J. Goree, Plasma Sources Sci. Technol. 3, 400 (1994). 6V. E. Fortov et al. ,J. Exp. Theor. Phys. 87, 1087 (1998). 7V. W. Chow, D. A. Mendis, and M. Rosenberg, J. Geophys. Res. 98, 19065, doi:10.1029/93JA02014 (1993). 8A. Barkan, R. L. Merlino, and D. 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1.3076140.pdf	Microwave-assisted magnetization reversal and multilevel recording in composite media Shaojing Li,1,2, a/H20850Boris Livshitz,1,2H. Neal Bertram,2,3Eric E. Fullerton,1,2and Vitaliy Lomakin1,2 1Department of Electrical and Computer Engineering, University of California, San Diego, California, USA 2Center for Magnetic Recording Research, University of California, San Diego, California, USA 3Hitachi San Jose Research Center, San Jose, California, USA /H20849Presented 13 November 2008; received 20 September 2008; accepted 23 December 2008; published online 2 April 2009 /H20850 Microwave-assisted magnetic reversal /H20849MAMR /H20850is studied for media comprising exchange-coupled composite elements comprising soft and hard layers. Reversal in such elements occurs undersubstantially reduced reversal ﬁelds, microwave ﬁelds, and microwave resonant frequencies ascompared to those for homogeneous elements. Reversal can occur in uniform modes as well asnonuniform domain wall assisted modes depending on the soft layer thickness. In addition, amultilevel MAMR scheme is suggested where the recording media comprise multiple levels ofelements, with each level having a distinct resonant frequency. These levels are addressedindividually by tuning the frequency of the microwave ﬁeld. © 2009 American Institute of Physics . /H20851DOI: 10.1063/1.3076140 /H20852 I. INTRODUCTION A major limitation to increasing magnetic recording ar- eal densities is the superparamagnetic effect, which leads tospontaneous reversal in small magnetic particles become toosmall. 1,2Overcoming the superparamagnetic effect requires using materials with high anisotropy, which translates intohigh reversal ﬁelds. Several methods have been proposed tosolve this writability problem. 3–15Microwave-assisted mag- netic reversal /H20849MAMR /H20850signiﬁcantly reduces the reversal ﬁeld when the microwave ﬁeld frequency matches the mediaferromagnetic resonance /H20849FMR /H20850frequency. 10,12,13However, for ultrahigh densities the required reversal ﬁelds as well asmicrowave ﬁelds and frequencies can still be too high formedia with homogeneous elements. This paper studies MAMR in composite /H20849soft-hard /H20850 media. 5,16–18MAMR in such media occurs at signiﬁcantly lower reversal ﬁelds, microwave ﬁelds, and microwave fre-quencies compared to those of homogeneous elements. Inaddition, we show that MAMR schemes can be used formultilevel recording, in which each layer has a distinct FMRfrequency and is addressed by tuning the microwave fre-quency. II. MAMR IN COMPOSITE ELEMENTS Consider a dual-layer composite magnetic element com- posed of hard and soft sections. The /H20849bottom /H20850hard section is of size w,w, and dhin the x,y, and zdimensions, with vertical uniaxial anisotropy energy density Kh. The /H20849top/H20850soft section is of size w,w, and dswith vanishing anisotropy. The sections are coupled ferromagnetically through their com-mon interface with surface exchange energy density J s. Both layers have damping constants /H9251=0.1, saturation magnetiza-tionMs, and exchange length lex=/H20881A/Ms=w, where Ais the exchange constant. The element are subject to an externalﬁeld, which comprises an applied bias ﬁeld /H20849serving as a switching ﬁeld /H20850and a microwave ﬁeld /H20849serving as an assist- ing ﬁeld /H20850. The bias ﬁeld H ais applied at an angle of 45° to the easy axis. The microwave ﬁeld has a frequency fmw, amplitude Hmw, and it is applied along the xaxis. For given Hmwandfmw, there is a minimal Hr, referred to as reversal ﬁeld Hr, that leads to reversal over a reversal time tr, deﬁned as the time required for the magnetization vertical compo-nent to reach the opposite level of 0.9 M s. All results are obtained by numerically solving the Landau-Lifshitz-Gilbertequation, taking into account all effective ﬁelds and assuringnumerical accuracy. 6–8The elements in Fig. 1can be used to construct media for high-density magnetic recording. For ex-ample, a bit patterned media with pitch of 8 nm and w=d h =5 nm results in a recording density of 10 Tbit /in2with thermal stability above 70 kBTwith the Boltzmann constant kBand temperature T=400 K. The chosen parameters are representative of practical materials, such as FePt. The reversal behavior of composite and homogeneous elements is studied and compared for HK=2Kh/Ms =60 kOe, Ms=1250 emu /cm3,Js=11.25 ergs /cm2, and dh=w=5 nm. Figure 1shows the reversal time tras a func- tion of Hrand fmwfor homogeneous and composite ele- ments. Dark regions represent nonreversal while brighter re-gions show ﬁnite reversal times /H20849in picoseconds /H20850. Figure 1/H20849a/H20850 depicts the results for a homogeneous element of thicknessd h=2wforHmw=0.14 HK. A resonance dip with the minimal reversal ﬁeld Hrres=10 kOe /H110610.17Hkis obtained for the reso- nant frequency of around fmwres=120 GHz. These Hrresandfrres are very high and may be hard to achieve in practical record- ing systems. For composite element, however, the situation isdifferent. Figures 1/H20849b/H20850and1/H20849c/H20850show the results for compos- ite elements with d s=w/2 and ds=w, respectively. The mi-a/H20850Electronic mail: sli@ucsd.edu.JOURNAL OF APPLIED PHYSICS 105, 07B909 /H208492009 /H20850 0021-8979/2009/105 /H208497/H20850/07B909/3/$25.00 © 2009 American Institute of Physics 105 , 07B909-1crowave ﬁeld for the results in Figs. 1/H20849b/H20850and 1/H20849c/H20850was Hmw=0.07 HK, which is half of that used in Fig. 1/H20849a/H20850. The resonant frequency decreases signiﬁcantly being around fmwres=42 GHz and fmwres=23 GHz for ds=w/2 and ds=w cases, respectively. The corresponding Hrresalso decrease sig- niﬁcantly to Hrres=6.6 kOe /H110610.11HKand Hrres=4.8 kOe /H110610.08HK, respectively. Moreover, trfor composite elements are about half of that for the homogeneous element and varysmoothly as a function of H aand frres. These substantially lower resonant frequencies, microwave ﬁelds, reversal ﬁelds,and reversal times are important for practical MAMR appli-cations. The slowly changing and narrowly distributed rever-sal time suggests a more stable switching for composite ele-ments. It should be noted that the energy barriers for the homo- geneous and composite elements in Fig. 1are mostly deter- mined by the domain wall energy E dw=4w2/H20881AK h=70kBT /H20849with T=400 K/H20850since the element’s size is greater than the domain wall length ddw=4/H20881A/Kh/H110154.2 nm. Therefore, the gains for the composite elements are obtained without sacri-ﬁcing the thermal stability. 19,20 The resonant behavior in Fig. 1reﬂects the FMR prop- erties of the considered elements. Two reversal mechanismsare observed as shown schematically in Fig. 2. For thin com- posite elements with thickness below the domain wall length,precession is ﬁrst enhanced coherently in the soft layer, andit, assists reversal in the hard layer. For thick composite el-ements with layers thicker than the domain wall length, the reversal in the soft layer is incoherent. The reversal starts inthe top part of the soft section and then a domain wall isformed in the soft section. The domain wall propagatesthough the soft and subsequently through the hard section.The resonant frequency in this case is mainly determined bythe external ﬁeld with the exchange ﬁeld. These two ﬁeldsare much smaller than the anisotropy ﬁeld H K, thus leading to a signiﬁcant FMR frequency reduction. III. MAMR FOR MULTILEVEL RECORDING From Fig. 1it is clear that the FMR frequencies can be tuned in a wide range by either changing the anisotropy ﬁeld50 100 150 200 250 3000510152025 fmw(GHz) 600800100012001400160018002000 15 20 25 30 35 40 45 50 5524681012 fmw(GHz) 500550600650700750800850900950100020 30 40 50 60 70 8024681012 fmw(GHz)Ha(KOe) 500100015002000(a)( b ) (c) homogenous compositehd ww sd hd ww sJ250 ps500 ps 500 ps 150300 200250350400450 150300 200250350400450 200 150Ha,(kOe) Ha,( k O e )Ha,( k O e ) FIG. 1. /H20849Color online /H20850Color map of the reversal time /H20849given in picoseconds in the color bar /H20850vs the microwave frequency fmwand applied bias ﬁeld Hafor HK=60 kOe, Ms=1250 emu /cm3,Js=11.25 ergs /cm2,/H9270=0.1 ns, and /H9251=0.1. /H20849a/H20850homogeneous element with dh=2w;/H20849b/H20850composite element with dh=w, ds=w/2;/H20849c/H20850composite element with dh=w,ds=w. ()a ()b FIG. 2. Schematic representation of the spin time evolution in the regime of /H20849a/H20850uniform and /H20849b/H20850nonuniform /H20849microwave-assisted domain wall /H20850reversal.07B909-2 Li et al. J. Appl. Phys. 105 , 07B909 /H208492009 /H20850in the case of homogeneous elements or by changing the anisotropy ﬁeld, coupling, and geometrical parameters in thecase of composite elements. The possibility to tune the FMRand reduce the reversal ﬁeld near this frequency suggests anovel multilevel recording scheme. The proposed mediacomprise several layers, where each layer has a differentFMR frequency /H20851Fig. 3/H20849a/H20850/H20852. The microwave ﬁeld is used to assist reversing elements in different levels by tuning themicrowave frequency to the FMR frequency of the layer be-ing recorded. This method is anticipated to result in a multi-level recording scheme with a number of advantages overother multilevel recording methods. For example, there ex-pected to be no need in multipass recording since every levelcan be addressed independently. This scheme does not re-quire addressing the elements in different layers by differentstrength of the reversal ﬁeld. In addition, a recording systemthat can generate microwave ﬁelds at several frequencies po-tentially can address several levels simultaneously, thus in-creasing the recording speed. To demonstrate the possibility of recording elements with different FMR frequencies independently, we consideran example of a two-level system comprised of homoge-neous elements /H20851Fig. 3/H20849a/H20850/H20852. In this system, the element in Layer 1 and Layer 2 have anisotropy H K1=15 KOe and HK2=12 KOe, respectively. All elements are of size w/H11003w /H11003wwith w=10 nm and have Ms=500 emu /cm3. The sepa-ration between the layers is w. The microwave and bias ﬁelds are applied simultaneously to both layers. Figure 3/H20849b/H20850shows the ﬁnal magnetization states in the two layers as a functionof microwave frequency and the bias ﬁeld. Area I and II,respectively, represent regimes of nonreversal and reversal ofboth layers. Area III and Area IV , respectively, represent re-gimes where Layer 1 and Layer 2 can be reversed individu-ally. From Fig. 3it is evident that the ﬁeld and element parameters can be found that lead to individual switching ofthe layers with different resonant frequency. Various mediaelements can be used. For example, composite elements inFig. 1offer a great ﬂexibility in tuning the structure param- eters. Similar study of MAMR using composite media and multilevel recording were also presented recently. 21,22 IV. SUMMARY We showed that MAMR in exchange-coupled composite elements is allowed for signiﬁcantly reduced reversal biasﬁelds, microwave ﬁelds, microwave frequencies, and rever-sal times. Reversal mode can be uniform or nonuniform. Inthe latter case, domain walls in the soft section of the com-posite elements initiated by the assisting microwave ﬁeldplay an important role. Utilizing the ability to tune the FMR frequency, we sug- gested a multilayer recording scheme. In this scheme, ele-ments at different levels are designed to support FMR atdifferent frequencies and are addressed by a properly tunedmicrowave ﬁeld. 1H. J. Richter, J. Phys. D 40, R149 /H208492007 /H20850. 2M. P. Sharrock, J. Appl. Phys. 76,6 4 1 3 /H208491994 /H20850. 3G. Bertotti, C. Serpico, and I. D. Mayergoyz, Phys. Rev. Lett. 86,7 2 4 /H208492001 /H20850. 4K.-Z. Gao, E. D. Boerner, and H. Neal Bertram, Appl. Phys. Lett. 81, 4008 /H208492002 /H20850. 5K.-Z. Gao and J. Fernandez-de-Castro, J. Appl. Phys. 99, 08K503 /H208492006 /H20850. 6B. Livshitz, R. Choi, A. Inomata, N. H. Bertram, and V . Lomakin, J. Appl. Phys. 103, 07C516 /H208492008 /H20850. 7B. Livshitz, A. Inomata, N. H. Bertram, and V . Lomakin, Appl. Phys. Lett. 91, 182502 /H208492007 /H20850. 8V . Lomakin, R. Choi, B. Livshitz, S. Li, A. Inomata, and H. N. Bertram, Appl. Phys. Lett. 92, 022502 /H208492008 /H20850. 9V . Lomakin, S. Li, B. Livshitz, A. Inomata, and N. Bertram, IEEE Trans. Magn. 44, 3454 /H208492008 /H20850. 10K. Rivkin and J. B. Ketterson, Appl. Phys. Lett. 89, 252507 /H208492006 /H20850. 11J. J. M. Ruigrok, R. Coehoorn, S. R. Cumpson, and H. W. Kesteren, J. Appl. Phys. 87, 5398 /H208492000 /H20850. 12W. Scholz and S. Batra, J. Appl. Phys. 103, 07F539 /H208492008 /H20850. 13Z. Z. Sun and X. R. Wang, Phys. Rev. B 74, 132401 /H208492006 /H20850. 14C. Thirion, W. Wernsdorfer, and D. Mailly, Nat. Mater. 2,5 2 4 /H208492003 /H20850. 15J. G. Zhu, X. C. Zhu, and Y . H. Tang, IEEE Trans. Magn. 44,1 2 5 /H208492008 /H20850. 16A. Y . Dobin and H. J. Richter, Appl. Phys. Lett. 89, 062512 /H208492006 /H20850. 17E. E. Fullerton, J. S. Jiang, M. Grimsditch, C. H. Sowers, and S. D. Bader, Phys. Rev. B 58, 12193 /H208491998 /H20850. 18M. Grimsditch, R. Camley, E. E. Fullerton, J. S. Jiang, S. D. Bader, and C. H. Sowers, J. Appl. Phys. 85, 5901 /H208491999 /H20850. 19D. Suess, T. Schreﬂ, S. Fahler, M. Kirschner, G. Hrkac, F. Dorfbauer, and J. Fidler, Appl. Phys. Lett. 87, 012504 /H208492005 /H20850. 20R. H. Victora and X. Shen, IEEE Trans. Magn. 41, 537 /H208492005 /H20850. 21M. A. Bashir et al. , 53rd Magn. Mag. Mat. Conf., Austin, TX, Abstract EC-09, 2008. 22M. A. Bashir, T. Schreﬂ, J. Dean, A. Goncharov, G. Hrkac, S. Bance, D.Allwood, and D. Suess, IEEE Trans. Magn. 44, 3519 /H208492008 /H20850.10 20 30 40 50 60 700.511.52 fmw(GHz)Ha(KOe) 00.511.522.53Head PoleMicrowave Generator 11,res anis K mwH Hff == 2, 2res anis K mwHH ff ==()cos 2mw mw a mwHH f t π = aH(a) (b) IIIIII IIII IVw w 1f2f FIG. 3. /H20849Color online /H20850/H20849a/H20850Schematic representation of a multilayer microwave-assisted magnetic recording system; /H20849b/H20850A reversal pattern of double-layer recording system. Four different areas represent different mag-netization states of in a two-layer structure comprising homogeneous ele-ments for different microwave frequencies. Area I corresponds to no switch-ing of any layer. Area II corresponds to switching of both layers. Area IIIcorresponds to switching of the lower layer only. Area IV corresponds to switching of the upper layer only. The results are given for H amw =2.25 kOe, /H9251=0.1, dh=w,HK1=15 kOe, and HK1=12 kOe.07B909-3 Li et al. J. Appl. Phys. 105 , 07B909 /H208492009 /H20850
1.469978.pdf	Modern He–He potentials: Another look at binding energy, effective range theory, retardation, and Efimov states A. R. Janzen and R. A. Aziz Citation: The Journal of Chemical Physics 103, 9626 (1995); doi: 10.1063/1.469978 View online: http://dx.doi.org/10.1063/1.469978 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/103/22?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Intraatomic correlation effects for the He–He dispersion and exchange–dispersion energies using explicitly correlated Gaussian geminals J. Chem. Phys. 86, 5652 (1987); 10.1063/1.452542 The possibility of a 4He2 bound state, effective range theory, and very low energy He–He scattering J. Chem. Phys. 76, 5069 (1982); 10.1063/1.442855 A study of the Efimov states and binding energies of the helium trimer through the Faddeev coordinate–momentum approach J. Chem. Phys. 68, 1006 (1978); 10.1063/1.435791 Another look at the energy shortage Phys. Today 26, 88 (1973); 10.1063/1.3128219 Calculated Scattering Cross Sections for He–He at Thermal Energies J. Chem. Phys. 40, 917 (1964); 10.1063/1.1725237 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.7 On: Wed, 03 Dec 2014 09:12:00Modern He–He potentials: Another look at binding energy, effective range theory, retardation, and Eﬁmov states A. R. Janzen and R. A. Aziza) Department of Physics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada ~Received 11 July 1995; accepted 1 September 1995 ! We compare a number of helium–helium potentials with respect to their predictions of dimer binding energy, scattering length, effective range and Eﬁmov states. We also study the effect ofretardation on the ‘‘best’’ potential. All realistic potentials support a weakly bound dimer, whilenone supports an Eﬁmov state. We agree with other authors that retardation decreases the bindingenergy by about 10%. Finally, we investigated the effect on the binding energy from the applicationof retardation over different ranges of separation. The precise effects of retardation at short range inrealistic potentials require further study. © 1995 American Institute of Physics. I. INTRODUCTION Modern interaction potentials predict a very weakly bound state for4He whether or not retardation forces are taken into account. Eﬁmov states are trimer states which areexpected to occur when a bound state exists close to disso-ciation. Effective-range theory ~ERT!in its modiﬁed form 1 can be employed to determine whether these Eﬁmov states occur and can be used to determine scattering lengths andeffective ranges. ERTmay also be used to compare estimatesof the binding energy with those obtained by a solution ofthe Schro¨dinger equation using Numerov’s method. 4All of these quantities are very sensitive in varying degrees to theshape of the interaction potential.The binding energy, effectsof retardation, scattering lengths, effective ranges, and thequestion of the number of Eﬁmov states are examined for anumber of recent and not-so-recent interaction potentials.Our ﬁndings suggest that: ~a!As reported by others, retardation decreases the well depth by 0.009 K 2and decreases the binding energy by about 10%.1,2 ~b!The effect on the binding energy is comparable from the application of retardation for separations above 10bohr as it is for separations below 10 bohr. ~c!The binding energy is roughly the same for all model andab initio potentials which possess modern disper- sion coefﬁcients and predict low-temperature virials. ~d!There appear to be no Eﬁmov states as predicted by ERT. II. THEORY The scattering length aand effective range rwere found from the effective range expansion3 1 a52kcot~h0!11 2rk2, ~1! wherek5(2mE)1/2/\andh0is the phase shift at energy E, calculated by numerical solution of Schro ¨dinger’s equation.4 Several pairs of kandh0were used to estimate slope and intercept of kcot~h0!vsk2/2 in Eq. ~1!, thus determining aandr.~In fact for k,0.01 cm21it was found that two pairs were sufﬁcient to give reliable values for the slope and in-tercept. ! The binding energy E bwas calculated directly from Schro¨dinger’s equation and was also computed from Eb5\2k2/(2m), where kis the appropriate root of the quadratic1 1 a5k21 2rk2. ~2! Hereaandrare the scattering length and effective range determined from Eq. ~1!. The number5of Eﬁmov states NEwas estimated from a andrusing the equation NE51 plnUa rU. ~3! Eﬁmov states are considered unlikely if NE,1. III. THE POTENTIALS A. Beck potential This potential6has a modiﬁed Buckingham–Corner form. The dispersion coefﬁcients are those of Dalgarno andKingston. 7The short range was determined by the calcula- tions of Phillipson8and the experiments of Amdur and co-workers;9other parameters were ﬁxed by appealing to experimental virial coefﬁcients available at that time. B. Bruch–McGee MDD-2 potential This potential10is piece-wise in form consisting of a Morse potential connected at short range to an exponentialfunction and at long range to a two-term van der Waals ex-pansion using older values of the dispersion coefﬁcients. 11–13 Parameters in the exponential function were obtained by a ﬁtto the theoretical results of Gilbert and Wahl. 14 C. Farrar–Lee ESMSV II potential This potential,15which is piece-wise exponential- Spline–Morse–Spline–van der Waals ~ESMSV !in form, was simultaneously ﬁtted to second virial coefﬁcients andelastic differential cross sections for 4He–4He. The two-terma!Author to whom correspondence should be addressed. 9626 J. Chem. Phys. 103(22), 8 December 1995 0021-9606/95/103(22)/9626/5/$6.00 © 1995 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.7 On: Wed, 03 Dec 2014 09:12:00multipole expansion used the dispersion coefﬁcients of Starkschall and Gordon ~C0!,16Davison ~C8!13and Dalgarno and Stewart ~C10!.17 D. Burgmans et al.ESMMSV potential Burgmans et al.18ﬁtted an exponential-Spline-Morse– Morse–Spline–van derWaals form of potential to elastic dif-ferential cross sections of a beam of 3He scattered from a beam of4He at the same energy. In this nonsymmetric case, the presence of both even and odd ‘‘1’’ partial waves wouldyield more structure and would lead to a less ambiguouspotential form. The short-range exponential parameters wereobtained by a ﬁt to the repulsive potential derived by Feltgenet al. 19from backward glory scattering data and the disper- sion coefﬁcients are identical to those used in the ESMSV IIpotential. E. Azizet al.HFDHE2 potential Azizet al.20proposed a potential of the Hartree–Fock dispersion form. It possesses nearly the correct Hartree–Fock repulsion 21as well as a long-range behavior based on dispersion coefﬁcients available at that time. Other param-eters were obtained by ﬁts to such experimental data as in-termediate temperature second virial coefﬁcients and high-temperature thermal conductivity and viscosity.Although thepotential is somewhat inconsistent with modern experimentaldata and theoretical results, its shallow well fortuitouslymakes it a useful ‘‘effective’’ pair-wise additive potential incondensed phase studies. 22 F. Feltgen et al.HFIMD potential A physically realistic ~but mathematically complicated ! two-parameter Hartree–Fock 1intra-atomic correlation cor- rection 1model dispersion ~HFIMD !potential model,23 which used all ab initio data available, was ﬁtted to mea- sured backward glory oscillations appearing in both integral 4He2and3He2scattering cross sections. Fitting to the4He2 data alone actually produces a more realistic potential with a deeper well ~10.935 K vs 10.741 K !. G. Tang–Toennies (TT) potential Tang and Toennies24presented a simple model potential which, like the HFD, partitions the total interaction energyinto correlated and uncorrelated energies. The model withindividual damping enables one to predict the interaction en-ergy from only a knowledge of the dispersion coefﬁcientsand well-known values of the energy and length parameters e andrm. H. Azizet al. HFD–B(HE) potential This potential25of the HFD–B form was ﬁtted to accu- rate low-temperature second virial data,26,27and recent room temperature viscosity data28while, at the same time, pinning the repulsive wall to the exact Born–Oppenheimer interac-tion energy calculated by Ceperley and Partridge 29at 1 bohr. The dispersion coefﬁcients are the ab initio values of Thakkar30and Koide et al.31I. Aziz and Slaman HFD–B2 potential This potential32is similar to the HFD–B ~HE!potential but more closely represents the accurate second virial coef-ﬁcients of 3He and4He measured by the various national standards laboratories in an effort to redeﬁne temperaturesbelow 18 K in terms of an ideal gas thermometer using he-lium gas. Its parameters eandrmare set close to the values of Liu and McLean, who used ab initio procedures.33 J. Aziz and Slaman LM2M2 compromise potential Both the HFD–B ~HE!and HFD–B2 potentials disagree with theab initioresults from 6.0 to 8.0 bohr, where Liu and McLean33consider them to be most accurate. The LM2M234 mimics the LM–2 ab initio results of Liu and McLean al- most to within their error bounds in such a way as to repro-duce fairly closely low-temperature virials and room-temperature viscosity. This ‘‘compromise’’ potential with awell depth of 10.97 K reproduces a variety of experimentaldata. K. Meath and co-workers XC (exchange–Coulomb) potentials The XC potential of Aziz et al.35partitions the total in- teraction energy into exchange and Coulomb interaction en-ergies. Two forms of potential are presented: simple overalldamped and individually damped models. The two-parameter overall damped ~XC–2 !and individually damped ~XCID–2 !closely agree with the best of the modern poten- tials. L. Azizet al.modiﬁcation of Tang–Toennies potential (AKSTT) In order to introduce more ﬂexibility into the original Tang–Toennies model, the Born–Mayer term Aexp~2ar! was replaced36byAexp~2ar1br2!and every other occur- rence of the quantity 82ar8is replaced by 82ar1br28.The construction of the helium potential involves incorporatingtheab initio dispersion coefﬁcients of Refs. 30 or 31 and adjusting bto ﬁt the room-temperature viscosity measure- ment of Ref. 28 and the ab initio results of Ref. 29 in the highly repulsive region. M. Anderson et al.quantum Monte Carlo potential (QMC) Anderson et al.37used the quantum Monte Carlo method to produce a potential which is ‘‘exact’’in that it requires nomathematical or physical approximations beyond those ofthe Schro¨dinger equation. The potential has no basis set su- perposition error, other systematic error, or experimental in-put.While the statistical errors are rather large, it nonethelessagrees with the LM2M2 potential remarkably well. For theconvenience of comparison, an HFD–B analytical functionwas ﬁtted to their numerical results.9627 A. R. Janzen and R. A. Aziz: He –He potentials J. Chem. Phys., Vol. 103, No. 22, 8 December 1995 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.7 On: Wed, 03 Dec 2014 09:12:00N. Tang–Toennies–Yiu perturbation theory potential (TTYPT) Tanget al.38derived a simple analytical expression for the entire potential energy curve from the perturbationtheory.The repulsive part assumes exchange of only one pairof electrons at any one time. The potential contains no ad-justable parameters and depends only on the known disper-sion coefﬁcients and the amplitude of the asymptotic wavefunction and the ionization energy of the atoms. O. Azizet al.HFD–B3–FCI1 potential This potential39is an analytical representation ~using the overall damped HFD–B form !of the best ab initioresults in the short-, intermediate- and long-range regions. The shortrange reproduces the quantum Monte Carlo calculations ofthe Born–Oppenheimer interaction energy of two helium at-oms with separations between 1.0 and 3.0 bohr. The interme-diate region mimics the full conﬁguration–interaction ~FCI! calculations of van Mourik and van Lenthe 40which possess very tight error bars. The long range of the potential uses thedispersion coefﬁcients of Refs. 30 and 31. We believe thispotential is the best characterization of the helium interac-tion. Further improvement may be achieved by incorporatingthe retardation coefﬁcient and the highly accurate C 6disper- sion coefﬁcient of Jamieson et al.41 P. Nitzet al. ‘‘reference’’ potential (NitzRef) Nitzet al.42ﬁtted a Buckingham–Corner potential to small-angle differential scattering cross sections43,44and joined it onto the HFD–B ~HE!potential ofAziz et al.25at a separation of r50.425 Å.At the same time, they adjusted the parameter Dof the HFD–B ~HE!potential so that the location of the zero crossing ~svalue !remained unchanged. The point of juncture is incorrectly stated in Ref. 42. Q. Nitzet al.BHFD potential Nitzet al.42ﬁtted a form similar to that of the NitzRef potential ~with the same dispersion coefﬁcients !to small- angle differential scattering cross sections43,44by adjusting parameters A,a,bandDas well as the Buckingham– Corner parameters. R. Nitzet al.BD potential (Nitz87) Nitzet al.42ﬁtted a Buckingham–Corner form to small- angle differential scattering cross sections.43,44To this they added an overall damped dispersion series with the disper-sion coefﬁcients of Refs. 30 and 31. The parameter Din the damping function was adjusted to produce nearly the same s value of the LM2M2 potential.This potential is referred to asNitz87 by Jamieson et al. 1Analysis of this potential shows that it has an unrealistically deep well ~e/k512.1433 K !, i.e., some 1.17 K deeper than the LM2M2 potential of Azizet al. 34Except for the dispersion series and the value of the parameter D, this potential differs from the 1991 LM2M2 potential of Aziz et al.34The increased depth leads to the inability of this potential to predict accurate low-temperaturevirial coefﬁcients.S. Jamieson et al.BDM (Nitz91) potential Jamieson et al.1modiﬁed the BD potential of Nitz et al. by including the sinusoidal ‘‘add-on’’ of the LM2M2 poten-tial of Aziz and Slaman. 34It is important to note that this potential, even with the add-on no longer reﬂects the ab ini- tioresults of Liu and McLean33from 3.5 to 7.5 bohr. Also, the BDM potential has the same unrealistically deep well asthe BD potential. Again, except for the dispersion series, thevalue of the parameter Dand the add-on component, this potential differs from the 1991 LM2M2 potential of Azizet al. 34 IV. RETARDATION The weakly bound state is large in spatial extent and has considerable probability beyond the outer classical turningpoint. Consequently, retardation must be taken into account. 2 Jamieson et al.41have recently computed the retarded dipole–dipole dispersion interaction in helium using a veryaccurate variational calculation of dipole transition frequen-cies and oscillator strengths. They present an r-dependent function in tabular form which multiplies the C 6dispersion coefﬁcient in such a way that the unretarded term C6r26, valid at short range, transforms into the retarded term C7r27 at long range. They also give analytical ﬁts to their table in the two intervals $10, 100 %bohr and $100, 200 %bohr. In those of our potentials which include retardation, namely HFD–B3–FCI1b, HFD–B3–FCI1b, and HFD–B3–FCI1d, we usetheir ﬁts in addition to our own ﬁts to their table in theintervals $0, 10 %,$200, 10 3%,$103,1 04%, and $104,1 05%bohr. Deviations from the table are less than 0.001% in the case ofthe ﬁts of Jamieson et al.and less than 0.005% in the case of our ﬁts. Retardation effects are not included in the dispersioncoefﬁcients beyond C 6. In the case of potential HFD–B3– FCI1b, the retardation factor of Jamieson et al.41is included in the nondamped region of potential HFD–B3–FCI1, i.e.,fromr5D3r m58.066–100,000 bohr. In the case of poten- tial HFD–B3–FCI1c, the retardation factor of Jamiesonet al. 41is included in both damped and nondamped regions, i.e., from 0 to 100,000 bohr. The effect of the inclusion ofretardation is demonstrated in Table II. V. RESULTS AND DISCUSSION The binding energy, scattering length, and effective range on the basis of each literature potential are presented inTable I.All modern potentials, which possess a realistic longrange and predict low-temperature virials, predict the exist-ence of a dimer. The values of the binding energy for anygiven potential agree whether they are obtained from a solu-tion of the Schro ¨dinger equation or from ERT. The binding energy of the potential ~HFD–B3–FCI1 !, which we consider to be most accurate, is about 1.6 mK without retardation.Theeffect of retardation is to reduce the binding energy by about10%. If a retardation factor is applied to the C 6coefﬁcient over all separations, then the well depth tends to be reducedin the process and in partas a result of the damping func- tion. To test the effect of a reduced well depth, a potential~HFD–B4 !is constructed so that it is almost identical to the HFD–B3–FCI1 potential at short and long range, but with a9628 A. R. Janzen and R. A. Aziz: He –He potentials J. Chem. Phys., Vol. 103, No. 22, 8 December 1995 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.7 On: Wed, 03 Dec 2014 09:12:00reduction in the well depth of about 0.009 K ~a slight change in well shape also occurs !and with no retardation included . This change in the potential produces a reduction in the bind-ing energy of about 3.2% ~see potential HFD–B4 in Table II!. If the retardation factor is applied to the HFD–B3–FCI1 potential only from 8 to 100,000 bohr ~i.e., in the nondamped region of the potential !, the reduction in the binding energy is only about 3.9%, i.e., from 1.59 4to 1.533mK~see poten- tial HFD–B3–FCI1b in Table II !.Application of the retarda- tion factor in this limited way introduces a minor disconti-nuity into the potential at 8 bohr of about 0.1%. If, inaddition, we include retardation in the damped region fromr50t oD3r m, the binding energy is further decreased by 5.4%, i.e., from 1.533to 1.446~see HFD–B3–FCI1c inTable II!.Although usually considered effective only at long range, we can conclude that retardation is found to have the short rangeeffect of decreasing the potential well depth, leading to a further decrease in the binding energy. If we include retardation to the HFD–B3–FCI1 potential fromr50 to 100,000 bohr, but decoupled from the damping function,w eﬁ n d ~see HFD–B3–FCI1d in Table II !that the binding energy is reduced from 1.594to 1.437mK compared with 1.446mK. That is, the damping function tends to de-TABLE I. Binding energies, scattering lengths, effective ranges, Eﬁmov states etc. Literature potentiale/k @K#rm @Å#Eb1 @mK#Eb2 @mK#Scattering length @Å#Effective range @Å#Eﬁmov statesLT virials predicted ?Deviation @ml.mol21# from Expt. at 2.6 K Beck@4# 10.3697 2.969 {{{ 0.051 2484.1 7.756 1.32 No 11.84 Bruch–McGee @10#10.7485 3.0238 {{{ 0.100 351.2 7.611 1.22 No 6.12 ESMSV II @15# 11.00 2.963 {{{ 0.272 214.7 7.520 1.07 No 5.25 ESMMSV @18# 10.57 2.97 {{{ 0.137 2293.6 7.845 1.15 No 10.79 HFDHE2 @20# 10.80 2.9673 0.835 0.835 124.3 7.396 0.89 No 2.59 HFIMD @19#a10.741 2.9788 0.520 0.521 156.4 7.463 0.97 No 3.81 HFIMD @19#b10.935 2.969 1.901 1.901 83.6 7.242 0.78 Nearly 20.45 TT@24# 10.8 2.967 0.736 0.736 132.2 7.419 0.92 No 3.73 HFD–B ~HE!@25# 10.948 2.963 1.691 1.691 88.5 7.277 0.80 Yes 0.02 HFD–B2 @32# 10.94 2.97 1.656 1.656 89.3 7.283 0.80 Yes 0.01 LM2M2 @34# 10.97 2.9695 1.310 1.310 100.0 7.326 0.83 Yes 0.95 XC–2 @35# 10.9845 2.9624 1.623 1.623 90.2 7.288 0.81 Yes 0.19 XCID–2 @35# 10.9819 2.9637 1.619 1.620 90.3 7.289 0.80 Yes 0.17 AKSTT @36# 10.94 2.97 1.410 1.410 96.5 7.315 0.82 Yes 0.71 QMC @37#c11.01 2.9634 1.815 1.815 85.5 7.264 0.78 Yes 20.31 TTYPT @38# 10.9847 2.9721 1.323 1.323 99.5 7.329 0.83 Yes 0.91 HFD–B3–FCI1 @39#10.956 2.96832 1.594 1.594 91.0 7.291 0.80 Yes 0.22 NitzRef. @42#d10.9271 2.9637 1.553 1.554 92.1 7.294 0.81 Yes 0.38 BHFD @42# 10.6534 2.9599 {{{ 0.148 290.0 7.571 1.16 No 6.07 BD~Nitz87 !@42#e12.1433 2.95 6.756 6.754 46.1 6.905 0.60 No 28.78 BDM ~Nitz91 !f12.1433 2.95 5.742 5.743 49.7 6.959 0.63 No 26.86 1The binding energy obtained by a solution of Schro ¨dinger’s equation. 2The binding energy obtained from ERT in its modiﬁed form. aFitted to both4He2and3He2cross sections. bFitted to only4He2cross sections. cMimic of the Anderson et al. ab initio potential. dPoint of juncture incorrectly speciﬁed in Ref. 42. eBD potential ~incorrectly referred to as Nitz87 in Ref. 1 !. fBDM potential ~incorrectly referred to as Nitz91 in Ref. 1 !. TABLE II. Binding energies, scattering lengths, effective ranges, Eﬁmov states etc. Modiﬁed potentiale/k @K#rm @Å#Eb1 @mK#Eb2 @mK#Scattering length @Å#Effective range @Å#Eﬁmov statesLT virials predicted ?Deviation @ml.mol21# from Expt. at 2.6 K HFD–B3–FCI1 @39#10.956 2.9683 1.59431.5944 91.0 7.291 0.80 Yes 0.22 HFD–B3–FCI1ag10.956 2.9683 1.59231.5925 91.0 7.291 0.80 Yes 0.22 HFD–B3–FCI1bh10.956 2.9683 1.53281.5336 92.7 7.293 0.81 Yes 0.48 HFD–B3–FCI1ci10.9473 2.9684 1.44611.4462 95.3 7.303 0.82 Yes 0.68 HFD–B3–FCI1dj10.9456 2.9684 1.43671.4368 95.6 7.304 0.82 Yes 0.74 HFD–B4k10.9473 2.9683 1.54351.5436 92.4 7.297 0.81 Yes 0.35 gHFD–B3–FCI1 potential with C651.460978 a.u. as calculated by Jamieson et al.41replacing C651.461 a.u. hHFD–B3–FCI1 potential with the retardation factor of Jamieson et al.41included from r5D3rm58.066 to 100 000 bohr. iHFD–B3–FCI1 potential with the retardation factor of Jamieson et al.41included from r50 to 100 000 bohr. jHFD–B3–FCI1 potential with the retardation factor of Jamieson et al.41included from r5D3rm58.066 to 100 000 bohr ~undamped region !and retardation added to the potential from r50t oD3rmdecoupled from the damping function. kPotential with short and long range very nearly equal to that of HFD–B3–FCI1, but with well modiﬁed to be equal to that of the overall retarded HFD–B3–FCI1c potential.9629 A. R. Janzen and R. A. Aziz: He –He potentials J. Chem. Phys., Vol. 103, No. 22, 8 December 1995 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.22.67.7 On: Wed, 03 Dec 2014 09:12:00crease the effect of retardation slightly. Further study appears warranted. In addition, we cannot agree with the conclusion of Jamieson et al.1that the repulsive wall inﬂuences ‘‘the scat- tering at ultralow temperatures’’or the binding energy in anysigniﬁcant way. The signiﬁcant change in the binding ener-gies of the potentials to which they refer as Nitz87 andNitz91 can be attributed to the substantial increase in thevalues of their well depth and accompanying change in wellshape. But these potentials are not able to predict accuratelow-temperature virials. Finally,noneof the recent realistic potentials predictthe existence of Eﬁmov states since the number of Eﬁmov statesis less than unity in each case. ACKNOWLEDGMENTS The research is supported in part by a grant from the Natural Sciences and Engineering Council of Canada ~RAA ! and the Faculty of Science of the University of Waterloo.The authors are indebted to Professor R. J. Le Roy for use ofhis Level 5.3 program, to M. J. Slaman for some of thepreliminary coding, and to Professor D. E. Nitz and Dr. M. J.Jamieson for helpful discussions regarding their potentials. 1M. J. Jamieson, A. Dalgarno, and M. Kimura, Phys. Rev. A 51, 2626 ~1995!. 2F. Luo, G. Kim, G. C. McBane, C. F. Giese, and W. R. Gentry, J. Chem. Phys.98, 9687 ~1993!. 3J. M. Blatt and J. D. Jackson, Phys. Rev. 26,1 8~1949!. 4R. J. Le Roy, University of Waterloo Chemical Physics Report No. CP-330R2, 1993 ~unpublished !. 5V. Eﬁmov, Phys. Lett. B 33, 563 ~1970!. 6D. E. Beck, Mol. Phys. 14,3 1 1 ~1968!. 7A. Dalgarno and A. E. 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1.5009739.pdf	Current-driven domain wall dynamics in ferromagnetic layers synthetically exchange- coupled by a spacer: A micromagnetic study Oscar Alejos , Victor Raposo , Luis Sanchez-Tejerina , Riccardo Tomasello , Giovanni Finocchio , and Eduardo Martinez Citation: Journal of Applied Physics 123, 013901 (2018); View online: https://doi.org/10.1063/1.5009739 View Table of Contents: http://aip.scitation.org/toc/jap/123/1 Published by the American Institute of Physics Articles you may be interested in Spin-phonon coupling in antiferromagnetic nickel oxide Applied Physics Letters 111, 252402 (2017); 10.1063/1.5009598 Role of hydrogen on the incipient crack tip deformation behavior in α-Fe: An atomistic perspective Journal of Applied Physics 123, 014304 (2018); 10.1063/1.5001255 Role of dislocations and carrier concentration in limiting the electron mobility of InN films grown by plasma assisted molecular beam epitaxy Journal of Applied Physics 123, 015701 (2018); 10.1063/1.5008903 Spatial dependence of the super-exchange interactions for transition-metal trimers in graphene Journal of Applied Physics 123, 013903 (2018); 10.1063/1.5007274 Edge enhanced growth induced shape transition in the formation of GaN nanowall network Journal of Applied Physics 123, 014302 (2018); 10.1063/1.5004496 Electrical and impedance spectroscopy analysis of sol-gel derived spin coated Cu 2ZnSnS 4 solar cell Journal of Applied Physics 123, 013101 (2018); 10.1063/1.5002619Current-driven domain wall dynamics in ferromagnetic layers synthetically exchange-coupled by a spacer: A micromagnetic study Oscar Alejos,1Victor Raposo,2Luis Sanchez-Tejerina,1Riccardo Tomasello,3 Giovanni Finocchio,4and Eduardo Martinez2,a) 1Dpto. Electricidad y Electronica, University of Valladolid, 47011 Valladolid, Spain 2Dpto. Fisica Aplicada, University of Salamanca, 37008 Salamanca, Spain 3Department of Engineering, University of Perugia, 06123 Perugia, Italy 4Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, 98122 Messina, Italy (Received 19 October 2017; accepted 13 December 2017; published online 3 January 2018) The current-driven domain wall motion along two exchange-coupled ferromagnetic layers with per- pendicular anisotropy is studied by means of micromagnetic simulations and compared to the con- ventional case of a single ferromagnetic layer. Our results, where only the lower ferromagnetic layer is subjected to the interfacial Dzyaloshinskii-Moriya interaction and to the spin Hall effect, indicatethat the domain walls can be synchronously driven in the presence of a strong interlayer exchange coupling, and that the velocity is signiﬁcantly enhanced due to the antiferromagnetic exchange cou- pling as compared with the single-layer case. On the contrary, when the coupling is of ferromagneticnature, the velocity is reduced. We provide a full micromagnetic characterization of the current- driven motion in these multilayers, both in the absence and in the presence of longitudinal ﬁelds, and the results are explained based on a one-dimensional model. The interfacial Dzyaloshinskii-Moriyainteraction, only necessary in this lower layer, gives the required chirality to the magnetization tex- tures, while the interlayer exchange coupling favors the synchronous movement of the coupled walls by a dragging mechanism, without signiﬁcant tilting of the domain wall plane. Finally, the domainwall dynamics along curved strips is also evaluated. These results indicate that the antiferromagnetic coupling between the ferromagnetic layers mitigates the tilting of the walls, which suggest these sys- tems to achieve efﬁcient and highly packed displacement of trains of walls for spintronics devices. Astudy, taking into account defects and thermal ﬂuctuations, allows to analyze the validity range of these claims. Published by AIP Publishing. https://doi.org/10.1063/1.5009739 I. INTRODUCTION Understanding and controlling the dynamics of domain walls (DWs) along ultrathin magnetic heterostructures consist-ing of a ferromagnetic (FM) strip sandwiched between a heavy metal (HM) and an oxide is nowadays the focus of intense research. 1–8These HM/FM/oxide multilayers exhibit high perpendicular magnetocrystalline anisotropy (PMA), and the broken inversion symmetry at the interfaces promotes chi- ral N /C19eel walls by the Dzyaloshinskii-Moriya interaction (DMI).4–11These DWs can be efﬁciently driven by current pulses due to the spin Hall effect (SHE) in the HM.4,5,12,13 Recent experimental studies14,15have shown that the DW dynamics can be even optimized in synthetic antiferro- magnetic heterostructures (SAFs), where antiferromagnetic coupling appears between two ferromagnetic layers isolatedby means of a non-magnetic spacer. The whole heterostruc- ture can be represented by HM/LFM/Spacer/UFM, where LFM and UFM stand for the lower and the upper FM layers,respectively. In these two-FM layers heterostructures, the DWs can be displaced even more efﬁciently and at much higher speeds if compared with the single-FM-layer stack(HM/FM/Oxide). This is due to a stabilization of the N /C19eel DW conﬁguration, and the exchange coupling torque that isdirectly proportional to the strength of the antiferromagnetic exchange coupling between the two FM layers. Moreover, because of the exchange coupling torque, the dependence ofthe DW velocity on the magnetic ﬁeld applied along thenanowire is different from that of the single-FM-layer heter- ostructure. These experimental results 14were explained within the framework of a one-dimensional model (1DM),which deals with the dynamics of the coupled DWs in theLFM and in the UFM layers by considering them as rigid objects. However, a more realistic analysis, taking into account the full three-dimensional dependence of the magne-tization in the two FM layers, is still missing and needed totest the validity of the 1DM. Accordingly, the current-driven DW (CDDW) dynamics in HM/LFM/Spacer/UFM multilayers is here investigated by means of full micromagnetic ( lM) simulations, and com- pared with the behavior of a single-FM-layer stack (HM/FM/Oxide). The two considered multilayer systems are sketchedin Figs. 1(a) and1(b), and the description of the geometry and dimensions are given in its caption. To elucidate the rel- evant aspects of this CDDW dynamics, simulations considerperfect strips as a ﬁrst approach, although additional simula-tions, which mimic realistic conditions by including disorder and thermal effects, have been also carried out. In order to collect detailed information about the acting mechanismsassociated to the coupling between the FM layers through a)Author to whom correspondence should be addressed: edumartinez@usal.es 0021-8979/2018/123(1)/013901/17/$30.00 Published by AIP Publishing. 123, 013901-1JOURNAL OF APPLIED PHYSICS 123, 013901 (2018) the spacer, which is determined by a certain interlayer exchange parameter Jex;16,17the magnetization state of both FM layers is simultaneously evaluated. Finally, both ferro- magnetic (FM) and antiferromagnetic (AF) coupling cases are considered, the former given by a positive Jex, and the latter by a negative Jex. The manuscript is structured as follows: Section II describes the details of the micromagnetic model ( lM) and the one-dimensional model (1DM). The current-driven DW dynamics along perfect samples, both in the absence and in the presence of in-plane longitudinal ﬁelds, is presented in Sec. III. Micromagnetic results for realistic samples are pre- sented in Sec. IVfor different multilayers where the thick- ness of the FM layers and the spacer ( tL FM,tU FM, and tS) and the saturation magnetization of the layers ( ML sandMU s) are varied. The current-driven DW motion along multilayers with curved parts is studied in Sec. V, and the main conclu- sions are discussed in Sec. VI. II. MODELS AND NUMERICAL DETAILS A. Micromagnetic model ( lM) Full micromagnetic ( lM) simulations have been per- formed by solving the Landau-Lifshitz-Gilbert equation aug-mented with the spin transfer torque ( ~s ST, STT) and Slonczewski-like spin-orbit torque ( ~sSO, SOT) due to the spin Hall torque18,19 d~m dt¼/C0c0~m/C2~Heffþ~Hth/C16/C17 þa~m/C2d~m dtþ~sSTþ~sSO;(1) where c0andaare the gyromagnetic ratio and the Gilbert damping constant, respectively. ~m~r;tðÞ/C17~mi~r;tðÞ¼~Mi ð~r;tÞ=Mi Sis the normalized local magnetization to its satura- tion value ( Mi s), deﬁned differently for each FM layer: Mi swhere i:L;Ufor the LFM and the UFM layers, respectively. ~Heffis the deterministic effective ﬁeld, which includes not only the intralayer exchange and the uniaxial anisotropy, but also the interlayer exchange17and the magnetostatic interac- tions adequately weighed to account for the different satura-tion magnetizations. The interlayer exchange contribution (~Hinter ex) to the effective ﬁeld ~Heff,a c t i n go ne a c hF M layer, is computed from the corresponding energy density (xinter ex¼/C0Jex tS~mL/C1~mU,w h e r e Jexis the interlayer exchange coupling parameter, tSis the thickness of the spacer between the LFM and the UFM layers, and ~mLand~mUrepresent the normalized magnetization in the Lower and in the Upperlayers, respectively) as ~Hinter ex;i¼/C01 l0Mi sdxinter ex d~mi¼Jex l0Mj stS~mj; (2) where i;j:L;U. Ferromagnetic (FM) and antiferromagnetic (AF) coupling cases are evaluated by a positive Jex, and by a negative Jex, respectively. The effective ﬁeld in the LFM layer requires an addi- tional term representing the interfacial DMI at the HM/LFMinterface. The rest of numerical details of other contributions to the effective ﬁeld can be found elsewhere. 18~Hthis the ther- mal ﬁeld, included as a Gaussian-distributed random ﬁeld.20,21 ~sSOrepresents the spin-orbit torque (SOT), which in the pre- sent work is solely acting on the LFM layer (the only one con- tacting the HM). This torque is given by the Slonczewski-like term ~sSO¼/C0c0~m/C2~HSL,w h e r e ~HSL¼H0 SL~m/C2~ris the Slonczewski-like effective ﬁeld. Here, ~r¼~uz/C2~uJis the unit vector along the direction of the polarization of the spin cur- rent generated by the spin Hall effect (SHE) in the HM, being orthogonal to both the direction of the electric current ~uJand the vector ~uzstanding for the normal to the HM/LFM inter- face. Finally, H0 SL¼/C22hhSHJHM=2l0ejjMstFM ðÞ determines the strength of the SHE,4where /C22his the Planck constant, eis the electron charge, l0is the vacuum permeability, hSHis the spin Hall angle, and Jis the magnitude of the current density ~JHMðtÞ¼JHMtðÞ~uJ. For straight samples ~uJ¼~ux, whereas for curved strips the direction and the local amplitude of cur- rent was previously computed by ﬁnite element method solv- ers.22On the other hand, Eq. (1)includes the spin transfer torques (STTs, ~sST) due to the electrical current ﬂowing across the FM layers ( ~Ji HMðtÞ¼Ji FMtðÞ~uJ,w i t h i¼U;Lfg ). This STTs21includes both adiabatic and non-adiabatic contribu- tions: ~sST¼bi ST~uJ/C1rðÞ ~m/C0nibi ST~m/C2~uJ/C1rðÞ ~m,w h e r e bi ST¼lBP ejjMisJi FMis the STT coefﬁcient, with Pis the polariza- tion factor, and Ji FMis the density current ﬂowing directly throw the FM layer i¼U;Lfg .niis the non-adiabatic coefﬁcient.21 Typical values of the parameters above have been chosen in our simulations. Except where the contrary is indicated, Ms values for the UFM and the LFM layers have been chosen, respectively, as ML s¼600 kA =ma n d MU s¼600 kA =m. The anisotropy constant, the intralayer exchange constant, and the Gilbert damping are Ku¼0:6M J=m3,A¼20pJ m;and a¼0:1 for both FM layers. The interfacial DMI in the lower FIG. 1. (a) Schematic representation of the multilayer structure with two FM layers. The relevant thicknesses for this study are marked on the ﬁgure, which are ﬁxed to tL FM¼tS¼tU FM¼0:8 nm, except otherwise indicated. The saturation magnetizations are also ﬁxed by default to ML S¼600 kA =m andMU S¼600 kA =m. The width is w¼100 nm. The anisotropy constant, the intralayer exchange constant, and the Gilbert damping are, respectively, K1¼0:6M J=m3,A¼20 pJ=m, and a¼0:1 for both FM layers. The inter- facial DMI in the lower FM is DL¼1:25 MJ =m2, and no DMI is considered for the UFM ( DU¼0). Along this work, the single-FM-layer (b) is consid- ered to have identical characteristics to those of the lower FM layer. The parameters used here can be found in the literature.4,7,14013901-2 Alejos et al. J. Appl. Phys. 123, 013901 (2018)FM is DL¼1:25 MJ =m2and null in the upper FM layer (DU¼0). The spin Hall angle representing the degree of polarization of the vertical spin current acting on the LFM ish SH¼/C00:12. Different values of the interlayer exchange parameter ( Jex) have been considered with magnitudes within the range 0 /C20Jexjj/C200:5m J=m2, but with Jextaking by default the values Jex¼60:5m J=m2for FM and AF coupling cases. For the study of the single-FM layer, thefollowing parameters were adopted: M S¼600 kA =m, Ku¼0:6M J=m3,A¼20 pJ=m,a¼0:1,D¼1:25 mJ =m2, andhSH¼/C00:12, which coincide with the ones chosen for the LFM layer in the two FM layer cases. Initially, we assumethat STT is negligible ( b i ST¼0) in the evaluated samples. We will show in Sec. III B that indeed the STT plays a marginal role on the current-driven dynamics evaluated in the present study. The dynamics equation of the magnetization over the full system was solved using MuMax323which was adapted to include the Ruderman–Kittel–Kasuya–Yosida interaction17 between non-adjacent FM layers separated by the spacer. Thein-plane side of the computational cells is Dx¼Dy¼3n m and different thicknesses Dz, depending on the thickness of the FM layers, were considered. A homemade micromagneticsolver was also used to verify the validity of the obtainedresults. Except the contrary is said, the presented results wereobtained at zero temperature. Simulations at room temperaturewere performed with a ﬁxed time step Dt¼0:1 ps. Several tests were performed with reduced cell sizes and time steps to assess the numerical validity of the presented results. Part of the simulations was carried out by considering perfect samples, without imperfections nor defects.However, other parts were computed under realistic condi-tions (see Secs. IVandV). In order to take into account the effects of disorder due to imperfections and defects in a real-istic way, we assume that the easy axis anisotropy direction is distributed among a length scale deﬁned by a “ grain size. ” The grains vary in size taking an average size of 10 nm. Thedirection of the uniaxial anisotropy of each grain is mainlydirected along the perpendicular direction ( z-axis) but with a small in-plane component which is randomly generated overthe grains. The maximum percentage of the in-plane compo-nent of the uniaxial anisotropy unit vector is varied from10% to 15%. The presented results correspond to an in-planemaximum deviation from the out-of-plane direction of 12%.Although other ways to account for imperfection could beadopted, we selected this one based on previous studies, which properly describe other experimental observations. 19 B. One-dimensional Model (1DM) The one dimensional model (1DM) assumes that the DW proﬁle can be described by the Bloch’s ansatz,9and therefore its dynamics can be described by means of the DWposition ( q) and the internal DW angle ( U). The 1DM has been developed by several authors to account for anddescribe the ﬁeld-driven and current-driven DW dynamics indifferent systems. 4,9,18Yang et al.14developed this 1DM to describe the DW dynamics in bi-layer FM systems in the presence of a strong interlayer exchange coupling betweenthe two FM layers. These equations assume that the DWs inthe LFM and the UFM layers move completely coupled to each other, and therefore, q L¼qU¼qrepresents the DW position along the longitudinal axis of the two walls. Thesame DW width in the two FM layers was also assumed ( D). On the other hand, the internal DW angle is different for each layer: U i, with the index i¼U;Lfg standing for the UFM and the LFM layers. We have derived the 1DM equa-tions, which can be written as a LML sþaUMU s/C0/C1 _q DþQLML s_ULþQUMU s_UU ¼c0p 2QUML sHL SLcosULþc0p 2QLMU sHU SLcosUU þc0QLML sþc0QUMU s/C0/C1 Hz/C0nLbL ST DML s/C0nUbU ST DMU s; (3) QL_q D/C0aL_UL¼c0p 2HxsinUL/C0p 2HycosUL/C0p 2QLHL DsinUL/C20 /C0HL ksin2UL 2þ2Jex l0ML stssinUL/C0UU ðÞ/C21 /C0QLbL ST D; (4) QU_q D/C0aU_UU¼c0p 2HxsinUU/C0p 2HycosUU/C0p 2QUHU DsinUU/C20 /C0HU ksin2UU 2/C02Jex l0MU stssinUL/C0UU ðÞ/C21 /C0QUbU ST D; (5) where the top dot notation represents the time derivative (_q/C17dq dt), and QL¼61(QU¼61) corresponds to an up- down (UD, upper sign) or to a down-up (DU, lower sign) DW conﬁguration in the LFM (UFM) layer. Hx;Hy;Hz ðÞ are the Cartesian components of the external magnetic ﬁeld. Hi D¼Di l0MisDis the effective DMI ﬁeld.18Hi k/C25Mi sNxis the magnetostatic shape anisotropy ﬁeld,19where Nx¼ti FMlog2ðÞ pD is the magnetostatic factor.24airepresents the Gilbert damp- ing term in each FM layer and c0is the gyromagnetic ratio. Besides, the assumption that both DW widths remain con- stant has been made ( D¼ﬃﬃﬃﬃﬃ ﬃ A Keffq , where Keff¼KL u/C01 2l0 ML s/C0/C12). The term Hi SL¼/C22hhi SHJ 2l0ejjMisti FMis the Slonczewskii-like term associated to the SHE, and bi ST¼lBP ejjMisJi FMis the STT coefﬁcient, with Pthe polarization factor and Ji FMthe density current ﬂowing directly along the FM layers i¼U;Lfg .niis the non-adiabatic parameter. Initially, we assume that STT is negligible ( bi ST¼0) in the evaluated samples. We will show in Sec. III B that indeed the STT plays a marginal role on the current-driven dynamics discussed in the present study. These 1DM Eqs. (3)–(5)can be directly expressed in the same manner as done in the supplementary information of Ref. 14. Indeed, we veriﬁed that using the inputs considered in the supplementary material of Ref. 14, we reproduce their results (not shown). Moreover, Eqs. (3)–(5)can be used to evaluate all the cases considered here solely by proper selec-tion of the inputs: FM coupling ( J ex>0, with QL¼QU¼61); AF coupling ( Jex<0, with QL¼61 and013901-3 Alejos et al. J. Appl. Phys. 123, 013901 (2018)QU¼71), and the single FM layer case ( Jex¼0, with QL¼61 and QU¼0¼MU S¼DU¼aU¼hU SH¼bU ST). In the present work, Eqs. (3)–(5)are numerically solved using a commercial software.25 III. RESULTS FOR PERFECT STRIPS A. Current-driven DW motion in the absence of longitudinal fields We ﬁrst describe the current-driven DW motion along perfect and straight systems. Representative snapshots of thelocal magnetization before and just at the end of a 2-ns longcurrent pulse with amplitude J¼J HM¼þ2:5T A=m2, are depicted in Fig. 2. In what follows, the units of the current density are given in TA =m2indicating 1012A=m2. Figure 2(b) shows the results for a single DW in the single-FM- layer stack. Figures 2(a) and2(c) correspond to a pair of DWs, one in each FM layer, equally located, in the case of the coupled multilayer system: Fig. 2(a) for FM coupling (Jex>0), and Fig. 2(c) for AF coupling ( Jex<0). Except the contrary is indicated the magnitude of interlayerexchange coupling is ﬁxed to J exjj¼0:5m J=m2. Initial chi- ral N /C19eel4conﬁgurations are stabilized in all cases, both in the single-FM-layer and in the coupled DWs of the FM cou-pling and AF coupling cases. In the latter case, the strongDMI at the HM/LFM interface along with the interlayerexchange interaction between the FM layers are sufﬁcientfor promoting that chiral magnetization textures. The FMcoupling ( J ex>0) makes domains in both FM layers to adopt equal orientation, leading to twin up-down (UD) DW transitions in both layers [Fig. 2(a)]. Conversely, the AF cou- pling ( Jex<0) promotes conﬁgurations where the upper and lower domains point in opposite directions. This fact resultsin the formation of paired DWs combining both types ofDWs, one UD in one FM layer and one DU in the other FMlayer [Fig. 2(c)].The dynamical behavior of DWs in the three presented cases shows noteworthy differences. The description of such a behavior requires the deﬁnitions of the internal DW ( U) and tilting ( v) angles depicted in the inset of Fig. 2. Figure 2(a) presents the results obtained for the FM coupling between the FM layers ( J ex>0). In this case, the SOT due to the SHE acts exclusively on the magnetization of the LFM layer, pushing forward the DW in this layer. The interlayerexchange FM coupling between both FM layers results in the simultaneous displacement of the DW in the UFM layer, which is dragged by the movement of its counterpart in theLFM layer. The behavior is rather similar to the single-FM case shown in Fig. 2(b), since the inner magnetization of both DWs rotates from the initial N /C19eel conﬁguration similar to the single-FM case. Note also that in these two cases, the DW plane is signiﬁcantly tilted due to the current. The tiltingincreases with Jand it is reduced in the FM coupling case with respect to the single FM layer case. The case in Fig. 2(c) corresponds to the interlayer AF coupling ( J ex<0). An antiparallel alignment of the magnet- izations in the LFM and the UFM layer occurs, both within the domains and inside the DWs. Now again the movement of the DW in the LFM drags the DW in the UFM due to the AF coupling, but the highest displacements are reached. Itcan be checked in Fig. 2(c)that the antiparallel alignment of the magnetizations within such paired DWs almost holds during the whole dynamics, then keeping the direction of themagnetizations along the longitudinal one, that is, the paral- lel/antiparallel alignment of the current ﬂow and the magne- tization within the DWs, depending on the type of the DW. Interestingly, no DW tilting is observed for this AF coupling case. The DW dynamics is, in all cases, determined by a set of terminal values of the DW velocity ( v), the DW angle ( U), the tilting angle ( v), and the DW width ( D). In other words, after a short transient all these observables reach a steady- FIG. 2. Micromagnetic snapshots of the initial ( t¼0) and ﬁnal (at t¼2 ns) states of the DWs in FM layers in the following cases: (a) FM coupling ( Jex>0), (b) single-FM-layer, and (c) AF coupling ( Jex<0). UFM and LFM layers are simultaneously shown in cases (a) and (c). The amplitude of the current pulse along the HM layer is J¼þ2:5T A=m2. No spin current is acting on the UFM layers. Other material parameters are given in the text and in the caption of Fig. 1. The thin red arrows show the DW displacements. The black thick arrows represent the orientation of the magnetization within the DW in the single- FM-layer case, while the blue and red thick arrows represent the orientation of the magnetization within the DWs in the UFM and LFM layers, respectivel y, for the coupled systems. Perfect samples and zero temperature are considered here. The inset depicts the deﬁnition of the DW angle Uand the tilting angle v as the angles formed, respectively, by the magnetization ~mDWwithin the DW, and the normal ~nDWto the DW, with respect to the longitudinal axis.013901-4 Alejos et al. J. Appl. Phys. 123, 013901 (2018)state (or terminal) regime with constant values. We checked that the terminal steady-state regime is completely reached within the ﬁrst 2 ns of the current application (see Sec. III C), and therefore, this time was adopted to evaluate the terminal values of the mentioned observables. Uandvare computed from the terminal magnetization snapshots (at t¼2 ns). The DW width Dis computed according to Thiele’s deﬁnition.21 The dependence of these terminal values on the current amplitude is shown in Fig. 3for the three considered cases: (a) FM coupling ( Jex>0), (b) single-FM-layer, and (c) AFcoupling ( Jex<0). In the ﬁrst two cases, Figs. 3(a)and3(b), the velocity asymptotically increases as the DW magnetiza- tion angle ( U) approaches a Bloch conﬁguration (i.e., a rota- tion of690/C14, depending on the wall type) as Jincreases. The variation of the internal DW angles is rather similar for both the FM coupling and the single-FM-layer cases [seegraphs in Figs. 3(a) and3(b)]. In fact, the results of the DW magnetization angle Uobtained for the latter case almost exactly overlap those corresponding to the LFM layer if theyare plotted within the same graph (not shown). Since the FIG. 3. lMresults of the current driven DW motion for the three evaluated cases: (a) FM coupling ( Jex>0), (b) single-FM layer, and (c) AF coupling (Jex<0). The dependence on the applied current Jof the terminal values of the DW velocity, the DW angle, tilting angle, and the DW width, which are shown from top to bottom graphs. These values were computed at t¼2 ns. The parameters are those given in the text. The term UD (DU) within the legends refer to the DW magnetization transition in the LFM from up-to-down (from down -to-up) along the positive direction of the longitudinal axis ( x-axis). Perfect samples and zero temperature conditions are considered here.013901-5 Alejos et al. J. Appl. Phys. 123, 013901 (2018)SHE acts exclusively on the LFM layer, and this layer and the single-FM layer have been chosen to exactly share thesame set of geometrical and intrinsic parameters, it can beconcluded that the driving force due to the SHE acquires arather similar magnitude in both cases. However, the termi-nal velocity ( v st) is much slower for the FM coupled system than in the latter case. This result can be understood as alower mobility of the paired DWs in the coupled system ascompared with that of the DW in the single-FM-layer. The1DM indeed provides a clue to satisfactorily explain thislower mobility. Micromagnetic ( lM) simulations show that after a short transient, the DWs adopt the steady-state regime(_U¼0) for the three evaluated cases. By imposing the steady state condition ( _U¼0) in the 1DM Eqs. (3)–(5),a n analytical expression is deduced for the terminal DW veloc-ity (v st) of the single FM layer18 vst¼p 2c0D a/C22hhSHJ 2el0MstFMcosUst; (6) where Ustis the terminal DW angle ( _U¼0, i.e., U!Ust). Equation (6)indicates that DW terminal velocity for a sin- gle-FM-layer monotonously increases with J, but with a decreasing slope as the absolute current is increased.9,18,19 This fact can be explained by the relative orientation of the magnetization within the DWs ( ~mi DWorUiwith i:L;U) and the direction of the electric current ﬂow ( ~JtðÞ¼JtðÞ~ux). In fact, the closer the direction of the magnetization within theDWs is to the direction of the current ﬂow, the more efﬁcientis the SHE pushing the DWs. 3,18,26However, the spin orbit torque (SOT) due to the SHE itself promotes the progressivemisalignment of both magnetization and current: as J increases, the angle U stasymptotically tends to 90/C14[see “DW angle vs. J ” graph in Fig. 3(b)], leading to the above- mentioned decrease of the slope of the DW speed depen-dence on the current amplitude. 18,19 Similarly, an analytical expression can be inferred from Eqs. (3)–(5)for the steady state (terminal) DW velocity in FM ( Jex>0) and AF ( Jex>0) coupling cases. In the cases being studied here, where aL¼aU¼a, no external ﬁeld is applied ( Hz¼0), no SHE is acting on the UFM ( HU SL¼0), no DMI on the UFM layer ( DU¼0), and by considering sta- tionary conditions (i.e., when _UL¼0 and _UU¼0, and vst/C17 _qðÞst), Eq. (3)leads to vst¼p 2c0D a/C22hhL SHJ 2el0ML sþMU s/C0/C1tL FMcosUst L; (7) which, except for the term ML sþMU s/C0/C1 , is equivalent to Eq. (6)for a single FM layer. Therefore, the DW velocity is pro- portional to the driving current and the cosine of the station-ary DW angle U st L, and inversely proportional to a weighed- up sum of the saturation magnetizations of both FM layers,i.e., the sum of M U sandML s. This explains the results for the FM coupling ( Jex>0) and single FM layer cases shown in Figs. 3(a)and3(b), where the variation of the DW angles are similar for both layers in the FM coupling case, and alsovery similar to the ones achieved for a single FM layer.However, the DW velocity is signiﬁcantly larger for thesingle FM case as only M L sappears in the denominator of Eq. (6), and not ML sþMU sas in Eq. (7). Another difference between cases (a) and (b) can be found in the DW tiltingangle ( v). The FM coupling reduces the tilting angle of the paired DWs, which is rather similar for both of them, as compared with the tilting angle of the DW in the single-FMstrip. A signiﬁcant contrast characterizes the CDDW dynam- ics in AF coupled systems ( J ex<0). The increase of the ter- minal velocity of the DWs with the current amplitude is in this case rather linear in the evaluated range, and higher speeds are reached for the highest currents in the AF cou-pling case [see Fig. 3(c)]. The key for this behavior resides in the fact that the N /C19eel conﬁguration of the DW in the LFM layer holds over a large range of applied currents Jjj: the AF coupling strongly supports the antiparallel alignment of the internal DW angles [see “DW angles vs. J ” graph in Fig. 3(c)], and the SOT is not sufﬁciently intense to promote a signiﬁcant misalignment between the current ﬂow and the magnetization within the DWs. Actually, the use of Eq. (7) derived from the 1DM, yields a rather good approach tocompute this terminal DW velocity, provided the DW angle in the LFM layer is set to U st L/C25180/C14, as it can be seen from fulllMsimulations [see “DW angles vs. J ” graph in Fig. 3(c)]. However, a slight but progressive slope reduction is obtained as the current is increased, which can be ascribed to the increasing misalignment between the magnetizationswithin the paired DW, as the same graph reveals for high currents. Another important characteristic of this CDDW dynamics is that DW tilting completely vanishes, so thatDWs hold perpendicular to the longitudinal direction ( x- axis). We have shown that the micromagnetic ( lM) results of the DW velocity vs. Jcan be qualitatively described by the analytical Eqs. (6)and(7)for the single FM layer and two coupled FM layers derived from the 1DM, respectively.However, it remains to check if the 1DM is also in quantita- tive agreement with the lMresults. To evaluate it, we have numerically solved the 1DM Eqs. (3)–(5)considering the same inputs parameters as for the lMstudy for the three evaluated cases: FM coupling, single-FM-layer, and AF cou- pling. First of all, it has to be noted that the micromagneti-cally computed DW width ( D) dependence on Jshown in the bottom graphs of Fig. 3indicates that Dalmost remains inde- pendent on J. The lMvalue agrees with the analytical pre- diction of the DW width, which can be estimated from D¼ﬃﬃﬃﬃﬃ ﬃ A Keffq , where Keff¼KL u/C01 2l0ML s/C0/C12, resulting in a value of D/C257:3 nm. The lMresults are compared to the 1DM predictions in Fig. 4. A ss h o w ni nF i g . 4, the 1DM predictions are in a good qualitative agreement with the lMresults, both for the DW velocity and the DW angles. The discrepancies between the lMand the 1DM results in the single FM layer [Fig. 4(b)] can be attributed to the approximated description providedby the 1DM, which neglects, among other aspects (such as the approximated description of the shape anisotropy ﬁeld, for instance), the DW tilting observed in the lMresults. Indeed, we notice that taking into account the DW tilting in the 1DM (see Refs. 10and18, for details) results also in a013901-6 Alejos et al. J. Appl. Phys. 123, 013901 (2018)good quantitative agreement with the lMresults for the sin- gle FM layer [see the blue curves in the graphs of Fig. 4(b)]. Regarding the FM coupling case [Fig. 4(a)], the quantitative disagreement between 1DM and lMresults should be additionally ascribed to the magnetostatic inter- action between the two FM layers, which is not taken into account in the 1DM Eqs. (3)–(5), and to the approximated description of the shape anisotropy ﬁelds ( Hi k/C25Mi sNx ¼ti FMlog2ðÞ pDMi s)24considered by the 1DM. Note that this 1DM description does not take into account the width wof the FM strips. On the other hand, the agreement between the lMand the 1DM results for the AF coupling case [Fig. 4(c)] looks remarkable also from a quantitative point of view. However, this ﬁt required to re-scale the DW width in the 1DM, which contrary to the other two cases, was settoD/C252:3 nm. Note that this value is not justiﬁed by the analytical prediction ( D/C257:3 nm) nor by the lMresults shown in bottom graphs of Fig. 3(c). It has to be also noticed that by imposing D/C257:3 nm as the input for AF coupling case, the DW velocity predicted by the 1DM over-estimates the lMresults of “DW velocity vs. J”b yaf a c t o r of/C243 (not shown), whereas the dependence of “DW angle vs. J” is hardly affected. For these reasons, we will continue analyzing hereafter the current-driven DW dynamics along multilayers with two FM layers adopting a full micromag-netic description, which naturally accounts for the 3D dependence of the magnetization, including the magneto- static interaction between them and the eventual DW tilting.B. The influence of the spin transfer torques on the current-driven DW motion In previous discussion, we have assumed that the most of the current ﬂows along the HM, so the only driving force on the DWs is due to the spin Hall effect (SHE), which drivesDWs along the current direction for the chiral DW nature con- sidered in the present study (left-handed chirality imposed by the DMI). However, the current could also partially ﬂow alongthe FM layers, and consequently the conventional adiabatic andnon-adiabatic spin transfer torques (STTs) could also contribute to the current-driven DW dynamics. In order to explore the inﬂuence of these STTs, we have evaluated the DW dynamicsalong the same systems studied before (Single-FM-layer stack, HM/FM/Oxide, and the multilayers with two FM layers HM/ LFM/Spacer/UFM, with AF coupling) by considering that theFM layers are also submitted to the same current density as the HM, J i FM¼JHMfori:L;U. The spin polarization factor of the STT is P¼Pi¼0:5 for both FM layers. The geometries and materials parameters considered in Sec. III A have been also adopted for this analysis. The results for the terminal DW velocities as a function of the current density J¼Ji FM¼JHM are shown in Figs. 5(a)and5(b) for the single-FM-layer stack and the multilayer HM/LFM/Spacer/UFM with AF coupling ( Jex¼/C00:5m J=m2), respectively. Three different values of the non-adiabatic parameter are considered:n¼n L¼nU:0;a¼0:1;and 2 a¼0:2, and the results are compared to the ones computed in the absence of STT ( P¼0), where the only driving mechanism is due to the SHE. FIG. 4. lMresults and 1DM predictions of the current driven DW motion for the three evaluated cases: (a) FM coupling ( Jex>0), (b) single-FM layer, and (c) AF coupling ( Jex<0). The dependence on the applied current Jof the terminal values of the DW velocity and the DW angle is shown from top to bottom graphs. The parameters are those given in the text. The DW width for the AF coupling ( Jex<0) was needed to be rescaled to D/C252:3 nm in the 1DM in order achieve quantitative agreement with lMresults. For the two other cases, FM coupling ( Jex>0) and single-FM layer, the input value of the DW width was D/C257:3 nm, as predicted by the analytical formula D¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A=Keffp . Perfect samples and zero temperature conditions are considered. The blue lines in (b) corre- spond to the 1DM results obtained taking into account the DW tilting (see Ref. 18for details). The inset in the bottom graph of (b) represents the DW tilting as function J.013901-7 Alejos et al. J. Appl. Phys. 123, 013901 (2018)The STT pushes the DW along the electron ﬂow (against the current). As commented, for the single-FM-layer stack[Fig. 5(a)], the DW velocity due to the SHE (which drives the DW along the current direction) increases monotonously up to asymptotic saturation with Jin the absence of STTs [P¼0, black dots in Fig. 5(a)]. When the STT is taken into account, the DW velocity decreases for a given current. This velocity reduction is larger as the non-adiabatic parametersincreases from n¼0 [open red symbols in Fig. 5(a)]t o n¼2a¼0:2 [open blue symbols in Fig. 5(b)]. Figure 5(b) also shows that the DW velocity reaches a maximum for agiven current, and for large currents the DW velocity starts to decrease again. These results indicate that the STTs act against the SHE, reducing the magnitude of the DW velocity,which is along the current direction. For the HM/LFM/Spacer/UFM with AF coupling, the perfect adiabatic STT [ P¼0:5;n¼0, red open symbols in Fig. 5(b)] does not signiﬁcantly modify the DW velocity. Under non-adiabatic conditions [ P¼0:5;n>0, ﬁlled green and blue open symbols in Fig. 5(b)], the DW velocity decreases with respect to the zero STT case ( P¼0). We conﬁrmed that the DWs in the LFM and in the UFM layersmove coupled even in the presence of STTs as due to the strong interlayer exchange coupling ( J exjj¼0:5m J=m2). Contrary to the single-FM-layer stack, the DW velocityincreases monotonously with J, and the slope of this increas- ing is reduced as the non-adiabatic parameter nincreases. This indicates that the main driving force in these AF cou-pled multilayers is still the SHE due to the current along the HM. In the rest of the discussion, we will neglect STTs for different reasons. Several experimental works, 27,28have shown that the STT is indeed negligible in these systems. Moreover, the experiments2showing DW motion along the electron ﬂow in high PMA systems are consistent with theperfect adiabatic conditions ( n¼0), and this case has shown to play a marginal role for multilayers with AF coupling [Fig. 5(b)]. On the other hand, we have also considered that the same current ﬂowing through the HM is also ﬂowing through the ultrathin FM layers. This is surely an exaggera- tion, as the electrical resistivity of the FM should be largerthan the one of the FM layers. In a more realistic case, the density current along the FM layer must be smaller than the one along the thicker and low-resistivity HM, andconsequently the STT should play a marginal role. For these reasons, we will not take into account the STTs in the rest of the manuscript. C. Inertia effect on the current-driven DW motion In Secs. III A andIII B, we have plotted the terminal DW velocity reached by the DWs after application of con- stant density current. Such values were obtained at tp¼2 ns, which was found sufﬁcient to achieve the steady-state termi- nal DW velocity. It is also interesting to evaluate the DW dynamics once the current pulse is turned off. The current-driven DW dynamics due to their own inertia has been stud- ied in systems with in-plane magnetization by Thomas et al. 29and Chauleau et al. ,30where the DW motion when the current pulse was turned off was essentially ascribed to the gyrotropic dynamics of the vortex DW conﬁgurations. Vogel et al.31shown that the DW motion induced by nano- second current pulses in Pt/Co/AlOx multilayers with per- pendicular magnetic anisotropy exhibits negligible inertia. More recent studies by Torrejon et al.19have shown that inertia effects result in a DW motion even when the current is switched off in high PMA systems with low damping. Our aim here is just to evaluate the inertia in HM/LFM/Spacer/ UFM stacks, with FM ( Jex>0) and AF ( Jex<0) coupling, and to compare this “after-effect” to the single-FM-layer stack. To do it, we applied the current pulse at t¼0, and monitor the temporal evolution of the DW position and the DW velocity along perfect samples (without disorder) and at zero temperature. The results are shown in Figs. 6(a) and 6(b). For a single-FM-layer [open circles in Figs. 6(a) and 6(b)], the DW takes some time to reach its terminal velocity from t¼0. It also takes some time to reduce its velocity to zero once the current pulse is switched off at t¼tp¼2 ns. As expected, these acceleration and deceleration times increase for the FM coupling case [black squares in Figs. 6(a) and6(b)] with respect the single FM layer stack. This is due to the larger effective DW mass of the FM coupled system as compared to the single-FM-layer stack.19Interestingly, the acceleration and deceleration times are signiﬁcantly short for the system with AF coupling [blue triangles in Figs. 6(a)and 6(b)], which constitutes an additional advantage of these sys- tems for some applications: DWs in these AF coupled stacks FIG. 5. lMresults showing the dependence on the applied current J¼Ji FM¼JHMof the terminal DW velocity the current driven DW motion for (a) single- FM layer stack and (b) AF coupling ( Jex<0) in the presence of STTs. The spin polarization factor of the STTs is P¼0:5 for both FM layers, and different values of the non-adiabatic parameter are evaluated: n¼nL¼nU:0;a;and 2 a. The duration of the current pulse is tp¼2 ns, and the velocity values corre- spond to the terminal state at t¼tp¼2 ns. Perfect samples and zero temperature conditions are considered.013901-8 Alejos et al. J. Appl. Phys. 123, 013901 (2018)can be accelerated and decelerated faster as their single-FM- layer and FM coupled counterparts. D. Current-driven DW motion under longitudinal fields Another revealing study of the consequences of the AF coupling between the two FM layers is the dependence ofDW motion on the application of an in-plane longitudinalﬁeld ( B x) for a given injected current. This lMstudy has been then performed by taking a 2-ns long current pulse of aﬁxed amplitude of J¼2:5T A=m 2, with either positive (J>0) or negative ( J<0) polarity. Again steady-state ter- minal values of DW observables are presented here. Withinthis context, positive (negative) ﬁelds mean applied ﬁeldsdirected along the positive (negative) x-axis. The three men- tioned cases: FM coupling ( J ex>0), single-FM-layer, and AF coupling ( Jex<0) have been also evaluated. The results are shown in Fig. 7. As it has been previously mentioned, the FM coupling between both FM layers ( Jex>0) leads to the formation of twin DWs in both FM layers, that is, with their magnetizationsperfectly aligned within the DW transition. This is a crucialpoint, since both magnetizations are similarly affected by theapplication of the longitudinal ﬁeld B x, either by reinforcing the alignment of the magnetization ( ~mDW) and the current ﬂow ( ~JtðÞ¼JtðÞ~ux), or by promoting their misalignment. This can be checked in the graphs of Fig. 7(a). In these graphs and the ones in Fig. 7(c), the terms UD and DU within the legends are used to refer to the magnetization transition asso-ciated to the DW in the LFM, which can go, respectively,from an up- d o m a i nt oa down -domain (UD) and from a down - domain to an up-domain (DU), along the positive direction of the longitudinal axis ( x-axis). According to the previous dis- cussion, the velocity of a DU (UD) DW increases (decreases)for positive ﬁelds ðB x>0Þand positive currents ( J>0). Conversely, the velocity of a UD (DU) DW increases (decreases) for Bx<0a n d J<0. The other cases combining different signs of the ﬁeld and the current ﬂow can be straight-forwardly derived. The cases when the application of the ﬁeldleads to an absolute decrease of the DW speed reach a pointwhere the DWs freeze and no displacement occurs. Note thatthe FM coupling reduces the longitudinal ﬁeld at which zeroDW velocity is achieved with respect to the single FM layercase [compare top graphs in Figs. 7(a) and7(b)]. This is a clear sign of the magnetostatic interaction between the inter-nal magnetic moments inside the DWs in the LFM and theUFM, which promotes their antiparallel alignment against theFM coupling. Further increase of the applied ﬁeld magnitude promotes the inversion of the chirality of the DWs and the subsequent inversion of the direction of DW displacement. This behavior is qualitatively similar to that of a DW in a sin- gle-FM-layer [Fig. 7(b)]. Different from this behavior, an absolute decrease of the DW velocity is obtained under the application of the longitu- dinal ﬁeld for the AF-coupled system ( J ex<0). As it has been shown, in the absence of driving force ( J¼0), the mag- netizations within the coupled DWs of the LFM and the UFM layers tend to be aligned antiparallel along the x-axis (UL/C25180/C14andUU/C250/C14). The longitudinal ﬁeld promotes the progressive misalignment with respect to the x-axis, inde- pendent of its sign. Therefore, due to the reduced SOT efﬁ- ciency for such a misalignment, the velocity decreases as Bxjj increases. In general, it can be observed that the DW tilting isnot null in the presence of in-plane ﬁelds [see graphs in Figs. 7(a),7(b) and7(c)]. Additionally, the DW width does not remain constant under B x(see bottom graphs in Fig. 7). We have also evaluated the 1DM predictions for the current-driven DW motion in the presence of longitudinal ﬁelds. The 1DM results are collected and compared to the lM results in Fig. 8. A good qualitative agreement is achieved for the three cases. The quantitative discrepancies are due to the same limitations discussed above for the pure current-driven case: the 1DM does not take into account the DW tilting angle nor the magnetostatic interaction between the two FM layers. Moreover, it assumes that the DW width is ﬁxed, which is not the case of the full lMresults shown in the bottom graphs of Fig.7. Nevertheless, the 1DM gives a good description of the lMresults provided that the DW width ( D¼2:3nm) is prop- erly selected for the AF coupling case. The agreement is also good for the FM coupling and single-FM-layer cases adopting a constant DW width as deduced from the analytical formula D¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A=Keffp ¼7:3 nm. E. Current-driven DW motion as a function of the interlayer exchange coupling Before discussing the case of realistic samples with imperfections, it is interesting to examine the current-driven DW dynamics for different values of the exchange coupling between the layers ( Jex). The lMresults of the DW veloci- ties of the lower and the upper FM layers are shown in Fig. 9 for two different current density amplitudes J, and for two different combinations of the saturation magnetization in the LFM and the UFM layers: (a) ML s¼MU s¼600 kA =m and FIG. 6. lMresults of the temporal evo- lution of the DW position (a) and the DW velocity (b) under a current pulse ofJ¼2T A=m2andtp¼2 ns for the three evaluated cases: FM coupling(J ex>0), single-FM layer, and AF cou- pling ( Jex<0). The same parameters as in Figs. 2and3are considered. The depicted results correspond to perfect samples at zero temperature.013901-9 Alejos et al. J. Appl. Phys. 123, 013901 (2018)(b)ML s¼600 kA =m and MU s¼800 kA =m. The gray rectan- gle indicates the range of Jex;where the DWs in the LFM and UFL move uncoupled from each other, i.e., DWs in the LFM and in the UFM depict different velocities. For stronginterlayer coupling, the DWs move coupled, but for small J exjj, they move uncoupled. The range of uncoupled DWmotion is different for both evaluated cases, and it is wider when the FM layers have different saturation magnetization. Note also that this uncoupled range is not symmetric with respect to Jex¼0. The fact that the threshold magnitudes of the interlayer exchange coupling needed for the coupled DW dynamics are different from the FM ( Jex>0) and AF FIG. 7. lMresults as a function of the in-plane longitudinal ﬁeld ( Bx) for the three evaluated cases: (a) FM coupling ( Jex>0), (b) single-FM layer, and (c) AF coupling ( Jex<0). The dependence on the applied ﬁeld of the terminal DW velocity, the DW angle, tilting angle, and the DW width is shown from top to bot- tom graphs. The parameters are those given in the text. The amplitude and the duration of the current pulse are J¼2:5T A=m2andtp¼2 ns, respectively, and the presented results were computed at t¼2 ns when the terminal regime was already reached. The term UD (DU) within the legends refer to the DW magneti- zation transition in the LFM from up-to-down (down -to-up) along the positive direction of the longitudinal axis ( x-axis). Representative snapshots are shown in the bottom graphs for the three evaluated cases. Perfect samples and zero temperature conditions are considered.013901-10 Alejos et al. J. Appl. Phys. 123, 013901 (2018)(Jex<0), coupling cases indicate that indeed the magneto- static coupling between the layers plays a role in the DW dynamics. This magnetostatic interaction between the mag- netization in the FM layers is complex in general. It includesdifferent contributions: Domain-Domain ( ~Hi!j d;D/C0D), Wall- Domain ( ~Hi!j d;W/C0D), and Wall-Wall ( ~Hi!j d;W/C0W) interactions where i;j:L;U. (for example, ~HL!U d;W/C0Wrepresents the mag- netostatic interaction generated by the internal DW magnetic FIG. 8. lMresults (dots) and 1DM (both solid and dashed lines) predictions of the current driven DW motion under longitudinal ﬁelds ( Bx) for the three evalu- ated cases: (a) FM coupling ( Jex>0), (b) single-FM layer, and (c) AF coupling ( Jex<0). The dependence on Bxof the DW velocity and the DW angle is shown from top to bottom graphs. The parameters are those given in the text. The amplitude and the duration of the current pulse are J¼2:5T A=m2 andtp¼2 ns, respectively, and the presented results were computed at t¼2 ns. The DW width for the AF coupling ( Jex<0) was needed to be rescaled to D/C252:3 nm in the 1DM in order achieve quantitative agreement with lMresults. For the two other cases, FM coupling ( Jex>0) and single-FM layer, the input value of the DW width was D/C257:3 nm, as predicted by the analytical formula D¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A=Keffp . The results correspond to perfect samples and zero temperature. FIG. 9. lMresults of the current driven DW motion for different values of the exchange coupling parameter Jexand two applied currents of ampli- tudes J¼1T A=m2orJ¼2T A=m2. Two different combinations of the sat- uration magnetization in the LFM and UFM are considered: (a) ML s¼MU s ¼600 kA =m and (b) ML s¼600 kA =m andMU s¼800 kA =m. The gray range indicates the range of Jexwhere the DWs in the LFM and UFL moveuncoupled from each other. The results were obtained at zero temperature. (c) and (d) show a schematic representa- tion of the magnetostatic ﬁeld created by magnetization in the ilayer on the j layer for the AF and FM coupling cases, respectively. Domain-Domain (~Hi!j d;D/C0D), Wall-Domain ( ~Hi!j d;W/C0D), and Wall-Wall ( ~Hi!j d;W/C0W) are shown.013901-11 Alejos et al. J. Appl. Phys. 123, 013901 (2018)moment in the Lower DW on the Upper DW). These interac- tions, which in general are difﬁcult to isolate from the globalmagnetostatic interaction in the system, are schematicallyshown in Figs. 9(c) and9(d) for and AF ( J ex<0) and FM (Jex>0) coupling cases. Note that even in the symmetric case ( ML s¼MU s¼600 kA =m), there is not a complete com- pensation for AF coupled layers [Fig. 9(c)]: although the Wall-Wall magnetostatic interaction ( ~Hi!j d;W/C0W) supports the AF coupling, and the Domain-Domain ( ~Hi!j d;D/C0D) interaction does not. For the FM coupling case [Fig. 9(d)], the magneto- static interaction between the wall moments ( ~Hi!j d;W/C0W) acts against the exchange interlayer coupling, which promotestheir parallel alignment. On the contrary, the parallel align-ment of the magnetization in the Lower and the UpperDomains is assisted by this magnetostatic interaction(~H i!j d;D/C0D). IV. MICROMAGNETIC RESULTS FOR REALISTIC AND ASSYMMETRIC MULTILAYERS Most of the former results were obtained for perfect sam- ples considering two FM layers with identical thickness(t L FM¼tU FM¼0:8 nm) and saturation magnetization ( ML S ¼MU S¼600 kA =m). The thickness of the spacer was also equal to the one of the FM layers ( tL FM¼tS¼tU FM¼0:8n m ) .In this section, we study the current-driven DW motion along realistic strips, i.e., with imperfections (details were given at theend of Sec. II A), and considering different FM layers, with dif- ferent thicknesses ( t L FMandtU FM) and saturation magnetization (ML SandMU S). The thickness of the spacer is also varied ( tS). Besides the FM coupling ( Jex>0), single-FM-layer and AF coupling ( Jex<0) cases, the lMresults collected in Fig. 10 also include the case where the two FM layers are not exchangecoupled ( J ex¼0, red circles). Note that in the absence of inter- layer exchange coupling, only the DW in the LFM is displaced as due to the SHE, which, as already mentioned, in the presentwork is only acting in the LFM layer. Therefore, the red circles in Fig. 10correspond to the DW velocity in the LFM layer, whereas the black squares (FM coupling) and blue triangles(AF coupling) represent the DW velocities in both the LFM and the UFM layers, where they move coupled. Several important conclusions can be extracted from the results shown in Fig. 10. First, the DW in the single-FM- layer case (green triangles) is less sensitive to the imperfec- tions. Such imperfections introduce a propagation threshold(J P) for the DW motion in the other cases (HM/LFM/Spacer/ UFM), where the DW dynamics is only driven by the SHE in the HM and the interlayer exchange coupling. In otherwords, the SHE, acting only in the LFM, must overcome the DW pinning in both FM layers. In the low current regime FIG. 10. lMresults of the current driven DW motion along realistic strips for different combinations of the saturation magnetization in the FM layers (a, b, and c, from top to bottom) and different thicknesses of the layers (1, 2, and 3, from left to right). The magnitude of the exchange coupling parameter is Jexjj ¼0:5m J=m2for the FM ( Jex>0) and AF ( Jex<0) coupling cases and zero for the no coupling case ( Jex¼0). These results were obtained at zero tempera- ture for realistic samples, with defects included as described in Sec. II.013901-12 Alejos et al. J. Appl. Phys. 123, 013901 (2018)[which is limited by the combination of thicknesses ( tL FM,tU FM andtS) and saturation magnetization values ( ML SandMU S)], the single-FM-layer case is the one depicting higher veloci-ties. However, in the high Jregime, the DW velocity satu- rates, as it was already explained here and elsewhere. 9,18For higher values of J, (except for a3 and b3, dominated by a strong magnetostatic interaction between the internal DWmoments), the largest velocity is achieved for the AF cou-pling case (blue triangles), and the smallest one in the FMcoupling case (black squares). Note, that the DW velocityincreases monotonously with Jin the AF coupling case. The inﬂuence of the disorder is evident for all cases, and in gen-eral, the propagation threshold is magniﬁed in the absence ofexchange coupling between the FM layers (see the redcircles for J ex¼0). This is expected again as due to the mag- netostatic interaction between the two dissimilar FM layers.In fact, this interaction promotes the antiparallel alignmentof the internal magnetic moments, and therefore it results inan attractive force between these two DWs, which naturallyexplains the larger propagation threshold of the DW in theLFM in the absence of exchange coupling ( J ex¼0). For theAF coupling case ( Jex<0) in the high current regime, the largest velocity is reached when the thickness of the spacer(t S) is reduced (compare cases 1 and 2 in Fig. 10). Once more, this is a consequence of the magnetostatic interaction:ast Sis reduced, the dipolar interaction between the internal magnetic moments supports their antiparallel alignment resulting in a larger DW velocity. Also, the DW velocity increases as the saturation magnetization of the FM layers isequal ( M L S¼MU S¼600 kA =m, compare cases a, b, and c in Fig.10). This later fact was already qualitative explained by Eq.(7), where the DW velocity scale with ðML SþMU SÞ/C01. V. CURRENT-DRIVEN MOTION ALONG CURVED STRIPS Apart from the larger velocity of the DWs, another important advantage of using AF coupled layers with respectto the single-FM-layer stacks is the absence of DW tilting (seeFigs. 2and 3). In a single-FM-layer stack, adjacent DWs depict opposite tilting of their DW plane, 32,33which imposes a limit in the density of information coded between adjacent FIG. 11. In-plane geometry to evaluate the current-driven DW dynamics along samples with curved parts. The strip contains three straight sections con-nected by two round-shaped sections. (a) The values of the geometrical parame- ters deﬁned therein are: w r¼512 nm, ri¼192 nm, ro¼256 nm, hx¼2ro þwr,a n d hy¼3roþri. (b) Spatial dis- tribution of the normalized current den- sity ( ~Jð~rÞ=JuÞalong the heavy metal (HM) under the lower FM layer. The color indicates the current density ( ~Jð~rÞ) normalized to the value in the straight part ( Ju), where the current is uniform Ju. FIG. 12. Displacement of a DW train along a curved strip in a single-FM strip. The initial state consists of a given DW train as deﬁned in the upper straight surrounded by a dashed green rectangle. The snapshots show the dis- placement in time of the set of DWs under the application of a current withamplitude J¼2T A=m 2at a tempera- ture of T¼300 K. Realistic samples with defects are considered here. A DW annihilation process starts at a time around t¼4 ns in the area sur- rounded by a solid red square.013901-13 Alejos et al. J. Appl. Phys. 123, 013901 (2018)DWs. Indeed, the DW tilting can result in the annihilation of adjacent DWs, leading to mischievous effects on the coded information in a DW-based device. Contrarily, our present study indicates that when two FM layers are exchange cou-pled by the synthetic antiferromagnet (AF coupling), the DWs are driven by the current without signiﬁcant tilting. For possi- ble applications, trains of DWs must be displaced by the action of the current not only along straight paths, but also along curved paths. Accordingly, the motion of trains of DWs along both a single-FM-layer and a multilayer with two AF coupled FM layers have been separately studied. The case of the DW displacement along FM curved strips constitutes one of the most interesting examples of application of the previous study. DW tilting limits in much cases the feasibility of these elements as racetrack memo- ries, 34since DWs in these strips move at different velocities at the curved sections, depending on their UD or DU conﬁg- uration. As an example of these curved geometries, a strip composed of three straight sections and two round-shaped sections, i.e., an inverse S-shaped element was evaluated. Its geometry is depicted in Fig. 11(a) . The evaluated dimensions are given in the caption of Fig. 11. The current density ~Jð~rÞbecomes non-uniform when it is forced to ﬂow along curved paths. The current distribution in the HM under the lower FM layer is shown in Fig. 11(b) , which clearly indicates a radial dependence: the current den- sity~Jð~rÞdepicts an inversely linear dependence on the radius when it is forced to ﬂow over semicircular arcs. These results were computed with COMSOL22and taken into account to evaluate the current driven DW motion. Realistic conditions have been considered, which include defects in the form of grains (see details at the end of Sec. II A) and thermal effects at room temperature ( T¼300 K). Two cases are considered, a single-FM-layer stack (Fig. 12) and a multilayer with AF coupling (Fig. 13). In both cases, a series of DWs is initially placed at one of their ends (see the areas surrounded by a green dashed rectangle at the upper ends of the strips), thendeﬁning a set of upanddown domains in an identical conﬁgu- ration along both strips. Currents of amplitude J¼2T A=m2 run and push forward the series of DWs in each strip. Snapshots of the displacement of the DWs are depicted in Fig. 12for a single-FM-layer. A DW annihilation event starts at a time t¼4 ns and takes place at the beginning of the lower curved section (see the down domain surrounded by the solid red square). The DWs limiting such a reduced down domain are moving from right to left in the preceding straight section, but they reach different velocities as theyenter the curved section. 34This fact, together with the inher- ent tilting of both DWs, result in the DWs making contact at their upper end, and then mutually annihilating. Ten different stochastic realizations of the thermal noise were evaluated, all of them showing similar annihilation events. Similar DW trains are also considered in the multilayer with AF coupling. The results are shown in Fig. 13. The ini- tial conﬁguration of the DW train holds in both FM layers along the whole dynamics, so that this DW train successfully reaches the bottom-lower end without annihilation. Differentfrom the preceding case, the formation of paired DWs in the LFM and UFM layers, containing both types of DWs, UD and DU, results in equalized velocities along the curved sec- tions for the two types of magnetization transitions in these strips. Additionally, the absence of DW tilting reduces thelikelihood of a contact between adjacent DWs. In principle, besides the high efﬁcient DW dynamics, AF-coupling systems can also improve the density of packedinformation, coded between adjacent walls. However, a deeper observation of the images shown in Fig. 13indicates that the second down domain in the UFM (second updomain in the LFM) is contracted when arriving at the ﬁrst curve (third image in Fig. 13,a tt¼1 ns). Then, this down domain extends a little bit on the straight line and again contracts at the second curve. Therefore, it seems that under realistic conditions (defects and thermal ﬂuctuations) the distance between adjacent DWs can also vary during the motion even FIG. 13. Displacement of a DW train along a curved strip corresponding to the AF-coupled multilayer. Two analo- gous DW trains as in Fig. 12are con- sidered as the initial state in the upper and lower FM layers, as shown within the dashed green rectangle in the upper ends of the UFM and LFM strips. The snapshots show the displacement in time of the set of DWs under the application of a current with amplitudeJ¼2T A=m 2at a temperature of T¼300 K. Realistic samples with defects are considered here. The train of DWs is displaced with the same velocity along the straight and curve parts of the strips, and no DW annihi- lation is observed.013901-14 Alejos et al. J. Appl. Phys. 123, 013901 (2018)for the AF coupling case, and consequently it is needed to evaluate the distance between adjacent DWs for realistic conditions. In order to get further insights into this behavior,we have also evaluated the dynamics of two DWs within each FM layer starting from different distances between them ( d 0). We monitor the evolution of the distance between these DWs at ﬁve different points along the curved sample. These points are labeled with letters in Fig. 14(a) , which cor- responds to an initial state where two DWs are initially sepa-rated by d 0/C25130 nm. The snapshots shown in Fig. 14(a) were obtained in the presence of a disorder (see disorder details in Sec. II A) but at zero temperature ( T¼0). It can be visually checked that the initial distance between the 2DWs does not change as they are driven along the track [see also Fig. 15(a) ]. However, in the presence of thermal noise atT¼300 K, we notice that the distance between the DWs slightly changes: see Figs. 14(b) and14(c) , which correspond to two different stochastic realizations and the same grain pattern. We recorded the temporal evolution of the out-of-plane component of the magnetization at the same ﬁve points indicated in Fig. 14(a) :m zi;tðÞ with i:A;B;C;D;andE. The FIG. 14. Displacement of two DWs along the AF-coupled multilayer under the application of a current with amplitude J¼2T A=m2. The snapshots correspond to the LFM and show the temporal displacement of two DWs: (a) Sample with disorder in the form of grains at zero temperature. (b) and (c) correspond to two different stochastic realizations computed at room temperature for the same grain pattern as in (a). FIG. 15. Temporal evolution of the out-of-plane magnetization ( mzði;tÞ) at ﬁve different points ( i:A, B, C, B, and E) along the LFM layer of the AF-coupled multilayer under the application of a current with amplitudeJ¼2T A=m 2for three cases: (a) sample with disorder and at T¼0. (b) and (c) correspond to the same grain pattern as in (a) but for two different stochastic realizations of the thermal noise at T¼3 0 0 K .( a ) ,( b ) ,a n d( c ) correspond to the snapshots shown in Fig. 14.( d )DtS¼tDU s/C0tUD sas deﬁned in the text at different points (A, B, C, B, and E) along the LFMlayer of the AF-coupled multilayer under the application of a current withamplitude J¼2T A=m 2. These points are marked in Fig. 14. The open symbols correspond to different grain patterns and different stochasticrealizations of the thermal noise. The red symbols depict the average overgrains patterns and thermal realizations and the blue symbols are the zerotemperature results.013901-15 Alejos et al. J. Appl. Phys. 123, 013901 (2018)results of mzi;tðÞ vs.tcorresponding to the cases depicted in Figs. 14(a) ,14(b) and14(c) are shown in Figs. 15(a) ,15(b) and15(c) . In order to provide a quantitative estimation of the temporal evolution of the distance between the DWs, we computed the difference in the switching (DW passage) times at the mentioned points: DtS/C17tDU S/C0tUD S, where tDU S andtUD Scorrespond to the times at which the left (DU) and the right (UD) DWs pass across the mentioned points. Thistemporal interval Dt Sconstitutes a measure of the distance between the DWs as they are driven along the track. As it isshown in Fig. 15(d) ,Dt Sdoes not vary from point to point at zero temperature [blue dots in Fig. 15(d) ]. In order to provide an statistically description of this thermally activated dynamics, we evaluated three differentgrain patterns and three different stochastic realizations ofthe thermal noise at T¼300 K. The corresponding results of Dt Sat the mentioned points are shown by open symbols in Fig. 15(d) , which indicates that the distance between the walls changes for different grains patterns and temperaturerealizations. However, the mean distance averaged overthese grains patterns and stochastic realizations [red squaresin Fig. 15(d) ] is hardly dependent on the point along the track. We have performed a similar study starting from two DWs initially separated by d 0/C2560 nm, and we veriﬁed that the DWs can collapse for some of the evaluated stochasticrealizations. Therefore, this imposes a limit in the density ofpacked information even for the AF coupling stacks.Although further studies are needed to evaluate others sam-ples with different strip width and curvature radius, our anal-ysis suggests that the AF coupled multilayers could be usedto efﬁciently drive trains of highly packed DWs. VI. CONCLUSIONS The current-driven DW motion has been studied by micromagnetic simulations in multilayers with two ferromag-netic layers separated by a spacer. These layers are coupledby the interlayer exchange coupling, which depending on itsmagnitude and sign, can generate ferromagnetic (FM) or anti-ferromagnetic (AF) coupling between them. The interfacialDzyalozinskii-Moriya interaction is only required at the inter-face between the heavy metal layer and the lower ferromag-netic layer, and provides the magnetization domain walltexture with adequate chirality. The results are compared tothe ones obtained for the single-ferromagnetic-layer case and qualitatively explained in terms of analytical expressions deduced from the one-dimensional model. However, thethree-dimensional space micromagnetic description allows forunraveling some details of such dynamics that are not fullyaccessible from a one-dimensional description, even thoughthe latter approach may draw rather good qualitative results.For low currents in perfect samples, the driving force resultingfrom spin-orbit torques (spin Hall effect) is not capable ofimpelling paired walls as efﬁciently as domain walls in thesingle-ferromagnetic-layer stack. Indeed, domain walls in theupper ferromagnetic layer are dragged by the moving walls in the lower ferromagnetic layer, because of the interlayer exchange coupling, which results in this lack of effectiveness.For higher currents, the coupled walls associated to the FMcoupling present an analogous behavior to that of domain walls in the single layer stack, i.e., the domain wall velocitysaturates as the current is increased. On the other hand, the AF coupling results in a high velocity of the coupled DWs, which are driven without sig-niﬁcant tilting by the spin Hall effect from the heavy metal.The antiferromagnetic coupling promotes the antiparallelalignment of the internal DW moments in the lower and inthe upper layers, both depicting a chiral N /C19eel conﬁguration. As a consequence of that, the DW increases monotonouslywith current density. The velocity of the AF coupled DWs isenhanced as the saturation magnetization of the layersbecomes similar in magnitude, and when their valuesdecrease, full realistic micromagnetic simulations indicate afaster coupled DW motion when the thickness of the spacerbetween the FM layers is reduced, and also when theselayers exhibit equal saturation magnetization. While thislater observation can be qualitatively described by the simpleone-dimensional model, the ﬁrst one is a direct consequenceof the magnetostatic interaction between the internal mag-netic moments of the DWs, which supports the antiparallelorientation between the internal magnetic moments in theAF coupling case. The conventional spin transfer torques do not signiﬁcantly perturb the current-driven DW dynamics generated by the Slonczewski-like spin-orbit torque in AFcoupled stacks, at least under perfect adiabatic conditions. Itwas also observed that inertia effects are signiﬁcantlyreduced in AF coupled stacks with respect to the single-FM-layer and FM coupling cases. The high efﬁciency of thecurrent-driven DW dynamics in these AF systems is alsocoherent with the results obtained under in-plane longitudi-nal applied ﬁelds, which are also presented here. Our micromagnetic simulations also indicate that up–down anddown –updomain walls move with different velocities along a single-FM-layer stack with curved parts.Moreover, domain wall tilting constitutes another importantissue that interferes with the proper working of DW-basedracetrack memories. This is particularly critical in the caseof single-FM-layer stacks with curved parts, since this tiltingmay give rise to domain wall annihilation, and consequently,imposes a limit for the high density packing of domain walls.Our micromagnetic simulations have also revealed antiferro-magnetic coupling as a sound ally to avoid tilting and, conse-quently, to help the safe displacement of domain walls alongsuch curved geometries. For these antiferromagnetic coupledstacks, up–down anddown –upwalls move with the same velocity along curved tracks at zero temperature. However,very close DWs can collapse even for AF coupling stacksunder realistic conditions. The variation of the relative dis-tance between adjacent walls is due to thermal ﬂuctuations. Therefore, further systematic theoretical and experimental studies are needed to evaluate this limitation for strips withdifferent widths and curvature radius. ACKNOWLEDGMENTS This work was supported by Project WALL, FP7- PEOPLE-2013-ITN 608031 from the European Commission,Project MAT2014-52477-C5-4-P from the Spanish government,013901-16 Alejos et al. J. Appl. Phys. 123, 013901 (2018)and Projects SA282U14 and SA090U16 from the Junta de Castilla y Leon. 1I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-Prejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, J. Vogel, M. Bonﬁm, A. Schuhl, and G. Gaudin,Nat. Mater. 10, 419 (2011). 2T. Koyama, D. Chiba, K. Ueda, K. Kondou, H. Tanigawa, S. Fukami, T. Suzuki, N. Ohshima, N. Ishiwata, Y. Nakatani, K. Kobayashi, and T. Ono,Nat. Mater. 10, 194 (2011). 3P. P. J. Haazen, E. Mure `, J. H. Franken, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans, Nat. 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5.0028918.pdf	J. Appl. Phys. 128, 220902 (2020); https://doi.org/10.1063/5.0028918 128, 220902 © 2020 Author(s).Spin-gapless semiconductors: Fundamental and applied aspects Cite as: J. Appl. Phys. 128, 220902 (2020); https://doi.org/10.1063/5.0028918 Submitted: 08 September 2020 . Accepted: 21 November 2020 . Published Online: 09 December 2020 Deepika Rani , Lakhan Bainsla , Aftab Alam , and K. G. Suresh Spin-gapless semiconductors: Fundamental and applied aspects Cite as: J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 View Online Export Citation CrossMar k Submitted: 8 September 2020 · Accepted: 21 November 2020 · Published Online: 9 December 2020 Deepika Rani,1,2 Lakhan Bainsla,1,3 Aftab Alam,1,a) and K. G. Suresh1,b) AFFILIATIONS 1Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, Maharashtra, India 2Department of Physics, Indian Institute of Technology Delhi, Delhi 110016, India 3Physics Department, University of Gothenburg, 412 96 Gothenburg, Sweden a)Electronic mail: aftab@iitb.ac.in b)Author to whom correspondence should be addressed: suresh@phy.iitb.ac.in ABSTRACT Spin-gapless semiconductors (SGSs) are new states of quantum matter, which are characterized by a unique spin-polarized band structure. Unlike conventional semiconductors or half-metallic ferromagnets, they carry a finite bandgap for one spin channel and a close (zero) gap forthe other and thus are useful for tunable spin transport applications. It is one of the latest classes of materials considered for spintronic devices.A few of the several advantages of SGS include (i) a high Curie temperature, (ii) a minimal amount of energy required to excite electrons fromthe valence to conduction band due to zero gap, and (iii) the availability of both charge carriers, i.e., electrons as well as holes, which can be 100% spin-polarized simultaneously. In this perspective article, the theoretical foundation of SGS is first reviewed followed by experimental advancements on various realistic materials. The first band structure of SGS was reported in bulk Co-doped PbPdO 2, using first-principles cal- culations. This was followed by a large number of ab initio simulation reports predicting SGS nature in different Heusler alloy systems. The first experimental realization of SGS was made in 2013 in a bulk inverse Heusler alloy, Mn 2CoAl. In terms of material properties, SGS shows a few unique features such as nearly temperature-independent conductivity ( σ) and carrier concentration, a very low temperature coefficient of resistivity, a vanishingly small Seebeck coefficient, quantum linear magnetoresistance in a low temperature range, etc. Later, several other systems, including 2-dimensional materials, were reported to show the signature of SGS. There are some variants of SGSs that can show aquantum anomalous Hall effect. These SGSs are classic examples of topological (Chern) insulators. In the later part of this article, we havetouched upon some of these aspects of SGS or the so-called Dirac SGS systems as well. In general, SGSs can be categorized into four different types depending on how various bands corresponding to two different spin channels touch the Fermi level. The hunt for these different types of SGS materials is growing very fast. Some of the recent progress along this direction is also discussed. Published under license by AIP Publishing. https://doi.org/10.1063/5.0028918 I. INTRODUCTION Spintronics has emerged as the most important topic of research in the field of magnetic materials today. This is becauseof the immense application potential that this field offers.Chronologically, the first and foremost type of materials identified as spintronic materials is the half-metallic ferromagnets (HMFs). Subsequently, novel classes of materials such as dilute magneticsemiconductors (DMSs), spin-gapless semiconductors (SGSs), andvery recently spin semimetals, which are the spin analogs of gaplesssemiconductors (GSs) and semimetals, respectively, emerged. The main role of all these materials is to provide spin-polarized charge carriers or spin currents. These are primarily ferromagnetic innature and hence the interest in applications. However, due to the disadvantages such as stray fields of ferromagnets, spin-polarized antiferromagnets or ferrimagnets are also being investigated these days. This has given birth to the field of antiferromagnetic spintronics. Often the difference in the electronic band structure of various mate-rial classes as mentioned above is such that we can “tune ”the system from one to another with the help of external parameters such as c o m p o s i t i o n ,p r e s s u r e ,t e m p e r a t u r e ,m a g n e t i cf i e l d ,e t c .A l s o ,au n i q u eand exact identification of the nature of such materials is often difficultin the absence of direct probing of the band structural details. In thisarticle, we focus only on various types of spin-gapless semiconductors, both in a bulk and thin-film form. We also highlight the experimentalJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-1 Published under license by AIP Publishing.and theoretical findings made on various SGS materials with a mention about their application potential. We present a detailed account of the work done on Heusler alloy based materials and a fewother systems, which have been identified as promising SGS systems.In the light of the recent interest in topological materials, efforts arealso devoted to make use of such properties in achieving potential SGS materials. This aspect is also covered in this article. As mentioned above, spin-gapless semiconductors are a special type of magnetic material, which exhibits a closed gap for one spinband and a finite, non-zero gap for the other. 1,2They can be consid- ered a combination of gapless semiconductors3and half-metallic ferromagnets. The latter itself is an interesting class of materials in the spintronics family with zero density of states (DOS) for one spinchannel and finite DOS for the other at the Fermi level, giving riseto 100% spin polarization. Figure 1 shows the schematic pictures of the density of states for half-metallic ferromagnets and spin-gapless semiconductors along with gapless semiconductors. Due to the unique electronic structure of SGS materials, the conducting (free)carriers are not only fully spin polarized but are also easily excitedbecause of the gapless nature of one of the spin channels. Also, ascompared to the conventional semiconductors, the charge carriers have high mobility in these materials. At the same time, their elec- trical conductivity is lower than that of HMFs and hence advanta-geous with regard to spin injection to semiconductors (i.e., a betterconductivity match). These features lead to peculiar transport prop- erties and thus offer novel functionalities in the field of spintronics. SGS materials exhibit novel properties such as (i) only a very smallamount of energy is required to excite electrons from the valenceband (VB) to the conduction band (CB), (ii) the excited charge car-riers, both electrons and holes, can be 100% spin-polarized simulta- neously, (iii) fully spin-polarized electrons and holes can be easily separated using the Hall effect, and (iv) the spin-up and spin-downelectrons and holes, with both full spin polarization and tunablecarrier concentrations, can be easily manipulated in their respectivechannels by shifting the Fermi level through gate voltage control. 2 In general, SGSs can be categorized into four different types depending on how various bands corresponding to two differentspin channels touch the Fermi level.1Schematics of these four SGS types are shown in Fig. 2 . In type I SGSs [ Fig. 2(a) ], the valence band maximum (VBM) and the conduction band minimum(CBM) are in the same spin channel (i.e., either spin up or spindown), while there is a gap in the opposite spin channel; i.e., onespin channel is gapless and the other is semiconducting. This cor- responds to the conventional SGS as mentioned before. In type II SGSs, the VBM and CBM are in the opposite spin channels [seeFig. 2(b) ]. In this case, there is a gap between the conduction and valence bands for both the majority and minority electrons, butthere is no gap between the majority electrons in the VB and the minority electrons in the CB. Also, depending on how the VBM and CBM touch each other, the bandgap can be direct (if VBMand CBM touch each other at the same k point) or indirect (if theytouch each other at different k points). In type III (type IV) SGSs,the VBM (CBM) is of one spin character, while the CBMs (VBMs) arise from both spin channels. Thus, in type III SGSs, one spin channel is gapless and the other one is gapped with the top of VBbeing lower than the Fermi level [see Fig. 2(c) ]. In type IV SGSs, one spin channel is gapless, while the top of the valence band forthe opposite spin channel touches the Fermi level, which is sepa- rated from its corresponding CB by a gap [see Fig. 2(d) ]. GS behavior was first predicted in a dilute magnetic semicon- ductor, PbPdO 2, by X. L. Wang based on first-principles calcula- tions using local density approximation (LDA).1Also, based on first-principles calculations, the authors found that Co-substituted PbPdO 2exhibits the band structure of a type II spin-gapless semi- conductor [as shown in Fig. 2(b) ].1This was the first report on spin-gapless semiconductors. Later, SGS nature was confirmedexperimentally in a Co-substituted PbPdO 2thin film based on magneto-transport properties.4From the applied point of view, this realization was significant because many of these SGS materialshave a higher Curie temperature (T C) compared to the other known spintronic, semiconducting family, namely, DMS.5 However, later based on more accurate hybrid functional calcula- tions,6it was realized that the ground state of PbPdO 2is rather a semiconductor (small bandgap). Another breakthrough happenedwith the identification of Heusler alloys with SGS properties. Thesealso have advantages over DMS and other reported magnetic semi-conductors. Many Heusler-based alloys have a stable structure, high T C, and high spin-polarization, which make them more suit- able for applications in spintronics. Theoretically, many Heusleralloys have been identified as SGSs, 1,7–10but only a few have been confirmed experimentally. The first experimental confirmation of SGS behavior in the Heulser family was reported in full-Heusler alloy Mn 2CoAl.11The almost temperature-independent conductiv- ity, i.e., a very small temperature coefficient of resistivity (TCR)value, a negligible change in the carrier concentration with temper-ature, and a negligible Seebeck coefficient were considered to be the signatures of a SGS material. Later, SGS nature was confirmed on the basis of theory and experiment in quaternary Heusler alloys,CoFeMnSi 12and CoFeCrGa,13by Bainsla et al. Fully compensated ferrimagnetic SGS nature was reported in quaternary Heusler alloyCrVTiAl 14by Venkateswara et al. Most of the SGS materials predicted so far possess parabolic energy dispersions. However, for spintronic devices, in comparisonwith parabolic-like SGSs, Dirac-like SGSs (DSGS) would be a better FIG. 1. Schematic picture of the density of states for a half-metal (HM), a spin- gapless semiconductor (SGS), and a gapless semiconductor (GS).Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-2 Published under license by AIP Publishing.choice because they can lead to low energy consumption and ultra- fast transport because of their unique linear band dispersion.15,16 In Dirac-like SGSs, the dispersion between energy and momentum is linear. The effective mass of the electrons can be very low (madesignificantly smaller compared to that of the normal carriers) in a Dirac-like SGS due to its graphene-like linear dispersion. Thus, it is highly desirable to find novel, stable DSGSs with an intrinsic 100%spin-polarized Dirac state. The same four types of the gapless para-bolic bands as described above can be realized for DSGS as well as shown in Fig. 3 . In addition to graphene, the Dirac state was found to exist in some similar two-dimensional materials such asphosphorene, 17TiB 2monolayers,18and silicene.19DSGS should not be confused with a Dirac half-metal (DHM). The band structures ofDSGS and DHM are shown in Fig. 4 , and it can be seen that the Dirac point of DHM is located above (or possibly below) the Fermilevel and does not intersect with the Fermi level, clearly indicating half-metallic nature. On the other hand, DSGS is essentially defined by the observation that its conducting band and valence band exactlymeet each other at the Fermi level (called as the Dirac-like gaplessstate) in one spin channel (Sec. III). 20In other words, the DSGS band structure is more specific than that of DHM in the sense that its Dirac point should intersect the Fermi leve l rather than simply being located n e a rt h eF e r m il e v e l .T h eo v e r v i e wo ft h ea r t i c l ei nt h ef o r mo faf l o wchart is shown in Fig. 5 . II. EXPERIMENTAL SIGNATURES OF SGS MATERIALS To accurately establish SGS behavior in a material, one needs careful ab initio calculations and experimental measure- ments. If not done appropriately, this may sometimes lead to misleading predictions. One such example is a CoFeMnSi qua- ternary Heusler alloy, which was earlier predicted to be a half-metal based on simple magnetic measurements and hard x-rayphotoelectron spectroscopy. 21It should be noted that merely sat- isfying the Slater –Pauling rule does not guarantee the existence of half-metallic or SGS nature. Also, no concrete idea about the metallicity or semiconducting behavior of a material can beobtained using x-ray photoelectron spectroscopy measurements.Even though a large number of compounds have been predicted to show SGS nature, only a few among them have been verified experimentally. Therefore, it is important to understand the FIG. 2. Energy band diagrams for four types of spin-gapless semiconductors (SGSs) with parabolic dispersion between energy and momentum: (a) type I, (b) typ e II, (c) type III, and (d) type IV . FIG. 3. Energy band diagrams for four types of spin-gapless semiconductors (SGS) with linear dispersion between energy and momentum: (a) type I, (b)type II, (c) type III, and (d) type IV .Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-3 Published under license by AIP Publishing.experimental signatures that directly or indirectly confirm the SGS nature. To be precise, it is not just one experimental observ-able that can guarantee the SGS behavior of a material but rather a combination of many properties that help in qualifying a mate- rial to be SGS. Transport measurements give a relatively morereliable information about the SGS behavior of a given material.Qualitatively, the main experimental signatures of a SGS mate-rial include(i) nearly temperature-independent low conductivity ( /difference10 5S/cm), (ii) relatively low and almost temperature independent charge carrier concentration ( /difference1019cm/C03), (iii) a vanishingly small Seebeck coefficient, (iv) quantum linear magneto-resistance (MR) at low tempera- tures, and (v) low anomalous Hall conductivity (approximately up to a couple of hundred S/cm). FIG. 4. Schematic band structures of DHM and DSGS. Reproduced with permission from Wang et al ., Appl. Phys. Rev. 5, 041103 (2018). Copyright 2018 AIP Publishing LLC. FIG. 5. Flow chart presenting the overview of the article.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-4 Published under license by AIP Publishing.Though most of the SGS materials show semiconducting nature with a very low temperature coefficient of resistivity values, in some cases such as Mn 2CoAl22and Ti 2MnAl films,23the resistivity shows metallic type behavior in a low temperature regime, which is mainlyattributed to impurity levels originating from atomic disorder,defects, or off-stoichiometry. Thus, a careful analysis of transport properties is essential to understand the difference between a half- metal and a SGS. Another important difference between a half-metaland SGS is the value of anomalous Hall conductivity at room tem-perature. SGSs generally have about one order less anomalous Hallconductivity compared to that of half-metals. III. THEORETICAL TOOLS The theoretical predictions of SGS materials are mostly based on first-principles calculations using Density Functional Theory(DFT). A number of materials, including both monolayer and bulk materials, have been predicted to exhibit SGS characteristics. Various DFT packages used so far include Vienna Ab initio Simulation Package (VASP), 24Spanish Initiative for Electronic Simulations with Thousands of Atoms (SIESTA),25Cambridge Sequential Total Energy Package (CASTEP),26DMol3,27ABINIT,28 etc. Among these, the VASP, CASTEP, and ABINIT are based on plane-wave basis sets for periodic systems, whereas DMol3and SIESTA are based on atomic orbital basis sets. Approximationssuch as the local density approximation (LDA) 29and the general- ized gradient approximation (GGA)30are widely used to investigate 2D and 3D materials. Among numerous GGA, the Perdew –Wang (PW91)31and the Perdew –Burke –Ernzerhof (PBE)32functionals are routinely used. Although GGA-PBE gives a fair description ofthe electronic properties 33for parabolic-type SGS, its main draw- back is the inaccurate prediction of bandgaps, which often makes it inadequate for very complex systems. The situation is more compli-cated in systems in which the electrons tend to be localized andstrongly correlated. To solve this problem, an additional orbital-dependent interaction (the so-called Hubbard U) is taken into account. In the GGA+U method, GGA-calculations are coupled with an additional interaction, which is considered only for highlylocalized atomic-like orbitals on the given site, i.e., similar to the“U”interaction in the Hubbard model. 34More recently, hybrid functional based on a screened Coulomb potential by Heyd, Scuseria, and Ernzerhof (HSE)35is also used frequently, which yields a pretty accurate prediction of the magnetic effects and bandgaps forDSGSs. Moreover, in the case of materials containing heavy ele-ments, the spin –orbit coupling effect 36has been considered. IV. PARABOLIC-TYPE SGS In this section, we will discuss the theoretically and experi- mentally proposed parabolic-type SGSs including oxides,half-Heusler, full-Heusler, LiMgPdSn-type quaternary Heusler, diamond like quaternary compound CuMn 2InSe 4, graphene-like ZnO, and BiFe 0:83Ni0:17O3. A. Oxide gapless semiconductor, PbPdO 2 The concept of spin-gapless semiconductors was first verified in Co-substituted PbPdO 2by Wang based on first-principlescalculations.1PbPdO 2is the first oxide-based gapless semiconduc- tor. However, pure PbPdO 2is a non-magnetic gapless semicon- ductor. Wang proposed to replace Pd by Co in one unit cell (which corresponds to a 25% substitution of Co with Pd) to intro-duce electron spin into the system. Figure 6 shows the spin- resolved band structure for the Co-doped PbPdO 2.T h eb a n d structure shows that the valence band maximum of the majority electrons touches the Fermi level at the Γpoints and the conduc- tion band minimum of the minority electrons touches the Fermilevel as well but at the U point and between the T and Y points.Thus, the valence band of majority electrons and the conductionband of minority electrons is gapless but indirectly. This band structure resembles type II spin-gapless band structures, as shown inFig. 2(b) . It was also found that the d-orbitals from both Co a n dP da n dt h ep - o r b i t a l sf r o mo x y g e nc o n t r i b u t et os u c ha nindirect spin gapless band str ucture in the Co-substituted PbPdO 2. With such substitution, one can expect that with proper elemental substitutions, the spin-gapless features can be realized in various other gapless or narrow band oxides andnon-oxide semiconductors, ferromagnetic or antiferromagneticsemiconductors, or in conductive ferromagnetic oxides and non- oxides. Bulk PbPdO 2was experimentally found to be a p-type gapless semiconductor based on x-ray photoemission and x-rayabsorption spectroscopy. 37In Co-doped Pb(Pd 1/C0xCox)O2,aS G S state was verified for x ¼0:125 based on the same experimental probe. On the other hand, as Co concentration increases (x¼0:25), a gap opens up and causes spin splitting at both the top of the valence band and the bottom of the conductionband. 38One, however, needs further measurements to confirm the SGS behavior in this material. B. Half-Heusler alloys The half-Heulser alloys having the general formula XYZ crys- tallize in a non-centrosymmetric C1 bcubic structure with space FIG. 6. Spin-resolved band structure of Co-substituted PbPdO 2. Majority spin (solid lines) and minority spin (dashed lines). Reproduced with permission from Wang, Phys. Rev. Lett. 100, 156404 (2008). Copyright 2008 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-5 Published under license by AIP Publishing.group F /C2243m (#216). This structure can be derived by filling the octahedral sites (X) in the tetrahedral ZnS-type (YZ) structure. The crystal structure consists of three interpenetrating fcc sub-lattices located at Wyckoff positions 4a(0,0,0), 4b(1/2,1/2,1/2),and 4c(1/4,1/4,1/4), respectively, each of which is filled by X, Y,and Z atoms. 39 1. FeMnGa/Al/In Very recently, Zhang et al.40investigated the structural, elec- tronic, and magnetic properties of half-Heusler FeMnAl/Ga/In alloys using first-principles calculations. The authors show that theFeMnGa alloy manifests a band structure of SGS with a finite gapin one of the spin channels and a zero gap in the other, as showninFig. 7 . The strong hybridization between the d-states of the tran- sition metal atoms is responsible for the gap. Also, the SGS charac- ter arises due to the redistribution of the electronic states of the Feand Mn atoms originating from the interaction of FeMn-d andGa-p states. The alloy also exhibits fully compensated ferrimagne-tism (FCF) with Fe and Mn moments aligned antiparallel to each other. Similar band dispersion and compensated moments were found in the half-Heusler FeMnAl 0:5In0:5alloy.40Thus, both FeMnGa and FeMnAl 0:5In0:5alloys were found to exhibit fully compensated ferrimagnetic SGS nature.2.Mn2Si Zhang et al.41in 2015 predicted fully compensated ferrimagnetic SGS nature in the Mn 2Si alloy. The band structure of Mn 2Si at its equilibrium lattice constant calculated by them is shown in Fig. 8 .I ti s clearly seen that the VBM of the spin-up electrons touches the Fermi level at the L-point, whereas the CBM of the spin-up electrons andthat of the spin-down electrons touches the Fermi level at the X-pointand thus, there is an indirect zero gap in the spin-up channel. InSec.IV C , we will discuss full-Heusler based SGSs. C. Full-Heusler based SGSs Full-Heulser alloys having the general formula X 2YZ crys- tallize in the L2 1cubic structure with space group Fm/C223m (#225). The additional X atom, in this case, occupies the remaining tetrahedral sites in the XYZ structure. Thus, the two X atoms occupy sites with a tetrahedral symmetry, and the Yand Z atoms occupy sites with an octahedral symmetry. Thus,the full Heusler structure consists of four interpenetrating face-centered cubic (fcc) sublattices located at (0,0,0), (1/4,1/4,1/4), (1/2,1/2,1/2), and (3/4,3/4,3/4). 42,43In addition to the L2 1type structure, full-Heusler alloys crystallize in an inverse Heuslerstructure if the atomic number of Y is higher than that of X. 44 In this case, X atoms do not form a simple cubic lattice, and FIG. 7. (a) Band structure for the majority-spin channel, (b) the spin-resolved density of states (DoS), and (c) the band structure for the minority-spin cha nnel for FeMnGa. Reproduced with permission from Zhang et al., IUCrJ 6, 610 (2019). Copyright 2019 IUCr Journals.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-6 Published under license by AIP Publishing.they occupy the Wyckoff positions 4a(0,0,0) and 4d(3/4,3/4,3/4). The prototype of this structure is CuHg 2Ti with space group F/C2243m(# 216). There have been various theoretical and experi- mental reports on SGS prediction in full-Heusler alloys. In Subsections IV C 1 –IV C 4 , we discuss the most important ones in detail. Jakobsson et al.45studied various inverse Heusler alloys including the spin -gapless semiconductors such as Mn 2CoAl, Ti 2MnAl, Cr 2ZnSi, Ti 2CoSi, and Ti 2VAs and found that the inter-sublattice exchange interactions play an essential role in the formation of the magnetic ground state andin determining the T C. It was also found that due to the finite energy gap in one spin channel, the exchange interactionsdecay sharply with the distance, and hence, magnetism of these SGSs can be described using nearest- and next-nearest-neighbor exchange interactions alone. W ithin the density functional theory (DFT), people have also employed GW-approximationsto accurately capture the electron –electron correlation effects and make a more reliable prediction about the band structure of the concerned systems. 331. Mn 2CoAl The electronic and magnetic properties of Mn 2CoAl were studied by Liu et al.46in 2008 based on first-principles calculations. The electronic band structure of Mn 2CoAl as predicted by Liu et al. is shown in Fig. 9 . It is clearly seen that the alloy exhibits a band structure of a spin-gapless semiconductor with gapless nature in theminority subband. However, at that time, the concept of SGS wasnot known and they classified the alloy as a normal half-metal.Later, Ouardi et al. 11confirmed the SGS characteristics in polycrys- talline bulk Mn 2CoAl from their experimental findings, and this was the very first report on the confirmation of SGS characteristicsin Heusler alloys. In this report, the authors found that the alloycrystallizes in the inverse Heusler structure with a saturation magne-tization of 1.93 μ B/f.u. and a Curie temperature of 720 K. The con- ductivity behavior of Mn 2CoAl was very unusual and was quite different from a normal metal or a semiconductor. The conductivitywas found to increase almost linearly with temperature (see Fig. 10 ), indicating a non-metallic behavior; however, the temperature coeffi-cient of resistivity value was quite low at /C01:4/C210 /C09Ωm/K. Thus, the conductivity can be considered nearly temperature independentup to 300 K and has a value of 2440 S/cm at 300 K. Also, theauthors found a vanishingly small Seebeck coefficient over a widerange of temperatures (5 K ,T,150 K), which was attributed to the electron and hole compensation. At 300 K, the alloy exhibits avery small value of the Seebeck coefficient of 2 μV/K (see Fig. 10 ), unexpected for a normal semiconductor. Figure 10(c) shows the temperature dependence of carrier concentration, which shows that it is almost temperature independent, a signature of gapless systems.The authors also found an exceptional effect in magnetoresistanceof bulk Mn 2CoAl, where a sign change was observed at around 150 K. The results of the MR measurements at different tempera-tures as obtained by Ouardi et al. are displayed in Fig. 11 .A tl o w e r temperatures, the MR is non-saturating and nearly linear even inhigh fields. This nature is similar to that observed in gapless semi-conductors, which are known to exhibit a linear MR. The low tem-perature MR is positive and has a value of /difference10% at 2 K, and above 200 K, it was found to be negative (with a low value) and shows asaturating tendency on the applied field. Thus, temperature inde-pendent conductivity and carrier concentration, a very low Seebeckcoefficient, and linear MR at low temperatures were considered tobe signatures of SGS materials. In 2013, Jamer et al. 22studied the magnetic and transport properties of Mn 2CoAl thin films grown epitaxially on the GaAs (001) substrate using molecular beam epitaxy (MBE). In thisreport, as grown Mn 2CoAl films were found to exhibit tetragonal distortion and become cubic with postgrowth annealing. Thesefilms were not fully chemically ordered, which was indicated byXRD and magnetization. The films were found to exhibit a metal-like resistivity at low temperatures (T ,200 K) (see Fig. 12 ), which was assumed to arise from constant carrier concentration, n(T),and decreasing mobility, μ(E) [because ρ(T)=1/n(T) μ(T)] due to increasing carrier-phonon scattering with increasing temperature.At high temperatures, it decreases with increasing temperature witha maximum at 200 K. The inset of Fig. 12 shows the variation of MR with temperature at different fields. 22It can be seen that the FIG. 8. (a) Calculated band structure and (b) the total and atom/spin projected density of states for Mn 2Si. Reproduced with permission from Zhang et al ., Europhys. Lett. 111, 37009 (2015). Copyright 2015 IOP Publishing.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-7 Published under license by AIP Publishing.MR exhibits a negative value, which arises due to reduction in random spin-flip scattering with field. In 2014, Xu et al.47deposited Mn 2CoAl films on a thermally oxidized Si substrate using a magnetron sputtering system. Unlike Jamer et al.,22in this report, the authors found that the electrical resistivity exhibits semiconducting nature in the complete temper-ature regime (5 K –300 K) [see Fig. 13(a) ]. Also, the MR was found to follow a linear trend as shown in Fig. 13(b) , and this behavior is similar to that observed in bulk Mn 2CoAl and other gapless systems. However, unlike bulk Mn 2CoAl, these films do not show any sign change in MR and have a comparatively low MR. Theabsence of the low temperature positive MR may be due to theshifting of the Fermi level with the deviation in the composition in films. The inset of Fig. 13(b) shows the variation of MR with temperature in a field of 5 T, which shows a maximum at around50 K. This behavior is similar to that observed in MBE-grownMn 2CoAl films reported by Jamer et al.22The unusual maximum in the temperature dependence of MR can be considered a com- petitive effect between the enhancement of negative MR arising from spin-dependent scattering and reduction in MR due to sig-nificant impurity scattering at low temperatures.Later, Galanakiset al. 48on the basis of first-principles calculations, found that to get SGS nature during Mn 2CoAl growth, the occurrence of defects should be minimized as SGS is destroyed by atomic swaps evenbetween sites with different local symmetry as well as presence of extra Co. However, SGS nature is retained even in the presence oftetragonal distortion and thus, small lattice deformations arising due to lattice mismatch will not affect the SGS nature. Another theoretical report by Chen et al. 49studied the effect of pressure and found that the SGS nature of Mn 2CoAl is destroyed when the external pressure is beyond about 25 GPa. Ludbrook et al.50in 2017 reported the observation of topo- l o g i c a lH a l ls i g n a li nM n 2CoAl thin films capped by a thin layer of Pd over a broad temperature range with perpendicular mag-netic anisotropy. This indicates the existence of skyrmions inthese films. Figure 14 s h o w st h a tt h et o p o l o g i c a lH a l ls i g n a le x i s t s over a wide range of temperature, from 3 K to ambient tempera- ture. The authors also found that the topological Hall effect van- ishes at around 280 K which corresponds to the transition toin-plane magnetic anisotropy. In 2018, Chen et al. 51studied the low temperature transport properties of epitaxial Mn 2CoAl films grown on MgO(001) sub- strates by molecular beam epitaxy. This study showed that the resis- tivity at low temperatures shows T1=2dependence which originated from disorder-enhanced three-dimensional electron –electron inter- action. Very recently, Buckley et al.52studied the effect of disorder in DC magnetron sputtered Mn 2CoAl films. The authors claimed that the films can be best interpreted as disordered metals rather FIG. 9. Band structure for (a) the majority-spin channel and (b) the minority-spin channel of Mn 2CoAl. Reproduced with permission from Liu et al ., Phys. Rev. B 77, 014424 (2008). Copyright 2008 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-8 Published under license by AIP Publishing.than spin-gapless semiconductors. This is because the DC resistiv- ity at 300 K was found to be 200 μΩcm with a negative TCR value of 0 :7/C210/C07Ω/C0cm=K as expected for a disordered metal. Also, the conductivity was well described by a weak localization model.To further study the effect of disorder the authors also carried outband structure calculations using DFT and found that as the disor- der is induced, new states are added in the zero width bandgap region of majority-spin channel whereas, the minority channel hasa gap making Mn 2CoAl a half-metal. Similar results were also reported by Xu et al.53based on detailed structural analysis. 2. Ti 2MnAl Skaftouros et al.8investigated the band structure of inverse Heusler alloy Ti 2MnAl using first-principles calculations where they found that the alloy exhibits SGS character, with zero totalmagnetic moment. Thus, the alloy was found to be a fully compen-sated ferrimagnetic SGS. The band structure of Ti 2MnAl is shown inFig. 15 . Later Feng et al.23studied the magnetic and transport properties of Ti 2MnAl film grown on Si(001) substrate using FIG. 10. T emperature dependence of electrical conductivity, Seebeck coeffi- cient, and carrier concentration for Mn 2CoAl. Reproduced with permission from Ouardi et al ., Phys. Rev. Lett. 110, 100401 (2013). Copyright 2013 American Physical Society. FIG. 11. Variation of magnetoresistance with field for Mn 2CoAl. Reproduced with permission from Ouardi et al ., Phys. Rev. Lett. 110, 100401 (2013). Copyright 2013 American Physical Society. FIG. 12. Resistivity vs temperature for the Mn 2CoAl film grown epitaxially on the GaAs(001) substrate. The inset shows the magnetoresistance MR(T) at several magnetic fields. Reproduced with permission from Jamer et al ., Appl. Phys. Lett. 103, 142403 (2013). Copyright 2013 AIP Publishing LLC.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-9 Published under license by AIP Publishing.magnetron sputtering. The variation of resistivity with temperature as reported by these authors is shown in Fig. 16 . Above 70 K, the resistivity was found to decrease with increasing temperature, thusexhibiting semiconducting behavior; however, at low temperaturesthe alloy exhibits metallic nature. A similar behavior has beenobserved in MBE-grown Mn 2CoAl film as well. In general, signchange in the temperature coefficient of resistivity is observed in semimetals or semiconductors with narrow bandgap in which low temperature conduction arises due to impurity levels originatingfrom atomic disorder, defects or non-stoichiometry. Figure 17 shows the dependence of magnetoresistance on field at differenttemperatures for Ti 2MnAl film. The MR was found to be negative FIG. 13. (a) Longitudinal resistivity as a function of temperature measured at zero magnetic field. (b) MR measured with the magnetic field perpendicular to t he film plane. The inset shows the variation of the absolute MR value with temperature. Reproduced with permission from Xu et al ., Appl. Phys. Lett. 104, 242408 (2014). Copyright 2014 AIP Publishing LLC. FIG. 14. T emperature range of the topological Hall effect. Hall effect measurements on a trilayer with a compensation temperature of 270 K. At 3 K, where the AHE domi- nates, the THE persists as a shoulder in (a) and (b). The THE peaks are clear closer to the compensation temperature in (c) and (d), and persists up to almo st ambient temperature. Reproduced from Ludbrook et al., Sci. Rep. 7, 13620 (2017). Copyright 2017 Springer Nature.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-10 Published under license by AIP Publishing.at room temperature which was due to the reduced scattering centers with increasing magnetic field. At low temperatures, the MR was found to be positive which may arise due to spin fluctua- tions. Also, a sign change in MR is observed at 15 K. This trendis similar to that observed in Mn 2CoAl.11Later, Ti 2MnAl was investigated using full potential linearized augmented plane-wave (FP-LAPW) method implemented in WIEN2k crystal program by Singh et al.54They found that the alloy retains its spin-gaplessstate within a uniform strain between /C015% to 10%. Also, for /C05% of tetragonal strain, the alloy retains its SGS character whereas the positive 5% tetragonal strain destroys the SGS char-acter completely. Recently Shi et al. 55theoretically predicted magnetic Weyl semimetallic nature and large intrinsic anomalous Hall effect (AHE) in Ti 2MnAl. Despite the vanishing net mag- netic moment of the system, such a large AHE (300 S/cm) arisesout of large Berry curvature from the Weyl points. Also, unlike Co-based Heusler alloys, the Weyl nodes do not derive from nodal lines in this case, due to the lack of mirror symmetries inthe inverse Heusler structure. 3. Ti 2CrSi Wang and Jin10found that FCF-SGS nature can be achieved in antiferromagnetic semiconductor Ti 2CrSi by lattice distortion. It was found that Ti 2CrSi achieves SGS character at /C02:0% and þ11:4% uniform strains and at +1:8% tetragonal distortions. The occurrence of the SGS feature was primarily attributed to the increased d-dhybridization between the transition metal atoms and the anti- bonding s-states of Si atoms for negative and positive uniform strains, respectively. In the case of tetragonal distortions, the increased cova-lent interactions between the next-nearest neighboring Ti(A) and Cratoms are responsible for the occurrence of SGS behavior. 4. Ti 2CrSn Ti2CrSn was predicted to be a fully compensated ferrimagnetic semiconductor with different band gaps in spin-up and spin-down channels. Jia et al.56suggested a way to induce SGS nature by substi- tuting Si or Ge for Sn in Ti 2CrSn. It was found that the bandgap in the spin-up channel decreases continuously with increase in Si or Ge content. The band structures of Ti 2CrSn 0:5Si0:5and Ti 2CrSn 0:5Ge0:5 are shown in Fig. 18 . It was found that in the case of the FIG. 15. Spin-resolved band structure for Ti 2MnAl (color scheme: solid lines, upspin; dashed lines, down-spin). Reproduced with permission from Skaftouros et al. , Appl. Phys. Lett. 102, 022402 (2013). Copyright 2013 AIP Publishing LLC. FIG. 16. Variation of resistance as a function of temperature under zero mag- netic field for Ti 2MnAl. The inset 1 (left) shows the measured conductance data along with a fitted curve governing the equation, σ(T)¼σ0þσaexp(/C0Eg=kBT). The inset 2 (right) shows the measured resistance data along with the fitted curve governing the equation, ρ¼aþbTþcT2. Reproduced with permission from Feng et al., Phys. Status Solidi RRL 9, 641 (2015). Copyright 2015 John Wiley & Sons. FIG. 17. Magnetoresistance vs applied field at different temperatures for Ti2MnAl film. Reproduced with permission from Feng et al., Phys. Status Solidi RRL 9, 641 (2015). Copyright 2015 John Wiley & Sons.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-11 Published under license by AIP Publishing.Ti2CrSn 0:5Si0:5alloy, the spin-up channel has a zero bandgap, whereas the spin-down channel has a bandgap of 0.57 eV and thealloy exhibits SGS character. On the other hand, Ti 2CrSn 0:5Ge0:5 shows nearly SGS character with a very small bandgap of 0.07 in the spin-up channel and 0.56 eV in the spin-down channel. Also, theF C Fn a t u r ei sr e t a i n e dw i t hS io rG es u b s t i t u t i o n .F o rx .0:75, Ti 2CrSn 1/C0xZx(Z¼Si, Ge) alloys become ordinary nonmagnetic semiconductors. It should be noted that although most of the tita- nium based full-Heusler alloys prefer to crystallize with L2 1order- ing,57Ti2MnAl,9strained-Ti 2CrSi,10and doped-Ti 2CrSn56alloys exhibit SGS behavior in their inverse Heusler (X-type) structure. InSubsection IV D , we will discuss quaternary Heusler-based SGSs. D. LiMgPdSn-type quaternary Heusler-based SGSs Equiatomic quaternary Heusler alloys result when one of the X atoms in a full-Heusler alloy (X 2YZ) is replaced by a different transition metal element (X0). The prototype crystal structure is LiMgPdSn with space group F/C2243m(#216) and is also called a Y-type structure. Due to translational and rotational symmetries in this structure, shifting the atomic positions by (1/4,1/4,1/4), (1/2,1/ 2,1/2), or (3/4,3/4,3/4) in a unit cell will only change the origin ofthe structure but not the configuration. Depending on the different possibilities of crystallographic positions of X, X0, Y, and Z ele- ments, there can be three non-degenerate configurations for equia- tomic quaternary Heusler alloys.58If the Z atom is fixed at the 4a (0,0,0) Wyckoff position, the three possible energetically non-degenerate configurations are the following: (a) Type I !X at 4d(3/4,3/4,3/4), X 0at 4c(1/4,1/4,1/4), and Y at 4b(1/2,1/2,1/2) sites. (b) Type II !X at 4b(1/2,1/2,1/2), X0at 4c(1/4,1/4,1/4), and Y at 4d(3/4,3/4,3/4) sites. (c) Type III !X at 4d(3/4,3/4,3/4), X0at 4b(1/2,1/2,1/2, and Y at 4c(1/4,1/4,1/4) sites. The SGS nature in quaternary Heusler alloys was first pre- dicted by Ozdogan et al. based on first-principles calculations.59 Later, the same was predicted in various quaternary Heusler alloys and experimentally verified in a few systems, which is discussed indetail in Subsections IV D 1 –IV D 4 . 1. CoFeMnSi There are various experimental and theoretical reports21,60,61 in the past, which predicted half-metallic nature for CoFeMnSi. It FIG. 18. Spin-resolved band structures for (a) Ti 2CrSn 0:5Si0:5and (b) Ti 2CrSn 0:5Ge0:5compounds at their respective equilibrium lattice parameters. Reproduced with per- mission from Jia et al., J. Magn. Magn. Mater. 367, 33 (2014). Copyright 2014 Elsevier Publications.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-12 Published under license by AIP Publishing.was in the year 2013, when Xu et al.62predicted SGS nature for CoFeMnSi based on first-principles calculations with a half-metallic gap in one spin channel and zero gap in the other. Later, SGS naturewas verified experimentally and theoretically by Bainsla et al. in 2015 12by carefully probing the structural, magnetization, magneto- transport, Hall effect and spin polarization measurements. This was the first quaternary Heusler alloy, which was experimentally verified to show SGS signatures. The alloy was found to crystallize in aDO 3-type structure with a Curie temperature of 620 K. Figure 19(a) shows the temperature dependence of the electrical conductivity,σ xx(T), and the carrier concentration, n(T), of CoFeMnSi. The elec- trical conductivity behavior reflects the non-metallic nature as the electrical conductivity increases with increasing temperature. Theconductivity was found to vary linearly with temperature in thehigh-temperature region, whereas in the low temperature region, anon-linear trend was observed, which may be due to the disorder- enhanced coherent scattering of conduction electrons. The conduc- tivity value at 300 K was found to be 2980 Scm /C01, which is slightly higher than the SGS Mn 2CoAl (2440 Scm/C01).11The variation of Hall conductivity with field at 5 K is shown in Fig. 19(b) . The value of anomalous Hall conductivity σxy0was found to be 162 Scm/C01. This value is higher than that observed in Mn 2CoAl (22 Scm/C01)11 but smaller than that of half-metallic Heusler alloys including Co2FeSi (200 Scm/C01)63and Co 2MnAl (2000 Scm/C01).64From Fig. 19(a) , it was found that the carrier concentration was almost temperature independent in the range of 5 –300 K, strongly sup- porting the existence of SGS behavior in this alloy. The transportspin polarization (P) measured by the point contact Andreevreflection technique was found to be 64% for CoFeMnSi. Later, in 2017, Bainsla et al. 65studied CoFeMnSi thin films grown on a Cr-buffered MgO(001) substrate. It was found that thefilms exhibit a B2-type structure for T a/C21500/C14C, where T ais the postannealing temperature. The Gilbert constant was found to be 0.0046 for T a¼600/C14C, which is smaller than that of a permalloy. The total magnetic moment as deduced from an x-ray magneticcircular dichroism technique was found to be 4 μ B/f.u. for Ta¼600/C14C. In the same year, Kushwaha et al.66reported a possi- bility of SGS nature in CoFeMnSi epitaxial thin films deposited on the MgO (001) substrate. Based on first-principles calculations, Han et al.67studied the spin transport properties of the GaAs/CoFeMnSi heterostructureand the CoFeMnSi/GaAs/CoFeMnSi magnetic tunnel junction (MTJ) and found that the heterostructure exhibits an excellent spin diode effect and a spin filtering effect and the MTJ has a largetunnel magnetoresistance ratio (up to 2 /C210 3). Very recently, the structural stability, half-metallicity, and magnetism of theCoFeMnSi/GaAs(001) interface have been studied in detail by Feng et al., 68which shows that the CoFeMnSi/GaAs heterostructure with an MnMn-terminated interface in the top-type structure in whichthe termination of nine CoFeMnSi layers is connected to the top ofthe As-terminated GaAs layer preserves 100% spin polarization,and thus, it is believed that CoFeMnSi would be useful for tunable spin transport based applications and spin injection. In 2018, Bainsla et al . 69studied the tunnel magnetoresistance (TMR) in MgO-based magnetic tunnel junctions (MTJs) with equiatomicquaternary CoFeMnSi Heusler and CoFe alloy electrodes. It was found that maximum TMR ratios of 101% and 521% were observed at room temperature and 10 K, respectively, as shown in Fig. 20 . The sensitivity of magnetic tunnel junctions (MTJs) is determinedby the TMR ratio. A large TMR ratio is useful in spintronic appli-cations. Ideally, a MTJ with half-metallic ferromagnets should show infinite TMR; however, due to the interface disorder of the FIG. 19. (a) T emperature dependence of the electrical conductivity (left-hand scale), variation of carrier concentration, n(T), with temperature (right- hand scale). (b) Anomalous Hall effect (AHE): Field dependence of Hall conductivity, σxy(T), at 5 K. The inset in (b) shows the magnetization isotherm obtained at 5 K. Reproduced with permission from Bainsla et al., Phys. Rev. B 91, 104408 (2015). Copyright 2015 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-13 Published under license by AIP Publishing.half-metals that reduces the spin polarization significantly, lower TMR ratios have been observed. Thus, it is very important to maintain the high spin polarization at the interface. Recently,Zhang et al. 70demonstrated that strong perpendicular magnetic anisotropy can be achieved in Ta/Pd/CoFeMnSi (2.3 nm)/MgO(1.3 nm)/Pd films annealed at 300 /C14C with the effective anisotropy constant K effof 5 :6/C2105erg/cm3. It was also found that the mag- netic properties of the films are sensitive to hydrogenation as theresidual magnetization (M r) decreased by 75% under the atmo- sphere with an H 2of 5%. Recently, Bainsla et al.71reported the Gilbert damping constant ( α) for single crystalline CoFeMnSi films grown on (001)-oriented single crystalline MgO substrates. Figure 21 shows the thickness dependence of the Gilbert damping constant. The lowest value of α(0:0027+0:0001) was found for a 10 nm-thick CoFeMnSi film annealed at 600/C14C. This value is much smaller than those observed in typical transition metal fer- romagnets but higher than those in ultralow damping metals. The Gilbert damping constant was found to be weakly dependent onthe thickness. The low Gilbert damping constant value observedin this material is attractive for the application in spin-transfertorque devices. 2. CoFeCrGa Bainsla et al.13predicted SGS nature in another quaternary Heusler alloy CoFeCrGa by combined theoretical and experimentalstudies. The spin-resolved band structure as shown in Fig. 22 clearly reflects SGS nature with a closed band-gap character in the majority-spin band and a small finite bandgap in the minority-spinband. Under pressure, CoFeCrGa was found to transform from SGS to HMF. A clear evidence of SGS nature was observed from the variation of electrical conductivity and carrier concentrationwith temperature, as shown in Fig. 23 . The electrical conductivity, σ xx, was found to decrease with increasing temperature and hence show non-metallic behavior. Also, the conductivity value at 300 K was found to be 3233 Scm/C01, which is slightly higher than that of Mn 2CoAl (2440 Scm/C01)11and CoFeMnSi (2980 Scm/C01).12The observations in CoFeCrGa were similar to that of CoFeMnSi exceptfor an abrupt increase in the carrier concentration [n(T)] at 250 K. The authors have attributed such behavior in n(T) to the onset of thermal excitations, which dominate over the half-metallic gap inthe minority-spin band. The value of anomalous Hall conductivityσ xy0as deduced from the field-dependent transport measurements was found to be 185 Scm/C01, which is comparable to that observed in CoFeMnSi (165 Scm/C01).12 3. CrVTiAl Galanakis et al.72studied the quaternary Heusler compounds CrVXAl (X ¼Ti, Zr, Hf) and suggested them to be a potential material for room temperature spin filter devices. In 2013, Stephenet al. 73carried out an experimental investigation on CrVTiAl. The authors found it to be a magnetic semiconductor, but the resistivity behavior was found to be a combination of metallic and semicon- ducting contributions as shown in Fig. 24 . Also, the magnetization FIG. 20. (a) The TMR curves for the MTJs with TCoFeMnSi a ¼873 K and TMTJ a¼723 K with different measurement temperatures T . (b) The TMR values as a function of T . Reproduced with permission from Bainsla et al., Appl. Phys. Lett. 112, 052403 (2018). Copyright 2018 AIP Publishing LLC. FIG. 21. Gilbert damping constant α(solid circle) and relaxation rate G (open circle) for the CoFeMnSi films annealed at 600/C14C as a function of the CoFeMnSi thickness t. The values for α(G) of the 30 nm-thick films annealed at 500/C14C and 700/C14C are also shown with the squares and triangles, respectively. Reproduced with permission from Bainsla et al ., J. Phys. D: Appl. Phys. 51, 495001 (2018). Copyright 2018 IOP Publishing.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-14 Published under license by AIP Publishing.was found to be linearly dependent on field and independent of temperature (see Fig. 25 ), as expected for polycrystalline antiferro- magnets. Later in 2018, Venkateswara et al.14reported some strik- ing differences in the experimental results for the same alloy, wherethey reported semiconducting nature for the complete temperaturerange (5 –300 K). Based on first-principles calculations, the ground- state configuration (type III) was found to be a fully compensated ferrimagnet with bandgaps of 0.58 and 0.30 eV for the spin-up and-down bands, respectively. The next-higher-energy structural con-figuration (type II) was also found to be a fully compensated ferri- magnetic but has a spin-gapless semiconducting nature, whereas the highest-energy configuration (type III) corresponds to a non-magnetic, gapless semiconductor (see Fig. 26 ). Because of the small (,1 mRy/atom) energy differences among these configurations, the authors claimed that at finite temperatures, the alloy can exist in a disordered phase, which is a mixture of the three configurations.Based on their theoretical and experimental findings, they con-cluded that CrVTiAl is a fully compensated ferrimagnet with a pre- dominantly spin-gapless semiconducting nature. Very recently, Stephen et al. 74studied CrVTiAl thin films grown on Si/SiO2 substrates capped with a 1 nm of Al. The authors FIG. 22. (a) Band structure for the majority-spin channel, (b) spin-resolved density of states (DoS), and (c) the band structure for the minority-spin channe l for bulk CoFeCrGa. Reproduced with permission from Bainsla et al., Phys. Rev. B 92, 045201 (2015). Copyright 2015 American Physical Society. FIG. 23. Variation of electrical conductivity ( σxx) and carrier concentration n(T) with temperature for CoFeCrGa. The inset shows the carrier concentration around 250 –280 K. Reproduced with permission from Bainsla et al., Phys. Rev. B 92, 045201 (2015). Copyright 2015 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-15 Published under license by AIP Publishing.have shown the presence of two parallel conducting channels by means of σ(T ) and MR(B) measurements. The temperature depen- dence of electrical conductivity σ(T) for CrVTiAl films is shown in Fig. 27 . Here F400, F600, and F700 represent the films grown at 400/C14C and subsequently annealed to 600/C14C and 700/C14C, respec- tively. The conductivity was found to increase linearly for moderatetemperatures, a characteristic of SGS nature, whereas at higher tem- peratures, the conductivity follows an exponential dependence, indicating the presence of a semiconducting gap. The authorsdescribed the σ(T) dependence using the two carrier conduction model, which assumes two parallel conducting channels: onegapless and one gapped, and thus, σ(T) can be written as σ(T)¼σ SGSþσSCe/C0ΔE KBT, (1) where σSCis the zero-temperature contribution to the semicon- ducting component and σSGSis the conductivity in the gapless channel. Figure 28 shows the field variation of magnetoresistance for CrVTiAl films. It can be seen that for the F600 film, a sharp MR cusp is observed for B ,1:5 T and temperatures less than 5 K, which is a characteristic of weak localization (WL) arising fromquantum interference of the wave-like nature of the scattering carri-ers. Also, the dominant positive MR found in all of the films isslightly suppressed by an additional small negative MR at higher temperatures and is related to the saturating magnetization observed in the anomalous Hall effect. Another interesting featureobserved in MR is the small ripples for all the films at the highestfield and the lowest temperature. This can be attributed to theonset of Shubnikov –de Haas quantum oscillations. 4. Co 1+xFe1−xCrGa Very recently, Rani et al.75proposed a way to design new SGS materials based on known SGS materials with the aim of improvingother properties such as Curie temperature and spin polarizationbased on their findings on the effect of Co substitution for Fe inspin-gapless semiconductor CoFeCrGa. The authors found that the alloys Co 1þxFe1/C0xCrGa crystallize in the Y-type Heusler structure, and the saturation magnetization was in fair agreement with theSlater –Pauling rule, which is a prerequisite for spintronic materials. An important observation is that the alloy retained the SGS charac- ter when 40% Fe is replaced by Co after which it becomes a half- metal. Also, with an increase in Co concentration, the transitiontemperature was found to increase. For x/C200:4, the absence of exponential dependence of resistivity on temperature indicates thesemiconducting nature but with spin-gapless behavior. The con- ductivity value ( σ xx) at 300 K lies in the range of 2289 S/cm – 3294 S/cm, which is close to other reported SGS materials such asCoFeCrGa, Mn 2CoAl, and CoFeMnSi. The order of magnitude of anomalous Hall conductivity ( σAHE) was also found to be similar to other SGS materials. The negligible Seebeck coefficient along with the conductivity behavior supported the SGS nature in these alloys. The authors have also studied the anomalous Hall effect andfound that the intrinsic contribution to the anomalous Hall con-ductivity increases with x, which can be correlated with the enhancement in chemical order. Other than the above-mentioned quaternary Heusler alloys, Gao et al. 76demonstrated that all four types of SGSs can be realized in quaternary Heusler alloys depending on the spin characters ofthe bands around the Fermi energy. They studied a series of XX 0YZ alloys (X, X0, and Y are transition metal elements except Tc, and Z is one of B, Al, Ga, In, Si, Ge, Sn, Pb, P, As, Sb, and Bi) and found FIG. 25. Variation of magnetization with field and temperature for the CrVTiAl alloy. Reproduced with permission from Stephen et al ., Appl. Phys. Lett. 109, 242401 (2016). Copyright 2016 AIP Publishing LLC. FIG. 24. T emperature dependence of the electrical resistivity for a bulk CrVTiAl alloy. Reproduced with permission from Stephen et al ., Appl. Phys. Lett. 109, 242401 (2016). Copyright 2016 AIP Publishing LLC.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-16 Published under license by AIP Publishing.70 new stable SGSs on the basis of first-principles calculations. Table I tabulates the outcome of this report. It should be noted that Heusler compounds are well known to show antisite disorder, and disorder can greatly affect the propertiesof these materials and many times this can even lead to destruction of the SGS nature because of emergence of new states at the Fermi level. There have been a few studies on disorder calculations of these materials, which indeed help us to understand the properties FIG. 26. Spin-resolved density of states (DoS) and the band structure for the (a) type I, (b) type II, and (c) type III configuration of CrVTiAl. Reproduced with permission from Venkateswara et al., Phys. Rev. B 97, 054407 (2018). Copyright 2018 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-17 Published under license by AIP Publishing.of an experimentally observed structure rather than a traditionally used ideal structure.52,77–79There is a lot of scope in studying the disorder effects in these systems. E. Diamond like quaternary compound CuMn 2InSe 4 Han et al.80studied the electronic band structures and mag- netic properties of diamond like quaternary compound (DLQC) CuMn 2InSe 4and found that the compound exhibits SGS behaviorbased on first-principles calculations. The CuMn 2InSe 4compound was already synthesized by Delgado and Sagredo,81and it was found to crystallize with a stannite structure, in the tetragonalspace group I42m (No. 121), with Cu, Mn, In, Se occupying 2a (0,0, 0), 4d (0, 0.5, 0.25), 2b (0, 0, 0.5), and 8i (0.2390, 0.2390, 0.1261)Wyckoff positions, respectively [see Fig. 29(b) ].Figure 29(a) shows the calculated electronic band structures of the CuMn 2InSe 4com- pound at the equilibrium lattice parameter. It can be clearly seenthat there is a zero gap for a spin-up channel and an indirectbandgap (1.01 eV) for a spin-down channel. Thus, theCuMn 2InSe 4compound is classified as type I SGSs. Experimental verification of this behavior is yet to be reported. F. CO –Mn –g-ZnO Graphene-like ZnO (g-ZnO) is a newly found crystalline form of ZnO,82,83which has a two-dimensional graphene-like hexagonal structure and a high surface-to-volume ratio as com- pared to ZnO in the bulk, film, or nanostructure forms.Topsakal et al. 84predicted that the band structure of the g-ZnO monolayer is similar to ZnO in the wurtzite phase, and it pos-sesses non-magnetic semiconduc ting nature. Intrinsic defect introduction, 84non-metal decoration,85and Al decoration86 have been used to tune the electronic and magnetic properties of pristine g-ZnO. Interestingly, transition metal (TM) elementincorporation has been proved to be an efficient way to alter the properties of g-ZnO. 87–89The 3d elements prefer to hexa- coordinate with the host atoms in elemental 2D materials, suchas graphene, 90,91silicene,92and germanene.93I nt h ec a s eo f g-ZnO, because of the strong interaction between TM and O ascompared to that between TM and Zn, the most possible adsorption position for 3d TM elements on g-ZnO is the top of the O atom. 94Leiet al.95studied the effect of adsorption of t r a n s i t i o nm e t a le l e m e n t s( C r ,M n ,F e ,C o ,N i ,a n dC u )o nt h eg-ZnO electronic structure and found that the interactionbetween the TM element and g-ZnO is strengthened by employ- ing the CO molecule, and with the decoration of the CO –Mn FIG. 27. Zero-field conductivity vs temperature for three different films of CrVTiAl (F400, F600, and F700). The increasing conductivity indicates semicon-ducting behavior. The points are experimental data, and the solid curves are fitsto Eq. (1). Reproduced with permission from Stephen et al ., Phys. Rev. B 99, 224207 (2019). Copyright 2019 American Physical Society. FIG. 28. Magnetoresistance as a function of the applied field for CrVTiAl films. Reproduced with permission from Stephen et al ., Phys. Rev. B 99, 224207 (2019). Copyright 2019 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-18 Published under license by AIP Publishing.complex on g-ZnO, the spin-gapless semiconducting state in the CO–Mn –g-ZnO system has been observed. Figure 30 shows the calculated density of states for various TM in CO-TM –g-ZnO. One can notice that for CO-Mn –g-ZnO, there are gaps betweenthe occupied and unoccupied bands for both the spin-up and spin-down channels, and there is no gap between the unoccu- pied spin-up band and the occupied spin-down band [see Fig. 30(b) ] and thus exhibiting SGS nature.TABLE I. Optimized lattice parameters, total magnetic moment (M tot), formation enthalpy ( ΔH), convex hull energy ( ΔEcon), and type of SGS for various quaternary Heusler alloys XX0YZ. Alloys with a distance to the convex hull (E con) less than 0.10 eV/atom are highlighted in bold. The compounds with symbol yare either dynamically or mechani- cally unstable. Reproduced with permission from Gao et al., Phys. Rev. Mater. 3, 024410 (2019). Copyright 2019 American Physical Society. XX’YZ (4a,4b,4c,4d)latt. (Å)Msat (μB)ΔH(eV/ atom)ΔEcon (eV/ atom)SGS typeXX’YZ (4a,4b,4c,4d)latt. (Å) Msat(μB)ΔH(eV/ atom)ΔEcon(eV/ atom)SGS type Nv=2 1 Nv=2 6 IrVYSn 6.720 3.00 −0.0942 0.5628 SOC-I CoOsTiSb 6.255 2.00 −0.1635 0.3515 I CoVYSn 6.620 3.00 −0.0862 0.3848 II CoFeHfSb 6.232 2.00 −0.2847 0.2523 I CoVScSn 6.402 3.00 −0.2049 0.2221 III CoOsZrSb 6.453 2.00 −0.1075 0.4645 I IrVScSn 6.518 3.00 −0.2488 0.4052 SOC-II RhFeTiSb 6.259 1.95 −0.3896 0.1104 I RhVScSn 6.518 3.00 −0.2773 0.3527 I CoFeTiSb 6.074 2.00 −0.2948 0.2202 I CoVYGe 6.377 3.00 −0.0763 0.4697 II IrFeTiSb 6.287 1.99 −0.2932 0.3108 III CoVScGe 6.145 3.00 −0.2749 0.2931 II CoRuTiSb 6.228 2.00 −0.3261 0.1889 I IrVScGe 6.300 3.00 −0.3025 0.4045 II CoFeNbGe 5.961 2.00 −0.2374 0.1506 I RhVScGe 6.290 3.00 −0.3318 0.4502 II CoOsNbSny6.352 2.00 −0.0609 0.2091 I RhVYGe 6.512 3.00 −0.1377 0.5663 III CoRuTaSny6.303 2.00 −0.1268 0.1852 I CoVYSi 6.297 3.00 −0.1077 0.4701 II IrFeTaSn 6.354 1.98 −0.1782 0.2328 I CoVScSi 6.058 3.00 −0.3550 0.2990 II CoOsTaGe 6.143 2.00 −0.0702 0.3048 I IrVScSi 6.215 3.00 −0.4254 0.4096 SOC-II CoOsTaSiy6.064 1.99 −0.2546 0.2234 I RhVScSi 6.210 3.00 −0.4242 0.4628 II CoOsTaSn 6.332 2.00 −0.007 0.2413 I RhVYSi 6.438 3.00 −0.1862 0.6398 III CoFeTaGe 5.938 2.00 −0.2475 0.1275 I PtVScAl 6.369 3.00 −0.4431 0.2869 SOC-I CoFeTaSi 5.856 2.00 −0.4222 0.1275 I PtVYAl 6.608 3.00 −0.2477 0.5013 I CoFeTaSn 6.154 2.00 −0.1522 0.0898 I PtVYGa 6.600 3.00 −0.1867 0.5733 I IrCoNbAl 6.162 1.99 −0.5563 0.0277 I FeCrHfAl 6.142 3.00 −0.2456 0.0504 II IrCoNbGa 6.173 2.00 −0.4043 0.0097 I OsCrHfAl 6.299 3.00 −0.403 0.0530 II IrCoNbIn 6.360 2.00 −0.1326 0.1544 I RuCrHfAl 6.284 3.00 −0.4544 0.0666 II IrCoTaAl 6.140 2.00 −0.5579 0.0631 I FeCrTiAl 5.964 3.00 −0.292 0.0504 II IrCoTaGay6.150 2.00 −0.4200 0.0370 I FeCrZrAl 6.194 3.00 −0.2156 0.0914 III IrCoTaIny6.336 2.00 −0.1622 0.1768 I OsCrZrAl 6.374 3.00 −0.3543 0.0617 SOC-II CoCoNbAly5.970 2.00 −0.4312 0.0082 I RuCrZrAl 6.335 3.00 −0.4154 0.0626 III CoCoNbGay5.968 2.00 −0.3299 0.0001 I FeCrScSi 5.992 3.00 −0.279 0.2400 II CoCoNbIny6.179 2.00 −0.0869 0.0331 I FeCrScSn 6.364 3.00 −0.0891 0.2309 II IrCoTiPb 6.380 2.00 −0.0571 0.3829 I FeCrYSi 6.236 3.00 −0.00811 0.4739 III IrCoTiSn 6.276 2.00 −0.3789 0.1461 I OsCrYSi 6.386 3.00 −0.0246 0.4860 SOC-III IrCoTiSi 5.965 2.00 −0.6805 0.0785 I CoVHfAl 6.211 3.00 −0.2896 0.1134 I CoRuCrAly5.848 2.01 −0.2802 0.0558 II IrVHfAl 6.346 3.00 −0.4634 0.1596 II NiCrMnAl 5.809 2.00 −0.2127 0.1173 III RhVHfAl 6.342 3.00 −0.3855 0.2355 II NiReCrAl 5.920 1.97 −0.1633 0.2177 II CoVZrAl 6.258 3.00 −0.2662 0.1408 I CoOsCrAl 5.866 2.00 −0.2412 0.0688 II CoVZrGa 6.238 3.00 −0.2317 0.2233 I NV=28 IrTiZrSny6.651 2.98 −0.3335 0.3965 II NiFeMnAl 5.731 4.00 −0.2773 0.0577 IV IrTiZrSi 6.385 2.96 −0.4322 0.4778 II Continue with NV=21 FeVNbAl 6.117 2.99 −0.2012 0.1238 II MnCrNbAl 6.077 3.00 −0.1912 0.0228 II FeVTaAl 6.097 2.99 −0.2202 0.0958 II MnCrTaAl 6.053 2.99 −0.2124 0.0256 II MnCrZrGe 6.157 2.99 −0.1473 0.2687 II FeVHfGe 6.158 3.00 −0.2094 0.2646 II MnCrZrSi 6.076 3.00 −0.2569 0.2621 II FeVHfSi 6.079 3.00 −0.3187 0.2753 II MnCrZrSn 6.393 3.00 −0.0593 0.2317 II FeVHfSn 6.386 3.00 −0.129 0.1580 IIJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-19 Published under license by AIP Publishing.G. BiFe 0.83Ni0.17O3 Rajan et al.96studied the effect of oxygen vacancies (OVs) on the electronic structure and magnetic properties of multiferroicBiFe 0:83Ni0:17O3. They found that depending on the location of OVs, the material can exhibit half-metallic, spin-gapless semicon-ductor and bipolar magnetic conductor behavior. The authors found that when 1 OV is nearer to Ni, the electronic density of states shows spin-gapless semiconducting behavior where theup-spin channel is semiconducting and the down-spin channel isgapless. An almost zero bandgap in the spin-down channel and a bandgap of 1.5 eV was found in the spin-up channel for the above-mentioned configuration with an OV concentration of 5.56at. %. Figure 31 shows the total density of states calculated after the relaxation of six modeled hexagonal cells of BiFe 0:83Ni0:17O3 with 1 OV nearer to Ni. V. DIRAC-TYPE SGS In general, Dirac-type SGSs can be divided into two classes: (1) d-state DSGSs in which the Dirac state is contributed by d-orbitals of transition metal) atoms and (2) p-state DSGSs in whichthe Dirac state is contributed by p-orbitals of main-group atoms. A. The d-state type DSGSs 1. Mn-intercalated epitaxial graphene on SiC (0001) Dirac-like SGS nature was proposed in Mn-intercalated epitax- ial graphene by Li et al.97Figures 32(a) and 32(b) show thehomogeneous lattice model that consists of Mn atoms sandwiched between the SiC(0001) substrate and a graphene monolayer. Theelectronic structure and magnetism in this material depend on the p-orbital hybridization between Mn d- and Co p-orbitals, which is affected by the change in the Mn coverage defined by a factor v.This factor is defined as the ratio of one Mn atom to the corre-sponding surface Si atoms per unit cell. On the basis of first- principles simulations, the authors found that the DSGS state of this system will occur at 1 3ML,v,vmax, while5 12ML,vmax,1 2ML, where 1 ML corresponds to one Mn atom per surface Si atom.Thus, the Dirac state in this material can be tuned by substratemodulation and was explained on the basis of the Mn –SiC interac- tion and its quasi-2D inversion symmetry. For example, the band structures and total density of states of the v ¼ 5 12ML model calcu- lated by Li et al. are shown in Fig. 32(c) , which shows the Dirac cone emerging in the minority-spin channel having gapless natureand a gap of 190 meV in the majority-spin channel. It was also found that the main contribution in density of states near the Fermi level is from the Mn atoms, confirming the d-state DSGS nature inthis material. In 2012, Gao et al. 98had reported experimental cover- age tuning on the basis of Mn intercalated epitaxial graphene onSiC (0001), and the intrinsic Dirac states were confirmed explicitly. InFig. 33 , it can be clearly seen that the Dirac point shifts down under the effect of Mn coverage, and at an extreme coverage ofv¼0:6 ML, the Dirac state vanishes. Interestingly, its Dirac state recovers the native state (without Mn intercalation) in parallel with the desorption of Mn when sample [in Fig. 33(d) ] is heated to 1200 /C14C for a short time. FIG. 29. (a) Calculated band structures and (b) the crystal structure of the DLQC CuMn 2InSe 4compound. Reproduced with permission from Han et al., Results Phys. 10, 301 (2018). Copyright 2018 Elsevier Publications.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-20 Published under license by AIP Publishing.2. CrO 2/TiO 2heterostructures Rutile CrO 2and Cr-doped titanium dioxide were found to show half-metallic characteristics.100,101Caiet al.99studied CrO 2/ TiO 2heterostructures, a superlattice of CrO 2/TiO 2, and ultrathin films of CrO 2coupled to a TiO 2substrate. The authors considered a series of (CrO 2)n/(TiO 2)mthin films and superlattice models with indices (n –m) in their first-principles studies. The crystal structure models and corresponding band structures of the (4 /C010) superlat- tice and (5 /C09) thin films are shown in Figs. 34 and35, respectively.InFig. 35(a) , one can clearly see two symmetric Dirac cones across the Fermi level along the diagonals of the Brillouin zone (BZ). For the (5 /C09) thin film, there are four asymmetric Dirac cones as shown in Fig. 35(b) . Thus, both systems were found to be DSGSs. Also, the Dirac states were mainly contributed by the 2D CrO 2 layers, dominated by Cr-d orbitals and they belong to d-state DSGS. One should note that the Dirac points in these materials are single- spin Dirac species and are field-tunable; i.e., depending on the fieldalignment, they can be massive or massless. A gap opening by SOC FIG. 30. Total density of states (DoS) of CO –TM–g-ZnO for various choice of transition metals [TM ¼(a) Cr, (b) Mn, (c) Fe, (d) Co, (e) Ni, and (f) Cu]. The dashed line indicates the Fermi level. Reproduced with permission from Lei et al., Appl. Surf. Sci. 416, 681 (2017). Copyright 2017 Elsevier Publications.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-21 Published under license by AIP Publishing.is responsible for the field-tunability of the Dirac points. The mirror symmetry is broken when the crystallographic c-direction is thespin quantization axis. In the (4 /C010) superlattice, degeneracy is lifted by SOC, and identical gaps of 3.7 meV are opened up at each Dirac node. On the other hand, for the (5 /C09) thin film, SOC opens gaps differently for Dirac points on the diagonal (k a=k b) and the antidiagonal (k a¼/C0kb) of the Brilliouin zone. On the antidiagonal, the gap was found to be over 5 meV, whereas on the diagonal, it was about 0.5 meV. 3. Transition metal halides Transition metal trihalides (TMHs) encompass a family of materials with the general formula (TMX 3) (TM = Ti, V, Cr, Mn, Fe, Mo, Ru and X = Cl, Br, I).102,103The existence of Dirac states in a TM honeycomb spin lattice (such as the graphene structure) sug- gests that the Dirac states can also arise due to the honeycomb spinlattice geometry. 1. Vanadium trihalide monolayer .H e et al. 104investigated the geometry, stability, electronic, and magnetic properties of VCl 3and VI3monolayers using first-principles calculations within the self- consistent Hubbard U approach (DFT+U scf) together with the Monte Carlo simulations. Monolayer VCl 3and VI 3structures are shown in Fig. 36 . The ferromagnetic states were found to be the most stable magnetic configurations for VCl 3and VI 3monolayers. The electronic band structures and density of states of VCl 3and VI3are depicted in Fig. 37 . It can be clearly seen that the spin-up channels are gapless and spin-down channels have a large bandgap for both VCl 3and VI 3. Thus, both are DHM, and the Dirac pointslocated at K for VCl 3and VI 3are just 20 and 106 meV above the Fermi level, respectively. The calculated Fermi velocities, v F, were approximately 0 :16/C2106m/s and 0 :1/C2106m/s, for the VCl 3and VI3monolayers, respectively, higher than those of many other reported Dirac-type materials. To get a deeper insight into theorigin of Dirac states, the authors also calculated the partial elec- tronic density of states and found that the Dirac states are mainly contributed by the d-states of V atoms, whereas the X-p z(X¼Cl, I) does not contribute significantly, and thus, both of them belongto d-state DSGS. The T Cwas found to be 80 and 98 K for VCl 3 and VI 3monolayers, respectively, far below the room temperature. To tune the T Cvalues and to shift the Dirac states exactly at the Fermi level, He et al. studied the doping effects. They found that the electron and hole doping are effective ways to improve the fer-romagnetism of VCl 3and VI 3monolayers. Figure 38 shows the variation of exchange energies with the carrier doping concentra- tion. The T Cwas found to improve with doping with a value of 353 K and 246 K for doped VCl 3and VI 3, respectively. Also, at the electron doping levels of 0.1 and 0.7 per unit cell for VCl 3and VI3monolayers, respectively, the Dirac states are exactly located at the Fermi level. Under doping conditions, the Dirac half-metallic states were transformed to DSGS states (see the inset of Fig. 38 ), which suggests that doping can be considered an effective methodto obtain DSGSs by tuning the band structures of previously iden-tified DHMs. 2. Nickel chloride monolayer . Using first-principles calcula- tions, He et al. 105proposed the NiCl 3monolayer as a new candi- date for DSGSs. The T Cvalue for NiCl 3monolayers was found to be about 400 K. The band structures calculated usingPerdew –Burke –Ernzerhof (PBE) and hybrid HSE06 functional methods are shown in Figs. 39(a) and 39(b) , respectively. The spin-down channels of NiCl 3possess a 1.22 eV and 4.09 eV bandgap at PBE and HSE06 levels, respectively, whereas thespin-up channel manifests a gapless semiconductor feature(having a zero gap) with a linear dispersion relation around the Fermi level. The corresponding 3D band structures are also shown in Fig. 39(c) to further understand the distribution of the linear dispersion relation in the Brillouin zone. The calculatedFermi velocity was found to be 4 /C210 5m/s at the HSE06 level, w h i c hi sh a l fo ft h a to fg r a p h e n e .106The band structures of NiCl 3with the inclusion of SOC using PBE and HSE06 func- tional methods are shown in Figs. 39(d) and 39(e) ,r e s p e c t i v e l y . The NiCl 3monolayer becomes an intrinsic Chern insulator with a large non-trivial bandgap of /difference24 meV. The specific distribu- tion of the Berry curvature in the momentum space is displayed inFig. 40 , and it can be seen that the nontrivial edge states con- necting the valence and conduction bands cross the insulatinggap of the Dirac cone. Integrating the Berry curve across theentire BZ revealed a Chern number (C) value of /C01 with a non- trivial topological state, which is consistent with the appearance of only one chiral edge state [see Fig. 40(a) ]. When the Fermi level is located in the insulating gap of the spin-up Dirac cone,the anomalous Hall conductivity shows a quantized charge Hallplateau of σ xy¼Ce2=h, as expected from the non-zero Chern number [see Fig. 40(b) ]. The authors also calculated the edge states of the NiCl 3monolayer with zigzag and armchair geometry FIG. 31. Total density of states for the relaxed structure of six modeled hexago- nal cells of BiFe 0:83Ni0:17O3having 1 oxygen vacancy (OV) nearer to Ni. Reproduced with permission from Rajan et al ., R. Soc. Open Sci. 4, 170273 (2017). Copyright 2017 Royal Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-22 Published under license by AIP Publishing.using Green ’sf u n c t i o n s ,a ss h o w ni n Figs. 40(c) and 40(d) , respectively. The 2D mirror symmetry breaking in TMCl 3materi- als results in Rashba SOC, which may lead to the valley-polarizededge state. 107However, there is no signature of the valley-polarized edge state in NiCl 3, and thus, it would not be a suitable alternative material for valleytronic applications. 3. Manganese halides . The MnF 3crystal has been synthesized for many years,108,109but there had been no theoretical work of its electronic band structure. In 2017, Jiao et al.110demonstrated the DSGS features of MnF 3by means of first-principles calculations. This is the first time that the DSGS state was demonstrated to existin bulk materials. It was found that the ferromagnetic state is energetically most stable out of all the magnetic configurations including the antiferromagnetic state and the non-magnetic state.Figures 41(a) and 41(b) show the calculated PBE band structures and the Brillouin zone of MnF 3. Without SOC, it can be seen that, in the spin-up channel, there are a total of eight Dirac coneslocated at/around the Fermi level. The large number of Dirac cones and cone degeneracy are unusual and unique among all reported Dirac materials. Because of Pauli repulsion, the valence bandmaximum and the conduction band minimum of the Dirac conescannot absolutely touch at the Fermi level since they are both con-tributed by electrons of spin-up orientation. Thus, a small gap opens for every cone, ranging from 1.4 to 33.8 meV. This gapped Dirac cone feature has been proposed to be a unique feature forvalley current transport. 111Also, an energy gap of 4.1 eV was found in the spin-down channel. The authors also studied the 3D band dispersion plot to get more insight into the Dirac feature of MnF 3 in the whole BZ, as shown in Fig. 42 . A new type of 3D band FIG. 32. Geometric and electronic structures of Mn-intercalated epitaxialgraphene on SiC(0001). (a) and (b)show the top and side views of the optimized geometry for Mn intercalation coverage of v ¼5=12 ML, respectively. The blue rhombus in (a) represents thesupercell. The five different Mn atoms are marked as numbers (1) –(5). It should be noted that for adding Mn5into a 2 /C22 supercell with v¼1=3 ML, there is only one configu- ration. Here, Mn3, Mn4, and Mn5 atoms form a trimer. The correspond-ing spin-resolved band structure alongthe high-symmetry lines for (c) majority and (d) minority-spin channels. (e) Total density of states and the partialdensity of states. Reproduced with per-mission from Li et al., Phys. Rev. B 92, 201403 (2015). Copyright 2015 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-23 Published under license by AIP Publishing.structure plots has been observed, with two rings of Dirac nodes in the M –K–Γplane of the BZ. Also, if the symmetry effect is consid- ered, an additional Dirac ring will be present in the A –H–L plane of the BZ. Multiple Dirac ring materials are very rare, and it differsfrom the known Dirac materials such as graphene. Such a uniquemultiple Dirac ring feature in MnF 3is expected to bring about more fascinating electronic properties and applications as com- pared to other Dirac materials with the single cone or ring. B. The p-state-type DSGSs 1. Graphitic carbon nitrides Carbon nitrides have been widely investigated over the past 100 years.112 –117However, they recently attracted a lot of atten- tion due to the possibility of occurrence of graphitic 2D likeproperties. Zhang et al., 118based on first-principles simulations, found that a honeycomb lattice of a modified tri-s-triazine (C7N6) unit has a spin-polarized Dirac cone in the band struc- tures. The authors considered two types of graphitic carbonnitride frameworks g-C 14N12and g-C 10N9(see Fig. 43 )w i t h s-triazine and carbon- rich tri-s-triazine (C 7N6)a sb u i l d i n g blocks. Figure 44 shows the band structures and DoS. In the spin-down channel, g-C 14N12possesses a linear energy disper- sion, whereas for g-C 10N9, the spin-down channel possesses a parabolic one. In the spin-up channel, the gaps between thebands are 2.47 and 2.07 eV, respectively. Thus, g-C 14N12can be FIG. 34. Structure and symmetry of rutile oxides and the heterostructures of CrO 2and TiO 2. (a) Rutile structure cleaved along the (001) plane, showing the XO 6octahe- dra with two distinctive orientations. (b) Symmetry and local coordinates of a XO 2unit cell. Here, we use x, y, and z for the local coordinates and a, b, and c for the global orientation. (c) Structural model of the superlattice of CrO 2and TiO 2. (d) Structural model of the CrO 2thin film on the TiO 2substrate. Cr, Ti, and O are represented by orange, blue, and gray balls, respectively. Reproduced with permission from Cai et al., Nano Lett. 15, 6434 (2015). Copyright 2015 American Chemical Society. FIG. 33. (a) Electronic structure of epitaxial graphene along Γ–K with a linear energy dispersion around the Dirac point indicated by crossings, (b) after 0.1 ML Mn intercalation, (c) after 0.6 ML Mn intercalation, and (d) recovering ofthe graphene energy band by heating the sample up to 1200 /C14C. Reproduced with permission from Gao et al ., ACS Nano 6, 6562 (2012). Copyright 2012 American Chemical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-24 Published under license by AIP Publishing.referred to as a DSGS. However, the hybrid honeycomb lattice of C7N6and s-triazine (C 3N3) units gives rise to a parabolic-type SGS. Both frameworks were found to be energetically stable with respect to g-C 6N6. Also, using Monte Carlo simulations com- bined with heat capacity values, ferromagnetic ordering with aT Cvalue of /difference830 K was obtained for g-C 14N12.2. YN 2monolayer Using first-principles simulations, p-state DSGS nature is proposed in a YN 2monolayer with octahedral coordination by Liuet al.119This work was motivated by the successful synthesis of MoN 2bulk120and the increasing interest in TMN 2com- pounds.121The TMN 2(TM¼Y, Zr, Nb) monolayers can exist in 1H, 1T, and 1T0polymorphs, among which 1T is most stable and is shown in Fig. 45(a) . The absence of the imaginary frequency indicates the dynamical stability of the 1T-YN2 monolayer, as shown in Fig. 45(b) . The spin-resolved band structure of FM 1T-YN 2is shown in Fig. 46(a) . It can be seen that in the spin- down channel, the band structure possesses a gapless feature withVBM and CBM touching the Fermi level (Dirac points), whereas alarge gap of 4.57 eV is there in the spin-up channel. A similar feature has been produced within DFT+U and HSE06 calculations as well. To understand the origin of the single-spin Dirac state inthe YN 2monolayer, the authors also calculated the projected density of states (PDOS) and found that mainly the N atoms con-tribute to its Dirac state near the Fermi level, confirming the p-state DSGS feature. The Fermi velocity was found to be 3:74/C210 5m/s, which is very high as compared to other reported Dirac materials [see Fig. 46(b) ]. Also, the monolayer exhibits a robust ferromagnetic ground state with a Curie temperature above332 K. There are other TM nitride monolayers, such as scandium dinitrides and lanthanide dinitrides, which are proposed to be investigated for such rich properties. 3. Half-Heusler MnPK Dehghan and Davatolhagh122introduced a new class of a d0-d Dirac half-Heusler compound where d refers to the 3d-transition metal element and d0stands for the metal element with the valence electron configuration ns1,2,( n/C01)d0. FIG. 35. Single-spin Dirac points of the CrO 2/TiO 2inter- face. (a) and (b) show the non-relativistic DFT band struc-ture of superlattice and thin-film models, respectively. Here, the blue and red bands are for two spins, and the Fermi level is set to zero energy. Reproduced with per-mission from Cai et al ., Nano Lett. 15, 6434 (2015). Copyright 2015 American Chemical Society. FIG. 36. The top (a) and side (b) views of the VX 3(X¼Cl, I) monolayer. Reproduced with permission from He et al., J. Mater. Chem. C 4, 2518 (2016). Copyright 2016 Royal Society of Chemistry.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-25 Published under license by AIP Publishing.A systematic investigation on the spin-gapless behavior of MnPK using first-principles calculation within both GGA andGGA + U methods, under uniform and tetragonal strain condi-tions, has been carried out by You et al. 123This band structure is shown in Fig. 47 . The authors also found that Hubbard U mainly affects the band structure in the spin-down channelwhile keeping the SGS nature intact in the spin-up channel.The SGS nature was found to be quite robust against uniform and tetragonal strains. Althoug h a larger number of materials are theoretically predicted to show the 2D DSGS feature, noneof these materials are successfully synthesized experimentally. As such, this is the need of the hour to either synthesize these predicted materials or find new approaches to search stableand easily synthesizable DSGS materials. FIG. 37. Spin-polarized band structure and total density of states (DoS) for (a) VCl 3and (b) VI 3. The red and blue lines/areas represent the spin-up and spin-down chan- nels, respectively. Reproduced with permission from He et al., J. Mater. Chem. C 4, 2518 (2016). Copyright 2016 Royal Society of Chemistry. FIG. 38. The spin exchange parameters as a function of carrier concentration calculated for (a) VCl 3and (b) VI 3. The positive and negative values are for electron and hole doping, respectively. The calculated band structures for electron doping of 0.1 and 0.7 are presented in the insets for VCl 3and VI 3, respectively. Reproduced with per- mission from He et al., J. Mater. Chem. C 4, 2518 (2016). Copyright 2016 Royal Society of Chemistry.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-26 Published under license by AIP Publishing.VI. FUTURE PROSPECTS OF SGS MATERIALS From an application perspective, recently, the non-volatile magnetoresistive random access memory (MRAM) devices attracted a lot of attention because they can replace the dynamic random access memory (DRAM). This can lead to a significant reduction of electrical power consumption in commercial infor-mation technology devices. To realize the potential application ofMRAM, it is crucial to improve the performance of magnetictunnel junctions (MTJs), which is the basic element of MRAM.There are three fundamental issues regarding the performance ofMTJs: (i) a large TMR ratio for robust reading of digital datastored in MRAM cells, (ii) magnetization switching with lowconsumption power, and (iii) high perpendicular magnetic anisot- ropy of the magnetization direction of the free (memory) layer for a long retention of digital data. 124The magnetization switching of the free layer in MTJs could be achieved in different ways, such asusing the spin –orbit torque generated by heavy elements and topological materials, by spin-transfer torque due to the spin-polarized current, and also by applying the electric field. 125 –127 The electric-field driven magnetization switching is drawing much attention because it reduces the energy dissipation by afactor of 100 when compared with that in spin-transfer-torquedevices. This makes it comparable to that of the semiconductor based field-effect transistors but with added non-volatile function- ality. 126,127The electrical manipulation of magnetic anisotropy FIG. 39. Band structures of 2D NiCl 3with and without SOC calculated at the PBE and HSE06 exchange correlation levels. The insets of (a) and (b) show the details of Dirac states near the Fermi level where VB and CB are marked by green and black lines, respectively. The Fermi level is indicated by the horizo ntal dotted lines. The spin-up and spin-down channels are represented by red and blue lines, respectively. The 3D band structures around the Fermi level in the 2D k-space with and without SOC are shown in (c) and (f) panels, respectively. Reproduced with permission from He et al ., Nanoscale 9, 2246 (2017). Copyright 2017 Royal Society of Chemistry.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-27 Published under license by AIP Publishing.and switching has been achieved across a number of different material systems.128Due to their relatively low carrier concentra- tion, SGS materials are suitable candidates for the electric-fielddriven magnetization switching and MTJ applications. SGS mate-rials could make a bridge between half-metals and diluted mag- netic semiconductors. SGSs have a high Curie temperature compared to diluted magnetic semiconductors, which make them apotential candidate for room temperature applications. However,there are some challenges to demonstrate electric-field driven mag- netization switching in SGS materials such as growth of thin films (less than 5 nm) and sustaining the SGS band structure for thesethin films. In most cases, thin-film growth of these materials hasissues with surface roughness and chemical ordering. At criticallylow thicknesses, chemical disorder increases, which results in the loss of SGS band topology. The spin-polarized current passing through the ferromagnet causes spin-transfer-torque (STT) in nano-meter scale magnetic devices. 129,130Using this STT, a magnetization reversal or steady-state magnetization precession could be achieved,which is utilized in nonvolatile MRAM with low power consump- tion and in radio-frequency ( r f) oscillators and diode devices.131 The theoretical current density JCrequired for the magnetization switching and oscillation is proportional to the Gilbert dampingconstant ( α) multiplied by the saturation magnetization M Sof the ferromagnetic free layer.129Thus, the materials with low saturation magnetization and a low Gilbert damping constant are required to achieve STT magnetization switching at low JC. The Gilbert damping constant is primarily due to the intrinsic mechanism ofspin relaxation inherent in magnetic metals, and its origin is the spin –orbit interaction in the electronic band structure of ferromag- netic metals. The Gilbert damping constant, α, can be understoodby Kambersky ’s classic simple argument equation, 132 α¼1 γMSμB2D(EF)(g/C02)2 τ(EF), (2) where μBis the Bohr magneton, D(EF) is the total density of states at the Fermi energy E F,gis the Lande g factor, γis the gyromag- netic ratio, and τis the electron momentum scattering time. Clearly, αis directly proportional to D(E F). Therefore, αwill be smaller for magnetic alloys with lower D(E F), such as half-metals, as noted by Mizukami et al.133 The typical value of the Gilber t damping constant for Fe is 2/C210/C03,133,134while even smaller values were reported in half- metallic Heusler alloys with a half-metallic gap in minority states, e.g., 1.0 –1:5/C210/C03for Co 2FeAl films.133,135 –137Schoen et al.138 studied the CoFe binary alloys and reported a Gilbert damping cons- tant value of 5 /C210/C04with the lowest D(E F). Most of the studies on the Gilbert damping constant explained that low density of states at Fermi energy is the origin of the low value of α.H o w e v e r ,t h el o w e s t value of αin ferromagnetic metals is many times higher than that observed for insulating ferromagnets ( α/difference10/C05), such as yttrium – iron –garnet, with zero-D(E F).139 –141The ideal SGS materials possess a bandgap for one spin state and a gapless-semi-metallic state for the other spin state at the E F. Thus, SGSs have negligible D(E F) and are expected to have αeven lower than the half-metallic fer- romagnetic systems. One of the authors71studied the Gilbert damping constant in equiatomic quaternary Heusler alloy CoFeMnSi thin films. The 10 nm thick CoFeMnSi shows the Gilbert damping constant value of α¼2:7/C210/C03,w h i c hi st h e FIG. 40. (a) The distribution of the Berry curvature in the momentum space for NiCl 3. (b) Anomalous Hall conductivity vs energy as the Fermi level is shifted from its origi- nal position. The calculated local density of states of edge states for (c) zigzag and (d) armchair insulators. Reproduced with permission from He et al., Nanoscale 9, 2246 (2017). Copyright 2017 Royal Society of Chemistry.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-28 Published under license by AIP Publishing.FIG. 41. Spin-polarized band structures of MnF 3without SOC calculated at the PBE (top) and HSE06 (bottom) exchange correlation levels. The inset of (d) displays the corresponding Brillouin zone. Reproduced with permission from Jiao et al., Phys. Rev. Lett. 119, 016403 (2017). Copyright 2017 American Physical Society. FIG. 42. (a) 3D electronic band plot of MnF 3along the M –K–Γplane. (b) 3D band view of VBM and its corresponding projection. Reproduced with permission from Jiao et al., Phys. Rev. Lett. 119, 016403 (2017). Copyright 2017 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-29 Published under license by AIP Publishing.smallest among the transition metal ferromagnets. They also studied the effect of chemical ordering and the spin-gapless electronic struc- ture and further discussed the possibility of reducing the αto the ultralow damping regime of /difference10/C05using the ab initio calculations and experimental observations. One way to achieve a ultralow α value is to synthesize highly crystalline and ordered films, whichcan help in sustaining the spin-gapless electronic structure and hence the ultralow density of states at E F. Very recently, a reconfig- urable magnetic tunnel diode and transistor based on half-metallicmagnets (HMMs) and spin-gapless semiconductors have been pro-posed (employing first-principles calculations), which consist of a HMM and a SGS electrode separated by a thin insulating tunnel barrier. This new spintronic device based on SGSs and HHMscombines reconfigurability and nonvolatility on the diode and tran- sistor level. Also, the unique band structure of the SGS base elec- trode limits the base-collector leakage current and allows dual-mode operation of the transistor. 142,143 It should be noted that the spin-gapless semiconductors have almost 100% spin polarization only at very low drain biases.144 Also, they are much less sensitive to the conductivity mismatch problem due to their low conductivity and are nominally free from it if the inject carriers are of only one spin sign. Thus, they can actas very efficient spin injectors into semiconductors. 145On the con- trary, SGS are not useful as spin-polarized channels for spin-MOSFETs because the zero energy required for the excitation nullifies any possible gain by the field effect. FIG. 43. Schematic representation of (a) g-C 14N12and (c) g-C 10N9with the unit cells indicated by the light yellow region. The corresponding phonon spectrums along high symmetry points of the BZ are plotted in (b) and (d), respectively. Reproduced with permission from Zhang et al ., Carbon 84, 1 (2015). Copyright 2015 Elsevier Publications.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-30 Published under license by AIP Publishing.FIG. 44. Spin-resolved band structure, electronic density of states (DoS), and spatial distribution of spin-polarizedelectron density for (a) g-C 14N12and (b) g-C 10N9. The insets show the enlarged views of the degenerate bands near the Fermi level. The energy at the Fermi level wasset to zero, and the isosurface value of spin-polarizedelectron density was set to 0.003 A /C143. Reproduced with permission from Zhang et al ., Carbon 84, 1 (2015). Copyright 2015 Elsevier Publications. FIG. 45. Structure of the 1T-YN 2monolayer in top and side views. The inset shows the octahedral structure unit (top right corner). The shaded area with Bravais lattice vectors a and b marks the primitive unit cell. (b) Phonon dispersion for the 1T-YN 2monolayer. The absence of the imaginary frequency indicates its dynamical stability. Reproduced with permission from Liu et al., Nano Research 10, 1972 (2017). Copyright 2017 Springer Publications.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-31 Published under license by AIP Publishing.FIG. 46. (a) Spin-polarized band structure of the 1T-YN 2monolayer. (b) Comparison of the values of the Fermi velocity ( vF) between previously reported Dirac materials. Reproduced with permission from Liu et al., Nano Research 10, 1972 (2017). Copyright 2017 Springer Publications. FIG. 47. Spin-polarized simulated electronic band structure of half-Heusler compound MnPK at the equilibrium lattice constant. For the GGA + U method, an ons ite Coulomb energy of 4 eV is applied. Reproduced from You et al., Materials 12(19), 3117 (2019). Copyright 2019 MDPI.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-32 Published under license by AIP Publishing.VII. CONCLUSION Spin-gapless semiconductors are members of zero-gap materi- als with unique electronic properties and open up various prospects for practical applications in the spintronics, electronics, and optics fields. The properties of SGS materials can be tuned by external factors such as electric fields, magnetic fields, pressure, impurities,etc. The tunable spin direction characteristic may be useful to design qubits for quantum computing, data storage, and coding/ decoding. As compared to the diluted magnetic semiconductors, inthe SGS, the speed of the 100% polarized spin electrons could be much larger. The excited carriers can be 100% spin polarized with tunable capabilities and can have superior performance comparedto half-metals and diluted magnetic semiconductors. From the applied point of view, this realization was significant because many of these SGS materials have a higher Curie temperature comparedto the other known spintronic, semiconducting family, namely,DMS. The band structures shown in Fig. 3 having linear dispersion with a finite gap may not really exist 1because such a dispersion can be destroyed by any interaction opening up the bandgap. Thus,a deviation from linear dispersion [shown by the dotted curves in Fig. 3(a) ] takes place. However, according to the model for quantum resistance, 146with proper tuning induced by pressure and doping, the linear dispersion may still hold at some random point in the reciprocal space or can be realized in low-dimensional mate- rials such as graphene. This article discusses the recent progress on the material realiza- tion of parabolic and Dirac-type spin-gapless semiconductors. The concept of SGSs was first verified in Co-substituted PbPdO 21by band structure calculations. A major breakthrough happened withthe identification of Heusler alloys with SGS properties. Many Heusler-based alloys have a stable structure, high T C,a n dh i g hs p i n polarization. Among Heusler alloys, various systems have beenreported to show SGS character, however, mainly because of the diffi- culties in their synthesis and growth, and only a few of them have been experimentally verified, too indirectly in most of the cases.These include half-Heusler, full-Heusler, and quaternary Heusler alloys. The first experimental verification of SGS characteristics among Heusler alloys was reported in polycrystalline Mn 2CoAl11fol- lowed by CoFeMnSi,12CoFeCrGa,13etc. An important class of mate- rials among Heusler-based SGS is FCF-SGS materials. As compared to the conventional MTJs with magnetic materials, in SGS-FCF mate-rials, there would be no stray magnetic fields that prevent the distor-tion of the domain structure of the materials. CrVTiAl 14is one such material where SGS and FCF characters can be realized simultane- ously. Furthermore, to realize the application of the new SGS materi-als in devices, the first step is to fabricate well-ordered SGS films on semiconducting substrates. It is challenging to maintain the same sto- ichiometry structure in the thin-film form as in the poly-crystallinebulk. A few Heusler-based SGSs have been synthesized in a thin-film form, which include Mn 2CoAl,51Ti2MnAl,23CoFeMnSi,65and CrVTiAl.74Another important aspect is to tune the properties of existing SGSs with suitable substitutions that may help in further improving their properties, as reported in Co 1þxFe1/C0xCrGa.75As mentioned in the beginning, based on the few experimental reports,almost temperature-independent conductivity, a low value of temper- ature coefficient of resistivity, a very small Seebeck coefficient,temperature-independent carri er concentration, and linear MR have been considered signatures of SGSs. However, a direct evi- dence of zero gap in such materials is yet to be studied, which isone of the main future challenges in these materials. Also, in thefuture, it will be interesting to study the properties of MJT basedon SGSs and their realization in real time devices. Other parabolic-type SGSs include diamond like quaternary compound CuMn 2InSe 480and CO –Mn complex decorated graphene such as ZnO95and BiFe 0:83Ni0:17O3.96Experimental verification of this behavior is yet to be reported. We also discussed recent progress in DSGSs. For spintronic devices, in comparison with parabolic-like SGSs, these DSGSs would be a better choice because they can lead to low energy con-sumption and ultra-fast transport because of their unique linearband dispersion. DSGSs exhibit 100% spin polarization, masslessfermions around the Fermi level, and ideal dissipationless proper- ties. The prediction of DSGSs is mainly based on theoretical calcu- lations. DSGSs can be divided into two classes: (1) d-state DSGSsin which the Dirac state is contributed by d-orbitals of transitionmetal atoms and (2) p-state DSGSs in which it is contributed by p-orbitals of main-group atoms. The p-state DSGSs gain more atten- tion due to their applications in high-speed spintronics. The p-state DSGSs include 2D graphitic nitrides 118and some TM nitride layered materials.119The search of DSGSs was mainly focused on layered materials until DSGS nature was predicted in MnF 3in a bulk form with excellent properties.110Until now, among the pre- dicted DSGSs, only bulk MnF 3has been synthesized. In MnF 3,i t was observed that in the spin-up channel, there are a total of eightDirac cones located at/around the Fermi level. The large number ofDirac cones and cone degeneracy are unusual and unique among all reported Dirac materials. Also, a small gap opens for every cone, ranging from 1.4 to 33.8 meV. This gapped Dirac cone feature hasbeen proposed to be a unique feature for valley current transport.The 3D band structure of MnF 3was observed to have two rings of Dirac nodes in the M –K–Γplane of the BZ. Such a unique multiple Dirac ring feature in this material is expected to bring about more fascinating electronic properties and applications as compared toother Dirac materials with the single cone or ring. On the otherhand, under different coverage of Mn atoms, Dirac states were real-ized in Mn-intercalated epitaxial graphene, which was demon- strated experimentally to be in the DHM state instead of the DSGS state. However, later, it was confirmed that substrate modulationcan transform its DHM state to the DSGS state. To demonstratethe feasibility of various theoretically predicted DSGSs, more emphasis on their synthesize and characterization is essential. Therefore, it is quite clear that the spintronics community is verykeenly watching the development of different kinds of SGS materi-als in view of their superior place in devices. It is hoped that futureactivities in this topic will give rise to better understanding of the physics of these materials, thereby allowing their full exploitation in practical applications. ACKNOWLEDGMENTS A.A. acknowledge DST-SERB (Grant No. CRG/2019/002050) for funding to support this research. T he authors thank Y. Venkateswara, J .N a g ,a n dA .I .M a l l i c kf o ru s e f u ld i s c u s s i o n s .Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 220902 (2020); doi: 10.1063/5.0028918 128, 220902-33 Published under license by AIP Publishing.DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1X. L. Wang, Phys. Rev. Lett. 100, 156404 (2008). 2X. Wang, D. Shi-Xue, and Z. 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1.2712324.pdf	Model of phase locking in spin-transfer-driven magnetization dynamics R. Bonin, G. Bertotti, C. Serpico, I. D. Mayergoyz, and M. d’Aquino Citation: Journal of Applied Physics 101, 09A506 (2007); doi: 10.1063/1.2712324 View online: http://dx.doi.org/10.1063/1.2712324 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/101/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Micromagnetic study of phase-locking in spin-transfer nano-oscillators driven by currents and ac fields J. Appl. Phys. 109, 07C914 (2011); 10.1063/1.3559476 Spin-torque driven magnetic vortex self-oscillations in perpendicular magnetic fields Appl. Phys. Lett. 96, 102508 (2010); 10.1063/1.3358387 Thermal dynamics in symmetric magnetic nanopillars driven by spin transfer Appl. Phys. Lett. 92, 172501 (2008); 10.1063/1.2918012 Trends in spin-transfer-driven magnetization dynamics of Co Fe ∕ Al O ∕ Py and Co Fe ∕ Mg O ∕ Py magnetic tunnel junctions Appl. Phys. Lett. 89, 262509 (2006); 10.1063/1.2425017 Spin-transfer effects in nanoscale magnetic tunnel junctions Appl. Phys. Lett. 85, 1205 (2004); 10.1063/1.1781769 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 07:30:19Model of phase locking in spin-transfer-driven magnetization dynamics R. Bonina/H20850and G. Bertotti INRIM, 10135 Torino, Italy C. Serpico Dipartimento di Ingegneria Elettrica, Università di Napoli “Federico II,” 80125 Napoli, Italy I. D. Mayergoyz Electrical and Computer Engineering Department, University of Maryland, College Park, Maryland 20742 and UMIACS, University of Maryland, College Park Maryland 20742 M. d’Aquino Dipartimento di Ingegneria Elettrica, Universitá di Napoli “Federico II,” 80133 Napoli, Italy /H20849Presented on 9 January 2007; received 6 November 2006; accepted 21 December 2006; published online 4 May 2007 /H20850 A simpliﬁed model of phase locking is discussed, which can be fully solved in analytical terms with no limitations as to the intensity of the coupling mechanism responsible for the locking. Ananomagnet with uniaxial symmetry is considered, jointly driven by a spin-polarized current, a dcmagnetic ﬁeld along the symmetry axis, and a radio-frequency circularly polarized magnetic ﬁeld.The conditions are determined under which locking occurs between current-induced oscillations andthe action of the rf ﬁeld. The locking effect exhibits hysteresis as a function of the current. © 2007 American Institute of Physics ./H20851DOI: 10.1063/1.2712324 /H20852 Signiﬁcant efforts are under way to study the effect of a spin-polarized current in three-layer structures, composed bya “ﬁxed” layer with pinned magnetization, a nonmagneticspacer, and a “free” layer whose magnetization is subject tothe torque exerted by the spin-polarized current. In this con-nection, the recent discovery of phase locking effects in spin-transfer devices has generated considerable interest. 1–3The key point is to comprehend in detail how phase locking willemerge under different dynamical regimes induced by thecurrent and the ﬁeld. In this paper, we discuss phase locking between current- induced magnetization precession and a circularly polarizedradio-frequency /H20849rf/H20850ﬁeld applied in the free-layer plane. We consider the case of a system with uniaxial symmetry, wherethe free-layer and ﬁxed-layer easy axes as well as the exter-nal dc magnetic ﬁeld are all perpendicular to the layerplanes, 4,5and we restrict our analysis to uniformly magne- tized layers. Under these assumptions, the phase-lockingproblem can be fully solved in analytical terms. We introduce a system of Cartesian unit vectors /H20849e x,ey,ez/H20850. The free layer is parallel to the /H20849x,y/H20850plane, and the electron current ﬂows along the zdirection, which repre- sents the symmetry axis of the problem. The magnetizationdynamics in the free layer is described by the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation with the addition of the spin-transfer term. 6,7We consider the simplest case where no asymmetry exists between forward and backward spin-transfer effects. In dimensionless form, the equation for thiscase isdm dt−/H9251m/H11003dm dt=−m/H11003/H20849heff−/H9252m/H11003ez/H20850, /H208491/H20850 where the free-layer magnetization mand the effective ﬁeld heffare normalized by the saturation magnetization Ms, time is measured in units of /H20849/H9253Ms/H20850−1/H20849/H9253is the absolute value of the gyromagnetic ratio /H20850,/H9251is the damping parameter, and the unit vector ezgives the direction of the spin polarization. The parameter /H9252is expressed as7,8/H9252=bpJe/Jp, where Jeis the current density, taken as positive when the electrons ﬂowfrom the free into the ﬁxed layer, and J p=/H92620Ms2/H20841e/H20841d//H6036/H20849/H92620is the vacuum permeability, eis the electron charge, dis the thickness of the free layer, and /H6036is the reduced Planck con- stant /H20850. The parameter bpis model dependent but always smaller than unity. Therefore, since typically Jp /H11229109Ac m−2,/H9252/H112701 for the typical current densities em- ployed in spin-transfer experiments, Je/H11351108Ac m−2. The effective ﬁeld appearing in Eq. /H208491/H20850is given by heff =−/H11509gL//H11509m, where gL/H20849m,ha/H20850=1 2/H20851N/H11036/H20849mx2+my2/H20850+Nzmz2/H20852−/H9260 2mz2−ha/H20849t/H20850·m /H208492/H20850 is the energy density of the free layer normalized by /H92620Ms2, NzandN/H11036are the demagnetizing factors along the symmetry axis and in the plane perpendicular to it /H20849Nz+2N/H11036=1/H20850,/H9260 =2K1//H92620Ms2is the dimensionless anisotropy constant, and K1being the physical anisotropy constant. The external ﬁeld ha/H20849t/H20850is composed by a dc component aligned to the zaxis and a rf component of angular frequency /H92750in the /H20849x,y/H20850 plane, that is, ha/H20849t/H20850=ha/H11036/H20849cos/H92750tex+sin/H92750tey/H20850+hazez.I no u r discussion, we will use the dimensionless parameter /H9260eff=/H9260 +N/H11036−Nz.a/H20850Electronic mail: bonin@inrim.itJOURNAL OF APPLIED PHYSICS 101, 09A506 /H208492007 /H20850 0021-8979/2007/101 /H208499/H20850/09A506/3/$23.00 © 2007 American Institute of Physics 101, 09A506-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 07:30:19Equation /H208491/H20850describes a dynamical system evolving on the surface of the unit sphere /H20841m/H208412=1. Complete analytical solutions for this dynamics were previously obtained in thecases where only the current or the rf ﬁeld are present, thatis, when h a/H11036=0 and /H9252/HS110050 or when ha/H11036/HS110050 and /H9252=0. The ﬁrst case was studied for uniaxial systems in Ref. 5.I tw a s proved that the system has two ﬁxed points, mz= ±1, which are always present for all values of dc ﬁeld and injectedcurrent. Besides, for currents and ﬁelds satisfying the condi-tion −1 /H33355/H20849 /H9252//H9251−haz/H20850//H9260eff/H333551, a limit cycle appears in the dynamics corresponding to a self-oscillatory regime for the magnetization. Magnetization dynamics under ha/H11036/HS110050 and /H9252=0 was studied in Ref. 9by using methods of nonlinear dynamical system theory and bifurcation theory.10These same methods can be used to study the general case, ha/H11036 /HS110050 and/H9252/HS110050. The starting point is to describe the magneti- zation dynamics in the rotating frame where the rf ﬁeld isstationary. Thanks to symmetry conditions, in the rotatingframe Eq. /H208491/H20850becomes dm dt−/H9251m/H11003dm dt=−m/H11003/H20851/H20849haz/H11032−/H9275/H11032+/H9260effmz/H20850ez+ha/H11036ex +/H9251/H9275/H11032m/H11003ez/H20852, /H208493/H20850 where haz/H11032=haz−/H9252//H9251,/H9275/H11032=/H92750−/H9252//H9251. /H208494/H20850 Equation /H208493/H20850shows no explicit dependence on the current anymore. In other words, the analysis of the dynamics drivenby the combined action of the spin-polarized current and therf ﬁeld is identical to the analysis of the dynamics driven bythe rf ﬁeld only, 9once the dc ﬁeld and the angular frequency are redeﬁned as expressed by Eq. /H208494/H20850. These conclusions in- dicate that the results obtained in Ref. 9should be immedi- ately applicable to the phase-locking problem of interesthere. It was shown in Ref. 9that time-persistent solutions of Eq. /H208493/H20850can only be ﬁxed points or limit cycles. Fixed points in the rotating frame are observed in the laboratory frame asperiodic modes, termed P-modes, in which the magnetization precesses around the symmetry axis in synchronism with therf ﬁeld. On the other hand, a limit cycle in the rotating frameis observed in the laboratory frame as a quasiperiodic mode/H20849Q-mode /H20850, resulting from the combination of the periodic motion along the limit cycle with the rotation of the rotatingframe. Position and stability of ﬁxed points are analyticallyobtained by standard methods. 10In particular, as shown in Ref. 11, only two or four ﬁxed points can be present in the dynamics of Eq. /H208493/H20850. In the second case, one of the ﬁxed points must be a saddle /H20849Poincaré index theorem10/H20850. Limit cycles can be analytically studied by taking advantage of thefact that /H9251/H112701 and/H9252/H112701. The dynamics described by Eq. /H208493/H20850 can then be viewed as a perturbation of the conservative one,obtained for /H9251=/H9252=0, which permits one to apply Poincaré- Melnikov theory for slightly dissipative systems10,11to deter- mine limit cycles. One introduces the Melnikov function M/H20849g˜0/H20850=−/H20859C/H20849g˜0/H20850/H20849m/H11003heff/H20850·dm, where C/H20849g˜0/H20850is the mtrajec- tory of constant gL/H20849m,ha/H20850+/H92750mz=g˜0. Then, the equation M/H20849g˜0/H20850=0 represents the necessary and sufﬁcient conditionfor the existence of a limit cycle. The limit cycle is /H9251close to the trajectory C/H20849g˜0/H20850and is stable /H20849unstable /H20850if/H11509M/H20849g˜0/H20850//H11509g˜0 /H110220/H20849/H110210/H20850. The control parameters which determine the dynamical response of the system described by Eq. /H208493/H20850are /H20849haz/H11032,ha/H11036,/H9275/H11032/H20850, where haz/H11032and/H9275/H11032are given by Eq. /H208494/H20850.I nt h e case of interest here, ha/H11036and/H92750are given and one is inter- ested in the dependence on hazand/H9252//H9251. Therefore, the dy- namical response of the system can be represented in the/H20849h az,/H9252//H9251/H20850control plane. In Fig. 1/H20849a/H20850, we show the various dynamical regimes in the /H20849haz,/H9252//H9251/H20850plane for ha/H11036=0.003 and/H92750=1. In more detail, the bold symbols identify the re- gions where P-modes and Q-modes are present. The slash notation, that is, P/PandP/Q, indicates the coexistence of two different P-modes or a P-mode and a Q-mode. When a variation of external conditions induces qualitative changesin a given dynamical regime, we are in the presence of abifurcation. Three types of bifurcations 11are present in the diagram of Fig. 1:/H20849i/H20850saddle-node bifurcation /H20849labeled by d/H20850, when a saddle-node pair is either created or annihilated; /H20849ii/H20850 Hopf bifurcation /H20849labeled by h/H20850, when one of the ﬁxed points of the dynamics changes from stable to unstable or vice versawith the simultaneous appearance or disappearance of a limitcycle; and /H20849iii/H20850saddle-connection bifurcation /H20849labeled by c/H20850, when a limit cycle appears or disappears in the vicinity of aseparatrix. In this picture, one can determine the conditions under which phase locking will appear. Let us ﬁrst consider thecase, discussed in Ref. 5, where no rf ﬁeld is present. By applying a positive dc ﬁeld sufﬁcient to saturate the magne- FIG. 1. /H20849a/H20850Stability diagram in the /H20849haz,/H9252//H9251/H20850control plane. System param- eters are ha/H11036=0.003, /H92750=1,/H9251=0.02, and /H9260eff=−1. Bifurcation lines are d: saddle node, h: Hopf, and c: saddle connection. States are P: stable P-mode andQ: stable Q-mode. /H20849b/H20850\\\ shading: region of phase locking under ﬁxed ﬁeld and increasing current and /// shading: region of phase locking underﬁxed ﬁeld and decreasing current. Frequency behavior along vertical line Ais shown in Fig. 2.09A506-2 Bonin et al. J. Appl. Phys. 101, 09A506 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 07:30:19tization, in our case haz/H11022/H20841/H9260eff/H20841=1, the magnetization state under zero current is mz=1. When the current is increased, a Hopf bifurcation occurs for /H9252//H9251=haz+/H9260eff, where current- induced magnetization precession of angular frequency /H9275 =/H9252//H9251appears. This regime persists until the hbifurcation line/H9252//H9251=haz−/H9260effis reached, where the state mz=−1 be- comes stable. Figures 1and 2 show how this simple picture is modiﬁed when one introduces the rf ﬁeld. A complex bi-furcation structure appears for currents /H9252//H9251close to the rf ﬁeld frequency /H92750. Let us consider, for example, what hap- pens when we move along vertical line A in Fig. 1/H20849b/H20850/H20849see Fig.2/H20850, that is, the dc ﬁeld is ﬁxed and the current varies. At zero current, the system is in the stable P-mode of the P regime. The magnetization is almost aligned to the zaxis and precesses around this axis in synchronism with the rf ﬁeld atthe angular frequency /H92750. Then, when the current is in- creased, the hbifurcation line is reached and the system jumps to the Q-mode of the Qregime. In this regime, the system executes a quasiperiodic motion resulting from thecombination of the current-induced precession and the rfﬁeld action. Therefore, two angular frequencies are presentin the system response /H20849see Fig. 2/H20850. This situation persists also when the saddle-node bifurcation line dis crossed and a further stable P-mode, coexisting with the Q-mode /H20849P/Q regime /H20850, is created. When the ﬁrst saddle-connection bifur- cation cline is crossed, the limit cycle disappears and the system jumps to stable P-mode of the Pregime. The appear- ance of this P-mode means that the current-induced preces- sion starts to proceed in synchronism with the rf ﬁeld. Inother words, phase locking has taken place. When the second cbifurcation line is reached, the Q-mode of the P/Qregime appears, but the system remains in the P-mode and the phase locking is observed until the system crosses the second d bifurcation line and jumps to the Q-mode previously created, which is the only available stable state. Again two frequen-cies are observed, /H92750and the frequency of the current- induced precession. For higher currents, a the second hbi- furcation line is crossed, where the Q-mode disappears and the system reaches the large-current Pregime. In this regime, the magnetization is almost antiparallel to the zaxis and precesses around the zaxis with frequency /H92750. By decreasing the current from this state, an hysteretic behavior is observeddue to the coexistence of P-modes and Q-modes in the P/Q regimes between the dandcbifurcation lines /H20851see Figs. 2/H20849a/H20850 and2/H20849b/H20850/H20852. A qualitatively different behavior is observed if the bi- furcation lines are crossed in a different order. Let us con-sider, for example, vertical line B in Fig. 1/H20849b/H20850/H20849h az=1.9 /H20850. The dynamics under increasing current is similar to the one at haz=1.2. However, when we decrease the current, after crossing the second cbifurcation line, we reach the hbifur- cation line before the dbifurcation. This means that at the h bifurcation a second stable P-mode is created /H20849P/Pregime /H20850 but the system stays in the previous stable P-mode locked to the rf ﬁeld at the angular frequency /H92750. When the dbifurca- tion is reached, the system jumps to the unlocked initialP-mode of the Pregime, but no quantitative effects are ob- served in the frequency response since the magnetizationprecesses around the zaxis in synchronism with the rf ﬁeld both before and after the bifurcation. Finally, we remark that if we initially saturate the system with a dc ﬁeld h az/H11407−/H9260eff+/H92750, no phase-locking effects oc- cur and the reversible dynamics described by Ref. 5takes place. 1M. R. Pufall, W. H. Rippard, S. Kaka, T. J. Silva, and S. E. Russek, Appl. Phys. Lett. 86, 082506 /H208492005 /H20850. 2S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, Nature /H20849London /H20850437, 389 /H208492005 /H20850. 3F. B. Mancoff, N. D. Rizzo, B. N. Engel, and S. Tehrani, Nature /H20849London /H20850 437, 393 /H208492005 /H20850. 4S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, and E. E. Fullerton, Nat. Mater. 5, 210 /H208492006 /H20850. 5Y . B. Bazaliy, B. A. Jones, and S. C. Zhang, Phys. Rev. B 69, 094421 /H208492004 /H20850. 6J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 7J. C. Slonczewski, J. Magn. Magn. Mater. 247,3 2 4 /H208492002 /H20850. 8G. Bertotti, C. Serpico, I. D. Mayergoyz, R. Bonin, and M. d’Aquino, J. Magn. Magn. Mater. /H20849to be published /H20850. 9G. Bertotti, C. Serpico, and I. D. Mayergoyz, Phys. Rev. Lett. 86,7 2 4 /H208492001 /H20850. 10L. Perko, Differential Equations and Dynamical Systems /H20849Springer, New York, 1996 /H20850. 11G. Bertotti, I. D. Mayergoyz, and C. Serpico, in The Science of Hysteresis , edited by G. Bertotti and I. D. Mayergoyz /H20849Elsevier, Oxford, 2006 /H20850, V ol. 2, Chap. 7. FIG. 2. Bold lines represent angular frequencies present in the system re- sponse to current for ﬁxed dc ﬁeld haz=1.2 /H20851vertical line A in Fig. 1/H20849b/H20850/H20852./H20849a/H20850 Increasing current and /H20849b/H20850decreasing current. d: saddle-node bifurcation, h: Hopf bifurcation, and c: saddle-connection bifurcation. Thin line: /H9275=/H9252//H9251.09A506-3 Bonin et al. J. Appl. Phys. 101, 09A506 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 07:30:19
5.0012734.pdf	Appl. Phys. Lett. 116, 209902 (2020); https://doi.org/10.1063/5.0012734 116, 209902 © 2020 Author(s).Erratum: “Controlling acoustic waves using magnetoelastic Fano resonances” [Appl. Phys. Lett. 115, 082403 (2019)] Cite as: Appl. Phys. Lett. 116, 209902 (2020); https://doi.org/10.1063/5.0012734 Submitted: 05 May 2020 . Accepted: 06 May 2020 . Published Online: 20 May 2020 O. S. Latcham , Y. I. Gusieva , A. V. Shytov , O. Y. Gorobets , and V. V. Kruglyak ARTICLES YOU MAY BE INTERESTED IN Substrate dependent terahertz response of monolayer WS 2 Applied Physics Letters 116, 203108 (2020); https://doi.org/10.1063/5.0006617 Raman spectroscopy of GaSb 1−xBix alloys with high Bi content Applied Physics Letters 116, 202103 (2020); https://doi.org/10.1063/5.0008100 Visible and near-infrared dual-band photodetector based on gold–silicon metamaterial Applied Physics Letters 116, 203107 (2020); https://doi.org/10.1063/1.5144044Erratum: “Controlling acoustic waves using magnetoelastic Fano resonances” [Appl. Phys. Lett. 115, 082403 (2019)] Cite as: Appl. Phys. Lett. 116, 209902 (2020); doi: 10.1063/5.0012734 Submitted: 5 May 2020 .Accepted: 6 May 2020 . Published Online: 20 May 2020 O. S. Latcham,1 Y. I.Gusieva,2 A. V. Shytov,1 O. Y. Gorobets,2 and V. V. Kruglyak1,a) AFFILIATIONS 1University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom 2Igor Sikorsky Kyiv Polytechnic Institute, 37 Prosp. Peremohy, Kyiv 03056, Ukraine a)Electronic mail: V.V.Kruglyak@exeter.ac.uk https://doi.org/10.1063/5.0012734 In the originally published article,1the material of the nonmag- netic matrix was mistakenly quoted as silicon nitride. In fact, siliconnitride has a value of the shear modulus that is different from the value of 298 GPa used for plotting graphs. Furthermore, the effect of refraction was not accounted for in Ref. 1. Hence, although Eq. (8) in Ref. 1is valid for a wave propagating at angle hin an inﬁnite sample, it must be replaced by k 2 x;x¼q Cx2x2/C0~xx~xy/C0/C1/C0k2 x;yx2/C0~xx~xyþcB2 MsC~xx/C18/C19 x2/C0~xx~xyþcB2 MsC~xy/C20/C21 ;(8) where kx;yis equal to that of the incident wave. The branch with Imkx;x>0 describes a forward wave decaying into the slab. Equation (9)for the impedance must be replaced by ZðF=BÞ x;ME¼Ckx;x x1þcB2 CM s~xy7ixkx;y kx;x x2/C0~xx~xy0 B@1 CA; (9) where “–” and “ þ” signs correspond to the impedance values for the forward [superscript “(F)”] and backward [superscript “(B)”] propa-gating waves, respectively. Thus, the impedance is non-reciprocal for ﬁnite values of h. The magnetoelastic resonance frequency is deﬁned by ReZðF=BÞ x;ME¼0, and so Eq. (10)must be replaced by xME¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ xxxy/C0cB2 MsCxys : (10)This frequency is no longer angle dependent, and so the state- ment “Note also the hdependence of the resonant frequency xMEas reﬂected in Eq. (10)” must be disregarded. Equations (11) and(12) for the reﬂection and transmission coef- ﬁcients, respectively, must be replaced by Rx¼ð~gxþ1Þð1/C0gxÞsinðkx;xdÞ ð~gxgxþ1Þsinðkx;xdÞþiðgxþ~gxÞcosðkx;xdÞ; (11) Tx¼iðgxþ~gxÞ ð~gxgxþ1Þsinðkx;xdÞþiðgxþ~gxÞcosðkx;xdÞ; (12) FIG. 3. The reﬂection coefﬁcient, R(f), in the oblique incidence geometry is deter- mined by the interplay between the enhancement of the magnetoelastic coupling and a non-monotonic variation of the background reﬂectivity. Colored curves repre- sent speciﬁc incidence angles sweeping from 0/C14to 45/C14. Moderate Gilbert damping ofa¼10/C03is assumed. The dashed vertical line corresponds to the magnetoelas- tic resonance frequency. Appl. Phys. Lett. 116, 209902 (2020); doi: 10.1063/5.0012734 116, 209902-1 Published under license by AIP PublishingApplied Physics Letters ERRATUM scitation.org/journal/aplw h e r ew eh a v ed e n o t e d gx¼ZðFÞ ME=Z0and~gx¼ZðBÞ ME=Z0andZ0is the impedance of the nonmagnetic matrix. Equations (11) and(12) given here coincide with their counter- parts from the original article at normal incidence. However, Fig. 3 is altered and must be replaced by the version here. The amended Fig. 3 reveals interplay between the enhancement of the resonant reﬂectivity at oblique angles and the angular dependence of the non-resonant reﬂection amplitude. The latter vanishes at h/C2530/C14and changes its sign at larger angles. Equation (13)must be replaced by Rx¼iCR=2 ðx/C0xMEÞþiCR=2ei/þR0; (13) where /¼/C02 arctan ½C C0ﬃﬃﬃﬃ xx xyq tanh/C138is an extra phase acquired by the resonant contribution at ﬁnite hvalues. The phase rapidly reaches p already for relatively small values of h, changing the sign of the Fano interference contribution. Equations (14) and (15) for the linewidth and ﬁgure of merit, respectively, must be replaced byCR¼cB2 2MsC2coshﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q0C0p xycos2hþC2 C2 0xxsin2h ! d; (14) !¼CR CFMR¼cdB2 2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q0C0pHBcos2hþC2 C2 0Mssin2h ! aC2M2 scosh: (15) This enhances both the linewidth CRand the ﬁgure of merit ! at oblique incidence by a factor of 1 =cosh.I ft h er a t i o C2=C2 0 is large, !may be somewhat reduced at large hbut only slightly. Table I andFig. 4 must be replaced by the versions here. The corrections described here do not change the primary ﬁnd- ings of Ref. 1: the magnetoelastic coupling can be manifested in reso- nant behavior of the acoustic reﬂectivity, which is enhanced in the oblique incidence geometry. The amended version of the manuscript can be found at arXiv.org:1906.07297. REFERENCE 1O. S. Latcham, Y. I. Gusieva, A. V. Shytov, O. Y. Gorobets, and V. V. Kruglyak, Appl. Phys. Lett. 115, 082403 (2019).TABLE I. Comparison of the ﬁgure of merit !for different materials, assuming d¼20 nm, l0HB¼50 mT, and C0¼298 GPa. Parameters YIG Co Py !ðh¼0/C14Þ 4.3/C210/C021.7/C210/C032.7/C210/C04 CR(ns–1) 1.9 /C210/C047.5/C210/C032.0/C210/C04 CFMR(ns–1) 4.4 /C210/C034.3 0.74 !ðh¼30/C14Þ 4.1/C210/C022.5/C210/C032.8/C210/C04 CR(ns–1) 1.8 /C210/C041.1/C210/C022.1/C210/C04 CFMR(ns–1) 4.4 /C210/C034.3 0.74 fME¼xME=2p(GHz) 2.97 7.14 6.26 B(MJ m–3) 0.55 10 –0.9 C(GPa) 74 80 50 q(kg m–3) 5170 8900 8720 a 0.9/C210/C041.8/C210/C024.0/C210/C03 Ms(kA m–1) 140 1000 760FIG. 4. Both ﬁgure of merit !and radiative linewidth CRare enhanced in the obli- que incidence geometry ( h>0/C14). Ferromagnetic linewidth CFMR remains unchanged. Co is assumed with a¼10/C03:Applied Physics Letters ERRATUM scitation.org/journal/apl Appl. Phys. Lett. 116, 209902 (2020); doi: 10.1063/5.0012734 116, 209902-2 Published under license by AIP Publishing
1.168409.pdf	Possible sources of coercivity in thin films of amorphous rare earth-transition metal alloys Roscoe Giles , and Masud Mansuripur Citation: Computers in Physics 5, 204 (1991); doi: 10.1063/1.168409 View online: https://doi.org/10.1063/1.168409 View Table of Contents: https://aip.scitation.org/toc/cip/5/2 Published by the American Institute of Physics ARTICLES YOU MAY BE INTERESTED IN Bessel Functions of Fractional Order Computers in Physics 5, 244 (1991); https://doi.org/10.1063/1.4822982 Abelian sandpiles Computers in Physics 5, 198 (1991); https://doi.org/10.1063/1.168408 Vortex motion in amorphous ferrimagnetic thin film elements AIP Advances 7, 056001 (2017); https://doi.org/10.1063/1.4973295 On the limits of coercivity in permanent magnets Applied Physics Letters 111, 072404 (2017); https://doi.org/10.1063/1.4999315 Computational Astrophysical Fluid DynamicsAdvances in Computer Hardware and Software have Created New Opportunities for Studying the “Fluid Cosmos” Computers in Physics 5, 138 (1991); https://doi.org/10.1063/1.4822976 Use of the Macintosh Oscilloscope in Undergraduate Science Education Computers in Physics 5, 220 (1991); https://doi.org/10.1063/1.4822980Possible sources of coercivity in thin films of amorphous rare earth-transition metal alloys Roscoe Giles College of Engineering, Boston University, Boston, Massachusetts 02215 Masud Mansuripur Optical Sciences Center, University of Arizona, Tucson, Arizona 85721 (Received 25 June 1990: accepted 27 November 1990) Computer simulations of a two-dimensional lattice of magnetic dipoles are performed on the Connection Machine. The lattice is a discrete model for thin films of amorphous rare earth- transition metal alloys, which have application as the storage media in erasable optical data storage systems. In these simulations the dipoles follow the dynamic equation of Landau- Lifshitz-Gilbert under the influence of an effective field arising from local anisotropy, near- neighbor exchange, classical dipole-dipole interactions, and an externally applied field. The effect of random axis anisotropy on the coercive field is studied and it is found that the fields required for the nucleation of reverse-magnetized domains are generally higher than those observed in the experiments. Various “defects” are then introduced in the magnetic state of the lattice and the values of coercivity corresponding to different types, sizes, and strengths of these “defects” are computed. It was found, for instance, that voids have insignificant effects on the value of the coercive field, but that reverse-magnetized seeds of nucleation, formed and stabilized in areas with large local anisotropy, can substantially reduce the coercivity. Magnetization reversal in thin films of amorphous rare earth-transition metal alloys is of considerable importance in erasable optical data storage.‘-‘* The success of thermo- magnetic recording and erasure depends on the reliable and repeatable reversal of magnetization in micron-size areas within the storage medium. A major factor entering the thermomagnetic process is the coercivity of the mag- netic medium and its temperature dependence. The pur- pose of this paper is to investigate coercivity at the submi- crometer scale using large scale computer simulations. There exists a substantial literature addressing the various aspects and mechanisms of coercivity in thin films; the in- terested reader may consult Refs. 13-26. Our computer simulations were performed on a two- dimensional hexagonal lattice of magnetic dipoles follow- ing the Landau-Lifshitz-Gilbert equation. In addition to interacting with an externally applied field, the dipoles were subject to effective fields arising from local uniaxial anisotropy, nearest neighbor exchange, and long range di- pole-dipole interactions. Details of the micromagnetic model have been previously published27-32 and will not be repeated here. Suffice it to say that the massive parallelism of the Connection Machine on which these simulations were performed, together with the fast Fourier transform algorithm,303’ which was used to compute the demagnetiz- ing fields, enabled us to accurately simulate a large (256 x 256) hexagonal lattice of dipoles. Since the lattice constant was chosen to be 10 A in these simulations, the total area of the lattice corresponds to a section of the mag- netic film with dimensions 0.256 X 0.222 pm. The reported results in this paper utilize a color cod- ing scheme for representing the state of magnetization. Since the magnitude of the magnetization vector m will be fixed throughout the lattice, the color sphere is used to rep- resent its local orientation. The color sphere is white at its north pole, black at its south pole, and covers the visible spectrum on its equator in the manner shown in Fig. 1. As one moves from the equator to the north pole on a great circle, the color pales, i.e., it mixes with increasing amounts of white, until it becomes white at the pole. Moving toward the south pole has the opposite effect as the color mixes with increasing amounts of black. Thus when the magneti- zation vector at a given site is perpendicular to the plane of the lattice and along the positive (negative) Z axis, its cor- responding pixel will be white (black). For m in the plane of the lattice the pixel is red when pointing along + X, light green along + Y, blue along - X, and purple along - Y. In the same manner, other orientations of m map onto the corresponding color on the color sphere. The organization of this paper is as follows. In Sec. I we describe the nucleation coercivity and the influence of random axis anisotropy and/or defects on nucleation. Sec- tion II is devoted to the structure and motion of domain walls, where we discuss certain aspects of wall coercivity. Also presented in this section are several results concerning demagnetization that involve wall motion. Concluding re- 204 COMPUTERS IN PHYSICS, MAR/APR 1991 number seed should disappear when the simulated lattice becomes sufficiently large. Frame (a) in Fig. 2 shows the initial phase of the re- versal process (i.e., the nucleation phase) for the basic sample with 0 = 45” under an applied field of Hext = 12.6 kOe, which happens to be just above the coercive field for the sample. The nucleation site is in the lower central part FtG. 1. The color circle shown in thi$ figure may be used to encode the direction of magnetiratton tn the planeofthe lattice. In this scheme a red pixel is associated with local magncttration direction along + X, light green corresponds to + Y, blue to - X. and purple to - 1’. When a vector is not completely in the plane of the lattice, but has a perpendicular component along + Z (or - Z), its associated color is obtained by mixing the color of its in-plane component with a certain amount of white(orbla~k),thestrengthofHhite(orblack)dependingonthemagnitudeofthe vertical component. A vector fully aligned with the + 2 direction is shown by a white pixel. while a vector in the - Z direction is displayed as black. (a) marks are the subject of Sec. III. A generalized version of the Stoner-Wohlfarth theory of magnetization reversal by coherent rotation is described in the Appendix. We com- pare the predictions of this model with some of the results obtained by computer simulation, and show the excellent agreement between them. I. NUCLEATION COERCIVITY, RANUOM AXIS ANISOTROPY, AND VARIOUS DEFECT MECHANISMS In order to gain an understanding of the possible sources of nucleation coercivity, we chose a lattice with the following set of parameters: saturation magnetization M, = 100 emu/cm”, anisotropy energy constant K, = 10h erg/cm’, exchange stiffness coefficient A, = lo-’ erg/cm, film (n) thickness h = 500 A, damping coefftcient a = 0.5, and gyromagnetic ratio y = - 10’ Hz/Oe. This set of param- eters shall be referred to as the set corresponding to the basic sample. The axes of anisotropy were distributed ran- domly and independently among the lattice cells in such a way as to keep their deviation from the Z axis below a certain maximum angle 0. In the following discussions 0 will be referred to as the cone angle. The first set of simulations concerned the relationship between thecoercive field H, and the cone angle 0. Hyster- esis loops were traced for several cone angles in the range of 2U-45”. The loops were always square (i.e., nucleation co- ercivity dominated the wall motion coercivity) and H, de- creased monotonically from 17 kOe at 0 = 20” to 12.5 kOe at 0 = 45”. H, for a given cone angle showed a slight de- pendence on the choice of seed for the random number generator. For instance, in the case of 0 = 45”, different seeds gave rise to coervicities between 12.5 and 12.80 kOe. (‘) Similar variations in the nucleation coercivity of real mate- FIG. 2. Early stage of nucleation for three samples with M, = IM)emu/cm’. rials can be expected, provided that small areas of these K, = IOb erg/cm’, h = 500 A, and cone angle 0 = 45’. The applied field in all cases films are subjected to the external field. Alternatively, the is 12.6 kOe. In frame (a) the sample has exchange stiffness coefiicient A, = 10-‘erg/cm. For the sample in frame (b) the exchange parameter iS dependence of the computed coercivity on the random A, =0.8X IO ‘, while frame (c) represents a sample with A, = 0.6~ IO-‘. COMPUTERS IN PHYSICS, MAR/APR 1991 205 1 OO I I I 25 50 7.5 100 125 t-l, (kOe) FIG. 3. Average components of magnetization along Xand Z versus magnitude of the in-plane applied field H,. The parameters of the lattice are those of the basic sample with 0 = 45’. and the system is relaxed to the steady state for each value of the applied field. of the frame and its periodic continuation, due to the boundary conditions and the hexagonal symmetry of the lattice, appears at the upper left corner of the frame. In order to understand the relative significance of ani- sotropy and exchange in the nucleation process, we varied the exchange parameter while keeping all other parameters (including the random number seed) fixed at their pre- vious values corresponding to frame (a). Frame (b) shows the early nucleation stage for a sample whose exchange parameter A, has been reduced to 80% of the original val- ue. Within the accuracy of calculations, the value of coer- civity for this sample was found to be the same as the origi- nal sample’s coercivity, although the nucleation site at the applied field of 12.6 kOe appears to be different. Further reduction of A, to 60% of its original value does not make any difference either [see frame (c)l; the coercivity re- mains the same and even the site of nucleation remains the same as in frame (b). From the above observations we con- clude that nucleation coercivity is controlled by the anisot- ropy field Hk = 2K,/M, as well as by the spread in the distribution of the axes of anisotropy. The nucleation site must have an “average” anisotropy field close to the ap- plied field (i.e., H,,, z2(K,)/M, - 4rrMs)), but the strength of exchange is not of primary significance in this respect. It must be emphasized at this point that K,, as used in our model, is different from the bulk anisotropy as mea- sured for a real magnetic film by, say, torque magneto- metry. Bulk anisotropy should be denoted by (K,), where brackets indicate spatial averaging, whereas K, itself rep- resents the strength of local anisotropy associated with each dipole. Thus when the cone angle 0 is increased while K, is being kept constant, the bulk anisotropy (K, ) should decrease. To clarify the distinction between K, and (K, ), we simulated an experiment in which the bulk anisotropy of the sample could be measured. In the experiment, one ap- plies an in-plane field, say along the X axis, and monitors the normal component of magnetization (M,) as a func- tion of the strength of the applied field. The value of bulk anisotropy (K, ) is then obtained from the curvature of the plot of (M, ) vs H, (Ref. 33 ). Figure 3 shows plots of (M, ) and (M, ) as functions of H,, obtained by simulation for the basic sample with cone angle 0 = 45”. For each value of H, the lattice was relaxed to the steady-state before (M,) and (M,) were computed. It is seen in this figure that for values of H, below 12 kOe the magnetic moments move coherently and reversibly toward the direction of the ap- plied field. Using either the slope of (M, ) or the curvature of (M,) it is rather straightforward to show that 2 (K, )/MT - 4rMs N 13 kOe, in agreement with our previous results concerning the nu- cleation field. Although not relevant to the present discussion, it is interesting to know what happens when H, in the preced- ing experiment is increased beyond the critical value of 12 kOe. At the critical field some moments flip over to the other side and create regions of reverse magnetization. Frame (a) in Fig. 4 shows the distribution of M, across the (a) FIG.4. DistributionofM, (a) andexchangeenergy (b) acrossthelatticein thesteadystate, undertheappliedin-plane field H, = I2 kOe.Thecolorcodingschemeherediffers from that used in all theotherfiguresofthepaper,and isapplicableonly whenascalar function (such as the local valueof M,across the lattice) is to bedisplayed. In thisscheme the color red is assigned to the minimum value of the function, while the color purple is used to represent the maximum value. All the other values are then mapped onto the color circle in a linear fashion, starting from red and moving counterclockwise to purple. 206 COMPUTERS IN PHYSICS, MAR/APR 1991 “E < i 5 5 .- ‘ij .g E 0” z -0.853 ';i P w -0 856 B -0.859 Iii 5 -0.862 I t I 0.3 1.0 1.5 Time (ns) FIG. 5. Average magnetization and energy during the relaxation process that leads to thesteady statein Fig. 4. (a) (M,) versus time. (b) (M,) versus time. (c) Total energy of the lattice versus time. The inset shows separate plots of exchange, anisot- ropy. and demagnetization energies. lattice in the steady state, under the applied field H, = 12 kOe. (Note: The color code used for Fig. 4 differs from the code described in the Introduction and used for all the oth- er figures in this paper. The caption to Fig. 4 describes this particular coloring scheme.) The blue regions in Fig. 4(a) have a small positive value of M,, while M, for the yellow regions is small and negative. The same coloring scheme is also used in frame (b) to show the distribution of exchange energy in the steady state. Plots of the exchange energy distribution emphasize domain walls and enhance regions with rapid spatial variation of magnetization. Behavior of the average magnetization and energy during the relaxa- tion process under the applied field H, = 12 kOe are shown in Fig. 5. The rapid drop in (M,) is due to the onset of demagnetization, which also causes a small drop in (M,). The reduction in energy is attributable to a lowered anisotropy energy, as is readily observed from the inset in Fig. 5(c). Going back to the subject of magnetization reversal under a perpendicularly applied field, it appears that the nucleation coercivity is always about 2(K,)/M,. On the other hand, one can make the following assumptions about a real sample: (i) The bulk of the material has very little dispersion in its local easy axes, and (ii) only a few isolated submicron-size regions have large values of 0. Under these circumstances (K, ) N K, while, at the same time, since nucleation takes place in regions of large dispersion, the resulting coercivity is significantly below 2 (K, )/MS. In contrast to these results, the experimentally ob- served values of coercivity for real samples (with param- eters similar to those of the basic sample) are only a few kilo Oersteds. Nucleation in real materials therefore can- not be attributed to random axis anisotropy alone. We be- lieve that the reveral process has its origins in what may be termed “defects,” be they structural or magnetic in nature. Results of simulation studies pertaining to several different types of defects are outlined in the following subsections. A. Defects of type 1 These defects are small regions of the sample from which the magnetic material has been removed and, as such, they may be characterized as voids. Figure 6 corresponds to a circular void with diameter D = 500 A in the basic sample. Random axis anisotropy with a cone angle of 0 = 45” has also been assumed throughout the lattice. The simulation results indicate that, within the accuracy of calculations, the value of coercivity has been unaffected by the void (i.e., H, = 12.6 kOe). Due to the long range dipole-dipole inter- actions, however, the nucleation site has moved from the (a) (b) FIG. 6, Nucleation in the basic sample with a defect of type I (void) at the center. (a) The state of magnetization of the lattice at an applied field of H,,, = 12.5 kOr, just below coercivity. (b) Nucleation and growth ofa reverse magnetized domain under an applied field of H,,, = 12.6 kOe. COMPUTERS IN PcIYStCS. MdRldPR ,001 907 lower central part of the lattice in the absence of the void [see Fig. 2, frame (a) ] to the lower left corner of the lattice in Fig. 6. It should be noted that the assumed defect does not influence the magnetic properties of the void boundary; in particular, the saturation magnetization MS, the anisotro- py constant K,, and the distribution of the anisotropy axes at the periphery of the void have been left intact. In reality, one expects the presence of the void to alter these param- eters, albeit to an extent which is not well understood at the present time. Thus, despite the above result, the possibility that real voids could act as nucleation centers should not be completely ruled out. B. Defects of type 2 The second type of defect is a small region with large ani- sotropy constant K, and with reverse magnetization. In simulation results displayed in Fig. 7 the basic sample had the same axes of anisotropy as in the preceding cases, but (al) (a21 (a31 (a41 FIG. 7. Nucleation in thebasicsamplewithadefect oftype2. (al ) Adefect withdiameter D = 2tXlA in theremanent state. (a2) Growthoftheiniti$domain under anapplied field of He., = 3.46 kOe. The state shown in this frame corresponds to f = I89 ps after the application of the field. (a3) Continued growth under Hex, = 3.46 kOe. The state shown here was obtained at = 500 ps. (a4) State of the lattice under H,., Growth of the initial domain under the applied field of H.,, = I .98 kOe. = 3.46 kOe at I = 797 ps. (bl) Defect of initial diameter D = 300 .& in the remanem state. (b2) 208 COMPUTERS IN PHYSICS, MAR/APR 1991 K, within the defective region at the center of the lattice was increased tenfold to 10’ erg/cm3. A defect with diame- ter D = 100 A was not stable in the remanent state and collapsed. A defect with 200 A diameter, however, was stable; the remanent pattern ofmagnetization in this case is shown in frame (a 1) . The required field for the expansion of this defect is only 3.45 kOe, which is substantially below the value of coercivity for the same sample without defect (12.57 kOe). Frames (a2), (a3), and (a4) in Fig. 7 show the growth of this nucleus under the applied field of 3.46 kOe. Similar results were obtained for a defect diameter of 300 A, as shown in frames (bl ) and (b2). The coercivity in this case was 1.98 kOe. When the defect diameter was in- creased to 500 A, the coercivity dropped to 1.28 kOe. These results clearly indicate that defects of type 2 can control the coercivity in a major way. The preceding numerical results are in good agree- ment with predictions based on a relatively simple theory. Consider a circular domain of radius r in a film of thick- ness h, saturation magnetization M,, and domain wall en- ergy density (T,,. Let an external field H,,, be applied per- pendicular to the plane of the film, favoring the direction of magnetization inside the domain. Assuming that O<r 5 h, the energy of the system (relative to the saturated state with no domains) is written E- - 2n?hkfyH,,, -I- 2rrrha, - n-(r + 1.5h)2h(2m14x2). (1) The approximation in Eq. ( 1) is caused by the last term, which corresponds to demagnetization. The implicit as- sumption here is that, upon the formation of the domain, the demagnetizing field in and around the domain within a radius of r + 1.5h vanishes. Of course, if the domain radius r is much less than the film thickness h, the above approxi- mation fails, because in that case the demagnetizing field cannot be reduced in a substantial way in locations that are as far away as r + 1.5h from the center. Similarly, when r happens to be much larger than h, the approximation fails once again because now the demagnetizing tield is not di- minished within the domain; only the annular region between r - 1.5h and r + 1.5h may now be assumed to have zero demagnetizing field. These are the reasons be- hind the restrictions imposed on Eq. ( 1) . When He,, is sufftciently small, the net pressure on the wall will be inwards and the domain tends to collapse (ex- cept that in the case of interest here the large value of K, within the domain opposes this tendency). At the onset of expansion, when He,, is large enough to begin to push the wall outwards, the net pressure is zero, that is, dE /dr = 0. One can readily derive the expression for the critical value of He,, as follows: Hex, = [ u’m - 2n-(r + l.5h)Ms2]/2rM,. (2) Although in subsequent discussions the numerical value of u, will be obtained from the formula uw = 4JX, (3) (a) (d) (4 FIG. 8. Nucleation and growth in the basic sample with isolated regions of large anisotropy. Six small areas ofdiameter 400 A within the lattice were assigned a value of K. which wastivetimesgreaterthan K, for therestofthelattice. Themagnetizationoftheentirelatticewas thensaturatedandrelaxed totheremanent state, Frame (a) shows the statrofthelatticeunder If,., = 12.5 kOe.just belowcoercivity. In frame (b) theapplied field is 12.6 kOeand there is nucleation. Frame (c) shows how thegrowth ofthe initial nucleus is hampered by three of the defects. The remaining frames follow the growth process in time and show the way in which the magnetization manages to reverse the high K, regions. COMPUTERS IN PHYSICS, MAR/APR 1991 209 it should be remembered that, due to random anisotropy and the presence of vertical Bloch lines, the actual value of LT, in our simulations is somewhat different from the value of 1.265 $rg/cm predicted by Eq. (3). For r = 100, 150, and 250 A, corresponding to the simulated defects of type 2, the calculated coercivities from Eq. (2) are 3.65, 2.33, and 1.27 kOe, respectively. Of course, regions with large K, are not necessarily reverse magnetized in every situation. Consider, for in- stance, the case of a completely saturated sample with six regularly spaced defects shown in Fig. 8. The defects are cylindrical regions of diameter D = 400 A and K, = 5 x IO6 erg/cm3. Otherwise, the lattice has parameters of the basic sample with cone angle 0 = 45”. Frame (a) shows the state of the lattice under an applied field of Hz = 12.5 kOe, which is slightly below the coercivity for the sample. Frames (b)-(f) show the nucleation and growth of a reverse domain under the applied field of Hz = 12.6 kOe. Although the defects act as temporary barriers to the growing nucleus, the walls eventually sweep through the entire sample. At the end, the magnetization of the sample is fully saturated in the reverse direction, and defects of type 2 (which could have formed around the regions of high Ku ) do not materialize. In contrast to the preceding results, Fig. 9 shows a case where defects of type 2 with either polarity can be stable. In this case Jhere are seven cylindrical regions of diameter D = 200 A and K, = 10’ erg/cm3. The central region is initially reverse magnetized and thus constitutes a defect of type 2. The rest of the sample is saturated along + 2 and then relaxed to the remanent state, as shown in frame (a). Under an external field Hz = - 3.5 kOe (just above coercivity), the central nucleus expands and covers the rest of the sample with the exception of the high K, regions. Frames (b)-(f) follow the growth process in time under the applied field. The six unreversed regions in frame (f) may now act as defects of type 2 for future reversals. Finally, one must recognize that defects of type 2 are inherently unstable and could be eliminated by applying sufficiently large magnetic fields. The required field for destroying a particular defect, of course, depends on its size and on the strength of its anisotropy. In reality, if coercivity is controlled by this type of defect, then one expects to find a dependence of H, on the history of saturation and, in particular, on the value of the largest field applied to satu- rate the sample. Such dependencies have indeed been ob- served in practice for some RE-TM thin film samples.33 C. Defects of type 3 Here, we assumed that the anisotropy constant K, within the central region of the sample is only half the value of K, elsewhere. All other parameters were the same as in the previous cases. The entire sample (including the defect) was initially magnetized along + Z and the system was allowed to relax and settle down into the remanent state. The various frames in Fig. 10 correspond to defects of dif- ferent sizes and show the state of magnetization early on in the process of reversal, under an applied field which is only slightly above the computed coercive field. Frames (a)- (a) (e) FIG. 9. Growth from a defect of the second type in the basic sample containing seven isolated regions of large anisotropy. Each region has diameter D = 200 w and Ku = IO’ erg/cm’. Thecentraldefect was initially reverse magnetizedand thereforeconstitutesadefect oftype2. The rest ofthe lattice wassaturatedalong + Zand relaxed to the remanent state, as shown in frame (a). Frame (b) shows the state of the lattice under a reverse field of 3.5 kOe, which is only slightly above the coercivity for this sample. The remaining frames (c)-(f) follow the growth process in time and show how the magnetization fails to reverse in high K, areas. The unreversed regions now become defects of the second type for future reversals. 210 COMPUTERS IN PHYSICS, MAR/APR 1991 (b) (c) (d) FIG. IO. Nucleation in thebasicsample withdefectsoftype3 at thecenterofthe lattice. Thevalueof K, within thedefect isonly halfitsvalueelsewhere. (a) Defect ofdiameter ZOO,& under an applied tield of 12.64 kQr. (b) Defect ofdiameter 600 .&subjected to theapplied field of I I .75 kOe. (c) Thedefect diameter is 800 .&and theexternal field is 9.4 kOe. (d) The defect diameter is ICMXI A and the applied field is 8.7 kOe. (d) correspoond to defect diameters of D = 200, 600, 800, and 1000 A, respectively. The corresponding coercive fields for these samples were computed a,s H, = 12.64, 11.75,9.4, and 8.7 kOe. Except for the 200 A defect which does not help much in reducing coercivity [although one of the initial nuclei in frame (a) is centered on this defect], the other defects have an appreciable effect on the value of L?, and nucleation always begins at the defect. 0. Defects of type 4 In this type of defect the axes of anisotropy within the de- fective region are uniformly tilted away from the normal by a fixed angle. For several defects of this type the various frames of Fig. 11 show the states of the lattice both before and after nucleation. Except for the directions of local easy axes within the defects, all other parameters in these simu- lations were the same as before. Frames (al) and ($2) in Fig. 11 correspond to a defect diameter of D = 1000 A and a uniform tilt angle of lo” from normal within the defect. In (al) the applied field is 12.32 kOe, which is just below coercivity, whereas in (a2) the applied field is 12.34 kOe. Compared to the basic sample with no defects, the coercivi- ty has dropped only slightly, but the nucleation site is now on the boundary of the defect. Frames (bl ) and (b2) cor- respond to a similar defect with a tilt angle of 20”. The coercive field in this case has dr?pped to 10.45 kOe. For a smaller defect of diameter 400 A and 20” tilt angle, shown in frames (cl) and (c2), the coercivity was about 11.33 kOe. Apparently, in order to affect coercivity significantly, a defect of type 4 must be relatively large and have a sub- stantial tilt angle. The Stoner-Wohlfarth theory of magnetization rever- sal by coherent rotation” is applicable to this type of defect provided that the defect is not too small. A generalized version of this theory which includes the effects of demag- netization is described in the Appendix. It is shown in the Appendix that one of the preceding simulation results con- cerning a 1000 A defect of type 4 with tilt angle of 20” is in good agreement with the theory. II. STRUCTURE AND MOTION OF DOMAIN WALLS IN THE PRESENCE OF EXTERNAL AND/OR DEMAGNETIZING FIELDS Having studied the process of nucleation in some detail, we now turn to the subject of domain wall structure and its associated coercivity. Figure 12 shows the structure of do- COMPUTERS IN PHYSICS, MAR/APR 1991 211 (all (a21 I t” m:i:L+ ::.I., . . _‘YL ..;; -;>+--,‘+ / ._. L,‘: TV.-.. :. ‘, ?_._ ;.+ . _ .b.;Y-A. -;> - , ,: / b _,: (,“. ;.J.“. : :_ _, ,,i,.,” ,A., ,,a : 11 b I :, -“p <;. li -. -, j** i .I-. ,,-, (bl) (b2) (c2) FIG. I I. Nucleation in the basic sample with defectsoftype 4. (al) Defect with diameter of lKKl.& and anisotropyaxis tilt of lo”, subject to an external field of 12.32 kOe. The defect is visible as the orange colored region in the center ofthe lattice. Although a red spot near the left boundary has formed at this stage, the applied field is not strong enough to reverse the magnetization of the sample. (a2) Same as (al) but with an applied field of 12.34 kOe. The state shown in this frame is a snap shot of the reversal process. The nucleated domain continues to grow until the entire sample is reversed. (bl ) Defect with diameter of loo0 .& and anisotropy axis tilt of 20’. subject to an applied field of 10.44 kOe. (b2) Same as (bl) but with an applied field of 10.46 kOe. This is a snap shot of the reversal process. (cl) Defect with diameter of400 A and anisotropy axis tilt of 20”, subject to an applied field of II.32 kOe. (~2) Same as (cl) but with an applied field of II.34 kOe. Again. this is a snap shot of the reversal process. main walls in a medium with random axis anisotropy (cone in frame (b) was obtained. Notice that there are three ver- angle 0 = 45”) and with the same parameters as the basic tical Bloch lines (2~ VBLs) in each wall and that the walls sample. Initially the central band of the lattice was magne- tized in the + Z direction while the remaining part was are no longer straight. By allowing the lattice to relax for magnetized in the - Z direction, as shown in frame (a). another 0.9 ns we obtain the pattern of frame (c), which When the lattice was allowed to relax for 0.8 ns, the pattern shows significant VBL movements along the walls. Finally, frame (d) shows the steady-state situation at t = 4.56 ns. 212 COMPUTERS IN PHYSICS, MAR/APR 1991 (a) (b) (d) FIG, 12. Formationofdomain wallsin thrbasic sample with aconeangleof45”andin theabsrnceofan applied field. (a) Dipolesin the white regionareinitializedalong + Z, whilcdipolesin thedark region areinitializedalong - Z. (b) Thebtateofthelatticeat f = 0.8 ns. Each wallcontains threevertical Bloch linesat thisstage. (c) Thestateofthe lattice at f = I .7 ns. The number of VBLs has not changed since the previous frame, but they have moved along the walls. (d) The steady state of the lattice at f = 4.56 ns. The number of VBLs in each wall is still 3. Both walls are now straightened considerably, but the number of VBLs in each wall has not changed; no amount of relaxation can unwind a 2rr Bloch line. The curves in Fig. 13 show average magnetization (M,) and total energy E,,,, of the system during the relaxa- tion process which was depicted in the previous figure. The inset in Fig. 13 (b) shows the various components of ener- gy. Obviously, the demagnetization energy does not change much during the process of wall formation. This result should be expected since, in this particular example, tilm thickness h is several times greater than the wall thick- ness. On the other hand, anisotropy energy drops sharply in the early phase as the moments throughout the lattice move closer to the local easy axes. In fact, this reduction is large enough to overwhelm the modest increase in the ani- sotropy energy at the walls. For the same reasons the ex- change energy of the entire system rises, albeit very slight- ly, despite a sharp reduction of the exchange energy at the walls. A perpendicular field Hz = - 200 Oe moves the two walls in Fig. 12 (d) somewhat closer to each other, but fails to eliminate the stripe of reverse magnetization. The steady state of the lattice under this applied field is shown in Fig. 14. The corresponding curves of (44,) and E,,, in Fig. 15 0 y -1 \” -2 z cu -3 3 -4 ‘ji 1.2 F W e I.1 z a 1.0 0-J b 5 0.9 0 2 3 Time (ns) 4 FIG. 13. Plots of average magnetization and energy in the process of domain wall formation corrresponding to Fig. 12. (a) CM,) versus time. (b) Total energy of the lattice versus time. The imet shows the evolution of exchange, amsotropy, and de- magnetization energies during the initial phase of the process. COMPUTERS IN PHYSICS. MAR/APR 1991 213 FIG. 14. Steady state of the lattice (shown here at I = 3.66 ns) when the stripe domain of Fig. 12(d) is subjected to an external field H: = - 2M) Oe. indicate that the time needed to arrive at the steady state is about 2 ns. In this experiment, the force ofdemagnetization opposes the external field in collapsing the reverse-magne- tized stripe. The stripe domain shown in Fig. 12(d) will collapse under the applied field of HZ = - 1000 Oe, as shown in Fig. 16. Frames (a) and (b) in this figure correspond to f = 0.96 ns and t = 3.58 ns, respectively. The curves of W, > and -%, in Fig. 17 show the rate of reduction of the average magnetization and energy during this collapse pro- cess. In the remaining investigations we used a different set of parameters for the lattice. These parameters were: M, = 175 emu/cm3, K, = 0.5 x lo6 erg/cm3, A, = 0.5 X lo-’ erg/cm, and cone angle 0 = 20“. In one sim- ulation experiment we initialized the state of magnetiza- tion randomly, with each dipole moment being equally likely to be set either parallel or antiparallel to the Z axis. After about 900 steps (corresponding to 5 ps) in which the state of the lattice was relaxed following the LLG equation, (a) 55 P 0.868 W g 0.866 c - 0.864 s is 0.862 15 0.860 L-A E tot b) I 3 0 2 Time (ns) FIG. 15. Plots ofaverage magnetization and energy when the stripe domain of Fig. 12(d) shrinks under an external field Hz = - 200 Oe. 4 the system arrived at the state shown in frame (a) of Fig. 18. Small domains had clearly formed at this stage, but the system was far from equilibrium. Twenty-thousand itera- tions and 1.2 ns later, the system arrived at the equilibrium state shown in frame (b). The final state is demagnetized with stripe domains containing several vertical Bloch lines in their walls. This experiment is similar to rapid cooling of a real sample in zero field from above the Curie point to the room temperature. In another simulation experiment we applied a reverse external field of 3.16 kOe (just above coercivity ) to initiate the reversal. Once the nuclei had formed, the field was re- duced to zero and the domains were left to themselves to FIG. 16. Collapseofthestripedomain ofFig. 12(d) under an external field H, = - ICkXOe. Frames (a) and (b) show thestateofthe latticeat f = 0.96 ns and t = 3.58 ns, respectively. 214 COMPUTERS IN PHYSICS, MAR/APR IWl 6 0.7- $ 5 (W 0.6 I , I 0 I 2 3 4 Time (ns) FIG. 17. Plots ofaverage magnetization and energy when the stripe domain of Fig. 12(d) collapses under an external field H, = - loo0 Oe. develop under the pressure of the wall energy and the de- magnetizing force. The various frames of Fig. 19 show sev- eral states of this development. In frame (a) the field has just been turned off, leaving behind three nuclei that are clearly visible in the picture. Since the force of demagnet- ization for this sample is larger than that of the wall energy, the nuclei expand and eventually cause the sample to de- magnetize. First the two nuclei in the lower part of the frame merge; then the remaining domains expand as shown in frames (b) and (c). Soon, however, the larger domain begins to push the smaller one toward collapse, as shown in frames (d) and (e). Eventually, the small domain disap- pears and the lattice reaches equilibrium as shown in frame (f). Figure 20(a) shows the average lattice magnetization (M,) versus time for this experiment. The initial sharp drop in (il4,) occurs when the early nuclei merge and ex- pand. The plateau corresponds to the time during which one dcmain expands at the expense of the other. At the end of the plateau, the sudden collapse of the small domain (similar to a bursting bubble) causes a rapid drop in (M, ). Soon afterwards the magnetization reaches an equilibrium value near zero, and the lattice begins to stabilize. The plot of energy versus time in Fig. 20(b) shows a similar behav- ior. The inset in Fig. 20 shows the various contributions to energy, namely, the energies due to exchange, anisotropy, and demagnetization. Note how the burst of the small bub- ble, at around t = 23 ns, causes the demagnetizing energy to rise, while at the same time both exchange and anisotro- py energies (which are associated with domain walls) drop. Ill. CONCLUOING REMARKS Several hypothetical mechanisms of coercivity in thin films of amorphous rare earth-transition metal alloys were ex- amined in this paper. Using computer simulations, we found that regions as small as a few hundred angstroms in diameter with unusually large or small magnetic param- eters could act as nucleation centers and initiate the rever- sal process. Values of the coercive field obtained by simula- tion are comparable to those observed in practice. Whether or not these hypothetical sources exist in real materials is a question whose answer must await further progress in ex- perimental “nanomagnetics.” Among the existing tools for observation of the magnetic state in thin films, Lorentz electron microscopy34 and magnetic force microscopy3’ have the potential to clarify the situation in the near future. (a) (b) FIG. 18. Relaxation of the lattice starting from a random initial state and in the absence of an applied field. The parameters for this simulation are: M, = 175 emu/cm’, K, =O.SXlbrrg/cm’.A, =0.5X10 ’ erg/cm, h = 5CXl A, a = 0.5, y = - IO’ Hz/Oe, and cone angle 0 = 20”. (a) The state of the lattice at I = 5.06 ps. (b) The state of the lattice at I = 1.2 ns. COMPUTERS IN PHYSICS, MAR/APR 1991 215 (a) (d) (e) FIG. 19. Demagetization in theabsenceofan applied field, forasample with thesame set ofparametersas in Fig, 18. Thesampleis initially saturated, then briefly exposed toan external field of 3.16 kOe in order to create several small nuclei. The field is then turned off and the domains allowed to evolve under internal forces. (a) The state ofthe lattice immediately after the external field has been turned off. (b) The situation at I = I ns. The two nuclei in the lower pan of frame (a) have merged. (c) The stateofmagnetization ofthelatticeat f = 5 ns. Thedomainin thecenteroftheframehasnow reacheditsmaximumsizeand, from nowon, it willshrink. (d) At t = 20”s. Thesmalldomaininthecen- ter is shrinking, while the big domain continues to expand. (e) At t = 24 ns the small central domain is about to burst. (f) The tinal state. The domain is now steady and the net magnetization of the lattice is close to zero. 0.65 0.60 o.!Jo- o.!Jo- I I '!. ‘!. 8 8 I I 0.40 -i 0.40 -i 0.30 - i. 0.30 - i. E E dmag dmag ,.>. . ,.>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . _ . . . . . _ / / 0.20 - 0.20 - ‘Q- . . . . . . . . . . . .“...---C’ ‘Q- . . . . . . . . . . . .“...---C’ /._.-. -.----.,, /._.-. -.----.,, E E ani, ani, L -.-.-.-.-.-._.. L -.-.-.-.-.-._.. ,* ---------- -- ,* ---------- -- I I I IO 20 30 1 Time (ns) FIG. 20. Plots of average magnetization and energy during the relaxation process described in Fig. 19. (a) (M,) versus time. (b) Averageenergy ofthe latticeand its various components versus time. ACKNOWLEDGMENTS This work has been made possible by grants from the IBM Corporation and, in part, by support from the Optical Data Storage Center at the University of Arizona. APPENDIX The theory of magnetization reversal by coherent rotation was developed by Stoner and Wohlfarth in the context of elongated fine particles.“’ Their theory has since been adapted and applied to reversal in thin films.‘7-“9 In this appendix we generalize the Stoner-Wohlfarth theory to ac- count for the demagnetizing effects in thin films. The re- sults will then be applied to the basic sample with defects of type 4 (see Sec. I) in order to determine the dependence of coercivity on the tilt angle. Consider a uniform film with magnetization M, and anisotropy energy constant K,, as shown in Fig. 21. The axis of anisotropy makes angle 0, with the Z axis, and assuming that the magnetization processes are coherent, we denote by 0, the angle between the magnetization vec- tor and Z. The applied field H,,, is also uniform and its angle with Z is denoted by 0,. All angles are to be mea- sured clockwise from the positive Z axis, as indicated in the figure. The question we are about to address is the follow- ing: For a fixed set of values of K, ,O, ,M,, and O,, how does the angle 0, vary with the magnitude of the applied field H,,,? In particular, what is the equilibrium value of 216 COMPUTERS IN PHYSICS, MAR/APR 1991 c between 0 and n: Note that both O,,f and H,, are con- stants, depending only on the internal parameters of the / /I / film M,, K,, and O,d. Later, we will show that in the ab- sence of an external field, the equilibrium orientation of ( \+lsfhq 44 t\T M L Direction of Unioxial Anisotropy magnetization is along the direction of this effective field. Using the above definitions for H,.,, and Oe,Y, one can rewrite the expression for energy in Eq. (A2) as E,, = $(K, + 2&f) - MJI,,, cos(O,,, - O,,) - p4,Hefy cos[2(0, - O,,)]. (A51 FIG. ?I. Cross-sectional view ofa thin magnetic film with uniaxial anisotropy and uniform magnrtizatmn under an externally applied field Hk>, All angles are mea- sured cloclwiw from the positive 2 axis. M, is the saturation magnetization of the film and its angle with Z is denoted by 0,. The angle of the applied field with Z is O.,whilethranglebeturm Zand theaxisofanisotropyisO,.Thefield, themagnr- tization. the axis of anisotropy, and the Z axis are coplanar, Since Heff and O,, are constants, independent of the mag- nitude and the direction of the applied field, one defines the relative values of O,, O,, and H,,, as follows: 6, = 0, - Oer, (‘46) 6, = 0, - Oeff, (A7) ii,,, = Hex, /Herr- (A81 In terms of these relative parameters, Eq. (A5) is now written O,w when He,, = 0, and how does 0, change as He,, in- E, = q(K, + 2rrMf) - JM,H,,[2ji,,, co&, - &,) creases from zero to infinity in the fixed direction given by Cl,? + 1 cos(2$,)]. (A9) To answer the above question, we consider the mag- netic energy E,, of the system consisting of the external The proklem is now reducedLo finding the equilibri- field energy, the demagnetizing energy, and the anisotropy u,m value of 0, as a function of He,, for a fixed valuz of energy, as follows: 0,. To this end, we differentiate E,, with respect to a,+,, and set the derivative equal to zero in order to find the E,,, = - M,<H,,, cos(0, - 0,) -I- 2rrMf cos” 0, minima and maxima of the energy function. We find + K, sin’(O, - 0, ). (Al) The second and third terms in Eq. (A 1) can be combined to yield, 4, = - WH,,, cost@,,, - 0,) + i(K, + h-M:) - ,/(K,/2)’ - mkffK, cos(20,) + (7~kff)~ K,, sin( 20, ) ’ K, cos(20,) - 27i~; (A21 Next we define an effective internal field Hen and its asso- ciated angle Oeff as follows: He, = @L/M, 1” - 16~K, cos(20, ) + (4mvs f, (A3) O,, = +- tan- ’ ( 2K, /MS ) cos ( 20, ) - ~?TM, (A4) (2K,/M,)sin(20,) . In evaluating Oeff from Eq. (A4) it is imperative that one take into consideration the signs of both the numerator and the denominator of the arctangent’s argument. The value thus obtained for the arctangent should be somewhere in the interval between 0 and 277, resulting in a value of CD,, aE, T=$ikf5H,,[2@e,, sin(6, - &I, 6,) + sin(26,)]. C-410) Aside from an irrelevant constant coefticient, the right- hand side of Eq. (AlO) contains the following sinusoidal functions: F(GM) = 2ii,,, sin(6, - 6$,), (All) G(G,) = - sin(26,). t.412) Figure 22 shows plots of these functions with 6, arbitrar- ily set to 45”, with several values of&,, choFn from 0 to 1 in s:eps of 0.1. Fzr given values of H,,, and O,, the curves F( 0, ) and G( 0, ) cross in at most four points, at which points the derivative of E, is zero. To determine those crossing points that correspond to actual minima of ener- gy, we note in Fig. 22 that as one moves from the left to the right of a crossing point corresponding to a minimum, the slope of E,,, , which is proportional to F( 0) - G( 0)) goes from a negative value to zero, then to a positive value. In other words, before the crossing point F( 0) must be less than G(O), whereas after the crossing point F( 0) must be greater than G( 0). Those crossing points that satisfy this criterion,are marked with a small circle in Fig. 22. At HeXf = 0 there are always two stable values for 6,) namely, 0 and 180”, corresponding to 0, = O,, and I-nUDII~CmC I” DYYClre ..AD,IDD ,oo. 0.7 -2’ 1 b--./I I 0 50 100 150 200 250 300 350 6, (degrees) FIG. 22. Plots of the functions F(6,) and_G(G,) defined in Eqs. (All) and (Al2). Thevar$usF(O,) sho_wn herehaveOX, = 45’and H,,, = Oto I instepsof 0. I. The points 0, at which F(0, ) crosses G(O, ) from below correspond to mini- maofenergy Em. These crossingpointsare identifiedon the figure with small circles 0. 8M = O,, + 180”. For the situation depicted in Fig. 22, 0, = 45”, that is, 0, = O,, + 45”. Now, if the system happens to be in the stable stat: with 0, = O,, when the applied field is zero, then, as He,, incKeases, the crossing Roint m^oves toward larger values of 9!M until it reaches 0, = 0, = 45” for infinitely large H,,,. On th,e other hand, if zriginally 0, = O,, + 180”, then, as He,* in- Fesses, 0, decreases until it reaches aAcritical value of 0, = 135” at the critical field value of H, = 0.5. At the critical point, the minimum state of energy in which the system has bgen residing becomes a saddle point. Further increases in He,, eliminate this minimum, forcing the sys- tem to jump to the only remaining stataof minimum energy which, &-t the case of Fig. 22, is at 0, = 15”. Aftekthe jump, 0, increases continyusly with increasing He,,, asymptotically approaching 0, = 45”. h Qualitatively, the behavior just described for the case of 0, = 45” applies to all othzr values of 0, as well, hut the values of the critical field H, azd the critical angle 0, will depend on the exact value of CD,, of course. To deter- mine these critical parameters one notes that at the critical point the two curves F(a) and G(O) become tangent to each other, that is, F(6,) = G(&), F’(&) = G’(&.). Solving these equations, one obtains (A13a) (A13b) tan 6, = - (tan gH)‘13, (A141 i& = - cos3 &Jcos 6,. (A151 Now, assuming that the equilibriim state in the ab- sence of the external field occurs at 0, = 0,Jhere exist only two pos$bilities. In the first instance OgOHA<90”, in which case 0, increases continuously toward 0, with increasing H,,, ; no critical fields will be reached in this case and no disco$inuous jumps will oc%ur. In the second in- stance 90”<0, < 180”. In lhis c%e 0, jnitially increases with &,, until it reaches 0, at He,, = Hc. At the critical field 0, jumps to the other side and suddenly becomes greater than QH. The process then resumes its/\continuous nature, with O,M asymptotically approaching 0,. As an example, consider the following set of param- eters corresponding to a defect of type 4 studied in Sec. I: &U = 10” erg/cm3, 0, = 20”, M, = 100 emu/cm3, and 0, = 180”. From Eq. (A3) we find He%= 19.055 kOe and, from Eq,(A4), O,, = 21.215”. Thus 0, = 158.785”, resulting in 0, = 36.11” and H, = 0.566. The critical (i.e., switcking) field is thus given by He,, = He, X H, = 10.78 kOe, in good agreement with the value of 10.450k0e which was arrived at numerically in Sec. I for a 1000 A defect. Figure 23 shows several hysteresis loops for a thin film sample with K, = 10” erg/cm3 and M, = 100 emu/cm3. ‘O - (a) (b) C 05 0 -05 -10 ;:;::,:,-3, -20 -10 0 IO 20 -20 -10 0 IO 20 -20 -10 0 IO 20 FIG. 23. Calculated hysteresis loops for a thin film sample according to the Stoner-Wohlfarth theory, in- cluding the effects of demagnetization. The external field is parallel to Z, the tilm parameters are M, = 100emu/cm’ and K, = lO”erg/cm’, and the loops in (a)-(f) correspond to 0, = 0”. 2V, 45’. 70’, 85’, and 9(P, respectively. 218 COMPUTERS IN PHYSICS, MAR/APR 1991 The external field is assumed to be along the Z axis, that is 0, = 0” or 180”. The values of 0, corresponding to differ- ent loops in Fig. 23 are O”, 20”, 45”, 70”, 85”, and 90”. When 0, = 0” we tind from Eq. (A4) that Oeff = 0” provided that K, > 2rM:, which happens to be the case here. We also find He, = 2K,/M, - 4z-Ms = 18.744 kqe from Eq. kA3).FromEqs. (A14) and (A15) onefinds@, =O”and H, = 1, leading to a perfectly square loop with a coercivity of 18.744 kOe, as shown in the figure. The lowest value of coercivity is around 10 kOe, and is reached when 0, N 45”. The loop at 0, = 85” has a curious shape: The jump in 0, has caused a drop (rather than an increase) in the Z-com- ponent of magnetization. Finally, for 0, = 90” we have Oefl. = 90” and H,, = 2K, /MS + 477M, = 2 1.256 kOe. In this case there are no jumps but there is a discontinuity of slope at He,, = Hen, where the magnetization comes into alignment with the direction of the applied field. REFERENCES 1. P. Hansen and H. Heitmann, IEEE Trans. Magnet. 25.4390 (1989). 2. P. Chaudhari, J. J. Cuomo, and R. J. Gambino, Appl. Phys. Lett. 22, 337 (1973). 3. R. J. Gambino, P. Chaudhari, and J. J. Cuomo, AIP Conf. Proc. 18 (I), 578- 592 (1973). 4. T. Chen. D. Cheng, and G. B. Charlan, IEEE Trans. Magnet. 16,1194 (1980). 5. Y. Mimura, N. Imamura, and T. Kobayashi, IEEE Trans. Magnet. 12, 779 (1976). 6. Y. Mimura, N. Imamura, T. Kobayashi, A. Okada, and Y. Kushiro, J. Appl. Phys. 49, 1208 (1978). 7. F. E. Luborsky, J. Appl. Phyr. 57, 3592 (1985). 8. H. Tsujimoto, M. Shouji, A. Saito, S. Matsushita, and Y. Sakurai, J. Magnet. Magnet. Mat. 35. 199 (1983). 9. G. A. N. Connell. R. Allen, and M. Mansuripur, J. Appl. Phys. 53,7759 ( 1982). IO. M. Umer-Wille. P. Hansen, and K. Wit&, IEEE Trans. Magnet. 16, 1188 (1980). Il. T. C. Anthony, J. Burg, S. Naberhuis, and H. Birecki, J. Appl. Phys. 59. 213 (1986). 12. Y. Sakurai and K. Onishi. J. Magnet. Magnet. Mat. 35, 183 (1983). 13. R. Harris, M. Plischke, and M. J. Zuckermann, Phys. Rev. Lett. 31, 160 ( 1973). 14. R. Harris, S. H. Sung, and M. J. Zuckermann, IEEE Trans. Magnet. 14. 725 (1978). 15. R. Friedberg and D. I. Paul, Phys. Rev. Lett. 34, 1234 (1975). 16. D. I. Paul, Phys. Lett. A 64,485 (1978). 17. D. 1. Paul, J. Appl. Phys. 53, 2362 ( 1982). 18. B. K. Middelton, “Magnetic Thin Films and Devices,” m Acriue and Passive Thin Film Deoices. edited by T. J. Coutts (Academic, New York, 1978), Chap. Il. 19. A. Sukiennicki and E. Della Terre, J. Appl. Phys. 55, 3739 ( 1984). 20. K. Ohashi, H. Tcuji, S. Tsunashima. and S. Uchiyama, Jpn. J. Appl. Phys. 19, 1333 (1980). 21. K. Ohashi, H. Takagi, S. Tsunashima, S. Uchiyama, and T. Fujii, J. Appl. Phys. 50, 1611 (1979). 22. M. C. Chi and R. Alben, J. Appl. Phys. 48,2987 ( 1977). 23. J. M. D. Coey, J. Appl. Phys. 49, 1646 (1978). 24. J. M. D. Coey and D. H. Ryan, IEEE Trans. Magnet. 20, 1278 ( 1984). 25. E. Callen, Y. J. Liu, and J. R. Cullen, Phys. Rev. B 16, 263 ( 1977). 26. R. C. O’Handley, J. Appl. Phys. 62, RI5 (1987). 27. M. Mansuripur and R. Giles, Comput. Phys. 4,291 ( 1990). 28. M. Mansuripur, J. Appl. Phys. 63,5809 (1988). 29. M. Mansuripur and T. W. McDaniel, J. Appl. Phyb. 63,383l (1988). 30. M. Mansuripur and R. Giles, IEEE Trans. Magnet. 24,2326 (1988). 31. M. Mansuripur, J. Appl. Phys. 66.3731 (1989). 32. M. Mansuripur and M. F. Ruane, IEEE Trans. Magnet. 22, 33 ( 1986). 33. P. Wolniansky, S. Chase, R. Rosenvold, M. Ruane, and M. Mansuripur, J. Appl. Phys. 60,346 (1986). 34. C. J. Lin and D. Rugar, IEEE Trans. Magnet. 24,231 I ( 1988). 35. D. Rugar, H. J. Mamin, and P. Guthner, Appl. Phys. Lett. 55, 2588 (1989). 36. E. C. Stonerand E. P. Wohlfarth, Phil. Trans. R. Sot. A 240,599 (1948). 37. D. 0. Smith, J. Appl. Phys. 29,264 (1958). 38. E. M. Bradley and M. Prutton, J. Electron. Control 6, 81 (1959) 39. S. Middelhoek, Ph.D. thesis, University of Amsterdam, 1961. COMPUTERS IN PHYSICS, MARlAPR 1991 219
1.1722262.pdf	On the Minimum of Magnetization Reversal Time Ryoichi Kikuchi Citation: J. Appl. Phys. 27, 1352 (1956); doi: 10.1063/1.1722262 View online: http://dx.doi.org/10.1063/1.1722262 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v27/i11 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 10 May 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsJOURNAL OF APPLIED PHYSICS VOLUME 27, NUMBER 11 NOVEMBER, 1956 On the Minimum of Magnetization Reversal Time R YOICHI KIKUCHI* Armour Research Foundation of Illinois Institute of Technology, Chicago, Illinois (Received March 30,1956; revised manuscript received July 25, 1956) . A modifi~d Landau-Lifshitz,equation is solved for a single-domain sphere and an infinitely-wide thin smgle-domam sheet of ferromagnetic material neglecting anisotropy. The external magnetic field is switched from o~e. direc~ion to its opposite i~stantaneously at the initial time and the behavior of the magnetization ~ector IS mv~s~lgated thereafter. It IS sh?~n that there is a critical value of the damping constant correspond mg to the ~1~lmum value of the (repetl~IVe) magnetization reversal time. If the damping constant is larger than the crItical value, the magnetIzatIOn vector moves slower; if it is smaller, the magnetization vector m?~es faster but oscillates so that it takes longer time until it comes to a rest at the final position. The CrItIcal values of the Landau-Lifshitz damping constant A are yJ1 for the sphere and 0.013yM for the thin sheet, w~ere. 'Y a~d M are the ~yromagnetic ratio and the magnetization, respectively. The computed mini mum sWltchmg time for the thm sheet of 4-79 molybdenum Permalloy is of the order of 10-9 sec. I. INTRODUCTION CORE loss is known to be an important factor in determining the useful upper frequency limit or repetition rate of ferromagnetic core devices. Until recently, losses in available materials were so large that these upper limits were almost entirely determined by . the core loss, and any decrease in core loss could be expected to bring about an improvement in perform ance. Recent advances in the technology of ferro magnetic materials, (e.g., the development of low-loss ferrite materials and ultrathin low-loss magnetic tapes) gives reason to consider the question, "What is the limit to the performance which can be realized by reducing core losses?" This question about the existence of any performance limit for the repetition rate of pulse-operated ferro magnetic devices (e.g., magnetic storage register units) has never been seriously considered. It is the purpose of this paper to present arguments to support the con tention that: (1) there is a limit to the repetition rate which can be achieved, and (2) that the repetition rate (minimum remagnetization time) occurs for a particular optimum value of the core loss, so that any decrease or increase of the core loss from this optimum value in creases the remagnetization time. These results are speculative in that they are based on a particular form of the damped gyro magnetic equation, viz., dM/dt='YMX[~- (a/YM)dM/dtJ, (1) where M is the magnetization vector, ~ the total effective field (the external magnetic field, demagnet izing field and the field due to the eddy current), a a phenomenological damping constant, and l' the gyromagnetic ratio. Equation (1) reduces to the com monly used Landau-Lifshitz equation: dM/dt='YMX~- (A/M2)[MX (MX~)], (2) ---- * Present address: Department of Physics, Wayne State University, Detroit, Michigan. when a2«1 if we set a= A/yM. (This can be established by substituting the entire right side of Eq. (1) for the term dM/ dt in the brackets, using the triple vector product identity, and dropping terms of order a2 or higher.) When a2> 1, Eqs. (1) and (2) differ in a way which is essential to the arguments presented herein. Use of the Landau-Lifshitz equation would yield the implausible result that the remagnetization time ap proaches zero as A-700, i.e., the greater the damping, the shorter the remagnetization time. For the particular case of the reversal of the magnetization in a single domain sphere, the Landau-Lifshitz equation would yield the result that the reversal time is proportional to 1/;\. Justification for modifying the form of the damping term and for the particular form used in Eq. (1) has been given by Gilbert,I·2 The need for using a different form stems from the fact that the Landau-Lifshitz equation predicts an upper limit to the torque exerted on a ferro magnetic disk by a strong rotating field, which torque is smaller than the observed torque. I Gilbert justifies the use of the particular form used in Eq. (1) by noting that, if one reformulates the undamped equation of motion in Lagrangian form and introduces the damping in a consistent way by means of a Rayleigh dissipation function, the resulting damped equation of motion is that given by Eq. (1).2 This result does not exclude the possibility of other kinds of damping; however, in the absence of experimental evidence, it is the best guide we have for choosing a particular form of the damping term from the myriad conceivable forms. Additional experimental evidence will, of course, be needed to establish conclusively the correctness of the choice. The value of a which minimizes the remagnetization time will, of course, depend upon the geometry of the domain configurations during remagnetization, and cannot be calculated for complicated multidomain models. The calculations presented herein are there- 1 T. L. Gilbert an~ J. M. Kelly, Proceedings of the Pittsburgh Conference on Magnehsm and Magnetic Materials June 14-16 1955 (Am. lnst. Electr. Engrs., October, 1955), p. 253: ' 2 T. L. Gilbert, Phys. Rev. 100, 1243 (1955). 1352 Downloaded 10 May 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsMINIMUM MAGNETIZATION REVERSAL TIME 1353 fore limited to two simple single-domain models: a single-domain sphere and a single-domain sheet. These models could be realized in practice by using small spheres or thin oblate spheroids, which are known to maintain a single domain configuration, (although it would be exceedingly difficult to make measurements because the observed signals would be extremely small). The gyromagnetic equation (1) for these models re duces to a set of two ordinary nonlinear differential equations which can be solved, at least approximately. In massive specimens with multidomain configurations, the gyro magnetic equation reduces to a set of non linear partial differential equations which are com pletely intractable. Although the quantitative results for the minimum remagnetization time and the value of a for which this minimum occurs in the single-domain models are inapplicable to multidomain models, the qualitative conclusions regarding the existence of an optimum damping should remain valid. The purpose for analyzing the models chosen is largely didactic.3 II. A SINGLE-DOMAIN SPHERE An isotropic sphere is assumed to be fully saturated. If the external magnetic field He is rotated slowly, the magnetization vector M will follow it pointing the direction of He at each moment.4 But if He is switched from one direction to another very quickly, M cannot follow it exactly, due to gyro magnetic and damping effects for M. In the models of the present paper, He is assumed to be switched from one direction to its opposite instantaneously at t=O and we examine the behavior of M for t>O. We assume that the motion of M is described by Eq. (1). It is convenient first to transform Eq. (1) into the Landau-Lifshitz form: dM/dt='Y(1 +(2)-1 X {MXSJ- (a/M)[MX (MXSJ)]}. (3) When a2«1, this reduces to the commonly used Landau Lifshitz equation (2) as was pointed out in the pre vious section. For the sake of simplicity, the time scale is changed so that Eq. (3) is written as M2dM/dT=MMXSJ-a[MX (MXSJ)] (4) where (5) and M is the magnitude of M. If one neglects the eddy current contribution to the internal magnetic field, SJ is the sum of two parts: the external field He with a magnitude of He and the de magnetizing field Hd= -(4'7I'/3)M. He is assumed to be switched from the -z direction to the +z at t=O. Inserting SJ=He+Hd into Eq. (4) and writing it in 3 The analysis of the present paper is done only for isotropic material, the presence and influence of anisotropy field being neglected, i Gilbert, Kelly, and Ekstein, Phys, Rev. 98, 1200 (1955). FIG. 1. Contour of the magnetization vector M(schematic) for a sphere. I is the initial direction. The curve (1) is for high damping, a»1; (2) for low damping, a«1. I , He components, one obtains the following simultaneous equations: M2dMx/dT=MMyHe-aMxM.He, (6a) M2dMy/dT= -MMxH.-aMyM.He, (6b) M2dM ./dT=a(M x2+My2)He. (6c) As expected, these equations have the constant M2 = M ,,2+ M i+ M.z as a special integraL Using this constant, Eq. (6c) reduces to: (7) which can be integrated to give the time TF required to flip M. from M Zi to M Zf. Going back to the actual time tF through Eq. (5), one obtains If M at t=O had exactly the -z direction, Mzi=-M and tF would become infinite. Therefore, in order to get tractable results, one has to assume that the direc tion of M(t=O) is slightly different from the -z axis so that M .i~ -M. Similarly, one assumes M 'f~M. When M zi and M./ are fixed, Eq. (8) states that tF is proportional to (1+a2)/a in its dependence on the damping constant a. Hence, the minimum remagnetiza tion occurs when (9) For a>amin, the magnetization vector M moves slower, and for a<amin, it moves faster but rotates around the external magnetic field so that the net traveling time between the two fixed values of Mz becomes longer. Behavior of the magnetization vector M is shown schematically in Fig. 1 for a large and a small vaues of a. III. A SINGLE-DOMAIN SHEET As a next model, we treat an infinitely-wide isotropic sheet. It is assumed that the sheet behaves as a single Downloaded 10 May 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions1354 RYOICHI KIKUCHI domain, which is the minimum energy configuration for an ideal sheet of infinite extent. There is some evi dences that this idealized structure may be approxi mated in ultrathin films.4,6 It is also assumed that the sheet is either nonconducting or very thin so that eddy current damping may be neglected. Equation (4) is the starting point for this case also. The coordinate axes are so chosen that the x and y axes are in the plane of the sheet and the z axis is perpen dicular to it. If one neglects the eddy current contribu tion to the magnetic field as before, ~ consists of two parts: the external field He = iH e and the demagnet izing field Hd= -47rM .k, the unit vectors in the x, y and z directions being denoted by i, j, and k, respec tively. In the following, the fields M and .~ are measured in units of M. We define a=M/M, b=He/M. (10) Inserting ~=He+Hd into Eq. (4) and using Eq. (10), one obtains the following simultaneous equations: dax/dr= -47ra ya.-a(bai-47ra xal-b), (11 a) day/ dr= 47raxa.+ba.-a(baxall-47raya.2), (11 b) da./dr= -ball-a(47raz-41ra.3+baxaz). (l1c) As one expects, these equations imply a relation stating that the vector a has a constant magnitude, which is unity. It is convenient to use the magnetic energy as a parameter: (12) where the first term is the energy of interaction with the external field and the second is the demagnetizing energy. Expressed in terms of the variables used in Eq. (11), one has u= U /MHe= -ax+21raNb. (13) It can be shown, as it is naturally expected, that be cause of the damping equation (4), u decreases mono tonically in time. A second parameter <I> is introduced by the relation so that u=-a x+<I>2. Eliminating all from Eqs. (11), one arrives at 6 R. L. Conger, Phys. Rev. 98,1752 (1955). (14) (15) (17) In the process of transformation, 1-bu/47r-b¢2/81r and I-bu/21r-b¢2/ 1r, which should be multiplied with the second terms of Eqs. (16) and (17), respec tively, have been replaced by unity. This replacement is justified because b=He/M is of the order of 10-2 or 10-3 experimentally and, from Eq. (15), one knows that u and <I> are of the order of unity or less. On the basis of Eqs. (16) and (17) we discuss the behavior of u and <I> or of the vector a = M/ M. As the external field is on the positive x axis, the initial direction of the magnetization vector a is assumed to be close to the negative x axis, so that a.=i cosll;+j sinOi (18) where 0, is slightly larger than -1r. This equation im plies that at t= 0 <1>,=0, U,= -cosO,. Equation (11c) gives the initial value for dcp/dr: (d<l» = _ (21rb)1 sinO,. dr , (19) (20) The flipping (reversal) time is defined as the time during which u changes from u, to Uf defined by (21) so that if a.=O at this final moment, axf= -ax;. It should be remembered that Oi is close to -1r and there fore cosO, is negative, and Ui>O and Uf<O. Case 1. Solution for a-HO When a is very large, Eqs. (16) and (17) state that U and <I> depend on a only through the combination aT. This is the case for the initial conditions, too, because Eq. (20) gives lim(~) =0. ....... '" d(ar) i (22) Therefore, one can conclude the following relation for the time of flipping, rF: arF= constant independent of a. (23) Combining this with Eq. (5) one obtains the result that the actual time of flipping tF has the following form arp 1+a2 a tF=---r"'V-x constant. (24) "1M a "1M Hence tF is proportional to a. The value of the constant arF is determined as follows. Combining Eqs. (22) and (17), one obtains (25) Downloaded 10 May 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsMINIMUM MAGNETIZATION REVERSAL TIME 1355 Repeated differentiations of Eq. (17) show that all the derivatives of cp with respect to aT vanish at t=O for large values of a, so that one can conclude cp=O for all T. (26) Then Eq. (16) is solved, and yields U= -tanh(baT-e), (27) with a constant e defined by Ui = tanhe = -cosO i. (28) The flipping time TF is defined from Uf=COSO;= -tan(baTF-e). (29) From Eqs. (28) and (29), one obtains baTF= 2e. (30) Therefore Eq. (24) becomes finally tF=a(2e/b,-M). (31) Case 2. Solution for cy---?O First we solve Eqs. (16) and (17) for a=O. From the former, one knows that U= Ui= constant independent of T, (32) which transforms Eq. (17) into (33) Multiplication of this equation by d¢/ dT and integration give (::r = 27rb[ sin20i-2(4: + cOSOi}pL cp4]' (34) where the initial conditions (19) and (20) have been taken into account. The solution cp of this equation can be expressed in terms of elliptic functions. In order to make the calculation simpler, one neglects b/47r in comparison to cosO;. Then the solution of Eq. (34) is (35) where K i is defined by (36) and en is an elliptic function.6 The quantity cp of Eq. (35) is a periodic function of T with a period of 2Ki/ (7rb)!. In order to extend the calculation to a finite but small 6 See, for instance, Whittaker and Watson, Modern Analysis (Cambridge University Press, New York, 1935), Chap. 22. value of a, one takes into account the fact that U varies very little during one periodic motion of cp, so that U can be assumed constant for a period. Under this assump tion, for one periodic motion of cp, Eq. (35) can be used: where K is defined by Eq. (36) with Oi replaced by 0 and the period is given by 2K/(7rb)i. For these results, the initial conditions at T=T are Eqs. (19) and (20) with Oi replaced by o. The change of U during one period, t:.u, is approximated using Eq. (16) and assuming U in the last term to be a constant, -cosO. Then the integra tion gives (b)I 0 t:.u= -2Ka ; sin20-16a7r sin2; T+2K!C1rb)+ X i cn2«47rb)t(T-T)+K)dT. (38) T Using the relation (39) where E is given by (40) Eq. (38) becomes fJ.u = -2a{~sin20+87r(E -COS2~)}. fJ.T 2 K 2 (41) Letting U= -cosO, one has fJ.8 = _ 16a7r{b sin2 0 + E -COS2~}. t:.T sinO 167r K 2 (42) This can be looked upon as a derivative of 0 with respect to T and is integrated to give the time of flipping: where (43) 1 fei ( 0 E b )-1 1=- cos2-----sin28 sinOd8 167r ei 2 K 167r (44) is independent of a. Equation (37) is not accurate when \ cosO \ """ h/47r, but as this holds for an interval of 0 negligibly small compared to the total range of integra tion, 0f-f)i, the error caused by assuming Eq. (37) for all values of 0 is not appreciable. Going back to the acutal time t through the relation Downloaded 10 May 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions1356 RYOICHI KIKUCHI -u 4.3---\ 2.5--1 ! I I I I I , I I I I I I I I I t I (a) -U~+--L _ .7 4.9--1 5-1 1 J I I I I I I I I 1 I I 1 I I I I I I I I I I I I I (b) FIG. 2. Examples of analog computor recordings for values of a=0.OO9 and 0.014. The curves are cp, -U, a. and da./dr from top to bottom, and the abscissa is 4.".ar, the unit time marking at the bottom being 0.5. The horizontal lines indicate the zero levels and the vertical lines the initial time. (5), we have tF""'IhMa for a«1. (45) The integral (44) was computed numerically. Of course, it depends on the choice of Oi and Of. Two examples of the results are as follows: (46) Case 3. Solution for Intermediate 0: A rough estimate is made as follows. Using the solu tions (31) and (45), this region may be interpolated as .. i >- I' <t 800 600 400 200 o (47) • a FIG. 3. Plot of 411"'YMtF read from the analog computor results against a (black circles). The dotted curve is due to the inter polation formula (47). This function has a minimum (48) corresponding to a value of a (49) When one assumes b=81rXlO-4 and uses (46), one ob tains amin= {0.0160 for Of= _1~O, 0.0150 for (}f= -2 . (50) It should be noted that amin does not vary much for the change of Of. In order to obtain more precise knowledge of amin, one has to solve Eqs. (16) and (17) for intermediate values of a. This was done numerically by using an Downloaded 10 May 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsMINIMUM MAGNETIZATION REVERSAL TIME 1357 analog computor. Two examples of the results for a=0.009 and 0.014 are shown in Fig. 2, in which the curves are from top to bottom, cp, -u, -(U+cp2) = ax and dax/dr. The abscissa is 4ITar=47r'YMta(1+a2)-\ a dimensionless quantity proportional to time, the unit time marking at the bottom being 0.5. It may be noticed that oscillatory motions of a become more apparent as a is lowered passing through the a=0.01 region. In Fig. 3 the time of flipping tF' read from the meas ured curves is plotted against a. For the numerical computations, the values {Ji= -162° and {Jf= -18° were used. It may seem that one would take a value of {Ji closer to -180°, but if it is closer, the errors in read ing the results increases. The values of {Ji and (Jf adopted are not so bad as it might appear because the value of cos(18°) is 0.95, being fairly close to its maximum value, and also because the value of amiu is not expected to depend on the choice of {Ji very much, as was mentioned in connection with Eq. (SO). As a is lowered, oscillations of a around the x axis increase its amplitude, and when the amplitude of a last oscillation reaches a certain amount the reading of tF' suddenly jumps discontin uously as shown in Fig. 3. In Fig. 4 the contour of the vector a is shown schematically. This is for a small value of a so that a oscillates around the external magnetic field He. For a large value of a, the vector a moves almost in the xy plane and approaches to the direction of He monotonically without oscillation. The reason for the existence of the minimum reversal time is understood from Figs. 2, 3, and 4 and is sum marized as follows. When Ci is larger than alllin the mag netization vector M moves slower, and on the other hand when a is smaller M moves faster but oscillates around the external field as is shown in Fig. 4 so that the actual time for M to come to a rest near He IS longer. One sees from Fig. 2 that amin occurs around Cimin ""0.013, or Amin ""O.OByM. (51) The values of tF' calculated from the interpolation formula (47) are also plotted in the figure. Although Eq. (47) did not take into account the discontinuous jump of tF, the estimate of amin from it is very close to the computed one. This indicates the correctness of the numerical computation by the analog computor. The value of the minimum flipping time, read from Fig. 2, is approximately 41T'YMtF,min=300. Using the values, 'Y=1.76X107 sec! oe-! and 47rM=7400 oe for z y FIG. 4. Schematic picture of a contour of the magnetization vector M (or a) for a thin sheet. This is for a small value of the damping constant a. The sheet is in the xy plane and the point I indicates the initial direction of the vector. 4--79 molybdenum Permalloy, one obtains IF, min"" 2.3 X 10-9 Sec. (52) The dependence of tF', min on the external field He is estimated from Eq. (48) by using the definition b=H./ M: (53) This indicates that tF', min is inversely proportional to (He)~. The value of (52) is based on the assumption that b=81TX10--4• This means He= 1.5 oe if one takes 41TM = 7400 oe. But it seems more reasonable to choose He=0.2 oe. One can estimate how much (52) changes for the new value of He, without repeating the actual numerical computation, if one accepts the (He)-l dependence of tF,min as shown in (53). The modified value is tF, min"'" 6.3 X 10--9 sec. (54) ACKNOWLEDGMENTS The author is deeply indebted to H. Ekstein and T. L. Gilbert who suggested the problem to him and gave stimulating discussions. Thanks are also due to C. J. Moore of the computing group of Armour Re search Foundation for his help in operating the analog computer. Downloaded 10 May 2013 to 128.206.9.138. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions