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Co 2FeAl thin films grown on MgO substrates: Correlation between static, dynamic and structural properties M. Belmeguenai1, H. Tuzcuoglu1, M. S. Gabor2, T. Petrisor jr2, C. Tiusan2;3, D. Berling4, F. Zighem1, T. Chauveau1, S. M. Chérif1and P. Moch1 1Laboratoire des Sciences des Procédés et des Matériaux, CNRS - Université Paris 13, France 2Center for Superconductivity, Spintronics and Surface Science, Technical University of Cluj-Napoca, Romania 3Institut Jean Lamour, CNRS-Université de Nancy, France and 4Institut de Science des Matériaux de Mulhouse, CNRS-Université de Haute-Alsace, France Co2FeAl (CFA) thin films with thickness varying from 10 nm to 115 nm have been deposited on MgO(001)substratesbymagnetronsputteringandthencappedbyTaorCrlayer. X-raysdiffraction (XRD) revealed that the cubic [001]CFA axis is normal to the substrate and that all the CFA films exhibit full epitaxial growth. The chemical order varies from the B2phase to the A2phase when decreasing the thickness. Magneto-optical Kerr effect (MOKE) and vibrating sample magnetometer measurements show that, depending on the field orientation, one or two-step switchings occur. Moreover, the films present a quadratic MOKE signal increasing with the CFA thickness, due to the increasing chemical order. Ferromagnetic resonance, MOKE transverse bias initial inverse susceptibility and torque (TBIIST) measurements reveal that the in-plane anisotropy results from the superposition of a uniaxial and of a fourfold symmetry term. The fourfold anisotropy is in accord with the crystal structure of the samples and is correlated to the biaxial strain and to the chemical order present in the films. In addition, a large negative perpendicular uniaxial anisotropy is observed. Frequency and angular dependences of the FMR linewidth show two magnon scattering and mosaicity contributions, which depend on the CFA thickness. A Gilbert damping coefficient as low as 0.0011 is found. I. INTRODUCTION The performances of spintronic devices depend on the spin polarization of the current. Therefore, half metallic materials should be ideal compounds as high spin polar- ized current sources to realize a very large giant magne- toresistance, a low current density for current induced magnetization reversal, and an efficient spin injection intosemiconductors. Theoretically, differentkindsofma- terials, such as Fe 3O4[1, 2], CrO 2[3], mixed valence per- ovskites [4] and Heusler alloys [5, 6], have been predicted to be half metals. Moreover, the half metallic proper- ties in these materials have been experimentally demon- strated at low temperature. However, oxide half metals havelowCurietemperature( TC)andthereforetheirspin polarization is miserably low at room temperature. From this point of view, Heusler alloys are promising materials for spintronics applications, because a number of them have generally high TC[7] and therefore they may offer an alternative material choice to obtain half metallicity even at room temperature. Furthermore, their structural and electronic properties strongly depend on the crystal structure. Recently, Heusler compounds have attracted considerable experimental and theoretical interest, not only because of their half metallic behaviour but also due to magnetic shape memory and inverse magneto-caloric properties that they exhibit. One of the most important Co-based full-Heusler alloys is Co 2FeAl (CFA). It has a highTC(TC= 1000K) [7] and, therefore, it is promis- ing for practical applications. Indeed, it can provide giant tunnelling magnetoresistance ( 360%at RT) [8,9] when used as an electrode in magnetic tunnel junctions. Furthermore, as we illustrate in our present study, CFApresents the lowest magnetic damping parameter among Heuslers. This low damping should provide significantly lower current density required for spin-transfer torque (STT) switching, particularly important in prospective STT devices. However, the integration of CFA as a ferromagnetic electrode in spintronic devices requires a good knowledge allowing for a precise control of its mag- netic properties, such as its saturation magnetization, its magnetic anisotropy, the exchange stiffness parameter, the gyromagnetic factor and the damping mechanisms monitoring its dynamic behaviour. In this paper we used X-rays diffraction (XRD), ferromagnetic resonance in microstrip line (MS-FMR) under in-plane and out of plane applied magnetic field, combined with transverse biased initial inverse susceptibility and torque (TBIIST) method, in order to perform a complete correlated analy- sis between structural and magnetic properties of epitax- ial Co 2FeAl thin films grown on MgO(001) substrates. In addition, a detailed study of the different relaxation mechanisms leading to the linewidth broadening is pre- sented. II. SAMPLES PREPARATION AND EXPERIMENTAL METHODS CFA films were grown on MgO(001) single-crystal sub- strates using a magnetron sputtering system with a base pressure lower than 3109Torr. Prior to the deposi- tion of the CFA films, a 4 nm thick MgO buffer layer was grown at room temperature (RT) by rf sputtering from a MgO polycrystalline target under an Argon pressure of 15 mTorr. Next, the CFA films, with variable thick-arXiv:1305.0714v1 [cond-mat.mtrl-sci] 3 May 20132 30 40 50 60 70050100150200250300 42.5 45.0 47.50100200 115 nm 70 nm 45 nm 20 nm 2(degrees)Intensity (arb. units)(002) CFA (004) CFA 10 nm(a) (220) CFA 30 40 50 60 70 80 90050100150200250300350400450500(b) 20 nm50 nmCFA(002) CFA(004)Intensity (arb. units) 2 (degrees) Figure 1: (Colour online) (a) X-ray 2!(out-of-plane) diffractionpatternusing(CuX-rayssource)fortheCr-capped and (b)2pattern (Co X-rays source) for the Ta-capped Co2FeAl of different thicknesses. The inset shows selected area in plane diffraction patterns around (220) Co 2FeAl re- flection. nesses (10 nmd115nm), were deposited at RT by dc sputtering under an Argon pressure of 1 mTorr, at a rate of 0.1 nm/s. Finally, the CFA films were capped with a MgO(4nm)/Cr(10nm) or with a MgO(4nm)/Ta(10nm) bilayer. Afterthegrowthofthestack, thestructureswere ex-situ annealed at 600oC during 15 minutes in vacuum (pressure lower than 3108Torr). The structural prop- erties of the samples have been characterized by XRD us- ing a four-circle diffractometer. Their magnetic dynamic properties have been studied by microstrip ferromagnetic resonance (MS-FMR). The MS-FMR characterization was done with the help of a field modulated FMR setup using a vector network analyzer (VNA) operating in the 0.1-40 GHz frequency range. The sample (with the film side in direct con- tact) is mounted on 0.5 mm microstrip line connected to the VNA and to a lock-in amplifier to derive the field modulated measurements via a Schottky detector. Thissetup is piloted via a Labview program providing flexi- bility of a real time control of the magnetic field sweep direction, step and rate, real time data acquisition and visualization. It allows both frequency and field-sweeps measurements with magnetic fields up to 20 kOe applied parallel or perpendicular to the sample plane. In-plane angular dependence of resonance frequencies and fields are used to measure anisotropies. The complete analy- sis of in-plane and perpendicular field resonance spectra exhibiting uniform precession and perpendicular stand- ing spin wave (PSSW) modes leads to the determination of most of the magnetic parameters: effective magneti- zation, gyromagnetic factor, exchange stiffness constant and anisotropy terms. In addition, the angular and the frequency dependences of the FMR linewidth are used in order to identify the relaxation mechanisms responsi- ble of the line broadening and allow us for evaluating the parameters which monitor the intrinsic damping (Gilbert constant) and the extrinsic one (two magnon scattering, inhomogeneity, mosaïcity). Magnetization at saturation and hysteresis loops for each sample were measured at room temperature using a vibrating sample magnetometer (VSM) and a magneto- optical Kerr effect (MOKE) system. Transverse biased initialinversesusceptibilityandtorquemethod(TBIIST) [10] has been used to study the in-plane anisotropy for comparison with MS-FMR. In this technique both a lon- gitudinal magnetic sweep field HL(parallel to the inci- denceplane)andastatictransversefield HB(perpendic- ulartotheincidenceplane)areappliedintheplaneofthe film and the longitudinal reduced magnetization compo- nentmLis measured versus HLfor various directions of HLwithconventionalmagneto-opticalKerrsetup. From the measured hysteresis loops mL(HL)under transverse biased field, the initial inverse susceptibility ( 1) and the field offset ( H) which are related to the second and first angle-derivative of the magnetic anisotropy, respec- tively, are derived. Fourier analysis of 1andHversus the applied field direction then easily resolves contribu- tions to the magnetic anisotropy of different orders and gives the precise corresponding values of their amplitude and of their principal axes. In order to obtain the desirable accuracy or even sim- ply meaningful results higher-order nonlinear in mLcon- tributions (quadratic or Voigt effect) as well as polar or other contributions to the Kerr signal are carefully deter- mined and corrected [10]. TBIIST method surely does not have the same recognition than FMR techniques but seems to be complementary, especially for samples with a weak magnetic signal detectable with difficulty by FMR methods. III. STRUCTURAL CHARACTERIZATION Figure 1 shows the X-rays 2!diffraction patterns for CFA of different thicknesses. These XRD patterns show that, in addition to the feature arising from the3 Figure 2: (Colour online) Pole figures around the Co 2FeAl (022) type reflection, for the 45 nm thick film, indicat- ing the growth of Co 2FeAl on MgO with the Co 2FeAl (001)[110]kMgO (001)[100] epitaxial relation. The 0 and 90 degrees axis of the graph correspond to the MgO [100]and [010] crystalline directions. (002) peak of the MgO substrate, the Cr-capped samples (Fig. 1a: Cu X-rays source ( = 0:15406nm)) exhibit only two peaks which are attributed to the (002) and (004) diffraction lines of CFA. The Ta-capped films (Fig. 1b: Co-X-rays source ( = 1:7902)) show an additional peak (around 2= 63 °) arising from the (002) line is- sued from the Ta film. Pole figures (Fig. 2) allow to assert an epitaxial growth of the CFA films according to the expected CFA(001)[110]//MgO(001)[100] epitax- ial relation. Using scans of various different orientations we evaluated the out-of-plane ( a?) and the in-plane ( ak) lattice parameters (Fig. 3). A simple elastic model al- lowed us for deriving the unstrained a0 cubic parameter aswellasthein-plane "kandtheout-of-plane "?strains: a0= C11a?+ 2C12ak
(C11+ 2C12); "k=C11 (C11+ 2C12) aka?
a0; "?=2C12 (C11+ 2C12) aka?
a0(1) where the values of the elastic constants C11= 253 GPa andC12= 165GPa have been calculated previously [11]. Introducing the Poisson coefficient =C12=(C11+ C12)the above parameters write as: 05 0 1 0 00.5650.5700.575 out-of-plane in-planelattice parameter (nm) thickness (nm)-10-50510152025 A002/A004 ratio (%)Figure 3: (Colour online) Evolution of the out-of-plane and in-plane lattice parameters and of the ratio of the integral in- tensitiesofthe (002)and(004)Co2FeAlpeaksA(002)/A(004) with respect to the Co 2FeAl films thickness. a0= (1)a?+ 2ak
+ 2ak (1 +); "?=(1) (1 +) aka?
a0; "k=2 1 + aka?
a0(2) Thecubiclatticeconstant a0doesnotdependuponthe thickness, except for the thinner 10 nm film (Fig. 4a), which shows a significant reduction; its value, 0:5717 0:0005nm, is slightly smaller than the reported one in the bulk compound with the L2 1structure (0.574 nm). The in-plane strain "kreveals a tensile stress originat- ing from the mismatch with the lattice of the MgO sub- strate: however, its value does not exceed a few°/°°, well below the Heussler/MgO mismatch, thus excluding an efficient planar clamping. The strain "kdecreases versus the thickness, at least above 40 nm (Fig. 4b). Odd Miller indices (e.g.: (111);(311);...) are allowed for diffraction in the L2 1phase [12]. In contrast, they are forbidden in the B2 phase, which is characterized by a total disorder between Al and Fe atoms but a regular occupation of the Co sites. In the A2 phase the chemical disorder between Fe, Co and Al sites is complete: (hkl) diffraction is only allowed for even indices subjected to h+k+l= 4n. We do not observe (111)or(311)lines and then conclude to the absence of the L2 1phase in the studied films. In contrast, a (002)peak is observed, thus indicating that the samples do not belong to the A2 phase. However, the ratio I002=I004of the integrated in- tensities of the (002)and of the (004)peaks increases ver- sus the film thickness (Fig. 3). This ratio is proportional to(12c)2, wherecis the chemical disorder. Assuming that the thickest film belongs to the B2 phase ( c= 0) the dependence of cupon the film thickness is shown in4 0 2 04 06 08 0 1 0 0 1 2 00.00.20.40.0050.0060.0070.5700.5710.572 (c)c Film thickness (nm)(b)(a)a0 (nm) Figure 4: (Colour online) Thickness dependence of (a) the lattice cubic parameter a0, the in-plane strain "kand (c) the chemical order cof Co 2FeAl thin films. figure 4c: the A2 phase ( c= 0:5) is almost completely achieved for the 10 nm thick sample. The reduction of a0in the thinner sample is probably due to its previously noticed [13] smaller value in the A2 phase compared to the B2 one. IV. MAGNETIC PROPERTIES The experimental magnetic data have been analyzed considering a magnetic energy density which, in addition toZeeman, demagnetizingandexchangeterms, ischarac- terized by the following effective anisotropy contribution [14]: Eanis: =1 2(1 +cos(2('M'u))Kusin2M+ K?sin2M1 8(3 + cos 4('M'4))K4sin4M(3) In the above expression, Mand'Mrespectively rep- resent the out-of-plane and the in-plane (referring to the substrate edges) angles defining the direction of the mag- netizationMS.'uand'4define the angles between an easy uniaxial planar axis or an easy planar fourfold axis, respectively, with respect to this substrate edge. Ku,K4 andK?are in-plane uniaxial, fourfold and out-of-plane uniaxialanisotropyconstants, respectively. Weintroduce the effective magnetization Meff=Heff=4obtained by: 4Meff=Heff= 4MS2K? MS= 4MSH?(4) As experimentally observed, the effective perpendicu- lar anisotropy term K?(and, consequently, the effective perpendicular anisotropy field H?), is thickness depen- dent:K?describes an effective perpendicular anisotropyterm which writes as: K?=K?V+2K?S d(5) whereK?Srefers to the perpendicular anisotropy term of the interfacial energy density. Finally we define Hu= 2Ku=MSandH4= 4K4=Msas the in-plane uniaxial and the fourfold anisotropy fields respectively. The resonance expressions for the frequency of the uniform and PSSW modes assuming in-plane or perpendicular applied mag- neticfieldsaregivenbyequations(6)and(7)respectively [14, 15]. Fn:= 2(Hcos('H'M) +2K4 MScos 4('M'4) +2Ku MScos 2('M'u) +2Aex: MSn d2 ) (Hcos('H'M) + 4Meff+K4 2MS(3 + cos 4('M'4)) +Ku MS(1 + cos 2('M'u)) +2Aex: MS(n d)2)(6) F?:= 2(H4Meff+2Aex: MSn d2 )(7) In the above expressions =2=g1:397106 s1.Oe1is the gyromagnetic factor, nis the index of the PSSW and Aexis the exchange stiffness constant. The experimental results concerning the measured peak-to-peak FMR linewidths
HPPare analyzed in this work taking account of both intrinsic and extrinsic mechanisms. Therefore, in the most FMR experiments, the observed magnetic field linewidth (
HPP) is usu- ally analyzed by considering four different contributions as given by equation (8) [16-21].
HPP=
HGi+ (
Hmos+
Hinh+
H2mag)(8) When the applied field and the magnetization are paral- lel, the intrinsic contribution is not angular dependent; it derives from the Gilbert damping and is given by:
HGi=2p 3 2f (9) (9) wherefis the driven frequency and is the Gilbert coefficient. The relevant mechanisms [16] describing the extrinsic contributions are: 1- Mosaicity: the orientation spread of the crystallites contributes to the linewidth. Its contribution is given by:
Hmos=@Hres @'H
'H=@H @'H
'H res(10)5 Where
'His the average spread of the easy axis anisotropy direction in the film plane. It is worth to men- tion that for frequency dependent measurements along the easy and hard axes the partial derivatives are zero and thus the mosaicity contribution vanishes. The suffix “res” indicates that equation (10) should be evaluated at the resonance. Therefore, using equation (6) for uniform mode (n= 0), the expression of@H @'His found and then calculated using the corresponding value of Hand'M at the resonance. 2- Inhomogeneous residual linewidth
Hinhpresent at zero frequency. This contribution is frequency and angle independent inhomogeneity related to various local fluctuations such as the value of the film thickness. 3- Two magnon scattering contribution to the linewidth. This contribution is given by [22-24]:
H2mag= 0+ 2cos 2('H'2)+ 4cos 4('H'4) arcsin fp f2+f2 0+f0! (11) with:f0= Meff. The expected fourfold symmetry induces the 0and4coefficients; the coefficient 2is phenomenogically introduced. Theanalysisofthevariationoftheresonancelinewidth
HPPversus the frequency and the in-plane field ori- entation allows for evaluating ,
'H,
Hinh,0,2 (and'2) and 4(and'4which, from symmetry consid- erations, is expected to have a 0°or45°value, depending upon the chosen sign of 4). A. Static properties The magnetization at saturation measured by VSM, averaged upon all the samples has been found to be MS= 100050emu/cm3, thus providing a magnetic moment of 5.05 ±0.25 Bohr magneton ( B) per unit for- mula, in agreement with the previously published values for the B2 phase [7]. For all the studied films the hystere- sis loops were obtained by VSM and MOKE with an in- plane magnetic field applied along various orientations. Figure 5 shows representative behaviors of different CFA films. The observed shape mainly depends on the field orientation, in agreement with the expected characteris- tics of the magnetic anisotropy. As confirmed below, in all the studied samples this anisotropy consists into the superposition of a fourfold and of a uniaxial term show- ing parallel easy axes: this common axis coincides with one of the substrate edges and, consequently, with one of the<110>crystallographic directions of the CFA phase. It results that if an orientation (say 'H= 0re- lated to [110]) is the easiest, the perpendicular direction (('H= 90) related to [110]) is less easy. A similar sit- uation was studied and interpreted previously [25]: it is expected to provide square hysteresis loops for 'H= 0, -60 -40 -20 0 20 40 60-1.0-0.50.00.51.0(a) d=115 nmM/Ms Magnetic field (oe) H=0° H=45° H=90° -60 -40 -20 0 20 40 60-1.0-0.50.00.51.0(b) d=50 nmM/Ms Magnetic field (oe) H=0° H=45° H=90° 0.00 0.02 0.04 0.06 0.08 0.1001020304050607080Coercive field (Oe) 1/d(nm) Cr-capped Ta-capped(c)Figure 5: (Colour online) MOKE hysteresis loops of the (a) 115 nm Cr-capped and (b) 50 nm Ta-capped Co 2FeAl thin films. Themagneticfieldisappliedparalleltothefilmsurface, at various angles ( 'H) with a respect to edges of the MgO substrate ( [100]or[010]). (c) Thickness dependence of the coercive field, deduced from hysteresis loops along the easy axis, of Co 2FeAl Cr- and Ta-capped thin films.6 0 50 100 150 200 250 300 350-2.50.022.525.0-505253035 ML2-MT2MLMTML CFA(50nm)/MgOKerr rotation (mdeg) Sample orientation (degrees) Fit FitMLMT ML2-MT2ML CFA(115nm)/MgOKerr rotation (mdeg) Fit Fit 0 2 04 06 08 0 1 0 0 1 2 002420222426283032Amplitude, offset (mdeg) Film thickness (nm) ML Amplitude of MLMT Amplitude of ML2-MT2 Offset of MLMT Figure 6: (Colour online) (a) Separated quadratic MOKE contributions as a function of the sample orientation at 46° incidence. The fits are obtained using equation (12). (b) The MLcontribution (at angle of incidence of 46°), the amplitudes andoffsetofthe MLMTcontributionandtheamplitudeofthe (M2 LM2 T) as a function of the Co 2FeAl thickness. as evidenced in figure 5, while in contrast, for 'H= 90 , it leads to a two steps reversal, as can be seen in figure 5. The intermediate step leads to a magnetization nearly perpendicular to the applied field. For all the studied films a two steps loop is observed for 'Hranging in the f55130°ginterval. In figure 5c the deduced coercive fields (HC) from hysteresis loops along the easy direction ('H= 0) are compared for different thicknesses (10, 20, 45, 50, 70, and 115 nm). For both Cr-capped and Ta- capped films HC increases linearly with the inverse of the filmthickness. TheCr-cappedsamplespresenthigherco- ercive fields due to the different interface quality. One can also observe that MOKE hysteresis loops are not strictly centrosymmetrical (see for example Fig. 5b for'H= 90) indicating the superposition of symmet- rical (even function of applied sweep field HL) and anti- symmetrical (odd in HL) components in the Kerr signal. It has been shown and confirmed [26, 27] that, for in- plane magnetized thin films, the antisymmetrical part observed in the mL(HL)loops arises from the second or-der magneto-optical effects quadratic in magnetization. Therefore, the present study was not limited to the usual linear MOKE. We have also investigated this quadratic contribution through the study of the Kerr signal depen- dence upon the film orientation under a saturating in- plane field. Within the cubic approximation for a (001) surface, the Kerr rotation angle writes as [27]: K=a1ML+a2(M2 LM2 T) sin(4 )+ (b2+ 2a2cos(4 ))MLMT(12) WhereMLandMTstand for the longitudinal (i.e.: within the incidence plane) and the transverse (i.e.: nor- mal to the incidence plane) component of the magneti- zation, respectively, and where is the angle of a cubic <110>axis with the plane of incidence. The first term describes the usual linear contribution while the follow- ing ones correspond to the quadratic MOKE (QMOKE). The experimental study was performed under an angle of incidence of 46°using a field magnitude large with re- spect to the anisotropy field. The different contributions to the Kerr signal, as functions of the film orientation are extracted by applying a rotating field technique [10]. Representative results obtained with 115 Cr- and 50 nm Ta-capped films are shown in figure 6. Beside the lon- gitudinal component ( ML) of the Kerr rotation, which is dominant, the QMOKE signal, which is most proba- bly due to the second order spin-orbit coupling [26], is present. The derived ( M2 LM2 T) andMLMTangular variations show the behaviour expected from the above equation. The values for the amplitudes of the 2MLMTand of the (M2 LM2 T) contributions are the same within the experimental error for each sample suggesting that the applied cubic model is correct. The offset of the MLMT contributionissmallerthantheamplitudes,butgenerally it follows the same trend as the amplitudes. As the thick- ness decreases the amplitudes and the offset decrease, suggesting that the chemical order progressively changes from the B2 to the A2 phase, as discussed above. More- over, the amplitudes and offset values of CFA are compa- rable to those measured for Co 2MnSi, which presents the L21phase [28]. The TBIIST results are discussed in the following section, in order to allow for a comparison with the data derived from the FMR study of the dynamic properties. B. Dynamic properties 1. Exchange stiffness and effective magnetization TheuniformprecessionandthefirstPSSWmodeshave been observed in perpendicular and in-plane applied field configurations for samples thicknesses down to 50 nm. For the thickest film (115 nm) it was even possible to ob- serve the second PSSW. For lower sample thickness, the7 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6246810121416d=115 nm Fit Fit Fit FitFrequency (GHz) Magnetic field (kOe) Uniform mode: H=90° PSSW mode: H=90° Uniform mode: H=45° PSSW mode: H=45° 14 15 16 17 18 1924681012141618Frequency (GHz) Magnetic field (kOe) Uniform mode PSSW1 PSSW2 Fit Fit Fitd=115 nm 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6246810121416182022 Fit Fit Fitd=50 nmFrequency (GHz) Magnetic field (kOe) Uniform mode: H=0° Uniform mode: H=45° PSSW mode: H=0° 14 15 16 17 18 19 202468101214161820d=50nmFrequency (GHz) Magnetic field (kOe) Uniform mode PSSW mode Fit Fit Figure 7: (Colour online) Field dependence of the resonance frequency of the uniform precession and of the two first perpendic- ular standing spin wave excited (PSSW) mode of 115 nm Cr-capped and 50 nm Ta-capped Co 2FeAl films. The magnetic field is applied perpendicular or in the film plane. The fits are obtained using equations (6) and (7) with the parameters indicated in the Table I. 0.00 0.02 0.04 0.06 0.08 0.101213141516171819204Meff (kOe) 1/d (nm-1) Cr-capped films Ta-capped films Fit Figure 8: (Colour online) Thickness dependence of the effec- tive magnetization ( 4Meff) extracted from the fit of FMR measurements. The solid lines are the linear fits. PSSW modes are not detected due their high frequencies over-passing the available bandwidth (0-24 GHz). Typ- ical in-plane and perpendicular field dependences of theresonance frequencies of the uniform and PSSW modes are shown on figure 7 for the 115 nm Cr- and the 50 nm Ta-capped films. By fitting the data in figure 7 to the above presented model, the gyromagnetic factor ( ), the exchange stiffness constant ( Aex) and the effective magnetization (4Meff) are extracted. The fitted and Aexvalues are 2.92 GHz/kOe and 1.5 µerg/cm, respec- tively: they do not depend of the studied sample. The derived exchange constant is in good agreement with the reported one by Trudel et al. [7]. Meff=Heff=4 Figure 8 plots out the extracted effective magnetiza- tion 4Meffversus the film thickness 1=d. It can be seen thatMefffollows a linear variation. This allows to derive the perpendicular surface anisotropy coefficient K?S:K?S=1:8erg/cm2. The limit of 4Meffwhen 1=dtends to infinity is equal to 12.2 kOe: within the above mentioned experimental precision about the mag- netization at saturation it does not differ from 4MS. We conclude that the perpendicular anisotropy field de- rives from a surface energy term; being negative, it pro- vides an out-of-plane contribution. It may originate from the magneto-elastic coupling arising from the interfacial stress due to the substrate.8 0 50 100 150 200 250 300 35051015203690200400600800 F=6GHz MeasurementsF=9GHzHPP (Oe) Applied field direction H (degrees) MeasurementsH=255OeH=597Oe Fit Measurements Measurements F(GHz) FitF=6GHzF=9GHz Measurements Measurementsd=50 nmHr (Oe)Fit Fit Fit Fit 0 50 100 150 200 250 300 3500100200300-40-2002040 Measurements Fit(Oe) Applied field direction H (degrees)d=50 nm Measurements FitH (Oe) 0 50 100 150 200 250 300 35012151821681012200400600 F=8GHzd=20 nmHPP (Oe) Applied field direction H (degrees) Measurements H=828 Oe H=591 OeMeasurements Fit FitF(GHz)Measurements F=8GHz FitHr (Oe) Measurements Fit 0 50 100 150 200 250 300 3500200400600-50050 Measurements Fit(Oe) Applied field direction H (degrees) Measurements FitH (Oe)d=20 nm Figure 9: (Colour online) Angular dependence of the resonance frequency ( Fr), resonance field ( Hr), peak to peak field FMR linewidth (
HPP), inverse susceptibility ( 1) and the field offset ( H) of 50 nm and 20 nm thick Co 2FeAl Ta-capped thin films. The TBIIST measurements were obtained using transverse static bias field HB= 200Oe and 225 Oe respectively for 50 nm and 20 nm thick Co 2FeAl films. The solid lines refer to the fit suing the above mentioned models. 2. Magnetic anisotropy Figure 9 shows the angular dependences of the reso- nance field (at fixed frequency) and of the resonance fre- quency (at fixed applied field) compared to the static TBIIST measurements for three different CFA films. Both FMR and TBIIST data show that the angular be- havior is governed by a superposition of uniaxial and fourfold anisotropy terms with the above-mentioned easy axes. As noticed above, the symmetry properties of the epitaxial observed films agree with the principal direc- tions of the fourfold contribution. The fourfold and uni- axialanisotropyfieldsextractedfromthefitoftheexperi- mentalTBIISTandFMRdatausingtheabove-presented model are drawn on figure 10 and summarized in Table I: the compared results issued from the two techniques are in excellent agreement. For all the samples the fourfold anisotropy is dominant. While the uniaxial anisotropy field (H2) of the Cr-capped films is small and does notseem to depend upon the thickness, in the Ta-capped filmsH2is higher, maybe due to interface effects, and is a decreasing function of the thickness (Figure 10). As sug- gested previously, we believe that the uniaxial anisotropy is induced by the stepping of the substrates, probably resulting from a small miscut along their [100]crystallo- graphicdirectioncorrespondingtothe [110]studiedfilms. The reduced effect of the steps of the substrate when the thickness increases could then explain the thickness de- pendence of H2. However, up to now we have no com- pletelysatisfyinginterpretationofthepresenceof H2and of its variations versus the nature of the film capping. The fourfold anisotropy fields ( H4) are comparable for Cr- and Ta-capped films and decrease when their thick- ness increases, as seen in figure 10. For large values of d(45nm or higher) H4lies around 200 Oe and shows a small linear variation versus the in-plane strain "k, as shown in the insert of figure 10. This evolution confirms a direct correlation between the H4 field and the in-plane biaxial strain for the films with thicknesses above 45 nm.9 0.00 0.02 0.04 0.06 0.08 0.10 0.120102030402004006008001000 %H4 (Oe)Anistropy fields (Oe) 1/d(nm-1) H2:Cr-capped H4:Cr-capped H2:Ta-capped H4:Ta-capped B2 A20.5 0.6 0.7200205210215220225 Figure 10: (Colour online) Thickness dependence of the uni- axial (H2) and the fourfold anisotropy fields ( H4) extracted from the fit of FMR measurements. The solid lines are the linear fits. The inset shows the evolution of the H4field, for the 45, 70 and 115 nm thick samples, with the in-plane biaxial strain. At smaller values of d(10 or 20 nm) a large increase ofH4, up to 920 Oe, is observed. It is presumably re- lated to the B2)A2 phase transition observed through X-rays diffraction. The observed symmetry argues for a magneto-crystalline contribution, which, as previously observed [29, 30], would be higher in phase A2 than in phase B2. 3. FMR linewidth In figure 9, the FMR peak to peak linewidth ((
HPP) is plotted as a function of the field angle 'Hfor the 50 nm and 20 nm Ta-capped CFA films using three driv- ing frequencies: 6, 8, and 9 GHz.
HPPis defined as the field difference between the extrema of the sweep- field measured FMR spectra. All the other samples show a qualitatively similar behaviour to one of the samples presented here. The positions of the extrema depend on the sample. The observed pronounced anisotropy of the linewidth cannot be due to the Gilbert damping contri- bution, which is expected to be isotropic, and must be due to additional extrinsic damping mechanisms. In the 50 nm thick sample, the
HPPangular variation shows a perfect fourfold symmetry (in agreement with the vari- ation of the resonance position). Such behaviour is char- acteristic of two magnon scattering. This effect is cor- related to the presence of defects preferentially oriented along specific crystallographic directions, thus leading to anasymmetry(seeequation(11)). Concerningthe20nm thick film, the in-plane angular dependence of
HPPis less simple and shows eight maxima, that is expected from a mosaicity driven linewidth broadening. It can be 4 6 8 1 01 21 41 61 82 02 22 410203040506070 10 nm, H=90°10 nm, H=45°20 nm 50 nm70 nmFMR linewidth HPP (Oe) Frequency (GHz)Figure 11: (Colour online) Frequency dependence of the easy axis ('H= 0) peak to peak field FMR linewidth (
HPP) for Co2FeAl thin. The solid lines refer to the fit using equations (8-11). seen that a smaller fourfold symmetry (four maxima) is superimposed on the eight maxima, indicating that two magnon scattering is still present. Therefore, the entire angular dependence of the FMR linewidth in our samples can be explained as resulting of the four contributions appearing in equation (8). In figure 11,
HPPfor the field parallel to an easy axis and a hard axis ( 'H= 45 °for 10 nm thick sample) of the fourfold anisotropy is plotted as a function of driving frequency for all samples. An apparently extrinsic contri- bution to linewidth was observed, which increased with decreasing film thickness. It should be mentioned that the observed linear increase of the linewidth with fre- quency in figure 11 maybe due to Gilbert damping but other relaxation mechanisms can lead to such linear be- haviour. Therefore, only an effective damping parameter effcan be extracted from the slope of the curves and ranges between about 0.00154 for the easy axis of the 50 nm thick film and 0.0068 for easy axis the thinnest film. The pertinent parameters could thus be, in principle de- rived from the conjointly analysis of the frequency and angular dependence of
HPP. However, due to the lim- ited experimental precision, some additional hypotheses are necessary in order to allow for a complete determi- nation of the whole set of parameters describing the in- trinsic Gilbert damping and the two magnon damping. A detailed analysis is presented in the appendix. Using the previously reported value: = 1:1103[31], which is in agreement with our experimental results, we were able to 0for each film. 0,2,4,'2,'4are listed in Table II which also contains the parameters describing the damping effects of the mosaïcity (
'H) and of the inhomogeneity contribution (
Hinh). The two magnon and the mosaïcity (
'H) contribu- tions to
HPPincrease when the thickness decreases, probably due to the progressive above reported loss of10 chemical order. The increase of the residual inhomo- geneities linewidth (
Hinh) with the thickness is most probably due the increase of defects and roughness. The uniaxial term 2is observed only in the thinnest (20 and 10 nm) samples. As expected, '4= 0, but the sign of 4is sample dependent. Finally, it is important to no- tice that the very low value of the intrinsic damping in the studied samples allows for investigating the extrinsic contributions. V. CONCLUSION Co2FeAl films of various thicknesses (10 nm d115 nm)) were prepared by sputtering on a (001) MgO sub- strate. They show full epitaxial growth with chemical order changing from B2 to A2 phase as thickness de- creases. MOKE and VSM hysteresis loops obtained with different field orientations revealed that, depending on the direction of the in-plane applied field, two or one jump switching occur, due to the superposition of uni- axial and fourfold anisotropies. The samples present a quadratic MOKE contribution with decreasing ampli- tudes as the CFA thickness decreases. The microstrip ferromagnetic resonance (MS-FMR) and the transverse biased initial inverse susceptibility and torque (TBIIST) methods have been used to study the dynamic proper- ties and the anisotropy. The in-plane anisotropy presents two contributions, showing a fourfold and a twofold ax- ial symmetry, respectively. A good agreement concern- ing the relevant in-plane anisotropy parameters deduced from the fit of MS-FMR and TBIIST measurements has been obtained. The fourfold in-plane field shows a thick- ness dependence behavior correlated to the thickness dependence of the chemical order and strain in sam- ples. The angular and frequency dependences of the FMR linewidth are governed by two magnon scattering, mosaïcity and by a sample independent Gilbert damping equal to 0.0011 Appendix In the section dealing with the discussion of the FMR linewidth measurements we stated that the conjointly analysis of the frequency and angular dependence of
HPPdoes not allow for the determination of all the parameters given in equation (8) and additional hypoth- esisshouldbedone. Theaimofthisappendixistoclarify themannerinwhichtheparameterssummarizedinTable II is done. For most of the exploitable measurements the mi- crowave frequency f during the
HPPmeasurements is not larger than f0and generally smaller ( f0varies from 18.5 to 28.5 GHz, depending on the film thickness). It then results that the two magnon damping is practically proportional to f and that the sum of the Gilbert and ofthe two magnon damping terms reads as (see equations (9) and (11)):
HGi+2mag=((p 3+0 2Heff)+2 2Heffcos 2('H'2) +4 2Heffcos 4('H'4))4 f(13) It is not possible to completely identify the respec- tive contributions of the Gilbert and of the two magnon damping, only according to equation (13). The quasi- linear variation versus the frequency (Fig. 11) observed for
HPPallows for defining an effective damping pa- rametereff, which, is angle dependent due to two magnon scattering. The experimentally derived coeffi- cienteff, from the linear fit of data presented in figure 11, varies from 0.0068 to 0.00154. Furthermore, the mea- sured angular variation of the linewidth allows for evalu- ating ( 2,'2) and ( 4,'4) but, concerning the isotropic terms appearing in equation (13), only the sum +p 30 2Heff can be derived. However, remembering that cannot be negative, the maximal available value of 0(correspond- ing to= 0) is easily found. Moreover, a lowest value can be obtained for 0noticing that equation (13) can also be written:
HGi+2mag=((p 3+0j2jj4j 2Heff)+ j2j 2Heff(1cos 2('H'2))+ j4j 2Heff(1cos 4('H'4)))4 f(14) where the adequate third and the fourth terms rep- resent the twofold and the fourfold contributions, which take into account that both of them are necessarily non- negative for any value of 'H. The additional residual two magnon isotropic contribution cannot be negative. Hence: 0>j2j+j4j. Introducingthisminimalaccessiblevalueof 0, (j2j+ j4j), the maximal value of the Gilbert coefficient is then easily obtained. To summarize, for each sample the experimental data provide the allowed intervals for and for0, respectively [0, min] and [(j2j+j4j),0Max], and indeed, the chosen value of within [0,min] allows for deducing 0. The smallest calculated interval for , equal to [0, 1:4103] is obtained for the 70 nm film. A previous publication by Mizukami el al. [31] has con- cluded to a Gilbert coefficient equal to: = 1:1103. We then stated that = 1:1103and, consequently, we were able to deduce 0for each film. 0,2,4,'2, '4are listed in Table II which also contains the parame- ters describing the damping effects of the mosaïcity and of the inhomogeneity.11 [1] [1] A. Yanase and K. Siratori, J. Phys. Soc. Jpn. 53, 312 (1984) [2] [2] Z. Zhang and S. Satpathy, Phys. Rev. B 44, 13319 (1991) [3] [3] K. Schwarz, J. Phys. F: Met. Phys. 16, L211 (1986) [4] [4] J. H. Park, E. Vescovo, H. J. Kim, C. Kwon, R. Ramesh, and T. Venkatesan, Nature (London) 392, 794 (1998) [5] [5] H. C. Kandpal, G. H. Fecher, and C. Felser, J. Phys. D 40, 1507 (2007) [6] [6] R. A. de Groot, F. M. Mueller, P. G. van Engen and K. H. J. Buschow, Phys. Rev. Lett. 50, 2024 (1983) [7] [7] S. Trudel, O. Gaier, J. Hamrle, and B. Hillebrands, J. Phys. D 43, 193001 (2010) [8] [8] W. H. Wang, E. Liu, M. Kodzuka, H. Sukegawa, M. Wojcik, E. Jedryka, G. H. Wu, K. Inomata, S. Mitani, and K. Hono, Phys. Rev. B 81, 140402 (R) (2010) [9] [9] W. H. Wang, H. Sukegawa, and K. Inomata, Phys. Rev. B 82, 092402 (2010) [10] [10] D. Berling, S. Zabrocki, R. Stephan, G. Garreau, J.L. Bubendorff, A. Mehdaoui, D. Bolmont, P. Wetzel, C. Pirri and G. Gewinner, J. Magn. Magn. Mat. 297, 118 (2006) [11] [11]M.S.Gabor, T.PetrisorJr., andC.Tiusan, M.Hehn and T. Petrisor, Phys. Rev. B 84, 134413 (2011) [12] [12] D. Fruchart, R. Fruchart, Ph. L’Héritier, K. Kane- matsu,R.Madar,S.Misawa,Y.Nakamura,P.J.Ziebeck, and K. R. A. Webster, in Magnetic Properties of Metals, edited by H. P. J. Wijn, Landolt- Börnstein, New Series, Group III, Vol. 19c, Pt. 2 (Springer-Verlag, Berlin, 1986). [13] [13] K. Inomata, N. Ikeda, N. Tezuk, R. Goto, S. Sug- imoto, M. Wojcik and E. Jedryka, Sci. Technol. Adv. Mater. 9 (2008) 014101 [14] [14] M. Belmeguenai, F. Zighem, Y. Roussigné, S-M. Chérif, P. Moch, K. Westerholt, G. Woltersdorf, and G. Bayreuther Phys. Rev. B 79, 024419 (2009) [15] [15]M.Belmeguenai, F.Zighem, T.Chauveau, D.Faurie, Y. Roussigné, S-M. Chérif, P. Moch, K.Westerholt and P. Monod, J. Appl. Phys. 108, 063926 (2010) [16] [16] Kh. Zakeri, J. Lindner, I. Barsukov, R. Mecken- stock, M. Farle, U. von Hörsten, H. Wende, W. Keune,J. Rocker, S. S. Kalarickal, K. Lenz, W. Kuch, and K. Baberschke, Phys. Rev. B 76, 104416 (2007) [17] [17] H. Lee, Y.-H. A. Wang, C. K. A. Mewes, W. H. Butler, T. Mewes, S. Maat, B. York, M. J. Carey, and J. R. Childress, Appl. Phys. Lett. 95, 082502 (2009) [18] [18] H. Kurebayashi, T. D. Skinner, K. Khazen, K. Ole- jnik, D. Fang, C. Ciccarelli, R. P. Campion, B. L. Gal- lagher, L. Fleet, A. Hirohata, and A. J. Ferguson, Appl. Phys. Lett 102, 062415 (2013) [19] [19] K D Sossmeier, F Beck, R C Gomes, L F Schelp and M Carara, J. Phys. D: Appl. Phys. 43, 055003 (2010) [20] [20] Y. Y. Zhou, X. Liu, J. K. Furdyna, M. A. Scarpulla and O. D. Dubon, Phys. Rev. B 80, 224403 (2009) [21] [21] W. Platow, A. N. Anisimov, G. L. Dunifer, M. Farle, and K. Baberschke, Phys. Rev. B 58, 5611 (1998) [22] [22] R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999) [23] [23] D. L. Mills and R. Arias, Physica B 384, 147 (2006) [24] [24] R. Arias and D. L. Mills, J. Appl. Phys. 87, 5455 (2000). [25] [25] R. P. Cowburn, S. J. Gray, J. Ferré, J. A. C. Bland, and J. Miltat, J. Appl. Phys. 78, 7210 (1995). [26] [26] K. Postava, D. Hrabovsky, J. Pistora, A. R. Fert, S. Visnovsky, and T. Yamaguchi, J. Appl. Phys. 91, 7293 (2002). [27] [27] R.M. Osgood III, S.D. Bader, B. M. Clemens, R.L. White, H. Matsuyama, J. Magn. Magn. Mater. 182, 297 (1998). [28] [28] G. Wolf, J. Hamrle, S. Trudel, T. Kubota, Y. Ando and B. Hillebrands, J. Appl. Phys.110, 043904 (2011) [29] [29] O. Gaier, J. Hamrle, S. J. Hermsdoerfer, H. Schultheiß, B. Hillebrands, Y. Sakuraba, M. Oogane, and Y. Ando, J. Appl. Phys. 103, 103910 (2008) [30] [30] S.Trudel, G. Wolf, J. Hamrle, B. Hillebrands, P. Klaer, M. Kallmayer, H.-J. Elmers, H. Sukegawa, W. Wang, and K. Inomata, Phys. Rev. B 83, 104412 (2011) [31] [31] S. Mizukami, D. Watanabe, M. Oogane, Y. Ando, Y. Miura, M. Shirai and T. Miyazaki, J. Appl. Phys.105, 07D306 (2009)

2013-05-03

Co2FeAl (CFA) thin films with thickness varying from 10 nm to 115 nm have been deposited on MgO(001) substrates by magnetron sputtering and then capped by Ta or Cr layer. X-rays diffraction (XRD) revealed that the cubic $[001]$ CFA axis is normal to the substrate and that all the CFA films exhibit full epitaxial growth. The chemical order varies from the $B2$ phase to the $A2$ phase when decreasing the thickness. Magneto-optical Kerr effect (MOKE) and vibrating sample magnetometer measurements show that, depending on the field orientation, one or two-step switchings occur. Moreover, the films present a quadratic MOKE signal increasing with the CFA thickness, due to the increasing chemical order. Ferromagnetic resonance, MOKE transverse bias initial inverse susceptibility and torque (TBIIST) measurements reveal that the in-plane anisotropy results from the superposition of a uniaxial and of a fourfold symmetry term. The fourfold anisotropy is in accord with the crystal structure of the samples and is correlated to the biaxial strain and to the chemical order present in the films. In addition, a large negative perpendicular uniaxial anisotropy is observed. Frequency and angular dependences of the FMR linewidth show two magnon scattering and mosaicity contributions, which depend on the CFA thickness. A Gilbert damping coefficient as low as 0.0011 is found.

Co2 FeAl thin films grown on MgO substrates: Correlation between static, dynamic and structural properties

1305.0714v1

1 Laser Controlled Spin Dynamics of Ferromagnetic Thin Film from Femtosecond to Nanosecond Timescale Sucheta Mondal and Anjan Barman * Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block JD, Secto r III, Salt Lake, Kolkata 700 106, India. * abarman@bose.res.in Key words: ( Thin Film Heterostructures, Ultraf a st Demagnetization, Gilbert Damping, Time - resolved Magneto - optical Kerr Effect ) Laser induced modulatio n of the magnetization dynamics occurring over various time - scales have been unified here for a Ni 80 Fe 20 thin film excited by amplified femtosecond laser pulses. The weak correlation between demagnetization time and pump fluence with substantial enhancemen t in remagnetization time is demo n strated using three - temperature model considering the temperatures of electron, spin and lattice. The picosecond magnetization dynamics is modeled using the L andau - Lifshitz - Gilbert equation. W ith increasing pump fluence th e Gilbert damping parameter shows significant enhancement from its intrinsic value due to increment in the ratio of electronic temperature to Curie temperature within very short time scale. The precessional frequency experiences noticeable red shift with i ncreasing pump fluence. The changes in the local magnetic properties due to accumulation and dissipation of thermal energy within the probed volume are described by the evolution of temporal chirp parameter in a comprehensive manner. A unification of ultra fast magnetic processes and its control over broad timescale would enable the integration of various magnetic processes in a single device and use one effect to control another. 2 I. INTRODUCTION Recent development in magnetic storage [1] and memory [2 ] device s heavily relies up on increasing switching speed and coherent switching of magnetic states in ferromagnetic thin films and patterned structures . O perating speeds of information storage devices have progressed into the sub - gig ahertz regime and controlled switching in individual layers of magnetic multilayers and hetero structures has become important . The relaxation processes involved in magnetization dynamics set natural limit s for these switching times and data transfer rates. In the context of precessional magnetization dynamics the natural relaxation rate against the small perturbation is expressed as Gilbert damping ( α ) according to the Landau - Liftshiz - Gilbert (LLG) equation [3, 4] . This is analogous to viscous damping of the mechanical frictional torque and leads to the direct dissipation of energy from the uniform precessional mode to thermal bath in case of zero wav e - vector excitation. Gilbert damping originates from spin - orbit coupling and depends on the coupling strength and d - band width of the 3 d ferromagnet [5] . Th e damping can be varied by various intrinsic and extrinsic mechanisms including phonon drag [6] , Edd y current [7], doping [8] or capping [9] with other material , injection of spin current [10] , magnon - magnon scattering [11] and controlling temperature of the system [12] . Intri n sic and extrinsic nature of Gilbert damping were primarily studied by using fe rromagnetic resonance (FMR) technique. When the magnetization is aligned with either in - plane or out - of - plane applied magnetic field, the linewidth is proportional to the frequency with a slope determined by damping co ef f icient. This is the homogeneous or intrinsic contribution to the FMR linewidth. However, experiments show an additional frequency - independent contribution to the linewidth corresponds to inhomogeneous line broadening [13, 14] . However , state - of - the - art technique based on pump - probe geomet ry has been developed and rigorously exploited for measuring ultrafast magnetization dynamics of ferromagnetic thin films during last few decades [15, 16] . Using time - resolved magneto - optical Kerr effect (TR - MOKE) technique one can directly address the pro cesses which are responsible for the excitation and relaxation of a magnetic system on their characteristic time scales [17 - 19] . Generally during the pump - probe measurements pump fluence is kept low to avoid nonlinear effects and sample surface degradation . Some recent experiments reveal that nonlinear spin waves play a crucial role in high power thin film precession al dynamics by introducing spin - wave instability [20] similar to FMR experiments by appl ication of high rf power [21] . The coercivity and aniso tropy of the ferromagnetic thin films can also be lowered by pump fluence , which may have potential application in heat assisted magnetic recording (HAMR) [22] . Recent report s reveal that damping coefficient can be increased or decreased noticeably in the higher excitation regime due to opening of further energy dissipation channels beyond a threshold pump power [23 - 25] . Not only relaxation parameters but also frequency shift due to enhancement in pump power has been observed [20] . However, the experimental evidence for large modulation of Gilbert damping along with frequency shift and temporal chirping of the uniform precessio n al motion is absent in the literature . This investigation demands suitable choice of material, and here we have chosen Permalloy (Ni 80 Fe 20 3 or Py here on) because of its high permeability, negligible magneto - crystalline anisotropy, very low coercivity, large anisotropic magnetoresistance with reasonably low damping. Also, due to its negligible magnetostriction P y is less sensitive to st rain and stress exerted during the thermal treatment in HAMR [22] . In this article , we have used femto - second amplified laser pulses for excitation and detection of ultrafast magnetization dynamics in a P y thin film. Pump fluence dependent ultrafast demag netization is investigated along with fast and slow remagnetization . Our comprehensive study of the picosecond dynamics reveals transient nature of enhanced Gilbert damping in presence of high pump fluence . Also , the time - varying precession is subjected to temporal chirping which occurs due to enhancement of temperature of the probed volume within a very short time scale being followed by successive heat dissipation. This fluence dependent modulation of magnetization dynamics will undoubtedly found suitable application in spintronic and magnonic devices . II. SAMPLE PREPARATION AND CHARACTERIZATION 20 - nm - thick Permalloy (Ni 80 Fe 20 , Py hereafter) film was deposited by using electron - beam evaporation technique (SVT Associates, model: Smart Nano Tool AVD - E01) (ba se pressure = 3 × 10 −8 Torr, deposition rate = 0.2 Å/S) on 8 × 8 mm 2 silicon (001) wafer coated with 300 - nm - thick SiO 2 . Subsequently, 5 - nm - thick SiO 2 is deposited over the Ni 80 Fe 20 using rf sputter - deposition technique (base pressure = 4.5 × 10 −7 Torr, Ar pressure = 0.5 mTorr , deposition r ate = 0.2 Å/S, rf power = 60 W). This capping layer protects the surface from environmental degradation, oxidation and laser ablation during the pump - probe experiment using femtosecond laser pulses. From the vibrating sample magnetometry (VSM ) we have obtained the saturation magnetization (M s ) and Curie temperature (T c ) to be 850 emu/cc and 86 3 K , respectively [26] . To study the ultrafast magnetization dynamics of this sample, we have used a custom - built time resolved magneto optical Kerr eff ect (TRMOKE) magnetometer based on optical pump - probe technique as shown in Fig. 1 (a). Here, the second harmonic (λ = 400 nm, repetition rate = 1 kHz, pulse width > 40 fs) of amplified femtosecond laser pulse generated from a regenerative amplifier system (Libra, Coherent) is used to excite the dynamics while the fundamental laser pulse (λ = 800 nm, repetition rate = 1 kHz, pulse width ≈ 40 fs) is used as probe to detect the time - resolved polar Kerr signal from the sample. The temporal resolution of the me asurement is limited by the cross - correlation between the pump and probe pulses ( ≈120 fs). The probe beam having diameter of about 100 µm is normally incident on the sample whereas the pump beam is kept slightly defocused (spot size is about 300 µm) and is obliquely ( ≈ 30 ◦ with normal to the sample plane) incident on the sample maintaining an excellent spatial overlap with the probe spot. Time - resolved Kerr signal is collected from the uniformly excited part of the sample and slight misalignment during the course of the experiment does not affect the pump - probe signal significantly. A large magnetic field of 3.5 kOe is first applied at a small angle of about 10° to 4 the sample plane to saturate its magnetization. This is followed by reduction of the magnetic field to the bias field value ( H = in - plane component of the bias field), which ensures that the magnetization remains saturated along the bias field direction. The tilt of magnetization from the sample plane ensures a finite demagnetizing field along the direction of the pump pulse, which is further modified by the pump pulse to induce a precessional dynamics within the sample [17] . In our experiment a 2 - ns time window has been used, which gave a damped uniform precession of magnetization. The pump beam is chopped at 373 Hz frequency and the dynamic signal in the probe pulse is detected by using a lock - in amplifier in a phase sensitive manner. Simultaneous time - resolved reflectivity and Kerr rotation data were measured and no significant breakthrough of one into another has been found [26] . The probe fluence is kept constant at 2 mJ / cm 2 during the measurement to avoid additional contribution to the modulation of spin dynamics via laser heating. Pump fluence ( F ) was varied from 10 to 55 mJ / cm 2 to study the fl uence dependent modulation in magnetization dynamics. All the experiments were performed under ambient condition and room temperature. III. RESULTS AND DISCUSSIONS A. Laser induced ultrafast demagnetization When a femtosecond laser pulse interacts with a ferromagnetic thin film in its saturation condition , the magnetization of the system is partially or fully lost within hundreds of femtosecond as measured by the time - resolved Kerr rotation or ellipticity. This is known as ultrafast demagnetization of the ferromagnet and was first observed by Beaurepire et al. in 1996 [ 27 ] . This is generally followed by a fast recovery of the magnetization within sub - picosecond to few picosecond s and a slower recovery within tens to hundreds of picoseconds, known as the fa st and slow remagnetization . In many cases the slower recovery is accompanied by a coherent magnetization precession and damping [ 17 ]. In our pump - probe experiment, the sample magnetization is maintained in the saturated state by application of a magnetic field H = 2.4 kOe before zero delay . Right after the zero - delay and the interaction of the pump pulse with the electrons in the ferromagnetic metal, ultrafast demagnetization takes place . The local magnetization is immediately quenched within first few hun dreds of fs followed by a subsequent fast remagnetization in next few ps [27] . Figure 1 (b) shows ultrafast demagnetization obtained for different pump fluences. Several models have been proposed over two decades to explain the ultrafast demagnetization [16 , 28 - 31] . Out of those a phenomenological thermodynamic model , called three temperature m odel [ 27, 32, 33] has been most widely used , where the dynamics of these spin fluctuations can be describes as: ) ( ) )( ( 0 ) ( 2 1 ) ( 1 2 1 t M e A A e A A A t M lat el sp el t lat el sp el lat el t sp el lat el sp el lat el (1) . This is an approximated form based on the assumption that the electron temperature rises instantaneously upon laser excitation and can be applied to fit time - resolved Kerr rotation data taken within few picoseconds timescale . T he whole system i s divided into three subsystems: 5 el ectron, spin and lattice system. On laser excitation the hot electrons are created above Fermi level. Then during energy rebalancing between the subsystems , quenched magnetization relaxes back to the initial state. The two exponential functions in the abov e equation mirror the demagnetization given by demagnetization time (τ el - sp ) for energy transfer between electron - spin and the decay of electron temperature ( τ el - lat ) owing to the tr ansfer of energy to the lattice. In addition to these characteristics time constants, the spin - lattice relaxation time also can be extracted by including another exponential term in the above equation if the spin specific heat is taken into account [ 34 ] . θ is the Heaviside step function and Γ(t) stands for the Gaussian function to be convoluted with the laser pulse envelope determining the temporal resolution (showing the cross c orrelation between the probe an d pump pulse) . The constant, A 1 indicates the ratio between amount of magnetization after equilibrium between electrons, s pins, and lattice is restored and the initial magnetization. A 2 is proportional to the initial electronic temperature rise. We have plotted A1 and A2, normalized with their values at the highest fluence, as a function of pump fluence in Fig. 3S of the supp lemental material which shows that magnitude of both parameters increases with laser fluence [26] . We have observed that with increasing fluence the demagnetization time has been negligibly varied within a range of 250 ±40 fs. The weak or no correlation bet ween the pump fluence and the demagnetization rate describes the intrinsic nature of the spin scattering , governed by various mechanisms including Elliott - Yafet mechanism [ 35 ] . Another important observation here is that the delay of demagnetization process es which is the time delay between pump pulse (full width at half maxima, FWHM ≈ 130±20 fs) and starting point of the ultrafast demagnetization, becomes shorter due to increase in pump fluences. A plausible explanation for this is the dependence of delay o f demagnetization on the electron - thermaliz ation time which is eventually proportional to electron density or pump fluences [ 3 6 ] . On the other hand , fast remagnetization time has been found to be increased noticeably from 0.40 ± 0.05 ps to 0.8 0 ± 0.05 ps w ithin the experimental fluence range of 10 - 55 mJ/cm 2 . The larger is the pump fluence, the higher is the electron temperature or further the spin temperature. Therefore, it is reasonable that magnetization recovery time increases with the pump fluence. B. Pump fluence dependent modulation in Gilbert damping F ig ure 1 (c) shows the representat ive Kerr rotation data for F = 25 mJ/cm 2 consisting of three temporal regions , i.e. ultrafast demagnetization , fast remagnetization and slow remagnetization superposed with damped precession within the time window of 2 ns . We process the magnetization precession part after subtracting a bi - exponential background to estimate the damping and its modulation . The slower remagnetization is mainly due to heat diffusion from th e lattice to the substrate and surrounding. Within our experimental fluence range the slow remagnetization time has increased from ≈0.4 ns to ≈1.0 ns. The precessiona l dynamics is described by phenomenological Landau - Lifshitz - Gilbert (LLG) equation, dt M d M M H M dt M d s eff (2) 6 where γ is the gyromagnetic ratio, M is magnetization , α is Gilbert damping constant and H eff is the effective magnetic field consisting of several field components. The variation of precessional frequency with the angle between sample plane and bias magnetic field direction is plotted in F ig. 1 (d ), which reveals that there is no uniaxial anisotropy present in this sample. The energy deposit ed by the pump pulse, in terms of heat within the probed volume, plays a very crucial role in modification of local magnetic properties , i.e. magnetic moment, anisotropy, coercivity, magnetic susceptibility , etc. With increasing fluence the precessional fr equency experienced a red shift [20, 25]. Thus, at the onset of the precessional dynamics (about 10 ps from zero delay), for relatively high fluence, the initial frequency ( f i ) will be smaller than its intrinsic value (in absence of any significant heat di ssipation). As time progresses and the sample magnetization gradually attains its equilibrium value, the precessional frequency continuously changes, causing a temporal chirping of the damped oscillatory Kerr signal. The frequency shift can be estimated fr om the amount of temporal chirping [ 3 7 ]. Figure 2 (a) shows the background subtracted time - resolved Kerr rotation data ( precessional part) for different pump fluences fitted with a damped sinusoidal function with added temporal chirping , ) ) ( 2 sin( / t bt f Ae i t k where A , τ , f i , b and Φ are the amplitude of the magnetization precession, the relaxation time, the initial precessional frequency, chirp parameter and initial phase, respectively. At this point, we are unsure of the exact nature of the damping, i.e. it may consis t of both intrinsic and extrinsic mechanisms and hence we term it as effective damping parameter ( α eff ) which can be extracted using the following formula [3 8 ] , ) 2 4 ( 1 eff eff M H ( 3 ) γ = 1. 83 ×10 7 Hz/ Oe for Py an d M eff is the effecti ve magnetization including pump - induced changes at H = 2.4 kOe . This formula is exploited to extract effective damping parameter precisely in the moderate bias fi eld regime. The variation of relaxation time and effective damping are plotted with pump fluence in F ig. 2 (b) and (c). Here, τ decreases with fluence while damping increases significantly with respect to its intrinsic value within this fluence range. We h ave repeated the experiment for two different field values (2.4 and 1.8 kOe). The slope of fluence dependent damping remains unaltered for both the field values . We have also observed increase in relative amplitude s of precession with pump fluence as shown in the inset of Fig. 2 (c). To verify the transient nature of damping we have performed another set of experiment where the probed area is exposed to different pump fluences ( F i ) for several minutes. After the irradiation, the precessioanl dynamics is mea sured from that area with fixed probe and pump fluences 2 and 10 mJ/cm 2 , respectively. We found that damping remains almost constant for all the measurements (as shown in Fig. 2 (d)) . These results demonstrate that the enhancement of 7 damping is transient a nd only exists in the presence of high pump fluence but dropped to its original value when the pump laser was set to initial fluence. The bias field dependence of precessional dynamics at four different pump fluence s is studied to gain more insight about the origin of fluence dependent damping. First, we plotted the average frequency ( f FFT ) with bias field which is obtained from the fast Fourier transformation (FFT) of the precessional data in F ig. 3 (a). The experimental data points are fitted with the Ki ttel formula, ) 4 ( 2 eff FFT M H H f ( 4 ) M eff is the effective magne tization of the sample. Figure 3 (b) shows that effective magnetization does not v ary much within the applied fluence range. So , we infer that with increasing fluence there is no induced anisotropy developed in the system which can modify the effective damping up to this extent [23] . The variation of relaxation time with bias field for four different pump fluences are plotted in F ig. 3 (c). Relaxation time is increased with decreasing field for each case but for the higher fluence regime, those value s seem to be fluctuating. This depend ence of τ on field was fitted with eq uation 3 to extract damping coefficient at different fluence values. We have further plotted the damping coefficient as a function of precession frequency ( f FFT ) [see supplementa l material , F ig. 4 S ] [26] , which shows a n invariance of α eff with f FFT . From that we can infer that the damping coefficient in our sample within the experimental field and fluence regime are intrinsic in nature and hence, we may now term it as the intrinsic damping coefficient α 0 . The extrinsic contributions to damping mainly come from magnetic anisotropy field, two - magnon scattering, multimodal dephasing for excitation of several spin - wave modes, etc, which are negligible in our present case. F igure 3 (d) shows the variation of α 0 with pump flue nce, which shows that even the intrinsic damping is significantly increasing with pump fluence [20, 3 9 ] . For generation of perpendicular standing spin - wave modes the film needs to be thick enough. Though the film thickness is 20 nm here, but within the app lied bias field range we have not found any other magnetic mode appearing with the uniform Kittel mode within the frequency window of our interest (as shown in F ig. 5 S of suppleme n tal material ) [26] . Also , for 20 - nm - thick Py film, the effect of eddy curre nt will be negligible [ 40 ] . The overlap between spatial profile of focused probe and pump laser spot may lead to the generation of magnons that propagate away from the region that is being probed. Generally , enhancement of nonlocal damping by spin - wave emi ssion becomes significant when the excitation area is less than 1 µm . Recently J. Wu et al. showed that propagation of magnetostatic spin waves could be significant even for probed regions of tens of microns in size [ 4 1 ] . Also , by generating spin - wave trap in the pump - probe experiment modification of precessional frequency in ferromagnetic thin film due to accumulation and dissipation of thermal energy within the probed volume has been reported [ 4 2 ] . D uring our experiment the overlap between probe and pump spot is maintai ned carefully and Kerr signal is 8 collected from the uniformly excited part of the sample so that slight misalignment during the course of experiment does not introduce any nonlocal effects . We will now substantiate our results with some theo retical arguments which involve the calculation of electronic temperature rise in the system due to application of higher pump fluence. The electronic temperature ( T e ) is related to absorbed laser energy per unit volume ( E a ) according to the following equa tion [ 4 3 ] , 2 / ) ( 2 0 2 T T E e a ( 5 ) where, ξ is the electronic specific heat of the system and T 0 is the initial electronic temperature (room temperature here). First, we have estimated E a according to the optical parameters of the sample by using the following equation, ] / ) 1 ( ) 1 [( d R F e E d a ( 6 ) where , d is sample thickness, Ψ is optical penetration depth ( ~ 17 nm for 400 - nm pump laser in 20 - nm - thick Py film ), R is the reflectivity of the sample (0. 5 measured for the Py film ) and F is applied pump fluence. By solving equations ( 5 ) and ( 6 ) we have observed that T e increases from ≈ 1800 to 4 5 00 K within our experimental fluence range of 10 to 55 mJ/cm 2 . Decay time of the electron temperature and other r elevant parameters (i.e. E a , T e at various flue nce s ) are described in the supplementa l material [26] . The sample remains in its magnetized state even if the electronic temperature exceeds the Curie temperature T c . Importantly, ratio of the system temperat ure , T (as decay of electronic temperature is strongly correlated with rise of lattice temperature) to T c is affecting the magnetization relaxation time which fundamentally depends on susceptibility . Accordingly damping should be proportional to susceptibi lity which is strongly temperature dependent [ 40 ] . Various procedures for exciting precessional dynamics in ferromagnets show the different mechanisms to be responsible for exploration of different energy dissipation channels. The spin - phonon interaction m echanism, which historically has been thought to be the main contribution to magnetization damping, is important for picosecond - nanosecond applications at high temperatures such as spin caloritronics. But for laser - induced magnetization dynamics, where spi n - flips occur mainly due to electron scattering, quantum Landau - Lifshitz - Bloch equation is sometimes exploited to explain the temperature dependence of damping by considering a simple spin - electron interaction as a source for magnetic relaxation [4 4 ] . This approach suggests that increasing ratio between system temperature and Curie temperature induces electron - impurity led spin - dependent scattering. Even slightly below T c a pure change in the magnetization magnitude oc c urs which causes the enhancement of da mping . Also our experimental results revea l that the precession amplitude and damping have been subjected to a sudden change for F > 30 mJ/cm 2 . Energy density deposited in the probed volume is proportional to pump fluence. For higher fluence, the temperatu re dependence of the electronic specific heat plays major role . The increase in the electronic specific heat value with temperature 9 may lead to longer thermal - relaxation time . We infer that relative balance between the energy depo sited into the lattice and electron system is also different for higher fluence regime compared to that in the lower fluence regime. Thus , the system temperature remains well above Curie temperature for F > 30 mJ/cm 2 , during the onset of precession for t ≥ 10 ps. This may open up additional energy dissipation channel for the magnetization relaxation process over nanoseconds time scale. Sometimes within very short time scale the spin temperature can go beyond the Curie temperature leading towards formation o f paramagnetic state but that is a highly non - equilibrium case [ 45 ] . However we believe that in our experiment, even for the high fluence limit and in local thermal equilibrium the ferro magnetic to paramagnetic tra n sition is not observed . R epetitive measur ements established the reversibility of the damping parameter and bias - magnetic - field dependence of precessional frequency confirms ferromagnetic nature of the sample . C. Frequency modulation and temporal chirping Pump fluence also eventua lly modulates the precessional frequency by introducing temporal chirping in the uniform precession. After immediate arrival of pump pulse, due to enhancement of the surface temperature, the net magnetization is reduced in picosecond time scale which resul ts in chirping of the precessional oscillation . The initial frequency ( f i ) is reduced with respect to its intrinsic value at a constant field. But when the probed volume cools with time, the spins try to retain their original precessional frequency. Thus , within a fixed time window, the average frequency ( f FFT ) also undergoes slight modification. In the high fluence regime, significant red shift is observed in both f FFT and f i . For H = 2.4 and 1.8 kOe, modulation of frequency is found to be 0.020 GHz.cm 2 /mJ for f FFT and 0.028 GHz.cm 2 /mJ for f i , from the slope of linear fit (as shown in F ig. 4(a)). The f FFT is redu ced by 7.2 % of the extrapolated value at zero pump fluence for both the fields. On the other hand, f i is decreased by 8.7% of its zero pump value f or the highest pump fluence . The temporal chirp parameter , b shows giant enhancement within the experimental fluence range ( F ig. 4 (b)). For H = 2.4 kOe, b has increased up to ten times (from 0.03 GHz/ns to 0.33 GHz/ns) in this fluence limit which implies a n increase in frequency of 0.66 GHz. Within our experimental scan window (2 ns) , the maximum frequency shift is found to be 4.5% for F = 55 mJ/cm 2 . For another bias field ( H = 1.8 kOe), the enhancement of chirp parameter follows the similar trend. This ult rafast modulation is attributed to the thermal effect on the local magnetic properties within the probe d volume and is inferred to be reversible [ 3 7 ] . We have also plotted the variation of b with applied bias field for four different pump fluencies . I t see ms to be almost constant for all the field values in moderate fluence regime (as shown in F ig. 4 (c)) . But for F = 40 mJ/cm 2 , data points are relatively scattered and large errors have been considered to take care of those fluctuations. 10 IV. CONCLUSION In essence, fluence dependent study of ultrafast magnetization dynamics in Ni 80 Fe 20 thin film reveals very weak correlation between ultrafast demagnetization time and Gilbert damping within our experimental fluence range. W e have reported large enhancement o f damping with pump fluence. F rom the bias field as well as pump fluence dependence of experimentally obtained dynamic al parameters we have excluded all the possible extrinsic contributions and observed a pump - induced modulation of intrinsic Gilbert dampin g. Also , from repetitive measurements with different pump irradiat ion we have shown that the pump - induced changes are reversible in nature. Enha n cement of the system temperature to Curie temperature ratio is believed to be responsible for increment in rema gnetization times and damping. The temporal chirp parameter has been found to be increased by up to ten times within the experimental fluence range , while the frequency experiences a significant red shift. F rom application point of view, as increasing dema nd for faster and efficient magnetic memory devices, has led the scientific community in the extensive research field of ultrafast magnetization dynamics, our results will further enlighten the understanding of modulation of magnetization dynamics in ferro magnetic systems in presence of higher pump fluence. Usually l ow damping materials are preferred because it is easier to switch their magnetization in expense of smaller energy , lower write current in STT - MRAM devices and longer propagation length of spin waves im magnonic devices . On the other hand, higher damping is also required to stop the post switching ringing of the signal. The results also have important implications on the emergent field of all - optical helicity dependent switching [4 6 - 4 8 ]. In thi s context, the transient modulation of Gilbert damping and other dynamical parameters in ferromagnetic materials is of fundamental interest for characterizing and controlling ultrafast responses in magnetic structures. Acknowledgements: We gratefully ack nowledge the financial support from S. N. Bose National Centre for Basic Sciences (grant no.: SNB/AB/12 - 13/96 and SNB/AB/18 - 19/211 ) and Department of Science and Technology (DST), Government of India (grant no.: SR/NM/NS - 09/2011 ). We also gratefully ackno wledge the technical assistance of Dr. Jaivardhan Sinha and Mr. Samiran Choudhury for preparation of the sample. SM acknowledges DST for INSPIRE fellowship. References : [ 1 ] O. Hellwig, A. Berger, T. Thomson, E. Dobisz, Z. Z Bandic, H. Yang, D. S. Kercher and E. E. Fullerton, Separating dipolar broadening from the intrinsic switching field distribution in perpendicular patterned media , Appl. Phys. Lett. 90 , 162516 (2007) . 11 [ 2 ] S. Tehrani, E. Chen, M. 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Cahill, Optical - helicity - driven magnetiza tion dynamics in metallic ferromagnets, Nat. Comm. 8 ,15085 (2017). [4 7 ] T. D. Cornelissen, R. Córdoba and B. Koopmans, Microscopic model for all optical switching in ferromagnets, Appl. Phys. Lett. 108 , 142405 (2016). [4 8 ] Md. S, El Hadri, M. Hehn, G. Ma linowski and S. Mangin, J. Phys. D: Appl. Phys. Materials and devices for all - optical helicity dependent switching, 50 , 133002 (2017). 15 Figure 1 : (a) Schematic of experimental geometry . In the inset, φ is shown as in - plane rotational angle of H, (b) pump fluence dependence of ultrafast demagnetization; Solid lines are fit ting line s. P ump fluences ( F ) having unit of mJ/cm 2 are mentioned in numerical figure. The Gaussian envelope of laser pulse is presented to describe the convolution. (c) Repre sentative time resolved Kerr rotation data with three distinguished temporal regions for F = 25 mJ/cm 2 . (d) Angular variation of precessional frequency at H = 1.1 kOe for 20 - nm - thick Py film. φ is presented in degree. 16 Figure 2 : (a) Background subtracted time - resolved Kerr rotation data for different pump fluences at H = 2.4 kOe. F having unit of mJ/cm 2 is mentioned in numerical figure. Solid lines are fit ting lines . Pump f luence dependen ce of (b) relaxation time ( τ ) and (c) effective damping ( α eff ). Black and blue symbols represent the variation of these parameters at two different field values, H = 2.4 and 1.8 kOe, respectively. A mplitude of precession is also plotted with pump fluence for H = 2.4 kOe , (d) Variation of effective damping with irradiation fluence ( F i ) at H = 2.4 kOe. In order to check the possible damage in the sample as high fluence values the pump fluence was taken up to the targeted value of F i for several minut es followed by reduction of the pump fluence to a constant value of 10 mJ/cm 2 and the pump - probe measurement was performed. The damping coefficient is found to be unaffected by the irradiation fluence as shown in (d). 17 Figure 3 : (a) Bias field dependence of precessional frequency for F = 10 mJ/cm 2 . The red solid line indicates the Kittel fit. (b) Pump fluence dependence of effective magnetization (M eff ) of the probed volume. (c) Bias field dependence of relaxation time ( τ ) for four differ ent fluences. F having unit of mJ/cm 2 is mentioned in numerical figures. Solid lines are the fitted data. (d) Variation of intrinsic Gilbert damping ( α 0 ) with pump fluence. 18 Figure 4 : (a) Pump - fluence dependence of precessional fre quencies for H = 2.4 and 1.8 kOe. Red and black symbols represent the variation of average frequency ( f FFT ) and initial frequency ( f i ) respectively. (b) Variation of temporal chirp parameter ‘ b’ with pump fluence for two different magnetic field values. (c ) Variation of temporal chirp parameter with bias field for four different pump fluences. F having unit of mJ/cm 2 is mentioned in numerical figure. Dotted lines are guide to eye.

2018-12-20

Laser induced modulation of the magnetization dynamics occurring over various time-scales have been unified here for a Ni80Fe20 thin film excited by amplified femtosecond laser pulses. The weak correlation between demagnetization time and pump fluence with substantial enhancement in remagnetization time is demonstrated using three-temperature model considering the temperatures of electron, spin and lattice. The picosecond magnetization dynamics is modeled using the Landau-Lifshitz-Gilbert equation. With increasing pump fluence the Gilbert damping parameter shows significant enhancement from its intrinsic value due to increment in the ratio of electronic temperature to Curie temperature within very short time scale. The precessional frequency experiences noticeable red shift with increasing pump fluence. The changes in the local magnetic properties due to accumulation and dissipation of thermal energy within the probed volume are described by the evolution of temporal chirp parameter in a comprehensive manner. A unification of ultrafast magnetic processes and its control over broad timescale would enable the integration of various magnetic processes in a single device and use one effect to control another.

Laser Controlled Spin Dynamics of Ferromagnetic Thin Film from Femtosecond to Nanosecond Timescale

1812.08404v1

arXiv:1502.00176v1 [math.RA] 31 Jan 2015BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES WITH INTEGER COEFFICIENTS ON CYCLES NEALY BOWDEN, SARAH HAGEN, MELANIE KING, AND STEPHANIE REIN DERS Abstract. Aninteger generalized spline is a set of vertex labels on an edge- labeled graph that satisfy the condition that if two vertices are join ed by an edge, the vertex labels are congruent modulo the edge label. Foundationa l work on these objects comes from Gilbert, Polster, and Tymoczko, who generaliz e ideas from geometry/topology (equivariant cohomology rings) and algebra (a lgebraic splines) to develop the notion of generalized splines . Gilbert, Polster, and Tymoczko prove that the ring of splines on a graph can be decomposed in terms of splin es on its subgraphs (in particular, on trees and cycles), and then fully an alyze splines on trees. Following Handschy-Melnick-Reinders and Rose, we analyz e splines on cycles, in our case integer generalized splines. The primary goal of this paper is to establish two new bases for the m odule of integer generalized splines on cycles: the triangulation basis and t he King ba- sis. Unlike bases in previous work, we are able to characterize each b asis element completely in terms of the edge labels of the underlying cycle. As an ap plication we explicitly construct the multiplication table for the ring of integer g eneralized splines in terms of the King basis. 1.Introduction Aninteger generalized spline is a set of vertex labels on an edge-labeled graph that satisfy the condition that if two vertices are joined by an edge, the vertex labels are congruent modulo the edge label. (See Definition 2.1 for a precise sta tement.) Figure 1 shows examples of splines on a three-cycle. Theterm“spline”comesfromthenameofthethinstripsofwooduse dbyengineersto model larger constructions like ships or cars. Mathematicians later adopted the term We are extremely grateful to Julianna Tymoczko, Elizabeth Drellich, and Yue Cao for their insight and contributions to this paper. We would also like to thank Rut h Haas and Joshua Bowman for valuable discussions on these topics, and Michael DiPasq uale for his thorough review and comments. This work was supported by Smith College and the Nat ional Science Foundation through the Center for Women in Mathematics [DMS-1143716]. 12 BOWDEN, HAGEN, KING, AND REINDERS 25 3 111 25 3 0212 25 3 0015 Figure 1. The edge labels are t2,5,3uand the sets of vertex labels t1,1,1u,t0,2,12u, and t0,0,15ueach form a spline on the cycle. torefertopiecewisepolynomialsonpolytopeswiththepropertytha tthepolynomials on the faces agree at their shared edges up to a given degree of sm oothness. These mathematical splines are also used for object-modeling purposes, hence the use of the name. Billera pioneered the algebraic study of splines, especially looking into q uestions regarding thedimension ofthe moduleof splines [2]. Many peoplecont inued Billera’s work, including among others, Rose [12, 13] and Haas [7] who worked on identifying dimension and bases for the module of splines. Splinetheorydevelopedindependently intopologyandgeometry. Go resky, Kottwitz, and MacPherson [6], Payne [11], and Bahri, Franz, and Ray [1] constr ucted equivari- ant cohomology rings using splines, although they did not use that na me. Gilbert, Polster, and Tymoczko generalize the notion of splines that we use here to what they call generalized splines [4] . These generalized splines are built on the dual graph of the polytopes found in classical splines. The work of B illera and Rose shows that the two constructions (on polytopes or their duals) ar e equivalent in most cases, including the cases of classical interest [3]. Cycles turn out to be a particularly important family of graphs to stu dy. Indeed Gilbert, Polster, and Tymoczko show that the ring of generalized sp lines on a graph Gcan be decomposed in terms of splines on certain trees and cycles in G[4]. They completely describe splines on trees, while leaving open the investigat ion of splines on cycles. Similarly, Rose showed that cycles play a key role in the relat ions defining modules of splines [13]. Handschy, Melnick, and Reinders begin analysis of integer generalize d splines on cycles [9]. They prove the existence of a certain flow-up basis (see D efinition 2.3), what we call the smallest-value basis, for splines on cycles, and thus prove that suchBASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 3 spline modules are free. They define their basis for arbitrary cycles , but only have formulas for the leading nonzero elements. Inthispaperwe introducetwo newbases forthemoduleofinteger g eneralized splines on cycles: the triangulation basis and the King basis. Each of these b ases is fully expressible in terms of the edge labels of the cycle, and each has its o wn strengths. The triangulation basis, so called because it is constructed from tria ngulated cycles, is useful because it exists on arbitrary cycles (Theorem 4.2). The a dvantage of the King basis lies in the fact that it is relatively simple to calculate, with the e ntries almost constant (Definition 5.1). Although the King basis only exists o n cycles with a pair of relatively prime adjacent edge labels, this restriction is not u ncommon in applications. Infactanevengreaterrestrictionthatalledgelabe lsberelativelyprime is commonly used [5, 10]. The results of our work naturally generalize t o principle ideal domains, which include classical univariate splines and Pr¨ ufer d omains; see forthcoming work [8]. As an application we present the multiplication table of splines on cycles where the products of splines are expressed in terms of the King basis. Finding multiplica- tion tables of equivariant cohomology rings in terms of Schubert bas es is the central problem of Schubert calculus. We view this work as a step in that geom etric direc- tion. The rest of this paper is organized as follows. In Section 2 we summar ize the im- portant definitions and theorems that we use in our work. In Sectio n 3 we provide a criterion for the existence of flow-up bases. Sections 4 and 5 are dedicated to proving the existence of the triangulation basis and King basis respe ctively. In the final section we give the multiplication table for the King basis and end w ith an open question. 2.Preliminaries 2.1.Results from Handschy, Melnick, and Reinders. Handschy, Melnick, and Reinders proved a number of results about splines on cycles [9]. Many of their propositionsandtheorems play key rolesinour proofsregarding tr iangulationsplines and King splines. We also use their notation, which we describe in this se ction. 2.1.1.Basic Definitions. The foundational combinatorial object we study is an edge- labeled graph, defined here:4 BOWDEN, HAGEN, KING, AND REINDERS Definition 2.1 (Edge-Labeled Graphs) .LetGbe a graph with kedges ordered e1,e2,...,e kandnvertices ordered v1,...,vn. Letℓibe a positive integer label on edgeeiand letL“ tℓ1,...,ℓkube the set of edge labels. Then pG,L qis an edge- labeled graph. Withthisnotationforedge-labeledgraphswehavetheformaldefin itionofsplines: Definition 2.2 (Splines).A spline on the edge-labeled graph pG,L qis a vertex- labeling as follows: if two vertices are connected by an edge eithen the two vertex labels are equivalent modulo ℓi. We denote a spline G“ pg1,...,gnqwheregiis the label on vertex vifor1ďiďn. In this paper we assume the labels giPZ. 2.1.2.Flow-Up Classes and the Smallest-Value Basis. Flow-up classes are a partic- ularly nice class of splines on cycles. They arise geometrically ([5], [10], [1 4]) and are an analogue of upper triangular matrices. Definition 2.3 (Flow-Up Classes) .Fix a cycle with edge labels pCn,Lqand fixk with1ďkăn. A flow-up class GkonpCn,Lqis a spline with kleading zeros. We say that a basis whose elements are flow-up classes is a flow-up basis . The simplest flow-up class is the trivial spline; It exists on any edge-labele d cycle. Proposition 2.4 (Trivial Splines [9, Prop 2.5]) .Fix a cycle with edge labels pCn,Lq. The smallest flow-up class on pCn,LqisG0“ p1,...,1q. Moreover, any multiple of G0is also a spline. We call the multiples of G0trivial splines. The following theorem establishes that flow-up classes exist on any e dge-labeled cycle. Theorem 2.5 (Flow-Up Classes on n-cycles [9, Thrm 4.3]) .Fix a cycle with edge labels pCn,Lq. Letně3and1ďkăn. There exists a flow-up class GkonpCn,Lq. The next definition introduces smallest flow-up classes. Definition 2.6 (Smallest Flow-Up Class) .Fix a cycle with edge labels pCn,Lq. The smallest flow-up class Gk“ p0,...,0,gk`1,...,gnqonpCn,Lqis the flow-up class whose nonzero entries are positive and if G1 k“ p0,...,0,g1 k`1,...,g1 nqis another flow-up class with positive entries then g1 iěgifor all entries. By convention we consider G0“ p1,...,1qthe smallest flow-up class G0.BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 5 The following theorem gives an explicit formula for the smallest leading e lement of flow-up classes. Theorem 2.7 (Smallest Leading Element of Gk[9, Thrm 4.5]) .Fix a cycle with edge labels pCn,Lq. Fixně3andksuch that 2ďkăn. LetGk´1“ p0,...,0,gk...,gnqbe a flow-up classon pCn,Lq. Theleadingelement gkis amultiple of lcmpℓk´1,gcdpℓk,...,ℓnqq and there is a flow-up class Gk´1withgk“lcmpℓk´1,gcdpℓk,...,ℓnqq. The smallest flow-up classes exist and form a basis for the set of splin es given any edge-labeled cycle. Theorem 2.8 (Basisfor n-Cycles [9, Thrm4.7]) .Fix a cycle with edge labels pCn,Lq. The smallest flow-up classes G0,G1,...,Gn´1exist on pCn,Lqand form a basis over the integers for the Z-module of splines on pCn,Lq. 2.2.Useful Computational Tool. For reasons related to finding an explicit basis for splines on cycles, we want to find a formula for the value of the va riablexin the following pair of congruences: # x”ymoda x”0 modb We note the conditions for when such a solution exists and we give an e xplicit formulation for xin terms of y,a, andbprovided a solution does exist. Proposition 2.9. Consider the system of congruences # x”ymoda x”0 modb. If this system has a solution then one solution is given by the following formula: ‚Ifa gcdpa,bq“1thenx“bis a solution to the system. ‚Ifa gcdpa,bq‰1then x“yˆb gcdpa,bq˙ ˆb gcdpa,bq˙´1 modpa gcdpa,bqq is a solution to the system.6 BOWDEN, HAGEN, KING, AND REINDERS Proof.The Chinese Remainder Theorem tells us that this system of congrue nces is satisfied if and only if y”0 mod gcd pa,bq. In what follows we will assume that a solution exists, and thus that y”0 mod gcd pa,bq. Case 1: Let’s deal first with the case wherea gcdpa,bq“1. This condition implies that gcd pa,bq “aand sob“anfor some nPZ. Because y”0 mod gcd pa,bqby assumption and gcd pa,bq “awe have y”0 moda. In other words, y“amfor somemPZ. Thenx“bsatisfies the system of congruences because bis congruent to zero modulo bandb“anis congruent to y“ammoduloa. Case 2:Now supposea gcdpa,bq‰1. We can rewrite the system of congruences as # x“y`as x“bt Equate both expressions. bt“y`as Recall that y”0 mod gcd pa,bq. This allows us to divide both sides by gcd pa,bqand get an integer as the result.ˆb gcdpa,bq˙ t“y gcdpa,bq`ˆa gcdpa,bq˙ s Putting this back into modular form we haveˆb gcdpa,bq˙ t“y gcdpa,bqmodˆa gcdpa,bq˙ . The integers´ b gcdpa,bq¯ and´ a gcdpa,bq¯ are relatively prime so we can take the inverse of the first modulo the second. t”y gcdpa,bqˆb gcdpa,bq˙´1 modˆa gcdpa,bq˙ . Plug this expression for tinto the equation x“bt: x“yˆb gcdpa,bq˙ ˆb gcdpa,bq˙´1 modpa gcdpa,bqq. This value is a solution to the original system of congruences. /square Notice that this second case simplifies enormously if gcdpa,bq “1. In this situation xreduces to: x“ybrb´1smodaBASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 7 3.Basis Condition LetpG,L qbe an arbitrary graph on nvertices with an arbitrary edge-labeling. Con- sider a set of flow-up classes G0...Gn´1onpG,L q. In this section we give a necessary and sufficient condition for this set to form a basis for the module of t he splines on pG,L q. Any set G0,...,Gn´1that meets this basis condition is called a flow-up basis . Such a basis is useful because linear independence is trivially verified. LetG0...Gn´1be a set of flow-up classes and for each idenote Gi“ p0,...,0,gpiq i`1,...,gpiq nq. The subscript of each gpiqindicates the entry-position of gpiqin the spline Gi. The superscript piqis to keep track of the fact that we are working with the flow-up clas s Gi. In much of this paper and in previous work the superscript is suppr essed when the flow-up class in question is obvious. Theorem 3.1 (Basis Condition) .The following are equivalent: ‚The set tG0,...,Gn´1uforms a flow-up basis. ‚For each flow-up spline Ai“ p0,...,0,ai`1,...,a nqthe entry ai`1ofAiis an integer multiple of the entry gpiq i`1ofGi. Proof.Suppose that G0,...,Gn´1forms a flow-up basis for the module of splines on a graph pG,L q. Suppose that Ai“ p0,...,0,ai`1,...,a nqis a spline on pG,L qwith exactlyileading zeros. We will show that ai`1“cgpiq i`1for some cPZ. SinceG0,...,Gn´1form a basis, we can write Aias a linear combination of the splinesG0,...,Gn´1. The fact that Aihasileading zeros implies that the coeffi- cients of G0,...,Gi´1must be 0. Thus we have Ai“ciGi`...`cn´1Gn´1for some ci,...,c n´1PZ. Consider the pi`1qthentry of the splines on the right-hand side of this equation. Note that Giis the only element of Gi,...,Gn´1with a nonzero entry in this position. Considering the pi`1qthentry on each side of the equation, we have ai`1“cigpiq i`1`ci`10`...`cn´10“cigpiq i`1. Now we prove the converse. Let A“ pa1,...,a nqbe an arbitrary spline on pG,L q. We prove by induction that A“A1 j`j´1ÿ k“0ckGk8 BOWDEN, HAGEN, KING, AND REINDERS for all 1 ďjďnwhereA1 jis a spline with (at least) jleading zeros. For our base case, note that by hypothesis we have A“¨ ˚˚˚˝an´c0gp0q n ... a2´c0gp0q 2 0˛ ‹‹‹‚`c0G0 sincea1“c0gp0q 1. Letting A1 1“ p0,a2´c0gp0q 2,...,a n´c0gp0q nqgivesA“A1 1`ř0 k“0ckGk. Thus our claim holds for j“1. Suppose as our induction hypothesis that we have A“A1 i`ři´1 k“0ckGkfor some 1ďiďn´1. We can write this as A“¨ ˚˚˚˚˚˚˚˝a1 n... a1 i`1 0 ... 0˛ ‹‹‹‹‹‹‹‚`i´1ÿ k“0ckGk. By hypothesis we have that a1 i`1“cigpiq i`1for some ciPZ. So we can write A“¨ ˚˚˚˚˚˚˚˚˚˚˝a1 n´cigpiq n ... a1 i`2´cigpiq i`2 0 0 ... 0˛ ‹‹‹‹‹‹‹‹‹‹‚`iÿ k“0ckGk. LettingA1 i`1“ p0,...,0,0,a1 i`2´cigpiq i`2,...,a1 n´cigpiq nqgivesusA“A1 i`1`ři k“0ckGk. By induction we have A“A1 j`řj´1 k“0ckGkfor all 1 ďjďn. In particular we have A“A1 n`řn´1 k“0ckGk. ButA1 nis a spline with nleading zeros. So A1 n“ p0,...,0q. ThusA“řn´1 k“0ckGk. We conclude that every spline can be written as a linear combination of G0,...,Gn´1as desired. /squareBASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 9 One important observation is that the basis condition is only a conditio n on the first nonzero entry of each spline in a set of flow-up classes G0,...,Gn´1. This gives us the following useful corollary: Corollary 3.2. Suppose the set of flow-up classes tG0,...,Gn´1uforms a basis for the module of splines. Suppose tG1 0,...,G1 n´1uis a set of flow-up classes for which for eachithe first nonzero entry of G1 iequals the first nonzero entry of Gi. Then the settG1 0,...,G1 n´1ualso forms a basis for the module of splines. 4.The Triangulation Splines Triangulationsplinesformanotherbasisofflow-upclassesforcycle s. Theyaresimilar toHandschy, Melnick, andReinders’ smallest-valueflow-upclasses inthattheleading nonzero elements of both are the same. However we give a formula f or every entry of the triangulation splines, unlike the smallest-value flow-up classes . Definition 4.1 (Triangulation Splines) .Fix an edge-labeled cycle pCn,Lq. For 1ďkďn´1the vector Hk“ p0,...,0,hk`1,...,hnqhas entries as follows: ‚hk`1“lcmpℓk,gcdpℓk`1,...,ℓnqq ‚Fork`1ăiďnifℓi´1 gcdpℓi´1,...,ℓnq“1thenhi“gcdpℓi,...,ℓnq. ‚Fork`1ăiďnifℓi´1 gcdpℓi´1,...,ℓnq‰1then hi“hi´1ˆgcdpℓi,...,ℓnq gcdpℓi´1,...,ℓnq˙ ˆgcdpℓi,...,ℓnq gcdpℓi´1,...,ℓnq˙´1 modℓi´1 gcdpℓi´1,...,ℓnq The next theorem establishes that triangulation splines exist on any edge-labeled cycle. Theorem 4.2 (ExistenceofTriangulationSplines) .Fix an edge-labeledcycle pCn,Lq. For1ďkďn´1the vector Hkis a spline on pCn,Lq. Proof.Start with an edge-labeled cycle pCn,Lq. For 3 ďkďn´1 add an edge between vertices v1andvkas shown in Figure 2. Label the edge between v1andvk with gcd pℓk,...,ℓnq. We will show the vector Hksatisfies all of the edge conditions represented by this graph, which implies it satisfies the cycle’s edge c onditions in particular.10 BOWDEN, HAGEN, KING, AND REINDERS ℓ1ℓ2ℓ3ℓn´1 ℓn gcdpℓn´1,ℓnq gcdpℓ4,...,ℓnq gcdpℓ3,...,ℓnq (a)Add edgesℓ1ℓ2ℓ3ℓn´1 ℓn gcdpℓn´1,ℓnq gcdpℓ4,...,ℓnq gcdpℓ3,...,ℓnq 0h2h3 (b)Base case Figure 2. Triangulated Cycle Label vertices v1,...,vkzero. Label vertex vk`1with hk`1“lcmpℓk,gcdpℓk`1,...,ℓnqq. The integer hk`1satisfies the edge conditions on the downward edges (edges with lower-indexed vertices) at vertex vk`1by construction: # hk`1”0 modℓk hk`1”0 mod gcd pℓk`1,...,ℓnq This is our base case, and we will label vertices from hk`2tohn´1inductively. Our induction hypothesis is that hk`1,...,hifork`1ďiďn´1 satisfy the edge conditions for downward edges. Consider the system of congruen ces at vertex vi`1 represented by the edges labeled ℓiand gcd pℓi`1,...,ℓnq: # hi`1”himodℓi hi`1”0 mod gcd pℓi`1,...,ℓnq By the Chinese Remainder Theorem a solution hi`1exists if and only if hi”0 mod gcdpℓi,gcdpℓi`1,...,ℓnqq. In other words a solution exists if and only if hi”0 mod gcdpℓi,...,ℓnq. By our induction hypothesis hisatisfies the downward edge conditions at vertex viso in particular hi”0 mod gcd pℓi,...,ℓnq. Thus a solution hi`1exists.BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 11 This means hi`1“$ & %hi´ gcdpℓi`1,...,ℓnq gcdpℓi,...,ℓnq¯ ´ gcdpℓi`1,...,ℓnq gcdpℓi,...,ℓnq¯´1 modℓi gcdpℓi`1,...,ℓnqifℓi gcdpℓi,...,ℓnq‰1 gcdpℓi`1,...,ℓnq ifℓi gcdpℓi,...,ℓnq“1 is a solution by Proposition 2.9. In conclusion we can label each vertex vifork`1ăiďn´1 with hi“$ & %hi´1´ gcdpℓi,...,ℓnq gcdpℓi´1,...,ℓnq¯ ´ gcdpℓi,...,ℓnq gcdpℓi´1,...,ℓnq¯´1 modℓi´1 gcdpℓi´1,...,ℓnqifℓi´1 gcdpℓi´1,...,ℓnq‰1 gcdpℓi,...,ℓnq ifℓi´1 gcdpℓi´1,...,ℓnq“1 andhiwill satisfy the edge conditions represented by the edges labeled ℓi´1and gcdpℓi,...,ℓnq. Lastly for an integer hnto satisfy the edge conditions at vertex vnit must satisfy the following system of congruences: # hn”hn´1modℓn´1 hn”0 modℓn The Chinese Remainder Theorem tells us that a solution hnexists to this system if and only if hn´1”0 mod gcd pℓn´1,ℓnq. We showed by induction that our choice of hn´1satisfies the edge conditions of the downward edges at the pn´1q-th vertex. In particular this means hn´1”0 mod gcd pℓn´1,ℓnqbecause this is the edge condition represented by the edge labeled gcd pℓn´1,ℓnq. Therefore hn“$ & %hn´1´ ℓn gcdpℓn´1,ℓnq¯ ´ ℓn gcdpℓn´1,ℓnq¯´1 modℓn´1 gcdpℓn´1,ℓnqifℓn´1 gcdpℓn´1,ℓnq‰1 ℓn ifℓn´1 gcdpℓn´1,ℓnq“1 satisfies the vertex vnedge conditions by Proposition 2.9. Choose this integer to label the n-th vertex. All of the congruences represented by the graph are accounted for so the vector Hk“ p0,...,0,hk`1,...,hnqis a spline on the graph. In particular Hkis a spline on the cycle pCn,Lqas desired. /square12 BOWDEN, HAGEN, KING, AND REINDERS The Corollary to the Basis Condition Theorem allows us to succinctly co nclude that the set of triangulation splines H0,...,Hn´1forms a basis for the set of splines on an edge-labeled cycle. Theorem 4.3. Fix an edge-labeled cycle pCn,Lq. The set of triangulation splines H0,...,Hn´1form a basis for the set of splines on pCn,Lq. Proof.Thesetofsmallestflow-upclasses G0,...,Gn´1formabasisforthesetofsplines onpCn,Lqby Theorem 2.8. The leading entry of Hkequals the leading entry of Gk by construction for 0 ďkďn´1. Thus the set of triangulation splines H0,...,Hk forms a basis for the set of splines on pCn,Lqby Corollary 3.2. /square As an example, we calculate the triangulation basis for the 4-cycle wit h edge labels t2,6,10,15u. 2615 10 The first basis element H0is, as always, the trivial spline p1,1,1,1q. The nonzero entries of the second basis element H1are calculated as follows: hp1q 2“lcmp2,gcdp6,10,15qq “2 hp1q 3“2ˆgcdp15,10q gcdp6,15,10q˙ ˆgcdp15,10q gcdp6,15,10q˙´1 mod6 gcdp6,15,10q“2¨5¨ p5q´1 mod 6 “50 hp1q 4“50ˆgcdp10q gcdp15,10q˙ ˆgcdp10q gcdp15,10q˙´1 mod15 gcdp15,10q“50¨2¨ p2q´1 mod 3 “200 The nonzero entries of the third basis element H2are calculated as follows: hp2q 3“lcmp6,gcdp10,15qq “30 hp2q 4“30ˆgcdp10q gcdp15,10q˙ ˆgcdp10q gcdp15,10q˙´1 mod15 gcdp15,10q“50¨2¨ p2q´1 mod 3 “120 The only nonzero element of the final basis element H3ishp3q 4“lcmp15,10q “ 30. Thus we have the following triangulation basis for the 4-cycle with edge la- bels t2,6,10,15u:H0“ p1,1,1,1q,H1“ p0,2,15,200q,H3“ p0,0,30,120q, and H4“ p0,0,0,30q.BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 13 5.The King Splines In this section we define King splines on n-cycles and prove that they form a basis for the set of splines. Definition 5.1 (King splines) .Fix a cycle with edge-labels pCn,Lqand assume ℓn´1 andℓnrelatively prime. The King splines on pCn,Lqare the vectors K0“¨ ˚˚˚˚˚˚˝1 1 ... 1 1 1˛ ‹‹‹‹‹‹‚,K1“¨ ˚˚˚˚˚˚˝k1 ℓ1... ℓ1 ℓ1 0˛ ‹‹‹‹‹‹‚,K2“¨ ˚˚˚˚˚˚˝k2 ℓ2... ℓ2 0 0˛ ‹‹‹‹‹‹‚,...,K n´1“¨ ˚˚˚˚˚˚˝kn´1 0 ... 0 0 0˛ ‹‹‹‹‹‹‚ where ki“# ℓi¨ℓnrℓ´1 nsmodℓn´1for1ďiďn´2 ℓn´1ℓn fori“n´1. By convention, we call K0the trivial King spline. As our terminology suggests, the King splines are in fact splines. Theorem 5.2. Letně3. Fix a cycle with edge-labels pCn,Lqwithℓn´1andℓn relatively prime. The King splines K0,...,K n´1are splines on pCn,Lq. Proof.First we note that the trivial King spline K0is the same as the trivial spline G0which is indeed a spline on pCn,Lqby Proposition 2.4. ConsideranarbitraryKingspline Ki“ p0,...,0,ℓi,...,ℓ i,kn´1qwhere1 ďiďn´2. It has zero for its first ientries,ℓifor entries i`1 ton´1, andkn´1for its last entry. We want to show that Kiis a spline on pCn,Lq. Note that zero is congruent to itself modulo any integer, so in particular the following congruence s are satisfied: ! 0”0 modℓjfor 1 ďjďi´1 (1) Also, since the integer ℓiis congruent to zero modulo ℓiwe have14 BOWDEN, HAGEN, KING, AND REINDERS ℓi”0 modℓi (2) The integer ℓiis congruent to itself modulo any integer, so in particular the following congruences are satisfied: ! ℓi”ℓimodℓjfori`1ďjďn´2 (3) Finally we know ki“ℓi¨ℓnrℓ´1 nsmodℓn´1satisfies the following two congruences # ki”ℓimodℓn´1 ki”0 modℓn(4) by Proposition 2.9. Collect the congruences in 1, 2, 3, and 4 into a sing le system of congruences. This system represents the edge conditions on pCn,Lq. The vector Ki satisfies all of these congruences so Kiis a spline on pCn,Lq. Now consider the vector Kn´1“ p0,...,0,kn´1q. Zero is congruent to itself modulo any integer, so the following system of congruences is satisfied: ! 0”0 modℓjfor 1 ďjďn´2. (5) Sincekn´1“ℓn´1ℓnwe know # kn´1”0 modℓn´1 kn´1”0 modℓn(6) Collect the congruences in 5 and 6 into a single system. This system re presents the edge conditions on pCn,Lq. The vector Kn´1satisfies all of these congruences so Kn´1is a spline on pCn,Lq. Thus we have that Kiis a spline for all 0 ďiďn´1 as desired. /square Now that we know the King splines are splines, we confirm that they fo rm a ba- sis. Theorem 5.3. Fix a cycle with edge labels pCn,Lqwithℓn´1andℓnrelatively prime. The set of King splines K0,...,K n´1forms a basis for the set of splines on pCn,Lq.BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 15 Proof.The set of smallest flow-up classes G0,...,Gn´1form a basis for the set of splines on pCn,Lqby Theorem 2.8. We constructed the King splines so that the leading entry Kiequals the leading entry of Gifor 0 ďiďn´1. Thus the set of King splines K0,...,K n´1forms a basis for the set of splines on pCn,Lqby Corollary 3.2. /square 6.Multiplication Tables The fact that we have simple explicit formulas for the entries of the K ing basis is a powerful computational tool. In this section we use the King basis to write the product of any pair of basis elements as a linear combination of basis e lements. This kind of calculation is important in geometry and topology, which use sp lines over polynomial rings to describe cohomology rings. 6.1.Multiplication Tables for n-Cycles on the King Basis. When multiplying splines the operation is performed component-wise. Consider the K ing basis on a given n-cycle. Since the entries in the trivial spline K0are all ones, multiplying any spline Ki(with 0ďiďn´1) byK0simply yields Ki. The following theorem gives us the product of any pair of non-trivial King splines. Theorem 6.1. For arbitrary Ki,Kjwithi,j‰0andiďj, we have the product KiKj“liKj`kjpki´liq kn´1Kn´1. Proof.We give a proof by construction. Consider arbitrary basis elements KiandKjwithi,j‰0 andiďj. Their product KiKjhas zeros up to the jthentry. The entries numbered j`1 through n´1 are ℓi¨ℓj. The last entry is ki¨kj. Note that ℓi¨Kjhas zeros for the first jentries,ℓi¨ℓjfrom entries j`1 ton´1, andℓi¨kjfor thenthentry. This is almost exactly the product KiKj. However we want this last entry to be ki¨kj. Addingkjpki´liq kn´1Kn´1gives the desired result. Thus for KiKjwithi,j‰0 andiďjwe have16 BOWDEN, HAGEN, KING, AND REINDERS KiKj“ℓiKj`kikj´likj kn´1Kn´1“ℓiKj`kjpki´liq kn´1Kn´1 Since we are working in the integers, our last step is to prove that th e coefficient kjpki´ℓiq kn´1 is indeed an integer. We know ki”ℓimodℓn´1because Kiis a spline. Say ki´ℓi“pℓn´1for some pPZ. Similarly, we know kj”0 modℓnbecauseKjis a spline. Say kj“qℓnfor some qPZ. By definition we have kn´1“ℓn´1ℓn. Plugging these values into the expressionkikj´likj kn´1yields the following: kjpki´ℓiq kn´1“pqℓnqppℓn´1q ℓn´1ℓn“pq Thuskjpki´ℓiq kn´1is always an integer. /square Note that the product KiKn´1for anyiďn´1 simplifies significantly. Corollary 6.2. Choose any i‰0. ThenKiKn´1“kiKn´1. Proof.We apply the formula for the product KiKjto the particular case where j“n´1 and simplify: KiKn´1“ℓiKn´1`kn´1pki´ℓiq kn´1Kn´1“kiKn´1 /square For example consider the 5-cycle with edge labels t3,4,8,2,5u. The King basis on a 5-cycle with these labels looks like the following: K0 5348 2 1111 1K1 5348 2 033 3 15K2 5348 2 004 4 20K3 5348 2 000 8 40K4 5348 2 000 0 10BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 17 Let’s multiply the elements K1andK3. We obtain K1K3“ K1 5348 2 033 3 15ˆ K3 5348 2 000 8 40“ 5348 2 000 24 600 By the formula given above K1K3“3K3`40p15´3q 10K4“3K3`48K4. Pictorially this solution is shown below. 3K3`48K4“3K3 5348 2 000 8 40`48 K4 5348 2 000 0 10“ 5348 2 000 24 600 Remark 6.3. The same argument can be used to give the multiplication tabl e for arbitrarily labeled 3-cycles using the triangulation basi s (Def 4.1, Thrm 4.3). Given the basis elements H0,H1,andH2we have the following table H0“¨ ˝1 1 1˛ ‚,H1“¨ ˝hp1q 3 hp1q 2 0˛ ‚,H2“¨ ˝hp2q 3 0 0˛ ‚ H0H1 H2 H0H0H1 H2 H1H1hp1q 2H1`ΦH2hp1q 3H2 H2H2hp1q 3H2hp2q 3H2 whereΦ“hp1q 3php1q 3´hp1q 2q hp2q 3. Unlike with the King basis, we do not have nice formulas for entries of t he triangu- lation basis. This leads to the following open question.18 BOWDEN, HAGEN, KING, AND REINDERS Question 6.4. Is there a positive or combinatorial formula for the multipl ication table of general n-cycles (i.e.not alternating sums from successively correcting each spline entry)? References [1] A. Bahri, M. Franz, and N. Ray, The equivariant cohomology ring of weighted projective spa ce, Math. Proc. Cambridge Philos. Soc. 146(2009), no. 2, 395-405. MR 2475973 [2] L. Billera, Homology of smooth splines: generic triangulations and a co njecture of Strang , Trans. Amer. Math. Soc. 310(1998), no. 1, 325340. MR 965757 [3] L. Billera and L. Rose, A dimension series for multivariate splines , Discrete Comput. Geom. 6 (1991), no. 2, 107-128. MR 1083627 [4] S.Gilbert, S.Polster,andJ.Tymoczko, Generalized splines on arbitrary graphs , arXiv:1306.0801 (2013) [5] R. Goldin and S. Tolman, Towards generalizing Schubert calculus in the symplectic c ategory. J. Symplectic Geom. 7(2009), no. 4, 449-473. MR 2552001 [6] M. Goresky, R. Kottwitz, and R. MacPherson, Homology of affine Springer fibers in the unram- ified case . Duke Math. J. 121(2004), no. 3, 509-561. MR 2040285 [7] R. Haas, Module and vector space bases for spline spaces , J. Approx. Theory 65(1991), no. 1, 73-89 MR 1098832 [8] Hagen, S., Tymoczko, J.: A constructive algorithm to find a basis for splines over prin ciple rings and Pr¨ ufer domains. In process. [9] M. Handschy, J. Melnick, S. Reinders, Integer Generalized Splines on Cycles. arXiv:1409.1481 (2014) [10] A. Knutson and T. Tao, Puzzles and (equivariant) cohomology of Grassmannians . Duke Math. J. 119 (2003), no. 2, 221-260 . MR 1997946 [11] S. Payne, Equivariant Chow cohomology of toric varieties , Math. Res. Lett. 13(2006), no. 1, 29-41. MR 2199564 [12] L. Rose, Combinatorial and topological invariants of modules of pie cewise polynomials . Adv. Math.116(1995), no. 1, 3445. MR 1361478 [13] L. Rose, Graphs, syzygies and multivariate splines , Discrete Comput. Geom, 32(2004), no. 4, 623637 [14] J. Tymoczko, An Introduction to Equivariant Cohomology and Homology, Fo llowing Goresky, Kottwitz, and MacPherson , arXiv:math/0503369 (2005)

2015-01-31

An integer generalized spline is a set of vertex labels on an edge-labeled graph that satisfy the condition that if two vertices are joined by an edge, the vertex labels are congruent modulo the edge label. Foundational work on these objects comes from Gilbert, Polster, and Tymoczko, who generalize ideas from geometry/topology (equivariant cohomology rings) and algebra (algebraic splines) to develop the notion of generalized splines. Gilbert, Polster, and Tymoczko prove that the ring of splines on a graph can be decomposed in terms of splines on its subgraphs (in particular, on trees and cycles), and then fully analyze splines on trees. Following Handschy-Melnick-Reinders and Rose, we analyze splines on cycles, in our case integer generalized splines. The primary goal of this paper is to establish two new bases for the module of integer generalized splines on cycles: the triangulation basis and the King basis. Unlike bases in previous work, we are able to characterize each basis element completely in terms of the edge labels of the underlying cycle. As an application we explicitly construct the multiplication table for the ring of integer generalized splines in terms of the King basis.

Bases and Structure Constants of Generalized Splines with Integer Coefficients on Cycles

1502.00176v1

1 Thermally induced all-optical ferromagnetic resonance in thin YIG films E. Schmoranzerová1*, J. Kimák1, R. Schlitz3, S.T. B. Goennenwein3,6, D. Kriegner2,3, H. Reichlová2,3, Z. Šobáň2, G. Jakob5, E.-J. Guo5, M. Kläui5, M. Münzenberg4, P. Němec1, T. Ostatnický1 1Faculty of Mathematics and Physics, Charles University, Prague, 12116, Czech Republic 2Institute of Physics ASCR v.v.i , Prague, 162 53, Czech Republic 3Technical University Dresden, 01062 Dresden, Germany 4Institute of Physics, Ernst-Moritz-Arndt University, 17489, Greifswald, Germany 5Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany 6 Department of Physics, University of Konstanz, 78457 Konstanz, Germany Laser-induced magnetization dynamics is one of the key methods of modern opto-spintronics which aims at increasing the spintronic device speed1,2. Various mechanisms of interaction of ultrashort laser pulses with magnetization have been studied, including ultrafast spin-transfer3, ultrafast demagnetization4, optical spin transfer and spin orbit torques 5,6,7 , or laser-induced phase transitions8,9. All these effects can set the magnetic system out of equilibrium, which can result in precession of magnetization. Laser-induced magnetization precession is an important research field of its own as it enables investigating various excitation mechanisms and their ultimate timescales2. Importantly, it also represents an all-optical analogy of a ferromagnetic resonance (FMR) experiment, providing valuable information about the fundamental parameters of magnetic materials such as their spin stiffness, magnetic anisotropy or Gilbert damping10. The “all-optical FMR” (AO-FMR) is a local and non-invasive method, with spatial resolution given by the laser spot size, which can be focused to the size of few micrometers. This makes it particularly favourable for investigating model spintronic devices. Magnetization precession has been induced in various classes of materials including ferromagnetic metals11, semiconductors10, 12, or even in materials with a more complex spin structure, such as non- collinear antiferromagnets13. Ferrimagnetic insulators, with Yttrium Iron Garnet (YIG, Y 3Fe5O12) as the prime representative14, are of particular importance for spintronic applications owing to their high spin pumping efficiency15 and the lowest known Gilbert damping16. However, inducing magnetization dynamics in ferrimagnetic garnets using optical methods is quite challenging, as it requires large photon energies 2 (bandgap of YIG is Eg ≈ 2.8 eV)17. This spectral region is rather difficult to access with most common ultrafast laser systems, which are usually suited for near-infrared wavelengths. Therefore, methods based mostly on non-thermal effects, such as inverse Faraday18,19 and Cotton-Mouton effect20 or photoinduced magnetic anisotropy21, 22 have been used to trigger the magnetization precession in YIG so far. For these phenomena to occur, large laser fluences of tens of mJ/cm2 are required23. In contrast, laser fluences for a thermal excitation of magnetization precession usually do not exceed tens of J/cm2 (Refs. 12, 21, 13). Using the low fluence excitation regime allows for the determination of quasi-equilibrium material parameters, not influenced by strong laser pulses. In magnetic garnets, an artificial engineering of the magnetic anisotropy via the inclusion of bismuth was necessary to achieve thermally-induced magnetization precession21. In this paper, we show that magnetization precession can be induced thermally by femtosecond laser pulses in a thin film of pure YIG only by adding a metallic capping layer. The laser pulses locally heat the system, which sets the magnetization out of equilibrium due to the temperature dependence of its magnetocrystalline anisotropy. This way we generate a Kittel (n = 0, homogeneous precession) FMR mode, with a precession frequency corresponding to the quasi-equilibrium magnetic anisotropy of the thin YIG film10. We thus prove that the AO-FMR method is applicable for determining micromagnetic parameters of thin YIG films. Using the AO-FMR technique we revealed that at low temperature the Kittel mode damping is significantly faster than at room-temperature, in accord with previous FMR experiments24,25. Our experiments were performed on a 50 nm thick layer of pure YIG grown by pulsed-laser deposition on a gadolinium-gallium-garnet (GGG) (111)-oriented substrate. One part of the film was covered by 8 nm of Au capping layer, the other part by Pt capping, both being prepared by ion-beam sputtering. Part of the sample was left uncapped as a reference. X-ray diffraction confirmed the excellent crystal quality of the YIG film with a very low level of growth-induced strain, as described in detail in Ref. 26. The magnetic properties were further characterized using SQUID magnetometry and ferromagnetic resonance experiments, showing the in-plane orientation of magnetization (see Supplementary Material, Part 1 and Figs. S1 and S2). The deduced low-temperature (20 K) saturation magnetization µ0Ms 180 mT is in agreement with results published on qualitatively similar samples27 again confirming a good quality of the studied YIG film. Magnetic anisotropy of the system at 20 K was established from an independent magneto-optical experiment (Ref. 28), the corresponding anisotropy constants for cubic anisotropy of the first and second order are Kc1 = 4680 J/m3 and Kc2 = 223 J/m3, while the overall uniaxial out-of-plane anisotropy is vanishingly small. 3 Laser-induced dynamics was studied in a time-resolved magneto-optical experiment in transmission geometry, as schematically shown in Fig. 1(a). An output of a Ti:Sapphire oscillator generating 200 fs laser pulses was divided into a strong pump beam, with fluences tuned between 70 and 280 µJ/cm2, and a 20- times weaker probe beam. The beams were focused on a 30 m spot on the sample, which was placed in a cryostat and kept at cryogenic temperatures (typically 20 K). An external magnetic field (up to 550 mT) generated by an electromagnet was applied in y direction (see Fig. 1). The wavelength of pump pulses (800 nm) was set well below the absorption edge of the YIG layer, as indicated in the transmission spectrum of the sample in Fig. 1(b). The wavelength of probe pulses (400 nm) was tuned to match the maximum of the magneto-optical response of bulk YIG [see inset in Fig. 1(b) and Ref. 29]. The detected time-resolved magnetooptical (TRMO) signal corresponding to the rotation of polarization plane of the probe beam Δβ, was measured as a function of the time delay Δt between pump and probe pulses. In Fig. 1(c), we show an example of TRMO signals observed in uncapped YIG and two YIG/metal heterostructures. Clearly, in the presence of the metallic capping layer an oscillatory TRMO signal is observed, whose amplitude depends on the capping metal used. Frequency and damping of the oscillations, on the other hand, remain virtually unaffected by the type of the capping layer, while no oscillations are observed in the uncapped YIG sample. The TRMO signals can be phenomenologically described by a damped harmonic function after removing a slowly varying background (see Supplementary Material, Part 2 and Fig. S3),12 ∆𝛽(Δ𝑡)=𝐴cos(2𝜋𝑓𝛥𝑡+𝜑)exp(−𝛥𝑡 𝜏⁄ ), (1) where A is the amplitude of precession, f its frequency, φ the phase and τ the damping time. The fits are shown in Fig. 1(c) as solid lines. In order to demonstrate that the TRMO signals result from (laser-induced) magnetization dynamics, we varied the external magnetic field Hext and extracted the particular precession parameters by fitting the detected signals by Eq. (1). As depicted in Fig. 2(a), the experimentally observed dependence of the precession frequency on the applied field is in excellent agreement with the solution of Landau-Lifshitz- Gilbert (LLG) equation, using the free energy of a [111] oriented cubic crystal [see Supplementary, section 5, Eq. (S5) and Ref. 28]. This correspondence with the LLG model proves that our oscillatory signals reflect indeed the precession of magnetization in uniform (Kittel) mode in YIG. We stress that the precession frequency is inherent to the YIG layer and does not depend on the type of the capping layer. 4 The detection of the uniform Kittel mode can be further confirmed by comparing the frequency of the oscillatory TRMO signal with the frequency of resonance modes observed in a conventional, microwave- driven ferromagnetic resonance (MW-FMR) experiment. The MW-FMR experiment was performed in the in-plane ( H = 0°) and out-of-plane ( H = 90°) geometry of the external field. We measured the TRMO signals in YIG/Au sample in a range of magnetic field angles H and modelled the angular dependency of f by LLG equation with the same parameters that were used in Fig. 2(a). The output of the model is presented in Fig. 2(b), together with precession frequencies obtained from TRMO and FMR experiments. The MW-FMR data fit well to the overall trend, confirming the presence of uniform magnetization precession [Ref. 30] To find the exact physical mechanism that triggers laser-induced magnetization precession in our YIG/metal bilayers, we measured the TRMO signals at different sample temperatures T. For comparison we calculated also the dependence of f on the first order cubic anisotropy constant Kc1 from the LLG equation, which is shown in the inset of Fig. 2(c). This graph reveals that f should be directly proportional to Kc1 in the studied range of temperatures . In Fig 2(c) we plot f as a function of T, together with the temperature dependence of Kc1 (T) obtained from Ref. 28 and Ref. 32. Clearly, both Kc1 and f show a similar trend in temperature. Considering also the temperature dependence of the precession amplitude [see Fig. S5 (a) and Section 4 of Supplementary Material], we identify the pump pulse-induced heating and consequent modification of the magnetocrystalline anisotropy constant Kc1 as the dominant mechanism driving laser-induced magnetization precession. In order to estimate the pump-induced increase in quasi-equilibrium temperature of the sample, we first fit the temperature dependence of the parameter Kc1 reported in literature by a second order polynomial [Fig. 2(c)]. Owing to the linear relation between f and Kc1 and the known temperature dependence of f, the measured dependence of f on pump fluence I can be converted to the intensity dependence of the temperature increase T(I), which is shown Fig. 2(d). As expected, higher fluence leads to more pronounced heating, which results in a decrease of the precession frequency. Note that for the highest intensity of 300 J/cm2, the sample temperature can increase by almost 80 K. Nature of the observed laser-induced magnetization precession was further investigated by comparing samples with different capping layers. In Fig. 3(a) we show the amplitude A of the oscillatory signal in the YIG/Pt and YIG/Au layers as a function of I. The difference between the samples is apparent both in the absolute amplitude of the precession and in its increase with I, the YIG/Pt showing stronger precession. Furthermore, precession damping is stronger in YIG/Au than in YIG/Pt, as apparent from Fig 3 (b) where effective Gilbert damping parameter eff is presented as a function of Hext. These values of eff were 5 obtained by fitting the TRMO data by the LLG equation, as described in the Supplementary Material (Section 5). Despite the relatively large fitting error, we can still see that YIG/Pt shows slightly lower 0.020, while the YIG/Au has 0.025. To understand these differences, we modeled the propagation of laser-induced heat in GGG/YIG/Pt and GGG/YIG/Au multilayers by using the heat equation (see Supplementary Material, Section 7). In Fig. 3(c), T is presented as a function of time delay t after pump excitation for selected depths from the sample surface. In Fig. 3(d), the same calculations are presented for variable depths and fixed t. The model clearly demonstrates that a significantly higher T can be expected in the Pt-capped layer simply due to its smaller reflection coefficient as compared to Au-capping (see Supplementary Material, Section 7). This in turn leads to a higher amplitude of the laser-induced magnetization precession in YIG/Pt compared to the YIG/Au, as apparent in Fig. 3(a). According to our model, an extreme increase in temperature is induced in the first few picoseconds after excitation, which acts as a trigger of magnetization precession. After approximately 10 ps, precession takes place in quasi-equilibrium conditions. The system returns to equilibrium on a timescale of nanoseconds, which shows also in the TRMO signals as the slowly varying background (Fig. S3). The precession frequency we detect reflects the quasi-equilibrium state of the system. Therefore, the temperature increase T deduced from the TRMO signal can be compared with our model for large time delays after the excitation (t 10 ps). In YIG/Au sample, the experimental values of T = (25 10) K for excitation intensity of 150 J/cm2 [see Fig. 2(d)], while the model gives us T 5K [Fig. 3 (c)]. Clearly, the values match in the order of magnitude but there is a factor of 5 difference. This difference results from the boundary conditions of the model that assumes ideal heat transfer between the sample and the holder, which is experimentally realized using a silver glue with less than perfect performance at cryogenic conditions. From Fig. 3(d) it also follows that large thermal gradients are generated across the 50 nm layer. This could lead to significant inhomogeneity in magnetic properties of the layer, that would increase the damping parameter by an extrinsic term. In our TRMO measurements, is indeed very large for a typical YIG sample (TRMO 2-2.5 x10-2) and exceeds the value obtained from room-temperature MW-FMR by almost an order of magnitude ( FMR 1x10-3, see Supplementary Material, Section 1b). As the modeled thermal gradient alone cannot account for such a large change in Gilbert damping (see Supplementary Material, Section 6), we attribute this increase in Gilbert damping to the difference in the ambient temperatures. Large change of Gilbert damping (by a factor of 30) between room and cryogenic (20 K) temperature has recently been reported on a seemingly high quality YIG thin film24. It was explained in terms of the presence of rare earth or Fe2+ impurities that are activated at cryogenic temperatures. It is likely that the same 6 process occurs in our sample. Even though other mechanisms related to the optical excitation can also contribute to the increase in TRMO (see Supplementary Material, Section 6), the all-optical and standard FMR generated Kittel modes correspond very well [see Fig. 2(b)]. Furthermore, also the observed sample- dependent Gilbert damping is consistent with this explanation. The YIG/Pt sample is heated to higher temperature by the pump laser pulse [Fig. 3(c), (d)] than the YIG/Au sample, which according to Ref. 24 corresponds to a lower Gilbert damping. It is worth noting that damping parameter can be increased also by spin-pumping from YIG to the metallic layer. However, this effect is expected to be significantly higher when Pt is used as a capping, which does not agree with our observations. In conclusion, we demonstrated the feasibility of the all-optical ferromagnetic resonance method in 50- nm thin films of plain YIG. Magnetization precession can be triggered by laser-induced heating of a metallic capping layer deposited on top of the YIG film. The consequent change of sample temperature modifies its magnetocrystalline anisotropy, which sets the system out of equilibrium and initiates the magnetization precession. Based on the field dependence of precession frequency, we identify the induced magnetization dynamics as the fundamental (Kittel) FMR mode, which is virtually independent of the type of capping and reflects the quasi-equilibrium magnetic anisotropy. The Gilbert damping parameter is influenced by line- broadening mechanism due to low-temperature activation of impurities, which is an important aspect to be taken into account for low-temperature spintronic device applications. Regarding the efficiency of the optical magnetization precession trigger, it was found that the type of capping layer strongly influences the precession amplitude. The precession in YIG/Pt attained almost twice the amplitude of that in YIG/Au under the same conditions. This indicates that a suitable choice of capping layer should be considered in an optimization of this local non-invasive magnetometric method. Acknowledgments: This work was supported in part by the INTER-COST grant no. LTC20026 and by the EU FET Open RIA grant no. 766566. We also acknowledge CzechNanoLab project LM2018110 funded by MEYS CR for the financial support of the measurements at LNSM Research Infrastructure and the German Research Foundation (DFG SFB TRR173 Spin+X projects A01 and B02 #268565370). 7 LITERATURE [1] A. Hirohata et al., JMMM 509, 16671 (2020) [2] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010) [3] F. Siegries et al., Nature 571, 240–244 (2019) [4] E. 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Lett. 101, 262406 (2012) [20] M. Montazeri et al., Nat. Comm. 6, 8958 (2015) [21] L. A. Shelukhin, et al., Phys. Rev. B 97, 014422 (2018) [22] A. Stupakiewicz et al., Nature 542, 71 (2017) [23] F. Atoneche et al., Phys. Rev. B 81, 214440 (2010) [24] C. L. Jermain et al., PRB 95, 174411 (2017) 8 [25] H. Maier-Flaig et al., Phys. Rev. B 95, 214423 (2017) [26] B. Bhoi et al.: J. Appl.Phys. 123, 203902 (2018) [27] J. Mendil et al.: Phys. Rev. Mat. 3, 034403 (2019) [28] E. Schmoranzerova et al., ArXiv XXX (2021) [29] E. Lišková Jakubisova et al., Appl. Phys. Lett. 108, 082403 (2016) [30] We note that the FMR data were obtained at room temperature while the TRMO experiment was performed at 20K. However, as apparent from Fig. 2(c), the precession frequency varies by less than 10% between 20 K and 300 K, which is well below the experimental error of (H). This justifies comparison of the precession frequencies obtained from the TRMO experiment with the FMR data. [31] M. Haider et al., J. Appl. Phys. 117, 17D119 (2015) [32] N. Beaulieu et al., IEEE Magnetics Letters 9, 3706005 (2018) FIGURES 9 Fig. 1: (a) Schematic illustration of the pump&probe experimental setup, where Eprobe is the probe beam linear polarization orientation which is rotated by an angle after transmission through the sample with respect to the orientation E’probe. An external magnetic field H ext is applied at an angle H. (b) Absorption spectrum of the studied YIG sample, where OD stands for the optical density defined as minus the decadic logarithm of sample transmittance. The red arrow indicates the wavelength of the pump beam PUMP = 800 nm. Inset: Spectrum of Kerr rotation K of bulk YIG crystal29. The blue arrow shows the wavelength of the probe beam PROBE = 400 nm. (c) Typical time-resolved magneto-optical signals of a plain 50 nm YIG film (black dots), YIG /Pt (green dots) and YIG/Au bilayer (blue dots) at 20 K and 0Hext = 100 mT, applied at an angle H = 40°. Lines indicate fits by Eq. (1). The data were offset for clarity. Fig. 2: (a) Frequency f of magnetization precession as a function of magnetic field applied at an angle H = 40°, for YIG/Pt (blue dots) and YIG/Au (green triangles) at T = 20 K and I = 150 J/cm2. The line is calculated from LLG equation (Eq. S3) with the free energy given by (Eq.S5) (b) Field-angle dependence of f in YIG/Au sample for 0Hext = 300 mT (blue dots), compared to a model by LLG model (line) and to frequencies measured by MW-FMR (red stars)32. (c) Temperature dependence of f in YIG/Au sample (black points), where 0Hext = 300 mT was applied at H = 40°. The temperature dependence of cubic anisotropy constant Kc1 was obtained from Ref. 28 (red dots) and Ref. 32 (red star, T = 20 K). The data were fitted by an inverse polynomial dependence 𝐾ଵ(𝑇)= ଵ (ା்ା்మ), with parameters: a = 0.18 m2/kJ; b= 9 x 10-4 m2/kJ.K; c = 9 x 10-6m2/kJ.K2. Inset: Dependence f(Kc1) obtained from the LLG equation. (d) f as a function of pump pulse fluence I, from which the increase of sample temperature T for the used pump fluences was evaluated using the f(T) dependence. 10 Fig. 3: Comparison of magnetization precession in YIG/Pt and YIG/Au samples. (a) Precession amplitude A as a function of pump fluence I (dots) with the corresponding linear fits 𝐴 = 𝑠∙𝐼. The parameter s Pt = (1.05 0.09)x10-2 rad.cm2/J in the YIG/Pt, and s Au = (0.50.1)x10-2 rad.cm2/J in YIG/Au. These dependencies were measured for 0Hext = 300 mT and T 0 = 20 K. In YIG/Pt sample the as-measured data obtained for H = 40° are shown. In the YIG/Au sample, the A(I) dependence was originally measured for H = 21° and recalculated to H = 40° according to the measured angular dependence, as described in detail in Supplementary Material, Section 3. (b) Gilbert damping eff for Hext applied at an angle H = 40°. The values of eff result from fitting the TRMO signals to LLG equation; I = 140 J/cm2. (c) and (d) Increase in lattice temperature as a function of time delay between pump and probe pulses for selected depths from the sample surface (c) and as a function of depth for fixed time delays (d). I = 140 J/cm2, T0 = 20 K. The heat capacities and conductivities of individual layers are provided in the Supplementary Material, Section 7. 11 Thermally induced all-optical ferromagnetic resonance in thin YIG films: Supplementary Material E. Schmoranzerová1*, J. Kimák1, R. Schlitz3 , S.T. B. Goennenwein3, D. Kriegner2,3, H. Reichlová2,3, , Z. Šobáň2, G. Jakob5, E.-J. Guo5, M. Kläui5, M. Münzenberg4, P. Němec1 , T. Ostatnický1 1Faculty of Mathematics and Physics, Charles University, Prague, 12116, Czech Republic 2Institute of Physics ASCR v.v.i , Prague, 162 53, Czech Republic 3Technical University Dresden, 01062 Dresden, Germany 4Institute of Physics, Ernst-Moritz-Arndt University, 17489, Greifswald, Germany 5Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany 6 Department of Physics, University of Konstanz, 78464 Konstanz, Germany 1. Magnetic characterization A. SQUID magnetometry A superconducting quantum device magnetometer (SQUID) was used to characterize the magnetic properties of the thin YIG film at several sample temperatures. The magnetic hysteresis loops, detected with magnetic field applied in [2-1-1] crystallographic direction of the YIG layer, are shown in Fig. S1. As expected [t26], the saturation magnetization increases at low temperatures, which is accompanied by a slight increase in coercive field. At room temperature, the effective saturation magnetization is estimated to be Ms = 95 kA/m. This value is in good agreement with the effective magnetization Meff obtained from the ferromagnetic resonance (FMR) measurement (see Section 1b), which indicates only a weak out-of- plane magnetic anisotropy [s1]. However, as discussed in detail in Ref. 26, the Ms from our SQUID measurement is burdened by a relatively large error. Therefore, mere comparison of SQUID and FMR experiment is not sufficient to evaluate the size of the out-of-plane magnetic anisotropy. An additional experiment such as static magneto-optical measurement [28] is needed in order to get more precise estimation of the out-of plane magnetic anisotropy. B. FMR measurement The SQUID magnetometry was complemented by so-called broad band ferromagnetic resonance measurements using a co-planar waveguide to apply electromagnetic radiation of a variable frequency f =/2 to the sample. The measurement was performed at room temperature and further details on the method can be found in Ref. s2. An exemplary set of spectra showing the normalized microwave transmission | S21|norm obtained at different external fields magnitudes applied in the sample plane, is shown in Fig. S2(a). The set of Lorentzian-shape resonances can be fitted by the equation: 12 |𝑆ଶଵ|୬୭୰୫=ቀഘ మቁమ ቀഘ మഏିഘబ మഏቁାቀഘ మቁమ+𝑦 (S1) Where f0 =0/2 is the FMR resonance frequency, /2 is the half width half maximum line width, B the amplitude of the FMR line and y0 a frequency independent offset. From an automated fitting of the set of lines obtained at different Hext, we extract the magnetic field dependence of the resonance frequency 0/2 (Hext) [Fig. S2(b)] and linewidth (Hext) [Fig. S2(c)]. Clearly, the resonance frequencies correspond to the fundamental (Kittel) mode, and can correspondingly be fitted by the Kittel formula [s3]: ఠబ ଶగ=ఊ ଶగඥ𝜇𝐻ୣ୶୲(𝜇𝐻ୣ୶୲+𝜇𝑀ୣ) (S2) Where Meff is the effective saturation magnetization that includes the out-of-plane anisotropy term, and is gyromagnetic ratio. From this fit, it is possible to evaluate Meff , Kittel = 94.9 kA/m From the linewidth dependence (Hext)=2 + 0 we can extract both the inhomogeneous line broadening and the Gilbert damping parameter, as shown in Fig. S2(c) [s2]. In our experiment, the inhomogeneous linewidth broadening is 0 = 55.8 MHz, and the Gilbert damping parameter = 0.001. Both values are on a higher side compared e.g. with YIG prepared by liquid phase epitaxy [s8] but in good agreement with typical YIG thin films similar to our layers, which were prepared by pulsed laser deposition [27]. This again confirms the good quality of the studied thin YIG films. 2. Processing of time-resolved magneto-optical data In order to extract the parameters describing the precession of magnetization correctly from the time- resolved magneto-optical (TRMO) signals, it is first necessary to remove the slowly varying background on which the oscillatory signals are superimposed. For this purpose, we fitted the measured data by the second-order polynomial. The fitted curve was then subtracted from the measured signals, as demonstrated in Fig. (S3). From the physical point of view, the background can be attributed to a slow return of magnetization to its equilibrium state after the pump beam induced heating, which can take place on the timescale of tens of nanoseconds [10]. Since both saturation magnetization Ms and magnetocrystalline anisotropy Kc are temperature-dependent, their temporal variation can in principle contribute to the background signal. However, as explained later in Section 4, the variation of Ms is very weak at cryogenic temperatures. The heat-induced modification of Kc, and the resulting change of the magnetization quasi-equilibrium orientation, is, therefore, more probable origin of the slowly varying background, which is detected in the MO experiment by the Cotton-Mouton effect [28]. 3. Angular dependence of precession amplitude In order to mutually compare the values of precession amplitudes measured in YIG/Pt and YIG/Au samples at different angles of the external magnetic field H, it is necessary to correct their values for the value of H. The following procedure was used to correct the data presented in Fig. 3 of the main text . 13 First, we measured in detail angular dependence of the precession amplitude in the YIG/Au layer, which is presented in Fig. S4. Amplitude of the oscillatory signal detected in our experiment does not depend solely on the amplitude of the magnetization precession but also on the size of the magneto-optical (MO) effect. In our experimental setup, the change of H was achieved by tilting the sample relative to the position of electromagnet poles [see Fig. 1(a)]. The MO response, however, varies also with the angle of incidence which is modified simultaneously with a change of H [see Fig. 1(a)] . Therefore, it is not straightforward to describe the A(H) analytically. Instead, we fitted the measured dependence A(H) by a rational function in a form of y = 1/(A+Bx2), which is the lowest order polynomial function that can describe the signal properly. From the fit we derived a correction factor of 1.7 by which the amplitudes A measured at H =21° has to be multiplied to correspond to that measured at H =40°. This factor was then used to recalculate the intensity dependence of the precession amplitude A(I) in YIG/Au measured at H =21° to the A(I) at H =40°, which could be directly compared to the A(I) dependence detected at YIG/Pt for H =40° - see Fig. S4(b). 4. Temperature dependence of precession amplitude In order to further investigate the origin of the laser-induced magnetization precession, the amplitude of the oscillatory MO signal was measured as a function of the sample temperature in YIG/Pt sample, see Fig. S5(a). In Fig. S5 (b), we show temperature dependence of saturation magnetization Ms, as obtained from Ref. 32 The only parameter changed within this experiment was the sample temperature. It is reasonable to expect that the size of the magneto-optical effect is not strongly temperature dependent in the studied temperature range between 20 and 50 K (see Ref. 28) Therefore, the dependence A(T) presented in Fig S5 corresponds directly to the temperature dependence of magnetization precession amplitude. By comparing the Ms(T) and A(T) data, it is immediately apparent that the laser-induced heating would not modify Ms enough to account for the large change of the magnetization precession amplitude with the sample temperature. Even assuming the most extreme laser-induced temperature increase T 80 K shown in Fig. 3(c), the laser-induced Ms variation would be less than 5%, while the precession amplitude changes by more than 50% between 20 and 50 K. In contrast, the magnetocrystalline anisotropy Kc1 changes drastically even in this relatively narrow temperature range [see Fig. 2(c)]. Consequently, the change of Kc1, which leads to a significant change of the position of quasi-equilibrium magnetization orientation in the studied sample (see Section 5) provides a more plausible explanation for the origin of the laser-induced magnetization precession in the YIG/metal layer. 5. LLG equation model The data were modelled by numerical solution of the Landau-Lifshitz-Gilbert (LLG) equation, as defined in [s9]: ௗ𝑴(௧) ௗ௧= −𝜇𝛾ൣ𝑴(𝑡)×𝑯𝒆𝒇𝒇(𝑡)൧+ఈ ெೞቂ𝑴(𝑡)×ௗ𝑴(௧) ௗ௧ቃ, (S3) 14 where is the gyromagnetic ratio, is the Gilbert damping constant, and MS is saturated magnetization.The effective magnetic field Heff is given by: 𝑯𝒆𝒇𝒇(𝑡)=డி డ𝑴 (S4) where F is energy density functional that contains contributions from the external magnetic field Hext, demagnetizing field and the magnetic anisotropy of the sample. We consider the form of F including first- and second-order cubic terms as defined in Ref. [t24]. The polar angle is measured with respect to the crystallographic axis [111] and the azimuthal angle = 0 corresponds to the direction [21ത1ത], with an appropriate index referring to the magnetization position (index M) or the direction of the external magnetic field (index H). The resulting functional takes the form (in the SI units): 𝐹= −𝜇𝐻𝑀[sin𝜃ெsin𝜃ு+cos𝜃ெcos𝜃ுcos(𝜑ு−𝜑ெ)]+ቀଵ ଶ𝜇𝑀ଶ−𝐾uቁsinଶ𝜃ெ +𝐾c1 12ൣ7cosସ𝜃ெ−8cosଶ𝜃ெ+4−4√2cosଷ𝜃ெsin𝜃ெcos3𝜑ெ൧ +c2 ଵ଼ൣ−24cos𝜃ெ+45cosସ𝜃ெ−24cosଶ𝜃ெ+4−2√2cosଷ𝜃ெsin𝜃ெ(5cosଶ𝜃ெ− 2)cos3𝜑ெ+cos𝜃ெcos6𝜑ெ൧ , (S5) where 0 is the vacuum permeability and we consider the following values of constants: magnetization M = 174 kA/m, first-order cubic anisotropy constant Kc1=4.68 kJ/m3, second-order cubic anisotropy constant Kc2 = 222 J/m3 [t24]. For modelling the dependence of precession frequency on the external magnetic field Hext [Fig. 2 (a)] and on the angle H [Fig. 2 (a)], we assumed that in a steady state magnetization direction is parallel to Hext, i.e. M = H, and M = H. This is surely fulfilled for large enough magnitude of Hext. Since the coercive field is very small, we can assume the procedure to be correct. Further correspondence to experimental data Evaluation of the Gilbert damping factor from the as-measured magneto-optical oscillatory data was done by fitting signals by a theoretical curve calculated by solving numerically LLG equation [Eq. (S3)]. We considered the magnetization free energy density in a form of Eq. (S5) using magnitude and direction of the external magnetic field from the experiment. The electron g-factor was set to 2.0 and then the Gilbert factor and five parameters of the fourth-order polynomial to remove the background MO signal were the fitting parameters. The resulting dependence of fitted effective Gilbert factors αeff on external magnetic field is displayed in Fig. 3(b) in the main text, from which the field-independent Gilbert factor α can be evaluated. 6. Comparison of Gilbert damping parameter from MW-FMR and TRMO experiments The Gilbert damping from the room-temperature FMR measurement on the YIG film 1∙10-3 and the results from fits of the low-temperature pump&probe data 2∙10-2, differ by an order of magnitude. As detailed in the main text, we attribute this difference to the different sample temperatures in the AO-FMR and MW-FMR measurements. However, one might also argue that the increased damping in the optical experiments is caused either by a spatial inhomogeneity of the magnetization oscillations or it is the result of the perturbation of the YIG surface. 15 In the former case, we expect that the spatial inhomogeneity of the temperature distribution [see Fig. 3(d)] causes the magnetization to oscillate in a form of a superposition of harmonic waves with well- defined in-plane wavevectors. Considering the dispersion of the allowed oscillatory modes [s4] and including the relevant value of the exchange stiffness [s5], we revealed that neither the inhomogeneity due to the finite cross section of the excitation laser beam nor the temperature gradient perpendicular to the sample surface can cause such a strong decrease of the Gilbert damping factor that is observed experimentally. Here, we provide an estimate on which time scales the mode dispersion influences the decay of the signal if the exchange stiffness is taken into account. Following [s5], the mode dispersion is described by the additive exchange field in the form: 𝜇𝐻ex=𝐷ቈ𝜋ଶ 𝑑ଶ𝑛ଶ+𝑘∥ଶ , where D ≈ 5∙10-17 T.m2 is the exchange stiffness, n is the order of the confined magnon mode, d is the YIG layer thickness and k‖ is the in-plane magnon wave vector. We consider here only the n = 0 case since this is the only visible harmonic mode observed in the experimental MO data, as proven by the numerical fitting. Note that the frequency shift ∆𝜔/2𝜋=|𝛾|𝜇𝐻ex/2𝜋, where the symbol γ stands for the electron gyromagnetic ratio, of the n = 1 mode would be 5.5 GHz, which would be then clearly distinguishable from the basic n = 0 mode in the lowest external magnetic fields. The in-plane wave vector k‖ can be calculated from the FWHM (full width at half maximum) width of the laser spot on the sample L, which is about 30 µm in our case, that leads to the order of magnitude k‖ ≈ (2π/L) ≈ 105 m-1. The frequency increase due to the finite laser spot size can be estimated as ∆𝜔/2𝜋=|𝛾|𝐷𝑘∥ଶ/2𝜋≈14 kHz. Inverse of this value ( 0.1 ms) determines the typical time scale at which the magnon dynamics is influenced by their dispersion due to the finite laser spot size, which is clearly out of the range of the experimental time scale. The presence of a metallic layer on the top of the YIG sample surface can result into two significant damping processes. First, the magnetization oscillations (and thus oscillations of the macroscopic magnetic field) are coupled to electromagnetic modes which penetrate the surrounding material and can be eventually radiative for small magnon wave vectors. Penetration into conductive material in turn causes energy dissipation through finite conductivity of such material. We checked the magnitudes of the additional damping caused by the radiative field and energy dissipation in a thin metallic layer and we found that these processes exist but the additional energy loss cannot explain the observed magnitude of the Gilbert damping parameter. The second possible explanation of the increased precession damping due to the presence of the metal/YIG surface may be that there is an additional perturbation to otherwise homogeneous sample due to some inhomogeneity through surface roughness or spatially inhomogeneous local spin pinning. Since both the surface roughness and spin pinning can depend on the composition of the capping layer, it can also cause a minor difference in the resulting damping factor, as observed in Fig. 3(b). Overall, we attribute the experimentally observed difference in Gilbert damping measured by FMR and pump&probe techniques to the difference in ambient temperatures that were used in these experiments, which is in accord with the results of Ref. t22. 7. Heat propagation in YIG/Pt and YIG/Au 16 Heat propagation in our sample structures was modelled in terms of the heat equation: డ் డ௧=ఒ ∆𝑇 , (S4) where T is the local temperature, λ is the local thermal conductivity, c is the heat capacity and the symbol Δ denotes the Laplace operator. The spatio-temporal temperature distribution in the studied sample has been calculated by a direct integration of Eq. (S4) in a time domain, assuming excitation of the metallic layer by an ultrashort optical pulse [with a temporal duration of 100 fs (FWHM)]. We have taken the whole structure profile of vacuum/metal/YIG/GGG into consideration, assuming that the GGG substrate had a perfect heat contact with the cold finger of the cryostat, which has been held on a constant temperature. The respective heat conductivities ( λ) and heat capacities ( c) were set to the following values. Au: λ = 5 W/m K [s6], c = 1.3∙104 J/cm3, Pt: λ = 10 W/m K [s7], c = 1.2∙104 J/cm3, YIG: λ = 60 W/m K, c = 6.7∙103 J/cm3, GGG: λ = 300 W/m K, c = 2.1∙104 J/m3. To evaluate the initial heat transfer from the optical pulses to the capping metallic layer, we considered the proper geometry of our experiment, i.e. a 8 nm thick metallic layer deposited on the YIG sample, the incidence angle of the laser beam of 45 degrees and its p-polarization. We then used optical constants of gold and platinum in order to calculate transmission and reflection coefficients of a nanometer-thick metallic layers by means of the transfer matrix method. From those, we estimated the efficiency of power conversion from the optical field to heat to be 3% for gold and 6.5% for platinum. The total amount of heat density was then calculated by multiplication of the pump pulse energy density and the above-mentioned efficiency. The data shown in Fig. 3(c)-(d) were then extracted from the full spatio-temporal temperature distribution. Clearly, the temperature increase in the YIG/Pt sample is approximately twice larger than that of the YIG/Au sample as a consequence of twice larger efficiency of the light-heat energy conversion in favour of platinum. Correspondingly, also the amplitudes of the MO oscillations in Fig. 3(a) reveal the ratio 2:1. LITERATURE [s1] B. Bhoi et al.: J. Appl.Phys. 123, 203902 (2018) [s2] H. Maier-Flaig et al,.PRB 95, 214423 (2017) [s3] Ch. Kittel: “Introduction to solid state physics (8th ed.)”. New Jersey: Wiley. (2013). [s4] D. D. Stancil, A. Prabhakar: Spin waves – theory and applications (Springer, New York, 2009). [s5] S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. Hillebrands, and A. Conca, J. Phys. D: Appl. Phys. 48, 015001 (2014). [s6] G. K. White, Proc. Phys. Soc. A 66, 559 (1953). [s7] X. Zhang, H. Xie, M. Fujii, H Ago, K. Takahashi, T. Ikuta, T. Shimizu, Appl. Phys. Lett. 86, 171912 (2005) 17 [s8] C. Dubs et al., Phys. Rev. Materials 4, 024416 (2020) [s9] J. Miltat, G. Albuquerque, and A. Thiaville, An introduction to micromagnetics in the dynamic regime, in Spin dynamics in confined magnetic structures I, edited by B. Hillebrands and K. Ounadjela, Springer, Berlin, 2002, vol. 83 of Topics in applied physics. FIGURES Fig. S1: Magnetic hysteresis loops measured by SQUID magnetometry with magnetic field Hext applied in direction [2-1-1] at several sample temperatures. The saturation magnetization obtained from SQUID magnetometry measurement at room temperature is roughly Ms = 95 kA/m , assuming a YIG layer thickness of 50 nm. 18 Fig. S2: (a) Ferromagnetic resonance spectra measured at room temperature for several different external magnetic field magnitudes µ0Hext from 0 to 540 mT applied in the sample plane. Resonance peaks were fitted by Eq. (S1) and the obtained resonance frequencies and linewidths are plotted as points in panels (b) and (c), respectively. The lines correspond to fit by Kittel formula [Eq. (S2)], which enables to evaluate effective magnetization Meff = 94.9 kA/m and Gilbert damping parameter of = 0.001. 19 Fig. S3: Removal of slowly varying background from time-resolved magneto-optical signals. The red dots correspond to as-measured signals, line indicates the polynomial background that is subtracted from the raw signals. Black dots show the signal after background subtraction, black line representing the fit by Eq. (1) of the main text. The data were taken at external field of 0Hext = 300 mT, temperature 20 K and pump fluence I = 140 J/cm2.. Fig. S4: (a) Dependence of the amplitude A of oscillatory magneto-optical signal on the sample tilt (different field angles of magnetic field H ) measured in YIG/Au sample . 0Hext = 300 mT, temperature T = 20 K and pump fluence I = 150 J/cm2.. (b) Pump intensity dependence of A measured for YIG/Au sample at H =21° (red points), the same dependence recalculated to correspond to H =40 (blue points) where A(I) was measured for YIG/Pt sample (green points); T = 20 K. Fig. S5: (a) Temperature dependence of amplitude of the time-resolved magneto-optical signals measured for external field 0Hext = 300 mT applied at an angle H = 30°. (b) Temperature dependence of saturation magnetization Ms obtained from Ref. 32.

2021-10-31

All-optical ferromagnetic resonance (AO-FMR) is a powerful tool for local detection of micromagnetic parameters, such as magnetic anisotropy, Gilbert damping or spin stiffness. In this work we demonstrate that the AO-FMR method can be used in thin films of Yttrium Iron Garnet (YIG) if a metallic capping layer (Au, Pt) is deposited on top of the film. Magnetization precession is triggered by heating of the metallic layer with femtosecond laser pulses. The heating modifies the magneto-crystalline anisotropy of the YIG film and shifts the quasi-equilibrium orientation of magnetization, which results in precessional magnetization dynamics. The laser-induced magnetization precession corresponds to a uniform (Kittel) magnon mode, with the precession frequency determined by the magnetic anisotropy of the material as well as the external magnetic field, and the damping time set by a Gilbert damping parameter. The AO-FMR method thus enables measuring local magnetic properties, with spatial resolution given only by the laser spot size.

Thermally induced all-optical ferromagnetic resonance in thin YIG films

2111.00586v1

arXiv:1505.07248v1 [math.AP] 27 May 2015LOGARITHMIC STABILITY IN DETERMINING A BOUNDARY COEFFICIE NT IN AN IBVP FOR THE WAVE EQUATION KA¨IS AMMARI AND MOURAD CHOULLI Abstract. In [2] we introduced a method combining together an observab ility inequality and a spectral de- composition to get a logarithmic stability estimate for the inverse problem of determining both the potential and the damping coefficient in a dissipative wave equation fro m boundary measurements. The present work deals with an adaptation of that method to obtain a logarithm ic stability estimate for the inverse problem of determining a boundary damping coefficient from boundary m easurements. As in our preceding work, the different boundary measurements are generated by varyin g one of the initial conditions. Keywords : inverse problem, wave equation, boundary damping coefficie nt, logarithmic stability, boundary measurements. MSC: 35R30. Contents 1. Introduction 1 1.1. The IBVP 2 1.2. Main result 2 2. Preliminaries 3 2.1. Extension lemma 3 2.2. Observability inequality 4 3. The inverse problem 5 3.1. An abstract framework for the inverse source problem 5 3.2. An inverse source problem for an IBVP for the wave equation 7 3.3. Proof of Theorem 1.1 7 Appendix A. 9 Appendix B. 10 References 11 1.Introduction We are concerned with an inverse problem for the wave equation whe n the spatial domain is the square Ω = (0,1)×(0,1). To this end we consider the following initial-boundary value problem (abbreviated to IBVP in the sequel) : (1.1) ∂2 tu−∆u= 0 in Q= Ω×(0,τ), u= 0 on Σ 0= Γ0×(0,τ), ∂νu+a∂tu= 0 on Σ 1= Γ1×(0,τ), u(·,0) =u0, ∂tu(·,0) =u1. Here Γ0= ((0,1)×{1})∪({1}×(0,1)), Γ1= ((0,1)×{0})∪({0}×(0,1)) 12 KA¨IS AMMARI AND MOURAD CHOULLI and∂ν=ν·∇is the derivative along ν, the unit normal vector pointing outward of Ω. We note that νis everywhere defined except at the vertices of Ω and we denote by Γ = Γ0∪Γ1. The boundary coefficient ais usually called the boundary damping coefficient. In the rest of this text we identify a|(0,1)×{0}bya1=a1(x),x∈(0,1) anda|{0}×(0,1)bya2=a2(y), y∈(0,1). In that case it is natural to identify a, defined on Γ 1, by the pair ( a1,a2). 1.1.The IBVP. We fix 1/2<α≤1 and we assume that a∈A, where A={b= (b1,b2)∈Cα([0,1])2, b1(0) =b2(0), bj≥0}. This assumption guarantees that the multiplication operator by aj,j= 1,2, defines a bounded operator on H1/2((0,1)). The proof of this fact will be proved in Appendix A. LetV={u∈H1(Ω);u= 0 on Γ 0}and we consider on V×L2(Ω) the linear unbounded operator Agiven by Aa= (w,∆v), D(Aa) ={(v,w)∈V×V; ∆v∈L2(Ω) and∂νv=−awon Γ1}. One can prove that Aais a m-dissipative operator on the Hilbert space V×L2(Ω) (for the reader’s convenience we detail the proof in Appendix B). Therefore, Aais the generator of a strongly continuous semigroup of contractions etAa. Hence, for each ( u0,u1), the IBVP (1.1) possesses a unique solution denoted byua=ua(u0,u1) so that (ua,∂tua)∈C([0,∞);D(Aa))∩C1([0,∞),V×L2(Ω)). 1.2.Main result. For 0<m≤M, we set Am,M={b= (b1,b2)∈A∩H1(0,1)2;m≤bj,/ba∇dblbj/ba∇dbl2 H1(0,1)≤M}. LetU0given by U0={v∈V; ∆v∈L2(Ω) and∂νv= 0 on Γ 1}. We observe that U0×{0} ⊂D(Aa), for anya∈A. LetCa∈B(D(Aa);L2(Σ1)) defined by Ca(u0,u1) =∂νua(u0,u1)|Γ1. We define the initial to boundary operator Λa:u0∈ U0−→Ca(u0,0)∈L2(Σ1). ClearlyCa∈B(D(Aa);L2(Σ1)) implies that Λ a∈B(U0;L2(Σ1)), when U0is identified to a subspace of D(Aa) endowed with the graph norm of Aa. Precisely the norm in U0is the following one /ba∇dblu0/ba∇dblU0=/parenleftBig /ba∇dblu0/ba∇dbl2 V+/ba∇dbl∆u0/ba∇dbl2 L2(Ω)/parenrightBig1/2 . Henceforth, for simplicity sake, the norm of Λ a−Λ0inB(U0;L2(Σ1)) will denoted by /ba∇dblΛa−Λ0/ba∇dbl. Theorem 1.1. There exists τ0>0so that for any τ > τ0, we find a constant c >0depending only on τ such that (1.2) /ba∇dbla−0/ba∇dblL2((0,1))2≤cM/parenleftBig/vextendsingle/vextendsingleln/parenleftbig m−1/ba∇dblΛa−Λ0/ba∇dbl/parenrightbig/vextendsingle/vextendsingle−1/2+m−1/ba∇dblΛa−Λ0/ba∇dblL2(Σ1)/parenrightBig , for eacha∈ Am,M. We point out that our choiceofthe domain Ω is motivated by the fact t he spectral analysisofthe laplacian under mixed boundary condition is very simple in that case. However t his choice has the inconvenient that the square domain Ω is no longer smooth. So we need to prove an obse rvability inequality associated to this non smooth domain. This is done by adapting the existing results. We n ote that the key point in establishing this observability inequality relies on a Rellich type identity for the doma in Ω. The inverse problem we discuss in the present paper remains largely o pen for an arbitrary (smooth) domain as well as for the stability around a non zero damping coefficien t. Uniqueness and directional Lipschitz stability, around the origin, was established by the author s in [3].LOGARITHMIC STABILITY IN DETERMINING A BOUNDARY COEFFICIE NT 3 The determination of a potential and/or the sound speed coefficien t in a wave equation from the so-called Dirichlet-to-Neumann map was extensively studied these last decad es. We refer to the comments in [2] for more details. 2.Preliminaries 2.1.Extension lemma. We decompose Γ 1as follows Γ 1= Γ1,1∪Γ1,2, where Γ 1,1= (0,1)× {0}and Γ1,2={0}×(0,1). Similarly, we write Γ 0= Γ0,1∪Γ0,2, with Γ 0,1={1}×(0,1) and Γ 0,2= (0,1)×{1}. Let (g1,g2)∈L2((0,1))2. We say that the pair ( g1,g2) satisfies the compatibility condition of the first order at the vertex (0 ,0) if (2.1)/integraldisplay1 0|g1(t)−g2(t)|2dt t<∞. Similarly, we can define the compatibility condition of the first order at the other vertices of Ω. We need also to introduce compatibility conditions of the second orde r. Let (fj,gj)∈H1((0,1))× L2((0,1)),j= 1,2. We say that the pair [( f1,g1),(f2,g2)] satisfies the compatibility conditions of second order at the vertex (0 ,0) when (2.2) f1(0) =f2(0),/integraldisplay1 0|f′ 1(t)−g2(t)|2dt t<∞and/integraldisplay1 0|g1(t)−f′ 2(t)|2dt t<∞. The compatibility conditions ofthe second orderat the other vertic es ofΩ aredefined in the same manner. The following theorem is a special case of [4, Theorem 1.5.2.8, page 50 ]. Theorem 2.1. (1) The mapping w−→(w|Γ0,1,w|Γ0,2,w|Γ1,1,w|Γ1,2) = (g1,...,g 4), defined on D(Ω)is extended from H1(Ω)onto the subspace of H1/2((0,1))4consisting in functions (g1,...,g 4) so that the compatibility condition of the first order is sati sfied at each vertex of Ωin a natural way with the pairs(gj,gk). (2) The mapping w→(w|Γ0,1,∂xw|Γ0,1,w|Γ0,2,∂yw|Γ0,2w|Γ1,1,−∂yw|Γ1,1,w|Γ1,2,−∂xw|Γ1,2) = ((f1,g1),...(f4,g4)) defined on D(Ω)is extended from H2(Ω)onto the subspace of [H3/2((0,1))×H1/2((0,1))]4of functions ((f1,g1),...(f4,g4))so that the compatibility conditions of the second order are satisfied at each vertex of Ω in a natural way with the pairs [(fj,gj),(fk,gk)]. Lemma 2.1. (Extension lemma) Let gj∈H1/2((0,1)),j= 1,2, so that (g1,g2),(g1,0)and(g2,0)satisfy the first order compatibility condition respectively at the vertices(0,0),(1,0)and(0,1). Then there exists u∈H2(Ω)so thatu= 0onΓ0and∂νu=gjonΓ1,j,j= 1,2. Proof.(i) We define f1(t) =/integraltextt 0g2(s)dsandf2(t) =/integraltextt 0g1(s)ds. Then (f1,g1) and (f2,g2) satisfy the com- patibility conditions of the second order at the vertex (0 ,0). (ii) Let/tildewideg1∈H1/2((0,1)) be such that/integraltext1 0|/tildewideg1(t)|2 tdt<∞. Let/tildewidef1(t) =/integraltextt 0g2(s)ds. Hence, it is straightfor- ward to check that ( /tildewidef1,/tildewideg1) and (0,g2) satisfy the compatibility conditions of the second order at (0 ,0). (iii) From steps (i) and (ii) we derive that the pairs [( f1,g1),(f2,g2)], [(f1,g1),(0,g2)] and [(0,g1),(f2,g2)] satisfy the second order compatibility conditions respectively at th e vertices (0 ,0), (1,0) and (0,1). We see that unfortunately the pair [(0 ,g1),(0,g2)] doesn’t satisfy necessarily the compatibility conditions of the second order at the vertex (1 ,1). We pick χ∈C∞(R) so thatχ= 1 in a neighborhood of 0 and χ= 0 in a neighborhood of 1. Then [(0 ,χg1),(0,χg2)] satisfies the compatibility condition of the second order at the vertex (1,1). Since this construction is of local character at each vertex, t he cutoff function at the vertex (1,1) doesn’t modify the construction at the other vertices. In othe r words, the compatibility conditions of the second order are preserved at the other vertices. We comple te the proof by applying Theorem 2.1. /square4 KA¨IS AMMARI AND MOURAD CHOULLI Corollary 2.1. Leta= (a1,a2)∈Aandgj∈H1/2((0,1)),j= 1,2, so that (g1,g2),(g1,0)and(g2,0) satisfy the first order compatibility condition respective ly at the vertices (0,0),(1,0)and(0,1). Then there existsu∈H2(Ω)so thatu= 0onΓ0and∂νu=ajgjonΓ1,j,j= 1,2. Proof.It is sufficient to prove that ( a1g1,a2,g2) and (ajgj,0),j= 1,2, satisfy the first order compatibility condition at (0 ,0) witha1(0) =a2(0) for the first pair and without any condition on ajfor the second pair. Usinga1(0) =a2(0), we get t−1|a1(t)−a2(t)|2≤2t−1|a1(t)−a1(0)|2+2t−1|a2(t)−a2(0)|2 ≤2t−1+2α([a1]2 α+[a2]2 α) ≤2([a1]2 α+[a2]2 α). This estimate together with the following one |a1(t)g1(t)−a2(t)g2(t)|2≤2|a1(t)−a2(t)|2|g1(t)|2+2|a2(t)|2|g1(t)−g2(t)|2 yield /integraldisplay1 0|a1(t)g1(t)−a2(t)g2(t)|2dt t≤4([a1]2 α+[a2]2 α)/ba∇dblf/ba∇dblL2((0,1))+2/ba∇dbla2/ba∇dblL∞((0,1))/integraldisplay1 0|g1(t)−g2(t)|2dt t. Hence/integraldisplay1 0|g1(t)−g2(t)|2dt t<∞=⇒/integraldisplay1 0|a1(t)g1(t)−a2(t)g2(t)|2dt t<∞. If (gj,0) satisfies the first compatibility at the vertex (0 ,0). Then /integraldisplay1 0|gj(t)|2dt t<∞. Therefore/integraldisplay1 0|ajgj(t)|2dt t≤ /ba∇dblaj/ba∇dbl2 L∞((0,1))/integraldisplay1 0|gj(t)|2dt t<∞. Thus (ajgj,0) satisfies also the first compatibility at the vertex (0 ,0). /square 2.2.Observability inequality. We discuss briefly how we can adapt the existing results to get an obs erv- ability inequality corresponding to our IBVP. We first note that Γ0⊂ {x∈Γ;m(x)·ν(x)<0}, Γ1⊂ {x∈Γ;m(x)·ν(x)>0}, wherem(x) =x−x0,x∈R2, andx0= (α,α) withα>1. The following Rellich identity is a particular case of identity [5, (3.5), pag e 227]: for each 3 /2< s <2 andϕ∈Hs(Ω) satisfying ∆ ϕ∈L2(Ω), (2.3) 2/integraldisplay Ω∆ϕ(m·∇ϕ)dx= 2/integraldisplay Γ∂νϕ(m·∇ϕ)dσ−/integraldisplay Γ(m·ν)|∇ϕ|2dσ. Lemma 2.2. Let(v,w)∈D(Aa). Then 2/integraldisplay Ω∆v(m·∇v)dx= 2/integraldisplay Γ∂νv(m·∇v)dσ−/integraldisplay Γ(m·ν)|∇v|2dσ. Proof.Let (v,w)∈D(Aa). By Corollary 2.1, there exists /tildewidev∈H2(Ω) so that/tildewidev= 0 on Γ 0and∂ν/tildewidev=−aw on Γ1. In light of the fact that z=v−/tildewidevis such that ∆ z∈L2(Ω),z= 0 on Γ 0and∂νz= 0 on Γ 1, we get z∈Hs(Ω) for some 3 /2< s <2 by [5, Theorem 5.2, page 237]. Therefore v∈Hs(Ω). We complete the proof by applying Rellich identity (2.3). /square Lemma2.2athand, wecan mimic the proofof[7, Theorem7.6.1,page2 52]in orderto obtainthe following theorem:LOGARITHMIC STABILITY IN DETERMINING A BOUNDARY COEFFICIE NT 5 Theorem 2.2. We assume that a≥δonΓ1, for some δ >0. There exist M≥1andω >0, depending only onδ, so that /ba∇dbletAa(v,w)/ba∇dblV×L2(Ω)≤Me−ωt/ba∇dbl(v,w)/ba∇dblV×L2(Ω),(v,w)∈D(Aa), t≥0. An immediate consequence of Theorem 2.2 is the following observability inequality. Corollary 2.2. We fix0<δ0<δ1. Then there exist τ0>0andκ, depending only on δ0andδ1so that for anyτ≥τ0anda∈Asatisfyingδ0≤a≤δ1onΓ1, /ba∇dbl(u0,u1)/ba∇dblV×L2(Ω)≤κ/ba∇dblCa(u0,u1)/ba∇dblL2(Σ1). Moreover,Cais admissible for etAaand(Ca,Aa)is exactly observable. We omit the proof of this corollary. It is quite similar to that of [7, Coro llary 7.6.5, page 256]. 3.The inverse problem 3.1.An abstract framework for the inverse source problem. In the present subsection we consider an inverse source problem for an abstract evolution equation. The result of this subsection is the main ingredient in the proof of Theorem 1.1. LetHbe a Hilbert space and A:D(A)⊂H→Hbe the generator of continuous semigroup ( T(t)). An operatorC∈B(D(A),Y),Yis a Hilbert space which is identified with its dual space, is called an admiss ible observation for ( T(t)) if for some (and hence for all) τ >0, the operator Ψ ∈B(D(A),L2((0,τ),Y)) given by (Ψx)(t) =CT(t)x, t∈[0,τ], x∈D(A), has a bounded extension to H. We introduce the notion of exact observability for the system z′(t) =Az(t), z(0) =x, (3.1) y(t) =Cz(t), (3.2) whereCis an admissible observation for T(t). Following the usual definition, the pair ( A,C) is said exactly observable at time τ >0 if there is a constant κsuch that the solution ( z,y) of (3.1) and (3.2) satisfies /integraldisplayτ 0/ba∇dbly(t)/ba∇dbl2 Ydt≥κ2/ba∇dblx/ba∇dbl2 H, x∈D(A). Or equivalently (3.3)/integraldisplayτ 0/ba∇dbl(Ψx)(t)/ba∇dbl2 Ydt≥κ2/ba∇dblx/ba∇dbl2 H, x∈D(A). Letλ∈H1((0,τ)) such that λ(0)/ne}ationslash= 0. We consider the Cauchy problem (3.4) z′(t) =Az(t)+λ(t)x, z(0) = 0 and we set (3.5) y(t) =Cz(t), t∈[0,τ]. We fixβin the resolvent set of A. LetH1be the space D(A) equipped with the norm /ba∇dblx/ba∇dbl1=/ba∇dbl(β−A)x/ba∇dbl and denote by H−1the completion of Hwith respect to the norm /ba∇dblx/ba∇dbl−1=/ba∇dbl(β−A)−1x/ba∇dbl. As it is observed in [1, Proposition 4.2, page 1644] and its proof, when x∈H−1(which is the dual space of H1with respect to the pivot space H) andλ∈H1((0,T)), then, according to the classical extrapolation theory of semig roups, the Cauchy problem (3.4) has a unique solution z∈C([0,τ];H). Additionally ygiven in (3.5) belongs to L2((0,τ),Y). Whenx∈H, we have by Duhamel’s formula (3.6) y(t) =/integraldisplayt 0λ(t−s)CT(s)xds=/integraldisplayt 0λ(t−s)(Ψx)(s)ds.6 KA¨IS AMMARI AND MOURAD CHOULLI Let H1 ℓ((0,τ),Y) =/braceleftbig u∈H1((0,τ),Y);u(0) = 0/bracerightbig . We define the operator S:L2((0,τ),Y)−→H1 ℓ((0,τ),Y) by (3.7) ( Sh)(t) =/integraldisplayt 0λ(t−s)h(s)ds. IfE=SΨ, then (3.6) takes the form y(t) = (Ex)(t). LetZ= (β−A∗)−1(X+C∗Y). Theorem 3.1. We assume that (A,C)is exactly observable at time τ. Then (i)Eis one-to-one from HontoH1 ℓ((0,τ),Y). (ii)Ecan be extended to an isomorphism, denoted by /tildewideE, fromZ′ontoL2((0,τ);Y). (iii) There exists a constant /tildewideκ, independent on λ, so that (3.8) /ba∇dblx/ba∇dblZ′≤/tildewideκ|λ(0)|e/bardblλ′/bardbl2 L2((0,τ)) |λ(0)|2τ/ba∇dbl/tildewideEx/ba∇dblL2((0,τ),Y). Proof.(i) and (ii) are contained in [1, Theorem 4.3, page 1645]. We need only t o prove (iii). To do this, we start by observing that S∗:L2((0,τ),Y)→H1 r((0,τ);Y) =/braceleftbig u∈H1((0,τ),Y);u(τ) = 0/bracerightbig , the adjoint of S, is given by S∗h(t) =/integraldisplayτ tλ(s−t)h(s)ds, h∈H1 r((0,τ);Y). We fixh∈H1 r((0,τ);Y) and we set k=S∗h. Then k′(t) =λ(0)h(t)−/integraldisplayτ tλ′(s−t)h(s)ds. Hence [|λ(0)|/ba∇dblh(t)/ba∇dbl]2≤/parenleftbigg/integraldisplayτ t|λ′(s−t)| |λ(0)|[|λ(0)|/ba∇dblh(s)/ba∇dbl]ds+/ba∇dblk′(t)/ba∇dbl/parenrightbigg2 ≤2/parenleftbigg/integraldisplayτ t|λ′(s−t)| |λ(0)|[|λ(0)|/ba∇dblh(s)/ba∇dbl]ds/parenrightbigg2 +2/ba∇dblk′(t)/ba∇dbl2 ≤2/ba∇dblλ′/ba∇dbl2 L2((0,τ)) |λ(0)|2/integraldisplayt 0[|λ(0)|/ba∇dblh(s)/ba∇dbl]2ds+2/ba∇dblk′(t)/ba∇dbl2. The last estimate is obtained by applying Cauchy-Schwarz’s inequality . A simple application of Gronwall’s lemma entails [|λ(0)|/ba∇dblh(t)/ba∇dbl]2≤2e2/bardblλ′/bardbl2 L2((0,τ)) |λ(0)|2τ/ba∇dblk′(t)/ba∇dbl2. Therefore, /ba∇dblh/ba∇dblL2((0,τ);Y)≤√ 2 |λ(0)|e/bardblλ′/bardbl2 L2((0,τ)) |λ(0)|2τ/ba∇dblk′/ba∇dblL2((0,τ);Y). This inequality yields (3.9) /ba∇dblh/ba∇dblL2((0,τ);Y)≤√ 2 |λ(0)|e/bardblλ′/bardbl2 L2((0,τ)) |λ(0)|2τ/ba∇dblS∗h/ba∇dblH1r((0,τ);Y).LOGARITHMIC STABILITY IN DETERMINING A BOUNDARY COEFFICIE NT 7 The adjoint of S∗, acting as a bounded operator from [ Hr((0,1);Y)]′intoL2((0,τ);Y), gives an extension ofS. We denote by /tildewideSthis operator. By [1, Proposition 4.1, page 1644] S∗defines an isomorphism from [Hr((0,1);Y)]′ontoL2((0,τ);Y). In light of the fact that /ba∇dbl/tildewideS/ba∇dblB([Hr((0,1);Y)]′;L2((0,τ);Y))=/ba∇dblS∗/ba∇dblB(L2((0,τ);Y);Hr((0,1);Y)), (3.9) implies (3.10)|λ(0)|√ 2e−/bardblλ′/bardbl2 L2((0,τ)) |λ(0)|2τ≤ /ba∇dbl/tildewideS/ba∇dblB([Hr((0,1);Y)]′;L2((0,τ);Y)). On the other hand, according to [1, Proposition 2.13, page 1641], Ψ p ossesses a unique bounded extension, denoted by/tildewideΨ fromZ′into [Hr((0,1);Y)]′and there exists a constant c>0 so that (3.11) /ba∇dbl/tildewideΨ/ba∇dblB(Z′;[Hr((0,1);Y)]′)≥c. Consequently, /tildewideE=/tildewideS/tildewideΨ gives a unique extension of Eto an isomorphism from Z′ontoL2((0,τ);Y). We end up the proof by noting that (3.8) is a consequence of (3.9) an d (3.11). /square 3.2.An inverse source problem for an IBVP for the wave equation. In the present subsection we are going to apply the result of the preceding subsection to H=V×L2(Ω),H1=D(Aa) equipped with its graph norm and Y=L2(Γ1). We consider the the IBVP (3.12) ∂2 tu−∆u=λ(t)w inQ, u= 0 on Σ 0, ∂νu+a∂tu= 0 on Σ 1, u(·,0) = 0, ∂tu(·,0) = 0. Let (0,w)∈H−1andλ∈H1((0,τ)). From the comments in the preceding subsection, (3.12) has a un ique solutionuwso that (uw,∂tuw)∈C([0,τ];V×L2(Ω)) and∂νuw|Γ1∈L2(Σ1). We consider the inverse problem consisting in the determination of w, so that (0,w)∈H−1, appearing in the IBVP (3.12) from the boundary measurement ∂νuw|Σ1. Here the function λis assumed to be known. Taking into account that {0}×V′⊂H−1, whereV′is the dual space of V, we obtain as a consequence of Corollary 2.1: Proposition 3.1. There exists a constant C >0so that for any λ∈H1((0,τ))andw∈V′, (3.13) /ba∇dblw/ba∇dblV′≤C|λ(0)|e/bardblλ′/bardbl2 L2((0,τ)) |λ(0)|2τ/ba∇dbl∂νu/ba∇dblL2(Σ1). 3.3.Proof of Theorem 1.1. We start by observing that uais also the unique solution of /braceleftbigg/integraltext Ωu′′(t)vdx=/integraltext Ω∇u(t)·∇vdx−/integraltext Γ1au′(t)v,for allv∈V. u(0) =u0, u′(0) =u1. Letu=ua−u0. Thenuis the solution of the following problem (3.14)/braceleftbigg/integraltext Ωu′′(t)vdx=/integraltext Ω∇u(t)·∇vdx−/integraltext Γ1au′(t)v−/integraltext Γ1au′ 0(t)v,for allv∈V. u(0) = 0, u′(0) = 0. Fork,ℓ∈Z, we set λkℓ= [(k+1/2)2+(ℓ+1/2)2]π2 φkℓ(x,y) = 2cos((k+1/2)πx)cos((ℓ+1/2)πy). We check in a straightforward manner that u0= cos(√λkℓt)φkℓwhen (u0,u1) = (φkℓ,0). In the sequel k,ℓare arbitrarily fixed. We set λ(t) = cos(√λkℓt) and we define wa∈V′by wa(v) =−/radicalbig λkℓ/integraldisplay Γ1aφkℓv.8 KA¨IS AMMARI AND MOURAD CHOULLI In that case (3.14) becomes /braceleftbigg/integraltext Ωu′′(t)vdx=/integraltext Ω∇u(t)·∇vdx−/integraltext Γ1au′(t)v+λ(t)wa(v),for allv∈V. u(0) = 0, u′(0) = 0. Consequently, uis the solution of (3.12) with w=wa. Applying Proposition 3.1, we find (3.15) /ba∇dblwa/ba∇dblV′≤Ceλkℓτ2/ba∇dbl∂νu/ba∇dblL2(Σ1). But (3.16) a1(0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Γ1(aφkℓ)2dσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle=1√λkℓ|wa((a1⊗a2)φkℓ)| ≤1√λkℓ/ba∇dblwa/ba∇dblV′/ba∇dbl(a1⊗a2)φkℓ/ba∇dblV, where we used a1(0) =a2(0), and (3.17) /ba∇dbl(a1⊗a2)φkℓ/ba∇dblV≤C0/radicalbig λkl/ba∇dbla1⊗a2/ba∇dblH1(Ω). HereC0is a constant independent on aandφkℓ. We note (a1⊗a2)φkℓ∈Veven ifa1⊗a2/ne}ationslash∈V. Now a combination of (3.15), (3.16) and (3.17) yields a1(0)/parenleftBig /ba∇dbla1φk/ba∇dbl2 L2((0,1))+/ba∇dbla2φℓ/ba∇dbl2 L2((0,1))/parenrightBig ≤C/ba∇dbla1/ba∇dblH1(0,1)/ba∇dbla2/ba∇dblH1(0,1)eλkℓτ2/2/ba∇dbl∂νu/ba∇dblL2(Σ1), whereφk(s) =√ 2cos((k+1/2)πs). This and the fact that m≤aj(0) and/ba∇dblaj/ba∇dblH1((0,1))≤Mimply /ba∇dbla1φk/ba∇dbl2 L2((0,1))+/ba∇dbla2φℓ/ba∇dbl2 L2((0,1))≤CM2 meλkℓτ2/2/ba∇dbl∂νu/ba∇dblL2(Σ1). Hence, where j= 1 or 2, /ba∇dblajφk/ba∇dbl2 L2((0,1))≤CM2 mek2τ2π2/ba∇dbl∂νu/ba∇dblL2(Σ1). Let ak j=/integraldisplay1 0aj(x)φk(x)dx, j= 1,2. Since |ak j|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1 0aj(x)φk(x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ /ba∇dblajφk/ba∇dblL1((0,1))≤ /ba∇dblajφk/ba∇dblL2((0,1)), we get (ak j)2≤CM2 mek2τ2π2/ba∇dbl∂νu/ba∇dblL2(Σ1). On the other hand /ba∇dbl∂νu/ba∇dblL2(Σ1)=/ba∇dblΛa(φkl)−Λ0(φkl)/ba∇dblL2(Σ)≤Ck2/ba∇dblΛa−Λ0/ba∇dbl. Hence (3.18) ( ak j)2≤CM2 mek2(τ2π2+1)/ba∇dblΛa−Λ0/ba∇dbl. Letq=M2 mandα=τ2π2+2. We obtain in a straightforward manner from (3.18) /summationdisplay |k|≤N(ak j)2≤CqeαN2/ba∇dblΛa−Λ0/ba∇dbl.LOGARITHMIC STABILITY IN DETERMINING A BOUNDARY COEFFICIE NT 9 Consequently, /ba∇dblaj/ba∇dbl2 L2((0,1))≤/summationdisplay |k|≤N(ak j)2+1 N2/summationdisplay |k|>Nk2(ak j)2 ≤C/parenleftBigg qeαN2/ba∇dblΛa−Λ0/ba∇dbl+/ba∇dblaj/ba∇dbl2 H1((0,1)) N2/parenrightBigg ≤C/parenleftbigg qeαN2/ba∇dblΛa−Λ0/ba∇dbl+M2 N2/parenrightbigg ≤CM2/parenleftbigg1 meαN2/ba∇dblΛa−Λ0/ba∇dbl+1 N2/parenrightbigg . That is (3.19) /ba∇dblaj/ba∇dbl2 L2((0,1))≤CM2/parenleftbigg1 meαN2/ba∇dblΛa−Λ0/ba∇dbl+1 N2/parenrightbigg . Assume that /ba∇dblΛa−Λ0/ba∇dbl ≤δ=me−α. Let thenN0≥1 be the greatest integer so that C meαN2 0/ba∇dblΛa−Λ0/ba∇dbl ≤1 N2 0. Using 1 meα(N0+1)2/ba∇dblΛa−Λ0/ba∇dbl ≤1 (N0+1)2, we find (2N0)2≥(N0+1)2≥1 α+1ln/parenleftbiggm /ba∇dblΛa−Λ0/ba∇dbl/parenrightbigg . This estimate in (3.19) with N=N0gives (3.20) /ba∇dblaj/ba∇dblL2((0,1))≤2C√ α+1M/vextendsingle/vextendsingleln/parenleftbig m−1/ba∇dblΛa−Λ0/ba∇dbl/parenrightbig/vextendsingle/vextendsingle−1/2. When/ba∇dblΛa−Λ0/ba∇dbl ≥δ, we have (3.21) /ba∇dblaj/ba∇dblL2((0,1))≤M δ/ba∇dblΛa−Λ0/ba∇dbl. In light of (3.20) and (3.21), we find a constants c>0, that can depend only on τ, so that /ba∇dblaj/ba∇dblL2((0,1))≤cM/parenleftBig/vextendsingle/vextendsingleln/parenleftbig m−1/ba∇dblΛa−Λ0/ba∇dbl/parenrightbig/vextendsingle/vextendsingle−1/2+m−1/ba∇dblΛa−Λ0/ba∇dbl/parenrightBig . Appendix A. We prove the following lemma Lemma A.1. Let1/2<α≤1anda∈Cα([0,1]). Then the mapping f/ma√sto→afdefines a bounded operator onH1/2((0,1)). Proof.We recall that H1/2((0,1)) consists in functions f∈L2((0,1)) with finite norm /ba∇dblf/ba∇dblH1/2((0,1))=/parenleftbigg /ba∇dblf/ba∇dbl2 L2((0,1))+/integraldisplay1 0/integraldisplay1 0|f(x)−f(y)|2 |x−y|2dxdy/parenrightbigg1/2 . Leta∈Cα([0,1]). We have |a(x)f(x)−a(y)f(y)|2 |x−y|2≤ /ba∇dbla/ba∇dbl2 L∞(0,1)|f(x)−f(y)|2 |x−y|2+|f(y)|2[a]2 α |x−y|2(1−α), where [a]α= sup{|a(x)−a(y)||x−y|−α;x,y∈[0,1], x/ne}ationslash=y}.10 KA¨IS AMMARI AND MOURAD CHOULLI Using that 1 /2<α≤1, we find that x→ |x−y|−2(1−α)∈L1((0,1)),y∈[0,1], and /integraldisplay1 0dx |x−y|2(1−α)≤1 2α−1, y∈[0,1]. Henceaf∈H1/2((0,1)) with /ba∇dblaf/ba∇dblH1/2((0,1))≤1 2α−1/ba∇dbla/ba∇dblCα([0,1])/ba∇dblf/ba∇dblH1/2((0,1)). Here /ba∇dbla/ba∇dblCα([0,1])=/ba∇dbla/ba∇dblL∞((0,1))+[a]α. /square Appendix B. We give the proof of the following lemma Lemma B.1. Leta∈AandAabe the unbounded operator defined on V×L2(Ω)by Aa= (w,∆v), D(Aa) ={(v,w)∈V×V; ∆v∈L2(Ω)and∂νv=−awonΓ1}. ThenAais m-dissipative. Proof.Let/an}b∇acketle{t·,·/an}b∇acket∇i}htbe scalar product in V×L2(Ω). That is /an}b∇acketle{t(v1,w1),(v2,w2)/an}b∇acket∇i}ht=/integraldisplay Ω∇v1·∇v2dx+/integraldisplay Ωw1w2dx,(vj,wj)∈V×L2(Ω), j= 1,2. For (v1,w1)∈D(Aa), we have /an}b∇acketle{tAa(v1,w1),(v1,w1)/an}b∇acket∇i}ht=/an}b∇acketle{t(w1,∆v1),(v1,w1)/an}b∇acket∇i}ht (B.1) =/integraldisplay Ω∇w1·∇v1dx+/integraldisplay Ω∆v1w1dx Applying twice Green’s formula, we get /integraldisplay Ω∇w1·∇v1dx=−/integraldisplay Ωw1∆v1dx+/integraldisplay Γ1w1∂νv1dσ, (B.2) /integraldisplay Ω∆v1w1dx=−/integraldisplay Ω∇v1·∇w1dx−/integraldisplay Γ1aw1w1dσ. (B.3) We take the sum side by side of identities (B.2) and (B.3). Using that ∂νv1=−aw1on Γ1we obtain /integraldisplay Ω∇w1·∇v1dx+/integraldisplay Ω∆v1w1dx=−/integraldisplay Ωw1∆v1dx−/integraldisplay Ω∇v1·∇w1dx−2/integraldisplay Γ1a|w1|2dσ =−/an}b∇acketle{t(v1,w1),Aa(v1,w1)/an}b∇acket∇i}ht−2/integraldisplay Γ1a|w1|2dσ. This and (B.1) yield ℜ/an}b∇acketle{tAa(v1,w1),(v2,w2)/an}b∇acket∇i}ht=−/integraldisplay Γ1a|w1|2dσ≤0. In other words, Aais dissipative. We complete the proof by showing that Aais onto implying that Aais m-dissipative. To this end we are going to show that for each ( f,g)∈V×L2(Ω), the problem w=f,−∆v=g. has a unique solution ( v,w)∈D(Aa).LOGARITHMIC STABILITY IN DETERMINING A BOUNDARY COEFFICIE NT 11 In light of the fact ψ→/parenleftbig/integraltext Ω|∇ψ|2dx/parenrightbig1/2defines an equivalent norm on V, we can apply Lax-milgram’s lemma. We get that there exists a unique v∈Vsatisfying/integraldisplay Ω∇v·∇ψdx=/integraldisplay Ωgψdx−/integraldisplay Γ1awψdσ, ψ∈V. From this identity, we deduce in a standard way that −∆v=gand∂νv=−awon Γ1. The proof is then complete /square References [1]C. Alves, A.-L. Silvestre, T. Takahashi and M. Tucsnak, Solving inverse source problems using observability. Appl i- cations to the Euler-Bernoulli plate equation , SIAM J. Control Optim. 48 (2009), 1632-1659. [2]K. Ammari andM. Choulli ,Logarithmic stability in determining two coefficients in a di ssipative wave equation. Exten- sions to clamped Euler-Bernoulli beam and heat equations , to appear in J. Diff. Equat. [3]K. Ammari andM. Choulli ,Determining a boundary coefficient in a dissipative wave equa tion: uniqueness and directional Lipschitz stability , arXiv:1503.04528. [4]P. Grisvard ,Elliptic problems in nonsmooth domains , Pitman Publishing Inc., 1985. [5]P. Grisvard ,Contrˆ olabilit´ e des solutions de l’´ equation des ondes en pr´ esence de singularit´ es , J. Math. Pures Appl. 68 (1989), 215-259. [6]V. Komornik andE. Zuazua ,A direct method for the boundary stabilization of the wave eq uation, J. Math Pures Appl. 69 (1990), 33-54. [7]M. Tucsnak and G. Weiss ,Observation and control for operator semigroups. Birkh¨ auser Advanced Texts, Birkh¨ auser Verlag, Basel, 2009. UR Analysis and Control of PDEs, UR 13ES64, Department of Mathem atics, Faculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisia E-mail address :kais.ammari@fsm.rnu.tn Institut ´Elie Cartan de Lorraine, UMR CNRS 7502, Universit ´e de Lorraine, Boulevard des Aiguillettes, BP 70239, 54506 Vandoeuvre les Nancy cedex - Ile du Saulcy, 57045 Metz cedex 01, France E-mail address :mourad.choulli@univ-lorraine.fr

2015-05-27

In [2] we introduced a method combining together an observability inequality and a spectral decomposition to get a logarithmic stability estimate for the inverse problem of determining both the potential and the damping coefficient in a dissipative wave equation from boundary measurements. The present work deals with an adaptation of that method to obtain a logarithmic stability estimate for the inverse problem of determining a boundary damping coefficient from boundary measurements. As in our preceding work, the different boundary measurements are generated by varying one of the initial conditions.

Logarithmic stability in determining a boundary coefficient in an ibvp for the wave equation

1505.07248v1

Magnon-phonon interactions in magnetic insulators Simon Streib,1Nicolas Vidal-Silva,2, 3, 4Ka Shen,5and Gerrit E. W. Bauer1, 5, 6 1Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands 2Departamento de Física, Universidad de Santiago de Chile, Avda. Ecuador 3493, Santiago, Chile 3Center for the Development of Nanoscience and Nanotechnology (CEDENNA), 917-0124 Santiago, Chile 4Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile 5Department of Physics, Beijing Normal University, Beijing 100875, China 6Institute for Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan (Dated: June 4, 2019) We address the theory of magnon-phonon interactions and compute the corresponding quasi- particle and transport lifetimes in magnetic insulators with focus on yttrium iron garnet at inter- mediate temperatures from anisotropy- and exchange-mediated magnon-phonon interactions, the latter being derived from the volume dependence of the Curie temperature. We find in general weak effects of phonon scattering on magnon transport and the Gilbert damping of the macrospin Kittel mode. The magnon transport lifetime differs from the quasi-particle lifetime at shorter wavelengths. I. INTRODUCTION Magnons are the elementary excitations of magnetic order, i.e. the quanta of spin waves. They are bosonic andcarryspinangularmomentum. Ofparticularinterest are the magnon transport properties in yttrium iron gar- net (YIG) due to its very low damping ( <104), which makes it one of the best materials to study spin-wave or spin caloritronic phenomena [1–6]. For instance, the spin Seebeck effect (SSE) in YIG has been intensely studied in the past decade [7–13]. Here, a temperature gradi- ent in the magnetic insulator injects a spin current into attached Pt contacts that is converted into a transverse voltage by the inverse spin Hall effect. Most theories ex- plain the effect by thermally induced magnons and their transport to and through the interface to Pt [7, 14–19]. However, phonons also play an important role in the SSE through their interactions with magnons [20–22]. Magnetoelastic effects in magnetic insulators were ad- dressed first by Abrahams and Kittel [23–25], and by Kaganov and Tsukernik [26]. In the long-wavelength regime, the strain-induced magnetic anisotropy is the most important contribution to the magnetoelastic en- ergy, whereas for shorter wavelengths, the contribution from the strain-dependence of the exchange interaction becomes significant [27–29]. Rückriegel et al.[28] com- puted very small magnon decay rates in thin YIG films due to magnon-phonon interactions with quasi-particle lifetimesqp?480 ns;even at room temperature. How- ever, these authors do not consider the exchange interac- tion and the difference between quasi-particle and trans- port lifetimes. Recently, it has been suggested that magnon spin transport in YIG at room temperature is driven by the magnon chemical potential [3, 30]. Cornelissen et al. [3] assume that at room temperature magnon- phonon scattering of short-wavelength thermal magnons is dominated by the exchange interaction with a scat- tering time of qp1 ps, which is much faster than the anisotropy-mediated magnon-phonon coupling con-sidered in Ref. [28] and efficiently thermalizes magnons and phonons to equal temperatures without magnon de- cay. Recently, the exchange-mediated magnon-phonon interaction [31] has been taken into account in a Boltz- mann approach to the SSE, but this work underestimates the coupling strength by an order of magnitude, as we will argue below. In this paper we present an analytical and numeri- cal study of magnon-phonon interactions in bulk ferro- magnetic insulators, where we take both the anisotropy- and the exchange-mediated magnon-phonon interactions into account. By using diagrammatic perturbation the- ory to calculate the magnon self-energy, we arrive at a wave-vector dependent expression of the magnon scat- tering rate, which is the inverse of the magnon quasi- particle lifetime qp. The magnetic Grüneisen parameter m=@lnTC=@lnV[32, 33], where TCis the Curie tem- perature and Vthe volume of the magnet, gives direct access to the exchange-mediated magnon-phonon inter- action parameter. We predict an enhancement in the phonon scattering of the Kittel mode at the touching points of the two-magnon energy (of the Kittel mode and a finite momentum magnon) and the longitudinal and transverse phonon dispersions, for YIG at around 1:3 T and4:6 T. We also emphasize the difference in magnon lifetimesthatbroadenlightandneutronscatteringexper- iments, and the transport lifetimes that govern magnon heat and spin transport. The paper is organized as follows: in Sec. II we briefly review the theory of acoustic magnons and phonons in ferro-/ferrimagnets, particularly in YIG. In Sec. III we derive the exchange- and anisotropy-mediated magnon- phonon interactions for a cubic Heisenberg ferromagnet with nearest neighbor exchange interactions in the long- wavelength limit. In Sec. IV we derive the magnon decay rate from the imaginary part of the magnon self-energy in a diagrammatic approach and in Sec. V we explain the differences between the magnon quasi-particle and transport lifetimes. Our numerical results for YIG are discussed in Sec. VI. Finally in Sec. VII we summarizearXiv:1906.01042v1 [cond-mat.str-el] 3 Jun 20192 and discuss the main results of the present work. The va- lidity of our long-wavelength approximation is analyzed in Appendix A and in Appendix B we explain why sec- ond order magnetoelastic couplings may be disregarded. In Appendix C we briefly discuss the numerical methods used to evaluate the k-space integrals. II. MAGNONS AND PHONONS IN FERROMAGNETIC INSULATORS Without loss of generality, we focus our treatment on yttrium iron garnet (YIG). The magnon band structure of YIG has been determined by inelastic neutron scatter- ing [34–36] and by ab initio calculation of the exchange constants [37]. The complete magnon spectral function hasbeencomputedforalltemperaturesbyatomisticspin simulations [38], taking all magnon-magnon interactions into account, but not the magnon-phonon scattering. The pure phonon dispersion is known as well [29, 39]. In the following, we consider the interactions of the acoustic magnons from the lowest magnon band with transverse and longitudinal acoustic phonons, which allows a semi- analytic treatment but limits the validity of our results to temperatures below 100 K. Since the low-temperature values of the magnetoelastic constants, sound velocities, and magnetic Grüneisen parameter are not available for YIG, we use throughout the material parameters under ambient conditions. A. Magnons Spinsinteractwitheachotherviadipolarandexchange interactions. We disregard the former since at the energy scaleEdip0:02 meV [28] it is only relevant for long- wavelength magnons with wave vectors k.6107m1 and energies Ek=kB.0:2 K, which are negligible for the thermal magnon transport in the temperature regime we are interested in. The lowest magnon band can then be described by a simple Heisenberg model on a course- grained simple cubic ferromagnet with exchange interac- tionJ Hm=J 2X hi6=jiSi
SjX igBBSz i;(2.1) where the sum is over all nearest neighbors and ~Siis the spin operator at lattice site Ri. The lattice constant of the cubic lattice or YIG is a= 12:376Aand the effective spin per unit cell ~S=~Msa3=(gB)14:2~at room temperature [28] ( S20forT.50 K[40]), where the g-factorg2,Bis the Bohr magneton and Msthe sat- uration magnetization. The parameter Jis an adjustable parameter that can be fitted to experiments or computed from first principles. Bis an effective magnetic field that orients the ground-state magnetization vector to the z axis and includes the (for YIG small) magnetocrystallineanisotropyfield. The 1=Sexpansionofthespinoperators in terms of Holstein-Primakoff bosons reads [41], S+ i=Sx+iSyp 2S[bi+O(1=S)];(2.2) S i=SxiSyp 2Sh by i+O(1=S)i ;(2.3) Sz i=Sby ibi; (2.4) whereby iandbiare the magnon creation and annihilation operators with boson commutation ruleh bi;by ji =i;j. Then Hm!X kEkby kbk; (2.5) where the magnon operators by kandbkare defined by bi=1p NX keik
Ribk; (2.6) by i=1p NX keik
Riby k; (2.7) andNthe number of unit cells. The dispersion relation Ek=gBB+ 4SJX =x;y;zsin2(ka=2) (2.8) becomes quadratic in the long-wavelength limit ka1: Ek=gBB+Eexk2a2; (2.9) whereEex=SJ. WithEex=kB40 K = 3:45 meV the latter is a good approximation up to k0= 1=a 8108m1[34]. The effective exchange coupling is thenJ0:24 meV. The lowest magnon band does not depend significantly on temperature [38], which implies thatEex=SJdoes not depend strongly on temper- ature. The temperature dependence of the saturation magnetization and effective spin Sshould therefore not affect the low-energy exchange magnons significantly. By using Eq. (2.9) in the following, our theory is valid for k.k0(see Fig. 1) or temperatures T.100 K. In this regime the cut-off of an ultraviolet divergence does not affect results significantly (see Appendix A). We disre- gard magnetostatic interactions that affect the magnon spectrum only for very small wave vectors since at low temperatures the phonon scattering is not significant. B. Phonons We expand the displacement Xiof the position riof unit cellifrom the equilibrium position Ri Xi=riRi; (2.10) into the phonon eigenmodes Xq, X i=1p NX q;e qXqeiq
Ri;(2.11)3 where2 fx;y;zgandqa wave vector. We define polarizations 2f1;2;3gfor the elastic continuum [42] eq1= (cosqcosq;cosqsinq;sinq);(2.12) eq2=i(sinq;cosq;0); (2.13) eq3=i(sinqcosq;sinqsinq;cosq);(2.14) where the angles qandqare the spherical coordinates of q=q(sinqcosq;sinqsinq;cosq);(2.15) which is valid for YIG up to 3 THz(12 meV) [29, 39]. The phonon Hamiltonian then reads Hp=X qPqPq 2m+m 2~2"2 qXqXq ; =X q"q ay qaq+1 2 ; (2.16) wherethecanonicalmomenta Pqobeythecommutation relations [Xq;Pq00] =i~q;q00and the mass of the YIG unit cell m=a3= 9:81024kg[27]. The phonon dispersions for YIG then read "q=~cjqj; (2.17) wherec1;2=ct= 3843 m=sis the transverse sound ve- locity andc3=cl= 7209 m=sthe longitudinal velocity at room temperature [27]. In terms of phonon creation and annihilation operators Xq=aq+ay qp 2m"q=~2; P q=1 irm"q 2 aqay q ; (2.18) andh aq;ay q00i =q;q0;0. In Fig. 1 we plot the longitudinal and transverse phonon and the acoustic magnon dispersion relations for YIG at zero magnetic field. The magnon-phonon inter- action leads to an avoided level crossing at points where magnon and phonon dispersion cross, as discussed in Refs. [27] and [28]. III. MAGNON-PHONON INTERACTIONS We derive in this section the magnon-phonon interac- tions due to the anisotropy and exchange interactions for a cubic lattice ferromagnet. A. Phenomenological magnon-phonon interaction In the long-wavelength/continuum limit ( k.k0) the magnetoelastic energy to lowest order in the deviations of magnetization and lattice from equilibrium reads [23– 26, 28] 0.0 0.5 1.0 1.5 k[109m−1]024681012Ek[meV]magnon model parabolic approximation longitudinal acoustic phonon transverse acoustic phononFigure 1. Dispersion relations of the acoustic phonons and magnons in YIG at zero magnetic field. Eme=n M2sZ d3rX [BM(r)M(r) +B0 @M(r) @r
@M(r) @r X(r);(3.1) wheren= 1=a3. The strain tensor Xis defined in terms of the lattice displacements X, X(r) =1 2@X(r) @r+@X(r) @r ;(3.2) with, for a cubic lattice [28], B=Bk+ (1)B?; (3.3) B0 =B0 k+ (1)B0 ?: (3.4) Bis caused by magnetic anisotropies and B0 by the exchange interaction under lattice deformations. For YIG at room temperature [27, 33] Bk=kB47:8 K = 4:12 meV;(3.5) B?=kB95:6 K = 8:24 meV;(3.6) B0 k=a2=kB2727 K = 235 meV ;(3.7) B0 ?=a20: (3.8) We discuss the values for B0 kandB0 ?in Sec. IIIC. B. Anisotropy-mediated magnon-phonon interaction The magnetoelastic anisotropy (3.1) is described by the Hamiltonian [28],4 Han mp=X q qbqXq+  qby qXq +1p NX q;k;k0kk0q;0X an kk0;by kbk0Xq +1p NX q;k;k0k+k0+q;0X bb kk0;bkbk0Xq +1p NX q;k;k0k+k0q;0X bb kk0;by kby k0Xq;(3.9) with interaction vertices q=B?p 2Sh iqzex q+qzey q + (iqx+qy)ez q ; (3.10) an kk0;=Ukk0;; (3.11) bb kk0;=Vkk0;; (3.12) bb kk0;=V kk0;; (3.13) and Uq;=iBk Sh qxex q+qyey q2qzez qi ;(3.14) Vq;=iBk Sh qxex qqyey qi +B? Sh qyex q+qxey qi : (3.15) The one magnon-two phonon process is of the same order in the total number of magnons and phonons as the two magnon-one phonon processes, but its effect on magnon transport is small, as shown in Appendix B. C. Exchange-mediated magnon-phonon interaction The exchange-mediated magnon-phonon interaction is obtained under the assumption that the exchange inter- actionJijbetween two neighboring spins at lattice sites riandrjdepends only on their distance, which leads to the expansion to leading order in the small parameter (jrirjja) Jij=J(jrirjj)J+J0
(jrirjja);(3.16) whereais the equilibrium distance and J0=@J=@a. With ri=Ri+XRi;the Heisenberg Hamiltonian (2.1) is modulated by Hex mp=J0X iX =x;y;z X Ri+aeX Ri
SRi
SRi+ae; (3.17)where eis a unit vectors in the direction. Expanding the displacements in terms of the phonon and magnon modes Hex mp=1p NX q;k;k0kk0q;0X ex kk0;by kbk0Xq;(3.18) with interaction ex kk0;= 8iJ0SX e kk0;sinka 2 sink0 a 2 sin(kk0 )a 2 iJ0a3SX e kk0;kk0(kk0 );(3.19) where the last line is the long-wavelength expansion. The magnon-phonon interaction bb k;k0;= ex k;k0;+ an k;k0; (3.20) conserves the magnon number, while (3.12) and (3.13) do not. Phonon numbers are not conserved in either case. The value of J0for YIG is determined by the magnetic Grüneisen parameter [32, 33] m=@lnTC @lnV=@lnJ @lnV=J0a 3J;(3.21) whereV=Na3is the volume of the magnet. The only assumption here is that the Curie temperature TCscales linearly with the exchange constant J[43]. mhas been measured for YIG via the compressibility to be m= 3:26[32], and via thermal expansion, m=3:13[33], so we set m=3:2. For other materials the magnetic Grüneisen parameter is also of the order of unity and in many cases m10=3[32, 33, 44]. A recent ab initio study of YIG finds m=3:1[45]. Comparing the continuum limit of Eq. (3.17) with the classical magnetoelastic energy (3.1) B0 k= 3mJS2a2=2; (3.22) whereforYIG B0 k=a2235 meV . Wedisregard B0 ?since it vanishes for nearest neighbor interactions by cubic lat- tice symmetry. The coupling strength of the exchange-mediated magnon-phonon interaction can be estimated from the exchange energy SJ0aEex=SJ[31, 46] following Akhiezer et al.[47, 48]. Our estimate of SJ0a= 3mSJ is larger by 3m, i.e. one order of magnitude. Since the scattering rate is proportional to the square of the in- teraction strength, our estimate of the scattering rate is a factor 100larger than previous ones. The assumption J0aJis too small to be consistent with the experi- mental Grüneisen constant [32, 33]. Ref. [3] educatedly guessedJ0a100J;which we now judge to be too large.5 Figure2. Feynmandiagramsofinteractionsbetweenmagnons (solid lines) and phonons (dashed lines). The arrows indicate the energy-momentum flow. (a) magnon-phonon interconver- sion, (b) magnon number-conserving magnon-phonon inter- action, (c) and (d) magnon number non-conserving magnon- phonon interactions. D. Interaction vertices The magnon-phonon interactions in the Hamiltonian (3.9) are shown in Fig. 2 as Feynman diagrams. Fig. 2(a) illustrates magnon and phonon interconversion, which is responsible for the magnon-phonon hybridization and level splitting at the crossing of magnon and phonon dis- persions [27, 28]. The divergence of this diagram at the magnon-phonon crossing points is avoided by either di- rect diagonalization of the magnon-phonon Hamiltonian [42] or by cutting-off the divergence by a lifetime param- eter [31]. This process still generates enhanced magnontransport that is observable as magnon polaron anoma- lies in the spin Seebeck effect [22] or spin-wave excitation thresholds [49, 50], but these are strongly localized in phase space and disregarded in the following, where we focus on the magnon scattering rates to leading order in 1=Sof the scattering processes in Fig. 2(b)-(d). IV. MAGNON SCATTERING RATE Here we derive the magnon reciprocal quasi-particle lifetime1 qp= as the imaginary part of the wave vector dependent self-energy, caused by acoustic phonon scat- tering [28], (k) =2 ~Im(k;Ek=~+i0+):(4.1) This quantity is in principle observable by inelastic neu- tron scattering. The total decay rate = c+ nc+ other(4.2) is the sum of the magnon number conserving decay rate cand the magnon number non-conserving decay rate nc, which are related to the magnon-phonon scattering timempand the magnon-phonon dissipation time mr by mp=1 c; mr=1 nc: (4.3) otheris caused by magnon-magnon and magnon disorder scattering, thereby beyond the scope of this work. The self-energy to leading order in the 1=Sexpansion is of second order in the magnon-phonon interaction [28], 2(k;i!) =1 NX k0~2bb k;k0;2 2m"kk0;nB("kk0;)nB(Ek0) i~!+"kk0;Ek0+1 +nB("kk0;) +nB(Ek0) i~!"kk0;Ek0 1 NX k0~2bb k;k0;2 2m"kk0;1 +nB("k+k0;) +nB(Ek0) i~!+"k+k0;+Ek0+nB("k+k0;)nB(Ek0) i~!"k+k0;+Ek0 ; (4.4) where the magnon number conserving magnon-phonon scattering vertex bb k;k0;= ex k;k0;+ an k;k0;and the Planck (Bose) distribution function nB(") = (e"1)1 with inverse temperature = 1=(kBT). The Feynman diagrams representing the magnon number conserving and non-conserving contributions to the self-energy areshown in Fig. 3. We write the decay rate in terms of four contributions (k) = c out(k) + nc out(k) c in(k) nc in(k);(4.5) whereoutandindenote the out-scattering and in- scattering parts. The contributions to the decay rate read [28]6 k qk-q k kk qq-k k k(a) (b) Figure 3. Feynman diagrams representing the self-energy Eq. (4.4) due to (a) magnon number-conserving magnon- phonon interactions and (b) magnon number non-conserving magnon-phonon interactions. c out(k) =~ mNX q;bb k;kq;2 "q[(1 +nB(Ekq))nB("q)(EkEkq+"q) + (1 +nB(Ekq))(1 +nB("q))(EkEkq"q)]; (4.6) c in(k) =~ mNX q;bb k;kq;2 "q[nB(Ekq)(1 +nB("q))(EkEkq+"q) +nB(Ekq)nB("q)(EkEkq"q)]; (4.7) nc out(k) =~ mNX q;bb k;qk;2 "q[nB(Eqk)(1 +nB("q))(Ek+Eqk"q)]; (4.8) nc in(k) =~ mNX q;bb k;qk;2 "q[(1 +nB(Eqk))nB("q)(Ek+Eqk"q)]; (4.9) where the sum is over all momenta qin the Brillouin zone. Here the magnon/phonon annihilation rate is pro- portional to the Boson number nB, while the creation ratescaleswith 1+nB. Forexample,intheout-scattering rate c out(k)theincomingmagnonwithmomentum kgets scattered into the state kqand a phonon is either ab- sorbedwithprobability nBoremittedwithprobability (1 +nB). The out- and in-scattering rates are related by the detailed balance c in(k)= c out(k) = nc in(k)= nc out(k) =eEk:(4.10) For high temperatures kBTEk, we may expand the Bose functions nB(Ek)kBT=E kand we find in outT2and = out inT. For low temperatures kBTEk, the out-scattering rate out!const:and the in-scattering rate ineEk!0. The scattering processes (c) and (d) in Fig. 2 conserve energy and linear momentum, but not angular momentum. A loss of an-gular momentum after integration over all wave vectors corresponds to a mechanical torque on the total lattice that contributes to the Einstein-de Haas effect [51]. V. MAGNON TRANSPORT LIFETIME Inthissectionwecomparethetransportlifetime tand the magnon quasi-particle lifetime qpthat can be very different [52–54], but, to the best of our knowledge, has not yet been addressed for magnons. The magnon decay rate is proportional to the imaginary part of self energy, as shown in Eq. (4.1). On the other hand, the transport is governed by transport lifetime tin the Boltzmann equation that agrees with qponly in the relaxation time approximation. The stationary Boltzmann equation for7 the magnon distribution can be written as [3, 42] @fk(r) @r
@Ek @(~k)= in[f]out[f];(5.1) wherefk(r)is the magnon distribution function. The in andoutcontributions to the collision integral are related to the previously defined in- and out-scattering rates by in[f] = (1 +fk) in[f]; (5.2) out[f] =fk out[f]; (5.3) where the equilibrium magnon distribution nB(Ek)is re- placed by the non-equilibrium distribution function fk. The factor (1 +fk)corresponds to the creation of a magnon with momentum kin the in-scattering process and the factor fkto the annihilation in the out-scattering process. The phonons are assumed to remain at thermal equilibrium, so we disregard the phonon drift contribu- tion that is expected in the presence of a phononic heat current. Magnon transport is governed by three linear response functions, i.e. spin and heat conductivity and spin See- beck coefficient [42]. These can be obtained from the ex- pansion of the distribution function in terms of temper- ature and chemical potential gradients and correspond to two-particle Green functions with vertex corrections, that reflect the non-equilibrium in-scattering processes, captured by a transport lifetime tthat can be different from the quasi-particle (dephasing) lifetime qpdefined by the self-energy. We define the transport life time of a magnon with momentum kin terms of the collision integral out[f]in[f] =1 k;t[f](fk(r)f0;k);(5.4) withf0;k=nB(Ek)and we assume a thermalized quasi- equilibrium distribution function fk(r) =nBEk(r) kBT(r) ; (5.5) whereis the magnon chemical potential. We linearize the function fkin terms of small deviations fkfrom equilibrium f0;k, fk=fkf0;k: (5.6) leading to [3] fk=k;t[f]@f0;k @Ek@Ek @(~k)
 r+Ek TrT ;(5.7) where the gradients of chemical potential rand tem- perature rTdrive the magnon current. In the relax- ation time approximation we disregard the dependence ofk;t[f]onfand recover the quasi-particle lifetime k;t!k;qp.Tofirstorderinthephononoperatorsandsecondorder in the magnon operators the collision integral for magnon number non-conserving processes, nc out[f]nc in[f] =~ mNX qjbb k;qk;j2 "q(Ek+Eqk"q) [(1 +nq)fkfqknq(1 +fqk)(1 +fk)]; (5.8) where the interaction vertex bb k;k0;is given by Eq. (3.12) andnq=nB("q). By using the expansion (5.6) in the collision integral that vanishes at equilibrium, out[f0]in[f0] = 0; (5.9) we arrive at 1 nc k;t=~ mNX qjbb k;qk;j2 "q(Ek+Eqk"q)  nB(Ekq)nq+fqk fk(nB(Ek)nq) : (5.10) For the magnon number conserving process the deriva- tion is similar and we find 1 c k;t=~ mNX qjbb k;kq;j2 "q" (EkEkq+"q)  nqnB(Ekq)fkq fk(nB(Ek) +nq+ 1) +(EkEkq"q)  1 +nB(Ekq) +nq+fkq fk(nB(Ek)nq)# ; (5.11) with interaction vertex bb k;k0;given by Eq. (3.20). Due to thefkq=fkterm this is an integral equation. It can be solved iteratively to generate a geometric series referred to as vertex correction in diagrammatic theo- ries. By simply disregarding the in-scattering with terms fkq=fkthetransportlifetimereducestothethequasi- particle lifetime of the self-energy. We leave the general solution of this integral equation for future work, but argue in Sec. VID that the vertex corrections are not important in our regime of interest. VI. NUMERICAL RESULTS A. Magnon decay rate In the following we present and analyze our results for the magnon decay rates in YIG. We first consider8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 k[109m−1]01020304050γc(k) [106s−1] (100) (001) (110) (111) (011)0.0 0.5 1.0−0.10.00.10.2 Figure 4. Magnon decay rate in YIG due to magnon-phonon interactions for magnons propagating along various directions atT= 50 KandB= 0. We denote the propagation direction by(lmn), i.e.lex+mey+nez. The inset shows the relative deviation c= cfrom the (100) direction. 0.00 0.01 0.02 0.03 0.04 0.05 kx[109m−1]0.000.050.100.150.200.250.300.350.40γc(k) [103s−1]γc, total γc, anisotropy γc, exchange Figure 5. Comparison of the contributions from exchange- mediated and anisotropy-mediated magnon-phonon interac- tions to the magnon number conserving scattering rate cat T= 50 KandB= 0. the case of vanishing effective magnetic field ( B= 0) and discuss the magnetic field dependence in Sec. VIC. Since our model is only valid in the long-wavelength ( k< 8×108m1) and low-temperature ( T.100 K) regime, we focus first on T= 50 Kand discuss the temperature dependence in Sec. VIB. InFig.4weshowthemagnonnumberconservingdecay rate c(k), which is on the displayed scale dominated by the exchange-mediated magnon-phonon interaction and is isotropic for long-wavelength magnons. In Fig. 5 we compare the contribution from the exchange-mediated magnon-phonon interaction ( c k4) and from the anisotropy-mediated magnon-phonon interaction ( ck2). We observe a cross-over at k4107m1: for much smaller wave numbers, the exchange contribution can be disregarded and for larger wave numbers the exchange contribution becomes domi- nant. The magnon number non-conserving decay rate ncin Fig. 6 is much smaller than the magnon-conserving one. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 k[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1](100) (001) (110) (111) (011)Figure 6. Magnon decay rate in YIG due to magnon num- ber non-conserving magnon-phonon interactions for magnons propagating along various directions at T= 50 KandB= 0. This is consistent with the low magnetization damping of YIG, i.e. the magnetization is long-lived. We observe divergent peaks at the crossing points (shown in Fig. 1) with the exception of the (001) direction. These diver- gences occur when magnons and phonons are degenerate atk= 0:48109m1(1:2 meV) andk= 0:9109m1 (4:3 meV), respectively, at which the Boltzmann formal- ism does not hold; a treatment in the magnon-polaron basis [42] or a broadening parameter [31] would get rid of the singular behavior. The divergences are also sup- pressed by arbitrarily small effective magnetic fields (see Sec. VIC). There are no peaks along the (001) direc- tion because in the (001) direction the vertex function Vq;(see Eq. (3.15)) vanishes for q= (0;0;kz). For k >~cl=(D(p 82)) = 1:085109m1the decay rate ncvanishes because the decay process does not conserve energy ((Ek+Eqk"q) = 0). B. Temperature dependence Above we focused on T= 50 Kand explained that we expect a linear temperature dependence of the magnon decay rates at high, but not low temperatures. Fig. 7 showsourresultsforthetemperaturedependenceat kx= 108m1. Deviations from the linear dependence at low temperatures occurs when quantum effects set in, i.e. the Rayleigh-Jeans distribution does not hold anymore, 1 e"=(kBT)16kBT ": (6.1) C. Magnetic field dependence The numerical results presented above are for a mono- domain magnet in the limit of small applied magnetic fields. A finite magnetic field Balong the magnetization directioninducesanenergygap gBBinthemagnondis- persion, which shifts the positions of the magnon-phonon9 0 2 4 6 8 10 T[K]0.00.10.20.30.40.50.60.7γ[103s−1]γc γnc Figure7. Temperaturedependenceofthemagnondecayrates ncand catB= 0,kx= 108m1andky=kz= 0, i.e. along (100). 0.0 0.2 0.4 0.6 0.8 1.0 1.2 kx[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1]B= 0 T B= 0.1 T B= 0.5 T B= 1 T B= 2 T Figure 8. Magnetic field dependence of the magnon number non-conserving magnon decay rate in YIG at T= 50 Kwith magnon momentum along (100). crossingpointsto longerwavelengths. Themagneticfield suppresses the (unphysical) sharp peaks at the crossing points (see Fig. 8) that are caused by the divergence of thePlanckdistributionfunctionforavanishingspinwave gap. In the magnon number conserving magnon-phonon interactions, the magnetic field dependence cancels in the delta function and enters only in the Bose func- tion vianB(magnetic freeze-out). Fig. 9 shows that the magnetic field mainly affects magnons with energies .2gBB= 0:23(B=T) meV. As shown in Fig. 10 the magnon decay by phonons does not vanish for the k= 0Kittel mode, but only in the presence of a spin wave gap E0=gBB. Both magnon conserving and non-conserving scattering pro- cesses contribute. The divergent peaks at B1:3 Tand B4:6 Tin ncare caused by energy and momentum conservation in the two-magnon-one-phonon scattering process, (Ek=0+Eq"q) =(2gBB+Eexq2a2~cq);(6.2) when the gradient of the argument of the delta function 0.0 0.1 0.2 0.3 0.4 0.5 kx[109m−1]0246810δγc/γcB= 1 T B= 10 TFigure 9. Relative deviation  c= cfrom theB= 0result of the magnon number conserving magnon decay rate in YIG at T= 50 Kwith magnon momentum along (100). vanishes, rq(Ek=0+Eq"q) = 0; (6.3) i.e., the two-magnon energy Ek=0+Eqtouches either the transverse or longitudinal phonon dispersion "q. The total energy of the two magnons is equivalent to the en- ergy of a single magnon with momentum qbut in a field 2B, resulting in the divergence at fields that are half of those for the magnon-polaron observed in the spin See- beck effect [31, 42]. The two-magnon touching condition can be satisfied in all directions of the phonon momen- tumq, which therefore contributes to the magnon decay rate when integrating over the phonon momentum q. For k6= 0this two-magnon touching condition can only be fulfilled for phonons along a particular direction and the divergence is suppressed. The magnon decay rate is related to the Gilbert damp- ingkas~ k= 2kEk[55]. We find that phonons contribute only weakly to the Gilbert damping, nc 0= ~ nc 0=(2E0)108atT= 50 K, which is much smaller than the total Gilbert damping 105in YIG, but the peaks at 1:3 Tand4:6 Tmight be observable. The phonon contribution to the Gilbert damping scales lin- early with temperature, so is twice as large at 100 K. At low temperatures ( T.100 K) Gilbert damping in YIG has been found to be caused by two-level systems [56] and impurity scattering [40], while for higher tempera- tures magnon-phonon [57] and magnon-magnon scatter- ing involving optical magnons [34] have been proposed to explain the observed damping. Enhanced damping as a function of magnetic field at higher temperatures might reveal other van Hove singularities in the joint magnon- phonon density of states. D. Magnon transport lifetime We do not attempt a full solution of the integral equa- tions (5.10) and (5.11) for the transport lifetime. How- ever, we can still estimate its effect by the observation10 0 1 2 3 4 5 6 Magnetic field [T]01020304050γ[103s−1]γc(k= 0) γnc(k= 0) Figure 10. Magnetic field dependence of the magnon decay rates in YIG at k= 0andT= 50 K. that the ansatz 1 k;tkncan be an approximate solu- tion of the Boltzmann equation with in-scattering. Our results for the magnon number conserving interac- tion are shown in Fig. 11 (for rT= 0and finite rjjex), where t=1 t. We consider the cases n= 0;2;4, wheren= 0ork;t= const:would be the solution for a short-range scattering potential. For very long wave- lengths (k.4107m1) the inverse quasi-particle life- time1 k;qpk2and for shorter wavelengths 1 k;qpk4. n= 2is a self-consistent solution only for very small k.4107m1, while1 k;qpk4is a good ansatz up tok.0:3109m1. We see that the transport life- time approximately equals the quasi-particle lifetime in the regime of the validity of the n= 4power law. For the magnon number non-conserving processes in Fig. 12 the quasi-particle lifetime behaves as 1 k;qpk2. The ansatz n= 2turns out to be self-consistent and we see deviations of the transport lifetime from the quasi- particlelifetimefor k&5107m1. Theplotonlyshows our results for k<1108m1because our assumption of an isotropic lifetime is not valid for higher momenta in this case. We conclude that for YIG in the long-wavelength regime the magnon transport lifetime (due to magnon- phonon interactions) should be approximately the same as the quasi-particle lifetime, but deviations at shorter wavelengths require more attention. VII. SUMMARY AND CONCLUSION We calculated the decay rate of magnons in YIG induced by magnon-phonon interactions in the long- wavelength regime ( k.1109m1). Our model takes only the acoustic magnon and phonon branches into account and is therefore valid at low to intermedi- ate temperatures ( T.100 K). The exchange-mediated magnon-phonon interaction has been recently identified as a crucial contribution to the overall magnon-phonon interaction in YIG at high temperatures [3, 29, 45]. We emphasize that its coupling strength can be derived from 0.0 0.1 0.2 0.3 0.4 0.5 kx[109m−1]0.00.51.01.52.0γc t(k) [106s−1]quasi-particle 1/τ∼k0 1/τ∼k2 1/τ∼k4Figure 11. Inverse of the magnon transport lifetime in YIG (with magnon momentum along (100)) due to magnon num- ber conserving magnon-phonon interactions at T= 50 Kand B= 0for magnons along the (100) direction. 0.00 0.02 0.04 0.06 0.08 0.10 kx[109m−1]0.00.20.40.60.81.01.21.41.6γnc t(k) [103s−1]quasi-particle 1/τ∼k0 1/τ∼k2 1/τ∼k4 Figure 12. Inverse of the magnon transport lifetime in YIG (with magnon momentum along (100)) due to magnon num- ber non-conserving interactions at T= 50 KandB= 0. experimental values of the magnetic Grüneisen parame- term=@lnTC=@lnV[32, 33]. In previous works this interaction has been either disregarded [28], underesti- mated [29, 46], or overestimated [3]. In the ultra-long-wavelength regime the wave vector dependent magnon decay rate (k)is determined by the anisotropy-mediated magnon-phonon interaction with (k)k2, while for shorter wavelengths k&4107m1 the exchange-mediated magnon-phonon interaction be- comes dominant, which scales as (k)k4. The magnon number non-conserving processes are caused by spin- orbit interaction, i.e., the anisotropy-mediated magnon- phonon interaction, and are correspondingly weak. In a finite magnetic field the average phonon scatter- ingcontribution, fromthemechanismunderstudy, tothe Gilbert damping of the k= 0macrospin Kittel mode is about three orders of magnitude smaller than the best values for the Gilbert damping 105. However, we predict peaks at 1:3 Tand4:6 T, that may be experi- mentally observable in high-quality samples. The magnon transport lifetime, which is given by the balance between in- and out-scattering in the Boltz-11 mann equation, is in the long-wavelength regime approx- imately the same as the quasi-particle lifetime. However, the magnon quasi-particle and transport lifetime differ more significantly at shorter wavelengths. A theory for magnon transport at room temperature should therefore include the “vertex corrections”. A full theory of magnon transport at high temperature requires a method that takes the full dispersion relations of acoustic and optical phonons and magnons into ac- count. This would also require a full microscopic descrip- tion of the magnon-phonon interaction, since the magne- toelastic energy used here only holds in the continuum limit. ACKNOWLEDGMENTS N. V-S thanks F. Mendez for useful discussions. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) as well as a Grant-in-Aid for Scientific Research on Innovative Area, ”Nano Spin Conversion Science” (Grant No. 26103006), CONICYT-PCHA/Doctorado Nacional/2014-21140141, Fondecyt Postdoctorado No. 3190264, and Fundamen- tal Research Funds for the Central Universities. Appendix A: Long-wavelength approximation The theory is designed for magnons with momen- tumk < 0:8109m1and phonons with momen- tumq < 2:5109m1(corresponding to phonon en- ergies/frequencies 12 meV/3 THz), but relies on high- momentum cut-off parameters kcbecause of the assump- tion of quadratic/linear dispersion of magnon/phonons. We see in Fig. 13 that the scattering rates only weakly depend onkc. The dependence of the scattering rate on the phonon momentum cut-off qcis shown in Fig. 14. qc= 3:15 109m1corresponds to an integration over the whole Brillouin zone, approximated by a sphere. From these considerations we estimate that the long-wavelength ap- proximation is reliable for k.8108m1. Opti- cal phonons (magnons) that are thermally excited for T?100 K (300 K) are not considered here. Appendix B: Second order magnetoelastic coupling The magnetoelastic energy is usually expanded only to first order in the displacement fields. Second order terms can become important e.g. when the first order terms vanish. Thisisthecaseforone-magnontwo-phononscat-tering processes. The first order term X q qbqXq+  qby qXq (B1) onlycontributeswhenphononandmagnonmomentaand energies cross, giving rise to magnon polaron modes [42]. In other areas of reciprocal space the second order term should therefore be considered. Eastman [58, 59] derived the second-order magnetoelastic energy and determined thecorrespondingcouplingconstantsforYIG.Inmomen- tum space, the relevant contribution to the Hamiltonian is of the form H2p1m=1p NX k;q1;1;q2;2 q1+q2+k;0b q11;q22Xq11Xq22bk +q1+q2k;0b q11;q22Xq11Xq22by k ;(B2) where the interaction vertices are symmetrized, b q11;q22=1 2 ~b q11;q22+~b q22;q11 ;(B3) and obey b q11;q22= b q11;q22 : (B4) The non-symmetrized vertex function is ~b q11;q22=1 a2p 2S[B144(iI1I1;x$y) +B155(iI2I2;x$y) +B456(iI3I3;x$y)]; (B5) with I1=a2ex q11qx 1h ey q22qz 2+ez q22qy 2i ;(B6) I2=a2h ey q11qy 1+ez q11qz 1i h ey q22qz 2+ez q22qy 2i ; (B7) I3=a2 ex q11qz 1+ez q11qx 1 h ex q22qy 2+ey q22qx 2i ; (B8) andx$ydenotes an exchange of xandy. The relevant coupling constants in YIG are [58, 59] B144=648 meV; (B9) B155=446 meV; (B10) B456=328 meV: (B11) The magnon self-energy (see Fig. 15) reads 2p1m(k;i!) =2 NX q1;1;q2;21 X q1+q2+k;0 b q11;q222F1(q1; )F2(q2; !): (B12)12 0.0 0.2 0.4 0.6 0.8 1.0 kx[109m−1]0102030405060γc(k) [106s−1](a) kc= 3.15×109m−1 kc= 0.8×109m−1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 kx[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1](b) kc= 3.15×109m−1 kc= 0.8×109m−1 Figure 13. Dependence the magnon decay rate along (100) on the high magnon momentum cut-off kcfor the (a) magnon number conserving ( c) and (b) non-conserving ( nc) contributions at T= 50 KandB= 0. 0.0 0.2 0.4 0.6 0.8 1.0 kx[109m−1]0102030405060γc(k) [106s−1](a) qc= 3.15×109m−1 qc= 2.5×109m−1 qc= 2×109m−1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 kx[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1](b) qc= 3.15×109m−1 qc= 2×109m−1 Figure 14. Dependence the magnon decay rate along (100) on the high phonon momentum cut-off qcfor the (a) magnon number conserving ( c) and (b) non-conserving ( nc) contributions at T= 50 KandB= 0. with phonon propagator F(q; ) =~2 m1 ~2 2+"2 q: (B13) and leads to a magnon decay rate nc 2p(k) =2 ~Im 2p1m(k;i!!Ek=~+i0+) =~3 m2NX q1;1;q2;2q1+q2+k;01 "1"2b q11;q222 f2(Ek+"1"2) [n1n2] +(Ek"1"2) [1 +n1+n2]g; (B14) where n1=nB("q11); n2=nB("q22); (B15) "1="q11; "2="q22: (B16) The first term in curly brackets on the right-hand-side of Eq. (B14) describes annihilation and creation of a phonon as a sum of out-scattering minus in-scattering contributions, n1(1 +n2)(1 +n1)n2=n1n2;(B17)while the second term can be understood in terms of out-scattering by the creation of two phonons and the in-scattering by annihilation of two phonons, (1 +n1)(1 +n2)n1n2= 1 +n1+n2:(B18) For this one-magnon-two-phonon process the quasi- particle and the transport lifetimes are the same, t=qp; (B19) since this process involves only a single magnon that is either annihilated or created. The collision integral is then independent of the magnon distribution of other magnons and the transport lifetime reduces to the quasi- particle lifetime. The two-phonon contribution to the magnon scatter- ing rate in YIG at T= 50 Kand along (100) direction as shown in Fig. 16 is more than two orders of magni- tude smaller than that from one-phonon processes and therefore disregarded in the main text. The numerical results depend strongly on the phonon momentum cutoff qc, even in the long-wavelength regime, which implies that the magnons in this process dominantly interact13 k k q'q Figure 15. Feynman diagram representing the self-energy Eq. (B12) due to one-magnon-two-phonon processes. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 kx[109m−1]0.00.10.20.30.40.5γnc 2p(k) [103s−1]qc= 3.15×109m−1 qc= 2.5×109m−1 qc= 2×109m−1 Figure 16. Two-phonon contribution to the magnon number non-conserving magnon scattering rate with magnon momen- tum along (100) for different values of the phonon momentum cutoffqcatT= 50 KandB= 0. with short-wavelength, thermally excited phonons. In- deed, the second order magnetoelastic interaction (B5) is quadratic in the phonon momenta, which favors scatter- ingwithshort-wavelengthphonons. Ourlong-wavelength approximation therefore becomes questionable and the results may be not accurate at T= 50 K, but this should not change the main conclusion that we can disregard these diagrams. Our finding that the two-phonon contributions are so small can be understood in terms of the dimension- ful prefactors of the decay rates (Eqs. (4.8-4.9) and (B14)): The one-phonon decay rate is proportional to ~=(ma2)7106s1, while the two-phonon decay rate is proportional to ~3=(m2a4")33 s1, where "1 meVis a typical phonon energy. The coupling con- stants for the magnon number non-conserving processes areBk;?5 meVwhile the strongest two phonon cou- pling which enhances the two-phonon process by about a factor 100, but does not nearly compensate the pref- actor. The two phonon process is therefore three orders of magnitudes smaller than the contribution of the one phonon process. The physical reason appears to be the large mass density of YIG, i.e. the heavy yttrium atoms. Appendix C: Numerical integration The magnon decay rate is given be the weighted den- sity of statesI=Z BZd3qf(q)("(q)); (C1) that contain the Dirac delta function (")that can be eliminated to yield I=X qiZ Aid2qf(q) jr"(q)j; (C2) where the qiare the zeros of "(q)andAithe surfaces inside the Brillouin zone with "(q) ="(qi). The calcu- lation these integrals is a standard numerical problem in condensed matter physics. For aspherical Brillouin zone of radius qcand spherical coordinates (q;; ), I=Z 0dZ2 0dZqc 0dqq2sin()f(q;; )("(q;; )): (C3) When"(qi;;) = 0 ("(q;; )) =X qi(;)(qqi(;)) j"0(qi(;);;)j;(C4) where"0=@"=@qand I=Z 0dZ2 0dX qi(;)<qcq2 i(;) sin() f(qi(;);;) j"0(qi(;);;)j;(C5) which is particularly useful when the zeros of "(q;; ) can be calculated analytically for linear and quadratic dispersion relations. 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2019-06-03

We address the theory of magnon-phonon interactions and compute the corresponding quasi-particle and transport lifetimes in magnetic insulators with focus on yttrium iron garnet at intermediate temperatures from anisotropy- and exchange-mediated magnon-phonon interactions, the latter being derived from the volume dependence of the Curie temperature. We find in general weak effects of phonon scattering on magnon transport and the Gilbert damping of the macrospin Kittel mode. The magnon transport lifetime differs from the quasi-particle lifetime at shorter wavelengths.

Magnon-phonon interactions in magnetic insulators

1906.01042v1

arXiv:2108.07676v1 [math.SP] 17 Aug 2021Spectral enclosures for the damped elastic wave equation Biagio Cassano1, Lucrezia Cossetti2and Luca Fanelli3 1Dipartimento di Matematica e Fisica, Universit` a degli Stu di della Campania “Luigi Vanvitelli”, Viale Lincoln 5, 81100 Caserta, Italy; biagio.cassano@unicampania.it 2Fakult¨ at f¨ ur Mathematik, Institut f¨ ur Analysis, Karlsr uher Institut f¨ ur Technologie (KIT), Englerstraße 2, 7613 1 Karlsruhe, Germany; lucrezia.cossetti@kit.edu 3Ikerbasque &Departamento de Matematicas, Universidad del Pa´ ıs Vasco/ Euskal Herriko Unibertsitatea (UPV/EHU), Aptdo. 644, 48080, Bilbao, Spain; luca.fanelli@ehu.es 17 August 2021 Abstract In this paper we investigate spectral properties of the damp ed elastic wave equation. Deducing a correspon- dence between the eigenvalue problem of this model and the on e of Lam´ e operators with non self-adjoint perturbations, we provide quantitative bounds on the locat ion of the point spectrum in terms of suitable norms of the damping coefficient. 1 Introduction This paper is concerned with the damped elastic wave equation utt+a(x)ut−∆∗u= 0,(x,t)∈Rd×(0,∞), (1) Herea:Rd→Cd×ddenotes the damping coefficient assumed to be a (possibly) non herm itian matrix. We shall make the standard assumption of a bounded damping, i.e.a∈L∞(Rd)d.The symbol −∆∗is used to denote the Lam´ e operator of elasticity which is a matrix-valued differential operator acting, w.r.t. the spacial variable x∈Rdon smooth vector fields as −∆∗u=−µ∆u−(λ+µ)∇divu, u ∈C∞ 0(Rd)d:=C∞ 0(Rd;Cd). (2) The material-dependent Lam´ e parameters λ,µ∈Rare assumed to satisfy the ellipticity condition µ>0, λ+µ≥0. (3) It is customarily to write the second-order evolution system ( 1) as a doubled first-order system introducing the vector field U= (u,ut)T.Then (1) can be rewritten as Ut=A∗ aU,whereA∗ ais the 2d×2dmatrix-valued damped elastic wave operator defined as A∗ a:=/parenleftbigg0 1 ∆∗−a/parenrightbigg ,D(A∗ a) :=H2(Rd)d×˙H1(Rd)d. (4) The damped elastic wave equation ( 1) and the corresponding damped operator ( 4) have attracted considerable attention in the last decades. In the constant coefficient case, na melya(x) =α, α >0,Bocanegra-Rodr´ ıguez et al. [10] considered the longtime dynamics of this semilinear model in the pres ence of nonlinear structural forcing terms and external forces: they proved well-posedness ` a laHadamard and established the existence of finite dimensional global attractors together with the upper semic ontinuity thereof. Energy decay results in relation with stability properties of solutions to this elastic model hav e been also deeply investigated. In [ 6] 1Bchatnia and Daoulatli obtained a general energy decay estimate in a three dimensional bounded domain in the presence of localized nonlinear damping and an external force. By a dding viscoelastic dissipation of memory type Bchatnia and Guesmia [ 8] established a more general energy decay. Different viscoelastic d issipations have been considered in [ 31,32]. Strong stability of Lam´ e systems with fractional order boundar y damping were studied by Benaissa and Gaouar in [ 9]. For the undamped elastic wave equation, more commonly known as Navier equation , a more varied bibli- ography is available. In [ 3] Barcel´ o et al. proved uniform resolvent estimates (Limiting Absor ption Principle) for this model. With this stationary tool at hands they also proved a priori averaged estimates for the corre- sponding Cauchy problem. The resolvent estimates in [ 3] were generalized in [ 16] and then improved in [ 26], where a sharp result (analogous to the one available for the Laplacia n [25]) was proved. Surprisingly, differently from the Laplacian, in [ 26] the authors also showed the failure of uniform Sobolev and Carlema n inequalities for the Lam´ e operator. In [ 23] it was proved that if spacial lower-order perturbations are repla ced by temporal ones,i.e.if one considers the damped equation, then those estimates becom e available again. In [ 2] the authors generalized the results in [ 3] proving Agmon-H¨ ormander type estimates of the Navier equatio n when this is perturbed by small 0-th order matrix-valued potential. From thes e results Strichartz estimates for the evolution equation followed (in the same manner as for classical wave equation , see [11,12]). These Strichartz estimates were then generalized in [ 23,24]. In particular in [ 23] the endpoint case is deduced. The Navier equation got also attention of the inverse problem’s comm unity. In particular, inverse scat- tering was studied in [ 4,5], whereas inverse boundary problems were considered in [ 5,7,18,22,33]. Boundary determination of Lam´ e parameters has been studied in [ 13,30,35]. In this paper we are interested in spectral properties of the damp ed elastic wave equation ( 1), or equivalently of the elastic wave operator ( 4). More precisely, we aim at deducing quantitative bounds on the loca tion of the point spectrum of A∗ ain terms of suitable norms of the damping coefficient. In order to do t hat we establish a correspondence (see Lemma 2.1) between the eigenvalue problem associated to ( 4) and the one corresponding to suitable Lam´ e operators with non self-adjoint perturbations, that is operators of the form −∆∗+V, (5) whereVdenotes the operator of multiplication by a (possibly) non hermitian m atrix-valued function V:Rd→ Cd×d. The study of the spectrum of ( 5) has already a bibliography. It is well known that the free Lam´ e ope rator −∆∗is self-adjoint on H2(Rd)dandσ(−∆∗) =σac(−∆∗) = [0,∞).It is a natural question [ 14–16,26] to ask whether and how the spectrum changesunder 0th-orderpertur bations,i.e.considering the operator( 5). In [15], adapting to the elasticity setting the method of multipliers developed for non self-adjoint Schr¨ odinger operators in [20] (see also [ 21] for similar problems on the plane), the author showed that the poin t spectrum of the perturbed Lam´ e operator ( 5) remains empty (as in the free case) under suitable variational sma ll perturbations (inverse-square Hardy potential with small coupling constant is co vered). Later, in [ 14] we showed that full spectral stability, i.e.σ(−∆∗+V) =σ(−∆∗) = [0,∞),can be proved in three dimensions d= 3 under perturbations which satisfy a smallness condition of Hardy-type (s ee [14, Thm. 1.4]). Focusing on the point spectrum only, if no stability can be proved a priori, an interesting qu estion is related to provide quantitative bounds on the location in the complex plane of this part of the spectr um which, in the perturbed setting, is possibly no longer empty. In this direction, some preliminary result va lid for the discretespectrum can be found in [16] (see also [ 26]). Later, these results have been extended in [ 14] to cover embedded eigenvalues as well. More precisely in [ 14] the following result was proved. Theorem 1.1 (Thm. 1.1, [ 14]).Letd≥2,0< γ≤1/2ifd= 2and0≤γ≤1/2ifd≥3andV∈ Lγ+d 2(Rd;Cd×d).Then there exists a universal constant cγ,d,λ,µ>0independent on Vsuch that σp(−∆∗+V)⊂/braceleftbigg z∈C:|z|γ≤cγ,d,λ,µ/ba∇dblV/ba∇dblγ+d 2 Lγ+d 2(Rd)/bracerightbigg . (6) In the self-adjoint case, namely for real-valued perturbations, t he result above holds for a larger class of indices γ.More precisely, the following result holds true. 2Theorem 1.2 (Thm. 3.1, [ 16]).Letd≥2, γ >0ifd= 2andγ≥0ifd≥3andV∈Lγ+d 2(Rd;R).Then there exists a universal constant cγ,d,λ,µ>0independent of Vsuch that any negative eigenvalue z(if any) of the self-adjoint perturbed Lam´ e operator −∆∗+VIRdsatisfies |z|γ≤cγ,d,λ,µ/ba∇dblV−/ba∇dblγ+d 2 Lγ+d 2(Rd), (7) whereV−is the negative part of V,i.e.V−(x) := max {−V(x),0}. Making use of Theorem 1.1and Theorem 1.2and the correspondencebetween the eigenvalueproblem associat ed to the damped elastic wave operator and the one of the perturbed Lam´ e operator ( 5) (see Lemma 2.1below) we shall prove the following two results valid in the self-adjoint and th e non self-adjoint setting. Theorem 1.3. Letd≥2and assume γsatisfies the hypotheses of Theorem 1.2anda∈L∞(Rd;R).Then there exists a universal constant cγ,d,λ,µ>0independent of the damping asuch that for any positive (respectively negative) eigenvalue zof the damped elastic wave operator A∗ aanda−∈Lγ+d 2(Rd)(respectively a+∈Lγ+d 2(Rd)) satisfies (±z)γ−d 2≤cγ,d,λ,µ/ba∇dbla∓/ba∇dblγ+d 2 Lγ+d 2(Rd), (8) Settingγ=d/2 in (8), the previous theorem provides sufficient condition on the size of t he damping coefficient to guarantee absence of positive (respectively negative) eigenva lues. Corollary 1.1. Ifd≥2and cd 2,d,λ,µ/ba∇dbla∓/ba∇dbld Ld(Rd)<1, thenA∗ ahas no positive (respectively negative) eigenvalues. In the non self-adjoint setting we shall prove the following result. Theorem 1.4. Letd≥2and assume γsatisfies the hypotheses of Theorem 1.1anda∈L∞(Rd;Cd×d)is a (possibly) non hermitian matrix. Then there exists a univer sal constant cγ,d,λ,µ>0independent of the damping asuch that σp(A∗ a)⊂/braceleftBig z∈C:|z|γ−d 2≤cγ,d,λ,µ/ba∇dbla/ba∇dblγ+d 2 Lγ+d 2/bracerightBig . (9) Remark 1.1.Notice that in the non self-adjoint case, due to the more restrictiv e class of indices for which Theorem 1.1is valid compared to Theorem 1.2, no analogous of Corollary 1.1holds true ( γ=d/2 is not admissible). The main motivation behind our project relies on the following simple obs ervation: the ellipticity condition ( 3) allows taking λ+µ= 0 in the definition of the Lam´ e operator ( 2). This choice turns the Lam´ e operator ( 2) into a vector Laplacian and consequently the damped elastic wave equat ion (1) into a system of classical damped wave equations. For the (scalar) damped wave equation, results in the spirit of Theorem 1.3and Theorem 1.4 have been recently proved in [ 27]. Thus, Theorem 1.3and Theorem 1.4can be seen as a generalization of the results in [ 27, Thm. 1, Thm. 5 and Thm. 6] in the sense that they recover∗them when λ+µ= 0. Theorem 1.3and Theorem 1.4are not stated for d= 1,as a matter of fact the one dimensional case is rather special and it is treated separately. In d= 1 the Lam´ e operator −∆∗turns into a scalar differential operator, more precisely it is simply a multiple of the Laplacian −∆∗:=−µd2 dx2−(λ+µ)d2 dx2=−(λ+2µ)d2 dx2. As a straightforward consequence of the celebrated result of Ab ramov, Aslanian and Davies for 1D-Schr¨ odinger operators (see [ 1, Thm. 4]), in [ 16] the following result for the one dimensional non self-adjoint Lam´ e operator was proved. ∗the constants involved slightly differ due to the presence of the coefficient µof the vector Laplacian and due to the vectorial form of the wave equation once λ+µ= 0 in (1). 3Theorem 1.5 (Thm. 1.1, [ 16]).Letd= 1andV∈L1(R;C).Then σp(−∆∗+V)⊂/braceleftBig z∈C:|z|1/2≤1 2√λ+2µ/ba∇dblV/ba∇dblL1(R)/bracerightBig . Remark 1.2.We stress that Theorem 1.1 in [ 16] was stated only for eigenvalues outside the essential spectrum, namely for z∈C\[0,∞).Nevertheless, it is easy to show that embedded eigenvalues can be c overed as well (see [17, Cor. 2.16]). In the self-adjoint case, as an immediate consequence of the Lieb- Thirring inequalities ([ 28,29]) valid for the Schr¨ odinger operators, one has the following result. Theorem 1.6. Letd= 1andV−∈L1(R;R).Then σp(−∆∗+V)⊂/braceleftBig z∈C:|z|1/2≤1 2√λ+2µ/ba∇dblV−/ba∇dblL1(R)/bracerightBig . (10) Theorem 1.5and Theorem 1.6together with Lemma 2.1below allow to deduce properties on the point spectrum of the one dimensional damped elastic wave operator A∗ a.Differently from the higher dimensional setting, in d= 1Theorem 1.5doesnotentailanyquantitativeboundonthe locationinthe complex planeoftheeigenvalues ofA∗ a,on the other hand it provides an explicit smallness condition on the size of theL1-norm of the damping such thatA∗ adoes not have eigenvalues. More precisely we have the following resu lt. Theorem 1.7. Letd= 1anda∈L1(R;C).If/ba∇dbla/ba∇dblL1(R)<2√λ+2µ,thenσp(A∗ a) =∅.Moreover, the constant 2√λ+2µis optimal. In the self-adjoint situation it holds true a slightly different result co mpared to the ones introduced so far. Theorem 1.8. Letd= 1and assume that ais real-valued and satisfies /integraldisplay R|x||a(x)|dx<∞and lim R→∞/ba∇dbla/ba∇dblL∞(R\BR(0))= 0. (11) Letzbe a real eigenvalue of A∗ a.Ifz>0and/integraltext Ra<−4√λ+2µ(orz <0and/integraltext Ra>4√λ+2µ), then |z| ≥(λ+2µ)/parenleftBigg/integraldisplay R|x||a(x)|dx/parenrightBigg−1 . Moreover the following quantitative bound on the location of eigenva lues holds. Theorem 1.9. Letd= 1and assume that ais real-valued and satisfies (11). Moreover, assume |z|<(λ+2µ)/parenleftBigg/integraldisplay R|x||a(x)|dx/parenrightBigg−1 . Ifz>0and/integraltext Ra<0(respectively, z <0and/integraltext Ra>0), then there exists exactly one α>0satisfying 2/parenleftBigg/integraldisplay Ra−(x)dx/parenrightBigg−1 ≤α≤ −4/parenleftBigg/integraldisplay Ra(x)dx/parenrightBigg−1/parenleftBigg respectively, 2/parenleftBigg/integraldisplay Ra+(x)dx/parenrightBigg−1 ≤α≤4/parenleftBigg/integraldisplay Ra(x)dx/parenrightBigg−1/parenrightBigg such thatz/αis an eigenvalue of A∗ a. The rest of the paper is divided as follows. In the next Section we pro vide the proof of the preliminary Lemma2.1establishing the correspondence between the eigenvalue problem a ssociated to the damped elastic waveoperatorand the perturbed Lam´ eoperator. Afterwards , in Section 2.1we showthe validity of Theorem 1.3 and Theorem 1.4which hold in higher dimension d≥2.The one dimensional case, that is Theorem 1.7- Theorem 1.9, is treated separately in Section 2.2. 42 Proofs As a starting point we show how the eigenvalue problem associated to the damped elastic wave operator A∗ a defined in ( 4) is related to the one of a perturbed Lam´ e operator of the form ( 5). Lemma 2.1. Letd≥1and assume a∈L∞(Rd;Cd×d).For everyz∈C, z∈σp(A∗ a)⇐⇒ −z2∈σp(−∆∗+za). Proof.Assumez∈σp(A∗ a),then there exists a non-trivial Ψ = ( ψ1,ψ2)T∈ D(A∗ a) such that A∗ aΨ =zΨ.In other words, ψ1∈H2(Rd)d, ψ2∈˙H1(Rd)dandψ2=zψ1,∆∗ψ1−aψ2=zψ2.Plugging the first equation in the second one gives −∆∗ψ1+zaψ1=−z2ψ1.Sinceψ1/\e}atio\slash= 0,then−z2∈σp(−∆∗+za).Conversely, assume −z2∈σp(−∆∗+za),then there exists a non-trivial ψ∈H2(Rd)dsuch that ( −∆∗+za)ψ=−z2ψ.Defining Ψ := (ψ,zψ)T,then Ψ∈ D(A∗ a) and (A∗ aΨ)T= (zψ,∆∗ψ−zaψ) =z(ψ,zψ) =zΨT.Therefore,z∈σp(A∗ a). Remark 2.1.From the validity of Lemma 2.1, one has that 0 /∈σp(A∗ a) as the spectrum of the unperturbed Lam´ e operator −∆∗+0a=−∆∗is purely continuous. 2.1 Higher dimensions d≥2 :Proof of Theorem 1.3and Theorem 1.4 With Lemma 2.1at hands we now show that Theorem 1.3and Theorem 1.4are consequence of Theorem 1.2 and Theorem 1.1, respectively. Proof of Theorem 1.3.From Lemma 2.1we know that z∈σp(A∗ a) if and only if −z2∈σ(−∆∗+za).From Theorem 1.2there exists cγ,d,λ,µ>0 such that |z|2γ≤cγ,d,λ,µ/ba∇dbl(za)−/ba∇dblLγ+d 2 Lγ+d 2(Rd), (12) where (za)−is the negative part of za,i.e.(za)−=za+ifz∈(−∞,0) and (za)−=za−ifz∈(0,∞).Using this fact in ( 12) and dividing both sides of ( 12) by|z|γ+d/2(z/\e}atio\slash= 0,see Remark 2.1) we obtain ( 8). Now we consider the non self-adjoint situation. Proof of Theorem 1.4.The proof of Theorem 1.4is analogous to the one of Theorem 1.3. Letz∈σp(A∗ a),then by Lemma 2.1−z2∈σp(−∆∗+za).Using the eigenvalue bound ( 6) then one has |z|2γ≤cγ,d,λ,µ|z|γ+d 2/ba∇dbla/ba∇dblγ+d 2 Lγ+d 2(Rd), which gives ( 9) and concludes the proof. 2.2 1D: Proof of Theorem 1.7, Theorem 1.8and Theorem 1.9 We startconsideringtheself-adjointsituation. Let z∈Randlet{λ∗ n(za)}N n=1denotethe sequenceofeigenvalues of−∆∗+za,then the following preliminary lemma on the sum of the square root of t he eigenvalues holds. Lemma 2.2. Letd= 1.Then N/summationdisplay n=1|λ∗ n(za)|1/2≥ −z 4√λ+2µ/integraldisplay Ra(x)dx. (13) Moreover if/integraltext R|x||a(x)|dx<∞,then the following bound on the number Nof eigenvalues λ∗ n(za) N≤1+|z| λ+2µ/integraldisplay R|x||a(x)|dx (14) holds. 5Proof.Ifλ∗ n(za) is an eigenvalue of −∆∗+za,then there exists ψ∈H2(R) such that −(λ+2µ)∆ψ+zaψ= λ∗ n(za)ψor, equivalently,/parenleftBig −∆+za λ+2µ/parenrightBig ψ=λ∗ n(za) λ+2µψ. (15) Denoting by λn(V) the eigenvalues of the Schr¨ odinger operator −∆ +V,then we conclude that λ∗ n(za) is an eigenvalue of −∆∗+zaif and only if there exists n∈Nsuch thatλ∗ n(za) is a multiple of an eigenvalue λn(za/(λ+2µ))ofthe Schr¨ odingeroperator −∆+za/(λ+2µ),moreprecisely λn(za/(λ+2µ)) =λ∗ n(za)/(λ+2µ). In particular the number of eigenvalues coincides. The Buslaev-Fad deev-Zakharov trace formula ( cf.[19]) for 1D-Schr¨ odinger operator −∆+Vstates that N/summationdisplay n=1|λn(V)|1/2≥ −1 4/integraldisplay RV(x)dx, this and the correspondence above give immediately ( 13). The Bargmann bound [ 34, Pb. 22] provides a control from above of the number of eigenvalu es of the 1D- Schr¨ odinger operator −∆+Vunder the condition/integraltext R|x||V(x)dx<∞.More precisely, N≤1+/integraldisplay R|x||V(x)|dx. (16) Similarly as above (that is using the correspondence between eigenv alues of the Lam´ e operator −∆∗+zaand of the Schr¨ odinger operator −∆+za/(λ+2µ)) from ( 16) one easily gets ( 14). This concludes the proof. Proof of Theorem 1.8.Letzbe a real eigenvalue of A∗ a,in order to prove Theorem 1.8we will show that if |z|<(λ+ 2µ)/parenleftBig/integraltext R|x||a(x)|dx/parenrightBig−1 then/integraltext Ra≥ −4√λ+2µforz >0 and/integraltext Ra≤4√λ+2µforz <0.First of all notice that ( 13) is non-trivial only if z/integraltext Ra(x)dx <0.This last condition, in particular is known to be a sufficient condition which guarantees that inf σ(−∆∗+za)<0.From the decay assumption ( 11), then it follows that −∆∗+zaposses at least one negative eigenvalue. From the upper bound ( 14) it follows that if |z|<(λ+2µ)/parenleftBig/integraltext R|x||a(x)|dx/parenrightBig−1 then−∆∗+zahas exactly one negative eigenvalue λ∗ 1(za).Thus, from ( 13) and the correspondence in Lemma 2.1one has |z|=|λ1(za)|1/2≥ −z 4√λ+2µ/integraldisplay Ra(x)dx. (17) This implies/integraltext Ra(x)dx≥ −4√λ+2µforz>0 and/integraltext Ra(x)dx≤4√λ+2µforz <0. Proof of Theorem 1.9.From the hypotheses, as above, one has that −∆∗+zaposses exactly one negative eigenvalue. The Lieb-Thirring type bound ( 10) in Theorem 1.6and the estimate in ( 17) give −z 4√λ+2µ/integraldisplay Ra(x)dx≤ |λ1(za)|1/2≤z 2√λ+2µ/integraldisplay Ra−(x)dx, /parenleftBig respectively −z 4√λ+2µ/integraldisplay Ra(x)dx≤ |λ1(za)|1/2≤|z| 2√λ+2µ/integraldisplay Ra+(x)dx/parenrightBig . Using the correspondence in Lemma 2.1the result follows. Proof of Theorem 1.7.Ifz∈Cis an eigenvalue of A∗ a,then from Lemma 2.1−z2∈σp(−∆∗+za).Thus, from Theorem 1.5we have |z| ≤1 2√λ+2µ|z|/ba∇dbla/ba∇dblL1(R). Dividing by |z|,which cannot be zero (see Remark 2.1), one has 1 ≤1 2√λ+2µ/ba∇dbla/ba∇dblL1(R).If theL1-norm ofais small, namely if /ba∇dbla/ba∇dblL1(R)<2√λ+2µ,then we get a contradiction. Thus, σp(A∗ a) =∅.The optimality of the result can be proved as in [ 27, Thm. 4]. 6References [1] A.A. Abramov, A. Aslanyan, and E.B. 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2021-08-17

In this paper we investigate spectral properties of the damped elastic wave equation. Deducing a correspondence between the eigenvalue problem of this model and the one of Lam\'e operators with non self-adjoint perturbations, we provide quantitative bounds on the location of the point spectrum in terms of suitable norms of the damping coefficient.

Spectral enclosures for the damped elastic wave equation

2108.07676v1

The domain wall spin torque-meter I.M. Miron, P.-J. Zermatten, G. Gaudin, S. Auret, B. Rodmacq, and A. Schuhl SPINTEC, CEA/CNRS/UJF/GINP, INAC, 38054 Grenoble Cedex 9, France (Dated: September 15, 2021) Abstract We report the direct measurement of the non-adiabatic component of the spin-torque in domain walls. Our method is independent of both the pinning of the domain wall in the wire as well as of the Gilbert damping parameter. We demonstrate that the ratio between the non-adiabatic and the adiabatic components can be as high as 1, and explain this high value by the importance of the spin- ip rate to the non-adiabatic torque. Besides their fundamental signicance these results open the way for applications by demonstrating a signicant increase of the spin torque eciency. PACS numbers: 72.25.Rb,75.60.Ch,75.70.Ak,85.75.-d 1arXiv:0810.4633v1 [cond-mat.other] 25 Oct 2008The possibility of manipulating a magnetic domain wall via spin torque eects when passing an electrical current through it opens the way for conceptually new devices such as domain wall shift register memories[1]. Early spin-torque theories[2, 3, 4] were based on a so called adiabatic approximation which assumed that the incoming electron's spin follows exactly the magnetization as it changes direction within the domain wall. Nevertheless, the observed critical currents needed to trigger the domain wall motion were lower than the value predicted within this framework[5]. As rst predicted by Zhang[6], the existence of a non-adiabatic term in the extended Landau-Lifshitz-Gilbert equation leads to the vanishing of the intrinsic critical current. The action of this non-adiabatic torque on a DW is expected to be identical to that of an easy axis magnetic eld. Micromagnetic simulations have been used to predict the velocity dependence on current for a DW submitted to the action of the two components of the spin-torque[7]. The quantitative measurement of this non-adiabatic torque can be achieved either by demonstrating the equivalence of eld and current in a static regime, or by observing the complex dynamic behavior[7]. The main diculty of these measurements comes from the pinning of the DW by material imperfections. It masks the existence of the intrinsic critical current, and in addition, above the depinning current, obscures the DW velocity dependence on current. Moreover, most of the DW velocity measurements were done using materials with in-plane magnetization[5, 8, 9, 10, 11], where the velocity can also depend on the micromagnetic structure of the wall[12] (transverse wall or vortex wall). Despite the simpler micromagnetic structure of the DWs, very few results were reported[13] for Perpendicular Magnetic Anisotropy (PMA) materials. In this case the intrinsic pinning is much stronger, probably due to a local variation of the perpendicular anisotropy. Up to now, none of the measurements were able to clearly evidence the equivalence between eld and current, nor to reproduce the predicted dynamic behavior; hence the value of the non-adiabatic torque is still under debate. In this letter we use a novel approach for the measurement of the non-adiabatic component of spin-torque. Instead of measuring the DW velocity, we perform a quasistatic measurement of its displacement under current and magnetic eld. In principle this method is similar to any quasi-static force measurement: a small displacement is created, rst with the unknown force and then with a known reference force. In our case the unknown force is caused by the electric current passing through the DW while the reference force is due to an applied magnetic eld. By comparing the two displacements one directly compares the applied 2FIG. 1: Schematic representation of the experimental setup. The inset shows an SEM picture of a sample. forces. Due to the high sensitivity of our method (able to detect DW motion down to 102nm[14]) we can study the displacement of the DW inside its pinning center. Since the measurement relies on the comparison to a reference force, the method is independent of the strength of the pinning. Moreover, as the eld and current are applied quasi-statically, the damping parameter does not play any role. According to recent theories[6, 15, 16] that derived the value of the spin torque, (the ratio between the non-adibatic and adiabatic torques) is given by the ratio between the rate of the spin- ip of the conduction electrons and that of the s-d exchange interaction. Generally, two conditions must be fullled to obtain a high spin- ip rate. First it is necessary to have a strong crystalline eld inside the material. The electric elds will yield a magnetic eld in the rest frame of the moving electrons. Second, a breaking of the inversion symmetry is needed. Otherwise the total torque of the magnetic eld on the electron spin averages out, and the spin- ip may only occur during momentum scattering[17]. In order to highlight these eects we have patterned samples from a Pt 3nm/ Co0.6nm/(AlO x)2nmlayer[18]. In this case the symmetry is broken by the presence of the 3AlO xon one side of the Co layer, and of the heavy Pt atoms on the other[19, 20]. We will emphasize the importance of the spin- ip interaction to spin torque by comparing results from these samples with those for samples fabricated from a symmetric Pt 3nm/ Co0.6nm/Pt 3nmlayer[21], where a much smaller spin- ip rate is expected. As the only dier- ence between the two structures is the upper layer, we expect similar growth properties for the Co layer. Both samples exhibit PMA and a strong Anomalous Hall Eect (AHE)[22]. The lms are patterned into the shape depicted in Figure 1. This shape is well suited for a quasi static measurement as a constriction is created by the presence of the four wires used for the AHE measurement (gure 1 inset). This way a DW can be pinned in a position where changes in the out of plane component of the magnetization (i.e. DW motion) can be detected by electrical measurements. A current is passed through the central wire. This current will serve to push the domain wall as well as to probe the eventual displacement. In the case where the DW does not move under the action of the current, the transverse resistance remains unchanged and the voltage measured across the side wires (AHE) will be linear with the current. If the DW moves due to the electric current, the exciting force will create resistance variations, causing a nonlinear relationship between the measured voltage and the applied current. A simple way to detect such nonlinearities is to apply a perfectly harmonic low frequency (10 Hz) ac current, and look at the rst harmonic in the Fast Fourier Transform (FFT) of the measured voltage. Its value is a measure of the amplitude of the DW displacement at the frequency of the applied current. To quantitatively compare the action of a magnetic eld to that of an electric current, the magnetic eld is applied at the same frequency and in phase (or opposition of phase) with the electric current. By applying current and eld simultaneously, we ensure that their corresponding torques act on the same DW conguration. In addition to the displacement provoked by the current, the eld induced displacement will add to the value of the rst harmonic, which can be either increased if the eld and current push the wall in the same direction, or decreased if they act in opposite directions. Figure 2 shows the dependence of the resistance variation at the frequency of the current (
R f) on the current amplitude for dierent values of the eld amplitude. First, at low current and eld amplitudes the displacement is almost linear ( 107A/cm2), but for higher values, the
R fvaries more rapidly. A simple estimation based on the value of the resistance variation compared to the total Hall resistance of a cross (1 ) yields 1 nm for the 4FIG. 2: (a) Dependence of the resistance variation on the current amplitude for several eld amplitudes (Pt/Co/AlO xsample). The inset shows a possible nonlinear and asymmetric potential well. The energy landscape can be modeled by an eective out of plane magnetic eld that has negative values on one side of the equilibrium position and positive values on the other. (b) A zoom on the small amplitude regime. The inset shows the perfect superposition obtained by shifting the curves horizontaly with 1.25
105Acm2Oe1.5maximum amplitude of the DW motion in the rst regime and 7 nm for the second regime. This behavior can be explained by the anatomy of the local pinning. The local potential well trapping the DW can be considered as a superposition of the geometric pinning [23] and intrinsic pinning caused by defects randomly distributed inside the material[13, 21]. Because the potential well for the small scale displacements (below 10nm) is dominated by the random intrinsic pinning rather than geometric pinning (the increase of the length of the DW is small 1%) in the general case it should be asymmetric. We have veried the supposed asymmetry of the eective potential well by applying alongside the ac current and eld, a dc bias eld that changes the local potential well (inset of gure 3). By varying this eld we observed a reduction of the current amplitude needed to access this strongly non-linear regime (gure 3). When the magnetic bias eld was reversed this second regime was no longer attained with the available current densities (not shown). The observed dependence of
R fon current and eld (gures 2 and 3) is in perfect agreement with the characteristic features of the non-adiabatic component of the spin-torque. First, we do not observe any critical current down to the lowest current value (106A/cm2- gure 4 in [14]). Futhermore, by extrapolating the amplitude of the DW displacement (gure 2), when the current is reduced, the displacement goes to zero as the current goes to zero, in agreement with the absence of the critical current. However, the most important feature of the
R fbehavior is that the curves obtained for any eld amplitude can be obtained from the curve corresponding to zero eld just by shifting it horizontally (in current): towards the lower current values when the eld and current act in the same direction on the DW and towards higher values when their actions are opposed. This means that any displacement of the DW can also be achieved with a dierent current if a magnetic eld is added. The dierence in current is compensated by the magnetic eld. The value of this horizontal shift gives the eld to current correspondence. The inset of gure 1b shows that all the curves corresponding to dierent eld amplitudes have the same shape; by shifting them horizontally (using the eld-current correspondence), they all collapse on the zero eld amplitude curve. This shows that independently of the direction or strength of the applied current and eld, as predicted by the theories, their eect on the DW is fundamentally similar. Moreover, further evidence that this correspondence is intrinsic and not in uenced by pinning is that its value remains the same within the dierent amplitude regimes as well as when the local potential well is tuned by a constant bias eld. 6FIG. 3: The nonlinear regime (Pt/Co/AlO xsample). When an external bias eld is added, the eective pinning eld changes (inset) and the nonlinear regime is reached for dierent current and eld amplitudes. However, this does not cause any change in the eld to current correspondence: the horizontal distance between the curves remains the same. 7Since the motion of the DW is quasi-static the magnetization can be considered to be at equilibrium during motion. In this case the sum of all torques must be zero. In order for the DW to remain at rest, the torque from the applied current must be compensated by the torque generated by the magnetic eld. The upturn observed on the -60 Oe curve (gure 2b) determines the position of the zero amplitude point. Note that the position of this point is in perfect agreement with the eld to current correspondence obtained from the horizontal shifting of the curves. By taking into account the micromagnetic structure of the DW (very thin 5nm Bloch wall) the two torques are integrated over the width of the wall, and by comparing their values (the eld torque is easily calculated; [14]) the non-adiabatic term of the spin-torque is determined. In the case of Pt/Co/AlO xstacks the current-eld correspondence is approximately 1.25 105A/cm2to 1Oe, corresponding to a value of = 1. Similar measurements (gure 1 in [14]) were also performed in the saturated state (with- out the DW). They conrm that there is no contribution to the signal from the ordinary Hall eect, but indicate a small contribution from thermoelectric eects - the Nernst- Ettingshausen Eect(NEE)[24]. The contribution from DW motion to
R fis much higher than the NEE for the Pt/Co/AlO xstack. In the case of Pt/Co/Pt layers we nd that the amplitude of the current induced DW motion is much smaller and entirely masked by the NEE. When a DW is moving inside the perfectly harmonic region at the bottom of the po- tential well, its displacement depends linearly on the applied force. In such a scenario, the current induced DW motion and the NEE are indistinguishable. They both lead to a linear dependence of the
R fresponse on current. The only possibility to separate these eects, for the Pt/Co/Pt layer, is to attain the high amplitude nonlinear regime of DW motion. This is done by keeping the current amplitude constant and varying the eld amplitude. When the current and eld push the wall in the same direction, the nonlinear regime should be reached for smaller eld amplitudes, than if their actions were opposed. In the presence of current induced displacements, the nonlinearities observed in the
R fversus eld amplitude curve should be asymmetric. Moreover the asymmetry should depend on the current value. Such an asymmetry is observed (inset of gure 4) in the case of Pt/Co/AlO xsamples. In contrast to this behavior, a fully symmetric dependence that does not depend on the current amplitude is measured for the Pt/Co/Pt samples (gure 4). We conclude that in this case the spin torque induces DW displacements smaller than the resolution limit of this method. This limit value leads to (supplementary notes) 0.02. 8FIG. 4: The nonlinear response of a DW to magnetic eld. (a)
R fvs. the amplitude of the eld for three dierent current densities in the case of Pt/Co/Pt layers (inset Pt/Co/AlO x).(b) Derivative of
R fvs. the eld amplitude for a Pt/Co/Pt sample (inset Pt/Co/AlO x). Theoretical estimations[6] based on a spin- ip frequency of 1012Hz yield a value =0.01. To clarify the dierence of the spin-torque eciency in the two samples, the symmetry breaking due to the presence of the AlO xsurface must be taken into account. As a metallic lm gets thinner, the conduction electron's behavior resembles more and more to that of a two-dimensional electron gas. When such a gas is trapped in an asymmetric potential well, 9the spin-orbit coupling is much stronger than in the case of a symmetric potential due to the Rashba interaction[25]. This eect was rst evidenced in nonmagnetic materials where this interaction leads to a band splitting (0.15 eV for the surface states of Au (111)[26]). In the case of ferromagnetic metals this eect was already proposed to contribute as an eective magnetic eld[27] for certain DW micromagnetic structures, but should not have any eect for Bloch walls in PMA materials. The simple 1D representation used in this case[27] to model the DW accounts for the coherent rotation of the spins of the incoming electrons around the eective eld, but excludes any de-coherence between electrons having dierent k-vector directions on the Fermi sphere (dierent directions of the Rashba eective eld) as well as possible spatial inhomogeneities of this eld (surface roughness). Since the spin-torque is caused by the cumulative action of all conduction electrons [6], the relevant parameter is not the spin- ip rate of a single electron but the relaxation rate of the out of equilibrium spin-density [6]. In a more realistic 2D case, in the presence of the above mentioned strong decoherence eects, the relaxation rate of the out of equilibrium spin- density approaches the rate of spin precession around the Rashba eective eld. The above value of the measured spin-orbit splitting (0.15 eV) will yield in this case an eective spin- ip rate of 30
1012Hz, which is in excellent agreement with the order of magnitude of the measured non-adiabatic parameter, supporting this scenario. In summary, a technique that allows the direct measurement of the torque from an electric current on a DW was developed. We have pointed out the importance of spin- ip interactions to spin torque by comparing its eciency between two dierent systems. We show that the Pt/Co/AlO xsample with the required symmetry properties to increase the spin- ip frequency (breaking of the inversion symmetry) shows an enhanced spin torque eect. A value of the order of 1 was measured for the parameter approaching the maximum value predicted by existing theories. This value can be explained by order of magnitude considerations on the Rashba eect observed on surface states of metals. Obtaining a high eciency spin torque in a low coercivity material would make possible the development of nanoscale devices whose magnetization could be switched at low current densities. The order of magnitude of the current densities would be similar to the one observed for magnetic 10semiconductors[28], but, as the resistance is smaller, the supplied power will be lower. [1] S. Parkin, US Patent 309,6,834,005 (2004). [2] L. Berger, J. Appl. Phys. 55, 1954 (1984). [3] A. Thiaville, Y. Nakatani, J. Miltat, and N. Vernier, J. Appl. Phys. 95, 70497051 (2004). [4] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [5] M. Klui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, L. J. Heyderman, F. Nolting, and U. Rudiger, Phys. Rev. Lett. 94, 106601 (2005). [6] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). [7] A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett 69, 990996 (2005). [8] G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L. Erskine, Phys. Rev. Lett. 97, 057203 (2006). [9] M. Hayashi, L. Thomas, C. Rettner, R. Moriya, Y. B. Bazaliy, and S. S. P. Parkin, Phys. Rev. Lett. 98, 037204 (2007). [10] P.-O. Jubert, M. Klui, and A. Bischof, J. Appl. Phys. 99, 08G523 (2006). [11] N. Vernier, D. A. Allwood, D. Atkinson, M. D. Cooke, and R. P. Cowburn, Europhys. Lett. 65, 526532 (2004). [12] Y. Nakatani, A. Thiaville, and J. Miltat, Nature Mater 2, 521 (2003). [13] D. Ravelosona, D. Lacour, J. A. Katine, B. D. Terris, and C. Chappert, Phys. Rev. Lett. 95, 117203 (2005). [14] See supplementary notes . [15] A. Vanhaverbeke and M. Viret, Phys. Rev. B 75, 024411 (2007). [16] P. Pichon and A. Thiaville, Phys. Rev. B 75, 174414 (2007). [17] J. Fabian and S. Das Sarma, J.Vac. Sci. Technol. B 17 , 1708 (1999). [18] A. Manchon, C. Ducruet, L. Lombard, S. Auret, B. Rodmacq, B. Dieny, S. Pizzini, J. Vogel, V. Uhlir, M. Hochstrasser, et al., arXiv:0712.1905v1 (2007). [19] O. Krupin, G. Bihlmayer, K. Starke, S. Gorovikov, J. E. Prieto, K. Dbrich, S. Blgel, and G. Kaindl, Phys. Rev. B 71, 201403(R) (2005). [20] H. Cercellier, C. Didiot, Y. Fagot-Revurat, B. Kierren, L. Moreau, and D. Malterre, Phys. Rev. B 73, 195413 (2006). 11[21] P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier, J. Ferr, V. Baltz, B. Rodmacq, B. Di- eny,and R. L. Stamps, Phys. Rev. Lett. 99, 217208 (2007). [22] C. L. Canedy, X. W. Li, and G. Xiao, Appl. Phys. Lett. 81, 5367 (1997). [23] J. Wunderlich, D. Ravelosona, C. Chappert, F. Cayssol, V. Mathet, J. Ferr, J.-P. Jamet, and A. Thiaville, IEEE Transactions on Magnetics 37, 2104 (2001). [24] B. J. Thaler, R. Fletcher, and J. Bass, J. Phys. F: Met. Phys. 8, 131 (1978). [25] L. Petersen and P. Hedegard, Surface science 459, 49 (2000). [26] S. LaShell, B. A. McDougall, and E. Jensen, Phys. Rev. Lett. 77, 3419 (1996). [27] K. Obata and G. Tatara, Phys. Rev. B 77, 214429 (2008). [28] M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno, Nature 428, 539 (2004). 12

2008-10-25

We report the direct measurement of the non-adiabatic component of the spin-torque in domain walls. Our method is independent of both the pinning of the domain wall in the wire as well as of the Gilbert damping parameter. We demonstrate that the ratio between the non-adiabatic and the adiabatic components can be as high as 1, and explain this high value by the importance of the spin-flip rate to the non-adiabatic torque. Besides their fundamental significance these results open the way for applications by demonstrating a significant increase of the spin torque efficiency.

The domain wall spin torque-meter

0810.4633v1

1 Ultrafast optical control of magnetization in EuO thin films T. Makino1,*, F. Liu2,3, T. Yamasaki4, Y. Kozuka2, K. Ueno5,6, A. Tsukazaki2, T. Fukumura6,7, Y. Kong3, and M. Kawasaki1,2 1 Correlated Electron Research Group (CERG) and Cr oss-Correlated Materials Research Group (CMRG), RIKEN Advanced Science Institut e, Wako 351-0198, Japan 2 Quantum Phase Electronics Ce nter and Department of Applied Physics, Un iversity of Tokyo, Tokyo 113-8656, Japan, 3 School of Physics, Nankai Un iversity, Tianjin 300071, China 4 Institute for Materials Research, T ohoku University, Sendai, 980-8577, Japan 5Graduate School of Arts and Sciences, University of Tokyo, Tokyo 153-8902, Japan 6PRESTO, Japan Science and Technology Agency, Tokyo 102-0075, Japan, 7Department of Chemistry, Univers ity of Tokyo, Tokyo 113-0033, Japan, All-optical pump-probe detection of magnetization precession has been performed for ferromagnetic EuO thin films at 10 K. We demonstrate that the circ ularly-polarized light can be used to control the magnetization precession on an ultrafast time scale. This takes place within the 100 fs duration of a single laser pulse, through combined contribution from two nonthermal photomagnetic effects, i.e., enhancement of the magnetization and an inverse Faraday effect. From the magnetic field dependences of the frequency and the Gilbert damping parameter, the intrinsic Gilbert damping coefficient is evaluated to be α ≈ 3×10-3. PACS numbers: 78.20.Ls, 42 .50.Md, 78.30.Hv, 75.78.J 2 Optical control of the spin in magnetic materials has been one of the major issues in the field of spintronics, magnetic storage technology, and quantum computing1. One type of the spin controls is based on the directional manipulation in the spin moments 2. This yields in observations of spin precession (reorientation) in antiferromagnets and ferromagne ts when magnetization is canted with respect to an external field3–14. In many previous reports, the spin precession has been driven with the thermal demagnetization induced with the photo-irradiation. Far more intriguing is the ultrafast nonthermal control of magnetization by light 8,10,14, which involves triggering and suppression of the precession. The precession-related anisotropy is expected to be manipulated through laser-induced modulation of electronic state because the anisotropy field originates from the magnetorcrystalline anisotropy based on the spin-orbit coupling. Recently, the spin precession with the non-thermal origin has been observed in bilayer manganites due to a hole-concentration-dependent anisotropic field in competing magnetic phases 15. Despite the success in triggering the reorientation by ultrafast laser pulses, the authors have not demonstrated the possibility of the precessional stoppage. On the other hand, photomagnetic switch of the precession has been reported in ferrimagnetic garnets with use of helicity in light 8,10. The authors attributed the switching behavior to long-lived photo-induced modification of the magnetocrystalline anisotropy16 combined with the inverse Faraday effects17,18. The underlying mechanism for the former photo-induced effect is believed to be redistribution in doped ions16. This is too unique and material-dependent, which is not observed in wide variety of magnets. For establishing the universal scheme of such “helicity-controllable” precession, it should be more useful to rely on more generalized mechanisms such as the carrier-induced ferromagnetism and the magnetic polarons 19. A ferromagnet should be a better choice than a ferrimagnet or an antiferromagnet, e.g., for aiming a larger-amplitude modulation by making use of its larger polarization-rotation angle per unit length . We have recently reported the optically-induced enhancement of magnetization in ferromagnetic EuO associated with the optical transition from the 4 f to 5 d states 20. This enhancement was attributed to the strengthened collective magnetic ordering, mediated with the magnetic polarons. The helicity-controllable precession is expected to be observed in EuO by combining the photo-induced magnetization enhancement20 with the inverse Faraday effect17,18 because the magnetization is related to the magnetic anisotropy. The occurrence of the inverse Faraday effects is expected because of the high crystalline symmetry in EuO17,18. The magnetic properties of EuO are represented by the saturation magnetization of 6.9 μB/Eu, the Curie temperature of 69 K, and the strong in-plane anisotropy21,22. In this article, we report observation of the photomagnetic switch of the spin precession with the nonthermal origin in a EuO thin film for the first time to the best of our knowledge. Due to the above-mentioned reasons, our findings deserve the detailed investigations such as the dependence on the circularly polarized lights, the frequency of precession, the Gilbert damping constants, and the magnitudes of the photo-induced anisotropic field. EuO films were deposited on YAlO 3 substrate using a pulsed laser deposition system with a base pressure lower than 8×10-10 Torr22. The EuO films were then capped with AlO x films in-situ . EuO and AlO x layers have thicknesses of 310 and 30 nm, respectively. The film turned out to be too insulating to be quantified by a conventional transport measurement method. The all-optical experiments have been performed using a standard optical set-up with a Ti:sapphire laser combined with a regenerative amplifier (accompanied with optical parametric amplifier). The wavelength, width, and repetition rate of the output pulse were 650 nm, ≈100 fs, and 1 kHz, respectively. The pump and probe pulses were both incident on the film at angles of θ H ≈ 45 degree from the direction normal to the film plane as shown in inset of Fig. 1. The direction of the probe beam is slightly deviated from that of the pump so as to ensure the sufficient spatial separation of the reflected beams. The angle between the sample plane and the external field is approximately 45 degree. The polarization rotation of the reflected probe pulses due to the Kerr effect was detected using a Wollaston prism and a balanced photo-receiver. The pump fluence was approximately 0.5 mJ/cm 2. A magnetic field was applied using a superconducting electromagnet cryostat. The maximum applied magnetic field was μ 0H ≈ 3 T. All the measurements were performed at 10 K. Figure 1 shows a magneto-optical Kerr signal as a function of the pump-probe delay time for a EuO film at μ 0H = 3.2 T under the irradiation of right-circularly 3 polarized ( σ+) light. Its time trace is composed of instantaneous increase and d ecay of the Kerr rotation, and superimposed oscillation20. The oscillatory structure corresponds to the precession of magnetization. A solid (black) curve in Fig. 1 shows the result of fit to the experimental data using an exponentially decaying function and a damped oscillatory function. The precession is observed even wi th the linearly polarized light, which is consistent with the fact that EuO is a ferromagnet at this temperature. FIG. 1 (color online). Time-resolved Kerr signals recorded for a EuO thin film at a magnetic field of 3.2 T, and a temperature of 10 K for ri ght circularly-polarized ( σ +) light. The inset schematically shows the experimental arrangement. Experimental data are shown by (red) symbols, while the result of fit was shown by a full (blue) line. For the detailed discussion of the precession properties, we subtracted the non-oscillatory part from the Kerr signal as a background. The results are shown in Fig. 2 for nine magnetic fields and for σ + and left-circularly polarization (σ¯). The subtracted data were then fitted with the damped harmonic function in the form of Aexp(−t/τ) sin(2πft+φ), where A and φ are the amplitude and the phase of oscillation, respectively. The amplitude of the precession was not found to depend on the plane of the linear polarization of the pump pulse. There is a linear relationship between the amplitude of precession and the pump fluence for the excitation intensity range measured. It is also noticed in Fig. 2 that the precession amplitudes are different each other for the two helicities ( σ + and σ¯) even at the same magnetic fields. The magnetic field dependence of the amplitude is summarized in Fig. 3(d). The minimum precession amplitude appears at around μ0H = +0.4 T for the σ¯, while the minimum is observed at μ0H = −0.4 T for the σ+ as indicated by the shaded regions. To explain such disappearance of the precession and the triggering of precession even with a linearly-polarized light, it is necessary to take two effects into account. One of the effects that we seek should be odd with respect to the helicity of light. An effective magnetic field through the inverse Faraday effect is plausible to interpret this phenomenon because this satisfies the above requirements [H F // (black arrows) in Figs. 3(a) and 3(b)]. While the normal Faraday effect causes difference in the refractive indices for the left and right circularly polarized lights propagating in a magnetized medium, it is also possible to induce the inverse process where circularly polarized lights create a magnetization or an effective field 17,18. The field associated with the inverse Faraday effect changes its sign when the circular polarization is changed from left-handed to right-handed. FIG. 2 (color online). A series of precession signals under various magnetic fields for right- and left-circularly polarized ( σ+ and σ¯) lights. Solid circles show the e xperimental data for which the non-oscillatory background is s ubtracted, while solid curves represent the calculated data as described in the text. The other effect involved is considered to be the photoinduced enhancement of the anisotropic field (magnetization) associated with the 4 f →5d optical transition [ ΔM (purple arrows) in Figs. 3(a) and 3(b)]20. Our previous work quantified the photoinduced 4 enhancement of the magnetization to be ΔM/M ≈ 0.1%20. The amplitude of precession is determined from combination of ΔM with the component of the inverse-Faraday field ( HF //) approximately projected onto the easy-axis direction. For example, no precession is triggered for μ0H of +0.4 T ( −0.4 T) and σ¯ (σ+), which is due to the balance of these two effects [Fig. 3(a)]. On the other hand, constructive contribution of these effects leads to a change in the direction of the magnetization [two dashed lines and a red arrow in Fig. 3(b)], which enhances the precession amplitude. The strength of the photoinduced field H F can be estimated to be approximately 0.2 T at the laser fluence of 0.5 mJ/cm2. The derivation was based on Eq. (17) of Ref. 10. For more quantitative discussion for the suppression and enhancement of precession, the effect of the perpendicular component of inverse Faraday field is necessary to be taken into account. Such analysis is not performed here because this goes beyond the scope of our work. FIG. 3 (color online). Graphical illustrations of the magnetic precession; its suppression (a) and enhancement (b). M is a magnetization (green), H the external magnetic field (blue), Heff the effective magnetic field (red), ΔM a photo-induced magnetization enhancement (purple), and the HF // the inverse Faraday field (black). The situations of suppression correspond to the conditions of 0.4 T for σ¯ and −0.4 T for σ+. The situations of enhancement are for opposite cases. Magnetic field depe ndences of the magnetization precession related quantities for σ+ and σ¯; precession frequency f (c), amplitude (d), and effective Gilbert damping αeff (e) (f). For the derivation of the precession-related parameters, we plot the frequency ( f) and the amplitude of the magnetization precession for two different helicities as a function of H with closed symbols in Figs. 3(c) and 3(d). To deduce the Landé g-factor g, we calculated f(H) using a set of Kittel equations for taking the effect of tilted geometry into account as12,23: 12 f HH ( 1 ) 2 1e f f cos( ) cosH HH M ( 2 ) 2e f f cos( ) cos 2H HH M ( 3 ) Here, γ is the gyromagnetic ratio ( gμB/h), μB the Bohr magneton, h Planck’s constant, and θH an angle between the magnetic field and direction normal to the plane. Meff is the effective demagnetizing field given as Meff = MS-2K ⊥/MS, where MS is the saturation magnetization and K⊥ is the perpendicular magnetic anisotropy constant. θ is an equilibrium angle for the magnetization, which obeys the following equation: eff sin 2 (2 / )sin( )H HM ( 4 ) A solid (black) line in Fig. 3( c) corresponds to the result of the least-square fit for the frequency f. The values of parameters are g ≈ 2 and μ0Meff ≈ 2.4 T. The g value is consistent with the one derived from the static ferromagnetic resonance measurement 24. Having evaluated the precession-related parameters such as g and Meff, we next discuss H dependence of an effective Gilbert damping parameter αeff. This quantity is defined as: eff1 2f ( 5 ) Figures 3(e) and 3(f) show the effective Gilbert damping parameter αeff derived from the decay time constant ( τ) for σ+ and σ¯, respectively. Despite relatively strong ambiguity shown with ba rs in Figs. 3(e) and 3(f), the damping parameters αeff is not independent of the magnetic field. It is rather a ppropriate to interpret that for 5 αeff for low fields are larger than those at higher fields. Such dependence on magnetic field is consistent with those in general observed for a wide range of the ferrimagnets and ferromagnet s. Two-magnon scattering has been adopted for the explanation of this trend25. When the magnitude or direction of the magnetic anisotropy fluctuates microscopically, magnons can couple more efficiently to the precessional motion 25. Such may cause an additional channel of relaxation. Due to the suppressed influence of the abovementioned two-magnon scattering, the higher-field data correspon d to an intrinsic Gilbert damping constant α ≈ 3×10 -3, as shown with a dashed (black) line in Figs. 3(e) and 3(f). This value is comparable with that reported in Fe26,27,28,29 and significantly larger than that of yttrium iron garnet, which is known for intrinsically low magnetic damping8,10,14. In conclusion, we have reported the observation of magnetization precession and the dependence on light helicity in ferromagnetic EuO films. We attribute it to the photo-induced magnetization enhancement combined with the inverse Faraday effect. The magnetic field dependence of the precession properties al lowed us the evaluation of the Gilbert damping constant to be ≈3×10 -3. Acknowledgements—the authors thank K. Katayama, M. Ichimiya, and Y. Takagi for helpful discussion. This research is granted by the Japan Society for the Promotion of Science (JSPS) throug h the “Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program),” initiated by the Council for Science and Technology Policy (CSTP) and in part supported by KAKENHI (Grant Nos. 23104702 and 24540337) from MEXT, Japan (T. M.). REFERENCES * tmakino@riken.jp 1 A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pisarev, A. M. Balbashov, and Th. Rasing, Nature 435, 655 (2005). 2 A. Kirilyuk, A. V. Kimel, and Th. Rasing, Rev. Mod. Phys. 82, 2731 (2010). 3 C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, D. Weller, E. L. Garwin, and H. C. Siegmann, Science 285, 864 (1999). 4 Q. Zhang, A. V. Nurmikko, A. Anguelouch, G. Xiao, and A. Gupta, Phys. Rev. Lett. 89, 177402 (2002). 5 M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W. J. M. de Jonge, and B. Koopmans, Phys. Rev. Lett. 88, 227201 (2002). 6 R. J. Hicken, A. Barman, V. V. Kruglyak, S. Ladak, J. Phys. D 36, 2183 (2003). 7 I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann, J. Stöhr, G. Ju, B. Lu, and D. Weller, Nature 428, 831 (2004). 8 F. Hansteen, A. Kimel, A. Kirilyuk, and Th. Rasing, Phys. Rev. Lett. 95, 047402 (2005). 9 B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, and W. J. M. de Jonge, Phys. Rev. Lett. 95, 267207 (2005). 10 F. Hansteen, A. Kimel, A. Kirilyuk, and Th. Rasing, Phys. Rev. B 73, 014421 (2006). 11 J. Stöhr, and H. C. Siegmann, Magnetism: From Fundamentals to Nanoscale Dynamics; (Dordrecht Springer, 2006). 12 S. Mizukami, H. Abe, D. Watanabe, M. Oogane, Y. Ando, and T. Miyazaki, Appl. Phys. Express 1, 121301 (2008). 13 M. C. Langner, C. L. S. Kantner, Y. H. Chu, L. M. Martin, P. Yu, J. Seidel, R. Ramesh, and J. Orenstein, Phys. Rev. Lett. 102, 177601 (2009). 14 F. Atoneche, A. M. Kalashnikova, A. V. Kimel, A. Stupakiewicz, A. Maziewski, A. Kirilyuk, and Th. Rasing, Phys. Rev. B 81, 214440 (2010). 15 S. Tomimoto, M. Matsubar a, T. Ogasawara, H. Okamoto, T. Kimura, and Y. Tokura, Phys. Rev. Lett. 98, 017402 (2007). 16 R. W. Teale and D. W. Temple, Phys. Rev. Lett. 19, 904 (1967). 17 J. P. Van der Ziel, P. S. Pershan, and L. D. Malmstrom, Phys. Rev. Lett. 15, 190 (1965). 18 P. S. Pershan, J. P. Van der Ziel, and L. D. Malmstrom, Phys. Rev. 143, 574 (1966). 19 S. Koshihara, A. Oiwa, M. Hirasawa, S. Katsumoto, Y. Iye, C. Urano, H. Takagi, and H. Munekata, Phys. Rev. Lett. 78, 4617 (1997). 20 F. Liu, T. Makino, T. Yamazaki, K. Ueno, A. Tsukazaki, T. Fukumura, Y. Kong, and M. Kawasaki, Phys. Rev. Lett. 108, 257401 (2012). 21 A. Mauger and C. Godart, Phys. Rep. 141, 51 (1986). 22 T. Yamasaki, K. Ueno, A. Tsukazaki, T. Fukumura, and M. Kawasaki, Appl. Phys. Lett. 98, 082116 (2011). 23 L. D. Landau and E. M. Li fshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960). 24 J. F. Dillon and C. E. Olsen, Phys. Rev. 135, A434 (1964). 25 N. Mo, J. Hohlfeld, M. Islam, C. S. Brown, E. Girt, P. Krivosik, W. Tong, A. Rebei, C. E. Patton, Appl. Phys. Lett. 92, 022506 (2008). 26 D. Bastian and E. Biller, Phys. Status Solidi A 35, 113 (1976). 27 F. Schreiber, J. Pflaum, Z. Frait, Th. Mühge, J. Pelzl, Solid State Commun. 93, 965 (1995). 28 J. Pelzl, R. Meckenstock, D. Spoddig, F. Schreiber, J. Pflaum and Z. Frait, J. Phys. Condens. Matter 15, S451 (2003). 29 E. Carpene, C. Piovera, C. Dallera, E. Mancini, E. Puppin, Phys. Rev. B 84, 134425 (2011)

2012-07-28

All-optical pump-probe detection of magnetization precession has been performed for ferromagnetic EuO thin films at 10 K. We demonstrate that the circularly-polarized light can be used to control the magnetization precession on an ultrafast time scale. This takes place within the 100 fs duration of a single laser pulse, through combined contribution from two nonthermal photomagnetic effects, i.e., enhancement of the magnetization and an inverse Faraday effect. From the magnetic field dependences of the frequency and the Gilbert damping parameter, the intrinsic Gilbert damping coefficient is evaluated to be {\alpha} \approx 3\times10^-3.

Ultrafast optical control of magnetization in EuO thin films

1207.6686v1

1 Investigation of the temp erature-dependence of fe rromagnetic resonance and spin waves in Co 2FeAl 0.5Si0.5 Li Ming Loong1, Jae Hyun Kwon1, Praveen Deorani1, Chris Nga Tung Yu2, Atsufumi Hirohata3,a), and Hyunsoo Yang1,b) 1Department of Electrical and Computer Engine ering, National University of Singapore, 117576 Singapore 2Department of Physics, The Univers ity of York, York, YO10 5DD, UK 3Department of Electronics, The Unive rsity of York, York, YO10 5DD, UK Co 2FeAl 0.5Si0.5 (CFAS) is a Heusler compound th at is of interest for sp intronics applications, due to its high spin polarization a nd relatively low Gilbert dampi ng constant. In this study, the behavior of ferromagnetic resonance as a functi on of temperature was investigated in CFAS, yielding a decreasing trend of damping constant as the temperature was increased from 13 to 300 K. Furthermore, we studied spin waves in CF AS using both frequency domain and time domain techniques, obtaining group velocities and atte nuation lengths as high as 26 km/s and 23.3 m, respectively, at room temperature. a) Electronic mail: atsufumi.hirohata@york.ac.uk b) Electronic mail: eleyang@nus.edu.sg 2 Half-metallic Heusler compounds with low Gilbert damping constant ( ) are promising candidates for spin transfer torque-based (STT) spintronic devices,1-3 spin-based logic systems,4 as well as spin wave-based data comm unication in microelectronic circuits.5 Hence, a deeper fundamental understanding of the magnetiza tion dynamics, such as the behavior of ferromagnetic resonance (FMR) and spin waves in Heusler compounds, could enable better engineering and utilization of these compounds fo r the aforementioned applications. In previous work, FMR has been investigated in several Heusler compounds, such as Co 2FeAl (CFA),6 Co2MnSi (CMS),7 and Co 2FeAl 0.5Si0.5 (CFAS).8 In addition, the variation of with temperature has been studied for other material s, such as Co, Fe, Ni, and CoFeB.9-11 However, the temperature-dependence of in Heusler compounds has not be en reported yet. Furthermore, while there have been some studies of spin waves in Heusler compounds, such as CMS and Co2Mn 0.6Fe0.4Si (CMFS),7,12 these studies have focused on frequency domain measurements. Thus, time domain measurements remain scarce, and mainly consist of time-resolved magneto-optic Kerr effect (TR-MOKE) experiments. 13 In this work, we investigate the temperature- dependence of in CFAS, a half-metallic Heusler compound.14,15 Moreover, we utilize both frequency domain and pulsed inductive micr owave magnetometry (PIMM) time domain measurements to study the magnetiza tion dynamics in CFAS. We obtain of 0.0025 at room temperature, which is 6 times lower than the va lue at 13 K. In addition, we evaluate the group velocity ( vg) and the attenuation length ( ) in CFAS, leading to values as high as 26 km/s and 23.3 m respectively, at room temperature. CFAS (30 nm thick) was grown by ultrah igh vacuum (UHV) molecular beam epitaxy (MBE) on single crystal MgO (001) substrates and capped with 5 nm of Au. The base pressure was 1.210-8 Pa and the pressure during deposition was typically 1.6 10-7 Pa. The substrates 3 were cleaned with acetone, IPA and deionised wate r in an ultrasonic bath before being loaded into the chamber. After the film growth, the samples were in-situ annealed at 600 °C for 1 hour. CFAS alloy and Au pellets were used as targ ets for electron-beam bombardment. Figure 1(a) shows the vibrating sample magnetometry (V SM) results, from which the saturation magnetization ( Ms) was extracted. The measurement was also repeated at different temperatures to extract the corresponding values of Ms for subsequent data fitting. The Ms value increases from 1100 emu/cc at 300 K, to 1160 emu/cc at 13 K. From the VSM data, we verify a hard axis along [100] and an easy axis along [110] , consistent with earlier reports.3,8 In addition, the -2 XRD data shown in Fig. 1(b) verified the presen ce of the characteristic (004) peak, indicating that the CFAS film wa s at least B2-ordered.1,14 As shown in Fig. 1(c), the film was patterned into mesas, which were integrated with asymme tric coplanar waveguides (ACPW). The ACPWs were electrically isolated from the mesa by 50 nm of Al 2O3, which was deposited by RF sputtering. Vector network analy zer (VNA) and PIMM techniques were used to excite and detect ferromagnetic resonance (FMR) as well as spin waves in CFAS. The former technique allows frequency domain measurements, while the la tter technique was us ed for time domain measurements. The experimental setup enable d the excitation of Damon-Eshbach-type (DE) modes, as the external magnetic field was applied along the ACPWs, shown in Fig. 1(c).16 A VNA was connected to the AC PWs, and reflection as well as transmission signals were measured to study the FMR and spin wave pr opagation, respectively. Background subtraction was performed to obtain the resonance peaks. Figure 2(a) shows th e FMR frequency as a function of applied magnetic fiel d at different temperatures, with the corresponding fits using the Kittel formula,17 ݂ൌఊ ଶగඥሺܪܪሻሺܪܪ4ܯߨ ௦ሻ, (1)4 where f is the resonance frequency, is the gyromagnetic ratio, H is the applied magnetic field, and Ha is the anisotropy field. The g factor, which was extr acted using the equation ߛ ൌ 2ߤ݃ߨ/݄ ,where B is the Bohr magneton and h is Planck’s constant, was found to be 2.03 0.02, while Ha generally decreased from 130 Oe at 13 K to 70 Oe at 300 K. The ( g – 2) value is lower than those of Co and Ni, but comparable to th ose of other Heusler comp ounds, such as CMS and Co2MnAl (CMA).18 The deviation of the g factor from the free electron value of 2 is correlated with the spin-orbit interaction in a material, where a smaller deviation indicates weaker spin- orbit interaction, and lower .18 The inset of Fig. 2(a) show s the resonance frequency at H = 1040 Oe as a function of temperature, with a Bloch fitting, indicating a Curie temperature of approximately 1000 K. The Bloch fitting was perf ormed by substituting the following equation19 into Eq. (1): ܯ௦ൌܽ൫1െܽ ଵܶଷ/ଶെܽଶܶହ/ଶെܽଷܶ/ଶ൯, ( 2 ) where T is temperature, and a0, a1, a2, and a3 are positive coefficients. As shown in Fig. 2(b), the extracted FMR field linewidths were fitted with the linear equation20 ΔH ൌ ΔH0 4αf/, where H is the field linewidth and H0is the extrinsic field linewidth . This enabled the extraction of the intrinsic Gilbert damping ( ) from the fit line slopes. Figure 2(c) shows that increases as the temperature decreases. The value of at room temperature was found to be 0.0025, which is comp arable with the previously reported room temperature value for CFAS.8 The trend of with temperature is consistent with previous first- principle calculations,9 and could be attributed to longe r electron scattering time at lower temperatures, due to a reduction in phonon-elec tron scattering. Consequently, the angular momentum transfer at low temperatures occu rs predominantly by direct damping through intraband transitions.11 Similar temperature-dependence of has also been observed 5 experimentally. For example, the of Co 20Fe60B20 has been found to increase by a factor of 3 from 0.007 at 300 K, to 0.023 at 5 K.11 This is comparable to our results, where increases by a factor of almost 6 from 0.0025 at 300 K to 0.014 at 13 K. It shoul d be noted that spin pumping into the Au cap layer could have contributed to the measured resonance linewidth, thus causing the extracted to be higher than its actual value ( CFAS). Thus, = CFAS + sp, where sp denotes the spin pumping c ontribution to the damping.21 While an investigation of sp in the CFAS/Au system would exceed the scope of this work, sp values for a Fe/Au system21,22 have nonetheless been included in Fig. 2(c) to prov ide a gauge of the temperature dependence of sp, as well as a rough estimation of the magnitude of sp in the CFAS/Au system. Figure 2(d) shows an increase in H0 as temperature increases. This could be due to the effect of temperature on the interaction between magnetic precession and sample inhomogeneities, or on magnon-magnon scattering, as these f actors contribute to H0.20,23 In both Fig. 2(c) and 2(d), room temperature values of and H0 for sputter-deposited CFAS were in cluded, for comparison with the MBE sample. It can be seen that the is higher for the sputter-deposite d sample, consistent with lower half-metallic character due to greater structural disorder.1,6 We have also measured the time domain PI MM data at 300 K as shown in Fig. 3(a), where SW15 and SW30 denote edge-to-edge signal line separations of 15 and 30 m, respectively. The width of all the signal lines was fixed at 10 m. Using the temporal positions of the centers of the Gaussian wavepackets ( t15 and t30, respectively), the group velocity ( vg) was calculated with the equation5,24 vg ൌ 1 5 m/ሺt30 – t15ሻ. Fast Fourier transform (FFT) was performed on the PIMM data, as shown in Fig. 3(b), verifying the presence of multiple modes, where each mode manifested as a dark-light-dark oscillation. The vg decreases from 26 km/s at 50 Oe to 11 km/s at 370 Oe, as shown in Fig. 3( c). Moreover, from the VNA transmission data, 6 which is another measure of spin wave propagation, attenuation length ( ) and were extracted as a function of magnetic fiel d at room temperature, using the method reported elsewhere.24,25 The spin wave amplitude was extracted from Lorentzian fittings of the VNA transmission resonance peaks, which were measured using wa veguides with different center-to-center signal line-signal line (S-S) spacings. Then, was extracted using the equation24 A1expሺ x1/ሻ ൌ A2expሺ x2/ሻ, where A1 and A2 denote the measured spin wave amplitudes, while x1 and x2 denote the different S-S spacings for the corresponding waveguides. The decreases from 23.3 m at 460 Oe, to 12.1 m at 1430 Oe, as shown in Fig. 3(c). Using the following equation,25 was calculated at different magnetic fields, as shown in Fig. 3(d) ߙൌఊሺଶగெೞሻమௗషమೖ ଶగሺுାଶగெ ೞሻ ( 3 ) where d is the film thickness and k is a spin wave vector, which can be estimated by 2 /(signal line width).5 The values (0.0026 – 0.0031) are consistent with the room temperature value (0.0025) obtained from the FMR measurements. As shown in Fig. 3(c), and vg decreased as the applie d magnetic field increased, consistent with previous experimental11 and theoretical5 results. This trend can be understood in terms of the following equation,5,26 ݒൌఊమఓబమெೞమௗ ଼గ݁ିଶௗ, ( 4 ) where μ0 is the permeability of free space. As the a pplied magnetic field increases, the resonance frequency increases, thus vg decreases. In addition, for a given value of , the magnetic precession will decay within a ce rtain amount of time. Hence, the distance travelled by the precessional disturbance within that amount of time depends on its propagation velocity, vg. Consequently, the higher the vg, the longer the distance travelled, and thus, the higher the .The 7 obtained values of and vg are comparable to those of other ferromagnetic materials for the Damon-Eshbach surface spin wave mode.5,11,12 For example, of 18.95 m was extracted for CFA by micromagnetic simulations,5 while and vg values as high as 23.9 m and 25 km/s, respectively, were experimentally observed in CoFeB.11 Furthermore, as high as 16.7 m was experimentally observed in CMFS.12 In conclusion, we have found a decreasing trend of with increasing temperature for MBE-grown Co 2FeAl 0.5Si0.5, in the temperature range of 13 – 300 K. The room temperature value of was found to be 0.0025, which was approximately 6 times lower than that at 13 K. We have also investigated vg and in CFAS, obtaining values as high as 26 km/s and 23.3 m respectively, at room temperature. This work was supported by the Singa pore NRF CRP Award No. NRF-CRP 4-2008-06. 8 References 1 T. Graf, C. Felser, and S. S. P. Parkin, Prog. Solid State Chem. 39, 1 (2011). 2 S. Ikeda, H. Sato, M. Yamanouchi, H. Ga n, K. Miura, K. Mizunuma, S. Kanai, S. Fukami, F. Matsukura, N. Kasai, and H. Ohno, SPIN 2, 1240003 (2012). 3 H. Sukegawa, Z. C. Wen, K. Kondou, S. Kasai, S. Mitani, and K. Inomata, Appl. Phys. Lett. 100, 042508 (2012). 4 M. Jamali, J. H. Kwon, S. M. Seo, K. J. Lee, and H. Yang, Scientific Reports 3, 3160 (2013). 5 J. H. Kwon, S. S. Mukherjee, P. Deorani, M. Hayashi, and H. Yang, Appl. Phys. A: Mater. Sci. Process. 111, 369 (2013). 6 S. Mizukami, D. Watanabe, M. Oogane, Y. A ndo, Y. Miura, M. Shirai, and T. Miyazaki, J. Appl. Phys. 105, 07D306 (2009). 7 H. Pandey, P. C. Joshi, R. P. Pant, R. Pras ad, S. Auluck, and R. C. Budhani, J. Appl. Phys. 111, 023912 (2012). 8 L. H. Bai, N. Tezuka, M. Kohda, and J. Nitta, Jpn. J. Appl. Phys. 51, 083001 (2012). 9 K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 (2007). 10 B. Heinrich, D. J. Meredith, and J. F. Cochran, J. Appl. Phys. 50, 7726 (1979). 11 H. M. Yu, R. Huber, T. Schwarze, F. Bra ndl, T. Rapp, P. Berberich, G. Duerr, and D. Grundler, Appl. Phys. Lett. 100, 262412 (2012). 12 T. Sebastian, Y. Ohdaira, T. Kubota, P. Pi rro, T. Bracher, K. Vogt , A. A. Serga, H. Naganuma, M. Oogane, Y. Ando, and B. Hillebrands, Appl. Phys. Lett. 100, 112402 (2012). 9 13 Y. Liu, L. R. Shelford, V. V. Kruglyak, R. J. Hicken, Y. Sakuraba, M. Oogane, and Y. Ando, Phys. Rev. B 81, 094402 (2010). 14 B. Balke, G. H. Fecher, and C. Felser, Appl. Phys. Lett. 90, 242503 (2007). 15 R. Shan, H. Sukegawa, W. H. Wang, M. Kodzuka, T. Furubayashi, T. Ohkubo, S. Mitani, K. Inomata, and K. Hono, Phys. Rev. Lett. 102, 246601 (2009). 16 R. W. Damon and J. R. Eshbach, J. Appl. Phys. 31, S104 (1960). 17 C. Kittel, Physical Review 73, 155 (1948). 18 B. Aktas and F. Mikailov, Advances in Nanoscale Magnetism: Proceedings of the International Conference on Nanoscale Magnetism ICNM-2 007, June 25 -29, Istanbul, Turkey . (Springer, 2008). p. 63. 19 S. T. Lin and R. E. Ogilvie, J. Appl. Phys. 34, 1372 (1963). 20 P. Krivosik, N. Mo, S. Kalarickal , and C. E. Patton, J. Appl. Phys. 101, 083901 (2007). 21 E. Montoya, B. Kardasz, C. Burrowes, W. Hu ttema, E. Girt, and B. Heinrich, J. Appl. Phys. 111, 07C512 (2012). 22 M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 71, 064420 (2005). 23 M. Farle, Reports on Progress in Physics 61, 755 (1998). 24 K. Sekiguchi, K. Yamada, S. M. Seo, K. J. Le e, D. Chiba, K. Kobayashi, and T. Ono, Appl. Phys. Lett. 97, 022508 (2010). 25 K. Sekiguchi, K. Yamada, S. M. Seo, K. J. Le e, D. Chiba, K. Kobayashi, and T. Ono, Phys. Rev. Lett. 108, 017203 (2012). 26 D. D. Stancil, Theory of Magnetostatic Waves . (Springer, Berlin, 1993). 10 Figure captions FIG. 1. (a) Normalized magnetic hysteresis data along the crystallographic hard [100] and easy [110] axes of MBE-grown CFAS. Ms is the saturation magnetization. (b) -2 XRD data of the MBE-grown CFAS sample. (c) Optical microscopy image of the CFAS me sa integrated with asymmetric coplanar waveguides (ACPW). The or ientation of the in-plane magnetic field ( H) is indicated. FIG. 2. (a) FMR frequency at different magnetic fields. Inset: FMR frequency for a fixed field (1040 Oe) at different temperat ures. (b) Resonance linewidth as a function of frequency at different temperatures (symbols), with correspo nding fit lines. (c) Gilbert damping parameter ( ) at different temperatures. The spin pumping contribution to damping ( sp) for a Fe/Au system has been included, where all sp values were obtained from literature, except those at 13 K and room temperature, which were obtained by extra polating the literature valu es. (d) Extrinsic field linewidth (H0) at different temperatures. FIG. 3. (a) PIMM data from two differ ent signal line-signal line spacings at H = 50 Oe for 300 K. (b) Fast Fourier transform (FFT) of room temp erature PIMM data. (c) Room temperature group velocity ( vg, axis: left and bottom) and attenuation length ( , axis: top and right) at different magnetic fields. (d) Room temperature Gilbert damping parameter ( ) at different magnetic fields. 11 FIG. 1 12 FIG. 2 800 1000 1200 140091011121314 (d) (c)(b) (a) 13 K 90 K 210 K 294 KResonance frequency (GHz) Magnetic field (Oe)10 11 12 13 1450100150200250300 90 K 210 K 294 K H (Oe) Resonance frequency (GHz) 0 100 200 3000.0000.0050.0100.0150.020 Temperature (K) MBE Sputtered sp Au-Fe 0 100 200 300050100150200 H0 (Oe) Temperature (K) MBE Sputtered0 200 800 100012000412 fR (GHz) Temperature (K)13 FIG. 3

2014-10-02

Co2FeAl0.5Si0.5 (CFAS) is a Heusler compound that is of interest for spintronics applications, due to its high spin polarization and relatively low Gilbert damping constant. In this study, the behavior of ferromagnetic resonance as a function of temperature was investigated in CFAS, yielding a decreasing trend of damping constant as the temperature was increased from 13 to 300 K. Furthermore, we studied spin waves in CFAS using both frequency domain and time domain techniques, obtaining group velocities and attenuation lengths as high as 26 km/s and 23.3 um, respectively, at room temperature.

Investigation of the temperature-dependence of ferromagnetic resonance and spin waves in Co2FeAl0.5Si0.5

1410.0439v1

Ferromagnetic resonance modulation in d-wave superconductor/ferromagnetic insulator bilayer systems Yuya Ominato,1Ai Yamakage,2Takeo Kato,3and Mamoru Matsuo1, 4, 5, 6 1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China. 2Department of Physics, Nagoya University, Nagoya 464-8602, Japan 3Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan 4CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China 5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan 6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan (Dated: May 6, 2022) We investigate ferromagnetic resonance (FMR) modulation in d-wave superconductor (SC)/ferromagnetic insulator (FI) bilayer systems theoretically. The modulation of the Gilbert damping in these systems re ects the existence of nodes in the d-wave SC and shows power-law decay characteristics within the low-temperature and low-frequency limit. Our results indicate the eectiveness of use of spin pumping as a probe technique to determine the symmetry of unconven- tional SCs with high sensitivity for nanoscale thin lms. I. INTRODUCTION Spin pumping (SP)1,2is a versatile method that can be used to generate spin currents at magnetic junctions. While SP has been used for spin accumulation in vari- ous materials in the eld of spintronics3,4, it has recently been recognized that SP can also be used to detect spin excitation in nanostructured materials5, including mag- netic thin lms6, two-dimensional electron systems7{9, and magnetic impurities on metal surfaces10. Notably, spin excitation detection using SP is sensitive even for such nanoscale thin lms for which detection by con- ventional bulk measurement techniques such as nuclear magnetic resonance and neutron scattering experiment is dicult. Recently, spin injection into s-wave superconductors (SCs) has been a subject of intensive study both theoret- ically11{20and experimentally21{34. While the research into spin transport in s-wave SC/magnet junctions is ex- pected to see rapid development, expansion of the devel- opment targets toward unconventional SCs represents a fascinating research direction. Nevertheless, SP into un- conventional SCs has only been considered in a few recent works35,36. In particular, SP into a d-wave SC, which is one of the simplest unconventional SCs that can be real- ized in cuprate SCs37, has not been studied theoretically to the best of our knowledge, although experimental SP in ad-wave SC has been reported recently38. In this work, we investigate SP theoretically in a bi- layer magnetic junction composed of a d-wave SC and a ferromagnetic insulator (FI), as shown in Fig. 1. We apply a static magnetic eld along the xdirection and consider the ferromagnetic resonance (FMR) experiment of the FI induced by microwave irradiation. In this setup, the FMR linewidth is determined by the sum of the in- trinsic contribution made by the Gilbert damping of the bulk FI and the interface contribution, which originates from the spin transfer caused by exchange coupling be- Microwavex yz Spin current Ferromagnetic resonanceInteractionFIG. 1. Schematic of the d-wave SC/FI bilayer system. The two-dimensional d-wave SC is placed on the FI. Precessional motion of the magnetization is induced by microwave irradia- tion. The spins are injected and the magnetization dynamics are modulated because of the interface magnetic interaction. tween thed-wave SC and the FI. We then calculate the interface contribution to the FMR linewidth, which is called the modulation of the Gilbert damping hereafter, using microscopic theory based on the second-order per- turbation39{41. We show that the temperature depen- dence of the modulation of the Gilbert damping exhibits a coherent peak below the transition temperature that is weaker than that of s-wave SCs11,13{15. We also show that because of the existence of nodes in the d-wave SCs, the FMR linewidth enhancement due to SP remains even at zero temperature. The paper is organized as follows. In Sec. II, we in- troduce the model Hamiltonian of the SC/FI bilayer sys- tem. In Sec. III, we present the formalism to calculate the modulation of the Gilbert damping. In Sec. IV, we present the numerical results and explain the detailed behavior of the modulation of the Gilbert damping. In Sec. V, we brie y discuss the relation to other SC sym- metries, the proximity eect, and the dierence between d-wave SC/FI junctions and d-wave SC/ferromagnetic metal junctions. We also discuss the eect of an eectivearXiv:2201.06060v2 [cond-mat.mes-hall] 5 May 20222 Zeeman eld due to the exchange coupling. In Sec. VI, we present our conclusion and future perspectives. II. MODEL The model Hamiltonian of the SC/FI bilayer system His given by H=HFI+HdSC+HT: (1) The rst term HFIis the ferromagnetic Heisenberg model, which is given by HFI=JX hi;jiSi
Sj~ hdcX jSx j; (2) whereJ>0 is the exchange coupling constant, hi;ji represents summation over all the nearest-neighbor sites, Sjis the localized spin at site jin the FI, is the gy- romagnetic ratio, and hdcis the static magnetic eld. The localized spin Sjis described as shown using the bosonic operators bjandby jof the Holstein-Primako transformation42 S+ j=Sy j+iSz j= 2Sby jbj1=2 bj; (3) S j=Sy jiSz j=by j 2Sby jbj1=2 ; (4) Sx j=Sby jbj; (5) where we require [ bi;by j] =i;jto ensure that S+ j,S j, andSx jsatisfy the commutation relation of angular mo- mentum. The deviation of Sx jfrom its maximum value S is quantied using the boson particle number. It is conve- nient to represent the bosonic operators in the reciprocal space as follows bk=1p NX jeik
rjbj; by k=1p NX jeik
rjby j;(6) whereNis the number of sites. The magnon opera- tors with wave vector k= (kx;ky;kz) satisfy [bk;by k0] = k;k0. Assuming that the deviation is small, i.e., that hby jbji=S1, the ladder operators S jcan be approx- imated asS+ j(2S)1=2bjandS j(2S)1=2by j, which is called the spin-wave approximation. The Hamiltonian HFIis then written as HFIX k~!kby kbk; (7) where we assume a parabolic dispersion ~!k=Dk2+ ~ hdcwith a spin stiness constant Dand the constant terms are omitted. The second term HdSCis the mean-eld Hamiltonian for the two-dimensional d-wave SC, and is given by HdSC=X k(cy k";ck#) k
k
kkck" cy k# ;(8)wherecy kandckdenote the creation and annihilation operators, respectively, of the electrons with the wave vectork= (kx;ky) and thexcomponent of the spin =";#, andk=~2k2=2mis the energy of conduc- tion electrons measured from the chemical potential . We assume that the d-wave pair potential has the form
k=
cos 2kwith the phenomenological temperature dependence
= 1:76kBTctanh 1:74r Tc T1! ; (9) wherek= arctan(ky=kx) denotes the azimuth angle of k. Using the Bogoliubov transformation given by ck" cy k# = ukvk vkuk k" y k# ; (10) where y kand kdenote the creation and annihilation operators of the Bogoliubov quasiparticles, respectively, andukandvkare given by uk=r Ek+k 2Ek; vk=r Ekk 2Ek; (11) with the quasiparticle energy Ek=p 2 k+
2 k, the mean- eld Hamiltonian can be diagonalized as HdSC=X k( y k"; k#) Ek 0 0Ek k" y k# :(12) The density of states of the d-wave SC is given by43 D(E)=Dn= Re2 K
2 E2 ; (13) whereDn=Am= 2~2is the density of states per spin of the normal state, Ais the system area, and K(x) is the complete elliptic integral of the rst kind in terms of the parameterx, where K(x) =Z=2 0dp 1xcos2: (14) D(E) diverges at E=
= 1 and decreases linearly when E=
1 because of the nodal structure of
k. The density of states for an s-wave SC, in contrast, has a gap forjEj<
. This dierence leads to distinct FMR modulation behaviors, as shown below. The third term HTdescribes the spin transfer between the SC and the FI at the interface HT=X q;k Jq;k+ qS k+J q;k qS+ k
; (15) whereJq;kis the matrix element of the spin transfer pro- cesses, and  q= (y qiz q)=2 andS k=Sy kiSz kare3 (a) Spin transfer process (b) Self-energy Jq,kJ*q,k p/uni2191p+q/uni2193 p/uni2191p+q/uni2193 −k −k /uni03A3R k(/uni03C9)= FIG. 2. (a) Diagrams of the bare vertices of the spin transfer processes at the interface. (b) Self-energy within the second- order perturbation. the Fourier components of the ladder operators and are given by + q=X pcy p"cp+q#;  q=X pcy p+q#cp"; (16) S k(2S)1=2by k; S+ k(2S)1=2bk: (17) Using the expressions above, HTcan be written as HTp 2SX p;q;k Jq;kcy p"cp+q#by k+J q;kcy p+q#cp"bk : (18) The rst (second) term describes a magnon emission (absorption) process accompanying an electron spin- ip from down to up (from up to down). A diagrammatic representation of the interface interactions is shown in Fig. 2 (a). In this work, we drop a diagonal exchange coupling at the interface, whose Hamiltonian is given as HZ=X q;kJq;kx qSx k: (19) This term does not change the number of magnons in the FI and induces an eective Zeeman eld on electrons in the two-dimensional d-wave SC. We expect that this term does not aect our main result because the coupling strength is expected to be much smaller than the super- conducting gap and the microwave photon energy. We will discuss this eect in Sec. V brie y. III. FORMULATION The coupling between the localized spin and the mi- crowave is given by V(t) =~ hacX i(Sy icos!tSz isin!t); (20)wherehacis the amplitude of the transverse oscillating magnetic eld with frequency !. The microwave irra- diation induces the precessional motion of the localized spin. The Gilbert damping constant can be read from the retarded magnon propagator dened by GR k(t) =1 i~(t)h[S+ k(t);S k(0)]i; (21) where(t) is a step function. Second-order perturbation calculation of the magnon propagator with respect to the interface interaction was performed and the expression of the self-energy was derived in the study of SP39{41. Fol- lowing calculation of the second-order perturbation with respect to Jq;k, the Fourier transform of the retarded magnon propagator is given by GR k(!) =2S=~ !!k+i!(2S=~)R k(!); (22) whereis the intrinsic Gilbert damping constant that was introduced phenomenologically44{46. The diagram of the self-energy R k(!) is shown in Fig. 2 (b). From the expressions given above, the modulation of the Gilbert damping constant is given by =2SIm R k=0(!) ~!: (23) Within the second-order perturbation, the self-energy is given by R k(!) =X qjJq;kj2R q(!); (24) whereR q(!) represents the dynamic spin susceptibility of thed-wave SC dened by R q(!) =1 i~Z dtei(!+i0)t(t)h[+ q(t); q(0)]i:(25) Substituting the ladder operators in terms of the Bogoli- ubov quasiparticle operators into the above expression and performing a straightforward calculation, we then obtain434 R q(!) =X pX =1X 0=1(p+Ep)(p+q+0Ep+q) +
p
p+q 4Ep0Ep+qf(Ep)f(0Ep+q) Ep0Ep+q+~!+i0; (26) wheref(E) = 1=(eE=kBT+ 1) is the Fermi distribution function. In this paper, we focus on a rough interface modeled in terms of the mean J1and variance J22of the distribution ofJq;k(see Appendix A for detail). The congurationally averaged coupling constant is given by jJq;k=0j2=J12q;0+J22: (27) In this case, is written as =2SJ12 ~!ImR q=0(!) +2SJ22 ~!X qImR q(!):(28) The rst term represents the momentum-conserved spin- transfer processes, which vanish as directly veried from Eq. (26). This vanishment always occurs in spin-singlet SCs, including sandd-wave SCs, since the spin is conserved43. Consequently, the enhanced Gilbert damp- ing is contributed from spin-transfer processes induced by the roughness proportional to the variance J22 =2SJ22 ~!X qImR q(!): (29) The wave number summation can be replaced as X q(


)!Dn 2Z1 1dZ2 0d(


): (30) Changing the integral variable from toEand substi- tuting Eq. (26) into Eq. (29), we nally obtain =2SJ 22D2 n ~!Z1 1dE[f(E)f(E+~!)] Re2 K
2 E2 Re2 K
2 (E+~!)2 : (31) Note that the coherence factor vanishes in the above ex- pression by performing the angular integral. The en- hanced Gilbert damping in the normal state is given by n= 2SJ 22D2 n; (32) for the lowest order of !. This expression means that  is proportional to the product of the spin-up and spin- down densities of states at the Fermi level7.IV. GILBERT DAMPING MODULATION Figure 3 shows the enhanced Gilbert damping constant as a function of temperature for several FMR frequen- cies, where is normalized with respect to its value in the normal state. We compare in thed-wave SC shown in Figs. 3 (a) and (c) to that in the s-wave SC shown in Figs. 3 (b) and (d). The enhanced Gilbert damping for thes-wave SC is given by13 =2SJ 22D2 n ~!Z1 1dE[f(E)f(E+~!)]  1 +
2 E(E+~!) RejEjp E2
2 Re" jE+~!jp (E+~!)2
2# ; (33) where the temperature dependence of
is the same as that for the d-wave SC, given by Eq. (9). Note that the BCS theory we are based on, which is valid when the Fermi energy is much larger than
, is described by only some universal parameters, including Tc, and inde- pendent of the detail of the system in the normal state. When ~!=k BTc= 0:1,shows a coherence peak just below the transition temperature Tc. However, the co- herence peak of the d-wave SC is smaller than that of thes-wave SC. Within the low temperature limit, in thed-wave SC shows power-law decay behavior described by/T2. This is in contrast to in thes-wave SC, which shows exponential decay. The dierence in the low temperature region originates from the densities of states in thed-wave ands-wave SCs, which have gapless and full gap structures, respectively. When the FMR frequency increases, the coherence peak is suppressed, and de- cays monotonically with decreasing temperature. has a kink structure at ~!= 2
, where the FMR frequency corresponds to the superconducting gap. Figure 4 shows atT= 0 as a function of !. In thed-wave SC,grows from zero with increasing !as /!2. When the value of becomes comparable to the normal state value, the increase in is suppressed, andthen approaches the value in the normal state. In contrast, in thes-wave SC vanishes as long as the condition that ~! < 2
is satised. When ~!exceeds 2
,then increases with increasing !and approaches the normal state value. This dierence also originates from the distinct spectral functions of the d-wave and s-wave SCs. Under the low temperature condition that T= 0:1Tc, the frequency dependence of does not5 0.1 5.0 T/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.01.2 0.20.40.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1n0.1 0.5 1.0 1.5 2.03.04.05.0/uni210F/uni03C9/kBTc T/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.02.0 0.51.01.5/uni03B4/uni03B1//uni03B4/uni03B1n(a) d-wave (b) s-wave (c) d-wave (d) s-wave T/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.01.2 0.20.40.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.02.0 0.51.01.5/uni03B4/uni03B1//uni03B4/uni03B1n0.1 0.5 1.0 1.5 2.03.04.05.0 FIG. 3. Enhanced Gilbert damping as a function of tem- peratureT. The left panels (a) and (c) show in thed- wave SC in the low and high frequency cases, respectively. The right panels (b) and (d) show in thes-wave SC in the low and high frequency cases, respectively. nis the normal state value. change for the s-wave SC, and it only changes in the low-frequency region where ~!.kBTfor thed-wave SC (see the inset in Fig. 4). V. DISCUSSION We discuss the modulation of the Gilbert damping in SCs with nodes other than the d-wave SC consid- ered in this work. Other SCs with nodes are expected to exhibit the power-law decay behavior within the low- temperature and low-frequency limit as the d-wave SCs. However, the exponent of the power can dier due to the dierence of the quasiparticle density of states. Fur- thermore, in the p-wave states, two signicant dierences arise due to spin-triplet Cooper pairs. First, the uni- form spin susceptibility R q=0(!) can be nite in the spin- triplet SCs because the spin is not conserved. Second, the enhanced Gilbert damping exhibits anisotropy and the value changes by changing the relative angle between the Cooper pair spin and localized spin35. In our work, proximity eect between FIs and SCs was not taken into account because the FMR modula- tion was calculated by second-order perturbation based on the tunnel Hamiltonian. Reduction of superconduct- /uni210F/uni03C9=2/uni0394(T =0) s-waved-wave 00.2 2 4 6 81.2 0.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1n /uni210F/uni03C9/kBTc0.4 0.0 100 10.05 0.000.1T/Tc=0.0FIG. 4. Enhanced Gilbert damping as a function of fre- quency!. The vertical dotted line indicates the resonance frequency ~!= 2
(T= 0). The inset shows an enlarged view in the low-frequency region. ing gap due to the proximity eect15and eect of the subgap Andreev bound states that appear in the ab-axis junction47would also be an important problem left for future works. Physics of the FMR modulation for d-wave SC/ferromagnetic metal junctions is rather dier- ent from that for d-wave SC/FI junctions. For d-wave SC/ferromagnetic metal junctions, spin transport is described by electron hopping across a junction and the FMR modulation is determined by the product of the density of states of electrons for a d-wave SC and a ferromagnetic metal. (We note that the FMR modulation is determined by a spin susceptibility of d-wave SC, which in general includes dierent informa- tion from the density of states of electrons.) While the FMR modulation is expected to be reduced below a SC transition temperature due to opening an energy gap, its temperature dependence would be dierent from results obtained in our work. Finally, let us discuss eect of the diagonal exchange coupling given in Eq. (19) (see also the last part of Sec. II). This term causes an exchange bias, i.e., an eec- tive Zeeman eld on conduction electrons in the d-wave SC, which is derived as follows. First, the x-component of the localized spin is approximated as hSx ji S, which gives Sx kSp Nk;0. Next, the matrix element Jq;k=0is replaced by the congurationally averaged value Jq;k=0=J1q;0. Consequently, the eective Zeeman eld term is given by HZEZX p(cy p"cp"cy p#cp#); (34) where we introduced a Zeeman energy as EZ=J1Sp N. This term induces spin splitting of conduction electrons6 in thed-wave SC and changes the spin susceptibility of the SC. The spin-splitting eect causes a spin excitation gap and modies the frequency dependence in Fig. 4, that will provide additional information on the exchange cou- pling at the interface. In actual experimental setup for thed-wave SC, however, the Zeeman energy, that is less than the exchange bias between a magnetic insulator and a metal, is estimated to be of the order of 0 :1 erg=cm2. This leads to the exchange coupling that is much less thanJ0:1 meV for YIG48. Therefore, we expect that the interfacial exchange coupling is much smaller than the superconducting gap and the microwave photon en- ergy though it has not been measured so far. A detailed analysis for this spin-splitting eect is left for a future problem. VI. CONCLUSION In this work, we have investigated Gilbert damping modulation in the d-wave SC/FI bilayer system. The enhanced Gilbert damping constant in this case is pro- portional to the imaginary part of the dynamic spin sus- ceptibility of the d-wave SC. We found that the Gilbert damping modulation re ects the gapless excitation that is inherent in d-wave SCs. The coherence peak is sup- pressed in the d-wave SC when compared with that in thes-wave SC. In addition, the dierences in the spec- tral functions for the d-wave ands-wave SCs with gap- less and full-gap structures lead to power-law and ex- ponential decays within the low-temperature limit, re- spectively. Within the low-temperature limit, in the d-wave SC increases with increasing !, whilein the s-wave SC remains almost zero as long as the excitation energy ~!remains smaller than the superconducting gap 2
. Our results illustrate the usefulness of measurement of the FMR modulation of unconventional SCs for determi- nation of their symmetry through spin excitation. We hope that this fascinating feature will be veried exper- imentally in d-wave SC/FI junctions in the near future. To date, one interesting result of FMR modulation in d-wave SC/ferromagnetic metal structures has been re- ported38. This modulation can be dependent on metallic states, which are outside the scope of the theory pre- sented here. The FMR modulation caused by ferromag- netic metals is another subject that will have to be clar- ied theoretically in future work. Furthermore, our work provides the most fundamental basis for application to analysis of junctions with vari- ous anisotropic SCs. For example, some anisotropic SCs are topological and have an intrinsic gapless surface state. SP can be accessible and can control the spin excitation of the surface states because of its interface sensitivity. The extension of SP to anisotropic and topological supercon- ductivity represents one of the most attractive directions for further development of superconducting spintronics. Acknowledgments.| This work is partially supportedby the Priority Program of Chinese Academy of Sciences, Grant No. XDB28000000. We acknowledge JSPS KAK- ENHI for Grants (No. JP20H01863, No. JP20K03835, No. JP20K03831, No. JP20H04635, and No.21H04565). Appendix A: Magnon self-energy induced by a rough interface The roughness of the interface is taken into account as an uncorrelated (white noise) distribution of the ex- change couplings35, as shown below. We start with an exchange model in the real space Hex=X jZ d2rJ(r;rj)(r)
Sj =X q;kJq;kq
Sk: (A1) The spin density (r) in the SC and the spin Sjin the FI are represented in the momentum space as (r) =1 AX qeiq
rq; (A2) Sj=1p NX keik
rjSk; (A3) whereAdenotes the area of the system and Nis the number of sites. The exchange coupling constant is also obtained to be Jq;k=1 Ap NX jZ d2rei(q
r+k
rj)J(r;rj): (A4) The exchange model Hexis decomposed into the spin transfer term HTand the eective Zeeman eld term HZ asHex=HT+HZ. Now we consider the roughness eect of the interface. Uncorrelated roughness is expressed by the mean J1and varianceJ22as 1p NX jJ(r;rj) =J1; (A5) 1 NX jj0J(r;rj)J(r0;rj0)J12=J22A2(rr0);(A6) whereOis the congurational average of Oover the roughness. The above expressions lead to the congu- rationally averaged self-energy R k=0(!) =X qjJq;k=0j2R q(!) =J12R q=0(!)J22X qR q(!); (A7) which coincides with the model Eq. (27) in the main text. This model provides a smooth connection between the7 specular (J12R q=0) and diuse ( J22P qR q) limits. The uncorrelated roughness case introduced above is a simplelinear interpolation of the two. Extensions to correlated roughness can be made straightforwardly. 1Y. Tserkovnyak, A. Brataas, and G. E. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 2F. Hellman, A. Homann, Y. Tserkovnyak, G. S. D. Beach, E. E. Fullerton, C. 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2022-01-16

We investigate ferromagnetic resonance (FMR) modulation in $d$-wave superconductor (SC)/ferromagnetic insulator (FI) bilayer systems theoretically. The modulation of the Gilbert damping in these systems reflects the existence of nodes in the $d$-wave SC and shows power-law decay characteristics within the low-temperature and low-frequency limit. Our results indicate the effectiveness of use of spin pumping as a probe technique to determine the symmetry of unconventional SCs with high sensitivity for nanoscale thin films.

Ferromagnetic resonance modulation in $d$-wave superconductor/ferromagnetic insulator bilayer systems

2201.06060v2

1 First and second order magnetic anisotropy and damping of europium iron garnet under high strain Víctor H. Ortiz1, Bassim Arkook1, Junxue Li1, Mohammed Aldosary1, Mason Biggerstaff1, Wei Yuan1, Chad Warren2, Yasuhiro Kodera2, Javier E. Garay2, Igor Barsukov1*, and Jing Shi1* 1Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA 2 Department of Mechanical and Aerospace Engineering, University of California, San Diego, CA 92093, USA Understanding and tailoring static and dynamic properties of magnetic insulator thin films is important for spintronic device applications. Here, we grow atomically flat epitaxial europium iron garnet (EuIG) thin films by pulsed laser deposition on (111) -oriented garnet sub strates with a range of lattice parameters. By controlling the lattice mismatch between EuIG and the substrates, we tune the strain in EuIG films from compressive to tensile regime, which is characterized by X - ray diffraction. Using ferromagnetic resonance , we find that in addition to the first -order perpendicular magnetic anisotropy which depends linearly on the strain, there is a significant second -order one that has a quadratic strain dependence. Inhomogeneous linewidth of the ferromagnetic resonance inc reases notably with increasing strain, while the Gilbert damping parameter remains nearly constant (≈ 2× 10-2). The se results provide valuable insight into the spin dynamics in ferrimagnetic insulators and useful guidance for material synthesis and engineer ing of next -generation spintronics applications. *: Corresponding authors: Igor Barsukov ( igorb@ucr.edu ) and Jing Shi ( jing.shi@ucr.edu ) 2 Ferrimagnetic insulators (FMIs) have played an important role in uncovering a series of novel spintronic effects such as spin Seebeck effect (SSE) and spin Hall magnetoresistance (SMR). In addition, FMI thin films have proved to be an excellent source of proximity -induced ferromagnetism in adjacent layers (e.g., heavy metals [1], graphene [1] and topological insulators [2]) and of pure spin currents [3–6]. FMIs have also been shown to be a superb medium for magnon spin currents with a long decay length [7,8]. Among FMIs, rare earth iron garnets (REIGs) in particular have a plethora of desirable properties for practical applications: high Curie temperature (T c > 550 K), strong chemical stability, and relatively large band gaps (~ 2.8 eV). Compared to other magnetic materials, REIGs are distinct owing to their magnetoelastic effect with the magnetostriction coefficient ranging from -8.5×106 to +21 ×106 at room temperature [9] and up to two orders of magnitude increases at low temperatures [10]. This unique feature allows for tailoring ma gnetic anisotropy in REIG thin films via growth, for example, by means of controlling lattice mismatch with substrates, film thickness, oxygen pressure, and chemical substitution. In thin films, the magnetization usually prefers to be in the film plane due to magnetic shape anisotropy; however, the competing perpendicular magnetic anisotropy (PMA) can be introduced by utilizing magneto -crystalline anisotropy or interfacial strain, both of which have been demonstrated through epitaxial growth [11–14]. In the study of Tb 3Fe5O12 (TbIG) and Eu3Fe5O12 (EuIG) thin films, the PMA field H2ꓕ was found to be as high as 7 T under interfacial strain [11], much stronger than the demagnetizing field. While using strain is proven to be an effective way of manipulating magn etic anisotropy, it often comes at a cost of increasing magnetic inhomogeneity and damping of thin films [15,16]. In this work, we investigate the effect of strain on magnetic properties of (111) -oriented EuIG thin films for the following reasons: (1) The spin dynamics in EuIG bulk crystals is particularly interesting but has not been studied thoroughly in the thin film form. Compared to other REIGs, the Eu3+ ions occupying the dodecahedral sites (c -site) should have the J = 0 ground state according to the Hund’s rules, which do not contribute to the total magnetic moment; therefore, EuIG thin films can potentially have a ferromagnetic resonance (FMR) linewidt h as narrow as that of Y 3Fe5O12 (YIG) [17,18] or Lu 3Fe5O12 (LuIG) [19]. In EuIG crystals, a very narrow linewidth (< 1 Oe) [20] was indeed observed at low temperatures, but it showed a nearly two orders of magnitude increase at high temperatures, which ra ises fundamental questions regarding the damping mechanism responsible for this precipitous change. (2) Although it has 3 been shown that the uniaxial anisotropy can be controlled by moderate strain for different substrate orientations and even in polycrysta lline form [21], the emergence of the higher -order anisotropy at larger strain, despite its technological significance, has remained elusive. We grow EuIG films by pulsed laser deposition (PLD) from a target densified by powders synthesized using the meth od described previously [22]. The films are deposited on (111) -oriented Gd3Sc2Ga3O12 (GSGG), Nd 3Ga5O12 (NGG), Gd 2.6Ca0.4Ga4.1Mg 0.25Zr0.65O12 (SGGG), Y3Sc2Ga3O12 (YSGG), Gd3Ga5O12 (GGG), Tb 3Ga5O12 (TGG) and Y 3Al5O12 (YAG) single crystal substrates, with the lattice mismatch 𝜂=𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 −𝑎𝐸𝑢𝐼𝐺 𝑎𝐸𝑢𝐼𝐺 (where 𝑎 represents the lattice parameter of the referred material) ranging from +0.45% (GSGG) to -3.95% (YAG) in the decreasing order (see Table I). After the standard solvent cleaning process, the substrates are annealed at 220 °C inside the PLD chamber with the base pressure lower than 10-6 Torr for 5 hours prior to deposition . Then the temperature is increased to ~ 600 °C in the atmosphere of 1.5 mT orr oxygen mixed with 12% (wt.) ozone for 30 minutes. A 248 nm KrF excimer pulsed laser is used to ablate the target with a power of 156 mJ and a repetition rate of 1 Hz. We crystalize the films by ex situ annealing at 800 °C for 200 s in a steady flow of oxygen using rapid thermal annealing (RTA) . Reflection high energy electron diffraction (RHEED) is used to evaluate the crystalline structural properties of the EuIG films grown on various substrates (Fig. 1a). Immediately after the deposition, RHEED dis plays the absence of any crystalline order. After ex situ rapid thermal annealing, all EuIG films turn into single crystals. We carry out atomic force microscopy (AFM) on all samples and find that they show atomic flatness and good uniformity with root -mean-square (RMS) roughness < 2 Å (Fig. 1b). In addition, we perform X -ray diffraction (XRD) on all samples using a Rigaku SmartLab with Cu K α radiation with a Ni filter and Ge(220) mirror as monochromators, at room temperature in 0.002° steps over the 2 range from 10° to 90° [23]. In a representative XRD spectrum (Fig. 1c), two (444) Bragg peaks are present, one from the 50 nm thick EuIG film and the other from the YSGG substrate, which confirms the epitaxial growth and single crystal structure of the fi lm without evidence of any secondary phases. Other REIG films grown under similar conditions , i.e., by PLD in oxygen mixed with ozone at ~600 °C during followed by RTA, have shown no observable interdiffusion across the interface from high resolution trans mission electron microscopy and energy dispersive X -ray spectroscopy (Fig. S1 , [24]). The EuIG Bragg peak ( a0 = 12.497 Å) is shifted with respect to the expected peak position of unstrained bulk crystal, indicating a change in the EuIG lattice parameter pe rpendicular to the 4 surface ( aꓕ). For the example shown in Fig. 1c, the EuIG (444) peak shifts to left with respect to its bulk value, indicating an out -of-plane tensile strain and therefore an in -plane compressive strain in the EuIG lattice. A common app roach for inferring the in -plane strain ε|| of thin films from the standard −2 XRD measurements involves the following equation [23], 𝜀∥= −𝑐11+2 𝑐12+4 𝑐44 2𝑐11+4 𝑐12−4 𝑐44 𝜀⊥, with 𝜀⊥=𝑎⊥−𝑎𝑜 𝑎𝑜, (1) where a0 is the lattice parameter of the bulk material, and aꓕ can be calculated using 𝑎⊥= 𝑑ℎ𝑘𝑙√ℎ2+𝑘2+𝑙2 from the interplanar distance 𝑑ℎ𝑘𝑙 obtained from the XRD data (Fig. S 2, [25]), and cij are the elastic stiffness constants of the crystal which in most cases can be found in the literature [9]. However, due to the wide range of strain values studied in this work and the possibility that the films may contain different amounts of crystalline defects, we perform reciprocal space mapping (RSM) measurements on a subset of our EuIG samples (Fig. S 3, [26]) and compared the measured in -plane lattice parameters with the calculated ones using Eq. 1. We observe that the average in -plane strain s measured by RSM has a systematic difference of 40% from the calculated values based on the elastic properties (Fig. S 4, [26]). Given this nearly constant factor for all measured films, we find that the elastic stiffness constants of our EuIG films may deviate from the literature reported bulk values , possibly due to stochiometric deviations or slight unit cell distortion in thin films . Here we adopt the reported lattice parameter value ( a0 = 12.497 Å) as the reference due to the difficulty of grow ing sufficiently thick, unstrained EuIG films usin g PLD . In the thickness -tuned magnetic anisotropy study [11], the anisotropy field in REIG films is found to be proportional to η/(t+t o), which was attributed to the relaxation of strain as the film thickness t increases. Here in EuIG samples with small lattice mismatch η (e.g., NGG/EuIG), the strain is mostly preserved in 50 nm thick films (pseudomorphic regime), whereas for larger η (e.g., YAG/EuIG ), the lattice parameter of EuIG films shows nearly complete structural relaxation to the bulk value. For this reason, in the samples with larger η (YAG = -3.95 %, GSGG = 0.45%), we grow thinner EuIG films (20 nm) in order to retain a larger in -plane strain (compressive for 5 YAG, tensile for GSGG). For EuIG films gr own on TGG and GGG substrates, the paramagnetic background of the substrates is too large to obtain a reliable magnetic moment measurement of the EuIG films; therefore, the results of thinner films on these two substrates are not included in this study. Room-temperature magnetic hysteresis curves for YSGG/EuIG sample are shown in Fig. 1d with the magnetic field applied parallel and perpendicular to the film [26]. The saturation field for the out -of-plane loop (~1100 Oe) is clearly larger than that for the i n-plane loop, indicating that the magnetization prefers to lie in the film plane. Moreover, since the demagnetizing field 4π Ms (≈ 920 Oe) is less than the saturation field in the out -of-plane loop (Fig. S 5, [27]), it suggests the presence of additional easy -plane anisotropy result ing from the magnetoelastic effect due to interfacial strain. As shown in this example, we can qualitatively track the evolution of the magnetic anisotropy in samples with different strains. However, this approach cannot provide a quantitative description when high -order anisotropy contributions are involved. To quantitatively determine magnetic anisotropy in all EuIG films, we perform polar angle (H)-dependent FMR measurements using an X -band microwave cavity with f requency f = 9.32 GHz and field modulation. The samples are rotated from H = 0° to H = 180° in 10° steps, where H = 90° corresponds to the field parallel to the sample plane (Fig. 2a). The spectra at Η = 0° for all samples are displayed in Fig. 2b and show a single resonance peak which can be well fitted by a Lorentzian derivative. Despite different strains in all s amples, the resonance field Hres is lower for the in -plane direction ( H = 90°) than for the out -of-plane direction ( H = 0°). A quick inspection reveals that the out -of-plane Hres shifts to larger values as η increases in the positive direction (e.g., fro m YAG/EuIG to GSGG/EuIG), corresponding to stronger easy -plane anisotropy. Furthermore, the Hres values at θΗ = 0° show a large spread among the samples. Fig. 2c shows a comparison of FMR spectra at different polar angles between two representative samples: NGG/EuIG (small η) and YAG/EuIG (large η). Figs. 3a -c show Hres vs. θH for three representative EuIG films . To evaluate magnetic anisotropy, we fit the data using the Smit -Beljers formalism by considering the first -order −𝐾1cos2𝜃 and the second -order −1 2𝐾2cos4𝜃 uniaxial anisotropy energy terms [28]. From this fitting, we extract the parameters 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠− 2𝐾1 𝑀𝑠= 4𝜋𝑀𝑠- 𝐻2⊥ and 𝐻4⊥= 2𝐾2 𝑀𝑠 (see Table I), her e 𝐻2⊥ and 𝐻4⊥ being the first- and second - order anisotropy fields, respectively , and favoring 6 out-of-plane (in -plane) orientation of magnetization when they are positive (negative). The spectroscopic g-factor is treated as a fitted parameter which is found as a nearly constant , g = 1.40 (Fig. S 6, [28]), in accordance to the previous results obtained by Miyadai [31]. In Figs. 3d and 3e, we present 𝐻2⊥ and 𝐻4⊥ as functions of the measured out -of-plane strain 𝜀⊥ and in -plane strain 𝜀∥. Clear ly, the magnitude of 4𝜋𝑀𝑒𝑓𝑓 is greater than the demagnetizing field for EuIG 4𝜋𝑀𝑠=920 𝑂𝑒; therefore, 𝐻2⊥ is negative for all samples, i.e., favoring the in -plane orientation. As shown in Fig. 3d, |𝐻2⊥| increases linearly with increasing in -plane strain η. This is consistent with the magnetoelastic effect in (111) -oriented EuIG films [9]. As briefly discussed earlier, due to the constant scaling factor between the calculated and measured 𝜀∥, we rewrite t he magnetoelastic contribution to the first -order perpendicular anisotropy as −9𝛯 3𝑀𝑠𝜀⊥, with the parameter 𝛯 containing the information related to the magnetoelastic constant λ111 and elastic stiffness cii. We fit the magnetoelastic equation in Ref. [11] using the parameter 𝛯 and obtain 𝛯= −(7.06±0.95)×104 𝑑𝑦𝑛𝑒 𝑐𝑚2 from the slope. On the other hand, based on the reported literature values ( 𝜆111=+1.8×10−6, c11 = 25.10 ×1011 dyne/cm2, c12 = 10.70 ×1011 𝑑𝑦𝑛𝑒 𝑐𝑚2, c44 = 7.62 ×1011 𝑑𝑦𝑛𝑒 𝑐𝑚2) [10], we obtain 𝛯𝑙𝑖𝑡=−6.12×104 𝑑𝑦𝑛𝑒 𝑐𝑚2. This result suggests that even though the actual elastic properties of our EuIG films may be different from the ones reported in for EuIG crystals due to the thin film unit cell distortion (Table S1 , [32]), the pertaining parameter 𝛯 appears to be relatively insensitive to variations of stoichiometry . The intercept of the straight -line fit should give the magneto -crystalline anisotropy coefficient of EuIG Kc. We find Kc = (+62.76 ± 0.18 ) × 103 erg/cm3, which is differ ent from the previously reported values for EuIG bulk crystals in both the magnitude and sign ( Kc = -38 × 103 erg/cm3) [31]. Similar growth -modified magneto - crystalline anisotropy was observed in EuIG films grown with relatively lo w temperatures (requiring post -deposition annealing to crystalize) [10]. In the absence of interfacial interdiffusion, the anomalous anisotropy may be related to partial deviation from the chemical ordering of the garnet structure [31]. By comparing the first - and second -order anisotropy fields 𝐻2⊥ and 𝐻4⊥ vs. 𝜀∥ plotted in Figs. 3d and 3e, we find that the former dominates over the entire range of 𝜀∥ (except for YAG/EuIG). In contrast to the linear dependence for 𝐻2⊥, 𝐻4⊥ can be fitted well with a quadratic 𝜀∥ dependence , which is not surprising for materials with large magnetostriction constants (such 7 as EuIG) under large strains. For relatively small 𝜀∥, the linear strain term in the magnetic anisotropy energy dictates . For large 𝜀∥, higher -order strain terms may not be neglected. By including the ( 𝜀∥cos2θ)2 term, we obtain excellent fitting to the FMR data, indicating that the second -order expansion in 𝜀∥ is adequate. In contrast to 𝐻2⊥, 𝐻4⊥ is always positive, thus favoring out-of-plane magnetization orientation. It is worth pointing out that for YAG and TGG, the magnitude of the 𝐻2⊥ becomes comparable with that of the 𝐻4⊥, but the sign differ s. Comparison of 𝐻4⊥ with 4𝜋𝑀𝑒𝑓𝑓 reveals that a coexistence (bi -stable) magnetic state can be realized when 𝐻4⊥>4𝜋𝑀𝑒𝑓𝑓 [31, 33 -35]. The results are summarized in Table I. The above magnetic anisotropy energy analysis only deals with the polar angle dependence , but in principle, it can also vary in the film plane and therefore depend on the azimuthal angle. To understand the latter, w e perform azimuthal angle dependent FMR measurements on all samples. We indeed observe a six -fold in -plane anisotropy in Hres due to the crystalline symmetry of EuIG (111). However, the amplitude of the six -fold Hres variation is less than 15 Oe, about two orders of magnitude smaller than the average value of Hres for most samples, thus we omit the in-plane anisotropy in our analysis. Besides the Hres information, t he FMR spectra in Fig. 2 c reveal s significant variations in FMR linewidth, which contains information of magnetic inhomogeneity and Gilbert damping. To investigate these properties systematically, we perform broad -band (up to 15 GHz) FMR measurements with m agnetic field applied in the film plane, using a coplanar waveguide setup. From the frequency dependence of Hres, we obtain 4𝜋𝑀𝑒𝑓𝑓 and g independently via fitting the data with the Kittel equation. These values agree very well with those previously found from the polar angle dependence. We plot the half width at half maximum, ∆𝐻, as a function of frequency f in Fig. 4a. While ∆𝐻 varies significantly across the samples, the data for each sample fall approximately on a straight line and the slope of ∆𝐻 vs. 𝑓 appears to be visibly close to each other. For a quantitative evaluation of ∆𝐻, we consider the following contributions: the Gilbert damping ∆𝐻𝐺𝑖𝑙𝑏𝑒𝑟𝑡 , two -magnon scatt ering ∆𝐻𝑇𝑀𝑆, and the inhomogeneous linewidth ∆𝐻0 [36], ∆𝐻=∆𝐻𝐺𝑖𝑙𝑏𝑒𝑟𝑡 +∆𝐻𝑇𝑀𝑆 +∆𝐻0 . (3) 8 The Gilbert term, ∆𝐻𝐺𝑖𝑙𝑏𝑒𝑟𝑡 =2𝜋𝛼𝑓 |𝛾|, depends linearly on f, where α is the Gilbert damping parameter; the two -magnon term is described through ∆𝐻𝑇𝑀𝑆 =𝛤0𝑎𝑟𝑐𝑠𝑖𝑛 √√𝑓2+(𝑓𝑜 2)2 −𝑓𝑜 2 √𝑓2+(𝑓𝑜 2)2 +𝑓𝑜 2 [37], where 𝛤0 denotes the magnitude of the two -magnon scattering, f0 = 2γMeff; and ∆𝐻0, the inhomogeneous linewidth w hich is frequency independent. By fitting Eq. (3) to the linewidth data, we obtain quantitative information on magnetic damping through the Gilbert parameter and two -magnon scattering magnitude as well as the magnetic inhomogeneity [39–40]. In Fig. 4a, the overall linear behavior for all samples is an indication of a relatively small two -magnon scattering contribution ∆𝐻𝑇𝑀𝑆 which therefore may be disregarded in the fitting process. Figs. 4b and 4c show both ∆𝐻0 and α vs. 𝜀∥. It is cl ear that four of the samples with the smallest ∆𝐻0 (~ 10 Oe) are those with relatively low in -plane strain (|𝜀∥|<0.30% ). In the meantime, the XRD spectra of these samples show fringes characteristic of well conformed crystal planes (Fig. S 2), and moreover, the RSM plots (Fig. S 3) reveal a uniform strain distribution in the films [41]. On the compressive strain side, ∆𝐻0 increases steeply to 400 Oe at 𝜀∥ ~ -0.40 %, and their XRD spectra show no fringes and the RSM graphs indicate non- uniform strain relaxation in the samples (Figs. S2 and S3 ). In sharp contrast to the ∆𝐻0 trend, the Gilbert damping α remains about 2 ×10-2 over the entire range of 𝜀∥, sugges ting that the intrinsic magneti c damping of EuIG films is nearly unaffected by the inhomogeneity. In fact, the magnitude of α is significantly larger than that of YIG [17,18] or LuIG films [19], which is somewhat unexpected for Eu3+ in EuIG with J = 0. A possible reason for this enhance d damping is that other valence states of Eu such as Eu2+ (J =7/2) may be present, which leads to non -zero magnetic moments of Eu ions in the EuIG lattice and thus results in a larger damping constant, common to other REIG with non -zero 4f -moments [42]. The X -ray photoelectron spectroscopy data taken on YSGG(111)/EuIG(50 nm) (Fig. S7 , [43]) indicates such a possibility. While the FMR linewidth presents large variations across the sample set, we have identified that the non-uniform strain relaxation process caused by large lattice mismatch with the substrate is a main source of the inhomogeneity linewidth ∆𝐻0, but it does not affect the Gilbert damp ing α. The results raise interesting questions on the mechanisms of intrinsic damping and the origin of magnetic inhomogeneity in EuIG thin films , both of which warrant further investigations. 9 In summary, we find that uniaxial magnetic anisotropy in PLD -grown EuIG(111) thin films can be tuned over a wide range via magnetostriction and lattice -mismatch induced strain. The first - order anisotropy field depends linearly on the strain and the second order anisotropy field has a quadratic dependence. While non -uniform strain relaxation significantly increases the magnetic inhomogeneity, the Gilbert damping remains nearly constant over a wide range of in -plane strain. The results demonstrate broad tunab ility of magnetic properties in REIG films and provide guidance for implementation of EuIG for spintronic applications. Further studies to elucidate the role of Eu2+ sites in magnetic damping are called upon. We thank Dong Yan and Daniel Borchardt for the ir technical assistance. This work was supported as part of the SHINES, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences under Award No. SC0012670. J.S. acknowledges support by DOE BES Award No. DE -FG02 -07ER46351 and I.B. acknowledges support by the National Science Foundation under grant number NSF -ECCS -1810541. 10 References [1] Z. Wang, C. Tang, R. Sachs, Y. Barlas, and J. Shi, Proximity -Induced Ferromagnetism in Graphene Revealed by the Anomalous Hall Effect , Phys. Rev. Lett. 114, 016603 (2015). [2] Z. Jiang, C. -Z. Chang, C. Tang, P. Wei, J. S. Moodera, and J. Shi, Independent Tuning of Electronic Properties and Induced Ferromagnetism in Topological Insulators with Heterostructure Approach , Nano Lett. 15, 5835 (2015). [3] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Spin Seebeck Insulator , Nat. Mater. 9, 894 (2010). [4] Y. 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Garay, An Integrated Approach to Doped Thin Films with Strain Tunable Magnetic Anisotropy: Powder Synthesis, Target Prepara tion and Pulsed Laser Deposition of Bi:YIG , 3831 , 1 (2016). [23] E. Anastassakis, Strained Superlattices and Heterostructures: Elastic Considerations , J. Appl. Phys. 68, 4561 (1990). [24] See Supplemental Information figure S1 at http://placeholder.html fo r the HRTEM image and EDS mapping. [25] See Supplemental Information figure S2 at http://placeholder.html for the θ−2θ HRXRD of the samples. [26] See Supplemental Information figure S3 and S 4 at http://placeholder.html for the XRD and strain analysis of the samples. [27] See Supplemental Information figure S 5 at http://placeholder.html for the M vs H hysteresis loops of the samples. [28] I. Barsukov, Y. Fu, A. M. Gonçalves, M. Spasova, M. Farle , L. C. Sampaio, R. E. Arias, and I. N. Krivorotov, Field -Dependent Perpendicular Magnetic Anisotropy in CoFeB Thin Films , Appl. Phys. Lett. 105, 152403 (2014). [29] See Supplemental Information figure S6 at http://placeholder.html for the g -factor compari son of the samples. [30] T. Miyadai, Ferrimagnetic Resonance in Europium -Iron Garnet , J. Phys. Soc. Japan 15, 2205 (1960). [31] F. B. Hagedorn, Annealing Behavior and Temperature Dependence of the Growth - Induced Magnetic Anisotropy in Epitaxial Sm -YIGG , J. Appl. Phys. 45, 3123 (1974). [32] See Supplemental Information Table S1 at http://placeholder.html for the calculated parameters of the rhombohedral distorted unit cell . [33] Y. Fu, I. Barsukov, J. Li, A. M. Gonçalves, C. C. Kuo, M. Farle, and I. N. Krivorotov, Temperature Dependence of Perpendicular Magnetic Anisotropy in CoFeB Thin Films , Appl. Phys. Lett. 108, (2016). [34] R. Skomski, H. P. Oepen, and J. Kirschner, Unidirect ional Anisotropy in Ultrathin Transition - Metal Films , Phys. Rev. B 58, 138 (1998). [35] J. M. Shaw, H. T. Nembach, M. Weiler, T. J. Silva, M. Schoen, J. Z. Sun, and D. C. Worledge, Perpendicular Magnetic Anisotropy and Easy Cone State in 13 Ta/Co 60Fe20B20/MgO, IEEE Magn. Lett. 6, 1 (2015). [36] I. Barsukov, P. Landeros, R. Meckenstock, J. Lindner, D. Spoddig, Z. A. Li, B. Krumme, H. Wende, D. L. Mills, and M. Farle, Tuning Magnetic Relaxation by Oblique Deposition , Phys. Rev. B - Condens . Matter Mater. Phys. 85, 1 (2012). [37] J. Lindner, K. Lenz, E. Kosubek, K. Baberschke, D. Spoddig, R. Meckenstock, J. Pelzl, Z. Frait, and L. Mills, Non-Gilbert -Type Damping of the Magnetic Relaxation in Ultrathin Ferromagnets: Importance of Magnon -Magno n Scattering , Phys. Rev. B - Condens. Matter Mater. Phys. 68, 6 (2003). [38] A. Navabi, Y. Liu, P. Upadhyaya, K. Murata, F. Ebrahimi, G. Yu, B. Ma, Y. Rao, M. Yazdani, M. Montazeri, L. Pan, I. N. Krivorotov, I. Barsukov, Q. Yang, P. Khalili Amiri, Y. Tserk ovnyak, and K. L. Wang, Control of Spin -Wave Damping in YIG Using Spin Currents from Topological Insulators , Phys. Rev. Appl. 11, 1 (2019). [39] A. Etesamirad, R. Rodriguez, J. Bocanegra, R. Verba, J. Katine, I. N. Krivorotov, V. Tyberkevych, B. Ivanov, an d I. Barsukov, Controlling Magnon Interaction by a Nanoscale Switch , ACS Appl. Mater. Interfaces 13, 20288 (2021). [40] I. Barsukov, H. K. Lee, A. A. Jara, Y. J. Chen, A. M. Gonçalves, C. Sha, J. A. Katine, R. E. Arias, B. A. Ivanov, and I. N. Krivorotov, Giant Nonlinear Damping in Nanoscale Ferromagnets , Sci. Adv. 5, 1 (2019). [41] X. Guo, A. H. Tavakoli, S. Sutton, R. K. Kukkadapu, L. Qi, A. Lanzirotti, M. Newville, M. Asta, and A. Navrotsky, Cerium Substitution in Yttrium Iron Garnet: Valence State, Structure, and Energetics , (2013). [42] C. Tang, P. Sellappan, Y. Liu, Y. Xu, J. E. Garay, and J. Shi, Anomalous Hall Hysteresis in Tm 3Fe5O12/Pt with Strain -Induced Perpendicular Magnetic Anisotropy , Phys. Rev. B 94, 140403 (2016). [43] See Supplemental Information figure S7 at http://placeholder.html for the XPS spectra and analysis. 14 Figures Figure SEQ Figure \* ARABIC 1 : Structural and magnetic property characterization of EuIG 50 nm film grown on YSGG(111) substrate. (a) Reflection high energy electron diffraction (RHEED) pattern along the direction, displaying single crystal structure after rapid thermal anneal ing process. (b) 2 mm 2 mm atomic force microscope (AFM) surface morphology scan, demonstrating a root -mean -square (RMS) roughness of 1.7 Å. (c) Intensity semi -log plot of - 2 XRD scan. The dashed line corresponds to the XRD peak for bulk EuIG. (d) Mag netization hysteresis loops for field out -of-plane and in -plane directions. Figure 1: Structural and magnetic property characterization of EuIG 50 nm film grown on TGG(111) substrate . (a) Reflection high energy electron diffraction (RHEED ) pattern along the ⟨112⟩ direction, displaying single crystal structure after rapid thermal annealing process. (b) 5 mm 5 mm atomic force microscope ( AFM ) surface morphology scan, demonstrating a root-mean - square (RMS) roughness of 1.8 Å. (c) Intensity semi -log plot of - 2 XRD scan. The dashed line corresponds to the XRD peak for bulk EuIG. (d) Magnetization hysteresis loops for field out -of- plane and in -plane directions. 15 Figure 2 Polar angle dependent ferromagnetic resonance (FMR). (a) Coordinate system used for the FMR measurement. (b) Room temperature FMR derivative absorption spectra for θH = 0° (out - of-plane configuration) for EuIG on different (111) substrates. (c) FMR derivative absorption spectra for 50 nm EuIG grown on NGG(111) ( 𝜀∥ ≈ 0) and 20 nm EuIG on YAG(111) (𝜀∥< 0) with polar angle θH ranging from 0° (out -of-plane) to 90° ( in-plane) at 300 K, where 𝜀∥ is in-plane strain between the EuIG film and substrate. 16 Figure 3 Polar angle dependent ferromagnetic resonance field Hres for (a) tensile in -plane strain (𝜀∥ > 0), (b) in -plane strain close to zero ( 𝜀∥ ≈ 0), and (c) compressive in -plane strain ( 𝜀∥ < 0). Solid curves represent the best fitting results. In -plane strain dependence of the anisotropy fields H 2ꓕ (d) and H 4ꓕ (e). 17 Figure 4 FMR linewidth and magnetic damping of EuIG films as a function of in -plane strain. (a) Half width at half maximum ∆𝐻 vs. frequency f for EuIG films grown on different substrates, with the corresponding fitting according to Eq. (3). In -plane strain depen dence of inhomogeneous linewidth ΔH0 (b) and Gilbert parameter α (c). 18 Substrate asubstrate (Å) η (%) t (nm) 𝜀∥ (%) 𝜀⊥ (%) g H2ꓕ (Oe) H4ꓕ (Oe) α (×10-2) ΔHo (Oe) Γo (Oe) GSGG 12.554 0.45 50 0.34 -0.16 1.40 -1394.2 ± 44.9 339.79 ± 6.59 2.46 ± 0.03 21.4 ± 1.3 2.61 25 0.46 -0.21 1.41 -1543.6 ± 39.7 709.47 ± 27.5 1.58 ± 0.06 10.2 ± 1.7 6.05 NGG 12.508 0.06 50 0.12 -0.06 1.38 -1224.4 ± 5.7 18.34 ± 0.05 2.41 ±0.01 8.9 ± 0.7 0.20 SGGG 12.480 - 0.14 50 -0.13 0.06 1.40 -909.6 ± 15.2 164.8 ± 1.36 2.13 ± 0.01 5.6 ± 0.4 0.50 YSGG 12.426 - 0.57 50 -0.27 0.12 1.37 -709.4 ± 22.0 377.3 ± 5.09 2.47 ± 0.03 9.9 ± 1.8 2.47 GGG 12.383 - 0.92 50 -0.45 0.21 1.38 -1015.0 ± 81.3 887.2 ± 37.27 2.20 ± 0.14 412.2 ± 8.4 3.35 TGG 12.355 - 1.14 50 -0.38 0.18 1.38 -393.4 ± 53.6 245.0 ± 10.00 2.29 ± 0.20 253.4 ± 11.8 0.20 YAG 12.004 - 3.95 20 -0.42 0.20 1.37 -36.8 ± 47.1 424.8 ± 20.91 1.86 ± 0.20 217.0 ± 22.6 0.20 Table 1 Structural and magnetic parameters for the EuIG thin films grown on different substrates.

2021-11-30

Understanding and tailoring static and dynamic properties of magnetic insulator thin films is important for spintronic device applications. Here, we grow atomically flat epitaxial europium iron garnet (EuIG) thin films by pulsed laser deposition on (111)-oriented garnet substrates with a range of lattice parameters. By controlling the lattice mismatch between EuIG and the substrates, we tune the strain in EuIG films from compressive to tensile regime, which is characterized by X-ray diffraction. Using ferromagnetic resonance, we find that in addition to the first-order perpendicular magnetic anisotropy which depends linearly on the strain, there is a significant second-order one that has a quadratic strain dependence. Inhomogeneous linewidth of the ferromagnetic resonance increases notably with increasing strain, while the Gilbert damping parameter remains nearly constant (~ 2x10^-2). These results provide valuable insight into the spin dynamics in ferrimagnetic insulators and useful guidance for material synthesis and engineering of next-generation spintronics applications.

First and second order magnetic anisotropy and damping of europium iron garnet under high strain

2111.15142v1

1Nonlinear magnetization dyna mics of antiferromagnetic spin resonance induced by inte nse terahertz magnetic field Y Mukai1,2,4,6, H Hirori2,3,4,7, T Yamamoto5, H Kageyama2,5, and K Tanaka1,2,4,8  1 Department of Physics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan 2 Institute for Integrated Cell-Material Scien ces (WPI-iCeMS), Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan 3 PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan 4 CREST, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan 5 Department of Energy and Hydrocarbon Chemistr y, Graduate School of Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan E-mail: 6 mukai@scphys.kyoto-u.ac.jp 7 hirori@icems.kyoto-u.ac.jp 8 kochan@scphys.kyoto-u.ac.jp We report on the nonlinear magnetization dynamics of a HoFeO 3 crystal induced by a strong terahertz magnetic field resonantly enhanced with a split ring resonator and measured with magneto-optical Kerr effect microscopy. The terahertz magnetic field induces a large change (~40%) in the spontaneous magnetization. The frequency of the antife rromagnetic resonance decreases in proportion to the square of the magnetization change. A modified Landau-Lifshitz-Gilbert equation with a phenomenological nonlinear damping term quantitatively reproduced the nonlinear dynamics. PACS: 75.78.Jp, 76.50.+g, 78.47.-p, 78.67.Pt 21. Introduction Ultrafast control of magnetization dynamics by a femtosecond optical laser pulse has attracted considerable attention from the persp ective of fundamental physics and technological applications of magnetic recording and inform ation processing [1]. The first observation of subpicosecond demagnetization of a fe rromagnetic nickel film demonstrated that a femtosecond laser pulse is a powerful stimulus of ultrafast magnetization dynamics [2], and it has led to numerous theoretical and experimental inves tigations on metallic and semiconducting magnets [3-8]. The electronic state created by the laser pulse has a strongly nonequilibrium distribution of free electrons, which consequently leads to demagnetization or even magnetic reversal [1,2,9-11]. However, the speed of the magnetizat ion change is limited by the slow thermal relaxation and diffusion, and an alternative t echnique without the limits of such a thermal control and without excessive thermal energy would be desirable. In dielectric magnetic media, carrier heating hardly occurs, since no free electrons are present [12]. Consequently, great effort has been devoted to clarifying the spin dynamics in magnetic dielectrics by means of femtosecond laser pulses. A typical method for nonthermal optical control of magnetism is the inverse Faraday effect, where circularly polarized intense laser irradiation induces an effective magnetic field in the medium. Recently, new optical excitation methods avoiding the thermal effect such as the ma gneto-acoustic effect is also reported [13,14]. In particular, these techniques have been used in many studies on antiferromagnetic dielectrics because compared with ferromagne ts, antiferromagnets have inhe rently higher spin precessional frequencies that extend into the terahertz (THz) regime [12,15]. Additionally, ultrafast manipulation of the antiferromagnetic order parame ter may be exploited in order to control the magnetization of an adjacent ferromagnet through the exchange interaction [16]. The THz wave generation technique is possibly a new way of optical spin control through direct magnetic excitation without undesirable thermal effects [17-19]. As yet however, no technique has been successful in driving magnetic motion excited directly by a magnetic field into a nonlinear dynamics regime that would presumably be fo llowed by a magnetization reversal [20-22]. In our previous work [23], we demonstrated that the THz magnetic field can be resonantly enhanced with a split ring resonator (SRR) and may become a tool for the efficient excitation of a magnetic resonance mode of antiferromagnetic dielectric HoFeO 3. We applied a Faraday rotation technique to detect the magnetization ch ange but the observed Faraday signal averaged 3the information about inhomogeneous magnetiza tion induced by localized THz magnetic field of the SRR over the sample thickness [23]. In th is Letter, we have developed a time-resolved magneto-optical Kerr effect (MOKE) micr oscopy in order to access the extremely field-enhanced region, sample surface near th e SRR structure. As a result, the magnetic response deviates from the linear response in the strong THz magnetic field regime, remarkably showing a redshift of the antiferromagnetic r esonance frequency that is proportional to the square of the magnetization change. The observe d nonlinear dynamics could be reproduced with a modified Landau-Lifshitz-Gilbert (LLG) e quation having an additional phenomenological nonlinear damping term. 2. Experimental setup Figure 1 shows the experimental setup of MOKE microscopy with a THz pump pulse excitation. Intense single-cycle THz pulses were generated by optical rectification of near-infrared (NIR) pulses in a LiNbO 3 crystal [24-26]; the maximum peak electric field was 610 kV/cm at focus. The sample was a HoFeO 3 single crystal polished to a thickness of 145 µm, with a c-cut surface in the Pbnm setting [27]. (The x-, y-, and z-axes are parallel to the crystallographic a-, b-, and c-axes, respectively. ) Before the THz pump excitation, we applied a DC magnetic field to the sample to saturate its magnetization along the crystallographic c-axis. We fabricated an array of SRRs on the crystal surface by using gold with a thickness of 250 nm. The incident THz electric field, parallel to the metallic arm with the SRR gap (the x-axis), drove a circulating current that resulted in a strong magnetic near-field normal to the crystal surface [23,28,29]. The SRR is essentially subwavelength LC circuit, and the current induces magnetic field B nr oscillating with the LC resonance frequency (the Q-factor is around 4). The right side of the inset in figure 1 shows the spatial distribut ion of the magnetic field of the SRR at the LC resonance frequency as calculated by the fin ite-difference time-domain (FDTD) method. Around the corner the current density in the metal is very high, inducing the extremely enhanced magnetic field in the HoFeO 3 [29]. At room temperature, the two magnetizations mi (i=1,2) of the different iron sublattices in HoFeO 3 are almost antiferromagnetically aligned along the x-axis with a slight canting angle 0(=0.63°) owing to the Dzyaloshinskii fiel d and form a spontaneous magnetization MS along the z-axis [30]. In the THz region, ther e are two antiferromagnetic resonance modes (quasiantiferromagnetic (AF) and quasiferroma gnetic (F) mode [31]). The magnetic field Bnr 4generated along the z-axis in our setup causes AF-mode motion; as illustrated in figure 2(a), the Zeeman torque pulls the spins along the y-ax is, thereby triggering precessional motions of mi about the equilibrium directions. The precessional motions cause the macroscopic magnetization M=m 1+m2 to oscillate in the z-direction [32,33]. The resultant magnetization change Mz(t) modulates the anti-symmetric off-diagonal element of the dielectric tensor εxyaሺൌ െεyxaሻ and induces a MOKE signal (Kerr ellipticity change  [34,35] (see Appendix A for the detection scheme of the MOKE measuremen t). The F-mode oscillation is also excited by THz magnetic field along the x or y-axis. Howe ver, the magnetization deviations associated with the F-mode, Mx and My, do not contribute to the MOKE in our experimental geometry, where the probe light was incident no rmal to the c-cut surface of HoFeO 3 (the xy-plane) [34,35]. In addition, the amplitude of the F-mode is much smaller than AF-mode because the F-mode resonance frequency ( F~0.37 THz) differs from the LC resonance frequency ( LC~0.56 THz). -10010x position (µm) -10 0 10 y position (µm) THz pump HoFeO3 z y x SRR Objective lens Nonpolarized beam splitter Quarter wave plateWollaston prism Lens Balanced photodiodes Visible probe Bin Bnr Ein -10 0 10 y position (µm) 120 80 40 0 Figure 1. Schematic setup of THz pump-visible MOKE measurement. The left side of the inset shows the photograph of SRR fabricated on the c-cut surface of the HoFeO 3 crystal and the white solid line indicat es the edge of the SRR. The red soli d and blue dashed circles indicate the probe spots for the MOKE measurement. The right side of the inset shows the spatia l distribution of the enhancement facto r calculated by the FDTD method, i.e., the ratio between the Fourier amplitude at LC of the z-component of Bnr (at z=0) and the incident THz pulse Bin. 5To detect the magnetization change induced onl y by the enhanced magnetic field, the MOKE signal just around the corner of the SRR (indicated by the red circle in figure 1’s inset), where the magnetic field is enhanced 50-fold at the LC resonance frequency, was measured with a 400 nm probe pulse focused by an objective lens (spot diameter of ~1.5 µm). Furthermore, although the magnetic field reaches a maximum at the surface and decreases along the z-axis with a decay length of lTHz~5 µm, the MOKE measurement in refl ection geometry, in contrast to the Faraday measurement in transmission [23], can evaluate the magnetization change induced only by the enhanced magnetic field around the sample surface since the penetration depth of 400 nm probe light for typical orthoferrites is on the orde r of tens of nm [35]. (The optical refractive indices of rare-earth orthoferrites in th e near ultraviolet region including HoFeO 3 are similar to each other, regardless of the rare-earth ion speci es, because it is mostly determined by the strong optical absorption due to charge transfer and orbital promotion transitions inside the FeO 6 tetragonal cluster [35].) All experiments in this study were performed at room temperature. 3. Results and discussions Figure 2(a) (upper panel) shows the calculated temporal magnetic waveform together with the incident magnetic field. The maximum peak am plitude is four times that of the incident THz pulse in the time domain and reaches 0.91 T. Th e magnetic field continues to ring until around 25 ps after the incident pulse has decayed away. The spectrum of the pulse shown in figure 2(c) has a peak at the LC resonance frequency ( LC=0.56 THz) of the SRR, which is designed to coincide with the resonance frequency of the AF-mode ( νAF0=0.575 THz). Figure 2(a) (lower panel) shows the time development of the MOKE signal for the highest THz excitation intensity (pump fluence I of 292 µJ/cm2 and maximum peak magnetic field Bmax of 0.91 T). The temporal evolution of is similar to that of the Faraday rotation measured in the previous study and the magnetization oscillates harmonically with a period of ~2 ps [23], implying that the THz magnetic field coherently drives the AF-mode motion. As shown in figure 2(b), as th e incident pump pulse intensity increases, the oscillation period becomes longer. The Fourier transform spectra of the MOKE signals for different pump intensities are plotted in figure 2(c). As the ex citation intensity increases, the spectrum becomes asymmetrically broadened on the lower freque ncy side and its peak frequency becomes redshifted. Figure 2(d) plots the center-of-mass fre quency (open circles) and the integral (closed circles) of the power spectrum P( ) as a function of incident pulse fluence. The center 6frequency monotonically redshifts and P() begins to saturate. As shown in figure 2(c), the MOKE spectra obtained at the center of the SRR (indicated in the inset of figure 1) does not show a redshift even for the highest intensity excitation, suggesting that the observed redshift originates from the nonlinearity of the precessional spin motion rather than that of the SRR response. We took the analytic signal approach (ASA) to obtain the time development of the instantaneous frequency (t) (figure 3(c)) and the envelope amplitude 0(t) (figure 3(d)) from the measured magnetization change (t)=Mz(t)/|MS| (figure 3(b)) (see Appendix B for the details of the analytic signal approach). As is described in the Appendix C, the MOKE signal 6 4 2 0 Integral of P( ) (arb. units) 300 200 100 0 Fluence (µJ/cm2)0.575 0.570 0.565Frequency (THz)1.0 0.8 0.6 0.4 0.2 0.0Intensity P( ) (arb. units) 0.60 0.58 0.56 0.54 Frequency (THz) 50% 100% 100% (x 3.7) (center) 10% |Bnr|2 z m2 m1 M x Bnr y (a) (c) (d) (b) -0.020.000.02∆(degrees) 40 30 20 10 0 Time (ps)1.0 0.5 0.0 -0.5B (T) Bnr Bin (x 3) 0.08 0.06 0.04 0.02 0.00∆(degrees) 24 20 16 12 8 Time (ps)10%100% Figure 2. (a) Upper panel: Incident magnetic field of the THz pump pulse Bin estimated by electro-optic sampling (dashed line) and the THz magnetic near-field Bnr calculated by the FDTD method (solid line). The illustration s hows the magnetization motion for the AF-mode. Lower panel: The MOKE signal for a pump fluence of 292 µJ/cm2 (100%). (b) Comparison o f two MOKE signals for different pump fluences, vertically offset for clarity. (c) The FFT powe r spectrum of the magnetic near-field Bnr (black solid line). The spectra P() of the MOKE signals for a series of pump fluences obtained at th e corner (solid lines) and at the center (blue dashed circle in the inset of figure 1) for a pump fluence of 100% (dashed line). Each spectru m of the MOKE signal is normalized by the peak amplitude at the corner for a pump fluence o f 100%. (d) Intensity dependence of the center-of-m ass frequency (open circles) and the integral (closed circles) of the P(). 7(t) is calibrated to the magnetization change (t) by using a linear relation, i.e., (t)=g(t), where g (=17.8 degrees−1) is a conversion coefficient. The tim e resolved experiment enables us to separate the contributions of the applied magnetic field and magnetization change to the frequency shift in the time domain. A comparison of the temporal profiles between the driving magnetic field (figure 3(a)) and the frequency e volution (figure 3(c)) shows that for the low pump fluence (10%, closed blue circles), the frequency is redshifted only when the magnetic field persists ( t < 25 ps), and after that, it recovers to the constant AF mode frequency (νAF0=0.575 THz). This result indicates that the signals below t = 25 ps are affected by the persisting driving field and the redshift may orig inate from the forced oscillation. As long as the 0.575 0.570 0.565 0.560 0.555Frequency (THz) 50 40 30 20 10 0 Time (ps)-0.4-0.20.00.20.4Magnetization change 0.5 0.0 -0.5Bnr (T) 0.4 0.3 0.2 0.1 0.0Amplitude 0 Experiment 100% 10% Experiment 100% Simulation 100% (1=0) 100% 10% Simulation(a) (b) (c) (d) Figure 3. (a) FDTD calculated magnetic field Bnr for pump fluence of 100%. (b) Temporal evolution of the magnetization change obtained from the experimental data (gray circles) an d the LLG model (red line). (c) Instantaneous frequencies and (d) envelope amplitudes fo r pump fluences of 100% and 10% obtained by the analytic signals calculated from the experimental data (circles) and the LLG simulation with nonlinear damping paramete r (1=1×10−3, solid lines) and without one ( 1=0, dashed line). 8magnetic response is under the linear regime, the instantaneous frequency is independent on the pump fluence. However, for the high pump fluenc e (100%) a redshift (a maximum redshift of ~15 GHz relative to the constant frequency νAF0) appears in the delay time ( t < 25 ps) and even after the driving field decays away ( t > 25 ps) the frequency continues to be redshifted as long as the amplitude of the magnetization change is large. These results suggest that the frequency redshift in the high intensity case depends on the magnitude of the magnetization change, implying that its origin is a nonlinear precessional spin motion with a large amplitude. The temperature increase due to the THz absorption (for HoFeO 3 T=1.7×10−3 K, for gold SRR T=1 K) is very small (see Appendix D). In ad dition, the thermal relaxation of the spin system, which takes more than a nanosecond [36], is much longer than the frequency modulation decay (~50 ps) in figure 3(c). Therefore, laser heating can be ignored as the origin of the redshift. Figure 4 shows a parametric plot of the instantaneous frequency (t) and envelope amplitude 0(t) for the high pump fluence (100%). The instantaneous frequency shift for t > 25 ps has a square dependence on the amplitude, i.e., νAF=νAF0(1െCζ02). To quantify the relationship between the redshift and magnetization change, it would be helpful to have an analytical expression of the AF mode frequency AF as a function of the magnetization change, which is derived from the LLG equation based on the two- lattice model [32,33]. The dynamics of the sublattice magnetizations mi (i=1,2), as shown in the inset of figure 2(a), are described by dRi dt=െγ (1+α2)ቀRi×[B(t)+Beff,i]െαRi×൫Ri×[B(t)+Beff,i]൯ቁ, (1) where Ri=mi/m0 (m0=|mi|) is the unit directional vector of the sublattice magnetizations, =1.76×1011 s−1T−1 is the gyromagnetic ratio, V(Ri) is the free energy of the iron spin system normalized with m0, and Beff,i is the effective magnetic field given by െ∂V/∂Ri (i=1,2) (see Appendix E). The second term represents the ma gnetization damping with the Gilbert damping constant  Since Beff,i depends on the sublattice magnetizations mi and the product of these quantities appears on the right side of Eq. (1), the LLG e quation is intrinsically nonlinear. If the angle of 9the sublattice magnetization precession is sufficien tly small, Eq. (1) can be linearized and the two fixed AF- and F-modes for the weak excitation can be derived. However, as shown in figure 3(b), the deduced maximum magnetization change reaches ~0.4, corresponding to precession angles of 0.25° in the xz-plane and 15° in th e xy-plane. Thus, the magnetization change might be too large to use the linear approximation. For such a large magnetization motion, assuming the amplitude of the F-mode is zero and =0 in Eq. (1), the AF mode frequency AF in the nonlinear regime can be deduced as νAF =νAF0ට1ିζ02tan2β0 K(D), ( 2 ) D =ඨ ζ02(rAF2ି1) tan2β0 1ିζ02tan2β0, ( 3 ) 0.575 0.570 0.565Frequency (THz) 0.4 0.3 0.2 0.1 0.0 Amplitude 0Experiment t > 25 ps t < 25 ps Analytic Solution 2nd order expansion Figure 4. Relation between instantaneous frequency and envelope amplitude 0 obtaine d from the magnetization change; for t < 25 ps (open circles) and for t > 25 ps (closed circles), the analytic solution (blue line) and second orde r expansion of the analytic solution (gree n dashed line). Errors are estim ated from the spatial inhomogeneity of the driving magnetic field (see Appendix H). 10where K(D) is the complete elliptic integral of the first kind, rAF(≈60) is the ellipticity of the sublattice magnetization precession trajectory of the AF-mode (see Appendix F), and 0 is the amplitude of the (t). As shown in figure 4, the analytic solution can be approximated by the second order expansion νAF≈νAF0(1െtan2β0(rAF2െ1)ζ024⁄) and matches the observed redshift for t > 25 ps, showing that the frequency appr oximately decreases with the square of (t). The discrepancy of the experimental data from the theoretical curve ( t < 25 ps) may be due to the forced oscillation of the AF-mode caused by the driving field. To elaborate the nonlinear damping effects, we compared the measured (t) with that calculated from the LLG equation with the damping term. As shown in figures 3(c) and 3(d), the experiment for the high intensity excitation devi ates from the simulation with a constant Gilbert damping (dashed lines) even in the t > 25 ps time region, suggesting nonlinear damping becomes significant in the large amplitude region. To describe the nonlinear damping phenomenologically, we modified the LLG equa tion so as to make the Gilbert damping parameter depend on the displacement of th e sublattice magnetization from its equilibrium position, (Ri)=0+1Ri. As shown in figures 3(b)-(d), the magnetization change (t) derived with Eq. (1) (solid line) with the damping parameters ( 0=2.27×10−4 and 1=1×10−3) nicely reproduces the experiments for both the high (100%) and low (10%) excitations.1 These results suggest that the nonlinear damping plays a signifi cant role in the large amplitude magnetization dynamics. Most plausible mechanism for the nonlinear damping is four magnon scattering process, which has been introdu ced to quantitatively evaluate the magnon mode instability of ferromagnet in the nonlinear response regime [37]. 4. Conclusions In conclusion, we studied the nonlinear magnetization dynamics of a HoFeO 3 crystal excited by a THz magnetic field and measured by MOKE microscopy. The intense THz field can induce the large magnetization change (~40%), and the ma gnetization change can be kept large enough 1 The damping parameter 0 (=2.27×10−4) and conversion coefficient g (=17.8 degrees−1) are determined from the least-squares fit of the calculated result without the nonlinear damping parameter 1 to the experimental MOKE signal for the low pump fluence of 29.2 µJ/cm2. The nonlinear damping parameter 1 (=1×10−3) is obtained by fitting the experimental result for the high intensity case ( I=292 µJ/cm2) with the values of 0 and g obtained for the low excitation experiment. The estimated value of g is consistent with the stat ic MOKE measurement; the Kerr ellipticity induced by the spontaneous magnetization MS is ~0.05 degrees ( g~20 degrees−1). See Appendix G for details on the static Kerr measurement. 11to induce the redshift even after the field has gone , enabling us to separate the contributions of the applied magnetic field and ma gnetization change to the frequency shift in the time domain. The resonance frequency decreases in proportion to the square of the magnetization change. A modified LLG equation with a phenomenologi cal nonlinear damping term quantitatively reproduced the nonlinear dynamics. This suggest s that a nonlinear spin relaxation process should take place in a strongly driven regime. Th is study opens the way to the study of the practical limits of the speed and efficiency of magnetization reversal, which is of vital importance for magnetic recording and information processing technologies. 12Acknowledgments We are grateful to Shintaro Takayoshi, Masah iro Sato, and Takashi Oka for their discussions with us. This study was supported by a J SPS grants (KAKENHI 26286061 and 26247052) and Industry-Academia Collaborative R&D grant fro m the Japan Science and Technology Agency (JST). 13Appendix A. Detection sche me of MOKE measurement We show the details of the detection scheme of the MOKE measurement. A probe pulse for the MOKE measurement propagates along the z direction. By using the Jones vector [38], an electric field E 0 of the probe pulse polarized linearly along the x-axis is described as E0 =ቀ1 0ቁ. ( A . 1 ) The probe pulse E1 reflected from the HoFeO 3 surface becomes elliptically polarized with a polarization rotation angle and a ellipticity angle . It can be written as E1 =R(െ߶)MR(θ)E0ൌ൬cos θ cos ߶െ ݅ sin θ sin ߶ cos θ sin ߶ ݅ sin θ cos ߶൰, (A.2) where M is the Jones matrix describing  phase retardation of the y component with respect to the x component M=ቀ10 0െiቁ, ( A . 3 ) and R(ψ) is the rotation matrix R(ψ)=൬cosψ sinψ െsinψcosψ൰. (A.4) The reflected light passes through the quarter wave plate, which is arranged such that its fast axis is tilted by an angle of 45° to the x-axis. The Jones matrix of the wave plate is given by Rቀെπ 4ቁMRቀπ 4ቁ. ( A . 5 ) Thus, the probe light E 2 after the quarter wave plate is described as follows, 14E2 = ൬E2,x E2,y൰=Rቀെπ 4ቁMRቀπ 4ቁE1 =1 2൬cosሺθ߶ሻെsin (θെ߶+)i(െcosሺθെ߶ሻsin (θ߶)) cosሺθെ߶ሻsin (θϕ)+i(c o sሺθ߶ሻsin (θെ߶))൰. (A.6) The Wollaston prism after the quarter wave plat e splits the x and y-polarization components of the probe light E2. The spatially separated two pulses are incident to the balanced detector and the detected probe pulse intensity ratio of the di fferential signal to the total corresponds to the Kerr ellipticity angle as follows, 〈หாమ,ೣหమ〉ି〈หாమ,หమ〉 〈หாమ,ೣหమ〉ା〈หாమ,หమ〉ൌെsin2θ. ( A . 7 ) In the main text, we show the Kerr ellipticity change =w−wo, where the ellipticity angles (w and wo) are respectively obtained with and without the THz pump excitation. Appendix B. Analytic signal approach and short time Fourier transform The Analytic signal approach (ASA) allows the extraction of the time evolution of the frequency and amplitude by a simple procedure and assumes that the signal contains a single oscillator component. In our study, we measure only the MOKE signal originating from the AF-mode and it can be expected that the single oscillator assumption is valid. In the ASA, the time profile of the magnetization change (t) is converted into an analytic signal (t), which is a complex function defined by using the Hilbert transform [39]; ψ(t)=ζ0(t)exp( i߶(t))=ζ(t)+i ζ෨(t), (B.1) ζ෨(t)ൌ1 π pζ(t) tିτ∞ -∞ dτ. ( B . 2 ) where the p is the Cauthy principal value. The real part of (t) corresponds to (t). The real function 0(t) and (t) represent the envelope amplitude and instantaneous phase of the magnetization change. The instantaneous frequency (t)(=2(t)) is given by (t)=d(t)/dt. In the analysis, we averaged 0(t) and (t) over a ten picosecond time range. To confirm whether the ASA gives appropriate results, as shown in figure B.1 we compare 15them with those obtained by the short time Fourie r transform (STFT). As shown in figure B.1(a), the time-frequency plot shows only one oscillato ry component of the AF-mode. As shown in figures B.1(b) and (c), the instantaneous freque ncies and amplitudes obtained by the ASA and the STFT are very similar. Because the ASA provides us the instantaneous amplitude with a simple procedure, we showed the time evolu tions of frequency and amplitude derived by the ASA in the main text. Appendix C. Determination of conversion coefficient g and linear damping parameter 0 The conversion coefficient g and the linear damping parameter 0(=) in Eq. (1) are determined by fitting the experimental MOKE signal (t) for the low pump fluence of 29.2 µJ/cm2 with the LLG calculation of the magnetization change (t). Figure C.1 shows the MOKE signal (t) (circle) and the calculated magnetization change (t) (solid line). From the least-squares fit of the calculated result to th e experiment by using a linear relation, i.e., (t)=g(t), we obtained the parameters g(=17.8 degrees−1) and 0(=2.27×10−4). 0.575 0.570 0.565 0.560 0.555 0.550Frequency (THz) 50 40 30 20 100 Time (ps)ASA 100% 10% STFT 100% 10% 1.0 0.8 0.6 0.4 0.2 0.0 Fourier am plitude (arb. units) 50 40 30 20 100 Time (ps)0.4 0.3 0.2 0.1 0.0Amplitude 0ASA 100% 10% STFT 100% 10% (a) (b) (c) 1.2 1.0 0.8 0.6 0.4 0.2 0.0Frequency (THz) 5040302010 Time (ps)(arb. units) 1.0 0.0 Figure B.1. (a) Time-dependence of the power spectrum of the magnetization oscillation for the highest THz excitation ( I=292 µJ/cm2) obtained by the STFT. Comparison of (b) instantaneous frequencies and (c) amplitudes obtained by the ASA and STFT with a time window with FWHM of 10 ps. 16 Appendix D. Laser heating effect The details of the calculation of the temperature change are as follows: For HoFeO 3: The absorption coefficient abs of HoFeO 3 at 0.5 THz is ~4.4 cm−1 [40]; the fluence IHFO absorbed by HoFeO 3 can be calculated as IHFO=I(1−exp(−absd)), where d (=145 µm) is the sample thickness and I is the THz pump fluence. For the highest pump fluence, I=292 µJ/cm2, IHFO is 18.1 µJ/cm2. Since the sample thickness is much smaller than the penetration depth, d≪abs−1, we assume that the heating of the sample due to the THz absorption is homogeneous. By using the heat capacity Cp of 100 J mol−1 K−1 [27], and the molar volume v of ~1.4×102 cm3/mol [27], the temperature change T can be estimated as T=IHFOv/Cpd ~1.7×10−3 K. For gold resonator (SRR): The split ring resonator has an absorption band (center frequency ~0.56 THz, band width ~50 GHz) originated from the LC resonance (figure 2( c)). Assuming the SRR absorbs all incident THz light in this frequency band, the absorbed energy accounts for 3 % of the total pulse energy. Hence, for the highest THz pump fluence, I=292 µJ/cm2, the fluence absorbed by the SRR is Igold=8.76 µJ/cm2. By using the heat capacity Cp of 0.13 J g−1 K−1 [41], the number of the SRRs per unit area N of 4×104 cm−2, and the mass of the SRR m of 1.6×10−9 g, the temperature change -10x10-3-50510degrees) 50 40 30 20 10 0 Time (ps)-0.10.00.1 Magnetization change Experiment Simulation Figure C.1. Experimentally observed MOKE signal (circle) and LLG simulatio n result of the magnetization change (solid line) for the pump fluence of 29.2 µJ/cm2. 17T can be estimated as T=Igold/CpNm ~ 1 K Appendix E. Free energy of HoFeO 3 The free energy F of the iron spin (Fe3+) system based on the two-lattice model is a function of two different iron sublattice magnetizations mi, and composed of the exchange energy and one-site anisotropy energy [32,33]. The free en ergy normalized by the sublattice magnetization magnitude, V=F/m0 (m0=|mi|), can be expanded as a power series in the unit directional vector of the sublattice magnetizations, Ri=mi/m0=(Xi,Yi,Zi). In the magnetic phase 4 (T > 58K), the normalized free energy is given as follows [32,33]: V=ER1·R2+D(X1Z2െX2Z1)െAxx(X12+X22)െAzz(Z12+Z22), (E.1) where E(=6.4×102 T) and D(=1.5×10 T) for HoFeO 3 are respectively the symmetric and antisymmetric exchange field [42]. Axx and Azz are the anisotropy constants. As mentioned in Appendix F, the temperature dependent values of the anisotropy constants can be determined from the antiferromagnetic resonance frequencies. The canting angle of Ri to the x-axis β0 under no magnetic field is given by tan 2β0=D E+AxxିAzz. ( E . 2 ) Appendix F. Linearized resonance modes and anisotropy constants ( Axx and Azz) The nonlinear LLG equation of Eq. (1) can be linearized and the two derived eigenmodes correspond to the AF and F-mode. The sublatti ce magnetization motion for each mode is given by the harmonic oscillation of mode coordinates; for the AF-mode ( QAF, PAF)=((X1−X2)s i nβ0+(Z1+Z2)c o sβ0, Y1−Y2), and for the F-mode ( QF, PF)=((X1+X2)sinβ0−(Z1−Z2)cosβ0, Y1+Y2), QAF=AAFcosωAFt, ( F . 1 ) PAF=AAFrAFsinωAFt, ( F . 2 ) QF=AFcosωFt, ( F . 3 ) PF=AFrAFsinωFt, ( F . 4 ) 18 where AAF,F represents the amplitude of each mode. AF,F, and rAF,F are the resonance frequencies and ellipticities, which are given by ωAF=γට(b+a)(d-c), ( F . 5 ) ωF=γට(b-a)(d+c), ( F . 6 ) rAF=γටሺௗିሻ (b+a), ( F . 7 ) rF=γටሺௗାሻ (b-a), ( F . 8 ) where =1.76×1011 s−1T−1 is the gyromagnetic ratio, and a=െ2Axxcos2β0െ2Azzsin2β0െEcos 2β0െDsin 2β0, (F.9) b=E, ( F . 1 0 ) c=2Axxcos2β0െ2Azzcos2β0+Ecos 2β0+Dsin 2β0, (F.11) d=െEcos 2β0െDsin 2β0. ( F . 1 2 ) Substituting the literature values of the exchange fields ( E=6.4×102 T and D=1.5×10 T [42]) and the resonance frequencies at room temperature ( AF/2=0.575 THz and F/2=0.37 THz) to Eqs. (F.5) and (F.6), Axx and Azz can be determined to 8.8×10−2 T and 1.9×10−2 T. Appendix G. MOKE measurement for the spontaneous magnetization Figure G.1 shows time-development of the MOKE signals for the different initial condition with oppositely directed magnetization. We applied the static magnetic field (~0.3 T) to saturate the magnetization along the z-axis before the TH z excitation. The spontaneous magnetization of single crystal HoFeO a can be reversed by the much smaller magnetic field (~0.01 T) because of the domain wall motion [27]. Then, we separately measured the static Kerr ellipticity angle  and THz induced ellipticity change for different initial magnetization Mz=±Ms without the static magnetic field  In figure G.1 we plot the summation of the time resolved MOKE signal and the static Kerr ellipticity  The sings of the ellipticity offset angle 19 for the different spontaneous magnetization (±M S) are different and their magnitudes are ~0.05 degrees. The conversion coefficient g(=1/~/0.05 degrees) is estimated to be ~20 degrees−1, which is similar to the value dete rmined by the LLG fitting (~17.8 degrees−1). In the case of the AF-mode excitation, the phases of the magnetization oscillations are in-phase regardless of the direction of the spontaneous magnetization M=±Ms, whereas they are out-of-phase in the case of the F-mode excitation. We can explain this claim as follows: In the case of AF-mode excitation, the external THz magne tic field is directed along the z-direction as shown in the inset of figure 2(a), the signs of the torques acting on the sublattice magnetization mi (i=1,2) depends on the direction of mi, however, the resultant oscillation of the macroscopic magnetization M= m1+m2 along the z-direction has same phase for the different initial condition M=±Ms. In the case of the F-mode excitation with the external THz magnetic field along the x or y-direction, the direction of the torques acting on the magnetization M depends on the initial direction and the phase of the F-mode osc illation changes depending on the sign of the spontaneous magnetization ±Ms. Appendix H. Influence of the spatial distri bution of magnetic field on magnetization change As shown in the inset of figure 1, the pump magnetic field strongly localizes near the metallic arm of the SRR and the magnetic field strength significantly depends on the spatial position r within the probe pulse spot area. The intensity distribution of the probe pulse Iprobe(r) has an 0.05 0.00 -0.05Kerr ellipticity (degrees) 25 20 15 10 5 Time (ps) +MS -MS Figure G.1. The MOKE signals, the temporal change of the Kerr ellipticity , measured for different initial conditions with oppositely directed magnetizations. 20elongated Gaussian distribution with spatial widt hs of 1.1 µm along the x-axis and 1.4 µm along the y-axis [full width at half maximum (FWHM) intensity]. The maximum magnetic field is 1.2 times larger than the minimum one in the spot diameter, causing the different magnetization change dynamics at different positions. To take into account this spatial inhomogeneity to the simulation, the spatially weighted average of magnetization change ζ̅(t) has to be calculated as follows: ζ̅(t)=ζ(r,t)Iprobe(r)dr Iprobe(r)ௗr , ( H . 1 ) where (r,t) is a magnetization change at a position r and time t. Figure H.1(a) shows the simulation result of the spatially averaged magnetization change ζ̅(t) and the non-averaged (r0,t) without the nonlinear damping term ( 1=0), where r0 denotes the peak position of Iprobe(r). For the low excitation intensity (10%), ζ̅(t) is almost the same as (r0,t) as shown in figure H.1(a). On the other hand, for the high excitation intensity, the spatial inhomogeneity of magnetization change dyna mics induces a discrepancy between the ζ̅(t) and (a) (b) (c) -0.100.000.10Magnetization change 50 40 30 20 100 Time (ps)-0.6-0.4-0.20.00.20.4 Averaged Non-averaged Averaged Non-averaged100% 10% 0.575 0.570 0.565 0.560 0.55550 40 30 20 100 Time (ps)0.575 0.570 0.565 0.560 0.555Frequency (THz) Averaged Non-averaged Experiment Averaged Non-averaged Experiment 100% 10% 0.4 0.3 0.2 0.1 0.0 50 40 3020 100 Time (ps)0.12 0.08 0.04 0.00Amplitude Averaged Non-averaged Experiment Averaged Non-averaged Experiment100% 10% Figure H.1. Comparison of the spatially averag ed and non-averaged magnetization change for the different pump fluences of 10% and 100%. (a) Temporal evolutions of the magnetization change, (b) instantaneous frequencies and (c) normalized envelope amplitudes. Open circles show the experimental results. 21(r0,t). This discrepancy is caused by the quasi-i nterference effect between the magnetization dynamics with different frequencies and amplit udes at different positions. Figures H.1(b) and (c) show the instantaneous freque ncy and envelope amplitude obt ained from the data shown in figure H.1(a) by using analytic signal approach with the experimental result. For the averaged magnetization change, the frequency redshift is more emphasized (figure H.1(b)) and the decay time becomes shorter (figure H.1(c)). Nonetheless, neither spatially averaged nor non-averaged simulation reproduces the experimental result of the instantaneous frequency (figure H.1(b)) without nonlinear damping term. 22References [1] Kirilyujk A, Kimel A V and Rasing T 2010 Rev. Mod. Phys. 82 2731 [2] Beaurepaire E, Merle J-C, Daunois A and Bigot J-Y 1996 Phys. Rev. Lett. 76 4250 [3] Barman A, Wang S, Maas J D, Hawkins A R, Kwon S, Liddle A, Bokor J and Schmidt H 2006 Nano Lett. 6 2939 [4] Bovensiepen U 2007 J. 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2016-01-23

We report on the nonlinear magnetization dynamics of a HoFeO3 crystal induced by a strong terahertz magnetic field resonantly enhanced with a split ring resonator and measured with magneto-optical Kerr effect microscopy. The terahertz magnetic field induces a large change (~40%) in the spontaneous magnetization. The frequency of the antiferromagnetic resonance decreases in proportion to the square of the magnetization change. A modified Landau-Lifshitz-Gilbert equation with a phenomenological nonlinear damping term quantitatively reproduced the nonlinear dynamics.

Nonlinear magnetization dynamics of antiferromagnetic spin resonance induced by intense terahertz magnetic field

1601.06213v1

Spin-Torque-driven Terahertz Auto Oscillations in Non-Collinear Coplanar Antiferromagnets Ankit Shuklaand Shaloo Rakhejay Holonyak Micro and Nanotechnology Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801 (Dated: January 19, 2022) We theoretically and numerically study the terahertz auto oscillations, or self oscillations, in thin- lm metallic non-collinear coplanar antiferromagnets (AFMs), such as Mn 3Sn and Mn 3Ir, under the eect of antidamping spin torque with spin polarization perpendicular to the plane of the lm. To obtain the order parameter dynamics in these AFMs, we solve three Landau-Lifshitz-Gilbert equa- tions coupled by exchange interactions assuming both single- and multi-domain (micromagnetics) dynamical processes. In the limit of a strong exchange interaction, the oscillatory dynamics of the order parameter in these AFMs, which have opposite chiralities, could be mapped to that of two damped-driven pendulums with signicant dierences in the magnitude of the threshold currents and the range of frequency of operation. The theoretical framework allows us to identify the in- put current requirements as a function of the material and geometry parameters for exciting an oscillatory response. We also obtain a closed-form approximate solution of the oscillation frequency for large input currents in case of both Mn 3Ir and Mn 3Sn. Our analytical predictions of threshold current and oscillation frequency agree well with the numerical results and thus can be used as com- pact models to design and optimize the auto oscillator. Employing a circuit model, based on the principle of tunnel anisotropy magnetoresistance, we present detailed models of the output power and eciency versus oscillation frequency of the auto oscillator. Finally, we explore the spiking dynamics of two unidirectional as well as bidirectional coupled AFM oscillators using non-linear damped-driven pendulum equations. Our results could be a starting point for building experimen- tal setups to demonstrate auto oscillations in metallic AFMs, which have potential applications in terahertz sensing, imaging, and neuromorphic computing based on oscillatory or spiking neurons. I. INTRODUCTION Terahertz (THz) radiation, spanning from 100 Giga- hertz (GHz) to 10 THz, are non-ionizing, have short wavelength, oer large bandwidth, scatter less, and are absorbed or re ected dierently by dierent materials. As a result, THz electronics can be employed for safe biomedical applications, sensing, imaging, security, qual- ity monitoring, spectroscopy, as well as for high-speed and energy-ecient non-von Neumann computing (e.g., neuromorphic computing). THz electronics also has po- tential applications in beyond-5G communication sys- tems and Internet of Things. Particularly, the size of the antennae for transmitting the electromagnetic signal could be signicantly miniaturized in THz communica- tion networks [1{8]. These aforementioned advantages and applications have led to an intense research and de- velopment in the eld of THz technology with an aim to generate, manipulate, transmit, and detect THz sig- nals [3, 9]. Therefore, the development of ecient and low power signal sources and sensitive detectors that op- erate in the THz regime is an important goal [3]. Most coherent THz signal sources can be categorized into three types | particle accelerator based sources, solid state electronics based sources, and photonics based sources [3, 9]. Particle accelerator based signal genera- tors include free electron lasers [10], synchrotrons [11], ankits4@illinois.edu yrakheja@illinois.eduand gyrotrons [12]. While particle accelerator sources have the highest power output, they require a large and complex set-up [13]. Solid state generators include diodes [14{16], transistors [17, 18], frequency multipli- ers [19], and Josephson junctions [20], whereas photonics based signal sources include quantum cascade lasers [21], gas lasers [22], and semiconductor lasers [23]. Solid-state generators are ecient at microwave frequencies whereas their output power and eciency drop signicantly above 100 GHz [13]. THz lasers, on the other hand, provide higher output power for frequencies above 30 THz [24], however, their performance for lower THz frequencies is plagued by noise and poor eciency [13]. Here, we present the physics, operation, and performance bench- marks of a new type of nanoscale THz generator based on the ultra-fast dynamics of the order parameter of an- tiferromagnets (AFMs) when driven by spin torque. Spin-transfer torque (STT) [25, 26] and spin-orbit torque (SOT) [27] enable electrical manipulation of ferro- magnetic order in emerging low-power spintronic radio- frequency nano-oscillators [28]. When a spin current greater than a certain threshold (typically around 108 109A=cm2[28, 29]) is injected into a ferromagnet (FM) at equilibrium, the resulting torque due to this current pumps in energy which competes against the intrinsic Gilbert damping of the material. When the spin torque balances the Gilbert damping, the FM magnetization un- dergoes a constant-energy steady-state oscillation around the spin polarization of the injected spin current. Such oscillators are nonlinear, current tunable with frequencies in the range of hundreds of MHz to a few GHz with out- put power in the range of nano-Watt (nW). They are alsoarXiv:2106.08528v2 [cond-mat.mes-hall] 15 Jan 20222 compatible with the CMOS technology [28]; however, the generation of the THz signal using FMs would require prohibitively large amount of current, which would lead to Joule heating and degrade the reliability of the elec- tronics. It would also lead to electromigration and hence irreversible damage to the device set-up [30]. AFM materials, which are typically used to exchange bias [31] an adjacent FM layer in spin-valves or magnetic tunnel junctions for FM memories and oscillators, have resonant frequencies in the THz regime [32{35] due to their strong exchange interactions. It was suggested that STT could, in principle, be used to manipulate the mag- netic order in conducting AFMs [36], leading to either stable precessions for their use as high-frequency oscilla- tors [37, 38] or switching of the AFM order [39] for their use as magnetic memories in spin-valve structures. The SOT-based spin Hall eect (SHE), on the other hand, could enable the use of both conducting [40{44] and in- sulating [29, 43, 45, 46] AFMs in a bilayer comprising an AFM and a non-magnetic (NM) layer, for its use as a high frequency auto-oscillator [47]. Table I lists the salient results from some of the re- cently proposed AFM oscillators. These results, how- ever, are reported mainly for collinear AFMs, while de- tailed analyses of the dynamics of the order parameter in the case of non-collinear AFMs is lacking. In this paper, rstly, we ll this existing knowledge gap in the model- ing of auto oscillations in thin-lm non-collinear copla- nar AFMs like Mn 3Ir;Mn3Sn, or Mn 3GaN under the ac- tion of a dc spin current. Secondly, we compare their performance (generation and detection) against that of collinear AFMs such as NiO for use as a THz signal source. In the case of NiO, inverse spin Hall eect (iSHE) is employed for signal detection, whereas in this work we utilize the large magnetoresistance of metallic AFMs. Fi- nally, we investigate these auto oscillators as possible can- didates for neuron emulators. Considering that the spin polarization is perpendicular to the plane of the AFM thin-lm, three possible device geometries are identied and presented in Fig. 1 for the generation and detection of auto oscillations in metallic AFMs. Figure 1(a) is based on the phenomena of spin injec- tion and accumulation in a local lateral spin valve struc- ture. Charge current, I write, injected into the structure is spin-polarized along the magnetization of FM 1(FM 2) and gets accumulated in the NM. It then tunnels into the AFM with the required perpendicular spin-polarization (adapted from Ref. [50]). The bottom MgO layer used here would reduce the leakage of charge current into the metallic AFMs considered in this work. This would reduce chances of Joule heating in the AFM thin-lm layer. On the other hand, in Fig. 1(b), spin lter- ing [51] technique is adopted wherein, a conducting AFM is sandwiched between two conducting FMs. Dier- ent scattering rates of the up-spins and down-spins of the injected electron ensemble at the two FM interfaces results in a perpendicularly polarized spin current as shown. The structure in Fig. 1(c) generates spin polar- FM1 Iwrite FM2NMJs npAFMMgOPtLoadIdeal Bias Tee FM1FM2 IwriteAFM MgOIdeal Bias Tee Load JsnpIread FM1 NMLoadIdeal Bias Tee(a) (b) (c)xyzIread IwriteIread Pt MgO JsnpAFMMgOLbtCbt Cbt Lbt LbtCbtFIG. 1. Device geometries to inject perpendicularly polar- ized spin current in thin-lm metallic AFMs. In all the cases, Iwrite is the charge current injected to generate spin current, whereas, I readis the charge current injected to extract the oscillations as a transduced voltage signal using the princi- ples of tunnel anisotropy magnetoresistance (TAMR). (a) Lat- eral spin valve structure leads to spin accumulation in NM followed by injection into the AFM. (b) Perpendicular spin valve structure spin lters the injected charge current. (c) FM/NM/AFM trilayer structure generates spin current due to interfacial spin-orbit torque. ization perpendicular to the interface due to the interfa- cial SOTs generated at the FM/NM interface (adapted from Ref. [52, 53]). In this case, the spin current injected into the AFM has polarization along both yandzdirec- tion; however, the interface properties could be tailored to suppress the spin polarization along y[52]. In order to extract the THz oscillations of the order parameter as a measurable voltage signal, the tunnel anisotropy mag- netoresistance (TAMR) measurements are utilized [54]. In this work, we establish the micromagnetic model for non-collinear coplanar AFMs with three sublattices along with the boundary conditions in terms of both the sublattice magnetizations (Section II A), as well as the N eel order parameter (Section II B). We show that in the macrospin limit the oscillation dynamics correspond to that of a damped-driven pendulum (Section III A and Section III B). The oscillation dynamics of AFM mate- rials with two dierent chiralities in then compared in Section III C. We use the TAMR detection scheme to extract the oscillations as a voltage signal and present models of the output power and eciency as a function of the oscillator's frequency (Section IV). This is followed by a brief investigation of the eect of inhomogeneity due to the exchange interaction on the dynamics of the AFM order (Section V). Finally, we discuss the implication of our work towards building coherent THz sources in Sec-3 TABLE I. Recent numerical studies on electrically controlled AFM THz oscillators. The investigated AFM materials, the direction of their uniaxial anisotropy axis ueand that of the spin polarization of the injected spin current npare listed. Salient results along with the schemes to extract the oscillation as a voltage signal are also brie y stated. Ref. [44] does not provide the name of a specic AFM, however, an AFM with uniaxial anisotropy is considered. Ref. AFM material ue np Salient Features Detection Schemes [45] NiO x x a) THz oscillations for current above a threshold iSHE b) Feedback in AFM/Pt bilayer sustains oscillation [29] NiO x -z a) Hysteretic THz oscillation in a biaxial AFM iSHE b) Threshold current dependence on uniaxial anisotropy [40]Fe2O3 a)z y a) Monodomain analysis of current driven oscillations in - AFM insulators with DMI b)y y Similar to [45] - [41]Fe2O3 x y a) Canted net magnetization due to DMI Dipolar radiation b) Small uniaxial anisotropy leads to low power THz frequency [42] CuMnAs ;Mn2Auy z a) Low dc current THz signal generation due to N eel SOT - b) Phase locked detector for external THz signal [43] NiO x Varied a) Comparison of analytical solutions to micromagnetic results AMR/SMR b) Eect of DMI on hysteretic nature of dynamics [48] Mn 2RuxGa z y a) Generation of spin current in single AFM layer AMR b) Oscillation dependence on reactive and dissipative torques [44] Uniaxial Ani. z Varied a) Non-monotonic threshold current variation with np - b) Eects of anisotropy and exchange imperfections [49] Mn 3Sn x-y plane z a) Eective pendulum model based on multipole theory AHE [46] NiO;Cr2O3 x Varied a) General eective equation of a damped-driven pendulum iSHE b) Analytic expression of threshold current and frequency Our Mn 3Sn;Mn3Ir x-y plane z a) Dierent numerical and analytic models TAMR Work b) Inclusion of generation current for TAMR eciency c) Non-linear dynamics of bidirectional coupled oscillators tion VI, and towards hardware neuron emulators for neu- romorphic computing architecture in Section VII. Some of the salient results from this work are listed in Table I. II. THEORY A. Magnetization Dynamics We consider a micromagnetic formalism in the con- tinuum domain [43, 55] under which a planar non- collinear AFM is considered to be composed of three equivalent interpenetrating sublattices, each with a con- stant saturation magnetization Ms[56]. Each sublattice, i(= 1, 2 or 3), is represented as a vector eld mi(r;t) such that for an arbitrary r=r0,kmi(r0;t)k= 1. The dynamics of the AFM under the in uence of magnetic elds, damping, and spin torque is assumed to be gov- erned by three Landau-Lifshitz-Gilbert (LLG) equations coupled by exchange interactions. For sublattice i, the LLG is given as [57] @mi @t= 0 miHe i
+i mi@mi @t !smi(minp)!s(minp);(1) wheretis time in seconds, He iis the position dependent eective magnetic eld on i,iis the Gilbert dampingparameter for i, and !s=~ 2e Js Msda(2) is the frequency associated with the input spin current density,Js, with spin polarization along np. Here,dais the thickness of the AFM layer, ~is the reduced Planck's constant,0is the permeability of free space, eis the el- ementary charge, and = 17:61010T1s1is the gyro- magnetic ratio. For all sublattices, the spin polarization, np, is assumed to be along the zaxis. Finally, is a mea- sure of the strength of the eld-like torque as compared to the antidamping-like torque. The eect of eld-like torque on the sublattice vectors here is the same as that of an externally applied magnetic eld|canting towards the spin polarization direction. Results presented in the main part of this work do not include the eect of the eld-like torque; however, a small discussion on the same is presented in the supplementary material [58]. The eective magnetic eld, He iat each sublattice, includes contributions from internal elds as well as ex- ternally applied magnetic elds and is obtained as He i(r;t) =1 0MsF mi(r;t); (3) where mi=@ @mir
@ @(rmi), andFis the energy density of the AFM, considered in our work. It is given as4 F=X hi;ji Jmi
mj+Aijrmi
rmj
+Aii3X i=1(rmi)2+3X i=1Khm2 i;zKe(mi
ue;i)2 +DX hi;jiz
(mimj) +Dii3X i=1(mi;zr
mi (mi
r)mi;z) +DijX hi;ji((mi;zr
mj (mi
r)mj;z)(mj;zr
mi(mj
r)mi;z)) 3X i=10MsHa
mi;(4) wherehi;jirepresents the sublattice ordered pairs (1 ;2), (2;3) and (3;1). The rst three terms in Eq. (4) represent exchange en- ergies. HereJ(>0) is the homogeneous inter-sublattice exchange energy density whereas Aii(>0) andAij(< 0) are the isotropic inhomogeneous intra- and inter- sublattice exchange spring constants, respectively. The next two terms in Eq. (4) represent magnetocrystalline anisotropy energy for biaxial symmetery upto the low- est order withKe(>0) andKh(>0) being the easy and hard axes anisotropy constants, respectively. We assume that the easy axes of sublattices 1, 2 and 3 are along ue;1=(1=2)x+ (p 3=2)y,ue;2=(1=2)x(p 3=2)y andue;3=x, respectively, and an equivalent out of plane hard axis exists along the zaxis. The next three terms represent the structural symmetry breaking interfacial Dzyaloshinskii-Moriya Interaction (iDMI) energy density in the continuum domain. Its origin lies in the interaction of the antiferromagnetic spins with an adjacent heavy metal with a large spin-orbit coupling [59, 60]. Here, we assume the AFM crystal to have Cnvsymmetry [61] such that the thin-lm AFM is isotropic in its plane, and D, Dii, andDijrepresent the eective strength of homoge- neous and inhomogeneous iDMI, respectively, along the zdirection. Finally, the last term in Eq. (4) represents the Zeeman energy due to an externally applied mag- netic eld Ha. Now, using Eq. (4) in Eq. (3) we get the eective eld for sublattice ias He i=X j j6=i J 0Msmj+Aij 0Msr2mj +2Aii 0Msr2mi 2Kh 0Msmi;zz+2Ke 0Ms(mi
ue;i)ue;i +Dz(mjmk) 0Ms2Dii 0Ms((r
mi)zrmi;z) Dij 0Ms((r
(mjmk))zr(mj;zmk;z)) +Ha; (5)where (i;j;k ) = (1;2;3);(2;3;1);or (3;1;2), respectively. In order to explore the dynamics of the AFM, we adopt a nite dierence discretization scheme and discretize the thin-lm of dimension LWdainto smaller cells of sizesLsWsd. Each of these cells is centered around position rsuch that mi(r;t) denotes the average mag- netization of the spins within that particular cell [43]. Finally, we substitute Eq. (5) in Eq. (1) and use fourth- order Runge-Kutta rule along with the following bound- ary conditions for sublattice iof the thin-lm considered (see supplementary material [58]): 2Aii@mi @+AijX j j6=imi@mj @mi +Diimi(z) +Dijmi(mi((z)(mkmj))) = 0;(6) whereis the normal vector perpendicular to a surface parallel to xory. The above equation ensures that the net torque due to the internal elds on the boundary magnetizations of each sublattice is zero in equilibrium as well as under current injection [62]. For the energy density presented in Eq. (4), the elds at the boundary are non-zero only for inhomogeneous inter- and intra- sublattice exchange, and Dzyaloshinskii-Moriya interac- tions. Finally, in the absence of DMI, we have the Neu- mann boundary condition@mi @= 0, which implies that the boundary magnetization does not change along the surface normal . For all the numerical results presented in this work we solve the system of Eqs. (1), (5), and (6) with the equilibrium state as the starting point. The equilibrium solution in each case was arrived at by solving these three equations for zero external eld and zero current with a large Gilbert damping of 0 :5. B. N eel Order Dynamics The aforementioned micromagnetic modeling ap- proach assuming three sublattices is extremely useful in exploring the physics of the considered AFM systems. It is, however, highly desirable to study an eective dynam- ics of the AFMs under the eect of internal and external stimuli in order to gain fundamental insight. Therefore, we consider an average magnetization vector mand two staggered order parameters n1andn2to represent an equivalent picture of the considered AFMs. These vec- tors are dened as [37, 56, 63] m1 3(m1+m2+m3); (7a) n11 3p 2(m1+m22m3); (7b) n21p 6(m1+m2); (7c)5 such thatkmk2+kn1k2+kn2k2= 1. The energy land- scape (Eq. (4)) can then be represented as F 3=3J 2m2+Am(rm)2+An 2 (rn1)2+ (rn2)2 +Kh m2 z+n2 1;z+n2 2;z
Ke 23 2(n1;xn2;y)2 +1 2(n1;y+n2;x)2+mx mxp 2(n1;xn2;y) +my my+p 2(n1;y+n2;x) + 4n1;xn2;y +p 3Dz
(n1n2) +Dii(mzr
m(m
r)mz +n1;zr
n1(n1
r)n1;z+n2;zr
n2 (n2
r)n2;z) +p 3Dij(n1;zr
n2(n1
r)n2;z n2;zr
n1+ (n2
r)n1;z)0MsHa
m; (8) whereAm= Aii+Aij
andAn= 2AiiAij
. An equation of motion involving the staggered order parameters can be obtained by substituting Eq. (1) in the rst-order time derivatives of Eq. (7) and evaluating each term carefully (see supplementary material [58]). How- ever, an analytical study of such an equation of motion that consists of contributions from all the energy terms of Eqs. (4) or (8) would be as intractable as the dynam- ics of individual sublattices itself. Therefore, we consider the case of AFMs with strong inter-sublattice exchange interaction such that J jDjK e. This corresponds to systems with ground state conned to the easy-plane (xyplane) and those that host kmk1 (weak ferro- magnetism), n1?n2, andkn1kkn2k1=p 2 [56, 63]. However, when an input current is injected in the system, the sublattice vectors cant towards the spin polarization direction leading to an increase in the magnitude of m while decreasing that of n1andn2. Spin polarization along the zdirection and an equal spin torque on each sublattice vector ensures that n1andn2have negligible zcomponents at all times (Eqs. (7b), (7c)). Therefore, we consider n1(r;t) =0 @p 1n2 1zcos'(r;t)p 1n2 1zsin'(r;t) n1z(r;t)1 A; (9a) n2(r;t) =0 @p 1n2 2zcos('(r;t)=2)p 1n2 2zsin('(r;t)=2) n2z(r;t)1 A;(9b) where'is the azimuthal angle from the xaxis and jn1zj;jn2zj1. The two choices for n2correspond to two dierent classes of materials|one with a positive (+ =2) chiral- ity and the other with a negative ( =2) chirality [56]. Materials that have a negative (positive) value of Dcor- respond to + =2(=2) chirality because the respective conguration reduces the overall energy of the system. L12phase of AFMs like Mn 3Ir;Mn3Rh, or Mn 3Pt is ex- pected to host + =2 chirality whereas the hexagonalphase of AFMs like Mn 3Sn;Mn3Ge, or Mn 3Ga is ex- pected to host=2 chirality [56, 64]. III. SINGLE DOMAIN ANALYSIS A. Positive Chirality Dening n3= (n1n2)=, and considering the case of +=2 chirality, it can be shown that mis just a de- pendent variable of the N eel order dynamics. To a rst order, mcould be expressed as [56, 63] m=1 !E(n1_ n1+n2_ n2+n3_ n3 0Ha !snp); (10) where!E= 3 J=Ms. One can then arrive at the equa- tion of motion for the N eel vectors as n1 n1c2r2n1!E!Kn1+!E!Kh(n1
z)z +!E!D(n2z) +!E!ii Dn1+!E!ij Dn2+!E_n1 !E!s(n1np)] +n2 n2c2r2n2!E!Kn2 +!E!Kh(n2
z)z!E!D(n1z) +!E!ii Dn2 !E!ij Dn1+!E_n2!E!s(n2np)i +n3n3 + 0(n1_n1+n2_n2+n3_n3)Ha 0_Ha= 0; (11) wherec=p !E An=Ms,!Kh= 2 Kh=Ms,!K;n1= !K 4 n1;y+n2;x+p 2my
^ x+ (n1;x+ 3n2;y +p 2mx
^ y
,!D=p 3 D Ms,!K;n2=!K 4((n1;y+n2;x +p 2my
^ x+ n1;x+ 3n2;y+p 2mx
^ y
, !ii D;ni=2 Dii Ms((r
ni)zrni;z), and!ij D;ni=p 3 Dij Ms((r
ni)zrni;z). The equations of motion (Eqs. (10) and (11)) derived here are useful in the numerical study of textures like domain walls, skyrmions, and spin-waves in AFMs with biaxial anisotropy under the eect of external magnetic eld and spin current. However, here we are interested in analytically studying oscillatory dynamics of the order parameter in thin-lm AFMs, therefore, we neglect in- homogeneous interactions compared to the homogeneous elds. Using Eq. (9) in Eq. (11) and neglecting the time derivative of n1zandn2z, we have '+!E_'+!E!K 2sin 2'+!E!s= 0; (12) where!K= 2 Ke=Ms. This indicates that in the limit of strong exchange interaction, the dynamics of the stag- gered order parameters is identical to that of a damped- driven non-linear pendulum [65]. This equation is iden- tical to the case of collinear AFMs such as NiO when the6 (c) (d) (a) (b)Negative Chirality Positive Chirality FIG. 2. Stationary solution for AFMs with positive (a, b) and negative chirality (c, d). (a) Sublattice magnetization for currents below the threshold current Jth1 s. WhenJs= 0 (equilibrium state), the sublattice vectors micoincide with the easy axes ue;i, whereas for a non-zero current smaller than Jth1 s, the macrospins have stationary solutions other than the equilibrium solution, as depicted by dashed and dotted line. The zcomponent of these vectors is zero. (b) An equivalent representation of (a) through the staggered order parameters n1andn2. They are perpendicular to each other and have zero out-of-plane component. The thinner dash-dotted gold arrows through the thicker arrows of n1represent the analytic expression of the stationary solution '=1 2sin1 2!s !K . The average magnetization mis vanishingly small, as can also be noticed from Eq. (10). (c) Sublattice magnetization at equilibrium ( Js= 0). Thezcomponent of these vectors is zero. Here, only the sublattice vector m3coincides with its corresponding easy axis. On the other hand, m1andm2are oriented such that the energy due to DMI is dominant over anisotropy. (d) An equivalent representation of (c) through the staggered order parameters n1andn2, and average magnetization m.n1andn2are almost perpendicular to each other with a negative chirality, as assumed in Eq.(9). A small in-plane net magnetization (shown by the magnied green arrow) also exists in this case [56]. (a) (b) (c) (d) (e) (g) (f) (h)Positive Chirality Negative Chirality α = 0.01α = 0.01 α = 0.01 α = 0.01da = 4 nm da = 4 nm da = 4 nm da = 4 nm FIG. 3. Upper panel (a-d) shows the time averaged frequency as a function of input spin current, whereas the lower panel (e-h) shows the FFT of the oscillations corresponding to the cases marked by the dashed red boxes above. In the time averaged frequency response in the upper panel, the dashed black lines denote the analytic expression of frequency (Eq. (17)). (a), (c) Frequency response for dierent lm thicknesses for = 0:01. (e), (g) FFT of the signal corresponding to Js=Jth2 s. (b), (d) Frequency response for dierent damping constants for da= 4 nm. (f), (h) FFT of the signal corresponding to = 0:1 andJth1 s, respectively. Positive chirality: The numerical values of the average frequency match very well against the analytic expression for lower damping and large current. On the other hand, non-linearity and, hence, higher harmonics are observed for small current and large damping. Negative chirality: The numerical values of the average frequency exactly match against the analytic expression for all values of damping and input current considered here. direction of spin polarization is perpendicular to the easy- plane [29, 44, 66]. However, the dynamics of the non- collinear coplanar AFMs discussed here is signicantlydierent in the direction of the spin torques, magnitude of threshold currents as well as the range of possible fre- quencies. Here, the sin 2 'dependence signies a two-fold7 anisotropy symmetric system. B. Negative Chirality For the case of=2 chirality, it can be shown that m is a dependent variable of the N eel order; however, in this case there are additional in-plane terms that arise due to a competition between the DMI, exchange coupling and magnetocrystalline anisotropy. To a rst order, mis ex- pressed as [56, 67] (also see supplementary material [58]) m=1 !E(n1_ n1+n2_ n2+n3_ n3 0Ha !snp)!K 2!E(cos'xsin'y); (13) which is used to arrive at the equation of motion for the N eel vectors as n1 n1c2r2n1!E!Kn1+!E!Kh(^ n1
z)z +!E!D(n2z) +!E!ii Dn1+!E!ij Dn2+!E_n1 !E!s(n1np)] +n2 n2c2r2n2!E!Kn2 +!E!Kh(^ n2
z)z!E!D(n1z) +!E!ii Dn2 !E!ij Dn1+!E_n2!E!s(n2np)i +n3n3 + 0(n1_n1+n2_n2+n3_n3)Ha 0_Ha !K 2(sin'x+ cos'y) _' 0!K 2(Ha;zsin'x +Ha;zcos'y(Ha;xsin'+Ha;ycos')z) = 0: (14) Similar to the previous case, we are interested in a the- oretical analysis of the oscillation dynamics in thin lm AFMs with negative chirality. Therefore, we use Eq. (9) in Eq. (14) and neglect all the inhomegeneous interac- tions to arrive at a damped-driven linear pendulum equa- tion given as '+!E_'+!E!s= 0: (15) Here the dependence of the dynamics on anisotropy is not zero but very small, and it scales proportional to !3 K !2 Ecos 6'[67]. However, for a rst-order approxima- tion in mand dynamics in the THz regime, it can be safely ignored. The cos 6 'dependence implies that these materials host a six-fold anisotropic symmetry. Though this equation is similar to that obtained for the case of a collinear AFM with spin polarization along the easy axis [44], the dynamics is signicantly dierent from that of the collinear AFM. C. Comparison of Dynamics for Positive and Negative Chiralities Here, we contrast the dynamics of AFM order param- eter for positive and negative chiralities. The numericalresults presented in this section are obtained in the single- domain limit assuming thickness da= 4 nm,= 0:01, Ms= 1:63 T,Ke= 3 MJ=m3,J= 2:4108J=m3, D=20 MJ=m3for positive chirality or 20 MJ =m3for negative chirality [56], unless specied otherwise. Figure 2 shows the stationary solutions of the thin- lm AFM system with dierent chiralities. For the case of positive chirality, it can be observed from Fig. 2(a) that in equilibrium the sublattice vectors micoincide with the easy axes ue;i. When a non-zero spin current is applied, the equilibrium state is disturbed; however, be- low a certain threshold, Jth1 s, the system dynamics con- verge to a stationary solution in the easy-plane of the AFM, indicated by dashed blue, and dotted red set of arrows. An equivalent representation of the stationary solutions in terms of the staggered order parameters is presented in Fig. 2(b). n1andn2are perpendicular to each other with zero out-of-plane component for all val- ues of the input currents. The gold dash-dotted arrows passing through n1correspond to the stationary solu- tions given as '=1 2sin1 2!s !K , obtained analytically by setting both _ 'and 'as zero in Eq. (12). In positive chirality material, the average magnetization mis van- ishingly small in the stationary state. This can also be perceived from Eq. (10) as we do not consider any exter- nal eld. Since these materials have a two-fold symmetry, they also host '=1 2sin1 2!s !K stationary states. For the case of negative chirality, it can be observed from Fig. 2(c) that in equilibrium only the sublattice vector m3coincides with the its corresponding easy axis, whereas both m1andm2are oriented such that the en- ergy due to DMI is dominant over anisotropy, which in turn lowers the overall energy of the system. It can be observed from Fig. 2(d) that n1andn2are almost per- pendicular to each other. A small in-plane net magneti- zation exists in this case and is shown here as a zoomed in value (zoom factor = 100x) for the sake of compari- son to staggered order parameters. Due to the six-fold anisotropy dependence other equilibrium states, wherein either of m1orm2coincide with their easy axis while the other two sublattice vectors do not, also exist. Finally, due to the small anisotropy dependence, non-equilibrium stationary states exist for much lower currents [68] than those considered here, and therefore are not shown. For materials with positive chirality, the system be- comes unstable when the input spin current exceeds the threshold,Jth1 s. The resultant spin torque pushes the N eel vectors out of the easy-plane, and they oscillate around the spin polarization axis, np=z;with THz fre- quency due to strong exchange. This threshold current is given as [29, 44, 46] Jth1 s=da2e ~Ms !K 2=da2e ~Ke; (16) while the frequency of oscillation in the limit of large input current (neglecting the sin 2 'term) from Eq. (12)8 (a) (b) (c) (d)Positive Chirality Negative Chirality FIG. 4. The out-of-plane (z) component of the average magnetization, m, and n3for dierent values of input currents for both positive and negative chirality. Positive chirality: (a) When current is increased from zero but to a value below the threshold (0:95Jth1 s),mzis zero. However, it increases to a large value when Js=Jth1 s.mzdecreases again to a smaller value when the current is decreased to Js= 0:86Jth1 s. Finally, when the current is further reduced below the lower threshold to 0 :9Jth2 s, mzbecomes zero again. (b) n3is initially equal to 1 =p 2, but decreases in magnitude during the AFM dynamics since the magnitude of mincreases when the sublattice vectors move out of the plane. As soon as the current is lowered below Jth2 s, the system goes to a stationary state and n3= 1=p 2. Negative chirality: (c) Since the threshold current in this case is small, non-zeromzis observed for all values of current considered here. (d) n3decreases in magnitude when current increases but approaches1=p 2 for lower values of current. Here = 0:01, andda= 4 nm. is given as[44, 46] f=1 2!s =1 2~ 2e Js Msda1 : (17) Additionally, for small damping, there exists a lower threshold,Jth2 s< Jth1 s, which is equal to the current that pumps in the same amount of energy that is lost in one time period due to damping [29, 44, 46]. The lower threshold current is given as Jth2 s=da2e ~Ms 2 p!E!K=da2e ~2 p 6JKe:(18) The presence of two threshold currents enables energy- ecient operation of the THz oscillator in the hysteretic region [29]. The average frequency response as a function of the input spin current and the Fourier transform of the os- cillation dynamics is plotted in the left panel of Fig. 3 for materials with positive chirality. It can be observed from Fig. 3(a) that the fundamental frequency for dif- ferent lm thicknesses scales as predicted by Eq. (17) except for low currents near Jth2 swhere non-linearity in the form of higher harmonics appears as seen from the FFT response in Fig. 3(e). Next, Figs. 3(b), (f) show that the non-linearity in the frequency response for low input current increases as the value of the damping co- ecient increases. This is expected as the contribution from the uniaxial anisotropy (sin 2 'term) becomes sig- nicant owing to large damping and low current making the motion non-uniform ( '6= 0) [29]. In the case of materials with negative chirality, small equivalent anisotropy suggests that the threshold current for the onset of oscillations is very small, while the fre- quency of oscillations increases linearly with the spin cur- rent considered here and is given by Eq. (17). Indeed the same can be observed from Figs. 3(c), (d) where the re- sults of numerical simulations exactly match the analyticexpression. The FFT signal in Figs. 3(g), (h) contains only one frequency corresponding to uniform rotation of the order ( '= 0). This coherent rotation of the order parameter with such small threshold current [68] in AFM materials with negative chirality opens up the possibil- ity of operating such AFM oscillators at very low en- ergy for frequencies ranging from MHz-THz. It can also be observed from Fig. 3(b), (d) that for lower values of damping, such as = 0:005, the frequency of oscillations saturates for input current slightly above Jth1 s. This is because the energy pumped into the system is larger than that dissipated by damping. As a result the sublattice vectors move out of the easy-plane and get oriented along the spin polarization direction (slip- op). For larger val- ues of damping, the same would be observed for larger values of current. Finally, we would like to point out that the values of both Jth1 sandJth2 sobserved from numerical simulations were slightly dierent from their analytical values for dierent damping constants, similar to that reported in Ref. [43] for collinear AFMs. Figure 4 shows the out-of-plane (z) components of m andn3for non-zero currents for AFMs with dierent chi- ralities. For negative chirality materials, the steady-state zcomponent of both mandn3does not oscillate with time ( '0), whereas, for the case of positive chirality, the steady-state zcomponents of both mandn3show small oscillations with time ( '6= 0) similar to the case of NiO with spin polarization along the hard axis [29]. It can be observed from Fig. 4(a) that for positive chi- rality, as current increases from below the upper thresh- old current (0.95 Jth1 s) toJth1 s, the out-of-plane compo- nent of magnetization vectors and hence the average mag- netization mincreases from zero to a larger value. Due to the hysteretic nature of the AFM oscillator, the mag- nitude of mreduces when current is lowered but is non- zero as long as the input current is above Jth2 s. Similarly, it can be observed from Fig. 4(b) that the out-of-plane9 component of n3, which was initially 1 =p 2, decreases as the current increases above Jth1 s. When the current is lowered to a value below Jth1 s(0:86Jth1 shere) , the magnitude of the out-of-plane component of n3increases again and eventually saturates to 1 =p 2 when the current is lowered further below Jth2 s(0:9Jth2 s, here). It can also be observed from Fig. 4(c) that for negative chirality AFMs, the out-of-plane component of the av- erage magnetization although small is non-zero even for small currents due to the lower value of threshold current. On the other hand, nz 3in Fig. 4(d) decreases in magnitude from an initial value of = 1=p 2 to<1=p 2 as current increases sincekmkincreases. The values of current are assumed to be the same for both positive and negative chirality AFMs for the sake of comparison. Next, using (a) (b) (c) (d) FIG. 5.mzfor four dierent values of input current Js.mz increases with current and so does the frequency of oscillation. Here (a)Js= 1:6Jth2 s, (b)Js= 0:9Jth1 s. They are both inside the hysteretic region bounded by Jth2 sandJth1 s. (c)Js=Jth1 s, and (d)Js= 1:5Jth1 slies outside the hysteretic region. These results correspond to = 0:01, andda= 4 nm. Eq. (9) in Eq. (10), it can be shown that _ 'is directly pro- portional to mz[56]. Therefore, to present the features of angular velocity with input current, we show mzfor four dierent values of input current Jsfor positive chi- rality material in Fig. 5 . Here, Figs. 5(a)-(c) correspond to the hysteretic region, whereas Fig. 5(d) is for current outside the hysteretic region. As mentioned previously, an increase in current increases the spin torque on the sublattice vectors which leads to an increase in mzand hence _'. IV. SIGNAL EXTRACTION An important requirement for the realization of an AFM-based auto-oscillator is the extraction of the gen- erated THz oscillations as measurable electrical quanti-ties viz. voltage and current. It is expected for the ex- tracted voltage signal to oscillate at the same frequency as that of the N eel vector and contain substantial out- put power ( >1W) [69]. In this regard, the landmark theoretical work on NiO based oscillator [29] suggested the measurement of spin pumped [70] time varying in- verse spin Hall voltage [71] across the heavy metal (Pt) of a NiO=Pt heterostructure. However, the time varying voltage at THz frequency requires an AFM with signif- icant in-plane biaxial anisotropy [29, 69], thus limiting the applicability of this scheme to only select AFM ma- terials. In addition, the output power of the generated signal is sizeable (above 1 W) only for frequencies below 0:5 THz [69]. A potential route to overcoming the afore- mentioned limitations is coupling the AFM signal genera- tor to a high-Q dielectric resonator, which would enhance the output power even for frequencies above 0 :5 THz [41]. This method, however, requires devices with sizes in the 10's micrometers range for frequencies above 2 THz and for the AFMs to possess a tilted net magnetization in their ground state [69]. A more recent theoretical work on collinear AFM THz oscillators [43] suggested employ- ing Anisotropy Magnetoresistance (AMR) or Spin Mag- netoresistance (SMR) measurements in a four terminal AFM/HM spin Hall heterostructure. This would enable the extraction of the THz oscillations as longitudinal or transverse voltage signals. However, the reported values of both AMR and SMR at room temperatures in most AFMs is low and would, in general, require modulating the band structure for higher values [72]. Ideal Bias RLC LR(t) A TJ CbtLbt Pac(a) RLPac (b) Zth UthIread FIG. 6. (a) An equivalent circuit representation of Fig. 1 (adapted from [69]). The generation (write) current is not shown in the circuit, although its eect is included as a varia- tion in the resistance R(t) through its frequency dependence. (b) Thevenin equivalent of (a). A recent theoretical work [69] proposed employing a four terminal AFM tunnel junction (ATJ) in a spin Hall bilayer structure with a conducting AFM to eectively generate and detect THz frequency oscillations as vari- ations in the tunnel anisotropy magnetoresistance [54]. A DC current passed perpendicularly to the plane of the ATJ generates an AC voltage, which is measured across an externally connected load. It was shown that both the output power and its eciency decrease as fre- quency increases, nevertheless, it was suggested that this scheme could be used for signal extraction in the fre- quency range of 0 :110 THz, although the lateral size10 of the tunnel barrier required for an optimal performance depends on the frequency of oscillations (size decreases as the frequency increases) [69]. The analysis presented in Ref. [69], however, neglects the generation current com- pared to the read current while evaluating the eciency of power extraction. But it can be observed from the results in Section III C that the threshold current, and, therefore, the generation current, depend on AFM ma- terial properties, such as damping, anisotropy, and ex- change constants, and could be quite large. Therefore, in our work we include the eect of the generation current to accurately model the power eciency of the TAMR scheme. (a) (b) FIG. 7. (a) Output power and (b) eciency dependence on the area of cross-section of the tunnel barrier for dierent fre- quencies. The thickness of the barrier is xed to db= 1 nm. The eect of write current and the input power associated with it is not considered here, therefore, these results are in- dependent of the choice of the AFM material. In order to evaluate the performance of the TAMR scheme, an equivalent circuit representation (adapted from Ref. [69]) of the device setup of Fig. 1 is shown in Fig. 6(a), while its Thevenin equivalent representation is shown in Fig. 6(b). The circuits in Fig. 6 only repre- sent the read component, while the THz generation com- ponent is omitted for the sake of clarity. In Fig. 6(a), the dashed red box encloses a circuit representation of the ATJ, comprising a series combination of an oscillat- ing resistance R(t) =R0+
Rcos!tand inductance L=0db, connected in parallel to a junction capacitor C=Ac0=db(assumed parallel plate). The constant component, R0, in the oscillating resistance, R(t), is the equilibrium resistance of the MgO barrier and is given asR0=RA(0) exp(db) Ac. Here,RA(0) is the resistance- area product of a zero-thickness tunnel barrier, is the tunneling parameter, dbis the barrier thickness, and Ac is the cross-sectional area. The pre-factor,
R, of the time varying component of R(t) is the resistance variation due to the oscillation of the magnetization vectors with respect to the polarization axis.
R= (=(2 +))R0, whereis the TAMR ratio of the barrier and depends on the temperature and material properties. Due to the ow of the DC current, Iread, an alter- nating voltage develops across the ATJ, which is mea- sured across an externally connected load RL, separated from the ATJ via an ideal bias tee (enclosed in the green dashed box). The bias-tee, characterized by an induc- tanceLbtand a capacitance Cbt, and assumed to have noTABLE II. List of common antiferromagnetic materials Mn 3X and their associated parameters. Here Msis in Tesla,Keis in kJ=m3, andJis in MJ=m3. Sign ofDwhich decides the chirality is also mentioned. X M sKeJ D Ref. Ir 0:01 1:63 3000 240 - [56] Pt 0:013 1:37 10 280 - [73] Rh 0:013 2:00 10 230 - [73{75] Ga 0:008 0:54 100 110 + [75{78] Sn 0:003 0:50 110 59 + [67, 75, 77, 79] Ge 0:0009 0:28 1320 77 + [75, 77, 78, 80] GaN 0:1 0:69 10 280 - [81, 82] NiN 0:1 1:54 10 177 - [81, 82] voltage drop across it, blocks any DC current from ow- ing into the external load. Therefore, the AC voltage of the ATJ is divided only into its impedance (a combina- tion ofR0,L, andC) and that of the load RL[69]. Next, we simplify the ATJ circuit into a Thevenin impedance Zthand voltage Uthas shown in Fig. 6(b). They are evaluated as Zth=R0+j!L (1) +j; (19) and Uth=Uac (1) +j; (20) wherej=p1,!= 2f,=!2LC,=!R0C, and Uac=Iread
R. The output voltage and average power across the load can then be obtained as UL=UthRL Zth+RL=Uacr 1 +jp+r(1+j);(21) and PL=1 2jULj2 RL=U2 ac 2RLr2 1 +qr2+ 2r+p2; (22) wherer=RL=R0,q= (1)2+2, andp=!L R0. Finally, the eciency of the power extraction can be obtained as =PL Pin =0:5r 1 +qr2+ 2r+p21 I2 writeRGenR0=U2ac+ 1;(23) whereRGenis the resistance faced by the generation cur- rent. It can be observed from Eqs. (21)-(23) that the output voltage, output power, and the eciency of power extraction decrease with an increase in frequency since , ,q, andpincrease with ![69, 83]. Considering that the load impedance is xed to 50 by the external circuit, one can only optimize the source impedance to achieve PL>1W andUL>1 mV. In this regard, the resistance of the source tunnel barrier can11 TABLE III. Material Parameters of the NM, and at the NM/AFM interface. Parameters Values Ref. gM 3:81010S/m2[50] gm 3:8109S/m2[50] Cu 6109 m2[50] tCu 5 nm [50] be altered by either varying the thickness of the tunnel barrier,db, or its cross-sectional area, Ac. However, the optimum values of dbandAcfor the desired output sig- nals is frequency dependent, and, therefore, tunnel barri- ers of dierent sizes would be required for dierent oper- ating frequencies [69, 83]. For all estimates, we consider db= 1 nm,= 1:3,= 5:6 nm1,RA(0) = 0:14 m2, and= 9:8 [69]. For reliable operation of the tunnel bar- rier, we consider the electric eld across the barrier to be E= 0:3 V=nm [69], which is below the barrier break- down eld. Ignoring the eect of the generation current in Eq. (23), as suggested in Ref. [69], we deduce from Fig. 7 that the optimal cross-sectional area Ac0:36 m2forf= 0:1 THz,Ac0:25m2forf= 1 THz, Ac0:16m2forf= 10 THz. IrSnPtRhGeGaGaNNiN X10-710-510-3ζ(%) f=2.0THz FIG. 8. Power eciency for dierent materials Mn 3X listed in Table II. The dashed horizontal line shows the expected e- ciency of= 0:011% for the optimized geometry ( db= 1 nm, andAc= 0:24m2) if write current is neglected. The e- ciency, however, decreases signicantly due to the inclusion of write current. Here the AFM thin-lm thickness dais as- sumed to be 4 nm. Table II lists the material properties of various con- ducting AFMs. Depending on the sign of their DMI constant, these AFMs could host moments with either a positive or a negative chirality. The closed-form model presented in Eq. (17) can be used to evaluate the re- quired spin current for frequency f= 2 THz, regardless of the chirality since frequency scales linearly with theinput current in this region (see Fig. 3). For a given spin current density ( Js), the charge current density ( Jwrite) for the lateral spin-valve structure of Fig. 1(a) is given as Jwrite =gM+gm gMgmJs: (24) wheregMandgmare the conductance of the majority- and minority-spin electrons at the NM (Cu)/FM inter- face. The input power required to start the oscilla- tions is given as ( JwriteAc)2RCu, whereRGen=RCu= CuLCu ACu=CupAc tCupAcis the resistance of the copper (NM) underneath the bottom MgO. In order to evaluate the resistance of the copper layer, we have assumed its length and width to be the same as MgO and the AFM thin-lm. The eciency of power extraction for the listed AFM materials is presented in Fig. 8. The dashed horizon- tal line denotes the expected eciency of = 0:011% if the eect of generation current is neglected and the area of cross-section of MgO is optimized for f= 2:0 THz. However, it can be observed that the eciency decreases signicantly i.e. by a few orders when the input power due to the generation current in included in the analy- sis. For materials with large damping and large uniaxial anisotropy constants, the required generation current is higher leading to lower eciency. This result shows that further optimization of the device geometry for dierent materials is required to increase the eciency. This method of power extraction could be more suit- able for materials with negative chirality. We can observe from Fig. 9 that the output power as well as the eciency for both Mn 3Sn and Mn 3Ge for frequencies between 0.1 THz and 2.0 THz are signicant. The required genera- tion current for Mn 3Ge is smaller than that for Mn 3Sn, therefore, the eciency is higher for the former. Also, the eciency of power extraction increases with decrease in area of cross-section in both the cases but this is ac- companied by a decrease in output power. It might be possible to increase the output power and overall eciency of the system if the material properties of the tunnel barrier such as ;, andRA(0) could be altered. Large room temperature tunneling magnetore- sistance in an ATJ is feasible either by using a tunnel barrier other than MgO [84] or inserting a diusion bar- rier to enhance magneto-transport [85]. Here we adopted the TAMR extraction scheme because we have consid- ered metallic AFMs so a DC current through the ATJ structure can be easily applied. In addition, the three- or four-terminal compact ATJ structure along with its small lateral size enables dense packing of several such THz oscillators on a chip accompanied with a net in- crease in the output power and eciency of the oscillator array [72]. For example, with an array of 10 10 such AFM oscillators excited in parallel, the output power and eciency could be scaled up by 100 compared to the results presented in Figs. 7 and 8.12 (a) (b) (c) (d) FIG. 9. Upper panel: (a) Output power and (b) eciency for Mn3Sn. Lower panel: (c) Output power and (d) eciency for Mn3Ge. Two dierent cross-section size of the MgO barrier is considered. The output power depends only on the frequency of oscillation and therefore is same for both the materials. The eciency of power extraction depends on the generation current, which is lower for Mn 3Ge, leading to a higher value of eciency in that case. V. EFFECTS OF INHOMOGENEITY DUE TO EXCHANGE INTERACTION The results presented in Section III C correspond to the case of a single-domain AFM particle and are, there- fore, independent of the lateral dimensions of the thin- lm. This can also be deduced from the equations of the threshold current and the average oscillation frequency. However, when the lateral dimensions of the AFM thin- lm exceed several 10's of nm, micromagnetic analysis must be carried out. In this section, we analyze the dy- namics in thin-lm AFMs of varying dimensions within a micromagnetic simulation framework. We consider AFM thin-lms of dimensions 50 nm 50 nm and investigate the eect of the inhomogeneity due to exchange interac- tions. In each case, the thin-lm was divided into smaller cubes, each of size 1 nm 1 nmdanm, since the do- main wall width
0=p (2AiiAij)=(2Ke)>1 nm for Kecorresponding to Mn 3Ir as listed in Table II. It can be observed from Fig. 10 that for materials with positive chirality the eects of inhomogeneity becomes important for low currents. On the other hand, for materials with negative chirality, inhomogeneities do not appear to have any eect. For positive chirality materials, the numerical values of frequency for dierent spring constants deviates signicantly from that obtained from the single domain solution, as well as analytic results. In this case, the hys- teretic region reduces in size since the lower threshold current increases in magnitude as compared to the the- oretical prediction as can be observed from Fig. 10(a). Positive Chirality Negative Chirality (a) (b) (c) (d)FIG. 10. Frequency vs. input current for dierent values of inhomogeneous exchange constants (intra-sublattice (a, c), inter-sublattice (b, d)) for both positive and negative chirality. In all cases = 0:01, andda= 4 nm. Other parameters correspond to those of Mn 3Ir as listed in Table II for both positive and negative chirality materials with the exception of the sign ofDfor the latter. While we have not included the eect of inhomogeneous DMI in our work, we expect such interactions to lead to the formation of domain walls in the thin-lm similar to the case of collinear AFMs [43]. A more detailed analy- sis of the dynamics of the positive chirality materials due to variation in exchange interaction as well as inhomoge- neous DMI would be carried out in a future publication. VI. DISCUSSION We focused on the dynamics of the order parameters in exchange dominant non-collinear coplanar AFMs with both positive (+ =2) and negative (=2) chiralities as- sociated to the orientation of equilibrium magnetization vectors. In both these classes of AFMs, the exchange en- ergy is minimized for a 2 =3 relative orientation between the sublattice vectors. Next, the negative (positive) sign of the iDMI coecient minimizes the system energy for counterclockwise (clockwise) ordering of m1;m2, andm3 in the xyplane leading to positive (negative) chiral- ity. Finally, all the sublattice vectors coincide with their respective easy axis only in the case of the positive chiral- ity materials due to the relative anticlockwise orientation of the easy axes. On the other hand, the negative chi- rality materials have a six-fold symmetry wherein only one of the sublattice vectors can coincide with its respec- tive easy axis. As a result, these AFM materials with dierent chiralities have signicantly distinct dynamics in the presence of an input spin current. For AFM ma- terials with + =2 chirality, oscillatory dynamics are ex-13 cited only when the injected spin current overcomes the anisotropy, thus indicating the presence of a larger cur- rent threshold. Moreover, the dynamics in such AFMs is hysteretic in nature. Therefore, it is possible to sustain oscillations by lowering the current below that required to initiate the dynamics as long the energy pumped in by the current overcomes that dissipated by damping. On the other hand, in the case of =2 chirality AFMs, os- cillations can be excited when signicantly smaller spin current with appropriate spin polarization is injected into the AFM. Hence, =2 chirality AFMs may be more amenable to tuning the frequency response over a broad frequency range, from the MHz to the THz range [68]. The oscillation of the AFM N eel vectors can be mea- sured as a coherent AC voltage with THz frequencies across an externally connected resistive load through the tunnel anisotropic magnetoresistance measurements for both +=2 and=2 chirality materials. In general, as the frequency increases, the magnitude of both the out- put power and the eciency of power extraction decrease, however, it is possible to enhance both these quantities by optimizing the cross-sectional area of the tunnel junc- tion. This, however, is limited due to larger threshold current requirement for materials with large damping. Therefore, a hybrid scheme of electrically synchronized AFM oscillators on a chip could be used to further en- hance the power and eciency [86, 87]. (a) (b) FIG. 11.mzfor larger damping, = 0:1, andda= 4 nm. (a) Non-coherent (spike-like) signals near the threshold current Js= 1:1Jth1 s. (b) Coherent signal for larger current Js= 1:5Jth1 s. The angular frequency is directly proportional to mz, and therefore it would show the exact same features (in the absence of any external eld) for the chosen values of current. Metallic AFMs such as Mn 3Ir and Mn 3Sn could be considered as examples of + =2 and=2 chiralities, respectively. Recently, thin-lms with dierent thickness ranging from 1 nm to 5 nm of both these materials have been grown using UHV magnetron sputtering [88{91]. In addition, dierent values of damping constants have been reported for Mn 3Sn [56, 77]. Therefore, we expect the results presented in Sections III C, IV and V to be useful for benchmarking THz dynamics in experimental set-ups with such thin lms metallic antiferromagnets. (a) (b) (c) (d)FIG. 12. Time dynamics (single and train of spikes) of a single \neuron" for dierent input currents and frequencies. The net input current should be greater than the threshold current ( > 0:2) for a non-zero dynamics. For an input current above the threshold, as the external frequency increases the dynamics changes from (a) bursts of spikes to (c) single spikes to (d) no spikes ( = 0:3). As the input current increases to = 0:4 the range of external frequency where the spiking behaviour is observed increases. VII. POTENTIAL APPLICATIONS Neurons in the human brain could be thought of as a network of coupled non-linear oscillators, while the stim- uli to excite neuronal dynamics is derived from the neigh- boring neurons in the network [8, 92{94]. For materials with +=2 chirality, a non-linear behaviour was observed for large damping, and input currents near the threshold current,Jth1 s, in Fig. 3(b), (f). This non-linearity cor- responds to Dirac-comb-like magnetization dynamics, as shown in Fig. 11(a), and is similar to the dynamics of biological neurons in their spiking behaviour as well as a dependence on the input threshold. However, unlike a biological neuron which shows various dynamical modes such as spiking, bursting, and chattering [95], the dy- namics here shows only spikes and does not show any refractory (\resting") period. Recent works [96, 97] have shown that it is possible to generate single spiking as well as bursting behaviours using NiO-based AFM oscil- lators by considering an input DC current below Jth1 s, and superimposing it with an AC current. As the AC current changes with time, the total current could either go above the threshold, thereby triggering a non-linear response, or below the threshold current resulting in a \resting" period. Here we explore the possibility of spik- ing behaviours in + =2 chirality materials such as Mn 3Ir under the eect of an input spin current. We use the non- linear pendulum model of Eq. (12) and study the possible dynamics in case of a single oscillator, two unidirectional coupled oscillators, and two bidirectional coupled oscil-14 lators. A. Ultra-fast Hardware Emulator of Neurons We consider a large damping of = 0:1 while the other material parameters correspond to that of Mn 3Ir as listed in Table II. Next we choose an input current Js(t) = Jdc s+Jac s(t), whereJdc s= 0:8Jth1 sis the dc component of the input current, superimposed with a smaller ac signal Jac s(t) =Jth1 scos(2fact). The time dynamics of this non-linear oscillator is governed by '+!E_'+!E!K 2sin 2'+!E!s(t) = 0; (25) where!s(t)(/Js(t)) is the time varying input current. (a) (b) FIG. 13. The dynamics of two neuron system with unidirec- tional coupling at fac= 60 GHz, and = 0:3. The dotted blue curve corresponds to the rst neuron. (a) Second neuron shows no spike for = 0:028 but a single spike for = 0:032. (b) The single spiking behaviour changes to bursts with three spikes asincreases and coupling strengthens. Figure 12 presents the dynamics of Eq. (25) for dif- ferent input current and frequencies. Firstly, it can be observed that the input current must be greater than the threshold current to excite any dynamics viz.  must be greater than 0.2 (dotted line corresponding to = 0:2 shows no spikes for any value of external fre- quency). Secondly, for input currents above the thresh- old viz.=f0:3;0:4g, a train of spikes is observed for lower frequency of 20 GHz in Fig. 12(a). However, as frequency of the input excitation increases the number of observed spikes decreases for both values of current con- sidered here (Fig. 12(b, c)). Finally, it can be observed from Fig. 12(d) that for very large frequency the spiking behaviour vanishes for lower current ( = 0:3) but per- sists for higher current ( = 0:4). For higher values of current, the cut-o frequency is higher. This observed spiking behaviour is indeed similar to that of biological neurons [95]. Here, however, the observed dynamics is very fast in the THz regime and thus the AFM oscilla- tors could be used as the building blocks of an ultra-high throughput brain-inspired computing architecture. (a) (b) (c) (d)FIG. 14. The dynamics of two neuron system with unidirec- tional coupling at fac= 180 GHz. The dashed blue curve corresponds to the rst neuron. = 0:3: (a) Single spike for= 0:04 but not for = 0:036. (b) The single spik- ing behaviour become prominent as the coupling strengthens. = 0:4: (c) Single spike for = 0:032 in response to a double spiking behaviour of the rst neuron. (d) For larger second neuron shows bursting dynamics with two spikes. B. Two unidirectional coupled articial neurons A network composed of interacting oscillators forms the basis of the oscillatory neurocomputing model pro- posed by Hoppensteadt and Izhikevich [98]. In such a network, the dynamics of an oscillating neuron (or a \node") is controlled by the incoming input signal as well as its coupling to neighboring neurons. To inves- tigate this coupling behaviour we consider a system of two unidirectional coupled neurons. The rst neuron is driven by an external signal and its dynamics is governed by Eq. (25). The dynamics of the second neuron, on the other hand, depends on the output signal of the rst neu- ron as well as the coupling between the two neurons. It is governed by 'j+!E_'j+!E!K 2sin 2'j+!E!s ij!E_'isgn(!s) = 0; (26) whereij=is the unidirectional coupling coecient from neuron i= 1 toj= 2. There is no feedback from the second neuron to the rst and therefore ji= 0. In addition to the input from the rst neuron, the second neuron is also driven by a constant DC current Jdc s2(/!s in Eq. (26)). We choose this DC current to be the same as that for the rst neuron viz. Jdc s2= 0:8Jth1 s. The dynamics of the second neuron for two dierent external input currents ( =f0:3;0:4g) and frequencies ( fac= f60;180gGHz) is presented in Figs. 13 and 14.15 Firstly, it can be observed that in all the cases the sec- ond neuron shows a spiking behaviour only for above a certain value. Secondly, for = 0:3 andfac= 60 GHz, wherein the rst neuron shows bursting behaviour con- sisting of three spikes, the second neuron shows a single spike (Fig. 13(a)) for lower value of , and three spikes for stronger coupling (Fig. 13(b)). This behaviour is due to the threshold dependence of the second neuron as well as due to its inertial dynamics. Similar behaviour is also observed for = 0:4, andfac= 180 GHz in Figs. 14(c), (d). Thirdly, for = 0:3 andfac= 180 GHz, wherein the rst neuron shows a single spike, Fig. 14(a) shows that compared to the case of fac= 60 GHz a slightly higher value ofis now required to excite the second neuron. The single spiking behaviour of the second neuron be- comes more prominent as the coupling strength increases because of a stronger input as shown in Fig. 14(b). Re- cently, it was suggested that this coupled behaviour of THz articial neurons could be used to build ultra-fast multi-input AND, OR, and majority logic gates [96]. (a) (b) (c) (d) FIG. 15. The dynamics of two neuron system with bidirec- tional coupling at fac= 180 GHz, and = 0:3. First neu- ron shows bursting behaviour in this system while the second neuron follows the rst neuron for all values of . As the cou- pling between the two neurons increase the number of spikes for both the neurons increases. C. Two bidirectional coupled articial neurons In some circuits it is possible that the coupling be- tween any two neurons is bidirectional. In such cases, in addition to a forward coupling from the rst neuron to the second, a feedback exists from the second neuron to the rst. The dynamics of each neuron of this coupled system is governed by Eq. (26), however, !s=!s(t) for the rst neuron, as discussed previously. We consider 12=21=. Figures 15 and 16 show the dynamics of the two neurons of this coupled system with the couplingatfac= 180 GHz for = 0:3 and 0.4, respectively. (a) (b) (c) (d) FIG. 16. The dynamics of two neuron system with bidirec- tional coupling at fac= 180 GHz, and = 0:4. Second neuron res when the coupling is above a certain threshold which in turn leads to another spike for the rst neuron. As the coupling between the two neurons increase the number of spikes for both the neurons increases. Firstly, Fig. 15(a) shows that for = 0:04 the dynam- ics of both _'1and _'2are almost similar to that presented in Fig. 14(a), viz. the eects of coupling is very small. However, as the coupling between the two neurons in- creases (Fig. 15(b)-(d)), a positive feedback is established between the two neuron leading to dynamics with two or more spikes, in general. This is observed after the sec- ond neuron has red, at least once, because the positive feedback leads to a net input greater than the threshold current to the rst neuron, even though the external in- put has reduced below the threshold. Similar behavior is also observed in the case of = 0:4, although at lower values of coupling, as presented in Fig. 16. The results allude to the threshold behaviour of the neurons, inertial nature of the dynamics, and a dependence of the dynam- ics on the phase dierence between the two neurons. The dynamics of two bidirectional coupled articial neurons presented here could be the rst step towards building AFM-based recurrent neural networks or reservoir com- puting [99], instead of the slower FM-based coupled os- cillator systems [100, 101]. VIII. CONCLUSION In this work, we numerically and theoretically explore the THz dynamics of thin-lm metallic non-collinear coplanar AFMs such as Mn 3Ir and Mn 3Sn, under the action of an injected spin current with spin polarization perpendicular to the plane of the lm. Physically, these two AFM materials dier in their spin conguration viz. positive chirality for Mn 3Ir, and negative chirality for16 Mn3Sn. In order to explore the dynamics numerically, we solve three LLG equations coupled to each other via inter-sublattice exchange interactions. We also analyze the dynamics theoretically in the limit of strong exchange and show that it can be mapped to that of a damped- driven pendulum if the eects of inhomogeneity in the material are ignored. We nd that the dynamics of Mn 3Ir is best described by a non-linear pendulum equation and has a hysteretic behaviour, while that of Mn 3Sn in the THz regime is best described by a linear pendulum equa- tion and has a signicantly small threshold for oscillation. The hysteretic dynamics in the case of Mn 3Ir allows for possibility of energy ecient THz coherent sources. On the other hand, a small threshold current requirement in the case of Mn 3Sn indicates the possibility of e- cient coherent signal sources from MHz to THz regime. We employ the TAMR detection scheme to extract the THz oscillations as time-varying voltage signals across an external resistive load. Including inhomogeneous ef- fects leads to a variation in the dynamics | the lowerthreshold current for sustaining the dynamics increases, the hysteretic region reduces, and the frequency of oscil- lation decreases for lower current levels. Finally, we also show that the non-linear behaviour of positive chirality materials with large damping could be used to emulate articial neurons. An interacting network of such oscil- lators could enable the development of neurocomputing circuits for various cognitive tasks. The device setup and the results presented in this paper should be useful in designing experiments to further study and explore THz oscillations in thin-lm metallic AFMs. ACKNOWLEDGEMENTS This research is funded by AFRL/AFOSR, under AFRL Contract No. FA8750-21-1-0002. The authors also acknowledge the support of National Science Foun- dation through the grant no. CCF-2021230. Ankit Shukla is also grateful to Siyuan Qian for fruitful dis- cussions. [1] M. Tonouchi, Cutting-edge terahertz technology, Nature photonics 1, 97 (2007). [2] J. Walowski and M. M unzenberg, Perspective: Ultra- fast magnetism and thz spintronics, Journal of Applied Physics 120, 140901 (2016). [3] D. M. 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2021-06-16

We theoretically and numerically study the terahertz auto oscillations in thin-film metallic non-collinear coplanar antiferromagnets (AFMs), such as $\mathrm{Mn_{3}Sn}$ and $\mathrm{Mn_{3}Ir}$, under the effect of anti-damping spin-torque with spin polarization perpendicular to the plane of the film. To obtain the order parameter dynamics in these AFMs, we solve three Landau-Lifshitz-Gilbert equations coupled by exchange interactions assuming both single- and multi-domain (micromagnetics) dynamical processes. In the limit of strong exchange interaction, the oscillatory dynamics of the order parameter in these AFMs, which have opposite chiralities, could be mapped to that of a linear damped-driven pendulum in the case of $\mathrm{Mn_{3}Sn}$, and a non-linear damped-driven pendulum in case of $\mathrm{Mn_{3}Ir}$. The theoretical framework allows us to identify the input current requirements as a function of the material and geometry parameters for exciting an oscillatory response. We also obtain a closed-form approximate solution of the oscillation frequency for large input currents in case of both $\mathrm{Mn_{3}Ir}$ and $\mathrm{Mn_{3}Sn}$. Our analytical predictions of threshold current and oscillation frequency agree well with the numerical results and thus can be used as compact models to design and optimize the auto oscillator. Employing a circuit model, based on the principle of tunnel anisotropy magnetoresistance, we present detailed models of the output power and efficiency versus oscillation frequency of the auto oscillator. Finally, we explore the spiking dynamics of two unidirectional as well as bidirectional coupled AFM oscillators using non-linear damped-driven pendulum equations.

Spin-Torque-driven Terahertz Auto Oscillations in Non-Collinear Coplanar Antiferromagnets

2106.08528v2

STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING KAIS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL Abstract. In this paper we study the stability problem of a tree of elastic strings with local Kelvin-Voigt damping on some of the edges. Under the compatibility condition of displacement and strain and continuity condition of damping coecients at the vertices of the tree, exponential/polynomial stability are proved. Our results generalizes the cases of single elastic string with local Kelvin-Voigt damping in [21, 24, 5]. Contents 1. Introduction 1 2. Well-posedness of the system 4 3. Asymptotic behaviour 6 4. Further comments: graph case 14 References 16 1.Introduction In this paper, we investigate the asymptotic stability of a tree of elastic strings with local Kelvin-Voigt damping. We rst introduce some notations needed to formulate the problem under consideration. Let Tbe a tree ( i.e.Tis a planar connected graph without closed paths). degree of a vertex { number of incident edges at that vertex R{ root ofT, a designated vertex with degree 1 exterior vertex { vertex with degree 1 interior vertex { vertex with degree greater than 1 e{ the edge incident the root R O{ the vertex ofTother thanR { multi-index of length k, = (1;


;k) 2010 Mathematics Subject Classication. 35B35, 35B40, 93D20. Key words and phrases. Tree, dissipative wave operator, Kelvin-Voigt damping, Frequency approach. 1arXiv:1803.07280v1 [math.AP] 20 Mar 20182 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL O{ vertex ofTwith index   e{ edge ofTwith index   M{ set of the interior vertices of T S{ set of the exterior vertices of T, excludingR IM{ index set ofM IS{ index set ofS We choose empty index for edge eand vertexO. Assume there are medges, dierent from e;that branch out from O;we denote these edges by eo; = 1;:::;m and the other vertex of the edge eobyOo, i.e. the interior vertex O, contained in the edge e;has multiplicity equal tom+ 1. Furthermore, length of the edge eis denoted by `. Then,emay be parametrized by its arc length by means of the functions , dened in [0 ;`] such that (`) =Oand(0) is the other vertex of this edge. •R •O• O1 •O2•O1,1 • O2,2e•O1,2 • O2,1Dirichlet boundary condition Dirichlet boundary conditions Figure 1. A Tree-Shaped network Now, we are ready to introduce a planar tree-shaped network of Nelastic strings, where N2, see [20, 22, 26, 17, 19] and [16] concerning the model. More precisely, we consider the following initial and boundary value problem : (1.1)@2u @t2(x;t)@ @x@u @x+a(x)@2u @x@t (x;t) = 0;0<x<` ; t> 0;2I:=IM[IS; (1.2) u(0;t) = 0; u(`;t) = 0;2IS; t> 0; (1.3) u(0;t) =u(`;t); t> 0; = 1;2;:::;m ;2IM; (1.4) mX =1@u @x(0;t) +a(0)@2u @x@t(0;t) =@u @x(`;t) +a(`)@2u @x@t(`;t); t> 0;2IM;STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 3 (1.5) u(x;0) =u0 (x);@u @t(x;0) =u1 (x);0<x<` ;2I; whereu: [0;`](0;+1)!R;2I;be the transverse displacement with index  ,a2 L1(0;`) and, either ais zero, that is, eis a purely elastic edge, or a(x)>0, on (0;`). Such edge will be called a K-V edge. This setting also includes the case that a(x)>0 only on subintervals of (0 ;`), since we then can consider this edge as the union of pure elastic edges and K-V edges. We assume thatTcontains at least one K-V edge. Furthermore, we suppose that every maximal subgraph of purely elastic edges is a tree, whose leaves are attached to K-V edges. Models of the transient behavior of some or all of the state variables describing the motion of exible structures have been of great interest in recent years, for details about physical motivation for the models, see [16], [20] and the references therein. Mathematical analysis of transmission partial dierential equations is detailed in [20]. For the feedback stabilization problem the wave or Schr odinger equations in networks, we refer the readers to references [7]-[11], [20]. Our aim is to prove, under some assumptions on damping coecients a, 2I, exponential and polynomial stability results for the system (1.1)-(1.5). We dene the natural energy E(t) of a solution u= (u)2Iof (1.1)-(1.5) by (1.6) E(t) =1 2X 2IZ` 0 @u @t(x;t)2 +@u @x(x;t)2! dx: It is straightforward to check that every suciently smooth solution of (1.1)-(1.5) satises the following dissipation law (1.7)d dtE(t) =X 2IZ` 0a(x)@2u @x@t(x;t)2 dx0; and therefore, the energy is a nonincreasing function of the time variable t. The main results of this paper then concern the precise asymptotic behavior of the solutions of (1.1)-(1.5). Our technique is a special frequency domain analysis of the corresponding operator. This paper is organized as follows: In Section 2, we give the proper functional setting for system (1.1)-(1.5) and prove that the system is well-posed. In Section 3, we analyze the resolvent of the wave operator associated to the dissipative system (1.1)-(1.5) and prove the asymptotic behavior of the corresponding semigroup. In the last section we give some comments on the cases of more general graph for the network.4 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL 2.Well-posedness of the system In order to study system (1.1)-(1.5) we need a proper functional setting. We dene the following space H=VH whereH=Y 2IL2(0;`) andV=( u2Y 2IH1(0;`) :u(0) = 0; u(`) = 0;2IS;satises (2.8)) (2.8) u(0) =u(`); = 1;:::;m ;2IM; and equipped with the inner products (2.9) <u;~u>H=X 2IZ` 0(u(x)~u(x) +u0 (x)~u0(x))dx: System (1.1)-(1.5) can be rewritten as the rst order evolution equation (2.10)8 >>>< >>>:@ @t0 @u @u @t1 A=Ad0 @u @u @t1 A; u(0) =u0;@u @t=u1 where the operator Ad:D(Ad)H!H is dened by Ad0 @u v1 A:=0 @v (u0+av0)01 A; with a:= (a)2Iandav0:= (av0 )2I; and D(Ad) :=( (u;v)2H; v2V;(u0+av0)2Y 2IH1(0;`) : (u;v) satises (2 :11)) ; (2.11)mX =1u(0) +a(0)v0 (0) =u0 (`) +a(`)v0 (`);2IM: Lemma 2.1. The operatorAdis dissipative,f0;1g(Ad) :the resolvent set of Ad: Proof. For (u;v)2D(Ad);we have Re(hAd(u;v);(u;v)iH) =ReX 2I Z` 0v0 u0 dx+Z` 0(u0 +av0 )0vdx! : Performing integration by parts and using transmission and boundary conditions, a straightfor- ward calculations leads to Re(hAd(u;v);(u;v)iH) =X 2IZ` 0a(x)jv0 (x)j2dx0STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 5 which proves the dissipativeness of the operator AdinH: Next, using Lax-Milgram's lemma, we prove that 1 2(Ad):For this, let ( f;g)2H and we look for (u;v)2D(Ad) such that (IAd)(u;v) = (f;g) which can be written as uv=f;2I; (2.12) v(u0 +av0 )0=g;2I: (2.13) Letw2V; multiplying (2.13) by w, then summing over  2I, we obtain X 2IZ` 0vwdxX 2I[(u0 +av0 ) (x)w(x)]` 0+ (2.14)X 2IZ` 0(u0 +av0 )w0 dx=X 2IZ` 0gwdx: Replacingvin the last equality by (2.12), we get (2.15) '(u;w) = (w) where '(u;w) =X 2I Z` 0uwdx+Z` 0(1 +a)u0 w0 ! and (w) =X 2I Z` 0(f+g)wdx+Z` 0af0 w0 dx! : The fuction 'is a continuous sesquilinear form on VVand is a continuous anti-linear form onV; hereVis equipped with the inner product f;g =X 2I Z` 0uwdx+Z` 0u0 w0 ! : Since'is coercive on V;the conclusion is deduced by the Lax-Milgram lemma : By the same why we prove that 0 2(Ad).  By the Lumer-Phillip's theorem (see [27, 29]), we have the following proposition. Proposition 2.2. The operatorAdgenerates aC0-semigroup of contraction (Sd(t))t0on the Hilbert spaceH. Hence, for an initial datum (u0;u1)2H, there exists a unique solution u;@u @t 2C([0;+1);H) to problem (2.10). Moreover, if (u0;u1)2D(Ad), then  u;@u @t 2C([0;+1);D(Ad)):6 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL Furthermore, the solution ( u;@u @t) of (1.1)-(1.5) with initial datum in D(Ad) satises (1.7). There- fore the energy is decreasing. 3.Asymptotic behaviour In order to analyze the asymptotic behavior of system (1.1)-(1.5), we shall use the following characterizations for exponential and polynomial stability of a C0-semigroup of contractions: Lemma 3.1. [18, 28] AC0-semigroup of contractions (etA)t0dened on the Hilbert space H is exponentially stable if and only if (3.16) iR(A) and (3.17) lim sup jj!+1 (iIA )1 L(H)<1 Lemma 3.2. [14]AC0-semigroup of contractions (etA)t0on the Hilbert space Hsatises etAy0 C t1 ky0kD(A) for some constant C > 0and for>0if and only if (3.16) holds and (3.18) lim sup jj!+11  (iIA )1 L(H)<1: Lemma 3.3 (Asymptotic stability) .The operatorAdveries (3.16) and then the associated semigroup (Sd(t))t0is asymptotically stable on H. Proof. Since 02(Ad) we only need here to prove that ( iIAd) is a one-to-one correspondence in the energy space Hfor all2R. The proof will be done in two steps: in the rst step we will prove the injective property of ( iIAd) and in the second step we will prove the surjective property of the same operator. Suppose that there exists 2Rsuch thatKer(iIAd)6=f0g. So=iis an eigenvalue ofAd;then let (u;v) an eigenvector of D(Ad) associated to :For every in Iwe have v=iu; (3.19) (u0 +av0 )0=iv: (3.20) We have hAd(u;v);(u;v)iH=X 2IZ` 0ajv0 j2dx= 0: Thenav0 = 0 a.e. on (0 ;`).STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 7 Letea K-V edge. According to (3.19) and the fact that av0 = 0 a.e. on (0 ;`); we haveu0 = 0 a.e. on !:Using (3.20), we deduce that v= 0 on!:Return back to (3.19), we conclude that u= 0 on!: Puttingy=u0 +av0 = (1 + ia)u0 ;we havey2H2(0;`) andy0=2u: Henceysatises the Cauchy problem y00+2 1 +iay= 0; y(z0) = 0; y0(z0) = 0 for somez0in!:Thenyis zero on (0 ;`) and hence u0 anduare zero on (0 ;`). Moreoveruandu0 +av0 vanish at 0 and at `. Ifeis a purely elastic edge attached to a K-V edge at one of its ends, denoted by x;thenu(x) = 0; u0 (x) = 0:Again, by the same way we can deduce that u0 and uare zero in L2(0;`) and at both ends of e. We iterate such procedure on every maximal subgraph of purely elastic edges of T(from leaves to the root), to obtain nally that (u;v) = 0 inD(Ad);which is in contradiction with the choice of ( u;v): Now given ( f;g)2H, we solve the equation (iIAd)(u;v) = (f;g) or equivalently, (3.21)8 < :v=iuf 2u+u00+i(au0)0= (af0)0ifg: Let's dene the operator Au=u00i(au0)0;8u2V: It is easy to show that Ais an isomorphism from VontoV0(whereV0is the dual space ofVobtained by means of the inner product in H). Then the second line of (3.21) can be written as follow (3.22) u2A1u=A1 g+if(af0)0
: Ifu2Ker(I2A1), then2uAu= 0. It follows that (3.23) 2u+u00+i(au0)0= 0: Multiplying (3.23) by uand integrating over T, then by Green's formula we obtain 2X 2IZ` 0ju(x)j2dxX 2IZ` 0ju0 (x)j2dxiX 2IZ` 0a(x)ju0 (x)j2dx= 0: This shows that X 2IZ` 0a(x)ju0 (x)j2dx= 0;8 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL which imply that au0= 0 inT. Inserting this last equation into (3.23) we get 2u+u00= 0; inT: According to the rst step, we have that Ker( I2A1) =f0g. On the other hand thanks to the compact embeddings V ,!HandH ,!V0we see that A1is a compact operator in V. Now thanks to Fredholm's alternative, the operator ( I2A1) is bijective in V, hence the equation (3.22) have a unique solution in V, which yields that the operator ( iIAd) is surjective in the energy space H. The proof is thus complete.  Before stating the main results, we dene a property (P) on aas follows (P)82I; a0 ;a00 2L1(0;`) and82IM; a0 (`)mX =1a0 (0)0: Theorem 3.4. Suppose that the function asatises property (P), then (i)Ifais continuous at every inner node of Tthen (Sd(t))t0is exponentially stable on H. (ii)Ifais not continuous at least at an inner node of Tthen (Sd(t))t0is polynomially stable onH, in particular there exists C > 0such that for all t>0we have eAt(u0;u1) HC t2 (u0;u1) D(A);8(u0;u1)2D(A): Proof. According to Lemma 3.1, Lemma 3.2, and Lemma 3.3, it suces to prove that for = 0, whenais continous at every inner node, or = 1=2, whenais not continuous at an inner node, there exists r>0 such that (3.24) inf k(u;v)kH;2R k(iIAd)(u;v)kHr: Suppose that (3.24) fails. Then there exists a sequence of real numbers n, withn!1 (without loss of generality, we suppose that n>0 ), and a sequence of vectors ( un;vn) in D(Ad) withk(un;vn)kH= 1 such that (3.25)  nk(inIAd)(un;vn)kH!0: We shall prove that k(un;vn)kH=o(1);which contradict the hypotheses on ( un;vn): Writing (3.25) in terms of its components, we get for every  2I;  n(inu;nv;n) =:f;n=o(1) inH1(0;`); (3.26)  n(inv;n(u0 ;n+av0 ;n)0) =:g;n=o(1) inL2(0;`): (3.27)STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 9 Note that  nX 2IZ` 0a(x)jv0 (x)j2dx=Re(h n(inIAd)(un;vn);(un;vn)iH) =o(1): Hence, for every  2I (3.28)  2n a1 2 v0 ;n L2(0;`)=o(1): Then from (3.26), we get that (3.29)  2n a1 2 nu0 ;n L2(0;`)=o(1): DeneT;n= (u0 ;n+av0 ;n) and multiplying (3.27) by  nqT;nwhereqis any real function inH2(0;`), we get (3.30) ReZ` 0inv;nqT;ndxReZ` 0T0 ;nqT;ndx=o(1): Using (3.26) we have ReZ` 0inv;nqT;ndx =ReZ` 0v;nq(v0 ;n+ nf0 ;n)dx+ReZ` 0inv;nqav0 ;ndx =1 2h q(x)jv;n(x)j2i` 0+1 2Z` 0q0jv;nj2dxImZ` 0qanv;nv0 ;ndx+o(1): (3.31) On the other hand, integrating the second term in (3.30) by parts, yields (3.32) ReZ` 0T0 ;nqT;ndx=1 2h q(x)jT;n(x)j2i` 01 2Z` 0q0jT;nj2dx: Hence, by substituing (3.31) and (3.32) into (3.30), we obtain 1 2Z` 0q0jv;nj2dx+1 2Z` 0q0jT;nj2dxImZ` 0qanv;nv0 ;ndx 1 2h q(x)jv;n(x)j2i` 0+h q(x)jT;n(x)j2i` 0 =o(1): (3.33) Lemma 3.5. The following property holds (3.34) ImZ` 0qanv;nv0 ;ndx=o(1): Proof. Since 2na1 2 v0 ;n!0 inL2(0;`) andqa1 2 2L1(0;`);it suces to prove that (3.35) 1 2n a1 2 v;n L2(0;`)=O(1): For this, taking the inner product of (3.27) by i12 nav;nleads to (3.36)2 n a1 2 v;n 2 L2(0;`)=i1 nZ` 0T0 ;nav;ndxi12 nZ` 0g;nav;ndx:10 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL Sincea2L1(0;`) andg;n!0 inL2(0;`) we can deduce the inequality (3.37)Re(i12 nZ` 0g;nav;ndx)1 42 n a1 2 v;n 2 L2(!)+o(1): On the other hand, we have Re i1 nZ` 0T0 ;nav;ndx! =Re i1 nT;n(x)a(x)v;n(x)` 0 +Re" i1 nZ` 0 a0 u0 ;nv;n+aa0 v0 ;nv;n+au0 ;nv0 ;n dx# : (3.38) Using (3.28) and (3.29) we have (3.39) Re i1 nZ` 0au0 ;nv0 ;ndx! =o(1): Using again (3.28) and the fact that a0 2L1(0;`);we conclude that (3.40) Re i1 nZ` 0aa0 v0 ;nv;ndx! 1 42 n a1 2 v;n 2 L2(0;`)+o(1): Now by (3.27), we obtain after integrating by parts that Re" i1 nZ` 0a0 u0 ;nv;ndx# =Re"  nZ` 0a0 (v0 ;n+ nf0 ;n)v;ndx# =1 2h  na0 (x)jv;n(x)j2i` 01 2 nZ` 0a00 jv;nj2dx+o(1): Furthermore, using that a00 2L1(0;`) and thatv;nis bounded, we deduce (3.41) Re" i1 nZ` 0a0 u0 ;nv;ndx# 1 2h  na0 (x)jv;n(x)j2i` 0+O(1): Combining (3.39), (3.40), (3.41) with (3.38), we get Re(i1 nZ` 0T0 ;nav;ndx) Re i1 nT;n(x)a(x)v;n(x)` 0 +1 2h  na0 (x)jv;n(x)j2i` 0+1 42 n a1 2 v;n 2 L2(0;`)+O(1): (3.42) Thus, substituting (3.37) and (3.42) into (3.36) leads to 1 22 n a1 2 v;n 2 L2(0;`) Re i1 nT;n(x)a(x)v;n(x)` 0 +X 2I a0 (`)jv;n(`)j2a0 (0)jv;n(0)j2 +O(1) (3.43) Case (i): Here ais continuous in all nodes. For = 0, it follows from (3.43) that (3.44)X 2I2 n a1 2 v;n 2 L2(0;`)2X 2I a0 (`)jv;n(`)j2a0 (0)jv;n(0)j2 +O(1):STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 11 We have used the continuity condition of vandaand the compatibilty condition (1.5) at inner nodes and the Dirichlet condition of uandvat externel nodes. To conclude, notice that from the property (P) we deduce that X 2I a0 (`)jv;n(`)j2a0 (0)jv;n(0)j2 0: Then, (3.44), yields 2 n a1 2 v;n 2 L2(0;`)=O(1) for every 2I;and the proof of Lemma 3.5 is complete for case (i). Case (ii): Recall that here the function ais not continuous at some internal nodes. For =1 2, we want estimate the rst term in the right hand side of (3.43). To do this it suces to estimate Re(i1 nT;n(x)a(x)v;n(x)) at an inner node x=xwhena(x)6= 0. For simplicity and without loss of generality we suppose that xis the end of e;nidentied to 0 via . Sinceais continuous on [0 ;`], there exists a positive number k<`such thata(x)6= 0 on [0;`]. We rst prove (3.45) nkv;nk2 L2(0;k)=o(1): We need the following Gagliardo-Nirenberg inequality [25] in estimation: There exists two positives constants C1andC2such that, for any winH1(0;k), (3.46) kwkL1(0;k)C1kwk1 2 L2(0;k)kw0k1 2 L2(0;k)+C2kwkL2(0;k): Multiplying (3.27) by i1 nv;ninL2(0;`) and integrating by parts, we obtain 1 2nkv;nk2 L2(0;`)=i1 2n[T;n(x)v;n(x)]k 0+i1 2nZk 0T;nv0 ;ndx+o(1): By (3.28) and (3.29) we have i1 2nRk 0T;nv0 ;ndx=o(1): Using Gagliardo-Nerenberg inequality (3.46), (3.28), (3.29) and the boundedness of v;n, kv;nkL1(0;k)C1kv;nk1 2 L2(0;k) v0 ;n 1 2 L2(0;k)+C2kv;nkL2(0;k)=O(1); 3 8nkT;nkL1(0;k)C1 1 4nT;n 1 2 L2(0;k) 1 nT0 ;n 1 2 L2(0;k)+C23 8nkT;nkL2(0;k)=o(1): It follows that i1 2n[T;n(x)v;n(x)]k 0=o(1) and then 1 2nkv;nk2 L2(0;k)=o(1): Then, we multiply (3.27) by i1 2nv;nand we repeat exactly the same strategy as before, using (3.27) and1 2nkv;nk2=o(1), we obtain (3.45). We are now ready to estimate Re(i1 2nT;n(0)v;n(0)):12 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL Applying Gagliardo-Nerenberg inequality (3.46) to w=v;nwe obtain, using (3.28), 1 2nkv;nkL1(0;k)C1 3 4nv;n 1 2 L2(0;k) 1 4nv0 ;n 1 2 L2(0;k)+C21 2nkv;nkL2(0;k) o(1) + 3 4nv;n o(1): Using again the Gagliardo-Nerenberg inequality (3.46) with w=T;n, kT;nkL1(0;k)C1 1 4nT;n 1 2 L2(0;k) 1 4nT0 ;n 1 2+C2kT;nkL2(0;k) o(1) 1 4nT0 ;n 1 2 L2(0;k)+o(1) o(1) +o(1) 3 4nv;n L2(0;k): Here, we have used (3.27),(3.28) and (3.29). Then (3.47)jRe(i1 2nT;n(0)v;n(0))j1 2nkv;nkL1(0;`)kT;nkL1(0;`)1 43 2nkv;nk2 L2(0;k)+o(1) and (3.48) Reh i1 2nT;n(x)v;n(x)ik 021 2nkv;nkL1(0;`)kT;nkL1(0;`)1 23 2nkv;nk2 L2(0;k)+o(1) Multiplying (3.27) by iv;ninL2(0;k) and integrating by parts, we obtain (3.49) 3 2nkv;nk2 L2(0;k)=i1 2n[T;n(x)v;n(x)]k 0+i1 2nZk 0T;nv0 ;ndx+o(1): Using (3.28) and (3.29), the second term on the left hand side of (3.49) converge to zero. We conclude, using (3.48) that 3 2nkv;nk2 L2(0;k)=o(1): Return back to (3.47) which yields Re(i1 2nT;n(0)v;n(0)) =o(1): We obtain the same result if we suppose that xis the end of e;nidentied to `via, that is Re(i1 2nT;n(`)v;n(`)) =o(1); and we then conclude that the rst term on the right hand side of (3.43) converge to zero. Now, summing over  2Iin (3.43) by taking into account the estimate of Reh i1 2nT;n(x)v;n(x)i` 0 and the inequality in (P), we obtain that X 2I1 2n a1 2 nv;n 2 L2(0;`)=O(1); then 3 2n a1 2 v;n 2 L2(0;`)=O(1) for every 2I;and the proof of Lemma 3.5 is complete for case (ii). STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 13 Return back to the general case. Substituting (3.34) in (3.33) leads to (3.50)1 2Z` 0q0jv;nj2dx+1 2Z` 0q0jT;nj2dx1 2h q(x) jv;n(x)j2+jT;n(x)j2i` 0=o(1) for every 2I: Let 2Isuch thateis a K-V string. First, note that from (3.35), we deduce that a1 2 v;n 2 L2(0;`)=o(1): Then, we take q(x) =Rx 0a(s)dsin (3.50) to obtain (3.51)1 2Z` 0ajT;nj2dx1 2 Z` 0a(s)ds! jv;n(`)j2+jT;n(`)j2 =o(1): Since1 2R` 0ajT;nj2dx=o(1) andR` 0a(s)ds> 0, then (3.51) implies (3.52) jT;n(`)j2+jv;n(`)j2=o(1): Therefore (3.50) can be rewritten as 1 2Z` 0q0jv;nj2dx+1 2Z` 0q0jT;nj2dx +1 2 q(0)jv;n(0)j2+q(0)jT;n(0)j2 =o(1): (3.53) Takingq=xin (3.53) implies that kv;nkL2(0;`)=o(1) andkT;nkL2(0;`)=o(1):Moreover, u0 ;n L2(0;`)=kT;nav;nkL2(0;`)=o(1): By lettingq=`xin (3.53) and by taking into account the convergence of v;nandT;nin L2(0;`);we get (3.54) v;n(0) =o(1) andT;n(0) =o(1): Finally, notice that (3.52) signies that (3.55) v;n(`) =o(1) andT;n(`) =o(1): To conclude, it suces to prove that (3.56)kv;nkL2(0;`)=o(1) and u0 ;n L2(0;`)=kT;nkL2(0;`)=o(1) for every 2Isuch thateis purely elastic. To do this, starting by a string eattached at one end to only K-V strings. Using continuity condition of vand the compatibility condition at inner nodes, implies that esatises (3.54) or (3.55). Moreover, by taking q= 1 in (3.50), we conclude that esatises (3.54) and (3.55). Then using again (3.50) with q=x;we deduce that (3.56) is satised by e:We iterate such procedure on each maximally connected subgraph of purely elastic strings (from leaves to the root). Thusk(un;vn)kH=o(1);which contradicts the hypoyhesis k(un;vn)kH= 1: 14 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL Remark 3.6. (1)If for every 2I,ais continuous on [0;`]and not vanish in such interval then we don't need the property (P) in the Theorem 3.4. Indeed (P) is used only to estimate Re i1 nZ` 0T0 ;nav;ndx! in(3.36) , according to 1 2n a1 2 v;n L2(0;`). This is equivalent to estimate Re i1 nZ` 0T0 ;nv;ndx! according to 1 2nkv;nkL2(0;`): Re i1 nZ` 0T0 ;nv;ndx! =Re i1 nT;nv;n` 0+Re i1 nZ` 0T;nv0 ;ndx! = Re i1 nT;n(x)v;n(x)` 0+o(1) as in case (ii) (proof of Theorem 3.4) we prove without using (P) that Re i1 nT;n(x)v;n(x)` 02 n 4kv;nk2 L2(0;`)+o(1): (2)We nd here the particular cases studied in [23, 24, 1, 17, 22] . Note that concerning the result of polynomial stability in [1, 17] the authors proved that the1 t2decay rate of solution is optimal when the damping coecient is a characteristic function. 4.Further comments: graph case In this section we want generalize the previous results to a general graph. Then we suppose that Tis a connected graph G(thenGcan contains some circuits).STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 15 •• •• ••• •• • Figure 2. A Graph We conserve the same notations as in T, we just replace SbyS0which is the set of all the external nodes, and IS0denote the set of such nodes. Then I=IM[IS0. We denote by Jthe setf1;:::;Ngand fork2Iwe will denote by Jkthe set of indices of edges adjacent to sk. If k2IS0;then the index of the unique element of Jkwill be denoted by jk: We suppose that the graph is directed, then we need to dene the incidence matrix D= (dkj)pN;p=jIj;as follows, dkj=8 >>< >>:1 ifj(`j) =sk; 1 ifj(0) =sk; 0 otherwise, The system (1.1)-(1.5) is rewritten as follows (4.57)@2uj @t2(x;t)@ @x@uj @x+aj(x)@2uj @x@t (x;t) = 0;0<x<`j; t> 0; j2J; (4.58) ujk(sk;t) = 0; k2IS0; t> 0; (4.59) uj(sk;t) =ul(sk;t); t> 0; j;l2Jk; k2IM; (4.60)X j2Jkdkj@uj @x(sk;t) +aj(sk)@2uj @x@t(sk;t) = 0; t> 0; k2IM; (4.61) uj(x;0) =u0 j(x);@uj @t(x;0) =u1 j(x);0<x<`j; j2J: As in the case of a tree, we suppose that Gcontains at least a K-V edge and that every max- imal subgraph of purely elastic edges is (a tree), the leaves of which K-V edges are attached. Furthermore, we suppose that S06=;:16 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL Finally the property (P) is rewritten as follows, (P0)8j2J; a0 j;a00 j2L1(0;`j) and8k2IM;X j2Jkdkja0 j(sk)0: Under the property (P') the system (4.57)-(4.61) is polynomially stable, and it is exponentially stable if and only if ais continuous at each inner nodes, i.e., uj(sk) =ul(sk); t > 0;;j;l2 Jk; k2IM. References [1] M. Alves, J. M. Revera, M. Sep ulveda, O. V. Villagr an and M. Z. Gary, The asymptotic behavior of the linear transmission problem in viscoelasticity, Math. Nachr., 287(2014), 483-497. [2] K. Ammari and S. Nicaise, Stabilization of elastic systems by collocated feedback, Lecture Notes in Mathe- matics, 2124, Springer, Cham, 2015. [3] K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings, Evolu- tion Equations and Control Theory, 1(2015), 1-19. [4] K. Ammari, D. Mercier and V. R egnier, Spectral analysis of the Schr odinger operator on binary tree-shaped networks and applications, Journal of Dierential Equations, 259(2015), 6923-6959. [5] K. Ammari, D. Mercier, V. R egnier and J. Valein, Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings, Commun. Pure Appl. Anal., 11(2012), 785-807. [6] K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force, SIAM Journal on Control and Optimization ,39(2000), 1160-1181. [7] K. Ammari, A. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptotic Analysis, 28(2001), 215-240. [8] K. Ammari and M. Jellouli, Remark in stabilization of tree-shaped networks of strings, Appl. Maths., 4 (2007), 327-343. [9] K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., ESAIM Control Optim. Calc. Var, 6(2001), 361-386. [10] K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Di. Integral. Equations, 17 (2004), 1395-1410. [11] K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, J. Dyn. Cont. Syst., 11 (2005), 177-193. [12] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc., 305(1988), 837{852. [13] H. T. Banks, R. C. Smith and Y. Wang, Smart Materials Structures , Wiley, 1996. [14] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347(2010), 455{478. [15] H. Brezis, Analyse Fonctionnelle, Th eorie et Applications , Masson, Paris, 1983. [16] R. D ager and E. Zuazua, Wave propagation, observation and control in 1-d exible multi-structures , volume 50 of Math ematiques & Applications (Berlin), Springer-Verlag, 2006. [17] F. Hassine, Stability of elastic transmission systems with a local Kelvin-Voigt damping, Eur. J. Control., 23(2015), 84-93. [18] F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert space, Ann. Dierential Equations ,1(1985), 43-56.STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 17 [19] S. Chen, K. Liu and Z. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping, SIAM J. Appl. Math., 59(1999), 651-668. [20] J. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis of dynamic elastic multi-link struc- tures , Birkh auser, Boston-Basel-Berlin, 1994. [21] Z. Liu and B. Rao, Frequency domain characterization of rational decay rate for solution of linear evolution equations, Z. Angew. Math. Phys., 56(2005), 630{644. [22] Z. Liu and Q. Zhang, Stability of a string with local Kelvin-Voigt damping and non-smooth coecient at interface, ESAIM Control Optim. Calc. Var., 23(2017), 443-454. [23] K. Liu and Z. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin- Voigt damping, SIAM J. Control Optim., 36(1998), 1086-1098. [24] K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys., 53(2002), 265-280. [25] Z. Liu and S. Zheng, Semigroups associated with dissipative systems , Chapman & Hall/CRC Research Notes in Mathematics, 398. Chapman & Hall/CRC, Boca Raton, FL, 1999. [26] K. Liu, Z. Liu and Q. Zhang, Eventual dierentiability of a string with local Kelvin-Voigt damping, SIAM Journal on Control and Optimization, 54(2016), 1859-1871. [27] A. Pazy, Semigroups of linear operators and applications to partial dierential equations , Springer, New York, 1983. [28] J. Pr uss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc. ,248(1984), 847-857. [29] M. Tucsnak and G. Weiss, Observation and control for operator semigroups, Birkh auser Advanced Texts: Basler Lehrb ucher , Birkh auser Verlag, Basel, 2009. UR Analysis and Control of PDEs, UR 13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Tunisia E-mail address :kais.ammari@fsm.rnu.tn Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812-3000, United States E-mail address :zliu@d.umn.edu UR Analysis and Control of PDEs, UR 13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Tunisia E-mail address :farhat.shel@ipeit.rnu.tn

2018-03-20

In this paper we study the stability problem of a tree of elastic strings with local Kelvin-Voigt damping on some of the edges. Under the compatibility condition of displacement and strain and continuity condition of damping coefficients at the vertices of the tree, exponential/polynomial stability are proved.

Stability of the wave equations on a tree with local Kelvin-Voigt damping

1803.07280v1

1 Comparison of micromagnetic parameters of ferromagnetic semiconductors (Ga,Mn)(As,P) and (Ga,Mn)As N. Tesařová1, D. Butkovi čová1, R. P. Campion2, A.W. Rushforth2, K. W. Edmonds, P. Wadley2, B. L. Gallagher2, E. Schmoranzerová,1 F. Trojánek1, P. Malý1, P. Motloch4, V. Novák3, T. Jungwirth3, 2, and P. N ěmec1,* 1 Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 3, 121 16 Prague 2, Czech Republic 2School of Physics and Astronomy, University of Nottingham, Nottingham NG72RD, United Kingdom 3 Institute of Physics ASCR, v.v.i., Cukrovar nická 10, 16253 Prague 6, Czech Republic 4 University of Chicago, Chicago, IL 60637, USA We report on the determination of microm agnetic parameters of epilayers of the ferromagnetic semiconductor (Ga,Mn)As, which has easy axis in the sample plane, and (Ga,Mn)(As,P) which has easy axis perpendicula r to the sample plane. We use an optical analog of ferromagnetic resonance where the laser-pulse-induced precession of magnetization is measured directly in the time domain. By the analysis of a single set of pump-and-probe magneto-optical data we determined the magnetic anisotropy fields, the spin stiffness and the Gilbert damping consta nt in these two materials. We show that incorporation of 10% of phosphorus in (Ga, Mn)As with 6% of manganese leads not only to the expected sign change of the perpendicu lar-to-plane anisotropy field but also to an increase of the Gilbert damping and to a reduction of the spin stiffness. The observed changes in the micromagnetic parameters upon incorporating P in (Ga,Mn)As are consistent with the reduced hole density, conductivity, and Curie temperature of the (Ga,Mn)(As,P) material. We report that th e magnetization precession damping is stronger for the n = 1 spin wave resonance mode than for the n = 0 uniform magnetization precession mode. PACS numbers: 75.50.Pp, 75.30.Gw, 75.70.-i, 78.20.Ls, 78.47.D- I. INTRODUCTION (Ga,Mn)As is the most widely studied dilute d magnetic semiconductor (DMS) with a carrier- mediated ferromagnetism.1 Investigation of this material system can provide fundamental insight into new physical phenomen a that are present also in ot her types of magnetic materials – like ferromagnetic metals – where they can be exploited in spintronic applications.2-5 Moreover, the carrier concentration in DMSs is several orders of ma gnitude lower than in conventional FM metals which enables manipul ation of magnetization by external stimuli – e.g. by electric6,7 and optical8,9 fields. Another remarkable propert y of this material is a strong sensitivity of the magnetic anisotropy to the ep itaxial strain. (Ga,Mn)A s epilayers are usually prepared on a GaAs substrate where the growth-i nduced compressive strain leads to in-plane orientation of the easy axis (EA) for Mn concentrations ≥2%.10 However, for certain experiments – e.g., for a visualization of magn etization orientation by the magneto-optical polar Kerr effect11-17 or the anomalous Hall effect12,18 – the EA orientation in the direction perpendicular to the sample plane is more suitable. To achieve this, (Ga,Mn)As layers have been grown on relaxed (In,Ga)As buffer laye rs that introduce a tensile strain in (Ga,Mn)As. 11,12,14,16-18 However, the growth on (In,Ga)As la yers can result in a high density of line defects that can lead to high coerci vities and a strong pinning of domain walls 2 (DW).16,17 Alternatively, tensile strain and perpendicular-to-plane or ientation of the EA can be achieved by incorporation of small amount s of phosphorus in (Ga,Mn)(As,P) layers.19,20 In these epilayers, the EA can be in the sample plane for the as-grown material and perpendicular to the plane for fully annealed (Ga,Mn)(As,P). 21 The possibility of magnetic anisotropy fine tuning by the thermal annealing tu rns out to be a very favorable property of (Ga,Mn)(As,P) because it enables the preparation of materials with extr emely low barriers for magnetization switching.22,23 Compared to tensile-stained (Ga,Mn)As/(In,Ga)As films, (Ga,Mn)(As,P)/GaAs epilayers show weaker DW pinning, which allows observation of the intrinsic flow regimes of DW propagation.13,15,24 Preparation of uniform (Ga,Mn)As epilayers with minimized dens ity of unintentional extrinsic defects is a rather challenging task which requires optimized growth and post- growth annealing conditions.25 Moreover, the subsequent determination of material micromagnetic parameters by the standard char acterization techniques, such as ferromagnetic resonance (FMR), is complicated by the fact th at these techniques require rather thick films, which may be magnetically inhomogeneous.25,26 Recently, we have reported the preparation of high-quality (Ga,Mn)As epila yers where the individually optimized synthesis protocols yielded systematic doping trends, whic h are microscopically well understood.25 Simultaneously with the optimization of the ma terial synthesis, we developed an optical analog of FMR (optical-FMR)25, where all micromagnetic pa rameters of the in-plane (Ga,Mn)As were deduced from a single magneto -optical (MO) pump-and-probe experiment where a laser pulse induces precession of magnetization.27,28 In this method the anisotropy fields are determined from the dependence of the precession frequency on the magnitude and the orientation of the external magnetic field, the Gilbert damping cons tant is deduced from the damping of the precession signal, and the sp in stiffness is obtained from the mutual spacing of the spin wave resonance modes observe d in the measured MO signal. In this paper we apply this all optical-FMR to (Ga,Mn)(As,P) . We demonstrate the applicability of this method also for the determination of microma gnetic parameters in DMS materials with a perpendicular-to-plane orientation of the EA. By this method we show that the incorporation of P in (Ga,Mn)As leads not only to the expect ed sign change of the perpendicular-to-plane anisotropy field but also to a considerable in crease of the Gilbert damping and to a reduction of the spin stiffness. Moreover, we illustrate that the all optical-FMR can be very effectively used not only for an investig ation of the uniform magnetizati on precession but also for a study of spin wave resonances. II. EXPERIMENTAL In our previous work we reported in de tail on the preparation and micromagnetic characterization of (Ga,Mn)A s epilayers prepared in MBE laboratory in Prague. 25 We also pointed out that the preparati on of (Ga,Mn)As by this highly non-equilibrium synthesis in two distinct MBE laboratories in Prague and in No ttingham led to a growth of epilayers with micromagnetic parameters that showed the same doping trends.25 Nevertheless, the preparation of epilayers with identical paramete rs (e.g., thickness, nominal Mn content, etc.) in two distinct MBE machines is still a nontrivial task. Therefore, in this study of the role of the phosphorus incorporation to (Ga,Mn)As we opted for a dire ct comparison of materials prepared in one MBE mach ine. The investigated Ga 1-xMn xAs and Ga 1-xMn xAs1-yPy epilayers were prepared in Nottingham20 with the same nominal amount of Mn (x = 6%) and the same growth time on a GaAs substrate (with 50 nm thick GaAsP buffer layer in the case of (Ga,Mn)(As,P)]. They differ only in the incorpor ation of P (y = 10%) in the latter epilayer. 3 The inferred epilayer thicknesses are (24.5 േ 1.0) nm for both (Ga,Mn)As and (Ga,Mn)(As,P).29 The as-grown layers, wh ich both had the EA in th e epilayer plane, were thermally annealed (for 48 hours at 180°C). This led to an increase in Curie temperature and to a rotation of the EA to the perpendicular-to-plane orientation for (Ga,Mn)(As,P).20,21 The magnetic anisotropy of the samples was studied using a superconducting quantum interference device (SQUID) magneto meter and by the all-optical FMR.25 The hole concentration was determined by fitting to Hall effect measurements at low temperatures (1.8 K) for external magnetic fields from 2 T to 6 T. In this range the magnetization is saturated and one can obtain th e normal Hall coefficient af ter correction for the field dependence of the anomalous Hall du e to the weak magnetoresistance.30 The time-resolved pump-and-probe MO experiments were performe d using a titanium sapphire pulsed laser (pulse width 200 fs) with a repetition rate of 82 MHz, which was tuned ( hυ = 1.64 eV) above the GaAs band gap. The energy fl uence of the pump pulses was around 30 μJcm-2 and the probe pulses were at least ten times weak er. The pump pulses were circularly polarized (with a helicity controlled by a quarter wave plate) and the probe pulses were linearly polarized (in a direction perpendicular to the external magnetic field). The time-resolved MO data reported here correspond to the polariz ation-independent part of the pump-induced rotation of probe polarization plane, which was computed from the measured data by averaging the signals obtained for the opposite helicities of circularly polarized pump pulses. 27, 28 The experiment was performed close to the normal-incidence geometry, where the angles of incidence were 9° and 3° (measured from the sample normal) for the probe and the pump pulses, respectively. The rotation of the probe polarization plane is caused by two MO effects – the polar Kerr effect and the magnetic linear dichroism, which are sensitiv e to perpendicular-to-plane and in-plane components of magnetization, respectively. 31-33 For all MO experiments, samples were mounted in a cryostat and cooled down to ≈ 15 K. The cryostat was placed between the poles of an electromagnet and the external magnetic field Hext ranging from ≈ 0 to 585 mT was applied in the sample plane, either in the [010] or [110] crystallographic di rection of the sample (see inset in Fig. 1 for a definition of the coordinate system). Prior to all measurements, we always prepared the magnetiza tion in a well-defined state by first applying a strong saturating magnetic field and then reducing it to the desired magnitude of Hext. III. RESULTS AND DISCUSSION A. Sample characterization The hysteresis loops measured by SQUID magnetometry for external magnetic field applied along the in-plane [-110] and perpendicu lar-to-plane [001] crystallographic directions in (Ga,Mn)As and (Ga,Mn)(As,P) samples are s hown in Fig. 1(a) and Fig. 1(c), respectively. These data confirm the expected in-plane and perp endicular-to-plane orient ations of the EA in (Ga,Mn)As and (Ga,Mn)(As,P), respectively. Moreover, they reveal that for the (Ga,Mn)(As,P) sample, an external magnetic field of 250 mT is needed to rotate the magnetization into the sample plane. In Fig. 1(b) and Fig. 1(d) we show the temperature dependences of the remanent magnetization of the samples from which the Curie temperature T c of 130 K and 110 K can be deduced. The measur ed saturation ma gnetization also indicates very similar density of Mn moments contributing to the ferromagnetic state in the two samples. 4 Fig. 1 (Color online): Magnetic characterization of samples: (a), (b) (Ga,Mn)As and (c), (d) (Ga,Mn)(As,P). (a), (c) Hysteresis loops measured in at 2 K for the external magnetic field applied in the sample plane (along the crystallographic direction [-110]) and perpendicular to sample plane (along the crystallographic direction [001]). (b), (d) Temperature dependence of the remanent magnetization. Inset: Definition of the coordinate system. The electrical characterization of the samp les is shown in Fig. 2. The measured data show a sharp Curie point singula rity in the temperature derivative of the resistivity which confirms the high quality of the samples.25 The hole densities inferred from Hall measurements are (1.3 0.2) 1021 cm-3 and (0.8 0.2) 1021 cm-3 for (Ga,Mn)As and (Ga,Mn)(As,P), respectively. The hole density obtained for (Ga, Mn)As is in agreement with our previous measurements for simila r films in magnetic fields up 14 T.30 The reduction of the density of itinerant holes quantitatively correlates with the observed increase of the resistivity of the (Ga,Mn)(As,P) film as compared to the (Ga,Mn)As sample. 5 Fig. 2 (Color online): Electrical char acterization of samples. Temperature dependence of the resistivity (a) and its temperature derivative (b). B. Time-resolved magnet o-optical experiment In Fig. 3(a) and 3(b) we show the measur ed MO signals that reflect the magnetization dynamics in (Ga,Mn)As and (Ga,Mn)(As,P) sa mples, respectively. Th ese signals can be decomposed into the oscillatory parts [Figs. 3(c) and 3(d)] and the non-oscillatory pulse-like background [Fig. 3(e) and 3(f)].27, 28 The oscillatory part arises from the precessional motion of magnetization around the quasi-equilibrium EA and the pulse-like function reflects the laser-induced tilt of the EA and the laser-induced demagnetization.25,31 The pump polarization-independent MO data reported here, which were measured at a relatively low excitation intensity of 30 μJcm-2, can be attributed to the ma gnetization precession induced by a transient heating of the sample due to the absorption of the laser pulse.8,9 Before absorption of the pump pulse the magnetization is along th e EA direction. Absorptio n of the laser pulse leads to a photo-injection of electron-hole pa irs. The subsequent fast non-radiative recombination of photo-injected electrons induces a transi ent increase of the lattice temperature (within tens of picoseconds afte r the impact of the pu mp pulse). The laser- induced change of the lattice temperature then leads to a change of the EA position.34 As a result, magnetization starts to follow th e EA shift by the precessional motion. Finally, dissipation of the heat leads to a return of the EA to the equilibrium position and the precession of magnetization is stopped by a Gilbert damping.25 It is apparent from Fig. 3 that the measured MO signals are strongly dependent on a magnit ude of the external magnetic field, which was applied in the epilayer plan e along the [010] crystall ographic direction in both samples. In particular, absorption of the laser pulse does not induce precession of magnetization in (Ga,Mn)(As,P) unless magnetic field stronger than 20 mT is applied [see Fig. 3(d)]. 6 Fig. 3 (Color online): Time-resolved magneto-optical (MO) signals measured in (Ga,Mn)As (a) and (Ga,Mn)(As,P) (b) for two magnitudes of the external magnetic field applied along the [010] crystallographic direction. The measured MO signals were decomposed in to oscillatory parts [(c) and (d]), which correspond to the magnetization precession, and to non-oscillatory part s [(e) and (f)], which are connected with the quasi- equilibrium tilt of the easy axis and with the demagnetization. Note different x-scales in the left and in the right columns. The magnetization dynamics is describe d by the Landau-Lifshitz-Gilbert (LLG) equation that is usually expressed in the form35,36: ௗሺ௧ሻ ௗ௧ൌെ ߛൣ ሺݐሻൈሺݐሻ൧ఈ ெೞቂሺݐሻൈௗሺ௧ሻ ௗ௧ቃ, ( 1 ) where = (gμB)/ћ is the gyromagnetic ratio, g is the Landé g-factor, μB is the Bohr magneton, ħ is the reduced Planck constant, is the Gilbert damping constant, and Heff is the effective magnetic field. Nevertheless, it is more conve nient to express this equation in spherical coordinates where the directi on of the magnetization vector M is given by the polar angle θ and azimuthal angle φ and where Heff can be directly connected w ith angular derivatives of the free energy density functional F (see the Appendix).37 For small deviations δ and δ of magnetization from its equilibrium position (given by 0 and 0), the solution of LLG equation can be written in the form (t) = 0 + δ(t) and (t) = 0 + δ(t) as ߠሺݐሻൌߠܣఏ݁ି௧ݏܿሺ2ݐ݂ߨΦ ఏሻ, ( 2 ) ߮ሺݐሻൌ߮ܣఝ݁ି௧ݏܿ൫2ݐ݂ߨΦ ఝ൯, ( 3 ) where the constants A (A) and () represent the initial amplitude and phase of (), respectively, f is the magnetization precession frequency, and kd is the precession damping rate (see the Appendix). The pr ecession frequency reflects the in ternal magnetic anisotropy of the sample that can be characterized by the cubic ( KC), in-plane uniaxial ( Ku) and out-of-plane uniaxial ( Kout) anisotropy fields (see Eq. (A4) in the Appendix).10 Moreover, f depends also on the magnitude and on the orientation of Hext (see the Appendix) and, therefore, the magnetic 7 field dependence of f can be used to evalua te the magnetic anisotropy fields in the sample. If the applied in-plane magnetic field is strong e nough to align the magnetiz ation parallel with Hext (i.e., for Hext exceeding the saturation field in the sa mple for a particular orientation of Hext), = H = π/2 and = H and if the precession damping is relatively slow , i.e. α2 ≈ 0 f can be expressed as ݂ൌఓಳ ඩ൬ܪ௫௧െ2ܭ௨௧ሺଷା௦ସఝ ሻ ଶ2ܭ௨݊݅ݏଶቀ߮ுെగ ସቁ൰ ൈሺܪ௫௧2ܭݏܿ4߮ ுെ2ܭ௨݊݅ݏ2߮ுሻ, (4) Fig. 4 (Color online): Fourier spectrum of the oscillatory part of the MO signal measured in (Ga,Mn)As for external magnetic fields applied alon g the [010] crystallo graphic direction. f0 and f1 indicate the frequencies of the uniform magnetization precession and the fi rst spin wave resona nce, respectively. In Fig. 4 we show the fast Fourier transfor m (FFT) spectra of the oscillatory parts of the MO signals measured in the (Ga,Mn )As sample for different values of Hext. This figure clearly reveals that for all external magnetic fields there are two distinct oscillatory frequencies present in the measured data . These precession modes are the spin wave resonances (SWRs) – i.e., spin waves (or magno ns) that are selectively amplified by fulfilling the boundary conditions: In a homogeneous thin magnetic film with a thickness L, only the perpendicular standing waves with a wave vector k fulfilling the resonant condition kL = n (where n is the mode number) are amplified.25,38-41 In our case – using the ferromagnetic films with a thickness around 25 nm – we detect only42 the uniform magnetiza tion precession with zero k vector (i.e. the precession where at any instant of time all magnetic moments are parallel over the entire sample; n = 0 at frequency f0) and the first SWR (i.e. n = 1 at frequency f1). See the inset in Fig. 8 for a schematic de piction of the modes. In Fig. 5 we plot the amplitudes of the uniform magnetization precession ( A0) and of the first SWR ( A1) as a function of the exte rnal magnetic field Hext. In the (Ga,Mn)As sample, the oscillations are present even when no magnetic field is applied and the precession amplitude increases slightly with an increasing Hext (up to 20 mT for A0 and up to 60 mT for A1). Above this value, a further increase of Hext leads to a suppression of the oscillations, but the suppression of the first SWR is slower than that of the uniform magnetization precession [see Fig. 5(c)]. In 8 (Ga,Mn)(As,P), the oscillatory signal starts to appear at 50 mT, reaches its maximum for Hext 175 mT, and a further increase of Hext leads to its monotonic d ecrease, like in the case of (Ga,Mn)As. The observed field dependence of the precession amplitude, which expresses the sensitivity of the EA position on the laser- induced sample temperature change, can be qualitatively understood as follows. In (Ga,Mn)As, the position of the EA in the sample plane is given by a competition between the cubic and the in-plane uniaxial magnetic anisotropies.10,25 The laser-induced heating of the sa mple leads to a reduction of the magnetization magnitude M and, consequently, it enhances th e uniaxial anisotropy relative to the cubic anisotropy.9 This is because the uniaxial anisotropy component scales with magnetization as ~ M2 while the cubic component scales as ~ M4. The application of Hext along the [010] crystallographic di rection deepens the minimum in the [010] direction in the free energy density functional F (due to the Zeeman term in F, see Eq. (A4) in the Appendix). Measured data shown in Fig. 5 reve al that in the (Ga,Mn)As sample, Hext initially (for Hext up to 20 mT) destabilizes the posit ion of EA but stabilizes it for large values of Hext (where the position of the energy minimum in F is dominated by the Zeeman term, which is not temperature dependent). In the case of (Ga,Mn)( As,P), the position of the EA is determined by the strong perpendicular-to-p lane anisotropy. Therefore, w ithout an external magnetic field, the laser-induced heating of the sample doe s not change significantl y the position of EA and, consequently, does not initiate the pr ecession of magnetization [see Fig. 5(b)]. The application of an in-plane fi eld moves the energy minimum in F towards the sample plane [see Fig. 1(c)] which makes the EA position more sensitive to the laser-induced temperature change. Finally, for a sufficiently strong Hext, the sample magnetic anisotropy is dominated by the temperature-independent Zeeman term, wh ich again suppresses the precession amplitude. The markedly different ratio A1/A0 in the (Ga,Mn)As and (Ga,Mn )(As,P) samples is probably connected with a different surface magnetic anis otropy and/or a slight difference in magnetic homogeneity in these two samples.43,44 Fig. 5 (Color online): Dependence of the amplitude of the uniform magnetization precession ( A0) and the first spin wave resonance ( A1) on the magnitude of the external magnetic field ( Hext) applied along the [010] crystallographic direction in (Ga,Mn )As (a) and (Ga,Mn)(As,P) (b). (c) and (d) Dependence of the ratio A1 / A0 on Hext. 9 C. Determination of magnetic anisotropy In Fig. 6 we plot the ma gnetic field dependences of f0 and f1 for two different orientations of Hext. The frequency f0 of the spatially uniform precession of magnetization is given by Eq. (4). For the SWRs, where the local moments are no longer para llel (see the inset in Fig. 8), restoring torques due to exchange interaction and internal magnetic dipolar interaction have to be included in the analysis.39-41,45 For Hext along the [010] crystallographic direction (i.e., for φH = /2) Eq. (4) can be written as ݂ൌఓಳ ඥሺܪ௫௧െ2ܭ௨௧ܭ∆ܪሻሺܪ௫௧െ2ܭെ2ܭ௨∆ܪሻ , (5) where Hn is the shift of the resonant field for the nth spin-wave mode with respect to the n = 0 uniform precession mode. Analogically, for Hext applied in the [110] crystallographic direction (i.e., for φH = /4) ݂ൌఓಳ ඥሺܪ௫௧െ2ܭ௨௧2ܭܭ௨∆ܪሻሺܪ௫௧2ܭ∆ܪሻ. (6) The lines in Fig. 6 represent the fits of all four measured dependencies fn = fn (Hext, H) [where n = 0; 1 and H = /4; /2] with a single set of anisotropy constants for each of the samples, which confirms the credibility of the fitting pr ocedure. The obtained an isotropy constants at ≈ 15 K are: KC = (17 ± 3) mT, Ku = (11 ± 5) mT, Kout = (-200 ± 20) mT for (Ga,Mn)As and KC = (14 ± 3) mT, Ku = (11 ± 5) mT, Kout = (90 ± 10) mT for (Ga,Mn)(As,P), respectively (in both cases we considered the Mn g-factor of 2). For (Ga,Mn)As, we can now compare these anisotropy constants with those obtained by the same fitting procedure for samples prepared in a different MBE laboratory (in Prague) – see Fig. 4 in Ref. 25. We see that the previously reported25 doping trends of KC and Kout predict for a sample with nominal Mn doping x = 6% the anisotropy fields which are the same as thos e reported in this pape r for the sample grown in Nottingham. This observation is in accord with the current microscopic understanding of their origin – KC reflects the zinc-blende crystal st ructure of the host semiconductor and Kout Fig. 6 (Color online): Magnetic field dependence of the precession frequencies f0 and f1 for two different orientations of the external magnetic field (points) measured in (Ga,Mn)As (a) and (Ga,Mn)(As,P) (b). Lines are the fits by Eqs. (5) and (6). ΔH 1 indicates the shift of the resonant field for the first spin-wave mode with respect to the uniform precession mode. 10 is a sum of the anisotropy due to the growth-induced lattice-ma tching strain and of the thin- film shape anisotropy, which should be the sa me for equally doped and optimally synthesized samples, independent of the growth chamber. On the other hand, the micr oscopic origin of in- plane uniaxial anisotropy field K u is still not established10,25 and our data reveal that it is considerably smaller in the sample grown in Nottingham. Th e incorporation of phosphorus does not change significa ntly the values of KC and Ku but it strongly modi fies the magnitude and changes the sign of Kout, which is in agreement with the previous results obtained by FMR experiment.22 D. Determination of spin stiffness The observation of a higher-o rder SWR enables us to also determine the exchange spin stiffness constant D, which is a parameter that is rather difficult to extract from other experiments in (Ga,Mn)As.25,46 In homogeneous thin films, Hn is given by the Kittel formula43 Δܪ ܪെܪൌ݊ଶ ఓಳగమ మ, ( 7 ) where L is the thickness of the magnetic film. By fitting the data in Fig. 6, we obtained H1 = (363 ± 2) mT for (Ga,Mn)As and (271 ± 2) mT for (Ga,Mn)(As,P) which correspond to D = (2.5 ± 0.2) meVnm2 and (1.9 ± 0.2) meVnm2 for (Ga,Mn)As and (Ga,Mn)(As,P), respectively (note that the relatively large experimental error in D is given mainly by the uncertainty of the epilayer thickness).29 The value obtained for (Ga,Mn)As is again in agreement with that reported previously for samples grown in Prague,25 which also confirms the consistent determination of the epilayer thicknesses in both MBE laboratories.29 The incorporation of phosphorus leads to a reduction of D which correlates with the decrease of the hole density,47 and the reduced Tc in (Ga,Mn)(As,P), as compared to its (Ga,Mn)As counterpart. E. Determination of Gilbert damping The Gilbert damping constant α can be determined by fitting the measured dynamical MO signals by the LLG equation. 35,36,48 For a relatively slow precession damping and a sufficiently strong external magnetic field, the analytical solution of the LLG equation gives (see the Appendix) ݇ ௗൌߙఓಳ ଶ൬2ܪ௫௧െ2ܭ௨௧ሺଷାହ௦ସఝ ಹሻ ଶܭ௨ሺ1െ3݊݅ݏ2߮ ுሻ൰. (8) Eq. (8) shows not only that kd is proportional to but also that for obtaining a correct value of from the measured MO precession signal damp ing it is necessary to take into account a realistic magnetic anisotropy of the investigated sample. Nevert heless, the correct dependence of kd on magnetic anisotropy was not cons idered in the previous studies35,36,48 where only one effective magnetic field was used, which is probably one of the reasons why mutually inconsistent results were obtained for Ga 1-xMn xAs with a different Mn content x. An increase of from 0.02 to 0.08 for an increase of x from 3.6% to 7.5% was reported in Ref. 36. On 11 the contrary, in Ref. 48 values of from 0.06 to 0.19 – without any apparent doping trend – were observed for x from 2% to 11%. For numerical modeling of the measured MO data, we first computed from the LLG equation (Eqs. (A1) and (A2) in the Appendix with th e measured magnetic anisotropy fields) the time-dependent deviations of the spherical angles [ (t) and (t)] from the corresponding equilibrium values ( 0, 0). Then we calculated how such changes of and modify the static magneto-optical response of the samp le, which is the signal that we detect experimentally31 0 00 2sin2 2cos2 ,MLD s MLD PKEPMt MPt Pt t MO . (9) The first two terms in Eq. (9) are connected wi th the out-of-plane and in-plane movement of magnetization, and the last term describes a change of the sta tic magneto-optical response of the sample due to the laser-induced demagnetization.31 PPKE and PMLD are MO coefficients that describe the MO response of the sample which we measured independently in a static MO experiment,32,33 and β is the probe polarization orientation with respect to the crystallographic direction [100].31 To further simplify the fitting procedure, we can extract the oscillatory parts from the measured MO data (cf. Fig. 3), which effectively removes the MO signals due to the laser-induced demagnetization [i .e., the last term in Eq. (9)] and due to the in-plane movement of the easy axis [i.e., a part of the MO signal desc ribed by the second term in Eq. (9)].31 Examples of the fitting of the precessional MO optical data are shown in Fig. 7(a) and (b) for (Ga,Mn)As and (Ga,Mn)(As,P), respectively. We stress that in our case the only fitting parameters in the modeling are the damping coefficient and the initial deviations of the spherical angles from the corresponding equilibrium values. By this numerical modeling we deduced a de pendence of the damping factor on the external magnetic field for two different orientations of Hext. At smaller fields, the dependences obtained show a strong anisotropy w ith respect to the field angle th at can be fully ascribed to the field-angle dependence of the precession frequency.25 However, when plotted as a function of the precession frequency, the de pendence on the field-an gle disappears – see Fig. 7(c) and (d) for (Ga,Mn)As and (Ga,Mn )(As,P), respectively. For both materials, initially decreases monotonously with f and finally it saturates at a certain value for f ≥ 10 GHz. A frequency-dependent (or magnetic field-dependent) damping parameter was reported in various magnetic materials and a va riety of underlying mechanisms responsible for it were suggested as an explanation.49-51 In our case, the most probable explanation seems to be the one that was used by Walowski et al. to explain the experimental results obtained in thin films of nickel.49 They argued that in the low field range small magnetization inhomogeneities can be formed – the magnetizati on does not align parallel in an externally applied field, but forms ripples.49 Consequently, the measured MO signal which detects sample properties averaged over the laser spot size, which is in our case about 30 m wide (FWHM), experiences an apparent oscillation damping because the magnetic properties (i.e., the precession frequencies) are slightly differing within the spot size (see Fig. 6 and 7 in Ref. 49). On the other hand, for stronger external fields the sample is fully homogeneous and, therefore, the precession damping is not de pendent on the applied field (the precession frequency), as expected for the in trinsic Gilbert damping coefficient.52,53 We note that the observed monotonous frequency decrease of α is in fact a signature of a magnetic homogeneity of the studied epilayers.25 The obtained frequency-independent values of α are (0.9 ± 0.2) 10-2 for (Ga,Mn)As and (1.9 ± 0.5) 10-2 for (Ga,Mn)(As,P), respectively. The 12 observed enhancement of the magnetization pre cession damping due to the incorporation of phosphorus is also clearly apparent directly from Figs. 7(a) and 7(b) where the MO data with similar precession frequencies are shown for (G a,Mn)As and (Ga,Mn)(As,P), respectively. In (Ga,Mn)As the value of α obtained is again fully in accord with the reported Mn doping trend in α in this material.25 In (Ga,Mn)(As,P), the determined α is similar to the value 1.2 10-2 which was reported by Cubukcu et al. for (Ga,Mn)(As,P) with a si milar concentration of Mn and P.22 Comparing to the doping trends in the se ries of optimized (Ga,Mn)As materials,25 the value of α i n o u r ( G a , M n ) ( A s , P ) s a m p l e i s c o n s i s tent with the measured Gilbert damping constant in lower Mn-doped (Ga,Mn)As epilayers with similar hole densities and resistivities to those of the (Ga,Mn)(As,P) film. Fig. 7 (Color online): Determination of the Gilbert da mping. (a) and (b) Oscillatory part of the MO signal (points) measured in (Ga,Mn)As for the external magnetic field 100 mT (a) and in (Ga,Mn)(As,P) for 350 mT (b); magnetic field applied along the [010] crystallographic direction leads to a similar frequency ( f0 7.5 GHz) in both cases. Lines are fits by the Landau-Lifshitz-G ilbert equation. (c) and (d) Dependence of the damping factor () on the precession frequency for two different orienta tions of the external magnetic field in (Ga,Mn)As (c) and (Ga,Mn)(As,P) (d). The high quality of our MO data enables us to evaluate not only the damping of the uniform magnetization precession, which is addresse d above, but also the damping of the first SWR. To illustrate this procedure, we show in Fig. 8(a) the MO data measured for Hext = 13 250 mT applied along the [010] crystallographic direction in (Ga,Mn)As. The experimental data (points) obtained can be fitted by a sum of two expone ntially damped cosine functions (line) which enables us to separate, directly in a time domain, the contributions of the individual precession modes to the measured MO signal. In this particular case, the uniform magnetization precession occurs at a frequency f 0 = 12.2 GHz and this precession mode is damped with a rate constant kd0 = 0.79 ns-1. Remarkably, the first SWR, which has a frequency f1 = 23.0 GHz, has a considerably la rger damping rate constant kd1 = 1.7 ns-1 – see Fig. 8(b) where the contribution of individual modes ar e directly compared and also Fig. 8(c) where Fourier spectra computed from the measured MO data for two diffe rent ranges of time delays are shown. To convert the damping rate constant kdn obtained to the damping constant Fig. 8 (Color online): Comparison of the Gilbert damping of the uniform magnetization precession and of the first spin wave resonance. (a) Oscillatory part of the MO signal (points) measured in (Ga,Mn)As for the external magnetic field 250 mT applied along the [010] crystallographic direction. The solid line is a fit by a sum of two exponentially damped cosine functions that are shown in (b). Inset: Schematic illustration 39 of the spin wave resonances with n = 0 (uniform magnetization precession with zero k vector) and n = 1 (perpendicular standing wave with a wave vector k fulfilling the resonant condition kL = ) in a magnetic film with a thickness L. (c) Normalized Fourier spectra computed fo r the depicted ranges of time delays from the measured MO data, which are shown in (a). (d) Dependence of the damping factor ( n) on the precession frequency for the uniform magnetization precession ( n = 0) and the first spin wave resonance ( n = 1). 14 n for the n-th mode, we can use the ge neralized analytic al solution of the LLG equation. For a sufficiently strong Hext along the [010] crystallographic direction (i.e., when φ φH = /2), Eq. (8) can be written as ݇ௗൌߙఓಳ ଶሺ2ܪ௫௧2∆ܪ െ2ܭ௨௧2ܭܭ௨ሻ. (10) For the case of MO data measured at Hext = 250 mT, the damping constants obtained for modes with n = 0 and 1 are 0 = 0.009 and 1 = 0.011, respectively. [We note that the value of 0 obtained from the analytical solution of LLG equation is identical to that determined by the numerical fitting and shown in Fig. 7(c), which confirms the consistency of this procedure.] In Fig. 8(d) we show the dependence of 0 and 1 on the precession frequency. These data clearly show that even if the modes with n = 0 and 1 were oscillating with the same frequency, the SWR mode with n = 1 would have a larger damping coefficient. However, for sufficiently high fr equencies (i.e., external magnetic fields) the damping of the two modes is nearly equal [see Fig. 8(d)]. This feature can be ascribed to the presence of an extrinsic contribution to the damping coeffici ent for the SWR modes. The extrinsic damping probably originates from small variations of the sample thickness (< 1 nm) within the laser spot size54 and/or from the presence of a weak bulk inhomogeneity,43 which is apparent as small variations of ΔHn. The frequency spacing and the (Ki ttel) character of the SWR modes is insensitive to such small variations of ΔHn but the resulting frequency variations (see Eq. 5) can still strongly affect the observed damping of the oscillations. For high enough external magnetic fields, the variations of ΔHn have a negligible role a nd the damping of the SWR modes is governed solely by the intrinsic Gilbert damping parameter. IV. CONCLUSIONS We used the optical analog of FMR, wh ich is based on a pump-and-probe magneto- optical technique, for the determination of micromagnetic parameters of (Ga,Mn)As and (Ga,Mn)(As,P) DMS materials. The main advantage of this technique is that it enables us to determine the anisotropy constants, the spin s tiffness and the Gilbert damping parameter from a single set of the experimental magneto-optical data measured in films with a thickness of only several tens of nanometers. To addres s the role of phosphorus incorporation in (Ga,Mn)As, we measured simultaneously proper ties of (Ga,Mn)As and (Ga,Mn)(As,P) with 6% Mn-doping which were grown under identical conditions in the sa me MBE laboratory. We have shown that the laser-i nduced precession of magnetization is closely connected with a magnetic anisotropy of the samples. In partic ular, in (Ga,Mn)As with in-plane magnetic anisotropy the laser-pulse-induced precession of magnetization was observed even when no external magnetic field was applied. On the cont rary, in (Ga,Mn)(As,P) with perpendicular-to- plane magnetic EA the precession of magnetizat ion was observed only when the EA position was destabilized by an external in-plane ma gnetic field. From the measured magneto-optical data we deduced the anisotropy constants, spin stiffness, and Gilber t damping parameter in both materials. We have shown that the incorp oration of 10% of P in (Ga,Mn)As leads not only to the expected sign change of the perpendi cular-to-plane anisotropy field but also to a considerable increase of the G ilbert damping which correlates with the increased resistivity and reduced itinerant hole density in the (Ga,Mn)(As,P) material. We also observed a reduction of the spin stiffness consistent with the suppression of T c upon incorporating P in 15 (Ga,Mn)As. Finally, we found that in small exte rnal magnetic fields the damping of the first spin wave resonance is sizably stronger than that of the uniform magnetization precession. ACKNOWLEDGEMENTS This work was supported by the Grant Agency of the Czech Republic grant no. P204/12/0853 and 202/09/H041, by the Grant Agency of Charles University in Prague grant no. 1360313 and SVV-2013-267306, by EU grant ERC Advanced Grant 268066 - 0MSPIN, and by Praemium Academiae of the Academy of Sciences of the Czech Republic, from the Ministry of Education of the Czech Re public Grant No. LM2011026, and from the Czech Science Foundation Grant No. 14-37427G. APPENDIX Due to symmetry reasons, it is conveni ent to rewrite the LLG equation given by Eq. (1) in spherical coordinates where M S describes the magnetiza tion magnitude and polar θ and azimuthal φ angles characterize its orientation. We define the perpendicular-to-plane angle θ (in-plane angle φ) in such a way that it is counted from the [001] ([100]) crystallographic direction and it is positive wh en magnetization is tilted towards the [100] ([010]) direction (see inset of Fig. 1 for the co ordinate system definition). The time evolution of magnetization is given by37 ௗெೞ ௗ௧ൌ0, ( A 1 ) ௗఏ ௗ௧ൌെఊ ሺଵାఈమሻெೞቀߙ ∙ܣ ௦ఏቁൌΓఏሺߠ,߮ሻ , ( A 2 ) ௗఝ ௗ௧ൌఊ ሺଵାఈమሻெೞ௦ఏቀܣെఈ∙ ௦ఏቁൌΓఝሺߠ,߮ሻ , ( A 3 ) where A = dF/d and B = dF/d are the derivatives of the free energy density functional F with respect to and , respectively. We express F in a form10 ܨൌܯ ௌܭ݊݅ݏଶߠቀଵ ସ݊݅ݏଶ2݊݅ݏ߮ଶߠ ݏܿଶߠቁെܭ ௨௧ݏܿଶߠെೠ ଶ݊݅ݏଶߠሺ1െ݊݅ݏ2߮ ሻെ െܪ௫௧൫ߠݏܿߠݏܿ ுߠ݊݅ݏߠ݊݅ݏ ுݏܿሺ߮െ߮ுሻ൯൩, (A4) where KC, Ku and Kout are the constants that characterize the cubic, uniaxial and out-of-plane magnetic anisotropy fields in (Ga,Mn)As, respectively. Hext is the magnitude of the external magnetic field whose orientati on is described by the angles θH and φH, which are again counted from the [001] and [100] crystallographic direc tions, respectively. For small deviations δθ and δφ from the equilibrium values θ0 and φ0, the Eqs. (A2) and (A3) can be written in a linear form as ௗఏ ௗ௧ൌܦଵሺߠെߠሻܦଶሺ߮െ߮ሻ, ( A 5 ) ௗఝ ௗ௧ൌܦଷሺߠെߠሻܦସሺ߮െ߮ሻ, ( A 6 ) where 16 ܦଵൌௗഇ ௗୀబ,ୀబ , ( A 7 a ) ܦଶൌௗഇ ௗୀబ,ୀబ , ( A 7 b ) and analogically for D3, D4. The solution of Eqs. (A5) and (A6) is expressed by Eqs. (2) and (3) where the magnetizati on precession frequency f and the damping rate kd are given by ݂ൌඥସሺభరିమయሻିሺభାరሻమ ସగ, ( A 8 ) ݇ௗൌെభାర ଶ. ( A 9 ) Eqs. (A8) and (A9) for F in the form (A4) can be simplified when the geometry of our experiment – i.e., the in-plane orientation of the external magnetic field ( θH = π/2) – is taken into account. The equilibrium orientation of magnetization is in the sample plane for (Ga,Mn)As ( θ0 = π/2) and the same applies for (Ga,Mn)(As, P) if sufficiently strong external magnetic field (see Fig. 1) is applied ( θ0 ≈ θH = π/2). In such conditi ons, the precession frequency f and the damping rate kd are given by the following equations ݂ൌఓಳ ଶగሺଵାఈమሻ ۣളളളളളളളളളളളളളളളለ ൬ܪ௫௧ݏܿሺ߮െ߮ுሻെ2ܭ௨௧ሺଷା௦ସఝ ሻ ଶ2ܭ௨݊݅ݏଶቀ߮െగ ସቁ൰ൈ ൈሺܪ௫௧ݏܿሺ߮െ߮ுሻ2ܭݏܿ4߮െ2ܭ ௨݊݅ݏ2߮ሻ ߙଶ ەۖ۔ۖۓ൬ܪ௫௧ݏܿሺ߮െ߮ுሻെ2ܭ௨௧ሺଷା௦ସఝ ሻ ଶ2ܭ௨݊݅ݏଶቀ߮െగ ସቁ൰ൈ ൈሺܪ௫௧ݏܿሺ߮െ߮ுሻ2ܭݏܿ4߮െ2ܭ ௨݊݅ݏ2߮ሻെ െቀܪ௫௧ݏܿሺ߮െ߮ுሻെܭ௨௧ሺଷାହ௦ସఝ ሻ ସೠሺଵିଷ௦ଶఝ ሻ ଶቁଶ ۙۖۘۖۗ( A10) ݇ௗൌߙఓಳ ଶሺଵାఈమሻ൬2ܪ௫௧ݏܿሺ߮െ߮ுሻെ2ܭ௨௧ሺଷାହ௦ସఝ ሻ ଶܭ௨ሺ1െ3݊݅ݏ2߮ ሻ൰. (A11) 17 REFERENCES * Corresponding author; nemec@karlov.mff.cuni.cz 1 T. Jungwirth, J. Sinova, J. Mašek, J. Ku čera, and A. H. MacDonald, Rev. Mod. Phys. 78, 809 (2006). 2 Editorial , Nature Materials 9, 951 (2010). 3 H. Ohno, Nature Materials 9, 952 (2010). 4 T. Dietl, Nature Materials 9, 965 (2010). 5 T. Jungwirth, J. Wunderlich, V. Novak, K. Ol ejnik, B. L. Gallagher, R. P. Campion, K. W. Edmonds, A. W. Rushforth, A. J. Ferguson, P. Nemec, Rev. Mod. Phys. in press (2014). 6 H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. 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Rev. Lett. 28, 100 (1972).

2014-05-19

We report on the determination of micromagnetic parameters of epilayers of the ferromagnetic semiconductor (Ga,Mn)As, which has easy axis in the sample plane, and (Ga,Mn)(As,P) which has easy axis perpendicular to the sample plane. We use an optical analog of ferromagnetic resonance where the laser-pulse-induced precession of magnetization is measured directly in the time domain. By the analysis of a single set of pump-and-probe magneto-optical data we determined the magnetic anisotropy fields, the spin stiffness and the Gilbert damping constant in these two materials. We show that incorporation of 10% of phosphorus in (Ga,Mn)As with 6% of manganese leads not only to the expected sign change of the perpendicular to plane anisotropy field but also to an increase of the Gilbert damping and to a reduction of the spin stiffness. The observed changes in the micromagnetic parameters upon incorporating P in (Ga,Mn)As are consistent with the reduced hole density, conductivity, and Curie temperature of the (Ga,Mn)(As,P) material. We report that the magnetization precession damping is stronger for the n = 1 spin wave resonance mode than for the n = 0 uniform magnetization precession mode.

Comparison of micromagnetic parameters of ferromagnetic semiconductors (Ga,Mn)(As,P) and (Ga,Mn)As

1405.4677v1

Microscopic analysis of sound attenuation in low-temperature amorphous solids reveals quantitative importance of non-ane eects Grzegorz Szamel1,a)and Elijah Flenner1 Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523, USA (Dated: 4 April 2022) Sound attenuation in low temperature amorphous solids originates from their disordered structure. However, its detailed mechanism is still being debated. Here we analyze sound attenuation starting directly from the microscopic equations of motion. We derive an exact expression for the zero-temperature sound damping coecient. We verify that the sound damping coecients calculated from our expression agree very well with results from independent simulations of sound attenuation. The small wavevector analysis of our expression shows that sound attenuation is primarily determined by the non-ane displacements' contribution to the sound wave propagation coecient coming from the frequency shell of the sound wave. Our expression involves only quantities that pertain to solids' static congurations. It can be used to evaluate the low temperature sound damping coecients without directly simulating sound attenuation. I. INTRODUCTION The physics of sound attenuation in amorphous solids is drastically dierent than in crystalline solids. At low temperatures, when thermal eects can be neglected, sound is attenuated due to the inherent disorder of amor- phous solids, whereas the attenuation is absent in crys- talline solids. To understand the physical mechanism be- hind sound attenuation one can examine its wavevector kdependence. Sound attenuation in amorphous solids has a complicated dependence on the wavevector1, but small wavevector k4scaling of sound damping coe- cients has long been conjectured on an experimental basis2,3. An initial interpretation was that this small wavevector behavior originates from Rayleigh scattering of sound waves from the solid's inhomogeneities. Recent computer simulations4{6veried that in classical three- dimensional zero-temperature amorphous solids at the smallest wavevectors sound damping coecients scale as k4, although a logarithmic correction to this scaling was also claimed7. The specic physical mechanism of sound attenuation in low temperature amorphous solids is still debated. Zeller and Pohl2obtained the Rayleigh scattering law us- ing an \isotopic scattering"3model in which every atom of the glass is an independent source of scattering. Sev- eral recent experimental and simulational results were analyzed within the framework of the uctuating elastic- ity theory of Schirmacher8{10. This theory posits that an amorphous solid can be modeled as a continuous medium with spatially varying elastic constants. The inhomo- geneity of the elastic constants causes sound scattering and attenuation. In the limit of the wavelength being much larger than the characteristic spatial scale of the inhomogeneity this mechanism is equivalent to Rayleigh scattering and the theory predicts that sound damping a)Email: grzegorz.szamel@colostate.educoecients scale with the wavevector as k4. If the elastic constant variations have slowly decaying, power-law-like correlations, the theory predicts a logarithmic correction to Rayleigh scattering7,15. Other physical approaches, e.g.local oscillator9,11{14and random matrix16{18mod- els, can also be used to derive the Rayleigh scattering law. For this reason, Rayleigh scaling cannot serve to distinguish between dierent models9, and other model predictions must be used to determine the mechanism behind sound attenuation. Three recent studies came to very dierent conclu- sions regarding the applicability of the uctuating elas- ticity theory for sound attenuation. First, Caroli and Lema^ tre19analyzed a version of the theory derived from microscopic equations of motion. They obtained allthe parameters needed to calculate sound attenuation from the theory from the same simulations that were used to test the theoretical predictions. Caroli and Lema^ tre showed that this version of the theory underestimates sound damping coecients by about two orders of mag- nitude. Second, Kapteijns et al.20analyzed the dependence of sound attenuation in a two-dimensional glass on a param- eter, which \resembles" changing the stability of the amorphous solid. To calculate the disorder parameter8 of the uctuating elasticity theory they replaced uctu- ations of local elastic constants (which are used in the theoretical description) by the sample-to-sample uctu- ations of bulk elastic constants. In this way they were able to sidestep the issue of the denition of local elastic constants21and of the correlation volume. While Kaptei- jnset al. showed that the disorder parameter and the sound damping coecient have the same dependence on , they left the calculation of the pre-factor for the scaling for further research. Finally, Mahajan and Pica Ciamarra22argued that sound attenuation is proportional to the square of the disorder parameter according to a version of uctuating elasticity theory that incorporates an elastic correlation length9,23. They relied upon a relation between the bo-arXiv:2107.14254v2 [cond-mat.dis-nn] 31 Mar 20222 son peak, the speed of sound, and an elastic correlation length to show that the speed of sound and the boson peak frequency can be used to infer the change of the sound damping coecient. Again, the magnitude of the sound damping coecient was not addressed. The results described above show that it is dicult to distinguish between and to validate dierent semi- phenomenological models invoked to describe sound at- tenuation in zero-temperature amorphous solids. One of the reasons is that most of these approaches involve an adjustable parameter (or parameters) and therefore are able to predicts trends rather than absolute values of sound damping coecients. For example, neither Kaptei- jnset al. nor Mahajan and Pica Ciamarra calculated the values of sound damping coecients (in contrast to Car- oli and Lema^ tre), but rather investigated the variation of the sound attenuation between dierent glasses. Limited range of glasses that can be created in silico makes it dif- cult to distinguish between trends predicted by dierent models or dierent versions of a model. Our goal is to understand the microscopic origin of the sound attenuation. We derive an exact expression for the sound damping coecient in terms of quanti- ties that can be calculated from static congurations of amorphous solids, without the need to directly simulate sound attenuation. Our expression is analogous to well- known Green-Kubo formulae for transport coecients24. The latter expressions allow one to calculate transport coecients without explicitly simulating transport pro- cesses. While both our expression and Green-Kubo for- mulae need to be evaluated numerically, they can also serve as starting points for approximate analyses and treatments that can shed light at the validity of semi- phenomenological models. We hope that the results of one such analysis, which we present at the end of the pa- per, can inspire new models or be incorporated into the existing ones. In Sec. II we start from the microscopic equations of motion for harmonic vibrations. We derive an ex- act equation of motion for an auto-correlation function that has been used to determine sound attenuation. We identify the self-energy and show that its real part repro- duces the non-Born contribution to the zero-temperature wave propagation coecients. The imaginary part of the self-energy is the origin of sound attenuation. We show that sound damping coecients calculated this way agree very well with those obtained from direct simulations of sound attenuation in zero-temperature glasses with dif- ferent stability. In Sec. III we present the small wavevec- tor expansion of our expression for the sound damping co- ecient. It shows that the limiting k4sound attenuation originates from the same physics as the non-Born con- tribution to the elastic constants and wave propagation coecients, i.e.from the forces inducing non-ane dis- placements, which appear due to the amorphous solids' disordered structure. More precisely, attenuation of the sound wave is primarily determined by the contribution to the non-Born part of the wave propagation coecientfrom a shell of frequencies around the frequency of the sound wave. We thus show the common origin of the renormalization of the elastic constants and of sound at- tenuation. In Sec. IV we discuss the results of an approx- imate evaluation of our expression for the sound damp- ing coecient which assumes that the exact eigenvectors of the Hessian matrix can be replaced by plane waves. These results allow us to critically evaluate the relation between our exact expression and the uctuating elas- ticity theory. We end the paper with a discussion of our results and related descriptions of the sound attenuation. II. MICROSCOPIC ANALYSIS OF SOUND ATTENUATION We start from microscopic equations of motion for small displacements of Nspherically symmetric particles of unit mass comprising our model amorphous solid, @2 tui=X jHij
uj: (1) Hereuiis the displacement of the ith particle from its in- herent structure (potential energy minimum) position Ri andHis the Hessian calculated at the inherent structure, Hij=X l6=i@2V(Ril) @Ri@Rj(2) whereV(r) is the pair potential and Hijis a 3x3 tensor. To derive an expression for the sound damping coe- cient we use a slightly modied procedure proposed by Gelin et al.7. We assume that at t= 0 the particles are displaced from their equilibrium positions according to ui(t= 0) = ^eexp(ik
Ri),_ui(t= 0) = 0, where ^eis a unit vector and wavevector kis one of the wavevectors allowed by periodic boundary conditions. We then ana- lyze the time dependence of the auto-correlation func- tion of the single-particle displacement averaged over the whole system, C(t) =N1P iu i(t= 0)
ui(t). We anticipate that in the limit of small wavevectors k the auto-correlation function will exhibit damped oscilla- tions,C(t)/cos(vkt) exp((k)t=2), and we will iden- tifyvas the speed of sound and ( k) as the damping coecient. Solving Eqs. (1) with our initial conditions is equiva- lent to solving the following equations @2 tai=X jHij(k)
aj; (3) whereH(k) is the wavevector-dependent Hessian, Hil(k) =Hilexp[ik
(RiRl)], with initial conditions ai(t= 0) = ^e,_ai(t= 0) = 0. In terms of the new variables,C(t) =N1P iai(t= 0)
ai(t). To analyze C(t) we use the standard projection op- erator approach25. First, we dene a scalar product of two displacement vectors, aiandbj,i;j= 1;:::;N , hajbi=P ia i
bi. Next, we dene a unit vector j1iwith3 components 1i=N1=2^e, and projection operator Pon the unit vector,P=j1ih1jand orthogonal projection Q,Q=Ij 1ih1jwhereIis the identity matrix. Using the projection operator approach we obtain the following expression for the Fourier transform C(!) =
1 0C(t) exp(i(!+i))dtof the displacement auto- correlation function, C(!) =i(!+i) (!+i)2h1jH(k)j1i+(k;!); (4) where the self-energy (k;!) reads (k;!) = 1H(k)Q1 (!+i)2+QH(k)QQH(k)1 : (5) Equations (4-5) are exact. While it is straightforward to calculateh1jH(k)j1i, evaluation of the self-energy re- quires inversion of a large-dimensional matrix for each allowed wavevector. To make the numerical eort man- ageable, in the denominator in Eq. (5) we approximate H(k) byH. As argued in Appendix A, this approxima- tion does not in uence the small wavevector dependence of the sound damping coecients. In the small wavevector limit, the rst non-trivial term in the denominator in Eq. (4), h1jH(k)j1i, can be ex- pressed in terms of the Born contributions to the zero- temperature wave propagation coecients26, h1jH(k)j1i=1^eABorn  k^e k+o(k2) (6) where=N=V is the number density, Greek indices denote Cartesian components, the Einstein summation convention for Greek indices is hereafter adopted, and ABorn   is the Born wave propagation coecient, which can be expressed as the average of the local Born wave propagation coecients ABorn j;  , ABorn j;  = 2X l6=j@2V(Rjl) @Rj;@Rj; Rjl;Rjl; (7) over the whole system, ABorn   =N1X jABorn j; : (8) For example, if the coordinate system is chosen such that ^eis in the x direction, and we are interested in a trans- verse wave and choose kin theydirection, then the right hand side of (6) becomes 1ABorn xyxyk2. In the absence of the self-energy term, (4) predicts the Born value of the speed of sound and no sound damp- ing. Both the renormalization of the sound speed and the sound attenuation originate from the self-energy. The self-energy can be calculated using the eigenvalues and eigenvectors of the Hessian. In the thermodynamic limit27, when the spectrum of the Hessian becomes con- tinuous, we can use the Plemelj identity to identify the Born Term Deform Full Theory (a) (b)vT 2.02.53.0vL 5.05.5 Tp0.1 0.2Tp0.1 0.2 Tp = 0.200 Tp = 0.085 Tp = 0.062 Theory; T p = 0.200 Theory; T p = 0.085 Theory; T p = 0.062(c) (d) e ||k e 㾉 k k4k4ΓT 10−410−310−210−11ΓL 10−410−310−210−11 k0.1 1 k0.1 1FIG. 1. Upper panels: the transverse (a) and longitudi- nal (b) speed of sound obtained from the theory, Eq. (13), (red squares) and calculated from the elastic constants (black circles) as a function of parent temperature Tp. The black triangles are the Born values of the speed of sound. Lower panels: the transverse (c) and longitudinal (d) damping co- ecients obtained from the theory, Eq. (14), (squares) and obtained from sound attenuation simulations5(circles) for dif- ferent parent temperatures. Rayleigh scaling /k4is recov- ered at small wavevectors. The error bars for the theoretical calculation represent the uncertainty due to using dierent bin sizes. imaginary component of the self-energy, which is respon- sible for sound attenuation. The real 0(k;!) and imag- inary 00(k;!) parts of the self-energy read, 0(k;!) = d (k; ) 2!2
1; (9) 00(k;!) = 2!(k;j!j); (10) where denotes the Cauchy principal value. The func- tion ( k; ) is dened through the sum over eigenvectors Epof the Hessian matrix with non-zero28eigenvalues 2 p such that p2[ ; + d ], where d is the bin size, (k; ) = (1=d )X p2[ ; +d ]jh1jH(k)QjEpij2:(11) The key conceptual issue in writing Eq. (11) (and closely related equations (18,21)) is that the thermodynamic4 limit is implied for the expression at the right-hand-side. In this limit the spectrum becomes dense and phonon bands are not distinguishable. Thus, to calculate ( k; ) from the analysis of nite-size simulations we need to choose bin size d such that phonon bands are not re- solved. In the numerical calculations described below we tried a few bin sizes between 0.1 and 0.2 and found that within this range the results were not very sensitive to the bin size. To evaluate the displacement auto-correlation function we need to nd complex poles of the denominator at the right-hand-side of Eq. (4). In the small wavevector limit this can be done perturbatively, using kas the small pa- rameter. This leads to the following pair of poles, !=vki00(k;vk)=(2vk) (12) where the renormalized speed of sound vis given by v2= lim k!0k2 h1jH(k)j1i0(k; 0) : (13) The last term in Eq. (12) is our main result. It says that the sound damping in zero-temperature amorphous solids is determined by ( k; )= 2calculated at the wave's fre- quency, = vk, (k) =00(k;vk) vk= 2(k;vk) (vk)2: (14) We emphasize that ( k; )= 2is the same function that, after integration over the whole frequency spectrum, de- termines the renormalization of the wave propagation coecients. Note that v, (k), and related quantities dened below depend on the angle between the polariza- tion of the initial condition ^eand the direction of the wavevector ^k. To verify Eqs. (13-14) we calculated vand (k) for model zero-temperature glasses analyzed in Ref.5. These glasses were obtained by instantaneously quenching su- percooled liquids equilibrated using the swap Monte Carlo algorithm29at dierent parent temperatures Tpto their inherent structures using the fast inertial relaxation engine minimization30. The glasses consist of spherically symmetric, polydisperse particles which interact via a po- tential/r12, with a smooth cuto, see Appendix B and Refs.5,29for details. The parent temperature controls the glass's stability and thus its properties5,31,32. We calculated eigenvalues and eigenvectors of the Hes- sian using ARPACK33and Intel Math Kernel Library34. Then, using Eqs. (13-14) we evaluated vand (k) for the longitudinal, ^ek^k, and the transverse, ^e?^k, sound. To calculate ( k; ) one chooses a kncompatible with periodic boundary conditions. Then one calculates jh1jH(kn)QjEp( p)ijand bins the results according to the square root of the eigenvalue of jEpito determine (kn; ). Note that 2 pis the eigenvalue corresponding tojEpi. The damping is given by (14) where ( kn; ) is evaluated at = jknjv. Fig. 1 shows results for vand for three parent tem- peratures. Tp= 0:2; glasses obtained by quenching liq- uid samples equilibrated at 0 :2 are much less stable thantypical laboratory glasses. Tp= 0:085, which is be- tween the mode-coupling temperature Tc:108 and the estimated laboratory glass transition temperature Tg0:072; glasses obtained by quenching samples equili- brated at 0:085 are about as stable as typical laboratory glasses.Tp= 0:062, which is well below estimated Tg; glasses obtained by quenching liquid samples equilibrated at 0:062 are as stable as laboratory ultrastable glasses ob- tained by the vapor deposition method35,36. We previ- ously showed5that sound damping coecients decrease by more than an order of magnitude over this range of stability. For all three parent temperatures there is excellent agreement between results of Eqs. (13-14) and transverse and longitudinal sound speeds, vTandvL, and transverse and longitudinal sound damping coecients, Tand L obtained previously5from direct simulations of sound attenuation. At small wavevectors we recover Rayleigh scaling, /k4, but the theory also accurately predicts sound damping for wavevectors outside the Rayleigh scal- ing regime. The predicted damping coecients depart from the simulation results for larger wavevectors, but at larger wavevectors the assumptions used to nd the poles, Eq. (12), become invalid. III. THE ORIGIN OF SOUND ATTENUATION: NON-AFFINE EFFECTS To get some physical insight into the origin of sound attenuation in zero-temperature amorphous solids we ex- amine the small wavevector expansion of h1jH(k)QjEpi, h1jH(k)QjEpi=iN1=2X jj; ^e k
Ep j (15) +1N1=2X j ABorn j; ^e^eABorn   ^e kkEp j;+o(k2): In Eq. (15) Ep jdenotes the component of the pth eigen- vector of the Hessian corresponding to particle jandEp j; denotes its Cartesian component . Furthermore, j; denotes the vector eld describing forces due ane de- formations, j; =X l6=j@2V(Rjl) @Rj; @RjRjl;: (16) Specically, j; is proportional to the force on parti- clejresulting from a deformation along the direction that linearly depends on the coordinates. Finally, the 2nd term at the right-hand-side of Eq. (15) accounts for the spatial variation of the local Born wave propagation coecients. As discussed in the literature37,38, forces encoded in vector eld j; do not seem to posses any longer- range correlations. In contrast, non-ane displace- ments given byH1
exhibit characteristic vortex-like structures and correlations extending over many particle5 Tp = 0.200 vTk1Γ(vTk1)(a)ΓT terms 10−410−21 Ω 0 0.5 πΥ/[2(vTk1)2] πΘ/(2v2T) k21 πΔ/(2v2T)Tp = 0.062 vTk1ΓT(vTk1)(b)ΓT terms 10−410−21 Ω0.5 FIG. 2. The terms that contribute to the transverse sound attenuation for k1= 2=L = 0:13722, where Lis the length of the simulation box, given by Eq. 20 for Tp= 0:200 (a) andTp= 0:062 (b). The vertical lines marks the frequency =vTk1, wherevTis the transverse speed of sound, and the horizontal line is the sound attenuation calculated in simula- tions from Ref.5. The value of (k1; =vTk1)=[2(vTk1)2] gives the damping for k1. diameters37{40. The characteristic length of these cor- relations determines the minimal length scale on which a macroscopic elastic approach can be used to describe the response of amorphous solids39. It follows from the combination of Eqs. (9), (13) and (15) that the renor- malization of the wave propagation coecients originates from the rst term in Eq. (15), lim k!0k20(k; 0) =N1 d ( ) 2(17) where ( ) is dened analogously to ( k; ), ( ) = (1=d )X p2[ ; +d ]p ^e ^k2 (18) with p =N1=2X jj; 
Ep j: (19) Equations (17-18) reproduce the exact expression for the non-Born contribution to the wave propagation co- ecients derived from the analysis of the non-ane displacements37,38. We note that function ( ) is closely related to function (!) introduced and evaluated by Lema^ tre and Maloney, see Eq. (32) of Ref.38.While only the rst term in Eq. (15) determines the renormalization of the wave propagation coecients, both terms contribute to sound attenuation, (k) = 2v2 (vk) +k2
(vk) (20) where
( ) is dened analogously to ( k; ),
( ) = (1=d )X p2[ ; +d ]Ap  ^k^e ^k2 (21) with Ap  =1N1=2X j ABorn j; ^e^eABorn   Ep j;: (22) We note that the second term in Eq. (20) is expressed in term of the uctuations of the local Born wave propaga- tion coecients, see Eq. (22). Thus, the physical content of the second term resembles that of the uctuating elas- ticity theory. We will discuss this correspondence further in the next section. It is the rst term in Eq. (20) that makes the dominant contribution to the damping coecient, see Fig. 2. This implies that the sound damping is primarily determined by function ( ), which is the same function that also determines the renormalization of the wave propagation coecients, Eq. (17). While previous studies suggested41 and analyzed approximately42the importance the non- ane eects for the sound attenuation, we have presented the rst approach that accounts for these eects exactly. IV. SOUND DAMPING IN THE PLANE-WAVE APPROXIMATION The most recent version of the uctuating elastic- ity theory discussed by Mahajan and Pica Ciammarra22 posits that \amorphous materials can be described as ho- mogeneous isotropic elastic media punctuated by quasilo- calized modes acting as elastic heterogeneities." This suggests that plane waves should be a reasonable zeroth order approximation for the eigenvectors of the Hessian matrix describing an amorphous solid. To check this sup- position we calculated  and
contributions in Eq. (20) approximating the exact eigenvectors by plane waves, Ep j/^eqeiq
Rj, see Appendix C for details. For the contributions to transverse wave damping coecient we obtained the following expressions  2v2(vTk)1 60k4 3v2 T v5 L+4 v3 T ABorn xyABorn xy +v2 T v5 L1 v3 T ABorn xyABorn xy+ABorn xyABorn xy ;(23) k2 2v2
(vTk)1 12k4 31 v3 L+2 v3 ThD
ABorn xyxy
2E +D ABorn yyxy
2E +D ABorn zyxy
2Ei ; (24)6 where we implicitly assumed analyticity of the correlation functions of local wave propagation coecients at the vanishing wavevector. For example, we assumed that at q!0, D
ABorn xyxy
2E = lim q!0N1X jeiq
Rj ABorn j;xyxyABorn xyxy
2 (25) and other similar equalities, as discussed in Appendix C. We note that while the exact formula (18) for  contri- bution involves non-ane forces , approximate formula (23) is expressed in terms of correlations of local wave propagation coecients. This follows from the fact that, as shown in Appendix C, for small wavevectors q X jj; 
^eqeiq
Rj=i X jABorn j;  ^eqqeiq
Rj+o(q): (26) Furthermore, we note that formulae (23-24) are reminis- cent of Zeller and Pohl's \isotopic scattering" model in that every atom jis a source of scattering of a plane wave, with the amplitude depending on its local wave propa- gation coecient ABorn j;  . Importantly, our approximate formulae involve correlations of local wave propagation coecients that vanish at the macroscopic level and thus do not appear in the semi-phenomenological uctuating elasticity theory, e.g.ABorn j;yyxy . The plane wave approximation recovers analytically the Rayleigh scattering k4scaling of the sound damp- ing coecient. However, it is quantitatively quite inac- curate, see Fig. 3. This implies that at least for the pur- pose of calculating sound damping, eigenvectors of the Hessian are not well approximated by plane waves. We note that the plane-wave approximation becomes more accurate with decreasing parent temperature or increas- ing glass stability. Finally, we note that the rst term in square brackets in Eq. (24), which involves correlations of the uctua- tions of the local shear modulus,
ABorn j;xyxy , represents the result of the microscopic, isotopic scattering-like, version of the uctuating elasticity theory. As shown in Fig. 3, this term is about 2.5-4 times smaller than the complete plane-wave result, and thus it severely underestimates sound attenuation. In Fig. 3 we also show the result of a semi- phenomenological uctuating elasticity theory. To cal- culate this result we started from the celebrated formula of Rayleigh43that predicts the attenuation of a trans- verse wave due to inclusions of volume Vdand number densityn, R(k) =nVdvT k4=(6), where is the dis- order parameter. In Rayleigh's calculation character- ized the variation of \optical density". To adopt his calculation to the present problem we expressed in terms of the variation of the square of the transverse speed of sound, = (v2 T)2Vd=v4 T. Next, we added to Tp = 0.062 Tp = 0.2Simulation TheoryPlane WaveFET(ΔABorn xyxy)2ΓT 10−410−310−210−11ΓT 10−410−310−210−11 k0.1 1 k0.1 1FIG. 3. Comparison of the sound damping coecients ob- tained from the theory (squares) and evaluated from sound attenuation simulations (circles) with predictions of the plane- wave approximation, Eqs. (23-24) (red symbols) and micro- scopic version of the uctuating elasticity theory, i.e. the contribution due toD
ABorn xyxy
2E term in Eq. (24) (black line). Also shown is the result of a semi-phenomenological uctuating elasticity theory (FET, green line). Rayleigh's expression the the contribution of the longitu- dinal waves excited due to the presence of the inclusions, (k) =nVdvT k4(v3 T=(2v3 L))=(6). The complete for- mula of the semi-phenomenological uctuating elasticity theory thus reads FET(k) =k4vT 6 1 +1 2v3 T v3 L nVd : (27) We note that if one makes the identicationD
ABorn xyxy
2E =(3v4 T) =nVd , the contribution to sound attenuation due to the rst term in square brack- ets in Eq. (24) becomes identical to expression (27). To calculate the value of FETwe need the disorder parameter and the volume fraction of the inclusions nVd. For we used previously obtained results for the uctuations of local elastic constants44. We recall that disorder parameters calculated this way increase slightly with increasing box size used to dene local elastic constants, thus we used the largest box size considered in Ref.44. Furthermore, we note that Mahajan and Pica Ciamarra's formulation of uctuating elasticity theory assumesnVd1, see the SI of Ref.22. To calculate the upper bound for FETwe substituted nVd= 1. Figure 3 shows that the result of this procedure signicantly underestimates sound attenuation. We note that in addition to the microscopic ver- sion of the uctuating elasticity theory, originally de- rived by Caroli and Lema^ tre and embodied in the rst term in square brackets in Eq. (24), and the semi- phenomemonogical approach resulting in expression (27) one could compare our results to predictions of more so- phisticated versions of the uctuating elasticity theory, e.g. the version relying upon the self consistent Born approximation8. This comparison is left for future work.7 V. DISCUSSION According to our microscopic analysis, sound attenu- ation in zero-temperature amorphous solids is primarily determined by internal forces induced by initial ane displacements of the particles, i.e.by the physics of non- ane displacement elds. Quantitatively, the damping coecient is proportional to the non-ane contribution to the wave propagation coecients from the frequency equal to the frequency of the sound wave. It is not trivial that our exact calculation (as opposed to the plane-wave approximation discussed in the previous section) repro- duces the Rayleigh scaling of sound damping coecients. This fact results from the frequency dependence of  and
, which deserves further theoretical study. The mechanism of the attenuation revealed by our mi- croscopic analysis was mentioned by Caroli and Lema^ tre in Ref.19. It was investigated in Ref.41, where Caroli and Lema^ tre considered separately the eects of the long- wavelength, elastic continuum-like, and small-scale, pri- marily non-ane, motions with the small-scale motions being the scatterers for the long-wavelength ones. An earlier study by Wang, Szamel and Flenner5found a strong correlation between the sound attenuation co- ecient and the amplitude of the vibrational density of states of quasilocalized modes. The latter modes were dened using a cuto in the participation ratio, folow- ing Mizuno et al.45and Wang et al.31. We attempted to quantify the relative contributions of the extended and quasi-localized modes by separating the contributions of modes with small and large participation ratio. We did not nd convincing evidence for the dominance of small participation ratio modes versus larger participation ra- tio modes. We note in this context that local oscillator models11{14express the sound attenuation coecient in terms of the contributions from localized \defects"46,47 referred to as \soft modes". The formulas derived in these approaches are similar to our Eqs. (14) and (20). The details of expressions of Refs.11{14and our Eqs. (14) and (20) dier; in particular we express the self-energy in terms of all the exact eigenvectors and eigenvalues of the Hessian matrix. In order to evaluate the local oscillator based sound attenuation coecient formulas one needs to characterize the properties of the soft modes. In practical applications one may parametrize the soft modes' prop- erties and t the parameters to the experimental results. Such a procedure was used by Schober14and resulted in a good agreement between the theory and experiment. In view of both the previously found5correlation be- tween the sound attenuation coecient and the ampli- tude of the vibrational density of states of quasilocalized modes and the success of local oscillator approach14we believe that future work should investigate whether dom- inant contributions to the sound atteanuation coecient formulas (14) and (20) originate from well dened regions that can be identied as \defects". Damart et al.40demonstrated that the non-ane dis-placement eld was responsible for high-frequency har- monic dissipation in a simulated amorphous SiO 2. There- fore, it appears that non-ane displacements are respon- sible for dissipation over the full frequency range. Fur- ther theoretical development is needed to connect the low-frequency and high-frequency theories. Recently, Baggioli and Zaccone developed an approxi- mate microscopic theory for the sound attenuation that takes into account non-ane displacements42. This the- ory shares physical insight with our approach but it is quantitatively as inaccurate as the plane-wave version of our exact formula. As we mentioned in the introduction, Gelin et al.7 found a logarithmic correction to the Rayleigh scattering scaling of the sound damping coecients, which within the uctuating elasticity theory could originate from the slowly decaying correlations between local values of the elastic constants7,15. Within our approach, a logarithmic correction could originate from a logarithmic dependence of ( ) or
( ) on frequency . Our present numerical data are consistent with the absence of such a logarith- mic dependence but it would be interesting to investigate this issue farther. Within the plane-wave approximation a logarithmic correction could result from a logarithmic small wavevec- tor divergence of the correlation functions of local Born wave propagation coecients. We did not observe such a divergence but we note that our systems were signif- icantly smaller that those discussed in Ref.7. We note that if the correlation functions of local wave propagation coecients are singular, additional terms in the plane- wave approximation will appear. These terms will origi- nate from the anisotropic small wavevector character of the correlation functions of local wave propagation coef- cients. Our approach arrives at the physical picture of sound attenuation dierent from that postulated in the uctuat- ing elasticity theory. While the latter theory can predict trends20,22, it is quantitatively very inaccurate, as noted earlier by Caroli and Lema^ tre19. Our analysis revealed that the uctuating elasticity theory misses the domi- nant non-ane eects. In addition, it does not include the contributions due to uctuations of local microscopic wave propagation coecients that vanish at the macro- scopic level. Most importantly, the uctuating elastic- ity theory uses plane-wave-like picture of sound in low- temperature amorphous solids. The comparison of the results obtained using the full theoretical expression and adopting the plane-wave approximation, shown in Fig. 3, suggests that this leads to large quantitative discrepan- cies. Finally, we note that calculating sound attenuation us- ing Eq. (14) or (20) is somewhat numerically demanding but more straightforward than analyzing the time de- pendence of the velocity or displacement auto-correlation functions. The latter analysis suers from large nite-size eects5,48that make the evaluation of the sound damping coecients at the smallest wavevectors allowed by the pe-8 riodic boundary conditions dicult. Our approach oers an attractive alternative way to evaluate sound damping coecients of low temperature elastic solids that does not suer from nite size eects. ACKNOWLEDGMENTS We thank A. Ninarello for generously providing equi- librated congurations at very low parent temperatures and E. Bouchbinder for comments on the manuscript. We gratefully acknowledge the support of NSF Grant No. CHE 1800282.AUTHOR DECLARATIONS Con ict of interest The authors have no con icts to disclose. DATA AVAILABILITY The data that support the ndings of this study are available from the corresponding author upon reasonable request. Appendix A: Approximation H(k)H in Eq. (5) of the main text First, we examine the small wavevector expansion of QH(k)Q. Thei;jelement, which is a 3x3 tensor, reads [QH(k)Q]ij=Hijeik
(RiRj)N1X lHileik
(RiRl)
^e^eN1^eX l^e
Hljeik
(RlRj) +N2^eX l;m^e
Hlm
^eeik
(RlRm)^e= [QH(k= 0)Q]ij+ HQ1(k) ij+ HQ2(k) ij+o(k2); (A.1) where the matrix elements of the terms of the rst and second order in k,HQ1(k) andHQ2(k), read  HQ1(k) ij=i(1ij)( @2V(Rij) @R2 ik
(RiRj)N1X l^e
" @2V(Ril) @R2 ik
(RiRl)@2V(Rlj) @R2 jk
(RjRl)# ^e) ; (A.2)  HQ2(k) ij=1 2(1ij)@2V(Rij) @R2 i(k
(RiRj))21 2N1X l6=i@2V(Ril) @R2 i
^e(k
(RiRl))2^e 1 2N1^eX l6=j^e
@2V(Rlj) @R2 l(k
(RlRj))2+1 2N2^eX l6=m^e
@2V(Rlm) @R2 l
^e(k
(RlRm))2^e: (A.3) Next, we assume that for small wavevectors kwe can treat terms HQ1(k) andHQ2(k) in the denominator of Eq. (5) of the main text perturbatively. Due to the symmetry, the term of the rst order in k,HQ1(k), willcontribute in the second order of the perturbation ex- pansion. In contrast, the term of the second order in k, HQ2(k), will contribute in the rst order. Here we will show the contribution of HQ2(k) term. It reads Q2(k;!) (A.4) =X eigenvec. p;qh1jH(k)QjEpiD Ep (!+i)2+p
1EpE EpHQ2(k)EqD Eq (!+i)2+q
1EqE hEqjQH(k)j1i: Counting powers of kin the expression above shows that, at least perturbatively, term HQ2(k) results in a correc-9 tion that is higher order in kthan the dominant small wavevector result of approximation H(k)H in the de- nominator of Eq. (5). Appendix B: Simulation details We obtained zero-temperature glasses by instanta- neously quenching supercooled liquids of unit number density,= 1:0, equilibrated through the swap Monte Carlo algorithm29. The constituent particles of these liquids have unit mass and diameters selected using distribution P() =A 3, where2[0:73;1:63] and Ais a normalization factor. The cross-diameter ij is determined according to a non-additive mixing rule, ij=i+j 2(1jijj) with= 0:2. The interac- tion between two particles iandjis given by the inverse power law potential, V(rij) = (ij=rij)12+Vcut(rij), when the separation rijis smaller than the potential cut- orc ij= 1:25ij, and zero otherwise. Here, Vcut(rij) = c0+c2(rij=ij)2+c4(rij=ij)4, and the coecients c0,c2 andc4are chosen to guarantee the continuity of V(rij) atrc ijup to the second derivative. The number of particles Nvaried between 48000 and 192000. The largest systems had to be analyzed to de- termine sound attenuation at the lowest wavevectors re-ported. Appendix C: Plane-wave approximation We assume that for small wavevectors we can approxi- mate eigenvectors of the Hessian matrix by plane waves. We note that strictly speaking, for our amorphous solids the normalization factor is conguration-dependent. We checked that this dependence is weak and for this reason we use the following approximation, Ep jN1=2^eqeiq
Rj: (C.1) Approximate plane-wave eigenvectors are labeled by their wavevector qand their polarization ^eq. For each wavevector qwe have one longitudinal and two trans- verse modes. We assume that the associated eigenvalues are given by ( vLq)2and (vTq)2for the longitudinal and transverse modes, respectively. Here we will present the derivation of approximate for- mula for the contribution to the transverse sound damp- ing coecient originating from , Eq. (23) of the main text. The contribution originating from
, Eq. (24) of the main text and the approximate expression for the longitudinal sound damping can be derived in a similar way. First, we need to calculate iN1=2X jj; ^e k
Ep jiN1X jj; ^e k
^eqeiq
Rj=iN1X jX l6=j@2V(Rjl) @Rj; @RjRjl;^e k
^eqeiq
Rj:(C.2) Using thei$jsymmetry we get iN1X jX l6=j@2V(Rjl) @Rj; @RjRjl;^e k
^eqeiq
Rj=i 2NX jX l6=j@2V(Rjl) @Rj;@Rj; Rjl;^eq^e kh 1eiq
(RjRl)i eiq
Rj =1 2NX jX l6=j@2V(Rjl) @Rj;@Rj; Rjl;Rjl;^eqq^e keiq
Rj+o(q) = (N)1X jABorn j;  ^eqq^e keiq
Rj+o(q):(C.3) Next, we need to take the square of the absolute value of expression (C.3) for a given wavevector qand polar- ization ^eqand then integrate over spherical shell with frequencyqvL=kvTfor longitudinal and qvT=kvTfor transverse modes. We shall note that since the spher- ical shell is specied in the frequency space, there will be additional factors, 1 =vLfor longitudinal and 1 =vTfor transverse modes. Finally, we need to multiply the result by=(2v2 Tk2) to get the contribution to the transverse sound damping coecient. To perform these calculations we assume that wavevec- torkis parallel to the yaxis and the sound polarization ^e is along the xaxis. Furthermore, we specify the polariza- tion vector for the approximate plane-wave eigenvectors as^eL q=^q(sincos;sinsin;cos) for the longi-tudinal modes and ^eT1 q= (coscos;cossin;sin) and ^eT2 q= (sin;cos;0) for the two transverse modes. The contribution of the longitudinal modes reads  2 (vTk)2v4 T v5 LVk6 (2)3(C.4)  d^q(N)1X jABorn j;xy ^eL q^qei^q(kvT=vL)
Rj2 : Guided by our numerical calculations we assume that the following small wavevector limit is nite and does not10 depend on the direction lim q!0N1X jABorn j;xyeiq
RjX lABorn l; xyeiq
Rl  ABorn xyABorn xy : (C.5) Expression (C.4) becomes  2 (vTk)2v4 T v5 LVk6 (2)31 2N ABorn xyABorn xy d^q^eL q^q^eL q ^q = 2 (vTk)2v4 T v5 LVk6 (2)31 2N ABorn xyABorn xy 4 15( + + ) =1 60k4 3v2 T v5 L ABorn xyABorn xy + ABorn xyABorn xy + ABorn xyABorn xy : (C.6) Assuming again that the small wavevector limit of the correlation functions of local wave propagation coe- cients is nite and does not depend on the direction, the contribution of the two transverse modes reads  2 (vTk)21 vTVk6 (2)31 2N ABorn xyABorn xy  d^q ^eT1 q^q^eT1 q ^q+ ^eT2 q^q^eT2 q ^q
= 2 (vTk)21 vTVk6 (2)31 2N ABorn xyABorn xy 4 15(4   ) =1 60k4 31 v3 T 4 ABorn xyABorn xy ABorn xyABorn xy ABorn xyABorn xy (C.7) Adding expressions (C.6) and (C.7) we get Eq. (23) of the main text. REFERENCES 1See, e.g., C. Masciovecchio, G. Baldi, S. Caponi, L. Comez, S. Di Fonzo, D. Fioretto, A. Fontana, A. Gessini, S.C. Santucci, F. Sette, G. Viliani, P. Vilmercati and G. Ruocco, \Evidence for a Crossover in the Frequency Dependence of the Acoustic Attenu- ation in Vitreous Silica", Phys. Rev. Lett. 97, 035501 (2006). 2R.C. Zeller and R.O. Pohl, \Thermal conductivity and specic heat of noncrystalline solids", Phys. Rev. B 4, 2029 (1971). 3M. P. Zaitlin and A. C. Anderson, \Phonon thermal transport in noncrystalline materials", Phys. Rev. B 12, 4475 (1975). 4H. Mizuno and A. Ikeda, \Phonon transport and vibrational ex- citations in amorphous solids", Phys. Rev. E 98, 062612 (2018). 5L. Wang, L. Berthier, E. Flenner, P. Guan and G. 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2021-07-29

Sound attenuation in low temperature amorphous solids originates from their disordered structure. However, its detailed mechanism is still being debated. Here we analyze sound attenuation starting directly from the microscopic equations of motion. We derive an exact expression for the zero-temperature sound damping coefficient. We verify that the sound damping coefficients calculated from our expression agree very well with results from independent simulations of sound attenuation. The small wavevector analysis of our expression shows that sound attenuation is primarily determined by the non-affine displacements' contribution to the sound wave propagation coefficient coming from the frequency shell of the sound wave. Our expression involves only quantities that pertain to solids' static configurations. It can be used to evaluate the low temperature sound damping coefficients without directly simulating sound attenuation.

Microscopic analysis of sound attenuation in low-temperature amorphous solids reveals quantitative importance of non-affine effects

2107.14254v2

arXiv:2205.06399v1 [cond-mat.mes-hall] 13 May 2022Precession dynamics of a small magnet with non-Markovian da mping: Theoretical proposal for an experiment to determine the correlation tim e,✩✩ Hiroshi Imamura, Hiroko Arai, Rie Matsumoto, Toshiki Yamaj i, Hiroshi Tsukahara✩ National Institute of Advanced Industrial Science and Tech nology (AIST), Tsukuba, Ibaraki 305-8568, Japan Abstract Recent advances in experimental techniques have made it pos sible to manipulate and measure the magnetization dynamics on the femtosecond time scale which is the same order as the corr elation time of the bath degrees of freedom. In the equations of motion of magnetization, the correlation of the bath is repr esented by the non-Markovian damping. For development of th e science and technologies based on the ultrafast magnetization dyna mics it is important to understand how the magnetization dyn amics depend on the correlation time. It is also important to deter mine the correlation time experimentally. Here we study the precession dynamics of a small magnet with the non-Markovian damping. E xtending the theoretical analysis of Miyazaki and Seki [J. C hem. Phys. 108, 7052 (1998)] we obtain analytical expressions of the prece ssion angular velocity and the e ffective damping constant for any values of the correlation time under assumption of small Gilbert damping constant. We also propose a possible experi ment for determination of the correlation time. Keywords: non-Markovian damping, generalized Langevin equation, LL G equation, ultrafast spin dynamics, correlation time 1. Introduction Dynamics of magnetization is the combination of precession and damping. The precession is caused by the torque due to the internal and external magnetic fields. Typical time scal e of the precession around the external field and the anisotrop y field is nanosecond. The damping is caused by the coupling with the bath degrees of freedom such as conduction electron s and phonons. The typical time scale of the relaxation of con- duction electrons and phonons is picosecond or sub-picosec ond which is much faster than precession. In typical experiment al situations such as ferromagnetic resonance and magnetizat ion process, the time correlation of the bath degrees of freedom can be neglected and the magnetization dynamics is well repr e- sented by the Landau-Lifshitz-Gilbert (LLG) equation with the Markovian damping term[1–3]. Recent advances in experimental techniques such as fem- tosecond laser pulse and time-resolved magneto-optical Ke rr effect measurement have made it possible to manipulate and measure magnetization dynamics on the femtosecond time scale[4–11]. In 1996, Beaurepaire et al. observed the femto sec- ond laser pulse induced sub-picosecond demagnetization of a Ni thin film[4], which opens the field of ultrafast magnetiza- tion dynamics. The all-optical switching of magnetization in a ✩Permanent address: High Energy Accelerator Research Organ ization (KEK), Institute of Materials Structure Science (IMSS), Ts ukuba, Ibaraki 305- 0801, Japan ✩✩This work is partly supported by JSPS KAKENHI Grant Numbers JP19H01108 and JP18H03787. Email addresses: h-imamura@aist.go.jp (Hiroshi Imamura), arai-h@aist.go.jp (Hiroko Arai)ferrimagnetic GdFeCo alloy was demonstrated by Stanciu et a l. using a 40 femtosecond circularly polarized laser pulse[5] . The helicity-dependent laser-induced domain wall motion in Co /Pt multilayer thin films was reported by Quessab et al.[11]. To understand the physics behind the ultrafast magnetizati on dynamics it is necessary to take into account the time correl a- tion of bath in the equations of motion of magnetization. The first attempt was done by Kawabata in 1972[12]. He derived the Bloch equation and the Fokker-Planck equation for a classic al spin interacting with the surroundings based on the Nakajim a- Zwanzig-Mori formalism[13–15]. In 1998, Miyazaki and Seki constructed a theory for the Brownian motion of a classical spin and derived the integro-di fferential form of the generalized Langevin equation with non-Markovian damping[16]. They also showed that the generalized Langevin equation reduces to the LLG equation with modified parameters in a certain limit. Atxitia et al. applied the theory of Miyazaki and Seki to the atomistic model simulations and showed that materials with smaller correlation time demagnetized faster[17]. Despite the experimental and theoretical progresses to dat e little attention has been paid to how to determine the correl a- tion time experimentally. For development of the science an d technologies based on the ultrafast magnetization dynamic s it is important to determine the correlation time experimenta lly as well as to understand how the magnetization dynamics de- pend on the correlation time. In this paper the precession dynamics of a small magnet with non-Markovian damping is theoretically studied based on th e macrospin model. The magnet is assumed to have a uniaxial anisotropy and to be subjected to an external magnetic field parallel to the magnetization easy axis. The non-Markovian ity Preprint submitted to Journal of Magnetism and Magnetic Mat erials May 16, 2022enhances the precession angular velocity and reduces the da mp- ing. Assuming that the Gilbert damping constant is much smaller than unity, the analytical expressions of the prece ssion angular velocity and the e ffective damping constant are derived for any values of the correlation time by extending the analy sis of Miyazaki and Seki[16]. We also propose a possible exper- iment for determination of the correlation time. The correl a- tion time can be determined by analyzing the external field at which the enhancement of the precession angular velocity is maximized. The paper is organized as follows. Section 2 explains the theoretical model and the equations of motion. Section 3 giv es the numerical and theoretical analysis of the precession dy nam- ics in the absence of an anisotropy field. The e ffect of the anisotropy field is discussed in Sec. 4. A possible experimen t for determination of the correlation time is proposed in Sec . 5. The results are summarized in Sec. 6. 2. Theoretical model We calculate the magnetization dynamics in a small mag- net with a uniaxial anisotropy under an external magnetic fie ld based on the macrospin model. The magnetization easy axis is taken to be z-axis and the magnetic field is applied in the positive z-direction. In terms of the magnetization unit vector, m=(mx,my,mz), the energy density is given by E=K(1−m2 z)−µ0MsH m z, (1) where Kis the effective anisotropy constant including the crys- talline, interfacial, and shape anisotropies. µ0is the vacuum permeability, Msis the saturation magnetization, His the exter- nal magnetic field. The e ffective field is obtained as Heff=(Hkmz+H)ez, (2) where ezis the unit vector in the positive zdirection and Hk= 2K/(µ0Ms) is the effective anisotropy field. The magnetization precesses around the e ffective field with damping. The energy and angular momentum are absorbed by the bath degrees of freedom such as conduction electrons and phonons until the magnetization becomes parallel to the e ffec- tive field. The equations of motion of mcoupled with the bath is given by the Langevin equation with the stochastic field re p- resenting the bath degrees of freedom. If the time scale of th e bath is much smaller than the precession frequency the stoch as- tic field can be treated as the Wiener process[18] as shown by Brown[3]. Since we are interested in the ultrafast magnetization dyna m- ics of which time scale is the same order as the correlation ti me of the bath degrees of freedom, the stochastic field should be treated as the Ornstein-Uhlenbeck process[18, 19]. As show n by Miyazaki and Seki [16] the equations of motion of mtakes the following integro-di fferential form: ˙m=−γm×(Heff+r)+αm×/integraldisplayt −∞ν(t−t′) ˙m(t′)dt′,(3)whereγis the gyromagnetic ratio, αis the Gilbert damping con- stant, and ris the stochastic field. The first term represents the precession around the sum of the e ffective field and the stochas- tic field, and the second term represents the non-Markovian damping. The memory function in the non-Markovian damping term is defined as ν(t−t′)=1 τcexp/parenleftigg −|t−t′| τc/parenrightigg , (4) whereτcis the correlation time of the bath degrees of freedom. The stochastic field, r, satisfies/angbracketleftri(t)/angbracketright=0 and /angbracketleftrj(t)rk(t′)/angbracketright=µ 2δj,kν(t−t′), (5) where/angbracketleft/angbracketrightrepresents the statistical mean, and µ=2αkBT γMsV. (6) The subscripts jandkstand for x,y, orz,kBis the Boltzmann constant, Tis the temperature, Vis the volume of the mag- net, andδj,kis Kronecker’s delta. The LLG equation with the Markovian damping derived by Brown [3] is reproduced in the limit ofτc→0 because lim τc→0ν(t−t′)=2δ(t−t′), where δ(t−t′) is Dirac’s delta function. Equation (3) is equivalent to the following set of the first order di fferential equations, ˙m=−γm×[Heff+δH] (7) ˙δH=−1 τcδH−α τ2cm−γ τcR, (8) where Rrepresents the stochastic field due to thermal agita- tion. Equations (7), (8) are used for numerical simulations . The stochastic field, R, satisfies/angbracketleftRj(t)/angbracketright=0 and /angbracketleftRj(t)Rk(t′)/angbracketright=µδj,kδ(t−t′). (9) 3. Precession dynamics in the absence of an anisotropy field In this section the precession dynamics in the absence of an anisotropy field, i.e. Hk=0, is considered. The initial di- rection of magnetization is assumed to be m=(1,0,0). The numerical simulation shows that the non-Markovian damping enhances the precession angular velocity and reduces the da mp- ing. The numerical results are theoretically analyzed assu ming thatα≪1. The analytical expressions of the precession an- gular velocity and the e ffective damping constant are obtained. The case with Hk/nequal0 will be discussed in Sec. 4. 3.1. numerical simulation results We numerically solve Eqs. (7), (8) for H=10 T,α=0.05, andτc=1 ps. The temperature is assumed to be low enough to set R=0 in Eq. (8). Figure 1(a) shows the trajectory of m on a unit sphere. The initial direction is indicated by the fil led circle. The plot of the temporal evolutions of mx,my, and mzare shown in Fig. 1(b). The magnetization relaxes to the positiv ez direction with precessing around the external field. The res ults 2t [ps] 0.00 0.01 0.03 0.02 0.04 100 200 0 t [ps] a) b) c) d) z x yHφ [rad / ps] 1.76 1.77 1.79 1.78 1.80 100 200 0 100 200 0-1 1 0mx, m y, m z t [ps] φ00.05 αmzmymx yαeff Figure 1: (a) Trajectory of mon a unit sphere. The external field of H= 10 T is applied in the positive zdirection. The initial direction is assumed to bem=(1,0,0) as indicated by the filled circle. The other parameters are τc=1 ps, andα=0.05. (b) Temporal evolution of mx,my,mz. (c) Temporal evolution of the precession angular velocity, ˙φ. The solid red curve shows the simulation result. The dotted black line indicates the resu lt of the Markovian LLG equation, i.e. ˙φ0=γH/(1+α2). (d) Temporal evolution of the e ffective damping constant, αeff. The solid red curve shows the simulation result. The dotted black line indicates α=0.05. are quite similar to that of the Markovian LLG equation, whic h implies that the non-Markovianity in damping causes renorm al- ization of the gyromagnetic ratio and the Gilbert damping co n- stant in the Markovian LLG equation. The renormalized value of the gyromagnetic ratio can be observed as a variation of the precession angular velocity, ˙φ, where the polar and azimuthal angles are defined as m= (sinθcosφ,sinθsinφ,cosθ). Figure 1(c) shows that temporal evolution of ˙φ(solid red) together with the precession angular velocity without non-Makovianity, ˙φ0=γH/(1+α2), (dotted black). The precession angular velocity increases with inc rease of time and saturates to a certain value around 1.798. The sha pe of the time dependence of ˙φis quite similar to that of mzshown in Fig. 1(b), which suggests that the non-Markovian damping acts as an effective anisotropy field in the precession dynamics. The renormalization of the Gilbert damping constant can be observed as a variation of the temporal evolution of the pola r angle, ˙θ. Rearranging the LLG equation for ˙θ, the effective damping constant can be defined as αeff=−˙θ/(γHsinθ). (10) In Fig. 1(d)αeffis shown by the red solid curve as a function of time. The effective damping constant is reduced to about one- fifth of the original value of α=0.05 (dotted black). Contrary to˙φ,αeffdoes not show clear correlation with the dynamics of m. During the precession, αeffis kept almost constant. The enhancement of the precession angular velocity and the reduction of the Gilbert damping constant due to the non- Markovian damping will be explained by deriving the e ffectiveLLG equation that is valid up to the first order of αin the next subsection. 3.2. Theoretical analysis Since the Gilbert damping constant, α, of a conventional magnet is of the order of 0 .01∼0.1, it is natural to take the first order ofαto derive the effective equations of motion for m. The other parameters related to the motion of mareγ,H, andτc. Multiplying these parameters we can obtain the dimen- sionless parameter, ξ=γHτc, which represents the increment of the precession angle during the correlation time. In the case ofξ<1 Miyazaki and Seki dereived the e ffective LLG equation using time derivative series expansion[16]. W e first briefly review their analysis. Then we derive the e ffective LLG equation forξ>1 using the time-integral series expansion and show that the e ffective LLG equation has the same form for bothξ< 1 andξ> 1. Therefore, it is natural to assume that the derived effective LLG equation is valid for any values of ξ includingξ=1. 3.2.1. Brief review of Miyazaki and Seki’s derivation of the ef- fective LLG equation for ξ<1 In Ref. 16, Miyazaki and Seki derived the e ffective LLG equation with renormalized parameters using the time deriv a- tive series expansion. Similar analysis of the LLG equation was also done by Shul in the study of the damping due to strain[20, 21]. The following is the brief summary of the der iva- tion. Successive application of the integration by parts using ν(t− t′)=τc[dν(t−t′)/dt′] gives the following time derivative se- ries expansion: /integraldisplayt −∞ν(t−t′) ˙m(t′)dt′=∞/summationdisplay n=1(−τc)n−1dnm dtn. (11) Then the non-Markovian damping term in Eq. (3) is expressed as α∞/summationdisplay n=1(−τc)n−1/parenleftigg m×dnm dtn/parenrightigg . (12) The first derivative, n=1, is given by ˙m=−γHm×ez+O(α), (13) where Ois the Bachmann–Landau symbol. For n=2, substi- tution of Eq. (13) into the time derivative of Eq. (13) gives ¨m=(−γH)2(m×ez)×ez+O(α). (14) The n-th order time derivative is obtained by using the same algebra as dn dtnm=(−γH)n/bracketleftbig(m×ez)×ez.../bracketrightbig+O(α), (15) where ezappears ntimes. Expanding the vector products we obtain for even order time derivatives d2nm dt2n=(−1)n(γH)2n/bracketleftbigm−mzez/bracketrightbig+O(α), (16) 3and for odd order time derivatives d2n+1m dt2n+1=(−1)n(γH)2n˙m+O(α). (17) Substituting Eqs. (16) and (17) into Eq. (12) the non- Markovian damping term is expressed as −∞/summationdisplay n=1γ2nm×ez+∞/summationdisplay n=0α2n+1m×˙m, (18) where γ2n=αγH m z(−1)n−1ξ2n−1(19) α2n+1=α(−1)nξ2n. (20) The sums in Eq. (18) converge for ξ<1. Introducing ˜γ=γ/parenleftigg 1+αmzξ 1+ξ2/parenrightigg (21) ˜α=α 1+ξ2, (22) Eq. (3) can be expressed as the following e ffective LLG equa- tion with renormalized gyromagnetic ratio, ˜ γ, and damping constant, ˜α: ˙m=−˜γm×(H+r)+˜αm×˙m+O(α2). (23) 3.2.2. Derivation of the e ffective LLG equation for ξ>1 Forξ>1 we expand Eq. (3) in power series of 1 /ξusing the time integral series expansion approach. Using the integra tion by parts with dν(t−t′)/dt′=ν(t−t′)/τcthe integral part of the non-Markovian damping can be written as /integraldisplayt −∞ν(t−t′) ˙m(t′)dt′=1 τc/integraldisplayt −∞˙m(t′)dt′ −1 τc/integraldisplayt −∞ν(t−t′)/bracketleftigg/integraldisplayt′ −∞˙m(t′′)dt′′/bracketrightigg dt′. (24) Successive application of the integration by parts gives /integraldisplayt −∞ν(t−t′) ˙m(t′)dt′=−∞/summationdisplay n=1/parenleftigg −1 τc/parenrightiggn Jn, (25) where Jnis the nth order multiple integral defined as Jn=/integraldisplayt −∞/integraldisplayt1 −∞···/integraldisplaytn−1 −∞˙m(tn)dtn···dt2dt1. (26) From Eq. (17), on the other hand, ˙ mis expressed as ˙m=1 (−1)n(γH)2nd2n dt2n˙m+O(α). (27) Substituting Eq. (27) into Eq. (26) the multiple integrals a re calculated as J2n=1 (−1)n(γH)2n˙m (28) J2n−1=1 (−1)n(γH)2n¨m. (29)Then Eq. (25) becomes /integraldisplayt −∞ν(t−t′) ˙m(t′)dt′=∞/summationdisplay n=11 (−1)n−1ξ2n˙m +∞/summationdisplay n=1τc (−1)nξ2n¨m. (30) Substituting Eq. (30) into the second term of Eq. (3) the non- Markovian damping term is expressed as α∞/summationdisplay n=11 (−1)n−1ξ2nm×[ ˙m+τc¨m]+O(α2). (31) From Eq. (16) ¨ mis expressed as ¨m=(−1)(γH)2/bracketleftbigm−mzez/bracketrightbig. (32) Substituting Eqs. (31) and (32) into Eq. (3) we obtain ˙m=−γH∞/summationdisplay n=1/bracketleftigg 1+αmz (−1)n−1ξ2n−1/bracketrightigg m×ez−γm×r +α∞/summationdisplay n=11 (−1)n−1ξ2nm×˙m+O(α2). (33) The sums converge for ξ>1, and the effective LLG equation forξ>1 has the same form as ξ<1, i.e. Eq. (23). Since the effective LLG equation has the same form for both ξ< 1 and ξ>1, it is natural to Eq. (23) is valid for any values of ξ. As pointed out by Miyazaki and Seki, and independently by Suhl the effect of the non-Markovian damping on the precession can be regarded as the renormalization of the e ffective field [16, 20, 21]. Equation (23) can be expressed as ˙m=−γm×/parenleftigg H+αHξ 1+ξ2mz/parenrightigg ez−γm×r +˜αm×˙m+O(α2). (34) The second term in the bracket represents the fictitious unia xial anisotropy field originated from the non-Markovian damping . The fictitious anisotropy field increases with increase of ξfor ξ<1 and takes the maximum value of αH m z/2 atξ=1, i.e. γHτc=1. Forξ>1 the fictitious anisotropy field decreases with increase ofξand vanishes in the limit of ξ→∞ because the non-Markovian damping term vanishes in the limit of τc→ ∞. The precession angular velocity, ˙φ, is expected to have the sameξdependence as the fictitious anisotropy field and to have the same temporal evolution as mzas shown in Figs. 1(b) and 1(c). 3.2.3. The Correlation time dependence of the precession an - gular velocity, and e ffective damping constant Equation (21) tells us that up to the first order of αthe pre- cession angular velocity can be approximated as ˙φ≃˜γH=γH/bracketleftigg 1+αmzγHτc 1+(γHτc)2/bracketrightigg , (35) 4τc’ =1/( γH) 0.1 1 10 0.01 τc [ps] τc [ps] a) b) φ [rad / ps] 1.76 1.77 1.79 1.78 1.80 0.1 1 10 0.01 α, αeff ~ αeffα0.04 0.00 0.02 0.05 0.01 0.03 ααα effeffeffαeffαeff~sim. approx. Figure 2: (a) The correlation time, τc, dependence of the precession angular velocity,δ˙φ, atθ=5◦forH=10 T. The solid yellow curve shows the ap- proximation result, ˜ γH. The dotted black curve shows the simulation results obtained by numerically solving Eqs. (7) and (8). The thin ve rtical dotted line indicates the critical value of the correlation time, τ′ c=1/(γH). (b)τcdepen- dence of ˜α(solid yellow) andαeff(dotted black). The parameters and the other symbols are the same as panel (a). where the second term in the square bracket represents the en - hancement due to the fictitious anisotropy field. In Fig. 2(a) the approximation result of Eq. (35) at θ=5◦ where ˙φis almost saturated is plotted as a function of τcby the solid yellow curve. The external field and the Gilbert damp- ing constant are assumed to be H=10 T andα=0.05, re- spectively. The corresponding simulation results obtaine d by numerically solving Eqs. (7) and (8) are shown by the dotted black curve. Both curves agree well with each other because αis as small as 0.05. The precession angular velocity is maxi- mized at the critical value of the correlation time τ′ c=1/(γH). Figure 2(b) shows the τcdependence of ˜α(solid yellow) and αeff(dotted black) for the same parameters as panel (a). Both curves agree well with each other and are monotonic decreasi ng functions ofτc. They vanish in the limit of τc→∞ similar to the non-Markovian damping term. 4. Effect of an anisotropy field on precession dynamics The theoretical analysis given in the previous section can be applied to the case with Hk/nequal0 by replacingξwithξk= γ(H+Hkmz)τc. Following the same procedure as for Hk=0 Eq. (3) can be expressed as ˙m=−γm×/parenleftig H+αHξk 1+ξ2 kmz+αHkξk 1+ξ2 km2 z/parenrightig ez −γm×r+α 1+ξ2 km×˙m+O(α2). (36) The second and the third terms in the bracket can be regarded as the fictitious uniaxial and unidirectional anisotropy fie lds caused by the non-Markovian damping. Similar to the re- sults for Hk=0 the precession angular velocity is maximized atξk=1. The renormalized damping constant is given by α/(1+ξ2 k) which is a monotonic decreasing function of ξkand vanishes in the limit of ξk→∞ .c) d) a) b) 24 6 14 12 10 8 0 H [T] δφ /φ0 [%] 2 013 δφ /φ0 [%] 2 01 24 6 14 12 10 8 0 H [T] Hk = 0 H’=1/( γτ c) τc = 1 ps θ = 5 oθ = 5 oHk = 1 T τc = 1 ps H’=1/( γτ c) − HkmzH = 6, 7, 8, 9 T H = 2, 3, 4, 5 T = 1 ps = 5 = 0 = 1 ps τ θH τ = 1 ps = 1 ps = 1 ps τc = 1 ps τ = 1 ps τ = 1 ps θτH τ = 5 = 1 ps = 1 T = 1 ps = 1 ps = 1 T = 1 ps = 1 ps τc = 1 ps ττ = 1 ps = 1 ps 3δφ /φ0 [%] 2 013 δφ /φ0 [%] 2 013 t [ps] 100 200 0 t [ps] 100 200 0Hk = 0 τc = 1 ps Hk = 0 τc = 1 ps Figure 3: (a)τcdependence ofδ˙φ/˙φ0atθ=5◦. From top to bottom the external field is H=2,3,4,5 T. The parameters are Hk=0, andτc=1 ps. (b) The same plot as panel (a) for H≥5 T. From top to bottom the external field is H=6,7,8,9 T. (c) Hdependence ofδ˙φ/˙φ0atθ=5◦obtained by solving Eqs. (7) and (8). The parameters are Hk=0, andτc=1 ps. The critical value of the external field, H′=1/(γτc), is indicated by the thin vertical dotted line. (d) The same plot as panel (c) for Hk=1 T. The thin vertical dotted line indicates the critical value of the external field, H′=1/(γτc)−Hkmz. 5. A possible experiment to determine the correlation time Based on the results shown in Secs. 3 and 4 we propose a possible experiment to determine the correlation time, τc. Sim- ilar to the previous sections we first discuss the case withou t anisotropy field, i.e. Hk=0, and then extend the discussion to the case with Hk/nequal0. In Figs. 3(a) and 3(b) we show the temporal evolution of the enhancement of angular velocity, δ˙φ/˙φ0, obtained by the solv- ing Eqs. (7) and (8) for various values of H. The increment of the precession angular velocity is defined as δ˙φ=˙φ−˙φ0. The initial state and the correlation time are assumed to be m=(1,0,0) andτc=1 ps, respectively. As shown in Fig. 3(a),δ˙φ/˙φ0increases with increase of HforH≤5T. Once the external field exceeds the critical value of 1 /(γτc)=5.7 T, δ˙φ/˙φ0decreases with increase of Has shown in Fig. 3 (b). The results suggest that correlation time can be determined by a na- lyzing the external field that maximizes the enhancement of t he precession angular velocity. Figure 3(c) shows the Hdependence ofδ˙φ/˙φ0atθ=5◦ whereδ˙φ/˙φ0is almost saturated. The enhancement is maxi- mized at the critical value of the external field, H′=5.7 T. The correlation time is calculated as τc=1/(γH′)=1 ps. If the system has a uniaxial anisotropy field, Hk, the en- hancement of the precession angular velocity is maximized a t H′=1/(γτc)−Hkmzas shown in Fig. 3(d). The correlation time is obtained as τc=1/γ(H′+Hkmz). The above analysis is expected to be performed experimen- 5tally using the time resolved magneto optical Kerr e ffect mea- surement technique. In the practical experiments the analy sis can be simplified as follows. The polar angle of the initial st ate is not necessarily large. It can be small as far as the preces- sion angular velocity can be measured. Instead of analyzing δ˙φ/˙φ0, one can analyze ˙φ/Hor˙φ/(H+Hkmz) because they are maximized at the same value of Hasδ˙φ/˙φ0. Since the required magnetic field is as high as 10 T, a superconducting magnet [22 ] is required. 6. Summary In summary we theoretically analyze the ultrafast precessi on dynamics of a small magnet with non-Markovian damping. As- sumingα≪1, we derive the effective LLG equation valid for any values ofτc, which is a direct extension of Miyazaki and Seki’s work[16]. The derived e ffective LLG equation reveals the condition for maximizing ˙φin terms of Handτc. Based on the results we propose a possible experiment for determinat ion ofτc, whereτccan be determined from the external field that maximizesδ˙φ/˙φ0. References [1] L. Landau, E. Lifshits, ON THE THEORY OF THE DISPER- SION OF MAGNETIC PERMEABILITY IN FERROMAGNETIC BODIES, Physikalische Zeitschrift der Sowjetunion 8 (1935 ) 153. doi:10.1016/B978-0-08-010586-4.50023-7 . [2] T. Gilbert, Classics in Magnetics A Phenomenological Th eory of Damp- ing in Ferromagnetic Materials, IEEE Transactions on Magne tics 40 (6) (2004) 3443–3449. doi:10.1109/TMAG.2004.836740 . [3] W. F. Brown, Thermal Fluctuations of a Single-Domain Par ticle, Physical Review 130 (5) (1963) 1677–1686. doi:10.1103/PhysRev.130.1677 . [4] E. Beaurepaire, J.-C. Merle, A. Daunois, J.-Y . Bigot, Ul trafast Spin Dy- namics in Ferromagnetic Nickel, Physical Review Letters 76 (22) (1996) 4250–4253. doi:10.1103/PhysRevLett.76.4250 . [5] C. D. Stanciu, F. Hansteen, A. V . Kimel, A. Kirilyuk, A. Ts ukamoto, A. Itoh, T. Rasing, All-Optical Magnetic Recording with Cir cu- larly Polarized Light, Physical Review Letters 99 (4) (2007 ) 047601. doi:10.1103/PhysRevLett.99.047601 . [6] G. P. Zhang, W. H¨ ubner, G. Lefkidis, Y . Bai, T. F. George, Paradigm of the time-resolved magneto-optical Kerr e ffect for femtosecond magnetism, Nature Physics 5 (7) (2009) 499–502. doi:10.1038/nphys1315 . [7] J.-Y . Bigot, M. V omir, E. Beaurepaire, Coherent ultrafa st magnetism in- duced by femtosecond laser pulses, Nature Physics 5 (7) (200 9) 515–520. doi:10.1038/nphys1285 . [8] A. Kirilyuk, A. V . Kimel, T. Rasing, Ultrafast optical ma nipulation of magnetic order, Reviews of Modern Physics 82 (3) (2010) 2731 –2784. doi:10.1103/RevModPhys.82.2731 . [9] J.-Y . Bigot, M. V omir, Ultrafast magnetization dynamic s of nanostruc- tures: Ultrafast magnetization dynamics of nanostructure s, Annalen der Physik 525 (1-2) (2013) 2–30. doi:10.1002/andp.201200199 . [10] J. Walowski, M. M¨ unzenberg, Perspective: Ultrafast m agnetism and THz spintronics, Journal of Applied Physics 120 (14) (2016) 140901. doi:10.1063/1.4958846 . [11] Y . Quessab, R. Medapalli, M. S. El Hadri, M. Hehn, G. Mali nowski, E. E. Fullerton, S. Mangin, Helicity-dependent all-optical dom ain wall motion in ferromagnetic thin films, Physical Review B 97 (5) (2018) 0 54419. doi:10.1103/PhysRevB.97.054419 . [12] A. Kawabata, Brownian Motion of a Classical Spin, Progr ess of Theoret- ical Physics 48 (6) (1972) 2237–2251. doi:10.1143/PTP.48.2237 . [13] S. Nakajima, On Quantum Theory of Transport Phenomena: Steady Diffusion, Progress of Theoretical Physics 20 (6) (1958) 948–95 9. doi:10.1143/PTP.20.948 .[14] R. Zwanzig, Ensemble Method in the Theory of Irreversib il- ity, The Journal of Chemical Physics 33 (5) (1960) 1338–1341 . doi:10.1063/1.1731409 . [15] H. Mori, Transport, Collective Motion, and Brownian Mo - tion, Progress of Theoretical Physics 33 (3) (1965) 423–455 . doi:10.1143/PTP.33.423 . [16] K. Miyazaki, K. Seki, Brownian motion of spins revisite d, The Journal of Chemical Physics 108 (17) (1998) 7052–7059. doi:10.1063/1.476123 . [17] U. Atxitia, O. Chubykalo-Fesenko, R. W. Chantrell, U. N owak, A. Rebei, Ultrafast Spin Dynamics: The E ffect of Col- ored Noise, Physical Review Letters 102 (5) (2009) 057203. doi:10.1103/PhysRevLett.102.057203 . [18] C. W. Gardiner, Stochastic Methods: A Handbook for the N atural and Social Sciences, 4th Edition, no. 13 in Springer Series in Sy nergetics, Springer, Berlin Heidelberg, 2009. [19] G. E. Uhlenbeck, L. S. Ornstein, On the Theory of the Brownian Motion, Physical Review 36 (5) (1930) 823–841. doi:10.1103/PhysRev.36.823 . [20] H. Suhl, Theory of the magnetic damping constant, IEEE T ransactions on Magnetics 34 (4) (1998) 1834–1838. doi:10.1109/20.706720 . [21] H. Suhl, Relaxation Processes in Micromagnetics, Oxfo rd University Press, 2007. doi:10.1093/acprof:oso/9780198528029.001.0001 . [22] H. W. Weijers, U. P. Trociewitz, W. D. Markiewicz, J. Jia ng, D. My- ers, E. E. Hellstrom, A. Xu, J. Jaroszynski, P. Noyes, Y . Viou chkov, D. C. Larbalestier, High field magnets with HTS conductors, I EEE Transactions on Applied Superconductivity 20 (3) (2010) 57 6–582. doi:10.1109/TASC.2010.2043080 . 6

2022-05-13

Recent advances in experimental techniques have made it possible to manipulate and measure the magnetization dynamics on the femtosecond time scale which is the same order as the correlation time of the bath degrees of freedom. In the equations of motion of magnetization, the correlation of the bath is represented by the non-Markovian damping. For development of the science and technologies based on the ultrafast magnetization dynamics it is important to understand how the magnetization dynamics depend on the correlation time. It is also important to determine the correlation time experimentally. Here we study the precession dynamics of a small magnet with the non-Markovian damping. Extending the theoretical analysis of Miyazaki and Seki [J. Chem. Phys. 108, 7052 (1998)] we obtain analytical expressions of the precession angular velocity and the effective damping constant for any values of the correlation time under assumption of small Gilbert damping constant. We also propose a possible experiment for determination of the correlation time.

Precession dynamics of a small magnet with non-Markovian damping: Theoretical proposal for an experiment to determine the correlation time

2205.06399v1

arXiv:1211.2900v1 [math.AP] 13 Nov 2012CRITICAL EXPONENT FOR THE SEMILINEAR WAVE EQUATION WITH SCALE INVARIANT DAMPING YUTA WAKASUGI Abstract. In this paper we consider the critical exponent problem for t he semilinear damped wave equation with time-dependent coeffic ients. We treat the scale invariant cases. In this case the asymptotic behav ior of the solution is very delicate and the size of coefficient plays an essential ro le. We shall prove that if the power of the nonlinearity is greater than the Fuji ta exponent, then there exists a unique global solution with small data, provi ded that the size of the coefficient is sufficiently large. We shall also prove some b low-up results even in the case that the coefficient is sufficiently small. 1.Introduction We consider the Cauchy problem for the semilinear damped wave equa tion (1.1)/braceleftBigg utt−∆u+µ 1+tut=|u|p,(t,x)∈(0,∞)×Rn, (u,ut)(0,x) = (u0,u1)(x), x∈Rn, whereµ >0, (u0,u1)∈H1(Rn)×L2(Rn) have compact support and 1 < p≤ n n−2(n≥3),1< p <∞(n= 1,2). Our aim is to determine the critical exponent pc, which is the number defined by the following property: Ifpc<p, all small data solutions of (1.1) are global; if 1 <p≤pc, the time-local solution cannot be extended time-globally for some data regardless of smallness. We note that the linear part of (1.1) is invariant with respect to the h yperbolic scaling ˜u(t,x) =u(λ(1+t)−1,λx). In this case the asymptotic behavior of solutions is very delicate. It is known that the size of the damping term µplays an essential role. The damping term µ/(1+t) is known as the borderline between the effective andnon-effective dissipation, here effective means that the solution behaves like that of the corresponding par abolic equation, and non-effective means that the solution behaves like that of the free wave equation. Concretely, for linear damped wave equation (1.2) utt−∆u+(1+t)−βut= 0, if−1<β <1, then the solution uhas the same Lp-Lqdecay rates as those of the solution of the corresponding heat equation (1.3) −∆v+(1+t)−βvt= 0. Moreover, if −1/3<β <1 then the asymptotic profile of uis given by a solution of (1.3) (see [14]). This is called the diffusion phenomenon . In particular, the constant Key words and phrases. damped wave equation; time dependent coefficient; scale inva riant damping; critical exponent. 12 YUTA WAKASUGI coefficient case β= 0 has been investigated for a long time. We refer the reader to [8, 9]. On the other hand, if β >1 then the asymptotic profile of the solution of (1.2) is given by that of the free wave equation /squarew= 0 (see [13]). Wirth [12] considered the linear problem (1.4)/braceleftBigg utt−∆u+µ 1+tut= 0, (u,ut)(0,x) = (u0,u1)(x). He proved several Lp-Lqestimates for the solutions to (1.4). For example, if µ>1 it follows that /ba∇dblu(t)/ba∇dblLq/lessorsimilar(1+t)max{−n−1 2(1 p−1 q)−µ 2,−n(1 p−1 q)}(/ba∇dblu0/ba∇dblHsp+/ba∇dblu1/ba∇dblHs−1 p), /ba∇dbl(ut,∇u)(t)/ba∇dblLq/lessorsimilar(1+t)max{−n−1 2(1 p−1 q)−µ 2,−n(1 p−1 q)−1}(/ba∇dblu0/ba∇dblHs+1 p+/ba∇dblu1/ba∇dblHs p), where 1< p≤2, 1/p+ 1/q= 1 ands=n(1/p−1/q). This shows that if µ is sufficiently large then the solution behaves like that of the corresp onding heat equation (1.5)µ 1+tvt−∆v= 0 ast→ ∞, and ifµis sufficiently small then the solution behaves like that of the free wave equation. We mention that for the wave equation with spa ce-dependent damping /squareu+V0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−1ut= 0, a similar asymptotic behavior is obtained by Ikehata, Todorova and Yordanov [6]. The Gauss kernel of (1.5) is given by Gµ(t,x) =/parenleftbiggµ 2π((1+t)2−1)/parenrightbiggn 2 e−µ|x|2 2((1+t)2−1). We can obtain the Lp-Lqestimates of the solution of (1.5). In fact, it follows that /ba∇dblv(t)/ba∇dblLq/lessorsimilar(1+t)−n(1 p−1 q)/ba∇dblv(0)/ba∇dblLp for 1≤p≤q≤ ∞. In particular, taking q= 2 andp= 1, we have /ba∇dblv(t)/ba∇dblL2/lessorsimilar(1+t)−n 2/ba∇dblv(0)/ba∇dblL1. From the point of view of the diffusion phenomenon, we expect that t he same type estimate holds for the solution of (1.4) when µis large. To state our results, we introduce an auxiliary function ψ(t,x) :=a|x|2 (1+t)2, a=µ 2(2+δ) with a positive parameter δ. We have the following linear estimate: Theorem 1.1. For anyε >0, there exist constants δ >0andµ0>1such that for anyµ≥µ0the solution of (1.4)satisfies /integraldisplay Rne2ψu2dx≤Cµ,ε(1+t)−n+ε(1.6) /integraldisplay Rne2ψ(u2 t+|∇u|2)dx≤Cµ,ε(1+t)−n−2+ε(1.7) fort≥0, whereCµ,εis a positive constant depending on µ,εand/ba∇dbl(u0,u1)/ba∇dblH1×L2.SEMILINEAR DAMPED WAVE EQUATION 3 Remark 1.1. The constant µ0depends on ε. The relation is µ0∼ε−2. Therefore, as εsmaller,µ0must be larger. We can expect that εcan be removed and the same result holds for much smaller µ. However, we have no idea for the proof. We also consider the critical exponent problem for (1.1). For the co rresponding heat equation (1.5) with nonlinear term |u|p, the critical exponent is given by pF:= 1+2 n, which is well known as the Fujita critical exponent (see [4]). Thus, w e can expect that the critical exponent of (1.1) is also given by pFifµis sufficiently large. For the damped wave equation with constant coefficient utt−∆u+ut=|u|p, Todorovaand Yordanov[11] provedthe critical exponent is given b ypF. Lin, Nishi- hara and Zhai [7] (see also Nishihara[10]) extended this result to time -dependent coefficient cases utt−∆u+(1+t)−βut=|u|p with−1<β <1. Theyprovedthat pFisstillcritical. Recently, D’abbicco, Lucente and Reissig [3] extended this result to more general effective b(t) by using the linear decay estimates, which are established by Wirth [14]. For the scale-in variant case (1.1), very recently, D’abbicco [1] proved the existence of the glob al solution with small data for (1.1) in the case n= 1,2,µ≥n+2 andpF< p. He also obtained the decay rates of the solution without any loss ε. Our main result is following: Theorem 1.2. Letp > pFand0< ε <2n(p−pF)/(p−1). Then there exist constantsδ>0andµ0>1having the following property: if µ≥µ0and I2 0:=/integraldisplay Rne2ψ(0,x)(u2 0+|∇u0|2+u2 1)dx is sufficiently small, then there exists a unique solution u∈C([0,∞);H1(Rn))∩ C1([0,∞);L2(Rn))of(1.1)satisfying /integraldisplay Rne2ψu2dx≤Cµ,εI2 0(1+t)−n+ε(1.8) /integraldisplay Rne2ψ(u2 t+|∇u|2)dx≤Cµ,εI2 0(1+t)−n−2+ε(1.9) fort≥0, whereCµ,εis a positive constant depending on µandε. Remark 1.2. Similarly as before, we note that µ0depends on ε. The relation is µ0∼ε−2∼(p−pF)−2. Therefore, as pis closer to pF,µ0must be larger. Thus, we can expect that εcan be removed and the same result holds for much smaller µ. As mentioned above, D’abbicco [1]obtained an affirmative result for this expectation when n= 1,2. However, we have no idea for the higher dimensional cases.4 YUTA WAKASUGI We prove Theorem 1.2 by a multiplier method which is essentially develope d in [11]. Lin, Nishihara and Zhai [7] refined this method to fit the damping term b(t) = (1+t)−βwith−1<β <1. They used the property β <1 and so we cannot apply their method directly to our problem (1.1). Therefore, we nee d a further modification. Instead of the property β <1, we assume that µis sufficiently large and modify the parameters used in the calculation. Remark1.3. We can also treat other nonlinear terms, for example −|u|p,±|u|p−1u. We also have a blow-up result when µ>1 and 1<p≤pF. Theorem 1.3. Let1<p≤pFandµ>1. Moreover, we assume that/integraldisplay Rn(µ−1)u0+u1dx>0. Then there is no global solution for (1.1). Remark 1.4. Theorem 1.3 is essentially included in a recent work by D’abb icco and Lucente [2]. In this paper we shall give a much simpler proof. One of our novelty is blow-up results for the non-effective damping c ases. We also obtain blow-up results in the case 0 <µ≤1. Theorem 1.4. Let0<µ≤1and 1<p≤1+2 n+(µ−1). We also assume /integraldisplay Rnu1(x)dx>0. Then there is no global solution for (1.1). Remark 1.5. In Theorem 1.4, we do not put any assumption on the data u0, and the blow-up results hold even for the case p≥pF. We can interpret this phenomena as that the equation (1.1)loses the parabolic structure and recover the hyperbolic structure if µis sufficiently small. We prove this theorem by a test-function method developed by Qi S . Zhang [15]. In the same way of the proof of Theorem 1.4, we can treat the damp ing terms (1+t)−βwithβ >1 (see Remark 3.1). In the next section, we give a proof of Theorem 1.2. We can prove Th eorem 1.1 by the almost same way, and so we omit the proof. In Section 3, we sh all prove Theorem 1.3 and Theorem 1.4. 2.Proof of Theorem 1.2 In this section we prove our main result. First, we prepare some not ation and terminology. We put /ba∇dblf/ba∇dblLp(Rn):=/parenleftbigg/integraldisplay Rn|f(x)|pdx/parenrightbigg1/p . ByH1(Rn) we denote the usual Sobolev space. For an interval Iand a Banach spaceX, we define Cr(I;X) as the Banach space whose element is an r-times continuously differentiable mapping from ItoXwith respect to the topology in X. The letter Cindicates the generic constant, which may change from a line toSEMILINEAR DAMPED WAVE EQUATION 5 the next line. We also use the symbols /lessorsimilarand∼. The relation f/lessorsimilargmeansf≤Cg with some constant C >0 andf∼gmeansf/lessorsimilargandg/lessorsimilarf. We first describe the local existence: Proposition 2.1. For anyp >1,µ >0andδ >0, there exists Tm∈(0,+∞] depending on I2 0such that the Cauchy problem (1.1)has a unique solution u∈ C([0,Tm);H1(Rn))∩C1([0,Tm);L2(Rn)), and ifTm<+∞then we have liminf t→Tm/integraldisplay Rneψ(t,x)(u2 t+|∇u|2+u2)dx= +∞. We can provethis propositionby standard arguments(see [5]). We p rovea priori estimate for the following functional: M(t) := sup 0≤τ≤t/braceleftbigg (1+τ)n+2−ε/integraldisplay Rne2ψ(u2 t+|∇u|2)dx+(1+τ)n−ε/integraldisplay Rne2ψu2dx/bracerightbigg . We putb(t) =µ 1+tandf(u) =|u|p. By a simple calculation, we have −ψt=2 1+tψ,∇ψ=2ax (1+t)2,|∇ψ|2 −ψt=b(t) 2+δ and ∆ψ=n 2+δb(t) 1+t=:/parenleftBign 2−δ1/parenrightBigb(t) 1+t. Here and after, δi(i= 1,2,...) denote a positive constant depending only on δsuch that δi→0+asδ→0+. Multiplying (1.1) by e2ψut, we obtain ∂ ∂t/bracketleftbigge2ψ 2(u2 t+|∇u|2)/bracketrightbigg −∇·(e2ψut∇u) (2.1) +e2ψ/parenleftbigg b(t)−|∇ψ|2 −ψt−ψt/parenrightbigg u2 t+e2ψ −ψt|ψt∇u−ut∇ψ|2 /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright T1 =∂ ∂t/bracketleftbig e2ψF(u)/bracketrightbig +2e2ψ(−ψt)F(u), whereFis the primitive of fsatisfyingF(0) = 0. Using the Schwarz inequality, we can calculate T1≥e2ψ/parenleftbigg1 5(−ψt)|∇u|2−b(t) 4(2+δ)u2 t/parenrightbigg . From this and integrating (2.1), we have d dt/integraldisplay Rne2ψ 2(u2 t+|∇u|2)dx+/integraldisplay Rne2ψ/braceleftbigg/parenleftbiggb(t) 4−ψt/parenrightbigg u2 t+−ψt 5|∇u|2/bracerightbigg dx (2.2) ≤d dt/integraldisplay Rne2ψF(u)dx+2e2ψ(−ψt)F(u)dx.6 YUTA WAKASUGI On the other hand, by multiplying (1.1) by e2ψu, it follows that ∂ ∂t/bracketleftbigg e2ψ/parenleftbigg uut+b(t) 2u2/parenrightbigg/bracketrightbigg −∇·(e2ψu∇u) +e2ψ/braceleftBig |∇u|2+/parenleftbigg −ψt+1 2(1+t)/parenrightbigg b(t)u2+2u∇ψ·∇u/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright T2−2ψtuut−u2 t/bracerightBig =e2ψuf(u). We calculate e2ψT2= 4e2ψu∇ψ·∇u−2e2ψu∇ψ·∇u = 4e2ψu∇ψ·∇u−∇·(e2ψu2∇ψ)+2e2ψu2|∇ψ|2+e2ψ(∆ψ)u2 and have ∂ ∂t/bracketleftbigg e2ψ/parenleftbigg uut+b(t) 2u2/parenrightbigg/bracketrightbigg −∇·(e2ψ(u∇u+u2∇ψ)) (2.3) +e2ψ/braceleftBig |∇u|2+4u∇u·∇ψ+((−ψt)b(t)+2|∇ψ|2)u2 /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright T3 +(n+1−2δ1)b(t) 2(1+t)u2−2ψtuut−u2 t/bracerightBig =e2ψuf(u). T3is estimated as T3=/parenleftbigg 1−4 4+δ/2/parenrightbigg |∇u|2+δ 2|∇ψ|2u2+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/radicalbig 4+δ/2∇u+/radicalbig 4+δ/2∇ψ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ≥δ2(|∇u|2+b(t)(−ψt)u2). Thus, we can rewrite (2.3) as ∂ ∂t/bracketleftbigg e2ψ/parenleftbigg uut+b(t) 2u2/parenrightbigg/bracketrightbigg −∇·(e2ψ(u∇u+u2∇ψ)) +e2ψ/braceleftbigg δ2(|∇u|2+b(t)(−ψt)u2)+(n+1−2δ2)b(t) 2(1+t)u2−2ψtuut−u2 t/bracerightbigg ≤e2ψuf(u). Integrating the above inequality and then multiplying by a large param eterνand adding (1+ t)×(2.2), we obtain d dt/bracketleftbigg/integraldisplay e2ψ/braceleftbigg1+t 2(u2 t+|∇u|2)+νuut+νb(t) 2u2/bracerightbigg dx/bracketrightbigg +/integraldisplay e2ψ/braceleftBig/parenleftBigµ 4−ν−1 2/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright T4+(−ψt)(1+t)/parenrightBig u2 t+/parenleftBig νδ2−1 2/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright T5+(−ψt)(1+t) 5/parenrightBig |∇u|2 +νδ2b(t)(−ψt)u2+(n+1−2δ1)νb(t) 2(1+t)u2+2ν(−ψt)uut/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright T6/bracerightBig dx ≤d dt/bracketleftbigg (1+t)/integraldisplay e2ψF(u)dx/bracketrightbigg +C/integraldisplay e2ψ(1+(1+t)(−ψt))|u|p+1dx.SEMILINEAR DAMPED WAVE EQUATION 7 We put the condition for µandνas µ 4−ν−1 2>0 (2.4) νδ2−1 2>0. (2.5) Then the terms T4andT5are positive. Using the Schwarz inequality, we can estimateT6as |T6| ≤1 2(−ψt)(1+t)u2 t+2ν2 1+t(−ψt)u2. Now we put an another condition (2.6) µ≥2ν δ2. Then we obtain the following estimate. d dtˆE(t)+H(t)+(n+1−2δ1)νb(t) 2(1+t)J(t;u2) (2.7) ≤d dt[(1+t)J(t;F(u))]+C(J(t;|u|p+1)+(1+t)Jψ(t;|u|p+1)), where ˆE(t) :=/integraldisplay e2ψ/braceleftbigg1+t 2(u2 t+|∇u|2)+νuut+νb(t) 2u2/bracerightbigg dx, H(t) =/integraldisplay e2ψ/braceleftbigg/parenleftbiggµ 4−ν−1 2/parenrightbigg u2 t+/parenleftbigg νδ2−1 2/parenrightbigg |∇u|2/bracerightbigg dx J(t;u) =/integraldisplay e2ψudx, Jψ(t;u) =/integraldisplay e2ψ(−ψt)udx. Multiplying (2.7) by (1+ t)n+1−ε, we have d dt[(1+t)n+1−εˆE(t)]−(n+1−ε)(1+t)n−εˆE(t)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright T7 +(1+t)n+1−εH(t)+(n+1−2δ1)(1+t)n+1−ενb(t) 2(1+t)J(t;u2) ≤d dt[(1+t)n+2−εJ(t;F(u))] +C((1+t)n+1−ε(J(t;|u|p+1)+(1+t)Jψ(t;|u|p+1)). Now we estimate the bad term T7. First, by the Schwarz inequality, one can obtain |νuut| ≤ν 4δ3b(t)u2 t+δ3νb(t)u2, whereδ3determined later. From this, T7is estimated as T7≤(n+1−ε)(1+t)n−ε ×/integraldisplay e2ψ/braceleftbigg/parenleftbigg1+t 2+ν(1+t) 4δ3µ/parenrightbigg u2 t+1+t 2|∇u|2+νb(t) 2(1+2δ3)u2/bracerightbigg dx.8 YUTA WAKASUGI We strengthen the conditions (2.4) and (2.5) as µ 4−ν−1 2−(n+1−ε)/parenleftbigg1 2+ν 4δ3µ/parenrightbigg >0, (2.8) νδ2−1 2(n+2−ε)>0. (2.9) Moreover, we take ε= 3δ1and then choose δ3such that (n+1−2δ1)−(n+1−3δ1)(1+2δ3)>0. Under these conditions, we can estimate T7and obtain d dt[(1+t)n+1−εˆE(t)]≤d dt[(1+t)n+2−εJ(t;F(u))] +C(1+t)n+1−ε(J(t;|u|p+1)+(1+t)Jψ(t;|u|p+1)). By integrating the above inequality, it follows that (1+t)n+1−εˆE(t)≤CI2 0+(1+t)n+2−εJ(t;|u|p+1) +C/integraldisplayt 0(1+τ)n+1−ε(J(τ;|u|p+1)+(1+t)Jψ(τ;|u|p+1))dτ. By a simple calculation, we have (1+t)E(t)+1 1+tJ(t;u2)≤CˆE(t), where E(t) :=/integraldisplay e2ψ(u2 t+|∇u|2)dx. Thus, we obtain (1+t)n+2−εE(t)+(1+t)n−εJ(t;u2) (2.10) ≤CI2 0+(1+t)n+2−εJ(t;|u|p+1) +C/integraldisplayt 0(1+τ)n+1−ε(J(τ;|u|p+1)+(1+t)Jψ(τ;|u|p+1))dτ. Now we turn to estimate the nonlinear terms. We need the following lem ma: Lemma 2.2 (Gagliardo-Nirenberg) .Letp,q,r∈[1,∞]andσ∈[0,1]satisfy 1 p=σ/parenleftbigg1 r−1 n/parenrightbigg +(1−σ)1 q except forp=∞orr=nwhenn≥2. Then for some constant C=C(p,q,r,n)> 0, the inequality /ba∇dblh/ba∇dblLp≤C/ba∇dblh/ba∇dbl1−σ Lq/ba∇dbl∇h/ba∇dblσ Lrfor anyh∈C1 0(Rn) holds. Noting that J(t;|u|p+1) =/integraldisplay/vextendsingle/vextendsingle/vextendsinglee2 p+1ψu/vextendsingle/vextendsingle/vextendsinglep+1 dxSEMILINEAR DAMPED WAVE EQUATION 9 and∇(e2 p+1ψu) =2 p+1e2 p+1ψ(∇ψ)u+e2 p+1ψ∇u, we apply the above lemma to J(t;|u|p+1) withσ=n(p−1) 2(p+1)and have J(t;|u|p+1)≤C/parenleftbigg/integraldisplay e4 p+1ψu2dx/parenrightbigg1−σ 2(p+1) ×/parenleftbigg/integraldisplay e4 p+1ψ|∇ψ|2u2dx+/integraldisplay e4 p+1ψ|∇u|2dx/parenrightbiggσ 2(p+1) . We note that e4 p+1ψ|∇ψ|2=4a2|x|2 (1+t)4e4 p+1ψ≤C1 (1+t)2e2ψ and obtain J(t;|u|p+1)≤C/parenleftbigg/integraldisplay e2ψu2dx/parenrightbigg1−σ 2(p+1) ×/parenleftbigg1 (1+t)2/integraldisplay e2ψu2dx+/integraldisplay e2ψ|∇u|2dx/parenrightbiggσ 2(p+1) . Therefore, we can estimate (1+t)n+2−εJ(t;|u|p+1)≤(1+t)n+2−ε{(1+t)−(n−ε)M(t)}1−σ 2(p+1) ×{(1+t)−(n+2−ε)M(t)}σ 2(p+1). By a simple calculation, if (2.11) ε<2n/parenleftbig p−/parenleftbig 1+2 n/parenrightbig/parenrightbig p−1 then we have (1+t)n+2−εJ(t;|u|p+1)≤CM(t)p+1. We note that the conditions (2.6), (2.8), (2.9), (2.11) are fulfilled by the determi- nation of the parameters in the order p→ε→δ→ν→µ. In a similar way, we can estimate the other nonlinear terms. Consequ ently, we obtain the a priori estimate M(t)≤CI2 0+CM(t)p+1. This proves Theorem 1.2. 3.Proof of Theorem 1.3 and Theorem 1.4 In this section we first give a proof of Theorem 1.3. We use a method b y Lin, Nishihara and Zhai [7] to transform (1.1) into divergence form and t hen a test- function method by Qi S. Zhang [15]. Letµ>1. We multiply (1.1) by a positive function g(t)∈C2([0,∞)) and obtain (gu)tt−∆(gu)−(g′u)t+(−g′+gb)ut=g|u|p. We now choose g(t) as the solution of the initial value problem for the ordinary differential equation (3.1)/braceleftBigg−g′(t)+g(t)b(t) = 1, t>0, g(0) =1 µ−1.10 YUTA WAKASUGI The solution g(t) is explicitly given by g(t) =1 µ−1(1+t). Thus, we obtain the equation in divergence form (3.2) ( gu)tt−∆(gu)−(g′u)t+ut=g|u|p. Next, we apply a test function method. We first introduce test fun ctions having the following properties: η(t)∈C∞ 0([0,∞)),0≤η(t)≤1, η(t) =/braceleftbigg1,0≤t≤1 2, 0, t≥1,(t1) φ(x)∈C∞ 0(Rn),0≤φ(x)≤1, φ(x) =/braceleftbigg1,|x| ≤1 2, 0,|x| ≥1,(t2) η′(t)2 η(t)≤C/parenleftBig1 2≤t≤1/parenrightBig ,|∇φ(x)|2 φ(x)≤C/parenleftBig1 2≤ |x| ≤1/parenrightBig . (t3) LetRbe a large parameter in (0 ,∞). We define the test function ψR(t,x) :=ηR(t)φR(x) :=η/parenleftbiggt R/parenrightbigg φ/parenleftBigx R/parenrightBig , Letqbe the dual of p, that isq=p p−1. Suppose that uis a global solution with initial data ( u0,u1) satisfying /integraldisplay Rn((µ−1)u0+u1)dx>0. We define IR:=/integraldisplay QRg(t)|u(t,x)|pψR(t,x)qdxdt, whereQR:= [0,R]×BRandBR:={x∈Rn;|x| ≤R}. By the equation (3.2) and integration by parts one can calculate IR=−/integraldisplay BR((µ−1)u0+u1)φq Rdx +/integraldisplay QRgu∂2 t(ψq R)dxdt+/integraldisplay QR(g′u−u)∂t(ψq R)dxdt−/integraldisplay QRgu∆(ψq R)dxdt =:−/integraldisplay BR((µ−1)u0+u1)φq Rdx+J1+J2+J3. By the assumption on the data ( u0,u1) it follows that IR<J1+J2+J3 for largeR. We shall estimate J1,J2andJ3, respectively. We use the notation ˆQR:= [R/2,R]×BR,˜QR:= [0,R]×(BR\BR/2).SEMILINEAR DAMPED WAVE EQUATION 11 We first estimate J3. By the conditions (t1)-(t3) and the H¨ older inequality we have the following estimate: |J3|/lessorsimilarR−2/integraldisplay ˜QRg(t)|u|ψq−1 Rdxdt /lessorsimilarR−2/parenleftbigg/integraldisplay ˜QRg(t)|u|pψq R(t,x)dxdt/parenrightbigg1/p/parenleftbigg/integraldisplay ˜QRg(t)dxdt/parenrightbigg1/q /lessorsimilar˜I1/p RRn+2 q−2, where ˜IR:=/integraldisplay ˜QRg(t)|u|pψq R(t,x)dxdt. In a similar way, we can estimate J1andJ2as |J1|+|J2|/lessorsimilarˆI1/p RRn+2 q−2,ˆIR:=/integraldisplay ˆQRg(t)|u|pψq R(t,x)dxdt. Hence, we obtain (3.3) IR/lessorsimilar(˜I1/p R+ˆI1/p R)Rn+2 q−2, in particular I1−1/p R/lessorsimilarRn+2 q−2. If 1<p<pF, by letting R→ ∞we haveIR→0 and henceu= 0, which contradicts the assumption on the data. If p=pF, we have onlyIR≤Cwith some constant Cindependent of R. This implies that g(t)|u|pis integrable on (0 ,∞)×Rnand hence lim R→∞(˜IR+ˆIR) = 0. By (3.3), we obtain lim R→∞IR= 0. Therefore, umust be 0. This also leads a contradiction. Proof of Theorem 1.4. The proof is almost same as above. Let 0 <µ≤1. Instead of (3.1), we consider the ordinary differential equation (3.4) −g′(t)+g(t)b(t) = 0 withg(0)>0. We can easily solve this and have g(t) =g(0)(1+t)µ. Then we have (3.5) ( gu)tt−∆(gu)−(g′u)t=g|u|p. Using the same test function ψR(t,x) as above, we can calculate IR:=/integraldisplay QRg(t)|u|pψq Rdxdt =−/integraldisplay BRg(0)u1φq Rdx+3/summationdisplay k=1Jk, where J1=/integraldisplay QRgu∂2 t(ψq R)dxdt, J 2=/integraldisplay QRg′u∂t(ψq R)dxdt, J 3=−/integraldisplay QRgu∆(ψq R)dxdt.12 YUTA WAKASUGI We note that the term of u0vanishes and so we put the assumption only for u1. We first estimate J2. Notingg′(t) =µg(0)(1+t)µ−1, we have |J2|/lessorsimilar1 R/integraldisplay ˆQR(1+t)µ−1|u|ψq−1 Rdxdt. By noting that (1+ t)µ−1∼g(t)1/p(1+t)µ/q−1and the H¨ older inequality, it follows that |J2|/lessorsimilar1 R/parenleftbigg/integraldisplay ˆQRg|u|pψq Rdxdt/parenrightbigg1/p/parenleftBigg/integraldisplayR R/2/integraldisplay BR(1+t)µ−qdxdt/parenrightBigg1/q /lessorsimilarˆI1/p RR−1+(n+(µ−q+1))/q, whereˆIRis defined as before. A simple calculation shows −1+(n+(µ−q+1))/q≤0⇔p≤1+2 n+(µ−1). In the same way, we can estimate J1andJ3as |J1|+|J3|/lessorsimilar(ˆI1/p R+˜I1/p R)R−2+(n+µ+1)/q, where˜IRis same as before. It is also easy to see that −2+(n+µ+1)/q≤0⇔p≤1+2 n+(µ−1). Finally, we have IR/lessorsimilar˜I1/p R+ˆI1/p R ifp≤1+2/(n+(µ−1)). The rest of the proof is same as before. /square Remark 3.1. We can apply the proof of Theorem 1.4 to the wave equation with non-effective damping terms /braceleftbigg utt−∆u+b(t)ut=|u|p, (u,ut)(0,x) = (u0,u1)(x), where b(t) = (1+t)−β withβ >1. We can easily solve (3.4)and have g(t) =g(0)exp/parenleftbigg1 −β+1((1+t)−β+1−1)/parenrightbigg . We note that g(t)∼1. The same argument implies that if 1<p≤1+2 n−1,/integraldisplay u1dx>0, then there is no global solution. We note that the exponent 1+2/(n−1)is greater than the Fujita exponent. This shows that when β >1, the equation loses the parabolic structure even in the nonlinear cases. One can exp ect that the critical ex- ponent is given by the well-known Strauss critical exponent . However, this problem is completely open as far as the author’s knowledge.SEMILINEAR DAMPED WAVE EQUATION 13 Acknowledgement The author has generous support from Professors Tatsuo Nishit ani and Kenji Nishihara. In particular, Prof. Nishihara gave the author an essen tial idea for the proof of Theorem 1.4. References [1]M. D’abbicco ,Semilinear scale-invariant wave equations with time-depe ndent speed and damping , arXiv:1211.0731v1. [2]M. D’abbicco, S. Lucente ,A modified test function method for damped wave equations , arXiv:1211.0453v1. [3]M. D’abbicco, S. Lucente, M. Reissig ,Semi-Linear wave equations with effective damping , arXiv:1210.3493v1. [4]H. Fujita ,On the blowing up of solutions of the Cauchy problem for ut= ∆u+u1+α, J. Fac. Sci. Univ. Tokyo Sec. I, 13(1966), 109-124. [5]R. Ikehata, K. Tanizawa ,Global existence of solutions for semilinear damped wave eq uations inRNwith noncompactly supported initial data , Nonlinear Anal., 61(2005), 1189-1208. [6]R. Ikehata, G. Todorova, B. Yordanov ,Optimal decay rate of the energy for wave equa- tions with critical potential , J. Math. Soc. Japan (in press). [7]J. Lin, K. Nishihara, J. Zhai ,Critical exponent for the semilinear wave equation with tim e- dependent damping , Discrete Contin. Dyn. Syst., 32(2012), 4307-4320. [8]A. Matsumura ,On the asymptotic behavior of solutions of semi-linear wave equations , Publ. Res. Inst. Math. Sci. Kyoto Univ., 12(1976), 169-189. [9]K. Nishihara ,Lp−Lqestimates of solutions to the damped wave equation in 3-dimensional space and their application , Math. Z., 244(2003), 631-649. [10]K. Nishihara ,Asymptotic behavior of solutions to the semilinear wave equ ation with time- dependent damping , Tokyo J. Math., 34(2011), 327-343. [11]G. Todorova, B. Yordanov ,Critical exponent for a nonlinear wave equation with dampin g, J. Differential Equations, 174(2001), 464-489. [12]J. Wirth ,Solution representations for a wave equation with weak diss ipation, Math. Meth. Appl. Sci., 27(2004), 101-124. [13]J. Wirth ,Wave equations with time-dependent dissipation I. Non-effe ctive dissipation , J. Differential Equations, 222(2006), 487-514. [14]J. Wirth ,Wave equations with time-dependent dissipation II. Effecti ve dissipation , J. Dif- ferential Equations 232(2007), 74-103. [15]Qi S. Zhang ,A blow-up result for a nonlinear wave equation with damping: the critical case , C. R. Acad. Sci. Paris S´ er. I Math., 333(2001), 109-114. E-mail address :y-wakasugi@cr.math.sci.osaka-u.ac.jp Department of Mathematics, Graduate School of Science, Osa ka University, Toy- onaka, Osaka, 560-0043, Japan

2012-11-13

In this paper we consider the critical exponent problem for the semilinear damped wave equation with time-dependent coefficients. We treat the scale invariant cases. In this case the asymptotic behavior of the solution is very delicate and the size of coefficient plays an essential role. We shall prove that if the power of the nonlinearity is greater than the Fujita exponent, then there exists a unique global solution with small data, provided that the size of the coefficient is sufficiently large. We shall also prove some blow-up results even in the case that the coefficient is sufficiently small.

Critical exponent for the semilinear wave equation with scale invariant damping

1211.2900v1

arXiv:1605.04543v1 [cond-mat.mtrl-sci] 15 May 2016Propagation of Thermally Induced Magnonic Spin Currents Ulrike Ritzmann, Denise Hinzke, and Ulrich Nowak Fachbereich Physik, Universit¨ at Konstanz, D-78457 Konst anz, Germany (Dated: 19.12.2013) The propagation of magnons in temperature gradients is inve stigated within the framework of an atomistic spin model with the stochastic Landau-Lifshitz- Gilbert equation as underlying equation of motion. We analyze the magnon accumulation, the magnon te mperature profile as well as the propagation length of the excited magnons. The frequency di stribution of the generated magnons is investigated in order to derive an expression for the influ ence of the anisotropy and the damping parameter on the magnon propagation length. For soft ferrom agnetic insulators with low damping a propagation length in the range of some µm can be expected for exchange driven magnons. PACS numbers: 75.30.Ds, 75.30.Sg, 75.76.+j I. INTRODUCTION Spin caloritronics is a new, emerging field in mag- netism describing the interplay between heat, charge and spin transport1,2. A stimulation for this field was the dis- covery of the spin Seebeck effect in Permalloy by Uchida et al.3. Analog to the Seebeck effect, where in an elec- tric conductor an electrical voltage is created by apply- ing a temperature gradient, in a ferromagnet a temper- ature gradient excites a spin current leading to a spin accumulation. The generated spin accumulation was de- tected by measuring the spin current locally injected into a Platinum-contact using the inverse spin Hall effect3,4. A first explanation of these effect was based on a spin- dependent Seebeckeffect, wherethe conductionelectrons propagate in two different channels and, due to a spin dependent mobility, create a spin accumulation in the system5. Interestingly, it was shown later on that this effect also appears in ferromagnetic insulators6. This shows that in addition to conduction-electron spin-currents, chargeless spin-currents exist as well, where the angular momentum is transported by the magnetic excitations of the system, so-called magnons. A first theoretical description of such a magnonic spin Seebeck effect was developed by Xiao et al.7. With a two temperature model including the local magnon (m) and phonon (p) temperatures the measured spin Seebeck voltage is calculated to be linearly depen- dent on the local difference between magnon and phonon temperature, ∆ Tmp=Tm−Tp. This temperature dif- ference decays with the characteristic lengthscale λ. For the ferromagnetic material YIG they estimate the length scale in the range of several millimeters. The contribution of exchange dominated magnons to the spin Seebeck effect was investigated in recent experi- ments by Agrawalet al.8. Using Brillouin lightscattering the difference between the magnon and the phonon tem- perature in a system with a linear temperature gradient was determined. They found no detectable temperature difference and estimate a maximal characteristic length scale of the temperature difference of 470 µm. One pos- sible conclusion from this results might be be that in- stead of exchangemagnons, magnetostatic modes mainlycontribute to the spin Seebeck effect and are responsible for the long-range character of this effect. Alternatively, phononsmightcontributetothemagnonaccumulationas well viaspin-phonon drag9,10. A complete understanding of these different contributions to the spin Seebeck effect is still missing. In this paper thermally excited magnonic spin currents and their length scale of propagation are investigated. Usingatomisticspinmodelsimulationwhichdescribethe thermodynamics of the magnetic system in the classical limit including the whole frequency spectra of excited magnons,wedescribespincurrentsbyexchangemagnons in the vicinity of a temperature step. After introducing our model, methods and basic definitions in Section II we determine the magnon accumulation as well as the corresponding magnon temperature and investigate the characteristic lengthscale of the decay of the magnon ac- cumulation in Section III. In Section IV we introduce an analytical description which is supported by our sim- ulations shown in Section V and gives insight into the material properties dependence of magnon propagation. II. MAGNETIZATION PROFILE AND MAGNON TEMPERATURE For the investigationofmagnonic spin currentsin tem- perature gradients we use an atomistic spin model with localized spins Si=µi/µsrepresenting the normalized magnetic moment µsof a unit cell. The magnitude of the magnetic moment is assumed to be temperature in- dependent. Wesimulateathree-dimensionalsystemwith simple cubic lattice structure and lattice constant a. The dynamics of the spin system are described in the classical limit by solving the stochastic Landau-Lifshitz-Gilbert (LLG) equation, ∂Si ∂t=−γ µs(1+α2)Si×(Hi+α(Si×Hi)), (1) numerically with the Heun method11withγbeing the gyromagnetic ratio. This equation describes a preces- sion of each spin iaround its effective field Hiand the2 coupling with the lattice by a phenomenological damp- ing term with damping constant α. The effective field Hiconsists of the derivative of the Hamiltonian and an additional white-noise term ζi(t), Hi=−∂H ∂Si+ζi(t) . (2) The Hamiltonian Hin our simulation includes exchange interaction of nearest neighbors with isotropic exchange constant Jand an uniaxial anisotropy with an easy axis inz-direction and anisotropy constant dz, H=−J 2/summationdisplay <i,j>SiSj−dz/summationdisplay iS2 i,z. (3) The additional noise term ζi(t) of the effective field Hi includes the influence of the temperature and has the following properties: /angbracketleftζ(t)/angbracketright= 0 (4) /angbracketleftBig ζi η(0)ζj θ(t)/angbracketrightBig =2kBTpαµs γδijδηθδ(t) . (5) Herei,jdenote lattice sites and ηandθCartesian com- ponents of the spin. We simulate a model with a given phonon temperature Tpwhich is space dependent and includes a temperature step inz-direction in the middle of the system at z= 0 fromatemperature T1 pinthehotterareato T2 p= 0K(see Fig. 1). We assume, that this temperature profile stays constant during the simulation and that the magnetic excitationshavenoinfluence onthe phonontemperature. The system size is 8 ×8×512, large enough to minimize finite-size effects. All spins are initialized parallel to the easy-axis in z- direction. Due tothetemperaturestepanon-equilibrium in the magnonic density of states is created. Magnons propagate in every direction of the system, but more magnons exist in the hotter than in the colder part of the system. This leads to a constant net magnon current from the hotter towards the colder area of the system. Due to the damping of the magnons the net current ap- pears around the temperature step with a finite length scale.After an initial relaxation time the system reaches a steadystate. In this steady state the averagedspin cur- rent from the hotter towardsthe colderregionis constant and so the local magnetization is time independent. We can now calculate the local magnetization m(z) depend- ing on the space coordinate zas the time average over all spins in the plane perpendicular to the z-direction. We use the phonon temperature T1 p= 0.1J/kBin the heated area, the anisotropy constant dz= 0.1Jand vary the damping parameter α. The resulting magnetization versus the space coordinate zfor different damping pa- rameters in a section around the temperature step is shown in Fig. 1. For comparison the particular equi- librium magnetization m0of the two regions is also cal- culated and shown in the figure.m0α= 1α= 0.1α= 0.06Tp space coordinate z/a phonon temperature kBTp/Jmagnetization m0.1 0 403020100-10-20-30-401 0.995 0.99 0.985 0.98 0.975 FIG. 1. Steady state magnetization mand equilibrium mag- netization m0over space coordinate zfor a given phonon temperature profile and for different damping parameters α in a small section around the temperature step. Far away from the temperature step on both sides the amplitudes of the local magnetization m(z) converge to the equilibrium values, only in the vicinity of the tem- perature step deviations appear. These deviations de- scribe the magnon accumulation, induced by a surplus of magnons from the hotter region propagating towards the colder one. This leads to a less thermal excitation in the hotter area and the value of the local magneti- zation increases. In the colder area the surplus of in- coming magnons decrease the value of the local magneti- zation. For smaller values of αthe magnons can propa- gate overlargerdistances before they are finally damped. This leads to a damping-dependent magnon accumula- tion which increases with decreasing damping constant α. Forafurtheranalysisinthecontextofthespin-Seebeck effect we define a local magnon temperature Tm(z) via the magnetization profile m(z). For that the equilib- rium magnetization m0(T) is calculated for the same model but homogeneous phonon temperature Tp. In equilibrium magnon temperature Tmand the phonon temperature Tpare the same and we can determine the fitequilibrium data magnon temperature kBTm/Jmagnetization m0 10.90.80.70.60.50.40.30.20.101 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 FIG. 2. Equilibrium magnetization m0over the magnon temperature Tm. Red points show the simulated equilibrium magnetization and the black line shows a fit of the data.3 T0 mα= 1α= 0.1α= 0.06Tp space coordinate z/a phonon temperature kBTp/Jmagnon temperature kBTm/J 0.1 0 403020100-10-20-30-400.1 0.08 0.06 0.04 0.02 0 FIG. 3. Magnon temperature Tmover the space coordinate zfor different damping parameters αcorresponding to the results in Fig. 1. (magnon) temperature dependence of the equilibrium magnetization m0(Tm) of the system. The magnetiza- tion of the equilibrium system decreases for increasing magnontemperatureasshownin Fig. 2 andthe behavior can be described phenomenologically with a function12 m0(T) = (1−Tm/Tc)βwhereTcistheCurietemperature. Fitting ourdatawefind Tc= (1.3326±0.00015)J/kBand for the exponent we get β= 0.32984±0.00065. This fit of the data is also shown in Fig. 2 and it is a good approximation over the whole temperature range. The inverse function is used in the following to determine the magnon temperature for a given local magnetization and with that a magnon temperature profile Tm(z). The resulting magnon temperature profiles are shown in Fig. 3. Far away from the temperature step the magnon temperature Tm(z) coincides with the given phonon temperature Tp, and deviations — dependent on the damping constant α— appear only around the tem- perature step. These deviations correspond to those of the local magnetization discussed in connection with Fig. 1. III. MAGNON PROPAGATION LENGTH To describe the characteristic lengthscale of the magnon propagationaround the temperature step we de- fine the magnon accumulation ∆ m(z) as the difference between the relative equilibrium magnetization m0(z) at the given phonon temperature Tp(z) and the calculated local magnetization m(z): ∆m(z) =m0(z)−m(z) . (6) Weinvestigatethemagnonpropagationinthecolderpart of the system, where Tp(z) = 0. For a small magnon temperature, the temperature dependence of the magne- tization can be approximated as m(Tm)≈1−β TcTm. (7)α= 1.00α= 0.50α= 0.10α= 0.08α= 0.06 space coordinate z/amagnon accumulation ∆m 2502001501005001 10−2 10−4 10−6 10−8 10−10 FIG. 4. Magnon accumulation ∆ mover space coordinate z in the colder region of the system at Tp= 0K for different damping constants αshows exponential decay with magnon propagation length ξ. The points show the data from our simulation and the lines the results from an exponential fit. These linear equation is in agreement with an analytical solution for low temperatures presented by Watson et al.12. For low phonon temperatures one obtains for the difference between phonon and magnon temperature ∆T=Tm−Tp=β Tc∆m. (8) Note, that the proportionality between magnon accumu- lation and temperature difference holds for higher tem- peratures as well as long as magnon and phonon temper- ature are sufficiently close so that a linear approximation applies,thoughtheproportionalityfactorincreases. Note also, that this proportionalitywas determined in theoret- ical descriptions of a magnonic spin Seebeck effect7. Our results for the magnon accumulation should hence be rel- evantfortheunderstandingofthemagnonicspinSeebeck where the temperature difference between the magnons in the ferromagnet and the electrons in the non-magnet plays a key role. We further investigate our model as before with a tem- perature in the heated area of T1 p= 0.1J/kB, anisotropy constant dz= 0.1Jand different damping parameters. The magnon accumulation ∆ mversus the space coordi- natezin the colder region of the system at Tp= 0K is shown in Fig. 4. Apart from a sudden decay close to the temperature step the magnon accumulation ∆ m(z) then decays exponentially on a length scale that depends on the damping constant α. To describe this decay we fit the data with the function ∆m(z) = ∆m(0)·e−z ξ. (9) We define the fitting parameter ξas the propagation length of the magnons. Here, the deviations from the exponential decay at the beginning of the system are ne- glected. The fits for the data are shown in Fig. 4 as continuous lines. The propagation length dependence on the damping parameter αis shown in Fig. 5. The values of the prop- agation length from our simulations, shown as points,4 dz= 0.01Jdz= 0.05Jdz= 0.10Jdz= 0.50J damping constant αpropagation length ξ/a 1 0.1100 10 1 FIG. 5. Magnon propagation length ξover the damping con- stantαfor different anisotropy constant dz. Numerical data is shown as points and the solid lines are from Eq. (19). are inversely proportional to the damping constant α and, furthermore, show also a strong dependence on the anisotropy constant dz. This behavior will be discussed in the next two sections with an analytical analysisof the magnon propagation and an investigation of the frequen- cies of the propagating magnons. A simple approxima- tion for the propagation length leads to Eq. (19) which is also shown as solid lines in Fig. 5. IV. ANALYTICAL DESCRIPTION WITH LINEAR SPIN-WAVE THEORY For the theoretical description of the magnon accu- mulation, excited by a temperature step in the system, we solve the LLG equation (Eq. (1)), analytically in the area with Tp= 0K. We consider a cubical system with lattice constant awhere all spins are magnetized in z-direction parallel to the easy-axis of the system. As- suming only small fluctuations in the x- andy-direction we have Sz i≈1 andSx i,Sy i≪1. In that case we can linearize the LLG-equation and the solution of the re- sulting equation consists of a sum over spin waves with wavevectors qand the related frequency ωqwhich decay exponentially in time dependent on their frequency and the damping constant αof the system, S± i(t) =1√ N/summationdisplay qS± q(0)e∓iqri±iωqt·e−αωqt. (10) The frequency ωqof the magnons is described by the usual dispersion relation ¯hωq=1 (1+α2)/parenleftBig 2dz+2J/summationdisplay θ(1−cos(qθaθ))/parenrightBig . (11) The dispersion relation includes a frequency gap due to the anisotropy constant and a second wavevector depen- dent term with a sum over the Cartesian components13. Considering now the temperature step, magnons from the hotter area propagate towards the colder one. Weinvestigate the damping process during that propagation inordertodescribethe propagatingfrequenciesaswellas to calculate the propagation length ξof the magnons for comparisonwiththeresultsfromsectionIII.Themagnon accumulation will depend on the distance to the temper- ature step and — for small fluctuations of the SxandSy components — can be expressed as ∆m(z) = 1−/angbracketleftSz(z)/angbracketright ≈1 2/angbracketleftbig Sx(z)2+Sy(z)2/angbracketrightbig , (12) where the brackets denote a time average. We assume that the local fluctuations of the SxandSycomponents can be described with a sum over spin waves with differ- ent frequencies and damped amplitudes aq(z), Sx(z) =/summationdisplay qaq(z)cos(ωqt−qr) , (13) Sy(z) =/summationdisplay qaq(z)sin(ωqt−qr) . (14) In that case for the transverse component of the magne- tization one obtains /angbracketleftbig Sx(z)2+Sy(z)2/angbracketrightbig =/angbracketleftBigg/summationdisplay qaq(z)2/angbracketrightBigg , (15) where mixed terms vanish upon time averaging. The magnon accumulation can be written as: ∆m(z) =1 2/angbracketleftBigg/summationdisplay qaq(z)2/angbracketrightBigg . (16) The amplitude aq(z) of a magnon decays exponentially as seen in Eq. (10) dependent on the damping constant and the frequency of the magnons. In the next step we describe the damping process during the propagation of the magnons. In the one-dimensional limit magnons only propagate in z-direction with velocity vq=∂ωq ∂q. Then the propagation time can be rewritten as t=z/vqand we can describe the decay of the amplitude with aq(z) = aq(0)·f(z) with a damping function f(z) = exp/parenleftBig −αωqz ∂ωq ∂qz/parenrightBig . (17) The amplitudes are damped exponentially during the propagation which defines a frequency dependent propa- gation length ξωq=/radicalbigg J2−/parenleftBig 1 2(1+α2)(¯hωq−2dz)−J/parenrightBig2 α(1+α2)¯hωq,(18) where we used γ=µs/¯h. In the low anisotropy limit this reduces to ξωq=λ/παwhereλ= 2π/qis the wave length of the magnons.5 The total propagation length is then the weighted av- erage over all the excited frequencies. The minimal fre- quency is defined by the dispersion relation with a fre- quency gap of ωmin q= 1/(¯h(1 +α2))2dz. For small fre- quencies above that minimum the velocity is small, so the magnons are damped within short distances. Due to the fact that the damping process is also frequency de- pendent higher frequencies will also be damped quickly. In the long wave length limit the minimal damping is at the frequency ωmax q≈4dz/(¯h(1 +α2)) which can be determined by minimizing Eq. (17) . In a three-dimensional system, besides the z- component of the wavevector, also transverse compo- nents of the wavevector have to be included. The damping of magnons with transverse components of the wavevector is higher than described in the one- dimensionalcase,becausethe additionaltransverseprop- agationincreasethepropagationtime. Inoursimulations the cross-section is very small, so that transverse compo- nents of the wave-vectors are very high and get damped quickly. Thisfact andthe highdamping forhighfrequen- cies described in Eq. (17) can explain the very strong damping at the beginning of the propagation shown in Fig. 4. V. FREQUENCIES AND DAMPING OF PROPAGATING MAGNONS In this section we investigate the frequency distribu- tionofthemagnonicspincurrentwhilepropagatingaway from the temperature step. First we determine the fre- quencies of the propagating magnons in our simulations with Fourier transformation in time to verify our as- sumptions from the last section. As before a system of 8×8×512spins with a temperature step in the center of the system is simulated with an anisotropy of dz= 0.1J. The temperature of the heated area is T1 p= 0.1J/kBand the damping constant is α= 0.1. After an initial relax- ation to a steady-state the frequency distribution of the propagating magnons in the colder area is determined by Fourier transformation in time of S±(i) =Sx(i)±iSy(i). The frequency spectra are averaged over four points in thex-y-plane and analyzed depending on the distance z of the plane to the temperature step. The results for small values of zare shown in Fig. 6(a) and for higher values of z, far away from the temper- ature step, for the regime of the exponential decay, in Fig. 6(b). For small values of z, near the temperature step, the frequency range of the propagating magnons is very broad. The minimum frequency is given by ωmin q= 2dz/(¯h(1+α2)) and far away from the temperature step the maximum peak is around ωmax q= 4dz/(¯h(1 +α2)). These characteristic frequencies are in agreement with our findings in section IV. Furthermore, a stronger damping for higher frequen- cies can be observed. This effect corresponds to the strong damping of magnons with wavevector compo-10864204·10−3 3·10−3 2·10−3 1·10−3 0z= 20az= 10az= 1a(a) frequency ¯hωq/Jamplitude |S+(ωq)| 1.41.210.80.60.40.208·10−5 6·10−5 4·10−5 2·10−5 0ωminz= 100az= 90az= 80a(b) frequency ¯hωq/Jamplitude |S+(ωq)| FIG.6. Amplitude |S+(ωq)|versusthefrequency ωqfor asys- tem with 8 ×8×512 spins. (a):after propagation over short distances form 1 to 20 lattice constants. (b): after propaga - tion over longer distances from 80 to 100 lattice constants. nents transverse to the z-direction and it explains the higher initial damping, which was seen in the magnon accumulation in Fig. 4. A much narrower distribu- tion propagates over longer distances and reaches the area shown in Fig. 6(b). In that area the damping can be described by one-dimensional propagation of the magnons in z-direction with a narrow frequency distri- bution around the frequency with the lowest damping ωmax q= 4dz/(¯h(1+α2)).The wavelength and the belong- ing group velocity of the magnons depending on their frequency in the one-dimensional analytical model are shown in Fig.7(a). In the simulated system magnons with the longest propagation length have a wavelength ofλ= 14a. Depending on the ratio dz/Jthe wave- length increases for systems with lower anisotropy. As discussed in the last chapter, magnons with smaller fre- quencies are less damped in the time domain, but due to their smaller velocity the magnons very close to the min- imum frequency also have a smaller propagation length. To investigate the frequency-dependent damping- process during the propagation of the magnons we calcu- latetheratiooftheamplitudeofthemagnons |S+(ωq,x)| forz= 80aandz= 80a+∆ with ∆ = 10 a,20a,50aand normalize it to a damping per propagation of one spin. The resulting ratios ( |S+(ωq,x)|/|S+(ωq,x−∆)|)1/∆are shown in Fig. 7 in comparison with the frequency-6 2 1.5 1 0.5 0 43.532.521.510.50100 80 60 40 20 0ωmin qvqλ(a) frequency ¯hωq/J velocityvq¯h/(Ja)wave length λ/a damping function∆ = 10a∆ = 20a∆ = 50a(b) frequency ω[J/¯h]damping ratio 10.90.80.70.60.50.40.30.21 0.98 0.96 0.94 0.92 0.9 FIG. 7. (a): Wavelength λand group velocity vqof the magnons in a one-dimensional model dependent on the fre- quencyωq. (b):Damping ratio as explained in the text versus the frequency ωqfor different distances ∆ and compared to the damping function (Eq. (17)). dependent damping-function (Eq. (17)). The figure shows a good agreement between simulation and our an- alytical calculations. These results explain the dependence of the magnon propagation length on the model parameters. The fre- quency with the maximal amplitude is determined by the anisotropy constant. Under the assumption that the frequency with the lowest damping is dominant and the contribution of other frequencies can be neglected the propagation length can be calculated as ξ=a 2α/radicalbigg J 2dz, (19) where the square-root term is the domain wall width of the model. This formula is also plotted in Fig. 5. The comparison with our simulations shows good agreement though the equation above gives only the propagation of those magnons with the smallest damp- ingduringthe propagation.Inthe consideredsystemwith α= 0.1 anddz= 0.1Jwe get a propagation length of aboutξ= 11aat a wavelength of the magnons λ= 14a. For smaller values of the anisotropy and smaller damp- ing parameters the frequency distribution of the thermal magnons is broader and Eq. (19) is an overestimation of the real propagation length since the magnon accu-mulation is no longer exponentially decaying due to the broader spectrum of propagating frequencies. However we would expect for soft ferromagnetic insulators with a smalldamping constantof10−4−10−3and ananisotropy constant in the range of 10−3J−10−2Ja propagation length of 103a−105awhich would be in the micrometer- range. VI. SUMMARY AND DISCUSSION Using the frameworkof an atomistic spin model we de- scribe thermally induced magnon propagationin a model containing a temperature step. The results give an im- pression of the relevant length scale of the propagation of thermally induced exchange magnons and its depen- dence on system parameters as the anisotropy, the ex- change and the damping constant. In the heated area magnons with a broad frequency distribution are gener- atedandbecauseoftheverystrongdampingformagnons with high frequency, especially those with wave-vector components transverse to the propagation direction in z-direction, most of the induced magnons are damped on shorter length scales. Behind this region of strong damping near the temperature step, the propagation of magnons is unidirectional and the magnon accumula- tion decays exponentially with the characteristic prop- agation length ξ. This propagation length depends on the damping parameter but also on system properties as the anisotropy of the system, because of the dependence on the induced frequencies. In contrast to long range magnetostatic spin waves, which can propagate over distances of some mm14,15, we find that for exchange magnons the propagation length is considerably shorter and expect from our findings for soft ferromagnetic insulators with a low damping con- stant a propagation length in the range of some µm for those magnons close to the frequency gap and the lowest damping. These findings will contribute to the under- standing of length scale dependent investigations of the spin Seebeck effect8,16–18. Recent experiments investigate the longitudinal spin Seebeck effect, where the generated spin current longitudinal to the applied temperature gradient is measured19–22. In this configuration Kehlberger et al. show that the measured spin current is dependent on the thickness of the YIG layer and they observe a saturation ofthespincurrentonalengthscaleof100 nm16. Thissat- uration can be explained by the lengthscale of the prop- agation of the thermally excited magnons. Only those magnons reaching the YIG/Pt interface of the sample contribute to the measured spin current and — as shown here — exchange magnons thermally excited at larger distances are damped before they can reach the inter- face. In this paper, we focus on the propagation length of those magnons with the lowest damping, however the lengthscale of the magnon accumulation at the end of a temperature gradient is dominated by a broad range7 of magnons with higher frequencies which are therefore damped on shorter length scales. ACKNOWLEDGMENTS The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG) for financial support viaSPP 1538 “Spin Caloric Transport” and the SFB 767 “Controlled Nanosystem: Interaction and Interfacing to the Macroscale”. 1G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nature Mater. 11, 391 (2012). 2G. E. W. Bauer, A. H. MacDonald, and S. Maekawa, Solid State Commun. 150, 459 (2010). 3K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. 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2016-05-15

The propagation of magnons in temperature gradients is investigated within the framework of an atomistic spin model with the stochastic Landau-Lifshitz-Gilbert equation as underlying equation of motion. We analyze the magnon accumulation, the magnon temperature profile as well as the propagation length of the excited magnons. The frequency distribution of the generated magnons is investigated in order to derive an expression for the influence of the anisotropy and the damping parameter on the magnon propagation length. For soft ferromagnetic insulators with low damping a propagation length in the range of some $\mu$m can be expected for exchange driven magnons.

Propagation of Thermally Induced Magnonic Spin Currents

1605.04543v1

Anisotropic Magnetization Relaxa tion in Ferromagnetic GaMnAs Thin Films Kh.Khazen, H.J.von Bardeleben, M.Cubukcu, J.L.Cantin Institut des Nanosciences de Paris, Université Paris 6, UMR 7588 au CNRS 140, rue de Lourmel, 75015 Paris, France V.Novak, K.Olejnik, M.Cukr Institut of Physics, Academy of Sciences, Cukrovarnicka 10, 16253 Praha, Czech Republic L.Thevenard, A. Lemaître Laboratoire de Photonique et des Nanostructures, CNRS Route de Nozay, 91460 Marcoussis, France Abstract: The magnetic properties of annealed, epitaxial Ga 0.93Mn 0.07As layers under tensile and compressive stress have been investigat ed by X-band (9GHz) and Q-band (35GHz) ferromagnetic resonance (FMR) spectroscopy. From the analysis of the linewidths of the uniform mode spectra the FMR Gilbert damping factor α has been determined. At T=4K we obtain a minimum damping factor of α = 0.003 for the compressively stressed layer. Its value is not isotropic. It has a minimum value for th e easy axes orientations of the magnetic field and increases with the measuring temperature. It s average value is for both type of films of the order of 0.01 in spite of strong differences in the inhomogeneous linewidth which vary between 20 Oe and 600 Oe for the layers grown on GaAs and GaInAs substrates respectively . PACS numbers: 75.50.Pp, 76.50.+g, 71.55.Eq Introduction: The magnetic properties of ferromagnetic Ga 1-xMn xAs thin films with Mn concentrations between x=0.03 and 0.08 have been studied in great detail in the recent years both theoretically and experimentally. For recent reviews see references [1, 2]. A particularity of GaMnAs ferro magnetic thin films as comp ared to conventional metal ferromagnetic thin films is the predominance of the magnetocrystalline anisotropy fields over the demagnetization fields. The strong anisotropy fields are not directly related to the crystal structure of GaMnAs but are induced by the la ttice mismatch between the GaMnAs layers and the substrate material on which they are grow n. When grown on (100) GaAs substrates the difference in the lattice constants induces biaxial strains of ≈ 0.2% which give rise to anisotropy fields of several 103 Oe. The low value of the de magnetization fields (~300Oe) is the direct consequence of the small spin conc entration in diluted magnetic semiconductors (DMS) which for a 5% Mn doping leads to a saturation magnetization of only 40 emu/cm3. As the strain is related to the lattice mismat ch it can be engineer ed by choosing different substrate materials. The two systems which have been investigated most often are (100) GaAs substrates and (100)GaInAs pa rtially relaxed buffer layers. These two cases correspond to compressive and tensile strained Ga MnAs layers respectively [3]. The static micro-magnetic pr operties of GaMnAs layers can be determined by magnetization, transport, magneto-optical and ferromagnetic resonance techniques. For the investigation of the ma gnetocrystalline anisotropies the ferromagnetic resonance spectroscopy (FMR) technique has been shown to be partic ularly well adapted [2, 4]. The dynamics and relaxation processes of the magnetization of such layers have hardly been investigated up to now [5-7]. The previous FMR studies on this subject concerned either unusually low doped GaMnAs layers [5, 7] or employed a single microwave frequency [6] which leads to an overestimation of the damping factor. The knowle dge and control of the relaxation processes is in particular important for device applications as they de termine for example the critical currents necessary for current induced magne tization switching. It is thus important to determine the damping factor for state of the art samples with high Curie temperatures of T C ≈ 150K, such as those used in this work. Anothe r motivation of this work is the search for a potential anisotropy of the ma gnetization relaxation in a dilu ted ferromagnetic semiconductor in which the m agnetocrystalline anisotro pies are strong and dom inant over the demagnetization contribution. The intr insic sm all angle m agnetization re laxation is generally described by one param eter, the Gilbert d amping coefficient α, which is defined by the Landau Lifshitz Gilbert (LLG) equation of m otion for the m agnetization: ⎥⎦⎤ ⎢⎣⎡× +⎥⎦⎤ ⎢⎣⎡× −= ⋅dtsdMeffH MdtMdrr rrr γα γ1 eq.1 with M the m agnetization, H eff the effective m agnetic field, α the dam ping fa ctor, γ the gyrom agnetic ratio and s the uni t vector parallel to M. The dam ping factor α is generally assum ed to be a scal ar quantity [8, 9] . It is defined for sm all angle precess ion relaxatio n which is the case of FMR experim ents. This param eter can be experim entally determ ined by FMR spectr oscopy either from the angular variation of the linewidth or from the variati on of the uniform mode linewidth ∆Hpp with th e microwave frequency. In this second case the linewidth is given by: ω γω ⋅ ⋅⋅ + ∆= ∆+ ∆= ∆ MGH H H Hin inpp 2 hom hom hom32)( eq.2 With ∆Hpp the first derivative p eak-to -peak linewid th of the uniform mode of Lorentzian lineshape, ω the angular m icrowave frequency an d G the Gil bert dam ping factor from which the m agnetization independent damping factor α can be deduced as α=G/γM. In eq. 2 it is assum ed that the m agnetiz ation and th e applied magnetic f ield are collin ear which is fulfilled for high symm etry direction s in GaMnAs such as [001], [110] and [100]. Ot herwise a 1/cos ( θ-θH) term has to be added to eq.2 [8]. ∆Hinhom is the inhom ogeneous, frequency indepe ndent linewidth; it can be further decom posed in three con tributions, re lated to the crysta lline imperf ection of the f ilm [10]: int inthom HHH H HHr H Hr H Hr in ∆⋅ + ∆⋅ + ∆⋅∂= ∆δδφδφδθθδ eq.3 These three term s were introduced to take in to account a slight m osaic structure of the metallic thin f ilms def ined by the polar angles (θ, φ) and their distributions ( ∆θ, ∆φ) - expressed in the first two term s in eq.3- and a distribution of the internal anisotropy fields H int – the last term of eq.3. In the case of homo epitaxial III-V films obtained by MBE growth like GaMnAs on GaAs, films of high crystalline qu ality are obtained [LPN] and only the third component ( ∆Hint) is expected to play an important role. Practically, the variation of the FMR linewid th with the microwave frequency can be measured with resonant cavity systems at different frequencies between 9GHz and 35GHz; the minimum requirement -used also in this work - is the use of two frequencies. We disposed in this work of 9GHz and 35GHz spectrometers. The linewidth is decomposed in a frequency independent inhomogeneous part and a linear fr equency dependent homogeneously broadened part. For most materials the inhomogeneous fr action of the linewidth is strongly sample dependent and depends further on the interface quali ty and the presence of cap layers. It can be smaller but also much larger than the intrinsic linewidth. In Ga 0.95Mn 0.05As single films total X-band linewidths be tween 100Oe and 1000Oe have been encountered. These observations indicate already the impor tance of inhomogeneous broadening. The homogeneous linewidth will depend on the intrinsic sample properties. This approach supposes that the inhomogeneous linewidth is frequency independent and the homogenous linewidth linear dependent on the frequency, two assumptions generally valid for high symmetry orientations of the a pplied field for which the magne tization is parallel to the magnetic field. It should be underlined that in diluted magnetic semiconductor (DMS) materials like GaMnAs the damping parameter is not only determined by the sample composition x Mn [5]. It is expected to depend as we ll on (i) the magnetic compensati on which will vary with the growth conditions, (ii) the (hol e) carrier concentration respon sible for the ferromagnetic Mn- Mn interaction which is influenced by the presence of native donor defects like arsenic antisite defects or Mn interstitial ions [11] and (iii) the valence bandstructure, sensitive to the strain in the film. Due to the high out-of –plane and in-plane anisotropy of the magnetic parameters [12] which further vary with the applied field and the temperature a rather complex situation with an anisotropic and te mperature dependent da mping factor can be expected in GaMnAs. Whereas the FMR Gilbert damping factor has been determined for many metallic ferromagnetic thin films [8] only three experimental FMR studies have been published for GaMnAs thin films up to now [5-7]. In ref.[ 5,7] low doped GaMnAs laye rs with a critical temperature of 80K which do not correspond to the high quality, standa rd layers available today were studied. In the ot her work [6] higher doped layers were investigated but the experiments were limited to a single microw ave frequency (9GHz) and thus no frequency dependence could be studied. In this work we present the results of FMR studies at 9GHz and 35 GHz on two high quality GaMnAs layers with optimum critical temperatures: one is a compressively strained layer grown on a GaAs buffer layer and the othe r a tensile strained layer grown on a (Ga,In)As buffer layer. Due to the opposite sign of the strains the easy axis of magnetization is in-plane [ 100] in the first case and out-o f-plane [001] in the second. The GaMnAs layers have been annealed ex-situ after their growth in order to reduce the electrical and magnetic compensation, to homogenize the laye rs and to increase the Curie temperature to ≈ 130K. Such annealings have become a st andard procedure for improving the magnetic properties of low temperature molecular b eam epitaxy (LTMBE) grown GaMnAs films. Indeed, the low growth temperature required to incorporate the high Mn concentration without the formation of precipitates gives rise to native defect the conc entration of which can be strongly reduced by the annealing. Experimental details A first sample consisting of a Ga 0.93Mn 0.07As layer of 50nm thickness has been grown at 250° C by low temperature molecular beam epitaxy on a semi-insulating (100) oriented GaAs substrate. A thin GaAs buffer layer has been grown before the deposition of the magnetic layer. The second sample, a 50 nm thick Ga 0.93Mn 0.07As layer have been grown under very similar conditions on a partially relaxed (100) Ga 0.902In0.098As buffer layer; for more details see ref. [13]. After the growth the structure was thermally annealed at 250° C for 1h under air or nitrogen gas fl ow. The Curie temperatures were 157K and 130K respectively. Based on conductivity measuremen ts the hole concentratio n is estimated in the 1020cm-3 range. The FMR measurements were performed with Bruker X-band and Q-band spectrometers under standard conditions: mW microwave power and 100 KHz field modulation. The samples were measured at te mperatures between 4K and 170K. The angular variation of the FMR spectra was measured in the two rotation planes (110) and (001). The peak-to peak linewidth of the first derivati ve spectra were obtained from a lineshape simulation. The value of the st atic magnetization M(T) had been determined by a commercial superconducting quantum interference device (SQUID) magnetometer. A typical hysteresis curve is shown in the inset of fig.8. Experimental results: The saturation magnetizations of the two laye rs and the magneto crystalline anisotropy constants which had been previously de termined by SQUID and FMR measurements respectively are given in table I. The anisotropy constants had been determined in the whole temperature range but for clarity only its values at T=55K and T=80K are given in table I. We see that the dominant anisotropy constant K 2⊥ are of different sign with -55000 erg/cm3 to +91070 erg /cm3 and that the other three constants ha ve equally opposite signs in the two types of layers. The easy axes of magnetization are the in-pla ne [100] and the out-of-plane [001] direction respectively. Howe ver the absolute values of th e total effective perpendicular anisotropy constant Ku=K 2⊥ +K 4⊥ are less different for the two samples: -46517erg/cm3 and +57020erg/cm3 respectively. More detailed inform ation on the measurements of these micromagnetic parameters will be published elsewhere. For the GaMnAs/GaAs layers the peak-to-peak linewidth of the first derivative uniform mode spectra has been strongly re duced by the thermal annealing; in the non annealed sample the X-band linewidth was highl y anisotropic with va lues between 150Oe and 500Oe at T=4K. After annealing it is reduced to an quasi isotropic average value of 70Oe at X-band. Quite differently, for the GaMnAs/GaInA s system the annealing process decreases the linewidth of the GaMnAs layers only marginally. Although full angular dependencies have been measured by FMR we will analyze only the linewidth of the four high symmetry field orientations H//[001], H //[100], H//[1-10], H//[ 110] corresponding to the hard and easy axes of magnetization. As will be shown below, in spite of rather similar high critical temperatures (157K/130K) the linewidth are drastically di fferent for the two cases. 1. GaMnAs on GaAs In fig. 1a and 1b we show typical low te mperature FMR spectra at X-band and Q-band frequencies for the hard [001] /intermediate [100] axis orientation of the applied magnetic field. The spectra are characterized by excelle nt signal to noise ra tios and well defined lineshapes. We see that at both frequencies the lineshape is close to a Lorentzian. In addition to the main mode one low intensity spin wave resonance is observed at both frequencies at lower fields (not shown). The linewidth at X-band (fig.2) is of the order of 50Oe to 75Oe with a weak orientation and temperature dependence. Above T>130K, close to the criti cal temperature, the linewidth increase strongly. At Q-band we observe a systematic increase by a factor of two of the total linewidth (fig.3) with an increase d temperature and orient ation dependence. As generally observed in GaMnAs, the easy axis orie ntation gives rise to th e lowest linewidth. At Q-band the lineshape is perfectly Lorentzian (f ig.1b). These linewidth are among the smallest ever reported for GaMnAs thin films, which re flects the high crystal line and magnetic quality of the film. To determine the damping factor α we have plotted the frequency dependence of the linewidth for the different orientations and at various temperatures. An example is given in fig. 4 for T=80K; this allows us to determin e the inhomogeneous linewidth obtained from a linear extrapolation to zero frequency and the damping factor from the slope. The inhomogeneous linewidth at T=80K is of the order of 30 Oe, i.e. 50% of the total linewidth at X-band. This shows that the approximation ∆Hinhom<< ∆Hhomo which had been previously used [5] to deduce the damping factor from a single (X-band) frequenc y measurement is not fulfilled here. The temperature dependence of the inhomogeneous linewidth is shown in fig.5. Similar trends as for the total linewidth in the non annealed films are observed: the linewidth is high at the lowest temperatures, decreases with increasing temperat ures up to 120K and increases again close to T C. From the slope of the linewidth variati on with microwave frequency we obtain the damping factor α (fig.6). Its high temperature value is of the order of 0.010 but we observe a systematic, linear variation with the temperatur e and a factor two difference between the easy axis orientation [100] and the hard axis orientation [001]. 2. GaMnAs on GaInAs Similar measurements have been performed on the annealed tensile strained layer. In tensile strained GaMnAs films the easy axis of magnetization ([001]) coincides with the strong uniaxial second order anisotropy directio n. For that reason no FMR resonance can be observed at temperatures below T=80K for the easy axis orientation H// [001] at X-band. For the other three orientations the resonances can be observed at X-band in the whole temperature range 4K to T C. Due to the strong temperature dependence of the anisotropy constants and the parallel decr ease of the internal anisotropy fields the easy axis resonance becomes observable at X-band for temperatures above 80K. In the films on GaInAs much higher linewidth are encountered th an in films on GaAs, the values are up to ten times higher indicating a strong inhomogeneity in this film. A second low fiel d resonance is systematically observed at X-band and Q-band; it is equally attributed to a spin wave resonance. Figures 7a and 7b show typical FMR spectra at X- and Q-band re spectively. At both frequencies the lineshape can no longer be simu lated by a Lorentzian but has changed into a Gaussian lineshape. Contrary to the first cas e of GaMnAs/GaAs the X-band linewidth varies monotonously in the whole temperature region (fig.8). We observe a linewidth of ~600Oe at T=4K, which decreases only slowly with temperature; the linewidth becomes minimal in the 100 K to 140K range. The Curie temperature “s een” by the FMR spectroscopy is s lightly higher as compared to the one measured by SQUID due to th e presence of the applied magnetic field. At low temperature the Q-ba nd linewidth vary strongly w ith the orientation of the applied field with values be tween 500Oe and 700Oe. The lowest value is observed for the easy axis orientation. They decrease as at X-band only slowly with increasing temperature and increase once again when approaching the Curie temperature. At Q-band the easy axis FMR spectrum, which is also accompanied by a str ong spin wave spectrum at lower fields, is observable in the whole temperature range. For this sample we observe especially at Q-band a systematic difference between the cubic axes [100], [001] linewidth and the one for the in-plane [110] and [1-10] field orientations (fig.8). The most surprising observation is that for temperatures below T<100K the linewidth for H//[100] and H//[110] are co mparable at X-band and Q-band and thus an analysis in the simple model discussed above is not possible. We attribute this to much higher crystallographic/magnetic inhomogeneities, which mask the homogenous linewidth. The origin of the strong inhomogeneity is still unclear. The only orientation for which in the whole temperature range a systematic increas e in the linewidth between X-and Q-band is observed is the H//[1-10] orienta tion. We have thus analyzed th is variation (fig.10) according to eq.1. In spite of important differences in the lin ewidth the slope varies only weakly which indicates that the inhomogeneous linewidth is very temperature dependent and decreases monotonously with increasing temperature from 570Oe to 350Oe. In the high temperature range (T ≥100K) the easy axis orientation could also be analyzed (fig.11). The inhomogeneous linewidth are lower than for the hard axis orientation at the same temperatures and are in the 300Oe range (fig.12). The homogenous linewidth at 9Ghz is in the 50Oe range which is close to the values determined in the first case of GaMnAs/GaAs. From the slope (fig.13) we obtain the da mping factor which for the hard axis orientation is α=0.010 in the whole temperature range. Th is value is comparable to the one measured for the GaMnAs/GaAs film for H//[110]. The damping factor for the easy axis orientation is lower but increases close to T C as in the previous case. Discussion: An estimation of the FMR intrinsic damping factor in a ferromagnetic GaMnAs thin film has been made within a model of localiz ed Mn spins coupled by p-d kinetic exchange with the itinerant-spin of holes treated by the 6-band Kohn-Luttinger Hamiltonian [5]. Note, that these authors take for the effective kinetic exchange field the value in the mean- field approximation, i.e. H eff=JN<S>, so that their calculation are made within the random phase approximation (RPA). RPA calculations of α have been made by Heinrich et al. [14] and have recently been used by Tserkovnyak et al .[15] for numerical app lications to the case of Ga 0.95Mn 0.05As. Both models however, are phenomenological and include an adjustable parameter: the quasiparticle lifetime Γ for the holes in [5] and the spin-flip relaxation T 2 in [15]. These models do not take into account neither multi-magnon scattering nor any damping beyond the RPA. It has been argued elsewhere [16], that in diluted magnetic semiconductors such affects are only impor tant at high temperatur e (i.e. at T>Tc). In particular, the increase of α in the vicinity of Tc may be attributed to such effects that are beyond the scope of the models of references [5] and [14]. At low temperatures T<<Tc however, where the corrections to the RPA are expected to be negligib le, the models of [5, 14,15] provide us with a numerical value of α in agreement with our experiments if we introduce reasonable values of these parameters. For GaMnAs films with metallic conductivity, Mn concentrations of x= 0.05 and hole concentr ations of 0.5 nm-3 (5x1020cm-3) Sinova et al [5] predict an isotropi c low temperature damping factor α between 0.02 and 0.03 depending on the quasiparticle life time broa dening. Tserkovnyak et al [15] found a similar value of α ≈ 0.01 for the isotropic damping factor fo r a typical GaMnAs film with 5% Mn doping and full hole polarization. Both predicted values are of the same orde r of magnitude as the experimental values determined in this study. Our results for GaMn As/GaAs show further that the damping factor is not isotropic as generally a ssumed but is anisotropic with a lowest value for the in-plane easy axis orientations of the applied magnetic field H//[100], H//[1-10] and an increase of up to a factor of two for the hardest axis orientat ion H// [001]. Intrinsic anisotropic damping is related to the fact that the free energy density depends on the orientation of the magnetization which in the case of GaMnAs is related to th e anisotropy of the p-hole Fermi surface. We have shown (table I) that the anisotropy of the magnetocristalline constants and the related fields are important in these strained layers and it is thus not surprisi ng to find also anisotropy of the damping factor. For further discussions on this subject see reference [9]. The system might also contain extrinsic an isotropies related to the pres ence of lattice defects. Their influence can be deduced from the value and anisotropy of the inhomogeneous linewidth. In the case of the compressively strained layers (G aMnAs/GaAs) we see that their value is small and rather isotropic quite differen tly from the tensile strained film. It is in the first case that our measurements show a factor of two anisot ropy of the Gilbert damping factors. A further indication for the intrinsic charac ter of the anisotropy is the fact that the damping factor for the perpendicular orientation has the highest value. In this case any contribution from two magnon scattering will be minimized. Anyway, such contributions are generally only important at low frequency measurements in the 2-6GHz range but even there they were found to be negligible [7]. Additional material related parameters are expected to further influence the damping factor. As the spin flip relaxation times will depend on the sample properties and in particular the presence of scattering centers we will not expect to find a unique damping factor even for GaMnAs/GaAs samples with the same Mn composition x. More likely, different damping factors are expected to be found in real films and their values might be used to assess the film quality. In this sense the GaMnAs/GaAs film studi ed here is of course “better” than the one on GaInAs in line with the strong difference in the sample inhomogeneities. The inhomogeneous linewidth originates from spatial inhomogeneities in the local magnetic anisotropy fields and inhomogeneities in the local exchange interactions. Given the particular growth conditions of these films, low temper ature molecular beam epitaxy, inhomogeneities can not be expected to be neglig ible in these materials. If we had analyzed our X-band results of the GaMnAs/GaAs film in the spirit of ref. [5], i.e. assuming a negligible inhomoge neous linewidth - ( ∆H inhom=0) -, we would have obtained artificially increased damping factors. A further contribution might be expected from the intrinsic disorder in these films: as GaMnAs is a diluted magnetic semiconductor with random distribution of the Mn ions, this disorder will even for crysta llographically perfect crystals give some importance to this term. In the previous studies of the FMR damping factor in GaMnAs/GaAs single films higher values have been reported. Matsuda et al [6] found damping factors between 0.02 and 0.06 in the T=10K to T=20K temperature range. They observed the same tend encies as in this work concerning the anisotropy and temperature dependence of α: the lowest damping factor is seen for the easy axis orientation and its va lues increases with increasing temperatures. The films of their study were however significantly different: (i) the Mn doping concentration was lower, x=0.03 and thus the hole concentratio n was equally lower and (ii) the critical temperature of the annealed film was only T=80K. The anisotropy in the inhomogeneous linewidth at T=20K was equally much higher, varying between 30Oe for the easy axis to 250Oe for the hardest axis [110]. In a second study Sinova et al [5] have measured an annealed GaMnAs/GaAs sample with a similar composition (x= 0.08) and critical temperature (TC=130K) as the one studied here. They deduced a damping factor of the order of α=0.025 with only a slight temperature dependence betw een 4K and 80K and an increase close to T C. However, these measurements were done at one (X-band) microwave frequency only and the numerical value of α was obtained by assuming a neglig ible inhomogeneous linewidth. As explained above, the value of α can be expected to be overestimated in this case.In the photovoltage measurements of ref.7 the dampi ng factor of a low (x=0.02) doped GaMnAs layer has been determined with microwave fr equencies from 2 to 19.6GHz but for one field orientation H//[001] (h ard axis) and one temperature (T =9K) only. Interestingly, their measurements show a linear behavior even in the low frequency range down to 4GHz which demonstrates the negligible contribution from two magnon scattering in this case. The intrinsic damping factor α plays also an important role in the critical currents required to switch the magnetization in FM/NM/ FM trilayers [5]. However, in trilayer structures interface and spin pumping effects wi ll add to the intrinsic damping factor of the ferromagnetic material and give rise to an incr eased effective damping fa ctor. Sinova et al [5] have estimated the critical current for real istic GaMnAs layers: based on a value of α= 0.02, they estimated the critica l current density to J C=105A/cm2. It should be noted that the damping factor involved in the domain wall motion [ 17, 18] is by definition different form the FMR damping factor. Both are however linearly related with αFMR<αDW [17, 18]. First observations of current induced magne tization switching in Ga 0.956Mn 0.044As/GaAs/Ga 0.967Mn 0.033As trilayers confirm these theoretical predictions [19]. These authors observed a critical current density of ≈ 105A/cm2 which would have been predicted from Slonczewski’s formula [20] for a domain wall damping factor of αDW=0.002. Conclusion: We have determined the intrinsic FM R Gilbert damping factor for annealed Ga0.93Mn 0.07As thin films with high critical temperatures. To evaluate the influence of the strain the two prototype cases of compressive and tensile strained layers were studied. In both cases we find an average damping factor of the order of 0.01. We thus see that the sign of the strain does not seem to influence the damping factor strongly. The homogeneity of the films as judged from the inhomogeneous linewid th is much higher in the case of GaAs substrates than for GaInAs substrates. This must be attributed to the high dislocation density in the GaInAs layer [13]. In the case of the GaMnAs/GaAs layers, where the small linewidth allows a finer analysis of the data, we observe an anisotropy of the da mping factor, which has the lowest value for the easy axis orientation. This value of α[1−10]=0.003 is an order of magnitude lower than the previous reported va lues. The few experiment al results available seem to indicate that the damping factor decr eases with increasing Mn concentration. This corresponds well to the th eoretical predictions by Sinova et al [5] for the case of small quasi- particle lifetime broadening. It wi ll be interesting to test this behavior further in more highly doped layers (x ≈0.15), which become now available. Acknowledgment: We thank Alain Mauger of the IMPMC laboratory of the University Paris 6 and Bret Heinrich from the Simon Frazer University in Vancouver for many helpful discussions. References : [1] T. Jungwirth, J. Sinova, J. Masek, J. Kucera, and A.H. MacDonald, Rev.Mod.Phys. 78, 809 (2006). [2] X. Liu, and J. 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Rev. Lett., 93, 086107 (2004) [12] L.Thevenard, L.Largeau,, O.Mauguin, A.Lemaitre, K.Khazen and H.J.von Bardeleben, Phys. Rev.B75 ,195218 (2006) [13] L. Thevenard, L. Largeau, O. Mauguin, G. Patriarche, A. Lemaître, N. Vernier and J. Ferré, Phys. Rev. B 73, 195331 (2006) [14] B.Heinrich, D.Fraitova, V.Kambersky , Phys. Stat. Solidi 23, 501 (1967). [15] T.Y.Tserkovnyak, G.A.Fiete, B.I.Halperin, Appl.Phys.Lett. 84, 5234 (2004) [16] A.Mauger, D.L. Mills, Phys.Rev. B29, 3815 (1984) [17] S.C.Chen and HL.Huang, I EEE Transactions on Magnetics 33, 3978 (1997) [18] H.L.Huang, V.L.Sobolev, and S.C.Chen, J.Appl.Phys. 81, 4066(1997) [19] D.Chiba, Y.Sato, T.Kita, F. Matsukura, and H.Ohno, Phys.Rev.Lett. 93, 216602(2004) [20] J.C.Slonczewski, J.Magn.Magn.Mater. 159, L1(1996) Figure Captions: Figure 1a: X-band FMR spectrum of the GaMnAs /GaAs film taken at T=20K and for H// [001]; the peak-to-peak linewidth is 60Oe. The experimental spectrum is shown by circles and the Lorentzian lineshape simulation by a line. Figure 1b: Q-band FMR spectrum of the GaMnAs /GaInAs film taken at T=80K and for H// [100]; the peak to peak linewidth is 120Oe. The experimental spectrum is shown by circles and the Lorentzian lineshape simulation by a line Figure 2 (color on line): X-band peak-to-peak li ne widths for the GaMnAs/GaAs film for the four main field orientations: H//[001] black s quares, H//[1-10] olive lower triangles, H//[110] red circles, H//[100] blue upper triangles. Figure 3 color on line): Q-band peak-to-peak line widths for the GaMnAs/GaAs film for the four main field orientations: H//[001] black s quares, H//[1-10] olive lower triangles, H//[110] red circles, H//[100] blue upper triangles. Figure 4 (color on line): Peak-t o-peak linewidth at 9GHz and 35GHz for the GaMnAs/GaAs film; T=80K and H//[001] black squares, H//[1-10] olive lowe r triangles, H//[110] red circles, H//[100] blue upper triangles Figure 5 (color on line): GaMnAs/GaAs inhomogeneous linewidth as a function of temperature for four orientations of the applie d field: H//[001] black squares, H//[1-10] olive lower triangles, H//[110] red circles, H//[100] blue upper triangles Figure 6 (color on line): damping factor α as a function of temperature and magnetic field orientation; H//[001] black s quares, H//[1-10] olive lower triangles, H//[110] blue upper triangles, H//[100] red circles; the maximum erro r in the determination of the linewidth is estimated to 10G which co rresponds to an error in α of 0.001 Figure 7a: X-band FMR spectra for the GaMnAs /GaInAs film at 25K and H// [110] (hard axis); the low field spin wave resonance (SW) is of high intensity in this case; circles: experimental points, line: simulation with Gaussian lineshape Figure 7b: Q-band FMR spectra for the GaMnAs /GaInAs film at 25K and H// [110] (hard axis); the low field spin wave resonance (SW) is of high intensity in this case; circles: experimental points, line: simulation with Gaussian lineshape Figure 8 (color on line): GaMnAs/GaInAs: X-band FMR linewidth as a function of temperature for the four orientations of the applied magnetic field: H//[001] black squares, H//[1-10] olive lower triangles, H//[110] red circ les, H//[100] blue upper triangles; the easy axis FMR spectrum is not observable below T=100K ; a typical hysteresis curve as measured by SQUID is shown in the inset. Figure 9 (color on line): GaMnAs/GaInAs : Q-band FMR linewidth as a function of temperature for the four orientations of the applied magnetic field : H//[001] black squares, H//[1-10] olive lower triangl es, H//[110] red circles, H//[100] blue upper triangles Figure 10 (color on line): GaMnAs/GaInAs : FM R linewidth as a function of microwave frequency for H//[1-10] at different temper atures; T=10K (black squares), T=25K (red circles), T=55K(black spades), T=80K( blue stars), T=115K (red triangles) Figure 11 (color on line): GaMnAs/GaInAs: FMR linewidth as a function of microwave frequency for H//[001] (easy ax is) at different temperatur es; T=100K (red circles), T=120K (blue spades),T=130K(black squares) Figure 12: GaMnAs/GaInAs inhomogeneous linewi dth as a function of temperature for two orientations of the applied field: H//[001] squares, H//[1-10] lower triangles, Figure 13 : GaMnAs/GaInAs damping factor α as a function of temperature and magnetic field orientation; H//[001] squa res, H//[1-10] lower triangles. Tables Ga 0.93Mn 0.07As/GaAs G a0.93Mn 0.07As/Ga 0.902In0.098As Ms(T=4K)= 47e mu/cm3 Ms(T=4K)= 38 e mu/cm3 TC =157K (SQUID) TC=130K (SQUID) Anisotropy constants (T=80K) Anisotropy constants (T=55K) K2⊥ = - 55000 e rg/cm3 K 2⊥ = +91070 e rg/cm3 K2// =2617 er g/cm3 K 2// = -2464 erg/cm3 K4⊥ = 8483 erg/cm3 K 4⊥ = -34050 erg/cm3 K2// =2590 er g/cm3 K 2// = -1873er g/cm3 Easy axis of magnetization Easy axis of magnet ization [100] 4K<T<T C [001] 4K<T< T C Table I Table caption Table I: M icrom agnetic param eters of the two sam ples studied in this work: s aturation magnetization M s at T=4K, critical tem perature T C, magneto crystalline anisotropy constants of second and fourth order K 2⊥ , K 2//, K 4⊥, K 4// at T=55K and T=80K respectively and the orientation of the easy axis for m agnetization. 7500 7600 7700 7800 7900 8000SW FMR Signal (arb.u.) Magnetic Field H ( Oe) Figure 1a 10000 1020 0 10400 1060 0 10800 1100 0 FMR Signal (arb.u.) Magnetic Field H (Oe) Figure 1b 0 20 40 60 80 100 120140 160050100150200250300Linew idth ∆H (Oe) Tempera ture (K) 001 110 100 1-10 Figure 2 0 20 40 60 80 100 120 140160050100150200250300Linew idth ∆H (Oe) Temperature ( K) 001 110 100 1-10 Figure 3 0 5 10 15 20 25 30 35 4004080120160200Linewi dth ( G) Freq uency (GHz) Figure 4 20 40 60 80 100 120 140 160020406080100120∆Hinhom (Oe) Temperature (K) 001 110 100 1-10 Figure 5 0 20 40 60 80 100 120140 1600.0040.0060.0080.0100.0120.0140.0160.0180.020Damping factor α Temperature (K ) 001 110 100 1-10 Figure 6 4000 5000 6000 7000 8000SW FMR S ignal (arb. u.) Magnetic Field (Oe ) Figure 7a 13000 14000 15000 16000 17000SW FMR Signal (arb. u.) Magnet ic Field (Oe) Figure 7b 0 20 406 08 0 100120140300400500600700800 Linewidth ∆H (Oe) Temperature (K) [110] [100] [1-10] [001]-500 0 500-40040 M (emu/cm2) Magnetic F ield (O e) Figure 8 0 204 0 608 0 100 120 140300400500600700800 Linewidth ∆H (Oe) Temperat ure (K ) [110] [100] [1-10] [001] Figure 9 0 5 10 15 20 25 30 35 40350400450500550600650700750 115K80K55K25K10K Linewidth ∆H (Oe) Freque ncy (GHz) Figure 10 0 5 10 15 20 25 30 35 40200250300350400450500 100K120K130K Linewidth ∆H (Oe) Freque ncy (GHz) Figure 11 0 204 0 608 0 100120140200300400500600 ∆Hinhom (Oe) Temperat ure (K ) [001] [1-10] Figure 12 0 204 06 0 80 100 120 1400.0040.0060.0080.0100.0120.0140.0160.0180.020Dam ping factor α Temperature (K) Figure 13

2008-09-26

The magnetic properties of annealed, epitaxial Ga0.93Mn0.07As layers under tensile and compressive stress have been investigated by X-band (9GHz) and Q-band (35GHz) ferromagnetic resonance (FMR) spectroscopy. From the analysis of the linewidths of the uniform mode spectra the FMR Gilbert damping factor "alpha" has been determined. At T=4K we obtain a minimum damping factor of "alpha" = 0.003 for the compressively stressed layer. Its value is not isotropic. It has a minimum value for the easy axes orientations of the magnetic field and increases with the measuring temperature. Its average value is for both type of films of the order of 0.01 in spite of strong differences in the inhomogeneous linewidth which vary between 20 Oe and 600 Oe for the layers grown on GaAs and GaInAs substrates respectively.

Damping and magnetic anisotropy of ferromagnetic GaMnAs thin films

0809.4644v2

Gilbert damping in two-dimensional metallic anti-ferromagnets R. J. Sokolewicz,1, 2M. Baglai,3I. A. Ado,1M. I. Katsnelson,1and M. Titov1 1Radboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, the Netherlands 2Qblox, Delftechpark 22, 2628 XH Delft, the Netherlands 3Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden (Dated: March 29, 2024) A finite spin life-time of conduction electrons may dominate Gilbert damping of two-dimensional metallic anti-ferromagnets or anti-ferromagnet/metal heterostructures. We investigate the Gilbert damping tensor for a typical low-energy model of a metallic anti-ferromagnet system with honeycomb magnetic lattice and Rashba spin-orbit coupling for conduction electrons. We distinguish three regimes of spin relaxation: exchange-dominated relaxation for weak spin-orbit coupling strength, Elliot-Yafet relaxation for moderate spin-orbit coupling, and Dyakonov-Perel relaxation for strong spin-orbit coupling. We show, however, that the latter regime takes place only for the in-plane Gilbert damping component. We also show that anisotropy of Gilbert damping persists for any finite spin-orbit interaction strength provided we consider no spatial variation of the N´ eel vector. Isotropic Gilbert damping is restored only if the electron spin-orbit length is larger than the magnon wavelength. Our theory applies to MnPS 3monolayer on Pt or to similar systems. I. INTRODUCTION Magnetization dynamics in anti-ferromagnets con- tinue to attract a lot of attention in the context of possible applications1–4. Various proposals utilize the possibility of THz frequency switching of anti- ferromagnetic domains for ultrafast information storage and computation5,6. The rise of van der Waals magnets has had a further impact on the field due to the pos- sibility of creating tunable heterostructures that involve anti-ferromagnet and semiconducting layers7. Understanding relaxation of both the N´ eel vector and non-equilibrium magnetization in anti-ferromagnets is recognized to be of great importance for the function- ality of spintronic devices8–13. On one hand, low Gilbert damping must generally lead to better electric control of magnetic order via domain wall motion or ultrafast do- main switching14–16. On the other hand, an efficient con- trol of magnetic domains must generally require a strong coupling between charge and spin degrees of freedom due to a strong spin-orbit interaction, that is widely thought to be equivalent to strong Gilbert damping. In this paper, we focus on a microscopic analysis of Gilbert damping due to Dyakonov-Perel and Elliot-Yafet mechanisms. We apply the theory to a model of a two- dimensional N´ eel anti-ferromagnet with a honeycomb magnetic lattice. Two-dimensional magnets typically exhibit either easy-plane or easy-axis anisotropy, and play crucial roles in stabilizing magnetism at finite temperatures17,18. Easy-axis anisotropy selects a specific direction for mag- netization, thereby defining an axis for the magnetic or- der. In contrast, easy-plane anisotropy does not select a particular in-plane direction for the N´ eel vector, allowing it to freely rotate within the plane. This situation is anal- ogous to the XY model, where the system’s continuous symmetry leads to the suppression of out-of-plane fluc- tuations rather than fixing the magnetization in a spe- cific in-plane direction19,20. Without this anisotropy, themagnonic fluctuations in a two-dimensional crystal can grow uncontrollably large to destroy any long-range mag- netic order, according to the Mermin-Wagner theorem21. Recent density-functional-theory calculations for single-layer transition metal trichalgenides22, predict the existence of a large number of metallic anti-ferromagnets with honeycomb lattice and different types of magnetic order as shown in Fig. 1. Many of these crystals may have the N´ eel magnetic order as shown in Fig. 1a and are metallic: FeSiSe 3, FeSiTe 3, VGeTe 3, MnGeS 3, FeGeSe 3, FeGeTe 3, NiGeSe 3, MnSnS 3, MnSnS 3, MnSnSe 3, FeSnSe 3, NiSnS 3. Apart from that it has been predicted that anti-ferromagnetism can be induced in graphene by bringing it in proximity to MnPSe 323or by bringing it in double proximity between a layer of Cr 2Ge2Te6and WS224. Partly inspired by these predictions and recent technological advances in producing single-layer anti- ferromagnet crystals, we propose an effective model to study spin relaxation in 2D honeycomb anti-ferromagnet with N´ eel magnetic order. The same system was studied by us in Ref. 25, where we found that spin-orbit cou- pling introduces a weak anisotropy in spin-orbit torque and electric conductivity. Strong spin-orbit coupling was shown to lead to a giant anisotropy of Gilbert damping. Our analysis below is built upon the results of Ref. 25, and we investigate and identify three separate regimes of spin-orbit strength. Each regime is characterized by qualitatively different dependence of Gilbert damping on spin-orbit interaction and conduction electron transport time. The regime of weak spin-orbit interaction is dom- inated by exchange field relaxation of electron spin, and the regime of moderate spin-orbit strength is dominated by Elliot-Yafet spin relaxation. These two regimes are characterized also by a universal factor of 2 anisotropy of Gilbert damping. The regime of strong spin-orbit strength, which leads to substantial splitting of electron Fermi surfaces, is characterized by Dyakonov-Perel relax- ation of the in-plane spin component and Elliot-Yafet re-arXiv:2311.16268v2 [cond-mat.dis-nn] 28 Mar 20242 FIG. 1. Three anti-ferromagnetic phases commonly found among van-der-Waals magnets. Left-to-right: N´ eel, zig-zag, and stripy. laxation of the perpendicular-to-the-plane Gilbert damp- ing which leads to a giant damping anisotropy. Isotropic Gilbert damping is restored only for finite magnon wave vectors such that the magnon wavelength is smaller than the spin-orbit length. Gilbert damping in a metallic anti-ferromagnet can be qualitatively understood in terms of the Fermi surface breathing26. A change in the magnetization direction gives rise to a change in the Fermi surface to which the conduction electrons have to adjust. This electronic re- configuration is achieved through the scattering of elec- trons off impurities, during which angular momentum is transferred to the lattice. Gilbert damping, then, should be proportional to both (i) the ratio of the spin life-time and momentum life-time of conduction electrons, and (ii) the electric conductivity. Keeping in mind that the con- ductivity itself is proportional to momentum life-time, one may conclude that the Gilbert damping is linearly proportional to the spin life-time of conduction electrons. At the same time, the spin life-time of localized spins is inversely proportional to the spin life-time of conduc- tion electrons. A similar relation between the spin life- times of conduction and localized electrons also holds for relaxation mechanisms that involve electron-magnon scattering27. Our approach formally decomposes the magnetic sys- tem into a classical sub-system of localized magnetic mo- ments and a quasi-classical subsystem of conduction elec- trons. A local magnetic exchange couples these sub- systems. Localized magnetic moments in transition- metal chalcogenides and halides form a hexagonal lat- tice. Here we focus on the N´ eel type anti-ferromagnet that is illustrated in Fig. 1a. In this case, one can de- fine two sub-lattices A and B that host local magnetic moments SAandSB, respectively. For the discussion of Gilbert damping, we ignore the weak dependence of both fields on atomic positions and assume that the modulus S=|SA(B)|is time-independent. Under these assumptions, the magnetization dynamics of localized moments may be described in terms of two fields m=1 2S SA+SB
,n=1 2S SA−SB
, (1) which are referred to as the magnetization and staggeredmagnetization (or N´ eel vector), respectively. Within the mean-field approach, the vector fields yield the equations of motion ˙n=−Jn×m+n×δs++m×δs−, (2a) ˙m=m×δs++n×δs−, (2b) where dot stands for the time derivative, while δs+and δs−stand for the mean staggered and non-staggered non- equilibrium fields that are proportional to the variation of the corresponding spin-densities of conduction electrons caused by the time dynamics of nandmfields. The en- ergy Jis proportional to the anti-ferromagnet exchange energy for localized momenta. In Eqs. (2) we have omitted terms that are propor- tional to easy axis anisotropy for the sake of compact- ness. These terms are, however, important and will be introduced later in the text. In the framework of Eqs. (2) the Gilbert damping can be computed as the linear response of the electron spin- density variation to a time change in both the magneti- zation and the N´ eel vector (see e. g. Refs.25,28,29). In this definition, Gilbert damping describes the re- laxation of localized spins by transferring both total and staggered angular momenta to the lattice by means of conduction electron scattering off impurities. Such a transfer is facilitated by spin-orbit interaction. The structure of the full Gilbert damping tensor can be rather complicated as discussed in Ref. 25. However, by taking into account easy axis or easy plane anisotropy we may reduce the complexity of relevant spin configurations to parameterize δs+=α∥ m˙m∥+α⊥ m˙m⊥+αmn∥×(n∥×˙m∥),(3a) δs−=α∥ n˙n∥+α⊥ n˙n⊥+αnn∥×(n∥×˙n∥), (3b) where the superscripts ∥and⊥refer to the in-plane and perpendicular-to-the-plane projections of the corre- sponding vectors, respectively. The six coefficients α∥ m, α⊥ m,αm,α∥ n,α⊥ n, and αnparameterize the Gilbert damp- ing. Inserting Eqs. (3) into the equations of motion of Eqs. (2) produces familiar Gilbert damping terms. The damping proportional to time-derivatives of the N´ eel vec- tornis in general many orders of magnitude smaller than that proportional to the time-derivatives of the magneti- zation vector m25,30. Due to the same reason, the higher harmonics term αmn∥×(n∥×∂tm∥) can often be ne- glected. Thus, in the discussion below we may focus mostly on the coefficients α∥ mandα⊥ mthat play the most important role in the magnetization dynamics of our system. The terms proportional to the time-derivative of ncorrespond to the transfer of angular momentum between the sub- lattices and are usually less relevant. We refer to the results of Ref. 25 when discussing these terms. All Gilbert damping coefficients are intimately related to the electron spin relaxation time. The latter is rel- atively well understood in non-magnetic semiconductors3 with spin-orbital coupling. When a conducting electron moves in a steep potential it feels an effective magnetic field caused by relativistic effects. Thus, in a disordered system, the electron spin is subject to a random magnetic field each time it scatters off an impurity. At the same time, an electron also experiences precession around an effective spin-orbit field when it moves in between the collisions. Changes in spin direction between collisions are referred to as Dyakonov-Perel relaxation31,32, while changes in spin-direction during collisions are referred to as Elliot-Yafet relaxation33,34. The spin-orbit field in semiconductors induces a char- acteristic frequency of spin precession Ω s, while scalar disorder leads to a finite transport time τof the con- ducting electrons. One may, then, distinguish two limits: (i) Ω sτ≪1 in which case the electron does not have sufficient time to change its direction between consec- utive scattering events (Elliot-Yafet relaxation), and (ii) Ωsτ≫1 in which case the electron spin has multiple pre- cession cycles in between the collisions (Dyakonov-Perel relaxation). The corresponding processes define the so-called spin relaxation time, τs. In a 2D system the spin life-time τ∥ s, for the in-plane spin components, appears to be dou- ble the size of the life-time of the spin component that is perpendicular to the plane, τ⊥ s32. This geometric ef- fect has largely been overlooked. For non-magnetic 2D semiconductor one can estimate35,36 1 τ∥ s∼( Ω2 sτ,Ωsτ≪1 1/τ, Ωsτ≫1, τ∥ s= 2τ⊥ s. (4) A pedagogical derivation and discussion of Eq. 4 can be found in Refs. 35 and 36. Because electrons are con- fined in two dimensions the random spin-orbit field is always directed in-plane, which leads to a decrease in the in-plane spin-relaxation rate by a factor of two compared to the out-of-plane spin-relaxation rate as demonstrated first in Ref. 32 (see Refs. 36–40 as well). The reason is that the perpendicular-to-the-plane component of spin is influenced by two components of the randomly changing magnetic field, i. e. xandy, whereas the parallel-to-the- plane spin components are only influenced by a single component of the fluctuating fields, i. e. the xspin pro- jection is influenced only by the ycomponent of the field and vice-versa. The argument has been further general- ized in Ref. 25 to the case of strongly separated spin-orbit split Fermi surfaces. In this limit, the perpendicular-to- the-plane spin-flip processes on scalar disorder potential become fully suppressed. As a result, the perpendicular- to-the-plane spin component becomes nearly conserved, which results in a giant anisotropy of Gilbert damping in this regime. In magnetic systems that are, at the same time, con- ducting there appears to be at least one additional energy scale, ∆ sd, that characterizes exchange coupling of con- duction electron spin to the average magnetic moment of localized electrons. (In the case of s-d model descriptionit is the magnetic exchange between the spin of conduc- tionselectron and the localized magnetic moment of d orfelectron on an atom.) This additional energy scale complicates the simple picture of Eq. (4) especially in the case of an anti-ferromagnet. The electron spin precession is now defined not only by spin-orbit field but also by ∆sd. As the result the conditions Ω sτ≪1 and ∆ sdτ≫1 may easily coexist. This dissolves the distinction between Elliot-Yafet and Dyakonov-Perel mechanisms of spin re- laxation. One may, therefore, say that both Elliot-Yafet and Dyakonov-Perel mechanisms may act simultaneously in a typical 2D metallic magnet with spin-orbit coupling. The Gilbert damping computed from the microscopic model that we formulate below will always contain both contributions to spin-relaxation. II. MICROSCOPIC MODEL AND RESULTS The microscopic model that we employ to calculate Gilbert damping is the so-called s–dmodel that couples localized magnetic momenta SAandSBand conducting electron spins via the local magnetic exchange ∆ sd. Our effective low-energy Hamiltonian for conduction electrons reads H=vfp·Σ+λ 2 σ×Σ z−∆sdn·σΣzΛz+V(r),(5) where the vectors Σ,σandΛdenote the vectors of Pauli matrices acting on sub-lattice, spin and valley space, respectively. We also introduce the Fermi velocity vf, Rashba-type spin-orbit interaction λ, and a random im- purity potential V(r). The Hamiltonian of Eq. (5) can be viewed as the graphene electronic model where conduction electrons have 2D Rashba spin-orbit coupling and are also cou- pled to anti-ferromagnetically ordered classical spins on the honeycomb lattice. The coefficients α∥ mandα⊥ mare obtained using linear response theory for the response of spin-density δs+to the time-derivative of magnetization vector ∂tm. Impu- rity potential V(r) is important for describing momen- tum relaxation to the lattice. This is related to the an- gular momentum relaxation due to spin-orbit coupling. The effect of random impurity potential is treated pertur- batively in the (diffusive) ladder approximation that in- volves a summation over diffusion ladder diagrams. The details of the microscopic calculation can be found in the Appendices. Before presenting the disorder-averaged quantities α∥,⊥ m, it is instructive to consider first the contribution to Gilbert damping originating from a small number of electron-impurity collisions. This clarifies how the num- ber of impurity scattering effects will affect the final re- sult. Let us annotate the Gilbert damping coefficients with an additional superscript ( l) that denotes the number of scattering events that are taken into account. This4 01234(i) ?["] (0) ?(1) ?(2) ? (1) ? 102101100101 01234(i) k["] (0) k(1) k(2) k(1) k FIG. 2. Gilbert damping in the limit ∆ sd= 0. Dotted (green) lines correspond to the results of the numerical evaluation of ¯α(l) m,⊥,∥forl= 0,1,2 as a function of the parameter λτ. The dashed (orange) line corresponds to the diffusive (fully vertex corrected) results for ¯ α⊥,∥. m. means, in the diagrammatic language, that the corre- sponding quantity is obtained by summing up the ladder diagrams with ≤ldisorder lines. Each disorder line cor- responds to a quasi-classical scattering event from a sin- gle impurity. The corresponding Gilbert damping coeffi- cient is, therefore, obtained in the approximation where conduction electrons have scattered at most lnumber of times before releasing their non-equilibrium magnetic moment into a lattice. To make final expressions compact we define the di- mensionless Gilbert damping coefficients ¯ α∥,⊥ mby extract- ing the scaling factor α∥,⊥ m=A∆2 sd πℏ2v2 fS¯α∥,⊥ m, (6) where Ais the area of the unit cell, vfis the Fermi ve- locity of the conducting electrons and ℏ=h/2πis the Planck’s constant. We also express the momentum scat- tering time τin inverse energy units, τ→ℏτ. Let us start by computing the coefficients ¯ α∥,⊥(l) m in the formal limit ∆ sd→0. We can start with the “bare bub- ble” contribution which describes spin relaxation without a single scattering event. The corresponding results read ¯α(0) m,⊥=ετ1−λ2/4ε2 1 +λ2τ2, (7a) ¯α(0) m,∥=ετ1 +λ2τ2/2 1 +λ2τ2−λ2 8ε2 , (7b) where εdenotes the Fermi energy which we consider pos- itive (electron-doped system).In all realistic cases, we have to consider λ/ε≪1, while the parameter λτmay in principle be arbitrary. For λτ≪1 the disorder-induced broadening of the electron Fermi surfaces exceeds the spin-orbit induced splitting. In this case one basically finds no anisotropy of “bare” damping: ¯ α(0) m,⊥= ¯α(0) m,∥. In the opposite limit of substan- tial spin-orbit splitting one gets an ultimately anisotropic damping ¯ α(0) m,⊥≪¯α(0) m,∥. This asymptotic behavior can be summarized as ¯α(0) m,⊥=ετ( 1 λτ≪1, (λτ)−2λτ≫1,(8a) ¯α(0) m,∥=ετ( 1 λτ≪1, 1 2 1 + (λτ)−2
λτ≫1,(8b) where we have used that ε≫λ. The results of Eq. (8) modify by electron diffusion. By taking into account up to lscattering events we obtain ¯α(l) m,⊥=ετ( l+O(λ2τ2) λτ≪1, (1 +δl0)/(λτ)2λτ≫1,(9a) ¯α(l) m,∥=ετ( l+O(λ2τ2) λτ≪1, 1−(1/2)l+1+O((λτ)−2)λτ≫1,(9b) where we have used ε≫λagain. From Eqs. (9) we see that the Gilbert damping for λτ≪1 gets an additional contribution of ετfrom each scattering event as illustrated numerically in Fig. 2. This leads to a formal divergence of Gilbert damping in the limit λτ≪1. While, at first glance, the divergence looks like a strong sensitivity of damping to impurity scatter- ing, in reality, it simply reflects a diverging spin life-time. Once a non-equilibrium magnetization mis created it becomes almost impossible to relax it to the lattice in the limit of weak spin-orbit coupling. The formal diver- gence of α⊥ m=α∥ msimply reflects the conservation law for electron spin polarization in the absence of spin-orbit coupling such that the corresponding spin life-time be- comes arbitrarily large as compared to the momentum scattering time τ. By taking the limit l→ ∞ (i. e. by summing up the entire diffusion ladder) we obtain compact expressions ¯α⊥ m≡¯α(∞) m,⊥=ετ1 2λ2τ2, (10a) ¯α∥ m≡¯α(∞) m,∥=ετ1 +λ2τ2 λ2τ2, (10b) which assume ¯ α⊥ m≪¯α∥ mforλτ≫1 and ¯ α⊥ m= ¯α∥ m/2 forλτ≪1. The factor of 2 difference that we observe when λτ≪1, corresponds to a difference in the elec- tron spin life-times τ⊥ s=τ∥ s/2 that was discussed in the introduction32. Strong spin-orbit coupling causes a strong out-of-plane anisotropy of damping, ¯ α⊥ m≪¯α∥ mwhich corresponds to5 a suppression of the perpendicular-to-the-plane damping component. As a result, the spin-orbit interaction makes it much easier to relax the magnitude of the mzcompo- nent of magnetization than that of in-plane components. Let us now turn to the dependence of ¯ αmcoefficients on ∆sdthat is illustrated numerically in Fig. 3. We consider first the case of absent spin-orbit coupling λ= 0. In this case, the combination of spin-rotational and sub- lattice symmetry (the equivalence of A and B sub-lattice) must make Gilbert damping isotropic (see e. g.25,41). The direct calculation for λ= 0 does, indeed, give rise to the isotropic result ¯ α⊥ m= ¯α∥ m=ετ(ε2+∆2 sd)/2∆2 sd, which is, however, in contradiction to the limit λ→0 in Eq. (10). At first glance, this contradiction suggests the exis- tence of a certain energy scale for λover which the anisotropy emerges. The numerical analysis illustrated in Fig. 4 reveals that this scale does not depend on the values of 1 /τ, ∆sd, orε. Instead, it is defined solely by numerical precision. In other words, an isotropic Gilbert damping is obtained only when the spin-orbit strength λis set below the numerical precision in our model. We should, therefore, conclude that the transition from isotropic to anisotropic (factor of 2) damping occurs ex- actly at λ= 0. Interestingly, the factor of 2 anisotropy is absent in Eqs. (8) and emerges only in the diffusive limit. We will see below that this paradox can only be re- solved by analyzing the Gilbert damping beyond the in- finite wave-length limit. One can see from Fig. 3 that the main effect of finite ∆sdis the regularization of the Gilbert damping diver- gency ( λτ)−2in the limit λτ≪1. Indeed, the limit of weak spin-orbit coupling is non-perturbative for ∆ sd/ε≪ λτ≪1, while, in the opposite limit, λτ≪∆sd/ε≪1, the results of Eqs. (10) are no longer valid. Assuming ∆sd/ε≪1 we obtain the asymptotic expressions for the results presented in Fig. 3 as ¯α⊥ m=1 2ετ(2 3ε2+∆2 sd ∆2 sdλτ≪∆sd/ε, 1 λ2τ2 λτ≫∆sd/ε,(11a) ¯α∥ m=ετ(2 3ε2+∆2 sd ∆2 sdλτ≪∆sd/ε, 1 +1 λ2τ2λτ≫∆sd/ε,(11b) which suggest that ¯ α⊥ m/¯α∥ m= 2 for λτ≪1. In the op- posite limit, λτ≫1, the anisotropy of Gilbert damping grows as ¯ α∥ m/¯α⊥ m= 2λ2τ2. The results of Eqs. (11) can also be discussed in terms of the electron spin life-time, τ⊥(∥) s = ¯α⊥(∥) m/ε. For the inverse in-plane spin life-time we find 1 τ∥ s= 3∆2 sd/2ε2τ λτ ≪∆sd/ε, λ2τ ∆sd/ε≪λτ≪1, 1/τ 1≪λτ,(12) that, for ∆ sd= 0, is equivalent to the known result of Eq. (4). Indeed, for ∆ sd= 0, the magnetic exchange 103102101100101 101101103105m;k;?["]
sd="= 0:1
sd="= 0m;k m;?FIG. 3. Numerical results for the Gilbert damping compo- nents in the diffusive limit (vertex corrected)as the function of the spin-orbit coupling strength λ. The results correspond toετ= 50 and ∆ sdτ= 0.1 and agree with the asymptotic expressions of Eq. (11). Three different regimes can be dis- tinguished for ¯ α∥ m: i) spin-orbit independent damping ¯ α∥ m∝ ε3τ/∆2 sdfor the exchange dominated regime, λτ≪∆sd/ε, ii) the damping ¯ α∥ m∝ε/λ2τfor Elliot-Yafet relaxation regime, ∆sd/ε≪λτ≪1, and iii) the damping ¯ α∥ m∝ετfor the Dyakonov-Perel relaxation regime, λτ≫1. The latter regime is manifestly absent for ¯ α⊥ min accordance with Eqs. (12,13). plays no role and one observes the cross-over from Elliot- Yafet ( λτ≪1) to Dyakonov-Perel ( λτ≫1) spin relax- ation. This cross-over is, however, absent in the relaxation of the perpendicular spin component 1 τ⊥s= 2( 3∆2 sd/2ε2τ λτ ≪∆sd/ε, λ2τ ∆sd/ε≪λτ,(13) where Elliot-Yafet-like relaxation extends to the regime λτ≫1. As mentioned above, the factor of two anisotropy in spin-relaxation of 2 Dsystems, τ∥ s= 2τ⊥ s, is known in the literature32(see Refs.36–38as well). Unlimited growth of spin life-time anisotropy, τ∥ s/τ⊥ s= 2λ2τ2, in the regime λτ≪1 has been described first in Ref. 25. It can be qual- itatively explained by a strong suppression of spin-flip processes for zspin component due to spin-orbit induced splitting of Fermi surfaces. The mechanism is effective only for scalar (non-magnetic) disorder. Even though such a mechanism is general for any magnetic or non- magnetic 2D material with Rashba-type spin-orbit cou- pling, the effect of the spin life-time anisotropy on Gilbert damping is much more relevant for anti-ferromagnets. In- deed, in an anti-ferromagnetic system the modulus of m is, by no means, conserved, hence the variations of per- pendicular and parallel components of the magnetization vector are no longer related. In the regime, λτ≪∆sd/εthe spin life-time is de- fined by exchange interaction and the distinction between Dyakonov-Perel and Elliot-Yafet mechanisms of spin re- laxation is no longer relevant. In this regime, the spin- relaxation time is by a factor ( ε/∆sd)2larger than the momentum relaxation time. Let us now return to the problem of emergency of the6 106410541044103410241014 12k=?n= 32 n= 64n= 96 n= 128 FIG. 4. Numerical evaluation of Gilbert damping anisotropy in the limit λ→0. Isotropic damping tensor is restored only ifλ= 0 with ultimate numerical precision. The factor of 2 anisotropy emerges at any finite λ, no matter how small it is, and only depends on the numerical precision n, i.e. the number of digits contained in each variable during computa- tion. The crossover from isotropic to anisotropic damping can be understood only by considering finite, though vanishingly small, magnon qvectors. factor of 2 anisotropy of Gilbert damping at λ= 0. We have seen above (see Fig. 4) that, surprisingly, there ex- ists no energy scale for the anisotropy to emerge. The transition from the isotropic limit ( λ= 0) to a finite anisotropy appeared to take place exactly at λ= 0. We can, however, generalize the concept of Gilbert damping by considering the spin density response function at a finite wave vector q. To generalize the Gilbert damping, we are seeking a response of spin density at a point r,δs+(r) to a time derivative of magnetization vectors ˙m∥and ˙m⊥at the point r′. The Fourier transform with respect to r−r′ gives the Gilbert damping for a magnon with the wave- vector q. The generalization to a finite q-vector shows that the limits λ→0 and q→0 cannot be interchanged. When the limit λ→0 is taken before the limit q→0 one finds an isotropic Gilbert damping, while for the oppo- site order of limits, it becomes a factor of 2 anisotropic. In a realistic situation, the value of qis limited from below by an inverse size of a typical magnetic domain 1/Lm, while the spin-orbit coupling is effective on the length scale Lλ= 2πℏvf/λ. In this picture, the isotropic Gilbert damping is characteristic for the case of suffi- ciently small domain size Lm≪Lλ, while the anisotropic Gilbert damping corresponds to the case Lλ≪Lm. In the limit qℓ≪1, where ℓ=vfτis the electron mean 2 0 2 k[a.u.]2:50:02:5energy [a.u.]=
sd= 4 2 0 2 k[a.u.]=
sd= 2 2 0 2 k[a.u.]=
sd= 1FIG. 5. Band-structure for the effective model of Eq. (5) in a vicinity of Kvalley assuming nz= 1. Electron bands touch for λ= 2∆ sd. The regime λ≤2∆sdcorresponds to spin-orbit band inversion. The band structure in the valley K′is inverted. Our microscopic analysis is performed in the electron-doped regime for the Fermi energy above the gap as illustrated by the top dashed line. The bottom dashed line denotes zero energy (half-filling). free path, we can summarize our results as ¯α⊥ m=ετ ε2+∆2 sd 2∆2 sdλτ≪qℓ≪∆sd/ε, 1 3ε2+∆2 sd ∆2 sdqℓ≪λτ≪∆sd/ε, 1 2λ2τ2 λτ≫∆sd/ε,, (14a) ¯α∥ m=ετ ε2+∆2 sd 2∆2 sdλτ≪qℓ≪∆sd/ε, 2 3ε2+∆2 sd ∆2 sdqℓ≪λτ≪∆sd/ε, 1 +1 λ2τ2λτ≫∆sd/ε,(14b) which represent a simple generalization of Eqs. (11). The results of Eqs. (14) correspond to a simple behav- ior of Gilbert damping anisotropy, ¯α∥ m/¯α⊥ m=( 1 λτ≪qℓ, 2 1 +λ2τ2
qℓ≪λτ,(15) where we still assume qℓ≪1. III. ANTI-FERROMAGNETIC RESONANCE The broadening of the anti-ferromagnet resonance peak is one obvious quantity that is sensitive to Gilbert damping. The broadening is however not solely defined by a particular Gilbert damping component but depends also on both magnetic anisotropy and anti-ferromagnetic exchange. To be more consistent we can use the model of Eq. (5) to analyze the contribution of conduction electrons to an easy axis anisotropy. The latter is obtained by expanding the free energy for electrons in the value of nz, which has a form E=−Kn2 z/2. With the conditions ε/λ≫1 and ε/∆sd≫1 we obtain the anisotropy constant as K=A 2πℏ2v2( ∆2 sdλ 2∆sd/λ≤1, ∆sdλ2/2 2∆ sd/λ≥1,(16)7 where Ais the area of the unit cell. Here we assume both λand ∆ sdpositive, therefore, the model natu- rally gives rise to an easy axis anisotropy with K > 0. In real materials, there exist other sources of easy axis or easy plane anisotropy. In-plane magneto-crystalline anisotropy also plays an important role. For example, N´ eel-type anti-ferromagnets with easy-axis anisotropy are FePS 3, FePSe 3or MnPS 3, whereas those with easy plane and in-plane magneto-crystalline anisotropy are NiPS 3and MnPSe 3. Many of those materials are, how- ever, Mott insulators. Our qualitative theory may still apply to materials like MnPS 3monolayers at strong elec- tron doping. The transition from 2∆ sd/λ≥1 to 2∆ sd/λ≤1 in Eq. (16) corresponds to the touching of two bands in the model of Eq. (5) as illustrated in Fig. 5. Anti-ferromagnetic magnon frequency and life-time in the limit q→0 are readily obtained by linearizing the equations of motion ˙n=−Jn×m+Km×n⊥+n×(ˆαm˙m), (17a) ˙m=Kn×n⊥+n×(ˆαn˙n), (17b) where we took into account easy axis anisotropy Kand disregarded irrelevant terms m×(ˆαn˙n) and m×(ˆαm˙m). We have also defined Gilbert damping tensors such as ˆαm˙m=α∥ m˙m∥+α⊥ m˙m⊥, ˆαn˙n=α∥ n˙n∥+α⊥ n˙n⊥. In the case of easy axis anisotropy we can use the lin- earized modes n=ˆz+δn∥eiωt,m=δm∥eiωt, hence we get the energy of q= 0 magnon as ω=ω0−iΓ/2, (18) ω0=√ JK, Γ =Jα∥ n+Kα∥ m (19) where we took into account that K≪J. The expression forω0is well known due to Kittel and Keffer42,43. Using Ref. 25 we find out that α∥ n≃α⊥ m(λ/ε)2and α⊥ n≃α∥ m(λ/ε)2, hence Γ≃α∥ m K+J/2 ε2/λ2+ε2τ2 , (20) where we have simply used Eqs. (10). Thus, one may often ignore the contribution Jα∥ nas compared to Kα∥ m despite the fact that K≪J. In the context of anti-ferromagnets, spin-pumping terms are usually associated with the coefficients α∥ nin Eq. (3b) that are not in the focus of the present study. Those coefficients have been analyzed for example in Ref. 25. In this manuscript we simply use the known results forαnin Eqs. (17-19), where we illustrate the effect of both spin-pumping coefficient αnand the direct Gilbert damping αmon the magnon life time. One can see from Eqs. (19,20) that the spin-pumping contributions do also contribute, though indirectly, to the magnon decay. The spin pumping contributions become more important in magnetic materials with small magnetic anisotropy. The processes characterized by the coefficients αnmay also be 103102101100101 0:000:010:021=k m="= 0:04 ="= 0:02 ="= 0:01FIG. 6. Numerical evaluation of the inverse Gilbert damping 1/¯α∥ mas a function of the momentum relaxation time τ. The inverse damping is peaked at τ∝1/λwhich also corresponds to the maximum of the anti-ferromagnetic resonance quality factor in accordance with Eq. (21). interpreted in terms of angular momentum transfer from one AFM sub-lattice to another. In that respect, the spin pumping is specific to AFM, and is qualitatively differ- ent from the direct Gilbert damping processes ( αm) that describe the direct momentum relaxation to the lattice. As illustrated in Fig. 6 the quality factor of the anti- ferromagnetic resonance (for a metallic anti-ferromagnet with easy-axis anisotropy) is given by Q=ω0 Γ≃1 α∥ mr J K. (21) Interestingly, the quality factor defined by Eq. (21) is maximized for λτ≃1, i. e. for the electron spin-orbit length being of the order of the scattering mean free path. The quantities 1 /√ Kand 1 /¯α∥ mare illustrated in Fig. 6 from the numerical analysis. As one would ex- pect, the quality factor vanishes in both limits λ→0 andλ→ ∞ . The former limit corresponds to an over- damped regime hence no resonance can be observed. The latter limit corresponds to a constant α∥ m, but the reso- nance width Γ grows faster with λthan ω0does, hence the vanishing quality factor. It is straightforward to check that the results of Eqs. (20,21) remain consistent when considering systems with either easy-plane or in-plane magneto-crystalline anisotropy. Thus, the coefficient α⊥ mnormally does not enter the magnon damping, unless the system is brought into a vicinity of spin-flop transition by a strong external field. IV. CONCLUSION In conclusion, we have analyzed the Gilbert damping tensor in a model of a two-dimensional anti-ferromagnet on a honeycomb lattice. We consider the damping mech- anism that is dominated by a finite electron spin life-time8 due to a combination of spin-orbit coupling and impu- rity scattering of conduction electrons. In the case of a 2D electron system with Rashba spin-orbit coupling λ, the Gilbert damping tensor is characterized by two com- ponents α∥ mandα⊥ m. We show that the anisotropy of Gilbert damping depends crucially on the parameter λτ, where τis the transport scattering time for conduction electrons. For λτ≪1 the anisotropy is set by a geo- metric factor of 2, α∥ m= 2α⊥ m, while it becomes infinitely large in the opposite limit, α∥ m= (λτ)2α⊥ mforλτ≫1. Gilbert damping becomes isotropic exactly for λ= 0, or, strictly speaking, for the case λ≪ℏvfq, where qis the magnon wave vector. This factor of 2 is essentially universal, and is a geomet- ric effect: the z-component relaxation results from fluctu- ations in two in-plane spin components, whereas in-plane relaxation stems from fluctuations of the z-component alone. This reflects the subtleties of our microscopic model, where the mechanism for damping is activated by the decay of conduction electron momenta, linked to spin-relaxation through spin-orbit interactions. We find that Gilbert damping is insensitive to mag- netic order for λ≫∆sd/ετ, where ∆ sdis an effective exchange coupling between spins of conduction and local- ized electrons. In this case, the electron spin relaxation can be either dominated by scattering (Dyakonov-Perel relaxation) or by spin-orbit precession (Elliot-Yafet re- laxation). We find that the Gilbert damping component α⊥ m≃ε/λ2τis dominated by Elliot-Yafet relaxation irre- spective of the value of the parameter λτ, while the other component crosses over from α∥ m≃ε/λ2τ(Elliot-Yafet relaxation) for λτ≪1, to α∥ m≃ετ(Dyakonov-Perel re- laxation) for λτ≫1. For the case λ≪∆sd/ετthe spin relaxation is dominated by interaction with the exchange field. Crucially, our results are not confined solely to the N´ eel order on the honeycomb lattice: we anticipate a broader applicability across various magnetic orders, including the zigzag order. This universality stems from our focus on the large magnon wavelength limit. The choice of the honeycomb lattice arises from its unique ability to main- tain isotropic electronic spectra within the plane, coupled with the ability to suppress anisotropy concerning in- plane spin rotations. Strong anisotropic electronic spec- tra would naturally induce strong anisotropic in-plane Gilbert damping, which are absent in our results. Finally, we show that the anti-ferromagnetic resonance width is mostly defined by α∥ mand demonstrate that the resonance quality factor is maximized for λτ≈1. Our microscopic theory predictions may be tested for systems such as MnPS 3monolayer on Pt and similar heterostruc- tures.ACKNOWLEDGMENTS We are grateful to O. Gomonay, R. Duine, J. Sinova, and A. Mauri for helpful discussions. This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 873028. Appendix A: Microscopic framework The microscopic model that we employ to calculate Gilbert damping belongs to a class of so-called s–dmod- els that describe the physical system in the form of a Heisenberg model for localized spins and a tight-binding model for conduction electrons that are weakly coupled by a local magnetic exchange interaction of the strength ∆sd. Our effective electron Hamiltonian for a metallic hexagonal anti-ferromagnet is given by25 H0=vfp·Σ+λ 2[σ×Σ]z−∆sdn·σΣzΛz,(A1) where the vectors Σ,σandΛdenote the vectors of Pauli- matrices acting on sub-lattice, spin and valley space re- spectively. We also introduce the Fermi velocity vf, Rashba-type spin-orbit interaction λ. To describe Gilbert damping of the localized field n we have to add the relaxation mechanism. This is pro- vided in our model by adding a weak impurity potential H=H0+V(r). The momentum relaxation due to scat- tering on impurities leads indirectly to the relaxation of Heisenberg spins due to the presence of spin-orbit cou- pling and exchange couplings. For modeling the impurity potential, we adopt a delta- correlated random potential that corresponds to the point scatter approximation, where the range of the im- purity potential is much shorter than that of the mean free path (see e.g. section 3.8 of Ref. 44), i.e. ⟨V(r)V(r′)⟩= 2πα(ℏvf)2δ(r−r′), (A2) where the dimensionless coefficient α≪1 characterizes the disorder strength. The corresponding scattering time for electrons is obtained as τ=ℏ/παϵ , which is again similar to the case of graphene. The response of symmetric spin-polarization δs+to the time-derivative of non-staggered magnetization, ∂tm, is defined by the linear relation δs+ α=X βRαβ|ω=0˙mβ, (A3) where the response tensor is taken at zero frequency25,45. The linear response is defined generally by the tensor Rαβ=A∆2 sd 2πSZdp (2πℏ)2 Tr GR ε,pσαGA ε+ℏω,pσβ ,(A4)9 where GR(A) ε,pare standing for retarded(advanced) Green functions and the angular brackets denote averaging over disorder fluctuations. The standard recipe for disorder averaging is the diffu- sive approximation46,47that is realized by replacing the bare Green functions in Eq. (A4) with disorder-averaged Green functions and by replacing one of the vertex op- erators σxorσywith the corresponding vertex-corrected operator that is formally obtained by summing up ladder impurity diagrams (diffusons). In models with spin-orbit coupling, the controllable dif- fusive approximation for non-dissipative quantities may become, however, more involved as was noted first in Ref. 48. For Gilbert damping it is, however, sufficient to consider the ladder diagram contributions only. The disorder-averaged Green function is obtained by including an imaginary part of the self-energy ΣR(not to be confused here with the Pauli matrix Σ 0,x,y,z) that is evaluated in the first Born approximation Im ΣR= 2παv2 fZdp (2π)2Im1 ε−H0+i0. (A5) The real part of the self-energy leads to the renormaliza- tion of the energy scales ε,λand ∆ sd. In the first Born approximation, the disorder-averaged Green function is given by GR ε,p=1 ε−H0−iIm ΣR. (A6) The vertex corrections are computed in the diffusive approximation. The latter involves replacing the vertex σαwith the vertex-corrected operator, σvc α=∞X l=0σ(l) α, (A7) where the index lcorresponds to the number of disorder lines in the ladder. The operators σ(l) αcan be defined recursively as σ(l) α=2ℏv2 f ετZdp (2π)2GR ε,pσ(l−1) αGA ε+ℏω,p, (A8) where σ(0) α=σα. The summation in Eq. (A7) can be computed in the full operator basis, Bi={α,β,γ}=σαΣβΛγ, where each index α,βandγtakes on 4 possible values (with zero standing for the unity matrix). We may always normalize TrBiBj= 2δijin an analogy to the Pauli matrices. The operators Biare, then, forming a finite-dimensional space for the recursion of Eq. (A8). The vertex-corrected operators Bvc iare obtained by summing up the matrix geometric series Bvc i=X j1 1− F ijBj, (A9)where the entities of the matrix Fare given by Fij=ℏv2 f ετZdp (2π)2Tr GR ε,pBiGA ε+ℏω,pBj .(A10) Our operators of interest σxandσycan always be de- composed in the operator basis as σα=1 2X iBiTr (σαBi), (A11) hence the vertex-corrected spin operator is given by σvc α=1 2X ijBvc iTr(σαBi). (A12) Moreover, the computation of the entire response tensor of Eq. (A4) in the diffusive approximation can also be expressed via the matrix Fas Rαβ=α0ετ 8ℏX ij[TrσαBi]F 1− F ij[TrσβBj],(A13) where α0=A∆2 sd/πℏ2v2 fSis the coefficient used in Eq. (6) to define the unit of the Gilbert damping. It appears that one can always choose the basis of Bioperators such that the computation of Eq. (A13) is closed in a subspace of just three Bioperators with i= 1,2,3. This enables us to make analytical computa- tions of Eq. (A13). Appendix B: Magnetization dynamics The representation of the results can be made some- what simpler by choosing xaxis in the direction of the in-plane projection n∥of the N´ eel vector, hence ny= 0. In this case, one can represent the result as δs+=c1n∥×(n∥×∂tm∥) +c2∂tm∥+c3∂tm⊥+c4n, where ndependence of the coefficients cimay be param- eterized as c1=r11−r22−r31(1−n2 z)/(nxnz) 1−n2z, (B1a) c2=r11−r31(1−n2 z)/(nxnz), (B1b) c3=r33, (B1c) c4= (r31/nz)∂tmz+ζ(∂tm)·n. (B1d) The analytical results in the paper correspond to the evaluation of δs±up to the second order in ∆ sdusing perturbative analysis. Thus, zero approximation corre- sponds to setting ∆ sd= 0 in Eqs. (A1,A5). The equations of motion on nandmare given by Eqs. (2), ∂tn=−Jn×m+n×δs++m×δs−, (B2a) ∂tm=m×δs++n×δs−, (B2b)10 It is easy to see that the following transformation leaves the above equations invariant, δs+→δs+−ξn, δ s−→δs−−ξm, (B3) for an arbitrary value of ξ. Such a gauge transformation can be used to prove that the coefficient c4is irrelevant in Eqs. (B2). In this paper, we compute δs±to the zeroth order in |m|– the approximation which is justified by the sub- lattice symmetry in the anti-ferromagnet. A somewhat more general model has been analyzed also in Ref. 25 to which we refer the interested reader for more technical details. Appendix C: Anisotropy constant The anisotropy constant is obtained from the grand po- tential energy Ω for conducting electrons. For the model of Eq. (A1) the latter can be expressed as Ω =−X ς=±1 βZ dε g(ε)νς(ε), (C1) where β= 1/kBTis the inverse temperature, ς=±is the valley index (for the valleys KandK′),GR ς,pis the bare retarded Green function with momentum pand in the valley ς. We have also defined the function g(ε) = ln (1 + exp[ β(µ−ε)]), (C2) where µis the electron potential, and the electron density of states in each of the valleys is given by, νς(ε) =1 πZdp (2πℏ)2Im Tr GR ς,p, (C3) where the trace is taken only over spin and sub-lattice space, In the metal regime considered, the chemical potential is assumed to be placed in the upper electronic band. In this case, the energy integration can be taken only for positive energies. The two valence bands are always filled and can only add a constant shift to the grand potential Ω that we disregard. The evaluation of Eq. (C1) yields the following density of states ντ(ε) =1 2πℏ2v2 f 0 0 < ε < ε 2 ε/2 +λ/4ε2< ε < ε 1, ε ε > ε 1,(C4)where the energies ε1,2correspond to the extremum points (zero velocity) for the electronic bands. These energies, for each of the valleys, are given by ε1,ς=1 2 +λ+p 4∆2+λ2−4ς∆λnz
, (C5a) ε2,ς=1 2 −λ+p 4∆2+λ2+ 4ς∆λnz
(C5b) where ς=±is the valley index. In the limit of zero temperature we can approximate Eq. (C1) as Ω =−X ς=±1 βZ∞ 0dε(µ−ε)νς(ε). (C6) Then, with the help of Eq. (C1) we find, Ω =−1 24πℏ2v2 fX ς=± (ε1,ς−µ)2(4ε1,ς−3λ+ 2µ) +(ε2,ς−µ)2(4ε2,ς+ 3λ+ 2µ) . (C7) By substituting the results of Eqs. (C5) into the above equation we obtain Ω =−1 24πℏ2v2 fh (4∆2−4nz∆λ+λ2)2/3 +(4∆2+ 4nz∆λ+λ2)2/3−24∆µ+ 8µ3i .(C8) A careful analysis shows that the minimal energy cor- responds to nz=±1 so that the conducting electrons prefer an easy-axis magnetic anisotropy. By expanding in powers of n2 zaround nz=±1 we obtain Ω = −Kn2 z/2, where K=1 2πℏ2v2( |∆2λ| | λ/2∆| ≥1, |∆λ2|/2|λ/2∆| ≤1.(C9) This provides us with the easy axis anisotropy of Eq. (16). 1S. A. Siddiqui, J. Sklenar, K. Kang, M. J. Gilbert, A. Schleife, N. Mason, and A. Hoff- mann, Journal of Applied Physics 128, 040904 (2020), https://pubs.aip.org/aip/jap/article-pdf/doi/10.1063/5.0009445/15249168/040904 1online.pdf. 2V. V. Mazurenko, Y. O. Kvashnin, A. I. Lichtenstein, and M. I. Katsnelson, Journal of Experimental and Theoretical Physics 132, 506 (2021).11 3L.ˇSmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 040501 (2022). 4B. A. Bernevig, C. Felser, and H. Beidenkopf, Nature 603, 41 (2022). 5T. G. H. Blank, K. A. Grishunin, B. A. Ivanov, E. A. Mashkovich, D. Afanasiev, and A. V. Kimel, Phys. Rev. Lett. 131, 096701 (2023). 6W. Wu, C. Yaw Ameyaw, M. F. Doty, and M. B. Jungfleisch, Journal of Applied Physics 130, 091101 (2021), https://pubs.aip.org/aip/jap/article- pdf/doi/10.1063/5.0057536/13478272/091101 1online.pdf. 7M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S. Novoselov, Nature Nanotechnology 14, 408 (2019). 8K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, Physical Review Letters 106, 107206 (2011), publisher: American Physical Society. 9R. Cheng, D. Xiao, and A. Brataas, Physical Review Let- ters116, 207603 (2016), publisher: American Physical So- ciety. 10S. Urazhdin and N. 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2023-11-27

A finite spin life-time of conduction electrons may dominate Gilbert damping of two-dimensional metallic anti-ferromagnets or anti-ferromagnet/metal heterostructures. We investigate the Gilbert damping tensor for a typical low-energy model of a metallic anti-ferromagnet system with honeycomb magnetic lattice and Rashba spin-orbit coupling for conduction electrons. We distinguish three regimes of spin relaxation: exchange-dominated relaxation for weak spin-orbit coupling strength, Elliot-Yafet relaxation for moderate spin-orbit coupling, and Dyakonov-Perel relaxation for strong spin-orbit coupling. We show, however, that the latter regime takes place only for the in-plane Gilbert damping component. We also show that anisotropy of Gilbert damping persists for any finite spin-orbit interaction strength provided we consider no spatial variation of the N\'eel vector. Isotropic Gilbert damping is restored only if the electron spin-orbit length is larger than the magnon wavelength. Our theory applies to MnPS3 monolayer on Pt or to similar systems.

Gilbert damping in two-dimensional metallic anti-ferromagnets

2311.16268v2

arXiv:2203.01632v1 [math.AP] 3 Mar 2022STABILITY RESULTS OF LOCALLY COUPLED WAVE EQUATIONS WITH LO CAL KELVIN-VOIGT DAMPING: CASES WHEN THE SUPPORTS OF DAMPING AN D COUPLING COEFFICIENTS ARE DISJOINT MOHAMMAD AKIL1, HAIDAR BADAWI1, AND SERGE NICAISE1 Abstract. In this paper, we study the direct/indirect stability of loc ally coupled wave equations with local Kelvin-Voigt dampings/damping and by assuming that the sup ports of the dampings and the coupling coeffi- cients are disjoint. First, we prove the well-posedness, st rong stability, and polynomial stability for some one dimensional coupled systems. Moreover, under some geometr ic control condition, we prove the well-posedness and strong stability in the multi-dimensional case. Contents 1. Introduction 1 2. Direct and Indirect Stability in the one dimensional case 4 2.1. Well-Posedness 4 2.2. Strong Stability 5 2.3. Polynomial Stability 9 2.3.1. Proof of Theorem 2.6 9 2.3.2. Proof of Theorem 2.7 13 3. Indirect Stability in the multi-dimensional case 16 3.1. Well-posedness 16 3.2. Strong Stability 17 Appendix A. Some notions and stability theorems 20 References 21 1.Introduction The direct and indirect stability of locally coupled wave equations with lo cal damping arouses many interests in recent years. The study of coupled systems is also motivated by se veralphysicalconsiderationslike Timoshenko and Bresse systems (see for instance [ 10,6,3,2,1,15,14]). The exponential or polynomial stability of the wave equation with a local Kelvin-Voigt damping is considered in [ 20,23,13], for instance. On the other hand, the direct and indirect stability of locally and coupled wave equations with lo cal viscous dampings are analyzed in [8,18,16]. In this paper, we are interested in locally coupled wave equations wit h local Kelvin-Voigt dampings. Before stating our main contributions, let us mention similar results f or such systems. In 2019, Hayek et al.in [17], studied the stabilization of a multi-dimensional system of weakly cou pled wave equations with one or two locally Kelvin-Voigt damping and non-smooth coefficient at the interfa ce. They established different stability 1Universit ´e Polytechnique Hauts-de-France, CERAMATHS/DEMAV, Valencien nes, France E-mail address :Mohammad.Akil@uphf.fr, Haidar.Badawi@uphf.fr, Serge.N icaise@uphf.fr . Key words and phrases. Coupled wave equations, Kelvin-Voigt damping, strong stab ility, polynomial stability . 1results. In 2021, Akil et al.in [24], studied the stability of an elastic/viscoelastic transmission problem of locally coupled waves with non-smooth coefficients, by considering: utt−/parenleftbig aux+b0χ(α1,α3)utx/parenrightbig x+c0χ(α2,α4)yt= 0,in (0,L)×(0,∞), ytt−yxx−c0χ(α2,α4)ut= 0, in (0,L)×(0,∞), u(0,t) =u(L,t) =y(0,t) =y(L,t) = 0, in (0,∞), wherea,b0,L >0,c0/\e}atio\slash= 0, and 0 < α1< α2< α3< α4< L. They established a polynomial energy decay rate of type t−1. In the same year, Akil et al.in [5], studied the stability of a singular local interaction elastic/viscoelastic coupled wave equations with time delay, by consid ering: utt−/bracketleftbig aux+χ(0,β)(κ1utx+κ2utx(t−τ))/bracketrightbig x+c0χ(α,γ)yt= 0,in (0,L)×(0,∞), ytt−yxx−c0χ(α,γ)ut= 0, in (0,L)×(0,∞), u(0,t) =u(L,t) =y(0,t) =y(L,t) = 0, in (0,∞), wherea,κ1,L >0,κ2,c0/\e}atio\slash= 0, and 0 < α < β < γ < L . They proved that the energy of their system decays polynomially in t−1. In 2021, Akil et al.in [4], studied the stability of coupled wave models with locally memory in a past history framework via non-smooth coefficients on t he interface, by considering: utt−/parenleftbigg aux+b0χ(0,β)/integraldisplay∞ 0g(s)ux(t−s)ds/parenrightbigg x+c0χ(α,γ)yt= 0,in (0,L)×(0,∞), ytt−yxx−c0χ(α,γ)ut= 0, in (0,L)×(0,∞), u(0,t) =u(L,t) =y(0,t) =y(L,t) = 0, in (0,∞), wherea,b0,L >0,c0/\e}atio\slash= 0, 0< α < β < γ < L , andg: [0,∞)/ma√sto−→(0,∞) is the convolution kernel function. They established an exponential energy decay rate if the two wave s have the same speed of propagation. In case of different speed of propagation, they proved that the ene rgy of their system decays polynomially with ratet−1. In the same year, Akil et al.in [7], studied the stability of a multi-dimensional elastic/viscoelastic transmission problem with Kelvin-Voigt damping and non-smooth coeffi cient at the interface, they established some polynomial stability results under some geometric control con dition. In those previous literature, the authors deal with the locally coupled wave equations with local dampin g and by assuming that there is an intersection between the damping and coupling regions. The aim of th is paper is to study the direct/indirect stability of locally coupled wave equations with Kelvin-Voigt dampings/d amping localized via non-smooth coefficients/coefficient and by assuming that the supports of the d ampings and coupling coefficients aredisjoint. In the first part of this paper, we consider the following one dimensio nal coupled system: utt−(aux+butx)x+cyt= 0,(x,t)∈(0,L)×(0,∞), (1.1) ytt−(yx+dytx)x−cut= 0,(x,t)∈(0,L)×(0,∞), (1.2) with fully Dirichlet boundary conditions, (1.3) u(0,t) =u(L,t) =y(0,t) =y(L,t) = 0, t∈(0,∞), and the following initial conditions (1.4) u(·,0) =u0(·), ut(·,0) =u1(·), y(·,0) =y0(·) and yt(·,0) =y1(·), x∈(0,L). In this part, for all b0,d0>0 andc0/\e}atio\slash= 0, we treat the following three cases: Case 1 (See Figure 1): (C1)/braceleftiggb(x) =b0χ(b1,b2)(x), c(x) =c0χ(c1,c2)(x), d(x) =d0χ(d1,d2)(x), where 0< b1< b2< c1< c2< d1< d2< L. Case 2 (See Figure 2): (C2)/braceleftiggb(x) =b0χ(b1,b2)(x), c(x) =c0χ(c1,c2)(x), d(x) =d0χ(d1,d2)(x), where 0< b1< b2< d1< d2< c1< c2< L. 2Case 3 (See Figure 3): (C3)/braceleftiggb(x) =b0χ(b1,b2)(x), c(x) =c0χ(c1,c2)(x), d(x) = 0, where 0< b1< b2< c1< c2< L. While in the second part, we consider the following multi-dimensional co upled system: b1b2c1c2d1d2Lb0c0 0d0 Figure 1. Geometric description of the functions b,canddin Case 1. b1b2 0 d1d2c1c2Lb0d0c0 Figure 2. Geometric description of the functions b,canddin Case 2. 0b1b2c1c2Lb0c0 Figure 3. Geometric description of the functions bandcin Case 3. utt−div(∇u+but)+cyt= 0 in Ω ×(0,∞), (1.5) ytt−∆y−cyt= 0 in Ω ×(0,∞), (1.6) with full Dirichlet boundary condition (1.7) u=y= 0 on Γ ×(0,∞), and the following initial condition (1.8) u(·,0) =u0(·), ut(·,0) =u1(·), y(·,0) =y0(·) andyt(·,0) =y1(·) in Ω, 3where Ω ⊂Rd,d≥2 is an open and bounded set with boundary Γ of class C2. Here,b,c∈L∞(Ω) are such thatb: Ω→R+is the viscoelastic damping coefficient, c: Ω→Ris the coupling function and (1.9) b(x)≥b0>0 inωb⊂Ω, c(x)≥c0/\e}atio\slash= 0 inωc⊂Ω and c(x) = 0 on Ω \ωc and (1.10) meas( ωc∩Γ)>0 and ωb∩ωc=∅. In the first part of this paper, we study the direct and indirect sta bility of system ( 1.1)-(1.4) by consider- ing the three cases ( C1), (C2), and (C3). In Subsection 2.1, we prove the well-posedness of our system by using a semigroup approach. In Subsection 2.2, by using a general criteria of Arendt-Batty, we prove the stron g stability of our system in the absence of the compactness of the re solvent. Finally, in Subsection 2.3, by using a frequency domain approach combined with a specific multiplier metho d, we prove that our system decay polynomially in t−4or int−1. In the second part of this paper, we study the indirect stability of s ystem (1.5)-(1.8). In Subsection 3.1, we prove the well-posedness of our system by using a semigroup app roach. Finally, in Subsection 3.2, under some geometric control condition, we prove the strong stability of this system. 2.Direct and Indirect Stability in the one dimensional case In this section, we study the well-posedness, strong stability, and polynomial stability of system ( 1.1)-(1.4). The main result of this section are the following three subsections. 2.1.Well-Posedness. In this subsection, we will establish the well-posedness of system ( 1.1)-(1.4) by using semigroup approach. The energy of system ( 1.1)-(1.4) is given by E(t) =1 2/integraldisplayL 0/parenleftbig |ut|2+a|ux|2+|yt|2+|yx|2/parenrightbig dx. Let (u,ut,y,yt) be a regular solution of ( 1.1)-(1.4). Multiplying ( 1.1) and (1.2) byutandytrespectively, then using the boundary conditions ( 1.3), we get E′(t) =−/integraldisplayL 0/parenleftbig b|utx|2+d|ytx|2/parenrightbig dx. Thus, if ( C1) or (C2) or (C3) holds, we get E′(t)≤0. Therefore, system ( 1.1)-(1.4) is dissipative in the sense that its energy is non-increasing with respect to time t. Let us define the energy space Hby H= (H1 0(0,L)×L2(0,L))2. The energy space His equipped with the following inner product (U,U1)H=/integraldisplayL 0vv1dx+a/integraldisplayL 0ux(u1)xdx+/integraldisplayL 0zz1dx+/integraldisplayL 0yx(y1)xdx, for allU= (u,v,y,z)⊤andU1= (u1,v1,y1,z1)⊤inH. We define the unbounded linear operator A:D(A)⊂ H −→ H by D(A) =/braceleftbig U= (u,v,y,z)⊤∈ H;v,z∈H1 0(0,L),(aux+bvx)x∈L2(0,L),(yx+dzx)x∈L2(0,L)/bracerightbig and A(u,v,y,z)⊤= (v,(aux+bvx)x−cz,z,(yx+dzx)x+cv)⊤,∀U= (u,v,y,z)⊤∈D(A). Now, ifU= (u,ut,y,yt)⊤is the state of system ( 1.1)-(1.4), then it is transformed into the following first order evolution equation (2.1) Ut=AU, U(0) =U0, whereU0= (u0,u1,y0,y1)⊤∈ H. 4Proposition 2.1. If (C1) or (C2) or (C3) holds. Then, the unbounded linear operator Ais m-dissipative in the Hilbert space H. Proof.For allU= (u,v,y,z)⊤∈D(A), we have ℜ/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH=−/integraldisplayL 0b|vx|2dx−/integraldisplayL 0d|zx|2dx≤0, which implies that Ais dissipative. Now, similiar to Proposition 2.1 in [ 24] (see also [ 5] and [4]), we can prove that there exists a unique solution U= (u,v,y,z)⊤∈D(A) of −AU=F,∀F= (f1,f2,f3,f4)⊤∈ H. Then 0∈ρ(A) andAis an isomorphism and since ρ(A) is open in C(see Theorem 6.7 (Chapter III) in [ 19]), we easily get R(λI−A) =Hfor a sufficiently small λ >0. This, together with the dissipativeness of A, imply thatD(A) is dense in Hand that Ais m-dissipative in H(see Theorems 4.5, 4.6 in [ 22]). /square According to Lumer-Phillips theorem (see [ 22]), then the operator Agenerates a C0-semigroup of contrac- tionsetAinHwhich gives the well-posedness of ( 2.1). Then, we have the following result: Theorem 2.2. For allU0∈ H, system ( 2.1) admits a unique weak solution U(t) =etAU0∈C0(R+,H). Moreover, if U0∈D(A), then the system ( 2.1) admits a unique strong solution U(t) =etAU0∈C0(R+,D(A))∩C1(R+,H). 2.2.Strong Stability. In this subsection, we will prove the strong stability of system ( 1.1)-(1.4). We define the following conditions: (SSC1) ( C1) holds and |c0|<min/parenleftbigg√a c2−c1,1 c2−c1/parenrightbigg , (SSC3) ( C3) holds, a= 1 and |c0|<1 c2−c1. The main result of this section is the following theorem. Theorem 2.3. Assume that ( SSC1) or(C2) or(SSC3) holds. Then, the C0-semigroupofcontractions/parenleftbig etA/parenrightbig t≥0 is strongly stable in H; i.e. for all U0∈ H, the solution of ( 2.1) satisfies lim t→+∞/ba∇dbletAU0/ba∇dblH= 0. According to Theorem A.2, to prove Theorem 2.3, we need to prove that the operator Ahas no pure imaginary eigenvalues and σ(A)∩iRis countable. Its proof has been divided into the following Lemmas. Lemma 2.4. Assume that ( SSC1) or (C2) or (SSC3) holds. Then, for all λ∈R,iλI−Ais injective, i.e. ker(iλI−A) ={0}. Proof.From Proposition 2.1, we have 0 ∈ρ(A). We still need to show the result for λ∈R∗. For this aim, suppose that there exists a real number λ/\e}atio\slash= 0 and U= (u,v,y,z)⊤∈D(A) such that AU=iλU. Equivalently, we have v=iλu, (2.2) (aux+bvx)x−cz=iλv, (2.3) z=iλy, (2.4) (yx+dzx)+cv=iλz. (2.5) Next, a straightforward computation gives (2.6) 0 = ℜ/a\}b∇acketle{tiλU,U/a\}b∇acket∇i}htH=ℜ/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH=−/integraldisplayL 0b|vx|2dx−/integraldisplayL 0d|zx|2dx. 5Inserting ( 2.2) and (2.4) in (2.3) and (2.5), we get λ2u+(aux+iλbux)x−iλcy= 0 in (0 ,L), (2.7) λ2y+(yx+iλdyx)x+iλcu= 0 in (0 ,L), (2.8) with the boundary conditions (2.9) u(0) =u(L) =y(0) =y(L) = 0. •Case 1: Assume that ( SSC1) holds. From ( 2.2), (2.4) and (2.6), we deduce that (2.10) ux=vx= 0 in ( b1,b2) andyx=zx= 0 in ( d1,d2). Using (2.7), (2.8) and (2.10), we obtain (2.11) λ2u+auxx= 0 in (0 ,c1) and λ2y+yxx= 0 in ( c2,L). Deriving the above equations with respect to xand using ( 2.10), we get (2.12)/braceleftiggλ2ux+auxxx= 0 in (0 ,c1), ux= 0 in ( b1,b2)⊂(0,c1),and/braceleftiggλ2yx+yxxx= 0 in ( c2,L), yx= 0 in ( d1,d2)⊂(c2,L). Using the unique continuation theorem, we get (2.13) ux= 0 in (0 ,c1) and yx= 0 in ( c2,L) Using (2.13) and the fact that u(0) =y(L) = 0, we get (2.14) u= 0 in (0 ,c1) and y= 0 in ( c2,L). Now, ouraim is to provethat u=y= 0 in (c1,c2). For this aim, using ( 2.14) and the fact that u,y∈C1([0,L]), we obtain the following boundary conditions (2.15) u(c1) =ux(c1) =y(c2) =yx(c2) = 0. Multiplying ( 2.7) by−2(x−c2)ux, integrating over ( c1,c2) and taking the real part, we get (2.16) −/integraldisplayc2 c1λ2(x−c2)(|u|2)xdx−a/integraldisplayc2 c1(x−c2)/parenleftbig |ux|2/parenrightbig xdx+2ℜ/parenleftbigg iλc0/integraldisplayc2 c1(x−c2)yuxdx/parenrightbigg = 0, using integration by parts and ( 2.15), we get (2.17)/integraldisplayc2 c1|λu|2dx+a/integraldisplayc2 c1|ux|2dx+2ℜ/parenleftbigg iλc0/integraldisplayc2 c1(x−c2)yuxdx/parenrightbigg = 0. Multiplying ( 2.8) by−2(x−c1)yx, integrating over ( c1,c2), taking the real part, and using the same argument as above, we get (2.18)/integraldisplayc2 c1|λy|2dx+/integraldisplayc2 c1|yx|2dx+2ℜ/parenleftbigg iλc0/integraldisplayc2 c1(x−c1)uyxdx/parenrightbigg = 0. Adding ( 2.17) and (2.18), we get (2.19)/integraldisplayc2 c1|λu|2dx+a/integraldisplayc2 c1|ux|2dx+/integraldisplayc2 c1|λy|2dx+/integraldisplayc2 c1|yx|2dx≤2|λ||c0|(c2−c1)/integraldisplayc2 c1(|y||ux|+|u||yx|)dx. Using Young’s inequality in ( 2.19), we get (2.20)/integraldisplayc2 c1|λu|2dx+a/integraldisplayc2 c1|ux|2dx+/integraldisplayc2 c1|λy|2dx+/integraldisplayc2 c1|yx|2dx≤c2 0(c2−c1)2 a/integraldisplayc2 c1|λy|2dx +a/integraldisplayc2 c1|ux|2dx+c2 0(c2−c1)2/integraldisplayc2 c1|λu|2dx+/integraldisplayc2 c1|yx|2dx, consequently, we get (2.21)/parenleftbigg 1−c2 0(c2−c1)2 a/parenrightbigg/integraldisplayc2 c1|λy|2dx+/parenleftbig 1−c2 0(c2−c1)2/parenrightbig/integraldisplayc2 c1|λu|2dx≤0. Thus, from the above inequality and ( SSC1), we get (2.22) u=y= 0 in ( c1,c2). 6Next, we need to prove that u= 0 in (c2,L) andy= 0 in (0 ,c1). For this aim, from ( 2.22) and the fact that u,y∈C1([0,L]), we obtain (2.23) u(c2) =ux(c2) = 0 and y(c1) =yx(c1) = 0. It follows from ( 2.7), (2.8) and (2.23) that (2.24)/braceleftiggλ2u+auxx= 0 in ( c2,L), u(c2) =ux(c2) =u(L) = 0,and/braceleftiggλ2y+yxx= 0 in (0 ,c1), y(0) =y(c1) =yx(c1) = 0. Holmgren uniqueness theorem yields (2.25) u= 0 in ( c2,L) andy= 0 in (0 ,c1). Therefore, from ( 2.2), (2.4), (2.14), (2.22) and (2.25), we deduce that U= 0. •Case 2: Assume that ( C2) holds. From ( 2.2), (2.4) and (2.6), we deduce that (2.26) ux=vx= 0 in ( b1,b2) andyx=zx= 0 in ( d1,d2). Using (2.7), (2.8) and (2.26), we obtain (2.27) λ2u+auxx= 0 in (0 ,c1) and λ2y+yxx= 0 in (0 ,c1). Deriving the above equations with respect to xand using ( 2.26), we get (2.28)/braceleftiggλ2ux+auxxx= 0 in (0 ,c1), ux= 0 in ( b1,b2)⊂(0,c1),and/braceleftiggλ2yx+yxxx= 0 in (0 ,c1), yx= 0 in ( d1,d2)⊂(0,c1). Using the unique continuation theorem, we get (2.29) ux= 0 in (0 ,c1) and yx= 0 in (0 ,c1). From (2.29) and the fact that u(0) =y(0) = 0, we get (2.30) u= 0 in (0 ,c1) and y= 0 in (0 ,c1). Using the fact that u,y∈C1([0,L]) and (2.30), we get (2.31) u(c1) =ux(c1) =y(c1) =yx(c1) = 0. Now, using the definition of c(x) in (2.7)-(2.8), (2.26) and (2.31) and Holmgren theorem, we get u=y= 0 in ( c1,c2). Again, using the fact that u,y∈C1([0,L]), we get (2.32) u(c2) =ux(c2) =y(c2) =yx(c2) = 0. Now, using the same argument as in Case 1, we obtain u=y= 0 in (c2,L), consequently, we deduce that U= 0. •Case 3: Assume that ( SSC3) holds. Using the same argument as in Cases 1 and 2, we obtain (2.33) u= 0 in (0 ,c1) and u(c1) =ux(c1) = 0. Step 1. The aim of this step is to prove that (2.34)/integraldisplayc2 c1|u|2dx=/integraldisplayc2 c1|y|2dx. 7For this aim, multiplying ( 2.7) byyand (2.8) byuand using integration by parts, we get /integraldisplayL 0λ2uydx−/integraldisplayL 0uxyxdx−iλc0/integraldisplayc2 c1|y|2dx= 0, (2.35) /integraldisplayL 0λ2yudx−/integraldisplayL 0yxuxdx+iλc0/integraldisplayc2 c1|u|2dx= 0. (2.36) Adding ( 2.35) and (2.36), taking the imaginary part, we get ( 2.34). Step 2. Multiplying ( 2.7) by−2(x−c2)ux, integrating over ( c1,c2) and taking the real part, we get (2.37)−ℜ/parenleftbigg/integraldisplayc2 c1λ2(x−c2)(|u|2)xdx/parenrightbigg −ℜ/parenleftbigg/integraldisplayc2 c1(x−c2)/parenleftbig |ux|2/parenrightbig xdx/parenrightbigg +2ℜ/parenleftbigg iλc0/integraldisplayc2 c1(x−c2)yuxdx/parenrightbigg = 0, using integration by parts in ( 2.37) and (2.33), we get (2.38)/integraldisplayc2 c1|λu|2dx+a/integraldisplayc2 c1|ux|2dx+2ℜ/parenleftbigg iλc0/integraldisplayc2 c1(x−c2)yuxdx/parenrightbigg = 0. Using Young’s inequality in ( 2.38), we obtain (2.39)/integraldisplayc2 c1|λu|2dx+/integraldisplayc2 c1|ux|2dx≤ |c0|(c2−c1)/integraldisplayc2 c1|λy|2dx+|c0|(c2−c1)/integraldisplayc2 c1|ux|2dx. Inserting ( 2.34) in (2.39), we get (2.40) (1 −|c0|(c2−c1))/integraldisplayc2 c1/parenleftbig |λu|2+|ux|2/parenrightbig dx≤0. According to ( SSC3) and (2.34), we get (2.41) u=y= 0 in ( c1,c2). Step 3. Using the fact that u∈H2(c1,c2)⊂C1([c1,c2]), we get (2.42) u(c1) =ux(c1) =y(c1) =yx(c1) =y(c2) =yx(c2) = 0. Now, from ( 2.7), (2.8) and the definition of c, we get /braceleftbiggλ2u+uxx= 0 in ( c2,L), u(c2) =ux(c2) = 0,and/braceleftbiggλ2y+yxx= 0 in (0 ,c1)∪(c2,L), y(c1) =yx(c1) =y(c2) =yx(c2) = 0. From the above systems and Holmgren uniqueness Theorem, we get (2.43) u= 0 in ( c2,L) and y= 0 in (0 ,c1)∪(c2,L). Consequently, using ( 2.33), (2.41) and (2.43), we get U= 0. The proof is thus completed. /square Lemma 2.5. Assume that ( SSC1) or (C2) or (SSC3) holds. Then, for all λ∈R, we have R(iλI−A) =H. Proof.See Lemma 2.5 in [ 24] (see also [ 4]). /square Proof of Theorems 2.3. From Lemma 2.4, we obtain that the operator Ahas no pure imaginary eigenvalues (i.e.σp(A)∩iR=∅). Moreover, from Lemma 2.5and with the help of the closed graph theorem of Banach, we deduce that σ(A)∩iR=∅. Therefore, according to Theorem A.2, we get that the C 0-semigroup ( etA)t≥0 is strongly stable. The proof is thus complete. /square 82.3.Polynomial Stability. In this subsection, we study the polynomial stability of system ( 1.1)-(1.4). Our main result in this section are the following theorems. Theorem 2.6. Assume that ( SSC1) holds. Then, for all U0∈D(A), there exists aconstant C >0 independent ofU0such that (2.44) E(t)≤C t4/ba∇dblU0/ba∇dbl2 D(A), t >0. Theorem 2.7. Assume that ( SSC3) holds . Then, for all U0∈D(A) there exists a constant C >0 independent ofU0such that (2.45) E(t)≤C t/ba∇dblU0/ba∇dbl2 D(A), t >0. According to Theorem A.3, the polynomial energy decays ( 2.44) and (2.45) hold if the following conditions (H1) iR⊂ρ(A) and (H2) limsup λ∈R,|λ|→∞1 |λ|ℓ/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble L(H)<∞withℓ=/braceleftigg1 2for Theorem 2.6, 2 for Theorem 2.7, aresatisfied. Sincecondition( H1)isalreadyprovedin Subsection 2.2. We stillneedtoprove( H2), let usproveit byacontradictionargument. Tothisaim,supposethat( H2)isfalse,thenthereexists/braceleftbig/parenleftbig λn,Un:= (un,vn,yn,zn)⊤/parenrightbig/bracerightbig n≥1⊂ R∗ +×D(A) with (2.46) λn→ ∞asn→ ∞and/ba∇dblUn/ba∇dblH= 1,∀n≥1, such that (2.47) ( λn)ℓ(iλnI−A)Un=Fn:= (f1,n,f2,n,f3,n,f4,n)⊤→0 inH,asn→ ∞. For simplicity, we drop the index n. Equivalently, from ( 2.47), we have iλu−v=f1 λℓ, f1→0 inH1 0(0,L), (2.48) iλv−(aux+bvx)x+cz=f2 λℓ, f2→0 inL2(0,L), (2.49) iλy−z=f3 λℓ, f3→0 inH1 0(0,L), (2.50) iλz−(yx+dzx)x−cv=f4 λℓ, f4→0 inL2(0,L). (2.51) 2.3.1.Proof of Theorem 2.6.In this subsection, we will prove Theorem 2.6by checking the condition ( H2), by finding a contradiction with ( 2.46) by showing /ba∇dblU/ba∇dblH=o(1). For clarity, we divide the proof into several Lemmas. By taking the inner product of ( 2.47) withUinH, we remark that /integraldisplayL 0b|vx|2dx+/integraldisplayL 0d|zx|2dx=−ℜ(/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH) =λ−1 2ℜ(/a\}b∇acketle{tF,U/a\}b∇acket∇i}htH) =o/parenleftig λ−1 2/parenrightig . Thus, from the definitions of bandd, we get (2.52)/integraldisplayb2 b1|vx|2dx=o/parenleftig λ−1 2/parenrightig and/integraldisplayd2 d1|zx|2dx=o/parenleftig λ−1 2/parenrightig . Using (2.48), (2.50), (2.52), and the fact that f1,f3→0 inH1 0(0,L), we get (2.53)/integraldisplayb2 b1|ux|2dx=o(1) λ5 2and/integraldisplayd2 d1|yx|2dx=o(1) λ5 2. Lemma 2.8. The solution U∈D(A) of system ( 2.48)-(2.51) satisfies the following estimations (2.54)/integraldisplayb2 b1|v|2dx=o(1) λ3 2and/integraldisplayd2 d1|z|2dx=o(1) λ3 2. 9Proof.We give the proof of the first estimation in ( 2.54), the second one can be done in a similar way. For this aim, we fix g∈C1([b1,b2]) such that g(b2) =−g(b1) = 1,max x∈[b1,b2]|g(x)|=mgand max x∈[b1,b2]|g′(x)|=mg′. The proof is divided into several steps: Step 1. The goal of this step is to prove that (2.55) |v(b1)|2+|v(b2)|2≤/parenleftigg λ1 2 2+2mg′/parenrightigg/integraldisplayb2 b1|v|2dx+o(1) λ. From (2.48), we deduce that (2.56) vx=iλux−λ−1 2(f1)x. Multiplying ( 2.56) by 2gvand integrating over ( b1,b2), then taking the real part, we get /integraldisplayb2 b1g/parenleftbig |v|2/parenrightbig xdx=ℜ/parenleftigg 2iλ/integraldisplayb2 b1guxvdx/parenrightigg −ℜ/parenleftigg 2λ−1 2/integraldisplayb2 b1g(f1)xvdx/parenrightigg . Using integration by parts in the left hand side of the above equation , we get (2.57) |v(b1)|2+|v(b2)|2=/integraldisplayb2 b1g′|v|2dx+ℜ/parenleftigg 2iλ/integraldisplayb2 b1guxvdx/parenrightigg −ℜ/parenleftigg 2λ−1 2/integraldisplayb2 b1g(f1)xvdx/parenrightigg . Using Young’s inequality, we obtain 2λmg|ux||v| ≤λ1 2|v|2 2+2λ3 2m2 g|ux|2and 2λ−1 2mg|(f1)x||v| ≤mg′|v|2+m2 gm−1 g′λ−1|(f1)x|2. From the above inequalities, ( 2.57) becomes (2.58) |v(b1)|2+|v(b2)|2≤/parenleftigg λ1 2 2+2mg′/parenrightigg/integraldisplayb2 b1|v|2dx+2λ3 2m2 g/integraldisplayb2 b1|ux|2dx+m2 g mg′λ−1/integraldisplayb2 b1|(f1)x|2dx. Inserting ( 2.53) in (2.58) and the fact that f1→0 inH1 0(0,L), we get ( 2.55). Step 2. The aim of this step is to prove that (2.59) |(aux+bvx)(b1)|2+|(aux+bvx)(b2)|2≤λ3 2 2/integraldisplayb2 b1|v|2dx+o(1). Multiplying ( 2.49) by−2g/parenleftbig aux+bvx/parenrightbig , using integration by parts over ( b1,b2) and taking the real part, we get |(aux+bvx)(b1)|2+|(aux+bvx)(b2)|2=/integraldisplayb2 b1g′|aux+bvx|2dx+ ℜ/parenleftigg 2iλ/integraldisplayb2 b1gv(aux+bvx)dx/parenrightigg −ℜ/parenleftigg 2λ−1 2/integraldisplayb2 b1gf2(aux+bvx)dx/parenrightigg , consequently, we get (2.60)|(aux+bvx)(b1)|2+|(aux+bvx)(b2)|2≤mg′/integraldisplayb2 b1|aux+bvx|2dx +2λmg/integraldisplayb2 b1|v||aux+bvx|dx+2mgλ−1 2/integraldisplayb2 b1|f2||aux+bvx|dx. By Young’s inequality, ( 2.52), and (2.53), we have (2.61) 2 λmg/integraldisplayb2 b1|v||aux+bvx|dx≤λ3 2 2/integraldisplayb2 b1|v|2dx+2m2 gλ1 2/integraldisplayb2 b1|aux+bvx|2dx≤λ3 2 2/integraldisplayb2 b1|v|2dx+o(1). Inserting ( 2.61) in (2.60), then using ( 2.52), (2.53) and the fact that f2→0 inL2(0,L), we get ( 2.59). 10Step 3. The aim of this step is to prove the first estimation in ( 2.54). For this aim, multiplying ( 2.49) by −iλ−1v, integrating over ( b1,b2) and taking the real part , we get (2.62)/integraldisplayb2 b1|v|2dx=ℜ/parenleftigg iλ−1/integraldisplayb2 b1(aux+bvx)vxdx−/bracketleftbig iλ−1(aux+bvx)v/bracketrightbigb2 b1+iλ−3 2/integraldisplayb2 b1f2vdx/parenrightigg . Using (2.52), (2.53), the fact that vis uniformly bounded in L2(0,L) andf2→0 inL2(0,1), and Young’s inequalities, we get (2.63)/integraldisplayb2 b1|v|2dx≤λ−1 2 2[|v(b1)|2+|v(b2)|2]+λ−3 2 2[|(aux+bvx)(b1)|2+|(aux+bvx)(b2)|2]+o(1) λ3 2. Inserting ( 2.55) and (2.59) in (2.63), we get /integraldisplayb2 b1|v|2dx≤/parenleftbigg1 2+mg′λ−1 2/parenrightbigg/integraldisplayb2 b1|v|2dx+o(1) λ3 2, which implies that (2.64)/parenleftbigg1 2−mg′λ−1 2/parenrightbigg/integraldisplayb2 b1|v|2dx≤o(1) λ3 2. Using the fact that λ→ ∞, we can take λ >4m2 g′. Then, we obtain the first estimation in ( 2.54). Similarly, we can obtain the second estimation in ( 2.54). The proof has been completed. /square Lemma 2.9. The solution U∈D(A) of system ( 2.48)-(2.51) satisfies the following estimations (2.65)/integraldisplayc1 0/parenleftbig |v|2+a|ux|2/parenrightbig dx=o(1) and/integraldisplayL c2/parenleftbig |z|2+|yx|2/parenrightbig dx=o(1). Proof. First, let h∈C1([0,c1]) such that h(0) =h(c1) = 0. Multiplying ( 2.49) by 2a−1h(aux+bvx), integrating over (0 ,c1), using integration by parts and taking the real part, then using ( 2.52) and the fact that uxis uniformly bounded in L2(0,L) andf2→0 inL2(0,L), we get (2.66) ℜ/parenleftbigg 2iλa−1/integraldisplayc1 0vh(aux+bvx)dx/parenrightbigg +a−1/integraldisplayc1 0h′|aux+bvx|2dx=o(1) λ1 2. From (2.48), we have (2.67) iλux=−vx−λ−1 2(f1)x. Inserting ( 2.67) in (2.66), using integration by parts, then using ( 2.52), (2.54), and the fact that f1→0 in H1 0(0,L) andvis uniformly bounded in L2(0,L), we get (2.68)/integraldisplayc1 0h′|v|2dx+a−1/integraldisplayc1 0h′|aux+bvx|2dx= 2ℜ/parenleftbigg λ−1 2/integraldisplayc1 0vh(f1)xdx/parenrightbigg /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =o(λ−1 2) +ℜ/parenleftigg 2iλa−1b0/integraldisplayb2 b1hvvxdx/parenrightigg /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =o(1)+o(1) λ1 2. Now, we fix the following cut-off functions p1(x) := 1 in (0 ,b1), 0 in ( b2,c1), 0≤p1≤1 in (b1,b2),andp2(x) := 1 in ( b2,c1), 0 in (0 ,b1), 0≤p2≤1 in (b1,b2). Finally, take h(x) =xp1(x)+(x−c1)p2(x) in (2.68) and using ( 2.52), (2.53), (2.54), we get the first estimation in (2.65). By using the same argument, we can obtain the second estimation in (2.65). The proof is thus completed. /square Lemma 2.10. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimations (2.69) |λu(c1)|=o(1),|ux(c1)|=o(1),|λy(c2)|=o(1)and|yx(c2)|=o(1). 11Proof.First, from ( 2.48) and (2.49), we deduce that (2.70) λ2u+auxx=−f2 λ1 2−iλ1 2f1in (b2,c1). Multiplying ( 2.70) by 2(x−b2)¯ux, integrating over ( b2,c1) and taking the real part, then using the fact that uxis uniformly bounded in L2(0,L) andf2→0 inL2(0,L), we get (2.71)/integraldisplayc1 b2λ2(x−b2)/parenleftbig |u|2/parenrightbig xdx+a/integraldisplayc1 b2(x−b2)/parenleftbig |ux|2/parenrightbig xdx=−ℜ/parenleftbigg 2iλ1 2/integraldisplayc1 b2(x−b2)f1uxdx/parenrightbigg +o(1) λ1 2. Using integration by parts in ( 2.71), then using ( 2.65), and the fact that f1→0 inH1 0(0,L) andλuis uniformly bounded in L2(0,L), we get (2.72) 0 ≤(c1−b2)/parenleftbig |λu(c1)|2+a|ux(c1)|2/parenrightbig =ℜ/parenleftig 2iλ1 2(c1−b2)f1(c1)u(c1)/parenrightig +o(1), consequently, by using Young’s inequality, we get |λu(c1)|2+|ux(c1)|2≤2λ1 2|f1(c1)||u(c1)|+o(1) ≤1 2|λu(c1)|2+2 λ|f1(c1)|2+o(1). Then, we get (2.73)1 2|λu(c1)|2+|ux(c1)|2≤2 λ|f1(c1)|2+o(1). Finally, from the above estimation and the fact that f1→0 inH1 0(0,L), we get the first two estimations in (2.69). By using the same argument, we can obtain the last two estimation s in (2.69). The proof has been completed. /square Lemma 2.11. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimation (2.74)/integraldisplayc2 c1|λu|2+a|ux|2+|λy|2+|yx|2dx=o(1). Proof.Inserting ( 2.48) and (2.50) in (2.49) and (2.51), we get −λ2u−auxx+iλc0y=f2 λ1 2+iλ1 2f1+c0f3 λ1 2in (c1,c2), (2.75) −λ2y−yxx−iλc0u=f4 λ1 2+iλ1 2f3−c0f1 λ1 2in (c1,c2). (2.76) Multiplying ( 2.75) by 2(x−c2)uxand (2.76) by 2(x−c1)yx, integrating over ( c1,c2) and taking the real part, then using the fact that /ba∇dblF/ba∇dblH=o(1) and/ba∇dblU/ba∇dblH= 1, we obtain (2.77)−λ2/integraldisplayc2 c1(x−c2)/parenleftbig |u|2/parenrightbig xdx−a/integraldisplayc2 c1(x−c2)/parenleftbig |ux|2/parenrightbig xdx+ℜ/parenleftbigg 2iλc0/integraldisplayc2 c1(x−c2)yuxdx/parenrightbigg = ℜ/parenleftbigg 2iλ1 2/integraldisplayc2 c1(x−c2)f1uxdx/parenrightbigg +o(1) λ1 2 and (2.78)−λ2/integraldisplayc2 c1(x−c1)/parenleftbig |y|2/parenrightbig xdx−/integraldisplayc2 c1(x−c1)/parenleftbig |yx|2/parenrightbig xdx−ℜ/parenleftbigg 2iλc0/integraldisplayc2 c1(x−c1)uyxdx/parenrightbigg = ℜ/parenleftbigg 2iλ1 2/integraldisplayc2 c1(x−c1)f3yxdx/parenrightbigg +o(1) λ1 2. Using integration by parts, ( 2.69), and the fact that f1,f3→0 inH1 0(0,L),/ba∇dblu/ba∇dblL2(0,L)=O(λ−1),/ba∇dbly/ba∇dblL2(0,L)= O(λ−1), we deduce that (2.79) ℜ/parenleftbigg iλ1 2/integraldisplayc2 c1(x−c2)f1uxdx/parenrightbigg =o(1) λ1 2andℜ/parenleftbigg iλ1 2/integraldisplayc2 c1(x−c1)f3yxdx/parenrightbigg =o(1) λ1 2. 12Inserting ( 2.79) in (2.77) and (2.78), then using integration by parts and ( 2.69), we get /integraldisplayc2 c1/parenleftbig |λu|2+a|ux|2/parenrightbig dx+ℜ/parenleftbigg iλc0/integraldisplayc2 c1(x−c2)yuxdx/parenrightbigg =o(1), (2.80) /integraldisplayc2 c1/parenleftbig |λy|2+|yx|2/parenrightbig dx−ℜ/parenleftbigg iλc0/integraldisplayc2 c1(x−c1)uyxdx/parenrightbigg =o(1). (2.81) Adding ( 2.80) and (2.81), we get /integraldisplayc2 c1/parenleftbig |λu|2+a|ux|2+|λy|2+|yx|2/parenrightbig dx=ℜ/parenleftbigg 2iλc0/integraldisplayc2 c1(x−c1)uyxdx/parenrightbigg −ℜ/parenleftbigg 2iλc0/integraldisplayc2 c1(x−c2)yuxdx/parenrightbigg +o(1) ≤2λ|c0|(c2−c1)/integraldisplayc2 c1|u||yx|dx+2λ|c0| a1 4(c2−c1)a1 4/integraldisplayc2 c1|y||ux|dx+o(1). Applying Young’s inequalities, we get (2.82) (1 −|c0|(c2−c1))/integraldisplayc2 c1(|λu|2+|yx|2)dx+/parenleftbigg 1−1√a|c0|(c2−c1)/parenrightbigg/integraldisplayc2 c1(a|ux|2+|λy|2)dx≤o(1). Finally, using ( SSC1), we get the desired result. The proof has been completed. /square Lemma 2.12. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimations (2.83)/integraldisplayc1 0/parenleftbig |z|2+|yx|2/parenrightbig dx=o(1)and/integraldisplayL c2/parenleftbig |v|2+a|ux|2/parenrightbig dx=o(1). Proof.Using the same argument of Lemma 2.9, we obtain ( 2.83). /square Proof of Theorem 2.6.Using (2.53), Lemmas 2.8,2.9,2.11,2.12, we get /ba∇dblU/ba∇dblH=o(1), which contradicts (2.46). Consequently, condition (H2) holds. This implies the energy decay estimation ( 2.44). 2.3.2.Proof of Theorem 2.7.In this subsection, we will prove Theorem 2.7by checking the condition ( H2), that is by finding a contradiction with ( 2.46) by showing /ba∇dblU/ba∇dblH=o(1). For clarity, we divide the proof into several Lemmas. By taking the inner product of ( 2.47) withUinH, we remark that /integraldisplayL 0b|vx|2dx=−ℜ(/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH) =λ−2ℜ(/a\}b∇acketle{tF,U/a\}b∇acket∇i}htH) =o(λ−2). Then, (2.84)/integraldisplayb2 b1|vx|2dx=o(λ−2). Using (2.48) and (2.84), and the fact that f1→0 inH1 0(0,L), we get (2.85)/integraldisplayb2 b1|ux|2dx=o(λ−4). Lemma 2.13. Let0< ε <b2−b1 2, the solution U∈D(A)of the system (2.48)-(2.51)satisfies the following estimation (2.86)/integraldisplayb2−ε b1+ε|v|2dx=o(λ−2). Proof.First, we fix a cut-off function θ1∈C1([0,c1]) such that (2.87) θ1(x) = 1 if x∈(b1+ε,b2−ε), 0 if x∈(0,b1)∪(b2,L), 0≤θ1≤1 elsewhere . Multiplying ( 2.49) byλ−1θ1v, integrating over (0 ,c1), using integration by parts, and the fact that f2→0 in L2(0,L) andvis uniformly bounded in L2(0,L), we get (2.88) i/integraldisplayc1 0θ1|v|2dx+1 λ/integraldisplayc1 0(ux+bvx)(θ′ 1v+θvx)dx=o(λ−3). 13Using (2.84) and the fact that /ba∇dblU/ba∇dblH= 1, we get 1 λ/integraldisplayc1 0(ux+bvx)(θ′ 1v+θvx)dx=o(λ−2). Inserting the above estimation in ( 2.88), we get the desired result ( 2.86). The proof has been completed. /square Lemma 2.14. The solution U∈D(A)of the system (2.48)-(2.51)satisfies the following estimation (2.89)/integraldisplayc1 0(|v|2+|ux|2)dx=o(1). Proof.Leth∈C1([0,c1]) such that h(0) =h(c1) = 0. Multiplying ( 2.49) by 2h(ux+bvx), integrating over (0,c1) and taking the real part, then using integration by parts and the fact that f2→0 inL2(0,L), we get (2.90) ℜ/parenleftbigg 2/integraldisplayc1 0iλvh(ux+bvx)dx/parenrightbigg +/integraldisplayc1 0h′|ux+bvx|2dx=o(λ−2). Using (2.84) and the fact that vis uniformly bounded in L2(0,L), we get (2.91) ℜ/parenleftbigg 2/integraldisplayc1 0iλvh(ux+bvx)dx/parenrightbigg = 2/integraldisplayc1 0iλvhuxdx+o(1). From (2.48), we have (2.92) iλux=−vx−/parenleftbig f1/parenrightbig x λ2. Inserting ( 2.92) in (2.91), using integration by parts and the fact that f1→0 inH1 0(0,L), we get (2.93) ℜ/parenleftbigg 2/integraldisplayc1 0iλvh(ux+bvx)dx/parenrightbigg =/integraldisplayc1 0h′|v|2dx+o(1). Inserting ( 2.93) in (2.90), we obtain (2.94)/integraldisplayc1 0h′/parenleftbig |v|2+|ux+bvx|2/parenrightbig dx=o(1). Now, we fix the following cut-off functions θ2(x) := 1 in (0 ,b1+ε), 0 in ( b2−ε,c1), 0≤θ2≤1 in (b1+ε,b2−ε),andθ3(x) := 1 in ( b2−ε,c1), 0 in (0 ,b1+ε), 0≤θ3≤1 in (b1+ε,b2−ε). Takingh(x) =xθ2(x)+(x−c1)θ3(x) in (2.94), then using ( 2.84) and (2.85), we get (2.95)/integraldisplay (0,b1+ε)∪(b2−ε,c1)|v|2dx+/integraldisplay (0,b1)∪(b2,c1)|ux|2dx=o(1). Finally, from ( 2.85), (2.86) and (2.95), we get the desired result ( 2.89). The proof has been completed. /square Lemma 2.15. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimations (2.96) |λu(c1)|=o(1)and|ux(c1)|=o(1), (2.97)/integraldisplayc2 c1|λu|2dx=/integraldisplayc2 c1|λy|2dx+o(1). Proof.First, using the same argument of Lemma 2.10, we claim ( 2.96). Inserting ( 2.48), (2.50) in (2.49) and (2.51), we get λ2u+(ux+bvx)x−iλcy=−f2 λ2−if1 λ−cf3 λ2, (2.98) λ2y+yxx+iλcu=−f4 λ2−if3 λ+cf1 λ2. (2.99) 14Multiplying ( 2.98) and (2.99) byλyandλurespectively, integrating over(0 ,L), then using integration by parts, (2.84), and the fact that /ba∇dblU/ba∇dblH= 1 and /ba∇dblF/ba∇dblH=o(1), we get λ3/integraldisplayL 0u¯ydx−λ/integraldisplayL 0ux¯yxdx−ic0/integraldisplayc2 c1|λy|2dx=o(1), (2.100) λ3/integraldisplayL 0y¯udx−λ/integraldisplayL 0yx¯uxdx+ic0/integraldisplayc2 c1|λu|2dx=o(1) λ. (2.101) Adding ( 2.100) and (2.101) and taking the imaginary parts, we get the desired result ( 2.97). The proof is thus completed. /square Lemma 2.16. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following asymptotic behavior (2.102)/integraldisplayc2 c1|λu|2dx=o(1),/integraldisplayc2 c1|λy|2dx=o(1)and/integraldisplayc2 c1|ux|2dx=o(1). Proof.First, Multiplying ( 2.98) by 2(x−c2)¯ux, integrating over ( c1,c2) and taking the real part, using the fact that /ba∇dblU/ba∇dblH= 1 and /ba∇dblF/ba∇dblH=o(1), we get (2.103) λ2/integraldisplayc2 c1(x−c2)/parenleftbig |u|2/parenrightbig xdx+/integraldisplayc2 c1(x−c2)/parenleftbig |ux|2/parenrightbig xdx=ℜ/parenleftbigg 2iλc0/integraldisplayc2 c1(x−c2)y¯uxdx/parenrightbigg +o(1). Using integration by parts in ( 2.103) with the help of ( 2.96), we get (2.104)/integraldisplayc2 c1|λu|2dx+/integraldisplayc2 c1|ux|2dx≤2λ|c0|(c2−c1)/integraldisplayc2 c1|y||ux|+o(1). Applying Young’s inequality in ( 2.104), we get (2.105)/integraldisplayc2 c1|λu|2dx+/integraldisplayc2 c1|ux|2dx≤ |c0|(c2−c1)/integraldisplayc2 c1|ux|2dx+|c0|(c2−c1)/integraldisplayc2 c1|λy|2dx+o(1). Using (2.97) in (2.105), we get (2.106) (1 −|c0|(c2−c1))/integraldisplayc2 c1/parenleftbig |λu|2+|ux|2/parenrightbig dx≤o(1). Finally, from the above estimation, ( SSC3) and (2.97), we get the desired result ( 2.102). The proof has been completed. /square Lemma 2.17. Let0< δ <c2−c1 2. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimations (2.107)/integraldisplayc2−δ c1+δ|yx|2dx=o(1). Proof.First, we fix a cut-off function θ4∈C1([0,L]) such that (2.108) θ4(x) := 1 if x∈(c1+δ,c2−δ), 0 if x∈(0,c1)∪(c2,L), 0≤θ4≤1 elsewhere . Multiplying ( 2.99) byθ4¯y, integrating over (0 ,L) and using integration by parts, we get (2.109)/integraldisplayc2 c1θ4|λy|2dx−/integraldisplayL 0θ4|yx|2dx−/integraldisplayL 0θ′ 4yx¯ydx+iλc0/integraldisplayc2 c1θ4u¯ydx=o(1) λ2. Using (2.102) and the definition of θ4, we get (2.110)/integraldisplayc2 c1θ4|λy|2dx=o(1),/integraldisplayL 0θ′ 4yx¯ydx=o(λ−1), iλc 0/integraldisplayc2 c1θ4u¯ydx=o(λ−1). Finally, Inserting ( 2.110) in (2.109), we get the desired result ( 2.111). The proof has been completed. /square Lemma 2.18. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimations (2.111)/integraldisplayc1+ε 0|λy|2dx,/integraldisplayc1+ε 0|yx|2dx,/integraldisplayL c2−ε|λy|2dx,/integraldisplayL c2−ε|yx|2dx,/integraldisplayL c2|λu|2dx,/integraldisplayL c2|ux|2dx=o(1). 15Proof.Letq∈C1([0,L]) such that q(0) =q(L) = 0. Multiplying ( 2.98) by 2q¯yxintegrating over (0 ,L), using (2.102), and the fact that yxis uniformly bounded in L2(0,L) and/ba∇dblF/ba∇dblH=o(1), we get (2.112)/integraldisplayL 0q′/parenleftbig |λy|2+|yx|2/parenrightbig dx=o(1). Now, take q(x) =xθ5(x)+(x−L)θ6(x) in (2.112), such that θ5(x) := 1 in (0 ,c1+ε), 0 in ( c2−ε,L), 0≤θ1≤1 in (c1+ε,c2−ε),andθ2(x) 1 in ( c2−ε,L), 0 in (0 ,c1+ε), 0≤θ2≤1 in (c1+ε,c2−ε). Then, we obtain the first four estimations in ( 2.111). Now, multiplying ( 2.98) by 2q/parenleftbig ux+bvx/parenrightbig integrating over (0,L) and using the fact that uxis uniformly bounded in L2(0,L), we get (2.113)/integraldisplayL 0q′/parenleftbig |λu|2+|ux|2/parenrightbig dx=o(1). By taking q(x) = (x−L)θ7(x), such that θ7(x) = 1 in ( c2,L), 0 in (0 ,c1), 0≤θ7≤1 in (c1,c2), we get the the last two estimations in ( 2.111). The proof has been completed. /square Proof of Theorem 2.7.Using (2.85), Lemmas 2.14,2.16,2.17and2.18, we get /ba∇dblU/ba∇dblH=o(1), which contradicts ( 2.46). Consequently, condition (H2) holds. This implies the energy decay estimation ( 2.45) 3.Indirect Stability in the multi-dimensional case In this section, we study the well-posedness and the strong stabilit y of system ( 1.5)-(1.8). 3.1.Well-posedness. In this subsection, we will establish the well-posednessof ( 1.5)-(1.8) by usinf semigroup approach. The energy of system ( 1.5)-(1.8) is given by (3.1) E(t) =1 2/integraldisplayL 0/parenleftbig |ut|2+|∇u|2+|yt|2+|∇y|2/parenrightbig dx. Let (u,ut,y,yt) be a regular solution of ( 1.5)-(1.8). Multiplying ( 1.5) and (1.7) byutandytrespectively, then using the boundary conditions ( 1.9), we get (3.2) E′(t) =−/integraldisplay Ωb|∇ut|2dx, using the definition of b, we get E′(t)≤0. Thus, system ( 1.5)-(1.8) is dissipative in the sense that its energy is non-increasing with respect to time t. Let us define the energy space Hby H=/parenleftbig H1 0(Ω)×L2(Ω)/parenrightbig2. The energy space His equipped with the inner product defined by /a\}b∇acketle{tU,U1/a\}b∇acket∇i}htH=/integraldisplay Ωvv1dx+/integraldisplay Ω∇u∇u1dx+/integraldisplay Ωzz1dx+/integraldisplay Ω∇y·∇y1dx, for allU= (u,v,y,z)⊤andU1= (u1,v1,y1,z1)⊤inH. We define the unbounded linear operator Ad:D(Ad)⊂ H −→ H by D(Ad) =/braceleftbig U= (u,v,y,z)⊤∈ H;v,z∈H1 0(Ω),div(ux+bvx)∈L2(Ω),∆y∈L2(Ω)/bracerightbig and AdU= v div(∇u+b∇v)−cz z ∆y+cv ,∀U= (u,v,y,z)⊤∈D(Ad). 16/tildewideΩ Ωωc ωb •x0 Γ0Γ1 •xν Figure 4. Geometric description of the sets ωbandωc IfU= (u,ut,y,yt) is a regular solution of system ( 1.5)-(1.8), then we rewrite this system as the following first order evolution equation (3.3) Ut=AdU, U(0) =U0, whereU0= (u0,u1,y0,y1)⊤∈ H. For all U= (u,v,y,z)⊤∈D(Ad), we have ℜ/a\}b∇acketle{tAdU,U/a\}b∇acket∇i}htH=−/integraldisplay Ωb|∇v|2dx≤0, which implies that Adis dissipative. Now, similar to Proposition 2.1 in [ 7], we can prove that there exists a unique solution U= (u,v,y,z)⊤∈D(Ad) of −AdU=F,∀F= (f1,f2,f3,f4)⊤∈ H. Then 0∈ρ(Ad) andAdis an isomorphism and since ρ(Ad) is open in C(see Theorem 6.7 (Chapter III) in [19]), we easily get R(λI−Ad) =Hfor a sufficiently small λ >0. This, together with the dissipativeness of Ad, imply that D(Ad) is dense in Hand that Adis m-dissipative in H(see Theorems 4.5, 4.6 in [ 22]). According to Lumer-Phillips theorem (see [ 22]), then the operator Adgenerates a C0-semigroup of contractions etAdin Hwhich gives the well-posedness of ( 3.3). Then, we have the following result: Theorem 3.1. For allU0∈ H, system ( 2.1) admits a unique weak solution U(t) =etAdU0∈C0(R+,H). Moreover, if U0∈D(A), then the system ( 2.1) admits a unique strong solution U(t) =etAdU0∈C0(R+,D(Ad))∩C1(R+,H). 3.2.Strong Stability. In this subsection, we will prove the strong stability of system ( 1.5)-(1.8). First, we fix the following notations /tildewideΩ = Ω−ωc,Γ1=∂ωc−∂Ω and Γ 0=∂ωc−Γ1. Letx0∈Rdandm(x) =x−x0and suppose that (see Figure 4) (GC) m·ν≤0 on Γ 0= (∂ωc)−Γ1. 17The main result of this section is the following theorem Theorem 3.2. Assume that (GC)holds and (SSC) /ba∇dblc/ba∇dbl∞≤min/braceleftigg 1 /ba∇dblm/ba∇dbl∞+d−1 2,1 /ba∇dblm/ba∇dbl∞+(d−1)Cp,ωc 2/bracerightigg , whereCp,ωcis the Poincarr´ e constant on ωc. Then, the C0−semigroup of contractions/parenleftbig etAd/parenrightbig is strongly stable inH; i.e. for all U0∈ H, the solution of (3.3)satisfies lim t→+∞/ba∇dbletAdU0/ba∇dblH= 0. Proof.First, let us prove that (3.4) ker( iλI−Ad) ={0},∀λ∈R. Since 0∈ρ(Ad), then we still need to show the result for λ∈R∗. Suppose that there exists a real number λ/\e}atio\slash= 0 and U= (u,v,y,z)⊤∈D(Ad), such that AdU=iλU. Equivalently, we have v=iλu, (3.5) div(∇u+b∇v)−cz=iλv, (3.6) z=iλy, (3.7) ∆y+cv=iλz. (3.8) Next, a straightforward computation gives 0 =ℜ/a\}b∇acketle{tiλU,U/a\}b∇acket∇i}htH=ℜ/a\}b∇acketle{tAdU,U/a\}b∇acket∇i}htH=−/integraldisplay Ωb|∇v|2dx, consequently, we deduce that (3.9) b∇v= 0 in Ω and ∇v=∇u= 0 in ωb. Inserting ( 3.5) in (3.6), then using the definition of c, we get (3.10) ∆u=−λ2uinωb. From (3.9) we get ∆ u= 0 inωband from ( 3.10) and the fact that λ/\e}atio\slash= 0, we get (3.11) u= 0 in ωb. Now, inserting ( 3.5) in (3.6), then using ( 3.9), (3.11) and the definition of c, we get (3.12)λ2u+∆u= 0 in/tildewideΩ, u= 0 in ωb⊂/tildewideΩ. Using Holmgren uniqueness theorem, we get (3.13) u= 0 in/tildewideΩ. It follows that (3.14) u=∂u ∂ν= 0 on Γ 1. Now, our aim is to show that u=y= 0 inωc. For this aim, inserting ( 3.5) and (3.7) in (3.6) and (3.8), then using (3.9), we get the following system λ2u+∆u−iλcy= 0 in Ω , (3.15) λ2y+∆y+iλcu= 0 in Ω , (3.16) u= 0 on ∂ωc, (3.17) y= 0 on Γ 0, (3.18) ∂u ∂ν= 0 on Γ 1. (3.19) 18Let us prove ( 3.4) by the following three steps: Step 1. The aim of this step is to show that (3.20)/integraldisplay Ωc|u|2dx=/integraldisplay Ωc|y|2dx. For this aim, multiplying ( 3.15) and (3.16) by ¯yand ¯urespectively, integrating over Ω and using Green’s formula, we get λ2/integraldisplay Ωu¯ydx−/integraldisplay Ω∇u·∇¯ydx−iλ/integraldisplay Ωc|y|2dx= 0, (3.21) λ2/integraldisplay Ωy¯udx−/integraldisplay Ω∇y·∇¯udx+iλ/integraldisplay Ωc|u|2dx= 0. (3.22) Adding ( 3.21) and (3.22), then taking the imaginary part, we get ( 3.20). Step 2. The aim of this step is to prove the following identity (3.23) −d/integraldisplay ωc|λu|2dx+(d−2)/integraldisplay ωc|∇u|2dx+/integraldisplay Γ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u ∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dΓ−2ℜ/parenleftbigg iλ/integraldisplay ωccy(m·∇¯u)dx/parenrightbigg = 0. For this aim, multiplying ( 3.15) by 2(m·∇¯u), integrating over ωcand taking the real part, we get (3.24) 2 ℜ/parenleftbigg λ2/integraldisplay ωcu(m·∇¯u)dx/parenrightbigg +2ℜ/parenleftbigg/integraldisplay ωc∆u(m·∇¯u)dx/parenrightbigg −2ℜ/parenleftbigg iλ/integraldisplay ωccy(m·∇¯u)dx/parenrightbigg = 0. Now, using the fact that u= 0 in∂ωc, we get (3.25) ℜ/parenleftbigg 2λ2/integraldisplay ωcu(m·∇¯u)dx/parenrightbigg =−d/integraldisplay ωc|λu|2dx. Using Green’s formula, we obtain (3.26)2ℜ/parenleftbigg/integraldisplay ωc∆u(m·∇¯u)dx/parenrightbigg =−2ℜ/parenleftbigg/integraldisplay ωc∇u·∇(m·∇¯u)dx/parenrightbigg +2ℜ/parenleftbigg/integraldisplay Γ0∂u ∂ν(m·∇¯u)dΓ/parenrightbigg = (d−2)/integraldisplay ωc|∇u|2dx−/integraldisplay ∂ωc(m·ν)|∇u|2dx+2ℜ/parenleftbigg/integraldisplay Γ0∂u ∂ν(m·∇¯u)dΓ/parenrightbigg . Using (3.17) and (3.19), we get (3.27)/integraldisplay ∂ωc(m·ν)|∇u|2dx=/integraldisplay Γ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u ∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dΓ andℜ/parenleftbigg/integraldisplay Γ0∂u ∂ν(m·∇¯u)dΓ/parenrightbigg =/integraldisplay Γ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u ∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dΓ. Inserting ( 3.27) in (3.26), we get (3.28) 2 ℜ/parenleftbigg/integraldisplay ωc∆u(m·∇¯u)dx/parenrightbigg = (d−2)/integraldisplay ωc|∇u|2dx+/integraldisplay Γ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u ∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dΓ. Inserting ( 3.25) and (3.28) in (3.24), we get ( 3.23). Step 3. In this step, we prove ( 3.4). Multiplying ( 3.15) by (d−1)u, integrating over ωcand using ( 3.17), we get (3.29) ( d−1)/integraldisplay ωc|λu|2dx+(1−d)/integraldisplay ωc|∇u|2dx−ℜ/parenleftbigg iλ(d−1)/integraldisplay ωccy¯udx/parenrightbigg = 0. Adding ( 3.23) and (3.29), we get /integraldisplay ωc|λu|2dx+/integraldisplay ωc|∇u|2dx=/integraldisplay Γ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u ∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dΓ−2ℜ/parenleftbigg iλ/integraldisplay ωccy(m·∇¯u)dx/parenrightbigg −ℜ/parenleftbigg iλ(d−1)/integraldisplay ωccy¯udx/parenrightbigg = 0. Using (GC), we get (3.30)/integraldisplay ωc|λu|2dx+/integraldisplay ωc|∇u|2dx≤2|λ|/integraldisplay ωc|c||y||m·∇u|dx+|λ|(d−1)/integraldisplay ωc|c||y||u|dx. 19Using Young’s inequality and ( 3.20), we get (3.31) 2 |λ|/integraldisplay ωc|c||y||m·∇u|dx≤ /ba∇dblm/ba∇dbl∞/ba∇dblc/ba∇dbl∞/integraldisplay ωc/parenleftbig |λu|2+|∇u|2/parenrightbig dx and (3.32) |λ|(d−1)/integraldisplay ωc|c(x)||y||u|dx≤(d−1)/ba∇dblc/ba∇dbl∞ 2/integraldisplay ωc|λu|2dx+(d−1)/ba∇dblc/ba∇dbl∞Cp,ωc 2/integraldisplay ωc|∇u|2dx. Inserting ( 3.32) in (3.30), we get /parenleftbigg 1−/ba∇dblc/ba∇dbl∞/parenleftbigg /ba∇dblm/ba∇dbl∞+d−1 2/parenrightbigg/parenrightbigg/integraldisplay ωc|λu|2dx+/parenleftbigg 1−/ba∇dblc/ba∇dbl∞/parenleftbigg /ba∇dblm/ba∇dbl∞+(d−1)Cp,ωc 2/parenrightbigg/parenrightbigg/integraldisplay ωc|∇u|2dx≤0. Using (SSC) and (3.20) in the above estimation, we get (3.33) u= 0 and y= 0 in ωc. In order to complete this proof, we need to show that y= 0 in/tildewideΩ. For this aim, using the definition of the function cin/tildewideΩ and using the fact that y= 0 inωc, we get (3.34)λ2y+∆y= 0 in/tildewideΩ, y= 0 on ∂/tildewideΩ, ∂y ∂ν= 0 on Γ 1. Now, using Holmgren uniqueness theorem, we obtain y= 0 in/tildewideΩ and consequently ( 3.4) holds true. Moreover, similar to Lemma 2.5 in [ 7], we can prove R(iλI− Ad) =H,∀λ∈R. Finally, by using the closed graph theorem of Banach and Theorem A.2, we conclude the proof of this Theorem. /square Let us notice that, under the sole assumptions ( GC) and (SSC), the polynomial stability of system ( 1.5)-(1.8) is an open problem. Appendix A.Some notions and stability theorems In order to make this paper more self-contained, we recall in this sh ort appendix some notions and stability results used in this work. Definition A.1. Assume that Ais the generator of C0−semigroup of contractions/parenleftbig etA/parenrightbig t≥0on a Hilbert space H. TheC0−semigroup/parenleftbig etA/parenrightbig t≥0is said to be (1) Strongly stable if lim t→+∞/ba∇dbletAx0/ba∇dblH= 0,∀x0∈H. (2) Exponentially (or uniformly) stable if there exists two positive co nstantsMandεsuch that /ba∇dbletAx0/ba∇dblH≤Me−εt/ba∇dblx0/ba∇dblH,∀t >0,∀x0∈H. (3) Polynomially stable if there exists two positive constants Candαsuch that /ba∇dbletAx0/ba∇dblH≤Ct−α/ba∇dblAx0/ba∇dblH,∀t >0,∀x0∈D(A). /square To show the strong stability of the C0-semigroup/parenleftbig etA/parenrightbig t≥0we rely on the following result due to Arendt-Batty [9]. Theorem A.2. Assume that Ais the generatorof a C 0−semigroup of contractions/parenleftbig etA/parenrightbig t≥0on a Hilbert space H. IfAhas no pure imaginary eigenvalues and σ(A)∩iRis countable, where σ(A) denotes the spectrum of A, then the C0-semigroup/parenleftbig etA/parenrightbig t≥0is strongly stable. /square Concerning the characterization of polynomial stability stability of a C0−semigroup of contraction/parenleftbig etA/parenrightbig t≥0we rely on the following result due to Borichev and Tomilov [ 12] (see also [ 11] and [21]) 20Theorem A.3. Assume that Ais the generator of a strongly continuous semigroup of contractio ns/parenleftbig etA/parenrightbig t≥0 onH. IfiR⊂ρ(A), then for a fixed ℓ >0 the following conditions are equivalent (A.1) limsup λ∈R,|λ|→∞1 |λ|ℓ/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble L(H)<∞, (A.2) /ba∇dbletAU0/ba∇dbl2 H≤C t2 ℓ/ba∇dblU0/ba∇dbl2 D(A),∀t >0, U0∈D(A),for some C >0. /square References [1] F. Abdallah, M. Ghader, and A. Wehbe. Stability results o f a distributed problem involving Bresse system with histor y and/or Cattaneo law under fully Dirichlet or mixed boundary condit ions.Math. Methods Appl. Sci. , 41(5):1876–1907, 2018. [2] M. Akil and H. Badawi. The influence of the physical coeffici ents of a Bresse system with one singular local viscous dampi ng in the longitudinal displacement on its stabilization. Evolution Equations & Control Theory , 2022. [3] M. Akil, H. Badawi, S. Nicaise, and A. Wehbe. On the stabil ity of Bresse system with one discontinuous local internal Kelvin–Voigt damping on the axial force. Zeitschrift f¨ ur angewandte Mathematik und Physik , 72(3):126, May 2021. [4] M. Akil, H. Badawi, S. Nicaise, and A. Wehbe. Stability re sults of coupled wave models with locally memory in a past his tory framework via nonsmooth coefficients on the interface. Mathematical Methods in the Applied Sciences , 44(8):6950–6981, 2021. [5] M. Akil, H. Badawi, and A. Wehbe. Stability results of a si ngular local interaction elastic/viscoelastic coupled wa ve equations with time delay. Communications on Pure & Applied Analysis , 20(9):2991–3028, 2021. [6] M. Akil, Y. Chitour, M. Ghader, and A. Wehbe. Stability an d exact controllability of a Timoshenko system with only one fractional damping on the boundary. Asymptotic Analysis , 119:221–280, 2020. 3-4. [7] M. Akil, I. Issa, and A. Wehbe. A N-dimensional elastic/v iscoelastic transmission problem with Kelvin-Voigt dampi ng and non smooth coefficient at the interface, 2021. [8] F. Alabau-Boussouira and M. L´ eautaud. Indirect contro llability of locally coupled wave-type systems and applica tions.J. Math. Pures Appl. (9) , 99(5):544–576, 2013. [9] W. Arendt and C. J. K. Batty. Tauberian theorems and stabi lity of one-parameter semigroups. Trans. Amer. Math. Soc. , 306(2):837–852, 1988. [10] M. Bassam, D. Mercier, S. Nicaise, and A. Wehbe. Polynom ial stability of the Timoshenko system by one boundary dampi ng. Journal of Mathematical Analysis and Applications , 425(2):1177 – 1203, 2015. [11] C. J. K. Batty and T. Duyckaerts. Non-uniform stability for bounded semi-groups on Banach sp aces.J. Evol. Equ. , 8(4):765– 780, 2008. [12] A. Borichev and Y. Tomilov. Optimal polynomial decay of functions and operator semigroups. Math. Ann. , 347(2):455–478, 2010. [13] N. Burq and C. Sun. Decay rates for Kelvin-Voigt damped w ave equations II: The geometric control condition. Proc. Amer. Math. Soc. , 150(3):1021–1039, 2022. [14] L. H. Fatori and R. N. Monteiro. The optimal decay rate fo r a weak dissipative Bresse system. Applied Mathematics Letters , 25(3):600 – 604, 2012. [15] L. H. Fatori, R. N. Monteiro, and H. D. F. Sare. The Timoshenko system with history and Cattaneo law .Applied Mathematics and Computation , 228:128–140, Feb. 2014. [16] S. Gerbi, C. Kassem, A. Mortada, and A. Wehbe. Exact cont rollability and stabilization of locally coupled wave equa tions: theoretical results. Z. Anal. Anwend. , 40(1):67–96, 2021. [17] A. Hayek, S. Nicaise, Z. Salloum, and A. Wehbe. A transmi ssion problem of a system of weakly coupled wave equations wi th Kelvin-Voigt dampings and non-smooth coefficient at the inte rface.SeMA J. , 77(3):305–338, 2020. [18] C. Kassem, A. Mortada, L. Toufayli, and A. Wehbe. Local i ndirect stabilization of N-d system of two coupled wave equa tions under geometric conditions. C. R. Math. Acad. Sci. Paris , 357(6):494–512, 2019. [19] T. Kato. Perturbation Theory for Linear Operators . Springer Berlin Heidelberg, 1995. [20] K. Liu and B. Rao. Exponential stability for the wave equ ations with local Kelvin-Voigt damping. Z. Angew. Math. Phys. , 57(3):419–432, 2006. [21] Z. Liu and B. Rao. Characterization of polynomial decay rate for the solution of linear evolution equation. Z. Angew. Math. Phys., 56(4):630–644, 2005. [22] A. Pazy. Semigroups of linear operators and applications to partial differential equations , volume 44 of Applied Mathematical Sciences . Springer-Verlag, New York, 1983. [23] L. Tebou. Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping. Discrete Contin. Dyn. Syst. , 36(12):7117–7136, 2016. [24] A. Wehbe, I. Issa, and M. Akil. Stability results of an el astic/viscoelastic transmission problem of locally coupl ed waves with non smooth coefficients. Acta Applicandae Mathematicae , 171(1):23, Feb 2021. 21

2022-03-03

In this paper, we study the direct/indirect stability of locally coupled wave equations with local Kelvin-Voigt dampings/damping and by assuming that the supports of the dampings and the coupling coefficients are disjoint. First, we prove the well-posedness, strong stability, and polynomial stability for some one dimensional coupled systems. Moreover, under some geometric control condition, we prove the well-posedness and strong stability in the multi-dimensional case.

Stability results of locally coupled wave equations with local Kelvin-Voigt damping: Cases when the supports of damping and coupling coefficients are disjoint

2203.01632v1

arXiv:2009.05372v1 [math.AP] 10 Sep 2020Blow – up results for semilinear damped wave equations in Einstein – de Sitter spacetime Alessandro Palmieri Abstract We prove by using an iteration argument some blow-up results for a semilinear damped wave equation in generalized Einstein-de Sitter spacetime with a time-de pendent coefficient for the damping term and power nonlinearity. Then, we conjecture an expression for t he critical exponent due to the main blow- up results, which is consistent with many special cases of th e considered model and provides a natural generalization of Strauss exponent. In the critical case, w e consider a non-autonomous and parameter dependent Cauchy problem for a linear ODE of second order, wh ose explicit solutions are determined by means of special functions’ theory. Keywords Semilinear damped wave equation, Einstein – de Sitter space time, power nonlinearity, generalized Strauss exponent, lifespan estimates, modifie d Bessel functions AMS Classification (2020) Primary: 35B44, 35L05, 35L71; Secondary: 35B33, 33C10 1 Introduction In recent years, the wave equation in Einstein – de Sitter spa cetime has been considered in [9, 10] in the linear case and in [11, 12, 27] in the semilinear case. Let us c onsider the semilinear wave equation with power nonlinearity in a generalized Einstein – de Sitter spacetime , that is, the equation with singular coefficients ϕtt−t−2k∆ϕ+2t−1ϕt=|ϕ|p, (1) wherek∈[0,1)andp >1. This model is the semilinear wave equation in Einstein – de S itter spacetime with power nonlinearity for k= 2/3andn= 3. It has been proved in [12, 27] that for 1<p/lessorequalslantmax/braceleftbig p0/parenleftbig k,n+2 1−k/parenrightbig ,p1(k,n)/bracerightbig a local in time solution to the corresponding Cauchy problem (with initial data prescribed at the initial time t= 1) blows up in finite time, provided that the initial data fulfil l certain integral sign conditions. More specifically, in [12] the subcritical case for (1) is investi gated, while in [27] the critical case and the upper bound estimates for the lifespan are studied. Here and throu ghout the paper p0(k,n)is the positive root of the quadratic equation /parenleftBig n−1 2−k 2(1−k)/parenrightBig p2−/parenleftBig n+1 2+3k 2(1−k)/parenrightBig p−1 = 0, (2) when the coefficient for p2is not positive, we set formally p0(k,n).=∞, while p1(k,n).= 1+2 (1−k)n. (3) Note thatp1(k,n)is related to the Fujita exponent pFuj(n).= 1+2 n. Indeed, according to this notation, it holdsp1(k,n) =pFuj/parenleftbig (1−k)n/parenrightbig andp1(0,n) =pFuj(n). On the other hand, p0(k,n)is a generalization of the Strauss exponent for the classical semilinear wave equa tion, sincep0(0,n) =pStr(n), wherepStr(n)is the positive root of the quadratic equation (n−1)p2−(n+1)p−2 = 0. In this paper, we generalize the model (1) with a general mult iplicative constant µfor the damping term. More specifically, we investigate the blow – up dynamic for th e Cauchy problem utt−t−2k∆u+µt−1ut=|u|px∈Rn, t∈(1,T), u(1,x) =εu0(x) x∈Rn, ut(1,x) =εu1(x) x∈Rn,(4) 1wherek∈[0,1),p>1,µis the nonnegative multiplicative constant in the time – dep endent coefficient for the damping term and ε >0describes the size of the initial data. Let us point out that t he not damped caseµ= 0can be treated as well via our approach. More precisely, we will focus on proving blow-up results whe never the exponent pbelongs to the range 1<p/lessorequalslantmax/braceleftbig p0/parenleftbig k,n+µ 1−k/parenrightbig ,p1(k,n)/bracerightbig , clearly, under suitable sign assumptions for u0,u1. According to (2), the shift p0/parenleftbig k,n+µ 1−k/parenrightbig ofp0(k,n)is nothing but the positive root to the quadratic equation /parenleftBig n−1 2+µ−k 2(1−k)/parenrightBig p2−/parenleftBig n+1 2+µ+3k 2(1−k)/parenrightBig p−1 = 0. (5) Therefore, the critical exponent p0/parenleftbig k,n+µ 1−k/parenrightbig for (4) is obtained by the corresponding exponent in the not damped case via a formal shift in the dimension of magnitudeµ 1−k. Let us provide an overview on the methods that we are going to u se to prove the main results in this paper. In the subcritical case 1< p <max/braceleftbig p0/parenleftbig k,n+µ 1−k/parenrightbig ,p1(k,n)/bracerightbig , we employ a standard iteration argument based on a multiplier argument (see also [17, 18, 19, 21] for f urther details on the multiplier argument). This approach is based on the employment of two time – dependent fu nctionals related to a local solution uto (4) and generalizes the method from [34] for the semilinear w ave equation with scale – invariant damping. The first functional is the space average of uand its dynamic will be considered for the iterative argumen t. On the other hand, we will work with a positive solution of the adjoint linear equation in order to prove the positivity of the second auxiliary functional. Hence, t his second functional will also provide a first lower bound estimate for the first functional, allowing us to begin with the iteration procedure. In the critical case we should sharpen our iteration frame by considering a d ifferent time – dependent functional, so that a slicing procedure may be applied. In comparison to what hap pens in the subcritical case, a more precise analysis of the adjoint linear equation is necessary in the c ritical case p=p0/parenleftbig k,n+µ 1−k/parenrightbig . This approach follows the one developed in [27] which is in turn a generaliz ation of the ideas introduced by Wakasa and Yordanov in [36, 37] an developed in different frameworks in [ 29, 30, 21, 3, 4]. Whereas in the other critical casep=p1(k,n), we can still work with the space average of a local in time sol ution as functional, although a slicing procedure has to be applied in order to deal with log arithmic factors in the lower bound estimates. 1.1 Notations Throughout this paper we use the following notations: φk(t).=t1−k 1−kdenotes a distance function produced by the speed of propagation ak(t) =t−k, while the amplitude of the light cone is given by the functio n Ak(t).=/integraldisplayt 1τ−kdτ=φk(t)−φk(1); (6) the ball in Rnwith radius Raround the origin is denoted BR;f/lessorsimilargmeans that there exists a positive constantCsuch thatf/lessorequalslantCgand, similarly, for f/greaterorsimilarg;IνandKνdenote the modified Bessel function of first and second kind of order ν, respectively; finally, as in the introduction, p0(k,n)is the positive solution to (2) andp1(k,n)is defined by (3). 1.2 Main results Before stating the main theorems, let us introduce a suitabl e notion of energy solution to the semilinear Cauchy problem (4). Definition 1.1. Letu0∈H1(Rn)andu1∈L2(Rn). We say that u∈C/parenleftbig [1,T),H1(Rn)/parenrightbig ∩C1/parenleftbig [1,T),L2(Rn)/parenrightbig ∩Lp loc/parenleftbig [1,T)×Rn/parenrightbig is an energy solution to (4) on [1,T)ifufulfillsu(1,·) =εu0inH1(Rn)and the integral relation /integraldisplay Rn∂tu(t,x)ψ(t,x)dx−ε/integraldisplay Rnu1(x)ψ(1,x)dx−/integraldisplayt 1/integraldisplay Rn∂tu(s,x)ψs(s,x)dxds +/integraldisplayt 1/integraldisplay Rns−2k∇u(s,x)·∇ψ(s,x)dxds+/integraldisplayt 1/integraldisplay Rnµs−1∂tu(s,x)ψ(s,x)dxds =/integraldisplayt 1/integraldisplay Rn|u(s,x)|pψ(s,x)dxds (7) for any test function ψ∈C∞ 0([1,T)×Rn)and anyt∈(1,T). 2We point out that performing a further step of integration by parts in (7), we find the integral relation /integraldisplay Rn∂tu(t,x)ψ(t,x)dx−/integraldisplay Rnu(t,x)ψs(t,x)dx+/integraldisplay Rnµt−1u(t,x)ψ(t,x)dx −ε/integraldisplay Rnu1(x)ψ(1,x)dx+ε/integraldisplay Rnu0(x)ψs(1,x)dx−ε/integraldisplay Rnµu0(x)ψ(1,x)dx +/integraldisplayt 1/integraldisplay Rnu(s,x)/parenleftbig ψss(s,x)−s−2k∆ψ(s,x)−µs−1ψs(s,x)+µs−2ψ(s,x)/parenrightbig dxds =/integraldisplayt 1/integraldisplay Rn|u(s,x)|pψ(s,x)dxds (8) for anyψ∈C∞ 0([1,T)×Rn)and anyt∈(1,T). Remark 1.Let us point out that if the Cauchy data have compact support, saysuppuj⊂BRforj= 0,1 and for some R>0, then, for any t∈(1,T)a local solution uto (4) the support condition suppu(t,·)⊂BR+Ak(t) is satisfied, where Akis defined by (6). Consequently, in Definition 1.1 it is possib le to consider test functions which are not compactly supported, i.e., ψ∈C∞([1,T)×Rn). Theorem 1.2 (Subcritical case) .Letµ/greaterorequalslant0and let the exponent of the nonlinear term psatisfy 1<p<max/braceleftBig p0/parenleftbig k,n+µ 1−k/parenrightbig ,p1(k,n)/bracerightBig . Let us assume that u0∈H1(Rn)andu1∈L2(Rn)are nonnegative, nontrivial and compactly supported functions with supports contained in BRfor someR>0. Let u∈C/parenleftbig [1,T),H1(Rn)/parenrightbig ∩C1/parenleftbig [1,T),L2(Rn)/parenrightbig ∩Lp loc/parenleftbig [1,T)×Rn/parenrightbig be an energy solution to (4)according to Definition 1.1 with lifespan T=T(ε)and satisfying the support condition suppu(t,·)⊂BAk(t)+Rfor anyt∈(1,T). Then, there exists a positive constant ε0=ε0(u0,u1,n,p,k,µ,R )such that for any ε∈(0,ε0]the energy solutionublows up in finite time. Moreover, the upper bound estimate for the lifespan T(ε)/lessorequalslant/braceleftBigg Cε−p(p−1) θ(n,k,µ,p ) ifp<p0/parenleftbig k,n+µ 1−k/parenrightbig , Cε−(2 p−1−(1−k)n)−1 ifp<p1(k,n),(9) holds, where the positive constant Cis independent of εand θ(n,k,µ,p).= 1−k+/parenleftBig (1−k)n+1 2+µ+3k 2/parenrightBig p−/parenleftBig (1−k)n−1 2+µ−k 2/parenrightBig p2. In order to properly state the results in the critical case, l et us explicit provide the threshold for µwhich yields the transition from a dominant p0/parenleftbig k,n+µ 1−k/parenrightbig to the case in which p1(k,n)is the highest exponent. Due to the fact that p0/parenleftbig k,n+µ 1−k/parenrightbig is the biggest solution of (5), we have that p1(k,n)>p0/parenleftbig k,n+µ 1−k/parenrightbig if and only if /parenleftBig n−1 2+µ−k 2(1−k)/parenrightBig p1(k,n)2−/parenleftBig n+1 2+µ+3k 2(1−k)/parenrightBig p1(k,n)−1>0. By straightforward computations, it follows that p1(k,n)>p0/parenleftbig k,n+µ 1−k/parenrightbig forµ>µ0(k,n), where µ0(k,n).=(1−k)2n2+(1−k)(1+2k)n+2 n(1−k)+2. (10) Note that for k= 0the splitting value µ0(k,n)does coincide with the one for the semilinear wave equation with scale – invariant damping in the flat case from the work [1 6]. Theorem 1.3 (Critical case: part I) .Let0/lessorequalslantµ/lessorequalslantµ0(k,n)such thatµ/lessorequalslantkorµ/greaterorequalslant2−k. We consider p=p0/parenleftbig k,n+µ 1−k/parenrightbig . Let us assume that u0∈H1(Rn)andu1∈L2(Rn)are nonnegative, nontrivial and compactly supported functions with supports contained in BRfor someR>0. Let u∈C/parenleftbig [1,T),H1(Rn)/parenrightbig ∩C1/parenleftbig [1,T),L2(Rn)/parenrightbig ∩Lp loc/parenleftbig [1,T)×Rn/parenrightbig 3be an energy solution to (4)according to Definition 1.1 with lifespan T=T(ε)and satisfying the support condition suppu(t,·)⊂BAk(t)+Rfor anyt∈(1,T). Then, there exists a positive constant ε0=ε0(u0,u1,n,p,k,µ,R )such that for any ε∈(0,ε0]the energy solutionublows up in finite time. Moreover, the upper bound estimate for the lifespan T(ε)/lessorequalslantexp/parenleftBig Cε−p(p−1)/parenrightBig holds, where the positive constant Cis independent of ε. Remark 2.It seems that the assumption in Theorem 1.3 for the multiplic ative constant µ/lessorequalslantkorµ/greaterorequalslant2−k is technical, since it is due to the method we are going to appl y for the proof. Theorem 1.4 (Critical case: part II) .Letµ/greaterorequalslantµ0(k,n)andp=p1(k,n). Let us assume that u0∈H1(Rn) andu1∈L2(Rn)are nonnegative, nontrivial and compactly supported functio ns with supports contained in BRfor someR>0. Let u∈C/parenleftbig [1,T),H1(Rn)/parenrightbig ∩C1/parenleftbig [1,T),L2(Rn)/parenrightbig ∩Lp loc/parenleftbig [1,T)×Rn/parenrightbig be an energy solution to (4)according to Definition 1.1 with lifespan T=T(ε)and satisfying the support condition suppu(t,·)⊂BAk(t)+Rfor anyt∈(1,T). Then, there exists a positive constant ε0=ε0(u0,u1,n,p,k,µ,R )such that for any ε∈(0,ε0]the energy solutionublows up in finite time. Moreover, the upper bound estimate for the lifespan T(ε)/lessorequalslantexp/parenleftBig Cε−(p−1)/parenrightBig holds, where the positive constant Cis independent of ε. The remaining part of the paper is organized as follows: the p roof of the result in the subcritical case (cf. Theorem 1.2) is carried out in Section 2; in Section 3 we prove Theorem 1.3 by generalizing the approach introduced in [36]; finally, we show the proof of Theorem 1.4 i n Section 4 via a standard slicing procedure. 2 Subcritical case In this section we are going to prove Theorem 1.2. Let ube a local in time solution to (4) and let us assume that the assumptions from the statement of Theorem 1.2 on pand on the data are fulfilled. We will follow the multiplier approach introduced by [20] and then improve d by [34], to derive a suitable iteration frame for the time – dependent functional U0(t).=/integraldisplay Rnu(t,x)dx. (11) In order to obtain a first lower bound estimate for U0we will introduce a second time – dependent functional, following the main ideas of the pioneering paper [38] and ada pting them to the case with time – depend coefficients as in [13, 12, 34, 31]. The section is organized as follows: in Section 2.1 we determ ine a suitable positive solution to the adjoint homogeneous linear equation with separate variabl es, then, we use this function to derive a lower bound estimate for U0in Section 2.3; in Sections 2.2 and 2.4 the derivation of the i teration frame and its application in an iterative argument are dealt with, respec tively. 2.1 Solution of the adjoint homogeneous linear equation In this section, we shall determine a particular positive so lution to the adjoint homogeneous linear equation Ψss−s−2k∆Ψ−µs−1Ψs+µs−2Ψ = 0. (12) First of all, we recall the remarkable function ϕ(x).=/braceleftBigg/integraltext Sn−1ex·ωdσωifn/greaterorequalslant2, coshx ifn= 1,(13) introduced in [38] for the study of the critical semilinear w ave equation. The main properties of this function that will used throughout this paper are the following: ϕis a positive and smooth function that satisfies ∆ϕ=ϕand asymptotically behaves like cn|x|−n−1 2e|x|as|x| → ∞ . 4If we look for a solution to (12) with separate variables, tha t is, we consider the ansatz Ψ(s,x) =̺(s)ϕ(x), then, it suffices to find a positive solution to the ODE ̺′′−s−2k̺−µs−1̺′+µs−2̺= 0. (14) We perform the change of variable τ=φk(s). By using ̺′=t−kd̺ dτ, ̺′′=t−2kd2̺ dτ2−kt−1−kd̺ dτ, it follows with straightforward computations that ̺solves (14) if and only if d2̺ dτ2−k+µ 1−k1 τd̺ dτ+/parenleftbiggµ (1−k)21 τ2−1/parenrightbigg ̺= 0. (15) To further simplify the previous equation, we carry out the t ransformation ̺(τ) =τσζ(τ), whereσ.=1+µ 2(1−k). Hence, using d̺ dτ(τ) =στσ−1ζ(τ)+τσdζ dτ(τ),d2̺ dτ2=σ(σ−1)τσ−2ζ(τ)+2στσ−1dζ dτ(τ)+τσd2ζ dτ2(τ), we get that ̺is a solution to (15) if and only if ζsolves τ2d2ζ dτ2−/parenleftbigg 2σ−k+µ 1−k/parenrightbigg τdζ dτ+/bracketleftbigg σ/parenleftbigg σ−1−k+µ 1−k/parenrightbigg +µ (1−k)2−τ2/bracketrightbigg ζ= 0. (16) Due to the choice of the parameter σ, equation (16) is nothing but a modified Bessel equation of or der γ.=µ−1 2(1−k), that is, (16) can be rewritten as τ2d2ζ dτ2−τdζ dτ−(γ2+τ2)ζ= 0. If we pick the modified Bessel function of the second kind Kγas solution to the previous equation, then, up to a negligible multiplicative constant, we found ρ(s).=s1+µ 2Kγ/parenleftbig φk(s)/parenrightbig (17) as a positive solution to (14) and, in turn, Ψ(s,x).=ρ(s)ϕ(x) =s1+µ 2Kγ/parenleftbig φk(s)/parenrightbig ϕ(x) (18) as a positive solution of the adjoint equation (12). In the next sections, we will need to employ the asymptotic be havior of the function ̺=̺(t)fort→ ∞. SinceKγ(z) =/radicalbig π/(2z)/parenleftbig e−z+O(z−1)/parenrightbig asz→ ∞ (cf. [23]), then, the following asymptotic estimate holds ̺(t) =/radicalbiggπ 2tk+µ 2e−φk(t)/parenleftbig 1+O(t−1+k)/parenrightbig fort→ ∞. (19) The solution Ψof the adjoint equation (12) that we determined in this secti on will be employed in Section 2.3 to introduce a second time – dependent functiona l with the purpose to establish a first lower bound estimate for U0. 2.2 Derivation of the iteration frame In this section we are going to determine the iteration frame for the functional U0=U0(t)defined in (11). Let us choose as test function ψ=ψ(s,x)in the integral relation (7) such that ψ= 1on the forward cone {(s,x)∈[1,t]×Rn:|x|/lessorequalslantR+Ak(s)}. Then, /integraldisplay Rn∂tu(t,x)dx−ε/integraldisplay Rnu1(x)dx+/integraldisplayt 1/integraldisplay Rnµs−1∂tu(s,x)dxds=/integraldisplayt 1/integraldisplay Rn|u(s,x)|pdxds which can be rewritten as U′ 0(t)−U′ 0(1)+/integraldisplayt 1µs−1U′ 0(s)ds=/integraldisplayt 1/integraldisplay Rn|u(s,x)|pdxds. 5Differentiating the last identity with respect to t, we get U′′ 0(t)+µt−1U′ 0(t) =/integraldisplay Rn|u(t,x)|pdx. Multiplying the previous equation by tµ, it follows tµU′′ 0(t)+µtµ−1U′ 0(t) =d dt/parenleftbig tµU′ 0(t)/parenrightbig =tµ/integraldisplay Rn|u(t,x)|pdx. Integrating twice this relation over [1,t], we find U0(t) =U0(1)+U′ 0(1)/integraldisplayt 1τ−µdτ+/integraldisplayt 1τ−µ/integraldisplayτ 1sµ/integraldisplay Rn|u(s,x)|pdxdsdτ. (20) On the one hand , from (20) we derive the lower bound estimate U0(t)/greaterorsimilarε, (21) where the unexpressed positive multiplicative constant de pends onu0,u1due to the nonnegativeness of u0,u1 andU(j)(1) =ε/integraltext Rnuj(x)dxforj∈ {0,1}. On the other hand, we obtain the estimate U0(t)/greaterorequalslant/integraldisplayt 1τ−µ/integraldisplayτ 1sµ/integraldisplay Rn|u(s,x)|pdxdsdτ (22) /greaterorsimilar/integraldisplayt 1τ−µ/integraldisplayτ 1sµ(R+Ak(s))−n(p−1)(U0(s))pdsdτ, where in the second step we applied Jensen’s inequality and t he support property for u(s,·). Therefore, we proved the following iteration frame for U0 U0(t)/greaterorequalslantC/integraldisplayt 1τ−µ/integraldisplayτ 1sµ−(1−k)n(p−1)(U0(s))pdsdτ (23) for a suitable positive constant C=C(n,p,k)and fort/greaterorequalslant1. In Section 2.2 we will employ (23) to derive iteratively a sequence of lower bound estimates for U0. However, we shall first derive in Section 2.3 another lower bound estimate for U0that will provide, together with (21), the starting point fo r the iteration procedure. 2.3 First lower bound estimate for the functional LetΨ = Ψ(t,x)be the function defined by (18). Since this function is smooth and positive, by applying the integral relation (8) to Ψand using the fact that Ψsolves the adjoint equation (12), we get 0/lessorequalslant/integraldisplayt 1/integraldisplay Rn|u(s,x)|pΨ(s,x)dxds =/integraldisplay Rn∂tu(t,x)Ψ(t,x)dx−/integraldisplay Rnu(t,x)Ψs(t,x)dx+/integraldisplay Rnµt−1u(t,x)Ψ(t,x)dx −ε/integraldisplay Rn(̺(1)u1(x)+(µ̺(1)−̺′(1))u0(x))ϕ(x)dx. If we introduce the auxiliary functional U1(t).=/integraldisplay Rnu(t,x)Ψ(t,x)dx, (24) then, from the last estimate we have U′ 1(t)−2̺′(t) ̺(t)U1(t)+µt−1U1(t)/greaterorequalslantε/integraldisplay Rn/parenleftbig ̺(1)u1(x)+(µ̺(1)−̺′(1))u0(x)/parenrightbig ϕ(x)dx, (25) where we applied the relation U′ 1(t) =/integraldisplay Rn∂tu(t,x)Ψ(t,x)dx+/integraldisplay Rnu(t,x)ψs(t,x)dx=/integraldisplay Rn∂tu(t,x)Ψ(t,x)dx+̺′(t) ̺(t)U1(t). 6Let compute more explicitly the term on the right – hand side o f (25) and show its positiveness. By using the recursive identity K′ γ(z) =−Kγ+1(z)+γ zKγ(z) for the derivative of the modified Bessel function of the seco nd kind and γ=µ−1 2(1−k), it follows ̺′(t) =1+µ 2tµ−1 2Kγ/parenleftbig φk(t)/parenrightbig +t1+µ 2−kK′ γ/parenleftbig φk(t)/parenrightbig =1+µ 2tµ−1 2Kγ/parenleftbig φk(t)/parenrightbig +t1+µ 2−k/parenleftBig −Kγ+1/parenleftbig φk(t)/parenrightbig +µ−1 2t−1+kKγ/parenleftbig φk(t)/parenrightbig/parenrightBig =µtµ−1 2Kγ/parenleftbig φk(t)/parenrightbig −t1+µ 2−kKγ+1/parenleftbig φk(t)/parenrightbig . In particular, the following relations hold µ̺(1)−̺′(1) = K γ+1/parenleftbig φk(1)/parenrightbig >0, ̺(1) = K γ/parenleftbig φk(1)/parenrightbig >0, so that we may rewrite (25) as U′ 1(t)−2̺′(t) ̺(t)U1(t)+µt−1U1(t)/greaterorequalslantε/integraldisplay Rn/parenleftbig Kγ/parenleftbig φk(1)/parenrightbig u1(x)+Kγ+1/parenleftbig φk(1)/parenrightbig u0(x)/parenrightbig ϕ(x)dx /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright.=Ik,µ[u0,u1]. (26) Multiplying (26) by tµ/̺2(t), we have d dt/parenleftbiggtµ ̺2(t)U1(t)/parenrightbigg =tµ ̺2(t)U′ 1(t)−2̺′(t) ̺3(t)tµU1(t)+µtµ−11 ̺2(t)U1(t)/greaterorequalslantεIk,µ[u0,u1]tµ ̺2(t). Integrating the previous inequality over [1,t]and using the sign assumption on u0, we get U1(t)/greaterorequalslant̺2(t)t−µ ̺2(1)U1(1)+εIk,µ[u0,u1]̺2(t) tµ/integraldisplayt 1sµ ̺2(s)ds /greaterorequalslantεIk,µ[u0,u1]̺2(t) tµ/integraldisplayt 1sµ ̺2(s)ds. Thanks to (19), there exists T0=T0(k,µ)>1such that U1(t)/greaterorsimilarεIk,µ[u0,u1]tke−2φk(t)/integraldisplayt T0s−ke2φk(s)ds fort/greaterorequalslantT0. Consequently, for for t/greaterorequalslant2T0, shrinking the domain of integration in the last inequality , we have U1(t)/greaterorsimilarεIk,µ[u0,u1]tke−2φk(t)/integraldisplayt t/2s−ke2φk(s)ds= 2−1εIk,µ[u0,u1]tke−2φk(t)/parenleftBig e2φk(t)−e2φk(t 2)/parenrightBig = 2−1εIk,µ[u0,u1]tk/parenleftBig 1−e2φk(t 2)−2φk(t)/parenrightBig = 2−1εIk,µ[u0,u1]tk/parenleftBig 1−e−2 1−k(1−2k−1)t1−k/parenrightBig /greaterorequalslant2−1εIk,µ[u0,u1]tk/parenleftBig 1−e−2 1−k(21−k−1)T1−k 0/parenrightBig /greaterorsimilarεtk. (27) By repeating exactly the same computations as in [28, Sectio n 3] (which are completely independent of the amplitude function Ak), we obtain /integraldisplay BR+Ak(t)(Ψ(t,x))p′dx= (̺(t))p′/integraldisplay BR+Ak(t)(ϕ(x))p′dx/lessorsimilar(̺(t))p′ep′(R+Ak(t))(R+Ak(t))n−1−n−1 2p′. Therefore, by using (19), for t/greaterorequalslantT0we get /integraldisplay BR+Ak(t)(Ψ(t,x))p′dx/lessorsimilarep′(R−φk(1))tk+µ 2p′(R+Ak(t))n−1−n−1 2p′ /lessorsimilart(1−k)(n−1)+[k+µ 2−(1−k)n−1 2]p′. (28) Then, combining Hölder’s inequality, (27) and (28), it foll ows /integraldisplay Rn|u(t,x)|pdx/greaterorequalslant(U1(t))p/parenleftBigg/integraldisplay BR+Ak(t)(Ψ(t,x))p′dx/parenrightBigg−(p−1) /greaterorsimilarεptkp−(1−k)(n−1)(p−1)+[(1−k)n−1 2−k+µ 2]p /greaterorsimilarεpt(1−k)(n−1)+k 2p−((1−k)n−1 2+µ 2)p(29) 7fort/greaterorequalslantT1.= 2T0. Finally, plugging (29) in (20), for t/greaterorequalslantT1it holds U0(t)/greaterorequalslant/integraldisplayt T1τ−µ/integraldisplayτ T1sµ/integraldisplay Rn|u(s,x)|pdxdsdτ/greaterorsimilarεp/integraldisplayt T1τ−µ/integraldisplayτ T1sµ+(1−k)(n−1)+k 2p−((1−k)n−1 2+µ 2)pdsdτ /greaterorsimilarεpt−((1−k)n−1 2+µ 2)p−µ/integraldisplayt T1/integraldisplayτ T1(s−T1)µ+(1−k)(n−1)+k 2pdsdτ /greaterorsimilarεpt−((1−k)n−1 2+µ 2)p−µ(t−T1)µ+(1−k)(n−1)+k 2p+2. Summarizing we proved the lower bound estimate for the funct ionalU0 U0(t)/greaterorequalslantKεpt−a0(t−T1)b0(30) fort/greaterorequalslantT1, whereK=K(n,k,µ,p,R,u 0,u1)is a suitable positive constant and a0.=/parenleftbig (1−k)n−1 2+µ 2/parenrightbig p+µ, b 0.=µ+(1−k)(n−1)+k 2p+2. (31) 2.4 Iteration argument In this section we will use the iteration frame (23) to prove t hatU0blows up in finite time under the assumptions of Theorem 1.2. More precisely, we are going to p rove the sequence of lower bound estimates U0(t)/greaterorequalslantDjt−aj(t−T1)bj(32) fort/greaterorequalslantT1, where{Dj}j∈N,{aj}j∈Nand{bj}j∈Nare sequences of nonnegative real numbers that will be determined iteratively during the proof. Clearly, for j= 0the estimate in (32) is nothing but (30) with D0=Kεpanda0,b0defined by (31). In order to prove (32) via an inductive argument, it remains jus t to prove the inductive step. Let us assume the validity of (32) for j. We prove now its validity for j+1too. Plugging (32) into (23), for t>T1we get U0(t)/greaterorequalslantC/integraldisplayt T1τ−µ/integraldisplayτ T1sµ−(1−k)n(p−1)(U0(s))pdsdτ /greaterorequalslantCDp j/integraldisplayt T1τ−µ/integraldisplayτ T1sµ−(1−k)n(p−1)−ajp(s−T1)bjpdsdτ /greaterorequalslantCDp jt−(1−k)n(p−1)−µ−ajp/integraldisplayt T1/integraldisplayτ T1(s−T1)µ+bjpdsdτ =CDp j (1+µ+bjp)(2+µ+bjp)t−(1−k)n(p−1)−µ−ajp(t−T1)2+µ+bjp, which is exactly (32) for j+1provided that Dj+1.=CDp j (1+µ+bjp)(2+µ+bjp), (33) aj+1.= (1−k)n(p−1)+µ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright.=α+paj, bj+1.= 2+µ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright.=β+pbj. (34) Employing recursively (34), we may express explicitly ajandbjas follows aj=α+paj−1=···=αj−1/summationdisplay k=0pk+a0pj=/parenleftBig α p−1+a0/parenrightBig pj−α p−1, (35) bj=β+pbj−1=···=βj−1/summationdisplay k=0pk+b0pj=/parenleftBig β p−1+b0/parenrightBig pj−β p−1. (36) Combining (34) and (36), we find bj= 2+µ+pbj−1</parenleftBig β p−1+b0/parenrightBig pj, 8that implies, in turn, Dj/greaterorequalslantCDp j−1 (2+µ+pbj−1)2=CDp j−1 b2 j/greaterorequalslantC /parenleftBig β p−1+b0/parenrightBig2 /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright.=/tildewideCDp j−1p−2j=/tildewideCDp j−1p−2j. Applying the logarithmic function to both sides of the last i nequality and using the resulting inequality iteratively, we get logDj/greaterorequalslantplogDj−1−2jlogp+log/tildewideC /greaterorequalslantp2logDj−2−2(j+(j−1)p)logp+(1+p)log/tildewideC /greaterorequalslant···/greaterorequalslantpjlogD0−2logpj−1/summationdisplay k=0(j−k)pk+log/tildewideCj−1/summationdisplay k=0pk. Using the formulas j−1/summationdisplay k=0(j−k)pk=1 p−1/parenleftbiggpj+1−p p−1−j/parenrightbigg andj−1/summationdisplay k=0pk=pj−1 p−1, (37) that can be shown via an inductive argument, we obtain logDj/greaterorequalslantpjlogD0−2logp p−1/parenleftbiggpj+1−p p−1−j/parenrightbigg +(pj−1)log/tildewideC p−1 =pj/parenleftBigg logD0−2plogp (p−1)2+log/tildewideC p−1/parenrightBigg +2jlogp p−1+2plogp (p−1)2−log/tildewideC p−1. Let us denote by j0=j0(n,p,k,µ)∈Nthe smallest integer greater thanlog/tildewideC 2logp−p p−1. Then, for any j/greaterorequalslantj0 we have logDj/greaterorequalslantpj/parenleftBigg logD0−2plogp (p−1)2+log/tildewideC p−1/parenrightBigg =pjlog/parenleftBig Kp−(2p)/(p−1)2/tildewideC1/(p−1)εp/parenrightBig =pjlog(E0εp),(38) whereE0.=Kp−(2p)/(p−1)2/tildewideC1/(p−1). Combining (32), (35), (36) and (38), for j/greaterorequalslantj0andt/greaterorequalslantT1it holds U0(t)/greaterorequalslantexp/parenleftbig pjlog(E0εp)/parenrightbig t−aj(t−T1)bj = exp/parenleftBig pj/parenleftBig log(E0εp)−/parenleftBig α p−1+a0/parenrightBig logt+/parenleftBig β p−1+b0/parenrightBig log(t−T1)/parenrightBig/parenrightBig tα/(p−1)(t−T1)−β/(p−1). Fort/greaterorequalslant2T1, we have log(t−T1)/greaterorequalslantlog(t/2), so forj/greaterorequalslantj0 U0(t)/greaterorequalslantexp/parenleftBig pj/parenleftBig log(E0εp)+/parenleftBig β−α p−1+b0−a0/parenrightBig logt−/parenleftBig β p−1+b0/parenrightBig log2/parenrightBig/parenrightBig tα/(p−1)(t−T1)−β/(p−1) = exp/parenleftBig pj/parenleftBig log/parenleftBig 2−b0−β/(p−1)E0εptθ(n,k,µ,p ) p−1/parenrightBig/parenrightBig/parenrightBig tα/(p−1)(t−T1)−β/(p−1), (39) where for the exponent of tin the last equality we used β−α p−1+b0−a0=2 p−1−(1−k)n+(1−k)(n−1)+k 2p+2−/parenleftbig (1−k)n−1 2+µ 2/parenrightbig p =2p p−1−(1−k)−/parenleftBig (1−k)n−1 2+µ−k 2/parenrightBig p =1 p−1/braceleftBig 1−k+/parenleftBig (1−k)n+1 2+µ+3k 2/parenrightBig p−/parenleftBig (1−k)n−1 2+µ−k 2/parenrightBig p2/bracerightBig =θ(n,k,µ,p) p−1.(40) Note thatθ(n,k,µ,p)is a positive quantity for p < p0/parenleftbig k,n+µ 1−k/parenrightbig . Let us fix ε0>0sufficiently small so that ε−p(p−1) θ(n,k,µ,p ) 0 /greaterorequalslant21−(b0(p−1)+β)/θ(n,k,µ,p)T1. Then, for any ε∈(0,ε0]and fort/greaterorequalslant2(b0(p−1)+β)/θ(n,k,µ,p)ε−p(p−1) θ(n,k,µ,p )it results t/greaterorequalslant2T1and2−b0−β/(p−1)E0εptθ(n,k,µ,p ) p−1>1, 9also, letting j→ ∞ in (39) it turns out that U0(t)blows up. Consequently, we proved the blowing – up of U0in finite time for any ε∈(0,ε0]wheneverp<p0/parenleftbig k,n+µ 1−k/parenrightbig and, moreover, as byproduct we found the upper bound estimate for the lifespan T(ε)/lessorsimilarε−p(p−1) θ(n,k,µ,p )as well. So far we applied only the lower bound estimate in (30) for U0. Nevertheless, we also proved another lower bound estimate for U0, namely, (21). Using (21) instead of (30), the initial value s for the parameters in (32) are a0=b0= 0andD0≈ε. Repeating the computations analogously as in the previous case and using logDj/greaterorequalslantpjlog(E1ε) forj/greaterorequalslantj1, wherej1is a suitable nonnegative integer and E1is a suitable positive constant, in place of (38) and β−α p−1+b0−a0=2 p−1−(1−k)n instead of (40), we obtain immediately the blow – up of U0in finite time for p < p1(k,n)and the corre- sponding upper bound estimate for the lifespan in (9). 3 Critical case: part I In order to study the critical case p=p0/parenleftbig k,n+µ 1−k/parenrightbig , we will follow an approach which is based on the technique introduced in [36] and subsequently applied to di fferent frameworks in [37, 29, 30, 21, 3, 4, 27]. From (39) it is clear that we can no longer employ U0as functional to study the blow – up dynamic. Therefore, we need to sharpen the choice of the functional. M ore precisely, we are going to consider a weighted space average of a local in time solution to (4). Hence, the bl ow – up result will be proved by applying the so – called slicing procedure in an iteration argument to show a sequence of lower bound est imates for the above mentioned functional. Throughout this section we wor k under the assumptions of Theorem 1.3. The section is organized as follows: in Section 3.1 we determ ine a pair of auxiliary functions which have a fundamental role in the definition of the time – dependent fu nctional and in the determination of the iteration frame, while in Section 3.2 we establish some fund amental properties for these functions; finally, in Section 3.3 we determine the iteration frame for the weigh ted space average whose dynamic provides the blow – up result. 3.1 Auxiliary functions In this section, we introduce two auxiliary functions (see ξqandηqbelow). These auxiliary functions represent a generalization of the solution to the classical free wave e quation given in [39] and are defined by using the remarkable function ϕintroduced in [38], that we have already used in the section f or the subcritical case (the definition of this function is given in (13)). According to our purpose of introducing the auxiliary funct ions, we begin by determining the solutions yj=yj(t,s;λ,k,µ),j∈ {0,1}of the non – autonomous, parameter – dependent, ordinary Cau chy problems ∂2 tyj(t,s;λ,k,µ)−λ2t−2kyj(t,s;λ,k,µ)+µt−1yj(t,s;λ,k,µ) = 0, t>s, yj(s,s;λ,k,µ) =δ0j, ∂tyj(s,s;λ,k,µ) =δ1j,(41) whereδijdenotes the Kronecker delta, s/greaterorequalslant1is the initial time and λ >0is a real parameter. To find a system of independent solutions to d2y dt2−λ2t−2ky+µt−1dy dt= 0 (42) we start by performing the change of variable τ=τ(t;λ,k).=λφk(t). By the straightforward relations dy dt=λt−kdy dτ,d2y dt2=λ2t−2kd2y dτ2−λkt−k−1dy dτ, it follows that ysolves (42) if and only if τd2y dτ2+µ−k 1−kdy dτ−τy= 0. (43) 10Carrying out the transformation y(τ) =τνw(τ)withν=ν(k,µ).=1−µ 2(1−k), it turns out that ysolves (43) if and only if wsolves the modified Bessel equation of order ν τ2d2w dτ2+τdw dτ−/parenleftbig ν2+τ2/parenrightbig w= 0. (44) Employing the modified Bessel function of first and second kin d of orderν, denoted, respectively, by Iν(τ) andKν(τ), as independent solutions to (44), then, we obtain V0(t;λ,k,µ).=τνIν(τ) = (λφk(t))νIν(λφk(t)), V1(t;λ,k,µ).=τνKν(τ) = (λφk(t))νKν(λφk(t)) as basis for the space of solutions to (42). Proposition 3.1. The functions y0(t,s;λ,k,µ).=λφk(s)sµ−1 2t1−µ 2/bracketleftbig Iν−1(λφk(s))Kν(λφk(t))+K ν−1(λφk(s))Iν(λφk(t))/bracketrightbig , (45) y1(t,s;λ,k,µ).= (1−k)−1s1+µ 2t1−µ 2/bracketleftbig Kν(λφk(s))Iν(λφk(t))−Iν(λφk(s))Kν(λφk(t))/bracketrightbig , (46) solve the Cauchy problems (41)forj= 0andj= 1, respectively, where ν=1−µ 2(1−k)andIν,Kνdenote the modified Bessel function of order νof the first and second kind, respectively. Proof. Since we proved that V0,V1form a system of independent solutions to (42), we may expres s the solutions to (41) as linear combinations of V0,V1in the following way yj(t,s;λ,k,µ) =aj(s;λ,k,µ)V0(t;λ,k,µ)+bj(s;λ,k,µ)V1(t;λ,k,µ) (47) for suitable coefficients aj(s;λ,k,µ),bj(s;λ,k,µ), withj∈ {0,1}. We can describe the initial conditions ∂i tyj(s,s;λ,k) =δijthrough the system /parenleftbiggV0(s;λ,k,µ)V1(s;λ,k,µ) ∂tV0(s;λ,k,µ)∂tV1(s;λ,k,µ)/parenrightbigg/parenleftbigga0(s;λ,k,µ)a1(s;λ,k,µ) b0(s;λ,k,µ)b1(s;λ,k,µ)/parenrightbigg =I, whereIdenotes the identity matrix. Also, to determine the coefficie nts in (47), we calculate the inverse matrix /parenleftbiggV0(s;λ,k,µ)V1(s;λ,k,µ) ∂tV0(s;λ,k,µ)∂tV1(s;λ,k,µ)/parenrightbigg−1 = (W(V0,V1)(s;λ,k,µ))−1/parenleftbigg∂tV1(s;λ,k,µ)−V1(s;λ,k,µ) −∂tV0(s;λ,k,µ)V0(s;λ,k,µ)/parenrightbigg , (48) whereW(V0,V1)denotes the Wronskian of V0,V1. Next, we compute explicitly the function W(V0,V1). Thanks to ∂tV0(t;λ,k,µ) =ν(λφk(t))ν−1λφ′ k(t)Iν(λφk(t))+(λφk(t))νI′ ν(λφk(t))λφ′ k(t), ∂tV1(t;λ,k,µ) =ν(λφk(t))ν−1λφ′ k(t)Kν(λφk(t))+(λφk(t))νK′ ν(λφk(t))λφ′ k(t), recallingφ′ k(t) =t−kand2ν−1 =k−µ 1−k, we can express W(V0,V1)as follows: W(V0,V1)(t;λ,k,µ) = (λφk(t))2ν(λφ′ k(t))/braceleftbig K′ ν(λφk(t))Iν(λφk(t))−I′ ν(λφk(t))Kν(λφk(t))/bracerightbig = (λφk(t))2ν(λφ′ k(t))W(Iν,Kν)(λφk(t)) =−(λφk(t))2ν−1(λφ′ k(t)) =−λ2ν(φk(t))2ν−1φ′ k(t) =−c−1 k,µλ2νt−µ, whereck,µ.= (1−k)k−µ 1−kand in the third equality we used the value of the Wronskian of Iν,Kν W(Iν,Kν)(z) = Iν(z)∂Kν ∂z(z)−Kν(z)∂Iν ∂z(z) =−1 z. Plugging the previously determined representation of W(V0,V1)in (48), we have /parenleftbigg a0(s;λ,k,µ)a1(s;λ,k,µ) b0(s;λ,k,µ)b1(s;λ,k,µ)/parenrightbigg =ck,µλ−2νsµ/parenleftbigg −∂tV1(s;λ,k,µ)V1(s;λ,k,µ) ∂tV0(s;λ,k,µ)−V0(s;λ,k,µ)/parenrightbigg . 11Let us begin by showing (45). Using the above representation ofa0(s;λ,kµ),b0(s;λ,k,µ)in (47), we find y0(t,s;λ,k,µ) =ck,µλ−2νsµ/braceleftbig ∂tV0(s;λ,k,µ)V1(t;λ,k,µ)−∂tV1(s;λ,k,µ)V0(t;λ,k,µ)/bracerightbig =ck,µνsµφ′ k(s)(φk(s))ν−1(φk(t))ν/braceleftbig Iν(λφk(s))Kν(λφk(t))−Kν(λφk(s))Iν(λφk(t))/bracerightbig +ck,µλsµφ′ k(s)(φk(s))ν(φk(t))ν/braceleftbig I′ ν(λφk(s))Kν(λφk(t))−K′ ν(λφk(s))Iν(λφk(t))/bracerightbig . Using the following recursive relations for the derivative s of the modified Bessel functions ∂Iν ∂z(z) =−ν zIν(z)+Iν−1(z), ∂Kν ∂z(z) =−ν zKν(z)−Kν−1(z), there is a cancellation in the last relation, so, we arrive at y0(t,s;λ,k,µ) =ck,µλsµφ′ k(s)(φk(s)φk(t))ν/braceleftbig Iν−1(λφk(s))Kν(λφk(t))+K ν−1(λφk(s))Iν(λφk(t))/bracerightbig .(49) Thanks to ck,µsµφ′ k(s)(φk(s)φk(t))ν= (1−k)−1sµ−k(st)1−µ 2=φk(s)sµ−1 2t1−µ 2, from (49) it follows immediately (45). Let us show now the rep resentation for y1. Plugging the above determined expressions for a1(s;λ,k,µ),b1(s;λ,k,µ)in (47), we get y1(t,s;λ,k,µ) =ck,µλ−2νsµ/braceleftbig V1(s;λ,k,µ)V0(t;λ,k,µ)−V0(s;λ,k,µ)V1(t;λ,k,µ)/bracerightbig =ck,µλ−2νsµ(λφk(s))ν(λφk(t))ν/braceleftbig Kν(λφk(s))Iν(λφk(t))−Iν(λφk(s))Kν(λφk(t))/bracerightbig =ck,µsµ(φk(s)φk(t))ν/braceleftbig Kν(λφk(s))Iν(λφk(t))−Iν(λφk(s))Kν(λφk(t))/bracerightbig . (50) Hence, due to ck,µsµ(φk(s)φk(t))ν= (1−k)−1s1+µ 2t1−µ 2, from (50) it results (46). The proof is complete. Lemma 3.2. Lety0,y1be the functions defined in (45)and(46), respectively. Then, the following identities are satisfied for any t/greaterorequalslants/greaterorequalslant1 ∂y1 ∂s(t,s;λ,k,µ) =−y0(t,s;λ,k,µ)+µs−1y1(t,s;λ,k,µ), (51) ∂2y1 ∂s2(t,s;λ,k,µ)−λ2s−2ky1(t,s;λ,k,µ)−µs−1∂y1 ∂s(t,s;λ,k,µ)+µs−2y1(t,s;λ,k,µ) = 0.(52) Remark 3.As the operator ∂2 s−λ2s−2k−µs−1∂s+µs−2is the formal adjoint of ∂2 t−λ2t−2k+µt−1∂t, in particular, (51) and (52) tell us that y1solves also the adjoint problem to (42) with final conditions (0,−1). Proof. Let us introduce the pair of independent solutions to (42) z0(t;λ,k,µ).=y0(t,1;λ,k,µ), z1(t;λ,k,µ).=y1(t,1;λ,k,µ). Since the Wronskian W(z0,z1)(t;λ,k,µ)solves the differential equation W′(z0,z1) =−µt−1W(z0,z1)with initial condition W(z0,z1)(1;λ,k,µ) = 1 , then,W(z0,z1)(t;λ,k,µ) =t−µ. Therefore, repeating similar computations as in the proof of Proposition 3.1, we may show t he representations y0(t,s;λ,k,µ) =sµ{z′ 1(s;λ,k,µ)z0(t;λ,k,µ)−z′ 0(s;λ,k,µ)z1(t;λ,k,µ)}, y1(t,s;λ,k,µ) =sµ{z0(s;λ,k,µ)z1(t;λ,k,µ)−z1(s;λ,k,µ)z0(t;λ,k,µ)}. Let us prove (51). Differentiating the second one of the previ ous representations with respect to s, we find ∂y1 ∂s(t,s;λ,k) =µsµ−1{z0(s;λ,k,µ)z1(t;λ,k,µ)−z1(s;λ,k,µ)z0(t;λ,k,µ)} +sµ{z′ 0(s;λ,k,µ)z1(t;λ,k,µ)−z′ 1(s;λ,k,µ)z0(t;λ,k,µ)} =µs−1y1(t,s;λ,k,µ)−y0(t,s;λ,k,µ). 12On the other hand, due to the fact that z0,z1satisfy (42), then, ∂2y1 ∂s2(t,s;λ,k) =sµ{z′′ 0(s;λ,k,µ)z1(t;λ,k,µ)−z′′ 1(s;λ,k,µ)z0(t;λ,k,µ)} +2µsµ−1{z′ 0(s;λ,k,µ)z1(t;λ,k,µ)−z′ 1(s;λ,k,µ)z0(t;λ,k,µ)} +µ(µ−1)sµ−2{z0(s;λ,k,µ)z1(t;λ,k,µ)−z1(s;λ,k,µ)z0(t;λ,k,µ)} =sµ/braceleftbig/bracketleftbig λ2s−2kz0(s;λ,k,µ)−µs−1z′ 0(s;λ,k,µ)/bracketrightbig z1(t;λ,k,µ) −/bracketleftbig λ2s−2kz1(s;λ,k,µ)−µs−1z′ 1(s;λ,k,µ)/bracketrightbig z0(t;λ,k,µ)/bracerightbig +2µsµ−1{z′ 0(s;λ,k,µ)z1(t;λ,k,µ)−z′ 1(s;λ,k,µ)z0(t;λ,k,µ)} +µ(µ−1)sµ−2{z0(s;λ,k,µ)z1(t;λ,k,µ)−z1(s;λ,k,µ)z0(t;λ,k,µ)} =λ2s−2ksµ/braceleftbig z0(s;λ,k,µ)z1(t;λ,k,µ)−z1(s;λ,k,µ)z0(t;λ,k,µ)/bracerightbig +µsµ−1{z′ 0(s;λ,k,µ)z1(t;λ,k,µ)−z′ 1(s;λ,k,µ)z0(t;λ,k,µ)} +µ(µ−1)sµ−2{z0(s;λ,k,µ)z1(t;λ,k,µ)−z1(s;λ,k,µ)z0(t;λ,k,µ)} =λ2s−2ky1(t,s;λ,k,µ)−µs−1y0(t,s;λ,k,µ)+µ(µ−1)s−2y1(t,s;λ,k,µ). Applying (51), from the last chain of equalities we get ∂2y1 ∂s2(t,s;λ,k) =λ2s−2ky1(t,s;λ,k,µ)+µs−1/parenleftbigg∂y1 ∂s(t,s;λ,k)−µs−1y1(t,s;λ,k,µ)/parenrightbigg +µ(µ−1)s−2y1(t,s;λ,k,µ) =λ2s−2ky1(t,s;λ,k,µ)+µs−1∂y1 ∂s(t,s;λ,k)−µs−2y1(t,s;λ,k,µ). Thus, we proved (52) too. This completes the proof. Proposition 3.3. Letu0∈H1(Rn)andu1∈L2(Rn)be functions such that suppuj⊂BRforj= 0,1 and for some R>0and letλ>0be a parameter. Let ube a local in time energy solution to (4)on[1,T) according to Definition 1.1. Then, the following integral id entity is satisfied for any t∈[1,T) /integraldisplay Rnu(t,x)ϕλ(x)dx=εy0(t,1;λ,k)/integraldisplay Rnu0(x)ϕλ(x)dx+εy1(t,1;λ,k)/integraldisplay Rnu1(x)ϕλ(x)dx +/integraldisplayt 1y1(t,s;λ,k)/integraldisplay Rn|u(s,x)|pϕλ(x)dxds, (53) whereϕλ(x).=ϕ(λx)andϕis defined by (13). Proof. Assumingu0,u1compactly supported, we can consider a test function ψ∈C∞([1,T)×Rn)in Definition 1.1 according to Remark 1. Hence, we take ψ(s,x) =y1(t,s;λ,k,µ)ϕλ(x)(heret,λcan be treated as fixed parameters). Consequently, ψsatisfies ψ(t,x) =y1(t,t;λ,k,µ)ϕλ(x) = 0, ψ(1,x) =y1(t,1;λ,k,µ)ϕλ(x), ψs(t,x) =∂sy1(t,t;λ,k,µ)ϕλ(x) =/parenleftbig µt−1y1(t,t;λ,k,µ)−y0(t,t;λ,k,µ)/parenrightbig ϕλ(x) =−ϕλ(x), ψs(1,x) =∂sy1(t,1;λ,k,µ)ϕλ(x) = (µy1(t,1;λ,k,µ)−y0(t,1;λ,k,µ))ϕλ(x), and ψss(s,x)−s−2k∆ψ(s,x)−µ∂s(s−1ψ(s,x)) =/parenleftbig ∂2 s−λ2s−2k−µs−1∂s+µs−2/parenrightbig y1(t,s;λ,k,µ)ϕλ(x) = 0, where we used (51), (52) and the property ∆ϕλ=λ2ϕλ. Then, employing the above defined ψin (8), we find immediately (53). This completes the proof. Proposition 3.4. Lety0,y1be the functions defined in (45)and(46), respectively. Then, the following estimates are satisfied for any t/greaterorequalslants/greaterorequalslant1 y0(t,s;λ,k,µ)/greaterorequalslantsµ−k 2tk−µ 2cosh/parenleftbig λ(φk(t)−φk(s))/parenrightbig ifµ∈[2−k,∞), (54) y1(t,s;λ,k,µ)/greaterorequalslantsµ+k 2tk−µ 2sinh/parenleftbig λ(φk(t)−φk(s))/parenrightbig λifµ∈[0,k]∪[2−k,∞). (55) 13Proof. The proof of the inequalities (54) and (55) is based on the fol lowing minimum type principle: letw=w(t,s;λ,k,µ)be a solution of the Cauchy problem /braceleftBigg ∂2 tw−λ2t−2kw+µt−1∂tw=h,fort>s/greaterorequalslant1, w(s) =w0, ∂tw(s) =w1,(56) whereh=h(t,s;λ,k,µ)is a continuous function; if h/greaterorequalslant0andw0=w1= 0(i.e.wis asupersolution of the homogeneous problem with trivial initial conditions), t hen,w(t,s;λ,k,µ)/greaterorequalslant0for anyt>s. In order to prove this minimum principle, we apply the contin uous dependence on initial conditions (note that fort/greaterorequalslant1the functions t−2kandµt−1are smooth). Indeed, if we denote by wǫthe solution to (56) with w0=ǫ>0andw1= 0, then,wǫsolves the integral equation wǫ(t,s;λ,k,µ) =ǫ+/integraldisplayt sτ−µ/integraldisplayτ sσµ/parenleftbig λ2σ−2kwǫ(σ,s;λ,k,µ)+h(σ,s;λ,k,µ)/parenrightbig dσdτ. By contradiction, one can prove easily that wǫ(t,s;λ,k,µ)>0for anyt > s. Hence, by the continuous dependence on initial data, letting ǫ→0, we find that w(t,s;λ,k,µ)/greaterorequalslant0for anyt>s. Let us prove the validity of (55). Denoting by w1=w1(t,s;λ,k,µ)the function on the right – hand side of (55), we find immediately w1(s,s;λ,k,µ) = 0 and∂tw1(s,s;λ,k,µ) = 1. Moreover, ∂2 tw1(t,s;λ,k,µ) =λ−1sk+µ 2tk−µ 2/bracketleftBig k−µ 2/parenleftBig k−µ 2−1/parenrightBig t−2sinh/parenleftbig λ(φk(t)−φk(s))/parenrightbig +(k−µ)t−1cosh/parenleftbig λ(φk(t)−φk(s))/parenrightbig λφ′ k(t) +sinh/parenleftbig λ(φk(t)−φk(s))/parenrightbig (λφ′ k(t))2+cosh/parenleftbig λ(φk(t)−φk(s))/parenrightbig λφ′′ k(t)/bracketrightBig =/bracketleftBig k−µ 2/parenleftBig k−µ 2−1/parenrightBig t−2+λ2t−2k/bracketrightBig w1(t,s;λ,k,µ)−µsk+µ 2t−1−k+µ 2cosh/parenleftbig λ(φk(t)−φk(s))/parenrightbig and ∂tw1(t,s;λ,k,µ) =λ−1sk+µ 2tk−µ 2/bracketleftBig k−µ 2t−1sinh/parenleftbig λ(φk(t)−φk(s))/parenrightbig +λt−kcosh/parenleftbig λ(φk(t)−φk(s))/parenrightbig/bracketrightBig =k−µ 2t−1w1(t,s;λ,k,µ)+sk+µ 2t−k+µ 2cosh/parenleftbig λ(φk(t)−φk(s))/parenrightbig imply that ∂2 tw1(t,s;λ,k,µ)−λ2t−2kw1(t,s;λ,k,µ)+µt−1∂tw1(t,s;λ,k,µ) =k−µ 2/parenleftBig k+µ 2−1/parenrightBig w1(t,s;λ,k,µ)/lessorequalslant0, where in the last step we employ the assumption µ /∈(k,2−k)to guarantee that the multiplicative constant is negative. Therefore, y1−w1is a supersolution of (56) with h= 0andw0=w1= 0. Thus, applying the minimum principle we have that (y1−w1)(t,s;λ,k)/greaterorequalslant0for anyt>s, that is, we showed (55). In a completely analogous way, one can prove (54), repeating the previous argument based on the mini- mum principle with w0(t,s;λ,k,µ).=sµ−k 2tk−µ 2cosh/parenleftbig λ(φk(t)−φk(s))/parenrightbig in place of w1(t,s;λ,k,µ)andy0in place ofy1, respectively. However, in order to guarantee that w0(s,s;λ,k,µ) = 1 and∂tw0(s,s;λ,k,µ)/lessorequalslant0, we are forced to require µ/greaterorequalslantk, which provides, together with the condition µ /∈(k,2−k)that is necessary to ensure that w0is actually a subsolution of the homogeneous equation, the r ange forµin (54). Remark 4.Although (54) might be restrictive from the viewpoint of the range forµin the statement of Theorem 1.3, we can actually overcome this difficulty by showi ng a transformation which allows to link the caseµ∈[0,k]to the case µ∈[2−k,2], when a lower bound estimate for y0is available. Indeed, if we perform the transformation v=v(t,x).=tµ−1u(t,x), then,uis a solution to (4) if and only if vsolves vtt−t−2k∆v+(2−µ)t−1vt=t(1−µ)(p−1)|v|px∈Rn, t∈(1,T), v(1,x) =εu0(x) x∈Rn, ut(1,x) =εu1(x)+ε(1−µ)u0(x) x∈Rn.(57) Let us point out that in (57) a time – dependent factor which de cays with polynomial order appears in the nonlinear term on the right – hand side. Therefore, we will re duce the case µ∈[0,k]to the case µ/greaterorequalslant2−k, up to the time – dependent factor t(1−µ)(p−1)in the nonlinearity. We can introduce now for t/greaterorequalslants/greaterorequalslant1andx∈Rnthe definition of the following auxiliary function ξq(t,s,x;k,µ).=/integraldisplayλ0 0e−λ(Ak(t)+R)y0(t,s;λ,k,µ)ϕλ(x)λqdλ, (58) ηq(t,s,x;k,µ).=/integraldisplayλ0 0e−λ(Ak(t)+R)y1(t,s;λ,k,µ) φk(t)−φk(s)ϕλ(x)λqdλ, (59) 14whereq>−1,λ0>0is a fixed parameter and Akis defined by (6). Combining Proposition 3.3 and (58) and (59), we establish a f undamental equality, whose role will be crucial in the next sections in order to prove the blow – up res ult. Corollary 3.5. Letu0∈H1(Rn)andu1∈L2(Rn)such that suppuj⊂BRforj= 0,1and for some R>0. Letube a local in time energy solution to (4)on[1,T)according to Definition 1.1. Let q >−1and let ξq(t,s,x;k),ηq(t,s,x;k)be the functions defined by (58)and(59), respectively. Then, /integraldisplay Rnu(t,x)ξq(t,t,x;k,µ)dx=ε/integraldisplay Rnu0(x)ξq(t,1,x;k,µ)dx+ε(φk(t)−φk(1))/integraldisplay Rnu1(x)ηq(t,s,x;k,µ)dx +/integraldisplayt 1(φk(t)−φk(s))/integraldisplay Rn|u(s,x)|pηq(t,s,x;k,µ)dxds (60) for anyt∈[1,T). Proof. Multiplying both sides of (53) by e−λ(Ak(t)+R)λq, integrating with respect to λover[0,λ0]and applying Fubini’s theorem, we get easily (60). 3.2 Properties of the auxiliary functions In this section, we establish lower and upper bound estimate s for the auxiliary functions ξq,ηqunder suitable assumptions on q. In the lower bound estimates, we may restrict our considera tions to the case µ/greaterorequalslant2−k thanks to Remark 4, even though the estimate for ηqthat will be proved thanks to (55) clearly would be true also for µ∈[0,k]. Lemma 3.6. Letn/greaterorequalslant1,k∈[0,1),µ/greaterorequalslant2−kandλ0>0. If we assume q >−1, then, fort/greaterorequalslants/greaterorequalslant1and |x|/lessorequalslantAk(s)+Rthe following lower bound estimates are satisfied: ξq(t,s,x;k,µ)/greaterorequalslantB0sµ−k 2tk−µ 2/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−q−1; (61) ηq(t,s,x;k,µ)/greaterorequalslantB1sµ+k 2tk−µ 2/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−q. (62) HereB0,B1are positive constants depending only on λ0,q,R,k and we employ the notation /an}b∇acketle{ty/an}b∇acket∇i}ht.= 3+|s|. Proof. We adapt the main ideas in the proof of Lemma 3.1 in [36] to our m odel. Since /an}b∇acketle{t|x|/an}b∇acket∇i}ht−n−1 2e|x|/lessorsimilarϕ(x)/lessorsimilar/an}b∇acketle{t|x|/an}b∇acket∇i}ht−n−1 2e|x|(63) holds for any x∈Rn, there exists a constant B=B(λ0,R,k)>0independent of λandssuch that B/lessorequalslant inf λ∈/bracketleftBigλ0 /angbracketleftAk(s)/angbracketright,2λ0 /angbracketleftAk(s)/angbracketright/bracketrightBiginf |x|/lessorequalslantAk(s)+Re−λ(Ak(s)+R)ϕλ(x). Let us begin by proving (61). Using the lower bound estimate i n (54), shrinking the domain of integration in (58) to/bracketleftBig λ0 /angbracketleftAk(s)/angbracketright,2λ0 /angbracketleftAk(s)/angbracketright/bracketrightBig and applying the previous inequality, we arrive at ξq(t,s,x;k,µ)/greaterorequalslantsµ−k 2tk−µ 2/integraldisplay2λ0//angbracketleftAk(s)/angbracketright λ0//angbracketleftAk(s)/angbracketrighte−λ(Ak(t)−Ak(s))cosh/parenleftbig λ(φk(t)−φk(s))/parenrightbig e−λ(Ak(s)+R)ϕλ(x)λqdλ /greaterorequalslantBsµ−k 2tk−µ 2/integraldisplay2λ0//angbracketleftAk(s)/angbracketright λ0//angbracketleftAk(s)/angbracketrighte−λ(Ak(t)−Ak(s))cosh/parenleftbig λ(φk(t)−φk(s))/parenrightbig λqdλ =B 2sµ−k 2tk−µ 2/integraldisplay2λ0//angbracketleftAk(s)/angbracketright λ0//angbracketleftAk(s)/angbracketright/parenleftBig 1+e−2λ(φk(t)−φk(s))/parenrightBig λqdλ /greaterorequalslantB 2sµ−k 2tk−µ 2/integraldisplay2λ0//angbracketleftAk(s)/angbracketright λ0//angbracketleftAk(s)/angbracketrightλqdλ=B(2q+1−1)λq+1 0 2(q+1)/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−q−1. 15Repeating similar steps as before, thanks to (55) we obtain ηq(t,s,x;k,µ)/greaterorequalslantsµ+k 2tk−µ 2/integraldisplay2λ0//angbracketleftAk(s)/angbracketright λ0//angbracketleftAk(s)/angbracketrighte−λ(Ak(t)−Ak(s))sinh/parenleftbig λ(φk(t)−φk(s))/parenrightbig λ(φk(t)−φk(s))e−λ(Ak(s)+R)ϕλ(x)λqdλ /greaterorequalslantB 2sµ+k 2tk−µ 2/integraldisplay2λ0//angbracketleftAk(s)/angbracketright λ0//angbracketleftAk(s)/angbracketright1−e−2λ(φk(t)−φk(s)) φk(t)−φk(s)λq−1dλ /greaterorequalslantB 2sµ+k 2tk−µ 21−e−2λ0φk(t)−φk(s) /angbracketleftAk(s)/angbracketright φk(t)−φk(s)/integraldisplay2λ0//angbracketleftAk(s)/angbracketright λ0//angbracketleftAk(s)/angbracketrightλq−1dλ =B(2q−1)λq 0 2qsµ+k 2tk−µ 2/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−q1−e−2λ0φk(t)−φk(s) /angbracketleftAk(s)/angbracketright φk(t)−φk(s), with obvious modifications in the case q= 0. The previous inequality implies (62), provided that we sho w the validity of the inequality 1−e−2λ0φk(t)−φk(s) /angbracketleftAk(s)/angbracketright φk(t)−φk(s)/greaterorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1. Hence, we need to prove this inequality. For φk(t)−φk(s)/greaterorequalslant1 2λ0/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht, it holds 1−e−2λ0φk(t)−φk(s) /angbracketleftAk(s)/angbracketright/greaterorequalslant1−e−1 and, consequently, 1−e−2λ0φk(t)−φk(s) /angbracketleftAk(s)/angbracketright φk(t)−φk(s)/greaterorsimilar/parenleftbig φk(t)−φk(s)/parenrightbig−1/greaterorequalslantAk(t)−1/greaterorequalslant/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1. On the other hand, when φk(t)−φk(s)/lessorequalslant1 2λ0/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht, using the estimate 1−e−σ/greaterorequalslantσ/2forσ∈[0,1], we get easily 1−e−2λ0φk(t)−φk(s) /angbracketleftAk(s)/angbracketright φk(t)−φk(s)/greaterorequalslantλ0 /an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/greaterorequalslantλ0 /an}b∇acketle{tAk(t)/an}b∇acket∇i}ht. Therefore, the proof of (62) is completed. Next we prove an upper bound estimate in the special case s=t. Lemma 3.7. Letn/greaterorequalslant1,k∈[0,1),µ/greaterorequalslant0andλ0>0. If we assume q >(n−3)/2, then, for t/greaterorequalslant1and |x|/lessorequalslantAk(t)+Rthe following upper bound estimate holds: ξq(t,t,x;k,µ)/lessorequalslantB2/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−n−1 2/an}b∇acketle{tAk(t)−|x|/an}b∇acket∇i}htn−3 2−q. (64) HereB2is a positive constant depending only on λ0,q,R,k and/an}b∇acketle{ty/an}b∇acket∇i}htdenotes the same function as in the statement of Lemma 3.6. Proof. Due to the representation ξq(t,t,x;k,µ) =/integraldisplayλ0 0e−λ(Ak(t)+R)ϕλ(x)λqdλ, the proof is exactly the same as in [27, Lemma 2.7]. 3.3 Derivation of the iteration frame In this section, we define the time – dependent functional who se dynamic is studied in order to prove the blow – up result. Then, we derive the iteration frame for this functional and a first lower bound estimate of logarithmic type. Fort/greaterorequalslant1we introduce the functional U(t).=tµ−k 2/integraldisplay Rnu(t,x)ξq(t,t,x;k,µ)dx (65) for someq>(n−3)/2. 16From (60), (61) and (62), it follows U(t)/greaterorsimilarB0ε/integraldisplay Rnu0(x)dx+B1εAk(t) /an}b∇acketle{tAk(t)/an}b∇acket∇i}ht/integraldisplay Rnu1(x)dx. As we assume both u0,u1nonnegative and nontrivial, then, we find that U(t)/greaterorsimilarε (66) for anyt∈[1,T), where the unexpressed multiplicative constant depends on u0,u1. In the next proposition, we derive the iteration frame for the functional Ufor a given value of q. Proposition 3.8. Letn/greaterorequalslant1,k∈[0,1)andµ∈[0,k]∪[2−k,∞). Let us consider u0∈H1(Rn)and u1∈L2(Rn)such that suppuj⊂BRforj= 0,1and for some R >0and letube a local in time energy solution to (4)on[1,T)according to Definition 1.1. If Uis defined by (65)withq= (n−1)/2−1/p, then, there exists a positive constant C=C(n,p,R,k,µ )such that U(t)/greaterorequalslantC/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt 1φk(t)−φk(s) s/parenleftbig log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)(U(s))pds (67) for anyt∈(1,T). Proof. By (65), applying Hölder’s inequality we find sk−µ 2U(s)≤/parenleftbigg/integraldisplay Rn|u(s,x)|pηq(t,s,x;k,µ)dx/parenrightbigg1/p /integraldisplay BR+Ak(s)/parenleftbig ξq(s,s,x;k,µ)/parenrightbigp′ /parenleftbig ηq(t,s,x;k,µ)/parenrightbigp′/pdx 1/p′ . Hence, /integraldisplay Rn|u(s,x)|pηq(t,s,x;k,µ)dx/greaterorequalslant/parenleftbig sk−µ 2U(s)/parenrightbigp/parenleftBigg/integraldisplay BR+Ak(s)/parenleftbig ξq(s,s,x;k,µ)/parenrightbigp/(p−1) /parenleftbig ηq(t,s,x;k,µ)/parenrightbig1/(p−1)dx/parenrightBigg−(p−1) . (68) Let us determine an upper bound for the integral on the right – hand side of (68). By using (64) and (62), we obtain /integraldisplay BR+Ak(s)/parenleftbig ξq(s,s,x;k,µ)/parenrightbigp/(p−1) /parenleftbig ηq(t,s,x;k,µ)/parenrightbig1/(p−1)dx /lessorequalslantB−1 p−1 1Bp p−1 2s−µ+k 2(p−1)t−k−µ 2(p−1)/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−n−1 2p p−1+q p−1/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht1 p−1/integraldisplay BR+Ak(s)/an}b∇acketle{tAk(s)−|x|/an}b∇acket∇i}ht(n−3 2−q)p p−1dx /lessorequalslantB−1 p−1 1Bp p−1 2s−µ+k 2(p−1)tµ−k 2(p−1)/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht1 p−1/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht1 p−1(−n−1 2p+n−1 2−1 p)/integraldisplay BR+Ak(s)/an}b∇acketle{tAk(s)−|x|/an}b∇acket∇i}ht−1dx /lessorequalslantB−1 p−1 1Bp p−1 2s−µ+k 2(p−1)tµ−k 2(p−1)/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht1 p−1/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht1 p−1(−n−1 2p+n−1 2−1 p)+n−1log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht, where in the second inequality we used value of qto get exactly −1as power for the function in the integral. Consequently, from (68) we have /integraldisplay Rn|u(s,x)|pηq(t,s,x;k,µ)dx/greaterorsimilar/parenleftbig sk−µ 2U(s)/parenrightbigpsµ+k 2tk−µ 2/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−n−1 2(p−1)+1 p/parenleftbig log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1) /greaterorsimilartk−µ 2/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1sk 2(p+1)+µ 2(1−p)/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−n−1 2(p−1)+1 p/parenleftbig log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)/parenleftbig U(s)/parenrightbigp. Combining the previous lower bound estimate and (60), we arr ive at U(t)/greaterorequalslanttµ−k 2/integraldisplayt 1(φk(t)−φk(s))/integraldisplay Rn|u(s,x)|pηq(t,s,x;k,µ)dxds /greaterorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt 1(φk(t)−φk(s))sk 2(p+1)+µ 2(1−p)/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−n−1 2(p−1)+1 p/parenleftbig log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)/parenleftbig U(s)/parenrightbigpds /greaterorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt 1(φk(t)−φk(s))/an}b∇acketle{tAk(s)/an}b∇acket∇i}htk(p+1) 2(1−k)−µ(p−1) 2(1−k)−n−1 2(p−1)+1 p/parenleftbig log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)/parenleftbig U(s)/parenrightbigpds /greaterorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt 1(φk(t)−φk(s))/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−(n−1 2+µ−k 2(1−k))p+(n−1 2+µ+k 2(1−k))+1 p/parenleftbig log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)/parenleftbig U(s)/parenrightbigpds, 17where in third step we used s= (1−k)1 1−k(Ak(s)+φk(1))1 1−k≈ /an}b∇acketle{tAk(s)/an}b∇acket∇i}ht1 1−kfors/greaterorequalslant1. Sincep=p0/parenleftbig k,n+µ 1−k/parenrightbig from (5) it follows −/parenleftBig n−1 2+µ−k 2(1−k)/parenrightBig p+/parenleftBig n−1 2+µ+k 2(1−k)/parenrightBig +1 p=−1−k 1−k=−1 1−k, (69) then, plugging (69) in the above lower bound estimate for U(t)it yields U(t)/greaterorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt 1(φk(t)−φk(s))/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−1 1−k/parenleftbig log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)/parenleftbig U(s)/parenrightbigpds /greaterorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt 1φk(t)−φk(s) s/parenleftbig log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)/parenleftbig U(s)/parenrightbigpds, which is exactly (67). Therefore, the proof is completed. Lemma 3.9. Letn/greaterorequalslant1,k∈[0,1)andµ∈[0,k]∪[2−k,∞). Let us consider u0∈H1(Rn)andu1∈L2(Rn) such that suppuj⊂BRforj= 0,1and for some R>0and letube a local in time energy solution to (4) on[1,T)according to Definition 1.1. Then, there exists a positive co nstantK=K(u0,u1,n,p,R,k,µ )such that the lower bound estimate /integraldisplay Rn|u(t,x)|pdx/greaterorequalslantKεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht(n−1)(1−p 2)+(k−µ)p 2(1−k) (70) holds for any t∈(1,T). Proof. We modify of the proof of Lemma 5.1 in [36] accordingly to our m odel. Let us fix q>(n−3)/2+1/p′. Combining (65), (66) and Hölder’s inequality, it results εtk−µ 2/lessorsimilartk−µ 2U(t) =/integraldisplay Rnu(t,x)ξq(t,t,x;k,µ)dx /lessorequalslant/parenleftbigg/integraldisplay Rn|u(t,x)|pdx/parenrightbigg1/p/parenleftBigg/integraldisplay BR+Ak(t)/parenleftbig ξq(t,t,x;k,µ/parenrightbigp′ dx/parenrightBigg1/p′ . Hence, /integraldisplay Rn|u(t,x)|pdx/greaterorsimilarεptk−µ 2p/parenleftBigg/integraldisplay BR+Ak(t)/parenleftbig ξq(t,t,x;k,µ/parenrightbigp′ dx/parenrightBigg−(p−1) . (71) Let us determine an upper bound estimates for the integral of ξq(t,t,x;k,µ)p′. By using (64), we have /integraldisplay BR+Ak(t)/parenleftbig ξq(t,t,x;k,µ/parenrightbigp′ dx/lessorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−n−1 2p′/integraldisplay BR+Ak(t)/an}b∇acketle{tAk(t)−|x|/an}b∇acket∇i}ht(n−3)p′/2−p′qdx /lessorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−n−1 2p′/integraldisplayR+Ak(t) 0rn−1/an}b∇acketle{tAk(t)−r/an}b∇acket∇i}ht(n−3)p′/2−p′qdr /lessorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−n−1 2p′+n−1/integraldisplayR+Ak(t) 0/an}b∇acketle{tAk(t)−r/an}b∇acket∇i}ht(n−3)p′/2−p′qdr. Performing the change of variable Ak(t)−r=̺, one gets /integraldisplay BR+Ak(t)/parenleftbig ξq(t,t,x;k,µ/parenrightbigp′ dx/lessorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−n−1 2p′+n−1/integraldisplayAk(t) −R(3+|̺|)(n−3)p′/2−p′qd̺ /lessorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−n−1 2p′+n−1 because of (n−3)p′/2−p′q<−1. If we combine this upper bound estimates for the integral of ξq(t,t,x;k,µ)p′, the inequality (71) and we employ t≈ /an}b∇acketle{tAk(t)/an}b∇acket∇i}ht1 1−kfort/greaterorequalslant1, then, we arrive at (70). This completes the proof. In Proposition 3.8, we derive the iteration frame for U. In the next result, we shall prove a first lower bound estimate of logarithmic type for U, as base case for the iteration argument. 18Proposition 3.10. Letn/greaterorequalslant1,k∈[0,1)andµ∈[0,k]∪[2−k,∞). Let us consider u0∈H1(Rn)and u1∈L2(Rn)such that suppuj⊂BRforj= 0,1and for some R >0and letube a local in time energy solution to (4)on[1,T)according to Definition 1.1. Let Ube defined by (65)withq= (n−1)/2−1/p. Then, fort/greaterorequalslant3/2the functional U(t)fulfills U(t)/greaterorequalslantMεplog/parenleftbig2t 3/parenrightbig , (72) where the positive constant Mdepends on u0,u1,n,p,R,k,µ . Proof. From (60) it results U(t)/greaterorequalslanttµ−k 2/integraldisplayt 1(φk(t)−φk(s))/integraldisplay Rn|u(s,x)|pηq(t,s,x;k,µ)dxds. Consequently, applying (62) first and then (70), we find U(t)/greaterorequalslantB1/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt 1(φk(t)−φk(s))sµ+k 2/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−q/integraldisplay Rn|u(s,x)|pdxds /greaterorequalslantB1Kεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt 1(φk(t)−φk(s))sµ+k 2/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−q+(n−1)(1−p 2)+(k−µ)p 2(1−k)ds /greaterorsimilarεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt 1(φk(t)−φk(s))/an}b∇acketle{tAk(s)/an}b∇acket∇i}htµ+k 2(1−k)−n−1 2+1 p+(n−1)(1−p 2)+(k−µ)p 2(1−k)ds /greaterorsimilarεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt 1(φk(t)−φk(s))/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−(n−1 2+µ−k 2(1−k))p+(n−1 2+µ+k 2(1−k))+1 pds /greaterorsimilarεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt 1(φk(t)−φk(s))/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−1 1−kds/greaterorsimilarεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt 1φk(t)−φk(s) sds. Integrating by parts, we obtain /integraldisplayt 1φk(t)−φk(s) sds=/parenleftbig φk(t)−φk(s)/parenrightbig logs/vextendsingle/vextendsingle/vextendsingles=t s=1+/integraldisplayt 1φ′ k(s)logsds =/integraldisplayt 1s−klogsds/greaterorequalslantt−k/integraldisplayt 1logsds. Consequently, for t/greaterorequalslant3/2 U(t)/greaterorsimilarεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1t−k/integraldisplayt 1logsds/greaterorequalslantεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1t−k/integraldisplayt 2t/3logsds/greaterorequalslant(1/3)εp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1t1−klog(2t/3) /greaterorsimilarεplog(2t/3), where in the last line we applied t≈ /an}b∇acketle{tAk(t)/an}b∇acket∇i}ht1 1−kfort/greaterorequalslant1. Thus, the proof is over. In order to conclude the proof of Theorem 1.3 it remains to use an iteration argument together with a slicing procedure for the domain of integration. This proce dure consists in determining a sequence of lower bound estimates for U(t)(indexes by j∈N) and, then, proving that U(t)may not be finite for tover a certainε– dependent threshold by taking the limit as j→ ∞. Since the iteration frame (67) and the first lower bound estimate (72) are formally identical to those in [27, Section 2.3] (of course, for different values of the critical exponent p), the iteration argument can be rewritten verbatim as in [27 , Section 2.4]. Finally, we show how the previous steps can be adapted to the t reatment of the case µ∈[0,k]. According to Remark 4 , through the transformation v(t,x) =tµ−1u(t,x), we may consider the transformed semilinear Cauchy problem (57) for v. Note that v0.=u0andv1.=u1+(1−µ)u0satisfies the same assumptions for u0 andu1in the statement of Theorem 1.3 in this case (nonnegativenes s and nontriviality, compactly supported and belongingness to the energy space H1(Rn)×L2(Rn)). Of course, we may introduce the auxiliary function ξq(t,s,x;k,2−µ),ξq(t,s,x;k,2−µ)as in (58), (59) replacing µby2−µ. In Corollary 3.5, nevertheless, we have to replace the fundamental identity (53) by /integraldisplay Rnv(t,x)ξq(t,t,x;k,2−µ)dx=ε/integraldisplay Rnv0(x)ξq(t,1,x;k,2−µ)dx+εAk(t)/integraldisplay Rnv1(x)ηq(t,s,x;k,2−µ)dx +/integraldisplayt 1(φk(t)−φk(s))s(1−µ)(p−1)/integraldisplay Rn|v(s,x)|pηq(t,s,x;k,2−µ)dxds. 19As we have already pointed out in Remark 4, the estimates in (5 4) and (55) holds true in this case with 2−µinstead ofµ(we recall that this was the actual reason to consider the tra nsformed problem in place of the original one). Moreover, also the lower bound estimate i n (70) is valid for v, provided that we replace µ by2−µ. Accordingly to what we have just remarked, the suitable tim e – dependent functional to study for the transformed problem is V(t).=t1−µ+k 2/integraldisplay Rnv(t,x)ξq(t,t,x;k,2−µ)dx. In fact, Vsatisfies V(t)/greaterorsimilarεfort∈[1,T)and, furthermore, it is possible to derive for Vcompletely analogous iteration frame and first logarithmic lower bound , respectively, as the ones for Uin (67) and (72), respectively. We point out that both for the iteration frame and for the first logarithmic lower bound estimate the time – dependent factor t(1−µ)(p−1)in the nonlinearity compensates the modifications due to the replacement of µby2−µin the proofs of Propositions 3.8 and 3.10. 4 Critical case: part II In Section 2, we derived the upper bound for the lifespan in th e subcritical case, whereas in Section 3 we studied the critical case p=p0/parenleftbig k,n+µ 1−k/parenrightbig . It remains to consider the critical case p=p1(k,n), that is, whenµ/greaterorequalslantµ0(k,n). In this section, we are going to prove Theorem 1.4. In this cr itical case, our approach will be based on a basic iteration argument combined with the slicing procedure introduced for the first time in the paper [1]. The parameters characterizing the slicing procedure are given by the sequence {ℓj}j∈N, whereℓj.= 2−2−(j+1). As time – depending functional we consider the same one studi ed in Section 2, namely, U0defined in (11). Hence, since p=p1(k,n)is equivalent to the condition (1−k)n(p−1) = 2, (73) we can rewrite (23) as U0(t)/greaterorequalslantC/integraldisplayt 1τ−µ/integraldisplayτ 1sµ−2(U0(s))pdsdτ (74) for anyt∈(1,T)and for a suitable positive constant C >0. Let us underline that (74) will be the iteration frame in the iteration procedure for the critical case p=p1(k,n). We know that U0(t)/greaterorequalslantKεfor anyt∈(1,T)and for a suitable positive constant K, provided that u0,u1 are nonnegative, nontrivial and compactly supported (cf. t he estimate in (21)). Thus, U0(t)/greaterorequalslantCKpεp/integraldisplayt 1τ−µ/integraldisplayτ 1sµ−2dsdτ/greaterorequalslantCKpεp/integraldisplayt 1τ−µ−2/integraldisplayτ 1(s−1)µdsdτ =CKpεp µ+1/integraldisplayt 1τ−µ−2(τ−1)µ+1dτ/greaterorequalslantCKpεp µ+1/integraldisplayt ℓ0τ−µ−2(τ−1)µ+1dτ /greaterorequalslantCKpεp 3µ+1(µ+1)/integraldisplayt ℓ0τ−1dτ/greaterorequalslantCKpεp 3µ+1(µ+1)log/parenleftbiggt ℓ0/parenrightbigg (75) fort/greaterorequalslantℓ0= 3/2, where we used τ/lessorequalslant3(τ−1)forτ/greaterorequalslantℓ0in the second last step. Therefore, by using recursively (74), we prove now the seque nce of lower bound estimates U0(t)/greaterorequalslantKj/parenleftbigg log/parenleftbiggt ℓj/parenrightbigg/parenrightbiggσj fort/greaterorequalslantℓj (76) for anyj∈N, where{Kj}j∈N,{σ}j∈Nare sequences of positive reals that we determine afterward s in the inductive step. Clearly (76) for j= 0holds true thanks to (75), provided that K0= (CKpεp)/(3µ+1(µ+1)) andσ0= 1. Next we show the validity of (76) by using an inductive argume nt. Assuming that (76) is satisfied for some j/greaterorequalslant0, we prove (76) for j+1. According to this purpose, we plug (76) in (74), so, after sh rinking the domain of integration, we get U0(t)/greaterorequalslantCKp j/integraldisplayt ℓjτ−µ/integraldisplayτ ℓjsµ−2/parenleftBig log/parenleftBig s ℓj/parenrightBig/parenrightBigσjp dsdτ 20fort/greaterorequalslantℓj+1. If we shrink the domain of integration to [(ℓj/ℓj+1)τ,τ]in thes– integral (this operation is possible for τ/greaterorequalslantℓj+1), we find U0(t)/greaterorequalslantCKp j/integraldisplayt ℓj+1τ−µ−2/integraldisplayτ ℓjτ ℓj+1sµ/parenleftBig log/parenleftBig s ℓj/parenrightBig/parenrightBigσjp dsdτ /greaterorequalslantCKp j/integraldisplayt ℓj+1τ−µ−2/parenleftBig log/parenleftBig τ ℓj+1/parenrightBig/parenrightBigσjp/integraldisplayτ ℓjτ ℓj+1/parenleftBig s−ℓj ℓj+1τ/parenrightBigµ dsdτ =CKp j(µ+1)−1/parenleftBig 1−ℓj ℓj+1/parenrightBigµ+1/integraldisplayt ℓj+1τ−1/parenleftBig log/parenleftBig τ ℓj+1/parenrightBig/parenrightBigσjp dτ /greaterorequalslant2−(j+3)(µ+1)CKp j(µ+1)−1(1+pσj)−1/parenleftBig log/parenleftBig t ℓj+1/parenrightBig/parenrightBigσjp+1 fort/greaterorequalslantℓj+1, where in the last step we applied the inequality 1−ℓj/ℓj+1>2−(j+3). Hence, we proved (76) forj+1provided that Kj+1.= 2−(j+3)(µ+1)C(µ+1)−1(1+pσj)−1Kp jandσj+1.= 1+σjp. Let us establish a suitable lower bound for Kj. Using iteratively the relation σj= 1 +pσj−1and the initial exponent σ0= 1, we have σj=σ0pj+j−1/summationdisplay k=0pk=pj+1−1 p−1. (77) In particular, the inequality σj−1p+1 =σj/lessorequalslantpj+1/(p−1)yields Kj/greaterorequalslantL/parenleftbig 2µ+1p/parenrightbig−jKp j−1 (78) for anyj/greaterorequalslant1, whereL.= 2−2(µ+1)C(µ+1)−1(p−1)/p. Applying the logarithmic function to both sides of (78) and using the resulting inequality iteratively, we obt ain logKj/greaterorequalslantplogKj−1−jlog/parenleftbig 2µ+1p/parenrightbig +logL /greaterorequalslant.../greaterorequalslantpjlogK0−/parenleftBiggj−1/summationdisplay k=0(j−k)pk/parenrightBigg log/parenleftbig 2µ+1p/parenrightbig +/parenleftBiggj−1/summationdisplay k=0pk/parenrightBigg logL =pj/parenleftBigg log/parenleftbiggCKpεp 3µ+1(µ+1)/parenrightbigg −plog/parenleftbig 2µ+1p/parenrightbig (p−1)2+logL p−1/parenrightBigg +/parenleftbiggj p−1+p (p−1)2/parenrightbigg log/parenleftbig 2µ+1p/parenrightbig −logL p−1, where we applied again the identities in (37). Let us define j2=j2(n,p,k,µ)as the smallest nonnegative integer such that j2/greaterorequalslantlogL log/parenleftbig 2µ+1p/parenrightbig−p p−1. Consequently, for any j/greaterorequalslantj2the following estimate holds logKj/greaterorequalslantpj/parenleftBigg log/parenleftbiggCKpεp 3µ+1(µ+1)/parenrightbigg −plog/parenleftbig 2µ+1p/parenrightbig (p−1)2+logL p−1/parenrightBigg =pjlog(Nεp), (79) whereN.= 3−(µ+1)CKp(µ+1)−1/parenleftbig 2µ+1p/parenrightbig−p/(p−1)2 L1/(p−1). Combining (76), (77) and (79), we arrive at U0(t)/greaterorequalslantexp/parenleftbig pjlog(Nεp)/parenrightbig/parenleftBig log/parenleftBig t ℓj/parenrightBig/parenrightBigσj /greaterorequalslantexp/parenleftbig pjlog(Nεp)/parenrightbig/parenleftbig1 2logt/parenrightbig(pj+1−1)/(p−1) = exp/parenleftBig pjlog/parenleftBig 2−p/(p−1)Nεp(logt)p/(p−1)/parenrightBig/parenrightBig/parenleftbig1 2logt/parenrightbig−1/(p−1) fort/greaterorequalslant4and for any j/greaterorequalslantj2, where we employed the inequality log(t/ℓj)/greaterorequalslantlog(t/2)/greaterorequalslant(1/2)logtfort/greaterorequalslant4. Introducing the notation H(t,ε).= 2−p/(p−1)Nεp(logt)p/(p−1), the previous estimate may be rewritten as U0(t)/greaterorequalslantexp/parenleftbig pjlogH(t,ε)/parenrightbig/parenleftbig1 2logt/parenrightbig−1/(p−1)(80) 21fort/greaterorequalslant4and anyj/greaterorequalslantj2. If we fixε0=ε0(n,p,k,µ,R,u 0,u1)such that exp/parenleftBig 2N−(1−p)/pε−(p−1) 0/parenrightBig /greaterorequalslant4, then, for any ε∈(0,ε0]and fort>exp/parenleftbig 2N−(1−p)/pε−(p−1)/parenrightbig we havet/greaterorequalslant4andH(t,ε)>1. Therefore, for anyε∈(0,ε0]and fort>exp/parenleftbig 2N−(1−p)/pε−(p−1)/parenrightbig lettingj→ ∞ in (80) we see that the lower bound for U0(t)blows up and, consequently, U0(t)may not be finite as well. Summarizing, we proved that U0blows up in finite time and, moreover, we showed the upper bound esti mate for the lifespan T(ε)/lessorequalslantexp/parenleftBig 2N−(1−p)/pε−(p−1)/parenrightBig . Hence the proof of Theorem 1.4 in the critical case p=p1(k,n)is complete. 5 Final remarks According to the results we obtained in Theorems 1.2, 1.3 and 1.4 it is quite natural to conjecture that max/braceleftbig p0/parenleftbig k,n+µ 1−k/parenrightbig ,p1(k,n)/bracerightbig is the critical exponent for the semilinear Cauchy problem ( 4), although the global existence of small data solutions is completely open in the supercritical case. Fur thermore, this exponent is consistent with other models studied in the literature. In the flat case k= 0, this exponent coincide with max{pStr(n+µ),pFuj(n)}which in many subcases has been showed to be optimal in the case of semilinear wave equat ion with time – dependent scale – invariant damping, see [5, 8, 7, 22, 16, 34, 28, 31, 24, 25, 6] and referen ces therein for further details. On the other hand, in the undamped case µ= 0(that is, for the semilinear wave equation with speed of propagation t−k) the exponent max{p0(k,n),p1(k,n)}is consistent with the result for the generalized semilinear Tricomi equation (i.e., the semilinear wave equ ation with speed of propagation tℓ, whereℓ >0) obtained in the recent works [13, 14, 15, 21]. Clearly, in the very special case µ= 0andk= 0, our result is nothing but a blow-up result for the classical semilinear wave equation for exponents below pStr(n), which is well – known to be optimal (for a detailed historical overview on Strauss’ conjecture and a c omplete list of references we address the reader to the introduction of the paper [33]). As we have already explained in the introduction, for µ= 2andk= 2/3the equation in (4) is the semi- linear wave equation in the Einstein – de Sitter spacetime. I n particular, our result is a natural generalization of the results in [12, 27]. Furthermore, we underline explicitly the fact that the expo nentp0/parenleftbig k,n+µ 1−k/parenrightbig for (4) is obtained by the corresponding exponent in the not damped case µ= 0via a formal shift in the dimension of magnitudeµ 1−k. This phenomenon is due to the threshold nature of the time – de pendent coefficient of the damping term and it has been widely observed in the special case k= 0not only for the semilinear Cauchy problem with power nonlinearity but also with nonlinarity of derivative type|ut|p(see [32]) or weakly coupled system (see [2, 26, 32]). Finally, we have to point out that after the completion of the final version of this work, we found out the existence of the paper [35], where the same model is consider ed. We stress that the approach we used in the critical case is completely different, and that we slightly i mproved their result, by removing the assumption on the size of the support of the Cauchy data (cf. [35, Theorem 2.3]), even though we might not cover the full rangeµ∈[0,µ0(k,n)]in the critical case due to the assumption µ/ne}ationslash∈(k,2−k). Acknowledgments A. Palmieri is supported by the GNAMPA project ‘Problemi sta zionari e di evoluzione nelle equazioni di campo nonlineari dispersive’. The author would like to ackn owledge Karen Yagdjian (UTRGV), who first introduced him to the model considered in this work. References [1] Agemi, R., Kurokawa, Y., Takamura, H.: Critical curve fo rp-qsystems of nonlinear wave equations in three space dimensions. J. Differential Equations 167(1) (2000), 87–133. 22[2] Chen, W., Palmieri, A.: Weakly coupled system of semilin ear wave equations with distinct scale- invariant terms in the linear part. Z. Angew. Math. Phys. 70: 67, (2019) [3] Chen, W., Palmieri, A.: Nonexistence of global solution s for the semilinear Moore – Gibson – Thompson equation in the conservative case. Discrete and Continuous Dynamical Systems - A 40(9) (2020), 5513– 5540. doi: 10.3934/dcds.2020236 [4] Chen, W., Palmieri, A.: Blow-up result for a semilinear w ave equation with a nonlinear memory term. To appear in Springer INdAM Series. [5] D’Abbicco, M.: The threshold of effective damping for sem ilinear wave equations. Math. Meth. Appl. Sci.38(6) (2015), 10032–1045. [6] D’Abbicco, M.: The semilinear Euler-Poisson-Darboux e quation: a case of wave with critical dissipation. Preprint, arXiv:2008.08703v2 (2020). [7] D’Abbicco, M., Lucente, S.: NLWE with a special scale inv ariant damping in odd space dimension. In:Dynamical Systems, Differential Equations and Application s 10th AIMS Conference, Suppl. 2015, 312–319. [8] D’Abbicco, M., Lucente, S., Reissig, M.: A shift in the St rauss exponent for semilinear wave equations with a not effective damping. J. Differential Equations 259(10) (2015), 5040–5073. [9] Galstian, A., Kinoshita, T., Yagdjian, K.: A note on wave equation in Einstein and de Sitter space-time. J. Math. Phys. 51(5) (2010), 052501 [10] Galstian, A., Yagdjian, K.: Microlocal analysis for wa ves propagating in Einstein & de Sitter spacetime. Math. Phys. Anal. Geom. 17(1-2) (2014), 223–246. [11] Galstian, A., Yagdjian, K.: Global solutions for semil inear Klein-Gordon equations in FLRW spacetimes. Nonlinear Anal. 113(2015), 339–356. [12] Galstian, A., Yagdjian, K.: Finite lifespan of solutio ns of the semilinear wave equation in the Einstein-de Sitter spacetime. Reviews in Mathematical Physics (2019), doi: 10.1142/S0129055X2050018X [13] He, D., Witt, I., Yin, H.: On the global solution problem for semilinear generalized Tricomi equations, I.Calc. Var. Partial Differential Equations 56(2) (2017), Art. 21 [14] He, D., Witt, I., Yin, H.: On semilinear Tricomi equatio ns with critical exponents or in two space dimensions. J. Differential Equations 263(12) (2017), 8102–8137. 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[20] Lai, N.A., Takamura, H., Wakasa, K.: Blow-up for semili near wave equations with the scale invariant damping and super-Fujita exponent. J. Differential Equatio ns.263(2017), 5377–5394. [21] Lin, J., Tu, Z.: Lifespan of semilinear generalized Tri comi equation with Strauss type exponent. Preprint, arXiv:1903.11351v2 (2019). [22] Nunes do Nascimento, W., Palmieri, A., Reissig, M.: Sem i-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation. Math. Nachr. 290(11-12) (2017), 1779–1805. [23] Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C. W. (Eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, New York, NY (2010). 23[24] Palmieri, A.: Global existence results for a semilinea r wave equation with scale-invariant damping and mass in odd space dimension. In M. D’Abbicco et al. (eds.), New Tools for Nonlinear PDEs and Application , Trends in Mathematics, https://doi.org/10.1007/978-3- 030-10937-0_12 [25] Palmieri, A.: A global existence result for a semilinea r wave equation with scale-invariant damping and mass in even space dimension. Math Meth Appl Sci. 42(8) (2019), 2680–2706. [26] Palmieri, A.: A note on a conjecture for the critical cur ve of a weakly coupled system of semi- linear wave equations with scale-invariant lower order ter ms.Math Meth Appl Sci. (2020) 1–30. https://doi.org/10.1002/mma.6412 [27] Palmieri, A.: Lifespan estimates for local solutions t o the semilinear wave equation in Einstein – de Sitter spacetime. Preprint, arXiv:2009.04388 (2020). [28] Palmieri, A., Reissig, M.: A competition between Fujit a and Strauss type exponents for blow-up of semi-linear wave equations with scale-invariant damping a nd mass. J. Differential Equations 266(2-3) (2019), 1176–1220. 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2020-09-10

We prove by using an iteration argument some blow-up results for a semilinear damped wave equation in generalized Einstein-de Sitter spacetime with a time-dependent coefficient for the damping term and power nonlinearity. Then, we conjecture an expression for the critical exponent due to the main blow-up results, which is consistent with many special cases of the considered model and provides a natural generalization of Strauss exponent. In the critical case, we consider a non-autonomous and parameter-dependent Cauchy problem for a linear ODE of second-order, whose explicit solutions are determined by means of special functions' theory.

Blow-up results for semilinear damped wave equations in Einstein-de Sitter spacetime

2009.05372v1

arXiv:1609.01063v2 [math.AP] 23 Nov 2016REMARKS ON AN ELLIPTIC PROBLEM ARISING IN WEIGHTED ENERGY ESTIMATES FOR WAVE EQUATIONS WITH SPACE-DEPENDENT DAMPING TERM IN AN EXTERIOR DOMAIN MOTOHIRO SOBAJIMA AND YUTA WAKASUGI Abstract. This paper is concerned with weighted energy estimates and d if- fusion phenomena for the initial-boundary problem of the wa ve equation with space-dependent damping term in an exterior domain. In this analysis, an el- liptic problem was introduced by Todorova and Yordanov. Thi s attempt was quite useful when the coefficient of the damping term is radial ly symmetric. In this paper, by modifying their elliptic problem, we establi sh weighted energy estimates and diffusion phenomena even when the coefficient of the damping term is not radially symmetric. 1.Introduction LetN≥2. We consider the wave equation with space-dependent damping te rm in an exterior domain Ω ⊂RNwith a smooth boundary: utt−∆u+a(x)ut= 0, x∈Ω, t >0, u(x,t) = 0, x ∈∂Ω, t >0, (u,ut)(x,0) = (u0,u1)(x), x∈Ω,(1.1) where we denote by ∆ the usual Laplacian in RNand byutanduttthe first and second derivative of uwith respect to the variable t, andu=u(x,t) is a real-valued unknown function. The coefficient of the damping term a(x) satisfies a∈C2(Ω), a(x)>0 onΩ and lim |x|→∞/parenleftBig /a\}b∇acketle{tx/a\}b∇acket∇i}htαa(x)/parenrightBig =a0 (1.2) with some constants α∈[0,1) anda0∈(0,∞), where /a\}b∇acketle{ty/a\}b∇acket∇i}ht= (1+|y|2)1 2fory∈RN. In this moment, the initial data ( u0,u1) are assumed to have compact supports in Ω and to satisfy the compatibility condition of order k≥1: (uℓ−1,uℓ)∈(H2∩H1 0(Ω))×H1 0(Ω),for allℓ= 1,...,k (1.3) whereuℓis successively defined by uℓ= ∆uℓ−2−a(x)uℓ−1(ℓ= 2,...,k). We note that existence and uniqueness of solution to the problem (1.1) have been discussed (see e.g., Ikawa [2, Theorem 2]). It is proved in Matsumura [4] that if Ω = RNanda(x)≡1, then the solution u of (1.1) satisfies the energy decay estimate/integraldisplay RN(|∇u(x,t)|2+|ut(x,t)|2)dx≤C(1+t)−N 2−1/ba∇dbl(u0,u1)/ba∇dbl2 H1×L2, Key words and phrases. Damped wave equation; elliptic problem; exterior domain; w eighted energy estimates; diffusion phenomena. 12 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI where the constant Cdepends on the size of the supprot of initial data. Moreover, it is shown in Nishihara [7] that uhas the same asymptotic behavior as the one of the problem/braceleftbigg vt−∆v= 0, x ∈RN, t >0, v(x,0) =u0(x)+u1(x), x∈RN.(1.4) In particular, we have /ba∇dblu(·,t)−v(·,t)/ba∇dblL2=o(t−N 4) ast→ ∞. Energy decay properties of solutions to (1.1) for general cases with a(x)≥ /a\}b∇acketle{tx/a\}b∇acket∇i}ht−α(0≤α≤1) have been dealt with by Matsumura [5]. On the other hand, Mochizuki [6] proved that if 0 ≤a(x)≤C/a\}b∇acketle{tx/a\}b∇acket∇i}ht−αfor some α >1, then the energy of the solution to (1.1) does not vanish as t→ ∞for suitable initial data. (The solution has an asymptotic behavior similar to the solution of the usual wave equation without damping). Therefore one can expect that diffusio n phenomena occur only when a(x)≥C/a\}b∇acketle{tx/a\}b∇acket∇i}ht−αforα≤1. In this paper, we discuss precise decay rates of the weighted ener gy/integraldisplay RN(|∇u(x,t)|2+|ut(x,t)|2)Φ(x,t)dx with a special weight function Φ(x,t) = exp/parenleftbigg βA(x) 1+t/parenrightbigg (for some A∈C2(RN) andβ >0) which is introduced by Todorova and Yordanov [12] based on the ideas in [11] and in [3]. They proved weighted energy e stimates/integraldisplay RNa(x)|u(x,t)|2Φ(x,t)dx≤C(1+t)−N−α 2−α+ε, /integraldisplay RN(|∇u(x,t)|2+|ut(x,t)|2)Φ(x,t)dx≤C(1+t)−N−α 2−α−1+ε whena(x) is radially symmetric and satisfies (1.2). After that, Radu, Todoro va and Yordanov [8] extended it to higher-order derivatives. In [13], t he second author proved diffusion phenomena for (1.1) with Ω = RNanda(x) =/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α(α∈[0,1)) by comparing the solution of the following problem a(x)vt−∆v= 0, x ∈RN, t >0, v(x,0) =u0(x)+1 a(x)u1(x), x∈RN.(1.5) In [10], diffusion phenomena for (1.1) with an exterior domain and for g eneral radially symmetric damping term are obtained. However, the weighte d energy esti- matesand diffusion phenomenafor (1.1) with non-radially symmetric damping are still remaining open. The difficulty seems to come from the choice o f auxiliary function Ain the weighted energy, which strongly depends on the existence of positive solution to the Poisson equation ∆ A(x) =a(x). In fact, an example of non-existence of positive solution to ∆ A=afor non-radial a(x) is shown in [10]. Radu, Todorova and Yordanov [9] considered the case Ω = RNand used a solu- tionA∗(x) of ∆A∗=a1(1 +|x|)−αwitha1>0 satisfying a1(1 +|x|)−α≥a(x) forx∈RN, that is, A∗(x) is a subsolution of the equation ∆ A=a. In general one cannot obtain the optimal decay estimate via this choice becaus e of the luck of the precise behavior of a(x) at the spatial infinity which can be expected toWAVE EQUATION WITH SPACE-DEPENDENT DAMPING 3 determine the precise decay late of weighted energy estimates. Ou r main idea to overcome this difficulty is to weaken the equality ∆ A=aand consider the in- equality (1 −ε)a≤∆A≤(1+ε)a, and to construct a solution having appropriate behavior, we employ a cut-off argument. The aim of this paper is to give a proof of Ikehata–Todorova–Yorda nov type weighted energy estimates for (1.1) with non-radially symmetric dam ping and to obtain diffusion phenomena for (1.1) under the compatibility condition of order 1 and the condition (1.2) (without any restriction). This paper is originated as follows. In Section 2, we discuss related ellip tic and parabolic problems. The weighted energy estimates for (1.1) ar e established in Section 3 (Proposition 3.5). Section 4 is devoted to show diffusion ph enomena (Proposition 4.1). 2.Related elliptic and parabolic problems 2.1.An elliptic problem for weighted energy estimates. As we mentioned above, in general, existence of positive solutions to the Poisson equ ation ∆A(x) = a(x) is false for non-radial a(x). Thus, we weaken this equation and consider the following inequality (1−ε)a(x)≤∆A(x)≤(1+ε)a(x), x∈Ω, (2.1) whereε∈(0,1) is a parameter. Here we construct a positive solution Aof (2.1) satisfying A1ε/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α≤A(x)≤A2ε/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α, (2.2) |∇A(x)|2 a(x)A(x)≤2−α N−α+ε (2.3) for some constants A1ε,A2ε>0. Lemma 2.1. For every ε∈(0,1), there exists Aε∈C2(Ω)such that Aεsatisfies (2.1)–(2.3). Proof.Firstly, we extend a(x) as a positive function in C2(RN); note that this is possible by virtue of the smoothness of ∂Ω. To simplify the notation, we use the same symbol a(x) as a function defined on RN. We construct a solution of approximated equation ∆Aε(x) =aε(x), x∈RN for some aε∈C2(RN) satisfying (1−ε)a(x)≤aε(x)≤(1+ε)a(x), x∈RN. (2.4) Noting (1.2), we divide a(x) asa(x) =b1(x)+b2(x) with b1(x) = ∆/parenleftbigga0 (N−α)(2−α)/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α/parenrightbigg =a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α+a0α N−α/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α−2, b2(x) =a(x)−a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α−a0α N−α/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α−2. Then we have lim |x|→∞/parenleftbiggb2(x) a(x)/parenrightbigg = lim |x|→∞/bracketleftbigg1 /a\}b∇acketle{tx/a\}b∇acket∇i}htαa(x)/parenleftbigg /a\}b∇acketle{tx/a\}b∇acket∇i}htαa(x)−a0−a0α N−α/a\}b∇acketle{tx/a\}b∇acket∇i}ht−2/parenrightbigg/bracketrightbigg = 0.(2.5)4 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI Letε∈(0,1) be fixed. Then by (2.5) there exists a constant Rε>0 such that|b2(x)| ≤εa(x) forx∈RN\B(0,Rε). Here we introduce a cut-off function ηε∈C∞ c(RN,[0,1]) such that ηε≡1 onB(0,Rε). Define aε(x) :=b1(x)+ηε(x)b2(x) =a(x)−(1−ηε(x))b2(x), x∈RN. Thenaε(x) =a(x) onB(0,Rε) and for x∈RN\B(0,Rε), /vextendsingle/vextendsingle/vextendsingle/vextendsingleaε(x) a(x)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle= (1−ηε(x))|b2(x)| a(x)≤ε and therefore (2.4) is verified. Next we define B1ε(x) :=a0 (N−α)(2−α)/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α, x∈RN, B2ε(x) :=−/integraldisplay RNN(x−y)ηε(y)b2(y)dy, x∈RN, whereNis the Newton potential given by N(x) = 1 2πlog1 |x|ifN= 2, Γ(N 2+1) N(N−2)πN 2|x|2−NifN≥3. Then we easily see that ∆ B1ε(x) =b1(x) and ∆ B2ε=ηε(x)b2(x). Moreover, noting that supp( ηεb2) is compact, we see from a direct calculation that there exist a constant Mε>0 such that |B2ε(x)| ≤/braceleftBigg Mε(1+log/a\}b∇acketle{tx/a\}b∇acket∇i}ht) ifN= 2, Mε/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−NifN≥3,|∇B2ε(x)| ≤Mε/a\}b∇acketle{tx/a\}b∇acket∇i}ht1−N, x∈RN. This yields that Bε:=B1ε+B2εis bounded from below and positive for x∈RN with sufficiently large |x|. Moreover, we have lim |x|→∞/parenleftBig /a\}b∇acketle{tx/a\}b∇acket∇i}htα−2Bε(x)/parenrightBig =a0 (N−α)(2−α) and lim |x|→∞/parenleftbigg|∇Bε(x)|2 a(x)Bε(x)/parenrightbigg = lim |x|→∞/parenleftBigg 1 /a\}b∇acketle{tx/a\}b∇acket∇i}htαa(x)·1 /a\}b∇acketle{tx/a\}b∇acket∇i}htα−2Bε(x)/vextendsingle/vextendsingle/vextendsingle/vextendsinglea0 N−α/a\}b∇acketle{tx/a\}b∇acket∇i}ht−1x+/a\}b∇acketle{tx/a\}b∇acket∇i}htα−1∇B2ε(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg =2−α N−α. Using the same argument as in the proof of [10, Lemma 3.1], we can see that there exists a constant λε≥0 such that Aε(x) :=λε+Bε(x) satisfies (2.1)-(2.3). /squareWAVE EQUATION WITH SPACE-DEPENDENT DAMPING 5 2.2.A parabolic problem for diffusion phenomena. Here we consider Lp-Lq type estimates for solutions to the initial-boundary value problem of the following parabolic equation a(x)wt−∆w= 0, x∈Ω, t >0, w(x,t) = 0, x ∈∂Ω, t >0, w(x,0) =f(x), x∈Ω.(2.6) Here we introduce a weighted Lp-spaces Lp dµ:=/braceleftBigg f∈Lp loc(Ω) ;/ba∇dblf/ba∇dblLp dµ:=/parenleftbigg/integraldisplay Ω|f(x)|pa(x)dx/parenrightbigg1 p <∞/bracerightBigg ,1≤p <∞ which is quite reasonable because the corresponding elliptic operato ra(x)−1∆ can be regarded as a symmetric operator in L2 dµ. TheLp-Lqtype estimates for the semigroup associated with the Friedrichs’ extension −L∗(inL2 dµ) of−a(x)−1∆ are stated in [10]. The proof is based on Beurling–Deny’s criterion and Gagliardo–Nirenberg inequality. Proposition 2.2 ([10, Proposition 2.6]) .LetetL∗be a semigroup generated by L∗. For every f∈L1 dµ∩L2 dµ, we have /ba∇dbletL∗f/ba∇dblL2 dµ≤Ct−N−α 2(2−α)/ba∇dblf/ba∇dblL1 dµ(2.7) and /ba∇dblL∗etL∗f/ba∇dblL2 dµ≤Ct−N−α 2(2−α)−1/ba∇dblf/ba∇dblL1 dµ. (2.8) 3.Weighted energy estimates In this section we establish weighted energy estimates for solutions of (1.1) by introducing Ikehata–Todorova–Yordanov type weight function w ith an auxiliary function Aεconstructed in Subsection 2.1. To begin with, let us recall the finite speed propagation property of the wave equation (see [2]). Lemma 3.1 (Finite speed of propagation) .Letube the solution of (1.1)with the initial data (u0,u1)satisfying supp(u0,u1)⊂B(0,R0) ={x∈Ω;|x| ≤R0}. Then, one has suppu(·,t)⊂ {x∈Ω ;|x| ≤R0+t} and therefore |x|/(R0+1+t)≤1fort≥0andx∈suppu(·,t). Before introducing a weight function, we also recall two identities fo r partial energy functionals proved in [10]. Lemma 3.2 ([10, Lemma 3.7]) .LetΦ∈C2(Ω×[0,∞))satisfyΦ>0and∂tΦ<0 and letube a solution of (1.1). Then d dt/bracketleftbigg/integraldisplay Ω/parenleftBig |∇u|2+|ut|2/parenrightBig Φdx/bracketrightbigg =/integraldisplay Ω(∂tΦ)−1/vextendsingle/vextendsingle∂tΦ∇u−ut∇Φ/vextendsingle/vextendsingle2dx +/integraldisplay Ω/parenleftBig −2a(x)Φ+∂tΦ−(∂tΦ)−1|∇Φ|2/parenrightBig |ut|2dx.6 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI Lemma 3.3 ([10, Lemma 3.9]) .LetΦ∈C2(Ω×[0,∞))satisfyΦ>0and∂tΦ<0 and letube a solution to (1.1). Then, we have d dt/bracketleftbigg/integraldisplay Ω/parenleftBig 2uut+a(x)|u|2/parenrightBig Φdx/bracketrightbigg = 2/integraldisplay Ωuut(∂tΦ)dx+2/integraldisplay Ω|ut|2Φdx−2/integraldisplay Ω|∇u|2Φdx +/integraldisplay Ω/parenleftbig a(x)∂tΦ+∆Φ/parenrightbig |u|2dx. Here we introduce a weight function for weighted energy estimates , which is a modification of the one in Todorova-Yordanov [12]. Definition 3.4. Defineh:=2−α N−αand forε∈(0,1), Φε(x,t) = exp/parenleftbigg1 h+2εAε(x) 1+t/parenrightbigg , (3.1) whereAεis given in Lemma 2.1. And define for t≥0, E∂x(t;u) :=/integraldisplay Ω|∇u|2Φεdx, E ∂t(t;u) :=/integraldisplay Ω|ut|2Φεdx, (3.2) Ea(t;u) :=/integraldisplay Ωa(x)|u|2Φεdx, E ∗(t;u) := 2/integraldisplay ΩuutΦεdx, (3.3) and also define E1(t;u) :=E∂x(t;u)+E∂t(t;u)andE2(t;u) :=E∗(t;u)+Ea(t;u). Now we are in a position to state our main result for weighted energy e stimates for solutions of (1.1). Proposition 3.5. Assume that (u0,u1)satisfies supp(u0,u1)⊂B(0,R0)and the compatibility condition of order k0≥1. Letube a solution of the problem (1.1). For every δ >0and0≤k≤k0−1, there exist ε >0andMδ,k,R0>0such that for every t≥0, (1+t)N−α 2−α+2k+1−δ/parenleftBig E∂x(t;∂k tu)+E∂t(t;∂k tu)/parenrightBig +(1+t)N−α 2−α+2k−δEa(t;∂k tu) ≤Mδ,k,R0/ba∇dbl(u0,u1)/ba∇dbl2 Hk+1×Hk(Ω). To prove, this, we prepare the following two lemmas. Lemma 3.6. Fort≥0, we have 1−ε h+2ε1 1+tEa(t;u)≤E∂x(t;u). (3.4) Proof.As in the proof of [10, Lemma 3.6], by integration by parts we have /integraldisplay Ω∆(logΦ ε)|u|2Φεdx=/integraldisplay Ω/parenleftbigg ∆Φε−|∇Φε|2 Φε/parenrightbigg |u|2dx≤/integraldisplay Ω|∇u|2Φεdx. Noting that ∆(logΦ ε(x)) =1 h+2ε∆Aε(x) 1+t≥1−ε h+2εa(x) 1+t, we have (3.4). /square In order to clarify the effect of the finite propagation property, w e now put a1:= inf x∈Ω/parenleftBig /a\}b∇acketle{tx/a\}b∇acket∇i}htαa(x)/parenrightBig . ThenWAVE EQUATION WITH SPACE-DEPENDENT DAMPING 7 Lemma 3.7. Fort≥0, we have E∂t(t;u)≤1 a1(R0+1+t)αEa(t;∂tu), (3.5) /integraldisplay ΩAε(x) a(x)|ut|2Φεdx≤A2ε a1(R0+1+t)2E∂t(t;u), (3.6) |E∗(t;u)| ≤2√a1(R0+1+t)α 2/radicalbig Ea(t;u)E∂t(t;u).(3.7) Proof.Bya(x)−1≤a−1 1/a\}b∇acketle{tx/a\}b∇acket∇i}htα≤a−1 1(1+|x|)αand the finite propagation property we have/integraldisplay Ω|ut|2Φεdx=/integraldisplay Ωa(x) a(x)|ut|2Φεdx≤1 a1(R0+1+t)αEa(t;∂tu). Using the Cauchy-Schwarz inequality and the above inequality yields ( 3.6): /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay ΩuutΦεdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ≤/parenleftbigg/integraldisplay Ω|u|2Φεdx/parenrightbigg/parenleftbigg/integraldisplay Ω|ut|2Φεdx/parenrightbigg ≤(R0+1+t)α a1/parenleftbigg/integraldisplay Ωa(x)|u|2Φεdx/parenrightbigg E∂t(t;u) ≤(R0+1+t)α a1Ea(t;u)E∂t(t;u). We can prove (3.7) in a similar way. /square Lemma 3.8. (i) For every t≥0, we have d dtE1(t;u)≤ −Ea(t;∂tu). (3.8) (ii)For every ε∈(0,1 3)andt≥0, d dtE2(t;u)≤ −1−3ε 1−εE∂x(t;u)+/parenleftbigg2 a1+A2ε(R0+1)2 εa2 1/parenrightbigg (R0+1+t)αEa(t;∂tu). (3.9) Proof.Noting (2.3), we have −2a(x)Φε+∂tΦε−(∂tΦε)−1|∇Φε|2 =/parenleftbigg −2a(x)−Aε(x) (h+2ε)(1+t)2+1 h+2ε|∇Aε(x)|2 Aε(x)/parenrightbigg Φε ≤/parenleftbigg −2a(x)+h+ε h+2εa(x)/parenrightbigg Φε ≤ −a(x)Φε. This implies (3.8). On the other hand, from (2.3) and (2.1) we see a(x)∂tΦε+∆Φε=1 h+2ε/parenleftbigg −a(x)Aε(x) (1+t)2+|∇Aε(x)|2 (h+2ε)(1+t)2+∆Aε(x) 1+t/parenrightbigg Φε ≤1 h+2ε/parenleftbigg −a(x)Aε(x) (1+t)2+(h+ε)a(x)Aε(x) (h+2ε)(1+t)2+(1+ε)a(x) 1+t/parenrightbigg Φε ≤/parenleftbigg −ε (h+2ε)2a(x)Aε(x) (1+t)2+1+ε h+2εa(x) 1+t/parenrightbigg Φε.8 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI Therefore combining it with Lemma 3.6, we have /integraldisplay Ω/parenleftbig a(x)∂tΦε+∆Φε/parenrightbig |u|2dx ≤1+ε 1−ε/integraldisplay Ω|∇u|2Φεdx−ε (h+2ε)21 (1+t)2/integraldisplay Ωa(x)Aε(x)|u|2Φεdx. Using (3.6), we have 2/integraldisplay Ωuut(∂tΦε)dx =−2 h+2ε1 (1+t)2/integraldisplay ΩuutAε(x)Φεdx ≤2 h+2ε1 (1+t)2/parenleftbigg/integraldisplay Ωa(x)Aε(x)|u|2Φεdx/parenrightbigg1 2/parenleftbigg/integraldisplay ΩAε(x) a(x)|ut|2Φεdx/parenrightbigg1 2 ≤2(R0+1) h+2ε1 1+t/parenleftbigg/integraldisplay Ωa(x)Aε(x)|u|2Φεdx/parenrightbigg1 2/parenleftbiggA2ε a1E∂t(t;u)/parenrightbigg1 2 ≤ε (h+2ε)21 (1+t)2/integraldisplay Ωa(x)Aε(x)|u|2Φεdx+A2ε(R0+1)2 εa1E∂t(t;u). Applying (3.5), we obtain (3.9). /square Lemma 3.9. The following assertions hold: (i)Sett∗(R0,α,m) := max/braceleftbigg/parenleftBig 2m a1/parenrightBig1 1−α,R0+1/bracerightbigg . Then for every t,m≥0and t1≥t∗(R0,α,m), d dt/parenleftBig (t1+t)mE1(t;u)/parenrightBig ≤m(t1+t)m−1E∂x(t;u)−1 2(t1+t)mEa(t;∂tu).(3.10) (ii)for every t,λ≥0andt2≥R0+1, d dt/parenleftBig (t2+t)λE2(t;u)/parenrightBig ≤λ(1+ε)(t2+t)λ−1Ea(t;u)−1−3ε 1−ε(t2+t)λE∂x(t;u) +/parenleftbigg2 a1+A2ε(R0+1)2 εa2 1+λ 2εa2 1t1−α 2/parenrightbigg (t2+t)λ+αEa(t;∂tu).(3.11) (iii)In particular, setting ν:=4 a1+2A2ε(R0+1)2 εa2 1+1 4εa1, t∗∗(ε,R0,α,λ) := max/braceleftBigg/parenleftbigg(1−ε)(λ+α)ν ε/parenrightbigg1 1−α ,/parenleftbigg2(λ+α) a1/parenrightbigg1 1−α ,R0+1/bracerightBigg , one has that for t,λ≥0andt3≥t∗∗(ε,R0,α,λ), d dt/parenleftBig ν(t3+t)λ+αE1(t;u)+(t3+t)λE2(t;u)/parenrightBig ≤ −1−4ε 1−ε(t3+t)λE∂x(t;u)+λ(1+ε)(t3+t)λ−1Ea(t;u).(3.12)WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 9 Proof.(i)Letm≥0 be fixed and let t1≥t∗(R0,α,m). Using (3.8) and (3.5), we have (t1+t)−md dt/parenleftBig (t1+t)mE1(t;u)/parenrightBig ≤m t1+tE∂x(t;u)+m t1+tE∂t(t;u)+d dtE1(t;u) ≤m t1+tE∂x(t;u)+m t1+tE∂t(t;u)−Ea(t;∂tu) ≤m t1+tE∂x(t;u)+/parenleftbiggm(R0+1+t)α a1(t1+t)−1/parenrightbigg Ea(t;∂tu). Therefore we obtain (3.10). (ii)Fort≥0, andt≥R0+1, (t2+t)−λd dt/parenleftBig (t2+t)λE2(t;u)/parenrightBig ≤λ t2+tE∗(t;u)+λ t2+tEa(t;u)+d dtE2(t;u) ≤λ t2+tE∗(t;u)+λ t2+tEa(t;u)−1−3ε 1−εE∂x(t;u) +/parenleftbigg2 a1+A2ε(R0+1)2 εa2 1/parenrightbigg (R0+1+t)αEa(t;∂tu). Noting that by (3.7) and (3.5), λ t2+tE∗(t;u)≤2λ(R0+1+t)α a1(t2+t)/radicalbig Ea(t;u)Ea(t;∂tu) ≤λε t2+tEa(t;u)+λ εa2 1(R0+1+t)2α t2+tEa(t;∂tu) ≤λε t2+tEa(t;u)+λ εa2 1t1−α 2(t2+t)αEa(t;∂tu), we deduce (3.11). (iii)Combining (3.10) with m=λ+αand (3.11), we have for t3≥t∗∗(ε,R0,α,λ) andt≥0, d dt/parenleftBig ν(t3+t)λ+αE1(t;u)+(t3+t)λE2(t;u)/parenrightBig ≤/parenleftbigg ν(λ+α)(t3+t)α−1−1−3ε 1−ε/parenrightbigg (t3+t)λE∂x(t;u)+λ(1+ε)(t3+t)λ−1Ea(t;u) +/parenleftbigg2 a1+A2ε(R0+1)2 εa2 1+λ 2εa2 1t1−α 3−ν 2/parenrightbigg (t3+t)λ+αEa(t;∂tu) ≤ −1−4ε 1−ε(t3+t)λE∂x(t;u)+λ(1+ε)(t3+t)λ−1Ea(t;u). This proves the assertion. /square10 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI Proof of Proposition 3.5. Firstly, by (3.7) we observe that ν(t3+t)αE1(t;u)+E2(t;u)≥4 a1(t3+t)αE1(t;u)−|E∗(t;u)|+Ea(t;u) ≥4 a1(t3+t)αE∂t(t;u) −2√a1(t3+t)α 2/radicalbig Ea(t;u)E∂t(t;u)+Ea(t;u) ≥3 4Ea(t;u). By using the above estimate, we prove the assertion via mathematic al induction. Step 1 ( k= 0).By (3.12) using Lemma 3.6 implies that d dt/parenleftBig ν(t3+t)λ+αE1(t;u)+(t3+t)λE2(t;u)/parenrightBig ≤/parenleftbigg −1−4ε 1−ε+λ(1+ε)(h+2ε) 1−ε/parenrightbigg (t3+t)λE∂x(t;u). Therefore taking λ0=(1−ε)(1−4ε) (1+ε)(h+2ε), (λ0↑h−1asε↓0) we have d dt/parenleftBig ν(t3+t)λ0+αE1(t;u)+(t3+t)λ0E2(t;u)/parenrightBig ≤ −ε(1−4ε) 1−ε(t3+t)λ0E∂x(t;u). Integrating over (0 ,t) with respect to t, we see 3 4(t3+t)λ0Ea(t;u)+ε(1−4ε) 1−ε/integraldisplayt 0(t3+s)λ0E∂x(s;u)ds ≤ν(t3+t)λ0+αE1(t;u)+(t3+t)λ0E2(t;u)+ε(1−4ε) 1−ε/integraldisplayt 0(t3+s)λ0E∂x(s;u)ds ≤νtλ0+α 3E1(0;u)+tλ0 3E2(0;u). Using (3.10) with m=λ0+1 and integrating over (0 ,t), we obtain (t3+t)λ0+1E1(t;u)+1 2/integraldisplayt 0(t3+s)λ0+1Ea(s;∂tu)ds ≤tλ0+1 3E1(0;u)+(λ0+1)/integraldisplayt 0(t3+s)λ0E∂x(s;u)ds ≤tλ0+1 3E1(0;u)+(λ0+1)(1−ε) ε(1−4ε)/parenleftBig νtλ0+α 3E1(0;u)+tλ0 3E2(0;u)/parenrightBig . This proves the desired assertion with k= 0 and also the integrability of ( t3+ s)λ0+1Ea(s;∂tu). Step 2 ( 1< k≤k0−1).Suppose that for every t≥0, (1+t)λ0+2k−1E1(t;∂k−1 tu)+(1+t)λ0+2k−2Ea(t;∂k−1 tu)≤Mε,k−1/ba∇dbl(u0,u1)/ba∇dbl2 Hk×Hk−1(Ω) and additionally, /integraldisplayt 0(1+s)λ0+2k−1Ea(s;∂k tu)ds≤M′ ε,k−1/ba∇dbl(u0,u1)/ba∇dbl2 Hk×Hk−1(Ω). Since the initial value ( u0,u1) satisfies the compatibility condition of order k,∂k tu is also a solution of (1.1) with replaced ( u0,u1) with (uk−1,uk). Applying (3.12)WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 11 withλ=λ0+ 2k, putting t3k=t∗∗(ε,R0,α,λ0+ 2k) (see Lemma 3.9 (iii)) and integrating over (0 ,t), we have 3 4(t3k+t)λ0+2kEa(t;∂k tu)+1−4ε 1−ε/integraldisplayt 0(t3k+s)λ0+2kE∂x(s;∂k tu)ds ≤ν(t3k+t)λ0+2k+αE1(t;∂k tu)+(t3k+t)λ0+2kE2(t;∂k tu) +1−4ε 1−ε/integraldisplayt 0(t3k+s)λ0+2kE∂x(s;∂k tu)ds ≤νtλ0+2k+α 3kE1(0;∂k tu)+tλ0+2k 3kE2(0;∂k tu) +(λ0+2k)(1+ε)/integraldisplayt 0(t3k+s)λ0+2k−1Ea(s;∂k tu)ds ≤νtλ0+2k+α 3kE1(0;∂k tu)+tλ0+2k−1 3kE2(0;∂k tu) +(λ0+2k)(1+ε)M′ ε,k−1/ba∇dbl(u0,u1)/ba∇dbl2 Hk×Hk−1(Ω). Moreover, from (3.10) with m=λ0+2k+1 we have (t3k+t)λ0+2k+1E1(t;∂k tu)+1 2/integraldisplayt 0(t3k+s)λ0+2k+1Ea(s;∂k+1 tu)ds ≤tλ0+2k+1 3kE1(0;∂k tu)+(λ0+2k+1)/integraldisplayt 0(t3k+s)λ0+2kE∂x(s;∂k tu)ds ≤M′′ ε,k/parenleftBig E1(0;∂k tu)+E2(0;∂k tu)+/ba∇dbl(u0,u1)/ba∇dbl2 Hk×Hk−1(Ω)/parenrightBig with some constant M′′ ε,k>0. By induction we obtain the desired inequalities for allk≤k0−1. /square 4.Diffusion phenomena as an application of weighted energy estimates Proposition 4.1. Assume that (u0,u1)∈(H2∩H1 0(Ω))×H1 0(Ω)and suppose that supp(u0,u1)⊂B(0,R0). Letube the solution of (1.1). Then for every ε >0, there exists a constant Cε,R0>0such that /vextenddouble/vextenddouble/vextenddoubleu(·,t)−etL∗[u0+a(·)−1u1]/vextenddouble/vextenddouble/vextenddouble L2 dµ≤Cε,R0(1+t)−N−α 2(2−α)−1−α 2−α+ε/ba∇dbl(u0,u1)/ba∇dblH2×H1. To prove Proposition 4.1 we use the following lemma stated in [10, Sectio n 4]. Lemma 4.2. Assume that (u0,u1)∈(H2∩H1 0(Ω))×H1 0(Ω)and suppose that supp(u0,u1)⊂ {x∈Ω;|x| ≤R0}. Then for every t≥0, u(x,t)−etL∗[u0+a(·)−1u1] =−/integraldisplayt t/2e(t−s)L∗[a(·)−1utt(·,s)]ds −et 2L∗[a(·)−1ut(·,t/2)] −/integraldisplayt/2 0L∗e(t−s)L∗[a(·)−1ut(·,s)]ds, (4.1) whereL∗is the (negative) Friedrichs extension of −L=−a(x)−1∆inL2 dµ.12 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI Proof of Proposition 4.1. First we show the assertion for ( u0,u1) satisfying the compatibility condition of order 2. Taking L2 dµ-norm of both side, we have /vextenddouble/vextenddouble/vextenddoubleu(x,·)−etL∗[u0+a(·)−1u1]/vextenddouble/vextenddouble/vextenddouble L2 dµ≤ J1(t)+J2(t)+J3(t), where J1(t) :=/integraldisplayt t/2/vextenddouble/vextenddoublee(t−s)L∗[a(·)−1utt(·,s)]/vextenddouble/vextenddouble L2 dµds, J2(t) :=/vextenddouble/vextenddoubleet 2L∗[a(·)−1ut(·,t/2)]/vextenddouble/vextenddouble L2 dµ, J3(t) :=/integraldisplayt/2 0/vextenddouble/vextenddoubleL∗e(t−s)L∗[a(·)−1ut(·,s)]/vextenddouble/vextenddouble L2 dµds. Noting that for x∈Ω, a(x)−1Φε(x,t)−1≤1 a1/a\}b∇acketle{tx/a\}b∇acket∇i}htαexp/parenleftbigg −A1ε h+2ε/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α 1+t/parenrightbigg ≤1 a1/parenleftbiggα(h+2ε) (2−α)eA1ε/parenrightbiggα 2−α (1+t)α 2−α, we see that for k= 0,1, /vextenddouble/vextenddoublea(·)−1∂k+1 tu(·,s)/vextenddouble/vextenddouble2 L2 dµ=/integraldisplay Ωa(x)−1|∂k+1 tu(·,s)|2dx ≤ /ba∇dbla(·)−1Φε(·,t)−1/ba∇dblL∞(Ω)/integraldisplay Ω|∂k+1 tu(·,s)|2Φεdx ≤/tildewideC(1+t)α 2−αE∂t(t,∂k tu) ≤/tildewideCMε,k(1+t)−λ0−2−2α 2−α−2k/ba∇dbl(u0,u1)/ba∇dbl2 Hk+1×Hk. Therefore from Proposition 3.5 with k= 1 and k= 0 we have J1(t)≤/integraldisplayt t/2/vextenddouble/vextenddoublea(·)−1utt(·,s)/vextenddouble/vextenddouble L2 dµds ≤/radicalBig /tildewideCM1/ba∇dbl(u0,u1)/ba∇dblH2×H1/integraldisplayt t/2(1+s)−λ0 2−1−α 2−α−1ds ≤2(2−α) λ0(2−α)+1−α/radicalBig /tildewideCMε,1(1+t)−λ0 2−1−α 2−α/ba∇dbl(u0,u1)/ba∇dblH2×H1 and J2(t)≤/vextenddouble/vextenddoublea(·)−1ut(·,t/2)/vextenddouble/vextenddouble L2 dµ≤/radicalBig /tildewideCMε,0(1+t)−λ0 2−1−α 2−α/ba∇dbl(u0,u1)/ba∇dblH1×L2. Moreover, by Lemma 2.2, we see by Cauchy–Schwarz inequality that fort≥1, J3(t)≤C/integraldisplayt/2 0(t−s)−N−α 2(2−α)−1/vextenddouble/vextenddoublea(·)−1ut(·,s)/vextenddouble/vextenddouble L1 dµds ≤C/parenleftbiggt 2/parenrightbigg−N−α 2(2−α)−1/integraldisplayt/2 0/radicalBig /ba∇dblΦ−1ε(·,s)/ba∇dblL1(Ω)E∂t(s;u)ds.WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 13 Since /ba∇dblΦ−1(·,t)/ba∇dblL1(Ω)≤/integraldisplay RNexp/parenleftbigg −A1ε h+2ε|x|2−α 1+t/parenrightbigg dx = (1+t)N 2−α/integraldisplay RNexp/parenleftbigg −A1ε h+2ε|y|2−α/parenrightbigg dy, we deduce J3(t)≤C′(1+t)−N−α 2(2−α)−1/ba∇dbl(u0,u1)/ba∇dblH1×L2/integraldisplayt/2 0(1+s)N−α 2(2−α)−λ0 2−1−α 2−αds ≤C′/parenleftbiggN−α 2(2−α)−λ0 2+1 2−α/parenrightbigg (1+t)−N−α 2(2−α)−1(1+t/2)N−α 2(2−α)−λ0 2−1−α 2−α+1 ×/ba∇dbl(u0,u1)/ba∇dblH1×L2 ≤C′′(1+t)−λ0 2−1−α 2−α/ba∇dbl(u0,u1)/ba∇dblH1×L2. Consequently, we obtain /vextenddouble/vextenddouble/vextenddoubleu(·,t)−etL∗[u0+a(·)−1u1]/vextenddouble/vextenddouble/vextenddouble L2 dµ≤C′′′(1+t)−λ0 2−1−α 2−α/ba∇dbl(u0,u1)/ba∇dblH2×H1. Next we show the assertion for ( u0,u1) satisfying ( u0,u1)∈(H2×H1 0(Ω))× H1 0(Ω) (the compatibility condition of order 1) via an approximation argu ment. Fixφ∈C∞ c(RN,[0,1]) such that φ≡1 onB(0,R0) andφ≡0 onRN\B(0,R0+1) and define for n∈N, /parenleftbigg u0n u1n/parenrightbigg =/parenleftbigg φ˜u0n φ˜u1n/parenrightbigg ,/parenleftbigg ˜u0n ˜u1n/parenrightbigg =/parenleftbigg 1+1 nA/parenrightbigg−1/parenleftbigg u0 u1/parenrightbigg , whereAis anm-accretive operator in H=H1 0(Ω)×L2(Ω) associated with (1.1), that is, A=/parenleftbigg 0−1 −∆a(x)/parenrightbigg endowed with domain D(A) = (H2∩H1 0(Ω))×H1 0(Ω). Then ( u0n,u1n) satisfies supp(u0n,u1n)⊂B(0,R0+1) and the compatibility condition of order 2. Let vn be a solution of (1.1) with ( u0n,u1n). Observe that /ba∇dbl(u0n,u1n)/ba∇dbl2 H2×H1≤C2/ba∇dblφ/ba∇dbl2 W2,∞/ba∇dbl(˜u0,˜u1)/ba∇dbl2 H2×H1 ≤C′2/ba∇dblφ/ba∇dbl2 W2,∞(/ba∇dbl(˜u0,˜u1)/ba∇dbl2 H+/ba∇dblA(˜u0,˜u1)/ba∇dbl2 H) ≤C′2/ba∇dblφ/ba∇dbl2 W2,∞(/ba∇dbl(u0,u1)/ba∇dbl2 H+/ba∇dblA(u0,u1)/ba∇dbl2 H) ≤C′′2/ba∇dblφ/ba∇dbl2 W2,∞/ba∇dbl(u0,u1)/ba∇dbl2 H2×H1 with suitable constants C,C′,C′′>0, and /parenleftbiggu0n u1n/parenrightbigg →/parenleftbiggφu0 φu1/parenrightbigg =/parenleftbiggu0 u1/parenrightbigg inH asn→ ∞and also u0n+a−1u1n→u0+a−1u1inL2 dµasn→ ∞. Using the result of the previous step, we deduce /vextenddouble/vextenddouble/vextenddoublevn(·,t)−etL∗[u0n+a(·)−1u1n]/vextenddouble/vextenddouble/vextenddouble L2 dµ≤˜C(1+t)−λ0 2−1−α 2−α/ba∇dbl(u0,u1)/ba∇dblH2×H114 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI with some constant ˜C >0. Letting n→ ∞, by continuity of the C0-semigroup e−tAinHwe also obtain diffusion phenomena for initial data in ( H2∩H1 0(Ω))∩ H1 0(Ω). /square Acknowledgments This work is supported by Grant-in-Aid for JSPS Fellows 15J01600 of Japan Society for the Promotion of Science and also partially supported by Grant-in-Aid for Young Scientists Research (B), No. 16K17619. The authors w ould like to thank the referee for giving them valuable comments and suggestions. References [1] M. Ikawa, Mixed problems for hyperbolic equations of sec ond order, J. Math. Soc. Japan 20(1968), 580–608. [2] M. Ikawa, Hyperbolic partial differential equations and wave phenomena, American Math- ematical Society (2000). [3] R. Ikehata, Some remarks on the wave equation with potent ial type damping coefficients, Int. J. Pure Appl. Math. 21(2005), 19–24. [4] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci. 12(1976), 169–189. [5] A. Matsumura, Energy decay of solutions of dissipative w ave equations, Proc. Japan Acad., Ser. A53(1977), 232–236. [6] K. Mochizuki, Scattering theory for wave equations with dissipative terms, Publ. Res. Inst. Math. Sci. 12(1976), 383–390. [7] K. Nishihara, Lp-Lqestimates of solutions to the damped wave equation in 3-dime nsional space and their application, Math. Z. 244(2003), 631–649. [8] P. Radu, G. Todorova, B. Yordanov, Higher order energy de cay rates for damped wave equations with variable coefficients, Discrete Contin. Dyn. Syst. Ser. S 2(2009), 609–629. [9] P. Radu, G. Todorova, B. Yordanov, Decay estimates for wa ve equations with variable coefficients, Trans. Amer. Math. Soc. 362(2010), 2279–2299. [10] M. Sobajima and Y. Wakasugi, Diffusion phenomena for the wave equation with space- dependent damping in an exterior domain, J. Differential Equations 261(2016), 5690–5718. [11] G. Todorova, B. Yordanov, Critical exponent for a nonli near wave equation with damping, J. Differential Equations 174(2001), 464–489. [12] G. Todorova, B. Yordanov, Weighted L2-estimates for dissipative wave equations with vari- able coefficients, J. Differential Equations 246(2009), 4497–4518. [13] Y. Wakasugi, On diffusion phenomena for the linear wave e quation with space-dependent damping, J. Hyp. Diff. Eq. 11(2014), 795–819. (M.Sobajima) Departmentof Mathematics, Faculty of Science andTechnolo gy, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba-ken 27 8-8510, Japan E-mail address :msobajima1984@gmail.com (Y.Wakasugi) Graduate School of Mathematics, NagoyaUniversity, Furocho, Chikusaku, Nagoya 464-8602 Japan E-mail address :yuta.wakasugi@math.nagoya-u.ac.jp

2016-09-05

This paper is concerned with weighted energy estimates and diffusion phenomena for the initial-boundary problem of the wave equation with space-dependent damping term in an exterior domain. In this analysis, an elliptic problem was introduced by Todorova and Yordanov. This attempt was quite useful when the coefficient of the damping term is radially symmetric. In this paper, by modifying their elliptic problem, we establish weighted energy estimates and diffusion phenomena even when the coefficient of the damping term is not radially symmetric.

Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain

1609.01063v2

1 Magnetic Damping in Epitaxial Fe Alloyed with Vanadium and Aluminum David A. Smith1, Anish Rai2,3, Youngmin Lim1, Timothy Hartnett4, Arjun Sapkota2,3, Abhishek Srivastava2,3, Claudia Mewes2,3, Zijian Jiang1, Michael Clavel5, Mantu K. Hudait5, Dwight D. Viehland6, Jean J. Heremans1, Prasanna V. Balachandran4,7, Tim Mewes2,3, Satoru Emori1 1Department of Physics, Virginia Tech, Blacksburg, VA 24061, U.S.A. 2Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, U.S.A. 3Center for Materials for Information Technology (MINT), University of Alabama, Tuscaloosa, AL 35487, U.S.A . 4Department of Material Science and Engineering, University of Virginia, Charlottesville, VA 22904, U.S.A. 5Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24061, U.S.A. 6Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061, U.S.A. 7Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, U.S.A. 2 To develop low -moment, low -damping metallic ferromagnets for power -efficient spintronic devices, it is crucial to understand how magnetic relaxation is impacted by the addition of nonmagnetic elements. Here, we compare magnetic relaxation in epitaxial Fe films alloyed with light nonmagnetic elements of V and Al. FeV alloys exhibit lower intrinsic damping compared to pure Fe, reduced by nearly a factor of 2, whereas damping in FeAl alloys increases with Al content . Our experimental and computat ional results indicate that reducing the density of states at the Fermi level , rather than the average atomic number, has a more significant impact in lowering damping in Fe alloyed with light elements . Moreover, FeV is confirmed to exhibit an intrinsic Gi lbert damping parameter of ≃0.001, among the lowest ever reported for ferromagnetic metals. I. INTRODUCTION The relaxation of magnetization dynamics (e.g., via Gilbert damping) plays important roles in many spintronic applications, including those based on magnetic switching1,2, domain wall motion3,4, spin wave propagation5,6, and su perfluid -like spin transport7,8. For devices driven by spin -torque precessional dynamics1,9,10, the critical current density for switching is predicted to scale with the produ ct of the Gilbert damping parameter and the saturation magnetization 2,11. Thus, it is desirable to engineer magnetic materials that possess both low damping and low moment for energy -efficient operation . While some electrically insulating magnetic oxides have been considered for certain applications5,12,13, it is essential to engineer low -damping, low - moment metallic ferromagnets for robust electrical readout via giant magnetoresistance and tunnel magnetoresistance. Fe is the elemental ferromagnet with the lowest intrinsic Gilbert damping parameter ( ≃0.002)14,15, albeit with the highest saturation magnetization ( ≃2.0 T). 3 Recent experiments have reported that Gilbert damping can be further reduced by alloy ing Fe with Co (also a ferromagnetic element), with Fe 75Co25 yielding an ultralow intrinsic Gilbert damping parameter of ≃0.00116,17. However, Fe 75Co25 is close to the top of the Slater -Pauling curve , such that its saturation magnetization is greater than that of Fe by approximately 20 %18. There is thus an unmet need to engineer ferromagnetic alloys tha t simultaneously exhibit lower damping and lower moment than Fe. A promising approach towards low -damping, low -moment ferromagnetic metals is to introduce nonmagnetic elements into Fe . In addition to diluting the magnetic moment, nonmagnetic elements int roduced into Fe could influence the spin -orbit coupling strength ξ, which underlies spin relaxation via orbital and electronic degrees of freedom19–21. Simple atomic physics suggests that ξ is related to the average atomic number <Z> of the alloy so that, conceivably, damping might be lowered by alloying Fe with lighter (lower -Z) elements. Indeed, motivated by the premise of lowering damping through a reduced <Z> and presumably ξ, prior experiments have explored Fe thin films alloyed with V20,22,23, Si24, and Al25. However, the experimentally reported damping parameters for these alloys are often a factor of >2 higher22,23 ,25 than the theoretically predicted intrinsic Gilbert damping parameter of ≃0.002 in Fe26 and do not exhibit a significant dependence on the alloy composition20,23,24. A possible issue is that the reported damping parameters – obtained from the frequency dependence of ferromagnetic resonance (FMR) linewidth with the film magnetized in -plane – may include contributions from non-Gilbert relaxation induced by inhomogeneity and defects (e.g., two -magnon scattering)27–36, which can be affected by the alloying. Therefore, how Gilbert damping in Fe is impacted by alloying with low -Z elements remains an open question. 4 Here, we investigate the compositiona l dependence of magnetic relaxation at room temperature in epitaxial thin films of ferromagnetic FeV and FeAl alloys. Both alloys are crystalline bcc solid solutions and hence constitute excellent model systems. We employ two configurations of FMR measurem ents to gain complementary insights: (1) FMR with samples magnetized in the film plane (similar to the prior experiments) to derive the “effective” Gilbert damping parameter, 𝛼𝑒𝑓𝑓𝐼𝑃, which is found to include extrinsic magnetic relaxation due to two - magnon scattering, and (2) FMR with samples magnetized perpendicular to the film plane to quantify the intrinsic Gilbert damping parameter, 𝛼𝑖𝑛𝑡, which is free of the two -magnon scattering contribution. Since Al ( Z = 13) is a much lighter element than V ( Z = 23), we might expect lower magnetic relaxation in FeAl than FeV, if the smaller < Z> lowers intrinsic Gilbert damping via reduced ξ. Instead, we find a significant decrease in magnetic relaxation by alloying Fe w ith V – i.e., yielding an intrinsic Gilbert damping parameter of ≃0.001, on par with the lowest values reported for ferromagnetic metals – whereas damping in FeAl alloys increases with Al content . These experimental results , combined with density functi onal theory calculations, point to the density of states at the Fermi level D(EF) as a plausible dominant factor for the lower (higher) Gilbert damping in FeV (FeAl). We thus find that incorporating a low -Z element does not generally lower damping and that, rather, reducing D(EF) is an effective route for lower damping in Fe alloyed wi th a nonmagnetic element. Our findings confirm that FeV is an intrinsically ultralow -damping alloy, as theoretically predicted by Mankovsky et al.26, which also possesses a lower saturation magnetization than Fe and FeCo. The combination of low damping and low moment makes FeV a highly promising material for practical metal -based spintronic applications. 5 II. FILM DEPOSITION AND STRUCTURAL PROPERTIES Epitaxial Fe 100-xVx and Fe 100-xAlx thin films were grown using dc magnetron sputtering on (001) -oriented MgO substrates. Prior to deposition, the substrates were annealed at 600 oC for 2 hours37. The base pressure prior to deposition was < 5×10-8 Torr, and all film s were grown with an Ar pressure of 3 mTorr. Fe and V (Al) 2” targets were dc co -sputtered to deposit Fe 100-xVx (Fe 100-xAlx) films at a substrate temperature of 200 oC. By adjusting the deposition power, we tuned the deposition rate of each material (calibrated by X -ray reflectivity) to achieve the desired atomic percentage x of V (Al). All FeV and FeAl films had a thickness of 25 nm, which is well above the thickness regime where interfacial effects dominate31,38. The FeV (FeAl) films were capped with 3 -nm-thick V (Al) deposited at room temperature to protect against oxidation, yielding a film structure of MgO/Fe 100-xVx(25nm)/V(3nm) or MgO/Fe 100-xAlx(25nm)/Al(3nm). We confirmed the epitaxial bcc structure of our thi n films using high resolution X -ray diffraction. 2θ -ω scans show only the (002) peak of the film and the (002) and (004) peaks of the substrate, as shown in Fig ure 1. Rocking curve scans of the film peaks show similar full -width - at-half-maximum values of ≃ 1.3o irrespective of composition . The epitaxial relation between bcc Fe and MgO is well known16,39: the bcc film crystal is rotated 45o with respect to the substrate crystal , such that the [100] axis of the film lies parallel to the [110] axis of the substrate. The absence of the (001) film peak indicates that our epitaxial FeV and FeAl films are solid sol utions rather than B2 -ordered compounds40. 6 III. MAGNETIC RELAXATION 3.1. In -Plane Ferromagnetic Resonance Many spintronic devices driven by precessional magnetization dynamics are based on in - plane magnetized thin films. The equilibrium magnetization also lies in -plane for soft ferromagnetic thin films dominated by shape anisotropy (i.e., negligible perpendicular magnetic anisotropy), as is the case for our epitaxial FeV and FeAl films. We therefore first discuss FMR results w ith films magnetized in -plane. The in -plane FMR results further provide a basis for comparison with previous studies20,22,23,25. Samples were placed with the film side facing a coplanar waveguid e (maximum frequency 50 GHz) and magnetized by an e xternal field H (from a conventional electromagnet, maximum field 1.1 T) along the in -plane [100] and [110] axes of the films. Here, unless otherwise stated, we show results for H || [110] of the film. FMR spectra were acquired via field modulation by sweeping H and fixing the microwave excitation frequency. Exemplary spectra for Fe, Fe 80V20, and Fe 80Al20 are shown in Fig ure 2, where we compare the peak -to-peak linewidths at a microwave excitation frequency of 20 GHz. We see that the linewidth for Fe 80V20 shows a ≃ 25 % reduction compared to Fe. We further note that the linewidth for the Fe 80V20 sample here is a factor of ≃ 2 narrower than that in previously reported FeV20; a possible origin of the narrow linewidth is discussed later . In contrast, Fe 80Al20 shows an enhancement in linewidth over Fe, which is contrar y to the expectation of lower magnetic relaxation with a lower average atomic number. The FMR linewidth is generally governed not only by magnetic relaxation, but also by broadening contributions from magnetic inhomogeneities28,41,42. To disentangle the magnetic 7 relaxation and inhomogeneous broadening contributions to the linewidth, the typical prescription is to fit the frequency f dependence of linewidth ∆𝐻𝑝𝑝𝐼𝑃 with the linear relation41 ∆𝐻𝑝𝑝𝐼𝑃=∆𝐻0𝐼𝑃+ℎ 𝑔𝜇𝐵𝜇02 √3𝛼𝑚𝑒𝑎𝑠𝐼𝑃𝑓, (1) where h is the Planck constant, 𝜇𝐵 is the Bohr magneton, 𝜇0 is the permeability of free space, and 𝑔 is the g-factor obtained from the frequency dependence of the resonance field (see Section IV and Supplementa l Material). In Eq. (1), the slope is attributed to viscous magnetic damping, captured by the measured damping parameter 𝛼𝑚𝑒𝑎𝑠𝐼𝑃, while t he zero -frequency linewidth ∆𝐻0𝐼𝑃 is attributed to inhomogeneo us broadening. The fitting with Eq. (1) was carried out for f 10 GHz, where H was sufficiently large to saturate the films. As is evident from the results in Fig ure 3, Fe80V20 has lower linewidths across all frequencies and a slightly lower slope, i.e., 𝛼𝑚𝑒𝑎𝑠𝐼𝑃. On the other hand, Fe 80Al20 shows higher linewidths and a higher slope. The measured viscous damping includes a small contribution from eddy currents, parameter ized by 𝛼𝑒𝑑𝑑𝑦 (Supplemental Material) , and a contribution due to radiative damping43, given by 𝛼𝑟𝑎𝑑 (Supplemental Material). Together these contributions make up ≃20 % of the total 𝛼𝑚𝑒𝑎𝑠𝐼𝑃 for pure Fe and decrease in magnitude with increasing V or Al content . We subtract these to obtain the effective in -plane Gilbert damping parameter, 𝛼𝑒𝑓𝑓𝐼𝑃=𝛼𝑚𝑒𝑎𝑠𝐼𝑃−𝛼𝑒𝑑𝑑𝑦 − 𝛼𝑟𝑎𝑑. (2) As shown in Fig ure 4a, 𝛼𝑒𝑓𝑓𝐼𝑃 remains either invariant or slightly decreases in Fe 100-xVx up to x = 25, whereas we observe a monotonic enhancement of 𝛼𝑒𝑓𝑓𝐼𝑃 with Al content in Figure 4b . These results point to lower (higher) damping in FeV (FeAl) and suggest a factor other than the average atomic number governing magnetic relaxation in these alloys. However, such a conclusion assumes that 𝛼𝑒𝑓𝑓𝐼𝑃 is a reliable measure of intrinsic Gilbert damping . In reality, 𝛼𝑒𝑓𝑓𝐼𝑃 may include 8 a contribution from defect -induced two -magnon scattering27–31,35,36, a well -known non -Gilbert relaxation mechanism in in -plane magnetized epitaxial films27,32 –34,44. We show in the next subsection that substantial two -magnon scattering is indeed present in our FeV and FeAl alloy thin films. Although Eq. (1) is not necessarily the correct framework for quantifying Gilbert damping in in -plane magnetized thin films, we can gain insight into the quality (homogeneity) of the films from ∆𝐻0𝐼𝑃. For our samples, μ0∆𝐻0𝐼𝑃 is below ≈ 1 mT (see Fig ure 4c,d), which implies higher film quality for our FeV samples than previously reported20. For example, Fe 73V27 in Scheck et al. exhibits μ0∆𝐻0𝐼𝑃 ≃ 2.8 mT20, whereas Fe 75V25 in our study exhibits μ0∆𝐻0𝐼𝑃 ≃ 0.8 mT. Although 𝛼𝑒𝑓𝑓𝐼𝑃 is comparable between Scheck et al. and our study, the small ∆𝐻0𝐼𝑃 leads to overall much narrower linewidths in our FeV films (e.g., as shown in Figs. 2 and 3) . We speculate that the annealing of the MgO substrate prior to film deposition37 – a common practice for molecular beam epitaxy – facilitates high -quality epitaxial film growth and hence small ∆𝐻0𝐼𝑃 even by sputtering. 3.2. Out -of-Plane Ferromagnetic Resonance To quantify intrinsic Gilbert damping, we performed broadband FMR with the film magnetized out -of-plane, which is the configuration that suppresses two -magnon scattering28–31. Samples were placed in side a W-band shorted -waveguide spectrometer (frequency range 70 -110 GHz) in a superconducting electromagnet that enabled measurements at fields > 4 T. This high field range is well above the shape anisotropy field of ≤2 T for our films and hence sufficient to completely saturate the film out -of-plane. 9 The absence of two -magnon scattering in broadband out -of-plane FMR allows us to reliably obtain the measured viscous damping parameter 𝛼𝑚𝑒𝑎𝑠𝑂𝑃 by fitting the linear frequency dependence of the linewidth ∆𝐻𝑝𝑝𝑂𝑃, as shown in Figure 5, with ∆𝐻𝑝𝑝𝑂𝑃=∆𝐻0𝑂𝑃+ℎ 𝑔𝜇𝐵𝜇02 √3𝛼𝑚𝑒𝑎𝑠𝑂𝑃𝑓. (3) We note that the zero -frequency linewidth for the out -of-plane configuration ∆𝐻0𝑂𝑃 (Figure 6c,d) is systematically g reater than that for the in -plane configuration ∆𝐻0𝐼𝑃 (Figure 4c,d). Such a trend of ∆𝐻0𝑂𝑃>∆𝐻0𝐼𝑃, often seen in epitaxial films15,33,45, may be explained by the stronger contribution of inhomogeneity to the FMR field when the magnetic precessional orbit is circular, as is the case for out -of-plane FMR, compared to the case of the highly elliptical precession in in-plane FMR41; however, the detailed mechanisms contributing to the zero -frequency linewidth remain the subject of future work . The larger ∆𝐻0𝑂𝑃 at high V and Al concentrations may be due to broader distributions o f anisotropy fields and saturation magnetization, or the presence of a secondary crystal phase that is below the resolution of our X -ray diffraction results. The absence of two -magnon scattering in out -of-plane FMR allows us to quantify the intrinsic Gilbert damping parameter, 𝛼𝑖𝑛𝑡=𝛼𝑚𝑒𝑎𝑠𝑂𝑃−𝛼𝑒𝑑𝑑𝑦, (4) by again subtracting the eddy current contribution 𝛼𝑒𝑑𝑑𝑦. Since we utilize a shorted waveguide, the contribution due to radiative damping does not apply. From the compositional dependence of 𝛼𝑖𝑛𝑡 as summarized in Figure 6a1, a reduction in intrinsic Gilbert damping is evidenced with V alloying. Our observation is in contrast to the previous experiments on FeV alloys20,22,23 where the reported damping parameters remain >0.002 1 We were unable to carry out out -of-plane FMR measurements for FeV with x = 20 (Fig. 2(c,d )) as the sample had been severely damaged during transit. 10 and depend weakly on the V concentration. In particular, the observed minimum of 𝛼𝑖𝑛𝑡≃0.001 at x ≃ 25-30 is approximately half of the lowest Gilbert damping parameter previously reported for FeV20 and that of pure Fe15. The low 𝛼𝑖𝑛𝑡 here is also comparable to the lowest damping parameters reported for ferromagnetic metals, such as Fe75Co2516,17 and Heusler compounds46–48. Moreover, t he reduced intrinsic damping by alloying Fe w ith V is qualitatively consistent with the computational prediction by Mankov sky et al.26, as shown by the curve in Figure 6a. Our experimental finding therefore confirms that FeV is indeed an intrinsically ultralow -damping ferromagnet that possesses a smaller saturation magnetization than Fe. In contrast to the reduction of 𝛼𝑖𝑛𝑡 observed in FeV alloys, FeAl shows an increase in intrinsic damping with increasing Al concentration, as seen in Figure 6b. Recalling that Al has an atomic number of Z = 13 that is lower than Z = 23 for V, this trend clashes with the expectation that lower < Z> red uces the intrinsic Gilbert damping through a reduction of the atomic spin -orbit coupling. Thus, we are required to consider an alternative mechanism to explain the higher (lower) damping in FeAl (FeV), which we discuss further in Section V. 3.3. Magnetic Relaxation: Practical Consideration s For both FeV and FeAl alloys, 𝛼𝑖𝑛𝑡 derived from out -of-plane FMR (Figure 6a,b) is consistently lower than 𝛼𝑒𝑓𝑓𝐼𝑃 derived from in -plane FMR (Fig ure 4a,b). Th is discrepancy between 𝛼𝑖𝑛𝑡 and 𝛼𝑒𝑓𝑓𝐼𝑃 implies a two-magnon scattering contribution to magnetic relaxation in the in-plane configuration (Figure 4a,b). For many applications including spin -torque oscillators and magnonic devices , it is crucial to minimize magnetic relaxation in in-plane magnetized thin films. While the in -plane magnetic relaxation ( 𝛼𝑒𝑓𝑓𝐼𝑃≃0.002) is already quite low for the FeV alloys shown here, the low intrinsic Gilbert damping ( 𝛼𝑖𝑛𝑡≃0.001) points to the possibility of 11 even lower relaxation and narrow er FMR linewidths by minimizing two -magnon scattering and inhomogeneous linewidth broadening. Such ultralow magnetic relaxation in FeV alloy thin films may be achieved by optimizing structural properties through growth conditions16 or seed layer engineering49. While ultralow intrinsic Gilbert damping values have been confirmed in high -quality epitaxial FeV, it would be desirable for device integration to understand how magnetic relaxation in FeV would be impacted by the presence of grain boundaries, i.e. in polycrystalline thin films. Reports on polycrystalline FeCo49 suggest intrinsic damping values comparable to those seen in epitaxial FeCo16,17. While beyond the scope of this study, our future work will explore the possibility of low damping in polycrystalline FeV thin films. IV. SPECTROSCOPIC PARAMETERS The results presented so far reveal that magnetic relaxation is reduced by alloying Fe with V, whereas it is increased by alloying Fe with Al. On the other hand, FeV and FeAl alloys exhibit similar compositional dependence of the spectroscopic parameters: effective magnetization Meff (here, equivalent to saturation magnetization Ms), magnetocrystalline anisotropy field Hk, and the g-factor 𝑔 – all of which are quantified by fitting the frequency dependence of resonance field (Supplemental Material) . As shown in Fig ure 7a, there is a systematic reduction in Meff with increasing concentration of V and Al. We also note in Fig ure 7b a gradual reduction in magnitude of the in -plane cubic anisotropy. Both of these trends are expected as magnetic Fe atoms are r eplaced with nonmagnetic atoms of V and Al. The reduction of Meff by ≃20% in the ultralow -damping Fe 100-xVx alloys with x = 25-30, compared to pure Fe, is of particular practical interest. The saturation magnetization of these FeV alloys is on par with 12 commonly used soft ferromagnetic alloys (e.g., Ni 80Fe2050, CoFeB51), but the damping parameter of FeV is several ti mes lower. Further, w hile FeV and FeCo in the optimal composition window show similarly low intrinsic damping parameters, FeV provides the advantage of lower moment . With the product 𝛼𝑖𝑛𝑡𝑀𝑒𝑓𝑓 approximately proportional to the critical current densi ty to excite precessional dynamics by spin torque2,11, FeV is expected to be a superior material platform for low-power spin tronic devices . The g-factor 𝑔=2(1+𝜇𝐿/𝜇𝑆) is related to the orbital moment 𝜇𝐿 and spin moment 𝜇𝑆; the deviation from the spin -only value of 𝑔= 2.00 provides insight into the strength of spin -orbit coupling ξ52. As seen in Figure 7c, 𝑔 increases by 1-2% with both V and Al alloying, which suggests that ξ increases slightly with the addition of these low -Z elements. This finding verifies that < Z> is not necessarily a good predictor of ξ in a solid. Moreover, the higher 𝑔 for FeV is inconsistent with the scenario for lower damping linked to a reduced spin -orbit coupling. Thus, spin-orbit coupling alone cannot explain the observed behavior of Gilbert damping in Fe alloyed with low -Z elements. V. DISCUSSION In contrast to what has been suggested by prior experimental studies20,22 –25, we have shown that the reduction of average atomic number by alloying with a light element (e.g., Al in this case) does not generally lower the intrinsic Gilbert dampin g of Fe. A possible source for the qualitatively distinct dependencies of damping on V and Al contents is the density of states at the Fermi level, D(EF): it has been predicted theoretically that the intrinsic Gilbert damping parameter is reduced with decr easing D(EF), since D(EF) governs the availability of states for spin-polarized electrons to scatter into21,26,53 –55. Such a correlation between lower damping and 13 smaller D(EF) has been reported by recent experiments on FeCo alloys17,50, FeRh alloys40, CoNi alloys56, and Heusler compounds46,48,57. The similarity in the predicted composition dependence of the Gilbert damping parameter for FeCo and FeV26 suggests that the low damping of FeV may be correlated with reduced D(EF). However, no prior experiment has corroborated this correlation for FeV or other alloys of Fe and light elements. We therefore e xamine whether the lower (higher) damping in FeV (FeAl) compared to Fe can be qualitatively explained by D(EF). Utilizing the Quantum ESPRESSO58 package to perform density functional theory calculations (details in Supplemental Material) , we calculated the density of states for Fe, Fe 81.25V18.75, and Fe 81.25Al18.75. It should be recalled that although FeV and FeAl films measured experimentally her e are single -crystalline, they are solid solutions in which V or Al atoms replace Fe atoms at arbitrary bcc lattice sites. Therefore, f or each of the binary alloys, we computed 6 distinct atomic configurations in a 2×2×2 supercell , as shown in Figure 8 . The spin -split density of states for each unique atomic configuration is indicated by a curve in Figure 9. Here, D(EF) is the sum of the states for the spin -up and spin -down bands, averaged over results from the 6 distinct atomic configurations. As summari zed in Fig ure 9 and Table 1, FeV has a smaller D(EF) than Fe, whereas FeAl has a larger D(EF). These calculation results confirm a smaller (larger ) availability of states for spin-polarized electrons to scatter into in FeV (FeAl), qualitatively consistent with the lower (higher) intrinsic Gilbert damping in FeV (FeAl). We remark that this correlation between damping and D(EF) is known to hold parti cularly well in the limit of low electronic scattering rates 𝜏−1, where intra band scattering dominates21,54. Gilmore et al. have pointed out that at sufficiently high electronic scattering rates, i.e., when ℏ𝜏−1 is large enough that inter band scattering is substantial, the simple correlation between the 14 strength of Gilbert damping and D(EF) breaks down. It is unclear whether our FeV and FeAl alloy films at room temperature are in the intraband - or interband -dominated regime. Schoen et al. have argued that polycrystalline FeCo alloy films – with higher degree of structural disorder and likely higher electronic scattering rates than our epitaxial films – at room temperature are still well within the intraband -dominated regime17. On the other hand, a recent temperature - dependent study on epitaxial Fe suggests coexistence of the intraband and interband contributions at room temperature15. A consistent explanation for the observed room -temperature intrinsic damping in our alloy films is that the interband contribution depends weakly on alloy composition; it appears re asonable to conclude that D(EF), primarily through the intraband contribution, governs the difference in intrinsic Gilbert damping among Fe, FeV, and FeAl . VI. SUMMARY We have experimentally in vestigated magnetic relaxation in epitaxial thin films of Fe alloyed with low -atomic -number nonmagnetic elements V and Al . We observe a reduction in the intrinsic Gilbert damping parameter to 𝛼𝑖𝑛𝑡≃0.001 in FeV films , comparable to the lowest - damping ferromagnetic metals reported to date. In contrast, an increase in damping is observed with the addition of Al, demonstrating that a smaller average atomic number does not necessarily lower intrinsic damping in an alloy . Furthermore, our results on FeV and FeAl cannot be explained by the change in spin -orbit coupling through alloying . Instead, we conclude that the density of states at the Fermi level plays a larger role in determining the magnitude of damping in Fe alloyed w ith lighter elements. Our work also confirms FeV alloys as promising ultra low- damping , low-moment metallic materials for practical power -efficient spin -torque devices. 15 Acknowledgements: This research was funded in part by 4 -VA, a collaborative partnership for advancing the Commonwealth of Virginia, as well as by the ICTAS Junior Faculty Program. D.A.S. acknowledges support of the Virginia Tech Graduate School Doctoral Assistantship. A. Sapkota and C. M . would like to acknowledge support by NSF -CAREER Award No. 1452670, A.R. and T.M. would like to acknowledge support by DARPA TEE Award No. D18AP00011, and A. Srivastava would like to acknowledge support by NASA Award No. CAN80NSSC18M0023. We thank M.D. Stiles for helpful input regarding intrinsic damping mechanisms in alloys. The data that support the findings of this study are available from the corresponding author upon reasonable request. Number of Spin -Up States (eV-1) at EF Number of Spin -Down States (eV-1) at EF Fe 10.90 3.44 Fe81.25V18.75 6.28 ± 1.80 4.61 ± 0.43 Fe81.25Al18.75 6.81 ± 1.58 10.20 ± 3.03 Table 1: Number of spin -up and spin -down states at EF. For Fe81.25V18.75 and Fe81.25Al18.75, the average and standard deviation of values for the 6 distinct atomic configurations (cf. Figure 8) are shown. 16 References: 1 A. Brataas, A.D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012). 2 J.Z. Sun, Phys. Rev. B 62, 570 (2000). 3 T. Weindler, H.G. Bauer, R. Islinger, B. Boehm, J.Y. Chauleau, and C.H. Back, Phys. Rev. Lett. 113, 237204 (2014). 4 A. Mougin, M. Cormier, J.P. Adam, P.J. Metaxas, and J. Ferré, Europhys. Lett. 78, 57007 (2007). 5 A. V. Chumak, V.I. Vasyuchka, A.A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015). 6 J. Han, P. Zhang, J.T. Hou, S.A. Siddiqui, and L. 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Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. De Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin -Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A.P. Seitsonen, A. Smogunov, P. Umari, and R.M. Wentzcovitch, J. Phys.Condens. Matter 21, 395502 (2009). 21 15 30 45 60 75 90 MgO (004) MgO (004) MgO (004)MgO (002) MgO (002) MgO (002)BCC Fe (002) Log(Intensity) (arb. units)BCC Fe80V20 (002) 2q (deg)BCC Fe80Al20 (002) Figure 1: (a) 2θ-ω X-ray diffraction scans showing (00 2) and (004) substrate and (002) film peaks for bcc Fe, Fe 80V20, and Fe 80Al20. 22 -15 -10 -5 0 5 10 15 Fe 2.70 mT FMR Signal (arb. units)Fe80V20 2.04 mT m0(H - HFMR) (mT)Fe80Al203.20 mT Figure 2: FMR spectra at f = 20 GHz with the magnetic field H applied in the film plane, fitted using a Lorentzian derivative (solid curve ) for Fe, Fe 80V20 and Fe 80Al20. 23 0 10 20 30 40 5002468 Fe Fe80V20 Fe80Al20 Scheck et al.m0DHIP PP (mT) Frequency (GHz) Figure 3: FMR linewidths versus microwave frequency for the magnetic field applied within the plane of the film for three distinct alloys. The solid lines are linear fit s, described by Eq. (1), from which the effective damping parameter and zero frequency linewidth are determined. The dashed line represents the result for Fe 73V27 from Scheck et al.20 24 0 10 20 30 40246 0 10 20 302468 0 10 20 30 40024 0 5 10 15 20 25 3001 Fe100-xVx Scheck et al.aIP eff x 103 aIP eff x 103 Fe100-xAlxm0DHIP 0 (mT) Alloy Composition, x (%)(a) (b) (c) (d)m0DHIP 0 (mT) Alloy Composition, x (%) Figure 4: The effective damping parameter 𝛼𝑒𝑓𝑓𝐼𝑃 for (a) Fe 100-xVx and (b) Fe 100-xAlx and zero frequency linewidth 𝜇0Δ𝐻0𝐼𝑃 for (c) Fe 100-xVx and (d) Fe 100-xAlx, obtained from in -plane FMR. The solid symbols in (a) and (c) represent results reported by Scheck et al.20 25 0 20 40 60 80 100 12001020304050 Fe Fe70V30 Fe70Al30m0DHOP PP (mT) Frequency (GHz) Figure 5: FMR linewidths versus applied microwave frequency for the magnetic field applied perpendicular to the plane of the film for three distinct alloys. The line is a linear fit, described by Eq. (3), from which the intrinsic Gilbert damping parameter and zero frequency linewidth are determined. 26 0 10 20 30 400123 0 10 20 300246 0 10 20 30 4001020 0 10 20 3001020 Fe100-xVx Mankovsky et al.aint x 103 Fe100-xAlxaint x 103(a) (b) (c) (d)m0DHOP 0 (mT) Alloy Composition, x (%) m0DHOP 0 (mT) Alloy Composition, x (%) Figure 6: The intrinsic Gilbert damping parameter 𝛼𝑖𝑛𝑡 for (a) Fe 100-xVx and (b) Fe 100-xAlx and zero frequency linewidth 𝜇0Δ𝐻0𝑂𝑃 for (c) Fe 100-xVx and (d) Fe 100-xAlx, obtained from out -of- plane FMR. In (a), the dashed curve show s the predicted intrinsic damping parameter computed by Mankovsky et al.26 27 0.81.21.62.02.4 204060 0 10 20 30 402.082.102.122.14 Fe Fe100-xVx Fe100-xAlx m0Meff (T)(a) |m0Hk| (mT)(b) g-factor Alloy Composition, x (%)(c) Figure 7: (a) Effective magnetization, (b) in -plane cubic anisotropy field, and (c) g-factor versus V and Al concentration. The solid (open) markers represent data from in -plane (out -of-plane) measurements . 28 Figure 8: The six unique atomic configurations from the supercell program for mimicking the Fe81.25V18.75 or Fe81.25Al18.75 solid solution. 29 -10010-10010 -1.0 -0.5 0.0 0.5 1.0-10010 (a) Fe81.25V18.75 Density of States (eV-1) (b)Fe E - EF (eV)(c)Fe81.25Al18.75 Figure 9: Calculated spin-up (positive) and spin -down (negative) densit ies of states for (a) Fe, (b) Fe 81.25V18.75 and (c) Fe 81.25Al18.75. Results from the 6 distinct atomic configurations are shown in (b,c); the average densities of states at EF for Fe81.25V18.75 and Fe81.25Al18.75 are shown in Table 1.

2020-04-09

To develop low-moment, low-damping metallic ferromagnets for power-efficient spintronic devices, it is crucial to understand how magnetic relaxation is impacted by the addition of nonmagnetic elements. Here, we compare magnetic relaxation in epitaxial Fe films alloyed with light nonmagnetic elements of V and Al. FeV alloys exhibit lower intrinsic damping compared to pure Fe, reduced by nearly a factor of 2, whereas damping in FeAl alloys increases with Al content. Our experimental and computational results indicate that reducing the density of states at the Fermi level, rather than the average atomic number, has a more significant impact in lowering damping in Fe alloyed with light elements. Moreover, FeV is confirmed to exhibit an intrinsic Gilbert damping parameter of $\simeq$0.001, among the lowest ever reported for ferromagnetic metals.

Magnetic Damping in Epitaxial Fe Alloyed with Vanadium and Aluminum

2004.04840v3

Magnetic damping anisotropy in the two-dimensional van der Waals material Fe3GeTe 2from rst principles Pengtao Yang, Ruixi Liu, Zhe Yuan, and Yi Liu The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, 100875 Beijing, China (Dated: October 18, 2022) Magnetization relaxation in the two-dimensional itinerant ferromagnetic van der Waals ma- terial, Fe 3GeTe 2, below the Curie temperature is fundamentally important for applications to low-dimensional spintronics devices. We use rst-principles scattering theory to calculate the temperature-dependent Gilbert damping for bulk and single-layer Fe 3GeTe 2. The calculated damp- ing frequency of bulk Fe 3GeTe 2increases monotonically with temperature because of the dominance of resistivitylike behavior. By contrast, a very weak temperature dependence is found for the damp- ing frequency of a single layer, which is attributed to strong surface scattering in this highly conned geometry. A systematic study of the damping anisotropy reveals that orientational anisotropy is present in both bulk and single-layer Fe 3GeTe 2. Rotational anisotropy is signicant at low tem- peratures for both the bulk and a single layer and is gradually diminished by temperature-induced disorder. The rotational anisotropy can be signicantly enhanced by up to 430% in gated single-layer Fe3GeTe 2. I. INTRODUCTION Newly emerged intrinsic two-dimensional (2D) ferro- magnetic (FM) van der Waals (vdW) materials1{6have become the subject of intense research. Weak vdW bonding facilitates the extraction of thin layers down to atomic thicknesses, whereas strong magnetocrystalline anisotropy protects long-range magnetic order. These materials provide an exciting arena to perform funda- mental investigations on 2D magnetism and promis- ing applications of low-dimensional spintronics devices. Among these materials, Fe 3GeTe 2(FGT) is especially attractive for its itinerant ferromagnetism and metal- licity, such that both spin and charge degrees of free- dom can be exploited for designing functional devices. Bulk FGT has a relatively high Curie temperature ( TC) of approximately 220-230 K.7{11Atomically thin lay- ers of FGT have lower TCs, which, however, have been raised to room temperature (by ionic gating4) and be- yond (by patterning12). As a FM metal at reasonably high temperature, FGT opens up vast opportunities for applications.13{23 The dynamical properties of FGT critically aect the applicability and performance of these proposed low- dimensional spintronics devices. The most salient of these properties is the dynamical dissipation of mag- netization. It is usually described using a phenomeno- logical parameter called Gilbert damping, which char- acterizes the eciency of the instantaneous magneti- zation to align eventually with the eective magnetic eld during its precessional motion. Although this pa- rameter has been extensively studied in conventional FM materials, such as 3 dtransition metals and alloys, two key issues with the Gilbert parameter of FGT re- main to be addressed: the temperature dependence and anisotropy (one naturally expects anisotropic damping in FGT because of its layered structure and the strong magnetocrystalline anisotropy). Temperature-dependentGilbert damping was rst observed in Fe24and later more systematically in Fe, Co and Ni.25{27A nonmonotonic temperature dependence has been found, for which a so- called \conductivitylike" component decreases with in- creasing temperature, usually at low temperatures, and a \resistivitylike" component increases with temperature, usually at high temperatures. This nonmonotonic be- havior has been successfully described by the torque- correlation model28and reproduced by rst-principles computations.29{32Anisotropic damping was rst theo- retically predicted in FM metals33and in noncollinear magnetic textures.34With dierent orientation of the equilibrium magnetization with respect to the crystal- lographic axes, the damping parameter can be quanti- tatively dierent in general. This is referred to as the orientational anisotropy. Even for the same equilibrium magnetization orientation in a single crystalline lattice, the magnetization may precess instantaneously along dierent directions resulting in the so-called rotational anisotropy.33The orientational anisotropy of damping has been observed in recent experiments on single-crystal FM alloys,35{37but the underlying physical mechanism remains unclear. The dimensionless Gilbert damping parameter can be expressed in terms of a frequency via= M ,38 whereM=jMjis the magnetization magnitude and is the gyromagnetic ratio. Despite of the dierent dimen- sions, these two parameters are equivalent39and both present in literature for experimental24{27,35{37and the- oretical studies.28,29,31,33,34,40{42 In this study, we systematically investigate temperature-dependent Gilbert damping in single- layer (SL) and bulk FGT using rst-principles scattering theory. Considering that the magnetization perpen- dicular to the 2D atomic planes is favored by the strong magnetocrystalline anisotropy, we calculate the damping as a function of temperature below TCand nd nearly temperature-independent damping in thearXiv:2210.08429v1 [cond-mat.mes-hall] 16 Oct 20222 (a) FeⅠ FeⅡ Ge Te(b) FIG. 1. (a) Side and (b) top view of the lattice structure for bulk Fe 3GeTe 2. The black dashed frame delineates the in-plane unit cell. SL and damping dominated by resistivitylike behavior in the bulk. Varying the equilibrium direction of the FGT magnetization produces a twofold symmetry in damping. When the magnetization is aligned inside the 2D planes, a remarkable rotational anisotropy in the Gilbert damping is present for in- and out-of-plane rotating magnetization. This paper is organized as follows. The crystalline structure of SL and bulk FGT is brie y introduced in Sec. II, followed by a description of our theoretical methods and computational details. The calculated temperature- dependent damping in SL and bulk FGT is presented in Sec. III. The two types of damping anisotropy, i.e., orien- tational and rotational anisotropy, are analyzed in Sect. IV. Conclusions are drawn in Sec. V. II. GEOMETRIC STRUCTURE OF FGT AND COMPUTATIONAL METHODS The lattice structure of FGT is shown in Fig. 1. Two dierent types of Fe atoms occupy inequivalent Wycko sites and are denoted as FeI and FeII. Five atomic layers stack along the caxis to form an SL of FGT: Ge and FeII constitute the central atomic layer perpendicular to thecaxis, and two FeI layers and two Te layers are lo- cated symmetrically above and beneath the central layer, respectively. Single layers with ABAB :::stacking form the bulk FGT, where Layer A is translated in plane with respect to Layer B, such that the Ge atoms in Layer A lie on top of the Te and FeII atoms in Layer B. The electronic structure of bulk and SL FGT has been determined using the linear augmented plane wave method43within the local density approximation (LDA). Dierent types of exchange-correlation functionals have been investigated in the literature, among which LDA was found to yield satisfactory structural and magnetic properties for FGT.44We employ experimentally ob- tained lattice constants7for bulk FGT calculations and obtain magnetic moments of 1.78 Band 1.13Bfor the two types of Fe, respectively. The initial structure of a single layer is taken from the bulk lattice and fully relaxed, resulting in an in-plane constant a= 3:92A. A vacuum spacing of 11.76 A is chosen to exclude theinterlayer interaction under periodic boundary condi- tions. The magnetic moments for the Fe atoms in SL FGT are obtained as 1.72 Band 1.01B. All the calculated magnetic moments are in good agreement with experimental7,8,45,46and calculated values44,47,48re- ported in the literature. The Gilbert damping calculation is performed using the scattering theory of magnetization dissipation pro- posed by Brataas et al.49Within this theory, a single do- main FM metal is sandwiched between two nonmagnetic (NM) metal leads. The Gilbert damping that charac- terizes the energy dissipation during magnetization dy- namics can be expressed in terms of a scattering ma- trix and its derivative with respect to the magnetiza- tion direction. We thus construct a two-terminal trans- port structure as Au jFGTjAu, where the Au lattice is slightly deformed to match that of FGT: we use 3 1 and 41 unit cells (UCs) of Au (001) to match the UCs of SL and bulk FGT, respectively. To investigate the ef- fect of temperature on Gilbert damping, we use a frozen thermal lattice and spin disorder31,40,50to mimic lattice vibration and spin uctuation at nite temperatures in FGT. The measured Debye temperature  D= 232 K and temperature-dependent magnetization for the bulk8 and SL3are employed to model the lattice and spin disor- der. In the scattering calculations, lateral supercells are employed to satisfy periodic boundary conditions perpen- dicular to the transport direction. The electronic poten- tials required for the transport calculation are calculated self-consistently using a minimal basis of tight-binding linear mun-tin orbitals (TB-LMTOs), and the result- ing band structures for SL and bulk FGT eectively re- produce those obtained using the linear augmented plane wave method. Then, the scattering matrices consisting of re ection and transmission probability amplitudes for the Bloch wave functions incident from the NM leads are determined by the so-called \wave function matching" method, which is also implemented using TB-LMTOs.40 Other computational details can be found in our previous publications.31,40{42In this work, we focus on the damp- ing with collective magnetization dynamics in the long- wave limit corresponding to the reported values in ex- periment via ferromagnetic resonance and time-resolved magneto-optical Kerr eect. The damping with a - nite wavelength can be determined in our framework of scattering calculation42or using the torque-correlation model,51but the wavelength dependence of damping is beyond the scope of the current study. III. TEMPERATURE-DEPENDENT DAMPING The strong magnetocrystalline anisotropy of FGT re- sults in the equilibrium magnetization being naturally perpendicular to the atomic layers. Slightly excited mag- netization deviates from the plane normal (denoted as ^ z) and relaxes back by dissipating energy and angular mo- mentum, as schematized in the inset of Fig. 2(a). The3 𝛂∥𝛂∥𝑴(𝑡)𝑥𝑦𝑧 5101520F|| (10-3) 0 0.2 0.4 0.6 0.8 1T/TC468Q|| (108 Hz)Lattice disorder only(b)(a) BulkSingle layer BulkSingle layer FIG. 2. The calculated dimensionless Gilbert damping pa- rameterk(a) and corresponding damping frequency k(b) for single-layer and bulk Fe 3GeTe 2as a function of tempera- ture. The relaxation of the instantaneous magnetization M(t) results in a change in the in-plane magnetization component, which is parallel to the atomic planes, as schematized in the inset of (a). The empty symbols in (b) denote the damping frequencies that are calculated considering only thermal lat- tice disorder. The green line indicates the linear temperature dependence. Gilbert damping parameter kdescribes the eciency of such a dissipative process. The calculated kof SL and bulk FGT is plotted in Fig. 2(a) as a function of temperature. The damping for both increases monoton- ically with the temperature. This behavior resembles the so-called \resistivitylike" damping observed in many single-crystal FM metals.24{26However, the damping k for the bulk tends to diverge as the temperature ap- proachesTC. This divergence originates from vanishing magnetization, as has been found in three-dimensional FM alloys.42Therefore, as temperatures approaching TC, it is more appropriate to use the damping frequency pa- rameter= M . The calculated damping frequencies are shown in Fig. 2(b). The damping of a SL FGT, S k, is larger than the damping of the bulk, B k, especially at low temperatures. This dierence can be attributed to the strong surface eect of highly conned SL FGT. The lowered symmetry at the surface signicantly enhances spin-orbit coupling (SOC),52which enables the dissipa- tion of angular momentum from electronic spins to the orbital degree of freedom and then into the lattice reser-voir. In addition, as the thickness of a single layer is considerably smaller than the electronic mean free path, conduction electrons in FGT are strongly scattered by the surface. Therefore, the two necessary ingredients for Gilbert damping, namely, SOC and electronic scattering, are both enhanced in the SL compared with the bulk, re- sulting in a larger damping for the SL. The calculated damping frequency S kremains nearly constant with increasing temperature, except for a mi- nor increase at T > 0:6TC. To gain further insight into the temperature eect, we perform the damping calcu- lation considering only lattice disorder, where the calcu- latedS latare plotted as red empty circles in Fig. 2(b). Lattice-disorder-induced damping in the SL FGT, S lat, exhibits a very weak temperature dependence, indicating that increasing lattice vibration does not in uence the damping frequency. The dierence between S latandS k increases slightly only near TC, which can be attributed to the strong spin uctuation. The overall weak tem- perature dependence in the damping for a single layer indicates that a non-thermal disorder scattering mecha- nism is dominant: the strong surface scattering in such a thin layer (only a few A) combined with the enhanced SOC at the surfaces is the main channel for the magnetic damping in the SL FGT instead of spin uctuation and lattice vibration. Gilbert damping with a similarly weak temperature dependence has also been found in a permal- loy,40,53where chemical disorder scattering overwhelms thermally induced disorder. The temperature dependence of the bulk damping fre- quency is signicantly dierent from that of the SL. The calculated bulk damping, B k, (shown by the black solid diamonds in Fig. 2(b)) increases linearly with the temper- ature. This typical resistivitylike behavior suggests that the interband transition in bulk FGT is the dominant damping mechanism.54We also calculate the damping frequencyB latconsidering only lattice disorder, as shown as the black empty diamonds in Fig. 2(b). Comparing the results corresponding to the solid and empty diamonds leads us to conclude that both lattice and spin disorder substantially contribute to damping in bulk FGT. As the temperature approaches TC, the bulk damping is compa- rable with that in the single layer. IV. ANISOTROPIC DAMPING The damping torque exerted on the magnetization in the Landau-Lifshitz-Gilbert equation has the general form of M(t)[~
_M(t)], where the Gilbert damping parameter ~or the corresponding frequency is a tensor. This tensor and its elements depend on both the instan- taneous M(t) and its time derivative _M(t), where the anisotropy has been extensively analyzed using theoret- ical models55and rst-principles calculations.33,34Fol- lowing the denition given by Gilmore et al. ,33we call the anisotropic damping that depends on the equilibrium orientation of Meqthe orientational anisotropy and that4 𝑴𝐞𝐪𝑥𝑦𝑧𝜃 -U/2 -U/40U/4U/2V81216F|| (10-3)Single layerBulk FIG. 3. The calculated Gilbert damping parameter kfor SL (red circles) and bulk FGT (black diamonds) as a function of the angle between the equilibrium magnetization Meqand the atomic layer normal (^ z) of Fe 3GeTe 2. The lines are tted usingC0+C2cos 2. depending on _M(t) the rotational anisotropy. Consider- ing the layered structure of vdW materials, the lowered symmetry should result in remarkable anisotropy for the magnetization relaxation. Both the orientational and ro- tational anisotropy in bulk and SL FGT have been sys- tematically analyzed in this section. Notably, the damp- ing tensor is reduced to a scalar for the conguration shown in Fig. 2. Under a large in-plane magnetic eld, the perpendicu- lar magnetization of FGT can be tilted toward the exter- nal eld direction, which is dened as the y-axis without loss of generality. Thus, the angle between the equilib- rium magnetization Meqand the plane normal ^ zis re- ferred to as , as shown in the inset of Fig. 3. At = 0, as studied in Sec. III, xx=yy=k. For6= 0, xx=kstill holds, whereas the other diagonal element yydepends on specic values of . Here, we focus on k to study the orientational anisotropy of damping. The calculated in-plane damping kis plotted as a function ofin Fig. 3 for a SL at 77 K and bulk FGT at 100 K. The temperature is chosen in this way to obtain the same relative magnetization for the two systems, namely, M=M s= 88%, according to the experimentally measured magnetization as a function of temperature.3,8The same twofold symmetry is found for the damping parameters of both SL and bulk FGT, which can be eectively tted using a cos 2 term. As the magnetization rotates away from the easy axis, kincreases and reaches a maximum when the magnetization aligns inside the FGT layer. The changes, [(==2)(= 0)]=(= 0), are 62% for the SL and 39% for the bulk. A similar dependence of the damping on the magnetization orientation has been recently observed in single-crystal CoFe alloys.35,36The predicted anisotropic damping of FGT shown in Fig. 3 should analogously be experimentally observable. The rotational anisotropy of damping33in FGT is most signicant when the equilibrium magnetization lies in- side the atomic plane of FGT (along the hard axis), i.e., 0 0.20.40.6 0.81T/TC120150180QC/Q|| (%) 0 0.2 0.4 0.6 0.8 1T/TC10152025Q (108 Hz)Bulk Single layerBulkQCQ||Single layer BulkBulk 𝛂∥𝑴(𝐭) 𝑥𝑦𝑧(a)𝜶"(b)FIG. 4. (a) Schematic of damping with the equilibrium mag- netization Meqlying inside the atomic plane. Then, the in- stantaneous magnetization M(t) dissipates both the in- and out-of-plane spin angular momentum. The two types of dis- sipation are denoted as k(k) and?(?). (b) The calcu- lated Gilbert damping frequency k(?)as a function of tem- perature for single-layer and bulk Fe 3GeTe 2. The inset shows the ratio of the two frequencies ?=k. ==2. As schematized in Fig. 4(a), the magne- tization M(t) loses its in- or out-of-plane components depending on the instantaneous precessional direction _M(t). In this case, one has xx=kandzz=?, whereas the o-diagonal elements of the damping ten- sor are guaranteed to remain zero by symmetry.40The calculatedkand?for SL and bulk FGT are shown as a function of temperature in Fig. 4(b). For SL FGT, k(as shown by the circles with horizontal hatching) is nearly independent of temperature, which is the same as forMeqalong the easy axis. This result suggests that de- spite the sizable orientational anisotropy in the damping of SL FGT, the temperature has very little in uence on the specic values of the damping frequency. The calcu- lated?for the SL (shown by the red circles with vertical hatching) is considerably larger than kat low tempera- tures but decreases with increasing temperature. ?be- comes comparable with knear the Curie temperature, indicating that the rotational anisotropy is signicantly diminished by temperature. The calculated kfor bulk FGT with Meqalong the hard axis (shown by the black diamonds with horizontal hatching) is temperature-independent, in sharp contrast to the linear temperature dependence of kwith Meq along the easy axis shown in Fig. 2(b). This result sug- gests that the damping is already saturated in this case5 -0.4 -0.2 0 0.2 0.4 E-EF (eV)100200300400500λ⊥/λ|| (%)50 K 77 K 100 KSingle layer FIG. 5. The calculated rotational damping anisotropy for single-layer Fe 3GeTe 2as a function of the Fermi energy at dierent temperatures. at a suciently large scattering rate, where saturated damping has also been found in FM Ni.25The calculated ?of bulk FGT is also larger than kat low tempera- tures and slightly decreases with increasing temperature. We summarize the results for the rotationally anisotropic damping frequency by plotting the ratio between ?and kin the inset of Fig. 4(b). The ratio for both SL and bulk FGT decreases with increasing temperature and approaches unity near TC. This behavior is consistent with the results calculated using the torque-correlation model,33where rotationally anisotropic damping disap- pears gradually as the scattering rate increases. In highly disordered systems, the damping is more isotropic, as in- tuitively expected. We emphasize that the calculated ?values are dis- tinct from those reported in previous studies in the literature,55that is,?was found to vanish in single- crystal monoatomic FM layers based on the breathing Fermi surface model.56{58Interband scattering is ne- glected in the breathing Fermi surface model. However, the resistivitylike behavior of our calculated kfor bulk FGT shows that interband scattering plays an important role in this vdW FM material. One of the unique advantages of 2D vdW materials is the tunability of the electronic structure via electri- cal gating.4,59To simulate such a scenario, we slightly adjust the Fermi level EFof SL FGT without changing the band structure for simplicity. The calculated rota- tional anisotropy in the damping ?=kof SL FGT is shown as a function of the Fermi energy in Fig. 5. At all the temperatures considered, the anisotropy ratio ?=k increases dramatically as EFis lowered by 0.3 eV, es- pecially at low temperatures, and only exhibits minor changes when EFis increased. At 50 K, the ratio ?=k becomes as high as 430%, which is almost three times larger than that obtained without gating. This result suggests that a small quantity of holes doped into SLFGT at low temperatures remarkably enhances the rota- tional damping anisotropy. V. CONCLUSIONS We have systematically studied Gilbert damping in a 2D vdW FM material Fe 3GeTe 2by using rst-principles scattering calculations where the temperature-induced lattice vibration and spin uctuation are modeled by frozen thermal lattice and spin disorder. When the mag- netization is perpendicular to the 2D atomic plane, the damping frequency of bulk FGT increases linearly with the temperature, whereas that of SL FGT exhibits a weak temperature dependence. The dierence can be attributed to surface scattering (which is absent in the bulk) dominating scattering due to temperature-induced disorder in SLs, which have a thickness smaller than the electronic mean free path. The anisotropy of Gilbert damping in this 2D vdW material has also been thor- oughly investigated. The orientational anisotropy, which depends on the direction of the equilibrium magnetiza- tion with respect to the atomic planes, exhibits twofold rotational symmetry in both the bulk and SL. When the equilibrium magnetization is parallel to the atomic plane, the damping is signicantly enhanced compared to that with the magnetization perpendicular to the atomic plane. The rotational anisotropic damping depending on the direction of motion of the instantaneous magnetiza- tion is remarkable with the equilibrium magnetization ly- ing inside the atomic plane. With an out-of-plane compo- nent in the timederivative of the precessional magnetiza- tion, the damping frequency ( ?) is much larger than the one where only in-plane magnetization is varying ( k). The ratio?=kis larger than unity for both the bulk and a single layer and decreases with increasing temper- ature. In SL FGT, ?=kcan be enhanced up to 430% by slight holedoping at 50 K. Antiferromagnetic order has recently been discovered in 2D vdW materials (as reviewed in Ref. 60 and the ref- erences therein) and some intriguing properties are found in their damping behaviors.61,62Owing to the more com- plex magnetic order, more than a single parameter is necessary in describing the damping in antiferromagnetic dynamics.63,64It would be very interesting to study the magnetization relaxation in these 2D materials with more complex magnetic order. ACKNOWLEDGMENTS The authors are grateful to Professor Xiangang Wan at Nanjing University for his support and helpful dis- cussions. Financial support for this study was provided by the National Natural Science Foundation of China (Grants No. 11734004 and No. 12174028).6 yiliu@bnu.edu.cn 1Cheng Gong, Lin Li, Zhenglu Li, Huiwen Ji, Alex Stern, Yang Xia, Ting Cao, Wei Bao, Chenzhe Wang, Yuan Wang, Z. Q. Qiu, R. J. Cave, Steven G. 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2022-10-16

Magnetization relaxation in the two-dimensional itinerant ferromagnetic van der Waals material Fe$_3$GeTe$_2$, below the Curie temperature, is fundamentally important for applications to low-dimensional spintronics devices. We use first-principles scattering theory to calculate the temperature-dependent Gilbert damping for bulk and single-layer Fe$_3$GeTe$_2$. The calculated damping frequency of bulk Fe$_3$GeTe$_2$ increases monotonically with temperature because of the dominance of resistivitylike behavior. By contrast, a very weak temperature dependence is found for the damping frequency of a single layer, which is attributed to strong surface scattering in this highly confined geometry. A systematic study of the damping anisotropy reveals that orientational anisotropy is present in both bulk and single-layer Fe3GeTe2. Rotational anisotropy is significant at low temperatures for both the bulk and a single layer and is gradually diminished by temperature-induced disorder. The rotational anisotropy can be significantly enhanced by up to 430% in gated single-layer Fe$_3$GeTe$_2$.

Magnetic damping anisotropy in the two-dimensional van der Waals material Fe$_3$GeTe$_2$ from first principles

2210.08429v1

arXiv:1802.02415v1 [cond-mat.mes-hall] 7 Feb 2018Breaking the current density threshold in spin-orbit-torq ue magnetic random access memory Yin Zhang,1,2H. Y. Yuan,3,∗X. S. Wang,4,1and X. R. Wang1,2,† 1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 2HKUST Shenzhen Research Institute, Shenzhen 518057, China 3Department of Physics, Southern University of Science and T echnology of China, Shenzhen 518055, China 4School of Microelectronics and Solid-State Electronics, University of Electronic Science and Technology of China, C hengdu, Sichuan 610054, China (Dated: March 25, 2022) Spin-orbit-torque magnetic random access memory (SOT-MRA M) is a promising technology for the next generation of data storage devices. The main bot tleneck of this technology is the high reversal current density threshold. This outstanding problem of SOT-MRAM is now solved by using a current density of constant magnitude and varying flow direction that reduces the reversal current density threshold by a factor of more than t he Gilbert damping coefficient. The Euler-Lagrange equation for the fastest magnetization rev ersal path and the optimal current pulse are derived for an arbitrary magnetic cell. The theore tical limit of minimal reversal current density and current density for a GHz switching rate of the ne w reversal strategy for CoFeB/Ta SOT-MRAMs are respectively of the order of 105A/cm2and 106A/cm2far below 107A/cm2and 108A/cm2in the conventional strategy. Furthermore, no external mag netic field is needed for a deterministic reversal in the new strategy. Subject Areas: Magnetism, Nanophysics, Spintronics I. INTRODUCTION Fast and efficient magnetization reversal is of not only fundamentally interesting, but also technologically im- portant for high density data storage and massive in- formation processing. Magnetization reversal can be in- duced by magnetic field [1–3], electric current through direct [4–9] and/or indirect [10–22] spin angular mo- mentum transfer from polarized itinerant electrons to magnetization, microwaves [23], laser light [24], and even electric fields [25]. While the magnetic field in- duced magnetization reversal is a matured technology, it suffers from scalability and field localization problems [8, 26] for nanoscale devices. Spin transfer torque mag- netic random-access memory is an attractive technol- ogy in spintronics [26] although Joule heating, device durability and reliability are challenging issues [11, 26]. In an spin-orbit-torque magnetic random access mem- ory (SOT-MRAM) whose central component is a heavy- metal/ferromagnet bilayer, an electric current in the heavy-metal layer generates a pure spin current through the spin-Hall effect [10, 11] that flows perpendicularly into the magnetic layer. The spin current, in turn, pro- duces spin-orbit torques (SOT) through spin angular momentum transfer [4, 5] and/or Rashba effect [16–22]. SOT-MRAM is a promising technology because writing charge current does not pass through the memory cells so that the cells do not suffer from the Joule heating and associated device damaging. In principle, such de- vices are infinitely durable due to negligible heating from ∗[Corresponding author:]yuanhy@sustc.edu.cn †[Corresponding author:]phxwan@ust.hkspin current [11]. However, the reversal current density threshold (above 107A/cm2[14, 15] for realistic materi- als) in the present SOT-MRAM architecture is too high. To have a reasonable switching rate (order of GHz), the current density should be much larger than 108A/cm2 [14, 15] that is too high for devices. In order to lower the minimalreversalcurrentdensityaswellastoswitchmag- netization states at GHz rate at a tolerable current den- sity in SOT-MRAM, it is interesting to find new reversal schemes (strategies) that can achieveabove goals. In this paper, we show that a proper current density pulse of time-dependent flow direction and constant magnitude, much lower than the conventional threshold, can switch a SOT-MRAM at GHz rate. Such a time-dependent cur- rent pulse can be realized by using two perpendicular currentspassingthrough the heavy-metallayer. The the- oretical limit of minimal reversal current density of the new reversal strategy for realistic materials can be of the order of 105A/cm2, far below 107A/cm2in the con- ventional strategy that uses a direct current (DC), both based on macrospin approximation. The validity of the macrospin model is also verified by micromagnetic simu- lations. II. MACROSPIN MODEL AND RESULTS A. Model Our new reversal strategy for an SOT-MRAM, whose central component is a ferromagnetic/heavy-metal bi- layer lying in the xy-plane with initial spin along the +z-direction as shown in Fig. 1, uses a current den- sityJ=JcosΦˆx+JsinΦˆygenerated from two time-2 m FM HM JxJx JyJy FIG. 1. Schematic illustration of new reversal scheme for SOT-MRAMs. Two perpendicular currents flowin the heavy- metal layer of a ferromagnet/heavy-metal bilayer to genera te a current whose direction can vary in the xy-plane. dependent electric currents flowing along the x- and the y-directions, where Φ is a time-dependent angle between Jand the x-axis and Jis a constant total current den- sity. The magnetic energy density is ε=−Kcos2θwith Kbeing the anisotropy coefficient and θbeing the polar angle of the magnetization. In the absence of an electric current, the system has two stable states m= +ˆzand m=−ˆzwheremis the unit direction of magnetization M=Mmof magnitude M. The electric current gen- erates a transverse spin current perpendicularly flowing into the ferromagnetic layer via the spin-Hall effect [10], andthenproducesaneffectiveSOTonthemagnetization[4, 5, 16], i.e. /vector τ=−am×(m׈s)+βam׈s, (1) where the first term on the right-hand-side is the Slonczewski-liketorquewhile thesecondtermis thefield- like torque. The spin-polarization direction is ˆ s=ˆJ׈z (for other type of spin-Hall effect, see Note [27]) with ˆJ being the unit vector of current density. a=¯h 2edθSHJ measures SOT where ¯ h,e, anddare respectively the Plank constant, the electron charge, and the sample thickness. θSHis the spin Hall angle which measures the conversion efficiency between the spin current and charge current. βmeasures the field-like torque and can be an arbitrary real number since this torque may also be directly generated from the Rashba effect [16]. Themagnetizationdynamicsunderanin-planecurrent densityJis governed by the generalized dimensionless Landau-Lifshitz-Gilbert (LLG) equation, ∂m ∂t=−m×heff+αm×∂m ∂t+/vector τ, (2) whereαis the Gilbert damping constant that is typically much smaller than unity. The effective field is heff= −∇mεfrom energy density ε. Time, magnetic field and energy density are respectively in units of ( γM)−1,M andµ0M2, where γandµ0are respectively the gyro- magnetic ratio and vacuum magnetic permeability. In this unit system, a=¯h 2edµ0M2θSHJbecomes dimension- less. The magnetization mcan be conveniently described by a polar angle θand an azimuthal angle φin thexyz- coordinate. In terms of θandφ, the generalized LLG equation becomes (1+α2)˙θ=−αKsin2θ+a(1−αβ)cosθsin(Φ−φ)+a(α+β)cos(Φ−φ)≡F1, (3a) (1+α2)˙φsinθ=Ksin2θ−a(1−αβ)cos(Φ−φ)+a(α+β)cosθsin(Φ−φ)≡F2. (3b) B. Derivation of the Euler-Lagrange equation The goal is to reverse the initial state θ= 0 to the target state θ=πby SOT. There are an infinite number of paths that connect the initial state θ= 0 with the target state θ=π, and each of these paths can be used as a magnetization reversal route. For a given reversal route, there are an infinite number of current pulses that can reverse the magnetization. The theoretical limit of minimal current density Jcis defined as the smallest val- ues of minimal reversal current densities of all possible reversal routes. Then it comes two interesting and im- portant questions: 1) What is Jcabove which there is at least one reversal route that the current density can re- verse the magnetization along it? 2) For a given J > Jc,what are the optimal reversal route and the optimal cur- rent pulse Φ( t) that can reverse the magnetization at the highest speed? Dividing Eq. (3b) by Eq. (3a), one can obtain the following constraint, G≡∂φ ∂θsinθF1−F2= 0. (4) The magnetization reversal time Tis T=/integraldisplayπ 0dθ ˙θ=/integraldisplayπ 01+α2 F1dθ. (5) The optimization problem here is to find the optimal reversal route φ(θ) and the optimal current pulse Φ( t)3 such that Tis minimum under constraint (4). Using the Lagrange multiplier method, the optimal reversal route and the optimal current pulse satisfy the Euler-Lagrange equations [28, 29], ∂F ∂φ=d dθ(∂F ∂(∂φ/∂θ)),∂F ∂Φ=d dθ(∂F ∂(∂Φ/∂θ)),(6) whereF= (1 +α2)/F1+λGandλis the Lagrange multipliers which can be determined self-consistently by Eq. (6) and constrain (4). Given a current density of constant magnitude J, Eq. (6) may or may not have a solution of φ(θ) that continuously passing through θ= 0 andθ=π. If such a solution exists, then φ(θ) is the optimal path for the fastest magnetization reversal and the corresponding solution of Φ( t) is the optimal current pulse. The theoretical limit of minimal reversal current density is then the smallest current density Jc below which the optimal reversal path does not exist. C. The optimal current pulse and theoretical limit of minimal reversal current density From Eqs. (3a), (3b) and (4) as well as F= (1 + α2)/F1+λG, theEuler-Lagrangeequationof (6)becomes λd dθ(F1sinθ) = 0, (7a) 1+α2 F2 1∂F1 ∂φ−λ∂G ∂φ=−1+α2 F2 1∂F1 ∂Φ+λ∂G ∂Φ= 0.(7b) From Eq. (7a), one has λ/ne}ationslash= 0 orλ= 0. Ifλ/ne}ationslash= 0,F1must beF1=C/sinθ(C/ne}ationslash= 0) so that (1+ α2)˙θ=C/sinθ→ ∞asθ→0 orπ. This solution is not physical, and shouldbe discarded. Therefore, the only allowedsolution must be λ= 0, and one has ∂F1/∂Φ = 0 according to Eq. (7b). Interestingly, this is exactly the condition of maximal ˙θ=F1/(1+α2) as Φ varies. Φ satisfies tan(Φ − φ) =1−αβ α+βcosθ, or Φ = tan−1(1−αβ α+βcosθ)+φ+π(β <−α) (8a) Φ = tan−1(1−αβ α+βcosθ)+φ (β >−α).(8b) Substituting Eq. (8) into the LLG equation (3), θ(t) andφ(t) are determined by the following equations, ˙θ=1 1+α2[aP(θ)−αKsin2θ], (9a) ˙φ=1 1+α2[2Kcosθ−a(α+β)(1−αβ)sinθ P(θ)],(9b) whereP(θ) =/radicalbig (α+β)2+(1−αβ)2cos2θ. To reverse magnetization from θ= 0 toθ=π,amust satisfy a > αKsin(2θ)/P(θ) according to Eq. (9a) so that ˙θis no negative for all θ. Obviously, ˙θ= 0 atθ=π/2 whenFIG. 2. The log α-dependence of Jcfor various βare plotted as the solid curves for model parameters of M= 3.7×105 A/m,K= 5.0×103J/m3,θSH= 0.084 and d= 0.6 nm. As a comparison, Jdc cis also plotted as the dashed lines. β=−α. The magnetization reversal is not possible in this case, and β=−αisasingularpoint. The theoretical limit of minimal reversal current density Jcforβ/ne}ationslash=−α is Jc=2αeKd θSH¯hQ, (10) whereQ≡max{sin2θ/P(θ)}forθ∈[0,π]. In comparison with the current density threshold [13, 14, 18] (Jdc c) in the conventional strategy for β= 0, Jdc c=2eKd θSH¯h(1−H√ 2K), (11) the minimal reversal current density is reduced by more than a factor of α. HereH(≃22 Oe in experiments) is a small external magnetic needed for a deterministic reversal in conventional strategy. Using CoFeB/Ta pa- rameters of M= 3.7×105A/m,K= 5.0×103J/m3, θSH= 0.084 and d= 0.6 nm [11, 14, 15], Fig. 2 shows logα-dependenceof Jc(solidlines)and Jdc c(dashedlines) forβ= 0 (black), 0 .3 (red) and −0.3 (blue), respectively. BothJdc candJcdepend on β. The lower the damping of a magnetic material is, the smaller our minimum switch- ing current density will be. For a magnetic material of α= 10−5, the theoretical limit of minimal reversal cur- rent density can be five order of magnitude smaller than the value in the conventional strategy. For a given J > Jc, the shortest reversal time is given by Eqs. (5) and (9a): T=/integraldisplayπ 01+α2 aP(θ)−αKsin2θdθ. (12) The optimal reversal path is given by φ(θ) =/integraltextθ 0˙φ ˙θdθ′ where˙θand˙φare given by Eqs. (9a) and (9b). Eq. (9a) gives t(θ) =/integraltextθ 0(1+α2)/(aP(θ)−αKsin2θ)dθ′and thenθ(t) is just θ(t) =t−1(θ). Thus, Φ( θ,φ),φ(θ) andθ(t) giveφ(t) =φ(θ(t)) and Φ( t) = Φ(θ(t),φ(t)).4 (d) (e) mz mxmymz my mx(a) (b) (c) (f) mymz mx FIG. 3. Model parameters of M= 3.7×105A/m,K= 5.0×103J/m3,θSH= 0.084,α= 0.008 and d= 0.6 nm are used to mimic CoFeB/Ta bilayer, and β= 0.3 for (a), (c), (d) and (f) while β= 0.1 for (b) and (e). The theoretical limit of minimum reversal current density is Jc= 1.56×105A/cm2forβ= 0.1 andJc= 1.28×105A/cm2forβ= 0.3. Optimal current pulses ((a)-(c)) and fastest reversal routes ((d)-(f)) are for J= 1.92×106A/cm2((a) and (d)), and for J= 9.0×106A/cm2((b), (c)), (e) and (f). Using the same parameters as those for Fig. 2 with α= 0.008 and various β, Fig. 3 shows the optimal current pulses ((a)-(c)) and the corresponding fastest magnetization reversal routes ((d)-(f)) for β= 0.3 and J= 1.92×106A/cm2≈15Jc((a) and (d)), for β= 0.1 andJ= 9.0×106A/cm2≈58Jc((b) and (e)), and for β= 0.3 andJ= 9.0×106A/cm2≈70Jc((c) and (f)). It isknownthatTahaslesseffecton α[11]. Theminimalre- versal current density Jcunder the optimal current pulse is 1.56×105A/cm2forβ= 0.1 and 1.28×105A/cm2 forβ= 0.3 which is far below Jdc c= 9.6×106A/cm2for the same material parameters [15]. The multiple oscilla- tions ofmxandmyreveal that the reversal is a spinning process and optimal reversal path winds around the two stable states many times. Correspondingly, the driving current makes also many turns as shown by the multiple oscillations of JxandJy. The number of spinning turns dependsonhowfar JisfromJc. Thecloser JtoJcis, the number of turns is larger. The number of turns is about 5 in Figs. 3(a) and 3(d) for J≈15Jcand one turn for J >50Jcas shown in Figs. 3(b), 3(c), 3(e) and 3(f), so that the reversal is almost ballistic. The reversaltime for β= 0.3 andJ= 1.92×106A/cm2is about 10 nanosec- onds, for β= 0.1 andJ= 9.0×106A/cm2is about 3.3 nanoseconds, and for β= 0.3 andJ= 9.0×106A/cm2is about 2.1 nanoseconds. Figure 4 is the reversaltime Tas a function of current density Junder the optimal current pulse for the same parameters as those for Fig. 2. TheFIG. 4. Magnetization reversal time Tunder the optimal current pulses as a function of Jfor various αandβ. reversaltime quickly decreases to nanoseconds as current density increases. In a real experiment, there are many uncertainties so that the current pulse may be different from the optimal one. To check whether our strategy is robust again small fluctuations, we let the current pulse in Fig. 3(c) deviate from its exact value. Numerical simulations show that the magnetization reversal is not significantly influenced at least when the deviation be- tween the real current and optimal current is less than five percents.5 III. VERIFICATION OF MACROSPIN MODEL BY MICROMAGNETIC SIMULATION In our analysis, the memory cell is treated as a macrospin. A nature question is how good the macrospin model is for a realistic memory device. To answer this question, we carried out micromagnetic simulations by using Newton-Raphson algorithm [30] for two memory cells of 150 nm ×150nm×0.6 nm (Figs. 5(a), (b), (d) and (e)) and 250 nm ×250 nm×0.6 nm (Figs. 5(c) and (f)). To model the possible edge pinning effect due to mag- netic dipole-dipole interaction, we consider square-shape devices instead of cylinder shape device whose edge pin- ning is negligible. To make a quantitative comparison, the material parameters are the same as those used in Fig. 3. In our simulations, the unit cell size is 2 nm ×2 nm×0.6 nm. For a fair comparison, the optimal current pulses shown in Figs. 3(a) and (c) of respective current densityJ= 1.92×106A/cm2andJ= 9.0×106A/cm2 were applied to the memory cell of 150 nm ×150 nm×0.6 nm. The symbols in Figs. 5(a) and (b) are the time evo- lution of averaged magnetization mx,myandmzwhile thesolidlinesarethetheoreticalpredictionsofmacrospin model shown in Figs. 3(d) and (f). The perfect agree- ments prove the validity of the macrospin approximation for our device of such a size. To further verify that the memory device can be treated as a macrospin, Figs. 5(d) and (e) are the spin configurations in the middle of the reversal at t= 5.5 ns for Fig. 5(a) and at t= 1.2 ns for Fig. 5(b). Thefactthatallspinsalignalmostinthesame direction verifies the validity of the macro spin model. In real experiments, non-uniformity of current density is in- evitable. To demonstrate the macrospin model is still valid, we let current density linearly varies from 9 .5×106 A/cm2on the leftmost column of cells to 8 .5×106A/cm2 onthe rightmostcolumn ofcells. As expected, thereis no noticeable difference with the data shown in Figs. 5(b) and (e). For the large memory device of 250 nm ×250 nm×0.6 nm, the optimal current pulse shown in Fig. 3(c) of cur- rent density J= 9.0×106A/cm2was considered. The time evolution of averaged magnetization mx,myand mzare plotted in Fig. 5(c), with the symbols for simula- tions andsolid linesfor the macrospinmodel. They agree very well although there is a small deviation for device of such a large size. Figure 5(f) is the spin configurations in the middle of the reversal at t= 1.2 ns for Fig. 5(c). The marcospin model is not too bad although all spins are not perfectly aligned in this case. In summary, for a normal SOT-MRAM device of size less than 300 nm [11, 15], macrospin model describes magnetization reversal well. However, for a larger sam- ple size and lower current density ( J <106A/cm2for the same material parameters as those used in Fig. 3), only the spins in sample center can be reversed while the spins near sample edges are pinned.(a) (b) (c) (d) (e) (f) t = 5.5 ns t = 1.2 ns t = 1.2 ns FIG. 5. (a)-(c) Time evolution of the average magnetization : cycles for micromagnetic simulations and solid lines are th e- oretical predictions from macrospin model. (a) and (b) are for the memory cell of 150 nm ×150 nm×0.6 nm and optimal current pulse of current density of J= 1.92×106A/cm2and J= 9.0×106A/cm2, respectively. (c) is for the memory cell of 250 nm ×250 nm×0.6 nm and optimal current pulse of current density of J= 9.0×106A/cm2. (d)-(f) Spin configu- rations respectively corresponding to (a)-(c) in the middl e of magnetization reversal at t= 5.5 ns and 1.2 ns. The cell size in micromagnetic simulation is 2 nm ×2 nm×0.6 nm. IV. DISCUSSION Obviously, the strategy present here can easily be gen- eralized to the existing spin-transfer torque MRAM. The mathematics involved are very similar, and one expects a substantial current density reduction is possible there if a proper optimal current pulse is used. Of course, how to generate such a current pulse should be much more challenge than that for SOT-MRAM where two perpen- dicular currents can be used. In the conventional strat- egy that uses a DC-current, a static magnetic field along current flow is required for a deterministic magnetiza- tion reversal [13, 14, 18]. Although several field-free de- signs have been proposed [19, 20], an antiferromagnet is needed to create an exchange bias which plays the role of an applied magnetic field. As we have shown, such a requirement or complication is not needed in our strat- egy. Ourstrategydoesnothaveanotherproblemexisting in the conventional strategy in which the magnetization can only be placed near θ=π/2 [13, 14, 18] so that the system falls into the target state by itself through the damping. Therefore, one would like to use materials with largerdamping in the conventionalstrategyin order to speed up this falling process. In contrast, our strategy preferslowdampingmaterials,andreversalisalmostbal- listic when current density is large enough ( >50Jcin the current case). To reverse the magnetization from θ=π toθ= 0, one only needs to reverse the current direction of the optimal current pulse. One should notice that the Euler-Lagrange equation allows us to easily obtain the optimal reversal current pulse and theoretical limit6 of the minimal reversal current density for an arbitrary magnetic cell such as in-plane magnetized layer [11] and biaxial anisotropy. V. CONCLUSION In conclusion, weinvestigatedthe magnetizationrever- sal of SOT-MRAMs, and propose a new reversal strat- egy whose minimal reversal current density is far be- low the existing current density threshold. For popular CoFeB/Ta system, it is possible to use a current densityless than 106A/cm2to reversethe magnetizationat GHz rate, in comparison with order of J≈108A/cm2in the conventional strategy. ACKNOWLEDGMENTS This work was supported by the National Natural Sci- ence Foundation of China (Grant No. 11774296 and No. 61704071) as well as Hong Kong RGC Grants No. 16300117 and No. 16301816. X.R.W. acknowledges the hospitalities of Beijing Normal University and Beijing ComputationalScience ResearchCenterduringhisvisits. [1] C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, D. Weller, E. L. Garwin, H. C. Siegmann, Minimum field strength in precessional magnetization reversal , Science 285, 864 (1999). [2] M. Bauer, J. Fassbender, B. Hillebrands, and R. L. Stamps, Switching behavior of a Stoner particle beyond the relaxation time limit , Phys. Rev. B 61, 3410 (2000). [3] Z. Z. Sun and X. R. Wang, Theoretical Limit of the Mini- mal Magnetization Switching Field and the Optimal Field Pulse for Stoner Particles , Phys. Rev. Lett. 97, 077205 (2006). [4] J. C. Slonczewski, Current-driven excitation of magnetic multilayers , J. Magn. Magn. Mater. 159, L1 (1996). [5] L. 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[12] SatoruEmori, UweBauer, Sung-MinAhn, EduardoMar- tinez and Geoffrey S. D. Beach, Current-driven dynamics of chiral ferromagnetic domain walls , Nat. Mat. 12, 611- 616 (2013). [13] Ki-Seung Lee, Seo-Won Lee, Byoung-Chul Min, and Kyung-Jin Lee, Threshold current for switching of a per- pendicular magnetic layer induced by spin Hall effect , Appl. Phys. Lett. 102, 112410 (2013). [14] S. Fukami, T. Anekawa, C. Zhang and H. Ohno, A spin- orbit torque switching scheme with collinear magneticeasy axis and current configuration , Nat. Nanotech. 11, 621-625 (2016). [15] Jun-Yang Chen, Mahendra DC, Delin Zhang, Zhengyang Zhao, Mo Li, and Jian-Ping Wang, Field-free spin- orbit torque switching of composite perpendicular CoFeB/Gd/CoFeB layers utilized for three-terminal mag- netic tunnel junctions , Appl. Phys. Lett. 111, 012402 (2017). [16] A. Manchon and S. Zhang, Theory of spin torque due to spin-orbit coupling , Phys. Rev. B 79, 094422 (2009). [17] Ioan Mihai Miron, Gilles Gaudin, Stphane Auffret, Bernard Rodmacq, Alain Schuhl, Stefania Pizzini, Jan Vogel and Pietro Gambardella, Current-driven spin torque induced by the Rashba effect in a ferromagnetic metal layer , Nat. Mat. 9, 230-234 (2010). [18] X. Zhang, C. H. Wan, Z. H. Yuan, Q. T. Zhang, H. Wu, L. Huang, W. J. Kong, C. Fang, U. Khan, and X. F. Han,Electrical control over perpendicular magnetization switching driven by spin-orbit torques , Phys. Rev. B 94, 174434 (2016). [19] A. van den Brink, G. Vermijs, A. Solignac, J. Koo, J. T. Kohlhepp, H.J.M.SwagtenandB.Koopmans, Field-free magnetization reversal by spin-Hall effect and exchange bias, Nat. Commun. 7, 10854 (2016). [20] ShunsukeFukami, Chaoliang Zhang, Samik DuttaGupta, Aleksandr Kurenkov and Hideo Ohno, Magnetization switching by spin-orbit torque in an antiferromagnet- ferromagnet bilayer system , Nat. Mater. 15, 535-541 (2016). [21] A. Manchon, Spin Hall effect versus Rashba torque: a Diffusive Approach , arXiv:1204.4869. [22] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Bl¨ ugel, S. Auffret, O. Boulle, G. Gaudin, and P. Gambardella, Symmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures , Nat. Nanotech. 8, 587 (2013). [23] Z. Z. Sun and X. R. Wang, Magnetization reversal through synchronization with a microwave , Phys. Rev. B74, 132401 (2006). [24] J.-Y. Bigot, L. Guidoni, E. Beaurepaire, and P. N. Saeta , Femtosecond Spectrotemporal Magneto-optics , Phys.Rev. Lett. 93, 077401 (2004). [25] Fumihiro Matsukura, Yoshinori Tokura and Hideo Ohno, Control of magnetism by electric fields , Nat. Nanotech. 10, 209-220 (2015).7 [26] E. Chen, D. Apalkov, Z. Diao, A. Driskill-Smith, D. Druist, D. Lottis, V. Nikitin, X. Tang, S. Watts, S. Wang, S. A. Wolf, A. W. Ghosh, J.W. Lu, S. J. Poon, M. Stan, W. H. Butler, S. Gupta, C. K. A. Mewes, Tim Mewes, and P. B. 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2018-02-07

Spin-orbit-torque magnetic random access memory (SOT-MRAM) is a promising technology for the next generation of data storage devices. The main bottleneck of this technology is the high reversal current density threshold. This outstanding problem of SOT-MRAM is now solved by using a current density of constant magnitude and varying flow direction that reduces the reversal current density threshold by a factor of more than the Gilbert damping coefficient. The Euler-Lagrange equation for the fastest magnetization reversal path and the optimal current pulse are derived for an arbitrary magnetic cell. The theoretical limit of minimal reversal current density and current density for a GHz switching rate of the new reversal strategy for CoFeB/Ta SOT-MRAMs are respectively of the order of $10^5$ A/cm$^2$ and $10^6$ A/cm$^2$ far below $10^7$ A/cm$^2$ and $10^8$ A/cm$^2$ in the conventional strategy. Furthermore, no external magnetic field is needed for a deterministic reversal in the new strategy.

Breaking the current density threshold in spin-orbit-torque magnetic random access memory

1802.02415v1

Ferromagnetic resonance study of Co/Pd/Co/Ni multilayers with perpendicular anisotropy irradiated with Helium ions J-M. L. Beaujour, A. D. Kent,1D. Ravelosona,2and I. Tudosa and E. E. Fullerton3 1Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA 2Institut d'Electronique Fondamentale, UMR CNRS 8622, Universite Paris Sud, 91405 Orsay Cedex, France 3University of California, San Diego, Center for Magnetic Recording Research, La Jolla, CA 92093-0401, USA (Dated: April 24, 2022) We present a ferromagnetic resonance (FMR) study of the eect of Helium ion irradiation on the magnetic anisotropy, the linewidth and the Gilbert damping of a Co/Ni multilayer coupled to Co/Pd bilayers. The perpendicular magnetic anisotropy decreases linearly with He ion uence, leading to a transition to in-plane magnetization at a critical uence of 5 1014ions/cm2. We nd that the damping is nearly independent of uence but the FMR linewidth at xed frequency has a maximum near the critical uence, indicating that the inhomogeneous broadening of the FMR line is a non- monotonic function of the He ion uence. Based on an analysis of the angular dependence of the FMR linewidth, the inhomogeneous broadening is associated with spatial variations in the magnitude of the perpendicular magnetic anisotropy. These results demonstrate that ion irradiation may be used to systematically modify the magnetic anisotropy and distribution of magnetic anisotropy parameters of Co/Pd/Co/Ni multilayers for applications and basic physics studies. PACS numbers: Co/Ni multilayers are of great interest in informa- tion technology and in spin-transfer devices because they combine high spin polarization with large perpendicu- lar magnetic anisotropy (PMA) [1{3]. Perpendicular anisotropy was predicted in Co/Ni multilayers and has been shown experimentally to be a function of layer com- position and thin lm growth conditions [4]. Recently it has been shown that the coercivity and perpendicu- lar anisotropy of a multilayer can be tailored by Helium ion irradiation [5], making it possible to modify lms af- ter growth to tune their magnetic properties. This is of great interest for applications and basic physics stud- ies, as in many cases the perpendicular anisotropy of a structure sets importance device metrics, and ion irradi- ation oers the possibility of changing these properties, both locally and globally, after device fabrication. For instance, in spin-transfer magnetic random access mem- ories (STT-MRAM) the current threshold for switching is proportional to the PMA [2, 6]. Light ion irradiation has been used to vary the mag- netic properties of multilayer lms in many earlier studies [7{11]. For instance, the coercivity of Co/Pt multilayers was found to decrease with ion dose [8]. This behavior was attributed to interface mixing and strain relaxation reducing the PMA. Very recently, it was reported that the coercive eld of Co/Ni multilayers decreases linearly with increasing He+irradiation uence up to F= 1015 ions/cm2, suggesting changes in the magnetic anisotropy of the lm [10]. The eect of ion irradiation on the FMR linewidth has also been studied in Au/Fe multilayer lms with PMA [11]. The PMA is reduced by He+irradi- ation and the authors explained this by a reduction of the inhomogeneous contribution to the FMR linewidth. In a recent paper, we presented a FMR study of the anisotropy and the linewidth of a Co/Ni multilayer lmexposed to a relatively high He+irradiation uence ( F = 1015ions/cm2) [12]. In addition to a strong decrease of the PMA, the contribution to the linewidth from spa- tial variation of the anisotropy, was reduced compared to that of a non irradiated Co/Ni multilayer. Further- more, a correlation between the anisotropy distribution and the linewidth broadening from two-magnon scatter- ing (TMS) mechanism was observed. However, a system- atic study of the eect of the He+irradiation on the FMR spectra as a function of uence has yet to be reported. In this paper, we present a FMR study of a Co/Ni multilayer coupled to Co/Pd bilayers exposed to Helium ion irradiation of uence up to 1015ions/cm2. The PMA and the contributions to the FMR linewidth, including those from Gilbert damping ( ), are studied as a function of uence. The samples had the following layer structure: jj3 Taj1 Pdj0.3 Coj1 Pdj0.14 Coj[0.8 Nij0.14 Co]3j1 Pdj0.3 Coj1 Pdj3 Tajj(layer thickness in nm) and was fabricated by dc magnetron sputtering. The Co/Ni multilayer is embed- ded between Co/Pd bilayers to enhance the overall PMA of the lm and to have resonance frequencies in which the full angular dependence of the FMR response could be investigated in a 1 T electromagnet. The substrate was cleaved into several pieces that were then exposed to dif- ferent doses of Helium ion irradiation of energy 20 keV with uence in the range 1014F1015ions/cm2. FMR measurements were conducted at room tempera- ture using a coplanar waveguide (CPW). Details of the experimental setup can be found in [13]. The eld swept CPW transmission signal ( S21) was recorded as a func- tion of frequency for dc magnetic elds normal to the lm plane and as a function of the out-of-plane eld angle at 20 GHz. The magnetization density of the lm at room temperature was measured with a SQUID magnetome-arXiv:1010.0268v2 [cond-mat.mes-hall] 5 Oct 20102 03 06 09 00.40.60.8K K0 5 1 001020H /s615490Hres (T) 0 1 2.5 F ield angle /s61542H (deg)20 GHz 5 7.5 10HH -0.40.00.40.80 102030f (GHz)(a) (b) /s615490Hres (T) /s61549 0Meff ( c) F (1014 ions/cm2)K (105 J/m3)21 FIG. 1: a) Angular dependence of the resonance eld for dif- ferent irradiation uence ( 1014ions/cm2). The solid lines are guides to the eye. b) Frequency dependence of the reso- nance eld when the applied eld is normal to the lm plane (H= 90o) at selected uences. The solid lines are the ts to Eq. 1. The zero frequency intercept gives the eective de- magnetization eld, 0Me. c) The second and fourth order perpendicular anisotropy constants, K1andK2, versus u- ence. ter:Ms'4:75105A/m. Within the measurement uncertainty, Msremains unchanged after irradiation. Fig. 1a shows the out of plane angular dependence of the resonance eld at 20 GHz for dierent uences. For the non-irradiated lm, the resonance eld when H is normal to the lm plane ( H= 90o) is smaller than that when the eld is in the lm plane ( H= 0o). This shows that the magnetic easy axis is normal to the lm plane. As the uence increases, HresatH= 90oin- creases whereas that at H= 0odecreases. In the high uence range, F>51014ions/cm2,Hresis the larger when the eld is normal to the lm plane, i.e. the mag- netic easy axis is in the lm plane. Fig. 1b shows the frequency dependence of the resonance eld for dierent uences when the dc eld is normal to the lm plane. This data is tted to the resonance condition [14]: f=1 2 0(HresMe); (1) where is the gyromagnetic ratio. 0Me, the eec- tive easy plane anisotropy, is given by: Me=Ms 2K1=(0Ms), whereK1is the second order anisotropy constant. We nd that 0Meis negative at low u- ence which implies that the PMA is sucient to over- come the demagnetizing energy and hence the easy axis is normal to the lm plane. As the uence is further increased,0Mebecomes positive. These results con- rm that there is a re-orientation of the easy axis, as was inferred indirectly through magnetic hysteresis loop mea- surements in Ref. [10]. 0Mechanges sign for uence between 5 and 7.5 1014ions/cm2. Therefore, by expos- ing the lm to a specic uence, it is posible to engineer 01000 1000 3 06 09 001000 102030020406080 F=0Δ Hα ΔHinh ΔHtot F=5/s615490ΔH (mT)f /s61472 ( GHz ) F=10F ield angle /s61542H ( deg )HH /s615490ΔH (mT) F=7.5H FIG. 2: On the left, the linewidth as a function of frequency for the lm irradiated at 7.5 1014ions/cm2. The solid line is a linear t to the experimental data. On the right, the angular dependence of the linewidth at 20 GHz for a selec- tion of uences. The solid lines represent the ts to the total linewidth
Htot=
H+
Hinh, , where the intrinsic damp- ing and the inhomogeneous contribution are represented by the dashed line and the dotted line respectively. the anisotropy so that the PMA eld just compensates the demagnetization eld. The second order perpendicular anisotropy constant K1decreases linearly with uence (Fig. 1c). The lm irradiated at 1015ions/cm2has an anisotropy constant 40% smaller than that of the non-irradiated lm. The 4thorder anisotropy constant K2is determined from the angular dependence of Hresfor magnetization angles 4590o[13].K2is smaller than K1by a factor 10, and is nearly independent of uence. The FMR linewidth 0
Hwhen the dc eld is ap- plied normal the lm plane was measured as a function of frequency. Fig. 2 shows 0
Hversusffor the lm irradiated at F= 7:51014ions/cm2. The linewidth in- creases linearly with frequency, characteristic of Gilbert damping, an intrinsic contribution to the linewidth
H [15]:
H=4 0 f: (2) From a linear t to the experimental data, the magnetic damping constant is estimated from the slope of the line: = 0:0370:004. The lms irradiated at F= 0;1 and 101014ions/cm2shows a similar frequency dependence of the linewidth and have about the same damping con- stant,0:04. At intermediate uence F= 2:5, 51014 ions/cm2, the linewidth is enhanced and is frequency in- dependent, i.e. the linewidth is dominated by an inhomo- geneous contribution,
Hinh. The angular dependence of the linewidth measured at 20 GHz is shown in Fig. 2 for lms irradiated at selected uences. For the non- irradiated lm and the lm irradiated at 1015ions/cm2, the linewidth is practically independent of the eld an-3 05 1 01 53060901200 5100.030.04µ0 Δ/s61512/s61472 ⊥ (mT)F (1014 ions/cm2)ΔK1 ( 105 J/m3 )H 0.00.10.20.30.4 α-dampingF (1014 ions/cm2) FIG. 3: The uence dependence of the linewidth at 20 GHz when the dc eld is normal to the lm plane (squares). The solid circles represent the uence dependence of the distribu- tion in the PMA constant K1determined from tting
Hvs. H. The inset shows the Gilbert damping constant as a function of uence. gle from about 30oup to 90o. For the lm irradiated at 51014ions/cm2,
His clearly angular dependent and shows a minimum at an intermediate eld angle. The angular dependence of the linewidth was t to a sum of the intrinsic linewidth
Hand an inhomo- geneous contribution
Hinhfor magnetization angles 45o90o, an angular range in which TMS does not contribute to the linewidth [13]. The inhomogeneous linewidth is given by:
Hinh:(H) =j@Hres=@K 1j
K1+j@Hres=@j
;(3) where
K1is the width of the distribution of anisotropies and
is the distribution of the angles of the magnetic easy axis relative to the lm normal. The computed linewidth contributions are shown for the lm irradiated atF= 51014ions/cm2in Fig. 2. Note that the intrin- sic contribution
His practically independent of eld angles, as expected when the angle between the magne- tization and the applied eld is small. For this sample, the maximum angle is about 5oand it is due to the fact that the resonance eld ( Hres'0:6 T) is much larger than the eective demagnetization eld ( Me'0). Theinhomogeneous contribution from the distribution in the anisotropy eld directions does not signicantly aect the t. For the lm irradiated at the lower and upper uence range, the angular dependence of the intrinsic linewidth is computed xing the value of to that obtained from the t of the frequency dependence of the linewidth. For the other lms ( F=2.5 and 51014ions/cm2),was a tting parameter. The uence dependence of
K1and the linewidth at 20 GHz are shown in Fig. 3. The inset shows the Gilbert damping constant as a function of uence. The linewidth at 20 GHz when the eld is normal to the lm plane is a non monotonic function of uence.
Hincreases as the uence increases, reaching a maximum value at F 51014ions/cm2. Then, as the uence is further in- creased,
Hdecreases and falls slightly below the range of values at the lower uence range. Interestingly, the larger linewidth is observed just at the uence for which 0Me= 0. The magnetic damping is practically not aected by irradiation within the error bars: 0:04. The distribution of PMA constants,
K1, shows a similar uence dependence as the total linewidth, with a maxi- mum at F51014ions/cm2, clearly indicating that this is at the origin of the uence dependence of the mea- sured linewidth. The distribution in PMA anisotropy is almost zero when the uence is above 7 1014ions/cm2. The largest value of
K1corresponds to variation of K1 ofabout 8%, which is much larger than that of non irradi- ated lm and the highly irradiated lm,
K1=K12% and 0.3% respectively. In summary, irradiation of Co/Pd/Co/Ni lms with Helium ions leads to clear changes in its magnetic char- acteristics, a signicant decrease in magnetic anisotropy and a change in the distribution of magnetic anisotropies. Importantly, this is achieved without aecting the lm magnetization density and magnetic damping, which re- main virtually unchanged. It would be of interest to have a better understanding of the origin of the maximum in the distribution of magnetic anisotropy at the critical uence, the uence needed to produce a reorientation of the magnetic easy axis. Nonetheless, these results clearly demonstrate that ion irradiation may be used to system- atically tailor the magnetic properties of Co/Pd/Co/Ni multilayers for applications and basic physics studies. [1] S. Mangin et al: , Nat. Mater. 5, 210 (2006). [2] S. Mangin et al: , Appl. Phys. Lett. 94, 012502 (2009). [3] D. Bedau et al: , Appl. Phys. Lett. 96, 022514 (2010). [4] G. H. O. Daalderop et al: , Phys. Rev. Lett. 68, 682 (1992); S. Girod et al: , Appl. Phys. Lett. 94, 262504 (2009). [5] C. Chappert et al: , Science 280, 1919 (1998). [6] J. Z. Sun, Phys. Rev. B 62, 570 (2000). [7] A. Traverse et al: , Europhysics Letters 8, 633 (1989). [8] T. Devolder, Phys. Rev. B 62,5794 (2000). [9] J. Fassbender etal: , J. Phys. D: J. Appl. 37, R179 (2004).[10] D. Stanescu et al: , J. Appl. Phys. 103, 07B529 (2008). [11] C. Bilzer et al: , J. Appl. Phys. 103, 07B518 (2008). [12] J.-M. L. Beaujour et al: , Phys. Rev. B 80, 180415(R) (2009). [13] J.-M. L. Beaujour etal: , Eur. Phys. J. B 59, 475 (2007). [14] S. V. Vonsovskii, Ferromagnetic Resonance (Pergamon, Oxford, 1966). [15] Spin Dynamics in Conned Magnetic Structures II (edited by B. Hillebrands and K. Ounadjela (Springer, Heidelberg, 2002)).

2010-10-01

We present a ferromagnetic resonance (FMR) study of the effect of Helium ion irradiation on the magnetic anisotropy, the linewidth and the Gilbert damping of a Co/Ni multilayer coupled to Co/Pd bilayers. The perpendicular magnetic anisotropy decreases linearly with He ion fluence, leading to a transition to in-plane magnetization at a critical fluence of 5x10^{14} ions/cm^2. We find that the damping is nearly independent of fluence but the FMR linewidth at fixed frequency has a maximum near the critical fluence, indicating that the inhomogeneous broadening of the FMR line is a non-monotonic function of the He ion fluence. Based on an analysis of the angular dependence of the FMR linewidth, the inhomogeneous broadening is associated with spatial variations in the magnitude of the perpendicular magnetic anisotropy. These results demonstrate that ion irradiation may be used to systematically modify the magnetic anisotropy and distribution of magnetic anisotropy parameters of Co/Pd/Co/Ni multilayers for applications and basic physics studies.

Ferromagnetic resonance study of Co/Pd/Co/Ni multilayers with perpendicular anisotropy irradiated with Helium ions

1010.0268v2

Green’s function formalism for spin transport in metal-insulator-metal heterostructures Jiansen Zheng,1Scott Bender,1Jogundas Armaitis,2Roberto E. Troncoso,3,4and Rembert A. Duine1,5 1Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands 2Institute of Theoretical Physics and Astronomy, Vilnius University, Saul˙ etekio Ave. 3, LT-10222 Vilnius, Lithuania 3Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 4Departamento de Física, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile 5Department of Applied Physics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (Dated: September 13, 2017) We develop a Green’s function formalism for spin transport through heterostructures that contain metallic leads and insulating ferromagnets. While this formalism in principle allows for the inclusion of various magnonic interactions, we focus on Gilbert damping. As an application, we consider ballistic spin transport by exchange magnons in a metal-insulator-metal heterostructure with and without disorder. For the former case, we show that the interplay between disorder and Gilbert damping leads to spin current fluctuations. For the case without disorder, we obtain the dependence of the transmitted spin current on the thickness of the ferromagnet. Moreover, we show that the results of the Green’s function formalism agree in the clean and continuum limit with those obtained from the linearized stochastic Landau-Lifshitz-Gilbert equation. The developed Green’s function formalism is a natural starting point for numerical studies of magnon transport in heterostructures that contain normal metals and magnetic insulators. PACS numbers: 05.30.Jp, 03.75.-b, 67.10.Jn, 64.60.Ht I. INTRODUCTION Magnons are the bosonic quanta of spin waves, oscil- lations in the magnetization orientation in magnets1,2. Interest in magnons has recently revived as enhanced ex- perimental control has made them attractive as potential data carriers of spin information over long distances and withoutOhmicdissipation3. Ingeneral, magnonsexistin two regimes. One is the dipolar magnon with long wave- lengths that is dominated by long-range dipolar interac- tions and which can be generated e.g. by ferromagnetic resonance4,5. The other type is the exchange magnon6, dominated by exchange interactions and which generally has higher frequency and therefore perhaps more poten- tial for applications in magnon based devices3. In this paper, we focus on transport of exchange magnons. Thermally driven magnon transport has been widely investigated, and is closely related to spin pumping of spin currents across the interface between insulating fer- romagnets (FMs) and normal metals (NM)7–9and de- tection of spin current by the inverse spin Hall Effect10. The most-often studied thermal effect in this context is the spin Seebeck effect, which is the generation of a spin current by a temperature gradient applied to a magnetic insulator that is detected in an adjacent normal metal via the inverse spin Hall effect11,12. Here, thermal fluc- tuations in the NM contacts drive spin transport into the FM, while the dissipation of spin back into the NM by magnetic dynamics is facilitated by the above mentioned spin-pumping mechanism. The injection of spin into a FM can also be accom-plished electrically, via the interaction of spin polarized electrons in the NM and the localized magnetic mo- ments of the FM. Reciprocal to spin-pumping is the spin- transfer torque, which, in the presence of a spin accu- mulation (typically generated by the spin Hall effect) in the NM, drives magnetic dynamics in the FM13,14. Spin pumping likewise underlies the flow of spin back into the NM contacts, which serve as magnon reservoirs. In two-terminal set-ups based on YIG and Pt, the char- acteristic length scales and device-specific parameter de- pendence of magnon transport has attracted enormous attention, both in experiments and theory. Cornelis- senet al.15studied the excitation and detection of high- frequency magnons in YIG and measured the propagat- ing length of magnons, which reaches up to 10m in a thin YIG film at room temperature. Other experi- ments have shown that the polarity reversal of detected spins of thermal magnons in non-local devices of YIG are strongly dependent on temperature, YIG film thick- ness, and injector-detector separation distance16. That the interfaces are crucial can e.g. be seen by changing the interface electron-magnon coupling, which was found to significantly alter the longitudinal spin Seebeck effect17. A linear-response transport theory was developed for diffusive spin and heat transport by magnons in mag- netic insulators with metallic contacts. Among other quantities, this theory is parameterized by relaxation lengths for the magnon chemical potential and magnon- phonon energy relaxation18,19. In a different but closely- related development, Onsager relations for the magnon spin and heat currents driven by magnetic field andarXiv:1709.03775v1 [cond-mat.mes-hall] 12 Sep 20172 temperature differences were established for insulating ferromagnet junctions, and a magnon analogue of the Wiedemann-Franz law was is also predicted20,21. Wang et al.22consider ballistic transport of magnons through magnetic insulators with magnonic reservoirs — rather thanthemoreexperimentallyrelevantsituationofmetal- lic reservoirs considered here — and use a nonequilib- rium Green’s function formalism (NEGF) to arrive at Landuaer-Bütikker-type expressions for the magnon cur- rent. Theabove-mentionedworksareeitherinthelinear- response regime or do not consider Gilbert damping and/or metallic reservoirs. So far, a complete quantum mechanical framework to study exchange magnon trans- port through heterostructures containing metallic reser- voirs that can access different regimes, ranging from bal- listic to diffusive, large or small Gilbert damping, and/or small or large interfacial magnon-electron coupling, and that can incorporate Gilbert damping, is lacking. Figure 1: Illustration of the system where magnon transport in a ferromagnet (orange region) is driven by a spin accu- mulation difference
L
Rand temperature difference TLTRbetween two normal-metal leads (blue regions). Spin- flip scattering at the interface converts electronic to magnonic spin current. Here, Sis the local spin density in equilibrium. In this paper we develop the non-equilibrium Green’s functionformalism23forspintransportthroughNM-FM- NM heterostructures (see Fig. 1). In principle, this for- malism straightforwardly allows for adding arbitrary in- teractions, such as scattering of magnons with impuri- ties and phonons, Gilbert damping, and magnon-magnon interactions, and provides a suitable platform to study magnon spin transport numerically, in particular beyond linear response. Here, we apply the formalism to ballistic magnon transport through a low-dimensional channel in the presence of Gilbert damping. For that case, we com- pute the magnon spin current as a function of channel length both numerically and analytically. For the clean case in the continuum limit we show how to recover our results from the linearized stochastic Landau-Lifshitz- Gilbert (LLG) equation24used previously to study ther- mal magnon transport in the ballistic regime25that ap- plies to to clean systems at low temperatures such that Gilbert damping is the only relaxation mechanism. Us- ing this formalism we also consider the interplay between Gilbert damping and disorder and show that it leads to spin-current fluctuations. This paper is organized as follows. In Sec. II, we discuss the non-equilibrium Green’s function approachto magnon transport and derive an expression for the magnon spin current. Additionally a Landauer-Büttiker formula for the magnon spin current is derived. In Sec. III, we illustrate the formalism by numerically con- sidering ballistic magnon transport and determine the dependence of the spin current on thickness of the ferro- magnet. To further understand these numerical results, we consider the formalism analytically in the continuum limit in Sec. IV, and also show that in that limit we ob- tain the same results using the stochastic LLG equation. We give a further discussion and outlook in section V. II. NON-EQUILIBRIUM GREEN’S FUNCTION FORMALISM In this section we describe our model and, using Keldysh theory, arrive at an expression for the density matrix of the magnons from which all observables can be calculated. The reader interested in applying the final re- sult of our formalism may skip ahead to Sec. IIE where we give a summary on how to implement it. A. Model j j/primej j/primej j/prime∆µL TL∆µR TR TFMNNM FM NM JLJRρj,j/prime G(±) j,j/prime(t,t/prime) Gk,k/prime(t,t/prime) Gk,k/prime(t,t/prime) ΣFM,(±)ΣL,(±)ΣR,(±) Self-energy Figure 2: Schematic for the NM-FM-NM heterostructure and notationfortheGreen’sfunctionsandself-energies. Thearray of circles denotes the localized magnetic moments, while the two regions outside the parabolic lines denote the leads, i.e., reservoirs of polarized electrons. Moreover, JL=R j;kk0denotes the interface coupling, and TL=Rand
L=Rdenote the temper- ature and spin accumulation for the leads. The properties of the magnons are encoded in G(+) j;j0(t;t0), the retarded magnon Green’s function, and the magnon density matrix j;j0. The number of sites in the spin-current direction is N. The self- energies FM; (),L;(),R;()are due to Gilbert damping, and the left and right lead, respectively. We consider a magnetic insulator connected to two nonmagnetic metallic leads, as shown in Fig. 2. For3 our formalism it is most convenient to consider both the magnons and the electrons as hopping on the lattice for the ferromagnet. Here, we consider the simplest versions of such cubic lattice models; extensions, e.g. to multi- ple magnon and/or electron bands, and multiple leads are straightforward. The leads have a temperature TL=R and a spin accumulation
L=Rthat injects spin cur- rent from the non-magnetic metal into the magnetic in- sulator. This nonzero spin accumulation could, e.g., be established by the spin Hall effect. The total Hamiltonian is a sum of the uncoupled magnon and lead Hamiltonians together with a coupling term: ^Htot=^HFM+^HNM+^HC: (1) Here, ^HFMdenotesthefreeHamiltonianforthemagnons, ^HFM=X <j;j0>Jj;j0by j0bj+X j
jby jbjX <j;j0>hj0;jby j0bj: (2) wherebj(by j)is a magnon annihilation (creation) opera- tor. This hamiltonian can be derived from a spin hamil- tonian using the Holstein-Primakoff transformation26,27 and expanding up to second order in the bosonic fields. Eq. (2) describes hopping of the magnons with amplitude Jj;j0between sites labeled by jandj0on the lattice, with an on-site potential energy
jthat, if taken to be homo- geneous, would correspond to the magnon gap induced by a magnetic field and anisotropy. We have taken the external field in the zdirection, so that one magnon, created at site jby the operator ^by j, corresponds to spin +~. The Hamiltonian for the electrons in the leads is ^HNM=X r2fL;RgX <k;k0>X 2";#tr^ y kr^ k0r+h:c:(3) where the electron creation ( y kr) and annihilation ( kr) operators are labelled by the lattice position k, spin, and an index rdistinguishing (L)eft and (R)ight leads. The hopping amplitude for the electrons is de- notedbytrandcouldinprinciplebedifferentfordifferent leads. Moreover, terms to describe hopping beyond near- estneighborcanbestraightforwardlyincluded. Belowwe will show that microscopic details will be incorporated in a single parameter per lead that describes the coupling between electrons and magnons. Finally, the Hamiltonian that describes the coupling between metal and insulator, ^HC, is given by28 ^HC=X r;j;kk0 Jr j;kk0^by j^ y k#r^ k0"r+ h:c: ;(4) with the matrix elements Jr j;kk0that depend on the mi- croscopic details of the interface. An electron spin that flips from up to down at the interface creates one magnon withspin +~inthemagneticinsulator. Thisformofcou- pling between electrons and magnons derives from inter- face exchange coupling between spins in the insulators with electronic spins in the metal28. Gk/prime,k/prime/prime;↑ Gk/prime/prime/prime,k;↓t/prime, j/primet, jFigure 3: Feynman diagram for the spin-flip processes emit- ting and absorbing magnons that are represented by the wavy lines. The two vertices indicate the exchange coupling at one of the interfaces of the magnetic insulator (sites j;j0) and nor- mal metal (sites k;k0;k00;k000).Gk0k00;"andGk000k;#denotes the electron Keldysh Green’s function of one of the leads. B. Magnon density matrix and current Our objective is to calculate the steady-state magnon Green’s function iG< j;j0(t;t0) =h^by j0(t0)^bj(t)i, from which all observables are calculated (note that time-dependent operators refer to the Heisenberg picture). This “lesser” Green’s function follows from the Keldysh Green’s func- tion iGj;j0(t;t0)Trh ^(t0)TC1 ^bj(t)^by j0(t0)i ;(5) with ^(t0)the initial (at time t0) density matrix, and C1the Keldysh contour, and Tr[:::]stands for perform- ing a trace average. The time-ordering operator on this contour is defined by TC1 ^O(t)^O0(t0) (t;t0)^O(t)^O0(t0)(t0;t)^O0(t0)^O(t); (6) with(t;t0)the corresponding Heaviside step function and the +()sign applies when the operators have bosonic (fermionic) commutation relations. In Fig. 2 we schematically indicate the relevant quantities enter- ing our theory. Att= 0, the spin accumulation in the two leads is We compute the magnon self energy due the coupling between magnons and electrons to second order in the coupling matrix elements Jj;kk0. This implies that the magnons acquire a Keldysh self-energy due to lead r given by ~r j;j0(t;t0) =i ~X kk0k00k000Jr j;kk0(Jr) j0;k00k000 Gk0k00;r"(t;t0)Gk000k;r#(t0;t);(7) whereGk0k00;r(t;t0)denotes the electron Keldysh Green’s function of lead r, that reads Gkk0;r(t;t0) =ihTC1^ kr(t)^ y k0r(t0)i:(8)4 The Feynman diagram for this self-energy is shown in Fig. 3. While this self-energy is computed to second order inJr j;kk0, the magnon Green’s function and the magnon spin current, both of which we evaluate below, contain all orders in Jr j;kk0, which therefore does not need to be small. In this respect, our approach is different from the work of Ohnuma et al.[29], who evaluate the interfacial spin current to second order in the electron- magnon coupling. Irreducible diagrams other than that in Fig. 3 involve one or more magnon propagators as in- ternal lines and therefore correspond to magnon-magnon interactions at the interface induced by electrons in the normal metal. For the small magnon densities of interest to use here these can be safely neglected and the self- energy in Eq. (7) thus takes into account the dominant process of spin transfer between metal and insulator. The lesser and greater component of the electronic Green’s functions can be expressed in terms of the spec- tral functions Akk0;r()via iG< kk0;r=Akk0;r()NFr kBTr ; iG> kk0;r=Akk0;r() 1NFr kBTr ;(9) withNF(x) = [ex+ 1]1the Fermi distribution function, Trthe temperature of lead r(kBbeing Boltzmann’s con- stant) and;rthe chemical potential of spin projection in leadr. As we will see later on, the lead chemical potential are taken spin-dependent to be able to imple- ment nonzero spin accumulation. The spectral function is related to the retarded Green’s function via Akk0;r() =2Imh G(+) kk0;r()i ; (10) which does not depend on spin as the leads are taken to be normal metals. While the retarded Green’s function of the leads can be determined explicitly for the model that we consider here, we will show below that such a level of detail is not needed but that, instead, we can pa- rameterize the electron-magnon coupling by an effective interface parameter. As mentioned before, all steady-state properties of the magnon system are determined by the magnon lesser Green’s function leading to the magnon density matrix. It is ultimately given by the kinetic equation23,30 j;j0h^by j0(t)^bj(t)i=Zd (2)h G(+)()i~<()G()()i j;j0; (11) where ~j;j0(t;t0)is the total magnon self-energy dis- cussed in detail below, of which the "lesser" component enters in the above equation. In the above and what follows, quantities with suppressed site indexes are in- terpreted as matrices, and matrix multiplication applies for products of these quantities. The retarded (+)and advanced ()magnon Green’s functions satisfy h h~()()i G()() = 1;(12)where=i0. The magnon self-energies have con- tributions from the leads, as well as a contribution from the bulk denoted by ~FM: ~() =~FM() +X r2fL;Rg~r():(13) From Eq. (7) we find that for the retarded and advanced component, the contribution due to the leads is given by ~r;() j;j0() =X kk0k00k000Jr j;kk0(Jr) j0;k00k000Zd0 (2)Zd00 (2) Ak0k00;r(0)Ak000k;r(00)NF 0r" kBTr NF 00r# kBTr +000 ; (14) whereas the "lesser" self-energy can be shown to be of the form: ~r;< j;j0() = 2iNB
r kBTr Imh ~r;(+) j;j0()i ;(15) withNB(x) = [ex1]1the Bose-Einstein distribution function and
r=r"r#the spin accumulation in leadr. Having established the contributions due to the leads, we consider the bulk self-energy ~FM, which in princi- ple could include various contributions, such as magnon conserving and nonconsering magnon-phonon interac- tions, or magnon-magnon interactions. Here, we consider magnon non-conserving magnon-phonon coupling as the source of the bulk self-energy and use the Gilbert damp- ing phenomenology to parameterize it by the constant which for the magnetic insulator YIG is of the order of104. Gilbert damping corresponds to a decay of the magnons into phonons with a rate proportional to their energy. This thus leads to the contributions ~FM;< j;j0() = 2NB kBTFM ~FM;(+) j;j0() ; ~FM;(+) j;j0() =ij;j0; (16) whereTFMis the temperature of phonon bath. We note that in principle the temperature could be taken position dependent to implement a temperature gradient, but we do not consider this situation here. With the results above, the density-matrix elements j;j0can be explicitly computed from the magnon re- tarded and advanced Green’s function and the “lesser” component of the total magnon self-energy using Eq. (11). The magnon self-energy is evaluated using the explicitexpressionfortheretardedandadvancedmagnon self-energies due to leads and Gilbert damping ~FM, see Eq. (16). We are interested in the computation of the magnon spin currenthjm;jj0iin the bulk of the FM from site j5 to sitej0, which in terms of the magnon density matrix reads, hjm;jj0i=i(hj;j0j0;jc:c:); (17) and follows from evaluating the change in time of the lo- cal spin density, ~dh^by j^bji=dt, using the Heisenberg equa- tions of motion. The magnon spin current in the bulk thus follows straightforwardly from the magnon density matrix. While the formalism presented so far provides a com- plete description of the magnon spin transport driven by metallic reservoirs, we discuss two simplifying develop- ments below. First, we derive a Landauer-Bütikker-like formula for the spin current from metallic reservoirs to the magnon system. Second, we discuss how to replace the matrix elements Jr j;k;k0by a single phenomenologi- cal parameter that characterizes the interface between metallic reservoirs and the magnetic insulator. C. Landauer-Büttiker formula In this section we derive a Landauer-Büttiker formula for the magnon transport. Using the Heisenberg equa- tions of motion for the local spin density, we find that the spin current from the left reservoir into the magnon system is given by jL s~ 2* d dtX k ^ y k"L k"L y k#L k#L+ =2 ~X j;kk0Re[ JL
 j;kk0g< j;kk0(t;t0)];(18) in terms of the Green’s function g< j;kk0(t;t0)ih^ y k0"L(t0)^ k#L(t0)^bj(t)i:(19) This “lesser” coupling Green’s function g< j;kk0(t;t0)is cal- culated using Wick’s theorem and standard Keldysh methods as described below. We introduce the spin-flip operator for lead r ^dy kk0;r(t) =^ y k0"r(t)^ k#r(t); (20) so that the coupling Green’s function becomes g< j;kk0(t;t0)ih^dy kk0;L(t0)^bj(t)i: (21) The Keldysh Green’s function for the spin-flip operator is given by r kk0k00k000(t;t0) =ihTC1^dkk0;r(t)^dy k00k000;r(t0)i(22) and using Wick’s theorem we find that r;> kk0k00k000(t;t0) =iG> kk000;r#(t;t0)G< k0k00;r"(t0;t) ; r;< kk0k00k000(t;t0) =iG> k0k00;r"(t0;t)G< kk000;r#(t;t0) ; r;(+) kk0k00k000(t;t0) =i(tt0)h G> kk000;r#(t;t0)G< k0k00;r"(t0;t) G> k0k00;r"(t0;t)G< kk000;r#(t;t0)i ;(23)where we used the definition for the electron Green’s function in Eq. (8). Applying the Langreth theorem30and Fourier trans- forming, we write down the lesser coupling Green’s function in terms of the spin-flip Green’s function and magnon Green’s function g< j;kk0() =X j0;k00k000JL j0;k00k000 G(+) j;j0()L;< kk0k00k000() +G< j0;j()L;() kk0k00k000() ; (24) where the retarded and “lesser" magnon Green’s function are given by Eq. (11) and Eq. (12). Using these results, we ultimately find that jL s=Zd 2 NB
L kBTL NB
R kBTR T() +Zd 2 NB
L kBTL NB kBTFM Trh ~L()G(+)()~FM()G()()i ; (25) with the transmission function T()Trh ~L()G(+)()~R()G()()i :(26) In the above, the rates ~L=R()are defined by ~r()2Imh ~r;(+)()i ; (27) and ~FM()2Imh ~FM;(+)()i ;(28) and correspond to the decay rates of magnons with en- ergydue to interactions with electrons in the normal metal at the interfaces, and phonons in the bulk, respec- tively. This result is similar to the Laudauer-Büttiker formalism23for electronic transport using single-particle scattering theory. In the present context, a Landauer- Büttiker-like for spin transport was first derived by Ben- deret al.[28] for a single NM-FM interface. In the absence of Gilbert damping, the spin current would cor- respond to the expected result from Landauer-Bütikker theory, i.e., the spin current from left to the right lead is then given by the first line of Eq. (25). The presence of damping gives leakage of spin current due to the coupling with the phononic reservoir, as the second term shows. Finally,wenotethatthespincurrentfromtherightreser- voir into the system is obtained by interchanging labels L and R in the first term, and the label L replaced by R in the second one. Due to the presence of Gilbert damping, however, we have in general that jL s6=jR s. D. Determining the interface coupling We now proceed to express the magnon spin current (Eq.(25))intermsofamacroscopic,measurablequantity6 ratherthantheinterfacialexchangeconstants Jr j;k;k0. For
rF(withFthe Fermi energy of the metallic leads), which is in practice always obeyed, we have for low energies and temperatures that ~r;() j;j0()'i1 4X kk0k00k000Jr j;kk0(Jr) j0;k00k000 Ak0;k00;r(F)Ak000;k;r(F)(
r):(29) Here, we also neglected the real part of this self-energy which provides a small renormalization of the magnon energies but is otherwise unimportant. The expansion for small energies in Eq. (29) is valid as long as F, which applies since is a magnon energy, and therefore at most on the order of the thermal energy. Typically, the above self-energy is strongly peaked for j;j0at the interface because the magnon-electron interactions occur at the interface. For j;j0at the interface we have that the self-energy depends weakly on varying j;j0along the interface provided that the properties of the interface do not vary substantially from position to position. We can thus make the identification: ~r;() j;j0()'ir(
r)j;j0j;jr;(30) withjrthe positions of the sites at the r-th interface, andrparametrizing the coupling between electrons and magnons at the interface. Note that rcan be read off from Eq. (29). Rather than evaluating this parameter in terms of the matrix elements Jr j;kk0and the electronic spectral functions of the leads Ak;k0;r(), we determine it in terms of the real part of the spin-mixing conductance g"#;r, a phenomenological parameter that characterizes thespin-transferefficiencyattheinterface31. Thiscanbe donebynotingthatintheclassicallimittheself-energyin Eq. (30) leads to an interfacial contribution, determined by the damping constant r=N, to the Gilbert damping ofthehomogeneousmode, where Nisthenumberofsites of the system perpendicular to the leads, as indicated in Fig. 2. In terms of the spin-mixing conductance, we have that this contribution is given by32g"#;r=4srN, withsr the saturation spin density per area of the ferromagnet at the interface with the r-th lead. Hence, we find that r=g"#;r 4sr; (31) which is used to express the reservoir contributions to the magnon self-energies in terms of measurable quantities. The spin-mixing conductance can be up to 5~nm2for YIG-Pt interfaces33, leading to the conclusion that can be of the order 110for that case. E. Summary on implementation We end this section with some summarizing remarks on implementation that may facilitate the reader who is interested in applying the formalism presented here.Table I: Parameters chosen for numerical calculations based on the NEGF formalism (unless otherwise noted). Quantity Value J 0:05eV
L=J 2:0105
R=J 0:0  8
=J 2:0103 kBTFM=J0:60 First, one determines the retarded and advanced magnon Green’s functions. This can be done given a magnon hamiltonian characterized by matrix ele- mentshj;j0in Eq. (2), mixing conductances for the metal-insulator interfaces g"#;r, and a value for the Gilbert damping constant , from which one computes the retarded self-energies at the interfaces in Eq. (30) with Eq. (31), and Eq. (16). The retarded and ad- vanced magnon Green’s functions are then computed via Eq. (12), which amounts to a matrix inversion. The next stepistocalculatethedensitymatrixforthemagnonsus- ing Eq. (11), with as input the expressions for the “lesser” self-energies in Eqs. (15) and (16). Finally, the spin cur- rent is evaluated using Eq. (17) in the bulk of the FM or Eq. (25) at the NM-FM interface. In the next sections, we discuss some applications of our formalism. III. NUMERICAL RESULTS In this section, we present results of numerical calcula- tions using the formalism presented in the previous sec- tion. A. Clean system For simplicity, we consider now the situation where the leads and magnetic insulators are one dimensional. The values of various parameters are displayed in Ta- ble I, where we take the hopping amplitudes Jj;j0= J(j;j0+1+j;j01), i.e.,Jj;j0is equal toJbetween near- est neighbours, and zero otherwise. We focus on trans- port driven by spin accumulation in the leads and set all temperatures equal, i.e., TL=TR=TFMT. We also assume both interfaces to have equal proper- ties, i.e., for the magnon-electron coupling parameters to obeyL=R. First we consider the case without disorder and take
j=
. We are interested in how the Gilbert damping affects the magnon spin current. In particular, we calculate the spincurrentinjectedintherightreservoirasafunctionof system size. The results of this calculation are shown in Fig. 4 for various temperatures, which indicates that for7 a certain fixed spin accumulation, the injected spin cur- rent decays with the thickness of the system for N > 25, for the parameters we have chosen. We come back to the various regimes of thickness dependence when we present analyticalresultsforcleansystemsinthecontinuumlimit in Sec. IV. From these results we define a magnon relax- 0 20 40 60 80 100 120 140 160 system/uni00A0size/uni00A0(d/a)10/uni00AD410/uni00AD310/uni00AD210/uni00AD1100magnon/uni00A0spin/uni00A0current/uni00A0(J)kBT/J=0.012 kBT/J=0.024 kBT/J=0.048 kBT/J=0.108 kBT/J=0.192 Figure 4: System-size dependence of spin current ejected in the right reservoir for = 6:9102;= 8:0and various temperatures. ation length drelaxusing the definition jm(d)/exp(d=drelax); (32) applied to the region N > 25and where d= Nawith athe lattice constant. The magnon relaxation length depends on system temperature and is shown in Fig. 5. We attempt to fit the temperature dependence with drelax(T) =a( 0+ 1p T+ 2 T);(33) with 0; 1; 2constantsand Tdefinedasthedimension- less temperature TkBT=J. The term proportional to 1is expected for quadratically dispersing magnons with Gilbert damping as the only relaxation mechanism15,25. The terms proportional to 0and 2are added to charac- terize the deviation from this expected form. Our results show that the relaxation length has not only 1=p T behaviour. This is due to the finite system size, the con- tact resistance that the spin current experiences at the interface between metal and magnetic insulator, and the deviation of the magnon dispersion from a quadratic one due to the presence of the lattice. B. Disordered system We now consider the effects of disorder on the spin current as a function of the thickness of the FM. We con- sider a one-dimensional system with a disorder potential drelax/a=γ0+γ1 T*+γ2 T* Numerical result Fitted curve 0.0 0.2 0.4 0.6 0.8115120125130135140145150 T*=kBT/Jdrelax/aFigure 5: Magnon relaxation length as a function of dimen- sionlesstemperature Tfor= 6:9102;= 8:0. Thefitted parameters are obtained as 0= 114:33; 1= 0:96; 2= 0:32. implemented by taking
j=
(1+j), wherejis a ran- dom number evenly distributed between and(with 1andpositive)thatisuncorrelatedbetweendifferent sites. In one dimension, all magnon states are Anderson localized34. Since this is an interference phenomenon, it is expected that Gilbert damping diminishes such local- ization effects. The effect of disorder on spin waves was investigated using a classical model in Ref. [35], whereas Ref. [36] presents a general discussion of the effect of dissipation on Anderson localization. Very recently, the effect of Dzyaloshinskii-Moriya interactions on magnon localization was studied37. Here we consider how the in- terplay between Gilbert damping and the disorder affects the magnon transport. For a system without Gilbert damping the spin current carried by magnons is conserved and therefore indepen- dent of position regardless of the presence or absence of disorder. DuetothepresenceofGilbertdampingthespin current decays as a function of position. Adding disorder on top of the dissipation due to Gilbert damping causes the spin current to fluctuate from position to position. For large Gilbert damping, however, the effects of dis- order are suppressed as the Gilbert damping suppresses the localization of magnon states. In Fig. 6 we show nu- merical results of the position dependence of the magnon current for different combinations of disorder and Gilbert damping constants. The plots clearly show that the spin current fluctuates in position due to the combined ef- fect of disorder and Gilbert damping, whereas it is con- stant without Gilbert damping, and decays in the case with damping but without disorder. Note that for the two cases without Gilbert damping the magnitude of the spin current is different because the disorder alters the conductance of the system and each curve in Fig. 6 cor- responds to a different realization of disorder. To characterize the fluctuations in the spin current, we8 0 20 40 60 80 site j1.01.52.02.53.03.54.04.5magnon spin current (J)1e7 =0.0,=0.0 =0.0,=0.0015 =0.0069,=0.0 =0.0069,=0.0015 Figure 6: Spatial dependence of local magnon current for the case without Gilbert damping and disorder ( = 0;= 0), without disorder ( = 6:9103;= 0), without Gilbert damping (= 0;= 1:5103), and both disorder and Gilbert damping ( = 6:9103;= 1:5103). The inter- face coupling parameter is taken equal to = 0:8. define the correlation function Cj=vuut jm;j;j+1jm;j;j+1
2 jm;j;j+1
2; (34) where the bar stands for performing averaging over the realizations of disorder. Fig. (7) shows this correlation function for j=N1as a function of Gilbert damp- ing for various strengths of the disorder. As we expect, basedonthepreviousdiscussion, thefluctuationsbecome small as the Gilbert damping becomes very large or zero, leaving an intermediate range where there are sizeable fluctuations in the spin current. IV. ANALYTICAL RESULTS In this section we analytically compute the magnon transmission function in the continuum limit a!0 for a clean system. We consider again the situation of a magnon hopping amplitude Jj;j0that is equal to Jand nonzero only for nearest neighbors, and a con- stant magnon gap
j=
. We compute the magnon density matrix, denoted by (x;x0), and retarded and advanced Green’s functions, denoted by G()(x;x00;). Here, the spatial coordinates in the continuum are de- noted byx;x0;x00;


. We take the system to be trans- lationally invariant in the yz-plane and the current direction as shown in Fig. 1 to be x. In the continuum limit, the imaginary part of the vari- 0.000 0.005 0.010 0.015 0.020 α012345CN−1 δ=0.0005 δ=0.001 δ=0.0015Figure 7: Correlation function Cjthat characterizes the fluc- tuations in the spin curent for j=N1as a function of the Gilbert damping constant, for three strengths of the disorder potential. The curves are obtained by performing averaging over 100 realizations of the disorder. The interface coupling parameter is taken equal to = 0:8. ous self-energies acquired by the magnons have the form: Imh ~r;(+)(x;x0;)i = ~r(r)(xxr)(xx0) ; Imh ~FM;(+)(x;x0;)i =(xx0);(35) wherexris the position of the r-th lead, and where ~ris the parameter that characterizes the interfacial coupling between magnons and electrons. We use a different nota- tion for this parameter as in the continuum situation its dimension is different with respect to the discrete case. To express ~rin terms of the spin-mixing conductance we have that ~r=g"#=4~srwhere ~sris now the three- dimensional saturated spin density of the ferromagnet. We proceed by evaluating the magnon transmission function from Eq. (26). We compute the rates in Eq. (27) from the self-energies Eqs. (35) and find for the transmis- sion function in the first instance that T() = 4~L~R(
L)(
R) Zdq (2)2g(+)(xL;xR;q;)g()(xR;xL;q;);(36) where qis the two-dimensional momentum that results fromFouriertransforminginthe yz-plane. TheGreen’s functionsg()(x;x0;q;)obey [compare Eq. (12)]  (1i)+Ad2 dx2Aq2
iX r2fL;Rg~r(
r)(xxr)3 5g()(x;x0;q;) =(xx0); (37)9 whereA=Ja2. This Green’s function is evaluated using standard techniques for inhomogeneous boundary value problems (see Appendix A) to ultimately yield T() = 4~2(
L)(
R)Zdq (2)2jt(q;)j2;(38) with t(q;) =A A22~2(
L)(
R)
sinh(d) iA~(2
L
R) cosh(d)]1; (39) with=p (Aq2+
i)=Aand whered=xR xL. Note that we have at this point taken both interfaces equal for simplicity, so that ~L= ~R~. In terms of an interfacial Gilbert damping parameter 0we have that ~=d0. Let us identify the magnon decay length l ; where=p A=kBTis proportional to the thermal de Broglie wavelength. Equipped with a closed, analytic expression, we may now, in an analogous way as Hoffman et al.[25], investigate the behavior of Eq. (38) in the thin FM (dl) and thick FM ( dl) regimes. In order to do so, we take L= 0so that the second term in Eq. (25) vanishes and the spin current is fully determined by the transmission coefficient T(). Before analyzing the result for the spin current more closely, we remark that the result for the transmission function may also be obtained from the linearized stochastic Landau-Lifshitz- Gilbert equation, as shown in Appendix B. A. Thin film regime ( dl) In the thin film regime, the transmission coefficient T()exhibits scattering resonances near =nqfor given q, where nq A=q2+1 2+n22 d2 andnisanintegerandwhere =p A=
isthecoherence length of the ferromagnet. When the ferromagnet is suf- ficiently thin ( d=1=2=pl), one finds that these peaks are well separated, and the transmission coefficient is approximated as a sum of Lorentzians: T =P1 n=0Tn, where: Tn()AnqL nR n Ln+ Rn+ FMn(40) with Anq() =n (nq)2+ (n=2)2(41)as the spin wave spectral density. The broadening rates are given by FM n= 2,L 0= 20,R 0= 20(R), L n6=0= 40,R n6=0= 40(R)andn= FM n+ L n+ R n. Intheextremesmalldissipationlimit(i.e. neglecting spectral broadening by the Gilbert damping), one has: Anq()!2(nq); (42) and the current has the simple form, jL s=P1 n=0jn, where jn=a2Zd2q (2)2L nR n n NBnq kBT NBnq
R kBT (43) where L n,R nand FM nare all evaluated at =nq. Eq. (43) allows one to estimate the thickness dependence of the signal. Supposing R.nq, whendg"#=s, then0, and L nR n=FM njL s;cl1=d; whend g"#=s, then0, andjL s;cl1=d2. The enhancement of the spin current for small dis in rough agreement with our numerical results in the previous section as shown in Fig. 4. B. Thick film regime ( dl) In the thick film regime, the transmission function be- comes T()(4Ad)2L xR xp (0q)2+ (FMx=2)2e2rd j(4A)2(d)2LxRxi4dARxS(r)j2 where L=R= FM x = L=R= FM n6=0,r= Re[], andS(r)isthe sign ofr. For1, we have=ikx 1 +i=2Ak2 x
, wherekx=p q2+2=A. For energies  > A (q2+ 2),kxis imaginary, and the contribution to the spin current decays rapidly with d. When, however,  < A(q2+2),kxis real, and r= q2+2
=2kx =(for thermal magnons), so that the signal decays over a length scale l/1=p T, in agreement with our numerical results as shown in Fig. 5. C. Comparison with numerical results In order to compare the numerical with the analyti- cal results we plot in Fig. 8 the transmission function as a function of energy. Here, the numerical result is evaluated for a clean system using Eq. (26) while the an- alytical result is that of Eq. (38). While they agree in the appropriate limit ( N!1;a!0), for finite Nthere are substantial deviations that are due to the increased importance of interfacing coupling relative to the Gilbert damping for small systems and the deviations of the dis- persion from a quadratic one.10 Δ/J=0.2N=20 Analytic Numerical 0 1 2 3 4 50.000.020.040.060.08 ϵ/JT(ϵ) Figure 8: Magnon transmission function as a function of en- ergy. The parameters are chosen to be
=J= 0:2;= 0:069;= 8:0. V. DISCUSSION AND OUTLOOK We have developed a NEGF formalism for exchange magnon transport in a NM-FM-NM heterostructure. We have illustrated the formalism with numerical and ana- lytical calculations and determined the thickness depen- dence of the magnon spin current. We have also con- sidered magnon disorder scattering and shown that the interplay between disorder and Gilbert damping leads to spin-current fluctuations. Wehavealsodemonstratedthatforacleansystem,i.e., without disorder, in the continuum limit the results ob- tained from the NEGF formalism agree with those fromthe stochastic LLG formalism. The latter is suitable for a clean system in the continuum limit where the vari- ous boundary conditions on the solutions of the stochas- tic equations are easily imposed. The NEGF formal- ism is geared towards real-space implementation, such that, e.g., disorder scattering due to impurities are more straightforwardly included as illustrated by our example application. The NEGF formalism is also more flexi- ble for systematically including self-energies due to ad- ditional physical processes, such as magnon-conserving magnon-phonon scattering and magnon-magnon scatter- ing, or, for example, for treating strong-coupling regimes intowhichthestochasticLandau-Lifshitz-Gilbertformal- ism has no natural extension. Using our formalism, a variety of mesoscopic transport features of magnon transport can be investigated includ- ing, e.g., magnon shot noise38. The generalization of our formalism to elliptical magnons and magnons in antifer- romagnets is an attractive direction for future research. Acknowledgments This work was supported by the Stichting voor Funda- menteel Onderzoek der Materie (FOM), the Netherlands Organization for Scientific Research (NWO), and by the European Research Council (ERC) under the Seventh Framework Program (FP7). J. Z. would like to thank the China Scholarship Council. J. A. has received fund- ing from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska- Curie grant agreement No 706839 (SPINSOCS). Appendix A: Evaluation of magnon Green’s function in the continuum limit In this appendix we evaluate the magnon Green’s function in the continuum limit that is determined by Eq. (37). For simplicity we take the momentum qequal to zero and suppress it in the notation, as it can be trivially restored afterwards. The Green’s function is then determined by 2 4iiX r2fL;Rg~r(
r)(xxr) +Ad2 dx2H3 5g()(x;x0;) =(xx0): (A1) To determine this Green’s function we first solve for the states (x)that obey: 2 4iiX r2fL;Rg~r(
r)(xxr) +Ad2 dx2H3 5(x) = 0: (A2) Integrating this equation across x=xLandx=xRleads to the boundary conditions: x=xL:i~L(
L)(xL) +Ad(x) dxjx=xL= 0; (A3) x=xR:i~R(
R)(xR)Ad(x) dxjx=xR= 0: (A4)11 ForxL<x<xR, the general solution is: (x) =Beikx+Ceikx; (A5) withk=p (iH)=A. We write the solution obeying the boundary condition at x=xLas  L(x) =eikx+Ceikx; (A6) With the boundary condition at x=xL( Eq.A4), we find that C=Ak~L(
L) Ak~L(
L) e2ikxL: For the solution obeying the boundary condition at x=xR, we write  R(x) =Beikx+eikx: (A7) With the boundary condition at x=xR( Eq.A4), we have: A(iBkeikxRikeikxR) =i~R(R)(BeikxR+eikxR); so that B=Ak~R(
R) Ak~R(
R) e2ikxR: The Green’s function is now given by39 g()(x;x0;) =8 < :() L(x0)() R(x) AW()(x0)forx>x0; () L(x)() R(x0) AW()(x0)forx<x0:(A8) with the Wronskian W(x0) = L(x0)d R(x0) dx0 R(x0)d L(x0) dx0: Inserting the result for the Green’s function in Eq. (36) and using that k+=iandk= (k+), we obtain Eqs. (38) and (39) after restoring the q-dependence and taking ~R= ~L= ~. Appendix B: Stochastic Formalism for Spin Transport in a Ferromagnet Here, we show how to recover our analytical results from the stochastic Landau-Lifshitz-Gilbert equation, general- izing the results of Ref. [25] to the case of nonzero spin accumulation in the metallic reservoirs. The dynamics of the spin density unit vector nis governed by: (1 +n)~_n+n(H+h)Anr2n= 0; (B1) where H=
^zis the effective applied magnetic field (in units of energy) and his the bulk stochastic field24. We assume a spin accumulation 0=
Rzin the right normal metal, while the spin accumulation in the left lead is taken zero. The boundary condition at x= 0reads js(x= 0) =A~sn@xnjx=0 =g"# 4(n(n0) +n~_n) +nh0 L x=0(B2) and atx=d: js(x=d) =A~sn@xnjx=d =g"# 4(n~_n) +nh0 R x=d: (B3)12 Defining (x;t) =n(x;t)p ~s=2, wherennxiny, we linearize the dynamics around the equilibrium orientation n=z. Fourier transforming: (x;q;) =Zdt 2~Zd2r? 2eit=~eir?
q (x;r?;t); the bulk equation of motion reads: A @2 x2
=hp ~s : (B4) The bulk transformed stochastic field h=hxihyobeys the fluctuation dissipation theorem: hh(x;q;)h(x0;q0;0)i= 2 (2)3(~2=~s) (xx0)(qq0)(0) tanh [=2kBT]: (B5) The boundary conditions, Eqs. (B2) and (B3), become respectively: A@x ig"# 4~s(
R) =hRp 2~s(B6) atx= 0and A@x +ig"# 4~s =hLp 2~s(B7) atx=d, where we have taken the coupling at both interfaces equal. Similarly, the interfacial stochastic fields obey: D h0 R(q;)h0 R(q0;0)E =2 (2)30d~2~s(
R)(qq0)(0) tanh [(
R)=2kBT](B8) and D h0 L(q;0)h0 L(q0;)E =2 (2)30d~2~s(qq0)(0) tanh [=2kBT]: (B9) Using Eqs. (B4)-(B9), one finds the current on the left side of the structure: jL sz
js(x= 0)to be of the form: jL s=Zd 2 NB kBT NB
R kBT T() (B10) where T()is the transmission coefficient in Eq. (38). Hence, for a clean system and in the continuum limit the results of the stochastic Landau-Lifshitz-Gilbert equation coincide with those of the NEGF formalism given by Eq. (25). 1Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz. Statistical physics. 1958. 2Charles Kittel. Introduction tosolidstate, volume 162. John Wiley & Sons, 1966. 3AV Chumak, VI Vasyuchka, AA Serga, and B Hillebrands. Magnon spintronics. NaturePhysics, 11(6):453–461, 2015. 4T Kasuya and RC LeCraw. Relaxation mechanisms in fer-romagnetic resonance. Physical ReviewLetters, 6(5):223, 1961. 5M Sparks. Ferromagnetic resonance in thin films. i. theory of normal-mode frequencies. Physical ReviewB, 1(9):3831, 1970. 6Christian W Sandweg, Yosuke Kajiwara, Andrii V Chu- mak, Alexander A Serga, Vitaliy I Vasyuchka, Mat-13 tias Benjamin Jungfleisch, Eiji Saitoh, and Burkard Hille- brands. Spin pumping by parametrically excited exchange magnons. Physical reviewletters, 106(21):216601, 2011. 7Nynke Vlietstra, BJ van Wees, and FK Dejene. Detection of spin pumping from yig by spin-charge conversion in a au/ni 80 fe 20 spin-valve structure. Physical ReviewB, 94(3):035407, 2016. 8ATalalaevskij,MDecker,JStigloher,AMitra,HSKörner, OCespedes, CHBack, andBJHickey. Magneticproperties of spin waves in thin yttrium iron garnet films. Physical ReviewB, 95(6):064409, 2017. 9J Holanda, O Alves Santos, RL Rodríguez-Suárez, A Azevedo, and SM Rezende. Simultaneous spin pump- ing and spin seebeck experiments with thermal control of the magnetic damping in bilayers of yttrium iron garnet and heavy metals: Yig/pt and yig/irmn. Physical Review B, 95(13):134432, 2017. 10E Saitoh, M Ueda, H Miyajima, and G Tatara. Conver- sion of spin current into charge current at room temper- ature: Inverse spin-hall effect. Applied PhysicsLetters, 88(18):182509, 2006. 11K Uchida, S Takahashi, K Harii, J Ieda, W Koshibae, Kazuya Ando, S Maekawa, and E Saitoh. Observation of the spin seebeck effect. Nature, 455(7214):778–781, 2008. 12Jiang Xiao, Gerrit EW Bauer, Ken-chi Uchida, Eiji Saitoh, Sadamichi Maekawa, et al. Theory of magnon-driven spin seebeck effect. Physical ReviewB, 81(21):214418, 2010. 13L Berger. Emission of spin waves by a magnetic multilayer traversed by a current. Physical ReviewB, 54(13):9353, 1996. 14John C Slonczewski. Current-driven excitation of mag- netic multilayers. Journal ofMagnetism andMagnetic Materials, 159(1-2):L1–L7, 1996. 15LJ Cornelissen, J Liu, RA Duine, J Ben Youssef, and BJ Van Wees. Long-distance transport of magnon spin information in a magnetic insulator at room temperature. NaturePhysics, 11(12):1022–1026, 2015. 16XJZhou, GYShi, JHHan, QHYang, YHRao, HWZhang, LL Lang, SM Zhou, F Pan, and C Song. Lateral transport properties of thermally excited magnons in yttrium iron garnetfilms. AppliedPhysicsLetters, 110(6):062407, 2017. 17Er-Jia Guo, Joel Cramer, Andreas Kehlberger, Ciaran A Ferguson, Donald A MacLaren, Gerhard Jakob, and Math- ias Kläui. Influence of thickness and interface on the low- temperature enhancement of the spin seebeck effect in yig films.Physical ReviewX, 6(3):031012, 2016. 18Ludo J Cornelissen, Kevin JH Peters, GEW Bauer, RA Duine, and Bart J van Wees. Magnon spin transport driven by the magnon chemical potential in a magnetic insulator. Physical ReviewB, 94(1):014412, 2016. 19Benedetta Flebus, SA Bender, Yaroslav Tserkovnyak, and RA Duine. Two-fluid theory for spin superfluidity in mag- netic insulators. Physical reviewletters, 116(11):117201, 2016. 20Kouki Nakata, Pascal Simon, and Daniel Loss.Wiedemann-franz law for magnon transport. Physical ReviewB, 92(13):134425, 2015. 21Kouki Nakata, Pascal Simon, and Daniel Loss. Spin cur- rentsandmagnondynamicsininsulatingmagnets. Journal ofPhysicsD:AppliedPhysics, 50(11):114004, 2017. 22Baigeng Wang, Jian Wang, Jin Wang, and DY Xing. Spin current carried by magnons. Physical ReviewB, 69(17):174403, 2004. 23Massimiliano Di Ventra. Electrical transport innanoscale systems, volume 14. Cambridge University Press Cam- bridge, 2008. 24William Fuller Brown Jr. Thermal fluctuations of a single- domain particle. Physical Review, 130(5):1677, 1963. 25Silas Hoffman, Koji Sato, and Yaroslav Tserkovnyak. Landau-lifshitz theory of the longitudinal spin seebeck ef- fect.Physical ReviewB, 88(6):064408, 2013. 26T Holstein and Hl Primakoff. Field dependence of the intrinsic domain magnetization of a ferromagnet. Physical Review, 58(12):1098, 1940. 27Assa Auerbach. Interacting electrons andquantum magnetism. Springer Science & Business Media, 2012. 28Scott A Bender, Rembert A Duine, and Yaroslav Tserkovnyak. Electronic pumping of quasiequilibrium bose-einstein-condensed magnons. Physical reviewletters, 108(24):246601, 2012. 29Y. Ohnuma, M. Matsuo, and S. Maekawa. Theory of spin Peltier effect. ArXive-prints, June 2017. 30Joseph Maciejko. An introduction to nonequilibrium many-body theory. LectureNotes, 2007. 31Arne Brataas, Yu V Nazarov, and Gerrit EW Bauer. Finite-element theory of transport in ferromagnet–normal metalsystems. Physical ReviewLetters, 84(11):2481, 2000. 32Yaroslav Tserkovnyak, Arne Brataas, and Gerrit EW Bauer. Enhanced gilbert damping in thin ferromagnetic films.Physical reviewletters, 88(11):117601, 2002. 33Jia, Xingtao, Liu, Kai, Xia, Ke, and Bauer, Gerrit E. W. Spin transfer torque on magnetic insulators. EPL, 96(1):17005, 2011. 34P. W. Anderson. Absence of diffusion in certain random lattices. Phys.Rev., 109:1492–1505, Mar 1958. 35Martin Evers, Cord A. Müller, and Ulrich Nowak. Spin- wave localization in disordered magnets. Phys.Rev.B, 92:014411, Jul 2015. 36I. Yusipov, T. Laptyeva, S. Denisov, and M. Ivanchenko. Localization in open quantum systems. Phys.Rev.Lett., 118:070402, Feb 2017. 37M. Evers, C. A. Müller, and U. Nowak. Weak localiza- tion of magnons in chiral magnets. ArXive-prints, August 2017. 38AkashdeepKamraandWolfgangBelzig. Magnon-mediated spin current noise in ferromagnet| nonmagnetic conductor hybrids. Physical ReviewB, 94(1):014419, 2016. 39Russell. L. Herman. Introduction to partial differential equations. 2015.

2017-09-12

We develop a Green's function formalism for spin transport through heterostructures that contain metallic leads and insulating ferromagnets. While this formalism in principle allows for the inclusion of various magnonic interactions, we focus on Gilbert damping. As an application, we consider ballistic spin transport by exchange magnons in a metal-insulator-metal heterostructure with and without disorder. For the former case, we show that the interplay between disorder and Gilbert damping leads to spin current fluctuations. For the case without disorder, we obtain the dependence of the transmitted spin current on the thickness of the ferromagnet. Moreover, we show that the results of the Green's function formalism agree in the clean and continuum limit with those obtained from the linearized stochastic Landau-Lifshitz-Gilbert equation. The developed Green's function formalism is a natural starting point for numerical studies of magnon transport in heterostructures that contain normal metals and magnetic insulators.

Green's function formalism for spin transport in metal-insulator-metal heterostructures

1709.03775v1

2011-01-28

The entanglement between two atoms in a damping Jaynes-Cummings model is investigated with different decay coefficients of the atoms from the upper level to other levels under detuning between the atomic frequency and the quantized light field frequency. The results indicate that the larger the decay coefficient is, the more quickly the entanglement decays. The detuning enhances the entanglement's average value at long times. More importantly, the results show that the so-called sudden death effect can be avoided by enhancing the detuning or the decay coefficient.

Entanglement between two atoms in a damping Jaynes-Cummings model

1101.5522v1

Dynamical depinning of chiral domain walls Simone Moretti,Michele Voto, and Eduardo Martinez Department of Applied Physics, University of Salamanca, Plaza de los Caidos, Salamanca 37008, Spain. The domain wall depinning eld represents the minimum magnetic eld needed to move a domain wall, typically pinned by samples' disorder or patterned constrictions. Conventionally, such eld is considered independent on the Gilbert damping since it is assumed to be the eld at which the Zeeman energy equals the pinning energy barrier (both damping independent). Here, we analyse numerically the domain wall depinning eld as function of the Gilbert damping in a system with per- pendicular magnetic anisotropy and Dzyaloshinskii-Moriya interaction. Contrary to expectations, we nd that the depinning eld depends on the Gilbert damping and that it strongly decreases for small damping parameters. We explain this dependence with a simple one-dimensional model and we show that the reduction of the depinning eld is related to the nite size of the pinning barriers and to the domain wall internal dynamics, connected to the Dzyaloshinskii-Moriya interaction and the shape anisotropy. I. INTRODUCTION Magnetic domain wall (DW) motion along ferromag- netic (FM) nanostructures has been the subject of in- tense research over the last decade owing to its po- tential for new promising technological applications1,2 and for the very rich physics involved. A consider- able eort is now focused on DW dynamics in systems with perpendicular magnetic anisotropy (PMA) which present narrower DWs and a better scalability. Typ- ical PMA systems consist of ultrathin multi-layers of heavy metal/FM/metal oxide (or heavy metal), such as Pt=Co=Pt3,4or Pt=Co=AlOx5{7, where the FM layer has a thickness of typically 0 :61 nm. In these systems, PMA arises mainly from interfacial interactions between the FM layer and the neighbouring layers (see Ref.8and references therein). Another important interfacial ef- fect is the Dzyaloshinskii-Moriya interaction (DMI)9,10, present in systems with broken inversion symmetry such as Pt/Co/AlOx. This eect gives rise to an internal in- plane eld that xes the DW chirality (the magnetization rotates always in the same direction when passing from up to down and from down to up domains) and it can lead to a considerably faster domain wall motion10and to new magnetic patterns such as skyrmions11or helices12. Normally, DWs are pinned by samples' intrinsic disorder and a minimum propagation eld is needed in order to overcome such pinning energy barrier and move the DW. Such eld is the DW depinning eld ( Hdep) and it repre- sents an important parameter from a technological point of view since a low depinning eld implies less energy required to move the DW and, therefore, a energetically cheaper device. From a theoretical point of view, DW motion can be described by the Landau-Lifshitz-Gilbert (LLG) equa- tion13which predicts, for a perfect sample without dis- order, the velocity vseld curve depicted in Fig. 1 and labelled as Perfect . In a disordered system, experi- ments have shown that a DW moves as a general one- dimensional (1D) elastic interface in a two-dimensional disordered medium3,4and that it follows a theoreticalvelocityvsdriving force curve, predicted for such inter- faces14,15(also shown in Fig. 1 for T= 0 andT= 300K). Moreover, this behaviour can be reproduced by including disorder in the LLG equation16{18. At zero temperature (T= 0) the DW does not move as long as the applied eld is lower than Hdep, while, at T6= 0, thermal ac- tivation leads to DW motion even if H < H dep(the so called creep regime). For high elds ( H >> H dep) the DW moves as predicted by the LLG equation in a per- fect system. Within the creep theory, the DW is con- sidered as a simple elastic interface and all its internal dynamics are neglected. Conventionally, Hdepis consid- ered independent of the Gilbert damping because it is as- sumed to be the eld at which the Zeeman energy equals the pinning energy barrier19,20(both damping indepen- dent). Such assumption, consistently with the creep the- ory, neglects any eects related to the internal DW dy- namics such as DW spins precession or vertical Bloch lines (VBL) formation21. The damping parameter, for its part, represents another important parameter, which controls the energy dissipation and aects the DW veloc- ity and Walker Breakdown22. It can be modied by dop- ing the sample23or by a proper interface choice as a con- sequence of spin-pumping mechanism24. Modications of the DW depinning eld related to changes in the damping parameter were already observed in in-plane systems23,25 and attributed to a non-rigid DW motion23,25. Oscilla- tions of the DW depinning eld due to the internal DW dynamics were also experimentally observed in in-plane similar systems26. Additional dynamical eects in soft samples, such as DW boosts in current induced motion, were numerically predicted and explained in terms of DW internal dynamics and DW transformations27,28. Here, we numerically analyse the DW depinning eld in a system with PMA and DMI as function of the Gilbert damping. We observe a reduction of Hdepfor low damp- ing and we explain this behaviour by adopting a simple 1D model. We show that the eect is due to the nite size of pinning barriers and to the DW internal dynam- ics, related to the DMI and shape anisotropy elds. This article is structured as follows: in Section II we present the simulations method, the disorder implementation andarXiv:1705.07489v2 [cond-mat.mes-hall] 25 Aug 20172 theHdepcalculations. The main results are outlined and discussed in Section III, where we also present the 1D model. Finally, the main conclusions of our work are summarized in Section IV. ●●●● ● ● ● ●●●●●●●●●●�=�� �=���� ●������� ������� ������� �������� ����★ FIG. 1. DW velocity vsapplied eld as predicted by the LLG equation in a perfect system and by the creep law atT= 0 andT= 300K. II. MICROMAGNETIC SIMULATIONS We consider a sample of dimensions (102410240:6) nm3with periodic bound- ary conditions along the ydirection, in order to simulate an extended thin lm. Magnetization dynamics is analysed by means of the LLG equation13: dm dt= 0 1 +2(mHe) 0 1 +2[m(mHe)]; (1) where m(r;t) =M(r;t)=Msis the normalized magneti- zation vector, with Msbeing the saturation magnetiza- tion. 0is the gyromagnetic ratio and is the Gilbert damping. He=Hexch+HDMI+Han+Hdmg+Hz^uz is the eective eld, including the exchange, DMI, uni- axial anisotropy, demagnetizing and external eld con- tributions13respectively. Typical PMA samples param- eters are considered: A= 171012J=m,Ms= 1:03 106A=m,Ku= 1:3106J=m3andD= 0:9 mJ=m2, whereAis the exchange constant, Dis the DMI constant andKuis the uniaxial anisotropy constant. Disorder is taken into account by dividing the sample into grains by Voronoi tessellation29,30, as shown in Fig. 2(a). In each grain the micromagnetic parameters fMs;Dc;Kug change in a correlated way in order to mimic a normally distributed thickness31: tG=N(t0;)!8 < :MG= (MstG)=t0 KG= (Kut0)=tG DG= (Dct0)=tG; (2) where the subscript Gstands for grain, t0is the aver- age thickness ( t0= 0:6nm) andis the standard devi- ation of the thickness normal distribution. The sample is discretized in cells of dimensions (2 20:6)nm3,smaller than the exchange length lex5nm. Grain size is GS=15 nm, reasonable for these materials, while the thickness uctuation is = 7%. Eq. (1) is solved by the nite dierence solver MuMax 3.9.329. A DW is placed and relaxed at the center of the sample as depicted in Fig. 2(b). Hdepis calculated by applying a sequence of elds and running the simulation, for each eld, until the DW is expelled from the sample, or until the system has reached an equilibrium state (i.e. the DW remains pinned): max<().maxindicates the maxi- mum torque, which rapidly decreases when the system is at equilibrium. It only depends on the system parame- ters and damping. For each value of , we choose a spe- cic threshold, (), in order to be sure that we reached an equilibrium state (see Supplementary Material32for more details). The simulations are repeated for 20 dif- ferent disorder realizations. Within this approach, Hdep corresponds to the minimum eld needed to let the DW propagate freely through the whole sample. In order to avoid boundaries eects, the threshold for complete de- pinning is set tohmzi>0:8, wherehmziis averaged over all the realizations, i.e. hmzi=PN i=1hmzii=N, where N= 20 is the number of realizations. We checked that, in our case, this denition of Hdepcoincides with tak- ingHdep= MaxfHi depg, withHi depbeing the depinning eld of the single realization. In other words, Hdepcor- responds to the minimum eld needed to depin the DW from any possible pinning site considered in the 20 real- izations33. Following this strategy, the DW depinning eld is nu- merically computed with two dierent approaches: (1) by Static simulations, which neglect any precessional dynamics by solving dm dt= 0 1 +2[m(mHe)]: (3) This is commonly done when one looks for a minimum of the system energy and it corresponds to the picture in whichHdepsimply depends on the balance between Zeeman and pinning energies.34 (2) by Dynamic simulations, which include precessional dynamics by solving the full Eq. (1). This latter method corresponds to the most realistic case. Another way to estimate the depinning eld is to calculate the DW veloc- ityvseld curve at T= 0 and look for minimum eld at which the DW velocity is dierent from zero. For these simulations we use a moving computational region and we run the simulations for t= 80ns (checking that longer simulations do not change the DW velocity, meaning that we reached a stationary state). This second setup re- quires more time and the calculations are repeated for only 3 disorder realizations. Using these methods, the depinning eld Hdepis cal- culated for dierent damping parameters .3 (a) (b) xy (c) FIG. 2. (a) Grains structure obtained by Voronoi tassellation. (b) Initial DW state. (c) Sketch of the internal DW angle . III. RESULTS AND DISCUSSION A. Granular system Our rst result is shown in Fig. 3(a)-(b), which depicts the nal average magnetization hmzias function of the applied eld for dierent damping parameters. In the Static simulations (Fig. 3(a)) Hdepdoes not depend on damping, so that a static depinning eld can be dened. Conversely, in the Dynamic simulations (Fig. 3(b)), Hdep decreases for low damping parameters. The depinning eld is indicated by a star in each plot and the static depinning eld is labelled as Hs. The same result is ob- tained by calculating Hdepfrom the DW velocity vsap- plied eld plot, shown in Fig. 3(c). The stars in Fig. 3(c) correspond to the depinning elds calculated in the pre- vious simulations and they are in good agreement with the values predicted by the velocity vseld curve. The dynamical depinning eld 0Hd, normalized to the static depinning eld 0Hs= (871)mT, with 0being the vacuum permeability, is shown in Fig. 3(d) as function of the damping parameter .Hdsaturates for high damp- ing (in this case 0:5) while it decreases for low damp- ing untilHd=Hs0:4 at= 0:02. This reduction must be related to the precessional term, neglected in the static simulations. The same behaviour is observed with dier- ent grain sizes (GS=5 and 30 nm) and with a dierent disorder model, consisting of a simple variation of the Ku module in dierent grains. This means that the eect is not related to the grains size or to the particular disorder model we used. Additionally, Fig. 4 represents the DW energy35as function of DW position and damping parameter for 0Hz= 70 mT. At high damping, the average DW en- ergy density converges to 110 mJ=m2, in good agree- ment with the analytical value 0= 4pAK0D= 10:4 mJ=m2, whereK0is the eective anisotropy K0= Ku0M2 s=2. On the contrary, for low damping, the DW energy increases up to (0:02)14 mJ=m2. This increase, related to DW precessional dynamics, reduces the eective energy barrier and helps the DW to over- ●●●●●●●●●●●●●●●●● ○○○○○○○○○○○○○○○○○ ■■■■■■■■■■■■■■■■■ ●α=���� ○α=��� ■α=��� ���������������������<��> ★(�) ������ �� ●●●●●●●●●●●●●●●●● ○○○○○○○○○○○○○○○○○ ■■■■■■■■■■■■■■■■■ �� �� �� �������������������������� ������� �����(��)<��>★★★(�) ������� ●●●●●●●●●○○○○○○○○○■■■■■■■■■●α=���� ○α=��� ■α=��� �� �� �� �������������� ������� �����(��)�� ��������(�/�)★★★(�) ●●●●● ● ● ������������������������������������ �������α��/��(�)FIG. 3. Average hmzias function of applied eld for dif- ferent damping parameters for the (a) Static simulations and (b)Dynamic simulations. (c) DW velocity vs applied eld for dierent damping. (d) Dynamical depinning eld, normalized toHs, as function of damping. come the pinning barriers. Fig. 4(c) shows the total en- ergy of the system (including Zeeman). As expected36, the energy decreases as the DW moves. Finally, Fig. 5 shows the DW motion as function of time for= 0:02 and= 0:5, along the same grain pattern (and therefore along the same pinning barriers). The applied eld is 0Hz= 70mT, which satises Hd(0:02)<Hz<Hd(0:5). The initial DW conguration is the same but, for = 0:02, VBL start to nucleate and the DW motion is much more turbulent (see Supplemen- tary Material32for a movie of this process). At t= 4 ns the DW has reached an equilibrium position for = 0:5, while it has passed through the (same) pinning barriers for= 0:02. Thus, one might think that the reduction of the depinning eld could be related to the presence4 ●●●●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ●α=���� ○α=��� ■α=��� �������������������������� �� ��������(μ�)σ��(��/��)(�) ● ●● ����������������������������� �������α<σ��> (��/��)(�) ●●●●●●●●●●●● ●●●●●●●●●●●●○○○○○○○○○○○○○○○○○○○○○○○■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ���������������-��-������� �� ��������(��)����� ������ �������(��/��) (�) FIG. 4. (a) DW energy density as function of DW posi- tion for dierent damping. The nal drop corresponds to the expulsion of the DW. (b) Average DW density as funci- ton of damping. Dashed line represents the analytical value 110 mJ=m2. (c) Total energy density of the system as function of DW position for dierent damping parameters. of VBL and their complex dynamics21. Further insights about this mechanism are given by analysing the DW depinning at a single energy barrier as described in the next subsection. B. Single barrier In order to understand how the DW precessional dy- namics reduces Hdep, we micromagnetically analysed the DW depinning from a single barrier as sketched in Fig. 6. We considered a strip of dimensions (1024 2560:6)nm3 and we divided the strip into two regions, R1andR2, which are assumed to have a thickness of t1= 0:58 and t2= 0:62 nm respectively. Their parameters vary ac- cordingly (see Sec. II), generating the DW energy bar- rier () shown in Fig. 6(b). A DW is placed and re- laxed just before the barrier. The nite size of the DW (
DW15 nm, with
DWbeing the DW width pa- rameter) smooths the abrupt energy step and, in fact, the energy prole can be successfully tted by using theBloch prole22 DW=0+ + 2 1 + cos 2 arctan expx0x
DW ; (4) wherex0= 20 nm is the step position, while 0and 1are the DW energies at the left and right side of the barrier as represented in Fig. 6(b). This means that the pinning energy barrier has a spatial extension which is comparable to the DW width. By performing the same static and dynamic simulations, we obtain a static depinning eld of 0Hs= 120 mT and, when decreasing the damping parameter, we observe the same reduction of the depinning eld as in the granular system (see Fig. 6(c)). In this case the DW behaves like a rigid object whose spins precess coherently and no VBL nucleation is observed. Hence, Hdepreduction does not depend directly on the presence of VBL but on the more general mechanism of spins' precession already present in this simplied case. Nevertheless, an important characteristic of these single barrier simulations is that the barrier is localized and it has a nite size which is of the order of the DW width. Note that the same holds for the granular system: despite a more complex barrier structure, the dimension of the single barrier between two grains has the size of the DW width. Thus, in order to understand the interplay between the DW precessional dynamics and the nite size of the bar- rier, we considered a 1D collective-coordinate model with a localized barrier. The 1D model equations, describing the dynamics of the DW position qand the internal angle (sketched in Fig. 2(c)), are given by16 (1 +2)_= 0[(Hz+Hp(q))  HKsin 2 2 2HDMIsin |{z } Hint()];(5) (1 +2)_q
DW= 0[(Hz+Hp(q)) + HKsin 2 2 2HDMIsin ;(6) whereHK=MsNxis the shape anisotropy eld, favour- ing Bloch walls, with Nx=t0log 2=(
DW)37being the DW demagnetizing factor along the xaxis.HDMI = D=(0Ms
DW) is the DMI eld. Hint() represents the internal DW eld, which includes DMI and shape anisotropy. Hintfavours Bloch ( ==2) or N eel wall (= 0 or=) depending on the relative strength ofHKandHDMI. In our system, the DMI dominates over shape anisotropy since 0HDMI170 mT while 0HK30 mT. Hence, the DW equilibrium angle is5 Out[64]= Out[60]= … (a) 𝜶=𝟎.𝟎𝟐time 0 0.1 ns 0.2 ns 0.3 ns 4 ns time 0 0.1 ns 0.2 ns 0.3 ns 4 ns(b) 𝜶=𝟎.𝟓 … Out[395]=mx Out[395]=mx Out[62]= Out[65]= FIG. 5. (a) Snapshots of the magnetization dynamics at subsequent instants under 0Hz= 70mT, for two dierent damping: (a)= 0:02 and (b) = 0:5. The grains pattern, and therefore the energy barrier, is the same for both cases. In order to let the DW move across more pinning sites, these simulations were performed on a larger sample with Lx= 2048 nm. =(= 0 or=additionally depends on the sign of the DMI). Hp(q) is the DW pinning eld, obtained from the DW energy prole (Eq. (4)) as follows: the max- imum pinning eld is taken from the static simulations while the shape of the barrier is taken as the normalized DW energy gradient (see Supplementary Material32for more details), Hp(q) =Hs@DW(x) @x N= = 2Hsexp x0q
DW sinh 2 arctan exp x0q
DWi 1 + exp 2(x0q)
DW :(7) The corresponding pinning eld is plotted in Fig. 7(a).38 The results for the dynamical Hdep, obtained with this modied 1D model, are plotted in Fig. 6(c) and they show a remarkable agreement with the single barrier mi- cromagnetic simulations. This indicates that the main factors responsible for the reduction of Hdepare already included in this simple 1D model. Therefore, additional insights might come from analysing the DW dynamics within this 1D model. Fig. 7(b) and (c) represents the DW internal angle and the DW position qas function of time for dierent damping. The plots are calculated with0Hz= 55 mT which satises Hdep(0:02)< Hz< Hdep(0:1)< H dep(0:5). As shown in Fig. 7(b) and (c), below the depinning eld ( = 0:1,= 0:5), both the internal angle and the DW position oscillate before reach- ing the same nal equilibrium state. However, the am-plitude of these oscillations (the maximum displacement) depends on the damping parameter. Fig. 7(d) shows the nal equilibrium position as function of the applied eld for dierent damping. The equilibrium position is the same for all damping and it coincides with the position at whichHz=Hp(q). Conversely, the maximum dis- placement, shown in Fig. 7(e), strongly increases for low damping parameters. For applied eld slightly smaller than the depinning eld, the DW reaches the boundary of the pinning barrier, meaning that a further increase of the eld is enough to have a maximum displacement higher than the barrier size and depin the DW. In other words, the decrease of the depinning eld, observed in the single barrier simulations, is due to DW oscillations that depend on and that can be larger than the bar- rier size, leading to DW depinning for lower eld. The DW dynamics and the depinning mechanism are further claried in Fig. 7(f) and Fig. 7(g). Fig. 7(f) represents the DW coordinates fq;gfor0Hz= 55 mT and dif- ferent damping. Before reaching the common equilib- rium state, the DW moves in orbits (in the fq;gspace) whose radius depends on the damping parameter. For = 0:5 (black line) the DW rapidly collapse into the - nal equilibrium state. Conversely, for = 0:1 (red open circles), the DW orbits around the equilibrium state be- fore reaching it. If the radius of the orbit is larger than the barrier size the DW gets depinned, as in the case of= 0:02 (blue full circles). This mechanism is also represented in Fig. 7(g), where the DW orbits are placed in the energy landscape. The energy is calculated as6 ●●●●●●●●●●●●●●●●●μ� ����������� ��●�� �������������������������������������������� �������α��/��○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○�� �������������(�)-���������������������������������������(��)σ��(��/��)��δσσ�σ�(b) (c) R1R2yx(a) FIG. 6. (a) Sketch of the two regions implemented for the single barrier (SB) micromagnetic simulations. (b) DW en- ergy as function of DW position along the strip. Blue solid line represents the analytical value, red points the DW con- voluted energy (due to the nite size of the DW) while black dashed line a t using Eq. 4. (c) Dynamical depinning eld, normalized to the static depinning eld, for the single bar- rier simulations as function of damping, obtained from full micromagnetic simulations and the 1D model. (q;) =DW(q;)20MsHzq, whereDWis given by Eq. (4). Fig. 7(g) shows that the equilibrium state cor- responds to the new minimum of the energy landscape. Furthermore, it conrms that the applied eld is below the static depinning eld, at which the pinning barrier would have been completely lifted. Nevertheless, while reaching the equilibrium state, the DW moves inside the energy potential and, if the radius of the orbit is larger than the barrier size, the DW can overcome the pinning barrier, as shown for = 0:02 in Fig. 7(g). At this point we need to understand why the amplitude of the DW oscillations depends on damping. By solving Eq. (5) and Eq.(6) for the equilibrium state ( _ q= 0, _= 0) we obtain _q= 0)jHp(q)j=Hz+Hint()  Hz 2HDMI sin; (8) _= 0)jHp(q)j=HzHint() Hz+ 2HDMIsin; (9) since0HDMI0HKand, therefore, Hint (=2)HDMIsin. These equations have a single com- mon solution which corresponds to jHp(q)j=Hzand =0=(at whichHint() = 0). However, at t= 0,the DW starts precessing under the eect of the applied eld and, if 6=whenjHp(q)j=Hz, the DW does not stop at the nal equilibrium position but it continues its motion, as imposed by Eq. (8) and (9). In other words, the DW oscillations in Fig. 7(b) are given by oscillations of the DW internal angle , around its equilibrium value 0=. These oscillations lead to a modication of the DW equilibrium position due to the DW internal eld (Hint()), which exerts an additional torque on the DW in order to restore the equilibrium angle. As previously commented, if the amplitude of these oscillations is large enough, the DW gets depinned. From Eq. (8) we see that the new equilibrium position (and therefore the am- plitude of the oscillations) depends on the DMI eld, the value of the DW angle and the damping parameter. In particular, damping has a twofold in uence on this dynamics: one the one hand, it appears directly in Eq. (8), dividing the internal eld, meaning that for the same deviation of from equilibrium, we have a stronger internal eld for smaller damping. On the other hand, the second in uence of damping is on the DW internal angle: once the DW angle has deviated from equilibrium, the restoring torque due to DMI is proportional to the damping parameter (see Eq. (9)). Hence, a lower damp- ing leads to lower restoring torque and a larger deviation offrom equilibrium. The maximum deviation of from equilibrium ( =max0) is plotted in Fig. 8(b) as function of damping for 0Hz= 40 mT. As expected, a lower damping leads to a larger deviation . In this latter section, the DW was set at rest close to the barrier and, therefore, the initial DW velocity is zero. Nevertheless, one might wonder what happens when the DW reaches the barrier with a nite velocity. We simu- lated this case by placing the DW at an initial distance d1= 200 nm from the barrier. The depinning is further reduced in this case (see Supplementary Material32for more details). However, in the static simulations, the de- pinning eld remains constant, independently from the velocity at which the DW reaches the barrier, meaning that the reduction of Hdepis again related to the DW precession. When the DW starts from d1it reaches the barrier precessing, thus with a higher deviation from its equilibrium angle, leading to a higher eect of the inter- nal eld. C. Dierent DMI and pinning barriers Finally, by using the 1D model it is possible to ex- plore the dependence of Hdepon the pinning potential amplitudeHs(related to the disorder strength) and on the DMI constant D. The depinning eld as function of damping for dierent values of Hsis plotted in Fig. 9(a). The reduction of Hdepis enhanced for larger values of Hs(strong disorder). This is consistent with our expla- nation, since for strong disorder we need to apply larger elds that lead to larger oscillations of . Fig. 9(b) represents the dynamical Hdepas function of7 ●●●●●●●●●●●●●●●●●●●●●●●● ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼�����������������������������������(��)�� ��������(��)●●●●●●●●■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆◆▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼○○○○○○○○○○○○○○○○○○○○○○●α=����■α=���◆α=���▲α=����▼α=���○α=������������������������������������������������� ������� �����μ���(��)���������������(��)������������(��)●●●●●●●■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼○○○○○○○○○○○○○○○○○○○○○●α=����■α=���◆α=���▲α=����▼α=���○α=������������������������������������������������� ������� �����μ���(��)�����������(��)������������(��)-�������������-���-��-�������������(��)μ���(��) Max Displacement Eq. 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▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼●α=����○α=���α=������������������������������������ �(��)ϕ(°) FIG. 7. (a) Pinning eld obtained from Eq. (7) as function of DW position. DW position internal angle as function of time for dierent damping parameter and 0Hz= 55 mT. (c) DW position qas function of time for dierent damping and 0Hz= 55 mT. (d) Equilibrium position as function of applied eld for dierent damping. (e) Maximum DW displacement as function of the applied eld for dierent damping. (f) DW coordinates fq;gfor0Hz= 55 mT and dierent damping. (g) DW coordinatesfq;ginside the energy landscape: =DW(q;)20MsHzq. ������������������������������ �������αδϕ(°) FIG. 8. Maximum deviation of from its equilibrium posi- tion as function of damping. damping for 0Hs= 120 mT and dierent DMI con- stants (expressed in term of the critical DMI constant Dc= 4pAK0== 3:9 mJ=m2)39. In this case, the reduc- tion ofHdepis enhanced for low DMI, until D= 0:05Dc, but a negligible reduction is observed for D= 0. This non-monotonic behaviour can be explained by looking at the dependence of andHinton the DMI constant. Fig. 10(a) shows the maximum uctuation as func- tion of DMI for 0Hz= 30 mT. increases for low DMI and it has a maximum at HDMI =HK, which in our case corresponds to D= 0:014Dc. The increase offor small values of Dis due to the smaller restor- ing torque in Eq. (9). This holds until HDMI =HK, where shape anisotropy and DMI are comparable and they both aect the DW equilibrium conguration. As a consequence, the reduction of Hdepis enhanced by de-creasingDuntilD0:014Dc, while it is reduced if 0< D < 0:014Dc. Another contribution is given by the amplitude of the internal eld, Hint. Fig. 10(b) de- picts0Hintas function of andD. The maximum , obtained at 0Hz= 30 mT, is additionally marked in the plot. The internal eld decreases with the DMI but this reduction is compensated by an increase in , which leads to an overall increase of 0Hint, as discussed in the previous part. However, at very low DMI, the in- ternal eld is dominated by shape anisotropy and, inde- pendently on the DW angle displacement, it is too small to have an eect on the depinning mechanism. Note, however, that the amplitude of Hintshould be compared with the amplitude of the pinning barrier Hs. Fig. 9(b) is calculated with 0Hs= 120 mT and the internal eld, given by shape anisotropy ( HK=215 mT), has indeed a negligible eect. However, larger eects are observed, in the caseD= 0, for smaller Hs, with reduction of Hdep up toHd=Hs0:6, as shown in Fig. 9(c), which is calcu- lated with0Hs= 30 mT. In other words, the reduction of the depinning eld depends on the ratio between the pinning barrier and the internal DW eld. Finally, it is interesting to see what happens for weaker disorder and dierent DMI in the system with grains. Fig. 11 shows the dynamical Hdep, for dierent pinning potential and dierent DMI, obtained in the granular system. The results are in good agreement with what predicted by the 1D model for dierent disorder strengths. However, we observe a smaller dependence on the DMI parameter. This is due to two reasons:8 ●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆◆◆◆◆◆◆◆◆◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ●μ���=�� �� ■μ���=�� �� ◆μ���=�� �� ▲μ���=��� �� ������������������������������������ �������α��/��(�) ●●●●●●●●●●●●●●●●●●●● ■■■■■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ○○○○○○○○○○○○○○○○○○○○ □□□□□□□□□□□□□□□□□□□□ ●�=���� � �■�=��� � � ◆�=��� � �▲�=��� � � ▼�=��� � �○�=��� � � □�=��� ��������������������������������������� �������α��/��(�) μ���=��� �� ●●●●●●●●●●●●●●●●●●●● ■■■■■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ○○○○○○○○○○○○○○○○○○○○ □□□□□□□□□□□□□□ □□□□□□ ●�=���� � �■�=���� � � ◆�=���� � �▲�=���� � � ▼�=���� � �○�=���� � � □�=��� ��������������������������������������� �������α��/��(�) μ���=�� �� FIG. 9. (a) Dynamical Hdepas function of damping for dier- entHs(disorder strength). (b) Dynamical Hdepas function of damping for dierent DMI constant and 0Hs= 120 mT. (c) Dynamical Hdepas function of damping for dierent DMI constant and 0Hs= 30 mT. (1) in the system with grains the static pinning barrier is0Hs= 87 mT and the dependence of the depinning eld with DMI is smaller for smaller barriers, as shown in Fig. 9(c). (2) The DW motion in the granular system presents the formation of VBL which might also contribute to the reduction of the depinning eld. The mechanism is the same: a VBL is a non-equilibrium conguration for the DW (as a deviation of from equilibrium) that generates additional torques on the DW, which contribute to the DW depinning. ����� ���� ��� ������������� �/��δϕ(°)(�) π����=�� μ�����(��) � ��� ��� ��� ��� ������������������������������ �/��δϕ(°)(�)FIG. 10. (a) Max DW angle uctuation =maxeq as function of DMI for 0Hz= 30 mT. (b) Internal DW eld0Hintas function of DMI and . The green points correspond the max uctuation plotted in (a). Note that the scale is logarithmic in (a). ●●●●● ● ● □□□□□ □ ●μ���=�� �� □μ���=�� ����������������/��(�) ●●●●● ● ● ◇◇◇◇◇ ◇ ○○○○○ ○ ●�=��� ��/��~���� � ◇�=��� ��/��~���� � ○�=� ������������������������������ �������α��/��(�) FIG. 11. (a) Dynamical Hdepas function of damping for dif- ferentHs(disorder strength). (b) Dynamical Hdepas function of damping for dierent DMI constants.9 IV. CONCLUSIONS To summarize, we have analysed the DW depinning eld in a PMA sample with DMI and we found that Hdep decreases with the damping parameter with reductions up to 50%. This decrease is related to the DW inter- nal dynamics and the nite size of the barrier: due to DW precession, the DW internal angle ( ) deviates from equilibrium and triggers the internal DW eld (DMI and shape anisotropy) which tries to restore its original value. At the same time, the internal eld pushes the DW above its equilibrium position within the energy barrier. This mechanism leads to DW oscillations and, if the ampli- tude of the oscillations is higher than the barrier size, the DW gets depinned for a lower eld. Deviations of  from equilibrium and DW oscillations are both damping dependent and they are enhanced at low damping. In the system with grains the mechanism is the same but deviations from the internal DW equilibrium include the formation of VBL with more complex dynamics. The eect is enhanced for low DMI (providing thatHDMI> H K) and for stronger disorder since we need to apply larger external elds, which lead to larger DW oscillations. These results are relevant both from a tech- nological and theoretical point of view, since they rstly suggest that a low damping parameter can lead to a lowerHdep. Furthermore, they show that micromagnetic calculations of the depinning eld, neglecting the DW precessional dynamics can provide only an upper limit forHdep, which could actually be lower due to the DW precessional dynamics. V. ACKNOWLEDGEMENT S.M. would like to thank K. Shahbazi, C.H. Mar- rows and J. Leliaert for helpful discussions. This work was supported by Project WALL, FP7- PEOPLE-2013- ITN 608031 from the European Commission, Project No. MAT2014-52477-C5-4-P from the Spanish government, and Project No. SA282U14 and SA090U16 from the Junta de Castilla y Leon. Corresponding author: simone.moretti@usal.es 1D. Allwood, Science 309, 1688 (2005). 2S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 3P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier, J. Ferr e, V. Baltz, B. Rodmacq, B. Dieny, and R. L. Stamps, Physical Review Letters 99, 217208 (2007). 4J. Gorchon, S. Bustingorry, J. Ferr e, V. Jeudy, A. B. Kolton, and T. Giamarchi, Physical Review Letters 113, 027205 (2014), arXiv:1407.7781. 5T. A. Moore, I. M. Miron, G. Gaudin, G. Serret, S. Auf- fret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and M. Bonm, Applied Physics Letters 93, 262504 (2008), arXiv:0812.1515. 6I. M. Miron, T. Moore, H. Szambolics, L. D. Buda- Prejbeanu, S. Auret, B. Rodmacq, S. Pizzini, J. Vogel, M. Bonm, A. Schuhl, and G. Gaudin, Nature materials 10, 419 (2011). 7E. Ju e, A. Thiaville, S. Pizzini, J. Miltat, J. Sampaio, L. D. Buda-Prejbeanu, S. Rohart, J. Vogel, M. Bonm, O. Boulle, S. Auret, I. M. Miron, and G. Gaudin, Phys- ical Review B 93, 014403 (2016). 8F. Hellman, A. Homann, Y. Tserkovnyak, G. S. Beach, E. E. Fullerton, C. Leighton, A. H. MacDonald, D. C. Ralph, D. A. Arena, H. A. D urr, P. Fischer, J. Grollier, J. P. Heremans, T. Jungwirth, A. V. Kimel, B. Koop- mans, I. N. Krivorotov, S. J. May, A. K. Petford-Long, J. M. Rondinelli, N. Samarth, I. K. Schuller, A. N. Slavin, M. D. Stiles, O. Tchernyshyov, A. Thiaville, and B. L. Zink, Reviews of Modern Physics 89, 025006 (2017), arXiv:1607.00439. 9A. Cr epieux and C. Lacroix, Journal of Magnetism and Magnetic Materials 182, 341 (1998). 10A. Thiaville, S. Rohart, E. Ju e, V. Cros, and A. Fert, EPL (Europhysics Letters) 100, 57002 (2012).11S. Rohart and A. Thiaville, Physical Review B 88, 184422 (2013). 12N. Perez, E. Martinez, L. Torres, S.-H. Woo, S. Emori, and G. S. D. Beach, Applied Physics Letters 104, 092403 (2014). 13G. Bertotti, Hysteresis in Magnetism (Academic Press, 1998). 14A. B. Kolton, A. Rosso, and T. Giamarchi, Physical Re- view Letters 94, 047002 (2005). 15E. Agoritsas, V. Lecomte, and T. Giamarchi, Physica B: Condensed Matter 407, 1725 (2012), arXiv:1111.4899. 16E. Martinez, L. Lopez-Diaz, L. Torres, C. Tristan, and O. Alejos, Physical Review B 75, 174409 (2007). 17M. Voto, L. Lopez-Diaz, and L. Torres, Journal of Physics D: Applied Physics 49, 185001 (2016). 18M. Voto, L. Lopez-Diaz, L. Torres, and S. Moretti, Phys- ical Review B 94, 174438 (2016). 19P. J. Metaxas, Solid State Physics - Advances in Research and Applications , Vol. 62 (2010) pp. 75{162. 20J. H. Franken, M. Hoeijmakers, R. Lavrijsen, and H. J. M. Swagten, Journal of Physics: Condensed Matter 24, 024216 (2012), arXiv:1112.1259. 21Y. Yoshimura, K.-j. Kim, T. Taniguchi, T. Tono, K. Ueda, R. Hiramatsu, T. Moriyama, K. Yamada, Y. Nakatani, and T. Ono, Nature Physics 12, 157 (2015). 22B. Hillebrands and A. Thiaville, Spin Dynamics in Con- ned Magnetic Structures III (Springer, 2005). 23T. A. Moore, P. M ohrke, L. Heyne, A. Kaldun, M. Kl aui, D. Backes, J. Rhensius, L. J. Heyderman, J.- U. Thiele, G. Woltersdorf, A. Fraile Rodr guez, F. Nolt- ing, T. O. Mente s, M. A. Ni~ no, A. Locatelli, A. Potenza, H. Marchetto, S. Cavill, and S. S. Dhesi, Physical Review B82, 094445 (2010). 24Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Physical Review Letters 88, 117601 (2002).10 25U.-H. Pi, Y.-J. Cho, J.-Y. Bae, S.-C. Lee, S. Seo, W. Kim, J.-H. Moon, K.-J. Lee, and H.-W. Lee, Physical Review B84, 024426 (2011). 26L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, and S. S. P. Parkin, Nature 443, 197 (2006). 27H. Y. Yuan and X. R. Wang, Physical Review B 92, 054419 (2015). 28H. Yuan and X. Wang, The European Physical Journal B 88, 214 (2015). 29A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, AIP Ad- vances 4, 107133 (2014), arXiv:1406.7635. 30J. Leliaert, B. Van de Wiele, A. Vansteenkiste, L. Laurson, G. Durin, L. Dupr e, and B. Van Waeyenberge, Journal of Applied Physics 115, 233903 (2014). 31A. Hrabec, J. Sampaio, M. Belmeguenai, I. Gross, R. Weil, S. M. Ch erif, A. Stashkevich, V. Jacques, A. Thiaville, and S. Rohart, Nature Communications 8, 15765 (2017), arXiv:1611.00647. 32See Appendices. 33This denition is preferred over the average of Hi depsince it is more independent on the sample size. In fact, by in-creasing the sample dimension along the xdirection, we increase the probability of nding the highest possible hj in the single realization and the average of Hi depwill tend to the maximum. 34This is solved by the Relax solver of MuMax with the as- sumption=(1 +2) = 1. 35The DW energy is calculated as the energy of the system with the DW minus the energy of the system without the DW (uniform state). The prole is obtained by moving the DW with an external applied eld and then subtracting the Zeeman energy. 36X. Wang, P. Yan, J. Lu, and C. He, Annals of Physics 324, 1815 (2009), arXiv:0809.4311. 37S. Tarasenko, a. Stankiewicz, V. Tarasenko, and J. Ferr e, Journal of Magnetism and Magnetic Materials 189, 19 (1998). 38The same results are obtained with a Gaussian barrier, meaning that the key point is the nite size of the barrier rather than its shape. 39ForD >D c, DW have negative energies and the systems spontaneously breaks into non-uniform spin textures. Appendix A: Maximum torque and equilibrium state In this section we show in more detail how the maximum torque represents an indicator of the equilibrium state. Maximu torque is dened as max 0= Maxf1 1 +2miHe;i 1 +2mi(miHe;i)g=1 0Maxdmi dt ; (A1) over all cells with label i=f1;:::;N =Nx
Nyg. MuMax3.9.329can provide this output automatically if selected. We perform the same simulations as indicated in the main text, without any stopping condition, but simply running fort= 20 ns. Fig. 12(a) shows the average mzcomponent for = 0:2 andBz= 10 mT, while Fig. 12(b) depicts the corresponding maximum torque. We can see that, once the system has reached equilibrium, the maximum torque has dropped to a minimum value. The same results is obtained for dierent damping but the nal maximum torque is dierent. Numerically this value is never zero since it is limited by the code numerical precision and by the system parameters, in particular by damping. Fig. 12(c) represents the maximum torque as function of applied eld for dierent damping. The maximum torque is clearly independent on the applied eld but depends on the damping value. Finally, Fig. 12(d) shows the max torque as function of damping. The maximum torque decreases with damping and it saturates for 0:5 since we have reached the minimum numerical precision of the code29. For higher damping the maximum torque oscillates around this minimum sensibility value, as shown in the inset of Fig. 12(d). The value obtained with these preliminary simulations is used to set a threshold () for the depinning eld simulations in order to identify when the system has reached an equilibrium. Furthermore, additional tests were performed, without putting any max torque condition, but simply running the simulations for a longer time ( t= 80;160 ns) and calculating the depinning eld in order to ensure that the results obtained with these two method were consistent, i.e., that we have actually reached an equilibrium state with the maximum torque condition.11 ������������������-���-���-��� ����(��)<��>⨯��-�α=���(�) ��������������������������� ����(��)��� ������/γ �(��)α=���(�) ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ ◇◇◇◇◇◇◇◇ ○α=����□α=���◇α=��� � �� �� �� �������������������� μ���(��)��� ������/γ �(��)(�) ● ●● ��������������������-������������������� �������α��� ������/γ �(��)(�) ��������������������� ����(��)τ���/γ�(��)α=��� FIG. 12. (a) average mzas function of time. (b) Max torque/ 0(max) as function of time. maxrapidly decreases when the system is at equilibrium. (c) Max torque as function of applied eld for dierent damping. (d) Max torque at equilibrium as function of damping. The inset shows the max torque as function of time for = 0:5. Appendix B: 1D energy barrier As commented in the main text, the pinning eld implemented in the 1D model simulations is obtained by using the shape of the DW energy prole derivative @(x)=@x(beingxthe DW position) and the amplitude of the depinning eld obtained in the full micromagnetic simulations Hsfor the single barrier case. Namely Hdep=Hs@(x) @x N; (B1) where we recall that Nstands for the normalized value. This choice might sound unusual and needs to be justied. In fact, having the DW energy prole, the depinning eld could be simply calculated as20 Hdep=1 20Ms@(x) @x: (B2) This expression is derived by imposing that the derivative of the total DW energy E(x) = 20MsHzx+(x) (Zeeman + internal energy) must be always negative. However, in our case also Ms(x) depends on the DW position and the results obtained with Eq. B2 is dierent from the depinning eld measured in the static single barrier simulations. For this reason we use Eq. B1 which keep the correct barrier shape and has the measured static value. Finally, we recall that equivalent results are obtained by using a simple Gaussian shape for the pinning eld, meaning that the key point is the localized shape of the barrier, rather than its exact form. Appendix C: Dynamical depinning for a moving Domain Wall In this section we show the results for the dynamical depinning eld when the DW is placed at an initial distance of d1= 200 nm from the barrier. In this way the DW hits the pinning with an initial velocity. The d0case corresponds to the DW at rest relaxed just before the barrier and extensively analysed in the main text. Also for this conguration we performed static and dynamic simulations, neglecting or including the DW precessional dynamics respectively. The depinning eld for the d1case is further reduces at small damping, reaching Hd=Hs0:08 (Hd= 9 mT and Hs= 120 mT) at = 0:02. Nevertheless, the depinning eld remains constant in the static simulations independently on the velocity at which the DW hits the barrier. This suggests that, rather than related to the DW velocity, the reduction is again related to the DW precession. When the DW starts from d1it reaches the barrier precessing, thus with a higher displacement from its equilibrium angle, leading to a higher eect of the internal eld.12 ●●●●●● ○○○○○○ ◆◆◆◆◆ ◆ □□□□□ □ ●��(�������) ○��(�������)◆��(������) □��(������) ��� ��� ��� ��� ������������������������ �������α��/�� FIG. 13. Dynamical depinning eld as function of damping for static and dynamic simulations for the d0andd1cases.

2017-05-21

The domain wall depinning field represents the minimum magnetic field needed to move a domain wall, typically pinned by samples' disorder or patterned constrictions. Conventionally, such field is considered independent on the Gilbert damping since it is assumed to be the field at which the Zeeman energy equals the pinning energy barrier (both damping independent). Here, we analyse numerically the domain wall depinning field as function of the Gilbert damping in a system with perpendicular magnetic anisotropy and Dzyaloshinskii-Moriya interaction. Contrary to expectations, we find that the depinning field depends on the Gilbert damping and that it strongly decreases for small damping parameters. We explain this dependence with a simple one-dimensional model and we show that the reduction of the depinning field is related to the internal domain wall dynamics, proportional to the Dzyaloshinskii-Moriya interaction, and the finite size of the pinning barriers.

Dynamical depinning of chiral domain walls

1705.07489v2

arXiv:1409.2340v1 [math.AP] 8 Sep 2014Self-similar solutions of the one-dimensional Landau–Lifshitz–Gilbert equation Susana Gutiérrez1and André de Laire2 Abstract We consider the one-dimensional Landau–Lifshitz–Gilbert (LLG) equation, a model des- cribing the dynamics for the spin in ferromagnetic material s. Our main aim is the analytical study of the bi-parametric family of self-similar solution s of this model. In the presence of damping, our construction provides a family of global solut ions of the LLG equation which are associated to a discontinuous initial data of infinite (t otal) energy, and which are smooth and have finite energy for all positive times. Special emphas is will be given to the behaviour of this family of solutions with respect to the Gilbert dampi ng parameter. We would like to emphasize that our analysis also includes th e study of self-similar so- lutions of the Schrödinger map and the heat flow for harmonic m aps into the 2-sphere as special cases. In particular, the results presented here re cover some of the previously known results in the setting of the 1d-Schrödinger map equation. Keywords and phrases: Landau–Lifshitz–Gilbert equation, Landau–Lifshitz equa tion, ferro- magnetic spin chain, Schrödinger maps, heat-flow for harmon ic maps, self-similar solutions, asymptotics. Contents 1 Introduction and statement of results 2 2 Self-similar solutions of the LLG equation 10 3 Integration of the Serret–Frenet system 12 3.1 Reduction to the study of a second order ODE . . . . . . . . . . . . . . . . . . . 12 3.2 The second-order equation. Asymptotics . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 The second-order equation. Dependence on the parameter s . . . . . . . . . . . . 28 3.3.1 Dependence on α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.2 Dependence on c0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4 Proof of the main results 34 5 Some numerical results 37 6 Appendix 41 1School of Mathematics, University of Birmingham, Edgbasto n, Birmingham, B15 2TT, United Kingdom. E-mail:s.gutierrez@bham.ac.uk 2Laboratoire Paul Painlevé, Université Lille 1, 59655 Ville neuve d’Ascq Cedex, France. E-mail: andre.de-laire@math.univ-lille1.fr 11 Introduction and statement of results In this work we consider the one-dimensional Landau–Lifshi tz–Gilbert equation (LLG) ∂t/vectorm =β/vectorm×/vector mss−α/vectorm×(/vectorm×/vectormss), s∈R, t>0, (LLG) where/vectorm = (m 1,m2,m3) :R×(0,∞)−→S2is the spin vector, β≥0,α≥0,×denotes the usual cross-product in R3, andS2is the unit sphere in R3. Here we have not included the effects of anisotropy or an exter nal magnetic field. The first term on the right in (LLG) represents the exchange interaction, w hile the second one corresponds to the Gilbert damping term and may be considered as a dissipative t erm in the equation of motion. The parameters β≥0andα≥0are the so-called exchange constant and Gilbert damping coe fficient, and take into account the exchange of energy in the system and the effect of damping on the spin chain respectively. Note that, by considering the time -scaling/vectorm(s,t)→/vectorm(s,(α2+β2)1/2t), in what follows we will assume w.l.o.g. that α, β∈[0,1] andα2+β2= 1. (1.1) The Landau–Lifshitz–Gilbert equation was first derived on p henomenological grounds by L. Lan- dau and E. Lifshitz to describe the dynamics for the magnetiz ation or spin /vectorm(s,t)in ferromag- netic materials [24, 11]. The nonlinear evolution equation (LLG) is related to several physical and mathematical problems and it has been seen to be a physica lly relevant model for several magnetic materials [19, 20]. In the setting of the LLG equati on, of particular importance is to consider the effect of dissipation on the spin [27, 7, 6]. The Landau–Lifshitz family of equations includes as specia l cases the well-known heat-flow for harmonic maps and the Schrödinger map equation onto the 2-sphere. Precisely, when β= 0 (and therefore α= 1) the LLG equation reduces to the one-dimensional heat-flow equation for harmonic maps ∂t/vectorm =−/vectorm×(/vectorm×/vectormss) =/vectormss+|/vectorms|2/vectorm (HFHM) (notice that |/vectorm|2= 1, and in particular /vectorm·/vectormss=−|/vectorms|2). The opposite limiting case of the LLG equation (that is α= 0, i.e. no dissipation/damping and therefore β= 1) corresponds to theSchrödinger map equation onto the sphere ∂t/vectorm =/vectorm×/vectormss. (SM) Both special cases have been objects of intense research and w e refer the interested reader to [21, 14, 25, 13] for surveys. Of special relevance is the connection of the LLG equation wi th certain non-linear Schrödinger equations. This connection is established as follows: Let u s suppose that /vectormis the tangent vector of a curve in R3, that is/vectorm =/vectorXs, for some curve /vectorX(s,t)∈R3parametrized by the arc-length. It can be shown [7] that if /vectormevolves under (LLG) and we define the so-called filament funct ionu associated to /vectorX(s,t)by u(s,t) = c(s,t)ei/integraltexts 0τ(σ,t)dσ, (1.2) in terms of the curvature cand torsion τassociated to the curve, then usolves the following non-local non-linear Schrödinger equation with damping iut+(β−iα)uss+u 2/parenleftbigg β|u|2+2α/integraldisplays 0Im(¯uus)−A(t)/parenrightbigg = 0, (1.3) whereA(t)∈Ris a time-dependent function defined in terms of the curvatur e and torsion and their derivatives at the point s= 0. The transformation (1.2) was first considered in the 2undamped case by Hasimoto in [18]. Notice that if α= 0, equation (1.3) can be transformed into the well-known completely integrable cubic Schröding er equation. The main purpose of this paper is the analytical study of self -similar solutions of the LLG equation of the form /vectorm(s,t) =/vector m/parenleftbiggs√ t/parenrightbigg , (1.4) for some profile /vector m:R→S2, with emphasis on the behaviour of these solutions with resp ect to the Gilbert damping parameter α∈[0,1]. Forα= 0, self-similar solutions have generated considerable inte rest [22, 21, 4, 15, 9]. We are not aware of any other study of such solutions for α >0in the one dimensional case (see [10] for a study of self-similar solutions of the harmonic map flow in higher dimensions). However, Lipniacki [26] has studied self-similar solutions for a rel ated model with nonconstant arc-length. On the other hand, little is known analytically about the effe ct of damping on the evolution of a one-dimensional spin chain. In particular, Lakshmanan and Daniel obtained an explicit solitary wave solution in [7, 6] and demonstrated the dampin g of the solution in the presence of dissipation in the system. In this setting, we would like t o understand how the dynamics of self-similar solutions to this model is affected by the intro duction of damping in the equations governing the motion of these curves. As will be shown in Section 2 self-similar solutions of (LLG) of the type (1.4) constitute a bi-parametric family of solutions {/vector mc0,α}c0,αgiven by /vectormc0,α(s,t) =/vector mc0,α/parenleftbiggs√ t/parenrightbigg , c 0>0, α∈[0,1], (1.5) where/vector mc0,αis the solution of the Serret–Frenet equations /vector m′=c/vector n, /vector n′=−c/vector m+τ/vectorb, /vectorb′=−τ/vector n,(1.6) with curvature and torsion given respectively by cc0,α(s) =c0e−αs2 4, τc0,α(s) =βs 2, (1.7) and initial conditions /vector mc0,α(0) = (1,0,0), /vector nc0,α(0) = (0,1,0),/vectorbc0,α(0) = (0,0,1). (1.8) The first result of this paper is the following: Theorem 1.1. Letα∈[0,1],c0>01and/vector mc0,αbe the solution of the Serret–Frenet system (1.6)with curvature and torsion given by (1.7)and initial conditions (1.8). Define/vectormc0,α(s,t)by /vectormc0,α(s,t) =/vector mc0,α/parenleftbiggs√ t/parenrightbigg , t> 0. Then, 1The case c0= 0corresponds to the constant solution for (LLG), that is /vectormc0,α(s,t) =/vector m/parenleftbiggs√ t/parenrightbigg = (1,0,0),∀α∈[0,1]. 3(i) The function /vectormc0,α(·,t)is a regular C∞(R;S2)-solution of (LLG) fort>0. (ii) There exist unitary vectors /vectorA± c0,α= (A± j,c0,α)3 j=1∈S2such that the following pointwise convergence holds when tgoes to zero: lim t→0+/vectormc0,α(s,t) = /vectorA+ c0,α,ifs>0, /vectorA− c0,α,ifs<0,(1.9) where/vectorA− c0,α= (A+ 1,c0,α,−A+ 2,c0,α,−A+ 3,c0,α). (iii) Moreover, there exists a constant C(c0,α,p)such that for all t>0 /bardbl/vectormc0,α(·,t)−/vectorA+ c0,αχ(0,∞)(·)−/vectorA− c0,αχ(−∞,0)(·)/bardblLp(R)≤C(c0,α,p)t1 2p, (1.10) for allp∈(1,∞). In addition, if α>0,(1.10) also holds for p= 1. Here,χEdenotes the characteristic function of a set E. The graphics in Figure 1 depict the profile /vector mc0,α(s)for fixedc0= 0.8and the values of α= 0.01,α= 0.2, andα= 0.4. In particular it can be observed how the convergence of /vector mc0,α to/vectorA± c0,αis accelerated by the diffusion α. m1m2m3 (a)α= 0.01 m1m2m3 (b)α= 0.2 m1m2m3 (c)α= 0.4 Figure 1: The profile /vector mc0,αforc0= 0.8and different values of α. Notice that the initial condition /vectormc0,α(s,0) =/vectorA+ c0,αχ(0,∞)(s)+/vectorA− c0,αχ(−∞,0)(s), (1.11) has a jump singularity at the point s= 0whenever the vectors /vectorA+ c0,αand/vectorA− c0,αsatisfy /vectorA+ c0,α/ne}ationslash=/vectorA− c0,α. In this situation (and we will be able to prove analytically t his is the case at least for certain ranges of the parameters αandc0, see Proposition 1.5 below), Theorem 1.1 provides a bi-para metric family of global smooth solutions of (LLG) associated to a di scontinuous singular initial data (jump-singularity). 4As has been already mentioned, in the absence of damping ( α= 0), singular self-similar solutions of the Schrödinger map equation were previously o btained in [15], [22] and [4]. In this framework, Theorem 1.1 establishes the persistence of a jum p singularity for self-similar solutions in the presence of dissipation. Some further remarks on the results stated in Theorem 1.1 are in order. Firstly, from the self-similar nature of the solutions /vectormc0,α(s,t)and the Serret–Frenet equations (1.6), it follows that the curvature and torsion associated to these solution s are of the self-similar form and given by cc0,α(s,t) =c0√ te−αs2 4t andτc0,α(s,t) =βs 2√ t. (1.12) As a consequence, the total energy E(t)of the spin /vectormc0,α(s,t)found in Theorem 1.1 is expressed as E(t) =1 2/integraldisplay∞ −∞|/vectorms(s,t)|2ds=1 2/integraldisplay∞ −∞c2 c0,α(s,t)ds =1 2/integraldisplay∞ −∞/parenleftbiggc0√ te−αs2 4t/parenrightbigg2 ds=c2 0/radicalbiggπ αt, α> 0, t>0. (1.13) It is evident from (1.13) that the total energy of the spin cha in at the initial time t= 0is infinite, while the total energy of the spin becomes finite for all posit ive times, showing the dissipation of energy in the system in the presence of damping. Secondly, it is also important to remark that in the setting o f Schrödinger equations, for fixed α∈[0,1]andc0>0, the solution /vectormc0,α(s,t)of (LLG) established in Theorem 1.1 is associated through the Hasimoto transformation (1.2) to the filament fu nction uc0,α(s,t) =c0√ te(−α+iβ)s2 4t, (1.14) which solves iut+(β−iα)uss+u 2/parenleftbigg β|u|2+2α/integraldisplays 0Im(¯uus)−A(t)/parenrightbigg = 0,withA(t) =βc2 0 t(1.15) and is such that at initial time t= 0 uc0,α(s,0) = 2c0/radicalbig π(α+iβ)δ0. Hereδ0denotes the delta distribution at the point s= 0and√zdenotes the square root of a complex number zsuch that Im(√z)>0. Notice that the solution uc0,α(s,t)is very rough at initial time, and in particular uc0,α(s,0) does not belong to the Sobolev class Hsfor anys≥0. Therefore, the standard arguments (that is a Picard iteration scheme based on Strichartz estimates a nd Sobolev-Bourgain spaces) cannot be applied at least not in a straightforward way to study the l ocal well-posedness of the initial value problem for the Schrödinger equations (1.15). The exi stence of solutions of the Scrödinger equations (1.15) associated to an initial data proportiona l to a Dirac delta opens the question of developing a well-posedness theory for Schrödinger equa tions of the type considered here to include initial data of infinite energy. This question was ad dressed by A. Vargas and L. Vega in [29] and A. Grünrock in [12] in the case α= 0and whenA(t) = 0 (see also [2] for a related problem), but we are not aware of any results in this setting w henα >0(see [14] for related well-posedness results in the case α >0for initial data in Sobolev spaces of positive index). Notice that when α>0, the solution (1.14) has infinite energy at the initial time, however the 5energy becomes finite for any t>0. Moreover, as a consequence of the exponential decay in the space variable when α>0,uc0,α(t)∈Hm(R), for allt>0andm∈N. Hence these solutions do not fit into the usual functional framework for solutions of t he Schrödinger equations (1.15). As already mentioned, one of the main goals of this paper is to study both the qualitative and quantitative effect of the damping parameter αand the parameter c0on the dynamical behaviour of the family {/vectormc0,α}c0,αof self-similar solutions of (LLG) found in Theorem 1.1. Pre cisely, in an attempt to fully understand the regularization of the solut ion at positive times close to the initial timet= 0, and to understand how the presence of damping affects the dyn amical behaviour of these self-similar solutions, we aim to give answers to the f ollowing questions: Q1: Can we obtain a more precise behaviour of the solutions /vector mc0,α(s,t)at positive times tclose to zero? Q2: Can we understand the limiting vectors /vectorA± c0,αin terms of the parameters c0andα? In order to address our first question, we observe that, due to the self-similar nature of these solutions (see (1.5)), the behaviour of the family of soluti ons/vectormc0,α(s,t)at positive times close to the initial time t= 0is directly related to the study of the asymptotics of the ass ociated profile /vector mc0,α(s)for large values of s. In addition, the symmetries of /vector mc0,α(s)(see Theorem 1.2 below) allow to reduce ourselves to obtain the behaviour of the profi le/vector mc0,α(s)for large positive values of the space variable. The precise asymptotics of the profile is given in the following theorem. Theorem 1.2 (Asymptotics) .Letα∈[0,1],c0>0and{/vector mc0,α,/vector nc0,α,/vectorbc0,α}be the solution of the Serret–Frenet system (1.6)with curvature and torsion given by (1.7)and initial conditions (1.8). Then, (i) (Symmetries). The components of /vector mc0,α(s),/vector nc0,α(s)and/vectorbc0,α(s)satisfy respectively that •m1,c0,α(s)is an even function, and mj,c0,α(s)is an odd function for j∈ {2,3}. •n1,c0,α(s)andb1,c0,α(s)are odd functions, while nj,c0,α(s)andbj,c0,α(s)are even func- tions forj∈ {2,3}. (ii) (Asymptotics). There exist an unit vector /vectorA+ c0,α∈S2and/vectorB+ c0,α∈R3such that the following asymptotics hold for all s≥s0= 4/radicalbig 8+c2 0: /vector mc0,α(s) =/vectorA+ c0,α−2c0 s/vectorB+ c0,αe−αs2/4(αsin(/vectorφ(s))+βcos(/vectorφ(s))) −2c2 0 s2/vectorA+ c0,αe−αs2/2+O/parenleftBigg e−αs2/4 s3/parenrightBigg , (1.16) /vector nc0,α(s) =/vectorB+ c0,αsin(/vectorφ(s))+2c0 s/vectorA+ c0,ααe−αs2/4+O/parenleftBigg e−αs2/4 s2/parenrightBigg , (1.17) /vectorbc0,α(s) =/vectorB+ c0,αcos(/vectorφ(s))+2c0 s/vectorA+ c0,αβe−αs2/4+O/parenleftBigg e−αs2/4 s2/parenrightBigg . (1.18) Here,sin(/vectorφ)andcos(/vectorφ)are understood acting on each of the components of /vectorφ= (φ1,φ2,φ3), with φj(s) =aj+β/integraldisplays2/4 s2 0/4/radicalbigg 1+c2 0e−2ασ σdσ, j∈ {1,2,3}, (1.19) 6for some constants a1,a2,a3∈[0,2π), and the vector /vectorB+ c0,αis given in terms of /vectorA+ c0,α= (A+ j,c0,α)3 j=1by /vectorB+ c0,α= ((1−(A+ 1,c0,α)2)1/2,(1−(A+ 2,c0,α)2)1/2,(1−(A+ 3,c0,α)2)1/2). As we will see in Section 2, the convergence and rate of conver gence of the solutions /vectormc0,α(s,t) of the LLG equation established in parts (ii)and(iii)of Theorem 1.1 will be obtained as a con- sequence of the more refined asymptotic analysis of the assoc iated profile given in Theorem 1.2. With regard to the asymptotics of the profile established in p art(ii)of Theorem 1.2, it is important to mention the following: (a) The errors in the asymptotics in Theorem 1.2- (ii)depend only on c0. In other words, the bounds for the errors terms are independent of α∈[0,1]. More precisely, we use the notationO(f(s))to denote a function for which exists a constant C(c0)>0depending on c0, but independent on α, such that |O(f(s))| ≤C(c0)|f(s)|,for alls≥s0. (1.20) (b) The terms /vectorA+ c0,α,/vectorB+ c0,α,B+ jsin(aj)andB+ jcos(aj),j∈ {1,2,3}, and the error terms in Theorem 1.2- (ii)depend continuously on α∈[0,1](see Subsection 3.3 and Corollary 3.14). Therefore, the asymptotics (1.16)–(1.18) show how the profi le/vector mc0,αconverges to /vector mc0,0as α→0+and to/vector mc0,1asα→1−. In particular, we recover the asymptotics for /vector mc0,0given in [15]. (c) We also remark that using the Serret–Frenet formulae and the asymptotics in Theorem 1.2- (ii), it is straightforward to obtain the asymptotics for the der ivatives of/vectormc0,α(s,t). (d) Whenα= 0and for fixed j∈ {1,2,3}, we can write φjin (1.19) as φj(s) =aj+s2 4+c2 0ln(s)+C(c0)+O/parenleftbigg1 s2/parenrightbigg , and we recover the logarithmic contribution in the oscillat ion previously found in [15]. Moreover, in this case the asymptotics in part (ii)represents an improvement of the one established in Theorem 1 in [15]. Whenα>0,φjbehaves like φj(s) =aj+βs2 4+C(α,c0)+O/parenleftBigg e−αs2/2 αs2/parenrightBigg , (1.21) and there is no logarithmic correction in the oscillations i n the presence of damping. Consequently, the phase function /vectorφdefined in (1.19) captures the different nature of the oscillatory character of the solutions in both the absence a nd the presence of damping in the system of equations. (e) Whenα= 1, there exists an explicit formula for /vector mc0,1,/vector nc0,1and/vectorbc0,1, and in particular we have explicit expressions for the vectors /vectorA± c0,1in terms of the parameter c0>0in the asymptotics given in part (ii). See Appendix. 7(f) At first glance, one might think that the term −2c2 0/vectorA+ c0,αe−αs2/2/s2in (1.16) could be included in the error term O(e−αs2/4/s3). However, we cannot do this because e−αs2/2 s2>e−αs2/4 s3, for all2≤s≤/parenleftbigg2 3α/parenrightbigg1/2 , α∈(0,1/8], (1.22) and in our notation the big- Omust be independent of α. (The exact interval where the inequality in (1.22) holds can be determined using the so-ca lled Lambert Wfunction.) (g) Let/vectorB+ c0,α,sin= (Bjsin(aj))3 j=1,/vectorB+ c0,α,cos= (Bjcos(aj))3 j=1. Then the orthogonality of /vector mc0,α,/vector nc0,αand/vectorbc0,αtogether with the asymptotics (1.16)–(1.18) yield /vectorA+ c0,α·/vectorB+ c0,α,sin=/vectorA+ c0,α·/vectorB+ c0,α,cos=/vectorB+ c0,α,sin·/vectorB+ c0,α,cos= 0, which gives relations between the phases. (h) Finally, the amplitude of the leading order term control ling the wave-like behaviour of the solution/vector mc0,α(s)around/vectorA± c0,αfor values of ssufficiently large is of the order c0e−αs2/4/s, from which one observes how the convergence of the solution t o its limiting values /vectorA± c0,αis accelerated in the presence of damping in the system. See Fig ure 1. We conclude the introduction by stating the results answeri ng the second of our questions. Pre- cisely, Theorems 1.3 and 1.4 below establish the dependence of the vectors /vectorA± c0,αin Theorem 1.1 with respect to the parameters αandc0. Theorem 1.3 provides the behaviour of the limiting vector/vectorA+ c0,αfor a fixed value of α∈(0,1)and “small” values of c0>0, while Theorem 1.4 states the behaviour of /vectorA+ c0,αfor fixedc0>0andαclose to the limiting values α= 0andα= 1. Recall that/vectorA− c0,αis expressed in terms of the coordinates of /vectorA+ c0,αas /vectorA− c0,α= (A+ 1,c0,α,−A+ 2,c0,α,−A+ 3,c0,α) (1.23) (see part (ii)of Theorem 1.1). Theorem 1.3. Letα∈[0,1],c0>0, and/vectorA+ c0,α= (A+ j,c0,α)3 j=1be the unit vector given in Theorem 1.2. Then /vectorA+ c0,αis a continuous function of c0>0. Moreover, if α∈(0,1]the following inequalities hold true: |A+ 1,c0,α−1| ≤c2 0π α/parenleftbigg 1+c2 0π 8α/parenrightbigg , (1.24) /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA+ 2,c0,α−c0/radicalbig π(1+α)√ 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤c2 0π 4+c2 0π α√ 2/parenleftBigg 1+c2 0π 8+c0/radicalbig π(1+α) 2√ 2/parenrightBigg +/parenleftbiggc2 0π 2√ 2α/parenrightbigg2 ,(1.25) /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA+ 3,c0,α−c0/radicalbig π(1−α)√ 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤c2 0π 4+c2 0π α√ 2/parenleftBigg 1+c2 0π 8+c0/radicalbig π(1−α) 2√ 2/parenrightBigg +/parenleftbiggc2 0π 2√ 2α/parenrightbigg2 .(1.26) The following result provides an approximation of the behav iour of/vectorA+ c0,αfor fixedc0>0and values of the Gilbert parameter close to 0and1. Theorem 1.4. Letc0>0,α∈[0,1]and/vectorA+ c0,αbe the unit vector given in Theorem 1.2. Then /vectorA+ c0,αis a continuous function of αin[0,1], and the following inequalities hold true: |/vectorA+ c0,α−/vectorA+ c0,0| ≤C(c0)√α|ln(α)|,for allα∈(0,1/2], (1.27) |/vectorA+ c0,α−/vectorA+ c0,1| ≤C(c0)√ 1−α,for allα∈[1/2,1]. (1.28) Here,C(c0)is a positive constant depending on c0but otherwise independent of α. 8As a by-product of Theorems 1.3 and 1.4, we obtain the followi ng proposition which asserts that the solutions /vectormc0,α(s,t)of the LLG equation found in Theorem 1.1 are indeed associate d to a discontinuous initial data at least for certain ranges o fαandc0. Proposition 1.5. With the same notation as in Theorems 1.1 and 1.2, the followi ng statements hold: (i) For fixed α∈(0,1)there exists c∗ 0>0depending on αsuch that /vectorA+ c0,α/ne}ationslash=/vectorA− c0,α for allc0∈(0,c∗ 0). (ii) For fixed c0>0, there exists α∗ 0>0small enough such that /vectorA+ c0,α/ne}ationslash=/vectorA− c0,α for allα∈(0,α∗ 0). (iii) For fixed 0<c0/ne}ationslash=k√πwithk∈N, there exists α∗ 1>0with1−α∗ 1>0small enough such that /vectorA+ c0,α/ne}ationslash=/vectorA− c0,α for allα∈(α∗ 1,1). Remark 1.6. Based on the numerical results in Section 5, we conjecture th at/vectorA+ c0,α/ne}ationslash=/vectorA− c0,αfor allα∈[0,1)andc0>0. We would like to point out that some of our results and their pr oofs combine and extend several ideas previously introduced in [15] and [16]. The ap proach we use in the proof of the main results in this paper is based on the integration of the S erret–Frenet system of equations via a Riccati equation, which in turn can be reduced to the stu dy of a second order ordinary differential equation given by f′′(s)+s 2(α+iβ)f′(s)+c2 0 4e−αs2 2f(s) = 0 (1.29) when the curvature and torsion are given by (1.7). Unlike in the undamped case, in the presence of damping no exp licit solutions are known for equation (1.29) and the term containing the exponential in the equation (1.29) makes it difficult to use Fourier analysis methods to study analytical ly the behaviour of the solutions to this equation. The fundamental step in the analysis of the be haviour of the solutions of (1.29) consists in introducing new auxiliary variables z,handydefined by z=|f|2, y= Re(¯ff′)andh= Im(¯ff′) in terms of solutions fof (1.29), and studying the system of equations satisfied by t hese key quantities. As we will see later on, these variables are the “ natural” ones in our problem, in the sense that the components of the tangent, normal and binorma l vectors can be written in terms of these quantities. It is important to emphasize that, in or der to obtain error bounds in the asymptotic analysis independent of the damping parameter α(and hence recover the asymptotics whenα= 0andα= 1as particular cases), it will be fundamental to exploit the c ancellations due to the oscillatory character of z,yandh. The outline of this paper is the following. Section 2 is devot ed to the construction of the family of self-similar solutions {/vectormc0,α}c0,αof the LLG equation. In Section 3 we reduce the study of the properties of this family of self-similar solutions to that of the properties of the solutions of the complex second order complex ODE (1.29). This analysis is of independent interest. Section 4 contains the proofs of the main results of this paper as a cons equence of those established in 9Section 3. In Section 5 we give provide some numerical result s for/vectorA+ c0,α, as a function of α∈[0,1] andc0>0, which give some inside for the scattering problem and justi fy Remark 1.6. Finally, we have included the study of the self-similar solutions of t he LLG equation in the case α= 1 in Appendix. Acknowledgements. S. Gutiérrez and A. de Laire were supported by the British proj ect “Singular vortex dynamics and nonlinear Schrödinger equat ions” (EP/J01155X/1) funded by EPSRC. S. Gutiérrez was also supported by the Spanish projec ts MTM2011-24054 and IT641- 13. Both authors would like to thank L. Vega for many enlightening conversations and for his continuous support. 2 Self-similar solutions of the LLG equation First we derive what we will refer to as the geometric represe ntation of the LLG equation. To this end, let us assume that /vectorm(s,t) =/vectorXs(s,t)for some curve /vectorX(s,t)inR3parametrized with respect to the arc-length with curvature c(s,t)and torsion τ(s,t). Then, using the Serret–Frenet system of equations (1.6), we have /vectormss= cs/vectorn+c(−c/vectorn+τ/vectorb), and thus we can rewrite (LLG) as ∂t/vectorm =β(cs/vectorb−cτ/vectorn)+α(cτ/vectorb+cs/vectorn), (2.1) in terms of intrinsic quantities c,τand the Serret–Frenet trihedron {/vectorm,/vectorn,/vectorb}. We are interested in self-similar solutions of (LLG) of the f orm /vectorm(s,t) =/vector m/parenleftbiggs√ t/parenrightbigg (2.2) for some profile /vector m:R−→S2. First, notice that due to the self-similar nature of /vectorm(s,t)in (2.2), from the Serret–Frenet equations (1.6) it follows that the u nitary normal and binormal vectors and the associated curvature and torsion are self-similar a nd given by /vectorn(s,t) =/vector n/parenleftbiggs√ t/parenrightbigg ,/vectorb(s,t) =/vectorb/parenleftbiggs√ t/parenrightbigg , (2.3) c(s,t) =1√ tc/parenleftbiggs√ t/parenrightbigg andτ(s,t) =1√ tτ/parenleftbiggs√ t/parenrightbigg . (2.4) Assume that /vectorm(s,t)is a solution of the LLG equation, or equivalently of its geom etric version (2.1). Then, from (2.2)–(2.4) it follows that the Serret–Fr enet trihedron {/vector m(·),/vector n(·),/vectorb(·)}solves −s 2c/vector n=β(c′/vectorb−cτ/vector n)+α(cτ/vectorb+c′/vector n), (2.5) As a consequence, −s 2c=αc′−βcτ andβc′+αcτ= 0. Thus, we obtain c(s) =c0e−αs2 4andτ(s) =βs 2, (2.6) 10for some positive constant c0(recall that we are assuming w.l.o.g. that α2+β2= 1). Therefore, in view of (2.4), the curvature and torsion associated to a se lf-similar solution of (LLG) of the form (2.2) are given respectively by c(s,t) =c0√ te−αs2 4tandτ(s,t) =βs 2t, c 0>0. (2.7) Notice that given (c,τ)as above, for fixed time t>0one can solve the Serret–Frenet system of equations to obtain the solution up to a rigid motion in the sp ace which in general may depend ont. As a consequence, and in order to determine the dynamics of t he spin chain, we need to find the time evolution of the trihedron {/vectorm(s,t),/vectorn(s,t),/vectorb(s,t)}at some fixed point s∗∈R. To this end, from the above expressions of the curvature and t orsion associated to /vectorm(s,t)and evaluating the equation (2.1) at the point s∗= 0, we obtain that /vectormt(0,t) =/vector0. On the other hand, differentiating the geometric equation (2.1) with res pect tos, and using the Serret–Frenet equations (1.6) together with the compatibility condition /vectormst=/vectormts, we get the following relation for the time evolution of the normal vector c/vectornt=β(css/vectorb+c2τ/vectorm−cτ2/vectorb)+α((cτ)s/vectorb−ccs/vectorm+csτ/vectorb). The evaluation of the above identity at s∗= 0together with the expressions for the curvature and torsion in (2.7) yield /vectornt(0,t) =/vector0. The above argument shows that /vectormt(0,t) =/vector0, /vectornt(0,t) =/vector0and/vectorbt(0,t) = (/vectorm×/vectorn)t(0,t) =/vector0. Therefore we can assume w.l.o.g. that /vectorm(0,t) = (1,0,0), /vectorn(0,t) = (0,1,0)and/vectorb(0,t) = (0,0,1), and in particular /vector m(0) =/vectorm(0,1) = (1,0,0), /vector n(0) =/vectorn(0,1) = (0,1,0),and/vectorb(0) =/vectorb(0,1) = (0,0,1).(2.8) Givenα∈[0,1]andc0>0, from the theory of ODE’s, it follows that there exists a uniq ue {/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)} ∈/parenleftbig C∞(R;S2)/parenrightbig3solution of the Serret–Frenet equations (1.6) with curvature and torsion (2.6) and initial conditions (2.8) su ch that /vector mc0,α⊥/vector nc0,α, /vector mc0,α⊥/vectorbc0,α, /vector nc0,α⊥/vectorbc0,α and |/vector mc0,α|2=|/vector nc0,α|2=|/vectorbc0,α|2= 1. Define/vectormc0,α(s,t)as /vectormc0,α(s,t) =/vector mc0,α/parenleftbiggs√ t/parenrightbigg . (2.9) Then,/vector mc0,α(·,t)∈ C∞/parenleftbig R;S2/parenrightbig for allt>0, and bearing in mind both the relations in (2.3)–(2.4) and the fact that the vectors {/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)}satisfy the identity (2.5), a straightfor- ward calculation shows that /vector mc0,α(·,t)is a regular C∞(R;S2)-solution of the LLG equation for allt>0. Notice that the case c0= 0yields the constant solution /vector m0,α(s,t) = (1,0,0). Therefore in what follows we will assume that c0>0. The rest of the paper is devoted to establish analytical prop erties of the solutions {/vectormc0,α(s,t)}c0,α defined by (2.9) for fixed α∈[0,1]andc0>0. As already mentioned, due to the self-similar nature of these solutions, it suffices to study the properties of the associated profile /vector mc0,α(·)or, equivalently, of the solution {/vector mc0,α,/vector nc0,α,/vectorbc0,α}of the Serret–Frenet system (1.6) with curvature and torsion given by (2.6) and initial conditions (2.8). As w e will continue to see, the analysis of the profile solution {/vector mc0,α,/vector nc0,α,/vectorbc0,α}can be reduced to the study of the properties of the solutions of a certain second order complex differential equ ation. 113 Integration of the Serret–Frenet system 3.1 Reduction to the study of a second order ODE Classical changes of variables from the differential geomet ry of curves allow us to reduce the nine equations in the Serret–Frenet system into three complex-v alued second order equations (see [8, 28, 23]). Theses changes of variables are related to ster eographic projection and this approach was also used in [15]. However, their choice of stereographi c projection has a singularity at the origin, which leads to an indetermination of the initial con ditions of some of the new variables. For this reason, we consider in the following lemma a stereog raphic projection that is compatible with the initial conditions (2.8). Although the proof of the lemma below is a slight modification of that in [23, Subsections 2.12 and 7.3], we have included it s proof here both for the sake of completeness and to clarify to the unfamiliar reader how the integration of the Frenet equations can be reduced to the study of a second order differential equa tion. Lemma 3.1. Let/vector m= (mj(s))3 j=1,/vector n= (nj(s))3 j=1and/vectorb= (bj(s))3 j=1be a solution of the Serret– Frenet equations (1.6)with positive curvature cand torsion τ. Then, for each j∈ {1,2,3}the function fj(s) =e1 2/integraltexts 0c(σ)ηj(σ)dσ,withηj(s) =(nj(s)+ibj(s)) 1+mj(s), solves the equation f′′ j(s)+/parenleftbigg iτ(s)−c′(s) c(s)/parenrightbigg f′ j(s)+c2(s) 4fj(s) = 0, (3.1) with initial conditions fj(0) = 1, f′ j(0) =c(0)(nj(0)+ibj(0)) 2(1+mj(0)). Moreover, the coordinates of /vector m,/vector nand/vectorbare given in terms of fjandf′ jby mj(s) = 2/parenleftBigg 1+4 c(s)2/vextendsingle/vextendsingle/vextendsingle/vextendsinglef′ j(s) fj(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg−1 −1, nj(s)+ibj(s) =4f′ j(s) c(s)fj(s)/parenleftBigg 1+4 c(s)2/vextendsingle/vextendsingle/vextendsingle/vextendsinglef′ j(s) fj(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg−1 . (3.2) The above relations are valid at least as long as mj>−1and|fj|>0. Proof. For simplicity, we omit the index j. The proof relies on several transformations that are rather standard in the study of curves. First we define the com plex function N= (n+ib)ei/integraltexts 0τ(σ)dσ. (3.3) ThenN′=iτN+ (n′+ib′)ei/integraltexts 0τ(σ)dσ. On the other hand, the Serret–Frenet equations imply that n′+ib′=−cm−iτNe−i/integraltexts 0τ(σ)dσ. Therefore, setting ψ=cei/integraltexts 0τ(σ)dσ, we get N′=−ψm. (3.4) Using again the Serret–Frenet equations, we also obtain m′=1 2(ψN+ψN). (3.5) 12Let us consider now the auxiliary function ϕ=N 1+m. (3.6) Differentiating and using (3.4), (3.5) and (3.6) ϕ′=N′ 1+m−Nm′ (1+m)2 =N′ 1+m−ϕm′ 1+m =−ϕ2ψ 2−ψ 2(1+m)(2m+ϕN). Noticing that we can recast the relation m2+n2+b2= 1asNN= (1−m)(1+m)and recalling the definition of ϕin (3.6), we have ϕN= 1−m, so that ϕ′+ϕ2ψ 2+ψ 2= 0. (3.7) Finally, define the stereographic projection of (m,n,b)by η=n+ib 1+m. (3.8) Observe that from the definitions of Nandϕ, respectively in (3.3) and (3.6), we can rewrite η as η=ϕe−i/integraltexts 0τ(σ)dσ, and from (3.7) it follows that ηsolves the Riccati equation η′+iτη+c 2(η2+1) = 0, (3.9) (recall that ψ=cei/integraltexts 0τ(σ)dσ). Finally, setting f(s) =e1 2/integraltexts 0c(σ)η(σ)dσ, (3.10) we get η=2f′ cf(3.11) and equation (3.1) follows from (3.9). The initial conditio ns are an immediate consequence of the definition of ηandfin (3.8) and (3.10). A straightforward calculation shows that the inverse trans formation of the stereographic pro- jection is m=1−|η|2 1+|η|2, n=2Reη 1+|η|2, b=2Imη 1+|η|2, so that we obtain (3.2) using (3.11) and the above identities . Going back to our problem, Lemma 3.1 reduces the analysis of t he solution {/vector m,/vector n,/vectorb}of the Serret–Frenet system (1.6) with curvature and torsion give n by (2.6) and initial conditions (2.8) to the study of the second order differential equation f′′(s)+s 2(α+iβ)f′(s)+c2 0 4e−αs2/2f(s) = 0, (3.12) 13with three initial conditions: For (m1,n1,b1) = (1,0,0)the associated initial condition for f1is f1(0) = 1, f′ 1(0) = 0, (3.13) for(m2,n2,b2) = (0,1,0)is f2(0) = 1, f′ 2(0) =c0 2, (3.14) and for(m3,n3,b3) = (0,0,1)is f3(0) = 1, f′ 3(0) =ic0 2. (3.15) It is important to notice that, by multiplying (3.12) by ¯f′and taking the real part, it is easy to see that d ds/bracketleftbigg1 2/parenleftbigg eαs2 2|f′|2+c2 0 4|f|2/parenrightbigg/bracketrightbigg = 0. Thus, E(s) :=1 2/parenleftbigg eαs2 2|f′|2+c2 0 4|f|2/parenrightbigg =E0,∀s∈R, (3.16) withE0a constant defined by the value of E(s)at some point s0∈R. The conservation of the energyE(s)allows us to simplify the expressions of mj,njandbjforj∈ {1,2,3}in the formulae (3.2) in terms of the solution fjto (3.12) associated to the initial conditions (3.13)–(3.1 5). Indeed, on the one hand notice that the energies associated t o the initial conditions (3.13)– (3.15) are respectively E0,1=c2 0 8, E 0,2=c2 0 4andE0,3=c2 0 4. (3.17) On the other hand, from (3.16), it follows that /parenleftBigg 1+4 c2 0e−αs2 2|f′ j|2(s) |fj|2(s)/parenrightBigg−1 =c2 0 8E0,j|fj|2(s), j∈ {1,2,3}. Therefore, from (3.17), the above identity and formulae (3. 2) in Lemma 3.1, we conclude that m1(s) = 2|f1(s)|2−1, n 1(s)+ib1(s) =4 c0eαs2/4¯f1(s)f′ 1(s), (3.18) mj(s) =|fj(s)|2−1, n j(s)+ibj(s) =2 c0eαs2/4¯fj(s)f′ j(s), j∈ {2,3}. (3.19) The above identities give the expressions of the tangent, no rmal and binormal vectors in terms of the solutions {fj}3 j=1of the second order differential equation (3.12) associated to the initial conditions (3.13)–(3.15). By Lemma 3.1, the formulae (3.18) and (3.19) are valid as long a smj>−1, which is equivalent to the condition |fj| /ne}ationslash= 0. As shown in Appendix, for α= 1there is˜s>0such thatmj(˜s) =−1 and then (3.18) and (3.19) are (a priori) valid just in a bound ed interval. However, the trihedron {/vector m,/vector n,/vectorb}is defined globally and fjcan also be extended globally as the solution of the linear equation (3.12). Then, it is simple to verify that the functi ons given by the l.h.s. of formulae (3.18) and (3.19) satisfy the Serret–Frenet system and henc e, by the uniqueness of the solution, the formulae (3.18) and (3.19) are valid for all s∈R. 143.2 The second-order equation. Asymptotics In this section we study the properties of the complex-value d equation f′′(s)+s 2(α+iβ)f′(s)+c2 0 4f(s)e−αs2/2= 0, (3.20) for fixedc0>0,α∈[0,1),β >0such thatα2+β2= 1. We begin noticing that in the caseα= 0, the solution can be written explicitly in terms of paraboli c cylinder functions or confluent hypergeometric functions (see [1]). Another anal ytical approach using Fourier analysis techniques has been taken in [15], leading to the asymptotic s f(s) =C1ei(c2 0/2)ln(s)+C2e−is2/4 se−i(c2 0/2)ln(s)+O(1/s2), (3.21) ass→ ∞, where the constants C1,C2andO(1/s2)depend on the initial conditions and c0. Forα= 1, equation (3.20) can be also solved explicitly and the solut ion is given by f(s) =2f′(0) c0sin/parenleftbiggc0 2/integraldisplays 0e−σ2/4dσ/parenrightbigg +f(0)cos/parenleftbiggc0 2/integraldisplays 0e−σ2/4dσ/parenrightbigg . In the case α∈(0,1), one cannot compute the solutions of (3.20) in terms of known functions and we will follow a more analytical analysis. In contrast wi th the situation when α= 0, it is far from evident to use Fourier analysis to study (3.20) when α>0. For the rest of this section we will assume that α∈[0,1). In addition, we will also assume that s>0and we will develop the asymptotic analysis necessary to est ablish part (ii)of Theorem 1.2. At this point, it is important to recall the expressions give n in (3.18)–(3.19) for the coordinates of the tangent, normal and binormal vectors associated to ou r family of solutions of the LLG equation in terms f. Bearing this in mind, we observe that the study of the asympto tic behaviour of these vectors are dictated by the asymptotic behaviour of the variables z=|f|2, y= Re(¯ff′),andh= Im(¯ff′) (3.22) associated to the solution fof (3.20). As explained in the remark (a) after Theorem 1.2, we need to wo rk with remainder terms that are independent of α. To this aim, we proceed in two steps: first we found uniform es timates forα∈[0,1/2]in Propositions 3.2 and 3.3, then we treat the case α∈[1/2,1)in Lemma 3.6. In Subsection 3.3 we provide some continuity results that allo ws us to take α→1−and give the full statement in Corollary 3.14. Finally, notice that thes e asymptotics lead to the asymptotics for the original equation (3.20) (see Remark 3.9). We begin our analysis by establishing the following: Proposition 3.2. Letc0>0,α∈[0,1),β >0such thatα2+β2= 1, andfbe a solution of (3.20). Define z,yandhasz=|f|2andy+ih=¯ff′. Then (i) There exists E0≥0such that the identity 1 2/parenleftbigg eαs2 2|f′|2+c2 0 4|f|2/parenrightbigg =E0 holds true for all s∈R. In particular, f,f′,z,yandhare bounded functions. Moreover, for alls∈R |f(s)| ≤√8E0 c0,|f′(s)| ≤/radicalbig 2E0e−αs2/4, (3.23) |z(s)| ≤8E0 c2 0and|h(s)|+|y(s)| ≤8E0 c0e−αs2/4. (3.24) 15(ii) The limit z∞:= lim s→∞z(s) exists. (iii) Letγ:= 2E0−c2 0z∞/2ands0= 4/radicalbig 8+c2 0. For alls≥s0, we have z(s)−z∞=−4 s(αy+βh)−4γ s2e−αs2/2+R0(s), (3.25) where |R0(s)| ≤C(E0,c0)e−αs2/4 s3. (3.26) Proof. Part(i)is just the conservation of energy proved in (3.16). Next, us ing the conservation law in part (i), we obtain that the variables {z,y,h}solve the first-order real system z′= 2y, (3.27) y′=βs 2h−αs 2y+e−αs2/2/parenleftbigg 2E0−c2 0 2z/parenrightbigg , (3.28) h′=−βs 2y−αs 2h. (3.29) To show (ii), plugging (3.27) into (3.29) and integrating from 0to somes>0we obtain z(s)−1 s/integraldisplays 0z(σ)dσ=−4 βs/parenleftbigg h(s)−h(0)+α 2/integraldisplays 0σh(σ)dσ/parenrightbigg . (3.30) Also, using the above identity, d ds/parenleftbigg1 s/integraldisplays 0z(σ)dσ/parenrightbigg =−4 βs2/parenleftbigg h(s)−h(0)+α 2/integraldisplays 0σh(σ)dσ/parenrightbigg . (3.31) Now, since from part (i)|h(s)| ≤8E0 c0e−αs2/4, bothhandα/integraltexts 0σh(σ)dσare bounded functions, thus from (3.31) it follows that the limit of1 s/integraltexts 0zexists, ass→ ∞. Hence (3.30) and previous observations conclude that the limit z∞:= lims→∞z(s)exists and furthermore z∞:= lim s→∞z(s) = lim s→∞1 s/integraldisplays 0z(σ). (3.32) We continue to prove (iii). Integrating (3.31) between s>0and+∞and using integration by parts, we obtain z∞−1 s/integraldisplays 0z(σ)dσ=−4 β/integraldisplay∞ sh(σ) σ2dσ+4 βh(0) s−2α β/bracketleftbigg1 s/integraldisplays 0σh(σ)dσ+/integraldisplay∞ sh(σ)dσ/bracketrightbigg .(3.33) From (3.30) and (3.33), we get z(s)−z∞=−4 βh(s) s+2α β/integraldisplay∞ sh(σ)dσ+4 β/integraldisplay∞ sh(σ) σ2. (3.34) In order to compute the integrals in (3.34), using (3.27) and (3.28), we write h=2 β/parenleftbiggy′ s+α 4z′−2E0 se−αs2/2+c2 0 2sze−αs2/2/parenrightbigg . 16Then, integrating by parts and using the bound for yin (3.24), /integraldisplay∞ sh(σ) =2 β/parenleftBigg −y s+/integraldisplay∞ sy σ2+α 4(z∞−z)−2E0/integraldisplay∞ se−ασ2/2 σ+c2 0 2/integraldisplay∞ sz σe−ασ2/2/parenrightBigg .(3.35) Also, from (3.27) and (3.34), we obtain /integraldisplay∞ sh(σ) σ2=2 β/parenleftBigg/integraldisplay∞ sy′ σ3+α 2/integraldisplay∞ sy σ2−2E0/integraldisplay∞ se−ασ2/2 σ3+c2 0 2/integraldisplay∞ sz σ3e−ασ2/2/parenrightBigg .(3.36) Multiplying (3.34) by β2, using (3.35), (3.36) and the identity α/integraldisplay∞ se−ασ2/2 σn=e−αs2/2 sn+1−(n+1)/integraldisplay∞ se−ασ2/2 σn+2,for allα≥0, n≥1, we conclude that (α2+β2)(z−z∞) =−4 s(αy+βh)−8E0 s2e−αs2/2 +8α/integraldisplay∞ sy σ2+8/integraldisplay∞ sy′ σ3+2c2 0/integraldisplay∞ se−ασ2/2z/parenleftbiggα σ+2 σ3/parenrightbigg . (3.37) Finally, using (3.27) and the boundedness of zandy, an integration by parts argument shows that 8α/integraldisplay∞ sy σ2+8/integraldisplay∞ sy′ σ3=−4αz s2−8y s3−12z s4+8/integraldisplay∞ sz/parenleftbiggα σ3−6 σ5/parenrightbigg . (3.38) Bearing in mind that α2+β2= 1, from (3.37) and (3.38), we obtain the following identity z−z∞=−4 s(αy+βh)−8E0 s2e−αs2/2−4αz s2−8y s3−12z s4+8/integraldisplay∞ sz/parenleftbiggα σ3+6 σ5/parenrightbigg dσ +2c2 0/integraldisplay∞ se−ασ2/2z/parenleftbiggα σ+2 σ3/parenrightbigg dσ,(3.39) for alls>0. In order to prove (iii), we first write z=z−z∞+z∞and observe that 8α/integraldisplay∞ sz σ3= 8α/integraldisplay∞ sz−z∞ σ3+4αz∞ s2, /integraldisplay∞ sz σ5=/integraldisplay∞ sz−z∞ σ5+z∞ 4s4and /integraldisplay∞ se−ασ2/2z/parenleftbiggα σ+2 σ3/parenrightbigg =/integraldisplay∞ se−ασ2/2(z−z∞)/parenleftbiggα σ+2 σ3/parenrightbigg +z∞ s2e−αs2/2. Therefore, we can recast (3.39) as (3.25) with R0(s) =−4α(z−z∞) s2−8y s3−12(z−z∞) s4+8/integraldisplay∞ s(z−z∞)/parenleftbiggα σ3+6 σ5/parenrightbigg dσ +2c2 0/integraldisplay∞ se−ασ2/2(z−z∞)/parenleftbiggα σ+2 σ3/parenrightbigg dσ.(3.40) Let us take s0≥1to be fixed in what follows. For t≥s0, we denote /bardbl · /bardbltthe norm of L∞([t,∞)). From the definition of R0in (3.40) and the elementary inequalities α/integraldisplay∞ se−ασ2/2 σn≤e−αs2/2 sn+1,for allα≥0, n≥1, (3.41) 17and/integraldisplay∞ se−ασ2/2 σn≤e−αs2/2 (n−1)sn−1,for allα≥0, n>1, (3.42) we obtain /bardblR0/bardblt≤8/bardbly/bardblt t3+4 t2/parenleftBig 8+c2 0e−αt2/2/parenrightBig /bardblz−z∞/bardblt. Hence, choosing s0= 4/radicalbig 8+c2 0, so that4 t2/parenleftBig 8+c2 0e−αt2/2/parenrightBig ≤1/2, from (3.24) and (3.25) we conclude that there exists a constant C(E0,c0)>0such that /bardblz−z∞/bardblt≤C(E0,c0) te−αt2/4,for allα∈[0,1)andt≥s0, which implies that |z(s)−z∞| ≤C(E0,c0) se−αs2/4,for allα∈[0,1), s≥s0. (3.43) Finally, plugging (3.24) and (3.43) into (3.40) and bearing in mind the inequalities (3.41) and (3.42), we deduce that |R0(s)| ≤C(E0,c0)e−αs2/4 s3,∀s≥s0= 4/radicalBig 8+c2 0, (3.44) and the proof of (iii)is completed. Formula (3.25) in Proposition 3.2 gives zin terms of yandh. Therefore, we can reduce our analysis to that of the variables yandhor, in other words, to that of the system (3.27)–(3.29). In fact, a first attempt could be to define w=y+ih, so that from (3.28) and (3.29), we have thatwsolves/parenleftBig we(α+iβ)s2/4/parenrightBig′ =e(−α+iβ)s2/4/parenleftbigg γ−c2 0 2(z−z∞)/parenrightbigg . (3.45) From (3.43) in Proposition 3.2 and (3.45), we see that the lim itw∗= lims→∞w(s)e(α+iβ)s2/4 exists (at least when α/ne}ationslash= 0), and integrating (3.45) from some s>0to∞we find that w(s) =e−(α+iβ)s2/4/parenleftbigg w∗−/integraldisplay∞ se(−α+iβ)σ2/4/parenleftbigg γ−c2 0 2(z−z∞)/parenrightbigg dσ/parenrightbigg . In order to obtain an asymptotic expansion, we need to estima te/integraltext∞ se(−α+iβ)σ2/4(z−z∞), fors large. This can be achieved using (3.43), /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞ se(−α+iβ)σ2/4(z−z∞)dσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)/integraldisplay∞ se−ασ2/2 σdσ (3.46) and the asymptotic expansion /integraldisplay∞ se−ασ2/2 σdσ=e−αs2/2/parenleftbigg1 αs2−2 α2s4+8 α3s6+···/parenrightbigg . However this estimate diverges as α→0. The problem is that the bound used in obtaining (3.46) does not take into account the cancellations due to th e oscillations. Therefore, and in order to obtain the asymptotic behaviour of z,yandhvalid for all α∈[0,1), we need a more refined analysis. In the next proposition we study the system (3.27)–(3.29), where we consider the cancellations due the oscillations (see Lemma 3.5 below ). The following result provides estimates that are valid for s≥s1, for somes1independent of α, ifαis small. 18Proposition 3.3. With the same notation and terminology as in Proposition 3.2 , let s1= max/braceleftBigg 4/radicalBig 8+c2 0,2c0/parenleftbigg1 β−1/parenrightbigg1/2/bracerightBigg . Then for all s≥s1, y(s) =be−αs2/4sin(φ(s1;s))−2αγ se−αs2/2+O/parenleftBigg e−αs2/2 β2s2/parenrightBigg , (3.47) h(s) =be−αs2/4cos(φ(s1;s))−2βγ se−αs2/2+O/parenleftBigg e−αs2/2 β2s2/parenrightBigg , (3.48) where φ(s1;s) =a+β/integraldisplays2/4 s2 1/4/radicalbigg 1+c2 0e−2αt tdt, a∈[0,2π)is a real constant, and bis a positive constant given by b2=/parenleftbigg 2E0−c2 0 4z∞/parenrightbigg z∞. (3.49) Proof. First, notice that plugging the expression for z(s)−z∞in (3.25) into (3.28), the system (3.28)–(3.29) for the variables yandhrewrites equivalently as y′=s 2(βh−αy)+2c2 0 se−αs2/2(βh+αy)+γe−αs2/2+R1(s), (3.50) h′=−s 2(βy+αh), (3.51) where R1(s) =−c2 0 2e−αs2/2R0(s)+2c2 0γe−αs2 s2, (3.52) andR0is given by (3.40). Introducing the new variables, u(t) =eαty(2√ t), v(t) =eαth(2√ t), (3.53) we recast (3.50)–(3.51) as /parenleftbiggu v/parenrightbigg′ =/parenleftbiggαK β(1+K) −β0/parenrightbigg/parenleftbiggu v/parenrightbigg +/parenleftbiggF 0/parenrightbigg , (3.54) with K=c2 0e−2αt t, F=γe−αt √ t+e−αt √ tR1(2√ t), whereR1is the function defined in (3.52). In this way, we can regard (3 .54) as a non-autonomous system. It is straightforward to check that the matrix A=/parenleftbiggαK β(1+K) −β0/parenrightbigg is diagonalizable, i.e. A=PDP−1, with D=/parenleftbiggλ+0 0λ−/parenrightbigg , P=/parenleftbigg−αK 2β−i∆1/2−αK 2β+i∆1/2 1 1/parenrightbigg , 19λ±=αK 2±iβ∆1/2,and∆ = 1+K−α2K2 4β2. (3.55) At this point we remark that the condition t≥t1, witht1:=s2 1/4ands1≥2c0(1 β−1)1/2, implies that 0<K/parenleftbigg1 β−1/parenrightbigg ≤1,∀t≥t1, (3.56) so that ∆ = 1+K−(1−β2) 4β2K2=/parenleftbigg 1+K 2+K 2β/parenrightbigg/parenleftbigg 1+K 2/parenleftbigg 1−1 β/parenrightbigg/parenrightbigg ≥1 2,∀t≥t1.(3.57) Thus, defining w= (w1,w2) =P−1(u,v), (3.58) we get/parenleftBig e−/integraltextt t1Dw/parenrightBig′ =e−/integraltextt t1D/parenleftBig (P−1)′Pw+P−1˜F/parenrightBig , (3.59) with˜F= (F,0). From the definition of wand taking into account that uandvare real functions, we have that w1= ¯w2and therefore the study of (3.59) reduces to the analysis of t he equation: /parenleftBig e−/integraltextt t1λ+w1/parenrightBig′ =e−/integraltextt t1λ+G(t), (3.60) with G(t) =iαK′ 4β∆1/2(w1+ ¯w1)−∆′ 4∆(w1−¯w1)+iF 2∆1/2. From (3.60) we have w1(t) =e/integraltextt t1λ+/parenleftbigg w1(t1)+w∞−/integraldisplay∞ te−/integraltextτ t1λ+G(τ)dτ/parenrightbigg , (3.61) with w∞=/integraldisplay∞ t1e−/integraltextτ t1λ+G(τ). Since w1=iu 2∆1/2+v 2+iαKv 4β∆1/2, (3.62) we recastGasG=i(G1+G2+G3)with G1=αK′v 4β∆1/2−∆′ 4∆3/2/parenleftbigg u+αKv 2β/parenrightbigg , G2=γe−αt 2t1/2∆1/2andG3=e−αt 2t1/2∆1/2R1(2t1/2). Now, from the definition of Kand∆, we have K′=−K/parenleftbigg 2α+1 t/parenrightbigg , K′′=K/parenleftBigg/parenleftbigg 2α+1 t/parenrightbigg2 +1 t2/parenrightBigg , ∆′=K′/parenleftbigg 1−α2K 2β2/parenrightbigg and∆′′=K/parenleftBigg/parenleftbigg 2α+1 t/parenrightbigg2 +1 t2/parenrightBigg/parenleftbigg 1−α2K 2β2/parenrightbigg −α2 2β2K2/parenleftbigg 2α+1 t/parenrightbigg2 . Also, since s1= max{4/radicalbig 8+c2 0,2c0(1/β−1)1/2}, for allt≥t1=s2 1/4, we have in particular thatt≥8+c2 0andt≥c2 0(1/β−1), hence c2 0 tβ=c2 0 t/parenleftbigg1 β−1/parenrightbigg +c2 0 t≤2 (3.63) 20and /vextendsingle/vextendsingle/vextendsingle/vextendsingle1−α2K 4β2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1+1 4β/parenleftbiggc2 0 tβ/parenrightbigg ≤2 β. (3.64) Therefore |K′| ≤c2 0e−2αt/parenleftbigg2α t+1 t2/parenrightbigg ,|∆′| ≤2c2 0e−2αt β/parenleftbigg2α t+1 t2/parenrightbigg (3.65) and |∆′′| ≤24c2 0 βe−2αt/parenleftbiggα t+1 t2/parenrightbigg . (3.66) From Proposition 3.2, uandvare bounded in terms of the energy. Thus, from the definition o f G1and the estimates (3.56), (3.57) and (3.65), we obtain |G1(t)| ≤C(E0,c0)e−2αt β2/parenleftbiggα t+1 t2/parenrightbigg . Since /vextendsingle/vextendsingle/vextendsinglee±/integraltextτ t1λ+/vextendsingle/vextendsingle/vextendsingle≤2, (3.67) we conclude that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞ te−/integraltextτ t1λ+G1(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0) β2/integraldisplay∞ te−2ατ/parenleftbiggα τ+1 τ2/parenrightbigg ≤C(E0,c0)e−2αt β2t. (3.68) Here we have used the inequality α/integraldisplay∞ te−2ασ σndσ≤e−2αt 2tn, n≥1, (3.69) which follows by integrating by parts. In order to handle the terms involving G2andG3, we need to take advantage of the oscillatory character of the involved integrals, which is exploited in L emma 3.5. From (3.57), (3.65) and (3.66), straightforward calculations show that the functi on defined by f=γ/(2t1/2∆1/2)satisfies the hypothesis in part (ii)of Lemma 3.5 with a= 1/2andL=C(E0,c0)/β. Thus invoking this lemma with f=γ/(2t1/2∆1/2)and noticing that 1 ∆1/2= 1+/parenleftbigg1 ∆1/2−1/parenrightbigg and that /vextendsingle/vextendsingle/vextendsingle/vextendsingle1 ∆1/2−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−∆ ∆1/2(∆1/2+1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤|K|/vextendsingle/vextendsingle/vextendsingle1−α2K 4β2/vextendsingle/vextendsingle/vextendsingle |∆1/2(∆1/2+1)|≤2√ 2c2 0 βt, where we have used (3.57) and (3.64), we conclude that /integraldisplay∞ te−/integraltextτ t1λ+G2(τ)dτ=γ 2(α+iβ)t1/2e−/integraltextt t1λ+e−αt+R2(t), (3.70) with |R2(t)| ≤C(E0,c0)e−αt β2t3/2. ForG3, we first write explicitly (recall the definition of R1in (3.52)) G3(t) =−c2 0R0(2√ t)e−3αt 4t1/2∆1/2+c2 0γe−5αt 4t3/2∆1/2:=G3,1(t)+G3,2(t). (3.71) 21Using (3.44) and (3.57), we see that |G3,1(t)| ≤C(E0,c0)e−4αt/t2, so that we can treat this term as we did for G1to obtain /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞ te−/integraltextτ t1λ+G3,1(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)e−4αt t. (3.72) For the second term, using (3.57), (3.65) and (3.63), it is ea sy to see that the function fdefined byf= (c2 0γ)/(4t3/2∆1/2)satisfies |f(t)| ≤C(E0,c0) t3/2and|f′(t)| ≤C(E0,c0)/parenleftbiggα t3/2+1 t5/2/parenrightbigg , as a consequence, invoking part (i)of Lemma 3.5, we obtain /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞ te−/integraltextτ t1λ+G3,2(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)e−5αt βt3/2. (3.73) From (3.61), (3.67), (3.68), (3.72) and (3.73), we deduce th at w1(t) =e/integraltextt t1λ+(w1(t1)+w∞)−γ(β+iα) 2t1/2e−αt+R3(t)with|R3(t)| ≤C(E0,c0)e−αt β2t.(3.74) Now we claim that e/integraltextt t1λ+=Cα,c0eiβI(t)+H(t),withI(t) =/integraldisplayt t1/radicalbig 1+K(σ)dσ,|H(t)| ≤3c2 0e−2αt t,(3.75) and Cα,c0= exp/parenleftbiggα 2/integraldisplay∞ t1Kdσ/parenrightbigg exp/parenleftbigg −iα2 4β/integraldisplay∞ t1K2 ∆1/2+(1+K)1/2dσ/parenrightbigg . Indeed, recall that λ+=αK 2+iβ∆1/2so that e/integraltextt t1λ+=eα/integraltextt t1K 2eiβ/integraltextt t1∆1/2 . (3.76) First, we notice that α/integraldisplayt t1K 2=c2 0α/integraldisplay∞ t1e−2ασ 2σ−c2 0α/integraldisplay∞ te−2ασ 2σ, where both integrals are finite in view of (3.69). Moreover, b y combining with the fact that |1−e−x| ≤x, forx≥0, we can write exp/parenleftbigg −c2 0α/integraldisplay∞ te−2ασ 2σ/parenrightbigg = 1+H1(t), with |H1(t)| ≤c2 0e−2αt 4t,for allt≥c2 0/4. (3.77) The above argument shows that eα/integraltextt t1K 2=eα/integraltext∞ t1K 2(1+H1(t)), (3.78) withH1(t)satisfying (3.77). 22For the second term of the eigenvalue, using the definition of ∆in (3.55), we write iβ/integraldisplayt t1∆1/2=iβ/integraldisplayt t1/parenleftBig ∆1/2−√ 1+K/parenrightBig +iβ/integraldisplayt t1√ 1+K =−iα2 4β/integraldisplayt t1K2 ∆1/2+(1+K)1/2+iβ/integraldisplayt t1√ 1+K. Proceeding as before and using that |1−eix| ≤ |x|, forx∈R, and that α/integraldisplay∞ tK2 ∆1/2+√ 1+K≤α/integraldisplay∞ tK2(σ)dσ=αc4 0/integraldisplay∞ te−4ατ τ2≤c4 0e−4αt 4t2, we conclude that eiβ/integraltextt t1∆1/2 =eiβI(t)e−iα2 4β/integraltext∞ t1K2 ∆1/2+(1+K)1/2(1+H2), (3.79) with |H2(t)| ≤c4 0e−4αt 16βt2≤c2 0e−4αt 8t, bearing in mind (3.63). Therefore, from (3.76), (3.78) and ( 3.79), e/integraltextt t1λ+=Cα,c0eiβI(t)(1+H1(t))(1+H2(t)). The claim follows from the above identity, the bounds for H1andH2, and the fact that Cα,c0 satisfies that |Cα,c0|=|e/integraltext∞ t1λ+| ≤2(see (3.67)). From (3.74), the claim and writing Cα,c0(w1(t1)+w∞) = (beia)/2 (3.80) for some real constants aandbsuch thatb≥0anda∈[0,2π), it follows that w1(t) =b 2ei(βI(t)+a)−γ(β+iα) 2t1/2+Rw1(t)with|Rw1(t)| ≤C(E0,c0)e−αt β2t. (3.81) The above bound for Rw1(t)easily follows from the bounds for R3(t)andH(t)in (3.74) and (3.75) respectively, and the fact that |w1(t)| ≤C(E0,c0),∀t≥t1. (3.82) This last inequality is a consequence of (3.53), (3.57), (3. 62), (3.63) and the bounds for yandh established in (3.24) in Proposition 3.2. Going back to the definition of win (3.58), we have (u,v) =P(w1,w2), that is u=−αK 2β(w1+ ¯w1)−i∆1/2(w1−¯w1) = 2Im(w1)+R4(t), v= (w1+ ¯w1) = 2Re(w1),(3.83) with |R4(t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle−αK βRe(w1)+2(∆1/2−1)Im(w1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤K β|Re(w1)|+2|∆−1| ∆1/2+1|Im(w1)| ≤2c2 0e−2αt βt(|Re(w1)|+|Im(w1)|)≤C(E0,c0)e−2αt βt, 23where we have used (3.57), (3.64), and (3.82). From (3.81) an d (3.83), we obtain u(t) =bsin(βI(t)+a)−αγ t1/2e−αt+R5(t), v(t) =bcos(βI(t)+a)−βγ t1/2e−αt+R6(t), with |R5(t)|+|R6(t)| ≤C(E0,c0)e−αt/(β2t). The asymptotics for yandhgiven in (3.47) and (3.48) are a direct consequence of (3.53) and the above identities and bounds. Finally, we compute the value of b. In fact, from (3.47) and (3.48) lim s→∞(y2(s)+h2(s))eαs2/2=b2. On the other hand, since y+ih=¯ff′and using the conservation of energy (3.16) /parenleftbig y2(s)+h2(s)/parenrightbig eαs2/2=|y+ih|2(s)eαs2/2=|f′|2|f|2eαs2/2= (2E0−c2 0 4|f|2)|f|2, so that, taking the limit as s→ ∞ and recalling that z=|f|2, (3.49) follows. Remark 3.4. From the definitions of bin(3.49), andbeiain(3.80) (in terms of Cα,c0,w1(t1) andw∞in(3.80)), it is simple to verify that bandbeiadepend continuously on α∈[0,1), provided that z∞is a continuous function of α. In Subsection 3.3 we will prove that z∞depends continuously on α, forα∈[0,1], and establish the continuous dependence of the constants band beiawith respect to the parameter αin Lemma 3.13 above. In the proof of Proposition 3.3, we have used the following ke y lemma that establishes the control of certain integrals by exploiting their oscillato ry character. Lemma 3.5. With the same notation as in the proof of Proposition 3.2. (i) Letf∈C1((t1,∞))such that |f(t)| ≤L/taand|f′(t)| ≤L/parenleftbiggα ta+1 ta+1/parenrightbigg , for some constants L,a>0. Then, for all t≥t1andl≥1 /integraldisplay∞ te−/integraltextτ t1λ+e−lατf(τ)dτ=1 (α+iβ)e−/integraltextt t1λ+e−lαtf(t)+F(t), with |F(t)| ≤C(l,a,c0)Le−lαt βta. (3.84) (ii) If in addition f∈C2((t1,∞)), |f′(t)| ≤L/ta+1and|f′′(t)| ≤L/parenleftbiggα ta+1+1 ta+2/parenrightbigg , (3.85) then |F(t)| ≤C(l,a,c0)Le−lαt βta+1. (3.86) 24HereC(l,a,c0)is a positive constant depending only on l,aandc0. Proof. Defineλ=λ+. Recall (see proof of Proposition 3.2) that λ+=αK 2+iβ∆1/2and∆ = 1+K−α2K2 4β2,withK=c2 0e−2αt t. SettingRλ= 1/λ−1/(iβ)and integrating by parts, we obtain /parenleftbigg 1+lα iβ/parenrightbigg/integraldisplay∞ te−/integraltextτ t1λe−lατf(τ)dτ=e−/integraltextt t1λe−lαtf(t)/parenleftbigg1 iβ+Rλ/parenrightbigg +/integraldisplay∞ te−/integraltextτ t1λe−lατ/parenleftbigg −lαfRλ+f′ λ−fλ′ λ2/parenrightbigg dτ, or, equivalently, /integraldisplay∞ te−/integraltextτ t1λe−ατf(τ)dτ=1 lα+iβe−/integraltextt t1λe−αtf(t)+F(t), with F(t) =iβ lα+iβ/parenleftbigg e−/integraltextt t1λe−lαtRλf+/integraldisplay∞ te−/integraltextτ t1λe−lατ/parenleftbigg −lαfRλ+f′ λ−fλ′ λ2/parenrightbigg dτ/parenrightbigg . Using (3.57), (3.63) and (3.65), it is easy to check that for a llt≥t1 |λ| ≥β√ 2and|λ′| ≤3c2 0/parenleftbigg2α t+1 t2/parenrightbigg . (3.87) On the other hand, |Rλ|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleiβ−λ iβλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤√ 2 β2/parenleftbigg β|1−∆1/2|+αK 2/parenrightbigg , with, using the definition of ∆in (3.57) and (3.63), αK 2≤c2 0 2tand|1−∆1/2|=|1−∆| 1+∆1/2≤ |1−∆| ≤c2 0 t+c2 0 4βt/parenleftbiggc2 0 βt/parenrightbigg ≤2c2 0 βt. Previous lines show that |Rλ| ≤10c2 0 β2t. (3.88) The estimate (3.84) easily follows from the bounds (3.67), ( 3.69), (3.87), (3.88) and the hypothe- ses onf. To obtain part (ii)we only need to improve the estimate for the term /integraldisplay∞ te−/integraltextτ t1λe−lατf′ λdτ in the above argument. In particular, it suffices to prove that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞ te−/integraltextτ t1λe−lατf′ λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(l,c0,a)Le−lαt β2ta+1. Now, consider the function g=f′/λ. Notice that from (3.63), (3.87) and the hypotheses on f in (3.85), we have |g(t)| ≤√ 2L βta+1 25and |g′(t)| ≤√ 2 βL/parenleftbiggα ta+1+1 ta+2/parenrightbigg +6L β/parenleftbiggc2 0 βt/parenrightbigg/parenleftbigg2α ta+1+1 ta+2/parenrightbigg ≤14L β/parenleftbigg2α ta+1+1 ta+2/parenrightbigg . Therefore, from part (i), we obtain /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞ te−/integraltextτ t1λe−lατf′ λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(l,c0,a)Le−lαt/parenleftbigg1 βta+1+1 β2ta+1/parenrightbigg ≤C(l,c0,a)Le−lαt β2ta+1, as desired. We remark that if α∈[0,1/2], the asymptotics in Proposition 3.3 are uniform in α. Indeed, max α∈[0,1/2]/braceleftBigg 4/radicalBig 8+c2 0,2c0/parenleftbigg1 β−1/parenrightbigg1/2/bracerightBigg = 4/radicalBig 8+c2 0=s0. Therefore in this situation we can omit the dependence on s1in the function φ(s1;s), because the asymptotics are valid with φ(s) :=φ(s0;s) =a+β/integraldisplays2/4 s2 0/4/radicalbigg 1+c2 0e−2αt tdt. (3.89) We continue to show that the factor 1/β2in the big-Oin formulae (3.47) and (3.48) are due to the method used and this factor can be avoided if αis far from zero. More precisely, we have the following: Lemma 3.6. Letα∈[1/2,1). With the same notation as in Propositions 3.2 and 3.3, we hav e the following asymptotics: for all s≥s0, y(s) =be−αs2/4sin(φ(s))−2αγ se−αs2/2+O/parenleftBigg e−αs2/2 s2/parenrightBigg , (3.90) h(s) =be−αs2/4cos(φ(s))−2βγ se−αs2/2+O/parenleftBigg e−αs2/2 s2/parenrightBigg . (3.91) Here, the function φis defined by (3.89) and the bounds controlling the error terms depend on c0, and the energy E0, and are independent of α∈[1/2,1) Proof. Letα∈[1/2,1)and define w=y+ih. From Proposition 3.3 and (1.21), we have that for allα∈[1/2,1) lim s→∞we(α+iβ)s2/4=bie−i˜a, (3.92) where˜a:=a+C(α,c0),aandbare the constants defined in Proposition 3.3 and C(α,c0)is the constant in (1.21). Then, since wsatisfies /parenleftBig we(α+iβ)s2/4/parenrightBig′ =e(−α+iβ)s2/4/parenleftbigg γ−c2 0 2(z−z∞)/parenrightbigg , (3.93) integrating the above identity between sand infinity, we(α+iβ)s2/4=ibe−i˜a−/integraldisplay∞ se(−α+iβ)σ2/4/parenleftbigg γ−c2 0 2(z−z∞)/parenrightbigg dσ. 26Now, integrating by parts and using (3.41) (recall that 1≤2α), we see that /integraldisplay∞ se(−α+iβ)σ2/4dσ= 2(α+iβ)e(−α+iβ)s2/4 s+O/parenleftBigg e−αs2/4 s3/parenrightBigg ,∀s≥s0. Next, notice that from (3.43) in Proposition 3.2, we also obt ain /integraldisplay∞ se(−α+iβ)σ2/4(z−z∞)dσ=O/parenleftBigg e−αs2/2 s2/parenrightBigg ,∀s≥s0. The above argument shows that for all s≥s0 w(s) =ibe−αs2/4e−i(˜a+βs2/4)−2(α+iβ)γ se−αs2/2+O/parenleftBigg e−αs2/2 s2/parenrightBigg . (3.94) The asymptotics for yandhin the statement of the lemma easily follow from (3.94) beari ng in mind thatw=y+ihand recalling that the function φbehaves like (1.21) when α>0. In the following corollary we summarize the asymptotics for z,yandhobtained in this section. Precisely, as a consequence of Proposition 3.2- (iii), Proposition 3.3 and Lemma 3.6, we have the following: Corollary 3.7. Letα∈[0,1). With the same notation as before, for all s≥s0= 4/radicalbig 8+c2 0, y(s) =be−αs2/4sin(φ(s))−2αγ se−αs2/2+O/parenleftBigg e−αs2/2 s2/parenrightBigg , (3.95) h(s) =be−αs2/4cos(φ(s))−2βγ se−αs2/2+O/parenleftBigg e−αs2/2 s2/parenrightBigg , (3.96) z(s) =z∞−4b se−αs2/4(αsin(φ(s))+βcos(φ(s)))+4γe−αs2/2 s2+O/parenleftBigg e−αs2/4 s3/parenrightBigg , (3.97) where φ(s) =a+β/integraldisplays2/4 s2 0/4/radicalbigg 1+c2 0e−2αt tdt, for some constant a∈[0,2π), b=z1/2 ∞/parenleftbigg 2E0−c2 0 4z∞/parenrightbigg1/2 , γ= 2E0−c2 0 2z∞ andz∞= lim s→∞z(s). Here, the bounds controlling the error terms depend on c0and the energy E0, and are independent ofα∈[0,1). Remark 3.8. In the case when s<0, the same arguments to the ones leading to the asymptotics in the above corollary will lead to an analogous asymptotic b ehaviour for the variables z,hand yfors<0. As mentioned at the beginning of Subsection 3.2, here we hav e reduced ourselves to the case of s >0when establishing the asymptotic behaviour of the latter qu antities due to the parity of the solution we will be applying these results to. 27Remark 3.9. The asymptotics in Corollary 3.7 lead to the asymptotics for the solutions fof the equation (3.20), at least if |f|∞:=z1/2 ∞is strictly positive. Indeed, this implies that there exist s s∗≥s0such thatf(s)/ne}ationslash= 0for alls≥s∗. Then writing fin its polar form f=ρexp(iθ), we haveρ2θ′= Im(¯ff′). Hence, using (3.22), we obtain ρ=z1/2andθ′=h/z. Therefore, for all s≥s∗, θ(s)−θ(s∗) =/integraldisplays s∗h(σ) z(σ)dσ. (3.98) Hence, using the asymptotics for zandhin Corollary 3.7, we can obtain the asymptotics for f. In the case that α∈(0,1], we can also show that the phase converges. Indeed, the asymp totics in Corollary 3.7 yield that the integral in (3.98) converges as s→ ∞forα>0, and we conclude that there exists a constant θ∞∈Rsuch that f(s) =z(s)1/2exp/parenleftbigg iθ∞−i/integraldisplay∞ sh(σ) z(σ)dσ/parenrightbigg ,for alls≥s∗. The asymptotics for fis obtained by plugging the asymptotics in Corollary 3.7 int o the above expression. 3.3 The second-order equation. Dependence on the parameter s The aim of this subsection is to study the dependence of the f,z,yandhon the parameters c0>0andα∈[0,1]. This will allow us to pass to the limit α→1−in the asymptotics in Corollary 3.7 and will give us the elements for the proofs of T heorems 1.3 and 1.4. 3.3.1 Dependence on α We will denote by f(s,α)the solution of (3.20) with some initial conditions f(0,α),f′(0,α)that are independent of α. Indeed, we are interested in initial conditions that depen d only onc0(see (3.13)–(3.15)). Moreover, in view of (3.17), we assume that the energyE0in (3.16) is a function ofc0. In order to simplify the notation, we denote with a subindex αthe derivative with respect toαand by′the derivative with respect to s. Analogously to Subsection 3.2, we define z(s,α) =|f(s,α)|2, y(s,α) = Re(¯f(s,α)f′(s,α)), h(s,α) = Im(¯f(s,α)f′(s,α)) (3.99) and z∞(α) = lim s→∞|f(s,α)|2. Observe that in Proposition 3.2- (ii), we proved the existence of z∞(α), forα∈[0,1). For α∈(0,1], the estimates in (3.24) hold true and hence z(s,α)is a bounded function whose derivative decays exponentially. Therefore, it admits a li mit at infinity for all α∈[0,1]and z∞(1)is well-defined. The next lemma provides estimates for zα,hαandyα. Lemma 3.10. Letα∈(0,1). There exists a constant C(c0), depending on c0but not onα, such that for all s≥0, |zα(s,α)| ≤C(c0)min/braceleftBigg s2 √1−α+s3,s2 /radicalbig α(1−α),1 α2√1−α/bracerightBigg , (3.100) |yα(s,α)|+|hα(s,α)| ≤C(c0)e−αs2/4min/braceleftBigg s2 √1−α+s3,s2 /radicalbig α(1−α)/bracerightBigg . (3.101) 28Proof. Differentiating (3.12) with respect to α, f′′ α+s 2(α+iβ)f′ α+c2 0 4fαe−αs2/2=g, (3.102) where g(s,α) =−/parenleftbigg 1−iα β/parenrightbiggs 2f′+c2 0s2 8fe−αs2/2. Also, since the initial conditions do not depend on α, fα(0,α) =f′ α(0,α) = 0. (3.103) Using the estimates in (3.23) and that α2+β2= 1, we obtain |g| ≤C(c0)/parenleftbiggs βe−αs2/4+s2e−αs2/2/parenrightbigg ,for alls≥0. (3.104) Multiplying (3.102) by ¯f′ αand taking real part, we have 1 2/parenleftbig |f′ α|2/parenrightbig′+αs 2|f′ α|2+c2 0 8/parenleftbig |fα|2/parenrightbig′e−αs2/2= Re(g¯f′ α). (3.105) Multiplying (3.105) by 2eαs2/2and integrating, taking into account (3.103), |f′ α|2eαs2/2+c2 0 4|fα|2= 2/integraldisplays 0eασ2/2Re(g¯f′ α)dσ. (3.106) Let us define the real-valued function η=|f′ α|eαs2/4. Then (3.106) yields η2(s)≤2/integraldisplays 0eασ2/4|g|ηdσ, for alls≥0. Thus, by the Gronwall inequality (see e.g. [3, Lemma A.5]), η(s)≤/integraldisplays 0eασ2/4|g|,dσ, for alls≥0. (3.107) From (3.104), (3.106) and (3.107), we conclude that (|f′ α|eαs2/4+c0 2|fα|)2≤2(|fα|2eαs2/2+c2 0 4|fα|2) ≤4/integraldisplays 0eασ2/4|g|ηdσ≤4/parenleftBigg sup σ∈[0,s]η(σ)/parenrightBigg/parenleftbigg/integraldisplays 0eασ2/4|g|dσ/parenrightbigg ≤/parenleftbigg/integraldisplays 0eασ2/4|g|dσ/parenrightbigg2 . Thus, using (3.104), from the above inequality it follows |f′ α|eαs2/4+c0 2|fα| ≤C(c0)/integraldisplays 0/parenleftbiggσ β+σ2e−ασ2/4/parenrightbigg dσ, for alls≥0. (3.108) In particular, for all s≥0, |fα(s)| ≤C(c0)min/braceleftBigg s2 √1−α+s3,s2 /radicalbig α(1−α)/bracerightBigg , |f′ α(s)| ≤C(c0)e−αs2/4min/braceleftBigg s2 √1−α+s3,s2 /radicalbig α(1−α)/bracerightBigg ,(3.109) 29where we have used that /integraldisplays 0σ2e−ασ2/4dσ≤s2/integraldisplays 0e−ασ2/4dσ≤s2/radicalbig π/α. Notice that from (3.103) and (3.109), |fα(s)| ≤/integraldisplays 0|f′ α|dσ≤C(c0)/radicalbig α(1−α)/integraldisplays 0σ2e−ασ2/4dσ, and /integraldisplay∞ 0σ2e−ασ2/4dσ=2√π α3/2, (3.110) so that |fα(s)| ≤C(c0) α2√1−α. (3.111) On the other hand, differentiating the relations in (3.99) wi th respect to α, |zα| ≤2|fα||f|,|yα+ihα| ≤ |fα||f′|+|f||f′ α|. (3.112) By putting together (3.23), (3.109), (3.111) and (3.112), we obtain (3.100) and (3.101). Lemma 3.11. The function z∞is continuous in (0,1]. More precisely, there exists a constant C(c0)depending on c0but not onα, such that |z∞(α2)−z∞(α1)| ≤C(c0) L(α2,α1)|α2−α1|,for allα1,α2∈(0,1], (3.113) where L(α2,α1) :=α2 1α3/2 2/parenleftBig α3/2 1√ 1−α2+α3/2 2√ 1−α1/parenrightBig . In particular, |z∞(1)−z∞(α)| ≤C(c0)√ 1−α,for allα∈[1/2,1]. (3.114) Proof. Letα1,α2∈(0,1],α1< α2. By classical results from the ODE theory, the functions y(s,α),h(s,α)andz(s,α)are smooth in R×[0,1)and continuous in R×[0,1](see e.g. [5, 17]). Hence, integrating (3.27) with respect to s, we deduce that z∞(α2)−z∞(α1) = 2/integraldisplay∞ 0(y(s,α2)−y(s,α1))ds= 2/integraldisplay∞ 0/integraldisplayα2 α1dy dµ(s,µ)dµds. (3.115) To estimate the last integral, we use (3.101) /integraldisplayα2 α1|dy dµ(s,µ)|dµ≤C(c0)s2 √α1/integraldisplayα2 α1e−µs2/4 √1−µdµ. (3.116) Now, integrating by parts, /integraldisplayα2 α1e−µs2/4 √1−µdµ= 2/parenleftBig√ 1−α1e−α1s2/4−√ 1−α2e−α2s2/4/parenrightBig −s2 2/integraldisplayα2 α1/radicalbig 1−µe−µs2/4dµ. Therefore, by combining with (3.115) and (3.116), |z∞(α2)−z∞(α1)| ≤C(c0)√α1/parenleftbigg√ 1−α1/integraldisplay∞ 0s2e−α1s2/4ds−√ 1−α2/integraldisplay∞ 0s2e−α2s2/4ds/parenrightbigg , 30and bearing in mind (3.110), we conclude that |z∞(α2)−z∞(α1)| ≤C(c0)√α1/parenleftBigg√1−α1 α3/2 1−√1−α2 α3/2 2/parenrightBigg , which, after some algebraic manipulations and using that α1,α2∈(0,1], leads to (3.113). The estimate for z∞near zero is more involved and it is based in an improvement of the estimate for the derivative of z∞. Lemma 3.12. The function z∞is continuous in [0,1]. Moreover, there exists a constant C(c0)>0, depending on c0but not onαsuch that for all α∈(0,1/2], |z∞(α)−z∞(0)| ≤C(c0)√α|ln(α)|. (3.117) Proof. As in the proof of Lemma 3.11, we recall that the functions y(s,α),h(s,α)andz(s,α) are smooth in any compact subset of R×[0,1). From now on we will use the identity (3.39) fixings= 1. We can verify that the two integral terms in (3.39) are conti nuous functions at α= 0, which proves that z∞is continuous in 0. In view of Lemma 3.11, we conclude that z∞is continuous in [0,1]. Now we claim that /vextendsingle/vextendsingle/vextendsingle/vextendsingledz∞ dα(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)|ln(α)|√α,for allα∈(0,1/2]. (3.118) In fact, once (3.118) is proved, we can compute |z∞(α)−z∞(0)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayα 0dz∞ dµ(µ)dµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)/integraldisplayα 0|ln(µ)|√µdµ= 2C(c0)√α(|ln(α)|+2), which implies (3.117). It remains to prove the claim. Differentiating (3.39) (recal l thats= 1) with respect to α, and using that y(1,·),h(1,·)andz(1,·)are continuous differentiable in [0,1/2], we deduce that there exists a constant C(c0)>0such that /vextendsingle/vextendsingle/vextendsingle/vextendsingledz∞ dα(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)+8|I1(α)|+2c2 0|I2(α)|, (3.119) with I1(α) =/integraldisplay∞ 1z σ3+α/integraldisplay∞ 1zα σ3+6/integraldisplay∞ 1zα σ5(3.120) and I2(α) =−α 2/integraldisplay∞ 1e−ασ2/2zσ+α/integraldisplay∞ 1e−ασ2/2zα σ+2/integraldisplay∞ 1e−ασ2/2zα σ3. (3.121) By (3.24) and (3.100), zis uniformly bounded and zαgrows at most as a cubic polynomial, so that the first and the last integral in the r.h.s. of (3.120) are bounded independently of α∈[0,1/2]. In addition, (3.100) also implies that |zα|=|zα|1/2|zα|1/2≤C(c0)(s3)1/2/parenleftbigg1 α2/parenrightbigg1/2 =C(c0)s3/2 α, (3.122) which shows that the remaining integral in (3.120) is bounde d. 31Thus, the above argument shows that |I1(α)| ≤C(c0)for allα∈[0,1/2]. (3.123) The same arguments also yield that the first two integrals in t he r.h.s. of (3.121) are bounded byC(c0)α−1/2.Using once more that |zα| ≤C(c0)s2α−1/2, we obtain the following bounds for the remaining two integrals in (3.121) /vextendsingle/vextendsingle/vextendsingle/vextendsingleα/integraldisplays 1e−ασ2/2zα σ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)√α/integraldisplay∞ 1ασe−ασ2/2dσ=C(c0)√αe−α/2≤C(c0)√α and /vextendsingle/vextendsingle/vextendsingle/vextendsingle2/integraldisplay∞ 1e−ασ/2zα σ3dσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)√α/integraldisplay∞ 1e−ασ2/2 σdσ≤C(c0)|ln(α)|√α. In conclusion, we have proved that |I2(α)| ≤C(c0)|ln(α)|√α, which combined with (3.119) and (3.123), completes the proo f of claim. We end this section showing that the previous continuity res ults allow us to “pass to the limit” α→1−in Corollary 3.7. Using the notation b(α) =banda(α) =afor the constants defined for α∈[0,1)in Proposition 3.3 in Subsection 3.2, we have Lemma 3.13. The valueb(α)is a continuous function of α∈[0,1]and the value b(α)eia(α)is continuous function of α∈[0,1)that can be continuously extended to [0,1]. The function a(α) has a (possible discontinuous) extension for α∈[0,1]such thata(α)∈[0,2π). Proof. By Lemma 3.12, we have the continuity of z∞in [0,1]. Therefore, in view of Remark 3.4, the function beiais a continuous function of α∈[0,1)and by (3.49) bis actually well-defined and continuous in α∈[0,1]. It only remains to prove that the limit L:= lim α→1−b(α)eia(α)(3.124) exists. Ifb(1) = 0 , it is immediate that L= 0and we can give any arbitrary value in [0,2π)to a(1). Let us suppose that b(1)>0. Integrating (3.93), we get w(s)e(α+iβ)s2/4=w(s0)e(α+iβ)s2 0/4+/integraldisplays s0e(−α+iβ)σ2/4/parenleftbigg γ−c2 0 2(z−z∞)/parenrightbigg dσ, and this relation is valid for any α∈(0,1]. Letα∈(0,1). In view of (3.92), letting s→ ∞, we have ibei(a+C(α,c0))=w(s0)e(α+iβ)s2 0/4+/integraldisplay∞ s0e(−α+iβ)σ2/4/parenleftbigg γ−c2 0 2(z−z∞)/parenrightbigg dσ, (3.125) whereC(α,c0)is the constant in (1.21). Notice that the r.h.s. of (3.125) i s well-defined for any α∈(0,1]and by the arguments given in the proof of Lemma 3.11 and the do minated convergence theorem, the r.h.s. is also continuous for any α∈(0,1]. Therefore, the limit Lin (3.124) exists and is given by the r.h.s. of (3.125) evaluated in α= 1and divided by ieiC(1,c0). Moreover, lim α→1−eia(α)=L b(1), so that by the compactness of the the unit circle in C, there exists θ∈[0,2π)such thateiθ= L/b(1)and we can extend aby defining a(1) =θ. 32The following result summarizes an improvement of Corollar y 3.7 to include the case α= 1 and the continuous dependence of the constants appearing in the asymptotics on α. Precisely, we have the following: Corollary 3.14. Letα∈[0,1],β≥0withα2+β2= 1andc0>0. Then, (i) The asymptotics in Corollary 3.7 holds true for all α∈[0,1]. (ii) Moreover, the values bandbeiaare continuous functions of α∈[0,1]and each term in the asymptotics for z,yandhin Corollary 3.7 depends continuously on α∈[0,1]. (iii) In addition, the bounds controlling the error terms de pend onc0and are independent of α∈[0,1]. Proof. Lets≥s0fixed. As noticed in the proof of Lemma 3.11, the functions y(s,α),h(s,α), z(s,α)are continuous in α= 1. In addition, by Lemma 3.13 beiais continuous in α= 1, using the definition of φ, it is immediate that bsin(φ(s))andbcos(φ(s))are continuous in α= 1. Therefore the big- Oterms in (3.95), (3.96) and (3.97) are also are continuous in α= 1. The proof of the corollary follows by letting α→1−in (3.95), (3.96) and (3.97). 3.3.2 Dependence on c0 In this subsection, we study the dependence of z∞as a function of c0, for a fixed value of α. To this aim, we need to take into account the initial conditio ns given in (3.13)–(3.15). More generally, let us assume that fis a solution of (3.20) with initial conditions f(0)andf′(0)that depend smoothly on c0, for anyc0>0, and that E0>0is the associated energy defined in (3.16). To keep our notation simple, we omit the parameter c0in the functions fandz∞. Under these assumptions, we have Proposition 3.15. Letα∈[0,1]andc0>0. Thenz∞is a continuous function of c0∈(0,∞). Moreover if α∈(0,1], the following estimate hold /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglez∞−/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(0)+f′(0)√π√α+iβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤√2E0c0π α/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(0)+f′(0)√π√α+iβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/parenleftbigg√2E0c0π 2α/parenrightbigg2 . (3.126) Proof. Since we are assuming that the initial conditions f(0)andf′(0)depend smoothly on c0, by classical results from the ODE theory, the functions f,y,handzare smooth with respect to s andc0. From (3.39) with s= 1, we have that z∞can be written in terms of continuous functions ofc0(the continuity of the integral terms follows from the domin ated convergence theorem), so thatz∞depends continuously on c0. To prove (3.126), we multiply (3.20) by e(α+iβ)s2/4, so that (f′e(α+iβ)s2/4)′=−c2 0 4f(s)e(−α+iβ)s2/4. Hence, integrating twice, we have f(s) =f(0)+G(s)+F(s), (3.127) with G(s) =f′(0)/integraldisplays 0e−(α+iβ)σ2/4dσandF(s) =−c2 0 4/integraldisplays 0e−(α+iβ)σ2/4/integraldisplayσ 0e(−α+iβ)τ2/4f(τ)dτdσ. 33Since by Proposition 3.2 |f(s)| ≤2√2E0 c0, we obtain |F(s)| ≤√2E0c0 2/integraldisplays 0e−ασ2/4/integraldisplayσ 0e−ατ2/4dτdσ≤√2E0c0 2·π α. (3.128) Using (3.127) and the identity, |z1+z2|2=|z1|2+2Re(¯z1z2)+|z2|2, z1,z2∈C, we conclude that z(s) =|f(s)|2satisfies z(s) =|f(0)+G(s)|2+2Re(¯F(s)(f(0)+G(s)))+|F(s)|2. Therefore, for all s≥0, |z(s)−|f(0)+G(s)|2| ≤2|F(s)||f(0)+G(s)|+|F(s)|2. Hence we can use the bound (3.128) and then let s→ ∞. Noticing that lim s→∞G(s) =f′(0)/integraldisplay∞ 0e−(α+iβ)σ2/4dσ=f′(0)√π√α+iβ, the estimate (3.126) follows. 4 Proof of the main results In Section 3 we have performed a careful analysis of the equat ion (3.12), taking also into con- sideration the initial conditions (3.13)–(3.15). Therefo re, the proofs of our main theorem consist mainly in coming back to the original variables using the ide ntities (3.18) and (3.19). For the sake of completeness, we provide the details in the followin g proofs. Proof of Theorem 1.2 .Letα∈[0,1],c0>0and{/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)}be the unique C∞(R;S2)-solution of the Serret–Frenet equations (1.6) with curvat ure and torsion (2.6) and initial conditions (2.8). In order to simplify the notation , in the rest of the proof we drop the subindexes c0andαand simply write {/vector m(·),/vector n(·),/vectorb(·)}for{/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)}. First observe that if we define {/vectorM,/vectorN,/vectorB}in terms of {/vector m,/vector n,/vectorb}by /vectorM(s) = (m(−s),−m(−s),−m(−s)), /vectorN(s) = (−n(−s),n(−s),n(−s)), /vectorB(s) = (−b(−s),b(−s),b(−s)), s∈R, then{/vectorM,/vectorN,/vectorB}is also a solution of the Serret system (1.6) with curvature a nd torsion (2.6). Notice also that {/vectorM(0),/vectorN(0),/vectorB(0)}={/vector m(0),/vector n(0),/vectorb(0)}. Therefore, from the uniqueness of the solution we conclude t hat /vectorM(s) =/vector m(s),/vectorN(s) =/vector n(s)and/vectorB(s) =/vectorb(s),∀s∈R. This proves part (i)of Theorem 1.2. 34Second, in Section 3 we have seen that one can write the compon ents of the Frenet trihedron {/vector m,/vector n,/vectorb}as m1(s) = 2|f1(s)|2−1, n 1(s)+ib1(s) =4 c0eαs2/4¯f1(s)f′ 1(s), (4.1) mj(s) =|fj(s)|2−1, n j(s)+ibj(s) =2 c0eαs2/4¯fj(s)f′ j(s), j∈ {2,3}, (4.2) withfjsolution of the second order ODE (3.12) with initial conditi ons (3.13)-(3.15) respectively, and associated initial energies (see (3.17)) E0,1=c2 0 8andEj,1=c2 0 8,forj∈ {2,3}. (4.3) Notice that the identities (4.1)–(4.2) rewrite equivalent ly as m1,c0,α= 2z1−1, n1,c0,α=4 c0eαs2/4y1, b1,c0,α=4 c0eαs2/4h1, mj,c0,α=zj−1, nj,c0,α=2 c0eαs2/4yj, bj,c0,α=2 c0eαs2/4hj, j∈ {2,3},(4.4) in terms of the quantities {zj,yj,hj}defined by zj=|fj|2, yj= Re(¯fjf′ j)andhj= Im(¯fjf′ j). Denote by zj,∞,aj,bj,γjandφjthe constants and function appearing in the asymptotics of {yj,hj,zj}proved in Section 3 in Corollary 3.14. Taking the limit as s→+∞in (4.1)–(4.2), and since |/vector m(s)|= 1, we obtain that there exists /vectorA+= (A+ j)3 j=1∈S2with A+ 1= 2z1,∞−1, A+ j=zj,∞−1,forj∈ {2,3}. (4.5) The asymptotics stated in part (ii)of Theorem 1.2 easily follows from formulae (4.1)–(4.2) and the asymptotics for {zj,yj,hj}established in Corollary 3.14. Indeed, it suffices to observe that from the formulae for bjandγjin terms of the initial energies E0,jandzj,∞given in Corollary 3.14, (4.3) and (4.5) we obtain b2 1=c2 0 16(1−(A+ 1)2), b2 2=c2 0 4(1−(A+ 2)2), b2 3=c2 0 4(1−(A+ 3)2), (4.6) γ1=−c2 0 4A+ 1, γ2=−c2 0 2A+ 2, γ3=−c2 0 2A+ 3. (4.7) Substituting these constants in (3.95), (3.96) and (3.97) i n Corollary 3.14, we obtain (1.16), (1.17) and (1.18). This completes the proof of Theorem 1.2- (ii). Proof of Theorem 1.1 .Letα∈[0,1], andc0>0. As before, dropping the subindexes, we will denote by {/vector m,/vector n,/vectorb}the unique solution of the Serret–Frenet equations (1.6) wi th curvature and torsion (2.6) and initial conditions (2.8). Define /vectorm(s,t) =/vector m/parenleftbiggs√ t/parenrightbigg . (4.8) 35As has been already mentioned (see Section 2), part (i)of Theorem 1.1 follows from the fact that the triplet {/vector m,/vector n,/vectorb}is a regular- (C∞(R;S2))3solution of (1.6)-(2.6)-(2.8) and satisfies the equation −s 2c/vector n=β(c′/vectorb−cτ/vector n)+α(cτ/vectorb+c′/vector n). Next, from the parity of the components of the profile /vector m(·)and the asymptotics established in parts(i)and(ii)in Theorem 1.2, it is immediate to prove the pointwise conver gence (1.9). In addition,/vectorA−= (A+ 1,−A+ 2,−A+ 3)in terms of the components of the vector /vectorA+= (A+ j)3 j=1. Now, using the symmetries of /vector m(·), the change of variables η=s/√ tgives us /bardbl/vectorm(·,t)−/vectorA+χ(0,∞)(·)−/vectorA−χ(−∞,0)(·)/bardblLp(R)=3/summationdisplay j=1/parenleftbigg 2t1/2/integraldisplay∞ 0|mj(η)−A+ j|pdη/parenrightbigg1/p .(4.9) Therefore, it only remains to prove that the last integral is finite. To this end, let s0= 4/radicalbig 8+c2 0. On the one hand, notice that since /vector mand/vectorA+are unitary vectors, /integraldisplays0 0|mj(s)−Aj|pds≤2ps0. (4.10) On the other hand, from the asymptotics for /vector m(·)in (1.16), (1.20), and the fact that the vectors /vectorA+and/vectorB+satisfy|/vectorA+|2= 1and|/vectorB+|2= 2, we obtain /parenleftbigg/integraldisplay∞ s0|mj(s)−A+ j|pds/parenrightbigg1/p ≤2√ 2c0(α+β)/parenleftBigg/integraldisplay∞ s0e−αs2p/4 sp/parenrightBigg1/p +2c2 0/parenleftBigg/integraldisplay∞ s0e−αs2p/2 s2p/parenrightBigg1/p +C(c0)/parenleftBigg/integraldisplay∞ s0e−αs2p/4 s3p/parenrightBigg1/p . (4.11) Since the r.h.s. of (4.11) is finite for all p∈(1,∞)ifα∈[0,1], and for all p∈[1,∞)if α∈(0,1], inequality (1.10) follows from (4.9), (4.10) and (4.11). T his completes the proof of Theorem 1.1. Proof of Theorem 1.3 .The proof is a consequence of Proposition 3.15. In fact, reca ll the relations (4.5) and (3.17), that is A+ 1= 2z1,∞−1,andA+ j=zj,∞−1,forj∈ {2,3}, and E0,1=c2 0 8, E 0,j=c2 0 4,forj∈ {2,3}, Thus the continuity of /vectorA+ c0,αwith respect to c0, follows from the continuity of z∞in Proposi- tion 3.15. Using the initial conditions (3.13)–(3.15), the values for the energies E0,jforj∈ {1,2,3}, and the identity√π√α+iβ=√π√ 2/parenleftbig√ 1+α−i√ 1−α/parenrightbig , we now compute /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglefj(0)+f′ j(0)√π√α+iβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 = 1, ifj= 1, 1+c2 0π 4+c0√π√ 2√1+α,ifj= 2, 1+c2 0π 4+c0√π√ 2√1−α,ifj= 3.(4.12) 36Then, substituting the values (4.12) in (3.126) and using th e above relations together with the inequality√1+x≤1+x/2forx≥0, we obtain the estimates (1.24)–(1.26). Proof of Theorem 1.4 .Recall that the components of /vectorA+ c0,αare given explicitly in (4.5) in terms of the functions zj,∞, forj∈ {1,2,3}. The continuity on [0,1]ofA+ j,c0,αas a function ofαforj∈ {1,2,3}follows from that of zj,∞established in Lemma 3.12. Notice also that the estimates (1.27) and (1.28) are an immediate consequence of (3.117) in Lemma 3.12 and (3.114) in Lemma 3.11, respectively. Before giving the proof of Proposition 1.5, we recall that whe nα= 0orα= 1, the vector /vectorA+ c0,α= (Aj,c0,α)3 j=1is determined explicitly in terms of the parameter c0(see [15] for the case α= 0and Appendix for the case α= 1). Precisely, A1,c0,0=e−πc2 0 2, (4.13) A2,c0,0= 1−e−πc2 0 4 8πsinh(πc2 0/2)|c0Γ(ic2 0/4)+2eiπ/4Γ(1/2+ic2 0/4)|2, (4.14) A3,c0,0= 1−e−πc2 0 4 8πsinh(πc2 0/2)|c0Γ(ic2 0/4)−2e−iπ/4Γ(1/2+ic2 0/4)|2(4.15) and /vectorA+ c0,1= (cos(c0√π),sin(c0√π),0). (4.16) Proof of Proposition 1.5 .Recall that (see Theorem 1.1) /vectorA− c0,α= (A+ 1,c0,α,−A+ 2,c0,α,−A+ 3,c0,α), (4.17) withA+ j,c0,αthe components of /vectorA+ c0,α. Therefore /vectorA+ c0,α/ne}ationslash=/vectorA− c0,αiffA+ 1,c0,α/ne}ationslash= 1or−1. Parts (ii)and(iii)follow from the continuity of A+ 1,c0,αin[0,1]established in Theorem 1.4 bearing in mind that, from the expressions for A+ 1,c0,0in (4.13) and A+ 1,c0,1in (4.16), we have that A+ 1,c0,0/ne}ationslash=±1for allc0>0andA+ 1,c0,1/ne}ationslash=±1ifc0/ne}ationslash=k√πwithk∈N. In order to proof part (i), we will argue by contradiction. Assume that for some α∈(0,1), there exists a sequence {c0,n}n∈Nsuch thatc0,n>0,c0,n−→0asn→ ∞ and/vectorA+ c0,n,α=/vectorA− c0,nα. Hence from (4.17) the second and third component of /vectorA+ c0,n,αare zero. Thus the estimate (1.25) in Theorem 1.3 yields c0,n/radicalbig π(1+α)√ 2≤c2 0,nπ 4+c2 0,nπ α√ 2/parenleftBigg 1+c2 0,nπ 8+c0,n/radicalbig π(1+α) 2√ 2/parenrightBigg +/parenleftBigg c2 0,nπ 2√ 2α/parenrightBigg2 . Dividing by c0,n>0and letting c0,n→0asn→ ∞, the contradiction follows. 5 Some numerical results As has been already pointed out, only in the cases α= 0andα= 1we have an explicit formula for/vectorA+ c0,α(see (4.13)–(4.16)). Theorems 1.3 and 1.4 give information about the behaviour of /vectorA+ c0,α for small values of c0for a fixed valued of α, and for values of αnear to 0 or 1 for a fixed valued of c0. The aim of this section is to give some numerical results tha t allow us to understand the map 37(α,c0)∈[0,1]×(0,∞)/ma√sto→/vectorA± c0,α∈S2. For a fixed value of α, we will discuss first the injectivity and surjectivity (in some appropriate sense) of the map c0/ma√sto→/vectorA± c0,αand second the behaviour of /vectorA+ c0,αasc0→ ∞. For fixedα, defineθc0,αto be the angle between the unit vectors /vectorA+ c0,αand−/vectorA− c0,αassociated to the family of solutions /vectormc0,α(s,t)established in Theorem 1.1, that is θc0,αsuch that cos(θc0,α) = 1−2(A+ 1,c0,α)2. (5.1) It is pertinent to ask whether θc0,αmay attain any value in the interval [0,π]by varying the parameterc0>0. In Figure 2 we plot the function θc0,αassociated to the family of solutions /vectormc0,α(s,t)estab- lished in Theorem 1.1 for α= 0,α= 0.4andα= 1, as a function of c0>0. The curves θc0,0 andθc0,1are exact since we have explicit formulae for A+ 1,c0,αwhenα= 0andα= 1(see (4.13) and (4.16)). We deduce that in the case α= 0, there is a bijective relation between c0>0and the angles in (0,π). In the case α= 1, there are infinite values of c0>0that allow to reach any angle in [0,π]. Ifα∈(0,1), numerical simulations show that there exists θ∗ α∈(0,π)such that the angles in (θ∗ α,π)are reached by a unique value of c0, but for angles in [0,θ∗ α]there are at least two values of c0>0that produce them (See θc0,0.4in Figure 2). θc0,0 π c0 θc0,0.4 π c0 θc0,1 π c0 Figure 2: The angles θc0,αas a function of c0forα= 0,α= 0.4andα= 1. These numerical results suggest that, due to the invariance of (LLG) under rotations2, for a fixedα∈[0,1)one can solve the following inverse problem: Given any disti nct vectors /vectorA+,/vectorA−∈ S2there exists c0>0such that the associated solution /vectormc0,α(s,t)given by Theorem 1.1 (possibly multiplied by a rotation matrix) provides a solution of (LLG ) with initial condition /vectorm(·,0) =/vectorA+χ(0,∞)(·)+/vectorA−χ(−∞,0)(·). (5.2) Note that in the case α= 1the restriction /vectorA+/ne}ationslash=/vectorA−can be dropped. In addition, Figure 2 suggests that /vectorA+ c0,α/ne}ationslash=/vectorA− c0,αfor fixedα∈[0,1)andc0>0. Indeed, notice that /vectorA+ c0,α/ne}ationslash=/vectorA− c0,αif and only if A1/ne}ationslash=±1or equivalently cosθc0,α/ne}ationslash=−1, that isθc0,α/ne}ationslash=π, which is true if α∈[0,1)for anyc0>0(See Figure 2). Notice also that when α= 1, then the valueπis attained by different values of c0. The next natural question is the injectivity of the applicat ionc0−→θc0,α, for fixed α. Precisely, can we generate the same angle using different val ues ofc0? In the case α= 0, the 2In fact, using that (M/vector a)×(M/vectorb) = (det M)M−T(/vector a×/vectorb),for allM∈ M3,3(R), /vector a,/vectorb∈R3, it is easy to verify that if /vectorm(s,t)is a solution of (LLG) with initial condition /vectorm0, then/vectormR:=R/vectormis a solution of (LLG) with initial condition /vectorm0 R:=R/vectorm0, for any R∈SO(3). 38plot ofθc0,0in Figure 2 shows that the value of c0is unique, in fact one has following formula sin(θc0,0/2) =A1,c0,0=e−c2 0 2π(see [15]). In the case α= 1, we have sin(θc0,1/2) =A1,c0,1= cos(c0/radicalbig π), moreover /vectorA+ c0,1=/vectorA+ c0+2k√π,1,for anyk∈Z. (5.3) As before, if α∈(0,1)we do not have an analytic answer and we have to rely on numeric al simulations. However, it is difficult to test the uniqueness o fc0numerically. Using the command FindRoot in Mathematica, we have found such values. For instance, for α= 0.4, we obtain that c0≈2.1749andc0≈6.6263give the same value of /vectorA+ c0,0.4. The respective profiles /vector mc0,0.4(·)are shown in Figure 3. This multiplicity of solutions suggests t hat the Cauchy problem for (LLG) with initial condition (5.2) is ill-posed, at least for cert ain values of c0. This interesting problem will be studied in a forthcoming paper. m1m2m3 (a)/vector mc0,0.4(·), withc0≈2.1749 m1m2m3 (b)/vector mc0,0.4(·), withc0≈6.6263 Figure 3: Two profiles /vector mc0,0.4(·), with the same limit vector /vectorA+ c0,0.4. The rest of this section is devoted to give some numerical res ults on the behaviour of the limiting vector /vectorA+ c0,α. In particular, the results below aim to complement those es tablished in Theorem 1.3 on the behaviour of /vectorA+ c0,αfor small values of c0, whenαis fixed. We start recalling what it is known in the extremes cases α= 0andα= 1. Precisely, if α= 0, the explicit formulae (4.13)–(4.15) for /vectorA+ c0,0allow us to prove that lim c0→0+A+ 3,c0,0= 0 andlim c0→∞A+ 3,c0,1= 1, (5.4) and also that {A+ 3,c0,0:c0∈(0,∞)}= (0,1). Whenα= 1the picture is completely different. In factA+ 3,c0,1= 0for allc0>0, and the limit vectors remain in the equator plane S1×{0}. The natural question is what happens with /vectorA+ c0,αwhenα∈(0,1)as a function of c0. Although we do not provide a rigorous answer to this question , in Figure 4 we show some numerical results. Precisely, Figure 4 depicts the curves /vectorA+ c0,0.01,/vectorA+ c0,0.4and/vectorA+ c0,0.8as functions ofc0, forc0∈[0,1000]. We see that the behaviour of /vectorA+ c0,αchanges when αincreases in the sense that the first and second coordinates start oscillating more and more as αgoes to 1. In all the cases the third component remains monotonically increasin g withc0, but the value of A+ 3,1000,α seems to be decreasing with α. At this point it is not clear what the limit value of A+ 3,c0,αas 39c0→ ∞ is. For this reason, we perform a more detailed analysis of A+ 3,c0,αand we show the curvesA+ 3,1,α,A+ 3,10,α,A+ 3,1000,α(for fixedα∈[0,1]) in Figure 5. From these results we conjecture that{A+ 3,c0,·}c0>0is a pointwise nondecreasing sequence of functions that con verges to 1for any α<1asc0→ ∞. This would imply that, for α∈(0,1)fixed,A1,c0,α→0asc0→ ∞, and since A1,c0,α→1asc0→0(see (1.24)), we could conclude by continuity (see Theorem 1 .3) that for any angleθ∈(0,π)there exists c0>0such thatθis the angle between /vectorA+ c0,αand−/vectorA+ c0,α(see (5.1)). This provides an alternative way to justify the surj ectivity of the map c0/ma√sto→/vectorA+ c0,α(in the sense explained above). A+ 1A+ 2A+ 3 (a)/vectorA+ c0,0.01 A+ 1A+ 2A+ 3 (b)/vectorA+ c0,0.4 A+ 1A+ 2A+ 3 (c)/vectorA+ c0,0.8 Figure 4: The curves /vectorA+ c0,0.01,/vectorA+ c0,0.4and/vectorA+ c0,0.8as functions of c0, forc0∈[0,1000]. 01 1αA+ 3,1,αA+ 3,10,αA+ 3,1000,α Figure 5: The curves A+ 3,1,α,A+ 3,10,α,A+ 3,1000,αas functions of α, forα∈[0,1]. The curves in Figure 5 also allow us to discuss further the res ults in Theorem 1.4. In fact, whenαis close to 1 the slope of the functions become unbounded and, roughly speaking, the behaviour of A+ 3,c0,αis in agreement with the result in Theorem 1.4, that is A+ 3,c0,α∼C(c0)√ 1−α,asα→1−. Numerically, the analysis is more difficult when α∼0, because the number of computations needed to have an accurate profile of A+ 3,c0,αincreases drastically as α→0+. In any case, Figure 5 suggests that A+ 3,c0,αconverges to A+ 3,c0,0faster than√α|ln(α)|. We think that this rate of convergence can be improved to α|ln(α)|. In fact, in the proof of Lemma 3.10 we only used energy estimates. Probably, taking into account the oscill ations in equation (3.102) (as did in Proposition 3.3), it would be possible to establish the nece ssary estimates to prove the following conjecture: |/vectorA+ c0,α−/vectorA+ c0,0| ≤C(c0)α|ln(α)|,forα∈(0,1/2]. 406 Appendix In this appendix we show how to compute explicitly the soluti on/vectormc0,α(s,t)of the LLG equation in the case α= 1. As a consequence, we will obtain an explicit formula for the limiting vector /vectorA+ c0,1and the other constants appearing in the asymptotics of the a ssociated profile established in Theorem 1.2 in terms of the parameter c0in the case when α= 1. We start by recalling that if α= 1thenβ= 0. We need to find the solution {/vector m,/vector n,/vectorb}of the Serret–Frenet system (1.6) with c(s) =c0e−s2/4,τ≡0and the initial conditions (1.8). Hence, it is immediate that m3=n3≡0, b1=b2≡0andb3≡1. To compute the other components, we use the Riccati equation (3.9) satisfied by the stereographic projection of {mj,nj,bj} ηj=nj+ibj 1+mj,forj∈ {1,2}, (6.1) found in the proof of Lemma 3.1. For the values of curvature an d torsionc(s) =c0e−s2/4and τ(s) = 0 the Riccati equation (3.9) reads η′ j+iβs 2ηj+c0 2e−αs2/4(η2 j+1) = 0. (6.2) We see that when α= 1, and thusβ= 0, (6.2) is a separable equation that we write as: dηj η2 j+1=−c0 2e−αs2/4, so integrating, we get ηj(s) = tan/parenleftBig arctan(ηj(0))−c0 2Erf(s)/parenrightBig , (6.3) whereErf(s)is the non-normalized error function Erf(s) =/integraldisplays 0e−σ2/4dσ. Also, using (1.8) and (6.1) we get the initial conditions η1(0) = 0 andη2(0) = 1 . In particular, ifc0is small (6.3) is the global solution of the Riccati equation , but it blows-up in finite time if c0is large. As long as ηjis well-defined, by Lemma 3.1, fj(s) =ec0 2/integraltexts 0e−ασ2/4ηj(σ)dσ. The change of variables µ= arctan(ηj(0))−c0 2Erf(s) yields/integraldisplays 0e−ασ2/4ηj(σ)dσ=2 c0ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos/parenleftbig arctan(ηj(0))−c0 2Erf(s)/parenrightbig cos(arctan( ηj(0)))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, and after some simplifications, we obtain f1(s) =/vextendsingle/vextendsingle/vextendsinglecos/parenleftBigc0 2Erf(s)/parenrightBig/vextendsingle/vextendsingle/vextendsingleandf2(s) =/vextendsingle/vextendsingle/vextendsinglecos/parenleftBigc0 2Erf(s)/parenrightBig +sin/parenleftBigc0 2Erf(s)/parenrightBig/vextendsingle/vextendsingle/vextendsingle. In view of (3.18) and (3.19), we conclude that m1(s) = 2|f1(s)|2−1 = cos(c0Erf(s))andm2(s) =|f2(s)|2−1 = sin(c0Erf(s)).(6.4) 41A priori, the formulae in (6.4) are valid only as long as ηis well-defined, but a simple verification show that these are the global solutions of (1.6), with n1(s) =−sin(c0Erf(s))andn2(s) = cos(c0Erf(s)). In conclusion, we have proved the following: Proposition 6.1. Letα= 1, and thusβ= 0. Then, the trihedron {/vector mc0,1,/vector nc0,1,/vectorbc0,1}solution of(1.6)–(1.8)is given by /vector mc0,1(s) = (cos(c0Erf(s)),sin(c0Erf(s)),0), /vector nc0,1(s) =−(sin(c0Erf(s)),cos(c0Erf(s)),0), /vectorbc0,1(s) = (0,0,1), for alls∈R. In particular, the limiting vectors /vectorA+ c0,1and/vectorA− c0,1in Theorem 1.2 are given in terms ofc0as follows: /vectorA± c0,1= (cos(c0√π),±sin(c0√π),0). Proposition 6.1 allows us to give an alternative explicit pr oof of Theorem 1.2 when α= 1. Corollary 6.2. [Explicit asymptotics when α= 1] With the same notation as in Proposition 6.1, the following asymptotics for {/vector mc0,1,/vector nc0,1,/vectorbc0,1}holds true: /vector mc0,1(s) =/vectorA+ c0,1−2c0 s/vectorB+ c0,1e−s2/4sin(/vector a)−2c2 0 s2/vectorA+ c0,1e−s2/2+O/parenleftBigg e−s2/4 s3/parenrightBigg , /vector nc0,1(s) =/vectorB+ c0,1sin(/vector a)+2c0 s/vectorA+ c0,1e−s2/4−2c2 0 s2/vectorB+ c0,1e−s2/2sin(/vector a)+O/parenleftBigg e−s2/4 s3/parenrightBigg , /vectorbc0,1(s) =/vectorB+ c0,1cos(/vector a), where the vectors /vectorA+ c0,1,/vectorB+ c0,1and/vector a= (aj)3 j=1are given explicitly in terms of c0by /vectorA+ c0,1= (cos(c0√π),sin(c0√π),0),/vectorB+ c0,1= (|sin(c0√π)|,|cos(c0√π)|,1), a1=/braceleftBigg 3π 2,ifsin(c0√π)≥0, π 2,ifsin(c0√π)<0,a2=/braceleftBigg π 2,ifcos(c0√π)≥0, 3π 2,ifcos(c0√π)<0,anda3= 0. Here, the bounds controlling the error terms depend on c0. Proof. By Proposition 6.1, /vector mc0,1(s) = (cos(c0√π−c0Erfc(s)),sin(c0√π−c0Erfc(s)),0), /vector nc0,1(s) =−(sin(c0√π−c0Erfc(s)),cos(c0√π−c0Erfc(s)),0), /vectorbc0,1(s) = (0,0,1),(6.5) where the complementary error function is given by Erfc(s) =/integraldisplay∞ se−σ2/4dσ=√π−Erf(s). It is simple to check that sin(c0Erfc(s)) =e−s2/4/parenleftbigg2c0 s−4c0 s3+24c0 s5+O/parenleftBigc0 s7/parenrightBig/parenrightbigg , cos(c0Erfc(s)) = 1+e−s2/2/parenleftbigg −2c2 0 s2+8c2 0 s4−56c2 0 s6+O/parenleftbiggc2 0 s8/parenrightbigg/parenrightbigg , 42so that, using (6.5), we obtain that m1(s) =n2(s) = cos(c0√π)+2c0 se−s2/4sin(c0√π)−2c2 0 s2e−s2/2cos(c0√π)+O/parenleftBigg e−s2/4 s3/parenrightBigg , m2(s) =−n1(s) = sin(c0√π)−2c0 se−s2/4cos(c0√π)−2c2 0 s2e−s2/2sin(c0√π)+O/parenleftBigg e−s2/4 s3/parenrightBigg . The conclusion follows from the definitions of /vectorA+ c0,1,/vectorB+ c0,1and/vector a. Remark 6.3. Notice that /vector ais not a continuous function of c0, but the vectors (B+ jsin(aj))3 j=1 and(B+ jcos(aj))3 j=1are. References [1] M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables , volume 55 of National Bureau of Standards Applied Mathematics Series . For sale by the Superintendent of Documents, U.S. Governme nt Printing Office, Washington, D.C., 1964. [2] V. Banica and L. Vega. On the Dirac delta as initial conditi on for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire , 25(4):697–711, 2008. [3] H. Brézis. Opérateurs maximaux monotones et semi-groupes de contract ions dans les espaces de Hilbert . North-Holland Publishing Co., Amsterdam, 1973. North-Ho lland Mathematics Studies, No. 5. Notas de Matemática (50). [4] T. F. Buttke. A numerical study of superfluid turbulence in the self-induction approxima- tion.Journal of Computational Physics , 76(2):301–326, 1988. [5] E. A. Coddington and N. Levinson. Theory of ordinary differential equations . McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. [6] M. Daniel and M. Lakshmanan. Soliton damping and energy l oss in the classical continuum heisenberg spin chain. Physical Review B , 24(11):6751–6754, 1981. [7] M. Daniel and M. Lakshmanan. Perturbation of solitons in the classical continuum isotropic Heisenberg spin system. Physica A: Statistical Mechanics and its Applications , 120(1):125– 152, 1983. [8] G. Darboux. Leçons sur la théorie générale des surfaces. I, II . Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics]. Éd itions Jacques Gabay, Sceaux, 1993. Généralités. Coordonnées curvilignes. Surfaces minima. [ Generalities. Curvilinear coordi- nates. Minimum surfaces], Les congruences et les équations linéaires aux dérivées partielles. Les lignes tracées sur les surfaces. [Congruences and linea r partial differential equations. Lines traced on surfaces], Reprint of the second (1914) edit ion (I) and the second (1915) edition (II), Cours de Géométrie de la Faculté des Sciences. [Course on Geometry of the Faculty of Science]. [9] F. de la Hoz, C. J. García-Cervera, and L. Vega. A numerica l study of the self-similar solutions of the Schrödinger map. SIAM J. Appl. Math. , 70(4):1047–1077, 2009. 43[10] P. Germain and M. Rupflin. Selfsimilar expanders of the h armonic map flow. Ann. Inst. H. Poincaré Anal. Non Linéaire , 28(5):743–773, 2011. [11] T. L. Gilbert. A lagrangian formulation of the gyromagn etic equation of the magnetization field.Phys. Rev. , 100:1243, 1955. [12] A. Grünrock. Bi- and trilinear Schrödinger estimates in one space dimension with applica- tions to cubic NLS and DNLS. Int. Math. Res. Not. , (41):2525–2558, 2005. [13] M. Guan, S. Gustafson, K. Kang, and T.-P. Tsai. Global qu estions for map evolution equations. In Singularities in PDE and the calculus of variations , volume 44 of CRM Proc. Lecture Notes , pages 61–74. Amer. Math. Soc., Providence, RI, 2008. [14] B. Guo and S. Ding. Landau-Lifshitz equations , volume 1 of Frontiers of Research with the Chinese Academy of Sciences . World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. [15] S. Gutiérrez, J. Rivas, and L. Vega. Formation of singul arities and self-similar vortex motion under the localized induction approximation. Comm. Partial Differential Equations , 28(5- 6):927–968, 2003. [16] S. Gutiérrez and L. Vega. Self-similar solutions of the localized induction approximation: singularity formation. Nonlinearity , 17:2091–2136, 2004. [17] P. Hartman. Ordinary differential equations . John Wiley & Sons Inc., New York, 1964. [18] H. Hasimoto. A soliton on a vortex filament. J. Fluid Mech , 51(3):477–485, 1972. [19] A. Hubert and R. Schäfer. Magnetic domains: the analysis of magnetic microstructure s. Springer, 1998. [20] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev. Magnetic s olitons. Physics Reports , 194(3-4):117–238, 1990. [21] M. Lakshmanan. The fascinating world of the Landau-Lif shitz-Gilbert equation: an overview. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 369(1939):1280–1300, 2011. [22] M. Lakshmanan, T. W. Ruijgrok, and C. Thompson. On the dy namics of a continuum spin system. Physica A: Statistical Mechanics and its Applications , 84(3):577–590, 1976. [23] G. L. Lamb, Jr. Elements of soliton theory . John Wiley & Sons Inc., New York, 1980. Pure and Applied Mathematics, A Wiley-Interscience Publicatio n. [24] L. Landau and E. Lifshitz. On the theory of the dispersio n of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion , 8:153–169, 1935. [25] F. Lin and C. Wang. The analysis of harmonic maps and their heat flows . World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. [26] T. Lipniacki. Shape-preserving solutions for quantum vortex motion under localized induc- tion approximation. Phys. Fluids , 15(6):1381–1395, 2003. [27] M. Steiner, J. Villain, and C. Windsor. Theoretical and experimental studies on one- dimensional magnetic systems. Advances in Physics , 25(2):87–209, 1976. 44[28] D. J. Struik. Lectures on Classical Differential Geometry . Addison-Wesley Press, Inc., Cambridge, Mass., 1950. [29] A. Vargas and L. Vega. Global wellposedness for 1D non-l inear Schrödinger equation for data with an infinite L2norm. J. Math. Pures Appl. (9) , 80(10):1029–1044, 2001. 45

2014-09-08

We consider the one-dimensional Landau-Lifshitz-Gilbert (LLG) equation, a model describing the dynamics for the spin in ferromagnetic materials. Our main aim is the analytical study of the bi-parametric family of self-similar solutions of this model. In the presence of damping, our construction provides a family of global solutions of the LLG equation which are associated to a discontinuous initial data of infinite (total) energy, and which are smooth and have finite energy for all positive times. Special emphasis will be given to the behaviour of this family of solutions with respect to the Gilbert damping parameter. We would like to emphasize that our analysis also includes the study of self-similar solutions of the Schr\"odinger map and the heat flow for harmonic maps into the 2-sphere as special cases. In particular, the results presented here recover some of the previously known results in the setting of the 1d-Schr\"odinger map equation.

Self-similar solutions of the one-dimensional Landau-Lifshitz-Gilbert equation

1409.2340v1

arXiv:1104.1625v1 [cond-mat.mes-hall] 8 Apr 2011Magnetization Dissipation in Ferromagnets from Scatterin g Theory Arne Brataas∗ Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Yaroslav Tserkovnyak Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA Gerrit E. W. Bauer Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan and Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands The magnetization dynamicsofferromagnets are often formu lated intermsof theLandau-Lifshitz- Gilbert (LLG) equation. The reactive part of this equation d escribes the response of the magnetiza- tion in terms of effective fields, whereas the dissipative par t is parameterized by the Gilbert damping tensor. We formulate a scattering theory for the magnetizat ion dynamics and map this description on the linearized LLG equation by attaching electric contac ts to the ferromagnet. The reactive part can then be expressed in terms of the static scattering matri x. The dissipative contribution to the low-frequency magnetization dynamics can be described as a n adiabatic energy pumping process to the electronic subsystem by the time-dependent magnetiz ation. The Gilbert damping tensor depends on the time derivative of the scattering matrix as a f unction of the magnetization direction. By the fluctuation-dissipation theorem, the fluctuations of the effective fields can also be formulated in terms of the quasistatic scattering matrix. The theory is formulated for general magnetization textures and worked out for monodomain precessions and doma in wall motions. We prove that the Gilbert damping from scattering theory is identical to the r esult obtained by the Kubo formalism. PACS numbers: 75.40.Gb,76.60.Es,72.25.Mk I. INTRODUCTION Ferromagnets develop a spontaneous magnetization below the Curie temperature. The long-wavelengthmod- ulations of the magnetization direction consist of spin waves, the low-lying elementary excitations (Goldstone modes) of the ordered state. When the thermal energy is much smaller than the microscopic exchange energy, the magnetization dynamics can be phenomenologically ex- pressed in a generalized Landau-Lifshitz-Gilbert (LLG) form: ˙ m(r,t) =−γm(r,t)×[Heff(r,t)+h(r,t)]+ m(r,t)×/integraldisplay dr′[˜α[m](r,r′)˙ m(r′,t)],(1) where the magnetization texture is described by m(r,t), the unit vector along the magnetization direction at po- sitionrand timet,˙ m(r,t) =∂m(r,t)/∂t,γ=gµB//planckover2pi1is the gyromagnetic ratio in terms of the g-factor (≈2 for free electrons) and the Bohr magneton µB. The Gilbert damping ˜αis a nonlocal symmetric 3 ×3 tensor that is a functional of m. The Gilbert damping tensor is com- monly approximated to be diagonal and isotropic (i), lo- cal (l), and independent of the magnetization m, with diagonal elements αil(r,r′) =αδ(r−r′). (2) The linearized version of the LLG equation for small- amplitude excitations has been derived microscopically.1It has been used very successfully to describe the mea- sured response of ferromagnetic bulk materials and thin films in terms of a small number of adjustable, material- specific parameters. The experiment of choice is fer- romagnetic resonance (FMR), which probes the small- amplitude coherent precession of the magnet.2The Gilbertdampingmodelinthelocalandtime-independent approximationhasimportantramifications, suchasalin- ear dependence of the FMR line width on resonance fre- quency, that have been frequently found to be correct. The damping constant is technologically important since it governs the switching rate of ferromagnets driven by external magnetic fields or electric currents.3In spatially dependent magnetization textures, the nonlocal charac- ter of the damping can be significant as well.4–6Moti- vated by the belief that the Gilbert damping constant is animportantmaterialproperty, weset outheretounder- stand its physical origins from first principles. We focus on the well studied and technologically important itiner- ant ferromagnets, although the formalism can be used in principle for any magnetic system. The reactive dynamics within the LLG Eq. (1) is de- scribed by the thermodynamic potential Ω[ M] as a func- tional of the magnetization. The effective magnetic field Heff[M](r)≡ −δΩ/δM(r) is the functional derivative with respect to the local magnetization M(r) =Msm(r), including the external magnetic field Hext, the magnetic dipolar field Hd, the texture-dependent exchange energy, and crystal field anisotropies. Msis the saturation mag- netization density. Thermal fluctuations can be included by a stochastic magnetic field h(r,t) with zero time av-2 left reservoirF N Nright reservoir FIG. 1: Schematic picture of a ferromagnet (F) in contact with a thermal bath (reservoirs) via metallic normal metal leads (N). erage,/an}b∇acketle{th/an}b∇acket∇i}ht= 0, and white-noise correlation:7 /an}b∇acketle{thi(r,t)hj(r′,t′)/an}b∇acket∇i}ht=2kBT γMs˜αij[m](r,r′)δ(t−t′),(3) whereMsis the magnetization, iandjare the Cartesian indices, and Tis the temperature. This relation is a con- sequence ofthe fluctuation-dissipation theorem (FDT) in the classical (Maxwell-Boltzmann) limit. The scattering ( S-) matrix is defined in the space of the transport channels that connect a scattering region (the sample) to real or fictitious thermodynamic (left and right) reservoirs by electric contacts with leads that are modeled as ideal wave guides. Scattering matri- ces are known to describe transport properties, such as the giant magnetoresistance, spin pumping, and current- inducedmagnetizationdynamicsinlayerednormal-metal (N)|ferromagnet (F).8–10When the ferromagnet is part of an open system as in Fig. 1, also Ω can be expressed in terms of the scattering matrix, which has been used to express the non-local exchange coupling between fer- romagnetic layers through conducting spacers.11We will show here that the scattering matrix description of the effective magnetic fields is valid even when the system is closed, provided the dominant contribution comes from the electronic band structure, scattering potential disor- der, and spin-orbit interaction. Scattering theory can also be used to compute the Gilbert damping tensor ˜ αfor magnetization dynamics.15 The energy loss rate of the scattering region can be ex- pressedin termsofthe time-dependent S-matrix. To this end, the theory of adiabatic quantum pumping has to be generalizedtodescribedissipationinametallicferromag- net. The Gilbert damping tensor is found by evaluating the energy pumping out of the ferromagnet and relat- ing it to the energy loss that is dictated by the LLG equation. In this way, it is proven that the Gilbert phe- nomenology is valid beyond the linear response regime of small magnetization amplitudes. The key approxima- tion that is necessary to derive Eq. (1) including ˜ αis the (adiabatic) assumption that the ferromagnetic resonance frequencyωFMRthat characterizesthe magnetizationdy- namics is small compared to internal energy scale set by the exchange splitting ∆ and spin-flip relaxation rates τs. The LLG phenomenology works well for ferromag- nets for which ωFMR≪∆//planckover2pi1, which is certainly the case for transition metal ferromagnets such as Fe and Co. Gilbert damping in transition-metal ferromagnets is generally believed to stem from the transfer of energy fromthemagneticorderparametertotheitinerantquasi-particle continuum. This requires either magnetic disor- der or spin-orbit interactions in combination with impu- rity/phonon scattering.2Since the heat capacitance of the ferromagnet is dominated by the lattice, the energy transferred to the quasiparticles will be dissipated to the lattice as heat. Here we focus on the limit in which elas- tic scattering dominates, such that the details of the heat transfer to the lattice does not affect our results. Our ap- proachformallybreaks down in sufficiently clean samples at high temperatures in which inelastic electron-phonon scattering dominates. Nevertheless, quantitative insight can be gained by our method even in that limit by mod- elling phonons by frozen deformations.12 In the present formulation, the heat generated by the magnetization dynamics can escape only via the contacts to the electronic reservoirs. By computing this heat cur- rent through the contacts we access the total dissipa- tion rate. Part of the heat and spin current that es- capes the sample is due to spin pumping that causes energy and momentum loss even for otherwise dissipa- tion less magnetization dynamics. This process is now wellunderstood.10For sufficiently largesamples, the spin pumping contribution is overwhelmed by the dissipation in the bulk of the ferromagnet. Both contributions can be separated by studying the heat generation as a func- tion of the length of a wire. In principle, a voltage can be added to study dissipation in the presence of electric cur- rents as in 13,14, but we concentrate here on a common and constant chemical potential in both reservoirs. Although it is not a necessity, results can be simpli- fied by expanding the S-matrix to lowest order in the amplitude of the magnetization dynamics. In this limit scattering theory and the Kubo linear response formal- ism for the dissipation can be directly compared. We will demonstrate explicitly that both approaches lead to identical results, which increases our confidence in our method. The coupling to the reservoirs of large samples is identified to play the same role as the infinitesimals in the Kubo approach that guarantee causality. Our formalism was introduced first in Ref. 15 lim- ited to the macrospin model and zero temperature. An extension to the friction associatedwith domain wall mo- tion was given in Ref. 13. Here we show how to handle general magnetization textures and finite temperatures. Furthermore, we offer an alternative route to derive the Gilbert damping in terms of the scattering matrix from the thermal fluctuations of the effective field. We also explain in more detail the relation of the present theory to spin and charge pumping by magnetization textures. Our paper is organized in the following way. In Sec- tion II, we introduce our microscopic model for the fer- romagnet. In Section III, dissipation in the Landau- Lifshitz-Gilbert equation is exposed. The scattering the- ory of magnetization dynamics is developed in Sec. IV. We discuss the Kubo formalism for the time-dependent magnetizationsin Sec. V, before concluding our article in Sec. VI. The Appendices provide technical derivations of spin, charge, and energy pumping in terms of the scat-3 tering matrix of the system. II. MODEL Our approach rests on density-functional theory (DFT), which is widely and successfully used to describe the electronic structure and magnetism in many fer- romagnets, including transition-metal ferromagnets and ferromagnetic semiconductors.16In the Kohn-Sham im- plementation of DFT, noninteracting hypothetical par- ticles experience an effective exchange-correlationpoten- tial that leads to the same ground-statedensity as the in- teractingmany-electronsystem.17Asimpleyetsuccessful scheme is the local-densityapproximationto the effective potential. DFT theory can also handle time-dependent phenomena. We adopt here the adiabatic local-density approximation (ALDA), i.e. an exchange-correlationpo- tential that is time-dependent, but local in time and space.18,19As the name expresses, the ALDA is valid when the parametric time-dependence of the problem is adiabatic with respect to the electron time constants. Here we consider a magnetization direction that varies slowly in both space and time. The ALDA should be suited to treat magnetization dynamics, since the typical time scale ( tFMR∼1/(10 GHz) ∼10−10s) is long com- paredtothethat associatedwith theFermi andexchange energies, 1 −10 eV leading to /planckover2pi1/∆∼10−13s in transition metal ferromagnets. In the ALDA, the system is described by the time- dependent effective Schr¨ odinger equation ˆHALDAΨ(r,t) =i/planckover2pi1∂ ∂tΨ(r,t), (4) where Ψ( r,t) is the quasiparticle wave function at posi- tionrand timet. We consider a generic mean-field elec- tronic Hamiltonian that depends on the magnetization direction ˆHALDA[m] and includes the periodic Hartree, exchange and correlation potentials and relativistic cor- rectionssuchasthe spin-orbitinteraction. Impurityscat- tering including magnetic disorder is also represented by ˆHALDA.The magnetization mis allowed to vary in time and space. The total Hamiltonian depends additionally on the Zeeman energy of the magnetization in external Hextand dipolar Hdmagnetic fields: ˆH=ˆHALDA[m]−Ms/integraldisplay drm·(Hext+Hd).(5) For this general Hamiltonian (5), our task is to de- duce an expression for the Gilbert damping tensor ˜ α. To this end, from the form of the Landau-Lifshitz-Gilbert equation (3), it is clear that we should seek an expansionin terms of the slow variations of the magnetizations in time. Such an expansion is valid provided the adiabatic magnetization precession frequency is much less than the exchange splitting ∆ or the spin-orbit energy which gov- erns spin relaxation of electrons. We discuss first dissi- pation in the LLG equation and subsequently compare it with the expressions from scattering theory of electron transport. This leads to a recipe to describe dissipation by first principles. Finally, we discuss the connection to the Kubo linear response formalism and prove that the two formulations are identical in linear response. III. DISSIPATION AND LANDAU-LIFSHITZ-GILBERT EQUATION The energy dissipation can be obtained from the solu- tion of the LLG Eq. (1) as ˙E=−Ms/integraldisplay dr[˙ m(r,t)·Heff(r,t)] (6) =−Ms γ/integraldisplay dr/integraldisplay dr′˙ m(r)·˜α[m](r,r′)·˙ m(r′).(7) Thescatteringtheoryofmagnetizationdissipationcanbe formulated for arbitrary spatiotemporal magnetization textures. Much insight can be gained for certain special cases. In small particles or high magnetic fields the col- lective magnetization motion is approximately constant in space and the “macrospin” model is valid in which all spatial dependences are disregarded. We will also consider special magnetization textures with a dynamics characterized by a number of dynamic (soft) collective coordinates ξa(t) counted by a:20,21 m(r,t) =mst(r;{ξa(t)}), (8) wheremstis the profile at t→ −∞.This representation has proven to be very effective in handling magnetiza- tion dynamics of domain walls in ferromagnetic wires. The description is approximate, but (for few variables) it becomes exact in special limits, such as a transverse domain wall in wires below the Walker breakdown (see below); it becomes arbitrarily accurate by increasing the number of collective variables. The energy dissipation to lowest (quadratic) order in the rate of change ˙ξaof the collective coordinates is ˙E=−/summationdisplay ab˜Γab˙ξa˙ξb, (9) The (symmetric) dissipation tensor ˜Γabreads4 ˜Γab=Ms γ/integraldisplay dr/integraldisplay dr′∂mst(r) ∂ξaα[m](r,r′)·∂mst(r′) ∂ξb. (10) The equation of motion of the collective coordinates un- der a force F=−∂Ω ∂ξ(11) are20,21 ˜η˙ξ+[F+f(t)]−˜Γ˙ξ= 0, (12) introducing the antisymmetric and time-independent gy- rotropic tensor: ˜ηab=Ms γ/integraldisplay drmst(r)·/bracketleftbigg∂mst(r) ∂ξa×∂mst(r) ∂ξb/bracketrightbigg .(13) We show below that Fand˜Γ can be expressed in terms of the scattering matrix. For our subsequent discussions it is necessary to include a fluctuating force f(t) (with /an}b∇acketle{tf(t)/an}b∇acket∇i}ht= 0),which has not been considered in Refs. 20,21. From Eq. (3) if follows the time correlation of fis white and obeys the fluctuation-dissipation theorem: /an}b∇acketle{tfa(t)fb(t′)/an}b∇acket∇i}ht= 2kBT˜Γabδ(t−t′). (14) In the following we illustrate the collective coordinate description of magnetization textures for the macrospin model and the Walker model for a transverse domain wall. The treatment is easily extended to other rigid textures such as magnetic vortices. A. Macrospin excitations When high magnetic fields are applied or when the system dimensions are small the exchange stiffness dom- inates. In both limits the magnetization direction and its low energy excitations lie on the unit sphere and its magnetization dynamics is described by the polar angles θ(t) andϕ(t): m= (sinθcosϕ,sinθsinϕ,cosθ).(15) The diagonal components of the gyrotropic tensor vanish by (anti)symmetry ˜ ηθθ= 0, ˜ηϕϕ= 0.Its off-diagonal components are ηθϕ=MsV γsinθ=−ηϕθ. (16) Vis the particle volume and MsVthe total magnetic moment. We now have two coupled equations of motion MsV γ˙ϕsinθ−∂Ω ∂θ−/parenleftBig ˜Γθθ˙θ+˜Γθϕ˙ϕ/parenrightBig = 0,(17) −MsV γ˙θsinθ−∂Ω ∂ϕ−/parenleftBig ˜Γϕθ˙θ+˜Γϕϕ˙ϕ/parenrightBig = 0.The thermodynamic potential Ω determines the ballistic trajectories of the magnetization. The Gilbert damping tensor˜Γabwill be computed below, but when isotropic and local, ˜Γ =˜1δ(r−r′)Msα/γ, (18) where˜1 is a unit matrix in the Cartesian basis and α is the dimensionless Gilbert constant, Γ θθ=MsVα/γ, Γθϕ= 0 = Γ ϕθ, and Γ ϕϕ= sin2θMsVα/γ. B. Domain Wall Motion We focus on a one-dimensional model, in which the magnetization gradient, magnetic easy axis, and external magnetic field point along the wire ( z) axis. The mag- netic energy of such a wire with transverse cross section Scan be written as22 Ω =MsS/integraldisplay dzφ(z), (19) in terms of the one-dimensional energy density φ=A 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 −Hamz+K1 2/parenleftbig 1−m2 z/parenrightbig +K2 2m2 x,(20) whereHais the applied field and Ais the exchange stiff- ness. Here the easy-axis anisotropy is parametrized by an anisotropy constant K1. In the case of a thin film wire, there is also a smaller anisotropy energy associated with the magnetization transverse to the wire governed byK2. In a cylindrical wire from a material without crystal anisotropy (such as permalloy) K2= 0. When the shape of such a domain wall is pre- served in the dynamics, three collective coordinates characterize the magnetization texture: the domain wall position ξ1(t) =rw(t), the polar angle ξ2(t) = ϕw(t), and the domain wall width λw(t). We con- sider a head-to-head transverse domain wall (a tail- to-tail wall can be treated analogously). m(z) = (sinθwcosϕw,sinθwsinϕw,cosθw), where cosθw= tanhrw−z λw(21) and cscθw= coshrw−z λw(22) minimizes the energy (20) under the constraint that the magnetization to the far left and right points towardsthe5 domain wall. The off-diagonal elements are then ˜ ηrl= 0 = ˜ηlrand ˜ηrϕ=−2Ms/γ=−˜ηϕr.The energy (20) reduces to Ω =MsS/bracketleftbig A/λw−2Har+K1λw+K2λwcos2ϕw/bracketrightbig . (23) Disregarding fluctuations, the equation of motion Eq. (12) can be expanded as: 2˙rw+αϕϕ˙ϕ+αϕr˙rw+αϕλ˙λw=γK2λwsin2ϕw, (24) −2 ˙ϕ+αrr˙rw+αrϕ˙ϕ+αrλ˙λw= 2γHa, (25) A/λ2 w+αλr˙rw+αλϕ˙ϕ+αλλ˙λw=K1+K2cos2ϕw, (26) whereαab=γΓab/MsS. When the Gilbert dampingtensorisisotropicandlocal in the basis of the Cartesian coordinates, ˜Γ =˜1δ(r− r′)Msα/γ αrr=2α λw;αϕϕ= 2αλw;αλλ=π2α 6λw.(27) whereas all off-diagonal elements vanish. Most experiments are carried out on thin film ferro- magnetic wires for which K2is finite. Dissipation is es- pecially simple below the Walker threshold, the regime in which the wall moves with a constant drift velocity, ˙ϕw= 0 and23 ˙rw=−2γHa/αrr. (28) The Gilbert damping coefficient αrrcan be obtained di- rectly from the scattering matrix by the parametric de- pendence of the scattering matrix on the center coordi- nate position rw. When the Gilbert damping tensor is isotropic and local, we find ˙ rw=λwγHa/α. The domain wall width λw=/radicalbig A/(K1+K2cos2ϕw) and the out- of-plane angle ϕw=1 2arcsin2γHa/αK2. At the Walker- breakdownfield ( Ha)WB=αK2/(2γ) the sliding domain wall becomes unstable. In a cylindrical wire without anisotropy, K2= 0,ϕwis time-dependent and satisfies ˙ϕw=−(2+αϕr) αϕϕ˙rw (29) while ˙rw=2γHa 2/parenleftBig 2+αϕr αϕϕ/parenrightBig +αrr. (30) For isotropic and local Gilbert damping coefficients,22 ˙rw λw=αγHa 1+α2. (31) Inthe nextsection, weformulatehowthe Gilbert scatter- ing tensor can be computed from time-dependent scat- tering theory.IV. SCATTERING THEORY OF MESOSCOPIC MAGNETIZATION DYNAMICS Scattering theory of transport phenomena24has proven its worth in the context of magnetoelectronics. It has been used advantageously to evaluate the non- local exchange interactions multilayers or spin valves,11 the giantmagnetoresistance,25spin-transfertorque,9and spin pumping.10We first review the scattering theory of equilibrium magnetic properties and anisotropy fields and then will turn to non-equilibrium transport. A. Conservative forces Considering only the electronic degrees of freedom in our model, the thermodynamic (grand) potential is de- fined as Ω =−kBTlnTre−(ˆHALDA−µˆN), (32) whileµis the chemical potential, and ˆNis the number operator. The conservative force F=−∂Ω ∂ξ. (33) can be computed for an open systems by defining a scat- teringregionthat isconnectedby idealleadstoreservoirs at common equilibrium. For a two-terminal device, the flow of charge, spin, and energy between the reservoirs can then be described in terms of the S-matrix: S=/parenleftbigg r t′ t r′/parenrightbigg , (34) whereris the matrix of probability amplitudes of states impinging from and reflected into the left reservoir, while tdenotes the probability amplitudes of states incoming from the left and transmitted to the right. Similarly, r′andt′describes the probability amplitudes for states that originate from the right reservoir. r,r′,t, andt′are matricesin the space spanned by eigenstates in the leads. We areinterested in the free magnetic energymodulation by the magnetic configuration that allows evaluation of the forces Eq. (33). The free energy change reads ∆Ω =−kBT/integraldisplay dǫ∆n(ǫ)ln/bracketleftBig 1+e(ǫ−µ)/kBT/bracketrightBig ,(35) where ∆n(ǫ)dǫis the change in the number of states at energyǫand interval dǫ, which can be expressed in terms of the scattering matrix45 ∆n(ǫ) =−1 2πi∂ ∂ǫTrlnS(ǫ). (36) Carrying out the derivative, we arrive at the force F=−1 2πi/integraldisplay dǫf(ǫ)Tr/parenleftbigg S†∂S ∂ξ/parenrightbigg ,(37)6 wheref(ǫ) is the Fermi-Dirac distribution function with chemical potential µ. This established result will be re- producedandgeneralizedtothedescriptionofdissipation and fluctuations below. B. Gilbert damping as energy pumping Here we interpretGilbert damping asan energypump- ing process by equating the results for energy dissipa- tion from the microscopic adiabatic pumping formalism with the LLG phenomenology in terms of collective co- ordinates, Eq. (9). The adiabatic energy loss rate of a scattering region in terms of scattering matrix at zero temperature has been derived in Refs. 26,27. In the ap- pendices, we generalize this result to finite temperatures: ˙E=/planckover2pi1 4π/integraldisplay dǫ/parenleftbigg −∂f ∂ǫ/parenrightbigg Tr/bracketleftbigg∂S(ǫ,t) ∂t∂S†(ǫ,t) ∂t/bracketrightbigg .(38) Since we employ the adiabatic approximation, S(ǫ,t) is the energy-dependent scattering matrix for an instanta- neous (“frozen”)scattering potential at time t. In a mag- netic system, the time dependence arises from its magne- tization dynamics, S(ǫ,t) =S[m(t)](ǫ). In terms of the collective coordinates ξ(t),S(ǫ,t) =S(ǫ,{ξ(t)}) ∂S[m(t)] ∂t≈/summationdisplay a∂S ∂ξa˙ξa, (39) where the approximate sign has been discussed in the previous section. We can now identify the dissipation tensor (10) in terms of the scattering matrix Γab=/planckover2pi1 4π/integraldisplay dǫ/parenleftbigg −∂f ∂ǫ/parenrightbigg Tr/bracketleftbigg∂S(ǫ) ∂ξa∂S†(ǫ) ∂ξb/bracketrightbigg .(40)In the macrospin model the Gilbert damping tensor can then be expressed as ˜αij=γ/planckover2pi1 4πMs/integraldisplay dǫ/parenleftbigg −∂f ∂ǫ/parenrightbigg Tr/bracketleftbigg∂S(ǫ) ∂mi∂S†(ǫ) ∂mj/bracketrightbigg ,(41) wheremiis a Cartesian component of the magnetization direction.. C. Gilbert damping and fluctuation-dissipation theorem At finite temperatures the forces acting on the mag- netization contain thermal fluctuations that are related to the Gilbert dissipation by the fluctuation-dissipation theorem, Eq. (14). The dissipation tensor is therefore ac- cessible via the stochastic forces in thermal equilibrium. The time dependence of the force operators ˆF(t) =−∂ˆHALDA(m) ∂ξ(42) is caused by the thermal fluctuations of the magneti- zation. It is convenient to rearrange the Hamiltonian ˆHALDAinto an unperturbed part that does not de- pend on the magnetization and a scattering potential ˆHALDA(m) =ˆH0+ˆV(m). In the basis of scattering wave functions of the leads, the force operator reads ˆF=−/integraldisplay dǫ/integraldisplay dǫ′/an}b∇acketle{tǫα|∂ˆV ∂ξ|ǫ′β/an}b∇acket∇i}htˆa† α(ǫ)ˆaβ(ǫ′)ei(ǫ−ǫ′)t//planckover2pi1, (43) where ˆaβannihilates an electron incident on the scatter- ing region, βlabels the lead (left or right) and quantum numbers of the wave guide mode, and |ǫ′β/an}b∇acket∇i}htis an associ- ated scatteringeigenstateat energy ǫ′. We takeagainthe left and rightreservoirsto be in thermal equilibrium with the same chemical potentials, such that the expectation values /angbracketleftbig ˆa† α(ǫ)ˆaβ(ǫ′)/angbracketrightbig =δαβδ(ǫ−ǫ′)f(ǫ).(44) Therelationbetweenthematrixelementofthescattering potential and the S-matrix /bracketleftbigg S†(ǫ)∂S(ǫ) ∂ξ/bracketrightbigg αβ=−2πi/an}b∇acketle{tǫα|∂ˆV ∂ξ|ǫβ/an}b∇acket∇i}ht(45)follows from the relation derived in Eq. (61) below as well as unitarity of the S-matrix,S†S= 1. Taking these relationsintoaccount,the expectationvalueof ˆFisfound to be Eq. (37). We now consider the fluctuations in the forceˆf(t) =ˆF(t)− /an}b∇acketle{tˆF(t)/an}b∇acket∇i}ht, which involves expectation values /angbracketleftbig ˆa† α1(ǫ1)ˆaβ1(ǫ′ 1)ˆa† α2(ǫ2)ˆaβ2(ǫ′ 2)/angbracketrightbig −/angbracketleftbig ˆa† α1(ǫ1)ˆaβ1(ǫ′ 1)/angbracketrightbig/angbracketleftbig ˆa† α2(ǫ2)ˆaβ2(ǫ′ 2)/angbracketrightbig =δα1β2δ(ǫ1−ǫ′ 2)δβ1α2δ(ǫ′ 1−ǫ2)f(ǫ1)[1−f(ǫ2)], (46) where we invoked Wick’s theorem. Putting everything7 together, we finally find /an}b∇acketle{tfa(t)fb(t′)/an}b∇acket∇i}ht= 2kBTδ(t−t′)Γab, (47) where Γ abhas been defined in Eq. (40). Comparing with Eq. (14), we conclude that the dissipation tensor Γ ab governingthe fluctuationsisidentical tothe oneobtained from the energy pumping, Eq. (40), thereby confirming the fluctuation-dissipation theorem. V. KUBO FORMULA The quality factor of the magnetization dynamics of most ferromagnets is high ( α/lessorsimilar0.01). Damping can therefore often be treated as a small perturbation. In the presentSectionwedemonstratethat the dampingob- tained from linear response (Kubo) theory agrees28with that ofthe scattering theory ofmagnetization dissipation in this limit. At sufficiently low temperatures or strong elastic disorder scattering the coupling to phonons may be disregarded and is not discussed here. The energy dissipation can be written as ˙E=/angbracketleftBigg dˆH dt/angbracketrightBigg , (48) where/an}b∇acketle{t/an}b∇acket∇i}htdenotes the expectation value for the non- equilibrium state. We are interested in the adiabatic response of the system to a time-dependent perturba- tion. In the adiabatic (slow) regime, we can at any time expand the Hamiltonian around a static configuration at the reference time t= 0, ˆH=ˆHst+/summationdisplay aδξa(t)/parenleftBigg ∂ˆH ∂ξa/parenrightBigg m(r)→mst(r).(49) The static part, ˆHst, is the Hamiltonian for a magneti- zation for a fixed and arbitrary initial texture mst, as, without loss of generality, described by the collective coordinates ξa. Since we assume that the variation of the magnetization in time is small, a linear expansion in terms of the small deviations of the collective coordinate δξi(t) is valid for sufficiently short time intervals. We can then employ the Kubo formalism and express the energy dissipation as ˙E=/summationdisplay aδ˙ξa(t)/parenleftBigg ∂ˆH ∂ξa/parenrightBigg m(r)→mst(r),(50) where the expectation value of the out-of-equilibrium conservative force /parenleftBigg ∂ˆH ∂ξa/parenrightBigg m(r)→mst(r)≡∂aˆH (51)consists of an equilibrium contribution and a term linear in the perturbed magnetization direction: /angbracketleftBig ∂aˆH/angbracketrightBig (t) =/angbracketleftBig ∂aˆH/angbracketrightBig st+/summationdisplay b/integraldisplay∞ −∞dt′χab(t−t′)δξb(t′). (52) Here, we introduced the retarded susceptibility χab(t−t′) =−i /planckover2pi1θ(t−t′)/angbracketleftBig/bracketleftBig ∂aˆH(t),∂bˆH(t′)/bracketrightBig/angbracketrightBig st,(53) where/an}b∇acketle{t/an}b∇acket∇i}htstis the expectation value for the wave functions of the static configuration. Focussing on slow modula- tions we can further simplify the expression by expand- ing δξa(t′)≈δξa(t)+(t′−t)δ˙ξa(t), (54) so that /angbracketleftBig ∂aˆH/angbracketrightBig =/angbracketleftBig ∂aˆH/angbracketrightBig st+/integraldisplay∞ −∞dt′χab(t−t′)δξb(t)+ /integraldisplay∞ −∞dt′χab(t−t′)(t′−t)δ˙ξb(t). (55) The first two terms in this expression, /an}b∇acketle{t∂aˆH/an}b∇acket∇i}htst+/integraltext∞ −∞dt′χab(t−t′)δξb(t),correspond to the energy vari- ation with respect to a change in the static magnetiza- tion. These terms do not contribute to the dissipation since the magnetic excitations are transverse, ˙ m·m= 0. Only the last term in Eq. (55) gives rise to dissipation. Hence, the energy loss reduces to29 ˙E=i/summationdisplay ijδ˙ξaδ˙ξb∂χS ab ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=0, (56) whereχS ab(ω) =/integraltext∞ −∞dt[χab(t)+χba(t)]eiωt/2. The symmetrized susceptibility can be expanded as χS ab=/summationdisplay nm(fn−fm) 2/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}ht+(a↔b) /planckover2pi1ω+iη−(ǫn−ǫm), (57) where|n/an}b∇acket∇i}htis an eigenstate of the Hamiltonian ˆHstwith eigenvalueǫn,fn≡f(ǫn),f(ǫ) is the Fermi-Dirac distri- bution function at energy ǫ, andηis a positive infinites- imal constant. Therefore,8 i/parenleftbigg∂χS ab ∂ω/parenrightbigg ω=0=π/summationdisplay nm/parenleftbigg −∂fn ∂ǫ/parenrightbigg /an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}htδ(ǫn−ǫm), (58) and the dissipation tensor Γab=π/summationdisplay nm/parenleftbigg −∂fn ∂ǫ/parenrightbigg /an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}htδ(ǫn−ǫm). (59) We nowdemonstratethatthe dissipationtensorobtained from the Kubo linear response formula, Eq. (59), is identical to the expression from scattering theory, Eq. (40), following the Fisher and Lee proof of the equiv- alence of linear response and scattering theory for the conductance.36 The static Hamiltonian ˆHst(ξ) =ˆH0+ˆV(ξ) can be decomposed into a free-electron part ˆH0=−/planckover2pi12∇2/2m and a scattering potential ˆV(ξ). The eigenstates of ˆH0 are denoted |ϕs,q(ǫ)/an}b∇acket∇i}ht,with eigenenergies ǫ, wheres=± denotes the longitudinal propagation direction along the system (say, to the left or to the right), and qa trans- verse quantum number determined by the lateral con- finement. The potential ˆV(ξ) scatters the particles be-tween the propagating states forward or backward. The outgoing (+) and incoming ( −) scattering eigenstates of the static Hamiltonian ˆHstare written as/vextendsingle/vextendsingle/vextendsingleψ(±) s,q(ǫ)/angbracketrightBig , whichform anothercomplete basiswith orthogonalityre- lations/angbracketleftBig ψ(±) s,q(ǫ)/vextendsingle/vextendsingle/vextendsingleψ(±) s′,q′(ǫ′)/angbracketrightBig =δs,s′δq,q′δ(ǫ−ǫ′).33These wave functions can be expressed as/vextendsingle/vextendsingle/vextendsingleψ(±) s,q(ǫ)/angbracketrightBig = [1 + ˆG(±) stˆV]|ϕs,q/an}b∇acket∇i}ht, where the retarded (+) and advanced ( −) Green’s functions read ˆG(±) st(ǫ) = (ǫ±iη−ˆHst)−1. By expanding Γ abin the basis of outgoing wave functions, |ψ(+) s,q/an}b∇acket∇i}ht, the energy dissipation (59) becomes Γab=π/summationdisplay sq,s′q′/integraldisplay dǫ/parenleftbigg −∂fs,q ∂ǫ/parenrightbigg/angbracketleftBig ψ(+) s,q/vextendsingle/vextendsingle/vextendsingle∂aˆH/vextendsingle/vextendsingle/vextendsingleψ(+) s′,q′/angbracketrightBig/angbracketleftBig ψ(+) s′,q′/vextendsingle/vextendsingle/vextendsingle∂bˆH/vextendsingle/vextendsingle/vextendsingleψ(+) s,q/angbracketrightBig , (60) where wave functions should be evaluated at the energy ǫ. Let us now compare this result, Eq. (60), to the direct scattering matrix expression for the energy dissipation, Eq. (40). The S-matrix operator can be written in terms of the T-matrix as ˆS(ǫ;ξ) = 1−2πiˆT(ǫ;ξ), where the T-matrix is defined recursively by ˆT=ˆV[1+ˆG(+) stˆT]. We then find ∂ˆT ∂ξa=/bracketleftBig 1+ˆVˆG(+) st/bracketrightBig ∂aˆH/bracketleftBig 1+ˆG(+) stˆV/bracketrightBig . The change in the scattering matrix appearing in Eq. (40) is then ∂Ss′q′,sq ∂ξa=−2πi/an}b∇acketle{tϕs,q|/bracketleftBig 1+ˆVˆG(+) st/bracketrightBig ∂aˆH/bracketleftBig 1+ˆG(+) stˆV/bracketrightBig |ϕs′,q′/an}b∇acket∇i}ht=−2πi/angbracketleftBig ψ(−) s′,q′/vextendsingle/vextendsingle/vextendsingle∂aˆH/vextendsingle/vextendsingle/vextendsingleψ(+) s′,q′/angbracketrightBig . (61) Since /angbracketleftBig ψ(−) s,q(ǫ)/vextendsingle/vextendsingle/vextendsingle=/summationdisplay s′q′Ssq,s′q′/angbracketleftBig ψ(+) s′q′(ǫ)/vextendsingle/vextendsingle/vextendsingle(62) andSS†= 1, we can write the linear response result, Eq. (60), as energy pumping (40). This completes our proof of the equivalence between adiabatic energy pump- ingintermsofthe S-matrixandtheKubolinearresponse theory.VI. CONCLUSIONS We have shown that most aspects of magnetization dynamics in ferromagnets can be understood in terms of the boundary conditions to normal metal contacts, i.e. a scattering matrix. By using the established numerical methods to compute electron transport based on scatter- ing theory, this opens the way to compute dissipation in ferromagnets from first-principles. In particular, our for-9 malism should work well for systems with strong elastic scattering due to a high density of large impurity poten- tials or in disordered alloys, including Ni 1−xFex(x= 0.2 represents the technologically important “permalloy”). The dimensionless Gilbert damping tensors (41) for macrospin excitations, which can be measured directly in terms of the broadening of the ferromagnetic reso- nance, havebeen evaluated for Ni 1−xFexalloysby ab ini- tiomethods.42Permalloy is substitutionally disordered and damping is dominated by the spin-orbit interaction in combination with disorder scattering. Without ad- justable parameters good agreement has been obtained with the available low temperature experimental data, which is a strong indication of the practical value of our approach. In clean samples and at high temperatures, the electron-phonon scattering importantly affects damping. Phonons are not explicitly included here, but the scat- tering theory of Gilbert damping can still be used for a frozen configuration of thermally displaced atoms, ne- glecting the inelastic aspect of scattering.12 While the energy pumping by scattering theory has been applied to described magnetization damping,15it can be used to compute other dissipation phenomena. This has recently been demonstrated for the case of current-induced mechanical forces and damping,43with a formalism analogous to that for current-induced mag- netization torques.13,14 Acknowledgments We would like to thank Kjetil Hals, Paul J. Kelly, Yi Liu, Hans Joakim Skadsem, Anton Starikov, and Zhe Yuan for stimulating discussions. This work was sup- ported by the EC Contract ICT-257159 “MACALO,” theNSFunderGrantNo.DMR-0840965,DARPA,FOM, DFG, and by the Project of Knowledge Innovation Pro- gram(PKIP) of Chinese Academy of Sciences, Grant No. KJCX2.YW.W10 Appendix A: Adiabatic Pumping Adiabatic pumping is the current response to a time- dependent scattering potential to first order in the time- variation or “pumping” frequency when all reservoirsare at the same electro-chemical potential.38A compact for- mulation of the pumping charge current in terms of the instantaneous scattering matrix was derived in Ref. 39. In the same spirit, the energy current pumped out of the scattering region has been formulated (at zero tempera- ture) in Ref. 27. Some time ago, we extended the charge pumping concept to include the spin degree of free- domandascertainedits importancein magnetoelectronic circuits.10More recently, we demonstrated that the en- ergyemitted byaferromagnetwith time-dependentmag- netizations into adjacent conductors is not only causedby interface spin pumping, but also reflects the energy loss by spin-flip processes inside the ferromagnet15and therefore Gilbert damping. Here we derive the energy pumping expressions at finite temperatures, thereby gen- eralizing the zero temperature results derived in Ref. 27 and used in Ref. 15. Our results differ from an earlier ex- tension to finite temperature derived in Ref. 40 and we point out the origin of the discrepancies. The magneti- zation dynamics must satisfy the fluctuation-dissipation theorem, which is indeed the case in our formulation. We proceed by deriving the charge, spin, and energy currentsintermsofthetimedependenceofthescattering matrix of a two-terminal device. The transport direction isxand the transverse coordinates are ̺= (y,z). An arbitrary single-particle Hamiltonian can be decomposed as H(r) =−/planckover2pi12 2m∂2 ∂x2+H⊥(x,̺), (A1) where the transverse part is H⊥(x,̺) =−/planckover2pi12 2m∂2 ∂̺2+V(x,̺).(A2) V(̺) is an elastic scattering potential in 2 ×2 Pauli spin space that includes the lattice, impurity, and self-consistent exchange-correlation potentials, including spin-orbit interaction and magnetic disorder. The scat- teringregionisattachedtoperfect non-magneticelectron wave guides (left α=Land rightα=R) with constant potential and without spin-orbit interaction. In lead α, the transverse part of the 2 ×2 spinor wave function ϕ(n) α(x,̺) and its corresponding transverse energy ǫ(n) α obey the Schr¨ odinger equation H⊥(̺)ϕ(n) α(̺) =ǫ(n) αϕ(n) α(̺), (A3) wherenis the spin and orbit quantum number. These transverse wave guide modes form the basis for the ex- pansion of the time-dependent scattering states in lead α=L,R: ˆΨα=/integraldisplay∞ 0dk√ 2π/summationdisplay nσϕ(n) α(̺)eiσkxe−iǫ(nk) αt//planckover2pi1ˆc(nkσ) α,(A4) where ˆc(nkσ) αannihilates an electron in mode nincident (σ= +) or outgoing ( σ=−) in leadα. The field opera- tors satisfy the anticommutation relation /braceleftBig ˆc(nkσ) α,ˆc†(n′k′σ′) β/bracerightBig =δαβδnn′δσσ′δ(k−k′). The total energy is ǫ(nk) α=/planckover2pi12k2/2m+ǫ(n) α. In the leads the particle, spins, and energy currents in the transport10 direction are ˆI(p)=/planckover2pi1 2mi/integraldisplay d̺Trs/parenleftBigg ˆΨ†∂ˆΨ ∂x−∂ˆΨ† ∂xˆΨ/parenrightBigg ,(A5a) ˆI(s)=/planckover2pi1 2mi/integraldisplay d̺Trs/parenleftBigg ˆΨ†σ∂ˆΨ ∂x−∂ˆΨ† ∂xσˆΨ/parenrightBigg ,(A5b) ˆI(e)=/planckover2pi1 4mi/integraldisplay d̺Trs/parenleftBigg ˆΨ†H∂ˆΨ ∂x−∂ˆΨ† ∂xHˆΨ/parenrightBigg +H.c., (A5c) where we suppressed the time tand lead index α,σ= (σx,σy,σz) is a vector of Pauli matrices, and Tr sdenotes the trace in spin space. Note that the spin current Is flows in the x-direction with polarization vector Is/Is. To avoid dependence on an arbitrary global potential shift, it is convenient to work with heat ˆI(q)rather than energy currents ˆI(ǫ): ˆI(q)(t) =ˆI(ǫ)(t)−µˆI(p)(t), (A6) whereµis the chemical potential. Inserting the waveg-uide representation (A4) into (A5), the particle current reads41 ˆI(p) α=/planckover2pi1 4πm/integraldisplay∞ 0dkdk′/summationdisplay nσσ′(σk+σ′k′)× ei(σk−σ′k′)xe−i/bracketleftBig ǫ(nk) α−ǫ(nk′) α/bracketrightBig t//planckover2pi1ˆc†(nk′σ′) αˆc(nkσ) α.(A7) Weareinterestedinthelow-frequencylimitoftheFourier transforms I(x) α(ω) =/integraltext∞ −∞dteiωtI(x) α(t). Following Ref. 41 we assume long wavelengths such that only the inter- vals withk≈k′andσ=σ′contribute. In the adiabatic limitω→0 this approach is correct to leading order in /planckover2pi1ω/ǫF,whereǫFis the Fermi energy. By introducing the (current-normalized) operator ˆc(nσ) α(ǫ(nk) α) =1/radicalBig dǫ(nkσ) α dkˆc(nkσ) α, (A8) which obey the anticommutation relations /braceleftBig ˆc(nσ) α(ǫα),ˆc†(n′σ′) β(ǫβ)/bracerightBig =δαβδnn′δσσ′δ(ǫα−ǫβ). (A9) The charge current can be written as ˆI(c) α(t) =1 2π/planckover2pi1/integraldisplay∞ ǫ(n) αdǫdǫ′/summationdisplay nσσe−i(ǫ−ǫ′)t//planckover2pi1ˆc†(nσ) α(ǫ′)ˆc(nσ) α(ǫ). (A10) Weoperateinthe linearresponseregimeinwhichapplied voltages and temperature differences as well as the exter- nally induced dynamics disturb the system only weakly. Transport is then governed by states close to the Fermi energy. We may therefore extend the limits of the en- ergy integration in Eq. (A10) from ( ǫ(n) α,∞) to (−∞ to∞). We relabel the annihilation operators so that ˆa(nk) α= ˆc(nk) α+denotes particles incident on the scattering region from lead αandˆb(nk) α= ˆc(nk) α−denotes particles leavingthe scatteringregionbylead α. Using the Fourier transforms ˆc(nσ) α(ǫ) =/integraldisplay∞ −∞dtˆc(nσ) α(t)eiǫt//planckover2pi1, (A11) ˆc(nσ) α(t) =1 2π/planckover2pi1/integraldisplay∞ −∞dǫˆc(nσ) α(ǫ)e−iǫt//planckover2pi1,(A12) we obtain in the low-frequency limit41 ˆI(p) α(t) = 2π/planckover2pi1/bracketleftBig ˆa† α(t)ˆaα(t)−ˆb† α(t)ˆbα(t)/bracketrightBig ,(A13) whereˆbαis a column vector of the creation operators forall wave-guidemodes {ˆb(n) α}. Analogouscalculations lead to the spin current ˆI(s) α= 2π/planckover2pi1/parenleftBig ˆa† ασˆaα−ˆb† ασˆbα/parenrightBig (A14) and the energy current ˆI(e) α=iπ/planckover2pi12/parenleftBigg ˆa† α∂ˆaα ∂t−ˆb† α∂ˆbα ∂t/parenrightBigg +H.c..(A15) Next, we express the outgoing operators ˆb(t) in terms of the incoming operators ˆ a(t) via the time-dependent scattering matrix (in the space spanned by all waveguide modes, including spin and orbit quantum number): ˆbα(t) =/summationdisplay β/integraldisplay dt′Sαβ(t,t′)ˆaβ(t′).(A16) When the scattering region is stationary, Sαβ(t,t′) only depends on the relative time difference t−t′, and its Fourier transform with respect to the relative time is energy independent, i.e.transport is elastic and can11 be computed for each energy separately. For time- dependent problems, Sαβ(t,t′) also depends on the total timet+t′and there is an inelastic contribution to trans- port as well. An electron can originate from a lead with energyǫ, pick up energy in the scattering region and end up in the same or the other lead with different energy ǫ′. The reservoirs are in equilibrium with controlled lo- cal chemical potentials and temperatures. We insert the S-matrix (A16) into the expressions for the currents,Eqs. (A13), (A14), (A15), and use the expectation value at thermal equilibrium /angbracketleftBig ˆa†(n) α(t2)ˆa(m) β(t1)/angbracketrightBig eq=δnmδαβfα(t1−t2)/2πℏ,(A17) wherefβ(t1−t2) = (2π/planckover2pi1)−1/integraltext dǫ−iǫ(t1−t2)//planckover2pi1fα(ǫ) and fα(ǫ) is the Fermi-Dirac distribution of electrons with energyǫin theα-th reservoir. We then find 2π/planckover2pi1/angbracketleftBig ˆb† α(t)ˆbα(t)/angbracketrightBig eq=/summationdisplay β/integraldisplay dt1dt2S∗ αβ(t,t2)Sαβ(t,t1)fβ(t1−t2), (A18) 2π/planckover2pi1/angbracketleftBig ˆb† α(t)σˆbα(t)/angbracketrightBig eq=/summationdisplay β/integraldisplay dt1dt2S∗ αβ(t,t2)σSαβ(t,t1)fβ(t1−t2), (A19) 2π/planckover2pi1/angbracketleftBig /planckover2pi1∂tˆb† α(t)ˆbα(t)/angbracketrightBig eq=/summationdisplay β/integraldisplay dt1dt2/bracketleftbig /planckover2pi1∂tS∗ αβ(t,t2)/bracketrightbig Sαβ(t,t1)fβ(t1−t2). (A20) Next, we use the Wigner representation (B1): S(t,t′) =1 2π/planckover2pi1/integraldisplay∞ −∞dǫS/parenleftbiggt+t′ 2,ǫ/parenrightbigg e−iǫ(t−t′)//planckover2pi1, (A21) and by Taylor expanding the Wigner represented S-matrix S((t+t′)/2,ǫ) aroundS(t,ǫ), S((t+t′)/2,ǫ) =/summationtext∞ n=0∂n tS(t,ǫ)(t′−t)n/(2nn!), we find S(t,t′) =1 2π/planckover2pi1/integraldisplay∞ −∞dǫe−iǫ(t−t′)//planckover2pi1ei/planckover2pi1∂ǫ∂t/2S(t,ǫ) (A22) and /planckover2pi1∂tS(t,t′) =1 2π/planckover2pi1/integraldisplay∞ −∞dǫe−iǫ(t−t′)//planckover2pi1ei/planckover2pi1∂ǫ∂t/2/parenleftbigg1 2/planckover2pi1∂t−iǫ/parenrightbigg S(t,ǫ). (A23) The factor 1 /2 scaling the term /planckover2pi1∂tS(t,ǫ) arises from commuting ǫwithei/planckover2pi1∂ǫ∂t/2. The currents can now be evaluated as I(c) α(t) =−1 2π/planckover2pi1/summationdisplay β/integraldisplay∞ −∞dǫ/bracketleftBig/parenleftBig e−i∂ǫ∂t/planckover2pi1/2S† βα(ǫ,t)/parenrightBig/parenleftBig ei∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig fβ(ǫ)−fα(ǫ)/bracketrightBig (A24a) I(s) α(t) =−1 2π/planckover2pi1/summationdisplay β/integraldisplay∞ −∞dǫ/bracketleftBig/parenleftBig e−i∂ǫ∂t/planckover2pi1/2S† βα(ǫ,t)/parenrightBig σ/parenleftBig ei∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig fβ(ǫ)/bracketrightBig (A24b) I(ǫ) α(t) =−1 4π/planckover2pi1/summationdisplay β/integraldisplay∞ −∞dǫ/bracketleftBig/parenleftBig e−i∂ǫ∂t/2/planckover2pi1(−i/planckover2pi1∂t/2+ǫ)S† βα(ǫ,t)/parenrightBig/parenleftBig e+i∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig fβ(ǫ)−ǫfα(ǫ)/bracketrightBig −1 4π/planckover2pi1/integraldisplay∞ −∞dǫ/bracketleftBig/parenleftBig e−i∂ǫ∂t/2/planckover2pi1S† βα(ǫ,t)/parenrightBig/parenleftBig ei∂ǫ∂t/2/planckover2pi1(i/planckover2pi1∂t/2+ǫ)Sαβ(ǫ,t)/parenrightBig fβ(ǫ)−ǫfα(ǫ)/bracketrightBig ,(A24c) where the adjoint of the S-matrix has elements S†(n′,n) βα=S∗(n,n′) αβ. We are interested in the average (DC) currents, where simplified ex pressions can be found by partial integration over energy and time intervals. We will consider the total DC curren tsout ofthe scattering region, I(out)=−/summationtext αIα, when the electrochemical potentials in the reservoirs are equal, fα(ǫ) =f(ǫ) for allα. The averaged pumped spin and12 energy currents out of the system in a time interval τcan be written compactly as I(c) out=1 2π/planckover2pi1τ/integraldisplayτ 0dt/integraldisplay dǫTr/braceleftbigg/bracketleftbigg f/parenleftbigg ǫ−i/planckover2pi1 2∂ ∂t/parenrightbigg S/bracketrightbigg S†−f(ǫ)/bracerightbigg , (A25a) I(s) out=1 2π/planckover2pi1τ/integraldisplayτ 0dt/integraldisplay dǫTr/braceleftbigg σ/bracketleftbigg f/parenleftbigg ǫ−i/planckover2pi1 2∂ ∂t/parenrightbigg S/bracketrightbigg S†/bracerightbigg , (A25b) I(ǫ) out=1 2π/planckover2pi1τ/integraldisplayτ 0dt/integraldisplay dǫTr/braceleftbigg/bracketleftbigg/parenleftbigg ǫ−i/planckover2pi1 2∂ ∂t/parenrightbigg f/parenleftbigg ǫ−i/planckover2pi1 2∂ ∂t/parenrightbigg S/bracketrightbigg S†−ǫf(ǫ)/bracerightbigg +1 2π/planckover2pi1τ/integraldisplayτ 0dt/integraldisplay dǫTr/braceleftbigg/bracketleftbigg f/parenleftbigg ǫ−i/planckover2pi1 2∂ ∂t/parenrightbigg S/bracketrightbigg/parenleftbigg −i/planckover2pi1∂S† ∂t/parenrightbigg/bracerightbigg , (A25c) where Tr is the trace over all waveguide modes (spin and orbital quantum numbers). As shown in Ap- pendix C the charge pumped into the reservoirs vanishes for a scattering matrix with a periodic time dependence when,integrated over one cycle: I(p) out= 0. (A26) This reflects particle conservation; the number of elec- trons cannot build up in the scattering region for peri- odic variations ofthe system. We can showthat a similar contribution to the energy current, i.e.the first line in Eq. (A25c), vanishes, leading to to the simple expression I(e) out=−i 2π/integraldisplayτ 0dt τ/integraldisplay dǫTr/braceleftbigg/bracketleftbigg f/parenleftbigg ǫ−i/planckover2pi1 2∂ ∂t/parenrightbigg S/bracketrightbigg∂S† ∂t/bracerightbigg . (A27) Expanded to lowest order in the pumping frequency the pumped spin current (A25b) becomes I(s) out=1 2π/planckover2pi1/integraldisplayτ 0dt τ/integraldisplay dǫTr/braceleftbigg/parenleftbigg SS†f−i/planckover2pi1 2∂S ∂tS†∂ǫf/parenrightbigg σ/bracerightbigg (A28) This formula is not the most convenient form to com- pute the current to specified order. SS†also contains contributions that are linear and quadratic in the pre- cession frequency since S(t,ǫ) is theS-matrix for a time- dependent problem. Instead, wewouldliketoexpressthe current in terms of the frozenscattering matrix Sfr(t,ǫ). The latter is computed for an instantaneous, static elec- tronic potential. In our case this is determined by a mag- netization configuration that depends parametrically on time:Sfr(t,ǫ) =S[m(t),ǫ]. Using unitarity of the time-dependentS-matrix (as elaborated in Appendix C), ex- pand it to lowest order in the pumping frequency, and insert it into (A28) leads to39 I(s) out=i 2π/summationdisplay β/integraldisplayτ 0dt τ/integraldisplay dǫ/parenleftbigg −∂f ∂ǫ/parenrightbigg Tr/braceleftbigg∂Sfr ∂tS† frσ/bracerightbigg . (A29) We evaluate the energy pumping by expanding (A27) to second order in the pumping frequency: I(e) out=/planckover2pi1 4π/integraldisplayτ 0dt τ/integraldisplay dǫTr/braceleftbigg −ifS∂S† ∂t−(∂ǫf)1 2∂S ∂t∂S† ∂t/bracerightbigg . (A30) As a consequence of unitarity of the S-matrix (see Ap- pendix C), the first term vanishes to second order in the precession frequency: I(e) out=/planckover2pi1 4π/integraldisplayτ 0dt τ/integraldisplay dǫ/parenleftbigg −∂f ∂ǫ/parenrightbigg Tr/braceleftBigg ∂Sfr ∂t∂S† fr ∂t/bracerightBigg ,(A31) where,at this point , we may insert the frozen scattering matrix since the current expression is already propor- tional to the square of the pumping frequency. Further- more, since there is no net pumped charge current in one cycle (and we are assuming reservoirs in a common equilibrium), the pumped heat current is identical to the pumped energy current, I(q) out=I(e) out. Our expression for the pumped energy current (A31) agrees with that derived in Ref. 27 at zero temperature. Our result (A31) differs from Ref. 40 at finite tempera- tures. The discrepancy can be explained as follows. In- tegration by parts over time tin Eq. (A27), using /bracketleftbigg f/parenleftbigg ǫ−i/planckover2pi1 2∂ ∂t/parenrightbigg i/planckover2pi1∂S ∂t/bracketrightbigg S†= 2/bracketleftbigg ǫf/parenleftbigg ǫ−i/planckover2pi1 2∂ ∂t/parenrightbigg S/bracketrightbigg S†−2/bracketleftbigg/parenleftbigg ǫ−i/planckover2pi1 2∂ ∂t/parenrightbigg f/parenleftbigg ǫ−i/planckover2pi1 2∂ ∂t/parenrightbigg S/bracketrightbigg S†,(A32) and the unitarity condition from Appendix C, /integraldisplayτ 0dt τ/integraldisplay dǫ/bracketleftbigg/parenleftbigg ǫ−i/planckover2pi1 2∂ ∂t/parenrightbigg f/parenleftbigg ǫ−i/planckover2pi1 2∂ ∂t/parenrightbigg S/bracketrightbigg S†=/integraldisplayτ 0dt τ/integraldisplay dǫǫf(ǫ), (A33)13 the DC pumped energy current can be rewritten as I(ǫ) out=1 π/planckover2pi1/integraldisplayτ 0dt τ/integraldisplay dǫTr/braceleftbigg/bracketleftbigg ǫf/parenleftbigg ǫ−i/planckover2pi1 2∂ ∂t/parenrightbigg S/bracketrightbigg S†−ǫf(ǫ)/bracerightbigg . (A34) Next, we expand this to the second order in the pumping frequency and find I(ǫ) out=1 π/planckover2pi1/integraldisplayτ 0dt τ/integraldisplay dǫTr/braceleftbigg ǫf(ǫ)/parenleftbig SS†−1/parenrightbig −ǫ(∂ǫf)i/planckover2pi1 2∂S ∂tS†−ǫ(∂2 ǫf)/planckover2pi12 8∂2S ∂t2S†/bracerightbigg . (A35) This form of the pumped energy current, Eq. (A35), agrees with Eq. (10) in Ref. 40 if one ( incorrectly ) as- sumesSS†= 1. Although for the frozen scattering ma- trix,SfrS† fr= 1, unitarity does not hold for the Wigner representation of the scattering matrix to the second or- der in the pumping frequency. ( SS†−1) therefore does not vanish but contributes to leading order in the fre- quency to the pumped current, which may not be ne- glected at finite temperatures. Only when this term is included our new result Eq. (A31) is recovered. Appendix B: Fourier transform and Wigner representation There is a long tradition in quantum theory to trans- form the two-time dependence of two-operator correla- tion functions such as scattering matrices by a mixed (Wigner)representationconsistingofaFouriertransform over the time difference and an average time, which has distinct advantages when the scattering potential varies slowlyintime.44Inordertoestablishconventionsandno- tations, we present here a short exposure how this works in our case. The Fourier transform of the time dependent annihi- lation operators are defined in Eqs. (A11) and (A12).Consider a function Athat depends on two times t1 andt2,A=A(t1,t2). The Wigner representation with t= (t1+t2)/2 andt′=t1−t2is defined as: A(t1,t2) =1 2π/planckover2pi1/integraldisplay∞ −∞dǫA(t,ǫ)e−iǫ(t1−t2)//planckover2pi1,(B1) A(t,ǫ) =/integraldisplay∞ −∞dt′A/parenleftbigg t+t′ 2,t−t′ 2/parenrightbigg eiǫt′//planckover2pi1.(B2) We also need the Wigner representation of convolutions, (A⊗B)(t1,t2) =/integraldisplay∞ −∞dt′A(t1,t′)B(t′,t2).(B3) By a series expansion, this can be expressed as44 (A⊗B)(t,ǫ) =e−i(∂A t∂B ǫ−∂B t∂A ǫ)/2A(t,ǫ)B(t,ǫ) (B4) which we use in the following section. Appendix C: Properties of S-matrix Here we discuss some general properties of the two- point time-dependent scattering matrix. Current conser- vation is reflected by the unitarity of the S-matrix which can be expressed as /summationdisplay βn′s′/integraldisplay dt′S(α1β) n1s1,n′s′(t1,t′)S(α2β)∗ n2s2,n′s′(t′,t2) =δn1n2δs1s2δα1α2δ(t1−t2). (C1) Physically, this means that a particle entering the scattering region from a lead αat some time tis bound to exit the scattering region in some lead βat another (later) time t′. Using Wigner representation (B1) and integrating over the local time variable, this implies (using Eq. (B4)) 1 =/parenleftbig S⊗S†/parenrightbig (t,ǫ) =e−i/parenleftBig ∂S t∂S† ǫ−∂S† t∂S ǫ/parenrightBig /2S(t,ǫ)S†(t,ǫ), (C2) where 1 is a unit matrix in the space spanned by the wave guide modes ( labelled by spin sand orbital quantum numbern). Similary, we find 1 =/parenleftbig S†⊗S/parenrightbig (t,ǫ) =e+i/parenleftBig ∂S t∂S† ǫ−∂S† t∂S ǫ/parenrightBig /2S†(t,ǫ)S(t,ǫ). (C3) To second order in the precession frequency, by respectively sub tracting and summing Eqs. (C2) and (C3) give Tr/braceleftbigg∂S ∂t∂S† ∂ǫ−∂S ∂ǫ∂S† ∂t/bracerightbigg = 0 (C4)14 and Tr/braceleftbig SS†−1/bracerightbig = Tr/braceleftbigg∂2S ∂t2∂2S† ∂ǫ2−2∂2S ∂t∂ǫ∂2S† ∂t∂ǫ+∂2S ∂ǫ2∂2S† ∂t2/bracerightbigg . (C5) Furthermore, foranyenergydependent function Z(ǫ)andarbitrarymatrixin thespacespannedbyspinandtransverse waveguide modes Y, Eq. (C2) implies 1 τ/integraldisplayτ 0dt/integraldisplay dǫZ(ǫ)Tr/braceleftbigg/bracketleftbigg e−i/parenleftBig ∂S t∂S† ǫ−∂S† t∂S ǫ/parenrightBig /2S(t,ǫ)S†(t,ǫ)−1/bracketrightbigg Y/bracerightbigg = 0. (C6) Integration by parts with respect to tandǫgives 1 τ/integraldisplayτ 0dt/integraldisplay dǫTr/braceleftbigg/bracketleftbigg e−i/parenleftBig ∂S t∂S† ǫ−∂S t∂ZS† ǫ/parenrightBig /2S(t,ǫ)Z(ǫ)S†(t,ǫ)−Z(ǫ)/bracketrightbigg Y/bracerightbigg = 0, (C7) which can be simplified to 1 τ/integraldisplayτ 0dt/integraldisplay dǫTr/braceleftbigg/parenleftbigg/bracketleftbigg Z/parenleftbigg ǫ+i 2∂ ∂t/parenrightbigg S(t,ǫ)/bracketrightbigg S†(t,ǫ)−Z(ǫ)/parenrightbigg Y/bracerightbigg = 0. (C8) Similarly from (C3), we find 1 τ/integraldisplayτ 0dt/integraldisplay dǫTr/braceleftbigg/parenleftbigg S†(t,ǫ)/bracketleftbigg Z/parenleftbigg ǫ−i 2∂ ∂t/parenrightbigg S(t,ǫ)/bracketrightbigg −1/parenrightbigg Y/bracerightbigg = 0. (C9) Using this result for Y= 1 andZ(ǫ) =f(ǫ) in the expression for the DC particle current (A25a), we see that unitarity indeed implies particle current conserva- tion,/summationtext αI(c) α= 0 for a time-periodic potential. However, such a relation does not hold for spins. Choosing Y=σ, we cannot rewrite Eq. (C9) in the form (A25b), unless theS-matrix and the Pauli matrices commute. 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2011-04-08

The magnetization dynamics of ferromagnets are often formulated in terms of the Landau-Lifshitz-Gilbert (LLG) equation. The reactive part of this equation describes the response of the magnetization in terms of effective fields, whereas the dissipative part is parameterized by the Gilbert damping tensor. We formulate a scattering theory for the magnetization dynamics and map this description on the linearized LLG equation by attaching electric contacts to the ferromagnet. The reactive part can then be expressed in terms of the static scattering matrix. The dissipative contribution to the low-frequency magnetization dynamics can be described as an adiabatic energy pumping process to the electronic subsystem by the time-dependent magnetization. The Gilbert damping tensor depends on the time derivative of the scattering matrix as a function of the magnetization direction. By the fluctuation-dissipation theorem, the fluctuations of the effective fields can also be formulated in terms of the quasistatic scattering matrix. The theory is formulated for general magnetization textures and worked out for monodomain precessions and domain wall motions. We prove that the Gilbert damping from scattering theory is identical to the result obtained by the Kubo formalism.

Magnetization Dissipation in Ferromagnets from Scattering Theory

1104.1625v1

arXiv:1409.6900v2 [cond-mat.mes-hall] 17 Apr 2015Dissipationless Multiferroic Magnonics Wei Chen1and Manfred Sigrist2 1Max-Planck-Institut f ¨ur Festk¨orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, G ermany 2Theoretische Physik, ETH-Z¨ urich, CH-8093 Z¨ urich, Switz erland (Dated: October 15, 2018) We propose that the magnetoelectric effect in multiferroic i nsulators with coplanar antiferromag- netic spiral order, such as BiFeO 3, enables electrically controlled magnonics without the ne ed of a magnetic field. Applying an oscillating electric field in th ese materials with frequency as low as household frequency can activate Goldstone modes that ma nifests fast planar rotations of spins, whose motion is essentially unaffected by crystalline aniso tropy. Combining with spin ejection mech- anisms, such a fast planar rotation can deliver electricity at room temperature over a distance of the magnetic domain, which is free from energy loss due to Gil bert damping in an impurity-free sample. PACS numbers: 85.75.-d, 72.25.Pn, 75.85.+t Introduction.- A primary goal of spintronic research is to seek for mechanisms that enable electric ( E) field controlled spin dynamics, since, in practice, Efields are much easier to manipulate than magnetic ( B) fields. As spinsdonotdirectlycoupleto Efield, incorporatingspin- orbit coupling seems unavoidable for this purpose. Along this line came the landmark proposals such as spin field effect transistor [1] and spin-orbit torque [2–5], the real- izations of which suggest the possibility of spin dynamics with low power consumption. On the other hand, in an- other major category of spintronics, namely magnonics, which aims at the generation, propagation, and detection of magnons, a mechanism that enables electrically con- trolled magnonics without the aid of a magnetic field has yet been proposed. Raman scattering experiments [6, 7] on the room tem- perature multiferroic BiFeO 3(BFO) shed light on this issue. The magnetic order of BFO is a canted antiferro- magnetic(AF) spiralontheplanespannedbythe electric polarization Palong[111]andoneofthethreesymmetry- equivalent wave vectors on a rhombohedral lattice [8, 9]. The spins have only a very small out-of-plane component [10, 11]. Applying a static Efield∼100kV/cm signif- icantly changes the cyclon (in-plane) and extra-cyclon (out-of-plane) magnons because of the magnetoelectric effect [7]. Indeed, spin-orbit coupling induced magneto- electric effects are a natural way to connect Efield to the spin dynamics of insulators [12, 13]. Motivated by the Raman scattering experiments on BFO, in this Let- ter we propose that applying an oscillating Efield to a coplanar multiferroic insulator (CMI) that has AF spiral order can achieve electrically controlled dissipationless magnonics, which can deliver electricity with frequency as low as household frequency up to the range of mag- netic domains. Compared to the magnonics that uses B field, microwave, or spin torques to generate spin dynam- ics in prototype Y 3Fe5O12(YIG) [14–16], the advantage of using CMI is that a single domain sample up to mm size is available [17], and Raman scattering data indicatewell-defined magnons in the absence of Bfield [7], so an external Bfield is not required in the proposed mecha- nism. Spin dynamics in CMI.- We start from the AF spiral on a square lattice shown in Fig. 1 (a), described by H=/summationdisplay i,αJSi·Si+α−Dα·(Si×Si+α) (1) whereα={a,c}are the unit vectors defined on thexz-plane,J >0, andDα=Dαˆ y>0 is the Dzyaloshinskii-Moriya (DM) interaction. The staggered moment ( −1)iSiin the ground state shown in Fig. 1 (a) is characterized by the angle θα=Q·α= −sin−1/parenleftig Dα/˜Jα/parenrightig betweenneighboringspins,where ˜Jα= /radicalbig J2+D2α. The DM interaction Dα=D0 α+wE×α (2) can be controlled by an Efield [18], where D0 αrepresents the intrinsic value due to the lack of in version symmetry of theα-bond. In the rotated reference frame S′defined by S′z i=Sz icosQ·ri+Sx isinQ·ri, S′x i=−Sz isinQ·ri+Sx icosQ·ri,(3) andS′y i=Sy i, the Hamiltonian is H=/summationdisplay i,α˜Jα/parenleftbig S′x iS′x i+α+S′z iS′z i+α/parenrightbig +JS′y iS′y i+α.(4) Since˜Jα>J, the spins have collinear AF order and all S′z i= (−1)iSlie inxz-plane. Thespin dynamicsin the absenceof Bfieldisgoverned by the Landau-Lifshitz-Gilbert (LLG) equation dS′ i dt=∂H ∂S′ i×S′ i+αGS′ i×/parenleftbigg∂H ∂S′ i×S′ i/parenrightbigg (5)2 expressed in the S′frame, where αGis the phenomeno- logical damping parameter. Eq. (5) can be solved by the spin wave ansatz for the even ( e) and odd ( o) sites [19] /parenleftbiggS′x e,o S′y e,o/parenrightbigg =/parenleftbiggux e,o vy e,o/parenrightbigg ei(k·re,o−ωt). (6) Ignoringthe dampingterm inEq.(5) yieldseigenenergies ω± k 2S= /parenleftigg/summationdisplay α˜Jα±γα−(k)/parenrightigg2 −/parenleftigg/summationdisplay αγα+(k)/parenrightigg2 1/2 ,(7) whereγα±(k) =/parenleftig ˜Jα/2±J/2/parenrightig cosk·α. Their eigenval- ues and eigenvectors near k= (0,0) andk= (π,π) are summarized below /braceleftig ω+ k→(0,0),ω− k→(π,π)/bracerightig = 2S/radicalbig 2(D2a+D2c), ue ve uo vo ∝ 0 1 0 ∓1 +O/parenleftbiggD J/parenrightbigg . /braceleftig ω− k→(0,0),ω+ k→(π,π)/bracerightig = 0, ue ve uo vo ∝ 1 0 ∓1 0 .(8) The in-plane magnon dS′ i/dt= (dS′x i/dt,0,0) is gapless, while the out-of-plane magnon dS′ i/dt= (0,dS′y i/dt,0) develops a gap, as displayed in Fig. 1 (c). Even includ- ing the damping term in Eq. (5), the in-plane magnons very near the Goldstone modes ω− k→(0,0)andω+ k→(π,π)re- mainunchangedanddamping-free. Awayfromthe Gold- stone limit, the eigenenergies become complex, hence the magnons are subject to the damping and decay within a time scale set by α−1 G. Spin dynamics induced by oscillating Efield.-We an- alyze now the spin dynamics in the damping-free in- plane magnonchannel induced bymagnetoelectriceffects (Eq. (8)). Unlike the spin injection by using the spin Hall effect (SHE) to overcome the damping torque [16], our design does not require an external Bfield, and is fea- sible over a broad range of frequencies. Consider the device shown in Fig. 2, where an oscillating electric field E=E0cosωtis applied parallel to the ferroelectric mo- ment over a region of length L=Na, such that the DM interaction in Eq. (2) oscillates in this region. Thus, the wave length of the spiral changes with time yielding an oscillation of the number of spirals inside this region, nQ=L 2π/|Q|≈N 2πJ/bracketleftbig D0 a+wE0acosωt/bracketrightbig ,(9) assumingDa=D0 a+wE0a≪J,Dc= 0, and E⊥a. Suppose the spin S0at one boundary is fixed by, for FIG. 1: (color online) Schematics of 2D AF spiral in the (a) originalS-frame and the (b) rotated S′-frame. Red and blue arrows indicate the spins on the two sublattices. (c) Spin wave dispersion ω+ k(dashed line) and ω− k(solid line) solved in theS′frame, with Da/J= 0.14,Dc= 0. Inserts show their eigen modes in the S′frame near k= (0,0) and (π,π), where the spin dynamics dS′ i/dtis indicated by black arrows or symbols. instance, surface anisotropy because of specific coating. ThenSNat the other boundary rotates by ∂θN ∂t=−N JwE0aωsinωt, (10) because whenever the number of waves nQchanges by 1, SNrotates 2πin orderto to wind or unwind the spin tex- ture in the Efield region. The significance of this mecha- nism is that although the Efield is driven by a very small frequencyω, the spin dynamics ∂tθNat the boundary is manyordersofmagnitude enhanced because of the wind- ing process. The rotation of SNserves as a driving force for the spin dynamics in the field-free region from SNto SN+M. As long as the spin dynamics is slower than the energy scale of the DM interaction ∂tθi<|D0|//planckover2pi1∼THz, one can safely consider the Efield region as adiabatically changing its wave length but remaining in the ground state. The spins in the field-free region rotate coherently ∂tθN=∂tθN+1=...=∂tθN+M, synonymous to exciting theω− k→(0,0)mode in Eq.(8), hence the spin dynamics in the field-free region remains damping-free in an ideal situation. In real materials, crystalline anisotropy and impuri- ties are the two major sources to spoil the spin rota- tional symmetry implicitly assumed here. In the supple- mentary material[20], their effects are discussed by draw- ing analogy with similar situations in the atom absorp- tion on periodic substrates and the impurity pinning of charge density wave states. It is found that crystalline3 quantity symbol magnitude lattice constant a nm s−dexchange Γ 0.1eV s−dexchange time τex 10−14s spin relaxation time τsf 10−12s spin diffusion length λN 10nm spin density n01027/m3 spin Hall angle θH 0.1 intrinsic DM D0 α10−3eV superexchange J 0.1eV Eq. (2) w 10−19C electric flux quantum ˜Φ0 E 1V TABLE I: List of material parameters and their order of mag- nitude values. anisotropy remains idle because of the long spiral wave length and the smallness of crystalline anisotropy com- pared to exchange coupling. The impurities that tend to pin the spins alongcertaincrystalline directionopen up a gap in the Goldstone mode and cause energy dissipation, which nevertheless do not obstruct the coherent rotation of spins generated by Eq. (10). FIG. 2: (color online) Experimental proposal of using oscil - latingEfield to induce spin dynamics in CMI. The AF spiral order is shown in the S′frame. The Efield is applied between S′ 0andS′ N, causing dynamics in the whole spin texture. Two ways for spin ejection out of S′ N+Mare proposed: (a) Using SHE to converted it into a charge current. (b) Using time- varying spin accumulation and inductance. Spin ejection and delivery of electricity.- We now ad- dress the spin ejection from the CMI to an attached normal metal (NM). A spin current is induced in the NM when a localized spin Siat the NM/CMI inter- face rotates [16, 21]. Defining the conduction electron spinm(r,t) =−∝an}b∇acketle{tσ∝an}b∇acket∇i}ht/2, thes-dcoupling at the interface Hsd= Γσ·Sidefines a time scale τex=/planckover2pi1/2S|Γ|, withΓ<0 [21]. The Bloch equation in the NM reads ∂m ∂t+∇·Js=1 τexm׈Si−δm τsf(11) whereJs=JNM s/varotimesσ/planckover2pi1/2 is the spin current tensor, and τsfis the spin relaxation time in the NM. In equilib- rium, we assume mhybridizes with each Sion the spi- ral texture locally. If the dynamics of Siis slow com- pared to 1/τex, which is true for the proposed mechanism and also for other usual means such as ferromagnetic resonance[16], mfollows−ˆSiat any time with a very small deviation m=m0+δm=−n0ˆSi+δm, wheren0 is the local equilibrium spin density. The spin current tensorJs=−D0∇δmis obtained from the diffusion of δm, whereD0is the spin diffusion constant. Under such an adiabatic process, the small deviation is[21] δm=τex 1+ξ2/braceleftigg −ξn0∂ˆSi ∂t−n0ˆSi×∂ˆSi ∂t/bracerightigg ,(12) whereξ=τex/τsf<1 so one can drop the first term on the right hand side, and replace ˆSi×∂tˆSi→δ(r)ˆSi×∂tˆSi sinceˆSiis located at the NM/CMI interface r= 0 (ras coordinate perpendicular to the interface). The resulting equation solves the time dependence of δm. Away from r= 0, Eq. (11) yields D0∇2δm=δm/τsf, which solves the spatial dependence of δm. The spin current caused by a particular Sithen follows JNM sδˆm=δmD0 λN=−τexn0D0 (1+ξ2)λNˆSi×∂ˆSi ∂te−r/λN,(13) whereλN=/radicalbig D0τsf, similar to results obtained previ- ously[16]. Ifonlythein-planeGoldstonemodeisexcited, as shown in Fig. 2, it is equivalent to a global rotation of spinsˆSi= (−1)i(sin(θ(t) +Q·ri),0,cos(θ(t) +Q·ri)) in the field-free region. Thus the time dependence in Eq. (13), ˆSi×∂tˆSi=ˆy∂θ/∂t, is that described by Eq. (10), and is the same for every Siat the NM/CMI interface, even though each Sipoint at a different polar angle. In other words, the spin current ejected from each Siof the AF spiral, described by Eq. (13), is the same, so a uniform spin current flows into the NM. We propose two setups to convert the ejected spin cur- rent into an electric signal. The first device uses inverse SHE[16]inaNMdepositedatthesideofthespiralplane, yieldingδˆmperpendicular to JNM sand consequently a voltage in the transverse direction, as shown in Fig. 2 (a). The second design ejects spin into a NM film de- posited on top of the spiral plane, as shown in Fig. 2 (b), causing δˆmparallel to JNM s. A spin accumulation in the NM develops and oscillates with time, producing an oscillatingmagnetic flux Φ Bthrougha coil that wraps around the NM, hence a voltage E=−∂ΦB/∂t. Experimental realizations.- TheRamanscatteringdata on BFO [7] show that applying |E| ∼100kV/cm can4 change the spin wave velocity by δv0/v0∼1%. We can make use of this information to estimate the field- dependence win Eq. (2). The ω− kmode in Eq. (7) near k= (0,0) is ω− k→0= 2√ 2SJka/bracketleftbigg 1+5 16/parenleftbiggD2 ak2 a+D2 ck2 c J2k2/parenrightbigg/bracketrightbigg = (v0+δv0)k, (14) wherev0= 2√ 2SJais the spin wave velocity in the absence of DM interaction. Assuming Da∝ne}ationslash= 0,Dc= 0, andE⊥a, the Raman scattering data gives w∼ 10−19C∼ |e|. We remark that a coplanar magnetic order can be mapped into a spin superfluid [36, 37] ψiby ∝an}b∇acketle{tSi∝an}b∇acket∇i}ht=S(sinθi,0,cosθi) =√v(Imψi,0,Reψi),(15) wherevis the volumeofthe 3Dunit cell. Within this for- malism, the Efield can induce quantum interference of the spin superfluid via magnetoelectric effect, in which the electric flux vector ΦE=/contintegraltext E×dlis quantized [24, 25]. The flux quantum is ˜Φ0 E= 2πJ/w, which is ˜Φ0 E∼1V for BFO, close to that ( ∼10V) obtained from current-voltage characteristics of a spin field-effect tran- sistor [24], indicating that strong spin-orbit interaction reduces the flux quantum to an experimentally accessi- ble regime. For instance, BFO has a spiral wave length 2π/Q∼100nm, so in a BFO ring of µm size, the num- ber of spirals at zero field is nQ∼10, and applying |E| ∼1kV/cm can change nQby 1. Besides changing the winding number, we remarkthat the magnetoelectric effect can also be used to affect the topological proper- ties of a magnet in a different respect[26]. Table I lists the parameters and their order of magnitude values by assuming CMI has similar material properties as other magnetic oxide insulators such as YIG, and we adopt lattice constant a∼1nm for both CMI and the NM for simplicity. For the device in Fig. 2, consider the field |E0| ∼ 100kV/cm oscillating with a household frequency ω∼ 100Hz is applied to a range L∼1mm. This region covers N=L/a∼106sites with a number of spirals nQ∼104 at zero field. The Efield changes the number of spi- rals tonQ∼105within time period 1 /ω∼0.01s, so the spins at the boundary SNwind with angular speed ∂tθN∼107sinωtwhich is enhanced by 5 orders of mag- nitude from the driving frequency ω. To estimate the ejected spin current in Eq. (13), we use the typical spin relaxation time τsf∼10−12s and length λN∼10nm for heavy metals [16]. The s-dcoupling can range be- tween [16] 0 .01eV to 1eV. We choose Γ ∼0.1eV, which givesτex∼10−14s. The spin Hall angle θH∼0.1 has been achieved [27, 28]. To estimate n0, we use the fact that thes-dhybridization Γ σ·Siis equivalent to ap- plying a magnetic field H= 2ΓSi/µ0gµBlocally at the interface atomic layer of the NM. Given the typical mo- lar susceptibility χm∼10−4cm3/mol and molar volumeVm∼10cm3/mol, the interface magnetization of the NM isn0µB=χmH/Vm∼104C/sm, thus n0∼1027/m3. Theoscillating Efieldgives ˆSi×∂tˆSi=∂tθNˆy∼ˆy107Hz, so the ejected spin current is JNM s∼1024/planckover2pi1/m2s. Using the design in in Fig. 2(a) to convert JNM sinto a charge current via inverse SHE yields JNM c∼104A/m2, hence a voltage ∼µV oscillating with ωin a mm-wide sample. To use the setup in Fig. 2(b), a NM film of area ∼1 mm2 and thickness ∼10nm yields E ∼mV oscillating with ω. In summary, we propose that for multiferroics that have coplanar AF spiral order, such as BFO, applying an oscillating Efield with frequency as low as house- hold frequency generates a coherent planar rotation of the spin texture whose frequency is many orders of mag- nitude enhanced. This coherent rotation activates the Goldstone mode of multiferroic insulators that remains unaffected by the crystalline anisotropy. The Goldstone mode can be used to deliver electricity at room tempera- ture up to the extensions of magnetic domains, in a way that is free from the energy loss due to Gilbert damping if the sample is free from impurities. The needlessness ofBfield greatly reduces the energy consumption and increases the scalability of the proposed device, pointing to its applications in a wide range of length scales. We thank exclusively P. Horsch, J. Sinova, H. Naka- mura, Y. Tserkovnyak, D. Manske, M. Mori, C. Ulrich, J. Seidel, and M. Kl¨ aui for stimulating discussions. Supplementary material I. Crystalline anisotropy in multiferroics First we demonstrate that because the wave length of the spiral order in multiferroics is typically 1 ∼2 orders longer than the lattice constant, and the exchange cou- pling is typically few orders larger than the crystalline anisotropy energy, the spiral order remains truly incom- mensurate and very weakly affected by the crystalline anisotropy. For simplicity, we consider a spiral state with wave vector Q∝ba∇dbl(1,0) and translationally invariant per- pendicular to Qsuch that the geometry can be reduced to a 1D problem. The classical elastic energy for a 1D antiferromagnetic (AF) spiral is E0=/summationdisplay n−˜JaS2cos(θn+1−θn−θa) ≈ −N˜JaS2+/summationdisplay n1 2˜JaS2(θn+1−θn−θa)2,(16) whereθn=Q·rnis the angle relative to the staggered spin (−1)iSi, andθa=Q·ais the natural pitch an- gle between neighboring spins ( a= (a,0)). The square lattice symmetry of our model yields a 4-fold degener- ate crystalline spin anisotropy[38], leading to the total5 energy E=/summationdisplay n1 2˜JaS2(θn+1−θn−θa)2 +/summationdisplay nVani(1−cos4θn), (17) whereVaniis the anisotropy energy per site. This is the well-known Frenkel-Kontorowa(FK) model[30, 31] that has been discussed extensively owing to its rich physics. FIG. 3: (color online) Schematics of mapping the AF spi- ral order in the presence of crystalline anisotropy into FK model. The angles θiof staggered spins ( −1)iSi(blue ar- rows) are mapped into displacements xiof particles (orange dots). The width of the 4-fold degenerate pinning potential V(1−cos2πxi/b) isb=π/2, and the spacing of particles in the absence of the pinning potential is a0=Q·a. We consider the limit of weak anisotropy V= Vani/˜JaS2a2≪1 and the case of long wavelength of the spiral, θa≪π/2 whereπ/2 is the angle between two minima of the anisotropy potential. In the spirit of Ref.[32, 33] we assume now that there are prime num- bers,MandLwithM˜θa=Lπ/2 andM≫Lwhich is the average pitch in the ground state of Eq.(17). Then we introduce the parametrization θn=n˜θa+ϕn 4(18) and the misfit parameter δ= 4(θa−˜θa). Turning to the continuous limit one can derive the effective en- ergy functional based on expanding the first harmonic approximation[32–34], ˜E[ϕ] =/integraldisplay dx/bracketleftigg 1 2/parenleftbiggdϕ dx−δ/parenrightbigg2 +VMcos(Mϕ)/bracketrightigg (19) withVM∼VMwhich can become extremely small for M≫1. The commensurate-incommensurate transition happens ifδis large enough to stabilize the formation of solitonsδ > δc(M)∼4√VM/π. Deep inside the incom- mensurate phase, ϕ(x)≈δxsuch thatθn≈θanfollows esentially the natural spiral pitch. In our system, BFO, the spiral wave length ℓ≈ 60nm∼100awhich yields M∼100/4 = 25, i.e. every 25thspin could be pinned along one of the 4 anisotropy minima (assuming L= 1). Typical anisotropy ener- gies for ferrites[35] lead to Vani∼10−3eV while theexchange energy is Ja∼0.1eV, from which we obtain V∼Vani/Ja∼10−2and consequently VM∼10−50is a negligible number. The misfit parameter may be as large asδ= 4(θa−˜θa)∼π/M2such thatδ≫δc(M) is well satisfied, even if by an electrical field Mshrinks by one order of magnitude. Thus, the electric field-driven oscillations of the spin spiral remains most likely unaf- fected by the spin anisotropy. The small VMrenders the energy gap due to the anisotropy energy irrelevant, hence the in-plane magnon mode remains essentially un- damped. Another important consequence of this analy- sis is that although the concept of spin superfluidity, i.e., treating the spin texture as a quantum condensate, has been proposed long ago, its realization in collinear mag- nets is problematic because of the crystalline anisotropy and subsequently the formation of domain walls. We demonstrate explicitly that multiferroics are not sub- ject to these problems because of the noncollinear or- der, hence a room temperature macroscopic condensate of mm size can be realized. II. Phase-pinning impurities in multiferroics We proceed to show that dilute, randomly distributed impurities, exist either in the bulk of the multiferroic or at the metal/multiferroic interface, do not obstruct the proposed electrically controlled multiferroic magnonics. Drawing analogyfrom the FK model, impurities that pin the spins along certain crystalline directions, denoted by phase-pinningimpurities, arethe impuritiesto be consid- ered because they tend to impede the coherent motion of spins[39]. Since we propose to use an oscillating Efield to drive the spin rotation from the boundary, each cross section channel is equivalent, which reduces the problem from 2D to 1D. This leads us to consider the following 1D classical model similar to Eq. (17) for the field-free region (S′ N+1toS′ N+Min the Fig. 2 of the main text). E=/summationdisplay i1 2˜JaS2(θi+a−θi−θa)2−/summationdisplay i∈impVimpcos4θi,(20) whereVimp>0 is the pinning potential, and/summationtext i∈imp sums over impurity sites. The total length of the chain is L′=MawithManinteger. Inthepresenceofoscillating Efield that causes the winding of boundary spins ( S′ N in the Fig. 2 of the main text), the angle of spins in the disordered field-free region has three contributions θi=θ0 i+∆θi+ηi, (21) whereθ0 irepresents the spiral texture in the unstretched cleanlimitsatisfying θ0 i+a−θ0 i−θa= 0,∆θiisthestretch- ing of the spin texture caused by winding of boundary spins, and ηiis the distortion due to impurities. Only the later two contribute to the elastic energy, so Eq. (20)6 becomes E=/summationdisplay i1 2˜JaS2(∆θi+a−∆θi+ηi+a−ηi)2 −/summationdisplay i∈impVimpcos4θi. (22) In this analysis we consider weak impurities Vimp≪ ˜JaS2, andassumethatthe windingofthe boundaryspins is slow such that the winding spreads through the whole field-free region evenly, causing every pair of neighboring spins to stretch by the same amount ∆ θi+a−∆θi= ∆θ. For the electrically driven magnonics proposed in the main text, which can achieve winding of boundary spins byθN∼nQ∼105within half-period, a field-free region of lengthL′∼mm has ∆θ∼0.1, so our numerics is done with ∆θlimited within this value. In the weak impurity limit, the length scale L0over whichηichanges by O(1) can be calculated in the fol- lowing way. The elastic energy part in Eq. (22) within L0is, in the continuous limit, K(L0) =1 a/integraldisplayL0 0dx1 2˜JaS2a2/parenleftbigg∆θ a+∂xη/parenrightbigg2 =L0 2a˜JaS2∆θ2+˜JaS2∆θ α1+˜JaS2a 2α0L0,(23) whereα0andα1are numerical constants of O(1), and are set to be unity without loss of generality. Denoting impurity density as nimp=Nimp/L′whereNimpis the total number of impurities in the sample, the impurity potential energy within L0is calculated by V(L0) =−VimpRe /summationdisplay i∈impe4i(θ0 i+∆θ+η) =−Vimp/radicalbig nimpL0. (24) Note that the contribution comes not from the zeroth order impurity averaging, but its fluctuation that mimics a random walk in the complex plane[40]. The phase ηis assumed to be constant within L0and chosen to give Eq. (24) and hence the total energy E(L0) =K(L0) + V(L0) withinL0. Minimizing the total energy per site E(L0)/L0gives the most probable pinning length L0. In the unstretched case ∆ θ= 0, L0=/parenleftigg˜JaS2a α0Vimpn1/2 imp/parenrightigg2/3 (25) is similar to the Fukuyama-Lee-Rice (FLR) length that characterizes the impurity pinning of a charge density wave ground state[40, 41]. Putting Eq. (25) back to Eqs. (24) and (23), the corresponding E(L0)a/L0<0 can be viewed as the pinning energy per site that im- pedes the coherent rotation of spins, and equivalently represents the gap opened at the Goldstone mode.00.020.040.060.08 0.10246810 /CapDΕltaΘLog10/LParen1L0/Slash1a/RParen1/LParen1a/RParen1 Aimp/EΘual10/Minus5 10/Minus4 10/Minus3 10/Minus2 00.020.040.060.08 0.1/Minus1/Minus0.500.51 /CapDΕltaΘΕ/Multiply103/LParen1b/RParen1 Aimp/EΘual10/Minus4 10/Minus3 5/Multiply10/Minus3 10/Minus2 FIG. 4: (color online) (a) The logrithmic of the dimensionle ss FLR length log10(L0/a) versus winding angle per site ∆ θ, in several values of the dirtiness parameter Aimp. Dashed line indicates the threshold when L0∼mm. (b) The dimensionless pinning energy ǫversus winding per site. In the presence of the stretching ∆ θ, the expression ofL0is rather lengthy. It is convenient to define two dimensionless parameters Aimp=Vimp ˜JaS2/radicalbigg Nimpa L′, ǫ=E(L0) ˜JaS2/parenleftbigga L0/parenrightbigg , (26) whereAimp(the ”dirtiness parameter”) is the impurity potential measured in unit of the elastic constant times the square root of the impurity density, and ǫis the to- tal energy per site measured in unit of the elastic con- stant. Figure 4 shows the logrithmic of the dimensionless pinning length L0/aand the dimensionless total energy ǫ, plotted as functions of the stretching ∆ θ. There are two evidences showing that the spin texture, originally pinned by impurities with the pinning length in Eq. (25), is depinned by the stretching ∆ θ: Firstly, the pinning lengthL0increases as increasing ∆ θ. For a particular sample size, for instance L′∼mm, the spin texture is depinned when the pinning length exceeds the sample sizeL0> L′, or equivalently when ∆ θis greater than a certain threshold (intercept of the dashed line and the coloredlinesinFig.4(a)). Secondly, thepinningenergy ǫ becomes positive at large ∆ θ, indicating that the elastic energy from stretching overcomes the impurity pinning energy, so the spin texture is depinned. From Fig. 4, it is also evident that the cleaner is the sample, the easier it is to depin the spins by stretching, as smaller Aimpre- quires smaller threshold value of ∆ θ. We conclude that thephasepinning impuritiesdonot hampertheproposed electrically driven multiferroic magnonics as long as the dirtiness of the sample is limited, the winding speed of the boundary spin is sufficient, and the sample size is short enough. [1] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990). [2] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008); and Phys. Rev. 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Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Appl. Phys. 43, 264002 (2010). [16] Y. Kajiwara et al., Nature 464, 262 (2010). [17] R. D. Johnson, P. Barone, A. Bombardi, R. J. Bean, S. Picozzi, P. G. Radaelli, Y. S. Oh, S.-W. Cheong, and L. C. Chapon, Phys. Rev. Lett. 110, 217206 (2013). [18] K. Shiratori and E. Kita, J. Phys. Soc. Jpn. 48, 1443 (1980). [19] R. de Sousa and J. E. Moore, Phys. Rev. B 77, 012406 (2008). [20] The Supplementary Material is included in this arXiv version. [21] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). [22] B. I. Halperin and P. C. Hohenberg, Phys. Rev. 188, 898(1969). [23] P. Chandra, P. Coleman, and A. I. Larkin, J. Phys.: Con- dens. Matter 2, 7933 (1990). [24] W. Chen, P. Horsch, and D. Manske, Phys. Rev. B 87, 214502 (2013). [25] W. Chen and M. Sigrist, Phys. Rev. B 89, 024511 (2014). [26] W.-L. You, G.-H. Liu, P. Horsch, and A. M. Ole´ s, Phys. Rev. B90, 094413 (2014). [27] T. Seki, Y. Hasegawa, S. Mitani, S. Takahashi, H. Ima- mura, S. Maekawa, J. Nitta, and K. Takanashi, Nature Mater.7, 125 (2008). [28] C.-F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 101, 122404 (2012). [29] E. B. Sonin, Adv. Phys. 59, 181 (2010). [30] Y. I. Frenkel and T. Kontorowa, Zh. Eksp. Theor. Fiz. 8, 1340 (1938). [31] F. C. Frank and J. H. Van der Merwe, Proc. R. Soc. 198, 205 (1949). [32] P. Bak, Rep. Prog. Phys. 45, 587 (1982). [33] G. Theodorou and T. M. Rice, Phys. Rev. B 18, 2840 (1978). [34] V. L. Prokovskii and A. L. Talapov, Zh. Eksp. Theor. Fiz.75, 1151 (1978). [35] K. Yosida and M. Tachiki, Prog. Theo. Phys. 17, 331 (1957). [36] B. I. Halperin and P. C. Hohenberg, Phys. Rev. 188, 898 (1969). [37] P. Chandra, P. Coleman, and A. I. Larkin, J. Phys.: Con- dens. Matter 2, 7933 (1990). [38] E. B. Sonin, Adv. Phys. 59, 181 (2010). [39] P. M. Chaikin and T. C. Lubensky, Principles of condensed matter physics , Cambridge University Press (1995), p. 616. [40] H. Fukuyamaand P. A. 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2014-09-24

We propose that the magnetoelectric effect in multiferroic insulators with coplanar antiferromagnetic spiral order, such as BiFeO$_{3}$, enables electrically controlled dissipationless magnonics. Applying an oscillating electric field in these materials with frequency as low as household frequency can activate Goldstone modes that manifests fast planar rotations of spins, whose motion is not obstructed by crystalline anisotropy. Combining with spin ejection mechanisms, such a fast planar rotation can deliver electricity at room temperature over a distance of the magnetic domain, which is free from the energy loss due to Gilbert damping.

Dissipationless Multiferroic Magnonics

1409.6900v2

Consistent microscopic analysis of spin pumping eects Gen Tatara RIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama, 351-0198 Japan Shigemi Mizukami WPI - Advanced Insitute for Materials Research, Tohoku University Katahira 2-1-1, Sendai, Japan (Dated: October 20, 2018) Abstract We present a consistent microscopic study of spin pumping eects for both metallic and insulating ferromagnets. As for metallic case, we present a simple quantum mechanical picture of the eect as due to the electron spin ip as a result of a nonadiabatic (o-diagonal) spin gauge eld. The eect of interface spin-orbit interaction is brie y discussed. We also carry out eld-theoretic calculation to discuss on the equal footing the spin current generation and torque eects such as enhanced Gilbert damping constant and shift of precession frequency both in metallic and insulating cases. For thick ferromagnetic metal, our study reproduces results of previous theories such as the correspondence between the dc component of the spin current and enhancement of the damping. For thin metal and insulator, the relation turns out to be modied. For the insulating case, driven locally by interfacesdexchange interaction due to magnetic proximity eect, physical mechanism is distinct from the metallic case. Further study of proximity eect and interface spin-orbit interaction would be crucial to interpret experimental results in particular for insulators. 1arXiv:1706.01185v1 [cond-mat.mes-hall] 5 Jun 2017I. INTRODUCTION Spin current generation is of a fundamental importance in spintronics. A dynamic method using magnetization precession induced by an applied magnetic eld, called the spin pumping eect, turns out to be particularly useful1and is widely used in a junction of a ferromagnet (F) and a normal metal (N)(Fig. 1). The generated spin current density (in unit of A/m2) has two independent components, proportional to _nandn_n, wherenis a unit vector describing the direction of localized spin, and thus is represented phenomenologically as js=e 4(Arn_n+Ai_n); (1) whereeis the elementally electric charge and ArandAiare phenomenological constants having unit of 1 =m2. Spin pumping eect was theoretically formulated by Tserkovnyak et al.2 by use of scattering matrix approach3. This approach, widely applied in mesoscopic physics, describes transport phenomena in terms of transmission and re ection amplitudes (scattering matrix), and provides quantum mechanical pictures of the phenomena without calculating explicitly the amplitudes. Tserkovnyak et al. applied the scattering matrix formulation of general adiabatic pumping4,5to the spin-polarized case. The spin pumping eect was described in Ref.2in terms of spin-dependent transmission and re ection coecients at the FN interface, and it was demonstrated that the two parameters, ArandAi, are the real and the imaginary part of a complex parameter called the spin mixing conductance. The spin mixing conductance, which is represented by transmission and re ection coecients, turned out to be a convenient parameter for discussing spin current generation and other eects like the inverse spin-Hall eect. Nevertheless, scattering approach hides microscopic physical pictures of what is going on, as the scattering coecients are not fundamental material parameters but are composite quantities of Fermi wave vector, electron eective mass and the interface properties. Eects of slowly-varying potential is described in a physically straightforward and clear manner by use of a unitary transformation that represents the time-dependence. (See Sec. II A for details.) The laboratory frame wave function under time-dependent potential, j (t)i, is written in terms of a static ground state ('rotated frame' wave function) jiand a unitary matrixU(t) asj (t)i=U(t)ji. The time-derivative @tis then replaced by a covariant derivative,@t+ (U1@tU), and the eects of time-dependence are represented by (the time- component of) an eective gauge eld, A  i(U1@tU) (See Eq. (12)). In the same 2FIG. 1. Spin pumping eect in a junction of ferromagnet (F) and normal metal (N). Dynamic magnetization n(t) generates a spin current jsthrough the interface. manner as the electromagnetic gauge eld, the eective gauge eld generates a current if spatial homogeneity is present (like in junctions) and this is a physical origin of adiabatic pumping eect in metals. In the perturbative regime or in insulators, a simple picture instead of eective gauge eld can be presented. Let us focus on the case driven by an sdexchange interaction, Jsdn(t)
, whereJsdis a coupling constant and is the electron spin. Considering the second-order eect of the sdexchange interaction, the electron wave function has a contribution of a time-dependent amplitude U(t1;t2) = (Jsd)2(n(t1)
)(n(t2)
) = (Jsd)2[(n(t1)
n(t2)) +i[n(t1)n(t2)]
];(2) wheret1andt2are the time of the interactions. The rst term on the right-hand side, representing the amplitude for charge degrees of freedom, is neglected. The spin contribution vanishes for static spin conguration, as is natural, while for slowly varying case, it reads U(t1;t2)'i(t1t2)(Jsd)2(n_n)(t1)
: (3) As a result of this amplitude, spin accumulation and spin current is induced proportional to n_n. The fact indicates that n_nplays a role of an eective scalar potential or voltage in electromagnetism, as we shall demonstrate in Sec. VII B for insulators. (The factor of time dierence is written in terms of derivative with respect to energy or angular frequency in a rigorous derivation. See for example, Eqs. (129)(132).) The essence of spin pumping eect is therefore the non-commutativity of spin operators. The above picture in the perturbative regime naturally leads to an eective gauge eld in the strong coupling limit6. 3The same scenario applies for cases of spatial variation of spin, and an equilibrium spin current proportional to nrinemerges, where idenotes the direction of spatial variation7. The spin pumping eect is therefore the time analog of the equilibrium spin current induced by vector spin chirality. Moreover, charge current emerges from the third-order process from the identity6 tr[(n1
)(n2
)(n3
)] = 2in1
(n2n3); (4) and this factor, a scalar spin chirality, is the analog of the spin Berry phase in the pertur- bative regime. The spin pumping eect and spin Berry's phase and spin motive force have the same physical root, namely the non-commutative spin algebra. From the scattering matrix theory view point the cases of metallic and insulating fer- romagnet make no dierence as what conduction electrons in the normal metal see is the interface. From physical viewpoints, such treatment appears too crude. Unlike the metallic case discussed above, in the case of insulator ferromagnet, the coupling between the mag- netization and the conduction electron in normal metal occurs due to a magnetic proximity eect at the interface. Thus the spin pumping by an insulator ferromagnet seems to be a locally-induced perturbative eect rather than a transport induced by a driving force due to a generalized gauge eld. We therefore need to apply dierent approaches for the two cases as brie y argued above. In the insulating case, one may think that magnon spin current is generated inside the ferromagnet because magnon itself couples to an eective gauge eld8 similarly to the electrons in metallic case. This is not, however, true, because the gauge eld for magnon is abelian (U(1)). Although scattering matrix approach apparently seems to apply to both metallic and insulating cases, it would be instructive to present in this paper a consistent microscopic description of the eects to see dierent physics governing the two cases. A. Brief overview of theories and scope of the paper Before carrying out calculation, let us overview history of theoretical studies of spin pumping eect. Spin current generation in a metallic junction was originally discussed by Silsbee9before Tserkovnyak et al. It was shown there that dynamic magnetization induces spin accumulation at the interface, resulting in a diusive ow of spin in the normal metal. 4Although of experimental curiosity at that time was the interface spin accumulation, which enhances the signal of conduction electron spin resonance, it would be fair to say that Silsbee pointed out the `spin pumping eect'. In Ref.2, spin pumping eect was originally argued in the context of enhancement of Gilbert damping in FN junction, which had been a hot issue after the study by Berger10, who studied the case of FNF junction based on a quantum mechanical argument. Berger discussed that when a normal metal is attached to a ferromagnet, the damping of ferromagnet is enhanced as a result of spin polarization formed in the normal metal, and the eect was experimentally conrmed by Mizukami11. Tserkovnyak et al. pointed out that the eect has a dierent interpretation of the counter action of spin current generation, because the spin current injected into the normal metal indicates a change of spin angular momentum or a torque on ferromagnet. In fact, the equation of motion for the magnetization of ferromagnet reads _n= Bnn_na3 eSdjs; (5) where is the gyromagnetic ratio, is the Gilbert damping coecient, dis the thickness of the ferromagnet, Sis the magnetude of localized spin, and ais the lattice constant. Spin current of Eq. (1) thus indicates that the gyromagnetic ratio and the the Gilbert damping coecient are modied by the spin pumping eect to be2 ~=+a3 4SdAr ~ =  1 +a3 4SdAi1 : (6) The spin pumping eect is therefore detected by measuring the eective damping constant and gyromagnetic ratio. The formula (6) is, however, based on a naive picture neglecting the position-dependence of the damping torque and the relation between the pumped spin current amplitude and damping or would not be so simple in reality. (See Sec. V.) The issue of damping in FN junction was formulated based on linear-response theory by Simanek and Heinirch12,13. They showed that the damping coecient is given by the rst- order derivative with respect to the angular frequency !of the imaginary part of the spin correlation function and argued that the damping eect is consistent with the Tserkovnyak's spin pumping eect. Recently, a microscopic formulation of spin pumping eect in metallic junction was provided by Chen and Zhang14and one of the author15by use of the Green's 5functions, and a transparent microscopic picture of pumping eect was provided. Scattering representation and Green's function one are related14because the asymptotic behaviors of the Green's functions at long distance are governed by transmission coecient16. In the study of Ref.15, the uniform ferromagnet was treated as a dot having only two degrees of freedom of spin. Such simplication neglects the dependence on electron wave vectors in ferromagnets and thus cannot discuss the the case of inhomogeneous magnetization or position-dependence of spin damping. The aim of this paper is to provide a microscopic and consistent theoretical formula- tion of spin pumping eect for metallic and insulating ferromagnets. We do not rely on the scattering approach. Instead we provide elementary quantum mechanical argument to demonstrated that spin current generation is a natural consequence of magnetization dy- namics (Sec. II). Based on the formulation, the eect of interface spin-orbit interaction is discussed in Sec. III. We also provide a rigorous formulation based on eld-theoretic ap- proach emploied in Ref.15in Sec. IV. We also reproduce within the same framework Berger's result10that the spin pumping eect is equivalent to the enhancement of the spin damping (Sec. V). Eect of inhomogeneous magnetization is brie y discussed in Sec. VI. Case of insulating ferromagnet is studied in Sec. VII assuming that the pumping is induced by an interface exchange interaction between the magnetization and conduction electron in normal metal, namely, by magnetic proximity eect. The interaction is treated perturbatively similarly to Refs.17,18. The dominant contribution to the spin current, the one linear in the interface exchange interaction, turns out to be proportional to _n, while the one proportional to n_nis weaker if the proximity eect is weak. The contribution from the magnon, magnetization uctuation, is also studied. As has been argued8, a gauge eld for magnon emerges from magnetization dynamics. It is, however, an adiabatic one diagonal in spin, which acts as chemical potential for magnon giving rise only to adiabatic spin polarization proportional to n. This is in sharp contrast to the metallic case, where electrons are directly driven by spin- ip component of spin gauge eld, resulting in perpendicular spin accumulation, i.e., along _nandn_n. The excitation in ferromagnet when magnetization is time-dependent is therefore dierent for metallic and insulating cases. We show that magnon excitation nevertheless generates perpendicular spin current,n_n, in the normal metal as a result of annihilation and creation at the interface, which in turn ips electron spin. The result of magnon-driven contribution agrees with the 6one in previous study19carried out in the context of thermally-driven spin pumping ('spin Seebeck' eect). It is demonstrated that the magnon-induced spin current depends linearly on the temperature at high temperature compared to magnon energy. The amplitude of magnon-driven spin current provides the magnitude of magnetic proximity eect. In our analysis, we calculate consistently the pumped spin current and change of the Gilbert damping and resonant frequency and obtain the relations among them. It is shown that the spin mixing conductance scenario saying that the magnitude of spin current pro- portional ton_nis given by the enhancement factor of the Gilbert damping constant2, applies only the case of thick ferromagnetic metal. For thin metallic case and insulator case, dierent relations hold (See Sec. VIII.). II. QUANTUM MECHANICAL DESCRIPTION OF METALLIC CASE In this section, we derive the spin current generated by the magnetization dynamics of metallic ferromagnet by a quantum mechanical argument. It is sometimes useful for intuitive understanding, although the description may lack clearness as it cannot handle many-particle nature like particle distributions. In Sec. IV we formulate the problem in the eld-theoretic language. A. Electrons in ferromagnet with dynamic magnetization The model we consider is a junction of metallic ferromagnet (F) and a normal metal (N). The magnetization (or localized spins) in the ferromagnet is treated as spatially uniform but changing with time slowly. As a result of strong sdexchange interaction, the conduction electron's spin follows instantaneous directions of localized spins, i.e., the system is in the adiabatic limit. The quantum mechanical Hamiltonian for the ferromagnet is HF=r2 2mFMn(t)
; (7) wheremis the electron's mass, is a vector of Pauli matrices, Mrepresents the energy split- ting due to the sdexchange interaction and n(t) is a time-dependent unit vector denoting the localized spin direction. The energy is measured from the Fermi energy F. 7As a result of the sdexchange interaction, the electron's spin wave function is given by20 jnicos 2j"i+ sin 2eij#i (8) wherej"iandj#irepresent the spin up and down states, respectively, and ( ;) are polar coordinates for n. To treat slowly varying localized spin, we switch to a rotating frame where the spin direction is dened with respect to instantaneous direction n7. This corresponds to diagonalizing the Hamiltonian at each time by introducing a unitary matrix U(t) as jn(t)iU(t)j"i; (9) where U(r) =0 @cos 2sin 2ei sin 2eicos 21 A; (10) where states are in vector representation, i.e., j"i=0 @1 01 Aandj#i=0 @0 11 A. The rotated Hamiltonian is diagonalized as (in the momentum representation) eHFU1HFU=kMz; (11) wherekk2 2mFis the kinetic energy in the momentum representation (Fig. 2). In general, FIG. 2. Unitary transformation Ufor conduction electron in ferromagnet converts the original Hamiltonian HFinto a diagonalized uniformly spin-polarized Hamiltonian eHFand an interaction with spin gauge eld, As;t
. when a statej ifor a time-dependent Hamiltonian H(t), satisfying the Schr odinger equation i@ @tj i=H(t)j i, is written in terms of a state j iconnected by a unitary transformation jiU1j i, the new state satises a modied Schr odinger equation  i@ @t+iU1@ @tU ji=~Hji; (12) 8where ~HU1HU. Namely, there arises a gauge eld iU1@ @tUin the new frame ji. In the present case of dynamic localized spin, the gauge eld has three components (sux t denotes the time-component); As;tiU1@ @tUAs;t
; (13) explicitly given as7 As;t=1 20 BBB@@tsinsincos@t @tcossinsin@t (1cos)@t1 CCCA: (14) Including the gauge eld in the Hamiltonian, the eective Hamiltonian in the rotated frame reads eHe FeHF+As;t
=0 @kMAz s;tA s;t A+ s;tk+M+Az s;t1 A (15) whereA s;tAx s;tiAy s;t. We see that the adiabatic ( z) component of the gauge eld, Az s;t, acts as a spin-dependent chemical potential (spin chemical potential) generated by dynamic magnetization, while non-adiabatic ( xandy) components causes spin mixing. In the case of uniform magnetization we consider, the mixing is between the electrons with dierent spin"and#but having the same wave vector k, because the gauge eld A s;tcarries no momentum. This leads to a mixing of states having an excitation energy of Mas shown in Fig. 3. In low energy transport eects, what concern are the electrons at the Fermi energy; The wave vector kshould be chosen as kF+andkF, the Fermi wave vectors for "and# electrons, respectively. (Eects of nite momentum transfer is discussed in Sec. VI. ) The Hamiltonian Eq. (15) is diagonalized to obtain energy eigenvalues of ~ k=k q (M+Az s;t)2+jA? s;tj2, wherejA? s;tj2A+ s;tA s;tand=represents spin ("and#cor- respond to + and , respectively). We are interested in the adiabatic limit, and so the contribution lowest-order, namely, the rst order, in the perpendicular component, A? s;t, is sucient. In the present rotating-frame approach, the gauge eld is treated as a static po- tential, since it already include time-derivative to the linear order (Eq. (14)). Moreover, the adiabatic component of the gauge eld, Az s;t, is neglected, as it modies the spin pumping only at the second-order of time-derivative. The energy eigenvalues, k'kM, are 9thus unaected by the gauge eld, while the eigenstates to the linear order read jk"iFjk"iA+ s;t Mjk#i jk#iFjk#i+A s;t Mjk"i; (16) corresponding to energy of k+andk, respectively. For low energy transport, states we need to consider are the following two having spin-dependent Fermi wave vectors, kFfor =";#, namely jkF""iF=jkF""iA+ s;t MjkF"#i jkF##iF=jkF##i+A s;t MjkF#"i: (17) FIG. 3. For uniform magnetization, the non-adiabatic components of the gauge eld, A s;t, induces a spin ip conserving the momentum. B. Spin current induced in the normal metal Spin pumping eect is now studied by taking account of the interface hopping eects on states in Eq. (17). The interface hopping amplitude of electron in F to N with spin is denoted by ~tand the amplitude from N to F is ~t . We assume that the spin-dependence of electron state in F is governed by the relative angle to the magnetization vector, and hence the spin is the one in the rotated frame. Assuming moreover that there is no spin ip scattering at the interface, the amplitude ~tis diagonal in spin. (Interface spin-orbit interaction is considered in Sec. III.) The spin wave function formed in the N region at the 10interface as a result of the state in F (Eq. (17)) is then jkF"iN~tjkF"i=~t"jkF"i ~t#A+ s;t MjkF#i jkF#iN~tjkF#i=~t#jkF#i+~t"A s;t MjkF"i; (18) wherekFis the Fermi wave vector of N electron. The spin density induced in N region at the interface is therefore es(N)=1 2(NhkF"jjkF"iN"+NhkF#jjkF#iN#) (19) whereis the spin-dependent density of states of F electron at the Fermi energy. It reads es(N)=1 2X T^z"# M Re[T"#]A? s;t+ Im[T"#](^zA? s;t)
(20) whereA? s;t= (Ax s;t;Ay s;t;0) =As;t^zAz s;tis the transverse (non-adiabatic) components of spin gauge eld and T0~t ~t0: (21) Spin density of Eq. (20) is in the rotated frame. The spin polarization in the laboratory frame is obtained by a rotation matrix Rij, dened by U1iURijj; (22) as s(N) i=Rijes(N) j: (23) Explicitly,Rij= 2mimjij, wherem sin 2cos;sin 2sin;cos 2
7. Using Rij(A? s;t)j=1 2(n_n)i Rij(^zA? s;t)j=1 2_ni; (24) andRiz=ni, the induced interface spin density is nally obtained as s(N)=s 0n+ Re[s](n_n) + Im[s]_n (25) 11where s 01 2X T s"# 2MT"#: (26) Since the N electrons contributing to induced spin density is those at the Fermi energy, the spin current is simply proportional to the induced spin density as jsN=kF ms(N), resulting in j(N) s=kF ms 0n+kF mRe[s](n_n) +kF mIm[s]_n: (27) This is the result of spin current at the interface. The pumping eciency is determined by the product of hopping amplitudes t"andt #. The spin mixing conductance dened in Ref.2corresponds to iT"#. If spin mixing eects due to spin-orbit interaction is neglected at the interface, the hopping amplitudes tare chosen as real, and Im[ s] = 0. If spin current proportional to _nis measured, it would be useful tool to estimate the strength of interface spin-orbit interaction, as discussed in Sec. III. It should be noted that the spin pumping eect at the linear order in time-derivative is mapped to a static problem of spin polarization formed by a static spin-mixing potential in the rotated frame as was mentioned in Ref.15. The rotate frame approach employed here provides clear physical picture, as it grasps the low energy dynamics in a mathematically proper manner. In this approach, as we have seen, it is clearly seen that pumping of spin current arises as a result of o-diagonal components of the spin gauge eld that cause electron spin ip. Important role of nonadiabaticity is also indicated in a recent analysis based on the full counting statistics21. In the strict sense, spin pumping eect is a result of a non-adiabatic process including state change. The same goes for general adiabatic pumping; Some sort of state change is necessary for current generation, although the nonadiabaticity is obscured in the conventional \adiabatic\ argument focusing on the wave function in the laboratory frame. In the case of slowly-varying external potential with frequency acting on electrons, the state change is represented by the Fermi distribution dierence, f(!+ )f(!)' f0(!), where!is the electron frequency3,4. The existence of a factor of f0clearly indicates that a state change or nonadiabaticity is necessary for current pumping. 12III. EFFECTS OF INTERFACE SPIN-ORBIT INTERACTION In this section, we discuss the eect of spin-orbit interaction at the interface, which modies hopping amplitude ~t. We particularly focus on that linear in the wave vector, namely the interaction represented in the continuum representation by a Hamiltonian Hso=a2(x)X ij ijkij; (28) where ijis a coecient having the unit of energy representing the spin-orbit interaction, ais the lattice constant, and the interface is chosen as at x= 0. Assuming that spin- orbit interaction is weaker than the sdexchange interaction in F, we carry out a unitary transformation to which diagonalize the sdinteraction to obtain Hso=a2(x)X ije ijkij; (29) wheree ijP l ilRlj, withRijbeing a rotation matrix dened by Eq. (22). This spin- orbit interaction modies diagonal hopping amplitude ~tiin the direction iat the interface to become a complex as eti=~t0 iiX je ijj: (30) (In this section, we denote the total hopping amplitude including the interface spin-orbit interaction by etand the one without by et0.) We consider the hopping amplitude perpendic- ular to the interface, i.e., along the xdirection, and suppress the sux irepresenting the direction. In the matrix representation for spin the hopping amplitude is et(etx) =0 @et"et"# et#"et#1 A; (31) where et"=~t0 "ie xzet#=~t0 #+ie xz et"#=i(e xx+ie xy) et#"=i(e xxie xy): (32) Let us discuss how the spin pumping eect discussed in Sec. II B is modied when the hopping amplitude is a matrix of Eq. (31). The spin pumping eciency is written as in Eqs. 13(21)(26). In the absence of spin-orbit interaction hopping amplitude ~tis chosen as real, and thus the contribution proportional to n_nin Eq. (27) is dominant. Spin-orbit interaction enhances the other contribution proportional to _nbecause it gives rise to an imaginary part. Moreover, it leads to spin mixing at the interface, modifying the spin accumulation formed in the N region at the interface. The electron states in the N region at the interface are now given instead of Eq. (18) by the following two states (choosing basis as0 @jkF"i jkF#i1 A) jkF"iNetjkF""iF=0 @et"et"#A+ s;t M et#"et#A+ s;t M1 A jkF#iNetjkF##iF=0 @et"#+et"A s;t M et#+et#"A s;t M1 A: (33) The pumped (i.e., linear in the gauge eld) spin density for these two states are NhkF"jjkF"iN=2 M(A? s;tRe[Ttot "#] + (^zA? s;t)Im[Ttot "#] +Re[(et"#)et#"]0 BBB@Ax s;t Ay s;t 01 CCCA+ Im[(et"#)et#"]0 BBB@Ay s;t Ax s;t 01 CCCA1 CCCA ^z(Ax s;tRe[(et")et"#et#(et#")]Ay s;tIm[(et")et"#et#(et#")]) (34) NhkF#jjkF#iN=2 M(A? s;tRe[Ttot "#] + (^zA? s;t)Im[Ttot "#] +Re[(et"#)et#"]0 BBB@Ax s;t Ay s;t 01 CCCA+ Im[(et"#)et#"]0 BBB@Ay s;t Ax s;t 01 CCCA1 CCCA +^z(Ax s;tRe[(et")et"#et#(et#")]Ay s;tIm[(et")et"#et#(et#")]) (35) We here focus on the linear eect of interface spin-orbit interaction and neglect the spin polarization along the magnetization direction, n. The expression for the pumped spin current then agrees with Eq. (27) with the amplitude swritten in terms of hopping including the interface spin-orbit, T"#= ((~t0 ")+i(e xz))(~t0 #+ie xz): (36) 14If bulk spin-orbit interaction is neglected, bare hopping amplitude ~t0 is real and we may reasonably assume that e ijis real. The interface spin-orbit then leads to an imaginary part as (usinge xz=ni xi) Im[s] ="# 2M(~t0 "+~t0 #) xini: (37) The amplitude of spin current proportional to _nthus works as a probe for interface spin-orbit interaction strength, xi. Let us discuss some examples. Of recent particular interest is the interface Rashba interaction, represented by antisymmetric coecient (R) ij=ijkR k; (38) whereRis a vector representing the Rashba eld. In the case of interface, Ris perpen- dicular to the interface, i.e., Rk^x. Therefore the interface Rashba interaction leads to (R) xj= 0 and does not modify spin pumping eect at the linear order. (It contributes at the second order as discussed in Ref.14.) In other words, vector coupling between the wave vector and spin in the form of kexists only along the x-direction, and does not aect the interface hopping (i.e., does not include kx). In contrast, a scalar coupling (D)(k
) ((D)is a coecient), called the Dirac type spin-orbit interaction, leads to (D) ij=(D)ij. The spin current along _nthen reads j_n s=(D)kF("#) 2mM(~t0 "+~t0 #)nx_n: (39) For the case of in-plane easy axis along the zdirection and magnetization precession given byn(t) = (sincos!t;sinsin!t;cos), whereis the precession angle and !is the angular frequency, we expect to have a dc spin current along the ydirection, as nx_n=! 2sin2^y (nx_ndenotes time average). IV. FIELD THEORETIC DESCRIPTION OF METALLIC CASE Here we present a eld-theoretic description of spin pumping eect of metallic ferromag- net. The many-body approach has an advantage of taking account of particle distributions automatically. Moreover, it describes propagation of particle density in terms of the Green's 15functions, and thus is suitable for studying spatial propagation as well as for intuitive under- standing of transport phenomena. All the transport coecients are determined by material constants. The formalism presented here is essentially the same as in Ref.15, but treating the fer- romagnet of a nite size and taking account of electron states with dierent wave vectors. Interface spin-orbit interaction is not considered here. Conduction electron in ferromagnetic and normal metals are denoted by eld operators d,dyandc,cy, respectively. These operators are vectors with two spin components, i.e., d(d";d#). The Hamiltonian describing the F and N electrons is HF+HN, where HFZ Fd3rdy r2 2mFMn(t)
 d HNZ Nd3rcy r2 2mF c: (40) We set the Fermi energies for ferromagnet and normal metal equal. The hopping through the interface is described by the Hamiltonian HIZ IFd3rZ INd3r0 cy(r0)t(r0;r;t)d(r) +dy(r)t(r0;r;t)c(r0)
; (41) wheret(r0;r;t) represents the hopping amplitude of electron from rin ferromagnetic regime to a siter0in the normal region and the integrals are over the interface (denoted by I F and I Nfor F and N regions, respectively). The hopping amplitude is generally a matrix depending on magnetization direction n(t), and thus depends on time t. Hopping is treated as energy-conserving. Assuming sharp interface at x= 0, the momentum perpendicular to the interface is not conserved on hopping. We are interested in the spin current in the normal region, given by j s;i(r;t) =1 4m(r(r)r(r0))itr[G< N(r;t;r0;t)jr0=r; (42) whereG< N(r;t;r0;t0)i c(r;t)cy(r0;t0) denotes the lesser Green's function for the normal region. It is calculated from the Dyson's equation for the path-ordered Green's function dened for a complex time along a complex contour C GN(r;t;r0;t0) =gN(rr0;tt0) +Z cdt1Z cdt2Z d3r1Z d3r2gN(rr1;tt1)N(r1;t1;r2;t2)GN(r2;t2;r0;t0); (43) 16whereg< Ndenotes the Green's function without interface hopping and  N(r1;t1;r2;t2) is the self-energy for N electron, given by the contour-ordered Green's function in the ferromagnet as N(r1;t1;r2;t2)Z IFd3r3Z IFd3r4t(r1;r3;t1)G(r3;t1;r4;t2)t(r2;r4;t2): (44) Herer1andr2are coordinates at the interface I Nin N region and r3andr4are those in IFfor F.Gis the contour-ordered Green's function for F electron in the laboratory frame including the eect of spin gauge eld. We denote Green's functions of F electron by G andgwithout sux and those of N electron with sux N. The lesser component of the normal metal Green's function is obtained from Eq. (43) as (suppressing the time and space coordinates) G< N= (1 +Gr Nr N)g< N(1 + a NGa N) +Gr N< NGa N: (45) For pumping eects, the last term on the right-hand side is essential, as it contains the information of excitation in F region. We thus consider the second term only; G< N'Gr N< NGa N; (46) and neglect spin-dependence of the normal region Green's functions, Gr NandGa N. The contribution is diagramatically shown in Fig. 4. A. Rotated frame To solve for the Green's function in the ferromagnet, rotated frame we used in Sec. II A is convenient. In the eld representation, the unitary transformation is represented as (Fig. 5(c)) d=U~d; c =U~c; (47) whereUis the same 22 matrix dened in Eq. (10). We rotate N electrons as well as F electrons, to simplify the following expressions. The hopping interaction Hamiltonian reads HI=Z IFd3rZ INd3r0 ~cy(r0)~t(r0;r)~d(r) +~dy(r)~t(r0;r)~c(r0) ; (48) 17FIG. 4. (a) Schematic diagramatic representations of the lessor Green's function for N electron connecting the same position r,G< N(r;r)'Gr N< NGa Nrepresenting propagation of electron density. It is decomposed into a propagation of N electron from rto the interface at r2, then hopping to r4 in the F side, a propagation inside F, followed by a hopping to N side (to r1) and propagation back tor. (Position labels are as in Eqs. (43)(44).) (b): The self energy < Nrepresents all the eects of the ferromagnet. (c) Standard Feynman diagram representation of lessor Green's function for N atr, Eqs. (46) and (44). FIG. 5. Unitary transformation Uof F electron converts the original system with eld operator d(shown as (a)) to the rotated one with eld operator ~dU1d(b). The hopping amplitude for representation in (b) is modied by U. If N electrons are also rotated as ~ cU1c, hopping becomes ~tU1tU, while the N electron spin rotates with time, as shown as (c). where ~t(r0;r)Uy(t)t(r0;r;t)U(t); (49) is the hopping amplitude in the rotated frame. The rotated amplitude (neglecting interface spin-orbit interaction) is diagonal in spin; ~t=0 @~t"0 0~t#1 A: (50) Including the interaction with spin gauge eld, the Hamiltonian for F and N electrons in 18the momentum representation is HF+HN=X k~dy k0 @kMAz s;tA s;t A+ s;tk+M+Az s;t1 A~dk+X k(N) k~cy k~ck (51) As for the hopping, we consider the case the interface is atomically sharp. The hopping Hamiltonian is then written in the momentum space as HI=X kk0 ~cy(k)~t(k;k0)~d(k0) +~dy(k0)~t(k;k0)~c(k) ; (52) wherek= (kx;ky;kz),k0= (k0 x;ky;kz), choosing the interface as the plane of x= 0. Namely, the wave vectors parallel to the interface are conserved while kxandk0 xare uncorrelated. B. Spin density induced by magnetization dynamics in F Pumped spin current in N is calculated by evaluating < Nand using Eqs. (42)(45)(46). Before discussing the spin current, let us calculate spin density in ferromagnet induced by magnetization dynamics neglecting the eect of interface, HI. (Eects of HIare discussed in Sec. V.) The spin accumulation in N is discussed by extending the calculation here as shown in Sec. IV C. The lessor Green's function in F in the rotated frame including the spin gauge eld to the linear order is calculated from the Dyson's equation G<=g<+gr(As;t
)g<+g<(As;t
)ga; (53) whereg(=<;r,a) represents Green's functions without spin gauge eld. The lessor Green's function satises for static case g<=F(gagr), whereF0 @f"0 0f#1 Ais spin- dependent Fermi distribution function. We thus obtain the Green's function at the linear order as15 G<=gr[As;t
;F]ga+gaF(As;t
)gagr(As;t
)Fgr: (54) The last two terms of the right-hand side are rapidly oscillating as function of position and are neglected. The commutator is calculated as (sign denotes spin"and#) [As;t
;F] = (f+f)X ()A s;t: (55) 19FIG. 6. Feynman diagram for electron spin density of ferromagnet induced by magnetization dynamics (represented by spin gauge eld As) neglecting the eect of normal metal. The amplitude is essentially given by the spin ip correlation function (Eq. (58)). In the rotated frame, the spin density in F pumped by the spin gauge eld is therefore (diagrams shown in Fig. 6) ~s(F) (k;k0)iZd! 2tr[G<(k;k0;!)] =iZd! 2X k00(fk00+fk00)X ()A s;ttr[gr(k;k00;!)ga(k00;k0;!)] =8 < :iRd! 2P k00(fk00+fk00)A s;tgr (k;k00;!)ga (k00;k0;!) (=) 0 ( =z): (56) Let us here neglect the eects of interface in dicussing spin polarization of F electrons; Then the Green's functions are translationally invariant, i.e., ga(k;k0) =k;k0ga(k) (a= r;a). Using the explicit form of the free Green's function, ga (k;!) =1 !k;i0, and Zd! 2gr (k;k00;!)ga (k00;k0;!) =i k;k;+i0; (57) the spin density in the rotated frame then reduces to ~s(F) (k) =A s;t; (58) where X kfk;fk; k;k;+i0; (59) is the spin correlation function with spin ip, + i0 meaning an innitesimal positive imaginary part. Since we focus on adiabatic limit and spatially uniform magnetization, the correlation function is at zero momentum- and frequency-transfer. We thus easily see that =n+n 2M; (60) 20wheren=P kfkis spin-resolved electron density. The spin polarization of Eq. (58) in the rotated frame is proportional to A? s;t, and represents a renormalization of total spin in F. In fact, it corresponds in the laboratory frame tos(F)/n_n, and exerts a torque proportional to _nonn. It may appear from Eq. (60) that a damping of spin, i.e., a torque proportional to n_n, arises when the imaginary part for the Green's function becomes nite, because 1 Mis replaced by1 Mii, whereiis the imaginary part. This is not always the case. For example, nonmagnetic impurities introduce a nite imaginary part inversely proportional to the elastic lifetime ( ),i 2. They should not, however, cause damping of spin. The solution to this apparent controversy is that Eq. (56) is not enough to discuss damping even including lifetime. In fact, there is an additional process called vertex correction contributing to the lesser Green's function, and it gives rise to the same order of small correction as the lifetime does, and the sum of the two contributions vanishes. Similarly, we expect damping does not arise from spin-conserving component of spin gauge eld, Az s;t. This is indeed true as we explicitly demonstrate in Appendix A. We shall show in Sec. V that damping arises from the spin- ip components of the self energy. C. Spin polarization and current in N FIG. 7. Feynman diagram for electron spin density of normal metal driven by the spin gauge eld of ferromagnetic metal, As. The spin current is represented by the same diagram but with spin current vertex. The spin polarization of N electron lesser Green's function including the self-energy to 21the linear order is calculated from Eqs. (46) (54)(55) as (diagram shown in Fig. 7) itr[G< N(r;t;r0;t)] =iX kk0k00eik
reik0
r0gr N(k;!)ga N(k0;!) X (fk00fk00)A s;t~t(k;k00)~t (k00;k0)gr (k00;!)ga (k00;!): (61) We assume that dependence of N Green's functions on !is weak and useP keik
rgr N(k;!) = iNeikFxejxj=`gr N(r), where`is elastic mean free path, NandkFare the density of states at the Fermi energy and Fermi wave vector, respectively, whose !-dependences are ne- glected. (For innitely wide interface, the Green's function becomes one-dimensional.) As a result of summation over wave vectors, the product of hopping amplitudes ~t(k;k00)~t (k00;k0) is replaced by the average over the Fermi surface, ~t~t T, i.e., ~t(k;k00)~t (k00;k0)!T: (62) The spin polarization of N electron induced by magnetization dynamics (the spin gauge eld) is therefore obtained in the rotated frame as ~s(N) (r;t) =jgr N(r)j2X A s;tT; (63) or using += ~s(N)(r;t) =2jgr N(r)j2 A? s;tRe[+T+] + (^zA? s;t)Im[+T+] : (64) In the laboratory frame, we have (using s(N) i=Rij~s(N) j) s(N)(r;t) =jgr N(r)j2 Re[+T+](n_n) + Im[+T+]_n : (65) The spin current induced in N region is similarly given by (neglecting the contribution proportional to n) js(r;t) =kF mjgr N(r)j2 Re[+T+](n_n) + Im[+T+]_n =ejxj=`(Re[s](n_n) + Im[s]_n); (66) where s2kF2 N 2mM(n+n)T+: (67) 22The coecient sis essentially the same as the one in Eq. (27) derived by quantum mechan- ical argument, as quantum mechanical dimensionless hopping amplitude corresponds to N~t of eld representation. For 3d ferromagnet, we may estimate the spin current by approximating roughly M 1=NF1eV andnkF3. The hopping amplitude jT+jin metallic case would be order ofF. The spin current density then is of the order of (including electric charge eand recovering ~),jse~kF mh~! F51011A/m2if precession frequency is 10 GHz. V. SPIN ACCUMULATION IN FERROMAGNET The spin current pumping is equivalent to the increase of spin damping due to magne- tization precession, as was discussed in Refs.2,10. In this section, we conrm this fact by calculating the torque by evaluating the spin polarization of the conduction electron spin in F region. There are several ways to evaluate damping of magnetization. One way is to calculate the spin- ip probability of the electron as in Ref.10, which leads to damping of localized spin in the presence of strong sdexchange interaction. The second is to estimate the torque on the electron by use of equation motion22. The relation between the damping and spin current generation is clearly seen in this approach. In fact, the total torque acting on conduction electron is ( ~times) the time-derivative of the electron spin density, ds dt=i [H;dy]d + dy;d]
: (68) At the interface, the right-hand side arises from the interface hopping. Using the hopping Hamiltonian of Eq. (41), we have ds dt interface=i cytd dytyc
; (69) as the interface contribution. As is natural, the the right -hand side agrees with the denition of the spin current passing through the interface. Evaluating the right-hand side, we obtain in general a term proportional to n_n, which gives the Gilbert damping, and term proportional to_n, which gives a renormalization of magnetization. In contrast, away from the interface, the commutator [ H;d] arises from the kinetic term H0R d3rjrdj2 2mdescribing electron 23propagation, resulting in ds dt=i [H0;dy]d + dy;d]
=r
j s (70) wherej s(r)i 2m(rrr r0) dy(r0)d(r) jr0=ris the spin current. Away from the inter- face, the damping therefore occurs if the spin current has a source or a sink at the site of interest. Here we use the third approach and estimate the torque on the localized spin by calculat- ing the spin polarization of electrons as was done in Refs.7,23. The electron spin polarization at positionrin the ferromagnet at time tiss(F)(r;t) dyd , which reads in the rotated frames(F) =R~s(F) , where ~s(F) (r;t) =itr[G<(r;r;t;t)]; (71) whereG< 0(r;r0;t;t0)iD ~dy 0~dE is the lesser Green's function in F region, which is a matrix in spin space ( ;0=). We are interested in the eect of the N region arising from the hopping. We must note that the hopping interaction of Eq. (48) is not convenient for integrating out N electrons, since the ~ celectrons' spins are time-dependent as a result of a unitary transformation, U(t). We thus use the following form (Fig. 5(b)), HI=Z IFd3rZ INd3r0 cy(r0)U~t(r0;r)~d(r) +~dy(r)~t(r0;r)Uyc(r0) ; (72) namely, the hopping amplitude between ~dandcelectrons includes unitary matrix U. Let us argue in the rotated frame why the eect of damping arising from the interface. In the totally rotated frame of Fig. 5(c), the spin of F electron is static, while that of N electron varies with time. When F electron hops to N region and comes back, therefore, electron spin gets rotated with the amount depending on the time it stayed in N region. This eect is in fact represented by a retardation eect of the matrices UandU1in Eq. (72). If o-diagonal nature of UandU1are neglected, the interface eects are all spin-conserving and do not induce damping for F electron (See Sec. A). We now proceed calculation of induced spin density in the ferromagnetic metal. Diagra- matic representation of the contribution is in Fig. 8. Writing spatial and temporal positions explicitly, the self-energy of F electron arising from the hopping to N region reads ( r1and 24FIG. 8. Diagramatic representation of the spin accumulation in ferromagnetic metal induced as a result of coupling to the normal metal (Eqs. (71)(73)). Conduction electron Green's functions in ferromagnet and normal metal are denoted by gandgN, respectively. Time-dependent matrix U(t), dened by Eq. (10), represents the eect of dynamic magnetization. Expanding UandU1 with respect to slow time-dependence of magnetization, we obtain gauge eld representation, Eq. (75). r2are in F) a(r1;r2;t1;t2) =Z INd3r0 1Z INd3r0 2~t(r1;r0 1)U1(t1)ga N(r0 1;r0 2;t1t2)U(t2)~ty(r2;r0 2) (73) wherea= r;a;<. We assume the Green's function in N region is spin-independent; i.e., we neglect higher order contribution of hopping. Moreover, we treat the hopping to occur only at the interface, i.e., at x= 0. The self-energy is then represented as a(r1;r2;t1;t2) =a2(x1)(x2)~tU1(t1)U(t2)~tyX kga N(k;t1t2); (74) whereais the interface thickness, which we assume to be the order of the lattice constant. Diagramatic representation of Eqs. (71)(73) are in Fig. 8. Expanding the matrix using spin gauge eld as U1(t1)U(t2) = 1i(t1t2)As;t+O((As;t)2), we obtain the gauge eld contribution of the self-energy as a(r1;r2;t1;t2) =a2(x1)(x2)Zd! 2dei!(t1t2) d!~tAs;t~tyX kga N(k;!) =a2(x1)(x2)Zd! 2ei!(t1t2)~tAs;t~tyX kd d!ga N(k;!) (75) 25The linear contribution of the lessor component of the o-diagonal self-energy is G<(r;t;r0;t) =grrga+gr<ga+g<aga =a2Zd! 2X k gr(r;!)dgr N(k;!) d!~tAs;t~tyg<(r;!) +gr(r;!)dg< N(k;!) d!~tAs;t~tyga(r;!) +g<(r;!)dga N(k;!) d!~tAs;t~tyga(r;!) (76) For nite distance from the interface, r, dominant contribution arises from the terms contain- ing bothgr(r;!) andga(r;!), as they do not contain a rapid oscillation like ei(kF++kF)r ande2ikFr. Using an approximationP kgr N(k;!)iNand partial integration with respect to!, Eq. (76) reduces to G<(r;t;r0;t) = 2iNa2Zd! 2f0 N(!)gr(r;!)~tAs;t~tyga(r;!) (77) For damping, o-diagonal contributions, A s;t, are obviously essential. The result of the spin density in F in the rotated frame, Eq. (71), is ~s(F) (r;t) = 2iNa2Zd! 2f0 N(!)A s;ttr[gr(r;!)~t~tyga(r;!)] = 2iNa2Zd! 2f0 N(!)A s;tX kk0ei(kk0)
rtr[gr(k;!)~t~tyga(k0;!)] (78) Evaluating the trace in spin space, we obtain ~s(F)(r;t) =N A? s;t 1(r) + ( ^zA? s;t) 2(r) (79) where 1(r)X ~t~ty gr (r)ga (r) 2(r)X (i)~t~ty gr (r)ga (r): (80) We consider an interface with innite area and consider spin accumulation averaged over the plane parallel to the interface. The wave vector contributing is then those with nite kx but withky=kz= 0 and Green's function become one-dimensional like X keik
rgr (k) =im kFeikFjxjejxj=(2`); (81) 26where`vF(vFkF=m) is electron mean free path for spin . The induced spin density in the ferromagnet is nally obtained from Eq. (79) as s(F)(r;t) =m2Na2 2kF+kFX  (n_n)T;ei(kF+kF)x+_n(i)T;ei(kF+kF)x =m2Na2 2kF+kFX  (n_n) Re[T";#] cos((kF+kF)x) + Im[T";#] sin((kF+kF)x) +_n Im[T";#] cos((kF+kF)x)Re[T";#] sin((kF+kF)x) (82) and the torque on localized spin, Mns(F), is (r;t) =m2Na2M 2kF+kFX  _n Re[T";#] cos((kF+kF)x) + Im[T";#] sin((kF+kF)x) + (n_n) Im[T";#] cos((kF+kF)x)Re[T";#] sin((kF+kF)x) : (83) A. Enhanced damping and spin renormalization of ferromagnetic metal The total induced spin accumulation density in ferromagnet is s(F)1 dZ0 ddxs(F)(x) =1 M (n_n)h Im[](1cos~d) + Re[] sin~di +_nh Re[](1cos~d) + Im[] sin~di ; (84) where ~d(kF+kF)d,dis the thickness of ferromagnet and m2Na2M kF+kF(kF+kF)dT";#: (85) As a result of this induced electron spin density, s(F), the equation of motion for the averaged magnetization is modied to be10 _n=n_n BnMns(F); (86) whereBis the external magnetic eld. Let us rst discuss thin ferromagnet case, djkF+kFj1, where oscillating part with respect to ~dis neglected. The spin density then reads s(F)'1 M(Im[](n_n) + Re[]_n) and the equation of motion becomes (1 + Im)_n=~n_n Bn; (87) 27where ~+ Re; (88) is the Gilbert damping including the enhancement due to the spin pumping eect. The precession angular frequency !Bis modied by the imaginary part of T";#, i.e., by the spin current proportional to _n, as !B= B 1 + Im: (89) This is equivalent to the modication of the gyromagnetic ratio ( ) or theg-factor. For most 3d ferromagnets, we may approximatem2NaMF2 2kF+kF(kF+kF)'O(1) (askF+ kF/M), resulting in a dT";#. As discussed in Sec. III, when interface spin-orbit interaction is taken into account, we have T";#=~t0 "~t0 #+ie xz(~t0 "+~t0 #) +O((e )2), where ~t0  ande xzare assumed to be real. Moreover, ~t0 can be chosen as positive and thus T";#>0. (~t0 here is eld-representaion, and has unit of energy.) Equations (88) and (89) indicates that the strength of the hopping amplitude ~t0 and interface spin-orbit interaction e xzare experimentally accessible by measuring Gilbert damping and shift of resonance frequency as has been known2. A signicant consequence of Eq. (88) is that the enhancement of the Gilbert damping, a d1 F2~t0 "~t0 #; (90) can exceed in thin ferromagnets the intrinsic damping parameter , as the two contributions are governed by dierent material parameters. In contrast to the positive enhancement of damping, the shift of the resonant frequency or g-factor can be positive or negative, as it is linear in the interface spin-orbit parameter e xz. Experimentally, enhancement of the Gilbert damping and frequency shift has been mea- sured in many systems11. In the case of Py/Pt junction, enhancement of damping is observed to be proportional to 1 =din the range of 2nm <d< 10nm, and the enhancement was large, ='4 atd= 2 nm11. These results appears to be consistent with our analysis. Same 1 =d dependence was observed in the shift of g-factor. The shift was positive and magnitude is about 2% for Py/Pt and Py/Pd with d= 2nm, while it was negative for Ta/Pt11. The exis- tence of both signs suggests that the shift is due to the linear eect of spin-orbit interaction, and the interface spin-orbit interaction we discuss is one of possible mechanisms. 28For thin ferromagnet, ~d.1, the spin accumulation of Eq. (84) reads s(F)=1 M((n_n)Re[thin] +_nIm[thin]); (91) where thin~d=m2Na2M 2kF+kFT";#: (92) Equation (91) indicates that the roles of imaginary and real part of T";#are interchanged for thick and thin ferromagnet, resulting in ~=+ Imthin !B= B 1Rethin; (93) for thin ferromagnet. Thus, for weak interface spin-orbit interaction, positive shift of reso- nance frequency is expected (as Re thin>0). Signicant feature is that the damping can be smallened or even be negative if strong interface spin-orbit interaction exists with negative sign of Imthin. Our result indicates that 'spin mixing conductance' description of Ref.2 breaks down in thin metallic ferromagnet (and insulator case as we shall see in Sec. VII D). In this section, we have discussed spin accumulation and enhanced Gilbert damping in ferromagnet attached to a normal metal. In the eld-theoretic description, the damping enhancement arises from the imaginary part of the self-energy due to the interface. Thus a randomness like the interface scattering changing the electron momentum is essential for the damping eect, which sounds physically reasonable. The same is true for the reaction, namely, spin current pumping eect into N region, and thus spin current pumping requires randomness, too. (In the quantum mechanical treatment of Sec. II, change of electron wave vector at the interface is essential.) The spin current pumping eect therefore ap- pears dierent from general pumping eects, where randomness does not play essential roles apparently3. Spin accumulation and enhanced Gilbert damping was discussed by Berger10based on a quantum mechanical argument. There 1 =ddependence was pointed out and the damping eect was calculated by evaluating the decay rate of magnons. Comparison of enhanced Gilbert damping with experiments was carried out in Ref.2but in a phenomenological man- ner. 29VI. CASE WITH MAGNETIZATION STRUCTURE Field theoretic approach has an advantage that generalization of the results is straightfor- ward. Here we discuss brie y the case of ferromagnet with spatially-varying magnetization. The excitations in metallic ferromagnet consist of spin waves (magnons) and Stoner excita- tion. While spin waves usually have gap as a result of magnetic anisotropy, Stoner excitation is gapless for nite wave vector, ( kF+kF)<jqj<(kF++kF), and it may be expected to have signicant contribution for magnetization structures having wavelength larger than kF+kF. Let us look into this possibility. Our result of spin accumulation in ferromagnet, represented in the rotated frame, Eq. (63), indicates that when the magnetization has a spatial prole, the accumulation is deter- mined by the spin gauge eld and spin correlation function depending on the wave vector q as X qA s;t(q)(q;0); (94) where (q; )X kfk+q;fk; k+q;k;+ +i0; (95) is the correlation function with nite momentum transfer qand nite angular frequency . For the case of free electron with quadratic dispersion, the correlation function is24 (q; ) =Aq+i Bqst(q) +O( 2); (96) where Aq=ma3 82 (kF++kF) 1 +(kF+kF)2 q2 +1 2q3((kF++kF)2q2)(q2(kF+kF)2) lnq+ (kF++kF) q(kF++kF) Bq=m2a3 4jqj; (97) and st(q)8 < :1 (kF+kF)<jqj<(kF++kF) 0 otherwise: (98) 30describes the wave vectors where Stoner excitation exists. As we see from Eq. (96), the Stoner excitation contribution vanishes to the lowest order in , and thus the spin pumping eect in the adiabatic limit ( !0) is not aected. Moreover, the real part of the correlation function,Aq, is a decreasing function of qand thus the spin pumping eciency would decrease when ferromagnet has a structure. However, for rigorous argument, we need to include the spatial component of the spin gauge eld arising form the spatial derivative of the magnetization prole. As for the eect of the Stoner excitation on spin damping (Gilbert damping), it was demonstrated for the case of a domain wall that the eect is negligibly small for a wide wall with thickness (kF+kF)1(Refs.24,25). Simanek and Heinrich presented a result of the Gilbert damping as the linear term in the frequency of the imaginary part of the spin correlation function integrated over the wave vector12. The result is, however, obtained for a model where ferromagnet is atomically thin layer (a sheet), and would not be applicable for most experimental situations. Discussion of Gilbert damping including nite wave vector and the impurity scattering was given in Ref.26. Inhomogenuity eects of damping of a domain wall was studied recently in detail27. VII. INSULATOR FERROMAGNET In this section, we discuss the case of ferromagnetic insulator. It turns out that the generation mechanisms for spin current in the insulating and metallic cases are distinct. A. Magnon and adiabatic gauge eld The Lagrangian for the insulating ferromagnet is LIF=Z d3r S_(cos1)J 2(rS)2 HK; (99) whereJis the exchange interaction between the localized spin, S, andHKdenotes the magnetic anisotropy energy. We rst study low energy magnon dynamics induced by slow magnetization dynamics. For separating the classical variable and uctuation (magnon), rotating coordinate descrip- tion used in the metallic case is convenient. For magnons described by the Holstein-Primakov 31boson, the unitary transformation is a 3 3 matrix dened as follows28. S=UeS; (100) where U=0 BBB@coscossinsincos cossincossinsin sin 0 cos1 CCCA= een : (101) The diagonalized spin eSis represented in terms of annihilation and creation operators for the Holstein-Primakov boson, bandby, as29 eS=0 BBB@q S 2(by+b) iq S 2(byb) Sbyb1 CCCA: (102) We neglect the terms that are third- and higher-order in boson operators. Derivatives of the localized spin then read @S=U(@+iAU;)eS; (103) where AU;iU1rU; (104) is the spin gauge eld represented as a 3 3 matrix. The spin Berry's phase of the Lagrangian (99) is written in terms of magnon as (derivation is in Sec. B) Lm= 2S 2Z d3ri[by(@t+iAz s;t)bby( @tiAz s;t))b]; (105) namely, magnons interacts with the adiabatic component of the same spin gauge eld for electrons,Az s;t, dened in Eq. (14). As magnon is a single-component eld, the gauge eld is also single-component, i.e., a U(1) gauge eld. This is a signicant dierence between insulating and metallic ferromagnet; In the metallic case, conduction electron couples to an SU(2) gauge eld with spin- ip components, which turned out to be essential for spin current generation. In contrast, in the insulating case, the magnon has diagonal gauge eld, i.e., a spin chemical potential, which simply induces diagonal spin polarization. Pumping of magnon was discussed in a dierent approach by evaluating magnon source term in Ref.30. 32The exchange interaction at the interface is represented by a Hamiltonian HI=JIZ d3rIS(r)
cyc; (106) whereJIis the strength of the interface sdexchange interaction and the integral is over the interface. We consider a sharp interface at x= 0. Using Eq. (100), the interaction is represented in terms of magnon operators up to the second order as HI=JIZ d3rI" (Sbyb)cy(n
)c+r S 2 bycy
c+bcy
c# ; (107) where e+ie=0 BBB@coscosisin cossin+icos sin1 CCCA: (108) Equation (107) indicates that there are two mechanisms for spin current generation; namely, the one due to the magnetization at the interface (the term proportional to n) and the one due to the magnon spin scattering at the interface (described by the term linear in magnon operators). Let us brie y demonstrate based on the expression of Eq. (107) that spin- ip processes due to magnon creation or annihilation lead to generation of spin current in the normal metal. At the second order, the interaction induces a factor on the electron wave function ((t)
)((t0)
) for magnon creation and ( (t)
)((t0)
) for annihilation (we allow an innitesimal dierence in time tandt0). The factor for the creation has charge and spin contributions, ( (t)
)((t0)
) =(t)
(t0) +i
((t)(t0)). For magnon annihilation, we have ( (t)(t0)), and thus the sum of the magnon creation and annihilation processes give arise to a factor X q[(nq+ 1)((t)(t0)) +nq((t)(t0))] =X q[(2nq+ 1)Re[ (t)(t0)] +iIm[(t)(t0)]: (109) For adiabatic change the amplitude is expanded as ((t)(t0)) = 2i(1 +i(tt0) cos_)n(tt0)(n_ni_n) +O((@t)2); (110) 33where we see that an retardation eect from the adiabatic change of magnetization (rep- resented by the second term on the right-hand side) gives rise to a magnon state change proportional to n_nand _n. The retardation contribution for the spin part (Eq. (109)) is (tt0)X q[(2nq+ 1)(n_n) +i_n]: (111) We therefore expect that a spin current proportional to n_nemerges proportional to the magnon creation and annihilation number,P q(2nq+ 1). (As we shall see below, the factor tt0reduces to a derivative with respect to angular frequency of the Green's function.) A rigorous estimation using Green's function method is presented in Sec. VII C. In Eq. (111), the last term proportional to _nis an imaginary part arising from the dierence of magnon creation and annihilation probabilities of vacuum, nq+ 1 andnq. The term is, however, unphysical one corresponding to a real energy shift due to magnon interaction, and is removed by redenition of the Fermi energy. B. Spin current pumped by the interface exchange interaction Here we study the spin current pumped by the classical magnetization at the interface, namely, the one driven by the term proportional to Snin Eq. (107). We treat the ex- change interaction perturbatively to the second order as the exchange interaction between conduction electron and insulator ferromagnet is localized at the interface and is expected to be weak. The weak coupling scheme employed here is in the opposite limit as the strong coupling (adiabatic) approach used in the metallic ferromagnet (Sec. IV). In the perturbative regime, the issue of adiabaticity needs to be argued carefully. In the strong sdcoupling limit, the adiabaticity is trivially satised, as the time needed for the electron spin to follow the localized spin is the fastest timescale. In the weak coupling limit, this timescale is long. Nevertheless, the adiabatic condition is satised if the electron spin relaxation is strong so that the electron spin relaxes quickly to the local equilibrium state determined by the localized spin. Thus the adiabatic condition is expected to be MIsf=~1, whereMIandsfare the interface spin splitting energy, and conduction electron spin relaxation time, respectively. In the following calculation, we consider the case ofFsf=~1, i.e., ~(sf)1F, as the spin ip lifetime is by denition longer than the elastic electron lifetime , which satises F=~1 in metal. Our results therefore cover 34both adiabatic and nonadiabatic limits. The calculation is carried out by evaluating Feynmann diagrams of Fig. 9, similar to the study of Refs.17,18. A dierence is that while Refs.17,18assumed a smooth magnetization structure and used a gradient expansion, the exchange interaction we consider is localized. FIG. 9. T he Feynmann diagrams for spin current pumped by interface sdexchange interaction. The lesser Green's function for normal metal including the interface exchange interaction to the linear order is G(1)< N(r;t;r;t) =MIZd! 2Zd 2X kk0ei tei(k0k)
r (f(!+ )f(!))gr k0;!+ ga k!f(!)gr k0;!+ gr k!+f(!+ )ga k0;!+ ga k! (n
); (112) whereMIJISis the local spin polarization at the interface. Expanding the expression with respect to and keeping the dominant contribution at long distance, i.e., the terms containing both gaandgr. UsingP kga k!eik
r'im kFeikrejxj `(ga(r)), the result of spin current is j(1) s(r;t) =MIm kF_nejxj=`: (113) The second-order contribution is similarly calculated to obtain G(2)< N(r;t;r;t)'(MI)2Zd! 2Zd 1 2Zd 2 2 X kk0k00ei( 1+ 2)tei(k0k)
rf0(!)gr k0;!ga k!( 1ga k00!+ 2gr k00!)(n 1
)(n 2
) =2i(MI)2jgr(r)j2(n_n)
: (114) The corresponding spin current at the interface ( x= 0) is thus j(2) s(x= 0;t) =(MI)2m kF(n_n); (115) 35and the total spin current reads js(x= 0;t) =MIm kF_n2(MI)2m kF(n_n): (116) In the perturbation regime, the spin current proportional to _nis dominant (larger by a factor of (M I)1) compared to the one proportional to n_n. Expression of spin current induced by the interface exchange interaction was presented in Ref.31in the limit of strong spin relaxation, MIsf1, wheresfis the spin relaxation time of electron. By solving the Landau-Lifshitz-Gilbert equation for the electron spin, they obtained an result of Eq. (116) with M Ireplaced by MIsf. C. Calculation of magnon-induced spin current Here magnon-induced spin current due to the magnon interaction in Eq. (107) is cal- culated. As magnon is a small uctuation of magnetization, the contribution here is a small correction to the contribution of Eq. (116). Nevertheless, the magnon contribution has a typical linear dependence on the temperature, and is expected to be experimentally identied easily. Spin current induced in normal metal is evaluated by calculating the self-energy arising from the interface magnon scattering of Eq. (107). The contribution to the path-ordered Green's function of N electron from the magnon scattering to the second order is GN(r;t;r0:t0) =Z Cdt1Z Cdt2X r1r2gN(r;t;r1;t1)I(r1;t1;r2;t2)gN(r2;t2;r0;t0); (117) where I(r1;t1;r2;t2)iSJ2 I 2D(r1;t1;r2;t2)gN(r1;t1;r2;t2); (118) represents the self energy. Here D(r1;t1;r2;t2)ihTCB(r1;t1)B(r2;t2)i; (119) is the Green's function for magnon dressed by the magnetization structure ( is dened in Eq. (108)), B(r;t)(t)by(r;t) + y (t)b(r;t): (120) 36FIG. 10. T he Feynmann diagrams for spin current pumped by magnons at the interface. Green's functions for magnons and electrons in the normal metal are denoted by DandgN, respectively.  represents the eects of magnetization dynamics (Eq. (108)). Diagramatic representation is in Fig. 10. In the present approximation including the inter- face scattering to the second order, the electron Green's function in Eq. (118) is treated as spin-independent, resulting in a self energy (dened on complex time contour) I(r1;t1;r2;t2) =iSJ2 I 2(+i  )D(r1;t1;r2;t2)gN(r1;t1;r2;t2): (121) We focus on the spin-polarized contribution, I; (r1;t1;r2;t2)SJ2 I 2eD (r1;t1;r2;t2)gN(r1;t1;r2;t2); (122) whereeD  D. The spin-dependent contribution of lessor Green's function, Eq. (117), reads (time and spatial coordinates partially suppressed) G< N(r;t;r0;t0) = Z1 1dt1Z1 1dt2 gr N(tt1)r I; (t1;t2)g< N(t2t0) +gr N< I; ga N+g< Na I; ga N  G< N; (r;t;r0:t0): (123) For the self energy type of the Green's functions, depending on two time as g(t1t2)D(t1t2) (Eq. (122)), real-time components are written as (suppressing time and sux of N) (See Sec. C) [g(t1t2)D(t1t2)]r=grD<+g>Dr=g<Dr+grD> [g(t1t2)D(t1t2)]a=gaD>+g<Da=gaD<+g>Da [g(t1t2)D(t1t2)]<=g<D<: (124) The Green's function eDis that of a composite eld Bdened in Eq. (120), and is decom- 37posed to elementary magnon Green's function, D, as eD (r1;t1;r2;t2) = [y(t1)(t2)] D(r1;t1;r2;t2)[y(t2)(t1)] D(r2;t2;r1;t1); (125) where D(r1;t1;r2;t2)i TCb(r1;t1)by(r2;t2) : (126) The spin-dependent factor in Eq. (125) is calculated for adiabatic dynamics as y(t1)(t2) = 2in(t1) + (t2t1)[ +i_n] +O((@t)2); (127) where 2 cos_n+n_n: (128) The real-time Green's functions are therefore ( D(1;2)D(r1;t1;r2;t2)) eD< (r1;t1;r2;t2) = 2in(t1)[D<(r1;t1;r2;t2)D>(r2;t2;r1;t1)] + (t2t1) [D<(r1;t1;r2;t2) +D>(r2;t2;r1;t1)] +i_n[D<(r1;t1;r2;t2)D>(r2;t2;r1;t1)] eDr (1;2) =(t1t2)(eD< (1;2)eD> (1;2)) eDa (1;2) =(t2t1) (D< (1;2)D> (1;2)); (129) andeD< is obtained by exchanging <and>ineD< . Elementary Green's functions are calculated as D<(r1;t1;r2;t2) =iX qeiq
(r1r2)nqei!q(t1t2) D>(r1;t1;r2;t2) =iX qeiq
(r1r2)(nq+ 1)ei!q(t1t2); (130) where!qis magnon energy and nq1 e!q1. In our model, the interface is atomically at and has an innite area, and thus ri(i= 1;2) are atx= 0. Fourier components dened as (a= r;a;<;> ) eDa (x1= 0;t1;x2= 0;t2)X qZd 2ei (t1t2)eDa (q; ); (131) 38are calculated from Eq. (129) as eD< (q; ) =i 2n(D< D> +) +d d  (D< +D> +) +i_n(D< D> +) eDr (q; ) =i 2n(Dr +Dr +) +d d  (Dr Dr +) +i_n(Dr +Dr +) eDa (q; ) =i 2n(Da +Da +) +d d  (Da Da +) +i_n(Da +Da +) ; (132) where Da 1 !qi0; Dr 1 !q+i0 D< nq(Da Dr ); D> +(1 +nq)(Da +Dr +): (133) The spin part of the Green's function, Eq. (123), is G< N; (r;t;r0;t) =SJ2 I 2Zd! 2Zd 2X kk0X k00q gr N;k! eDr (q; )g> N;k00;! +eD< (q; )gr N;k00;!  g< N;k0! +gr N;k!eDr (q; )g> N;k00;! ga N;k0!+g< N;k! eDa (q; )g> N;k00;! +eD< (q; )ga N;k00;!  ga N;k0! : (134) The contribution survives at long distance is the one containing gr N;!(r) andga N;!(r), i.e., G< N; (r;t;r0;t)'Zd! 2X kk0gr N;k!ga N;k0!eik
reik0
r0eI; ; (135) where eI; SJ2 I 2Zd 2X k00q fk0eDr (q; )fkeDa (q; ) (fk001)(ga N;k00;! gr N;k00;! ) +eD< (q; )(fk0gr N;k00;! fkga N;k00;! +fk00(ga N;k00;! gr N;k00;! )) : (136) We focus on the pumped contribution, containing derivative with respect to in Eq. (132). The result is, using partial integration with respect to ( eIis a vector representation of eI; ), eI'iSJ2 I 2Zd 2X k00q fk0[ (Dr Dr +) +i_n(Dr +Dr +)]fk[ (Da Da +) +i_n(Da +Da +)]
(fk001)d d (ga N;k00;! gr N;k00;! ) + [ (D< +D> +) +i_n(D< D> +)]d d [(fk00fk)ga N;k00;! (fk00fk0)gr N;k00;! ] : (137) 39Usingd d ga k00;! = (ga k00;!)2+O( ) and an approximation, we obtainP k00(ga k00;!)2'i 2F, eI' FSJ2 I 2Zd 2X qk00  (fk001)[fk0(Dr Dr +)fk(Da Da +)] +1 2(2fk00fkfk0)(D< +D> +) +i_n (fk001)[fk0(Dr +Dr +)fk(Da +Da +)] +1 2(2fk00fkfk0)(D< D> +) : (138) As argued for Eq. (111), only the imaginary part of self energy contributes to the induced spin current, as the real part, the shift of the chemical potential, is compensated by redis- tribution of electrons. The result is thus eI'i  FSJ2 I 2X qk00(1 + 2nq)(2fk00fkfk0): (139) We further note that the component of proportional to n(Eq. (128)) does not contribute to the current generation, as a result of gauge invariance. (In other words, the contribution cancels with the one arising from the eective gauge eld for magnon.) The nal result of the spin current pumped by the magnon scattering is therefore jm s(r;t) = FSJ2 I 2jgr(r)j2X q(1 + 2nq)(n_n): (140) At high temperature compared to magnon energy, !q1, 1 + 2nq'2kBT !q, and the magnon-induced spin current depends linearly on temperature. The result (140) agrees with previous study carried out in the context of thermally-induced spin current19. D. Correction to Gilbert damping in the insulating case In this subsection, we calculate the correction to the Gilbert damping and g-factor of insulating ferromagnet as a result of spin pumping eect. We study the torque on the ferromagnetic magnetization arising from the eect of conduction electron of normal metal, given by I=BIn=MI(nsI); (141) 40where BIHI n=MIsI; (142) is the eective magnetic eld arising from the interface electron spin polarization, sI(t) itr[G< N(0;t)]. The contribution to the electron spin density linear in the interface ex- change interaction, Eq. (106), is s(1); I(t) =iZ dt1MIn(t1)tr[gN(t;t1)gN(t1;t)]<; (143) where the Green's functions connect positions at the interface, i.e., from x= 0 tox= 0, and are spin unpolarized. (The Feynman diagrams for the spin density are the same as the one for the spin current, Fig. 9 with the vertex jsreplaced by the Pauli matrix.) Pumped contribution proportional to the time variation of magnetization is obtained as s(1) I(t) =MI_nZd! 2X kk0f0(!)(ga N;k0gr N;k0)(ga N;kgr N;k) =MI()2_n: (144) The second order contribution similarly reads s(2); I(t) =i 2Z dt1Z dt2(MI)2n(t1)n (t2)tr[gN(t;t1)gN(t1;t2) gN(t2;t)]< '2(MI)2()3(n_n): (145) The interface torque is therefore I=(MI)2(n_n) + 2(MI)3_n: (146) Including this torque in the LLG equation, _n=n_n Bn+, we have (1I)_n=I(n_n) Bn; (147) where I= 2d(M I)3 I=+d(M I)2; (148) whereddmp=dis the ratio of the length of magnetic proximity ( dmp) and thickness of the ferromagnet, d. The Gilbert damping constant therefore increases as far as the interface 41spin-orbit interaction is neglected. The resonance frequency is !B= B 1I, and the shift can have both signs depending on the sign of interface exchange interaction, MI. There may be a possibility that magnon excitation induce torque that corresponds to eective damping. In fact, such torque arises of hbior by are nite, i.e., if magnon Bose condensation glows, as seen from Eq. (102). Such condensation can in principle develop from the interface interaction of magnon creation or annihilation induced by electron spin ip, Eq. (107). However, conventional spin relaxation processes arising from the second order of random spin scattering do not contribute to such magnon condensation and additional damping. Comparing the result of pumped spin current, Eq. (116), and that of damping coecient, Eq. (148), we notice that the 'spin mixing conductance' argument2, where the coecients for the spin current component proportional to n_nand the enhancement of the Gilbert damping constant are governed by the same quantity (the imaginary part of 'spin mixing conductance') does not hold for the insulator case. In fact, our result indicates that the spin current component proportional to n_narises from the second order correction to the interaction (the second diagram of Fig. 9), while the damping correction arises from the rst order process (the rst diagram of Fig. 9). Although the magnitudes of the two eects happen to be both second order of interface spin splitting, MI, physical origins appear to be distinct. From our analysis, we see that the 'spin mixing conductance' description is not general and applies only to the case of thick metallic ferromagnet (see Sec. V A for metallic case). VIII. DISCUSSION Our results are summarized in table I. Let us discuss experimental results in the light of our results. In the early ferromagnetic resonance (FMR) experiments, consistent studies of g-factor and the Gilbert damping were carried out on metallic ferromagnets11. The results appear to be consistent with theories (Refs.2,10and the present paper). Both the damping constant and the gfactor have 1 =d-dependence on the thickness of ferromagnet in the range of 2nm<d< 10nm11. The maximum additional damping reaches 0:1 atd=2nm, which exceeds the original value of 0:01. Theg-factor modulation is about 1% at d= 2nm, and its sign depends on the material; the g-factor increases for Pd/Py/Pd and Pt/Py/Pt while 42Ferromagnet(F) Ai Ar  ! B Assumption Equations Metal ImT+ ReT+ReT+ ImT+ImT+ ReT+Thick F Thin F(27)(66) (88)(89) (93) Insulator MI (MI)2(MI)2(MI)3Weak spin relaxation(116) (148) - (MI)2P q(1 + 2nq) - - Magnon (140) TABLE I. Summary of essential parameters determining spin current js, corrections to the Gilbert dampingand resonance frequency shift !Bfor metallic and insulating ferromagnets. Coe- cientsAiandArare for the spin current, dened by Eq. (1). Label indicates that it is not discussed in the present paper.: For strong spin relaxation case, the density of states is replaced by inverse of electron spin- ip time, sf.31 decreases for Ta/Py/Ta. These results appear consistent with ours, because !Bis governed by ImT+, whose sign depends on the sign of interface spin-orbit interaction. In contrast, damping enhancement proportional to Re T+is positive for thick metals. However, other possibilities like the eect of a large interface orbital moment playing a role in the gfactor, cannot be ruled out at present. Recently, inverse spin Hall measurement has become common for detecting the spin cur- rent. In this method, however, only the dc component proportional to n_nis accessible so far and there remains an ambiguity for qualitative estimates because another phenomeno- logical parameter, the conversion eciency from spin to charge, enters. Qualitatively, the values ofArobtained by the inverse spin Hall measurements32and FMR measurements are consistent with each other. The cases of insulating ferromagnets have been studied recently. In the early experiments, orders of magnitude smaller value of Arcompared to metallic cases were reported31, while those small values are now understood as due to poor interface quality. In fact, FMR measurements on epitaxially grown samples like YIG/Au/Fe turned out to show Arof 1 51018m2(Refs.33,34), which is the same order as in the metallic cases. Inverse spin Hall measurements on YIG/Pt reports similar values35, and the value is consistent with the rst principles calculation36. Systematic studies of YIG/NM with NM=Pt, Ta, W, Au, Ag, Cu, Ti, V, Cr, Mn etc. were carried out with the result of Ar10171018m2(Refs.37{40). If we use naive phenomenological relation, Eq. (6), Ar= 1018m2corresponds to = 3104if 43a= 2A,S= 1 andd= 20A. Assuming interface sdexchange interaction, the value indicates MI0:01, which appears reasonable at least by the order of magnitude from the result of x-ray magnetic circular dichroism (XMCD) suggesting spin polarization of interface Pt of 0:05B41. On the other hand, FMR frequency shift of insulators cannot be explained by our theory. In fact, the shift for YIG/Pt is !B=!B1:6102, which is larger than 2103, while our perturbation theory assuming weak interface sdinteraction predicts !B=!B< . We expect that the discrepancy arises from the interface spin-orbit interaction that would be present at insulator-metal interface, which modies the magnetic proximity eect and damping torque signicantly. It would be necessary to introduce anomalous sdcoupling at the interface like the one discussed in Ref.42. Experimentally, in uence of interface spin- orbit interaction43and proximity eect needs to be carefully characterized by using the microscopic technique, such as MCD, to compare with theories. IX. SUMMARY We have presented a microscopic study of spin pumping eects, generation of spin cur- rent in ferromagnet-normal metal junction by magnetization dynamics, for both metallic and insulating ferromagnets. As for the case of metallic ferromagnet, a simple quantum mechan- ical picture was developed using a unitary transformation to diagonal the time-dependent sdexchange interaction. The problem of dynamic magnetization is thereby mapped to the one with static magnetization and o-diagonal spin gauge eld, which mixes the electron spin. In the slowly-varying limit, spin gauge eld becomes static, and the conventional spin pumping formula is derived simply by evaluating the spin accumulation formed in the nor- mal metal as a result of interface hopping. The eect of interface spin-orbit interaction was discussed. Rigorous eld-theoretical derivation was also presented, and the enhancement of spin damping (Gilbert damping) in the ferromagnet as a result of spin pumping eect was discussed. The case of insulating ferromagnet was studied based on a model where spin current is driven locally by the interface exchange interaction as a result of magnetic prox- imity eect. The dominant contribution turns out to be the one proportional to _n, while magnon contribution leads to n_n, whose amplitude depends linearly on the temperature. Our analysis clearly demonstrate the dierence in the spin current generation mechanism 44for metallic and insulating ferromagnet. The in uence of atomic-scale interface structure on the spin pumping eect are open and urgent issues, in particular for the case of ferrimagnetic insulators which have two sub-lattice magnetic moments. ACKNOWLEDGMENTS GT thanks H. Kohno, C. Uchiyama, K. Hashimoto and A. Shitade for valuable discus- sions. This work was supported by a Grant-in-Aid for Exploratory Research (No.16K13853) and a Grant-in-Aid for Scientic Research (B) (No. 17H02929) from the Japan Society for the Promotion of Science and a Grant-in-Aid for Scientic Research on Innovative Areas (No.26103006) from The Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. Appendix A: Eect of spin-conserving spin gauge eld on spin density Here we calculate contribution of spin-conserving spin gauge eld, Az s;t, on the interface eects of spin density in F. It turns out that spin-conserving spin gauge eld combined with interface eects does not induce damping. This result is consistent with a naive expectation that only the nonadiabatic components of spin current should contribute to damping. FIG. 11. Diagramatic representation of the contribution to the lessor Green's function for F electron arising from the interface hopping (represented by tandt) and spin gauge eld ( As;t). The diagram (c) includes the spin gauge eld implicitly in unitary matrices UandU1. The contribution to the lesser Green's function in F from the interface hopping (lowest, the second-order in the hopping) at the linear order in the spin gauge eld reads (diagra- 45matically shown in Fig. 11) G<=G< (a)+G< (b)+G< (c) G< (a)=gr(As;t
)grr 0g<+gr(As;t
)gr< 0ga+gr(As;t
)g<a 0ga+g<(As;t
)gaa 0ga G< (b)=grr 0gr(As;t
)g<+gr< 0g<(As;t
)ga+gra 0ga(As;t
)ga+g<a 0ga(As;t
)ga G< (c)=grrg<+gr<ga+g<aga: (A1) Here a~tU1ga NU~ty;(a= a;r;<) a 0~tga N; (A2) are self energy due to the interface hopping, where ais the full self energy including the time-dependent unitary matrix U, which includes spin gauge eld. a 0is the contribution of awith the spin gauge eld neglected. We here focus on the contribution of the adiabatic ( z) component,Az s;t. Usingg<=F(gagr) for F (Fis a 22 matrix of the spin-polarized Fermi distribution function) and g< N=fN(ga Ngr N) and noting that all the angular frequencies of the Green's function are equal, we obtain G< (a)+G< (b)'Az s;tz 2F[(gr)3r 0(ga)3a 0](FfN)[(gr)2ga+gr(ga)2](a 0r 0) : (A3) The contribution G< (c)is calculated noting that ~tU1ga NU~ty=ga N~t~tydga N d!~t(As;t
)~ty+O((As;t)2): (A4) The linear contribution with respect to the zcomponent of the gauge eld turns out to be G< (c)'Az s;tz F (gr)2@ @!r 0(ga)2@ @!a 0 + (FfN)grga@ @!(a 0r 0) : (A5) We therefore obtain the eect of spin-conserving gauge eld as G<=Az s;tz@ @! F (gr)2r 0(ga)2a 0 + (FfN)grga(a 0r 0) ; (A6) which vanishes after integration over !. Therefore, contribution from spin-conserving gauge eld and interface hopping vanishes in the spin density, leaving the damping unaected. 46Appendix B: Magnon representation of spin Berry's phase term Here we derive the expression for the spin Berry's phase term of the Lagrangian (99) in terms of magnon operator. The time-integral of the term is written by introducing an articial variable uas44 Z dtL B=SZ dt_(cos1) =S2Z dtZ1 0duS
(@tS@uS); (B1) whereS(t;u) is extended to a function of tandu, but onlyS(t;u= 1) is physical. Noting that the unitary transformation matrix element of Eq. (101) is written as Uij= (ej)i; (B2) wherer1e,e2eande3n, we obtain S
(@tS@uS) =eS
[(@t+iAU;t)eS(@u+iAU;u)eS)]: (B3) Evaluating to the second order in the magnon operators, we have @teS@ueS= 2i ^z[(@uby)(@tb)(@tby)(@ub)]: (B4) Using the explicit form of AU;, the gauge eld contribution is @ueS
[eSiAU;teS)] =S2 [(@uby)(sin_+i_) + (@ub)(sin_i_)]2S 2cos(@t)@u(byb): (B5) The terms linear in the boson operators vanish by the equation of motion, and the second- order contribution is S
(@tS@uS) = 2S 2 i@u[by(@tb)(@tby)b] @u[cos(@t)byb] +@t[cos(@u)byb] + sin((@t)(@u)(@u)(@t))byb : (B6) Integrating over tandu, the term total derivative with respect to tof Eq. (B6) vanishes, resulting in Z dtZ1 0duS
(@tS@uS) = 2S 2Z dt i[by(@tb)(@tby)b]cos(@t)byb + sin((@t)(@u)(@u)(@t))byb : (B7) 47The last term of Eq. (B7) represents the renormalization of spin Berry's phase term, i.e., the eectS!Sbyb, which we neglect below. The Lagrangian for magnon thus reads Lm= 2S 2Z d3ri[by(@t+iAz s;t)bby( @tiAz s;t))b]; (B8) namely, magnons interacts with the adiabatic component of spin gauge eld, Az s;t. Appendix C: Decomposition of contour-ordered self energy Here we summarize decomposition formula of self energy. Obviously, we have [gD]<=g<D<: (C1) Retarded component is dened as [gD]r[gD]t[gD]<; (C2) where the time-ordered one is [g(t1t2)D(t1t2)]t(t1t2)g>D>+(t2t1)g<D< =grDr+grD<+g<Dr+g<D<: (C3) We thus obtain [gD]r=grDr+grD<+g<Dr: (C4) Noting that grDa= 0, we can write it as [gD]r=grD<+g>Dr=g<Dr+grD>: (C5) The advanced component is similarly written as [gD]a=gaDa+gaD<+g<Da =gaD>+g<Da=gaD<+g>Da: (C6) 1E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). 482Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 3M. V. Moskalets, Scattering matrix approach to non-stationary quantum transport (Imperial College Press, 2012). 4M. B uttiker, H. Thomas, and A. Pretre, Zeitschrift f ur Physik B Condensed Matter 94, 133 (1994). 5P. W. Brouwer, Phys. Rev. B 58, R10135 (1998). 6G. Tatara and H. Kohno, Phys. Rev. B 67, 113316 (2003). 7G. Tatara, H. Kohno, and J. Shibata, Physics Reports 468, 213 (2008). 8V. K. Dugaev, P. Bruno, B. Canals, and C. Lacroix, Phys. Rev. B 72, 024456 (2005). 9R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19, 4382 (1979). 10L. Berger, Phys. Rev. B 54, 9353 (1996). 11S. Mizukami, Y. Ando, and T. Miyazaki, Japanese J. Appl. Phys. 40, 580 (2001). 12E.Sim anek and B. Heinrich, Phys. Rev. B 67, 144418 (2003). 13E.Sim anek, Phys. Rev. B 68, 224403 (2003). 14K. Chen and S. Zhang, Phys. Rev. Lett. 114, 126602 (2015). 15G. Tatara, Phys. Rev. B 94, 224412 (2016). 16D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981). 17A. Takeuchi and G. Tatara, J. Phys. Soc. Japan 77, 074701 (2008). 18K. Hosono, A. Takeuchi, and G. Tatara, Journal of Physics: Conference Series 150, 022029 (2009). 19H. Adachi, J.-i. Ohe, S. Takahashi, and S. Maekawa, Phys. Rev. B 83, 094410 (2011). 20J. J. Sakurai, Modern Quantum Mechanics (Addison Wesley, 1994). 21K. Hashimoto, G. Tatara, and C. Uchiyama, arXiv:1706.00583 (2017). 22G. Tatara and P. Entel, Phys. Rev. B 78, 064429 (2008). 23H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Japan 75, 113706 (2006). 24G. Tatara and H. Fukuyama, J. Phys.l Soc. Japan 63, 2538 (1994). 25G. Tatara and H. Fukuyama, Phys. Rev. Lett. 72, 772 (1994). 26N. Umetsu, D. Miura, and A. Sakuma, J. Phys. Soc. Japan 81, 114716 (2012). 27H. Y. Yuan, Z. Yuan, K. Xia, and X. R. Wang, Phys. Rev. B 94, 064415 (2016). 28G. Tatara, Phys. Rev. B 92, 064405 (2015). 29C. Kittel, Quantum theory of solids (Wiley, New York, 1963). 4930K. Nakata and G. Tatara, J. Phys. Soc. Japan 80, 054602 (2011). 31Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). 32F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Althammer, I.-M. Imort, G. Reiss, A. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 107, 046601 (2011). 33B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y.-Y. Song, Y. Sun, and M. Wu, Phys. Rev. Lett. 107, 066604 (2011). 34C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya, E. Girt, Y. Sun, Y.-Y. Song, and M. Wu, Applied Physics Letters 100, 092403 (2012). 35Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Takahashi, H. Nakayama, T. An, Y. Fujikawa, and E. Saitoh, Applied Physics Letters 103, 092404 (2013). 36X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, EPL (Europhysics Letters) 96, 17005 (2011). 37H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014). 38C. Du, H. Wang, F. Yang, and P. C. Hammel, Phys. Rev. Applied 1, 044004 (2014). 39C. Du, H. Wang, F. Yang, and P. C. Hammel, Phys. Rev. B 90, 140407 (2014). 40H. Wang, C. Du, P. C. Hammel, and F. Yang, Applied Physics Letters 110, 062402 (2017). 41Y. M. Lu, Y. Choi, C. M. Ortega, X. M. Cheng, J. W. Cai, S. Y. Huang, L. Sun, and C. L. Chien, Phys. Rev. Lett. 110, 147207 (2013). 42K. Xia, W. Zhang, M. Lu, and H. Zhai, Phys. Rev. B 55, 12561 (1997). 43M. Caminale, A. Ghosh, S. Auret, U. Ebels, K. Ollefs, F. Wilhelm, A. Rogalev, and W. E. Bailey, Phys. Rev. B 94, 014414 (2016). 44A. Auerbach, Intracting Electrons and Quantum Magnetism (Springer Verlag, 1994). 50

2017-06-05

We present a consistent microscopic study of spin pumping effects for both metallic and insulating ferromagnets. As for metallic case, we present a simple quantum mechanical picture of the effect as due to the electron spin flip as a result of a nonadiabatic (off-diagonal) spin gauge field. The effect of interface spin-orbit interaction is briefly discussed. We also carry out field-theoretic calculation to discuss on the equal footing the spin current generation and torque effects such as enhanced Gilbert damping constant and shift of precession frequency both in metallic and insulating cases. For thick ferromagnetic metal, our study reproduces results of previous theories such as the correspondence between the dc component of the spin current and enhancement of the damping. For thin metal and insulator, the relation turns out to be modified. For the insulating case, driven locally by interface $sd$ exchange interaction due to magnetic proximity effect, physical mechanism is distinct from the metallic case. Further study of proximity effect and interface spin-orbit interaction would be crucial to interpret experimental results in particular for insulators.

Consistent microscopic analysis of spin pumping effects

1706.01185v1

Control of magnon-photon coupling by spin torque Anish Rai1,and M. Benjamin Jung eisch1,y 1Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, United States (Dated: February 20, 2023) We demonstrate the in uence of damping and eld-like torques in the magnon-photon coupling process by classically integrating the generalized Landau-Lifshitz-Gilbert equation with RLC equa- tion in which a phase correlation between dynamic magnetization and microwave current through combined Amp ere and Faraday eects are considered. We show that the gap between two hybridized modes can be controlled in samples with damping parameter in the order of 103by changing the direction of the dc current density Jif a certain threshold is reached. Our results suggest that an experimental realization of the proposed magnon-photon coupling control mechanism is feasible in yttrium iron garnet/Pt hybrid structures. I. INTRODUCTION Coherent magnon-photon coupling in hybrid cavity- spintronics contributed to the advancement of magnon- based quantum information and technologies [1{15]. The collective excitations of an electron spin system in mag- netically ordered media called magnons can couple to mi- crowave photons via dipolar interaction, demonstrating level repulsion and Rabi oscillations [3]. Strongly cou- pled magnon-photon systems have been explored to bring many exotic eects into the limelight, some of which in- clude the manipulation of spin currents [16], and bidi- rectional microwave-to-optical transduction [17, 18]. In addition to the coherent magnon-photon coupling, there exists an exciting domain of dissipative magnon-photon coupling where level attraction can be observed, which is characterized by a coalescence of the hybridized magnon- photon modes [19{25]. The theoretical framework of magnon-photon coupling is given by the following dispersion relation [26] of the hybridized modes: e!=1 2 (e!m+e!c)q (e!me!c)2+ 4g2 ;(1) wheree!m=!mi!mande!c=!ci!care the complex resonance frequencies of the magnon and pho- ton (cavity) modes, respectively. gis the coupling be- tween the two modes. andare the intrinsic damping rates of the magnon and photon modes, respectively. The real and imaginary parts of e!represent the dispersion shape and the linewidth of the coupled modes, respec- tively. The second term of the square root in Eq. (1) not only gives the strength of the coupling but also reveals the nature of the coupling. Harder and co-workers [19] carefully introduced a coupling term based on the cavity Lenz eect to mitigate the Amp ere eect. However, the on-demand manipulation of the magnon-photon polari- ton by spin torques has not been addressed so far. arai@udel.edu ymbj@udel.eduIn this work, we examine the in uence of damping- and eld-like torques in the magnon-photon coupling pro- cess. Our results indicate that the magnitude of the level repulsion (manifested by the frequency gap of the hybridized modes) and, hence, the magnon-photon cou- pling strength can eciently be controlled by varying the magnitude and the direction of dc current density Jfor realistic parameters of the magnetic properties. By cou- pling the generalized Landau-Lifshitz Gilbert equation with the RLC equation of the cavity, we show that an on-demand manipulation of the magnon-photon coupling strength can be achieved for current densities of the order as small as 105A/cm2. This article is structured in the following fashion. In section II, we discuss the classical description to model our system, in which the ferromagnetic resonance of the magnetic system is strongly coupled to photon resonator mode of the microwave cavity. In section III, we intro- duce the parameters used for the analysis followed by a detailed discussion of our ndings. In section IV, we summarize our work. FIG. 1. The schematic of the experimental setup. A pat- terned YIG/platinum(Pt) bilayer is the sample under consid- eration. The dc current is passed through the platinum layer. The microwave current is passed through the cavity and an- alyzed using a Vector Network Analyzer (VNA). Here, the external magnetic eld is applied along bzdirection.arXiv:2302.08910v1 [cond-mat.mes-hall] 17 Feb 20232 FIG. 2. Magnon-photon coupling control for an intermediate value of the Gilbert damping parameter and a continuous, low current density J: The dispersion ( !!c) in (a-c) and the linewidth (
!) in (d-f) are plotted as a function of the eld detuning (!m!c) for= 2:27103. The hybridization of magnon and photon modes is compared for dierent dc current densities J: (a), (d) J = 5105A=cm2, (b), (e) J = 0 A =cm2and (c), (f) J =5 105A=cm2. The blue and red line represent the two hybridized modes. The inset in (b) shows that for larger eld detuning ( !m!c), the uncoupled photon mode approaches !c making (!!c) approach zero. II. CLASSICAL DESCRIPTION The magnetization dynamics in ferromagnetic systems can be described by the generalized Landau-Lifshitz- Gilbert equation [27{29] given by: d~M dt= ~M~He Ms ~Md~M dt! + aJ Ms ~M ~M~ p bJ ~M~ p ;(2) where~Mis the magnetization vector, Msis the satu- ration magnetization, ~Heis the eective magnetic eld including external eld ~H, anisotropy, microwave, and demagnetization elds, is the gyromagnetic ratio, is the Gilbert damping parameter, ~ pis the spin polarization unit vector. Furthermore, the terms proportional to aJ andbJare the damping-like torque and eld-like torque, respectively. The coecients aJandbJare dened as [30]: aJ=aJ~ 2eMsd;bJ=bJ~ 2eMsd; (3)whereaandbare the damping-like torque eciency and eld-like torque eciency, respectively. Jis the dc current density, whose polarity determines the direc- tions of eld-like and damping-like torque terms through Eqs. (2) and (3), ~is the reduced Planck's constant, e is the electron charge, and dis the thickness of the fer- romagnetic sample. We dene the magnetic eld, mag- netization, and spin polarization unit vectors as ~Ht= hx(t)bx+hy(t)by+Hbz,~M=mx(t)bx+my(t)by+Msbzand ~ p=bz, whereHandMsare the dc magnetic eld and saturation magnetization, respectively, and hx;y(t) and mx;y(t) are the dynamic magnetic eld and magnetiza- tion. If we dene the dynamic components, h=hx+ihy andm=mx+imy, then Eq. (2) can be reduced to: (!~!m+ ~cJ)m+!sh= 0; (4) where ~!mis the complex ferromagnetic resonance fre- quency dened by ~ !m=!mi!(where!m' His the ferromagnetic resonance frequency), !s= Ms, and ~cJ=bJiaJ. The eective RLC circuit for the cavity3 FIG. 3. Magnon-photon coupling control for an intermediate value of the Gilbert damping parameter and a pulsed, high current density J: The dispersion ( !!c) in (a-c) and the linewidth (
!) in (d-f) are plotted as a function of the eld detuning (!m!c) for= 2:5103. The hybridization of magnon and photon modes is compared for dierent dc current densities J: (a), (d) J = 5106A=cm2, (b), (e) J = 0 A =cm2and (c), (f) J =5 106A=cm2. The blue and red line represent the two hybridized modes. can be written as [19]: Rjx;y(t) +1 CZ jx;y(t)dt+Ldjx;y(t) dt=V0x;y(t);(5) where R, L, and C represent the resistance, inductance, and capacitance, respectively. V0x;yis the voltage that drives the microwave current. For j=jx+ijyandV0= V0x+iV0y, we have [19] !2!2 c+i2!!c
j=i! LV0; (6) where!c= 1=p LCis the cavity resonance frequency and = (R=2)p C=L is the intrinsic damping of the cavity- photon mode. The microwave magnetic eld will exert a torque on the magnetization through Amp ere's law. The relation can be expressed as: hx=KAjy;hy=KAjx; (7) whereKAis the positive coupling term associated with a phase relation between jx;yandhx;y. In a similar way, the precessional magnetization will induce a voltage inthe RLC circuit through Faraday induction: Vx=KFLdmy dt;Vy=KFLdmx dt; (8) whereKFis the positive coupling term associated with a phase relation between Vx;yandmx;y. Combining Eqs. (4)-(8) gives us the coupled equations of the form:  !2!2 c+i2!c! i!2KF i!sKA!~!m+ ~cJ j m = i!!cj0 0 ;(9) wherej0=V0p C=L. The hybridized eigenmodes are calculated by solving the determinant of Eq. (9). This yields the following analytical form [see Supplemental Material (SM)]: ~!= !c 1+i+!m 1+i r !c 1+i!m 1+i2 +2!c!sKFKA (1+i)(1+i) 2; (10) where= ~cJ. Here, is the gyromagnetic ratio and ~ cJ is a complex term associated with bJandaJdened by ~cJ=bJiaJ.4 FIG. 4. Variation of coherent magnon-photon coupling (minimum frequency gap between two hybridized modes) for dierent values ofandJ. For (a), (b), and (c) is varied from 3 103to 5105andJ(continuous) is varied from 5105A=cm2 to 5105A=cm2and for (d), (e), and (f) is varied from 4 103to 5105andJ(pulsed) is varied from 5106A=cm2to 5106A=cm2. Based on our model, we can distinguish between eld-like contribution (a) and (d), damping-like contribution (b) and (e), and a combination of eld-like and damping-like contribution to the manipulation of the anticrossing gap (c) and (f). For (a) and (d) a= 0 andb= 0:05 (pure eld-like torque eect), for (b) and (e) a= 0:2 andb= 0 (pure damping-like torque eect), and for (c) and (f) a= 0:2 andb= 0:05 (combination of damping-like and eld-like torque eects). III. RESULTS AND DISCUSSION For our model we choose the following realistic pa- rameters [18, 31{34]. The frequency of the cavity mode is selected at !c=2= 10 GHz with a cavity damping = 1104(corresponding to quality factor Q5000). The reduced gyromagnetic ratio ( =2), damping-like torque eciency ( a), and eld-like torque eciency ( b) are taken as 2 :8106Hz=Oe, 0:2, and 0:05, respectively. For a Pt/FM bilayer, the typical range of damping-like torque eciency ( a) is 0.10 to 0.20 [32, 35, 36] and the typical value of eld-like torque eciency ( b) is0.05 [33, 37{39]. Due to its low Gilbert damping parame- ter and high spin density, we choose yttrium iron garnet (YIG) as magnetic material. In particular, we consider a YIG lm with a thickness t= 2105cm (smallest thickness available commercially) and saturation magne- tization,Ms= 144 emu =cm3[18]. For the calculation, the termKFKAis taken as 5106[19]. For YIG lms, depending upon the thickness and preparation method,  varies from order 103to 105[34, 38, 40{47]. Therefore, we varyin our model from 3 103to 5105. Further-more, we vary Jfrom5105A=cm2to 5105A=cm2. The maximum value of chosen current density is at least one order of magnitude smaller than what is used for magnetic tunnel junctions (MTJs) [48, 49]. Note that, a current density of this order of magnitude has previously been reported for YIG/Pt systems [50] to thermally con- trol magnon-photon coupling in experiment. Reference [50] reports that such current density leads to a rise of the system temperature above 40C. Negative eects of heat- ing on the magnetic properties can be drastically reduced by using a pulsed dc current through the Pt layer [51] in- stead of using a continuous current. For instance, using a pulsed current with duty cycle of 50%, heating eects can be mitigated while reaching reasonable high levels of cur- rent density between 5106A=cm2to 5106A=cm2 . Such a high value of current density will create an Oersted eld and, hence, modify the resonance condi- tion. The generated Oersted eld can be considered as a contribution to the eective magnetic eld presented in Eq. (2). Hence, this eld will modify the resonance posi- tion of the magnon modes in the following way: for one polarity of the current density (J), the resonance eld5 shifts up, while for the other, it shifts down. Experi- mentally, this aect can be compensated by tuning the biasing magnetic eld so the resonance frequency remains the same. In the following analysis, we consider two sce- narios: (1) a relatively low continuous current density and (2) a higher pulsed current density. The eects of both conditions on the magnon-photon coupling process are compared below. The proposed experimental set up and measurement conguration is shown in Fig. 1. A. Dispersion and Linewidth In Fig. 2 (intermediate value of Gilbert damping pa- rameterand continuous, low value of current density J) and Fig. 3 (intermediate value of and pulsed, high value ofJ), the hybridized mode frequency ( !!c) and linewidth (
!) are plotted as a function of the eld de- tuning (!m!c). We rst focus on the former, shown in Fig. 2, top pan- els: For= 2:27103and forJ= 0 A=cm2, we observe a level attraction of the real part of the eigenvalues [Fig. 2 (b)] in a small region. For J=5105A=cm2, a similar behavior is found [Fig. 2(a)]. However, the be- havior drastically changes for reversed current polarity: forJ= 5105(A=cm2), a gap (a prominent level repul- sion) is seen between the hybridized modes. This clearly shows that depending upon the strength and direction of the dc current density Jone can tune the gap between the hybridized modes, i.e., transitioning the system into the strong coupling regime. Let us now consider Fig. 3, top panels: A similar but enhanced behavior can be ob- served for higher values of J(i.e.,jJj= 5106A/cm2) and= 2:5103[Fig. 3]. As is obvious from Figs. 2 and 3, there is a shift in the position where the coher- ent coupling occurs. For negative and positive values of J, the resonance shifts towards the negative and positive sides of the eld detuning ( !m!c), respectively. This shift can be understood by the fact that dierent mag- nitudes of eld-like torques directly aect the resonance condition as will be discussed in Sec. III B. Next, we discuss the lower panels of Figs. 2 and 3. The linewidths of the two hybridized modes distinctly cross each other for J= 5105A=cm2andJ= 5106A=cm2 as is expected for a broad coupling region [Fig. 2(f) and Fig. 3(f)]. This eect is less distinct for the cases J= 5105A=cm2,J=5106A=cm2andJ= 0 A=cm2. However, despite the lower number of region in the cross- ing regime (the crossing is less spread), we emphasize that level crossings in the linewidths of the hybridized modes are are also observed here. We note that the coupling re- gion broadens as the current density increases from neg- ative values to positive values [Figs. 2(d,e,f) and Figs. 3 (d,e,f)] and nally a distinct crossing of linewidths is ob- served over a broad range [Fig. 2(f) and Fig. 3 (f)]. For special cases discussed in Sec. II of the SM, a level attraction [Fig. S1 (a,b)] in the real part and level repulsion [Fig. S1 (d,e)] in the imaginary part of theeigenvalues are observed along with exceptional points (EPs) [52{55]. For more details on the observed EP we refer to the SM. B. Anticrossing gap between hybridized modes Figure 4 shows the variation of the anticrossing gap between the hybridized modes for dierent values of  andJ. It is clear that the variation is nonlinear in nature. As is evident from the gure, the gap between the two hybridized modes becomes smaller for a larger value of . On the other hand, the gap also depends on the dc current density J: the value of for which the gap is very small increases if we go from from negative to positive value of the dc current density. For a positive value of J, we also observe the gap between the hybridized modes slowly increases as increases and becomes maximum for a particular value of , and then decreases if we further increase the value of , as shown in the inset of Fig 4(f). For the low regime, the anticrossing gap remains nearly the same for dierent orders of magnitude and directions of current density, as is shown in Figs. S2, S3 and S4 of the SM. However, for the high regime, we observe a level repulsion in the real part and a level crossing in the imaginary part of the eigenvalues for dierent orders of magnitude and directions of the current density, as is shown in Figs. S5, S6 and S7 of the SM. For a dierent coupling strength ( KFKA), we observe a similar trend. A positive current density is needed to increase the gap between the two hybridized modes as is shown in Fig. S8 (SM). In the following discussion, we chose Gilbert damping parameters of = 2:27103and= 2:5103for dierent orders of magnitude of Jwhere a very small anticrossing gap for zero current density is seen, as illus- trated in Fig. 4. For large (= 4103) and for very low (= 5105), the hybridized mode frequency ( !!c) and the linewidth (
!) plotted as a function of the eld detuning (!m!c) are shown in the Fig. S1 and Fig. S2 of the SM. Figure 5 shows the variation of the magni- tude of the gap between the two hybridized modes with respect to eld detuning ( !m!) for= 2:27103 and= 2:5103. For a pure eld-like torque eect, there is a horizontal shift [as shown in Figs. 5(a) and 5(d)] of the gap between the hybridized modes towards the positive value of eld detuning as we go from negative to positive values of the current density J. However, for a damping-like torque eect, there is a vertical shift [as shown in Figs. 5(b) and 5(e)] of the minimum gap (an- ticrossing gap): the anticrossing gap increases if we go from negative to positive values of J. For the combined eld-like and damping-like torques eect, there are both horizontal and vertical shifts as can be seen in Figs. 5(c) and 5(f). However, the horizontal shift due to the eect of eld-like torque, vertical shift due to damping-like torque and the combined shift due to both eld and damping- like torques are more pronounced for higher magnitudes6 FIG. 5. Variation of the gap between two hybridized modes with respect to the eld detunings ( !m!c) for= 2:27103 and continuous low Jfrom 5105A=cm2to 5105A=cm2[(a),(b) and (c)] and for = 2:5103and pulsed high J from 5106A=cm2to 5106A=cm2[(d),(e) and (f)]. There is (a),(d) a horizontal shift in the location of the gap for a= 0 andb= 0:05 (pure eld-like torque eect), (b), (e) vertical shift in the location of gap for a= 0:2 andb= 0 and (pure damping-like torque eect), and (c),(f) horizontal and vertical shifts for a= 0:2 andb= 0:05 (combined eect of damping-like and eld-like torques). ofJleading to an unusual behavior as is shown in panel (f). Finally, we note that introducing the eld-like and damping-like torques in the Landau-Lifshitz-Gilbert equation and coupling it with cavity mode through combined Amp ere and Faraday eects do not produce level attraction. The coupling term in our analysis is not aected by the term [see Eq. (10)], which is the parameter governed by spin torque. This means that transitioning the system from strong coupling to dissipative coupling and vice versa cannot be achieved by spin-transfer torques. IV. SUMMARY By coupling of the generalized LLG equation with the RLC equation of the cavity, we revealed the coupling be- tween magnon and photon modes under the in uence of damping and eld-like torques. Our results indicate that the magnitude of the level repulsion (manifested by the frequency gap of the hybridized modes) and, hence, themagnon-photon coupling strength can eciently be con- trolled by varying the magnitude and the direction of dc current density Jfor realistic parameters of the magnetic properties. Our model suggests that an on-demand ma- nipulation of the magnon-photon coupling strength can be achieved for current densities of the order as small as 105A/cm2and an intermediate Gilbert damping of the order 103. Higher values of Jcan denitely en- hance the eect of damping and eld-like torques on the magnon-photon coupling provided we use pulses of dc current to reduce possible heating eects. Therefore, the experimental realization of the proposed magnon-photon control mechanism should be feasible in YIG/Pt hybrid structures. ACKNOWLEDGMENT We acknowledge fruitful discussions with Dr. J. Skle- nar (Wayne State University) and Dr. J. Q. Xiao (Uni- versity of Delaware). 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2023-02-17

We demonstrate the influence of damping and field-like torques in the magnon-photon coupling process by classically integrating the generalized Landau-Lifshitz-Gilbert equation with RLC equation in which a phase correlation between dynamic magnetization and microwave current through combined Amp\`ere and Faraday effects are considered. We show that the gap between two hybridized modes can be controlled in samples with damping parameter in the order of $10^{-3}$ by changing the direction of the dc current density $J$ if a certain threshold is reached. Our results suggest that an experimental realization of the proposed magnon-photon coupling control mechanism is feasible in yttrium iron garnet/Pt hybrid structures.

Control of magnon-photon coupling by spin torque

2302.08910v1

A reduced model for precessional switching of thin-lm nanomagnets under the in uence of spin-torque Ross G. Lund1, Gabriel D. Chaves-O'Flynn2, Andrew D. Kent2, Cyrill B. Muratov1 1Department of Mathematical Sciences, New Jersey Institute of Technology , University Heights, Newark, NJ 07102, USA 2Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA (Dated: May 25, 2022) We study the magnetization dynamics of thin-lm magnetic elements with in-plane magnetization subject to a spin-current owing perpendicular to the lm plane. We derive a reduced partial dierential equation for the in-plane magnetization angle in a weakly damped regime. We then apply this model to study the experimentally relevant problem of switching of an elliptical element when the spin-polarization has a component perpendicular to the lm plane, restricting the reduced model to a macrospin approximation. The macrospin ordinary dierential equation is treated analytically as a weakly damped Hamiltonian system, and an orbit-averaging method is used to understand transitions in solution behaviors in terms of a discrete dynamical system. The predictions of our reduced model are compared to those of the full Landau{Lifshitz{Gilbert{Slonczewski equation for a macrospin. I. INTRODUCTION Magnetization dynamics in the presence of spin- transfer torques is a very active area of research with ap- plications to magnetic memory devices and oscillators1{3. Some basic questions relate to the types of magnetiza- tion dynamics that can be excited and the time scales on which the dynamics occurs. Many of the experimental studies of spin-transfer torques are on thin lm magnetic elements patterned into asymmetric shapes (e.g. an el- lipse) in which the demagnetizing eld strongly connes the magnetization to the lm plane. Analytic models that capture the resulting nearly in-plane magnetization dynamics (see e.g.4{8) can lead to new insights and guide experimental studies and device design. A macrospin model that treats the entire magnetization of the ele- ment as a single vector of xed length is a starting point for most analyses. The focus of this paper is on a thin-lm magnetic el- ement excited by a spin-polarized current that has an out-of-plane component. This out-of-plane component of spin-polarization can lead to magnetization precession about the lm normal or magnetization reversal. The for- mer dynamics would be desired for a spin-transfer torque oscillator, while the latter dynamics would be essential in a magnetic memory device. A device in which a perpen- dicular component of spin-polarization is applied to an in-plane magnetized element was proposed in Ref. [9] and has been studied experimentally10{12. There have also been a number of models that have considered the in u- ence of thermal noise on the resulting dynamics, e.g., on the rate of switching and the dephasing of the oscillator motion13{15. Here we consider a weakly damped asymptotic regime of the Landau{Lifshitz{Gilbert{Slonczewski (LLGS) equation for a thin-lm ferromagnet, in which the oscil- latory nature of the in-plane dynamics is highlighted. In this regime, we derive a reduced partial dierential equa- tion (PDE) for the in-plane magnetization dynamics un- der applied spin-torque, which is a generalization of theunderdamped wave-like model due to Capella, Melcher and Otto8. We then analyze the solutions of this equa- tion under the macrospin (spatially uniform) approxima- tion, and discuss the predictions of such a model in the context of previous numerical studies of the full LLGS equation16. The rest of this article is organized as follows. In Sec. II, we perform an asymptotic derivation of the reduced underdamped equation for the in-plane magnetization dynamics in a thin-lm element of arbitrary cross sec- tion, by rst making a thin-lm approximation to the LLGS equation, then a weak-damping approximation. In Sec. III, we then further reduce to a macrospin ordinary dierential equation (ODE) by spatial averaging of the underdamped PDE, and restrict to the particular case of a soft elliptical element. A brief parametric study of the ODE solutions is then presented, varying the spin-current parameters. In Sec. IV, we make an analytical study of the macrospin equation using an orbit-averaging method to reduce to a discrete dynamical system, and compare its predictions to the full ODE solutions. In Sec. V, we seek to understand transitions between the dierent so- lution trajectories (and thus predict current-parameter values when the system will either switch or precess) by studying the discrete dynamical system derived in Sec. IV. Finally, we summarize our ndings in Sec. VI. II. REDUCED MODEL We consider a domain R3occupied by a ferromag- netic lm with cross-section DR2and thickness d, i.e., =D(0;d). Under the in uence of a spin-polarized electric current applied perpendicular to the lm plane, the magnetization vector m=m(r;t), withjmj= 1 in and 0 outside, satises the LLGS equation (in SI units) @m @t= 0mHe+m@m @t+STT (1)arXiv:1605.08698v1 [cond-mat.mes-hall] 27 May 20162 in , with@m=@n= (n
r)m= 0 on@ , where nis the outward unit normal to @ . In the above,  > 0 is the Gilbert damping parameter, is the gyromagnetic ratio, 0is the permeability of free space, He=1 0MsE mis the eective magnetic eld, E(m) =Z  Ajrmj2+K(m)0MsHext
m) d3r +0M2 sZ R3Z R3r
m(r)r
m(r0) 8jrr0jd3rd3r0(2) is the micromagnetic energy with exchange constant A, anisotropy constant K, crystalline anisotropy function , external magnetic eld Hext, and saturation magne- tizationMs. Additionally, the Slonczewski spin-transfer torqueSTTis given by STT= ~j 2deMsmmp; (3) wherejis the density of current passing perpendicularly through the lm, eis the elementary charge (positive), pis the spin-polarization direction, and 2(0;1] is the spin-polarization eciency. We now seek to nondimensionalize the above system. Let `=s 2A 0M2s; Q =2K 0M2s;hext=Hext Ms:(4) We then rescale space and time as r!`r; t!t 0Ms; (5) obtaining the nondimensional form @m @t=mhe+m@m @tmmp;(6) where he=He=Ms, and =~j 2de0M2s(7) is the dimensionless spin-torque strength. Since we are interested in thin lms, we now assume thatmis independent of the lm thickness. Then, after rescaling E!0M2 sd`2E; (8) we have he'E m, whereEis given by a local energy functional dened on the (rescaled) two-dimensional do- mainD(see, e.g., Ref. [17]): E(m)'1 2Z D jrmj2+Q(m)2hext
m
d2r +1 2Z Dm2 ?d2r+1 4jlnjZ @D(m
n)2ds;(9)in which now m:D!S2,m?is its out-of-plane com- ponent,=d=`is the dimensionless lm thickness, and =d=L1 (whereLis the lateral size of the lm) is the lm's aspect ratio. The eective eld is given explicitly by he=
mQ 2rm(m)m?ez+hext; (10) andmsatises equation (6) in Dwith the boundary condition @m @n=1 2jlnj(m
n)(n(m
n)m) (11) on@D. We now parametrize min terms of spherical angles as m= (sincos;coscos;sin); (12) and the current polarization direction pin terms of an in-plane angle and its out-of-plane component p?as p=1p 1 +p2 ?(sin ;cos ;p?): (13) Writing==p 1 +p2 ?, after some algebra, one may then write equation (6) as the system @ @t=1 coshe
m+cos@ @t +(p?cossincos( ));(14) cos@ @t=he
m+@ @t+sin( );(15) where m=@m=@andm=@m=@formgiven by (12). Again, since we are working in a soft thin lm, we assume1 and that the out-of-plane component of the eective eld in equation (10) is dominated by the termhe
ez'm?=sin. Note that this assumes that the crystalline anisotropy and external eld terms in the out-of-plane directions are relatively small, so we assume the external eld is only in plane, though it is still possible to include a perpendicular anisotropy simply by renormalizing the constant in front of the m?term in he. We then linearize the above system in , yielding @ @t=E +@ @t+(p?cos( )); (16) @ @t=+sin( ) +(hxsin+hycos) +@ @t:(17) wherehx=he
exandhy=he
ey, andE() isE(m) evaluated at = 0.3 We now note that the last two terms in (17) are neg- ligible relative to wheneverjhxj;jhyjandare small, which is true of typical clean thin-lm samples of su- ciently large lateral extent. Neglecting these terms, one has @ @t=E +@ @t+(p?cos( )); (18) @ @t=sin( ) +: (19) Then, dierentiating (19) with respect to tand using the result along with (19) to eliminate and@ @tfrom (18), we nd a second-order in time equation for : 0 =@2 @t2+@ @t(+ 2cos( )) +E  +p?+2 sin( ) cos( );(20) where, explicitly, one has E =
+Q 2~0() +hext
(cos;sin); (21) and~() = ( m()). In turn, from the boundary condi- tion on min (11), we can derive the boundary condition foras n
r=1 2jlnjsin(') cos('); (22) where'is the angle parametrizing the normal nto@D vian= (sin';cos'). The model comprised of (20){(22) is a damped-driven wave-like PDE for , which coincides with the reduced model of Ref. [8] for vanishing spin-current density in an innite sample. This constitutes our reduced PDE model for magnetization dynamics in thin-lm elements under the in uence of out-of-plane spin currents. It is easy to see that all of the terms in (20) balance when the parameters are chosen so as to satisfy p?Q1=2jhextj1=2` Ljlnj:(23) This shows that it should be possible to rigorously obtain the reduced model in (20){(22) in the asymptotic limit ofL!1 and;;p?;Q;jhextj;!0 jointly, so that (23) holds. III. MACROSPIN SWITCHING In this section we study the behavior of the reduced model (20){(22) in the approximation that the magneti- zation is spatially uniform on an elliptical domain, and compare the solution phenomenology to that found by simulating the LLGS equation in the same physical situ- ation, as studied in Ref. [16].A. Derivation of macrospin model Integrating equation (20) over the domain Dand using the boundary condition (22), we have Z D@2 @t2+@ @t(+ 2cos( )) +p?+2 sin( ) cos( ) +Q 2~0() +hext
(cos;sin) d2r =1 2jlnjZ @Dsin(') cos(') ds:(24) Assume now that does not vary appreciably across the domainD, which makes sense in magnetic elements that are not too large. This allows us to replace (r;t) by its spatial average (t) =1 jDjR D(r;t) d2r, wherejDj stands for the area of Din the units of `2. Denoting time derivatives by overdots, and omitting the bar on  for notational simplicity, this spatial averaging leads to the following ODE for (t): +_(+ 2cos( )) +2 sin( ) cos( ) +p?+Q 2~0() +hext
(cos;sin) =jlnj 4jDjsin 2Z @Dcos(2') ds jlnj 4jDjcos 2Z @Dsin(2') ds:(25) Next, we consider a particular physical situation in which to study the macrospin equation, motivated by previous work10,11. As in Refs. [14{16], we consider an elliptical thin-lm element (recall that lengths are now measured in the units of `): D= (x;y) :x2 a2+y2 b2<1 ; (26) with no in-plane crystalline anisotropy, Q= 0, and no external eld, hext= 0. We take the long axis of the ellipse to be aligned with the ey-direction, i.e. b > a , with the in-plane component of current polarization also aligned along this direction, i.e., taking = 0. One can then compute the integral over the boundary in equation (25) explicitly, leading to the equation +_(+cos) +  sincos +2 sincos+p?= 0;(27) where we introduced the geometric parameter 0 <1 obtained by an explicit integration:  =jlnj 22abZ2 0b2cos2a2sin2p b2cos2+a2sin2d: (28)4 (d) (c) (a) (b) FIG. 1: Solutions of macrospin equation (30) for = 0:01,  = 0:1. In (a),p?= 0:2,= 0:03: decaying solution; in (b), p?= 0:2,= 0:06: limit cycle solution (the initial conditions in (a) and (b) are (0) = 3:5, to better visualize the behavior). In (c),p?= 0:3,= 0:08: switching solution; in (d), p?= 0:6,= 0:1: precessing solution. This may be computed in terms of elliptic integrals, though the expression is cumbersome so we omit it here. Importantly, up to a factor depending only on the eccen- tricity the value of  is given by d LlnL d: (29) For example, for an elliptical nanomagnet with dimen- sions 100302:5 nm (similar to those considered in Ref. [16]), this yields  '0:1. It is convenient to rescale time byp  and divide through by , yielding +1p _(+ 2 cos) + sincos +p?+2 sincos= 0;(30) where we introduced ==. We then apply this ODE to model the problem of switching of the thin-lm elements, taking the initial in-plane magnetization direc-tion to be static and aligned along the easy axis, an- tiparallel to the in-plane component of the spin-current polarization. Thus, we take (0) =; _(0) = 0; (31) and study the resulting initial value problem. B. Solution phenomenology Let us brie y investigate the solution phenomenology as the dimensionless spin-current parameters andp? are varied, with the material parameters, and , xed. We take all parameters to be constant in time for simplic- ity. We nd, by numerical integration, 4 types of solution to the initial value problem dened above. The sample solution curves are displayed in Fig. 1 below. The rst (panel (a)) occurs for small values of , and consists sim- ply of oscillations of around a xed point close to the long axis of the ellipse, which decay in amplitude towards the xed point, without switching.5 Secondly (panel (b)), still below the switching thresh- old, the same oscillations about the xed point can reach a nite xed amplitude and persist without switching. This behavior corresponds to the onset of relatively small amplitude limit-cycle oscillations around the xed point. Thirdly (panel (c)), increasing either ;p?or both, we obtain switching solutions. These have initial oscillations inabout the xed point near , which increase in ampli- tude, and eventually cross the short axis of the ellipse at ==2. Thenoscillates about the xed point near 0, and the oscillations decay in amplitude toward the xed point. Finally (panel (d)), further increasing andp?we obtain precessing solutions. Here, the initial oscillations about the xed point near quickly grow to cross =2, after which continues to decrease for all t, the magne- tization making full precessions around the out-of-plane axis. IV. HALF-PERIOD ORBIT-AVERAGING APPROACH We now seek to gain some analytical insight into the transitions between the solution types discussed above. We do this by averaging over half-periods of the oscil- lations observed in the solutions to generate a discrete dynamical system which describes the evolution of the energy of a solution (t) on half-period time intervals. Firstly, we observe that in the relevant parameter regimes the reduced equation (30) can be seen as a weakly perturbed Hamiltonian system. We consider both and  small, with .p , and assume = and p?.1. The arguments below can be rigorously jus- tied by considering, for example, the limit  !0 while assuming that =O() and that the values of and p?are xed. This limit may be achieved in the origi- nal model by sending jointly d!0 andL!1 , while keeping17 Ld `2lnL d.1: (32) The last condition ensures the consistency of the assump- tion thatdoes not vary appreciably throughout D. Introducing !(t) =_(t), (30) can be written to leading order as _=@H @!;_!=@H @; (33) where we introduced H=1 2!2+V(); V () =1 2sin2+p?: (34) At the next order, the eects of nite and  appear in the rst-derivative term in (30), while the other forc- ing term is still higher order. The behavior of (30) is therefore that of a weakly damped Hamiltonian system with Hamiltonian H, with the eects of andservingto slowly change the value of Has the system evolves. Thus, we now employ the technique of orbit-averaging to reduce the problem further to the discrete dynamics of H(t), where the discrete time-steps are equal (to the lead- ing order) to half-periods of the underlying Hamiltonian dynamics (which thus vary with H). Let us rst compute the continuous-in-time dynamics ofH. From (34), _H=!( _!+V0()); (35) which vanishes to leading order. At the next order, from (30), one has _H=!2 p (+ 2 cos): (36) We now seek to average this dynamics over the Hamil- tonian orbits. The general nature of the Hamiltonian orbits is either oscillations around a local minimum of V() (limit cycles) or persistent precessions. If the lo- cal minimum of Vis close to an even multiple of ,H cannot increase, while if it is close to an odd multiple thenHcan increase if is large enough. The switching process involves moving from the oscillatory orbits close to one of these odd minima, up the energy landscape, then jumping to oscillatory orbits around the neighbor- ing even minimum, and decreasing in energy towards the new local xed point. We focus rst on the oscillatory orbits. We may dene their half-periods as T(H) =Z +  d _; (37) where and +are the roots of the equation V() = Hto the left and right of the local minimum of V() about which (t) oscillates. To compute this integral, we assume that (t) follows the Hamiltonian trajectory: _=p 2(HV()): (38) We then dene the half-period average of a function f((t)) as hfi=1 T(H)Z +  f() dp 2(HV()); (39) which agrees with the time average over half-period to the leading order. Note that this formula applies irre- spectively of whether the trajectory connects  to + or +to . Applying this averaging to _H, we then have D _HE =1 T(H)Z +  (;H) d; (40) where we dened (;H) =(+ 2 cos)p 2(HV())p : (41)6 If the value ofHis such that either of the roots  no longer exist, this indicates that the system is now on a precessional trajectory. In order to account for this, we can dene the period on a precessional trajectory instead as T(H) =ZC Cd _; (42) whereCis a local maximum of V(). On the preces- sional trajectories, we then have D _HE =1 T(H)ZC C(;H) d: (43) In order to approximate the ODE solutions, we now decompose the dynamics of Hinto half-period time in- tervals. We thus take, at the n'th timestep,Hn=H(tn), tn+1=tn+T(Hn) and Hn+1=HnZ +(Hn)  (Hn)(;Hn) d; (44) ifHncorresponds to a limit cycle trajectory. The same discrete map applies to precessional trajectories, but with the integration limits replaced with CandC, re- spectively. A. Modelling switching with discrete map In order to model switching starting from inside a well ofV(), we can iterate the discrete map above, starting from an initial energy H0. We chooseH0by choosing a static initial condition (0) =0close to an odd multiple of(let us assume without loss of generality that we are close to), and computing H0=V(0). On the oscillatory trajectories, the discrete map then predicts the maximum amplitudes of oscillation (  (Hn)) at each timestep, by locally solving Hn=V() for each n. After some number of iterations, the trajectory will escape the local potential well, and one or both roots of Hn=V() will not exist. Due to the positive average slope ofV() the most likely direction for a trajectory to escape the potential well is _<0 (`downhill'). Assuming this to be the case, at some timestep tN, it will occur that the equationHN=V() has only one root = +>, implying that the trajectory has escaped the potential well, and will proceed on a precessional trajectory in a negative direction past ==2 towards= 0. To distinguish whether a trajectory results in switching or precession, we then perform a single half-period step on the precessional orbit from CtoC, and check whetherH< V (C): if this is the case, the tra- jectory moves back to the oscillatory orbits around the well close to = 0, and decreases in energy towards the xed point near = 0, representing switching. If how- everH> V(C) after the precessional half-period, the solution will continue to precess.In Fig. 2 below, we display the result of such an iter- ated application of the discrete map, for the same param- eters as the switching solution given in Fig. 1(c). In Fig. 2(a), the continuous curve represents the solution to (30), and the points are the predicted peaks of the oscillations, from the discrete map (44). Fig. 2(b) shows the energy of the same solution as a function of . Again the blue curve givesH(t) for the ODE solution, the green points are the prediction of the iterated discrete map, and the red curve is V(). The discrete map predicts the switch- ing behavior quite well, only suering some error near the switching event, when the change of His signicant on a single period. B. Modelling precession Here we apply the discrete map to a precessional solution|one in which the trajectory, once it escapes the potential well near , does not get trapped in the next well, and continues to rotate. Fig. 3(a) below dis- plays such a solution (t) and its discrete approximation, and Fig. 3(b) displays the energy of the same solution. Again, the prediction of the discrete map is excellent. V. TRANSITIONS IN TRAJECTORIES In this section we seek to understand the transi- tions between the trapping, switching, and precessional regimes as the current parameters andp?are varied. A. Escape Transition Firstly, let us consider the transition from states which are trapped in a single potential well, such as those in Figs. 1(a,b), to states which can escape and either switch or precess. Eectively, the absolute threshold for this transition is for the value of Hto be able to increase for some value close to the minimum of V() near. Thus, we consider the equation of motion (36) for H, and wish to nd parameter values such that _H>0 for somenear . This requires that !2 p (+ 2 cos)<0: (45) Assuming that !6= 0, we can see that the optimal value ofto hope to satisfy this condition is =, yield- ing a theoretical minimum =sfor the dimensionless current density for motion to be possible, with s= 2: (46) This is similar to the critical switching currents derived in previous work14. We then require >sfor the possi- bility of switching or precession. Note that this estimate is independent of the value of p?.7 (b) (a) FIG. 2: Switching solution (blue line) and its discrete approximation (green circles). Parameters: = 0:01,  = 0:1,p?= 0:3, = 0:08. Panel (a) shows the solution (t), and panel (b) shows the trajectory for this solution in the Hplane. The red line in (b) shows V(). B. Switching{Precessing Transition We now consider the transition from switching to pre- cessional states. This is rather sensitive and there is not in general a sharp transition from switching to precession. It is due to the fact that for certain parameters, the path that the trajectory takes once it escapes the potential well depends on how much energy it has as it does so. In fact, for a xed ;, and values of  > swe can sep- arate the (;p?)-parameter space into three regions: (i) after escaping the initial well, the trajectory always falls into the next well, and thus switches; (ii) after escaping, the trajectory may either switch or precess depending on its energy as it does so (and thus depending on its initial condition); (iii) after escaping, the trajectory completely passes the next well, and thus begins to precess. We can determine in which region of the parameter space a given point ( ;p?) lies by studying the discrete map (44) close to the peaks of V(). Assume that the trajectory begins at (0) =, and is thus initially in the potential well spanning the interval =23=2. Denote byCthe point close to ==2 at which V() has a local maximum. It is simple to compute C= 2+1 2sin1(2p?): (47) Moreover, it is easy to see that all other local maxima of V() are given by =C+k, fork2Z. We now consider trajectories which escape the initial well by crossing C. These trajectories have, for some value of the timestep nwhile still conned in the initial well, an energy value Hnin the range Htrap<Hn<V(C+); (48) where we deneHtrapto be the value of Hnsuch that the discrete map (44) gives Hn+1=V(C). We thushaveHn+1> V (C). In order to check whether the trajectory switches or precesses, we then compute Hn+2 and compare it to V(C). We may then classify the trajectories as switching if Hn+2V(C)<0, and precessional ifHn+2V(C)>0. Figure 4 displays a plot of HnV(C+) against Hn+2V(C). The blue line shows the result of applying the discrete map, while the red line is the iden- tity line. Values of HnV(C+) which are inside the range specied in (48) are thus on the negative x-axis here. We can classify switching trajectories as those for which the blue line lies below the x-axis, and precessing trajectories as those which lie above. In Fig. 4, the pa- rameters are such that both of these trajectory types are possible, depending on the initial value of Hn, and thus this set of parameters are in region (ii) of the parameter space. We note that, since the curve of blue points and the identity line intersect for some large enough value of H, this gure implies that if the trajectory has enough energy to begin precessing, then after several precessions the trajectory will converge to one which conserves en- ergy on average over a precessional period (indicated by the arrows). In region (i) of the parameter space, the portion of the blue line for HnV(C+)<0 would haveHn+2V(C)<0, while in region (iii), they would all haveHn+2V(C)>0. We can classify the parameter regimes for which switching in the opposite direction (i.e. switches from to 2) is possible in a similar way. It is not possible to have a precessional trajectory moving in this direction (_>0), though. We may then predict, for a given point ( ;p?) in pa- rameter space, by computing relations similar to that in Fig. 4, which region that point is in, and thus generate a theoretical phase diagram. In Fig. 5 below, we display the phase diagram in the8 (a) (b) FIG. 3: Precessing solution (blue line) and its discrete approximation (green circles). Parameters: = 0:01,  = 0:1,p?= 0:6, = 0:1. Panel (a) shows the solution (t), and panel (b) shows the trajectory for this solution in the Hplane. The red line in (b) shows V(). (;p?)-parameter space, showing the end results of solv- ing the ODE (30) as a background color, together with predictions of the bounding curves of the three regions of the space, made using the procedure described above. The predictions of the discrete map, while not perfect, are quite good, and provide useful estimates on the dif- ferent regions of parameter space. In particular, we note that the region where downhill switching reliably occurs (the portion of region (i) above the dashed black line) is estimated quite well. We would also note that we would expect the predictions of the discrete map to improve if the values of  and were decreased. VI. DISCUSSION We have derived an underdamped PDE model for mag- netization dynamics in thin lms subject to perpendic- ular applied spin-polarized currents, valid in the asymp- totic regime of small and , corresponding to weak damping and strong penalty for out-of-plane magnetiza- tions. We have examined the predictions of this model applied to the case of an elliptical lm under a macrospin approximation by using an orbit-averaging approach. We found that they qualitatively agree quite well with pre- vious simulations using full LLGS dynamics16. The benets of our reduced model are that they should faithfully reproduce the oscillatory nature of the in- plane magnetization dynamics, reducing computational expense compared to full micromagnetic simulations. In particular, in suciently small and thin magnetic ele- ments the problem further reduces to a single second- order scalar equation. The orbit-averaging approach taken here enables the investigation of the transition from switching to preces- sion via a simple discrete dynamical system. The regionsin parameter space where either switching or precession are predicted, as well as an intermediate region where the end result depends sensitively on initial conditions. It may be possible to further probe this region by includ- ing either spatial variations in the magnetization (which, in an earlier study16were observed to simply `slow down' the dynamics and increase the size of the switching re- gion), or by including thermal noise, which could result in instead a phase diagram predicting switching proba- bilities at a given temperature, or both. −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−0.05−0.04−0.03−0.02−0.0100.010.020.030.040.05 Hn−V(θC+π)Hn+2−V(θC−π)Switch Precess FIG. 4: Precession vs switching prediction from the discrete map. Parameters: = 0:01,  = 0:1,p?= 0:35,= 0:08. Values ofHnV(C+) to the left of the dashed line switch after the next period, the trajectory becoming trapped in the well around = 0. Values to the right begin to precess, and converge to a precessional xed point of the discrete map.9 σp⊥ 0 0.05 0.1 0.15 0.2 0.25 0.300.10.20.30.40.50.60.70.80.91 (i)(ii)(iii) FIG. 5: Macrospin solution phase diagram: = 0:01; = 0:1. The background color indicates the result of solving the ODE (30) with initial condition (31): the dark region to the left of the gure indicates solutions which do not escape their initial potential well, and the vertical dashed white line shows the computed value of the minimum current required to escape, s==(2). The black band represents solutions which decay, like in Fig. 1(a), while the dark grey band represents solutions like in Fig. 1(b). In the rest of the gure, the green points indicate switching in the negative direction like in Fig. 1(c), grey indicate switching in the positive direction, and white indicates precession like in Fig. 1(d). The solid black curves are the predictions of boundaries of the regions (as indicated in the gure) by using the discrete map, and the dashed line is the prediction of the boundary below which switching in the positive direction is possible.ACKNOWLEDGMENTS Research at NJIT was supported in part by NSF via Grant No. DMS-1313687. Research at NYU was sup- ported in part by NSF via Grant No. DMR-1309202. 1S. D. Bader and S. S. P. Parkin, Annu. Rev. Condens. Matter Phys. 1, 71 (2010). 2A. Brataas and A. D. Kent and H. Ohno, Nature Mat. 11, 372 (2012). 3A. D. Kent and D. C. Worledge, Nature Nanotechnol. 10, 187 (2015). 4C. J. Garc a-Cervera and W. E, J. Appl. Phys. 90, 370 (2001). 5A. DeSimone, R. V. Kohn, S. M uller and F. Otto Comm. Pure Appl. Math. 55, 1408 (2002). 6R. V. Kohn and V. V. Slastikov, Proc. R. Soc. Lond. Ser. A461, 143 (2005). 7C. B. Muratov and V. V. Osipov, J. Comput. Phys. 216, 637 (2006). 8A. Capella, C. Melcher and F. Otto, Nonlinearity 20, 2519 (2007). 9A. D. Kent, B. Ozyilmaz and E. del Barco Appl. Phys. Lett.84, 3897 (2004).10H. Liu, D. Bedau, D. Backes, J. A. Katine, J. Langer and A. D. Kent Appl. Phys. Lett. 97, 242510 (2010). 11H. Liu, D. Bedau, D. Backes, J. A. Katine, and A. D. Kent, Appl. Phys. Lett. 101, 032403 (2012). 12L. Ye, G. Wolf, D. Pinna, G. D. Chaves-O'Flynn and A. D. Kent, J. Appl. Phys. 117, 193902 (2015). 13K. Newhall and E. Vanden-Eijnden, J. Appl. Phys. 113, 184105 (2013). 14D. Pinna, A. D. Kent and D. L. Stein Phys. Rev. B 88, 104405 (2013). 15D. Pinna, D. L. Stein and A. D. Kent Phys. Rev. B 90, 174405 (2014). 16G. D. Chaves-O'Flynn, G. Wolf, D. Pinna and A. D. Kent, J. Appl. Phys. 117, 17D705 (2015). 17R. V. Kohn and V. V. Slastikov, Arch. Rat. Mech. Anal. 178, 227 (2005).

2016-05-27

We study the magnetization dynamics of thin-film magnetic elements with in-plane magnetization subject to a spin-current flowing perpendicular to the film plane. We derive a reduced partial differential equation for the in-plane magnetization angle in a weakly damped regime. We then apply this model to study the experimentally relevant problem of switching of an elliptical element when the spin-polarization has a component perpendicular to the film plane, restricting the reduced model to a macrospin approximation. The macrospin ordinary differential equation is treated analytically as a weakly damped Hamiltonian system, and an orbit-averaging method is used to understand transitions in solution behaviors in terms of a discrete dynamical system. The predictions of our reduced model are compared to those of the full Landau--Lifshitz--Gilbert--Slonczewski equation for a macrospin.

A reduced model for precessional switching of thin-film nanomagnets under the influence of spin-torque

1605.08698v1

arXiv:1503.01478v2 [cond-mat.mes-hall] 1 Aug 2015Critical current destabilizing perpendicular magnetizat ion by the spin Hall effect Tomohiro Taniguchi1, Seiji Mitani2, and Masamitsu Hayashi2 1National Institute of Advanced Industrial Science and Tech nology (AIST), Spintronics Research Center, Tsukuba 305-8568, Japan 2National Institute for Materials Science, Tsukuba 305-004 7, Japan (Dated: July 5, 2018) The critical current needed to destabilize the magnetizati on of a perpendicular ferromagnet via the spin Hall effect is studied. Both the dampinglike and field like torques associated with the spin current generated by the spin Hall effect is included in the La ndau-Lifshitz-Gilbert equation to model the system. In the absence of the fieldlike torque, the c ritical current is independent of the damping constant and is much larger than that of conventiona l spin torque switching of collinear magnetic systems, as in magnetic tunnel junctions. With the fieldlike torque included, we find that the critical current scales with the damping constant as α0(i.e., damping independent), α, and α1/2depending on the sign of the fieldlike torque and other parame ters such as the external field. Numerical and analytical results show that the critical cur rent can be significantly reduced when the fieldlike torque possesses the appropriate sign, i.e. wh en the effective field associated with the fieldlike torque is pointing opposite to the spin direction o f the incoming electrons. These results provideapathwaytoreducingthecurrentneededtoswitch ma gnetization usingthespin Hall effect. PACS numbers: 75.78.-n, 75.70.Tj, 75.76.+j, 75.40.Mg I. INTRODUCTION The spin Hall effect1–3(SHE) in a nonmagnetic heavy metal generates pure spin current flowing along the di- rection perpendicular to an electric current. The spin current excites magnetization dynamics in a ferromagnet attached to the nonmagnetic heavy metal by the spin- transfer effect4,5. There have been a number of exper- imental reports on magnetization switching and steady precession induced by the spin Hall effect6–9. These dy- namics have attracted great attention recently from the viewpoints ofboth fundamental physicsand practicalap- plications. An important issue to be solved on the magnetization dynamics triggered by the spin Hall effect is the reduc- tion of the critical current density needed to destabilize the magnetization from its equilibrium direction, which determines the current needed to switch the magneti- zation direction or to induce magnetization oscillation. The reported critical current density for switching8,10–13 or precession9is relatively high, typically larger than 107 A/cm2. One of the reasons behind this may be related to the recently predicted damping constant independent critical current when SHE is used14,15. This is in con- trast to spin-transfer-induced magnetization switching in a typical giant magnetoresistance (GMR) or magnetic tunnel junction (MTJ) device where the critical current is expected to be proportional to the Gilbert damping constant α. Here the magnetization dynamics is excited as a result of the competition between the spin torque and the damping torque16. Since the damping constant for typical ferromagnet in GMR or MTJ devices is rela- tively small ( α∼10−2−10−3)17,18, it can explain why the critical current is larger for the SHE driven systems. Thus in particular for device application purposes, it is crucial to find experimental conditions in which the mag-netization dynamics can be excited with lower current. Another factor that might contribute to the reduc- tion of the critical current is the presence of the field like torque19. In the GMR/MTJ systems, both the con- ventional spin torque, often referred to as the damp- inglike torque, and the fieldlike torque arise from the spin transfer between the conduction electrons and the magnetization4,19–23. Due to the short relaxation length of the transverse spin of the conduction electrons24,25, the damping like torque is typically larger than the field- like torque. Indeed, the magnitude of the field like torque experimentally found in GMR/MTJ systems has been reported to be much smaller than the damping like torque26–29. Because of its smallness, the fieldlike torque had notbeen consideredin estimatingthe criticalcurrent intheGMR/MTJsystems16,30–32,althoughitdoesplaya keyrolein particularsystems33,34. In contrast, recentex- periments found that the fieldlike torque associated with the SHE is larger than the damping like torque35–40. The physical origin of the large SHE-induced field- like torque still remains unclear. Other possible sources can be the Rashba effect36,41–44, bulk effect45, and the out of plane spin orbit torque46. Interestingly, the field like torque has been reported to show a large angu- lar dependence36,37,47(the angle between the current and the magnetization), which cannot be explained by the conventional formalism of spin-transfer torque in GMR/MTJsystems. Thefieldliketorqueactsasatorque duetoanexternalfieldandmodifiestheenergylandscape of the magnetization. As a result, a large fieldlike torque can significantly influence the critical current. However, the fieldlike torque had not been taken into account in considering the current needed to destabilize the magne- tization from its equilibrium direction and thus its role is still unclear. In this paper, we study the critical current needed to2 destabilize a perpendicular ferromagnet by the spin Hall effect. The Landau-Lifshitz-Gilbert(LLG) equationwith the dampinglike and fieldlike torques associated with the spin Hall effect is solved both numerically and analyti- cally. Wefindthatthecriticalcurrentcanbesignificantly reduced when the fieldlike torque possesses the appropri- ate sign with respect to the dampinglike torque. With the fieldlike torque included, the critical current scales with the damping constant as α0(i.e., damping indepen- dent),α, andα1/2, depending on the sign of the field- like torque and other parameters. Analytical formulas of such damping-dependent critical current are derived [Eqs. (19)-(21)], and they show good agreement with the numerical calculations. From these results, we find con- ditions in which the critical current can be significantly reduced compared to the damping-independent thresh- old, i.e., systems without the fieldlike torque. The paper is organized as follows. In Sec. II, we schematically describe the system under consideration. We discuss the definition of the critical current in Sec. III. Section IV summarizes the dependences of the crit- ical current on the direction of the damping constant, the in-plane field, and the fieldlike torque obtained by the numerical simulation. The analytical formulas of the critical current and their comparison to the numerical simulations are discussed in Sec. V. The condition at which damping-dependent critical current occurs is also discussed in this section. The conclusion follows in Sec. VI. II. SYSTEM DESCRIPTION The system we consider is schematically shown in Fig. 1, where an electric current flowing along the x-direction injects a spin current into the ferromagnet by the spin Hall effect. The magnetization dynamics in the ferro- magnet is described by the LLG equation, dm dt=−γm×H+αm×dm dt −γHsm×(ey×m)−γβHsm×ey,(1) whereγandαarethe gyromagneticratioandtheGilbert damping constant, respectively. We assume that the magnetization of the ferromagnet points along the film normal (i.e., along the zaxis), and an external in-plane magnetic field is applied along the xoryaxis. The total magnetic field His given by H=HapplnH+HKmzez, (2) whereHapplis the external field directed along the xor yaxis and HKis the uniaxial anisotropy field along the zaxis.nHandeiare unit vectors that dictate the di- rection of the uniaxial anisotropy field and the iaxis, respectively. Here we call the external field along the x andydirections the longitudinal and transverse fields, respectively. The third and fourth terms on the right- hand side of Eq. (1) are the damping like and fieldlikeHappl // y Happl // x m currentxz y FIG. 1. Schematic view of the spin-Hall system. The x axis is parallel to current, whereas the zaxis is normal to the film plane. The spin direction of the electrons entering the magnetic layer via the spin Hall effect points along the + yor −ydirection. torques associated with the spin Hall effect, respectively. Thetorquestrength Hscanbeexpressedwiththecurrent densityj, the spin Hall angle ϑ, the saturation magneti- zationM, and the thickness of the ferromagnet d, i.e., Hs=/planckover2pi1ϑj 2eMd. (3) The ratio of the fieldlike torque to the damping like torque is represented by β. Recent experiments found thatβis positive and is larger than 135–40. The magnetization dynamics described by the LLG equation can be regarded as a motion of a point particle on a two-dimensional energy landscape. In the presence of the fieldlike torque, the energy map is determined by the energy density given by34 E=−M/integraldisplay dm·H−βMHsm·ey.(4) Then, the external field torque and the fieldlike torque, which are the first and fourth terms on the right-hand- side of Eq. (1), can be expressed as −γm×B, where the effective field Bis B=−∂E ∂Mm. (5) The initial state of the numerical simulation is chosen to be the direction corresponding to the minimum of the effective energy density E. The explicit forms of the ini- tial state for the longitudinal and the transverse external fields are shown in Appendix A. We emphasize for the latter discussion in Sec. V that, using Eqs. (1), (4), and (5), the time change of the effec- tive energy density is described as dE dt=dEs dt+dEα dt. (6)3 Here the first and second terms on the right-hand side are the rates of the work done by the spin Hall torque and the dissipation due to damping, respectively, which are explicitly given by dEs dt=γMHs[ey·B−(m·ey)(m·B)],(7) dEα dt=−αγM/bracketleftBig B2−(m·B)2/bracketrightBig . (8) The sign of Eq. (7) depends on the current direction and the effective magnetic field, while that of Eq. (8) is always negative. The magnetic parameters used in this paper mimic the conditions achieved in CoFeB/MgO heterostructures48; M= 1500 emu/c.c., HK= 540 Oe, ϑ= 0.1,γ= 1.76×107rad/(Oe s), and d= 1.0 nm. The value of βis varied from −2, 0, to 2. Note that we have used a reducedHK(Refs.8,49) in ordertoobtain criticalcurrents that are the same order of magnitude with that obtained experimentally. We confirmed that the following discus- sions are applicable for a large value of HK(∼1T). III. DEFINITION OF CRITICAL CURRENT In this section, we describe how we determine the crit- ical current from the numerical simulations. In exper- iments, the critical current is determined from the ob- servation of the magnetization reversal8,12,41,46,48–50. As mentioned in Sec. II, in this paper, the initial state for calculation is chosen to be the minimum of the effective energy density. Usually, there are two minimum points above and below the xyplane because of the symmetry. Throughout this paper, the initial state is chosen to be the minimum point above the xyplane, i.e., mz(0)>0, for convention.” It should be noted that, once the mag- netization arrives at the xyplane during the current ap- plication, it can move to the other hemisphere after the current is turned off due to, for example, thermal fluc- tuation. Therefore, here we define the critical current as the minimum current satisfying the condition lim t→∞mz(t)< ǫ, (9) where a small positive real number ǫis chosen to be 0.001. The duration of the simulations is fixed to 5 µs, long enough such that all the transient effects due to the current application are relaxed. Figures 2(a) and 2(b) show examples of the magnetization dynamics close to the critical current, which are obtained from the numer- ical simulation of Eq. (1). As shown, the magnetization stays near the initial state for j= 3.1×106A/cm2, while it moves to the xyplane for j= 3.2×106A/cm2. Thus, the critical current is determined as 3 .2×106A/cm2in this case. We note that the choice of the definition of the criti- cal current has some arbitrariness. For comparison, weFIG. 2. Time evolution ofthe zcomponentof themagnetiza- tionmzin the presence of the transverse field of Happl= 200 with (a) j= 3.1×106A/cm2and (b)j= 3.2×106A/cm2. The value of βis zero. show numerically evaluated critical current with a differ- ent definition in Appendix B. The main results of this paper, e.g., the dependence of the critical current on the damping constant, are not affected by the definition. We also point out that the critical current defined by Eq. (9) focuses on the instability threshold, and does not guarantee a deterministic reversal. For example, in the case of Fig. 2(b), the reversal becomes prob- abilistic because the magnetization, starting along + z, stops its dynamics at the xyplane and can move back to its original direction or rotate to a point along −z resulting in magnetization reversal. Such probabilistic reversal can be measured experimentally using transport measurements8,12,41,46,49,50or by studying nucleation of magnetic domains via magnetic imaging48. On the other hand, it hasbeen reportedthat deterministicreversalcan take place when a longitudinal in-plane field is applied alongside the current41,49. It is difficult to determine the critical current analytically for the deterministic switch- ing for all conditions since, as in the case of Fig. 2(b), the magnetization often stops at the xyplane during the current application. This occurs especially in the pres- ence of the transverse magnetic field because all torques become zero at m=±eyand the dynamics stops. Here we thus focus on the probabilistic reversal.4 FIG. 3. Numerically evaluated mzatt= 5µs for (a)-(c) the longitudinal ( nH=ex) and (d)-(f) the transverse ( nH=ey) fields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0. The damping constant is α= 0.005. The color scale indicates the zcomponent of the magnetization ( mz) att= 5µs. The red/white boundary indicates the critical current fo r probabilistic switching, whereas the red/blue boundary gi ves the critical current for deterministic switching. IV. NUMERICALLY ESTIMATED CRITICAL CURRENT In this section, we show numerically evaluated critical current for different conditions. We solve Eq. (1) and apply Eq. (9) to determine the critical current. Figure 3 shows the value of mzatt= 5µs in the presence of (a)-(c) the longitudinal ( nH=ex) and (d)-(f) the trans- verse (nH=ey) fields. The value of βis 0 for Figs. 3(a) and 3(d), 2 .0 for Figs. 3(b) and 3(e), and −2.0 for Figs. 3(c) and 3(f), respectively. The damping constant isα= 0.005. The red/white boundary indicates the crit- ical current for the probabilistic switching, whereas the red and blue ( mz=−1) boundary gives the critical cur- rent for the deterministic switching. Using these results and the definition of the critical current given by Eq. (9), and performing similar calculations for different values of α, wesummarizethedependenceofthecriticalcurrenton the longitudinal and transverse magnetic fields in Fig. 4. The damping constant is varied as the following in each plot:α= 0.005, 0.01, and 0 .02. The solid lines in Fig. 4 represent the analytical formula derived in Sec. V.A. In the presence of longitudinal field In the case of the longitudinal field and β= 0 shown in Fig. 4(a), the critical current is damping- independent. Such damping-independent critical current has been reported previously for deterministic magneti- zation switching14,15. Similarly, in the case of the longi- tudinal field and negative β(β=−2.0) shown in Fig. 4(c), the critical current is damping-independent. In these cases, the magnitude of the critical current is rel- atively high. In particular, near zero field, the critical current exceeds ∼108A/cm2, which is close to the limit of experimentally accessible value. These results indicate that the useofthe longitudinal field with zeroornegative βis ineffective for the reduction of the critical current. On the other hand, when βis positive, the critical cur- rent depends on the damping constant, as shown in Fig. 4(b). Note that positive βis reported for the torques associated with the spin Hall effect or Rashba effect in the heterostructures studied experimentally35–37,39. The magnitude of the critical current, ∼10×106A/cm2, is relatively small compared with the cases of zero or neg- ativeβ. In this case, the use of a low damping material is effective to reduce the critical current. Interestingly, the critical current is not proportional to the damping constant, while that previously calculated for a GMR or MTJ system16is proportional to α. For example, the5 longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2) 0 50 100 150 2000 -10050 -150100150 β=0.0 -50 -100 -50 -150 -200: α=0.005: α=0.01: α=0.02 transverse magnetic field (Oe)critical current density (10 6 A/cm 2) 0 50 100 150 2000 -10050 -150100150 β=0.0 -50 -100 -50 -150 -200: α=0.005: α=0.01: α=0.02 transverse magnetic field (Oe)critical current density (10 6 A/cm 2) 0 50 100 150 2000 -4020 -6040 60 β=2.0 -20 -100 -50 -150 -200: α=0.005: α=0.01: α=0.02longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2) 0 50 100 150 2000 -4020 -5040 50 β=2.0 -20 -100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02-30-1010 30 transverse magnetic field (Oe)critical current density (10 6 A/cm 2) 0 50 100 150 2000 -10050 -150100150 β=-2.0 -50 -100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02(a) (b) (c) (d) (e) (f)longitudinal magnetic field (Oe)0 50 100 150 200β=-2.0 -100 -50 -150 -200: α=0.005: α=0.01: α=0.02critical current density (10 6 A/cm 2) 0 -10050 -150100150 -50 FIG. 4. Numerically evaluated critical currents in the pres ence of (a)-(c) the longitudinal ( nH=ex) and (d)-(f) the transverse (nH=ey) fields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0, respectively. The solid lines are analytically estimated critical current in Sec. V. critical current at zero longitudinal field in Fig. 4(b) is 12.3,17.2, and24 .0×106A/cm2forα= 0.005,0.01, and 0.02, respectively. These values indicate that the critical current is proportional to α1/2. In fact, the analytical formula derived in Sec. V shows that the critical current is proportional to α1/2for positive β[see Eq. (19)]. To summarize the case of the longitudinal field, the use of a heterostructure with positive β, which is found experimentally, has the possibility to reduce the critical current if a ferromagnet with low damping constant is used. In this case, the critical current is proportional to α1/2, which has not been found in previous works. B. In the presence of transverse field In the presence of the transverse field with β= 0, the critical current shows a complex dependence on the damping constant α, as shown in Fig. 4(d). When the current and the transversefield areboth positive (or neg- ative), the critical current is proportional to the damping constant αexcept near zero field. The numerically cal- culated critical current matches well with the analytical result, Eq. (20), shown by the solid lines. In this case, the use of the low damping material results in the reduc- tion of the critical current. On the other hand, when the current and the transversefield possessthe opposite sign, thecriticalcurrentisdampingindependent. Moreover,in this case, thecriticalcurrentisofthe orderof108A/cm2. Thus, it is preferable to use the current and field having the same sign for the reduction of the critical current. Itshould be noted that, in our definition, the same sign of current and field corresponds to the case when the direc- tion ofincoming electrons’spin (due to the SHE) and the transverse field are opposite to each other. The reason why the critical current becomes damping dependent in this situation will be explained in Sec. V. Whenβis positive the critical current depends on the damping constant for the whole range of the transverse field, as shown in Fig. 4(e). The critical current is roughly proportional to α1/2, in particular, close to zero field. The solid lines display the analytical formula, Eq. (21), and showgood agreementwith the numericalcalcu- lations. The damping dependence of the critical current becomes complex when the magnitude of the transverse field is increased [see Eq. (21)]. We note that the critical currentfor the positive βin Fig. 4(e) is smallerthan that forβ= 0 in Fig. 4(d) for the whole range of Happl. On the other hand, when βis negative, the critical current is almost independent of α, especially near zero field. However, when the transverse field is increased, there is a regime where the critical current depends on the damping constant. Such transition of the critical current with the transverse field is also predicted by the analytical solution, Eq. (21). Tosummarizethe caseofthe transversefield, the αde- pendence of the critical current can be categorized into the following: α0(damping independent), α,α1/2, or other complex behavior. As with the case of the longi- tudinal field, the use of a heterostructure with positive β allowsreductionofthe criticalcurrentwhen lowdamping ferromagnet is used. Overall, the most efficient condition6 to reduce the critical current is to use the transverse field with heterostructures that possess low αand positive β. In this case, the critical current is reduced to the order of 106A/cm2. V. ANALYTICAL FORMULA OF CRITICAL CURRENT In this section, we derive the analytical formula of the critical current from the linearized LLG equation51. The complex dependences ofthe critical currentonthe damp- ing constant αdiscussed in Sec. IV are well explained by the analytical formula. We also discuss the physical in- sight obtained from the analytical formulas. A. Derivation of the critical current To derive the critical current, we consider the stable condition of the magnetization near its equilibrium. It is convenient to introduce a new coordinate XYZin which theZaxis is parallel to the equilibrium direction. The rotationfromthe xyz-coordinatetothe XYZcoordinate is performed by the rotation matrix R= cosθ0−sinθ 0 1 0 sinθ0 cosθ cosϕsinϕ0 −sinϕcosϕ0 0 0 1 ,(10) where (θ,ϕ) are the polar and azimuth angles of the magnetization at equilibrium. The equilibrium magne- tization direction under the longitudinal and transverse magnetic field is given by Eqs. (A1) and (A2), respec- tively. Since we are interested in small excitation of the magnetization around its equilibrium, we assume that the components of the magnetization in the XYZcoor- dinate satisfy mZ≃1 and|mX|,|mY| ≪1. Then, the LLG equation is linearized as 1 γd dt/parenleftbigg mX mY/parenrightbigg +M/parenleftbigg mX mY/parenrightbigg =−Hs/parenleftbigg cosθsinϕ cosϕ/parenrightbigg ,(11) where the components of the 2 ×2 matrix Mare M1,1=αBX−Hssinθsinϕ, (12) M1,2=BY, (13) M2,1=BX (14) M2,2=αBY−Hssinθsinϕ. (15) Here,BXandBYare defined as BX=Happlsinθcos(ϕ−ϕH)+βHssinθsinϕ+HKcos2θ, (16)BY=Happlsinθcos(ϕ−ϕH)+βHssinθsinϕ+HKcos2θ, (17) whereϕHrepresents the direction of the external field within the xyplane:ϕH= 0 for the longitudinal field andπ/2 for the transverse field. The solution of Eq. (11) is mX,mY∝ exp{γ[±i/radicalbig det[M]−(Tr[M]/2)2−Tr[M]/2]t}, where det[M] and Tr[ M] are the determinant and trace of the matrix M, respectively. The imaginary part of the exponent determines the oscillation frequency around theZaxis, whereas the real part determines the time evolution of the oscillation amplitude. The critical current is defined as the current at which the real part of the exponent is zero. Then, the condition Tr[ M] = 0 gives α(BX+BY)−2Hssinθsinϕ= 0,(18) For the longitudinal field, Eq. (18) gives jLONG c=±2e√αMd /planckover2pi1ϑ/radicalBig 2H2 K−H2 appl/radicalbig β(2+αβ),(19) indicating that the critical current is roughly propor- tional to α1/2. This formula works for positive βonly52 if we assume 0 <2+αβ≃2, which is satisfied for typical ferromagnets. The critical current when the transverse field is applied reads jTRANS c=2αeMd /planckover2pi1ϑ(Happl/HK)HK/bracketleftBigg 1−1 2/parenleftbiggHappl HK/parenrightbigg2/bracketrightBigg ,(20) whenβ= 0, indicating that the critical current is pro- portional to α. The critical current for finite βis jTRANS c=2eMd /planckover2pi1ϑ ×−(1+αβ)Happl±/radicalBig H2 appl+2αβ(2+αβ)H2 K β(2+αβ). (21) Equation (21) works for the whole range of |Happl|(< HK) for positive β, while it only works when |Happl|> 2αβ(2 +αβ)HKfor negative β. For example, when β=−2.0, this condition is satisfied when |Happl|>108 Oe forα= 0.005 and |Happl|>152 Oe for α= 0.01. However the condition is not satisfied for the present range of Happlforα= 0.02. The solid lines in Fig. 4(f) show where Equation (21) is applicable. The zero-field limits of Eqs. (19) and (21) become identical, lim Happl→0jc=±2e√αMd /planckover2pi1ϑ√ 2HK/radicalbig β(2+αβ),(22) indicating that the critical current near zero field is pro- portional to α1/2whenβ >0.7 FIG. 5. Magnetization dynamics under the conditions of (a) nH=ey,Happl= 50 Oe, β= 0,α= 0.005, and j= 13.2×106 A/cm2, and (b) nH=ex,Happl= 50 Oe, β= 0,α= 0.005, andj= 90×106A/cm2. B. Discussions The solid lines in Fig. 4(b), 4(d), 4(e), and 4(f) show the analytical formulas, Eqs. (19), (20), and (21). As evident, these formulas agree well with the numerical re- sults in the regions where the critical currents depend on the dampingconstant. In this section, we discussthe rea- son why the critical current becomes damping dependent or damping independent depending on the field direction and the sign of β. It is useful for the following discussion to first study typical magnetization dynamics found in the numerical calculations. Figure 5 shows the time evolution of the x,yandzcomponents of the magnetization when the critical current depends on [Fig. 5(a)] or is independent of [Fig. 5(b)] the damping constant. For the former, the instability is accompanied with a precession of the magnetization. On the other hand, the latter shows that the instability takes place without the precession. We start with the case when the critical current be- comes damping dependent. To provide an intuitive pic- ture, we schematically show in Fig. 6(a) the torques ex- erted on the magnetization during one precession period when current is applied. The condition is the same with that described in Fig. 5(a), i.e., the transverse magnetic field is applied with β= 0. In Fig. 6(a), magnetization is shown by the large black arrow, while the directions of the spin Hall torque, the damping torque and the ex-ternal field torque are represented by the solid, dotted and dashed lines, respectively (the external field torque is tangent to the precession trajectory). As evident in Fig. 5(a), the precession trajectory is tilted to the posi- tiveydirection due to the transversefield. Depending on the direction of the magnetization the spin Hall torque has a component parallel, antiparallel, or normal to the damping torque. This means that the work done by the spin Hall torque, denoted by ∆ Esin Fig. 6 (a), is pos- itive, negative, or zero at these positions. This can be confirmed numerically when we calculate the work done by the spin Hall torque using Eq. (7). For an infinites- imal time ∆ t, the work done by the spin Hall torque is equal to the rate of its work ( dEs/dt), given in Eq. (7), times ∆ t, i.e. ∆Es= (dEs/dt)∆t. The solid line in Fig. 6(b) shows an example of the calculated rate of the work done by the spin Hall torque (solid line), dEs/dt in Eq. (7). As shown, dEs/dtis positive, negative, and zero, when the magnetization undergoes one precession period. Similarly, the energy dissipated by the damping torque,dEα/dt, can be calculated using Eq. (8) and is shown by the dotted line in Fig. 6(b). The calculated dissipation due to damping over a precession period is always negative. Details of how the rates, shown in Fig. 6, are calculated are summarized in Appendix C. Note that the strength of the spin Hall torque for ∆Es>0 is larger than that for ∆ Es<0 due to the an- gular dependence of the spin Hall torque, |m×(ey×m)|. Although it is difficult to see, thesolid line in Fig. 6(b) is slightly shifted upward. Thus the total energy supplied by the spin Hall torque during one precession, given by/contintegraltext dt(dEs/dt), does not average to zero and becomes posi- tive. When the current magnitude, |j|, is larger than |jc| in Eq. (20), the energy supplied by the spin Hall torque overcomes the dissipation due to the damping and con- sequently the precession amplitude grows, which leads to the magnetization instability shown in Fig. 5(a). The same picture is applicable when both directions of field and current are reversed. For this condition, the insta- bility of the magnetization is induced by the competition between the spin Hall torque and the damping torque. Therefore, the critical current depends on the damping constant α. When only the current direction is reversed in Figs. 6(a) and 6(b) (i.e., the sign of the magnetic field and current is opposite to each other), the sign of ∆ Esis reversed and thus the total energy supplied by the spin Hall torque becomes negative. This means that the spin Hall torque cannot overcome the damping torque to in- duce instability. Therefore, the critical current shown in Eq. (20) only applies to the case when the sign of the field and current is the same. As described in Sec. IV, the same sign of the current and field in our definition means that the incoming electrons’ spin direction, due to the spin Hall effect, is opposite to the transverse field direction. Next, we consider the case when the critical current is damping independent. Figure 6 (c) schematically shows the precession trajectory when the applied field points to8 FIG. 6. (a) A schematic view of the precession trajectory in the presence of the applied field in the positive y-direction. The solid and dotted arrows indicate the directions of the spin Hall torque and the damping torque, respectively. The dashed line, which is the tangent line to the precession tra- jectory, shows the field torque. The damping torque always dissipates energy from the ferromagnet. On the other hand, the spin Hall torque supplies energy (∆ Es>0) when its di- rection is anti-parallel to the damping torque, and dissipa tes energy (∆ Es<0) when the direction is parallel to the damp- ing torque. When the direction of the spin Hall torque is orthogonal to the damping torque, the spin Hall torque does not change the energy (∆ Es= 0). (b) Typical temporal vari- ation of the rates of the work done by the spin Hall torque, Eq. (7), (solid) and the dissipation due to damping, Eq. (8) (dotted) in the presence of the transverse field. The time is normalized by the period given by Eq. (C7). (c), (d) Similar figures with the longitudinal field. thexdirection and β= 0. The corresponding rate of work done by the spin Hall torque and the dissipation rate due to the damping torque are shown in Fig. 6 (d). Similar to the previous case, ∆ Escan be positive, nega- tive, or zero during one precession period. However, the total workdoneby the spin Hall torque,/contintegraltext dt(dEs/dt), be- comes zero in this case due to the symmetry of angular dependence of the spin Hall torque. This means that the spin Hall torque cannot compensate the damping torque, and thus, a steady precession assumed in the linearized LLG equation is not excited. This is evident in the nu- merically calculated magnetization trajectory shown in Fig. 5(b). For this case, the linearized LLG equation gives|jc| → ∞, indicating that the spin Hall torque can- not destabilize the magnetization. The same picture is alsoapplicable, forexample, in the absenceofthe applied field and β= 0. However, an alternative mechanism can cause destabi- lization of the magnetization. As schematically shown in Figs. 6(a) and 6(c), there is a component of the damping like spin Hall torque that is orthogonal to the damping torque when ∆ Es= 0. The spin Hall torque at this pointis parallel or antiparallel to the field torque depending on the position of the magnetization. When the spin Hall torqueissufficientlylargerthanthefieldtorque,themag- netization moves from its equilibrium position even if the total energy supplied by the spin Hall torque is zero or negative. This leads to an instability that occurs before one precession finishes. In this case, it is expected that the critical current is damping-independent because the instability is induced as a competition between the spin Hall torque and the field torque, not the damping torque. The time evolution of the magnetization shown in Fig. 5 (b) represents such instability. The work reported in Refs.14,49discusses a similar instability condition. The above physical picture is also applicable in the presence of the fieldlike torque. The fieldlike torque, which acts like a torque due to the transversefield, modi- fies the equilibrium direction ofthe ferromagnetand thus the precession trajectory. Consequently, the amount of energy supplied by the spin Hall torque and the dissipa- tion due to damping is changed when the fieldlike torque is present. Depending on the sign of β, the amount of the work done by the spin Hall torque increases or decreases compared to the case with β= 0. In our definition, posi- tiveβcontributes to the increase of the supplied energy, resulting in the reduction of the critical current. The complex dependence of the critical current on αarises when the fieldlike torque is present. To summarize the discussion, the critical current be- comes damping dependent when the energy supplied by the spin Hall torque during a precession around the equi- librium is positive. The condition that meets this criteria depends on the relative direction of the spin Hall torque and the damping torque, as briefly discussed above. To derive an analytical formula that describes the condition atwhichthe criticalcurrentbecomesdamping dependent is not an easy task except for some limited cases53. VI. CONCLUSION In summary, we have studied the critical current needed to destabilize a perpendicularly magnetized fer- romagnet by the spin Hall effect. The Landau-Lifshitz- Gilbert (LLG) equation that includes both the damping- like and fieldlike torques associated with the spin Hall effect is solved numerically and analytically. The criti- cal current is found to have different dependence on the damping constant, i.e., the critical current scales with α0 (damping-independent), α, andα1/2depending on the sign of the fieldlike torque. The analytical formulas of the damping-dependent critical current, Eqs. (19), (20), and (21), are derived from the linearized LLG equation, which explain well the numerical results. We find that systems with fieldlike torque having the appropriate sign (β >0 in our definition) are the most efficient way to re- duce the criticalcurrent. Fortypicalmaterialparameters found in experiment, the critical current can be reduced to the order of 106A/cm2when ferromagnets with rea-9 sonable parameters are used. ACKNOWLEDGMENTS The authorsacknowledgeT. Yorozu, Y. Shiota, and H. Kubota in AIST for valuable discussion sthey had with us. This workwassupported by JSPS KAKENHIGrant- in-AidforYoungScientists(B),GrantNo. 25790044,and MEXT R & D Next-Generation Information Technology. Appendix A: Initial state of the numerical simulation We assume that the magnetization in the absence of the applied field points to the positive zdirection. In the presence of the field, the equilibrium direction moves from the zaxis to the xyplane. Let us denote the zenith and azimuth angles of the initial state m(t= 0) asθand ϕ, i.e.,m(t= 0) = (sin θcosϕ,sinθsinϕ,cosθ). When the applied field points to the x-direction ( nH=ex), the initial state is /parenleftbigg θ ϕ/parenrightbigg nH=ex=/parenleftBigg sin−1[/radicalBig H2 appl+(βHs)2/HK] tan−1(βHs/Happl)/parenrightBigg ,(A1) where the value of ϕis 0< ϕ < π/ 2 forHappl>0 and βHs>0,π/2< ϕ < π forHappl<0 andβHs>0,π < ϕ <3π/2forHappl<0andβHs<0,and3π/2< ϕ <2π forHappl>0 andβHs<0. On the other hand, when the applied field points to the y-direction ( nH=ey), the initial state is /parenleftbigg θ ϕ/parenrightbigg nH=ey=/parenleftbigg sin−1[(Happl+βHs)/HK] π/2/parenrightbigg ,(A2) where the range of the inverse sine function is −π/2≤ sin−1x≤π/2. We note that the choice of the initial state does not affect the evaluation of the critical cur- rent significantly, especially in the small field and current regimes. Appendix B: Numerically evaluated critical current with different definition As mentioned in Sec. III, the definition of the critical current has arbitrariness. As an example, we show the time evolution of mzunder the conditions of nH=ex, Happl=−30 Oe,β= 0, and j= 110×106A/cm2in Fig. 7. In this case, the magnetization initially starts at mz= cos[sin−1(Happl/HK)]≃0.99, and finally moves to a pointmz→0.12. Since the final state does not satisfy Eq. (9), this current, j= 110×106A/cm2, should be regarded as the current smaller than the critical current in Sec. IV. However, from the analytical point of view, this current can be regarded as the current larger than magnetization 01 -1 -0.50.5 j=110×10 6 A/cm2 time (ns)0 2 4 6 8 10 Happl=-30 Oe FIG. 7. Time evolution of the zcomponent of the mag- netization mzin the presence of the longitudinal field with Happl=−30 Oe,β= 0, and j= 110×106A/cm2. The dotted line is a guide showing mz= 0. the critical current because the final state of the magne- tization is far away from the initial equilibrium. Regarding this point, we show the numerically eval- uated critical current with a different definition. The magnetic state can be regarded as unstable when it fi- nally arrives at a point far away from the initial state54. Thus, for example, one can define the critical current as a minimum current satisfying lim t→∞|mz(t)−mz(0)|> δ, (B1) where a small positive real number δis chosen to be 0.1 here. Figure 8 summarizes the numerically evalu- ated critical current with the definition of Eq. (B1). The analytical formulas, Eqs. (19)-(21), still fit well with the numerical results. The absolute values of the damping- dependent critical current are slightly changed when the definition of the critical current is changed. This is be- cause Eq. (B1) is more easily satisfied than Eq. (9), and thus the critical current in Fig. 8 is smaller than that shown in Fig. 4. However, the main results of this paper, such as the damping dependence of the critical current, are not changed by changing the definition of the critical current in the numerical simulations. Appendix C: Energy change during a precession As described in Sec. V, the linearized LLG equation assumes a steady precession of the magnetization due to the field torque when the current magnitude is close to the critical current. This is because the spin Hall torque compensates with the damping torque. Thus, Figs. 6(b) and 6(d) are obtained by substituting the solution of m precessing a constant energy curve of Einto Eqs. (7) and (8). When the transverse field is applied and β= 0, i.e., E=E, whereE=−M/integraltext dm·H, the precession trajec- tory on the constant energy curve of Eis given by55 mx(E) = (r2−r3)sn(u,k)cn(u,k),(C1)10 transverse magnetic field (Oe)critical current density (10 6 A/cm 2) 0 50 100 150 2000 -4020 -6040 60 β=2.0 -20 -100 -50 -150 -200: α=0.005: α=0.01: α=0.02 transverse magnetic field (Oe)critical current density (10 6 A/cm 2) 0 50 100 150 2000 -10050 -150100150 β=-2.0 -50 -100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02(a) (b) (c) (d) (e) (f)longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2) 0 50 100 150 2000 -4020 -5040 50 β=2.0 -20 -100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02-30-1010 30 longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2) 0 50 100 150 2000 -10050 -150100150 β=0.0 -50 -100 -50 -150 -200: α=0.005: α=0.01: α=0.02 longitudinal magnetic field (Oe)0 50 100 150 200β=-2.0 -100 -50 -150 -200: α=0.005: α=0.01: α=0.02 transverse magnetic field (Oe)critical current density (10 6 A/cm 2) 0 50 100 150 2000 -10050 -150100150 β=0.0 -50 -100 -50 -150 -200: α=0.005: α=0.01: α=0.02 critical current density (10 6 A/cm 2) 0 -10050 -150100150 -50 FIG. 8. Numerically evaluated critical currents with a diffe rent definition, Eq. (B1), in the presence of (a)-(c) the long itudinal (nH=ex) and (d)-(f) the transverse ( nH=ey) fields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0. The solid lines are the analytically estimated critical curren t described in Sec. V. my(E) =r3+(r2−r3)sn2(u,k),(C2) mz(E) =/radicalBig 1−r2 3−(r2 2−r2 3)sn2(u,k),(C3) whereu=γ/radicalbig HtHK/2√r1−r3t, andrℓare given by r1(E) =−E MHappl, (C4) r2(E) =Happl HK+/radicalBigg 1+/parenleftbiggHappl HK/parenrightbigg2 +2E MHK,(C5) r3(E) =Happl HK−/radicalBigg 1+/parenleftbiggHappl HK/parenrightbigg2 +2E MHK.(C6)The modulus of Jacobi elliptic functions is k=/radicalbig (r2−r3)/(r1−r3). The precession period is τ(E) =2K(k) γ/radicalbig HapplHK/2√r1−r3,(C7) whereK(k) is the first kind of complete elliptic inte- gral. The initial state is chosen to be my(0) =r3. Fig- ure 6(b) is obtained by substituting Eqs. (C1), (C2), and (C3) into Eqs. (7) and (8). We note that Eqs. (C1), (C2), and (C3) are functions of the energy den- sityE. Since we are interested in the instability thresh- old near the equilibrium, the value of Eis chosen close to the minimum energy Emin. In Fig. 6 (b), we use E=Emin+ (Emax−Emin)/NwithN= 100, where the minimum energy Eminand the maximum energy EmaxareEmin=−(MHK/2)[1 + (Happl/HK)2] and Emax=−MHappl, respectively. The value of Happlis chosen to be 50 Oe. Figure 6(d) is obtained in a similar way. 1M. I. Dyakonov and V. I. Perel, “Current-induced spin orientation of electrons in semiconductors,” Phys. Lett. A 35, 459 (1971). 2J. E. Hirsch, “Spin Hall Effect,” Phys. Rev. Lett. 83, 1834 (1999). 3Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, “Observation of the Spin Hall Effect in Semi- conductors,” Science 306, 1910 (2004).4J. C. Slonczewski, “Current-driven excitation of magnetic multilayers,” J. Magn. Magn. Mater. 159, L1 (1996). 5L. 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2015-03-04

The critical current needed to destabilize the magnetization of a perpendicular ferromagnet via the spin Hall effect is studied. Both the dampinglike and fieldlike torques associated with the spin current generated by the spin Hall effect is included in the Landau-Lifshitz-Gilbert equation to model the system. In the absence of the fieldlike torque, the critical current is independent of the damping constant and is much larger than that of conventional spin torque switching of collinear magnetic systems, as in magnetic tunnel junctions. With the fieldlike torque included, we find that the critical current scales with the damping constant as $\alpha^{0}$ (i.e., damping independent),$\alpha$, and $\alpha^{1/2}$ depending on the sign of the fieldlike torque and other parameters such as the external field. Numerical and analytical results show that the critical current can be significantly reduced when the fieldlike torque possesses the appropriate sign, i.e. when the effective field associated with the fieldlike torque is pointing opposite to the spin direction of the incoming electrons. These results provide a pathway to reducing the current needed to switch magnetization using the spin Hall effect.

Critical current destabilizing perpendicular magnetization by the spin Hall effect

1503.01478v2

Nearly isotropic spin-pumping related Gilbert damping in Pt/Ni 81Fe19/Pt W. Cao,1,L. Yang,1S. Auret,2and W.E. Bailey1, 2,y 1Materials Science and Engineering, Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA 2SPINTEC, Universit eGrenoble Alpes/CEA/CNRS, F-38000 Grenoble, France (Dated: July 12, 2021) A recent theory by Chen and Zhang [Phys. Rev. Lett. 114, 126602 (2015)] predicts strongly anisotropic damping due to interfacial spin-orbit coupling in ultrathin magnetic lms. Interfacial Gilbert-type relaxation, due to the spin pumping eect, is predicted to be signicantly larger for magnetization oriented parallel to compared with perpendicular to the lm plane. Here, we have measured the anisotropy in the Pt/Ni 81Fe19/Pt system via variable-frequency, swept-eld ferromag- netic resonance (FMR). We nd a very small anisotropy of enhanced Gilbert damping with sign opposite to the prediction from the Rashba eect at the FM/Pt interface. The results are contrary to the predicted anisotropy and suggest that a mechanism separate from Rashba spin-orbit coupling causes the rapid onset of spin-current absorption in Pt. INTRODUCTION The spin-transport properties of Pt have been studied intensively. Pt exhibits ecient, reciprocal conversion of charge to spin currents through the spin Hall eect (SHE)[1{4]. It is typically used as detection layer for spin current evaluated in novel congurations[5{7]. Even so, consensus has not yet been reached on the experi- mental parameters which characterize its spin transport. The spin Hall angle of Pt, the spin diusion length of Pt, and the spin mixing conductance of Pt at dierent inter- faces dier by as much as an order of magnitude when evaluated by dierent techniques[2, 3, 8{12]. Recently, Chen and Zhang [13, 14] (hereafter CZ) have proposed that interfacial spin-orbit coupling (SOC) is a missing ingredient which can bring the measurements into greater agreement with each other. Measurements of spin-pumping-related damping, particularly, report spin diusion lengths which are much shorter than those es- timated through other techniques[15, 16]. The introduc- tion of Rashba SOC at the FM/Pt interface leads to interfacial spin-memory loss, with discontinuous loss of spin current incident to the FM/Pt interface. The model suggests that the small saturation length of damping en- hancement re ects an interfacial discontinuity, while the inverse spin Hall eect (ISHE) measurements re ect the bulk absorption in the Pt layer[15, 16]. The CZ model predicts a strong anisotropy of the en- hanced damping due to spin pumping, as measured in ferromagnetic resonance (FMR). The damping enhance- ment for time-averaged magnetization lying in the lm plane ( pc-FMR, or parallel condition) is predicted to be signicantly larger than that for magnetization oriented normal to the lm plane ( nc-FMR, or normal condition). The predicted anisotropy can be as large as 30%, with pc-FMR damping exceeding nc-FMR damping, as will be shown shortly. In this paper, we have measured the anisotropy of the enhanced damping due to the addition of Pt in symmet-ric Pt/Ni 81Fe19(Py)/Pt structures. We nd that the anisotropy is very weak, less than 5%, and with the op- posite sign from that predicted in [13]. THEORY We rst quantify the CZ-model prediction for anisotropic damping due to the Rashba eect at the FM/Pt interface. In the theory, the spin-memory loss for spin current polarized perpendicular to the interfa- cial plane is always larger than that for spin current po- larized in the interfacial plane. The pumped spin po- larization=m_mis always perpendicular to the time-averaged or static magnetization hmit'm. For nc-FMR, the polarization of pumped spin current is always in the interfacial plane, but for pc-FMR, is nearly equally in-plane and out-of-plane. A greater damping enhancement is predicted in the pccondition than in the nccondition,
pc>
nc:
nc=Kh1 + 4(tPt) 1 +(tPt)i (1)
pc=Kh1 + 6(tPt) 1 +(tPt)+ 2[1 +(tPt)]2i (2) (tPt) =(1)coth(tPt=sd) (3) where the constant of proportionality K is the same for both conditions and the dimensionless parameters, and , are always real and positive. The Rashba parameter = (RkF=EF)2(4) is proportional to the square of the Rashba coecient R, dened as the strength of the Rashba potential,arXiv:1810.10595v4 [cond-mat.mtrl-sci] 22 Feb 20192 FIG. 1. Frequency-dependent half-power FMR linewidth
H1=2(!) of the reference sample Py(5 nm) (black) and sym- metric trilayer samples Pt(t)/Py(5 nm)/Pt(t) (colored). (a) pc-FMR measurements. (b) nc-FMR measurements. Solid lines are linear ts to extract Gilbert damping . (Inset): inhomogeneous broadening
H0inpc-FMR (blue) and nc- FMR (red). V(r) =R(z)(^k^z)
, where(z) is a delta function localizing the eect to the interface at z= 0 (lm plane isxy),kFis the Fermi wavenumber, and EFis the Fermi energy. The back ow factor is a function of Pt layer thickness, where the back ow fraction at innitely large Pt thickness dened as =(1)=[1 +(1)].= 0 (1) refers to zero (complete) back ow of spin current across the interface. sdis the spin diusion length in the Pt layer. To quantify the anisotropy of the damping, we dene Q: Q(
pc
nc)=
nc (5) as an anisotropy factor , the fractional dierence be- tween the enhanced damping in pc and nc conditions. Positive Q (Q >0) is predicted by the CZ model. A spin-memory loss factor of 0.90.1, corresponding to nearly complete relaxation of spin current at the in- terface with Pt, was measured through current perpen- dicular to plane-magnetoresistance (CPP-GMR)[8] Ac- cording to the theory[13, 14], the spin-memory loss can be related to the Rashba parameter by = 2, so we take0:45. The eect of variable  < 0:45 will be shown in Figure 3. To evaluate the thickness dependent back ow(tPt), we assume Pt sd= 14 nm, which is asso- ciated with the absorption of the spin current in the bulk of Pt layer, as found from CPP-GMR measurements[8] and cited in [13]. Note that this Pt sdis longer than that used sometimes to t FMR data[15, 16]; Rashba interfa- cial coupling in the CZ model brings the onset thickness down. The calculated anisotropy factor Q should then FIG. 2. Pt thickness dependence of Gilbert damping = (tPt) inpc-FMR (blue) and nc-FMR (red). 0refers to the reference sample ( tPt= 0). (Inset): Damping enhancement
(tPt) =(tPt)0due to the addition of Pt layers in pc-FMR (blue) and nc-FMR (red). Dashed lines refer to cal- culated
ncusing Equation 1 by assuming Pt sd= 14 nm and= 10%. The red dashed line ( = 0:15) shows a similar curvature with experiments; The black dashed line ( 0:25) shows a curvature with the opposite sign. be as large as 0.3, indicating that
pcis 30% greater than
nc(see Results for details). EXPERIMENT In this paper, we present measurements of the anisotropy of damping in the symmetric Pt( tPt)/Py(5 nm)/Pt(tPt) system, where \Py"=Ni 81Fe19. Because the Py thickness is much thicker than its spin coher- ence length[17], we expect that spin-pumping-related damping at the two Py/Pt interfaces will sum. The full deposited stack is Ta(5 nm)/Cu(5 nm)/Pt( tPt)/Py(5 nm)/Pt(tPt)/Al 2O3(3 nm),tPt= 1{10 nm, deposited via DC magnetron sputtering under computer control on ion-cleaned Si/SiO 2substrates at ambient temperature. The deposition rates were 0.14 nm/s for Py and 0.07 nm/s for Pt. Heterostructures deposited identically, in the same deposition chamber, have been shown to exhibit both robust spin pumping eects, as measured through FMR linewidth[18, 19], and robust Rashba eects (in Co/Pt), as measured through Kerr microscopy[20, 21]. The stack without Pt layers was also deposited as the ref- erence sample. The lms were characterized using vari- able frequency FMR on a coplanar waveguide (CPW) with center conductor width of 300 m. The bias mag- netic eld was applied both in the lm plane ( pc) and perpendicular to the plane ( nc), as previously shown in [22]. The nc-FMR measurements require precise align- ment of the eld with respect to the lm normal. Here,3 FIG. 3. Anisotropy factor Q for spin-pumping enhanced damping, dened in Equation 5. Solid lines are calculations using the CZ theory[13], Equations 1{3, for variable Rashba parameter 0 :010:45.Pt sdis set to be 14 nm. Back ow fraction is set to be 10% in (a) and 40% in (b). Black triangles, duplicate in (a) and (b), show the experimental values from Figure 2. samples were aligned by rotation on two axes to maxi- mize the resonance eld at 3 GHz. RESULTS AND ANALYSIS Figure 1 shows frequency-dependent half-power linewidth
H1=2(!) in pc- and nc-FMR. The measure- ments were taken at frequencies from 3 GHz to a cut-o frequency above which the signal-to-noise ratio becomes too small for reliable measurement of linewidth. The cuto ranged from 12{14 GHz for the samples with Pt (linewidth200{300 G) to above 20 GHz for tPt= 0. Solid lines stand for linear regression of the variable- frequency FMR linewidth
H1=2=
H0+2!= , where
H1=2is the full-width at half-maximum,
H0is the in- homogeneous broadening, is the Gilbert damping, ! is the resonance frequency and is the gyromagnetic ra- tio. The ts show good linearity with frequency !=2for all experimental linewidths
H1=2(!). The inset sum- marizes inhomogeneous broadening
H0inpc- and nc- FMR; its errorbar is 2 Oe. In Figure 2, we plot Pt thickness dependence of damp- ing parameters (tPt) extracted from the linear ts in Figure 1, for both pc-FMR and nc-FMR measurements. Standard deviation errors in the ts for are3104. The Gilbert damping saturates quickly as a function oftPtin both pc and nc conditions, with 90% of the ef- fect realized with Pt(3 nm). The inset shows the damp- ing enhancement
due to the addition of Pt layers
=0, normalized to the Gilbert damping 0of the reference sample without Pt layers. The Pt thickness dependence of
matches our previous study on Py/Pt heterostructures[19] reasonably; the saturation value of
Pt=Py=Pt is 1.7x larger than that measured for the single interface
Py=Pt [19] (2x expected). The dashed lines in the inset refer to calculated
ncusing Equation 1 (assuming Pt sd= 14 nm and = 10%).= 0:25 shows a threshold of Pt thickness dependence. When >0:25, the curvature of
(tPt) will have the opposite sign to that observed in experiments, so = 0:25 is the maxi- mum which can qualitatively reproduce the Pt thickness dependence of the damping. As shown in Figure 2 inset, the damping enhancement due to the addition of Pt layers is slightly larger in the ncgeometry than in the pcgeometry:
nc>
pc. This is opposite to the prediction of the model in [13]. The anisotropy factor Q(
pc
nc)=
ncfor the model (Q>0) and the experiment (Q <0) are shown to- gether in Figure 3 (a) and (b). The magnitude of Q for the experiment is also quite small, with -0.05 <Q<0. This very weak anisotropy, or near isotropy, of the spin- pumping damping is contrary to the prediction in [13], and is the central result of our paper. The two panels (a) and (b), which present the same experimental data (triangles), consider dierent model parameters, corresponding to negligible back ow ( = 0:1, panel a) and moderate back ow ( = 0:4, panel b) for a range of Rashba couplings 0 :010:45. A spin diusion length sd= 14 nm for Pt[8] was assumed in all4 cases. The choice of back ow fraction = 0:1 or 0:4 and the choice of spin diusion length of Pt sd= 14 nm follow the CZ paper[13] for better evaluation of their theory. For good spin sinks like Pt, the back ow fraction is usu- ally quite small. If = 0, then there will be no spin back ow. In this limit,
pc,
ncand the Q factor will be independent of Pt thickness. In the case of a short spin diusion length of Pt, e.g., sd= 3 nm, the anisotropy Q as a function of Pt thick- ness decreases more quickly for ultrathin Pt, closer to our experimental observations. However, we note that the CZ theory requires a long spin diusion length in or- der to reconcile dierent experiments, particularly CPP- GMR with spin pumping, and is not relevant to evaluate the theory in this limit. Leaving apart the question of the sign of Q, we can see that the observed absolute magnitude is lower than that predicted for = 0:05 for small back ow and 0.01 for moderate back ow. According to ref [13], a minimum level for the theory to describe the system with strong interfacial SOC is = 0:3. DISCUSSION Here, we discuss extrinsic eects which may result in a discrepancy between the CZ model (Q +0.3) and our experimental result (-0.05 <Q<0). A possible role of two- magnon scattering[23, 24], known to be an anisotropic contribution to linewidth
H1=2, must be considered. Two-magnon scattering is present for pc-FMR and nearly absent for nc-FMR. This mechanism does not seem to play an important role in the results presented. It is dicult to locate a two-magnon scattering contribution to linewidth in the pure Py lm: Figure 1 shows highly linear
H1=2(!), without oset, over the full range to !=2= 20 GHz, thereby re ecting Gilbert-type damp- ing. The damping for this lm is much smaller than that added by the Pt layers. If the introduction of Pt adds some two-magnon linewidth, eventually mistaken for intrinsic Gilbert damping , this could only produce a measurement of Q >0, which was not observed. One may also ask whether the samples are appropriate to test the theory. The rst question regards sample qual- ity. The Rashba Hamiltonian models a very abrupt inter- face. Samples deposited identically, in the same deposi- tion chamber, have exhibited strong Rashba eects, so we expect the samples to be generally appropriate in terms of quality. Intermixing of Pt in Ni 81Fe19(Py)/Pt[25] may play a greater role than it does in Co/Pt[26], although defocused TEM images have shown fairly well-dened in- terfaces for our samples[27]. A second question might be about the magnitude of the Rashba parameter in the materials systems of in- terest. Our observation of nearly isotropic damping isconsistent with the theory, within experimental error and apart from the opposite sign, if the Rashba parameter is very low and the back ow fraction is very low. Ab-initio calculations for (epitaxial) Co/Pt in the ref[28] have in- dicated= 0.02{0.03, lower than the values of 0.45 assumed in [13, 14] to treat interfacial spin-memory loss. The origin of the small, negative Q observed here is un- clear. A recent paper has reported that
pcis smaller than
ncin the YIG/Pt system via single-frequency, variable-angle measurements[7], which is contrary to the CZ model prediction as well. It is also possible that a few monolayers of Pt next to the Py/Pt interfaces are magnetized in the samples[19], and this may have an un- known eect on the sign, not taken into account in the theory. CONCLUSIONS In summary, we have experimentally demonstrated that in Pt/Py/Pt trilayers the interfacial damping at- tributed to spin pumping is nearly isotropic, with an anisotropy between lm-parallel and lm-normal mea- surements of <5%. The nearly isotropic character of the eect is more compatible with conventional descriptions of spin pumping than with the Rashba spin-memory loss model predicted in [13]. ACKNOWLEDGEMENTS We acknowledge support from the US NSF-DMR- 1411160 and the Nanosciences Foundation, Grenoble. wc2476@columbia.edu yweb54@columbia.edu [1] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Ap- plied Physics Letters 88, 182509 (2006). [2] O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Homann, Phys. Rev. Lett. 104, 046601 (2010). [3] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). [4] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014). [5] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). [6] D. Ellsworth, L. Lu, J. Lan, H. Chang, P. Li, Z. Wang, J. Hu, B. Johnson, Y. Bian, J. Xiao, R. Wu, and M. Wu, Nature Physics 12, 861 (2016). [7] H. Zhou, X. Fan, L. Ma, Q. Zhang, L. Cui, S. Zhou, Y. S. Gui, C.-M. Hu, and D. Xue, Phys. Rev. B 94, 134421 (2016). [8] H. Kurt, R. Loloee, K. Eid, W. P. Pratt, and J. Bass, Applied Physics Letters 81, 4787 (2002).5 [9] L. Vila, T. Kimura, and Y. Otani, Phys. Rev. Lett. 99, 226604 (2007). [10] K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008). [11] A. Azevedo, L. H. Vilela-Le~ ao, R. L. Rodr guez-Su arez, A. F. Lacerda Santos, and S. M. Rezende, Phys. Rev. B 83, 144402 (2011). [12] M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Gepr ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013). [13] K. Chen and S. Zhang, Phys. Rev. Lett. 114, 126602 (2015). [14] K. Chen and S. Zhang, IEEE Magnetics Letters 6, 1 (2015). [15] Z. Feng, J. Hu, L. Sun, B. You, D. Wu, J. Du, W. Zhang, A. Hu, Y. Yang, D. M. Tang, B. S. Zhang, and H. F. Ding, Phys. Rev. B 85, 214423 (2012). [16] J.-C. Rojas-S anchez, N. Reyren, P. Laczkowski, W. Savero, J.-P. Attan e, C. Deranlot, M. Jamet, J.-M. George, L. Vila, and H. Jar es, Phys. Rev. Lett. 112, 106602 (2014). [17] A. Ghosh, S. Auret, U. Ebels, and W. E. Bailey, Phys. Rev. Lett. 109, 127202 (2012).[18] A. Ghosh, J. F. Sierra, S. Auret, U. Ebels, and W. E. Bailey, Applied Physics Letters 98, 052508 (2011). [19] M. Caminale, A. Ghosh, S. Auret, U. Ebels, K. Ollefs, F. Wilhelm, A. Rogalev, and W. E. Bailey, Phys. Rev. B94, 014414 (2016). [20] I. M. Miron, G. Gaudin, S. Auret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella, Nature Materials 9, 230 (2010). [21] I. M. Miron, T. Moore, H. Szambolics, L. D. Buda- Prejbeanu, S. Auret, B. Rodmacq, S. Pizzini, J. Vogel, M. Bonm, A. Schuhl, and G. Gaudin, Nature Materials 10, 419 (2011). [22] H. Yang, Y. Li, and W. E. Bailey, Applied Physics Let- ters108, 242404 (2016). [23] R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999). [24] R. McMichael and P. Krivosik, IEEE Transactions on Magnetics 40, 2 (2004). [25] T. Golod, A. Rydh, and V. M. Krasnov, Journal of Ap- plied Physics 110, 033909 (2011). [26] G. Bertero and R. Sinclair, Journal of Magnetism and Magnetic Materials 134, 173 (1994). [27] W. E. Bailey, A. Ghosh, S. Auret, E. Gautier, U. Ebels, F. Wilhelm, and A. Rogalev, Phys. Rev. B 86, 144403 (2012). [28] S. Grytsyuk, A. Belabbes, P. M. Haney, H.-W. Lee, K.-J. Lee, M. D. Stiles, U. Schwingenschl ogl, and A. Manchon, Phys. Rev. B 93, 174421 (2016).

2018-10-24

A recent theory by Chen and Zhang [Phys. Rev. Lett. 114, 126602 (2015)] predicts strongly anisotropic damping due to interfacial spin-orbit coupling in ultrathin magnetic films. Interfacial Gilbert-type relaxation, due to the spin pumping effect, is predicted to be significantly larger for magnetization oriented parallel to compared with perpendicular to the film plane. Here, we have measured the anisotropy in the Pt/Ni$_{81}$Fe$_{19}$/Pt system via variable-frequency, swept-field ferromagnetic resonance (FMR). We find a very small anisotropy of enhanced Gilbert damping with sign opposite to the prediction from the Rashba effect at the FM/Pt interface. The results are contrary to the predicted anisotropy and suggest that a mechanism separate from Rashba spin-orbit coupling causes the rapid onset of spin-current absorption in Pt.

Nearly isotropic spin-pumping related Gilbert damping in Pt/Ni$_{81}$Fe$_{19}$/Pt

1810.10595v4

DAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER AND BOUSSINESQ EQUATIONS WILLIAM CHEN AND ALEJANDRO SARRIA Abstract. We revisit a family of innite-energy solutions of the 3D incompressible Euler equations proposed by Gibbon et al. [9] and shown to blowup in nite time by Constantin [6]. By adding a damping term to the momentum equation we examine how the damping coecient can arrest this blowup. Further, we show that similar innite-energy solutions of the inviscid 3D Boussinesq system with damping can develop a singularity in nite time as long as the damping eects are insucient to arrest the (undamped) 3D Euler blowup in the associated damped 3D Euler system. 1.Introduction We consider a family of exact innite-energy solutions of two three-dimensional (3D) uid models with a damping term, the incompressible Euler equations (1)( ut+u
ru+u=rp; divu= 0; and the inviscid Boussinesq system (2)8 >< >:ut+u
ru+u=rp+e3; t+u
r= 0; divu= 0: In (1)-(2), urepresents the uid velocity, pis the scalar pressure, is the scalar temperature in the context of thermal convection or the density in the modeling of geophysical uids, e3= (0;0;1)T, and2R+is a real parameter. For = 0, (1) reduces to the standard 3D Euler equations describing the motion of an ideal, incompressible homogeneous uid, while (2) becomes the standard 3D inviscid Boussinesq system modeling large scale atmospheric dynamics and oceanic ows [11, 16, 19]. If 0, (2) reduces to (1). When uin (1) is replaced by the diusion term 
u, we obtain the classical 3D Navier-Stokes equations. The global (in time) regularity problem for the aforementioned 3D models are long-standing open problems in mathematical uid dynamics. See Constantin [7] for a history and survey of results on the 3D Euler regularity problem and Feerman [8] for a more precise account of the Navier-Stokes regularity problem. In general, the main obstacle in obtaining global existence of smooth solutions of these 3D models for general initial data is controlling non- linear growth due to vortex stretching [14, 12]. To gain insight into this challenging problem, 2010 Mathematics Subject Classication. 35B44, 35B65, 35Q31, 35Q35. Key words and phrases. 3D Euler; 3D Boussinesq; Blowup; Damping; Innite-energy solutions. 1arXiv:1605.08965v3 [math.AP] 30 Jun 20162 WILLIAM CHEN AND ALEJANDRO SARRIA many researchers have turned their eorts to the 2D viscous Boussinesq equations (3)8 >< >:ut+u
ru=rp+
u+e2; t+u
r=
; divu= 0: System (3) can be shown to be formally identical to the 3D Euler or Navier-Stokes equations for axisymmetric swirling ows and retains key features of the 3D models such as the vortex stretching mechanism (see, e.g., [17]). The global regularity issue for (3) has been settled in the armative under various degrees of viscosity and dissipation: with full viscosity  >0 and >0, partial viscosity  >0 and= 0, or= 0 and >0, for anisotropic models [1, 4, 13, 15], and with fractional Laplacian dissipation (see [23] and references therein). In contrast, the question of global regularity for (3) in the inviscid case == 0 remains open; it is not apparent how to control vortex stretching when there is no dissipation ( = 0) and no thermal diusion ( = 0). Using a somewhat dierent approach, Adhikaria et al. [2] replaced 
uand
in (3) with damping terms uandfor > 0 and > 0 real parameters. Although the resulting damping eects are insucient to control vortex stretching for general initial data, the authors showed that a local (in time) solution will persist globally in time if the initial data is small enough in some homogeneous Besov space. The aim of this work is to examine how damping aects the global regularity of a particular class of innite-energy solutions of (1) and (2) which, in the absence of damping ( = 0), blowup in nite-time from smooth initial data. More particularly, the uid velocity and temperature considered here have the form (4) u(x;z;t) = (u(x;t);v(x;t);z (x;t));  (x;z;t) =z(x;t) for (x;z) = (x;y;z ). Our spatial domain will be the semi-bounded 3D channel (5)  f(x;z)2QRg of rectangular periodic cross-section Q[0;1]2withuandboth periodic in the xandy variables with period one. Note that the unbounded geometry of (5) in the vertical direction endows the uid under consideration with, at best, locally nite kinetic energy. Solutions of the 3D incompressible Euler equations (6)( ut+u
ru=rp divu= 0 of the form (4)i) were proposed by Gibbon et al. [9], and then shown to blowup in nite time numerically by Ohkitani and Gibbon [18], and analytically by Constantin [6]. See also [5, 22, 20, 21] for blowup results of other similar innite-energy solutions of the Euler and Boussinesq equations in two and three dimensions. For convenience of the reader, we now summarize Constantin's blowup result. The set-up used by Constantin is eectively the same as the one considered here. Imposing a velocity eld of the form (4)i) on (6) subject to periodicity in the xandyvariables of period one and with spatial domain (5), it follows that the vertical component of the velocity eld, z , satises the vertical component of the momentum equation (6)i) if the mean-zero functionDAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER AND BOUSSINESQ EQUATIONS 3 =(ux+vy) solves the nonlocal two-dimensional equation (7) t+u0
r = 2+ 2Z Q 2dx withu0= (u;v). For ( a;t)(a1;a2;t),a2Q, Constantin constructed the solution formula (Y(a;t);t) =0(t) (t)( 1 1 + 0(a)(t)Z Qda 1 + 0(a)(t)1Z Qda (1 + 0(a)(t))2) (8) for (a;0) = 0(a),satisfying the initial value problem (IVP) (9) 0(t) =Z Qda 1 + 0(a)(t)2 ;  (0) = 0; anda!Y(a;t) the 2D ow-map dened by dY dt=u0(Y(a;t);t); Y(a;0) =a: By comparing the blowup rates of the time integrals in (8) against the local term, Constantin proved the following blowup result for a large class of smooth initial data 0. Theorem 1.1 ([6]).Consider the initial boundary value problem for (7)with smooth mean- zero initial data 0and periodic boundary conditions. Suppose 0attains a negative minimum m0at a nite number of locations a02Qand, near these locations, 0has non-vanishing second-order derivatives. Set =1 m0and let (10) tE() =Z 0Z Qda 1 + 0(a)2 d: Then there exists a nite time TE>0, given by TElim %tE(); such that both the maximum and minimum values of diverge to positive and respectively negative innity as t%TE. The outline for the remainder of the paper is as follows. In Section 2 we introduce the damped two-dimensional equations (12)-(14) and summarize the main results of the paper. Then in Section 3 we derive the solution formulae (33)-(37), which we use in Section 4 to prove the Theorems. 2.Preliminaries 2.1.The Damped Two-dimensional Equations. As stated in the previous section, we are interested in the global regularity of solutions of (1) and (2) of the form (4) subject to periodic boundary conditions in the xandyvariables (period one), and with spatial domain (5). First note that, from incompressibility and periodicity, =(ux+vy) satises the mean-zero condition (11)Z Q (x;t)dx= 0:4 WILLIAM CHEN AND ALEJANDRO SARRIA Then imposing the ansatz (4) on the damped Boussinesq system (2), it is easy to check that the vertical component of the velocity eld, z , and the scalar temperature, z, satisfy the vertical component of (2)i) and equation (2)ii) if andsolve the nonlocal 2D system (12)( t+u0
r = 2 +I(t);x2Q; t> 0; t+u0
r= ; x2Q; t> 0 withu0= (u;v),>0 a real parameter, and (13) I(t) = 2Z Q 2(x;t)dxZ Q(x;t)dx: For0, (12)-(13) reduces to (14)( t+u0
r = 2 +I(t);x2Q; t> 0; I(t) = 2R Q 2(x;t)dx; which is just the associated 2D equation obtained from the vertical component of the damped 3D Euler system (1). For simplicity we will refer to equations (7) and (14), and the system (12)-(13), as the undamped Euler equation ,the damped Euler equation , and the damped Boussinesq system , respectively. Before summarizing the main results of the paper, we dene some notation that will be helpful in dierentiating among solutions of the various equations under consideration. 2.2.Notation. For > 0, we denote by ( B ;) and E the solution of the damped Boussinesq sys- tem (12)-(13) and the damped Euler equation (14), respectively. The undamped ( = 0) counterparts of ( B ;) and E will be denoted by dropping the subscript, i.e., ( B;) and respectively E, with the latter given (along characteristics) by formula (8). Other notation will be introduced in later sections in a similar manner. Lastly, by Cwe mean a generic positive constant that may change in value from line to line. 2.3.Summary of Results. For a smooth initial condition 0satisfying the conditions of Theorem 1.1, we determine in Theorem 2.1 below \how much" damping is required for the solution E of the damped Euler equation (14) to persist globally in time, or alternatively, for the nite-time blowup of the solution Eof the undamped Euler equation (7) to be suppressed. Theorem 2.1. Consider the damped Euler equation (14) with smooth mean-zero initial data 0and periodic boundary conditions. Suppose 0satises the conditions of Theorem 1.1, so the solution Eof the undamped Euler equation (7)blows up at a nite time TE. Then for 1=TE, the solution E of the damped Euler equation (14) exists globally in time. More particularly, for = 1=TE, E converges to a non-trivial steady state as t!+1, whereas, for > 1=TE, convergence is to the trivial steady state. In contrast, if 0<  < 1=TE, then there exists a nite time TE > TEsuch that the maximum and minimum values of E diverge to positive and respectively negative innity as t%TE . More particularly, let =1=m 0form0the negative minimum of 0attained at a nite number of points in Q,DAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER AND BOUSSINESQ EQUATIONS 5 and settE () =1 ln1tE()fortE()the undamped Euler time variable (10). Then for0<< 1=TE, the nite blowup time is TE lim%tE (). Before summarizing our next result, we note that Gibbon and Ohkitani [10] established a regularity criterion of BKM [3] type whereby a solution Eof (7) blows up at a nite time TEif and only if (15)ZTE 0k E(x;s)k1ds= +1 fork
k1the supremum norm. Let ( u;p) be a solution of the damped Euler system (1) for uas in (4)i). Since the vertical component !=vxuyof the vorticity != curl usatises !t+u0
r!= ( E )!; we may refer to E as the vorticity stretching rate. In the spirit of (15), a similar BKM-type criterion can be established for a solution of the damped Euler equation (14) where the blowup time TE is dened as the smallest time at which ZTE  0k E (x;s)k1ds= +1: Additionally, the argument used to prove Theorem 1.1 in [21], together with Theorem 2.2 below, shows that the same regularity criterion1is true for a solution of the damped Boussi- nesq system (12)-(13). Part one of our next Theorem shows that for smooth 0satisfying the conditions of Theorem 1.1 and 00, the existence of a nite blowup time TEfor the undamped Euler equation (7) leads to nite-time blowup of the damped Boussinesq system (12)-(13) if the damping coecient satises 0 <  < 1=TEand there is at least one point a12Qsuch that 0(a1) =m0and0(a1) = 0. In particular, we nd that the time inte- gral of B diverges to negative innity fora=a1, in agreement with the aforementioned BKM-type criterion. The second part of Theorem 2.2 shows that nite-time blowup is not restricted to nonnegative 0, although the blowup mechanism we uncover for nonpositive 0 is of a dierent nature and the singularity is eectively one dimensional. Brie y, we show the existence of smooth 0attaining its negative minimum m0at innitely many points a02Q, and00 satisfying 0(a0)6= 0, such that for TB>0 the nite blowup time of the associated undamped Boussinesq system (i.e. (12)-(13) with = 0), if 0<< 1=TB, then the time integral of B diverges to positive innity fora6=a0. Theorem 2.2. Consider the damped Bousinesq system (12)-(13) with periodic boundary conditions. (1)Suppose 0satises the conditions of Theorem 1.1, so the solution Eof the undamped Euler equation (7)blows up at a nite time TE. Further, let 0<< 1=TE, so that by Theorem 2.1 the solution E of the damped Euler equation (14) blows up at a nite time TE . Assume 0attains its negative minimum m0at nitely many points ai2Q,1in, and00is smooth with 0(aj) = 0 for some 1jn. Then there exists a nite time TB , satisfying 0<TB <TE , such that J(aj;t) = exp Zt 0 B (X(aj;s);s)ds !+1 1With E replaced by B 6 WILLIAM CHEN AND ALEJANDRO SARRIA ast%TB forJ= det@X @a the Jacobian of the 2D ow-map (see (16)). (2)There exist smooth mean-zero initial data 0attaining its negative minimum m0at innitely many points a02Q, and smooth 00with0(a0)6= 0, such that for TB>0the nite blowup time of the associated undamped Boussinesq system, if 0<< 1=TBanda6=a0, then J(a;t) = exp Zt 0 B (X(a;s);s)ds !0 ast%TB =1 ln 1TB
>TB. 3.Solution along characteristics In this section we derive the representation formula (33)-(34) for the general solution ( B (x;t);(x;t)) (along characteristics) of the damped Boussinesq system (12)-(13). Our approach is a natural generalization to higher dimensions of the arguments in [20, 21], and is somewhat more direct than the argument used in [6] to derive (8)-(9). For (a;t)(a1;a2;t),a2Q, and u0= (u;v), dene the 2D ow-map a!X(a;t) as the solution of the IVP (16)dX dt=u0(X(a;t);t); X(a;0) =a: Integrating equation (12)ii) along the ow-map yields (17) (X(a;t);t) =0(a) exp Zt 0 B (X(a;s);s)ds : Note that (18) (X(a;t);t) =0(a)J(a;t) forJ(a;t) = det@X @a the Jacobian determinant of Xsatisfying (19) J(a;t) = exp Zt 0 B (X(a;s);s)ds : Formula (18) follows directly from (17) and the fact that the Jacobian satises (20) Jt(a;t) =J(a;t) B (X(a;t);t); J (a;0)1: Next we use the above formulas to derive a second-order linear ODE for =J1. From (12)i), (18), and (20), it follows that @ @t( B (X(a;t);t)) =0(a)J(a;t)Jt(a;t) J(a;t)2 +Jt(a;t) J(a;t)+I(t): (21) Then dierentiating (20) with respect to t, and using (20) and (21) on the resulting expression leads to Jtt= 2J2 t J20JJt JI(t) J: (22) Setting=J1in (22) yields, after some rearranging, (23) tt(a;t) +t(a;t)I(t)(a;t) =0(a);DAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER AND BOUSSINESQ EQUATIONS 7 a second-order linear ODE parametrized by a2Q. Fixa2Qand setf(t) =(a;t). Then by (20) and (23), fsolves the IVP (24) f00(t) +f0(t)I(t)f(t) =0; f (0) = 1; f0(0) = 0(a) for0d dt. To nd a general solution of (24) we follow a standard variation of parameters argument. Consider the associated homogeneous equation (25) f00 h(t) +f0 h(t)I(t)fh(t) = 0: Let1(t) and2(t) be two linearly independent solutions of (25) satisfying 1(0) =0 2(0) = 1 and0 1(0) =2(0) = 0. Then by reduction of order, the general solution of (25) takes the formfh(t) =1(t)(c1(a) +c2(a)(t)) for (t) =Zt 0es 2 1(s)ds: Now consider a particular solution fp(t) =v1(a;t)1(t) +v2(a;t)2(t) of (24) for 2(t) = (t)1(t) andv1andv2to be determined. After some standard computations, we can write the general solution of (24) as (26) (a;t) =1(t) [ (a;t)0(a)(t)] for (a;t) = 1 + 0(a)(t) (27) and (t) =Zt 0es(s)1(s)ds(t)Zt 0es1(s)ds =Zt 0Zt s1(s) 2 1(z)e(sz)dzds:(28) The Jacobian is now obtained from (26) and =J1as (29) J(a;t) =1 1(t) (a;t)0(a)(t): To nd1note that for xed a2Qandc2Z2, the IVP (30) Z0(t) =u0(Z(t);t); Z(0) =a+c has a unique solution as long as u0= (u;v) stays smooth. Then by periodicity of u0and (16), Z1(t) =X(a;t) +candZ2(t) =X(a+c;t) both solve (30) with the same initial condition. Thus X(a;t) +c=X(a+c;t), which implies that (31)Z QJ(a;t)da1: Integrating (29) now yields (32) 1(t) =Z Qda (a;t)0(a)(t): Fori2Z+, set Ki(a;t) = 0(a) ( (a;t)0(a)(t))i; Ki(t) =Z Q 0(a) ( (a;t)0(a)(t))ida8 WILLIAM CHEN AND ALEJANDRO SARRIA and Li(a;t) =0(a) ( (a;t)0(a)(t))i; Li(t) =Z Q0(a) ( (a;t)0(a)(t))ida: Then using (18), (20), (29) and (32) we obtain, after a lengthy but straight-forward compu- tation, the solution formula B (X(a;t);t) =0 (t) K1(a;t)K2(t) 1(t) + L1(a;t)L2(t) 1(t)Zt 0es1(s)ds (33) for(t) satisfying (34) 0 (t) =et 2 1(t); (0) = 0: Moreover, the Jacobian (29) becomes (35) J(a;t) =1 (a;t)0(a)(t)Z Qda (a;t)0(a)(t)1 : Since (18) implies that if 00, then0 for as long as the solution is dened, setting 00 in (33)-(34) yields, after some rearranging, the general solution of the damped Euler equation (14) as E (X(a;t);t) =0 (t) (t)( 1 1 + 0(a)(t)Z Qda 1 + 0(a)(t)1Z Qda (1 + 0(a)(t))2)(36) for (37) 0 (t) =etZ Qda 1 + 0(a)(t)2 ; (0) = 0: 4.Proof of the Main Theorems The blowup result in Theorem 1.1 for the solution Eof the undamped Euler equation (7) is established by estimating blowup rates for the integral terms in (8) under the assumption that the smooth initial data 0behaves quadratically near the points where its minimum is attained [6]. Since we are interested in how damping can arrest this blowup, we consider the same class of initial data. In particular, this means that the blowup rates derived in [6] for the integral terms also hold here; however, and for convenience of the reader, we outline how to obtain these estimates in the proof of Theorem 2.1 below. Proof of Theorem 2.1 .Suppose the mean-zero initial data 0(a) is smooth and attains its negative minimum m0at a nite number of locations a02Q. Then the spatial term (38)1 1 + 0(a)(t) in (36) diverges to positive innity when a=a0asapproaches =1 m0;DAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER AND BOUSSINESQ EQUATIONS 9 and remains nite and positive for all 0 anda6=a0. Suppose 0has nonzero second-order partials near a0, so that locally, m0+1 22jaa0j2 0(a)m0+1 21jaa0j2 for 0jaa0jr,r>0 small, and 1> 2>0 the eigenvalues of the Hessian matrix of 0ata0.2This implies that (39) C1ln 1 +1r2 2 Z Dda + 0(a)m0C2ln 1 +2r2 2 for >0 small,Ci=2 i>0,i= 1;2, andDthe disk centered at a0of radiusr. Setting =1 +m0in (39), it follows that (40)Z Qda 1 + 0(a)Cln() for>0 small. By a similar argument, (41)Z Qda (1 + 0(a))2C : Next we need the corresponding behavior of the exponential term (42) et= exp tE () in (36)-(37) as %. In (42) we have introduced the notation t=tE to dierentiate the time variable in (36)-(37) from that in (33)-(34). For  > 0, we will refer to tE andtB as the damped Euler time variable and respectively the damped Boussinesq time variable . Since 0satises the conditions of Theorem 1.1, the limit TElim!tE(), for (43) tE() =Z 0Z Qda 1 + 0(a)2 d the associated undamped Euler time variable (10), is positive and nite, and represents the blowup time for Ein Theorem 1.1. That TEis nite follows from (40) and (43) which imply, for>0 small, the asymptotic relation (44) TEtE()(ln()1)2 whose right-hand side vanishes as %. Since (37) implies that the damped Euler time variable satises (45) tE () =1 ln1tE(); it follows that the behavior of the exponential (42) is determined by the limit (46) TE lim %tE (); which in turn depends on whether 0 << 1=TEor1=TE. 2If1=2=, then near a0, 0m0+1 2jaa0j2and the blowup rates (40)- (41) still hold.10 WILLIAM CHEN AND ALEJANDRO SARRIA Case 1 - Finite-time blowup for 0<< 1=TE. For 0<  < 1=TE, the argument of the logarithm in (45) satises 0 <1tE()1 for all 0. This implies that the limits lim %tE () =TE ; lim %exp tE () = exp TE  are both positive and nite. Using estimates (40) and (41) on (36), it follows that for a=a0 the spatial term dominates and E diverges to negative innity, E (X(a0;t);t)C () ln2()!1 ast%TE . If instead a6=a0, the second term in the bracket of (36) now dominates and the blowup is to positive innity, E (X(a;t);t)C () ln3()!+1: From (45) note that the blowup time TE approaches the undamped blowup time TEas the damping coecient vanishes. Further, for  >0 and0, we have that tE ()tE() with equality only at = 0. ThusTE >TE. Case 2 - Convergence to a nontrivial steady state for = 1=TE. For= 1=TE, (47) lim %tE () =TElim %1tE() TE= +1 and the exponential (42) vanishes, lim %exp tE () = 0: To determine how fast we use (44) and (45) to obtain (48) exp tE () =1 TE(TEtE())C()(ln()1)2 for>0 small. Using (40)-(41) and (47)-(48) on (36), we see that E converges to a mean-zero nontrivial steady state as t!+1, lim t!+1 E (X(a;t);t) =( C; a=a0; 0; a6=a0: Case 3 - Convergence to the trivial steady state for >1=TE. For > 1=TE, (45) implies the existence of 0 <  1< such thattE()%1 < TEas %1. ThentE ()!+1as%1, and lim %1exp tE () = 0: Since the spatial and integral terms in (36) stay positive and nite for 0 1, it follows that E (x;t)!0 ast!+1for all x2Q. DAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER AND BOUSSINESQ EQUATIONS 11 Before proving Theorem 2.2 we establish the following Lemma. Lemma 4.1. Let0(x)0and set=1=m 0. If0(t)< for allt2[0;T), 0<T+1, then 0< 1(t)<+1on. Proof. Suppose0(a)0 and 0(t)<for allt2[0;T), 0<T+1. The latter and (27) imply that ( a;t) = 1 + 0(a)(t)>0 for all ( a;t)2Q. Since1(t) satises 1(0)>0, there exists, by continuity, a positive t12 such that 0 <  1(t)<+1for all t2[0;t1). Note that 1(t) cannot diverge to positive innity at t1. Indeed, suppose (49) lim t%t11(t) = +1: Since0(a)0, (a;t)>0 for all ( a;t)2Q[0;t1], and(t)0 on [0;t1) (by (28)ii)), (a;t)0(a)(t) (a;t)>0 for all ( a;t)2Q[0;t1), and thus (50) 1 1(t)Z Qda (a;t)1 >0 fort2[0;t1). From (49) and (50) it follows that lim t%t1Z Qda (a;t)= +1; contradicting ( a;t)>0 for all ( a;t)2Q. Now suppose there exists t22 such that 1(t) vanishes as t%t2, namely, (51) lim t%t2Z Qda (a;t)0(a)(t)= 0: Since0(a)0 and 0< 1(t)<+1on [0;t2),(t)0 is continuous on [0 ;t2). Then boundedness of on [0;t2] and ( a;t)0(a)(t)>0 for all ( a;t)2Q[0;t2) imply that, for (51) to hold, (t)!1 ast%t2. From this and (28)i) it follows that lim t%t2 (t)Zt 0es1(s)ds = +1; or since 01<+1on [0;t2],(t)!+1ast%t2, contradicting 0<on .  Proof of Theorem 2.2 .Suppose 0(x) satises the conditions in Theorem 1.1 and 0(x) 0 for all x2Q. Further, and without loss of generality, assume there is one location a12Q such that0(a1) = 0 and 0(a1) =m0. Set=1=m 0. Then Lemma 4.1 and formulas (28) and (35) imply that, as long as 0 <, (52) J(a;t)1 1 + 0(a)(t)0(a)(t)Z Qda 1 + 0(a)(t)1 for all a2Q. Setting a=a1on (52) and using (40), it follows that J(a1;t)1 1 +m0(t)Z Qda 1 + 0(a)(t)1 C () ln()12 WILLIAM CHEN AND ALEJANDRO SARRIA for>0 small, and therefore J(a1;t)!+1 as%. Next, for 0 <  < 1=TEandTE>0 the nite blowup time of the undamped Euler equation (7), we prove the existence of a nite TB >0 such that t%TB as%. From (32), (34) and Lemma 4.1, (53) etdtZ Qda 1 + 0(a)2 d for 0<. Denote the damped Boussinesq time variable t=tB (). Then integrating (53) between 0 and tB yields the upper-bound (54) tB ()1 ln1tE()=tE () fortB in terms of the damped Euler time variable tE , which in turn depends on the undamped Euler time variable tEin (10) and the damping coecient  > 0. Since 0satises the conditions of Theorem 1.1, the limit TElim%tE() is positive and nite. Suppose 0<  < 1=TE. Then Theorem 2.1 implies that TE lim%tE () is also positive and nite. Thus letting %in (54), we see that the blowup time TB >0 is nite and satises TB lim %tB ()TE : This nishes the proof of the rst part of the Theorem. For the second part we adapt an argument used in [5, 21] to construct blowup and respectively global innite-energy solutions of the 2D Euler and inviscid Boussinesq equations. Let0(x) =sin2(2x). Then we look for a solution of (23) satisfying (x;0)1 and t(x;0) = 0(x). Suppose 1(t) and2(t) solve (55) 00 1+0 1I(t)1= 0; 00 2+0 2I(t)2= 1 with1(0) =0 1(0) = 1 and 2(0) = 0,0 2(0)6= 0. Then (56) (x;t) =1(t) +0(x)2(t) satises tt+tI(t)=0;  (x;0) = 1: Note that0 2(0)6= 0 must be such that 0(x) =t(x;0) = 1 +0(x)0 2(0) has mean zero over Q, as required by (11). Now, since Jsatises (31) it follows that 1 =Z Qdx 1sin2(2x)2; which yields the relation (57) 2=11 1: Using (57) we see that 0(x) = cos(4x), which satises the mean-zero condition (11). Next, using (57) on (55)ii) to eliminate the nonlocal term I(t) in (55)i) gives, after some simplication, d dt0 1 1 +0 1 1=1 2:DAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER AND BOUSSINESQ EQUATIONS 13 Dividing both sides by 1, dierentiating the resulting equation, and then setting N(t) = 0 1=1now leads to (58)d dt N01 2N2+N =N2: SinceN(0) = 1 and N0(0) =1 2, we integrate (58) and use a standard Gronwall-type argument to obtain (59) z01 2etz2; z (0) = 1 forz(t) =etN(t). Set (60) TB =1 ln (12) for 0<< 1=2. Note that TB is positive and nite. Then solving (59) for t2[0;TB ) gives z(t)2 et(12); or sincez(t) =etN(t) =etd dt(ln1(t)), (61) 1(t)42 (et(12))2: From (61) it follows that 1!+1ast%TB . Then by (56) and (57), 1 J(x;t)cos2(2x)1(t) +sin2(2x) 1(t); which implies that J(x;t)!0 ast%TB forx2Bf (x;y)2Qjx =2f1=4;3=4gg. Note thatBare precisely the points where 0(x) = cos(4x) does not equal its negative minimum. Lastly, by setting = 0 in (58) it is easy to see that the Jacobian of the associated undamped Boussinesq system vanishes as t%2, which agrees with TB !2 in (60) as !0. Thus we may write 0 << 1=2 as 0<< 1=TBforTB= 2 the blowup time for the solution of the undamped Boussinesq system with the same initial data.  Note that the blowup result in part 2 of Theorem 2.2 is eectively one dimensional in the sense that the initial data depends only on one of the two coordinate variables. Currently we do not know if this particular blowup is suppressed for 1=TB, or if a solution of the damped Boussinesq system (12)-(13) persist globally in time for 1=TEwhen the initial data satises the conditions of part one of Theorem 2.2 andis such that the associated undamped solution blows up in nite time. 5.Acknowledgments This work was partially supported by the Div III & P Research Funding Committee at Williams College.14 WILLIAM CHEN AND ALEJANDRO SARRIA References 1. D. Adhikari, C. Cao and J. Wu, Global regularity results for the 2D Boussinesq equations with vertical dissipation, J. Dierential Equations 251 (2011), 1637-1655. 2. D. Adhikari, C. Cao, J. Wu and X. Xu, Small global solutions to the damped two-dimensional Boussi- nesq equations, J. Dierential Equations 256 (2014), 3594-3613. 3. J.T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys. 94(1984), 61-66. 4. D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math. 203 (2006), 497-513. 5. S. Childress, G.R. Ierley, E.A. Spiegel and W.R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form, J. Fluid Mech. 203 (1989), 1-22. 6. P. Constantin, The Euler equations and nonlocal conservative Riccati equations, Int. Math. Res. Not. 9 (2000), 455-465. 7. P. Constantin, Singular, weak, and absent: Solutions of the Euler equations, Physica D 237 (2008), 1926-1931. 8. C.L. Feerman, Existence and smoothness of the Navier-Stokes equation , the millennium prize prob- lems, Clay Math. Inst., Cambridge, MA (2006), 57-67. 9. J.D. Gibbon, A.S. Fokas, C.R. Doering, Dynamically stretched vortices as solutions of the 3D Navier- Stokes equations, Physica D (1999), 497-510. 10. J.D. Gibbon, K. Ohkitani, Singularity formation in a class of stretched solutions of the equations for ideal magneto-hydrodynamics. Nonlinearity 14 (5) (2001), 1239-1264. 11. A.E. Gill, Atmosphere-Ocean Dynamics , Academic Press (London) (1982). 12. T.Y. Hou, Blowup or no blow-up? Unied computational and analytic approach to 3D incompressible Euler and NavierStokes equations, Acta Numerica (2009), 277-346. 13. T.Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations. Discrete Contin. Dyn. Syst. 12 (2005), 1-12. 14. T. Y. Hou and C. Li, Dynamic stability of the 3D axi-symmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math., 61 (5) (2008), 661-697. 15. M.J. Lai, R.H. Pan, K. Zhao, Initial boundary value problem for 2D viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760. 16. A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean , Courant Lecture Notes in Mathematics, vol. 9. AMS/CIMS, Providence (2003). 17. A.J. Majda and A.L. Bertozzi, Vorticity and incompressible ow , Cambridge Texts in Applied Math- ematics, Cambridge, 2002. 18. K. Ohkitani and J.D. Gibbon, Numerical study of singularity formation in a class of Euler and Navier- Stokes ows. Physics of Fluids 12 (2000), 3181-3194. 19. J. Pedlosky, Geophysical Fluid Dynamics , Springer, New York (1987). 20. A. Sarria, Regularity of stagnation-point form solutions of the two-dimensional incompressible Euler equations, Dierential and Integral Equations, 28 (3-4) (2015), 239-254. 21. A. Sarria and J. Wu, Blow-up in stagnation-point form solutions of the inviscid 2d Boussinesq equa- tions, J Dier Equations, 259 (2015), 3559-3576. 22. J.T. Stuart, Singularities in three-dimensional compressible Euler ows with vorticity, Theoret. Com- put. Fluid Dyn. 10 (1998), 385-391. 23. W. Yang, Q. Jiu and J. Wu, Global well-posedness for a class of 2D Boussinesq systems with fractional dissipation, J. Dierential Equations 257 (2014), 4188-4213. Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267 E-mail address :wyc1@williams.edu Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267 E-mail address :Alejandro.Sarria@williams.edu

2016-05-29

We revisit a family of infinite-energy solutions of the 3D incompressible Euler equations proposed by Gibbon et al. [9] and shown to blowup in finite time by Constantin [6]. By adding a damping term to the momentum equation we examine how the damping coefficient can arrest this blowup. Further, we show that similar infinite-energy solutions of the inviscid 3D Boussinesq system with damping can develop a singularity in finite time as long as the damping effects are insufficient to arrest the (undamped) 3D Euler blowup in the associated damped 3D Euler system.

Damped Infinite Energy Solutions of the 3D Euler and Boussinesq Equations

1605.08965v3

arXiv:1504.06042v1 [cond-mat.mes-hall] 23 Apr 2015Magnetizationdamping innoncollinearspinvalveswithant iferromagnetic interlayer couplings Takahiro Chiba1, Gerrit E. W. Bauer1,2,3, and Saburo Takahashi1 1Institute for Materials Research, Tohoku University, Send ai, Miyagi 980-8577, Japan 2WPI-AIMR, Tohoku University, Sendai, Miyagi 980-8577, Jap an and 3Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands (Dated: October 29, 2018) We study the magnetic damping in the simplest of synthetic an tiferromagnets, i.e. antiferromagnetically exchange-coupled spin valves in which applied magnetic fiel ds tune the magnetic configuration to become noncollinear. We formulate the dynamic exchange of spin cur rents in a noncollinear texture based on the spin- diffusiontheorywithquantum mechanicalboundaryconditionsa ttheferrromagnet|normal-metal interfacesand derive the Landau-Lifshitz-Gilbert equations coupled by t he static interlayer non-local and the dynamic ex- change interactions. We predict non-collinearity-induce d additional damping that can be sensitively modulated byanapplied magnetic field. The theoretical results compar e favorablywithpublished experiments. I. INTRODUCTION Antiferromagnets (AFMs) boast many of the functionali- tiesofferromagnets(FM)thatareusefulinspintroniccirc uits anddevices: Anisotropicmagnetoresistance(AMR),1tunnel- ing anisotropicmagnetoresistance(TAMR),2current-induced spintransfertorque,3–8andspincurrenttransmission9–11have all been found in or with AFMs. This is of interest because AFMshaveadditionalfeaturespotentiallyattractivefora ppli- cations. InAFMsthetotalmagneticmomentis(almost)com- pletely compensated on an atomic length scale. The AFM order parameter is, hence, robust against perturbations su ch as external magnetic fields and do not generate stray fields themselveseither. AspintronictechnologybasedonAFM el- ementsisthereforeveryattractive.12,13Drawbacksarethedif- ficulty to controlAFMs by magnetic fields and much higher (THz)resonancefrequencies,14–16whicharedifficulttomatch with conventional electronic circuits. Man-made magnetic multilayers in which the layer magnetizations in the ground state isorderedin anantiparallelfashion,17i.e. so-calledsyn- thetic antiferromagnets,donot su ffer fromthis drawbackand have therefore been a fruitful laboratory to study and modu- late antiferromagnetic couplings and its consequences,18but also found applications as magnetic field sensors.19Trans- port in these multilayers including the giant magnetoresis - tance (GMR)20,21are now well understood in terms of spin and charge diffusive transport. Current-induced magnetiza- tionswitchinginF|N|Fspinvalvesandtunneljunctions,22has been a game-changer for devices such as magnetic random access memories(MRAM).23A keyparameterof magnetiza- tiondynamicsisthemagneticdamping;asmalldampinglow- ersthethresholdofcurrent-drivenmagnetizationswitchi ng,24 whereasalargedampingsuppresses“ringing”oftheswitche d magnetization.25 Magnetization dynamics in multilayers generates “spin pumping”, i.e. spin current injection from the ferromagnet into metallic contacts. It is associated with a loss of an- gular momentum and an additional interface-related magne- tization damping.26,27In spin valves, the additional damp- ing is suppressed when the two magnetizations precess in- phase, while it is enhanced for a phase di fference ofπ(out- of-phase).27–30This phenomenon is explained in terms of a “dynamic exchange interaction”, i.e. the mutual exchange o fnon-equilibriumspin currents,which shouldbe distinguis hed from(butcoexistswith)theoscillatingequilibriumexcha nge- coupling mediated by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. The equilibriumcoupling is suppresse d whenthespacerthicknessisthickerthantheelasticmean-f ree path,31,32while the dynamiccouplingise ffective onthe scale oftheusuallymuchlargerspin-flipdi ffusionlength. Antiparallel spin valves provide a unique opportunity to study and control the dynamic exchange interaction between ferromagnets through a metallic interlayer for tunable mag - netic configurations.33,34An originallyantiparallel configura- tionisforcedbyrelativelyweakexternalmagneticfieldsin toa non-collinearconfigurationwith a ferromagneticcomponen t. Ferromagneticresonance(FMR)andBrillouinlightscatter ing (BLS) are two useful experimentalmethodsto investigateth e nature and magnitude of exchange interactions and magnetic damping in multilayers.35Both methods observe two reso- nances, i.e. acoustic (A) and optical (O) modes, which are characterizedbytheirfrequenciesandlinewidths.36,37 Timopheev etal.observedaneffectoftheinterlayerRKKY coupling on the FMR and found the linewidth to be a ffected by the dynamic exchange coupling in spin valves with one layerfixed by the exchange-biasof an inert AFM substrate.38 They measured the FMR spectrum of the free layer by tun- ing the interlayer coupling (thickness) and reported a broa d- ening of the linewidth by the dynamic exchange interaction. Taniguchi et al.addressed theoretically the enhancement of the Gilbert damping constant due to spin pumping in non- collinear F|N|F trilayer systems, in which one of the magne- tizations is excited by FMR while the other is o ff-resonant, butadoptaroleasspinsink.39Thedynamicsofcoupledspin valvesinwhichbothlayermagnetizationsarefreetomoveha s beencomputedby oneof us29and bySkarsvåg et al.33,49but only for collinear (parallel and antiparallel) configurati ons. Current-induced high-frequency oscillations without app lied magnetic field in ferromagnetically coupled spin valves has beenpredicted.40 Inthepresentpaper,wemodelthemagnetizationdynamics of the simplest of synthetic antiferromagnets, i.e. the ant i- ferromagnetically exchange-coupled spin valve in which th e (in-plane) ground state magnetizations are for certain spa cer thicknesses ordered in an antiparallel fashion by the RKKY interlayercoupling.41We focusonthecoupledmagnetization2 modes in symmetric spin valves in which in contrast to pre- vious studies, both magnetizations are free to move. We in- clude static magnetic fields in the film plane that deform the antiparallelconfigurationintoacantedone. Microwaveswi th longitudinal and transverse polarizations with respect to an externalmagneticfieldthenexciteAandOresonancemodes, respectively.31,42–46We develop the theory for magnetization dynamics and damping based on the Landau-Lifshitz-Gilbert equationwithmutualpumpingofspincurrentsandspintrans - fer torques based on the spin di ffusion model with quantum mechanical boundary conditions.27,47,48We confirm28,49that the additional damping of O modes is larger than that of the A modes. We report that a noncollinear magnetization con- figurationinducesadditionaldampingtorquesthat to the be st of ourknowledgehavenotbeen discussedin magneticmulti- layers before.50The external magnetic field strongly a ffects the dynamics by modulating the phase of the dynamic ex- change interaction. We compute FMR linewidths as a func- tion of applied magnetic fields and find good agreement with experimental FMR spectra on spin valves.31,32The dynam- ics of magnetic multilayers as measured by ac spin trans- fer torque excitation30reveals a relative broadening of the O modes linewidths that is well reproduced by our spin valve model. In Sec. II we present our model for noncollinear spin valves based on spin-di ffusiontheory with quantum mechan- ical boundary conditions. In Sec. III, we consider the mag- netization dynamics in antiferromagnetically coupled non - collinear spin valves as shown in Fig. 1(b). We derive the linearized magnetization dynamics, resonance frequencie s, and lifetimes of the acoustic and optical resonance modes in Sec. IV. We discuss the role of dynamicspin torqueson non- collinear magnetization configurations in relation to exte rnal magnetic field dependence of the linewidth. In Sec. V, we compare the calculated microwave absorption and linewidth with published experiments. We summarize the results and endwiththeconclusionsinSec. VI. II. SPINDIFFUSIONTRANSPORTMODEL We consider F1|N|F2 spin valves as shown in Fig. 1(a), in whichthemagnetizations MjoftheferromagnetsF j(j=1,2) are coupled by a antiparallel interlayer exchange interact ion and tilted towards the direction of an external magnetic fiel d. Applied microwaves with transverse polarizations with re- spect to an external magnetic field cause dynamics and, via spinpumping,spincurrentsandaccumulationsinthenormal - metal (NM) spacer. The longitudinal component of the spin accumulation diffuses into and generates spin accumulations inF thatwe showtobesmall later,butdisregardinitially. L et usdenotethepumpedspincurrent JP j,whileJB jisthediffusion (back-flow)spin currentdensity inducedby a spin accumula-NM zy x F1 F2 0 dN J1PJ2P J2BJ1Bµsθθ (c) Acoustic mode (d) Optical mode (a) m1 m2 H *OUFSMBZFS
DPVQMJOH (b) (b) H hx hy FIG.1. (a) Sketch of the sample withinterlayer exchange-co uplings illustrating the spin pumping and backflow currents. (b) Mag netic resonance in an antiferromagnetically exchange-coupled s pin valve with a normal-metal (NM) film sandwiched by two ferromagnets (F1,F2)subjecttoamicrowave magneticfield h. Themagnetization vectors (m1,m2) are tilted by an angle θin a static in-plane mag- neticfield Happliedalongthe y-axis. Thevectors mandnrepresent thesum anddifference ofthe twolayer magnetizations, respectively. (c) and (d): Precession-phase relations for the acoustic an d optical modes. tionµsjin NM,bothat theinterfaceF j, with27,51 JP j=Gr emj×/planckover2pi1∂tmj, (1a) JB j=Gr e/bracketleftBig/parenleftBig mj·µsj/parenrightBig mj−µsj/bracketrightBig , (1b) wheremj=Mj//vextendsingle/vextendsingle/vextendsingleMj/vextendsingle/vextendsingle/vextendsingleis the unit vector along the magnetic moment of F j(j=1,2). The spin current througha FM |NM interface is governed by the complex spin-mixing conduc- tance (per unit area of the interface) G↑↓=Gr+iGi.27The real component Grparameterized one vector component of the transverse spin-currentspumped and absorbed by the fer - romagnets. The imaginary part Gican be interpreted as an effective exchange field between magnetization and spin ac- cumulation, which in the absence of spin-orbit interaction is usuallymuchsmallerthantherealpart,forconductingaswe ll asinsultingmagnets.523 Thediffusionspin-currentdensityin NMreads Js,z(z)=−σ 2e∂zµs(z), (2) whereσ=ρ−1is the electrical conductivity and µs(z)= Ae−z/λ+Bez/λthe spin accumulationvectorthat is a solution ofthespindiffusionequation∂2 zµs=µs/λ2,whereλ=√Dτsf is the spin-diffusion length, Dthe diffusion constant, and τsf the spin-flip relaxation time. The vectors AandBare de-termined by the boundary conditions at the F1 |NM (z=0) and F2|NM (z=dN) interfaces: Js,z(0)=JP 1+JB 1≡Js1and Js,z(dN)=−JP 2−JB 2≡−Js2. Theresultingspin accumulation inN reads µs(z)=2eλρ sinh/parenleftBigdN λ/parenrightBig/bracketleftBigg Js1cosh/parenleftBiggz−dN λ/parenrightBigg +Js2cosh/parenleftbiggz λ/parenrightbigg/bracketrightBigg ,(3) withinterfacespin currents Js1=ηS 1−η2δJP 1+η2/parenleftBig m2·δJP 1/parenrightBig 1−η2(m1·m2)2m1×(m1×m2), (4a) Js2=−ηS 1−η2δJP 2+η2/parenleftBig m1·δJP 2/parenrightBig 1−η2(m1·m2)2m2×(m2×m1). (4b) Here δJP 1=JP 1+ηm1×(m1×JP 2), (5a) δJP 2=JP 2+ηm2×(m2×JP 1), (5b) S=sinh(dN/λ)/grandη=gr/[sinh(dN/λ)+grcosh(dN/λ)] are the efficiency of the back flow spin currents, and gr= 2λρGris dimensionless. The first terms in Eqs. (4a) and (4b) represent the mutual pumping of spin currents while the sec- ondtermsmaybeinterpretedasa spincurrentinducedbythe noncollinear magnetization configuration, including the b ack flowfromthe NMinterlayer. III. MAGNETIZATIONDYNAMICSWITH DYNAMIC SPINTORQUES We consider the magnetic resonance in the non-collinear spin valve shown in Fig. 1. The magnetization dynamics are describedbytheLandau-Lifshitz-Gilbert(LLG)equation, ∂tm1=−γm1×Heff1+α0m1×∂tm1+τ1,(6a) ∂tm2=−γm2×Heff2+α0m2×∂tm2−τ2.(6b) The first term in Eqs. (6a) and (6b) represents the torque in- ducedbytheeffectivemagneticfield Heff1(2)=H+h(t)−4πMsm1(2)zˆz+Jex MsdFm2(1),(7) which consists of an in-plane applied magnetic field H, a microwave field h(t), and the demagnetization field −4πMsm1(2)zˆzwith saturation magnetization Ms. The inter- layer exchange field is Jex/(MsdF)m2(1)with areal density of theinterlayerexchangeenergy Jex<0(forantiferromagnetic- coupling) and F layer thickness dF. The second term is the Gilbert dampingtorque that governsthe relaxationcharact er- ized byα0itowards an equilibrium direction. The third term,τ1(2)=γ/planckover2pi1/(2eMsdF)Js1(2), is the spin-transfertorque induced by the absorption of the transverse spin currents of Eqs. (4a ) and (4b), andγandα0are the gyromagnetic ratio and the Gilbert dampingconstant of the isolated ferromagneticfilm s, respectively. SometechnicaldetailsofthecoupledLLGequ a- tionsarediscussedinAppendixA.Introducingthetotalmag - netizationdirection m=(m1+m2)/2andthedifferencevector n=(m1−m2)/2,theLLG equationscanbewritten ∂tm=−γm×(H+h) +2πγMs(mzm+nzn)׈z +α0(m×∂tm+n×∂tn)+τm, (8a) ∂tn=−γn×/parenleftBigg H+h+Jex MsdFm/parenrightBigg +2πγMs(nzm+mzn)׈z +α0(m×∂tn+n×∂tm)+τn,(8b) where the spin-transfer torques τm=(τ1+τ2)/2 andτn= (τ1−τ2)/2become τm/αm=m×∂tm+n×∂tn +2ηm·(n×∂tn) 1−ηCm+2ηn·(m×∂tm) 1+ηCn,(9a) τn/αn=m×∂tn+n×∂tm −2ηm·(n×∂tm) 1+ηCm−2ηn·(m×∂tn) 1−ηCn,(9b) andC=m2−n2,while αm=α1gr 1+grcoth(dN/2λ), (10a) αn=α1gr 1+grtanh(dN/2λ), (10b) withα1=γ/planckover2pi12/(4e2λρMsdF).4 IV. CALCULATIONANDRESULTS We consider the magnetization dynamics excited by lin- early polarized microwaves with a frequency ωand in-plane magnetic field h(t)=(hx,hy,0)eiωtthat is much smaller than the saturation magnetization. For small angle magnetizati on precession the total magnetization and di fference vector may be separated into a static equilibrium and a dynamic com- ponent as m=m0+δmandn=n0+δn, respectively, wherem0=(0,sinθ,0),n0=(cosθ,0,0),C=−cos2θ, ands=−ˆzsin2θ. The equilibrium (zero torque) conditions m0×H=0 andn0×(H+Jex/(MsdF)m0)=0 lead to the relation sinθ=H/Hs, (11) whereHs=−Jex/(MsdF)=|Jex|/(MsdF) is the saturation field. TheLLGequationsread ∂tδm=−γδm×H−γm0×h +2πγMs(δmzm0+δnzn0)׈z +α0(m0×∂tδm+n0×∂tδn)+δτm,(12a) ∂tδn=−γδn×H−γn0×h +2πγMs(δnzm0+δmzn0)׈z −γHs(m0×δn−n0×δm) +α0(m0×∂tδn+n0×∂tδm)+δτn,(12b) withlinearizedspin-transfertorques δτm/αm=m0×∂tδm+n0×∂tδn −ηsin2θ 1+ηcos2θ∂tδnzm0+ηsin2θ 1−ηcos2θ∂tδmzn0,(13a) δτn/αn=m0×∂tδn+n0×∂tδm +ηsin2θ 1−ηcos2θδmzm0−ηsin2θ 1+ηcos2θδnzn0,(13b) To leading order in the small precessing components δmand δn,theLLG equationsinfrequencyspace become δmx=γhxγ(Hs+4πMs)+iω/parenleftBig α0+αm(1+η) 1−ηcos2θ/parenrightBig ω2−ω2 A−i∆Aωsin2θ,(14a) δny=−γhxγ(Hs+4πMs)+iω/parenleftBig α0+αn(1−η) 1−ηcos2θ/parenrightBig ω2−ω2 A−i∆Aωcosθsinθ, (14b) δmz=−γhxiω ω2−ω2 A−i∆Aωsinθ, (14c) δnx=−γhy4πγMs+iω/parenleftBig α0+αn(1−η) 1+ηcos2θ/parenrightBig ω2−ω2 O−i∆Oωcosθsinθ,(15a) δmy=γhy4πγMs+iω/parenleftBig α0+αm(1+η) 1−ηcos2θ/parenrightBig ω2−ω2 O−i∆Oωcos2θ, (15b) δnz=γhyiω ω2−ω2 O−i∆Oωcosθ. (15c)The A modes (δmx,δny,δmz) are excited by hx, while the O modes (δnx,δmy,δnz) couple to hy. The poles inδm(ω)and δn(ω)define the resonance frequencies and linewidths that do not depend on the magnetic field since we disregard anisotropyandexchange-bias. A. Acoustic andOpticalmodes An antiferromagnetically exchange-coupled spin valves generallyhave non-collinearmagnetizationconfiguration sby thepresenceofexternalmagneticfields. For H<Hs(0<θ< π/2),theacousticmode: ωA=γH/radicalbig 1+(4πMs/Hs), (16) ∆A=α0γ/parenleftBig Hs+4πMs+Hssin2θ/parenrightBig +αmγ(Hs+4πMs)+αA(θ)γHs,(17) andthe opticalmode: ωO=γ/radicalBig (4πMs/Hs)(H2s−H2), (18) ∆O=α0γ/parenleftBig 4πMs+Hscos2θ/parenrightBig +αn4πγMs+αO(θ)γHs, (19) where αA(θ)=α1grsin2θ 1+grtanh(dN/2λ)+2grsin2θ/sinh(dN/λ),(20) αO(θ)=α1grcos2θ 1+grtanh(dN/2λ)+2grcos2θ/sinh(dN/λ).(21) The additional broadeningin ∆Ais proportional toαmand αAwhile that in∆Oscales withαnandαOin. Figure 2 (a) showsαmandαnas a function of spacer layer thickness, in- dicating thatαnis always larger than αm, and thatαn(αm) strongly increases (decreases) with decreasing N layer thi ck- ness, especially for dN<λand large gr. Figure 2(b) shows thedependenceofαAandαOonthetiltedangleθfordifferent valuesof dN. Asθincreases,αAincreasesfrom0to αmwhile αOdecreasesαmto 0. The additional damping can be ex- plained by the dynamic exchange. When two magnetizations in spin valves precess in-phase, each magnet receives a spin current that compensates the pumped one, thereby reducing the interface damping. When the magnetizations precess out of phase, theπphase difference between both spin currents means that the moduli have to be added, thereby enhancing thedamping. When the magnetizations are tilted by an angle θas sketched in Fig. 1, we predict an additional damping torque expressedbythesecondtermsofEqs.(4a)and(4b). Figure3 shows the ratiosαA(θ)/αmandαO(θ)/αnas a function ofθ andgrfor different values ofλ/dN, thereby emphasizing the additional damping in the presence of noncollinear magneti - zations. In Fig. 3(a,b)with λ/dN=1, i.e. for a spin-di ffusive interlayer,the additionaldampingof both A- and O-modesis significant in a large region of parameter space. On the other5 0.1 1 10 100λ/d N012345αm/α1, αn/α1/α1 /α12 1 0.5 0 30 60 90 θ (degree)0.00.10.20.30.40.5αA, αOαO αAλ/d N=1 2 4 10(b)(a) αnαmgr=5 FIG.2. (a)αm(dashed line)andαn(solidline) as a functionof λ/dN for different values of the dimensionless mixing conductance gr. (b) αAC(dashed line)andαOP(solidline)as afunction oftiltangle θfor gr=5and different values ofλ/dN. hand, in Fig. 3(c,d) with λ/dN=10, i.e. for an almost spin- ballistic interlayer, the additional damping is more impor tant fortheA-mode,whiletheO-modeisa ffectedonlyclosetothe collinear magnetization. In the latter case the intrinsic d amp- ingα0dominates,however. B. In-phaseandOut-of-phasemodes Whentheappliedmagneticfieldislargerthanthesaturation field (H>Hs), both magnetizations point in the ˆydirection, and theδm(A) andδn(O) modes morph into in-phase and 180◦out-of-phase (antiphase) oscillations of δm1andδm2, respectively. The resonance frequency53and linewidth of the in-phasemodefor H>Hs(θ=π/2)are ωA=γ/radicalbig H(H+4πMs), (22) ∆A=2(α0+αm)γ(H+2πMs), (23) whilethoseoftheout-of-phasemodeare ωO=γ/radicalbig (H−Hs)(H−Hs+4πMs),(24) ∆O=2(α0+αn)γ(H−Hs+2πMs).(25)              B C D E HS HS θ (degree) θ (degree)     θ (degree)    θ (degree)   HS    HS0.1 0.3 0.5 0.7 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.55 0.65 0.75 0.85 0.95 0.01 0.02 0.03 0.04 0.05 FIG. 3. (a,c)αA(θ)/αmand (b,d)αO(θ)/αnas a function ofθandgr for different values ofλ/dN. (a,b) withλ/dN=1, (c,d) withλ/dN= 10 Figure 4(a) shows the calculated resonance frequencies of the A andO modesas a functionof an appliedmagneticfield Hwhile 4(b) displays the linewidths for α1/α0=1, which is representative for ferromagnetic metals, such as permal - loy (Py) with an intrinsic magnetic damping of the order of α0=0.01andacomparableadditionaldamping α1duetospin pumping. A value gr=4/5 corresponds toλ=20/200nm, ρ=10/2.5µΩcmfor N=Ru/Cu,54,55Gr=2/1×1015Ω−1m−2 for the N|Co(Py) interface56,57, anddF=1nm, for example. Thecolorsinthefigurerepresentdi fferentrelativelayerthick- nessesdN/λ. The linewidth of the A mode in Fig. 4(b) in- creases with increasing H, while that of the O mode starts to decrease until a minimum at the saturation field H=Hs. Figure 4(c) shows the linewidths for α1/α0=10, which de- scribes ferromagnetic materials with low intrinsic dampin g, such as Heusler alloys58and magnetic garnets.59In this case, the linewidth of the O modeis much largerthanthat of the A mode,especiallyforsmall dN/λ. In the limit of dN/λ→0 is easily established experimen- tally. The expressions of the linewidth in Eqs. (17) and (19) arethengreatlysimplifiedto ∆A=γ(Hs+4πMs+Hssin2θ)α0 and∆O=γ/parenleftBig 4πMs+Hscos2θ/parenrightBig α0+(4πγMs)grα1,and∆A≪ ∆Owhengrα1≫α0. The additional damping, Eq. (10b) re- duces toαm→0 andαn→2[γ/planckover2pi1/(4πMsdF)(h/e2)Gr] when themagnetizationsarecollinearandintheballisticspint rans- port limit.27In contrast to the acoustic mode, the dynamic exchange interaction enhances damping of the optical mode. ∆O≫∆AhasbeenobservedinPy |Ru|Pytrilayerspinvalves32 andCo|Cu multilayers30, consistentwiththepresentresults. For spin valves with ferromagnetic metals, the interface backflow spin-current [(1b)] reads JB j=(Gr/e)/bracketleftBig ξF/parenleftBig mj·µsj/parenrightBig mj−µsj/bracketrightBig ,whereξF=6 012ω/(4 πγ Ms)ωAωO 02468∆/(4 πMsα0) dN/λ= 0.3 1(a) (b) (c) 0 0.5 1 1.5 H/H s050dN/λ=0.01 0.1 0.3 1∆/(4 πγ Msα0)0.01γ A mode O mode 0.1 FIG. 4. (a) Resonance frequencies of the A and O modes as a func - tion of magnetic field for Hs/(4πMs)=1. (b), (c) Linewidths of the A (dashed line) and the O (solid line) modes for Hs/(4πMs)=1, gr=5, and different values of dN/λ. (b)α1/α0=1 and (c) α1/α0=10. 1−(G/2Gr)(1−p2)(1−ηF) (0≤ξF≤1),Gis the N|F interface conductance per unit area, and pthe conduc- tance spin polarization.51Here the spin diffusion efficiency is 1 ηF=1+σF GλFtanh(dF/λF) cosh(dF/λF), (26) whereσF,λF, anddFare the conductivity, the spin-flip dif- fusion length, and the layer thickness of the ferromagnets, respectively. For the material parameters of a typical fer- romagnet with dF=1 nm, the resistivity ρF=10µΩcm, G=2Gr=1015Ω−1m−2,λF=10nm,and p=0.7,ξF=0.95, whichjustifiesdisregardingthiscontributionfromtheout set. V. COMPARISONWITH EXPERIMENTS FMRexperimentsyieldtheresonantabsorptionspectraofa microwavefield ofa ferromagnet. Themicrowaveabsorptiont dP/dH ϕ=90 o 20 o 0o×5 ×5ϕHh(t)(a) 5 H (kOe) 22 4 6 0 0 0.4 0.8 1.2 H/H s Co(3)|Ru(1)|Co(3) Co(3)|Ru(1)|Co(3) Experiment (Ref. 11) Calculation 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 ∆/(4 πγ Msα₀)H/H s A mode O mode 0 2000 4000 6000 Field (Oe) (b) Experiment (Ref. 10): [Co(1)|Cu(1)] 10 Co(1) Calculation: Co(1)|Cu(1)|Co(1) FIG.5. (a)Derivativeofthemicrowaveabsorptionspectrum dP/dH at frequencyω/(2π)=9.22 GHz for different anglesϕbetween the microwavefieldandtheexternalmagneticfieldfor Hs/(4πMs)=0.5, ω/(4πγMs)=0.35,dN/λ=0.1,dF/λ=0.3α0=α1=0.02, and gr=4. The experimental data have been adopted from Ref. 31. (b) Computed linewidths of the A and O modes of a Co |Cu|Co spin valve (dashed line) compared with experiments on a Co |Cu multi- layer (solidline).30 powerP=2/angbracketlefth(t)·∂tm(t)/angbracketrightbecomesinourmodel P=1 4γ2Ms(Hs+4πMs)∆A (ω−ωA)2+(∆A/2)2h2 xsin2θ +1 4γ2Ms(4πMs)∆O (ω−ωO)2+(∆O/2)2h2 ycos2θ. (27) Pdepends sensitively on the character of the resonance, the polarization of the microwave, and the strength of the ap- plied magnetic field. In Figure 5(a) we plot the normalized derivative of the microwave absorption spectra dP/(P0dH) at different anglesϕbetween the microwave field h(t) and the external magnetic field H, where P0=γMsh2and h(t)=h(sinϕ,cosϕ,0)eiωt. Here we use the experimen- tal values Hs=5kOe, 4πMs=10kOe,dN=1nm, dF=3nm, and microwave frequency ω/(2π)=9.22GHz as found for a symmetric Co |Ru|Co trilayer.31λ=20nm for7 Ru,α0=α1=0.02, andgr=4 is adopted (correspond- ing toGr=2×1015Ω−1m−2).Whenh(t) is perpendicularto H(ϕ=90◦), only the A mode is excited by the transverse (δmx,δmz) component. When h(t) is parallel to H(ϕ=0◦), the O mode couples to the microwave field by the longitudi- nalδmycomponent. For intermediate angles ( ϕ=20◦), both modes are excited at resonance. We observe that the opti- cal mode signal is broader than the acoustic one, as calcu- lated. The theoretical resonance linewidths of the A and O modes as well as the absorption power as a function of mi- crowave polarization reproduce the experimental results f or Co(3.2nm)|Ru(0.95nm)|Co(3.2nm)well.31 Figure 5(b) shows the calculated linewidths of A and O modes as a function of an applied magnetic field for a Co(1nm)|Cu(1nm)|Co(1nm) spin valve. The experimental valuesλ=200nm andρ=2.5µΩcm for Cu,α0=0.01 and 4πMs=15kOe for Co, and gr=5 (corresponding to Gr=1015Ω−1m−2) for the interface have been adopted.57 We partially reproduce the experimental data for magnetic multilayers; for the weak-field broadenings of the observed linewidthsagreementisevenquantitative. Theremainingd is- crepanciesintheappliedmagneticfielddependencemightre - flect exchange-dipolar49and/or multilayer30spin waves be- yondourspinvalvemodelinthe macrospinapproximation. VI. CONCLUSIONS In summary, we modelled the magnetization dynamics in antiferromagnetically exchange-coupled spin valves as a model for synthetic antiferromagnets. We derivethe Landau - Lifshitz-Gilbert equations for the coupled magnetization s in- cluding the spin transfer torques by spin pumping based on the spin diffusion model with quantum mechanical boundary conditions. We obtain analytic expressionsfor the linewid ths of magnetic resonance modes for magnetizations canted byapplied magneticfields and achieve goodagreementwith ex- periments. We findthatthelinewidthsstronglydependonthe type of resonance mode (acoustic and optical) as well as the strength of magnetic fields. The magnetic resonance spectra reveal complex magnetization dynamics far beyond a simple precessionevenin the linear responseregime. Our calculat ed results compare favorably with experiments, thereby provi ng theimportanceofdynamicspincurrentsinthesedevices. Ou r model calculation paves the way for the theoretical design o f syntheticAFMmaterialthatisexpectedtoplayaroleinnext - generationspin-baseddata-storageandinformationtechn olo- gies. VII. ACKNOWLEDGMENTS TheauthorsthanksK.Tanaka,T.Moriyama,T.Ono,T.Ya- mamoto, T. Seki, and K. Takanashi for valuable discussions and collaborations. This work was supported by Grants-in- AidforScientificResearch(GrantNos. 22540346,25247056, 25220910,268063)fromtheJSPS,FOM(StichtingvoorFun- damenteel Onderzoek der Materie), the ICC-IMR, EU-FET Grant InSpin 612759, and DFG Priority Programme 1538 “Spin-CaloricTransport”(BA 2954 /2). AppendixA: CoupledLandau-Lifshitz-Gilbertequationsin noncollinearspinvalves Both magnets and interfaces in our NM |F|NM spin valves are assumed to be identical with saturation magnetization Ms andGrthe real part of the spin-mixing conductance per unit area (vanishing imaginary part). When both magnetizations are allowed to precess as sketched in Fig. 1 (a), the LLG equationsexpandedtoincludeadditionalspin-pumpandspi n- transfertorquesread ∂mi ∂t=−γmi×Heffi+α0imi×∂mi ∂t +αSPi/bracketleftBigg mi×∂mi ∂t−ηmj×∂mj ∂t+η/parenleftBigg mi·mj×∂mj ∂t/parenrightBigg mi/bracketrightBigg +αnc SPi(ϕ)mi×/parenleftBig mi×mj/parenrightBig , (A1) αnc SPi(ϕ)=αSPiη2 1−η2(mi·mj)2/bracketleftBigg mj·mi×∂mi ∂t+η/parenleftBigg mi·mj×∂mj ∂t/parenrightBigg (mj·mi)/bracketrightBigg , (A2) whereγandα0iare the gyromagnetic ratio and the Gilbert damping constant of the isolated ferromagnetic films labele d byiand thickness dFi. Asymmetric spin valves due to the thickness differencedFisuppress the cancellation of mutual spin-pumpinA-mode,whichmaybeadvantagetodetectboth modesintheexperiment. Thee ffectivemagneticfield Heffi=Hi+h(t)+Hdii(t)+Hexj(t) (A3)consistsoftheZeemanfield Hi,amicrowavefield h(t),thedy- namic demagnetization field Hdii(t), and interlayer exchange fieldHexj(t). 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B 67(R), 140404(2003)

2015-04-23

We study the magnetic damping in the simplest of synthetic antiferromagnets, i.e. antiferromagnetically exchange-coupled spin valves in which applied magnetic fields tune the magnetic configuration to become noncollinear. We formulate the dynamic exchange of spin currents in a noncollinear texture based on the spindiffusion theory with quantum mechanical boundary conditions at the ferrromagnet|normal-metal interfaces and derive the Landau-Lifshitz-Gilbert equations coupled by the static interlayer non-local and the dynamic exchange interactions. We predict non-collinearity-induced additional damping that can be sensitively modulated by an applied magnetic field. The theoretical results compare favorably with published experiments.

Magnetization damping in noncollinear spin valves with antiferromagnetic interlayer couplings

1504.06042v1

arXiv:2109.12071v1 [cond-mat.mtrl-sci] 24 Sep 2021Damping in yttrium iron garnet film with an interface Ravinder Kumar,1,2,∗B. Samantaray,1Shubhankar Das,3Kishori Lal,1D. Samal,2,4and Z. Hossain1,5,† 1Department of Physics, Indian Institute of Technology, Kan pur 208016, India. 2Institute of Physics, Bhubaneswar 751005, India. 3Institute of Physics, Johannes Gutenberg-University Main z, 55099 Mainz, Germany. 4Homi Bhabha National Institute, Anushakti Nagar, Mumbai 40 0085, India. 5Institute of Low Temperature and Structure Research, 50-42 2 Wroclaw, Poland. (Dated: September 27, 2021) We report strong damping enhancement in a 200 nm thick yttriu m iron garnet (YIG) film due to spin inhomogeneity at the interface. The growth-induced thin interfacial gadolinium iron garnet (GdIG) layer antiferromagnetically (AFM) exchange couple s with the rest of the YIG layer. The out-of-plane angular variation of ferromagnetic resonanc e (FMR) linewidth ∆ Hreflects a large in- homogeneous distribution of effective magnetization ∆4 πMeffdue to the presence of an exchange springlike moments arrangement in YIG. We probe the spin inh omogeneity at the YIG-GdIG inter- face by performing an in-plane angular variation of resonan ce fieldHr, leading to a unidirectional feature. The large extrinsic ∆4 πMeffcontribution, apart from the inherent intrinsic Gilbert co n- tribution, manifests enhanced precessional damping in YIG film. I. INTRODUCTION The viability of spintronics demands novel magnetic materials and YIG is a potential candidate as it ex- hibits ultra-low precessional damping, α∼3×10−5[1]. The magnetic properties of YIG thin films epitaxially grown on top of Gd 3Ga5O12(GGG) vary significantly due to growth tuning[ 2,3], film thickness[ 4], heavy met- als substitution[ 5–7] and coupling with thin metallic layers[8–10]. The growth processes may also induce the formation of a thin interfacial-GdIG layer at the YIG- GGG interface[ 11–13]. The YIG-GdIG heterostructure derived out of monolithic YIG film growth on GGG ex- hibits interestingphenomenasuchasall-insulatingequiv- alent of a synthetic antiferromagnet[ 12] and hysteresis loop inversion governed by positive exchange-bias [ 13]. The radio frequency magnetization dynamics on YIG- GdIG heterostructure still remains unexplored and need a detailed FMR study. The relaxation of magnetic excitation towards equi- librium is governed by intrinsic and extrinsic mecha- nisms, leading to a finite ∆ H[14,15]. The former mech- anism dictates Gilbert type relaxation, a consequence of direct energy transfer to the lattice governed by both spin-orbit coupling and exchange interaction in all mag- netic materials[ 14,15]. Whereas, the latter mechanism is a non-Gilbert-type relaxation, divided mainly into two categories[ 14,15]- (i) the magnetic inhomogeneity in- duced broadening: inhomogeneity in the internal static magnetic field, and the crystallographic axis orienta- tion; (ii) two-magnon scattering: the energy dissipates in the spin subsystem by virtue of magnon scattering with nonzero wave vector, k∝negationslash= 0, where, the uniform reso- nance mode couples with the degeneratespin waves. The ∗ravindk@iitk.ac.in †zakir@iitk.ac.inangular variation of Hrprovides information about the presence of different magnetic anisotropies[ 4,6]. Most attention has been paid towards the angular dependence ofHr[4,6], whereas, the angular variation of the ∆ H is sparsely investigated. The studies involving angular dependence of ∆ Hmay help to probe different contribu- tions to the precessional damping. Inthispaper, theeffectsofintrinsicandextrinsicrelax- ation mechanisms on precessionaldamping of YIG film is studied extensively using FMR technique. An enhanced value of α∼1.2×10−3is realized, which is almost two orders of magnitude higher than what is usually seen in YIGthinfilms, ∼6×10−5[1,2]. Theout-of-planeangular variation of ∆ Hshows an unusual behaviour where spin inhomogeneity at the interface plays significant role in defining the ∆ Hbroadening and enhanced α. In-plane angular variation showing a unidirectional feature, de- mandstheincorporationofanexchangeanisotropytothe free energy density, evidence of the presence of an AFM exchangecoupling at the YIG-GdIG interface. The AFM exchange coupling leads to a Bloch domain-wall-like spi- ralmoments arrangementin YIG and givesrise to a large ∆4πMeff. This extrinsic ∆4 πMeffcontribution due to spin inhomogeneity at the interface adds up to the inher- ent Gilbert contribution, which may lead to a significant enhancement in precessional damping. II. SAMPLE AND MEASUREMENT SETUPS We deposit a ∼200 nm thick epitaxial YIG film on GGG(111)-substrate by employing a KrF Excimer laser (Lambda Physik COMPex Pro, λ= 248 nm) of 20 ns pulse width. A solid state synthesized Y3Fe5O12target is ablated using an areal energy of 2.12 J.cm−2with a repetition frequency of 10 Hz. The GGG(111) substrate is placed 50 mm away from the target. The film is grown at 800oC temperature and in-situpost annealed at the same temperature for 60 minutes in pure oxygen envi-ronment. The θ−2θX-ray diffraction pattern shows epi- taxial growth with trails of Laue oscillations (Fig. 3(a) of ref[3]). FMR measurements are performed using a Bruker EMX EPR spectrometer and a broadband copla- nar waveguide (CPW) setup. The former technique uses a cavity mode frequency f≈9.60 GHz, and enables us to perform FMR spectra for various θHandφHangu- lar variations. The latter technique enables us to mea- sure frequency dependent FMR spectra. We define the configurations Hparallel ( θH= 90o) and perpendicular (θH= 0o) to the film plane for rf frequency and angu- lar dependent measurements. The resultant spectra are obtained as the derivative of microwave absorption w.r.t. the applied field H. III. RESULTS AND DISCUSSION A. Broadband FMR Fig.1(a) shows typical broadband FMR spectra in a frequency frange of 1.5 to 13 GHz for 200 nm thick YIG film at temperature T= 300 K and θH= 90o. The mode appearing at a lower field value is the main mode, whereas the one at higher field value represents surface mode. We discuss all these features in detail in the succeeding subsection IIIB. We determine the res- onance field Hrand linewidth (peak-to-peak linewidth) ∆Hfrom the first derivative of the absorption spectra. Fig.1(b) shows the rf frequency dependence of Hrat θH= 90oand 0o. We use the Kittel equation for fitting the frequency vs. Hrdata from the resonance condi- tion expressed as[ 10],f=γ[Hr(Hr+4πMeff)]1/2/(2π) forθH= 90oandf=γ(Hr−4πMeff)/(2π) for θH= 0o. Where, γ=gµB/ℏis the gyromagnetic ratio, 4πMeff= 4πMS−Haniis the effective magnetization consisting of 4 πMSsaturation magnetization (calculated using M(H)) and Hanianisotropy field parametrizing cu- bic and out-of-plane uniaxial anisotropies. The fitting gives 4πMeff≈2000 Oe, which is used to calculate the Hani≈ −370 Oe. Fig.1(c) shows the frequency dependence of ∆ Hat θH= 90o. The intrinsic and extrinsic damping contri- butions are responsible for a finite width of the FMR signal. The intrinsic damping ∆ Hintarises due to the Gilbert damping of the precessing moments. Whereas, the extrinsic damping ∆ Hextexists due to different non- Gilbert-type relaxations such as inhomogeneity due to the distribution of magnetic anisotropy ∆ Hinhom, or two-magnon scattering (TMS) ∆ HTMS. The intrinsic Gilbert damping coefficient ( α) can be determined using the Landau-Liftshitz-Gilbert equation expressed as[ 10], ∆H= ∆Hin+ ∆Hinhom= (4πα/√ 3γ)f+ ∆Hinhom. Considering the above equation where ∆ Hobeys lin- earfdependence, the slope determines the value of α, and ∆Hinhomcorresponds to the intercept on the ver- tical axis. We observe a very weak non-linearity in the fdependence of ∆ H, which is believed to be due to thecontribution of TMS to the linewidth ∆ HTMS. The non- linearfdependence of ∆ Hin Fig.1(c) can be described in terms of TMS, assuming ∆ H= ∆Hin+ ∆Hinhom+ ∆HTMS. We put a factor of 1 /√ 3 to ∆Hdue to the peak-to-peak linewidth value extraction[ 14]. The TMS induces non-linear slope at low frequencies, whereas a saturation is expected at high frequencies. TMS is in- duced by scattering centers and surface defects in the sample. The defects with size comparable to the wave- length of spin waves are supposed to act as scattering centres. The TMS term at θH= 90ocan be expressed as[16]- ∆HTMS(ω) = Γsin−1/radicalBigg/radicalbig ω2+(ω0/2)2−ω0/2/radicalbig ω2+(ω0/2)2+ω0/2,(1) withω= 2πfandω0=γ4πMeff. The prefactor Γ defines the strength of TMS. The extracted values are as follows: α= 1.2×10−3, ∆H0= 13 Oe and Γ = 2 .5 Oe. The Gilbert damping for even very thin YIG film is extremely low, ∼6×10−5. Whereas, the value we achieved is higher than the reported in the literature for YIG thin films[ 2]. Also, the value of Γ is insignificant, implying negligible contribution to the damping. B. Cavity FMR Fig.2(a) shows typical T= 300 K cavity-FMR (f≈9.6 GHz) spectra for YIG film performed at dif- ferentθH. The FMR spectra exhibit some universal features: (i) Spin-Wave resonance (SWR) spectrum for θH= 0o; (ii) rotating the Haway from the θH= 0o, the SWR modes successively start diminishing, and at certain critical angle θc(falls in a range of 30 −35o; shaded region in Fig. 2(b)), all the modes vanish except a single mode (uniform FMR mode). Further rotation ofHforθH> θc, the SWR modes start re-emerging. We observe that the SWR mode appearing at the higher field side for θH> θc, represents an exchange-dominated non-propagating surface mode[ 17–19]. The above dis- cussed complexity in HrvsθHbehaviour has already been realized in some material systems[ 19], including a µ-thick YIG film[ 18]. The localized mode or surface spin-wave mode appears for H∝bardblbut not⊥to the film- plane[17–19]. WeassigntheSWR modesforthesequence n= 1,2,3,...., as it provides the best correspondence toHex∝n2, where, Hex=Hr(n)−Hr(0) defines ex- change field[ 20]. The exchange stiffness can be obtained by considering the modified Schreiber and Frait classical approach using the mode number n2dependence of res- onance field (inset Fig. 3(c))[ 20]. For a fixed frequency, the exchange field Hexof thickness modes is determined by subtracting the highest field resonance mode ( n= 1) from the higher modes ( n∝negationslash= 1). In modified Schreiber and Frait equation, the Hexshows direct dependency on the exchange stiffness D:µ0Hex=Dπ2 d2n2(wheredis 2/s40/s99/s41/s40/s98/s41/s40/s97/s41 FIG. 1. Room temperature frequency dependent FMR measureme nts. (a) Representative FMR derivative spectra for differen t frequencies at θH= 90o. (b) Resonance field vs. frequency data for θH= 90oandθH= 0oare represented using red and blue data points, respectively. The fitting to both the data a re shown using black lines. (c) Linewidth vs. frequency data at θH= 90o. The solid red circles represent experimental data, wherea s the solid black line represents ∆ Hfitting. Inhomogeneous (∆Hinhom), Gilbert (∆ Hα) and two-magnon scattering (∆ HTMS) contributions to ∆ Hare shown using dashed green, solid yellow and blue lines, respectively. the film thickness). The linear fit of data shown in the inset of Fig. 2(b) gives D= 3.15×10−17T.m2. The ex- change stiffness constant Acan be determined using the relationA=D MS/2. The calculated value is A= 2.05 pJ.m−1, which is comparable to the value calculated for YIG,A= 3.7 pJ.m−1[20]. YIG thin films with in-plane easy magnetization ex- hibit extrinsic uniaxial magnetic and intrinsic magne- tocrystallinecubic anisotropies[ 21]. The total free energy density for YIG(111) is given by[ 21,22]: F=−HMS/bracketleftbigg sinθHsinθMcos(φH−φM) +cosθHcosθM/bracketrightbigg +2πM2 Scos2θM−Kucos2θM +K1 12/parenleftbigg7sin4θM−8sin2θM+4− 4√ 2sin3θMcosθMcos3φM/parenrightbigg +K2 108 −24sin6θM+45sin4θM−24sin2θM+4 −2√ 2sin3θMcosθM/parenleftbig 5sin2θM−2/parenrightbig cos3φM +sin6θMcos6φM (2) The Eq. 2consists of the following different energy terms; the first term is the Zeeman energy, the second term is the demagnetization energy, the third term is the out-of-plane uniaxial magnetocrystalline anisotropy energyKu, and the last two terms are the first and sec- ond order cubic magnetocrystalline anisotropy energies (K1andK2), respectively. The total free energy density equation is minimized by taking partial derivatives w.r.t. toθMandφMtoobtaintheequilibriumorientationofthe magnetization vector M(H), i.e.,∂F/∂θ M=∂F/∂φ M= 0. Theresonancefrequencyofuniformprecessionatequi- librium condition is expressed as[ 21,23,24]: ωres=γ MSsinθM/bracketleftBigg ∂2F ∂θ2 M∂2F ∂φ2 M−/parenleftbigg∂2F ∂θM∂φM/parenrightbigg2/bracketrightBigg1/2 (3) Mathematica is used to numerically solve the reso-nance condition described by Eq. 3for the energy den- sity given by Eq. 2. The solution for a fixed frequency is used to fit the angle dependent resonance data ( Hr vs.θH) shown in Fig. 2(b). The main mode data simulation is shown using a black line. The parame- ters obtained from the simulation are Ku=−1.45×104 erg.cm−3,K1= 1.50×103erg.cm−3, andK2= 0.13×103 erg.cm−3. The calculated uniaxial anisotropy field value isHu∼ −223 Oe. The ∆Hmanifests the spin dynamics and related re- laxation mechanisms in a magnetic system. The intrinsic contribution to ∆ Harises due to Gilbert term ∆ Hint≈ ∆Hα, whereas, the extrinsic contribution ∆ Hextconsists of line broadening due to ∆ Hinhomand ∆HTMS. The terms representing the precessional damping due to in- trinsic and extrinsic contributions can be expressed in different phenomenologicalforms. Figure 2(c) shows∆ H as a function of θH. TheθHvariation of ∆ Hshows distinct signatures due to different origins of magnetic damping. We consider both ∆ Hintand ∆Hextmag- netic damping contributions to the broadening of ∆ H, ∆H= ∆Hα+∆Hinhom+∆HTMS. The first term can be expressed as[ 14]- ∆Hα=α MS/bracketleftbigg∂2F ∂θ2 M+1 sin2θM∂2F ∂φ2 M/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂(2πf γ) ∂Hr/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1 .(4) The second term ∆ Hinhomhas a form[ 14]- ∆Hinhom=/vextendsingle/vextendsingle/vextendsingle/vextendsingledHr d4πMeff/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆4πMeff+/vextendsingle/vextendsingle/vextendsingle/vextendsingledHr dθH/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆θH.(5) Where, the dispersion of magnitude and direction of the 4πMeffare represented by ∆4 πMeffand ∆θH, re- spectively. The ∆ Hinhomcontribution arises due to a small spread of the sample parameters such as thickness, internal fields, or orientation of crystallites within the thin film. The third term ∆ HTMScan be written as[ 25]- 3/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s49/s50/s51/s32/s72 /s101/s120/s40/s107/s79/s101/s41 /s110/s50/s67/s114/s105/s116/s105/s99/s97/s108/s32/s97/s110/s103/s108/s101/s54/s53/s52/s51/s72 /s114/s32/s40/s32/s107/s79/s101/s32/s41/s50/s110/s32/s61/s32/s49 /s72/s32/s40/s32/s68/s101/s103/s114/s101/s101/s32/s41/s50 /s51 /s52 /s53 /s54/s68/s101/s114/s105/s118/s97/s116/s105/s118/s101/s32/s97/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41 /s72 /s32/s40/s107/s79/s101/s41/s48 /s49/s53 /s51/s48 /s52/s53 /s54/s48 /s55/s53 /s57/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48 /s32/s32 /s32/s69/s120/s112/s116/s46/s32/s68/s97/s116/s97 /s32 /s72 /s32 /s72 /s32 /s77 /s101/s102/s102 /s32 /s32 /s72 /s84/s77/s83/s72 /s32/s40/s79/s101/s41/s32 /s40/s100/s101/s103/s114/s101/s101 /s41/s40/s99/s41/s40/s98/s41/s40/s97/s41 FIG. 2. Room temperature out-of-plane angular θHdependence of FMR. (a) Derivative FMR spectra shown for diffe rentθH performed at ≈9.6 GHz. (b) θHvariation of uniform mode and SWR modes of resonance field Hr. Inset: Exchange field (Hex) vs mode number square ( n2). (c)θHvariation of the linewidth (∆ H), where, the experimental and simulated data are represented by solid yellow circles and black line, respect ively. The different contributions ∆ Hα, ∆4πMeff, ∆θHand ∆HTMS are represented by gray, purple, green and red lines, respec tively. ∆HTMS=/summationtext i=1Γout ifi(φH) µ0γΦsin−1/radicalbigg√ ω2+(ω0/2)2−ω0/2√ ω2+(ω0/2)2+ω0/2, Γout i= Γ0 iΦA(θ−π/4)dHr(θH) dω(θH)/slashbigg dHr(θH=0) dω(θH=0) (6) The prefactor Γout idefines the TMS strength and has aθHdependency in this case. The type and size of the defects responsible for TMS is difficult to characterize which makes it non-trivial to express the exact form of Γout i. Although, it mayhaveasimplified expressiongiven in Eq.6, where, Γ0 iis a constant; A(θ−π/4), a step function which makes sure that the TMS is deactivated forθH< π/4; anddHr(θH)/dω(θH), a normalization factor responsible for the θHdependence of the Γout i. In fig.2(c)the solid dark yellow circles and black solid line represent the experimental and simulated ∆ HvsθH data, respectively. We also plot contributions of different terms such as ∆ Hα(blue color line), ∆4 πMeff(purple colorline), ∆ θH(greencolorline) and ∆ HTMS(red color line). The fitting provides following extracted parame- ters,α= 1.3×10−3, ∆4πMeff= 58 Oe, ∆ θH= 0.29o and Γ0 i= 1.3 Oe. The precessional damping calculated fromthe∆ Hvs.θHcorroboratewiththevalueextracted from the frequency dependence of ∆ Hdata (shown in Fig.1(c));α= 1.2×10−3. The ∆ Hbroadening and the overwhelmingly enhanced precessional damping are thedirectconsequenceofcontributionsfromintrinsicand extrinsic damping. Usually, the Gilbert term and the inhomogeneity due to sample quality contribute to the broadening of ∆ Hand enhanced αin YIG thin films. If we interpret the ∆ HvsθHdata, it is clear that damping enhancement in YIG is arising from the extrinsic mag- netic inhomogeneity.The role of an interface in YIG coupled with metals or insulators leading to the increments in ∆ Handαhas been vastly explored. Wang et. al. [ 9] studied a variety of insulating spacers between YIG and Pt to probe the effect on spin pumping efficiency. Their results suggest the generation of magnetic excitations in the adjacent insulating layers due to the precessing magnetization in YIG at resonance. This happens either due to fluctu- ating correlated moments or antiferromagnetic ordering, via interfacial exchange coupling, leading to ∆ Hbroad- ening and enhanced precessional damping of the YIG[ 9]. Theimpurityrelaxationmechanismisalsoresponsiblefor ∆HbroadeningandenhancedmagneticdampinginYIG, but is prominent only at low temperatures[ 16]. Strong enhancement in magnetic damping of YIG capped with Pt has been observed by Sun et. al. [ 8]. They suggest ferromagneticorderingin an atomically thin Pt layerdue to proximity with YIG at the YIG-Pt interface, dynam- ically exchange couples to the spins in YIG[ 8]. In recent years, some research groups have reported the presence of a thin interfacial layer at the YIG-GGG interface[ 11– 13]. The 200 nm film we used in this study is of high quality with a trails of sharp Laue oscillations [see Fig 3(a) in ref.[ 3]]. Thus it is quite clear that the observed ∆Hbroadening and enhanced αis not a consequence of sample inhomogeneity. The formation of an interfacial GdIG layer at the YIG-GGG interface, which exchange couples with the YIG film may lead to ∆ Hbroadening and increased α. Considering the above experimental ev- idences leading to ∆ Hbroadening and enhanced Gilbert dampingdueto couplingwithmetals andinsulators[ 8,9], it is safe to assume that the interfacial GdIG layer at the interface AFM exchange couples with the YIG[ 11–13], and responsible for enhanced ∆ Handα. Fig.3shows in-plane φHangular variation of Hr. We 4/s32/s68/s97/s116/s97 /s32/s84/s111/s116/s97/s108 /s32/s69/s120/s99/s104/s97/s110/s103/s101/s72 /s114/s32/s40/s79/s101/s41/s50/s48/s48/s32/s110/s109/s40/s97/s41 /s40/s98/s41 /s49/s48/s48/s32/s110/s109 /s72/s32/s40/s68/s101/s103/s114/s101/s101/s41 FIG. 3. (a) In-plane angular φHvariation of Hr. The exper- imental data are represented by solid grey circles. Whereas , the simulated data for total and exchange (unidirectional) anisotropy are represented by black and red solid lines, re- spectively. (a) 200 nm thick YIG sample. (b) 100 nm thick YIG sample. simulate the in-plane HrvsφHangular variation using the free energy densities provided in ref. [ 26] and an additional term, −KEA.sinθM.cosφM, representing the exchange anisotropy ( KEA). Even though φHvaria- tion ofHrshown in Fig. 3(a) is not so appreciable as the film is 200 nm thick, a very weak unidirectional anisotropy trend is visible, suggesting an AFM exchange coupling between the interface and YIG. It has been shown that the large inhomogeneous 4 πMeffis a direct consequence of the AFM exchange coupling at the inter- face of LSMO and a growth induced interfacial layer[ 27]. The YIG thin film system due to the presence of a hard ferrimagnetic GdIG interfacial layer possesses AFM ex- change coupling[ 11–13]. A Bloch domain-wall-like spiral moments arrangement takes place due to the AFM ex- change coupling acrossthe interfacial GdIG and top bulk YIG layer[ 11–13]. An exchange springlike characteris- tic is found in YIG film due to the spiral arrangement of the magnetic moments [ 11–13]. The FMR measure- ment and the extracted value of ∆4 πMeffreflect inho- mogeneous distribution of 4 πMeffin YIG-GdIG bilayer system. The argument of Bloch domain-wall-like spiral arrangement of moments is conceivable, as this arrange- ment between the adjacent layers lowers the exchange interaction energy[ 27]. To further substantiate the pres-ence of an interfacial AFM exchange coupling leading to spin inhomogeneity at YIG-GdIG interface, we per- formed in-plane φHvariation of Hron a relatively thin YIG film ( ∼100 nm with growth conditions leading to the formation of a GdIG interfacial layer[ 13]). Fig.3(b) shows prominent feature of unidirectional anisotropydue to AFM exchange coupling in 100 nm thick film. It is evident that the interfacial layer exchange couples with the rest of the YIG film and leads to a unidirectional anisotropy. We observethatthe interfacialexchangecou- pling may cause ∆ Hbroadening and enhanced αdue to spin inhomogeneity at the YIG-GdIG interface, even in a 200 nm thick YIG film. IV. CONCLUSIONS The effects of spin inhomogeneity at the YIG and growth-induced GdIG interface on the magnetization dy- namics of a 200 nm thick YIG film is studied extensively using ferromagnetic resonance technique. The Gilbert damping is almost two orders of magnitude larger (∼1.2×10−3) than usually reported in YIG thin films. The out-of-plane angular dependence of ∆ Hshows an unusual behaviour which can only be justified after considering extrinsic mechanism in combination with the Gilbert contribution. The extracted parameters from the ∆HvsθHsimulation are, (i) α= 1.3×10−3from Gilbert term; (ii) ∆4 πMeff= 58 Oe and ∆ θH= 0.29o from the inhomogeneity in effective magnetization and anisotropy axes, respectively; (iii) Γ0 i= 1.3 Oe from TMS. The TMS strength Γ is not so appreciable, indicating high quality thin film with insignificant defect sites. The AFM exchange coupling between YIG and the interfacial GdIG layer causes exchange springlike behaviour of the magnetic moments in YIG, leading to a large ∆4 πMeff. The presence of large ∆4 πMeffimpels the quick dragging of the precessional motion towards equilibrium. A unidirectional behaviour is observed in the in-plane angular variation of resonance field due to the presence of an exchange anisotropy. This further reinforces the spin inhomogeneity at the YIG-GdIG interface due to the AFM exchange coupling. ACKNOWLEDGEMENTS We gratefully acknowledge the research support from IIT Kanpur and SERB, Government of India (Grant No. CRG/2018/000220). RK and DS acknowledge the financial support from Max-Planck partner group. ZH acknowledges financial support from Polish National Agency for Academic Exchange under Ulam Fellowship. 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Baber- schke, and Z. Frait, Phys. Rev. B 76, 104416 (2007) . [16] C. L. Jermain, S. V. Aradhya, N. D. Reynolds, R. A. Buhrman, J. T. Brangham, M. R. Page, P. C. Hammel, F. Y. Yang, and D. C. Ralph, Phys. Rev. B 95, 174411 (2017) . [17] C. Vittoria and J. H. Schelleng, Phys. Rev. B 16, 4020 (1977) . [18] J. T. Yu, R. A. Turk, and P. E. Wigen, Phys. Rev. B 11, 420 (1975) . [19] X. Liu, Y. Y. Zhou, and J. K. Furdyna, Phys. Rev. B 75, 195220 (2007) . [20] S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. Hillebrands, and A. Conca, Journal of Physics D: Applied Physics 48, 015001 (2014) . [21] S. Lee, S. Grudichak, J. Sklenar, C. C. Tsai, M. Jang, Q. Yang, H. Zhang, and J. B. Ketterson, Journal of Applied Physics 120, 033905 (2016) . [22] L.D.LandauandE.M.Lifshitz, Electrodynamics of Con- tinuous Media , 2nd ed. (Pergamon Press, Oxford, 1984). [23] H. Suhl, Phys. Rev. 97, 555 (1955) . [24] J. Smit and H. G. Beljers, Philips Res. Rep. 10, 113 (1955). 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2021-09-24

We report strong damping enhancement in a 200 nm thick yttrium iron garnet (YIG) film due to spin inhomogeneity at the interface. The growth-induced thin interfacial gadolinium iron garnet (GdIG) layer antiferromagnetically (AFM) exchange couples with the rest of the YIG layer. The out-of-plane angular variation of ferromagnetic resonance (FMR) linewidth $\Delta H$ reflects a large inhomogeneous distribution of effective magnetization $\Delta 4 \pi M_{eff}$ due to the presence of an exchange springlike moments arrangement in YIG. We probe the spin inhomogeneity at the YIG-GdIG interface by performing an in-plane angular variation of resonance field $H_{r}$, leading to a unidirectional feature. The large extrinsic $\Delta 4\pi M_{eff}$ contribution, apart from the inherent intrinsic Gilbert contribution, manifests enhanced precessional damping in YIG film.

Damping in yttrium iron garnet film with an interface

2109.12071v1

Atomistic spin dynamic method with both damping and moment of inertia eects included from rst principles S. Bhattacharjee, L. Nordstr om, and J. Fransson Department of Physics and Astronomy, Box 516, 75120, Uppsala University, Uppsala, Sweden (Dated: October 28, 2018) We consider spin dynamics for implementation in an atomistic framework and we address the feasibility of capturing processes in the femtosecond regime by inclusion of moment of inertia. In the spirit of an s-d-like interaction between the magnetization and electron spin, we derive a generalized equation of motion for the magnetization dynamics in the semi-classical limit, which is non-local in both space and time. Using this result we retain a generalized Landau-Lifshitz-Gilbert equation, also including the moment of inertia, and demonstrate how the exchange interaction, damping, and moment of inertia, all can be calculated from rst principles. PACS numbers: 72.25.Rb, 71.70.Ej, 75.78.-n In recent years there has been a huge increase in the in- terest in fast magnetization processes on a femto-second scale, which has been initialized by important develop- ments in experimental techniques [1{5], as well as po- tential technological applications [6]. From a theoretical side, the otherwise trustworthy spin dynamical (SD) sim- ulation method fails to treat this fast dynamics due to the short time and length scales involved. Attempts have been made to generalize the mesoscopic SD method to an atomistic SD, in which the dynamics of each individual atomic magnetic moment is treated [7, 8]. While this approach should in principle be well suited to simulate the fast dynamics observed in experiments, it has not yet reached full predictive power as it has inherited phe- nomenological parameters, e.g. Gilbert damping, from the mesoscopic SD. The Gilbert damping parameter is well established in the latter regime but it is not totally clear how it should be transferred to the atomic regime. In addition, very recently it was pointed out that the mo- ment of inertia, which typically is neglected, plays an im- portant role for fast processes [9]. In this Letter we derive the foundations for an atomistic SD where all the rele- vant parameters, such as the exchange coupling, Gilbert damping, and moment of inertia, can be calculated from rst principles electronic structure methods. Usually the spin dynamics is described by the phe- nomenological Landau-Lifshitz-Gilbert (LLG) equation [10, 11] which is composed of precessional and damping terms driving the dynamics to an equilibrium. By in- cluding the moment of inertia, we arrive at a generalized LLG equation _M=M( B+^G_M+^IM) (1) where ^Gand^Iare the Gilbert damping and the moment of inertia tensors, respectively. In this equation the eec- tive eld Bincludes both the external and internal elds, of which the latter includes the exchange coupling and anisotropy eects. Here, we will for convenience include the anisotropy arising from the classical dipole-dipole in- teraction responsible for the shape anisotropy as a partof the external eld. The damping term in the LLG equation usually consists of a single damping parame- ter, which essentially means that the time scales of the magnetization variables and the environmental variables arewell separated . This separation naturally brings a limitation to the LLG equation concerning its time scale which is restricting it to the mesoscopic regime. The addition of a moment of inertia term to the LLG equation can be justies as follows. A general process of a moment Munder the in uence of a eld Fis al- ways endowed with inertial eects at higher frequencies [12]. The eld Fand moment Mcan, for example, be stress and strain for mechanical relaxation, electric eld and electric dipole moment in the case of dielectric re- laxation, or magnetic eld and magnetic moment in the case of magnetic relaxation. In this Letter we focus on the latter case | the origin of the moment of inertia in SD. The moment of inertia leads to nutations of the mag- netic moments, see Fig. 1. Its wobbling variation of the azimuthal angle has a crucial role in fast SD, such as fast magnetization reversal processes. In the case of dielectric relaxation the inertial eects are quite thoroughly mentioned in the literature [13, 14], especially in the case of ferroelectric relaxors. Coey et al.[14] have proposed inertia corrected Debye's theory of dielectric relaxation and showed that by including inertial B precessionĜ dampingÎ nutation FIG. 1: The three contributions in Eq. (1), the bare preces- sion arising from the eective magnetic eld, and the super- imposed eects from the Gilbert damping and the moment of inertia, respectively.arXiv:1110.3387v2 [cond-mat.stat-mech] 1 Feb 20122 eects, the unphysical high frequency divergence of the absorption co-ecient is removed. Very recently Ciornei et al [9] have extended the LLG equation to include the inertial eects through a mag- netic retardation term in addition to precessional and damping terms. They considered a collection of uni- formly magnetized particles and treated the total angular momentum Las faster variable. They obtained Eq. (1) from a Fokker-Plank equation where the number den- sity of magnetized particles were calculated by integrat- ing a non-equilibrium distribution function over faster variables such that faster degrees of freedom appear as parameter in the calculation. The authors showed that at very short time scales the inertial eects become important as the precessional mo- tion of magnetic moment gets superimposed with nuta- tion loops due to inertial eects. It is pointed out that the existence of inertia driven magnetization dynamics open up a pathway for ultrafast magnetic switching [15] beyond the limitation [16] of the precessional switching. In practice, to perform atomistic spin dynamics simu- lations the knowledge of ^Gand^Iis necessary. There are recent proposals [17, 18] of how to calculate the Gilbert damping factor from rst principles in terms of Kubo- Greenwood like formulas. Here, we show that similar techniques may by employed to calculate the moment of inertia tensor ^I. Finally, we present a microscopical jus- tication of Eq. (1), considering a collective magnetiza- tion density interacting locally with electrons constitut- ing spin moments. Such a description would in principle be consistent with the study of magnetization dynam- ics where the exchange parameters are extracted from rst-principles electronic structure calculations, e.g den- sity functional theory (DFT) methods. We nd that in an atomistic limit Eq. (1) actually has to be general- ized slightly as both the damping and inertia tensors are naturally non-local in the same way as the exchange cou- pling included in the eective magnetic eld B. From our study it is clear that both the damping and the mo- ment of inertia eects naturally arise from the retarded exchange interaction. We begin by considering the magnetic energy E=M
B. Using that its time derivative is _E=M
_B+_M
B along with Eq. (1), we write _E=M
_B+1 _M
 ^G_M+^IM : (2) Relating the rate of change of the total energy to the HamiltonianH, through _E=hdH=dti, and expanding theHlinearly around its static magnetization M0, with M(t) =M0+(t), we can writeHH 0+(t)
rH0, whereH0=H(M0). Then the rate of change of the total energy equals _E= _
hrHito the rst order. Following Ref. [19] and assuming suciently slow dynamics such that(t0) =(t)_(t) +2(t)=2,=tt0, we canwrite the rate of change of the magnetic energy as _E= lim !!0_i[ij(!)j+i@!ij(!) _j@2 !ij(!)j=2]: (3) Here,ij(!) =R (i)()h[@iH0(t);@jH0(t0)]iei!dt0, =tt0, is the (generalized) exchange interaction tensor out of which the damping and moments of in- ertia can be extracted. Summation over repeated in- dices (i;j=x;y;z ) is assumed in Eq. (3). Equat- ing Eqs. (2) and (3) results in an internal contribu- tion to the eective eld about which the magnetiza- tion precesses Bint=lim!!0(!), the damping term ^G= lim!!0i@!(!) as well as the moment of inertia ^I= lim!!0@2 !(!)=2. For a simple order of magnitude estimate of the damp- ing and inertial coecients, ^Gand^I, respectively, we may assume for a state close to a ferromagnetic state that the spin resolved density of electron states (") corresponding to the static magnetization conguration H0is slowly varying with energy. At low temperatures we, then, nd ^G2 sp [h@iH0ih@jH0i]"="F; (4) in agreement with previous results [19]. Here, sp denotes the trace over spin 1/2 space. By the same token, the moment of inertia is estimated as ^I( =D) sp [h@iH0ih@jH0i]"="F; (5) where 2Dis the band width of density of electron states of the host material. Typically, for metallic systems the band width 2 D1|10 eV, which sets the time-scale of the inertial contribution to the femto second (1015 s) regime. It, therefore, denes magnetization dynamics on a time-scale that is one or more orders of magnitude shorter compared to e.g. the precessional dynamics of the magnetic moment. Next, we consider the physics leading to the LLG equa- tion given in Eq. (1). As there is hardly any microscopical derivation of the LLG equation in the literature, we in- clude here, for completeness the arguments that leads to the equation for the spin-dynamics from a quantum eld theory perspective. In the atomic limit the spin degrees of freedom are deeply intertwined with the electronic degrees of free- dom, and hence the main environmental coupling is the one to the electrons. In this study we are mainly con- cerned with a mean eld description of the electron structure, as in the spirit of the DFT. Then a natural and quite general description of the magnetic interac- tion due to electron-electron interactions on the atomic site around rwithin the material is captured by the s-d- like modelHint=R J(r;r0)M(r;t)
s(r0;t)drdr0, where J(r;r0) represents the interaction between the magneti- zation density Mand the electron spin s. From a DFT3 perspective the interaction parameter J(r;r0) is related to the eective spin dependent exchange-correlation func- tionalBxc[M(r0)](r). For generality we assume a fully relativistic treatment of the electrons, i.e. including the spin-orbit coupling. In this interaction the dichotomy of the electrons is displayed, they both form the magnetic moments and provide the interaction among them. Owing to the general non-equilibrium conditions in the system, we dene the action variable S=I CHintdt+SZ+SWZWN (6) on the Keldysh contour [20{22]. Here, the ac- tionSZ= H CR Bext(r;t)
M(r;t)dtdrrepresents the Zeeman coupling to the external eld Bext(r;t), whereas the Wess-Zumino-Witten-Novikov (WZWN) termSWZWN =RH CR1 0M(r;t;)
[@M(r;t;) @tM(r;t;)]ddtjM(r)j2drdescribes the Berry phase accumulated by the magnetization. In order to acquire an eective model for the magne- tization density M(r;t), we make a second order [23] ex- pansion of the partition function Z[M(r;t)]trTCeiS, and take the partial trace over the electronic degrees of freedom in the action variable. The eective action SM for the magnetization dynamics arising from the mag- netic interactions described in terms of Hint, can, thus, be written SM=I Z M(r;t)
D(r;r0;t;t0)
M(r0;t0)drdr0dtdt0; (7) whereD(r;r0;t;t0) =R J(r;r1)(i)hTs(r1;t)s(r2;t0)i J(r2;r0)dr1dr2is a dyadic which describes the electron mediated exchange interaction. Conversion of the Keldysh contour integrations into real time integrals on the interval ( 1;1) results in S=Z M(fast)(r;t)
[M(r;t)_M(r;t)]dtjM(r)j2dr +Z M(fast)(r;t)
Dr(r;r0;t;t0)
M(r0;t0)drdr0dtdt0 Z Bext(r;t)
M(fast)(r;t)dtdr; (8) withM(fast)(r;t) =Mu(r;t)Ml(r;t) and M(r;t) = [Mu(r;t) +Ml(r;t)]=2 which dene fast and slow vari- ables, respectively. Here, Mu(l)is the magnetization den- sity dened on the upper (lower) branch of the Keldysh contour. Notice that upon conversion into the real time domain, the contour ordered propagator Dis replaced by its retarded counterpart Dr. We obtain the equation of motion for the (slow) mag- netization variable M(r;t) in the classical limit by mini- mizing the action with respect to M(fast)(r;t), cross mul- tiplying by M(r;t) under the assumption that the totalmoment is kept constant. We, thus, nd _M(r;t) =M(r;t) Bext(r;t) +Z Dr(r;r0;t;t0)
M(r0;t0)dt0dr0 :(9) Eq. (9) provides a generalized description of the semi- classical magnetization dynamics compared to the LLG Eq. (1) in the sense that it is non-local in both time and space. The dynamics of the magnetization at some point rdepends not only on the magnetization locally at r, but also in a non-trivial way on the surrounding magne- tization. The coupling of the magnetization at dierent positions in space is mediated via the electrons in the host material. Moreover, the magnetization dynamics is, in general, a truly non-adiabatic process in which the information about the past is crucial. However, in order to make connection to the magne- tization dynamics as described by e.g. the LLG equa- tion as well as Eq. (1) above, we make the following consideration. Assuming that the magnetization dy- namics is slow compared to the electronic processes in- volved in the time-non-local eld D(r;r0;t;t0), we ex- pand the magnetization in time according to M(r0;t0) M(r0;t)_M(r0;t) +2M(r0;t)=2. Then for the inte- grand in Eq. (9), we get Dr(r;r0;t;t0)
M(r0;t0) = Dr(r;r0;t;t0)
[M(r0;t)_M(r0;t) +2 2M(r0;t)]:(10) Here, we observe that as the exchange coupling for the magnetization is non-local and mediated through D, this is also true for the damping (second term) and the inertia (third term). In order to obtain an equation of the exact same form as LLG in Eq. (1) we further have to assume that the magnetization is close to a uniform ferromag- netic state, then we can justify the approximations _M(r0;t)_M(r;t) and M(r0;t)M(r;t). When Bint=R D(r;r0;t;t0)
M(r0;t)dr0dt0= is included in the total eective magnetic eld B, the tensors of Eq. (1) ^Gand^Ican be identied with R D(r;r0;t;t0)dr0dt0 andR 2D(r;r0;t;t0)dr0dt0=2, respectively. From a rst principles model of the host materials we have, thus, de- rived the equation for the magnetization dynamics dis- cussed in Ref. 9, where it was considered from purely classical grounds. However it is clear that for a treatment of atomistic SD that allows for all kinds of magnetic or- ders, not only ferromagnetic, Eq. (1) is not sucient and the more general LLG equation of Eq. (9) together with Eq. (10) has to be used. We nally describe how the parameters of Eq. (1) can be calculated from a rst principles point of view. Within the conditions dened by the DFT system, the interaction tensor Dris time local which allows us to4 write lim "!0i@"Dr(r;r0;") =R Dr(r;r0;t;t0)dt0and lim"!0@2 "Dr(r;r0;") =R 2Dr(r;r0;t;t0)dt0, where Dr(r;r0;") = 4 spZ JrJ0r0f(!)f(!0) "!+!0+i ImGr 0(!)ImGr 0(!0)d! 2d!0 2dd0:(11) Here,Jrr0J(r;r0) whereas Gr rr0(!)Gr(r;r0;!) is the retarded GF, represented as a 2 2-matrix in spin- spaces. We notice that the above result presents a general expression for frequency dependent exchange interaction. Using Kramers-Kr onig's relations in the limit "!0, it is easy to see that Eq. (11) leads to Dr(r;r0; 0) =1 sp ImZ JrJ0r0f(!) Gr 0(!)Gr 0(!)d!dd0;(12) in agreement with e.g. Ref. [24]. We can make connection with previous results, e.g. Refs. 25, 26, and observe that Eq. (11) contains the isotropic Heisenberg, anisotropic Ising, and Dzyaloshinsky-Moriya exchange interactions between the magnetization densities at dierent points in space [22], as well as the onsite contribution to the magnetic anisotropy. Using the result in Eq. (11), we nd that the damping tensor is naturally non-local and can be reduced to ^G(r;r0) =1 spZ JrJ0r0f0(!) ImGr 0(!)ImGr 0(!)d!dd0;(13) which besides the non-locality is in good accordance with the results in Refs. [17, 25], and is closely connected to the so-called torque-torque correlation model [27]. With inclusion of the the spin-orbit coupling in Gr, it has been demonstrated that Eq. (13) leads to a local Gilbert damp- ing of the correct order of magnitude for the case of fer- romagnetic permalloys [17]. Another application of Kramers-Kr onig's relations leads, after some algebra, to the moment of inertia tensor ^I(r;r0) = spZ JrJ0r0f(!)mGr 0(!)@2 !ReGr 0(!) + ImGr 0(!)@2 !ReGr 0(!)]d! 2dd0;(14) where we notice that the moment of inertia is not sim- ply a Fermi surface eect but depends on the electronic structure as a whole of the host material. Although the structure of this expression is in line with the exchange coupling in Eq. (12) and the damping of Eq. (13), it is a little more cumbersome to compute due the presence of the derivatives of the Green's functions. Note that it is not possible to get completely rid of the derivatives through partial integration. These derivatives also makethe moment of inertia very sensitive to details of the elec- tronic structure, which has a few implications. Firstly the moment of inertia can take large values for narrow band magnetic materials, such as strongly correlated electron systems, where these derivatives are substantial. For such systems the action of moment of inertia can be im- portant for longer time scales too, as indicated by Eq. (5). Secondly, the moment of inertia may be strongly depen- dent on the reference magnetic ordering for which it is calculated. It is well known that already the exchange tensor parameters may depend on the magnetic order. It is the task of future studies to determine how trans- ferable the moment of inertia tensor as well as damping tensor are in-between dierent magnetic ordering. In conclusion, we have derived a method for atomistic spin dynamics which would be applicable for ultrafast (femtosecond) processes. Using a general s-d-like interac- tion between the magnetization density and electron spin, we show that magnetization couples to the surrounding in a non-adiabatic fashion, something which will allow for studies of general magnetic orders on an atomistic level, not only ferromagnetic. By showing that our method capture previous formulas for the exchange interaction and damping tensor parameter, we also derive a formula for calculating the moment of inertia from rst principles. In addition our results point out that all parameters are non-local as they enter naturally as bilinear sums in the same fashion as the well established exchange coupling. Our results are straight-forward to implement in existing atomistic SD codes, so we look on with anticipation to the rst applications of the presented theory which would be fully parameter-free and hence can take a large step towards simulations with predictive capacity. Support from the Swedish Research Council is ac- knowledged. We are grateful for fruitful and encouraging discussions with A. Bergman, L. Bergqvist, O. Eriksson, C. Etz, B. Sanyal, and A. Taroni. J.F. also acknowledges discussions with J. -X. Zhu. Electronic address: Jonas.Fransson@physics.uu.se [1] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Phys. Rev. Lett. 76, 4250 (1996). [2] G. P. Zhang and W. H ubner, Phys. Rev. Lett. 85, 3025 (2000). [3] M. G. M unzenberg, Nat. Mat. 9, 184 (2010); [4] S. L. Johnson et al. unpublished , arXiv:1106.6128v2. [5] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010). [6] J. Akerman, Science, 308, 508 (2005); [7] U. Nowak, O. N. Mryasov, R. Wieser, K. Guslienko, and R. W. Chantrell, Phys. Rev. B, 72, 172410 (2005). [8] B. Skubic, J. Hellsvik, L. Nordstr om, and O. Eriksson, Journ. of Phys.: Cond. Matter, 20, 315203 (2008). [9] M. -C. Ciornei, J. M. Rub , and J. -E. Wegrowe, Phys. Rev. B, 83, 020410(R) (2011).5 [10] L. Landau and E. Lifshitz, Phys. Z. Sowjetunion, 8, 153 (1935). [11] T. L. Gilbert, IEEE Trans. Mag. 40, 3443 (2004). [12] R. A. Sack, Proc. Phys. Soc. B, 70402 (1957). [13] W. T. Coey, Journal of Molecular Liquids, 114, 5 (2004). [14] W.T. Coey, Yu. P. Kalmykov and S.V. Titov, Phys. Rev. E, 65, 032102 (2002) [15] A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev, A. Kirilyuk, and Th. Rasing, Nature Phys. 5, 727 (2009). [16] W. F. Brown, Phys. Rev. 130, 1677 (1963). [17] H. Ebert, S. Mankovsky, D. Kodderitzsch, P. J. Kelly, Phys. Rev. Lett. 107, 066603 (2011). [18] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 (2007). [19] A. Brataas, Y. Tserkovnyak, and G. E.W. Bauer, Phys. Rev. Lett. 101, 037207 (2008).[20] J. -X. Zhu, Z. Nussinov, A. Shnirman, and A. V. Bal- atsky, Phys. Rev. Lett. 92, 107001 (2004). [21] J. Fransson and J. -X. Zhu, New J. Phys. 10, 013017 (2008). [22] J. Fransson, Phys. Rev. B, 82, 180411(R) (2010). [23] If higher order spin interaction terms than bilinear are needed it is straight forward to include higher order in this expansion. [24] V. P. Antropov, M. I. Katsnelson, M. van Schilfgaarde, and B. N. Harmon, Phys. Rev. Lett. 75, 729 (1995). [25] V. P. Antropov, M. I. Katsnelson, and A. I. Liechtenstein, Physica B, 237-238 , 336 (1997). [26] M. I. Katsnelson and A. I. Lichtenstein, J. Phys.: Con- dens. Matter, 16, 7439 (2004). [27] V. Kambersk y, Phys. Rev. B, 76, 134416 (2007).

2011-10-15

We consider spin dynamics for implementation in an atomistic framework and we address the feasibility of capturing processes in the femtosecond regime by inclusion of moment of inertia. In the spirit of an {\it s-d} -like interaction between the magnetization and electron spin, we derive a generalized equation of motion for the magnetization dynamics in the semi-classical limit, which is non-local in both space and time. Using this result we retain a generalized Landau-Lifshitz-Gilbert equation, also including the moment of inertia, and demonstrate how the exchange interaction, damping, and moment of inertia, all can be calculated from first principles.

Atomistic spin dynamic method with both damping and moment of inertia effects included from first principles

1110.3387v2

arXiv:0708.0463v1 [cond-mat.other] 3 Aug 2007Strong spin-orbit induced Gilbert damping and g-shift in ir on-platinum nanoparticles J¨ urgen K¨ otzler, Detlef G¨ orlitz, and Frank Wiekhorst Institut f¨ ur Angewandte Physik und Zentrum f¨ ur Mikrostruk turforschung, Universit¨ at Hamburg, Jungiusstrasse 11, D-20355 Hamburg , Germany (Dated: October 30, 2018) The shape of ferromagnetic resonance spectra of highly disp ersed, chemically disordered Fe0.2Pt0.8nanospheres is perfectly described by the solution of the La ndau-Lifshitz-Gilbert (LLG) equation excluding effects by crystalline anisotropy and su perparamagnetic fluctuations. Upon decreasing temperature , the LLG damping α(T) and a negative g-shift,g(T)−g0, increase pro- portional to the particle magnetic moments determined from the Langevin analysis of the magneti- zation isotherms. These novel features are explained by the scattering of the q→0 magnon from an electron-hole (e/h) pair mediated by the spin-orbit coup ling, while the sd-exchange can be ruled out. The large saturation values, α(0) = 0.76 andg(0)/g0−1 =−0.37, indicate the dominance of an overdamped 1 meV e/h-pair which seems to originate from th e discrete levels of the itinerant electrons in the dp= 3nmnanoparticles. PACS numbers: 76.50.+g, 78.67.Bf, 76.30.-v, 76.60.Es I. INTRODUCTION Theongoingdownscalingofmagneto-electronicdevices maintainstheyetintenseresearchofspindynamicsinfer- romagnetic structures with restricted dimensions. The effect of surfaces, interfaces, and disorder in ultrathin films1, multilayers, and nanowires2has been examined and discussed in great detail. On structures confined to the nm-scale in all three dimensions, like ferromagnetic nanoparticles, the impact of anisotropy3and particle- particle interactions4on the Ne´ el-Brown type dynamics, which controls the switching of the longitudinal magneti- zation, is now also well understood. On the other hand, the dynamics of the transverse magnetization, which e.g. determines the externally induced, ultrafast magnetic switching in ferromagnetic nanoparticles, is still a top- ical issue. Such fast switching requires a large, i.e. a critical value of the LLG damping parameter α5. This damping has been studied by conventional6,7and, more recently, by advanced8ferromagnetic resonance (FMR) techniques, revealing enhanced values of αup to the or- der of one. By now, the LLG damping of bulkferromagnets is al- most quantitatively explained by the scattering of the q= 0 magnon by conduction electron-hole (e/h) pairs due to the spin-orbit coupling Ω so9. According to recent ab initio bandstructure calculations10, the rather small values for α≈Ω2 soτresult from the small (Drude) re- laxation time τof the electrons. For nanoparticles, the Drude scattering and also the wave-vector conservation are ill-defined, and ab initio many-body approaches to the spin dynamics should be more appropriate. Numeri- cal workby Cehovin et al.11considersthe modification of the FMR spectrum by the discrete level structure of the itinerant electrons in the particle. However, the effect of the resulting electron-hole excitation, ǫp∼v−1 p, wherevp is the nanoparticle volume on the intrinsic damping has not yet been considered. Here we present FMR-spectra recorded atω/2π=9.1 GHz on Fe0.2Pt0.8nanospheres, the struc- tural and magnetic properties of which are summarized in Sect. II. In Sect. III the measured FMR-shapes will be examined by solutions of the LLG-equation of motion for the particle moments. Several effects, in particular those predicted for crystalline anisotropy12and superparam- agnetic (SPM) fluctuations of the particle moments13 will be considered. In Sect. IV, the central results of this study, i.e. the LLG-damping α(T) reaching values of almost one and a large g-shift,g(T)−g0, are presented. Since both α(T) and g(T) increase proportional to the particle magnetization, they can be related to spin-orbit damping torques, which, due to the large values of αand ∆gare rather strongly correlated. It will be discussed which features of the e/h-excitations are responsible for these correlations in a nanoparticle. A summary and the conclusions are given in the final section. II. NANOPARTICLE CHARACTERIZATION The nanoparticleassemblyhasbeen prepared14follow- ing the wet-chemical route by Sun et al.15. In order to minimize the effect of particle-particle interactions, the nanoparticles were highly dispersed14. Transmis- sion electron microscopy (TEM) revealed nearly spher- ical shapes with mean diameter dp= 3.1nmand a rather small width of the log-normal size distribution, σd= 0.17(3). Wide angle X-ray diffraction provided the chemically disordered fccstructure with a lattice con- stant a 0=0.3861 nm14. Themeanmagneticmomentsofthenanospheres µp(T) have been extracted from fits of the magnetization isotherms M(H,T), measured by a SQUID magnetome- ter (QUANTUM DESIGN, MPMS2) in units emu/g= 1.1·1020µB/g, to ML(H,T) =Npµp(T)L(µpH kBT). (1)2 Here are L(y) = coth( y)−y−1withy=µp(T)H/kBT the Langevin function and Npthe number of nanoparti- cles per gram. The fits shown in Fig.1(a) demanded for a small paramagnetic background, M−ML=χb(T)H, with a strong Curie-liketemperature variationof the sus- ceptibility χb, signalizing the presence of paramagnetic impurities. According to the inset of Fig. 1(b) this 1 /T- law turns out to agree with the temperature dependence of the intensity of a weak, narrow magnetic resonance withgi= 4.3 depicted in Figs. 2 and 3. Such narrow line with the same g-factor has been observed by Berger et al.16on partially crystallized iron-containing borate glass and could be traced to isolated Fe3+ions. The results for µp(T) depicted in Fig.1(b) show the moments to saturate at µp(T→0) = (910 ±30)µB. This yields a mean moment per atom in the fccunit cell ofµ(0) =µpa3 0/4vp= 0.7µBcorresponding to a spontaneous magnetization Ms(0) = 5.5kOe. Accord- ing to previous work by Menshikov et al.17on chemi- cally disordered FexPt1−xthis corresponds to an iron- concentration of x=0.20. Upon rising temperature the moments decrease rapidly, which above 40 K can be rather well parameterized by the empirical power law, µp(T≥40K)∼(1−T/TC)βrevealing β= 2 and for the Curie temperature TC= (320±20)K. This is con- sistent with TC= (310±10)KforFe0.2Pt0.8emerging from a slight extrapolation of results for TC(x≥0.25) ofFexPt1−x18. No quantitative argument is at hand for the exponent β= 2, which is much larger than the mean field value βMF= 1/2. We believe that β= 2 may arise from a reduced thermal stability of the magne- tization due to strong fluctuations of the ferromagnetic exchange between FeandPtin the disordered struc- ture and also to additional effects of the antiferromag- neticFe−FeandPt−Ptexchange interactions. In this context, it may be interesting to note that for low Feconcentrations, x≤0.3, bulkFexPt1−xexhibits fer- romagnetism only in the disordered structure17, while structural ordering leads to para- or antiferromagnetism. Recent first-principle calculations of the electronic struc- ture produced clear evidence for the stabilizing effect of disorder on the ferromagnetism in FePt19. From the Langevin fits in Fig.1(a) we obtain for the particle density Np= 3.5·1017g−1. Basing on the well known mass densities of Fe0.2Pt0.8and the organic ma- trix, we find by a little calculation20for the volume con- centration of the particles cp= 0.013 and, hence, for the mean inter-particle distance, dpp=dp/c1/3 p= 13.5nm. This implies forthe maximum (i.e. T=0) dipolar interac- tion between nearest particles, µ2 p(0)/4πµ0d3 pp= 0.20K, so that at the present temperatures, T≥20K, the sam- ple should act as an ensemble of independent ferromag- netic nanospheres. Since also the blocking temperature, Tb= 9K, as determined from the maximum of the ac- susceptibility at 0.1 Hz in zero magnetic field20, turned out to be low, the Langevin-analysis in Fig.1(a) is valid.FIG. 1: Fig. 1. (a) Magnetization isotherms of the nanospheres fitted to the Langevin model plus a small para- magnetic background χb·H; (b) temperature dependences of the magnetic moments of the nanoparticles µpand of the background susceptibility χb( inset units are emu/g kOe ) fitted to the indicated relations with TC= (320±20)K. The inset shows also data from the intensity of the paramagnetic resonance at 1.45 kOe, see Figs. 2 and 3. III. RESONANCE SHAPE Magnetic resonances at a fixed X-band frequency of 9.095 GHz have been recorded by a home-made mi- crowave reflectometer equipped with field modulation to enhancethesensitivity. Adouble-walledquartztubecon- taining the sample powder has been inserted to a mul- tipurpose, gold-plated VARIAN cavity (model V-4531). Keeping the cavity at room temperature, the sample couldbeeithercooledbymeansofacontinuousflowcryo- stat (Oxford Instruments, model ESR 900) down to 15 K orheatedupto500Kbyanexternal Pt-resistancewire20. At all temperatures, the incident microwave power was varied in order to assure the linear response. Someexamplesofthespectrarecordedbelowthe Curie3 FIG. 2: (a) Derivative of the microwave (f=9.095 GHz) ab- sorption spectrum recorded at T=52 K i.e. close to magnetic saturation of the nanospheres. The solid and dashed curves are based on fits to Eqs.(4) and (8), respectively, which both ignore SPM fluctuations and assume either a g-shift and zero anisotropy field HA(’∆g-FM’) or ∆ g= 0 and a randomly distributed HA=0.5 kOe (’a-FM’), Eq. (9). Also shown are fits to predictions by Eq.(11), which account for SPM fluctu- ations, with HA=0.5 kOe and ∆ g= 0 (’a-SPM’) and, using Eq.(11), for ∆ g/negationslash= 0 and HA=0 (’∆g-SPM’). The weak, nar- row resonance at 1.45 kOe is attributed to the paramagnetic background with g= 4.3±0.1 indicating Fe3+ 16impurities. temperature are shown in Figs.2 and 3. The spectra have been measured from -9.5 kOe to +9.5 kOe and proved to be independent of the sign of H and free of any hystere- sis. Thiscanbeexpectedduetothecompletelyreversible behavior of the magnetization isotherms above 20 K and the low blocking temperature of the particles. Lower- ing the temperature, we observe a downward shift of the mainresonanceaccompaniedbyastrongbroadening. On the other hand, the position and width of the weak nar- row line at (1 .50±0.05)kOecorresponding to gi∼=4.3 remain independent of temperature. This can be at- tributed to the previously detected paramagnetic Fe3+- impurities16and is supported by the integrated intensity of this impurity resonance Ii(T) evaluated from the am- plitudedifferenceofthederivativepeaks. Sincetheinten- sity of a paramagnetic resonance is given by the param- agnetic susceptibility, Ii(T)∼/integraltext dH χ′′ xx(H,T)∼χi(T) can be compared directly to the background suscepti- bilityχb(T), see inset to Fig.1(b). The good agree- ment between both temperature dependencies suggests to attribute χbto these Fe3+-impurities. An analysis of the fitted Curie-constant, Ci= 5emuK/g kOe , yields Ni= 164.1017g−1for theFe3+- concentration, which corresponds to 50 Fe3+per 1150 atoms of a nanosphere. With regard to the main intensity, we want to extract a maximum possible information, in particular, on theFIG. 3: Fig. 3. Derivative spectra at some representative temperatures and fits to Eq.(4). The LLG-damping, g-shift, and intensities are depicted in Fig. 4. intrinsic magnetic damping in nanoparticles. Unlike the conventional analysis of resonance fields and linewidths, as applied e.g. to Ni6and Co7nanoparticles, our ob- jective is a complete shape analysis in order to disen- tangle effects by the crystalline anisotropy12, by SPM fluctuations13, by an electronic g-shift21, and by differ- ent forms ofthe damping torque /vectorR22. Additional difficul- ties may enter the analysis due to non-spherical particle shapes, size distributions and particle interaction, all of which, however, can be safely excluded for the present nanoparticle assembly. The starting point of most FMR analyses is the phe- nomenological equation of motion for a particle moment ( see e.g. Ref. 13 ) d dt/vector µp=γ/vectorHeff×/vector µp−/vectorR , (2) using either the original Landau-Lifshitz (LL) damping with damping frequency λL /vectorRL=λL Ms(/vectorHeff×/vector µp)×/vector sp ,(3) ortheGilbert-dampingwiththe Gilbertdampingparam- eterαG, /vectorRG=αGd/vector µp dt×/vector sp , (4) where/vector sp=/vector µp/µpdenotes the direction of the particle moment. In Eq.(2), the gyromagneticratio, γ=g0µB/¯h, is determined by the regular g-factorg0of the precessing moments. It shouldperhapsbe notedthat the validityof the micromagnetic approximationunderlying Eq. (2) has been questioned23for volumes smaller than (2 λsw(T))3,4 whereλsw= 2a0TC/Tis the smallest wavelengthof ther- mally excited spin waves. For the present particles, this estimate leads to a fairly large temperature of ∼0.7TC up to which micromagnetics should hold. At first, we ignore the anisotropy being small in cubic FexPt1−x24,25, so that for the present nanospheres the effective field is identical to the applied field, /vectorHeff= /vectorH. Then, the solutions of Eq. (2) for the susceptibility of the two normal, i.e. circularly polarized modes, of Npindependent nanoparticles per gram take the simple forms χL ±(H) =Npµpγ1∓iα γH(1∓iα)∓ω(5) for/vectorR=/vectorRLwithα=λL/γMsand for the Gilbert torque /vectorRG χG ±(H) =Npµpγ1 γH∓ω(1+iαG).(6) For the LL damping, the experimental, transverse sus- ceptibility, χxx=1 2(χ++χ−), takes the form χL xx(H) =NpµpγγH(1+α2)−iαω (γH)2(1+α2)−ω2−2iαωγH. (7) As thesameshape isobtainedforthe Gilbert torquewith α=αG, the damping is frequently denoted as LLG pa- rameter. However, the gyromagnetic ratio in Eq.(7) has tobereplacedby γ/(1+α2), whichonlyfor α≪1implies also the same resonance field Hr. Upon increasing the damping up to α≈0.7 (the regime of interest here), the resonance field HrofχG xx(H) , determined by dχ′′/dH= 0,remains constant, HG r≈ω/γ, whileHL rdecreases rapidly. After renormalization γ/(1+α2) the resonance fields and also the shapes of χL(H) andχG(H) become identical. This effect should be observed when determin- ing theg-factor from the resonance fields of broad lines. It becomes even more important if the downward shift of Hris attributed to anisotropy, as done recently for the ratherbroadFMR absorptionof FexPt1−xnanoparticles with larger Fe-content, x≥0.326. In order to check here for both damping torques, we selected the shape measured at a low temperature, T=52 K, where the linewidth proved to be large (see Fig. 2) and the magnetic moment µp(T) was close to sat- uration (Fig. 1(b)). None of both damping terms could explain both, the observed resonance field Hrand the linewidth ∆ H=αω/γ, and, hence, the lineshape. By using/vectorRG, the shift of Hrfromω/γ= 3.00kOewas not reproduced by HG r=ω/γ, while for /vectorRLthe resonance fieldHL r, demanded by the line width, became signifi- cantly smaller than Hr. This result suggested to consider as next the effect of a crystallineanisotropyfield /vectorHAonthetransversesuscepti- bility, which has been calculated by Netzelmann from thefree energy of a ferromagnetic grain12. Specializing his general ansatz to a uniaxial /vectorHAoriented at angles ( θ,φ) withrespecttothedc-field /vectorH||/vector ezandthemicrowavefield, one obtains by minimizing F(θ,φ,ϑ,ϕ) =−µp[Hcosϑ+ 1 2HA(sinϑsinθ−cos(ϕ−φ)+cosθcosϑ)2] (8) the equilibrium orientation ( ϑ0,ϕ0) of the moment /vector µpof a spherical grain. After performing the trivial average overφ, one finds for the transverse susceptibility of a particle with orientation θ χL xx(θ,H) =γµp 2× (Fϑ0ϑ0+Fϕ0ϕ0/tan2ϑ0)(1+α2)−iαµpω(1+cos2ϑ0) (1+α2)(γHeff)2−ω2−iαωγ∆H. (9) HereH2 eff= (Fϑ0ϑ0Fϕ0ϕ0−F2 ϑ0ϕ0/(µpsinϑ0)2) and ∆H= (Fϑ0ϑ0+Fϕ0ϕ0/sin2ϑ0)/µpare given by the sec- ond derivatives of F at the equilibrium orientation of /vector µp. For the randomly distributed /vectorHAofNpindependent par- ticles per gram one has χL xx(H) =/integraldisplayπ/2 0d(cosθ)χL xx(θ,H).(10) In a strict sense, this result should be valid at fields larger than the so called thermal fluctuation field HT= kBT/µp(T), see e.g. Ref. 13, which for the present case amounts to HT= 1.0kOe. Hence, in Fig. 2 we fit- ted the data starting at high fields, reaching there an almost perfect agreement with the curve a-FM. The fit yields a rather small HA= 0.5kOewhich implies a small anisotropyenergyper atom, EA=1 2µp(0)HA= 1.0µeV. This number is smaller than the calculated value for bulk fcc FePt ,EA= 4.0µeV25, most probably due to the lowerFe-concentration (x=0.20) and the strong struc- tural disorder in our nanospheres. We emphasize, that the main defect of this a-FM fit curve arises from the finite value of dχ′′ xx/dHatH= 0. By means of Eq. (9) one finds χ′′ xx(H→0,θ)∼HAH/ω2, which remains fi- nite even after averaging over all orientations according toθ(Eq. (10)). Thefinite value ofthe derivativeof χ′′ xx(H→0)should disappear if superparamagnetic (SPM) fluctuations of the particles are taken into account. Classical work27 predicted the anisotropy field to be reduced by SPM, HA(y) =HA·(1/L(y)−3/y), which for y=H/HT≪1 impliesHA(y) =HA·y/5 and, therefore, χ′′ xx(H→0)∼ H2. A statistical theory for χL xx(H,T) which considers the effect of SPM fluctuations exists only to first order in HA/H21. The result of this linear model (LM) in HA/H which generalizes Eq. (4), can be cast in the form χLM ±(θ,H) =NpµpL(y)γ(1+A∓iαA) γ(1+B∓iαB)H∓ω. (11)5 The additional parameters are given in Ref. 21 and con- tain, depending on the symmetry of HA, higher-order Langevin functions Lj(y) and their derivatives. Observ- ing the validity of the LM for H≫HA= 0.5kOe, we fitted the data in Fig. 2 to Eq. 11 with χLM xx(θ,H) = 1 2(χLM ++χLM −) at larger fields. There one has also H≫ HT= 1.0kOeand the fit, denoted as a-SPM, agrees with the ferromagnetic result (a-FM). However, increas- ing deviations appear below fields of 4 kOe. By varying HAandα, we tried to improve the fit near the resonance Hr= 2.3kOeand obtained unsatisfying results. For low anisotropy, HA≤3kOe, the resonance field could be reproduced only by significantly lower values of α, which are inconsistent with the measured width and shape. For HA>3kOe, a small shift of Hroccurs, but at the same time the lineshape became distorted, tending to a two-peak structure also found in previous simulations13. Even at the lowest temperature, T= 22K, where the thermal field drops to HT= 0.4kOe, no signatures of such inhomogeneous broadening appear (see Fig. 3). Fi- nally, it should be mentioned that all above attempts to incorporate the anisotropy in the discussion of the line- shape were based on the simplest non-trivial, i.e. uni- axial symmetry, which for FePtwas also considered by the theory25. For cubic anisotropy, the same qualitative discrepancies were found in our simulations20. This in- sensitivitywith respecttothe symmetry of HAoriginates from the orientational averaging in the range of the HA- values of relevance here. As a finite anisotropy failed to reproduce Hr,∆H, and also the shape, we tried a novel ansatz for the magnetic resonanceofnanoparticlesbyintroducingacomplexLLG parameter, ˆα(T) =α(T)−i β(T). (12) According to Eq. (4) this is equivalent to a negative g- shift,g(T)−g0=−β(T)g0, which is intended to com- pensate the too large downward shift of HL rdemanded byχL xx(H) due to the large linewidth. In fact, insert- ing this ansatz in Eq. (5), the fit, denoted as ∆ g-FM in Fig. 2, provides a convincing description of the line- shape down to zero magnetic field. It may be interest- ing to note that the resulting parameters, α= 0.56 and β= 0.27, revealed the same shape as obtained by using the Gilbert-susceptibilities, Eq. (6). In spite of the agreement of the ∆ g-FM model with the data, we also tried to include here SPM fluctuations by using ˆ α(T,H) = ˆα(T)(1/L(y)−1/y)21forHA= 0. The result, designated as ∆ g-SPM in Fig. 2 agrees with the ∆g-FM curve for H≫HTwhere ˆα(T,H) = ˆα(T), but again significant deviations occur at lower fields. They indicate that SPM fluctuations do not play any role here, and this conclusion is also confirmed by the results at higher temperatures. There, the thermal fluc- tuation field, HT=kBT/µp(T), increases to values larger than the maximum measuring field, H= 10kOe, so that SPM fluctuations should cause a strong ther-/s48 /s51/s48/s48/s32/s75/s48/s55/s46/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s56/s49/s46/s48/s32 /s32 /s32/s97/s41 /s48/s46/s49/s56/s32/s43/s32/s48/s46/s53/s56/s32/s40/s32/s49/s45/s84/s47/s84 /s67/s32/s41/s32/s50 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s53/s48/s48/s48/s46/s48/s48/s46/s50 /s48/s46/s51/s57/s32/s40/s32/s49/s45/s84/s47/s84 /s67/s32/s41/s32/s50 /s32/s84/s32/s40/s32/s75/s32/s41/s103/s32/s47/s32/s103 /s48 /s32/s32 /s98/s41/s32 /s32/s73/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s126/s32/s40/s32/s49/s45/s84/s47/s84 /s67/s32/s41/s32/s50 FIG. 4: Temperature variation (a) of the LLG-damping αand (b) of the relative g-shifts with g0= 2.16 (following from the resonance fields at T > T C). Within the error margins, α(T) and ∆g(T) and also the fitted intensity of the LLG-shape (see inset) display the same temperature dependence as the particle moments in Fig. 1(b). mal, homogeneous broadening of the resonance due to ˆα(H≫HT) = ˆα·2HT/H. However, upon increasing temperature, the fitted linewidths, (Fig. 3) and damping parameters (Fig. 4) display the reverse behavior. IV. COMPLEX DAMPING In order to shed more light on the magnetization dy- namics of the nanospheres we examined the tempera- ture variation of the FMR spectra. Figure 3 shows some examples recorded below the Curie temperature, TC= 320K, together with fits to the a−FM model outlined in the last section. Above TC, the resonance fields and the linewidths are temperature independent revealing a mean g−factor,g0= 2.16±0.02, and a damping parameter ∆ H/Hr=α0= 0.18±0.01. Since g0is consistent with a recent report on g-values of FexPt1−xforx≥0.4321, we suspect that this resonance arises from small FexPt1−x-clusters in the inhomoge- neousFe0.2Pt0.8structure. Fluctuations of g0and of local fields may be responsible for the rather large width.6 This interpretation is supported by the observation that aboveTCthe lineshape is closer to a Gaussian than to the Lorentzian following from Eq.(7) for small α. The temperature variation for both components of the complex damping, obtained from the fits below TCto Eq. (7), are shown in Fig. 4. Clearly, they obey the same powerlaw as the moments, µp(T), displayed in Fig. 1(b), which implies ˆα(T) = (α−i β)ms(T)+α0.(13) Herems(T)=µp(T)/µp(0),α=0.58, and β=-∆g(0)/g0= 0.39 denote the reduced spontaneous magnetization and the saturation values for the complex damping, respec- tively. It should be emphasized that the fitted inten- sityI(T) of the spectra, shown by the inset to Fig. 4(b), exhibits the same temperature variation I(T)∼µp(T). This behavior is predicted by the ferromagnetic model, Eq. (7), and is a further indication for the absence of SPM effects on the magnetic resonance. If the reso- nance were dominated by SPM fluctuations, the inten- sity should decrease like the SPM Curie-susceptibility, ISPM(T)∼µ2 p(T)/T, following from Eq. (11), being much stronger than the observed I(T). At the beginning of a physical discussion of ˆ α(T), we should point out that the almost perfect fits of the line- shape to Eq. (7) indicate that the complex damping is related to an intrinsic mechanism and that eventual in- homogeneous effects by distributions of particle sizes and shapes in the assembly, as well as by structural disorder areratherunlikely. Sinceageneraltheoryofthemagneti- zation dynamics in nanoparticles is not yet available, we start with the current knowledge on the LLG-damping inbulkand thin film ferromagnets, as recently reviewed by B. Heinrich5. Based on experimental work on the archetypal metallic ferromagnets and on recent ab initio band structure calculations10there is now ratherfirm ev- idence that the damping ofthe q=0-magnonis associated with the torques /vectorTso=/vector ms×/summationtext j(ξj/vectorLj×/vectorS) on the spin /vectorS due to the spin-orbit interaction ξjat the lattice sites j. The action of the torque is limited by the finite lifetime τof an e/h excitation, the finite energy ǫof which may cause a phase, i.e. a g-shift. As a result of this magnon - e/h-pair scattering, the temperature dependent part of the LLG damping parameter becomes ˆα(T)−α0=λL(T) γMs(T) =(Ωso·ms(T))2 τ−1+i ǫ/¯h·1 γMs(T). (14) Forintraband scattering, ǫ≪¯h/τ, the aforementioned numerical work10revealed Ω so= 0.8·1011s−1and 0.3· 1011s−1as effective spin-orbit coupling in fcc Niand bcc Fe, respectively. Hence, the narrow unshifted (∆ g= 0)bulkFMR lines in pure crystals, where α≤10−2, are related to intraband scattering with ǫ≪¯h/τand toelectronic (momentum) relaxation times τsmaller than 10−13s. Basing on Eq. (14), we discuss at first the temperature variation,whichimpliesalineardependence, ˆ α(T)−α0∼ ms(T). Obviously, both, the real and imaginary part of ˆα(T)−α0, agree perfectly with the fits to the data in Fig. 4, if the relaxation time τremains constant. It may be interestingtonote herethat the observedtemperature variation of the complex damping λL(T) is not predicted by the classical model28incorporating the sd-exchange coupling Jsd. According to this model, which has been advanced recently to ferromagnets with small spin-orbit interaction29and ferromagnetic multilayers30,Jsdtrans- fers spin from the localized 3d-moments to the delocal- ized s-electron spins within their spin-flip time τsf. From the mean field treatment of their equations of motion by Turov31, we find a form analogous to Eq. (14) αsd(T) =Ω2 sdχs τ−1 sf+i/tildewideΩsd·1 γMs(T)(15) where Ω sd=Jsd/¯his the exchange frequency , χsthe Pauli-susceptibility of the s-electrons and /tildewideΩsd/Ωsd= (1 + Ω sdχs/γMd). The same form follows from more detailed considerations of the involved scattering process (see e.g. Ref. 5). As a matter of fact, the LLG-damping αsd=λsd/γMdcannot account for the observed tem- perature dependence, because Ω sdandχsare constants. The variation of the spin-torques with the spontaneous magnetization ms(T) drops out in this model, since the sd-scattering involves transitions between the 3d spin-up and -down bands due to the splitting by the exchange fieldJsdms(T). By passing from the bulk to the nanoparticle ferro- magnet, we use Eq. (14) to discuss our results for the complex ˆ α(T), Eq. (13). Recently, for Conanoparticles with diameters 1-4 nm, the existence of a discrete level structure near ǫFhas been evidenced32, which suggests to associate the e/h-energy ǫwith the level difference ǫp at the Fermi energy. From Eqs. (13),(14) we obtain rela- tions between ǫand the lifetime of the e/h-pair and the experimental parameters αandβ: τ−1=α βǫ ¯h, (16a) ǫ ¯h=β α2+β2Ω2 so γMs(0). (16b) Due toα/β= 1.5, Eq. (16a) reveals a strongly over- damped excitation, which is a rather well-founded con- clusion. The evaluation of ǫ, on the other hand, de- pends on an estimate for the effective spin-orbit cou- pling, Ω so=ηLχ1/2 eξso/¯hwhereηLrepresents the ma- trix element of the orbital angular momentum between7 the e/h states5. The spin-orbit coupling of the minor- ityFe-spins in FePthas been calculated by Sakuma24, ξso= 45meV, while the density of states D(ǫF)≈1/(eV atom)24,33yields a rather high susceptibility of the elec- trons,ηLχe=µ2 BD(ǫF) = 4.5·10−5. Assuming ηL=1, both results lead to Ω so≈3.5·1011s−1, which is by one order of magnitude larger than the values for Fe andNimentioned above. One reason for this enhance- ment and for a large matrix element, ηL=1, may be the strong hybridization between 3 dand 4d−Ptorbitals24in FexPt1−x. By inserting this result into Eq. (16b) we find ǫ= 0.8 meV. In fact, this value is comparable to an esti- mate for the level difference at ǫF32,ǫp= (D(ǫF)·Np)−1 which for ourparticles with Np= (2π/3)(dp/a0)3= 1060 atoms yields ǫp= 0.9 meV. Regarding the several in- volved approximations, we believe that this good agree- ment between the two results on the energy of the e/h excitation, ǫ≈ǫp, maybe accidental. However,wethink, that this analysisprovidesa fairlystrongevidence for the magnon-scattering by this excitation, i.e. for the gap in the electronic states due to confinement of the itinerant electrons to the nanoparticle. V. SUMMARY AND CONCLUSIONS The analysis of magnetization isotherms explored the mean magnetic moments of Fe0.2Pt0.8nanospheres (dp= 3.1nm) suspended in an organic matrix, their temperature variation up to the Curie temperature TC, the large mean particle-particle distance Dpp≫dpand the presence of Fe3+impurities. Above TC, the res- onance field Hrof the 9.1GHzmicrowave absorption yielded a temperature independent mean g-factor,g0= 2.16, consistent with a previous report21for paramag- neticFexPt1−xclusters. There, the lineshape proved to be closer to a gaussian with rather large linewidth, ∆H/Hr= 0.18, which may be associated with fluctua- tions of g0and local fields both due to the chemically disordered fccstructure of the nanospheres. Below the Curie temperature, a detailed discussion of the shape of the magnetic resonance spectra revealed a number of novel and unexpected features. (i) Starting at zero magnetic field, the shapes could be described almost perfectly up to highest field of 10 kOe by the solution of the LLG equation of motion for inde- pendentferromagneticsphereswithnegligibleanisotropy. Signatures of SPM fluctuations on the damping, which have been predicted to occur below the thermal field HT=kBT/µp(T), could not be realized. (ii) Upon decreasing temperature, the LLG damping in- creases proportional to µp(T), i.e. to the spontaneous magnetization of the particles, reaching a rather largevalueα= 0.7 forT≪TC. We suspect that this high intrinsic damping may be responsible for the absence of the predicted SPM effects on the FMR, since the under- lying statistical theory13has been developed for α≪1. This conjecturemayfurther be based onthe fact that the large intrinsic damping field ∆ H=α·ω/γ= 2.1kOe causes a rapid relaxation of the transverse magnetization (q= 0 magnon) as compared to the effect of statistical fluctuations of HTadded to Heffin the equation of mo- tion, Eq.(2)13. (iii) Along with the strong damping, the lineshape analy- sis revealed a significant reduction of the g-factor, which also proved to be proportional to µp(T). Any attempts to account for this shift by introducing uniaxial or cubic anisotropy fields failed, since low values of /vectorHAhad no effects on the resonance field due to the orientational av- eraging. On the other hand, larger /vectorHA’s, by which some small shifts of Hrcould be obtained, produced severe distortions of the calculated lineshape. The central results of this work are the temperature variation and the large magnitudes of both α(T) and ∆g(T). They were discussed by using the model of the spin-orbit induced scattering of the q= 0 magnon by an e/h excitation ǫ, well established for bulk ferromag- nets, where strong intraband scattering with ǫ≪¯h/τ proved to dominate5. In nanoparticles, the continuous ǫ(/vectork)-spectrum of a bulk ferromagnet is expected to be split into discrete levels due to the finite number of lat- tice sites creating an e/h excitation ǫp. According to the measured ratio between damping and g-shift, this e/h pair proved to be overdamped, ¯ h/τp= 1.5ǫp. Based on the free electron approximation for ǫp32and the den- sity of states D(ǫF) from band-structure calculations forFexPt1−x24,33, one obtains a rough estimate ǫp≈ 0.9 meV for the present nanoparticles. Using a reason- able estimate of the effective spin-orbit coupling to the minority Fe-spins, this value could be well reproduced by the measured LLG damping, α= 0.59. Therefore we conclude that the noveland unexpected results of the dy- namics of the transverse magnetization reported here are due to the presence of a broade/h excitation with energy ǫp≈1meV. Deeper quantitative conclusions, however, must await more detailed information on the real elec- tronic structure of nanoparticles near ǫF, which are also required to explain the overdamping of the e/h-pairs, as it is inferred from our data. The authors are indebted to E. Shevchenko and H. Weller (Hamburg) for the synthesis and the structural characterizationof the nanoparticles. One of the authors (J. K.) thanks B. Heinrich (Burnaby) and M. F¨ ahnle (Stuttgart) for illuminating discussions. 1G. Woltersdorf , M. Buess, B. Heinrich and C. H. Back, Phys. Rev. Lett. 95, 037401 (2005); B. Koopmans,J. J. M. Ruigrok, F. D. Longa, and W. J. M. de Jonge,8 ibid.95, 267207 (2005); Ultrathin magnetic structures III, IV, J. A. C. Bland, B. Heinrich (Eds.), Springer Verlag, Berlin Heidelberg New York (2005). 2Spin Dynamics in Confined Magnetic Structures III , B. Hillebrands, A. Thiaville (Eds.), Springer Verlag, Berlin, Heidelberg, New York (2006). 3W. Wernsdorfer, E. B. Orozco, K. Hasselbach, A. Benoit, B. Barbara, N. Demoncy, A. Loiseau, H. Pascard, and D. Mailly, Phys. Rev. Lett. 78, 1791 (1997); S. I. Woods, J. R. Kirtley, S. Sun, and R. H. Koch, ibid.87, 137205 (2001); 4C. Djurberg, P. Svedlindh, P. Nordblad, M. F. Hansen, F.Bødker, andS.Mørup, Phys.Rev.Lett. 79, 5154 (1997); F. Luis, F. Petroff, J. M. Torres, L. M. Garc´ ıa, J. Bar- tolom´ e, J. Carrey, and A. Vaur` es, ibid.88, 217205 (2002). 5B. Heinrich in Ultrathin magnetic structures III , J. A. C. Bland, B. Heinrich (Eds.), Springer Verlag, Berlin, Heidelberg, New York (2005), p.143. 6V. K. Sharma, A. Baiker, J. Chem. Phys. 75, 5596 (1981). 7M. Respaud, M. 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2007-08-03

The shape of ferromagnetic resonance spectra of highly dispersed, chemically disordered Fe_{0.2}Pt_{0.8} nanospheres is perfectly described by the solution of the Landau-Lifshitz-Gilbert (LLG) equation excluding effects by crystalline anisotropy and superparamagnetic fluctuations. Upon decreasing temperature, the LLG damping $\alpha(T)$ and a negative g-shift, g(T)-g_0, increase proportional to the particle magnetic moments determined from the Langevin analysis of the magnetization isotherms. These novel features are explained by the scattering of the $q \to 0$ magnon from an electron-hole (e/h) pair mediated by the spin-orbit coupling, while the sd-exchange can be ruled out. The large saturation values, $\alpha(0)=0.76$ and $g(0)/g_0-1=-0.37$, indicate the dominance of an overdamped 1 meV e/h-pair which seems to originate from the discrete levels of the itinerant electrons in the d_p=3 nm nanoparticles.

Strong spin-orbit induced Gilbert damping and g-shift in iron-platinum nanoparticles

0708.0463v1

arXiv:2002.03418v2 [math.AP] 9 Jan 2021Noname manuscript No. (will be inserted by the editor) Fujita modified exponent for scale invariant damped semiline ar wave equations. Felisia Angela Chiarello ·Giovanni Girardi · Sandra Lucente Received: date / Accepted: date Abstract The aim of this paper is to prove a blow up result of the solution for a se milinear scale invariant damped wave equation under a suitable decay conditio n on radial initial data. The admissible range for the power of the nonlinear term depends both o n the damping coefficient and on the pointwise decay order of the initial data. In addition we give an upper bound estimate for the lifespan of the solution. It depends not only on the exponent of the nonlinear term, not only on the damping coefficient but also on the size of the decay rate of th e initial data. Mathematics Subject Classification (2010) Primary 35B33 ·Secondary 35L70 1 Introduction In the recent years, the following Cauchy problem for the wave equ ation with scale invariant damping spreads a new line of research on variable coefficient type eq uations. More precisely, we are dealing with vtt(t,x)−∆v(t,x)+µ 1+tvt(t,x)+ν (1+t)2v(t,x) =|v(t,x)|p, t≥0, x∈Rn, v(0,x) = 0, vt(0,x) =εg(x),(1) withn≥2,µ, ν∈R,p >1 andga radial smooth function. In [DLR], [DL] and [Pe] and [Po] some results on the global existence of a solution for (1) with non co mpactly initial data appeared assuming a suitable decay behavior for g. Many other results concern blow-up and global existence for this equation, see [PR] for a summary of this problem. The main po int is to find a critical exponent, fixed a suitable space of data. More precisely, a level ¯ pis critical if for p >¯pone can prove that for ε >0 sufficiently small and for any gchosen in the fixed space there exists a unique global (in time) solution of the problem, and conversely if p∈(1,¯p), for any ε >0 there exist somegin this space such that the local solution cannot be prolonged over a finite time. Coming Felisia Angela Chiarello Department of Mathematical Sciences “G. L. Lagrange”, Poli tecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Supported by “Compagnia di San Paolo” (Torin o, Italy) E-mail: felisia.chiarello@polito.it Giovanni Girardi Department of Mathematics, University of Bari, Via Orabona n. 4, Italy. E-mail: giovanni.girardi@uniba.it Sandra Lucente Dipartimento Interateneo di Fisica, University of Bari, Vi a Orabona n. 4, Italy. Partially supported by PRIN 2017- linea Sud ”Qualitative and quantitative aspects of nonline ar PDEs.” E-mail: sandra.lucente@uniba.it2 Felisia Angela Chiarello et al. back to (1), in dependence on µ, νandn,a competition between two critical exponents appeared. In some cases the Strauss exponent is dominant; it is given by the wa ve equation theory, it will be denoted by pS(d) and it is the positive root of the quadratic equation (d−1)p2−(d+1)p−2 = 0. For other assumptions, the equation goes to an heat equation and a Fujita-type exponent pF(h) := 1+2 happears. In all known results, the quantities d >1,h >0 depend on ν, µandn. Changing the space of data, a change of critical exponent may appear. The novelty of our result consists in showing that if one takes into account the decay rate of the initial data then the Fujita type exponent depends also on such decay rate. In addition we give an up per bound estimate for the lifespan of the solution, in terms of the power of the nonlinear term, the size and the growth of the initial data. Let us recall that the lifespan of the solution is a fun ction of εwhich gives the maximal existence time: T(ε) := sup{T >0 such that the local solution uto (1) is defined on [0 ,T)×Rn}. Finally, we will prove the following. Theorem 1 Letn≥2. Letε >0andga radial smooth function satisfying g(|x|)≥M (1+|x|)¯k+1,with¯k >−1, (2) for some M >0and for any x∈Rn. Assuming in addition that ¯k+µ 2>0,µ 2/parenleftBigµ 2−1/parenrightBig ≥ν and 1< p < p F/parenleftBig ¯k+µ 2/parenrightBig , then the classical solution of (1)blows up. More precisely, the lifespan of the solution T(ε)>0is finite and satisfies T(ε)≤Cε−2(p−1) 4−(µ+2¯k)(p−1), (3) withC >0, independent of ε. Remark 1 Recently Ikeda, Tanaka, Wakasa in [ITW] consider a similar question f or cubic convo- lution nonlinearity and a critical decay appears. Remark 2 In [GL] we will also considera variantof problem (1), in which the nonline arity depends onv,t,vtcombined in a suitable way. Remark 3 The lifespan estimate for the same equation with compactly support ed data and ν/\e}atio\slash= µ 2/parenleftbigµ 2−1/parenrightbig has been considered in [PT]. If ν≤µ 2/parenleftbigµ 2−1/parenrightbig the lifespan estimate is different from (3) due to the compactness of the support of the initial data. The paper is organized as follows: in Section 2 we give an overview of th e known results and we state an auxiliary theorem; in Section 3 we prove the main results.Fujita modified exponent for scale invariant damped semilin ear wave equations. 3 2 Motivations 2.1 The case µ= 2,ν= 0 Let us start with a quite simple case vtt−∆v+2 1+tvt=|v|p, t≥0, x∈Rn, v(0,x) = 0, vt(0,x) =εg(x).(4) The global existence of small data solutions for this problem was firs t solved in [D] for a suitable range of nandp. Some non-existence results were also established for p < p F(n) := 1 +2 n. Except for the one-dimensional case a gap between this value and t he admissible exponents in [D] appeared. In [DLR] for dimension n= 2,3,this gap was covered with an unexpected result. Indeed, in that paper the Strauss exponent came into play. After wards, the global existence of small data solutions to (4) has been proved for any p > pS(n+2) also in odd dimension n≥5 in [DL] and in even dimension n≥4 in [Pe]. Moreover,we know that the exponent p2(n) := max {pS(n+2),pF(n)}is optimal; in fact, in [DLR] the authors prove the blow up of solutions of (4) for each 1 < p≤p2(n) in each dimension n∈N. In [DL,DLR,Pe], the authors provea global existence result not ne cessarilywhen the initial datum g=g(x) has compact support. More precisely, let n≥3, given a radial initial datum g(x) =g(|x|) withg∈C1(R), for any p > pS(n+ 2) it is possible to choose ¯k >0 andε0>0 such that (4) admits a radial global solution u∈C([0,∞)×Rn)∩C2([0,∞)×Rn\{0}) provided |g(h)(r)| ≤ε/a\}bracketle{tr/a\}bracketri}ht−(¯k+1+h)forh= 0,1. (5) In the present paper we discuss the dependence of ¯kfromnandp. In (5), the exponent ¯khas to belong to a suitable interval [ k1(n,p),k2(n,p)]. It is interesting to investigate the case of ¯k/\e}atio\slash∈ [k1(n,p),k2(n,p)]. In the sequel we will see that the bound k2(n,p) can be easily improved (see Remark 4). On the contrary if k < k1(n,p) then a new result appears. The known situation is the following: -k1(3,p) = max/braceleftbig3−p p−1,1 p−1/bracerightbig andk2(3,p) = 2(p−1), see [DLR]. -k1(n,p) = max/braceleftbig3−p p−1,n−1 2/bracerightbig andk2(n,p) = min/braceleftbig(n+1)p 2−2,n2−2n+13 2(n−3)/bracerightbig ifn≥5 odd, see [DL]. -k1(n,p) = max/braceleftbig3−p p−1,n−1 2/bracerightbig andk2(n,p) = min/braceleftBig (n+1)p 2−2,n−1/bracerightBig ifn≥4, see [Pe]. We can write in a different way the previous conditions. Firstly we conc entrate on the case n= 3. Forp∈(1,2) we have ¯k≥3−p p−1that is equivalent to p≥1+2 ¯k+1=pF(¯k+1). From above we have ¯k≤2(p−1) that is p≥¯k 2+1. The intersection of p=pF(¯k+1) and p= 1+¯k/2 is exactly in ¯k=−1+√ 17 2andp=pS(5). We summarize the situation in Figure 1. In the following graphs we denote in blue the zone of the known global existence results, in red the zone of the known blow-u p results. In this paper we want to cover the white zones.4 Felisia Angela Chiarello et al. p ¯k12 13 3+√ 17 4 −1+√ 17 2 Fig. 1:n= 3, µ= 2, ν= 0 Reading [DL] we see that the same situation appears for any odd n≥5. The critical curve p=pF(¯k+1) intersect the line p=2(¯k+2) n+1 in the Strauss couple /parenleftbigg ¯k0,2(¯k0+2) n+1/parenrightbigg =/parenleftBigg n−5+√ n2+14n+17 4,pS(n+2)/parenrightBigg . The only difference with the case n= 3 is that, in the global existence zone, a bound from above appears for pand this has some influence on k2(n,p). More precisely one can take p≤n+1 n−3,¯k≤n2−2n+13 2(n−3)ifn≥7, andp≤2,¯k≤3 ifn= 5. Hence, the result of such paper can be represented as in Figur e 2. Our aim is to prove blow up in the white zone below the Fujita curve. p ¯k1n+5 n+1n+1 n−3 n−1 2pS(n+2) ¯k0 Fig. 2:n≥5 odd,µ= 2, ν= 0 Even dimension is more delicate. In [Pe] the global existence result is e stablished in the blue zone below the line p=n+5 n+1except on the curve p=pF(¯k+1). For convenience of the reader, we precise that in the notation of [Pe] the role of ¯kis taken by the quantity k+n+1 2.Fujita modified exponent for scale invariant damped semilin ear wave equations. 5 p ¯k1n+5 n+1 n−1 2 n+1pS(n+2) ¯k0 Fig. 3:n≥4 even,µ= 2, ν= 0 2.2 The case µ >2 andν=µ 2(µ 2−1) In [Pe] and [Po] the author considers the Cauchy problem (1) for th e semilinear wave equation with scale invariant damping and mass terms, that is ν=µ 2(µ 2−1)≥0. We see that for µ= 2,it reduces to (4). Global existence of solutions to (1) holds under th e conditions µ∈[2,M(n)], M(n) =n−1 2/parenleftBigg 1+/radicalbigg n+7 n−1/parenrightBigg . In the even case [Pe], the initial data satisfies (5) for ¯k∈(k1(n,p,µ),k2(n,p,µ)] such that k1(n,p,µ) = max/braceleftBign−1 2,2 p−1−µ 2/bracerightBig ; (6) k2(n,p,µ) = min/braceleftBig n−1,n+µ−1 2p−µ+2 2/bracerightBig . (7) Rewriting these conditions in term of p, we find that p > pF/parenleftBig ¯k+µ 2/parenrightBig , p≥2¯k+µ+2 n+µ−1. The intersection of the curves those define the global existence z one gives p=pS(n+µ). Hence, the condition p > pS(n+µ) appears. Moreover, another bound from above appears: p <¯p:= min/braceleftbigg pF(µ),pF/parenleftbiggn+µ−1 2/parenrightbigg/bracerightbigg . This means that different results for large µand small µhold. This influences the positions of k1 andk2. For our purpose it is sufficient to say that for µ/\e}atio\slash= 2 and even nthe situation is similar to Figure 3. More precisely, in Figure 4 pS(n+µ) appears. The blow up result is indeed given in [NPR]. The zone between p=pS(n+µ) andp=pF/parenleftbig¯k+µ 2/parenrightbig is not covered by any known result. p ¯k1¯p k1 k2pS(n+µ) ¯k06 Felisia Angela Chiarello et al. Fig. 4:n≥4 even,ν=µ 2(µ 2−1)≥0 The corresponding global existence result for the Cauchy problem (1) in odd space dimension n≥1isstudiedin[Po]forradialandsmalldata,assumingcondition(5)wit h¯k∈[k1(n,p,µ),k2(n,p,µ)] wherek2satisfies (7) and it holds: k1(3,p,µ) = max/braceleftBig 1,2 p−1−µ 2,1 p−1/bracerightBig ; k1(n,p,µ) = max/braceleftBign−1 2,2 p−1−µ 2/bracerightBig , n≥5µ∈[2,n−1]; k1(n,p,µ) = max/braceleftBign−1 2,2 p−1−µ 2,1 p−1,/bracerightBig , n≥5µ∈(n−1,M(n)]. In anycase the condition p > pF(¯k+µ 2) appears.Hence in odd space dimension n≥5 the situation is not different from Figure 4. Reading Theorem 1 in the case ν=µ 2(µ 2−1), it is clear that the aim of this paper is to find blowing-up solutions to (1) even for p > pS(n+µ) by considering initial data with slow decay. More precisely, let us consider g(x)≃M (1+|x|)¯k+1,forn−1 2<¯k <¯k0, (8) where¯k0is such that pF/parenleftBig ¯k0+µ 2/parenrightBig =pS(n+µ). We will prove the blow up result in the left white side zones in Figures 1, 2 , 3, 4 where ¯k <¯k0, p > pS(n+µ) andp < pF(¯k0+µ 2). Under the same assumption on g, the quoted results assure that for p≥pF(¯k+µ 2) andp > pS(n+µ) there is global existence. Hence, p=pF(¯k+µ 2) is a critical curve for the Cauchy problem (1), provided ν=µ 2(µ 2−1)≥0. Remark 4 Still fixing ν=ν 2(ν 2−1)≥0, let us consider ¯k >¯k0andp > pS(n+µ). As discussed, the global existence results in the previous literature require pabove a line which depends on ¯k, because of a restriction of type ¯k≤k2(n,p,µ) which everytime appears. Actually, this restriction can be avoided; indeed, if the initial datum satisfies (8) with ¯k > k2(n,p,µ), then we can say that the initial datum also satisfies (5) with ¯k=k2(n,p,µ). Hence, the global existence of a solution to (1) follows from the known results. Remark 5 Forν=µ 2(µ 2−1)≥0,Theorem 1 provides some new information about the solution of (1) also when pbelongs to the red zone of Figure 1, 2, 3, 4, 5. In fact, for p <min/braceleftBig pS(n+µ),pF/parenleftBig ¯k+µ 2/parenrightBig/bracerightBig by the previous literature we know that the solution blows up in finite t ime, whereas Theorem 1 gives a life-span estimate in the case of radial initial data with non com pact support, relating this estimate with the decay rate of the data. 2.3 The case µ= 0 and ν= 0 In Figure 5 we summarize the wave equation case µ=ν= 0. The red blow-up zone was coveredby many authors, see [S] and the reference therein for the whole list o f blow up results. For µ=ν= 0 the global existence result has been completely solved in [GLS], wher e the interested reader can find a long bibliography of previous contributes. In particular the blu e zone, for radial solution without compact support assumption for the initial data has been e xploited by Kubo, see for example [K] and [KK]. Before these papers, Takamura obtained a blow up result in the green zone. In [T] the point is to find a critical decay level k0=2 p−1,equivalently p≤1 +2 k0.We underline that this a Fujita-type exponent.Fujita modified exponent for scale invariant damped semilin ear wave equations. 7 p ¯k13 1pS(n) 2 pS(n)−1 Fig. 5:µ=ν= 0 In Theorem 1, we generalize Takamura’s result when µ/\e}atio\slash= 0 and ν≤µ 2(µ 2−1). To this aim, it is sufficient to consider a peculiar wave equation with nonlinear term ha ving a decaying time- dependent variable coefficient. This means that we will deduce Theor em 1 from the following result. Theorem 2 Letn≥2. Given a smooth function g=g(|x|)withx∈Rn, we set r=|x|and we consider g=g(r)satisfying g(r)≥M (1+r)¯k+1,with¯k >−1, (9) for some M >0. Letu=u(t,r)be the radial local solution to utt−urr−n−1 rur= (1+t)−µ 2(p−1)|u|p, r >0, u(0,r) = 0, ut(0,r) =εg(r).(10) withp >1andp < pF(µ 2−1)ifµ >2. Assume in addition that −1<¯k <2 p−1−µ 2. (11) Then, given ε >0, the lifespan T(ε)>0of classical solutions to (10)satisfies T(ε)≤Cε−2(p−1) 4−(µ+2¯k)(p−1), (12) withC >0, independent of ε. Remark 6 The assumption p < pF(µ 2−1) ifµ >2 guarantees that the range of admissible ¯kin (11) is not empty. In the case µ= 0 Theorem 2 coincides with Takamura’s result in [T]. In the proof of T heorem 2, we will follows the same approach of that paper. 3 Proof of the main results 3.1 Proof of Theorem 2 We recall the crucial lemma of [T].8 Felisia Angela Chiarello et al. Lemma 1 Letn≥2andm= [n/2]. Given a smooth function g=g(|x|)withx∈Rn, we set r=|x|and we consider g=g(r). Let us denote by u0(t,r)the solution of the free wave problem /braceleftBigg /squareu0= 0 ( t,r)∈[0,∞)×[0,∞) u0(0,r) = 0, u0 t(0,r) =g(r). Letu=u(t,r)be a solution to utt−urr−n−1 rur=F(t,u) (13) with the initial condition u(0,r) = 0, ut(0,r) =εg(r), r∈[0,∞). (14) IfFis nonnegative, there exists a constant δm>0such that u(t,r)≥εu0(t,r)+1 8rm/integraldisplayt 0dτ/integraldisplayr+t+τ r−t+τλmF(t,u(t,λ))dλ, (15) u0(t,r)≥1 8rm/integraldisplayr+t r−tλmg(λ)dλ, (16) provided r−t≥2 δmt >0. The constant δmin the previous lemma is described in [T, Lemma 2.5]; it depends on the spa ce dimension, in particular it changes accordingly with the different repr esentations of the free wave solution in odd and even dimension. We are ready to prove that if (9) holds, then the solution of (10) blo ws up in finite time even for small ε. Let us fix δ >0; we define a blow-up set, Σδ=/braceleftBig (t,r)∈(0,∞)2:r−t≥max/braceleftbigg2 δmt,δ/bracerightbigg/bracerightBig , (17) whereδm>0 is the constant given in Lemma 1. Combining the assumption (10) with the formulas (15) and (16), for any ( t,r)∈Σδ,it holds u(t,r)≥εu0(t,r)≥ε 8rm/integraldisplayr+t r−tλmg(λ)dλ≥Mε 8rm/integraldisplayr+t r−tλm(1+λ)−(¯k+1)dλ. Then, (17) implies that u(t,r)≥Mε 8rm/parenleftbigg1+δ δ/parenrightbigg−(¯k+1)/integraldisplayr+t r−tλm−(¯k+1)dλ ≥Mε 8rm/parenleftbigg1+δ δ/parenrightbigg−(¯k+1) (r+t)−(¯k+1)/integraldisplayr+t r−tλmdλ≥Mε 8/parenleftbigg1+δ δ/parenrightbigg−(¯k+1)(r−t)m2t rm(r+t)¯k+1. Since (t,r)∈Σδ, we have u(t,r)≥C0tm+1 rm(r+t)¯k+1, where we set C0=ε2m−2M δmm/parenleftbiggδ 1+δ/parenrightbigg¯k+1 >0. (18)Fujita modified exponent for scale invariant damped semilin ear wave equations. 9 Now we assume an estimate of the form u(t,r)≥Cta rm(r+t)bfor (t,r)∈Σδ, (19) wherea,b, andCare positive constant. In particular, (19) holds true for a=m+1,b=¯k+1 andC=C0. Beingg≥0, from (16) we deduce u0≥0. Combining (15) and (19), for ( t,r)∈Σδ, we get u(t,r)≥1 8rm/integraldisplayt 0dτ/integraldisplayr+t−τ r−t+τλm (1+τ)µ 2(p−1)|u(τ,λ)|pdλ (20) ≥Cp 8rm/integraldisplayt 0τpa (1+τ)µ 2(p−1)dτ/integraldisplayr+t−τ r−t+τλm(1−p)(λ+τ)−pbdλ ≥Cp 8rm(r+t)pb+m(p−1)/integraldisplayt 0τpa (1+τ)µ 2(p−1)dτ/integraldisplayr+t−τ r−t+τdλ ≥Cp 4rm(r+t)pb+m(p−1)/integraldisplayt 0(t−τ) (1+τ)µ 2(p−1)τpadτ. By means of integration by parts, we obtain /integraldisplayt 0(t−τ)τpa (1+τ)µ 2(p−1)dτ≥1 (1+t)µ 2(p−1)/integraldisplayt 0(t−τ)τpadτ≥1 (1+t)µ 2(p−1)tpa+2 (pa+1)(pa+2). While searching a finite lifespan of a solution, it is not restrictive to ass umet >1. We have /integraldisplayt 0(t−τ)τpa (1+τ)p−1dτ≥tp(a−µ 2)+2+µ 2 2p−1(pa+1)(pa+2). (21) Let (t,r)∈Σδ, from (19)-(21), we can conclude u(t,r)≥C∗ta∗ rm(r+t)b∗for (t,r)∈Σδ, (22) with a∗=p/parenleftBig a−µ 2/parenrightBig +2+µ 2, b∗=pb+m(p−1), C∗=(C/2)p 2(pa+2)2. Let us define the sequences {ak},{bk},{Ck}fork∈Nby ak+1=p/parenleftBig ak−µ 2/parenrightBig +2+µ 2, a 1=m+1, (23) bk+1=pbk+m(p−1), b1=¯k+1, (24) Ck+1=(Ck/2)p 2(pak+2)2, C1=C0, (25) whereC0is defined by (18). Hence, we have ak+1=pk/parenleftbigg m+1−µ 2+2 p−1/parenrightbigg +µ 2−2 p−1, (26) bk+1=pk(¯k+1+m)−m, (27) Ck+1≥KCp k p2k(28)10 Felisia Angela Chiarello et al. for some constant K=K(p,µ,m)>0 independent of k. The relation (28) implies that for any k≥1 it holds Ck+1≥exp/parenleftbig pk(log(C0)−Sp(k))/parenrightbig , (29) Sp(k) =Σk j=0dj, (30) d0= 0 and dj=jlog(p2)−logK pjforj≥1. (31) We note that dj>0 for sufficiently large j.Since lim j→∞dj+1/dj= 1/p,the sequence Sp(k) converges for p >1 by using the ratio criterion for series with positive terms. Hence, t here is a positive constant Sp,K≥Sp(k) for any k∈N, so that Ck+1≥exp(pk(log(C0)−Sp,K)). (32) Therefore, by (22), (26)- (29), we obtain u(r,t)≥(r+t)m rmt−µ 2+2 p−1exp(pkJ(t,r)), (33) where J(t,r) := log(C0)−Sp,K+/parenleftBig m+1−µ 2+2 p−1/parenrightBig logt−(¯k+1+m)log(r+t). Thus if we prove that there exists ( t0,r0)∈Σδsuch that J(t0,r0)>0, then we can conclude that the solution to (10) blows up in finite time, in fact u(t0,r0)→ ∞fork→ ∞. By the definition of J=J(t,r), we find that J(t,r)>0 if /parenleftBig2 p−1−µ 2−¯k/parenrightBig logt >log/parenleftBigeSp,K C0/parenleftBig 2+r−t t/parenrightBig¯k+1+m/parenrightBig . In particular, we can take ( t,r) = (t,t+max{2t δm,δ})∈Σδ; then, it is enough to prove that /parenleftBig2 p−1−µ 2−¯k/parenrightBig logt >log/parenleftBigeSp,K C0/parenleftBig 2+2 δm/parenrightBig¯k+1+m/parenrightBig . Now, the crucial assumption (9) comes into play. The coefficient in th e left side is positive and by using (18) we find that J(t,r)>0 provided t > Cε−(2 p−1−µ 2−¯k)−1 , (34) where C=/parenleftBigeSp,Kδm m 2m−2M/parenleftBig1+δ δ/parenrightBig¯k+1/parenleftBig 2+2 δm/parenrightBig1+¯k+m/parenrightBig 1 2 p−1−µ 2−¯k, which is positive. As by-product, the inequality (34) gives the lifespa n estimate (12) and conclude the proof of Theorem 2.Fujita modified exponent for scale invariant damped semilin ear wave equations. 11 3.2 Proof of Theorem 1 We start rewriting the Cauchy problem (1) as a nonlinear wave equat ion with a time dependent potential. Let v=v(t,x) be a solution of (1); we define u(t,x) := (1+ t)µ 2v(t,x). Then the function u=u(t,x) is a solution of the Cauchy problem utt−∆u= (1+t)−µ 2(p−1)|u|p+(µ 2(µ 2−1)−ν)u (1+t)2, t≥0, x∈Rn, u(0,x) = 0, ut(0,x) =εg(x).(35) Ifgis radial, then uis radial and it satisfies equations (13) and (14) with F(t,u) = (1+ t)−µ 2(p−1)|u|p+/parenleftBigµ 2/parenleftBigµ 2−1/parenrightBig −ν/parenrightBigu (1+t)2. Let us fix δ >0 and use the same notation of the proof od Theorem 2. Since we are assuming µ 2(µ 2−1)−ν≥0, by comparison lemma, see [T, Lemma 2.9], we deduce u >0 inΣδ.Then it holds F(t,u)≥(1+t)−µ 2(p−1)|u|p; hence, by formula (15) in Lemma 1 we still derive the estimate (20). T hus, the proof of Theorem 2 guarantees the result of Theorem 1. References D. M. D’Abbicco. The threshold of effective damping for semilinear wave equat ions. Mathematical Methods in the Applied Sciences 38(2015), 1032–1045. DL. M. D’Abbicco, S. Lucente. NLWE with a special scale invariant damping in odd space dime nsion. 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A competition between Fujita and Strauss type exponents for blow-up of semi-linear wave equations with scale-invariant damping and mass , J. Differential Equations 266, (2019), 1176–1220. PT. A. Palmieri, Z. Tu. Lifespan of semilinear wave equation with scale invariant d issipation and mass and sub- Strauss power nonlinearity , Journal of Mathematical Analysis and Applications 470(1), (2019), 447-469. S. T. C. Sideris. Global behavior of solutions to nonlinear wave equations in three dimensions . Communications in Partial Differential Equations 8(1983), 1291-1323. T. H. Takamura. Blow-up for semilinear wave equations with s lowly decaying data in high dimensions. Differ- ential Integral Equations 8(1995), 647–661.

2020-02-09

The aim of this paper is to prove a blow up result of the solution for a semilinear scale invariant damped wave equation under a suitable decay condition on radial initial data. The admissible range for the power of the nonlinear term depends both on the damping coefficient and on the pointwise decay order of the initial data. In addition we give an upper bound estimate for the lifespan of the solution, in terms of the power of the nonlinearity, size and growth of initial data.

Fujita modified exponent for scale invariant damped semilinear wave equations

2002.03418v2

Comparative study of methodologies to compute the intrinsic Gilbert damping: interrelations, validity and physical consequences Filipe S. M. Guimar~ aes,J. R. Suckert, Jonathan Chico, Juba Bouaziz, Manuel dos Santos Dias, and Samir Lounis Peter Gr unberg Institut and Institute for Advanced Simulation, Forschungszentrum J ulich & JARA, 52425 J ulich, Germany (Dated: December 20, 2018) Relaxation eects are of primary importance in the description of magnetic excitations, leading to a myriad of methods addressing the phenomenological damping parameters. In this work, we consider several well-established forms of calculating the intrinsic Gilbert damping within a unied theoretical framework, mapping out their connections and the approximations required to derive each formula. This scheme enables a direct comparison of the dierent methods on the same footing and a consistent evaluation of their range of validity. Most methods lead to very similar results for the bulk ferromagnets Fe, Co and Ni, due to the low spin-orbit interaction strength and the absence of the spin pumping mechanism. The eects of inhomogeneities, temperature and other sources of nite electronic lifetime are often accounted for by an empirical broadening of the electronic energy levels. We show that the contribution to the damping introduced by this broadening is additive, and so can be extracted by comparing the results of the calculations performed with and without spin- orbit interaction. Starting from simulated ferromagnetic resonance spectra based on the underlying electronic structure, we unambiguously demonstrate that the damping parameter obtained within the constant broadening approximation diverges for three-dimensional bulk magnets in the clean limit, while it remains nite for monolayers. Our work puts into perspective the several methods available to describe and compute the Gilbert damping, building a solid foundation for future investigations of magnetic relaxation eects in any kind of material. I. INTRODUCTION Dynamical processes lie at the core of magnetic manip- ulation. From the torques acting on the magnetic mo- ments to how fast they relax back to their equilibrium orientations, a material-specic time-dependent theory is essential to describe and predict their behavior. In most cases, the description of the time evolution of the magnetization is done via micromagnetics1or atomistic spin dynamics (ASD)2,3approaches, in which the mag- netization is considered either as a classical continuous vector eld or as individual 3D vectors on a discrete lattice, respectively. They have been successfully used to describe a plethora of magnetic phenomena, ranging from spin waves in low dimensional magnets4, domain walls5and skyrmion6dynamics to thermal stability of magnetic textures7. These approaches model the mag- netization dynamics via a phenomenological equation of motion that contains both precessional and relaxation terms. A rst attempt to address these processes was per- formed by Landau and Lifshitz (LL), by considering a Larmor-like precessional torque and adding to it a (weaker) damping term of relativistic origin8. Since its phenomenological inception in 1935, the precise nature of the relaxation processes has been a source of intense debate. In particular, the original LL formulation was found to not properly describe situations in which the damping was large. This problem was addressed by Gilbert, who introduced a Rayleigh-like dissipation term into the magnetic Lagrangian, thus obtaining the now-ubiquitous Landau-Lifshitz-Gilbert (LLG) equation9, dM dt= MB+ MMdM dt =e MBe MM(MB):(1) where >0 is the gyromagnetic factor, Mis the (spin) magnetic moment, Bis the time-dependent eective magnetic eld acting on M, andis the scalar damping parameter named after Gilbert. The upper form of the LLG equation is due to Gilbert, and the lower one shows that it is equivalent to a LL equation with a renormalized gyromagnetic factor, e = =(1 +2). The rst term in the right-hand side of Eq. (1) describes the precession of the magnetic moments around the eective eld, while the second term is the Gilbert damping one, that de- scribes the relaxation of the magnetic moments towards B. This equation corrects the previously mentioned issue for large values of , for which the original LL equation is expected to fail10,11. The ferromagnetic resonance (FMR) technique is one of the most common procedures to probe magnetiza- tion dynamics12, in which the damping parameter is re- lated to the linewidth of the obtained spectra13. Al- though many measurements have been carried out in bulk materials12,14{18, their description at low temperatures is still controversial19{22. This can be attributed to the dif- ferent intrinsic and extrinsic mechanisms that can con- tribute to the relaxation processes23{36. When varying the temperature, two distinct regimes could be identi- ed in the measured relaxation parameters37. For high temperatures, a proportionality between the linewidth and the temperature was observed in most of the exper-arXiv:1807.11808v3 [cond-mat.mes-hall] 19 Dec 20182 iments. It was called resistivity-like, due to the simi- larity with the temperature dependence of this quantity. A conductivity-like regime (linewidth inversely propor- tional to the temperature) was identied at low temper- atures for certain materials such as Ni15,17, but not for Fe18,38. It was also seen that dierent concentrations of impurities aected this low-temperature regime, even suppressing it altogether16. From the theoretical point-of-view, the calculation of the Gilbert parameter is a challenging problem due to the many dierent mechanisms that might be at play for a given material39,40. Perhaps this is why most of the theoretical approaches have focused on contributions to the damping from electronic origin. The ultimate goal then becomes the development of a predictive theory of the Gilbert damping parameter, based on the knowledge of a realistic electronic structure of the target magnetic material. The ongoing eorts to complete this quest have resulted in the development of a myriad of tech- niques21,22,37,41{43. Comparisons between a few of these approaches are available44,45, including experimental val- idation of some methods24,46, but a complete picture is still lacking. We clarify this subject by addressing most of the well- established methods to calculate the Gilbert damping from rst principles. First, we connect the many dif- ferent formulas, highlighting the approximations made in each step of their derivations, determining what con- tributions to the damping they contain, and establish- ing their range of validity. These are schematically illus- trated in Fig. 1. Second, we select a few approaches and evaluate the Gilbert damping within a unied and con- sistent framework, making use of a multi-orbital tight- binding theory based on rst-principles electronic struc- ture calculations. FMR simulations and the mapping of the slope of the inverse susceptibility are used to bench- mark the torque correlation methods based on the ex- change and spin-orbit torques. We apply these dierent techniques to bulk and monolayers of transition metals (Fe, Ni and Co), for which the spin pumping mecha- nism is not present and only the spin-orbit interaction (SOI) contributes to the relaxation. Disorder and tem- perature eects are included by an empirical broadening of the electronic energy levels37,43,47,48. Third, we engage a longstanding question regarding the behavior of the damping in the low-temperature and low-disorder limits: should the intrinsic contribution to the Gilbert damping diverge for clean systems? Our results using the con- stant broadening model demonstrate that the divergence is present in the clean limit of 3D systems but not of the 2D ones49, which we attest by eliminating the pos- sibility of them being caused by numerical convergence issues or dierent anisotropy elds. Our results also in- dicate that the limit !!0 is not responsible for the divergence of the intrinsic damping, as it is commonly attributed19,37,43,50. Finally, we propose a new way to obtain the spin-orbit contribution that excludes the c- titious temperature/disorder contribution caused by thearticial broadening51,52: they can be discounted by sub- tracting the values of damping calculated without SOI. For bulk systems, this yields the total damping, while in layered materials this method should also discount part of the spin pumping contribution. In Ref. 20, where tem- perature and disorder are included via a CPA analogy, a similar articial increase of for high temperatures was removed by including vertex corrections. This work is organized as follows. We start, in Sec. II, with a brief overview of the dierent methods proposed in the literature. In Sec. III, we explain the theory used to calculate the response functions. We then turn to the distinct theoretical forms of calculating the damping: In Sec. IV, we analyze the dierent approaches related to the spin-spin responses, while in Sec. V, the torque methods are explored. We then discuss the obtained re- sults and conclude in Sec. VII. The Hamiltonian used in the microscopic theory is given in Appendix A, while the anisotropy elds for the 3D and 2D systems together with the transverse dynamical magnetic susceptibility given by the LLG equation are given in Appendix B. II. OVERVIEW OF METHODS ADDRESSING INTRINSIC GILBERT DAMPING We now focus on the dierent methods to describe the microscopic contributions to the Gilbert parameter, which encompasses eects that transfers energy and an- gular momentum out of the magnetic system. Within these mechanisms, the relativistic SOI comes to the fore. This is often referred to as the intrinsic contribution to the damping, and was rst identied by Landau and Lif- shitz8. The origin of this damping mechanism lies in the non-hermiticity of the relativistic corrections to the spin Hamiltonian when the magnetization precesses26,27. The elementary magnetic excitations, called magnons, can also be damped via Stoner excitations (electron-hole pairs with opposite spins)33,34,53. Alternatively, the con- duction electrons can carry spin angular momentum even in absence of the SOI. This leads to damping via the spin- pumping mechanism32,54{56. Early models proposed to describe these processes al- ready argued that the interaction between the magnetic moments and the conduction electrons is a key ingre- dient57. This led to the so-called breathing Fermi sur- face model, where the shape of the Fermi surface de- pends on the orientation of the magnetization through the SOI41. This approach, however, could only capture the conductivity-like regime, which diverges at low tem- peratures. The decay of magnons into Stoner excitations was also considered early on39, describing well the exper- imental behavior of Ni but also missing the increase at larger temperatures of other materials. An important progress was made by Kambersky us- ing the spin-orbit torque correlation function to calculate the damping parameter37. This approach captures both conductivity- and resistivity-like behaviors, which were3 ↵↵noSOI <latexit sha1_base64="QiCQ2bOsL/EV3Sx85lkyZMPneI4=">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</latexit><latexit sha1_base64="QiCQ2bOsL/EV3Sx85lkyZMPneI4=">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</latexit><latexit sha1_base64="QiCQ2bOsL/EV3Sx85lkyZMPneI4=">AAAEXnicbVPRbtMwFPXWso3CWAcvSAjJIpoE0qiSCgkkHpg6taxCZSWs20RTVa7jNlYdJ7IdWBX5G/gaXuE7eONTcNpubdNeKcrJPcc+917lDmJGpbLtv1vbheK9nd29+6UHD/cfHZQPH1/KKBGYdHDEInE9QJIwyklHUcXIdSwICgeMXA3Gpxl/9Z0ISSN+oSYx6YVoxOmQYqRMql9+5Q0i5stJaF6ph1gcIPgazkDfU+RGpTz6et7Uul+27Io9DbgOnDmwwDza/cPCc8+PcBISrjBDUnYdO1a9FAlFMSO65CWSxAiP0Yh0DeQoJLKXTnvS8MhkfDiMhHm4gtPs8okUhTIr2yhDpAKZ57LkJq6bqOG7Xkp5nCjC8cxomDCoIpgNCPpUEKzYxACEBTW1QhwggbAyY1xxkTThVN3oUukItowPlEEkFE6UkXHyA0dhiLifeq1T3XV6qZfVghFLLUfrnMJdKES4QZD4+o718yRNFiTNkxdiQRqcY5utBdsM86xbX7AuybP1hk7Tej+d/SUNrfOCTzUjGN8KauuCy/rSZAbDDX137iQBUiu6/AQ7tazWZNltVfHtY0N3q+YmRobKY4iPGFnBx5bz3qp6x56go0B5YpZd/sjd2D5rusY0Duitq7vm+vncbc0aYGYNFbQc6IkM6ZJZKCe/Puvgslpx7Irz5Y118mG+WnvgGXgBXgIHvAUn4Ay0QQdg8BP8Ar/Bn8K/4k5xv3gwk25vzc88AStRfPofnDSANQ==</latexit><latexit sha1_base64="QiCQ2bOsL/EV3Sx85lkyZMPneI4=">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</latexit> Inversion Inversion for low frequencies + sum ruleDyson equation of susceptibilityEquation of motion of susceptibility Spectral representation at T=0KSpectral representation at T=0KLow SOI Low spin pumpingNo spin pumpingComputational costsFull SOI Spin pumpingFMR linewidthSlope of inverse susceptibility Slope of inverse mean-field susceptibility Exchange torque correlation at Fermi surfaceSlope of mean-field SO-torque susceptibility with SOIProduct of spectral functions of opposite spinsSlope of mean-field susceptibilitySpin correlation at Fermi surfaceSO-torque correlation at Fermi surface with SOISpin response methodsTorque response methods Equation of motion of susceptibility + perturbation theory Dyson equation for Green function + Orbital quenchingSlope of SO-torque susceptibility without SOI Spectral representation at T=0KSlope of mean-field SO-torque susceptibility without SOISO-torque correlation at Fermi surface without SOILarge broadening Figure 1. Diagram exhibiting the dierent methods investigated in this work, their connections and range of validity. Two groups are identied: one related to the spin susceptibility (spin response methods), including the ferromagnetic resonance and the slope of the inverse susceptibility that involves a direct mapping of this quantity to the LLG equation; and the other associated with torque responses, for which approximations need to be taken. The steps indicated by solid lines represent exact connections, while dashed arrows involve some kind of approximation. The arrow on the left points from the methods that require less computational power (lower part) to the more demanding ones (upper part). Boxes are hyper-linked with the respective equations and sections. shown to originate from the intra- and interband transi- tions, respectively58. Recently, this so-called torque cor- relation method was re-obtained using a dierent per- turbative approach19, spurring discussions about the va- lidity of the obtained results, specially the divergence caused by the intraband transitions22. A similar method also based on torque correlation functions was developed using a scattering theory approach42involving the ex- change torque operator instead of the spin-orbit torque one. Results obtained in this way also present diverg- ing behavior in the clean limit of 3D structures20. A similar scattering framework was used to explain the enhancement of the Gilbert damping due to the spin pumping in thin lms32. Yet another method relating the Gilbert damping to the spin-spin response was pro- posed and related to the existing spin-orbit torque cor- relation method43. It also presented diverging intraband contributions when the parameter used to broaden the delta functions (which mimics the eect of disorder or temperature) was taken to zero59. The vertex correc- tions proposed in Ref. 59 did not remove this diver- gence. More recently, Costa and Muniz21showed that the damping parameters of layered structures remain -nite in the zero broadening limit, when extracted directly from the linewidth of the dynamical magnetic suscepti- bility (within the random phase approximation). Several of these methods have been implemented for material-specic calculations20,47,49,58,60{62, and some approaches were compared and related43{45,63. In this work, we start our analysis with the uniform frequency- dependent spin-spin susceptibility, which is measured ex- perimentally in FMR setups, to derive the other expres- sions for the damping parameter based on the spin- and torque-correlation methods. III. MICROSCOPIC THEORY We begin by setting the grounds of the theory we use to evaluate the dierent formulas of the Gilbert damping on equal footing. The electronic structure of the system is described by the mean-eld Hamiltonian ^H=^H0+^Hxc+^HSOI+^Hext: (2) The paramagnetic band structure is described by ^H0 within a multi-orbital tight-binding parametrization. An4 eective local electron-electron interaction within the mean-eld approximation is included in ^Hxc, which is re- sponsible for ferromagnetism. We also account for spin- orbit interaction through ^HSOI, and the interaction with external static magnetic elds via ^Hext. The explicit forms of all the terms are given in Appendix A. In this work, we investigate the dierent methods to compute the intrinsic Gilbert damping utilizing the pro- totypical bulk magnets Fe (bcc), Co (fcc) and Ni (fcc), and also square lattices corresponding to the (001) planes of those materials, with the same nearest-neighbor dis- tances as in its bulk forms. For simplicity, we consider the spin-orbit interaction and the local eective Coulomb interaction only on the dorbitals, with U= 1 eV64{66for all systems, and the spin-orbit strengths Fe SOI= 54 meV67,Co SOI= 70 meV68, andNi SOI= 133 meV68. The magnetic ground state is found by self-consistently enforcing charge neutrality for the bulk materials69. For the monolayer cases, the total number of electrons in the atomic plane is decreased to n= 7:3 (Fe),n= 8:1 (Co) and n= 9:0 (Ni), as the re- maining charge spills into the vacuum (which we are not explicitly taking into account within the model). The ground-state properties (spin moment M, orbital mo- mentM`and magnetic anisotropy energy K) obtained within this framework are listed in Table I. The easy axis for all the bulk systems and the monolayers were found to be along the (001) direction. We emphasize that our goal is not to achieve the most realistic description of the electronic structure of these materials, but rather to de- ne a concrete set of cases that allow us to compare the dierent methods to compute the Gilbert damping. The magnetic excitations are described using linear re- sponse theory, where the transverse magnetic response M(t) due to an oscillatory magnetic eld B(t) is given by70 M(t) =Z dt0(tt0)B(t0); (3) where the convention to sum over repeated indices of the components =fx;y;zgis used. This approach captures the orbitally-averaged part of the response. The bulk monolayer bcc Fe fcc Co fcc Ni Fe Co Ni M(B) 2.32 1.48 0.43 2.90 1.90 0.96 M`(B) 0.072 0.079 0.055 0.28 0.22 0.20 K(meV) 0.19 0.26 0.084 1.7 1.8 1.9 Table I. Ground state properties of the investigated systems. MandM`denotes the spin and orbital magnetic moments, respectively. Values obtained for = 1:36 meV. The mag- netic anisotropy constant Kis obtained from the anisotropy elds given by Eq. (B3).magnetic susceptibility is given by (tt0) =4 ^S(t);^S(t0) = 4i ^S(t);^S(t0); (4) in atomic units. ^S(t) is the-component of the spin op- erator. In the rst line of the equation above, we reprise the double-bracket notation of Zubarev for the spin-spin retarded Green function71. This notation is convenient for the derivations of Sec. V. For the crystal symmetries of the systems we are in- terested in, it is convenient to work in the circular ba- sis^S=^Sxi^Sy, which diagonalizes the susceptibil- ity matrix with components +(t) and+(t). The frequency- and wave vector-dependent transverse suscep- tibility+(q;!) is obtained within the random phase approximation (RPA), which captures the collective spin wave modes21,72, as well as the possible decay into particle-hole excitations (Stoner modes) described by the single-particle response function + 0(!). Considering matrices that take into account the orbital dependency, the two susceptibilities are related by [+]1= [+ 0]11 4U: (5) Here,U=Uis a matrix with the eective lo- cal Coulomb interaction strength within the dorbitals. It plays a similar role to the exchange-correlation ker- nel in the adiabatic local-density approximation of time- dependent DFT calculations73. We dene the transverse magnetic response of the system by summing the suscep- tibility matrix over all the dorbitals. The uniform single particle transverse susceptibility + 0(!) =+ 0(q= 0;!), obtained within the mean- eld approximation, is expressed in terms of the single- particle Green functions as + 0;(!) =1 NX kZF d" G"" (k;"+!) ImG## (k;") + ImG"" (k;") G## (k;"!)o : (6) The sum is over the wave vectors in the rst Brillouin zone, with Ntheir number. The indices ;represent orbitals and Fis the Fermi level. In the spirit of many preceding works37,43,47,48, the eect of temperature and disorder is modeled by in- troducing a constant band broadening on the en- ergy levels, such that G(!)!G(!+ i). The imag- inary part of the Green function is then dened as ImG(!) =1 2ifG(!+ i)G(!i)g. This ap- proach attempts to capture all the intrinsic eects origi- nated from the electronic structure of the system. The imaginary part of the susceptibility is related to the energy dissipation of the system74, encoding the relaxation mechanism of the magnetization towards equilibrium. The damping parameter is then obtained5 by mapping the transverse magnetic susceptibility ob- tained from the quantum mechanical calculation de- scribed above to the phenomenological form provided by the LLG, Eq. (1). On the following sections, we present dierent mapping procedures involving several approx- imations and explore their range of validity when the broadening is taken to zero (clean limit). IV. SPIN RESPONSE METHODS A. Ferromagnetic resonance Magnetic excitations can be investigated by applying time-dependent perturbations. This is done in FMR ex- periments where the magnetic sample is subjected to a static magnetic eld and an oscillatory radio-frequency one. By varying either the strength of the static compo- nent or the frequency of the oscillatory eld, the system can be driven through magnetic resonance. This setup yields the uniform mode of the transverse magnetic sus- ceptibility. As the Gilbert parameter describes the relax- ation mechanisms of the magnetization, it is related to the linewidth of the resonance peak21,75. We simulate this kind of experiments by calculating the transverse magnetic response relying on the linear response theory discussed in Sec. III, and mapping the imaginary part of the susceptibility into the result ob- tained from the LLG equation (see Appendix B), Im+(!) =2 !M [! (Bext+Ban)]2+ (!)2:(7) When xing the frequency and varying Bext;z, this func- tion presents a resonance at Bres= (! Ban;z)= with linewidth given by the full width at half maxi- mum
B= 2!= . On the other hand, when the eld is kept xed and the frequency is varied, the res- onance is located at !res= (Bext;z+Ban;z)=p 1 +2 with full width at half maximum approximately given by
!2 jBext;z+Ban;zj, in the limit 175. The Gilbert parameter can then be obtained either by tting Eq. (7) or through the ratio between the linewidth and the resonance position. In this sense, a divergence of the damping when !0 seems counter-intuitive, since this would imply that either the resonance position ( Bres or!res) goes to zero or that the corresponding linewidth increases drastically. In the presence of SOI, the SU(2) rotational symmetry is broken and the anisotropy eld Ban;zshifts the resonance position to a nite value | it costs a nite amount of energy to set the magnetization into precession76. Therefore, the divergence of the damp- ing parameter can only happen if the linewidth increases and goes to innity. To verify this claim, we simulate FMR experiments in fcc Co bulk by calculating the imaginary part of the transverse magnetic susceptibility as a function of the frequency!, in the presence of the spin-orbit interac- tion. In Fig. 2a, we present the obtained spectra fordierent values of the broadening . When a relatively large value of the broadening is used, = 13:6 meV (solid curve), the spectra displays a broad resonance peak, which can be characterized by a value = 1:3102, obtained by tting the linear response data with Eq. (7). When the broadening of the energy levels is decreased to= 4:1 meV (dashed curve), the peak shifts and be- comes sharper ( = 3:8103), as one intuitively ex- pects when disorder and/or temperature decreases. No- tice that most of the change in is due to the change in the peak width, while the resonance shift is relatively small. This can be viewed as a consequence of the smaller energy overlap between the bands, which decrease possi- ble interband transitions58. Surprisingly, by further de- creasing the broadening to = 0:41 meV (dotted curve), the peak becomes broader when compared to the pre- vious case, with = 5:6103. This counter-intuitive result represents a shorter lifetime of the magnetic excita- tion when the electronic lifetime (mean time between two successive scattering events) =1becomes longer. Obtaining the damping from the FMR curves is com- putationally demanding, though. The response function must be calculated for many frequencies (or magnetic elds) to resolve the peak. For the case of low broaden- ings that require many k-points in the Brillouin zone for a converged result, this task becomes prohibitive. In the next section, we provide alternative methods to obtain the Gilbert parameter based on the static limit of the susceptibility, and compare their outcomes with the ones obtained using the resonance approach. B. Inverse Susceptibility Method We proceed now to investigate a dierent mapping of the microscopic transverse susceptibility to the LLG equation and possible approximations to simplify the cal- culation of the Gilbert damping. From Eq. (B4), one can see thatdenes the slope of the imaginary part of the inverse susceptibility43, i.e., = 2 Mlim !!0Im[+(!)]1 !: (8) We will refer to this as the inverse susceptibility method (ISM). The mapping to the LLG model of the slope at small frequencies has a great advantage over the FMR one since it only requires a single frequency-point calcula- tion, instead of a full sweep over frequencies or magnetic elds for the tting procedure. In Fig. 2b, we display the damping parameter for bcc Fe, fcc Co and fcc Ni bulk systems calculated as a func- tion of the electronic energy broadening. We also include the results obtained from the FMR approach (solid sym- bols), which compare well with the ISM given in Eq. (8). Note that although Eq. (8) has an explicit linear depen- dence on the spin moment M, the susceptibility implic- itly depends on its value. The obtained curves are in- versely related to M: highest for Ni ( M0:45B), low-6 0.5 0.55 0.6 0.6502468·105 Frequencyω(meV)−Imχ−+(ω) (states/eV)η1= 13.6 meV→α= 1.3·10−2 η2= 4.1 meV→α= 3.8·10−3 η3= 0.41 meV→α= 5.6·10−3 1 10 10010−210−1100101102 Broadening η(meV)Gilbert damping αXC-TCM ISM Fe Fe 5·λSOI Fe 10·λSOI 1 10 10010−310−210−1100101 Broadening η(meV)Gilbert damping αISM Fe ISM Co ISM Co no SOI FMR Co ISM Ni 1 10 10010−310−210−1100101 Broadening η(meV)Gilbert damping αFe monolayer Co monolayer Ni monolayer10 100 1,000Temperature (K) 10 100 1,000Temperature (K) 10 100 1,000Temperature (K)(a) (b)(c) (d) Figure 2. Characteristics of the Gilbert damping in 3D and 2D systems in presence and absence of SOI. (a) Ferromagnetic resonance spectra for fcc Co, in presence of spin-orbit interaction and no external eld, calculated for three dierent decreasing broadenings 1= 13:6 meV (solid), 2= 4:1 meV (dashed) and 3= 0:41 meV (dotted). The values of the Gilbert damping given in the legend box, obtained by tting to Eq. (7), decrease from the rst case to the second, but increases again when is further decreased. (b) Gilbert damping in presence of spin-orbit interaction for bcc Fe (blue triangles), fcc Co (red circles, solid line) and fcc Ni (green squares) as a function of the broadening, obtained from the slope of the inverse susceptibility, Eq. (8). All values were computed with 108k-points in the full Brillouin zone. Solid red circles are the values obtained from the FMR spectra in (a), while the open red circles connected by dashed lines represent the damping parameter for fcc Co when SOI is not included in the calculations. (c) Damping parameter for bcc Fe for dierent SOI strenghts: SOI= 54:4 meV, 5 SOI, and 10 SOI. (d) Gilbert damping of Fe, Co and Ni monolayers in the presence of SOI. No increase in the Gilbert damping is seen when the broadening is decreased. est for Fe (M2:3B) and Co in-between ( M1:5B). This trend is conrmed by setting the SOI strength SOI to the same values for all the elements (not shown). The position of the minimum value of is connected with SOI, which determines when the intraband or interband transitions become more important58. To substantiate this claim, we employed the technique of articially scal- ing theSOI, as previously done in connection to the magnetic anisotropy energy77. The results are shown in Fig. 2c, where the SOI strength SOIof Fe bulk is mag- nied by factors of 5 and 10. Indeed, the minimum can clearly be seen to shift to larger values of .An important aspect to be considered is the conver- gence of Eq. (6) | failing to achieve numerical precision may give rise to spurious results49,78. This can be partly solved using sophisticated schemes to perform those cal- culations79,80. When the broadening is lowered, the con- vergence of the wave vector summation is aected by the increasingly dominant role of the poles of the Green functions in the vicinity of the Fermi energy. For that reason, to capture the intricacies of the electronic states | in particular, the important contributions from the small gaps opened by the weak SOI |, we calculated the slope of the response function using a very ne in-7 tegration mesh on the Brillouin zone reaching up to 109 k-points. The results in Fig. 2c also demonstrate that the divergence is not an issue of numerical convergence, since this behavior is shifted to larger values of broadenings, for which the convergence is more easily achieved. Nevertheless, such diverging eect only occurs in the presence of spin-orbit interaction. In Fig. 2b we also dis- play the values of for Co fcc obtained using the ISM when the SOI is not included in the calculations (cir- cles connected by dashed lines). In this case, noSOI lin- early goes to zero when the broadening is decreased21. The non-vanishing damping when SOI is not present can be interpreted as originating from the nite elec- tronic lifetimes introduced by the constant broadening parameter. As it stands, represents the coupling to a ctitious reservoir51,52providing dissipation mechanisms that physically should originate from disorder or temper- ature, for example. Obtaining the damping from the FMR spectra when SOI is not present requires an applied magnetic eld, such that the resonance frequency becomes nite and avoiding an innite response at zero frequency (repre- senting no cost of energy due to the rotational symmetry, i.e., the Goldstone mode). Nevertheless, the results pre- sented in Fig. 2d were obtained using the ISM without any applied eld. Calculations with an applied magnetic eld shifting the peak to the original anisotropy energy were indistinguishable from those values (with variations smaller than 3%). This is accordance to the phenomeno- logical expectations expressed through Eq. (B4), where the slope is independent of the magnetic eld. One can put our results for bulk ferromagnets into perspective by comparing with low dimensional systems. We investigated this case within our linear response ap- proach, using monolayers of Fe, Co and Ni. The calcu- lations follow the same procedure, except that the sum overkvectors in Eq. (6) is restricted to the 2D Brillouin zone. The results are presented as triangles (Fe), cir- cles (Co) and squares (Ni) connected by dotted lines in Fig. 2d, and once again exhibit a monotonous decay with the decrease of . We note that previous calculations of the damping parameter in thin lms also did not nd it to increase rapidly for decreasing broadening21,49. Besides the dimensionality, another main dierence from the bulk to the layered case is the larger anisotropy elds of the latter (see Table I). Nevertheless, this can- not explain the non-diverging behavior in the monolay- ers. We have already shown that by articially increas- ing the SOI strength of the bulk | and, consequently, its anisotropy eld |, the conductivity-like behavior of the damping occurs at even larger broadenings (see Fig. 2c). On the other hand, to rule out a possible divergence hap- pening at lower broadenings ( <0:1 meV, not reachable in our calculations), we have also scaled up SOIof the monolayers by one order of magnitude. This resulted in larger dampings, nonetheless, the same decreasing be- haviour with !0 was observed (not shown). There- fore, the divergence can only be attributed to the three-dimensionality of the ferromagnet. C. Approximate static limit methods We now look back to Fig. 1 and proceed to perform approximations on Eq. (8) in order to simplify the calcu- lations of the damping parameter. Here we follow Ref. 43. First, we use Eq. (5) that relates the RPA susceptibility matrix to the mean-eld response matrix 0, such that Im1Im1 0. Although Uis a real matrix, the sum over orbitals ( =P ) ends up mixing the real and imaginary parts of the matrix elements. Only when Re1 0=U=4 the relation above becomes an equality. This means that, within our model with Uacting only on thedorbitals,must also be dened by summing over those orbitals only. Under the previous assumption, we obtain 2 Mlim !!0Im[+ 0(!)]1 !: (9) This relation is only valid when + 0is decoupled from the other types of susceptibilities (transverse and longi- tudinal), as in the systems we investigate in this work. The damping parameter can therefore be obtained from the single-particle transverse susceptibility 0. For frequencies !in the meV range (where the col- lective spin excitations are located), + 0has a simple !-dependence81: + 0(!)Re0(0) + i!Im0 0(0): (10) where0 0(0) =d+ 0 d! !=0. These results are valid also in the presence of spin-orbit coupling. Using Eq. (10), the Gilbert damping can be written as 2 M Re+ 0(0)2lim !!0Im+ 0(!) !:(11) Although the expansion of the susceptibility for low fre- quencies was used, no extra approximation is employed, since Eq. (9) is calculated in the limit !!0. Re+ 0(0) can be obtained using the sum rule that relates the static susceptibility with the magnetic moments76. For 3dtransition metals, the external and the spin-orbit elds are three orders of magnitude smaller than U, and so the static susceptibility of the bulk systems reads Re+ 0(0)4=U. Thus,  MU2 8lim !!0Im+ 0(!) !: (12) Finally, from Eq. (6) it is possible to show that Eq. (12)8 simplies as  MU2 2NX k;TrfImG(k;F)^SImG(k;F)^S+g = 2MNX k;TrfImG(k;F)^T xcImG(k;F)^T+ xcg = MU2 8NX k;n# (k;F)n" (k;F): (13) wheren (k;F) =1 ImG (k;F) is the matrix el- ement of the spectral function of spin calculated at kandF. The second equation is written in terms of the \exchange-correlation torque operator", T xc= i^S;^Hxc =iUM^S. This result is equivalent to the one obtained in Ref. 42, which we reference as theexchange torque correlation method (XC-TCM) | although, in reality, it relates with the spin-spin re- sponse. The last step in Eq. (13) connects the damping with the product of spectral functions of opposite spins at the Fermi level, as shown theoretically in Ref. 81 and conrmed experimentally in Ref. 46. In Fig. 2c, we compare the results obtained with this approximated method with the ISM described before, for the dierent values of SOI scalings. For the bulk tran- sition metals we investigate, the approximation is very good, since the SOI is relatively small. In fact, even when the SOI is scaled one order of magnitude higher, the results of the XC-TCM are still very good. The formulas in Eq. (13) show that we have arrived at the bottom of the triangle in Fig. 1. These forms do not involve an integral over energy, which simplies substantially the calculation of . For that reason, they are suitable for rst-principles approaches (e.g., Refs. 20 and 62). This concludes our investigations of the spin response methods. In the next section, we take a dierent path to calculate the Gilbert damping. V. TORQUE RESPONSE METHODS Despite the simplicity of the methods based on the spin susceptibility discussed in the previous section, seminal work was based on a dierent type of response function. The main idea, rst proposed by Kambersky37, is to di- rectly relate to the spin-orbit interaction. Here, our aim is twofold. First, we connect the spin susceptibility with the spin-orbit torque response via the equation of motion, clarifying the damping mechanisms captured by this formalism. Second, we compare the results obtained with both types of methods. We start with the equation of motion for the spin-spin susceptibility. Its time-Fourier transform can be written as19 ! ^S;^S+ !=M+ ^S;^H ;^S+ !; (14)whereM=2 ^Sz . From the Hamiltonian given in Eq. (2), the commutator [ ^S;^H has four contributions: kinetic (spin currents, from ^H0), exchange torque (from ^Hxc), external torque (from ^Hext) and spin-orbit torque (from ^HSOI). In presence of SOI, the total spin magnetic moment is not a conserved quantity and spin angular momentum can be transferred to the orbital degrees of freedom. For bulk systems subjected to static external elds and in the present approximation for the electron- electron interaction, the only two non-vanishing torques are due to the external eld and the spin-orbit interac- tion. It also follows from these assumptions that the mechanisms that contribute to the relaxation arises then from the spin-orbit torques ^T SOI=i^S;^HSOI and from the broadening of the energy levels . It can be shown19that the inverse of the susceptibility +(!) = ^S;^S+ !is given by  +(!)1= + noSOI (!)1 1 ++ noSOI (!) (!)1  + noSOI (!)1(!): (15) Here,+ noSOI (!) is the susceptibility calculated excluding the SOI contribution to the Hamiltonian. The connection between the two susceptibilities in Eq. (15) is provided by the quantity M2(!) = i ^T SOI;^S+ + ^T SOI;^T+ SOI !:(16) Using Eq. (8), and noticing that the rst term on the right-hand side of the equation above does not contribute to the imaginary part, we nd =noSOI2 Mlim !!0Im ^T SOI;^T+ SOI ! !: (17) noSOI is the contribution obtained by inputting + noSOI (!) into Eq. (8), which is nite due to the broad- ening. Kambersky37rst obtained this same result following a dierent approach. In our framework, this would involve starting from Eq. (5) and exploiting the consequences of the fact that the collective spin excitations ( !meV) have low frequencies when compared to the exchange en- ergy (UeV). On the other hand, Hankiewicz et al.19 described the same expansion for low SOI, and justied its use for !. Bext. Finally, Edwards22shows that this formula is equivalent to a perturbation theory cor- rect to2 SOI(compared to Bext!). For that rea- son, he suggests that the states used in the calculation of ^T SOI;^T+ SOI !should not include SOI, since the op- erator ^T SOI/SOI. Due to the orbital quenching in the states without SOI, this leads to the absence of intra- band contributions and, consequently, of the divergent behavior for !082. In this approach, temperature and disorder eects are included in noSOI (shown in Fig. 2d for Co), while the9 spin-orbit intrinsic broadening is calculated by the sec- ond term in Eq. (17), which can also be obtained as noSOI . An extra advantage of calculating the damp- ing as the aforementioned dierence is that one explic- itly subtracts the contributions introduced by , provid- ing similar results to those obtained with vertex correc- tions20. Considering the torque-torque response within the mean-eld approximation (an exact result in the per- turbative approach22), we obtain, similarly to Eq. (13), noSOI = 2 MNX kTrfImG(k;F)^T SOIImG(k;F)^T+ SOIg: (18) In this formula, the involved quantities are matrices in spin and orbital indices and the trace runs over both. This is known as Kambersy's formula, commonly used in the literature43,44,47,49,58, which we refer to as spin-orbit torque correlation method (SO-TCM). As in Eq. (13), it relates the damping to Fermi level quantities only. When the SOI is not included in the calculation of the Green functionsG(k;F) and enters only through the torque operators, we name it perturbative SO-TCM22. These methods are placed at the bottom right of Fig. 1, with the main approximations required indicated by the long dashed arrows. We now proceed to compare these approaches with the ISM explained in Sec. IV B. Fig. 3 presents the calcula- tions of the SOI contribution to the damping parameter of bulk Fe (a), Co (b) and Ni (c) using the SO-TCM ob- tained in Eq. (18), when no external eld is applied. Both approaches, including SOI (red curve with squares) in the Green functions or not (green curve with triangles), are shown. For a meaningful comparison, we compute noSOI within the ISM. We rst note that the perturbative approach suggested by Edwards22describes reasonably well the large broad- ening range (i.e., mostly given by the interband transi- tions), but deviates from the other approaches for low . This is an expected behaviour since it does not include the intraband transitions that display the 1behav- ior within the constant broadening model. In the clean limit, the Gilbert damping computed from the pertur- bative SO-TCM approaches zero for all elements, in a very monotonic way for Co and Ni, but not for Fe. This method is thus found to be in agreement with the other ones only when SOI. The SO-TCM formula in- cluding the SOI in the states (i.e., Kambersky's formula) matches very well obtained within ISM in the whole range of broadenings. Finally, after demonstrating that the SO-TCM pro- vides very similar results to the ISM, we can use it to resolve the wave-vector-dependent contributions to the Gilbert parameter by planes in the reciprocal space, as (kmax z) =kmax zX jkzj(kz); (19) 1 10 10010−310−2Gilbert damping α(a)Fe ISM ( α−αnoSOI ) SO-TCM perturbative SO-TCM10 100 1,000Temperature [K] 1 10 10010−410−310−2Gilbert damping α(b)Co 10 100 1,000 1 10 10010−410−310−210−1100 Broadening η[meV]Gilbert damping α(c)Ni 10 100 1,000Figure 3. Comparison between noSOI for (a) Fe bcc, (b) Co fcc and (c) Ni fcc, obtained using the inverse susceptibility method (ISM) with the spin-orbit-torque correlation method (SO-TCM) with and without SOI in the states (perturbative SO-TCM). All the points were computed with 108k-points in the full Brillouin zone. where(kz) is given by the right-hand side of Eq. (18) summed over kx;ky. The result, displayed in Fig. 4, uses 100 million k-points for all curves and shows the expected divergence in presence of SOI and a decrease with when this interaction is absent. In every case, most of the con- tribution arises from the rst half ( kmax z<0:4). Note that when the broadening of the energy levels is low, the integrated alpha without SOI (Fig. 4b) displays step- like contributions, while when SOI is present, they are smoother. This is a consequence of the damping being10 01234Gilbert damping α(·10−2)(a) With SOI η(meV): 0.14 0.41 0.68 1.1 1.4 4.1 6.8 10.9 13.6 40.8 0 0.2 0.4 0.6 0.8 100.20.40.60.81 kmax zGilbert damping α(·10−3)(b) Without SOIDecreasingη Decreasingη Figure 4. Integrated Gilbert damping for fcc Co as a function ofkzplotted against the maximum value, kmax z(see Eq. (19)), with SOI (a) and without SOI (b). The curves were obtained using the SO-TCM given in Eq. (18). Colors represent dier- ent values of the broadening (in units of meV). The value of forkmax z= 0 (i.e., a single value of kzin the sum) represents a two-dimensional system, whilst for kmax z= 1 the sum covers the whole 3D Brillouin zone. In the latter case, the damping decreases when is decreased without SOI, while it increases drastically when SOI is present. For 2D systems, the caused by interband transitions in the former and intra- band in the latter. The convergence of the previous results for the smallest including SOI were tested with respect to the total number of k-points in the Brillouin zone in Fig. 5. By going from 10 million to 10 billion k-points, the results vary20%. However, compared with the result shown in Fig. 4a, the damping gets even larger, corroborating once more the divergent results. VI. DISCUSSIONS In this section, we make a few nal remarks on the pre- viously obtained results and we go beyond bulk systems to comment on the approximations taken and additional physical mechanisms that may come into play in other 0 0.2 0.4 0.6 0.8 1024 kmax zGilbert damping α(·10−2) k-points: 107 108 109 1010Figure 5. Integrated Gilbert damping for fcc Co as a func- tion ofkzplotted against the maximum value, kmax z, for = 0:14 meV and dierent amount of k-points (up to 10 billion) in the Brillouin zone. materials. We also provide a new analytical explanation for the divergence of the damping parameter within the constant broadening model. Our rst comment regards the application of static magnetic elds B. As described in Refs. 19 and 22, the approximations done in Eq. (15) to derive an expression forinvolves comparisons between the excitation en- ergy andB. However, all the results we have presented here were obtained in absence of static elds. We also performed calculations including external magnetic elds up toB7 T, and the computed damping parameter is weakly in uenced by their presence. We conclude that the validity of the SO-TCM formula given in Eq. (18) does not hinge on having a magnetic eld, supporting the arguments already given in Ref. 19. A further remark concerns the approximations made to obtain the mean-eld result in Eq. (12). We assumed that SOI is weak when using the magnetic sum rule. This approximation may break down when this is not the case. The spin pumping also aects the magnetic sum rule, which may worsen the agreement with the ISM results. Although this contribution is not present in the investigated (bulk-like) systems, it plays an important role in magnetic multilayers. This eect enhances the damping factor32,54,55. Furthermore, the SO-TCM ex- plicitly excludes spin pumping, as this is described by ^I S=i^S;^H0 , dropped from the equation of mo- tion. These validity conditions are indicated in Fig. 1 by the large blue rectangle (low SOI), red triangle (low spin pumping) and green rectangle (no spin pumping). Another mechanism that opens new spin relaxation channels is the coupling between transverse and longi- tudinal excitations induced by the SOI. This was one of the reasons raised in Ref. 21 to explain the divergence of the damping parameter. However, this is absent not only11 when the system has full spin rotational symmetry83, but also when rotational symmetry is broken by the SOI in 2D and 3D systems for the symmetries and materials we investigated. Even though the damping is nite in the rst two cases (as shown in Fig. 2d), the divergence is still present in the latter (Fig. 2b). We can also recognize that the mathematical expres- sion forin terms of the mean-eld susceptibility given in Eq. (12) is similar to the conductivity one (i.e., the slope of a response function)84| which leads to the same issues when approaching the clean limit ( !0). How- ever, the physical meaning is the exact opposite: While the divergence of the conductivity represents an innite acceleration of an ideal clean system, innite damping denotes a magnetic moment that is instantly relaxed in whichever direction it points (as d M=dt!0 for!1 ) | i.e., no dynamics10,11. This means that a clean 3D spin system is innitely viscous. Within the constant broadening model, the divergence of the Gilbert damp- ing can also be seen analytically by comparing Eq. (12) with the calculations of the torkance done in Ref. 48. By replacing the torque operator and the current density by the spin lowering and raising operators, respectively, the even contribution (in the magnetization) to the re- sponse function vanishes and only the odd one remains. In this approximation, it is also seen that only the Fermi surface quantities are left, while the Fermi sea does not contribute85. In the limit of low broadenings, this con- tribution is shown to diverge as 1. This divergence arises from intraband transitions which are still present in the clean limit, and originate from the nite electronic lifetimes introduced by the constant broadening approx- imation. The static limit ( !!0) is another reason that many authors considered to be behind the divergent damping behavior19,37,43,50. This limit is taken in Eq. (8) in or- der to eliminate the contribution of terms nonlinear in frequency from the inverse susceptibility (e.g., inertia ef- fects68,86). They can be present in the full microscopic calculation of the susceptibility but are not included in the phenomenological model discussed in Appendix B. Adding the quadratic term in frequency leads to an in- verse susceptibility given by Im[+(!)]1=! 2 M(!I) whereIis the o-diagonal element of the moment of iner- tia tensor86. The t to the expression linear in frequency then yields an eective e(!). In the vicinity of the res- onance frequency, e(!res) =!resI, which is clearly reduced in comparison to the one obtained in the static limit,e(0) =. According to Ref. 68, I=, which explains the discrepancy between the FMR and the ISM seen in Fig. 2b as !0. We can then conclude that the static limit is not the culprit behind the divergence of  in the clean limit.VII. CONCLUSIONS In this work, we presented a study of dierent meth- ods to calculate the intrinsic Gilbert damping , oering a panorama of how the approaches are related and their range of validity (see Fig. 1). They can be grouped into three main categories: the methods that directly employ the results of full microscopic calculations of the dynam- ical magnetic susceptibility (!) (FMR and ISM); the exchange-torque method (XC-TCM), which is also based on(!) but making use of the mean-eld approximation; and the spin-orbit torque-correlation method (SO-TCM), obtained from the (spin-orbit) torque-torque response via an equation of motion for (!). While the FMR, ISM and XC-TCM include all the contributions to the mag- netic relaxation, the SO-TCM provides only the intrinsic contribution due to the angular momentum transfer to the orbital degrees of freedom (not including, for exam- ple, the spin pumping mechanism). The XC- and SO- TCM, given by Eqs. (13) and (18), are predominant in the literature due to their simplicity in obtaining in terms of Fermi level quantities. It is important to note, however, that they rely on approximations that may not always be fulllled21. In order to implement and compare the dierent meth- ods, we constructed a unied underlying framework based on a multi-orbital tight-binding Hamiltonian using as case studies the prototypical bulk 3D systems: bcc Fe, fcc Co and fcc Ni. For this set of materials, the dierent methods lead to similar results for , showing that the corresponding approximations are well-founded. Even when the SOI strength is scaled up by one order of mag- nitude, this excellent agreement remains, as we explic- itly veried for bcc Fe. We found one method that falls out-of-line with the others in the clean limit, namely the perturbative form of the SO-TCM formula22,82. In this case, although the equation is identical to the well-known Kambersky formula, Eq. (18), the electronic states used to evaluate it do not include SOI. By comparison with the other methods, we conclude that the results obtained by the perturbative SO-TCM are only valid in the large broadening regime (compared to the SOI strength). Cen- tral to our analysis was a careful study of the convergence of our results with respect to the number of k-points, reaching up to 1010k-points in the full Brillouin zone. The behavior of is intimately connected with the con- stant broadening approximation for the electronic life- times. For high temperatures, the Gilbert damping in- creases with increasing temperature ( ), while for low temperatures it diverges for 3D ferromagnets (  1=), but not for 2D (ferromagnetic monolayers). Our calculations revealed that the high temperature values ofarise mostly from the broadening of the electronic states. In Ref. 20, the strongly increasing behaviour of for high temperatures was found to be spurious, and cured employing a more realistic treatment of disorder and temperature, and the so-called vertex corrections. We found that the contribution of the intrinsic SOI to 12 is additive to the one arising from the broadening, and can be easily extracted by performing a calculation of without SOI and subtracting this result from the SOI one,noSOI . Combined with the ISM, this provides a relatively simple and accurate way to obtain the in- trinsic damping, which discounts contributions from the additional broadening . This establishes an alternative way of accessing the high temperature regime of . The low-temperature divergence of when approach- ing the clean limit for 3D ferromagnets has also been the subject of much discussion. The rst diculty is in es- tablishing numerically whether this quantity actually di- verges or not. Our results consistently show an increase ofwith decreasing , down to the smallest achievable value of= 0:14 meV (Fig. 5), with no hints of a plateau being reached, but only when accounting for SOI. This divergence arises from the intraband contributions to , as discussed in Ref. 58. Refs. 22 and 82 used pertur- bation theory arguments to claim that such intraband contributions should be excluded. However, as we dis- cussed in Sec. IV B, adapting the formalism of Ref. 48 to the calculation of shows that these intraband terms are enabled by the constant broadening approximation, and so should be included in the calculations. Contrary to the high temperature regime, works that employ a more realistic treatment of disorder and temperature still nd the diverging behavior of 20,52. In real experiments, any kind of material disturbance such as disorder or temperature eects leads to a nite value of the damping. Besides that, a non-uniform com- ponent of the oscillatory magnetic eld (either from the apparatus itself or due to its limited penetration into the sample) induces excitations with nite wave vectors and nite linewidths39,87. A dierent way to determine the damping parameter is using the time-resolved Magneto- Optic Kerr Eect (TR-MOKE)40,88. It has the advan- tage that, as it accesses a smaller length scale ( 1µm) than FMR experiments (which probe the whole magnetic volume), the measured magnetic properties are more ho- mogeneous and thus the eect of linewidth broadening may be weaker. The magnetic excitations in nanomag- nets can also be probed by recent renements of FMR experimental setups89,90. Although the methods we described here are gen- eral, we did not explicitly addressed non-local sources of damping such as the spin-pumping32. As a future project, we plan to ascertain whether our conclusions have to be modied for systems where this mechanism is present. Systems that combine strong magnetic el- ements with heavy ones possessing strong SOI are ex- pected to have anisotropic properties, as well-known for the magnetic interactions91. It is then natural to explore when the Gilbert damping can also display signicant anisotropy, becoming a tensor instead of a scalar quan- tity47,78. Indeed, this has been observed experimentally in magnetic thin lms92,93. As the SOI, magnetic non- collinearity can also lead to other forms of damping in do- main walls and skyrmions50,94{98. From the microscopicpoint of view, the potential coupling between transverse and longitudinal degrees of freedom allowed by the non- collinear alignment should also be considered. Lastly, higher order terms in frequency, such as the moment of inertia68,86,99{101, might also become important in the dynamical magnetic susceptibility for large frequencies or for antiferromagnets, for instance. The description of magnetization dynamics of real ma- terials helps to design new spintronic devices able to con- trol the ow of information. Our work sheds light on fun- damental questions about the main relaxation descrip- tions used in the literature and sets ground for future theoretical predictions. Appendix A: Ground-state Hamiltonian In this Appendix, we give the explicit forms of the terms in the Hamiltonian written in Eq. 2. As the inves- tigated systems only have one atom in the unit cell, the site indices are omitted. The electronic hoppings in the lattice are described by ^H0=1 NX kX t(k)cy (k)c(k); (A1) withcy (k) andc(k) being the creation and annihila- tion operators of electrons with spin and wave vector kin the orbitals and, respectively. The tight-binding parameters t(k) were obtained by tting paramagnetic band structures from rst-principles calculations up to second nearest neighbors102, within the two-center ap- proximation103. The electron-electron interaction is characterized by a local Hubbard-like104interaction within the Lowde- Windsor approximation105, resulting in the mean-eld exchange-correlation term ^Hxc=X 2d U 2( M 0+X 2dn(200)) cy (k)c0(k): (A2) Here,Uis the local eective Coulomb interaction, M andare the-component of the magnetic moment vector (summed over the dorbitals) and of the Pauli matrix, respectively. nis the change in the occupation of orbitalcompared to the DFT calculations included in Eq. A1. Mandnare determined self-consistently. The atomic SOI is described by ^HSOI=X  0^L ^S 0cy (k)c0(k);(A3) whereLandSare thecomponents of the orbital and spin vector operators, respectively. The strength of the SOI,, is also obtained from rst-principles calculations.13 The interaction with a static magnetic eld Bextis described by ^Hext=B extX  0(^L 0+0)cy (k)c0(k); (A4) whereBis absorbed to B extand we used gL= 1 and gS= 2 as the Land e factors for the orbital and spin angular momentum. Appendix B: Phenomenology of FMR The semi-classical description of the magnetization is obtained using the Landau-Lifshitz-Gilbert (LLG) equa- tion (1)9. The eective eld acting on the magnetic mo- ment is obtained from the energy functional of the system asBe(t) =@E=@M. For the symmetries we investi- gate, the model energy106for the 3D cubic cases77can be written as E3D(M) =K4 M4(M2 xM2 y+M2 yM2 z+M2 xM2 z)M
Bext; (B1) while for 2D systems, E2D(M) =K2 M2M2 zM
Bext: (B2) Positive values of K4andK2yield easy magnetization direction along the (001) direction. We consider magnetic moments pointing along the easy axis, which denes the ^ zdirection. Static magnetic elds are applied along the same orientation. The magnetic moment is set into small angle precession, M=M^ z+Mx(t)^ x+My(t)^ y, by an oscillatory eld in the trans- verse plane, i.e., Bext(t) =Bext^ z+Bext(t). In this form, the eective eld (linear in the transverse components of the magnetization) is given by Be(t) =Ban(t)+Bext(t), with B3D an(t) =2K4 M2(Mx^ x+My^ y) , and B2D an=2K2 M^ z (B3) being the anisotropy elds for 3D and 2D systems, respec- tively. In the following expressions, K4andK2appear in the same way, so they are denoted by K. The Fourier transform of the linearized equation of mo- tion can be written using the circular components M= MxiMy. Within this convention, M=B= +=2 and +(!) =2 M [! (Bext+Ban)]i!; (B4) whereBan= 2K=M . ACKNOWLEDGMENTS We are very grateful to R. B. Muniz, A. T. Costa and D. M. Edwards for fruitful discussions. The authors also gratefully acknowledge the computing time granted through JARA-HPC on the supercomputers JURECA and JUQUEEN at Forschungszentrum J ulich, and the computing resources granted by RWTH Aachen Univer- sity under project jara0175. This work is supported by the European Research Council (ERC) under the Eu- ropean Union's Horizon 2020 research and innovation programme (ERC-consolidator grant 681405 { DYNA- SORE). f.guimaraes@fz-juelich.de 1Dan Wei, Micromagnetics and Recording Materials , SpringerBriefs in Applied Sciences and Technology (Springer Berlin Heidelberg, Berlin, Heidelberg, 2012). 2Corina Etz, Lars Bergqvist, A. Bergman, Andrea Taroni, and O. Eriksson, \Atomistic spin dynamics and surface magnons," J. Phys.: Condens. Matter 27, 243202 (2015). 3O. Eriksson, A. 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2018-07-31

Relaxation effects are of primary importance in the description of magnetic excitations, leading to a myriad of methods addressing the phenomenological damping parameters. In this work, we consider several well-established forms of calculating the intrinsic Gilbert damping within a unified theoretical framework, mapping out their connections and the approximations required to derive each formula. This scheme enables a direct comparison of the different methods on the same footing and a consistent evaluation of their range of validity. Most methods lead to very similar results for the bulk ferromagnets Fe, Co and Ni, due to the low spin-orbit interaction strength and the absence of the spin pumping mechanism. The effects of inhomogeneities, temperature and other sources of finite electronic lifetime are often accounted for by an empirical broadening of the electronic energy levels. We show that the contribution to the damping introduced by this broadening is additive, and so can be extracted by comparing the results of the calculations performed with and without spin-orbit interaction. Starting from simulated ferromagnetic resonance spectra based on the underlying electronic structure, we unambiguously demonstrate that the damping parameter obtained within the constant broadening approximation diverges for three-dimensional bulk magnets in the clean limit, while it remains finite for monolayers. Our work puts into perspective the several methods available to describe and compute the Gilbert damping, building a solid foundation for future investigations of magnetic relaxation effects in any kind of material.

Comparative study of methodologies to compute the intrinsic Gilbert damping: interrelations, validity and physical consequences

1807.11808v3

arXiv:2205.14166v2 [hep-ph] 27 Sep 2022Scalar field damping at high temperatures Dietrich B¨ odeker1, Jan Nienaber2 Fakult¨ at f¨ ur Physik, Universit¨ at Bielefeld, 33615 Biel efeld, Germany Abstract The motion of a scalar field that interacts with a hot plasma, l ike the inflaton during reheating, is damped, which is a dissipative process . At high tempera- tures the damping can be described by a local term in the effecti ve equation of motion. The damping coefficient is sensitive to multiple scat tering. In the loop expansion its computation would require an all-order resum mation. Instead we solve an effective Boltzmann equation, similarly to the compu tation of transport coefficients. For an interaction with another scalar field we o btain a simple re- lation between the damping coefficient and the bulk viscosity , so that one can make use of known results for the latter. The numerical prefa ctor of the damping coefficient turns out to be rather large, of order 104. 1bodeker@physik.uni-bielefeld.de 2jan.nienaber@uni-bielefeld.de 11 Introduction Scalar fields may play an important role in the early Universe. They can drive cosmic inflation, andtheirquantumfluctuationscanprovidetheseedofga laxyformation, they can cause phase transitions [ 1] and generate the baryon asymmetry of the Universe [ 2]. They can also be part or all of dark matter [ 3], or be responsible for today’s dark energy [4]. We consider a scalar field ϕwhich is (approximately) constant in space and which evolves in time. Important examples are the inflaton field which drives inflation or the axion field which can be dark matter. To be specific we will consider ϕbeing the inflaton, keeping in mind that our discussion applies to many other situ ations as well. When inflation ends, the inflaton field ϕstarts oscillating coherently around the minimum of its potential. It interacts with other fields leading to an ene rgy transfer thus creating a plasma and (re-)heating the Universe. At the same time, the motion ofϕgets damped. The plasma makes up an increasing fraction of the tot al energy density. For sufficiently strong interaction the plasma thermalizes. Eventually this thermal plasma dominates the energy density; the corresponding temperature is called reheat temperature TRH. This is, however, not the largest temperature of the plasma, which rises early during the reheating process and then decreases before it reaches TRH[5,6]. An oscillating inflaton field with frequency ω=mϕ, can be viewed as a state with high occupancy of inflaton particles with zero momentum and mass mϕ. When there are only few decay products present, the damping is dominated by in flaton decay into lighter particles [ 7]. If many particles have already been produced such that their oc- cupation numbers are of order one or larger, other effects come in to play. Parametric resonance can lead to very efficient particle production [ 8]. Then the decay products thermalize and acquire a thermal mass. At high temperature, ther mal masses can become larger than the inflaton mass such that the decay of an infla ton into plasma particlesiskinematically forbidden[ 9]. Thenotherprocessesthatinvolve multiplescat- terings become unsuppressed and open new channels for the ener gy transfer [ 10,11]. In this paper we consider the damping rate in the high-temperature regime where T is much larger than mϕand the mass of the plasma particles. We assume that the characteristic frequency ω∼˙ϕ/ϕand the damping rate γofϕare small compared to the thermalization rate of the plasma. Then the inflaton interacts n early adiabatically with an almost thermal plasma. In particular, there is no non-pertu rbative particle production through parametric resonance [ 8]. When the plasma is approximately ther- mal, its properties are fully specified by the temperature and by the instantaneous 2value ofϕ. Therefore the plasma ‘forgets’ about its past, and its effect on t he inflaton dynamics can be described by local terms. The effective equation of motion (without Hubble expansion) for the zero-momentum mode of ϕtakes the form [ 12] ¨ϕ+V′ eff+γ˙ϕ= 0, (1) where the prime denotes a derivative with respect to ϕ. The effective potential Veff and the damping coefficient γonly depend on the value of ϕand the temperature. For sufficiently slow evolution, higher derivative terms in Eq. ( 1) can be neglected. Note that the form of Eq. ( 1) follows from the separation of timescales alone. Ifωand the damping rate γare small compared to the thermalization rate, then γcan be obtained from a finite-temperature real-time correlation fu nction, evaluated in the the zero-momentum, zero-frequency limit [ 13,14]. E.g., the damping coefficient for the axion field is proportional to the Chern-Simons diffusion rate in QCD [ 13], the so-called strong sphaleron rate, which is non-perturbative and ha s been calculated on the lattice [ 15,16]. In many cases the required correlation functions can in principle be calculated perturbatively in thermal field theory. They are, how ever, sensitive to timescales much larger than the mean free time of the plasma particle s, so that one has to take into account multiple scatterings. This requires the res ummation of an infinite set of diagrams. Several authors have applied 1- or 2-loop a pproximations with resummed propagators containing a finite width (see e.g. [ 17–20]), which gives rise to a nonzero damping rate. However, proper treatment of the multiple interaction requires the resummation of a much larger class of diagrams [ 21,22]. Similar complications arise in the computation of transport coefficient s, such as bulk viscosity, which can be written as the zero-momentum zero-fr equency limit of a stress-tensor correlation function [ 23,24]. For viscosities, the required summation of diagrams has been performed. It was shown that this is equivalen t to solving an effective Boltzmann equation [ 23]. The physics behind bulk viscosity ζis closely related to the one of the damping coefficient γ. Both describe small deviations from thermal equilibrium. In the cas e of ζit is due to a uniform expansion of the system. The deviation of the tr ace of the stress tensor from the ideal-fluid form is proportional to ζ. When the field ϕchanges with time and interacts with the plasma particles, it changes their par ameters such as masses or couplings, driving the plasma out of equilibrium. Since ϕis spatially constant, this deviation from equilibrium is homogeneous and isotrop ic as well. For the model considered in Ref. [ 14] the damping coefficient could be related to a correlation function of the stress tensor. Thus there is a simple r elation between γ and the bulk viscosity ζof the thermal plasma, so that one can use the known result 3forζ. In this work we consider interactions of ϕwith another scalar field χthrough operators that cannot be expressed in terms the stress tensor , but which still allow for a perturbative treatment. We can proceed similarly to the comp utation of the bulk viscosity, for which the required resummation of diagrams is equ ivalent to solving an effective Boltzmann equation [ 23]. Damping coefficients have been computed from a Boltzmann equation long ago using a relaxation time approximation [ 12].3This approximation, however, does not give the correct result for the bulk viscosity in scalar theory [23,24]. Here we carefully treat the collision term as well as thermal effects by employing the effective Boltzmann equations which were used to pe rturbatively compute bulk viscosities in scalar theories [ 23,24], and in gauge theories [ 25,26]. This allows us to obtain the correct dependence on the coupling constan ts and explicitly compute γat leading order in perturbation theory. This paper is organized as follows. In Sec. 2we obtain the effective equation of motion for ϕand an expression for the damping coefficient in terms of the plasma- particle occupancy. The latter is computed in Sec. 3from an effective Boltzmann equation. In Sec. 4the solution to the Boltzmann equation is inserted into the effective equation of motion for ϕ, and the damping coefficients is expressed in terms of the known bulk viscosity of the plasma. Section 5contains conclusions and a brief outlook. Appendix Adeals with the thermodynamics of the plasma particles, and Appendix B describes the solution of the Boltzmann equation. 2 Effective equation of motion In this section, largely following Ref. [ 12], we obtain the effective equation of motion (1) from quantum field theory andrelate the coefficients therein to mic roscopic physics. We consider a scalar field Φ is coupled to another scalar field χthrough the interaction LΦχ=−A(Φ)χ2. (2) Restricting ourselves to renormalizable interactions we can have A(Φ) =µ 2Φ+λ 4Φ2(3) with coupling constants µandλ. Without Hubble expansion the equation of motion for Φ reads ¨Φ−∆Φ+V′(Φ)+A′(Φ)χ2= 0, (4) 3The damping coefficients computed in Refs. [ 18–20] are of the same form as in Ref. [ 12]. 4whereVis the part of the tree-level potential that depends only on Φ. Eq. (4) is still an equation for field operators. We want to write an equation of mot ion for the zero- momentum mode ϕof Φ, and we assume that ϕcan be approximated by a classical field. We write Φ =ϕ+ˆΦ (5) whereˆΦ contains the non-zero momentum modes of Φ. Through the intera ction,χ particles are produced. Once the χparticles are created, they can also produce Φ par- ticles which are represented by ˆΦ. Thus the production of Φ particles also contributes to the damping of ϕ. This effect is discussed in [ 27], where it was found that this contribution is subdominant unless the energy density in ϕis small compared to the energy density of the χparticles. In the context of reheating after inflation this would already be during radiation domination. We assume that ϕstill dominates the energy density and neglect this contribution. Then we can replace the fort h term in Eq. ( 4) byA′(ϕ)χ2. We assume that χinteracts rapidly with itself or other fields, so that it thermalizes on timescales which are short compared to the period of ϕoscillations. Furthermore, we assume that the interactions of χare weak enough, so that the typical mean free path ofχparticles is much larger than their typical de Broglie wavelength. The n χis made up of weakly interacting particles which can be described by th eir phase space density, or occupancy f(t,p). Since we consider a homogeneous system, it only depends on time tand on the particle momentum p. We may then replace χ2by its expectation value computed from the occupancy using the free-fi eld expression /angbracketleftbig χ2/angbracketrightbig =/integraldisplayd3p (2π)3f(t,p) E, (6) whereEis the one-particle energy (see below). Thus we arrive at the effect iveclassical equation of motion ¨ϕ+V′(ϕ)+A′(ϕ)∝angbracketleftχ2∝angbracketright= 0. (7) The deviations from equilibrium are assumed to be small, so that the oc cupancy in Eq. (6) can be written as f(t,p) =feq(t,p)+δf(t,p) (8) with the local equilibrium distribution feq(t,p) =1 exp/parenleftbig E/T)−1. (9) 5andδf≪feq. The temperature Tin Eq. (9) varies slowly with time. The mass of the χparticles depends on the value of ϕ, m2=m2 0+2A(ϕ) (10) wherem0is the zero-temperature mass at vanishing ϕ. Throughout this paper we assume that mis small compared to the temperature.4The mass appearing in the one- particle energy Ein Eqs. ( 6) and (9) also receives a thermal contribution m2 th∝T2, so thatE= (p2+m2 eff)1/2with m2 eff=m2+m2 th. (11) To avoid a tachyonic instability [ 29],m2 effmust be positive. With the help of Eq. ( 8), the expectation value in Eq. ( 6) becomes ∝angbracketleftχ2∝angbracketright=∝angbracketleftχ2∝angbracketrighteq+δ∝angbracketleftχ2∝angbracketright. (12) The first term inEq. ( 12)is nondissipative. It gives a thermal correction intheeffective potential in Eq. ( 1) [12], V′ eff=V′+A′∝angbracketleftχ2∝angbracketrighteq, (13) which is precisely the leading term in the high-temperature limit of the 1 -loop effective potential (see, e.g., [ 30]). The second term in Eq. ( 12) is dissipative and will give rise to the damping term in Eq. ( 1). 3 Boltzmann equation The occupancy of χparticles in Eq. ( 6) can be computed by solving a Boltzmann equation, because the timescale on which their mass changes is of or der 1/ωwhich is much larger than their typical de Broglie wavelength of order 1 /T. Due to the homogeneity, spatial momentum is conserved. Thus the Boltzmann equation takes the form ∂tf=C (14) whereCis the collision term. Now we insert Eq. ( 8) on the left-hand side of Eq. ( 14). We neglect ∂tδfbecause it is quadratic in small quantities, so that ∂tf≃ −feq(1+feq)∂t(E/T). (15) 4The opposite limit m≫Tis consideredin Ref. [ 28] with additional light degreesoffreedom. Then χcan be integrated out giving rise to an effective interaction of ϕwith the light plasma particles. 6The zero-momentum mode ϕdepends on time and changes the mass of the plasma particles through the interaction ( 2). If the oscillation is much slower than the ther- malization of the plasma, this is an adiabatic process that changes th e temperature in Eq. (9) at constant volume.5Thus the time dependence of the temperature is deter- mined by ∂tT=/parenleftbigg∂T ∂m2/parenrightbigg S,V∂tm2. (16) In the limit T≫mwe obtain (see Appendix A) /parenleftbigg∂T ∂m2/parenrightbigg S,V=T 4ρ/angbracketleftbig χ2/angbracketrightbig eq, (17) whereρistheenergydensityofthethermalplasma. Theone-particleener gyEdepends on time through the effective mass. We thus have ∂t(E/T) =1 2TE/bracketleftBigg 1−∝angbracketleftχ2∝angbracketrighteq 2ρ/parenleftbigg E2−T2∂m2 eff ∂T2/parenrightbigg/bracketrightBigg ∂tm2. (18) The third term in the square bracket is small compared to the first, both for hard (|p| ∼T) and for soft ( |p| ∼meff) momenta, and can be neglected, so that ∂tf≃ −feq(1+feq)Q 2T∂tm2(19) with Q(p)≡1 E−∝angbracketleftχ2∝angbracketrighteq 2ρE. (20) Now we insert Eq. ( 8) into the collision term. Since Cvanishes in equilibrium, its expansion in δfstarts at linear order, C≃/hatwideCδf. (21) Here we have neglected the contribution of Φ particles because the corresponding colli- sion term is quadratic in the Φ- χcouplings which we assume to be much smaller than the self-coupling of χentering /hatwideC. It is convenient to write the deviation from equilibrium as δf=−feq(1+feq)X. (22) 5This does not apply to the case ω>∼Twhich is considered in Refs. [ 31–33]. 7Similarly we write the linearized collision term as /hatwideCδf=feq(1+feq)/tildewideCX, (23) with the convolution [/tildewideCX](p)≡/integraldisplayd3p′ (2π)3/tildewideC(p,p′)feq(p′)/parenleftBig 1+feq(p′)/parenrightBig X(p′). (24) Then the kernel /tildewideCis symmetric [ 34], /tildewideC(p,p′) =/tildewideC(p′,p). (25) The Boltzmann equation thus turns into an equation for X, −∂tm2 2TQ=/tildewideCX. (26) Since the collision term vanishes in equilibrium for any temperature, th e linearized collision term has a zero mode X=X1associated with a shift of the temperature, which is given by X1(p) =E. Due to the symmetry of /tildewideCthe right-hand side of Eq. (26) is orthogonal to X1. For Eq. ( 26) to be consistent, the left-hand side must be orthogonal to X1as well. This is indeed the case when the second term in Eq. ( 20) is taken into account, which can be easily checked. Due to the zero mode the linear operator /tildewideCcannot be inverted. However, it can be inverted on the subspace orthogonal to the zero mode,6where orthogonality is defined with respect to the inner product (X,X′)≡/integraldisplayd3p (2π)3feq(1+feq)X(p)X′(p). (27) We can then write the solution as X=−∂tm2 2T/tildewideC−1Q, (28) and we finally obtain δf=feq(1+feq)∂tm2 2T/tildewideC−1Q. (29) 6This is equivalent to imposing the Landau-Lifshitz condition on the ene rgy density δρ= (2π)−3/integraltext d3pEδf= 0. 84 Damping coefficient and bulk viscosity Coming back to the effective equation of motion ( 7), we insert the solution ( 29) into Eq. (6) to obtain the second term in Eq. ( 12) as δ∝angbracketleftχ2∝angbracketright=1 2T/parenleftBig Q′,/tildewideC−1Q/parenrightBig ∂tm2. (30) Here we have introduced Q′(p)≡1/E. The factor ∂tm2is proportional to ˙ ϕ. Com- paring Eqs. ( 1) and (7) we see that the second term in Eq. ( 12) is indeed responsible for the damping, γ˙ϕ=A′δ∝angbracketleftχ2∝angbracketright. (31) Inserting Afrom Eq. ( 3) we obtain γ=1 4T/parenleftBig Q′,/tildewideC−1Q/parenrightBig (µ+λϕ)2(32) which is our main result. The computation of /tildewideC−1Qis described in Appendix B. However, at this point we do not need it explicitly,7because, as we shall see in a moment, the coefficient ( Q′,/tildewideC−1Q) also appears in the computation of the bulk viscosity ζof theχplasma. Therefore it can be read off directly from known results for ζ. To see this, we first recall that /tildewideC−1Q is orthogonal to X1=E, i.e., (E,/tildewideC−1Q) = 0. In Eq. ( 32) we may therefore replace Q′=Q+(∝angbracketleftχ2∝angbracketrighteq/2ρ)EbyQwithout changing our result for γ, which then reads γ=1 4T/parenleftBig Q,/tildewideC−1Q/parenrightBig (µ+λϕ)2. (33) Letusnowrecallsomepropertiesofthebulkviscosity, asdescribe d, e.g., inRef.[ 25]. When a plasma is uniformly compressed or rarified it leaves equilibrium, u nless this happens infinitely slowly. The pressure of the plasma then differs fro m the value it would have in the equilibrium state with the same energy density. This d eviation of the pressure from equilibrium is proportional to the bulk viscosity. In a plasma with scale invariance the bulk viscosity vanishes, for two d ifferent reasons. The first one is that a uniform expansion or rarefaction is a dilatation which is asymmetry transformationinascaleinvariant theory. Therefore suchatransformation does not take the system out of equilibrium. The second is that in a sc ale invariant theory the trace of the energy-momentum tensor Tµνalways vanishes. Therefore the pressure P=Tmm/3 equals ρ/3 even out of equilibrium. 7It will be usefull later, when we estimate the size of δfin order to check the accuracy of our approximations. 9Scale invariance is broken by zero-temperature masses and by the trace anomaly, i.e., by quantum effects. The bulk viscosity is then quadratic in the mea sure which controls the breaking of scale invariance. The bulk viscosity ζof the thermal plasma of scalar particles with mass mwas computed for scalar theory in Refs. [ 23,24]. Like in QCD [ 25] it can be written as ζ=1 T/parenleftBig q,/tildewideC−1q/parenrightBig , (34) with q(p) =−1 E/bracketleftbigg/parenleftbigg c2 s−1 3/parenrightbigg p2+c2 sm2 sub/bracketrightbigg . (35) Herecswithc2 s=∂P/∂ρis the speed of sound. In a scale invariant theory c2 sequals 1/3, so that the first term in the square bracket in Eq. ( 35) vanishes. Furthermore, msubwith m2 sub≡m2 eff−T2∂m2 eff ∂T2(36) is the so-called subtracted mass. In the massless limit m= 0,m2 effequalsT2times a function of the coupling constants (see Eqs. ( 10), (11)). Then the only contribution to the subtracted mass is from the running of the couplings renormaliz ed at the scale T. The subtracted mass is thus a measure of the deviation from scale in variance as well, because it vanishes when m= 0 and the couplings do not run. Since qappears twice in Eq. (34), the bulk viscosity is indeed quadratic in the measure of scale-invar iance violation. Now we replace p2byE2−m2 effin Eq. (35) which turns it into q(p) =/bracketleftbigg/parenleftbigg c2 s−1 3/parenrightbigg m2 eff−c2 sm2 sub/bracketrightbigg1 E−/parenleftbigg c2 s−1 3/parenrightbigg E. (37) Comparing Eqs. ( 20) and (37) we see that both Qandqconsist of a term proportional to 1/E, and one proportional to E. Furthermore, qappears on the left-hand side of a Boltzmann equation precisely like Qin Eq. (19),8and is thus orthogonal to X1as well. Therefore qmust be proportional to Q. Here we are interested in the limit T≫min which [23] |c2 s−1/3|=O(m2 sub/T2)≪1 (T≫m), (38) 8See Eq. (3.7) of Ref. [ 25]. 10andalsom2 eff≪T2. Thereforewecanapproximatethesquarebracketin( 37)by−m2 sub/3. This gives us the approximate factor of proportionality, so that q≃ −(m2 sub/3)Q. Then we obtain the following simple relation γ(ϕ,T) =9 4ζ m4 sub(µ+λϕ)2(39) of the damping coefficient in the effective equation of motion ( 1) and the bulk viscosity of theχplasma. Like in Ref. [ 14] the nontrivial dependence on the interaction of the plasma particles, on thermal masses, etc., is precisely the same for both quantities. Note that m4 subin the denominator of Eq. ( 39) removes the factors related to the breaking of scale invariance from ζ, which can also be seen explicitly in Eqs. ( 41) and (43) below. Thus, despite its similarity to the bulk viscosity, the damping c oefficient is not related to the breaking of scale invariance. The bulk viscosity for a self-interacting scalar field was computed in R ef. [23]. For the quartic self-interaction Lχχ=−g2 4!χ4(40) andT≫mthe leading-order result reads ζ=b 4m4 subm2 eff g8T3ln2/parenleftbiggκ2m2 eff T2/parenrightbigg (41) withb= 5.5×104andκ= 1.25. The effective mass for the χparticles is given by m2 eff=m2+g2 24T2, m2=m2 0+λ 2ϕ2. (42) Inserting Eq. ( 41) into Eq. ( 39) we obtain the damping coefficient γ(ϕ,T) =am2 eff g8T3ln2/parenleftbiggκ2m2 eff T2/parenrightbigg (µ+λϕ)2(43) with the remarkably large numerical prefactor a= 3.1×104. (44) In the temperature range T≫m/g2the form ( 41) and thus Eq. ( 43) remain valid when a cubic self-interaction is included in ( 40), while in the intermediate regime m≪ T≪m/g2the bulk viscosity depends nontrivially on the relative strengths of c ubic and quartic χself-couplings [ 23]. It is obvious from the dependence on the coupling constant gthat the result ( 43) cannot be obtained from a one-loop approximation to a 11correlation function, as anticipated in Refs. [ 22,28]. Instead, by solving the Boltzmann equation we have summed an infinite set of diagrams which all contribu te at leading order ing. We can compare our result with the one obtained in Ref. [ 12] for a single scalar field by putting χ=ϕ,µ= 0, and, up to an O(1) factor, g2=λ. In Ref. [ 12] the Boltzmann equation was solved in the collision-time approximation, i.e., b y replacing the linearized collision term on the right-hand side of Eq. ( 21) by a constant times δf. Such an approximation does not take into account the zero eigenv alues and the hierarchy of nonzero eigenvalues of /tildewideC. In Ref. [ 12] the collision time is determined by 2→2 scattering which changes momenta but not particle numbers. Bulk viscosity and the damping coefficient γare, however, determined by the slowest equilibration process, corresponding to the smallest eigenvalue of the linear collis ion operator, since it is the inverse of the collision operator that appears in Eqs. ( 32) and (34). In scalar field theory the slowest process is particle number equilibration. The refore the com- putation of Ref. [ 12] does not give the correct dependence on the coupling constant and underestimates the values of γandζ. Similarly, in Ref. [ 35] the rate for elastic scattering was used to estimate the damping coefficient. The importance of particle number changing processes for the bulk viscosity is well known. The reason why they are also important for the damping coe fficient is the following. When ϕevolves in time, it changes the mass of the χparticles, but not their momenta. This, in turn, changes the energy density of χparticles but leaves their number density unaffected. In order to relax to equilibrium, the χparticle number has to adjust to the equilibrium value corresponding to their new energy density. Let as finally discuss the range of validity of the effective equation of motion (1). There the damping term is linear in ˙ ϕ. This is related to the linearization of the Boltzmann equation, which requires that δf≪feq. In Appendix Bwe show that δf/feq∼meff g8T4∂tm2(45) for the interaction ( 40). When ϕoscillates with frequency ωand amplitude /tildewideϕ, the time derivative ∂tm2can be estimated as λω/tildewideϕ2. ForgT≫mwe thus need λω/tildewideϕ2≪g7T3(46) in order to be able to linearize the Boltzmann equation. We may also apply the condition ( 46) to a model with a single scalar field ϕwhich was considered in Ref. [ 12] by putting µ= 0 and λ∼g2. Then ( 46) turns into ω/tildewideϕ2≪g5T3. The energy density in ϕwould be ρϕ∼ω2/tildewideϕ2≪(ω/T)g5T4. Due to 12ω≪T, the energy carried by ϕwould be only a tiny fraction of the plasma energy densityρ∼T4. For the more interesting case that we have several fields, ϕcan give the dominant contribution to the total energy without violating the condition ( 46). 5 Conclusion A slowly moving homogeneous scalar field ϕinteracting with a thermal plasma drives it slightly out of equilibrium, giving rise to dissipation and damping. In the high- temperature regime the damping coefficient in the effective equation of motion for ϕ canbeefficientlycomputedbysolvinganappropriateBoltzmannequa tion, seeEq.( 32). We have considered a plasma made of a single species of scalar particle s. In this case we obtained a simple relation of the damping coefficient to the bulk visco sity of the plasma, Eq. ( 39). This extends a result [ 14] which was obtained for a scalar field with derivative interaction. Like in the computation of viscosity, the solution of the Boltzmann equation is dominated by the slowest process required fo r equilibration. This can be easily generalized to multicomponent plasmas, where again one has to identify the slowest process to solve the Boltzmann equation and th en use the resulting phase spacedensity tocomputethedissipative terms intheeffectiv e equationofmotion for the scalar field. Acknowledgements We thank Mikko Laine and Simona Procacci for comments and discuss ions. D.B. ac- knowledges support by the Deutsche Forschungsgemeinschaft ( DFG, German Research Foundation) through the CRC-TR 211 ’Strong-interaction matter under extreme con- ditions’– project number 315477589 – TRR 211. A Mass dependence of the temperature The free energy of an ideal gas has the high-temperature expans ion F(T,V,m2) =V(−aT4+bT2m2+···) (A.1) with positive constants aandb;mis the mass of one particle species. The coefficient acan also contain the contributions from other light species. At leadin g order our expansion is related to the energy density by ρ= 3aT4. For a scalar ∂F ∂m2=V 2∝angbracketleftχ2∝angbracketrighteq (A.2) 13which gives b=∝angbracketleftχ2∝angbracketrighteq/(2T2). The entropy is S=−∂F ∂T=V(4aT3−2bTm2+···). (A.3) This can be inverted to obtain the expansion for the temperature, T=T0+T2+···, for which we obtain T0=/parenleftbiggS 4aV/parenrightbigg1/3 , (A.4) T2=b 6am2 T0. (A.5) Differentiating T2with respect to m2then gives Eq. ( 17). B SolvingtheBoltzmannequationandestimating δf The linearization of the Boltzmann equation is only possible if the deviat ion from equilibrium is small, δf≪feq. This condition restricts the allowed values of the couplings and the amplitude of the zero-momentum mode ϕ. To estimate the size of δfin Eq. (29) we derive its explicit form, closely following Ref. [24]. For a self-interacting scalar field one has to include two contributio ns in the collision term, /tildewideC=/tildewideCel+/tildewideCinel. (B.1) /tildewideCeldescribes elastic 2 →2scattering which conserves particlenumber. Therefore ithas the additional zero mode X0= 1, associated with a shift of the chemical potential, and cannot be inverted on the subspace orthogonal to X1=E. One also has to include an inelastic contribution /tildewideCineldescribing particle number changing processes, even though its matrix element is higher order. /tildewideChas a single small eigenvalue con the subspace orthogonal to X1, with the approximate eigenvector X0⊥=X0−αX1 (B.2) whereα= (X1,X0)/(X1,X1). The small eigenvalue is approximately c=/parenleftbig X0⊥,/tildewideCinelX0⊥/parenrightbig (X0⊥,X0⊥)(B.3) while the other nonvanishing eigenvalues areof order /tildewideCel. Inthe numerator ofEq. ( B.3) we may replace X0⊥byX0because /tildewideCinelX1vanishes. This eigenvalue gives the leading 14contribution to /tildewideC−1, so that /tildewideC−1Q≃(X0⊥,Q) (X0,/tildewideCinelX0)X0⊥. (B.4) In the numerator of Eq. ( B.4) we can replace X0⊥byX0, because X1is orthogonal to Q. We insert this into Eq. ( 29), which finally gives δf(p) =feq(p)/bracketleftbig 1+feq(p)/bracketrightbig∂tm2 2T(X0,Q) (X0,/tildewideCinelX0)X0⊥(p). (B.5) We now use this to estimate the size of δf. We will encounter the integrals In≡/integraldisplayd3p (2π)3feq(1+feq)En(B.6) forn=−1,0, and 1. For n≥0 these are saturated at |p| ∼T, givingIn∼T3+n. Sincefeq≃T/EforE≪T, the integral I−1, is logarithmically infrared divergent in the massless limit and is cut off by meff. ThusI−1receives leading order contributions both from |p| ∼Tand from |p| ∼meff≪T, with the result I−1=T2 2π2ln/parenleftbigg2T meff/parenrightbigg . (B.7) The factor ( X0,Q) in the numerator of Eq. ( B.5) contains I−1andI1and is of order T2 modulo logarithms, because ∝angbracketleftχ2∝angbracketrighteq/ρ∼T−2. The denominator depends on the type of interaction (see below). Since the size of δf(p) depends on |p|, we need to know which values of |p|give the dominant contributionsto( Q′,/tildewideC−1Q)∝(Q′,X0⊥)whichentersthedampingcoefficient in Eq. (32). Using Eq. ( B.2) we find that the integrals ( B.6) appear in the combination I−1−αI0. The factor αis of order 1 /T. Thus|p| ∼Tand|p| ∼meffare equally important. In both regions X0⊥∼1. Due to the Bose factors in Eq. ( 22) the ratio δf/feqincreases with decreasing |p|, so that it takes its largest value when |p| ∼meff. Putting|p| ∼meff, collecting all factors and ignoring logarithms we obtain δf/feq∼T2∂tm2 meff(X0,/tildewideCinelX0). (B.8) Fortheχself-interaction( 40)andthe χ-ϕinteraction ( 2) withµ= 0,/tildewideCineldescribes scattering involving 6 particles. The corresponding squared matrix is proportional tog8. The momentum integral which enters the denominator in Eq. ( B.4) is saturated by soft momenta ( T∼meff)) [23]. It contains up to six Bose distributions, which for soft momenta satisfy feq(p)≃T/E, giving rise to a factor T6. By dimensional analysis one then finds ( X0,/tildewideCinelX0)∼g8T6/m2 eff, which yields the estimate ( 45). 15References [1] D. A. Kirzhnits and A. D. Linde, Macroscopic Consequences of the Weinberg Model,Phys. Lett. B 42(1972) 471–474 . [2] I. Affleck and M. Dine, A New Mechanism for Baryogenesis , Nucl. Phys. 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2022-05-27

The motion of a scalar field that interacts with a hot plasma, like the inflaton during reheating, is damped, which is a dissipative process. At high temperatures the damping can be described by a local term in the effective equation of motion. The damping coefficient is sensitive to multiple scattering. In the loop expansion its computation would require an all-order resummation. Instead we solve an effective Boltzmann equation, similarly to the computation of transport coefficients. For an interaction with another scalar field we obtain a simple relation between the damping coefficient and the bulk viscosity, so that one can make use of known results for the latter. The numerical prefactor of the damping coefficient turns out to be rather large, of order $ 10 ^ 4 $.

Scalar field damping at high temperatures

2205.14166v2

Coupled Spin-Light dynamics in Cavity Optomagnonics Silvia Viola Kusminskiy,1Hong X. Tang,2and Florian Marquardt1, 3 1Institute for Theoretical Physics, University Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany 2Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA 3Max Planck Institute for the Science of Light, Günther-Scharowsky-Straße 1, 91058 Erlangen, Germany Experiments during the past two years have shown strong resonant photon-magnon coupling in microwave cavities, while coupling in the optical regime was demonstrated very recently for the first time. Unlike with microwaves, the coupling in optical cavities is parametric, akin to optomechanical systems. This line of research promises to evolve into a new field of optomagnonics, aimed at the coherent manipulation of elementary magnetic excitations by optical means. In this work we derive the microscopic optomagnonic Hamiltonian. In the linear regime the system reduces to the well-known optomechanical case, with remarkably large coupling. Going beyond that, we study the optically induced nonlinear classical dynamics of a macrospin. In the fast cavity regime we obtain an effective equation of motion for the spin and show that the light field induces a dissipative term reminiscent of Gilbert damping. The induced dissipation coefficient however can change sign on the Bloch sphere, giving rise to self-sustained oscillations. When the full dynamics of the system is considered, the system can enter a chaotic regime by successive period doubling of the oscillations. I. INTRODUCTION The ability to manipulate magnetism has played his- torically an important role in the development of infor- mation technologies, using the magnetization of materi- als to encode information. Today’s research focuses on controlling individual spins and spin currents, as well as spin ensembles, with the aim of incorporating these sys- tems as part of quantum information processing devices. [1–4]. In particular the control of elementary excitations of magnetically ordered systems –denominated magnons or spin waves, is highly desirable since their frequency is broadly tunable (ranging from MHz to THz) [2, 5] while theycanhaveverylonglifetimes, especiallyforinsulating materials like the ferrimagnet yttrium iron garnet (YIG) [6]. The collective character of the magnetic excitations moreoverrendertheserobustagainstlocalperturbations. Whereas the good magnetic properties of YIG have beenknownsincethe60s, itisonlyrecentlythatcoupling andcontrollingspinwaveswithelectromagneticradiation in solid-state systems has started to be explored. Pump- probe experiments have shown ultrafast magnetization switching with light [7–9], and strong photon-magnon coupling has been demonstrated in microwave cavity ex- periments [10–18] –including the photon-mediated cou- pling between a superconducting qubit and a magnon mode [19]. Going beyond microwaves, this points to the tantalizing possibility of realizing optomagnonics : the coupled dynamics of magnons and photons in the op- tical regime, which can lead to coherent manipulation of magnons with light. The coupling between magnons and photons in the optical regime differs from that of the microwave regime, where resonant matching of fre- quencies allows for a linear coupling: one magnon can be convertedintoaphoton, andviceversa[20–22]. Intheop- tical case instead, the coupling is a three-particle process. This accounts for the frequency mismatch and is gener- z yzx optical mode optical shiftGSx0magnonmodeopticalmodeGˆ~SˆaabcFigure 1. (Color online) Schematic configuration of the model considered. (a)Optomagnoniccavitywithhomogeneousmag- netization along the z-axis and a localized optical mode with circular polarization in the y-z-plane. (b) The homogeneous magnon mode couples to the optical mode with strength G. (c) Representation of the magnon mode as a macroscopic spin on the Bloch sphere, whose dynamics is controlled by the cou- pling to the driven optical mode. ally called parametric coupling. The mechanism behind the optomagnonic coupling is the Faraday effect, where the angle of polarization of the light changes as it prop- agates through a magnetic material. Very recent first experiments in this regime show that this is a promising route, by demonstrating coupling between optical modes and magnons, and advances in this field are expected to develop rapidly [23–27]. In this work we derive and analyze the basic op- tomagnonic Hamiltonian that allows for the study of solid-state cavity optomagnonics. The parametric op- tomagnonic coupling is reminiscent of optomechanicalarXiv:1604.07053v3 [cond-mat.mes-hall] 19 Sep 20162 models. In the magnetic case however, the relevant oper- atorthatcouplestotheopticalfieldisthespin, insteadof the usual bosonic field representing a mechanical degree of freedom. Whereas at small magnon numbers the spin can be replaced by a harmonic oscillator and the ideas of optomechanics [28] carry over directly, for general trajec- tories of the spin this is not possible. This gives rise to rich non-linear dynamics which is the focus of the present work. Parametric spin-photon coupling has been studied previously in atomic ensembles [29, 30]. In this work we focus on solid-state systems with magnetic order and de- rive the corresponding optomagnonic Hamiltonian. After obtaining the general Hamiltonian, we consider a simple model which consists of one optical mode coupled to a homogeneous Kittel magnon mode [31]. We study the classical dynamics of the magnetic degrees of freedom and find magnetization switching, self-sustained oscilla- tions, and chaos, tunable by the light field intensity. The manuscript is ordered as follows. In Sec. (II) we present the model and the optomagnonic Hamiltonian which is the basis of our work. In Sec. (IIA) we discuss briefly the connection of the optomagnonic Hamiltonian derived in this work and the one used in optomechanic systems. In Sec. (IIB) we derive the optomagnonic Hamiltonian from microscopics, and give an expression for the optomagnonic coupling constant in term of ma- terial constants. In Sec. (III) we derive the classical coupled equations of motion of spin and light for a ho- mogeneous magnon mode, in which the spin degrees of freedomcanbetreatedasamacrospin. InSec. (IIIA)we obtain the effective equation of motion for the macrospin in the fast-cavity limit, and show the system presents magnetization switching and self oscillations. We treat the full (beyond the fast-cavity limit) optically induced nonlinear dynamics of the macrospin in Sec. (IIIB), and follow the route to chaotic dynamics. In Sec. (IV) we sketch a qualitative phase diagram of the system as a function of coupling and light intensity, and discuss the experimental feasibility of the different regimes. An out- look and conclusions are found in Sec. (V). In the Ap- pendix we give details of some of the calculations in the main text, present more examples of nonlinear dynamics asafunctionofdifferenttuningparameters, andcompare optomagnonic vs.optomechanic attractors. II. MODEL Further below, we derive the optomagnonic Hamilto- nian which forms the basis of our work: H=~
^ay^a~ ^Sz+~G^Sx^ay^a; (1) where ^ay(^a) is the creation (annihilation) operator for a cavity mode photon. We work in a frame rotating at the laser frequency !las, and
=!las!cavis the detuning with respect to the optical cavity frequency !cav. Eq. (1)assumes a magnetically ordered system with (dimension- less) macrospin S= (Sx;Sy;Sz)with magnetization axis along ^ z, and a precession frequency which can be con- trolled by an external magnetic field [32]. The coupling between the optical field and the spin is given by the last term in Eq. (1), where we assumed (see below) that light couples only to the xcomponent of the spin as shown in Fig. (1). The coefficient Gdenotes the para- metric optomagnonic coupling. We will derive it in terms of the Faraday rotation, which is a material-dependent constant. A. Relation to optomechanics Close to the ground state, for deviations such that SS(withS=jSj), we can treat the spin in the usual way as a harmonic oscillator, ^Sxp S=2(^b+^by), withh ^b;^byi = 1. Then the optomagnonic interaction ~G^Sx^ay^a~Gp S=2^ay^a(^b+^by)becomes formally equiv- alent to the well-known opto mechanical interaction [28], with bare coupling constant g0=Gp S=2. All the phe- nomena of optomechanics apply, including the “optical spring” (here: light-induced changes of the magnon pre- cession frequency) and optomagnonic cooling at a rate opt, and the formulas (as reviewed in Ref. [28]) can be taken over directly. All these effects depend on the light- enhanced coupling g=g0, where=pnphotis the cavity light amplitude. For example, in the sideband- resolved regime (  , whereis the optical cavity decay rate) one would have opt= 4g2=. Ifg > , one enters the strong-coupling regime, where the magnon mode and the optical mode hybridize and where coher- ent state transfer is possible. A Hamiltonian of the form of Eq. (1) is also encountered for light-matter interaction in atomic ensembles [29], and its explicit connection to optomechanics in this case was discussed previously in Ref. [30]. In contrast to such non-interacting spin en- sembles, the confined magnon mode assumed here can be frequency-separated from other magnon modes. B. Microscopic magneto-optical coupling G In this section we derive the Hamiltonian presented in Eq. (1) starting from the microscopic magneto-optical effect in Faraday-active materials. The Faraday effect is captured by an effective permittivity tensor that depends on the magnetization Min the sample. We restrict our analysis to non-dispersive isotropic media and linear re- sponseinthemagnetization, andrelegatemagneticlinear birefringence effects which are quadratic in M(denomi- nated the Cotton-Mouton or Voigt effect) for future work [5, 33]. In this case, the permittivity tensor acquires an antisymmetric imaginary component and can be written3 as"ij(M)="0("ijifP kijkMk), where"0(") is the vacuum (relative) permittivity, ijkthe Levi-Civita ten- sorandfamaterial-dependentconstant[33](hereandin what follows, Latin indices indicate spatial components). The Faraday rotation per unit length F=!fMs 2cp"; (2) depends on the frequency !, the vacuum speed of light c, and the saturation magnetization Ms. The magneto- optical coupling is derived from the time-averaged energy U=1 4
drP ijE i(r;t)"ijEj(r;t), using the complex representation of the electric field, (E+E)=2. Note that Uis real since "ijis hermitean [5, 33]. The magneto- optical contribution is UMO=i 4"0f dr M(r)
[E(r)E(r)]:(3) This couples the magnetization to the spin angular mo- mentum density of the light field. Quantization of this expression leads to the optomagnonic coupling Hamilto- nian. A similar Hamiltonian is obtained in atomic en- semble systems when considering the electric dipolar in- teraction between the light field and multilevel atoms, where the spin degree of freedom (associated with M(r) in our case) is represented by the atomic hyperfine struc- ture [29]. The exact form of the optomagnonic Hamil- tonian will depend on the magnon and optical modes. In photonic crystals, it has been demonstrated that opti- cal modes can be engineered by nanostructure patterning [34], and magnonic-crystals design is a matter of intense current research [3]. The electric field is easily quantized, ^E(+)(r;t) =P E(r)^a(t), where E(r)indicates the theigenmode of the electric field (eigenmodes are indi- cated with Greek letters in what follows). The magne- tization requires more careful consideration, since M(r) dependsonthelocalspinoperatorwhich, ingeneral, can- not be written as a linear combination of bosonic modes. There are however two simple cases: (i) small deviations of the spins, for which the Holstein-Primakoff representa- tion is linear in the bosonic magnon operators, and (ii) a homogeneous Kittel mode M(r) =Mwith macrospin S. In the following we treat the homogeneous case, to cap- ture nonlinear dynamics. From Eq. (3) we then obtain the coupling Hamiltonian ^HMO =~P j ^SjGj  ^ay ^a with Gj  =i"0fMs 4~SX mnjmn drE m(r)E n(r);(4) where we replaced Mj=Ms=^Sj=S, withSthe extensive total spin (scaling like the mode volume). One can diago- nalize the hermitean matrices Gj, though generically not simultaneously. In the present work, we treat the con- ceptually simplest case of a strictly diagonal coupling tosome optical eigenmodes ( Gj 6= 0butGj = 0). This is precludedonlyiftheopticalmodesarebothtime-reversal invariant ( Ereal-valued) and non-degenerate. In all the other cases, a basis can be found in which this is valid. For example, a strong static Faraday effect will turn op- tical circular polarization modes into eigenmodes. Al- ternatively, degeneracy between linearly polarized modes implies we can choose a circular basis. Consider circular polarization (R/L) in the yz-plane, such thatGxis diagonal while Gy=Gz= 0. Then we find Gx LL=Gx RR=G=1 ScF 4p"; (5) where we used Eq. (2) to express the coupling via the Faraday rotation F, and where is a dimensionless over- lap factor that reduces to 1if we are dealing with plane waves (see App. A). Thus, we obtain the coupling Hamil- tonianHMO=~G^Sx(^ay L^aL^ay R^aR). This reduces to Eq. (1) if the incoming laser drives only one of the two circular polarizations. The coupling Ggives the magnon precession frequency shift perphoton. It decreases for larger magnon mode volume, in contrast to GS, which describes the overall opticalshift for saturated spin ( Sx=S). For YIG, with"5andF200ocm1[5, 35], we obtain GS1010Hz(taking= 1), which can easily become comparable to the precession frequency . The ultimate limit for the magnon mode volume is set by the optical wavelength,(1m)3, which yields S1010. There- foreG1Hz, whereas the coupling to a single magnon would be remarkably large: g0=Gp S=20:1MHz. This provides a strong incentive for designing small mag- netic structures, by analogy to the scaling of piezoelectri- cal resonators [36]. Conversely, for a macroscopic volume of(1mm)3, withS1019, this reduces to G109Hz andg010Hz. III. SPIN DYNAMICS The coupled Heisenberg equations of motion are ob- tainedfromtheHamiltonianinEq. (1)byusing ^a;^ay = 1,h ^Si;^Sji =iijk^Sk. Wenextfocusontheclassicallimit, where we replace the operators by their expectation val- ues: _a=i(GSx
)a 2(amax) _S= (Gaaex ez)S+G S(_SS):(6) Here we introduced the laser amplitude maxand the in- trinsic spin Gilbert-damping [37], characterized by G, due to phonons and defects ( G104for YIG [38]). After rescaling the fields (see App.. B), we see that the4 classical dynamics is controlled by only five dimension- less parameters:GS ;G2 max ;
; ; G. These are inde- pendent of ~as expected for classical dynamics. In the following we study the nonlinear classical dy- namics of the spin, and in particular we treat cases where the spin can take values on the whole Bloch sphere and therefore differs significantly from a harmonic oscilla- tor, deviating from the optomechanics paradigm valid forSS. The optically induced tilt of the spin can be estimated from Eq. (6) as S=S =Gjaj2=  G2 max= =Bmax= , whereBmax=G2 maxis an op- tically induced effective magnetic field. We would ex- pect therefore unique optomagnonic behavior (beyond optomechanics) for large enough light intensities, such thatBmaxis of the order of or larger than the preces- sion frequency . We will show however that, in the case of blue detuning, even small light intensity can destabi- lize the original magnetic equilibrium of the uncoupled system, provided the intrinsic Gilbert damping is small. A. Fast cavity regime As a first step we study a spin which is slow compared to the cavity, where G_Sx2. In that case we can abyzx-0.10-0.0500.050.10 -0.2-0.100.10.2 Figure 2. (Color online) Spin dynamics (fast cavity limit) at blue detuning
= and fixedGS= = 2,= = 5, G= 0. The left column depicts the trajectory (green full line) of a spin (initially pointing near the north pole) on the Bloch sphere. The color scale indicates the optical damping opt. The right column shows a stereographic projection of the spin’s trajectory (red full line). The black dotted line indicates the equator (invariant under the mapping), while the north pole is mapped to infinity. The stream lines of the spin flow are also depicted (blue arrows). (a) Magnetization switching behavior for light intensity G2 max= = 0:36. (b) Limit cycle attractor for larger light intensity G2 max= = 0:64.expand the field a(t)in powers of _Sxand obtain an ef- fective equation of motion for the spin by integrating out the light field. We write a(t) =a0(t) +a1(t) +:::, where the subscript indicates the order in _Sx. From the equa- tion fora(t), we find that a0fulfills the instantaneous equilibrium condition a0(t) = 2max1  2i(
GSx(t));(7) from which we obtain the correction a1: a1(t) =1  2i(
GSx)@a0 @Sx_Sx:(8) To derive the effective equation of motion for the spin, we replacejaj2ja0j2+a 1a0+a 0a1in Eq. (6) which leads to _S=BeS+opt S(_SxexS) +G S(_SS):(9) HereBe= ez+Bopt, where Bopt(Sx) =Gja0j2ex acts as an optically induced magnetic field. The second term is reminiscent of Gilbert damping, but with spin- velocity component only along ex. Both the induced field Boptand dissipation coefficient optdepend explicitly on the instantaneous value of Sx(t): Bopt=G [( 2)2+ (
GSx)2] 2max2 ex(10) opt=2GSjBoptj(
GSx) [( 2)2+ (
GSx)2]2:(11) This completes the microscopic derivation of the optical Landau-Lifshitz-Gilbert equation for the spin, an impor- tant tool to analyze effective spin dynamics in different contexts [39]. We consider the nonlinear adiabatic dy- namics of the spin governed by Eq. (9) below. Two distinct solutions can be found: generation of new sta- ble fixed points (magnetic switching) and optomagnonic limit cycles (self oscillations). Given our Hamiltonian (Eq. (1)), the north pole is sta- ble in the absence of optomagnonic coupling – the se- lection of this state is ensured by the intrinsic damping G>0. By driving the system this can change due to the energy pumped to (or absorbed from) the spin, and the new equilibrium is determined by Beandopt, when optdominates over G. Magnetic switching refers to the rotation of the macroscopic magnetization by , to a new fixed point near the south pole in our model. This can be obtained for blue detuning
>0, in which case optis negative either on the whole Bloch sphere (when
> GS) or on a certain region, as shown in Fig. (2)a. Similar results were obtained in the case of spin opto- dynamics for cold atoms systems [30]. The possibility of switching the magnetization direction in a controlled way is of great interest for information processing with mag- netic memory devices, in which magnetic domains serve5 as information bits [7–9]. Remarkably, we find that for blue detuning, magnetic switching can be achieved for arbitrary small light intensities in the case of G= 0. This is due to runaway solutions near the north pole for
>0, as discussed in detail in App. C. In physical sys- tems, the threshold of light intensity for magnetization switching will be determined by the extrinsic dissipation channels. For higher intensities of the light field, limit cycle at- tractors can be found for j
j<GS, where the optically induced dissipation optcan change sign on the Bloch sphere (Fig. (2)b). The combination of strong nonlinear- ity and a dissipative term which changes sign, leads the system into self sustained oscillations. The crossover be- tween fixed point solutions and limit cycle attractors is determined by a balance between the detuning and the light intensity, as discussed in App. C. Limit cycle at- tractors require Bmax= >j
j=GS(note that from (11) BoptBmaxif(
GS)). We note that for both examples shown in Fig. (2), for the chosen parameters we have optGin the case of YIG, and hence taking G= 0is a very good approx- imation. More generally, from Eqs. (10) we estimate optGSB opt=3and therefore we can safely neglect Gfor(maxG)2SG3. The qualitative results (limit cycle, switching) survive up to opt&G, although quan- titatively modified as Gis increased: for example, the size of the limit cycle would change, and there would be a threshold intensity for switching. B. Full nonlinear dynamics The nonlinear system of Eq. (6) presents even richer behavior when we leave the fast cavity regime. For limit cycles near the north pole, when SS, the spin is well approximated by a harmonic oscillator, and the dy- namics is governed by the attractor diagram established for optomechanics [40]. In contrast, larger limit cycles will display novel features unique to optomagnonics, on which we focus here. Beyond the fast cavity limit, we can no longer give analytical expressions for the optically induced magnetic field and dissipation. Moreover, we can not define a coef- ficientoptsince an expansion in _Sxis not justified. We therefore resort to numerical analysis of the dynamics. Fig. (3) shows a route to chaos by successive period dou- bling, upon decreasingthe cavity decay . This route can be followed in detail as a function of any selected param- eter by plotting the respective bifurcation diagram. This is depicted in Fig. (4). The plot shows the evolution of the attractors of the system as the light intensity is increased. The figure shows the creation and expansion of a limit cycle from a fixed point near the south pole, followed by successive period doubling events and finally entering into a chaotic region. At high intensities, a limit t⌦ GSz⌦ yzxabc deIncreasing period of the limit cycle Chaos2⇡30.5 132.52 321Figure 3. (Color Online) Full non-linear spin dynamics and route to chaos for GS= = 3andG2 max= = 1(G= 0). The system is blue detuned by
= and the dynamics, after a transient, takes place in the southern hemisphere. The solid red curves represent the spin trajectory after the initial transient, on the Bloch sphere for (a) = = 3, (b)= = 2, (c)= = 0:9, (d)= = 0:5. (f)Szprojection as a function of time for the chaotic case = = 0:5. spin projectionGSz/⌦chaoslimit cycleperioddoublingcoexistence 1.0laser amplitudepG|↵max|2/⌦210-1-21.5 Figure 4. Bifurcation density plot for GS= = 3and= = 1 at
= (G= 0), as a function of light intensity. We plot theSzvalues attained at the turning points ( _Sz= 0). For other possible choices ( eg. _Sx= 0) the overall shape of the bifurcation diagram is changed, but the bifurcations and chaotic regimes remain at the same light intensities. For the plot, 30 different random initial conditions were taken. cycle can coexist with a chaotic attractor. For even big- ger light intensities, the chaotic attractor disappears and thesystemprecessesaroundthe exaxis, asaconsequence of the strong optically induced magnetic field. Similar bi- furcation diagrams are obtained by varying either GS= or the detuning
= (see App. D).6 11 2 xy-plane limit cycles"optomechanics"chaosoptomagnonic limit cyclesswitching chaos yz plane limit cycles⌦GS xy-plane limit cyclesB↵max⌦ Figure 5. Phase diagram for blue detuning with
= , as a function of the inverse coupling strength =GSand the op- tically induced field Bmax= =G2 max= . Boundaries are qualitative. Switching, in white, refers to a fixed point solu- tion with the spin pointing near the south pole. Limit cycles in thexyplane are shaded in blue, and they follow the op- tomechanical attractor diagram discussed in Ref. [40]. For higherBmax, chaos can ensue. Orange denotes the param- eter space in which limit cycles deviate markedly from op- tomechanical predictions. These are not in the xyplane and also undergo period doubling leading to chaos. In red is de- picted the area where pockets of chaos can be found. For largeBmax= , the limit cycles are in the yzplane. In the case of red detuning
= , the phase diagram remains as is, except that instead of switching there is a fixed point near the north pole. IV. DISCUSSION We can now construct a qualitative phase diagram for our system. Specifically, we have explored the qualitative behavior (fixed points, limit cycles, chaos etc.) as a func- tion of optomagnonic coupling and light intensity. These parameters can be conveniently rescaled to make them dimensionless. We chose to consider the ratio of magnon precessionfrequencytocoupling, intheform =GS. Fur- thermore, we express the light intensity via the maxi- mal optically induced magnetic field Bmax=G2 max. The dimensionless coupling strength, once the material of choice is fixed, can be tuned viaan external magnetic fieldwhichcontrolstheprecessionfrequency . Thelight intensity can be controlled by the laser. We start by considering blue detuning, this is shown in Fig. (5). The “phase diagram” is drawn for
= , and we set = andG= 0. We note that some of the transitions are rather crossovers (“optomechanical limit cycles” vs.“optomagnonic limit cycles”). In addition, the other “phase boundaries” are only approximate, obtained from direct inspection of numerical simulations. These are not intended to be exact, and are qualitatively validfor departures of the set parameters, if not too drastic – for example, increasing will lead eventually to the disappearance of the chaotic region. As the diagram shows, there is a large range of pa- rameters that lead to magnetic switching, depicted in white. This area is approximately bounded by the con- ditionBmax= .
=GS, which in Fig. (5) corresponds to the diagonal since we took
= . This condition is approximate since it was derived in the fast cavity regime, see App. C. As discussed in Sec. III, magnetic switching should be observable in experiments even for small light intensity in the case of blue detuning, pro- vided that all non-optical dissipation channels are small. The caveat of low intensity is a slow switching time. For Bmax= &
=GS, the system can go into self oscilla- tions and even chaos. For optically induced fields much smaller than the external magnetic field, Bmax we expect trajectories of the spin in the xyplane, precessing around the external magnetic field along ezand therefore the spin dynamics (after a transient) is effectively two- dimensional. This is depicted by the blue-shaded area in Fig.(5). These limit cycles are governed by the op- tomechanical attractor diagram presented in Ref. [40], as we show in App. E. There is large parameter region in which the optomagnonic limit cycles deviate from the optomechanical attractors. This is marked by orange in Fig.(5). As the light intensity is increased, for =GS1 the limit cycles remain approximately confined to the xy plane but exhibit deviations from optomechanics. This approximate confinement of the trajectories to the xy plane at large Bmax= (Bmax= &0:5for
= ) can be understood qualitatively by looking at the ex- pression of the induced magnetic field Boptdeduced in the fast cavity limit, Eq. (10). Since we consider
= , =GS1impliesGS
. In this limit, Bopt= can become very small and the spin precession is around the ezaxis. For moderate Bmax= and =GS, the limit cy- cles are tilted and precessing around an axis determined by the effective magnetic field, a combination of the opti- calinducedfieldandtheexternalmagneticfield. Bluede- tuning causes these limit cycles to occur in the southern hemisphere. Period doubling leads eventually to chaos. The region where pockets of chaos can be found is rep- resented by red in the phase diagram. For large light intensity, such that Bmax , the optical field domi- nates and the effective magnetic field is essentially along theexaxis. The limit cycle is a precession of the spin around this axis. According to our results optomagnonic chaos is at- tained for values of the dimensionless coupling GS=  110and light intensities G2 max= 0:11. This implies a number of circulating photons similar to the number of locked spins in the material, which scales with the cavity volume. This therefore imposes a condition on the minimum circulating photon density in the cavity. For YIG with characteristic frequencies 110GHz,7 theconditiononthecouplingiseasilyfulfilled(remember GS= 10GHz as calculated above). However the condi- tion on the light intensity implies a circulating photon density of108109photons/m3which is outside of the current experimental capabilities, limited by the power a typical microcavity can support (around 105 photons/m3). On the other hand, magnetic switching and self-sustained oscillations of the optomechanical type (but taking place in the southern hemisphere) can be at- tained for low powers, assuming all external dissipation channels are kept small. While self-sustained oscillations and switching can be reached in the fast-cavity regime, morecomplexnonlinearbehaviorsuchasperioddoubling and chaos requires approaching sideband resolution. For YIG the examples in Figs. 3, 4 correspond to a preces- sion frequency 3
109Hz(App. D), whereas can be estimated to be1010Hz, taking into account the light absorption factor for YIG ( 0:3cm1) [35]. For red detuning
<0, the regions in the phase dia- gram remain the same, except that instead of magnetic switching, the solutions in this parameter range are fixed points near the north pole. This can be seen by the sym- metry of the problem: exchanging
!
together withex!exandez!ezleaves the problem un- changed. The limit cycles and trajectories follow also this symmetry, and in particular the limit cycles in the xyplane remain invariant. V. OUTLOOK The observation of the spin dynamics predicted here will be a sensitive probe of the basic cavity optomagnonic model, beyond the linear regime. Our analysis of the op- tomagnonic nonlinear Gilbert damping could be general- ized to more advanced settings, leading to optomagnonic reservoir engineering (e.g. two optical modes connected by a magnon transition). Although the nonlinear dy- namics presented here requires light intensities outside of the current experimental capabilities for YIG, it should be kept in mind that our model is the simplest case for which highly non-linear phenomena is present. Increas- ing the model complexity, for example by allowing for multiple-mode coupling, could result in a decreased light intensity requirement. Materials with a higher Faraday constantwouldbealsobeneficial. Inthisworkwefocused on the homogeneous Kittel mode. It will be an interest- ing challenge to study the coupling to magnon modes at finite wavevector, responsible for magnon-induced dissi- pation and nonlinearities under specific conditions [41– 43]. The limit cycle oscillations can be seen as “opto- magnonic lasing”, analogous to the functioning principle of a laser where energy is pumped and the system set- tles in a steady state with a characteristic frequency, and also discussed in the context of mechanics (“cantilaser” [44]). These oscillations could serve as a novel sourceof traveling spin waves in suitable geometries, and the synchronization of such oscillators might be employed to improve their frequency stability. We may see the de- sign of optomagnonic crystals and investigation of opto- magnonic polaritons in arrays. In addition, future cav- ity optomagnonics experiments will allow to address the completely novel regime of cavity-assisted coherent op- tical manipulation of nonlinear magnetic textures, like domain walls, vortices or skyrmions, or even nonlinear spatiotemporal light-magnon patterns. In the quantum regime, prime future opportunities will be the conversion of magnons to photons or phonons, the entanglement be- tween these subsystems, and their applications to quan- tum communication and sensitive measurements. We note that different aspects of optomagnonic sys- tems have been investigated in a related work done simultaneously [45]. Our work was supported by an ERC-StG OPTOMECH and ITN cQOM. H.T. ac- knowledges support by the Defense Advanced Research ProjectsAgency(DARPA)MicrosystemsTechnologyOf- fice/Mesodynamic Architectures program (N66001-11-1- 4114) and an Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initia- tive grant (FA9550-15-1-0029). 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You, npj Quantum Information 1, 15014 (2015). [18] N. J. Lambert, J. A. Haigh, and A. J. Ferguson, Journal of Applied Physics 117, 41 (2015). [19] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya- mazaki, K. Usami, and Y. Nakamura, Science 349, 405 (2015), arXiv:1410.3781. [20] O. Ì. O. Soykal and M. E. Flatté, Physical Review Letters 104, 077202 (2010). [21] Y. Cao, P. Yan, H. Huebl, S. T. B. Goennenwein, and G. E. W. Bauer, Physical Review B 91, 094423 (2015). [22] B. Zare Rameshti, Y. Cao, and G. E. W. Bauer, Physical Review B 91, 214430 (2015). [23] X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, (2015), arXiv:1510.03545. [24] J. A. Haigh, S. Langenfeld, N. J. Lambert, J. J. Baum- berg, A. J. Ramsay, A. Nunnenkamp, and A. J. Fergu- son, Physical Review A 92, 063845 (2015). [25] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Science Advances 2, 1501286 (2016). [26] R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. 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Appendix A: Optomagnonic coupling Gfor plane waves In this section we calculate explicitly the optomagnonic coupling presented in Eq.. (5) for the case of plane waves mode functions for the electric field. We choose for definiteness the magnetization axis along the ^ zaxis, and consider the case Gx 6= 0. The Hamiltonian HMOis then diagonal in the the basis of circularly polarized waves, eR=L=1p 2(eyiez). The rationale behind choosing the coupling direction perpendicular to the magnetization axis, is to maximize the coupling to the magnon mode, that is to the deviations of the magnetization with respect to the magnetization axis. The relevant spin operator is therefore ^Sx, which represents the flipping of a spin. In the case of plane waves, we quantize the electric field according to ^E+()(r;t) = +()iP jejq ~!j 2"0"V^a(y) j(t)e+()ikj
r;whereV is the volume of the cavity, kjthe wave vector of mode jand we have identified the positive and negative frequency components of the field as E!^E+,E!^E. The factor of "0"in the denominator ensures the normalization ~!j="0"hjj
d3rjE(r)j2jji"0"h0j
d3rjE(r)j2j0i, which corresponds to the energy of a photon in state jjiabove the vacuumj0i. For two degenerate (R/L) modes at frequency !, using Eq. (2) we see that the frequency dependence cancels out and we obtain the simple form for the optomagnonic Hamiltonian HMO=~G^Sx(^ay L^aL^ay R^aR)with G=1 ScF 4p". Therefore the overlap factor = 1in this case.9 Appendix B: Rescaled fields and linearized dynamics To analyze Eq. (6) it is convenient to re-scale the fields such that a=maxa0,S=SS0and measure all times and frequencies in . We obtain the rescaled equations of motion (time-derivatives are now with respect to t0= t) _a0=i(GS S0 x
)a0 2 (a01) (B1) _S0=G2 max ja0j2exez S0+G S _S0S0 (B2) If we linearize the spin-dynamics (around the north-pole, e.g.), we should recover the optomechanics behavior. In this section we ignore the intrinsic Gilbert damping term. We set approximately S0(S0 x;S0 y;1)Tand from Eq. (B1) we obtain _S0 x=S0 y (B3) _S0 y=G2 max ja0j2S0 x (B4) We can now choose to rescale further, via S0 x= max=p S S00 xand likewise for S0 y. We obtain the following spin-linearized equations of motion: _S00 x=S00 y (B5) _S00 y=Gp Smax ja0j2S00 x (B6) _a0=i(Gp Smax S00 x
)a0 2 (a01) (B7) This means that the number of dimensionless parameters has been reduced by one, since the two parameters initially involving G, S, andmaxhave all been combined into Gp Smax (B8) In other words, for S0 x;y=Sx;y=S1, the dynamics should only depend on this combination, consistent with the optomechanicalanalogyvalidinthisregimeasdiscussedinthemaintext(wherewearguedbasedontheHamiltonian). Appendix C: Switching in the fast cavity limit From Eq. (9) in the weak dissipation limit ( G1) we obtain _Sx= Sy _Sy=SzBopt Sxopt S_SxSz; from where we obtain an equation of motion for Sx. We are interested in studying the stability of the north pole once the driving is turned on. Hence we set Sz=S, Sx= SBopt 2Sxopt _Sx; and we consider small deviations SxofSxfrom the equilibrium position that satisfies S0 x=SBopt= , whereBopt is evaluated at S0 x. To linear order we obtain Sx=  +S@Bopt @Sx Sx+ 2GS Bopt(
+GSB opt= ) h (= )2+ (
+GSB opt= )2i2_Sx: We see that the dissipation coefficient for blue detuning (
>0) is always negative, giving rise to runaway solutions. Therefore the solutions near the north pole are always unstable under blue detuning, independent of the light intensity.10 These trajectories run to a fixed point near the south pole, which accepts stable solutions for
>0(switching) or to a limit cycle. Near the south pole, Sz=S,S0 x=SBopt= and Sx=  S@Bopt @Sx Sx2GS Bopt(
GSB opt= ) h (= )2+ (
GSB opt= )2i2_Sx: Therefore for
> GSB opt= there are stable fixed points, while in the opposite case there are also runaway solutions that are caught in a limit cycle. For red detuning,
!
and the roles of south and north pole are interchanged. Appendix D: Nonlinear dynamics In this section we give more details on the full nonlinear dynamics described in the main text. In Figs. 3 and (4) of the main text we chose a relative coupling GS= = 3, around which a chaotic attractor is found. With our estimated GS1010Hzfor YIG, this implies a precession frequency 3
109Hz. In Fig. (3) the chaotic regime is reached at  =2withG2 max= = 1, which implies 2 maxS=3, that is, a number of photons circulating in the (unperturbed) 2.53 -0.5 -1Spin projectionSz/S GS/⌦Normalized coupling Figure 6. (Color online) Bifurcation density plot for G2 max= = 1and= = 1at
= (G= 0), as a function of the relative coupling strength GS= . The dotted blue line indicates GS= = 3, for comparison with Fig. (4). As in the main text, the points (obtained after the transient) are given by plotting the values of Szattained whenever the trajectory fulfills the turning point condition _Sz= 0, for 20 different random initial conditions.11 1.01.50-1-2Spin projectionGSz/⌦ /⌦Detuning Figure 7. (Color online) Bifurcation density plot for GS= = 3,G2 max= = 1and= = 1(G= 0), as a function of the detuning
= . The dotted blue line indicates
= = 1, for comparison with Fig. (4). cavity of the order of the number of locked spins and hence scaling with the cavity volume. Bigger values of the cavity decay rate are allowed for attaining chaos at the same frequency, at the expense of more photons in the cavity, as can be deduced from Fig. (4) where we took = . On the other hand we can think of varying the precession frequency by an applied external magnetic field and explore the nonlinearities by tuning GS= in this way (note that GSis a material constant). This is done in Fig. (6). Alternatively, the nonlinear behavior can be controlled by varying the detuning
, as shown in Fig. (7). Appendix E: Relation to the optomechanical attractors In this appendix we show that the optomagnonic system includes the higher order nonlinear attractors found in optomechanics as a subset in parameter space. In optomechanics, the high order nonlinear attractors are self sustained oscillations with amplitudes Asuch that the optomechanical frequency shift GAis a multiple of the mechanical frequency . Translating to our case, this meansGSn . SinceS=SGjmaxj2= =Bmax= we obtain the condition GS Bmax n (E1) for observing these attractors. We can vary Bmaxaccording to Eq. (E1). For =GS1we are in the limit of small Bmax= and we expect limit cycles precessing along ezas discussed in Sec. (IV). In Fig. 8 the attractor diagram12 5101520 2015105GS/⌦GSx/⌦ 10302020301040GS/⌦GSx/⌦ Figure 8. Attractor diagram for
= 1:5 and= = 1with condition G2Sjmaxj2=n 2. Top:n= 1, bottomn= 10. We plot theSxvalues attained at the turning points ( _Sx= 0) forSx>0. The diagram is symmetric for Sx<0as expected for a limit cycle on the Bloch sphere. The diagram at the left coincides to a high degree of approximation with the predictions obtained for optomechanical systems (i.e. replacing the spin by a harmonic oscillator). In contrast, this is no longer the case for the diagram on the right, which involves higher light intensities.13 obtained by imposing condition (E1) is plotted. Since the trajectories are in the xyplane, we plot the inflection point of the coordinate Sx. We expect GSx= evaluated at the inflection point, which gives the amplitude of the limit cycle, to coincide with the optomechanic attractors for small Bmax= and hence flat lines at the expected amplitudes (as calculated in Ref. [40]) as GS= increases. Relative evenly spaced limit cycles increasing in number as larger values ofGS= are considered are observed, in agreement with Ref. [40]. Remarkable, these limit cycles attractors are found on the whole Bloch sphere, and not only near the north pole where the harmonic approximation is strictly valid. These attractors are reached by allowing initial conditions on the whole Bloch sphere. For n= 1, (Fig. 8, top), switching is observed up to GS= 4and then perfect optomechanic behavior. For higher values of n, deviations from the optomechanical behavior are observed for small GS= (implying large Bmax= according to Eq. (E1)) and large amplitude limit cycles, as compared to the size of the Bloch sphere. An example is shown in Fig. 8, bottom, forn= 10.

2016-04-24

Experiments during the past two years have shown strong resonant photon-magnon coupling in microwave cavities, while coupling in the optical regime was demonstrated very recently for the first time. Unlike with microwaves, the coupling in optical cavities is parametric, akin to optomechanical systems. This line of research promises to evolve into a new field of optomagnonics, aimed at the coherent manipulation of elementary magnetic excitations by optical means. In this work we derive the microscopic optomagnonic Hamiltonian. In the linear regime the system reduces to the well-known optomechanical case, with remarkably large coupling. Going beyond that, we study the optically induced nonlinear classical dynamics of a macrospin. In the fast cavity regime we obtain an effective equation of motion for the spin and show that the light field induces a dissipative term reminiscent of Gilbert damping. The induced dissipation coefficient however can change sign on the Bloch sphere, giving rise to self-sustained oscillations. When the full dynamics of the system is considered, the system can enter a chaotic regime by successive period doubling of the oscillations.

Coupled Spin-Light dynamics in Cavity Optomagnonics

1604.07053v3

arXiv:0807.2901v1 [cond-mat.str-el] 18 Jul 2008Current-induced dynamics of spiral magnet Kohei Goto,1,∗Hosho Katsura,1,†and Naoto Nagaosa1,2,‡ 1Department of Applied Physics, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2Cross-Correlated Materials Research Group (CMRG), ASI, RI KEN, Wako 351-0198, Japan We study the dynamics of the spiral magnet under the charge cu rrent by solving the Landau- Lifshitz-Gilbert equation numerically. In the steady stat e, the current /vectorjinduces (i) the parallel shift of the spiral pattern with velocity v= (β/α)j(α,β: the Gilbert damping coefficients), (ii) the uniform magnetization Mparallel or anti-parallel to the current depending on the ch irality of the spiral and the ratio β/α, and (iii) the change in the wavenumber kof the spiral. These are ana- lyzed by the continuum effective theory using the scaling arg ument, and the various nonequilibrium phenomena such as the chaotic behavior and current-induced annealing are also discussed. PACS numbers: 72.25.Ba, 71.70.Ej, 71.20.Be, 72.15.Gd The current-induced dynamics of the magnetic struc- tureisattractingintensiveinterestsfromtheviewpointof the spintronics. A representative example is the current- driven motion of the magnetic domain wall (DW) in fer- romagnets [1, 2]. This phenomenon can be understood from the conservation of the spin angular momentum, i.e., spin torque transfer mechanism [3, 4, 5, 6, 7]. The memorydevicesusing this current-inducedmagneticDW motion is now seriously considered [8]. Another example is the motion of the vortex structure on the disk of a ferromagnet, where the circulating motion of the vortex core is sometimes accompanied with the inversion of the magnetization at the core perpendicular to the disk [9]. Therefore, the dynamics of the magnetic structure in- duced by the current is an important and fundamental issue universal in the metallic magnetic systems. On the otherhand, thereareseveralmetallicspiralmagnetswith the frustrated exchange interactions such as Ho metal [10, 11], and with the Dzyaloshinskii-Moriya(DM) inter- action suchas MnSi [12, 13, 14], (Fe,Co)Si [15], and FeGe [16]. Thequantumdisorderingunderpressureorthenon- trivial magnetic textures have been discussed for the lat- ter class of materials. An important feature is that the direction of the wavevector is one of the degrees of free- dom in addition to the phase of the screw spins. Also the non-collinear nature of the spin configuration make it an interesting arena for the study of Berry phase effect [17], whichappearsmostclearlyinthecouplingtothecurrent. However, the studies on the current-induced dynamics of the magnetic structures with finite wavenumber, e.g., an- tiferromagnet and spiral magnet, are rather limited com- pared with those on the ferromagnetic materials. One reason is that the observation of the magnetic DW has been difficult in the case of antiferromagnets or spiral magnets. Recently, the direct space-time observation of the spiral structure by Lorentz microscope becomes pos- sible for the DM induced spiralmagnets [15, 16] since the wavelength of the spiral is long ( ∼100nm). Therefore, the current-induced dynamics of spiral magnets is now an interesting problem of experimental relevance.In this paper, we study the current-induced dynamics of the spiral magnet with the DM interaction as an ex- plicit example. One may consider that the spiral magnet can be regarded as the periodic array of the DW’s in fer- romagnet,butithasmanynontrivialfeaturesunexpected from this naive picture as shown below. The Hamiltonian we consider is given by [18] H=/integraldisplay d/vector r/bracketleftBigJ 2(/vector∇/vectorS)2+γ/vectorS·(/vector∇×/vectorS)/bracketrightBig ,(1) whereJ >0 is the exchange coupling constant and γis the strength of the DM interaction. The ground state of His realized when /vectorS(/vector r) is a proper screw state such that /vectorS(/vector r) =S(/vector n1cos/vectork·/vector r+/vector n2sin/vectork·/vector r), (2) where the wavenumber /vectork=/vector n3|γ|/J, and/vector ni(i= 1,2,3) form the orthonormal vector sets. The ground state en- ergy is given by −VS2γ2/2JwhereVis the volume of the system. The sign of γis equal to that of( /vector n1×/vector n2)·/vector n3, determining the chirality of the spiral. The equation of motion of the spin under the current is written as ˙/vectorS=gµB ¯h/vectorBeff×/vectorS−a3 2eS(/vectorj·/vector∇)/vectorS+a3 2eSβ/vectorS×(/vectorj·/vector∇)/vectorS+α S/vectorS×˙/vectorS (3) where/vectorBeff=−δH/δ/vectorSis the effective magnetic field and α,βare the Gilbert damping constants introduced phe- nomenologically [19, 20]. We discretize the Hamiltonian Eq.(1) and the equation of motion Eq.(3) by putting spins on the chain or the square lattice with the lattice constant a, and replacing the derivative by the difference. The length of the spin |/vectorSi|is a constant of motion at each site i, and we can easily derive ˙H=δH δ/vectorS·˙/vectorS=−α|˙/vectorS|2from Eq.(3), i.e., the energy continues to decrease as the time evolution. We start with the one-dimensional case along x-axis as shown in Fig.1. The discretization means replacing ∂x/vectorS(x) by (/vectorSi+1−/vectorSi−1)/2a, and∂2 x/vectorS(x) by (/vectorSi+1−2/vectorSi+2 /vectorSi−1)/a2. We note that the wavenumber which mini- mizes Eq.(1) is k=k0= arcsin( γ/J) on the discretized one-dimensional lattice. Numerical study of Eq.(3) have been done with gµB/¯h= 1, 2e= 1,S= 1,a= 1 J= 2, and γ= 1.2. In this condition, the wavelength of the spiral λ= 2π/k0≈11.6 is long compared with the lattice constant a= 1, and we choose the time scale ∆t/(1 +α2) = 10−2. We have confirmed that the re- sults do not depend on ∆ teven if it is reduced by the factor 10−1or 10−2. The sample size Lis 104with the open boundary condition. As we will show later, the typical value of the current is j∼2γand in the real situation with the wavelength λ[nm], the exchange cou- pling constant J[eV] and the lattice constant a[nm], it is j≈3.2×1015J/(λa)[A/m2]. Substituting J= 0.02, λ= 100,a= 0.5 into above estimatation, the typi- cal current is 1012[A/m2], and the unit of the time is ∆t=J/¯h≈30[ps]. The Gilbert damping coefficients α,βare typically 10−3∼10−1in the realistic systems. In most of the cal- culations, however,wetake α= 5.0toaccelaratethecon- vergence to the steady state. The obtained steady state depends only on the ratio β/αexcept the spin configura- tions near the boundaries as confirmed by the simlations withα= 0.1. We employ the two types of initial condi- tion, i.e., the ideal proper screw state with the wavenum- berk0, and the random spin configurations. The differ- ence of the dynamics in these two cases are limited only in the early stage ( t <5000∆t). Now we consider the steady state with the constant velocity for the shift of the spiral pattern obtained after the time of the order of 105∆t. One important issue here is the current-dependence of the velocity, which has been discussed intensivelyfor the DW motion in ferromagnets. In the latter case, there appears the intrinsic pinning in the case of β= 0 [4], while the highly nonlinear behavior forβ/α/negationslash= 0 [20]. In the special case of β=α, the trivial solution corresponding to the parallel shift of the ground state configuration of Eq.(1) with the velocity v=jis considered to be realized [5]. Figure 2 shows the results for the velocity, the induced uniform magnetization Sx alongx-axis, and the wavevector kof the spiral in the steady state. The current-dependence of the velocity for the cases of β= 0.1,0.5α,αand 2αis shown in Fig. 2(a). Figure 2(b) shows the β/α-dependence of the ve- locity for the fixed current j= 1.2. It is seen that the velocity is almost proportional to both the current jand the ratio β/α. Therefore, we conclude that the velocity v= (β/α)jwithout nonlinear behavior up to the current j∼2γ, which is in sharp contrast to the case of the DW motion in ferromagnets. The unit of the velocity is given bya/∆t, which is of the order of 20[m /s] fora≈5[˚A] and ∆t≈30[ps]. In Fig. 2(c) shown the wavevector k of the spiral under the current j= 1.2 for various values ofβ/α. It shows a non-monotonous behavior with the maximum at β/α≈0.2, and is always smaller than the FIG. 1: Spin configurations in the spiral magnet (a) in the equilibrium state, and (b) under the current. Under the current /vectorj, the uniform magnetization Sxalong the spi- ral axis/current direction is induced together with the ro- tation of the spin, i.e., the parallel shift of the spiral pat - tern with the velocity v. Note that the magnetization is anti-parallel/parallel to the current direction with po si- tive/negative γforβ < α, while it is reversed for β > α, and the wavenumber kchanges from the equilibrium value. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5v jβ/α=2.0 β/α=1.0 β/α=0.5 β/α=0.1 (a) 0 0.5 1 1.5 2 0 0.5 1 1.5 2 v / j β / α(b) 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 k β / α(c) -1 -0.5 0 0.5 1 0 0.5 1 1.5 2 Sx β / α(d) FIG. 2: For the case of γ= 1.2, the numerical result is shown. (a) The steady state velocity vas a function of the current jwith the fixed values of β= 0.1,0.5α,α, and 2α. (b) The velocityvas a function of β/αfor a fixed value of the current j= 1.2.visalmost proportional to β/α. (c)Thewavenumber kas a function of β/α. The dotted line shows k0in the equilibrium. (d) The uniform magnetization Sxalong the current direction as a function of β/αfor a fixed value of j= 1.2. wavenumber k0in the equilibrium shown in the dotted line. Namely, the period of the spiral is elongated by the current. As shown in Fig. 2(d), there appears the uni- form magnetization Sxalong the x-direction. Sxis zero and changes the sign at β/α= 1. With the positive γ (as in the case of Fig. 2(d)), Sxis anti-parallel to the currentj//xwithβ < αand changes its direction for β > α. For the negative γ, the sign of Sxis reversed. As for the velocity /vector v, on the other hand, it is always parallel to the current /vectorj.3 Now we present the analysis of the above results in terms of the continuum theory and a scaling argument. For one-dimensional case, the modified LLG Eq.(3) can be recast in the following form: ˙/vectorS=−J/vectorS×∂2 x/vectorS−(2γSx+j)∂x/vectorS+/vectorS×(α˙/vectorS+βj∂x/vectorS).(4) It is convenientto introduceamovingcoordinates ˆξ(x,t), ˆη, andˆζ(x,t) (see Fig.1) [21]. They are explicitly defined through ˆ x, ˆyand ˆzas ˆζ(x,t) = cos( k(x−vt)+φ)ˆy+sin(k(x−vt)+φ)ˆz, ˆξ(x,t) =−sin(k(x−vt)+φ)ˆy+cos(k(x−vt)+φ)ˆz, and ˆη= ˆx. We restrict ourselves to the following ansatz: /vectorS(x,t) =Sxˆη+/radicalbig 1−S2xˆζ(x,t), (5) whereSxis assumed to be constant. By substituting Eq.(5) into Eq.(4), we obtain vas v=β αj, (6) by requiring that there is no force along ˆ η- andˆζ- directions acting on each spin. In contrast to the DW motioninferromagnets,thevelocity vbecomeszerowhen β→0 even for large value of the current. The numerical results in Fig. 2(a), (b) show good agreement with this prediction Eq.(6). On the other hand, the magnetization Sxalongx-axis is given by Sx=β/α−1 2γ−Jkj, (7) once the wavevector kis known. Here we note that the above solution is degenerate with respect to k, which needs to be determined by the numerical solution. From the dimensional analysis, the spiral wavenumber kis given by the scaling form, k=k0g(j/(2γ),β/α) with the dimensionless function g(x,y) and also is Sxthrough Eq.(7). Motivated by the analysis above, we study the γ- dependence of the steady state properties. In Fig.3, we show the numerical results for k/k0andSxas the functions of j/2γin the cases of β/α= 0.1,0.5 and 2. Roughly speaking, the degeneracies of the data are obtained approximately for each color points (the same β/αvalue) with different γvalues. The deviation from the scaling behavior is due to the discrete nature of the lattice model, which is relevant to the realistic situation. Forβ/α= 0.1(black points in Fig.3), kremainsconstant andSxis induced almost proportional to the current up j/2γ≈0.4, where the abrupt change of koccurs. For β/α= 0.5 (blue points) and β/α= 2.0 (red points), the changes of kandSxare more smooth. A remarkable result is that the spin Sion the lattice point iis well 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 k / k 0 j / 2γβ/α=2.0, γ=1.2 β/α=2.0, γ=1.0 β/α=2.0, γ=0.8 β/α=2.0, γ=0.6 β/α=0.5, γ=1.2 β/α=0.5, γ=1.0 β/α=0.5, γ=0.8 β/α=0.5, γ=0.6 β/α=0.1, γ=1.2 β/α=0.1, γ=1.0 β/α=0.1, γ=0.8 β/α=0.1, γ=0.6 (a) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Sx j / 2γβ/α=2.0, γ=1.2 β/α=2.0, γ=1.0 β/α=2.0, γ=0.8 β/α=2.0, γ=0.6 β/α=0.5, γ=1.2 β/α=0.5, γ=1.0 β/α=0.5, γ=0.8 β/α=0.5, γ=0.6 β/α=0.1, γ=1.2 β/α=0.1, γ=1.0 β/α=0.1, γ=0.8 β/α=0.1, γ=0.6 (b) FIG.3: Thescalingplotfor (a) k/k0wherek0isthewavenum- ber in the equilibrium without the current, and (b) Sxin the steady state as the function of j/2γ. The black, blue, and red color points correspond to β/α= 0.1, 0.5 and 2.0, respec- tively. The curves in (b) indicate Eq.(7) calculated from th e kvalues in (a), showing the good agreement with the data points. described by Eq.(5) at x=xi, and hence the relation Eq.(7) is well satisfied as shown by the curves in Fig. 3(b), even though the scaling relation is violated to some extent. For larger values of jbeyond the data points, i.e.,j/2γ >0.75 forβ/α= 0.1,j/2γ >1.5 forβ/α= 0.5 andj/2γ >0.9 forβ/α= 2.0, the spin configuration is disordered from harmonic spiral characterized by a sin- gle wavenumber k. The spins are the chaotic funtion of both space and time in this state analogousto the turbu- lance. This instability is triggered by the saturated spin Sx=±1, occuring near the edge of the sample. Next, we turn to the simulations on the two- dimensional square lattice in the xy-plane. In this case, the direction of the spiral wavevector becomes another important variable because the degeneracy of the ground state energy occurs. Starting with the random spin configuration, we sim- ulate the time evolution of the system without and with the current as shown in Fig.4. Calculation has been done with the same parameters as in the one-dimensional case4 (a)/vectorj= 0 (b)/vectorj= (0.3,0) (c)/vectorj= (0.3/√ 2,0.3/√ 2) color box of Sz(r) color box of |Sz(k)|2 FIG. 4: The time evolution of the zcomponent Szof the spin from the random initial configuration of the 102×102 section in the middle of the sample is shown in the case of (a)j= 0, (b) j= 0.3 along the x-axis, (c) j= 0.3 along the (1,1)-direction. From the left, t= 102∆t, 1.7×103∆t, 5×103∆t. The rightmost panels show the spectral intensity |Sz(/vectork,5×103∆t)|2from the whole sample of the size 210×210 in the momentum space /vectork= (kx,ky). whereγ= 1.2,β= 0, and the system size is 210×210. In the absence of the current, the relaxation of the spins into the spiral state is very slow, and many dislo- cations remain even after a long time. Correspondingly, the energy does not decrease to the ground state value but approaches to the higher value with the power-law like long-time tail. The momentum-resolved intensity is circularly distributed with the broad width as shown in Fig. 4(a) corresponding to the disordered direction of /vectork. This glassy behavior is distinct from the relaxation dynamics of the ferromagnet where the large domain for- mation occurs even though the DW’s remain. Now we put the current along the ˆ x(Fig. 4(b)) and (ˆ x+ˆy) (Fig. 4(c)) directions. It is seen that the direction of /vectorkis con- trolled by the current also with the radial distribution in the momentum space being narrower than that in the absence of j(Fig. 4(a)). This result suggests that the currentjwith the density ∼1012[A/m2] of the time du- ration∼0.1[µsec] can anneal the directional disorder of the spiral magnet. After the alignment of /vectorkis achieved, the simulations on the one-dimensional model described above are relevant to the long-time behavior.To summarize, we have studied the dynamics of the spiral magnet with DM interaction under the current j by solving the Landau-Lifshitz-Gilbert equation numeri- cally. In the steady state under the charge current j, the velocityvis given by ( β/α)j(α,β: the Gilbert-damping coefficients), the uniform magnetization is induced par- allel or anti-parallel to the current direction, and period of the spiral is elongated. The annealing effect especially on the direction of the spiral wavevector /vectorkis also demon- strated. TheauthorsaregratefultoN.FurukawaandY.Tokura for fruitful discussions. This work was supported in part by Grant-in-Aids (Grant No. 15104006, No. 16076205, and No. 17105002) and NAREGI Nanoscience Project fromtheMinistryofEducation, Culture, Sports, Science, andTechnology. HK wassupported bythe JapanSociety for the Promotion of Science. ∗Electronic address: goto@appi.t.u-tokyo.ac.jp †Electronic address: katsura@appi.t.u-tokyo.ac.jp ‡Electronic address: nagaosa@appi.t.u-tokyo.ac.jp [1] L. Berger, J. Appl. Phys. 49, 2156 (1978). [2] L. Berger, Phys. Rev. B 33, 1572 (1986). [3] J. C. Slonczewski, Int. J. Magn. 2, 85 (1972). [4] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601(2004). [5] E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204 (2005). [6] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205 (2004). [7] M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno, Nature428,539 (2004). [8] M. Hayashi, L. Thomas, C. Rettner, R. Moriya, and S. S. P. Parkin, Nat. Phys. 3, 21 (2007). [9] K. Yamada et al., Nat. Mat. 6, 269 (2007). [10] W. C. Koehler, J. Appl. Phys 36, 1078 (1965). [11] R. A. Cowley et al., Phys. Rev. B 57, 8394 (1998). [12] M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch, S. Blugel, and R. Wiesendanger, Nature (London) 447, 190 (2007). [13] C. Pfleiderer, S. R. Julian, and G. G. Lonzarich, Nature (London) 414, 427 (2001). [14] N. Doiron-Leyraud, I. R. Waker, L. Taillefer, M. J. Steiner, S. R. Julian, and G. G. Lonzarich, Nature (Lon- don) 425, 595 (2003). [15] M. Uchida, Y. Onose, Y. Matsui, and Y. Tokura, Science 311, 359 (2006). [16] M. Uchida et al., Phys. Rev. B 77, 184402 (2008). [17] M. V. Berry, Proc. R. Soc. London A 392, 45 (1984). [18] L. D. Landau, in Electrodynamics of Continuous Media (Pergamon Press, 1984), p178. [19] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). [20] A. Thiaville et al., Europhys. Lett. 69, 990 (2005). [21] T. Nagamiya, in Solid State Physics , edited by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic Press, New Yotk, 1967), Vol. 20, p. 305.

2008-07-18

We study the dynamics of the spiral magnet under the charge current by solving the Landau-Lifshitz-Gilbert equation numerically. In the steady state, the current ${\vec j}$ induces (i) the parallel shift of the spiral pattern with velocity $v=(\beta/\alpha)j$ ($\alpha$, $\beta$: the Gilbert damping coefficients), (ii) the uniform magnetization $M$ parallel or anti-parallel to the current depending on the chirality of the spiral and the ratio $\beta / \alpha $, and (iii) the change in the wavenumber $k$ of the spiral. These are analyzed by the continuum effective theory using the scaling argument, and the various nonequilibrium phenomena such as the chaotic behavior and current-induced annealing are also discussed.

Current-induced dynamics of spiral magnet

0807.2901v1

Effective damping enhancement in noncollinear spin structures Levente Rózsa,1,∗Julian Hagemeister,1Elena Y. Vedmedenko,1and Roland Wiesendanger1 1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany (Dated: August 29, 2021) Damping mechanisms in magnetic systems determine the lifetime, diffusion and transport prop- erties of magnons, domain walls, magnetic vortices, and skyrmions. Based on the phenomenological Landau–Lifshitz–Gilbert equation, here the effective damping parameter in noncollinear magnetic systems is determined describing the linewidth in resonance experiments or the decay parameter in time-resolved measurements. It is shown how the effective damping can be calculated from the elliptic polarization of magnons, arising due to the noncollinear spin arrangement. It is concluded that the effective damping is larger than the Gilbert damping, and it may significantly differ be- tween excitation modes. Numerical results for the effective damping are presented for the localized magnons in isolated skyrmions, with parameters based on the Pd/Fe/Ir(111) model-type system. Spinwaves(SW)ormagnonsaselementaryexcitations of magnetically ordered materials have attracted signifi- cant research attention lately. The field of magnonics[1] concerns the creation, propagation and dissipation of SWs in nanostructured magnetic materials, where the dispersion relations can be adjusted by the system ge- ometry. A possible alternative for engineering the prop- erties of magnons is offered by noncollinear (NC) spin structures[2] instead of collinear ferro- (FM) or antifer- romagnets (AFM). SWs are envisaged to act as informa- tion carriers, where one can take advantage of their low wavelengths compared to electromagnetic waves possess- ing similar frequencies[3]. Increasing the lifetime and the stability of magnons, primarily determined by the relax- ation processes, is of crucial importance in such applica- tions. The Landau–Lifshitz–Gilbert (LLG) equation[4, 5] is commonly applied for the quasiclassical description of SWs, where relaxation is encapsulated in the dimen- sionless Gilbert damping (GD) parameter α. The life- time of excitations can be identified with the resonance linewidth in frequency-domain measurements such as fer- romagnetic resonance (FMR)[6], Brillouin light scatter- ing (BLS)[7] or broadband microwave response[8], and with the decay speed of the oscillations in time-resolved (TR) experiments including magneto-optical Kerr effect microscopy (TR-MOKE)[9] and scanning transmission x- ray microscopy (TR-STXM)[10]. Since the linewidth is knowntobeproportionaltothefrequencyofthemagnon, measuring the ratio of these quantities is a widely ap- plied method for determining the GD in FMs[3, 6]. An advantage of AFMs in magnonics applications[11, 12] is their significantly enhanced SW frequencies due to the exchange interactions, typically in the THz regime, com- pared to FMs with GHz frequency excitations. However, it is known that the linewidth in AFM resonance is typ- ically very wide because it scales with a larger effective damping parameter αeffthan the GD α[13]. The tuning of the GD can be achieved in magnonic crystals by combining materials with different values of α. It was demonstrated in Refs. [14–16] that this leadsto a strongly frequency- and band-dependent αeff, based on the relative weights of the magnon wave functions in the different materials. Magnetic vortices are two-dimensional NC spin config- urations in easy-plane FMs with an out-of-plane magne- tized core, constrained by nanostructuring them in dot- orpillar-shapedmagneticsamples. Theexcitationmodes ofvortices, particularlytheirtranslationalandgyrotropic modes, havebeeninvestigatedusingcollective-coordinate models[17] based on the Thiele equation[18], linearized SW dynamics[19, 20], numerical simulations[21] and ex- perimental techniques[22–24]. It was demonstrated theo- retically in Ref. [21] that the rotational motion of a rigid vortex excited by spin-polarized current displays a larger αeffthan the GD; a similar result was obtained based on calculating the energy dissipation[25]. However, due to the unbounded size of vortices, the frequencies as well as the relaxation rates sensitively depend on the sample preparation, particularly because they are governed by the magnetostatic dipolar interaction. In magnetic skyrmions[26], the magnetic moment di- rections wrap the whole unit sphere. In contrast to vor- tices, isolated skyrmions need not be confined for stabi- lization, and are generally less susceptible to demagneti- zation effects[3, 27]. The SW excitations of the skyrmion lattice phase have been investigated theoretically[28–30] and subsequently measured in bulk systems[3, 8, 31]. It was calculated recently[32] that the magnon resonances measured via electron scattering in the skyrmion lattice phase should broaden due to the NC structure. Calcula- tions predicted the presence of different localized modes concentrated on the skyrmion for isolated skyrmions on a collinear background magnetization[33–35] and for skyrmions in confined geometries[20, 36, 37]. From the experimental side, the motion of magnetic bubbles in a nanodisk was investigated in Ref. [38], and it was pro- posed recently that the gyration frequencies measured in Ir/Fe/Co/Pt multilayer films is characteristic of a dilute array of isolated skyrmions rather than a well-ordered skyrmion lattice[6]. However, the lifetime of magnons in skyrmionic systems based on the LLG equation is appar-arXiv:1805.01815v2 [cond-mat.mtrl-sci] 15 Oct 20182 ently less explored. It is known that NC spin structures may influence the GD via emergent electromagnetic fields[29, 39, 40] or via the modified electronic structure[41, 42]. Besides deter- mining the SW relaxation process, the GD also plays a crucial role in the motion of domain walls[43–45] and skyrmions[46–48] driven by electric or thermal gradients, both in the Thiele equation where the skyrmions are assumed to be rigid and when internal deformations of the structure are considered. Finally, damping and de- formations are also closely connected to the switching mechanisms of superparamagnetic particles[49, 50] and vortices[51], as well as the lifetime of skyrmions[52–54]. Theαeffin FMs depends on the sample geometry due to the shape anisotropy[13, 55, 56]. It was demonstrated in Ref. [56] that αeffis determined by a factor describing the ellipticity of the magnon polarization caused by the shape anisotropy. Elliptic precession and GD were also investigated by considering the excitations of magnetic adatomsonanonmagneticsubstrate[57]. Thecalculation of the eigenmodes in NC systems, e.g. in Refs. [6, 20, 35], also enables the evaluation of the ellipticity of magnons, but this property apparently has not been connected to the damping so far. Although different theoretical methods for calculating αeffhave been applied to various systems, a general de- scriptionapplicabletoallNCstructuresseemstobelack- ing. Here it is demonstrated within a phenomenological description of the linearized LLG equation how magnons in NC spin structures relax with a higher effective damp- ing parameter αeffthan the GD. A connection between αeffand the ellipticity of magnon polarization forced by the NC spin arrangement is established. The method is illustrated by calculating the excitation frequencies of isolated skyrmions, considering experimentally deter- mined material parameters for the Pd/Fe/Ir(111) model system[58]. It is demonstrated that the different local- ized modes display different effective damping parame- ters, with the breathing mode possessing the highest one. The LLG equation reads ∂tS=−γ/primeS×Beff−αγ/primeS×/parenleftBig S×Beff/parenrightBig ,(1) withS=S(r)the unit-length vector field describing the spin directions in the system, αthe GD and γ/prime= 1 1+α2ge 2mthe modified gyromagnetic ratio (with gbeing theg-factor of the electrons, ethe elementary charge and mthe electron mass). Equation (1) describes the time evolution of the spins governed by the effective magnetic fieldBeff=−1 MδH δS, withHthe Hamiltonian or free energy of the system in the continuum description and Mthe saturation magnetization. The spins will follow a damped precession relaxing to a local minimum S0ofH, given by the condition S0×Beff=0. Note that generally the Hamiltonian rep- resents a rugged landscape with several local energy min- ima, corresponding to e.g. FM, spin spiral and skyrmionlattice phases, or single objects such as vortices or iso- lated skyrmions. The excitations can be determined by switching to a local coordinate system[20, 34, 47] with the spins along the zdirection in the local minimum, ˜S0= (0,0,1), and expanding the Hamiltonian in the variablesβ±=˜Sx±i˜Sy, introduced analogously to spin raising and lowering or bosonic creation and annihila- tion operators in the quantum mechanical description of magnons[59–61]. The lowest-order approximation is the linearized form of the LLG Eq. (1), ∂tβ+=γ/prime M(i−α)/bracketleftbig (D0+Dnr)β++Daβ−/bracketrightbig ,(2) ∂tβ−=γ/prime M(−i−α)/bracketleftbig D† aβ++ (D0−Dnr)β−/bracketrightbig .(3) For details of the derivation see the Supplemental Material[62]. The term Dnrin Eqs. (2)-(3) is respon- sible for the nonreciprocity of the SW spectrum[2]. It accounts for the energy difference between magnons propagating in opposite directions in in-plane oriented ultrathin FM films[63, 64] with Dzyaloshinsky–Moriya interaction[65, 66] and the splitting between clockwise and counterclockwise modes of a single skyrmion[20]. Here we will focus on the effects of the anomalous term[34]Da, which couples Eqs. (2)-(3) together. Equa- tions (2)-(3) may be rewritten as eigenvalue equations by assuming the time dependence β±(r,t) =e−iωktβ± k(r). (4) Forα= 0, the spins will precess around their equilib- riumdirection ˜S0. Iftheequationsareuncoupled, the ˜Sx and ˜Syvariables describe circular polarization, similarly to the Larmor precession of a single spin in an exter- nal magnetic field. However, the spins are forced on an elliptic path due to the presence of the anomalous terms. The effective damping parameter of mode kis defined as αk,eff=/vextendsingle/vextendsingle/vextendsingle/vextendsingleImωk Reωk/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (5) which is the inverse of the figure of merit introduced in Ref. [15]. Equation (5) expresses the fact that Im ωk, the linewidth in resonance experiments or decay coeffi- cient in time-resolved measurements, is proportional to the excitation frequency Re ωk. Interestingly, there is a simple analytic expression con- nectingαk,effto the elliptic polarization of the modes at α= 0. Forα/lessmuch1, the effective damping may be ex- pressed as αk,eff α≈/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0) k(r)/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingleβ+(0) k(r)/vextendsingle/vextendsingle/vextendsingle2 dr/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0) k(r)/vextendsingle/vextendsingle/vextendsingle2 −/vextendsingle/vextendsingle/vextendsingleβ+(0) k(r)/vextendsingle/vextendsingle/vextendsingle2 dr=/integraltext a2 k(r) +b2 k(r)dr/integraltext 2ak(r)bk(r)dr, (6)3 0.0 0.2 0.4 0.6 0.8 1.00246810 FIG. 1. Effective damping parameter αk,effas a function of inverseaspectratio bk/akofthepolarizationellipse, assuming constantakandbkfunctions in Eq. (6). Insets illustrate the precession for different values of bk/ak. where the (0)superscript denotes that the eigenvectors β± k(r)defined in Eq. (4) were calculated for α= 0, while ak(r)andbk(r)denote the semimajor and semiminor axes of the ellipse the spin variables ˜Sx(r)and ˜Sy(r) are precessing on in mode k. Details of the derivation are given in the Supplemental Material[62]. Note that an analogous expression for the uniform precession mode in FMs was derived in Ref. [56]. The main conclusion from Eq. (6) is that αk,effwill depend on the considered SW mode and it is always at least as high as the GD α. Although Eq. (6) was obtained in the limit of low α, numerical calculations indicate that the αk,eff/αratio tends to increase for increasing values of α; see the Sup- plementalMaterial[62]foranexample. Theenhancement of the damping from Eq. (6) is shown in Fig. 1, with the space-dependent ak(r)andbk(r)replaced by constants for simplicity. It can be seen that for more distorted po- larization ellipses the spins get closer to the equilibrium directionafterthesamenumberofprecessions, indicating a faster relaxation. Since the appearance of the anomalous terms Dain Eqs. (2)-(3) forces the spins to precess on an elliptic path, it expresses that the system is not axially sym- metric around the local spin directions in the equilib- rium state denoted by S0. Such a symmetry breaking naturally occurs in any NC spin structure, implying a mode-dependent enhancement of the effective damping parameter in NC systems even within the phenomeno- logical description of the LLG equation. Note that the NC structure also influences the electronic properties of the system, which can lead to a modification of the GD itself, see e.g. Ref. [42]. In order to illustrate the enhanced and mode- dependent αk,eff, we calculate the magnons in isolated chiralskyrmionsinatwo-dimensionalultrathinfilm. Thedensity of the Hamiltonian Hreads[67] h=/summationdisplay α=x,y,z/bracketleftBig A(∇Sα)2/bracketrightBig +K(Sz)2−MBSz +D(Sz∂xSx−Sx∂xSz+Sz∂ySy−Sy∂ySz),(7) withAthe exchange stiffness, Dthe Dzyaloshinsky– Moriya interaction, Kthe anisotropy coefficient, and B the external field. In the following we will assume D>0andB≥0 without the loss of generality, see the Supplemental Material[62] for discussion. Using cylindrical coordi- nates (r,ϕ)in real space and spherical coordinates S= (sin Θ cos Φ ,sin Θ sin Φ,cos Θ)in spin space, the equi- librium profile of the isolated skyrmion will correspond to the cylindrically symmetric configuration Θ0(r,ϕ) = Θ0(r)andΦ0(r,ϕ) =ϕ, the former satisfying A/parenleftbigg ∂2 rΘ0+1 r∂rΘ0−1 r2sin Θ 0cos Θ 0/parenrightbigg +D1 rsin2Θ0 +Ksin Θ 0cos Θ 0−1 2MBsin Θ 0= 0 (8) with the boundary conditions Θ0(0) =π,Θ0(∞) = 0. The operators in Eqs. (2)-(3) take the form (cf. Refs. [34, 35, 47] and the Supplemental Material[62]) D0=−2A/braceleftBigg ∇2+1 2/bracketleftbigg (∂rΘ0)2−1 r2/parenleftbig 3 cos2Θ0−1/parenrightbig (∂ϕΦ0)2/bracketrightbigg/bracerightBigg −D/parenleftbigg ∂rΘ0+1 r3 sin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg −K/parenleftbig 3 cos2Θ0−1/parenrightbig +MBcos Θ 0, (9) Dnr=/parenleftbigg 4A1 r2cos Θ 0∂ϕΦ0−2D1 rsin Θ 0/parenrightbigg (−i∂ϕ), (10) Da=A/bracketleftbigg (∂rΘ0)2−1 r2sin2Θ0(∂ϕΦ0)2/bracketrightbigg +D/parenleftbigg ∂rΘ0−1 rsin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg +Ksin2Θ0.(11) Equation (11) demonstrates that the anomalous terms Daresponsible for the enhancement of the effective damping can be attributed primarily to the NC arrange- ment (∂rΘ0and∂ϕΦ0≡1) and secondarily to the spins becoming canted with respect to the global out- of-plane symmetry axis ( Θ0∈ {0,π}) of the system. TheDnrtermintroducesanonreciprocitybetweenmodes with positive and negative values of the azimuthal quan- tum number (−i∂ϕ)→m, preferring clockwise rotat- ing modes ( m < 0) over counterclockwise rotating ones (m > 0) following the sign convention of Refs. [20, 34]. BecauseD0andDnrdepend onmbutDadoes not, it is expected that the distortion of the SW polarization el- lipse and consequently the effective damping will be more enhanced for smaller values of |m|. The different modes as a function of external field are shown in Fig. 2(a), for the material parameters de- scribing the Pd/Fe/Ir(111) system. The FMR mode at4 0.7 0.8 0.9 1.0 1.1 1.20255075100125150175 (a) 0.7 0.8 0.9 1.0 1.1 1.21.01.52.02.53.0 (b) FIG. 2. Localized magnons in the isolated skyrmion, with the interaction parameters corresponding to the Pd/Fe/Ir(111) system[58]:A = 2.0pJ/m,D =−3.9mJ/m2,K = −2.5MJ/m3,M= 1.1MA/m. (a) Magnon frequencies f= ω/2πforα= 0. Illustrations display the shapes of the excita- tion modes visualized on the triangular lattice of Fe magnetic moments, with red and blue colors corresponding to positive and negative out-of-plane spin components, respectively. (b) Effective damping coefficients αm,eff, calculated from Eq. (6). ωFMR =γ M(MB−2K), describing a collective in-phase precession of the magnetization of the whole sample, sep- arates the continuum and discrete parts of the spectrum, with the localized excitations of the isolated skyrmion located below the FMR frequency[34, 35]. We found a single localized mode for each m∈{0,1,−2,−3,−4,−5} value, so in the following we will denote the excita- tion modes with the azimuthal quantum number. The m=−1mode corresponds to the translation of the skyrmion on the field-polarized background, which is a zero-frequency Goldstone mode of the system and not shown in the figure. The m=−2mode tends to zero aroundB= 0.65T, indicating that isolated skyrmions become susceptible to elliptic deformations and subse- quently cannot be stabilized at lower field values[68]. The values of αm,effcalculated from Eq. (6) for the different modes are summarized in Fig. 2(b). It is impor- tant to note that for a skyrmion stabilized at a selected 0 20 40 60 80 100-0.04-0.020.000.020.040.06 -0.03 0.00 0.03-0.030.000.03FIG. 3. Precession of a single spin in the skyrmion in the Pd/Fe/Ir(111) system in the m= 0andm=−3modes at B= 0.75T, from numerical simulations performed at α= 0.1. Inset shows the elliptic precession paths. From fitting the oscillations with Eq. (4), we obtained |Reωm=0|/2π= 39.22GHz,|Imωm=0|= 0.0608ps−1,αm=0,eff= 0.25and |Reωm=−3|/2π= 40.31GHz,|Imωm=−3|= 0.0276ps−1, αm=−3,eff= 0.11. field value, the modes display widely different αm,effval- ues, with the breathing mode m= 0being typically damped twice as strongly as the FMR mode. The ef- fective damping tends to increase for lower field values, and decrease for increasing values of |m|, the latter prop- erty expected from the m-dependence of Eqs. (9)-(11) as discussed above. It is worth noting that the αm,eff parameters are not directly related to the skyrmion size. Wealsoperformedthecalculationsfortheparametersde- scribing Ir|Co|Pt multilayers[69], and for the significantly largerskyrmionsinthatsystemweobtainedconsiderably smaller excitation frequencies, but quantitatively similar effective damping parameters; details are given in the Supplemental Material[62]. The different effective damping parameters could pos- sibly be determined experimentally by comparing the linewidths of the different excitation modes at a selected field value, or investigating the magnon decay over time. An example for the latter case is shown in Fig. 3, dis- playing the precession of a single spin in the skyrmion, obtained from the numerical solution of the LLG Eq. (1) withα= 0.1. AtB= 0.75T, the frequencies of the m= 0breathing and m=−3triangular modes are close to each other (cf. Fig. 2), but the former decays much faster. Because in the breathing mode the spin is follow- ing a significantly more distorted elliptic path (inset of Fig. 3) than in the triangular mode, the different effective damping is also indicated by Eq. (6). In summary, it was demonstrated within the phe- nomenological description of the LLG equation that the effective damping parameter αeffdepends on the consid- ered magnon mode. The αeffassumes larger values if5 the polarization ellipse is strongly distorted as expressed by Eq. (6). Since NC magnetic structures provide an anisotropic environment for the spins, leading to a dis- tortion of the precession path, they provide a natural choice for realizing different αeffvalues within a single system. The results of the theory were demonstrated for isolated skyrmions with material parameters describing the Pd/Fe/Ir(111) system. The results presented here may stimulate further experimental or theoretical work on the effective damping in skyrmions, vortices, domain walls or spin spirals. The authors would like to thank U. Atxitia and G. Meier for fruitful discussions. Financial support by the Alexander von Humboldt Foundation, by the Deutsche Forschungsgemeinschaft via SFB 668, by the European Union via the Horizon 2020 research and innovation pro- gram under Grant Agreement No. 665095 (MAGicSky), and by the National Research, Development and Inno- vation Office of Hungary under Project No. K115575 is gratefully acknowledged. ∗rozsa.levente@physnet.uni-hamburg.de [1] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. Phys. 43, 264001 (2010). [2] M.Garst, J.Waizner, andD.Grundler, J.Phys.D:Appl. Phys. 50, 293002 (2017). [3] T. Schwarze, J. Waizner, M. Garst, A. Bauer, I. Stasinopoulos, H. Berger, C. Pfleiderer, and D. Grundler, Nat. Mater. 14, 478 (2015). [4] L. Landau and E. Lifshitz, Phys. 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Nanotechnol. 11, 444 (2016). [70] F. Romá, L. F. Cugliandolo, and G. S. Lozano, Phys. Rev. E 90, 023203 (2014). [71] A. O. Leonov, T. L. Monchesky, N. Romming, A. Ku-betzka, A. N. Bogdanov, and R. Wiesendanger, New J. Phys. 18, 065003 (2016). [72] J. Hagemeister, A. Siemens, L. Rózsa, E. Y. Vedme- denko, and R. Wiesendanger, Phys. Rev. B 97, 174436 (2018).Supplemental Material to Effective damping enhancement in noncollinear spin structures Levente Rózsa,1,∗Julian Hagemeister,1Elena Y. Vedmedenko,1and Roland Wiesendanger1 1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany (Dated: August 29, 2021) In the Supplemental Material the derivation of the linearized equations of motion and the effective damping parameter are discussed. Details of the numerical determination of the magnon modes in the continuum model and in atomistic spin dynamics simulations are also given. S.I. LINEARIZED LANDAU–LIFSHITZ–GILBERT EQUATION Here we will derive the linearized form of the Landau– Lifshitz–Gilbert equation given in Eqs. (2)-(3) of the maintextanddiscussthepropertiesofthesolutions. The calculation is similar to the undamped case, discussed in detail in e.g. Refs. [1–3]. Given a spin configuration sat- isfying the equilibrium condition S0×Beff=0, (S.1) the local coordinate system with ˜S0= (0,0,1)may be introduced, andtheHamiltonianbeexpandedinthevari- ables ˜Sxand˜Sy. Thelineartermmustdisappearbecause the expansion is carried out around an equilibrium state. The lowest-order nontrivial term is quadratic in the vari- ables and will be designated as the spin wave Hamilto- nian, HSW=/integraldisplay hSWdr, (S.2) hSW=1 2/bracketleftbig˜Sx˜Sy/bracketrightbig/bracketleftbiggA1A2 A† 2A3/bracketrightbigg/bracketleftbigg˜Sx ˜Sy/bracketrightbigg =1 2/parenleftBig ˜S⊥/parenrightBigT HSW˜S⊥. (S.3) The operator HSWis self-adjoint for arbitrary equi- librium states. Here we will only consider cases where the equilibrium state is a local energy minimum, mean- ing thatHSW≥0; the magnon spectrum will only be well-defined in this case. Since hSWis obtained as an expansion of a real-valued energy density around the equilibrium state, and the spin variables are also real- valued, fromtheconjugateofEq.(S.3)onegets A1=A∗ 1, A2=A∗ 2, andA3=A∗ 3. The form of the Landau–Lifshitz–Gilbert Eq. (1) in the main text may be rewritten in the local coordinates by simply replacing Sby˜S0everywhere, including the definitionoftheeffectivefield Beff. TheharmonicHamil- tonianHSWin Eq. (S.2) leads to the linearized equation of motion ∂t˜S⊥=γ/prime M(−iσy−α)HSW˜S⊥,(S.4) ∗rozsa.levente@physnet.uni-hamburg.dewithσy=/bracketleftbigg 0−i i0/bracketrightbigg the Pauli matrix. By replacing ˜S⊥(r,t)→˜S⊥ k(r)e−iωktas usual, for α= 0the eigenvalue equation ωk˜S⊥ k=γ MσyHSW˜S⊥ k (S.5) is obtained. If HSWhas a strictly positive spectrum, thenH−1 2 SWexists, and σyHSWhas the same eigenvalues asH1 2 SWσyH1 2 SW. Since the latter is a self-adjoint ma- trix with respect to the standard scalar product on the Hilbert space, it has a real spectrum, consequently all ωk eigenvalues are real. Note that the zero modes of HSW, which commonly occur in the form of Goldstone modes due to the ground state breaking a continuous symme- try of the Hamiltonian, have to be treated separately. Finally, we mention that if the spin wave expansion is performed around an equilibrium state which is not a local energy minimum, the ωkeigenvalues may become imaginary, meaning that the linearized Landau–Lifshitz– Gilbert equation will describe a divergence from the un- stable equilibrium state instead of a precession around it. Equations (2)-(3) in the main text may be obtained by introducing the variables β±=˜Sx±i˜Syas described there. The connection between HSWand the operators D0,Dnr, andDais given by D0=1 2(A1+A3), (S.6) Dnr=1 2i/parenleftBig A† 2−A2/parenrightBig , (S.7) Da=1 2/bracketleftBig A1−A3+i/parenleftBig A† 2+A2/parenrightBig/bracketrightBig .(S.8) An important symmetry property of Eqs. (2)-(3) in the main text is that if (β+,β−) =/parenleftbig β+ ke−iωkt,β− ke−iωkt/parenrightbig is an eigenmode of the equations, then (β+,β−) =/parenleftBig/parenleftbig β− k/parenrightbig∗eiω∗ kt,/parenleftbig β+ k/parenrightbig∗eiω∗ kt/parenrightBig is another solution. Following Refs. [1, 3], this can be attributed to the particle-hole symmetry of the Hamiltonian, which also holds in the presence of the damping term. From these two solutions mentioned above, the real-valued time evolution of the variables ˜Sx,˜Symay be expressed as ˜Sx k=eImωktcos (ϕ+,k−Reωkt)/vextendsingle/vextendsingleβ+ k+β− k/vextendsingle/vextendsingle,(S.9) ˜Sy k=eImωktsin (ϕ−,k−Reωkt)/vextendsingle/vextendsingleβ+ k−β− k/vextendsingle/vextendsingle,(S.10)arXiv:1805.01815v2 [cond-mat.mtrl-sci] 15 Oct 20182 withϕ±,k= arg/parenleftbig β+ k±β− k/parenrightbig . As mentioned above, the Imωkterms are zero in the absence of damping close to a local energy minimum, and Im ωk<0is implied by the fact that the Landau–Lifshitz–Gilbert equation de- scribes energy dissipation, which in the linearized case corresponds to relaxation towards the local energy min- imum. In the absence of damping, the spins will precess on an ellipse defined by the equation /parenleftBig ˜Sx k/parenrightBig2 /vextendsingle/vextendsingle/vextendsingleβ+(0) k+β−(0) k/vextendsingle/vextendsingle/vextendsingle2 cos2(ϕ+,k−ϕ−,k) +2˜Sx k˜Sy ksin (ϕ+,k−ϕ−,k)/vextendsingle/vextendsingle/vextendsingleβ+(0) k−β−(0) k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ+(0) k+β−(0) k/vextendsingle/vextendsingle/vextendsinglecos2(ϕ+,k−ϕ−,k) +/parenleftBig ˜Sy k/parenrightBig2 /vextendsingle/vextendsingle/vextendsingleβ+(0) k−β−(0) k/vextendsingle/vextendsingle/vextendsingle2 cos2(ϕ+,k−ϕ−,k)= 1,(S.11) where the superscript (0)indicatesα= 0. The semima- jor and semiminor axes of the ellipse akandbkmay be expressed from Eq. (S.11) as akbk=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ−(0) k/vextendsingle/vextendsingle/vextendsingle2 −/vextendsingle/vextendsingle/vextendsingleβ+(0) k/vextendsingle/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (S.12) a2 k+b2 k= 2/parenleftbigg/vextendsingle/vextendsingle/vextendsingleβ−(0) k/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingleβ+(0) k/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg .(S.13) Note thatβ+ kandβ− k, consequently the parameters of the precessional ellipse akandbk, are functions of the spatial position r. S.II. CALCULATION OF THE EFFECTIVE DAMPING PARAMETER FROM PERTURBATION THEORY Here we derive the expression for the effective damping parameter αeffgiven in Eq. (6) of the main text. By introducingβk=/parenleftbig β+ k,−β− k/parenrightbig , D=/bracketleftbiggD0+Dnr−Da −D† aD0−Dnr/bracketrightbigg ,(S.14) and using the Pauli matrix σz=/bracketleftbigg 1 0 0−1/bracketrightbigg , Eqs. (2)-(3) in the main text may be rewritten as −ωkσzβk=γ/prime M(D+iασzD)βk(S.15) in the frequency domain. Following standard perturba- tion theory, we expand the eigenvalues ωkand the eigen- vectorsβkin the parameter α/lessmuch1. For the zeroth-order terms one gets −ω(0) kσzβ(0) k=γ MDβ(0) k, (S.16) 0.0 0.1 0.2 0.3 0.4 0.50.00.51.01.52.02.5FIG. S1. Effective damping coefficients αm,effof the isolated skyrmion in the Pd/Fe/Ir(111) system at B= 1T, calcu- lated from the numerical solution of the linearized Landau– Lifshitz–Gilbert equation (S.15), as a function of the Gilbert damping parameter α. with realω(0) keigenvalues as discussed in Sec. S.I. The first-order terms read −ω(0) k/angbracketleftBig β(0) k/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleβ(1) k/angbracketrightBig −ω(1) k/angbracketleftBig β(0) k/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleβ(0) k/angbracketrightBig =γ M/angbracketleftBig β(0) k/vextendsingle/vextendsingle/vextendsingleD/vextendsingle/vextendsingle/vextendsingleβ(1) k/angbracketrightBig +iαγ M/angbracketleftBig β(0) k/vextendsingle/vextendsingle/vextendsingleσzD/vextendsingle/vextendsingle/vextendsingleβ(0) k/angbracketrightBig , (S.17) after taking the scalar product with β(0) k. The first terms on both sides cancel by letting Dact to the left, then using Eq. (S.16) and the fact that the ω(0) kare real. By applying Eq. (S.16) to the remaining term on the right- hand side one obtains ω(1) k=−iαω(0) k/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0) k/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingleβ+(0) k/vextendsingle/vextendsingle/vextendsingle2 dr/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0) k/vextendsingle/vextendsingle/vextendsingle2 −/vextendsingle/vextendsingle/vextendsingleβ+(0) k/vextendsingle/vextendsingle/vextendsingle2 dr,(S.18) by writing in the definition of the scalar product. By using the definition αk,eff=|Imωk/Reωk|≈/vextendsingle/vextendsingle/vextendsingleω(1) k/ω(0) k/vextendsingle/vextendsingle/vextendsingle and substituting Eqs. (S.12)-(S.13) into Eq. (S.18), one arrives at Eq. (6) in the main text as long as/vextendsingle/vextendsingle/vextendsingleβ−(0) k/vextendsingle/vextendsingle/vextendsingle2 − /vextendsingle/vextendsingle/vextendsingleβ+(0) k/vextendsingle/vextendsingle/vextendsingle2 does not change sign under the integral. It is worthwhile to investigate for which values of α does first-order perturbation theory give a good estimate forαk,effcalculated from the exact solution of the lin- earized equations of motion, Eq. (S.15). In the materials where the excitations of isolated skyrmions or skyrmion lattices were investigated, significantly different values of αhave been found. For example, intrinsic Gilbert damp- ing parameters of α= 0.02-0.04were determined experi- mentallyforbulkchiralmagnetsMnSiandCu 2OSeO 3[4], α= 0.28was deduced for FeGe[5], and a total damp- ing ofαtot= 0.105was obtained for Ir/Fe/Co/Pt mag- netic multilayers[6], where the latter value also includes3 various effects beyond the Landau–Lifshitz–Gilbert de- scription. Figure S1 displays the dependence of αm,eff onαfor the eigenmodes of the isolated skyrmion in the Pd/Fe/Ir(111) system, shown in Fig. 2 of the main text. Most of the modes show a linear correspondence between the two quantities with different slopes in the displayed parameter range, in agreement with Eq. (6) in the main text. For the breathing mode m= 0the convex shape of the curve indicates that the effective damping param- eter becomes relatively even larger than the perturbative expression Eq. (6) as αis increased. S.III. EIGENMODES OF THE ISOLATED SKYRMION Here we discuss the derivation of the skyrmion profile Eq. (8) and the operators in Eqs. (9)-(11) of the main text. The energy density Eq. (7) in polar coordinates reads h=A/bracketleftbigg (∂rΘ)2+ sin2Θ (∂rΦ)2+1 r2(∂ϕΘ)2 +1 r2sin2Θ (∂ϕΦ)2/bracketrightbigg +D/bracketleftbigg cos (ϕ−Φ)∂rΘ −1 rsin (ϕ−Φ)∂ϕΘ + sin Θ cos Θ sin ( ϕ−Φ)∂rΦ +1 rsin Θ cos Θ cos ( ϕ−Φ)∂ϕΦ/bracketrightbigg +Kcos2Θ−MBcos Θ. (S.19) The Landau–Lifshitz–Gilbert Eq. (1) may be rewritten as sin Θ∂tΘ =γ/primeBΦ+αγ/primesin ΘBΘ,(S.20) sin Θ∂tΦ =−γ/primeBΘ+αγ/prime1 sin ΘBΦ,(S.21) with Bχ=−1 MδH δχ =−1 M/bracketleftbigg −1 r∂r/parenleftbigg r∂h ∂(∂rχ)/parenrightbigg −∂ϕ∂h ∂(∂ϕχ)+∂h ∂χ/bracketrightbigg , (S.22) whereχstands for ΘorΦ. Note that in this form it is common to redefine BΦto include the 1/sin Θfactor in Eq. (S.21)[7]. The first variations of Hfrom Eq. (S.19)may be expressed as δH δΘ=−2A/braceleftbigg ∇2Θ−sin Θ cos Θ/bracketleftbigg (∂rΦ)2+1 r2(∂ϕΦ)2/bracketrightbigg/bracerightbigg −2Ksin Θ cos Θ +MBsin Θ −2Dsin2Θ/bracketleftbigg sin (ϕ−Φ)∂rΦ + cos (ϕ−Φ)1 r∂ϕΦ/bracketrightbigg , (S.23) δH δΦ=−2A/braceleftbigg sin2Θ∇2Φ + sin 2Θ/bracketleftbigg ∂rΘ∂rΦ +1 r2∂ϕΘ∂ϕΦ/bracketrightbigg/bracerightbigg + 2Dsin2Θ/bracketleftbigg sin (ϕ−Φ)∂rΘ + cos (ϕ−Φ)1 r∂ϕΘ/bracketrightbigg , (S.24) TheequilibriumconditionEq.(8)inthemaintextmay be obtained by setting ∂tΘ =∂tΦ = 0in Eqs. (S.20)- (S.21) and assuming cylindrical symmetry, Θ0(r,ϕ) = Θ0(r)and Φ0(r,ϕ) =ϕ. In the main text D>0 andB≥0were assumed. Choosing D<0switches the helicity of the structure to Φ0=ϕ+π, in which caseDshould be replaced by |D|in Eq. (8). For the background magnetization pointing in the opposite di- rectionB≤0, one obtains the time-reversed solutions with Θ0→π−Θ0,Φ0→Φ0+π,B→−B. Time rever- sal also reverses clockwise and counterclockwise rotating eigenmodes; however, the above transformations do not influence the magnitudes of the excitation frequencies. Finally, we note that the frequencies remain unchanged even if the form of the Dzyaloshinsky–Moriya interaction in Eq. (S.19), describing Néel-type skyrmions common in ultrathin films and multilayers, is replaced by an expres- sion that prefers Bloch-type skyrmions occurring in bulk helimagnets – see Ref. [3] for details. Fordeterminingthelinearizedequationsofmotion,one can proceed by switching to the local coordinate system as discussed in Sec. S.I and Refs. [1, 3]. Alternatively, they can also directly be derived from Eqs. (S.20)-(S.21) by introducing Θ = Θ 0+˜Sx,Φ = Φ 0+1 sin Θ 0˜Syand expanding around the skyrmion profile from Eq. (8) up to first order in ˜Sx,˜Sy– see also Ref. [2]. The operators in Eq. (S.3) read A1=−2A/parenleftbigg ∇2−1 r2cos 2Θ 0(∂ϕΦ0)2/parenrightbigg −2D1 rsin 2Θ 0∂ϕΦ0−2Kcos 2Θ 0+MBcos Θ 0, (S.25) A2=4A1 r2cos Θ 0∂ϕΦ0∂ϕ−2D1 rsin Θ 0∂ϕ,(S.26) A3=−2A/braceleftbigg ∇2+/bracketleftbigg (∂rΘ0)2−1 r2cos2Θ0(∂ϕΦ0)2/bracketrightbigg/bracerightbigg −2D/parenleftbigg ∂rΘ0+1 rsin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg −2Kcos2Θ0+MBcos Θ 0, (S.27)4 which leads directly to Eqs. (9)-(11) in the main text via Eqs. (S.6)-(S.8). The excitation frequencies of the ferromagnetic state may be determined by setting Θ0≡0in Eqs. (9)-(11) in the main text. In this case, the eigenvalues and eigenvec- tors can be calculated analytically[1], ωk,m=γ/prime M(1−iα)/bracketleftbig 2Ak2−2K+MB/bracketrightbig ,(S.28) /parenleftBig β+ k,m(r),β− k,m(r)/parenrightBig = (0,Jm−1(kr)),(S.29) withJm−1theBesselfunction ofthefirstkind, appearing due to the solutions being regular at the origin. Equa- tion (S.28) demonstrates that the lowest-frequency exci- tation of the background is the ferromagnetic resonance frequencyωFMR =γ M(MB−2K)atα= 0. Since the anomalous term Dadisappears in the out-of-plane mag- netized ferromagnetic state, all spin waves will be circu- larly polarized, see Eq. (S.29), and the effective damping parameterwillalwayscoincidewiththeGilbertdamping. Regarding the excitations of the isolated skyrmion, for α= 0the linearized equations of motion in Eq. (S.15) are real-valued; consequently, β± k,m(r)can be chosen to be real-valued. In this case Eqs. (S.9)-(S.10) take the form ˜Sx k,m= cos (mϕ−ωk,mt)/parenleftBig β+ k,m(r) +β− k,m(r)/parenrightBig ,(S.30) ˜Sy k,m= sin (mϕ−ωk,mt)/parenleftBig β+ k,m(r)−β− k,m(r)/parenrightBig .(S.31) This means that modes with ωk,m>0form> 0will rotate counterclockwise, that is, the contours with con- stant ˜Sx k,mand ˜Sy k,mwill move towards higher values of ϕastis increased, while the modes with ωk,m>0for m < 0will rotate clockwise. Modes with m= 0corre- spond to breathing excitations. This sign convention for mwas used when designating the localized modes of the isolated skyrmion in the main text, and the kindex was dropped since only a single mode could be observed be- low the ferromagnetic resonance frequency for each value ofm. S.IV. NUMERICAL SOLUTION OF THE EIGENVALUE EQUATIONS The linearized Landau–Lifshitz–Gilbert equation for the isolated skyrmion, Eqs. (2)-(3) with the operators Eqs.(9)-(11)inthemaintext, weresolvednumericallyby a finite-difference method. First the equilibrium profile was determined from Eq. (8) using the shooting method for an initial approximation, then obtaining the solution on a finer grid via finite differences. For the calculationswe used dimensionless parameters (cf. Ref. [8]), Adl= 1, (S.32) Ddl= 1, (S.33) Kdl=KA D2, (S.34) (MB)dl=MBA D2, (S.35) rdl=|D| Ar, (S.36) ωdl=MA γD2ω. (S.37) The equations were solved in a finite interval for rdl∈[0,R], with the boundary conditions Θ0(0) = π,Θ0(R) = 0. For the results presented in Fig. 2 in the main text the value of R= 30was used. It was confirmed bymodifying Rthattheskyrmionshapeandthefrequen- cies of the localized modes were not significantly affected by the boundary conditions. However, the frequencies of the modes above the ferromagnetic resonance frequency ωFMR =γ M(MB−2K)did change as a function of R, since these modes are extended over the ferromag- netic background – see Eqs. (S.28)-(S.29). Furthermore, in the infinitely extended system the equations of mo- tion include a Goldstone mode with/parenleftbig β+ m=−1,β− m=−1/parenrightbig =/parenleftbig −1 rsin Θ 0−∂rΘ0,1 rsin Θ 0−∂rΘ0/parenrightbig , corresponding to the translation of the skyrmion on the collinear background[1]. This mode obtains a finite frequency in the numerical calculations due to the finite value of R and describes a slow clockwise gyration of the skyrmion. However, this frequency is not shown in Fig. 3 of the main text because it is only created by boundary effects. In order to investigate the dependence of the effective damping on the dimensionless parameters, we also per- formed the calculations for the parameters describing the Ir|Co|Pt multilayer system[9]. The results are summa- rized in Fig. S2. The Ir|Co|Pt system has a larger di- mensionless anisotropy value ( −KIr|Co|Pt dl = 0.40) than the Pd/Fe/Ir(111) system ( −KPd/Fe/Ir(111) dl = 0.33). Al- though the same localized modes are found in both cases, the frequencies belonging to the m= 0,1,−3,−4,−5 modes in Fig. S2 are relatively smaller than in Fig. 2 compared to the ferromagnetic resonance frequency at the elliptic instability field where ωm=−2= 0. This agrees with the two limiting cases discussed in the lit- erature: it was shown in Ref. [1] that for Kdl= 0the m= 1,−4,−5modes are still above the ferromagnetic resonance frequency at the elliptic instability field, while in Ref. [2] it was investigated that all modes become soft withfrequenciesgoingtozeroat (MB)dl= 0inthepoint −Kdl=π2 16≈0.62,belowwhichaspinspiralgroundstate is formed in the system. Figure S2(b) demonstrates that the effective damping parameters αm,effare higher at the ellipticinstabilityfieldinIr|Co|PtthaninPd/Fe/Ir(111), showing an opposite trend compared to the frequencies. Regarding the physical units, the stronger exchange stiffness combined with the weaker Dzyaloshinsky–5 0.03 0.04 0.05 0.06 0.07 0.080246810 (a) 0.03 0.04 0.05 0.06 0.07 0.081.01.52.02.53.03.5 (b) FIG. S2. Localized magnons in the isolated skyrmion, with the interaction parameters corresponding to the Ir|Co|Pt multilayer system from Ref. [9]: A= 10.0pJ/m,D= 1.9mJ/m2,K=−0.143MJ/m3,M = 0.96MA/m. The anisotropy reflects an effective value including the dipolar in- teractions as a demagnetizing term, −K =−K 0−1 2µ0M2 withK0=−0.717MJ/m3. (a) Magnon frequencies f=ω/2π forα= 0. Illustrations display the shapes of the excitation modes visualized as the contour plot of the out-of-plane spin componentsona 1×1nm2grid,withredandbluecolorscorre- sponding to positive and negative Szvalues, respectively. (b) Effective damping coefficients αm,eff, calculated from Eq. (6) in the main text.Moriya interaction and anisotropy in the multilayer sys- tem leads to larger skyrmions stabilized at lower field val- ues and displaying lower excitation frequencies. We note that demagnetization effects were only considered here as a shape anisotropy term included in K; it is expected that this should be a relatively good approximation for the Pd/Fe/Ir(111) system with only a monolayer of mag- netic material, but it was suggested recently[6] that the dipolar interaction can significantly influence the excita- tion frequencies of isolated skyrmions in magnetic multi- layers. S.V. SPIN DYNAMICS SIMULATIONS For the spin dynamics simulations displayed in Fig. 3 in the main text we used an atomistic model Hamiltonian on a single-layer triangular lattice, H=−1 2/summationdisplay /angbracketlefti,j/angbracketrightJSiSj−1 2/summationdisplay /angbracketlefti,j/angbracketrightDij(Si×Sj)−/summationdisplay iK(Sz i)2 −/summationdisplay iµBSz i, (S.38) with the parameters J= 5.72meV for the Heisenberg exchange,D=|Dij|= 1.52meV for the Dzyaloshinsky– Moriya interaction, K= 0.4meV for the anisotropy, µ= 3µBfor the magnetic moment, and a= 0.271nm for the lattice constant. For the transformation be- tween the lattice and continuum parameters in the Pd/Fe/Ir(111) system see, e.g., Ref. [10]. The simula- tionswereperformedbynumericallysolvingtheLandau– Lifshitz–Gilbert equation on an 128×128lattice with periodic boundary conditions, which was considerably larger than the equilibrium skyrmion size to minimize boundary effects. The initial configuration was deter- mined by calculating the eigenvectors in the continuum model and discretizing it on the lattice, as shown in the insets of Fig. 2 in the main text. It was found that such a configuration was very close to the corresponding exci- tation mode of the lattice Hamiltonian Eq. (S.38), simi- larly to the agreement between the continuum and lattice equilibrium skyrmion profiles[10]. [1] C. Schütte and M. Garst, Phys. Rev. B 90, 094423 (2014). [2] V. P. Kravchuk, D. D. Sheka, U. K. Rössler, J. van den Brink, andYu.Gaididei, Phys.Rev.B 97, 064403(2018). [3] S.-Z. Lin, Phys. Rev. B 96, 014407 (2017). [4] T. Schwarze, J. Waizner, M. Garst, A. Bauer, I. Stasinopoulos, H. Berger, C. Pfleiderer, and D. Grundler, Nat. Mater. 14, 478 (2015). [5] M. Beg, M. Albert, M.-A. Bisotti, D. Cortés-Ortuño, W. Wang, R. Carey, M. Vousden, O. Hovorka, C. Ciccarelli, C. S. Spencer, C. H. Marrows, and H. Fangohr, Phys. Rev. B95, 014433 (2017). [6] B. Satywali, F. Ma, S. He, M. Raju, V. P. Kravchuk, M. Garst, A. Soumyanarayanan, and C. Panagopoulos,arXiv:1802.03979 (2018). [7] F. Romá, L. F. Cugliandolo, and G. S. Lozano, Phys. Rev. E90, 023203 (2014). [8] A. O. Leonov, T. L. Monchesky, N. Romming, A. Ku- betzka, A. N. Bogdanov, and R. Wiesendanger, New J. Phys.18, 065003 (2016). [9] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sam- paio, C. A. F. Vaz, N. Van Horne, K. Bouzehouane, K. Garcia, C. Deranlot, P. Warnicke, P. Wohlhüter, J.-M. George, M. Weigand, J. Raabe, V. Cros, and A. Fert, Nat. Nanotechnol. 11, 444 (2016). [10] J. Hagemeister, A. Siemens, L. Rózsa, E. Y. Vedme- denko, and R. Wiesendanger, Phys. Rev. B 97, 174436 (2018).

2018-05-04

Damping mechanisms in magnetic systems determine the lifetime, diffusion and transport properties of magnons, domain walls, magnetic vortices, and skyrmions. Based on the phenomenological Landau-Lifshitz-Gilbert equation, here the effective damping parameter in noncollinear magnetic systems is determined describing the linewidth in resonance experiments or the decay parameter in time-resolved measurements. It is shown how the effective damping can be calculated from the elliptic polarization of magnons, arising due to the noncollinear spin arrangement. It is concluded that the effective damping is larger than the Gilbert damping, and it may significantly differ between excitation modes. Numerical results for the effective damping are presented for the localized magnons in isolated skyrmions, with parameters based on the Pd/Fe/Ir(111) model-type system.

Effective damping enhancement in noncollinear spin structures

1805.01815v2

Optimal damping of vibrating systems: dependence on initial conditions K. Lelasa,∗, I. Nakićb aFaculty of Textile Technology, University of Zagreb, Croatia bDepartment of Mathematics, Faculty of Science, University of Zagreb, Croatia Abstract Common criteria used for measuring performance of vibrating systems have one thing in common: they do not depend on initial conditions of the system. In some cases it is assumed that the system has zero initial conditions, or some kind of averaging is used to get rid of initial conditions. The aim of this paper is to initiate rigorous study of the dependence of vibrating systems on initial conditions in the setting of optimal damping problems. We show that, based on the type of initial conditions, especially on the ratio of potential and kinetic energy of the initial conditions, the vibrating system will have quite different behavior and correspondingly the optimal damping coefficients will be quite different. More precisely, for single degree of freedom systems and the initial conditions with mostly potential energy, the optimal damping coefficient willbeintheunder-dampedregime, whileinthecaseofthepredominantkineticenergytheoptimaldamping coefficient will be in the over-damped regime. In fact, in the case of pure kinetic initial energy, the optimal damping coefficient is +∞! Qualitatively, we found the same behavior in multi degree of freedom systems with mass proportional damping. We also introduce a new method for determining the optimal damping of vibrating systems, which takes into account the peculiarities of initial conditions and the fact that, although in theory these systems asymptotically approach equilibrium and never reach it exactly, in nature and in experiments they effectively reach equilibrium in some finite time. Keywords: viscous damping, optimal damping, multi-degree of freedom, initial conditions 1. Introduction If we have an multi-degree of freedom (MDOF) linear vibrating system, i.e. a system of coupled damped oscillators, how to determine damping coefficients that ensure optimal evanescence of free vibrations? In the literature one finds several different criteria, typically based on frequency domain analysis of the system, although there are also approaches based on time domain analysis [1]. The tools used for designing the criteria include modal analysis [2], transfer functions [3], H2andH∞norms coming from systems theory [4, 5] and spectral techniques [6]. A general overview of the optimization tools for structures analysis can be found in e.g. [7]. Another optimization criterion used is to take as optimal the damping coefficients that minimize the (zero to infinity) time integral of the energy of the system, averaged over all possible initial conditions corresponding to the same initial energy [8]. This criterion was investigated widely, mostly by mathematicians in the last two decades, more details can be found, e.g., in references [8, 9, 10, 11, 12]. However, what is common to all these criteria is that they implicitly or explicitly ignore the dependence of the dynamics of the system on the initial conditions. Sometimes this is suitable, e.g. for systems with continuous excitation, but in some cases it make sense to study the free vibrations of the system with non- zero initial conditions. A prominent example where this is the case is the vibration control of buildings subjected to earthquake excitation [13, 14]. Indeed, depending on the initial conditions, MDOF systems can ∗Corresponding author Email addresses: klelas@ttf.unizg.hr (K. Lelas), nakic@math.hr (I. Nakić) Preprint submitted to Journal of Sound and Vibration January 31, 2024arXiv:2307.02352v2 [physics.class-ph] 30 Jan 2024exhibit oscillatory or non-oscillatory response [15], so it is clear that initial conditions can play an important role in the overall dynamics of the system. Implicitly, dependence of the behavior of system on initial conditions has been investigated in the context of time-optimal vibrations reduction [16] and transient response [17] in terms of computationally efficient methods for the calculation of the system response. Our aim with this paper is to start more systematic investigation of the role of initial conditions in the study of linear vibrating systems. Specifically, the dependence of the energy integral on the initial conditions has not been investigated, as far as we are aware, and therefore it is not clear how much information about the behavior of vibrating systems is lost by taking the average over all initial conditions or by assuming zero initial conditions and it is not clear how well the optimal damping obtained in this way works for a specific choice of initial conditions, e.g. for an experiment with initial conditions such that the initial energy consists only of potential energy, etc. We have chosen to study the particular criterion of minimizing time integral of the energy as in this case it is straightforward to modify it to take into account the initial conditions: instead of averaging over all possible initial conditions, we study the dependence of the time integral of the energy of the system on initial conditions. Specifically, for criteria based on frequency domain approach, which are designed for forced vibrations, it is not clear how to take into account the non-zero initial conditions in a systematic way. We will explore this dependence by considering free vibrations of single degree of freedom (SDOF), two-degree of freedom (2-DOF) and MDOF vibrating systems with mass proportional damping (MPD). In particular, for a SDOF, averaging over all initial conditions gives the critical damping as optimal [8, 10], and we show, by considering the minimization of the energy integral without averaging over initial conditions, that damping coefficients approximately 30%less than critical to infinite are obtained as optimal, depending on the initial conditions. We systematize all our results with respect to the relationship between initial potential and initial kinetic energy, e.g., for initial conditions with initial potential energy grater than initial kinetic energy the optimal damping coefficient is in the under-damped regime, while for initial conditions with initial kinetic energy grater than initial potential energy we find the optimal damping deep in the over-damped regime. We also consider the minimization of the energy integral averaged over a subset of initial conditions and obtain a significant dependence of the optimal damping coefficient on the selected subset. Qualitatively, we find the same behavior in 2-DOF and MDOF systems as well. Furthermore, we show that the minimization of the energy integral for certain types of initial conditions does not give a satisfactory optimal damping coefficient. Specifically, for SDOF systems, the obtained optimal damping coefficient does not distinguish between two initial states with the same magnitude of initial displacement and initial velocity, but which differ in the relative sign of initial displacement and initial velocity. These initial conditions differ significantly in the rate of energy dissipation as a function of the damping coefficient, i.e. it is not realistic for one damping coefficient to be optimal for both of these initial conditions. The same is true for each individual mode of MDOF systems with respect to the signs of initial displacements and velocities, expressed via modal coordinates. Another disadvantage of this criterion isthat, forinitialconditionswithpurelykineticinitialenergy, itgivesaninfiniteoptimaldampingcoefficient, which is not practical for experiments. Also, the energy integral is calculated over the entire time, due to the fact that these systems asymptotically approach equilibrium and never reach it exactly, but in nature and experiments they effectively reach equilibrium in some finite time. We introduce a new method for determining the optimal damping of MDOF systems, which practically solves the aforementioned problems and gives optimal damping coefficients that take into account the pecu- liarities of each initial condition and the fact that these systems effectively reach equilibrium in some finite time. We take that the system has effectively reached equilibrium when its energy drops to some small fraction of the initial energy, e.g., to the energy resolution of the measuring device with which we observe the system. Our method is based on the determination of the damping coefficient for which the energy of the system drops to that desired energy level the fastest. In this paper we focus on mass proportional damping so that we could analytically perform a modal analysis and present ideas in the simplest possible way, but, as we briefly comment at the end of the paper, everything we have done can be done in a similar fashion analytically for the case of Rayleigh damping [18] as well as for tuned mass damper [19, 20]. Also, it is possible to carry out this kind of analysis numerically 2for systems with damping that does not allow analytical treatment. This will be the subject of our further research. The rest of the paper is organized as follows: Section 2 is devoted to SDOF systems, in particular minimization of the energy integral and optimal damping is studied for the chosen set of initial conditions. In Section 3 we analyze 2-DOF systems with MPD. MDOF systems with MPD are the subject of Section 4. In Section 5 we propose a new optimization criterion and analyze its properties. Section 6 summarizes important findings of the paper. 2. SDOF systems Free vibrations of a SDOF linear vibrating system can be described by the equation ¨x(t) + 2γ˙x(t) +ω2 0x(t) = 0 , x(0) = x0,˙x(0) = v0, (1) where x(t)denotes the displacement from the equilibrium position (set to x= 0) as a function of time, the dots denote time derivatives, γ > 0is the damping coefficient, ω0stands for the undamped oscillator angular frequency (sometimes called the natural frequency of the oscillator) and (x0, v0)encode the initial conditions [21, 22]. The physical units of the displacement x(t)depend on the system being considered. For example, for a mass on a spring (or a pendulum) in viscous fluid, when it is usually called elongation , it is measured in [m], while for an RLC circuit it could either be voltage, or current, or charge. In contrast, the units of γandω0are[s−1]for all systems described with the SDOF model. The form of the solution to the equation (1) depends on the relationship between γandω0, producing three possible regimes [21, 22]: under-damped ( γ < ω 0), critically damped ( γ=ω0) and over-damped ( γ > ω 0) regime. Here we would like to point out that, although it is natural to classify the solution of SDOF into three regimes depending on the value of γ, we can actually take one form of the solution as a unique solution valid for all values of γ >0,γ̸=ω0, x(t) =e−γt x0cos(ωt) +v0+γx0 ωsin(ωt) , (2) where ω=p ω2 0−γ2is the (complex valued) damped angular frequency. In order to describe the critically damped regime, one can take the limit γ→ω0of the solution (2) to obtain the general solution of the critically damped regime xc(t) =e−ω0t(x0+ (v0+ω0x0)t). (3) Therefore, in order to calculate the energy and the time integral of the energy, we do not need to perform separate calculations for all three regimes, but a single calculation using the displacement given by (2) and the velocity given by ˙x(t) =e−γt v0cos(ωt)−γv0+ω2 0x0 ωsin(ωt) . (4) For simplicity, in this section we will refer to the quantity E(t) = ˙x(t)2+ω2 0x(t)2(5) as the energyof the system, and to the quantities EK(t) = ˙x(t)2andEP(t) =ω2 0x(t)2as the kinetic energy andpotential energy of the system respectively. The connection of the quantity (5) to the usual expressions for the energy is straightforward, e.g., for a mass mon a spring in viscous fluid E(t) =m 2E(t), (6) and similarly for other systems described with the SDOF model. Using (2) and (4) in (5), we obtain E(t) =e−2γt E0cos2(ωt) +γ ω2 0x2 0−v2 0
sin(2ωt) ω+ E0(ω2 0+γ2) + 4ω2 0γx0v0
sin2(ωt) ω2 (7) 3for the energy of the system, where E0=v2 0+ω2 0x2 0is the initial energy given to the system at t= 0. Accordingly, E0K=v2 0istheinitialkineticenergyand E0P=ω2 0x2 0istheinitialpotentialenergy. Expression (7) is valid for both under-damped and over-damped regimes, and to obtain the energy of the critically damped regime we take the γ→ω0limit of the energy (7), and obtain Ec(t) =e−2ω0t E0+ 2ω0 ω2 0x2 0−v2 0
t+ 2ω2 0(E0+ 2ω0x0v0)t2
. (8) 2.1. Minimization of the energy integral and optimal damping in dependence of initial conditions We consider the SDOF system with initially energy E0. All possible initial conditions that give this energy can be expressed in polar coordinates with constant radius r=√E0and angle θ= arctan v0 ω0x0 , i.e. we have ω0x0=rcosθ v0=rsinθ .(9) In Fig. 1 we sketch the circle given by (9), i.e. given by all possible initial conditions with the same energy E0. For clarity of the exposition, here we comment on a few characteristic points of the circle presented in Fig. 1: •Initial conditions ω0x0=±√E0andv0= 0, i.e. with purely potential initial energy (and zero initial kinetic energy), correspond to two points on the circle with θ={0, π}. •Initial conditions ω0x0=±p E0/2andv0=±p E0/2, i.e. with initial potential energy equal to initial kinetic energy, correspond to four points on the circle with θ={π/4,3π/4,5π/4,7π/4}. •Initial conditions ω0x0= 0andv0=±√E0, i.e. with purely kinetic initial energy (and zero initial potential energy), correspond to two points on the circle with θ={π/2,3π/2}. θ r=√E0(ω0x0, v0) ω0x0v0 Figure 1: Sketch of all possible initial conditions with the same initial energy E0in the (ω0x0, v0)coordinate system. Square of the coordinates corresponds to initial potential energy E0P=ω2 0x2 0and initial kinetic energy E0K=v2 0respectively. This representation gives us a useful visualization, e.g.: all initial conditions with E0P> E 0Kare represented by two arcs, i.e. points with θ∈(−π/4, π/4)∪(3π/4,5π/4)(blue dotted arcs); initial conditions with E0K=E0andE0P= 0are represented by two points on a circle with θ={π/2,3π/2}(two red filled circles); etc. Using (9) in (7) and (8), we obtain the energy of the under-damped and over-damped regime E(t, θ) =E0e−2γt cos2(ωt) +γcos 2θsin(2ωt) ω+ ω2 0+γ2+ 2ω0γsin 2θ
sin2(ωt) ω2 ,(10) and the energy of the critically damped regime Ec(t, θ) =E0e−2ω0t 1 + 2 ω0(cos 2 θ)t+ 2ω2 0(1 + sin 2 θ)t2
, (11) 40.1 0.71 1 2 3012345 γ/ω 0I(γ, θ)ω0/E0θ= 0 θ=π/4 θ=π/2 Figure 2: Integral (13) for three initial conditions θ={0, π/4, π/2}. as functions of θ, instead of x0andv0. Now we integrate energy (10) over all time, i.e. I(γ, θ) =Z∞ 0E(t)dt , (12) and obtain I(γ, θ) =E0 2ω0ω2 0+γ2 γω0+γ ω0cos 2θ+ sin 2 θ . (13) Integral (13) is valid for all three regimes, i.e. for any γ >0. We note here that the energy (see (7) and (8)) is invariant to a simultaneous change of the signs of the initial conditions, i.e. to the change (x0, v0)→(−x0,−v0)(but not to x0→ −x0orv0→ −v0separately). This change of signs corresponds to the change in angle θ→θ+π, therefore, functions (10), (11) and (13) are all periodic in θwith period π. In Fig. 2 we show the integral (13) for γ∈[0.1ω0,3ω0]for three different initial conditions, i.e. for θ={0, π/4, π/2}. We can see that I(γ, θ= 0)(red solid curve), with purely potential initial energy and zero initial kinetic energy, attains minimum for γ= 0.707ω0(rounded to three decimal places), i.e. well in the under-damped regime. For the initial condition with equal potential and kinetic energy, I(γ, θ=π/4) (black dotted curve) attains minimum for γ=ω0, i.e. at the critical damping condition. Interestingly, for the initial condition with purely kinetic energy and zero potential energy, I(γ, θ=π/2)(blue dashed curve) has no minimum in the displayed range of damping coefficients, therefore here we explicitly show this function I(γ, θ=π/2) =E0 2γ, (14) and it is clear that (14) has no minimum. This is easy to understand from a physical point of view, i.e. if all the initial energy is kinetic, the higher the damping coefficient, the faster the energy dissipation will be. If we consider the optimal damping as the one for which the integral (13) is minimal, we can easily determine the optimal damping coefficient γopt(θ)from the condition ∂I(γ, θ) ∂γ γopt= 0, (15) and we obtain γopt(θ) =r 1 2 cos2θω0. (16) In Fig. 3 we show the optimal damping coefficient (16) for θ∈[0,2π](function (16) has a period π, but here we choose this interval for completeness), and here we comment on the shown results with respect to the relationship between initial potential energy ( E0P=ω2 0x2 0) and initial kinetic energy ( E0K=v2 0) for any given initial condition, i.e. for any θ: 50 0.25 0.50.75 1 1.25 1.51.75 201234 γopt=ω0 θ/πγopt(θ)/ω0 Figure 3: Optimal damping coefficient (16) (solid red curve) as a function of all possible initial conditions, i.e. for θ∈[0,2π]. Below the dashed horizontal line, optimal damping coefficients are in the under-damped regime, above the line in the over- damped regime, and in the critically damped regime at the crossing points of the line and the solid red curve. •Initial conditions with E0P> E 0Kcorrespond to the set θ∈(−π/4, π/4)∪(3π/4,5π/4). For these initial conditions, optimal damping coefficients (16) are in the under-damped regime, i.e. γopt∈√ 2ω0/2, ω0
, with the minimum value γopt=√ 2ω0/2 = 0 .707ω0(rounded to three decimal places) obtained for θ={0, π}, i.e. for two initial conditions with E0=E0PandE0K= 0. •Initial conditions with E0P=E0Kcorrespond to four points θ={π/4,3π/4,5π/4,7π/4}with optimal damping coefficient (16) equal to critical damping, i.e. γopt=ω0. •Initial conditions with E0K> E 0Pcorrespond to the set θ∈(π/4,3π/4)∪(5π/4,7π/4). For these initial conditions, optimal damping coefficients (16) are in the over-damped regime, i.e. γopt∈(ω0,∞), where γoptdiverges for θ={π/2,3π/2}, i.e. for two initial conditions with E0K=E0andE0P= 0. Before closing this subsection, we would like to point out two more ways in which we can write relation (16) that will prove useful when dealing with MDOF systems. The ratio of the initial potential energy to the initial total energy is β=E0P E0= cos2θ , (17) where we used first of the relations (9) and E0P=ω2 0x2 0. Using (17), optimal damping coefficient (16) can be written as a function of the fraction of potential energy in the initial total energy, i.e. γopt(β) =r1 2βω0. (18) Thus, from (18) one can simply see that γoptis in the under-damped regime for β∈(1/2,1], in the critically damped regime for β= 1/2and in the over-damped regime for β∈[0,1/2). Using β=ω2 0x2 0/E0in (18) we can express the optimal damping coefficient in yet another way, as a function of the initial displacement x0, i.e. γopt(x0) =s E0 2x2 0=s v2 0+ω2 0x2 0 2x2 0, (19) where x0∈[−√E0/ω0,√E0/ω0]and for v0the condition v2 0=E0−ω2 0x2 0holds. One of the benefits of relation (19) is that it can be seen most directly that the optimal damping coefficient does not distinguish initial conditions (±x0,±v0)and(±x0,∓v0), which is a shortcoming of this optimization criterion, because the energy as a function of time is not the same for those two types of initial conditions (see (7) and (8)) and the energy decay may differ significantly depending on which of those initial conditions is in question. We will deal with these and other issues of energy integral minimization as an optimal damping criterion in the subsection 4.2. 60.1 0.781 1.66 2 3012345 γ/ω 0I(γ, ϕ1, ϕ2)ω0/E0ϕ1=−π/4,ϕ2=π/4 ϕ1=π/4,ϕ2= 3π/4 ϕ1=−π/4,ϕ2= 3π/4 Figure 4: Averaged integral (21) for three sets of initial conditions. 2.2. Minimization of the energy integral averaged over a set of initial conditions and optimal damping in dependence of the chosen set Now we calculate the average of the integral (12) over a set of initial conditions with θ∈[ϕ1, ϕ2], i.e. I(γ, ϕ1, ϕ2) =1 ϕ2−ϕ1Zϕ2 ϕ1I(γ, θ)dθ , (20) and we obtain I(γ, ϕ1, ϕ2) =E0 2ω0ω2 0+γ2 γω0+γ 2ω0(ϕ2−ϕ1)(sin 2 ϕ2−sin 2ϕ1) +1 2(ϕ2−ϕ1)(cos 2 ϕ1−cos 2ϕ2) .(21) In Fig. 4 we show averaged integral (21) for three different sets of initial conditions. For the set of initial conditions with ϕ1=−π/4andϕ2=π/4, i.e. with E0P≥E0K(where the equality holds only at the end points of the set), minimum of the averaged integral (solid red curve) is at γ= 0.781ω0(rounded to three decimal places). For the set of initial conditions with ϕ1=π/4andϕ2= 3π/4, i.e. with E0K≥E0P(where the equality holds only at the end points of the set), minimum of the averaged integral (dashed blue curve) is atγ= 1.658ω0(rounded to three decimal places). For the set of mixed initial conditions with ϕ1=−π/4 andϕ2= 3π/4, i.e. with E0P> E 0KandE0K> E 0Ppoints equally present in the set, minimum of the averaged integral (dotted black curve) is at the critical damping condition γ=ω0. If we consider the optimal damping as the one for which the averaged integral (21) is minimal, we can easily determine the optimal damping coefficient γopt(ϕ1, ϕ2)form the condition ∂I(γ, ϕ1, ϕ2) ∂γ γopt= 0, (22) and we obtain γopt(ϕ1, ϕ2) =s 2(ϕ2−ϕ1) 2(ϕ2−ϕ1) + sin 2 ϕ2−sin 2ϕ1ω0. (23) We note here that averaged integral (21) and optimal damping coefficient (23) are not periodic functions in variables ϕ1andϕ2, if we keep one variable fixed and change the other. But they are periodic, with period π, if we change both variables simultaneously. In Fig. 5 we show the optimal damping coefficient (23) as a function of ϕ2with fixed ϕ1= 0, and the results shown can be summarized as follows: •Forϕ1= 0andϕ2∈[0, π/2)∪(π,3π/2), the optimal damping coefficient (23) is in the under-damped regime. In this case, integral (20) is averaged over sets that have more points corresponding to initial conditions with E0P> E 0K, in comparison to the points corresponding to initial conditions with E0K> E 0P. 70 0.25 0.50.75 1 1.25 1.51.75 20.70.80.911.1 γopt=ω0 ϕ2/πγopt(0, ϕ2)/ω0 Figure 5: Optimal damping coefficient (23) (solid red curve) as a function of ϕ2∈[0,2π]for fixed ϕ1= 0. Below the dashed horizontal line, optimal damping coefficients are in the under-damped regime, above the line in the over-damped regime, and in the critically damped regime at the crossing points of the line and the solid red curve. •Forϕ1= 0andϕ2={π/2, π,3π/2,2π}, the optimal damping coefficient (23) is equal to critical damp- ing. In this case, integral (20) is averaged over sets that have equal amount of points corresponding to initial conditions with E0P> E 0Kand initial conditions with E0K> E 0P. •Forϕ1= 0andϕ2∈(π/2, π)∪(3π/2,2π), the optimal damping coefficient (23) is in the over-damped regime. In this case, integral (20) is averaged over sets that have more points corresponding to initial conditions with E0K> E 0P, in comparison to the points corresponding to initial conditions with E0P> E 0K. 3. 2-DOF systems with MPD Figure 6: Schematic figure of a 2-DOF system. Here we consider 2-DOF system shown schematically in Fig. 6. The corresponding equations of motion are m1¨x1(t) =−c1˙x1(t)−k1x1(t)−k2(x1(t)−x2(t)), m2¨x2(t) =−c2˙x2(t)−k3x2(t) +k2(x1(t)−x2(t)).(24) We will consider MPD [23], i.e. masses {m1, m2}, spring constants {k1, k2, k3}, and dampers {c1, c2}can in general be mutually different but the condition c1/m1=c2/m2holds. In this case we can use modal analysis [2, 21] and the system of equations (24) can be written via modal coordinates [21] as ¨q1(t) + 2γ˙q1(t) +ω2 01q1(t) = 0 ¨q2(t) + 2γ˙q2(t) +ω2 02q2(t) = 0 ,(25) where qi(t)andω0i, with i={1,2}, denote the modal coordinates and undamped modal frequencies of the two modes, while γ=ci/2miis the damping coefficient. In the analysis that we will carry out in this 8subsection, we will not need the explicit connection of modal coordinates qi(t)and mass coordinates, i.e. displacements xi(t), and we will deal with this in the next subsection when considering a specific example with given masses, springs and dampers. Similarly as in Section 2 (see (2)), the general solution for the i-th mode can be written as qi(t) =e−γt q0icos(ωit) +˙q0i+γq0i ωisin(ωit) , (26) where ωi=p ω2 0i−γ2is the damped modal frequency, and qi(0)≡q0iand ˙qi(0)≡˙q0iare the initial condi- tions of the i-th mode. Thus, the reasoning and the results presented in Section 2, with some adjustments, can by applied for the analysis of the 2-DOF system we are considering here. The energy of the system is E(t) =2X i=1mi˙xi(t)2 2+k1x1(t)2 2+k3x2(t)2 2+k2(x1(t)−x2(t))2 2, (27) and we take that the modal coordinates are normalised so that (27) can be written as E(t) =2X i=1Ei(t) =2X i=1 ˙qi(t)2+ω2 0iqi(t)2
(28) where Ei(t)in (28) denotes the energy of the i-th mode. Total energy at t= 0, i.e. the initial energy, is given by E0=2X i=1E0i=2X i=1(E0Ki+E0Pi) =2X i=1 ˙q2 0i+ω2 0iq2 0i
, (29) where E0idenotes the initial energy of the i-th mode, E0Ki= ˙q2 0iandE0Pi=ω2 0iq2 0idenote initial kinetic and initial potential energy of the i-th mode. All possible initial conditions with the same initial energy (29) can be expressed similarly as in the SDOF case (see (9) and Fig. 1) but with two pairs of polar coordinates, one pair for each mode. For the i-th mode we have radius ri=√E0iand angle θi= arctan ˙q0i ω0iq0i , i.e. we can write ω0iq0i=ricosθi ˙q0i=risinθi.(30) Thus, each initial condition with energy E0=E01+E02can be represented by points on two circles with radii r1=√E01andr2=√E02, for which condition r2 1+r2 2=E0holds, and with angles θ1andθ2that tell us how initial potential and initial kinetic energy are distributed within the modes. Using relation (10) for SDOF systems, we can write the energy of the i-th mode in polar coordinates (30) as Ei(t) =E0ie−2γt cos2(ωit) +γcos 2θisin(2ωit) ωi+ ω2 0i+γ2+ 2ω0iγsin 2θi
sin2(ωit) ω2 i (31) for the under-damped ( γ < ω 0i) and over-damped ( γ > ω 0i) regime, and the energy of the i-th mode in the critically damped regime is obtained analogously using the relation (11). Consequently, the integral of the energy (28) over the entire time, for some arbitrary initial condition, is simply calculated using relation (13) for each individual mode, we obtain I(γ,{E0i},{θi}) =2X i=1Z∞ 0Ei(t)dt=2X i=1E0i 2ω0iω2 0i+γ2 γω0i+γ ω0icos 2θi+ sin 2 θi . (32) Furthermore, initial energy of the i-th mode can be written as E0i=a2 iE0, where coefficient a2 i∈[0,1] denotes the fraction of the initial energy of the i-th mode in the total initial energy. Coefficients of the two 9modes satisfy a2 1+a2 2= 1and therefore can be parameterized as a1= cos ψ a2= sin ψ,(33) where ψ∈[0, π/2]. Taking (33) into account, we can write (32) as I(γ, ψ, θ 1, θ2) =E02X i=1a2 i 2ω0iω2 0i+γ2 γω0i+γ ω0icos 2θi+ sin 2 θi . (34) If we consider the optimal damping coefficient as the one for which the integral (34) is minimal, we can easily determine the optimal damping coefficient form the condition ∂I(γ, ψ, θ 1, θ2) ∂γ γopt= 0, (35) and we obtain γopt(ψ, θ1, θ2) =s ω2 01ω2 02 2ω2 02cos2ψcos2θ1+ 2ω2 01sin2ψcos2θ2. (36) It is easy to see that, for any fixed ψ, the function (36) has smallest magnitude for cos2θ1= cos2θ2= 1, which corresponds to the initial conditions with initial energy comprised only of potential energy distributed within the two modes, i.e E0=E0P1+E0P2. In that case we can write the denominator of (36) as f(ψ) =q 2ω2 02cos2ψ+ 2ω2 01sin2ψ=q 2(ω2 02−ω2 01) cos2ψ+ 2ω2 01, (37) where we used sin2ψ= 1−cos2ψ. Since ω01< ω 02, the function (37) has maximum for ψ= 0. Thus, the minimum value of the optimal damping coefficient (36) is√ 2ω01/2, and it is obtained for ψ= 0and θ1={0, π}, which corresponds to the initial conditions with initial energy comprised only of potential energy in the first mode, i.e. E0=E0P1. On the other hand, for any fixed ψ, the function (36) has singularities forcos2θ1= cos2θ2= 0, which corresponds to the initial conditions with initial energy comprised only of kinetic energy. Thus, the range of the optimal damping coefficient (36) is γopt∈h√ 2ω01/2,+∞ . (38) Now we calculate the average of the integral (34) over a set of all initial conditions, we obtain I(γ) =1 2π3Zπ/2 0dψZ2π 0dθ1Z2π 0dθ2I(γ, ψ, θ 1, θ2) =E0 42X i=1ω2 0i+γ2 γω2 0i , (39) and from the condition ∂I(γ) ∂γ γopt= 0, (40) we find that the optimal damping coefficient with respect to the averaged integral (39) is given by γopt=s 2ω2 01ω2 02 ω2 01+ω2 02. (41) Inordertomoreeasilyanalyzethebehaviorofthedampingcoefficient(36)withregardtothedistribution of the initial potential energy within the modes and its relationship with the damping coefficient (41), 10similarly as in subsection 2.1 (see (17) and (18)), we define the ratio of the initial potential energy of the i-th mode and the total initial energy, i.e. βi=E0Pi E0. (42) Since the initial potential energy satisfies E0P=E0P1+E0P2≤E0, we have βi∈[0,1]and the condition 0≤(β1+β2)≤1holds. Taking E0Pi=E0icos2θi(see (30)) and E0i=a2 iE0with (33) into account, we have β1= cos2ψcos2θ1 β2= sin2ψcos2θ2.(43) Using (43), relation (36) can be written as γopt(β1, β2) =s ω2 01ω2 02 2ω2 02β1+ 2ω2 01β2. (44) For clarity, we will repeat briefly, the minimum value of (44) is√ 2ω01/2, obtained for β1= 1andβ2= 0 (or in terms of the angles in (36), for ψ= 0andθ1={0, π}), while γopt→+∞forβ1=β2= 0(or in terms of the angles in (36), for any ψwith θ1={π/2,3π/2}andθ2={π/2,3π/2}). The benefit of relation (44) is that we expressed (36) through two variables instead of three, i.e. this way we lost information about the signs of the initial conditions and about distribution of initial kinetic energy within the modes, but the optimal damping coefficient (36) does not depend on those signs anyway, due to the squares of trigonometric functions in variables θ1andθ2, and, for a fixed distribution of initial potential energy within the modes, the optimal damping coefficient (36) is constant for different distributions of initial kinetic energy within the modes. By looking at relations (44) and (41), it is immediately clear that γopt(β1, β2) =γoptfor ω2 02β1+ω2 01β2=ω2 01+ω2 02 4, (45) while γopt(β1, β2)<γoptif the left hand side of relation (45) is greater than the right hand side, and γopt(β1, β2)>γoptif the left hand side of relation (45) is smaller than the right hand side. Again, similarly as in subsection 2.1 (see (19)), using βi=ω2 0iq2 0i/E0we can express the optimal damping coefficient (44) as a function of the initial modal coordinates as well, i.e. γopt(q01, q02) =s E0 2q2 01+ 2q2 02, (46) where q0i∈[−√E0/ω0i,√E0/ω0i]and the condition 0≤(ω2 01q2 01+ω2 02q2 02)≤E0holds. We can express condition (45) in terms of initial modal coordinates, i.e. γopt({q0i}) =γoptfor q2 01+q2 02 E0=ω2 01+ω2 02 4ω2 01ω2 02, (47) while γopt({q0i})<γoptif the left hand side of relation (47) is greater than the right hand side, and γopt({q0i})>γoptif the left hand side of relation (47) is smaller than the right hand side. We note here that we did not use explicit values of the undamped modal frequencies ω01andω02in the analysis so far, and relations presented so far are valid for any 2-DOF system with MPD. In the next subsection, we provide a more detailed quantitative analysis using an example with specific values of modal frequencies. 113.1. Quantitative example Here we consider the 2-DOF system as the one shown schematically in Fig. 6, but with m1=m2=m, k1=k2=k3=kandc1=c2=c. The corresponding equations of motion are m¨x1(t) =−c˙x1(t)−kx1(t)−k(x1(t)−x2(t)), m¨x2(t) =−c˙x2(t)−kx2(t) +k(x1(t)−x2(t)).(48) For completeness, we will investigate here the behavior of the optimal damping coefficient given by the minimization of the energy integral for different initial conditions, and its relationship with the optimal dampingcoefficientgivenbytheminimizationoftheaveragedenergyintegral, inallthreecoordinatesystems that we introduced in the previous subsection and additionally in the coordinate system defined by the initial displacements of the masses. System of equations (48) can be easily recast to the form (25) with the modal coordinates q1(t) =rm 4(x1(t) +x2(t)) q2(t) =rm 4(x1(t)−x2(t)),(49) and with the natural (undamped) frequencies of the modes ω01=ω0andω02=√ 3ω0, where ω0=p k/m. Normalisation factorsp m/4in (49) ensure that our expression (28) for the energy of the system corresponds to energy expressed over the displacements and velocities of the masses, i.e. E(t) =2X i=1 ˙qi(t)2+ω2 0iqi(t)2
=2X i=1m˙xi(t)2 2+kxi(t)2 2 +k(x1(t)−x2(t))2 2. (50) Using the specific values of undamped modal frequencies of this system, relations (36), (41) and (44) become γopt(ψ, θ1, θ2) =s 3 6 cos2ψcos2θ1+ 2 sin2ψcos2θ2ω0, (51) γopt=√ 6 2ω0, (52) γopt(β1, β2) =r3 6β1+ 2β2ω0. (53) Since ω01=ω0, the range of (53) is γopt∈[√ 2ω0/2,+∞)(see (38)). As examples of the behavior of the damping coefficient (51) as a function of the angles {ψ, θ1, θ2}and its relationship with the damping coefficient (52), in Fig. 7 we show γopt(ψ, θ1, θ2)/γoptforψ={π/3, π/6} andθi∈[0, π]. In Fig. 8 we show ratio of the damping coefficient (53) and the damping coefficient (52), i.e. γopt(β1, β2)/γopt. If the initial energy is comprised only of potential energy, in terms of initial modal coordinates we have E0=ω2 0q2 01+ 3ω2 0q2 02, thus, the initial modal coordinates satisfy q01s ω2 0 E0∈[−1,1], q02s ω2 0 E0∈h −√ 3/3,√ 3/3i , 0≤ω2 0 E0 q2 01+ 3q2 02
≤1.(54) 12Figure 7: Ratio γopt(ψ, θ1, θ2)/γoptof the optimal damping coefficients (51) and (52) for ψ={π/3, π/6}andθi∈[0, π]. (a) Forψ=π/3, the total initial energy E0is distributed within the modes as E01=E0/4andE02= 3E0/4. (b) For ψ=π/6, the total initial energy is distributed within the modes as E01= 3E0/4andE02=E0/4. Singularities for θ1=θ2=π/2 are indicated by infinity symbols, and the points around singularities for which γopt(ψ, θ1, θ2)/γopt>3are removed on both figures (central white areas). Black lines, on both figures, indicate the points for which γopt(ψ, θ1, θ2)/γopt= 1. On both figures, ratio attains minimum for the corner points, i.e. for (θ1, θ2) ={(0,0),(0, π),(π,0),(π, π)}. Figure 8: Ratio γopt(β1, β2)/γoptof the optimal damping coefficients (53) and (52) for β1∈[0,1],β2∈[0,1]and the constraint 0≤(β1+β2)≤1. Singularity for β1=β2= 0is indicated by the filled red circle, and the points near singularity, for which γopt(β1, β2)/γopt>3, are removed, thus, a small white triangle is formed with the right angle at the origin. Black line indicates the points for which γopt(β1, β2)/γopt= 1. The minimum value of the ratio is at the point (β1, β2) = (1 ,0). Furthermore, we can write the optimal damping coefficient (46) as γopt(q01, q02) =s E0 2ω2 0(q2 01+q2 02)ω0, (55) and the condition (47) as ω2 0 E0 q2 01+q2 02
=1 3. (56) 13In Fig. 9(a) we show the ratio of (55) and (52), i.e. γopt(q01, q02)/γopt. The domain of this function consists of points inside and on the ellipse, i.e. it is given by (54). Similarly as before, singularity at (q01, q02) = (0 ,0)is indicated by the infinity symbol, and the points for which γopt(q01, q02)/γopt>3are removed. For points inside the circle we have γopt(q01, q02)/γopt>1, and for points outside the circle we have γopt(q01, q02)/γopt<1. Minimum values of this ratio are√ 3/3≈0.58, obtained for the points (q01, q02) ={(−√E0/ω0,0),(√E0/ω0,0)}. Figure 9: (a) Ratio γopt(q01, q02)/γoptof the optimal damping coefficients (55) and (52). (b) Ratio γopt(x01, x02)/γoptof the optimal damping coefficients (57) and (52). Singularities, at points (0,0)on the both figures, are denoted by infinity symbols, and the points near singularities, for which γopt/γopt>3, are removed. Black circles on both figures indicate the points for which γopt/γopt= 1. Using (49) we can write the optimal damping coefficient (55) in terms of initial displacements xi(0)≡x0i as γopt(x01, x02) =s E0 m(x2 01+x2 02)=s E0 mω2 0(x2 01+x2 02)ω0. (57) If the initial energy is comprised only of potential energy, in terms of initial displacements we have E0= mω2 0(x2 01+x2 02−x01x02), thus, the initial displacements of the masses satisfy x0is mω2 0 E0∈[−1,1], 0≤mω2 0 E0 x2 01+x2 02−x01x02
≤1,(58) and the condition (56) is now mω2 0 E0 x2 01+x2 02
=2 3. (59) In Fig. 9(b) we show the ratio of (57) and (52), i.e. γopt(x01, x02)/γopt. The domain of this function consists of points given by (58). Similarly as before, singularity at (x01, x02) = (0 ,0)is indicated by the infinity symbol, and the points for which γopt(x01, x02)/γopt>3are removed. For points inside the circle γopt(x01, x02)/γopt>1, and for points outside the circle γopt(x01, x02)/γopt<1. Minimum values of this ratio are√ 3/3≈0.58, obtained for the points (x01, x02) = ±q E0 mω2 0,±q E0 mω2 0 . 14Figure 10: Schematic figure of a MDOF system with Ndegrees of freedom. 4. MDOF systems with MPD Here we consider the MDOF system with Ndegrees of freedom shown schematically in Fig. 10. As in the Section 3, we will consider MPD, i.e. masses {m1, m2, ..., m N}, spring constants {k1, k2, ..., k N+1}, and dampers {c1, c2, ..., c N}can in general be mutually different but the condition ci/mi= 2γholds for any i={1, ..., N}, where γis the damping coefficient. Therefore, the reasoning we presented in Section 3 can be applied here, with the main difference that now the system has Nmodes instead of two. Again, we can write each initial condition over polar coordinates, as in the 2-DOF case (see (30)), only now we have N pairs of polar coordinates instead of two. The energy of each mode is given by (31), and consequently, the integral of the total energy over the entire time, for some arbitrary initial condition, is simply calculated similarly as in (32), i.e. I(γ,{E0i},{θi}) =NX i=1Z∞ 0Ei(t)dt=NX i=1E0i 2ω0iω2 0i+γ2 γω0i+γ ω0icos 2θi+ sin 2 θi , (60) where, again, Ei(t)is the energy of the i-th mode, E0iis the initial energy of the i-th mode. Thus, each initial condition with energy E0=PN i=1E0iis represented by points on Ncircles with radii ri=√E0i, for which conditionPN i=1r2 i=E0holds, and with angles θithat tell us how initial potential and initial kinetic energy are distributed within the modes. Similarly as before, initial energy of the i-th mode can be written as E0i=a2 iE0, where coefficient a2 i∈[0,1]denotes the fraction of the initial energy of the i-th mode in the total initial energy E0, and the condition NX i=1a2 i= 1 (61) holds. Relation (61) defines a sphere embedded in N-dimensional space and we can express the coefficients aiover N-dimensional spherical coordinates ( N−1independent coordinates, i.e. angles, since the radius is equal to one), but for the sake of simplicity we will not do that here and we will stick to writing the expressions as a functions of the coefficients ai. Thus, we can write (60) as I(γ,{ai},{θi}) =NX i=1Z∞ 0Ei(t)dt=E0NX i=1a2 i 2ω0iω2 0i+γ2 γω0i+γ ω0icos 2θi+ sin 2 θi .(62) We differentiate relation (62) by γand equate it to zero and get γopt({ai},{θi}) = NX i=12a2 icos2θi ω2 0i!−1/2 (63) as the optimal damping coefficient for which integral (62) is minimal. For any fixed set of coefficients {ai}, the smallest magnitude of the function (63) is obtained for cos2θi= 1∀i, which corresponds to the 15initial conditions with initial energy comprised only of potential energy distributed within the modes, i.e E0=PN i=1E0Pi. In that case the denominator of (63) is f({ai}) = NX i=12a2 i ω2 0i!1/2 (64) and using a2 1= 1−PN i=2a2 i(see (61)) we can write (64) as f({ai}) = 2 ω2 01+NX i=22a2 i1 ω2 0i−1 ω2 01!1/2 . (65) Since ω01< ω 0ifor any i≥2, each term in the sum of relation (65) is negative, and we can conclude that the function (65) has maximum for the set {ai}={1,0, ...,0}. Thus, the minimum value of the optimal damping coefficient (63) is√ 2ω01/2, and it is obtained for a1= 1andθ1={0, π}, which corresponds to the initial conditions with initial energy comprised only of potential energy in the first mode, i.e. E0=E0P1. On the other hand, for any fixed set {ai}, the function (63) has singularities for cos2θi= 0∀i. Thus, the range of the optimal damping coefficient (63) is γopt∈h√ 2ω01/2,+∞ . (66) In Appendix A we have calculated the average of the integral (62) over a set of all initial conditions and obtained I(γ) =E0 2NNX i=1ω2 0i+γ2 γω2 0i . (67) We differentiate relation (67) by γand equate it to zero and obtain γopt=N1/2 NX i=11 ω2 0i!−1/2 (68) as the optimal damping coefficient with respect to the averaged integral (67). Since the ratio of the initial potential energy of the i-th mode and the total initial energy is βi=E0Pi E0=a2 icos2θi, (69) where βi∈[0,1]and the condition 0≤PN i=1βi≤1holds, we can write (63) as a function of the distribution of the initial potential energy over the modes, i.e. γopt({βi}) = NX i=12βi ω2 0i!−1/2 . (70) The minimum value of (70) is√ 2ω01/2, obtained for β1= 1andβi= 0fori≥2, while γopt→+∞for βi= 0∀i. Using βi=ω2 0iq2 0i/E0, we can write (63) as a function of initial modal coordinates as well, i.e. γopt({q0i}) =s E0 2PN i=1q2 0i, (71) where q0i∈[−√E0/ω0i,√E0/ω0i]and the condition 0≤PN i=1ω2 0iq2 0i≤E0holds. 164.1. Quantitative example Here we consider the MDOF system as the one shown schematically in Fig. 10 but with mi=m,ci=c fori={1, ..., N}, and with ki=kfori={1, ..., N + 1}. Such a system without damping, i.e with ci= 0∀i, is a standard part of the undergraduate physics/mechanics courses [22]. Therefore, for the MDOF system with Ndegrees of freedom we are considering here, the undamped modal frequencies are [21, 22] ω0i= 2ω0siniπ 2(N+ 1) ,with i={1, ..., N}, (72) and where ω0=p k/m. In Fig. 11(a) we show undamped modal frequencies ω01,ω0Nand damping coefficient γopt, i.e. (68), calculated with (72), as functions of N. We clearly see that the coefficient γoptis in the over-damped regime from the perspective of the first mode, and in the under-damped regime from the perspective of highest mode, for any N > 1, and in the case N= 1all three values match. In Fig. 11(b) we show ratios γopt/ω01andω0N/γoptand we see that both ratios increase with increasing N. Figure 11: (a) Undamped modal frequencies ω01(blue circles), ω0N(red x’s) and the damping coefficient γopt(black squares) as functions of the number of the masses N. (b) Ratios γopt/ω01(blue line) and ω0N/γopt(red line), shown as solid lines due to the high density of the shown points. We show in Appendix B that the following limits hold lim N→+∞γopt(N) = 0 , (73) lim N→+∞γopt(N) ω01(N)= +∞, (74) lim N→+∞ω0N(N) γopt(N)= +∞. (75) We note here that these limit values do not correspond to the transition from a discrete to a continuous system, but simply tell us the behavior of these quantities with respect to the increase in the number of masses, i.e. with respect to the increase in the size of the discrete system. From everything that has been said so far, it is clear that the damping coefficient γopt, obtained by minimizing the energy integral averaged over all initial conditions that correspond to the same initial energy, cannot be considered generally as optimal and that, by itself, it says nothing about optimal damping of the system whose dynamics started with some specific initial condition. Damping coefficient (63), which is given by the minimization of the energy integral for a specific initial condition, is of course a better choice for optimal damping of an MDOF system, than the damping coefficient γopt, if we want to consider how the system dissipates energy the fastest for a particular initial condition, but, as we argue in the subsection 4.2, this damping coefficient also has some obvious deficiencies. 174.2. Issues with the minimum of the energy integral as a criterion for optimal damping We can ask, for example, whether in an experiment, with known initial conditions, in which an MDOF system is excited to oscillate, a damping coefficient (63) would be the best choice if we want that the system settles down in equilibrium as soon as possible? Here, in three points, we explain why we think the answer to that question is negative: •From relation (62), we see that, due to the term sin 2θi, the energy integral is sensitive to changes θi→ −θiandθi→π−θi, which correspond to changes of initial conditions (q0i,˙q0i)→(q0i,−˙q0i)and (q0i,˙q0i)→(−q0i,˙q0i). When we differentiate (62) to determine γfor which the energy integral has a minimum, the term sin 2θicancels and as a result the coefficient (63) is not sensitive to this change in initial conditions. Such changes in the initial conditions lead to significantly different situations. For example, if q0i>0and ˙q0i>0, the i-th mode in the critical and over-damped regime (i.e. for γ≥ω0i) will never reach the equilibrium position, while for q0i>0and ˙q0i<0, and i-th mode initial kinetic energy grater than initial potential energy, it can go through the equilibrium position once, depending on the magnitude of the damping coefficient, and there will be the smallest damping coefficient in the over-damped regime for which no crossing occurs and for which the solution converges to equilibrium faster than for any other damping coefficient [24]. Therefore, the damping coefficient considered optimal would have to be sensitive to this change in initial conditions. •Damping coefficient (63) has singularities for cosθi= 0∀i, i.e. for initial conditions for which all initial energy is kinetic. For such initial conditions, the higher the damping coefficient, the higher and faster the dissipation. In other words, the higher the damping coefficient, the faster the energy integral decreases. Therefore, coefficient (63) diverges for that type of initial conditions. This would actually mean that, for this initial conditions, it is optimal to take the damping coefficient as high as possible, but in principle this corresponds to a situation in which all modes are highly over-damped, i.e. all masses reach their maximum displacements in a very short time and afterwards they begin to return to the equilibrium position almost infinitely slowly. Figuratively speaking, it is as if we immersed the system in concrete. This issue has recently been addressed in the context of free vibrations of SDOF [24] and was already noticed in [25]. Therefore, simply taking the highest possible damping coefficient, as suggested by relation (63) for this type of initial conditions, is not a good option. •The damping coefficient (63) is determined on the basis of the energy integral over the entire time and therefore it does not take into account that in nature and experiments these systems effectively return to the equilibrium state for some finite time. Because of the above points, in the next section we provide a new approach to determine the optimal damping of MDOF systems. 5. Optimal damping of an MDOF system: a new perspective From a theoretical perspective, systems with viscous damping asymptotically approach the equilibrium state and never reach it exactly. In nature and in experiments, these systems reach the equilibrium state which is not an exact zero energy state, but rather a state in which the energy of the system has decreased to the level of the energy imparted to the system by the surrounding noise, or to the energy resolution of the measuring apparatus. Following this line of thought, we will define a system to be in equilibrium for times t > τsuch that E(τ) E0= 10−δ, (76) where E(τ)is the energy of the system at t=τ,E0is the initial energy, and δ > 0is a dimensionless parameter that defines what fraction of the initial energy is left in the system. This line of thought has recently been used to determine the optimal damping of SDOF systems [24], and here we extend it to MDOF systems. Therefore, in what follows, we will consider as optimal the damping coefficient for which the systems energy drops to some energy level of interest, e.g. to the energy resolution of the experiment, the fastest and we will denote it with ˜γ. 185.1. Optimal damping of the i-th mode of a MDOF system with MPD Here we will consider the behavior of the energy of the i-th mode of the MDOF system with MPD and determine the optimal damping coefficient ˜γiof the i-th mode with respect to criterion (76). For any MDOF system with N≥1degrees of freedom with MPD, each mode behaves as a SDOF system studied in Section 2, with the damping coefficient γand the undamped (natural) frequency ω0i. Thus (see relation (31)), the ratio of the energy of the i-th mode, Ei(γ, t), and initial energy of the i-th mode, E0i, is given by Ei(γ, t) E0i=e−2γt cos2(ωit) +γcos 2θisin(2ωit) ωi+ ω2 0i+γ2+ 2ω0iγsin 2θi
sin2(ωit) ω2 i (77) for the under-damped ( γ < ω oi) and over-damped ( γ > ω 0i) regime. We will repeat here briefly for clarity, ωi=p ω2 0i−γ2is the damped angular frequency and θiis the polar angle which determines the initial conditions q0iand ˙q0iof the i-th mode and the distribution of the initial energy within the mode, i.e. initial potential and initial kinetic energy of the i-th mode are E0Pi=E0icos2θiandE0Ki=E0isin2θi respectively. Energy to initial energy ratio for the i-th mode in the critically damped regime ( γ=ω0i) is simply obtained by taking γ→ω0ilimit of the relation (77), and we obtain Ei(γ=ω0i, t) E0i=e−2ω0it 1 + 2 ω0i(cos 2 θi)t+ 2ω2 0i(1 + sin 2 θi)t2
. (78) In relations (77) and (78), we explicitly show that the energy depends on the damping coefficient and time, because in what follows we will plot these quantities as functions of these two variables for fixed initial conditions, i.e. fixed θi. We will investigate the behavior for several types of initial conditions, which of course will not cover all possible types of initial conditions, but will give us a sufficiently clear picture of the determinationandbehavioroftheoptimaldampingwithrespecttotheinitialconditionsandtheequilibrium state defined with condition (76). 5.1.1. Initial energy of the i-th mode comprised only of potential energy In Fig. 12 we show the base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi= 0, which corresponds to the initial energy of the i-th mode comprised only of potential energy. Four black contour lines denote points with Ei(γ, t)/E0i={10−3,10−4,10−5,10−6}respectively, as indicated by the numbers placed to the left of each contour line. Each contour line has a unique point closest to the γ axis, i.e. corresponding to the damping coefficient ˜γifor which that energy level is reached the fastest. As an example, we draw arrow in Fig. 12 that points to the coordinates (γ, t) = (0 .840ω0i,5.15ω−1 0i), i.e. to the tip of the contour line with points corresponding to Ei(γ, t) = 10−4E0i. Thus, for the initial condition θi= 0, ˜γi= 0.840ω0iis the optimal damping coefficient for the i-th mode to reach this energy level the fastest, and it does so at the instant τi= 5.15ω−1 0i. In Table 1 we show results for other energy levels corresponding to contour lines shown in Fig. 12. Here, and in the rest of the paper, we have rounded the results for the damping coefficient to three decimal places, and for the time to two decimal places. Ei(γ, t)/E0i˜γi[ω0i]τi[ω−1 0i] 10−30.769 4.18 10−40.840 5.15 10−50.885 6.16 10−60.915 7.20 Table 1: Optimal damping coefficient ˜γifor which the energy of the i-th mode drops to the level 10−δE0ithe fastest, with the initial condition θi= 0. Consider now, for example, a thought experiment in which we excite a MDOF system so that it vibrates only in the first mode and that all initial energy was potential, i.e. E01=E0andθ1= 0. Furthermore, 19Figure 12: The base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi= 0. For this initial condition, initial energy of the i-th mode is comprised only of potential energy. Four black contour lines denote points with Ei(γ, t)/E0i={10−3,10−4,10−5,10−6}respectively, as indicated by the numbers placed to the left of each contour line. As an example of determining the optimal damping for which the system reaches the desired energy level the fastest, i.e. with respect to the condition (76), we draw the arrow that points to the coordinates (γ, t) = (0 .840ω0i,5.15ω−1 0i)for which the i-th mode reaches the level Ei(γ, t)/E0i= 10−4the fastest. Thus, ˜γi= 0.840ω0iis the optimal damping coefficient to reach this energy level the fastest. Optimal values for other energy levels, denoted with contour lines, are given in Table 1. suppose that the system has effectively returned to equilibrium when its energy drops below 10−6E0, due to the resolution of the measuring apparatus. It is clear form the Table 1 that ˜γ1= 0.915ω01would be optimal in such a scenario. In the same scenario, optimal damping coefficient given by the minimization of the energy integral, i.e. (63), would be γopt=√ 2ω01/2 = 0 .707ω01, thus, a very bad choice in the sense that this damping coefficient would not be optimal even in an experiment with a significantly poorer energy resolution (see Table 1). This simple example illustrates that, from a practical point of view, one has to take into account both the initial conditions and the resolution of the measuring apparatus in order to determine the optimal damping coefficient. 5.1.2. Initial energy of the i-th mode comprised only of kinetic energy In Fig. 13(a) and (b) we show the base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi=π/2, which corresponds to the initial energy of the i-th mode comprised only of kinetic energy. In Fig. 13(b) we show results for larger data span than in Fig. 13(a), and only contour line for points corresponding to Ei(γ, t) = 10−3E0i. The left arrow in Fig. 13(b) indicates the same coordinates as the arrow in Fig. 13(a), and the right arrow in Fig. 13(b) points to the coordinates (γ, t) = (13 .316ω0i,4.66ω−1 0i) with Ei(γ, t) = 10−3E0i. Thus, for γ >13.316ω0ithe system comes sooner to the energy level 10−3E0ithan forγ= 0.722ω0i, but these highly over-damped damping coefficients would correspond to restricting the system to infinitesimal displacements from equilibrium, after which the system returns to the equilibrium state practically infinitely slowly [24]. Thus, for this initial condition we take the damping coefficient in the under-damped regime, i.e. ˜γi= 0.722ω0i, as optimal for reaching the level Ei(γ, t) = 10−3E0ithe fastest. For all energy levels the behaviour is qualitatively the same, and the results are given in Table 2. Consider now, for example, a thought experiment in which we excite a MDOF system so that it vibrates only in the first mode and that all initial energy was kinetic, i.e. E01=E0andθ1=π/2. Furthermore, suppose that the system has effectively returned to equilibrium when its energy drops below 10−6E0, due to the resolution of the measuring apparatus. It is clear form the Table 2 that ˜γ1= 0.892ω01would be optimal in such a scenario. In the same scenario, optimal damping coefficient given by the minimization of the energy integral, i.e. (63), would be γopt= +∞. 20Figure 13: The base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi=π/2. For this initial condition, initial energy of the i-th mode is comprised only of kinetic energy. (a) Four black contour lines denote points with Ei(γ, t)/E0i={10−3,10−4,10−5,10−6}respectively, andthearrowpointstothecoordinates (γ, t) = (0 .722ω0i,4.66ω−1 0i), with Ei(γ, t)/E0i= 10−3, for which this level of energy is reached in shortest time for the shown data span. (b) Contour line for points with Ei(γ, t) = 10−3E0iis shown for larger data span, left arrow points to the coordinates (γ, t) = (0 .722ω0i,4.66ω−1 0i), and the right arrow to the coordinates (γ, t) = (13 .316ω0i,4.66ω−1 0i), both with Ei(γ, t)/E0i= 10−3. Thus, for γ >13.316ω0i energy level 10−3E0iis reached faster than for γ= 0.7223ω0i. See text for details. Ei(γ, t)/E0i˜γi[ω0i]τi[ω−1 0i] 10−30.722 4.66 10−40.794 5.50 10−50.852 6.42 10−60.892 7.40 Table 2: Optimal damping coefficient ˜γifor which the energy of the i-th mode drops to the level 10−δE0ithe fastest, with the initial condition θi=π/2. Here we note that if in such an experiment we can set the damping coefficient to be in the over-damped regime in the first part of the motion, i.e. when the system is moving from the equilibrium position to the maximum displacement, and in the under-damped regime in the second part of the motion, i.e. when the system moves from the position of maximum displacement back towards the equilibrium position, then the fastest way to achieve equilibrium would be to take the largest experimentally available over-damped coefficient in the first part of the motion, and the under-damped coefficient optimised like in 5.1.1 in the second part of the motion, with the fact that we have to carry out the optimization with respect to the energy left in the system at the moment when the system reached the maximum displacement and with respect to the energy resolution of the experiment. 5.1.3. Initial energy of the i-th mode comprised of potential and kinetic energy In Fig. 14(a) we show the base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi=π/3, whichcorrespondstothe initialenergyofthe i-thmodecomprisedofkineticenergy E0Ki= 3E0i/4 and potential energy E0Pi=E0i/4, with both initial normal coordinate and velocity positive, i.e. with q0i>0and ˙q0i>0. The results for optimal damping are obtained by the same method as in 5.1.1 and are given in Table 3 for data shown in Fig. 14(a), and in Table 4 for data shown in Fig. 14(b). We see that the energy dissipation strongly depends on the relative sign between q0iand ˙q0i. It was recently shown, 21for free vibrations of SDOF, that for an initial condition with initial kinetic energy greater than initial potential energy and opposite signs between x0andv0, an optimal damping coefficient can be found in the over-damped regime [24], thus, the same is true when we consider any mode of a MDOF system with MPD. Figure 14: The base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), (a) for initial condition θi=π/3, and (b) for initial condition θi=−π/3. For both initial conditions, initial energy of the i-th mode is comprised of kinetic energy E0Ki= 3E0i/4 and potential energy E0Pi=E0i/4. For θi=π/3initial normal coordinate and velocity are of the same signs, i.e. q0i>0and ˙q0i>0. For θi=−π/3initial normal coordinate and velocity are of the opposite signs, i.e. q0i>0and ˙q0i<0. Ei(γ, t)/E0i˜γi[ω0i]τi[ω−1 0i] 10−30.751 4.66 10−40.825 5.58 10−50.875 6.55 10−60.908 7.58 Table 3: Optimal damping coefficient ˜γifor which the energy of the i-th mode drops to the level 10−δE0ithe fastest, with the initial condition θi=π/3.Ei(γ, t)/E0i˜γi[ω0i]τi[ω−1 0i] 10−31.075 1.87 10−41.112 2.42 10−51.135 3.02 10−61.145 3.64 Table 4: Optimal damping coefficient ˜γifor which the energy of the i-th mode drops to the level 10−δE0ithe fastest, with the initial condition θi=−π/3. Consider now, for example, a thought experiment in which we excite a MDOF system so that it vibrates only in the first mode and that 75%of initial energy was kinetic and 25%of initial energy was potential, and with q01>0and ˙q01>0, i.e. E01=E0andθ1=π/3. Furthermore, suppose that the system has effectively returned to equilibrium when its energy drops below 10−6E0, due to the resolution of the measuring apparatus. It is clear form the Table 3 that ˜γ1= 0.908ω01would be optimal in such a scenario. In the same scenario, but with q01>0and ˙q01<0, i.e. for θ1=−π/3, we see from Table 4 that ˜γ1= 1.145ω01 would be optimal. Optimal damping coefficient given by the minimization of the energy integral, i.e. (63), is insensitive to the change of the sign of ˙q01, and it would be γopt=√ 2ω01= 1.414ω01in both cases. We note here, that for the initial conditions of the i-th mode with initial kinetic energy much grater than initial potential energy, i.e. E0Ki>> E 0Pi, and with opposite signs of initial displacement and velocity, i.e.sgn(q0i)̸= sgn( ˙ q0i), the optimal damping coefficient is going to be deep in the over-damped regime and dissipation of initial energy will happen in a very short time. If, for any reason, this is not desirable in some particular application, one can always find damping coefficient in the under-damped regime, with that same initial condition, which can serve as an alternative. As an example of such a situation, in Fig. 15 we show the base 10 logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi=−9π/20, which corresponds to the initial energy of the i-th mode comprised of kinetic energy E0Ki≈0.98E0iand potential energy E0Pi≈0.02E0i, with q0i>0and ˙q0i<0. In Fig. 15 we see that the i-th mode will reach the energy 22level 10−6E0ithe fastest for γ= 3.222ω0i, and in case, e.g., that such damping coefficient is difficult to achieve experimentally, another choice for the optimal damping coefficient can be γ= 0.883ω0i. Figure 15: The base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi=−9π/20. Black contour line denotes the points with Ei(γ, t) = 10−6E0i. Left arrow points to the coordinates (γ, t) = (0 .883ω0i,7.30ω−1 0i)for which level 10−6E0iis reached the fastest in the under-damped regime, and the right arrow points to the coordinates (γ, t) = (3.222ω0i,0.87ω−1 0i)for which the same level is reached the fastest in the over-damped regime. 5.2. Optimal damping of a MDOF system with MPD If all modes of a MDOF system with Ndegrees of freedom are excited, the ratio of the energy of the system, E(γ, t), and initial energy of the system, E0, is given by E(γ, t) E0=NX i=1E0i E0e−2γt cos2(ωit) +γcos 2θisin(2ωit) ωi+ ω2 0i+γ2+ 2ω0iγsin 2θi
sin2(ωit) ω2 i ,(79) where the set of all initial energies of the modes, i.e. {E0i}, and the set of all polar angles, i.e. {θi}, determines the initial condition of the whole system. Since for MPD the damping of the system as a whole is determined by only one damping coefficient γ, we can calculate the base 10logarithm of the ratio (79), but using a unique units for γ,tandω0ifor all modes, and from these data determine the optimal damping coefficient ˜γ, for which the system will come to equilibrium in the sense of the condition (76) the fastest, in the same way as in subsubsections 5.1.1-5.1.3 where we showed how to determine the optimal damping of individual modes. One practical choice for the units might be ω01forγand for ω0i∀i, and ω−1 01fort. This way, we have the easiest insight into the relationship between the first mode and the optimal damping coefficient that we want to determine, in the sense that we can easily see whether the first mode is under- damped, over-damped or critically damped in relation to it, which is important since the first mode is often the dominant mode. If we apply this to the 2-DOF system studied in 3.1, we obtain E(γ, t) E0=2X i=1E0i E0e−2γt cos2(ωit) +γcos 2θisin(2ωit) ωi+ ω2 0i+γ2+ 2ω0iγsin 2θi
sin2(ωit) ω2 i ,(80) where ω01=ω0,ω02=√ 3ω0,ω1=p ω2 0−γ2,ω2=p 3ω2 0−γ2and we take that the damping coefficient is in ω0units, while the time is in ω−1 0units. We are now in a position to determine the optimal damping of this 2-DOF system for different initial conditions. Again, we will not investigate all possible types of the 23initial conditions, but two qualitatively different ones, one with initial energy comprised only of potential energy, and the other with initial energy comprised only of kinetic energy. These two examples will give us a picture of the procedure for determining the optimal damping coefficient ˜γfor this 2-DOF system. The same procedure for determining the optimal damping can be in principle carried out for any MDOF system with MPD, with any initial condition. 5.2.1. Optimal damping of the 2-DOF system with initial energy comprised only of potential energy Herewechooseinitialconditionwith E01=E02=E0/2andθ1=θ2= 0, i.e. withinitialpotentialenergy distributed equally between the two modes and zero initial kinetic energy. In Fig. 16 we show the base 10 logarithm of the ratio (80), i.e. log (E(γ, t)/E0), for the chosen initial condition. In Table 5 we show results for other energy levels corresponding to contour lines shown in Fig. 16. For this initial condition, optimal damping coefficient given by the minimization of the energy integral, i.e. (63), is γopt=p 3/4ω0= 0.866ω0. Figure16: Thebase 10logarithmoftheratio(80), i.e. log (E(γ, t)/E0), forinitialcondition E01=E02=E0/2andθ1=θ2= 0. For this initial condition, initial energy of the 2-DOF system is comprised only of potential energy distributed equally between the modes. Four black contour lines denote points with E(γ, t)/E0={10−3,10−4,10−5,10−6}respectively, as indicated by the numbers placed to the left of each contour line. As an example of determining the optimal damping for which the system reaches the desired energy level the fastest, i.e. with respect to the condition (76), we draw the arrow that points to the coordinates (γ, t) = (0 .859ω0,5.37ω−1 0)for which the energy of the system reaches the level E(γ, t)/E0= 10−4the fastest. Thus, ˜γ= 0.859ω0is the optimal damping coefficient to reach this energy level the fastest. E(γ, t)/E0˜γ[ω0]τ[ω−1 0] 10−30.817 4.36 10−40.859 5.37 10−50.893 6.27 10−60.924 7.55 Table 5: Optimal damping coefficient ˜γfor which the energy of the system drops to the level 10−δE0the fastest, with the initial condition E01=E02=E0/2andθ1=θ2= 0. 245.2.2. Optimal damping of the 2-DOF system with initial energy comprised only of kinetic energy Here we choose initial condition with E01=E02=E0/2andθ1=θ2=π/2, i.e. with initial kinetic energy distributed equally between the two modes and zero initial potential energy. In Fig. 17(a) and (b) we show the base 10logarithm of the ratio (80), i.e. log (E(γ, t)/E0), for the chosen initial condition. In Table 6 we show results for other energy levels corresponding to contour lines shown in Fig. 17(a). For this initial condition, optimal damping coefficient given by the minimization of the energy integral, i.e. (63), is γopt= +∞. Figure 17: The base 10logarithm of the ratio (80), i.e. log (E(γ, t)/E0), for initial condition E01=E02=E0/2andθ1= θ2=π/2. For this initial condition, initial energy of the 2-DOF system is comprised only of kinetic energy distributed equally between the modes. (a) Four black contour lines denote points with E(γ, t)/E0={10−3,10−4,10−5,10−6}respectively, and thearrowpointstothecoordinates (γ, t) = (0 .783ω0,4.60ω−1 0), with E(γ, t)/E0= 10−3, forwhichthislevelofenergyisreached in shortest time for the shown data span. (b) Contour line for points with E(γ, t) = 10−3E0is shown for larger data span, left arrow points to the coordinates (γ, t) = (0 .783ω0,4.60ω−1 0), and the right arrow to the coordinates (γ, t) = (15 .927ω0,4.60ω−1 0), both with Ei(γ, t)/E0i= 10−3. Thus, for γ >15.927ω0energy level 10−3E0is reached faster than for γ= 0.783ω0. E(γ, t)/E0˜γ[ω0]τ[ω−1 0] 10−30.783 4.60 10−40.838 5.72 10−50.861 6.47 10−60.909 7.78 Table 6: Optimal damping coefficient ˜γfor which the energy of the system drops to the level 10−δE0the fastest, with the initial condition E01=E02=E0/2andθ1=θ2=π/2. 6. Conclusion and outlook The main message of this paper is that the dissipation of the initial energy in vibrating systems signifi- cantly depends on the initial conditions with which the dynamics of the system started, and ideally it would be optimal to always adjust the damping to the initial conditions. We took one of the known criteria for optimal damping, the criterion of minimizing the (zero to infinity) time integral of the energy of the system, averaged over all possible initial conditions corresponding to the same initial energy, and modified it to take into account the initial conditions, i.e. instead of averaging over of all possible initial conditions, we studied 25the dependence of the time integral of the energy of the system on initial conditions and determined the optimal damping as a function of the initial conditions. We found that the thus obtained optimal damping coefficients take on an infinite range of values depending on the distribution of initial potential energy and initial kinetic energy within the modes. We also pointed out the shortcomings of the thus obtained optimal damping coefficients and introduced a new method for determining optimal damping. Our method is based on the determination of the damping coefficients for which the energy of the system drops the fastest be- low some energy threshold (e.g. below the energy resolution of the experiment). We have shown that our method gives, both quantitatively and qualitatively, different results from the energy integral minimization method. In particular, the energy integral minimization method gives infinite optimal damping for initial conditions with purely kinetic energy, i.e. this method overlooks the region of underdamped coefficients for which strong energy dissipation occurs with this type of initial conditions, while this region is clearly seen and taken into account if one looks the energy behaviour directly, as we did. Furthermore, the energy integral minimization method gives the optimal damping which does not depend on the signs of the initial conditions, and we have shown that energy dissipation can strongly depend on them, which is taken into account in our method. Although the paper is dedicated to the case of mass-proportional damping, the new method we propose for determining the optimal damping can be applied to the types of damping we did not study in this paper. For example, in the case of a system with Rayleigh damping, the energy can be determined analytically using modal analysis, and based on that analytical expression, it can be numerically investigated for which values of the mass and stiffness proportionality constants the energy of the system drops the fastest below some energy threshold which effectively corresponds to the equilibrium state. In the case of a system with damping that does not allow analytical treatment, energy, as a function of time and magnitudes of individual dampers, can be determined numerically, e.g. by studying the vibrating system as a first order ordinary differential equation with matrix coefficients and using modern numerical methods for finding a solution of such an equation. This approach allows one to numerically solve systems with many degrees of freedom. Thus, we can numerically analyze the time evolution of the energy and find a set of damping parameters for which the energy drops to a desired energy threshold the fastest. Of course, this approach can be applied only for systems with a moderate number of degrees of freedom and a small number of dampers, due to the rapid growth of the parameter space that needs to be searched. Despite these limitations, we believe that our approach to optimal damping can be useful because, as we have shown, it can provide insights that other approaches overlook. Therefore, in future work we will investigate in detail the application of our approach to systems with damping that does not allow modal analysis. Furthermore, real systems can respond to many different initial conditions in operating conditions. We envision that our approach can be used to provide an overall optimal damping with respect to all initial conditions or with respect to some expected range of initial conditions. For this purpose, one could consider the energy averaged over the initial conditions and find the damping for which this averaged energy drops to a desired energy threshold the fastest. This will be the topic of our next work. 7. Acknowledgments We are grateful to Bojan Lončar for making schematic figures of 2-DOF and MDOF systems, i.e. Fig. 6 and 10, according to our sketches. This work was supported by the QuantiXLie Center of Excellence, a projectco-financedbytheCroatianGovernmentandEuropeanUnionthroughtheEuropeanRegionalDevel- opment Fund, the Competitiveness and Cohesion Operational Programme (Grant No. KK.01.1.1.01.0004). The authors have no conflicts to disclose. Appendix A. Average of the integral (62)over a set of all initial conditions For reader’s convenience, we will repeat the integral (62) here I(γ,{ai},{θi}) =E0NX i=1a2 i 2ω0iω2 0i+γ2 γω0i+γ ω0icos 2θi+ sin 2 θi . (A.1) 26In order to calculate the average of (A.1) over a set of all initial conditions, one has to integrate (A.1) over all coefficients ai, which satisfyPN i=1a2 i= 1andai∈[−1,1], and over all angles θi∈[0,2π]. Due toR2π 0cos 2θidθi=R2π 0sin 2θidθi= 0, terms with sine and cosine functions don’t contribute to the average of (A.1). Integration over all possible coefficients aiamounts to calculating the average of a2 iover a sphere of radius one embedded in Ndimensional space. If we were to calculate the average of the equation of a spherePN i=1a2 i= 1over a sphere defined by that equation, we would get NX i=1a2 i= 1, (A.2) where a2 idenotes the average of a2 iover a sphere. Due to the symmetry of the sphere and the fact that we are integrating over the whole sphere, contribution of each a2 iin the sum (A.2) has to be the same, so we can easily conclude that a2 i=1 N, (A.3) for any i. Thus, the average of (A.1) over all possible initial conditions is I(γ) =E0 2NNX i=1ω2 0i+γ2 γω2 0i . (A.4) Appendix B. Limit values (73),(74)and(75) For reader’s convenience, we repeat here (68) and (72) γopt=N1/2 NX i=11 ω2 0i!−1/2 (B.1) ω0i= 2ω0siniπ 2(N+ 1) ,with i={1, ..., N}. (B.2) Using (B.2), we can write (B.1) as γopt= 2ω0N1/2 NX i=11 sin2ζi!−1/2 , (B.3) where ζi=iπ 2(N+1). Using the fact that sinx < xfor0< x < π/ 2, we obtain γopt<2ω0N1/2 4(N+ 1)2 π2NX i=11 i2!−1/2 = 2ω0N1/2 π 2(N+ 1) NX i=11 i2!−1/2 . Now taking N→ ∞and using the well-known formulaP∞ i=11 i2=π2 6, we obtain (73). Now we focus on the limit (74). We will use the following well-known inequality sinx > x/ 2for0< x < π/2(this can be easily seen by, e.g. using the fact that sinis a concave function on [0, π/2]). From (B.3) it follows γopt ω01=N1/2(sinζ1)−1 NX i=11 sin2ζi!−1/2 > N1/2ζ−1 1·1 2 NX i=11 ζ2 i!−1/2 =1 2N1/2 NX i=11 i2!−1/2 , hence we obtain (74). 27The limit (75) is also easy to prove. 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2023-07-05

Common criteria used for measuring performance of vibrating systems have one thing in common: they do not depend on initial conditions of the system. In some cases it is assumed that the system has zero initial conditions, or some kind of averaging is used to get rid of initial conditions. The aim of this paper is to initiate rigorous study of the dependence of vibrating systems on initial conditions in the setting of optimal damping problems. We show that, based on the type of initial conditions, especially on the ratio of potential and kinetic energy of the initial conditions, the vibrating system will have quite different behavior and correspondingly the optimal damping coefficients will be quite different. More precisely, for single degree of freedom systems and the initial conditions with mostly potential energy, the optimal damping coefficient will be in the under-damped regime, while in the case of the predominant kinetic energy the optimal damping coefficient will be in the over-damped regime. In fact, in the case of pure kinetic initial energy, the optimal damping coefficient is $+\infty$! Qualitatively, we found the same behavior in multi degree of freedom systems with mass proportional damping. We also introduce a new method for determining the optimal damping of vibrating systems, which takes into account the peculiarities of initial conditions and the fact that, although in theory these systems asymptotically approach equilibrium and never reach it exactly, in nature and in experiments they effectively reach equilibrium in some finite time.

Optimal damping of vibrating systems: dependence on initial conditions

2307.02352v2

Sliding Dynamics of Current-Driven Skyrmion Crystal and Helix in Chiral Magnets Ying-Ming Xie,1Yizhou Liu,1and Naoto Nagaosa1,∗ 1RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan (Dated: December 14, 2023) The skyrmion crystal (SkX) and helix (HL) phases, present in typical chiral magnets, can each be considered as forms of density waves but with distinct topologies. The SkX exhibits gyrodynamics analogous to electrons under a magnetic field, while the HL state resembles topological trivial spin density waves. However, unlike the charge density waves, the theoretical analysis of the sliding motion of SkX and HL remains unclear, especially regarding the similarities and differences in sliding dynamics between these two spin density waves. In this work, we systematically explore the sliding dynamics of SkX and HL in chiral magnets in the limit of large current density. We demonstrate that the sliding dynamics of both SkX and HL can be unified within the same theoretical framework as density waves, despite their distinct microscopic orders. Furthermore, we highlight the significant role of gyrotropic sliding induced by impurity effects in the SkX state, underscoring the impact of nontrivial topology on the sliding motion of density waves. Our theoretical analysis shows that the effect of impurity pinning is much stronger in HL compared with SkX, i.e., χSkX/χHL∼α2 (χSkX,χHL: susceptibility to the impurity potential, α(≪1) is the Gilbert damping). Moreover, the velocity correction is mostly in the transverse direction to the current in SkX. These results are further substantiated by realistic Landau-Lifshitz-Gilbert simulations. Introduction.— Density waves in solids represent a prevalent phenomenon, particularly in low-dimensional systems [1, 2]. They break the translational symmetry of the crystal, leading to the emergence of Goldstone bosons, i.e., phasons, which remain gapless when the pe- riod of density waves is incommensurate with the crystal periodicity. The sliding motion of density waves under an electric field Ehas been extensively studied. In this con- text, the impurity pinning of phasons results in a finite threshold field [1, 2]. In general, exploring the dynam- ics of pinning and depinning offers valuable insights into understanding the behavior of density waves. The skyrmion crystal (SkX) and helix (HL) phases in chiral magnets can be recognized as periodic density waves of spins, as depicted in Figs. 1 (a) and (b). The HL phase is stabilized in chiral magnet at small mag- netic field regions, with spins of neighboring magnetic moments arranging themselves in a helical pattern. SkX is a superposition of three phase-locked HL and comprises arrays of magnetic skyrmions, nanoscale vortex-like spin textures characterized by a non-zero skyrmion number Nsk=1 4πR R d2rs·(∂xs×∂ys) (sbeing the unit vec- tor of spin). Theoretically proposed magnetic skyrmions [3–5] were initially observed in the chiral magnet MnSi under magnetic fields [6–8], wherein the skyrmion lattice structure produces a six-fold neutron scattering pattern. Since then, the chiral magnetic states encompassing SkX and HL states have been the focus of extensive research [9–13]. The dynamics of SkX in a random environment, specif- ically the pinning effects from impurities, are manifested through the topological Hall effect. The current depen- dence of topological Hall resistivity ρxywas initially ex- plored theoretically by Zang et al [14] and experimentally ∗nagaosa@riken.jpby Schulz et al. [15]. To illustrate, a schematic plot is presented in Fig. 1(c). Typically, there are three distinct regions characterizing the dynamics of SkX: the pinned, creep, and flow regions. The topological Hall resistiv- ity decreases when SkX is depinned because the motion of SkX induces temporal changes in the emerging mag- netic fields Be, subsequently generating emergent elec- tric fields Eeand an opposing Hall contribution. Theo- retically, the pinning problem of both SkX and HL was investigated in terms of replica symmetry breaking [16], revealing a distinct difference in glassy states between SkX and HL. The key factor lies in the nontrivial topol- ogy of SkX, contrasting with the trivial topology in HL and most density wave states. However, this difference has not been theoretically explored in the context of slid- ing/moving density wave states for chiral magnets. In this work, we systemically study the current-driven sliding dynamics of the SkX and HL in chiral magnets. We employ the methodology proposed by Sneddon et al. [17] in their investigation of charge density waves and apply it to magnetic materials. This method allows us to investigate the current-driven dynamics of SkX and HL, considering both deformation and impurity pinning effects. Through this method, we reveal that the drift ve- locity correction ∆ vddue to the impurity pinning effects versus the current density jsin the flow region, follows ∆vd∝(vd0)d−2 2(−e∥+G αDe⊥) for the SkX phase, while ∆vd∝ −(vd0)d−2 2e∥for the HL phase with the spatial di- mension denoted as d. Here, e∥represents the direction of the intrinsic drift velocity vd0(the magnitude of vd0is proportional to the current density jsdue to the univer- sal linear current-velocity relation [18]), G= 4πNsk,Dis a form factor at the order of unity, α≪1 is the Gilbert damping parameter so that G/αD ≫1. Although the scaling relation ( vd0)d−2 2applies to both SkX and HL, we can see that the gyrodynamics of the SkX state inducedarXiv:2312.07116v2 [cond-mat.mes-hall] 13 Dec 20232 (a) (c) vd vs(b) (d) HL SkX ρxy vdCreep Flow pinned js FIG. 1. (a), (b) The current-driven motion of the SkX and HL, respectively. (c) Schematic of the Hall resistivity ρxy and drift velocity vdversus current density jswith pinned (yellow), creeping (green), and flowing (purple) highlighted. (d) The collective flow motion of the SkX, where the center of each skyrmion (red dots) and the impurities (black crosses) are highlighted. by its nontrivial topology results in its sliding dynamics more robust than in HL and mostly in the transverse di- rection. Finally, we explicitly conduct the micromagnetic simulations on both the SkX and HL systems, aligning well with our theoretical expectations. Our work demonstrates the unification of sliding dy- namics between spin density waves and charge density waves within the same theoretical framework. Our re- sults also vividly illuminate both the similarities and dif- ferences in the sliding dynamics between SkX and HL phases. This insight significantly enhances our under- standing of the sliding dynamics associated with topo- logical density wave phenomena, which possesses pos- sible applications in areas, such as skyrmion-based de- vices [19–21], depinning dynamics [22–27], Hall responses [14, 15, 28], and current-driven motion of Wigner crystals under out-of-plane magnetic fields [29–31]. Sliding dynamics for skyrmion crystals.— The current- driven motion of SkX is described by the Thiele equation assuming that its shape does not change [18, 32, 33]: G×(vs−vd) +D(βvs−αvd) +F= 0. (1) Here, the first term on the left represents the Magnus force, the second term is the dissipative force, and the last term arises from deformation and impurity-pinning effects. Here, vsis the velocity of conduction electrons, αis the damping constant of the magnetic system, and β describes the non-adiabatic effects of the spin-polarized current. The gyromagnetic coupling vector is denoted as G= (0,0,4πNsk), and the dissipation matrix Dij=δijD where i, j∈ {x, y}. It is noteworthy that the Thieleequation respects out-of-plane rational symmetry [Sup- plementary Material (SM) Sec. IA [34]]. To obtain the equation of motion of SkX, the displace- ment vector field of skyrmions is defined as u(r, t) so that the drift velocity vd=∂u(r,t) ∂t, where ris the posi- tion vector, tis the time. The force Fcan be expressed withu(r, t) asF(r, t) =Fimp+Fde, where the impurity pinning force Fimp=−P i∇U(r+u(r, t)−ri)ρ(r) = fimp(r+u(r, t))ρ(r) and the deformation force Fde=R dr′D(r−r′)u(r′, t′). Here, U(r−ri) is the impu- rity potential around site ri,D(r−r′) characterizes the restoration strength after deformation, and ρ(r) is the skyrmion density. Based on these definitions, the Thiele equation can be expressed as an equation of motion: ∂u(r, t) ∂t=ˆM0vs+ˆM1Z dr′D(r−r′)u(r′, t) +ˆM1fimp(r+u(r, t))ρ(r). (2) where ˆM0=1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 and ˆM1=1 G2+α2D2 αD G −G αD . Note that each skyrmion is now considered as a center-of-mass particle, and these skyrmions form a triangular lattice and move collec- tively with scatterings from impurities, as illustrated in Fig. 1(d). The displacement vector can be expanded around the uniform motion, u(r, t) =vdt+˜u(r, t). (3) Here,vdis the dominant uniform skyrmion motion veloc- ity,˜u(r, t) characterizes a small non-uniform part. Using the Green’s function approach to solve the differential equation Eq. (2), ˜u(r, t) can be obtained as [17, 29, 34] ˜u(r, t) =Z dr′Z dt′G(r−r′, t−t′){vd0−vd +M1fimp(r′+vdt′+˜u(r′, t′))ρ(r′)},(4) where the intrinsic drift velocity vd0=ˆM0vs, the Fourier component of the Green’s function Gis given by G−1(k, ω) =−iω−ˆM1D(k). (5) Here,D(k) arises from the Fourier transformation of de- formation D(k) =R dd(r)D(r)e−ik·r(the spatial dimen- sion is denoted as d). In the flow region, ˜u(r, t) in Eq. (4) can be solved perturbatively. Up to the second order, ˜u(r, t)≈ ˜u0(r, t) +˜u1(r, t) +˜u2(r, t), which, respectively, are ob- tained by replacing the terms in the brackets of Eq. (4) as M0vs−vd, M1fimp(r′+vdt′)ρ(r′), M1∇fimp(r′+vdt′)· ˜u1(r, t)ρ(r′). Based on this approximation and making use of ⟨˜u(r,t) ∂t⟩= 0, the self-consistent equation for the3 velocity reads (for details see SM Sec. IB [34]) vd=vd0+X gZddq (2π)d|ρ(g)|2Λ(q)ˆM1× q2 xqxqy qxqyq2 y Im [G(q−g,−q·vd)]ˆM1 qx qy .(6) where ρ(g) is the Fourier component of ρ(r) with gas the reciprocal skyrmion lattice vectors, and Λ( q) arises from the impurity average U(q1)U(q2) = (2 π)dΛ(q2)δ(q1+ q2). Note that the crucial aspects for the above method to be valid are (i) the impurity strength is weak, (ii) the drift velocity is large compared to the impurity effects and the SkX remains elastic, (iii) the deformation within each skyrmion is negligible so that each skyrmion can be regareded as a point object. To proceed further, we adopt the following approxi- mations. The current-driven distortion is expected to be weak so that D(k) would be dominant by the long-wave limit. In this case, D(k) can be expanded as Kxk2 x+Kyk2 y for the 2D case and as Kxk2 x+Kyk2 y+Kzk2 zfor the 3D case. On the other hand, the characterized frequency that enters into the Green’s function is q·vd∼vd/a. Using a reasonable parameter vd= 10 m/s, the skyrmion lattice constant a= 25 nm, we estimate vd/a∼0.4 GHz. This frequency is much smaller compared with the one of Kj, which is roughly the scale of exchange energy J∼1 meV∼240 GHz [14, 18]. As a result, the dominant contribution to the integral is given by the elastic modes vdgj≈ωk≈ D(k)/√ G2+α2D2withk=q−g→0, around which the imaginary part of Green’s function is the largest. With the above approximations, we perform the in- tegral in Eq. (6) and sum over the smallest gvectors: gj=√ 3κ0(sin(j−1)π 3,cos(j−1)π 3) with jas integers from 1 to 6 and κ0=4π 3a. Since the Thiele equation exhibits out-of-plane rotational symmetry, without loss of gen- erality, we set vd0along x-direction here. After some simplifications (for details see SM. Sec IB), we find the correction (∆ vd=vd−vd0) on the drift velocity due to the impurity and deformation are given by ∆vd≈χSkX d(vd0)d−2 2(−e∥+G αDe⊥), (7) where the susceptibility to the impurity potential χSkX d =9κ3 0|ρ1|2Λ0αD 4√ KxKy(G2+α2D2)ford= 2, while χSkX d = 9√ 3κ7/2 0Γ(G2+α2D2 4α2D2)|ρ1|2Λ0(αD)3/2 π2√ KxKyKz(G2+α2D2)ford= 3 (the function Γ(a) =R+∞ 0dxx6 (x4−a)2+x4). Note that we have replaced ρ(gj) =ρ1,Λ(gj) = Λ 0given the six-fold rotational sym- metry of the skyrmion lattice. The first important aspect in Eq. (7) is that the cor- rection ∆ vdis insensitive to vd0in 2D limit but follows a square-root scaling: ( vd0)1/2in 3D limit. Similar to many scaling phenomena, the dimension plays a critical 0 0.005 0.01-1.5-1-0.50 0 0.005 0.0100.510 0.005 0.01-1.5-1-0.50 0 0.005 0.0100.511.5 0 0.005 0.01-1.5-1-0.50 Skyrmion 3DSkyrmion 2D, G < 0 i=i=||Helix 2D Skyrmion 2D, G > 0(a) (b) (c) (d) (e) (f)-10 -8 -6 -4-2.5-2-1.5-1-0.50 Log(vd0/K)Log(|∆vd|) Slope ≈ 0.5 0°30°60°90° 120° 150° 180° 210° 240° 270°300°330°00.20.40.60.81 i=i=|| i=i=|| Helix 3DFIG. 2. (a) and (b) The correction ∆ vdversus vd0(in units ofK) for the 2D and 3D HL state, respectively. (c) and (d) The correction ∆ vdversus vd0(in units of K) for the 2D SkX with G= 4πandG=−4π, respectively, where the longitudinal (transverse) component is in blue (red). The 3D SkX case is plotted in (f) with G= 4π. (e) The angular- dependence of |∆vd(θ)|, where θis the angle of vd0. All the of ∆vdin these plots has been normalized. The parameters D= 5.577π,α= 0.04. role here. The second important aspect is that the cor- rection along the transverse direction directly reflects the skyrmion topological number Gwith the ratio compared to the longitudinal one as G/αD . These interesting as- pects embedded in the Eq. (7) will be further highlighted later. Helix case.— It is straightforward to generalize the above treatment to the helical spin order. The Thiele equation is reduced to one dimension: D(βvs−αvd) +F= 0. (8) The essential difference here is the absence of gyrotropic coupling ( G= 0). Following the same procedure [SM Sec. II], the self-consistent equation for the drift velocity is given by vd=vd0+Zddq (2π)dX g|ρ(g)|2 α2D2Λ(q)q3 xIm[G(q−g,−qxvd)]. (9) Here, vd0=β αvs, the flow direction of the HL is defined asx-direction. After adopting the approximation in the4 previous section, the analytical expression of the correc- tion ∆ vdof the helical magnetic state is ∆vd≈ −χHL d(vd0)d−2 2 (10) where χHL d =(KxKy)−1/2|ρ1|2Λ0g3 0 4αD,ford= 2 (KxKyKz)−1/2|ρ1|2Λ0g7/2 0 2√ 2π(αD)1/2 ,ford= 3 with g0=π a. Despite different magnetic state nature, the ∆ vdas a function vd0in Eq. (10) for the HL displays a consistent scaling behavior as the one of SkX shown in Eq. (7). Numerical evaluation.— To further justify our analyt- ical results, we calculate the ∆ vdnumerically according to Eqs. (6) and (9). For simplicity, we set the elastic coefficient Kjas isotropic with K≡Kj. Figs. 2(a) and (b) display the correction ∆ vdas a function of vd0of HL. Note that the zero drify velocity limit should be ig- nored since the impurity pinning effect would be dom- inant in practice. When the vdis beyond this limit so that the pinning effect can be treated as a perturbation, which is true for the flow region, the plots clearly indi- cate ∆ vd∝(vd0)d−2 2. The square root behavior in 3D (d= 3) is explicitly checked with the log-log plot (inset of Fig. 2(b)). Figs. 2(c) and (d) show the longitudinal component (blue) and transverse component (red) of the correction for the SkX case with positive Gand negative G, respec- tively. It is consistent with Eq. (7) that the transverse component is odd with respect to Gand is much larger than the longitudinal component as G/αD ≫1. This gyrotropic type correction is inherited from the Magnus forces in the Thiele equation, and this correction also implies that there exists a net change on the skyrmion Hall angle due to the impurities. Moreover, the angu- lar dependence of the total correction |∆vd|is shown in Figs. 2(e), where the anisotropy is very small. The ∆ vd as a function vd0also displays distinct scaling behavior between 2D [Figs. 2(c),(d)] and 3D case [Fig. 2(f)]. Overall, the scaling behavior of skyrmion similar to that of the HL in Fig. 2, as expected from our theoretical analysis. Moreover, the intrinsic drift velocity vd0is lin- early proportional to the current density jsfor both SkX and HL ( vd0∝js) at large js. As a result, we can replace vd0with jsin the scaling relation, i.e., ∆ vd∝(js)d−2 2. It is worth noting that the charge density wave also respects this scaling relation [17], despite its distinct microscopic nature. Physical interpretation.— Now we provide a physical interpretation of the observed scaling behavior: ∆ vd∝ (vd0)d−2 2. As we mentioned earlier, the dominant con- tribution to the drift velocity correction arises from the excitation of elastic modes. Hence, we expect the correc- tion to be proportional to the number of excited elastic modes at a fixed vd0. For the SkX case, these modes follow the dispersion: vd0|gj|=D(k)/√ G2+α2D2, which can be rewritten as vd0=kd/2m′with m′= 2π√ G2+α2D2/(√ 3aK). Next, the problem is mapped to evaluate the density of states of free fermions with 0.050.10.150.20.251234 0.10.20.30.40.52D HL2D SkX0102030051015202530vd,x clean js (1010 A/m2)vd (m/s)2D SkX, weak impuritiesvd,x weak impurities vd,y defect vd,y weak impurities 0102030051015202530cleanweak impurities2D HL, weak impuritiesvd (m/s) js (1010 A/m2)(a)(b) (d)∆vd (m/s)α(c)0102030012345|Δvd/<Δv∥>||Δv∥/<Δv∥>||Δv⊥/<Δv∥>|G/αD⃗vd⃗y⃗xΔ⃗v⊥Δ⃗v∥⃗vd0finalColumn 6Column 5|Δvd|FIG. 3. Simulation results of LLG equation. (a) and (c) The drift velocity vdversus current density js(in units of 1010A/m2) of the 2D SkX and HL at the clean and impurity case. (b) The magnitdue of longitutinal (∆ d,∥) and trans- verse (∆ d,⊥) drift velocity correction versus the current den- sity (normalized the avegare value of ∆ d,∥), where the G/αD ratio is highlighted (red dashed line). The coordinate relation between different vectors are shown in the inset. (d) the drift velocity correction as a function of the damping parameter α atjs= 2×1011A/m2(only vd,xis used for the SkX). For (a) to (c), α= 0.2 and β= 0.5αare employed in the simulations. an energy vd0. Recall that the density of states of free fermion N(E)∝Ed−2 2at energy E. Hence, it is expected that the correction follows the same scaling: ∆vd∝(vd0)d−2 2according to this argument. We em- phasize that the microscopic nature of the density waves in this argument are not essential, which mainly stems from the long-wave characteristic of elastic modes. This explains why the HL and charge density wave also follow the same scaling behavior. Micromagnetic simulation.— We now further validate our theory through solving the Landau–Lifshitz–Gilbert (LLG) equation with the spin transfer torque effect [35– 39] (for details see SM). The calculated drift velocity vd versus current density jscurves are shown in Fig. 3 for both the SkX and HL. For simplicity, we mainly focus on 2D SkX and HL with weak impurities here, where our analytical expressions from perturbation theory are applicable. Figs. 3(a) and (c) show vdin the clean and disordered case with α= 0.2. The correction between these two cases at both SkX and HL is indeed insensitive to the current density within the flow limit. It is noteworthy that due to the gyrodynamcs, the SkX exhibits a much smaller depinning critical current density. Fig. 3(b) is to show that the correction along the transverse direction is obviously larger than the longitudinal one with the ratio5 ∼G/αD , being consistent with Eq. (7). Interestingly, the longitudinal correction versus the damping parame- terαof 2D SkX and HL show a positive and negative correlation, respectively [Fig. 3(d)], which is also consis- tent with our analytical expressions [see χSkX dandχHL d in Eqs. (7) and (10)]. It can also be seen that the im- purity correction along longitudinal direction is typically much stronger in HL than in SkX as χSkX d/χHL d∼α2. These distinct features between SkX and HL highlight the importance of the nontrivial topology in the sliding dynamics of density waves. Discussion.— We have provided a thorough analy- sis of the sliding dynamics exhibited by the SkX and HL phases, highlighting both their similarities and dif- ferences in terms of density waves sliding with distinct topologies. Our theory could have broader applica- tions. For instance, one can explore the relationship be- tween the topological Hall effect and the current den- sity in the flow region. In the clean limit, the uni- versal linear current-velocity relation vd0∝jsimplies that the topological Hall resistivity ρxy, proportional to |(vs−vd0)×Be|/|vs|[15], is expected to exhibit a plateau in the flow region, as illustrated in Fig. 1(c). In thepresence of impurities, the topological Hall resistivity is modified to ρxy∝ |(vs−vd0−∆vd)×Be|/|vs|. Consid- ering that ∆ vd∝(vd0)(d−2)/2, we anticipate a modified relation ρxy=a+bj−2+d/2 s , where aandbremain inde- pendent of the current magnitude js. The second term, bj−2+d/2 s , represents the correction from impurities. Con- sequently, we expect that the ρxy-jsplateau in the flow region will gradually diminish with increasing disorder. Our theory can also be applied to investigate the slid- ing dynamics of a 2D Wigner crystal under out-of-plane magnetic fields [29–31]. 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The skyrmion dynamics and Thiele equation From the Landau-Lifshitz-Gilbert equation, it was obtained that the current-driven skyrmion dynamics are captured by the Thiele equation: G×(vs−vd) +D(βvs−αvd) +F= 0. (S1) One can rewrite the equation as −G(vsy−vdy) +D(βvsx−αvdx) +Fx= 0, (S2) G(vsx−vdx) +D(βvsy−αvdy) +Fy= 0. (S3) In the matrix form:  G αD αD−G vdx vdy = G βD βD−G vsx vsy + Fy Fx . (S4) Then,  vdx vdy = G αD αD−G−1 G βD βD−G vsx vsy + G αD αD−G−1 Fy Fx . (S5)  vdx vdy =1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy + G αD αD−G 0 1 1 0 Fx Fy , =1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy + αD G −G αD Fx Fy (S6) Without loss of generality, we can choose the current direction to be x-direction: vs= (vs,0). When the pinning force is set to be F= 0, one can solve vd∥,0=G2+αβD2 G2+α2D2vs,vd⊥,0=(α−β)GD G2+α2D2ˆz×vs. (S7) Therefore, the longitudinal drift velocity vdxis proportional to the electric current when the force Fis neglectable. In a general direction, we can write the intrinsic drift velocity as vd0=1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy (S8) =s 1 +β2γ2 1 +α2γ2 cosθSkH−sinθSkH sinθSkH cosθSkH vscosθs vssinθs (S9) =vd0 cos(θs+θSkH) sin(θs+θSkH) . (S10) Here, the angle θsis to characterize the applied current direction, the skyrmion Hall angle θSkH= atanγ(α−β) 1+αβγ2with γ=D G, and the magnitude of drift velocity vd0=|vd0|=vsq 1+β2γ2 1+α2γ2.2 Now we show that the Thiele equation respects rotational symmetry with the principal axis along z-direction from Eq. (S6). The rotational operator is defined as Rz= cosϕ−sin(ϕ) sin(ϕ) cos( ϕ) with ϕas the rotational angle. Under this rotational operation, Eq. (S6) becomes Rz vdx vdy =1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy + G αD αD−G 0 1 1 0 Fx Fy , =1 G2+α2D2 Rz G2+αβD2GD(β−α) GD(α−β)G2+αβD2 R−1 zRz vsx vsy +Rz αD G −G αD R−1 zRz Fx Fy (S11) It is easy to show Rz A B −B A R−1 z= A B −B A (S12) with AandBas constant. The Eq. (S13) is simplified as Rz vdx vdy =1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy + G αD αD−G 0 1 1 0 Fx Fy , =1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 Rz vsx vsy + αD G −G αD Rz Fx Fy (S13) Hence, we have shown that the Thiele equation respects out-of-plane rotational symmetry. B. The correction on the drifted velocity due to the pining and deformation effects Let us define the displacement of skyrmion lattice as u(r, t) so that the drift velocity vd(r, t) =∂u(r,t) ∂t. The force is given by F(r, t) = Fimp+Fde (S14) Fimp =−X i∇U(r+u(r, t)−ri)ρ(r) =fimp(r+u(r, t))ρ(r) (S15) Fde=Z dr′D(r−r′)u(r′, t′), (S16) where Fimdescribes the pining effect from impurities and Fdearises from the deformation of skyrmion lattice, U(r−ri) is the impurity potential around the site ri.ρ(r) is the skyrmion densities. ∂u(r, t) ∂t=1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy + αD G −G αDZ dr′D(r−r′) ux(r′, t) uy(r′, t) +1 G2+α2D2 αD G −G αD fimp(r+u(r, t))ρ(r). (S17) The displacement vector can be expanded around the uniform motion, u(r, t) =vdt+˜u(r, t). (S18) Here, vdis the dominant uniform skyrmion motion velocity, ˜u(r, t) characterizes a small non-uniform motion. Then the equation of motion is written as ∂ ∂t−1 G2+α2D2 αD G −G αDZ dr′D(r−r′) ˜ux(r′, t) ˜uy(r′, t) =1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy −vd +1 G2+α2D2 αD G −G αD fimp(r+vdt+˜u(r, t))ρ(r). (S19)3 Here, we have used D(q) = 0 in the long wave limit ( q→0) so thatR dr′D(r−r′) = 0. Let us try to solve the Green’s function of the operator at the left-hand side, which is given by ∂ ∂t−1 G2+α2D2 αD G −G αDZ dr′D(r−r′) G(r′, t) =δ(t)δ(r) 1 0 0 1 (S20) It is more economical to work in the momentum space with G(r, t) =Zdω 2πZddk (2π)de−iωt+ik·rG(k, ω). (S21) Let us define D(r) =Rddq (2π)deiq·rD(q), and then Z dr′Z D(r−r′)G(r′, t) =Z dr′Zddq (2π)deiq·(r−r′)D(q)Zddk (2π)dZdω 2πG(k, ω)ei(k·r′−ωt) =Zddk (2π)dZdω (2π)G(k, ω)D(k)ei(k·r−ωt). (S22) In the momentum space, we find G−1(k, ω) =−iω−1 G2+ (Dα)2 αD G −G αD D(k) (S23) Therefore, Eq. (S19) can be rewritten as ˜u(r, t) =Z dr′Z dt′G(r−r′, t−t′){1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy −vd (S24) +1 G2+α2D2 αD G −G αD fimp(r′+vdt′+˜u(r′, t′))ρ(r′)}. In the flow limit, the perturbation from the deformation and impurity can be regarded as small in comparison with the leading order term. As a result, the displacement vector can be expanded as ˜u0(r, t) =Z dr′Z dt′G(r−r′, t−t′)1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy −vd , (S25) ˜u1(r, t) =1 G2+α2D2Z dr′Z dt′G(r−r′, t−t′) αD G −G αD fimp(r′+vdt′)ρ(r′), (S26) ˜u2(r, t) =1 G2+α2D2Z dr′Z dt′G(r−r′, t−t′) αD G −G αD ∇fimp(r′+vdt′)·˜u1(r′, t′)ρ(r′).(S27) Here, u0,u1, and u2are the leading, first, and second order terms, respectively. Next, let us evaluate the volume- average velocity ∂˜u(r, t) ∂t =∂˜u0(r, t) ∂t +∂˜u2(r, t) ∂t (S28) Note the fact that under the impurity average fimp(r′+vt′) = 0 has been used so that u1(r, t) would not contribute directly. Since non-uniform motion must vanish over the volume average, we can obtain a self-consistent equation for the velocity vd. Next, let us work out the self-consistent equation for vd. The leading order ∂u0(r, t) ∂t=Z dr′Z dt′G(r−r′, t−t′) ∂t1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy −vd =Z dr′Z dt′Zddk (2π)dZdω 2πeik·(r−r′)−iω(t−t′)(−iω)G(k, ω)1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy −vd = lim ω→0−iωG(k= 0, ω)1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy −vd =1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy −vd (S29)4 As mentioned theD ∂˜u1(r,t) ∂tE would not contribute, now let us show it explicitly. Recall that ˜u1(r, t) =1 G2+α2D2Z dr′Z dt′G(r−r′, t−t′) αD G −G αD fimp(r′+vdt′)ρ(r′) (S30) Then, ∂˜u1(r, t) ∂t=Z dr′Z dt′ 1 G2+α2D2G(r−r′, t−t′) αD G −G αD fimp(r′+vdt′)ρ(r′) (S31) =−1 G2+α2D2Z dr′Z dt′Zddk (2π)dZdω 2πeik·(r−r′)−iω(t−t′)G(k, ω) αD G −G αD × (S32) Zddq (2π)d iqx iqy Uqeiq·(r′+vdt′)ρ(r′) (S33) = 0, (S34) because after averaging over the impurity configurations, Uk= 0. Now let us look at the second-order term ∂˜u2(r, t) ∂t=1 G2+α2D2Z dr′Z dt′∂G(r−r′, t−t′) ∂t αD G −G αD ∇fimp(r′+vdt′)·˜u1(r′, t′)ρ(r′).(S35) Note that ∇fimp(r′+vdt′)·˜u1(r′, t′) =−Zddq (2π)dU(q)eiq·(r′+vdt′)q2 xqxqy qxqyq2 y ˜u1x(r′, t′) ˜u1y(r′, t′) (S36) Substitute the form of ˜u1(r′, t), ∂˜u2(r, t) ∂t=−1 (G2+α2D2)2Z dr′Z dt′∂G(r−r′, t−t′) ∂t αD G −G αD ρ(r′)× Zddq1 (2π)dU(q1)eiq1(·r′+vdt′)q2 1xq1xq1y q1xq1yq2 1y × Z dt′′Z dr′′G(r′−r′′, t′−t′′) αD G −G αD ρ(r′′)Zddq2 (2π)d iq2x iq2y U(q2)eiq2·(r′′+vdt′′)(S37) Then write the terms at the right-hand side of the equation with their Fourier components, ∂˜u2(r, t) ∂t=−1 (G2+α2D2)2Z dt′Z dr′Zddk (2π)dZdω 2πG(k, ω)(−iω)eik·(r−r′)−iω(t−t′) αD G −G αD × Zddq1 (2π)dq2 1xq1xq1y q1xq1yq2 1y U(q1)eiq1·(r′+vdt′)X g1ρ(g1)eig1·r′× Z dt′′Z dr′′Zddk′ (2π)dZdω′ 2πG(k′, ω′)eik′·(r′−r′′)−iω′(t′−t′′) αD G −G αDX g2ρ(g2)eig2·r′′× Zddq2 (2π)d iq2x iq2y U(q2)eiq2·(r′′+vdt′′)(S38) We can take integrals with respect to the space and time, and take the average over disorders, several delta functions5 would appear on the right-hand side: U(q1)U(q2) = (2 π)dΛ(q2)δ(q1+q2), (S39)Z dt′eiωt′eiq1·vdt′e−iω′t′= 2πδ(ω−ω′+q1·vd), (S40) Z dr′e−ik·r′eiq1·r′eig1·r′eik′·r′= (2π)dδ(k′−k+q1+g1), (S41) Z dt′′eiω′t′′eiq2·vdt′′= 2πδ(ω′+q2·vd), (S42) Z dr′′e−ik′·r′′eig2·r′′eiq2·r′′= (2π)dδ(g2+q2−k′). (S43) Take the volume average, and consider constraints from delta functions: q2=−q1=q,g1=−g2=g,ω′=−q·vd, we find ∂˜u2(r, t) ∂t =1 (G2+α2D2)2X gZddq (2π)d|ρ(g)|2Λ(q) αD G −G αDq2 xqxqy qxqyq2 y × Im [G(q−g,−q·vd)] αD G −G αD qx qy . (S44) Therefore, ∂˜u(r, t) ∂t =1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy −vd + 1 (G2+α2D2)2X gZddq (2π)d|ρ(g)|2Λ(q) G αD αD−Gq2 xqxqy qxqyq2 y × Im [G(q−g,−q·vd)] G αD αD−G qx qy (S45) SetD ∂˜u(r,t) ∂tE = 0, the self-consistent equation for the velocity is vd=1 G2+α2D2 G2+αβD2GD(β−α) GD(α−β)G2+αβD2 vsx vsy + 1 (G2+α2D2)2X gZddq (2π)d|ρ(g)|2Λ(q) αD G −G αDq2 xqxqy qxqyq2 y × Im [G(q−g,−q·vd)] αD G −G αD qx qy . (S46) As argued in the main text, the largest imaginary part is contributed by k=q−gin long wave limit ( kis small). To further proceed, let us evaluate Im[ G(k, ω)]. G(k, ω) =1 −iω−1 G2+(Dα)2 αD G −G αD D(k)=−G2+α2D2 λ(k)iω−D(k) λ(k) αD−G G αD , (S47) where λ(k) = [D(k) +ω(iαD+G)][D(k) +ω(iαD−G)] =D(k)2−(G2+α2D2)ω2+ 2iαDωD(k). (S48) The imaginary part of Green’s function is given by Im[G(k, ω)] = −(G2+α2D2)[D2(k)−(G2+α2D2)ω2]ω [D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k) +2αDωD2(k) [D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k) αD−G G αD . (S49)6 The largest imaginary part is given by the real mode ωk=D(k)/√ G2+α2D2. As a result, the first term in Im[ G(k, ω)] can be negligible. In 2D, we can expand D(k) =Kxk2 x+Kyk2 y. (S50) Now we can show that Zd2k (2π)22αDωD2(k) [D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k) = (KxKy)−1/2Z+∞ 0dk′2πk′ (2π)22αDωk′4 (k′4−(G2+α2D2)ω2)2+ 4α2D2ω2k′4 =(KxKy)−1/2sgn(ω) 4. (S51) where the integralR+∞ 0dtt2 (t2−√ G2+α2D2 2αD)2+t2=π 2is used with t=k′2 2αD|ω|. Note that αD̸= 0 is taken. The multiplications between matrices give  αD G −G αDq2 xqxqy qxqyq2 y αD−G G αD αD G −G αD qx qy = (G2+α2D2)(q2 x+q2 y) Gqy+αDq x −Gqx+αDq y .(S52) Finally, we obtain δvd=(KxKy)−1/2 4(G2+α2D2)X g|ρ(g)|2Λ(g)sgn(−g·vd0)|g|2 Ggy+αDg x −Ggx+αDg y (S53) We have shown that δvdrespects out-of-plane rotational symmetry in the main text, which is inherited from the Theiele equation. Without loss of generality, let us set vd0to be along x-direction. In this case, after summing over the six smallest gvectors: gj=√ 3κ0(sin(j−1)π 3,cos(j−1)π 3) with κ0=4π 3a,jare integers from 1 to 6, we find δvd=9κ3 0|ρ1|2Λ0αD 4p KxKy(G2+α2D2)−1 G αD (S54) where ρ1=ρ(gj),Λ0= Λ(gj). In the 3D case, D(k) =Kxk2 x+Kyk2 y+Kzk2 z. (S55) Then, Zd3k (2π)32αDωD2(k) [D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k) = (KxKyKz)−1/2Z+∞ 04πk′2dk′ (2π)32αDωk′4 (k′4−(G2+α2D2)ω2)2+ 4α2D2ω2k′4 =(KxKyKz)−1/2Γ(G2+α2D2 4α2D2)sgn( ω)p 2αD|ω| 2π2. (S56) where Γ( a) =R+∞ 0dxx6 (x4−a)2+x4. Similarly, we can obtain δvd=(2αD)1/2(KxKyKz)−1/2Γ(G2+α2D2 4α2D2) 2π2(G2+α2D2)X g|ρ(g)|2Λ(g)sgn(−g·vd0)p |g·vd0||g|2 Ggy+αDg x −Ggx+αDg y .(S57) After summing over the six smallest gvectors, we find δvd=9√ 3κ7/2 0Γ(G2+α2D2 4α2D2)|ρ1|2Λ0αD√αDv d0 π2p KxKyKz(G2+α2D2)−1 G αD . (S58)7 II. HELICAL SPIN ORDER CASE In this section, we consider the helical spin order case. Without loss of generality, we denote the helical spin order has a variation along x-direction. The Thiele equation for the helical spin order would be D(βvs−αvd) +F= 0. (S59) For simplicity, we have omitted the index xin the following. The equation of motion becomes ∂u(r, t) ∂t=β αvs+1 αDZ dr′D(r−r′)u(r′, t′) +1 αDfimp(r+u(r, t))ρ(r). (S60) Similarly, by defining u(r, r) =vdt+ ˜u(r, t), the equation of motion can be rewritten as [∂ ∂t−1 αDZ dr′D(r−r′)]˜u(r′, t) =β αvs+1 αDfimp(r+u(r, t))ρ(r). (S61) Let us define the Green’s function G(r, t) so that [∂ ∂t−1 αDZ dr′D(r−r′)]˜u(r′, t)G(r′, t) =δ(r)δ(t). (S62) It is easy to obtain G−1(k, ω) =−iω−D(k) αD. (S63) The displacement ˜ u(r, t) is given by ˜u(r, t) =Z dr′Z dt′G(r−r′, t−t′)[β αvs−vd+1 αDfimp(r′+u(r′, t′))ρ(r′)]. (S64) Then up to the second order, ˜u0(r, t) =Z dr′dt′G(r−r′, t−t′)[β αvs−vd], (S65) ˜u1(r, t) =1 αDZ dr′Z dt′G(r−r′, t−t′)fimp(r′+vdt′)ρ(r′), (S66) ˜u2(r, t) =1 αDZ dr′Z dt′G(r−r′, t−t′)∂xfimp(r′+vdt′)˜u1(r′, t′)ρ(r′). (S67) Following a similar procedure in Sec. II, using ⟨∂u(r,t) ∂t⟩= 0, we obtain vd=vd0+1 α2D2X g|ρ(g)|2Zddq (2π)dΛ(q)q3 xIm[G(q−g,−qxvd)]. (S68) where the intrinsic drift velocity vd0=β αvs. Consider the Im[ G(k, ω)] is dominant in the long wave limit ( k→0) and expand D(k) =Kxk2 x+Kyk2 yind= 2, we find vd≈vd0−(KxKy)−1/2 8αDX g|ρ(g)|2Λ(g)|gx|3(S69) and similarly in d= 3 case, D(k) =Kxk2 x+Kyk2 y+Kzk2 z, we obtain vd≈vd0−(KxKyKz)−1/2 4√ 2π(αD)1/2X g|ρ(g)|2Λ(g)|gx|3p |gxvd0|. (S70) After summing over the smallest reciprocal lattice vectors for the helix: g= (±1,0)g0with g0=π a, the correction ∆vd=β αvs−χHL d(vd0)d−2 2 (S71) where χHL d= (KxKy)−1/2|ρ1|2Λ0g3 0 4αD, ford= 2 (KxKyKz)−1/2|ρ1|2Λ0g7/2 0 2√ 2π(αD)1/2 ,ford= 3(S72) Here, we have set ρ(g0) =ρ1and Λ( g0) = Λ 0in this case.8 III. NUMERICAL METHOD DETAILS A. Details for the main text Fig.2 The main text Fig.2 is obtained from the main text Eq. (6) and (9) by numerically integrating qand summing over the smallest reciprocal lattice gvectors. For the SkX, the 4000 ×4000 in-plane momentum grids of qare taken with a hexagonal boundary (the boundary length is 4 κ0); while for the HL, the 1000 ×1000 in-plane momentum grids are taken with a square boundary (the boundary length is 4 g0). In the 3D case, the 1000 out-of-plane momentum points ofqwithin [ −2g0,2g0] are used for both SkX and HL in evaluating the integral. Also, we set the elastic coefficients Kj= 10000, the lattice constant aas a natural unit of one, the damping parameter α= 0.04, the dissipative coefficient D= 5.577π, the additional parameter G=±4πfor the SkX. B. Micromagnetic simulations The micromagnetic simulations were performed using MuMax3 [35]. The Landau-Lifshitz-Gilbert (LLG) equation is numerically solved ˙s=−|γ0|s×Heff+αs×˙s+pa3 2eS(js· ∇)s−pa3β 2eM2ss×(js· ∇)s, (S73) where sis the unit vector of spin, γ0is the gyromagnetic constant, αis the Gilbert damping constant, Msis the saturation magnetization, and Heff=−1 µ0MsδH δSis the effective field. The spin transfer torque effect of the current is described by the last two terms [36–38]. βdescribes the non-adiabaticity of the spin transfer torque effect. The current is applied along the x-direction in the simulations. A typical chiral magnet can be described by the following Hamiltonian density H=A(∇S)2−DS·(∇ ×S)−µ0MsB·S (S74) The corresponding parameters and their values employed in the simulations are: the saturation magnetization Ms= 111 kA/m, the exchange stiffness A= 3.645 pJ/m, and the Dzyaloshinskii-Moriya interaction strength D= 0.567 mJ/m2. An external magnetic field B= 0.3 T (with its direction perpendicular to the skyrmion plane) is used in the simulations to stabilize the skyrmions. The simulations for helical state are performed at zero-field. The cell size is 1 nm ×1 nm×1 nm. We consider magnetic impurities with uniaxial magnetic anisotropy Himp=−KimpS2 z, where the easy-axis is perpendicular to the skyrmion 2D plane. For the weak impurity case, an impurity concentration x= 0.1% and impurity strength Kimp= 0.2A/l2(lis the cell size) were used. For the strong impurity case, an impurity concentration x= 0.5% and impurity strength Kimp= 0.6A/l2were used. The simulation results are averaged over 100 impurity distributions. The skyrmion velocity is extracted by using the emergent electric field method [39]. For each current density, the emergent electric field is also averaged over 100 time steps in order to get the skyrmion velocity. For the transverse correction, the damping value α= 0.2 is employed in the main text Figs. 3(a) and (b) for computational efficiency, as using smaller damping values results in significantly longer simulation times to obtain a reasonable correction along the transverse direction.

2023-12-12

The skyrmion crystal (SkX) and helix (HL) phases, present in typical chiral magnets, can each be considered as forms of density waves but with distinct topologies. The SkX exhibits gyrodynamics analogous to electrons under a magnetic field, while the HL state resembles topological trivial spin density waves. However, unlike the charge density waves, the theoretical analysis of the sliding motion of SkX and HL remains unclear, especially regarding the similarities and differences in sliding dynamics between these two spin density waves. In this work, we systematically explore the sliding dynamics of SkX and HL in chiral magnets in the limit of large current density. We demonstrate that the sliding dynamics of both SkX and HL can be unified within the same theoretical framework as density waves, despite their distinct microscopic orders. Furthermore, we highlight the significant role of gyrotropic sliding induced by impurity effects in the SkX state, underscoring the impact of nontrivial topology on the sliding motion of density waves. Our theoretical analysis shows that the effect of impurity pinning is much stronger in HL compared with SkX, i.e., $\chi^{SkX}/\chi^{HL}\sim \alpha^2$ ($\chi^{SkX}$, $\chi^{HL}$: susceptibility to the impurity potential, $\alpha$ ($\ll 1$) is the Gilbert damping). Moreover, the velocity correction is mostly in the transverse direction to the current in SkX. These results are further substantiated by realistic Landau-Lifshitz-Gilbert simulations.

Sliding Dynamics of Current-Driven Skyrmion Crystal and Helix in Chiral Magnets

2312.07116v2

arXiv:1308.3787v1 [cond-mat.mes-hall] 17 Aug 2013Thickness and power dependence of the spin-pumping effect in Y3Fe5O12/Pt heterostructures measured by the inverse spin Hall effect M. B. Jungfleisch,1,∗A. V. Chumak,1A. Kehlberger,2V. Lauer,1 D. H. Kim,3M. C. Onbasli,3C. A. Ross,3M. Kl¨ aui,2and B. Hillebrands1 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany 2Institute of Physics, Johannes Gutenberg-University Main z, 55099 Mainz, Germany 3Department of Materials Science and Engineering, MIT, Camb ridge, MA 02139, USA (Dated: September 17, 2018) The dependence of the spin-pumpingeffect on the yttrium iron garnet (Y 3Fe5O12, YIG) thickness detected by the inverse spin Hall effect (ISHE)has been inves tigated quantitatively. Due to the spin- pumping effect driven by the magnetization precession in the ferrimagnetic insulator Y 3Fe5O12film a spin-polarized electron current is injected into the Pt la yer. This spin current is transformed into electrical charge current by means of the ISHE. An increase o f the ISHE-voltage with increasing film thickness is observed and compared to the theoretically exp ected behavior. The effective damping parameter of the YIG/Pt samples is found to be enhanced with d ecreasing Y 3Fe5O12film thickness. The investigated samples exhibit a spin mixing conductance ofg↑↓ eff= (7.43±0.36)×1018m−2and a spin Hall angle of θISHE= 0.009±0.0008. Furthermore, the influence of nonlinear effects on the generated voltage and on the Gilbert damping parameter at hi gh excitation powers are revealed. It is shown that for small YIG film thicknesses a broadening of th e linewidth due to nonlinear effects at highexcitation powers is suppressedbecause of alack ofnon linear multi-magnon scatteringchannels. We have found that the variation of the spin-pumping efficienc y for thick YIG samples exhibiting pronounced nonlinear effects is much smaller than the nonlin ear enhancement of the damping. I. INTRODUCTION The generation and detection of spin currents have at- tracted much attention in the field of spintronics.1,2An effective method for detecting magnonic spin currents is the combination of spin pumping and the inverse spin Hall effect (ISHE). Spin pumping refers to the genera- tion of spin-polarized electron currents in a normal metal from the magnetization precession in an attached mag- netic material.3,4These spin-polarized electron currents are transformed into conventional charge currents by the ISHE, which allows for a convenient electric detection of spin-wave spin currents.5–7 After the discovery of the spin-pumping effect in fer- rimagnetic insulator (yttrium iron garnet, Y 3Fe5O12, YIG)/non-magnetic metal (platinum, Pt) heterosystems by Kajiwara et al.7, there was rapidly emerging inter- est in the investigation of these structures.6–13Since Y3Fe5O12is an insulator with a bandgap of 2.85 eV14no direct injection of a spin-polarized electron current into the Pt layer is possible. Thus, spin pumping in YIG/Pt structures can only be realized by exchange interaction between conduction electrons in the Pt layer and local- ized electrons in the YIG film. Spin pumping into the Pt layer transfers spin angular momentum from the YIG film thus reducing the mag- netization in the YIG. This angular momentum transfer results in turn in an enhancement of the Gilbert damp- ing of the magnetization precession. The magnitude of the transfer of angular momentum is independent of the ferromagnetic film thickness since spin pumping is an in- terface effect. However, with decreasing film thickness, the ratio between surface to volume increases and, thus,the interface character of the spin-pumping effect comes into play: the deprivation of spin angular momentum be- comes notable with respect to the precession ofthe entire magnetizationinthe ferromagneticlayer. Thus, theaver- age damping for the whole film increases with decreasing film thicknesses. It is predicted theoretically3and shown experimentallyinferromagneticmetal/normalmetalhet- erostructures (Ni 81Fe19/Pt) that the damping enhance- ment due to spin pumping is inversely proportional to the thickness of the ferromagnet.15,16 Since the direct injection of electrons from the insula- tor YIG into the Pt layer is not possible and spin pump- ing is an interface effect, an optimal interface quality is required in order to obtain a high spin- to charge cur- rent conversion efficiency.17,18Furthermore, Tashiro et al. have experimentally demonstrated that the spin mix- ing conductance is independent of the YIG thickness in YIG/Pt structures.11Recently, Castel et al. reported on the YIG thickness and frequency dependence of the spin-pumpingprocess.19Incontrasttoourinvestigations, they concentrate on rather thick ( >200 nm) YIG films, which are much thicker than the exchange correlation length in YIG20–22and thicker than the Pt thickness. Thus, the YIG film thickness dependence in the nanome- ter range is still not addressed till now. In this paper, we report systematic measurements of the spin- to charge-current conversion in YIG/Pt struc- tures as a function of the YIG film thickness from 20 nm to 300 nm. The Pt thickness is kept constant at 8.5 nm for all samples. We determine the effective damping as well as the ISHE-voltage as a function of YIG thickness and find that the thickness plays a key role. From these characteristics the spin mixing conductance and the spin2 FIG. 1. (Color online) (a) Schematic illustration of the ex- perimental setup. (b) Dimensions of the structured Pt layer on the YIG films. The Pt layer was patterned by means of optical lithography and ion etching. (c) Scheme of combined spin-pumping process and inverse spin Hall effect. Hall angle are estimated. The second part of this paper addresses microwave power dependent measurements of the ISHE-induced voltage UISHEand the ferromagnetic resonance linewidth for varying YIG film thicknesses. The occurrence of nonlinear magnon-magnon scattering processesonthe widening ofthe linewidth aswell astheir influence on the spin-pumping efficiency are discussed. II. SAMPLE FABRICATION AND EXPERIMENTAL DETAILS In Fig.1(a) a schematic illustration of the investigated samples is shown. Mono-crystalline Y 3Fe5O12samples of 20, 70, 130, 200 and 300 nm thickness were deposited by means of pulsed laser deposition (PLD) from a stoichio- metric target using a KrF excimer laser with a fluence of 2.6J/cm2andarepetition rateof10Hz.23In orderto en- sure epitaxial growth of the films, single crystalline sub- TABLE I. Variation of saturation magnetization MSand Gilbert damping parameter α0as a function of the YIG film thickness. Results are obtained using a VNA-FMR measure- ment technique. dYIG(nm)MS(kA/m) α0(×10−3) 20 161.7 ±0.2 2.169 ±0.069 70 176.4 ±0.1 0.489 ±0.007 130 175.1 ±0.2 0.430 ±0.015 200 176.4 ±0.1 0.162 ±0.008 300 176.5 ±0.1 0.093 ±0.007FIG. 2. Original Gilbert damping parameter α0measured by VNA-FMR technique. The increased damping at low sample thicknesses is explained by an enhanced ratio between surfa ce to volume, which results in an increased number of scatterin g centers and, thus, in an increased damping. The inset shows the saturation magnetization as a function of the YIG film thickness dYIG. The error bars are not visible in this scale. stratesofgadoliniumgalliumgarnet(Gd 3Ga5O12, GGG) in the (100) orientation were used. We achieved opti- mal deposition conditions for a substrate temperature of 650◦C±30◦C and an oxygen pressure of 6.67 ×10−3 mbar. Afterwards, each film was annealed ex-situ at 820◦C±30◦C by rapid thermal annealing for 300 s un- der a steady flow of oxygen. This improves the crystal- lographic order and reduces oxygen vacancies. We deter- mined the YIG thickness by profilometer measurements andthecrystallinequalitywascontrolledbyx-raydiffrac- tion(XRD). InordertodepositPtontothesamples,they were transferred at atmosphere leading to possible sur- face adsorbates. Therefore, the YIG film surfaces were cleaned in-situ by a low power ion etching before the Pt deposition.17We used DC sputtering under an ar- gon pressure of 1 ×10−2mbar at room temperature to deposit the Pt layers. XRR measurements yielded a Pt thickness of 8.5 nm, which is identical for every sample due to the simultaneously performed Pt deposition. The Pt layer was patterned by means of optical lithography and ion etching. In order to isolate the Pt stripes from the antenna we deposited a 300 nm thick square of SU-8 photoresist on the top. A sketch of the samples and the experimental setup is shown in Fig. 1(a), the dimensions of the structured Pt stripe are depicted in Fig. 1(b). In order to corroborate the quality of the fabricated YIG samples, we performed ferromagnetic resonance (FMR) measurements using a vector network analyzer (VNA).25Since the area deposited by Pt is small com- pared to the entire sample size, we measure the damp- ingα0of the bare YIG by VNA (this approach results in a small overestimate of α0), whereas in the spin- pumping measurement we detect the enhanced damp- ingαeffof the Pt covered YIG films. The VNA-FMR results are summarized in Tab. Iand in Fig. 2. Appar- ently, the 20 nm sample features the largest damping3 ofα20nm 0= (2.169±0.069)×10−3. With increasing film thickness α0decreasesto α300nm 0= (0.093±0.007)×10−3. There might be two reasons for the observed behavior: (1) The quality of the thinner YIG films might be worse due to the fabrication process by PLD. (2) For smaller YIG film thicknesses, the ratio between surface to vol- ume increases. Thus, the two-magnon scattering pro- cess at the interface is more pronounced for smaller film thicknesses and gives rise to additional damping.26The VNA-FMR technique yields the saturation magnetiza- tionMSfor the YIG samples (see inset in Fig. 2and Tab.I). The observed values for MSare larger than the bulkvalue,27,28butinagreementwiththevaluesreported for thin films.29The general trend of the film thickness dependence of MSis in agreement with the one reported in Ref.29,30and might be associated with a lower crystal quality after the annealing. Thespin-pumpingmeasurementsfordifferentYIGfilm thicknesses were performed in the following way. The samples were magnetized in the film plane by an exter- nal magnetic field H, and the magnetization dynamics was excited at a constant frequency of f= 6.8 GHz by an Agilent E8257D microwave source. The microwave signals with powers Pappliedof 1, 10, 20, 50, 100, 250 and 500mW wereapplied to a 600 µm wide 50Ohm-matched Cumicrostripantenna. Whiletheexternalmagneticfield was swept, the ISHE-voltage UISHEwas recorded at the edges of the Pt stripe using a lock-in technique with an amplitudemodulationatafrequencyof500Hz, aswellas the absorbed microwave power Pabs. All measurements were performed at room temperature. III. THEORETICAL BACKGROUND The equations describing the ferromagnetic resonance, the spin pumping and the inverse spin Hall effect are provided in the following and used in the experimental part of this paper. A. Ferromagnetic resonance In equilibrium the magnetization Min aferromagnetic material is aligned along the bias magnetic field H. Ap- plying an alternating microwave magnetic field h∼per- pendicularly to the external field Hresults in a torque onMand causes the magnetic moments in the sample to precess (see also Fig. 1(a)). In ferromagnetic resonance (FMR)themagneticfield Handtheprecessionfrequency ffulfill the Kittel equation31 f=µ0γ 2π/radicalbig HFMR(HFMR+MS), (1) whereµ0is the vacuum permeability, γis the gyromag- netic ratio, HFMRis the ferromagnetic resonance field andMSis the saturation magnetization (experimentallyobtained values of MSfor our samples can be found in Tab.I). The FMR linewidth ∆ H(full width at half maximum) is related to the Gilbert damping parameter αas16,18,27 µ0∆H= 4πfα/γ. (2) B. Spin pumping By attaching a thin Pt layer to a ferromagnet, the resonance linewidth is enhanced,3which accounts for an injection of a spin current from the ferromagnet into the normal metal due to the spin-pumping effect (see illus- tration in Fig. 1(c)). In this process the magnetization precession loses spin angular momentum, which gives rise to additional damping and, thus, to an enhanced linewidth. The effective Gilbert damping parameter αeff for the YIG/Pt film is described as16 αeff=α0+∆α=α0+gµB 4πMSdYIGg↑↓ eff,(3) whereα0is the intrinsic damping of the bare YIG film, gis the g-factor, µBis the Bohr magneton, dYIGis the YIG film thickness and g↑↓ effis the real part of the ef- fective spin mixing conductance. The effective Gilbert damping parameter αeffis inversely proportional to the YIG film thickness dYIG: with decreasing YIG thickness the linewidth and, thus, the effective damping parameter increases. When the system is resonantlydriven in the FMR con- dition, a spin-polarized electron current is injected from the magnetic material (YIG) into the normal metal (Pt). Inaphenomenologicalspin-pumpingmodel, theDCcom- ponent of the spin-current density jsat the interface, in- jected in y-direction into the Pt layer (Fig. 1(c)), can be described as15,16,32 js=f/integraldisplay1/f 0¯h 4πg↑↓ eff1 M2 S/braceleftBig M(t)×dM(t) dt/bracerightBig zdt,(4) whereM(t) is the magnetization. {M(t)×dM(t) dt}zis the z-component of {M(t)×dM(t) dt}, which is directed along the equilibrium axis of the magnetization (see Fig. 1(c)). Due to spin relaxation in the normal metal (Pt) the injected spin current jsdecays along the Pt thickness (y-direction in Fig. 1(c)) as15,16 js(y) =sinhdPt−y λ sinhdPt λj0 s, (5) whereλis the spin-diffusion length in the Pt layer. From Eq. (4) one can deduce the spin-current density at the interface ( y= 0)15 j0 s=g↑↓ effγ2(µ0h∼)2¯h(µ0MSγ+/radicalbig (µ0MSγ)2+16(πf)2) 8πα2 eff((µ0MSγ)2+16(πf)2). (6)4 FIG. 3. (Color online) ISHE-induced voltage UISHEas a func- tion of the magnetic field Hfor different YIG film thicknesses dYIG. Applied microwave power Papplied= 10 mW, ISHE- voltage for the 20 nm thick sample is multiplied by a factor of 5. Sincej0 sis inverselyproportionalto α2 effandαeffdepends inversely on dYIG(Eq. (3)), the spin-current density at the interface j0 sincreases with increasing YIG film thick- nessdYIG. C. Inverse spin Hall effect The Pt layer acts as a spin-current detector and trans- forms the spin-polarized electron current injected due to the spin-pumping effect into an electrical charge current by means of the ISHE (see Fig. 1(c)) as6,7,12,15,16 jc=θISHE2e ¯hjs×σ, (7) whereθISHE,e,σdenote the spin Hall angle, the elec- tron’s elementary charge and the spin-polarization vec- tor, respectively. Averaging the charge-current density over the Pt thickness and taking into account Eqs. ( 4) – (7) yields ¯jc=1 dPt/integraldisplaydPt 0jc(y)dy=θISHEλ dPt2e ¯htanh/parenleftbigdPt 2λ/parenrightbig j0 s.(8) Taking into account Eqs. ( 3), (6) and (8) we calcu- late the theoretically expected behavior of IISHE=A¯jc, whereAis the cross section of the Pt layer. Ohm’s law connects the ISHE-voltage UISHEwith the ISHE-current IISHEviaUISHE=IISHE·R, whereRis the electric resis- tance of the Pt layer. Rvaries between 1450 Ω and 1850 Ω for the different samples. IV. YIG FILM THICKNESS DEPENDENCE OF THE SPIN-PUMPING EFFECT DETECTED BY THE ISHE In Fig.3the magnetic field dependence of the gener- ated ISHE-voltage UISHEas a function of the YIG filmthickness is shown. Clearly, the maximal voltage UISHE at the resonance field HFMRand the FMR linewidth ∆ H vary with the YIG film thickness. The general trend shows, that the thinner the sample the smaller is the magnitude of the observed voltage UISHE. At the same time the FMR linewidth increases with decreasing YIG film thickness. In the following the ISHE-voltage generated by spin pumping is investigated as a function of the YIG film thickness. For these investigations we have chosen a rather small exciting microwave power of 1 mW. Thus, nonlinear effects like the FMR linewidth broadening due to nonlinear multi-magnon processes can be excluded (such processes will be discussed in Sec. V). Sec.IVA covers the YIG thickness dependent variation of the en- hanced damping parameter αeff. From these measure- ments the spin mixing conductance g↑↓ effis deduced. In Sec.IVBwe focus on the maximal ISHE-voltage driven by spin pumping as a function of the YIG film thickness. Finally, the spin Hall angle θISHEis determined. A. YIG film parameters as a function of the YIG film thickness As described in Sec. IIIB, the damping parameter is enhanced when a Pt layeris deposited onto the YIG film. ThisenhancementisinvestigatedasafunctionoftheYIG film thickness: the effective Gilbert damping parameter αeff(see Eq. ( 3)) is obtained from a Lorentzian fit to the experimental data depicted in Fig. 3and Eq. ( 2). The resultisshowninFig. 4. With decreasingYIG film thick- ness the linewidth and, thus, the effective damping αeff increases. This behavior is theoretically expected: ac- cording to Eq. ( 3)αeffis inversely proportional to dYIG. Since the Pt film is grown onto all YIG samples simulta- FIG. 4. (Color online) Enhanced damping parameter αeff of the YIG/Pt samples obtained by spin-pumping measure- ments. The red solid curve shows a fit to Eq. (3) taking the FMR measured values for MSand a constant value for g↑↓ eff into account. Papplied= 1 mW. The error bars for the mea- surement points at higher sample thicknesses are not visibl e in this scale.5 FIG. 5. (Color online) (a) ISHE-voltage UISHEas a function of the YIG film thickness dYIG. The black line is a linear in- terpolation as a guide tothe eye. (b) Corresponding thickne ss dependent charge current IISHE. The red curve shows a fit to Eqs. (6), (7), (8) with theparameters g↑↓ eff=(7.43±0.36)×1018 m−2andθISHE= 0.009±0.0008. The applied microwave power used is Papplied= 1 mW. neously, the spin mixing conductance g↑↓ effat the interface is considered to be constant for all samples.11Assum- ingg↑↓ effas constant and taking the saturation magne- tizationMSobtained by VNA-FMR measurements (see Fig.2and Tab. I) into account, a fit to Eq. ( 3) yields g↑↓ eff= (7.43±0.36)×1018m−2. The fit is depicted as a red solid line in Fig. 4. B. YIG thickness dependence of the ISHE-voltage driven by spin pumping Fig.5(a) shows the maximum voltage UISHEat the resonance field HFMRas a function of the YIG film thickness. UISHEincreases up to a YIG film thickness of around 200 nm when it starts to saturate (in the case of an applied microwave power of P applied= 1 mW). The corresponding charge current IISHEis shown in Fig.5(b). The observed thickness dependent behavior is in agreement with the one reported for Ni 81Fe19/Pt16 and for Y 3Fe5O12/Pt.11With increasing YIG film thick- ness the generated ISHE-current increases and tends to saturate at thicknesses near 200 nm (Fig. 5(b)). Accord- ing to Eq. ( 3), (6) and (8) it isIISHE∝j0 s∝1/α2 eff∝ (α0+c/dYIG)−2, wherecis a constant. Therefore, theISHE-current IISHEincreases with increasing YIG film thickness dYIGand goes into saturation at a certain YIG thickness. From Eqs. ( 3), (6) and (8) we determine the expected behavior of IISHE=A¯jcand compare it with our ex- perimental data. In order to do so, the measured values forMS(see Tab. I), the original damping parameter α0 determined by VNA-FMR measurements at 1 mW (see Tab.I) and the enhanced damping parameter αeffob- tained by spin-pumping measurements at a microwave power of 1 mW (see Fig. 4) are used. The Pt layer thick- ness isdPt= 8.5 nm and the microwave magnetic field is determined to be h∼= 3.2 A/m for an applied mi- crowavepower of 1 mW using an analytical expression.24 The spin-diffusion length in Pt is taken from literature as λ= 10 nm33,36and the damping parameter is assumed to be constant as α0= 6.68×10−4, which is the aver- age of the measured values of α0. The fit is shown as a red solid line in Fig. 5(b). We find a spin Hall angle of θISHE= 0.009±0.0008,which is in agreementwith litera- ture values varying in a range of 0.0037- 0.08.33–35Using FIG. 6. (Color online) (a) YIG thickness dependence of the ISHE-voltage driven by spin pumping for microwave powers in the range between 1 and 500 mW. The general thickness dependent behavior is independent of the applied microwave power. The error bars for the measurement at lower mi- crowave powers are not visible in this scale. (b) Deviation of the ISHE-voltage from the linear behavior with respect to the measured voltage U500mW ISHE. The inset shows experimental data for a YIG film thickness dYIG= 20 nm and the theoret- ically expected curve. The error bars of the 20 nm and the 70 nm samples are not visible in this scale.6 the fit we estimate the saturation value of the generated current. Although we observe a transition to saturation at sample thicknesses of 200 – 300 nm, we find that ac- cording to our fit, 90% of the estimated saturation level of 5 nA is reached at a sample thickness of 1.2 µm. V. INFLUENCE OF NONLINEAR EFFECTS ON THE SPIN-PUMPING PROCESS FOR VARYING YIG FILM THICKNESSES In order to investigate nonlinear effects on the spin- pumping effect for varying YIG film thicknesses, we per- formed microwavepower dependent measurements of the ISHE-voltage UISHEas function of the film thickness dYIG. For higher microwavepowersin the rangeof 1 mW to 500 mW we observe the same thickness-dependent behavior of the ISHE-voltage as in the linear case (Papplied= 1 mW, discussed in Sec. IVB): Near 200 nm UISHEstarts to saturate independently of the applied mi- crowave power, as it is shown in Fig. 6(a). Furthermore, it is clearly visible from Fig. 6(a) that for a constant film thickness the spin pumping driven ISHE-voltage in- creases with increasing applied microwave power. At high microwave powers the voltage does not grow lin- early and saturates. Fig. 6(b) shows the deviation of the ISHE-voltage ∆ UISHEfrom the linear behavior with respect to the measured value of U500mW ISHEat the excita- tion power Papplied= 500 mW. In order to obtain the relation between UISHEandPappliedfor each YIG film thickness dYIGthe low power regime up to 20 mW is fitted by a linear curve and extrapolated to 500 mW. Theinsetin Fig. 6(b) showsthe correspondingviewgraph for the case of the 20 nm thick sample. As it is visible from Fig. 6(b), the deviation from the linear behavior is drastically enhanced for larger YIG thicknesses. For the thin 20 nm and 70 nm samples we observe an almost linear behavior between UISHEandPappliedover the en- tire microwavepower range, whereas for the thicker sam- ples the estimated linear behavior and the observed non- linear behavior differ approximately by a factor of 2.5 (Fig.6(b)). We observe an increase of the ISHE-voltage as well as an broadening of the FMR linewidth with in- creasing microwave power. In Fig. 7(a) the normalized ISHE-voltage UISHEas function of the external magnetic fieldHis shown for different microwave powers Papplied in the range of 1 mW to 500 mW (YIG film thickness dYIG= 300 nm). The linewidth tends to be asymmet- ric at higher microwave powers. The shoulder at lower magnetic field is widened in comparison to the shoulder at higher fields. The reason for this asymmetry might be due to the formation of a foldover effect,37,38due to nonlinear damping or a nonlinear frequency shift.39,40 The results of the damping parameter αeffobtained by microwave power dependent spin-pumping measure- ments are depicted in Fig. 7(b). It can be seen, that with increasing excitation power the Gilbert damping for thicker YIG films is drastically increased. To present FIG. 7. (Color online) (a) Illustration of the linewidth bro ad- ening at higher excitation powers. The normalized ISHE- voltage spectra are shown as a function of the magnetic fieldHfor different excitation powers. Sample thickness: 300 nm. (b) Power dependent measurement of the damp- ing parameter αefffor different YIG film thicknesses dYIGob- tained by a Lorentzian fit to the ISHE-voltage signal. The error bars are omitted in order to provide a better readabil- ity of the viewgraph. (c) Nonlinear damping enhancement (α500mW eff−α1mW eff)/α1mW effas a function of the YIG film thick- nessdYIG. Due to a reduced number of scattering channels to other spin-wave modes for film thicknesses below 70 nm, the damping is only enhanced for thicker YIG films with increas- ing applied microwave powers. The error bars of the 200 nm and the 300 nm samples are not visible on this scale. this result more clearly the nonlinear damping enhance- ment (α500mW eff−α1mW eff)/α1mW effis shown in Fig. 7(c). The dampingparameterat asamplethicknessof20nm α20nm eff7 FIG. 8. (Color online) Dispersion relations calculated for each sample thickness taking into account the measured valu es of the saturation magnetization MS(see Tab. I). Backward volume magnetostatic spin-wave mode s as well as magnetostatic surface spin-wave modes (in red) and the first perpendicular standin g thickness spin-wave modes are depicted (in black and gray) . (a)–(e) show the dispersion relations for the investigated sample thicknesses of 20 nm – 300 nm. is almost unaffected by a nonlinear broadening at high microwave powers. With increasing film thickness the original damping α1mW effatPapplied= 1 mW increases by a factor of around 3 at Papplied= 500 mW. This factor is very close to the value of the deviation of the ISHE- voltage from the linear behavior (Fig. 6(b)). This behavior can be attributed to the enhanced prob- ability of nonlinear multi-magnon processes at larger sample thicknesses: In order to understand this, a fun- damental understanding of the restrictions for multi- magnon scattering processes can be derived from the en- ergy and momentum conservation laws: N/summationdisplay i¯hωi=M/summationdisplay j¯hωjandN/summationdisplay i¯hki=M/summationdisplay j¯hkj,(9) where the left/right sum of the equations runs over the initial/final magnons with indices i/j which exist be- fore/after the scattering process, respectively.41–43The most probable scattering mechanism in our case is the four-magnon scattering process with N= 2 and M= 2.43In Eq. (9) the wavevector ki/jand the frequency ωi/j are connected by the dispersion relation 2 πfi/j(ki/j) = ωi/j(ki/j). The calculated dispersion relations are shown in Fig.8(backward volume magnetostatic spin-wave modes with a propagation angle /negationslash(H,k) = 0◦as well as magnetostatic surface spin-wave modes /negationslash(H,k) = 90◦).44For this purpose, the measured values of MS (Tab.I) for each sample are used. In the case of the 20 nm sample thickness, the first perpendicular standing spin-wave mode (thickness mode) lies above 40 GHz, thesecond above 120 GHz. Thus, the nonlinear scattering probability obeying the energy- and momentum conser- vation is largely reduced. This means magnons cannot find a proper scattering partner and, thus, multi-magnon processes are prohibited or at least largely suppressed. With increasing film thickness the number of standing spin-wavemodesincreasesand, thus, thescatteringprob- ability grows. As a result, the scattering of spin waves from the initially excited uniform precession (FMR) to other modes is allowed and the relaxation of the original FMR mode is enhanced. Thus, the damping increases and we observe a broadening of the linewidth, which is equivalent to an enhanced Gilbert damping parameter αeffat higher YIG film thicknesses (see Fig. 7). In orderto investigatehowthe spin-pumping efficiency is affected by the applied microwave power, we measure simultaneously the generated ISHE-voltage UISHEand the transmitted ( Ptrans), as well as the reflected ( Prefl) microwave power, which enables us to determine the absorbed microwave power Pabs=Papplied−(Ptrans+ Prefl).17Since the 300 nm sample exhibits a strong non- linearity(largedeviationfromthelinearbehavior(Fig. 6) and large nonlinear linewidth enhancement (Fig. 7)), we analyze this sample thickness. In Fig. 9the normalized absorbed microwave power Pnorm=Pabs/PPapplied=1mW abs and the normalized ISHE-voltage in resonance Unorm= UISHE/UPapplied=1mW ISHEare shown as a function of the ap- plied power Papplied. Both curves tend to saturate at high microwave powers above 100 mW. The absorbed microwave power increases by a factor of 110 for applied microwave powers in the range between 1 and 500 mW,8 FIG. 9. (Color online) Normalized absorbed power Pnorm= Pabs/PPapplied=1mW abs (black squares) and normalized ISHE- voltageUnorm=UISHE/UPapplied=1mW ISHE (red dots) for varying microwave powers Papplied. The inset illustrates the indepen- dence of the spin-pumping efficiency UISHE/PabsonPapplied. YIG thickness illustrated: 300 nm. Error bars of the low power measurements are not visible in this scale. whereas the generated voltage increases by a factor of 80. The spin-pumping efficiency UISHE/Pabs(see inset in Fig.9) varies within a range of 30% for the differ- ent microwave powers Pappliedwithout clear trend. Since the 300 nm thick film shows a nonlinear deviation of the ISHE-voltageby afactorof2.3 (Fig. 6(b)) and the damp- ing is enhanced by a factor of 3 in the same range of Papplied(Fig.7(c)), we conclude that the spin-pumping process is only weakly dependent on the magnitude of the applied microwave power (see inset in Fig. 9). In our previous studies reported in Ref.12,13we show that sec- ondary magnons generated in a process of multi-magnon scattering contribute to the spin-pumping process and, thus, the spin-pumping efficiency does not depend on the applied microwave power. VI. SUMMARY The Y 3Fe5O12thickness dependence of the spin- pumping effect detected by the ISHE has been inves- tigated quantitatively. It is shown that the effective Gilbert damping parameter of the the YIG/Pt sam- ples is enhanced for smaller YIG film thicknesses, which is attributed to an increase of the ratio between sur- face to volume and, thus, to the interface character of the spin-pumping effect. We observe a theoretically expected increase of the ISHE-voltage with increasing YIG film thickness tending to saturate above thick- nesses near 200 – 300 nm. The spin mixing conductance g↑↓ eff= (7.43±0.36)×1018m−2as well as the spin Hall an-gleθISHE= 0.009±0.0008 are calculated and are found to be in agreement with values reported in the literature for our materials. The microwave power dependent measurements reveal the occurrence of nonlinear effects for the different YIG film thicknesses: for low powers, the induced voltage grows linearly with the power. At high powers, we ob- serve a saturation of the ISHE-voltage UISHEand a de- viation by a factor of 2.5 from the linear behavior. The microwave power dependent investigations of the Gilbert damping parameter by spin pumping show an enhance- ment by a factor of 3 at high sample thicknesses due to nonlinear effects. This enhancement of the damping is due to nonlinear scattering processes representing an additional damping channel which absorbs energy from the originally excited FMR. We have shown that the smaller the sample thickness, the less dense is the spin- wave spectrum and, thus, the less nonlinear scattering channels exist. Hence, the smallest investigated sample thicknesses (20 and 70 nm) exhibit a small deviation of the ISHE-voltage from the linear behavior and a largely reduced enhancement of the damping parameter at high excitation powers. Furthermore, we have found that the variation of the spin-pumping efficiencies for thick YIG samples which show strongly nonlinear effects is much smaller than the nonlinear enhancement of the damping. This is attributed to secondary magnons generated in a processofmulti-magnonscatteringthatcontributetothe spin pumping. It is shown, that even for thick samples (300 nm) the spin-pumping efficiency is only weakly de- pendent on the applied microwave power and varies only within a range of 30% for the different microwave powers without a clear trend. Our findings provide a guideline to design and create efficient magnon- to charge current converters. Further- more, the results are also substantial for the reversed effects: the excitation of spin waves in thin YIG/Pt bi- layers by the direct spin Hall effect and the spin-transfer torque effect.45 VII. ACKNOWLEDGMENTS We thank G.E.W. Bauer and V.I. Vasyuchka for valu- able discussions. Financial support by the Deutsche Forschungsgemeinschaft within the project CH 1037/1- 1 are gratefully acknowledged. AK would like to thank the Graduate School of Excellence Materials Science in Mainz (MAINZ) GSC 266. 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2013-08-17

The dependence of the spin-pumping effect on the yttrium iron garnet (Y3Fe5O12, YIG) thickness detected by the inverse spin Hall effect (ISHE) has been investigated quantitatively. Due to the spin-pumping effect driven by the magnetization precession in the ferrimagnetic insulator YIG film a spin-polarized electron current is injected into the Pt layer. This spin current is transformed into electrical charge current by means of the ISHE. An increase of the ISHE-voltage with increasing film thickness is observed and compared to the theoretically expected behavior. The effective damping parameter of the YIG/Pt samples is found to be enhanced with decreasing YIG film thickness. The investigated samples exhibit a spin mixing conductance of g=(7.43 \pm 0.36) \times 10^{18} m^{-2} and a spin Hall angle of theta_{ISHE} = 0.009 \pm 0.0008. Furthermore, the influence of nonlinear effects on the generated voltage and on the Gilbert damping parameter at high excitation powers are revealed. It is shown that for small YIG film thicknesses a broadening of the linewidth due to nonlinear effects at high excitation powers is suppressed because of a lack of nonlinear multi-magnon scattering channels. We have found that the variation of the spin-pumping efficiency for thick YIG samples exhibiting pronounced nonlinear effects is much smaller than the nonlinear enhancement of the damping.

Thickness and power dependence of the spin-pumping effect in Y3Fe5O12/Pt heterostructures measured by the inverse spin Hall effect

1308.3787v1

arXiv:1301.2114v1 [cond-mat.other] 10 Jan 2013APS/123-QED First-principles calculation of the Gilbert damping param eter via the linear response formalism with application to magnetic transition-metals and alloys S. Mankovsky1, D. K¨ odderitzsch1, G. Woltersdorf2, and H. Ebert1 1University of Munich, Department of Chemistry, Butenandtstrasse 5-13, D-81377 Munich, Germany and 2Department of Physics, Universit¨ at Regensburg, 93040 Reg ensburg, Germany (ΩDated: September 9, 2018) A method for the calculations of the Gilbert damping paramet erαis presented, which based on the linear response formalism, has been implemented within the fully relativistic Korringa-Kohn- Rostoker band structure method in combination with the cohe rent potential approximation alloy theory. To account for thermal displacements of atoms as a sc attering mechanism, an alloy-analogy model is introduced. This allows the determination of αfor various types of materials, such as elemental magnetic systems and ordered magnetic compounds at finite temperature, as well as for disordered magnetic alloys at T= 0 K and above. The effects of spin-orbit coupling, chemical a nd temperature induced structural disorder are analyzed. Cal culations have been performed for the 3dtransition-metals bcc Fe, hcp Co, and fcc Ni, their binary al loys bcc Fe 1−xCox, fcc Ni 1−xFex, fcc Ni 1−xCoxand bcc Fe 1−xVx, and for 5 dimpurities in transition-metal alloys. All results are in satisfying agreement with experiment. PACS numbers: 72.25.Rb 71.20.Be 71.70.Ej 75.78.-n I. INTRODUCTION Duringthe lastdecadesdynamicalmagneticproperties have attracted a lot of interest due to their importance in the development of new devices for spintronics, in par- ticular, concerning their miniaturization and fast time scale applications. A distinctive property of such devices is the magnetization relaxation rate characterizing the time scale on which a system being deviated from the equilibrium returns to it, or how fast the device can be switched from one state to another. In the case of dy- namics of a uniform magnetization /vecMthis property is associated with the Gilbert damping parameter ˜G(M) used first in the phenomenologicalLandau-Lifshitz (LL)1 and Landau-Lifshitz-Gilbert (LLG) theory2describing the magnetization dynamics processes by means of the equation: 1 γdM dτ=−M×Heff+M×/bracketleftBigg˜G(M) γ2M2sdM dτ/bracketrightBigg ,(1) whereMsis the saturation magnetization, γthe gy- romagnetic ratio and Heff=−∂MF[M(r)] being the effective magnetic field. Sometimes it is more conve- nient to use a dimensionless Gilbert damping parame- terαgiven byα=˜G/(γMs) (see, e.g.3–5). Safonov has generalized the Landau-Lifshitz equation by intro- ducing a tensorial form for the Gilbert damping parame- ter with the diagonal terms characterising magnetization dissipation6. Beingintroducedasaphenomenologicalpa- rameter, the Gilbert damping is normally deduced from experiment. In particular, it can be evaluated from the resonant line width in ferromagnetic-resonance (FMR) experiments. The difficulty of these measurements con- sists in the problem that there exist several different sources for the broadening of the line width, which havebeen discussed extensively in the literature7–13. The line width that is observed in ferromagnetic resonance spec- tra is usually caused by intrinsic and extrinsic relax- ation effects. The extrinsic contributions are a conse- quence of spatially fluctuating magnetic properties due to sample imperfections. Short range fluctuations lead to two magnon scattering while long range fluctuations lead to an inhomogeneous line broadening due a super- position of local resonances14. In order to separate the intrinsic Gilbert damping from the extrinsic effects it is necessary to measure the frequency and angular depen- dence of the ferromagnetic resonance line width, e. g. two magnon scattering can be avoided when the mag- netization is aligned along the film normal11(perpen- dicular configuration). Usually one finds a linear fre- quency dependence with a zero frequency offset and one can write ∆ H(ω) =αω γ+∆H(0). When such measure- ments are performed over a wide frequency range the slope of ∆ Has a function of frequency can be used to extract the intrinsic Gilbert damping constant. In metallic ferromagnets Gilbert damping is mostly caused by electron magnon scattering. In addition Gilbert-like damping can be caused by eddy currents. The magni- tude of the eddy current damping is proportional to d2, wheredis the sample thickness10. In sufficiently thin magnetic films ( d≤10 nm) the eddy current damping can be neglected10. However, for very thin films relax- ation mechanisms that occur at the interfaces can also increase and even dominate the damping. Such effects are spin pumping15,16and the modified electronic struc- ture at the interfaces. In the present work spin pumping and the modified interface electronic structure are not considered and we assume that bulk-like Gilbert damp- ing dominates. Much understanding of dynamical magnetic properties could in principle be obtained from the simulation of2 these processes utilizing time-dependent first-principles electronic structure calculations, that in turn would pave the way to developing and optimizing new materials for spintronic devices. In spite of the progress in the de- velopment of time-dependent density functional theory (TD-DFT) during the last decades17that allows to study variousdynamicalprocessesin atomsand molecules from firstprinciples, applicationstosolidsarerare. Thisisdue to a lack of universally applicable approximations to the exchange-correlation kernel of TD-DFT for solids. Thus, atthemoment, atractableapproachconsistsintheuseof the classical LLG equations, and employing parameters calculated within a microscopic approach. Note however that this approach can fail dealing with ultrafast mag- netization dynamics, which is discussed, for instance, in Refs. [18 and 19], but is not considered in the present work. Most of the investigations on the magnetization dissi- pation have been carried out within model studies. Here one has to mention, in particular, the so-called s-dor p-dexchange model20–23based on a separate considera- tion of the localized ’magnetic’ d-electrons and delocal- izeds- andp-electrons mediating the exchange interac- tions between localized magnetic moments and responsi- ble for the magnetization dissipation in the system. As was pointed out by Skadsem et al.24, the dissipation pro- cess in this case can be treated as an energy pumping out of thed-electron subsystem into the s-electron bath followed by its dissipation via spin-flip scattering pro- cesses. This model gave a rather transparent qualita- tive picture for the magnetization relaxation in diluted alloys, e.g. magnetic semiconductors such as GaMnAs. However, it fails to give quantitative agreement with ex- periment in the case of itinerant metallic systems (e.g. 3d-metalalloys),wherethe d-statesareratherdelocalised and strongly hybridized with the sp-electrons. As a con- sequence the treatment of allvalence electrons on the same footing is needed, which leads to the requirement of first-principles calculations of the Gilbert damping go- ing beyond a model-based evaluation. Various such calculations of the Gilbert damping pa- rameter are already present in the literature. They usu- ally assume a certain dissipation mechanism, like Kam- bersky’sbreathingFermisurface(BFS)25,26, ormoregen- eral torque-correlation models (TCM)3,27. These models include explicitly the spin-orbit coupling (SOC), high- lighting its key role in the magnetization dissipation pro- cesses. However, the latter methods used for electronic structure calculations cannot take explicitly into account disorder in the system that in turn is responsible for the aforementioned spin-flip scattering process. Therefore, this has to be simulated by using external parameters characterizing the finite lifetime of the electronic states. ThisweakpointwasrecentlyaddressedbyBrataas et al.4 who described the Gilbert damping by means of scatter- ing theory. This development supplied the formal basis for the first parameter-free investigations on disordered alloys for which the dominant scattering mechanism ispotential scattering caused by chemical disorder5. Theoretical investigations of the magnetization dissi- pation by means of first-principles calculations of the Gilbert damping parameter already brought much un- derstanding of the physical mechanisms responsible for this effect. First of all, key roles are played by two ef- fects: the SOC of the atomic species contained in the system and scattering on various imperfections, either impurities or structural defects, phonons, etc. Account- ing for the crucial role of scattering processes respon- sible for the energy dissipation, different types of scat- tering phenomena have to be considered. One can dis- tinguish between the ordered-compound or pure-element systems for which electron-phonon scattering is a very important mechanism for relaxation, and disordered al- loys with dominating scattering processes resulting from randomly distributed atoms of different types. In the first case, the Gilbert damping behavior is rather differ- ent at low and high temperatures. At high temperature atomic displacements create random potentials leading toSOC-inducedspin-flipscattering. At lowtemperature, where the magnetization dissipation is well described via the BFS (Breathing Fermi-surface) mechanism25,26, the spin-conserving electron-phonon scattering is required to bring the electronic subsystem to the equilibrium at ev- ery step of the magnetization rotation, i.e. to reoccupy the modified electronic states. In this contribution we describe a formalism for the calculation of the Gilbert damping equivalent to that of Brataas et al.4, however, based on the linear re- sponce theory28as implemented in fully relativistic mul- tiple scattering based Korringa-Kohn-Rostoker (KKR) formalism. It will be demonstrated that this allows to treat elegantly and efficiently the temperature depen- dence ofαin pure crystals as well as disordered alloys. II. THEORETICAL APPROACH To have direct access to real materials and to obtain a deeper understanding of the origin of the properties observed experimentally, the phenomenological Gilbert damping parameter has to be treated on a microscopic level. This implies to deal with the electrons responsible forthe energydissipation in the magnetic dynamicalpro- cesses. Thus, one equates the corresponding expressions for the dissipation rate obtained in the phenomenologi- cal and microscopic approaches ˙Emag=˙Edis. Although a temporal variation of the magnetization is a required condition for the energy dissipation to occur, the Gilbert damping parameter is defined in the limit ω→0 (see e.g., Ref. [24]) and therefore can be calculated within the adiabatic approximation. In the phenomenological LLG theory the time depen- dent magnetization M(t) is described by Eq. (1). Ac- cordingly, the time derivative of the magnetic energy is3 given by: ˙Emag=Heff·dM dτ=1 γ2(˙ˆm)T[˜G(M)˙ˆm](2) whereˆm=M/Msdenotes the normalized magnetiza- tion. To represent the Gilbert damping parameter in terms of a microscopic theory, following Brataas et al.4, the energy dissipation is associated with the electronic subsystem. The dissipation rate upon the motion of the magnetization ˙Edis=/angbracketleftBig dˆH dτ/angbracketrightBig , is determined by the under- lying Hamiltonian ˆH(τ). Assuming a small deviation of the magnetic moment from the equilibrium the normal- ized magnetization ˆm(τ) can be expanded around the equilibrium magnetization ˆm0 ˆm(τ) =ˆm0+u(τ), (3) resulting in the expression for the linearized time depen- dent Hamiltonian for the system brought out of equilib- rium: ˆH=ˆH0(ˆm0)+/summationdisplay µuµ∂ ∂uµˆH(ˆm0).(4) Due tothe smalldeviation from the equilibrium, ˙Ediscan be obtained within the linearresponseformalism, leading to the expression4: ˙Edis=−π/planckover2pi1/summationdisplay ij/summationdisplay µν˙uµ˙uν/angbracketleftBigg ψi/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH ∂uµ/vextendsingle/vextendsingle/vextendsingle/vextendsingleψj/angbracketrightBigg/angbracketleftBigg ψj/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH ∂uν/vextendsingle/vextendsingle/vextendsingle/vextendsingleψi/angbracketrightBigg ×δ(EF−Ei)δ(EF−Ej),(5) whereEFis the Fermi energy and the sums run over all eigenstates of the system. As Eq. (5) characterizes the rate of the energy dissipation upon transition of the system from the tilted state to the equilibrium, it can be identified with the corresponding phenomenological quantity in Eq. (2), ˙Emag=˙Edis. This leads to an ex- plicit expression for the Gilbert damping tensor ˜Gor equivalently for the damping parameter α=˜G/(γMs) (Ref. [4]): αµν=−/planckover2pi1γ πMs/summationdisplay ij/summationdisplay µν/angbracketleftBigg ψi/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH ∂uµ/vextendsingle/vextendsingle/vextendsingle/vextendsingleψj/angbracketrightBigg/angbracketleftBigg ψj/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH ∂uν/vextendsingle/vextendsingle/vextendsingle/vextendsingleψi/angbracketrightBigg ×δ(EF−Ei)δ(EF−Ej),(6) where the summation is running over all states at the Fermi surface EF. In full analogy to the problem of electric conductivity29, the sum over eigenstates |ψi/angbracketrightmay be expressed in terms of the retarded single-particle Green’s function Im G+(EF) =−π/summationtext i|ψi/angbracketright/angbracketleftψi|δ(EF−Ei). This leads for the parameter αto a Kubo-Greenwood-like equation: αµν=−/planckover2pi1γ πMsTrace /angbracketleftBigg ∂ˆH ∂uµImG+(EF)∂ˆH ∂uνImG+(EF)/angbracketrightBigg c(7)with/angbracketleft.../angbracketrightcindicating a configurational average in case of a disordered system. The most reliable way to account for spin-orbit cou- pling as the source of Gilbert damping is to evaluate Eq. (7) using a fully relativistic Hamiltonian within the framework of local spin density formalism (LSDA)30: ˆH=cα·p+βmc2+V(r)+βσ·ˆmB(r).(8) Hereαiandβare the standard Dirac matrices, σde- notes the vector of relativistic Pauli matrices, and pis the relativistic momentum operator31. The functions V(r) andB=σ·ˆmB(r) are the spin-averaged and spin-dependent parts, respectively, of the LSDA poten- tial. The spin density ms(r) as well as the effective ex- change field B(r) are assummed to be collinear within the unit cell and aligned along the z-direction in the equilibrium (i. e. ms,0(r) =ms(r)ˆm0=ms(r)ezand B0(r) =B(r)ˆm0=B(r)ez). Tilting of the magnetiza- tion direction by the angle θaccording to Eq. (3), i.e. ms(r) =ms(r)ˆm=ms(r)(sinθcosφ,sinθsinφ,cosθ) andB(r) =B(r)ˆmleads to a perturbation term in the Hamiltonian ∆V(r) =βσ·(ˆm−ˆm0)B(r) =βσ·uB(r),(9) with (see Eq. (4)) ∂ ∂uµˆH(ˆm0) =βσµB(r). (10) The Green’s function G+in Eq. (7) can be obtained in a very efficient way by using the spin-polarized relativis- tic version of multiple scattering theory30that allows us to treat magnetic solids: G+(r,r′,E) =/summationdisplay ΛΛ′Zn Λ(r,E)τnm ΛΛ′(E)Zm× Λ′(r′,E) −δnm/summationdisplay Λ/bracketleftbig Zn Λ(r,E)Jn× Λ′(r′,E)Θ(r′ n−rn) +Jn Λ(r,E)Zn× Λ′(r′,E)Θ(rn−r′ n)/bracketrightbig .(11) Herer,r′refer to site nandm, respectively, where Zn Λ(r,E) =ZΛ(rn,E) =ZΛ(r−Rn,E) is a function centered at site Rn. The four-component wave functions Zn Λ(r,E) (Jn Λ(r,E)) are regular (irregular) solutions to the single-site Dirac equation labeled by the combined quantum numbers Λ (Λ = ( κ,µ)), withκandµbeing the spin-orbit and magnetic quantum numbers31. The superscript ×indicates the left hand side solution of the Dirac equation. τnm ΛΛ′(E) is the so-called scattering path operator that transfers an electronic wave coming in at siteminto a wave going out from site nwith all possible intermediate scattering events accounted for. Using matrix notation with respect to Λ, this leads to the following expression for the damping parameter: αµµ=g πµtot/summationdisplay nTrace/angbracketleftbig T0µ˜τ0nTnµ˜τn0/angbracketrightbig c(12) with the g-factor 2(1 + µorb/µspin) in terms of the spin and orbital moments, µspinandµorb, respectively, the4 total magnetic moment µtot=µspin+µorb, ˜τ0n ΛΛ′= 1 2i(τ0n ΛΛ′−τ0n Λ′Λ) and with the energy argument EFomit- ted. The matrix elements in Eq. (12) are identical to those occurring in the context of exchange coupling32: Tnµ Λ′Λ=/integraldisplay d3rZn× Λ′(r)/bracketleftbigg∂ ∂uµˆH(ˆm0)/bracketrightbigg Zn Λ(r) =/integraldisplay d3rZn× Λ′(r) [βσµBxc(r)]Zn Λ(r).(13) The expression in Eq. (12) for the Gilbert-damping parameterαis essentially equivalent to the one obtained within the torque correlation method (see e.g. Refs. [33– 35]). However, in contrast to the conventional TCM the electronicstructureishererepresentedusingtheretarded electronic Green function giving the present approach much more flexibility. In particular, it does not rely on a phenomenological relaxation time parameter. The expression Eq. (12) can be applied straightfor- wardly to disordered alloys. This can be done by de- scribing in a first step the underlying electronic struc- ture (forT= 0 K) on the basis of the coherent po- tential approximation (CPA) alloy theory. In the next step the configurational average in Eq. (12) is taken fol- lowing the scheme worked out by Butler29when dealing with the electrical conductivity at T= 0 K or residual resistivity, respectively, of disordered alloys. This im- plies in particular that so-called vertex corrections of the type/angbracketleftTµImG+TνImG+/angbracketrightc− /angbracketleftTµImG+/angbracketrightc/angbracketleftTνImG+/angbracketrightcthat account for scattering-in processes in the language of the Boltzmann transport formalism are properly accounted for. One has to note that the factorg µtotin Eq. (12) is sep- arated from the configurational average /angbracketleft.../angbracketrightc, although both values, gandµtot, have to represent the averageper unit cell doing the calculations for compounds and al- loys. This approximation is rather reasonable in the case of compounds or alloys where the properties of the ele- ments of the system are similar (e.g. 3 d-element alloys), but can be questionable in the case of systems containing elements exhibiting significant differences (3 d-5d-, 3d-4f- compounds, etc), or in the case of non-uniform systems as discussed by Nibarger et al36. Thermal vibrations as a source of electron scattering can in principle be accounted for by a generalization of Eqs. (7) – (13) to finite temperatures and by including the electron-phonon self-energy Σ el−phwhen calculating the Green’s function G+. Here we restrict our considera- tion to elastic scattering processes by using a quasi-static representation of the thermal displacements of the atoms from their equilibrium positions. The atom displaced from the equilibrium position in the lattice results in a corresponding variation ∆ tn=tn−tn 0of the single-site scattering matrix in the global frame of reference37,38. A single-site scattering matrix tn(the underline denotes a matrix in an angular momentum representation Λ) for the atomndisplaced by the value sn νfrom the equilib- rium position in the lattice can be obtained using thetransformation matrices37,39 Un LL′(sν,E) = 4π/summationdisplay L′′il′′+l−l′ ×CLL′L′′jl′′(sn ν√ E)YL′′(ˆsn ν).(14) Heremeis the electron mass, jla spherical Bessel func- tion,CLL′L′′stands for the Gaunt coefficients, and a non-relativistic angular momentum representation with L= (l,ml) has been used. Performing a Clebsch-Gordon transformation for the transformation matrix Un LL′to the relativistic Λ representation, the tmatrixtnfor the shifted atom can be obtained from the non-shifted one tn 0from the expression tn ν= (Un ν)−1tn 0Un ν. (15) Treating for a discrete set of displacements sn νeach displacement as an alloy component, we introduce an alloy-analogy model to average over the set sn νthat is chosen to reproduce the thermal root mean square average displacement/radicalbig /angbracketleftu2/angbracketrightTfor a given temperature T. This in turn may be set according to /angbracketleftu2/angbracketrightT= 1 43h2 π2mkΘD[Φ(ΘD/T) ΘD/T+1 4] with Φ(Θ D/T) the Debye func- tion,hthe Planck constant, kthe Boltzmann constant andΘ DtheDebyetemperature40. Ignoringthezerotem- perature term 1 /4 and assuming a frozen potential for the atoms, the situation can be dealt with in full analogy to the treatment of disordered alloys on the basis of the CPA (see above). For small displacements the transformation Eq. (14) can be expanded with respect to sn ν(see Ref. [39]) re- sulting in a linear dependence on sn νfor non-vanishing contributions with ∆ l=|l−l′|=±1. This leads, in particular, in the presence of atomic displacements for transition-metals (TM), for which an angular momen- tum cut-off of lmax= 2 in the KKR multiple scattering expansion is in general sufficient for an undistorted lat- tice, to an angular momentum expansion up to at least lmax= 3. However, this is correct only under the assump- tion of very small displacements allowing linearisation of the transformation Uwith respect to the displacement amplitudes. Thus, since the temperature increase leads to a monotonous increaseof s, the cut-off lmaxshould also be increased. III. MODEL CALCULATIONS In the following we present results of calculations for which single parameters have artificially been manipu- lated in the first-principles calculations in order to sys- tematically reveal their role for the Gilbert-damping. This approach is used to disentangle competing influ- ences on the Gilbert-damping parameter.5 A. Vertex corrections The impact of vertex corrections is shown in Fig. 1 for two different cases: Fig. 1(a) represents the Gilbert damping parameter for an Fe 1−xVxdisordered alloy as a function of concentration, while Fig. 1(b) gives the cor- responding value for pure Fe in the presence of temper- ature induced disorder and plotted as a function of tem- perature. Both figures show results calculated with and without vertexcorrectionsallowingforcomparison. First of all, a significant effect of the vertex corrections is no- ticeable in both cases, although the variation depends on increasingconcentrationofVinthebinaryFe 1−xVxalloy and the temperature in the case of pure Fe, respectively. Some differences in their behavior can be explained by the differences of the systems under consideration. Deal- ing with temperature effects via the alloy analogy model, the system is considered as an effective alloy consisting of a fixed number of components characterizing different types of displacements. Thus, in this case the tempera- ture effect is associated with the increase of disorder in the system caused only by the increase of the displace- ment amplitude, or, in other words – with the strength of scattering potential experienced by the electrons rep- resented by tn(T)−tn 0. In the case of a random alloy theA1−xBxvariation of the scattering potential, as well as the difference tn A−tn B, upon changing the concentra- tions is less pronounced for small amounts of impurities Bandtheconcentrationdependenceisdeterminedbythe amount of scatterers of different types. However, when the concentration of impurities increases, the potentials of the components are also modified (this is reflected, e.g., in the shift of electronic states with respect to the Fermi level, that will be discussed below) and this can lead to a change of the concentration dependence of the vertex corrections. An important issue which one has to stress that neglect of the vertex corrections may lead to the unphysical result, α <0, as is shown in Fig. 1(a). In terms of the Boltzmann transport formalism, this is because of the neglect of the scattering-in term41lead- ing obviously to an incomplete description of the energy transfer processes. B. Influence of spin-orbit coupling Aswasalreadydiscussedabove,thespin-orbitcoupling for the electrons of the atoms composing the system is the main driving force for the magnetization relaxation, resultingintheenergytransferfromthemagneticsubsys- tem to the crystal lattice. Thus, the Gilbert damping pa- rameter should approach zero upon decreasing the SOC in the system. Fig. 2 shows the results for Py+15%Os, where√αis plotted as a function of the scaling param- eter of the spin-orbit coupling42applied to all atoms in the alloy. As one can see,√αhas a nearly linear depen- dence on SOC implying that αvaries in second order in the strength of the spin-orbit coupling43.00.10.2 0.3 0.4 0.5 concentration xV02040α × 103without vertex corrections with vertex correctionsFe1-xVx (a) 0100200 300 400 500 temperature (K)051015202530α × 103without vertex corrections with vertex correctionsbcc Fe (b) FIG. 1. The Gilbert damping parameter for (a) bcc Fe 1−xVx (T= 0 K) as a function of V concentration and (b) for bcc-Fe as a function of temperature. Full (open) symbols give resul ts with (without) the vertex corrections. 0 0.5 1 1.5 2 SOC scaling parameter00.10.20.30.4 α1/2 Py+15%Os FIG. 2. The Gilbert damping parameter for Py+15%Os as a function of the scaling parameter of spin-orbit coupling a p- plied to all atoms contained in the alloy. Red dashed line in plot – linear fit. The values 0 and 1 for the SOC scaling pa- rameter correspond to the scalar-relativistic Schr¨ oding er-like and fully relativistic Dirac equations, respectively.6 IV. RESULTS AND DISCUSSIONS A. 3dtransition-metals We have mentioned above the crucial role of scatter- ing processes for the energy dissipation in magnetiza- tion dynamic processes. In pure metals, in the absence of any impurity, the electron-phonon scattering mecha- nism is of great importance, although it plays a different role in the low- and high-temperature regimes. This was demonstrated by Ebert et al.28using the alloy analogy approach,aswellasbyLiu44et al.usingthe ’frozenther- mal lattice disorder’ approach. In fact both approaches are based on the quasi-static treatment of thermal dis- placements. However, while the average is taken by the CPA within the alloy analogy model the latter requires a sequence of super-cell calculations for this purpose. As a first example bcc Fe is considered here. The cal- culations have been performed accounting for the tem- perature induced atomic displacements from their equi- librium positions, according to the alloy analogy scheme described in section II. This leads, even for pure systems, to a scattering process and in this way to a finite value forα(see Fig. 3(a)). One can see that the experimen- tal results available in the literature are rather different, depending on the conditions of the experiment. In par- ticular, the experimental results Expt. 2 (Ref. [45]) and Expt. 3 (Ref. [46]) correspondto bulk while the measure- ments Expt. 1 (Ref. [47]) have been done for an ultrathin film with 2 .3 nm thickness. The Gilbert damping con- stant obtained within the present calculations for bcc Fe (circles,a= 5.44 a.u.) is compared in Fig. 3(a) with the experiment exhibiting rather good agreement at the temperature above 100 K despite a certain underestima- tion. One can also see a rather pronounced increase of the Gilbert damping observed in the experiment above 400 K (Fig. 3(a), Expt. 2 and Expt. 3), while the theo- retical value shows only little temperature dependent be- havior. Nevertheless, the increase of the Gilbert damp- ing with temperature becomes more pronounced when the temperature induced lattice expansion is taken into account, that can be seen from the results obtained for a= 5.45 a.u. (squares). Thick lines are used to stress the temperature regions for which corresponding lattice parameters are more appropriate. At low temperatures, below 100 K, the calculated Gilbert damping parameter goes up when the temperature decreases, that was ob- served only in the recent experiment47. This behavior is commonly denoted as a transition from low-temperature conductivity-like to high-temperature resistivity-like be- havior reflecting the dominance of intra- and inter-band transitions, respectively3. The latter are related to the increase of the smearing of electron energy bands caused by the increase of scattering events with temperature. Note that even a small amount of impurities reduces strongly the conductivity-like behavior28,45, leading to the more pronounced effect of impurity-scattering pro- cesses due to the increase of scattering events caused bychemical disorder. Large discrepancies between the lat- ter experimental data47and theoretical results of the α calculationsforbcc Fe arerelatedto the verysmall thick- ness of the film investigated experimentally, that leads to an increase of spin-transfer channels for magnetization dissipation as was discussed above. Results for the temperature dependent Gilbert- damping parameter αfor hcp Co are presented in Fig. 3(b) which shows, despite certain underestimation, a rea- sonable agreement with the experimental results45. The general trends at low and high temperatures are similar to those seen in Fe. The results for pure Ni are given in Fig. 3(c) that show in full accordance with experiment a rapid decrease of α with increasing temperature until a regime with a weak variation of αwithTis reached. Note that in the discussions above we have treated α as a scalar instead of a tensorial quantity ignoring a pos- sible anisotropy of the damping processes. This approx- imation is reasonable for the systems considered above with the magnetization directions along a three- or four- fold symmetry axis (see, e.g., the discussions in Ref. [48 and 49]). For a more detailed discussion of this issue the anisotropy of the Gilbert damping tensor α(M) has been investigated for bcc Fe. To demonstrate the depen- dence ofαon the magnetization direction M, the cal- culations have been performed for M= ˆz|M|with the ˆzaxis taken along the /angbracketleft001/angbracketright,/angbracketleft111/angbracketrightand/angbracketleft011/angbracketrightcrystallo- graphic directions. Fig. 4 presents the temperature de- pendence of the diagonal elements αxxandαyy. As to be expected for symmetry reasons, αxxdiffers from αyyonly in the case of ˆ z/bardbl/angbracketleft011/angbracketright. One can see that the anisotropic behavior of the Gilbert damping is pronounced at low temperatures. With an increase of the temperature the anisotropy nearly disappears, because of the smearing of the energy bands caused by thermal vibrations49. A sim- ilar behavior is caused by impurities with a random dis- tribution, aswasobservedforexamplefortheFe 0.95Si0.05 alloy system. The calculations of the diagonal elements αxxandαyyfor two different magnetization directions along/angbracketleft001/angbracketrightand/angbracketleft011/angbracketrightaxes giveαxx=αyy= 0.00123 in the first case and αxx= 0.00123 and αyy= 0.00127 in the second, i.e. the damping is nearly isotropic. The damping parameter αincreases very rapidly with decreasing temperature in the low temperature regime (T≤100 K) for all pure ferromagnetic 3 dmetals, Fe, Co, and Ni (see Fig. 3), leading to a significant discrep- ancy between theoretical and experimental results in this regime. The observed discrepancy between theory and experiment can be related to the exact limit ω= 0 taken intheexpressionfortheGilbertdampingparameter. Ko- renmann and Prange13have analyzedthe magnon damp- ing in the limit of small wave vector of magnons q→0, assuming indirect transitions in the electron subsystem and taking into account the finite lifetime τof the Bloch states due to electron-phonon scattering. They discuss the limiting cases of low and high temperatures showing the analogy of the present problem with the problem of7 0 200 400 600 Temperature (K)0246810α × 103Theory: a = 5.42 a.u. Theory: Fe+0.1% Vac. Theory: a = 5.45 a.u. Expt. 1 Expt. 2 Expt. 3bcc Fe (a) 0100200 300 400 500600 Temperature (K)05101520α × 103Expt Theory Theory: Co + 0.03% Vac Theory: Co + 0.1% Vachcp Co (b) 0100200 300 400 500 Temperature (K)00.050.10.150.20.25 αExpt Theory Theory: Ni + 0.1%Vacfcc Ni (c) FIG. 3. Temperature variation of the Gilbert damping pa- rameter of pure systems. Comparison of theoretical results with experiment: (a) bcc-Fe: circles and squares show the re - sults for ideal bcc Fe for two lattice parameters, a= 5.42 a.u. anda= 5.45 a.u.; stars show theoretical results for bcc Fe (a= 5.42 a.u.) with 0.1% of vacancies (Expt. 1 - Ref. [47], Expt. 2 - Ref. [45], Expt. 3 - Ref. [46]); (b) hcp-Co: circles show theoretical results for ideal hcp Co, stars - for Co with 0.03% of vacancies, and ’pluses’ - for Co with 0.1% of vacan- cies (Expt. Ref. [45]); and (c) fcc-Ni (Expt. Ref. [45]).extreme cases for the conductivity leading to the normal and anomalous skin effect. On the basis of their result, the authors point out that the expression for the Gilbert damping obtained by Kambersky25, withα∼τis cor- rect in the limit of small lifetime (i.e. qvFτ≪1, in their model consideration, where qis a magnon wave vector andvFis a Fermi velocity of the electron). In the low- temperature limit the lifetime τincreases with decreas- ingTand one has to use the expression corresponding to the ’anomalous’ skin effect for the conductivity, i.e. α∼tan−1(qvFτ)/qvF, leading to a saturation of αupon the increase of τ. Another possible reason for the low-temperature be- havior of the Gilbert damping observed experimentally can be structural defects present in the material. To simulate this effect, calculations have been performed for fcc Ni and bcc Fe with 0.1% of vacancies and for hcp Co with 0.1% and 0.03% of vacancies. Fig. 3(a)-(c) shows the corresponding temperature dependence of the Gilbert dampingparameterapproachingafinite value for T→0. The remaining difference in the T-dependent be- havior can be attributed to the non-linear dependence of the scattering cross section at low temperatures as is dis- cussed in the literature for transport properties of metals and is not accounted for within the present approxima- tion. B. 3dTransition-metal alloys As is mentioned above, the use of the linear response formalism within multiple scattering theory for the elec- tronicstructurecalculationsallowsustoperformthenec- essary configurational averaging in a very efficient way avoiding supercell calculations and to study with mod- erate effort the influence of varying alloy composition onα. The corresponding approach has been applied to 0100200 300 400 500 Temperature (K)051015202530α × 103Theory: z||<001>: αxx=αyy Theory: z||<011>: αxx Theory: z||<011>: αyy Theory: z||<111>: αxx=αyybcc Fe FIG. 4. Temperature variation of the αxxandαyycom- ponents of the Gilbert damping tensor of bcc Fe with the magnetization direction taken along different crystallogr aphic directions: M= ˆz|M|/bardbl/angbracketleft001/angbracketright(circles), M/bardbl/angbracketleft011/angbracketright(squares), M/bardbl/angbracketleft111/angbracketright(diamonds).8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 concentration xCo0123456 α × 103bcc, CPA CsCl: CPA (partial ord) bcc, NL CPA (ord) bcc, NL CPA (disord) n(EF)Fe1-xCox n(EF) n(EF) (sts./Ry) 102030405060 0 (a) -8-6-4 -2 0 2 energy (eV)00.511.52n↑tot(E) (sts./eV)00.511.52n↓tot(E) (sts./eV)x = 0.01 x = 0.5EF (b) FIG. 5. (a) Theoretical results for the Gilbert damping pa- rameter of bcc Fe 1−xCoxas a function of Co concentration: CPA results for the bcc structure (full circles) describing the random alloy system, results for the partially ordered system (opened square) for x= 0.5 (i.e. for Fe 1−xCox alloy with CsCl structure and alloy components randomly distributed in two sublattices in the following proportion s: (Fe0.9Co0.1)(Fe0.1Co0.9), the NL-CPA results for random al- loy with bcc structure (opened circles) and the NL-CPA re- sults for the the system with short-range order within the first-neighbor shell (opened diamonds). The dashed line rep - resents the DOS at the Fermi energy, EF, as a function of Co concentration. (b) spin resolved DOS for bcc Fe 1−xCoxfor x= 0.01 (dashed line) and x= 0.5 (solid line). the ferromagnetic 3 d-transition-metal alloy systems bcc Fe1−xCox, fcc Ni 1−xFex, fcc Ni 1−xCoxand bcc Fe 1−xVx. Fig. 5(a) shows as an example results for the Gilbert damping parameter α(x) calculated for bcc Fe 1−xCox forT= 0 K at different conditions. Full circles rep- resent the results of the single-cite CPA calculations characterizing the random Fe-Co alloy. These results are compared to those obtained employing the non-local CPA52,53(NL-CPA) assuming no short-range order in the system (opened circles). Dealing in both cases (CPA and NL-CPA), with completely disordered system, the NL-CPA maps the alloy problem on that of an impurity cluster embedded in a translational invariant effective medium determined selfconsistently, thereby accounting0 0.2 0.4 0.6 concentration xCo0246 α × 103Exp. TheoryFe1-xCox (a) 0 0.2 0.4 0.6 concentration xCo010203040 α × 103Exp. Theory (Starikov et al.) TheoryNi1-xCox (b) 0 0.2 0.4 0.6 concentration xFe01020 α × 103Theory (Starikov et al.) Theory Expt 1 Expt 2Ni1-xFex (c) 0 0.2 0.4 0.6 concentration xV012345 α × 103Expt. Theory, T = 0 K Theory, T = 300 KFe1-xVx (d) FIG. 6. The Gilbert damping parameter for Fe 1−xCox(a) Ni1−xCox(b) and Ni 1−xFex(c) as a function of Co and Fe concentration, respectively: present results within CPA ( full circles), experimental data by Oogane50(full diamonds). (d) Results for bcc Fe 1−xVxas a function of V concentration: T= 0 K (full circles) and T= 300 K (open circles). Squares: experimental data51. Open circles: theoretical results by Starikov et al.5.9 for nonlocal correlations up to the range of the cluster size. The present calculations have been performed for the smallestNL-CPAclusterscontainingtwositesforbcc based system, accountingfor the short-rangeorder in the first-neighborshell. As onecan see, this results in a small decrease of the αvalue in the region of concentrations aroundx= 0.5 (opened diamonds), that is in agreement with the results obtained for partially ordered system (opened square) for x= 0.5. The latter have been calcu- lated for the Fe 1−xCoxalloy having CsCl structure and alloycomponentsrandomlydistributed intwosublattices in the following proportions: (Fe 0.9Co0.1)(Fe0.1Co0.9). Because the moments and spin-orbit coupling strength do not differ very much for Fe and Co, the variation of α(x) should be determined in the concentrated regime essentially by the electronic structure at the Fermi en- ergyEF. As Fig. 5(a) shows, there is indeed a close correlation with the density of states n(EF) that may be seen as a measure for the number of available relaxation channels. The change of α(x) due to the increase of the Co concentration is primarily determined by an appar- ent shift of the Fermi energy also varying with concen- tration (Fig. 5(b)). The alloy systems considered have the common feature that the concentration dependence ofαis governed by the concentration dependent density of statesn(EF). A comparison of theoretical αvalues with the experi- mentforbccFe 1−xCoxisshowninFig. 6(a),demonstrat- ing satisfying agreement. In the case of Ni 1−xFexand Ni1−xCoxalloysshown in Fig. 6, (b) and (c), the Gilbert dampingdecreasesmonotonouslywith the increaseofthe Fe and Co concentration, in line with experimental data. At all concentrations the experimental results are under- estimated by theory approximately by a factor of 2. The calculated dampingparameter α(x) is found in verygood agreementwith theresultsbasedonthe scatteringtheory approach5demonstrating numerically the equivalence of the two approaches. An indispensable requirement to achieve this agreement is to include the vertex correc- tions mentioned above. As suggested by Eq. (12) the variation of α(x) with concentration xmay also reflect to some extent the variation of the averagemagnetic mo- ment of the alloy, µtot. ThepeculiarityoftheFe 1−xVxalloywhencomparedto those discussed above is that V is a non-magnetic metal and has only an induced spin magnetic moment. De- spite that, the concentration dependence of the Gilbert damping parameter at T= 0 K for small amounts of V (see Fig. 6(d)) displays the same trend as the pre- viously discussed alloys shown in Fig. 6(a)-(c). Taking into account a finite temperature of T= 300 K changes αvalue significantly at small V concentrations leading to an improved agreement with experiment for pure Fe, while it still compares poorly with the experimental data atxV= 0.27. One should stress once more that the con- centration dependent behavior of the Gilbert damping parameter of the alloys discussed above is different for an increased amount of impurities (more than 10%), as aresultofadifferentvariationoftheDOS n(EF)causedby a concentration dependent modification of the electronic states and shift of the Fermi level. C. 5dimpurities in 3 dtransition metals As discussed in our recent work28investigating the temperature dependent Gilbert damping parameter for pure Ni and for Ni with Cu impurities, αis primarily determined by the thermal displacement in the regime of small impurity concentrations. This behavior can also be seen in Fig. 7, where the results forFe with 5 d-impurities are shown. Solid lines represent results for T= 0 K for an impurity concentration of 1% (full squares) and 5% (full circles). As one can see, at smaller concentrations the maximum of the Gilbert damping parameter occurs for Pt. With increasing impurity content the αparame- ter decreases in such a way that at the concentration of 5% a maximum is observed for Os. The reason for this behavior lies in the rather weak scattering efficiency of Pt atoms due to a small DOS n(EF) of the Pt states when compared for example for Osimpurities (see Fig. 9). This results in a slowdecrease ofαat small Pt concentration when the BFS mechanism is mostly responsible for the energy dissipation. A con- sequence of this feature can be seen in the temperature dependence of α(T= 300 K, opened squares): a most pronounced temperature induced decrease of the αvalue is observed for Pt and Au. When the concentration of 5d-impurities is increased up to 5%, the maximum in α occurs for the element with the most efficient scatter- ing potential resulting in spin-flip scattering processes responsible for dissipation. The temperature effect at this concentration is very small. Considering in more detail the temperature dependent behavior of the Gilbert damping parameter for Fe with Os and Pt impurities, shown in Fig. 8, one can also ob- serve the consequence of the features mentioned above. At 1% of Pt impurities αdecreases much steeper upon increasing the temperature, as compared to the case of Os impurities. Therefore, in the first case the role of the scattering processes due to atomic displacements is much more pronounced than in the second case with rather strong scattering on the Os impurities. When the con- centration increases to 5% the dependence of αon the temperature in both cases becomes less pronounced. The previousresults can be comparedto the results for the 5d-impurities in the permalloy Fe 80Ni20(Py), which has been investigated also experimentally54. This system shows some difference in the concentration dependence when compared with pure Fe, because Py is a disordered alloy with a finite value of the αparameter. Therefore, a substitution of 5 dimpurities leads to a nearly linear increase of the Gilbert damping with impurity content, just as seen in experiment54. The total damping for 10% of 5 d-impurities shown in Fig.10(a)variesroughlyparabolicallyoverthe 5 dTMse-10 Ta W Re Os Ir Pt Au0246810 α × 103x = 0.05; T = 0 K x = 0.05; T = 300 K x = 0.01; T = 0 K x = 0.01; T = 300 K FIG. 7. Gilbert damping parameter for bcc Fe with 1% (squares) and 5% (circles) of 5 dimpurities calculated for T= 0K (full symbols) and for T= 300K (opened sysmbols). ries. This variation of αwith the type of impurity corre- lateswellwith the densityofstates n5d(EF) (Fig. 10(b)). Again the trend of the experimental data is well repro- duced by the calculated values that are however some- what too low. V. SUMMARY In summary, aformulationforthe Gilbert dampingpa- rameterαin terms of linear response theory was derived thatledtoaKubo-Greenwood-likeequation. Thescheme was implemented using the fully relativistic KKR band structuremethod incombinationwiththe CPAalloythe- ory. This allows to account for various types of scat- tering mechanisms in a parameter-free way, that might be either due to chemical disorder or to temperature- induced structural disorder (i.e. electron-phonon scat- tering effect). The latter has been described by using the so-called alloy-analogy model with the thermal dis- placement of atoms dealt with in a quasi-static manner. Corresponding applications to pure metals (Fe, Co, Ni) aswellastodisorderedtransition-metalalloysledto very good agreement with results based on the scattering the- ory approach of Brataas et al.4and well reproduces the experimental results. The crucial role of vertex correc- tions for the Gilbert damping is demonstrated both in the case of chemical as well as structural disorder and the accuracy of finite-temperature results is analyzed via test calculations. Furthermore, the flexibility and numerical efficiency of the present scheme was demonstrated by a study on metallic systems on a series of binary 3 d-alloys (Fe1−xCox, Ni1−xFex, Ni1−xCoxand Fe 1−xVx), 3d−5d TM systems, the permalloy-5 dTM systems. The agree- ment between the present theoretical and experimental results is quite satisfying, although one has to stress a systematic underestimation of the Gilbert damping by the numerical results. This disagreement could be caused either by the idealized system considered theoret- ically (e.g., the boundary effects are not accounted for0100200 300 400 500 temperature (K)12345α × 103Fe0.99Me0.01Pt Os (a) 0100200 300 400 500 temperature (K)22.533.54α × 103Fe0.95Me0.05 PtOs (b) FIG. 8. Gilbert damping parameter for bcc Fe 1−xMxwith M= Pt (circles) and M= Os (squares) impurities as a func- tion of temperature for 1% (a) and 5% (b) of the impurities. -8-6-4 -2 0 2 energy (eV)00.81.62.43.2n↑(E) (sts./eV)00.81.62.43.2n↓(E) (sts./eV)Pt OsEF EF FIG. 9. DOS for Pt in Fe 1−xPtx(full line) and Os in Fe1−xOsx(dashed line) for x= 0.01.11 Ta W Re OsIr Pt Au02468α × 102Expt. Theory (a) Ta W Re OsIr Pt Au051015n5d(EF) (Sts/Ry) (b) FIG. 10. (a) Gilbert damping parameter αfor Py/5d TM systems with 10 % 5d TM content in comparison with experiment54; (b) spin magnetic moment m5d spinand density of states n(EF) at the Fermi energy of the 5 dcomponent in Py/5d TM systems with 10 % 5d TM content.in present calculations) or because of additional intrin- sic dissipation mechanisms for bulk systems which have to be taken into account. These could be, for instance, effects of temperature induced spin disorder44. 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2013-01-10

A method for the calculations of the Gilbert damping parameter $\alpha$ is presented, which based on the linear response formalism, has been implemented within the fully relativistic Korringa-Kohn-Rostoker band structure method in combination with the coherent potential approximation alloy theory. To account for thermal displacements of atoms as a scattering mechanism, an alloy-analogy model is introduced. This allows the determination of $\alpha$ for various types of materials, such as elemental magnetic systems and ordered magnetic compounds at finite temperature, as well as for disordered magnetic alloys at $T = 0$ K and above. The effects of spin-orbit coupling, chemical and temperature induced structural disorder are analyzed. Calculations have been performed for the 3$d$ transition-metals bcc Fe, hcp Co, and fcc Ni, their binary alloys bcc Fe$_{1-x}$Co$_{x}$, fcc Ni$_{1-x}$Fe$_x$, fcc Ni$_{1-x}$Co$_x$ and bcc Fe$_{1-x}$V$_{x}$, and for 5d impurities in transition-metal alloys. All results are in satisfying agreement with experiment.

First-principles calculation of the Gilbert damping parameter via the linear response formalism with application to magnetic transition-metals and alloys

1301.2114v1

Discovery and regulation of chiral magnetic solitons: Exact solution from Landau-Lifshitz-Gilbert equation Xin-Wei Jin,1, 2Zhan-Ying Yang,1, 2,∗Zhimin Liao,3Guangyin Jing,1, †and Wen-Li Yang2, 4 1School of Physics, Northwest University, Xi’an 710127, China 2Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China 3School of Physics, Peking University, Beijing, 100871,China 4Insititute of Physics, Northwest University, Xi’an 710127, China (Dated: August 17, 2023) The Landau-Lifshitz-Gilbert (LLG) equation has emerged as a fundamental and indispensable framework within the realm of magnetism. However, solving the LLG equation, encompassing full nonlinearity amidst intricate complexities, presents formidable challenges. Here, we develop a precise mapping through geometric representation, establishing a direct linkage between the LLG equation and an integrable generalized nonlinear Schr ¨odinger equation. This novel mapping provides accessibility towards acquiring a great number of exact spatiotemporal solutions. Notably, exact chiral magnetic solitons, critical for stability and controllability in propagation with and without damping effects are discovered. Our formulation provides exact solutions for the long-standing fully nonlinear problem, facilitating practical control through spin current injection in magnetic memory applications. Introduction.— The seminal 1935 work by Landau and Lif- shitz, which laid down the foundational dynamical equation governing magnetization based on phenomenological insights [1–3], and the subsequent introduction of a damping term by Gilbert [4], the amalgamation of these concepts has given rise to the renowned Landau-Lifshitz-Gilbert (LLG) equation. Over the years, this equation has emerged as a fundamental and indispensable framework within magnetism field. Its con- temporary significance has been amplified through remarkable advancements, most notably the incorporation of an additional term that facilitates the explication of spin torque phenomena in spintronics [5–10], spin waves [11–18], magnetic solitons [19–28], spatio-temporal patterns [29, 30], and even chaotic behavior [31]. Further advancements have paved the way for applications in next-generation magnetic storage [32–34], neural networks [35–37], and logic gates [38–42]. Despite its deceptively simple appearance, solving the LLG equation poses an exceptional challenge [29, 30], rendering it a persistently unresolved problem for nearly nine decades. This complexity emanates from its intricate nature as a vector- based highly nonlinear partial differential equation. In real- world scenarios, the LLG equation encompasses a myriad of complex interactions among the components of the magneti- zation vector [4]. Consequently, solutions often necessitate recourse to linearization, approximations, and asymptotic techniques such as the Holstein-Primakoff (HP) transforma- tion [43, 44], reductive perturbation scheme [45, 46], and long wavelength approximation. Nonetheless, these techniques prove utterly ineffectual in regions of large amplitudes or strong nonlinearity. Therefore, exact solution of the LLG equation emerges as a potent bridge, overcoming these gaps and revealing profound revelations regarding magnetization dynamics, thereby furnishing insightful understandings for simulating and comprehending intricate magnetic systems. In this Letter, through a geometric representation [47], we establish an exact mapping of the LLG equation onto an integrable generalized nonlinear Schr ¨odinger equation, free of any approximation. This novel mapping provides accessibility towards acquiring a great number of exact spatiotemporal solutions of the original equation. Notably, we unveil an analytical formulation for chiral magnetic solitons,encompassing a spectrum ranging from left-handed, neutral to right-handed configurations, determined by a defined chirality factor. The derived exact solution indicates the potential for arbitrary manipulation of magnetic soliton motion through the injection of spin current —a discovery that aligns seamlessly with our numerical findings. To encapsulate the realism of dissipative devices, we incorporate Gilbert damping into the dynamics of these chiral magnetic solitons, thereby estimating their dynamic propagation. Modeling.— We consider an isotropic ferromagnetic nanowire with spin-polarized current flowing along the axis of nanowire as depicted in Fig. 1. A “nanowire” as defined here is a planar ferromagnetic stripe of length Lx, width Ly, and thickness Lzalong ˆx,ˆy, and ˆz, respectively, with Lx≫Ly>Lz. Figure 1: Schematic diagram of 1D ferromagnetic structure. Magnetic soliton excitation driven by spin-polarized currents. Here∆represents the width of magnetic solitons. The magnetization dynamics is described by the famous LLG equation ∂m ∂t=−γm×Heff+α/parenleftbigg m×∂m ∂t/parenrightbigg +τb, (1) where m=M/Ms= (mx,my,mz)is the unit magnetization vector with Msbeing the saturated magnetization. The first term on the right-hand side represents the torque contributed by the effective field Heff(including applied, demagnetizing, anisotropy, and exchange fields), γis the gyromagnetic constant. The second term describes the Gilbert damping torque, parameterized by a dimensionless damping factor α.arXiv:2308.08331v1 [nlin.PS] 16 Aug 20232 Figure 2: Spatial structure and classification of chiral magnetic solitons. (a) Vertical views of the left-handed and right-handed magnetic solitons. (b) Schematic plot of the chirality defined by the azimuth angle change. The pair of red arrows delineate the azimuthal directional changes of the left and right chiral magnetic solitons across the distribution axis. Their discrepancies in azimuthal variation are denoted by ∆ϕ′ LHand∆ϕ′ RH. (c)-(e) Spatial spin structures of left-handed magnetic soliton, neutral magnetic soliton, and right-handed magnetic soliton. (f)-(h) illustrate the azimuthal, polar angle, and phase gradient flow of the three kinds of chiral solitons. The last term τbrepresents the spin-transfer torque (STT), which comprises dual components that can be written as τb= −bJ(ˆJ·∇)M+βbJM×(ˆJ·∇)M. Here ˆJis the unit vector in the direction of the current. These two components are most commonly termed adiabatic and non-adiabatic spin torques, respectively, with bJ=P jeµB/(eMs)andβdefined as the non- adiabatic torque coefficient. Wherein, Prepresents the spin polarization of current, jeis the electric current density, µBis the Bohr magneton, and eis the magnitude of electron charge. In what follows, we take only adiabatic STT into consideration for two reasons: one is that the most widely agreed upon interaction between a spin-polarized current and a magnetic soliton is adiabatic STT; and the other is that the magnitude of the nonadiabatic spin torque is about 2 orders of magnitude smaller than adiabatic torque (β≈10−2). Let us begin by examining the most elementary effective field, encompassing solely exchange fields, i.e. Heff= (2A/Ms)∇2m, where Ais the exchange stiffness constant. The spatiotemporal transformation τ=γµ0Mst/(1+α2) andζ=λex·xare introduced to recast the LLG equation into the dimensionless Landau-Lifshitz form (Note that λex=/radicalbig 2A/(µ0M2s)is the exchange length): mτ=−m×mζζ−αm×/parenleftbig m×mζζ/parenrightbig +Qmζ, (2) whereQ=bJ(1+αβ)//radicalbig 2Aγ2µ0, a dimensionless number measuring the ratio of external spin current over exchange interaction strength. This dimensionless STT-LLG model (2) effectively describes the dynamics of nonlinear excita- tions, such as magnetic solitons, occurring in ferromagnetic nanowires upon spin injection. Moreover, it exhibits qualita-tive reproduction much of the behavior seen experimentally. For a permalloy nanowire, the standard material parameters are:γ=1.76×1011rad/s·T,Ms=8×105A/m,A=1.3× 10−11J/m,P=0.5. As a result, the units in time and space after rescaling are 1 τ≈5.70 ps ,1ζ≈5.68 nm. Chiral magnetic soliton.— The high nonlinearity of STT- LLG model (2) presents a great challenge for comprehensive analytical research and restricts the exploration of novel spin textures to the realm of micromagnetic simulation or weak nonlinearity. In this context, to obtain an analytical depiction of large-amplitude magnetic textures, we exactly map the STT-LLG equation (2) into a generalized nonlinear Schr ¨odinger (GNLS) equation (See Supplementary Material for further details on the spatial curve mapping procedure), devoid of any approximations. For no damping, the GNLS equation reads iΨτ+Ψζζ+2|Ψ|2Ψ−iQΨζ=0. (3) Exact cycloidal chiral magnetic soliton solutions can be constructed by applying the Darboux transformation (DT) [48, 49]. Indeed, using the mapping relationship between equation solutions, the specific expression of three components of magnetization are obtained (See Supplementary Material for3 detailed calculations): mx=2a a2+b2[asinh(Ξ)sin(Γ)−bcosh(Ξ)cos(Γ)]·sech2(Ξ), my=2a a2+b2[asinh(Ξ)cos(Γ)+bcosh(Ξ)sin(Γ)]·sech2(Ξ), mz=1−2a2 a2+b2sech2(Ξ), (4) with Ξ=2a/bracketleftbigg x+/integraldisplay (Q+4b)dt/bracketrightbigg , Γ=2b/bracketleftbigg x+/integraldisplay (Q−2(a2−b2)/b)dt/bracketrightbigg , where aandbdescribe the wave number and the velocity of the magnetic soliton. Fig. 2(a) depicts the spin textures of the obtained magnetic solitons. Evidently, these solitons showcase a mirror symmetry relative to the wave vector axis, indicating their inherent chirality. The chirality can be characterized by variations of the azimuth angle. To clarify, we denote the polar and azimuthal angles of mbyθandϕ, respectively (as shown in Fig. 1), such that M+=mx+imy=M0sin(θ)exp(iϕ),mz= M0cos(θ). Note that the azimuthal angle exhibits a periodic background, which arises from the variation of magnetization in space, as revealed by the solution (4). The oscillation structures present in the azimuthal angle profiles are related to the small oscillation of mν(ν=x,y), even though they may not be readily visible in Fig. 2(a). By eliminating the meaningless periodic background phase, the real phase jumps of the chiral magnetic solitons are obtained by calculating the intrinsic argument ϕ′(x) =argM′ +, where M′ +=M+exp(iΓ). As a result, the azimuthal angles of both chiral solitons are demonstrated in Fig. 2(b). Two red arrows span between the blue dashed line representing negative infinity and the corresponding positive infinity, delineating the azimuthal evolution of magnetic solitons across the distribution axis. The distinction in azimuthal variation for chiral magnetic solitons are denoted as ∆ϕ′ LHand∆ϕ′ RH. Notably, the two classes of chiral magnetic solitons exhibit opposite phase jumps, corresponding to two distinct chiralities. The total phase change is defined as ∆ϕ′=ϕ′(x→+∞)− ϕ′(x→ −∞). In general, the phase change of arbitrary magnetic solitons can be determined by integrating the phase gradient flow. Insight can be gained from combining both argument and the phase gradient flow ∇ϕ′(x). Starting from M′ +that constructed from exact solutions, we obtain ∇ϕ′(x) =2b[sech(2Ξ)+1]/parenleftig a2−b2 a2+b2/parenrightig sech(2Ξ)−1. (5) One can observe that the denominator of the aforementioned expression is consistently a non-positive value, which indi- cates that “ +” and “ −” families of phase gradient flow are characterized by the opposite signs of b. Here, we define a chirality factor C=sgn(b) =±1, which determines the chirality of magnetic solitons. It is straightforward to verify that the nonzero phase variation is characterized by a simple Figure 3: Coupling between chiral magnetic solitons and spin current injection. (a) Velocities of three distinct classes of chiral magnetic solitons plotted against spin-polarized currents ( j). (b)-(d) Controlled manipulation of right-handed magnetic solitons under varying current strengths, enabling forward, backward, and arrested motion. expression: ∆ϕ′=2Carctan (|a/b|).Thus, the chirality of the chiral magnetic soliton is entirely determined by this chirality factor C. When b=0, a special case naturally occurs, where the chirality factor cannot be defined, and chirality disappears, corresponding to the neutral cycloidal magnetic solitons. Finally, we can now classify the exact solution (4) into three categories based on the chirality factor, corresponding to neutral, left-handed, and right-handed chiral magnetic solitons. Figs. 2(c)–2(h) depict the typical spin textures, azimuthal angles, polar angles, and phase gradient flow at t=0 when no spin current is applied. Spin-current coupling and damping effect.— We now move to study the coupling between chiral magnetic soliton and the injection of spin current. It has been demonstrated that the spin transfer torque is capable of driving the domain wall or skyrmion [7, 26], eliciting their prompt movement at a considerable velocity upon the application of spin current. Here we report a comparable phenomenon on the chiral magnetic soliton from both theoretical and simulation results. The numerical simulation results depicting the relationship between the velocities of three categories of chiral magnetic solitons and the injected spin currents are illustrated in Fig. 3(a), and are in direct agreement with those obtained from the analytical solutions (4). These linear correlations can be realized from the equivalent GNLS equation (3), wherein the spin current term can be normalized to resemble the “driving velocity”, as supported by the dimensional analysis of Q. Figs. 3(b)–3(d) exemplify the current manipulation for right- handed magnetic soliton that comprise a series of transient snapshots captured during the magnetization evolution process (the model parameters are shown in the caption). The above4 Figure 4: Transmission of magnetic solitons in damped ferromagnetic nanowires and the anti-damping effect of non-adiabatic STT. (a) Schematic diagram of magnetization dissipation under damping. (b) Propagation of right-handed magnetic solitons in ferromagnetic nanowires with Gilbert-damping constant α=0.05. (c) Temporal evolution of the magnetization component mz in the absence of non-adiabatic STT, where damping constant α=0.01. (d) Temporal evolution of the magnetization component mzwith spin-polarized current j=3.7×107A·cm−2. (e) Gilbert-damping dependence of lifetime and moving distance. results highlight two notable aspects. Firstly, the chiral magnetic soliton possesses an inherent velocity linked to its initial magnetization state. Secondly, the solitons’ motion can be stimulated by a spin-polarized current, while preserving their chirality. The external injection of spin current offers a means to manipulate chiral magnetic solitons, granting control over their forward, backward, and frozen motion. Until now, our analysis is based on a perfect ferromagnetic wire in the absence of damping. Strictly speaking, in realistic nanowires, magnetic solitons cannot move over a large distance due to Gilbert damping. The existence of damping introduces a small torque field, which dissipates the energy of the system during magnetization dynamics, and leads to a helical precession of the magnetization towards the direction of the effective field, i.e. the minimum energy state (See Fig. 4(a)). To understand this damping effect in greater detail we performed numerical simulations of single- soliton dynamics. Fig. 4(b) shows the evolution of a right- handed magnetic soliton in a nanowire with Gilbert-damping constant α=0.05. It can be seen that the chiral magnetic soliton degenerates to a homogeneous magnetized state after propagating for about 223.8 ps. During the whole process, the magnetic soliton undergoes continuous deformation. This has two consequences: magnetic soliton spreading and slowing of internal oscillations. In order to characterize the presence of a chiral magnetic soliton, we define the soliton polarization Ps= [1−min(mz)] 2. The soliton is deemed to have dissipated when its polarization is lower than 5% in comparison to the maximum magnetization. The movement of magnetic solitons within ferromagnetic nanowires, subject to varying damping, resultsin distinct lifetimes. Fig. 4(e) depicts line graphs illustrating the relationship between the damping coefficient, lifetime, and moving distance. The dissipation of solitons due to damping is a challenge to circumvent, and one approach is to seek ferromagnetic materials with low damping coefficients. Here, we explored the potential anti-damping effect of non- adiabatic STT, as depicted in Figs. 4(c) and 4(d). In the absence of external spin current, the mzcomponent of the magnetic soliton diminishes during transmission. However, upon injecting an appropriate spin current, the incorporation of non-adiabatic STT enables the chiral magnetic soliton to propagate uniformly in its original velocity, resulting in a significantly extended lifetime. Conclusions.— In this Letter, we have shown that the dimensionless LLG equation containing STT is entirely equivalent to the generalized nonlinear Schr ¨odinger equation without any approximation. This remarkable integrable system enables us to predict novel exact spatiotemporal magnetic solitons. By applying the Darboux transformation, we obtain exact solution of chiral magnetic solitons, emerging within an isotropic ferromagnetic nanowire. Our analytical formulation establishes a distinct correlation between chiral magnetic solitons and the infusion of spin currents, corrobo- rating our numerical findings. This interrelation underscores the potential for arbitrary manipulation of magnetic soliton motion through spin current injection. The inherent chirality of the micromagnetic structure plays a pivotal role in soliton motion: a reversal in chirality leads to a shift in motion direction. To encapsulate the realism of dissipative devices, we investigate the influence of Gilbert damping on the motion5 of chiral magnetic solitons. The results reveal that in the presence of damping, chiral magnetic solitons gradually evolve toward a uniformly magnetized state. 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Thompson, Physica A: Statistical Mechanics and its Applications 84, 577 (1976). [48] F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, Physical Review Letters 109, 044102 (2012). [49] L. Ling, L.-C. Zhao, and B. Guo, Communications in Nonlinear Science and Numerical Simulation 32, 285 (2016).Supplementary Materials for “Discovery and regulation of chiral magnetic solitons: Exact solution from Landau-Lifshitz-Gilbert equation” Xin-Wei Jin and Zhan-Ying Yang∗ School of Physics, Northwest University, Xi’an 710127, China and Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China Zhimin Liao School of Physics, Peking University, Beijing, 100871,China Guangyin Jing† School of Physics, Northwest University, Xi’an 710127, China Wen-Li Yang Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China and Insititute of Physics, Northwest University, Xi’an 710127, ChinaarXiv:2308.08331v1 [nlin.PS] 16 Aug 20232 In this supplementary material, we will show more details on the exact geometric mapping between Landau-Lifshitz-Gilbert and generalized nonlinear Schr ¨odinger equation, and calculation details of solving the magnetic solitons. A. Exact mapping between LLG and GNLS equation: Geometric Representation We identify the magnetization state of ferromagnetic nanowire at any instant of time with a moving space curve in Euclidean three-dimensional space E3. This is achieved by mapping the unit magnetization vector m(x,t)on the unit tangent vector e1 associated with the curve. Thus the dimensionless STT-LLG equation (in the absence of damping) becomes e1t=e1×e1xx+Qe1x. (A.1) In the usual way, the normal and binormal vectors of the moving space curve are constructed by taking e2in the direction of e′1 ande3=e1×e2. The spatial variations of these orthogonal unit vectors is determined by the Serret-Frenet equations e1 e2 e3 x= 0κ0 −κ0τ 0−τ0 e1 e2 e3 , (A.2) where κ(x,t)andτ(x,t)are the curvature and torsion of the space curve. In view of (A.1) and (A.2) alongside the orthogonality of the three unit vectors, it is easy to obtain e1 e2 e3 t= 0 −κτ+Qκ κ x κτ−Qκ 0 −τ2+Qτ+κ−1κxx −κxτ2−Qτ−κ−1κxx 0 e1 e2 e3 . (A.3) The compatibility conditions∂ ∂t/parenleftig ∂ei ∂x/parenrightig =∂ ∂x/parenleftig ∂ei ∂t/parenrightig ,i=1,2,3, between Eqs. (A.2) and (A.3) lead to the following evolution equations for κandτ κt=−(κτx+2κxτ)−Qκx, (A.4a) τt=/parenleftbig κ−1κxx−τ2/parenrightbig x+κκx−Qτx. (A.4b) On making the complex transformation Ψ=1 2κexp/parenleftbig i/integraltext τdx/parenrightbig , we finally arrive at a generalized nonlinear Schr ¨odinger (GNLS) equation (it is easy to verify that the real and imaginary parts of (A.5) is equivalent to (A.4a) and (A.4b), respectively) iΨτ+Ψζζ+2|Ψ|2Ψ−iQΨζ=0. (A.5) Thus we have proved that the STT-LLG equation can be exactly mapped into the integrable GNLS equation. B. Lax Representation and Darboux Transformation We now turn to establish the connection between the solutions of the LLG equation and the GNLS equation. Using the Pauli matrices ( σ1,σ2,σ3), the LLG equation can be rewriten into the matrix form /hatwidemt=1 2i[/hatwidem,/hatwidemxx]+Q/hatwidemx, (B.1) where/hatwidem=mxσ1+myσ2+mzσ3and[·,·]denotes the Lie bracket of the matrices. For this equation, the boundary condition is given by lim x→±∞/hatwidem=σ3, i.e., lim x→±∞m= (0,0,1). Considering the Lax representation of the GNLS (A.5) ∂Φ ∂x=UΦ,∂Φ ∂t=VΦ, (B.2) where U=U0+λU1,V=V0+λV1+λ2V2andλis the spectral parameter, U0=/parenleftigg 0Ψ −Ψ∗0/parenrightigg ,U1=−σ3,V0=/parenleftigg i|Ψ|2iΨx+QΨ iΨ∗ x−QΨ∗−i|Ψ|2/parenrightigg ,V1=−2iU0−Qσ3,V2=2iσ3. (B.3)3 Suppose Φ1(x,t,λ)andΦ2(x,t,λ)are two linear independence eigenvectors of Lax pair (B.2), then Ω= (Φ1,Φ2)also satisfies Eq. (B.2). Let g(x,t) =Ω(x,t,λ)/vextendsingle/vextendsingle λ=0, we have gx=U0g,gt=V0g. From transformation /hatwideΦ=g−1Φ, we obtain ∂/hatwideΦ ∂x=/hatwideU/hatwideΦ,∂/hatwideΦ ∂t=/hatwideV/hatwideΦ, (B.4) where /hatwideU=g−1Ug−g−1gx=g−1(U−U0)g=λg−1U1g, /hatwideV=g−1Vg−g−1gt=g−1(V−V0)g=λg−1V1g+λ2g−1V2g,(B.5) Let/hatwidem=−g−1σ3gbe a solution of Eq. (B.1), then /hatwidem/hatwidemx=−g−1σ3U0σ3g+g−1σ2 3U0g=2g−1U0g, (B.6) Substitute (B.3) into (B.5), together with the definition of /hatwidemand (B.6), matrices (B.5) of new Lax pair (B.2) can be rewritten as /hatwideU=λ/hatwidem,/hatwideV=λQ/hatwidem−iλ/hatwidem/hatwidemx−2iλ2/hatwidem, (B.7) Using the factor /hatwidem2=I, the compatibility condition /hatwideUt−/hatwideVx+ [/hatwideU,/hatwideV] =0 exactly yields the matrix form LLG equation (B.1). Thus we proved the Lax gauge equivalence of the GNLS equation (A.5) and the dimensionless STT-LLG equation. Through the established gauge equivalence detailed above, it becomes evident that given a non-zero solution Φof the GNLS equation (A.5), the corresponding eigenfunctions can be derived via Lax pair (B.2). This process thereby elucidates the determination of the invertible matrices Ω(x,t,λ)andg(x,t). Further through the transformation /hatwideΦ=g−1Φand/hatwidem=−g−1σ3g, we are enabled to acquire the solution /hatwidemfor (B.1). Finally, the three components of magnetization, namely mx,my, and mz, can be obtained from the definition of /hatwidem, constituting the non-trivial solution to the original STT-LLG equation. To obtain the dynamical magnetic soliton in the ferromagnetic nanowire, we are going to construct the Darboux transformation of (A.5). Let Φ[0] 1(x,t,λ)andΦ[0] 2(x,t,λ)be the eigenfunction of the Lax pair (B.2) corresponding to the zero solution of the GNLS equation (A.5). Demonstrating the reciprocity of the Lax pair solution with respect to spectral parameters, it is straightforward to establish that if/parenleftbig Φ[0] 1(x,t,λ1),Φ[0] 2(x,t,λ1)/parenrightbigTrepresents the solution for Lax pair (B.2) corresponding to the spectral parameter λ1, then/parenleftbig Φ[0]∗ 2(x,t,λ∗ 1),−Φ[0]∗ 1(x,t,λ∗ 1)/parenrightbigTconstitutes the solution for the corresponding spectral parameter λ∗ 1. Denote H1=/parenleftigg Φ[0] 1Φ[0]∗ 2 Φ[0] 2−Φ[0]∗ 1/parenrightigg ,Λ1=/parenleftigg λ10 0λ∗ 1/parenrightigg , (B.8) where/parenleftbig Φ[0] 1,Φ[0] 2/parenrightbigT=/parenleftbig exp[−λ1x+2iλ2 1t−λ1/integraltext Qdt],exp[λ1x−2iλ2 1t+λ1/integraltext Qdt]/parenrightbigT. The Darboux matrix is acquired through the standard procedure T[1]=λI−H1Λ1H−1 1, (B.9) leading to the solution Φ[1]=T[1]Φof new spectral problem. Therefore the Darboux transformation is written as Ψ[1](x,t) =Ψ[0](x,t)−2(λ1+λ∗ 1)Φ[0] 1Φ[0]∗ 2 |Φ[0] 1|2+|Φ[0] 2|2. (B.10) Taking λ1=a+ibwe get the soliton solution of GNLS equation Ψ[1](x,t) =−2asech[2a(x+4bt+Qt)]exp/bracketleftbig 2i/bracketleftbig (2a2−2b2−Q)t−bx/bracketrightbig/bracketrightbig , (B.11) and the corresponding eigenfunction Φ[1](x,t,λ) = (Φ[1] 1,Φ[1] 2)T. Substitute the above results into Ω(x,t,λ) =/parenleftigg Φ[1] 1Φ[1]∗ 2 Φ[1] 2−Φ[1]∗ 1/parenrightigg , g(x,t) =Ω(x,t,λ)/vextendsingle/vextendsingle λ=0,/hatwidem=−g−1σ3gin sequence, we finally obtain exact cycloidal chiral magnetic soliton solution in ferromagnetic nanowires under the influence of spin current injection4 mx=/hatwidem12+/hatwidem21 2=2a a2+b2[asinh(Ξ)sin(Γ)−bcosh(Ξ)cos(Γ)]·sech2(Ξ), my=/hatwidem21−/hatwidem12 2i=2a a2+b2[asinh(Ξ)cos(Γ)+bcosh(Ξ)sin(Γ)]·sech2(Ξ), mz=/hatwidem11=1−2a2 a2+b2sech2(Ξ),(B.12) with Ξ=2a/bracketleftbigg x+/integraldisplay (Q+4b)dt/bracketrightbigg ,Γ=2b/bracketleftbigg x+/integraldisplay (Q−2(a2−b2)/b)dt/bracketrightbigg , where aandbdescribe the wave number and the velocity of the chiral soliton. To gain deeper insight into the interaction dynamics between two chiral magnetic solitons, we continue to utilize gauge transformation (2.2.67) to construct two-soliton solutions based on the above single soliton solutions. The second-order Darboux matrix is expressed as follows H2=/parenleftigg Φ[1] 1Φ[1]∗ 2 Φ[1] 2−Φ[1]∗ 1/parenrightigg ,Λ2=/parenleftigg λ20 0λ∗ 2/parenrightigg ,T[2]=λI−H2Λ2H−1 2, (B.13) and the specific Darboux transformation form of the two-soliton solution is subsequently obtained Ψ[2](x,t) =Ψ[1](x,t)−2(λ2+λ∗ 2)Φ[1] 1Φ[1]∗ 2 |Φ[1] 1|2+|Φ[1] 2|2. (B.14) Taking λ1=a1+ib1,λ2=a2+ib2, after tedious simplification, we get two-soliton solution of GNLS equation Ψ[2](x,t) =4η1eiβ2cosh(α2)+η2eiβ1cosh(α1)+iη3/parenleftbig eiβ1sinh(α1)−eiβ2sinh(α2)/parenrightbig η4cosh(α1+α2)+η5cosh(α1−α2)+η6cos(β1−β2), (B.15) where α1=2a1(x+4b1t+/integraldisplay Qdt),β1=4(a2−b2)t−2b2(x+/integraldisplay Qdt), α2=2a2(x+4b2t+/integraldisplay Qdt),β2=4(a2−b2)t−2b2(x+/integraldisplay Qdt), η1= [(a2 2−a2 1)−(b2−b1)2]a1,η4=−(a2−a1)2−(b2−b1)2,η3=2a1a2(b2−b1), η2= [(a2 1−a2 2)−(b1−b2)2]a2,η5=−(a2+a1)2−(b2−b1)2,η6=4a1a2. Continuing with the same approach in the previous text, we are able to provide a precise expression for the three-component of the magnetization mfor the dynamic chiral magnetic two-solitons. Owing to the complexity of its explicit expression, we opt to omit it and solely showcase the corresponding figure. Two typical solutions for the interaction between two chiral magnetic solitons are shown in Fig.(I). C. Chirality of cycloidal chiral magnetic soliton in Bloch sphere As shown in the main text, we denote the polar and azimuthal angles of the vector masθandϕ, respectively. This notation allows us to express M+=mx+imy=M0sin(θ)exp(iϕ)andmz=M0cos(θ). By employing the three-component analytical formulation, we can infer the inverse solution for θandϕ, which in turn can be mapped onto the Bloch unit sphere. This approach yields a trajectory map delineating the movement of chiral magnetic solitons across the unit sphere. Consequently, within the magnetization unit sphere, a chiral magnetic soliton traces a closed curve encompassing a single pole. The trajectories of motion for the two distinct types of chiral magnetic soliton solutions on the Bloch spheres can indirectly manifest their chirality. Commencing from negative infinity, which corresponds to the pole of ground state, the left and right-handed chiral5 (a) (b) Figure I: The interaction between two chiral magnetic solitons. (a) Interaction between left-handed and right-handed magnetic soliton. (b) Bound states formed by two right-handed magnetic Solitons. Figure II: Trajectories of chiral magnetic solitons on the Bloch sphere at time t=0. (a) Left-handed chiral magnetic soliton a=1,b=1, (b) Right-handed chiral magnetic soliton a=1,b=−1. magnetic solitons will give rise to enclosed paths, one proceeding in a clockwise direction and the other counterclockwise. This motion pattern eventually in mirror-symmetrical trajectories. ∗zyyang@nwu.edu.cn †jing@nwu.edu.cn

2023-08-16

The Landau-Lifshitz-Gilbert (LLG) equation has emerged as a fundamental and indispensable framework within the realm of magnetism. However, solving the LLG equation, encompassing full nonlinearity amidst intricate complexities, presents formidable challenges. In this context, we develop a precise mapping through geometric representation, establishing a direct linkage between the LLG equation and an integrable generalized nonlinear Schr\"odinger equation. This novel mapping provides accessibility towards acquiring a great number of exact spatiotemporal solutions. Notably, exact chiral magnetic solitons, critical for stability and controllability in propagation with and without damping effects are discovered. Our formulation provides exact solutions for the long-standing fully nonlinear problem, facilitating practical control through spin current injection in magnetic memory applications.

Discovery and regulation of chiral magnetic solitons: Exact solution from Landau-Lifshitz-Gilbert equation

2308.08331v1

Magnetoelastic Gilbert damping in magnetostrictive Fe 0.7Ga 0.3thin films W. K. Peria,1X. Wang,2H. Yu,2S. Lee,2, 3I. Takeuchi,2and P. A. Crowell1,∗ 1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA 2Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742, USA 3Department of Physics, Pukyong National University, Busan 48513, South Korea We report an enhanced magnetoelastic contribution to the Gilbert damping in highly magne- tostrictive Fe 0.7Ga0.3thin films. This effect is mitigated for perpendicular-to-plane fields, leading to a large anisotropy of the Gilbert damping in all of the films (up to a factor of 10 at room tem- perature). These claims are supported by broadband measurements of the ferromagnetic resonance linewidths over a range of temperatures (5 to 400 K), which serve to elucidate the effect of both the magnetostriction andphonon relaxation on the magnetoelastic Gilbert damping. Among the primary considerations in the design of spintronics devices is Gilbert damping. However, a full understanding of the mechanisms which cause damping of magnetization dynamics in ferromagnets remains elu- sive. Reports of anisotropy in the Gilbert damping have proven to be useful tools in the understanding of the un- derlying mechanisms involved [1–3], but there is much that is yet unclear. Studies of the temperature depen- dence also promise to be a uniquely powerful tool for a complete physical understanding [4, 5], however, there are few such reports in existence. Recently, it has been shown that spins can be co- herently coupled over large distances ( ∼1 mm) using magnon-phonon coupling [6–8]. It is also well known that magnetization dynamics can be excited elastically through this phenomenon [9], but its effect on Gilbert damping has been largely confined to theoretical calcu- lations [10–13] and lacks clear experimental validation. Furthermore, most studies have focused on yttrium iron garnet (YIG), which is weakly magnetostrictive. In this Letter, we observe a large and anisotropic mag- netoelastic contribution to the Gilbert damping in highly magnetostrictive Fe 0.7Ga0.3films through broadband measurements of the ferromagnetic resonance (FMR) linewidths over a wide range of temperatures. The perpendicular-to-plane linewidths exhibit a relatively low minimum in the Gilbert damping of approximately 0.004, similar to that of bcc Fe [14]. At room temperature, the Gilbert damping is as large as a factor of 10 greater with field applied in plane relative to out of plane. In fact, for any given sample and temperature, the anisotropy is, at minimum, about a factor of 2. We argue this is due to a mitigation of the magnetoelastic contribution for per- pendicular magnetization, arising from finite-thickness boundary conditions and weak elastic coupling to the substrate. The nonmonotonic temperature dependence of the Gilbert damping also shows the competing effects of the magnetostriction, which increases at low tempera- ture, and the phonon viscosity, which generally decreases at low temperature. The Fe 0.7Ga0.3films studied in this letter were de- posited on SiO 2/Si wafers at room temperature by dcmagnetron sputtering of an Fe 0.7Ga0.3target. The base pressure of the deposition chamber was 5 ×10−8torr, and the working pressure was kept at 5 ×10−3torr with Ar gas. The composition of the Fe 0.7Ga0.3films was quantitatively analyzed by energy dispersive spec- troscopy (EDS). Films were grown with thicknesses of 21 nm, 33 nm, 57 nm, and 70 nm (the 21 nm, 57 nm, and 70 nm belong to the same growth series). An addi- tional 33 nm film was grown at 200◦C. The 33 nm room temperature deposition was etched using an ion mill to obtain films with thicknesses of 17 nm and 26 nm. The thicknesses of the films were measured using x-ray reflec- tometry (see Supplemental Material). The FMR linewidths were measured using a setup in- volving a coplanar waveguide and modulation of the ap- plied magnetic field for lock-in detection as described in Ref. [15]. Measurements were done with the field applied in the plane (IP) and perpendicular to the plane (PP) of the film. The sample temperature was varied from 5 K to 400 K for both IP and PP configurations [16] with microwave excitation frequencies up to 52 GHz. The res- onance fields and linewidths were isotropic in the plane, and the absence of in-plane magnetic anisotropy was ver- ified with vibrating sample magnetometry (see Supple- mental Material). This is also consistent with the abun- dance of grain boundaries observed with atomic force mi- croscopy (AFM). In analyzing the FMR linewidths, we consider three contributions: Gilbert damping 4 παf/γ (αis the Gilbert damping coefficient, fis the microwave frequency, and γis the gyromagnetic ratio), inhomo- geneous broadening ∆ H0, and two-magnon scattering ∆HTMS (for IP fields). Eddy current damping and ra- diative damping contributions [17] are neglected because we expect them to be small ( <10−4) for these films. Linewidths of the 70 nm film at 300 K for both con- figurations of the applied field are shown in Fig. 1(a), and the IP linewidths with individual contributions to the linewidth plotted separately in Fig. 1(b). We fit the IP linewidths using a model of two-magnon scattering based on granular defects [15, 18, 19]. The fit for the 70 nm film is shown in Fig. 1(b), along with the two- magnon contribution alone given by the magenta curve.arXiv:2103.07008v1 [cond-mat.mtrl-sci] 11 Mar 20212 01 02030405005001000150020000500100015002000/s61508 H0/s61508HTMS/s61508H (Oe)F requency (GHz)H in-plane/s61508 HGilbertT = 300 K/s61537 = 0.0390 /s61617 0.0005/s61560 = 17 nm(a)( b)/s61508 H (Oe)T = 300 K/s61537 IP = 0.0390 /s61617 0.0005/s61537 PP = 0.0035 /s61617 0.0001/s61508HIP/s61508 HPP70 nm film FIG. 1. (a) FMR linewidths for IP (black squares) and PP (red circles) configurations for the 70 nm film. The IP linewidths are fit to a model of two-magnon scattering and the PP linewidths are fit using the standard Gilbert damping model. (b) Total linewidth (solid black), Gilbert linewidth (dotted blue), two-magnon scattering linewidth (dashed ma- genta), and inhomogeneous broadening (dashed/dotted red) for the 70 nm film with IP field. The fit parameters are the Gilbert damping α(indicated on the figure) and the RMS inhomogeneity field H/prime. The defect correlation length ξis fixed to 17 nm based on the structural coherence length obtained with x-ray diffrac- tion (XRD), which agrees well with the average grain di- ameter observed with AFM (see Supplemental Material). Furthermore, the high-frequency slope of the linewidths approaches that of the Gilbert damping since the two- magnon linewidth becomes constant at high frequencies [see Fig. 1(b)]. We now compare the IP and PP linewidths of the 70 nm film shown in Fig. 1(a). The two-magnon scat- tering mechanism is inactive with the magnetization per- pendicular to the plane [20], and so the PP linewidths are fit linearly to extract the Gilbert damping. We obtain a value of 0.0035±0.0001 for PP fields and 0 .039±0.0005 for IP fields, corresponding to an anisotropy larger than a factor of 10. Li et al. [3] recently reported a large anisotropy (∼factor of 4) in epitaxial Co 50Fe50thin films. First we discuss the dependence of the PP Gilbert dampingαPPon temperature for all of the films, shown in Fig. 2. We observe a significant temperature depen- 01 002 003 004 00024685 7 nm 17 nm2 6 nm3 3 nm (RT dep)/s61537 (×10-3)T emperature (K)21 nm 3 3 nm (200 °C dep)H perpendicular-to-plane70 nmFIG. 2. Gilbert damping αfor PP field shown as a func- tion of temperature for the 17 nm (orange), 21 nm (blue), 26 nm (green), 33 nm room temperature deposition (ma- genta), 33 nm 200◦C deposition (gold), 57 nm (red), and 70 nm (black) Fe 0.7Ga0.3films. dence in all cases (with the exception of the 33 nm room temperature deposition), characterized by a maximum at around 50 K. Then, at the lowest temperatures (5 to 10 K),αPPapproaches the same value for all of the films (/similarequal0.004). Now we turn to the temperature dependence of the IP Gilbert damping αIPshown in Fig. 3. The values obtained here were obtained by fitting the linewidths lin- early, but excluding the low-frequency points ( <∼20 GHz) since the two-magnon scattering becomes constant at high frequencies [21]. Here we note, upon comparison with Fig. 2, that a large anisotropy of the Gilbert damp- ing exists for all of the samples. In the 70 nm film, for instance,αIPis more than a factor of 10 larger than αPP at 300 K. In the temperature dependence of αIP, we ob- serve behavior which is similar to that seen in αPP(Fig. 2), namely, a maximum at around 50 K (with the excep- tion of the 21 nm film). Here, however, αIPdoes not approach a common value at the lowest temperatures in all of the samples as it does in the PP case. The IP Gilbert damping is larger than the PP Gilbert damping for all of the samples over the entire range of temperatures measured. This anisotropy of the Gilbert damping—along with the nonmonotonic temperature dependence—in all seven samples implies a contribution to the Gilbert damping in addition to Kambersk´ y damp- ing. We have verified that the orientation of FeGa(110)3 planes is completely random with XRD for the 33 nm (both depositions) and 70 nm films (see Supplemen- tal Material), and it is therefore not possible that the anisotropy is due to Kambersk´ y damping. Interface anisotropy has reportedly led to anisotropic Kambersk´ y damping in ultrathin ( ∼1 nm) films of Fe [2], but this is highly unlikely in our case due to the relatively large thicknesses of the films. In addition, the fact that the damping anisotropy shows no clear correlation with film thickness furthers the case that intrinsic effects, which tend to show a larger anisotropy in thinner films [2], cannot be the cause. The longitudinal resistivity ρxxof the 33 nm (both depositions) and 70 nm films (see Sup- plemental Material) shows very weak temperature de- pendence. In the Kambersk´ y model, the temperature dependence of the damping is primarily determined by the electron momentum relaxation time τ, and we would therefore not expect the Kambersk´ y damping to show a significant temperature dependence for samples where the residual resistivity ratio is approximately unity. It is plausible that the Kambersk´ y damping would still show a temperature dependence in situations where the spin polarization is a strong function of temperature, due to changes in the amount of interband spin-flip scattering. This kind of damping, however, would be expected to decrease at low temperature [22, 23]. The temperature dependence we observe for both αPPandαIPis therefore inconsistent with Kambersk´ y’s model, and the similarity between the two cases in this regard suggests that the enhanced Gilbert damping has a common cause that is mitigated in the PP configuration. It has been proposed that magnetoelastic coupling can lead to Gilbertlike magnetization damping through phonon relaxation processes [10, 12, 24]. Similar treat- ments calculate the magnetoelastic energy loss through interaction with the thermal population of phonons [11, 25]. The Kambersk´ y mechanism is often assumed to be the dominant Gilbert damping mechanism in metal- lic samples, so magnetoelastic Gilbert damping is usually studied in magnetic insulators, particularly yttrium iron garnet (YIG). There is the possibility, however, for the magnetoelastic damping to dominate in metallic samples where the magnetostriction is large, such as in Fe-Ga al- loys. Later we will discuss how magnetoelastic damping can be mitigated in thin films by orienting the magneti- zation perpendicular to the plane, and how the degree to which it is mitigated depends on the boundary conditions of the film. Here we outline a theory of magnetoelastic damping, which relies on the damping of magnetoelastic modes through phonon relaxation mechanisms. Figure 4 illus- trates the flow of energy through such a process. Analyt- ically, the procedure is to equate the steady-state heating rate due to Gilbert damping to the heating rate due to crystal viscosity, and solve for the Gilbert damping α in terms of the crystal shear viscosity ηand the mag- 01 002 003 004 000123453 3 nm (200 °C dep)17 nm2 6 nm/s61537 (×10-2)T emperature (K)33 nm (RT dep)57 nm2 1 nmH in-plane7 0 nmFIG. 3. Gilbert damping αfor IP field shown as a func- tion of temperature for the 17 nm (orange), 21 nm (blue), 26 nm (green), 33 nm room temperature deposition (ma- genta), 33 nm 200◦C deposition (gold), 57 nm (red), and 70 nm (black) Fe 0.7Ga0.3films. netostrictive coefficients λhkl. Shear strain uijresult- ing from the magnetoelastic interaction can be expressed asuij=λ111mimj[26], where mi≡Mi/Msare the reduced magnetizations. The leading-order shears thus have equations of motion given by ˙ uiz=λ111˙mi, where i=xory, andzis the direction of the static magnetiza- tion so that mz≈1. Longitudinal modes are quadratic in the dynamical component of the magnetization [24] and so will be neglected in this analysis. The heating rate due to Gilbert damping can be writ- ten as ˙Qα=Ms γα( ˙m2 x+ ˙m2 y), and the heating rate due to the damping of phonon modes as ˙Qη= 4η( ˙u2 xz+ ˙u2 yz) = 4ηλ2 111( ˙m2 x+ ˙m2 y) [12], with the factor of 4 accounting for the symmetry of the strain tensor. Equating the two, and solving for α(henceforward referred to as αme), we obtain αme=4γ Msηλ2 111. (1) We will restrict our attention to the case of isotropic mag- netostriction, and set λ111=λ. In order to use Eq. 1 to estimate αmein our films, we first estimate the shear viscosity, given for transverse phonons with frequency ωand relaxation time τas [27] η=2ρc2 t ω2τ, (2)4 (b) u(t) phonon pumpingM(t) H0 M(t) u(t)(a) kph dH0 FIG. 4. (a) Depiction of magnetoelastic damping process for magnetization in plane and (b) perpendicular to plane, where M(t) is the magnetization vector and u(t) is the lattice displacement. In panel (b), the magnon-phonon conversion process is suppressed when d < π/k ph, wheredis the film thickness and kphis the transverse phonon wavenumber at the FMR frequency. whereρis the mass density and ctis the transverse speed of sound. Using ω/2π= 10 GHz, τ= 10−11s, andct= 2.5 km/s, we obtain η≈2.3 Pa s. (The estimate of the phonon relaxation time is based on a phonon mean free path of the order of the grain size: ∼10 nm.) Furthermore, the magnetostriction of an equiv- alent sample has been measured to be ∼100 ppm at room temperature [28]. Then, with γ/2π= 29 GHz/T and Ms= 1123 emu/cc (extracted from FMR data taken at 300 K), we estimate αme≈0.016. This estimate gives us immediate cause to suspect that magnetoelastic Gilbert damping is significant (or even dominant) in these films. We now discuss why the magnetoelastic damping can be much weaker for PP magnetization in sufficiently thin films. We will start by assuming that there is no coupling between the film and substrate, and later we will relax this assumption. In this case the only phonons excited by the magnetization, to leading-order in the magneti- zations and strains, are transverse modes propagating in the direction of the static magnetization [24]. One may assume that the minimum allowable phonon wavenum- ber is given by π/d, wheredis the film thickness, since this corresponds to the minimum wavenumber for a sub- strate having much lower acoustic impedance than the film (requiring the phonons to have antinodes at the in- terfaces) [13]. (We also assume an easy-axis magnetic anisotropy at the interfaces, so that the dynamical mag- netizations have antinodes at the interfaces.) We expect then that the magnetoelastic damping will be suppressed for cases where the phonon wavelength, at the frequencyof the precessing magnetization, is greater than twice the film thickness [see Fig. 4(b)]. Thus, in sufficiently thin films (with weakly-coupled substrates), the magnetoelas- tic damping process can be suppressed when the mag- netization is perpendicular to the plane. However, the magnetoelastic damping can be active (albeit mitigated) when there is nonnegligible or “intermediate” coupling to the substrate. Before moving on, we briefly note the implications of Eq. (1) for the temperature dependence of the Gilbert damping. On the basis of the magnetostriction alone, αmewould be expected to increase monotonically as tem- perature is decreased ( λhas been shown to increase by nearly a factor of 2 from room temperature to 4 K in bulk samples with similar compositions [29]). However, the viscosity ηwould be expected to decrease at low tem- perature, leading to the possibility of a local maximum inαme. In polycrystalline samples where the grain size is smaller than the phonon wavelength, viscous damping of phonons due to thermal conduction caused by stress inhomogeneities can be significant [27, 30]. (In our case the phonon wavelengths are ∼100 nm and the grain sizes are∼10 nm.) This effect scales with temperature asη∼Tα2 T/Cχ [30], where αTis the thermal expansion coefficient, Cis the specific heat at constant volume, and χis the compressibility. At higher temperatures, αTand Cwill approach constant values, and χwill always de- pend weakly on temperature. We therefore expect that the viscosity is approximately linear in T. In this case, αmeis maximized where λ2(T) has an inflection point. We proceed to explain our data in terms of the mecha- nism described above, turning our attention again to the PP Gilbert damping for all of the films shown in Fig. 2. We previously argued that the magnetoelastic damp- ing mechanism will be suppressed for the case where the acoustic impedances of the film and substrate are mis- matched. However, the clear dependence on tempera- ture, which we have already shown is inconsistent with Kambersk´ y damping, appears to be consistent with the magnetoelastic damping mechanism. We estimate that the acoustic impedance of the film (defined as the product of mass density ρand transverse speed of sound ct[13]) is about a factor of 2 larger than the substrate. This sug- gests that the elastic coupling between the film and sub- strate, albeit weak, may be nonnegligible. Furthermore, experiments with YIG/GGG heterostructures (where the acoustic match is good) have demonstrated magnetic ex- citation of phononic standing waves that have boundary conditions dictated by the combined thickness of the film and substrate, rather than the film thickness alone (i.e., the wavelengths are much larger than the film thickness) [6, 31]. In this case, the Gilbert damping may contain some contribution from the magnetoelastic mechanism. A final point is that αPPapproaches/similarequal0.004 at 5 to 10 K for all of the films. Both the magnetostriction and the viscosity are quantities which could have significant5 01 0020030040002468/s61537me (×10-2)T emperature (K)Ref. [29]2 1 nm70 nm5 7 nm0 .000.250.500.751.00/s61548 2(T)//s615482(0)010020030001/s61544 (arb. units)T (K) FIG. 5. Magnetoelastic Gilbert damping αmefor the 21 nm (blue), 57 nm (red), and 70 nm (black) films (left ordinate) andλ2(T)/λ2(0) from Clark et al. [29] (magenta; right or- dinate) shown as a function of temperature. Inset shows the ratio ofαmeandλ2(T)/λ2(0), labeled as η(T), along with lin- ear fits for the 21 nm (blue), 57 nm (red), and 70 nm (black) films. variation between samples, leading to variations in αme. However, the viscosity becomes small at low temperature, which means that the Gilbert damping will approach the Kambersk´ y “limit,” a property that is determined by the electronic structure, implying that the Kambersk´ y damp- ing is/similarequal0.004 in these films and that it is the primary contribution to the Gilbert damping near T= 0. Now we revisit the IP Gilbert damping shown in Fig. 3. In this configuration, there is a strong temperature dependence of the Gilbert damping similar to that of the PP case, again implying the presence of magnetoe- lastic damping. However, the overall magnitude is much higher. That is because in this case arbitrarily long wave- length phonons can be excited regardless of the thick- ness of the film. Although we cannot directly measure the magnetostriction as a function of temperature, we estimate the scaling behavior of λby interpolating the data in Ref. [29] taken for bulk samples of similar com- position. In order to demonstrate that αIPscales with temperature as expected from the model, we have plot- ted the quantities αmeandλ2(T)/λ2(0) as functions of temperature in Fig. 5—where we define the quantity αme≡αIP−0.004—for the 21 nm, 57 nm, and 70 nm films (which are part of the same growth). The corre- lation between the two quantities is not completely con- vincing. There is, however, an additional temperature dependence in αmebesidesλ2(T), namely, the viscosity η(T). The inset of Fig. 5 shows the ratio of αmeand λ2(T), which [from Eq. (1)] is proportional to η(T). The linear fits provide strong evidence that the mechanism behind the viscosity is indeed the thermal conduction process that we have argued is approximately linear inT. It is noteworthy that the maximum in αme(∼50 to 75 K for all of the samples) coincides approximately with the inflection point in λ2(T). This was a consquence of our assumption that η(T) should be roughly linear. We also obtain a significant value for the zero-temperature viscosity, which is around 25 % of the value at 300 K. This is likely due to boundary-scattering processes which will preventαmefrom going to zero at low temperatures, par- ticularly for in-plane magnetization where αmeis much larger than 0.004 (our estimate for the Kambersk´ y damp- ing). For the PP case, αmeis much smaller due to limita- tions on the wavelengths of phonons that can be excited, so the Gilbert damping of all the samples approaches the Kambersk´ y limit of 0.004 near zero temperature. We also found that η(T) was linear for the 33 nm (200◦C depo- sition) film, but had a more complicated dependence on Tfor the 17 nm, 26 nm, and 33 nm (room temperature deposition) films (the latter three being notably of the same growth). The viscosity near zero temperature is within roughly a factor of 2 for all seven of the samples, however. Finally, we propose that this mechanism may be re- sponsible for a Gilbert damping anisotropy of similar magnitude reported in Ref. [3], observed in an epitax- ial Co 0.5Fe0.5thin film. The authors attributed the anisotropy to the Kambersk´ y mechanism [22, 23, 32, 33], arising from tetragonal distortions of the lattice. The magnetostriction is known to be highly anisotropic in bulk Co 0.5Fe0.5,viz.,λ100= 150 ppm and λ111= 30 ppm [34]. We therefore expect that the Gilbert damping aris- ing from the mechanism we have described may be much larger for M/bardbl(110) than M/bardbl(100), which is precisely what the authors observed. In summary, we observe large and anisotropic magne- toelastic Gilbert damping in Fe 0.7Ga0.3polycrystalline thin films (thicknesses ranging from 17 to 70 nm). At 300 K, the damping coefficient is more than a factor of 10 larger for field in plane than it is for field perpendicu- lar to the plane in the 70 nm film. The large anisotropy is caused by a mitigation of the magnetoelastic effect for perpendicular-to-plane fields due to a dependence on the elastic coupling of the film to the substrate, which in our case is weak. Finally, there is a nonmonotonic tempera- ture dependence of the Gilbert damping, which we show is consistent with our model. We acknowledge Rohit Pant and Dyland Kirsch for as- sistance with thin film deposition and characterization. This work was supported by SMART, a center funded by nCORE, a Semiconductor Research Corporation pro- gram sponsored by NIST. Parts of this work were carried out in the Characterization Facility, University of Min- nesota, which receives partial support from NSF through the MRSEC program, and the Minnesota Nano Cen- ter, which is supported by NSF through the National Nano Coordinated Infrastructure Network, Award Num- ber NNCI - 1542202.6 ∗Author to whom correspondence should be addressed: crowell@umn.edu [1] K. Gilmore, M. D. Stiles, J. Seib, D. Steiauf, and M. F¨ ahnle, Anisotropic damping of the magnetization dynamics in Ni, Co, and Fe, Phys. Rev. B 81, 174414 (2010). [2] L. Chen, S. Mankovsky, S. Wimmer, M. A. W. Schoen, H. S. K¨ orner, M. Kronseder, D. Schuh, D. Bougeard, H. Ebert, D. Weiss, and C. H. Back, Emergence of anisotropic Gilbert damping in ultrathin Fe layers on GaAs(001), Nat. Phys. 14, 490 (2018). [3] Y. Li, F. Zeng, S. S.-L. Zhang, H. Shin, H. Saglam, V. Karakas, O. Ozatay, J. E. Pearson, O. G. Heinonen, Y. Wu, A. Hoffmann, and W. 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Furuya, Combina- torial investigation of magnetostriction in Fe–Ga and Fe–Ga–Al, Appl. Phys. Lett. 93, 102507 (2008). [29] A. E. Clark, M. Wun-Fogle, J. B. Restorff, K. W. Dennis, T. A. Lograsso, and R. W. McCallum, Temperature de- pendence of the magnetic anisotropy and magnetostric- tion of Fe 100−xGax(x= 8.6, 16.6, 28.5), J. Appl. Phys. 97, 10M316 (2005). [30] C. Zener, Internal Friction in Solids II. General The- ory of Thermoelastic Internal Friction, Phys. Rev. 53, 90 (1938). [31] M. Ye, A. Brockmeyer, P. E. Wigen, and H. D¨ otsch, Magnetoelastic resonances in epitaxial garnet films, Le J. Phys. Colloq. 49, C8 (1988). [32] V. Kambersk´ y, On the Landau-Lifshitz relaxation in fer-7 romagnetic metals, Can. J. Phys. 48, 2906 (1970). [33] V. Kambersk´ y, On ferromagnetic resonance damping in metals, Czechoslov. J. Phys. 26, 1366 (1976). [34] R. C. Hall, Magnetic Anisotropy and Magnetostrictionof Ordered and Disordered Cobalt-Iron Alloys, J. Appl. Phys. 31, S157 (1960).Supplemental Material for “Magnetoelastic Gilbert damping in magnetostrictive Fe 0.7Ga 0.3thin films” W. K. Peria,1X. Wang,2H. Yu,2S. Lee,2, 3I. Takeuchi,2and P. A. Crowell1 1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA 2Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742, USA 3Department of Physics, Pukyong National University, Busan 48513, South Korea CONTENTS S1. Magnetization dynamics 1 S2. Ferromagnetic resonance linewidths of 70 nm film 2 S3. X-ray reflectivity 2 S4. X-ray diffraction 3 S5. Atomic force microscopy 3 S6. Vibrating sample magnetometry 3 S7. Longitudinal resistivity 3 References 4 S1. MAGNETIZATION DYNAMICS The treatment of magnetization dynamics begins with the Landau-Lifshitz-Gilbert equation of motion dM dt=−γM×Heff+α MsM×dM dt(S1) where the relaxation is characterized by the Gilbert damping parameter α. Upon linearizing this equation in the dynamic component of the magnetization, one obtains for the ac magnetic susceptibility of the uniform q= 0 mode χac(q= 0,ω)∝αω/γ (H−HFMR )2+ (αω/γ )2(S2) so that the field-swept full-width-at-half-maximum linewidth is given by ∆ HFWHM = 2αω/γ . Therefore, the Gilbert damping parameter αis obtained by measuring ∆ HFWHM as a function of ω. Relaxation of the uniform mode can include mechanisms which are not described by Gilbert damping. The most common of these is inhomogeneous broadening, which results from inhomogeneities in the system and is constant as a function of frequency. Another mechanism is two-magnon scattering, which is also extrinsic in nature. Two- magnon scattering originates from the negative group velocity at low qof the backward volume mode magnons for in-plane magnetization. The negative group velocity is due to a lowering of the magnetostatic surface charge energy for increasing q. The existence of negative group velocity at low qleads to the appearance of a mode at nonzero q that is degenerate with the uniform mode. Two-magnon scattering refers to the scattering of the uniform mode to the nonuniform degenerate mode. Much work has been done on the treatment of two-magnon scattering [S1–S3], and here we will simply give an expression for the contribution of two-magnon scattering to the field-swept linewidth ∆HTMS =γ2ξ2H/prime2 dω/dH/integraldisplay d2qΛ0q1 (1 + (qξ)2)3/21 πωα (ωα)2+ (ω−ωFMR )2(S3) withξthe defect correlation length, H/primethe RMS inhomogeneity field, and Λ 0qthe magnon-magnon coupling. In general, this leads to a nonlinear dependence of the linewidth on frequency. Eq. S3 is used to fit the IP linewidths.arXiv:2103.07008v1 [cond-mat.mtrl-sci] 11 Mar 20212 0102030405005001000150020002500(b) 5 K 50 K 150 K 225 K 300 K 400 K/s61508H (Oe)F requency (GHz)H in-plane 010203040500100200300400500 5 K 50 K 150 K 225 K 300 K 400 K/s61508H (Oe)F requency (GHz)H perpendicular-to-plane(a) FIG. S1. FMR linewidths of the 70 nm film with field PP (a) and IP (b) for sample temperatures of 5 K (blue), 50 K (gold), 150 K (black), 225 K (magenta), 300 K (red), and 400 K (orange). The solid lines are linear fits in both panels. In (b), the vertical dashed line indicates the lower bound of the points included in the fit. 12345610-1100101102103104105106(c)Intensity (arb. units)/s61553 /2/s61553 (degrees)d = 56.6 nm 12345610-1100101102103104105106(b)Intensity (arb. units)/s61553 /2/s61553 (degrees)200 °C depositiond = 33.1 nm 123410-1100101102103104105106(a)R T depositiond = 33.2 nm Intensity (arb. units)/s61553 /2/s61553 (degrees) FIG. S2. X-ray reflectivity data (black) overlaid with fits (red) for the (a) 33 nm (room temperature deposition), (b) 33 nm (200◦C deposition), and (c) 57 nm films. Thicknesses dobtained from the fits are indicated on the figure. S2. FERROMAGNETIC RESONANCE LINEWIDTHS OF 70 nm FILM The field-swept FMR linewidths of the 70 nm film are shown in Fig. S1 for field PP and IP. For the case of field IP, the data above 23 GHz were fit linearly to obtain the Gilbert damping. (This value varied between different samples since the characteristic roll-off frequency depends on both defect lengthscale and film thickness, but remained in the range 20 to 25 GHz.) It is safe to do this provided there are no inhomogeneities at lengthscales smaller than a few nm, which could cause the two-magnon scattering contribution to the linewidth to roll off at higher frequencies. We believe that defects at such small lengthscales are highly unlikely given the characterization performed on these samples. S3. X-RAY REFLECTIVITY In Fig. S2 we show x-ray reflectivity measurements at grazing incidence for 33 nm (room temperature and 200◦C depositions) and 57 nm films. The measurements were taken using a Rigaku SmartLab diffractometer. The thicknesses dyielded by the fits of the data are indicated on the figure.3 4243444546036912F e0.7Ga0.3(110)(a)I ntensity (arb. units)/s61553 /2/s61553 (degrees)33 nm (RT deposition)2 /s61553c= 44.14 /s61617 0.03 °F WHM = 0.78 /s61617 0.09 ° 4243444546036912F e0.7Ga0.3(110)(b)I ntensity (arb. units)/s61553 /2/s61553 (degrees)33 nm (200 °C deposition)2 /s61553c = 44.07 /s61617 0.05 °F WHM = 1.18 /s61617 0.17 ° 4243444546036912(c)F e0.7Ga0.3(110)Intensity (arb. units)/s61553 /2/s61553 (degrees)2/s61553c = 44.36 /s61617 0.02 °F WHM = 0.50 /s61617 0.07 °70 nm FIG. S3. X-ray diffraction symmetric θ/2θscans for (a) 33 nm room temperature deposition, (b) 33 nm 200◦C deposition, and (c) 70 nm films. Full width at half maxima (FWHM) and 2 θcenter positions are indicated on the figure. S4. X-RAY DIFFRACTION X-ray diffraction (XRD) measurements were performed in order to determine both the degree of orientation and the structural coherence length of the films. Symmetric θ/2θscans were taken with a Rigaku Smartlab diffractometer using Cu Kα 1(λ= 1.54˚A) radiation. The data for both samples are shown in Fig. S3. The grain size was estimated using the Scherrer formula for spherical grains [S4] as 13 nm, 9 nm, and 17 nm for the 33 nm (room temperature deposition), 33 nm (200◦C deposition), and 70 nm films respectively. Two-dimensional images were collected with a Bruker D8 Discover diffractometer using Co Kα 1(λ= 1.79˚A) radiation. Detector images showing the “ring” corresponding to the Fe 0.7Ga0.3(110) peak in four different samples are shown in Fig. S4. The ring indicates that the Fe 0.7Ga0.3(110) planes are randomly oriented over the range of the detector, which we take to be evidence that there is no texture over a macroscopic scale in these samples. Furthermore, the films were grown directly on top of amorphous SiO 2layers, so we do not expect an epitaxial relationship between the film and substrate. The Fe 0.7Ga0.3(110) peaks were the only measurable Bragg peaks since the structure factor is highest for this case. S5. ATOMIC FORCE MICROSCOPY Atomic force microscopy data are shown in Fig. S5 for the 33 nm (room temperature and 200◦C depositions), 57 nm, and 70 nm films. The field-of-view is 250 nm for the 33 nm films and 500 nm for the 57 nm and 70 nm films. The root-mean-square (RMS) roughness of the sample surfaces is 0.7 nm, 0.4 nm, 1.5 nm, and 1.3 nm for the 33 nm (room temperature deposition), 33 nm (200◦C deposition), 57 nm, and 70 nm films, respectively . S6. VIBRATING SAMPLE MAGNETOMETRY Vibrating sample magnetometry (VSM) data for the 33 nm (room temperature and 200◦C depositions) and 70 nm films are shown in Fig. S6. The magnetic field was applied in 3 different directions, with no discernible difference in the hysteresis loops. We conclude that there is no in-plane magnetocrystalline anisotropy over macroscopic lengthscales, which is consistent with the FMR measurements. S7. LONGITUDINAL RESISTIVITY Longitudinal resistivity ρxxwas measured as a function of temperature for the 33 nm (room temperature and 200◦C depositions) and 70 nm films (Fig. S7) by patterning Hall bars and performing 4-wire resistance measurements.4 65 60 55 50 45 40 35 2/s61553(°) 0100200300400Intensity (arb.units) 65 60 55 50 45 40 35 2/s61553(°) 0100200300400Intensity (arb.units) 65 60 55 50 45 40 35 2/s61553(°) 020406080Intensity (arb.units) 65 60 55 50 45 40 35 2/s61553(°) 020406080Intensity (arb.units)(b) (a) (c) (d)q xyz q xyz33 nm (RT) 33 nm (200 °C) 57 nm 70 nm FIG. S4. Two-dimensional detector images of the Fe 0.7Ga0.3(110) peak for (a) 33 nm (room temperature deposition), (b) 33 nm (200◦C deposition), (c) 57 nm, and (d) 70 nm films. The total scattering angle is 2 θand is shown on the abscissa. The measurement is conducted such that the symmetric configuration corresponds to the center of the detector, which is to say that the incident radiation is at an angle ω/similarequal26◦relative to the sample surface. In panel (a), the effect of moving vertically from the center of the detector on the scattering vector qis shown ( qis canted into the y-zplane). [S1] R. Arias and D. L. Mills, Extrinsic contributions to the ferromagnetic resonance response of ultrathin films, Phys. Rev. B 60, 7395 (1999). [S2] R. D. McMichael and P. Krivosik, Classical Model of Extrinsic Ferromagnetic Resonance Linewidth in Ultrathin Films, IEEE Trans. Magn. 40, 2 (2004). [S3] P. Krivosik, N. Mo, S. Kalarickal, and C. E. Patton, Hamiltonian formalism for two magnon scattering microwave relaxation: Theory and applications, J. Appl. Phys. 101, 083901 (2007).5 -2-1012Height(nm)(a) -2-1012Height(nm)(b) (c) (d) -2-1012Height(nm) -2-1012Height(nm)100 nm 100 nm100 nm 100 nm33 nm (RT) 57 nm33 nm (200 °C) 70 nm FIG. S5. Atomic force microscopy for (a) 33 nm (room temperature deposition), (b) 33 nm (200◦C deposition), (c) 57 nm, and (d) 70 nm films. RMS roughnesses are (a) 0.7 nm, (b) 0.4 nm, (c) 1.5 nm, and (d) 1.3 nm. [S4] M. Birkholz, Thin Film Analysis by X-Ray Scattering (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2006).6 -2000-1000010002000-101M/MsF ield (Oe) H || Si[100] H || Si[110] H || Si[010]70 nm(c) -200-1000100200-101 H || Si[100] H || Si[110] H || Si[010]M/MsF ield (Oe)33 nm ( 200 °C)(b) -50-2502550-101 H || Si[100] H || Si[110] H || Si[010]M/Ms F ield (Oe)33 nm (RT)(a) FIG. S6. Vibrating sample magnetometry of (a) 33 nm (room temperature deposition), (b) 33 nm (200◦C deposition), and (c) 70 nm films for H/bardblSi[100] (black), H/bardblSi[110] (red), and H/bardblSi[010] (blue). 0100200300400020040060080010007 0 nm(c)/s61554 xx (µΩ cm)T emperature (K) 0100200300400050100150200250/s61554xx (µΩ cm)T emperature (K)33 nm (200 °C deposition)(b) 0100200300400050100150200250(a)/s61554xx (µΩ cm)T emperature (K)33 nm (RT deposition) FIG. S7. Longitudinal resistivity ρxxas a function of temperature for the (a) 33 nm (room temperature deposition), (b) 33 nm (200◦C deposition), and (c) 70 nm films.

2021-03-11

We report an enhanced magnetoelastic contribution to the Gilbert damping in highly magnetostrictive Fe$_{0.7}$Ga$_{0.3}$ thin films. This effect is mitigated for perpendicular-to-plane fields, leading to a large anisotropy of the Gilbert damping in all of the films (up to a factor of 10 at room temperature). These claims are supported by broadband measurements of the ferromagnetic resonance linewidths over a range of temperatures (5 to 400 K), which serve to elucidate the effect of both the magnetostriction and phonon relaxation on the magnetoelastic Gilbert damping.

Magnetoelastic Gilbert damping in magnetostrictive Fe$_{0.7}$Ga$_{0.3}$ thin films

2103.07008v1

1 Predicting New Iron Garnet Thin Film s with Perpendicular Magnetic Anisotropy Saeedeh Mokarian Zanjani1, Mehmet C. Onbaşlı1,2,* 1 Graduate School of Materials Science and Engineering, Koç University , Sarıyer, 34450 Istanbul, Turkey . 2 Department of Electrical and Electronics Engineering, Koç University, Sarıyer, 34450 Istanbul, Turkey. * Corresponding Author: monbasli@ku.edu.tr Abstract: Perpendicular magnetic anisotropy (PMA) is a necessary condition for many spintronic applications like spin -orbit torques switching , logic and memory devices. An important class of magnetic insulators with low Gilbert damping at room temperature are iron garnets, which only have a few PMA types such as terbium and samarium iron garnet. More and stable PMA garnet options are necessary for researchers to be able to investigate new spintronic phenomena. In this study, we predict 20 new substrate/magnetic iron garnet film pairs with stable PMA at room temperature. The effective anisotropy energies of 10 different garnet films that are lattice -matched to 5 different commercially available garnet substrates have been calculated using shape, magnetoelastic and magnetocrystalline anisotropy terms . Strain type, tensile o r compressive depending on substrate choice, as well as the sign and the magnitude of the magnetostriction constants of garnets determine if a garnet film may possess PMA. We show the conditions in which Samarium, Gadolinium, Terbium, Holmium, Dysprosium a nd Thulium garnets may possess PMA on the investigated garnet substrate types . Guidelines for obtaining garnet films with low damping are presented. New PMA garnet film s with tunable saturation moment and field may improve spin - orbit torque memory and compensated magnonic thin film devices. 2 Introduction With the development of sputtering and pulsed laser deposition of high -quality iron garnet thin films with ultralow Gilbert damping, researchers have been able t o investigate a wide variety of magnetization switching and spin wave phenomena1-3. The key enabler in many of these studies has been Yttrium iron garnet (Y 3Fe5O12, YIG)4 which has a very low Gilbert damping allowing spin waves to propagate over multiple millimeters across chip. YIG thin films are useful for spin wave device applications, but since their easy axes lie along film plane, their utility cannot b e extended to different mechanisms such as spin -orbit torques, Rashba -Edelstein effect, logic devices, forward volume magnetostatic spin waves1. At the same time, to have reliable and fast response using low current densities as in spin -orbit torque switching, magnetization orientation needs to be perpendicular to the surface plane5. The possibility of having Dzyaloshinskii –Moriya interaction (DMI) in TmIG/GGG may enable stabilizing skyrmions and help drive skyrmion motion with pure spin currents6. There is a number of studies on tuning anisotropy or obtaining perpendicular magnetic anisotropy in insulator thin films7-12. Among the materials studied, insulating magnetic garnet s whose magnetic pr operties can be tuned have been a matter of interest over the past decades13-15 due to their low damping and high magnetooptical Faraday rotation. In order to obtain perpendicular magnetic anisotropy in magnetic garnets, one needs to engineer the anisotropy terms that give rise to out -of-plane easy axis. Angular dependence of total m agnetization energy density is called magnetic anisotropy energy and consists of contributions from shape anisotropy, strain -induced (magnetoelastic) and magnetocrystalline anisotropy. A magnetic material preferentially relaxes its magnetization vector tow ards its easy axis, which is the least energy axis , when there is no external field bias . Such energy minimization process drives magnetic switching rates as well as the stability of total magnetization v ector . Controlling magnetic anisotropy in thin film garnets not only offers researchers different testbeds for experimenting new PMA -based switching phenomena, but also allows the investigation of anisotropy -driven ultrafast dynamic magnetic response in thin film s and nanostructures. The most extensively studied garnet thin film is Yttrium Iron Garnet ( YIG). YIG films display in - plane easy axis because of their large shape anisotropy and negligible magnetocrystalline anisotropy3. Although PMA of ultrathin epitaxial YIG films has been reported16,17, the tolerance 3 for fabrication condition variations for PMA YIG is very limited and strain effects were found to change magnetocrystalline anisotropy in YIG. Strain -controlled anisotr opy has been observed in polycrystalline ultrathin YIG films17,18. In case of YIG thin film grown on Gadolinium Gallium Garnet (GGG), only partial anisotropy control has been possible through significant change in oxygen stoichiometry19, which increases damping. Since the fabrication of high -quality and highly PMA YIG films is not easy for practical thicknesses on gadolinium gallium garnet substrates (GGG), researchers have explored tuning magnetic anisotropy by substituting Yttrium sites with other rare earth elements20,21. New garnet thin films that can exhibit PMA with different coercivities, saturation fields, compensation points and tunable Gilbert damping values must be developed in order to evaluate the effect of these p arameters on optimized spintronic insulator devices . Since the dominant anisotropy energy term is s hape anisotropy in thin film YIG , some studies focus on reducing the shape anisotropy contribution by micro and nanopatterning22-24. Continuous YIG films were etched to form rectangular nanostrips with nanometer -scale thickness es, as schematic ally shown on Fig. 1(a) . Thus, least magnetic saturation field is needed along the longest dimension of YIG nanostrips . By growing ultrathin YIG, magnetoelastic strain contributions lead to a negative anisotropy field and thus PMA in YIG film s24. As the length -to-thickness ratio decreases , the effect of shape anisotropy is reduced and in-plane easy axis is converted to PMA17. While reducing the effect of shape anisotropy is necessary, one also needs to use m agnetoelastic anisotropy contribut ion to reorient magnetic easy axis towards out of film plane , as schematically shown on Fig. 1 (b). Strain-induced perpendicular magnetic anisotropy in rare earth (RE) iron garnets, especially in YIG, has been demonstrated to overcome shape anisotropy16,17,25,26. If magnetoelastic anisotropy term induced by crystal lattice mismatch is large r than shape anisotropy and has opposite sign, then magnetoelastic anisotropy overcome s shape . Thus, the easy axis of the film becomes perpendicular to the film plane and the hyster esis loop becomes square -shaped with low saturation field8. One can also achieve PMA in other RE magnetic iron garnets due to their lattice parameter mismatch with their substrates . PMA has previously been achieved using Substituted Gadolinium Gallium Garnet (SGGG) as substrate and a Samarium Gallium Garnet (SmGG) ultrathin film as buffer layer under (and on) YIG16. In case of thicker YIG films (40nm), the magnetic easy axis becomes in -plane again. An important case shown by Kubota et.al19 indicates that increasing in-plane strain (ε ||) or anisotropy field (H a) helps achieve perpendicular 4 magnetic anisotropy. In ref.8,19, they reported that if magnetostriction coefficient (λ 111) is negative and large eno ugh to overcome shape anisotropy, and tensile strain is introduced to the thin film sample (ε ||>0), the easy axis becomes perpendicular to the sample plane as in the case of Thul ium iron garnet (Tm 3Fe5O12, TmIG ). A different form of magnetoelastic anisotropy effect can be induced by using porosity in garnet thin films. Mesoporous Holmium Iron Garnet (Ho 3Fe5O12, HoIG) thin film on Si (001)27 exhibit s PMA due to reduced shape anisotropy, increased magnetostrictive and growth -induced anisotropy effects. Such combined effects lead to PMA in HoIG. In this porous thin film, the PMA was found to be independent of the substrate used, because the mechanical s tress does not result from a lattice or thermal expansion mismatch between the substrate and the film. Instead, the pore -solid architecture itself imposes an intrinsic strain on the solution processed garnet film. This example indicates that the film struc ture can be engineered in addition to the substrate choices in order to overcome shape anisotropy in thin film iron garnets. Another key method for controlling anisotropy is strain doping through substitutional elements and using their growth -induced aniso tropy effects , as schematically shown on Fig. 1(c) . Bi-doped yttrium iron garnet (Bi:YIG and Bi:GdIG) has been reported to possess perpendicular magnetic anisotropy due to the chemical composition change as the result of increased annealing temperature21,28. Another reason for PMA in these thin films is strain from GGG substrate29. Doping of oxides by Helium implantation was shown to reversibly and locally tune magnetic anisotropy30. For TbxY3-xFe5O12 (x=2.5, 2.0, 1.0, 0.37 ) samples grown by spontaneous nucleation technique31, magnetic easy axis was found to change from [111] to [100] direction as Tb concentration was decreased. The first -order anisotropy constant K 1 undergoes a change of sign near 190K . Another temperature -dependent lattice distortion effect that caused anisotropy chang e was also reported for YIG films32. These results indicate that temperature also plays an important role in both magnetic compensation, lattice distortion and change in anisotropy. In this study, we systematically calculate the anisotropy energ ies of 10 different types of lattice - matched iron garnet compounds epitaxially -grown as thin films (X 3Fe5O12, X = Y, Tm, Dy, Ho, Er, Yb, Tb, Gd, Sm, Eu) on commercial ly available (111) -oriented garnet substrates (Gd 3Ga5O12- GGG , Y 3Al5O12-YAG , Gd 3Sc2Ga3O12 -SGGG, Tb 3Ga5O12 –TGG, Nd 3Ga5O12 -NGG). Out of the 50 different film/substrate pairs, we found that 20 cases are candidates for room temperature PMA. Out of these 20 cases, 7 film/substrate pairs were experimentally tested and shown to exhibit 5 characteristics originating from PMA. The remaining 13 pairs, to the best of our knowledge, have not been tested for PMA experimentally. We indicate through systematic anisotropy calculations that large strain -induced magnetic anisotropy terms may overcome shape when the films are highly strained . We use only the room temperature values of λ 11133 and only report predictions for room temperature (300K) . Throughout the rest of this study, the films are labelled as XIG (X = Y, Tm, Dy, Ho , Er, Yb, Tb, Gd, Sm, Eu), i.e. TbIG (Terbium iron garnet) or SmIG (Samarium iron garnet) etc. to distinguish them based on the rare earth element. Our model could accurately predict the magnetic easy axis in almost all experimentally tested garnet film/substrate cases provided that the actual film properties are entered in the model and that the experimental film properties satisfy cubic lattice mat ching condition to the substrate. 6 Figure 1. Methods to achieve perpendicular magnetic anisotropy in iron garnet thin films. (a) Micro/nano -patterning reduces shape anisotropy and magnetoelastic anisotropies overcome shape. (b) Large strain -induced anisotropy must over come shape anisotropy to yield out -of-plane easy axis. (c) Substitutional doping in garnets overcome shape anisotropy by enhancing magnetocrystalline, growth -induced or magnetoelastic terms. Anisotropy energy den sity calculation s Total anisotropy energy density contains three main contributions; according the Equation 1, shape anisotropy ( Kshape ), first order cubic magnetocrystalline anisotropy ( K1), and strain -induced 7 (magnetoelastic) anisotropy ( Kindu) parameters determine the total effective anisotropy energy density16. Keff=Kindu +Kshape +K1 (1) In case of garnet film magnetized along [111] direction (i.e. on a 111 substrate) , the magneto -elastic anisotropy energy density, resulting from magneto -elastic coupling is calculated by Equation 2: Kindu =−3 2λ111σ|| (2) where λ111 is magnetostriction constant along [111] direction and it is usually negative at room temperature34. In Eqn. 2, σ|| is the in -plane stress induced in the material as a result of lattice mismatch between film and the substrate , and the in -plane stress is calculated from Equation 335: σ||=Y 1−νε|| (3) where Y is elastic mo dulus, and ν is P oisson’s ratio36. For calculation of in -plane strain, lattice parameter values obtaine d from the XRD characterization of the thin films are used . Equation 4 shows the strain relation as the lattice constant difference between the bulk form of the film and that of the substrate divided by the lattice constant for the bulk form of the film16. ε||=afilm −abulk afilm (4) Assuming the lattice parameter of the thin film matches with that of the substrate, the lattice constant of substrate can be used as the lattice constant of thin film for calculation of strain in Equation 537: ε||=asub−afilm afilm (5) The lattice constants used for the films and substrates examined for this study are presented on Table 1. Shape anisotropy energy density depends on the geometry and the intrinsic saturation magnetic moment of the iron garnet material. Shape anisotropy has a demagnetizing effect on the total 8 anisotropy energy density. These significant anisotropy effects can be observed in magnetic hysteresis loop s and FMR measurements24. The most common anisotropies in magnetic materials are shape anisotropy and magneto -crystalline anisotropy38,39. Considering that the film is continuous, the shape anisotropy is calculated as16 Kshape =2πMs2 (6) By obtaining the values of M s for rare earth iron garnets as a function of temperature40,41, the value for shape anisotropy energy density have been calculated using Equation 6. Intrinsic magnetic anisotropy42, so called magnetocrystalline anisotropy, has the weakest contribution to anisotropy energy densit y compared to shape, and strain -induced anisotropies9,11,16,19. The values for the first order magnetocrystalline anisotropy is calculated and reported previously for rare earth iron garnets at different temperatures43. A key consideration in magnetic thin films is saturation field. In anisotropic magnetic thin films, the anisotropy fiel ds have also been calculated using equation 7 as a measure of how much field the films need for magnetic saturation along the easy axis: HA=2Keff Ms (7) Table 1. List of magnetic iron garnet thin films and garnet substrates available off -the-shelf used for this study. The fourth column shows the lattice constants used for calculating the magnetoelastic anisotropy values of epitaxial garnets on the given substrates. Garnet material Chemical formula Purpose Bulk lattice constant (Å) GGG Gd3Ga5O12 Substrate 12.383 YAG Y3Al5O12 Substrate 12.005 SGGG Gd3Sc2Ga3O12 Substrate 12.480 TGG Tb3Ga5O12 Substrate 12.355 NGG Nd3Ga5O12 Substrate 12.520 YIG Y3Fe5O12 Film 12.376 TmIG Tm 3Fe5O12 Film 12.324 DyIG Dy3Fe5O12 Film 12.440 HoIG Ho3Fe5O12 Film 12.400 ErIG Er3Fe5O12 Film 12.350 YbIG Yb3Fe5O12 Film 12.300 9 TbIG Tb3Fe5O12 Film 12.460 GdIG Gd3Fe5O12 Film 12.480 SmIG Sm 3Fe5O12 Film 12.530 EuIG Eu3Fe5O12 Film 12.500 Results and Discussion Table s 1 and 2 list in detail the parameters used and the calculated anisotropy energy density terms for magnetic rare earth iron garnets at 300K . These tables show only the cases predicted to be PMA out of a total of 50 film/substrate pairs investigated . The extended version of Tables 1 and 2 for all calculated anisotropy energy density terms for all combinations of the 50 film/subst rate pairs are provided in the s upplementary tables. The tabulated values for saturation magnetization40,41 and lattice parameters44 have been used for the calculations . In this study , we assumed the value of Young’s modulus and Poisson ratio as 2.00×1012 dyne ·cm-2 and 0.29 for all garnet types , respectively, based on ref.36. We also assume that the saturation magnetization, used for calculation of shape anisotropy, does not change with the film thickness. The saturation magnetization (Ms) values and shape anisotropy for iron garnet films are presented in the third and fourth columns , respectively. The stress values for fully lattice -matched film s σ calculated using equation 3 and magnetostriction constants of the films , λ111, are presented on column s 6 and 7. Magnetoelastic anisotropy K indu, magnetocrystalline anisotropy energy density K 1, and the total magnetic anisotropy energy density K eff are calculated and listed on columns 8 , 9 and 10, respectively . H A on column 11 is the anisotropy field ( the fields required to saturate the film s). Table 2 . Anisotropy energy density parameters calculation results. Rare earth iron garnets on GGG (as=12.3 83Å), YAG ( Y3Al5O12, as=12.005Å) , SGGG (a s=12.48Å) and TGG (Tb 3Ga5O12, as=12.355Å), and NGG ( Nd3Ga5O12, as=12.509Å) substrates, with K eff < 0, are presented. Film Substr ate Ms (emu·cm- 3) Kshape (erg·cm- 3) (× 103) ε (× 10-3) σ (dyn·cm- 2) (×1010) λ111 (×10- 6) Kindu (erg·cm- 3) (×104) K1 (300K) (erg·cm- 3) (× 103) Keff (erg·cm- 3) (× 103) HA (Oe) (× 103) DyIG GGG 31.85 6.37 -4.58 -1.29 -5.9 -11.4 -5.00 -113 -7.09 HoIG GGG 55.73 19.5 -1.37 -0.386 -4 -2.3 -5.00 -8.66 -0.311 GdIG GGG 7.962 0.398 -7.77 -2.19 -3.1 -10.2 -4.10 -106 -26.5 SmIG GGG 140 123 -11.7 -3.30 -8.6 -42.6 -17.4 -321 -4.58 YIG YAG 141.7 126 -30.0 -8.44 -2.4 -30.4 -6.10 -184 -2.60 10 TmIG YAG 110.9 77.2 -25.9 -7.29 -5.2 -56.9 -5.80 -497 -8.97 DyIG YAG 31.85 6.37 -35.0 -9.85 -5.9 -87.2 -5.00 -870 -54.7 HoIG YAG 55.73 19.5 -31.9 -8.97 -4 -53.8 -5.00 -524 -18.8 ErIG YAG 79.62 39.8 -27.9 -7.87 -4.9 -57.8 -6.00 -545 -13.7 YbIG YAG 127.3 102 -24.0 -6.76 -4.5 -45.6 -6.10 -360 -5.66 GdIG YAG 7.962 0.398 -38.1 -10.7 -3.1 -49.9 -4.10 -502 -126 SmIG YAG 140 123 -41.9 -11.8 -8.6 -152.3 -17.4 -1420 -20.2 TbIG SGGG 15.92 1.59 1.61 0.452 12 -8.14 -8.20 -88.0 -11.1 GdIG SGGG 7.962 0.398 0.00 0.00 -3.1 0.00 -4.10 -3.70 -0.930 SmIG SGGG 140 123 -3.99 -1.12 -8.6 -14.5 -17.4 -39.3 -0.562 DyIG TGG 31.85 6.37 -6.83 -1.92 -5.9 -17.0 -5.00 -169 -10.6 HoIG TGG 55.73 19.5 -3.63 -1.02 -4 -6.13 -5.00 -46.8 -1.68 GdIG TGG 7.962 0.398 -10.0 -2.82 -3.1 -13.1 -4.10 -135 -33.9 SmIG TGG 140 123 -14.0 -3.93 -8.6 -50.8 -17.4 -402 -5.74 TbIG NGG 15.92 1.59 3.93 1.11 12 -19.9 -8.20 -206 -25.9 In this study, we take the same sign convention as in ref. 16 and the film s exhibit PMA when Keff < 0. So for obtaining PMA, negative and large values for anisotropy energy density are desired. As all the garnets (except TbIG ) possess negative magnetostriction constant s at room temperature, the sign of the strain (tensile or compressive) determines whether the induced anisotropy is negative or positive . In the literature16,20,45,46 however , we observe that PMA was defined for either positive or ne gative effective anisotropy energy density (K eff). This inconsistency may cause confusion among researchers . Thermodynamically, a higher energy means an unstable state with respect to lower energy cases. Easy axis, by definition, is the axis along which the magnetic material can be saturated with lowest external field or lowest total energy. A magnetic material would thus spontaneously minimize its energy and reorient it s magnetic moment along the easy axis. As a result, we use here Keff < 0 for out-of-plane easy axis . Due to the thermodynamic arguments mentioned above, we suggest researchers to use K eff < 0 definition for PMA. Effect of Substrate on Anisotropy Energy Density Changing the substrate alters the strain in the film, which also changes strain -induced anisotropy in the film. Figure 2 show s the calculated anisotropy energy density of rare earth iron garnet thin films grown on five commercially available differ ent substrates : Gadolinium Gallium Garnet (Gd3Ga5O12, GGG), Yttrium Aluminum Garnet ( Y3Al5O12, YAG), Substituted Gadolinium 11 Gallium Garnet ( Gd3Sc2Ga3O12, SGGG), Terbium Gallium Garnet (Tb 3Ga5O12, TGG ), and Neodymium Gallium Garnet (Nd 3Ga5O12, NGG) . As shown on Fig. 2(a), when gr own on GGG substrate ; Dysprosium Iron Garnet (DyIG), Holmium Iron Garnet (HoIG), Gadolinium Iron Garnet (GdIG), and Samarium Iron Garnet (SmIG) possess compressive strain (afilm>asubstrate ). Considering the large negative magnetostriction constant (λ 111) for each case, the strain -induced anisotropy energy densit ies are estimated to cause negative total effective anisotropy energy density . As a result, DyIG, HoIG, GdIG, and SmIG on GGG are predicted to be PMA cases. Based o n the shape, magnetoelastic and magnetocrystalline anisotropy terms (room temperature K1), Thulium iron garnet (TmIG) on GGG (111) is estimated to be in -plane easy axis although unambiguous experimental evidence indicates that TmIG grows with PMA on GGG (111) [1,19] . The fact that only considering shape, magnetocrystalline and magnetoelastic anisotropy terms does not verify this experimental result suggests that the PMA in TmIG/GGG (111) may originate from a different anisotropy term such as surface anisotropy , growth -induced or stoichiometry -driven anisotropy. Since the films used in the experiments are less than 10 nm or 5 -8 unit cells thick, surface effects may become more significant and may require density functional theory -based predictions to account for surface anisotropy effects . 12 Figure 2. Calculated effective anisotropy values for each rare earth iron garnet thin film when they are epitaxially grown on ( a) GGG, ( b) YAG, ( c) SGGG, ( d) TGG, (e) NGG substrates . Note that the scales of the axes are different in each part. 13 Yttrium Aluminum Garnet (YAG) is a substrate with smaller lattice parameter than all the rare earth iron garnet films considered . With a substrate lattice parameter of as=12.005Å, YAG causes significant and varying amounts of strain on YIG (af=12.376Å) , TmIG (af=12.324Å) , DyIG (af=12.440Å) , HoIG (af=12.400Å) , ErIG (af=12.350Å) , YbIG (af=12.300Å) , TbIG (af=12.460Å) , GdIG (af=12.480Å) , SmIG (af=12.530Å) and EuIG (a f=12.500Å) . Strain from YAG substrate yields negative strain -induced anisotropy energy density for these films . The strain -induced anisotropy term overcomes the shape anisotropy in these garnets when they are grown on YAG. Consequently , effective anisotropy energy densit ies become negative and these garnet films are estimated to possess perpendicular magnetic anisotropy . In the e xceptional case s of Terbium Iron Garnet (TbIG) and Europium Iron Garnet (EuIG) , compressive strain is not enough to induce negative strain anisotropy because the magnetostriction coefficient s of TbIG and EuIG are positive. So the strain -induced anisotropy term s are also positive for both TbIG (Kindu(TbIG) = 1.85×106 erg·cm-3) and EuIG (K indu(EuIG) = 3.01×105 erg·cm-3) and do not yield PMA. Other potential PMA garnets as a film on SGGG substrate are GdIG , TbIG , and SmIG. TbIG and GdIG cases are particularly interesting as growth conditions of these materials can be further optimized to achieve room temperature compensation and zero saturation magnetization. This property enables PMA garnet -based room temperature terahertz magnonics. The lattice parameters of GdIG (af=12.480Å) and SGGG (as=12.48Å) match exactly, so the in -plane strain value is zero and the effect of strain -induced anisotropy is eliminated completely. Consequently, due to small value for saturation magnetizat ion of GdIG, shape anisotropy (3.98 ×102 erg·cm-3) cannot compete with magnetocrystalline anisotropy ( -4.1×103 erg·cm-3). In other words, in this case, the influence of magnetocrystalline anisotropy is not negligible compared to the other anisotropy terms. Consequently , the anisotropy energy density is negative for GdIG when grown on SGGG due to the influence of magnetocrystalline anisotropy energy density. One other candidate for a PMA rare earth iron garnet on SGGG substrate is SmIG. Since the film lattice parameter is greater than that of the substrate, compressive strain ( -3.99 ×10-3) is induced in the film such that the resulting anisotropy energy density possesses a negative value of an order of magnitude ( -1.45×105 erg·cm-3) comparable to the sh ape anisotropy energy density (1.23×105 erg·cm-3). With its relatively large magnetocrystalline anisotropy energy density ( -1.74×104 erg·cm-3), SmIG has a perpendicular magnetic anisotropy due to negative value for effective 14 anisotropy energy density (-4.80×105 erg·cm-3). TbIG film on SGGG substrate is a PMA candidate with positive strain and this film was also recently experimentally demonstrated to have PMA47. Since TbIG’s lattice constant is smaller than that of the substrate, the film becomes subject to tensile strain ( +1.61 ×10-3). Since TbIG also has a positive λ 111 (in contrast to that of SmIG), the film’s magnetoelastic anisotropy term becomes large and negative and overcomes the shape anisotropy. In case of TbIG, magnetocrystalline anisotropy alone overcomes shape and renders the film PMA on SGGG. With the additio nal magnetoelastic anisotropy contribution ( -8.14×104 erg·cm-3), significant stability of PMA can be achieved. Terbium Gallium Garnet (TGG) is a substrate with lattice parameter (as=12.355Å ) such that it can induce tensile strain on TmIG, ErIG and YbIG and it induces compressive strain on the rest of the rare earth iron garnets (YIG, DyIG, HoIG, TbIG, GdIG, SmIG, EuIG) . In none of the tensile - strained cases, PMA can be achieved since the sign of the magnetoelastic anisotropy is positive and has the same sign as the shape anisotropy. Among the compressively strained cases, YIG, TbIG and EuIG are found to have weak magnetoelastic anisotropy terms which cannot overcome shape. As a result, YIG, TbIG and EuIG on TGG substrate are expected to have in -plane easy axis. DyIG, HoIG, GdIG and SmIG films on TGG achieve large and negative effective total anisotropy energy densities due to their negative λ 111 values . In addition, since the materials have compressive strain, the signs cancel and lead to large magnetoelastic anisotropy energy terms that can overcome shape in these materials. So these cases are similar to the conditions explained for GGG substrate, on which only DyIG, HoIG, GdIG, and SmIG films with compressive strain can gain a large negative strain -induced anisotropy energy density which can overcome shape a nisotropy . Neodymium gallium garnet (NGG) is a substrate 45 used for growing garnet thin films by pulsed laser deposition method. NGG has large lattice constant compared with the rest of the bulk rare earth garnets and yield compressive strain in all rare earth garnets investigated except for Samarium iron garnet (SmIG). For all cases other than SmIG, the sign of the magnetoelastic a nisotropy term is determined by the respective λ 111 for each rare earth iron garnet. YIG, TmIG, DyIG, HoIG, ErIG, YbIG cases have positive magnetoelastic anisotropy terms, which lead to easy axes along the ir film plane s. For SmIG and EuIG on NGG, magnetoel astic strain and anisotropy terms are not large enough to overcome large shape anistropy. For GdIG, the weak tensile strain on NGG substrates actually causes in -plane easy axes as magnetoelastic strain offsets the negative magnetocrystalline 15 anisotropy. The only rare earth iron garnet that can achieve PMA on NGG is Terbium Iron Garnet (TbIG) due to its large negative strain -induced anisotropy energy density ( -2.44×105 erg·cm-3). Its large and negative magnetoelastic anisotropy can offset shape (1.59×103 erg·cm-3) and first order magnetocrystalline anisotropy term ( -8.20×103 erg·cm-3), leading to a large negative effective magnetic anisotropy energy density ( -2.51×105 erg·cm-3). Consequently, we predict that growing TbIG on NGG substrate may yield PMA . Figure 3 show s the substrates on which one may expect PMA rare earth iron garnet films (or negative Keff) due to strain only. Figures 3(a)-(d) compare the calculated effective energy densities as a function of strain type and sign for YIG, TmIG, YbIG, TbIG. For comparing the calculation results presented here with the experimentally reported values for the anisotropy energy density of TmIG, we added the K eff directly from the experimental data in20 to Fig. 3(b). As shown on Fig. 3(b), the experimental TmIG thin film shows positive K eff as the result of tensile in -plane strain and large negative magnetostriction constant. Figure 4(a) -(f) shows the calculated effective energy densities as a function of strain type and sign for GdIG, SmIG, EuIG, HoIG, DyIG, ErIG, respectively. K eff may get a positive or a negative value in both compressive and tensile strain cases due to vary ing signs of λ 111 constants of rare earth iron garnets. In almost all cases that yield PMA on the given substrates, PMA iron garnets form under compressive lattice strain. The only exception s in which tensile strain can yield PMA in garnet thin films is Tb IG on SGGG and TbIG on NGG . In both of those cases, a small tensile strain enhances PMA but the magnetocrystalline anisotropy could already overcome shape and yield PMA without lattice strain. Therefore, experimental studies should target compressive latti ce strain. 16 Figure 3. Effect of substrate strain on the effective anisotropy energy densities of ( a) YIG, ( b) TmIG, the data inserted on the graph, with red square symbol , GGG (Exp.) , is the experimental value of effective anisotropy energy of TmIG on GGG based on ref.20 (c) YbIG, ( d) TbIG. Note that the axes scales are different in each part. 17 Figure 4. (a) GdIG, ( b) SmIG, ( c) EuIG, ( d) HoIG, ( e) DyIG, ( f) ErIG films on GGG, YAG, SGGG, TGG, and NGG substrates. Note that the axes scales are different in each part. 18 Based on Fig. 2 -4, the calculations in this paper numerically match with the reported values in the experimental demonstrations in literature both in sign and order of magnitude . However, since there are inconsistent sign conventions for predictin g the magnetic anisotropy state of the iron garnet samples in the literature so far, some of the previous studies draw dif ferent conclusions on the anisotropy despite the similar K eff. In case of TmIG, as shown in Fig. 3(b), our model p redicts that there is a tensile strain -induced anisotropy resulting from the difference in film and substrate lattice parameters and the film’ s negative magnetostriction constant. The experimental results of magnetic anisotropy in TmIG/GGG8,20,48 are consistent with our model predictions . Previous studies identify PMA , if the film Keff is positive. A shortcoming of this approach is that such a definition would also identify YIG/GGG as PMA although its in-plane easy axis behavior has been experimentally de monstrated in numerous studies3,32,49. Keff < 0 for PMA definition would be thermodynamically more appropriate and would also accurately explain almost all cases including YIG/GGG. Further explanation about TmIG exceptional case is included in the Supplementary Information. Sensitivity of Anisotropy to Variations in Saturation Magnetic Moment and Film Relaxation The films with predicted effective anisotropy energy and field may come out differently when fabricated due to unintentional variability in fabrication process conditions , film stoichiometry (rare earth ion to iron ratio and oxygen deficiency) as well as process -induced non -equilibrium phases in the garnet films. These changes may partially or completely relax the films or increase strain further due to secondary crystallin e phases. Practical minor changes in strain may dramatically alter both the sign and the magnitude of magnetoelastic ani sotropy energy and may cause a film predicted as PMA to come out with in -plane easy axis. On the other hand, o ff- stoichiometry may cause reduction in saturation magnetic moment. Reduction in saturation magnetic moment decreases shape anisotropy term quadratically (Kshape = 2πM s2), which implies that a 1 0% reduction in Ms leads to a 19% decrease in K shape and anisotropy field may increase (HA = 2K eff/Ms). Therefore, sample fabrication issues and the consequent changes in anisotropy terms may weaken or completely eliminate the PMA of a film/substrate pair and alter anisotropy field . While these effects may arise unintentionally, one can also use these effects deliberately for engineering garnet films for devices. Therefore , the sensitivity of anisotropy properties of garnet 19 thin films such as anisotropy field and effective anisotropy energy density need s to be evaluated with respect to changes in film strain and saturation magnetic moment. Figure 5 shows the sensitivity of the effective magnetic anisotropy energy density to deviation of both strain and saturation magnetization, M s for five PMA film/substrate combinations: (a) HoIG/GGG, (b) YIG/YAG, (c) SmIG/ SGGG, (d) HoIG/TGG , and (e) SmIG/NGG ). The negative sign of effective magnetic anisotropy energy indicates PMA . The change of anisotropy energy from negative to positive indicates a transition from PMA to in -plane easy axis . In these plots, calculated anisotropy energies are presented for saturation moments and strains scanned from 60% to 140% of tabulated bulk garnet Ms and of the strain s of fully lattice -matched films on the substrate s. The color scale indicates the anisotropy energy density in erg·cm-3. Although magnetocrystalline anisotropy energies are negative for all of the thin film rare earth garnets considered here, these terms are negligible with respect to shape anisotropy (K 1(300 K) ~ -5% of K shape). Therefore, magnetoelastic anisotropy term must be large enough to overcome shape anisotropy. A derivation of anisotropy energy as a function of Ms and strain in equations ( 8)-(10) shows that the negative λ111 values and negative strain states (compressive strain) for garnet films in Fig. 5(a)-(e) (HoIG, YIG, SmIG) enable these films to have PMA . To retain PMA state; λ 111 must be negative and large assuming elastic moduli and the Poisson’s ratio are constant . The necessary condition for maintai ning PMA is shown in equation 11 . Keff=Kindu +Kshape +K1 (8) Keff=−3 2λ111Y 1−vε||+2πMs2+K1 (9) Keff=3 2λ111Y 1−v|ε|||+2πMs2+K1 (10) Keff<0 if |3 2λ111Y 1−v|ε|||+K1|>2πMs2 (11) Relaxing each fully strained and lattice -matched thin film towards unstrained state (ε → 0 or moving from left to right on each plot in Fig. 5 causes the magnetoelastic anisotropy energy term to decrease in magnitude and gradually vanish . The total anisotropy energy decreases in intensity for decreasing strain and constant M s. When M s increases, shape anisotropy term also increases and overcomes magnetoelastic anisotropy term. As a result, higher M s for relaxed films (i.e. relatively thick and iron -rich garnets) may lose PMA. Therefore, one needs to optimize the film 20 stoichiometry and deposition process conditions, especially growth temp erature, oxygen partial pressure and film thickness, to ensure that the films are strained and stoichiometric. Since strains are less than 1% in Fig. 5(a), 5(c)-(e), these samples are predicted to be exp erimentally more reproducible. For YIG on YAG, as shown in Fig. 5 (b), the strains are around 3%, which may be challenging to reproduce. The cases presented in Fig. 5 (a)-(e) are the only cases among 50 film/substrate pairs where reasonable changes in M s and strain may lead to complete loss of PMA. The rest of the cases have not been found as sensitive to strain and M s variability and those predicted to be PMA are estimated to have stable anisotropy. Effective a nisotropy energy plots similar to Fig. 5(a) -(e) are presented in the supplementary figures for all 50 film/substrate pairs. While PMA is a useful metric for garnet films, the effective anisotropy energy of the films should also not be too high ( < a few 105 erg·cm-3) otherwise the saturation fields for these films would reach or exceed 0.5 Tesla (5000 Oe). Supplementary figures present the calculated effective anisotropy energy and anisotropy field values for all 50 film/substrate pairs for changing strain and Ms values. These figures indicate that one can span anisotropy fields of about 300 Oersteds up to 12.6 Tesla in PMA garnets. For practical integrated magnonic devices, the effective anisotropy energy should be large enough to have robust PMA although it shoul d not be too high such that effective anisotropy fields (i.e. saturation fields) would still be small and feasible. Engineered strain and M s through controlled oxygen stoichiometry may help keep anisotropy field low while retaining PMA. In addition, according to the recently published paper on magnetic anisotropy of HoIG50, the lattice matching in case of the thick samples becomes challenging to sustain, and the strain relaxes inside the film. Thus, the decrease in the anisotropy field is one consequences of the lower strain state, which is an advantage for magnonics or spin -orbit torque devices. Below a critical thickness, HoIG gr own on GGG has PMA. However, as the film reaches this critical thickness, the 40% or more strain relaxation is expected and the easy axis becomes in -plane. So thinner films are preferred to be grown in integrated device applications. 21 Figure 5. Effect of partial film relaxation and saturation magnetic moment variability on the effective anisotropy energy density of the films . Variation of effective magnetic anisotropy energy densities for (a) HoIG on GGG, ( b) YIG on YAG, ( c) SmIG on SGGG, ( d) HoIG on TGG and ( e) SmIG on NGG are presented when strain relaxation and magnetic saturation moments change independently. Film strain may vary from a completely lattice -matched state to the substrate to a relaxed state or a highly strained state due to microparticle nucleation . Strain variability alter s magnetoelastic anisotropy and cause a PMA film become in -plane easy axis. On the other hand, 22 magnetic saturation moments may deviate from the tabulated values because of process -induced off-stoich iometry in the films (i.e. rare earth ion to iron ratio or iron deficiency or excess , oxygen deficiency) . Relaxing the films reduces the magnetoelastic anisotropy term and diminishes PMA. Increasing M s strengthens shape anisotropy and eliminates PMA for lo w enough strains for all five cases presented. Minimizing Gilbert damping coefficient in garnet thin films is also an important goal for spintronic device applications. First principles predictions of physical origins of Gilbert damping 51 indicate that magnetic materials with lower M s tend to have lower damping. Based on this prediction, DyIG, HoIG and GdIG films are predicted to have lower Gilbert damping with respect to the others. Since the compen sation temperatures of these films could be engineered near room temperature, one may optimize their damping for wide bandwidths all the way up to terahertz (THz)52 spin waves or magnons . The first principles predictions also indicate that higher magnetic susceptibility (χm) in the films helps reduce damping (i.e. lower saturation field). Therefore, the PMA garnet films with lower anisotropy fields are estimated to have lower Gilbert dampi ng parameters with respect to PMA garnets with higher anisotropy fields . Conclusion Shape, magnetoelastic and magnetocrystalline m agnetic anisotropy energy terms have been calculated for ten different garnet thin films epitaxially grown on five different garnet substrates. Negative K eff (effective magnetic anisotropy energy) corresponds to perpendicular magnetic anisotropy in the convention used here . By choosing a substrate with a lattice parameter smaller than that of the film, one can induce compressive strain in the films to the extent that one can always overcome shape anisotropy and achieve PMA for large and negative λ 111. Among the PMA films predicted, SmIG possesses a high anisotropy energy density and this film is estimated to be a robus t PMA when grown on all five different substrates. In order to obtain PMA, magnetoelastic anisotropy term must be large enough to overcome shape anisotropy. Magnetoelastic anisotropy overcomes shape anisotropy when the strain type (compressive or tensile) and magnetoelastic anisotropy constants λ 111 of the garnet film have the correct signs (not necessarily opposite or same) and the magnetoelastic anisotropy term has a magnitude larger than shape anisotropy. Both compressive and tensile -strained films can , in principle, become PMA as long as shape anisotropy can be overcome with large magnetoelastic 23 strain effects. Here, in almost all cases that yield PMA on the given substrates, PMA iron garnets form under compressive lattice strain, except TbIG on SGGG and TbIG on NGG . These two cases have tensile strain and relatively large magnetocrystalline anisotropy, which could already overcome shape anisotropy without strain. Experiments are therefore suggested to target mainly compressive lattice strain. 20 different garnet film/substrate pairs have been predicted to exhibit PMA and their properties are listed on Table 2 . For 7 of these 20 potential PMA cases, we could find unambiguous experimental demonstration of PMA. Among the 20 PMA cases, HoIG/GGG, YIG/Y AG, SmIG/SGGG, HoIG/ TGG and SmIG/ NGG cases have been found to be sensitive to fabrication process or stoiochiometry -induced variations in M s and strain. In order to control effective anisotropy in rare earth iron garnets (RIGs), shape anisotropy could be tuned by doping garnet film s with Ce 53, Tb 31 and Bi 54 or by micro/nano -patterning . Saturation magnetization could also be increased significantly by doping, which results in increasing the shape anisotropy in the ma gnetic thin films. Among the cases predicted to possess PMA , anisotropy fields ranging from 310 Oe (0.31 T) to 12.6 T have b een calculated . Such a wide anisotropy field range could be spanned and engineered through strain state, stoichiometry as we ll as substrate choice. For integrated magnonic devices and circuits, garnets with low M s and lower anisotropy field s (HA < 0.5 T) would require less energy for switching and would be more appropriate due to their lower estimated Gilbert damping. Methods Calculation of anisotropy energy density. We used Keff = K indu + K shape + K 1 equation to calculate the total anisotropy energy density for each thin film rare earth iron garnet/substrate pair. Each anisotropy term consist of the following parameters: Keff=−3 2λ111Y 1−vε||+2πMs2+K1. The energy density is calculated based on the parameters reported in previous references16,34,36,40,41,43. First order magnetocrystalline anisotropy, K 1, is an intrinsic, temperature -dependent constant reported for each REIG material. Young’s modulus (Y), poison ratio (ν) and magnetostriction constant (λ111) parameters evolving in the magnetoelastic anisotropy energy density term (first term) are considered to be constant according to the values previous ly reported . For shape anisotropy energy calculations (second term), bulk s aturation magnetization (Ms) for each film was used. Since each film may exhibit variability in M s with respect to bulk, the model presented here yields the most accurate predictions when the actual film Ms, λ 111, Y, ν and K 1, and in -plane strain 24 are enter ed for each term. The original Microsoft Excel and MATLAB files used for generating the data for F igure s 1-5 are presented in the supplementary files . 25 References 1 Avci, C. O. et al. Current -induced switching in a magnetic insulator. Nature Materials 16, 309 (2017). 2 Chumak, A., Vasyuchka, V., Serga, A. & Hillebrands, B. Magnon spintronics. Nature Physics 11, 453 (2015). 3 Onbasli, M. et al. 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Static and Dynamic Magnetic Properties of Single ‐Crystalline Yttrium Iron Garnet Films Epitaxially Grown on Three Garnet Substrates. Advanced Electronic Materials 4, 1800106 (2018). 46 Li, G. et al. Tunable perpendicular magnetic anisotropy in epitaxial Y 3Fe5O12 films. APL Materials 7, 041104 (2019). 47 Rosenberg, E. R. et al. Magnetism and spin transport in rare -earth -rich epitaxial terbium and europium iron garnet films. Physical Review Materials 2, 094405 (2018). 48 Wu, C. et al. High -quality thulium iron garnet films with tunable perpendicular magnetic anisotropy by off -axis sputtering –correlation between magnetic properties and film strain. Scientific Reports 8, 11087 (2018). 49 Goto, T., Onbaşlı, M. C. & Ross, C. A. Magneto -optical properties of cerium substituted yttrium iron garnet films with reduced thermal budget for m onolithic photonic integrated circuits. Optics Express 20, 28507 -28517 (2012). 50 Zanjani, S. M. & Onbasli, M. C. Thin film rare earth iron garnets with perpendicular magnetic anisotropy for spintronic applications. AIP Advances 9, 035024 (2019). 51 Hickey , M. C. & Moodera, J. S. Origin of intrinsic Gilbert damping. Physical Review Letters 102, 137601 (2009). 52 Geprägs, S. et al. Origin of the spin Seebeck effect in compensated ferrimagnets. Nature Communications 7, 10452 (2016). 53 Baños -López, E., Cortés -Escobedo, C., Sánchez -De Jesús, F., Barba -Pingarrón, A. & Bolarín -Miró, A. Crystal structure and magnetic properties of cerium -doped YIG: Effect of doping concentration and annealing temperature. Journal of Alloys and Compounds 730, 127-134 (2018). 54 Lou, G., Kato, T., Iwata, S. & Ishibashi, T. Magneto -optical properties and magnetic anisotropy of Nd 0.5Bi2.5Fe5-yGayO12 thin films on glass substrates. Optical Materials Express 7, 2248 -2259 (2017). Acknowledgments M.C.O. acknowledges BAGEP 2017 Award and TUBITAK Grant No. 117F416. Competing interests There is no financial and non -financial competing interest among the authors. Author contributions M.C.O. designed the study. S.M.Z. performed the calculations and evaluated and analyzed the results with M.C.O. Both authors discussed the results and wrote the manuscript together.

2019-05-30

Perpendicular magnetic anisotropy (PMA) is a necessary condition for many spintronic applications like spin-orbit torques switching, logic and memory devices. An important class of magnetic insulators with low Gilbert damping at room temperature are iron garnets, which only have a few PMA types such as terbium and samarium iron garnet. More and stable PMA garnet options are necessary for researchers to be able to investigate new spintronic phenomena. In this study, we predict 20 new substrate/magnetic iron garnet film pairs with stable PMA at room temperature. The effective anisotropy energies of 10 different garnet films that are lattice-matched to 5 different commercially available garnet substrates have been calculated using shape, magnetoelastic and magnetocrystalline anisotropy terms. Strain type, tensile or compressive depending on substrate choice, as well as the sign and the magnitude of the magnetostriction constants of garnets determine if a garnet film may possess PMA. We show the conditions in which Samarium, Gadolinium, Terbium, Holmium, Dysprosium and Thulium garnets may possess PMA on the investigated garnet substrate types. Guidelines for obtaining garnet films with low damping are presented. New PMA garnet films with tunable saturation moment and field may improve spin-orbit torque memory and compensated magnonic thin film devices.

Predicting New Iron Garnet Thin Films with Perpendicular Magnetic Anisotropy

1905.13042v1

The fractional Landau-Lifshitz-Gilbert equation R.C. Verstraten1, T. Ludwig1, R.A. Duine1,2, C. Morais Smith1 1Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584CC Utrecht, The Netherlands 2Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands (Dated: November 24, 2022) The dynamics of a magnetic moment or spin are of high interest to applications in technology. Dissipation in these systems is therefore of importance for improvement of efficiency of devices, such as the ones proposed in spintronics. A large spin in a magnetic field is widely assumed to be described by the Landau-Lifshitz-Gilbert (LLG) equation, which includes a phenomenological Gilbert damping. Here, we couple a large spin to a bath and derive a generic (non-)Ohmic damping term for the low-frequency range using a Caldeira-Leggett model. This leads to a fractional LLG equation, where the first-order derivative Gilbert damping is replaced by a fractional derivative of orders≥0. We show that the parameter scan be determined from a ferromagnetic resonance experiment, where the resonance frequency and linewidth no longer scale linearly with the effective field strength. Introduction. — The magnetization dynamics of mate- rials has attracted much interest because of its techno- logical applications in spintronics, such as data storage or signal transfer [1–3]. The right-hand rule of magnetic forces implies that the basic motion of a magnetic mo- ment or macrospin Sin a magnetic field Bis periodic precession. However, coupling to its surrounding (e.g., electrons, phonons, magnons, and impurities) will lead to dissipation, which will align SwithB. Spintronics-based devices use spin waves to carry sig- nals between components [4]. Contrary to electronics, which use the flow of electrons, the electrons (or holes) in spintronics remain stationary and their spin degrees of freedom are used for transport. This provides a sig- nificant advantage in efficiency, since the resistance of moving particles is potentially much larger than the dis- sipation of energy through spins. The spin waves con- sist of spins precessing around a magnetic field and they are commonly described by the Landau-Lifshitz-Gilbert (LLG) equation [5]. This phenomenological description also includes Gilbert damping, which is a term that slowly realigns the spins with the magnetic field. Much effort is being done to improve the control of spins for practical applications [6]. Since efficiency is one of the main motivations to research spintronics, it is important to understand exactly what is the dissipation mechanism of these spins. Although the LLG equation was first introduced phe- nomenologically, since then it has also been derived from microscopic quantum models [7, 8]. Quantum dissipation is a topic of long debate, since normal Hamiltonians will always have conservation of energy. It can be described, for instance, with a Caldeira-Leggett type model [9–13], where the Hamiltonian of the system is coupled to a bath of harmonic oscillators. These describe not only bosons, but any degree of freedom of an environment in equilib- rium. These oscillators can be integrated out, leading toan effective action of the system that is non-local and ac- counts for dissipation. The statistics of the bath is cap- tured by the spectral function J(ω), which determines the type of dissipation. For a linear spectral function (Ohmic bath), the first-order derivative Gilbert damping is retrieved. The spectral function is usually very difficult to calcu- late or measure, so it is often assumed for simplicity that the bath is Ohmic. However, J(ω) can have any contin- uous shape. Hence, a high frequency cutoff is commonly put in place, which sometimes justifies a linear expan- sion. However, a general expansion is that of an sorder power-law, where scould be any positive real number. A spectral function with such a power-law is called non- Ohmic, and we refer to sas the “Ohmicness” of the bath. It is known that non-Ohmic baths exist [14–23] and that they can lead to equations of motion that include frac- tional derivatives [24–28]. Because fractional derivatives are non-local, these systems show non-Markovian dynam- ics which can be useful to various applications [29–31]. Here, we show that a macroscopic spin in contact with a non-Ohmic environment leads to a fractional LLG equation, where the first derivative Gilbert damping gets replaced by a fractional Liouville derivative. Then, we ex- plain how experiments can use ferromagnetic resonance (FMR) to determine the Ohmicness of their environ- ment from resonance frequency and/or linewidth. This will allow experiments to stop using the Ohmic assump- tion, and use equations based on measured quantities instead. The same FMR measurements can also be done with anisotropic systems. Aligning anisotropy with the magnetic field may even aid the realization of measure- ments, as this can help reach the required effective field strengths. In practice, the determination of the type of environment is challenging, since one needs to measure the coupling strength with everything around the spins. However, with the experiment proposed here, one canarXiv:2211.12889v1 [cond-mat.mes-hall] 23 Nov 20222 essentially measure the environment through the spin it- self. Therefore, the tools that measure spins can now also be used to determine the environment. This information about the dissipation may lead to improved efficiency, stability, and control of applications in technology. Derivation of a generalized LLG equation. — We con- sider a small ferromagnet that is exposed to an external magnetic field. Our goal is to derive an effective equa- tion of motion for the magnetization. For simplicity, we model the magnetization as one large spin (macrospin) ˆS. Its Hamiltonian (note that we set /planckover2pi1andkBto one) reads ˆHs=B·ˆS−KˆS2 z, where the first term (Zeeman) describes the coupling to the external mag- netic fieldB, and the second term accounts for (axial) anisotropy of the magnet. However, since a magnet con- sists of more than just a magnetization, the macrospin will be in contact with some environment. Following the idea of the Caldeira-Leggett approach [9–13, 32], we model the environment as a bath of harmonic oscillators, ˆHb=/summationtext αˆp2 α/2mα+mαω2 αˆx2 α/2, where ˆxαand ˆpαare position and momentum operators of the α-th bath oscil- lator with mass mαand eigenfrequency ωα>0. Further- more, we assume the coupling between the macrospin and the bath modes to be linear, ˆHc=/summationtext αγαˆS·ˆxα, where γαis the coupling strength between macrospin and the α-th oscillator. Thus, the full Hamiltonian of macrospin and environment is given by ˆH=ˆHs+ˆHc+ˆHb. Next, we use the Keldysh formalism in its path-integral version [33, 34], which allows us to derive an effective ac- tion and, by variation, an effective quasi-classical equa- tion of motion for the macrospin. For the path-integral representation of the macrospin, we use spin coherent states [34]|g/angbracketright= exp(−iφˆSz) exp(−iθˆSy) exp(−iψˆSz)|↑/angbracketright, whereφ,θ, andψare Euler angles and |↑/angbracketrightis the eigen- state of ˆSzwith the maximal eigenvalue S. Spin co- herent states provide an intuitive way to think about the macrospin as a simple vector S=/angbracketleftg|ˆS|g/angbracketright= S(sinθcosφ,sinθsinφ,cosθ) with constant length Sand the usual angles for spherical coordinates θandφ. For spins, the third Euler angle ψpresents a gauge freedom, which we fix as in Ref. [35] for the same reasons explained there. After integrating out the bath degrees of freedom, see Sup. Mat. [36] for details, we obtain the Keldysh partition functionZ=/integraltext Dgexp[iS], with the Keldysh action S=/contintegraldisplay dt/bracketleftbig S˙φ(1−cosθ)−Beff(Sz)·S/bracketrightbig −/contintegraldisplay dt/contintegraldisplay dt/primeS(t)α(t−t/prime)S(t/prime). (1) The first term, called Berry connection, takes the role of a kinetic energy for the macrospin; it arises from the time derivative acting on the spin coherent states (−i∂t/angbracketleftg|)|g/angbracketright=S˙φ(1−cosθ). The second term is the po- tential energy of the macrospin, where we introduced aneffective magnetic field, Beff(Sz) =B−KSzez, given by the external magnetic field and the anisotropy. The third term arises from integrating out the bath and accounts for the effect of the environment onto the macrospin; that is, the kernel function α(t−t/prime) contains informa- tion about dissipation and fluctuations. Dissipation is described by the retarded and advanced components αR/A(ω) =/summationtext α(γ2 α/2mαω2 α)ω2/[(ω±i0)2−ω2 α], whereas the effect of fluctuations is included in the Keldysh com- ponent,αK(ω) = coth(ω/2T) [αR(ω)−αA(ω)]. This is determined by the fluctuation-dissipation theorem, as we assume the bath to be in a high-temperature equilibrium state [33, 34, 37]. From the Keldysh action, Eq. (1), we can now de- rive an equation of motion for the macrospin by taking a variation. More precisely, we can derive quasi-classical equations of motion for the classical components of the anglesθandφby taking the variation with respect to their quantum components [38]. The resulting equations of motion can be recast into a vector form and lead to a generalized LLG equation ˙S(t) =S(t)×/bracketleftbigg −Beff[Sz(t)] +/integraldisplayt −∞dt/primeα(t−t/prime)S(t/prime) +ξ(t)/bracketrightbigg , (2) with the dissipation kernel [39] given by α(ω) =/integraldisplay∞ −∞dε πεJ(ε) (ω+i0)2−ε2, (3) where we introduced the bath spectral density J(ω) =/summationtext α(πγ2 α/2mαωα)δ(ω−ωα) [33, 36]. The last term in Eq. (2) contains a stochastic field ξ(t), which describes fluctuations (noise) caused by the coupling to the bath; the noise correlator for the components of ξ(t) is given by/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδmnαK(t−t/prime). Next, to get a better understanding of the generalized LLG equation, we consider some examples of bath spectral densities. Fractional Landau-Lifshitz-Gilbert equation. — For the generalized LLG equation (2), it is natural to ask: In which case do we recover the standard LLG equation? We can recover it for a specific choice of the bath spectral densityJ(ω), which we introduced in Eq. (3). Roughly speaking,J(ω) describes two things: first, in the delta functionδ(ω−ωα), it describes at which energies ωα the macrospin can interact with the bath; second, in the prefactor πγ2 α/2mαωα, it describes how strongly the macrospin can exchange energy with the bath at the fre- quencyωα. In our simple model, the bath spectral den- sity is a sum over δ-peaks because we assumed excitations of the bath oscillators to have an infinite life time. How- ever, also the bath oscillators will have some dissipation of their own, such that the δ-peaks will be broadened. If, furthermore, the positions of the bath-oscillator frequen- ciesωαis dense on the scale of their peak broadening, the bath spectral density becomes a continuous function in- stead of a collection of δ-peaks. In the following, we focus3 on cases where the bath spectral density is continuous. Since the bath only has positive frequencies, we have J(ω≤0) = 0. Even though J(ω) can have any pos- itive continuous shape, one might assume that it is an approximately linear function at low frequencies; that is, J(ω) =α1ωΘ(ω)Θ(Ωc−ω), (4) where Θ(ω) = 1 forω > 0 and Θ(ω) = 0 forω < 0 and Ωcis some large cutoff frequency of the bath such that we haveωsystem/lessmuchT/lessmuchΩc. Reservoirs with such a linear spectral density are also known as Ohmic baths. Insert- ing the Ohmic bath spectral density back into Eq. (3), while sending Ω c→∞ , we recover the standard LLG equation, ˙S(t) =S(t)×/bracketleftBig −Beff[Sz(t)] +α1˙S(t) +ξ(t)/bracketrightBig ,(5) where the first term describes the macrospin’s precession around the effective magnetic field, the second term— known as Gilbert damping—describes the dissipation of the macrospin’s energy and angular momentum into the environment, and the third term describes the fluctu- ations with/angbracketleftξm(t)ξn(t/prime)/angbracketright= 4α1Tδmnδ(t−t/prime), which are related to the Gilbert damping by the fluctuation- dissipation theorem. Note that the same results can be obtained without a cutoff frequency by introducing a counter term, which effectively only changes the zero- energy level of the bath, see Sup. Mat. [36] for details. The assumption of an Ohmic bath can sometimes be justified, but is often chosen out of convenience, as it is usually the simplest bath type to consider. To our knowl- edge, there has been little to no experimental verification whether the typical baths of magnetizations in ferromag- nets are Ohmic or not. To distinguish between Ohmic and non-Ohmic baths, we need to know how the mag- netization dynamics depends on that difference. Hence, instead of the previous assumption of a linear bath spec- tral density (Ohmic bath), we now assume that the bath spectral density has a power-law behavior at low frequen- cies, J(ω) = ˜αsωsΘ(ω)Θ(Ωc−ω), (6) where we refer to sas Ohmicness parameter [40]. It is convenient to define αs= ˜αs/sin(πs/2) and we should note that the dimension of αsdepends on s. Fors= 1 we recover the Ohmic bath. Correspondingly, baths with s < 1 are called sub-Ohmic and baths with s > 1 are called super-Ohmic. For 0 <s< 2 and Ωc→∞ , we find the fractional LLG equation ˙S(t) =S×[−Beff[Sz(t)] +αsDs tS(t) +ξ(t)],(7) whereDs tis a (Liouville) fractional time derivative of or- dersand the noise correlation is given by /angbracketleftξm(t)ξn(t/prime)/angbracketright= 2αsTδmn(t−t/prime)−s/Γ(1−s); for a detailed calculation,see the Sup. Mat. [36]. Indeed, in the limit of s→1, we recover the regular LLG equation. The fractional LLG equation (7) seems quite similar to the standard LLG equation (5). However, the first-order time derivative in the Gilbert damping is replaced by a fractional s-order time derivative in the fractional Gilbert damping; this has drastic consequences for the dissipative macrospin dynamics. Fractional Gilbert damping. — Fractional derivatives have a long history [29], and many different definitions exist for varying applications [26, 27, 29]. From our mi- croscopic model, we found the Liouville derivative [41], which is defined as Ds tS(t) =dn dtn1 Γ(n−s)/integraldisplayt −∞dt/prime(t−t/prime)n−1−sS(t/prime),(8) wherenis the integer such that n≤s<n + 1. This can be interpreted as doing a fractional integral followed by an integer derivative. The fractional integral is a direct generalization from the rewriting of a repeated integral, by reversing the order of integration into a single one, which leads to extra powers of ( t−t/prime). To provide some intuition to the effects of fractional friction, we propose a thought experiment. Suppose an object is traveling at constant speed, then x(t) =vt. Hence, the friction force acting on the object goes like Ds tx(t)∝vt1−s. Therefore, we find three regimes. For s= 1, the friction is constant in time. For s<1, the fric- tion force increases with time. Hence, longer movements will be less common. For s > 1, the friction decreases with time, so longer movements will be more likely once set in motion. Within the fractional LLG equation, we thus see two important new regimes. For s < 1 (sub-Ohmic), the friction is more likely to relax (localize) the spin (e.g. -1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.10.51510 [ωd-(B 0-KS)]/[α sS(B 0-KS)s][αsS(B 0-KS)s]2sin2θ/Ω2s0.20.40.60.81.1.21.41.61.8 FIG. 1. A lin-log plot of the amplitude sin2θas a function of driving frequency ωdplotted in dimensionless units for several values ofs. The resonance peaks change, depending on s. The resonance frequency ωresand linewidth ∆ H/2have been overlayed with crosses. The red dashed crosses have been calculated numerically, whereas the black solid crosses are the derived results from Eqs. (10) to (12).4 sub-diffusion) towards the B-field direction. For small movements, the friction could be very small, whereas it would greatly increase for bigger movements. This could describe a low dissipation stable configuration. For s > 1 (super-Ohmic), the friction could reduce as the spin moves further, which in other systems is known to cause L´ evy-flights or super-diffusion [24, 25, 42]. This might lead the system to be less stable, but can potentially also greatly reduce the amount of dissipation for strong signal transfer: In a similar way to the design of fighter-jets, unstable systems can be easily changed by small inputs, which leads to more efficient signal transfer. Ferromagnetic Resonance. — FMR is the phenomenon where the spin will follow a constant precession in a ro- tating external magnetic field. The angle θfrom thez- axis at which it will do so in the steady state will vary according to the driving frequency ωdof the magnetic field. Close to the natural frequency of the precession, one generally finds a resonance peak [43]. We assume a magnetic field of the form Beff(t) = Ω cos(ωdt) Ω sin(ωdt) B0−KSz , (9) where Ω is the strength of the rotating component, and we will neglect thermal noise. We search for a steady state solution of S(t) in the rotating frame where Beff(t) is constant. We will assume a small θapproximation where the ground state is in the positive zdirection, i.e. 0<Ω/lessmuchB0−KSandαsS/lessmuch(B0−KS)1−s. Then (see Sup. Mat. [36] for details of the calculations), we find that the resonance occurs at a driving frequency ωres≈(B0−KS) + (B0−KS)sαsScos/parenleftBigπs 2/parenrightBig .(10) It should be noted that this is different from what was to be expected from any scaling arguments, since the cosine term is completely new compared to previous results [43], and it vanishes precisely when s= 1. However, this new non-linear term scales as ( B0−KS)s, which is an easily controllable parameter. In the limit where B0−KSis small (resp. large), the linear term will vanish and the s-power scaling can be measured for the sub(resp. super)- Ohmic case. The amplitude at resonance is found to be sin2θres≈Ω2 /bracketleftbig αsS(B0−KS)ssin/parenleftbigπs 2/parenrightbig/bracketrightbig2, (11) and the Full Width at Half Maximum (FWHM) linewidth is given by ∆H/2≈2αsS(B0−KS)ssin/parenleftBigπs 2/parenrightBig . (12) Depending on the experimental setup, it might be eas- ier to measure either the resonance location or the width 0.0 0.5 1.0 1.5 2.001234 ωres/(B0-KS)ΔH/2/[αsS(B0-KS)s] 0.01 0.10 1 10 1000.0110s0.20.40.60.81.1.21.41.61.8FIG. 2. A plot of the linewidth in Eq. (13) as a function of res- onance frequency for several values of s. The inset shows the same plot in a log-log scale, where the slope of the linewidth is precisely the Ohmicness sof the bath. of the peak. Nevertheless, both will give the opportu- nity to see the sscaling inB0−KS. The presence of the anisotropy provides a good opportunity to reach weak or strong field limits. In fact, the orientation of the anisotropy can help to add or subtract from the magnetic field, which should make the required field strengths more reachable for experiments. Some setups are more suit- able for measuring the width as a function of resonance frequency. When s= 1, this relation can be directly derived from Eqs. (10) and (12). However, when s/negationslash= 1, the relation can only be approximated for strong or weak damping. For small αsS, we see that ∆H/2≈2αsS(ωres)ssin/parenleftBigπs 2/parenrightBig . (13) The resonance peaks have been calculated numerically in FIG. 1 in dimensionless values. The red dashed lines show the location of the numerically calculated peak and the FWHM line width. The black solid lines show the location of the analytically approximated result for the peak location and FWHM line width [Eqs. (10) and (12)]. For smallαsSand Ω, we see a good agreement be- tween the analytical results and the numerical ones, al- though sub-Ohmic seems to match more closely than super-Ohmic. This could be due to the greater sta- bility of sub-Ohmic systems, since the approximations might affect less a stable system. As one might expect from the thought experiment presented earlier, we can see in FIG. 1 that sub-Ohmic systems require higher, more energetic, driving frequencies to resonate, whereas super-Ohmic systems already resonate at lower, less en- ergetic, driving frequencies. In FIG. 2, we provide a plot of Eq. (13) to facilitate further comparison with experi- ments. If the assumption of Gilbert damping was correct, all that one would see is a slope of one in the log-log inset. Conclusion. — By relaxing the Ohmic Gilbert damp- ing assumption, we have shown that the low-frequency regime of magnetization dynamics can be modeled by a5 fractional LLG equation. This was done by coupling the macrospin to a bath of harmonic oscillators in the frame- work of a Caldeira-Leggett model. The Keldysh formal- ism was used to compute the out-of-equilibrium dynam- ics of the spin system. By analyzing an FMR setup, we found ans-power scaling law in the resonance frequency and linewidth of the spin, which allows for a new way to measure the value of s. This means that experiments in magnetization dynamics and spintronics can now avoid the assumption of Gilbert damping and instead measure the Ohmicness of the environment. This could aid in a better understanding of how to improve efficiency, stabil- ity, and control of such systems for practical applications. Acknowledgments. — This work was supported by the Netherlands Organization for Scientific Research (NWO, Grant No. 680.92.18.05, C.M.S. and R.C.V.) and (partly) (NWO, Grant No. 182.069, T.L. and R.A.D.). [1] I. D. Mayergoyz, G. Bertotti, and C. 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[38] In a straightforward variation with respect to quantum components, we would only obtain a noiseless quasi- classical equation of motion because the information about noise (fluctuations) is included in the Keldysh part ofα(t−t/prime), which appears in the action only with even powers of quantum components. However, there is a way [44] that allows us to retain information about noise in the quasi-classical equation of motion; see also [33, 34]. Namely, we perform a Hubbard-Stratonovich transforma- tion to linearize the contribution quadratic in quantum components. This linearization comes at the cost of in- troducing a new field, which takes the role of noise. [39] The dissipation kernel αis closely related to the retarded αRand advanced αAcomponents. Namely, it is given by α(ω) =−αR(ω)−αA(−ω). [40] From a mathematical perspective, any continuous but not smooth function can still be expanded in a power-law for small enough parameters. Hence, the only assumption that we make in this model is that the frequencies in the system are very small. Then, there will always exist an 0< s∈Rsuch that this expansion holds. In contrast,6 the Ohmic expansion can only be made for smooth con- tinuous functions. [41] This definition does not have any boundary conditions, as they would have to be at −∞ and would dissipate before reaching a finite time. One can, however, enforce boundary conditions by applying a very strong magnetic field for some time such that the spin aligns itself, and then quickly change to the desired field at t= 0. [42] A. A. Dubkov, B. Spagnolo, and V. V. Uchaikin, Interna- tional Journal of Bifurcation and Chaos 18, 2649 (2008). [43] T. Ludwig, I. S. Burmistrov, Y. Gefen, and A. Shnirman, Physical Review Research 2, 023221 (2020). [44] A. Schmid, Journal of Low Temperature Physics 49, 609 (1982).The fractional Landau-Lifshitz-Gilbert equation Supplementary Material R.C. Verstraten1, T. Ludwig1, R.A. Duine1,2, C. Morais Smith1 1Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584CC Utrecht, The Netherlands 2Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands (Dated: November 24, 2022) CONTENTS I. Keldysh microscopic model 1 A. Hamiltonian 1 B. Keldysh partition function 2 C. Quasi-classical equation of motion 4 D. Generalized Landau-Lifshitz-Gilbert equation 8 II. Fractional derivative from non-Ohmic spectral function 9 A. Calculating the effective Greens functions 9 B. Ohmic spectral function 11 C. Sub-Ohmic spectral function 13 D. Super-Ohmic spectral function 15 E. Comparison Ohmic versus non-Ohmic 17 III. FMR powerlaw derivation 18 A. Ferromagnetic Resonance 18 B. Resonance frequency and amplitude 20 C. Calculating the FWHM linewidth 21 IV. Dimensional analysis 22 References 22 I. KELDYSH MICROSCOPIC MODEL For pedagogical reasons we start with a microscopic derivation of the usual LLG equation before going into the fractional one. In this section, we combine spin coherent states with the Keldysh formalism [1, 2] to derive a stochastic Langevin-like equation of motion of a (macro) spin [3]. A. Hamiltonian In the main text, we introduced a spectral function J(ω) with a cutoff frequency Ω c. This was originally done from the perspective that any spectral function could be expanded to linear order; hence, the model would only be valid up to some highest frequency. However, the cutoff is also important for the model to be realistic, since any physical spectral function should vanish as ω→∞ . In the main text, we stated that the same results can be obtained by introducing a constant counter term in the Hamiltonian. This is a term which exactly completes the square of the coupling term and the harmonic potential of the bath and can be seen as a normalization of the zero-energy level. If we instead start the model with this counter term and drop the cutoff, we will get a Greens function αct(ω), which is precisely such that the original Greens function can be written as α(ω) =α(0) +αct(ω), i.e., the counter term in the Hamiltonian removes the zero frequency contribution of the Greens function. This α(ω= 0) generates a term in the equation of motion that goes as/integraltext∞ 0d/epsilon1J(/epsilon1) π/epsilon1[S(t)×S(t)]. Since the integral is finite, with a frequency cutoff in J(ω), the entire term is zero due to the cross product. This means that the equation of motion will be identical if we startarXiv:2211.12889v1 [cond-mat.mes-hall] 23 Nov 20222 either from the regular Hamiltonian with a frequency cutoff, or with a counter term and no cutoff. Here, we choose to show the method that includes a counter term, because then we do not need to calculate terms which would have canceled either way. The microscopic system that we describe is a large spin in an external magnetic field, where the spin is linearly coupled to a bath of harmonic oscillators in the same way as in Refs. [4–9]. Therefore, our Hamiltonian has the form of a system, coupling, bath, and counter term; H(t) =Hs+Hc+Hb+Hct, where Hs=B·ˆS−KS2 z, Hc=/summationdisplay αγαˆS·ˆxα, Hb=/summationdisplay αˆp2 α 2mα+mαω2 α 2ˆx2 α, Hct=/summationdisplay αγ2 α 2mαω2αˆS2. (1) Here,Bis the (effective) magnetic field, ˆSis the spin, Kis thez-axis anisotropy, γαis the coupling strength, and αis the index over all harmonic oscillators which have position ˆxα, momentum ˆpα, massmαand natural frequency ωα. Notice that the counter term is constant, since S2is a conserved quantity, and that we have indeed completed the square, such that H(t) =B·ˆS−KS2 z+/summationdisplay αˆp2 α 2mα+/bracketleftBigg/radicalbigg mαω2α 2ˆxα+/radicalBigg γ2α 2mαω2αˆS/bracketrightBigg2 . (2) B. Keldysh partition function We will use the Keldysh formalism to derive a quasi-classical equation of motion. Since this is an out-of-equilibrium system, a common choice would be to use the Lindblad formalism with a master equation [10]. However, Lindblad can only describe Markovian systems, which will not be the case when we introduce a non-Ohmic bath. In the Keldysh formalism, one starts with an equilibrium density matrix in the far past (effectively infinite on the relevant time scale). This then gets evolved with the time evolution operator as usual. However, in contrast to ordinary path integrals, once the present has been reached, one evolves back to the infinite past. Since there is infinite time for evolution, we can reach out-of-equilibrium states adiabatically. The benefit of integrating back to the infinite past is that we begin and end with the same in-equilibrium system, which means equilibrium techniques can be used, at the cost of having both the forward ( O+) and backward ( O−) quantities to take care of. To reach useful results, one can apply a Keldysh rotation to the classical ( Oc= (O++O−)/2) and quantum ( Oq=O+−O−) components with the added notation/vectorO=/parenleftbigg Oc Oq/parenrightbigg . To derive a quasi-classical equation of motion, the action can be expanded in all the quantum components, after which the Euler-Lagrange equation for the quantum components provides the equation of motion in terms of the classical components. FIG. 1. Figure extracted from Ref. [2]. The Keldysh contour starts at t=−∞, evolves forward to some time t, and then evolves backwards in time to t=−∞. To begin, we write down the Keldysh partition function Z= Tr/braceleftbigg TKexp/bracketleftbigg −i/contintegraldisplay KdtH(t)/bracketrightbigg ρ0/bracerightbigg , (3)3 whereTKis the Keldysh time ordering, ρ0is the density matrix at t=−∞, and the integral runs over the Keldysh contour, as shown in FIG. 1. After discretizing the Keldysh time integral in the way of FIG. 1, we can rewrite the trace as path-integrals over the spin coherent state |g/angbracketrightand the oscillators |ˆxα/angbracketrightand|ˆpα/angbracketright. This yields Z=/integraldisplay Dg/productdisplay α/integraldisplay Dˆxα/integraldisplay DˆpαeiS[g,{ˆxα},{ˆpα}], (4) with the Keldysh action S[g,{ˆxα},{ˆpα}] =/contintegraldisplay Kdt/bracketleftBigg (−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2 z,g +/summationdisplay α/parenleftBigg −γαSg·ˆxα+ˆpα·˙ˆxα−γ2 αS2 g 2mαω2α−ˆp2 α 2mα−mαω2 α 2ˆx2 α/parenrightBigg/bracketrightBigg , (5) where we defined Sg=/angbracketleftg|ˆS|g/angbracketright. The continuous path-integral seems to miss the boundary term /angbracketleftˆx1,α,g1|ρ0|ˆx2N,α,g2N/angbracketright/angbracketleftˆp2N,α|ˆx2N,α/angbracketright, but it is included in the Keldysh contour, as it connects the beginning and final contour time at t=−∞; see Ref. [2]. Now, we will integrate out the bath degrees of freedom, beginning by completing the square and performing the Gaussian integral over ˆpα. The Gaussian contribution in ˆpαwill act as a constant prefactor, so it will drop out of any calculation of an observable due to the normalization. Hence, we can effectively set it to one to find /integraldisplay Dˆpαexp/bracketleftbigg −i/contintegraldisplay Kdt/parenleftbiggˆp2 α 2mα−ˆpα·˙ˆxα/parenrightbigg/bracketrightbigg = exp/bracketleftbigg i/contintegraldisplay Kdt/parenleftBig −mα 2ˆxα∂2 tˆxα/parenrightBig/bracketrightbigg , (6) where we also did a partial integration in ˆxα. Next we will perform a similar approach for the positions, but it is useful to apply the Keldysh rotation first. Note that we can directly rewrite the integral over the Keldysh contour as a regular time integral over the quantum components. However, one must still rewrite the contents of the integral in terms of the quantum and classical parts of the variables, since the Keldysh rotation does not immediately work for products. The action can first be written as iS[g,{ˆxα}] =i/integraldisplay dt/parenleftBig [(−i∂t/angbracketleftg|)|g/angbracketright]q−[B·Sg]q+K/bracketleftbig S2 z,g/bracketrightbigq −/summationdisplay α/braceleftBigg [γαSg·ˆxα]q+γ2 α[S2 g]q 2mαω2α+/bracketleftBigmα 2ˆxα/parenleftbig ∂2 t+ω2 α/parenrightbigˆxα/bracketrightBigq/bracerightBigg/parenrightBigg . (7) We can then derive that −[γαSg·ˆxα]q=−γα/bracketleftbig S+ g·ˆx+ α−S− g·ˆx− α/bracketrightbig =−γα/bracketleftbigg/parenleftbigg Sc g+1 2Sq g/parenrightbigg ·/parenleftbigg ˆxc α+1 2ˆxq α/parenrightbigg −/parenleftbigg Sc g−1 2Sq g/parenrightbigg ·/parenleftbigg ˆxc α−1 2ˆxq α/parenrightbigg/bracketrightbigg =−γα/bracketleftbig Sc gˆxq α+Sq gˆxc α/bracketrightbig =−γα/bracketleftbigg/parenleftbigSc gSq g/parenrightbig τx/parenleftbigg ˆxc α ˆxq α/parenrightbigg/bracketrightbigg , (8) where we introduced τx=/parenleftbigg 0 1 1 0/parenrightbigg in the Keldysh (classical, quantum) space represented by an upper index candq respectively. Next, we want to derive a similar form for the part of the action that is quadratic in ˆxα. Since these are harmonic oscillators in equilibrium, we can refer the reader to Ref. [2], noting that a unit mass was used there, and conclude that /bracketleftBig −mα 2ˆxα/parenleftbig ∂2 t+ω2 α/parenrightbigˆxα/bracketrightBigq =/parenleftbigˆxc αˆxq α/parenrightbig/parenleftBigg 0/bracketleftbig G−1 α/bracketrightbigA /bracketleftbig G−1 α/bracketrightbigR/bracketleftbig G−1 α/bracketrightbigK/parenrightBigg/parenleftbigg ˆxc α ˆxq α/parenrightbigg , (9) where the retarded and advanced Greens functions read [G−1 α]R/A(t−t/prime) =δ(t−t/prime)mα 2[(i∂t±i0)2−ω2 α]. (10) The±i0 is introduced because we need an infinitesimal amount of dissipation on the bath for it to remain in equilibrium and the sign is tied to causality. This is because there is also an infinitesimal amount of energy transfer from the4 macroscopic spin to each of the oscillators. This results in an extra first-order derivative term, which is found by multiplying out the square with i0. One might want to set these terms to zero immediately, but as it turns out, these are very important limits, which shift away poles from integrals that we need to compute later. Once that is done, the limits are no longer important for the final result, and they may finally be put to zero. Since the bath is in equilibrium, we can use the fluctuation dissipation theorem to compute the Keldysh component using GK α(ω) =/bracketleftbig GR α(ω)−GA α(ω)/bracketrightbig coth/parenleftBigω 2T/parenrightBig . The ˆxdependent part of the action is now given by iSX=i/integraldisplay dt/bracketleftbigg −γα/parenleftbigSc gSq g/parenrightbig τx/parenleftbigg ˆxc α ˆxq α/parenrightbigg +/parenleftbigˆxc αˆxq α/parenrightbig G−1 α/parenleftbigg ˆxc α ˆxq α/parenrightbigg/bracketrightbigg , (11) which we can compute by completing the square to find iSX=i/integraldisplay dt/bracketleftbigg −γ2 α 4/vectorST g/parenleftbigg 0GA α GR αGK α/parenrightbigg /vectorSg/bracketrightbigg . (12) Before we write down the final effective action, we also have to rewrite the quadratic part in Sin a similar vector form, which is −γ2 α[S2 g]q 2mαω2α=−γ2 α 2mαω2α/parenleftbigSc gSq g/parenrightbig τx/parenleftbiggSc g Sq g/parenrightbigg . (13) Combining everything together, we find that the partition function of the system is given by Z=/integraltext Dg eiS[g], with the effective action iS[g] =i/integraldisplay dt/braceleftBigg /bracketleftbig (−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2 z,g/bracketrightbigq−/integraldisplay dt/prime/vectorST g(t)/parenleftbigg 0αA αRαK/parenrightbigg (t−t/prime)/vectorSg(t/prime)/bracerightBigg , (14) whereαA/R(t−t/prime) =/summationtext α/parenleftBig γ2 α 4GA/R α(t−t/prime) +γ2 α 2mαω2αδ(t−t/prime)/parenrightBig andαK(t−t/prime) =/summationtext αγ2 α 4GK α(t−t/prime). C. Quasi-classical equation of motion In the quasi-classical regime, we are interested in solutions where the quantum components ( q) are small compared to the classical components ( c). We can thus neglect terms of O[(q)3], but we must be careful with ( q)2. We can use a Hubbard-Stratonovich transformation to convert ( q)2terms into an expression with just ( q), but with a new fieldξadded to the path integral [3]. The action will then contain only terms of linear order in ( q), which means the partition function has the form Z∼/integraltext DcDq exp[if(c)q] =/integraltext Dc1 2πδ[f(c)]. Hence, only solutions that satisfy f(c) = 0 contribute to the path integral. Within that subset, we want to minimize the action. In order to derive the equation of motion of the system, we must understand the relation between |g/angbracketrightandSg= /angbracketleftg|S|g/angbracketright. Using the Euler angle representation [1], we can describe |g/angbracketrightas |g/angbracketright=g|↑/angbracketright=e−iφSze−iθSye−iψSz|↑/angbracketright=e−iφSze−iθSy|↑/angbracketrighte−iψS(15) and similarly /angbracketleftg|=eiψS/angbracketleft↑|eiθSyeiφSz. (16) Note that the ψangle is now independent of the quantum state |↑/angbracketright, since this angle is describing the rotation of the vector pointing in the spin direction, which is symmetric. Hence, this will yield a gauge symmetry. Using the Euler angle representation in the first terms of Eq. (14), we see that (−i∂t/angbracketleftg|)|g/angbracketright=/parenleftBig ˙ψSeiψS/angbracketleft↑|eiθSyeiφSz+eiψS/angbracketleft↑|˙θSyeiθSyeiφSz+eiψS/angbracketleft↑|eiθSy˙φSzeiφSz/parenrightBig e−iφSze−iθSy|↑/angbracketrighte−iψS =˙ψS+˙θ/angbracketleft↑|Sy|↑/angbracketright+˙φ/angbracketleft↑|eiθSySze−iθSy|↑/angbracketright. (17)5 We note that/angbracketleft↑|Sy|↑/angbracketright= 0, while the last term includes a rotation of the spin up state by θdegrees in the ydirection and then measures the Szcomponent of that state, which is Scosθ. Hence, (−i∂t/angbracketleftg|)|g/angbracketright=˙ψS+˙φScosθ. (18) We now define a new variable χsuch thatψ=χ−φ, which results in (−i∂t/angbracketleftg|)|g/angbracketright= ˙χS−˙φ(1−cosθ)S. (19) Making use of the Euler angle representation, we also see that Sg=S sinθcosφ sinθsinφ cosθ . (20) We see thatB·Sg=S[Bxsinθcosφ+Bysinθsinφ+Bzcosθ]. Similarly, KS2 z,g=KS2cos2θ. Now, we still have to compute the quantum parts of these quantities. We first note that Sq g,x/S= [sinθcosφ]q= 2 cosθcsinθq 2cosφccosφq 2−2 sinθccosθq 2sinφcsinφq 2; Sq g,y/S= [sinθsinφ]q= 2 sinθccosθq 2cosφcsinφq 2+ 2 cosθcsinθq 2sinφccosφq 2; Sq g,z/S= [cosθ]q=−2 sinθcsinθq 2; [cos2θ]q=−2 sinθccosθcsinθq. (21) Next, we will choose a gauge for χas in Ref. [11], which is ˙χc=˙φc(1−cosθc) χq=φq(1−cosθc). (22) Definingp= 1−cosθ, we see that [(−i∂t/angbracketleftg|)|g/angbracketright]q=/bracketleftBig ˙χS−˙φpS/bracketrightBig q=S/bracketleftBig φq˙pc−˙φcpq/bracketrightBig . Now,pq= 2 sinθcsinθq 2and ˙pc=˙θcsinθccosθq 2+˙θq 2cosθcsinθq 2,which leads to [(−i∂t/angbracketleftg|)|g/angbracketright]q=S/bracketleftBig φq˙pc−˙φcpq/bracketrightBig =S/bracketleftBigg φq˙θcsinθccosθq 2+φq˙θq 2cosθcsinθq 2−2˙φcsinθcsinθq 2/bracketrightBigg . (23) Next, we want to express B·Sq gin terms of Euler angles. We see that B·Sq g=S[Bxsinθcosφ+Bysinθsinφ+Bzcosθ]q = 2S/bracketleftBig Bx/parenleftbigg cosθcsinθq 2cosφccosφq 2−sinθccosθq 2sinφcsinφq 2/parenrightbigg +By/parenleftbigg sinθccosθq 2cosφcsinφq 2+ cosθcsinθq 2sinφccosφq 2/parenrightbigg −Bzsinθcsinθq 2/bracketrightBig , (24) where we used the results from Eq. (21). Similarly, we have K/bracketleftbig S2 z,g/bracketrightbigq=KS2[cos2θ]q=−2KS2sinθccosθcsinθq. (25) Combining these results, we conclude that /bracketleftbig (−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2 z,g/bracketrightbigq=S/bracketleftBigg φq˙θcsinθccosθq 2+φq˙θq 2cosθcsinθq 2−2(−Bz+KScosθc+˙φc) sinθcsinθq 2 −2Bx/parenleftbigg cosθcsinθq 2cosφccosφq 2−sinθccosθq 2sinφcsinφq 2/parenrightbigg −By/parenleftbigg sinθccosθq 2cosφcsinφq 2+ cosθcsinθq 2sinφccosφq 2/parenrightbigg/bracketrightBigg . (26)6 Remark that this expression only contains odd powers of ( q), so that we can neglect all higher-order terms to get /bracketleftBig (−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2 z,g/bracketrightBigq =S/bracketleftBig −θqsinθc(−Bz+KScosθc+˙φc) −θqcosθc(Bxcosφc+Bysinφc) +φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig . (27) Now, we focus on the part of the action in Eq. (14) that comes from the bath, given by iSb[g] =−i/integraldisplay dt/integraldisplay dt/prime/vectorST g(t)/parenleftbigg 0αA αRαK/parenrightbigg (t−t/prime)/vectorSg(t/prime). (28) Let us first consider what Sq gandSc gare in terms of φandθ. By performing some trigonometric operations on each of the components, we find that Sc g=S sinθccosθq 2cosφccosφq 2−cosθcsinθq 2sinφcsinφq 2 sinθccosθq 2sinφccosφq 2+ cosθcsinθq 2cosφcsinφq 2 cosθccosθq 2 (29) and Sq g= 2S cosθcsinθq 2cosφccosφq 2−sinθccosθq 2sinφcsinφq 2 sinθccosθq 2cosφcsinφq 2+ cosθcsinθq 2sinφccosφq 2 −sinθcsinθq 2 . (30) By expanding in the quantum components of Sc gandSq g, we see that Sc g= (q)0+O/parenleftbig (q)2/parenrightbig , Sq g= (q)1+O/parenleftbig (q)3/parenrightbig . Since the action only contains terms with at least one Sq g, we know that the only way to obtain a term of order ( q)2 is from (Sq g)2. Hence, we may neglect all terms beyond linear ( q) inS(c/q) g in the quasi-classical regime. This results in Sc g=S sinθccosφc sinθcsinφc cosθc , (31) Sq g=S θqcosθccosφc−φqsinθcsinφc φqsinθccosφc+θqcosθcsinφc −θqsinθc . (32) A useful remark for later is that this shows that Sq g=θq∂ ∂θcSc g+φq∂ ∂φcSc g. (33) Going back to iSb[g], we can rewrite this as a convolution, in the sense that iSb[g] =−i/integraldisplay dt/bracketleftbig Sc g(t)·/parenleftbig αA∗Sq g/parenrightbig (t) +Sq g(t)·/parenleftbig αR∗Sc g/parenrightbig (t) +Sq g(t)·/parenleftbig αK∗Sq g/parenrightbig (t)/bracketrightbig , (34) where (f∗g)(t) =/integraltext∞ −∞dt/primef(t−t/prime)g(t/prime). We see that the first two terms contain precisely one quantum component, but the last term has two quantum components. When writing down the Euler-Lagrange equation of motion, it is important to realize that the convolution operation will act as if it is a simple multiplication, since the convolution obeys d dx(f(x)∗g)(t) =/parenleftbiggdf dx∗g/parenrightbigg (t). (35)7 We now concentrate on the ( q)2part of this action, for which we would like to use a Hubbard-Stratonovich transfor- mation in order to reduce this to linear in ( q). Recall that a Hubbard-Stratonovich transformation is given by exp/bracketleftBig −a 2x2/bracketrightBig =/radicalbigg 1 2πa/integraldisplay Dξexp/bracketleftbigg −ξ2 2a−ixξ/bracketrightbigg . (36) However, we see that our action does not contain any purely quadratic terms, but rather a Greens functional shape asSq g(t)αK(t−t/prime)Sq g(t/prime). Hence, to use a Hubbard-Stratonovich like transformation, we must derive it from a Greens function exponential, similarly to Ref. [3]. Assuming that this is renormalizable and that αKcan be rewritten into a distribution, we have 1 =/integraldisplay Dξexp/bracketleftbigg −1 2/integraldisplay dt/integraldisplay dt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)/bracketrightbigg =/integraldisplay Dξexp/bracketleftbigg −1 2/integraldisplay dt/integraldisplay dt/prime/parenleftbigg ξ(t)−2/integraldisplay dt/prime/primeSq g(t/prime/prime)αK(t/prime/prime−t)/parenrightbigg [−2iαK]−1(t−t/prime)/parenleftbigg ξ(t/prime)−2/integraldisplay dt/prime/prime/primeαK(t/prime−t/prime/prime/prime)Sq g(t/prime/prime/prime)/parenrightbigg/bracketrightbigg =/integraldisplay Dξexp/bracketleftbigg −1 2/integraldisplay dt/integraldisplay dt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime) −iSq g(t)δ(t−t/prime)ξ(t/prime)−iξ(t)δ(t−t/prime)Sq g(t/prime)−2iSq g(t)αK(t−t/prime)Sq g(t/prime)/bracketrightbig =/integraldisplay Dξexp/bracketleftbigg −1 2/integraldisplay dt/integraldisplay dt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)−2iSq g(t)δ(t−t/prime)ξ(t/prime)−2iSq g(t)αK(t−t/prime)Sq g(t/prime)/bracketrightbigg , where we used that/integraltext dt/primeαK(t−t/prime)[αK]−1(t/prime−t/prime/prime) =δ(t−t/prime/prime) and that 2 iαKis positive real. Therefore, we find that exp/bracketleftbigg −i/integraldisplay dt/integraldisplay dt/primeSq g(t)αK(t−t/prime)Sq g(t/prime)/bracketrightbigg =/integraldisplay Dξexp/bracketleftbigg −1 2/integraldisplay dt/integraldisplay dt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)/bracketrightbigg ·exp/bracketleftbigg i/integraldisplay dtSq g(t)ξ(t)/bracketrightbigg . (37) The double integral in the first exponential signifies the statistical properties of ξ. For instance, if αKis delta-like, thenξwould have Gaussian statistics (e.g. white noise), but in general we will have time correlated noise defined by αK[3], such that /angbracketleftξ(t)ξ(t/prime)/angbracketright=−2iαK(t−t/prime). (38) Since there is no gdependence in the double ξexponential, we will leave it out of S[g] and only remember these statistics. Our partition function is then given by Z=/integraldisplay Dξexp (iSn[ξ])/integraldisplay Dgexp (iSsc[g,ξ]), (39) where the noise action is given by iSn[ξ] =−1 2/integraldisplay dt/integraldisplay dt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime) (40) and the semi-classical action is given by iSsc[g,ξ] =i/integraldisplay dtS/bracketleftBig −θqsinθc(−Bz+KScosθc+˙φc)−θqcosθc(Bxcosφc+Bysinφc) +φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig +i/integraldisplay dt/bracketleftbig ξ(t)Sq g(t)/bracketrightbig −i/integraldisplay dt/bracketleftbig Sc g(t)·/parenleftbig αA∗Sq g/parenrightbig (t) +Sq g(t)·/parenleftbig αR∗Sc g/parenrightbig (t)/bracketrightbig , (41) whereSc g(t) andSq g(t) include only up to first-order corrections in quantum components. Assuming that αA/Rcan be written in terms of distributions, we can define the distribution αdiss(t) =−αR(t)−αA(−t) and rewrite the semi-classical action as iSsc[g,ξ] =i/integraldisplay dtS/bracketleftBig −θqsinθc(−Bz+KScosθc+˙φc)−θqcosθc(Bxcosφc+Bysinφc) +φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig +i/integraldisplay dt/bracketleftbig/parenleftbig αdiss∗Sc g/parenrightbig (t) +ξ(t)/bracketrightbig Sq g(t). (42)8 Recall that, using the Euler angles, we have/integraltext Dg=/integraltext DθDφ sin(θ). Technically, the factor of sin( θ) would end up in the action. However, since one could define ρ= cos(θ) as a new variable in order to avoid this, we know that this term is not relevant to the physics. Hence, we can disregard it. Since all terms in iSsc[g,ξ] are either linear in θqorφq, we find two Euler-Lagrange equations of the form δLsc δθq= 0 andδLsc δφq= 0. (43) Remembering Eq. (33), we see thatδSq g(t) δθq=δSc g(t) δθcandδSq g(t) δφq=δSc g(t) δφc. Hence, the e.o.m. can be rearranged to yield ˙φc=1 Ssinθc/bracketleftbig −B(Sc z) +/parenleftbig αdiss∗Sc g/parenrightbig (t) +ξ(t)/bracketrightbig ·δSc g(t) δθc(44) and ˙θc=−1 Ssinθc/bracketleftbig −B(Sc z) +/parenleftbig αdiss∗Sc g/parenrightbig (t) +ξ(t)/bracketrightbig ·δSc g(t) δφc, (45) whereB(Sc z) = Bx By Bz−KSc z . D. Generalized Landau-Lifshitz-Gilbert equation We want to show that the equations found by the microscopic model are in fact precisely of the LLG form. For this, we will have to start from the LLG equation, and introduce the same two Euler angles θandφfor the spin, and show that this gives rise to the same set of equations as previously deduced. We begin with the generalized LLG equation ˙S(t) =S(t)×[−B(Sz) + (αdiss∗S) (t) +ξ(t)], (46) whereαdiss(t) =−αR(t)−αA(−t),/angbracketleftξ(t)ξ(t/prime)/angbracketright=−2iαK(t−t/prime) andB(Sz) = (Bx,By,Bz−KSz)T. Since the velocity ofSis always perpendicular to S, we know that the magnitude of Sis constant. Hence, we can go to spherical coordinates, such that S=S sinθcosφ sinθsinφ cosθ . (47) Inserting this into the LLG equation, we firstly see that ˙S=˙θ∂S ∂θ+˙φ∂S ∂φ=˙θS cosθcosφ cosθsinφ −sinθ +˙φS −sinθsinφ sinθcosφ 0 . Now, we notice that the RHS of the LLG equation can, without loss of generality, be written as S(t)×rwith r= (x,y,z )T. Working this out explicitly, we find that the LLG equation ˙S=S×rbecomes S ˙θcosθcosφ−˙φsinθsinφ ˙θcosθsinφ+˙φsinθcosφ −˙θsinθ =S zsinθsinφ−ycosθ xcosθ−zsinθcosφ ysinθcosφ−xsinθsinφ . (48) We note that the equation corresponding to the zcomponent can be written as ˙θ=−1 sinθr· −sinθsinφ sinθcosφ 0 =−1 Ssinθr·∂S ∂φ. (49)9 Now, we add up the ˆ xand ˆyequations, such that the ˙θcancels (i.e.−ˆxsinφ+ ˆycosφ). This yields ˙φsinθ(sin2φ+ cos2φ) =−zsinθ(sin2φ+ cos2φ) +ycosθsinφ+xcosθcosφ, which simplifies to ˙φ=1 sinθr· cosθcosφ cosθsinφ −sinθ =1 Ssinθr·∂S ∂θ. (50) By inserting r=−B(Sz) + (αdiss∗S) (t) +ξ(t), we see that this is identical to the equations derived from the microscopic model ˙φc=1 Ssinθc/bracketleftbig −B(Sz) +/parenleftbig αdiss∗Sc g/parenrightbig (t) +ξ(t)/bracketrightbig ·δSc g(t) δθc; (51) ˙θc=−1 Ssinθc/bracketleftbig −B(Sz) +/parenleftbig αdiss∗Sc g/parenrightbig (t) +ξ(t)/bracketrightbig ·δSc g(t) δφc. (52) Therefore, we may conclude that our microscopic model is described by the generalized LLG equation. For the fractional LLG equation, we are in particular interested in the case where αdiss∗S=αsDs tS, whereDs tis a fractional derivative. For instance, assuming 0 <s< 1, the Liouville fractional derivative is given by Ds tf(t) =1 Γ(1−s)/integraldisplayt −∞(t−t/prime)−sf/prime(t/prime)dt/prime. (53) So, ifαdiss=αsΘ(t) Γ(1−s)t−s∂t, then, because of the convolution with S, we would find a fractional LLG equation, whereas αdiss=α1δ(t)∂twould give the regular LLG equation. II. FRACTIONAL DERIVATIVE FROM NON-OHMIC SPECTRAL FUNCTION Here, we will compute the type of dissipation which comes from the spectral function. We will first derive the spectral function from microscopic quantities to see how it ends up in the Greens function. Then, we will calculate the dissipation for three different cases. A. Calculating the effective Greens functions We recall that αA/R(t−t/prime) =/summationdisplay α/parenleftbiggγ2 α 4GA/R α(t−t/prime) +δ(t−t/prime)γ2 α 2mαω2α/parenrightbigg , where [G−1 α]R/A(t−t/prime) =δ(t−t/prime)mα 2[(i∂t±i0)2−ω2 α]. By the fluctuation dissipation theorem, we also have αK(ω) =/bracketleftbig αR(ω)−αA(ω)/bracketrightbig coth/parenleftBigω 2T/parenrightBig . We are interested in finding closed forms for αR/A/K(t−t/prime). Using the relation /integraldisplay dt/primeG−1(t−t/prime)G(t/prime−t/prime/prime) =δ(t−t/prime/prime), (54) we note that mα 2[(i∂t±i0)2−ω2 α]GR/A α(t−t/prime/prime) =δ(t−t/prime/prime). (55)10 The Fourier transform1yields GR/A α(ω) =1 mα 2[(ω±i0)2−ω2α]. (56) We therefore find that αR/A(ω) =/summationdisplay α/parenleftbiggγ2 α 4GA/R α(ω) +γ2 α 2mαω2α/parenrightbigg =/summationdisplay αγ2 α 2mα/bracketleftbigg1 (ω±i0)2−ω2α+1 ω2α/bracketrightbigg =/summationdisplay αγ2 α 2mαω2αω2 (ω±i0)2−ω2α.(57) The spectral function is given by the imaginary part of the Fourier transform of the dynamical susceptibility χ(ω) =δ δS(ω)/summationdisplay αγαˆxα(ω). (58) The Hamiltonian e.o.m. for the bath can be found by going back to our starting Hamiltonian H=B·S+/summationdisplay αγαS·ˆxα+/summationdisplay αˆp2 α 2mα+mαω2 α 2ˆx2 α+/summationdisplay αγ2 α 2mαω2αS2. The e.o.m. reads ˙ˆxα=ˆpα mαand ˙ˆpα=−γαS−mαω2 αˆxα. (59) Combining both equations and taking the Fourier transform, we find ˆxα(ω) =γα mα[(ω+i0)2−ω2α]S(ω), (60) where we have taken an infinitesimal amount of dissipation + i0 on the oscillators into account. This leads to χ(ω) =/summationdisplay αγ2 α mα[(ω+i0)2−ω2α]=/summationdisplay αγ2 α 2mαωα/parenleftbigg1 ω+i0−ωα−1 ω+i0 +ωα/parenrightbigg . (61) We remark that Im1 x+i0=−πδ(x), (62) which leads to J(ω) = Imχ(ω) =−/summationdisplay απγ2 α 2mαωα[δ(ω−ωα)−δ(ω+ωα)] =−π 2/summationdisplay αγ2 α mαωαδ(ω−ωα), (63) where we used that all oscillator frequencies are positive. We can identify the spectral function in αR/Aas αR/A(ω) =/summationdisplay αγ2 α 2mαω2αω2 (ω±i0)2−ω2α=−/integraldisplay∞ 0dε πω2ε−1J(ε) (ω±i0)2−ε2. (64) Now, we will assume a particular shape for J(ε). This can be either Ohmic or non-Ohmic, but in general we may assume a power-law behavior as some J(ε) =αsεs. 1We use the convention f(ω) =/integraltext∞ −∞dteiωtf(t) andf(t) =1 2π/integraltext∞ −∞dωe−iωtf(ω), withδ(t) =1 2π/integraltext∞ −∞dωeiωt.11 B. Ohmic spectral function Beginning with the Ohmic case J(ε) =α1ε, we see that 2iImαR/A(ω) =αR/A(ω)−/bracketleftBig αR/A/bracketrightBig∗ (ω) =−α1/integraldisplay∞ 0dε π/bracketleftbiggω2 (ω±i0)2−ε2−ω2 (ω∓i0)2−ε2/bracketrightbigg =−α1/integraldisplay∞ −∞dε 2π/bracketleftbiggω2 (ω±i0)2−ε2−ω2 (ω∓i0)2−ε2/bracketrightbigg =−α1ω2/integraldisplay∞ −∞dε 2π/bracketleftbigg1 (ω±i0 +ε)(ω±i0−ε)−1 (ω∓i0 +ε)(ω∓i0−ε)/bracketrightbigg =±4i0ω3α1/integraldisplay∞ −∞dε 2π1 (ω±i0 +ε)(ω±i0−ε)(ω∓i0 +ε)(ω∓i0−ε), (65) which has four poles at ε=±1(ω±2i0). Since the integral scales as <1/|ε|, we can add an infinite radius half circle to complete a complex contour integral. Notice from symmetry that we will always have one of each of the four poles. We can thus drop the ±signs inside since this only changes the notation order in the fraction. We thus find that 2iImαR/A(ω) =±4i0ω3α1/integraldisplay∞ −∞dε 2π1 (ε+ω+i0)(ε−ω−i0)(ε+ω−i0)(ε−ω+i0). (66) Completing the contour along the top, we find poles at ε=±ω+i0, which yields 2iImαR/A(ω) =∓0·4ω3α1/bracketleftbigg1 (ω+i0 +ω+i0)(ω+i0 +ω−i0)(ω+i0−ω+i0) +1 (−ω+i0 +ω+i0)(−ω+i0−ω−i0)(−ω+i0−ω+i0)/bracketrightbigg =∓0·4ω3α1/bracketleftbigg1 2(ω+i0)(2ω)(2i0)+1 (2i0)(−2ω)2(−ω+i0)/bracketrightbigg =±i 2ω3α1/bracketleftbigg1 ω2+i0ω+1 ω2−i0ω/bracketrightbigg =±i 2ω3α1/bracketleftbiggω2−i0ω+ω2+i0ω ω4/bracketrightbigg =±iωα1. (67) Hence, Im αR/A(ω) =±α1ω/2. Similarly, 2 ReαR/A(ω) =αR/A(ω) +/bracketleftBig αR/A/bracketrightBig∗ (ω) =−α1/integraldisplay∞ 0dε π/bracketleftbiggω2 (ω±i0)2−ε2+ω2 (ω∓i0)2−ε2/bracketrightbigg =−α1/integraldisplay∞ −∞dε 2π/bracketleftbiggω2 (ω±i0)2−ε2+ω2 (ω∓i0)2−ε2/bracketrightbigg =−α1/integraldisplay∞ −∞dε 2π/bracketleftbiggω2 (ω±i0 +ε)(ω±i0−ε)+ω2 (ω∓i0 +ε)(ω∓i0−ε)/bracketrightbigg =α1/integraldisplay∞ −∞dε πω2(ε2−ω2) (ε+ω+i0)(ε−ω−i0)(ε+ω−i0)(ε−ω+i0). (68)12 Since the integral scales as 1 /|ε|, we can freely add the infinite circular contour along the top. Applying the residue theorem, we find 2 ReαR/A(ω) = 2iα1/bracketleftbiggω2((ω+i0)2−ω2) (ω+i0 +ω+i0)(ω+i0 +ω−i0)(ω+i0−ω+i0) +ω2((−ω+i0)2−ω2) (−ω+i0 +ω+i0)(−ω+i0−ω−i0)(−ω+i0−ω+i0)/bracketrightbigg = 2α1i/bracketleftbiggω2(2ωi0) 2(ω+i0)(2ω)(2i0)+ω2(−2ωi0) (2i0)(−2ω)2(−ω+i0)/bracketrightbigg =α1 2i/bracketleftbiggω2 ω+i0−ω2 ω−i0/bracketrightbigg =α1 2i/bracketleftbiggω2(−2i0) ω2/bracketrightbigg =α10, (69) which means that Re αR/A(ω) =α1 2·0 = 0. Hence, αR/A(ω) =±α1iω 2(70) and sinceαdiss(t) =−αR(t)−αA(−t) we have αdiss(ω) =−αR(ω)−αA(−ω) =−α1iω. (71) In the LLG equation, we thus find (αdiss∗S) (t) =1 2π/integraldisplay dωe−iωt(αdiss∗S) (ω) =1 2π/integraldisplay dωe−iωtαdiss(ω)S(ω) =1 2π/integraldisplay dωe−iωt(−α1iω)S(ω) =α1∂ ∂t1 2π/integraldisplay dωe−iωtS(ω) =α1˙S(t). (72) Furthermore, we have that αK(ω) = [αR(ω)−αA(ω)] coth/parenleftBigω 2T/parenrightBig =iα1ωcoth/parenleftBigω 2T/parenrightBig . (73) We have two regimes from the cotangent which crossover around ω≈2T. If the temperature is large enough that we can approximate coth/parenleftbigω 2T/parenrightbig ≈2T ω, then we have αK(ω)≈2iα1T. Therefore, we find αK(t) = 2iα1T1 2π/integraldisplay∞ −∞dωe−iωt= 2iα1Tδ(t), (74) and thus /angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδm,nαK(t−t/prime) = 4α1Tδm,nδ(t−t/prime). (75) We see that Eq. (72) and Eq. (75) combine to give the regular LLG equation with first-order dissipation and white noise fluctuation: ˙S(t) =S(t)×/bracketleftBig −B+α1˙S(t) +ξ(t)/bracketrightBig . (76)13 C. Sub-Ohmic spectral function We now consider the case where J(ε) =αssin/parenleftBigπs 2/parenrightBig εs, (77) with 0<s< 1. In this case, αR/A(ω) =−αssin/parenleftBigπs 2/parenrightBig/integraldisplay∞ 0dε πω2εs−1 (ω±i0)2−ε2. (78) Considering the LLG equation, the relevant dissipation term is ( αdiss∗S)(t). In terms of the Fourier transform of αdiss(t), we find that (αdiss∗S)(t) =/integraldisplay∞ −∞dt/primeαdiss(t−t/prime)S(t/prime) =−/integraldisplay∞ −∞dt/prime[αR(t−t/prime) +αA(t/prime−t)]S(t/prime) =−1 2π/integraldisplay∞ −∞dt/prime/integraldisplay∞ −∞dω/bracketleftBig e−iω(t−t/prime)αR(ω) +eiω(t−t/prime)αA(ω)/bracketrightBig S(t/prime) =−1 2π/integraldisplay∞ −∞dt/prime/integraldisplay∞ −∞dωe−iω(t−t/prime)/bracketleftbig αR(ω) +αA(−ω)/bracketrightbig S(t/prime) =αssin/parenleftbigπs 2/parenrightbig 2π2/integraldisplay∞ −∞dt/prime/integraldisplay∞ −∞dω/integraldisplay∞ 0dεe−iω(t−t/prime)/bracketleftbiggω2εs−1 (ω+i0)2−ε2+(−ω)2εs−1 (−ω−i0)2−ε2/bracketrightbigg S(t/prime) =−αssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ −∞dt/prime/integraldisplay∞ 0dε/integraldisplay∞ −∞dω/bracketleftbigg e−iω(t−t/prime)ω2εs−1 ε2−(ω+i0)2S(t/prime)/bracketrightbigg . (79) Notice that we have two poles at ω=±ε−i0, below the real axis. If t−t/prime<0, then the exponential will go to zero as ω→+i∞. Hence we could close the ωintegration with a complex contour as an infinite half-circle along the top, and get zero from the Cauchy theorem. If t−t/prime>0, however, we see that the exponential goes to zero when ω→−i∞. Hence, we can close the ωintegration along the bottom. Thus, using the residue theorem (reversing the integration direction), we find (αdiss∗S)(t) =−αssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ −∞dt/prime/integraldisplay∞ 0dε2πiΘ(t−t/prime) /bracketleftBig e−i(ε−i0)(t−t/prime)(ε−i0)2εs−1 (ε−i0 +i0 +ε)+e−i(−ε−i0)(t−t/prime)(−ε−i0)2εs−1 (−ε−i0 +i0−ε)/bracketrightBig S(t/prime) =−iαssin/parenleftbigπs 2/parenrightbig π/integraldisplayt −∞dt/prime/integraldisplay∞ 0dε/bracketleftbigg e−iε(t−t/prime)εs+1 ε+eiε(t−t/prime)εs+1 −ε/bracketrightbigg S(t/prime) =−iαssin/parenleftbigπs 2/parenrightbig π/integraldisplayt −∞dt/prime/integraldisplay∞ 0dε/bracketleftBig e−iε(t−t/prime)−eiε(t−t/prime)/bracketrightBig εsS(t/prime) =−2αssin/parenleftbigπs 2/parenrightbig π/integraldisplayt −∞dt/prime/integraldisplay∞ 0dεsin[ε(t−t/prime)]εsS(t/prime) =−2αssin/parenleftbigπs 2/parenrightbig π/integraldisplay∞ 0dε/braceleftbigg/bracketleftbig εs−1cos[ε(t−t/prime)]S(t/prime)/bracketrightbigt/prime=t t/prime=t0−/integraldisplayt t0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg =−2αssin/parenleftbigπs 2/parenrightbig π/integraldisplay∞ 0dε/braceleftbigg εs−1S(t)−εs−1cos[ε(t−t0)]S(t0)−/integraldisplayt t0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg .(80) The first term vanishes because of the cross product with S(t) in the LLG equation. The second term is where we had to be careful. Here, we should realize that the −∞ is physically only indicating that it is a time very far in the past. So, to avoid unphysical infinities, we introduced a finite initial time t0and we will take t0→−∞ later. For this, we need to introduce some fractional derivative notation. We define the Riemann-Liouville (RL) and Caputo (C)14 derivatives of order s, with an integer nsuch thatn≤s<n + 1, as RL t0Ds tf(t) =dn dtn1 Γ(n−s)/integraldisplayt t0dt/prime(t−t/prime)n−1−sf(t/prime), (81) C t0Ds tf(t) =1 Γ(n−s)/integraldisplayt t0dt/prime(t−t/prime)n−1−sf(n)(t/prime), (82) where we reserve the simpler Ds tnotation for the Liouville derivative that was used in the main text. Now, rescaling ε→ε/(t−t0) andε→ε/(t−t/prime) in the second and third terms of Eq. (80) respectively, we have (αdiss∗S)(t) = 2αssin/parenleftbigπs 2/parenrightbig π/integraldisplay∞ 0dεcos(ε)εs−1/bracketleftbigg (t−t0)−sS(t0) +/integraldisplayt t0dt/prime(t−t/prime)−s˙S(t/prime)/bracketrightbigg = 2αssin/parenleftbigπs 2/parenrightbig π/bracketleftBig cos/parenleftBigπs 2/parenrightBig Γ(s)/bracketrightBig/bracketleftbigg (t−t0)−sS(t0) +/integraldisplayt t0dt/prime(t−t/prime)−s˙S(t/prime)/bracketrightbigg = 2αssin/parenleftbigπs 2/parenrightbig πcos/parenleftBigπs 2/parenrightBig Γ(s)/bracketleftbig (t−t0)−sS(t0) + Γ(1−s)C t0Ds tS(t)/bracketrightbig = 2αssin/parenleftbigπs 2/parenrightbig πcos/parenleftBigπs 2/parenrightBig Γ(s)Γ(1−s)/bracketleftbigg(t−t0)−s Γ(1−s)S(t0) +C t0Ds tS(t)/bracketrightbigg = 2αssin/parenleftbigπs 2/parenrightbig πcos/parenleftBigπs 2/parenrightBigπ sin(πs)RL t0Ds tS(t) =αsRL t0Ds tS(t) =αsDs tS(t), (83) where we used several identities from Sec. 6 in the Sup. Mat. of Ref. [12] and in the last line we sent t0→−∞ . For the noise correlation, we have to compute the Keldysh component. This is αK(t) =1 2π/integraldisplay∞ −∞dωe−iωtαK(ω) =1 2π/integraldisplay∞ −∞dωe−iωt[αR(ω)−αA(ω)] coth/parenleftBigω 2T/parenrightBig =αssin/parenleftbigπs 2/parenrightbig 2π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωe−iωt/bracketleftbiggω2εs−1 ε2−(ω+i0)2−ω2εs−1 ε2−(ω−i0)2/bracketrightbigg coth/parenleftBigω 2T/parenrightBig (now sendω→−ωin the advanced part) =αssin/parenleftbigπs 2/parenrightbig 2π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dω/bracketleftbigge−iωtω2εs−1 ε2−(ω+i0)2+eiωtω2εs−1 ε2−(−ω−i0)2/bracketrightbigg coth/parenleftBigω 2T/parenrightBig =αssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωcos(ωt)ω2εs−1 ε2−(ω+i0)2coth/parenleftBigω 2T/parenrightBig . (84) Now, we send ω→ω/tandε→ε/t, which yields αK(t) =αst−1−ssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωcos(ω)ω2εs−1 ε2−(ω+i0)2coth/parenleftBigω 2tT/parenrightBig . (85) Taking the high temperature limit, we get αK(t) =αst−1−ssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωcos(ω)ω2εs−1 ε2−(ω+i0)22tT ω = 2Tαst−ssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωcos(ω)ωεs−1 ε2−(ω+i0)2 =−2Tαst−ssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωcos(ω)ωεs−1 (ω+i0−ε)(ω+i0 +ε). (86)15 Now, we want to close the integral over ωwith an infinite half-circle. For this, we need fast enough convergence of the integrand to zero. Splitting the cosine into two exponential parts cos( ω) = (eiω+e−iω)/2, we see that the first term goes to zero when ω→i∞and that the second term goes to zero when ω→−i∞. Since we have poles at ω=±ε−i0, the integral along the top half-plane vanishes. The integral along the bottom is then computed as αK(t) =−Tαst−ssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωe−iωωεs−1 (ω+i0−ε)(ω+i0 +ε) =−Tαst−ssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε−2πi/bracketleftbigge−i(ε−i0)(ε−i0)εs−1 ε−i0 +i0 +ε+e−i(−ε−i0)(−ε−i0)εs−1 −ε−i0 +i0−ε/bracketrightbigg = 2iTαst−ssin/parenleftbigπs 2/parenrightbig π/integraldisplay∞ 0dε/bracketleftbigge−iεεs 2ε+eiεεs 2ε/bracketrightbigg = 2iTαst−ssin/parenleftbigπs 2/parenrightbig π/integraldisplay∞ 0dεcos(ε)εs−1 = 2iTαst−ssin/parenleftbigπs 2/parenrightbig πcos/parenleftBigπs 2/parenrightBig Γ(s) =iTαst−ssin (πs) πΓ(s) =iTαst−s Γ(1−s). (87) We therefore find that /angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδm,nαK(t−t/prime) = 2αsTδm,n(t−t/prime)−s Γ(1−s), (88) where we assumed that t≥t/prime. Therefore, we have now found the fractional LLG equation ˙S(t) =S(t)×[−B+αsDs tS(t) +ξ(t)]. (89) D. Super-Ohmic spectral function We now consider the case where J(ε) =αssin/parenleftBigπs 2/parenrightBig εs, (90) with 1< s < 2. In this case, everything is equivalent to the sub-Ohmic case, up to Eq. (80), where we wanted to rewrite the dissipation into a fractional derivative. We had to introduce a finite initial time t0, which lead to (αdiss∗S)(t) =−2αssin/parenleftbigπs 2/parenrightbig π/integraldisplayt −∞dt/prime/integraldisplay∞ 0dεsin[ε(t−t/prime)]εsS(t/prime) =−2αssin/parenleftbigπs 2/parenrightbig π/integraldisplay∞ 0dε/braceleftbigg/bracketleftbig εs−1cos[ε(t−t/prime)]S(t/prime)/bracketrightbigt/prime=t t/prime=t0−/integraldisplayt t0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg =−2αssin/parenleftbigπs 2/parenrightbig π/integraldisplay∞ 0dε/braceleftbigg εs−1S(t)−εs−1cos[ε(t−t0)]S(t0)−/integraldisplayt t0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg .(91) The first term vanishes because of the cross product with S(t) in the LLG equation. However, the second term is more problematic compared to the sub-Ohmic case, since the identity used to rewrite it only holds for s <1. We solve this by writing it as a time derivative εs−1cos[ε(t−t0)]S(t0) =d dtεs−2sin[ε(t−t0)]S(t0), (92)16 and then switching the ordering of the derivative and integral. Performing also one more partial integration in t/prime, we get (αdiss∗S)(t) = 2αssin/parenleftbigπs 2/parenrightbig π/integraldisplay∞ 0dε/braceleftbiggd dtεs−2sin[ε(t−t0)]S(t0) +/integraldisplayt t0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg = 2αssin/parenleftbigπs 2/parenrightbig π/integraldisplay∞ 0dε/parenleftbiggd dtεs−2sin[ε(t−t0)]S(t0) +/bracketleftBig εs−2sin[ε(t−t/prime)]˙S(t/prime)/bracketrightBigt/prime=t t/prime=t0−/integraldisplayt t0dt/prime/braceleftBig −sin[ε(t−t/prime)]εs−2¨S(t/prime)/bracerightBig/parenrightbigg = 2αssin/parenleftbigπs 2/parenrightbig π/integraldisplay∞ 0dε/braceleftbiggd dtεs−2sin[ε(t−t0)]S(t0) +εs−2sin[ε(t−t0)]˙S(t0) +/integraldisplayt t0dt/primesin[ε(t−t/prime)]εs−2¨S(t/prime)/bracerightbigg . (93) Now, rescaling ε→ε/(t−t0) andε→ε/(t−t/prime) respectively, we have (αdiss∗S)(t) = 2αssin/parenleftbigπs 2/parenrightbig π/integraldisplay∞ 0dεsin(ε)εs−2/bracketleftbiggd dt(t−t0)1−sS(t0) + (t−t0)1−s˙S(t0) +/integraldisplayt t0dt/prime(t−t/prime)1−s¨S(t/prime)/bracketrightbigg = 2αssin/parenleftbigπs 2/parenrightbig π/bracketleftbigg sin/parenleftbiggπ(s−1) 2/parenrightbigg Γ(s−1)/bracketrightbigg/bracketleftbigg (1−s)(t−t0)−sS(t0) + (t−t0)1−s˙S(t0) +/integraldisplayt t0dt/prime(t−t/prime)1−s¨S(t/prime)/bracketrightbigg =−2αssin/parenleftbigπs 2/parenrightbig πcos/parenleftBigπs 2/parenrightBig Γ(s−1)/bracketleftBig (1−s)(t−t0)−sS(t0) + (t−t0)1−s˙S(t0) + Γ(2−s)C t0Ds tS(t)/bracketrightBig =−2αssin/parenleftbigπs 2/parenrightbig πcos/parenleftBigπs 2/parenrightBig Γ(s−1)Γ(1−(s−1))/bracketleftbigg(t−t0)−s Γ(1−s)S(t0) +(t−t0)1−s Γ(2−s)˙S(t0) +C t0Ds tS(t)/bracketrightbigg =−αssin (πs) ππ sin[π(s−1)]RL t0Ds tS(t) =αsDs tS(t), (94) where we used several identities from Sec. 6 in the Sup. Mat. of Ref. [12], Ref. [13, p.893], and in the last line we sentt0→−∞ . For the noise correlation, we have to compute the Keldysh component. This is αK(t) =1 2π/integraldisplay∞ −∞dωe−iωtαK(ω) =1 2π/integraldisplay∞ −∞dωe−iωt[αR(ω)−αA(ω)] coth/parenleftBigω 2T/parenrightBig =αssin/parenleftbigπs 2/parenrightbig 2π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωe−iωt/bracketleftbiggω2εs−1 ε2−(ω+i0)2−ω2εs−1 ε2−(ω−i0)2/bracketrightbigg coth/parenleftBigω 2T/parenrightBig . (95) Now, we send ω→−ωin the advanced part αK(t) =αssin/parenleftbigπs 2/parenrightbig 2π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dω/bracketleftbigge−iωtω2εs−1 ε2−(ω+i0)2+eiωtω2εs−1 ε2−(−ω−i0)2/bracketrightbigg coth/parenleftBigω 2T/parenrightBig =αssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωcos(ωt)ω2εs−1 ε2−(ω+i0)2coth/parenleftBigω 2T/parenrightBig . (96) Then, we insert cos( ωt) =d dtsin(ωt) ωand we send ω→ω/tandε→ε/t, which yields αK(t) =d dtαssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωsin(ωt)ωεs−1 ε2−(ω+i0)2coth/parenleftBigω 2T/parenrightBig =d dtαst−ssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωsin(ω)ωεs−1 ε2−(ω+i0)2coth/parenleftBigω 2tT/parenrightBig . (97)17 Taking the high-temperature limit, we get αK(t) =d dtαst−ssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωsin(ω)ωεs−1 ε2−(ω+i0)22tT ω =d dt2Tαst1−ssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωsin(ω)εs−1 ε2−(ω+i0)2 =−2Tαs(1−s)t−ssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωsin(ω)εs−1 (ω+i0−ε)(ω+i0 +ε). (98) Now, we want to close the integral over ωwith an infinite half-circle. For this, we need fast enough convergence of the integrand to zero. Splitting the cosine into two exponential parts sin( ω) = (eiω−e−iω)/2i, we see that the first term goes to zero when ω→i∞and that the second term goes to zero when ω→−i∞. Since we have poles at ω=±ε−i0, the integral along the top half-plane vanishes. The integral along the bottom is then computed as αK(t) =−iTαs(1−s)t−ssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε/integraldisplay∞ −∞dωe−iωεs−1 (ω+i0−ε)(ω+i0 +ε) =−iTαs(1−s)t−ssin/parenleftbigπs 2/parenrightbig π2/integraldisplay∞ 0dε−2πi/bracketleftbigge−i(ε−i0)εs−1 ε−i0 +i0 +ε+e−i(−ε−i0)εs−1 −ε−i0 +i0−ε/bracketrightbigg =−2Tαs(1−s)t−ssin/parenleftbigπs 2/parenrightbig π/integraldisplay∞ 0dε/bracketleftbigge−iεεs−2 2−eiεεs−2 2/bracketrightbigg =i2Tαs(1−s)t−ssin/parenleftbigπs 2/parenrightbig π/integraldisplay∞ 0dεsin(ε)εs−2 =−i2Tαs(1−s)t−ssin/parenleftbigπs 2/parenrightbig πcos/parenleftBigπs 2/parenrightBig Γ(s−1) =iTαst−ssin (πs) πΓ(s) =iTαst−s Γ(1−s). (99) We therefore find the same expression as in the sub-Ohmic case, which leads to /angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδm,nαK(t−t/prime) = 2αsTδm,n(t−t/prime)−s Γ(1−s). (100) Therefore, we have now also found the fractional LLG equation in the super-Ohmic case. E. Comparison Ohmic versus non-Ohmic In the Ohmic case, we started with J(ω) =α1ωand ended up with a α1˙S(t) friction term, and noise correlation 4α1kBTδ(t−t/prime). On the other hand, in the non-Ohmic case, we started with J(ε) =αssin/parenleftbigπs 2/parenrightbig εsand ended up with a frictionαsDs tS(t) and noise correlation 2 αskBT(t−t/prime)−s Γ(1−s). Although it is clear that both the non-Ohmic J(ω) and the friction term will go to the Ohmic case when s→1, the noise is less straightforward. We can see that, as s→1, the Gamma function will blow up, hence sending the correlation to zero, with the exception of t=t/prime. In this case, the numerator blows up even before taking the limit of s→1. Hence, we can expect this function to behave as a delta function. To find the correct prefactor, we integrate the distribution with a test-function f(t/prime) = 1 to find lim s→1/integraldisplay∞ −∞d(t−t/prime)(t−t/prime)−s Γ(1−s)·1 = lim s→1/bracketleftbigg(t−t/prime)1−s (1−s)Γ(1−s)/bracketrightbigg∞ (t−t/prime)=−∞ = lim s→1/bracketleftbigg(t−t/prime)1−s Γ(2−s)/bracketrightbigg∞ (t−t/prime)=−∞ =/bracketleftbigg(t−t/prime) |t−t/prime|Γ(1)/bracketrightbigg∞ (t−t/prime)=−∞ = 2. Hence, we see that the limit of the noise correlation becomes 4 α1kBTδ(t−t/prime), which is precisely as in the Ohmic case.18 III. FMR POWERLAW DERIVATION Here, we calculate the response of this spin-bath system to a rotating magnetic field. We will find the steady state solutions and calculate several quantities that experiments could measure. A. Ferromagnetic Resonance We will study the effects of a rotating magnetic field on the fractional LLG (FLLG) equation ˙S(t) =S(t)×/braceleftbig −Beff[t,S(t)] +αsDs tS(t) +ξ(t)/bracerightbig . (101) For simplicity, we assume that the temperature of the bath is low compared to the energy of the external fields, such that the thermal noise ξ(t) may be neglected. We apply a rotating magnetic field Beff[t,S(t)] = Ω cos(ωdt) Ω sin(ωdt) B0−KSz(t) (102) and use spherical coordinates S=S sinθcosφ sinθsinφ cosθ . (103) We will assume a small θapproximation, where the ground state is in the positive zdirection, i.e. 0 <Ω/lessmuchB0−KS andαsS/lessmuch(B0−KS)1−s. As shown in Section I D, we may rewrite the FLLG equation of this form in spherical coordinates as ˙φ=1 sinθ[−Beff[t,S(t)] +αsDs tS(t)]· cosθcosφ cosθsinφ −sinθ ; (104) ˙θ= [−Beff[t,S(t)] +αsDs tS(t)]· sinφ −cosφ 0 . (105) In the rotating frame, where B(t) is constant, we could expect the system to go to a steady state after some time. Hence, we introduce a new coordinate such, that φ=ωdt−ϕ. We may then set ˙ ϕ=˙θ= 0 to find the steady state in the rotating frame, where ωdsinθ= [−Beff[t,S(t)] +αsDs tS(t)]· cosθcos(ωdt−ϕ) cosθsin(ωdt−ϕ) −sinθ ; (106) 0 = [−Beff[t,S(t)] +αsDs tS(t)]· sin(ωdt−ϕ) −cos(ωdt−ϕ) 0 . (107) We note that S(t) is now only time dependent in the rotating-frame term, which means that we can explicitly calculate the fractional derivative. The Liouville derivative works well with Fourier transforms, hence the fractional derivative of a trigonometric function is given by Ds tsin(ωt) =|ω|ssin/parenleftBig ωt+ sign(ω)πs 2/parenrightBig , (108) and similarly for a cosine. We remark that the Liouville derivative of a constant can only be described by setting the initial time to some finite t0, in which case it becomes zero2. Combining this with the steady state expression for S, 2Since physically the infinite time only models a long time in the past, we will apply it as such. Formally, there are some restrictions on the functions that the Liouville derivative can be applied to. These include restrictions such as its integral over the whole domain being finite. Since this is not the case for a non-zero constant on an infinite interval, we have to regularize the lower integral boundary −∞ as a finitet0.19 we find αsDs tS(t) =αsDs tS sinθcos(ωdt−ϕ) sinθsin(ωdt−ϕ) cosθ =αsSsinθ|ωd|s cos/parenleftbig ωdt−ϕ+ sign(ωd)πs 2/parenrightbig sin/parenleftbig ωdt−ϕ+ sign(ωd)πs 2/parenrightbig 0 . (109) Hence, we find that ωdtanθ= [−Beff[t,S(t)] +αsDs tS(t)]· cos(ωdt−ϕ) sin(ωdt−ϕ) −tanθ = − Ω cos(ωdt) Ω sin(ωdt) B0−KScosθ +αsSsinθ|ωd|s cos/parenleftbig ωdt−ϕ+ sign(ωd)πs 2/parenrightbig sin/parenleftbig ωdt−ϕ+ sign(ωd)πs 2/parenrightbig 0 · cos(ωdt−ϕ) sin(ωdt−ϕ) −tanθ =−Ω cosϕ+B0tanθ−KSsinθ+αsSsinθ|ωd|scos/parenleftBig sign(ωd)πs 2/parenrightBig (110) and 0 = [−Beff[t,S(t)] +αsDs tS(t)]· sin(ωdt−ϕ) −cos(ωdt−ϕ) 0 = − Ω cos(ωdt) Ω sin(ωdt) B0−KScosθ +αsSsinθ|ωd|s cos/parenleftbig ωdt−ϕ+ sign(ωd)πs 2/parenrightbig sin/parenleftbig ωdt−ϕ+ sign(ωd)πs 2/parenrightbig 0 · sin(ωdt−ϕ) −cos(ωdt−ϕ) 0 = Ω sinϕ−αsSsinθ|ωd|ssin/parenleftBig sign(ωd)πs 2/parenrightBig . (111) With some rearranging, we have ϕ= arcsin/bracketleftbiggαsS Ω|ωd|ssin/parenleftBig sign(ωd)πs 2/parenrightBig sinθ/bracketrightbigg (112) and Ω cosϕ= Ω cos arcsin/bracketleftbiggαsS Ω|ωd|ssin/parenleftBig sign(ωd)πs 2/parenrightBig sinθ/bracketrightbigg =/radicalbigg Ω2−/bracketleftBig αsS|ωd|ssin/parenleftBig sign(ωd)πs 2/parenrightBig sinθ/bracketrightBig2 = (B0−KScosθ−ωd) tanθ+αsSsinθ|ωd|scos/parenleftBig sign(ωd)πs 2/parenrightBig . (113) Squaring this, we get Ω2= (B0−KScosθ−ωd)2tan2θ+ (αsS|ωd|s)2sin2θ+ 2αsS|ωd|s(B0−KScosθ−ωd) cos/parenleftBigπs 2/parenrightBigsin2θ cosθ, and multiplying by cos2θ= 1−sin2θ, we have (1−sin2θ)Ω2= (B0−KS/radicalbig 1−sin2θ−ωd)2sin2θ+/bracketleftBig (αsS|ωd|s)2−2αsS|ωd|sKScos/parenleftBigπs 2/parenrightBig/bracketrightBig sin2θ(1−sin2θ) + 2αsS|ωd|s(B0−ωd) cos/parenleftBigπs 2/parenrightBig sin2θ/radicalbig 1−sin2θ. (114) We could go further and make this into (effectively) a 4th order equation for sin2θ. However, since we are in a small θlimit, we will solve this equation up to first order in sin2θ, which yields Ω2= sin2θ/braceleftBig Ω2(B0−KS−ωd)2+/bracketleftBig (αsS|ωd|s)2−2αsS|ωd|sKScos/parenleftBigπs 2/parenrightBig/bracketrightBig + 2αsS|ωd|s(B0−ωd) cos/parenleftBigπs 2/parenrightBig/bracerightBig . (115)20 Hence, we find that sin2θ=Ω2 Ω2+ (B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftbigπs 2/parenrightbig =Ω2 (B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftbigπs 2/parenrightbig+O(Ω4). (116) Note that theO(Ω4) should formally be dimensionless, but we explain what we mean with terms being small in Section IV, as this is more subtle with fractional dimensions. B. Resonance frequency and amplitude Since we are studying ferromagnetic resonance, we want to find the driving frequency for which we get the largest response from the magnetic system. Since we know that the resonance for an Ohmic system is at ωd=B0−KS, we will expand the formula around this point to find the new maximum. To compute the resonance frequency ωres, we thus first assume that ωres≈(B0−KS)(1+y) withysmall, such that|ωres|s≈(B0−KS)s(1+sy) (forB0−KS > 0). This results in sin2θ=Ω2 (B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftbigπs 2/parenrightbig ≈Ω2 (B0−KS)2y2+ (αsS)2(B0−KS)2s(1 + 2sy)−2αsS(B0−KS)s+1(1 +sy)ycos/parenleftbigπs 2/parenrightbig. (117) Now, we put the derivative with respect to yequal to zero, to get (B0−KS)2y+s(αsS)2(B0−KS)2s−αsS(B0−KS)s+1(1 + 2sy) cos/parenleftBigπs 2/parenrightBig = 0. (118) Hence, we find that y=−s(αsS)2(B0−KS)2s+αsS(B0−KS)s+1cos/parenleftbigπs 2/parenrightbig (B0−KS)2−2sαsS(B0−KS)s+1cos/parenleftbigπs 2/parenrightbig =αsS(B0−KS)s−1cos/parenleftBigπs 2/parenrightBig +O(αsS)2, (119) which results in ωres≈(B0−KS)/bracketleftBig 1 +αsS(B0−KS)s−1cos/parenleftBigπs 2/parenrightBig/bracketrightBig = (B0−KS) +αsS(B0−KS)scos/parenleftBigπs 2/parenrightBig . (120) We see that the resonance frequency gets shifted by a small amount, depending on s, which scales non-linearly. Inserting this result into Eq. (116), we can now also find an approximation for the amplitude at resonance: sin2θres=Ω2 (B0−KS−ωres)2+ (αsS|ωres|s)2+ 2αsS|ωres|s(B0−KS−ωres) cos/parenleftbigπs 2/parenrightbig ≈Ω2/braceleftBig/bracketleftBig αsS(B0−KS)scos/parenleftBigπs 2/parenrightBig/bracketrightBig2 +/parenleftBig αsS/vextendsingle/vextendsingle/vextendsingle(B0−KS) +αsS(B0−KS)scos/parenleftBigπs 2/parenrightBig/vextendsingle/vextendsingle/vextendsingles/parenrightBig2 −2αsS/vextendsingle/vextendsingle/vextendsingle(B0−KS) +αsS(B0−KS)scos/parenleftBigπs 2/parenrightBig/vextendsingle/vextendsingle/vextendsingles αsS(B0−KS)scos2/parenleftBigπs 2/parenrightBig/bracerightBig−1 =Ω2 [αsS(B0−KS)s]2/bracketleftBig cos2/parenleftBigπs 2/parenrightBig +/vextendsingle/vextendsingle/vextendsingle1 +αsS(B0−KS)s−1cos/parenleftBigπs 2/parenrightBig/vextendsingle/vextendsingle/vextendsingle2s −2/vextendsingle/vextendsingle/vextendsingle1 +αsS(B0−KS)s−1cos/parenleftBigπs 2/parenrightBig/vextendsingle/vextendsingle/vextendsingles cos2/parenleftBigπs 2/parenrightBig/bracketrightBig−1 =Ω2 [αsS(B0−KS)s]2/bracketleftBig 1−cos2/parenleftBigπs 2/parenrightBig +O(αsS)/bracketrightBig−1 ≈Ω2 /bracketleftbig αsS(B0−KS)ssin/parenleftbigπs 2/parenrightbig/bracketrightbig2. (121) Since the sine function decreases as smoves away from one, we see that the amplitude actually increases for non-Ohmic environments.21 C. Calculating the FWHM linewidth Next, we are interested not only in the location of the resonance, but also how sensitive the resonance is to the driving frequency. One way to describe this is by using the Full Width at Half Maximum measure. This provides a well-defined line width independently of the shape of the peak. It is found by measuring the width of the peak at half the height of its maximum. This can be measured in the laboratories, but it can also be computed. Since our function of interest is of the form sin2θ(ωd) = Ω2/g(ωd), it makes sense to approximate the inverse function instead of the regular one. To this end, we will translate the FWHM measurement to the inverse function, and then Taylor expandg(ωd) near resonance as a parabola to solve for the new condition of this inverse function. Notice that from Eq. (116), we have g(ωd) = (B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftBigπs 2/parenrightBig . (122) The FWHM condition is Ω2 g(ωd)= sin2θ(ωd) =sin2θ(ωres) 2=Ω2 2g(ωres), (123) hence we must solve for 2 g(ωres) =g(ωd). To this end, let us assume that ωd=ωres+yand expand g(ωd) iny. We will use that |a+y|n≈an+nan−1y+1 2n(n−1)an−2y2 for smallyanda>0. Then, g(ωres+y) = (B0−KS−ωres−y)2+ (αsS|ωres+y|s)2+ 2αsS|ωres+y|s(B0−KS−ωres−y) cos/parenleftBigπs 2/parenrightBig ≈(B0−KS−ωres)2+ (αsSωs res)2+ 2αsSωs res(B0−KS−ωres) cos/parenleftBigπs 2/parenrightBig +y/parenleftBig −2(B0−KS−ωres) + 2s(αsS)2ω2s−1 res−2αsScos/parenleftBigπs 2/parenrightBig/braceleftbig ωs res+sωs−1 res[ωres−(B0−KS)]/bracerightbig/parenrightBig +y2/bracketleftBig 1 +s(2s−1)(αsS)2ω2s−2 res−2sαsSωs−1 rescos/parenleftBigπs 2/parenrightBig +s(s−1)αsSωs−2 res(B0−KS−ωres) cos/parenleftBigπs 2/parenrightBig/bracketrightBig =g(ωres) +y/parenleftBigg 2αsS(B0−KS)scos/parenleftBigπs 2/parenrightBig + 2s(αsS)2(B0−KS)2s−1/bracketleftBig 1 +αsS(B0−KS)s−1cos/parenleftBigπs 2/parenrightBig/bracketrightBig2s−1 −2αsS(B0−KS)scos/parenleftBigπs 2/parenrightBig/braceleftBigg/bracketleftBig 1 +αsS(B0−KS)s−1cos/parenleftBigπs 2/parenrightBig/bracketrightBigs +sαsScos/parenleftBigπs 2/parenrightBig (B0−KS)s−1/bracketleftBig 1 +αsS(B0−KS)s−1cos/parenleftBigπs 2/parenrightBig/bracketrightBigs−1/bracerightBigg/parenrightBigg +y2/braceleftBigg 1 +s(2s−1)(αsS)2(B0−KS)2s−2/bracketleftBig 1 +αsS(B0−KS)s−1cos/parenleftBigπs 2/parenrightBig/bracketrightBig2s−2 −2sαsS(B0−KS)s−1cos/parenleftBigπs 2/parenrightBig/bracketleftBig 1 +αsS(B0−KS)s−1cos/parenleftBigπs 2/parenrightBig/bracketrightBigs−1 −s(s−1)(αsS)2(B0−KS)2s−2cos2/parenleftBigπs 2/parenrightBig/bracketleftBig 1 +αsS(B0−KS)s−1cos/parenleftBigπs 2/parenrightBig/bracketrightBigs−2/bracerightBigg ≈g(ωres) +y/braceleftBig 2s(αsS)2(B0−KS)2s−1/bracketleftBig 1−2 cos2/parenleftBigπs 2/parenrightBig/bracketrightBig/bracerightBig +y2/braceleftBig 1−2s(αsS)(B0−KS)s−1cos/parenleftBigπs 2/parenrightBig + (αsS)2(B0−KS)2s−2/bracketleftBig s(2s−1)−3s(s−1) cos2/parenleftBigπs 2/parenrightBig/bracketrightBig/bracerightBig +O(αsS)3. (124) Now, we set 2 g(ωres) =g(ωres+y) =g(ωres) +by+ay2, and remark that g(ωres) = (αsS)2(B0−KS)2ssin2/parenleftbigπs 2/parenrightbig , in order to find that y=−b±/radicalbig b2+ 4ag(ωres) 2a=⇒∆FWHM =/radicalbig b2+ 4ag(ωres) a. (125)22 Hence, we find that the lowest-order contribution to the linewidth is given by ∆FWHM≈/radicalBig 4(αsS)2(B0−KS)2ssin2/parenleftbigπs 2/parenrightbig +O(αsS)3 1 +O(αsS) = 2(αsS)(B0−KS)ssin/parenleftBigπs 2/parenrightBig +O(αsS)2. (126) IV. DIMENSIONAL ANALYSIS The fractional derivative in the LLG equation has an impact on the dimensions of quantities. We can firstly see that in the chosen units, we have [ B0−KS] = [ωd] = time−1. Assuming Sto be dimensionless, then [ ωd] = [αsDs tS] = [αs][ωd]s, hence [αs] = [ωd]1−s. We can now start to understand what we mean when we say that certain quantities are small, since this has to be relative to something else. For instance, when we say αsSis small, we understand this asαsS/lessmuch(B0−KS)1−s. For Ω it is simpler, since there is no fractional derivative acting with it. Hence, for Ω small we simply mean Ω /lessmuchB0−KS. We can now also define some dimensionless variables, such as α/prime s=αsS(B0−KS)s−1 and Ω/prime= Ω/(B0−KS). We have used these variables in the figures to show the general behavior of the quantities. [1] A. Altland and B. D. Simons, Condensed Matter Field Theory (Cambridge University Press, 2010), 2nd ed. [2] A. Kamenev, Field theory of non-equilibrium systems (Cambridge University Press, 2011). [3] A. Schmid, Journal of Low Temperature Physics 49, 609 (1982). [4] A. O. Caldeira and A. J. Leggett, Physical Review Letters 46, 211 (1981). [5] A. O. Caldeira and A. J. Leggett, Physica A: Statistical Mechanics and its Applications 121, 587 (1983). [6] A. O. Caldeira and A. J. Leggett, Annals of Physics 149, 374 (1983). [7] A. O. Caldeira, An introduction to macroscopic quantum phenomena and quantum dissipation , vol. 9780521113755 (Cam- bridge University Press, 2012). [8] A. O. Caldeira and A. J. Leggett, Phys. Rev. A 31, 1059 (1985). [9] U. Weiss, Quantum dissipative systems (World scientific, 2012). [10] C. Gardiner and P. Zoller, Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics (Springer Science & Business Media, 2004). [11] A. Shnirman, Y. Gefen, A. Saha, I. S. Burmistrov, M. N. Kiselev, and A. Altland, Physical Review Letters 114, 176806 (2015). [12] R. C. Verstraten, R. F. Ozela, and C. M. Smith, Physical Review B 103, L180301 (2021). [13] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic press, 2014).

2022-11-23

The dynamics of a magnetic moment or spin are of high interest to applications in technology. Dissipation in these systems is therefore of importance for improvement of efficiency of devices, such as the ones proposed in spintronics. A large spin in a magnetic field is widely assumed to be described by the Landau-Lifshitz-Gilbert (LLG) equation, which includes a phenomenological Gilbert damping. Here, we couple a large spin to a bath and derive a generic (non-)Ohmic damping term for the low-frequency range using a Caldeira-Leggett model. This leads to a fractional LLG equation, where the first-order derivative Gilbert damping is replaced by a fractional derivative of order $s \ge 0$. We show that the parameter $s$ can be determined from a ferromagnetic resonance experiment, where the resonance frequency and linewidth no longer scale linearly with the effective field strength.

The fractional Landau-Lifshitz-Gilbert equation

2211.12889v1

Magnetic field gradient driven dynamics of isolated skyrmions and antiskyrmions in frustrated magnets J. J. Liang1, J. H. Yu1, J. C hen1, M. H. Qin1,*, M. Zeng1, X. B. Lu1, X. S. Gao1, and J. –M. Liu2,† 1Institute for Advanced Materials , South China Academy of Advanced Optoelectronics and Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, South China Normal University, Guangzhou 510006, China 2Laboratory of Solid State Microstr uctures and Innovative Center for Advanced Microstructures , Nanjing University, Nanjing 210093, China [Abstract] The study of skyrmion/antiskyrmion motion in magnetic materials is very important in particular for the spintronics applications. In this work , we stud y the dynamics of isolated skyrmions and antiskyrmions in frustrated magnets driven by magnetic field gradient , using the Landau -Lifshitz -Gilbert simulations on the frustrated classical Heisenberg model on the triangular lattice . A Hall-like motio n induced by the gradient is revealed in bulk system, similar to that in the well -studied chiral magnets. More interestingly, our work suggest s that the lateral confinement in nano -stripes of the frustrated system can completely suppress the Hall motion an d significantly speed up the motion along the gradient direction. The simulated results are well explained by the Thiele theory . It is demonstrated that t he acceleration of the motion is mainly determined by the Gilbert damping constant , which provides use ful information for finding potential materials for skyrmion -based spintronics . Keywords: skyrmion dynamics, field gradient, frustrated magnets PACS numbers: 12.39.Dc, 66.30.Lw .Kw, 75.10.Jm Email: *qinmh@scnu.edu.cn , † liujm@nju.edu.cn I. INTRODUCTION Magnetic skyrmions which are topological defects with vortex -like spin structures have attracted extensive attention since their discovery in chiral magnets due to their interesting physics and potential applications in spintronic devices.1-4 Specifically, the interesting characters of skyrmions such as the topological protection5, the ultralow critical currents required to drive skyrmions (~105 Am-2, several orders of smaller than that for domain -wall manipulation )3,6, and th eir nanoscale size make s them proposed to be promising candidates for low po wer consumption magnetic memories and high -density data processing devices. Theoretically, the cooperation of the energy competition among the ferromagnetic, Dzyaloshinskii -Moriya ( DM), and the Zeeman couplings and the thermal fluctuations is suggested to stabilize the skyrmions .7,8 Moreover, the significant effects of the uniaxial stress on the stabilization of the skyrmion lattice have been revealed in earlier works.9-12 On the skyrmion dynamics , it has been suggested that the skyrmions in chiral magnets can be effectively modulated by spin-polarized current,13-16 microwave fields,17 magnetic field gradients,18,19 electric field gradients,20,21 temperature gradient s22 etc. So far, some of these manipulations have been realized in experiments.23 Definitely , finding new magnetic systems with skyrmions is essential both in application potential and in basic physical research.24 More recently, frustrated magnets have been suggested theoretically to host skyrmion lattice phase . For example, skyrmion crystals an d isolated skyrmions have been reported in the frustrated Heisenberg model on the triangular lattice.25,26 In this system, i t is suggested that the skyrmion crystals are stabilized by the competing ferromagnetic nearest -neighbor (NN) and antifer romagnetic next-nearest -neighbo r (NNN) interaction s and thermal fluctuations at finite temperatures (T) under applied magnetic field h. Furthermore, the uniaxial anisotropy strongly affect s the spin orders in triangular antiferromagnets and stabilize s the isolated sk yrmions even at zero T.26,27 Compared with the skyrmions in chiral magnets, those in frustrated magnets hold two additional merits. On the one hand, the skyrmion lattice constant is typically an order of magnitude smaller than that of chiral magnets, and higher -density data processing devices are expected. On the other hand, the skyrmions are with two additional degrees -of-freedomvorticity and helicity ) due to the fact that the exchange interactions are insensitive to the direction of spin rotation . As a result, both skyrmion and anti skyrmion lattices are possible in frustrated magnets which keep the Z2 mirror symmetry in the xy spin component. Furthermore , the dynamics of skyrmions /antiskyrmions is probably different from that of chiral magnets , as revea led in earlier work which studied the current -induced dynamics in nanostripes of frustrated magnets.28 It has been demonstrated that the spin states formed at the edges create multiple edge channels and guide the skyrmion /antiskyrmion motion. It is noted that spin-polarized current may not drive the skyrmion well for insulating materials , and other control parameters such as field gradient are preferred . In chiral magnets, for example, the gradient can induce a Hall -like motion of skyrmions, i. e., the mai n velocity v (perpendicular to the gradient direction ) is induced by the gradient, and a low velocity v|| (parallel to the gradient direction ) is induced by the damping effect . Thus, the gradient -driven motion of skyrmions and antiskyrmions in frustrated systems is also expected . Furthermore , it has been suggested that the confined geometry suppress es the current -induced Hall motion of skyrmions and speed s up the motion along the current direction , which is instructive for future application s.29 In some ex tent, the gradient -driven motion could also be strongly affected by confining potential in narrow constricted geometries. Thus, as a first step, the field-gradient -induced dynamics of skyrmions and antiskyrmions in bulk frustrated magnets as well as in constricted geometries urgently deserves to be revealed theoretically . However, few works on this subject have been reported, as far as we know. In this work , we stud y the skyrmion /antiskyrmion dynamics in frustrated magnets induced by magnetic field gradien ts using Landau -Lifshitz -Gilbert (LLG) simulations and Thiele approach based on the frustrated classical Heisenberg model on two -dimensional triangular lattice . A Hall-like motion is revealed in bulk system, similar to that in chiral magnets. More interest ingly, our work demonstrates that the edge confinement in nanostripes of frustrated magnets c an completely suppress the Hall motion and significantly accelerate the motion along the gradient direction. The remainder of this manuscript is organized as foll ows: in Sec. II the model and the calculation method will be described. Sec. III is attributed to the results and discussion, and the conclusion is presented in Sec. IV . II. MODEL AND METHODS Following the earlier work,28 we consider the Hamiltonian 22' 12 , ,z z z i j i j i i i i i j i i i ijH J J h S D S D S S S S S , (1) where Si is the classical Heisenberg spin with unit length on site i. The first term is the ferromagnetic NN interaction with J1 = 1 (we use J1 as the energy unit, for simplicity) , and t he second term is the antiferromagnetic NNN interacti on with J2 = 0.5 , and the third term is the Zeeman coupling with a linear gradient field h = h0 + g·r (h0 = 0.4, r is the coordinate , and g is the gradient vector with a strength g) applied along the [001] direction ,28 and the fourth term is the bulk uniaxial anisotropy energy with D = 0.2 , and the last term is the easy plane anisotropy energy of the edges with D' = 2. D' is only consider ed at the edges for the nanostripes system , which may give rise to several types of edge states, as uncovered in earlier work .28 However, it has been confirmed that the skyrmion s/antiskyrmions in nanostripes move with the same speed when they are captured by one of these edge state s. In this work, we mainly concern the gradient -driven moti on of isolated skyrmions /antiskyrmion. We study the spin dynamics at zero T by numerically solving the LLG equation: ii i i idd dt dt SSS f S , (2) with the local effective field fi = (∂H/∂ Si). Here, γ = 6 is the gyromagnetic ratio , α is the Gilbert damping coefficient. We use the fourth -order Runge -Kutta method to solve the LLG equation. The initial spin configurations are obtained by solving the LLG equation at g = 0. Subsequently, t he spin dynamics are investigated under gradient fields. Furthermore, the simulated results are further explained using the approach proposed by Thiele.29 The displacement of the skyrmion/antiskyrmion is characteri zed by the position of its center (X, Y): (1 )d d (1 )d d ,. (1 )d d (1 )d dzz zzx S x y y S x y XY S x y S x y (3) Then, the velocity v = (vx, vy) is numerically calculated by d d , d d .xyv X t v Y t (4) At last, v and v|| are obtaine d through a s imple coordinate transformation . III. RESULTS AND DISCUSSION First, we investigate the spin configurations of possible isolated skyrmions and antiskyrmions with various vorticities and helicities obtained by LLG simulations of bulk system ( D' = 0) at zero g. Specifically, four typical isolated skyrmions with the topological charge Q = 1 have been observed in our simulations, as depicted in Fig 1(a). The first two skyrmions are N éel-type ones with different helicities, and the remaining two sk yrmions are Bloch -type ones. Furthermore, isolated antiskyrmoins are also possible in this system, and their spin configurations with Q = 1 are shown in Fig. 1(b). After the relaxation of the spin configurations at g = 0, the magnetic field gradient is applied along the direction of θ = /6 (θ is the angle between the gradient vector and the positive x axis, as shown in Fig. 2(a)) to study the dynamics of isolated skyrmions and antiskyrmions in bulk system . The LLG simulation is performed on a 28 × 28 triangular lattice with the periodic boundary condition applied along the y' direction perpendicular to the gradient . Furthermore , we constrain the spin directions at the edge s along the x direction by Sz = 1 (red circles in Fig. 2(a) ) to reduce the finite lat tice size effect . Similar to that in chiral magnets, the skyrmion /antiskyrmion motion can be also driven by the magnetic field gradients in frustrated magnets. Fig. 2 (b) and Fig. 2( c) give respectively the calculated v|| and v as functions of g at α = 0.04 . v|| of the skyrmion equals to that of the antiskyrmion , and both v|| and v increase linearl y with g. For a fixed g, the value of v is nearly an order of magnitude larger than that of v||, clearly exhibiting a Hall -like motion of the skyrmions/a ntiskyrmions. It is noted that v is caused by the gyromagnetic force which depends on the sign of the topological charge. Thus, along the y' direction, the skyrmion and antiskyrmion move oppositely under the field gradient , the same as earlier report.18 Moreover, v|| is resulted from the dissipative force which is associated with the Gilbert damping. For example , the linear dependence of v|| on the Gilbert damping constant α has been revealed in chiral magnets,18 which still hold s true for the frustrated magnets. The dependence of velocity on α at g = 103 is depicted in Fig. 3 , which clearly demonstrates that v|| increases linearly and v is almost invariant with the increa se of α. Subsequently , the simulated results are qualitatively explained by Thiele equations: || || '', and ,HHv Gv Gv vXY (5) with the skyrmion/antiskyrmoin center ( X', Y') in the x'y' coordinate system. Here, 2 2 2 S S S S S Sd S 4 , and d d . G r Q r rx y x x y y (6) For the frustrated bulk magnets with the magnetic field gradient applied along the x' direction, there are ''1 , and 0.z i iHHg S gqXY (7) For α << 1, q is almost invariant and the velocities can be estimated from || 2,. v gq v gqGG (8) Thus, a proportional relation between the velocity and field gradient is clearly demonstrated . Furthermore, v is inversely propor tional to G and/or the topological charge Q, resulting in the fact that the skyrmion and antiskyrmion move along the y' direction oppositely, as revealed in our simulations. For current -induced motion of skyrmions, the lateral confine ment can suppress the Hall motion and accelerate the motion along the current direction.29-31 The confinement effects on the h-gradient driven skyrmion/antiskyrmion motion are also investigated in the nanostripes of frustrated magnets. For this case, the L LG simulation is performed on an 84 × 30 triangular -lattice with an open boundary condition along the y direction. For convenience, the field gradient is applied along the x direction. The easy plane anisotropy with D' = 2 is considered at the lateral edges , which gives rise to the edge state and in turn confines the skyrmions/antiskyrmions . Fig. 4(a) gives the time dependence of the y coordinate of the skyrmion center for α = 0.04 and g = 103. It is clearly shown that t he isolated skyrmion jumps into the channel at Y = 17 and then moves with a constant speed along the gradient (negative x, for this case) direction . Furthermore, the position of the channel changes only a little due to the small range of the gradient consi dered in this work , which never affects our main conclusions. More interestingly, the skyrmion/antiskyrmion motion along the gradient direction can be significant accelerated by the lateral confinement, as shown in Fig. 4(b) which gives v|| (vx) as a fun ction of g at α = 0.04. For a fixed g, v|| of the nanostripes is almost two orders of magnitude larger than that of bulk system. When the skyrmion/antiskyrmion is captured by the edge state ( under which v = 0), the equation (5) gives v|| = gqγ/αΓ. It is s hown that v|| is inversely proportional to α in this confined geometry, and small α result s in a high speed of motion of the skyrmion/antiskyrmion . The inversely proportional relation between v|| and α has also been confirmed in our LLG simulations, as cle arly shown in Fig. 4(c) which gives the simulated v|| as a function of 1/α at g = 103. At last, we study the effect of the reversed gradient g on the skyrmion/antiskyrmion motion, and its trail is recorded in Fig. 4(d). It is clearly shown that the rever sed gradient moves the skyrmion/antiskyrmion out of the former channel near one lateral edge and drives it to the new channel near the other lateral edge. Subsequently, the skyrmion/antiskyrmion is captured by the n ew channel and moves reversely, resulting in the loop -like trail. As a result, our work suggests that one may modulate the moving channel by reversing the field gradient, which is meaningful for future applications such as in data eras ing/restoring . Anyway, it is suggested theoretically that the confined geometry in nanostripes of frustrated magnets could significantly speed up the field-driven motion of the isolated skyrmions/antiskyrmions, especially for system with small Gilbert damping constant. Furthermore , we would also like to point out th at th is acceleration is probably available in other confined materials such as chiral magnets, which deserves to be checked in future experiments. IV. CONCLUSION In conclusion, we have studied the magnetic -field-gradient -drive n motion of the isolated skyrmions and antiskyrmions in the frustrated triangular -lattice spin model using Landau -Lifshitz -Gilbert simulations and Thiele theory . The Hall -like motion is revealed in bulk system, similar to that in chiral magnets. More interestingly, it is suggested that the lateral confinement in the nanostripes of the frustrated system can suppress the Hall motion and significantly speed up the motion along the gradient direction. The acceleration of the motion is mainly determined by the Gilbert damping constant , whic h is helpful for finding potential materials for skyrmion -based spintronics . Acknowledgement s: The work is supported by the National Key Projects for Basic Research of China (Grant No. 2015CB921202 ), and the National Key Research Programme of China (Gra nt No. 2016YFA0300101), and the Natural Science Foundation of China ( Grant No. 51332007 ), and the Science and Technology Planning Project of Guangdong Province (Grant No. 2015B090927006) . X. Lu also thanks for the support from the project for Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme (2016). References: 1. A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. N iklowitz, and P. Bö ni 2009 Phys. Rev. Lett. 102 186602 2. H. Wilhelm, M. Baenitz, M. Schmidt, U.K. Rö ß ler, A. A. Leonov, and A. N. Bogdanov 2011 Phys. Rev. Lett. 107 127203 3. F. Jonietz et al 2010 Science 330 1648 4. S. Mü hlbauer et al 2009 Science 323 915 5. J. Hagemeister, N. Romming, K. von Bergmann, E.Y . Vedmedenko , and R. Wiesendanger 2015 Nat. Commun. 6 8455 6. X.Z. Yu, N. Kanazawa, W.Z. Zhang, T. Nagai, T. Hara, K. Kimoto, Y . Matsui, Y . Onose , and Y . Tokura 2012 Nat. Commun. 3 988 7. S. D. Yi, S . Onoda, N . Nagaosa, and J . H. Han 2009 Phy. Rev. B 80 054416 8. S. Buhrandt and L . Fritz 2013 Phy. Rev. B 88 195137 9. K. Shibata et al 2015 Nat. Nanotech. 10 589 10. Y . Nii et al 2015 Nat. Commun. 6 8539 11. A. Chacon et al 2015 Phys. Rev. Lett. 115 267202 12. J. Chen, W. P. Cai, M. H. Q in, S. Dong, X. B. Lu, X. S. Gao , and J. M. Liu 2017 Sci. Rep. 7 7392 13. J. Zang, M . Mostovoy, J . H. Han, and N. Nagaosa 2011 Phys. Rev. Lett. 107 136804 14. J. Sampaio, V . Cros, S. Rohart, A. Thiaville , and A. Fert 2013 Nat. Nanotech. 8 839 15. J. Iwasaki, M. Mochizuki, and N. Nagaosa, 2013 Nat. Commun. 4 1463 16. S. Huang, C. Zhou, G. Chen, H. Shen, A. K. Schmid, K. Liu, and Y . Wu 2017 Phy. Rev. B 96 144412 17. W. Wang, M. Beg, B. Zhang, W. Kuch, and H. Fangohr 2015 Phy. Rev. B 92 020403 18. C. Wang, D. Xiao , X. Chen, Y . Zhou, and Y . Liu 2017 New J. Phys 19 083008 19. S. Komineas and N. Papanicolaou, 2015 Phy. Rev. B 92 064412 20. P. Up adhyaya, G. Yu, P. K. Amiri, and K. L. Wang 2015 Phy. Rev. B 92 134411 21. Y.-H. Liu, Y . -Q. Li, and J. H. Han 2013 Phys . Rev. B 87 100402 22. L. Kong and J. Zang 2013 Phys. Rev. Lett. 111 067203 23. W. Legrand et al 2017 Nano Lett. 17 2703 -2712 24. Joseph Barker, Oleg A . Tretiakov 2016 Phys. Rev. Lett. 116 147203 25. T. Okubo, S. Chung, and H. Kawamura 2012 Phys. Rev . Lett. 108 017206 26. A. O. Leonov and M. Mostovoy, 2015 Nat. Commun. 6 8275 27. S. Hayami, S. -Z. Lin, and C. D. Batista 2016 Phy. Rev. B 93 184413 28. A. O. Leonov and M. Mostovoy 2017 Nat. Commun. 8 14394 29. J. Iwasaki, M. Mochizuki, and N. Nagaosa 2013 Nat. Nanotech. 8 742 30. I. Purnama, W. L. Gan, D. W. Wong, and W. S. Lew 2015 Sci. Rep. 5 10620 31. X. Zhang, G. P. Zhao, H. Fangohr, J. P. Liu, W. X. Xia, J. Xia, and F. J. Morvan 2015 Sci. Rep. 5 7643 FIGURE CAPTIONS Fig.1. Typical LLG snapshot of the spin configuration s of skyrmion s and antiskyrmion s. (a) skyrmion structure s and (b) antiskyrmion structure s with different helicit ies. Fig.2. (a) Effective model on the triangular lattice. (b) v|| and ( c) v as functions of g at α = 0.04 in bulk system . Fig.3. (a) v|| and ( b) v as functions of α at g = 103. Fig.4. (a) Time dependence of Y coordinate of the skyrmion center at α = 0.01 and g = 103. v|| as a function of (b) g at α = 0.04 , and (c) α at g = 103 in the nanostripes of frustrated magnets . (d) The trail of skyrmion /antiskyrmion . The red line records the motion with gradient g along the positive x direction , while the blu e line records the motion with the revers ed g. Fig.1. Typical LLG snapshot of the spin configuration s of skyrmion s and antiskyrmion s. (a) skyrmion structure s and (b) antiskyrmion structures with different helicit ies. Fig.2. (a) Effective model on the triangular lattice. (b) v|| and (c) v as functions of g at α = 0.04 in bulk system . Fig.3. (a) v|| and ( b) v as functions of α at g = 103. Fig.4. (a) Time dependence of Y coordinate of the skyrmion center at α = 0.01 and g = 103. v|| as a function of (b) g at α = 0.04, and (c) α at g = 103 in the nanostr ipes of frustrated magnets . (d) The trail of skyrmion /antiskyrmion . The red line records the motion with gradient g along the positive x direction , while the blu e line records the motion with the reversed g.

2017-12-10

The study of skyrmion/antiskyrmion motion in magnetic materials is very important in particular for the spintronics applications. In this work, we study the dynamics of isolated skyrmions and antiskyrmions in frustrated magnets driven by magnetic field gradient, using the Landau-Lifshitz-Gilbert simulations on the frustrated classical Heisenberg model on the triangular lattice. A Hall-like motion induced by the gradient is revealed in bulk system, similar to that in the well-studied chiral magnets. More interestingly, our work suggests that the lateral confinement in nano-stripes of the frustrated system can completely suppress the Hall motion and significantly speed up the motion along the gradient direction. The simulated results are well explained by the Thiele theory. It is demonstrated that the acceleration of the motion is mainly determined by the Gilbert damping constant, which provides useful information for finding potential materials for skyrmion-based spintronics.

Magnetic field gradient driven dynamics of isolated skyrmions and antiskyrmions in frustrated magnets

1712.03550v1

Spin Pumping in Ion-beam Sputtered Co2FeAl/Mo Bilayer s: Interfacial Gilbert Damping Sajid Husain, Vineet Barwal, and Sujeet Chaudhary* Thin Film Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016 (INDIA) Ankit Kumar, Nilamani Behera, Serkan Akansel, and Peter Svedlindh Ångström Laboratory, Department of Engineering Sciences, Box 534, SE -751 21 Uppsala, Sweden Abstract The spin pumping mechanism and associated interfacial Gilbert damping are demonstrated in ion-beam sputtered Co2FeAl (CFA)/Mo bilayer thin films employing ferromagnetic resonance spectroscopy . The d ependence of the net spin current transportation on Mo layer thickness, 0 to 10 nm, and the enhancement of the net effective Gilbert damping are reported . The experimental data has been analyzed using spin pumping theory in terms of spin current pumped through the ferromagnet/nonmagnetic metal interface to deduce the effective spin mixing conductance and the spin -diffusion length , which are estimated to be 1.16(±0.19 )×1019 m−2 and 3.50±0.35nm, respectively. The damping constant is found to be 8 .4(±0.3)×10-3 in the Mo(3.5nm) capped CFA(8nm) sample corresponding to a ~42% enhancement of the original Gilbert damping (6.0(± 0.3)×10-3) in the uncapped CFA layer. This is further confirm ed by insertin g a Cu dusting layer which reduce s the spin transport across the CFA/Mo interface. The Mo layer thickness dependent net spin current density is found to lie in the ra nge of 1-3 MAm-2, which also provides additional quantitative evidence of spin pumping in this bilayer thin film system . *Author for correspondence: sujeetc@physics.iitd.ac.in I. INTRODUCTION Magnetic damping is an exceedingly importan t property for spintronic devices due to its influence on power consumption and information writing in the spin-transfer torque random access memor ies ( STT-MRAMs) [1][2]. It is therefore of high importance to study the generation, manipulation , and detection of the flow of spin angular momentum to enable the design of efficient spin-based magneti c memories and logic devices [3]. The transfer of spin angular momentum known as spin pumping in ferromag netic (FM)/ nonmagnetic (NM) bilayer s provide s information of how the precession of the magnetization transfer s spin angular momentum into the adjacent nonmagnetic metallic layer [4]. This transfer ( pumping ) of spin angular momentum slows down the precession and leads to an enhance ment of the effective Gilbert damping constant in FM/NM bilayers . This enhancement has been an area of intensive research since the novel mechanism (theory) of spin pumping was proposed by Arne Brataas et al. [5] [6]. The amount of spin pumping is quantified by the magnitude of the spin current density at the FM/NM interface and theoretically [7] described as 4eff S effdgdt mJm where m is the magnetization unit vector, eff SJ is the effective spin current density pumped into the NM layer from the FM layer (portrayed in Fig. 1), and effg is the spin mixing conductance which is determined by the reflection coefficient s of conductance channels at FM/NM interface [5]. To date, a number of NM metals , such as Pt, Au, [5], Pd [8][9],-Ta [10] and Ru [11], etc. have been e xtensively investigated with regards to their performance as spin sink material when in contact with a FM . It is to be noted here that none of the Pt, Pd, Ru, and Au is an earth abundant material [12]. Thus , there is a natural need to search for new non-magnetic material s which could generate large spin current at the FM/NM in terface . In this study , we have explored the potential of the transition metal molybdenum (Mo) as a new candidate material for spin pumping owing to the fact that Mo possesses a large spin-orbit coupling [13]. To the best of our knowledge, Mo has not been used till date for the study of spin pumping effect in a FM/NM bilayer system . In a FM/NM bilayer and/or multilayer system s, there are several mechanisms for dissipation of the spin angular momentum which are categorized as intrinsic and extrinsic . In the intrinsic category , the magnon -electron coupling , i.e., spin -orbit coupling (SOC) contributes significant ly [14]. Among the extrinsic category , the two-magnon scattering (TMS ) mechanism is linked to the inhomogeneity and interface/surface roughness of the heterostructure , etc. [15] [16] [17]. For large SOC , interfacial d-d hybridization between the NM and FM layers is highly desirable [16]. Thus, the FM -NM interfacial hybridization is expected to result in enhancement of the transfer of spin angular momentum from the FM to the NM layer , and hence the NM layer can act as a spin reservoir (sink) [18]. But, the NM metallic layer does not always act as a perfect spin reservoir due to the spin accumulation effect which prevents transfer of angular momentum to some extent and a s a result , a backflow of spin-current towards the FM [6] is estabished . While the flow of spin angular momentum through the FM/NM interface is determine d by the effective spin-mixing conductance ()effg at the interface , the spin backflow is governed by the spin diffusion length ()d . It is emphasized here that t hese parameters ( effg and d ) are primarily tuned by appropriate selection of a suitable NM layer. In this work, we have performed ferromagnetic resonance (FMR) measurements to explore the spin pumping phenomenon and associated interfacial Gilbert damping enhancement in the Co2FeAl(8nm) /Mo( Mot) bilayer system , Mot is the thickness of Mo , which is varie d from 0 to 10 nm. The Mot dependent net spin current transfer across the interface and spin diffusion length of Mo are estimated . The choice of employing the Heusler alloy CoFe 2Al (CFA) as a thin FM layer lies in its half metallic character anticipated at room temperature [19] [20], a trait which is highly desirable in any spintronic device operating at room temperature. II. EXPERIMENTAL DETAILS The CFA thin films with fixed thickness of 8 nm were grown on naturally oxidized Si(100) substrate at 573K temperature using an ion-beam sputtering deposition system ( NORDIKO - 3450). The substrate temperature (573K) has been selected following the growth optimization reported in our previous reports [21] [20] [22]. On the top of the CFA layer , a Mo film with thickness Mot ( Mot=0, 0.5, 1.0, 1.5, 2.0, 3.0, 4, 5, 7, 8 and 10 nm) was deposited in situ at room temperature . In addition , a trilayer structure of CFA(8)/Cu(1)/Mo(5) was also prepared to understand and confirm the effect of an additional interface on the Gilbert damping (spin pumping ). Numbers in parenthesis are film thicknesses in nm. All the samples were prepared at a constant working pressure of ~8.5×10-5 Torr (base vacuum ~ 1.010-7 Torr); Ar gas was directly fed at 4 sccm into the rf-ion source operated at 75W with the deposition rate s of 0.03nm/s and 0.02nm/s for CFA and Mo, respectively . The deposition rate for Cu was 0.07nm/s at 80 W. The samples were then cut to 1×4 mm2 to record the FMR spectra employing a homebuilt FMR set- up [21] [23]. The data was collected in DC-magnetic field sweep mode by keeping the microwave frequency fixed . The saturation magnetization was measured using the Quantum Design make Physical Property Measurement System (Model PPMS Evercool -II) with the vibrating sample magnetometer option (QD PPMS -VSM). The film density, thickness and interface /surface roughness were estimated by simulating the specular X -ray reflectivity (XRR) spectra using the PANalytical X’Pert reflectivity software (Ver. 1.2 with segmented fit). To determine surface morphology /microstru cture (e.g., roughness) , topographical imaging was performed using the ‘Bruker dimension ICON scan assist’ atomic fo rce microscope (AFM). All measurements were performed at room temperature. III. RESULTS AND DISCUSSIONS A. X-ray Reflectivity and A tomic Force Microscopy : Interface/surface analysis Figure 2 shows the specular XRR spectra recorded on all the CFA(8)/Mo( Mot ) bilayer thin films . The fitting parameters were accurately determined by simu lating (red lines) the experimental curves (filled circles) and are presented in Table -I. It is evident that for the smallest NM layer thickness , Mo(0.5nm) , the estimated value s of the roughness from XRR and AFM are slightly larger in comparison to the thickness of the Mo layer which indicates that the surface coverage of Mo layer is not enough to cover all of the CFA surface in the CFA(8)/Mo(0.5) bilayer sample (modeled in Fig. 3(a)). For Mot ≥ 1nm, the film roughness is smaller than the thickness (indicating that t he Mo layer coverage is uniform as modeled in Fig s. 3(b)-(c)). For the thicker layer s of Mo ( Mot≥ 5nm) the estimated values of the surface roughness as estimated from both XRR and AFM are found to be similar ~0.6nm (c.f. the lowest right panel in Fig. 2). B. Ferromagnetic Resonance Study The FMR spectra were recorded on al l sample s in 5 to 11 GHz range of microwave frequenc ies. Figure. 4(a) shows the FMR spectra recorded on the CFA(8)/Mo( 5) bilayer thin film . The FMR spectra ()FMRI were fitted with the derivative of symmetric and anti -symmetric Lorentzian function s to extract the line -shape parameters, i.e., resonant field rH and linewidth H , given by [24] [21]: 22 2 2222 22() () ()22 2 ( ) ( )22FMR dc rdc r dcS ext A ext ext ext r dc rUIH HH HHH HH SA HF H F H HHSA HHHH H , (1) where S extFH and A extFH are the symmetric and anti -symmetric Lorentzian functions, respectively, with S and A being the corresponding coefficients. Symbol ‘U’ refers to the raw signal voltage from the VNA . The linewidth H is the full width at half maxim um (FWHM) , and dcH is the applied DC-magnetic field. The f vs. 0 rH plots are shown in Fig. 4(b). These are fitted using t he Kittel’s formul a [25]: 0()2r K r K eff f H H H H M , (2) where 𝛾 is the gyromagnetic ratio ; /Bg (1.76×1011s-1T-1) with g being the Lande’s splitting factor ; taken as 2, 0 effM is the effective saturation magnetization, and 0 KH is the uniaxial anisotropy field. The value s of 0 effM are comparable to the values of 0 SM (obtained from VSM measurements) as is shown in Fig . 4(c). Figure 4(e) shows the variation of 0 KH with Mot from which the decrease in 0 KH with increas e in Mot is clearly evident . This observed reduction in 0 KH could possibly stem from the spin a ccumulation increasing with increasing Mot [26]. The FMR spectra was also recorded on CFA (8)/Cu(1)/Mo(5) trilayer thin film for the comparison with the results of CFA (8)/Mo(5) bilayer. The magnitudes of 0 effM ( 0 kH ) for CFA (8)/Mo(5) and CFA (8)/Cu(1)/Mo(5) are found to be 1.33±0.08 T (0.55±0.15mT) and 1.30±0.04 T (3.21±0.13 mT) , respectivel y. Further, Fig. 4(d) shows the 0 rH vs. Mot behavior at different constant frequenc ies ranging between 5 to 11 GHz. The observed values of 0 rH are constant for all the Mo capped layers which clearly indicate s that the dominant contribution to the observed resonance spectra arises from the intrinsic effect, i.e., magnon -electron scattering [27]. C. Mo t hickness -dependen t spin pumping Figure 5(a) shows t he linewidth 0µH vs. f (for clarity, the results are shown only for a few sel ected film samples ). The frequency dependent linewidth can mainly have two contributions ; the intrinsic magnon -electron scattering contribution, and the extrinsic two- magnon scattering (TMS ) contribution. The extrinsic TMS contribution in linewidth has been analysed (not presented here) using the methods given by Arias and Mills [28]. A similar analysis was reported in one of our previous studies on the CFA/Ta system [21]. For the present case, t he linewidth analysis shows that inclusion of the TMS part does not affect the Gilbert damping , which means the TMS contribution is negligible in our case. Now , the effective Gilbert damping constant eff can be estimated using, 0 04efffHH . (2) Here, 0H is the frequency independent contribution from sample inhomogeneity , while the second term corresponds to the frequency dependent contribution associated with the intrinsic Gilbert relaxation . Here , eff , defined as eff SP CFA , is the effective Gilbert damping which includes the intrinsic value of CFA ()eff and a spin pumping contribution ( SP ) from the CFA/Mo bilayer . The extracted effective Gilbert damping constant values for different Mot are shown in Fig 5(b). An enhancement of the Gilbert damping constant with the increase of the Mo layer thickness is clearly observed , which is anticipated owing to the transfer of spin angular momenta by spin pumping from CFA to the Mo layer at the CFA /Mo interface . The value of eff is found to increase up to 8.4(± 0.3)×10-3 with the increase in Mot (≥ 3.5nm) , which corresponds to ~42% enhancement of the damping constant due to spin pumpin g. It is remarkable that such a large change in Gilbert damping is observed for the CFA/Mo bilayer ; the change is comparabl e to those reported when a high SOC NM such as Pt [8], Pd [29] [9], Ru [11], and Ta [30] is employed in FM/NM bilayer s. Here , we would like to mention that the enhancement of the Gilbert dampin g can , in principle , also be explained by extrinsic two-magnon scattering (TMS ) contribution s in CFA/Mo( Mot ) bilaye rs by considering the variation of rH with NM thickness [27]. In our case, the 0 rH is constant for all Mot (c.f. Fig . 4(d)). Thus the extrinsic contribution induced increase in eff is negligibly small and hence the enhancement of the damping is dominated by the spin pumping mechanism . The estimated values of 0 0µH are found to vary from 0.6 to 2.5 mT in the CFA/Mo( Mot ) thin films . The variation in 0 0µH is assigned to the finite, but small, statistical variations in sputtering conditions between samples with different Mot. Further, to affirm the spin pumping in the CFA/Mo bilayer system, a copper (Cu) dusting layer was inserted at the CFA/Mo interface. Fig ure 5(c) compares the linewidth vs. f plot of the CFA (8)/Mo(5) and CFA (8)/Cu(1)/Mo(5) heterostructures. The Gilbert damping was found to decrease from 8.4(± 0.3)×10-3 to 6.4(± 0.3)×10-3 after inserting the Cu (1) thin layer, which is comparable to the value of the uncapped CFA (3.5) sample. It may be noted that Cu has a very large spin diffusion length ( d~300nm) but weak SOC strength [32]. Due to the weak SOC, the asymmetry in the band structure at the FM/Cu interface would thus lead to a non -equilibrium spin accumulation at the CFA/ Cu interface [33]. This spin accumulation opposes the transfer of angular momentum into the Mo layer and hence the Gilbert damping value , after insertion of the dusting layer , is found very similar to that of the single layer CFA film. It is also known that enhancement of damping in the FM layer (when coupled to the NM layer ) can occur due to the magnetic proximity effect [34]. However, we did not find any evidence in favor of the proximity effect as the effective saturation magnetization did not show any increase on the inserti on of the ultrathin Cu dusting layer at CFA/Mo bilayer interface , which support s our claim of absence of spin pumping in the CFA/Cu/Mo trilayer sample . The flow of angular momentum across the FM/NM bilayer interface is determined by the effective complex spin-mixing conductance g Re(g ) Im(g )eff eff eff i , defined as the flow of angular momentum per unit area through the FM/NM metal interface created by the precessing moment s in the FM layer . The term effective spin-mixing conductance is being used because it contain s the forward and backflow of spin momentum at the FM/NM interface. The imaginary part of the spin-mixing conductance is usually assumed to be negligibly small Re(g ) Im(g )eff eff as compared to the real part [35] [36], and therefore, to determine the real part of the spin-mixing conductance , the obtained Mot dependent G ilbert damping is fit ted with the relation [29], 21Re(g ) 14Mo dt B eff CFA eff S CFAgeMt , (3) where CFAis the damping for a single layer CFA without Mo capping layer, Re(g )eff is given in unit s of m-2, B is the Bohr magneton, and CFAt is a CFA layer thickness . The exponential term describes the reflection of spin -current from Mo/air interface . Figure 5(b) shows the variation of the effective Gilbert damping constant with Mot and the fit using Eqn. (3) (red line) . The values of Re(g )eff and d are found to be 1. 16(±0.19 )×1019 m-2 and 3.5±0.35 nm, respectively. The value of the spin-mixing conductance is comparable to those recent ly reported in FM/Pt (Pd) thin films such as Co/Pt ( 1-4 ×1019 m-2) [8] [33], YIG/Pt (9.7 ×1018 m-2) [37], Fe/Pd (1×1020 m-2) [9], and Py/Pd(Pt) (1.4(3.2) ×1018 m-2) [34]. We now calculate the net intrinsic interfacial spin mixing conductance G which depends on the thickness and the nature of the NM layer as per the relation [9] [38], 11 4( ) Re(g ) 1 tanh3Mo Mo eff dtGt , (4) where 24( / )Z e c is a material dependent param eter (Z is the atomic number of Mo i.e., 42 and c is the speed of light) whose value for Mo is 0.0088 . Using Eq. (4), ()Mo Gt values have been compu ted for variou s Mot ; the results are shown in Fig. 6(a). The Mot dependence of G clearly suggest s that the spin mixing conductance critically depend s on the NM layer properties . For bilayers with Mot 6 nm, G attains its saturation value, which is quite comparable with those reported for Pd and Pt [34] [37]. Understandably, such a large value of the spin mixing conductance will yield a large spin current into the adjacent NM layer [6] [7] [37] [33]. In the next section, we have estimated the spin current from the experimental FMR data and discuss the same with regards to spin pumping in further detail . D. Spin current generation in Mo due to spin pumping The enhancement of the Gilbert damping observed in the CFA (8)/Mo( Mot) bilayers (Fig. 5(b)) is generally interpreted in terms of the spin -current generated in Mo layer by the spin pumping mechanism at the bilayer interface (Fig. 1). The associated net effective spin current density in Mo is described by the relation [38] [39]: 00 02 22 2 2 0224 2( ) G ( )8 4eff eff rf eff S Mo Mo effeffMM h eJ t t M , (5) where 2f and rfhis the rf-field (26 A/m) in the strip -line of our co-planar waveguide. G ( )Mot is the net intrinsic inte rfacial spin mixing conductance discussed in the previous section (Fig. 6). The estimated values of ()eff S MoJt for differen t microwave frequencies are shown in Fig. 7. It is clearly observed that the spin current density increase s with the increase in Mot, the increase becomes relatively less at higher Mot , which indicate s the progressive spin current generation in Mo . Such an appreciable change in current density directly provide s evidence of the interfacial enhancement of the Gilbert damping in these CFA/Mo bilayers . Further, it would be interesting to investigate the effect on the spin current generation in Mo layer if an ultra thin dusting layer of Cu is inserted at the CFA/Mo interface . In princip le, on insertion of a thin Cu layer , the spin pumping should cease because of the unmatched band structure between the CFA /Cu and Cu/M o interfaces owing to the insignificant SOC in Cu. This is in consonance with the observed decrease in Gilbert damping back to the value for the uncapped CFA layer (c.f. Fig. 5(c) and associated discussion ). The spi n-mixing conductance of the trilayer heterostructure can be evaluated by 0 g/eff B eff S CFAg M t [29], where sp eff CFA is the spin-pump ing induced Gilbert damping contribution which for the CFA/Cu/Mo trilayer is quite small , i.e., 4.0(±0.3) ×10-4 after Cu insertion. For the trilayer, geff is found to be 1.49 (±0.12) ×1017 m-2 which is two order s of magnitude small er compared to that of the CFA/Mo bilayers. Furthermore, u sing the values of geff , 0 effM and eff for the CFA/Cu/Mo trilayer hetero structure in Eqn. (5) and for f = 9GHz, the spin current density is found be 0.0278 (±0.001 3) MA/m2, which is two order of magnitude smaller than that in the CFA/Mo bilayers. Thus, t he reduction in eff and eff SJ subsequent to Cu dusting is quite comparable to previously reported results [33] [40]. IV. CONCLUSIONS We have systematically investigated the changes in the spin dynamics in the ion-beam sputtered Co2FeAl ( CFA )/Mo( Mot) bilayer s for various Mot at constant CFA thickness of 8nm . Increasing the Mo layer thickness to its spin diffusion length; CFA (8)/Mo( Mot = d), the effective Gilbert damping constant increases to 8.4(± 0.3)×10-3 which corresponds to about ~42% enhancement with respect to the eff value of 6.0(± 0.3)×10-3 for the uncapped CFA layer (i.e., without the top Mo layer ). We interpret our results based on the spin -pumping effect s where in the effective spin- mixing conductance , and spin -diffusion length are found to be 1.16(±0.19 )×1019 m−2 and 3.50±0.35nm, respectively. The spin pumping is further confirmed by inserting an ultrathin Cu layer at the CFA/Mo interface. The overall effect of the damping constant enhancement observed when Mo is deposited over CFA is remarkably comparable to the far less -abundant non- magnetic metals that are currently being used for spin pumping applications . From this view point , the demonstration of the new material , i.e., Mo, as a suitable spin pumping medium is indispensable for the development of novel STT spintronic devices . ACKNOWLEDGMENT S One of the authors SH acknowledge s the Department of Science and Technology , Govt. of India for providing the INSPIRE Fellowship. Authors thank the NRF facilit ies of IIT Delhi for AFM imaging . This work was in part supported by Knut and Alice Wallenberg (KAW) Foundation Grant No. KAW 2012.0031 . We also acknowledge the Ministry of Information Technology, Government of India for providing the financial grant . REFERENCES: [1] W. Kang, Z. Wang, H. Zhang, S. Li, Y. Zhang, and W. Zhao, Glsvlsi 299 (2017). [2] E. Eken, I. Bayram, Y. Zhang, B. Yan, W. Wu, and Y. Chen, Integr. VLSI J. 58, 253 (2017). [3] S. Bhatti, R. Sbiaa, A. Hirohata, H. Ohno, S. Fukami, and S. N. Piramanayaga m, Mater. Today http://dx.doi.org/10.1016/j.mattod.2017.07.007 1 (2017). [4] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). [5] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I. Halperin, Phys. Rev. B 66, 060404(R) (2002). [6] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002). [7] Y. Tserkovnyak, a Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). [8] S. Azzawi, A. Ganguly, M. Tokaç, R. M. Rowan -Robinson, J. 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Yoshino, R. Takahashi, Y. Kajiwara, K. Uchida, Y. Fujikawa, and E. Saitoh, Phys. Rev. B 85, 144408 (2012). [39] K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. Fujikawa, M. Matsuo, S. Maekawa, and E. S aitoh, J. Appl. Phys. 109, 103913 (2011). [40] T. Nan, S. Emori, C. T. Boone, X. Wang, T. M. Oxholm, J. G. Jones, B. M. Howe, G. J. Brown, N. X. Sun, and A. Samples, Phys. Rev. B 91, 214416 (2015). Table: 1 Summary of XRR simulated parameters , i.e., , FMt , Mot , and σ for the bilayer thin films [Si/CFA( 8)/Mo( Mot)]. Here , FMt , Mot, and σ refer to the density, thickness, and interface width of the individual layers , respectively. CFA (Nominal thickness = 8 nm) Mo MoOx S.No. 1 2 3 4 5 6 7 8 9 10 (gm/cc)±0.06 7.35 7.31 7.50 7.50 7.00 7.00 7.29 7.22 7.00 7.64 tFM(nm)±0.01 7.00 8.17 7.22 8.18 7.00 8.28 8.00 7.79 8.12 8.00 σ(nm) ±0.03 0.20 0.35 0.80 0.37 1.00 0.56 0.44 0.98 0.15 0.17 (gm/cc)±0.0 5 6.05 8.58 10.50 9.94 9.50 10.45 9.43 10.50 9.29 9.23 tMo(nm)±0.01 0.58 1.00 1.50 2.00 3.00 3.46 4.86 6.47 8.21 10.26 σ(nm) ±0.03 0.94 0.54 0.52 0.64 0.60 0.78 0.26 0.67 0.64 0.67 (gm/cc)±0.06 4.07 4.04 5.00 4.38 5.17 4.38 6.50 4.81 4.00 5.00 t(nm)±0.01 0.97 0.82 1.08 0.85 1.00 1.01 0.98 1.03 0.96 1.17 σ(nm) ±0.03 0.59 0.35 0.5 0.45 0.37 0.4 0.56 0.62 0.8 0.73 Figure captions FIG. 1. (color online) Schematic of the CFA/Mo bilayer structure used in our work portrayed for an example of spin current density eff SJ generated at the CFA/Mo interface by spin pumping . FIG. 2 XRR spectra and the AFM topographical images of Si/CFA( 8)/Mo( Mot ). In the respective XRR spectra, circles represent the recorded experimental data points, and lines represent the simulated profiles. The estimated values of the surface roughness in the entire sample series as obtained from XRR and AFM topographical measurements are compared in the lowest right panel. The simulated parameters are presented in the Table -I. All AFM images were recorded on a scan area of 10×10 m2. FIG.3 : The atomic representation (model) of the growth of th e Mo layer (yellow sphere) on top of the CFA (blue spheres) layer. The film changes from discontinuous to continuous as the thickness of the Mo layer is increased. Shown are the 3 different growth stages of the films: (a) least coverage (b) partial coverage and (c) full coverage . FIG. 4: (a) Typical FMR spectra recorded at various frequencies (numbers in graph are the microwave frequencies in GHz) for the Si/SiO 2/CFA(8)/Mo(5) bilayer sample (symbols correspond to experimental data and red lines are fits to the Eqn. (1)) Inset: FMR spectra of CFA single layer (filled circles) and CFA(8)/Mo(2) bilayer (open circles) samples measured at 5GHz showing the increase in linewidth due to spin pumping. (b) The resonance field 0 rH vs. f for all the samples ( red lines are the fits to the Eqn. (2). (c) Effective magnetization (scale on left) and saturation magnetization (scale on right) vs. Mot . The solid line represents the bulk value of the saturation magnetization of Co 2FeAl . (d) The resonance field 0 rH vs. Mot at different constant frequencies for CFA(8)/Mo( Mot ) bilayer thin films. (e) Anisotropy field 0 KH vs. Mot . (f) Comparison of 0 rH vs. f for the CFA(8)/Mo(5) and CFA(8)/Cu(1)/Mo(5) samples. FIG. 5: (a) Linew idth vs. frequency for Si/SiO 2/CFA(8)/Mo( Mot ) bilayer thin films. (b) Effective Gilbert damping constant vs. Mo layer thicknesses. (c) 0H vs. f for CFA(8)/ Mo(5) and CFA(8)/Cu(1)/Mo(5) films. FIG. 6 : Intrinsic s pin-mixing conductance vs. Mot of the CFA (8)/Mo( Mot) bilayers . FIG. 7. The effective spin current density (generated in Mo) vs. Mot at different microwave frequencies calculated using Eqn. (5) FIG. 1 FIG. 2 FIG. 3 FIG. 4 FIG. 5 FIG. 6 FIG. 7

2017-11-20

The spin pumping mechanism and associated interfacial Gilbert damping are demonstrated in ion-beam sputtered Co2FeAl (CFA) /Mo bilayer thin films employing ferromagnetic resonance spectroscopy. The dependence of the net spin current transportation on Mo layer thickness, 0 to 10 nm, and the enhancement of the net effective Gilbert damping are reported. The experimental data has been analyzed using spin pumping theory in terms of spin current pumped through the ferromagnet /nonmagnetic metal interface to deduce the effective spin mixing conductance and the spin-diffusion length, which are estimated to be 1.16(0.19)x10^19 m^-2 and 3.50(0.35)nm, respectively. The damping constant is found to be 8.4(0.3)x10^-3 in the Mo(3.5nm) capped CFA(8nm) sample corresponding to a ~42% enhancement of the original Gilbert damping (6.0(0.3)x10^-3) in the uncapped CFA layer. This is further confirmed by inserting a Cu dusting layer which reduces the spin transport across the CFA /Mo interface. The Mo layer thickness dependent net spin current density is found to lie in the range of 1-3 MAm^-2, which also provides additional quantitative evidence of spin pumping in this bilayer thin film system.

Spin Pumping in Ion-beam Sputtered Co_{2}FeAl/Mo Bilayers:Interfacial Gilbert Damping

1711.07455v1

arXiv:1703.03485v2 [math.AP] 10 Apr 2017LONG-TIME DYNAMICS OF THE STRONGLY DAMPED SEMILINEAR PLATE EQUATION IN Rn AZER KHANMAMEDOV AND SEMA YAYLA Abstract. We investigate the initial-value problem for the semilinea r plate equation containing local- ized strong damping, localized weak damping and nonlocal no nlinearity. We prove that if nonnegative damping coefficients are strictly positive almost everywher e in the exterior of some ball and the sum of these coefficients is positive a.e. in Rn, then the semigroup generated by the considered problem possesses a global attractor in H2(Rn)×L2(Rn). We also establish boundedness of this attractor in H3(Rn)×H2(Rn). 1.Introduction In this paper, our main purpose is to study the long-time dynamics (in terms of attractors) of the plate equation utt+γ∆2u−div(β(x)∇ut)+α(x)ut+λu−f(/ba∇dbl∇u(t)/ba∇dblL2(Rn))∆u+g(u) =h(x), (t,x)∈R+×Rn, (1.1) with initial data u(0,x) =u0(x),ut(0,x) =u1(x),x∈Rn, (1.2) whereγ >0,λ >0,h∈L2(Rn) and the functions α(·), β(·), f(·) andg(·) satisfy the following conditions: α, β∈L∞(Rn),α(·)≥0, β(·)≥0 a.e. in Rn, (1.3) α(·)≥α0>0 andβ(·)≥β0>0 a.e. in {x∈Rn:|x| ≥r0}, for somer0>0, (1.4) α(·)+β(·)>0 a.e. in Rn, (1.5) f∈C1(R+), f(z)≥0, for allz∈R+, (1.6) g∈C1(R),|g′(s)| ≤C/parenleftig 1+|s|p−1/parenrightig ,p≥1, (n−4)p≤n, (1.7) g(s)s≥0, for every s∈R. (1.8) The problem (1.1)-(1.2) can be reduced to the following Cauchy prob lem for the first order abstract differential equation in the space H2(Rn)×L2(Rn): /braceleftbiggd dtθ(t) =Aθ(t)+F(θ(t)), θ(0) =θ0, whereθ(t) = (u(t),ut(t)),θ0= (u0,u1),A(u, v) = (v,−γ∆2u+div(β(·)∇v)−α(·)v−λu),D(A) =/braceleftbig (u,v)∈H3(Rn)×H2(Rn) :γ∆2u−div(β(·)∇v)∈L2(Rn)/bracerightbig andF(u,v) = (0,f(/ba∇dbl∇u/ba∇dblL2(Rn))∆u −g(u)+h). Defining suitable equivalent norm in H2(Rn)×L2(Rn), it is easy to see that the operator A, thanks to (1.3), is maximal dissipative in H2(Rn)×L2(Rn) and consequently, due to Lumer-Phillips Theorem(see [1, Theorem4.3]), it generatesa linearcontinuoussem igroup/braceleftbig etA/bracerightbig t≥0. Also, by (1.6)-(1.7), we find that the nonlinear operator F:H2(Rn)×L2(Rn)→H2(Rn)×L2(Rn) is Lipschitz continuous on bounded subsets of H2(Rn)×L2(Rn). So, applying semigroup theory (see, for example [2, p. 56-58]), and taking advantage of energy estimates, we have the following we ll-posedness result. 2000Mathematics Subject Classification. 35B41, 35G20, 74K20. Key words and phrases. wave equation, plate equation, global attractor. 12 AZER KHANMAMEDOV AND SEMA YAYLA Theorem 1.1. Assume that the conditions (1.3), (1.6), (1.7) and (1.8) hol d. Then, for every (u0,u1)∈ H2(Rn)×L2(Rn), the problem (1.1)-(1.2) has a unique weak solution u∈C/parenleftbig [0,∞);H2(Rn)/parenrightbig ∩C1/parenleftbig [0,∞);L2(Rn)/parenrightbig , which depends continuously on the initial data and satisfie s the energy equality E(u(t))+/integraldisplay RnG(u(t,x))dx+1 2F/parenleftig /ba∇dbl∇u(t)/ba∇dbl2 L2(Rn)/parenrightig −/integraldisplay Rnh(x)u(t,x)dx +t/integraldisplay s/integraldisplay Rnα(x)|ut(τ,x)|2dxdτ+t/integraldisplay s/integraldisplay Rnβ(x)|∇ut(τ,x)|2dxdτ =E(u(s))+/integraldisplay RnG(u(s,x))dx+1 2F/parenleftig /ba∇dbl∇u(s)/ba∇dbl2 L2(Rn)/parenrightig −/integraldisplay Rnh(x)u(s,x)dx,∀t≥s≥0,(1.9) whereF(z) =z/integraltext 0f(√s)dsfor allz∈R+,G(z) =z/integraltext 0g(s)dsfor allz∈RandE(u(t)) =1 2/integraltext Rn(|ut(t,x)|2+ γ|∆u(t,x)|2+λ|u(t,x)|2)dx. Moreover, if (u0,u1)∈D(A), thenu(t,x)is a strong solution satisfying (u,ut)∈C([0,∞);D(A))∩C1[0,∞);H2(Rn)×L2(Rn). Thus, duetoTheorem1.1,theproblem(1.1)-(1.2)generatesastr onglycontinuoussemigroup {S(t)}t≥0 inH2(Rn)×L2(Rn) by the formula ( u(t),ut(t)) =S(t)(u0,u1), whereu(t,x) is a weak solution of (1.1)-(1.2) with the initial data ( u0,u1). Attractors for hyperbolic and hyperbolic like equations in unbounde d domains have been extensively studied by many authors over the last few decades. To the best of our knowledge, the first works in this area were done by Feireisl in [3] and [4], for the wave equations wit h the weak damping (the case γ= 0,β≡0 andf≡1 in (1.1)) . In those articles the author, by using the finite speed pr opagation property of the wave equations, established the existence of the global attractors in H1(Rn)×L2(Rn). The global attractors for the wave equations involving strong dam ping in the form −∆ut, besides weak damping, were investigated in [5] and [6], where the authors, by using splitting method, proved the existence of the global attractors in H1(Rn)×L2(Rn), under different conditions on the nonlinearities. Recently, in [7], the results of [5] and [6] have been improved for the w ave equation involving additional nonlocal nonlinear term in the form −(a+b/ba∇dbl∇u(t)/ba∇dbl2 L2(Rn))∆u(a≥0, b >0). For the plate equation with only weak damping and local nonlinearity (the case γ= 1,β≡0 andf≡0 in (1.1)), attractors were investigated in [8] and [9], where the author, inspired by the met hods of [10] and [11], proved the existence, regularityand finite dimensionality ofthe global attract orsinH2(Rn)×L2(Rn). The situation becomes more difficult when the equation contains localized damping te rms and nonlocal nonlinearities. Recently, in [12] and [13], the plate equation with localized weak damping (the caseβ≡0 in (1.1)) and involving nonlocal nonlinearities as −f(/ba∇dbl∇u/ba∇dblL2(Rn))∆uandf(/ba∇dblu/ba∇dblLp(Rn))|u|p−2uhave been considered. In these articles, the existence of global attractors has been pr oved when the coefficient α(·) of the weak damping term is strictly positive (see [12]) or, in addition to (1.3), is pos itive (see [13]) almost everywhere inRn. However, in the case when α(·) vanishes in a set of positive measure, the existence of the global attractor for (1.1) with β≡0 remained as an open question (see [12, Remark 1.2]). On the other hand, in the case when α≡0 and even β≡1, the semigroup {S(t)}t≥0generated by (1.1)-(1.2) does not possess a global attractor in H2(Rn)×L2(Rn). Indeed, if {S(t)}t≥0possesses a global attractor, then the linear semigroup/braceleftbig etA/bracerightbig t≥0decay exponentially in the real and consequently, complex space H2(Rn)×L2(Rn), which, due to Hille-Yosida Theorem (see [1, Remark 5.4]), implies neces sary condition iR⊂ρ(A). This condition is equivalent to the solvability of the equation ( iµI−A)(u,v) = (y,z) inH2(Rn)×L2(Rn), for every (y,z) inH2(Rn)×L2(Rn) andµ∈R. Choosing µ=√ λandy= 0, we have v=i√ λuand ∆(∆u−iu) =z. If the last equation for every z∈L2(Rn) has a solution u∈H3(Rn), then denoting ϕ= ∆u−iu, we can say that the equation ∆ ϕ=zhas a solution in H1(Rn), for every z∈L2(Rn). However, the last equation, as shown in [6], is not solvable in H1(Rn) for somez∈L2(Rn). Hence, the necessary condition iR⊂ρ(A) does not hold. Thus, in the case when α≡0 andβ≡1, the problem (1.1)-(1.2) does not have a global attractor, and in the case when β≡0 andα(·) vanishes in a set of positive measure, the existence of the global for (1.1)-(1.2) is an o pen question.LONG-TIME DYNAMICS 3 In this paper, we impose conditions (1.3)-(1.5) on damping coefficient sα(·) andβ(·), which, unlike the conditions imposed in the previous articles dealing with the wave and pla te equations involving strong damping and/or nonlocal nonlinearities, allow both of them to be vanis hed in the sets of positive measure such that in these sets the strong damping and weak damping comple te each other. Thus, our main result is as follows: Theorem 1.2. Under the conditions (1.3)-(1.8) the semigroup {S(t)}t≥0generated by the problem (1.1)- (1.2) possesses a global attractor AinH2(Rn)×L2(Rn)andA=Mu(N). HereMu(N)is unstable manifold emanating from the set of stationary points N(for definition, see [14, 359] ). Moreover, the global attractor Ais bounded in H3(Rn)×H2(Rn). The plan of the paper is as follows: In the next section, after the pr oof of two auxiliary lemmas, we establish asymptotic compactness of {S(t)}t≥0in the interior domain. Then, we prove Lemma 2.3, which plays a key role for the tail estimate, and thereby we show that the solutions of (1.1)-(1.2) are uniformly (with respect to the initial data) small at infinity for large time. This f act, together with asymptotic compactness in the interior domain, yields asymptotic compactness of{S(t)}t≥0in the whole space, and by applying the abstract result on the gradient systems, we estab lish the existence of the global attractor (see Theorem 2.3). In Section 3, by using the invariance of the globa l attractor, we show that it has an additional regularity. 2.Existence of the global attractor We begin with the following lemmas: Lemma 2.1. Assume that the condition (1.6) holds. Also, assume that the sequence {vm}∞ m=1is weakly star convergent in L∞/parenleftbig 0,∞;H2(Rn)/parenrightbig , the sequence {vmt}∞ m=1is bounded in L∞/parenleftbig 0,∞;L2(Rn)/parenrightbig and the sequence/braceleftig /ba∇dbl∇vm(t)/ba∇dblL2(Rn)/bracerightig∞ m=1is convergent, for all t≥0. Then, for every r>0andφ∈C1 0(B(0,r)) lim m→∞limsup l→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay 0/integraldisplay B(0,r)τ/parenleftig f(/ba∇dbl∇vm(τ)/ba∇dblL2(Rn))∆vm(τ,x)−f(/ba∇dbl∇vl(τ)/ba∇dblL2(Rn))∆vl(τ,x)/parenrightig ×φ(x)(vmt(τ,x)−vlt(τ,x))dxdτ|= 0,∀t≥0, whereB(0,r) ={x∈Rn:|x|<r}. Proof.Firstly, we have /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay 0/integraldisplay B(0,r)τ/parenleftig f(/ba∇dbl∇vm(τ)/ba∇dblL2(Rn))∆vm(τ,x)−f(/ba∇dbl∇vl(τ)/ba∇dblL2(Rn))∆vl(τ,x)/parenrightig ×φ(x)(vmt(τ,x)−vlt(τ,x))dxdτ| ≤1 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay 0τf/parenleftig /ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightigd dτ/integraldisplay B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingleKm,l r(t)/vextendsingle/vextendsingle, (2.1) whereKm,l r(t) =t/integraltext 0τ/parenleftig f(/ba∇dbl∇vm(τ)/ba∇dblL2(Rn))−f(/ba∇dbl∇vl(τ)/ba∇dblL2(Rn))/parenrightig/integraltext B(0,r)φ(x)∆vm(τ,x) ×(vmt(τ,x)−vlt(t,x))dxdτ−t/integraltext 0/integraltext B(0,r)τf/parenleftig /ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightig ∇φ(x)·∇(vm(τ,x)−vl(τ,x)) ×(vmt(τ,x)−vlt(τ,x))dxdτ. Applying [15, Corollary 4], we have that the sequence {vm}∞ m=1is rela- tively compact in C/parenleftbig [0,T];H2−ε(B(0,r))/parenrightbig , for everyε>0,T >0 andr>0. So, vm→vstrongly in C/parenleftbig [0,T];H2−ε(B(0,r))/parenrightbig , (2.2)4 AZER KHANMAMEDOV AND SEMA YAYLA for somev∈C/parenleftbig [0,T];H2−ε(B(0,r))/parenrightbig . Hence, we find lim m→∞limsup l→∞/vextendsingle/vextendsingleKm,l r(t)/vextendsingle/vextendsingle= 0,∀t≥0. (2.3) Now, denoting fε(u) =/braceleftbigg f(u), u≥ε f(ε),0≤u<εforε>0, we get /vextendsingle/vextendsingle/vextendsinglef/parenleftig /ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightig −fε/parenleftig /ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightig/vextendsingle/vextendsingle/vextendsingle≤max 0≤s1,s2≤ε|f(s1)−f(s2)|, and then, for the first term on the right hand side of (2.1), we obta in /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay 0τf/parenleftig /ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightigd dτ/integraldisplay B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay 0τfε/parenleftig /ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightigd dτ/integraldisplay B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle +c1t2max 0≤s1,s2≤ε|f(s1)−f(s2)|,∀t≥0. (2.4) Let us estimate the first term on the right hand side of (2.4). By usin g integration by parts, we have t/integraldisplay 0τfε/parenleftig /ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightigd dτ/integraldisplay B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ =tfε/parenleftig /ba∇dbl∇vl(t)/ba∇dblL2(Rn)/parenrightig/integraldisplay B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dx −t/integraldisplay 0fε/parenleftig /ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightig/integraldisplay B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ −t/integraldisplay 0τd dt/parenleftig fε/parenleftig /ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightig/parenrightig/integraldisplay B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ. (2.5) By the conditions of the lemma and the definition of fε, it follows that/braceleftig fε/parenleftig /ba∇dbl∇vm(·)/ba∇dblL2(Rn)/parenrightig/bracerightig∞ m=1is bounded in W1,∞(0,∞). Then, considering (2.2) in (2.5), we get lim m→∞limsup l→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay 0τfε/parenleftig /ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightigd dτ/integraldisplay B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.(2.6) Taking into account (2.3), (2.4) and (2.6) in (2.1), we obtain limsup m→∞limsup l→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay 0/integraldisplay B(0,r)τ/parenleftig f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn)))∆vm(t,x)−f(/ba∇dbl∇vl(t)/ba∇dblL2(Rn)))∆vl(t,x)/parenrightig ×φ(x)(vmt(τ,x)−vlt(τ,x))dxdτ| ≤c1t2max 0≤s1,s2≤ε|f(s1)−f(s2)|,∀t≥0, which yields the claim of the lemma, since ε>0 is arbitrary. /squareLONG-TIME DYNAMICS 5 Lemma 2.2. Assume that the condition (1.7) holds. Also, let the sequenc e{vm}∞ m=1be weakly star convergent in L∞/parenleftbig 0,∞;H2(Rn)/parenrightbig and the sequence {vmt}∞ m=1be bounded in L∞/parenleftbig 0,∞;L2(Rn)/parenrightbig . Then, for everyr>0andφ∈L∞(B(0,r)) lim m→∞lim l→∞t/integraldisplay 0/integraldisplay B(0,r)τ(g(vm(τ,x))−g(vl(τ,x)))φ(x)(vmt(τ,x)−vlt(τ,x))dxdτ= 0,∀t≥0. Proof.We have t/integraldisplay 0/integraldisplay B(0,r)τ(g(vm(τ,x))−g(vl(τ,x)))φ(x)(vmt(τ,x)−vlt(τ,x))dxdτ =t/integraldisplay 0/integraldisplay B(0,r)τφ(x)g(vm(τ,x))vmt(τ,x)dxdτ+t/integraldisplay 0/integraldisplay B(0,r)τφ(x)g(vl(τ,x))vlt(τ,x)dxdτ −t/integraldisplay 0/integraldisplay B(0,r)τφ(x)g(vm(τ,x))vlt(τ,x)dxdτ−t/integraldisplay 0/integraldisplay B(0,r)τφ(x)g(vl(τ,x))vmt(τ,x)dxdτ. (2.7) Let us estimate the first two terms on the right hand side of (2.7). A pplying integration by parts, we get t/integraldisplay 0/integraldisplay B(0,r)τφ(x)g(vm(τ,x))vmt(τ,x)dxdτ+t/integraldisplay 0/integraldisplay B(0,r)τφ(x)g(vl(τ,x))vlt(τ,x)dxdτ =t/integraldisplay 0τd dτ /integraldisplay B(0,r)φ(x)G(vm(τ,x))dx dτ+t/integraldisplay 0τd dτ /integraldisplay B(0,r)φ(x)G(vl(τ,x))dx dτ =t/integraldisplay B(0,r)φ(x)G(vm(t,x))dx+t/integraldisplay B(0,r)φ(x)G(vl(τ,x))dx −t/integraldisplay 0/integraldisplay B(0,r)φ(x)G(vm(τ,x))dxdτ−t/integraldisplay 0/integraldisplay B(0,r)φ(x)G(vl(τ,x))dxdτ. (2.8) By the conditions of the lemma, we obtain /braceleftbigg vm→vweakly star in L∞/parenleftbig 0,∞;H2(Rn)/parenrightbig , vmt→vtweakly star in L∞/parenleftbig 0,∞;L2(Rn)/parenrightbig ,(2.9) for somev∈L∞/parenleftbig 0,∞;H2(Rn)/parenrightbig ∩W1,∞/parenleftbig 0,∞;L2(Rn)/parenrightbig .Applying [15, Corollary 4], by (2.9), we have vm→vstrongly in C/parenleftbig [0,T];H2−ε(B(0,r))/parenrightbig , for everyε>0 andT >0. Hence, taking into account (1.7), we get G(vm)→G(v) strongly in C/parenleftbig [0,T];L1(B(0,r))/parenrightbig . (2.10) Then, passing to the limit in (2.8) and using (2.10), we obtain lim m→∞lim l→∞ t/integraldisplay 0/integraldisplay B(0,r)τφ(x)g(vm(τ,x))vmt(τ,x)dxdτ+t/integraldisplay 0/integraldisplay B(0,r)τφ(x)g(vl(τ,x))vlt(τ,x)dxdτ = 2t/integraldisplay B(0,r)φ(x)G(v(t,x))dx−2t/integraldisplay 0/integraldisplay B(0,r)φ(x)G(v(τ,x))dxdτ. (2.11)6 AZER KHANMAMEDOV AND SEMA YAYLA Now, for the last two terms on the right hand side of (2.7), consider ing (2.9), we get lim m→∞lim l→∞ −t/integraldisplay 0/integraldisplay B(0,r)τφ(x)g(vm(τ,x))vlt(τ,x)dxdτ−t/integraldisplay 0/integraldisplay B(0,r)τφ(x)g(vl(τ,x))vmt(τ,x)dxdτ =−2t/integraldisplay 0/integraldisplay B(0,r)τφ(x)g(v(τ,x))vt(τ,x)dxdτ =−2t/integraldisplay B(0,r)φ(x)G(v(t,x))dx+2t/integraldisplay 0/integraldisplay B(0,r)φ(x)G(v(τ,x))dxdτ. (2.12) Hence, considering (2.11)-(2.12) and passing to the limit in (2.7), we o btain the claim of the lemma. /square Now, we can prove the asymptotic compactness of {S(t)}t≥0in the interior domain. Theorem 2.1. Assume that the conditions (1.3)-(1.8) hold and Bis a bounded subset of H2(Rn)× L2(Rn). Then every sequence of the form {S(tk)ϕk}∞ k=1,where{ϕk}∞ k=1⊂ B,tk→ ∞,has a convergent subsequence in H2(B(0,r))×L2(B(0,r)), for every r>0. Proof.We will use the asymptotic compactness method introduced in [16]. Co nsidering (1.3), (1.6), (1.7) and (1.8) in (1.9), we have sup t≥0sup ϕ∈B/ba∇dblS(t)ϕ/ba∇dblH2(Rn)×L2(Rn)<∞. (2.13) Due to the boundedness of the sequence {ϕk}∞ k=1inH2(Rn)×L2(Rn), by (2.13), it follows that the sequence {S(·)ϕk}∞ k=1is bounded in L∞/parenleftbig 0,∞;H2(Rn)×L2(Rn)/parenrightbig . Then for any T≥1 there exists a subsequence {km}∞ m=1such thattkm≥T, and vm→vweakly star in L∞/parenleftbig 0,∞;H2(Rn)/parenrightbig , vmt→vtweakly star in L∞/parenleftbig 0,∞;L2(Rn)/parenrightbig , /ba∇dbl∇vm/ba∇dbl2 L2(Rn)→qweakly star in W1,∞(0,∞), vm→vstrongly in C/parenleftbig [0,T];H2−ε(B(0,r))/parenrightbig ,ε>0,(2.14) for somev∈L∞/parenleftbig 0,∞;H2(Rn)/parenrightbig ∩W1,∞/parenleftbig 0,∞;L2(Rn)/parenrightbig andq∈W1,∞(0,∞), where (vm(t),vmt(t)) = S(t+tkm−T)ϕkm. Now, taking into account (1.4) in (1.9), we find ∞/integraldisplay 0/ba∇dblvmt(t)/ba∇dbl2 L2(Rn\B(0,r0))dt+∞/integraldisplay 0/ba∇dbl∇vmt(t)/ba∇dbl2 L2(Rn\B(0,r0))dt≤c1. (2.15) By (1.1), we have vmtt(t,x)−div(β(x)∇vmt(t,x))+γ∆2vm(t,x)+α(x)vmt(t,x)+λvm(t,x) =f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn)))∆vm(t,x)−g(vm(t,x))+h(x). (2.16) Letη∈C∞(Rn), 0≤η(x)≤1,η(x) =/braceleftbigg 0,|x| ≤1 1,|x| ≥2andηr(x) =η/parenleftbigx r/parenrightbig . Multiplying (2.16) with η2 rvmand integrating the obtained equality over (0 ,T)×Rn, we get T/integraldisplay 0/parenleftig γ/ba∇dblηr∆vm(t)/ba∇dbl2 L2(Rn)+λ/ba∇dblηrvm(t)/ba∇dbl2 L2(Rn)/parenrightig dt =−1 2/integraldisplay Rnη2 r(x)β(x)|∇vm(T,x)|2dx+1 2/integraldisplay Rnη2 r(x)β(x)|∇vm(0,x)|2dxLONG-TIME DYNAMICS 7 −2 rn/summationdisplay i=1T/integraldisplay 0/integraldisplay Rnβ(x)vmtxi(t,x)ηrηxi/parenleftigx r/parenrightig vm(t,x)dxdt +T/integraldisplay 0/ba∇dblηrvmt(t)/ba∇dbl2 L2(Rn)dt−/integraldisplay Rnη2 r(x)vmt(T,x)vm(T,x)dx+/integraldisplay Rnη2 r(x)vmt(0,x)vm(0,x)dx −4γ rn/summationdisplay i=1T/integraldisplay 0/integraldisplay Rnηr(x)ηxi/parenleftigx r/parenrightig ∆vm(t,x)vmxi(t,x)dxdt−γT/integraldisplay 0/integraldisplay Rn∆/parenleftbig η2 r(x)/parenrightbig ∆vm(t,x)vm(t,x)dxdt −1 2/integraldisplay Rnη2 r(x)α(x)|vm(T,x)|2dx+1 2/integraldisplay Rnη2 r(x)α(x)|vm(0,x)|2dx −T/integraldisplay 0f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn)))/integraldisplay Rnη2 r(x)|∇vm(t,x)|2dxdt −2 rn/summationdisplay i=1T/integraldisplay 0f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn)))/integraldisplay Rnηrηxi/parenleftigx r/parenrightig vmxi(t,x)vmdxdt −T/integraldisplay 0/integraldisplay Rng(vm(t,x))η2 r(x)vm(t,x)dxdt +T/integraldisplay 0/integraldisplay Rnh(x)η2 r(x)vm(t,x)dxdt. (2.17) Taking into account (1.3), (1.6), (1.8), (1.9), (2.13) and (2.15) in (2 .17), we obtain lim sup m→∞T/integraldisplay 0/parenleftig γ/ba∇dbl∆vm(t)/ba∇dbl2 L2(Rn\B(0,2r))+λ/ba∇dblvm(t)/ba∇dbl2 L2(Rn\B(0,2r))/parenrightig dt ≤c2/parenleftigg 1+√ T r+T r+T/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightigg ,∀r≥r0. (2.18) Now, by (1.1), we have vmtt(t,x)−vltt(t,x)−div(β(x)·∇(vmt(t,x)−vlt(t,x)))+γ∆2(vm(t,x)−vl(t,x)) +α(x)(vmt(t,x)−vlt(t,x))+λ(vm(t,x)−vl(t,x)) =f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn)))∆vm(t,x)−f(/ba∇dbl∇vl(t)/ba∇dblL2(Rn)))∆vl(t,x)−g(vm)+g(vl). (2.19) Multiplying (2.19) by/summationtextn i=1xi(1−η4r)(vm−vl)xi+1 2(n−1)(1−η4r)(vm−vl), integrating the ob- tained equality over (0 ,T)×Rnand taking into account (2.13), we obtain 3γ 2T/integraldisplay 0/ba∇dbl∆(vm(t)−vl(t))/ba∇dbl2 L2(B(0,4r))dt+1 2T/integraldisplay 0/ba∇dblvmt(t)−vlt(t)/ba∇dbl2 L2(B(0,4r))dt ≤c3(1+T+rT)/ba∇dblvm−vl/ba∇dblC[0,T];H1(B(0,8r)) +c3/parenleftig√ T+r√ T/parenrightig/vextenddouble/vextenddouble/vextenddouble/radicalbig β(∇vmt−∇vlt)/vextenddouble/vextenddouble/vextenddouble L2((0,T)×B(0,8r)) +c3/parenleftig /ba∇dblvmt−vlt/ba∇dbl2 L2(0,T;L2(B(0,8r)\B(0,4r)))+/ba∇dblvm−vl/ba∇dbl2 L2(0,T;H2(B(0,8r)\B(0,4r)))/parenrightig .(2.20)8 AZER KHANMAMEDOV AND SEMA YAYLA Thus, considering (2.14), (2.15), (2.18) and passing to the limit in (2.2 0) , we get limsup m→∞limsup l→∞T/integraldisplay 0/bracketleftig /ba∇dbl∆(vm(t)−vl(t))/ba∇dbl2 L2(B(0,4r))+/ba∇dblvmt(t)−vlt(t)/ba∇dbl2 L2(B(0,4r))/bracketrightig dt ≤c4/parenleftigg 1+√ T r+T r+r√ T+T/ba∇dblh/ba∇dblL2(Rn\B(0,2r))/parenrightigg ,∀r≥r0. (2.21) Now,multiplying(2.19)by(1 −η2r)4t/bracketleftbig 2(vmt−vlt)+α0η4 r(vm−vl)/bracketrightbig andintegratingtheobtainedequal- ity over (0,T)×Rn,we obtain γT/ba∇dbl∆(vm(T)−vl(T))/ba∇dbl2 L2(B(0,2r))+T/ba∇dblvmt(T)−vlt(T)/ba∇dbl2 L2(B(0,2r))+ +Tλ/ba∇dblvm(T)−vl(T)/ba∇dbl2 L2(B(0,2r))≤T/integraldisplay 0/ba∇dblvmt(t)−vlt(t)/ba∇dbl2 L2(B(0,4r))dt +γT/integraldisplay 0/ba∇dbl∆(vm(t)−vl(t))/ba∇dbl2 L2(B(0,4r))dt+λT/integraldisplay 0/ba∇dblvm(t)−vl(t)/ba∇dbl2 L2(B(0,4r))dt +2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleT/integraldisplay 0/integraldisplay B(0,4r)t/parenleftig f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn))∆vm(t,x)−f(/ba∇dbl∇vl(t)/ba∇dblL2(Rn))∆vl(t,x))/parenrightig ×(1−η2r)4(vmt(t,x)−vlt(t,x))dxdt/vextendsingle/vextendsingle/vextendsingle +2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleT/integraldisplay 0/integraldisplay B(0,4r)t(g(vm(t,x))−g(vl(t,x)))(1−η2r)4(vmt(t,x)−vlt(t,x))dxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle +α0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleT/integraldisplay 0/integraldisplay B(0,4r)t/parenleftig f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn))∆vm(t,x)−f(/ba∇dbl∇vl(t)/ba∇dblL2(Rn))∆vl(t,x))/parenrightig ×(1−η2r)4η4 r(x)(vm(t,x)−vl(t,x))dxdt/vextendsingle/vextendsingle/vextendsingle +α0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleT/integraldisplay 0/integraldisplay B(0,4r)t(g(vm(t,x))−g(vl(t,x)))(1−η2r)4η4 r(x)(vm(t,x)−vl(t,x))dxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle +c5T rT/integraldisplay 0/integraldisplay B(0,4r)\B(0,r)|∇(vmt(t,x)−vlt(t,x))|2dxdt +c5T rT/integraldisplay 0/integraldisplay B(0,4r)\B(0,r)|vmt(t,x)−vlt(t,x)|2dxdt +c5T/ba∇dblvm−vl/ba∇dbl2 C([0,T];H1(B(0,4r))),∀r≥r0,∀T≥1. (2.22) Then, taking into account (2.14), (2.15), (2.21), Lemma 2.1 and Lem ma 2.2, and passing to the limit in (2.22), we find limsup m→∞limsup l→∞/parenleftig /ba∇dblvm(T)−vl(T)/ba∇dbl2 H2(B(0,2r))+/ba∇dblvmt(T)−vlt(T)/ba∇dbl2 L2(B(0,2r))/parenrightig ≤c6/parenleftbigg1 T+1√ Tr+1 r+r√ T+/ba∇dblh/ba∇dblL2(Rn\B(0,2r))/parenrightbigg ,∀r≥r0,∀T≥1. (2.23)LONG-TIME DYNAMICS 9 Thus, by the definition of vm, the inequality (2.23) yields limsup m→∞limsup l→∞/ba∇dblS(tkm)ϕkm−S(tkl)ϕkl/ba∇dbl2 H2(B(0,r))×L2(B(0,r)) ≤c7/parenleftbigg1 T+1√ Tr+1 r+r√ T+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg ,∀r≥2r0,∀T≥1. (2.24) Passing to the limit as T→ ∞in (2.24), we obtain liminf l→∞liminf m→∞/ba∇dblS(tk)ϕk−S(tm)ϕm/ba∇dbl2 H2(B(0,r))×L2(B(0,r)) ≤c7/parenleftbigg1 r+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg ,∀r≥2r0, which gives liminf l→∞liminf m→∞/ba∇dblS(tk)ϕk−S(tm)ϕm/ba∇dbl2 H2(B(0,r))×L2(B(0,r)) ≤c7/parenleftbigg1 /tildewider+/ba∇dblh/ba∇dblL2(Rn\B(0,/tildewider))/parenrightbigg ,∀/tildewider≥r≥2r0. (2.25) Consequently, by passing to the limit as /tildewider→ ∞in (2.25), we deduce liminf l→∞liminf m→∞/ba∇dblS(tk)ϕk−S(tm)ϕm/ba∇dblH2(B(0,r))×L2(B(0,r))= 0,∀r>0. (2.26) Letriր ∞asi→ ∞. Takingr=riin (2.26) and using the arguments at the end of the proof of [17, Lemma 3.4], we can say that there exist subsequences/braceleftig k(i) m/bracerightig such that /braceleftig k(1) m/bracerightig ⊃/braceleftig k(2) m/bracerightig ⊃...⊃/braceleftig k(i) m/bracerightig ⊃.. and /braceleftig S(tk(i) m)ϕk(i) m/bracerightig converges in H2(B(0,ri))×L2(B(0,ri)). Thus, the diagonal subsequence/braceleftig S(tk(m) m)ϕk(m) m/bracerightig converges in H2(B(0,r))×L2(B(0,r)), for every r>0. /square To establish the tail estimate, we need the following lemma. Lemma 2.3. Let the conditions (1.3)-(1.6) hold and Bbe a bounded subset of H2(Rn).Then for every ε >0there exist a constant δ≡δ(ε)>0and functions ψε∈L∞(Rn),ϕε∈C∞(Rn), such that 0≤ψε≤min/braceleftbig 1,δ−1β/bracerightbig a.e. inRn,0≤ϕε≤1inRn, supp(ϕε)⊂ {x∈Rn:α(x)≥δa.e. inRn}and /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglef/parenleftig /ba∇dbl∇u/ba∇dblL2(Rn)/parenrightig −fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig ψε∇u/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/ba∇dbl√ϕε∇u/ba∇dbl2 L2(Rn)/parenrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<ε, (2.27) for everyu∈B,wherefδis the function defined in the proof of Lemma 2.1. Proof.LetA0={x∈B(0,r0) :α(x) = 0}andAk=/braceleftbig x∈B(0,r0) : 0≤α(x)<1 k/bracerightbig . It is easy to see thatAk+1⊂Ak, andA0=∩ k>0Ak.Hence, lim k→∞mes(Ak) =mes(A0). So, forδ >0, there exists kδsuch that mes(Akδ\A0)<δ 3. (2.28) SinceAkδisameasurablesubsetof B(0,r0),thereexistsanopenset O(1) δ⊂B(0,r0)suchthatAkδ⊂O(1) δ and mes/parenleftig O(1) δ\Akδ/parenrightig <δ 3. (2.29) Now, letηδ∈C0(Rn) such that 0 ≤ηδ≤1, ηδ|O(1) δ= 1 and supp( ηδ)⊂O(2) δ,whereO(1) δ⋐O(2) δand mes/parenleftig O(2) δ\O(1) δ/parenrightig <δ 3. (2.30) Then setting ϕδ:= 1−ηδ, we haveϕδ∈C(Rn), 0≤ϕδ≤1,ϕδ|Rn\O(2) δ= 1 and supp( ϕδ)⊂Rn\O(1) δ.10 AZER KHANMAMEDOV AND SEMA YAYLA By (2.28)-(2.30), we obtain /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay O(2) δϕδ|∇u(x)|2dx−/integraldisplay O(2) δ\A0|∇u(x)|2dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay O(2) δ\O(1) δϕδ|∇u(x)|2dx−/integraldisplay O(2) δ\A0|∇u(x)|2dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤2/integraldisplay O(2) δ\A0|∇u(x)|2dx≤2c/ba∇dblu/ba∇dbl2 H2(Rn)/parenleftig mes/parenleftig O(2) δ\A0/parenrightig/parenrightign∗ <2cδn∗/ba∇dblu/ba∇dbl2 H2(Rn), (2.31) for everyu∈H2(Rn), wheren∗= 1, n = 1, q,0<q<1, n= 2, 2 n, n ≥3andc>0. Now, by (1.5), it follows that β >0 a.e. inA0. Hence, by Lebesgue dominated convergence theorem, there exis tsλδ>0 such that /integraldisplay A0λδ λδ+β(x)dx<δ, which yields/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay A0|∇u(x)|2dx−/integraldisplay A0β(x) λδ+β(x)|∇u(x)|2dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<cδn∗/ba∇dblu/ba∇dbl2 H2(Rn). (2.32) Thus, denoting ψδ=/braceleftigg β(x) λδ+β(x),x∈A0, 0,x∈Rn\A0,by (2.31) and (2.32), we get /vextendsingle/vextendsingle/vextendsingle/vextendsingle/ba∇dbl∇u/ba∇dbl2 L2(Rn)−/vextenddouble/vextenddouble/vextenddouble/radicalbig ψδ∇u/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)−/ba∇dbl√ϕδ∇u/ba∇dbl2 L2(Rn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle <3cδn∗/ba∇dblu/ba∇dbl2 H2(Rn), and consequently/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/ba∇dbl∇u/ba∇dblL2(Rn)−/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig ψδ∇u/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/ba∇dbl√ϕδ∇u/ba∇dbl2 L2(Rn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤/radicaligg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/ba∇dbl∇u/ba∇dbl2 L2(Rn)−/vextenddouble/vextenddouble/vextenddouble/radicalbig ψδ∇u/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)−/ba∇dbl√ϕδ∇u/ba∇dbl2 L2(Rn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle <√ 3cδ1 2n∗/ba∇dblu/ba∇dblH2(Rn). The last inequality, together with the differentiability of the function f, yields (2.27). /square Now, let us proof the following tail estimate. Theorem 2.2. Assume that the conditions (1.3)-(1.8) hold and Bis a bounded subset of H2(Rn)× L2(Rn). Then for any ε>0there existT≡T(B,ε)andR≡R(B,ε)such that /ba∇dblS(t)ϕ/ba∇dblH2(Rn\B(0,r))×L2(Rn\B(0,r))<ε, for everyt≥T,r≥Randϕ∈ B.LONG-TIME DYNAMICS 11 Proof.Let (u0,u1)∈ Band (u(t),ut(t)) =S(t)(u0,u1). Multiplying (1.1) with η2 rut, integrating the obtained equality over Rnand taking into account (2.13) ,we get 1 2d dt/parenleftig /ba∇dblηrut(t)/ba∇dbl2 L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2 L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2 L2(Rn)/parenrightig +d dt /integraldisplay Rnη2 r(x)G(u(t,x))dx +/vextenddouble/vextenddouble/vextenddouble/radicalbig βηr∇ut(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/vextenddouble/vextenddouble√αηrut(t)/vextenddouble/vextenddouble2 L2(Rn) −/integraldisplay Rnf(/ba∇dbl∇u(t)/ba∇dblL2(Rn))∆uη2 rutdx ≤c2/parenleftbigg1 r+1 r/vextenddouble/vextenddouble/vextenddouble/radicalbig βηr∇ut(t)/vextenddouble/vextenddouble/vextenddouble L2(B(0,2r))+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg ,∀r≥r0. (2.33) Now, let us estimate the last term on the left hand side of (2.33). By L emma 2.3, we have −f(/ba∇dbl∇u(t)/ba∇dblL2(Rn))/integraldisplay Rn∆uη2 rutdx ≥ −ε/ba∇dblηr∆u(t)/ba∇dblL2(Rn)/ba∇dblηrut(t)/ba∇dblL2(Rn)−c3 r +1 2fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig ψε∇u(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/vextenddouble/vextenddouble/vextenddouble/radicalbig φε∇u(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)/parenrightigg d dt/parenleftig /ba∇dblηr∇(u(t))/ba∇dbl2 L2(Rn)/parenrightig .(2.34) Moreover,forthe last termon the righthandside of(2.34), byusin gthe definition of fδand the properties ofψεandϕε, we obtain 1 2fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig ψε∇u(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/ba∇dbl√ϕε∇u(t)/ba∇dbl2 L2(Rn)/parenrightigg d dt/parenleftig /ba∇dblηr∇(u(t))/ba∇dbl2 L2(Rn)/parenrightig ≥1 2d dt/parenleftigg fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig ψε∇u(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/ba∇dbl√ϕε∇u(t)/ba∇dbl2 L2(Rn)/parenrightigg /ba∇dblηr∇(u(t))/ba∇dbl2 L2(Rn)/parenrightigg −c4/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig β∇ut(t)/vextenddouble/vextenddouble/vextenddouble L2(Rn)+/vextenddouble/vextenddouble√αut(t)/vextenddouble/vextenddouble L2(Rn)/parenrightbigg /ba∇dblηr∇(u(t))/ba∇dbl2 L2(Rn). (2.35) Considering (2.34) and (2.35) in (2.33), we obtain 1 2d dt/parenleftig /ba∇dblηrut(t)/ba∇dbl2 L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2 L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2 L2(Rn)/parenrightig +d dt /integraldisplay Rnη2 r(x)G(u(t,x))dx +/vextenddouble/vextenddouble/vextenddouble/radicalbig βηr∇ut(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/vextenddouble/vextenddouble√αηrut(t)/vextenddouble/vextenddouble2 L2(Rn) +1 2d dt/parenleftigg fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig ψε∇u(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/vextenddouble/vextenddouble/vextenddouble/radicalbig φε∇u(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)/parenrightigg /ba∇dblηr∇(u(t))/ba∇dbl2 L2(Rn)/parenrightigg −c4/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig β∇ut(t)/vextenddouble/vextenddouble/vextenddouble L2(Rn)+/vextenddouble/vextenddouble√αut(t)/vextenddouble/vextenddouble L2(Rn)/parenrightbigg /ba∇dblηr∇(u(t))/ba∇dbl2 L2(Rn). −ε/ba∇dblηr∆u(t)/ba∇dblL2(Rn)/ba∇dblηrut(t)/ba∇dblL2(Rn) ≤c2/parenleftbigg1 r+1 r/vextenddouble/vextenddouble/vextenddouble/radicalbig βηr∇ut(t)/vextenddouble/vextenddouble/vextenddouble L2(B(0,2r))+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg . (2.36) Multiplying (1.1) with µη2 ru, integrating the obtained equality over Rnand taking into account (1.6), (1.8) and (2.13), we get µγ/ba∇dblηr∆u(t)/ba∇dbl2 L2(Rn)+µλ/ba∇dblηru(t)/ba∇dbl2 L2(Rn)−µ/ba∇dblηrut(t)/ba∇dbl2 L2(Rn) +µ 2d dt/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig βηr∇u(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/vextenddouble/vextenddouble√αηru(t)/vextenddouble/vextenddouble2 L2(Rn)/parenrightbigg12 AZER KHANMAMEDOV AND SEMA YAYLA +µd dt /integraldisplay Rnη2 r(x)ut(t,x)u(t,x)dx ≤c5/parenleftbigg1 r+1 r/vextenddouble/vextenddouble/vextenddouble/radicalbig βηr∇ut(t)/vextenddouble/vextenddouble/vextenddouble L2(B(0,2r))+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg . (2.37) Summing (2.36) and (2.37), applying Young inequality and choosing εandµsmall enough, we obtain d dt /ba∇dblηrut(t)/ba∇dbl2 L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2 L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2 L2(Rn)+/integraldisplay Rn\B(0,r)η2 r(x)G(u(t,x))dx +1 2d dt/parenleftigg fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig ψε∇u(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/vextenddouble/vextenddouble/vextenddouble/radicalbig φε∇u(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)/parenrightigg /ba∇dblηr∇(u(t))/ba∇dbl2 L2(Rn)/parenrightigg +µ 2d dt/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig βηr∇u(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/vextenddouble/vextenddouble√αηru(t)/vextenddouble/vextenddouble2 L2(Rn)/parenrightbigg +c6/parenleftig /ba∇dblηrut(t)/ba∇dbl2 L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2 L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2 L2(Rn)/parenrightig ≤c7/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig β∇ut(t)/vextenddouble/vextenddouble/vextenddouble L2(Rn)+/vextenddouble/vextenddouble√αut(t)/vextenddouble/vextenddouble L2(Rn)/parenrightbigg /ba∇dblηr∇(u(t))/ba∇dbl2 L2(Rn) +c7/parenleftbigg1 r+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg ,∀r≥r0, whereci(i= 6,7) are positive constants. By denoting Φ(t) :=/ba∇dblηrut(t)/ba∇dbl2 L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2 L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2 L2(Rn)+/integraldisplay Rnη2 r(x)G(u(t,x))dx +1 2fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig ψε∇u(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/vextenddouble/vextenddouble/vextenddouble/radicalbig φε∇u(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)/parenrightigg /ba∇dblηr∇(u(t))/ba∇dbl2 L2(Rn) +µ 2/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig βηr∇u(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn\B(0,r))+/vextenddouble/vextenddouble√αηru(t)/vextenddouble/vextenddouble2 L2(Rn\B(0,r))/parenrightbigg , we get d dtΦ(t)+c6/parenleftig /ba∇dblηrut(t)/ba∇dbl2 L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2 L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2 L2(Rn)/parenrightig ≤c7/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig β∇ut(t)/vextenddouble/vextenddouble/vextenddouble L2(Rn)+/vextenddouble/vextenddouble√αut(t)/vextenddouble/vextenddouble L2(Rn)/parenrightbigg /ba∇dblηr∇(u(t))/ba∇dbl2 L2(Rn) +c7/parenleftbigg1 r+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg ,∀r≥r0. (2.38) Moreover, there exist /hatwidec≡/hatwidec(B)>0 such that /ba∇dblηrut(t)/ba∇dbl2 L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2 L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2 L2(Rn) ≤Φ(t)≤/hatwidec/parenleftig /ba∇dblηrut(t)/ba∇dbl2 L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2 L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2 L2(Rn)/parenrightig . (2.39) So, considering (2.39) in (2.38), we have d dtΦ(t)+H(t)Φ(t) ≤c7/parenleftbigg1 r+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg , whereH(t) =c8−c7/parenleftig/vextenddouble/vextenddouble√β∇ut(t)/vextenddouble/vextenddouble L2(Rn)+/ba∇dbl√αut(t)/ba∇dblL2(Rn)/parenrightig andc8>0. Then, by Gronwallinequality, we obtain Φ(t)≤e−/integraltextt 0H(τ)dτΦ(0)+c7/parenleftbigg1 r+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg/integraldisplayt 0e−/integraltextt τH(σ)dσdτ. (2.40)LONG-TIME DYNAMICS 13 Furthermore, applying Young inequality and taking into account (1.9 ), we have e−/integraltextt τH(σ)dσ≤e−1 2c8(t−τ)+c9/integraltextt τ/parenleftBig /ba∇dbl√β∇ut(t)/ba∇dbl2 L2(Rn)+/ba∇dbl√αut(t)/ba∇dbl2 L2(Rn)/parenrightBig dσ ≤c10e−1 2c8(t−τ),∀t≥τ≥0. (2.41) Therefore, considering (2.41) in (2.40), we get Φ(t)≤c10e−1 2c8tΦ(0)+c11/parenleftbigg1 r+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg ,∀t≥0, which completes the proof of the theorem. /square Now, we are in a position to prove the existence of the global attrac tor. Theorem 2.3. Let the conditions (1.3)-(1.8) hold. Then the semigroup {S(t)}t≥0generated by the problem (1.1)-(1.2) possesses a global attractor AinH2(Rn)×L2(Rn)andA=Mu(N). Proof.By Theorem 2.1 and Theorem 2.2, it follows that every sequence of th e form{S(tk)ϕk}∞ k=1, where {ϕk}∞ k=1⊂ B,tk→ ∞,andBis bounded subset of H2(Rn)×L2(Rn), has a convergent subsequence inH2(Rn)×L2(Rn). Since, by (1.6) and (1.8), the set N, which is the set of stationary points of {S(t)}t≥0is bounded in H2(Rn)×L2(Rn), to complete the proof, it is enough to show that the pair/parenleftbig S(t),H2(Rn)×L2(Rn)/parenrightbig is a gradient system (see [14]). Now, for (u(t),ut(t)) =S(t)(u0,u1), let the equality L(u(t),ut(t)) =L(u0,u1),∀t≥0, hold, where L(u,v) =1 2/integraltext Rn(|v(x)|2+γ|∆u(x)|2+λ|u(x)|2)dx+/integraltext RnG(u(x))dx+1 2F/parenleftig /ba∇dbl∇u/ba∇dbl2 L2(Rn)/parenrightig − /integraltext Rnh(x)u(x)dx.Then considering (1.3) and (1.9), we have αut(t,·) = 0 andβ∇ut(t,·) = 0 a.e. in Rn, fort≥0.Taking into account (1.5), from the above equalities, it follows that ut(t,·)utxi(t,·) = 0 a.e. in Rn, and consequently ∂ ∂xi/parenleftbig u2 t(t,·)/parenrightbig = 0 a.e. in Rn, fori=1,nandt≥0.The last equality means that u2 t(t,·) is independent of variable x, for everyt≥0. Hence, byut(t,·)∈L2(Rn), we have ut(t,·) = 0 a.e. in Rn, fort≥0. So, (u(t),ut(t)) = (ϕ,0),∀t≥0, where (ϕ,0)∈ N. Thus, the pair/parenleftbig S(t),H2(Rn)×L2(Rn)/parenrightbig is a gradient system. /square 3.Regularity of the global attractor We start with the following lemma. Lemma 3.1. Let the condition (1.7) hold and Kbe a compact subset of H2(Rn). Then for every ε>0 there exists a constant Cǫ>0such that /ba∇dblg(u1)−g(u2)/ba∇dblL2(Rn)≤ε/ba∇dblu1−u2/ba∇dblH2(Rn)+Cǫ/ba∇dblu1−u2/ba∇dblL2(Rn), (3.1) for everyu1,u2∈K.14 AZER KHANMAMEDOV AND SEMA YAYLA Proof.ByMeanValueTheorem,H¨ olderinequalityandtheembedding H2(Rn)֒→L2n (n−4)+(Rn)∩L2(Rn), we have /ba∇dblg(u)−g(v)/ba∇dbl2 L2(Rn)=/integraldisplay Rn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/integraldisplay 0g′(τu(x)+(1−τ)v(x))dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 |u(x)−v(x)|2dx ≤1/integraldisplay 0/integraldisplay {x∈Rn:|τu(x)+(1−τ)v(x)|>M}|g′(τu(x)+(1−τ)v(x))|2|u(x)−v(x)|2dxdτ +/ba∇dblg′/ba∇dblC[−M,M]/ba∇dblu−v/ba∇dbl2 L2(Rn) ≤c1/integraldisplay {x∈Rn:|u(x)|+|v(x)|>M}/parenleftig 1+|u(x)|2(p−1)+|v(x)|2(p−1)/parenrightig |u(x)−v(x)|2dx +/ba∇dblg′/ba∇dblC[−M,M]/ba∇dblu−v/ba∇dbl2 L2(Rn) ≤c2 /integraldisplay {x∈Rn:|u(x)|+|v(x)|>M}/parenleftig 1+|u(x)|2(p−1)q+|v(x)|2(p−1)q/parenrightig dx 1 q ×/ba∇dblu−v/ba∇dblH2(Rn)+/ba∇dblg′/ba∇dblC[−M,M]/ba∇dblu−v/ba∇dbl2 L2(Rn), (3.2) whereq= max/braceleftbig 1,n 4/bracerightbig . Since, by (1.7), H2(Rn)֒→L2(p−1)q(Rn), we have that Kis compact subset of L2(p−1)q(Rn).Hence, lim M→∞sup u,v∈K/integraldisplay {x∈Rn:|u(x)|+|v(x)|>M}/parenleftig 1+|u(x)|2(p−1)q+|v(x)|2(p−1)q/parenrightig dx= 0. (3.3) Thus, (3.2) and (3.3) give us (3.1). /square Theorem 3.1. The global attractor Ais bounded in H3(Rn)×H2(Rn). Proof.Letϕ∈ A. SinceAis invariant, there exists an invariant trajectory Γ = {(u(t),ut(t)) :t∈R} ⊂ Asuch that (u(0),ut(0)) =ϕ(see [18, p. 159]). Now, let us define v(t,x) :=u(t+σ,x)−u(t,x) σ,σ>0. Then, by (1.1), we get vtt(t,x)+γ∆2v(t,x)−div(β(x)∇vt)+α(x)vt(t,x)+λv(t,x) −f/parenleftig /ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightig ∆v(t,x)−f(/ba∇dbl∇u(t+σ)/ba∇dblL2(Rn))−f/parenleftig /ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightig σ∆u(t+σ,x) +g(u(t+σ,x))−g(u(t,x)) σ= 0, (t,x)∈R×Rn. (3.4) Multiplying (3.4) by vtand integrating the obtained equality over Rn, we find d dtE(v(t))+/vextenddouble/vextenddouble/vextenddouble/radicalbig β∇vt(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/vextenddouble/vextenddouble√αvt(t)/vextenddouble/vextenddouble2 L2(Rn) ≤ −1 2f/parenleftig /ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightigd dt/parenleftig /ba∇dbl∇v(t)/ba∇dbl2 L2(Rn)/parenrightig +f(/ba∇dbl∇u(t+σ)/ba∇dblL2(Rn))−f/parenleftig /ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightig σ ×/integraldisplay Rn∆u(t+σ,x)vt(t,x)dx−1 σ/integraldisplay Rn(g(u(t+σ,x))−g(u(t,x)))vt(t,x)dx ≤ −1 2f/parenleftig /ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightigd dt/parenleftig /ba∇dbl∇v(t)/ba∇dbl2 L2(Rn)/parenrightig +c1/ba∇dbl∇v(t)/ba∇dblL2(Rn)/ba∇dblvt(t)/ba∇dblL2(Rn)+1 σ/ba∇dblg(u(t+σ))−g(u(t))/ba∇dblL2(Rn)/ba∇dblvt(t)/ba∇dblL2(Rn).LONG-TIME DYNAMICS 15 Taking into account Lemma 3.1 in the last inequality, we obtain d dtE(v(t))+/vextenddouble/vextenddouble/vextenddouble/radicalbig β∇vt(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/vextenddouble/vextenddouble√αvt(t)/vextenddouble/vextenddouble2 L2(Rn) ≤ −1 2f/parenleftig /ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightigd dt/parenleftig /ba∇dbl∇v(t)/ba∇dbl2 L2(Rn)/parenrightig +c1/ba∇dbl∇v(t)/ba∇dblL2(Rn)/ba∇dblvt(t)/ba∇dblL2(Rn)+/parenleftig ε/ba∇dblv(t)/ba∇dblH2(Rn)+Cε/ba∇dblv(t)/ba∇dblL2(Rn)/parenrightig /ba∇dblvt(t)/ba∇dblL2(Rn),(3.5) for anyε>0. Moreover, by (2.13), we have /ba∇dblv(t)/ba∇dblL2(Rn)=/vextenddouble/vextenddouble/vextenddouble/vextenddoubleu(t+σ,x)−u(t,x) σ/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(Rn)≤sup 0≤t<∞/ba∇dblut(t)/ba∇dblL2(Rn)</hatwideC,∀t∈R. (3.6) Then, considering (3.6) in (3.5), we get d dtE(v(t))+/vextenddouble/vextenddouble/vextenddouble/radicalbig β∇vt(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/vextenddouble/vextenddouble√αvt(t)/vextenddouble/vextenddouble2 L2(Rn) ≤ −1 2f/parenleftig /ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightigd dt/parenleftig /ba∇dbl∇v(t)/ba∇dbl2 L2(Rn)/parenrightig +/parenleftig c2/ba∇dblv(t)/ba∇dbl1 2 H2(Rn)+ε/ba∇dblv(t)/ba∇dblH2(Rn)+/tildewiderCε/parenrightig /ba∇dblvt(t)/ba∇dblL2(Rn). (3.7) Now, let us estimate the first term on the right hand side of (3.7). By (2.13) and (3.6), we have −1 2f/parenleftig /ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightigd dt/parenleftig /ba∇dbl∇v(t)/ba∇dbl2 L2(Rn)/parenrightig ≤c3max 0≤s1,s2≤ε|f(s1)−f(s2)|/ba∇dblvt(t)/ba∇dblL2(Rn)/ba∇dbl∆v(t)/ba∇dblL2(Rn) −1 2fε/parenleftig /ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightigd dt/parenleftig /ba∇dbl∇v(t)/ba∇dbl2 L2(Rn)/parenrightig ≤c3max 0≤s1,s2≤ε|f(s1)−f(s2)|/ba∇dblvt(t)/ba∇dblL2(Rn)/ba∇dbl∆v(t)/ba∇dblL2(Rn) −1 2d dt/parenleftig fε/parenleftig /ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightig /ba∇dbl∇v(t)/ba∇dbl2 L2(Rn)/parenrightig +c4/ba∇dblv(t)/ba∇dblH2(Rn), (3.8) for anyε>0, wherefεis the function defined in the proof of Lemma 2.1. Considering (3.8) in ( 3.7), we obtain d dt/parenleftbigg E(v(t))+1 2fε/parenleftig /ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightig /ba∇dbl∇v(t)/ba∇dbl2 L2(Rn)/parenrightbigg +/vextenddouble/vextenddouble/vextenddouble/radicalbig β∇vt(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn)+/vextenddouble/vextenddouble√αvt(t)/vextenddouble/vextenddouble2 L2(Rn) ≤c3max 0≤s1,s2≤ε|f(s1)−f(s2)|/ba∇dblvt(t)/ba∇dblL2(Rn)/ba∇dbl∆u(t)/ba∇dblL2(Rn)+c4/ba∇dblv(t)/ba∇dblH2(Rn) +/parenleftig c2/ba∇dblv(t)/ba∇dbl1 2 H2(Rn)+ε/ba∇dblv(t)/ba∇dblH2(Rn)+/tildewiderCε/parenrightig /ba∇dblvt(t)/ba∇dblL2(Rn). (3.9) Letηr(x) be the cut-off function defined in the proof of Theorem 2.1. Multiply ing (3.4) by/summationtextn i=1xi(1−η2r0)vxi+1 2(n−1)(1−η2r0)v, and integrating over Rn, by (2.13) and (3.6), we get 3 2γ/ba∇dbl∆(v(t))/ba∇dbl2 L2(B(0,2r0))+1 2/ba∇dblvt(t)/ba∇dbl2 L2(B(0,2r0)) +d dt /summationdisplayn i=1/integraldisplay Rnxi(1−η2r0(x))vxi(t,x)vt(t,x)dx+1 2(n−1)/integraldisplay Rn(1−η2r0(x))vt(t,x)v(t,x)dx ≤c5/ba∇dblvt(t)/ba∇dbl2 L2(B(0,4r0)\B(0,2r0))+c5/ba∇dbl∆v(t)/ba∇dbl2 L2(B(0,4r0)\B(0,2r0))+c5/ba∇dblv(t)/ba∇dbl1 2 H2(B(0,4r0)) +c5/vextenddouble/vextenddouble/vextenddouble/radicalbig β∇vt(t)/vextenddouble/vextenddouble/vextenddouble L2(B(0,4r0))/parenleftig /ba∇dblv(t)/ba∇dbl1 2 H2(B(0,4r0))+/ba∇dblv(t)/ba∇dblH2(B(0,4r0))/parenrightig +c5/vextenddouble/vextenddouble√αvt(t)/vextenddouble/vextenddouble L2(B(0,4r0))/parenleftig /ba∇dblv(t)/ba∇dbl1 2 H2(B(0,4r0))+1/parenrightig +c5/ba∇dblv(t)/ba∇dbl3 2 H2(B(0,4r0))+c5. (3.10)16 AZER KHANMAMEDOV AND SEMA YAYLA Multiplying (3.4) by η2 r0vand integrating over Rn, we find d dt /integraldisplay Rnη2 r0v(t,x)vt(t,x)dx+1 2/vextenddouble/vextenddouble√αηr0v(t)/vextenddouble/vextenddouble2 L2(Rn)+1 2/vextenddouble/vextenddouble/vextenddouble/radicalbig βηr0∇v(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn) +/ba∇dbl∆v(t)/ba∇dbl2 L2(Rn\B(0,r0))+λ/ba∇dblv(t)/ba∇dbl2 L2(Rn\B(0,r0))−/ba∇dblvt(t)/ba∇dbl2 L2(Rn\B(0,r0)) ≤c6/ba∇dblv(t)/ba∇dbl3 2 H2(Rn)+c6/vextenddouble/vextenddouble/vextenddouble/radicalbig β∇vt(t)/vextenddouble/vextenddouble/vextenddouble L2(Rn)/parenleftig 1+/ba∇dblv(t)/ba∇dbl1 2 H2(B(0,4r0))/parenrightig +c6. (3.11) Multiplying (3.10) and (3.11) by δ2andδ, respectively, then summing the obtained inequalities with (3.9), choosing ε>0 andδ>0 sufficiently small and applying Young inequality, we get d dtΨ(t)+c7E(v(t))≤c8,∀t∈R, (3.12) where Ψ(t) :=E(v(t))+1 2fε/parenleftig /ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightig /ba∇dbl∇v(t)/ba∇dbl2 L2(Rn) +δ /integraldisplay Rnη2 r0v(t,x)vt(t,x)dx+1 2/vextenddouble/vextenddouble√αηr0v(t)/vextenddouble/vextenddouble2 L2(Rn)+1 2/vextenddouble/vextenddouble/vextenddouble/radicalbig βηr0∇v(t)/vextenddouble/vextenddouble/vextenddouble2 L2(Rn) +δ2 /summationdisplayn i=1/integraldisplay Rnxi(1−η2r0(x))vxi(t,x)vt(t,x)dx+1 2(n−1)/integraldisplay Rn(1−η2r0(x))vt(t,x)v(t,x)dx , and the positive constant c8, as the previous ci/parenleftbig i=1,7/parenrightbig , is independent of the trajectory Γ . Sinceδ>0 is sufficiently small, there exist constants c>0,/tildewidec>0 such that cE(v(t))≤Ψ(t)≤/tildewidecE(v(t)),∀t∈R. (3.13) Taking into account (3.13) in (3.12), we obtain d dtΨ(t)+c9Ψ(t)≤c8,∀t∈R, which yields Ψ(t)≤e−c9(t−s)Ψ(s)+c8 c9,∀t≥s. Passing to the limit as s→ −∞and considering (3.13), we get E(v(t))≤c10,∀t∈R. By using the definition of v, after passing to the limit as σ→0 in the last inequality, we find E(ut(t))≤c10,∀t∈R. (3.14) Considering (3.14) in (1.1), we obtain /ba∇dblu(t)/ba∇dblH3(Rn)≤c11,∀t∈R. Thus, the last inequality, together with (3.14), yields /ba∇dblϕ/ba∇dblH3(Rn)×H2(Rn)≤c12,∀ϕ∈ A, which completes the proof of the theorem. /squareLONG-TIME DYNAMICS 17 References [1] A. Pazy, Semigroups of linear operators and application s to partial di erential equations, Springer, New York, 1983 . [2] T. Cazenave and A. Haraux, An introduction to semilinear evolution equations, Oxford University Press, New York, 1998. [3] E. Feireisl, Attractors for semilinear damped wave equa tions on R3, Nonlinear Anal., 23 (1994) 187–195. [4] E. Feireisl, Asymptotic behavior and attractors for a se milinear damped wave equation with supercritical exponent , Proc. Roy. Soc. Edinburgh, 125 (1995) 1051–1062. [5] V. Belleri and V. Pata, Attractors for semilinear strong ly damped wave equations on R3, Discrete Contin. Dyn. Syst, 7 (2001) 719–735. [6] M. Conti, V. Pata and M. Squassina, Strongly damped wave e quations on R3with critical nonlinearities, Commun. Appl. Anal., 9 (2005) 161–176. [7] Z. Yang and P. Ding, Longtime dynamics of the Kirchhoff equ ation with strong damping and critical nonlinearity on RN, J. Math. Anal. Appl .,434 (2016) 1826-1851. [8] A.Kh. Khanmamedov, Existence of a global attractor for t he plate equation with a critical exponent in an unbounded domain, Applied Mathematics Letters, 18 (2005) 827-832. [9] A.Kh. Khanmamedov, Global attractors for the plate equa tion with a localized damping and a critical exponent in an unbounded domain, J.Differential Equations, 225 (2006) 528 -548. [10] B. Wang, Attractors for reaction diffusion equations in unbounded domains, Phys. D, 128 (1999) 41–52. [11] J.M. Ball, Global attractors for damped semilinear wav e equations, Discrete Contin. Dyn. Syst., 10 (2004) 31–52. [12] Z. Arat, A. Khanmamedov and S. Simsek, Global attractor s for the plate equation with nonlocal nonlinearity in unbounded domains, Dyn. Partial Differ. Equ., 11 (2014) 361– 379. [13] A. Khanmamedov and S. Simsek, Existence of the global at tractor for the plate equation with nonlocal nonlinearity i n Rn, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016) 151–172. [14] I. Chueshov and I. Lasiecka, Von Karman Evolution Equat ions:Well-posedness and long-time dynamics, Springer, Ne w York, 2010. [15] J. Simon, Compact sets in the space Lp(0,T;B), Annali Mat. Pura Appl., 146 (1987) 65-96. [16] A.Kh. Khanmamedov, Global attractors for von Karman eq uations with nonlinear interior dissipation, J. Math. Anal . Appl., 318 (2006) 92–101. [17] A.Kh.Khanmamedov, Global attractors for2-D wave equa tions with displacement dependent damping, Math. Methods Appl. Sci., 33 (2010) 177-187. [18] A.V. Babin and M.I. Vishik, Attractors of evolution equ ations, Amsterdam: North-Holland, 1992. Department of Mathematics, Faculty of Science, Hacettepe U niversity, Beytepe 06800 ,Ankara, Turkey E-mail address :azer@hacettepe.edu.tr Department of Mathematics, Faculty of Science, Hacettepe U niversity, Beytepe 06800 ,Ankara, Turkey E-mail address :semasimsek@hacettepe.edu.tr

2017-03-09

We investigate the initial-value problem for the semilinear plate equation containing localized strong damping, localized weak damping and nonlocal nonlinearity. We prove that if nonnegative damping coefficients are strictly positive almost everywhere in the exterior of some ball and the sum of these coefficients is positive a.e. in $% \mathbb{R} ^{n}$, then the semigroup generated by the considered problem possesses a global attractor in $H^{2}\left( \mathbb{R} ^{n}\right) \times L^{2}\left( \mathbb{R} ^{n}\right) $. We also establish boundedness of this attractor in $ H^{3}\left( \mathbb{R} ^{n}\right) \times H^{2}\left( \mathbb{R} ^{n}\right) $.

Long-time dynamics of the strongly damped semilinear plate equation in $\mathbb{R}^{n}$

1703.03485v2

1 Magnetic Skyrmion Transport in a Nanotrack With Spatially Varying Damping and Non-adiabatic Torque Xichao Zhang1,2, Jing Xia1, G. P. Zhao3, Xiaoxi Liu4, and Yan Zhou1 1School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172, China 2School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China 3College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China 4Department of Information Engineering, Shinshu University, Wakasato 4-17-1, Nagano 380-8553, Japan Reliable transport of magnetic skyrmions is required for any future skyrmion-based information processing devices. Here we present a micromagnetic study of the in-plane current-driven motion of a skyrmion in a ferromagnetic nanotrack with spatially sinusoidally varying Gilbert damping and/or non-adiabatic spin-transfer torque coefficients. It is found that the skyrmion moves in a sinusoidal pattern as a result of the spatially varying Gilbert damping and/or non-adiabatic spin-transfer torque in the nanotrack, which could prevent the destruction of the skyrmion caused by the skyrmion Hall effect. The results provide a guide for designing and developing the skyrmion transport channel in skyrmion-based spintronic applications. Index Terms —magnetic skyrmions, racetrack memories, micromagnetics, spintronics. I. I NTRODUCTION Magnetic skyrmions are quasiparticle-like domain-wall structures with typical sizes in the sub-micrometer regime [1]– [7]. They are theoretically predicted to exist in magnetic metals having antisymmetric exchange interactions [8], and confirmed by experiments [9], [10] just after the turn of the twenty- first century. Isolated skyrmions are expected to be used to encode information into bits [11], which might lead to the development of novel spintronic applications, such as the racetrack memories [12]–[19], storage devices [20]–[22], and logic computing devices [23]. The write-in and read-out processes of skyrmions in thin films are realizable and controllable at low temperatures [24]– [26]. A recent experiment has realized the current-induced creation and motion of skyrmions in Ta/CoFeB/TaO trilayers at room temperature [27]. Experimental investigations have also demonstrated the increased stability of skyrmions in mul- tilayers [28]–[30], which makes skyrmions more applicable to practical room-temperature applications. However, the skyrmion experiences the skyrmion Hall effect (SkHE) [31], [32], which drives it away from the longitudinal direction when it moves in a narrow nanotrack. As a con- sequence, in the high-speed operation, the transverse motion of a skyrmion may result in its destruction at the nanotrack edges [18], [33]–[36]. Theoretical and numerical works have proposed several intriguing methods to reduce or eliminate the detrimental transverse motion caused by the SkHE. For ex- ample, one could straightforwardly enhance the perpendicular magnetic anisotropy near the nanotrack edges to better confine the skyrmion motion [33]. An alternative solution is to trans- port skyrmions on periodic substrates [37]–[40], where the skyrmion trajectory can be effectively controlled. Moreover, by constructing antiferromagnetic skyrmions [34], [35] and anti- The first two authors contributed equally to this work. Corre- sponding authors: X. Liu (email: liu@cs.shinshu-u.ac.jp) and Y . Zhou (email: zhouyan@cuhk.edu.cn).ferromagnetically exchange-coupled bilayer skyrmions [18], [36], the SkHE can be completely suppressed. Recently, it is also found that the skyrmionium can perfectly move along the driving force direction due to its spin texture with a zero skyrmion number [41], [42]. In this paper, we propose and demonstrate that a skyrmion guide with spatially sinusoidally varying Gilbert damping and/or non-adiabatic spin-transfer torque (STT) coefficients can be designed for transporting skyrmions in a sinusoidal manner, which is inspired by a recent study on the magnetic vortex guide [43], where the vortex core motion is controlled via spatially varying Gilbert damping coefficient. The results provide a guide for designing and developing the skyrmion transport channel in future spintronic devices based on the manipulation of skyrmions. II. M ETHODS Our simulation model is an ultra-thin ferromagnetic nan- otrack with the length land the width w, where the thick- ness is fixed at 1nm. We perform the simulation using the standard micromagnetic simulator, i.e., the 1.2 alpha 5 release of the Object Oriented MicroMagnetic Framework (OOMMF) [44]. The simulation is accomplished by a set of built-in OOMMF extensible solver (OXS) objects. We employ the OXS extension module for modeling the interface-induced antisymmetric exchange interaction, i.e., the Dzyaloshinskii- Moriya interaction (DMI) [45]. In addition, we use the updated OXS extension module for simulating the in-plane current- induced STTs [46]. The in-plane current-driven magnetization dynamics is governed by the Landau-Lifshitz-Gilbert (LLG) equation augmented with the adiabatic and non-adiabatic STTs [44], [47] dM dt= 0MHeff+ MS(MdM dt) (1) +u M2 S(M@M @xM)u MS(M@M @x);arXiv:1607.04983v3 [cond-mat.mes-hall] 15 Dec 20162 Fig. 1. (a) The magnetic damping coefficient (x)and non-adiabatic STT coefficient(x)as functions of xin the nanotrack. (b) Trajectories of current- driven skyrmions with ==2 = 0:15,== 0:3, and= 2= 0:6. Dot denotes the skyrmion center. Red cross indicates the skyrmion destruction. (c) Skyrmion Hall angle as a function of xfor skyrmion motion with = =2 = 0:15,== 0:3, and= 2= 0:6. The dashed lines indicate  =14. (e) Real-space top-views of skyrmion motion with ==2 = 0:15,== 0:3, and= 2= 0:6.wandvdenote the nanotrack width and velocity direction, respectively. The dashed line indicates the central line of the nanotrack. The skyrmion is destroyed at t= 870 ps when= 2= 0:6. The out-of-plane magnetization component is represented by the red (z)-white ( 0)-green ( +z) color scale. where Mis the magnetization, MSis the saturation magne- tization,tis the time, 0is the Gilbert gyromagnetic ratio, is the Gilbert damping coefficient, and is the strength of the non-adiabatic STT. The adiabatic STT coefficient is given byu, i.e., the conduction electron velocity. The effective field Heffis expressed as Heff=1 0@E @M; (2) where0is the vacuum permeability constant. The average energy density Econtains the exchange, anisotropy, demag- netization, and DMI energies, which is given as E=A[r(M MS)]2K(n
M)2 M2 S0 2M
Hd(M) (3) +D M2 S(Mz@Mx @x+Mz@My @yMx@Mz @xMy@Mz @y); whereA,K, andDare the exchange, anisotropy, and DMI energy constants, respectively. nis the unit surface normal vector, and Hd(M)is the demagnetization field. Mx,My andMzare the three Cartesian components of M. The model is discretized into tetragonal volume elements with the size of 2nm2nm1nm, which ensures a good compromise between the computational accuracy and ef- ficiency. The magnetic parameters are adopted from Refs. [14], Fig. 2. (a)vx, (b)vy, and (c) as functions of andgiven by Eq. (11) and Eq. (12), respectively. vxandvyare reduced by u. [23]: 0= 2:211105m/(A
s),A= 15 pJ/m,D= 3mJ/m2, K= 0:8MJ/m3,MS= 580 kA/m. In all simulations, we assumeu= 100 m/s andw= 50 nm. The skyrmion is initially located at the position of x= 100 nm,y= 25 nm. The Gilbert damping coefficient is defined as a function of the longitudinal coordinate xas follows [Fig. 1(a)] (x) =amp
f1 + sin [2(x=)]g+min; (4) whereamp= (maxmin)=2is the amplitude of the  function.maxandminstand for the maximum and mini- mum values of the function, respectively. denotes the wavelength of the function. It is worth mentioning that the spatially varying can be achieved by gradient doping of lanthanides impurities in ferromagnets [43], [48], [49]. Exper- iments have found that is dependent on the interface [50]. Thus it is also realistic to construct the varying by techniques such as interface engineering. Indeed, as shown in Ref. [51], local control of in a ferromagnetic/non-magnetic thin-film bilayer has been experimentally demonstrated by interfacial intermixing induced by focused ion-beam irradiation. In a similar way, the non-adiabatic STT coefficient is also defined as a function of the longitudinal coordinate xas follows [Fig. 1(a)] (x) =amp
f1 + sin [2(x=)']g+min;(5) whereamp= (maxmin)=2is the amplitude of the  function.maxandminstand for the maximum and minimum values of the function, respectively. and'denote the wavelength and phase of the function, respectively. Since the value of depends on the material properties [52], it is expected to realize the spatial varying by constructing a superlattice nanotrack using different materials, similar to the model given in Ref. [43]. Note that the effect of varying  has also been studied in spin torque oscillators [53]. III. R ESULTS A. Nanotrack with spatially uniform and We first recapitulate the in-plane current-driven skyrmion motion in a nanotrack with spatially uniform and. As shown in Fig. 1(b), the skyrmion moves along the central line of the nanotrack when == 0:3. However, due to the SkHE, it shows a transverse shift toward the upper and lower edges when = 2= 0:6and==2 = 0:15, respectively.3 The skyrmion is destroyed by touching the upper edge when = 2= 0:6att= 870 ps. The skyrmion Hall angle , which characterizes the trans- verse motion of the skyrmion caused by the SkHE, is defined as  = tan1(vy=vx): (6) Figure 1(c) shows as a function of xfor the skyrmion motion with==2 = 0:15,== 0:3, and= 2= 0:6. It can be seen that  = 0when== 0:3, indicating the moving skyrmion has no transverse motion [Fig. 1(d)]. When==2 = 0:15,increases from15to0, indicating the moving skyrmion has a transverse shift toward the lower edge which is balanced by the transverse force due to the SkHE and the edge-skyrmion repulsive force [Fig. 1(d)]. When= 2= 0:6,decreases from 15to3within 870 ps, indicating the moving skyrmion shows a transverse motion toward the upper edge. At t= 870 ps, the skyrmion is destroyed as it touches the upper edge of the nanotrack [Fig. 1(d)]. It should be noted that the skyrmion profile is rigid before it touches the nanotrack edge. In order to better understand the transverse motion caused by the SkHE, we also analyze the in-plane current-driven skyrmion motion using the Thiele equation [54]–[57] by assuming the skyrmion moves in an infinite film, which is expressed as G(vu) +D(uv) =0; (7) where G= (0;0;4Q)is the gyromagnetic coupling vector with the skyrmion number Q=1 4Z m
@m @x@m @y dxdy: (8) m=M=M Sis the reduced magnetization and Dis the dissipative tensor D= 4DxxDxy DyxDyy : (9) u= (u;0)is the conduction electron velocity, and vis the skyrmion velocity. For the nanoscale skyrmion studied here, we have Q=1;Dxx=Dyy= 1;Dxy=Dyx= 0: (10) Hence, the skyrmion velocity is given as vx=u(+ 1) 2+ 1; vy=u() 2+ 1: (11) The skyrmion Hall angle is thus given as  = tan1(vy=vx) = tan1 + 1 : (12) By calculating Eq. (11), we show vxas functions of and in Fig. 2(a). vxranges between 0:5uand1:21u, indicating the skyrmion always moves in the +xdirection. When = 0:42 and= 1,vxcan reach the maximum value of vx= 1:21u. Similarly, we show vyas functions of andin Fig. 2(b). vyranges between0:5uandu, indicating the skyrmion can move in both the ydirections. When  <  ,vy>0, the skyrmion shows a positive transverse motion, while when > ,vy<0, the skyrmion shows a negative transverse motion. Fig. 3. (a) Trajectories of current-driven skyrmions with amp = 0:315;0:225;0:215.= 2wand= 0:3. (b) as a function of x for skyrmion motion with amp= 0:315;0:225;0:215.= 2wand = 0:3. (c) Trajectories of current-driven skyrmions with =w;2w;4w. amp= 0:225 and= 0:3. (d)as a function of xfor skyrmion motion with=w;2w;4w.amp= 0:225 and= 0:3. Fig. 4. (a) Trajectories of current-driven skyrmions with amp = 0:315;0:225;0:215.= 2w,'= 0, and= 0:3. (b) as a function ofxfor skyrmion motion with amp= 0:315;0:225;0:215.= 2w, '= 0 , and= 0:3. (c) Trajectories of current-driven skyrmions with =w;2w;4w.amp= 0:225,'= 0, and= 0:3. (d)as a function ofxfor skyrmion motion with =w;2w;4w.amp= 0:225,'= 0, and = 0:3. By calculating Eq. (12), we also show as functions of  andin Fig. 2(c), where varies between  = 45and  =45. Obviously, one has  = 0,<0, and >0 for=,> , and< , respectively, which agree with the simulation results for the nanotrack when the edge effect is not significant, i.e., when the skyrmion moves in the interior of the nanotrack. For example, using Eq. (12), the skyrmion has = 14and =14for= 2= 0:6and= =2 = 0:15, respectively, which match the simulation results att0ps where the edge effect is negligible [Fig. 1(c)]. B. Nanotrack with spatially varying or We first demonstrate the in-plane current-driven skyrmion motion in a nanotrack with spatially varying and spatially uniform, i.e.,is a function of x, as in Eq. (4), and = 0:3. Figure 3(a) shows the trajectories of the current-driven skyrmions with different (x)functions where = 2wand = 0:3. Formax= 0:75,min= 0:12, i.e.,amp= 0:315, the skyrmion moves in the rightward direction in a sinusoidal pattern. For max= 0:6,min= 0:15, i.e.,amp= 0:225, the maximum transverse shift of skyrmion is reduced in compared to that ofamp= 0:315. Formax= 0:45,min= 0:2, i.e., amp= 0:125, the amplitude of the skyrmion trajectory further4 Fig. 5. Trajectories of current-driven skyrmions with '= 02.amp= amp= 0:225 and== 2w. decreases. as a function of xcorresponding to Fig. 3(a) for different(x)functions are given in Fig. 3(b). Figure 3(c) shows the trajectories of the current-driven skyrmions with differentwhereamp= 0:225and= 0:3.as a function ofxcorresponding to Fig. 3(c) for different are given in Fig. 3(d). We then investigate the in-plane current-driven skyrmion motion in a nanotrack with spatially uniform and spatially varying, i.e.,is a function of x, as in Eq. (5), and = 0:3. Figure 4(a) shows the trajectories of the current-driven skyrmions with different (x)functions where = 2w,'= 0and= 0:3. The results are similar to the case with spatially varying. Formax= 0:75,min= 0:12, i.e.,amp= 0:315, the skyrmion moves in the rightward direction in a sinusoidal pattern. For max= 0:6,min= 0:15, i.e.,amp= 0:225, the maximum transverse shift of skyrmion is reduced in compared to that ofamp= 0:315. Formax= 0:45,min= 0:2, i.e., amp= 0:125, the amplitude of the skyrmion trajectory further decreases. as a function of xcorresponding to Fig. 4(a) for different(x)functions are given in Fig. 4(b). Figure 4(c) shows the trajectories of the current-driven skyrmions with differentwhereamp= 0:225and= 0:3.as a function ofxcorresponding to Fig. 4(c) for different are given in Fig. 4(d). From the skyrmion motion with spatially varying or spatially varying , it can be seen that the amplitude of trajectory is proportional to amporamp. The wavelength of trajectory is equal to ;, while the amplitude of trajectory is proportional to ;.also varies with xin a quasi-sinusoidal manner, where the peak value of (x)is proportional to amp, amp, and;. As shown in Fig. 2(c), when is fixed at a value between maxandmin, largerampwill lead to larger peak value of (x). On the other hand, a larger ;allows a longer time for the skyrmion transverse motion toward a certain direction, which will result in a larger amplitude of trajectory as well as a larger peak value of (x). Fig. 6. as a function of xfor skyrmion motion with '= 02. amp=amp= 0:225 and== 2w. C. Nanotrack with spatially varying and We also demonstrate the in-plane current-driven skyrmion motion in a nanotrack with both spatially varying and, i.e., bothandare functions of x, as given in Eq. (4) and Eq. (5), respectively. Figure 5 shows the trajectories of the current-driven skyrmions with spatially varying andwhereamp= amp= 0:225and== 2w. Here, we focus on the effect of the phase difference between the (x)and(x)functions. For'= 0 and'= 2, as the(x)function is identical to the(x)function, the skyrmion moves along the central line of the nanotrack. For 0<'< 2, as(x)could be different from(x)at a certainx, it is shown that the skyrmion moves toward the right direction in a sinusoidal pattern, where the phase of trajectory is subject to '. Figure 6 shows as a function of xcorresponding to Fig. 5 for '= 02where amp=amp= 0:225 and== 2w. It shows that  = 0when'= 0 and'= 2, while it varies with xin a quasi-sinusoidal manner when 0<'< 2. The amplitude of trajectory as well as the peak value of (x)reach their maximum values when '=. IV. C ONCLUSION In conclusion, we have shown the in-plane current-driven motion of a skyrmion in a nanotrack with spatially uniform and, where is determined by and, which can vary between  = 45and =45in principle. Then, we have investigated the in-plane current-driven skyrmion motion in a nanotrack with spatially sinusoidally varying or. The skyrmion moves on a sinusoidal trajectory, where the amplitude and wavelength of trajectory can be controlled by the spatial profiles of and. The peak value of (x)is proportional to the amplitudes and wavelengths of (x)and (x). In addition, we have demonstrated the in-plane current- driven skyrmion motion in a nanotrack having both spatially sinusoidally varying andwith the same amplitude and wavelength. The skyrmion moves straight along the central5 line of the nanotrack when (x)and(x)have no phase difference, i.e., '= 0. When'6= 0, the skyrmion moves in a sinusoidal pattern, where the peak value of (x)reaches its maximum value when '=. This work points out the possibility to guide and control skyrmion motion in a nan- otrack by constructing spatially varying parameters, where the destruction of skyrmion caused by the SkHE can be prevented, which enables reliable skyrmion transport in skyrmion-based information processing devices. ACKNOWLEDGMENT X.Z. was supported by JSPS RONPAKU (Dissertation Ph.D.) Program. G.P.Z. was supported by the National Natural Science Foundation of China (Grants No. 11074179 and No. 10747007), and the Construction Plan for Scientific Research Innovation Teams of Universities in Sichuan (No. 12TD008). Y .Z. was supported by the Shenzhen Fundamental Research Fund under Grant No. JCYJ20160331164412545. REFERENCES [1] H.-B. Braun, Adv. Phys. 61, 1 (2012). [2] N. Nagaosa and Y . Tokura, Nat. Nanotechnol. 8, 899 (2013). [3] Y .-H. Liu and Y .-Q. Li, Chin. Phys. B 24, 17506 (2015). [4] R. Wiesendanger, Nat. Rev. Mat. 1, 16044 (2016). [5] W. Kang, Y . Huang, X. Zhang, Y . Zhou, and W. Zhao, Proc. IEEE 104, 2040 (2016). [6] G. Finocchio, F. Büttner, R. Tomasello, M. Carpentieri, and M. Kläui, J. Phys. D: Appl. Phys. 49, 423001 (2016). [7] S. Seki and M. Mochizuki, Skyrmions in Magnetic Materials (Springer, Switzerland, 2016). [8] U. K. Roszler, A. N. Bogdanov, and C. Pfleiderer, Nature 442, 797 (2006). [9] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. 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2016-07-18

Reliable transport of magnetic skyrmions is required for any future skyrmion-based information processing devices. Here we present a micromagnetic study of the in-plane current-driven motion of a skyrmion in a ferromagnetic nanotrack with spatially sinusoidally varying Gilbert damping and/or non-adiabatic spin-transfer torque coefficients. It is found that the skyrmion moves in a sinusoidal pattern as a result of the spatially varying Gilbert damping and/or non-adiabatic spin-transfer torque in the nanotrack, which could prevent the destruction of the skyrmion caused by the skyrmion Hall effect. The results provide a guide for designing and developing the skyrmion transport channel in skyrmion-based spintronic applications.

Magnetic Skyrmion Transport in a Nanotrack With Spatially Varying Damping and Non-adiabatic Torque

1607.04983v3

Structural scale qderivative and the LLG-Equation in a scenario with fractionality J.Weberszpil Universidade Federal Rural do Rio de Janeiro, UFRRJ-IM/DTL and Av. Governador Roberto Silveira s/n- Nova Iguaçú, Rio de Janeiro, Brasil, 695014. J. A. Helayël-Netoy Centro Brasileiro de Pesquisas Físicas-CBPF-Rua Dr Xavier Sigaud 150, and 290-180, Rio de Janeiro RJ Brasil. (Dated: November 5, 2018) In the present contribution, we study the Landau-Lifshitz-Gilbert equation with two versions of structural derivatives recently proposed: the scale qderivative in the non-extensive statistical mechanics and the axiomatic met- ric derivative, which presents Mittag-Leffler functions as eigenfunctions. The use of structural derivatives aims to take into account long-range forces, possi- ble non-manifest or hidden interactions and the dimensionality of space. Hav- ing this purpose in mind, we build up an evolution operator and a deformed version of the LLG equation. Damping in the oscillations naturally show up without an explicit Gilbert damping term. Keywords: Structural Derivatives, Deformed Heisenberg Equation, LLG Equation, Non-extensive Statistics, Axiomatic Deformed Derivative I. INTRODUCTION In recent works, we have developed connections and a variational formalism to treat deformed or metric derivatives, considering the relevant space-time/ phase space as fractal or multifractal [1] and presented a Electronic address: josewebe@gmail.com yElectronic address: helayel@cbpf.br arXiv:1701.08076v2 [math-ph] 28 Feb 20172 variational approach to dissipative systems, contemplating also cases of a time-dependent mass [2]. The use of deformed-operators was justified based on our proposition that there exists an intimate relationship between dissipation, coarse-grained media and a limit energy scale for the interactions. Con- cepts and connections like open systems, quasi-particles, energy scale and the change in the geometry of space–time at its topological level, nonconservative systems, noninteger dimensions of space–time con- nected to a coarse-grained medium, have been discussed. With this perspective, we argued that deformed or, we should say, Metric or Structural Derivatives, similarly to the Fractional Calculus (FC), could allows us to describe and emulate certain dynamics without explicit many-body, dissipation or geometrical terms in the dynamical governing equations. Also, we emphasized that the paradigm we adopt was different from the standard approach in the generalized statistical mechanics context [3–5], where the modification of entropy definition leads to the modification of the algebra and, consequently, the concept of a derivative [1, 2]. This was set up by mapping into a continuous fractal space [6–8] which naturally yields the need of modifications in the derivatives, that we named deformed or, better, metric derivatives [1, 2]. The modifications of the derivatives, accordingly with the metric, brings to a change in the algebra involved, which, in turn, may lead to a generalized statistical mechanics with some adequate definition of entropy. The Landau-Lifshitz-Gilbert (LLG) equation sets out as a fundamental approach to describe physics in the field of Applied Magnetism. It exhibits a wide spectrum of effects stemming from its non-linear structure, and its mathematical and physical consequences open up a rich field of study. We pursue the investigation of the LLG equation in a scenario where complexity may play a role. The connection between LLG and fractionality, represented by an deformation parameter in the deformed differential equations, has not been exploited with due attention. Here, the use of metric derivatives aims to take into account long-range forces, possible non-manifest or hidden interactions and/or the dimensionality of space. In this contribution, considering intrinsically the presence of complexity and possible dissipative effects, and aiming to tackle these issues, we apply our approach to study the LLG equation with two metric or structural derivatives, the recently proposed scale qderivative [2] in the nonextensive statistical mechanics and, as an alternative, the axiomatic metric derivative (AMD) that has the Mittag-Leffler function as eigenfunction and where deformed Leibniz and chain rule hold - similarly to the standard calculus - but in the regime of low-level of fractionality. The deformed operators here are local. We3 actually focus our attention to understand whether the damping in the LLG equation can be connected to some entropic index, the fractionality or even dimensionality of space; in a further step, we go over into anisotropic Heisenberg spin systems in (1+1) dimensions with the purpose of modeling the weak anisotropy effects by means of some representative parameter, that depends on the dimension of space or the strength of the interactions with the medium. Some considerations about an apparent paradox in the magnetization or angular damping is given. Our paper is outlined as follows: In Section 2, we briefly present the scale qderivative in a nonex- tensive context, building up the qdeformed Heisenberg equation and applying to tackle the problem of the LLG equation; in Section 3, we apply the axiomatic derivative to build up the deformed Heisen- berg equation and to tackle again the problem of LLG equation. We finally present our Conclusions and Outlook in Section 4. II. APPLYING SCALE qDERIVATIVE IN A NONEXTENSIVE CONTEXT Here, in this Section, we provide some brief information to recall the main forms of scale qderivative. The readers may see ref. [1, 2, 6] for more details. Some initial claims here coincide with our work of Refs. [1, 2] and the approaches here are in fact based on local operators [1]. The local differential equation, dy dx=yq; (1) with convenient initial condition, yields the solution given by the q-exponential, y=eq(x)[3–5]. The key of our work here is the Scale qderivative (Sq-D) that we have recently defined as D (q)f(x)[1 + (1q)x]df(x) dx: (2) The eigenvalue equation holds for this derivative operator, as the reader can verify: D (q)f(x) =f(x): (3)4 A.qdeformed Heisenberg Equation in the Nonextensive Statistics Context With the aim to obtain a scale qdeformed Heisenberg equation, we now consider the scaleq derivative [2] dq dtq= (1 + (1q)xd dx(4) and the Scale - qDeformed Schrödinger Equation [2], i~D q;t =~2 2mr2 V =H ; (5) that, as we have shown in [2], is related to the nonlinear Schrödinger equation referred to in Refs. [10] as NRT-like Schrödinger equation (with q=q02compared to the qindex of the reference) and can be thought as resulting from a timescaleqdeformed-derivative applied to the wave function . Considering in eq.(5), (~r;t) =Uq(t;t0) (~r;t0), theqevolution operator naturally emerges if we take into account a timescaleqdeformed-derivative (do not confuse with formalism of discrete scale time derivative): Uq(t;t0) =e(i ~MqHqt) q: (6) Here,Mqis a constant for dimensional regularization reasons. Note that the q-deformed evolution operator is neither Hermitian nor unitary, the possibility of a qunitary asUy q(t;t0) qUq(t;t0) =1could be thought to come over these facts. In this work, we assume the case where the commutativity of Uqand Hholds, but the qunitarity is also a possibility. Now, we follow similar reasonings that can be found in Ref.[12] and considering the Sq-D. So, with these considerations, we can now write a nonlinear Scaleqdeformed Heisenberg Equation as D t;q^A(t) =i ~Mq[^A;H]; (7) where we supposed that UqandHcommute and Mqis some factor only for dimensional equilibrium.5 B.qdeformed LLG Equation To build up the scale qdeformed Landau-Lifshitz-Gilbert Equation, we consider eq.(7), with ^A(t) = ^Sq D t;q^Sq(t) =i ~Mq[^Sq;H]; (8) where we supposed that UqandHcommute. H=gqB ~Mq^Sq~Heff: (9) Here,~Heffis some effective Hamiltonian whose form that we shall clearly write down in the sequel. The scaleqdeformed momentum operator is here defined as bp q0=i~Mq0[1 +(1q0)x]@q @xq: Considering this operator, we obtain a deformed algebra, here in terms of commutation relation between coordinate and momentum  ^xq i;^pq j ={[1 +(1q0)x]~Mq0{jI (10) and, for angular momentum components, as h ^Lq i;^Lq ji ={[1 +(1q0)x]~Mq^Lq k: (11) Theq0factor in ^xq0 {;^pq0 j;^Lq0 i;^Lq0 j;Mq0is only an index and qis not necessarily equal to q0. The resulting scale qdeformed LLG equation can now be written as D t;q^Sq(t) =[1 +(1q0)x]gqB ~Mq^Sq~Heff: (12) Take ^mq q^Sq; q0[1+(1q0)x]gqB ~Mq. If we consider that the spin algebra is nor affected by any emergent effects, we can take q0= 1. Considering the eq.(7) with ^A(t) = ^Sqand ^mq=j qj^Sqandq0= 1; we obtain the qtime deformed LLG dynamical equation for magnetization as D t;q^mq(t) =j j^mq~Heff: (13)6 Considering ~Heff=H0^k;we have the solution: mx;q=cosq(0) cosq( H0t) +sinq(0) sinq( H0t): (14) In the figure, 0= 0: Figure 1: Increase/Damping- cosq(x) . III. APPLYING AXIOMATIC DERIVATIVE AND THE DEFORMED HEISENBERG EQUATION Now, to compare results with two different local operators, we apply the axiomatic metric derivative. Following the steps on [12] and considering the axiomatic MD [13], there holds the eigenvalue equation D xE(x) =E(x);whereE(x)is the Mittag-Leffler function that is of crucial importance to describe the dynamics of complex systems. It involves a generalization of the exponential function and several trigonometric and hyperbolic functions. The eigenvalue equation above is only valid if we consider very close to 1:This is what we call low-level fractionality [13]. Our proposal is to allow the use o Leibniz rule, even if it would result in an approximation. So, we can build up an evolution operator: U(t;t0) =E(i ~Ht); (15) and for the deformed Heisenberg Equation D tAH (t) =i ~[AH ;H]; (16)7 where we supposed that UandHcommute. To build up the deformed Landau-Lifshitz-Gilbert Equation, we use the eq. (16), and considering and spin operator ^S(t), in such a way that we can write the a deformed Heisenberg equation as D t^S(t) =i ~[^S;H]; (17) whith H=gB ~^S~Heff: (18) Here,~Heffis some effective Hamiltonian whose form that we will turn out clear forward. Now, consider the deformed momentum operator as [9, 11, 12] bp=i(~)Mx;@ @x: (19) Takingthisoperator, weobtainadeformedalgebra, hereintermsofcommutationrelationforcoordinate and momentum  ^x i;^p j ={(+ 1)~M{jI (20) and for angular momentum components as h ^L i;^L ji ={(+ 1)~M^L k: (21) The resulting the deformed LLG equation can now be written as J 0D t^S(t) =M(+ 1)gB ~^S~Heff: (22) If we take ^m ^S, M(+1)gB ~, we can re-write the equation as the deformed LLG J 0D t^m(t) =j j^m~Heff; (23) with~Heff=H0^k. We have the Solution of eq.(23): mx=Acos0E2(!2 0t2) +Asin0:x:E 2;1+(!2 0t2): (24)8 In the figure below, the reader may notice the behavior of the magnetization, considering 0= 0. Figure 2: a) Damping of oscillations. In the figure = 1. b) Increase of oscillations . For= 1;the solution reduces to mx=Acos(!0t+0), the standard Simple Harmonic Oscillator solution for the precession of magnetization. The presence of complex interactions and dissipative effects that are not explicitly included into the Hamiltonian can be seen with the use of deformed metric derivatives. Without explicitly adding up the Gilbert damping term, the damping in the oscillations could reproduce the damping described by the Gilbert term or could it disclose some new extra damping effect. Also, depending on the relevant parameter, the qentropic parameter or for , the increasing oscillations can signally that it is sensible to expect fractionality to interfere on the effects of polarized currents as the Slonczewski term describes. We point out that there are qualitative similarities in both cases, as the damping or the increasing of the oscillations, depending on the relevant control parameters. Despite that, there are also some interesting differences, as the change in phase for axiomatic derivative application case. Here, we cast some comments about an apparent paradox: If we make, as usually done in the literature for LLG, the scalar product in eq. (13) with, ^mq;we obtain an apparent paradox that the modulus of ^mqdoes not change. On the other hand, if instead of ^m;we proceed now with a scalar product with ~Heff and we obtain thereby the indications that the angle between ^mand~Heffdoes not change. So, how to explain the damping in osculations for ^mq?This question can be explained by the the following arguments. Even the usual LLG equation, with the term of Gilbert, can be rewritten in a form similar to eq. LLG9 without term of Gilbert. See eq. (2.7) in the Ref. [14]. The effective ~Hefffield now stores information about the interactions that cause damping. In our case, when carrying out the simulations, we have taken ~Heffas a constant effective field. Here, we can argue that the damping term, eq. (2.8) in Ref. [14] being small, this would cause the effective field ~Heff= !H(t) + !k( !S !H)to be approximately !H(t). In this way, the scalar product would make dominate over the term of explicit dissipation. This could, therefore, explain the possible inconsistency. IV. CONCLUSIONS AND OUTLOOK In short: Here, we tackle the problem of LLG equations considering the presence of complexity and dissipation or other interactions that give rise to the term proposed by Gilbert or the one by Slonczewski. With this aim, we have applied scale - qderivative and the axiomatic metric derivative to build up deformed Heisenberg equations. The evolution operator naturally emerges with the use of each case of the structural derivatives. The deformed LLG equations are solved for a simple case, with both structural or metric derivatives. Also, in connection with the LLG equation, we can cast some final considerations for future investiga- tions: Does fractionality simply reproduce the damping described by the Gilbert term or could it disclose some new effect extra damping? Is it sensible to expect fractionality to interfere on the effects of polarized currents as the Slonczewski term describes? These two points are relevant in connection with fractionality and the recent high precision measure- ments in magnetic systems may open up a new venue to strengthen the relationship between the fractional properties of space-time and Condensed Matter systems. [1] J. Weberszpil, Matheus Jatkoske Lazo and J.A. Helayël-Neto, Physica A 436, (2015) 399–404.10 [2] Weberszpil, J.; Helayël-Neto, J.A., Physica. A (Print), v. 450, (2016) 217-227; arXiv:1511.02835 [math-ph]. [3] C. Tsallis, J. Stat. Phys. 52, (1988) 479-487. [4] C. Tsallis, Brazilian Journal of Physics, 39, 2A, (2009) 337-356. [5] C. Tsallis, Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World (Springer, New York, 2009). [6] Alexander S. Balankin and Benjamin Espinoza Elizarraraz, Phys. Rev. E 85, (2012) 056314. [7] A. S. Balankin and B. Espinoza, Phys. Rev. E 85, (2012) 025302(R). [8] Alexander Balankin, Juan Bory-Reyes and Michael Shapiro, Phys A, in press, (2015) doi:10.1016/j.physa.2015.10.035. [9] Weberszpil, J. ; Helayël-Neto, J. A., Advances in High Energy Physics, (2014), p. 1-12. [10] F. D. Nobre, M. A. Rego-Monteiro, and C. Tsallis, Phys. Rev. Lett. 106, (2011) 140601. [11] J.Weberszpil, C.F.L.Godinho, A.ChermanandJ.A.Helayël-Neto, In: 7thConferenceMathematicalMethods in Physics - ICMP 2012, 2012, Rio de Janeiro. Proceedings of Science (PoS). Trieste, Italia: SISSA. Trieste, Italia: Published by Proceedings of Science (PoS), 2012. p. 1-19. [12] J. Weberszpil and J. A. Helayël-Neto, J. Adv. Phys. 7, 2 (2015) 1440-1447, ISSN 2347-3487. [13] J. Weberszpil, J. A. Helayël-Neto, arXiv:1605.08097 [math-ph] [14] M. Lakshmanan, Phil. Trans. R. Soc. A (2011) 369, 1280–1300 doi:10.1098/rsta.2010.0319

2017-01-27

In the present contribution, we study the Landau-Lifshitz-Gilbert equation with two versions of structural derivatives recently proposed: the scale $q-$derivative in the non-extensive statistical mechanics and the axiomatic metric derivative, which presents Mittag-Leffler functions as eigenfunctions. The use of structural derivatives aims to take into account long-range forces, possible non-manifest or hidden interactions and the dimensionality of space. Having this purpose in mind, we build up an evolution operator and a deformed version of the LLG equation. Damping in the oscillations naturally show up without an explicit Gilbert damping term.

Structural scale $q-$derivative and the LLG-Equation in a scenario with fractionality

1701.08076v2

Oscillation time and damping coecients in a nonlinear pendulum Jaime Arango March 24, 2022 Abstract We establish a relationship between the normalized damping co- ecients and the time that takes a nonlinear pendulum to complete one oscillation starting from an initial position with vanishing velocity. We establish some conditions on the nonlinear restitution force so that this oscillation time does not depend monotonically on the viscosity damping coecient. ASC2020: 34C15, 34C25 Keywords. oscillation time, damping, damped oscillations This paper is dedicated to the memory of Prof. Alan Lazer (1938-2020), University of Miami. It was my pleasure to discuss with him some of the results presented here 1 Introduction The pendulum is perhaps the oldest and fruitful paradigm for the study of an oscillating system. The apparent regularity of an oscillating mass going to and fro through the equilibrium position has fascinated the scientists well be- fore Galileo. There are plenty of mathematical models accounting for almost any observed behavior of the pendulum's oscillation. From the sheer amount of the literature on the subject, one would expect that there is no reasonable question regarding a pendulum that has no been already answered. And that might be true. Yet, for whatever reason, it is not impossible to take on a question whose answer does not seem to follow immediately from the classical sources. In a typical experimental setup with no noticeable damping, the oscilla- tions of a pendulum are periodic. Now, if the damping cannot be neglected, 1arXiv:2101.09400v2 [math.CA] 24 Jun 2021we still observe oscillations, even though they are non periodic. However, we can measure the time spent by a complete oscillation, and this time is a natural generalization of the period. But, how does depend this oscillation time on the characteristic of the medium, say on the viscosity of the sur- rounding atmosphere? It seems that there is no much information on how the damping aects the oscillation time. There are plenty of new publica- tions regarding damping and oscillations, ranging from analytical solutions ([5], [3],[6]), to very clever experimental setups (see for example [4]). The nature of the damping has been also extensively considered ([8], [2]), but the dependence of the oscillation time on the damping or on the non-linearity seems to be less investigated. For the sake of simplicity we analyze the oscillation time in the frame of a model that appear in almost any text book of ordinary dierential equations (see for example [1]): x+ 2_x+x(1 +f(x)) = 0; (1) wherex=x(t) measures the pendulum's deviation with respect to a vertical axis of equilibrium and 0 denote the viscous damping coecient. The termxf(x) models the nonlinear part of the restoring force. We've rescaled the time so that the period of the linear undamped oscillation is exactly 2 . The math of the solutions x=x(t) is classical. If fis smooth and x0 andv0are given real values, then there exists a unique solution satisfying the given conditions x(0) =x0and _x(0) =v0:Moreover, if f(0) = 0, then x= 0 is a stable equilibrium solution of (1). As a consequence, x(t) is dened for allt0 providedjx0j 1 andjv0j 1. Notice that the points of vanishing derivative of a solution x=x(t) to (1) are isolated and those points correspond, either to local maxima or to local minima. Denote by (x0;) the amount of time spent (by the mass) completing one oscillation starting from x0with vanishing velocity ( v0= 0). To be precise, if x=x(t) starts from x0 with vanishing velocity, then xreaches a local maximum at t= 0, and the oscillation is completed when xreaches the next local maximum. Certainly, the oscillation time generalizes the period of solutions for the undamped model (= 0). In this investigation we analyze the dependence of onx0 and onunder the following working hypothesis: Assumption 1.1. On smallneighborhood of 0the function fis even and for some constant a>0we have f(x) =ax2+O jxj4
; We shall show that for x0xed,reaches a positive minimum at some 0<  0<1:It does not seem obvious that an increase in the damping 20.0 0.1 0.2 0.3 0.4 0.5 Damping coefficient α2π6.46.97.5τ(x0,α,0) x0= 0.1 x0= 0.6 x0= 1.2 x0= 1.6Figure 1: Numerical simulation (x0;) depending on for several values of x0. The nonlinear term fwas chosen so that x(1 +f(x)) = sinx: coecientmight cause a decrease in . It is also worth noticing that the existence of a minimum of is a consequence the sign of the constant ain the above assumption. Indeed, according to numerical experiments carried out by the author, does not reach a positive minimum if a < 0. The author is not aware of a similar result in the current literature nor whether this phenomena has been experimentally addressed. The whole paper was written with the aim at the mathematical pendulum x(1 +f(x)) = sinx: In that case, Figure 1 summarize our ndings by picturing the numerically simulated value for (x0;). Interestingly, our qualitative analysis accurately re ects variations of that are not easy to spot numerically. For instance, the minimum of (x0;) forx0= 0:1 is not evident in Figure 1. The arguments and proofs in this paper are entirely based on well estab- lished techniques of ODE theory. However, the main result (Theorem 3.1) rests on delicate estimates involving a dierential equation describing the dependence of the solution x=x(t) with respect to . 2 Underdamped oscillations Denitions of underdamped oscillations in linear systems naturally carry over to solutions of (1). From now on, x(
;x0;) stands for the unique solution to (1) satisfying the initial condition x(0) =x0and _x(0) = 0:We also write (x0;) to highlight the dependence of the oscillation time on x0and. We 3will write simply orxwhen no confusion can arise. It is convenient to represent (1) in the phase space ( x;v) with _x=v: _x=v _v=2vxxf(x):(2) Equation (2) is explicitly solvable whenever f0, and in that case, its solution is given by xl(t) =et !(!cos!t+sin!t)x0 vl(t) =et !sin!tx 0(3) where!=p 12. Moreover, the oscillation time lis given by l=2 !=2p 12: Notice that lis an increasing function that solely depends on . Though a closed-form solution of (1) is either not known or impractical, we could express the relevant solutions implicitly. To that end, we rewrite (2) so that the nonlinear term xf(x) assumes the role of a non homogeneous forcing term. The expression for the solution ( x;v) is implicitly given by x(t) =xl(t)1 !Zt 0e(ts)sin!(ts)x(s)f(x(s))ds v(t) =vl(t)1 !Zt 0e(ts)(!cos!(ts)sin!(ts))x(s)f(x(s))ds (4) Next, we estimate the solutions of (2) in the conservative case ( = 0) in which all solutions are periodic and the period is given by (x0;0): Lemma 2.1. If(x;v)stands for the solution to (2)with= 0that satises (x(0);v(0)) = (x0;0), then there exists  >0so that for alljx0jand all 0twe have x(t) =x0cost+R1(t;x0); v(t) =x0sint+R2(t;x0); (5) where jRi(t;x0)jconstjx3 0j; i = 1;2: 4Proof. Letting= 0 in (4) we obtain R1(t;x0) =Zt 0cos(ts)x(s)f(x(s))ds: (6) Since (0;0) is a stable equilibrium solution to (2), there exists >0 and>0 so that any solution ( x;v) to (2) starting at ( x0;0), withjx0jsatises jx(t)j. Now write F(z) =zf(z) and notice that for some 2(;) we have F(x(s)) =F(x0coss+R1(s;x0)) =F(x0coss) +R1(s;x0)F0(): Next, identity (6), Assumption (1.1) and some standard estimations yield jR1(t;x0)j2ajx3 0j+c2Zt 0jR1(s;x0)jds wherec2= maxz2[;]jF0(z)j. The rst claim follows now from Gronwall's inequality. The proof of the estimation for R2is analogous. At this point it is appropriated to dene the half oscillation time ^= ^(x0;) to be the time spent by the solution x(t;x0;); t0;reaching the next local minimum. If = 0 andfis even, the symmetry of the solution (1) yields. 2^ =: Lemma 2.2. If^= ^(x0;)denote the half oscillation time and ais the constant of Assumption 1.1, then ^(x0;)>p 12and lim x0!0+^(x0;0) =+a 8x2 0+o(x3 0): Proof. We introduce introduce the polar coordinates r=p x2+v2;tan=x v; to obtain _= 1 +sin 2+ sin2f(x)
_r=v r(2v+xf(x))(7) As a consequence of equation (7) we obtain the following expression for the half oscillation time ^ = ^(x0;) ^=Z 0d 1 +sin 2+ sin2f(x()): (8) 5Now, the eect of the nonlinearity on the oscillation time is clear. By As- sumption 1.1 we obtain ^(x0;)>Z 0d 1 +sin 2=p 12: For= 0 we use estimation (5) to obtain ^(x0;0) =Z 0d 1ax2 0sin2cos2t()+o(x3 0): Now a straightforwards computation yields lim x0!0+^(x0;0) =;lim x0!0+@^ @x0(x0;0) = 0: Now, the expression for@2^ @x2 0(x0;0) is somewhat cumbersome. However, taking into account that lim x0!0t() =, we readily obtain lim x0!0+@2^ @x2 0(x0;0) =Z 02asin2cos2d=2a 8; and the second claim of the lemma follows by the second order Taylor ex- pansion of ^(x0;0) aroundx0 A reasoning analogous to that in the proof of the preceding lemma shows that (x0;)>2p 12l: This inequality is illustrated in Figure 2 when a= 1. Had we considered in Assumption 1.1 negative values for a, then the inequality would reverse to (x0;)<las it is depicted in Figure 2. 3 The role of the viscous damping It is not dicult at all to obtain a dierential equation describing the move- ment of the pendulum depending on the viscous damping coecient. Indeed, writing X(t;x0;) =@x @(t;x0;); V (t;x0;) =@v @(t;x0;): Derivation of equation (2) with respect to yields: _X=V _V=2VX2v(xf0(x) +f(x))X:(9) 6As for the initial conditions we have X(0;x0;) = 0; V (0;x0;) = 0: Let us write G(x) =d dx(xf(x)). Again, as we did with equation (2), equation (9) can be seen as a linear homogeneous part plus the forcing term 2v+G(x)X:The solution X;V is implicitly given by X(t) =1 !Zt 0e(ts)sin!(ts) 2v(s) +G(x(s))X(s)gds V(t) =1 !Zt 0e(ts)(!cos!(ts)sin!(ts)) 2v(s) +G(x(s))X(s)gds In particular, for = 0 the above expressions reduce to X(t) =Zt 0sin (ts) 2v(s) +G(x(s))X(s) ds V(t) =Zt 0cos (ts) 2v(s) +G(x(s))X(s) ds(10) The following lemma does the heavy lifting to deliver the main result of the paper. Lemma 3.1. Under Assumption 1.1, if ^= ^(x0;0)denotes the half oscil- lation time when = 0, then for 0<x 01we haveV(^;x0;0)>0: Proof. We start with an auxiliary estimate for X(t) in equation (10). By Lemma (2.1) and by Assumption 1.1, for 0 <twe have X(t) =x0(tcost+ sint) + 3ax2 0Zt 0sin (ts) cos2sX(s)ds+O(jx0j4) (11) Notice that X1(t)x0(tcost+ sint) does not vanish on (0 ;) and that G(x(s))>0 provided 0 <x 01. Further, the initial conditions for X(t) at t= 0 and equation (9) yield that X(0) = 0 = _X(0) = X(0) and... X(0) = 2x0(1 +f(x0))>0; meaning that X(t) is positive on an interval (0 ;) with>0. We claim that X(t)>0 for 0< t:On the contrary, there exists  < t 0<  such that X(t0) = 0 andX(t)>0 fort2(0;t0). Now, by Lemma 2.2 we know that 7^ >  . Therefore, the polar angle (t) in (7) satises <  (t)<0 for all 0<t< and a fortiori v(t)<0 on (0;]. But this is a contradiction to the rst equation of (10) evaluated at t=t0since fors2(0;t0) we have sin (t0s) 2v(s) +G(x(s))X(s) >0: Next, by the equation (11) it follows immediately that X(t) =X1(t) + O(jx0j3). Analogously, for V(t) we obtain V(t) =x0tsint+ 3ax2 0Zt 0cos (ts) cos2sX1(s)ds+O(jx0j4) V1(t) +V2(t) +O(jx0j4) whereV1(t)x0tsint. Now,V2(t) can be explicitly evaluated. For the reader's convenience, we write the complete expression for V2: V2(t) =3ax3 0 1 32(6t2+ 5) cost3 32tsin 3t 1 16tsint17 128cos 3t+37 128cost : Moreover, it is somewhat tedious but straightforward to show that V2is positive and increasing on a small neighborhood of . By Lemma 2.2 ^  > , therefore V2(^)>V 2() =9ax3 02 16: Again, by Lemma 2.2 we obtain V1(^) =V1() + (^)V0 1() +O(jx0j4) =ax3 02 8+O(jx0j4); so thatV(^) =V1(^) +V2(^)>0: Now we are in a position to show the main result of the paper Theorem 3.1. Under Assumption 1.1, there exists a  > 0such that for 0< x 0<  xed, the oscillation time (x0;), for 0<  < 1, reaches a positive minimum at some 0<< 1. Moreover, lim !1(x0;) =1: 8Proof. We let 0<x 01 xed by now and denote by ( x;v) be the solution of equation (2). By denition of ^ we havev(^;) = 0, so that the Implicit Function Theorem yields @^ @_v(^;) +V(^;) = 0; therefore @^ @=V(^;) x(^;) (1 +f(x(^;))): Sincex(^;) is negative, it follows from Lemma 3.1 that and@^ @j=0<0. Now we shall show that the last inequality holds for the oscillation time . To do that, write ^ x0=x(^(;x 0);x0) and see that (;x 0) = ^(;x 0) + ^(;^x0): That is to say, the half oscillation time depends on jx0jonly. Notice that ^x0x0and the equality holds in the conservative case = 0 only. Therefore @ @(;x 0) =@^ @(;x 0) +@^ @(;^x0)@^x0 @(;x 0)@^ @(;x 0) = 0: Moreover, since @^x0 @(;x 0) =v(^(;x 0);x0) = 0; we have that lim x0!0+@ @(;x 0) = 2 lim x0!0+@^x0 @(;x 0) Finally, by the rst claim of Lemma 2.2, (;x 0) must attain a minimum at some 0<< 1. 4 Conclusions and nal remarks An oscillating mass exhibits gradually diminishing amplitude in the presence of damping. The time spent by the mass completing one oscillation depends on several factors, as the model for the restoring force, how the oscillation starts, and the nature of the damping. For the sake of our discussion we consider a vertical pendulum with a nonlinear restoring force resembling the mathematical pendulum, letting the oscillation start at a small amplitude with vanishing velocity and a viscous damping model with a (normalized) viscosity coecient . We have proved that the oscillation time () does not depend monotonically on , meaning that there exists a threshold 9Figure 2: Numerical simulation of the oscillation time depending on the damping coecient with starting amplitude x0= 0:2 and non linear restoring term given by f(x) =ax2,a=1. The curve with the round marker (blue in the online version) corresponds to the oscillation time lof the linear case f0 0(which depends on the starting amplitude of the oscillation) such that  reaches a local minimum at 0(see Figure 1). It is worth noticing that this behavior cannot be observed if the restitution force is linear, i. e., what we report in this paper is essentially a nonlinear phenomenon. The proof of existence of a positive minimum for the oscillation time rests heavily on the fact that the constant ain Assumption 1.1 is positive. Just to experiment the eect of changing the sign of the constant a, we carried out some numerical simulations of with the nonlinear term f(x) =ax2for a= 1;1. The corresponding equations are particular cases of an unforced Dung oscillator [7]. The numerical results are shown in Figure 2. Just for the sake of the numerical experimentation we also considered negative values for. Ifa= 1 we see that reaches its minimum at a positive value for . By contrast, if a=1 no minimum seems to exist. The curve with the round marker (blue in the online version) corresponds to the oscillation time of the linear case l=2p 12. The numerical experimentation of the oscillation time (not shown in this paper) assuming a quadratic damping exhibits the same behavior as the graphics of Figure 2. If the readers are curious about the numerical experiments, they could take a look at the author's GitHub page https://github.com/arangogithub/Oscillation-time and download a Jupyter notebook with the python code featuring the results shown in Figures 1 and 2 10Acknowledgment The author would like to give the reviewer his very heartfelt thanks for carefully reading the manuscript and for pointing out several inaccuracies of the document. References [1] V. I. Arnold. Mathematical Methods of Classical Mechanics . Springer, 1989. [2] L. Cveticanin. Oscillator with strong quadratic damping force. Publ. Inst. Math. (Beograd) (N.S.) , 85(99):119{130, March 2009. [3] A Ghose-Choudhury and Partha Guha. An analytic technique for the solutions of nonlinear oscillators with damping using the abel equation arxiv:1608.02324 [nlin.si] , 2016. [4] Remigio Cabrera-Trujillo Niels C Giesselmann Dag Hanstorp Javier Tello Marmolejo, Oscar Isaksson. A fully manipulable damped driven harmonic oscillator using optical levitation. American Journal of Physics , 88(6):490{498, sep 2018. [5] Kim Johannessen. An analytical solution to the equation of motion for the damped nonlinear pendulum. European Journal of Physics , 35(3):035014, mar 2014. [6] D Kharkongor and Mangal C Mahato. Resonance oscillation of a damped driven simple pendulum. European Journal of Physics , 39(6):065002, sep 2018. [7] S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos . Springer, 1990. [8] L F C Zonetti, A S S Camargo, J Sartori, D F de Sousa, and L A O Nunes. A demonstration of dry and viscous damping of an oscillating pendulum. European Journal of Physics , 20(2):85{88, jan 1999. 11

2021-01-23

We establish a relationship between the normalized damping coefficients and the time that takes a nonlinear pendulum to complete one oscillation starting from an initial position with vanishing velocity. We establish some conditions on the nonlinear restitution force so that this oscillation time does not depend monotonically on the viscosity damping coefficient.

Oscillation time and damping coefficients in a nonlinear pendulum

2101.09400v2

1 Scaling of spin Hall angle in 3 d, 4d and 5d metals from Y3Fe5O12/metal spin pumping H. L. Wang†, C. H. Du†, Y . Pu, R. Adur, P. C. Hammel*, F. Y . Yang* Department of Physics, The Ohio State University, Columbus, OH 43210 ,USA †These authors made equal contributions to this work *E-mails:fyyang@physics.osu.edu; hammel@physics.osu.edu Abstract We have investigated spin pumping from Y3Fe5O12 thin films into Cu, Ag, Ta, W, Pt and Au with varying spin-orbit coupling strengths . From measurements of Gilbert damping enhance ment and inverse spin Hall signals spanning three orders of magnitude , we determine the spin Hall angles and interfacial spin mixing conductance s for the six metals . For noble metals Cu, Ag and Au (same d-electron counts) , the spin Hall angles vary as Z4 (Z: atomic number ), corroborating the role of spin -orbit coupling . In contrast, amongst the four 5 d metals, the variation of the spin Hall angle is dominated by the sensitivity of the d-orbital moment to the d-electron count, confirming theoretical predictions. PACS: 75.47.Lx, 76.50.+g, 75.70.Ak, 61.05.cp 2 Spin pumping of p ure spin c urrent s from a ferromagnet ( FM) into a nonmagnetic material ( NM) provide s a promising route toward energy -efficient spintronic devices . The inverse spin Hall effect (ISHE) in FM/Pt bilayer systems [1-13] is the most widely used tool for detecting s pin currents generated by either ferromagnetic resonance ( FMR ) or a thermal gradient. The intense interest in spin pumping emphasizes the pressing need for quantitative understanding of ISHE in normal metals other than Pt [10]. To date, spin Hall angles (SH) have been measured for several metals and alloys by spin Hall or ISHE measurements, mostly using metallic FMs [ 14]. Due to current shunting of the metallic FMs and potential confounding effects of anisotropic magnetoresistance ( AMR ) or anomalous Hall effect (AHE ), the reported values of SH vary significantly , sometimes by more than one order of magnitude for the same materials [14]. Here we report a systematic study of FMR spin pumping from insulating Y3Fe5O12 (YIG) epitaxial thin films grown by sputtering [15-22] into six normal metal s, Cu, Ag, Ta, W, Pt and Au , that span a wide range in two key parameters : a factor of ~50 in spin-orbit coupling strength [ 23] and over two orders of magnitude variation in spin diffusion length (SD) [4, 24-26]. Due to their weak spin-orbit coupling and relatively long spin diffusion length s, Cu and Ag present a significant challenge for ISHE detection of spin pumping . ISHE voltages (VISHE) exceeding 5 mV are generated in our YIG/Ta and YIG /W bilayer s and here we report ~1 V spin pumping signals in YIG/Cu and YIG /Ag bilayer s. The recently rep orted proximity effect in Pt [ 9, 13 ] should lead to at most V-level contribution to the measured VISHE, which is negligible compared with the observed mV-level VISHE in our YIG/Pt bilayer and should not be a factor in other five metals. The large 3 dynamic range that this sensitivity provides and the insulating nature of YIG films enable quantitative determination of spin mixing conductance s across the YIG/metal interfa ces [5, 12] and spin Hall angle s of these 3d, 4d and 5 d metals . We characterize the structural quality of our epitaxial YIG films deposited on (111) -oriented Gd3Ga5O12 (GGG) substrates using off-axis ultrahigh vacuum (UHV) sputtering [15-22] by high-resolution x -ray diffraction (XRD). A representative θ-2θ scan of a 20-nm YIG film shown in Fig. 1 a demonstrates phase purity and clear Laue oscillations , indicat ing high uniformity of the film . We find an out-of-plane lattice constant of the YIG film, c = 12.393 Å , very close to the bulk value of 12.376 Å . The XRD rocking curve in the left inset to Fig. 1a give s a full width at half maximum (FWHM) of only 0.0092, near the resolution limit of our high-resolution XRD, demonstrating the excellent crystalline quality of the YIG film . The atomic force microscopy (AFM) image in the right inset to Fig. 1a shows a smooth surface with a roughness of 0.15 nm. Figure 1b shows a representative FMR derivative absorption spectrum f or a 20-nm YIG film used in this study taken by a Bruker EPR spectrometer in a cavity at a radio -frequency (rf) f = 9.65 GHz and an input microwave power Prf = 0.2 mW with a magnetic field H applied in the film plane. The peak -to-peak linewidth ( H) obtained from the spectrum is 1 1.7 Oe and an effective saturation magnetization 4π𝑀eff = 1786 ± 36 Oe is extracted from fitt ing the angular dependence of resona nce field [27]. Due to the small magnetic anisotropy of YIG, the satura tion magnetization 4𝑀s can be approximated at 1786 Gauss which agrees well with the value reported for single crystal YIG , indicating the high magnetic quality of our YIG films [28]. In 4 this letter , all six YIG/metal bilayers are made from the same 20 -nm YIG film characterized in Figs. 1b and 1c. Our s pin pumping measurements are carried out in the center of the EPR cavity on the six YIG/ metal bilayer s at room temperature (approximate dimension s of 1.0 mm 5 mm). The thickness of the metal layers is 5 nm for Ag, Ta, W, Pt, Au and 10 nm for Cu, and all are made by UHV off-axis sputtering . Resistivity () measurements confirm that the Ta and W films are -phase [29, 30] (high resistivity , see Table I) . During the spin pumping measurements, a DC magnetic field H is applied in the xz-plane and the ISHE voltage is measured across the ~5-mm long metal layer along the y-axis, as illustrated in Fig. 1c. At resonan ce, the precessing YIG magnetization (M) transfers angular momentum to the conduction electrons in the normal metal. The resulting pure spin current Js is injected into the metal layer along the z-axis with spin polarization 𝜎 parallel to M and then converted to a charge current Jc SHJs𝜎 by the ISHE via the spin-orbit interaction . Figure 2 show s VISHE vs. H spectra of the six YIG/metal bilayers at θH = 90 and 270 (both with in -plane field) at Prf = 200 mW. For YIG/Ta and YIG /W bilayer s, |VISHE| exceeds 5 mV (1 mV/mm) . For YIG /Pt, YIG/Au and YIG /Ag, VISHE = 2.10 mV, 72.6 V and 1. 49 V, respectively . Due to the opposi te sign s of SH, Pt, Au and Ag give positive VISHE while Ta and W give negative VISHE at θH = 90. This agrees with the predicted signs of SH [31, 32] of the metals . When H is reversed from θH = 90 to 270, all the VISHE signal s change sign as expected from the ISHE . The rf-power dependenc ies of VISHE are shown in the upper insets to Figs. 2a -2f at θH = 90, each of which shows a linear dependence, indicat ing that the 5 observed spin pumping signals are in the linear regime. Furthermore, the large spin pumping signals provided by the YIG films e nable the observation of VISHE = 0.99 V in the YIG/Cu bilayer (Fig. 2f) . Due to the much weak er spin-orbit coupling [24] in Cu compared to th ose 5d metals , there is no previous report of ISHE detection of spin pumping in FM/Cu structure s. This first observation of VISHE in YIG/Cu enables the determination of spin Hall angle in Cu . When H is rotated from in -plane to out -of-plane , M remains essentially parallel to H at all angles since the FMR resonance field Hres (between 2500 and 5000 Oe ) always exceeds 4Meff (1786 Oe). The lower insets to Figs. 2a-2f show the angular dependenc ies of the normalized VISHE for the six bilayer s; all clearly exhibit the expected sinusoidal shape (VISHE Js𝜎 JsM JsH sinθH), confirming that the observed ISHE voltage s arise from FMR spin pumping. Since YIG is insulating we can rule out artifacts due to thermoelectric or magnetoelectric effects, such as AMR or AHE , enabling more straightforward measurement of the inverse spin H all effect than using metallic FMs. Figure s 3a-3f show the FMR derivative absorption spectr a (f = 9.65 GHz) of the 20-nm YIG film s before and after the deposition of the metal s. The FMR linewi dths are clearly enhanced in YIG/metal bilayers relative to the bare YIG films . The linewidth enhancement [10, 11] is a consequence of FMR spin pumping: the coupling that transfer s angular momentum from YIG to the metal adds to the damping of the precessing YIG magnetization , thus increas ing the linewidth. In order to accurately determine the enhance ment of Gilbert damping, we measured the frequency dependenc ies of the linewidth of a bare YIG film and the six YIG/ metal bilayers using a microstrip transmission line . In all 6 cases the linewidth increases linear ly with frequency (Fig. 3g). The Gilbert damping constant can be obtained using [ 33], Δ𝐻=Δ𝐻inh+4𝜋𝛼𝑓 √3𝛾, (1) where Hinh is the inhomog eneous broadening and is the gyromagnetic ratio . Table I shows the damping enhancement sp due to spin pumping : sp=YIG/NM−YIG, where YIG/NM and YIG = (9.1 ± 0.6) 10-4 are obtained from the least-squares fits in Fig. 3g . The observed ISHE voltages depend on several materials parameters [4, 11], 𝑉ISHE=−𝑒𝜃SH 𝜎N𝑡N+𝜎F𝑡F𝜆SDtanh(𝑡N 2𝜆SD)𝑔↑↓𝑓𝐿𝑃(𝛾ℎrf 2𝛼𝜔)2 , (2) where e is the electron charge , N (F) is the conductivity of the NM (FM), tN (tF) is the thickness of the NM (FM) layer, 𝑔↑↓ is the interfacial spin mixing conductance , = 2f is the FMR frequency , L is the sample length, and hrf = 0.25 Oe [34] in our FMR cavity at Prf = 200 mW. The factor P arises from the ellipticity of the magnetization precession [10], 𝑃=2𝜔[𝛾4𝜋𝑀s+√(𝛾4𝜋𝑀s)2+4𝜔2] (𝛾4𝜋𝑀s)2+4𝜔2= 1.21 (3) for all the FMR and spin pumping measurements. The spin mixing conductance can be determined from the damp ing enhancement [10-12], 𝑔↑↓=4𝜋𝑀s𝑡F 𝑔𝜇B(YIG/NM−YIG) (4) where 𝑔 and 𝜇B are the Landé 𝑔 factor and Bohr magneton , respectively. Although the reported spin diffusion length var ies from a few nm to a few hundred nm across the range of metals we have measured , the term 𝜆SDtanh(𝑡N 2𝜆SD) in Eq. (2) is rather insensitive to the value of SD for a given tN (e.g., 5 nm ) due to the limitation of film thickness ; for example, 𝜆SDtanh(𝑡N 2𝜆SD) = 1.70 nm for SD = 2 nm and 2.50 nm for SD = 7 [10]. In this calculation, w e assume 𝜆SD = 10 nm for Pt [4], 2 nm for W and Ta [25], 60 nm for Au [24], 700 nm for Ag [26] and 500 nm for Cu [24]. Electrical conduction in YIG can be neglected. From Eqs. (2)-(4), 𝑔↑↓ can be obtained from the Gilbert damping enhancement and SH can be calculated for the six metals (Table I) [ 35]. Consequently , the spin current density Js can be estimated using [11], 𝐽s=𝑡N𝜎N+𝑡F𝜎F 𝜃SH𝜆SDtanh(𝑡N 2𝜆SD)𝑉ISHE 𝐿. (5) The power of inverse spin Hall effect as a probe of spin pumping calls for a quantitative understanding to enable more precise and detailed experiments. Spin Hall angles have been measured in several normal metals by spin Hall or ISHE measurements, mostly using metallic FMs [10, 14, 32, 35]. Due to the impact of AMR or AHE in electrical ly conducting metallic FMs in the heterostructures , and the variation of sample quality among different groups, the reported values of SH vary significantly for the same materials, in some cases, by more than one order of magnitude [ 14]. Here, we report measurements of the spin Hall angle for various 3d, 4d and 5 d metals using Eq. (2) from the large ISHE signals and, independently, spin -pumping enhancement of Gilbert damping (to obtain 𝑔↑↓ and ) of the insulating YIG thin film . This set of experimental data can be compared to discern trends and uncover the roles of various materials parameters in spin -orbit coupling , including the atomic number as well as d-electron co unt in transition metals [ 23, 31]. We first show in Fig. 4a the linear dependence of SH on Z4 for Cu, Ag and Au , reflecting the key role of atomic number in spin-orbit coupling [23] and spin Hall physics in metals having a particular d-electron configuration . We note that the spin Hall angles of the four 5 d metals do not vary as Z4 at all , 8 indicating that the d-orbital filling plays the dominant role [ 31]. Figure 4b shows SH vs. Z for Ta, W, Pt and Au , which matches well with the theoretical calculations by Tanaka et al. [31], including the sign change and relative magnitude of spin Hall effect in 5d metals . These two results highlight and distinguish the roles of atomic number and d-orbital filling in spin Hall physics in transition metals , and clarify their relative importance . In conclusion , FMR spin pumping measurements on YIG/NM bilayers give mV-level ISHE voltages in YIG /Pt, YIG/Ta and YIG/W bilayer s and robust spin pumping signals in YIG/Cu and YIG/Ag . YIG/NM interfacial spin mixing conductance s are determined by the enhanced Gilbert damping which are measured by frequency dependence of FMR linewi dth before and after the deposition of metals. 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B 87, 174417 (2013). 26. T. Kimura and Y. Otani, Phys. Rev. Lett. 99, 196604 (2007). 27. M. Farley, Rep. Pro g. Phys. 61, 755 (1998). 28. P. Hansen, P. Röschmann and W. Tolksdorf , J. Appl. Phys . 45, 2728 (1974) . 29. L. Q. Liu, C. -F. Pai, Y . Li , H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). 30. C.-F. Pai, L. Q. Liu, Y . Li, H. W. Tsen g, D. C. Ralph, and R. A. Buhrman , Appl. Phys. Lett. 101, 122404 (2012). 31. T. Tanaka , H. Kontani , M. Naito , T. Naito , D. S. Hirashima , K. Yamada , and J. Inoue , Phys. Rev. B 77, 165117 (2008). 32. M. Morota , Y . Niimi , K. Ohnishi , D. H. Wei , T. Tanaka , H. Kontani , T. Kimura , and Y . Otani , Phys. Rev. B 83, 174405 (2011). 12 33. S. S. Kalarickal , P. Krivosik, M. Z. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, and J. P. Nibarger , J. Appl. Phys. 99, 093909 (2006). 34. See Supplementary Materials for detailed analysis of spin pumping parameters. 35. L. Q. Liu, R. A. Buhrman, and D. C. Ralph, arXiv:1111.3702 . 13 Table I . ISHE voltages at f = 9.65 GHz and Prf = 200 mW , FMR linewidth changes at f = 9.65 GHz , Gilbert damping enhancement due to spin pumping sp=YIG/NM−YIG (YIG= 9.1 ± 0.6 10-4) and resistivity of the six YIG/metal bilayers , and the calculated interfacial spin mixing conductance, spin Hall angle, and spin current density for each metal . Bilayer VISHE H change sp ( m) 𝑔↑↓(m-2) SH 𝐽𝑠 (A/m2) YIG/Pt 2.10 mV 24.3 Oe (3.6 ± 0.3) 10-3 4.8 10-7 (6.9 ± 0.6) 1018 0.10 ± 0.01 (2.0 ± 0.2) 107 YIG/Ta -5.10 mV 16.5 Oe (2.8 ± 0.2) 10-3 2.9 10-6 (5.4 ± 0.5) 1018 -0.069 ± 0.006 (1.6 ± 0.2) 107 YIG/W -5.26 mV 12.3 Oe (2.4 ± 0.2) 10-3 1.810-6 (4.5 ± 0.4) 1018 -0.14 ± 0.01 (1.4 ± 0.1) 107 YIG/Au 72.6 V 5.50 Oe (1.4 ± 0.1) 10-3 4.9 10-8 (2.7 ± 0.2) 1018 0.084 ± 0.007 (7.6 ± 0.7) 106 YIG/Ag 1.49 V 1.30 Oe (2.7 ± 0.2) 10-4 6.6 10-8 (5.2 ± 0.5) 1017 0.0068 ± 0.0007 (1.5 ± 0.1) 106 YIG/Cu 0.99 V 3.70 Oe (8.1 ± 0.6) 10-4 6.3 10-8 (1.6 ± 0.1) 1018 0.0032 ± 0.0003 (4.6 ± 0.4) 106 14 Figure Captions: Figure 1. (a) Semi -log θ-2θ XRD scan of a 20-nm thick YIG film near the YIG (444) peak (blue line) , which exhibits clear Laue oscillations corresponding to the film thickness. Left inset: rocking curve of the YIG film measured at 2θ = 50.639 for the first satellite peak (green arrow) to the left of the main YIG (444) peak gives a FWHM of 0. 0092. Righ t inset: AFM image of the YIG film with a roughness of 0.15 nm. (b) A representative room -temperature FMR derivative spectrum of a 20 -nm YIG film with an in -plane field at Prf = 0.2 mW, which gives a peak -to-peak linewidth of 11.7 Oe. (c) Schematic of experimental setup for ISHE voltage measurements. Figure 2 . VISHE vs. H spectra of (a) YIG/ Pt, (b) YIG/Ta, (c) YIG/W, (d) YIG/Au, (e) YIG/Ag, and (f) YIG/Cu bilayers at θH = 90(red) and 270 (blue) using Prf = 200mW . Top insets: rf-power dependencies of the corresponding VISHE at θH = 90. Bottom insets: a ngular dependenc ies (θH) of VISHE normalized by the magnitude of VISHE at θH = 90, where t he green curves are sin θH for Pt, Au, Ag, Cu, and -sinθH for Ta and W. Figure 3 . FMR derivative absorption spectr a of the 20-nm YIG films before (blue) and after (red) the deposition at f = 9.65 GHz of (a) Pt, (b) Ta, (c) W, (d) Au, (e) Ag, and (f) Cu. (g) Frequency dependence of peak -to-peak FMR linewi dth of a bare YIG film and the six YIG/ metal bilayers . Figure 4. (a) Spin Hall angles as a function of Z4 for Cu, Ag and Au , reflecting the Z4 dependence of SH for noble metals with the same d-orbital filling . (b) Spin Hall angles for 5d transition metals Ta, W, Pt and Au, of which b oth the signs and relative magnitudes agree well with the theoretical predictions in Ref. 3 1. 15 Figure 1. 100102104 49 50 51 52 53Intensity (c/s) 2 (deg)(a) GGG(444)YIG(444) 25.6 25.65 (deg)FWHM= 0.0092o 2400 2600 2800-2-1012dIFMR/dH (a.u.) H (Oe)H = 11.7 Oe Prf = 0.2 mW(b) (c) roughness: 0.15 nm16 Figure 2. -2000-1000010002000VISHE (V) (a)YIG/Pt H = 90o H = 270o010002000 0100 200Prf (mW) -6000-4000-20000200040006000VISHE (V) (b)YIG/Ta H = 90oH = 270o -6000-4000-20000 0100 200Prf (mW) -6000-4000-20000200040006000 -150 -100 -50 0 50 100 H - Hres (Oe)VISHE (V) (c)YIG/W H = 90oH = 270o -6000-4000-20000 0100 200Prf (mW)-80-4004080VISHE (V) (d)YIG/Au H = 90o H = 270o0204060 0100 200Prf (mW) -1.5-1-0.500.511.5 (e)YIG/Ag H = 90o H = 270oVISHE (V)00.511.5 0100 200Prf (mW) -1-0.500.51 -150 -100 -50 0 50 100 H - Hres (Oe)VISHE (V) (f)YIG/Cu H = 90o H = 270o00.51 0100 200Prf (mW)-101 0180 360 H (deg) -101 0180 360 H (deg) -101 0180 360H (deg)-101 0180 360 H (deg) -101 0180 360H (deg) -101 0180 360H (deg)17 Figure 3 . -2-1012 (a)YIG YIG/Pt -2-1012dIFMR/dH (a.u.) (b)YIG YIG/Ta -2-1012 -60-40-20 02040 H - Hres (Oe)(c)YIG YIG/W(d)YIG YIG/Au (e)YIG YIG/Ag -40-20 0204060 H -Hres (Oe)(f)YIG YIG/Cu 01020304050 0 5 10 15 20H (Oe) f (GHz)YIG/Pt YIG/Ta YIG/W YIG/Au YIG/Cu YIG/Ag YIG (g)18 Figure 4. 00.040.08 01x1072x1073x1074x107SH Z 4(a) CuAgAu -0.100.1 73747576777879 ZSH Ta WPtAu (b)

2013-07-10

Pure spin currents generated by spin pumping in ferromagnet/nonmagnet (FM/NM) bilayers produce inverse spin Hall effect (ISHE) voltages in the NM, from which spin pumping and transport characteristics of the NM can be extracted. Due to its exceptionally low damping, Y3Fe5O12 (YIG) is an important and widely used FM for microwave devices and ferromagnetic resonance (FMR) spin pumping. Here we report systematic investigation of spin pumping from 20-nm thick YIG thin films to a series of 3d, 4d and 5d normal metals (Cu, Ag, Ta, W, Pt and Au) with various spin-orbit coupling strengths. From enhanced Gilbert damping obtained from the frequency dependence of FMR linewidths and ISHE signals, the spin Hall angles and YIG/NM interfacial spin mixing conductances are quantitatively determined for these metals. The spin Hall angles largely vary as the fourth power of the atomic number, corroborating the dominant role of spin-orbit coupling across a broad range in the inverse spin Hall effect.

Scaling of spin Hall angle in 3d, 4d and 5d metals from Y3Fe5O12/metal spin pumping

1307.2648v2

Light-induced magnetization precession in GaMnAs E. Rozkotová, P. N ěmeca), P. Horodyská, D. Sprinzl, F. Trojánek, and P. Malý Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 3, 121 16 Prague 2, Czech Republic V. Novák, K. Olejník, M. Cukr, and T. Jungwirth Institute of Physics ASCR v.v.i., Cukrovarnická 10, 162 53 Prague, Czech Republic We report dynamics of the transient polar Kerr rotation (KR) and of the transient reflectivity induced by femtosecond laser pulses in ferromagnetic (Ga,Mn)As with no external magnetic field applied. It is s hown that the measured KR signal consist of several different contributions, among which only the oscillatory signal is directly connected with the ferromagnetic order in (Ga,Mn)As. The origin of the light-induced magnetization precession is discussed and the magnetization precession damping (Gilbert damping) is found to be strongly influenced by annealing of the sample. (Ga,Mn)As is the most intensively studied member of the family of diluted magnetic semiconductors with carrier-mediated ferromagn etism [1]. The sensitiv ity of ferromagnetism to concentration of charge carriers opens up th e possibility of magneti zation manipulation on the picosecond time scale using light pulses from ultrafast lasers [2]. Photoexcitation of a magnetic system can strongly disturb the equili brium between the mobile carriers (holes), localized spins (Mn ions), and the lattice. This in turn tr iggers a variety of dynamical processes whose characteristic time scales and st rengths can be investigated by the methods of time-resolved laser spectroscopy [2]. In part icular, the magnetization reversal dynamics in various magnetic materials attracts a significant attention because it is directly related to the speed of data storage in the magnetic reco rding [3]. The laser-induced precession of magnetization in ferromagnetic (Ga,Mn)As has been recently re ported by two research groups [4-6] but the physical processes responsible fo r it are still not well unde rstood. In this paper we report on simultaneous measurements of the light-induced magnetization precession dynamics and of the dynamics of photoinjected carriers. The experiments were performed on a 500 nm thick ferromagnetic Ga 1-xMn xAs film with x = 0.06 grown by the low temperature molecular beam epitaxy (LT-MBE) on a GaAs(001) substrate. We studied both the as-grown sample, with the Curie temperature TC ≈ 60 K and the conductivity of 120 Ω-1cm-1, and the sample annealed at 200°C for 30 hours, with TC ≈ 90 K and the conductivity of 190 Ω-1cm-1; using the mobility vs. hole density dependence typical for GaMnAs [7] we can r oughly estimate their hole densities as 1.5 x 1020 cm-3 and 3.4 x 1020 cm-3, respectively. Magnetic properties of the samples were measured using a superconducting quantum interference de vice (SQUID) with magnetic field of 20 Oe applied along different crysta llographic directions. The photo induced magnetization dynamics was studied by the time-resolved Kerr rotation (KR) technique [2] using a femtosecond titanium sapphire laser (Tsunami, Spectra Physics) . Laser pulses, with th e time width of 80 fs and the repetition rate of 82 MHz, were tune d to 1.54 eV. The energy fluence of the pump pulses was typically 15 μJ.cm-2 and the probe pulses were always at least 10 times weaker. The polarization of the pump pulses was either circular or linear, while the probe pulses were a) Electronic mail: nemec@karlov.mff.cuni.cz 1linearly polarized (typically along the [010] crystallographic direction in the sample, but similar results were obtained also for other orientations). The rotation angle of the polarization plane of the reflected probe pulses was obtained by taking the difference of signals measured by detectors in an optical br idge detection system [2]. Simultaneously, we measured also the sum of signals from the detectors, which corresponded to a probe intensity change due to the pump induced modification of the sample reflectivity. The experiment was performed with no external magnetic field applie d. However, the sample was cooled in some cases with no external magnetic field applied or alternatively with a ma gnetic field of 170 Oe applied along the [-110] direction. Fig. 1. Dynamics of photoinduced Kerr rotation angle (KR) measured for the as-grown sample at 10 K. (a) KR measured for σ + and σ - circularly polarized (CP) pump pulses; (b) KR measured for p and s linearly polarized (LP) pump pulses. Polarization-independent part (c) (polar ization-dependent part (d)) of KR signal, which was computed from the measured traces as an average of the signals (a half of the difference between the signals) detected for pump pulses with the opposite CP (LP). Inset: Fourier transform of the oscillations. No external magnetic field was applied during the sample cooling. In Fig. 1 we show typical te mporal traces of the transien t angles of KR measured for the as-grown sample at 10 K. The KR signal wa s dependent on the light polarization but there were certain features presen t for both the circular (Fig. 1 (a)) and linear (Fig. 1 (b)) polarizations. In Fig. 1 (c) we show the polari zation-independent part of the measured KR signal, which was the same for circular and line ar polarization of pump pulses. On the other hand, the amplitude of the polarization-dependent part of the signal (Fig. 1 (d)) was larger for the circular polarization. The interpretation of the polarization-dependent part of the signal is significantly complicated by the fact that the circularly polarized light generates spin- polarized carriers (electrons in particular), whose contribution to the measured KR signal can even exceed that of ferromagnetic ally coupled Mn spins [8]. In the following we concentrate on the polarization-independent part of the KR signal (Fig. 1 (c)). This signal can be fitted well (see Fig. 2) by an exponentially damped sine harmonic oscillation superimposed on a pulse-like function: () ( ) ( ) () [ ]()2 1 / exp / exp 1 sin / exp τ τ ϕωτ t t B t t A t KRD − −−+ + − = . (1) 2The oscillatory part of the KR signa l is characterized by the amplitude (A ), damping time ( τD), angular frequency ( fπω2= ), and phase ( ϕ). The pulse-like part of the KR signal is described by the amplitude ( B), rise time ( τ1), and decay time ( τ2). In the inset of Fig. 2 we show the dynamics of the sample reflectivity change ΔR/R. This signal monitored the change of the complex index of refraction of the sa mple due to carriers photoinjected by the pump pulse. From the dynamics of ΔR/R we can conclude that the population of photogenerated free carriers (electrons in par ticular [9]) decays within ≈ 50 ps after the photoinjection. This rather short lifetime of free electrons is similar to th at reported for the low temperature grown GaAs (LT-GaAs), which is generally interpreted as a consequence of a high concentration of nonradiative recombination centers induced by th e low temperature growth mode of the MBE [9]. It is also clearly apparent from the inset of Fig. 2 that the KR data can be fitted well by Eq. (1) only for time delays larger than ≈ 50 ps (i.e., just after th e population of photoinjected free electrons nonradiatively decayed). We will come back to this point later. Fig. 2. The fitting procedure applied to the polarization-independent part of KR signal. (a) The measured data from Fig. 1 (c) (points) are fitted (solid line) by a sum of the exponentially damped sine harmonic oscillation (solid line in part (b)) and the pulse-like KR signal (dashe d line in part (b)). Inset: Dynamics of the reflectivity change (thick solid line) and the detail of the fitted KR signal. In Fig. 3(a) we show the intensity dependence of A and B, and in Fig. 3(b) of ω and τD measured at 10 K. For the increasing inte nsity of pump pulses the magnitudes of A and B were increasing, ω was decreasing and the values of τD were not changing significantly. The application of magnetic field applied along the [-110] directi on during the sample cooling modified the value of ω. For 10 K (and pump intensity I0) the frequency decreased from 24.5 to 20 GHz (open and solid point in Fig. 3 (d ), respectively). The measured temperature dependence of A and B (Fig. 3 (c)) revealed that the oscillatory signal vanished above TC, while a certain fraction of the pulse-like KR signal persisted even above TC. This shows that only the oscillatory part of the KR si gnal was directly c onnected with the ferromagnetic order in (Ga,Mn)As. (It is worth noti ng that also the polarization-de pendent part of the KR signal was non-zero even above TC.) The frequency of oscillations was decreasing with the sample temperature (Fig. 3 (d)), but the values of τD were not changing sign ificantly (not shown here). 3 Fig. 3. Intensity dependence of ⎪A⎪and ⎪B⎪ (a), ω and τD (b) measured at 10 K; I0 = 15 μJ.cm-2, no external magnetic field was applied during the sample cooling. (c), (d) Temperature dependence of ⎪A⎪, ⎪B⎪ and ω (points) measured at pump intensity I0. The open point in (d) was obtained for the sample cooled with no external magnetic field applied and the data in (c) and the solid points in (d) were obtained for the sample cooled with magnetic field applied along the [-110]. The lines in (d) are the temperature dependence of the sample magnetization projections to different crystallographic directions measured by SQUID. The photoinduced magnetization precession was reported by A. Oiwa et al. , who attributed it to the precession of ferromagnetically coupled Mn spins induced by a change in magnetic anisotropy initiated by an increase in hole concentration [4]. It was also shown that the photoinduced magnetization precession a nd the ferromagnetic resonance (FMR) can provide similar information [4]. Magnetic an isotropy in (Ga,Mn)As is influenced by the intrinsic cubic anisotropy, which is arising from its zinc-blende symmetry, and by the uniaxial anisotropy, which is a result of a strain induced by different lattice constants of GaMnAs and the substrate. For the standard stressed GaMnAs films with Mn content above 2% grown on GaAs substrates the magnetic easy axes are in -plain. Consequently, the measured polar Kerr rotation is not sensitive to the steady state magn etization of the sample, but only to the light- induced transient out-of-plane magnetization due to the polar Kerr effect [2]. In our experiment, the pump pulses with a fluence I 0 = 15 μJ.cm-2 photoinjected electron-hole pairs with an estimated concentration Δp = Δn ≈ 8 x 1017 cm-3. This corresponded to Δp/p ≈ 0.5% and such a small increase in the hole concentrati on is highly improbable to lead to any sizable change of the sample anisotropy [1]. Another hypothesis about the origin of the light-induced magnetization precession was reported recently by J. Qi et al. [6]. The authors suggested that not only the transient increase in local hole concentration Δp but also the local temperature increase ΔT contributes to the change of anisotr opy constants. This modification of the sample anisotropy changes in turn the direction of the in-plane magnetic easy axis and, consequently, triggers a precessional motion of the magnetization around the altered magnetic anisotropy field . The magnitude of decreases as T (the sample temperature) or ΔT increases, primarily due to the decrease in the cubic anisotropy constant KMn anisHMn anisH 1c [6]. Our samples exhibit in-plane easy axis behavior typica l for stressed GaMnAs layers grown on GaAs substrates. To characterize their in-plane anis otropy we measured the temperature dependent magnetization projections to [110] , [010], and [-110] crystallographic directions – the results are shown in Fig. 3 (d) and in inset of Fig. 4 for the as-grown and the annealed sample, respectively. At low temperatures the cubi c anisotropy dominates (as indicated by the 4maximal projection measured along the [010] di rection) but the uniaxial in-plane component is not negligible and the sample magnetization is slightly tilted from the [010] direction towards the [-110] direction. Both samples exhibit rotation of magnetizat ion direction in the temperature region 10-25K, which is in agreement with the expected fast weakening of the cubic component with an increasing temperatur e. In our experiment, the excitation fluence I0 led to ΔT ≈ 10 K (as estimated from the GaAs specifi c heat of 1 mJ/g/K [6]) that can be sufficient for a change of the easy axis positi on. This temperature-ba sed hypothesis about the origin of magnetization precession is supported also by our observation that the oscillations were not fully developed immediately after the photoinj ection of carriers but only after ≈ 50 ps when phonons were emitted by the nonradiative decay of the population of free electrons (see inset in Fig. 2). We also point out that the measured precession frequency ω and the sample magnetization M (measured by SQUID) had very similar temperature dependence (see Fig. 3(d)). Fig. 4. Polarization-independent part of KR signal measured for the annealed sample at 10 K; I0 = 15 μJ.cm-2, the sample was cooled with magnetic field applied along the [-110]. Inset: Temperature dependence of the sample magnetization projections to different crystallographic directions measured by SQUID. An example of the results measured for the annealed sample is shown in Fig. 4. The analysis of the data revealed that at simila r conditions the precession frequency was slightly higher in the annealed sample (20 GHz and 24 GHz for the as-grown and the annealed sample, respectively). However, a major effect of the sample annealing was on the oscillation damping time τD, which increased from 0.4 ns to 1.1 ns. This prolongation of τD can be attributed to the improved quality of the ann ealed sample, which is indicated by the higher value of TC and by the more Brillouin-like temperature dependence of the magnetization (cf. Fig. 3 (d) and inset in Fig. 4). The damping of oscillations is connected with the precession damping in the Landau-Lifshitz-Gilbert equation [1]. The exact determination of the intrinsic Gilbert damping coefficient α from the measured data is not straightforward because it is difficult to decouple the contribution due to the inhomogeneous broadening [10]. In Ref. 6 the values of α from 0.12 to 0.21 were deduced for the as -grown sample from the analysis of the oscillatory KR signal. The time-domain KR should provide similar information as the frequency-domain based FMR, where the relaxa tion rate of the magnetization is connected with the peak-to-peak ferromagnetic resonance linewidth ΔHpp [10]. Indeed both methods showed that the relaxation rate of the magnetization is consider ably slower in the annealed samples (as indicated by the prolongation of τD in our experiment and by the reduction of ΔHpp in FMR [10]). 5 In conclusion, we studied the transient Ke rr rotation (KR) and the reflectivity change induced by laser pulses in (Ga ,Mn)As with no external magnetic field applied. We revealed that the measured KR signals consisted of seve ral different contributions and we showed that only the oscillatory KR signal was directly connected with the fe rromagnetic order in (Ga,Mn)As. Our data indicated that the phonons emitted by photoinjected carriers during their nonradiative recombination in (Ga,Mn)As can be re sponsible for the magnetic anisotropy change that was triggering the magnetization precession. We also observed that the precession damping was strongly suppressed in the ann ealed sample, which reflected its improved magnetic properties. This work was supported by Ministry of Education of the Czech Republic in the framework of the rese arch centre LC510, the research plans MSM0021620834 and AV0Z1010052, by the Grant Agen cy of the Charles University in Prague under Grant No. 252445, and by the Grant Agency of Academy of Sciences of the Czech Republic Grants FON/06/E 001, FON/06/E002, and KAN400100652. References [1] T. Jungwirth, J. Sinova, J. Maše k, A. H. MacDonald, Rev. Mod. Phys. 78, 809 (2006). [2] J. Wang, Ch. Sun, Y. Hashimoto, J. Kono, G.A. Khodaparast, L. Cywinski, L.J. Sham, G.D. Sanders, Ch.J. Stanton, H. Munekata, J. Phys.: Condens. Matter 18, R501 (2006). [3] A.V. Kimel, A. Kirilyuk, F. Hansteen, R.V. Pisarev, T. Ra sing, J. Phys.: Condens. Matter 19, 043201 (2007). [4] A. Oiwa, H. Takechi, H. Munekata, J. Supercond. 18, 9 (2005). [5] H. Takechi, A. Oiwa, K. Nomura, T. Kondo, H. Munekata, phys. stat. sol. (c) 3, 4267 (2006). [6] J. Qi, Y. Xu, N.H. Tolk, X. Liu, J.K. Furdyna, I.E. Perakis, Appl. Phys. Lett. 91, 112506 (2007). [7] T. Jungwirth et al., Phys. Rev. B 76, 125206 (2007). [8] A.V. Kimel, G.V. Astakhov, G.M. Schott, A. Kirilyuk, D. R. Yakovlev, G. Karczewski, W. Ossau, G. Schmidt, L.W. Molenkamp, Th. Rasing, Phys. Rev. Lett. 92, 237203 (2004). [9] M. Stellmacher, J. Nagle, J.F. Lampin, P. Santoro, J. Van eecloo, A. Alexandrou, J. Appl. Phys. 88, 6026 (2000). [10] X. Liu, J.K. Furdyna, J. Phys.: Condens. Matter 18, R245 (2006). 6

2008-02-14

We report dynamics of the transient polar Kerr rotation (KR) and of the transient reflectivity induced by femtosecond laser pulses in ferromagnetic (Ga,Mn)As with no external magnetic field applied. It is shown that the measured KR signal consist of several different contributions, among which only the oscillatory signal is directly connected with the ferromagnetic order in (Ga,Mn)As. The origin of the light-induced magnetization precession is discussed and the magnetization precession damping (Gilbert damping) is found to be strongly influenced by annealing of the sample.

Light-induced magnetization precession in GaMnAs

0802.2043v2

Mechanisms behind large Gilbert damping anisotropies I. P. Miranda1, A. B. Klautau2,∗A. Bergman3, D. Thonig3,4, H. M. Petrilli1, and O. Eriksson3,4 1Universidade de São Paulo, Instituto de Física, Rua do Matão, 1371, 05508-090, São Paulo, SP, Brazil 2Faculdade de Física, Universidade Federal do Pará, Belém, PA, Brazil 3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden and 4School of Science and Technology, Örebro University, Fakultetsgatan 1, SE-701 82 Örebro, Sweden (Dated: November 22, 2021) A method with which to calculate the Gilbert damping parameter from a real-space electronic structure method is reported here. The anisotropy of the Gilbert damping with respect to the magnetic moment direction and local chemical environment is calculated for bulk and surfaces of Fe 50Co50alloys from first principles electronic structure in a real space formulation. The size of the damping anisotropy for Fe 50Co50alloys is demonstrated to be significant. Depending on details of the simulations, it reaches a maximum-minimum damping ratio as high as 200%. Several microscopic origins of the strongly enhanced Gilbert damping anisotropy have been examined, where in particular interface/surface effects stand out, as do local distortions of the crystal structure. Although theory does not reproduce the experimentally reported high ratio of 400% [Phys. Rev. Lett. 122, 117203 (2019)], it nevertheless identifies microscopic mechanisms that can lead to huge damping anisotropies. Introduction: Magnetic damping has a critical impor- tanceindeterminingthelifetime,diffusion,transportand stability of domain walls, magnetic vortices, skyrmions, and any nano-scale complex magnetic configurations [1]. Given its high scientific interest, a possibility to obtain this quantity by means of first-principles theory [2] opens new perspectives of finding and optimizing materials for spintronic and magnonic devices [3–8]. Among the more promising ferromagnets to be used in spintronics devices, cobalt-iron alloys demonstrate high potentials due to the combination of ultralow damping with metallic conduc- tivity [4, 9]. Recently, Li et al.[10] reported an observed, gi- ant anisotropy of the Gilbert damping ( α) in epitaxial Fe50Co50thin films (with thickness 10 −20nm) reach- ing maximum-minimum damping ratio values as high as 400%. TheauthorsofRef. [10]claimedthattheobserved effect is likely due to changes in the spin-orbit coupling (SOC) influence for different crystalline directions caused by short-range orderings that lead to local structural dis- tortions. This behaviour differs distinctly from, for ex- ample, pure bcc Fe [11]. In order to quantitatively pre- dict the Gilbert damping, Kambersky’s breathing Fermi surface (BFS) [12] and torque-correlation (TC) [13] mod- els are frequently used. These methods have been ex- plored for elements and alloys, in bulk form or at sur- faces, mostly via reciprocal-space ab-initio approaches, in a collinear or (more recently) in a noncollinear con- figuration [14]. However, considering heterogeneous ma- terials, such as alloys with short-range order, and the possibility to investigate element specific, non-local con- tributions to the damping parameter, there are, to the best of our knowledge, no reports in the literature that rely on a real space method. In this Letter, we report on an implementation of ab initiodamping calculations in a real-space linear muffin-tin orbital method, within the atomic sphere approxi- mation (RS-LMTO-ASA) [15, 16], with the local spin density approximation (LSDA) [17] for the exchange- correlation energy. The implementation is based on the BFSandTCmodels, andthemethod(SupplementalMa- terial - SM, for details) is applied to investigate the re- ported, huge damping anisotropy of Fe50Co50(100)/MgO films [10]. A main result here is the identification of a microscopic origin of the enhanced Gilbert damping anisotropy of Fe50Co50(100) films, and the intrinsic rela- tionships to the local geometry of the alloy. Most signifi- cantly, wedemonstratethatasurfaceproducesextremely large damping anisotropies that can be orders of magni- tude larger than that of the bulk. We call the attention to the fact that this is the first time, as far as we know, that damping values are theoretically obtained in such a local way. Results: We calculated: i)ordered Fe50Co50in theB2 structure (hereafter refereed to as B2-FeCo) ii)random Fe50Co50alloysinbccorbctstructures, wherethevirtual crystal approximation (VCA) was applied; iii)Fe50Co50 alloys simulated as embedded clusters in a VCA matrix (host). In all cases VCA was simulated with an elec- tronic concentration corresponding to Fe50Co50. The ii) andiii)alloys were considered as in bulk as well as in the (001) surface, with bcc and bct structures (here- after correspondingly refereed as VCA Fe50Co50bcc, VCA Fe50Co50bct, VCA Fe50Co50(001) bcc and VCA Fe50Co50(001) bct). The effect of local tetragonal distor- tions was considered with a localc a= 1.09ratio (SM for details). All data for cluster based results, were obtained from an average of several different configurations. The total damping for a given site iin real-space ( αt, Eqs. S6 and S7 from SM) can be decomposed in non-local, αij (i/negationslash=j), and local (onsite), αonsite(orαii,i=j) contri- butions, each of them described by the tensor elements2 ανµ ij=g miπ/integraldisplay η(/epsilon1)Tr/parenleftBig ˆTν iImˆGijˆTµ jImˆGji/parenrightBig d/epsilon1,(1) wheremiis the total magnetic moment localized in the reference atomic site i,µ,ν={x,y,z},ˆTis the torque operator, and η(/epsilon1) =∂f(/epsilon1) ∂/epsilon1the derivative of the Fermi distribution. The scalar αijparameter is defined in the collinear regime as αij=1 2(αxx ij+αyy ij). To validate our methodology, the here obtained total damping for several systems (such as bcc Fe, fcc Ni, hcp and fcc Co and B2-FeCo) were compared with estab- lished values available in the literature (Table S1, SM), where an overall good agreement can be seen. Fig. 1 shows the non-local contributions to the damp- ing for bcc Fe and B2-FeCo. Although the onsite contri- butions are around one order of magnitude larger than the non-local, there are many αijto be added and total net values can become comparable. Bcc Fe and B2-FeCo have very different non-local damping contributions. El- ement resolved αij, reveal that the summed Fe-Fe in- teractions dominate over Co-Co, for distances until 2a inB2-FeCo. We observe that αijis quite extended in space for both bcc Fe and B2-FeCo. The different con- tributions to the non-local damping, from atoms at equal distance arises from the reduced number of operations in the crystal point group due to the inclusion of SOC in combination with time-reversal symmetry breaking. The B2-FeCo arises from replacing every second Fe atom in the bcc structure by a Co atom. It is interesting that this replacement (i.e. the presence of Co in the environment) significantly changes the non-local contributions for Fe- Fe pairs , what can more clearly be seen from the Insetin Fig. 1, where the non-local damping summed over atoms at the same relative distance for Fe-Fe pairs in bcc Fe andB2-FeCo are shown; the non-local damping of Fe-Fe pairs are distinctly different for short ranges, while long ranged (further than ∼2.25 Å) contributions are smaller and more isotropic. The damping anisotropy, i.e. the damping change, when the magnetization is changed from the easy axis to a new direction is1 ∆αt=/parenleftBigg α[110] t α[010] t−1/parenrightBigg ×100%, (2) whereα[110] tandα[010] tare the total damping obtained for magnetization directions along [110]and [010], re- spectively. Analogousdefinitionalsoappliesfor ∆αonsite. 1We note that this definition is different to the maximum- minimum damping ratio, defined asα[110] t α[010] t×100%, from Ref. [10].We investigated this anisotropy in surfaces and in bulk systems with (and without) tetragonal structural distor- tions. Our calculations for VCA Fe50Co50bcc show a damping increase of ∼13%, when changing the magne- tization direction from [010]to[110](Table S2 in the SM). The smallest damping is found for the easy magne- tization axis, [010], which holds the largest orbital mo- ment (morb) [18]. For VCA Fe50Co50bcc we obtained a small variation of ∼2%for the onsite contribution (α[010] onsite = 8.94×10−4andα[110] onsite = 8.76×10−4), what implies that the anisotropy comes mostly from the non-local contributions, particularly from the next- nearest neighbours. For comparison, ∆αt∼3%(with ∆αonsite∼0.4%) in the case of bcc Fe, what corrobo- rates the reported [11] small bcc Fe anisotropy at room temperature, andwiththebulkdampinganisotropyrates [19]. We also inspected the chemical inhomogeneity influ- ence on the anisotropy, considering the B2-FeCo alloy, where the weighted average damping (Eq. S7 of SM) was used instead. The B2-FeCo bcc (∼7%) and VCA Fe50Co50bcc (∼13%) anisotropies are of similar magni- tudes. Both B2structure and VCA calculations lead to damping anisotropies which are significantly lower than whatwasobservedintheexperiments, anditseemslikely thatthepresenceofdisorderincompositionand/orstruc- tural properties of the Fe/Co alloy would be important to produce large anisotropy effects on the damping. α ij × 10 -4 −3 −2 −1 0 1 2 3 4 Normalized distance 1.0 1.5 2.0 2.5 3.0 B2 Fe-Co B2 Fe-Fe B2 Co-Co bcc Fe-Fe −10 0 10 20 1.0 1.5 2.0 2.5 3.0 Figure 1. (Color online) Non-local damping contributions, αij, in (Fe-centered) bulk B2-FeCo and bcc Fe, as a function of the normalized distance in lattice constant units a.Inset: Non-local contributions from only Fe-Fe pairs summed, for each distance, in bcc Fe bulk (empty blue dots) and in the B2-FeCo (full red dots). The onsite damping for Fe (Co) in B2-FeCo isαFe onsite = 1.1×10−3(αCo onsite = 0.8×10−3) and for bcc Fe it is αFe onsite = 1.6×10−3. The magnetization direction isz([001]). Lines are guides for the eyes. Weanalyzedtheroleoflocaldistortionsbyconsidering3 a hypothetical case of a large, 15%(c a= 1.15), distortion on thez-axis of ordered B2-FeCo. We found the largest damping anisotropy ( ∼24%) when comparing the results with magnetization in the [001](α[001] t= 10.21×10−3) and in the [010](α[010] t= 7.76×10−3) directions. This confirms that, indeed, bct-like distortions act in favour of the∆αtenhancement (and therefore, of the maximum- minimum damping ratio), but the theoretical data are not large enough to explain the giant value reported ex- perimentally [10]. Nevertheless, in the case of an alloy, the local lattice distortions suggested in Ref. [10] are most to likely occur in an heterogeneous way [20], with different distortions for different local environments. To inspect this type of influence on the theoretical results, we investigated (Table S3, SM) clusters containing different atomic con- figurations embedded in a VCA Fe50Co50matrix (with Fe bulk lattice parameter); distortions were also consid- ered such that, locally in the clusters,c a= 1.15(Ta- ble S4, in the SM). Moreover, in both cases, two types of clusters have to be considered: Co-centered and Fe- centered. The αtwas then computed as the sum of the local and non-local contributions for clusters with a spe- cific central (Fe or Co) atom, and the average of Fe- and Co-centered clusters was taken. Fe-centered clus- tershaveshownlargeranisotropies, onaverage ∼33%for the undistorted (∼74%for the distorted) compared with ∼8%fortheundistortedCo-centeredclusters( ∼36%for the distorted). Although these results demonstrate the importance of both, local distortions as well as non-local contributions to the damping anisotropy, they are not still able to reproduce the huge observed [10] maximum- minimum damping ratio. We further proceed our search for ingredients that could lead to a huge ∆αtby inspecting interface effects, which are present in thin films, grain boundaries, stack- ing faults and materials in general. Such interfaces may influence observed properties, and in order to examine if they are relevant also for the reported alloys of Ref. [10], we considered these effects explicitly in the calcu- lations. As a model interface, we considered a surface, what is, possibly, the most extreme case. Hence, we per- formed a set of αtcalculations for the Fe50Co50(001), first on the VCA level. Analogous to the respective bulk systems, we found that the onsite contributions to the damping anisotropy are distinct, but they are not the main cause ( ∆αonsite∼18%). However, the lack of in- version symmetry in this case gives a surprisingly large enhancement of ∆αt, thus having its major contribution coming from the non-local damping terms, in particular from the next-nearest neighbours. Interestingly, negative non-local contributions appear when αtis calculated in the[010]direction. These diminish the total damping (the onsite contribution being always positive) and gives rise to a larger anisotropy, as can be seen by comparisonof the results shown in Table I and Table S5 (in the Sup- plemental Material). In this case, the total anisotropy was found to be more than ∼100%(corresponding to a maximum-minimum damping ratio larger than 200%). A compilation of the most relevant theoretical results obtained here is shown in Fig. 2, together with the ex- perimental data and the local density of states (LDOS) atEFfor each magnetization direction of a typical atom in the outermost layer (data shown in yellow). As shown in Fig. 2, the angular variation of αthas a fourfold ( C4v) symmetry, with the smallest Gilbert damping occurring at 90◦from the reference axis ( [100],θH= 0◦), for both surface and bulk calculations. This pattern, also found experimentally in [10], matches the in-plane bcc crys- tallographic symmetry and coincides with other mani- festations of SOC, such as the anisotropic magnetoresis- tance [10, 21]. Following the simplified Kambersky’s for- mula [13, 22], in which (see SM) α∝n(EF)and, there- fore, ∆α∝∆n(EF), we can ascribe part of the large anisotropy of the FeCo alloys to the enhanced LDOS dif- ferences at the Fermi level, evidenced by the close corre- lation between ∆n(EF)and∆αtdemonstrated in Fig. 2. Thus, as a manifestation of interfacial SOC (the so-called proximity effect [23]), the existence of ∆αtcan be under- stoodintermsofRashba-likeSOC,whichhasbeenshown to play an important role on damping anisotropy [24, 25]. Analogous to the bulk case, the higher morboccurs where the system presents the smallest αt, and the orbital moment anisotropy matches the ∆αtfourfold symme- try with a 90◦rotation phase (see Fig. S3, SM). Note that a lower damping anisotropy than Co50Fe50(001) is found for a pure Fe(001) bcc surface, where it is ∼49% (Table S2, SM), in accordance with Refs. [7, 26], with a dominant contribution from the onsite damping val- ues (conductivity-like character on the reciprocal-space [19, 27]). The VCA surface calculations on real-space allows to investigate the layer-by-layer contributions (intra-layer damping calculation), as shown in Table I. We find that themajorcontributiontothedampingsurfaceanisotropy comes from the outermost layer, mainly from the differ- ence in the minority 3dstates around EF. The deeper layers exhibit an almost oscillatory ∆αtbehavior, simi- lar to the oscillation mentioned in Ref. [28] and to the Friedel oscillations obtained for magnetic moments. The damping contributions from deeper layers are much less influenced by the inversion symmetry breaking (at the surface), as expected, and eventually approaches the typ- icalbulklimit. Therefore, changesintheelectronicstruc- tureconsiderednotonlytheLDOSoftheoutermostlayer but a summation of the LDOS of all layers (including the deeper ones), which produces an almost vanishing differ- ence between θH= 0◦andθH= 45◦(also approaching the bulk limit). The damping anisotropy arising as a sur- face effect agrees with what was observed in the case of Fe [7] and CoFeB [29] on GaAs(001), where the damping4 0o45o90o 135o 180o 225o 270o315o[100][110][010] [−110] 0.10.20.3 Δn(EF) (st./Ry−at.) 0.0050.0100.015 αt θH Figure 2. (Color online) Total damping and LDOS difference atEF,∆n(EF), as a function of θH, the angle between the magnetizationdirectionandthe [100]-axis. Squares: (redfull) VCAFe 50Co50(001)bcc. Triagles: (greenfull)averageover32 clusters (16 Fe-centered and 16 Co-centered), with bcc struc- ture at the surface layers (SM) embedded in a VCA medium; (gray open) similar calculations, but with a local lattice dis- tortion. Circles: (yellow open) ∆n(EF)betweenθH= 0◦and the current angle for a typical atom in the outermost layer of VCA Fe 50Co50(001) bcc; (blue full) experimental data [10] for a 10-nm Fe 50Co50/Pt thin film; (purple full) average bulk VCA Fe 50Co50bcc; and (brown full) the B2-FeCo bulk. Lines are guides for the eyes. anisotropy diminishes as the film thickness increases. Table I. Total intra-layer damping ( αt×10−3) and anisotropy, ∆αt(Eq. 2), of a typical (VCA) atom in each Fe 50Co50(001) bcc surface layer for magnetization along [010]and[110]di- rections. In each line, the sum of all αijin the same layer is considered. Outermost (layer 1) and deeper layers (2-5). Layerαt[010]αt[110] ∆αt 1 7.00 14.17 +102.4% 2 1.28 1.16 −9.4% 3 2.83 3.30 +16.6% 4 2.18 1.99 −8.7% 5 2.54 2.53 −0.4% We also studied the impact of bct-like distortions in thesurface, initiallybyconsideringtheVCAmodel. Sim- ilartothebulkcase,tetragonaldistortionsmaybeimpor- tant for the damping anisotropy at the surface, e.g. when local structural defects are present. Therefore, localized bct-like distortions of the VCA medium in the surface, particularly involving the most external layer were inves- tigated. The structural model was similar to what was used for the Fe50Co50bulk, consideringc a= 1.09(see SM). Our calculations show that tetragonal relaxations around a typical site in the surface induce a ∆αt∼75%, fromα[010] t= 8.94×10−3toα[110] t= 15.68×10−3. Themain effect of these distortions is an enhancement of the absolute damping values in each direction with respect to the pristine (bcc) system. This is due to an increase on αonsite, fromα[010] onsite = 7.4×10−3toα[010] onsite = 9.5×10−3, and fromα[110] onsite = 8.7×10−3toα[110] onsite = 11.7×10−3; the resulting non-local contributions remains similar to theundistortedcase. Theinfluenceofbct-likedistortions on the large damping value in the Fe50Co50surface is in line with results of Mandal et al.[30], and is related to the transition of minority spin electrons around EF. We then considered explicit 10-atom Fe50Co50clusters embedded in a VCA FeCo surface matrix. The results from these calculations were obtained as an average over 16 Fe-centered and 16 Co-centered clusters. We con- sidered clusters with undistorted bcc crystal structure (Fig. 2, yellow open circles) as well as clusters with lo- cal tetragonal distortions (Fig. 2, black open circles). As shown in Fig. 2 the explicit local tetragonal distortion influences the damping values ( α[010] t= 10.03×10−3and α[110] t= 14.86×10−3)andtheanisotropy, butnotenough toreproducethehugevaluesreportedintheexperiments. A summary of the results obtained for each undis- torted FeCo cluster at the surface is shown in Fig. 3: Co-centered clusters in Fig. 3(a) and Fe-centered clusters in Fig. 3(b). A large variation of αtvalues is seen from clustertocluster, dependingonthespatialdistributionof atomic species. It is clear that, αtis larger when there is a larger number of Fe atoms in the surface layer that sur- roundsthecentral, referenceclustersite. Thiscorrelation can be seen by the numbers in parenthesis on top of the blue symbols (total damping for each of the 16 clusters that were considered) in Fig. 3. We also notice from the figure that the damping in Fe-centered clusters are lower than in Co-centered, and that the [010]magnetization di- rection exhibit always lower values. In the Insetof Fig. 3 the onsite contributions to the damping, αonsite, and the LDOS atEFin the central site of each cluster are shown: a correlation, where both trends are the same, can be ob- served. The results in Fig. 3 shows that the neighbour- hood influences not only the local electronic structure at the reference site (changing n(EF)andαonsite), but also modifies the non-local damping αij, leading to the cal- culatedαt. In other words, the local spatial distribution affects how the total damping is manifested, something which is expressed differently among different clusters. This may open up for materials engineering of local and non-local contributions to the damping. Conclusions: We demonstrate here that real-space electronic structure, based on density functional theory, yield a large Gilbert damping anisotropy in Fe50Co50al- loys. Theory leads to a large damping anisotropy, when the magnetization changes from the [010]to the [110]di- rection, which can be as high as ∼100%(or200%in the minimum-maximum damping ratio) when surface calcu- lations are considered. This is in particular found for5

  










 
  
  
    
  


  
 
  




 
 
 
(0)(1)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(3)(3)(4)(4) (1)(1)(2)(2)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(4)(5) Figure 3. Damping for the [010](open circles) and [110](full circles) magnetization directions for distinct types of 10-atom Fe50Co50bcc clusters, embedded in VCA Fe 50Co50(001) bcc and without any distortion around the reference atom (for whichαtandαonsiteare shown). (a) Co-centered and (b) Fe-centered clusters. The quantity of Fe atoms in the surface layers (near vacuum) are indicated by the numbers in paren- thesis and the results have been ordered such that larger val- ues are to the left in the plots. Insets:αonsitefor the [010] (red open circles) and [110](blue filled circles) magnetization directions, and corresponding local density of states, n(EF), at the Fermi level (green filled and unfilled triangles) at the central atom (placed in the outermost layer) for both types of clusters. Lines are guides for eyes. contributions from surface atoms in the outermost layer. Hence the results presented here represents one more ex- ample, in addition to the well known enhanced surface orbital moment [31], of the so-called interfacial spin-orbit coupling. This damping anisotropy, which holds a bcc- like fourfold ( C4v) symmetry, has a close relation to the LDOS difference of the most external layer at EF(ma- jorly contributed by the minority dstates), as well as to the orbital moment anisotropy with a 90◦phase. As a distinct example of an interface, we consider explicitly the Fe50Co50cluster description of the alloy. In this case, besides an onsite contribution, we find that the damp- ing anisotropy is mostly influenced by non-local next- nearest-neighbours interactions. Several Gilbert damping anisotropy origins are also demonstrated here, primarily related to the presence of interfaces, alloy composition and local structural distor- tions (as summarized in Table S6, in the SM [32]). Pri- marily we find that: ( i) the presence of Co introduces anenhanced spin-orbit interaction and can locally modify the non-local damping terms; ( ii) the randomness of Co in the material, can modestly increase ∆αtas a total ef- fect by creating Co-concentrated clusters with enhanced damping; ( iii) at the surface, the spatial distribution of Fe/Co, increases the damping when more Fe atoms are present in the outermost layer; and ( iv) the existence of local, tetragonal distortions, which act in favour (via SOC) of the absolute damping enhancement, by modify- ing theαonsiteof the reference atom, and could locally change the spin relaxation time. Furthermore, in rela- tionship to the work in Ref. [10], we show here that bulk like tetragonal distortions, that in Ref. [10] were sug- gested to be the key reason behind the observed huge anisotropy of the damping, can in fact not explain the experimental data. Such distortions were explicitly con- sidered here, using state-of-the-art theory, and we clearly demonstrate that this alone can not account for the ob- servations. Although having a similar trend as the experimen- tal results of Ref. [10], we do not reproduce the most extreme maximum-minimum ratio reported in the ex- periment,∼400%(or∆αt∼300%). The measured damping does however include effects beyond the intrin- sic damping that is calculated from our electronic struc- turemethodology. Other mechanismsare knownto influ- ence the damping parameter, such as contributions from eddy currents, spin-pumping, and magnon scattering, to name a few. Thus it is possible that a significant part of the measured anisotropy is caused by other, extrin- sic, mechanisms. Despite reasons for differences between observation and experiment on films of Fe50Co50alloys, the advancements presented here provide new insights on the intrinsic damping anisotropy mechanisms, something which is relevant for the design of new magnetic devices. Acknowledgements: H.M.P. and A.B.K. acknowledge financial support from CAPES, CNPq and FAPESP, Brazil. The calculations were performed at the computa- tional facilities of the HPC-USP/CENAPAD-UNICAMP (Brazil), at the National Laboratory for Scientific Com- puting (LNCC/MCTI, Brazil), and at the Swedish Na- tional Infrastructure for Computing (SNIC). I.M. ac- knowledge financial support from CAPES, Finance Code 001, process n◦88882.332894/2018-01, and in the Insti- tutional Program of Overseas Sandwich Doctorate, pro- cess n◦88881.187258/2018-01. 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Eriksson3,4 1Universidade de São Paulo, Instituto de Física, Rua do Matão, 1371, 05508-090, São Paulo, SP, Brazil 2Faculdade de Física, Universidade Federal do Pará, Belém, PA, Brazil 3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden and 4School of Science and Technology, Örebro University, Fakultetsgatan 1, SE-701 82 Örebro, Sweden (Dated: November 22, 2021) I. Theory The torque-correlation model, first introduced by Kamberský [1], and later elaborated by Gilmore et al. [2], can be considered as both a generalization and an extended version of the breathing Fermi surface model, which relates the damping of the electronic spin orienta- tion, with the variation in the Fermi surface when the local magnetic moment is changed. In this scenario, and considering the collinear limit of the magnetic or- dering, due to the spin-orbit coupling (SOC), the tilting in magnetization ˆmby a small change δˆmgenerates a non-equilibrium population state which relaxes within a timeτtowards the equilibrium. We use an angle θ, to represent the rotation of the magnetization direction δˆm. IftheBlochstatesofthesystemsarecharacterizedbythe genericbandindex natwavevector k(withenergies /epsilon1k,n), it is possible to define a tensorfor the damping, that has matrix elements (adopting the isotropic relaxation time approximation) ανµ=gπ m/summationdisplay n,mdk (2π)3η(/epsilon1k,n)/parenleftbigg∂/epsilon1k,n ∂θ/parenrightbigg ν/parenleftbigg∂/epsilon1k,m ∂θ/parenrightbigg µτ ~ (S1) which accounts for both intraband ( n=m, conductivity- like) and interband ( n/negationslash=m, resistivity-like) contribu- tions [2]. Here µ,νare Cartesian coordinate indices, that will be described in more detail in the discussion below, while η(/epsilon1k,n) =∂f(/epsilon1) ∂/epsilon1/vextendsingle/vextendsingle/vextendsingle /epsilon1k,nis the derivative of the Fermi distribution, f, with respect to the energy /epsilon1, andn,mare band indices. Therefore, the torque- correlation model correlates the spin damping to vari- ations of the energy of single-particle states with respect to the variation of the spin direction θ, i.e.∂/epsilon1k,n ∂θ. Us- ing the Hellmann-Feynmann theorem, which states that ∂/epsilon1k,n ∂θ=/angbracketleftψk,n|∂H ∂θ|ψk,n/angbracketright, and the fact that only the spin- orbit Hamiltonian Hsochanges with the magnetization direction, the spin-orbit energy variation is given by ∂/epsilon1k,n(θ) ∂θ=/angbracketleftψk,n|∂ ∂θ/parenleftbig eiσ·ˆnθHsoe−iσ·ˆnθ/parenrightbig |ψk,n/angbracketright(S2) in which σrepresents the Pauli matrices vector, and ˆnis the direction around which the local moment hasbeen rotated. The expression in Eq. S2 can be eas- ily transformed into∂/epsilon1k,n(θ) ∂θ=i/angbracketleftψk,n|[σ·ˆn,Hso]|ψk,n/angbracketright and we call ˆT= [σ·ˆn,Hso]thetorqueoperator. In view of this, it is straightforward that, in the collinear case in which all spins are aligned to the zdirection, σ·ˆn=σµ(µ=x,y,z), originating the simplest {x,y,z}- dependent torque operator ˆTµ. Putting together the in- formation on Eqs. S1 and S2, and using the fact that the imaginary part of the Greens’ functions can be ex- pressed, in Lehmann representation, as Im ˆG(/epsilon1±iΛ) = −1 π/summationtext nΛ (/epsilon1−/epsilon1n)2+Λ2|n/angbracketright/angbracketleftn|, then it is possible to write in reciprocal-space [3]: ανµ=g mπ/integraldisplay /integraldisplay η(/epsilon1)Tr/parenleftBig ˆTνImˆGˆTµImˆG/parenrightBig d/epsilon1dk (2π)3.(S3) In a real-space formalism, the Fourier transformation of the Green’s function is used to find a very similar ex- pression emerges for the damping element ανµ ijrelative to two atomic sites iandj(at positions riandrj, respec- tively) in the material: ανµ ij=g miπ/integraldisplay η(/epsilon1)Tr/parenleftBig ˆTν iImˆGijˆTµ jImˆGji/parenrightBig d/epsilon1,(S4) where we defined mi= (morb+mspin)as the total mag- netic moment localized in the reference atomic site iin the pair{i,j}. The electron temperature that enters into η(ε)is zero and, consequently, the energy integral is per- formed only at the Fermi energy. In this formalism, then, the intraband and interband terms are replaced by onsite (i=j) and non-local ( i/negationslash=j) terms. After calculation of all components of Eq. S4 in a collinear magnetic back- ground, we get a tensor of the form αij= αxxαxyαxz αyxαyyαyz αzxαzyαzz , (S5) which can be used in the generalized atomistic Landau- Lifshitz-Gilbert (LLG) equation for the spin-dynamics of magnetic moment on site i[4]:∂mi ∂t=mi×/parenleftBig −γBeff i+/summationtext jαij mj·∂mj ∂t/parenrightBig . Supposing that all spins are parallel to the local zdirection, we can define the scalar2 αvalue as the average between components αxxandαyy, that is:α=1 2(αxx+αyy). Once one has calculated the onsite ( αonsite) and the non-local (αij) damping parameters with respect to the site of interest i, the total value, αt, can be defined as the sum of all these α’s: αt=/summationdisplay {i,j}αij. (S6) In order to obtain the total damping in an heteroge- neous atomic system (more than one element type), such as Fe50Co50(with explicit Fe/Co atoms), we consider the weighted average between the different total local damp- ing values ( αi t), namely: αt=1 Meff/summationdisplay imiαi t, (S7) wheremiis the local magnetic moment at site i, and Meff=/summationtext imiis the summed total effective magnetiza- tion. This equation is based on the fact that, in FMR ex- periments, the magnetic moments are excited in a zone- centered, collective mode (Kittel mode). In the results presented here, Eq. S7 was used to calculate αtofB2- FeCo, both in bcc and bct structures. II. Details of calculations The real-space linear muffin-tin orbital on the atomic- sphere approximation (RS-LMTO-ASA) [5] is a well- established method in the framework of the DFT to de- scribetheelectronicstructureofmetallicbulks[6,7], sur- faces [8, 9] and particularly embedded [10] or absorbed [11–14] finite cluster systems. The RS-LMTO-ASA is based on the LMTO-ASA formalism [15], and uses the recursion method [16] to solve the eigenvalue problem directly in real-space. This feature makes the method suitable for the calculation of local properties, since it does not depend on translational symmetry. The calculations performed here are fully self- consistent, and the spin densities were treated within the local spin-density approximation (LSDA) [17]. In all cases, we considered the spin-orbit coupling as a l·sterm included in each variational step [18–20]. The spin-orbit is strictly necessary for the damping calculations due to its strong dependence on the torque operators, ˆT. In the recursion method, the continued fractions have been truncated with the Beer-Pettifor terminator [21] after 22 recursion levels ( LL= 22). The imaginary part that comes from the terminator was considered as a natural choice for the broadening Λto build the Green’s func- tions ˆG(/epsilon1+iΛ), which led to reliable αparameters in comparison with previous results (see Table S1).To account for the Co randomness in the experimental Fe50Co50films [22], some systems were modeled in terms of the virtual crystal approximation (VCA) medium of Fe50Co50, considering the bcc (or the bct) matrix to have the same number of valence electrons as Fe50Co50(8.5 e−). However, we also investigated the role of the Co presence, as well as the influence of its randomness (or ordering), by simulating the B2(CsCl) FeCo structure (a=aFe). The VCA Fe50Co50andB2-FeCo bulks were simulated by a large matrix containing 8393 atoms in real-space, the first generated by using the Fe bcc lat- tice parameter ( aFe= 2.87Å) and the latter using the optimized lattice parameter ( a= 2.84Å). Thisachoice in VCA Fe50Co50was based on the fact that it is eas- ier to compare damping results for Fe50Co50alloy and pure Fe bcc bulk if the lattice parameters are the same, and the use of the aFehas shown to produce trustwor- thyαtvalues. On the other hand, bct bulk structures withc a= 1.15(B2-FeCo bct and VCA Fe50Co50bct) are based on even larger matrices containing 49412 atoms. The respective surfaces were simulated by semi-matrices of the same kind (4488 and 19700 atoms, respectively), considering one layer of empty spheres above the outer- most Fe50Co50(or pure Fe) layer, in order to provide a basis for the wave functions in the vacuum and to treat the charge transfers correctly. We notice that the investigations presented here are based on a (001)-oriented Fe50Co50film, in which only a small lattice relaxation normal to the surface is expected to occur (∼0.1%[23]). Damping parameters of Fe-centered and Co-centered clusters, embedded in an Fe50Co50VCA medium, have been calculated (explicitly) site by site. In all cases, these defects are treated self-consistently, and the po- tential parameters of the remaining sites were fixed at bulk/pristine VCA surface values, according to its envi- ronment. When inside the bulk, we placed the central (reference) atom of the cell in a typical site far away from the faces of the real-space matrix, avoiding any un- wanted surface effects. We considered as impurities the nearest 14 atoms (first and second nearest neighbours, up to 1a) from the central atom, treating also this sites self-consistently, in a total of 15 atoms. We calculated 10 cases with Fe and Co atoms randomly positioned: 5 with Fe as the central atom (Fe-centered) and 5 with Co as the central atom (Co-centered). An example (namely cluster #1 of Tables S3 and S4), of one of these clusters embedded in bulk, is represented in Fig. S1(a). As the self-consistent clusters have always a total of 15 atoms, the Fe (Co) concentration is about 47% (53%) or vice- versa. On the other hand, when inside the surface, we placed the central (reference) atom of the cluster in a typical site of the most external layer (near vacuum), since this has shown to be the layer where the damping anisotropy is larger. Therefore, we considered as impu- rities the reference atom itself and the nearest 9 atoms3 (up to 1a), in a total of 10 atoms (and giving a perfect 50% (50%) concentration). An example of one of these clusters embedded in a surface is shown in Fig. S1(b). (a) (b) Figure S1. (Color online) Schematic representation of an ex- ampleof: (a)Fe-centered15-atomclusterembeddedinaVCA Fe50Co50bcc bulk medium; (b) Co-centered 10-atom cluster embedded in a VCA Fe 50Co50(001) bcc surface medium. Yel- low and blue spheres represent Fe and Co atoms, respectively, while gray atoms represent the VCA Fe 50Co50sites (8.5 va- lence e−). The Fe(Co) concentration in the clusters are: (a) 53% (47%) and (b) 50% (50%) . The total number of atoms including the surrounding VCA sites are: (a) 339 and (b) 293. They were all accounted in the sum to obtain αtat the central (reference) Fe (a) and Co (b) site. To simulate a bct-like bulk distortion, the 8 first neigh- bours of the central atom were stretched in the cdi- rection, resulting in ac a= 1.15ratio. On the other hand, when embedded on the Fe50Co50(001) bcc surface, the central (reference) atom is placed in the outermost layer (near vacuum), and we simulate a bct distortion by stretching the 4 nearest-neighbours (on the second layer) to reproduce ac a= 1.09ratio (the maximum per- centage that the atoms, in these conditions, could be moved to form a bct-like defect). In this case, a total of 10 atoms (the nearest 9 atoms from the central one – up to 1a– and the reference atom) were treated self- consistently, analogous to as shown in Fig. S1(b). As in the case of the pristine bcc Fe50Co50clusters embedded in the VCA surface, we considered a total of 32 10-atom clusters with different Fe/Co spatial distributions, being 16 Fe-centered, and 16 Co-centered.III. Comparison with previous results Theab-initio calculation of the Gilbert damping, in the collinear limit, is not a new feature in the literature. Mainly, the reported theoretical damping results are for bulk systems [2, 4, 24–28], but, some of them even stud- ied free surfaces [29]. Therefore, in order to demonstrate the reliability of the on-site and total damping calcula- tions implemented here in real-space, a comparison of the presently obtained with previous (experimental and theoretical) results, are shown in Table S1. As can be seen, our results show a good agreement with previously obtainedαvalues, including some important trends al- readypredictedbefore. Forexample, thereducedGilbert damping of Co hcp with respect to the Co fcc due to the reduction of the density of states at the Fermi level [24, 28], (∼10.92states/Ry-atom in the hcp case and ∼16.14states/Ry-atom in the fcc case). IV. Details of the calculated damping values The damping values obtained for the systems studied here are shown in Tables (S2-S5). These data can be use- ful for the full understanding of the results presented in the main text. For easy reference, in Table S2 the αtof a typical atom in each system (bulk or surface) for different spin quantization axes are shown. These data are plot- ted in Fig. 2 of the main text. The obtained values show that, indeed, for bulk systems the damping anisotropies are not so pronounced as in the case of Fe50Co50(001) bcc surface. As observed in Table S2, the increase in αtwhen changing from the bcc Fe50Co50(c a= 1) to the bct Fe50Co50bulk structure (c a= 1.15) is qualitatively con- sistent to what was obtained by Mandal et al.[33] (from αt= 6.6×10−3in the bcc to αt= 17.8×10−3in the bct, withc a= 1.33[33]). Tables S3 and S4 refer to the damping anisotropies (∆αt) for all Fe-centered and Co-centered clusters stud- ied here, with different approaches: ( i) bcc clusters em- bedded in the VCA medium (Table S3) and ( ii) bct-like clusters embedded in the VCA medium (Table S4). In comparison with bct-like clusters, we found larger absoluteαtvalues but lower damping anisotropies. In all cases, Fe-centered clusters present higher ∆αtper- centages. In Table S5 the onsite damping anisotropies ( ∆αonsite) for each layer of the Fe50Co50(001) bcc surface ("1" repre- sents the layer closest to vacuum) are shown. In compar- ison with the total damping anisotropies (Table I of the main text), much lower percentages are found, demon- strating that the damping anisotropy effect comes ma- jorly from the non-local damping contributions. The most important results concerning the largest dampinganisotropiesaresummarizedinTableS6, below.4 Table S1. Total damping values ( ×10−3) calculated for some bulk and surface systems, and the comparison with previous literature results. The onsite contributions are indicated between parentheses, while the total damping, αt, are indicated without any symbols. All values were obtained considering the [001]magnetization axis. The VCA was adopted for alloys, except for the Fe 50Co50bcc in theB2structure (see Eq. S7). Also shown the broadening Λvalue considered in the calculations. Bulks a(Å) This work Theoretical Experimental Λ(eV) Fe bcc 2.87 4.2(1.6) 1 .3[2]a/(3.6)[4] 1.9[30]/2.2[31] Fe70Co30bcc 2.87 2.5(0.7) − 3−5[32]d Fe50Co50bcc 2.87 3.7(1.0)[VCA]/ 2.3(1.0)[B2]1.0[25]c[VCA]/ 6.6[33] [B2] 2.3[27] Ni fcc 3.52 27.8(57.7) 23 .7[34]/( 21.6[4])b26.0[31]/24.0[35] Ni80Fe20(Py) fcc 3.52 9.8(12.1) 3 .9[25]c8.0[30]/5.0[35] Co fcc 3.61 [3] 3.2(5.3) 5 .7[28]/(3.9[4])b11.0[30]∼5×10−2 Co hcp 2.48/4.04 [28] 2.1(6.2) 3 .0[28] 3.7[31] Co85Mn15bcc [36] 2.87 [28] 6.2(4.2) 6 .6[28] − Co90Fe10fcc 3.56 [37] 3.6(4.2) − 3.0[35]/4.8[37] Surfaces a(Å) This work Theoretical Experimental Fe(001) bcc [110] 2.87 5.8(5.4)e− 7.2[38]h/6.5[39]i Fe(001) bcc [100] 2.87 3.9(4.4)f∼4[29]g4.2[40]j Ni(001) fcc 3.52 80.0(129.6)∼10[29]g/12.7[41]m22.1[42]l PdFe/Ir(111) [43] fcc 3.84 3.9(2.7)n− − PdCo/Ir(111) [44] fcc 3.84 3.2(14.7)o− − aWith Λ∼2×10−2eV. bWith Λ = 5×10−3eV. cWith Λ∼1.4×10−4eV. dFor a 28%Co concentration, but the results do not significantly change for a 30%Co concentration. Range including results before and after annealing. eOf a typical atom in the more external surface layer (in contact with vacuum), in the [110] magnetization direction. fOf a typical atom in the more external surface layer (in contact with vacuum), in the [100] magnetization direction. gFor a (001) bcc surface with thickness of N= 8ML (the same number of slabs as in our calculations), and Λ = 10−2eV. hAnisotropic damping obtained for a 0.9 nm Fe/GaAs(001) thin film (sample S2 in Ref. [38]) in the [110] magnetization direction. iAnisotropic damping obtained for a 1.14 nm Fe/InAs(001) thin film in the in-plane [110] hard magnetization axis. jFor a 25-nm-thick Fe films grown on MgO(001). kFor epitaxial Fe(001) films grown on GaAs(001) and covered by Au, Pd, and Cr capping layers. lIntrinsic Gilbert damping for a free 4×[Co(0.2 nm)/Ni(0.6 nm)](111) multilayer. Not the same system as Ni(001), but the nearest system found in literature. mFor a Co | Ni multilayer with Ni thickness of 4 ML (fcc stacking). nOf a typical atom in the Fe layer. oOf a typical atom in the Co layer. The alloys with short-range orders (SRO) are described as FeCo clusters (with explicit Fe and Co atoms) embed- ded in the Fe50Co50VCA medium – with and without the bct-like distortion. In this case, the damping is cal- culated as a weighted average (Eq. S7). As discussed in the main text, it can be seen from Table S6 that distor- tions and disorder can increase the anisotropy but the major effect comes from the surface. We notice that the number of clusters considered is limited in the statistical average. IV. Kambersky’s simplified formula InordertoconnecttheanisotropyoftheGilbertdamp- ing to features in the electronic structure, we consider in the following Kambersky’s simplified formula for Gilbert damping [47, 48]α=1 γMs/parenleftBigγ 2/parenrightBig2 n(EF)ξ2(g−2)2 τ.(S8) Here,γis the gyromagnetic ratio, n(EF)represents the LDOS at the Fermi level, ξis the SOC strength, τis the electron scattering time, Msis the spin magnetic mo- ment, andgis the spectroscopic g-factor [35, 49]. Note that Eq. S8 demonstrates the direct relation between αandn(EF), often discussed in the literature, e.g., in Ref. [27]. Our first principles calculations have shown no significant change in ξ, upon variation of the mag- netization axis, for the FeCo systems ( ξCo= 71.02meV andξFe= 53.47meV). Hence, we can soundly relate the damping anisotropy ∆αtto∆n(EF). Figure S2 shows how the LDOS difference (per atom) ∆n(E)between the [010]and [110]magnetization di- rections is developed in pure Fe(001) bcc and in VCA Fe50Co50(001) bcc surfaces, respectively. In both cases,5 TableS2. Totaldamping( αt×10−3)ofatypicalatomin each system for the spin quantization axes [010](θH= 90◦) and [110](θH= 45◦); also shown for the [001]and [111]. Bulk and surface bct systems are simulated with c a= 1.15. Bulks Bulk αt[010]αt[110] ∆αt Fe bcc 4.18 4.31 +3.1%a B2-FeCo bcc 2.28 2.44 +7.2% B2-FeCo bct 7.76 8.85 +12.4% VCA Fe 50Co50bcc 3.70 4.18 +13.0% VCA Fe 50Co50bct 4.69 5.10 +8.7% αt[010]αt[001] ∆αt B2-FeCo bct 7.76 10.21 +24.1% VCA Fe 50Co50bct 4.69 5.75 +22.6% αt[010]αt[111] ∆αt Fe bcc 4.18 4.56 +9.1%b Surfaces Surface αt[010]αt[110] ∆αt Fe(001) bcc 3.85 5.75 +49.4% Fe/GaAs(001) bcc [38] 4.7(7) 7.2(7) +53(27)%c Fe/MgO(001) bcc [45] 3.20(25) 6.15(20) +92(14)%d VCA Fe 50Co50(001) bcc 7.00 14.17 +102.4% VCA Fe 50Co50(001) bct 15.20 14.80−2.6% αt[010]αt[001] ∆αt VCA Fe 50Co50(001) bct 15.20 15.56 +2.4% VCA Fe 50Co50(001) bcc 7.00 9.85 +40.7% aMankovsky et al.[24] find a damping anisotropy of ∼12% for bulk Fe bcc at low temperatures ( ∼50K) between [010] and [011] magnetization directions. For this result, the definition α=1 2(αxx+αyy)was used. bThis result agrees with Gilmore et al.[46], which find that the total damping of pure Fe bcc presents its higher value in the [111] crystallographic orientation and the lower value in the [001] direction, except for high scatter- ing rates. Also agrees with Mankovsky et al.[24] results. cAnisotropic damping obtained for a 0.9 nm Fe/GaAs(001) thin film (sample S2 in Ref. [38]) in the [010] and [110] magnetization directions. dFor a Fe(15 nm)/MgO(001) film at T= 4.5K in the high- est applied magnetic field, in which only intrinsic contri- butions to the anisotropic damping are left. the chosen layer,denoted as first, is the most external one (near vacuum). the VCA Fe50Co50(001) bcc we also calculated ∆n(E)for all layers summed (total DOS dif- ference). As can be seen, although in all cases the quantity ∆n(E)exhibits some oscillations, differently from what we observe forthe pureFe(001) surface case, at the Fermi energy, there is a non-negligible difference in the minor- ity spin channel ( 3dstates) for the VCA Fe50Co50(001). Considering the results presented in Table I (main text) the larger contribution to the damping anisotropy comes from the most external layer. The results by Li et al. [22] indicate a small difference (for two magnetization directions) of the total density of states at the FermiTable S3. Total damping anisotropy ( ×10−3) of all stud- ied Co-centered and Fe-centered bcc clusters for the spin- quantization axis [010]and [110], considering the 15-atom FeCo cluster together with the VCA medium in the summa- tion for total damping. Co-centered Cluster # αt[010]αt[110] ∆αt 1 10.11 9.65 4.8% 2 8.09 6.96 16.2% 3 7.81 7.02 11.3% 4 7.11 7.02 1.3% 5 7.48 6.88 8.7% Average 8.12 7.51 8.1% Fe-centered Cluster # αt[010]αt[110] ∆αt 1 2.68 2.03 32.0% 2 2.49 2.05 21.5% 3 2.56 1.86 37.6% 4 2.45 1.79 36.9% 5 2.76 2.01 37.3% Average 2.59 1.95 32.8% Table S4. Total damping anisotropy ( ×10−3) of all stud- ied Co-centered and Fe-centered bcc clusters for the spin quantization axis [010]and [110], with bct-like distortions/parenleftbigc a= 1.15/parenrightbig , considering the 15-atom FeCo cluster together with the VCA medium in the summation for total damping. Co-centered Cluster # αt[010]αt[110] ∆αt 1 5.85 4.37 33.9% 2 5.95 4.21 41.3% 3 5.88 4.35 35.2% 4 5.90 4.41 33.8% 5 5.86 4.34 35.0% Average 5.89 4.34 35.7% Fe-centered Cluster # αt[010]αt[110] ∆αt 1 2.36 1.39 69.8% 2 2.27 1.32 72.0% 3 2.22 1.26 76.2% 4 2.25 1.26 78.6% 5 2.42 1.38 75.4% Average 2.30 1.32 74.2% Table S5. Onsite damping ( αonsite×10−3) of a typical atom in each layer of the VCA Fe 50Co50(001) bcc for the spin quan- tization axis [010]and[110]. Layerαonsite[010]αonsite[110] ∆αonsite 1 7.36 8.70 +18.2% 2 0.63 0.69 +9.5% 3 1.41 1.44 +2.1% 4 0.87 0.86−1.1% 5 0.99 0.97−2.0%6 TableS6. SummaryofthemainFe 50Co50dampinganisotropy results for: pure ordered ( B2) alloy; pure random (VCA) bulk alloy; bcc bulk together with short-range order (SRO) clus- ters (see Table S3); bulk together with bct-like distorted clus- ters inside (see Table S4); surface calculations, in the pristine mode and with explicit bct-like clusters embedded (surface + distortion). The maximum-minimum ratio according to Ref. [22] isα[110] t α[010] t×100%. Structure ∆αtMax-min ratio Ordered alloy bcc 7.2% 107.2% Ordered alloy bct 24.1% 124.1% Random alloy bcc 13% 113% Random alloy bct 22.6% 122.6% Random alloy + SRO 14.9% 114.9% Random alloy + SRO + Distortion 47.2% 147.2% Surface (external layer) 102.4% 202.4% Surface (ext. layer) + Distortion 75.4% 175.4% 10-nm Co 50Fe50/Pt [22] (exp.) 281.3% 381.3% −2−1 0 1 2 −0.02 −0.01 0 0.01 0.02Δn(E)[010]−[110] (states/Ry−atom) Energy (E−EF) (Ry)Fe(001) bcc (first) VCA Fe50Co50(001) bcc (first) VCA Fe50Co50(001) bcc (all) Figure S2. LDOS difference (per atom), ∆n(E), between the [010]and[110]magnetization directions, for both spin chan- nels (full lines for majority spin and dashed lines for minority spin states), in the outermost layer in pure Fe(001) bcc (in black); outermost layer in VCA Fe 50Co50(001) bcc (in blue); and all layers summed in VCA Fe 50Co50(001) bcc (in red). level,N(EF), what the authors claim that could not ex- plain the giant maximum-minimum damping ratio ob- served. So, in order to clarify this effect in the VCA Fe50Co50(001) bcc, ∆n(E)was also calculated for the all layers summed, what is shown in Fig. S2 (in red). This difference is in fact smaller if we consider the DOS of the whole system, with all layers summed. However, if we consider only the most external layer, then the LDOS variation is enhanced. This is consistent with our theo- retical conclusions. As we mention in the main text, this do not rule out a role also played by local (tetragonal- like) distortions and other bulk-like factors in the damp- ing anisotropy.For the outermost layer of Fe(001) bcc, the calculated LDOSatEFis∼20.42states/Ry-atominthe[110]direc- tion and∼20.48states/Ry-atom in the [010] direction, which represents a difference of ∼0.3%and agrees with the calculations performed by Chen et al.[38]. V. Correlation with anisotropic orbital moment Besides the close relation exhibited between ∆αt and∆n(EF), we also demonstrate the existence of an anisotropic orbital moment in the outermost layer, in which the fourfold symmetry ( C4v) matches the damp- ing anisotropy with a 90◦phase. Fig. S3 shows this correlation between ∆αtand∆morbfor two situations: (i) for a typical atom in the outermost layer of VCA Fe50Co50(001) bcc (blue open dots); and ( ii) for a typi- cal atom in the VCA Fe50Co50bcc bulk, considering the same ∆morbscale. For case ( i) we find orbital moments differencesmorethanoneorderofmagnitudehigherthan case (ii). 0o45o90o 135o 180o 225o 270o315o[100][110][010] [−110] 0.51.52.5 Δmorb (µB/atom × 10−3) 0.0050.0100.015 αt θH Figure S3. (Color online) Total damping and orbital moment difference, ∆morbas a function of θH, the angle between the magnetizationdirectionandthe [100]-axis. Squares: (redfull) VCA Fe 50Co50(001) bcc. Circles: (blue open) morbdifference betweenθH= 90◦and the current angle for a typical atom in the outermost layer of VCA Fe 50Co50(001) bcc; and (yellow full) samemorbdifference but for a typical atom in the VCA Fe50Co50bcc bulk (in the same scale). Lines are guides for the eyes. VI. Contribution from next-nearest-neighbours Finally, we show in Fig. S4 the summation of all non- local damping contributions, αij, for a given normalized distance in the outermost layer of VCA Fe50Co50(001)7 bcc. As we can see, the next-nearest-neighbours from a reference site (normalized distanced a= 1) have very dis- tinctαijcontributions to αtfor the two different mag- netization directions ( [010]and[110]), playing an impor- tant role on the final damping anisotropy. We must note, however, that these neighbours in a (001)-oriented bcc surface are localized in the same layer as the reference site, most affected by the interfacial SOC. Same trend is observed ford a= 2, however less intense. This is con- sistent with our conclusions, about the relevance of the outermost layer on ∆αt. −3−2−1 0 1 2 0.5 1 1.5 2 2.5 3 3.5 4∑αij × 10−3 Normalized distance[110] [010] Figure S4. (Color online) Summation of all non-local Gilbert damping parameters ( αij,i/negationslash=j) in each neighboring normal- izeddistancebetweensites iandjfortheVCAFe 50Co50(001) bcc in the two most different directions for the damping anisotropy: [010](θH= 90◦), in blue open squares, and [110] (θH= 45◦), in red full circles. Lines are guides for the eyes. ∗aklautau@ufpa.br [1] V. Kamberský, Czech. J. Phys. 26, 1366 (1976). [2] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett.99, 027204 (2007). [3] D. Thonig and J. Henk, New J. Phys. 16, 013032 (2014). [4] D. Thonig, Y. Kvashnin, O. Eriksson, and M. Pereiro, Phys. Rev. Mater. 2, 013801 (2018). [5] S. Frota-Pessôa, Phys. Rev. B 46, 14570 (1992). [6] Y. O. Kvashnin, R. Cardias, A. Szilva, I. Di Marco, M. I. Katsnelson, A. I. Lichtenstein, L. 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2021-01-07

A method with which to calculate the Gilbert damping parameter from a real-space electronic structure method is reported here. The anisotropy of the Gilbert damping with respect to the magnetic moment direction and local chemical environment is calculated for bulk and surfaces of Fe$_{50}$Co$_{50}$ alloys from first principles electronic structure in a real space formulation. The size of the damping anisotropy for Fe$_{50}$Co$_{50}$ alloys is demonstrated to be significant. Depending on details of the simulations, it reaches a maximum-minimum damping ratio as high as 200%. Several microscopic origins of the strongly enhanced Gilbert damping anisotropy have been examined, where in particular interface/surface effects stand out, as do local distortions of the crystal structure. Although theory does not reproduce the experimentally reported high ratio of 400% [Phys. Rev. Lett. 122, 117203 (2019)], it nevertheless identifies microscopic mechanisms that can lead to huge damping anisotropies.

Mechanisms behind large Gilbert damping anisotropies

2101.02794v2

Hydrodynamic theory of coupled current and magnetization dynamics in spin-textured ferromagnets Clement H. Wong and Yaroslav Tserkovnyak Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA We develop the hydrodynamic theory of collinear spin currents coupled to magnetization dynamics in metallic ferromagnets. The collective spin density couples to the spin current through a U(1) Berry-phase gauge eld determined by the local texture and dynamics of the magnetization. We determine phenomenologically the dissipative corrections to the equation of motion for the electronic current, which consist of a dissipative spin-motive force generated by magnetization dynamics and a magnetic texture-dependent resistivity tensor. The reciprocal dissipative, adiabatic spin torque on the magnetic texture follows from the Onsager principle. We investigate the eects of thermal uctuations and nd that electronic dynamics contribute to a nonlocal Gilbert damping tensor in the Landau-Lifshitz-Gilbert equation for the magnetization. Several simple examples, including magnetic vortices, helices, and spirals, are analyzed in detail to demonstrate general principles. PACS numbers: 72.15.Gd,72.25.-b,75.75.+a I. INTRODUCTION The interaction of electrical currents with magnetic spin texture in conducting ferromagnets is presently a subject of active research. Topics of interest include current-driven magnetic dynamics of solitons such as do- main walls and magnetic vortices,1,2,3,4as well as the reciprocal process of voltage generation by magnetic dynamics.5,6,7,8,9,10,11,12This line of research has been fueled in part by its potential for practical applications to magnetic memory and data storage devices.13Funda- mental theoretical interest in the subject dates back at least two decades.5,6,14It was recognized early on6that in the adiabatic limit for spin dynamics, the conduction electrons interact with the magnetic spin texture via an eective spin-dependent U(1) gauge eld that is a local function of the magnetic conguration. This gauge eld, on the one hand, gives rise to a Lorentz force due to \ctitious" electric and magnetic elds and, on the other hand, mediates the so-called spin-transfer torque exerted by the conduction electrons on the collective magnetiza- tion. An alternative and equivalent view is to consider this force as the result of the Berry phase15accumulated by an electron as it propagates through the ferromagnet with its spin aligned with the ferromagnetic exchange eld.8,10,16In the standard phenomenological formalism based on the Landau-Lifshitz-Gilbert (LLG) equation, the low-energy, long-wavelength magnetization dynamics are described by collective spin precession in the eective magnetic eld, which is coupled to electrical currents via the spin-transfer torques. In the following, we develop a closed set of nonlinear classical equations governing current-magnetization dynamics, much like classical elec- trodynamics, with the LLG equation for the spin-texture \eld" in lieu of the Maxwell equations for the electro- magnetic eld. This electrodynamic analogy readily explains various interesting magnetoelectric phenomena observed recently in ferromagnetic metals. Adiabatic charge pumping bymagnetic dynamics17can be understood as the gener- ation of electrical currents due to the ctitious electric eld.18In addition, magnetic textures with nontrivial topology exhibit the so-called topological Hall eect,19,20 in which the ctitious magnetic eld causes a classical Hall eect. In contrast to the classical magnetoresis- tance, the ux of the ctitious magnetic eld is a topo- logical invariant of the magnetic texture.6 Dissipative processes in current-magnetization dynam- ics are relatively poorly understood and are of central interest in our theory. Electrical resistivity due to quasi- one-dimensional (1D) domain walls and spin spirals have been calculated microscopically.21,22,23More recently, a viscous coupling between current and magnetic dynam- ics which determines the strength of a dissipative spin torque in the LLG equation as well the reciprocal dis- sipative spin electromotive force generated by magnetic dynamics, called the \ coecient,"2was also calcu- lated in microscopic approaches.3,24,25Generally, such rst-principles calculations are technically dicult and restricted to simple models. On the other hand, the num- ber of dierent forms of the dissipative interactions in the hydrodynamic limit are in general constrained by sym- metries and the fundamental principles of thermodynam- ics, and may readily be determined phenomenologically in a gradient expansion. Furthermore, classical thermal uctuations may be easily incorporated in the theoretical framework of quasistationary nonequilibrium thermody- namics. The principal goal of this paper is to develop a (semi- phenomenological) hydrodynamic description of the dis- sipative processes in electric ows coupled to magnetic spin texture and dynamics. In Ref. 11, we drew the anal- ogy between the interaction of electric ows with quasis- tationary magnetization dynamics with the classical the- ory of magnetohydrodynamics. In our \spin magnetohy- drodynamics," the spin of the itinerant electrons, whose ows are described hydrodynamically, couples to the lo- cal magnetization direction, which constitutes the col- lective spin-coherent degree of freedom of the electronicarXiv:0905.4544v2 [cond-mat.mes-hall] 16 Nov 20092 uid. In particular, the dissipative coupling between the collective spin dynamics and the itinerant electrons is loosely akin to the Landau damping, capturing cer- tain kinematic equilibration of the relative motion be- tween spin-texture dynamics and electronic ows. In our previous paper,11we considered a special case of incom- pressible ows in a 1D ring to demonstrate the essential physics. In this paper, we establish a general coarse- grained hydrodynamic description of the interaction be- tween the electric ows and textured magnetization in three dimensions, treating the itinerant electron's degrees of freedom in a two-component uid model (correspond- ing to the two spin projections of spin-1 =2 electrons along the local collective magnetic order). Our phenomenology encompasses all the aforementioned magnetoelectric phe- nomena. The paper is organized as follows. In Sec. II, we use a Lagrangian approach to derive the semiclassical equation of motion for itinerant electrons in the adiabatic approx- imation for spin dynamics. In Sec. III, we derive the basic conservation laws, including the Landau-Lifshitz equation for the magnetization, by coarse-graining the single-particle equation of motion and the Hamiltonian. In Sec. IV, we phenomenologically construct dissipative couplings, making use of the Onsager reciprocity princi- ple, and calculate the net dissipation power. In particu- lar, we develop an analog of the Navier-Stokes equation for the electronic uid, focusing on texture-dependent eects, by making a systematic expansion in nonequi- librium current and magnetization consistent with sym- metry requirements. In Sec. V, we include the eects of classical thermal uctuations by adding Langevin sources to the hydrodynamic equations, and arrive at the central result of this paper: A set of coupled stochastic dier- ential equations for the electronic density, current, and magnetization, and the associated white-noise correlators of thermal noise. In Sec. VI, we apply our results to special examples of rotating and spinning magnetic tex- tures, calculating magnetic texture resistivity and mag- netic dynamics-generated currents for a magnetic spiral and a vortex. The paper is summarized in Sec. VII and some additional technical details, including a microscopic foundation for our semiclassical theory, are presented in the appendices. II. QUASIPARTICLE ACTION In a ferromagnet, the magnetization is a symmetry- breaking collective dynamical variable that couples to the itinerant electrons through the exchange interaction. Be- fore developing a general phenomenological framework, we start with a simple microscopic model with Stoner in- stability, which will guide us to explicitly construct some of the key magnetohydrodynamic ingredients. Within a low-temperature mean-eld description of short-ranged electron-electron interactions, the electronic action isgiven by (see appendix A for details): S=Z dtd3r^ y i~@t+~2 2mer2 2+
2m
^ ^ :(1) Here,
( r;t) is the ferromagnetic exchange splitting, m(r;t) is the direction of the dynamical order param- eter dened by ~h^ y^^ i=2 =sm,sis the local spin density, and ^ (r;t) is the spinor electron eld operator. For the short-range repulsion U > 0 discussed in ap- pendix A,
( r;t) = 2Us(r;t)=~and(r;t) =U(r;t), where=h^ y^ iis the local particle number density. For electrons, the magnetization Mis in the opposite di- rection of the spin density: M= sm, where <0 is the gyromagnetic ratio. Close to a local equilibrium, the magnetic order parameter describes a ground state con- sisting of two spin bands lled up to the spin-dependent Fermi surfaces, with the spin orientation dened by m. We will focus on soft magnetic modes well below the Curie temperature, where only the direction of the mag- netization and spin density are varied, while the uctu- ations of the magnitudes are not signicant. The spin density is given by s=~(+)=2 and particle den- sity by=++, whereare the local spin-up/down particle densities along m.scan be essentially constant in the limit of low spin susceptibility. Starting with a nonrelativistic many-body Hamilto- nian, the action (1) is obtained in a spin-rotationally invariant form. However, this symmetry is broken by spin-orbit interactions, whose role we will take into ac- count phenomenologically in the following. When the length scale on which m(r;t) varies is much greater than the ferromagnetic coherence length lc~vF=
, where vFis the Fermi velocity, the relevant physics is captured by the adiabatic approximation. In this limit, we start by neglecting transitions between the spin bands, treat- ing the electron's spin projection on the magnetization as a good quantum number. (This approximation will be relaxed later, in the presence of microscopic spin- orbit or magnetic disorder.) We then have two eec- tively distinct species of particles described by a spinor wave function ^ 0, which is dened by ^ =^U(R)^ 0. Here, ^U(R) is an SU(2) matrix corresponding to the local spa- tial rotationR(r;t) that brings the z-axis to point along the magnetization direction: R(r;t)z=m(r;t), so that ^Uy(^
m)^U= ^z. The projected action then becomes: S=Z dtZ d3r^ 0y" (i~@t+ ^a)(i~r^a)2 2me  2+
2^z ^ 0Z dtF[m];(2) where F[m] =A 2Z d3r(@im)2(3) is the spin-texture exchange energy (implicitly summing over the repeated spatial index i), which comes from the3 terms quadratic in the gauge elds that survive the pro- jection. In the mean-eld Stoner model, the ferromag- netic exchange stiness is A=~2=4me. To broaden our scope, we will treat it as a phenomenological constant, which, for simplicity, is determined by the mean electron density.26The spin-projected \ctitious" gauge elds are given by a(r;t) =i~hj^Uy@t^Uji; a(r;t) =i~hj^Uyr^Uji: (4) Choosing the rotation matrices ^U(m) to depend only on the local magnetic conguration, it follows from their denition that spin- gauge potentials have the form: a=@tm
amon (m); ai=@im
amon (m);(5) where amon (m) i~hj^Uy@m^Uji. We show in Ap- pendix B the well known result (see, e.g., Ref. 27) that amon is the vector potential (in an arbitrary gauge) of a magnetic monopole in the parameter space dened by m: @mamon (m) =qm; (6) whereq=~=2 is the monopole charge (which is ap- propriately quantized). By noting that the action (2) is formally identical to charged particles in electromagnetic eld, we can imme- diately write down the following classical single-particle Lagrangian for the interaction between the spin- elec- trons and the collective spin texture: L(r;_r;t) =me_r2 2+_r
a(r;t) +a(r;t); (7) where _ris the spin-electron (wave-packet) velocity. To simplify our discussion, we are omitting here the spin- dependent forces due to the self-consistent elds (r;t) and
( r;t), which will be easily reinserted at a later stage. See Eq. (29). The Euler-Lagrange equation of motion for v= _rderived from the single-particle Lagrangian (7), (d=dt)(@L=@_r) =@L=@r, gives me_v=q(e+vb): (8) The ctitious electromagnetic elds that determine the Lorentz force are qei=@ia@tai=qm
(@tm@im); qbi=ijk@jak=qijk 2m
(@km@jm):(9) They are conveniently expressed in terms of the tensor eld strength qf@a@a=qm
(@m@m) (10) byei=fi0andbi=ijkfjk=2.ijkis the antisymmet- ric Levi-Civita tensor and we used four-vector notation,dening@= (@t;r) anda= (a;a). Here and henceforth the convention is to use Latin indices to de- note spatial coordinates and Greek for space-time coor- dinates. Repeated Latin indices i;j;k are, furthermore, always implicitly summed over. III. SYMMETRIES AND CONSERVATION LAWS A. Gauge invariance The Lagrangian describing coupled electron transport and collective spin-texture dynamics (disregarding for simplicity the ordinary electromagnetic elds) is L(rp;vp;m;@m) =X p mev2 p 2+vp
a+a! A 2Z d3r(@im)2 =X p mev2 p 2+v pa! A 2Z d3r(@im)2:(11) v p(1;vp),vp=_r, andhere is the spin of indi- vidual particles labelled by p. The resulting equations of motion satisfy certain basic conservation laws, due to spin-dependent gauge freedom, space-time homogeneity, and spin isotropicity. First, let us establish gauge invariance due to an ambi- guity in the choice of the spinor rotations ^U(r;t)!^U^U0. Our formulation should be invariant under arbitrary di- agonal transformations ^U0=eifand ^U0=eig^z=2on the rotated fermionic eld ^ 0, corresponding to gauge transformations of the spin-projected theory: a=~@fanda=~@g=2; (12) respectively. The change in the Lagrangian density is given by L=j@fandL=j s@g; (13) respectively, where j=j++jandjs=~(j+j)=2 are the corresponding charge and spin gauge currents. The action S=R dtd3rLis gauge invariant, up to sur- face terms that do not aect the equations of motion, provided that the four-divergence of the currents vanish, which is the conservation of particle number and spin density: _+r
j= 0;_s+r
js= 0: (14) (The second of these conservation laws will be relaxed later.) Here, the number and spin densities along with the associated ux densities are =X pnp++; j=X pnpvpv; (15)4 and s=X pqnp~ 2(+); js=X pqnpvpsvs; (16) wherenp=(rrp) andp=for spins up and down. In the hydrodynamic limit, the above equations deter- mine the average particle velocity vand spin velocity vs, which allows us to dene four-vectors j= (;v) andj s= (s;svs). Microscopically, the local spin- dependent currents are dened, in the presence of electro- magnetic vector potential aand ctitious vector potential a, by mev= Reh y (i~raea) i; (17) wheree<0 is the electron charge. B. Angular and linear momenta Our Lagrangian (11) contains the dynamics of m(r) that is coupled to the current. In this regard, we note that the time component of the ctitious gauge poten- tial (B4),a=~@t'(1cos)=2, is a Wess-Zumino action that governs the spin-texture dynamics.4,6,28The variational equation mmL= 0 gives: s(@t+vs
r)m+mmF= 0: (18) To derive this equation, we used the spin-density con- tinuity equation (14) and a gauge-independent identity satised by the ctitious potentials: their variations with respect to mare given by ma(m;@m) =qm@m; (19) where m@ @mX @@ @(@m): (20) One recognizes that Eq. (18) is the Landau-Lifshitz (LL) equation, in which the spin density precesses about the eective eld given explicitly by hmF=A@2 im: (21) Equation (18) also includes the well-known reactive spin torque:= (js
r)m,3which is evidently the change in the local spin-density vector due to the spin angular momentum carried by the itinerant electrons. One can formally absorb this spin torque by dening an advective time derivative Dt@t+vs
r, with respect to the average spin drift velocity vs. Equation (18) may be written in a form that explicitly expresses the conservation of angular momentum:27,29 @t(smi) +@jij= 0; (22)where the angular-momentum stress tensor is dened by ij=svsjmiA(m@jm)i: (23) Notice that this includes both quasiparticle and collective contributions, which stem respectively from the trans- port and equilibrium spin currents. The Lorentz force equation for the electrons, Eq. (8), in turn, leads to a continuity equation for the kinetic momentum density.6To see this, let us start with the microscopic perspective: @t(vi) =@tX pnpvp=X p( _npvp+np_vp): (24) Using the Lorentz force equation for the second term, we have: meX pnp_vp=X pqnp(ei+ijkbkvpj) =X pqnpfiv p =sm
(@tm@im) +svsjm
(@jm@im) = (@im)
(mF) =A(@im)
(@2 jm); (25) utilizing Eq. (18) to obtain the last line. Coarse-graining the rst term of Eq. (24), in turn, we nd: X p_npvp=@jX p(rrp)vpivpj!@jX vivj: (26) Putting Eqs. (25) and (26) together, we can nally write Eq. (24) in the form: me@t(vi) +@j Tij+meX vivj! = 0;(27) where Tij=A (@im)
(@jm)ij 2(@km)2 (28) is the magnetization stress tensor.6 A spin-dependent chemical potential ^ =^K1^gov- erned by local density and short-ranged interactions can be trivially incorporated by redening the stress tensor as Tij!Tij+ij 2^T^K1^: (29) In our notation, ^ = (+;)T, ^= (+;)Tand ^Kis a symmetric 22 compressibility matrix in spin space, which includes the degeneracy pressure as well as self- consistent exchange and Hartree interactions. In general, Eq. (29) is valid only for suciently small deviations from the equilibrium density. Using the continuity equations (14), we can combine the last term of Eq. (27) with the momentum density rate of change: @t(vi) +@j(vivj) =(@t+v
r)vi;(30)5 which casts the momentum density continuity equation in the Euler equation form: meX (@t+v
r)vi+@jTij= 0: (31) We do not expect such advective corrections to @tto play an important role in electronic systems, however. This is in contrast to the advective-like time derivative in Eq. (18), which is rst order in velocity eld and is crucial for capturing spin-torque physics. C. Hydrodynamic free energy We will now turn to the Hamiltonian formulation and construct the free energy for our magnetohydrodynamic variables. This will subsequently allow us to develop a nonequilibrium thermodynamic description. The canon- ical momenta following from the Lagrangian (11) are pp@L @vp=mevp+ap; @L @_m=X pnp@a @_m=X pnpamon (m): (32) Notice that for our translationally-invariant system, the total linear momentum PX ppp+Z d3r(
r)m=meX pvp; (33) where we have used Eq. (5) to obtain the second equality, coincides with the kinetic momentum (mass current) of the electrons. The latter, in turn, is equivalent to the lin- ear momentum of the original problem of interacting non- relativistic electrons, in the absence of any real or cti- tious gauge elds. See appendix A. While Pis conserved (as discussed in the previous section and also follows now from the general principles), the canonical momenta of the electrons and the spin-texture eld, Eqs. (32), are not conserved separately. As was pointed out by Volovik in Ref. 6, this explains anomalous properties of the lin- ear momentum associated with the Wess-Zumino action of the spin-texture eld: This momentum has neither spin-rotational nor gauge invariance. The reason is that the spin-texture dynamics dene only one piece of the total momentum, which is associated with the coherent degrees of freedom. Including also the contribution as- sociated with the incoherent (quasiparticle) background restores the proper gauge-invariant momentum, P, which corresponds to the generator of the global translation in the microscopic many-body description. Performing a Legendre transformation to Hamiltonianas a function of momenta, we nd H[rp;pp;m;] =X pvp
pp+Z d3r_m
L =X p(ppa)2 2me+A 2Z d3r(@im)2 E+F; (34) whereEis the kinetic energy of electrons and Fis the exchange energy of the magnetic order. As could be expected,Eis the familiar single-particle Hamiltonian coupled to an external vector potential. According to a Hamilton's equation, the velocity is conjugate to the canonical momentum: vp=@H=@ pp. We note that ex- plicit dependence on the spin-texture dynamics dropped out because of the special property of the gauge elds: _m
@_ma=a. Furthermore, according to Eq. (19), we havemmE= (js
r)m, so the LL Eq. (18) can be written in terms of the Hamiltonian (34) as11 s_m+mmH= 0: (35) So far, we have included in the spin-texture equa- tion only the piece coupled to the itinerant electron de- grees of freedom. The purely magnetic part is tedious to derive directly and we will include it in the usual LL phenomenology.29To this end, we redene F[m(r)]!F+F0; (36) by adding an additional magnetic free energy F0[m(r)], which accounts for magnetostatic interactions, crystalline anisotropies, coupling to external elds, as well as energy associated with localized dorforbitals.30Then the to- tal free energy (Hamiltonian) is H=E+F, and we in general dene the eective magnetic eld as the thermo- dynamic conjugate of m:hmH. The LL equation then becomes %s_m+mh= 0; (37) where%sis the total eective spin density. To enlarge the scope of our phenomenology, we allow the possibility that%s6=s. For example, in the sdmodel, an extra spin density comes from the localized d-orbital electrons. Microscopically, %s@tmterm in the equation of motion stems from the Wess-Zumino action generically associ- ated with the total spin density. In the following, it may sometimes be useful to separate out the current-dependent part of the eective eld, and write the purely magnetic part as hmmF, so that h=hmm(js
r)m (38) and Eq. (37) becomes: %s_m+ (js
r)m+mhm= 0: (39)6 For completeness, let is also write the equation of motion for the spin- acceleration: me(@t+v
r)vi=q
(@tm@im) +vjm
(@jm@im)]r;(40) retaining for the moment the advective correction to the time derivative on the left-hand side and reinserting the force due to the spin-dependent chemical potential, ^=^K1^. These equations constitute the coupled re- active equations for our magneto-electric system. The Hamiltonian (free energy) in terms of the collective vari- ables is (including the elastic compression piece) H[;p;m] =X Z d3r(pa)2 2me +1 2Z d3r^T^K1^+F[m]; (41) where p=mev+ais the spin-dependent momentum that is locally averaged over individual particles. D. Conservation of energy So far, our hydrodynamic equations are reactive, so that the energy (41) must be conserved: P_H=_E+ _F= 0. The time derivative of the electronic energy Eis _E=Z d3rX  mev_v+ _mev2  2+ =Z d3rX  mevj_vj@j(vj)mev2  2+ =Z d3rX vj[me(@t+v
r)vj+@j] =Z d3rX qv
(e+vb) =Z d3rX qv
e=Z d3rjs
e: (42) The change in the spin-texture energy is given, according to Eq. (39), by _F=Z d3r_m
mF=Z d3r_m
hm =Z d3r_m
[%sm_m+m(js
r)m)] =Z d3rjs
e: (43) The total energy is thus evidently conserved, P= 0. When we calculate dissipation in the rest of the paper, we will omit these terms which cancel each other. The total energy ux density is evidently given by Q=X mev2  2+ v: (44)IV. DISSIPATION Having derived from rst principles the reactive cou- plings in our magneto-electric system, summed up in Eqs. (39)-(41), we will proceed to include the dissipa- tive eects phenomenologically. Let us focus on the lin- earized limit of small deviations from equilibrium (which may be spin textured), so that the advective correction to the time derivative in the Euler Eq. (40), which is quadratic in the velocity eld, can be omitted. To elimi- nate the quasiparticle spin degree of freedom, let us, fur- thermore, treat halfmetallic ferromagnets, so that =+ ands=q, whereq=~=2 is the electron's spin.31From Eq. (40), the equation of motion for the local (averaged) canonical momentum is:32 _p=q jbr; (45) in a gauge where a= 0, so that _p=me_vqe.33 ==K. The Lorentz force due to the applied (real) electromagnetic elds can be added in the obvious way to the right-hand side of Eq. (45). Note that since we are now interested in linearized equations close to equi- librium,in Eq. (45) can be approximated by its (ho- mogeneous) equilibrium value. Introducing relaxation through a phenomenological damping constant (Drude resistivity) =me ; (46) whereis the collision time, expressing the ctitious magnetic eld in terms of the spin texture, Eq. (45) be- comes: _pi=q (m@im)
(j
r)m@i ji: (47) Adding the phenomenological Gilbert damping34to the magnetic Eq. (37) gives the Landau-Lifshitz-Gilbert equation: %s(_m+m_m) =hm; (48) whereis the damping constant. Eqs. (47) and (48), along with the continuity equation, _ =r
j, are the near-equilibrium thermodynamic equations for (;p;m) and their respective thermodynamic conjugates (;j;h) = (H;pH;mH). This system of equations of motion may be written formally as @t0 @ p m1 A=b[m(r)]0 @ j h1 A: (49) The matrix ^ depends on the equilibrium spin texture m(r). By the Onsager reciprocity principle, ij[m] = sisjji[m], wheresi=if theith variable is even (odd) under time reversal.7 In the quasistationary description of a nonequilibrium thermodynamic system, the entropy S[;p;m] is for- mally regarded as a functional of the instantaneous ther- modynamic variables, and the probability of a given con- guration is proportional to eS=kB. If the heat conduc- tance is high and the temperature Tis uniform and con- stant, the instantaneous rate of dissipation P=T_Sis given by the rate of change in the free energy, P=_H=R d3rP: P=_h
_mj
_p=%s_m2+ j2; (50) where we used Eq. (47) and expressed the eective eld has a function of _mby taking mof Eq. (48): h=%sm_m%s_m: (51) Notice that the ctitious magnetic eld bdoes not con- tribute to dissipation because it does not do work. So far, there is no dissipative coupling between the current and the spin-texture dynamics, and the macro- scopic equations obey the global time-reversal symme- try. However, we know that dissipative couplings ex- ists due to the misalignment of the electron's spin with the collective spin texture and spin-texture resistivity.3,22 Following Ref. 11, we add these well-known eects phe- nomenologically by making an expansion in the equations of motion to linear order in the nonequilibrium quanti- ties _mandj. To limit the number of terms one can write down, we will only add terms that are spin-rotationally invariant and isotropic in real space (which disregards, in particular, such eects as the angular magnetoresis- tance and the anomalous Hall eect). To second order in the spatial gradients of m, there are only three dissipa- tive phenomenological terms with couplings ,0, and consistent with the above requirements, which could be added to the right-hand side of Eq. (47).35The momen- tum equation becomes: _pi=q (m@im)
(j
r)m@i ji (@km)2ji0@im
(j
r)mq_m
@im:(52) It is known that the \ term" comes from a misalignment of the electron spin with the collective spin texture, and the associated dephasing. It is natural to expect thus that the dimensionless parameter ~=s
, wheres is a characteristic spin-dephasing time.3The \terms" evidently describe texture-dependent resistivity, which is anisotropic with respect to the gradients in the spin texture along the local current density. Such term are also naturally expected, in view of the well-known giant- magnetoresistance eect,36in which noncollinear magne- tization results in electrical resistance. The microscopic origin of this term is due to spin-texture misalignment, which modies electron scattering. The total spin-texture-dependent resistivity can be putinto a tensor form: ij[m] =ij +(@km)2 +0@im
@jm +q m
(@im@jm): (53) The last term due to ctitious magnetic eld gives a Hall resistivity. Note that ^ [m] = ^ T[m], consistent with the Onsager theorem. We can nally write Eq. (47) as: _pi= ij[m]jj@iq_m
@im: (54) As was shown in Ref. 11, since the Onsager relations require thatb[m] =b[m]Twithin the current/spin- texture elds sector, there must be a counterpart to the term above in the magnetic equation, which is the well- known dissipative \ spin torque:" %s(_m+m_m) =hmqm(j
r)m:(55) The total dissipation Pis now given by P=%s_m2+ 2q_m
(j
r)m+ +(@km)2 j2 +0[(j
r)m]2 =%s _m+q %s(j
r)m2 + +(@km)2 j2 + 0(q)2 %s [(j
r)m]2: (56) The second law of thermodynamics requires the total dis- sipation to be positive, which puts some constraints on the allowed values of the phenomenological parameters. We can easily notice, however, that the dissipation (56) is guaranteed to be positive-denite if +0(q)2 %s; (57) which may serve as an estimate for the spin-texture re- sistivity due to spin dephasing. This is consistent with the microscopic ndings of Ref. 23. V. THERMAL NOISE At nite temperature, thermal agitation causes uc- tuations of the current and spin texture, which are cor- related due to their coupling. A complete description requires that we supplement the stochastic equations of motion with the correlators for these uctuations. It is convenient to regard these uctuations as being due to the stochastic Langevin \forces" ( ;j;h) on the right-hand side of Eq. (49). The complete set of nite- temperature hydrodynamic equations thus becomes: _=r
~j; _p+q_mirmi=^ [m]~jr~; %s(1 +m)_m=~hmqm(~j
r)m:(58)8 where (~;~j;~h) = (+;j+j;h+h). The simplest (while possibly not most realistic) case corresponds to a highly compressible uid, such that K!1 . In this limit,==K!0 and the last two equations com- pletely decouple from the rst, continuity equation. In the remainder of this section, we will focus on this special case. The correlations of the stochastic elds are given by the symmetric part of the inverse matrix b =b1,37 which is found by inverting Eq. (58) (reduced now to a system of two equations): ~j=^ 1(_p+q_mirmi); ~h=%sm_m%s_mq(~j
r)m: (59) Writing formally these equations as (after substituting ~j from the rst into the second equation) ~j ~h =b[m(r)] _p _m ; (60) we immediately read out for the matrix elements b(r;r0) =b(r)(rr0): ji;ji0(r) =(^ 1)ii0; ji;hi0(r) =q(^ 1)ik@kmi0; hi0;ji(r) =q(^ 1)ki@kmi0; hi;hi0(r) =%sii0+%sii0kmk (q)2(@kmi)(^ 1)kk0(@k0mi0):(61) According to the uctuation-dissipation theorem, we symmetrize b to obtain the classical Langevin correlators:37 hji(r;t)ji0(r0;t0)i=T=gii0; hji(r;t)hi0(r0;t0)i=T=qg0 ik@kmi0; hhi(r;t)hi0(r0;t0)i=T=%sii0 (q)2gkk0(@kmi)(@k0mi0); (62) whereT= 2kBT(rr0)(tt0) and ^g= [^ 1+ (^ 1)T]=2;^g0= [^ 1(^ 1)T]=2 (63) are, respectively, the symmetric and antisymmetric parts of the conductivity matrix ^ 1. The short-ranged, - function character of the noise correlations in space stems from the assumption of high electronic compressibility. Contrast this to the results of Ref. 11 for incompressible hydrodynamics. A presence of long-ranged Coulombic interactions and plasma modes would also give rise to nonlocal correlations. These are absent in our treatment, which disregards ordinary electromagnetic phenomena. Focusing on the microwave frequencies !characteris- tic of ferromagnetic dynamics, it is most interesting to consider the regime where !1. This means that we can employ the drift approximation for the rst of Eqs. (59): _pi=me_viqeiqei=q_m
(m@im): (64)Substituting this _pin Eq. (59), we can easily nd a closed stochastic equation for the spin-texture eld: %s(1 +m)_m+m$_m= (hm+h)m;(65) where we have dened the \spin-torque tensor" $=q2(^ 1)kk0(m@km@km) (m@k0m+@k0m): (66) The antisymmetric piece of this tensor modies the eec- tive gyromagnetic ratio, while the more interesting sym- metric piece determines the additional nonlocal Gilbert damping: $=$+$T 2%s=q2 %sG$; (67) where G$=gkk0 (m@km) (m@k0m)2@km @k0m +g0 kk0[(m@km) @k0m@km (m@k0m)]: (68) In obtaining Eq. (65) from Eqs. (59), we have separated the reactive spin torque out of the eective eld: h= hmqm(j
r)m. (The remaining piece hmthus re ects the purely magnetic contribution to the eective eld.) The total stochastic magnetic eld entering Eq. (65), h=h+qm(j
r)m; (69) captures both the usual magnetic Brown noise38h and the Johnson noise spin-torque contribution39hJ= qm(j
r)mthat arises due to the substitution j=~jj in the reactive spin torque q(j
r)m. Using correla- tors (62), it is easy to show that the total eective eld uctuations hare consistent with the nonlocal eec- tive Gilbert damping tensor (68), in accordance with the uctuation-dissipation theorem applied directly to the purely magnetic Eq. (65). To the leading, quadratic order in spin texture, we can replacegkk0!kk0= andg0 kk0!0 in Eq. (68). This ad- ditional texture-dependent nonlocal damping (along with the associated magnetic noise) is a second-order eect, physically corresponding to the backaction of the magne- tization dynamics-driven current on the spin texture.11 It should be noted that in writing the modied LLG equation (55), we did not systematically expand it to include the most general phenomenological terms up to the second order in spin texture. We have only included extra spin-torque terms, which are required by the On- sager symmetry with Eq. (52). The second-order Gilbert damping (68) was then obtained by solving Eqs. (52) and (55) simultaneously. (Cf. Refs. 11,40.) This means in particular, that this procedure does not capture second- order Gilbert damping eects whose physical origin is unrelated to the longitudinal spin-transfer torque physics studied here. One example of that is the transverse spin- pumping induced damping discussed in Refs. 41.9 VI. EXAMPLES A. Rigidly spinning texture To illustrate the resistivity terms in the electron's equation of motion (52), we rst consider 1D textures. Take, for example, the case of a 1D spin helix m(z) along thezaxis, whose spatial gradient prole is given by @zm=^ zm, whereis the wave vector of the spatial rotation and m?^ z. See Fig. 1. It gives anisotropic re- sistivity in the xyplane,r() ?, and along the zdirection, r() k: r() ?=(@zm)2=2; r() k= (+0)2: (70) FIG. 1: (Color online) The transverse magnetic helix, @zm= ^ zm, with texture-dependent anisotropic resistivity (70). We assume here translational invariance in the transverse ( xy) directions. Spinning this helix about the vertical zaxis gen- erates the dissipative electromotive forces f() z, which is spa- tially uniform and points everywhere along the zaxis. A magnetic spiral, @zm=^'m=^, spinning around the z axis, on the other hand, produces a purely reactive electromo- tive forceez, as discussed in the text, which is oscillatatory in space along the zaxis. The ctitious electric eld and dissipative force re- quire magnetic dynamics. A general texture globally ro- tating clockwise in spin space in the xyplane according to_m=!^ zm(which may be induced by applying a magnetic eld along the zdirection) generates an electric eld ei= (m_m)
@im=!(m^ zm)
@im =!@imz=!@icos (71)and aforce f() i=_m
@im=!^ z
(m@im) =!sin2@i'; (72) where (;') denote the position-dependent spherical an- gles parametrizing the spin texture. The reactive force (71) has a simple interpretation of the gradient of the Berry-phase15accumulation rate [which is locally deter- mined by the solid angle subtended by m(t)]. In the case of the transverse helix discussed above, ==2, '=z!t, so thatez= 0 whilef() z=! is nite. As an example of a dynamical texture that does not generate f()while producing a nite e, consider a spin spiral along the zaxis, described by @zm=^'m=^, and rotating in time in the manner described above. It is clear geometrically that the change in the spin texture in time is in a direction orthogonal to its gradients in space. Specically, =z,'=!t, so thatf() z= 0 while the electric eld is oscillatory, ez=!sin. B. Rotating spin textures We show here that a vortex rotating about its core in orbital space generates a current circulating around its core, as well as a current going radially with respect to the core. Consider a spin texture with a time depen- dence corresponding to the real-space rotation clockwise in thexyplane around the origin, such that m(r;t) = m(r(t);0) with _r=!^ zr=!r^, where we use polar co- ordinates (r;) on the plane normal to the zaxis in real space [to be distinguished from the spherical coordinates (;') that parametrize min spin space], we have _m= (_r
r)m=!@m: (73) Form(r;) in polar coordinates, the components of the electric eld are, er=!m
(@m@rm); e= 0; (74) while the components of the force are f() r=!(@rm)
(@m); f() =!(@m)2 r:(75) In order to nd the ctitious electromagnetic elds, we need to calculate the following tensors (which depend on the instantaneous spin texture): bijm
(@im@jm) = sin(@i@j'@j@i'); dij@im
@jm=@i@j+ sin2@i'@j': (76) As an example, consider a vortex centered at the ori- gin in thexyplane with winding number 1 and positive polarity, as shown in Fig. 2. Its angular coordinates are given by '= (+!t) + 2; =(r); (77)10 where= arg( r) andis a rotationally invariant func- tion such that !0 asr!0 and!=2 asr!1 . Evaluating the tensors in equation (76) for this vortex in polar coordinates gives drr= (@r)2,d= (sin=r)2, dr= 0, andbr=(@rcos)=r. The radial electric eld is then given by er=!rbr=!@rcos: (78) Theforce is in the azimuthal direction: f() r= 0; f() =!rd=!sin2 r: (79) We can interpret this force as the spin texture \dragging" the current along its direction of motion. Notice that the forces in Eqs. (78) and (79) are the negative of those in Eqs. (71) and (72), as they should be for the present case, since the combination of orbital and spin rotations of our vortex around its core leaves it invariant, producing no forces. FIG. 2: Positive-polarity magnetic vortex conguration pro- jected on the xyplane. mhas a positive (out-of-plane) z component near the vortex core. Rotating this vortex about the origin in real space generates the current in the xyplane shown in Fig. 3. The total resistivity tensor (53) is (in the cylindrical coordinates) ^ = +(drr+d) +0^d+q ^b= r ? ?  ;(80) where r= + (+0)(@r)2+sin r2 ; = +(@r)2+ (+0)sin r2 ; ?=q @rcos r: (81)Here, the two diagonal components, rand , describe the (dissipative) anisotropic resistivity, while the o- diagonal component, ?, captures what is called the topological Hall eect.19 In the drift approximation, Eq. (64), the current- density eld j=jr^ r+j^is given by j= ^ 1q(e+f()); jr j =q!^ 1@rcos sin2=r =q!sin r + 2 ?  ? ? r @r sin=r :(82) More explicitly, we may consider a prole =(1 er=a)=2, whereais the radius of the vortex core. The corresponding current (82) is sketched in Fig. 3. FIG. 3: We plot here the current in Eq. (82) (all parameters set to 1). Near the core, the current spirals inward and charges build up at the center (which is allowed for our compressible uid). We note that the ctitious magnetic eld ijkbjk=2 points everywhere in the zdirection, its total ux through the xyplane being given by F=Z ddr (rbr) =Z ddr (@'@rcos) = 2: (83) Note that the integrand is just the Jacobian of the map from the plane to the sphere dened by the spin-texture eld: ((r);'(r)) :R2!S2: (84) This re ects the fact that the ctitious magnetic ux is generally a topological invariant, corresponding to the 2 homotopy group of the mapping (84).6,42 C. Anisotropic resistivity of a 3D spiral Consider the texture described by @im=i^ zm, where the spatial rotation stays in the xyplane, but the11 wave vectorcan be in any direction. The spin texture forms a transverse helix in the zdirection and a planar spiral in the xandydirections. Fig. 4 shows such a conguration for pointing along ( x+y+z)=p 3. The ctitious magnetic eld bvanishes, but the anisotropic resistivity still depends nontrivially on the spin texture: ij= +(@km)2 ij+0@im
@jm = ( +2)ij+0ij; (85) which, according to j= ^ 1E, would give a transverse current signal for an electric eld applied along the Carte- sian axesx,y, orz. FIG. 4: (Color online) A set of spin spirals which is topo- logically trivial because r= 0 (and equivalent to the spin helix, Fig. 1, up to a global real-space rotation), hence the ctitious magnetic eld b, Eq. (76), is zero. There is, how- ever, an anisotropic texture-dependent resistivity with nite o-diagonal components, Eq. (85). VII. SUMMARY We have developed semi-phenomenologically the hy- drodynamics of spin and charge currents interacting with collective magnetization in metallic ferromagnets, gener- alizing the results of Ref. 11 to three dimensions and compressible ows. Our theory reproduces known re- sults such as the spin-motive force generated by mag- netization dynamics and the dissipative spin torque, al- beit from a dierent viewpoint than previous microscopic approaches. Among the several new eects predicted, we nd both an isotropic and an anisotropic texture- dependent resistivity, Eq. (53), whose contribution to theclassical (topological) Hall eect should be described on par with that of the ctitious magnetic eld. By calculat- ing the dissipation power, we give a lower bound on the spin-texture resistivity as required by the second law of thermodynamics. We nd a more general form, includ- ing a term of order , of the texture-dependent correction to nonlocal Gilbert damping, predicted in Ref. 11. See Eq. (68). Our general theory is contained in the stochastic hy- drodynamic equations, Eqs. (58), which we treated in the highly compressible limit. The most general situ- ation is no doubt at least as rich and complicated as the classical magnetohydrodynamics. A natural exten- sion of this work is the inclusion of heat ows and re- lated thermoelectric eects, which we plan to investigate in a future work. Although we mainly focused on the halfmetallic limit in this paper, our theory is in principle a two-component uid model and allows for the inclu- sion of a fully dynamical treatment of spin densities and associated ows.31Finally, our hydrodynamic equations become amenable to analytic treatments when applied to the important problem of spin-current driven dynamics of magnetic solitons, topologically stable objects that can be described by a small number of collective coordinates, which we will also investigate in future work. Acknowledgments We are grateful to Gerrit E. W. Bauer, Arne Brataas, Alexey A. Kovalev, and Mathieu Taillefumier for stimu- lating discussions. This work was supported in part by the Alfred P. Sloan Foundation and the NSF under Grant No. DMR-0840965. APPENDIX A: MANY-BODY ACTION We can formally start with a many-body action, with Stoner instability built in due to short-range repulsion between electrons:25 S[ (r;t); (r;t)] =Z CdtZ d3r  ^ + i~@t+~2 2mer2 ^ U " # # " ;(A1) where time truns along the Keldysh contour from 1 to1and back.  and are mutually independent Grassmann variables parametrizing fermionic coherent states and ^ += ( "; #) and ^ = ( "; #)T. The four- fermion interaction contribution to the action can be de- coupled via Hubbard-Stratonovich transformation, after12 introducing auxiliary bosonic elds and
: eiSU=~= exp i ~Z CdtZ d3rU " # # " =Z D[(r;t);
(r;t)] expi ~Z CdtZ d3r 2 4U
2 4U 2^ +^ +
2^ +^^  :(A2) In obtaining this result, we decomposed the interaction into charge- and spin-density pieces:  " # # "=1 4(^ +^ )21 4(^ +m
^^ )2; (A3) where mis an arbitrary unit vector. It is easy to show thath(r;t)i=Uh^ +(r;t)^ (r;t)iandh
(r;t)i= Uh^ +(r;t)^^ (r;t)i, when properly averaging over the coupled quasiparticle and bosonic elds. The next step in developing mean-eld theory is to treat the Hartree potential (r;t) and Stoner exchange
(r;t)
(r;t)m(r;t) elds in the saddle-point approx- imation. Namely, the eective bosonic action Se[(r;t);
(r;t)] =i~lnZ D[^ +;^ ]ei ~S(^ +;^ ;;
) (A4) is minimized, Se= 0, in order to nd the equations of motion for the elds and
. In the limit of suf- ciently low electron compressibility and spin suscepti- bility, the charge- and spin-density uctuations are sup- pressed, dening mean-eld parameters and 
. Since a constant only shifts the overall electrochemical po- tential, it is physically inconsequential. Our theory is de- signed to focus on the remaining soft (Goldstone) modes associated with the spin-density director m(r;t), while (r;t) and
( r;t) are in general allowed to uctuate close to their mean-eld values and 
, respectively. The saddle-point equation of motion for the collective spin direction m(r;t) follows from mSe[m] = 0, after integrating out electronic degrees of freedom. Because of the noncommutative matrix structure of the action (A2), it is still a nontrivial problem. The problem sim- plies considerably in the limit of large exchange split- ting
, where we can project spins on the local magnetic direction m. This lays the ground to the formulation dis- cussed in Sec. II, where the collective spin-density eld parametrized by the director m(r;t) interacts with the spin-up/down free-electron eld. The resulting equations of motion constitute the self-consistent dynamic Stoner theory of itinerant ferromagnetism. In the remainder of this appendix, we explicitly show that the semiclassical formalism developed in Secs. II- III B is equivalent to a proper eld-theoretical treatment. The equation of motion for the spin texture follows from extremizing the eective action with respect to variations inm. Because of the constraint on the magnitude of m, its variation can be expressed as m=m, withbeing an arbitrary innitesimal vector, so that the equation of motion is given by mmSe= 0: 0 =mmSe =1 ZZ D[^ +;^ ] (mmS)ei ~S[^ +;^ ;;
] =X (mma)@S @a mmF; (A5) whereZ=R D[^ +;^ ]ei ~S[^ +;^ ;;
]and we have used the path-integral representation of the vacuum expecta- tion value. aare the spin-dependent gauge potentials (4) andFthe spin exchange energy, appearing after we project spin dynamics on the collective eld
. Equa- tion (A5) may be expressed in terms of the hydrody- namic variables of the electrons. Dening spin-dependent charge and current densities, j = (;j), by =@S @a =h  i; j=@S @a =1 meRe  (i~ra)  =v; (A6) Eq. (A5) reduces to the Landau-Lifshitz Eq. (18). Min- imizing action (A4) with respect to the and
elds gives the anticipated self-consistency relations: (r;t) =Uh^ +(r;t)^ (r;t)i=U(++);
(r;t) =Uh^ +(r;t)^z^ (r;t)i=U(+):(A7) APPENDIX B: THE MONOPOLE GAUGE FIELD Let (;') be the spherical angles of m, the direction of the local spin density, and ^ be the spin up/down (=) spinors given by, up to a phase, ^+(;') = cos 2 ei'sin 2 ; ^(;') = ^+(;'+) = sin 2 ei'cos 2 :(B1) The spinors are related to the spin-rotation matrix ^U(m) by ^=^Uji. The gauge eld in mspace, which enters Eq. (5), is thus given by amon (;') =i~^y @m^=~ 21cos sin ^';(B2) where we used the gradient on a unit sphere: @m= ^@+^'@'=sin. The magnetic eld corresponding to this vector potential [extended to three dimensions by a(m)!a(;')=m] is given on the unit sphere by @mamon =@m(a'^') =m sin@(sina') =~ 2m: (B3)13 It follows from Eqs. (5) and (B2) that the spin-dependent real-space gauge elds are given by a=~ 2@'(1cos): (B4) Notice that the =monopole eld (B2), as well as the above gauge elds, are singular on the south/north pole(corresponding to the Dirac string). This is what allows a magnetic eld with nite divergence. Any other choice of the monopole gauge eld (B2) would correspond to a dif- ferent choice of the spinors (B1), translating into a gauge transformation of the elds (B4). This is immediately seeing by noticing that amon (m)!amon (m) +@mf(m) corresponds to a(r;t)!a(r;t) +@f(m(r;t)). 1L. Berger, J. Appl. Phys. 55, 1954 (1984); M. Kl aui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, and L. J. Heyderman, Appl. Phys. Lett. 83, 105 (2003); E. Saitoh, H. Miyajima, T. Yamaoka, and G. Tatara, Nature 432, 203 (2004); Z. Li and S. Zhang, Phys. Rev. Lett. 92, 207203 (2004); S. E. Barnes and S. Maekawa, ibid. 95, 107204 (2005); L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, and S. S. P. Parkin, Nature 443, 197 (2006); M. Yamanouchi, D. Chiba, F. Matsukura, T. Dietl, and H. Ohno, Phys. Rev. Lett. 96, 096601 (2006); M. Yamanouchi, J. Ieda, F. Matsukura, S. E. Barnes, S. Maekawa, and H. 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Shibata, J. Phys. Soc. Jpn. 75, 113706 (2006). 25R. A. Duine, A. S. N u~ nez, J. Sinova, and A. H. MacDonald, Phys. Rev. B 75, 214420 (2007). 26Coupling of this exchange energy to the electronic density uctuations can also be treated systematically. However, since it does not lead to any signicant eects, while at the same time unnecessarily complicating our discussion, we will disregard it in the following. 27Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B 57, R3213 (1998). 28H.-B. Braun and D. Loss, Phys. Rev. B 53, 3237 (1996). 29E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2, vol. 9 of Course of Theoretical Physics (Pergamon, Ox- ford, 1980), 3rd ed. 30Generally, the phenomenological free energy of a ferromag- net is given by F[m] =A(@im)2=2 +Uani(m)Mhext
m, whereAis the eective stiness, Uani(m) =Uani(m) is an anisotropic potential due to magnetostatic and crys- talline elds, Mis the equilibrium magnetization, and hext is an external magnetic eld. 31In the more general two-band model of spin-up and spin- down electrons, we would have to develop a two- uid ef- fective theory, with spin- ip scattering between the two uid components. The phenomenology simplies, however, reducing formally to the halfmetallic case, in the long- wavelength low-frequency limit: !sf1 andksd1,14 where!,kare the characteristic frequency, wave number of the magnetohydrodynamics and sf,sd/psfare the spin- ip time, spin-diusion length. In this limit, it may be possible to describe the hydrodynamic state of the system by the spin-texture eld, the charge-density distribution, and the charge-current eld. If any out-of-equilibrium spin imbalance decays suciently fast, therefore, we only need to retain a one- uid description for the charge ows. The key phenomenological modication is then to introduce a material-dependent dimensionless \spin-polarization" pa- rameterp, such that q!pqin the following equations of motion. Namely, the eective charge that couples the electronic particle-number ux densities jwith the spin- texture gauge eld is renormalized by p. While in the halfmetallic limit p= 1 and in normal metals p= 0, we may expect some intermediate value in realistic multiple- band ferromagnets with fast spin relaxation. 32To make contact with Ref. 11, dene the canonical current asJ=H Cp
dl, for an arbitrary closed curve C. Its equa- tion of motion is given by @tJ=qH Cdl
(vb). If the curveCcoincides with a quasi-1D wire, then vkdland we recover the reactive equation of Ref. 11: @tJ= 0. 33Similarly to qthat should generally be viewed as a material-dependent phenomenological parameters, the electron mass mefrom now on is also an eective parame-ter, which is not necessarily identical with the free-electron mass. 34T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 35We are not including the ordinary hydrodynamic viscosity in our treatment. 36M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petro, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988); G. Binasch, P. Gr unberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 (1989). 37L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, vol. 5 of Course of Theoretical Physics (Pergamon, Ox- ford, 1980), 3rd ed. 38W. F. Brown, Phys. Rev. 130, 1677 (1963). 39J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. B 78, 140402(R) (2008). 40S. Zhang, and S. S.-L. Zhang, Phys. Rev. Lett. 102, 086601 (2009). 41E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys. Rev. B 78, 020404(R) (2008); Y. Tserkovnyak, E. M. Han- kiewicz, and G. Vignale, ibid.79, 094415 (2009). 42A. A. Belavin and A. M. Polyakov, JETP Lett. 22, 503 (1975).

2009-05-28

We develop the hydrodynamical theory of collinear spin currents coupled to magnetization dynamics in metallic ferromagnets. The collective spin density couples to the spin current through a U(1) Berry-phase gauge field determined by the local texture and dynamics of the magnetization. We determine phenomenologically the dissipative corrections to the equation of motion for the electronic current, which consist of a dissipative spin-motive force generated by magnetization dynamics and a magnetic texture-dependent resistivity tensor. The reciprocal dissipative, adiabatic spin torque on the magnetic texture follows from the Onsager principle. We investigate the effects of thermal fluctuations and find that electronic dynamics contribute to a nonlocal Gilbert damping tensor in the Landau-Lifshitz-Gilbert equation for the magnetization. Several simple examples, including magnetic vortices, helices, and spirals, are analyzed in detail to demonstrate general principles.

Hydrodynamic theory of coupled current and magnetization dynamics in spin-textured ferromagnets

0905.4544v2

CONVERGENCE ANALYSIS OF AN IMPLICIT FINITE DIFFERENCE METHOD FOR THE INERTIAL LANDAU-LIFSHITZ-GILBERT EQUATION JINGRUN CHEN, PANCHI LI, AND CHENG WANG Abstract. The Landau-Lifshitz-Gilbert (LLG) equation is a widely used model for fast magnetization dynamics in ferromagnetic materials. Recently, the iner- tial LLG equation, which contains an inertial term, has been proposed to cap- ture the ultra-fast magnetization dynamics at the sub-picosecond timescale. Mathematically, this generalized model contains the rst temporal derivative and a newly introduced second temporal derivative of magnetization. Conse- quently, it produces extra diculties in numerical analysis due to the mixed hyperbolic-parabolic type of this equation with degeneracy. In this work, we propose an implicit nite dierence scheme based on the central dierence in both time and space. A xed point iteration method is applied to solve the im- plicit nonlinear system. With the help of a second order accurate constructed solution, we provide a convergence analysis in H1for this numerical scheme, in the`1(0;T;H1 h) norm. It is shown that the proposed method is second order accurate in both time and space, with unconditional stability and a natural preservation of the magnetization length. In the hyperbolic regime, signicant damping wave behaviors of magnetization at a shorter timescale are observed through numerical simulations. 1.Introduction The Landau-Lifshitz-Gilbert (LLG) equation [15, 19] describes the dissipative magnetization dynamics in ferromagnetic materials, which is highly nonlinear and has a non-convex constraint. Physically, it is widely used to interpret the experi- mental observations. However, recent experiments [5, 16, 17] conrm that its valid- ity is limited to timescales from picosecond to larger timescales for which the angular momentum reaches equilibrium in a force eld. At shorter timescales, e.g. 100 fs, the ultra-fast magnetization dynamics has been observed [17]. To account for this, the inertial Landau-Lifshitz-Gilbert (iLLG) equation is proposed [6, 10, 12]. As a result, the magnetization converges to its equilibrium along a locus with damping nutation simulated in [21], when the inertial eect is activated by a non-equilibrium initialization or an external magnetic eld. For a ferromagnet over 2Rd;d= 1;2;3, the observable states are depicted by the distribution of the magnetization in . The magnetization denoted by m(x;t) is a vector eld taking values in the unit sphere S2ofR3, which indicates that jmj= 1 in a point-wise sense. In micromagnetics, the evolution of mis governed by the LLG equation. In addition to experiment and theory, micromagnetics simulations Date : September 13, 2022. 2010 Mathematics Subject Classication. Primary 35K61, 65M06, 65M12. Key words and phrases. Convergence analysis, inertial Landau-Lifshitz-Gilbert equation, im- plicit central dierence scheme, second order accuracy. 1arXiv:2209.02914v2 [math.NA] 12 Sep 20222 JINGRUN CHEN, PANCHI LI, AND CHENG WANG have become increasingly important over the past several decades. Therefore, nu- merous numerical approaches have been proposed for the LLG equation and its equivalent form, the Landau-Lifshitz (LL) equation; see [9, 18] for reviews and ref- erences therein. In terms of time marching, the simplest explicit methods, such as the forward Euler method and Runge-Kutta methods, were favored in the early days, while small time step size must be used due to the stability restriction [22]. Of course, implicit methods avoid the stability constraint and these methods pro- duce the approximate solution in H1( ) [1, 2]. However, in order to guarantee the convergence of the schemes, a step-size condition k=O(h2) must be satised in both the theoretical analysis and numerical simulations. To obtain the weak solu- tion in the nite element framework, an intermediate variable vwith the denition v=@tmrepresenting the increment rate at current time is introduced, and to solve vin the tangent space of mwhere it satises v
m= 0 in a point-wise sense, then the conguration at the next time step can be obtained. Directly, the strong solu- tion can be obtained through solving the implicit mid-point scheme [4] and the im- plicit backward Euler scheme [13] using xed-point iteration methods. By contrast, the semi-implicit methods have achieved a desired balance between stability and eciency for the micromagnetics simulations. The Gauss-Seidel projection meth- ods [11, 20, 27], the linearized backward Euler scheme [8, 14], the Crank-Nicolson projection scheme [3], and the second order semi-implicit backward dierentiation formula projection scheme [7, 28] have been developed in recent years. In prac- tice, all these semi-implicit methods inherit the unconditional stability of implicit schemes, and achieve the considerable improvement in eciency. The LLG equation is a nonlinear parabolic system which consists of the gyro- magnetic term and the damping term. It is a classical kinetic equation that only contains the velocity; no acceleration is included in the equation. When relaxing the system from a non-equilibrium state or applying a perturbation, it is natural that an acceleration term will be present, resulting in the inertial term in the iLLG equation. More specically, the time evolution of m(x;t) is described by @tmand m@tmwith the addition of an inertial term m@ttm. Thus, the iLLG equa- tion is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy. To numerically study the hyperbolic behaviors of the magnetization, the rst-order accuracy tangent plane scheme (TPS) and the second-order accuracy angular mo- mentum method (AMM) are proposed in [23]. The xed-point iteration method is used for the implicit marching. These two methods aim to nd the weak solution. Furthermore, a second-order accurate semi-implicit method is presented in [21], and @ttmand@tmare approximated by the central dierence. In this work, we provide the convergence analysis of the implicit mid-point scheme on three time layers for the iLLG equation. Subject to the condition kCh2, it produces a unique second-order approximation in H1( T). Owing to the application of the mid-point scheme, it naturally preserves the magnetization length. Moreover, we propose a xed-point iteration method to solve the nonlinear scheme, which converges to a unique solution under the condition of kCh2. Numerical simulations are reported to conrm the theoretic analysis and study the inertial dynamics at shorter timescales. The rest of this paper is organized as follows. The iLLG equation and the numerical method are introduced in Section 2. The detailed convergence analysis is provided in Section 3. In addition, a xed-point iteration method for solvingCONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 3 the implicit scheme is proposed in Section 4, and the convergence is established upon the condition kCh2. Numerical tests, including the accuracy test and observation of the inertial eect, are presented in Section 5. Concluding remarks are made in Section 6. 2.The physical model and the numerical method The intrinsic magnetization of a ferromagnetic body m=m(x;t) : T:= (0;T)!S2is modeled by the conventional LLG equation: @tm=m
m+m@tm; (x;t)2 T; (2.1a) m(x;0) =m(0); x2 ; (2.1b) @m(x;t) = 0; (x;t)2@ [0;T]; (2.1c) whererepresents the unit outward normal vector on @ , and1 is the damping parameter. If the relaxation starts from a non-equilibrium state or a sudden perturbation is applied, the acceleration should be considered in the kinetic equation, which is the inertial eect observed in various experiments at the sub- picosecond timescale. In turn, its dynamics is described by the iLLG equation @tm=m(
m+He) +m(@tm+@ttm); (x;t)2 T; (2.2a) m(x;0) =m(0); x2 ; (2.2b) @tm(x;0) = 0; x2 ; (2.2c) @m(x;t) = 0; (x;t)2@ [0;T]; (2.2d) whereis the phenomenological inertia parameter, and Heis a perturbation of an applied magnetic eld. To ease the discussion, the external eld is neglected in the subsequent analysis and is only considered in micromagnetics simulations. An additional initial condition @tm(x;0) = 0 is added, which implies that the velocity is 0 att= 0 and it is a necessary condition for the well-posedness. Then the energy is dened as (2.3)E[m] =1 2Z  jrmj22m
He+j@tmj2 dx: For constant external magnetic elds, it satises the energy dissipation law (2.4)d dtE[m] =Z j@tmj2dx0: Therefore, under the condition of (2.2c), for almost all T02(0;T), we have (2.5)1 2Z  jrm(x;T0)j2+j@tm(x;T0)j2 dx1 2Z jrm(x;0)j2
dx: Before the formal algorithm is presented, here the spatial dierence mesh and the temporal discretization have to be stated. The uniform mesh for is con- structed with mesh-size hand a time step-size k > 0 is set. Let Lbe the set of nodesfxl= (xi;yj;zk)gin 3-D space with the indices i= 0;1;


;nx;nx + 1;j= 0;1;


;ny;ny +1 andk= 0;1;


;nz;nz +1, and the ghost points on the bound- ary of are denoted by ix= 0;nx+ 1,jy= 0;ny+ 1 andkz= 0;nz+ 1. We use the half grid points with mi;j;k=m((i1 2)hx;(j1 2)hy;(k1 2)hz). Here hx= 1=nx,hy= 1=ny,hz= 1=nzandh=hx=hy=hzholds for uniform4 JINGRUN CHEN, PANCHI LI, AND CHENG WANG spatial meshes. Due to the homogeneous Neumann boundary condition (2.2d), the following extrapolation formula is derived: (2.6)mix+1;j;k=mix;j;k;mi;jy+1;k=mi;jy;k;mi;j;k z+1=mi;j;k z; for any 1inx;1jny;1knz. Meanwhile, the temporal derivatives are discretized by the central dierence, with the details stated in the following denition. Denition 2.1. Forn+1=(x;tn+1)and n+1= (tn+1), dene d+ tn=n+1n k; d tn=nn1 k; and D+ t n= n+1 n k; D t n= n n1 k: Consequently, we denote dtn+1=1 2(d+ tn+d tn); Dt n+1=1 2(D+ t n+D t n): In particular, the second time derivative is approximated by the central dierence form (2.7) dtt=n+12n+n1 k2: Then for the initial condition (2.2c), there holds (2.8) m(xl;0) =m(xl;k);8l2L; whereL=f(i;j;k )ji= 1;


;nx;j= 1;


;ny;k= 1;


;nz:g. Denotemn h(n 0) as the numerical solution. Given grid functions fh;gh2`2( h;R3), we list denitions of the discrete inner product and norms used in this paper. Denition 2.2. The discrete inner product h
;
iin`2( h;R3)is dened by (2.9) hfh;ghi=hdX l2Lfh(xl)
gh(xl): The discrete `2norm andH1 hnorm ofmhare (2.10) kfhk2 2=hdX l2Lfh(xl)
fh(xl); and (2.11) kfhk2 H1 h=kfhk2 2+krhfhk2 2 withrhrepresenting the central dierence stencil of the gradient operator. Besides, the norm k
k1in`1( h;R3) is dened by (2.12) kfhk1= max l2Lkfh(xl)k1: Therefore, the approximation scheme of the iLLG equation is presented below.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 5 Algorithm 2.1. Givenm0 h;m1 h2W1;2( h;S2). Letmn1 h;mn h2W1;2( h;S2), we computemn+1 hby (2.13) dtmn+1 hmn h dtmn+1 h+dttmn h
=mn h
hmn h; where mn h=1 2(mn+1 h+mn1 h), and
hrepresents the standard seven-point stencil of the Laplacian operator. The corresponding fully discrete version of the above (2.13) reads as mn+1 hmn1 h 2kmn+1 h+mn1 h 2mn+1 hmn1 h 2k+mn+1 h2mn h+mn1 h k2 =mn+1 h+mn1 h 2
hmn+1 h+mn1 h 2 : (2.14) Within three time steps, there have not been many direct discretization methods to get the second-order temporal accuracy. Due to the mid-point approximation feature, this implicit scheme is excellent in maintaining certain properties of the original system. Lemma 2.1. Givenm0 h(xl)= 1, then the sequence fmn h(xl)gn0produced by (2.13) satises (i)jmn h(xl)j= 1;8l2L; (ii)1 2Dtkrhmn+1 hk2 2+kdtmn+1 hk2 2+1 2D tkd+ tmn hk2 2= 0. Proof. On account of the initial condition (2.2c), we see that m0(xl) =m1(xl) holding for all l2L. Taking the vector inner product with (2.13) by ( mn+1 h(xl) + mn1 h(xl)), it obvious that we can get jmn+1 hj=jmn hj=


=jm1 hj=jm0 hj= 1; in the point-wise sense. This conrms (i). In order to verify (ii), we take inner product with (2.13) by
hmn hand get 1 2Dtkrhmn+1 hk2 2hmn hdtmn+1 h;
hmn hihmn hdttmn h;
hmn hi= 0: Subsequently, taking inner products with dtmn+1 handdttmn+1 hseparately leads to the following equalities: kdtmn+1 hk2 2hmn hdttmn h;dtmn+1 hi=hmn hdtmn+1 h;
hmn hi; and 1 2D tkd+ tmn hk2 2+hmn hdttmn h;dtmn+1 hi=hmn hdttmn h;
hmn hi: A combination of the above three identities yields (ii).  In lemma 2.1, taking k!0 gives (2.15)d dt1 2krhmn+1 hk2 2+ 2k@tmn hk2 2 =k@tmn+1 hk2 2; which is consistent with the continuous energy law (2.4). Accordingly, in the ab- sence of the external magnetic eld, the discretized version energy dissipation law would be maintained with a modication (2.16)E(mn+1 h;mn h) = 2 mn+1 hmn h k 2 2+1 4(krhmn+1 hk2 2+krhmn hk2 2):6 JINGRUN CHEN, PANCHI LI, AND CHENG WANG Theorem 2.1. Givenmn1 h;mn h;mn+1 h2W1;2( h;S2), we have a discrete energy dissipation law, for the modied energy (2.16) : (2.17) E(mn+1 h;mn h)E(mn h;mn1 h): Proof. Denote a discrete function n:=mn+1 hmn1 h 2k+mn+1 h2mn h+mn1 h k2 1 2
h(mn+1 h+mn1 h): Taking a discrete inner product with (2.13) by ngives  4k2hmn+1 hmn1 h;mn+1 hmn1 hi+ 2k3hmn+1 hmn1 h;mn+1 h2mn h+mn1 hi(2.18) + 4kD mn+1 hmn1 h;
h(mn+1 h+mn1 h)E =D mn+1 h+mn1 h 2n;nE = 0: Meanwhile, the following estimates are available: hmn+1 hmn1 h;mn+1 hmn1 hi=kmn+1 hmn1 hk2 20; (2.19) hmn+1 hmn1 h;mn+1 h2mn h+mn1 hi =D (mn+1 hmn h) + (mn hmn1 h);(mn+1 hmn h)(mn hmn1 h)E ; =kmn+1 hmn hk2 2kmn hmn1 hk2 2;(2.20) D mn+1 hmn1 h;
h(mn+1 h+mn1 h)E =D rh(mn+1 hmn1 h);rh(mn+1 h+mn1 h)E =krhmn+1 hk2 2krhmn1 hk2 2 =(krhmn+1 hk2 2+krhmn hk2 2)(krhmn hk2 2+krhmn1 hk2 2):(2.21) Going back to (2.18), we arrive at  2k mn+1 hmn h k 2 2 mn hmn1 h k 2 2 (2.22) +1 4k (krhmn+1 hk2 2+krhmn hk2 2)(krhmn hk2 2+krhmn1 hk2 2) = mn+1 hmn1 h 2k 2 20; which is exactly the energy dissipation estimate (2.17). This nishes the proof of Theorem 2.1.  Meanwhile, it is noticed that, given the initial prole of matt= 0, namelym0, an accurate approximation to m1andm2has to be made. In more details, an O(k2+h2) accuracy is required for both m1,m2andm1m0 k,m2m1 k, which is needed in the convergence analysis. The initial prole m0could be taken as m0=m(
;0). This in turn gives a trivial zero initial error for m0. Form1andm2, a careful Taylor expansion revealsCONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 7 that m1=m0+k@tm0+k2 2@ttm0+O(k3) =m0+k2 2@ttm0+O(k3); (2.23) m2=m0+ 2k@tm0+ 2k2@ttm0+O(k3) =m0+ 2k2@ttm0+O(k3); (2.24) in which the initial data (2.2c), @tm(
;0)0, has been applied in the derivation. Therefore, an accurate approximation to m1andm2relies on a precise value of @ttmatt= 0. An evaluation of the original PDE (2.2a) implies that m0(@ttm0) =1 m0(
m0+H0 e); (2.25) in which the trivial initial data (2.2c) has been applied again. Meanwhile, motivated by the point-wise temporal dierentiation identity (2.26) m
@ttm=(@tm)2+1 2@tt(jmj2) =(@tm)2; and the fact that jmj1, we see that its evaluation at t= 0 yields (2.27) m0
@ttm0=(@tm0)2= 0: Subsequently, a combination of (2.26) and (2.27) uniquely determines @ttm0: (2.28) @ttm0=1 m0(m0(
m0+H0 e)); and a substitution of this value into (2.23), (2.24) leads to an O(k3) approximation tom1andm2. Moreover, with spatial approximation introduced, an O(k2+h2) accuracy is obtained for both m1,m2andm1m0 k,m2m1 k. This nishes the initialization process. 3.Convergence analysis The theoretical result concerning the convergence analysis is stated below. Theorem 3.1. Assume that the exact solution of (2.2) has the regularity me2 C3([0;T]; [C0( )]3)\C2([0;T]; [C2( )]3)\L1([0;T]; [C4( )]3). Denote a nodal interpolation operator Phsuch thatPhmh2C1( ), and the numerical solution mn h (n0) obtained from (2.13) with the initial error satisfying kep hk2+krhep hk2= O(k2+h2), whereep h=Phme(
;tp)mp h,p= 0;1;2, andkeq+1 heq h kk2=O(k2+h2), q= 0;1. Then the following convergence result holds for 2nT k ash;k!0+: kPhme(
;tn)mn hk2+krh(Phme(
;tn)mn h)k2C(k2+h2); (3.1) in which the constant C>0is independent of kandh. Before the rigorous proof is given, the following estimates are declared, which will be utilized in the convergence analysis. In the sequel, for simplicity of notation, we will use a uniform constant Cto denote all the controllable constants throughout this part.8 JINGRUN CHEN, PANCHI LI, AND CHENG WANG Lemma 3.1 (Discrete gradient acting on cross product) .[7]For grid functions fh andghover the uniform numerical grid, we have krh(fhgh)k2C kfhk2
krhghk1+kghk1
krhfhk2 : (3.2) Lemma 3.2 (Point-wise product involved with second order temporal stencil) .For grid functions fhandghover the time domain, we have fn+1 h2fn h+fn1 h k2
gn h=fn hfn1 h k
gn hgn1 h k +1 kfn+1 hfn h k
gn hfn hfn1 h k
gn1 h : (3.3) Now we proceed into the convergence estimate. First, we construct an approxi- mate solution m: (3.4) m=me+h2m(1); in which the auxiliary eld m(1)satises the following Poisson equation
m(1)=^Cwith ^C=1 j jZ @ @3 meds; (3.5) @zm(1)jz=0=1 24@3 zmejz=0; @zm(1)jz=1=1 24@3 zmejz=1; with boundary conditions along xandydirections dened in a similar way. The purpose of such a construction will be illustrated later. Then we extend the approximate prole mto the numerical \ghost" points, according to the extrapo- lation formula: (3.6) mi;j;0=mi;j;1;mi;j;nz +1=mi;j;nz; and the extrapolation for other boundaries can be formulated in the same man- ner. Subsequently, we prove that such an extrapolation yields a higher order O(h5) approximation, instead of the standard O(h3) accuracy. Also see the re- lated works [24, 25, 26] in the existing literature. Performing a careful Taylor expansion for the exact solution around the boundary sectionz= 0, combined with the mesh point values: z0=1 2h,z1=1 2h, we get me(xi;yj;z0) =me(xi;yj;z1)h@zme(xi;yj;0)h3 24@3 zme(xi;yj;0) +O(h5) =me(xi;yj;z1)h3 24@3 zme(xi;yj;0) +O(h5); (3.7) in which the homogenous boundary condition has been applied in the second step. A similar Taylor expansion for the constructed prole m(1)reveals that m(1)(xi;yj;z0) =m(1)(xi;yj;z1)h@zm(1)(xi;yj;0) +O(h3) =m(1)(xi;yj;z1) +h 24@3 zme(xi;yj;0) +O(h3); (3.8) with the boundary condition in (3.5) applied. In turn, a substitution of (3.7)-(3.8) into (3.4) indicates that (3.9) m(xi;yj;z0) =m(xi;yj;z1) +O(h5): In other words, the extrapolation formula (3.6) is indeed O(h5) accurate.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 9 As a result of the boundary extrapolation estimate (3.9), we see that the discrete Laplacian of myields the second-order accuracy at all the mesh points (including boundary points): (3.10)
hmi;j;k=
me(xi;yj;zk)+O(h2);80inx+1;0jny+1;0knz+1: Moreover, a detailed calculation of Taylor expansion, in both time and space, leads to the following truncation error estimate: mn+1 hmn1 h 2k=mn+1 h+mn1 h 2 mn+1 hmn1 h 2k+mn+1 h2mn h+mn1 h k2
hmn+1 h+mn1 h 2 +n; (3.11) whereknk2C(k2+h2). In addition, a higher order Taylor expansion in space and time reveals the following estimate for the discrete gradient of the truncation error, in both time and space: (3.12) krhnk2;knn1 kk2C(k2+h2): In fact, such a discrete k
kH1 hbound for the truncation comes from the regularity as- sumption for the exact solution, me2C3([0;T]; [C0( )]3)\C2([0;T]; [C2( )]3)\ L1([0;T]; [C4( )]3), as stated in Theorem 3.1, as well as the fact that m(1)2 C1([0;T]; [C1( )]3)\L1([0;T]; [C2( )]3), as indicated by the Poisson equation (3.5). We introduce the numerical error function en h=mn hmn h, instead of a di- rect comparison between the numerical solution and the exact solution. The error function between the numerical solution and the constructed solution mhwill be analyzed, due to its higher order consistency estimate (3.9) around the boundary. Therefore, a subtraction of (2.14) from the consistency estimate (3.11) leads to the error function evolution system: en+1 hen1 h 2k=mn+1 h+mn1 h 2~n h+en+1 h+en1 h 2n h+n;(3.13) n h:=mn+1 hmn1 h 2k+mn+1 h2mn h+mn1 h k2
hmn+1 h+mn1 h 2 ;(3.14) ~n h:=en+1 hen1 h 2k+en+1 h2en h+en1 h k2
hen+1 h+en1 h 2 :(3.15) Before proceeding into the formal estimate, we establish a W1 hbound forn h, which is based on the constructed approximate solution m(by (3.14)). Because of the regularity for me, the following bound is available: k` hk1;krh` hk1;kn hn1 h kk1C; ` =n;n1: (3.16) In addition, the following preliminary estimate will be useful in the convergence analysis.10 JINGRUN CHEN, PANCHI LI, AND CHENG WANG Lemma 3.3 (A preliminary error estimate) .We have ke` hk2 22ke0 hk2 2+ 2Tk`1X j=0kej+1 hej h kk2 2;8`
kT: (3.17) Proof. We begin with the expansion: e` h=e0 h+k`1X j=0ej+1 hej h k;8`
kT: (3.18) In turn, a careful application of the Cauchy inequality reveals that ke` hk2 22 ke0 hk2 2+k2k`1X j=0ej+1 hej h kk2 2 ; (3.19) k2k`1X j=0ej+1 hej h kk2 2k2
`
`1X j=0kej+1 hej h kk2 2Tk`1X j=0kej+1 hej h kk2 2; (3.20) in which the fact that `
kThas been applied. Therefore, a combination of (3.19) and (3.20) yields the desired estimate (3.17). This completes the proof of Lemma 3.3.  Taking a discrete inner product with the numerical error equation (3.13) by ~n h gives 1 2khen+1 hen1 h;~n hi=hmn+1 h+mn1 h 2~n h;~n hi +hen+1 h+en1 h 2n h;~n hi+hn;~n hi: (3.21) The analysis on the left hand side of (3.21) is similar to the ones in (2.19)-(2.21): 1 2khen+1 hen1 h;~n hi= 2k3hen+1 hen1 h;en+1 h2en h+en1 hi + 4k2hen+1 hen1 h;en+1 hen1 hi +1 4kD rh(en+1 hen1 h);rh(en+1 h+en1 h)E ; (3.22) hen+1 hen1 h;en+1 hen1 hi=ken+1en1 hk2 20; (3.23) hen+1 hen1 h;en+1 h2en h+en1 hi =ken+1 hen hk2 2ken hen1 hk2 2; (3.24) D en+1 hen1 h;
h(en+1 h+en1 h)E =D rh(en+1 hen1 h);rh(en+1 h+en1 h)E =krhen+1 hk2 2krhen1 hk2 2 =(krhen+1 hk2 2+krhen hk2 2)(krhen hk2 2+krhen1 hk2 2): (3.25) This in turn leads to the following identity: 1 2khen+1 hen1 h;~n hi=1 k(En+1e;hEne;h) + 4k2ken+1 hen1 hk2 2; (3.26) En+1e;h= 2ken+1 hen h kk2 2+1 4(krhen+1 hk2 2+krhen hk2 2): (3.27)CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 11 The rst term on the right hand side of (3.21) vanishes, due to the fact that mn+1 h+mn1 h 2~n his orthogonal to ~n h, at a point-wise level: (3.28) hmn+1 h+mn1 h 2~n h;~n hi= 0: The second term on the right hand side of (3.21) contains three parts: hen+1 h+en1 h 2n h;~n hi=I1+I2+I3; (3.29) I1=hen+1 h+en1 h 2n h;en+1 hen1 h 2ki; (3.30) I2=hen+1 h+en1 h 2n h;en+1 h2en h+en1 h k2i; (3.31) I3=hen+1 h+en1 h 2n h;
hen+1 h+en1 h 2 i: (3.32) The rst inner product, I1, could be bounded in a straightforward way, with the help of discrete H older inequality: I1=hen+1 h+en1 h 2n h;en+1 hen1 h 2ki  4ken+1 h+en1 hk2
kn hk1
ken+1 hen1 h kk2 Cken+1 h+en1 hk2
ken+1 hen1 h kk2 C(ken+1 hk2 2+ken1 hk2 2+ken+1 hen1 h kk2 2): (3.33) For the second inner product, I2, we denotegn h:=en+1 h+en1 h 2n h. An application of point-wise identity (3.3) (in lemma 3.2) reveals that I2=hgn h;en+1 h2en h+en1 h k2i =hen hen1 h k;gn hgn1 h ki + k hen+1 hen h k;gn hihen hen1 h k;gn1 hi : (3.34) Meanwhile, the following expansion is observed: gn hgn1 h k=1 4(en+1 hen h k+en1 hen2 h k)(n h+n1 h) +en+1 h+en h+en1 h+en2 h 4n hn1 h k: (3.35)12 JINGRUN CHEN, PANCHI LI, AND CHENG WANG This in turn indicates the associated estimate: kgn hgn1 h kk21 4(ken+1 hen h kk2+ken1 hen2 h kk2)
(kn hk1+kn1 hk1) +ken+1 hk2+ken hk2+ken1 hk2+ken2 hk2 4
kn hn1 h kk1 C ken+1 hen h kk2+ken1 hen2 h kk2 +ken+1 hk2+ken hk2+ken1 hk2+ken2 hk2 ; (3.36) in which the bound (3.16) has been applied. Going back to (3.34), we see that hen hen1 h k;gn hgn1 h kiken hen1 h kk2
kgn hgn1 h kk2 C ken+1 hen h kk2+ken1 hen2 h kk2+ken+1 hk2 +ken hk2+ken1 hk2+ken2 hk2 ken hen1 h kk2 C ken+1 hen h kk2 2+ken1 hen2 h kk2 2+ken+1 hk2 2 +ken hk2 2+ken1 hk2 2+ken2 hk2 2+ken hen1 h kk2 2 ; (3.37) I2C ken+1 hen h kk2 2+ken1 hen2 h kk2 2+ken+1 hk2 2 +ken hk2 2+ken1 hk2 2+ken2 hk2 2+ken hen1 h kk2 2 + k hen+1 hen h k;gn hihen hen1 h k;gn1 hi : (3.38) For the third inner product part, I3, an application of summation by parts formula gives I3=hen+1 h+en1 h 2n h;
hen+1 h+en1 h 2 i =hrhen+1 h+en1 h 2n h ;rhen+1 h+en1 h 2 i: (3.39) Meanwhile, we make use of the preliminary inequality (3.2) (in lemma 3.1) and get krhen+1 h+en1 h 2n h k2 C ken+1 h+en1 h 2k2
krhn hk1+kn hk1
krh(en+1 h+en1 h 2)k2 C ken+1 h+en1 h 2k2+krh(en+1 h+en1 h 2)k2 C ken+1 hk2+ken1 hk2+krhen+1 hk2+krhen1 hk2 : (3.40)CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 13 Again, the bound (3.16) has been applied in the derivation. Therefore, the following estimate is available for I3: I3krhen+1 h+en1 h 2n h k2
krhen+1 h+en1 h 2 k2 C ken+1 hk2+ken1 hk2+krhen+1 hk2+krhen1 hk2
 krhen+1 hk2+krhen1 hk2 C ken+1 hk2 2+ken1 hk2 2+krhen+1 hk2 2+krhen1 hk2 2 : (3.41) The estimate of I3can also be obtained by a direct application of discrete H older inequality: I3=hen+1 h+en1 h 2rhn h ;rhen+1 h+en1 h 2 i 1 4ken+1 h+en1 hk2
krhn hk1
krh(en+1 h+en1 h)k2 C ken+1 hk2 2+ken1 hk2 2+krhen+1 hk2 2+krhen1 hk2 2 : (3.42) A substitution of (3.33), (3.38) and (3.42) into (3.29) yields the following bound: hen+1 h+en1 h 2n h;~n hi=I1+I2+I3 C ken+1 hen h kk2 2+ken hen1 h kk2 2+ken1 hen2 h kk2 2 +ken+1 hk2 2+ken hk2 2+ken1 hk2 2+ken2 hk2 2+krhen+1 hk2 2+krhen1 hk2 2 + k hen+1 hen h k;gn hihen hen1 h k;gn1 hi : (3.43) The third term on the right hand side of (3.21) could be analyzed in a similar fashion: hn;~n hi=I4+I5+I6; (3.44) I4=hn;en+1 hen1 h 2ki; I 5=hn;en+1 h2en h+en1 h k2i; (3.45) I6=hn;
hen+1 h+en1 h 2 i; (3.46) I4=hn;en+1 hen1 h 2ki 2knk2
ken+1 hen1 h kk2  4(knk2 2+ken+1 hen1 h kk2 2); (3.47) I5=hn;en+1 h2en h+en1 h k2i =hen hen1 h k;nn1 ki + k hen+1 hen h k;nihen hen1 h k;n1i ; (3.48)14 JINGRUN CHEN, PANCHI LI, AND CHENG WANG hen hen1 h k;nn1 kiken hen1 h kk2
knn1 kk2 C(k2+h2)ken hen1 h kk2C(k4+h4) +1 2ken hen1 h kk2 2; (3.49) I5C(k4+h4) + 2ken hen1 h kk2 2 + k hen+1 hen h k;nihen hen1 h k;n1i ; (3.50) I6=hn;
hen+1 h+en1 h 2 i=hrhn;rhen+1 h+en1 h 2 i krhnk2
krhen+1 h+en1 h 2 k2C(k2+h2)krhen+1 h+en1 h 2 k2 C(k4+h4) +1 2 krhen+1 hk2 2+krhen1 hk2 2 : (3.51) Notice that the truncation error estimate (3.12) has been repeatedly applied in the derivation. Going back to (3.44), we obtain hn;~n hi=I4+I5+I6 C(k4+h4) + 2ken+1 hen h kk2 2+(+ 1) 2ken hen1 h kk2 2 +1 2 krhen+1 hk2 2+krhen1 hk2 2 + k hen+1 hen h k;nihen hen1 h k;n1i : (3.52) Finally, a substitution of (3.26)-(3.27), (3.28), (3.43) and (3.52) into (3.21) leads to the following inequality: 1 k(En+1e;hEne;h) + 4k2ken+1 hen1 hk2 2 C(k4+h4) +C ken+1 hen h kk2 2+ken hen1 h kk2 2+ken1 hen2 h kk2 2 +ken+1 hk2 2+ken hk2 2+ken1 hk2 2+ken2 hk2 2+krhen+1 hk2 2+krhen1 hk2 2 + k hen+1 hen h k;gn h+nihen hen1 h k;gn1 h+n1i : (3.53) Subsequently, a summation in time yields En+1e;hE2e;h+CT(k4+h4) +CknX j=0kej+1 hej h kk2 2+n+1X j=0(kej hk2 2+krhej hk2 2) + hen+1 hen h k;gn h+nihe2 he1 h k;g1 h+1i : (3.54)CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 15 For the term hen+1 hen h k;gn h+ni, the following estimate could be derived hen+1 hen h k;gn h+ni 4ken+1 hen h kk2 2+ 2(kgn hk2 2+knk2 2); (3.55) kgn hk2=ken+1 h+en1 h 2n hk2ken+1 h+en1 h 2k2
kn hk1 Cken+1 h+en1 h 2k2C(ken+1 hk2+ken1 hk2); (3.56) in which the bound (3.16) has been used again. Then we get hen+1 hen h k;gn h+ni 4ken+1 hen h kk2 2+ 2knk2 2 +C(ken+1 hk2 2+ken1 hk2 2) 1 2En+1e;h+ 2knk2 2+C(ken+1 hk2 2+ken1 hk2 2); (3.57) in which the expansion identity, En+1e;h= 2ken+1 hen h kk2 2+1 4(krhen+1 hk2 2+krhen hk2 2) (given by (3.27)), has been applied. Its substitution into (3.54) gives En+1e;h2E2e;h+CT(k4+h4) +CknX j=0kej+1 hej h kk2 2+n+1X j=0(kej hk2 2+krhej hk2 2) +C(ken+1 hk2 2+ken1 hk2 2) + 4knk2 22he2 he1 h k;g1 h+1i: (3.58) Moreover, an application of the preliminary error estimate (3.17) (in Lemma 3.3) leads to En+1e;h2E2e;h+CT(k4+h4) +C(T2+ 1)knX j=0kej+1 hej h kk2 2+CTke0 hk2 2 +Ckn+1X j=0krhej hk2 2+ 4knk2 22he2 he1 h k;g1 h+1i; (3.59) in which we have made use of the following fact: kn+1X j=0kej hk2 2k
(n+ 1) 2ke0 hk2 2+ 2TknX j=0kej+1 hej h kk2 2 2Tke0 hk2 2+ 2T2knX j=0kej+1 hej h kk2 2: (3.60)16 JINGRUN CHEN, PANCHI LI, AND CHENG WANG In addition, for the initial error quantities, the following estimates are available: E2e;h= 2ke2 he1 h kk2 2+1 4(krhe2 hk2 2+krhe1 hk2 2)C(k4+h4); (3.61) ke0 hk2 2C(k4+h4); (3.62) 4knk2 2C(k4+h4); (3.63) kg1 hk2=ke2 h+e0 h 21 hk2ke2 h+e0 h 2k2
k1 hk1C(k2+h2); (3.64) 2he2 he1 h k;g1 h+1i2ke2 he1 h kk2
(kg1 hk2+k1k2) C(k4+h4); (3.65) which comes from the assumption in Theorem 3.1. Then we arrive at En+1e;hC(T2+ 1)knX j=0kej+1 hej h kk2 2+Ckn+1X j=0krhej hk2 2+C(T+ 1)(k4+h4) C(T+ 1)(k4+h4) +C(T2+ 1)knX j=0Ej+1e;h; (3.66) in which the fact that Ej+1e;h= 2kej+1 hej h kk2 2+1 4(krhej+1 hk2 2+krhej hk2 2), has been used. In turn, an application of discrete Gronwall inequality results in the desired convergence estimate: En+1e;hCTeCT(k4+h4);for all (n+ 1) :n+ 1T k ; (3.67) ken+1 hen h kk2+krhen+1 hk2C(k2+h2): (3.68) Again, an application of the preliminary error estimate (3.17) (in Lemma 3.3) im- plies that ken+1 hk2 22ke0 hk2 2+ 2TknX j=0kej+1 hej h kk2 2C(k4+h4); so thatken+1 hk2C(k2+h2): (3.69) A combination of (3.68) and (3.69) nishes the proof of Theorem 3.1. 4.A numerical solver for the nonlinear system It is clear that Algorithm 2.1 is a nonlinear scheme. The following xed-point iteration is employed to solve it. Algorithm 4.1. Setmn+1;0 h= 2mn hmn1 handp= 0.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 17 (i)Computemn+1;p+1 hsuch that (4.1)mn+1;p+1 hmn1 h 2k=mn+1;p+1 h+mn1 h 2
h mn+1;p h+mn1 h 2! +mn+1;p+1 h+mn1 h 2 mn+1;p+1 hmn1 h 2k+mn+1;p+1 h2mn h+mn1 h k2! : (ii)Ifkmn+1;p+1 hmn+1;p hk2, then stop and set mn+1 h=mn+1;p+1 h. (iii) Setp p+ 1and go to (i). Denote the operator (4.2)Lp=Imn1 h2 kmn hk 2
h(mn+1;p h+mn1 h); and make the xed-point iteration solve the following equation (4.3)Lpmn+1;p+1 h=mn1 h+2 kmn hmn1 hk 2mn1 h
h(mn+1;p h+mn1 h); in its inner iteration. Under the condition kCh2withCa constant, the following lemma conrms the convergence of Algorithm 4.1. For any l2Land owing to the property ofjmh(xl)j= 1, it is clear that 0 <kmhk11. For the discretized `2 norm of ^m2C3([0;T]; [C0( )]3)\C2([0;T]; [C2( )]3)\L1([0;T]; [C4( )]3), we have (4.4) krh^mk22h1k^mk2: Then (4.5)k
h^mk2 2=hrh^m;rh
h^mi krh^mk2krh
h^mk22h1krh^mk2k
h^mk2; which in turn implies the following inverse inequality: (4.6) k
h^mk24h2k^mk2: Lemma 4.1. Letjmn1 hj=jmn hj= 1, there exists a constant c0such thatkmn1 hk1, kmn hk1c0. The solution mn+1;p hcalculated by (4.1) satisesjmn+1;p hj=jmn1 hj forp= 1;2;


, which means that we can still nd the constant c01satisfying kmn+1;p hk1c0. Then, for all p1, there exists a unique solution mn+1;p hin (4.1) and the following inequality is valid: (4.7)kmn+1;p+1 hmn+1;p hk24c0kh2kmn+1;pmn+1;p1k2: Proof. For anymh2S2, the following identity is clear: hmh;Lpmhi= 1; for allp1. Thus the operator Lpis positive denite for all p1, which provides the unique solvability of (4.1). Taking the discrete inner product with (4.1) bymn+1;p+1 h+mn1 h, we havejmn+1;p+1 hj= 1 in a point-wise sense, which means that the length of the magnetization is preserved at each step in the inner18 JINGRUN CHEN, PANCHI LI, AND CHENG WANG iteration. Thus, we can nd a constant c01 to control the `1norm ofmn1 h, mn handmn+1;p hforp= 1;2;


simultaneously. Subtraction of two subsequent equations in the xed-point iteration yields 1 2k(mn+1;p+1 hmn+1;p h) =1 4(mn+1;p+1 hmn+1;p h)
hmn+1;p h 1 4mn+1;p h
h(mn+1;p hmn+1;p1 h) 1 4mn h
h(mn+1;pmn+1;p1) 1 4(mn+1;p+1 hmn+1;p)
hmn1 h + 2kmn1 h(mn+1;p+1 hmn+1;p h) + 2k2mn h(mn+1;p+1 hmn+1;p h): Taking the inner product with ( mn+1;p+1 hmn+1;p h) by the above equation produces kmn+1;p+1 hmn+1;p hk2k 2kmn+1;p hk1k
h(mn+1;p hmn+1;p1 h)k2 +k 2kmn hk1k
h(mn+1;p hmn+1;p1 h)k2 c0kk
h(mn+1;pmn+1;p1)k2: In turn, the convergence result becomes (4.8)kmn+1;p+1 hmn+1;p hk24c0kh2kmn+1;p hmn+1;p1 hk2; which completes the proof of Lemma 4.1.  5.Numerical experiments 5.1.Accuracy tests. Consider the 1-D iLLG equation @tm=m@xxm+m(@tm+@ttm) +f: The exact solution is chosen to be me= (cos(x) sin(t2);sin(x) sin(t2);cos(t2))T with x=x2(1x)2, and the forcing term is given by f=@tme+me@xxme me(@tme+@ttme). Fixing the tolerance = 1.0e-07 for the xed-point iteration, we record the discrete `2and`1errors between the exact solution and numerical solution with a sequence of temporal step-size and spatial mesh-size. The parameters in the above 1-D equation are set as: = 0:1,= 10:0, and the nal timeT= 0:01. The temporal step-sizes and spatial mesh-sizes are listed in the Table 1 and Table 2. Table 1. The discrete `2and`1errors in terms of the temporal step-size. The spatial mesh-size is xed as h= 0:001 over = (0;1) and the nal time is T= 0:01. kkmhmek2kmhmek1 T/40 1.2500e-11 1.2584e-11 T/60 5.6887e-12 5.6024e-12 T/80 3.2008e-12 3.1525e-12 T/100 2.0487e-12 2.0174e-12 order 1.97 2.00CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 19 Table 2. The discrete `2and`1error in terms of the spatial mesh-size. The parameters are set as: the temporal step-size k= 2.0e-06, = (0 ;1) and the nal time T= 0:5. hkmhmek2kmhmek1 1/20 1.9742e-05 2.5320e-05 1/40 4.9846e-06 6.3459e-06 1/60 2.2340e-06 2.8201e-06 1/80 1.2720e-06 1.5853e-06 order 1.98 2.00 In addition, the 3-D iLLG equation is also considered: @tm=m
m+m(@tm+@ttm) +f: The exact solution is chosen to be me= (cos(xyz) sin(t2);sin(xyz) sin(t2));cos(t2))T with y=y2(1y)2and z=z2(1z)2, and the forcing term f=@tme+me
meme(@tme+@ttme). Similarly, we record the discrete `2and`1er- rors between exact and numerical solutions with a sequence of temporal step-sizes and spatial mesh-sizes. The corresponding parameters are set as: = 0:01 and = 1000:0. Besides, the nal time of this simulation is given by T= 0:01, with the temporal step-size and spatial mesh-size listed in Table 3 and Table 4. Table 3. The discrete `2and`1errors in terms of the temporal step-size. The spatial mesh-size is xed as h= 0:001 and nal time isT= 0:01. kkmhmek2kmhmek1 T/100 1.2678e-05 1.2765e-05 T/120 8.8067e-06 8.8830e-06 T/140 6.4725e-06 6.5419e-06 T/160 4.9576e-06 5.0224e-06 order 2.00 1.98 Table 4. The discrete `2and`1errors in terms of spatial mesh- size. The temporal step-size is xed as k= 2.0e-06. hkmhmek2kmhmek1 1/8 1.4392e-07 3.4940e-07 1/10 9.6832e-08 2.2864e-07 1/12 6.9825e-08 1.6079e-07 1/14 5.2828e-08 1.1895e-07 order 1.79 1.92 5.2.Micromagnetics tests. The inertial eect can be observed during the relax- ation of a system with a non-equilibrium initialization. To visualize this, we conduct micromagnetics simulations for both the LLG equation and the iLLG equation.20 JINGRUN CHEN, PANCHI LI, AND CHENG WANG In the following simulations, a 3-D domain = [0 ;1][0;1][0;0:4] is uniformly discretized into 10 104 cells, with uniform initialization m0= (p 2=2;p 2=2;0)T. For comparison, the LLG equation is discretized by the mid-point scheme proposed in [4] with the xed-point iteration solver proposed in this work. The damping pa- rameter is= 0:5 and the eld is xed as He= (10;0;0)T, which indicates that the system shall converge to m= (1;0;0)T. Here the relaxation of the magnetization behavior controlled by the LLG equation is visualized in Figure 1. Time0 1 2 3 4 5/angbracketleftm/angbracketright -0.500.51 /angbracketleftm1/angbracketright /angbracketleftm2/angbracketright /angbracketleftm3/angbracketright Figure 1. The relaxation of the spatially averaged magnetization controlled by the LLG equation. The nal time is T= 5:0 with k= 0:001, and the damping parameter is = 0:5. As for the counterpart of the LLG equation, wtih a given reference eld He, the discrete energy of the iLLG equation becomes (5.1) E[mn+1;mn] =1 4(krhmn+1 hk2 2+krhmn hk2 2)+  2 mn+1 hmn h k 2 21 2hmn+1 h+mn h;Hei: Setting the inertial parameter = 1:0, the spatially averaged magnetization is recorded to depict the inertial eect in Figure 2(a). Meanwhile, the energy decay is also numerically veried by Figure 2(b). The inertial eect is observed at shorter timescales for magnetization dynamics during the relaxation of the system with a non-equilibrium initialization. Furthermore, the inertial eect can also be activated by an external perturbation applied to an equilibrium state. Here we set the damping parameter = 0:02 and = 0:5, then the time step-size must be reduced to 0.001 with T= 3:0. For the equilibrium state m0= (1;0;0)T, the perturbation 4 :0sin(2ft) is applied along ydirection over the time interval [0 ;0:05], withf= 20. The relaxation of the iLLG equation, revealed by the evolution of the spatially averaged magnetization, is visualized in Figure 3.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 21 Time0 5 10 15 20/angbracketleftm/angbracketright -0.500.51 /angbracketleftm1/angbracketright /angbracketleftm2/angbracketright /angbracketleftm3/angbracketright (a)Spatially averaged magnetization Time0 2 4 6 8 10 12 14 16 18 20E[mn+1,mn] -4-3.8-3.6-3.4-3.2-3-2.8 (b)Energy decay Figure 2. The spatially averaged magnetization (A) and the en- ergy evolution (B) in the iLLG equation. Parameters setting: T= 20,k= 0:02,= 1:0 and= 0:5. 6.Conclusion In this work, an implicit mid-point scheme, with three time steps, is proposed to solve the inertial Landau-Lifshitz-Gilbert equation. This algorithm preserves the properties of magnetization dynamics, such as the energy decay and the constant length of magnetization and is proved to be second-order accurate in both space and time. In the convergence analysis, we rst construct a second-order approxi- mationmof the exact solution. It is found that mproducesO(h5) accuracy at the mesh points around the boundary sections, which simplies the estimation at boundary points. Then, by analyzing the error function between the numerical22 JINGRUN CHEN, PANCHI LI, AND CHENG WANG Time0 0.5 1 1.5 2 2.5 3/angbracketleftm/angbracketright -0.200.20.40.60.8/angbracketleftm1/angbracketright /angbracketleftm2/angbracketright /angbracketleftm3/angbracketright Figure 3. The response of the spatially averaged magnetization for the magnetic perturbation in the presence of the inertial eect. For the equilibrium initialization m0= (1;0;0)T, a perturbation 4:0sin(2ft) is applied along ydirection during time interval 00:05, withf= 20. The basic simulation parameters are: = 0:02,= 0:5,T= 3:0 andk= 0:001. solution and the constructed solution mh, we derive the convergence result in the H1( T) norm. Furthermore, a xed-point iteration method is proposed to solve this implicit nonlinear scheme under the time-step restriction kCh2. Numerical results conrm the theoretical analysis and clearly show the unique inertial eect in micromagnetics simulations. Acknowledgments This work is supported in part by the grant NSFC 11971021 (J. Chen), the program of China Scholarships Council No. 202106920036 (P. Li), the grant NSF DMS-2012269 (C. Wang). References 1. F. Alouges and P. Jaisson, Convergence ofanite element discretization forthe Landau-Lifshitz equations inmicromagnetism, Math. Models Methods Appl. Sci. 16(2006), no. 02, 299{316. 2. F. Alouges, E. Kritsikis, and J. Toussaint, Aconvergent nite element approximation for Landau-Lifschitz-Gilbert equation, Physica B: Condens. Matter 407(2012), no. 9, 1345{1349, 8th International Symposium on Hysteresis Modeling and Micromagnetics (HMM 2011). 3. R. An, H. Gao, and W. Sun, Optimal error analysis ofEuler andCrank-Nicolson projection nite dierence schemes forLandau-Lifshitz equation, SIAM J. Numer. Anal. 59(2021), no. 3, 1639{1662. 4. S. Bartels and A. Prohl, Convergence ofanimplicit nite element method forthe Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal. 44(2006), no. 4, 1405{1419. 5. E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Ultrafast spin dynamics in ferromagnetic nickel, Phys. Rev. Lett. 76(1996), 4250{4253. 6. S. Bhattacharjee, L. Nordstr om, and J. Fransson, Atomistic spin dynamic method with both damping andmoment ofinertia eects included from rst principles, Phys. Rev. Lett. 108 (2012), 057204.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 23 7. J. Chen, C. Wang, and C. 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Tsutsumi, Finite dierence scheme fortheLandau-Lifshitz equation, Japan J. Indust. Appl. Math. 29(2012), no. 1, 83{110. 14. H. Gao, Optimal error estimates ofalinearized Backward Euler FEM fortheLandau-Lifshitz equation, SIAM J. Numer. Anal. 52(2014), no. 5, 2574{2593. 15. T. Gilbert, ALagrangian formulation ofgyromagnetic equation ofthemagnetization eld, Phys. Rev. 100(1955), 1243{1255. 16. M. Kammerer, M. Weigand, M. Curcic, M. Noske, M. Sproll, A. Vansteenkiste, B. Van Waeyenberge, H. Stoll, G. Woltersdorf, C. H. Back, and G. Schuetz, Magnetic vortex core reversal byexcitation ofspin waves, Nat. Commun. 2(2011). 17. B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. F ahnle, T. Roth, M. Cinchetti, and M. Aeschlimann, Explaining the paradoxical diversity ofultrafast laser-induced demagnetization, Nature Mater. 9(2010), 259{265. 18. M. Kru z k and A. Prohl, Recent developments inthemodeling, analysis, andnumerics of ferromagnetism, SIAM Rev. 48(2006), no. 3, 439{483. 19. L. 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Wang, Surface pressure Poisson equation formulation oftheprimitive equations: Numerical schemes, SIAM J. Numer. Anal. 41(2003), 1163{1194. 25. C. Wang and J.-G. Liu, Convergence ofgauge method forincompressible ow, Math. Comp. 69(2000), 1385{1407. 26. C. Wang, J.-G. Liu, and H. Johnston, Analysis ofafourth order nite dierence method for incompressible Boussinesq equation, Numer. Math. 97(2004), 555{594. 27. X. Wang, C.J. Garc a-Cervera, and W. E, AGauss-Seidel projection method for micromagnetics simulations, J. Comput. Phys. 171(2001), no. 1, 357{372. 28. C. Xie, C. J. Garci a-Cervera, C. Wang, Z. Zhou, and J. Chen, Second-order semi-implicit projection methods formicromagnetics simulations, J. Comput. Phys. 404(2020), 109104.24 JINGRUN CHEN, PANCHI LI, AND CHENG WANG School of Mathematical Sciences and Suzhou Institute for Advanced Research, Uni- versity of Science and Technology of China, China Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou, Jiangsu 215123, China Email address :jingrunchen@ustc.edu.cn School of Mathematical Sciences, Soochow University, Suzhou, 215006, China Email address :lipanchi1994@163.com Mathematics Department, University of Massachusetts, North Dartmouth, MA 02747, USA Email address :cwang1@umassd.edu

2022-09-07

The Landau-Lifshitz-Gilbert (LLG) equation is a widely used model for fast magnetization dynamics in ferromagnetic materials. Recently, the inertial LLG equation, which contains an inertial term, has been proposed to capture the ultra-fast magnetization dynamics at the sub-picosecond timescale. Mathematically, this generalized model contains the first temporal derivative and a newly introduced second temporal derivative of magnetization. Consequently, it produces extra difficulties in numerical analysis due to the mixed hyperbolic-parabolic type of this equation with degeneracy. In this work, we propose an implicit finite difference scheme based on the central difference in both time and space. A fixed point iteration method is applied to solve the implicit nonlinear system. With the help of a second order accurate constructed solution, we provide a convergence analysis in $H^1$ for this numerical scheme, in the $\ell^\infty (0, T; H_h^1)$ norm. It is shown that the proposed method is second order accurate in both time and space, with unconditional stability and a natural preservation of the magnetization length. In the hyperbolic regime, significant damping wave behaviors of magnetization at a shorter timescale are observed through numerical simulations.

Convergence analysis of an implicit finite difference method for the inertial Landau-Lifshitz-Gilbert equation

2209.02914v2

1 Engineering Co 2MnAl xSi1-x Heusler compounds as a model system to correlate spin polarization, intrinsic Gilbert damping and ultrafast demagnetization C. Guillemard1,2, W. Zhang1*, G. Malinowski1, C. de Melo1, J. Gorchon1, S. Petit - Watelot1, J. Ghan baja1, S. Mangin1, P. Le Fèvre2, F. Bertran2, S. Andrieu1* 1 Institut Jean Lamour, UMR CNRS 7198, Université de Lorraine, 54500 Nancy France 2 Synchrotron SOLEIL -CNRS, Saint -Aubin, 91192 Gif -sur-Yvette, France Abstract: Engineering of magnetic materials f or developing better spintronic applications relies on the control of two key parameters: the spin polarization and the Gilbert damping responsible for the spin angular momentum dissipation. Both of them are expected to affect the ultrafast magnetization dyna mics occurring on the femtosecond time scale. Here, we use engineered Co2MnAl xSi1-x Heusler compounds to adjust the degree of spin polarization P from 60 to 100% and investigate how it correlates with the damping. We demonstrate experimentally that the damping decreases when increasing the spin polarization from 1.1 10-3 for Co 2MnAl with 63% spin polarization to an ultra -low value of 4.10-4 for the half -metal magnet Co 2MnSi. This allows us investigating the relation between these two parameters and the ultrafast demagnetization time characterizing the loss of magnetization occurring after femtosecond laser pulse excitation. The demagnetization time is observed to be inversely proportional to 1 -P and as a consequence to the magnetic damping, which can be attributed to the similarity of the spin angular momentum dissipation processes responsible for these two effects. Altogether, our high quality Heusler compounds allow controlling the band structure and therefore the channel for spin angular momentum dissipation. * corresponding authors : wei.z hang @univ -lorraine.fr stephane.andrieu@univ -lorraine.fr 2 I - INTRODUCTION During the last decades, extensive magnetic materials research has strived to engineer denser, faster and more energy efficient processing and data storage devices. On the one hand, a high spin polarization has been one of the most important ingredients th at have been seek [1]. For example, the spin polarization is responsible for a high readout signal in magnetic tunnel junction based devices [2,3] . Additionally, a high spin polarization results in a decrease of the threshold current for magnetization reve rsal by spin torques [4] required for the development of spin-transfer -torque magnetic random access memory devices [5] , for gyrotropic dynamics in spin -torque nano -oscillators [6] and for magnetic domain wall motion [7]. On the other hand, the intrinsic m agnetic energy dissipation during magnetization dynamics, which is determined by the Gilbert damping constant, needs to be low in order to build an energy efficient device. Fortunately, spin polarization and damping are usually closely related in magnetic materials. Nowadays, manipulation of the magnetization on the femtosecond timescale has become an outstanding challenge since the demonstration of subpicosecond magnetization quenching [8] and magnetization reversal on the picosecond timescale [9]. Despite the theoretical and experimental work that has been reported up to now , the relationship between the polarization at the Fermi level or the magnetic damping and the ultrafast demagnetization excited by femtosecond lasers, remains unclear [10-15]. Indeed, numerous mech anisms have been proposed but no consensus has yet been reached. In particular, efforts have been undertaken to unify the magnetization dynamics on the nanosecond timescale and the ultrafast demagnetization considering that the sp in-flip mechanisms involved in both phenomena could be the same [10-11,16] . Regarding the influence of the damping on the demagnetization time, different predictions have been reported both experimentally and theoretically . In this situation, the need for engineered samples in which the spin -polarization and magnetic damping are well controlled is of utmost importance to unveil their role on the ultrafast magnetization dynamics. Heusler compounds are a notable class of magnetic materials allowing for tunabl e spin-polarization and magnetic damping [ 17]. The absence of available electronic states in the minority band at the Fermi level leads to very high spin polarization and ultra -low damping due to a strong reduction of spin scattering [ 18-23]. Recently, ultra-low damping 3 coefficient associated with full spin polarization at the Fermi energy was reported in Co2Mn-based Heusler compounds , [22-23]. Among those alloys, Co 2MnSi has the smallest damping down to 4.1 x 10-4 with 100% spin -polarization while Co 2MnAl , which is not predicted to be a half -meta llic magnet, has a damping of 1.1 x 10-3 and a spin - polarization of 60 %. In the present work, we used Co 2MnAl xSi1-x quaternary Heusler compounds grown by Molecular Beam Epitaxy (MBE) to tune the spin -polarization at the Fermi energy . Controlling the amount of Al within the alloys allows tuni ng the spin -polarization from 60 to 100 % as measured by spin resolved photoemission. We show that the magnetic damping parameter for these alloys is among the lowest reported in the literature and decreases when the spin -polarization increases. Ultrafast magnetization dynamics experiments were thus performed on these prototype samples. This complete experimental characterization allows us to directly correlat e the ultrafast magnetization dynamics to these parameters and comparing our results to the different theory discussed above. The Co 2MnSi compound grows in the L2 1 structure whereas the Co 2MnAl com pound grows in the B2 phase as shown by STEM -HAADF analysis [22]. Such different structures are directly observable during the growth by Reflexion High Energy Electron Diffraction (RHEED ) since the surface lattice is different for bot h compounds. Ind eed, half streaks are observed along Co 2MnSi [110] azimuth due to the L2 1 chemical ordering [24] which is not the case for Co2MnAl [22]. The RHEED analysis on Co2MnAl xSi1-x films with x= 0, ¼ ,½ , ¾ ,1 reveals a regular decrease of these half -streaks intensity with x (Figure 1 a). This information that concerns only the surface is confirmed in the entire thickness of the films by using x -ray diffraction. Indeed, the (111) peak typical of the chemical ordering in the L2 1 structure clearly decreases and disappears with x ( Figure 1b). 4 Figure 1 : a) RHEED patterns along [110] showing the progressive vanishing of the half -streaks (observed on Co 2MnSi, x=0) at the surface with x. b) Confirmation of the transition from L2 1 to B2 chemical ordering in the entire film by the vanishing of the (111) peak and displacement of (220) peak with x as shown by x -ray diffraction. c) Spatial distribution of both chemical ordering in the films deduced from STEM -HAADF experiments: as the L 21 structure is observed in the entire Co 2MnSi film (x=0 ), and the B2 one i n Co 2MnAl (x=1 ), a mixing of both structure is clearly observed for x=0.5 . In addition, the displacement of the (220) peak with x allows us to extract a linear variation of the lattice constant (Figure 1b ), as observed in the case of a solid solution. This is an indication that the L2 1 chemical ordering progressively vanishes when increasing the Al substitution rate 𝑥. However, the chemical disorder distribution in the films cannot be easily determined by using the electron and x -ray diffraction analyses. To address this point, a STEM HAADF analysis has been carried on the Co 2MnAl ½Si½ films with a comparison with Co 2MnSi and Co 2MnAl. A clear mixing of both structures is 5 observ ed for x=½ where around 50% is L2 1 chemically ordered and 50%, B2, with typical domains size around 10nm along the growth axis (001) and a few nm in the plane of the film ( Figure 1c). The electronic properties of the Co 2MnAl xSi1-x(001) series were studied using spin - resolved photoemission (SR -PES) and ferromagnetic resonance (FMR). The SR -PES spectra were obtained by using the largest slit acceptance of the detector (+/ - 8°) at an angle of 8° of the normal axis of the surface. Such geometry allows us to analyze all the reciprocal space as confirmed by similar experiments but performed on similar polycrystalline films [23]. Getting the spin-polarization dependence with x using raw SR - PES spectra is however not obvious due to the existence of surface states systematically observed on Co 2MnSi but also on other Co 2Mn-based Heusler compounds [19, 22-23]. To get the bulk spin polarization, we thus used the S polarization of the photon beam. Indeed , we have shown that the surface states are no more detected due to their symmetry [19] without any loss of information on the bulk band structure [ 23]. The corresponding SR-PES spectra are shown in figure 2 . As expec ted, we thus obtain a tunable spin polarization at EF from 100% to 63% by substituting Si by Al, as shown in figure 3 . Figure 2 : spin -resolved photoemission spectra using P photon polarization (left), S photon polarization (middle) and resulting spin polarization curves (right) for the Co 2MnAl xSi1-x series, 6 The radiofrequency magnetic dynamics of the films were thus studied using ferromagnetic resonance (FMR) . The magnetic damping coefficient , the effective magnetic moment Ms (close to the true moment in our films due to very small anisotropy – see [ 22]), and the inhomogeneous linewidth f0 were thus extracted from the measurements performed on the Co 2MnAl xSi1-x(001) series. The results obtained on the same series used for photoemission experiments are shown in table I . As shown in figure 3, a clear correlation is observed between the spin polarization at EF and the magnetic damping coefficient , as theoretically expected. An ultra -low value was obtained for Co2MnSi (x= 0) due to the large spin gap [ 22]. By substituting Al by Si, the magnetic damping increase is explained by the decrease of the spin polarization. Co2MnAl xSi1-x Spin polarization (%) Ms (µB/f.u.) (x 10-3) f0 (MHz) g factor (0.01) x = 0 973 5.08 0.460.05 14.3 2.01 x = 0.25 903 4.85 0.730.15 21.7 1.99 x = 0.5 833 4.85 0.680.15 9 2.01 x = 0.75 703 4.8 1.000.05 81.5 2.00 x = 1 633 4.32 1.100.05 22 2.01 Table 1: data extracted from spin -resolved photoemission and ferromagnetic resonance experiments performed on the Co 2MnAl xSi1-x series. Figure 3: -top- spin polarization and magnetic damping dependence with Al content for the Co2MnAl xSi1-x series and –bottom - magnetic damping versus spin polarization . The lines are guide to the eyes. 7 In addition, t he magnetization is also observed to decrease with x in agreement with the Slater -Pauling description of the valence band electrons in Heusler compounds [25]. Indeed, as a 5 µ B magnetic moment per cell is expected for Co2MnSi (type IV valence electrons), it should decrease to 4 when replacing Si by Al (type III) as actually observed (Table I ). Finally, the FMR susceptibilities reach extremely small inhomogeneous linewidth f0, a proof of the excellent homogeneity of the magnetic properties (hence a high crystal quality) in our films. Figure 4 (a) shows the ultrafast demagnetization curves measured on the same Co2MnAl xSi1-x series with a maximum magnetization quenching ~1 5%. The temporal changes of the Kerr signals ∆𝜃𝑘(𝑡) were normalized by the saturation value 𝜃𝑘 just before the pump laser excitation. The time evolution of magnetization on sub -picosecond timescales c an be fitted according to Eq. (2 ) in terms of the three -Temperature M odel (3TM) [26], which describes the energy distribution among electrons, phonons, and spins after laser excitation. −∆𝑀(𝑡) 𝑀={[𝐴1 (𝑡𝜏0+1 ⁄ )0.5−𝐴2𝜏𝐸−𝐴1𝜏𝑀 𝜏𝐸−𝜏𝑀𝑒−𝑡 𝜏𝑀−𝜏𝐸(𝐴1−𝐴2) 𝜏𝐸−𝜏𝑀𝑒−𝑡 𝜏𝐸]Θ(𝑡)}∗𝐺(𝑡,𝜏𝐺) (2) where 𝐺(𝑡,𝜏𝐺) represents the convolution product with the Gaussian laser pulse profile, G is the full width at half maximum (FWHM) of the laser pulses. Θ(𝑡) is the Heavyside function . The constant A1 represents the amplitude of demagnetization obtained after equilibrium between the electrons, spins, and phonons is reestablished while A 2 is proportional to the initial electron temperature raise . The two critical time parameters 𝜏𝑀,𝜏𝐸 are the ultrafast demagnetization time and magnetization recovery time, respectively. In the low fluence regime, which corresponds to our measurements, 𝜏𝐸 becomes close to the electron -phonon relaxation time . A unique value of 𝜏𝐸=550 ± 20 𝑓𝑠 was used for fitt ing the demagnetization curves for all samples. T he ultrafast demagnetization time 𝜏𝑀 decrease s from 380 ±10 fs for Co 2MnSi to 165 ±10 fs for Co2MnAl (Figure 4b). The evolution of the demagnetization time with both spin polarization P and Gilbert damping 𝛼 is presented in figure 4c and 4d . A clear linear variation between 1𝜏𝑀⁄ and 1−𝑃 is observed in this series . As the magnetic damping 𝛼 is proportional to P here, this means that 1𝜏𝑀⁄ is proportional to 𝛼 too. A similar relation 8 between these two par ameters was proposed by Koopmans et al. [10]. However, they also predicted an influence of the Curie temperature . As the Curie temperature in Heusler compounds changes with the number of valence electron s and because the Co 2MnAl xSi1- x behave as solid solutions as indicated by the lattice spacing variation ( Figure 1b), we thus consider a linear decrease of 𝑇𝑐 with x going from 985 K to 697 K as exper imentally measured for x =0 and x=1, respectively. To test this possi ble influence of the Curie temperature on the ultrafast magnetization dynamics , we plot in figure 4d first the product 𝜏𝑀.𝛼 and second the product 𝜏𝑀.𝛼.𝑇𝑐(𝑥)𝑇𝑐(𝐶𝑜2𝑀𝑛𝑆𝑖 ) ⁄ . These results demonstrate that the Curie temperature does not influence the ultrafas t demagnetization in our samples . Figure 4 : (a) Ultrafast demagnetization curves obtained for different Al concentration x . The curves have been shifted vertically for sake of clarity. The solid lines represent fitted curves obtained using Eq. ( 2). (b) Ultrafast demagnetization time as a function of Al content x, (c) the inverse of 𝜏𝑀 as a function of 1-P, P being the spin polarization at E F, and d) test of Koopmans model with and without taking into account the Curie temperature of the films (see text). 9 One can now compare our experimental results with existing theoretica l models. We first discuss the dependence between the magnetic damping and the spin polarization. Ultra -low magnetic damping values are predicted in Half -Metal Magnet (HMM) Heusler compounds and explained by the lack of density of state at the Fermi energy for minority spin, or in other words by the full spin polarization [18,27,28] . Consequently, the magnetic damping is expected to increase when creating some states in the m inority band structure around the Fermi energy that is when decreasing the spin polarization [28]. If we confirmed in previous experimental works that ultra -low magnetic damping coefficients are actually observed especially on HMM Co2MnSi and Co 2MnGe [19,2 2- 23], we could not state any quantitative dependence between the damping values and the spin polarization. As prospected, the Co 2MnAl xSi1-x alloys are shown here to be ideal candidates to address this point . This allows us getting a clear experimental demonstration of these theoretical expectations. Furthermore, a linear dependence between the magnetic damping and the spin polarization is obtained. This behavior may be explained by the mixing of both L2 1 and B2 phases in the films. To the best of our knowledge, this experimental result is the first quantitative demonstration of the link between the magnetic damping and spin polarization. Second , the dependence between the magnetic damping and the demagnetization time observed here is a clear opportunity to test the different theoretical explanations proposed in the literature to explain ultrafast dynamics . In the last 15 years, t he influence of the damping on the ultrafast dynamics has been explored, both theoreticall y and experimentally. The first type of prediction we want to address is the link between the demagnetization time and the electronic structure via the spin polarization P. Using a basic approach considering the Fermi golden rule, several groups [12,13] proposed that the demagnetization process is linked to the population of minority and majority spin states at E F, leading to a dependence of the spin-scattering rate proportional to 1 -P [13]. As this spin scattering rate is linked to the inverse of the dem agnetization delay time , the 𝜏𝑀~(1−𝑃)−1 law was proposed . This law is clearly verified i n our samples series. One should note that this is a strong experimental demonstration since we compare samples grown in the same conditions , so with the same control of the stoichiometry and structural properties . 10 However, one point is still not clear since much larger demagnetization times in the picosecond timescale would be expected for large band gap and full spin -polarization. In the case of small band ga p of the order of 0.1 eV, Mann et al [13] showed that thermal effects from the heated electron system lead to a decrease of 𝜏𝑀. They calculated a reduction of the spin -flip suppression factor from 104 for a gap of 1 eV to 40 for a gap of 0.3 eV. However, the band gap of our Co 2MnSi was calculated to be around 0.8 eV with a Fermi energy in the middle of the gap [27,28] . This was corroborated by direct measurement using SR -PES [19, 22 ]. Therefore, according to their model, we should expect a much longer demagnetization time for Co 2MnSi. However, the largest values reported by several groups [13, 29] all on HMM materials are of the same order of magnitude, i.e. around 350 to 400fs . This probably means that a limitation exists due to another physical reason . One hypothesis should be to consider the 1.5eV photon energy which is much larger than the spin gap. During the excitation, the electrons occupying the top minority spin valence band can be directly excited into the conduction band. In a similar way, maj ority spin electrons are excited at energies higher than the spin band gap. Both of these effects may allow for spin flips scattering and only the majority electrons excited within the spin band gap energy range cannot flip their spins. Even if such photon energy influence is not considered based on the argument that the timescale for photon absorption followed by electronic relaxation is very fast compared to the magnetic relaxation process [16 ], performing experiments by changing the excitation wavelength to energies below the spin band gap would be very interesting to better understand ultrafast magnetization dynamics. Concerning the dependence between the demag netization time and the magnetic damping , different theoretical models have been proposed and two opposite trends were obtained; 𝛼 and 𝜏𝑀 being either directly [15] or inversely [10 ] proportional . From the experimental side, the inverse proportionality between 𝜏𝑀 and 𝛼 proposed by Koopmans et al. [10] could not be reproduced by doping a thin Permalloy film with rare -earth atoms [14]. However, the introduction of these rare -earth elements strongly modifies the magnetic relaxation properties and could induce different relaxation channels for 𝜏𝑀 and 𝛼 [30]. Zhang et al. performed a similar st udy using thin Co/Ni multilayers and observed a direct proportionality between 𝜏𝑀 and 𝛼 [15]. However, the damping extracted in their 11 study should be strongly influenced by the heavy metal Pt capping and seed layers which may induce strong spin pumping effect during the magnetization precession [30]. Furthermore, they did not take into account the influence of the Curie temperature. Therefore, in these studies, extrinsic effects might influence the magnetization dynamics in a different way on both time scales which makes more complex the comparison between theory and experiments. Therefore, o ur results offer a nice opportunity to disentangle the se different effects. According to different studies , the ultrafast demagnetization slows down when approaching the Curie temperature [ 10,16, 32,33]. In other words, a larger difference between the initial temperature and 𝑇𝑐 would lead to a faster demag netization . In our samples, 𝑇𝑐 goes up from Co2MnAl to Co2MnSi, whereas the demagnetization process becomes slower . Therefore, we conclude that, in the present case, the Curie temperatures of our samples are too high to affect 𝜏𝑀 which only depends on the intrinsic propertie s of the films, i.e. Gilbert damping and spin polarization. This also clarifies some points reported by Müller et al. work [ 12]. In their paper, they first reported a very fast demagnetization process in Co 2MnSi(110) and second a slow one in CrO 2 and LaSrMnO 3 films with 𝑇𝑐 values close to room temperature (390 K 360 K respectively). Therefore, it is not possible to state whether the very slow demag netization process in these compounds is due to a low 𝑇𝑐 or a large spin polarization. Furthermore, recent experimental results demonstrated a large decrease in the spin polarization at the Fermi level in CrO 2 as function of the temperature, resulting in less than 50% at 300 K [34]. In our samples we disentangle these two effects and the longest demagnetization time is found for Co 2MnSi (𝜏𝑀=380 𝑓𝑠), a true half -metal magnet with a 0.8 eV spin gap and a large 𝑇𝑐. In summary, we first demonstrate experimentally that substituting Si by Al in Co2MnAl xSi1-x Heusler compounds allows us to get a tunable spin polarization at E F from 60% in Co 2MnAl to 100% in Co 2MnSi, indicati ng the transition from metallic to half metallic behaviors. Second, a strong correlation between the spin polarization and the Gilbert magnetic damping is established in these films . This confirms the theoretical justification of ultra -low magnetic damping in Ha lf-Metal -Magnet s as a consequence of the spin gap. Third , the ultrafast spin dynamics results also nicely confirm that the spin gap is at the origin of the increase of the relaxation time. Our experiments allow us to go 12 further by establishing clear relati onships between the spin polarization, the magnetic damping and the demagnetization time. A n inverse relationship between demagnetization time and Gil bert damping is established in these alloys , which agrees well with the model proposed by Mann et al. [13] and with Koopmans et al. [10] but without considering any influence of Curie temperature much larger than room temperature in these films . Experimental section Co2MnAl xSi1-x(001) quaternary Heusler compounds are grown by Molecular Beam Epitaxy using an MBE machine e quipped with 24 materials. The s toichiometry is accurately controlled during the growth by calibration of the Co, Mn, Si and Al atomic fluxes using a quartz microbalance located at the pl ace of the sample. The error on each elem ent concentration is less than 1 % [23]. The films are grown directly on MgO(001) substrates, with the epitaxial relationship [100] (001) MgO // [110] (001) Heusler compound. The thickness is fixed to 20nm. The phot oemission experiments were done at the CASSIOPEE beamline at SOLEIL synchrotron source. The films were grown in a MBE connected to the beamline (see [19,22,35 ] for details). The SR -PES spectra were obtained by using the largest slit acceptance of the detec tor (+/ - 8°) at an angle of 8° of the normal axis of the surface. Such geometry allows us to analyze all the reciprocal space on similar polycrystalline films [23]. The radiofrequency magnetic dynamics of the films were thus studied using ferromagnetic resonance (FMR). A Vectorial Network Analyzor FMR set -up was used in the perpendicular geometry (see [ 22] for experimental details) where the static magnetic field is applied out of the plane of the film in order to avoid extrinsic bro adening of the linewidth due to the 2 -magnons scattering [ 36,37]. Ultrafast magnetization dynamics were investigat ed using polar time -resolved magneto -optical Kerr (TR -MOKE) experiments. An amplified Ti -sapphire laser producing 35 fs pulses at 800 nm with a repetition rate of 5 KHz is used . The pump beam is kept at the fundamental mode and is focused down to spot size of ~260 𝜇𝑚 while the probe is frequency doubled to 400 nm and focused to a spot size of ~60 𝜇𝑚. 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B 76, 104416 (2007)

2020-02-07

Engineering of magnetic materials for developing better spintronic applications relies on the control of two key parameters: the spin polarization and the Gilbert damping responsible for the spin angular momentum dissipation. Both of them are expected to affect the ultrafast magnetization dynamics occurring on the femtosecond time scale. Here, we use engineered Co2MnAlxSi1-x Heusler compounds to adjust the degree of spin polarization P from 60 to 100% and investigate how it correlates with the damping. We demonstrate experimentally that the damping decreases when increasing the spin polarization from 1.1 10-3 for Co2MnAl with 63% spin polarization to an ultra-low value of 4.10-4 for the half-metal magnet Co2MnSi. This allows us investigating the relation between these two parameters and the ultrafast demagnetization time characterizing the loss of magnetization occurring after femtosecond laser pulse excitation. The demagnetization time is observed to be inversely proportional to 1-P and as a consequence to the magnetic damping, which can be attributed to the similarity of the spin angular momentum dissipation processes responsible for these two effects. Altogether, our high quality Heusler compounds allow controlling the band structure and therefore the channel for spin angular momentum dissipation.

Engineering Co$_2$MnAl$_x$Si$_{1-x}$ Heusler compounds as a model system to correlate spin polarization, intrinsic Gilbert damping and ultrafast demagnetization

2002.02686v1

arXiv:2009.07777v1 [math.AP] 16 Sep 2020Exponential decay for semilinear wave equations with visco elastic damping and delay feedback Alessandro Paolucci∗Cristina Pignotti† Abstract In this paperwestudy aclassofsemilinearwavetype equationswith v iscoelasticdamping anddelayfeedbackwithtimevariablecoefficient.Bycombiningsemigro uparguments,careful energy estimates and an iterative approach we are able to prove, u nder suitable assumptions, a well-posedness result and an exponential decay estimate for solu tions corresponding to smallinitialdata.Thisextendsandconcludestheanalysisinitiatedin[16]an dthendeveloped in [13, 17]. 1 Introduction LetHbe a Hilbert space and let Abe a positive self-adjoint operator with dense domain D(A) inHand compact inverse in H. Let us consider the system: utt(t)+Au(t)−/integraldisplay+∞ 0µ(s)Au(t−s)ds+k(t)BB∗ut(t−τ) =∇ψ(u(t)), t∈(0,+∞) u(t) =u0(t), t∈(−∞,0], ut(0) =u1 B∗ut(t) =g(t), t∈(−τ,0),(1.1) whereτ >0 represents the time delay, Bis a bounded linear operator of Hinto itself, B∗ denotes its adjoint, and ( u0(·),u1,g(·)) are the initial data taken in suitable spaces. Moreover, the delay damping coefficient k: [0,+∞)→IR is a function in L1 loc([0,+∞)) such that /integraldisplayt t−τ|k(s)|ds<C∗,∀t∈(0,+∞), (1.2) for a suitable constant C∗,and the memory kernel µ: [0,+∞)→[0,+∞) satisfies the following assumptions: (i)µ∈C1(IR+)∩L1(IR+); (ii)µ(0) =µ0>0; ∗Dipartimento di Ingegneria e Scienze dell’Informazione e M atematica, Universit` a di L’Aquila, Via Vetoio, Loc. Coppito, 67010 L’Aquila Italy ( alessandro.paolucci2@graduate.univaq.it ). †Dipartimento di Ingegneria e Scienze dell’Informazione e M atematica, Universit` a di L’Aquila, Via Vetoio, Loc. Coppito, 67010 L’Aquila Italy ( pignotti@univaq.it ). 1(iii)/integraltext+∞ 0µ(t)dt= ˜µ<1; (iv)µ′(t)/lessorequalslant−δµ(t), for some δ>0. Furthermore, ψ:D(A1 2)→IR is a functional having Gˆ ateaux derivative Dψ(u) at every u∈D(A1 2).Moreover, in the spirit of [2], we assume the following hypot heses: (H1) For every u∈D(A1 2), there exists a constant c(u)>0 such that |Dψ(u)(v)|/lessorequalslantc(u)||v||H∀v∈D(A1 2). Then,ψcan be extended to the whole Hand we denote by ∇ψ(u) the unique vector representing Dψ(u) in the Riesz isomorphism, i.e. /an}b∇acketle{t∇ψ(u),v/an}b∇acket∇i}htH=Dψ(u)(v),∀v∈H; (H2) for all r>0 there exists a constant L(r)>0 such that ||∇ψ(u)−∇ψ(v)||H/lessorequalslantL(r)||A1 2(u−v)||H, for allu,v∈D(A1 2) satisfying ||A1 2u||H/lessorequalslantrand||A1 2v||H/lessorequalslantr. (H3)ψ(0) = 0,∇ψ(0) = 0 and there exists a strictly increasing continuous fun ctionhsuch that ||∇ψ(u)||H/lessorequalslanth(||A1 2u||H)||A1 2u||H, (1.3) for allu∈D(A1 2). We are interested in studying well–posedness and stability results, for small initial data, for the above model. Our results extend the ones of [16, 17] where abs tract evolution equations are analyzed and, in the specific case of memory damping, exponen tial decay is obtained essentially only in the linear case. Indeed, in the nonlinear setting, an extra standard frictional damping, not delayed, was needed in order to obtain existence and uniq ueness of global solutions with exponentially decaying energy for suitably small initial d ata. Moreover, in [16, 17] the delay damping coefficient k(t) is assumed to be constant and the results there obtained req uire a smallness assumption on /ba∇dblk/ba∇dbl∞.The analysis of [16, 17] has been extended in [13] by consider ing a time variable delay damping coefficient k(t) as in the present paper. However, also in [13] an extra frictional not delayed damping was needed, in the case of wave type equation with memory damping, when a locally Lipschitz continuous nonlinear ter m is included into the equation. Then, here, we focus on wave type equations with viscoelasti c damping, delay feedback and source term, obtaining well-posedness and stability resul ts for small initial data without adding extra frictional not delayed damping. So, we here improve an d conclude the analysis developed in [16, 17, 13] for the class of models at hand. Other models wi th viscoelastic damping and time delay are studied in recent literature. The first result is due to [12], in the linear setting. In that paper a standard frictional damping, not delayed, is included into the model in order to compensate the destabilizing effect of the delay feedback. Actually, at least in the linear case, the viscoelastic damping alone can counter the destab ilizing delay effect, under suitable assumptions, without needing other dampings. This has been shown, e.g., in [11, 3, 9, 23]. The case of viscoelastic wave equation with intermittent delay feedback has been studied in [20] 2while [14] deals with a model for plate equation with memory, source term, delay feedback and standard not delayed frictional damping. More extended is the literature in case of a frictional/stru ctural damping, instead of a viscoelastic term, which compensates the destabilizing effe ct of a time delay and, for specific models, mainly in the linear setting, several stability res ults have been quite recently obtained under appropriate assumptions (see e.g. [1, 4, 5, 7, 15, 18, 2 1, 22]). The paper is organized as follows. In Section 2 we give some pr eliminaries, writing system (1.1) in an abstract way. In Section 3 we prove the exponentia l decay of the energy associated to (1.1). Finally, in Section 4 some examples are illustrate d. 2 Preliminaries As in Dafermos [8], we define the function ηt(s) :=u(t)−u(t−s), s,t∈(0,+∞), (2.4) so that we can rewrite (1.1) in the following way: utt(t)+(1−˜µ)Au(t)+/integraldisplay+∞ 0µ(s)Aηt(s)ds+k(t)BB∗ut(t−τ) =∇ψ(u(t)), t∈(0,+∞), ηt t(s) =−ηt s(s)+ut(t), t,s∈(0,+∞), u(0) =u0(0), ut(0) =u1, B∗ut(t) =g(t), t∈(−τ,0), η0(s) =η0(s) =u0(0)−u0(−s)s∈(0,+∞).(2.5) LetL2 µ((0,+∞);D(A1 2)) be the Hilbert space of the D(A1 2)−valued functions in (0 ,+∞) endowed with the scalar product /an}b∇acketle{tϕ,ψ/an}b∇acket∇i}htL2µ((0,+∞);D(A1 2))=/integraldisplay∞ 0µ(s)/an}b∇acketle{tA1 2ϕ,A1 2ψ/an}b∇acket∇i}htHds and denote by Hthe Hilbert space H=D(A1 2)×H×L2 µ((0,+∞);D(A1 2)), equipped with the inner product /angbracketleftBigg u v w , ˜u ˜v ˜w /angbracketrightBigg H:= (1−˜µ)/an}b∇acketle{tA1 2u,A1 2˜u/an}b∇acket∇i}htH+/an}b∇acketle{tv,˜v/an}b∇acket∇i}htH+/integraldisplay∞ 0µ(s)/an}b∇acketle{tA1 2w,A1 2˜w/an}b∇acket∇i}htHds.(2.6) SettingU= (u,ut,ηt) we can restate (1.1) in the abstract form U′(t) =AU(t)−k(t)BU(t−τ)+F(U(t)), BU(t−τ) = ˜g(t) fort∈[0,τ], U(0) =U0,(2.7) 3where the operator Ais defined by A u v w = v −(1−˜µ)Au−/integraltext+∞ 0µ(s)Aw(s)ds −ws+v with domain D(A) ={(u,v,w)∈D(A1 2)×D(A1 2)×L2 µ((0,+∞);D(A1 2)) : (1−˜µ)u+/integraldisplay+∞ 0µ(s)w(s)ds∈D(A), ws∈L2 µ((0,+∞);D(A1 2))},(2.8) in the Hilbert space H,and the operator B:H → His defined by B u v w := 0 BB∗v 0 . Moreover, ˜g(t) = (0,Bg(t−τ),0), U0= (u0(0),u1,η0) andF(U) := (0,∇ψ(u),0)T.From (H2) and (H3) we deduce that the function Fsatisfies: (F1)F(0) = 0; (F2) for each r>0 there exists a constant L(r)>0 such that ||F(U)−F(V)||H/lessorequalslantL(r)||U−V||H (2.9) whenever ||U||H/lessorequalslantrand||V||H/lessorequalslantr. 3 Stability result In this section we want to prove an exponential stability res ult for the system (1.1) for small initial data. It’s well–known (see e.g. [10]) that the opera torAin the problem’s formulation (2.7) generates an exponentially stable semigroup {S(t)}t/greaterorequalslant0,namely there exist two costants M,ω>0 such that ||S(t)||L(H)/lessorequalslantMe−ωt. (3.10) Denoting /ba∇dblB/ba∇dblL(H)=/ba∇dblB∗/ba∇dblL(H)=b, (3.11) then/ba∇dblB/ba∇dblL(H)=b2.Our result will be obtained under an assumption on the coeffici entk(t) of the delay feedback which includes as particular cases kintegrable and kinL∞with/ba∇dblk/ba∇dbl∞ sufficiently small. More precisely, we assume (cf. [13]) that there exist two constants ω′∈[0,ω) andγ∈IR such that b2Meωτ/integraldisplayt 0|k(s+τ)|ds/lessorequalslantγ+ω′t,for allt/greaterorequalslant0. (3.12) Theorem 3.1. Assume (3.12). Moreover, suppose that 4(W) there exist ρ>0,Cρ>0, withL(Cρ)<ω−ω′ Msuch that if U0∈ Hand if˜g∈C([0,τ];H) satisfy ||U0||2 H+/integraldisplayτ 0|k(s)|·||˜g(s)||2 Hds<ρ2, (3.13) then the system (2.7)has a unique solution U∈C([0,+∞);H)satisfying ||U(t)||H/lessorequalslantCρ for allt>0. Then, for every solution Uof(2.7), with initial datum U0satisfying (3.13), ||U(t)||H/lessorequalslant˜M/parenleftbigg ||U0||H+/integraldisplayτ 0eωs|k(s)|·||˜g(s)||Hds/parenrightbigg e−(ω−ω′−ML(Cρ))t, t/greaterorequalslant0,(3.14) with˜M=Meγ. Proof.By Duhamel’s formula, using (3.10), we have ||U(t)||H/lessorequalslantMe−ωt||U0||H+Me−ωt/integraldisplayt 0eωs|k(s)|·||BU(s−τ)||Hds+ML(Cρ)e−ωt/integraldisplayt 0eωs||U(s)||Hds /lessorequalslantMe−ωt||U0||H+Me−ωt/integraldisplayτ 0eωs|k(s)|·||BU(s−τ)||Hds +Me−ωt/integraldisplayt τeωs|k(s)|·||BU(s−τ)||Hds+ML(Cρ)e−ωt/integraldisplayt 0eωs||U(s)||Hds. Hence, we obtain eωt||U(t)||H/lessorequalslantM||U0||H+M/integraldisplayτ 0eωs|k(s)|·||BU(s−τ)||Hds +/integraldisplayt 0/parenleftBig Meωτ|k(s+τ)|·||B|| L(H)+ML(Cρ)/parenrightBig eωs||U(s)||Hds, and then eωt||U(t)||H/lessorequalslantM||U0||H+M/integraldisplayτ 0eωs|k(s)|·||˜g(s)||Hds +/integraldisplayt 0/parenleftBig Mb2eωτ|k(s+τ)|+ML(Cρ)/parenrightBig eωs||U(s)||Hds. Therefore, using Gronwall’s inequality, eωt/ba∇dblU(t)/ba∇dblH/lessorequalslantM/parenleftbigg /ba∇dblU0/ba∇dblH+/integraldisplayτ 0eωs|k(s)|·/ba∇dbl˜g(s)/ba∇dblHds/parenrightbigg eMb2eωτ/integraltextt 0|k(s+τ)|ds+ML(Cρ)t and so, from (3.12), ||U(t)||H/lessorequalslantMeγ/parenleftbigg ||U0||H+/integraldisplayτ 0eωs|k(s)|·||˜g(s)||Hds/parenrightbigg e−(ω−ω′−ML(Cρ))t. This gives (3.14) with ˜Mas in the statement. 5In order to prove the stability result we need then to show tha t the well-posedness assump- tion (W) of Theorem 3.1 is satisfied for problem (1.1). For thi s, let us define the energy of the model (1.1) as E(t) :=E(u(t)) =1 2||ut(t)||2 H+1−˜µ 2||A1 2u(t)||2 H−ψ(u) +1 2/integraldisplayt t−τ|k(s+τ)|·||B∗ut(s)||2 Hds+1 2/integraldisplay+∞ 0µ(s)||A1 2ηt(s)||2 Hds.(3.15) The following lemma holds. Lemma 3.2. Letu: [0,T)→IRbe a solution of (1.1). Assume that E(t)/greaterorequalslant1 4||ut(t)||2 Hfor all t∈[0,T). Then, E(t)/lessorequalslant¯C(t)E(0), (3.16) for allt∈[0,T), where ¯C(t) =e2b2/integraltextt 0(|k(s)|+|k(s+τ)|)ds. (3.17) Proof.Differentiating E(t), we obtain dE dt=/an}b∇acketle{tut,utt/an}b∇acket∇i}htH+(1−˜µ)/an}b∇acketle{tA1 2u,A1 2ut/an}b∇acket∇i}htH−/an}b∇acketle{t∇ψ(u),ut/an}b∇acket∇i}htH+1 2|k(t+τ)|·||B∗ut(t)||2 H −1 2|k(t)|·||B∗ut(t−τ)||2 H+/integraldisplay+∞ 0µ(s)/an}b∇acketle{tA1 2ηt(s),A1 2ηt t(s)/an}b∇acket∇i}htHds. Then, from (1.1), dE dt=−/integraldisplay+∞ 0µ(s)/an}b∇acketle{tut(t),Aηt(s)/an}b∇acket∇i}htHds−k(t)/an}b∇acketle{tut,BB∗ut(t−τ)/an}b∇acket∇i}htH+1 2|k(t+τ)|·||B∗ut(t)||2 −1 2|k(t)|·||B∗ut(t−τ)||2+/integraldisplay+∞ 0µ(s)/an}b∇acketle{tAηt(s),ηt t(s)/an}b∇acket∇i}htds. Using the second equation of (2.5), we have that dE dt=−k(t)/an}b∇acketle{tut,BB∗ut(t−τ)/an}b∇acket∇i}htH+1 2|k(t+τ)|·||B∗ut||2 H−1 2|k(t)|·||B∗ut(t−τ)||2 H −/integraldisplay+∞ 0µ(s)/an}b∇acketle{tAηt(s),ηt s(s)/an}b∇acket∇i}htHds. Now, we claim that/integraldisplay+∞ 0µ(s)/an}b∇acketle{tηt s,Aηt(s)/an}b∇acket∇i}htHds/greaterorequalslant0. Indeed, integrating by parts and recalling assumption (iv) onµ(·),we deduce /integraldisplay+∞ 0µ(s)/an}b∇acketle{tηt s,Aηt(s)/an}b∇acket∇i}htHds=−1 2/integraldisplay+∞ 0µ′(s)||A1 2ηt(s)||2 Hds/greaterorequalslant0. Therefore, we have that dE(t) dt/lessorequalslant−k(t)/an}b∇acketle{tB∗ut,B∗ut(t−τ)/an}b∇acket∇i}htH+1 2|k(t+τ)|·||B∗ut(t)||2 H−1 2|k(t)|·||B∗ut(t−τ)||2 H. 6Now, using Cauchy-Schwarz inequality, we obtain the follow ing estimate: dE(t) dt/lessorequalslant1 2(|k(t)|+|k(t+τ)|)||B∗ut(t)||2 H /lessorequalslant1 2(|k(t)|+|k(t+τ)|)b2||ut||2 H = 2b2(|k(t)|+|k(t+τ)|)1 4||ut||2 H /lessorequalslant2b2(|k(t)|+|k(t+τ)|)E(t). Hence, the Gronwall Lemma concludes the proof. Before proving the well-posedness assumption (W) for solut ions to (2.7), we need the fol- lowing two lemmas. Lemma 3.3. Let us consider the system (2.7)with initial data U0∈ Hand˜g∈C([0,τ];H). Then, there exists a unique local solution U(·)defined on a time interval [0,δ), withδ/lessorequalslantτ. Proof.Sincet∈[0,τ], we can rewrite the abstract system (2.7) as an undelayed pr oblem: U′(t) =AU(t)−k(t)˜g(t)+F(U(t)), t∈(0,τ), U(0) =U0. Then, we can apply the classical theory of nonlinear semigro ups (see e.g. [19]) obtaining the existence of a unique solution on a set [0 ,δ), withδ/lessorequalslantτ. Lemma 3.4. LetU(t) = (u(t),ut(t),ηt)be a solution to (2.7)defined on the interval [0,δ),with δ/lessorequalslantτ.Then, 1. ifh(||A1 2u0(0)||H)<1−˜µ 2,thenE(0)>0; 2. ifh(||A1 2u0(0)||H)<1−˜µ 2andh/parenleftbigg 2 (1−˜µ)1 2¯C1 2(τ)E1 2(0)/parenrightbigg <1−˜µ 2,with¯C(τ)defined in (3.17), then E(t)>1 4||ut||2 H+1−˜µ 4||A1 2u||2 H +1 4/integraldisplayt t−τ|k(s+τ)|·||B∗ut(s)||2 Hds+1 4/integraldisplay+∞ 0µ(s)||A1 2ηt(s)||2 Hds,(3.18) for allt∈[0,δ). In particular, E(t)>1 4/ba∇dblU(t)/ba∇dbl2 H,for allt∈[0,δ). (3.19) Proof.We first deduce by assumption (H3) on ψthat |ψ(u)|/lessorequalslant/integraldisplay1 0|/an}b∇acketle{t∇ψ(su),u/an}b∇acket∇i}ht|ds /lessorequalslant||A1 2u||2 H/integraldisplay1 0h(s||A1 2u||H)sds/lessorequalslant1 2h(||A1 2u||H)||A1 2u||2 H.(3.20) 7Hence, under the assumption h(/ba∇dblA1 2u0(0)/ba∇dblH)<1−˜µ 2,we have that E(0) =1 2||u1||2 H+1−˜µ 2||A1 2u0(0)||2 H−ψ(u0(0))+1 2/integraldisplay0 −τ|k(s+τ)|·||B∗ut(s)||2 Hds +1 2/integraldisplay+∞ 0µ(s)||A1 2η0(s)||2 Hds /greaterorequalslant1 2||u1||2 H+1−˜µ 4||A1 2u0(0)||2 H+1 2/integraldisplay0 −τ|k(s+τ)|·||B∗ut(s)||2 Hds +1 2/integraldisplay+∞ 0µ(s)||A1 2η0(s)||2 Hds>0, obtaining 1 . In order to prove the second statement, we argue by contradic tion. Let us denote r:= sup{s∈[0,δ) : (3.18) holds ∀t∈[0,s)}. We suppose by contradiction that r<δ. Then, by continuity, we have E(r) =1 4||ut(r)||2 H+1−˜µ 4||A1 2u(r)||2 H+1 4/integraldisplayr r−τ|k(s+τ)|·||B∗ut(s)||2 Hds +1 4/integraldisplay+∞ 0µ(s)||A1 2ηr(s)||2 Hds.(3.21) Now, from (3.21) and Lemma 3.2 we can infer that h(||A1 2u(r)||H)/lessorequalslanth/parenleftBigg 2 (1−˜µ)1 2E1 2(r)/parenrightBigg <h/parenleftBigg 2 (1−˜µ)1 2¯C1 2(τ)E1 2(0)/parenrightBigg <1−˜µ 2. Hence, we have that E(r) =1 2||ut(r)||2 H+1−˜µ 2||A1 2u(r)||2 H−ψ(u(r))+1 2/integraldisplayr r−τ|k(s+τ)|·||B∗ut(s)||2 Hds +1 2/integraldisplay+∞ 0µ(s)||A1 2ηr(s)||2 Hds >1 4||ut(r)||2 H+1−˜µ 4||A1 2u(r)||2 H+1 4/integraldisplayr r−τ|k(s+τ)|·||B∗ut(s)||2 Hds +1 4/integraldisplay+∞ 0µ(s)||A1 2ηr(s)||2 Hds, which contradicts the maximality of r. Hence,r=δand this concludes the proof. Theorem 3.5. Problem (2.7), with initial data U0∈ Hand˜g∈C([0,τ];H),satisfies the well- posedness assumption (W). Then, for solutions of (2.7)corresponding to sufficiently small initial data the exponential decay estimate (3.14)holds. Proof.Let us fixN∈IN such that 2M2/parenleftBig 1+e2ωτC∗/parenrightBig e2γe−(ω−ω′)(N−1)τ<1 1+eωτb2C∗, (3.22) 8whereC∗is the constant defined in (1.2). Then, let ρbe a positive constant such that ρ/lessorequalslant(1−˜µ)1 2 2¯C1 2(Nτ)h−1/parenleftbigg1−˜µ 2/parenrightbigg . (3.23) Now, let us assume that the initial data ( u0(0),u1,η0) andB∗ut(s), s∈[−τ,0],satisfy the smallness assumption (1−˜µ)||A1 2u0(0)||2 H+||u1||2 H+/integraldisplay0 −τ|k(s+τ)|||B∗ut(s)||2 Hds +/integraldisplay+∞ 0µ(s)||A1 2η0(s)||2 Hds<ρ2.(3.24) Note that (3.24) is equivalent to /ba∇dblU0/ba∇dbl2 H+/integraldisplayτ 0|k(s)|·/ba∇dbl˜g(s)/ba∇dbl2 Hds<ρ2. (3.25) From Lemma 3.3 we know that there exists a local solution defin ed on a time interval [0 ,δ), withδ/lessorequalslantτ. From (3.24) and (3.23) we have that h(||A1 2u0(0)||H)<h/parenleftBigg ρ (1−˜µ)1 2/parenrightBigg /lessorequalslanth/parenleftBigg 1 2¯C1 2(Nτ)h−1/parenleftbigg1−˜µ 2/parenrightbigg/parenrightBigg <1−˜µ 2,(3.26) where we have used the fact that ¯C(Nτ)>1.This implies, from Lemma 3.4, E(0)>0. Fur- thermore, from (3.20) and (3.26) we get E(0)/lessorequalslant1 2||u1||2 H+3 4(1−˜µ)||A1 2u0(0)||2 H+1 2/integraldisplay0 −τ|k(s+τ)|·||B∗ut(s)||2 Hds +1 2/integraldisplay+∞ 0µ(s)||A1 2η0(s)||2 Hds<ρ2, which gives, recalling (3.23), h/parenleftBigg 2 (1−˜µ)1 2¯C1 2(Nτ)E1 2(0)/parenrightBigg <h/parenleftBigg 2 (1−˜µ)1 2¯C1 2(Nτ)ρ/parenrightBigg /lessorequalslanth/parenleftbigg h−1/parenleftbigg1−˜µ 2/parenrightbigg/parenrightbigg =1−˜µ 2.(3.27) Since¯C(Nτ)/greaterorequalslant¯C(τ), then h/parenleftBigg 2 (1−˜µ)1 2¯C1 2(τ)E1 2(0)/parenrightBigg /lessorequalslanth/parenleftBigg 2 (1−˜µ)1 2¯C1 2(Nτ)E1 2(0)/parenrightBigg <1−˜µ 2.(3.28) So, we can apply Lemma 3.4 and we obtain E(t)>1 4||ut(t)||2 H+1−˜µ 4||A1 2u(t)||2 H +1 4/integraldisplayt t−τ|k(s+τ)|·||B∗ut(s)||2 Hds+1 4/integraldisplay+∞ 0µ(s)||A1 2ηt(s)||2 Hds, for allt∈[0,δ).In particular we have that E(t)>1 4||ut(t)||2 H,fort∈[0,δ). 9Therefore, we can apply Lemma 3.2, obtaining E(t)/lessorequalslant¯C(τ)E(0)<¯C(τ)ρ2, for anyt∈[0,δ]. Since 0<1 4||ut(t)||2 H+1−˜µ 4||A1 2u(t)||2 H +1 4/integraldisplayt t−τ|k(s+τ)|·||B∗ut(s)||2 Hds+1 4/integraldisplay+∞ 0µ(s)||A1 2ηt(s)||2 Hds/lessorequalslantE(t)/lessorequalslant¯C(τ)E(0),(3.29) for allt∈[0,δ], then we can extend the solution to the entire interval [0 ,τ]. Now, observe that from (3.29) and (3.28) we have h(||A1 2u(τ)||H)/lessorequalslanth/parenleftBigg 2 (1−˜µ)1 2¯C1 2(τ)E1 2(0)/parenrightBigg <1−˜µ 2. (3.30) By continuity, (3.30) implies that there exists δ′>0 such that h(||A1 2u(t)||H)<1−˜µ 2,∀t∈[τ,τ+δ′). From this, arguing as before, we deduce E(t)>1 4||ut(t)||2 H+1−˜µ 4||A1 2u(t)||2 H+1 4/integraldisplayt t−τ|k(s+τ)|·||B∗ut(s)||2 Hds +1 4/integraldisplay+∞ 0µ(s)||A1 2ηt(s)||2 Hds, for anyt∈[τ,τ+δ′). In particular, also in such an interval we have E(t)>1 4||ut(t)||2 H. Hence, we can apply again Lemma 3.2 on the time interval [0 ,τ+δ′) obtaining 0<E(t)/lessorequalslant¯C(2τ)E(0). As before, we can then extend the solution the whole interval [0,2τ]. At timet= 2τwe have that h(/ba∇dblA1 2u(2τ)/ba∇dblH)/lessorequalslanth/parenleftBigg 2 (1−˜µ)1 2E1 2(2τ)/parenrightBigg /lessorequalslanth/parenleftBigg 2 (1−˜µ)1 2¯C1 2(2τ)E1 2(0)/parenrightBigg <1−˜µ 2, where we have used (3.28). Moreover, if 3 /lessorequalslantN, h/parenleftBigg 2 (1−˜µ)1 2¯C1 2(3τ)E1 2(0)/parenrightBigg /lessorequalslanth/parenleftBigg 2 (1−˜µ)1 2¯C1 2(Nτ)E1 2(0)/parenrightBigg <1−˜µ 2. Thus, one can repeat again the same argument. By iteration, w e then find a unique solution to the problem (2.7) on the interval [0 ,Nτ], whereNis the natural number fixed at the beginning of the proof. Moreover, the definition (3.23) of ρ,ensures that h(/ba∇dblA1 2u(Nτ)/ba∇dblH)/lessorequalslanth/parenleftBigg 2 (1−˜µ)1 2E1 2(Nτ)/parenrightBigg /lessorequalslanth/parenleftBigg 2 (1−˜µ)1 2¯C1 2(Nτ)E1 2(0)/parenrightBigg <1−˜µ 2, 10where we have used (3.27). Note that, by construction, (3.18 ) and (3.19) are satisfied in the whole [0,Nτ).Then, from (3.19), ||U(t)||2 H/lessorequalslant4E(t)/lessorequalslant4¯C(Nτ)E(0)<4¯C(Nτ)ρ2 and so ||U(t)||H/lessorequalslant2¯C1 2(Nτ)ρ,∀t∈[0,Nτ]. Thus, we have proved that, under the assumption (3.25) on the initial data, there exists a unique solutionU(·) to problem (2.7) defined on the time interval [0 ,Nτ].Moreover, ||U(t)||H/lessorequalslantCρ:= 2¯C1 2(Nτ)ρ. So far we have fixed ρsatisfying (3.23); now, eventually choosing a smaller ρ,we assume that ρ satisfies the additional assumption L(Cρ) =L(¯C1 2(Nτ)ρ)<ω−ω′ 2M. Then, the well–posedness assumption (W) of Theorem 3.1 is sa tisfied on [0 ,Nτ].Therefore, we obtain that Usatisfies the exponential decay estimate (3.14) and then ||U(t)||H/lessorequalslantM/parenleftbigg ||U(0)||H+/integraldisplayτ 0|k(s)|eωs||˜g(s)||Hds/parenrightbigg eγe−ω−ω′ 2t,∀t∈[0,Nτ].(3.31) In particular, ||U(Nτ)||H/lessorequalslantM/parenleftbigg ||U(0)||H+/integraldisplayτ 0eωs|k(s)|·||˜g(s)||Hds/parenrightbigg eγe−ω−ω′ 2Nτ.(3.32) Now, observe that /integraldisplayτ 0eωs|k(s)|·||˜g(s)||2 Hds/lessorequalslanteωτ/parenleftBigg/integraldisplayτ 0|k(s)|ds/parenrightBigg1/2/parenleftBigg/integraldisplayτ 0|k(s)|·||˜g(s)||2 Hds/parenrightBigg1/2 . Hence, ||U(t)||H/lessorequalslantMρ/parenleftBig 1+eωτC∗1/2/parenrightBig eγe−ω−ω′ 2t,∀t∈[0,Nτ], and then ||U(t)||2 H/lessorequalslant2M2ρ2/parenleftbig 1+e2ωτC∗/parenrightbig e2γe−(ω−ω′)t,∀t∈[0,Nτ], (3.33) whereC∗is the constant defined in (1.2). From (3.33) we deduce /ba∇dblU(Nτ)/ba∇dbl2 H+/integraldisplay(N+1)τ Nτeω(s−Nτ)|k(s)|·/ba∇dblB∗ut(s−τ)/ba∇dbl2 Hds /lessorequalslant2M2ρ2/parenleftbig 1+e2ωτC∗/parenrightbig e2γe−(ω−ω′)Nτ+eωτb2/integraldisplay(N+1)τ Nτ|k(s)|·/ba∇dblU(s−τ)/ba∇dbl2 Hds.(3.34) Now, observe that, for s∈[Nτ,(N+1)τ],it resultss−τ∈[(N−1)τ,Nτ],then from (3.33) we deduce /ba∇dblU(s−τ)/ba∇dbl2 H/lessorequalslant2M2ρ2/parenleftbig 1+e2ωτC∗/parenrightbig e2γe−(ω−ω′)(N−1)τ,∀s∈[Nτ,(N+1)τ]. 11This last estimate, used in (3.34), gives /ba∇dblU(Nτ)/ba∇dbl2 H+/integraldisplay(N+1)τ Nτeω(s−Nτ)|k(s)|·/ba∇dblB∗ut(s−τ)/ba∇dbl2 Hds /lessorequalslant2M2ρ2/parenleftbig 1+e2ωτC∗/parenrightbig e2γe−(ω−ω′)(N−1)τ/parenleftbig 1+eωτb2C∗/parenrightbig .(3.35) From (3.35) and (3.22), we then deduce ||U(Nτ)||2 H+/integraldisplay(N+1)τ Nτ|k(s)|·||B∗ut(s−τ)||2 Hds<ρ2. Thus (cf. with (3.25)), one can argue as before on the interva l [Nτ,2Nτ] obtaining a solution on [0,2Nτ].Iterating this procedure we get a global solution satisfyin g /ba∇dblU(t)/ba∇dblH<Cρ. Therefore, we have showed that the problem (2.7) satisfies th e well-posedness assumption (W) of Theorem 3.1. We have then proved that, for suitably small data, solutions to problem (2.7) are globally defined and their energies satisfy an exponential decay esti mate. Therefore, one can state the following theorem. Theorem 3.6. Let us consider (2.5). Then, there exists δ>0such that if (1−˜µ)||A1 2u0||2 H+||u1||2 H+/integraldisplay+∞ 0µ(s)||A1 2η0||2 Hds+/integraldisplayτ 0|k(s)|·||Bg(s−τ)||2 Hds<δ, (3.36) then the solution uis globally defined and it satisfies E(t)/lessorequalslantCe−βt, (3.37) whereCis a constant depending only on the initial data and β >0. Proof.By adirect application ofTheorem3.1 andTheorem3.5wehave that thereexist K,γ >0 such that ||U(t)||2 H/lessorequalslantKe−γt, (3.38) for allt/greaterorequalslant0, if the initial data are suitably small. Now, observe that t here exists a constant C >0 such that /integraldisplayt t−τ|k(s)|·||B∗ut(s)||2 Hds/lessorequalslantCC∗e−γ(t−τ). (3.39) Then, from (3.38) and (3.39) we obtain (3.37). 124 Examples 4.1 The wave equation with memory and source term Let Ω be a non-empty bounded set in IRn, with boundary Γ of class C2. Moreover, let O ⊂Ω be a nonempty open subset of Ω. We consider the following wave equation: utt(x,t)−∆u(x,t)+/integraldisplay+∞ 0µ(s)∆u(x,t−s)ds+k(t)χOut(x,t−τ) =|u(x,t)|σu(x,t),in Ω×(0,+∞), u(x,t) = 0,in Γ×(0,+∞), u(x,t) =u0(x,t) in Ω ×(−∞,0], ut(x,0) =u1(x),in Ω, ut(x,t) =g(x,t),in Ω×(−τ,0],(4.40) whereτ >0 is the time-delay, µ: (0,+∞)→(0,+∞) is a locally absolutely continuous memory kernel, which satisfies the assumptions (i)-(iv), σ >0 and the damping coefficient k(·) is a function in L1 loc([0,+∞)) satisfying (1.2). Then, system (4.40) falls in the form (1 .1) withA= −∆ andD(A) =H2(Ω)∩H1 0(Ω). Definingηt sas in (2.4), we can rewrite the system (4.40) in the following way: utt(x,t)−(1−˜µ)∆u(x,t)−/integraldisplay+∞ 0µ(s)∆ηt(x,s)ds+k(t)χOut(x,t−τ) =|u(x,t)|σu(x,t),in Ω×(0,+∞), ηt t(x,s) =−ηt s(x,s)+ut(x,t),in Ω×(0,+∞)×(0,+∞), u(x,t) = 0,in Γ×(0,+∞), ηt(x,s) = 0,in Γ×(0,+∞),fort/greaterorequalslant0, u(x,0) =u0(x) :=u0(x,0),in Ω, ut(x,0) =u1(x) :=∂u0 ∂t(x,t)/vextendsingle/vextendsingle/vextendsingle t=0,in Ω, η0(x,s) =η0(x,s) :=u0(x,0)−u0(x,−s),in Ω×(0,+∞), ut(x,t) =g(x,t),in Ω×(−τ,0).(4.41) As before, we introduce the Hilbert space L2 µ((0,+∞);H1 0(Ω)) endowed with the inner product /an}b∇acketle{tφ,ψ/an}b∇acket∇i}htL2µ((0,+∞);H1 0(Ω)):=/integraldisplay Ω/parenleftbigg/integraldisplay+∞ 0µ(s)∇φ(x,s)∇ψ(x,s)dx/parenrightbigg ds, and consider the Hilbert space H=H1 0(Ω)×L2(Ω)×L2 µ((0,+∞);H1 0(Ω)), equipped with the inner product /angbracketleftBigg u v w , ˜u ˜v ˜w /angbracketrightBigg H:= (1−˜µ)/integraldisplay Ω∇u∇˜udx+/integraldisplay Ωv˜vdx+/integraldisplay Ω/integraldisplay+∞ 0µ(s)∇w∇˜wdsdx. SettingU= (u,ut,ηt), we can rewrite (4.43) in the form (2.7), where A u v w = v (1−˜µ)∆u+/integraltext+∞ 0µ(s)∆w(s)ds −ws+v , 13with domain D(A) ={(u,v,w)∈H1 0(Ω)×H1 0(Ω)×L2 µ((0,+∞);H1 0(Ω)) : (1−˜µ)u+/integraldisplay+∞ 0µ(s)w(s)ds∈H2(Ω)∩H1 0(Ω), ws∈L2 µ((0,+∞);H1 0(Ω))}, B(u,v,ηt)T:= (0,χOv,0)TandF(U(t)) = (0,|u(t)|σu(t),0)T. For anyu∈H1 0(Ω) consider the functional ψ(u) :=1 σ+2/integraldisplay Ω|u(x)|σ+2dx. By Sobolev’s embedding theorem, we know that if 0 < σ <4 n−2, thenψis well-defined, and Gˆ ateaux differentiable at any point u∈H1 0(Ω), with Gˆ ateaux derivative given by Dψ(u)(v) =/integraldisplay Ω|u(x)|σu(x)v(x)dx, for anyv∈H1 0(Ω). Moreover, as in [2], if 0 <σ/lessorequalslant2 n−2, thenψsatisfies the assumptions (H1), (H2), (H3). Define the energy as follows: E(t) :=1 2/integraldisplay Ω|ut(x,t)|2dx+1−˜µ 2/integraldisplay Ω|∇u(x,t)|2dx−ψ(u(x,t)) +1 2/integraldisplayt t−τ/integraldisplay O|k(s+τ)|·|ut(x,s)|2dxds+1 2/integraldisplay+∞ 0µ(s)/integraldisplay Ω|∇ηt(x,s)|2dxds. Theorem 3.1 applies to this model giving well-posedness and exponential decay of the energy for suitably small initial data, provided that the condition (3 .12) on the time delay holds for every t/greaterorequalslant0. 4.2 The plate system with memory and source term Let Ω be a non-empty bounded set in IRn, with boundary Γ of class C2. Let us denote ν(x) the outward unit normal vector at any point x∈Γ.Moreover, let O ⊂Ω be a nonempty open subset of Ω. We consider the following viscoelastic plate sy stem: utt(x,t)+∆2u(x,t)−/integraldisplay+∞ 0µ(s)∆2u(x,t−s)ds+k(t)χOut(x,t−τ) =|u(x,t)|σu(x,t),in Ω×(0,+∞), u(x,t) =∂u ∂ν(x,t) = 0,in Γ×(0,+∞), u(x,t) =u0(x,t) in Ω ×(−∞,0], ut(x,0) =u1(x),in Ω, ut(x,t) =g(x,t),in Ω×(−τ,0],(4.42) whereτ >0 is the time-delay, µ: (0,+∞)→(0,+∞) is a locally absolutely continuous memory kernel, which satisfies the assumptions (i)-(iv), σ >0 and the damping coefficient k(·) is a function in L1 loc([0,+∞)) satisfying (1.2). This system again falls in (1.1) for A= ∆2with domainD(A) =H4(Ω)∩H1 0(Ω).Definingηt sas in (2.4), we can rewrite the system (4.42) in the 14following way: utt(x,t)+(1−˜µ)∆2u(x,t)+/integraldisplay+∞ 0µ(s)∆2ηt(x,s)ds+k(t)χOut(x,t−τ) =|u(x,t)|σu(x,t),in Ω×(0,+∞), ηt t(x,s) =−ηt s(x,s)+ut(x,t),in Ω×(0,+∞)×(0,+∞), u(x,t) =∂u ∂ν(x,t) = 0,in Γ×(0,+∞), ηt(x,s) = 0,in Γ×(0,+∞),fort/greaterorequalslant0, u(x,0) =u0(x) :=u0(x,0),in Ω, ut(x,0) =u1(x) :=∂u0 ∂t(x,t)/vextendsingle/vextendsingle/vextendsingle t=0,in Ω, η0(x,s) =η0(x,s) :=u0(x,0)−u0(x,−s),in Ω×(0,+∞), ut(x,t) =g(x,t),in Ω×(−τ,0).(4.43) Then, arguing analogously to the previous example, one can r ecast (4.43) in the form (2.7). Moreover, for ( n−4)σ/lessorequalslant4 (cf. e.g. [14]) the nonlinear source satisfies the required assumptions. 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Well–posedness and stability results for nonlinear abstra ct evolution equations with time delays. J. Evol. Equ. , 18:947–971, 2018. [18]R.L. Oliveira and H.P. Oquendo Stability and instability results for coupled waves with delay term . J. Math. Phys. , 61, Art. 071505, 2020. [19]A. Pazy . Semigroups of linear operators and applications to partia l differential equations. Vol. 44 of Applied Math. Sciences. Springer-Verlag , New York, 1983. [20]C. Pignotti. Stability results for second-order evolution equations wi th memory and switching time-delay. J. Dynam. Differential Equations 29:1309–1324, 2017. [21]B. Said-Houari and A. Soufyane. Stability result of the Timoshenko system with delay and boundary feedback. IMA J. Math. Control Inform. , 29:383–398, 2012. [22]G.Q. Xu, S.P. Yung and L.K. Li. Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim. Calc. Var. , 12:770–785, 2006. [23]Z. Yang . Existence and energy decay of solutions for the Euler-Bern oulli viscoelastic equation with a delay. Z. Angew. Math. Phys. , 66:727–745, 2015. 16

2020-09-16

In this paper we study a class of semilinear wave type equations with viscoelastic damping and delay feedback with time variable coefficient. By combining semigroup arguments, careful energy estimates and an iterative approach we are able to prove, under suitable assumptions, a well-posedness result and an exponential decay estimate for solutions corresponding to small initial data. This extends and concludes the analysis initiated in [16] and then developed in [13, 17].

Exponential decay for semilinear wave equations with viscoelastic damping and delay feedback

2009.07777v1

arXiv:1701.09110v1 [cond-mat.mes-hall] 31 Jan 2017Lack of correlation between the spin mixing conductance and the ISHE-generated voltages in CoFeB/Pt,Ta bilayers A. Conca,1,∗B. Heinz,1M. R. Schweizer,1S. Keller,1E. Th. Papaioannou,1and B. Hillebrands1 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany (Dated: June 21, 2021) We investigate spin pumping phenomena in polycrystalline C oFeB/Pt and CoFeB/Ta bilayers and the correlation between the effective spin mixing conduc tanceg↑↓ effand the obtained voltages generated by the spin-to-charge current conversion via the inverse spin Hall effect in the Pt and Ta layers. For this purpose we measure the in-plane angular dep endence of the generated voltages on the external static magnetic field and we apply a model to sepa rate the spin pumping signal from the one generated by the spin rectification effect in the magnetic layer. Our results reveal a dominating role of anomalous Hall effect for the spin rectification effect with CoFeB and a lack of correlation between g↑↓ effand inverse spin Hall voltages pointing to a strong role of th e magnetic proximity effect in Pt in understanding the observed increased damping . This is additionally reflected on the presence of a linear dependency of the Gilbert damping param eter on the Pt thickness. INTRODUCTION In spin pumping experiments,[1, 2] the magnetization of a ferromagnetic layer (FM) in contact with a non- magnetic one (NM) is excited by a microwave field. A spin current is generated and injected into the NM layer and its magnitude is maximized when the ferromagnetic resonance (FMR) condition is fulfilled. The spin cur- rent can be detected by using the inverse spin Hall effect (ISHE) for conversion into a charge current in appropri- ate materials. The injected spin current Jsin the NM layer has the form[1] Js=/planckover2pi1 4πg↑↓ˆm×dˆm dt(1) where ˆmis the magnetization unit vector and g↑↓is the realpartofthe spinmixing conductancewhich iscontrol- ling the intensity of the generated spin current. Its value is sensitive to the interface properties. The generation of the spin current opens an additional loss channel for the magnetic system and consequently causes an increase in the measured Gilbert damping parameter α: ∆αsp=γ/planckover2pi1 4πMsdFMg↑↓(2) This expression is only valid for thick enough NM lay- ers where no reflection of the spin current takes place at the interfaces. In principle, it allows the estimation ofg↑↓by measuring the increase in damping compared to the intrinsic value. However, other phenomena, like the magnetic proximity effect (MPE) in the case of Pt or interface effects depending on the exact material com- bination or capping layer material, can have the same influence, [7, 8] which challenges the measurement of the contribution from the spin pumping. In this sense, it is preferable to use an effective value g↑↓ eff. Still, if thespin pumping is the main contribution to the increase inα, a correlation between g↑↓ effand the measured ISHE voltages is expected. A suitable approach in order to un- derstand the weight of MPE on the value of g↑↓ effis the use of FM/NM with varying NM metals, with presence and absence of the MPE effect. The measurement of ∆ α andg↑↓ efftogether with the ISHE voltages generated by the spin current in the NM layer can bring clarity to the issue. However, the generation of an additional dc voltage by the spin rectification effect,[3–6] which adds to the voltagegeneratedbythe ISHE spin-to-chargeconversion, deters the analysis of the obtained data. The spin recti- fication originates from the precession of the magnetiza- tion in conducting layers with magnetoresistive proper- ties, mainly Anisotropic Magnetoresistance (AMR) and Anomalous Hall Effect (AHE). Information about the physics behind the measured voltage can only be ob- tained after separation of the different contributions. For this purpose, we made use of the different angulardepen- denciesofthecontributionsunderin-planerotationofthe external magnetic field. EXPERIMENTAL DETAILS Here, we report on results on polycrystalline Co40Fe40B20/Pt,Ta bilayers grown by rf-sputtering on Si substrates passivated with SiO 2. CoFeB is a material choice for the FM layer due to its low damping proper- ties and easy deposition.[9, 10] A microstrip-based VNA- FMR setup was used to study the damping properties. A more detailed description of the FMR measurement and analysis procedure is shown in previous work.[7, 10] A quadrupole-based lock-in setup described elsewhere[11] was used in order to measure the ISHE generated volt- age. The dependence of the voltage generated during the spin pumping experiment on the in-plane static external2 field orientation is recorded for a later separation of the pure ISHE signal from the spin rectification effect. GILBERT DAMPING PARAMETER AND SPIN MIXING CONDUCTANCE Figure1showsthedependenceoftheeffectivedamping parameter αeff(sum ofall contributions)onthe thickness dof the NM metal for a CoFeB layer with a fixed thick- ness of 11 nm. The case d= 0 nm represents the case of reference layers with Al capping. From previous studies it is known that the use of an Al capping layer induces a large increase of damping in Fe epitaxial layers.[7] For polycrystallineNiFe andCoFeBlayersthis is notthe case and it allows the measurement of the intrinsic value α0. [8] The observed behavior differs strongly for Pt and Ta. In the Pt case a large increase in damping is ob- served with a sharp change around d= 1 nm and a fast saturation for larger thicknesses. This is quali- tatively very similar to our previous report on Fe/Pt bilayers.[7] From the measured ∆ αwe extract the value g↑↓ eff= 6.1±0.5·1019m−2. This value is larger than the onereportedpreviouslyinourgroup[8]forthinnerCoFeB layers with larger intrinsic damping 4 .0±1.0·1019m−2 andalsolargerthanthevaluereportedbyKim et al.[12], 5.1·1019m−2. The impact of the Ta layer on damping is very reduced and, consequently, a low value for g↑↓ effof 0.9±0.3·1019m−2is obtained. This value is now smaller than the one reported by Kim et al.1.5·1019m−2) in- dicating that the difference between CoFeB/Pt and Ta is larger in our case. A reference has also to be made to the work of Liu et al.on CoFeB films thinner than in this work. [13] There, no value for the spin mixing conductance is provided, but the authors claim a vanish- FIG. 1. (Color online) Dependence of the effective Gilbert damping parameter αeffon the thickness of the NM metal. A large increase in damping is observed for the Pt case while a very small but not vanishing increase is observed for Ta. From the change ∆ αthe effective spin mixing conductance g↑↓ effis estimated using Eq. 2.ing impact on αfor the Ta case. On the contrary the increase due to Pt is almost three times larger than ours, pointing to a huge difference between both systems. In any case, the trend is similar, only the relative difference between Ta and Pt changes. A closer look to the data allows to distinguish a region in the Pt damping evolution prior to the sharp increase where a linear behavior is recognized ( d <1 nm). A lin- ear thickness dependence of αin spin-sink ferromagnetic films and in polarized Pt has been reported. [14, 15] The increasein damping due to spin currentabsorptionin the Pt with ferromagnetic order can then be described by: ∆α= ∆αMPE·dPt/dPt c (3) where ∆αMPEis the total increase in damping due only to the magnetic proximity effect in Pt, dPtis the thick- ness of the Pt layer and dPt cis a cutoff thickness which is in the order of magnitude of the coherence length in ferromagnetic layers.[15, 16] The inset in Fig. 1 shows a fit of Eq. 3 from where dPt c= 0.8nm isobtained assumingavalue ∆ αMPE= 1.2. The value is in qualitative agreement with the reported thickness where MPE is present in Pt, ( dPt MPE≤1 nm [17, 18]) and is lower than the one reported for Py/Pt systems.[14] /s45/s52/s48/s48/s52/s48/s56/s48/s49/s50/s48 /s40/s98/s41 /s80/s116/s32/s68/s97/s116/s97 /s32/s70/s105/s116 /s32/s83/s121/s109/s109/s101/s116/s114/s105/s99 /s32/s65/s110/s116/s105/s115/s121/s109/s109/s101/s116/s114/s105/s99/s86/s111/s108/s116/s97/s103/s101/s32/s40/s181/s86/s41 /s84/s97 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s45/s50/s48/s48/s50/s48 /s32/s32/s86/s111/s108/s116/s97/s103/s101/s32/s40/s181/s86/s41 /s181 /s48/s72/s32/s40/s109/s84/s41/s40/s97/s41 FIG. 2. (Color online) Voltage spectra measured for (a) CoFeB/Ta and (b) CoFeB/Pt at 13 GHz. The solid line is a fit to Eq. 4. The symmetric voltage Vsymand antisymmetric voltageVantisymcontributions are separated and plotted inde- pendently (dashed lines). The voltage signal is dominated b y Vantisymin the Pt case and by Vsymin the Ta case.3 The increase of damping due to spin pumping is de- scribed by an exponential dependence and explains the sharp increase at dPt= 1 nm. However, the fast increase does not allow for a deep analysis and it is pointing to a spin diffusion length in Pt not larger than 1 nm. In any case, this point has to been treated with care. The contribution of MPE to damping can be easily un- derestimated and consequently also the value for dPt c. In any case, the value can be interpreted as a lower limit for ∆αMPE. If this is substracted, under the assumption that the rest of increase is due to spin pumping, the spin mixing conductance due only to the this effect would be g↑↓ eff= 4.9±0.5·1019m−2. ELECTRICAL DETECTION OF SPIN PUMPING Figure 2(a),(b) shows two voltage measurements recorded at 13 GHz for a NM thickness of 3 nm and a nominal microwave power of 33 dBm. The measured voltage is the sum of the contribution of the ISHE effect and of spin rectification effect originating from the dif- ferent magnetoresistive phenomena in the ferromagnetic layer. While the spin rectification effect generates both a symmetric and an antisymmetric contribution, [3–5] the pure ISHE signal is only symmetric. For this reason a separation of both is carried out by fitting the voltage spectra (solid line) to Vmeas=Vsym(∆H)2 (H−HFMR)2+(∆H)2+ +Vantisym−2∆H(H−HFMR) (H−HFMR)2+(∆H)2(4) where ∆ HandHFMRare the linewidth and the reso- nance field, respectively. The dotted lines in Fig. 2 show the two contributions. When comparing the data for Pt and Ta some differences are observed. First of all, the absolute voltage values are smaller for the Pt cases and, moreimportant, therelativeweightofbothcontributions is different. While the first point is related to the differ- ent conductivity of Ta and Pt, the second one is related to the intrinsic effect causing the voltage. We calculate the ratio S/A = Vsym/Vantisymfor all the measurements and the results are shown in Fig. 3(a) as a function of the NM thickness. While the antisymmetric contribution is dominating in the Pt samples with a S/A ratio smaller than 1 for the samples with Pt, the opposite is true for the Ta case. Since the ISHE signal is contributing only toVsymit might be concluded that spin pumping is tak- ing place stronger in the Ta system. However, since also the spin rectificationeffect has a symmetric contribution, this conclusion cannot be supported. Furthermore, since the spin Hall angle θSHEhas opposite sign in these two materials, also the ISHE signal should have it. In appar-ent contradiction to this, we observe that both symmet- ric contributions have the same sign in (a) and (b). This points to the fact that for Pt, Vsymis dominated by the spin rectification effect, which does not change sign and overcompensates a smaller ISHE signal. All these con- siderations have the consequence that it is not possible to extract complete information of the origin of the mea- sured voltage by analyzing single spectra. For the same reason, the large increase in S/A for Ta for d= 5 nm or the change in sign for Pt with the same thickness cannot be correctly explained until the pure ISHE signal is not separated from the spin rectification effect. As already pointed out in recent papers[3–5, 11, 19], an analysis of the angular dependence (in-plane or out-of-plane) of the measured voltages can be used to separate the different contributions. In any case, before proceeding it has to be proven that allthe measurementswereperformed in the linearregime with small cone angles for the magnetization precession. The measurements performed out of this regime would have a large impact on the linewidth and a Gilbert-like dampingwouldnotbeguaranteed. Figure3(b) showsthe dependence of the voltage amplitude on the microwave nominal power proving indeed that the measurements were carried on in the linear regime. FIG. 3. (Color online) (a) Dependence of the ratio S/A =Vsym/Vantisymon the thickness of the NM layer. (b) Dependence of the total voltage on the applied microwave power proving the measurements were carried out in the lin- ear regime.4 SEPARATION OF THE ISHE SIGNAL FROM THE SPIN RECTIFICATION VOLTAGE We performed in-plane angular dependent measure- ments of the voltage and Eq. 4 was used to extract Vsym,antisymfor each value of the azimuthal angle φ spanned between the direction of the magnetic field and the microstrip antenna used to excite the magnetization. We used a model based on the work of Harder et al.[3] to fit the dependence. This model considers two sources for the spin rectification, which are the Anisotropic Mag- netoresistance (AMR) and the Anomalous Hall Effect (AHE): Vsym=Vspcos3(φ)+ +VAHEcos(Φ)cos( φ)+Vsym AMR−⊥cos(2φ)cos(φ) +Vsym AMR−/bardblsin(2φ)cos(φ) Vantisym=VAHEsin(Φ)cos( φ) +Vantisym AMR−⊥cos(2φ)cos(φ) +Vantisym AMR−/bardblsin(2φ)cos(φ) (5) Here,VspandVAHEare the contributions from spin pumping (pure ISHE) and from AHE, respectively. Φ is the phase between the rf electric and magnetic fields in the medium. The contribution from the AMR is di- vided in one generating a transverse ⊥(with respect to the antenna) or longitudinal /bardblvoltage. In an ideal case with perfect geometry and point-like electrical contacts Vsym,antisym AMR−/bardblshould be close to zero. Figure 4 shows the angular dependence of Vsym(top) andVantisym(bottom) for the samples with NM thick- ness of 3 nm. The lines are a fit to the model which is able to describe the dependence properly. From the data it can be clearly concluded that while the values ofVantisymare comparable, with the difference resulting from the different resistivity of Pt and Ta, the values ofVsymare much larger for Ta. The values obtained from the fits for the different contributions are plotted in Fig. 5 as a function of the thickness of the NM layer. The value of Φ is ruling the lineshape of the electrically measured FMR peak[20] which is always a combination of a dispersive ( D, antisymmetric) and a Lorentzian ( L, symmetric) contribution in the form D+iL. In order to compare the relative magnitudes of the different con- tributions independently of Φ we compute the quantities VAMR−/bardbl,⊥=/radicalbigg/parenleftBig Vantisym AMR−/bardbl,⊥/parenrightBig2 +/parenleftBig Vsym AMR−/bardbl,⊥/parenrightBig2 which it is equivalent to√ D2+L2and we show them together withVAHEandVsp. This step is important to allow for comparison of the different contributions independent of the value of Φ. Several conclusions can be extracted from Fig. 5. First of all, the spin rectification effect in CoFeB systems is al- most fully dominated by the AHE. AMR plays a veryFIG. 4. (Color online) Angular dependence of Vsym(top) andVantisym(bottom) for CoFeB/Pt,Ta samples with NM thickness of 3 nm. The lines are a fit to the model described in Eq. 5. minor role. This is a difference with respect to NiFe or Fe. [4, 11, 20] This is correlated with the very large AHE reported in CoFeB films. [21, 22] Second, the volt- ages generated by the spin pumping via the ISHE are larger in the case of Ta and of opposite sign as expected from the different sign of θSHEin both materials. This solves the apparent contradiction observed by the posi- tive symmetric contributions in both materials as shown in Fig. 2(a) and (b) and confirms the interpretation than inthecaseofPtthesymmetriccontributionisdominated by the spin rectification effect with opposite sign to the ISHE signal. Again, this shows that the interpretation using single spectra may lead to confusion and that angle dependent measurements are required. The evolution of the spin rectification voltages with NM thickness shows a saturation behavior in both cases for small thicknesses and a decrease with the NM layer thickness compatible with a dominant role of the re- sistance of the CoFeB layer. This is expected from the resistivity values for amorphous CoFeB layers, 300- 600µm·cm,[23] which are much larger than for β-Ta (6-10µm·cm) or sputtered Pt (100-200 µm·cm).[24, 25] However, the dependence does not completely agree with the expected behavior[19] 1 /dNMpointing out to addi- tional effects like a variation of the conductivity of Pt for the thinner layers. Concerning the correlation of the absolute values of the ISHE-generated voltages and the spin Hall angles in both materials, unfortunately the scatter in θSHEvalues in the literature is very large.[26] Howeverthis is reduced if we consider works were θSHEwas measured simultane- ously for Pt and Ta in similar samples. In YIG/Pt,Ta5 systems[27, 28] it was determined that |θPt SHE|>|θTa SHE| with a relative difference of around 30% which it is at odds with our results. On the contrary, in CoFeB/Pt,Ta bilayers|θTa SHE|= 0.15>|θPt SHE|= 0.07 is reported.[13] However the difference is not large enough to cover com- pletely the difference in our samples. In order to ex- plain this point together with the absolute low value in CoFeB/Pt we have to take into account the possibility of a certain loss of spin current at the interface FM/Pt or at the very first nanometer, the latter due to the presence of a static magnetic polarization due to the proximity ef- fect. With this the spin current effectively being injected in Pt would be lower than in the Ta case. The data does not allow for a quantitative estimation of the spin diffusion length λsd, but in any case the evo- lution is only compatible with a value for Pt not thicker than 1 nm, similarto reportedvalues forsputtered Pt[25] and a a value of a few nm for Ta, also compatible with literature.[28] An important point is the lack of correlation of g↑↓ eff and the expected generated spin current using Eq. 1 with the absolute measured ISHE voltage that results from the spin-to-charge current conversion, obtained after the separationfromthe overimposedspin rectificationsignal. This is true even if we substract the MPE contribution assumed for Eq. 3. The same non-mutually excluding explanations are possible here: ∆ αin Pt in mainly due to the MPE, or the spin current pumped into Pt van- ishes at or close to the interface. The first alternative would render Eq. 2 unuseful since most of the increase in damping is not due to spin pumping as long as the MPE is present. The second would reduce the validity of Eq. 1 to estimate the current injected in Pt and con- verted into a charge current by the ISHE. In any case, CoFeB/Ta shows very interesting properties, with strong spin pumping accompanied by only a minor impact on α. Let us discuss the limitations of the model defined in Eq. 5 and the suitability to describe the measurements. First of all, the model assumes a perfect isotropic mate- rial. The anisotropy in CoFeB is known to be small but not zero and a weak uniaxial anisotropy is present. The effect onthe angulardependenceisnegligible. Themodel assumes also a perfect geometry and point-like electrical contacts to measure the voltages. Our contacts are ex- tended (∼200µm) and a small misalignment is possible (angle between the antenna and the imaginary line con- necting the electrical contacts may not be exactly 90◦). This is the most probable reason for the non-vanishing small value for Vsym,antisym AMR−/bardbl. Nevertheless, the angular dependence of the measured voltage is well described by the model and no large deviations are observed. FIG. 5. (Color online) NM thickness dependence of the dif- ferent contributions to the measured voltages extracted fr om the angular dependence of VsymandVantisymfor Ta (top) and Pt (bottom). CONCLUSIONS In summary, we made use of in-plane angular de- pendent measurements to separate ISHE-generated from spin rectification voltages and we compare the absolute values and thickness dependence for Pt and Ta. Differ- ently to other materials, the spin rectification signal in CoFeB is almost fully dominated by AHE. No correlation between the observed spin mixing conductance via FMR measurement and the spin pumping signal is obtained pointing to a dominant role of the magnetic proximity effect in the increase in damping with Pt. ACKNOWLEDGEMENTS Financial support by M-era.Net through the HEUMEM project and by the Carl Zeiss Stiftung is gratefully acknowledged. ∗conca@physik.uni-kl.de [1] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, No. 4, 1375 (2005). [2] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). [3] M. Harder, Y. Gui, and C.-M. Hu, arXiv:1605.00710v1 [cond-mat.mtrl-sci] (2016).6 [4] W. T. Soh, B. Peng, and C. K. Ong, J. Phys. D: Appl. Phys. 47, 285001 (2014). [5] Y. Gui, L. Bai, and C. Hu, Sci. China-Phys. Mech. As- tron.56, 124 (2013). [6] W. Zhang, , B. Peng, F. Han, Q. Wang, W. T. Soh, C. K. Ong, and W. Zhang, Appl. Phys. Lett. 108, 102405 (2016). [7] A. Conca, S. Keller, L. Mihalceanu, T. Kehagias, G. P. Dimitrakopulos, B. Hillebrands, and E. Th. Pa- paioannou, Phys. Rev. B 93, 134405 (2016). [8] A. Ruiz-Calaforra, T. Br¨ acher, V. Lauer, P. Pirro, B. Heinz, M. Geilen, A. V. Chumak, A. Conca, B. Leven, and B. Hillebrands, J. Appl. Phys. 117, 163901 (2015). [9] A. Conca, J. Greser, T. Sebastian, S. Klingler, B. Obry, B. Leven, and B. Hillebrands, J. Appl.Phys. 113, 213909 (2013). [10] A. Conca, E. Th. Papaioannou, S. Klingler, J. Greser, T. Sebastian, B. Leven, J. L¨ osch, and B. Hillebrands, Appl. Phys. Lett. 104, 182407 (2014). [11] S. Keller et al., in preparation. [12] D.-J Kim, S.-I. Kim, S.-Y. Park, K.-D. Lee, and B.- G. Park, Current Appl. Phys. 14, 1344 (2014). Please note the different stoichiometry: Co 32Fe48B20. [13] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). [14] M. Caminale, A. Ghosh, S. Auffret, U. Ebels, K. Ollefs, F. Wilhelm, A. Rogalev, and W. E. Bailey, Phys. Rev. B. 94, 014414 (2016). [15] A. Ghosh, S. Auffret, U. Ebels, and W. E. Bailey, Phys. Rev. Lett. 109, 127202 (2012). [16] M. D. Stiles and A. Zangwill, Phys. Rev. B. 66, 014407 (2002).[17] M. Suzuki, H. Muraoka, Y. Inaba, H. Miyagawa, N. Kawamura, T. Shimatsu, H. Maruyanma, N. Ishi- matsu, Y. Isohama, and Y. Sonobe, Phys. Rev. B 72, 054430 (2005). [18] F. Wilhelm, P. Poulopoulos, G. Ceballos, H. Wende, K.Baberschke, P.Srivastava, D.Benea, H.Ebert, M.An- gelakeris, N. K. Flevaris, D. Niarchos, A. Rogalev, and N. B. Brookes, Phys. Rev. Lett. 85, 413 (2000). [19] R. Iguchi, and E. Saitoh, arXiv:1607.04716v1 [cond- mat.mtrl-sci] (2016). [20] M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C.-M. Hu, Phys. Rev. B. 84, 054423 (2011). [21] T. Zhu, P. Chen, Q. H. Zhang, R. C. Yu, and B. G. Liu, Appl. Phys. Lett. 104, 202404 (2014). [22] T. Zhu, Chin. Phys. B 23,No 4, 047504 (2014). [23] S. U. Jen, Y. D. Yao, Y. T. Chen, J. M. Wu, C. C. Lee, T. L. Tsai, Y. C. Chang, J. Appl. Phys. 99, 053701 (2006). [24] K. Stella, D. B¨ urstel, S. Franzka, O. Posth and D. Diesing, J. Phys. D: Appl. Phys. 42, 135417 (2009). [25] E. Sagasta, Y. Omori, M. Isasa, M. Gradhand, L. E. Hueso, Y. Niimi, Y. Otani, and F. Casanova, Phys. Rev. B. 44, 060412 (2016). [26] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, andT. Jungwirth, Reb.ofModern Phys. 87, 1213(2015). [27] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014). [28] C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Nale- tov, andJ. BenYoussef, Phys.Rev.B. 87, 174417 (2013).

2017-01-31

We investigate spin pumping phenomena in polycrystalline CoFeB/Pt and CoFeB/Ta bilayers and the correlation between the effective spin mixing conductance $g^{\uparrow\downarrow}_{\rm eff}$ and the obtained voltages generated by the spin-to-charge current conversion via the inverse spin Hall effect in the Pt and Ta layers. For this purpose we measure the in-plane angular dependence of the generated voltages on the external static magnetic field and we apply a model to separate the spin pumping signal from the one generated by the spin rectification effect in the magnetic layer. Our results reveal a dominating role of anomalous Hall effect for the spin rectification effect with CoFeB and a lack of correlation between $g^{\uparrow\downarrow}_{\rm eff}$ and inverse spin Hall voltages pointing to a strong role of the magnetic proximity effect in Pt in understanding the observed increased damping. This is additionally reflected on the presence of a linear dependency of the Gilbert damping parameter on the Pt thickness.

Lack of correlation between the spin mixing conductance and the ISHE-generated voltages in CoFeB/Pt,Ta bilayers

1701.09110v1

arXiv:2107.00982v3 [cond-mat.supr-con] 25 Aug 2021Anomalous Gilbert Damping and Duffing Features of the SFS ϕ0Josephson Junction Yu. M. Shukrinov1,2, I. R. Rahmonov1,3, A. Janalizadeh4, and M. R. Kolahchi4 1BLTP, JINR, Dubna, Moscow Region, 141980, Russia 2Dubna State University, Dubna, 141980, Russia 3Umarov Physical Technical Institute, TAS, Dushanbe 734063 , Tajikistan 4Department of Physics, Institute for Advanced Studies in Ba sic Sciences, P.O. Box 45137-66731, Zanjan, Iran (Dated: August 26, 2021) We demonstrate unusual features of phase dynamics, IV-char acteristics and magnetization dy- namics of the ϕ0Josephson junction at small values of spin-orbit interacti on, ratio of Josephson to magnetic energy and Gilbert damping. In particular, an anom alous shift of the ferromagnetic reso- nance frequency with an increase of Gilbert damping is found . The ferromagnetic resonance curves show the Duffing oscillator behaviour, reflecting the nonline ar nature of Landau-Lifshitz-Gilbert (LLG) equation. Based on the numerical analysis of each term in LLG equation we obtained an approximated equation demonstrated both damping effect and Duffing oscillator features. The re- sulting Duffing equation incorporates the Gilbert damping in a special way across the dissipative term and the restoring force. A resonance method for the dete rmination of spin-orbit interaction in noncentrosymmetric materials which play the role of barrie r inϕ0junctions is proposed. Introduction. The Josephson junctions (JJ) with the current-phaserelation I=Icsin(ϕ−ϕ0), wherethephase shiftϕ0is proportional to the magnetic moment of ferro- magneticlayerdetermined bythe parameterofspin-orbit interaction, demonstratea number ofunique featuresim- portant for superconducting spintronics, and modern in- formation technology [1–6]. The phase shift allows one to manipulate the internal magnetic moment using the Josephson current, and the reverse phenomenon which leads to the appearance of the DC component in the su- perconducting current [7–9]. Interactive fields can bring nonlinear phenomena of both classical, and quantum nature. A basic example is the magnons strongly interacting with microwave pho- tons [10]. As a result we could name Bose-Einstein con- densation of such quasiparticles, i.e. magnons [11, 12], and synchronization of spin torque nano-oscillators as they coherently emit microwave signals in response to d.c. current [13]. It is interesting that (semi)classical an- harmonic effects in the magnetodynamics described by the Landau-Lifshitz-Gilbert (LLG) model in thin films or heterostructures [14, 15], and the quantum anharmonic- ity in the cavity mangnonics [16] can well be modeled by so simple a nonlinear oscillator as Duffing. The cor- responding Duffing equation contains a cubic term and describesthe oscillationsofthe variousnonlinearsystems [17]. Despite the fact that nonlinear features of LLG are studied often during a long time and in different systems, manifestation of the Duffing oscillator behavior in the frameworkofthisequationisstill notcompletelystudied. Closer to our present investigation, in the study of the dynamics of antiferromagnetic bimeron under an alter- natingcurrent,Duffingequationformsagoodmodel, and this has applications in weak signal detection [14, 18, 19]. As another application with Duffing oscillator at work, we can mention the ultra thin Co 20Fe60B20layer, andits largeangle magnetizationprecessionunder microwave voltage. There are also ‘foldover’ features, characteris- tic of the Duffing spring, in the magnetization dynamics of the Co/Ni multilayer excited by a microwave current [15, 20, 21]. But nonlinear features of ϕ0Josephson junc- tions have not been carefully studied yet. In this Letter, we show that the Duffing oscillator helps in the under- standing of the nonlinear features of ϕ0Josephson junc- tions at small values of system parameters. Coupling of superconducting current and magnetiza- tion and its manifestation in the IV-characteristics and magnetizationdynamicsopensthedoorfortheresonance method determination of spin-orbit intensity in noncen- trosymmetric materials playing the role of barrier in ϕ0 junctions. As it is well known, the spin-orbit interaction plays an important role in modern physics, so any novel method for its determination in real materials would be very important. There are a series of recent experiments demonstrating the modification of Gilbert damping by the superconducting correlations (see Ref.[22] and cita- tionstherein). Inparticular, the pronouncedpeaksin the temperature dependence of Gilbert damping have been observed for the ferromagnetic insulator/superconductor multilayers [23] which might be explained by the pres- ence of spin relaxation mechanisms like the spin-orbit scattering [22]. Here, we use the noncentrosymmetric ferromagnetic material as a weak link in ϕ0junctions. The suitable candidates may be MnSi or FeGe, where the lack of inversion center comes from the crystalline structure [8]. The Gilbert damping determines the magnetization dynamics in ferromagnetic materials but its origin is not well understood yet. Effect of nonlinearity on damp- ing in the system is very important for application of these materials in fast switching spintronics devices. Our study clarifies such effects. In Ref.[24] the authors dis- cuss the experimental study of temperature-dependent2 Gilbert damping in permalloy (Py) thin films of varying thicknesses by ferromagnetic resonance, and provide an important insight into the physical origin of the Gilbert damping in ultrathin magnetic films. In this Letter we demonstrate an anomalous depen- dence of the ferromagnetic resonance frequency with an increase of the Gilbert damping. We find that the reso- nance curves demonstrate features of Duffing oscillator, reflecting the nonlinear nature of LLG equation. The damped precession of the magnetic moment is dynami- cally driven by the Josephson supercurrent, and the res- onance behavior is given by the dynamics of the Duffing spring. The resonance methods for the determination of spin-orbit interaction in the ϕ0junction are proposed. Model and Methods. In the considered SFS ϕ0junc- tion (see Fig.1) the superconducting phase difference ϕ and magnetization Mof the F layer are two coupled dy- namical variables. Based on the LLG equation for the Figure 1: Schematic view of SFS ϕ0Josephson junction. The external current applied along xdirection, ferromagnetic easy axis is along zdirection. magnetic moment Mwith effective magnetic field Heff, resistively capacitively shunted junction (RCSJ) model, and Josephson relation for the phase difference ϕ, we de- scribe dynamics of the SFS ϕ0junction by the system of equations in normalized variables dm dt=ωFheff×m+α/parenleftbigg m×dm dt/parenrightbigg , heff=Grsin(ϕ−rmy)/hatwidey+mz/hatwidez, (1) dV dt=1 βc[I−V+rdmy dt−sin(ϕ−rmy)], dϕ dt=V, wheremis vector of magnetization with components mx,y,z, normalized to the M0=/bardblM/bardbland and satisfy- ing the constraint/summationtext i=x,y,zm2 i(t) = 1,ωF= ΩF/ωc, ΩF=γK/νis ferromagnetic resonance frequency, γis the gyromagnetic ratio, Kis an anisotropic constant, ν is the volume of the ferromagnetic F layer, αis the phe- nomenologicaldamping constant(Gilbert damping), heff is the vector of effective magnetic field, normalized to theK/M0(heff=HeffM0/K),G=EJ/(Kν) relation of Josephson energy to magnetic one, ris a parameter of spin-orbit coupling, ϕis phase difference of JJ, Vis voltage normalized to the Vc=IcR,Iccritical current of JJ,Rresistance of JJ, βc= 2eIcCR2//planckover2pi1is McCumberparameter, Cis capacitance of JJ, Iis bias current nor- malized to the Ic. In this system of equation time tis normalized to the ω−1 c, whereωc= 2eIcR//planckover2pi1is character- istic frequency. In the chosen normalization, the average voltage corresponds to the Josephson frequency ωJ. Ferromagnetic resonance in ϕ0junction. The ferro- magnetic resonance features are demonstrated by aver- age voltage dependence of the maximal amplitude of the mycomponent ( mmax y), taken at each value of bias cur- rent. To stress novelty and importance of our finding, we first present the analytical results for average volt- age dependence of mmax yalong IV-characteristics in the ferromagnetic resonance region. As it was discussed in Refs.[8, 25, 26], in case Gr≪1,mz≈1, and neglecting quadratic terms mxandmy, we get /braceleftBigg ˙mx=ξ[−my+GrsinωJt−αmx] ˙my=ξ[mx−αmy],(2) whereξ=ωF/(1 +α2). This system of equations can be written as the second order differential equation with respect to the my ¨my=−2αξ˙my−ξ2(1+α2)my+ξ2GrsinωJt.(3) Corresponding solution for myhas the form my(t) =ω+−ω− rsinωJt−α++α− rcosωJt,(4) where ω±=Gr2ωF 2ωJ±ωF ((ωJ±ωF)2+(αωJ)2),(5) and α±=Gr2ωF 2αωJ ((ωJ±ωF)2+(αωJ)2).(6) So,mydemonstrates resonance with dissipation when Josephson frequency is approaching the ferromagnetic one (ωJ→ωF). The maximal amplitude mmax yas a function of voltage (i.e., Josephson frequency ωJ) at dif- ferentα, calculated using (4), is presented in Fig.2 (a). We see the usual characteristicvariation of the resonance curve with an increase in dissipation parameter when the maximal amplitude and position of resonance pick cor- responds to the damped resonance. We note that the analytical result (4) were obtained in the case Gr≪1. Presented in Fig.2(b) results of numerical simulations mmax y(V) dependence at different values of dissipation parameter αdemonstrate the essential differences with the results followedfrom the analytical consideration(4). We note also that the strong coupling of the supercon- ducting phase difference ϕand magnetization Mof the F layermanifests itself by appearanceof subharmonics of the resonance at ω= 1/2,1/3,1/4 demonstrated in the inset to Fig.2(b).3 Figure 2: (a) Analytical results for maximal amplitude mmax y in the ferromagnetic resonance region for different α; (b) Numerical results for maximal amplitude of magnetization my−component at each values of bias current and voltage along IV-characteristics of the ϕ0junction in the ferromag- netic resonance region for various α. Inset shows the man- ifestation of the resonance subharmonics. Parameters are: βc= 25, G=0.05, r=0.05, ωF= 0.5. We stress two important features followed from the presented results. First, the ferromagnetic resonance curves show the foldover effect, i.e., the features of Duff- ing oscillator. Different from a linear oscillator, the non- linear Duffing demonstrates a bistability under external periodic force [27]. Second, the ferromagnetic resonance curves demonstrate an unusual dependence of the reso- nance frequency as a function of Gilbert damping α. As shown in Fig. 3(a), an increase in damping leads to a nonuniform change in the resonant frequency, i.e., with an increase in damping the resonance maximum shifts toωFat small α, but then moves to the opposite side, demonstrating the usual damped resonance. So, with an increase in α, unusual dependence of the resonance voltage transforms to the usual one. For the parameters chosen, the critical value of this transformation is around α= 0.02−0.03. We call this unusual behaviour of the resonance maximum of mmax yas an “α-effect”. Both the α−effect and Duffing features in our system appear due to the nonlinear features of the system dynamics at small Figure 3: (a) α-dependence of the resonance curve mmax y(V) peak presented in Fig.2 in the damping parameter interval [0.006 – 0.2]. Dashed line indicates ferromagnetic resonan ce position; (b) Comparison of the resonance curves mmax y(V) calculated by full LLG equation (1) and the approximate equation (8). G,r,α≪1. To prove it, we have carried out the nu- merical analysis of each term of LLG full equation (first two equations in (1)) for the set of model parameters G= 0.05,r= 0.05α= 0.005. After neglecting the terms of order 10−6, we have ˙mx ξ=−mymz+Grmzsin(ϕ−rmy)−αmxm2 z, ˙my ξ=mxmz−αmym2 z, (7) ˙mz ξ=−Grmxsin(ϕ−rmy)+αmz(m2 x+m2 y), Inthisapproximationweobserveboththe“ α–effect”and Duffing oscillator features. Neglecting here the last term αmz(m2 x+m2 y) in third equationfor ˙ mz, which is orderof 10−4, leadstothe losingoftheDuffing oscillatorfeatures, but still keeps alpha-effect. We note that equation (7) keeps the time invariance of the magnetic moment, so that term plays an important role for manifestation of Duffing oscillator features by LLG equation. The generalized Duffing equation for ϕ0junction. The LLG is a nonlinear equation and in case of simple effective field it can be transformed to the Duffing equa- tion [14, 17]. Such transformation was used in Ref.[17] to demonstrate the nonlinear dynamics of the magnetic vortex state in a circular nanodisk under a perpendicular alternating magnetic field that excites the radial modes of the magnetic resonance. They showed Duffing-type nonlinear resonance and built a theoretical model corre- sponding tothe Duffing oscillatorfromthe LLG equation to explore the physics of the magnetic vortex core polar- ity switching for magnetic storage devices. The approximated LLG system of equations (7) demonstrates both α-effect and features of Duffing os- cillator. As demonstrated in the Supplemental Materials [28], the generalizedDuffing equation forthe ϕ0junction, ¨my+2ξα˙my+ξ2(1+α2)my −ξ2(1+α2)m3 y=ξ2GrsinωJt.(8)4 can be obtained directly from the LLG system of equa- tions. As we see, for small enough Gandr, it is only the dimensionless damping parameter αin LLG that plays a role in the dynamics of the system. We can think of a harmonic spring with a constant that is hardened or soft- ened by the nonlinear term. For a usual Duffing spring, with independent coefficients of the various terms, the resonancepeak relative to the harmonic (linear) resonant frequency folds over to the smaller (softening) or larger (hardening) frequencies. In the frequency response, the interplay of the specific dependence of each coefficient on αplays an important role and as Fig.3(a) shows, there is a particular αthat brings the resonant frequency closest to ferromagnetic resonance. Simulations of the mydynamics in the framework of Duffing equation can explain observed foldover effect in the frequency dependence of mmax y. Comparison the re- sults followed from analytical approximate equation (8) and results of full equation (1) for maximal amplitude of mmax yin the ferromagnetic resonance region is presented in Fig.3(b). So, the magnetization dynamics in the SFS ϕ0-junction due to the voltage oscillations can effectively be described by a scalar Duffing oscillator, synchronizing the precession of the magnetic moment with the Joseph- son oscillations. Effect of spin-orbit interactions. As we mentioned above, the spin-orbit interaction plays an important role in different fields of modern physics. Here we have sug- gested a novel method for its determination in real non- centrosymmetric ferromagnetic materials like MnSi or FeGe, where the lack of inversion center comes from the crystalline structure Ref.[8] and which play role a weak link in ϕ0junctions. Based on the obtained re- sults, presented in Fig.4, we propose different versions of the resonance method for the determination of spin-orbit interaction in these materials. Particularly, in Fig.4(a) we present the simulation results of maximal amplitude mmax ybased on (1) at G= 0.05,α= 0.01 at different values of spin-orbit parameter rin the ferromagnetic res- onance region. This case corresponds to the nonlinear approximation leading to the Duffing equation (8). The same characteristics calculated by equation (1) for larger valueα= 0.1, i.e. corresponding to the linear approxi- mation (3) are presented in Fig.4(b). As it was expected, in caseα= 0.01 the foldover effect is more distinct. In Fig.4(c) the r-dependence of the resonancepeak po- sition, obtained from the simulation results of full equa- tion atα= 0.01 andα= 0.1 for the same set of model and simulation parameters is demonstrated. We stress here that nonlinear features of LLG equation leading to the Duffing’s shift of the mmax ypeak of main harmonic with r presented in Fig.4(c) show the manifestation of nonlinearity. Despite the noted differences between results for α= 0.01 andα= 0.1 , we see in both cases a monotonic Figure 4: (a) Voltage dependence of mmax yin the ferromag- netic resonance region at different values of spin-orbit int er- action based on (1) at G= 0.05,α= 0.01. Inset enlarges the main harmonic; (b) The same as in (a) for α= 0.1; (c) Shift ofmmax ypeak as a function of spin-orbit interaction at two values of Gilbert damping; (d) r-dependence of the main harmonic and subharmonics peaks in case (a); (e) The same as in (d) for the case (b). linear increase of mmax ypeak of main harmonic and sub- harmonics with rdemonstrated in Fig.4(d) and Fig.4(e). Such lineardependence canbe noted fromEq. (6) ofRef. [14], but the authors did not discuss it. This dependence might serve as a calibrated curve for spin-orbit interac- tion intensity, thus creating the resonance methods for r determination. Conclusions. Based on the reported features of the ϕ0Josephson junction at small values of spin-orbit in- teraction, ratio of Josephson to magnetic energy and Gilbert damping, we have demonstrated that the cou- pled superconducting current and the magnetic moments in theϕ0-junction result in the current phase relation in-5 tertwining with the ferromagnetic LLG dynamics. The ferromagnetic resonance clearly shows this interplay. In particular, an anomalous shift of the ferromagnetic res- onance frequency with an increase of Gilbert damping is found. The ferromagnetic resonance curves demon- strate features of Duffing oscillator, reflecting the nonlin- ear nature of LLG equation. The obtained approximated equation demonstrates both damping effect and Duffing oscillator features. We have shown that due to the non- linearity, asmodeledbythe generalizedDuffing equation, the parameters of the system can compensate each other resulting in unusual response. The position of the maxi- mum can shift towards and then away from the expected resonant frequency, as the damping is decreased. There are also foldover effects that was explained by the pro- posed model. A resonance method for the determination of spin-orbit interaction in noncentrosymmetric materi- als which play the role of barrier in ϕ0junctions was proposed. The experimental testing of our results would in- volve SFS structures with ferromagnetic material having enough small value of Gilbert damping. Potential candi- date for experimental realization could be ferromagnetic metals or insulators which have small values of damping parameter ( α∼10−3−10−4). In Ref.[29] the authors report on a binary alloy of cobalt and iron that exhibits a damping parameterapproaching10−4, which is compa- rable to values reported only for ferrimagnetic insulators [30, 31]. Using superconductor-ferromagnetic insulator- superconductor on a 3D topological insulator might be a way to have strong spin-orbit coupling needed for ϕ0 JJ and small Gilbert dissipation for α-effect [5]. We note in this connection that the yttrium iron garnet YIG is especially interesting because of its small Gilbert damp- ing (α∼10−5). The interaction between the Joseph- son current and magnetization is determined by the ra- tio of the Josephson to the magnetic anisotropy energy G=EJ/(Kν) and spin-orbit interaction r. The value of the Rashba-type parameter rin a permalloy doped with Pt[32] and in the ferromagnets without inversion sym- metry, like MnSi or FeGe, is usually estimated to be in the range 0 .1−1. The value of the product Grin the ma- terialwith weakmagneticanisotropy K∼4×10−5KA−3 [33], and a junction with a relatively high critical current density of (3 ×105−5×106)A/cm2[34] is in the range 1−100. It givesthe set offerromagneticlayerparameters and junction geometry that make it possible to reach the values used in our numerical calculations for the possible experimental observation of the predicted effect. Numerical simulations were funded by the project 18- 71-10095oftheRussianScientificFund. A.J.andM.R.K. are grateful to IASBS for financial support.[1] Jacob Linder and W. A. Jason Robinson, Nature Physics 11, 307 (2015). [2] Yu. M. Shukrinov, Accepted for UFN. DOI:https://doi.org/10.3367/UFNe.2020.11.038894 [3] A.A. Mazanik, I.R. Rahmonov, A.E. Botha, and Yu.M. Shukrinov, Phys. Rev. Applied 14, 014003 (2020). [4] M. Nashaat and Yu. M. Shukrinov, Physics of Particles and Nuclei Letters, 17, 79. (2020). [5] I. V. Bobkova , A. M. Bobkov, I. R. Rahmonov, A. A. Mazanik , K. Sengupta, and Yu. M. Shukrinov, Phys. Rev. B102, 134505 (2020). [6] Yu. M. Shukrinov, I. R. Rahmonov, K. Sengupta and A. Buzdin, Applied Physics Letters, 110, 182407, (2017). [7] A. Buzdin, Phys. Rev. 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Appl. 14, 034022 (2020). [16] Mehrdad Elyasi, Yaroslav M. Blanter, and Gerrit E. W. Bauer, Phys. Rev. B 101, 054402 (2020). [17] K. W. Moon, B. S. Chun,W. Kim, Z. Q. Qiu, and C. Hwang, Sci. Rep. 4, 6170 (2015). [18] G. Wang, D. Chen, J. Lin, and X. Chen, IEEE Transac- tions on Industrial Electronics 46, 440 (1999). [19] R. Almog, S. Zaitsev, O. Shtempluck, E. Buks, Phys. Rev. Lett. 98, 078103 (2007). [20] A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, New York, 1979). [21] W. Chen, G. de Loubens, J.-M. L. Beaujour, J. Z. Sun, and A. D. Kent, Appl. Phys. Lett. 95, 172513 (2009). [22] M. A. Silaev, Phys. Rev. B 102, 144521 (2020). [23] Y. Yao, Q. Song, Y. Takamura, J. P. Cascales, W. Yuan, Y. Ma, Y. Yun, X. C. Xie, J. S. Moodera, and W. Han, Phys. Rev. B 97, 224414 (2018). [24] Yuelei Zhao, Qi Song, See-Hun Yang, Tang Su1, Wei Yuan, Stuart S. P. Parkin, Jing Shi and Wei Han, Sci. Rep.6, 22890 (2016). [25] Shukrinov, Y.M., Rahmonov, I.R., Phys. Part. Nuclei 51, 816 (2020). [26] Yu. M. Shukrinov, I. R. Rahmonov, and A. E. Botha., Low Temp. Phys. 46, 932 (2020). [27] IvanaKovacic, Michael JBrennan. The DuffingEquation : Nonlinear Oscillators and their Behaviour. — John Wi-6 ley and Sons, 2011. [28] See Supplemental Material for details of our procedure to get the generalized Duffing equation from Landau- Lifshitz-Gilbert system of equations. [29] M.A.W.Schoen, D.Thonig, M.L.Schneider, T. J.Silva, H. T. Nembach, O. Eriksson, O. Karis and J. M. Shaw, Nature Physics 12, 842 (2016). [30] O. A. Kelly, A. Anane, R. Bernard, J. B. Youssef, C. Hahn, A. H. Molpeceres, C. Carr´ et´ ero, E. Jacquet, C. Deranlot, P. Bortolotti, R. Lebourgeois, J.-C. Mage, G. de Loubens, O. Klein, V. Cros, and A. Fert, Appl. Phys. Lett.103, 082408 (2013).[31] M. C. Onbasli, A. Kehlberger, D. H. Kim, G. Jakob, M. Kl¨ aui, A. V. Chumak, B. Hillebrands, and C. A. Ross, APL Mater. 2, 106102 (2014). [32] A. Hrabec, F. J. T. Goncalves, C. S. Spencer, E. Aren- holz, A. T. N’Diaye, R. L. Stamps, and C. H. Marrows, Phys. Rev. B 93, 014432 (2016). [33] A. Yu. Rusanov, M. Hesselberth, J. Aarts, and A. I. Buzdin, Phys. Rev. Lett. 93, 057002 (2004). [34] J. W. A. Robinson, F. Chiodi, M. Egilmez, G. B. Hal´ asz and M. G. Blamire, Scientific Report 2, 699 (2012).arXiv:2107.00982v3 [cond-mat.supr-con] 25 Aug 2021Supplemental Material to “Anomalous Gilbert Damping and Du ffing Features of the SFSϕ0Josephson Junction” Yu. M. Shukrinov1,2, I. R. Rahmonov1,3, A. Janalizadeh4, and M. R. Kolahchi4 1BLTP, JINR, Dubna, Moscow Region, 141980, Russia 2Dubna State University, Dubna, 141980, Russia 3Umarov Physical Technical Institute, TAS, Dushanbe 734063 , Tajikistan 4Department of Physics, Institute for Advanced Studies in Ba sic Sciences, P.O. Box 45137-66731, Zanjan, Iran (Dated: August 26, 2021) Here, we demonstrate by numerical methods that a generalize d Duffing equation can be obtained directly from LLG system of equations, for small system para meters of S/F/S junction. Both the α−effect and Duffing features obtained by LLG system of equations appear due to the nonlinear features of its dynamics at small G,r,α≪1. To proveit, we have carried out the numerical analysis of each term of LLG full equation (first two equations in the equation (1) of the main text) for the set of model parameters G= 0.05,r= 0.05α= 0.005. After neglecting the terms of order 10−6, we have ˙mx ξ=−mymz+Grmzsin(ϕ−rmy)−αmxm2 z, ˙my ξ=mxmz−αmym2 z, (1) ˙mz ξ=−Grmxsin(ϕ−rmy)+αmz(m2 x+m2 y), The procedure is as follows. Expanding mn zin a series with the degree of ( mz−1) we can find mn z=nmz−(n−1). (2) From expression m2 x+m2 y+m2 z= 1 and (2), we obtain mz=2−m2 y 2. (3) Using approximation sin( ϕ−rmy) = sin(ωJt) in (1), differentiatingsecondequationofthe system(1) andsub- stituting ˙ mx,mxand ˙mzfrom first second and third equations of the system (1), respectively and using the expression (2), (3) and assuming mz= 1 only in denom- inators, we come to a second order differential equation with respect to my ¨my=a1˙m3 y+a2my˙m2 y+a3m4 y˙my+a4m2 y˙my+a5˙my +a6m5 y+a7m3 y+a8my−c1˙m2 ysinωJt (4) +c2m4 ysinωJt+c3m2 ysinωJt+AsinωJt.The numerical calculation for the used set of model parameters allows us to estimate each of the terms in the equation, as presented in Table I. Now, if we neglect those terms smaller than 10−4, the equation (4) takes on the form of Duffing equation with Table I: Numerical analysis of equation (4) terms. a1α ξa1˙m3 y∼1.76×10−5 a2 α2a2my˙m2 y∼3.4×10−8 a3ξα3a3m4 y˙my∼7.7×10−12 a4ξ(3α−α3)a4m2 y˙my∼2×10−5 a52ξα a5˙my∼6×10−4 a6ξ2(α2+2α4)a6m5 y∼5.56×10−9 a7ξ2(1+α2−α4)a7m3 y∼3.7×10−3 a8ξ2(1+α2)a8my∼6.1×10−2 c1 Gr c1˙m2 ysinϕ∼3.6×10−5 c22ξ2α2Grc2m4 ysinϕ∼5.3×10−11 c3ξ2Gr(α2−2)c3m2 ysinϕ∼4.5×10−5 Aξ2Gr AsinωJt∼6.25×10−4 damping dependent coefficients, i.e., we have a general- ization of the Duffing equation ¨my+2ξα˙my+ξ2(1+α2)my −ξ2(1+α2)m3 y=ξ2GrsinωJt.(5)

2021-07-02

We demonstrate unusual features of phase dynamics, IV-characteristics and magnetization dynamics of the $\varphi_0$ Josephson junction at small values of spin-orbit interaction, ratio of Josephson to magnetic energy and Gilbert damping. In particular, an anomalous shift of the ferromagnetic resonance frequency with an increase of Gilbert damping is found. The ferromagnetic resonance curves show the Duffing oscillator behaviour, reflecting the nonlinear nature of Landau-Lifshitz-Gilbert (LLG) equation. Based on the numerical analysis of each term in LLG equation we obtained an approximated equation demonstrated both damping effect and Duffing oscillator features. The resulting Duffing equation incorporates the Gilbert damping in a special way across the dissipative term and the restoring force. A resonance method for the determination of spin-orbit interaction in noncentrosymmetric materials which play the role of barrier in $\varphi_0$ junctions is proposed.

Anomalous Gilbert Damping and Duffing Features of the SFS {\boldmath $\varphi_0$} Josephson Junction

2107.00982v3

The following article has been accepted by Applied Physics Letters from AIP. Magnon -mediated spin currents in Tm 3Fe5O12/Pt with perpendicular magnetic anisotropy G. L. S. Vilela1,2, J. E. Abrao3, E. Santos3, Y. Yao4,5, J. B. S. Mendes 6, R. L. Rodríguez -Suárez7, S. M. Rezende3, W. Han4,5, A. Azevedo3, and J. S. Moodera1,8 1 Plasma Science and Fusion Center, and Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology , Cambridge, Massachusetts 02139, USA 2 Física de Materiais , Escola Politécnica de Pernambuco , Universidade de Pernambuco , Recife, Pernambuco 50720 -001, Bra sil 3 Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670 -901, Brasil 4 International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China. 5 Collaborative Innovation Center of Quantum Matter, Beijing 100871, China. 6 Depart amento de Física, Universidade Federal de Viçosa , Viçosa, Minas Gerais 36570 -900, Brasil 7Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile 8Department of Physics, Massachusetts Institute of Technology , Cambridge, Massachusetts 02139, USA Electronic mail: 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 films with PMA p resent labyrinth domain magnetic structures that enrich the magnetization dynamics, and could be employed in more efficient wave -based logic and memory computing devices. In MI/NM bilayers, where NM being a normal metal providing a strong spin -orbit coupli ng, the PMA benefits the spin -orbit torque (SOT) driven magnetization's switching by lowering the needed current and rendering the process faster, crucial for developing magnetic random -access memories (SOT -MRAM). In this work, w e investigated the magnetic anisotropies in thulium 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 microwave s and heat currents. TIG films presented a Gilbert damping constant 𝛼 ≈0.01, with resonance fields above 3.5 kOe and half linewidth s broader than 60 Oe , at 300 K and 9.5 GHz . The spin-to-charge current conversion through TIG/Pt was observed as a micro -voltage generated at the edges of the Pt film. The obtained spin Seebeck coefficient was 0.54 𝜇𝑉/K, confirm ing also the high interfacial spin transparenc y. Spin-dependent phenomena in systems composed by layers of magnetic insulators (MI) and non-magnetic heavy metals ( NM) with strong spin -orbit coupling have been extensively explored in the insulator -based spintronics [ 1-6]. Among th e MI materials, YIG (Y3Fe5O12) is widely employed in devices for generation and transmission of pure spin currents. The main reason is its very low magnetic damping with Gilbert parameter on the order of 10-5, and its large spin decay length which permits spin waves to travel distances of orders of centimeters inside it before they vanish [7-9]. When combined with heavy metals such as Pt, Pd, Ta, or W, many intriguing spin -current related phenomena emerge , such as the spin pumping effect (SPE) [10-14], spin Seebeck effect (SSE) [7, 15 -18], spin Hall effect (SHE) [19-21], and spin-orbit torque (SOT) [22-25]. The origin of these effects relies mainly on the spin diffusion length , and the quantum -mechanical exchange and spin -orbit interaction s at the interface and inside the heavy metal [26]. All of these effects turn out the MI/NM bilayer into a fascinating playground for exploring spin-orbit driven phenomena at interfaces [27-30]. Well investigated for many years, intrinsic YIG(111) films on GGG(111) , (GGG = Gd 3Ga5O12) exhibit in -plane anisotropy. To obtain YIG single -crystal films with perpendicular magnetic anisotropy (PMA) it is necessary to grow them on top of a different substra te or partially substitute the yttrium ions by rare-earth ions , to cause strain -induced anisotropy [31-33]. Even so, it is well -known that magnetic films with PMA play an important role in spintronic technology. 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 random access memory ( SOT-MRAM ) [34- 36]. Besides that, PMA increase s the information density in hard disk drives and magnetoresistive random access memories [37-39], and it is crucial for breaking the time - reversal symmetry in topological insulators (TIs) aiming towards quantized anomalous Hall state in MI/TI [ 40-42]. Recent ly, thin films of another rare-earth iron garnet, TIG (Tm 3Fe5O12), have caught the attention of researchers due to its large negative magnetostriction constant , which favors an out -of-plane easy axis [ 4, 43, 44 ]. TIG is a ferrimagnetic insulator with a critical temperature of 549 K, a crystal structure similar to YIG , and a Gilbert damping parameter on the order of 𝛼~ 10−2 [4, 45 ]. Investigations of spin transport effects have been reported in TIG/Pt [45, 46 ] and TIG/TI [42, 47], where the TIG was fabricated by 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 magnetoresistance, spin Seebeck , and spin -orbit tor que effects. In this paper, we first present a study of the magnetocrystalline and uni axial anisotropies , as well as the magnetic damping of sputtered epitaxial TIG thin films using the ferromagnetic resonance (FMR) technique . For obtaining the cubic and uniaxial anisotropy fields, w e analyze d the 2 dependence of the FMR spectr a 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 the ir 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 transport spin angular momentum without carrying charge currents . They are free of Joule heating and could lead to spin -wave based devices that are energetically more efficient . Employing the inverse spin Hall effect (ISHE) [12], we observed th e spin-to-charge conversion of these currents inside the Pt film which was detected as a developed micro -voltage . TIG films with thickness ranging from 15 to 60 nm were deposited by rf sputtering from a commercial target with the same nominal composition o f Tm 3Fe5O12, and a purity of 99.9 %. The deposition process was performed at room temperature, in pure argon working pressure of 2.8 mTorr , at a deposition rate of 1.4 nm/min. To improve the crystallinity and the magnetic ordering, the films were post -growth annealed for 8 hrs at 800 ℃ in a quartz tube in flowing oxygen. After the thermal treatment, the films yield ed a magnetization saturation of 100 emu/cm3, and an RMS roughness below 0.1 nm confirmed using a superconducting quantum interference device ( SQUID) and high -resolution X - ray diffraction measurements, as detailed in our recent article [44]. Moreover, the out-of-plane hysteresis loops showed curved shapes which might be related with labyrinth domain structures very common in garnet films with PMA [48]. The next step of sample preparation consist ed of an ex -situ deposition of a 4 nm -thick Pt film over the post -annealed TIG films using the dc sputtering technique . Platinum films were grown under a n Ar gas pressure of 3. 0 mTorr , at room temperature , and a deposition rate of 10 nm/min . The Pt films were not patterned. Ferromagnetic resonance (F MR) is a well - established technique for study of basic magnetic properties such as saturation magnetization, anisotropy energies and magnetic relaxation mechanisms . Furthermore , FMR has been central to the investigat ion of microwave -driven spin- pumping phenomena in FM/NM bilayers [11, 12, 49 ]. First, we used a homemade FMR spectrometer running at a fixed frequency of 9.5 GHz, at room temperature , where t he sample s were placed in the middle of the back wall of a rectangular microwave cavity operating in the TE102 mode with a Q factor of 2500. Field scan spectra of the derivative of the absorption power (𝑑𝑃𝑑𝐻⁄) were acquired by modulating the dc applied field 𝐻⃗⃗ 0 with a small sinusoidal field ℎ⃗ at 100 kHz and using lock-in amplifier detection . The resonance field 𝐻𝑅 was obtained as a function of the polar and azimuthal angles (θH,ϕ𝐻) of the applied magnetic field 𝐻⃗⃗ , as illustrated in Fig. 1(d), where 𝐻⃗⃗ =𝐻⃗⃗ 0+ℎ⃗ and ℎ≪ 𝐻0. The FMR spectra for TIG(t) films are shown in Figs. 1 (a, b and c) for thickness es t = 15, 30 and 60 nm respectively. The spectra were measured for 𝐻 applied along three different polar angles: θ𝐻=0° (blue), θ𝐻≅45° (green ) and θ𝐻= 90° (red) . The complete dependenc e of 𝐻𝑅, for each sample, 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 ( ∆𝐻) for TIG( 15 nm ) with 𝐻 applied along 𝜃𝐻=0°,50°, and 90° are 112 Oe, 74 Oe, and 72 Oe, r espectively. For TIG(30 nm), ∆𝐻 is 82 Oe, 72 Oe, and 65 Oe for 𝜃𝐻=0°,50°, and 90°, respectively. For TIG(60 nm), ∆𝐻 is 72 Oe, 75 Oe, and 61 Oe for 𝜃𝐻= 0°,45°, and 90°, 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), (f) and ( g) show the dependence of the resonance field 𝐻𝑅 with 𝜃𝐻 for different thickness of TIG. The red solid lines are theoretical fits obtained for the FMR condition. Magnetization curves are given in reference [44]. 3 as a function of the polar angle ( 0°≤𝜃𝐻≤90°) are shown in Figs. 1(e, f and g ). For all samples, 𝐻𝑅 was minimum for 𝜃𝐻= 0°, confirming that the perpendicular anisotropy field was strong enough to overcome the demagnetization field. While the films with t = 15 nm and 30 nm exhibited the maximum value of 𝐻𝑅 for 𝜃𝐻=90° (in-plane), the sample with t = 60 nm showed a maximum 𝐻𝑅at 𝜃𝐻~60°. To explain the behavior of 𝐻𝑅 as a function of the out -of-plane angle 𝜃𝐻, 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 density 𝜖 for GGG(111) / TIG(111) films , are: 𝝐=𝝐𝒁+𝝐𝑪𝑨+𝝐𝑫+𝝐𝑼, (1) where 𝝐𝒁 is the Zeeman energy density , 𝝐𝑪𝑨 is the cubic anisotropy energy density for (111) oriented thin films , 𝝐𝑫 is the demagnetization energy density , and 𝝐𝑼 is the uniaxial energy density . Taking into consideration the reference frame shown in Fig. 1(d), each energy density terms can be written as [50]: 𝝐𝒁=−𝑴𝑺𝑯(𝒔𝒊𝒏𝜽𝒔𝒊𝒏𝜽𝑯𝒄𝒐𝒔(𝝓−𝝓𝑯)+𝒄𝒐𝒔𝜽𝒄𝒐𝒔𝜽𝑯), (2) 𝝐𝑪𝑨=𝑲𝟏𝟏𝟐⁄(𝟑−𝟔𝒄𝒐𝒔𝟐𝜽+𝟕𝒄𝒐𝒔𝟒𝜽+ 𝟒 √𝟐𝒄𝒐𝒔𝜽𝒔𝒊𝒏𝟑𝝓𝒔𝒊𝒏𝟑𝜽), (3) 𝝐𝑫+𝝐𝑼=𝟐𝝅(𝑴⃗⃗⃗ ∙𝒆̂𝟑)𝟐−𝑲𝟐⊥(𝑴⃗⃗⃗ ∙𝒆̂𝟑𝑴𝑺 ⁄)𝟐 −𝑲𝟒⊥(𝑴⃗⃗⃗ ∙𝒆̂𝟑𝑴𝑺 ⁄)𝟒, (4) where 𝜽 and 𝝓 are the polar and azimuthal angles of the magnetization vector 𝑴⃗⃗⃗ , 𝑴𝑺 is the saturation magnetization , 𝑲𝟏 is the first order cubic anisotropy constant, and 𝑲𝟐⊥ and 𝑲𝟒⊥ are the first and second order uniaxial anisotropy constants. The uniaxial anisotropy terms come from two sources: growth induced and stress induced anisotropy. The relation between the resonance field and the excitation frequency 𝝎 can be obtained from [51, 52 ]: (𝝎𝜸⁄)𝟐=𝟏 𝑴𝟐𝒔𝒊𝒏𝟐𝜽[𝝐𝜽𝜽𝝐𝝓𝝓−(𝝐𝜽𝝓)𝟐], (5) where 𝜸 is the gyromagnetic ratio. The subscripts indicate partial derivatives with respect to the coordinates, 𝝐𝜽𝜽= 𝝏𝟐𝝐𝝏𝜽𝟐⁄|𝜽𝟎,𝝓𝟎, 𝝐𝝓𝝓=𝝏𝟐𝝐𝝏𝝓𝟐⁄|𝜽𝟎,𝝓𝟎 and 𝝐𝜽𝝓= 𝝏𝟐𝝐𝝏𝜽𝝏𝝓⁄|𝜽𝟎,𝝓𝟎, where 𝜽𝟎,𝝓𝟎 are the equilibrium angles of the magnetization determined by the energy density minimum conditions, 𝝏𝝐𝝏𝜽⁄|𝜽𝟎,𝝓𝟎=𝟎 and 𝝏𝝐𝝏𝝓⁄|𝜽𝟎,𝝓𝟎=𝟎. The best fits to the data obtained with the Eq. (5) are shown in Figs. 1 (e, f and g) by the solid red lines. The main physical parameters extracted from the fits , including the effective magnetization 4 𝝅𝑴𝒆𝒇𝒇, are summarized in Table 1 . Here 4𝝅𝑴𝒆𝒇𝒇= 4𝝅𝑴−𝟐𝑲𝟐⊥/𝑴𝑺 , where the second term is the out-of-plane uniaxial anisotropy field 𝑯𝑼𝟐=𝟐𝑲𝟐⊥/𝑴, also named as 𝑯⊥. It is important to notice that the large negative values of 𝑯𝑼𝟐 were sufficiently strong to saturate the magnetization along the direction perpendicular to the TIG film’s plane , thus overcoming the shape anisotropy . We used the saturation magnetization as the nominal value of 𝑴𝑺= 𝟏𝟒𝟎.𝟎 𝑮. As the thickness of the TIG film increase d, the magnitude of the perpendicular magnetic anisotropy field , 𝑯𝑼𝟐, decrease d due to the relaxation of the induced growth stresses as expected . Table 1. Physical parameters extracted from the theoretical fits of the FMR response of the TIG thin films with thickness 𝑡, performed at room 𝑇 and 9.5 GHz . 4𝜋𝑀𝑒𝑓𝑓 is the effective magnetization, H 1C is the cubic anisotropy f ield, H U2 and HU4 are the first and second order uniaxial anisotropy fields, respectively. H U2 is the out -of- plane uniaxial anisotropy field, also named as 𝐻⊥. TIG film’s thickness t 15 nm 30 nm 60 nm 4𝜋𝑀𝑒𝑓𝑓(G) -979 -799 - 383 𝐻1𝐶=2𝐾1𝑀𝑆⁄ (Oe) 31 26 -111 𝐻𝑈2=4𝜋𝑀𝑒𝑓𝑓−4𝜋𝑀𝑆 (Oe) -2,739 -2,559 -2,143 𝐻𝑈4=2𝐾4⊥𝑀𝑆⁄(Oe) 311 168 432 Fig. 2. (a) Ferromagnetic resonance spectra vs. in -plane applied field 𝐻 for a 30 nm -thick TIG film at frequencies ranging from 2 GHz to 14 GHz and temperature of 300 K, after normalization by background subtraction. (b) Half linewidth ∆𝐻 versus frequency for T IG(30 nm) at 300 K. The Gilbert damping parameter 𝛼 was extracted from the linear fitting of the data. (c) Damping 𝛼 versus temperature 𝑇 for TIG films with 30 nm and 60 nm of thickness. 4 To obtain the Gilbert damping parameter (𝜶) of the TIG thin films, we used the coplanar waveguide technique in the variable temperature insert of a physical property measurement system (PPMS) . A vector network analyzer measure d the amplitude of the forward complex transmission coefficients (𝑺𝟐𝟏) as a function of the in -plane magnetic field for different microwave frequencies (𝒇) and temperatures (𝑻). Figure 2(a) shows the FMR spectra (𝑺𝟐𝟏 versus 𝑯) for TIG(30 nm) corresponding to frequencies ranging from 2 GHz to 14 GHz at 300K , with a microwave power of 0 dBm , after normalization by background subtraction . Fitting each FMR spectra using the Lorentz function, we were able to extract the half linewidth ∆𝑯 for each frequency , as shown in Fig. 2(b). Then, 𝜶 was estimated based on the linear approximation ∆𝑯=∆𝑯𝟎+(𝟒𝝅𝜶𝜸⁄)𝒇, where ∆𝑯𝟎 reflects the contribution of magnetic inhomogeneities , the linear frequency part is caused by the intrinsic Gilbert damping mechanism , and 𝜸 is the gyromagnetic ratio [40]. The same analysis was performed for lower temperature data, and it was extended to TIG(60 nm). Due to the weak magnetization of the 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 , 𝜶=𝟎.𝟎𝟏𝟓 for TIG(60 nm) which is in agree ment with the values reported in the literature [4, 45 ], and it increases by 130 % as 𝑻 goes 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 propagati on through 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 coherent magnetization precession of the TIG inject ed a pure spin current 𝑱𝒔 into the Pt layer, which convert ed as a transverse charge current 𝑱𝒄 by means of the inverse spin Hall effect , expressed as 𝑱 𝒄 = 𝜽𝑺𝑯(𝛔̂ × 𝑱 𝒔), where 𝜽𝑺𝑯 is the spin Hall angle and 𝜎̂ is the spin polarization [55]. As the FMR was excited using the homemade spectrometer at 9.5 GHz , a spin pumping voltage (𝐕𝐒𝐏) was detected between the two silver painted electrodes Fig. 4. Spin Seebeck voltage (V SSE) excited by a thermal gradient in the longitudinal configuration ( 𝛻𝑇∥𝐽 𝑆) at room 𝑇, as shown in (a). (b) Field scan of V SSE for ∆𝑇=20𝐾 and different field polar angles 𝜃𝐻. (c) Field scan of V SSE for ∆𝑇=12,𝜃𝐻=90° and different azimuthal angles 𝜙𝐻. Spin voltage amplitude ∆𝑉𝑆𝑆𝐸 versus (d) 𝜃𝐻, (e) 𝜙𝐻, and (f) ∆𝑇. The solid red lines are theoretical fits of the sine (d), cosine (e) and linear (f) dependence of ∆𝑉𝑆𝑆𝐸 with 𝜃𝐻, 𝜙𝐻 and ∆𝑇, respectively . 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 SP for different microwave powers. (c) Linear dependence of the maximum VSP with the microwave power. ( d) 𝜃𝐻 scan of the charge current (I SP) generated by means of the inverse spin Hall effect in the Pt film. ( e) In-plane field s can of the FMR absorption derivative spectrum for 5 mW. 5 placed on the edges of the Pt film, as illustrated in Fig. 3(a). It is important to not e that when the magnetization vector was perpendicular to the sample ’s plane no V SP was detected. The sample TIG(60 nm)/Pt(4 nm) ha d dimension s of 3 x 4 mm2, and a resistance between the silver electrodes of 𝟒𝟖 𝛀 at zero field. The 𝑽𝑺𝑷 show ed a peak value of 𝟎.𝟖𝟓 𝛍𝐕 in the resonance magnetic field for an incident power of 185 mW , and an in -plane dc magnetic field ( 𝛉𝑯=𝟗𝟎°) as shown in Fig. 3(b). The signal rever sed when the field direction went through a 𝟏𝟖𝟎° rotation . The dependence of 𝑽𝑺𝑷 with the microwave incident power was linear , as shown in Fig. 3(c), whereas the spin pumping charge current ( 𝑰𝑺𝑷=𝑽𝑺𝑷𝑹⁄) had the dependence of 𝑽𝑺𝑷∝𝐬𝐢𝐧𝜽𝑯, showed in Fig. 3(d), for a fixed microwave power of 100 mW. The ratio between the microwave -driven voltage and the microwave power was 4µV/W. We also excited pure spin current s via the spin Seebeck effect (SSE) in TIG(60 nm) /Pt(4 nm) at room 𝑻. 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 subject ed 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 accumulation in the NM layer can be detected by measuring a transversal charge current due to the I SHE. To observe the SSE in our samples , the uncovered GGG surface was placed over a copper plate, acting as a thermal bath at room 𝑻, while the sample’s top was in thermal contact with a 𝟐× 𝟐 𝐦𝐦𝟐 commercial Peltier module through a thermal paste, as illustrate d 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 ( ∆𝑻) between the bottom and top of the sample was measured by a differential thermocouple. The ISHE voltage due to the SSE (𝑽𝑺𝑺𝑬) was detected between the two silver painted electrodes placed on the e dges of the Pt film . The behaviour of 𝑽𝑺𝑺𝑬 by sweeping the dc applied magnetic field ( 𝑯), while ∆𝑻, 𝜽𝑯 and 𝝓𝑯 were kept fixed was investigated . Fixing 𝛟𝑯=𝟎° and varying the magnetic field from out -of-plane (𝜽𝑯=𝟎°) to in -plane along x -direction (𝜽𝑯=𝟗𝟎°), 𝑽𝑺𝑺𝑬 went from zero to its maximum value of 5.5 𝝁𝑽 for ∆𝑻=𝟐𝟎 𝑲, as shown in F ig. 4(b). Around zero field, no matter the value of 𝜽𝑯, the TIG’s film magnetization tended to rely along its out -of-plane easy axis which zeroes 𝑽𝑺𝑺𝑬. For in -plane fields ( 𝜽𝑯=𝟗𝟎°) with ∆𝑻=𝟏𝟐 𝑲, 𝑽𝑺𝑺𝑬 was maximum when 𝛟𝑯=𝟎°, and it was zero for 𝛟𝑯=𝟗𝟎°. The reason 𝑽𝑺𝑺𝑬 went to zero for 𝛟𝑯=𝟗𝟎°, may be attributed to the generated charge flow along the x -direction while the silver electrodes were placed along y -direction , thus not enabling the current detection (see F ig. 4(c)). The analysis of the spin Seebeck amplitude ∆𝑽𝑺𝑺𝑬 versus 𝜽𝑯, 𝛟𝑯 and ∆𝑻 show ed a sine, cosine and linear dependence, respectively as can be seen in Fig. 4(d)-(e), where the red solid lines are theoretical fits . The Spin Seebeck coefficient (SSC) extracted from the linear fit of ∆𝑽𝑺𝑺𝑬 vs. ∆𝑻 was 0.54 𝝁𝑽/K. In conclusion , we used the FMR technique to probe the magnetic anisotropies and the Gilbert damping parameter of the sputtered TIG thin films with perpendicular magnetic anisotropy. The results showed higher resonance fields (> 3.5 kOe) and broader linewidth s (> 60 Oe) when comparing with YIG films at room 𝑇. Thinner TIG films (t = 15 nm and 30 nm) presented a well -defined PMA; on the other hand, the easy axis of thicker TIG film (60 nm) showed a de viation of 30 degrees from normal to the film plane . By numerically adjusting the FMR field dependence with the polar angle , we extracted the effective magnetization , the cubic (H1C) and the out-of-plane uniaxial anisotropy (HU2=H⊥) fields for the three TIG films . The thinnest film presented the highest intensity for H⊥ as expected, even so H⊥ was strong enough to overcome the shape anisotropy and g ave place to a perpendicular magnetic anisotropy in all the three thickness of TIG films . The Gilbert damping parameter (𝛼) for TIG(30 nm) and TIG(60 nm) films were estimated to be ≈ 10-2, by analyzing a set of FMR spectra using the coplanar waveguide technique at various microwave frequencies and temperatures. As 𝑇 went down to 150 K the damping increased monotonically 130 % . Furthermore , spin waves (magnons) were excited in TIG(60 nm)/Pt(4 nm) heterostructure through the spin pumping and spin Seebeck effects , at room 𝑇 and 9.5 GHz . The generated pure spin currents carried by the magnons were converted into charge current s once they reached the Pt film by means of the inverse spin Hall effect . The charge currents were detected as a micro -voltage measured at the edges of the Pt film , and they showed sine and cosine dependence with the polar and azimuthal angles , respectively, of the dc applied magnetic field . This voltage was linearly dependent on the microwave pow er for the SPE, and on the temperature gradient for the SSE. These results confirmed a good spin - mixing conductance in the interface TIG/Pt , and an efficient conversion of pure spin currents into charge currents inside the Pt film , which is crucial for the employment of TIG films with a robust PMA in the development of magnon -based spintronic devices for computing technologies. ACKNOWLEDGEME NTS This research is supported in the USA by Army Research Office (ARO W911NF -19-2-0041 and W911NF -19-2-0015 ), NSF (DMR 1700137), ONR (N00014 -16-1-2657), in Brazil by CAPES (Gilvania Vilela/POS -DOC -88881.120327/2016 - 01), FACEPE (APQ -0565 -1.05/14 and APQ -0707 -1.05/14), CNPq , UPE (PFA/PROGRAD/UPE 04/2017) and FAPEMIG - Rede de Pesquisa em Materiais 2D and Rede de Nanomagnetismo , in Chile by Fondo Nacional de Desarrollo Cientí fico y Tec nológico (FONDECYT) No. 1170723, and in China by the National Natural Science Foundation of China (11974025). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 6 1 P. Pirro, T. Brächer, A. V. Chumak, B. Lägel, C. Dubs, O. Surzhenko, P. Görnert, B. Leven, and B. Hillebrands, Applied Physics Letters 104, 012402 (2014). 2 A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nature Physics 11, 453 (2015). 3 L. J. 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2020-09-22

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 films with PMA present labyrinth domain magnetic structures that enrich the magnetization dynamics, and could be employed in more efficient wave-based logic and memory computing devices. In MI/NM bilayers, where NM being a normal metal providing a strong spin-orbit coupling, the PMA benefits the spin-orbit torque (SOT) driven magnetization's switching by lowering the needed current and rendering the process faster, crucial for developing magnetic random-access memories (SOT-MRAM). In this work, we investigated the magnetic anisotropies in thulium 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 {\alpha}~0.01, 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 micro-voltage generated at the edges of the Pt film. The obtained spin Seebeck coefficient was 0.54 {\mu}V/K, confirming also the high interfacial spin transparency.

Magnon-mediated spin currents in Tm3Fe5O12/Pt with perpendicular magnetic anisotropy

2009.10299v1

1 Control of magnetization dynamics by substrate orientation in YIG thin films Ganesh Gurjar1, Vinay Sharma3, S. Patnaik1*, Bijoy K. Kuanr2,* 1School of Physical Sciences, Jawaharlal Nehru University, New Delhi, INDIA 110067 2Special Centre for Nanosciences, Jawaharlal Nehru University, New Delhi, INDIA 110067 3Morgan State University, Department of Physics, Baltimore, MD, USA 21251 Abstract Yttrium Iron Garnet (YIG) and b ismuth (Bi) substituted YIG (Bi 0.1Y2.9Fe5O12, BYG) films are grown in-situ on single crystalline Gadolinium Gallium Garnet (GGG) substrates [with (100) and (111) orientation s] using pulsed laser deposition (PLD ) technique . As the orientation of the Bi- YIG film changes from (100) to (111) , the lattice constant is enhanced from 12.384 Å to 12.401 Å due to orientation dependent distribution of Bi3+ ions at dodecahedral sites in the lattice cell. Atomic force microscopy (AFM) images show smooth film surfaces with roughness 0.308 nm in Bi-YIG (111) . The change in substrate orientation leads to the modification of Gilbert damping which , in turn, gives rise to the enhancement of ferromagnetic resonance (FMR) line width . The best value s of Gilbert damping are found to be (0.54±0.06 )×10-4, for YIG (100) and (6.27±0.33) ×10-4, for Bi-YIG (111) oriented films . Angle variation ( ) measurements of the H r are also performed, that shows a four -fold symmetry for the resonance field in the (100) g rown film. In addition, the value of effective magnetization (4πM eff) and extrinsic linewidth (ΔH 0) are observed to be dependent on substrate orientation . Hence PLD growth can assist single -crystalline YIG and BY G films with a perfect interface that can be used for spintronics and related device applications. Keyword s: Pulse Laser Deposition, Epitaxial YIG thin films, lattice strain, ferromagnetic resonance, Gilbert damping, inhomogeneous broa dening Corresponding authors: bijoykuanr@mail.jnu.ac.in , spatnaik@mail.jnu.ac.in 2 1. Introduction One of the most widely studied material s for the realization of spintronic devices appears to be the iron garnets , particularly the yttrium iron garnet (YIG , Y3Fe5O12) [1,2] . In thin film form of YIG several potential applications have been envisaged that include spin-caloritronics [3,4] , magneto - optical (MO) devices, and microwave resonators, circulators, and filters [5–8]. The attraction of YIG over other ferroic materials is primarily due to their strong magnet o-crystalline anisotropy and low magnetization damping [2]. Furthermore, towards high frequency applications, YIG’s main advantage s are its electrically insulating behavior along with low ferromagnetic resonance line-width (H) and low Gilbert damping parameter [9–11]. These are important parameters for potential use in high fr equency filters and actuators [12–14]. In this paper, we report optimal growth parameters for pure and Bi -doped YIG on oriented subs trates and identify the conditions suitable for their prospective applications. In literature, YIG is known to be a room temperature ferrimagnetic insulator with a Tc near 560 K [15]. It has a cubic structure (space group Ia3̅d). The y ttrium (Y) ions occupy the dodecahedral 24c sites ( in the Wyckoff notation), two Fe ions at octahedral 16a and three at tetrahedral 24d sites, and oxygen the 96h sites [16,17] . The d site is resp onsible for the ferri magnetic nature of YIG. It is already reported that substitution of Bi in place of Y in YIG leads to substantial improvement in the magneto -optical response [7,18 –25]. It was also observed that MO performance increa ses linearly with Bi/Ce doping concentration [22]. Furthermore, substitution of Bi in YIG (BYG) is documented to provide growth -induced anisotropy that is useful in applications such as magnetic memory and logic devices [26–30]. The study of basic properties of Bi -substituted YIG materials is of great current interest due to their applications in magneto -optical devices , magnon -3 spintronics , and related fields such as caloritronics due to its high uniaxial anisotropy and faraday rotation [21,31 –35]. The structural and magnetic pr operties can be changed via change in Bi3+ concentration in YIG or via choosing a proper substrate orientation. Therefore, t he choice of perfect substrate orientation is crucial for the identification of the growth of Bi substituted YIG thin films. In this work, we have studied the structural and magnetic properties of Bi-substituted YIG [Bi0.1Y2.9Fe5O12 (BYG)] and YIG thin films with two different single crystalline Gadolinium Gallium Garnet (GGG) substrate orientation s: (100) and (111) . The YIG and BYG films of thickness ~150 nm were grown by pulsed laser deposit ion (PLD) method [23,36,37] on top of single -crystalline GGG substrates . The structural and magnetic properties of all grown films were carried out using x -ray diffraction (XRD), surface morphology by atomic force microscopy (AFM) , and magnetic properties via vibrating sample magnetometer (VSM) and ferromagnetic resonance (FMR) techniques. The FMR is the most useful technique to study the magnetization dynamics by measuring the properties of magnetic materials through evaluation of their damping parameter and linewidth . Furthermore, it provides insightful information on the static magnetic properties such as the saturation magnetization and the anisotropy field. FMR is also extremely helpful to study fundamentals of spin wave dynami cs and towards characte rizing the relaxation time and L ande g factor of magnetic material s [11]. 2. Experiment YIG a nd BY G target s were synthesized via the solid -state reaction method . Briefly, y ttrium oxide (Y2O3) and iron oxide (Fe 2O3) powder s were ground for ~14 hours before calcination at 1100 oC. 4 The calcined powders were pressed into pellets and sintered at 1300 oC. Using thes e YIG and BYG targets, thin films of thickness ~150 nm were grown in-situ on (100) - and (111) -oriented GGG substrate s by the PLD technique . The prepared samples have been labeled as YIG (100) , YIG (111) , BYG (100), and BYG (111). GGG substrates were cleaned using acetone and isopropanol. Before deposition, the deposition chamber was thoroughly cleaned and evacuated to a base vacuum of 2 ×10-6 mbar. We have used KrF excimer laser (248 nm), with pulse frequency 10 Hz to ablate the target s at 300mJ energy . During deposition , target to substrate distance, substrate temperature , and oxygen pressure w ere kept at ~4.8 cm, 825 oC, and 0.15 mbar , respectively. Best films were grown at a rate of 6 nm/min . The as -grown thin film s were annealed in-situ for 2 hours at 825 oC and cooled down to 300 oC in the presence of oxygen (0.15 mbar) throughout the process . The structural properties of the thin film were determined by XRD using Cu-Kα radiation (1.5406 Å) and surface morphology as well as the thickness of the film were calculated with atomic force microscopy by WITec Gmb H, Germany . Magnetic properties were studied using a 14 tesla PPMS (Cryogenic) . FMR measurements were carried out by the Vector Network Analyzer ( VNA ) (Keysight , USA) using a coplanar waveguide ( CPW ) in a flip -chip geometry with dc magnetic field applied parallel to the film plan e. 3. Results and Discussion 3.1 Structural properties The room temperature XRD data for the polycrystalline targets of YIG and BYG are plotted in figure 1 (a) and 1 (b) respectively. Rietveld refinement patterns after fitting XRD data are also included in the panel s. XRD peaks are indexed according to the JCPDS card no. ( # 43-0507) . Inset 5 of figure 1 (a) shows crystallographic sub -lattices of YIG that elucidates Fe13+ tetrahedral site, Fe23+ octahedral site , and Y3+ dodecahedral site. Inset (i) of figure 1 (b) shows evidence for successful incorporation of Bi into YIG ; the lattice constant increases when Bi is substituted into YIG due to larger ionic radii of Bi (1.170 Å) as compared with Y (1.019 Å) [19]. From Rietveld refinement we estimate the lattice constant of YIG and BYG to be 12.377 Å [38] and 12.401 Å respectively . Figure 2 (a) and 2 (b) show the XRD pattern of bare (100) and (111) oriented GGG substrates . This is followed by figure 2 (c) & 2(d) for YIG and figure 2 (e) & 2 (f) for BYG as grown thin films. XRD patterns confirm the single -crystalline grow th of YIG and BYG thin film s over GGG substrates . The l attice constant, lattice mismatch (with respect to substrate) , and lattice volume obtain ed from XRD data are listed in Table 1. Lattice cons tant a for the cubic structure is evaluated using the [39]. 𝒂=𝜆√ℎ2+𝑘2+𝑙2 2sin𝜃 (1) Where 𝜆 is the wavelength of Cu -Kα radiation , 𝜃 is the diffraction angle , and [h , k, l ] are the miller indices of the corresponding XRD peak. Lattice misfit (𝛥𝑎 𝑎) is evaluated using equation 2 [24,38] . 𝛥𝑎 𝑎=(𝑎𝑓𝑖𝑙𝑚− 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 ) 𝑎𝑓𝑖𝑙𝑚 100 (2) Where 𝑎𝑓𝑖𝑙𝑚 and 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 are the lattice constant of film and substrate respectively. Lattice constant of pure YIG bulk is 12.377 Å, whereas we have observed a larger value of lattice constants of YIG and BY G films than th at of bulk YIG as shown in T able 1 . Sim ilarly, to these obtained results, a larger value of lattice constants than that of bulk YIG has been reported as well [40–44]. 6 The obtained values of the lattice constant are in agreement with the previous reports [18,21,25,34,45] . In the case of BYG (111), the value of lattice constant slightly increases compared to BYG (100) because the distribution of Bi3+ in the dodecahedral site depends on the orientation of the substrate [28,46] . Inset (ii) of figure 1 (b) shows plane (111) has more contribution of Bi3+ ions [(ionic radius of bismuth (1.170 Å) is larger as compared with YIG (1.019 Å) [19]]. This slight increase in the lattice constant (in 111 direction) implies a more lattice mismatch (or strain ) in BYG films . Positive value of lattice mismatch indicates the slightly larger lattice constant of films (YIG and BYG) were observed as compared to substrates (GGG). We would like to emphasize that lattice plane dependence growth is important to signify the changes in the struc tural and magnetic properties. 3.2 Surface morphology study Room temperature AFM images with roughness are shown in figure 3 (a)-(d). Roughness plays an important role from the application prospective as it is related to Gilbert damping factor α. Lower roughness (root mean square height) is observed for the (111) oriented films of YIG and BYG compared to those grown on (100) oriented substrates. Available literature [61] indicate that roughness would depend more on variation on growth parameters ra ther than on substrate orientation. In this sense further study is needed to clarify substrate dependence of roughness. No significant change in the roughness is observed between YIG and BYG films [38,47] . Table 1 depicts a comparison between the roughne ss measured in YIG and the BY G thin films. 7 3.3 Static magnetization p roperties VSM magnetization measurements were performed at 300 K with magnetic field appl ied parallel to the film plane (in-plane) . Figure 3 (e) and 3 (f) show the magnetization plot s for YIG and BYG respectively after careful subtraction of paramagnetic contr ibution that is assigned to the substrate. The m easured saturation magnetization ( 4πM S) data are given in Table 1 which are in general agreement with the reported values [11,40,48] . Not much change in the measured 4πM S value of YIG and Bi -YIG films are observed . The ferrimagnetism nature of YIG arises from super - exchange interaction between the non -equivalent Fe3+ ions at octahedral and tetrahedral sites [49]. Bismuth located at dodecahedral site does not affect the tetrahedral and octahedral Fe3+ ions. So, Bismuth does not show a significant change in saturation magnetization at room temperature. I t is reported in literature th at Bi addition leads to increase in Curie temperature, so in t hat sense there is an decreasing trend in saturation magnetization in BYG films in contrast to YIG films [50,51] . Error bars in saturation magnetization relate to uncertainty in sample volume. 3.4 Ferromagnetic r esonance properties The FMR absorption spectroscopy is shown in figure 4. These measurements were performed at room temperature . The external dc magnetic field was appli ed parallel to the plane of the film . Lorentzian fit of the calibrated experimental data are used to calculate t he FMR linewidth (∆H) and resonance magnetic field (H r). From the e nsemble of all the FMR data at different resonance frequencies (f = 1 GHz -12 GHz ), we have calculated the gyromagnetic ratio (γ) , effective magnetization field ( 4𝜋𝑀𝑒𝑓𝑓) from the fitting of Kittel’s in-plane equation [52]. 8 In general, t he uniform precession of magnetization can be described by the Landau -Lifshitz - Gilbert (LLG) equation of motion; 𝜕𝑀⃗⃗ 𝜕𝑡=−𝛾(𝑀⃗⃗ ×𝐻⃗⃗ 𝑒𝑓𝑓)+𝐺 𝛾𝑀𝑠2[𝑀⃗⃗ ×𝜕𝑀⃗⃗ 𝜕𝑡] (3) Here, t he first term corresponds to the precessional torque in the effective magnetic field and the second term is the Gilbert damping torque. The gyromagnetic ratio is given by 𝛾=𝑔𝜇𝐵/ℏ , where 𝑔 is the Lande’s factor, 𝜇𝐵 is Bohr magnetron and ℏ is the Planck’s constant. Similarly, 𝐺=𝛾𝛼𝑀𝑠 is related to the intrinsic relaxation rate in the nanocomposites and 𝛼 represents the Gilbert damping constant. Ms (or 4πMs) is the saturation magnetization. It can be shown that t he solution for in -plane resonance frequency can be written as; 𝑓𝑟=𝛾′√(𝐻𝑟)(𝐻𝑟+4𝜋𝑀𝑒𝑓𝑓) (4), Where 𝛾′=𝛾/2𝜋, 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛𝑖 is the effective field and 𝐻𝑎𝑛𝑖=2𝐾1 𝑀𝑠 is the anisotropy field. Following through, we have obtained Gilbert damping parameter (α) and inhomogeneous broadening (∆H 0) linewidth from the fitting of Landau –Lifshitz –Gilbert equation (LLG) [53] 𝛥𝐻(𝑓)=𝛥𝐻0+4𝜋𝛼 √3𝛾𝑓 (5) Derived parameters from the FMR study are listed in T able 2 . The obtai ned Gilbert damping (α) is in agreement with the reported thin films used for the study of spin-wave propagation [2,27,54,55] . In the case of YIG no t much change in the value of α is seen . However , a substantial increase is observed in case of BYG with (111) orientation. Qualitatively this could be assigned to 9 the presence of Bi3+ ions which induce s spin-orbit coupling (SOC) [56–58] and also due to electron scattering inside the lattice as lattice mismatch (or strain ) increases [59]. We have seen more distribution of Bi3+ ions along (111) planes ( see inset (ii) of figure 1 (b) ) and also slightly larger lattice mismatch in BYG (111) from our XRD results. These results also explain higher value of Gilbert damping and ΔH 0 in case of BYG (111). The change in 4𝜋𝑀𝑒𝑓𝑓 could be attributed to uniaxial in -plane magnetic anisotropy . This is because no change in 4πM S is observed from magnetization measurements and 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛𝑖 [38,40,60] . The uniaxial inplane magnetic anisotropy is induced due to lattice mismatch between films and GGG substrates [38,40] . The calculated gyromagnetic ratio (γ) and ΔH 0 are also included in Table 2 . The magnitude of ΔH 0 is close to reported values for same substrate orientation [38]. In summary we find that YIG with (100) orientation yields lowest damping fact or and extrinsic contribution to linewidth. These are the r equired optimal parameters for spintronic s application with high spin diffusion length. However, MOKE signal is usual ly very low in bare YIG thin films because of its lower magnetic anisotropy and strain [61]. But previous reports suggest that magnetic anisotropy and magnetic domains formation can be achieved in YIG system by doping rare earth materials like Bi and Ce [18,61]. We have shown that anisotropic characteristic with Bi doping in YIG is more pron ounced along <111> direction which can lead to the enhanced MOKE signal in Bi -YIG films on <111> substrate. We have also recorded polar angle () data of resonance field ( Hr) versus magnetic field (H) at frequency 12 GHz for the BYG (100) and BYG (111) films (figure 5 (c) & 5(d) respectively where inset shows the azimuthal angle ( ) variation of Hr measured at frequency of 3 GHz ). The data are fitted with modified Kittel equation . From figure 5 (c) & (d), we can see that Hr increases up to 2.5 kOe in BYG (100) and 3.0 kOe in BYG (111) by varying the direction of H from 0 to 90 10 degree with respect to sample surface (inset of Fig 5 (a)) . Obtained parameters from angular variation of FMR magnetic field H r (θH) are listed in the inset of figure 5 (c) & (d) . From variation data (by varying the direction of H from 0 to 18 0 degree with respect to sample edge (Fig 5 (a) Inset ), we see clear four -fold and two -fold in-plane anisotropy in BYG (100) and BYG (111) films [61,62] . This further consolidates single -crystalline characteristics of our films. The change observed in Hr with respect to variation is 79.52 Oe in BYG (100) (H=0 to 45) and 19.25 Oe in BYG (111) ( H=0 to 45). Thus, during in-plane rotation, higher change in FMR field is observed along (100) orientation . 4. Conclusion In conclusion , we have grown high quality YIG and B i-YIG thin film s on GGG substrates with (100) and (111) orientation . The films were gr own by pulsed laser deposition. The optimal parameters i.e. target to substrate distance, substrate temperature, and oxygen pressure are determined to be ~ 4.8 cm, 825 oC, and 0.15 mbar, respectively. The as grown thin films have smooth surfaces and are found to be phase pure from AFM and XRD characterizations. From FMR measurements , we have found lower value of damping parameter in (100) YIG that indicates higher spin diffusion length for potential spintronics application. On the other -hand bismuth incorporation to YIG leads to dominance of anisotropic characteristics that augers well for application in magnetic bubble memory and magneto -optic devices . The enhanced value of α in Bi-YIG films is ascribed to the spin orbit coupled Bi3+ ions. We also ta bulate the values of magnetic parameters such as linewidth ( ∆H0), gyromagnetic ratio ( γ), and effective magnetization 4𝜋𝑀𝑒𝑓𝑓 with respect to substrate orientation. Unambiguous four-fold in -plane anisotropy is observed in (100) oriented films. We find high-quality magnetization dynamics and lower Gilbert 11 damping parameter is possible in Bi-YIG grown on (111) GGG in conjunction with enhanced magnetic anisotropy. The choice of perfect substrate orientation is therefore found to be crucial for the growth of YIG and Bi-YIG thin films for high frequency applications. Acknowledgments This work is supported by the MHRD -IMPRINT grant, DST (SERB, AMT , and PURSE -II) grant of Govt. of India. Ganesh Gurjar acknowledges CSIR, New Delhi for financial support . We acknowledge AIRF, JNU for access of PPMS facility. References [1] S.A. Manuilov, C.H. Du, R. Adur, H.L. Wang, V.P. Bhallamudi, F.Y. Yang, P.C. Hammel, Spin pumping from spinwaves in thin film YIG, Appl. Phys. Lett. 107 (2015) 42405. [2] C. Hauser, T. Richter, N. Homonnay, C. Eisenschmidt, M. Qaid, H. Deniz, D. Hesse, M. Sawicki, S.G. Ebbinghaus, G. 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Lett. 111 (2017) 192404. 20 List of Tables with caption Table 1: Lattice and magnetic p arameters obtained from XRD , AFM and VSM. S. No. Sample Lattice constant (Å) Lattice Mismatch (%) Lattice volume (Å3) Roughness (nm) 4πM S (Gauss) 1. YIG (100) 12.403 0.42 1907.81 0.801 1670.15±83.51 2. YIG (111) 12.405 0.40 1909.02 0.341 1654.06±82.70 3. BYG (100) 12.384 0.36 1899.11 0.787 1788.50±89.43 4. BYG (111) 12.401 0.65 1906.93 0.308 1816.31±90.82 Table 2: Damping and linewidth p arameters obtained from FMR S. No. Sample α (10-4) ΔH 0 (Oe) 4πM eff (Oe) γ' (GHz/kOe) 1. YIG (100) (0.54±0.06) 26.24±0.10 1938 .60±37.57 2.89±0.01 2. YIG (111) (1.05±0.13) 26.51±0.21 2331 .38±65.78 2.86±0.02 3. BYG (100) (1.66±0.10) 26.52 ±0.17 1701.67±31.87 2.89±0.11 4. BYG (111) (6.27±0.33) 29.28 ±0.62 2366 .85±62.60 2.86±0.02 21 Figure Captions Figure 1: XRD with Rietveld refinement pattern of (a) YIG target ( inset shows crystallographic sub-lattices, Fe 13+ tetrahedral site, Fe 23+ octahedral site and Y3+ dodecahedral site ) (b) BYG target (inset (i) shows effect of Bi doping into YIG , inset (ii) shows contribution of the Bi3+ along the (100) and (111) planes ). Figure 2: XRD pattern of (a) GGG (100) , (b) GGG (111), (c) YIG (100) , (d) YIG (111) , (e) BYG (100) , and (f) BYG (111) . Figure 3: AFM images of (a) YIG (100), (b ) YIG (111) , (c) BYG (100), (d) BYG (111) and static magnetization graph of (e ) YIG (100), YIG (111); and (f) BY G (100), BYG (111) . Figure 4: FMR absorption spectra of (a) YIG (100), (b) YIG (111), (c) BYG (100), and (d) BYG (111). Figure 5: (a) FMR magnetic field Hr is plotted as a function of frequency f. Experiment data fitted with Kittel equation for YIG and BYG oriented films. Inset shows how the applied field angle is measured from sample surface (b) Frequency -dependent FMR linewidth data fitted with LLG equation for YIG and BYG oriented films. Inset shows the magnified version to illustrate the effect of Bi doping in YIG . (c) and (d) show angular variation of FMR magnetic field (Hr (θH)) fitted with modified Kittel equation at 12 GHz frequency for BYG (100) and BYG (111) films . Insets show the FMR magnetic field (H r) as a function of azimuthal angle ( ). 22 Figure 1 23 Figure 2 24 Figure 3 25 Figure 4 26 Figure 5

2021-09-13

Yttrium Iron Garnet (YIG) and bismuth (Bi) substituted YIG (Bi0.1Y2.9Fe5O12, BYG) films are grown in-situ on single crystalline Gadolinium Gallium Garnet (GGG) substrates [with (100) and (111) orientations] using pulsed laser deposition (PLD) technique. As the orientation of the Bi-YIG film changes from (100) to (111), the lattice constant is enhanced from 12.384 {\AA} to 12.401 {\AA} due to orientation dependent distribution of Bi3+ ions at dodecahedral sites in the lattice cell. Atomic force microscopy (AFM) images show smooth film surfaces with roughness 0.308 nm in Bi-YIG (111). The change in substrate orientation leads to the modification of Gilbert damping which, in turn, gives rise to the enhancement of ferromagnetic resonance (FMR) line width. The best values of Gilbert damping are found to be (0.54)*10-4, for YIG (100) and (6.27)*10-4, for Bi-YIG (111) oriented films. Angle variation measurements of the Hr are also performed, that shows a four-fold symmetry for the resonance field in the (100) grown film. In addition, the value of effective magnetization (4{\pi}Meff) and extrinsic linewidth ({\Delta}H0) are observed to be dependent on substrate orientation. Hence PLD growth can assist single-crystalline YIG and BYG films with a perfect interface that can be used for spintronics and related device applications.

Control of magnetization dynamics by substrate orientation in YIG thin films

2109.05901v1

Spin transport and tunable Gilbert damping in a single-molecule magnet junction Milena Filipović,1Cecilia Holmqvist,1Federica Haupt,2and Wolfgang Belzig1 1Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany 2Institut für Theorie der Statistischen Physik, RWTH Aachen, D-52056 Aachen, Germany (Dated: October 24, 2019) We study time-dependent electronic and spin transport through an electronic level connected to two leads and coupled with a single-molecule magnet via exchange interaction. The molecular spin is treated as a classical variable and precesses around an external magnetic field. We derive expressions for charge and spin currents by means of the Keldysh nonequilibrium Green’s functions techniqueinlinearorderwithrespecttothetime-dependentmagneticfieldcreatedbythisprecession. The coupling between the electronic spins and the magnetization dynamics of the molecule creates inelastic tunneling processes which contribute to the spin currents. The inelastic spin currents, in turn, generate a spin-transfer torque acting on the molecular spin. This back-action includes a contribution to the Gilbert damping and a modification of the precession frequency. The Gilbert damping coefficient can be controlled by the bias and gate voltages or via the external magnetic field and has a nonmonotonic dependence on the tunneling rates. PACS numbers: 73.23.-b, 75.76.+j, 85.65.+h, 85.75.-d I. INTRODUCTION Single-molecule magnets (SMMs) are quantum mag- nets, i.e., mesoscopic quantum objects with a perma- nent magnetization. They are typically formed by paramagnetic ions stabilized by surrounding organic ligands.1SMMs show both classical properties such as magnetization hysteresis2and quantum properties such as spin tunneling,3coherence,4and quantum phase interference.2,5They have recently been in the center of interest2,6,7in view of their possible applications as in- formation storage8and processing devices.9 Currently, a goal in the field of nanophysics is to control and manipulate individual quantum systems, in particular, individual spins.10,11Some theoretical works have investigated electronic transport through a molecu- larmagnetcontactedtoleads.12–19Inthiscase, thetrans- port properties are modified due to the exchange inter- action between the itinerant electrons and the SMM,20 making it possible to read out the spin state of the molecule using transport currents. Conversely, the spin dynamics and hence the state of an SMM can also be controlled by transport currents. Efficient control of the molecule’s spin state can be achieved by coupling to fer- romagnetic contacts as well.21 Experiments have addressed the electronic trans- port properties through magnetic molecules such as Mn12and Fe 8,22,23which have been intensively stud- ied as they are promising candidates for memory devices.24Various phenomena such as large conduc- tance gaps,25switching behavior,26negative differen- tial conductance, dependence of the transport on mag- netic fields and Coulomb blockades have been exper- imentally observed.22,23,27,28Experimental techniques, including, for instance, scanning tunneling microscopy (STM),22,23,29–31break junctions,32and three-terminal devices,22,23,27have been employed to measure elec- tronic transport through an SMM. Scanning tunnelingspectroscopy and STM experiments show that quantum properties of SMMs are preserved when deposited on substrates.29The Kondo effect in SMMs with magnetic anisotropyhasbeeninvestigatedboththeoretically14and experimentally.33,34It has been suggested35and experi- mentally verified36that a spin-polarized tip can be used to control the magnetic state of a single Mn atom. In some limits, the large spin Sof an SMM can be treated as a classical magnetic moment. In that case, the spin dynamics is described by the Landau-Lifshitz- Gilbert (LLG) equation that incorporates effects of ex- ternal magnetic fields as well as torques originating from damping phenomena.37,38In tunnel junctions with mag- netic particles, LLG equations have been derived using perturbativecouplings39,40andthenonequilibriumBorn- Oppenheimer approximation.16Current-induced magne- tization switching is driven by a generated spin-transfer torque (STT)41–44as a back-action effect of the elec- tronic spin transport on the magnetic particle.16,45–47 A spin-polarized STM (Ref. 36) has been used to experimentally study STTs in relation to a molecular magnetization.48This experimental achievement opens new possibilities for data storage technology and appli- cations using current-induced STTs. The goal of this paper is to study the interplay be- tween electronic spin currents and the spin dynamics of an SMM. We focus on the spin-transport properties of a tunnel junction through which transport occurs via a single electronic energy level in the presence of an SMM. The electronic level may belong to a neighboring quan- tum dot (QD) or it may be an orbital related to the SMM itself. The electronic level and the molecular spin are coupled via exchange interaction, allowing for inter- actionbetweenthespinsoftheitinerantelectronstunnel- ing through the electronic level and the spin dynamics of the SMM. We use a semiclassical approach in which the magnetization of the SMM is treated as a classical spin whose dynamics is controlled by an external magneticarXiv:1211.3611v2 [cond-mat.mes-hall] 22 Jul 20132 field, while for the electronic spin and charge transport we use instead a quantum description. The magnetic field is assumed to be constant, leading to a precessional motion of the spin around the magnetic field axis. The electronic level is subjected both to the effects of the molecular spin and the external magnetic field, generat- ingaZeemansplitofthelevel. Thespinprecessionmakes additional channels available for transport, which leads to the possibility of precession-assisted inelastic tunnel- ing. During a tunnel event, spin-angular momentum may be transferred between the inelastic spin currents and the molecular spin, leading to an STT that may be used to manipulate the spin of the SMM. This torque includes the so-called Gilbert damping , which is a phenomenolog- ically introduced damping term of the LLG equation,38 and a term corresponding to a modification of the pre- cession frequency. We show that the STT and hence the SMM’s spin dynamics can be controlled by the external magnetic field, the bias voltage across the junction, and the gate voltage acting on the electronic level. The paper is organized as follows: We introduce our model and formalism based on the Keldysh nonequilib- rium Green’s functions technique49–51in Sec. II, where we derive expressions for the charge and spin currents in linear order with respect to a time-dependent magnetic field and analyze the spin-transport properties at zero temperature. In Sec. III we replace the general magnetic field of Sec. II by an SMM whose spin precesses in an external constant magnetic field, calculate the STT com- ponents related to the Gilbert damping, and the modifi- cationoftheprecessionfrequency, andanalyzetheeffects of the external magnetic field as well as the bias and gate voltages on the spin dynamics. Conclusions are given in Sec. IV. II. CURRENT RESPONSE TO A TIME DEPENDENT MAGNETIC FIELD A. Model and Formalism For the sake of clarity, we start by considering a junc- tion consisting of a noninteracting single-level QD cou- pled with two normal, metallic leads in the presence of an external, time-dependent magnetic field (see Fig. 1). The leads are assumed to be noninteracting and unaffected by the external field. The total Hamiltonian describing the junction is given by ^H(t) = ^HL;R+^HT+^HD(t). The Hamiltonian of the free electrons in the leads reads ^HL;R=P k;;2fL;Rgk^cy k^ck, wheredenotes the left (L) or right ( R) lead, whereas the tunnel cou- pling between the QD and the leads can be written as ^HT=P k;;2L;R[Vk^cy k^d+V k^dy ^ck]. The spin- independent tunnel matrix element is given by Vk. The operators ^cy k(^ck)and ^dy (^d)are the creation (an- nihilation) operators of the electrons in the leads and the QD, respectively. The subscript =";#denotes eVB(t) FIG. 1: (Color online) A quantum dot with a single electronic level0coupled to two metallic leads with chemical potentials LandRinthepresenceofanexternaltime-dependentmag- netic field~B(t). The spin-transport properties of the junction aredeterminedbythebiasvoltage eV=LR, theposition of the level 0, the tunnel rates LandR, and the magnetic field. the spin-up or spin-down state of the electrons. The electronic level 0of the QD is influenced by an ex- ternal magnetic field ~B(t)consisting of a constant part ~Bcand a time-dependent part ~B0(t). The Hamiltonian of the QD describing the interaction between the elec- tronic spin ^~ sand the magnetic field is then given by ^HD(t) = ^Hc D+^H0(t), where the constant and time- dependent parts are ^Hc D=P 0^dy ^d+gB^~ s~Bcand ^H0(t) =gB^~ s~B0(t). The proportionality factor gis the gyromagnetic ratio of the electron and Bis the Bohr magneton. The average charge and spin currents from the left lead to the electronic level are given by IL(t) =qd dt^NL =qi ~ ^H;^NL ;(1) where ^NL=P k;;0^cy kL(^)0^ck0Lis the charge and spin occupation number operator of the left contact. The index= 0corresponds to the charge current, while =x;y;zindicates the different components of the spin- polarized current. The current coefficients qare then q0=eandq6=0=~=2. In addition, it is useful to define the vector ^= (^1;^~ ), where ^1is the identity operator and ^~ consists of the Pauli operators with ma- trix elements (^~ )0. Using the Keldysh nonequilibrium Green’s functions technique, the currents can then be ob- tained as50,51 IL(t) =2q ~Re dt0Tr ^[^Gr(t;t0)^< L(t0;t)(2) +^G<(t;t0)^a L(t0;t)] ; where ^Gr;a;<are the retarded, advanced, and lesser Green’s functions of the electrons in the QD with the ma- trix elements Gr;a 0(t;t0) =i(tt0)hf^d(t);^dy 0(t0)gi andG< 0(t;t0) =ih^dy 0(t0)^d(t)i, while ^r;a;< L(t;t0) are self-energies from the coupling between the QD and the left lead. Their nonzero matrix elements3 are diagonal in the electronic spin space with re- spect to the basis of eigenstates of ^sz, given by r;a;< L(t;t0) =P kVkLgr;a;< kL(t;t0)V kL. The Green’s func- tionsgr;a;< kL(t;t0)are the retarded, advanced and lesser Green’s functions of the free electrons in the left lead. The retarded Green’s functions ^Gr 0of the electrons in the QD, in the presence of the constant magnetic field ~Bc, are found using the equation of motion technique,52 while the lesser Green’s functions ^G< 0are obtained from the Keldysh equation ^G< 0=^Gr 0^<^Ga 0, where multipli- cation implies internal time integrations.51The time- dependent part of the magnetic field can be expressed as~B0(t) =P !(~B!ei!t+~B !ei!t), where~B!is a com- plex amplitude. This magnetic field acts as a time- dependent perturbation that can be expressed as ^H0(t) =P !(^H!ei!t+^Hy !ei!t), where ^H!is an operator in the electronic spin space and its matrix representaton in the basis of eigenstates of ^szis given by ^H!=gB 2 B!zB!xiB!y B!x+iB!yB!z :(3) Applying Dyson’s expansion, analytic continuation rules and the Keldysh equation,51one obtains a first-order ap- proximation of the Green’s functions describing the elec- trons in the QD that can be written as ^Gr^Gr 0+^Gr 0^H0^Gr 0; (4) ^G<^Gr 0^<^Ga 0+^Gr 0^H0^Gr 0^<^Ga 0+^Gr 0^<^Ga 0^H0^Ga 0: The expression for the currents in this linear approxima- tion is given by IL(t) =2q ~Re Tr ^[^Gr 0^< L+^G< 0^a L (5) +^Gr 0^H0^Gr 0^< L+^Gr 0^H0^G< 0^a L+^G< 0^H0^Ga 0^a L] : Eq. (5) is then Fourier transformed in the wide- band limit, in which the level width function, () = 2 Imfr()g, is constant, Refr()g= 0, and one can hence write the retarded self-energy originating from the dot-lead coupling as r;a() =i=2. From this trans- formation, one obtains IL(t) =Idc L+X ![IL(!)ei!t+I L(!)ei!t]:(6) Using units in which ~= 1, the dc part of the currents51Idc Land the time-independent complex com- ponentsIL(!)are given by Idc L=qd LR [fL()fR()] Tr Imf^^Gr 0()g(7) and IL(!) =iqd 2LR n [fL()fR()] (8) Trf^[^Gr 0(+!)^H!^Gr 0() + 2iImf^Gr 0()g^H!^Ga 0(!)]g +X =L;R R[f(!)fL()] Tr[^^Gr 0()^H!^Ga 0(!)]o :In the above expressions, f() = [e()=kBT+ 1]1is the Fermi distribution of the electrons in lead , where kBis the Boltzmann constant. The retarded Green’s function ^Gr 0()is given by ^Gr 0() = [0r() (1=2)gB^~ ~Bc]1.16 The linear response of the spin current with respect to the applied time-dependent magnetic field can be ex- pressed in terms of complex spin-current susceptibilities defined as L j(!) =@IL(!) @B!j; j =x;y;z: (9) The complex components IL(!)are conversely given by IL(!) =P jL j(!)B!j. By taking into account that @^H!=@B!j= (1=2)gB^jand using Eq. (8), the current susceptibilities can be written as L j(!) =iqgBd 4LR n [fL()fR()](10) Trf^[^Gr 0(+!)^j^Gr 0() + 2iImf^Gr 0()g^j^Ga 0(!)]g +X  R[f(!)fL()]Tr[^^Gr 0()^j^Ga 0(!)]o : The components obey L j(!) =L j(!). In other words, they satisfy the Kramers-Kronig relations53that can be written in a compact form as L j(!) =1 iP1 1L j() !d; (11) withPdenoting the principal value. For anyi;j;k =x;y;zsuch thatj6=kandj;k6= i, whereiindicates the direction of the constant part of the magnetic field ~Bc=Bc~ ei, the complex current susceptibilities satisfy the relations L jj(!) =L kk(!) (12) andL jk(!) =L kj(!); (13) in addition to Eq. (11). The other nonzero compo- nents areL 0i(!)andL ii(!). In the absence of a constant magnetic field, the only nonvanishing components obey L xx(!) =L yy(!) =L zz(!). Finally, the average value of the electronic spin in the QD reads~ s(t) =h^~ s(t)i= (1=2)P 0~ 0h^dy (t)^d0(t)i= (i=2)P 0~ 0^G< 0(t;t)and the complex spin suscep- tibilities are defined as s ij(!) =@si(!) @B!j: (14) They represent the linear responses of the electronic spin componentstotheappliedtime-dependentmagneticfield and satisfy the relations similar to Eqs. (11), (12), and (13) given above.4 B. Analysis of the spin and current responses We start by analyzing the transport properties of the junction at zero temperature in response to the exter- nal time-dependent magnetic field ~B(t). The constant component of the magnetic field ~Bcgenerates a Zeeman split of the QD level 0, resulting in the levels ";#, where ";#=0gBBc=2in this section. The time-dependent periodic component of the magnetic field ~B0(t)then cre- ates additional states, i.e., sidebands, at energies "! and#!(see Fig. 2). These Zeeman levels and side- bandscontributetotheelastictransportpropertiesofthe junction when their energies lie inside the bias-voltage window ofeV=LR. However, energylevelsoutsidethebias-voltagewindow may also contribute to the electronic transport due to in- elastic tunnel processes generated by the time-dependent magnetic field. In these inelastic processes, an electron transmitted from the left lead to the QD can change its energy by!and either tunnel back to the left lead or out into the right lead. If this perturbation is small, as is assumed in this paper where we consider first-order cor- rections, the transport properties are still dominated by the elastic, energy-conserving tunnel processes that are associated with the Zeeman levels. The energy levels of the QD determine transport prop- erties such as the spin-current susceptibilities and the spin susceptibilities, which are shown in Fig. 3. The imaginary and real parts of the susceptibilities are plot- ted as functions of the frequency !in Figs. 3(a) and 3(c). In this case, the position of the unperturbed level 0is symmetric with respect to the Fermi surfaces of the leads and a peak or step in the spin-current and spin suscepti- bilities appears at a value of !, for which an energy level is aligned with one of the lead Fermi surfaces. In Figs. 3(b) and 3(d), the susceptibilities are instead plotted as functions of the bias voltage, eV. Here, each peak or step in the susceptibilities corresponds to a change in the number of available transport channels. The bias volt- age is applied in such a way that the energy of the Fermi surface of the right lead is fixed at R= 0while the en- ergy of the left lead’s Fermi surface is varied according toL=eV. III. SPIN-TRANSFER TORQUE AND MOLECULAR SPIN DYNAMICS A. Model with a precessing molecular spin Now we apply the formalism of the previous section to the case of resonant tunneling through a QD in the presence of a constant external magnetic field and an SMM [see Fig. 4(a)]. An SMM with a spin Slives in a(2S+ 1)-dimensional Hilbert space. We assume that the spinSof the SMM is large and neglecting the quan- tum fluctuations, one can treat it as a classical vector µLµR=00↑+ω↑↑−ω↓+ω↓−ω↓FIG. 2: (Color online) Sketch of the electronic energy levels of the QD in the presence of a time-dependent magnetic field. In a static magnetic field, the electronic level 0(solid black line) splits into the Zeeman levels ";#(solid red and blue lines). If the magnetic field in addition to the static component also in- cludes a time-dependent part with a characteristic frequency !, additionallevelsappearatenergies "!(dottedredlines) and#!(dotted blue lines). Hence, there are six channels available for transport. whose end point moves on a sphere of radius S. In the presence of a constant magnetic field ~Bc=Bc~ ez, the molecular spin precesses around the field axis according to~S(t) =S?cos(!Lt)~ ex+S?sin(!Lt)~ ey+Sz~ ez, where S?is the projection of ~Sonto thexyplane,!L=gBBc is the Larmor precession frequency and Szis the projec- tion of the spin on the zaxis [see Fig. 4(b)]. The spins of the electrons in the electronic level are coupled to the spin of the SMM via the exchange interaction J. The contribution of the external magnetic field and the pre- cessional motion of the SMM’s spin create an effective time-dependent magnetic field acting on the electronic level. The Hamiltonian of the system is now given by ^H(t) = ^HL;R+^HT+^HD(t) +^HS, where the Hamiltonians ^HL;R and ^HTare the same as in Sec. II. The Hamiltonian ^HS=gB~S~Bcrepresents the interaction of the molecu- lar spin~Swith the magnetic field ~Bcand consequently does not affect the electronic transport through the junc- tion but instead contributes to the spin dynamics of the SMM. The Hamiltonian of the QD in this case is given by ^HD(t) =^Hc D+^H0(t). Here, ^Hc D=P 0^dy ^d+gB^~ s~Bc e is the Hamiltonian of the electrons in the QD in the pres- ence of the constant part of the effective magnetic field, given by~Bc e= Bc+J gBSz ~ ez. The second term of the QD Hamiltonian, ^H0(t) =gB^~ s~B0 e(t), represents the in- teraction between the electronic spins of the QD, ^~ s, and the time-dependent part of the effective magnetic field, given by~B0 e(t) =JS? gB cos(!Lt)~ ex+ sin(!Lt)~ ey . The time-dependent effective magnetic field can be rewritten as~B0 e(t) =~B!Lei!Lt+~B !Lei!Lt, where~B!Lconsists of the complex amplitudes B!Lx=JS?=2gB,B!Ly=5 0.00.51.01.52.02.5-6-4-2024 w@e0DcijL@10-3mBD mL=e+wHaL mL=e¯+w ImczzLReczzLImcxyLRecxyLImcxxLRecxxL 0.00.51.01.52.0-1012 eV@e0DcijL@10-2mBD e¯-wHbL e¯ e¯+w e-w e e+w ImczzLReczzLImcxyLRecxyLImcxxLRecxxL 0.00.51.01.52.02.5-2-101 w@e0Dcijs@10-2mBe0-1DHcL ImczzsReczzsImcxysRecxysImcxxsRecxxs 0.00.51.01.52.0-4-20 eV@e0Dcijs@10-1mBe0-1DHdL ImczzsReczzsImcxysRecxysImcxxsRecxxs FIG. 3: (Color online) (a) Frequency and (b) bias-voltage dependence of the spin-current susceptibilities. (c) Frequency and (d) bias-voltage dependence of the spin susceptibilities. In (a) and (c), the chemical potential of the left lead is L= 20, while in (b) and (d) the frequency is set to != 0:160. All plots are obtained at zero temperature with ~Bc=Bc~ ez, and the other parameters set to R= 0; "= 1:480; #= 0:520; = 0:020, and L= R= 0:010. iJS?=2gB, andB!Lz= 0. The time-dependent pertur- bation can then be expressed as ^H0(t) = ^H!Lei!Lt+ ^Hy !Lei!Lt, where ^H!Lis an operator that can be written, using Eq. (3) and the above expressions for B!Li, as ^H!L=JS? 2 0 1 0 0 : (15) The time-dependent part of the effective magnetic field creates inelastic tunnel processes that contribute to the currents. The in-plane components of the spin current fulfill ILx(!L) =iILy(!L) (16) =JS? 2gB[L xx(!L) +iL xy(!L)]; where~Bcis replaced by ~Bc e. Thezcomponent vanishes to lowest order in H0(t).54Therefore, the inelastic spin current has a polarization that precesses in the xyplane. The inelastic spin-current components, in turn, exert an STT (Refs. 41-44) on the molecular spin given by ~T(t) =[~IL(t) +~IR(t)]; (17)thus contributing to the dynamics of the molecular spin through _~S(t) =gB~Bc~S(t) +~T(t): (18) Using expressions (6), (8), and (15), the torque of Eq. (17) can be calculated in terms of the Green’s functions ^Gr 0()and ^Ga 0()as Ti(t) =JS? 2d 2X  [f(!L)f()] (19) Imf(^i)#"Gr 0;""()Ga 0;##(!L)ei!Ltg; with=L;R. Here (^i)#",Gr 0;""(), andGa 0;##()are matrix elements of ^i,^Gr 0()and ^Ga 0()with respect to the basis of eigenstates of ^sz. This STT can be rewritten in terms of the SMM’s spin vector as ~T(t) = S_~S(t)~S(t) +_~S(t) + ~S(t):(20) The first term in this back-action gives a contribution to6 zBceVS⊥(t)(a) (b) FIG. 4: (Color online) (a) Resonant tunneling in the pres- ence of an SMM and an external, constant magnetic field. The electronic level of Fig. 1 is now coupled with the spin of an SMM via exchange interaction with the coupling con- stantJ. The dynamics of the SMM’s spin ~Sis controlled by the external magnetic field ~Bcthat also affects the electronic level. (b)PrecessionalmotionoftheSMM’sspininaconstant magnetic field ~Bcapplied along the zaxis. the Gilbert damping, characterized by the Gilbert damp- ing coefficient . The second term acts as an effective constant magnetic field and changes the precession fre- quency of the spin ~Swith the corresponding coefficient . The third term cancels the zcomponent of the Gilbert damping term, thus restricting the STT to the xyplane. The coefficient of the third term is related to by = =!LS2 ?=SSz. Expressing the coefficients and in terms of the current susceptibilities  xx(!L)and  xy(!L)results in =JSz gB!LSX [Ref xx(!L)gImf xy(!L)g]; (21) =J gB!LX [Imf xx(!L)g+ Ref xy(!L)g]:(22) By inserting the explicit expressions for Gr 0;""()and Ga 0;##(!L), one obtains =J2S2 z !LSd 8X [f(!L)f()](23) 1 [( 2)2+ (")2][( 2)2+ (#!L)2]; =J !Ld 4X [f(!L)f()](24) ( 2)2+ (")(#!L) [( 2)2+ (")2][( 2)2+ (#!L)2]; where";#=0gBBc e=2 =0(!L+JSz)=2are the energies of the Zeeman levels in this section. In the small precession frequency regime, !LkBT, !0and in the limit of Sz=S!1the expression for the coefficient is in agreement with Ref. 16. µLµR=0↑↓↑−ωL↓+ωLFIG. 5: (Color online) Sketch of the electronic energy levels of the QD in the presence of a molecular spin precessing with the frequency !Laround an external, constant magnetic field. The corresponding Zeeman levels are ";#. The precessional motion of the molecular spin results in emission (absorption) of energy corresponding to a spin flip from spin up (down) to spin down (up). Hence, there are only four channels available for transport. B. Analysis of the spin-transfer torque In the case of resonant tunneling in the presence of a molecular spin precessing in a constant external mag- netic field, one also needs to take the exchange of spin- angular momentum between the molecular spin and the electronic spins into account in addition to the effects of the external magnetic field. Due to the precessional motion of the molecular spin, an electron in the QD emit- ting (absorbing) an energy !Lalso undergoes a spin flip from spin up (down) to spin down (up), as indicated by the arrows in Fig. 5. As a result, the levels at en- ergies";#!Lare forbidden and hence do not con- tribute to the transport processes. Consequently, there are only four transport channels, which are located at energies";#!L. Also in this case, there are elastic and inelastic tunnel processes. Some of the possible in- elastic tunnel processes are shown in Fig. 6. These re- strictions on the inelastic tunnel processes are also vis- ible in Fig. 3(b), which identically corresponds to the case of the presence of a precessing molecular spin with !L= 0:160andJSz= 0:80. Namely, from Eq. (16), which is equivalent to RefILx(!L)g= ImfILy(!L)g= JS? 2gB[RefL xx(!L)gImfL xy(!L)g]andImfILx(!L)g= RefILy(!L)g=JS? 2gB[ImfL xx(!L)g+ Refxy(!L)g], and from the symmetries of the susceptibilities displayed in Fig. 3(b), it follows that there are no spin currents at eV=";#!L. As was mentioned, the spin currents generate an STT acting on the molecular spin. A necessary condition for the existence of an STT, and hence finite values of the coefficients andin Eqs. (23) and (24), is that ~IL(t)6=~IR(t)[see Eq. (17)]. This condition is met by the spin currents generated, e.g., by the inelastic tun- nel processes shown in Figs. 6(b) and 6(c). These tunnel processes occur when an electron can tunnel into the QD, undergo a spin flip, and then tunnel off the QD into ei-7 µLµL µLµLµL(a) (f) (e) (d) (c) (b) µL FIG. 6: (Color online) Sketch of the inelastic spin-tunneling processes in the QD in the presence of the precessing molec- ular spin in the field ~Bc=Bc~ ezfor different positions of the energy levels with respect to the chemical potentials of the leads,LandR. Only transitions between levels with the same color (blue or red) are allowed. Different colored curved arrows (magenta, brown, or green) represent different processes. ther lead. From these tunnel processes it is implied that the Gilbert damping coefficient and the coefficient  can be controlled by the applied bias or gate voltage as well as by the external magnetic field. If a pair of QD energy levels, coupled via spin-flip processes, lie within the bias-voltage window, the spin currents instead fulfill ~IL(t) =~IR(t), leading to a vanishing STT [see Fig. 6(d)]. In Figs. 6(e) and 6(f) the position of the energy levels of the QD are symmetric with respect to the Fermi levels of the leads, LandR. When the QD level with energy "is aligned with L, this simultaneously corresponds to the energy level #being aligned with R[see Fig. 6(f)]. As a result, a spin-up electron can now tunnel from the left lead into the level ", while a spin-down electron in the level#can tunnel into the right lead. These addi- tional processes enhance the STT compared to that of the case 6(e). The two spin-torque coefficients andexhibit a non- monotonic dependence on the tunneling rates , as can be seen in Figs. 7, 8, and 9. For !0, it is obvious that;!0. In the weak coupling limit !L, the coefficients andare finite if the Fermi surface energy of the lead ,fulfills either of the conditions ##+!L (25) or"!L" (26) insuchawaythateachconditionissatisfiedbytheFermi energy of maximum one lead. These conditions are re- laxed for larger tunnel couplings as a consequence of the broadening of the QD energy levels, which is also re- sponsible for the initial enhancement of andwith in- creasing . Notice, however, that andare eventually suppressed for !L, when the QD energy levels aresignificantly broadened and overlap so that spin-flip pro- cesses are equally probable in each direction and there is no net effect on the molecular spin. Physically, this suppression of the STT can be understood by noticing that for !La current-carrying electron perceives the molecular spin as almost static due to its slow precession compared to the electronic tunneling rates and hence the exchange of angular momenta is reduced. With increas- ing tunneling rates, the coefficient becomes negative before it drops to zero, causing the torque _~Sto oppose the rotational motion of the spin ~S. In Fig. 7, the Gilbert damping coefficient and the coefficientare plotted as functions of the applied bias voltage at zero temperature. We analyze the case of the smallestvalueof (redlines), assumingthat !L>0. For smalleV, all QD energy levels lie outside the bias-voltage window and there is no spin transport [see Fig. 6(a)]. Hence;!0. AteV=#the tunnel processes in Fig. 6(b) come into play, leading to a finite STT and the coefficientincreases while the coefficient has a local minimum. In the voltage region specified by Eq. (25) for L, the coefficient approaches a constant value while the coefficient increases. By increasing the bias voltage toeV=#+!Lthe tunnel processes in Fig. 6(c) occur, leading to a decrease of and a local maximum of . For #+!L<eV <"!L, the coefficients ;!0[see Fig 6(d)]. In the voltage region specified by Eq. (26) for L, approaches the same constant value mentioned above whiledecreases between a local maximum at eV= "!Land a local minimum at eV=", which approach the same values as previously mentioned extrema. With further increase of eV, all QD energy levels lie within the bias-voltage window and the STT consequently vanishes. Figure 8 shows the spin-torque coefficients andas functions of the position of the electronic level 0. An STT acting on the molecular spin occurs if the electronic level0is positioned in such a way that the inequalities (25) and (26) may be satisfied by some values of eV,0 and!L. Again, we analyze the case of the smallest value of(red curve). For the particular choice of parameters in Fig. 8, there are four regions in which the inequalities (25) and (26) are satisfied. Within these regions, ap- proaches a constant value while has a local maximum as well as a local minimum. These local extrema occur when one of the Fermi surfaces is aligned with one of the energy levels of the QD. For other values of 0, both andvanish. The coefficients andare plotted as functions of the precession frequency !Lin Fig. 9. Here, 0=eV=2and therefore the positions of the energy levels of the QD are symmetric with respect to the Fermi levels of the leads, LandR. Once more, we focus first on the case of the smallest value of (indicated by the red curve). The energies of all four levels of the QD depend on !L, i.e., ~Bc. For!L>0, when the magnitude of the external magnetic field is large enough, the tunnel processes in Fig. 6(f) take place due to the above-mentioned sym- metries. These tunnel processes lead to a finite STT,8 0.00.51.01.52.01234567 eV@e0Da@10-4D e¯+wLHaL e¯ ee-wLG=5e0G=3e0G=0.2e0G=0.02e0 0.00.51.01.52.02.5-8-6-4-2024 eV@e0Db@10-4D e¯+wLHbL e¯ ee-wLG=5e0G=3e0G=0.2e0G=0.02e0 FIG. 7: (Color online) (a) Gilbert damping coefficient and (b) coefficient as functions of the applied bias voltage eV= LR, withR= 0, for different tunneling rates at zero temperature. Other parameters are L= R= =2,"= 1:480, #= 0:520,S= 100,J= 0:010,JSz= 0:80, and!L= 0:160. In the case of the smallest value of (red lines), approaches a constant value when Llies within the energy range specified by Eqs. (25) and (26). The coefficient has one local minimum and one local maximum for the same energy range. -0.50.00.51.01.52.01234567 e0@eVDa@10-4D mR=eHaL mR=e-wL mR=e¯mR=e¯+wL mL=e-wLmL=e mL=e¯+wL mL=e¯G=2.5eVG=1.5eVG=0.1eVG=0.01eV -0.50.00.51.01.52.0-6-4-2024 e0@eVDb@10-4D mR=eHbL mR=e-wLmR=e¯+wL mR=e¯ mL=e mL=e-wL mL=e¯+wL mL=e¯G=2.5eVG=1.5eVG=0.1eVG=0.01eV FIG. 8: (Color online) (a) Gilbert damping coefficient and (b) coefficient as functions of the position of the electronic level0for different tunneling rates at zero temperature. The applied bias voltage is eV=LR, withR= 0. Other parameters are L= R= =2,"0= 0:24eV,S= 100,J= 0:005eV,JSz= 0:4eV, and!L= 0:08eV. In the case of the smallest value of (red lines), there are four regions in which the Gilbert damping and the change of the precession frequency occur. In each of these regions 0satisfies the inequalities (25) and (26), and approaches a constant value, while has one local maximum and one local minimum. a maximum for the Gilbert damping coefficient , and a negative minimum value for the coefficient. As !L increases, the inequalities of Eqs. (25) and (26) are sat- isfied and the tunnel processes shown in Fig. 6(e) may occur. Hence, there is a contribution to the STT, but as is shown in Eq. (23), the Gilbert damping decreases with increasing precession frequency. At larger values of !L, resulting in #+!L=L, the Gilbert damping co- efficient has a step increase towards a local maximum, while the coefficient has a local maximum, as a conse- quence of the enhancement of the STT due to additional spin-flip processes occurring in this case. For even larger value of!L, the conditions (25) and (26) are no longerfulfilled and both coefficients vanish. It is energetically unfavorable to flip the spin of an electron against the antiparallel direction of the effective constant magnetic fieldBc e. Hence, as !Lincreases, more energy is needed to flip the electronic spin to the direction of the field. This causes to decrease with increasing !L. Addition- ally, the larger the ratio !L=, the less probable it is that spin-angular momentum will be exchanged between the molecular spin and the itinerant electrons. For !L= 0, the molecular spin is static, i.e.,_~S= 0. In this case ~T(t) =~0. The coefficient then drops to zero while the coefficientreaches a negative local maximum which is close to 0. Both andreach an extremum value for9 -4-20240123 wL@e0Da@10-4DHaL mL=e-wL mL=emL=e¯ mL=e¯+wL G=5e0G=3e0G=0.2e0G=0.02e0 -4-2024-6-4-20 wL@e0Db@10-4DHbL mL=e¯+wLmL=emL=e¯ mL=e-wL G=5e0G=3e0G=0.2e0G=0.02e0 FIG. 9: (Color online) (a) Gilbert damping coefficient and (b) coefficient as functions of the precession frequency !L= gBBcof the spin ~Sof the SMM, with ~Bc=Bc~ ez, for different tunneling rates at zero temperature. The applied bias voltage iseV=LR= 20, withR= 0. The other parameters are the same as in Fig. 7. In the case of the smallest (red lines), the coefficient has a step increase towards a local maximum while the coefficient has a local maximum or minimum at a value of!Lcorresponding to a resonance of Lwith one of the levels in the QD. large values of at this point. For !L<0andj!Lj (red lines), at the value of !Lfor whichL="!L, the coefficienthasastepincreasetowardsalocalmaximum whilethecoefficient hasanegativelocalminimum. The coefficientthen decreases with a further decrease of !L as long as#L"!L. At the value of !Lfor whichL=#,has another step increase towards a local maximum while has a maximum value. Accord- ing to Eq. (23), the Gilbert damping also does not occur if~Sis perpendicular to ~Bc. In this case .0and the only nonzero torque component _~S(t)acts in the oposite direction than the molecular spin’s rotational motion. IV. CONCLUSIONS In this paper we have first theoretically studied time- dependent charge and spin transport through a small junction consisting of a single-level quantum dot cou- pled to two noninteracting metallic leads in the pres- ence of a time-dependent magnetic field. We used the Keldysh nonequilibrium Green’s functions method to de- rive the charge and spin currents in linear order with respect to the time-dependent component of the mag- netic field with a characteristic frequency !. We then focused on the case of a single electronic level coupled via exchange interaction to an effective magnetic field created by the precessional motion of an SMM’s spin in a constant magnetic field. The inelastic tunneling pro- cesses that contribute to the spin currents produce an STT that acts on the molecular spin. The STT con- sists of a Gilbert damping component, characterized bythe coefficient , as well as a component, characterized by the coefficient , that acts as an additional effective constant magnetic field and changes the precession fre- quency!Lof the molecular spin. Both anddepend on!Land show a nonmonotonic dependence on the tun- neling rates . In the weak coupling limit !L, can be switched on and off as a function of bias and gate voltages. The coefficient correspondingly has a local extremum. For !0, bothandvanish. Taking into account that spin transport can be controlled by the bias and gate voltages, as well as by external magnetic fields, our results might be useful in spintronic applica- tions using SMMs. Besides a spin-polarized STM, it may be possible to detect and manipulate the spin state of an SMM in a ferromagnetic resonance experiment56–59and thus extract information about the effects of the current- induced STT on the SMM. Our study could be com- plemented with a quantum description of an SMM in a single-molecule magnet junction and its coherent prop- erties, as these render the SMM suitable for quantum information storage. Acknowledgments We gratefully acknowledge discussions with Mihajlo VanevićandChristianWickles. Thisworkwassupported by Deutsche Forschungsgemeinschaft through SFB 767. We are thankful for partial financial support by an ERC Advanced Grant, project UltraPhase of Alfred Leiten- storfer.10 1G. Christou, D. Gatteschi, D. Hendrickson, and R. 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2012-11-15

We study time-dependent electronic and spin transport through an electronic level connected to two leads and coupled with a single-molecule magnet via exchange interaction. The molecular spin is treated as a classical variable and precesses around an external magnetic field. We derive expressions for charge and spin currents by means of the Keldysh non-equilibrium Green's functions technique in linear order with respect to the time-dependent magnetic field created by this precession. The coupling between the electronic spins and the magnetization dynamics of the molecule creates inelastic tunneling processes which contribute to the spin currents. The inelastic spin currents, in turn, generate a spin-transfer torque acting on the molecular spin. This back-action includes a contribution to the Gilbert damping and a modification of the precession frequency. The Gilbert damping coefficient can be controlled by the bias and gate voltages or via the external magnetic field and has a non-monotonic dependence on the tunneling rates.

Spin transport and tunable Gilbert damping in a single-molecule magnet junction

1211.3611v2

arXiv:0809.2910v1 [cond-mat.other] 17 Sep 2008Spin-transfer torque induced reversal in magnetic domains S. MurugeshaM. Lakshmananb aDepartment of Physics & Meteorology, IIT-Kharagpur, Kharagpu r 721 302, India bCentre for Nonlinear Dynamics, School of Physics, Bharathid asan University, Tiruchirappalli 620024, India Abstract Using the complex stereographic variable representation f or the macrospin, from a study of the nonlinear dynamics underlying the generalize d Landau-Lifshitz(LL) equation with Gilbert damping, we show that the spin-transf er torque is effectively equivalent to an applied magnetic field. We study the macrosp in switching on a Stoner particle due to spin-transfer torque on application of a spin polarized cur- rent. We find that the switching due to spin-transfer torque i s a more effective alternative to switching by an applied external field in the p resence of damping. We demonstrate numerically that a spin-polarized current in t he form of a short pulse can be effectively employed to achieve the desired macro-spin switching. Key words: Nonlinear spin dynamics, Landau-Lifshitz equation, Spin- transfer torque, Magnetization reversal PACS:75.10.Hk, 67.57.Lm, 75.60.Jk, 72.25.Ba 1 Introduction In recent times the phenomenon of spin-transfer torque has gained much at- tention in nanoscale ferromagnets[1,2,3]. Electromigration refers to the recoil linearmomentumimpartedontheatomsofametalorsemiconductor asalarge current is conducted across. Analogously, if the current is spin-p olarized, the transfer of a strong current across results in a transfer of spin angular momen- tum to the atoms. This has lead to the possibility of current induced s witch- ing of magnetization in nanoscale ferromagnets. With the success o f GMR, ∗Corresponding author. Tel: +91 431 2407093, Fax:+91 431 240 7093 Email address: lakshman@cnld.bdu.ac.in (M. Lakshmanan). Preprint submitted to Chaos, Solitons and Fractals 7 Septem ber 2021this has immense application potential in magnetic recording devices s uch as MRAMs[3,4,5,6]. The phenomenon has been studied in several nanomag netic pile geometries. The typical set up consists of a nanowire[3,7,8,9,10,11 ], or a spin-valve pillar, consisting of two ferromagnetic layers, one a long f erromag- neticpinnedlayer, and another small ferromagnetic layer or film, separated by a spacer conductor layer (see Figure 1). The pinned layer acts a s a reser- voir for spin polarized current which on passing through the conduc tor and on to the thin ferromagnetic layer induces an effective torque on th e spin magnetization in the thin film ferromagnet. A number of experiments have been conducted on this geometry and the phenomenon has been co nvincingly confirmed [12,13,14,15]. Although the microscopic quantum theory of the phe- nomenon is fairly well understood, interestingly the behavior of the average spin magnetization vector can be described at the semi-classical lev el by the LL equation with an additional term[16]. j xyz Pinned layer Conductor Thin film ConductorS S p Fig. 1. A schematic diagram of the spin-valve pillar. A thin fi lm ferromagnetic layer with magnetization Sis separated from long ferromagnetic layer by a conductor. ˆSp is the direction of magnetization in the pinned region, whic h also acts as a reservoir for spin polarized current. From a different point of view, several studies have focused on mag netic pulse induced switching of the macro-magnetization vector in a thin nanod ot un- der different circumstances [17,18,19,20]. Several experimental st udies have also focussed on spin-current induced switching in the presence of a magnetic field, switching behavior for different choices of the angle of the app lied field, variation in the switching time, etc., [12,21,22,23,24,25]. A numerical stu dy on the switching phenomenon induced by a spin current in the presen ce of a magnetic field pulse has also been investigated very recently in [26]. A s an extension to two dimensional spin configurations, the switching beh avior on a vortex has been studied in [27]. In this article, by investigating the nonlinear dynamics underlying the gener- alized Landau-Lifshitz equation with Gilbert damping, we look at the ex citing possibility of designing solid state memory devices at the nanoscale, w herein memory switching is induced using a spin polarized current alone, witho ut the reliance on an external magnetic field. We compare earlier studie d switch- ing behavior for the macro-magnetization vector in a Stoner partic le [17] in the presence of an external magnetic field, and the analogous cas e wherein the applied field is now replaced by a spin polarized current induced spin - 2transfer torque, i.e., with the thin film in the first case replaced by a s pin valve pillar. It will be shown that a pulse of spin polarized current is mor e effective in producing a switching compared to an applied field. In doing so we rewrite the system in terms of a complex stereographic variable inst ead of the macro-magnetization vector. This brings a significant clarity in unde rstanding the nonlinear dynamics underlying the macrospin system. Namely, it w ill be shown that, in the complex system, the spin-transfer torque is eff ectively an imaginary applied magnetic field. Thus the spin-transfer term can ac complish the dual task of precession of the magnetization vector and dissip ation. The paper is organized as follows: In Section 2 we discuss briefly the m odel system and the associated extended LL equation. In Section 3 we in troduce the stereographic mapping of the constant spin magnetization vec tor to a complex variable, and show that the spin-transfer torque is effect ively an imaginary applied magnetic field. In Section 4 we present results from our numerical study on spin-transfer torque induced switching pheno menon of the macro-magnetization vector, for a Stoner particle. In particular , we study two different geometries for the free layer, namely, (a) an isotropic sp here and (b) an infinite thin film. In applications to magnetic recording devices, the typ- ical read/write time period is of the order of a few nano seconds. We show that, in order to achieve complete switching in these scales, the spin -transfer torque induced by a short pulse of sufficient magnitude can be affirma tively employed. We conclude in Section 5 with a discussion of the results and their practical importance. 2 The extended LL equation The typical set up of the spin-valve pillar consists of a long ferromag netic element, or wire, with magnetization vector pinned in a direction indica ted by ˆSp, as shown in Figure 1. It also refers to the direction of spin polarizat ion of the spin current. A free conduction layer separates the pinned ele ment from the thin ferromagnetic film, or nanodot, whose average spin magne tization vectorS(t) (of constant magnitude S0) is the dynamical quantity of interest. The cross sectional dimension of the layers range around 70 −100nm, while the thickness of the conduction layer is roughly 2 −7nm[3,20]. The free layer thus acts as the memory unit, separated from the pinned layer cum reservoir by the thin conduction layer. It is well established that the dynamics of the magnetization vector Sin the film in the semiclassical limit is efficiently de- scribed by an extended LL equation[16]. If ˆ m(={m1,m2,m3}=S/S0) is the unit vector in the direction of S, then dˆ m dt=−γˆ m×/vectorHeff+λˆ m×dˆ m dt−γag(P,ˆ m·ˆSp)ˆ m×(ˆ m׈Sp),(1) 3a≡ℏAj 2S0Ve. (2) Here,γis the gyromagnetic ratio (= 0 .0176Oe−1ns−1) andS0is the satura- tion magnetization (Henceforth we shall assume 4 πS0= 8400, the saturation magnetization value for permalloy). The second term in (1) is the phe nomeno- logical dissipation term due toGilbert[28] with damping coefficient λ. The last term is the extension to the LL equation effecting the spin-transfe r torque, whereAis the area of cross section, jis the current density, and Vis the volume of the pinned layer.′a′, as defined in (2), has the dimension of Oe, and is proportional to the current density j.g(P,ˆ m·ˆSP) is given by g(P,ˆ m·ˆSp) =1 f(P)(3+ˆ m·ˆSp)−4;f(P) =(1+P)3 (4P3/2),(3) wheref(P)isthepolarizationfactorintroducedbySlonczewski [1],and P(0≤ P≤1) is the degree of polarization of the pinned ferromagnetic layer. F or simplicity, we take this factor gto be a constant throughout, and equal to 1. /vectorHeffisthe effective fieldacting onthespin vector due toexchange intera ction, anisotropy, demagnetization and applied fields: /vectorHeff=/vectorHexchange+/vectorHanisotropy+/vectorHdemagnetization +/vectorHapplied,(4) where /vectorHexchange=D∇2ˆ m, (5) /vectorHanisotropy=κ(ˆ m·ˆ e/bardbl)ˆ e/bardbl, (6) ∇·/vectorHdemagnetization =−4πS0∇·ˆ m. (7) Here,κis the strength of the anisotropy field. ˆ e/bardblrefers to the direction of (uniaxial) anisotropy, In what follows we shall only consider homogen eous spin states on the ferromagnetic film. This leaves the exchange inte raction term in (4) redundant, or D= 0, while (7) for /vectorHdemagnetization is readily solved to give /vectorHdemagnetization =−4πS0(N1m1ˆ x+N2m2ˆ y+N3m3ˆ z), (8) whereNi,i= 1,2,3 are constants with N1+N2+N3= 1, and {ˆ x,ˆ y,ˆ z}are the orthonormal unit vectors. Equation (1) now reduces to a dynamic al equation for a representative macro-magnetization vector ˆ m. In this article we shall be concerned with switching behavior in the film p urely induced by the spin-transfer torque term, and compare the resu lts with earlier studies on switching due to an applied field [17] in the presence of dissip ation. Consequently, it will be assumed that /vectorHapplied= 0 in our analysis. 43 Complex representation using stereographic variable It proves illuminating to rewrite (1) using the complex stereographic variable Ω defined as[29,30] Ω≡m1+im2 1+m3, (9) so that m1=Ω+¯Ω 1+|Ω|2;m2=−i(Ω−¯Ω) 1+|Ω|2;m3=1−|Ω|2 1+|Ω|2.(10) For the spin valve system, the direction of polarization of the spin-p olarized currentˆSpremains a constant. Without loss of generality, we chose this to be the direction ˆ zin the internal spin space, i.e., ˆSp=ˆ z. As mentioned in Sec. 2, we disregard the exchange term. However, for the purpose of illustration, we choose /vectorHapplied={0,0,ha3}for the moment but take ha3= 0 in the later sections. Defining ˆ e/bardbl={sinθ/bardblcosφ/bardbl,sinθ/bardblsinφ/bardbl,cosθ/bardbl} (11) and upon using (9) in (1), we get (1−iλ)˙Ω =−γ(a−iha3)Ω+im/bardblκγ/bracketleftBig cosθ/bardblΩ−1 2sinθ/bardbl(eiφ/bardbl− Ω2e−iφ/bardbl)/bracketrightBig −iγ4π S0 (1+|Ω|2)/bracketleftBig N3(1−|Ω|2)Ω−N1 2(1−Ω2−|Ω|2)Ω −N2 2(1+Ω2−|Ω|2)Ω−(N1−N2) 2¯Ω/bracketrightBig ,(12) wherem/bardbl=ˆ m·ˆ e/bardbl. Using (10) and (11), m/bardbl, and thus (12), can be written entirely in terms of Ω. It is interesting to note that in this representation the spin-trans fer torque (proportional to the parameter a) appears only in the first term in the right hand side of (12) as an addition to the applied magnetic field ha3but with a prefactor −i. Thus the spin polarization term can be considered as an effective applied magnetic field. Letting κ= 0, and N1=N2=N3in (12), we have (1−iλ)˙Ω =−γ(a−iha3)Ω, (13) which on integration leads to the solution Ω(t) = Ω(0) exp( −(a−iha3)γt/(1−iλ)) = Ω(0) exp( −a+λha3 1+|λ|2γt) exp(−iaλ−ha3 1+|λ|2γt). (14) 5The first exponent in (14) describes relaxation, or switching, while t he second term describes precession. From the first exponent in (14), we no te that the time scale ofswitching is given by 1 /(a+λha3).λbeing small, thisimplies that the spin-torque term is more effective in switching the magnetization vector. Further, letting ha3= 0, we note that in the presence of the damping term the spin transfer produces the dual effect of precession and diss ipation. To start with we shall analyze the fixed points of the system for the two cases which we shall be concerned with in this article: (i) the isotropic spher e char- acterized by N1=N2=N3= 1/3, and (ii) an infinite thin film characterized byN1= 0 =N3,N2= 1. (i) First we consider the case when the anisotropy field is absent, or κ= 0. From (12) we have (1−iλ)˙Ω =−aγΩ−iγ4πS0 1+|Ω|2/bracketleftBig N3(1−|Ω|2)Ω−N1 2(1−Ω2−|Ω|2)Ω −N2 2(1+Ω2−|Ω|2)Ω−(N1−N2) 2¯Ω/bracketrightBig .(15) In the absence of anisotropy ( κ= 0), we see from (15) that the only fixed point is Ω 0= 0.To investigate the stability of this fixed point we expand (15) up to a linear order in perturbation δΩ around Ω 0. This gives (1−iλ)δ˙Ω =−aγδΩ−iγ4πS0[N3−1 2(N1+N2)]δΩ+iγ2πS0(N1−N2)δ¯Ω.(16) For the isotropic sphere, N1=N2=N3= 1/3, (16) reduces to (1−iλ)δ˙Ω =−aγδΩ. (17) We find the fixed point is stable since a >0. For the thin film, N1= 0 = N3,N2= 1. (16) reduces to (1−iλ)δ˙Ω =−aγδΩ+iγ2πS0δΩ−iγπS0δ¯Ω. (18) This may be written as a matrix equation for Ψ ≡(δΩ,δ¯Ω)T, ˙Ψ =MΨ, (19) whereMis a matrix obtained from (18) and its complex conjugate, whose determinant and trace are |M|=(a2+3π2S2 0)γ2 1+λ2;Tr(M) =(−2a−4πS0λ)γ 1+λ2.(20) Since|M|is positive, the fixed point Ω 0= 0 is stable if Tr|M|<0, or, (a+2πS0λ)>0. 6The equilibrium point (a), Ω 0= 0, corresponds to ˆ m=ˆ z. Indeed this holds true even in the presence of an applied field, though we have little to d iscuss on that scenario here. (ii) Next we consider the system with a nonzero anisotropy field in the ˆ z direction. (12) reduces to (1−iλ)˙Ω =−aγΩ+iκγ(1−|Ω|2) (1+|Ω|2)Ω−iγ4πS0 (1+|Ω|2)/bracketleftBig N3(1−|Ω|2)Ω −N1 2(1−Ω2−|Ω|2)Ω−N2 2(1+Ω2−|Ω|2)Ω−(N1−N2) 2¯Ω/bracketrightBig .(21) Here again the only fixed point is Ω 0= 0.As in (i), the stability of the fixed point is studied by expanding (21) about Ω 0to linear order. Following the same methodology in (i) we find the criteria for stability of the fixe d point for the isotropic sphere is ( a+λκ)>0, while for the thin film it is (a+λ(κ+2πS0))>0. (iii) With nonzero κin an arbitrary direction the fixed point in general moves away from ˆ z. Finally, it is also of interest to note that a sufficiently large current lea ds to spin wave instabilities induced through spin-transfer torque [31,32]. In the present investigation, however, we have assumed homogeneous m agnetization over the free layer, thus ruling out such spin wave instabilities. Rece ntly we haveinvestigated spinwave instabilitiesoftheSuhltypeinduced bya napplied alternating field in thin film geometries using stereographic represen tation[30]. It will be interesting to investigate the role of a spin-torque on such instabil- ities in the spin valve geometry using this formulation. This will be pursu ed separately. 4 Spin-transfer torque induced switching Wenowlookattheinterestingpossibilityofeffectingcompleteswitchin g ofthe magnetization using spin-transfer torque induced by a spin curren t. Numerical studies on switching effected on a Stoner particle by an applied magne tic field, or in the presence of both a spin-current and applied field, in the pre sence of dissipation and axial anisotropy have been carried out recently and switching has been demonstrated [17,26]. However, the intention here is to ind uce the same using currents rather than the applied external fields. Also, achieving such localized magnetic fields has its technological challenges. Spin-t ransfer torque proves to be an ideal alternative to accomplish this task sinc e, as we have pointed out above, it can be considered as an effective (albeit c omplex) 7magnetic field. In analogy with ref. [17], where switching behavior due to an applied magnetic field has been studied we investigate here switching b ehav- ior purely due to spin-transfer torque, on a Stoner particle. Nume rical results in what follows have been obtained by directly simulating (12) and makin g use of the relations in (10), for appropriate choice of parameters . It should be remembered that (12) is equivalent to (1), and so the numerical results have been further confirmed by directly numerically integrating (1) also for the corresponding parameter values. We consider below two sample s differing in their shape anisotropies, reflected in the values of ( N1,N2,N3) in the de- magnetization field: a) isotropic sphere, N1=N2=N3= 1/3 and b) a thin filmN1= 0 =N3,N2= 1. The spin polarization ˆSpof the current is taken to be in the ˆ zdirection. The initial orientation of ˆ mis taken to be close to −ˆ z. In what follows this is taken as 170◦fromˆ zin the (z−x) plane. The orientation of uniaxial anisotropy ˆ e/bardblis also taken to be the initial direction ofˆ m. With these specified directions for ˆSpandˆ e/bardblthe stable fixed point is slightly away from ˆ z, the direction where the magnetization ˆ mis expected to switch in time. A small damping is assumed, with λ= 0.008. The magnitude of anisotropy κis taken to be 45 Oe. As stated earlier, for simplicity we have considered the magnetization to be homogeneous. 4.1 Isotropic sphere It is instructive to start by investigating the isotropic sphere, whic h is char- acterized by the demagnetization field with N1=N2=N3= 1/3. With these values for ( N1,N2,N3), (12) reduces to (1−iλ)˙Ω =−aγΩ+im/bardblκγ/bracketleftBig cosθ/bardblΩ−1 2sinθ/bardbl(eiφ/bardbl−Ω2e−iφ/bardbl)/bracketrightBig (22) A constant current of a= 10Oeis assumed. Using (2), for typical dimen- sions, this equals a current density of the order 108A/cm2. We notice that for the isotropic sample the demagnetization field does not play any r ole in the dynamics of the magnetization vector. In the absence of aniso tropy and damping the spin-transfer torque term leads to a rapid switching of Sto the ˆ zdirection. This is evident from (22), which becomes ˙Ω =−aγΩ, (23) with the solution Ω = Ω 0e−aγt, and the time scale for switching is given by 1/aγ. Figure 2.a shows the trajectory traced out by the magnetization vector S, for 5 ns, initially close to the −ˆ zdirection, switching to the ˆ zdirection. Figure 2.b depicts the dynamics with anisotropy but no damping, all ot her parameters remaining same. While the same switching is achieved, this is 8more smoother due to the accompanying precessional motion. Not e that with nonzero anisotropy, ˆ zis not the fixed point any more. The dynamics with damping but no anisotropy (Figure 2.c) resembles Figure 2.a, while Figu re 2.d shows the dynamics with both anisotropy and damping. -1 -0.5 0 0.5 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(d) xyz -1 -0.5 0 0.5 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(c) xyz-1 -0.5 0 0.5 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b) xyz -1 -0.5 0 0.5 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a) xyz Fig. 2. Trajectory of the magnetization vector m, obtained by simulating (12) for the isotropic sphere ( N1=N2=N3= 1/3), and using the relations in (10), for a= 10Oe(a)withoutanisotropyanddamping,(b)withanisotropybut nodamping, (c) without anisotropy but nonzero damping and (d) with both anisotropy and damping nonzero. The results have also been confirmed by nume rically integrating (1). The arrows point in the initial orientation (close to −ˆ z) and the direction of the spincurrent ˆ z. Evolution shownis for aperiodof 5 ns.Note that thefinal orientation is not exactly ˆ zin the case of nonzero anisotropy ((b) and (d)). It may be noticed that Figures 2.c and 2.d resemble qualitatively Figure s 2.a and 2.b, respectively, while differing mainly in the time taken for the switching. It is also noticed that switching in the absence of anisotro py is faster. Precession assisted switching has been the favored reco rding process in magnetic memory devices, as it helps in keeping the exchange interac tion at a minimum[18,19]. The sudden switching noticed in the absence of anisot ropy essentially refers to a momentary collapse of order in the magnetic m edia. 9This can possibly lead to strong exchange energy and a breakdown o f our assumption regarding homogeneity of the magnetization field. Howe ver, such rapid quenching assisted by short high intensity magnetic pulses has in fact been achieved experimentally [33]. A comparison with reference [17] is in order. There it was noted that with an applied magnetic field, instead of a spin torque, a precession assiste d switch- ing was possible only in the presence of a damping term. In Section 3 we pointed out how the spin transfer torque achieves both precessio n and damp- ing. Consequently, all four scenarios depicted in Figure 2 show switc hing of the magnetization vector without any applied magnetic field. 4.2 Infinite thin film Next we consider an infinite thin film, whose demagnetization field is give n by N1= 0 =N3andN2= 1. With these values (12) becomes (1−iλ)˙Ω =−aγΩ+im/bardblκγ/bracketleftBig cosθ/bardblΩ−1 2sinθ/bardbl(eiφ/bardbl−Ω2e−iφ/bardbl)/bracketrightBig −iγ4πS0/parenleftbigg1−|Ω|2 1+|Ω|2/parenrightbigg Ω.(24) Here againin the absence of anisotropy Ω = 0 is the only fixed point. Th us the spin vector switches to ˆ zin the absence of damping and anisotropy (Figure 3a). In order to achieve this in a time scale of 5 ns, we find that the value of a has to be of order 50 Oe. Again the behavior is in stark contrast to the case induced purely by an applied field[17], wherein the spin vector traces o ut a distorted precessional trajectory. As in Sec. 4.1, the trajecto ry traced out in the presence of damping is similar to that without damping (Figure 3c) . The corresponding trajectories traced out in the presence of anisot ropy are shown in Figures 3b and 3d. 4.3 Switching of magnetization under a pulsed spin-polariz ed current We noticed that in the absence of uniaxial anisotropy, the constan t spin polar- ized current can effect the desired switching to the orientation of ˆSp(Figure 2). This is indeed the fixed point for the system (with no anisotropy) . Fig- ure 2 traces the dynamics of the magnetization vector in a period of 5 ns, in the presence of a constant spin-polarized current. However, fo r applications in magnetic media we choose a spin-polarized current pulseof the form shown in Figure 4. It may be recalled here that, as was observed in 4.2, with a spin 10-1 -0.5 0 0.5 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(d) xyz -1 -0.5 0 0.5 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(c) xyz-1 -0.5 0 0.5 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b) xyz -1 -0.5 0 0.5 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a) xyz Fig. 3. Trajectory of the magnetization vector Sin a period of 5 ns, obtained as earlier by numerically simulating (12), and also confirming with (1), with demagne- tization field with by N1= 0 =N3,N2= 1 and a= 50Oe, and all the parameter values are as earlier. As in Figure 1, the ˆSand initial orientation are indicated by arrows. (a) Without anisotropy or damping, (b) nonzero an isotropy but zero damping, (c) without anisotropy but nonzero damping and (d) both anisotropy and damping nonzero. As earlier, in the presence of nonvanishin g anisotropy, the fixed point is not the ˆ zaxis. polarized current of sufficient magnitude, the switching time can inde ed be reduced. We choose a pulse, polarized as earlier along the ˆ zdirection, with rise time and fall time of 1 .5ns, and a pulse width, defined as the time interval between half maximum, of 4 ns. We assume the rise and fall phase of the pulse to be of a sinusoidal form, though, except for the smoothness, t he switching phenomenon is independent of the exact form of the rise or fall pha se. In Figures 5 and 6, we show trajectories of the spin vector for a pe riod of 25ns, for the two different geometries, the isotropic sphere and a thin fi lm. The action of the spin torque pulse, as in Figure 4, is confined to the fi rst 5ns. We notice that, with the chosen value of a, this time period is enough 11Rise time Fall time Pulse width 0 20 40 60 80 100 120 140 160 0 1 2 3 4 5a (Oe) time (ns) Fig. 4. Pulse form showing the magnitude of a, or effectively the spin-polarized current. The rise and fall phase are assumed to be of a sinusoi dal form. The rise and fall time are taken as 1 .5ns, and pulse width 4 ns. The maximum magnitude ofais 150Oe. to effect the switching. In the absence of anisotropy, the directio n ofˆSpis the fixed point. Thus a pulse of sufficient magnitude can effect a switc hing in the desired time scale of 5 ns. From our numerical study we find that in order for this to happen, the value of ahas to be of order 150 Oe, or, from (2), a current density of order 109A/cm2, a magnitude achievable experimentally (see for example [34]). Comparing with sections 4.1 and 4.2, we note th at the extra oneorder ofmagnitude in current density isrequired due to t he duration of the rise and fall phases of the pulse in Figure(4). Here again we co ntrast the trajectories with those induced by an applied magnetic field [17], w here the switching could be achieved only in the presence of a uniaxial aniso tropy. In Figure 5b for the isotropic sphere with nonzero crystal field anis otropy, we notice that the spin vector switches to the fixed point near ˆ zaxis in the first 5ns. However the magnetization vector precesses around ˆ zafter the pulse has been turned off. This is because in the absence of the spin- torque term, the fixed point is along ˆ e/bardbl, the direction of uniaxial anisotropy. Due to the nonzero damping term, the spin vector relaxes to the direction ofˆ e/bardblas time progresses. The same behavior is noticed in Figure 6b for the th in film, although the precessional trajectory is a highly distorted one due to the shape anisotropy. 12-1 -0.5 0 0.5 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b) xyz -1 -0.5 0 0.5 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a) xyz Fig. 5. Evolution of the magnetization vector Sin a period of 25 nsinduced by the spin-polarized current pulse in Figure 4, (a) with and (b) wi thout anisotropy for the isotropic sample, with N1=N2=N3= 1/3 all other parameters remaining same. A nonzero damping is assumed in both cases. The current pulse acts on the magnetization vector for the first 5 ns. In both cases switching happens in the first 5ns. In the presence of nonzero anisotropy field, (b), the magnet ization vector precesses to the fixed point near ˆ z. -1 -0.5 0 0.5 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b) xyz -1 -0.5 0 0.5 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a) xyz Fig. 6. Evolution of the magnetization vector Sin a period of 25 nsinduced by the spin-polarized current pulse in Figure 4, for a infinite t hin film sample, with N1= 0 =N3, andN2= 1, all other parameters remaining same, along with a nonzero damping. (a) Without anisotropy and (b) with anisot ropy. As in Figure 5, switching happens in the first 5 ns. 5 Discussion and conclusion We have shown using analytical study and numerical analysis of the n onlinear dynamics underlying the magnetization behavior in spin-valve pillars th at a very effective switching of macro-magnetization vector can be ach ieved by a spin transfer-torque, modeled using an extended LL equation. Re writing the 13extended LL equation using the complex stereographic variable, we find the spin-transfer torque term indeed acts as an imaginary applied field t erm, and can lead to both precession and dissipation. It has also been pointed out why the spin-torque term is more effective in switching the magnetization vector compared to the applied field. On application of a spin-polarized curre nt the average magnetization vector in the free layer was shown to switch to the direction of polarization of the spin polarized current. 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Cimpoesu, A. Spinu and L. Spinu. Switching be havior of a StonerWohlfarth particle subjected to spin-torque effect J. Appl. Phy. 2008; 103: 07B105. [27] J.-G. Caputo, Y. Gaididei, F. G. Mertens and D. D. Sheka. Vortex Polarity Switching by a Spin-Polarized Current Phys. Rev. Lett. 2007 ; 98: 056604. [28] T. L. Gilbert. A Phenomenological Theory of Damping in F erromagnetic Materials IEEE Trans. on Mag. 2004; 40: 3443-49. [29] M. Lakshmanan and K. Nakamura. Landau-Lifshitz Equati on of Ferromagnetism: Exact Treatment of the Gilbert Damping Phy s. Rev. Lett. 1984; 53: 2497-99. [30] C. Kosaka, K. Nakamura, S. Murugesh and M. Lakshmanan. P hysica D Equatorial and related non-equilibrium states in magnetiz ation dynamics of ferromagnets: Generalization of Suhls spin-wave instabil ities 2005; 203: 233-48. [31] M. L. Polianski and P. W. Brouwer. Current-Induced Tran sverse Spin-Wave Instability in a Thin Nanomagnet Phys. Rev. Lett 2004; 92: 02 6602. [32] S. Adam, M. L. Polianski and P. W. Brouwer. Current-indu ced transverse spin- wave instability in thin ferromagnets: Beyond linear stabi lity analysis Phys. Rev. B 2006; 73: 024425. [33] I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegman n, J. St¨ ohr, G. Ju, B. Lu and D. Weller. The ultimate speed of magnetic switch ing in granular recording media Nature 2004; 428: 831-33. [34] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, M . Tsoi and P. Wyder. Excitation of a Magnetic Multilayer by an Electric Current Phys. Rev. Lett. 1998; 80: 4281-84. 16

2008-09-17

Using the complex stereographic variable representation for the macrospin, from a study of the nonlinear dynamics underlying the generalized Landau-Lifshitz(LL) equation with Gilbert damping, we show that the spin-transfer torque is effectively equivalent to an applied magnetic field. We study the macrospin switching on a Stoner particle due to spin-transfer torque on application of a spin polarized current. We find that the switching due to spin-transfer torque is a more effective alternative to switching by an applied external field in the presence of damping. We demonstrate numerically that a spin-polarized current in the form of a short pulse can be effectively employed to achieve the desired macro-spin switching.

Spin-transfer torque induced reversal in magnetic domains

0809.2910v1

Antidamping spin-orbit torque driven by spin- ip re ection mechanism on the surface of a topological insulator: A time-dependent nonequilibrium Green function approach Farzad Mahfouzi,1,Branislav K. Nikoli c,2and Nicholas Kioussis1 1Department of Physics, California State University, Northridge, CA 91330-8268, USA 2Department of Physics and Astronomy, University of Delaware, Newark, DE 19716-2570, USA Motivated by recent experiments observing spin-orbit torque (SOT) acting on the magnetization ~ mof a ferromagnetic (F) overlayer on the surface of a three-dimensional topological insulator (TI), we investigate the origin of the SOT and the magnetization dynamics in such systems. We predict that lateral F/TI bilayers of nite length, sandwiched between two normal metal leads, will generate a large antidamping-like SOT per very low charge current injected parallel to the interface. The large values of antidamping-like SOT are spatially localized around the transverse edges of the F overlayer. Our analysis is based on adiabatic expansion (to rst order in @~ m=@t ) of time-dependent nonequilibrium Green functions (NEGFs), describing electrons pushed out of equilibrium both by the applied bias voltage and by the slow variation of a classical degree of freedom [such as ~ m(t)]. From it we extract formulas for spin torque and charge pumping, which show that they are reciprocal eects to each other, as well as Gilbert damping in the presence of SO coupling. The NEGF-based formula for SOT naturally splits into four components, determined by their behavior (even or odd) under the time and bias voltage reversal. Their complex angular dependence is delineated and employed within Landau-Lifshitz-Gilbert simulations of magnetization dynamics in order to demonstrate capability of the predicted SOT to eciently switch ~ mof a perpendicularly magnetized F overlayer. PACS numbers: 72.25.Dc, 75.70.Tj, 85.75.-d, 72.10.Bg I. INTRODUCTION The spin-orbit torque (SOT) is a recently discovered phenomenon1{4in ferromagnet/heavy-metal (F/HM) lateral heterostructures involves unpolarized charge cur- rent injected parallel to the F/HM interface induces switching or steady-state precession5of magnetization in the F overlayer. Unlike conventional spin-transfer torque (STT) in spin valves and magnetic tunnel junc- tion (MTJs),6{8where one F layer acts as spin-polarizer of electrons that transfer torque to the second F layer when its free magnetization is noncollinear to the direc- tion of incoming spins, heterostructures exhibiting SOT use a single F layer. Thus, in F/HM bilayers, spin-orbit coupling (SOC) at the interface or in the bulk of the HM layer is crucial to spin-polarized injected current via the Edelstein eect (EE)9,10or the spin Hall eect (SHE),11,12respectively. The SOT oers potentially more ecient magnetiza- tion switching than achieved by using MTJs underlying present STT-magnetic random access memories (STT- MRAM).13Thus, substantial experimental and theo- retical eorts have been focused on identifying physi- cal mechanisms behind SOT whose understanding would pave the way to maximize its value by using optimal materials combinations. For example, very recent ex- periments14{16have replaced HM with three-dimensional topological insulators (3D TIs).17The TIs enhance18{20 (by a factor ~vF=R, wherevFis the Fermi velocity on the surface of TI and Ris the Rashba SOC strength22? at the F/HM interface) the transverse nonequilibrium spin density driven by the longitudinal charge current, which is responsible for the large eld-like SOT compo- nent20,23observed experimentally.14{16 FIG. 1. (Color online) Schematic view of F/TI lateral bilayer operated by SOT. The F overlayer has nite length LF xand~ m is the unit vector along its free magnetization. The TI layer is attached to two N leads which are semi-innite in the x- direction and terminate into macroscopic reservoirs. We also assume that F and TI layers, as well as N leads, are innite in they-direction. The unpolarized charge current is injected by the electrochemical potential dierence between the left and the right macroscopic reservoirs which sets the bias voltage, LR=eVb. We mention that the results do not change if the TI surface is covered by the F overlayer partially or fully. Furthermore, recent experiments have also observed antidamping-like SOT in F/TI heterostructures with surprisingly large gure of merit (i.e., antidamping torque per unit applied charge current density) that sur- passes14{16those measured in a variety of F/HM het- erostructures. This component competes against the Gilbert damping which tries to restore magnetization to equilibrium, and its large gure of merit is, there- fore, of particular importance for increasing eciency of magnetization switching. Theoretical understanding ofarXiv:1506.01303v3 [cond-mat.mes-hall] 24 Jan 20162 the physical origin of antidamping-like SOT is crucial to resolve the key challenge for anticipated applications of SOT generated by TIs|demonstration of magnetization switching of the F overlayer at room temperature (thus far, magnetization switching has been demonstrated only at cryogenic temperature15). However, the microscopic mechanism behind its large magnitude14{16and ability to eciently (i.e., using as lit- tle dc current density as possible) switch magnetization15 remains under scrutiny. For example, TI samples used in these experiments are often unintentionally doped, so that bulk charge carriers can generate antidamping-like SOT via rather large24SHE (but not sucient to explain all reported values14,15). The simplistic picture,14in which electrons spin-polarized by the EE diuse into the F overlayer14to deposit spin angular momentum within it, cannot operate in technologically relevant F overlayers of'1 nm thickness16or explain complex angular depen- dence2,15,25typically observed for SOT. The Berry cur- vature mechanism25,26for antidamping-like SOT applied to lateral F/TI heterostructures predicts its peculiar de- pendence on the magnetization orientation,27vanishing when magnetization ~ mis parallel to the F/TI interface. This feature has thus far not been observed experimen- tally,15and, furthermore, it makes such antidamping-like SOT less ecient27(by requiring larger injected currents to initiate magnetization switching) than standard SHE- driven3,4antidamping-like SOT. We note that the recent experimental14{16and theo- retical14,27studies of SOT in lateral F/TI bilayer have focused on the geometry where an innite F overlayer covers an innite TI layer. Moreover, they assume14,27? purely two-dimensional transport where only the top sur- face of the TI layer is explicitly taken into account by the low-energy eective (Dirac) Hamiltonian supplemented by the Zeeman term due to the magnetic proximity ef- fect. On the other hand, transport in realistic TI-based heterostructures is always three-dimensional, with unpo- larized electrons being injected from normal metal con- tacts, re ected from the F/TI edge to ow along the sur- face of the TI in the yz-plane and then along the bottom TI surface in Fig. 1. In fact, electrons also ow within a thin layer (of thickness .2 nm in Bi 2Se3as the prototyp- ical TI material) underneath the top and bottom surfaces due to top and bottom metallic surfaces of the TI doping the bulk via evanescent wave functions.18Therefore, in this study we consider more realistic and experimentally relevant28F/TI bilayer geometries, illustrated in Fig. 1, where the TI layer of nite length LTI xand nite thick- nessLTI zis (partially or fully) covered by the F overlayer of lengthLF x. The two semi-innite ideal N leads are di- rectly attached to the TI layer. we should mention that the result does not depend on the length of TI layer that is covered by the FM. Our principal results are twofold and are summarized as follows: (i)Theoretical prediction for SOT: We predict that the geometry in Fig. 1 will generate large antidamping-likeSOT per low injected charge current. By studying spatial dependence of the SOT (see Fig. 4), we show that in a clean FM/TI interface the electrons exert anti-damping torque on the FM as they enter into the interface and un- less interfacial roughness or impurities are included the torque remains mainly concentrated around the edge of the interface. Although the exact results show strong nonperturbative features, based on second order pertur- bation we present two dierent interpretations showing that the origin of the antidamping SOT relies on the spin- ip re ection of the chiral electrons injected into the FM/TI interface. Its strong angular dependence (see Fig. 2), i.e., dependence on the magnetization direction ~ m, oers a unique signature that can be used to distin- guish it from other possible physical mechanisms. By numerically solving the Landau-Lifshitz-Gilbert (LLG) equation in the macrospin approximation, we demon- strate (see Figs. 5 and 6) that the obtained SOT is ca- pable of switching of a single domain magnetization of a perpendicularly magnetized F overlayer with bias voltage in the oder of the Magneto-Crystaline Anisotropy (MCA) energy. (ii)Theoretical formalism for SOT: The widely used quantum (such as the Kubo formula25{27,30) and semi- classical (such as the Boltzmann equation31) transport approaches to SOT are tailored for geometries where an innite F layer covers an innite TI or HM layer. Due to translational invariance, the nonequilibrium spin density ~Sinduced by the EE on the surface of TI or HM layer has uniform orientation ~S= (0;Sy;0) [in the coordinate system in Fig. 1], which then provides reference direc- tion for dening eld-like, f~ m^y, and antidamping- like,ad~ m(~ m^y), components of SOT. In order to analyze spatial dependence of SOT in the device geome- try of Fig. 1, while not assuming anything a priori about the orientation of eld-like and antidamping-like compo- nents of SOT, we employ adiabatic expansion32of time- dependent nonequilibrium Green functions (NEGFs)33,34 to derive formulas for torque, charge pumping35,36and Gilbert damping37in the presence of SOC. The NEGF- based formula for SOT naturally splits into four compo- nents, determined by their behavior (even or odd) under the time and bias voltage reversal. This gives us a general framework in quantum mechanics to analyze the dissi- pative (antidamping-like) and nondissipative (eld-like) force (torque) vector elds for a set of canonical variables (magnetization directions). Their angular (see Fig. 2) and spatial (see Fig. 4) dependence shows that although eld-like and antidamping-like SOTs are predominantly along the~ m^yand~ m(~ m^y) directions, respectively, they are not uniform and can exhibit signicant devia- tion from the trivial angular dependence dened by these cross products [see Fig. 2(h)]. The paper is organized as follows. In Sec. II, we present the adiabatic expansion of time-dependent NEGFs, in a representation that is alternative to Wigner representa- tion34(usually employed for this type of derivation32), and derive expressions for torque, charge pumping and3 Gilbert damping. In Sec. III, we decompose the NEGF- based expression for SOT into four components, deter- mined by their behavior (even or odd) under the time and bias voltage reversal, and investigate their angular de- pendence. Section IV discusses the angular dependence of the zero-bias transmission function which identies the magnetization directions at which substantial re ection occurs. In Sec. V, we study spatial dependence of SOT components and discuss their physical origin. Section VI presents LLG simulations of magnetization dynamics in the presence of predicted SOT, as well as a switching phase diagram of the magnetization state as a function of the in-plane external magnetic eld and SOT. We con- clude in Sec. VII. II. THEORETICAL FORMALISM We rst describe the time-dependent Hamiltonian model, H(t) =H0+U(t), of the lateral F/TI heterostruc- ture in Fig. 1. Here H0is the minimal tight-binding model for 3D TIs like Bi 2Se3on a cubic lattice of spacing awith four orbitals per site.38The thickness, LTI z= 8a of the TI layer is sucient to prevent hybridization be- tween its top and bottom metallic surface states.18The time-dependent potential U(t) =
surf1m~ m(t)
~=2; (1) depends on time through the magnetization of the F over- layer which acts as the slowly varying classical degree of freedom. Here ~ m(t) is the unit vector along the direc- tion of magnetization,
surf= 0:28 eV is the proximity induced exchange-eld term and 1mis a diagonal matrix with elements equal to unity for sites within the F/TI contact region in Fig. 1 and zero elsewhere. The semi- innite ideal N leads in Fig. 1 are taken into account through the self-energies33,34L;Rcomputed for a tight- binding model with one spin-degenerate orbital per site. The details of how to properly couple L;RtoH0, while taking into account that the spin operators for electrons on the Bi and Se sublattices of the TI are inequivalent,39 can be found in Ref. 40. Within the NEGF formalism33,34the advanced and lesser GFs matrix elements of the tight-binding Hamiltonian, H0, are dened by Gii0;oo0;ss0(t;t0) = i(tt0)hf^cios(t);^cy i0o0s0(t0)gi, andG< ii0;oo0;ss0(t;t0) = ih^cy i0o0s0(t0)^cios(t)i, respectively. Here, ^ cy ios(^cios) is the creation (annihilation) operator for an electron on site, i, with orbital, o, and spins, respectively,h:::idenotes the nonequilibrium statistical average, and ~= 1 to simplify the notation. These GFs are the matrix elements of the corresponding matrices GandG<used throughout the text. Under stationary conditions, the two GFs depend on the dierence of the time arguments, tt0, and can be Fourier transformed to energy. In the strictly adi- abatic limit one can employ41the same retarded GF,Gt(E) = [EH(t)LR]1, as under stationary conditions, but where the GF depends parametrically on time (denoted by the subscript t) and is computed for the frozen-in-time conguration of U(t). However, even for slow evolution of ~ m(t) corrections32to the adiabatic GF are needed to describe dissipation eects such as Gilbert damping or the charge current which can be pumped35 by the dynamics of ~ m(t). The so-called adiabatic expansion, which yields correc- tions beyond the strictly adiabatic limit, is traditionally performed using the Wigner representation34in which the fast and slow time scales are easily identiable.32The slow motion implies that the NEGFs vary slowly with the central time tc= (t+t0)=2 while they change fast with the relative time tr=tt0. By expanding the Wigner transformation of NEGFs G(<) W(E;tc) =Z1 1dtreiEtrG(<) tc+tr 2;tctr 2 ; (2) in the central time tcwhile keeping only terms contain- ing rst-order derivatives @=@tc(due to the slow varia- tion withtc) gives the rst-order correction beyond the strictly adiabatic limit.32This route requires to han- dle complicated expressions resulting from the Wigner transform applied to convolutions of the type C(t1;t2) =R dt3C1(t1;t3)C2(t3;t2). Here we provide an alternative derivation of the rst- order nonadiabatic correction. Namely, we consider t (observation time) and tt0(relative time) as the nat- ural variables to describe the time evolution of NEGFs and then perform the following Fourier transform42 G(t;t0) =Z1 1dE 2eiE(tt0)G(E;t): (3) The standard equations of motion for G(t;t0) and G<(t;t0) are cumbersome to manipulate42,43or solve numerically,44so they are usually transformed to some other representation.35Here we replace G(t;t0) in the standard equations of motion with the rhs of Eq. (3) to arrive at:  Ei@ @t 1H0U(t) Ei@ @t G(E;t) =1; (4) and G<(t;t0) =ZdE 2G(E;t)<(E)Gy(E;t0)eiE(tt0):(5) For the two-terminal heterostructure in Fig. 1 in the elastic transport regime,33,34(E) =L(E) +R(E) and<(E) =ifL(E)L(E) +ifR(E)R(E), where L;R=i(L;Ry L;R). The Fermi-Dirac distribution functions of electrons in the macroscopic reservoirs into which the left and right N leads terminate are fL;R(E) = f(EL;R), where the dierence between the electro- chemical potentials, L;R=EF+eVL;R, denes the bias voltageeVb=LR.4 Using the following identity X =L;RiGGy= (GGy) +i@ @E G@U @tGy +O@2U @t2 ; (6) the lesser GF to rst order in small @U(t)=@tand in the low bias Vbregime (i.e., the linear-response transport regime) can be expressed as, G<(t;t)'ZdE 20 @[G(E;t)Gy(E;t)]f(E) +X =L;Rf0eVGtGy t+if0Gt@U(t) @tGy t1 A; (7) where, the rst term corresponds to the density matrix of the equilibrium electrons occupying the time dependent single particle states, while the second and third terms describe the density matrix of the excited (nonequilibrium) electrons occupying the states close to the Fermi energy due to the bias voltage and time dependent term in the Hamiltonian, respectively. The retarded GF in the rst term in Eq. (7) expanded to rst order in @U(t)=@tis of the form G(E;t)'Gt+i@Gt @E@U(t) @tGt: (8) The lesser GF determines the time-dependent nonequilibrium density matrix (t) =1 iG<(t;t); (9) from which we determine the time-dependent expectation values of physical observables, A(t) = Tr [(t)^A]. In particular, the relevant quantities for the heterostructure in Fig. 1 are the charge current I(t) =eZdE 2iTr (E)G<(t;t) +f(E)(E)[G(E;t)Gy(E;t)] =e2 2Z dEf0(E)8 < :X (VV)T(E)
surf 2eX i@mi @tTi(E)9 = ;; (10) and the spin density si(t) =ZdE 2iTr i1mG< =ZdE 28 < :X f(E)Ti(E)
surf 2X jf0(E)@mj @tTij(E)9 = ;; (11) where;2fL;Rgandi;j2fx;y;zg. The \trace-formulas" in Eqs. (10) and (11) T(E) = Trh GtGy ti ; (12a) Ti(E) = Trh 1miGy tGti ; (12b) Ti(E) = Trh 1miGtGy ti ; (12c) Tij(E) = Trh 1mi(Gy tGt)1mj(GtGy t)i ;(12d) determine charge current45due toVb, charge current pumped35by the dynamics of ~ m(t) in the presence of SOC, the spin torque, and Gilbert damping tensor, re- spectively. In the expression for the spin density we ig- nore the antisymmetric part of Tijwhich corresponds to the renormalization of the precession frequency of mag-netization dynamics.32Note thatTiin Eq. (12)(b) and its time-reversal Tiin Eq. (12)(c) reveal a reciprocal46 relation between charge pumping by magnetization dy- namics in the presence of SOC and current-driven SOT ateach instant of time t. The spin density in Eq. (11) enters into the LL equa- tion for the magnetization dynamics @~ m @t=
surf 2~ m~ s(t): (13) The rst term in Eq. (11) generates the spin torque in Eq. (13) which has three contributions: ( i) an equilib- rium component responsible for the interlayer exchange interaction in the presence of a second F layer and/or magneto-crystalline anisotropy (MCA) in the presence of SOC; ( ii) a bias-induced eld-like torque modifying the5 equilibrium interlayer exchange and MCA elds; and ( iii) a damping (antidamping)-like torque describing angular momentum loss (gain) due to the ux of Fermi surface electrons. III. ANGULAR DEPENDENCE OF SOT COMPONENTS In order to understand the dierent contributions to Ti(E), we decompose it into even (e) or odd (o) terms under time-reversal Gt7!Gy t: Ti e(E) = [Ti(E) +Ti(E)]=2; (14) Ti o(E) = [Ti(E)Ti(E)]=2: (15) SinceP Ti o(E) = 0, the contribution of the odd com- ponent to the equilibrium spin density in Eq. (11) van- ishes identically, while its nonzero values appear only for Ewithin the bias window around EF. This motivates further splitting of Ti(E) into four components for the case of two-terminal devices Ti e;(E) =TiL (E) +TiR (E) 2; (16a) Ti o;(E) =TiL (E)TiR (E) 2; (16b) where the rst and second subscripts denote their be- havior (even or odd) under bias reversal Vb7!Vband time reversal, respectively. The corresponding four com- ponents of torque are determined by ~Te;=ZdE 2[fL(E) +fR(E)]~ e;(E); (16c) ~To;=ZdE 2[fL(E)fR(E)]~ o;(E);(16d) where the energy-resolved torque is given by ~ ;(E) =
surf 2~ m~T;(E); (17) and;2fe;og. The terms ~To;oand~To;eare non-zero only in nonequilibrium driven by Vb6= 0, and depend on electronic states in the bias voltage window around the Fermi energy (or on the Fermi surface states in the linear- response regime where integrals are avoided by multiply- ing integrand by eVb). The term ~Te;o0 is zero, while ~Te;eis nonzero also in equilibrium and, therefore, depends on all occupied electronic states. Figures 2(a-c) show the netvector eld (summed over all sites of the F overlayer) of ~ ;(EF) at zero temper- ature for dierent directions of ~ mon the unit sphere. Their angular behavior reveals that: ( i)~ o;oshown in Fig. 4(a) is the eld-like SOT generated by the EE, with predominant orientation along the ~ m^y-direction; ( ii) ~ o;ein Fig. 4(b) is the antidamping-like SOT with pre- dominant orientation along the ~ m(~ m^y)-direction; and ( iii)~ e;ein Fig. 4(c) along the ~ m^zdirection isthe eld-like component whose angular dependence be- haves approximately as 2( ~ m
^z)j~ m^zjsin(2) typical for torque components generated by the MCA eld.2 The corresponding angular dependence of the net i o;o(i2fx;y;zg),j~ o;ej, andi e;ealong the solid trajecto- ries shown in Figs. 2(a-c) are plotted in Figs. 2(d-f), respectively. We nd that the magnitude of the max- imum antidamping-like SOT is about a factor of four larger than that of the eld-like SOT. Additionally, the eld-like SOT peaks when the magnetization is in-plane. In contrast, the antidamping-like SOT peaks when the magnetization is out of plane, which can be attributed to the gap opening of the Dirac cone which in turn enhances the re ection (see Sec. IV) at the lateral boundaries of the F overlayer. Note that the magnitude of the netSOT components shown in Figs. 2(g-i) exhibits strong angular dependence because of the large SOC on the TI surface similar to that found in F/HM heterostructures when the Rashba SOC at the interface is suciently strong.25,26In particular, the signicant deviation of the angular dependence of the antidamping-like SOT from the trivial j~ m(~ m^y)jbe- havior in the limit of ~ m!^y, shown quantitatively in Fig. 2(h), indicates its nonperturbative variation with respect to the magnetization direction. A similar nonper- turbative angular behavior (i.e., strong deviation from the standard/sin2dependence on the precession cone angle) has been found for adiabatic charge pumping from a precessing F overlayer attached to the edge of 2D TIs47,48or to the surface of 3D TIs.36 IV. ANGULAR DEPENDENCE OF TRANSMISSION FUNCTION Figure 3 shows the transmission function TRL(EF) in Eq. (12a) for the heterostructure in Fig. 1 versus the ori- entation of ~ mon the unit sphere. We nd that the charge current determined by TRL(E) is smallest49when~ mk^z or~ mk^x. This is due to the re ection of Dirac elec- trons on the TI surface from the lateral boundaries of the F overlayer. Underneath the F overlayer, the ex- change eld,
surf~ m
~=2, induced by the magnetic proximity eect is superimposed on the Dirac cone sur- face dispersion. This opens an energy gap
surfwhen ~ mk^z(or smaller gap
surfcosformz6= 0) at the DP of the TI region underneath the F overlayer, which in turn gives rise to strong electronic re ection when ~ mk^z. For~ mk^x, there is no energy gap at the DP and the Dirac cone eectively shifts away from the center of the Brillouin zone due to proximity exchange eld. Neverthe- less electrons polarized by the EE along the y-axis re ect from the magnetization pointing along the x-axis.6 (a) (b) (c) (d) (e) (f) (g) (h) ( i) -0.20.00.2i e,e(EF)(1/a) x y z 0 45 90 135 1800.00.10.20.3|e,e(EF)|/sin 2 (1/a) Angle (deg) -5-4-3-2-1010-2i o,o(EF)(1/a) x y z 0 45 90 135 180024681010-2|o,o(EF)|/sin (1/a) Angle (deg) 0.00.10.2|o,e(EF)|(1/a) =0o =45o =90o 0 45 90 135 1800.00.10.20.3|o,e(EF)|/sin (1/a) Angle (deg)=90o=45o=0o FIG. 2. (Color online) (a){(c) The vector eld of SOT components ~ ;(EF), dened by Eq. (16), for dierent directions of ~ mon the unit sphere. The angular behavior in (a) and (c) shows that ~ o;oand~ e;ebehave as eld-like torques, while that in (b) shows that ~ o;ebehaves as antidamping-like torque. Cartesian components [(d)-(f)] and magnitude of ~ ;[(g){(i)] along the trajectories denoted by solid lines in the corresponding panels (a)-(c). The magnitude of ~ ;is divided by:j~ m^yjin (g); j~ m(~ m^y)jin (h) and 2( ~ m
^z)j~ m^zjin (i). V. SPATIAL DEPENDENCE OF SOT COMPONENTS AND PHYSICAL ORIGIN OF ANTIDAMPING-LIKE SOT Figure 4(a) demonstrates that large values of antidamping-like SOT from Fig. 2 are spatially localized around the transverse edges of the F overlayer, for Fermi energy inside and outside of the surface state gap induced by out-of-plane magnetic exchange coupling. While con- ductance in this system is close to zero at the Dirac point, we observe that the anti-damping torque does not de- pend strongly on the Fermi energy. This suggests a high eciency of SOT per injected current for the Fermi en- ergy close to the Dirac point. For the eld-like SOT in Fig. 4(b) we see the torque independent of the co- ordinate in the entire F/TI contact region, as expected from the phenomenology of the EE. In Fig. 4(c) we plot the contribution of the Fermi energy electrons to the FM/TI interface induced MCA eld. Even though to- tali e;o(EF)0, its spatially-resolved value plotted in Fig. 4(d) is nonzero which can be removed by perform- ing a proper gauge transformation. To understand the origin of the anti-damping SOT let us nd an expression for the average of SOT around axed axes~ m=~ m0+(~ m?cos()+~ m0~ m?sin())with small cone angle . By applying a rotation operator we can align the xed axes along z-axis such that ~ m0= ^ez and~ m?= ^ex. In this case we have hTzi=
surf 2=Zd 2ZdE 2X feiT;;(18) where,T;=TxiTyand=refers to the imaginary part. Expanding the GFs in Eq. 12(b) to the lowest order with respect to ei, we obtain, T;=
surf 4eiTrh 1mGt1m+GtGy t(19) +1mGtGy t1m+Gy ti : Plugging this expression into Eq.(18), and using the iden- tity,GtGy t=iP GtGy t, in linear bias voltage regime we obtain, hTzi=Vb
2 surf2 16Tr["" L1m## R1m"" R1m## L1m];(20) where,=iG< =GtGy t, corresponds to the density matrix inside the F overlayer for the electrons7 (a) TRL(EF)×10−2 4.5 4 3.5 3 𝑥𝑦𝑧 FIG. 3. (Color online) The transmission function TRL(EF) in Eq. (12a) for two-terminal heterostructure shown in Fig. 1 at dierent directions of magnetization ~ mon the unit sphere. The Fermi energy is set at EF= 3:1 eV (which is 0 :1 eV above the DP). 02 04 0-1010 2 04 0-505-10- 20τ x,ze ,e(EF)=0×10-4×10-2τye ,e(EF) (1/a)L ongitudinal Coordinate xF/TI (a)τxe ,o(EF) (1/a)m=(0,0,1)τ y,ze ,o(EF)=0τy,zo ,o(EF)=0τ x,zo ,e(EF)=0 EF=3.3eV EF=3.1eV( d)( c)(b)( a)× 10-2τyo ,e(EF) (1/a)×10-2τxo ,o(EF) (1/a) FIG. 4. (Color online) (a){(d) Spatial dependence of SOT components, i ;(EF) (i2fx;y;zgand;2fe;og), per unit length, for Fermi energy inside the magnetization in- duced gap around Dirac point ( EF= 3:1eV) and outside the gap (EF= 3:3eV), for~ mk^zin Fig. 1. Their physical meaning is explained in Fig. 2. The range of x-coordinate corresponds to the length LF x= 40aof the F overlayer, while the results are independent of the length of the TI layer un- derneath,LTI x. (holes) at the Fermi surface being injected from the lead (6=). The electron-hole analogy can be understood by dening the hole density matrix, iG> , from the iden- tityi(G< G> ) = 2=(G) =+P 6=. By con- sidering left-lead induced holes instead of right-lead in- duced electrons, we can interpret Eq.(20) as spin-resolved electron-hole recombination rate, where opposite spins have opposite contributions to the antidamping-like SOT. This picture focuses on the energy anti-dissipative aspect of the phenomena and, since  L( R) cor- FIG. 5. (Color online) SOT-induced magnetization trajecto- ries~ m(t) under dierent Vband~Bext= 0. Higher color inten- sity denotes denser bundle of trajectories which start from all possible initial conditions ~ m(t= 0) on the unit sphere. Solid curves show examples of magnetization trajectories, while the white circles denote attractors of trajectories. responds to the spin- right (left) moving electrons, Eq. (20) suggests that spin-momentum locking natu- rally has a signicant eect on the enhancement of the antidamping-like SOT magnitude. In particular, in the case of F/TI interface, the enhancement of the antidamping-like SOT occurs when the spin-up/down is along they-axis (~ m0k^y) which is the spin-polarization direction of electrons passing through the surface of the TI induced by the EE. Additionally, in this case the antidamping-like SOT gets smaller away from the F/TI transverse edge because the contribution of both of the leads to the spin density become identical. Therefore the anti-damping torque in this case is more localized around the edge. This eect is more signicant when the magne- tization is out of the plane and the Fermi energy is inside the
surfcosgap on the TI surface. A alternative interpretation of the results can be achieved by considering GtGy t=iP Gy tGt. In this case, the average of the antidamping-like SOT is ex- pressed by hTzi=Vb 4Tr[T"# LRT"# RL]; (21) where the F overlayer induced spin- ip transmission ma- trix is dened as T"# = (t"# )yt"# ; (22) and t"# =
surf 2p Gt+Gtp : (23) Although Eq. (21) is obtained from perturbative con- siderations, it looks identical to the Eq. (8) of Ref. 48 where a spin- ip re ection mechanism at the edge of the F/2D-TI interface was recognized to be responsible for the giant charge pumping (i.e., anti-damping torque) ob- served in the numerical simulation.48Eq. (23) describes a transmission event in which electrons injected from lead , get spin- ipped (from up to down) by the FM and then transmit to the lead . The path of the electrons describing this process is shown in Fig. 1. From the k- resolved results of the anti-damping torque (not shown8 here) we observe that while for the in-plane magnetiza- tion electrons moving in the same transverse direction (same sign for ky) on both left and right edges of the FM/TI interface contribute to the torque, in the case of out-of-plane magnetization for the left (right) edge of the interface the local anti-damping torque is induced mostly by the electrons with ky>0 (ky<0). It is worth mentioning that due to nonperturbative na- ture of the SOT induced by the chiral electrons, the ap- proximation presented in this section which can as well be obtained from the self energy corresponding to the vacuum polarization Feynman diagrams of the electron- magnon coupled system59, does not capture the phenom- ena accurately. This is evident in the angular dependence of the anti-damping torque which in the current section is considered up to second order eect ( 2), while the divergence-like behavior in Figs.2(h) suggest a linear de- pendence when the magnetization direction is close to the y-axis. This signies the importance of the higher order terms with respect to that can not be ignored. The ap- proximation presented in this section also suggests that blocking the lower surface leads to the reduction of the anti-damping torque. However, in this case an electron experiences multiple spin- ip re ections before transmit- ting to the next lead and in fact it turns out that the ex- act results stay intact even if the lower surface is blocked. This is similar to the conclusion made in Ref. 48 which shows the redundancy of blocking the lower edge of the 2D-TI to obtain a nonzero pumped charge current from precessing FM as proposed in Ref. 47. Although spin-momentum locking of the surface state of the TI resembles the 2D Rashba plane, in the case of TI surface state the cones with opposite spin-momentum locking reside on opposite surface sides of the TI slab while in the case of a Rashba plane they are only sepa- rated by the SOC energy. This means one can expect a smaller SOT for a FM on top of a 2D Rashba plane due to cancellation of the eects of the two circles with op- posite spin-momentum locking, where the nonzero anti- damping torque originates from the electron-hole asym- metry. VI. LLG SIMULATIONS OF MAGNETIZATION DYNAMICS IN THE PRESENCE OF SOT In order to investigate ability of predicted antidamping-like SOT to switch the magnetization direction of a perpendicularly magnetized F overlayer in the geometry of Fig. 1, we study magnetization dynamics in the macrospin approximation by numerically solving LLG equation at zero temperature supplemented by SOT components analyzed in Sec. III @~ m @t=1 2[~ o;e(~ m;EF) +~ o;o(~ m;EF)]eVb+ ~Bext~ m +~ m (~ m)
@~ m @t + (~ m
^z)(~ m^z)
MCA:(24) FIG. 6. (Color online) Phase diagram of the magnetization state in lateral F/TI heterostructure from Fig. 1 as a function of an in-plane external magnetic Bextk^xandVb(i.e., SOT /Vb). Thick arrows on each of the panels (a){(d) show the direction of sweeping of Bext xorVbparameter. The small- ness of central hysteretic region along the Vb-axis, enclosed by white dashed line in panel (b) and (d), shows that low currents are required to switch magnetization from mz>0 tomz<0 stable states. Here is the gyromagnetic ratio, (~ m)ij =
2 surfTij(~ m;EF)=8is the dimensionless Gilbert damp- ing tensor, and
MCA =
0 MCA +j~Te;ej=j(~ m
^z)(~ m^z)j, where
0 MCA represents the intrinsic MCA energy of the FM. We solve Eq. (24) by assuming that the Gilbert damping is a constant (its dependence on ~ mis relegated to future studies) and ignore the dependence of
MCA on~ mandVbwhile retaining its out-of-plane direction. Figure 5 shows the magnetization trajectories for all possible initial conditions ~ m(t= 0) on the unit sphere under dierent Vb. AtVb= 0, the two attractors are located as the north and south poles of the sphere. At niteVb, the attractors shift away from the poles along thez-axis within the xz-plane, while additional attractor appears on the positive (negative) y-axis under negative (positive)Vb. Note that the applied bias voltage Vbdrives dc current and SOT proportional to it in the assumed linear-response transport regime. Figure 6 shows the commonly constructed3,4,15,27 phase diagram of the magnetization state in the pres- ence of an external in-plane magnetic eld Bextk^xand the applied bias voltage Vb(i.e., SOT/Vb). The thick arrows in each panel of Fig. 6 denote the direction of the sweeping variable|in Fig. 6(a) [6(b)] we increase [de- crease]Vbslowly in time, and similarly in Fig. 6(c) [6(d)] we increase [decrease] the external magnetic eld gradu- ally. The size of hysteretic region in the center of these diagrams, enclosed by white dashed line in Figs. 6(b) and 6(d), measures the eciency of switching.3,4,15,27Since this region, where both magnetization states mz>0 and mz<0 are allowed, is relatively small in Figs. 6(a) and9 6(b), magnetization can be switched by low Bext xand smallVb(or, equivalently, small injected dc current), akin to the phase diagrams observed in recent experiments.15 Although we considered the FM a single domain, the fact that the anti-damping component of the SOT is mainly peaked around the edge of the FM/TI interface suggests that it is be more feasible in realistic cases to have the local magnetic moments at the edge of the FM switch rst and then the total magnetization switches by the propagation of the domain walls formed at the edge throughout the FM60,61. Therefore, a micromag- netic simulation of the system is required to investigate switching phenomena in large size systems which we rel- egate to future works. VII. CONCLUSIONS In conclusion, by performing adiabatic expansion of time-dependent NEGFs,33,34we have developed a frame- work which yields formulas for spin torque and charge pumping as reciprocal eects to each other connected by time-reversal, as well as Gilbert damping due to SOC. It also introduces a novel way to separate the SOT com- ponents, based on their behavior (even or odd) under time and bias voltage reversal, and can be applied to arbitrary systems dealing with classical degrees of free- dom coupled to electrons out of equilibrium. For the geometry28proposed in Fig. 1, where the F overlayercovers (either partially or fully) the top surface of the TI layer, we predict that low charge current owing solely on the surface of TI will induce antidamping-like SOT on the F overlayer via a physical mechanism that requires spin-momentum locking on the surface of TIs|spin- ip re ection at the lateral edges of a ferromagnetic island introduced by magnetic proximity eect onto the TI sur- face. This mechanism has been overlooked in eorts to understand why SO-coupled interface alone (i.e., in the absence of SHE current from the bulk of SO-coupled non- ferromagnetic materials) can generate antidamping-like SOT, where other explored mechanisms have included spin-dependent impurity scattering at the interface,55 Berry curvature mechanism,25,26as well as their com- bination.56 The key feature for connecting experimentally ob- served SOT and other related phenomena in F/TI het- erostructures (such as spin-to-charge conversion28,36,58) to theoretical predictions is their dependence2,24on the magnetization direction. The antidamping-like SOT pre- dicted in our study exhibits complex angular dependence, exhibiting \nonperturbative" change with the magnetiza- tion direction in Fig. 2(h), which should make it possible to easily dierentiate it from other competing physical mechanisms. ACKNOWLEDGMENTS F. M. and N. K. were supported by NSF PREM Grant No. 1205734, and B. K. 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2015-06-03

Motivated by recent experiments observing spin-orbit torque (SOT) acting on the magnetization $\vec{m}$ of a ferromagnetic (F) overlayer on the surface of a three-dimensional topological insulator (TI), we investigate the origin of the SOT and the magnetization dynamics in such systems. We predict that lateral F/TI bilayers of finite length, sandwiched between two normal metal leads, will generate a large antidamping-like SOT per very low charge current injected parallel to the interface. The large values of antidamping-like SOT are {\it spatially localized} around the transverse edges of the F overlayer. Our analysis is based on adiabatic expansion (to first order in $\partial \vec{m}/\partial t$) of time-dependent nonequilibrium Green functions (NEGFs), describing electrons pushed out of equilibrium both by the applied bias voltage and by the slow variation of a classical degree of freedom [such as $\vec{m}(t)$]. From it we extract formulas for spin torque and charge pumping, which show that they are reciprocal effects to each other, as well as Gilbert damping in the presence of SO coupling. The NEGF-based formula for SOT naturally splits into four components, determined by their behavior (even or odd) under the time and bias voltage reversal. Their complex angular dependence is delineated and employed within Landau-Lifshitz-Gilbert simulations of magnetization dynamics in order to demonstrate capability of the predicted SOT to efficiently switch $\vec{m}$ of a perpendicularly magnetized F overlayer.

Antidamping spin-orbit torque driven by spin-flip reflection mechanism on the surface of a topological insulator: A time-dependent nonequilibrium Green function approach

1506.01303v3

arXiv:1907.07470v2 [math.AP] 10 Dec 2019Inhomogeneous domain walls in spintronic nanowires L. Siemer∗I. Ovsyannikov†J.D.M. Rademacher‡ December 12, 2019 In case of a spin-polarized current, the magnetization dynamics in n anowires are governed by the classical Landau-Lifschitz equation with Gilbertdamp- ing term, augmented by a typically non-variational Slonczewski term. Tak- ing axial symmetry into account, we study the existence of domain w all type coherent structure solutions, with focus on one space dimen sion and spin-polarization, but our results also apply to vanishing spin-torqu e term. Using methods from bifurcation theory for arbitrary constant ap plied fields, we prove the existence of domain walls with non-trivial azimuthal pro file, referred to as inhomogeneous . We present an apparently new type of do- main wall, referred to as non-flat, whose approach of the axial magnetiza- tion has a certain oscillatory character. Additionally, we present th e leading order mechanism for the parameter selection of flatandnon-flat inhomoge- neous domain walls for an applied field below a threshold, which depends on anisotropy, damping, and spin-transfer. Moreover, numerical c ontinuation results of all these domain wall solutions are presented. 1 Introduction Magnetic domain walls (DWs) are of great interest both from a theor etical perspective and for applications, especially in the context of innovative magnetic storages [1]. Re- cent developments in controlled movement of DWs via spin-polarized c urrent pulses in nanomagnetic structures, in particular in nanowires, are thought to lead to a new class of potential non-volatile storage memories, e.g. racetrack memor y [1, 2, 3, 4]. These devices make use of the fact that spin-transfer driven effects ca n change the dynamics in sufficiently small ferromagnetic structures (e.g. nanowires), wh ere regions of uniform ∗Universit¨ at Bremen, lars.siemer@uni-bremen.de ; Corresponding author †Universit¨ at Hamburg, Lobachevsky State University of Nizhny No vgorod ‡Universit¨ at Bremen 1magnetization, separated by DWs, can appear [5, 6, 7]. This motivat es further studies of the existence of magnetic domains and their interaction with spin-po larized currents as a building block for the theory in this context. In this paper we take a mathematical per- spective and, in a model for nanomagnetic wires, rigorously study t he existence of DWs. This led us to discover an apparently new kind of DWs with a certain inho mogeneous and oscillatory structure as explained in more detail below. The description of magnetization dynamics in nanomagnetic structu res, governed by the Landau-Lifschitz-Gilbert (LLG) equation, is based on works by Berger and Slonczewski assuming a spin-polarized current [8, 9]. In the presence of a const ant applied field and a spin-polarized current, the dynamics driven by the joint action of magnetic field and spin torque can be studied by adding a spin-transfer term in the dire ction of the current (current-perpendicular-to-plane (CPP) configuration). In cas e of a spatially uniform magnetization, the resulting Landau-Lifschitz-Gilbert-Slonczewski (LLGS) equation for unit vector fields ( m1,m2,m3) =m=m(x,t)∈S2(cf. Figure 1) reads ∂tm−αm×∂tm=−m×heff+m×(m×J). (LLGS) with effective field heff, Gilbert damping factor α>0, and the last term is the so-called polarized spin transfer pseudotorque. Note that the above equation reduces to the LLG equation for J≡0, see§2 for more details. In this paper we consider the axially symmetric case and set heff:=∂2 xm+h−µm3e3,J:=β 1+ccpm3e3, (1) whereh=he3with a uniform and time-independent field strength h∈R, and m 3= /an}bracketle{tm,e3/an}bracketri}ht,e3∈S2. This effective field heffalso includes the diffusive exchange term ∂2 xm, the uniaxial anisotropy and demagnetization field. The specific here with parameter µ∈Rderives from a first order approximation in the thin film/wire limit for a u niformly magnetized body [6, 10]. In the axially symmetric structure, β≥0 andccp∈(−1,1) describe the strength of the spin-transfer and the ratio of the p olarization [7, 11]. The spin-transfer torque term may provide energy to the system und er certain conditions and counterbalance dissipation associated to the Gilbert damping te rm, which gives rise to coherent non-variational dynamics, see e.g. [12]. Notably, for β= 0 one obtains the LLG-equation that does not account for spin tr ansfer effects. Moreover, as shown in [12], this also holds up to parameter change in case ccp= 0. Hence, solutions to the LLGS equation for β= 0 orccp= 0 are also solutions to the LLG equation, so that all the analytical as well as numerical r esults forccp= 0 in this paper directly transfer to the LLG equation. A key ingredient for the separation of uniformly magnetized states in space are interfaces betweentwomagneticdomains. Themostcoherentformofsuchint erfacesintheuniaxial setting are relative equilibria with respect to translation and rotatio n symmetry of the form m(ξ,t) =m0(ξ)eiϕ(ξ,t),whereξ=x−standϕ(ξ,t):=φ(ξ)+Ωt, 2(a) (b)-5 5-101 xm1,m2 (c) Figure 1: Homogeneous DW profile ( q≡0) withα= 0.5,β= 0.1,µ=−1,h= 50,ccp= 0. (a) (m2,m3)-profile. (b) Projection onto S2. (c) Zoom-in on m1(blue solid) and m2 (red dashed). withspeedsandfrequency Ω. Here the complex exponential acts on m0∈S2by rotation about the e3-axis, i.e., the azimuth, and in spherical coordinates we can choose m0(ξ) = (sin(θ(ξ)),0,cos(θ(ξ))) with altitude angle θ. We refer to such solutions with m0(ξ)→ ±e3asξ→ ±∞orξ→ ∓∞asdomain walls . A first classification of DWs is based on the local wavenumber q:=φ′, which determines φuniquely due to the axial rotation symmetry and satisfies q(ξ) =/an}bracketle{t(m′ 1,m′ 2),(−m2,m1)/an}bracketri}ht 1−m2 3(ξ). (2) Definition 1. We call a DW with constant φhomogeneous (hom) , i.e.,q≡0, and inhomogeneous otherwise. Inhomogeneous DWs have a spatially inhomogeneous varying azimuth al angle, compare Figures 1 and 2. In the case of uniaxial symmetry and the LLG case β= 0, an explicit family of homo- geneous DWs was discovered in [13] for applied fields with arbitrary st rength and time dependence, cf. Figure 1. Furthermore, for constant applied fie lds and in case of ccp/ne}ationslash= 0 it was shown in [12] that DWs cannot be homogeneous, and the existe nce of inhomoge- neous DWs was proven, whose spatial profile slowly converges to ±e3and where |s| ≫1. This latter type of DWs is ‘weakly localized’ and has large ‘width’ in the se nse that the inverse slope of m3atm3(0) = 0 tends to infinity as |s| → ∞. An apparently thus far unrecognized distinction of DWs is based on t he convergence behavior of qasξ→ ±∞. Definition 2. We call a DW flatif|q(ξ)|has a limit on R∪ {∞}as|ξ| → ∞and non-flatotherwise. Note that homogeneous DWs are flat ones by definition (recall φ′=q). Moreover, for all DWsm0(ξ) converges to e3or−e3as|ξ| → ∞. 3small applied field (a)large applied field (b) -5 5-101 ξm1 (c)-5 5-101 ξm1,m2 (d) Figure 2: Shown are profiles of inhomogeneous DWs m(ξ) computed by parameter continua- tion, cf.§4, inccptoccp= 0.5 with fixed α= 0.5,β= 0.1,µ=−1. (a,c) codim2case h= 0.5,s= 0.112027,Ω = 0.447173, (b,d) codim 0 case h= 50,s= 19.92,Ω = 40.4. (c) magnification of the m1-component; note the change of frequency for small vs. largeξ. (d) Magnification of m1(blue solid) as well as m2(red dashed) component. The main result of this paper is an essentially complete understanding of the existence and the type of DWs near the aforementioned explicit solution family f or a nanowire geometry, i.e., µ <0. This includes the LLG case β·ccp= 0, but our focus is on the spintronic case β·ccp/ne}ationslash= 0 for which these results pertain 0 <|ccp| ≪1 and any value of the (constant) applied field h. The different types of DWs occur in parameter regimes close to ccp= 0 in the (spatial) ODE which results from the coherent structure ansatz. Since the parameters αandµ are material-dependent we take the applied field strength has the primary parameter. In brief, organized by stability properties of the steady states ±e3in the ODE, this leads to the following cases and existence results for localized DWs in nanow ires (µ<0): •‘codim-2’ (h∗<h<h∗) : existence of flat inhomogeneous DWs with s,Ω selected by the other parameters, •‘center’ (h=h∗orh=h∗) : existence of flat and non-flat inhomogeneous DWs, •‘codim-0’ (h<h∗orh>h∗) : existence of flat inhomogeneous DWs, whereh∗:=β/α+2µ α2(1+α2) as well as h∗:=β/α−2µ α2(1+α2). Note that h∗<h∗always (recallα>0 andµ<0). Due to symmetry reasons, we mainly discuss the existence of 4Figure 3: Stability diagram of homogeneous states ±e3inhandccpforα= 1,β= 0.5, and µ=−1. State + e3unstable to the left and stable to the right of Γ+,−e3stable to the left and unstable to the right of Γ−. Homogeneous DWs (hom) exist only on theh-axis, i.e., ccp≡0. See text for further explanations. Note that also negativ e applied fields are shown. right-moving DWs close to the explicit solution family and thus focus on an applied field β/α≤h(cf.§3). The main results can be directly transferred to the case of left -moving DWs (h≤β/α). Notably, the codim-0 case occurs for ‘large’ magnetic field habove a material dependent threshold. In the center and codim-2 cases t here is a selection of s and Ω by the existence problem. Thebasicrelationbetween thePDEandtheODEstabilitypropertiesw .r.t.handccpare illustrated in Figure 3 for α= 1,β= 0.5,µ=−1 fixed. Due to the fact that sand Ω are ODE parameters only, the diagram illustrates a slice in the four dimens ional parameter space with axes h,ccp,s, and Ω. Note that homogeneous DWs (hom) can occur only on the line ccp≡0 (see [12, Theorem 5] for details). The stability regions are defined as follows. monostable−(blue): + e3unstable and −e3stable,bistable(shaded blue): both +e3and−e3stable,monostable+(red): +e3stable and −e3unstable, unstable (shaded red): both + e3and−e3unstable. For a more detailed stability discussion, see Remark 5. Note that the transition from bistable to monostable in th e PDE does not coincide with the transition, of the homogeneous family, from codim- 2 to codim-0 in the ODE. In contrast, the analogous transitions occur simultaneo usly for example in the well-known Allen-Cahn orNagumo equation. In Figure 4 below, we present numerical evidence that inhomogeneo us DWs are indeed also dynamically selected states, especially for large applied fields, als o in the LLG case (β,ccp= 0). The understanding of DW selection by stability properties generally d epends on the exis- tence problem discussed in this paper, which is therefore a prerequ isite for the dynamical selection problem, cf. Remark 5. 5(a) (b) (c) Figure 4: Direct simulation of full PDE (LLGS) for α= 0.5,β= 0.1,µ=−1,h= 50, and ccp= 0 with dynamical selection of an inhomogeneous DW. Initial condition near homogeneous DW (9) in codim-0 regime ( h∗= 10.2,s0= 19.92, and Ω 0= 40.04). (a) Profile at t= 20 projected onto the sphere. (b) Speed and frequency of DW over time with asymptotic (selected) values s= 12.5 and Ω = 78 .28. (c) Space-time plots of DW components (without co-moving frame), range as i n black box in (b). Final profile is heteroclinic connection in (7), cf. Proposi tion 2. To our knowledge, existence results of DWs for ccp/ne}ationslash= 0 are new. In more detail, the existence of localised inhomogeneous, i.e. flat as well as non-flat, DW s forccp/ne}ationslash= 0 and especially for ccp= 0 are new results. Indeed, the existence proof of non-flat DWs is the most technical result and entails an existence proof of hetero clinic orbits in an ODE between an equilibrium and a periodic orbit. These solutions indicate th e presence of DWs in other regimes of spin driven phenomena and may be of interest for spin-torque transfer MRAM (Magnetoresistive random-access memory) syst ems [14]. This paper is organized as follows. In §2, the LLGS equation and coherent structures as well as first properties are discussed. Section 3 more precisely intr oduces homogeneous and inhomogeneous as well as flat and non-flat DWs and it also includes the main results of this paper (Theorem 1, 2, and 3). The technical proofs of Theorem 2 as well as Theorem 3 are deferred to Appendix 6.1 and 6.2. Section 4 pre sents results of 6numerical continuation in parameter ccpfor the three regimes of the applied field (codim- 2, center, and codim-0), where the center case is studied in more d etail. We conclude with discussion and outlook in §5. Acknowledgements L.S. and J.R. acknowledge support by the Deutsche Forschungsge meinschaft (DFG, Ger- manResearchFoundation)-Projektnummer 281474342/GRK222 4/1. J.R.alsoacknowl- edges support by DFG grant Ra 2788/1-1. I.O. acknowledges fund ing of a previous position by Uni Bremen, where most of this paper was written, as we ll as support by the recent Russian Scientific Foundation grant 19-11-00280. 2 Model equations and coherent structure form The classical model for magnetization dynamics was proposed by La ndau and Lifschitz based on gyromagnetic precession, and later modified by Gilbert [15, 16]. See [17] for an overview. The Landau-Lifschitz-Gilbert equation for unit vector fields m(x,t)∈S2 in one space dimension x∈Rand in terms of normalized time in dimensionless form is ∂tm−αm×∂tm=−m×heff. (LLG) Herem=M/MSrepresents thenormalizedmagnetization, heff=Heff/MStheeffective field, i.e.thenegativevariationalderivativeofthetotalmagneticf reeenergywithrespect tom, both normalized by the spontaneous magnetization MS. For gyromagnetic ratio γand saturation magnetization MSthe time is measured in units of ( γMS)−1, and it is assumed that the temperature of the magnetic body is constant and below the Curie temperature [5]. Finally, Gilbertdampingα>0 turnsmtowardsheffand both vectors are parallel in the static solution. In modern spin-tronic applications, e.g. Spin-Transfer Torque Mag netoresistive Random Access Memories (MRAM), the spin of electrons is flipped using a spin- polarized current. To take these effects into account, the LLG equation is supplement ed by an additional spin transfer torque term. Using a semiclassical approach, Sloncz ewski derived an ex- tended effective field Heff=heff−m×J, whereJ=J(m) depends on the magnetization and the second term is usually called Slonczewski term [9]. In contrast to the LLG equation, which can be written as the gradient of free ferromagnetic energy, this generalized form is no longer variational and the energy is no longer a Lyapunov functional. As to the specific form of Heff, including a leading order form of exchange interaction, uniaxial crystal anisotropy in direction e3, andZeemanas well as stray-field interactions with an external magnetic field, see e.g. [6], gives the well known form (1). In this paper we consider a constant applied magnetic field h∈Ralonge3and uniaxial anisotropy with parameter µ∈R, for which the anisotropy energy density is rotationally 7symmetric w.r.t. e3. According to the energetically preferred direction in the uniaxial case, minima of the anisotropy energy density correspond to easydirections, whereas saddles or maxima correspond to medium-hard orharddirections, respectively. There- fore, one refers to µ <0 aseasy-axis anisotropy and µ >0 aseasy-plane , both with regard to e3. As mentioned before, the LLG equation with its variational structu re appears as a special case of (LLGS) for β= 0 orccp= 0. While our main focus is the non-variational spintronic case β·ccp/ne}ationslash= 0, all results contain the case β·ccp= 0 and thus carry over to (LLG). It iswell-known that(LLGS) admitsanequivalent formasanexplicit ev olution equation of quasilinear parabolic type in the form, see e.g. [12], ∂tm=∂x(A(m)∂xm)+B(m,∂xm). As a starting point, we briefly note the existence of spatially homoge neous equilibrium solutions of (LLGS) for which m(x,t) is constant in xandt. Remark 1. The only (spatially)homogeneous equilibria of (LLGS)forβ >0are the constant up- and down magnetization states ±e3. Indeed, setting ∂tm=∂2 xm= 0in (LLGS), forβ/ne}ationslash= 0the last equation implies that m1=m2= 0and thus the only solutions m∗ ±∈S2arem∗ ±= (0,0,±1)T. Remark 2. In caseβ= 0as well as |h/µ|<1there exist a family of additional homogeneous solutions of (LLGS) given by m∗= (m1,m2,h/µ)T,withm2 1+m2 2= 1−(h/µ)2. Note that similar cases occur for symmetry axis being e1ande2, respectively (cf. Brown’s equations ). 2.1 Coherent structure ODE Duetotherotationsymmetryaroundthe e3-axisof (LLGS), itisnaturaltousespherical coordinates m= cos(ϕ)sin(θ) sin(ϕ)sin(θ) cos(θ) , whereϕ=ϕ(x,t) andθ=θ(x,t). This changes (LLGS) to /parenleftbigg α−1 1α/parenrightbigg/parenleftbigg ∂tϕsin(θ) −∂tθ/parenrightbigg =/parenleftbigg 2∂xϕ∂xθcos(θ) −∂2 xθ/parenrightbigg +sin(θ)/parenleftbigg ∂2 xϕ+β/(1+ccpcos(θ)) (∂xϕ)2cos(θ)+h−µcos(θ)/parenrightbigg (3) Note that the rotation symmetry has turned into the shift symmet ry in the azimutal angleϕ, as (3) depends on derivatives of ϕonly. 8Recall that DW solutions spatially connect the up and down magnetiza tion states ±e3 in a coherent way as relative equilibria with respect to the translation symmetry in x andφ, which yields the ansatz ξ:=x−st, θ=θ(ξ), ϕ=φ(ξ)+Ωt. (4) Such solutions are generalized travelling waves that move with const ant speeds∈R in space and rotate pointwise with a constant frequency Ω ∈Raround the e3-axis; solutions with Ω = 0 are classical travelling waves. As in [12], applying ansatz (4) to (3) leads to the so-called coherent structure ODE /parenleftbiggα−1 1α/parenrightbigg/parenleftbigg(Ω−sφ′)sin(θ) sθ′/parenrightbigg =/parenleftbigg2φ′θ′cos(θ) −θ′′/parenrightbigg +sin(θ)/parenleftbiggφ′′+β/(1+ccpcos(θ)) (φ′)2cos(θ)+h−µcos(θ)/parenrightbigg ,(5) where′=d/dξ. This system of two second-order ODEs does not depend on φand thus reduces to three dynamical variables ( θ,ψ=θ′,q=φ′). Following standard terminology for coherent structures, we refer to qas thelocal wavenumber . Writing (5) as a first-order three-dimensional system gives θ′=ψ ψ′= sin(θ)[h−Ω+sq+(q2−µ)cos(θ)]−αsψ q′=αΩ−β/(1+ccpcos(θ))−αsq−s+2qcos(θ) sin(θ)ψ, (6) and DWs in the original PDE are in 1-to-1-correspondence with the O DE solutions connecting θ= 0 andθ=π. 2.1.1 Blow-up charts and asymptotic states As in [12], the singularities at zeros of sin( θ) in (6) can be removed by the singular coordinate change ψ:=psin(θ), which is a blow-up transformation mapping the poles of the sphere ±e3to circles thus creating a cylinder. The resulting desingularized system reads θ′= sin(θ)p p′=h−Ω−αsp+sq−(p2−q2+µ)cos(θ) q′=αΩ−β/(1+ccpcos(θ))−sp−αsq−2pqcos(θ).(7) The coherent structure system (6) is equivalent to the desingular ized system (7) for θ/ne}ationslash=nπ,n∈Zand therefore also for maway from ±e3. Furthermore, the planar blow- up chartsθ= 0 andθ=πare invariant sets of (7), which are mapped to the single pointse3and−e3, respectively by the blow-down transformation. System (7) has a special structure (cf. Figure 6) that will be relevant for the subs equent DW analysis. In the remainder of this section we analyze this in some detail. 9Lemma 1. Consider the equations for pandqin(7)for an artificially fixed value of θ. In terms of z:=p+iqthis subsystem can be written as the complex (scalar) ODE z′=Az2+Bz+C, (8) whereA:=−cos(θ),B:=−(α+i)s,andCθ:=h−Ω+Aµ+i/parenleftig αΩ−β 1−Accp/parenrightig . ForA/ne}ationslash= 0the solution with z0=z(ξ0)away from the equilibria zθ +=−B 2A+ iγθ 2Aand zθ −=−B 2A−iγθ 2A, withγθ=γ(θ):=√ 4ACθ−B2, reads z(ξ) =γθ 2Atan/parenleftbiggγθ 2ξ+δ0/parenrightbigg −B 2A, (9) where δ0= arctan/parenleftbigg2Az0+B γθ/parenrightbigg −γθ 2ξ0. ForA= 0, the solution away from the equilibrium zπ/2=−Cπ/2/Bis given by z(ξ) =/parenleftbigg z0+Cπ/2 B/parenrightbigg eB(ξ−ξ0)−Cπ/2 B. Clearly, the solution of (8) relates only to those solutions of (7) for whichθis constant, i.e.,θ= 0,π. Although we are mostly interested in the dynamics on the blow-up ch arts, we consider θas a parameter in order to demonstrate the special behaviour of ( 7) forθ artificially fixed. Notably, the equilibria zθ ±of (8) forθ/ne}ationslash= 0,πare not equilibria in the full dynamics, due to the fact that (7) is only invariant for θat the blow-up charts. Proof.We readily verify the claimed form of the ODE and directly check the cla imed solutions. /squaresolid Remark 3. Lemma 1 states in particular that the desingularized ODE sys tem(7)can be solved explicitly on the invariant blow-up charts, where θ= 0,πand thusA=−1,1, respectively. System (7)possesses two real equilibria on each blow-up chart, Z0 ±:= (0,p0 ±,q0 ±)TandZπ ±:= (π,pπ ±,qπ ±)T. Herepθ σ:= Re(zθ σ),qθ σ:= Im(zθ σ)forθ= 0,π, σ=±and z0 +:= 1/2(B−iγ0), z0 −:= 1/2(B+iγ0) and analogously zπ +:= 1/2(−B+iγπ), zπ −:= 1/2(−B−iγπ), where we set γ0:=γ/vextendsingle/vextendsingle A=−1=√ −4C0−B2andγπ:=γ/vextendsingle/vextendsingle A=1=√ 4Cπ−B2 withC0:=C/vextendsingle/vextendsingle A=−1andCπ:=C/vextendsingle/vextendsingle A=1. Due to the analytic solution (9), we obtain the following more detailed r esult in case θ/ne}ationslash=π/2 (cf. Figure 6). 10Lemma 2. For each given 0≤θ≤πwithθ/ne}ationslash=π/2as a parameter, the fibers of (7)with constantθconsists entirely of heteroclinic orbits between zθ −andzθ +in caseIm(γθ)/ne}ationslash= 0, orγθ/ne}ationslash= 0, except for the equilibrium states. In case Im(γθ) = 0andRe(γθ)/ne}ationslash= 0, the fiber at fixed θis filled with periodic orbits away from the invariant line {q=s 2A}, for which the period of solutions close to it tends to infinity. Proof.Forθfixed in (8), consider the case Re( γθ) = 0 and also Im( γθ)/ne}ationslash= 0 forA/ne}ationslash= 0 which leads to z(ξ) = iIm(γθ) 2A·tan/parenleftbigg i/parenleftigIm(γθ) 2ξ+Im(δ0) /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =:ˇξ/parenrightig +Re(δ0)/parenrightbigg −B 2A =Im(γθ) 2A·isin(2Re(δ0))−sinh/parenleftbig 2ˇξ/parenrightbig cos(2Re(δ0))+cosh/parenleftbig 2ˇξ/parenrightbig−B 2A. For Re(γθ)/ne}ationslash= 0 as well as Im( γθ)/ne}ationslash= 0, we obtain z(ξ) =γθ 2Atan/parenleftbiggRe(γθ) 2ξ+Re(δ0) /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =:˜ξ+iIm(γθ) 2ξ+iIm(δ0)/parenrightbigg −B 2A =γθ 2A·sin(2˜ξ)+isinh/parenleftig 2/parenleftig Im(γθ) Re(γθ)˜ξ−Im(γθ) Re(γθ)Re(δ0)+Im(δ0)/parenrightig/parenrightig cos(2˜ξ)+cosh/parenleftig 2/parenleftig Im(γθ) Re(γθ)˜ξ−Im(γθ) Re(γθ)Re(δ0)+Im(δ0)/parenrightig/parenrightig−B 2A, The asymptotic states are Im/parenleftbig γθ/parenrightbig >0 : lim ξ→−∞z(ξ) =−iγθ 2A−B 2A,lim ξ→+∞z(ξ) = iγθ 2A−B 2A, as well as Im/parenleftbig γθ/parenrightbig <0 : lim ξ→−∞z(ξ) = iγθ 2A−B 2A,lim ξ→+∞z(ξ) =−iγθ 2A−B 2A, which simplify in case Re( γθ) = 0 to Im/parenleftbig γθ/parenrightbig >0 : lim ξ→−∞z(ξ) =Im(γθ)−B 2A,lim ξ→+∞z(ξ) =−Im(γθ)+B 2A, as well as Im/parenleftbig γθ/parenrightbig <0 : lim ξ→−∞z(ξ) =−Im(γθ)+B 2A,lim ξ→+∞z(ξ) =Im(γθ)−B 2A. Note that the asymptotic states coincide if γθ= 0. 11The last case to consider is Re/parenleftbig γθ/parenrightbig /ne}ationslash= 0 and Im/parenleftbig γθ/parenrightbig = 0, where the solutions are z(ξ) =Re/parenleftbig γθ/parenrightbig 2A·sin(2ˆξ)+isinh(2Im( δ0)) cos(2ˆξ)+cosh(2Im( δ0))−B 2A, withˆξ:=Re(γθ) 2ξ+Re(δ0) and which leads to periodic solutions of (8) iff Im(δ0)/ne}ationslash= 0⇔2AIm(z0)+Im(B)/ne}ationslash= 0⇔q0/ne}ationslash=s 2A, wherez0=p0+iq0and recall that B=−(α+i)s. /squaresolid Based on Lemma 2, explicitly onthe blow-up chart θ= 0 the heteroclinic orbits are from z0 −toz0 +in case Im( −4C0−B2)>0, or Im( −4C0−B2) = 0 and Re( −4C0−B2)≤0, and fromz0 +toz0 −if Im(−4C0−B2)<0 . Forθ=π, if Im(4Cπ−B2)>0, or Im(4Cπ−B2) = 0 and Re(4 Cπ−B2)≤0 they are connections from zπ −tozπ +, and if Im(4Cπ−B2)<0 fromzπ +tozπ −. ForA/ne}ationslash= 0, the case s= 0 is a special situation, which will be also discussed in the context of DWs in §3 later. It turns out that on the blow-up charts θ= 0 (orθ=π), the solution with appropriate initial conditions has a limit as |ξ| → ∞if and only if Im(√ −C0)/ne}ationslash= 0 (Im(√ Cπ)/ne}ationslash= 0). In terms of the parameters in (7) and with β−:=β 1−ccpandβ+:=β 1+ccp, this leads to the conditions for θ= 0 given by: Ω/ne}ationslash=β+ αor Ω =β+ αand Ω≤h−µ, (10) and forθ=πgiven by: Ω/ne}ationslash=β− αor Ω =β− αand Ω≥h+µ, (11) In caseccp= 0, i.e. for the LLG equation, the conditions in (10) and (11) reduce to Ω/ne}ationslash=β αor Ω =β αand 2µ≤β α, where the latter inequality always holds in case of a nanowire geometr y (µ<0). Hence standing domain walls in nanowires in case ccp= 0 can only connect equilibria, if they exist. Lemma 2 also states that the equilibria on the blow-up charts θ∈ {0,π}are surrounded by periodic orbits in case Im( γ0) = 0 and Re( γ0)/ne}ationslash= 0 (Im(γπ) = 0 and Re( γπ)/ne}ationslash= 0). In fact, system (7) is Hamiltonian (up to rescaling) on the blow-up ch arts for certain frequencies Ω, as follows 122p -11q (a)-5 5ξ -1 2 p q (b) Figure 5: (a) Phase plane streamplot with Mathematica of (14) around the equilibrium zπ −, i.e., (7) at θ=π, forα= 0.5,β= 0.1,µ=−1,h= 10.2,s= 4,Ω = 8.2 andccp= 0, which leads to/parenleftbig pπ −,qπ −/parenrightbigT= (1,0)T. The red solid line marks the trajectory with initial condition ( p0,q0) = (7/4,0) (cf. plot of solutions in b). (b) Plot of the profile for the solution highlighted in (a), where p(solid blue line) and q(dashed red line) are given by (9) for the parameter set as in (a). Proposition 1. The dynamics of (7)on the invariant blow-up chart θ= 0in case Ω =β+ α−s2 2possesses the invariant line {q=−s 2}and, after time-rescaling, for q/ne}ationslash=−s 2 the Hamiltonian H0(p,q) =−p2+q2+αsp+sq−h+β+/α+µ q+s 2 along solutions of (8). Analogously on the chart θ=π, in case Ω =β− α+s2 2(12) possesses the invariant line {q=s 2}and forq/ne}ationslash=s/2the Hamiltonian Hπ(p,q) =p2+q2−αsp−sq+h−β−/α+µ q−s 2. (13) Moreover, each half plane {θ= 0,q≤ −s 2},{θ= 0,q≥ −s 2}/parenleftbig {θ=π,q≤s 2}and {θ=π,q≥s 2}/parenrightbig is filled with periodic orbits encircling the equilibria at z0 ±/parenleftbig zπ ±/parenrightbig if addi- tionallyΩ>h−µ+s2 4(α2−1)/parenleftig Ω<h+µ+s2 4(1+α2)/parenrightig . Proof.For the sake of clarity, we will only present the computation for the blow-up chart θ=π; the computation on θ= 0 can be done in the same manner. With respect to the parameters of equation (7), the condition Im(4 C−B2) = 0 is equivalent to Ω =β− α+s2 2 and in this case the system (7) on {θ=π}is given by p′=p2−q2−αsp+sq+h+µ−β− α−s2 2 q′= 2pq−sp−αsq+αs2 2(14) 13We readily compute that for solutions of this dHπ dξ=∂Hπ ∂pp′+∂Hπ ∂qq′=q′·p′−p′·q′ (q−s 2)2= 0, which shows the canonical Hamiltonian structure of (14) up to time r escaling. If addi- tionally Ω< h+µ+s2 4(1+α2), it follows from Lemma 2 that each half plane is filled with periodic orbits. /squaresolid Proposition 1 concerns the special case that ccp∈(−1,1) andβ,Ω are such that (12) holds, whichishenceforthreferredtoasthe center-case . Inparticular, eachorbitexcept the lineq≡s/2 on the blow-up chart θ=πcan by identified via the quantity (13), and each equilibrium zπ ±has a neighborhood filled with periodic orbits if additionally h >β− α−µ+s2 4(1−α2) (cf. Figure 5). Note the relation between the conditions (10) and (11) and the conditions in Proposition 1 in case s= 0. Based on Lemma 2, we also state the following uniqueness result. Proposition 2. ForΩ<β+ α−s2 2/bracketleftig Ω>β+ α−s2 2/bracketrightig , orΩ =β+ α−s2 2andΩ≤h−µ+ s2 4(α2−1)there is a unique orbit with (θ,p,q)T(ξ)withθ(ξ)→0asξ→ −∞, and it holds that (p+iq)(ξ)→z0 −/bracketleftbig (p+iq)(ξ)→z0 +/bracketrightbig asξ→ −∞. Proof.The conditions on Ω are equivalent to those in Lemma 2. If the statem ent were false, it nevertheless follows from Lemma 2 that ( p+iq)(ξ)→z0 −asξ→ −∞. However, transverse to the blow-up chart, the equilibrium state Z0 −is stable for increasing ξand thus repelling for decreasing ξ. This contradicts the requirement θ(ξ)→0 asξ→ −∞. Together with the fact that Z0 −has a one-dimensional unstable manifold uniqueness follows. Analogously in case Ω >β+ α−s2 2. /squaresolid Domain walls are heteroclinic orbits between the blow-up charts and d ecisive for their bifurcation structure are the dimensions (and directions) of un/s table manifolds of the equilibria on these charts. Hence, we next discuss the equilibria Z0 ±andZπ ±and their stability. Transverse to the blow-up charts in θ-direction we readily compute the linearization ∂θ(sin(θ)p) = cos(θ)p, i.e., the transverse eigenvalue is −Aθ·Re(zθ ±) atθ= 0 andπ, respectively. The eigenvalues within the blow-up charts are determ ined by±iγ. With σ=±, respectively, the eigenvalues for Z0 σare ν0 1,σ=−σiγ0, ν0 2,σ=ν0 1,σ, ν0 3,σ= Re(z0 σ) (15) and forZπ σ νπ 1,σ=σiγπ, νπ 2,σ=νπ 1,σ, νπ 3,σ=−Re(zπ σ). (16) Therefore, the signs of the real parts within each blow-up chart a re opposite at Zπ + compared to Zπ −and determined by the sign of Re/parenleftbig ν0,π 1,+/parenrightbig . Hence, within the blow-up charts each equilibrium is either two-dimensionally stable, unstable or a linearly neutral center point. 142p -11q (a)2p -11q (b)2p -11q (c) 2p -11q (d)2p -11q (e)2p -11q (f) Figure 6: Phase plane streamplots (with Mathematica ) in blow-up charts near the equilib- riumzπ −= (1,0) forα= 0.5,β= 0.1,µ=−1,ccp= 0, i.e., the second and third equation of (7). (a-c) θ= 0 and (d-f) θ=π. (a,d) codim-2 regime, (b,e) center case, where Ω = β/α+s2/2 holds on the chart θ=π, and (c,f) codim-0 regime. The remaining parameters and equilibria in (a,d): h= 0.5,s0= 0.12, Ω0= 0.44, andz0 +=−1.06−0.12i,zπ +=−0.94+0.12i. In (b,e): h= 10.2,s0= 4, Ω 0= 8.2, andz0 +=−3−4i,zπ += 1 + 4i. In (c,f): h= 50.0,s0= 19.92, Ω0= 40.04, and z0 +=−10.96−19.92i,zπ += 8.96+19.92i. For completeness, we next notethat the equilibria onbothblow-up c harts can beneutral centers simultaneously (cf. Figure 3). However, this requires a ne gative spin polarization and a small Gilbert damping factor, and is not further studied in this p aper. Remark 4. The equilibria of both blow-up charts are centers simultane ously, if and only ifIm(±γ0,π) = 0andγ0,π/ne}ationslash= 0(compare Lemma 2). For example if α= 0.5,β= 0.1,µ= −1,ccp=−0.99,h= 10 s2=3960 199,Ω =β/α 1−ccp+s2 2=2000 199, we obtain γ0= 3.33551, γπ= 3.27469. 3 Domain Walls All domain walls between ±e3that we are aware of are of coherent structure type, and thus in one-to-one correspondence to heteroclinic connectio ns between the blow-up 15charts{θ= 0}and{θ=π}in(7). Typically we expect these to beheteroclinics between equilibria within the charts, but this is not necessary. Based on the p revious analysis, there are three options for heteroclinics between the charts: po int-to-point, point-to- cycle, and cycle-to-cycle. We study the first two in this section, fo r which Proposition 2 implies uniqueness oftheDW(uptotranslations/rotations) foragiv enset ofparameters. The third case can occur at most in a relatively small set of paramete rs (see Remark 4). Its analysis is beyond the scope of this paper. Note that in case of an existing connection between an equilibrium and a periodic orbit (see Proposition 1), the domain wall is automatically an inhomogeneou s non-flat one. Moreover, via the singular coordinate change any such heteroclinic solution is hetero- clinic between θ= 0,πin (6) and through the spherical coordinates it is a heteroclinic connection between ±e3in the sphere, possibly with unbounded ϕ. 3.1 Homogeneous Domain Walls It is known from [13] in case β= 0 and from [12] in case ccp= 0 (and arbitrary β) that (7) admits for µ<0 a family of explicit homogeneous DWs m0given by θ0 p0 q0 = 2arctan/parenleftbig eσ√−µξ/parenrightbig σ√−µ 0 (17) and parameterized by Ω =h+αβ 1+α2,s2=−(β−αh)2 µ(1+α2)2>0, andσ= 1 for positive speed s andσ=−1 for negative s; the family extends to s= 0 in the limit h→β αwith scaling of the frequency by Ω =β α+√−µ αs. Fors= 0 (standing) fronts with both orientations exist simultaneously ccp= 0 and are given by θ0 p0 q0 = 2arctan/parenleftbig e±√−µξ/parenrightbig ±√−µ 0 . Hence, the branches of left and right moving walls as parametrized b yseach have termination point at s= 0 (cf. Figure 13). The family of explicit DWs (17) have domain wall width√−µ, a profile independent of the applied field hand propagate along a nanowire ( µ <0) with velocity swhile precessing with azimuthal velocity Ω. Since these are unique up to sp atial reflection symmetry, the direction of motion is related to the spatial direction of connecting ±e3 throughσ, θ(−∞) = 0θ(+∞) =π⇔s>0 (wall moves to the right) θ(−∞) =π θ(+∞) = 0⇔s<0 (wall moves to the left) .(18) To simplify some notation we will focus on the case of right-moving walls including standing walls ( s≥0) and thus make the standing assumptions that h≥β/αas well as 16µ <0. We therefore have a 1-to-1 relation of parameters ( α,β,h,µ) and right-moving DWs from m(ξ,t) =m0(ξ,t;α,β,h,µ) with speed and frequency given by s0=s0(α,β,h,µ) :=αh−β√−µ(1+α2),Ω0= Ω0(α,β,h,µ) :=h+αβ 1+α2(19) where the subindex 0 emphasizes that ccp= 0. Sinces0is surjective on R≥0any velocity can be realised. Spatial reflection covers the case h≤β/α. Based on Lemma 1 as well as Remark 3 for ccp= 0 and (homogeneous) speed and frequency (19), one readily finds the asymptotic states of (7) giv en by E0:=Z0 −/vextendsingle/vextendsingle (s0,Ω0)=/parenleftbig 0,√−µ,0/parenrightbigTandEπ:=Zπ −/vextendsingle/vextendsingle (s0,Ω0)=/parenleftbig π,√−µ,0/parenrightbigT, with (spatial) eigenvalues (15), (16) given by ν0 k,−:=−αs0−2√−µ−(−1)kis0, ν0 3,−=√−µ, νπ k,−:=−αs0+2√−µ−(−1)kis0, νπ 3,−=−√−µ,(20) wherek= 1,2. Note that the above equilibria cannot be centers simultaneously ( recall µ <0), hence a cycle-to-cycle connection can not exist close to it (see Remark 4 for details). For this reason, we focus on point-to-point as well as poin t-to-cycle connections. 3.2 Inhomogeneous Domain Walls Homogeneous DWs exist only in case ccp= 0 [12, Theorem 5], are explicitly given by (17) and completely characterized by (19). By [12, Theorem 6], f ast inhomogeneous DW solutions with |s| ≫1 exist for any ccp∈(−1,1), but in contrast to (17), the gradient of these profiles is of order 1 /|s|and thus have a large ‘width’. The natural question arises what happens for any sin caseccp/ne}ationslash= 0. This section contains the main results of this paper: the existence, parameter selection and structure of inhomogeneous DW solutions in case of small |ccp|for any value of the applies field h, and thus any speed s. This will be achieved by perturbing away from the explicit solution m0given by (17), where the bifurcation structure is largely determine d by comparing the dimensions of the un/stable eigenspaces at the as ymptotic equilibrium states, which are determined by (20). LetW0 s/uandWπ s/udenote the stable and unstable manifolds associated to E0and Eπ, respectively, and w0 s/uas well aswπ s/ube the dimension of these manifolds so that w0 s+w0 u=wπ s+wπ u= 3. Notably w0 s= 2 andw0 u= 1 for all values of the parameters, andwπ sis either 1 or 3. Recall the standing assumption s0≥0. Ifwπ s= 1, the heteroclinic connection of E0andEπgenerically has codimension-2 , while forwπ s= 3 it has codimension-0 , and we refer to the transition point between 17these cases, following the discussion in §2.1.1, as the center case . From (20) we have wπ s= 1⇔0≤s0<2√−µ α,wπ s= 3⇔s0>2√−µ αand the center case at s0=2√−µ α. Hence, within the family of homogeneous DWs given by (17) and satisf ying (19), the different bifurcation cases have speed and frequency relations codim-0:s0>2√−µ αand Ω 0>β α−2µ α2, center:s0=2√−µ αand Ω 0=β α−2µ α2, codim-2: 0 ≤s0<2√−µ αandβ α≤Ω0<β α−2µ α2. Using (19) these can be written in terms of the parameters of (LLG S), which gives the characterization mentioned in the introduction §1. Remark 5. The case distinction is also related to the spectral stabili ty of the asymptotic statesm=±e3in the dynamics of the full PDE (LLGS)which is beyond the scope of this paper, but see Figure 3 for an illustration. In short, it follows from, e.g., [12, Lemma 1] that e3isL2-stable forh>β/α, while−e3isL2-stable forh<β/α −µand unstable for h>β/α −µ. Based on this, the stability curves in Figure 3 are defined as follows Γ+:=β/α h−µ−1,Γ−:= 1−β/α h+µ, which intersect at h=β 2α+/radicalbigg β2 4α2+µ2. Since the destabilisation of −e3ifβ >0corresponds to a Hopf-instability of the (purely essential)spectrum, it is effectively invisible in the coherent struct ure ODE, which detects changes in the linearization at zero temporal eigenvalue on ly. Visible from the PDE stability viewpoint is a transition of absolute spectrum th rough the origin in the complex plane of temporal eigenmodes, cf. [18]. Now in the center cas e, the state −e3is already L2-unstable since α>0as well asµ<0andh>β/α implies h=h∗=β α−2µ α2(1+α2)>β α−µ and therefore that Γ−never intersects the line ccp≡0ath=h∗. Moreover, it was shown in [19] that the family of explicit hom ogeneous DWs (9)is (linearly) stable for sufficiently small applied fields, actually for h <−µ/2, in case β= 0, hence in the bi-stable case where ±e3areL2-stable. As mentioned before, β= 0 is equivalent to ccp= 0in the LLGS equation with an additional shift in handβ, which leads to the (LLG)case. We expect these DWs are also stable for small perturbat ions in ccp, due to the properties of the operator established in [19], b ut further analysis also on the transition from convective/transient to absolute inst ability will be done elsewhere. 18With these preparations, we next state the mainresults, which con cern existence of DWs in the three regimes. Theorem 1. For any parameter set (α0,β0,h0,µ0)in the codim-0 case, i.e., µ0<0and h0> β0/α0−2µ0−2µ0/α2 0, the following holds. The explicit homogeneous DWs m0 in(9)lies in a smooth family mccpof DWs parameterized by (ccp,α,β,h,µ,s, Ω)near (0,α0,β0,h0,µ0,s0,Ω0)withs0,Ω0from(19)evaluated at (α0,β0,h0,µ0). Moreover, in caseccp= 0and(s,Ω)/ne}ationslash= (s0,Ω0)evaluated at (α,β,h,µ), orccp/ne}ationslash= 0, these are inhomogeneous flat DWs. Proof.As mentioned, in the codim-0 case we have wπ s= 3 and for all parameters w0 u= 1. Due to the existence of the heteroclinic orbit (17), this means W0 uintersectsWπ s transversely and non-trivially for ccp= 0 in a unique trajectory. Therefore, this DW perturbs to a locally unique family by the implicit function theorem for p erturbations of the parameters in (7). Forccp/ne}ationslash= 0 sufficiently small these are inhomogeneous DWs since the derivativ e of the third equation, the q-equation, in (7) with respect to ccpis nonzero in this case; hence already the equilibrium states move into the inhomogeneous regime. Forccp= 0 but (s,Ω)/ne}ationslash= (s0,Ω0) at (α,β,h,µ), it follows from [12, Theorem 5] that these DWs cannot be homogeneous. /squaresolid Next we consider the center case, where h=h∗=β/α−2µ−2µ/α2. We start with a result that follows from the same approach used in the codim-2 cas e and give refined results below. Corollary 1. The statement of Theorem 1 also holds for a parameter set in th e center case if the perturbed parameters (ccp,α,β,h,µ,s, Ω)satisfyΩ> s2/2 +β−/α. IfΩ = s2/2 +β−/αandΩ< h+µ+s2 4(1 +α2)the same holds except the DW is possibly non-flat. Proof.It follows from Proposition 1 and the discussion before that Ω >s2/2+β−/αfor the parameter perturbation implies that the eigenvalues of the per turbed equilibrium Zπ −≈Eπsatisfy Re(νπ k,−)<0,k= 1,2. Hence, the stable manifold at the target equi- librium is two-dimensional and lies in a smooth family with the center-sta ble manifold at the transition point. Then the proof is the same as in the codim-0 c ase. If the pertur- bation has Ω = s2/2+β−/αthen we consider as target manifold the three-dimensional stable manifold of a neighborhood of Zπ −within the blow-up chart θ= 0. This neigh- borhood consists of periodic orbits by Proposition 1 if Ω < h+µ+s2 4(1 +α2). By dimensionality the intersection with the unstable manifold of Z0 −persists and yields a heteroclinic orbit from the perturbed equilibrium at θ= 0 to the blow-up chart at θ=π. Perturbing ccpaway from zero moves the left-asymptotic state into the inhomoge neous regime and thus generates an inhomogeneous DWs. Note from Prop osition 1 that the right-asymptotic state is either an equilibrium with q/ne}ationslash= 0 or a periodic orbit along which q= 0 happens at most at two points. /squaresolid 19Next we present a refined result in which we show that typical pertu rbations indeed give non-flat DWs, i.e., heteroclinic connections with right-asymptotic st ate being a periodic orbit. The existence of flat DWs for ccp/ne}ationslash= 0 is severely constrained, but not ruled out by this result. Our numerical results, such as those presented in §4, always lead to a selected solution with a periodic asymptotic state. In addition, attempts to perform numerical continuation (see §4) of flat DWs to ccp/ne}ationslash= 0 failed. Here we added the constraint /tildewideH= 0 and allowed adjustment the parameters h ands, but the continuation process did not converge, which confirms nu merically the generic selection of a periodic orbit. As mentioned before, the right asymptotic state is e3in either case in the PDE coordi- nates; the difference between flat and non-flat lies in the finer deta ils of how the profile approaches e3in term ofpand alsoq, which relates to mthrough (2). Theorem 2. Consider the smooth family of DWs from Corollary 1 with ccp= 0for parameters satisfying (12)with fixedα >0,β≥0, andµ <0. Then there is a neighborhood (ccp,s,h)of(0,s0,h∗)suchthat the followingholds. FlatDWs occurat most on a surface in the (ccp,s,h)-parameter space and, for β/ne}ationslash= 0, satisfy|h−h0|2+|s−s0|2= O(|ccp|3), more precisely (31)holds, where h0=h∗ands0= 2√−µ/α. Otherwise DWs are non-flat, in particular all DWs not equal to m0forccp= 0orβ= 0are non-flat. Due to its more technical nature, the proof of this theorem is defe rred to Appendix 6.1. It remains to consider the codim-2 case. Theorem 3. For any parameter set (α0,β0,h0,µ0)in the codim-2 case, i.e., µ0<0and β0/α0≤h0< β0/α0−2µ0−2µ0/α2 0, the following holds. The explicit homogeneous DWsm0in(9)lies in a smooth family of DWs parameterized by (ccp,α,β,h,µ )near (0,α0,β0,h0,µ0). Here the values of (s,Ω)are functions of the parameters (ccp,α,β,h,µ ) and lie in a neighbourhood of (s0,Ω0)from(19). This family is locally unique near m0 and forccp/ne}ationslash= 0consists of inhomogeneous flat DWs. The proof of Theorem 3 is presented in Appendix 6.2 and is based on th e Melnikov method for perturbing from m0. As the unperturbed heteroclinic orbit has codimension two, thebifurcationisstudied inathree-parametricfamilywithpert urbationparameters η:= (ccp,s,Ω)T, which yields a two-component splitting function M(η) that measures the mutual displacement of the manifolds W0 uandWπ s. Due to the fact that Re( νπ j,−)<0 also forβ= 0 in the codim-0 regime and the case β= 0 is included in Theorem 2 as well as Theorem 3, we immediately get the f ollowing result. Corollary 2. Inhomogeneous flat DWs also exist in the LLG equation (β= 0), which can be flat or non-flat, respectively. Theorem3 completes theexistence study ofDWs. Therefore, for any valueoftheapplied fieldhthere exists a heteroclinic connection between the blow-up charts withccp/ne}ationslash= 0 andq/ne}ationslash≡0, thus an inhomogeneous (typically flat) DW. Recall that we have fo cused on 20right moving DWs, but all results are also valid for left moving walls due t o symmetry. Therefore inhomogeneous DWs exist with ccp/ne}ationslash= 0 for any value of the applied field h∈R. 4 Numerical Results Numerical continuation for ordinary differential equations is an est ablished tool for bi- furcation analysis in dynamical system. In this section we present c ontinuation results to illustrate the analytical results discussed in §3. In particular, we will focus on con- tinuation in the parameter ccpin the range of ( −0.5,0.5) as this perturbs away from the known family m0from (17) (cf. Figure 1b) with speed and frequency determined by (19) for a given applied field. Note that we also focus only on right-moving f ronts in this section for reasons of clarity. All results were produced by contin uation in AUTO-07P and graphics were created with Mathematica as well as MATLAB . Heteroclinic orbits were detected as solutions to the boundary valu e problem given by the desingularized system (7) plus a phase condition andboundary c onditions at ξ=−L andξ=Ltaken from the analytic equilibrium states in pandqon the blow-up charts (Remark 3). In the codim-2 case, the four required conditions are thep,qvalues at the charts. In the center case, the three required conditions are: ( 1,2) the two p,qvalues at the left chart and (3) the energy difference determined by the f unction (13). In the codim-0 case, the two required conditions are the pvalues at both charts. Moreover, we foundL= 50 was sufficiently large. In order to relate to (LLGS), we plot most of the profiles after blow ing down to the sphere rather than using the ODE phase space. (a)-5 5-0.04-0.02 ξq - 5
-101 ξ m1 (b) Figure 7: DWs obtained from continuation of m0in system (7) in the codim-2 regime h= 0.5 (h∗= 10.2) with initial speed and frequency s0= 0.12 as well as Ω 0= 0.44, and (ccp,s,Ω) = (−0.5,0.11221,0.44077). (a) Projection onto the sphere. (b) Zoom-in of corresponding q-profile (red) and m1component (blue). Following the standing assumption on positive speeds and using ccpas well ashas the main parameters, we keep the other parameters fixed with values α= 0.5,β= 0.1,µ=−1. 21(a) (b) Figure 8: DWs obtained from continuation of m0in system (7) projected onto the sphere in the codim-2 regime h= 10.1 (h∗= 10.2) with initial speed and frequency s0= 3.96 and Ω 0= 8.12. (a) ( ccp,s,Ω) = (−0.5,3.99541,8.05973). (b) ( ccp,s,Ω) = (0.5,4.08089,8.22402). The value of the applied field for the center case, given the fixed par ameters, is h∗= 10.2 (cf.§3.2), which leads to s0= 4.0 as well as Ω 0= 8.2 (cf. (19)). 4.1 Codim-2 case The lower boundary for values of the applied field hlies in the codim-2 regime and is given by h=β/α= 0.2. As a first numerical example we consider the slightly larger value h= 0.5. The results upon continuation in the negative as well as positive direction of ccpare presented in Figures 2a, 2c, and 7. The inhomogeneous nature of these solutions ( ccp/ne}ationslash= 0) is reflected in the significantly varying azimuthal angles, also visible in the oscillatory nature of the m1component in Figures 2c as well as 7b. The linear part of the splitting function (33) (see Theorem 3), which predicts the direc- tion of parameter variation for the existence of inhomogeneous DW s (ccp/ne}ationslash= 0) to leading order, reads in this example M(ccp,s,Ω) =/parenleftbigg −0.00147567 −0.499245 0.245945 −0.000577908 −0.245945 −0.499245/parenrightbigg · ccp s Ω , sothatM= (0,0)Tfor(s,Ω) = (−0.00283744 ·ccp,0.000240252 ·ccp). Fortheparameter values in Figure 7 and 2a we obtain, respectively, M(−0.5,−0.007788,0.000771) = (0 .00481558,0.00181945)T, M(0.5,−0.007973,0.007173) = (0 .00648248,−0.00133122)T. Note that here the splitting of the (1-dimensional) unstable manifold of the left equilib- rium and the (1-dimensional) stable manifold of the right equilibrium diffe r, i.e., are in opposite directions (signs) in frequency and speed for variations in ccp. 22In addition note the decrease in frequency in the m1component, and thus also in the m2as a result of the increase of the qcomponent towards zero, cf. Figure 7b. Here, the azimuthal angle decreases since φ=/integraltext qandq <0. As a further example in the codim-2 regime, we consider h= 10.1<10.2 =h∗near the upper boundary of the codim-2 regime in terms of the applied field h. The results of the continuation in ccpare presented in Figure 8. The linear approximation of the splitting in this case is given by M(−0.5,0.03541,−0.06027) = ( −0.000175537,−0.00104378)Tas wellasM(0.5,0.12089,0.10402) = ( −0.00149519,0.0014975)Tin(a)and(b),respectively. Note that the direction of splitting of the two components in this cas e is also dependent on the polarity sign, as in the previous example. In both cases ( h= 0.5 andh= 10.1), thecontinuationresultslookbasicallythesameinthecodim-2regime, wherethesolution is, roughly speaking, constantly spiraling down from the north to th e south pole. 4.2 Center case (a)-20 200.0450.055 ξq (b) Figure 9: DWs obtained from continuation of m0in system (7) in the center regime h=h∗= 10.2 withs=s0= 4,Ω = Ω 0= 8.2, andccp= 0.5. (a) Projection onto the sphere. (b) Profile of corresponding q-component. We perform computations in the center case with applied field h=h∗= 10.2 and fixed frequency Ω = β−/α+s2/2 (see Proposition 1 and its discussion details). Theorem 2 shows that the right asymptotic state is generically a periodic orbit a nd more precisely that in case ccp= 0, no constellation of handsexists, both not equal to zero, for which the right asymptotic state is the (shifted) equilibrium. The results o f continuation in ccp projected on the sphere look quite the same, which is why only the re sult forccp= 0.5 is presented in Figure 9. The fact that the right asymptotic state is a periodic orbit on the blow-up chart θ=πis reflected by the nearly constant oscillations in the q-profile forξclose to the right boundary (cf. Figure 9b). That the right state is not the equilibrium in the blow-up chart is furth er corroborated by computing the difference in energy /tildewideHbetween this equilibrium state (see Remark 3) and the approximate right asymptotic state obtained from continu ation. The analytic prediction of this difference up to second order is given by (31), whic h reads, for the 23-0.5 0 0.5 ccp-2-1010-6 (a)9.2 10.2 11.2 h-1010-4 (b)3.5 4 4.5 s-2010-3 (c) Figure 10: Continuation of m0insystem(7)inthecenter casewithappliedfield h=h∗= 10.2 and fixed frequency Ω = β−/α+s2/2; heres0= 4. Shown is the energy difference (solid blue line) between the equilibrium and asymptotic st ate from continuation on right boundary against the continuation parameter ccpin (a),hin (b), and sin (c). The red dashed curve in (b) and (c) is the quadratic appro ximation (21). chosen parameters, −0.006612+0.00673s−0.00183s2−0.00134h−0.000086h2+0.00077hs.(21) As this analytic prediction is independent of ccpthe dependence of /tildewideHonccp≈0 is of cubic or higher power. Indeed, the results plotted in Figure 10a sug gest an at least quartic dependence since a maximum lies at ccp= 0. The asymmetric nature of the graph suggests that odd powers appear in the expansion beyond o ur analysis, but also note the order of 10−6in/tildewideH. In addition to the dependence on ccp, continuations for ccp= 0 of/tildewideHinhwith fixeds=s0= 4 and in swith fixedh=h∗= 10.2 are plotted in Figure 10b and 10c, respectively. Here we also plot the quadratic pr ediction (21). 4.3 Codim-0 case Next, we consider an applied field h= 10.3 in the codim-0 regime, just above the applied field value for the center case h∗= 10.2. The results of continuation in ccpare plotted on the sphere in Figure 11. The azimuthal profile in φand hence in qare non-trivial as predicted for inhomogeneous DWs. In the ODE, qpossesses an oscillating profile and has a monotonically decreasing am - plitude in both cases. This is a consequence of the proximity to the ce nter case and the convergence to equilibria (see §3.2 for details). Recall that the speed and frequency are not selected by the existence problem during continuation in ccp, but are taken as the fixed parameters ( s0,Ω0) defined in (19). The final example is for a relatively large applied field h= 50 in the codim-0 regime, far away from the center case, and the results of continuation in ccpprojected on the sphere are presented in Figure 2b as well as Figure 12a. Moreover, the corresponding m1andm2profiles for ccp= 0.5 are presented in Figure 2d, and for ccp=−0.5, in Figure 12b. As in the previous example, the inhomogeneous nature is visible in the non-trivial azimuthal profile. In summary, switching on the parameter ccpleads to a variety of inhomogeneous flat as well as non-flat DW solutions, but also in case ccp= 0 there exist inhomogeneous DWs 24(a) (b) Figure 11: DWs obtained from continuation of m0in system (7) projected onto the sphere in the codim-0 regime h= 10.3 (h∗= 10.2) and initial speed and frequency s0= 4.04 and Ω0= 8.28. (a)ccp=−0.5. (b)ccp= 0.5. (a)-5 5-101 ξm1,m2 (b) Figure 12: DWs obtained from continuation of m0in system (7) in the codim-0 regime h= 50 (h∗= 10.2) withinitial speedandfrequency s0= 19.92,Ω0= 40.04, andccp=−0.5. (a) Projection onto the sphere. (b) Zoom-in of the correspon dingm1(solid blue) andm2(dashed red) component. (cf. Figure 4) which are much more complex than the homogeneous o ne given by (9) (cf. Figure 1b). Finally, recall from §3.1 that in the explicit family (9), the right moving DWs terminate ats= 0. The question arises what happens for ccp/ne}ationslash= 0 along the parameter s. To study this, we performed a continuation in the parameter sfor different values of ccp. For decreasing swe found that numerical continuation failed at some s>0 forccp/ne}ationslash= 0 (cf. Figure 13). The details of this apparent existence boundary a re beyond the scope of this paper. Note that the special role of sis reflected in the splitting function (34). In cases0=s= 0, the first column of (32) is zero (see (34)) and thus the parame terization ofccpcan not be written as a function in s. In detail, we continued the analytic solution (9) in ccpaway from zero for different initial values ofβ/α < h < h∗. This led to inhomogeneous ( ccp/ne}ationslash= 0) DWs, which we in turn continued in the parameter stowards zero for different fixed values of ccpuntil the con- tinuation process fails to converge. Based on this, there is numeric al evidence that DWs with opposite speed sign (counter-propagating fronts) can only e xist simultaneously for 25s= 0 (standing fronts). We took a polynomial fit on these points as an approximation of the existence boundary (cf. blue curve in Figure 13a). The continu ation process towards the boundary is indicated by red arrows in Figure 13a for positive ccp. Additionally, the corresponding results in the parameter space Ω and ccpis presented in 13b. 0 0.10.20.40.6 sc c (a)0.2 0.50.20.40.6 Ωc  (b) Figure 13: Continuation results in sand Ω for fixed ccpin the codim-2 parameter regime. Further parameters are α= 0.5,β= 0.1,µ=−1, andhfree. Blue solid line represents interpolated termination boundaryfor sin (a) and Ω in (b), respectively. Red arrows indicate continuation approach towards boundar y. As a last point, we briefly describe the numerical method for time-int egration near DWs, including freezing of speed and frequency (cf. Figure 4). All calcula tions were done with the (free) software package pde2path which is based on a Finite Element Method (FEM), cf. [20] and the references therein. Time-integration in pde2path with the so- called ‘freezing’ method is discussed in [21]. In addition to the phase co ndition for the speedweaddedaphaseconditionfortherotationandtime-integra tedvia asemi-implicit Euler scheme. 5 Discussion and Outlook We have presented results pertaining the existence of different ty pes of domain walls for the LLG as well as LLGS equation. Our main focus has been on a nonze ro polarisation parameterccp/ne}ationslash= 0 for any value of the applied field, including the high-field case, and thus for any domain wall speed. These results extend what is known in particular for inhomogeneously structured DWs, and we have discovered an appa rently new type of DWs with certain oscillatory tails, referred to as non-flat here. In detail, we have provided a classification of DWs based on co-dimens ion properties in a reduced (spatial) coherent structure ODE, which relates to st ability and selection properties that we review next. First, we have proven the existen ce of inhomogeneous flat DWs in case ccp= 0 as well as ccp/ne}ationslash= 0 for an applied field above a certain threshold, whichismainlymaterialdepending. Toourknowledge, theonlypreviou sexistence result forccp/ne}ationslash= 0 with ’large’ applied fields concerns less relevant non-localized DWs [ 12]. Here the existence problem does not select speed and frequency. Second, we have discussed the so-called center case, which is char acterized by non- hyperbolic equilibria in the underlying coherent structure ODE. In th is case, we have 26shown the existence of inhomogeneous DWs including the leading orde r selection mech- anism. These solutions are non-flat in case ccp= 0 and generically also non-flat for ccp away from zero, which was substantiated by numerical results. Th e fundamental obser- vation has been the existence of a Hamiltonian function in a certain pa rameter regime in the corresponding coherent structure ODE. Third, we have proven the existence of inhomogeneous DWs in the so -called codim-2 regime, which is a range of values for the applied field in which the speed sis between zero and the center case speed. In this regime, each solution in cas eccp/ne}ationslash= 0 is uniquely determined by its speed as well as frequency. Here we have also pre sented the leading order selection function in the coherent structure ODE variables pandq, which depends on the speed s, the frequency Ω, and is independent of ccpfor standing fronts. We believe that these results are not only interesting and relevant f rom a theoretical and mathematical viewpoint, but also from an application viewpoint. They could help to better understand the interfaces between different magnetic do mains in nanostructures, e.g. in the development of racetrack memories, which are a promising prospective high density storage unit that utilize a series of DWs by shifting at high spe ed along magnetic nanowires through nanosecond current pulses. In order to illustrate and corroborate these theoretical results , we have presented numer- ical computations for a variety of values for the applied field in §4. On the one hand, the examples in essence show that large applied fields lead to more com plex profiles of the DWs in case ccp/ne}ationslash= 0. On the other hand, while in the center case the DWs projected on the sphere appear similar to those for small applied fields, these s olutions approach the poles in a qualitatively different ‘non-flat’ manner – as predicted b y our analysis. Moreover, we compared the numerical and analytical results of th e selection mechanism in the center case, showing that the analytical leading order appro ximation predicts the effect of small perturbations in the parameters. Notably, for app lied fields above a cer- tain threshold, where the existence analysis does not provide a sele ction of speed and frequency, numerically the DWs selected in the PDE dynamics are in th e center case, both forccp= 0 as well as ccp/ne}ationslash= 0. Hence, it might be possible to detect these solutions in a high-field regime in real materials. One question concerning existence beyond our analysis is whether in homogeneous (flat or non-flat) solutions exist for any value of ccp∈(−1,1), and whether this class could be utilized in applications. A natural step towards the understanding of domain wall motion in n anowires beyond the question of existence concerns the dynamic stability. For inhom ogeneous solutions there appears to be no rigorous result in this direction. In particula r for larger applied fields, stability results would be an essential step towards underst anding the selection mechanismofsolutionsintermsofspeedandfrequency; ourfirstn umericalinvestigations show that solutions inthe center parameter regime areselected, i.e ., inhomogeneous non- flat DWs. Moreover, preliminary analytic results, for ccp= 0 as well as ccp/ne}ationslash= 0, show that selection mechanism is mainly determined by the value of the applied field, where in the bi-stable 27case (±e3linearly stable) homogeneous DWs are selected, and in the mono-sta ble case inhomogeneous non-flatDWsareselected, which will bestudied indet ail inanupcoming work. References [1] Stuart SP Parkin, Masamitsu Hayashi, and Luc Thomas. Magnetic domain-wall racetrack memory. Science, 320(5873):190–194, 2008. [2] William Reohr, Heinz Honigschmid, Raphael Robertazzi, Dietmar Gog l, Frank Pe- savento, Stefan Lammers, Kelvin Lewis, Christian Arndt, Yu Lu, Ha ns Viehmann, et al. Memories of tomorrow. IEEE circuits and devices magazine , 18(5):17–27, 2002. [3] CJ Lin, SH Kang, YJ Wang, K Lee, X Zhu, WC Chen, X Li, WN Hsu, YC Ka o, MT Liu, et al. 45nm low power cmos logic compatible embedded stt mram u tilizing a reverse-connection 1t/1mtj cell. In Electron Devices Meeting (IEDM), 2009 IEEE International , pages 1–4. IEEE, 2009. [4] Moinuddin K Qureshi, Vijayalakshmi Srinivasan, and Jude A Rivers. Scalable high performance main memory system using phase-change memory tec hnology. ACM SIGARCH Computer Architecture News , 37(3):24–33, 2009. [5] IsaakDMayergoyz, GiorgioBertotti, andClaudioSerpico. Nonlinear magnetization dynamics in nanosystems . Elsevier, 2009. [6] Alex Hubert and Rudolf Sch¨ afer. Magnetic domains: the analysis of magnetic microstructures . Springer Science & Business Media, 2008. [7] Giorgio Bertotti. Spin-transfer-driven magnetization dynamics .Magnetic Nanos- tructures in Modern Technology , pages 37–60, 2008. [8] L Berger. Emission of spin waves by a magnetic multilayer traverse d by a current. Physical Review B , 54(13):9353, 1996. [9] John C Slonczewski. Current-driven excitation of magnetic multila yers.Journal of Magnetism and Magnetic Materials , 159(1-2):L1–L7, 1996. [10] JA Osborn. Demagnetizing factors of the general ellipsoid. Physical review , 67(11- 12):351, 1945. [11] JC Slonczewski. Currents and torques in metallic magnetic multilay ers.Journal of Magnetism and Magnetic Materials , 247(3):324–338, 2002. [12] Christof Melcher and Jens DM Rademacher. Pattern formation in axially sym- metric landau–lifshitz–gilbert–slonczewski equations. Journal of Nonlinear Science , 27(5):1551–1587, 2017. 28[13] Arseni Goussev, JM Robbins, and Valeriy Slastikov. Domain-wall motion in fer- romagnetic nanowires driven by arbitrary time-dependent fields: A n exact result. Physical review letters , 104(14):147202, 2010. [14] Johan ˚Akerman. Toward a universal memory. Science, 308(5721):508–510, 2005. [15] LD Landau and EM Lifshitz. On the theory of the dispersion of ma gnetic perme- ability in ferromagnetic bodies, phy. z. sowjetunion 8: 153. Reproduced in Collected Papers of LD Landau , pages 101–114, 1935. [16] Thomas L Gilbert. A phenomenological theory of damping in ferro magnetic mate- rials.IEEE Transactions on Magnetics , 40(6):3443–3449, 2004. [17] M Lakshmanan. The fascinating world of the landau–lifshitz–gilbe rt equation: an overview. Philosophical Transactions of the Royal Society of London A : Mathemat- ical, Physical and Engineering Sciences , 369(1939):1280–1300, 2011. [18] Bj¨ orn Sandstede and Arnd Scheel. Absolute and convective in stabilities of waves on unbounded and large bounded domains. Physica D: Nonlinear Phenomena , 145(3- 4):233–277, 2000. [19] Yan Gou, Arseni Goussev, JM Robbins, and Valeriy Slastikov. St ability of precess- ing domain walls in ferromagnetic nanowires. Physical Review B , 84(10):104445, 2011. [20] T Dohnal, J Rademacher, H Uecker, and D Wetzel. pde2path 2.0. ENOC, 2014. [21] Jens DM Rademacher and Hannes Uecker. Symmetries, freezin g, and hopf bifurca- tions of traveling waves in pde2path, 2017. [22] Yuri A Kuznetsov. Elements of applied bifurcation theory , volume 112. Springer Science & Business Media, 2013. 6 Appendix 6.1 Proof of Theorem 2 We use the notation u=u(ξ;η,α,β,µ)= (θ(ξ;η,α,β,µ),p(ξ;η,α,β,µ),q(ξ;η,α,β,µ))T and bifurcation parameters η= (ccp,s,h)T, wheres0andh0=h∗are defined below (see §3.2 for details). The starting point for our perturbation analysis ar e the unperturbed parameters and explicit heteroclinic solution in the center case (12) , where the frequency is Ω0=s2 0/2+β/α. These are given by η0:= ccp0 s0 h0 := 0 2√−µ αβ α−2µ−2µ α2 29as well as u0=u0(ξ;η0,α,β,µ) := θ0(ξ;η0,α,β,µ) p0(ξ;η0,α,β,µ) q0(ξ;η0,α,β,µ) := 2arctan(exp(√−µξ))√−µ 0 . Unless stated otherwise, we suppress the explicit dependence of uonα,β, andµin the following discussion. Let us write Zπ:=Zπ −with the notation from Remark 3 so that the unperturbed right asymptotic state is given by Zπ(η0) =/parenleftbigg π,αs0 2,s0 2−/radicalbigg −µ α2/parenrightbiggT =/parenleftbig π,√−µ,0/parenrightbigT and its derivative with respect to ηis given by Zπ η(η0) = 0 0 0 0α 20 β 2√−µ2+α2 2−α 2√−µ . We write system (7) for brevity as u′=f(u;η), (22) sof(u;η) denotes the right side of (7). The linearization w.r.t. ηin the unperturbed heteroclinic connection u0, given by (17), is the non-autonomous linear equation u′ η=fu(u0;η0)uη+fη(u0;η0)η, (23) whereuη= (θη,pη,qη)T. Its homogeneous part is θ′ η=√−µcos(θ0)θη+pηsin(θ0) p′ η=−(αs0+2√−µcos(θ0))pη+s0qη q′ η=−s0pη−(αs0+2√−µcos(θ0))qη, (24) withθ0(ξ) = 2arctan(exp(√−µξ)) due to (17). We next solve (24) and determine its fundamental solution matrix. The first obvious vector-solution of it is U1=u′ 0= (θ′ 0,0,0) since the second and the third equation of (24) do not depend on θη. The other solutions can be obtained from U1and the result of Lemma 1. Changing to polar coordinates pη=rcosϕ, q η=rsinϕ, the equations for pηandqηbecome r′=−(αs0+2√−µcos(θ0))r ϕ′=−s0, 30whose general solution can be written as pη=r0r(ξ)cos(−s0ξ+ϕ0) qη=r0r(ξ)sin(−s0ξ+ϕ0) where r(ξ) = exp −αs0ξ−2√−µ/integraldisplay ξcos(θ0(τ))dτ =/parenleftig 1+e2√−µξ/parenrightig2 e(−2√−µ−αs0)ξ, andr0,ϕ0arearbitrary integrationconstants corresponding to suitable init ial conditions. Note that lim ξ→±∞r(ξ) =∞for 0≤s0<2√−µ/α. Next, the values of the integration constants have to be selected in order for the second and the third vector-solutions U2= θ1 1 r1r(ξ)cos(−s0ξ+ϕ1) r1r(ξ)sin(−s0ξ+ϕ1) , U3= θ2 1 r2r(ξ)cos(−s0ξ+ϕ2) r2r(ξ)sin(−s0ξ+ϕ2) (25) tobelinearlyindependent. Here θ1 1,θ2 1arenotrelevantforwhatfollows. Thedeterminant of the fundamental matrix reads detΦ(ξ) = det(U1(ξ),U2(ξ),U3(ξ)) =r1r2r2(ξ)θ′ 0(ξ)sin(ϕ2−ϕ1), which is non-zero for r1=r2= 1,ϕ1= 0 andϕ2=π/2, i.e. detΦ( ξ) =r2(ξ)θ′ 0(ξ). Together, we get the fundamental solution matrix of the homogen eous part as Φ(ξ) = θ′ 0(ξ)θ1 1(ξ) θ2 1(ξ) 0r(ξ)cos(−s0ξ)−r(ξ)sin(−s0ξ) 0r(ξ)sin(−s0ξ)r(ξ)cos(−s0ξ) . (26) The derivative of (22) with respect to ηis given by (23) and from the variation of constants formula we get for some ξ0that uη(ξ) = Φξ,ξ0uη(ξ0)+ξ/integraldisplay ξ0Φξ,τfη(u0(τ);η0)dτ, where Φ ξ,τ= Φ(ξ)·Φ−1(τ) is the evolution operator. Using (26) we find Φξ,τ(ξ,τ;η) = Θ1 Θ2 Θ3 0r(ξ) r(τ)cos(−s0(ξ−τ))−r(ξ) r(τ)sin(−s0(ξ−τ)) 0r(ξ) r(τ)sin(−s0(ξ−τ))r(ξ) r(τ)cos(−s0(ξ−τ)) ,(27) where the explicit forms of the functions Θ 1,2,3(ξ) are not relevant for the remainder of this proof. Since uη(ξ) tends to∂ηZ0 −forξ→ −∞the hyperbolicity of Z0 −(more 31precisely the resulting exponential dichotomy) implies Φ ξ,ξ0uη(ξ0)→0 asξ0→ −∞and so uη(ξ) =ξ/integraldisplay −∞Φξ,τfη(u(τ;η0);η0)dτ. (28) Regarding the limiting behavior as ξ→ ∞, recall that Corollary 1 states that the right asymptotic limit of the perturbed heteroclinic orbit is either the pert urbed equilibrium Zπ(η) or a periodic orbit around it in the blow-up chart at θ=π. The integral (28) distinguishes these case in the sense that either it has a limit as ξ→+∞so the heteroclinic orbit connects the two equilibria, or it does not and the h eteroclinic orbit connects to a periodic solution. We next determine uη(ξ) componentwise uη(ξ) =v:= v11v12v13 v21v22v23 v31v32v33 , wherevijare the components of (28) and index i= 1,2,3 relates to θ,p,qas well as j= 1,2,3 toccp,s,h. Towards this, we compute fη(u0(τ),η0) = 0 0 0 −β/α−√−µ α(2+α2) 1 2β 1+e2√−µ τ√−µ0 , and together with (28) and (27) we obtain v21=−β αIC−2βJS, v22=−√−µ α(2+α2)IC−√−µIS, v23=IC, v31=−β αIS+2βJC, v32=−√−µ α(2+α2)IS+√−µICv33=IS, where IC=IC(ξ):=ξ/integraldisplay −∞(1+exp( −2√−µξ))2 (1+exp( −2√−µτ))2cos(−s0(ξ−τ))dτ, IS=IS(ξ):=ξ/integraldisplay −∞(1+exp( −2√−µξ))2 (1+exp( −2√−µτ))2sin(−s0(ξ−τ))dτ, JC=JC(ξ):=ξ/integraldisplay −∞exp(−2√−µτ)(1+exp( −2√−µξ))2 (1+exp( −2√−µτ))3cos(−s0(ξ−τ))dτ, 32ξ ξ0iπ 2√−µ I1I2I3 I4 Figure 14: Contour CinCfor the integrals IandJ. JS=JS(ξ):=ξ/integraldisplay −∞exp(−2√−µτ)(1+exp( −2√−µξ))2 (1+exp( −2√−µτ))3sin(−s0(ξ−τ))dτ. Note that we do not provide explicit formulas for v11,v12andv13, because they are not needed for further computations. This is the reason why we ne glected the explicit expressions of Θ 1,Θ2, and Θ 3before. We now introduce the following complex-valued integrals for further computations: I(ξ):=IC(ξ)+iIS(ξ) =ξ/integraldisplay −∞/parenleftbig 1+e−2√−µξ/parenrightbig2 /parenleftbig 1+e−2√−µτ/parenrightbig2exp/parenleftbigg −i2√−µ α(ξ−τ)/parenrightbigg dτ, forICandISas well as J(ξ):=JC(ξ)+iJS(ξ) =ξ/integraldisplay −∞e−2√−µτ/parenleftbig 1+e−2√−µξ/parenrightbig2 /parenleftbig 1+e−2√−µτ/parenrightbig2exp/parenleftbigg −i2√−µ α(ξ−τ)/parenrightbigg dτ forJCandJS. We extend the above integrals to the complex plane and integrate a long the counter-clockwise oriented rectangular contour Cas illustrated in Figure 14, and letξ0→ −∞. We will provide the details of the computation of Ionly, asJcan be calculated in a fully analogous way. The complex integrand of Iis g(z;ξ):=/parenleftbig 1+e−2√−µξ/parenrightbig2 /parenleftbig 1+e−2√−µz/parenrightbig2exp/parenleftbigg −i2√−µ α(ξ−z)/parenrightbigg , with singularities in Cat the pointsi(π+2kπ) 2√−µ,k∈Z, one of which lies in the interior of C, namelyz0:=iπ 2√−µ. The contour integral Ican now be written via the residue theorem as I1(ξ)+I2(ξ)+I3(ξ)+I4(ξ) = 2πi/summationdisplay intCResg(z;ξ), whereI1,...,I 4are given by 33I1:z=x, I1(ξ0,ξ) =/parenleftig 1+e−2√−µξ/parenrightig2 exp/parenleftbigg −i2√−µ αξ/parenrightbiggξ/integraldisplay ξ0exp/parenleftig i2√−µ αx/parenrightig /parenleftbig 1+e−2√−µx/parenrightbig2dx I(ξ) = lim ξ0→−∞I1(ξ0,ξ) =/parenleftig 1+e−2√−µξ/parenrightig2 exp/parenleftbigg −i2√−µ αξ/parenrightbiggξ/integraldisplay −∞exp/parenleftig i2√−µ αx/parenrightig /parenleftbig 1+e−2√−µx/parenrightbig2dx, I2:z=ξ+iy, I2(ξ) =/parenleftig 1+e−2√−µξ/parenrightig2π√−µ/integraldisplay 0iexp/parenleftig −2√−µ αy/parenrightig /parenleftbig 1+e−2√−µ(ξ+iy)/parenrightbig2dy, I3:z=x+π√−µi, I3(ξ0,ξ) =/parenleftig 1+e−2√−µξ/parenrightig2 exp/parenleftbigg −i2√−µ αξ/parenrightbigg e−2π αξ0/integraldisplay ξexp/parenleftig i2√−µ αx/parenrightig /parenleftbig 1+e−2√−µξ/parenrightbig2dx =−e−2π αI1(ξ0,ξ), I4:z=ξ0+iy, I4(ξ0,ξ) =/parenleftig 1+e−2√−µξ/parenrightig2 exp/parenleftbigg i2√−µ α(ξ0−ξ)/parenrightbigg0/integraldisplay π√−µiexp/parenleftig −2√−µ αy/parenrightig /parenleftbig 1+e−2√−µ(ξ0+iy)/parenrightbig2dy, lim ξ0→−∞I4(ξ0,ξ) = 0. Utilizing the Laurent series of gwe obtain Resg(z;ξ)|z=z0=α+i 2α√−µe−π α/parenleftig 1+e−2√−µξ/parenrightig2 exp/parenleftbigg −i2√−µ αξ/parenrightbigg , which leads to I(ξ) =/parenleftig 1−e−2π α/parenrightig−1/parenleftbiggπi(α+i) α√−µe−π α/parenleftig 1+e−2√−µξ/parenrightig2 exp/parenleftbigg −i2√−µ αξ/parenrightbigg −I2(ξ)/parenrightbigg . Now we can write IC(ξ) = ReI(ξ) =π/parenleftbig 1+e−2√−µξ/parenrightbig2 α√−µ/parenleftbig eπ α−e−π α/parenrightbig/bracketleftbigg −cos/parenleftbigg −2√−µ αξ/parenrightbigg −αsin/parenleftbigg −2√−µ αξ/parenrightbigg/bracketrightbigg −1 1−e−2π αIr 2(ξ) 34as well as IS(ξ) = ImI(ξ) =π/parenleftbig 1+e−2√−µξ/parenrightbig2 α√−µ/parenleftbig eπ α−e−π α/parenrightbig/bracketleftbigg αcos/parenleftbigg −2√−µ αξ/parenrightbigg −sin/parenleftbigg −2√−µ αξ/parenrightbigg/bracketrightbigg −1 1−e−2π αIi 2(ξ), whereIr 2(ξ) andIi 2(ξ) are the real and imaginary part of I2(ξ), respectively. Studying the integral Jin a similar fashion, we obtain JC(ξ) = ReJ(ξ) =π/parenleftbig 1+e−2√−µξ/parenrightbig2 2α2√−µ/parenleftbig eπ α−e−π α/parenrightbig/bracketleftbigg αcos/parenleftbigg −2√−µ αξ/parenrightbigg −sin/parenleftbigg −2√−µ αξ/parenrightbigg/bracketrightbigg −1/parenleftig 1−e−2π α/parenrightigJr 2(ξ) as well as JS(ξ) = ImJ(ξ) =π/parenleftbig 1+e−2√−µξ/parenrightbig2 2α2√−µ/parenleftbig eπ α−e−π α/parenrightbig/bracketleftbigg cos/parenleftbigg −2√−µ αξ/parenrightbigg +αsin/parenleftbigg −2√−µ αξ/parenrightbigg/bracketrightbigg −1/parenleftig 1−e−2π α/parenrightigJi 2(ξ), where also here Jr 2(ξ) andJi 2(ξ) are the real and imaginary part of J2(ξ). Direct compu- tations show also that lim ξ→+∞Ii 2(ξ) =α 2√−µ/parenleftbigg 1−exp/parenleftbigg −2π α/parenrightbigg/parenrightbigg and lim ξ→+∞Ir 2(ξ) = lim ξ→+∞Jr 2(ξ) = lim ξ→+∞Ji 2(ξ) = 0. Summing up, the second and third component of uη(ξ)·ηfor sufficiently large ξare /parenleftigg α 2s β 2√−µccp+2+α2 2s−α 2√−µh/parenrightigg + π ρ /parenleftig −1 α√−µh+2 α2s/parenrightig cos/parenleftig −2√−µ αξ/parenrightig +/parenleftig −1√−µh+(3+α2) αs/parenrightig sin/parenleftig −2√−µ αξ/parenrightig /parenleftig 1√−µh−(3+α2) αs/parenrightig cos/parenleftig −2√−µ αξ/parenrightig +/parenleftig −1 α√−µh+2 α2s/parenrightig sin/parenleftig −2√−µ αξ/parenrightig + +O(e−2√−µξ), whereρ:= exp(π/α)−exp(−π/α). One readily verifies that the oscillatory part in the expression above vanishes if and only if sandhare zero and thus we infer that the heteroclinic connection cannot be between equilibria to first order in the parameters. In order to detect cancellations of these oscillatory parts for high er orders of sandh, we next consider the behavior of the quantity (13) with respect to pa rameter perturbations. With slight abuse of notation, for u= (θ,p,q)Twe writeH(u;η) :=H(p,q) evaluated 35at parameters η, and other parameters at some fixed value, and we always consider the heteroclinic solutions from Corollary 1. Our strategy in the following steps is as follows: we utilize the quantity Hbecause limξ→+∞Halways exists along these solutions. In order to distinguish whether this limit is an equilibrium or a periodic orbit, we consider /tildewideH(u;η):=H(u;η)−H(Zπ;η), i.e., the difference of the H-values of the (parameter dependent) equilibrium Zπand the limit ofuasξtends to infinity. Expanding /tildewideHin the limit ξ→ ∞with respect to the parameterηyields conditions for periodic asymptotics. In the following, subindice s of Hdenote partial derivatives, e.g. Hu=∂uH. Clearly,H(u0;η0) =H(Zπ(η0);η0), thus/tildewideH0= 0 and, since equilibria are critical points ofH, we have/tildewideHu(u0;η0) =/tildewideHη(u0;η0) = (0,0,0)T. The second derivative is given by d2 dη2/tildewideH=/an}bracketle{tuη,/tildewideHuuuη/an}bracketri}ht+/an}bracketle{tuη,/tildewideHuη/an}bracketri}ht+/an}bracketle{t/tildewideHηu,uη/an}bracketri}ht, (29) since/tildewideHηηis the zero matrix, /tildewideHuthe zero vector, and /tildewideHuη=/tildewideHT ηu. Thus /tildewideH(u0+uηη;η) =1 2(uηη)THuu(u0;η0)(uηη)+(uηη)THuη(u0;η0)η −1 2/parenleftbig Zπ η(η0)η/parenrightbigTHuu(Zπ(η0);η0)/parenleftbig Zπ η(η0)η/parenrightbig −/parenleftbig Zπ η(η0)η/parenrightbigTHuη(Zπ(η0);η0)η+O/parenleftbig /bardblη/bardbl3/parenrightbig .(30) With the derivatives Huu,Huηin (30) given by Huu(u;η) = 0 0 0 02 q−s/2−2p−αs (q−s/2)2 0−2p−αs (q−s/2)22p2−αsp+h−β−/α+µ−s2/4 (q−s/2)3 , Huη(u;η) = 0 0 0 0p−αq (q−s/2)2 0 β/α (1−ccp)2(q−s/2)2−p2−αpq−αs 2p−s 2q+h−β−/α+µ (q−s/2)3 −1 (q−s/2)2 , for the right hand side of (30) in the limit ξ→+∞we obtain 1 2(uηη)THuu(u0;η0)(uηη)+(uηη)THuη(u0;η0)η=−αβ2 4µ√−µc2 cp +(4+5α2+α4)(α2ρ2−4(1+α2)π2) 4α3ρ2√−µ(s−s0)2−α4ρ2−4(1+α2)π2 4αρ2µ√−µ(h−h0)2 −αβ(2+α2) 2µccp(s−s0)+α2β 2µ√−µccp(h−h0) +(2+α2)(α4ρ2−4(1+α2)π2) 2α2ρ2µ(s−s0)(h−h0), 36as well as 1 2/parenleftbig Zπ η(η0)η/parenrightbigTHuu(Zπ(η0);η0)/parenleftbig Zπ η(η0)η/parenrightbig +/parenleftbig Zπ η(η0)η/parenrightbigTHuη(Zπ(η0);η0)η= −αβ2 4µ√−µc2 cp+α(4+5α2+α4) 4√−µ(s−s0)2−α3 4µ√−µ(h−h0)2 −αβ(2+α2) 2µccp(s−s0)+α2β 2µ√−µccp(h−h0)+α2(2+α2) 2µ(s−s0)(h−h0). Therefore, the expansion in the limit ξ→+∞is independent of ccpand reads lim ξ→∞/tildewideH(u0+uηη;η) =−(1+α2)2(4+α2)π2 α3ρ2√−µ(s−s0)2 −2(1+α2)(2+α2)π2 α2ρ2µ(s−s0)(h−h0) +(1+α2)π2 αρ2µ√−µ(h−h0)2+O/parenleftbig /bardblη−η0/bardbl3/parenrightbig .(31) Recallρ= exp(π/α)−exp(−π/α). One readily verifies that the resulting (binary) quadratic form of (31) is negative definite for all α >0 so the only solution to the leading order problem d2 dη2/tildewideH(u0;η0) = 0 is thetrivial one( s,h) = (0,0), and anynon-trivial solution satisfies |s−s0|2+|h−h0|2= O(|ccp|3). In particular, for ccp= 0 there is a neighborhood of ( s0,h0) such that the only solution is the trivial one, which is therefore also the case in the LLG equation . In caseccp/ne}ationslash= 0, higher orders may lead to a solution with non-zero sand/orh, but there is numerical evidence that such solutions do not exist (see §4 for details). 6.2 Proof of Theorem 3 The idea of the proof is to apply Lyapunov-Schmidt reduction, i.e., to determine a bi- furcation equation whose solutions are in one-to-one correspond ence with heteroclinic connections between equilibria (7) near one of the explicit solutions u0from (17) con- necting the equilibria Z0 −andZπ −. In the present context this is known as Melnikov’s method, see for example [22]. Recallu0corresponds to a homogeneous DW for ccp= 0 with speed s0and rotation frequency Ω 0given by (19). In the present codim-2 parameter regime we will show that the bifurcation equation defines a codimension two bifurcation curve in the three- dimensional parameter space ( ccp,s,Ω), which passes through the point (0 ,s0,Ω0). The main part of the proof is to show the existence of certain integrals f or the considered parameter set. These integrals are almost identical to the ones st udied within the proof of Theorem 2 and we use the same approach. 37In this section we denote the parameter vector by η:= (ccp,s,Ω)T∈R3, with initial valueη0= (0,s0,Ω0)Tcorresponding to the unperturbed values. The solutions of the perturbedsystemcloseto u0hastheform u(ξ;η) =u0(ξ)+uη(ξ;η0)(η−η0)+O(/bardblη−η0/bardbl2), whereuη= (θη,pη,qη)T=O(/bardblη−η0/bardbl). As discussed in Appendix 6.1, the linearization (23) of system (7) aro und the unper- turbed heteroclinic connection u0has the fundamental solution matrix Φ( ξ) as defined in(26). Inthepresent codim-2casewithdim( W0 u) = dim(Wπ s) = 1inR3, thebifurcation equationM(η) = 0 entails two equations. Here M(η) measures the displacement of the manifoldsW0 uandWπ s, andwe willchoose thistobenearthepoint u0(0) =/parenleftbigπ 2,√−µ,0/parenrightbigT in the directions given by vectors v1(0) andv2(0) from adjoint solutions as detailed be- low. From the Taylor expansion M(η) =Mη(η0)(η−η0)+O(/bardblη−η0/bardbl2) we infer by the implicit function theorem that a full rank of Mη(η0) implies a one-to-one correspondence of solutions to the bifurcation equation with elements in the kernel o fM(η0). In order to compute Mη(η0) and its rank, we project onto the transverse directions to u0, which means to project the inhomogeneous part of equation (23) onto two linearly independent bounded solutions v1,v2of the adjoint variational equation v′=−AT·v, where AT= √−µcos(θ0) 0 0 sin(θ0)−αs0−2√−µcos(θ0) −s0 0 s0 −αs0−2√−µcos(θ0) . The solutions are given in terms of (25) by v1=U1×U2 detΦ= 0 −1 r(ξ)sin(−s0ξ) 1 r(ξ)cos(−s0ξ) andv2=U3×U1 detΦ= 0 1 r(ξ)cos(−s0ξ) 1 r(ξ)sin(−s0ξ) . Implementing the projection onto these, we obtain the so-called Melnikov integral Mη(η0):=+∞/integraldisplay −∞(v1,v2)T·fη(u0;η0)dξ =/parenleftbigg βICCα√−µIS−√−µICIS+αIC βICS−α√−µIC−√−µIS−IC+αIS/parenrightbigg ,(32) where ICC:=+∞/integraldisplay −∞/parenleftbig 1−e2√−µξ/parenrightbig eαs0ξ+2√−µξcos(−s0ξ) /parenleftbig 1+e2√−µξ/parenrightbig3dξ, ICS:=+∞/integraldisplay −∞/parenleftbig 1−e2√−µξ/parenrightbig eαs0ξ+2√−µξsin(−s0ξ) /parenleftbig 1+e2√−µξ/parenrightbig3dξ, 38IC:=+∞/integraldisplay −∞eαs0ξcos(−s0ξ)/parenleftbig 1+e2√−µξ/parenrightbig ·/parenleftbig 1+e−2√−µξ/parenrightbigdξ, IS:=+∞/integraldisplay −∞eαs0ξsin(−s0ξ)/parenleftbig 1+e2√−µξ/parenrightbig ·/parenleftbig 1+e−2√−µξ/parenrightbigdξ. We next show that the second and thirdcolumns in (32) have non-va nishing determinant so that the rank is always 2, in particular also for β= 0. For brevity, we present the calculations of ICCandICSonly, which are based on the same idea as the computations in Appendix 6.1. The solutions for ICandIScan be computed in an analogous way. From Appendix 6.1 we know that the following integral would not exist in cases0= 2√−µ/αand one readily verifies the existence for s0= 0. Therefore, we first assume 0<s0<2√−µ/αfor the moment and discuss the case s0= 0 later. We set I:=+∞/integraldisplay −∞g(ξ)dξ, g(ξ):=/parenleftbig 1−e2√−µξ/parenrightbig e(αs0+2√−µ)ξe−is0ξ /parenleftbig 1+e2√−µξ/parenrightbig3, so thatICC= Re(I) andICS= Im(I). Utilizing the same idea for the contour integral as in Appendix 6.1 (cf. Figure 14) and the residue theorem, we obtain I−eαs0iπ√−µ·es0π√−µ·I= 2πi/summationdisplay Res(g). The function ghas a pole of order three atiπ 2√−µand theLaurentseries gives Res(g) =−(α−i)2s2 0 8µ√−µexp/parenleftbigg (α−i)iπs0 2√−µ/parenrightbigg , and therefore I=2πs2 0eπs0 2√−µ/bracketleftig −2αcos/parenleftig παs0 2√−µ/parenrightig +(α2−1)sin/parenleftig παs0 2√−µ/parenrightig/bracketrightig 8µ√−µ/parenleftig 1−eπs0√−µcos/parenleftig παs0√−µ/parenrightig −ieπs0√−µsin/parenleftig παs0√−µ/parenrightig/parenrightig +i2πs2 0eπs0 2√−µ/bracketleftig (1−α2)cos/parenleftig παs0 2√−µ/parenrightig −2αsin/parenleftig παs0 2√−µ/parenrightig/bracketrightig 8µ√−µ/parenleftig 1−eπs0√−µcos/parenleftig παs0√−µ/parenrightig −ieπs0√−µsin/parenleftig παs0√−µ/parenrightig/parenrightig. Separating the real and imaginary parts of Iwe get ICC=πs2 0√−µeπs0 2√−µ 4µ2·2α/parenleftig 1−eπs0√−µ/parenrightig cos/parenleftig παs0 2√−µ/parenrightig +(1−α2)/parenleftig 1+eπs0√−µ/parenrightig sin/parenleftig παs0 2√−µ/parenrightig 1+e2πs0√−µ−2eπs0√−µcos/parenleftig παs0√−µ/parenrightig 39ICS=πs2 0√−µeπs0 2√−µ 4µ2·(1−α2)/parenleftig 1−eπs0√−µ/parenrightig cos/parenleftig παs0 2√−µ/parenrightig +2α/parenleftig 1+eπs0√−µ/parenrightig sin/parenleftig παs0 2√−µ/parenrightig 1+e2πs0√−µ−2eπs0√−µcos/parenleftig παs0√−µ/parenrightig . As mentioned before, one analogously gets IC=πs0eπs0 2√−µ 2µ·/parenleftig 1−eπs0√−µ/parenrightig cos/parenleftig παs0 2√−µ/parenrightig −α/parenleftig 1+eπs0√−µ/parenrightig sin/parenleftig παs0 2√−µ/parenrightig 1+e2πs0√−µ−2eπs0√−µcos/parenleftig παs0√−µ/parenrightig IS=πs0eπs0 2√−µ 2µ·α/parenleftig 1−eπs0√−µ/parenrightig cos/parenleftig παs0 2√−µ/parenrightig +/parenleftig 1+eπs0√−µ/parenrightig sin/parenleftig παs0 2√−µ/parenrightig 1+e2πs0√−µ−2eπs0√−µcos/parenleftig παs0√−µ/parenrightig . Having the explicit expressions for IC,IS,ICC, as well as ICS, we can study the rank of (32) in case 0 < s0<2√−µ/α. The determinant of the second and third column of (32) simplifies to (αIS−IC)2+(IS+αIC)2=/parenleftbig 1+α2/parenrightbig2π2s2 0eπs0√−µ/ne}ationslash= 0,∀s0/ne}ationslash= 0. Therefore,Mη(η0) has full rank and we obtain M(η) =/parenleftbiggβICCα√−µIS−√−µICIS+αIC βICS−α√−µIC−√−µIS−IC+αIS/parenrightbigg ·(η−η0)+O/parenleftbig /bardblη−η0/bardbl2/parenrightbig .(33) In the remaining case s0= 0, which means h=β/α, we haveICC=ICS=IS= 0, IC=1 2√−µ, and thus Mη(η0) =/parenleftigg 0−1 2α 2√−µ 0−α 2−1 2√−µ/parenrightigg , (34) whose rank is always 2. Thus the splitting directions are independent ofccpin first order in caseh=β/α. Moreover, note that the splitting in sis independent of the anisotropy µto first order in case h=β/α. This completes the proof of Theorem 2. 40

2019-07-17

In case of a spin-polarized current, the magnetization dynamics in nanowires are governed by the classical Landau-Lifschitz equation with Gilbert damping term, augmented by a typically non-variational Slonczewski term. Taking axial symmetry into account, we study the existence of domain wall type coherent structure solutions, with focus on one space dimension and spin-polarization, but our results also apply to vanishing spin-torque term. Using methods from bifurcation theory for arbitrary constant applied fields, we prove the existence of domain walls with non-trivial azimuthal profile, referred to as inhomogeneous. We present an apparently new type of domain wall, referred to as non-flat, whose approach of the axial magnetization has a certain oscillatory character. Additionally, we present the leading order mechanism for the parameter selection of flat and non-flat inhomogeneous domain walls for an applied field below a threshold, which depends on anisotropy, damping, and spin-transfer. Moreover, numerical continuation results of all these domain wall solutions are presented.

Inhomogeneous domain walls in spintronic nanowires

1907.07470v2

1 Nonlinear s ub-switching regime of magnetization dynamics in photo -magnetic garnets A. Frej, I. Razdolski, A. Maziewski, and A. Stupakiewicz Faculty of Physics, University of Bialystok, 1L Ciolkowskiego, 1 5-245 Bialystok, Poland Abstract. We analyze, both experimentally and numerically, the nonlinear regime of the photo -induced coherent magnetization dynamics in cobalt -doped yttrium iron garnet films. Photo -magnetic excitation with femtosecond laser pulses reveals a strongly nonlinear respo nse of the spin subsystem with a significant increase of the effective Gilbert damping. By varying both laser fluence and the external magnetic field, we show that this nonlinearity originates in the anharmonicity of the magnetic energy landscape. We numer ically map the parameter workspace for the nonlinear photo -induced spin dynamics below the photo - magnetic switching threshold. Corroborated by numerical simulations of the Landau -Lifshitz - Gilbert equation, our results highlight the key role of the cubic sy mmetry of the magnetic subsystem in reaching the nonlinear spin precession regime. These findings expand the fundamental understanding of laser -induced nonlinear spin dynamics as well as facilitate the development of applied photo -magnetism. 1. INTRODUCTION Recently, a plethora of fundamental mechanisms for magnetization dynamics induced by external stimul i at ultrashort time scale s has been actively d iscussed [1-5]. The main interest is not only in the excit ation of spin precession but in the switching of ma gnetization between multiple stable states, as it open s up rich possibilities for non-volatile magnetic data storage technology . One of t he most intriguing example s is the phenomenon of ultrafast switching of magnetization with laser pulses. Energy -efficie nt, non -thermal mechanisms of laser -induced magnetization switching require a theoretical understanding of coherent magnetization dynamics in a strongly non -equilibrium environment [6]. This quasiperiodic motion of magnetization is often mode led as an oscillator where the key parameters , such as frequency and damping , are considered within the framework of the Landau -Lifshit z-Gilbert (LLG) equation [1, 7] . Although it is inhe rently designed to describe small -angle spin precession with in the linear approximation, there are attempts to extend this formalism into the nonlinear regime where the precession parameters become angle -dependent [8]. This is particularly important in light of the discovery of the so -called p recessional switching , where magnetization , having been impulsively driven out of equilibrium, ends its precessional motion in a different minimum of the potential energy [6, 9 -11]. Obviously, such magnetization trajectories are characterized by very large precession angles (usually on the order of tens of degrees ). It is, however, generally believed that the magnetization excursion from the equilibrium of about 10 -20 degrees is already sufficient for the violat ion of the linear LLG approach [12, 13] . Thus, an intermediate regime under the switching stimulus threshold exists, taking a large area in the phase space and presenting an intriguing c hallenge in understanding fundamental spin dynamics. An impulsive optical stimulus often results in a thermal excitation mechanism, inducing concomitant temperature variations , which can impact the parameters of spin precession [14- 16]. This highlights the special role of the non -thermal optical mechanisms of switching [17-2 19]. Among those , we outline photo -magnetic excitation , which has been recently demonstrated in dielectric Co -doped YIG (YIG:Co) films [6, 11] . There, laser photons at a wavelength of 1300 nm resonantly excite the 5E → 5T2 electron transition s in Co -ions, resulting in an emerging photo -induced magnetic anisotropy and thus in a highly efficient excitation of the magnetic subsystem [6]. This photo -induced effective anisotropy field features a near ly instant aneous rise time (within the femtosecond pump laser pulse duration), shifting the equilibrium direction for the magnetization and thus triggering its l arge -amplitude precession. In the sub -switching regime (at excitation strengths just below the switching threshold ), the frequency of the photo -induced magnetization precession has been shown to depend on the excit ation wavelength [20]. However, nonlinearities in magnetization dynamics in the sub- switching regime have not yet been described in detail, and the underlying mechanism for the frequency variations is not understood. In this work , we systematically examine the intermediate sub -switching regime characterized by large angles of magnetization precession and the nonlinear response of the spin system to photo -magnetic excitations. We show a strong increase of the effective Gilbert damping at elevated lase r-induced excitation levels and quantify its nonlinearity within the existing phenomenological formalism [8]. We further map the nonlinear regime in the phase space formed by the effective photo -induced anisotropy field and the external magnetic field. Fig. 1. Sketch of m agnetization dynamics at various stimulus levels . Owing to the highly nonlinear magnetization dynamics in the switching regime, the nonlinearity onset manifests in the sub -switching regime too . This paper is organized in the following order: in the first part, we describe the details of the experiment for laser -induced large -amplitude magnetization precession. Next, we present the experimental results, followed by the fitting analysis . Then, we complement our findings with the results of numerical simulation of the photo -magnetic spin dynamics. Afterward , we discuss the workspace of parameters for the sub-switching regime of laser -induced magnetization precession. The paper ends with c onclusions. 3 2. EXPERIMENTAL DETAILS The experiments were done on a 7.5 μm -thick YIG:Co film with a composition of Y2CaFe 3.9Co0.1GeO 12. The Fe ions at the tetrahedral and octahedral sites are replaced by Co - ions [21]. The sample was grown by liquid -phase epitaxy on a 400 μm-thick gadolinium gallium garnet (GGG) substrate. It exhibits eight possible magnetization states along the garnet’s cubic cell diagonals due to its cubic magnetocrystalline anisotropy ( 𝐾1=−8.4×103 𝑒𝑟𝑔/𝑐𝑚3) dominating the energy landscape over the uniaxial anisotropy ( 𝐾𝑢=−2.5×103 𝑒𝑟𝑔/𝑐𝑚3). Owing to the 4 ° miscut, additional in -plane anisotropy is introduced, tilting the magnetization axes and resulting in slightly lower energy of half of the magnetiz ation states in comparison to the others. In the absence of the external magnetic field, the equilibrium magnetic state corresponds to the magnetization in the domains close to the <111> -type directions in YIG:Co film. Measurements of the Gilbert damping 𝛼 using the fe rromagnetic resonance technique resulted in 𝛼≈0.2. This relatively high damping is inextricably linked to the C o dopants [22- 24]. The n onlinearity of an oscillator is usually addressed by varying the intensity of the stimulus and comparing the response of the system under study. Here , we investigated the nonlinear magnetization dynamics by varying the optical pump fluence and , thus , the strength of the photo -magnetic effective field driving the magnetization out of the equilibrium. We perfor med systematic studies in various magnetic states of YIG:Co governed by the magnitude of the external magnetic field s. The magnetic field 𝐻⊥ was applied perpendicular to the sample plane and in -plane magnetic field 𝐻 was applied along the [110] direction of the YIG:Co crystal by means of an electromagnet. Owing to the introduced miscut, the studied YIG:Co exhibits four magnetic domains at 𝐻=0 [25]. The large jump at an in-plane magnetic f ield close to zero shows the magnetization switching in the domain structures between four magnetic phases. The optical spot size in this experiment was around 100 μm while the size of smaller domains was around 5 μm, resulting in the spatial averaging of the domains in the measurements. This behavior of magnetic domains was dis cussed and visualized in detail by magneto -optical Faraday effect in our previous papers [6, 25] . With an increase of the magnetic field up to a round 𝐻=0.4 kOe, larger and smaller domains are formed due to the domain wall motion, eventually resulting in a formation of a single domain in a noncollinear state. Upon further increase, the magnetization rotates towards the direction of the applied field until a collinear state with in -plane magnetization orientation is reached at about 2 kOe (see Fig. 2) . 4 Fig. 2. Magnetization reversal using static magneto -optical Faraday effect under perpendicular H (a) and in -plane H (b) magnetic fields. The grey area indicates the magnetization switching in magnetic domain structure [25]. The green area shows the saturation range with a collinear state of magnetization. Dynamic nonlinearities in the magne tic response were studied employing the pump -probe technique relying on the optical excitation of the spin precession in YIG :Co film. The pumping laser pulse at 1300 nm , with a duration of 50 fs and a repetition rate of 500 Hz , induce d spin dynamics through the photo -magnetic mechanism [6]. The transient Faraday rotation of the weak probe beam at 625 nm was used to monitor the dynamics of the out-of-plane magnetization component Mz. The diameter of the pump spot was around 1 40 μm , while the probe beam was focused within the pump spot with a size of around 50 μm . The fluence of the pump beam was varied in the range of 0.2 6.5 mJ/cm2, below the switching threshold of about 39 mJ/cm2 [20]. At 1300 nm pump wavelength, the optical absorption in our garnet is about 12%. An estimation of the temperature increase ΔT due to the heat load for the laser fluence of 6.5 mJ/cm2 results in ΔT <1 K (see Methods of Ref. 6). The polarization of both beams was linear and set along the [100] crystallographic direction in YIG:Co for the pump and the [010] direction for the probe pulse . The experiments were done at room temperature . At each magnetic field, we performed a series of laser fluence -dependent pump -probe experiments measuring the transients of an oscillating magnetization component normal to the sample plane . We then used a phenomenological damped oscillator response function to 5 fit the experimental data and retrieve the fit parameters such as a mplitude, frequency, lifetime and effective damping. In what follows, we analyze the obtained nonlinearities in the response of the magnetic system and employ numerical simulations to reproduce the experimental findings. 3. RESULTS A. Time -resolved photo -magnetic dynamics In order to determine the characteristics of the photo -magnetic precession , we carried out time -resolved measurements of a transient Faraday rotation ∆𝜃𝐹 in YIG:Co film. Fig. 3(a-d) exemplifies a few typical datasets obtained for four v arious pump fluences (between 1.7 and 6.5 mJ/cm2) in magnetic fields of various strength s. A general trend demonstrating a decrease of the precession amplitude and an increase of its frequency is seen upon the magnetic field increase . To get further insights into the magnetization dynamics, these datasets were fitted with a damped sine function on top of a non -oscillatory, exponentially decaying background : ∆𝜃𝐹(∆𝑡)=𝐴𝐹sin(2𝜋𝑓∆𝑡+𝜙)exp (−∆𝑡 𝜏1)+𝐵exp (−∆𝑡 𝜏2), (1) where 𝛥𝑡 is pump and probe time difference, 𝐴𝐹 is the amplitude, 𝑓 is the frequency , 𝜙 is the phase, 𝜏1 is the decay time of precession, and 𝜏2 is the decay time of the background with an amplitude 𝐵. Fig. 3. Time -resolved Faraday rotation at different magnetic fields H (a-d) and laser fluence s (I1-I4 correspond to 1.7, 3.2, 5.0, and 6.5 mJ/cm2, respectively) . The normalized MZ on the vertical axis is defined as ΔF/max, where max is obtained for saturation magnetization rotation at H (see. Fig. 2a). The curves are offset vertically without rescaling. The s olid lines are fittings with the damped sine function (Eq. 1) . 6 Fig. 4. Photo -magnetic precession parameters as a function of p ump fluence in different external magnetic field H: a) amplitude of the Faraday rotation AF, b) frequency of the precession , and c) effective damping. Different colors correspond to different external magnetic fields. The s olid lines are the linear fits where applicable , while the dashed lines are the visual guides. Some of the error bars are smaller than the data point symbols. At low applied fields 𝐻<1 kOe, where the photo -magnetic anisotropy field ( 𝐻𝐿) contribution to the total effective magnetic field is the strongest, the largest magnetization precession amplitude is observed. Figure 4 show s the most important parameters of the magnetiza tion precession, that is, amplitude, frequency and effective damping (Fig. 4a -c). The latter is obtained from the frequency and the lifetime as (2𝜋𝑓𝜏1)−1. Although the amplitude dependence on the pump fluence is mostly linear, the other two parameters exhibit a more complicated dependence, which is indicative of the noticeable nonlinearity in the magnetic system. In particular, at 𝐻=0.4 and 0.5 kOe, we observe d an increase in the effective damping with laser fluence, resulting in a faster decay of the magnetic precession. This is further corroborated by the frequency decrease seen in Fig. 4b. It is seen that the behavior of the magnetic subsystem is noticeably dissimilar at low ( below 1 kOe) and high (above 2 kOe) magnetic fields. At higher magnetic fie lds 𝐻>1 kOe we were unable to observe nonlinear magnetization response at pump fluences up to 10 mJ/cm2. This is indicative of a significant difference in the dynamic response in the collinear and noncollinear states of the magnetic subsystem. 4. Nonlinear precession of magnetization in anisotropic cubic crystal s The data shown in Fig. 4c clearly indicates the nonlinearity in the magnetic response manifesting in the increase of the effective damping with the excitation (laser) fluence. Previously, similar behavior was found in a number of metallic systems [26-29] and quickly attributed to laser heating. Interestingly, Chen et al . [30] found a decrease of the effective damping with laser fluence in FePt, while invoking the temper ature dependence of magnetic inhomogeneities to explain the results. There, the impact of magnetic inhomogeneity -driven damping contribution exhibits a similar response to laser heating and an increase in the static magnetic field. A more complicated mecha nism relying on the temperature -dependent 7 competition between the surface and bulk anisotropy contributions and resulting in the modification of the effective anisotropy field has been demonstrated in ultrathin Co/Pt bilayers [31, 32] . Nonlinear spin dynamics is a rapidly developing subfield enjoying rich prospects for ultrafast spintronics [33]. Importantly, all those works featured thermal excitation of magnetization dynamics in metallic, strongly absorptive systems. In stark contrast, we argue that the mechanism in the Co-doped YIG studied here is essentially non -thermal. This negligible temperature change ΔT is unable to induce significant variations of the parameters in the magnetic syst em of YIG:Co (T N=450 K), thus ruling out the nonlinearity mechanism discussed above. Rather, we note the work by M üller et al. [34], where the non -thermal nonlinear regime of magnetization dynamics in CrO 2 at high laser fluences was ascribed to the spin -wave instabilities at large precession amplitudes [35]. We also note the recently debated and physically rich mechanisms of magnetic nonlinearities, such as spin inertia [36-39] and relativistic effects [40, 41] . Yet, we argue that in our case of a cubic magnetic anisotropy - dominated energy landscape, a much simpler explanation for the nonlinear spin dynamics can be suggested. In particular, we attribute the amplitude -dependent effective dampin g to the anharmonicity of the p otential well for magnetization . Fig. 5. Energy landscape as a function of the polar angle 𝜃𝜑=45°in the linear (𝐻=2.5 kOe, green) and nonlinear (𝐻=0.4 kOe, red) precession regime s. The d ashed lines are the parabolic fits in the vicinity of the minima . 𝜃 is the polar angle of magnetization orientation measured from the normal to the sample plane along the [001] axis in YIG:Co . We performed numerical calculations of the energy density landscape 𝑊(𝜃,𝜑): 𝑊(𝜃,𝜑)=𝑊𝑐+𝑊𝑢+𝑊𝑑+𝑊𝑧 (2) taking into account the following terms in the free energy of the system: the Zeeman energy 𝑊𝑧=−𝑴∙𝑯, demagnetizing field term 𝑊𝑑=−2𝜋𝑀𝑠2sin2𝜃, cubic 𝑊𝑐=𝐾1∙ (sin4𝜃sin2𝜑cos2𝜃+sin2𝜃cos2𝜃cos2𝜑+sin2𝜃cos2𝜃sin2𝜑) and uniaxial anisotropy 𝑊𝑢=𝐾𝑢sin2𝜃 (𝜃 and 𝜑 are the polar and azimuthal angles, respectively ). In the calculations, we assume 𝐾1=−9∙ 103 erg/cm3, 𝐾𝑢=−3∙103 erg/cm3, and 𝑀𝑠 is the saturation 8 magnetization of 7.2 Oe [25]. Then, following [8] and [42], we calculate the precession frequency 𝑓 and the effective damping 𝛼𝑒𝑓𝑓: 𝑓=𝛾 2𝜋𝑀𝑠sin𝜃√𝛿2𝑊 𝛿𝜃2𝛿2𝑊 𝛿𝜑2−(𝛿2𝑊 𝛿𝜃𝛿𝜑)2 , (3) 𝛼𝑒𝑓𝑓=𝛼0𝛾(𝛿2𝑊 𝛿𝜃2+𝛿2𝑊 𝛿𝜑2sin−2𝜃) 8𝜋2𝑓𝑀𝑠, (4) where the 𝛾 is gyromagnetic ratio , and 𝛼0 is the Gilbert damping in YIG:Co [23, 24] . In Fig. 5, we only show the total energy as a function of the polar angle 𝜃, to illustrate the anharmonicity of the potential at small external in -plane magnetic fields. Experimental data and calculations of the energy 𝑊(𝜃,𝜑) have been published in Refs. [25, 43] . There, it is seen that at relative ly small external magnetic fields canting the magnetic state , the proximity of a neighboring energy minimum (to the right) effectively modifies the potential well for the corresponding o scillator (on the left) , introducing an anharmonicity . On the other hand, at sufficiently large magnetic fields, whic h, owing to the Zeeman energy term, modify the potential such that a single minimum emerges (shown in Fig. 5 in green), no nonlinearity is expected. This is also in line with the decreas ing impact of the cubic symmetry in the magnetic system, which is res ponsible for the anharmonicity of the energy potential. To get yet another calculated quantity that can be compare d to the experiment, we introduced the photo -magnetically in duced effective anisotropy term 𝐾𝐿. This contribution depends on the laser fluence I through the effective light -induced field 𝐻𝐿∝𝐼 as: 𝐾𝐿=−2𝐻𝐿𝑀𝑠cos2𝜃 (5) The presence of this term displaces the equilibrium for net magnetization. The equilibrium direction s can be obtained by minimizing the total energy with and without the photo - magn etic anisotropy term. Then, k nowing the angle between the perturbed and unperturbed equilibrium directions for the magnetization, we calculate d the precession amplitude 𝐴. We note the difference between the amplitudes 𝐴𝐹, which refers to the Faraday rotation of the probe beam, and 𝐴 standing for the opening angle of magnetization precession. Alth ough both are measured in degrees, their meaning is different. Having repeated this for a few levels of optical excitation, we obtain ed a linear slope of the amplitude vs excitation strength dependence. Figure 6 (a-c) illustrates the amplitude, frequency , and (linear ) effective damping as a function of the external magnetic field. The agreement between the calculated parameters and those obtained from fitting the experimental data is an impressive indication of the validity of our total energy approach. Further, the linear effective damping value of 𝛼≈0.2 obtained in the limit of strong field s, is in good agreement with the values known for our Co -doped YIG from previous works [6, 24] . In principle, the effective damping in garnets can increase towards lower magnetic fields. Conventionally attributed to the extrinsic damping contributions, this behavior has been observed in rare -earth iron garnets before as well and ascribed to the generation of the backward volume spin w ave mode by ultrashort laser pulses [44]. It is worth noting that there is no nonlinearity phenomenologically embedded in the approach given above. 9 Fig. 6. Photo -magnetic precession parameters at various magnetic fields: amplitude (a), frequency (b) , and (linear) effective damping (c). The points are from the experimental data, the solid lines are calculated as described in the text. The dark rectangular points are obtained in the FMR experimen ts. The g rey shaded area indicates the presence of a domain st ate (DS). The g reen shaded area show s the magnetization saturation state . Yet, the data presented in Fig. 4c indicates the persistent nonlinear behavior of the effective damping. To clarify the role of the potential anharmonicity, we fitted the potentials 𝑊(𝜃,𝜑) using a parabolic function with an anharmonic term : 𝑊(𝑥)=𝑊0+𝑘[(𝑥−𝑥0 )2+𝛽𝑥(𝑥−𝑥0)4] (6) Here 𝑥=𝜃 or 𝜑, and 𝛽𝑥 is the anharmonicity parameter. We calculated it independently for 𝜃 and 𝜑 for each dataset of 𝑊(𝜃,𝜑) obtained at different values of the external magnetic field 𝐻 by fitting the total energy with Eq. (6) in the vicinity of the energy minimum (Fig. 5) . This anharmonicity should be examined on equal footing with the no nlinear damping contribution. To quantify the latter, we follow the approach by Tiberkevich & Slavin [8] and analyze the effective damping dependencies on the precession amplitude by means of fitting a second -order polynomial to them : 𝛼=𝛼0+𝛼2𝐴2. (7) The examples of th e fit curves are shown in Fig. 7 a, demonstrating a good quality of the fit within a certain range of the amplitudes 𝐴 (below 45 ). It should, however, be noted that the model in Ref. [8] has been developed for the in -plane magnetic anisotropy, and thus its applicability for our case is limited. This is the re ason why we do not go beyond the amplitude dependence of the effective damping and do not analy ze the frequency dependence on 𝐴 in 10 the limit of strong effective fields. We note that the amplitude 𝐴, the opening angle of the precession, should be understood as a mathematical parameter only, and not as a true excursion angle of magnetization obtained in the real experimental conditions. There, large effective Gilbert damping values and a short decay t ime of the photo -magnetic anisotropy preclude the excursion of magnetization from its equilibrium to reach these 𝐴 values. Fig. 7. a) Effective damping in the linear and nonlinear precession regimes of the precession amplitude 𝐴. The lines are the second -order polynomial fits with Eq. (7). b ) Magnetic field dependence of the nonlinearity parameters: n onlinear damping coefficient 𝛼2 (points, obtained from experiments) and the 𝑊(𝜃) potential anharmonicity normalized 𝛽𝜃 (red line , calculated ). We note that the anharmonicity parameter 𝛽𝑥 calculated for the W(θ) profiles was found to be a few orders of magnitude larger than that obtained for W(𝜑). This difference in the anharmonicity justifies our earlier decision to focus on the shape of W(θ) potential only (cf. Fig. 5). This means that the potential for magnetization in the azimuthal plane is muc h closer to the parabolic shape and much larger amplit udes of the magnetization precession are required for it to start manifesting nonlinearities in dynamics. As such, we only consider the anharmonicity 𝛽𝑥 originating in the W(θ) potential energy. I n Fig. 7b, we compare the 𝛽𝜃 (red line) and 𝛼2 (points) dependencies on the external in -plane magnetic field. It is seen that its general shape is very similar, corroborating our assumption that the potential anharmonicity is the main driving force behind the obse rved nonlinearity. We argue that thanks to the c ubic magnetic anisotropy in YIG:Co film, the potential anharmonicity -related mechanism of nonlinearity allows for reaching the nonlinear regime at moderate excitation levels. 5. Simulation s of laser -induced magnetization dynamics 11 To further prove that the ob served nonlinearities in magnetization dynamics do not require introducing additional inertial or relativistic terms [33], we complemented our experimental findings with numerical simulations of the LLG equation: 𝑑𝐌 𝑑𝑡=−𝛾[𝐌×𝐇eff(𝑡)]+𝛼 𝑀𝑠(𝐌×𝑑𝐌 𝑑𝑡), (8) where 𝐻𝑒𝑓𝑓 is the effective field derived from Eq. (2) as : 𝐇eff(𝑡)=−∂𝑊𝐴 ∂𝑴+𝐇L(𝑡), (9) We employ ed the simulation model from Ref. [11] and added a term corresponding to the external magnetic field 𝐻. Calculations performed for a broad range of laser fluence s and external field values allowed us to obtain a set of traces of the magnetization dynamics . Figure 8 show s a great deal of similarity between simulations and experimen tal data (cf. Fig. 3). It is seen that t he frequency increases with increasing external field 𝐻 while the amplitude decreases (see Fig. 8a). The simulations for various stimulus strengths show the expected growth of the precession amplitude (see Fig. 8b). Fig. 8. Photo -magnetic precession obtained in numerical simulations of the LLG equation for: a) field dependence at moderate excitation level and b) power dependence (I=4, 10, 16, and 22 arb. units ) at 𝐻=0.4 kOe. We further repeated our fit procedure with Eq.(1) to obtain the precession parameters from these data. Figure 9 show s the values of the amplitude and frequency of the precession in the power regime. At a low field 𝐻=0.4 kOe (red) , the nonlinearity is clearly visible and comparable with experimental data, as seen in Fig. 4. Similarly, at high field s (green) , the behavior is mostly linear. Figure 9a shows a great deal of similarity between simulations (amplitude parameter) and experi mental data (normalize d value AF/max) (cf. Fig. 4a). The analysis of the damping parameter (Fig. 9c) also confirms the exp erimental findings (as in Fig. 7a), revealing the existence of two regimes, linear and nonlinear . The results of the s imulations confirm that the observation of the nonlinear response of the magnetic system can be attributed to the anharmonicity of the energy landscape. 12 Notably, in the simulations , as well as in the experimental data, we not only observe a second - order co rrection to the effective damping 𝛼2, but also a deviation from Eq.(7) at even larger amplitudes (cf. Fig. 7 a and Fig. 9c). The latter manifests as a reduction of the effective damping compared to the expected 𝛼0+𝛼2𝐴2 dependence shown with dashed lines. This higher -order effect is unlikely to originate in the multi -magnon scattering contribution since the latter would only further increase the effective damping [8]. We rather believe that th is is likely an artifact of the used damped oscillator model where in the range of 𝛼𝑒𝑓𝑓≈1 the quasiperiodic description of magnetization precession ceases to be physically justified. Fig. 9. Power dependence of the a) amplitude and b) frequency as obtained in the simulations for low (red dataset) and high (green dataset) external magnetic field s. c) Effective damping in the linear and nonlinear precession regimes . 6. Photo -induced phase diagram of sub -switching regime It is seen from both experimental and numerical results above that the cubic symmetry of the magnetic system is key for the observed nonlinear magnetization dynamics. To quantify the parameter space for the nonlinearity, we first estimate the realistic values of the effective light -induced magnetic field 𝐻𝐿. Throughout a number of works on photo -magnetism in Co - doped garnets, a single -ion approach to magnetic anisotropy is consistent ly utilized. We note 13 that in YIG:Co, it is the Co ions at tetrahedral sites that are predominantly responsible for the cubic anisotropy of the magnetic energy landscape [22]. In the near -IR range, these ions are resonantly excited at the 1300 nm wavelength, resulting in improved efficiency of the photo - magnetic stimulus , as compared to previous works [45]. Further, we note that at the magnetization switching threshold, about 90% of the Co3+ ions with a concentration on the order of 1020 cm-3 are excited with incident photons [11, 46] . Taking into account the single - ion contribution to the anisotropy 𝛥𝐾1~105 erg/cm3 [47], and assuming a linear relation between the absorbed laser power (or fluence) and the effective photo -magnetic field 𝐻𝐿, for the latter we find that 𝐻𝐿~1 kOe is sufficient for the magnetization switching. This means that the sub -switching regime of magnetiz ation dynamics (cf. Fig. 1) refers to the laser fluences (as well as wavelengths) , resulting in smaller effective fields. We reiterate that in previous works, the impact of the external magnetic field on the photo - magnetically driven magnetization precess ion has not been given detailed attention. To address this gap , we plotted the amplitude of the precession 𝐴 calculated in the same way as above in the sub-switching regime (Fig. 10) . As expected, the amplitude generally increases with 𝐻𝐿. However, we n ote a critical external field of about 0.5 kOe at which the desired amplitudes can be reached at smaller light -induced effective fields 𝐻𝐿. At this field, where the system enters a single domain state, the potential curvature around the energy minimum decreases, thus facilitating the large -angle precession. In other words, external magnetic field s can a ct as leverage for the effective field of the photo -induced anisotropy, thus reducing the magnetization switching threshold. An exhaustive study of magneti zation switching across the parameter space shown in Fig. 10 remains an attractive perspective for future studies. Fig. 10. Calculated amplitude m ap of the photo -induced magnetization precession in YIG:Co film. In our analysis, we only considered a truly photo -magnetic excitation and neglected the laser - induced effects of thermal origi n. It is, however, known that laser -driven heating can introduce an additional, long -lasting modification of magnetic anisotropy in iron garnets [48, 49] . The 14 relatively long relaxation times associated with cooling are responsible for the concomitant modulation of the precession parameters and thus facilitate nonlinearities in the response of the magnetic system. Yet, 1300 nm laser excitation of magnetization dynamics in YIG :Co film was shown to be highly polarization -dependent [6], thus indicating the dominant role of the non-thermal excitation mechanism. On the other hand, the unavoid able laser -induced heating with experi mental values of laser fluence in YIG:Co film has been estimated to not exceed 1 K [6]. As such, we do not expect modification of the Gilbert damping associated with the proximity of the ma gnetization compensation or N éel temperature in the ferromagnetic garnet [50]. However, a detailed investigation of the temperature -dependent nonlinear magnetization dynamics in the vicinity of the compensation point or a magnetic phase transition [51, 52] represents another promising research direction. Further , exploring the nonlinear regime in the response of the magnetic system to intense THz stimul i along the lines discussed in [33] enjoys a rich potentia l for spintronic applications. 7. CONCLUSIONS In summary, we studied, both experimentally and numerically, the nonlinear regime of magnetization dynamics in photo -magn etic Co -doped YIG film. After excitation with femtosecond laser pulses at fluences below the magnetization switching threshold, there is a range of external magnetic field where the magnetic system demonstrates strongly non linear precession characterized by a significant increase of t he effective Gilbert damping. We attribute this nonlinearity to the anharmonicity of the potential for the magnetic oscillator enhanced by the dominant role of the cubic magnetocrystalline anisotropy. The effective damping and its nonlinear contribution, a s obtained from numerical simulations, both demonstrate a very good agreement with the experimental findings. Simulations of the magnetization dynamics by means of the LLG equation further confirm the nonlinearity in the magnetic response below the switchi ng limit. Finally, we provide estimations for the realistic , effective photo -magnetic fields 𝐻𝐿 and map the workspace of the parameters in the sub - switching, nonlinear regime of photo -induced magnetization dynamics. 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2022-11-15

We analyze, both experimentally and numerically, the nonlinear regime of the photo-induced coherent magnetization dynamics in cobalt-doped yttrium iron garnet films. Photo-magnetic excitation with femtosecond laser pulses reveals a strongly nonlinear response of the spin subsystem with a significant increase of the effective Gilbert damping. By varying both laser fluence and the external magnetic field, we show that this nonlinearity originates in the anharmonicity of the magnetic energy landscape. We numerically map the parameter workspace for the nonlinear photo-induced spin dynamics below the photo-magnetic switching threshold. Corroborated by numerical simulations of the Landau-Lifshitz-Gilbert equation, our results highlight the key role of the cubic symmetry of the magnetic subsystem in reaching the nonlinear spin precession regime. These findings expand the fundamental understanding of laser-induced nonlinear spin dynamics as well as facilitate the development of applied photo-magnetism.

Nonlinear sub-switching regime of magnetization dynamics in photo-magnetic garnets

2211.08048v2

arXiv:1903.05415v2 [math.NA] 20 Mar 2020HIGHER-ORDER LINEARLY IMPLICIT FULL DISCRETIZATION OF THE LANDAU–LIFSHITZ–GILBERT EQUATION GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH Abstract. For the Landau–Lifshitz–Gilbert (LLG) equation of micromagnetics we study linearly implicit backward difference formula (BDF) time discre tizations up to order 5 combined with higher-order non-conforming finite elem ent space discretizations, which are based on the weak formulation due to Alou ges but use approximate tangent spaces that are defined by L2-averaged instead of nodal or- thogonality constraints. We prove stability and optimal-order erro r bounds in the situation of a sufficiently regular solution. For the BDF methods of or ders 3 to 5, this requires that the damping parameter in the LLG equations be ab ove a posi- tive threshold; this condition is not needed for the A-stable method s of orders 1 and 2, for which furthermore a discrete energy inequality irrespec tive of solution regularity is proved. 1.Introduction 1.1.Scope.In this paper we study the convergence of higher-order time and s pace discretizations of the Landau–Lifshitz–Gilbert (LLG) equation, wh ich is the basic model for phenomena in micromagnetism, such as in recording media [ 26, 36]. The main novelty of the paper lies in the construction and analysis of w hat is apparently the first numerical method for the LLG equation that is second-order convergent in both space and time to sufficiently regular solutions an d that satisfies, as an important robustness property irrespective of regularity, a discrete energy inequality analogous to that of the continuous problem. We study discretization in time by linearly implicit backward difference fo rmu- lae (BDF) up to order 5 and discretization in space by finite elements o f arbitrary polynomial degree. For the BDF methods up to order 2 we prove opt imal-order error bounds in the situation of a sufficiently regular solution and a dis crete energy inequality irrespective of solution regularity under very weak regula rity assumptions on the data. For the BDF methods of orders 3 to 5, we prove optima l-order error bounds in the situation of a sufficiently regular solution under the add itional condi- tionthatthedampingparameter intheLLGequationbeaboveameth od-dependent positive threshold. However, no discrete energy inequality irrespe ctive of solution regularity is obtained for the BDF methods of orders 3 to 5. The discretization in space is done by a higher-order non-conformin g finite ele- ment method based on the approach of Alouges [4, 5], which uses a p rojection to Date: March 23, 2020. 2010Mathematics Subject Classification. Primary 65M12, 65M15; Secondary 65L06. Key words and phrases. BDF methods, non-conforming finite element method, Landau– Lifshitz–Gilbert equation, energy technique, stability. 12 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH an approximate tangent space to the normality constraint. Contr ary to the point- wise orthogonality constraints in the nodes, which define the appro ximate tangent space in those papers and yield only first-order convergence also f or finite elements with higher-degree polynomials, we here enforce orthogonality ave raged over the finite element basis functions. With these modified approximate tang ent spaces we proveH1-convergence of optimal order in space and time under the assump tion of a sufficiently regular solution. Key issues in the error analysis are the properties of the orthogon al projection onto the approximate tangent space, the higher-order consiste ncy error analysis, and the proof of stable error propagation, which is based on non-s tandard energy estimates and uses both L2and maximum norm finite element analysis. 1.2.The Landau–Lifshitz–Gilbert equation. The standard phenomenological model for micromagnetism is provided by the Landau–Lifshitz (LL) e quation (1.1) ∂tm=−m×Heff−αm×(m×Heff) where the unknown magnetization field m=m(x,t) takes values on the unit sphereS2,α >0 is a dimensionless damping parameter, and the effective mag- netic fieldHeffdepends on the unknown m. The Landau–Lifshitz equation (1.1) can be equivalently written in the Landau–Lifshitz–Gilbert form (1.2) α∂tm+m×∂tm= (1+α2)/bracketleftbig Heff−/parenleftbig m·Heff/parenrightbig m/bracketrightbig . Indeed, in view of the vector identity a×(b×c) = (a·c)b−(a·b)c,fora,b,c∈R3, we have−m×/parenleftbig m×Heff/parenrightbig =Heff−/parenleftbig m·Heff/parenrightbig m,and taking the vector product of (1.1) withmand adding αtimes (1.1) then yields (1.2). Sincem×ais orthogonal to m,for anya∈R3,it is obvious from (1.1) that ∂tmis orthogonal to m:m·∂tm= 0; we infer that the Euclidean norm satisfies |m(x,t)|= 1 for allxand for allt, provided this is satisfied for the initial data. The term in square brackets on the right-hand side in (1.2) can be re written as P(m)Heff, where (with Ithe 3×3 unit matrix) P(m) =I−mmT is the orthogonal projection onto the tangent plane to the unit sp hereS2atm. In this paper we consider the situation (1.3) Heff=1 1+α2/parenleftbig ∆m+H/parenrightbig , whereH=H(x,t) is a given external magnetic field. The factor 1 /(1 +α2) is chosen for convenience of presentation, but is inessential for th e theory; it can be replaced by any positive constant factor. Withthischoiceof Heff, we arriveattheLandau–Lifshitz–Gilbert (LLG)equation in the form (1.4) α∂tm+m×∂tm=P(m)(∆m+H). Weconsiderthisequationasaninitial-boundaryvalueproblemonabou ndeddomain Ω⊂R3and a time interval 0 /lessorequalslantt/lessorequalslant¯t, with homogeneous Neumann boundaryHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 3 conditions and initial data m0taking values on the unit sphere, i.e., the Euclidean norm|m0(x)|equals 1 for all x∈Ω. We consider the following weak formulation, first proposed by Alouge s [4, 5]: Find the solution m:Ω×[0,¯t]→S2withm(·,0) =m0by determining, at m(t)∈H1(Ω)3, the time derivative ∂tm(omitting here and in the following the argumentt) as that function in the tangent space T(m) :=/braceleftbig ϕ∈L2(Ω)3:m·ϕ= 0 a.e./bracerightbig =/braceleftbig ϕ∈L2(Ω)3:P(m)ϕ=ϕ} that satisfies, for all ϕ∈T(m)∩H1(Ω)3, (1.5) α/parenleftbig ∂tm,ϕ/parenrightbig +/parenleftbig m×∂tm,ϕ/parenrightbig +/parenleftbig ∇m,∇ϕ/parenrightbig =/parenleftbig H,ϕ/parenrightbig , where the brackets ( ·,·) denote the L2inner product over the domain Ω. The numerical methods studied in this paper are based on this weak form ulation. 1.3.Previous work. There is a rich literature on numerical methods for Landau– Lifshitz(–Gilbert) equations; for the numerical literature up to 20 07 see the review by Cimr´ ak [17]. Alouges & Jaisson [4, 5] propose linear finite element discretizations in space and linearly implicit backward Euler in time for the LLG equation in the weak fo rmula- tion (1.5) and prove convergence withoutrates towards nonsmooth weak solutions, using a discrete energy inequality and compactness arguments. Co nvergence of this type was previously shown by Bartels & Prohl [11] for fully implicit meth ods that are based on a different formulation of the Landau–Lifshitz equatio n (1.1). In [6], convergence without rates towards weak solutions is shown for a m ethod that is (formally) of “almost” order 2 in time, based on the midpoint rule, for the LLG equation with an effective magnetic field of a more general type than (1.3). In a complementary line of research, convergence withrates has been studied under sufficiently strong regularity assumptions, which can, howev er, not be guar- anteed over a given time interval, since solutions of the LLG equation may develop singularities. A first-order error bound for a linearly implicit time discr etization of the Landau–Lifshitz equation (1.1) was proved by Cimr´ ak [16]. Op timal-order error bounds for linearly implicit time discretizations based on the bac kward Euler and Crank–Nicolson methods combined with finite element full discret izations for a different version of the Landau–Lifshitz equation (1.1) were obta ined under suf- ficient regularity assumptions by Gao [23] and An [7], respectively. In contrast to [4, 5, 6, 11], these methods do not satisfy an energy inequality irres pective of the solution regularity. Numerical discretizations for the coupled system of the LLG equat ion (1.5) with the eddy current approximation of the Maxwell equations are stud ied by Feischl & Tran [21], with first-order error bounds in space and time under suffi cient regularity assumptions. This also yields thefirst result offirst-order conver gence ofthemethod of Alouges & Jaisson [4, 5]. There are several methods for the LLG equations that are of for mal order 2 in time (thoughonlyoforder 1 inspace), e.g., [35, 31, 19], but noneof t hemcomes with an error analysis. Fully implicit BDF time discretizations for LLG equatio ns have4 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH been used successfully in the computational physics literature [37 ], though without giving any error analysis. To the authors’ knowledge, the second-order linearly implicit metho d proposed and studied here is thus the first numerical method for the LLG (or LL) equation thathasrigorousapriorierrorestimatesoforder2inbothspace andtimeunderhigh regularity assumptions and that satisfies a discrete energy inequa lity irrespective of regularity. We conclude this brief survey of the literature with a remark: The ex isting con- vergence results either give convergence of a subsequence witho ut rates to a weak solution(withoutimposingstrongregularityassumptions), orthey showconvergence with rates towards sufficiently regular solutions (as we do here). Bo th approaches yield insight into the numerical methods and have their merits, and th ey comple- ment each other. Clearly, neither approach is fully satisfactory, b ecause convergence without rates of some subsequence is nothing to observe inactual computations, and on the other hand high regularity is at best provable for close to con stant initial con- ditions [22] or over short time intervals. We regard the situation as a nalogous to the development of numerical methods and their analysis in other fie lds such as nonlinear hyperbolic conservation laws: second-order methods ar e highly popular in that field, even though they can only be shown to converge with ve ry low order (1/2 or less or only without rates) for available regularity properties; s ee, e.g., [32, Chapter 3]. Nevertheless, second-order methods arefavoredo ver first-order methods in many applications, especially if they enjoy some qualitative propert ies that give them robustness in non-regular situations. A similar situation occur s with the LLG equation, where the most important qualitative property appears to be the energy inequality. 1.4.Outline. InSection2we describe thenumerical methodsstudied inthispaper . They use time discretization by linearly implicit BDF methods of orders u p to 5 and space discretization by finite elements of arbitrary polynomial degr ee in a numerical scheme that is based on the weak formulation (1.5), with an approxim ate tangent space that enforces the orthogonality constraint approximately in anL2-projected sense. In Section 3 we state our main results: •For the full discretization of (1.5) by linearly implicit BDF methods of or ders 1 and 2 and finite element methods of arbitrary polynomial degree we g ive optimal- order error bounds in the H1norm, under very mild mesh conditions, in the case of sufficiently regular solutions (Theorem 3.1). For these methods w e also show a discrete energy inequality that requires only very weak regularity a ssumptions on the data (Proposition 3.1). This discrete energy inequality is of the s ame type as the one used in [5, 11] for proving convergence without rates to a weak solution. •For the linearly implicit BDF methods of orders 3 to 5 and finite element m ethods with polynomial degree at least 2, we have optimal-order error boun ds in theH1 norm only if the damping parameter αis larger than some positive threshold, which depends on the order of the BDF method (Theorem 3.2). Moreover , a stronger (but still mild) CFL condition τ/lessorequalslantchis required. A discrete energy inequality underHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 5 very weak regularity conditions is not available for the BDF methods o f orders 3 to 5, in contrast to the A-stable BDF methods of orders 1 and 2. In Section 4 we prove a perturbation result for the continuous pro blem by energy techniques, as a preparation for the proofs of our error bounds for the discretization. In Section 5 we study properties of the L2-orthogonal projection onto the discrete tangent space, which are needed to ensure consistency of the fu ll order and stability of the space discretization with the higher-order discrete tangen t space. In Section 6 we study consistency properties of the methods and p resent the error equation. In Sections 7 and 8 we prove Theorems 3.1 and 3.2, respectively. The higher- order convergence proofs are separated into consistency (Sec tion 6) and stability estimates. The stability proofs use the technique of energy estima tes, in an unusual version where the error equation is tested with a projection of the discrete time derivative of the error onto the discrete tangent space. These p roofs are different for the A-stable BDF methods of orders 1 and 2 and for the BDF met hods of orders 3 to 5. For the control of nonlinearities, the stability proofs also re quire pointwise error bounds, which are obtained with the help of finite element inver se inequalities from theH1error bounds of previous time steps. In Section 9 we illustrate our results by numerical experiments. In an Appendix we collect basic results on energy techniques for BDF methods that are needed for our stability proofs. 2.Discretization of the LLG equation We now describe the time and space discretization that is proposed a nd studied in this paper. 2.1.Time discretization by linearly implicit BDF methods. We shall dis- cretize the LLG equation (1.5) in time by the linearly implicit k-step BDF methods, 1/lessorequalslantk/lessorequalslant5, described by the polynomials δandγ, δ(ζ) =k/summationdisplay ℓ=11 ℓ(1−ζ)ℓ=k/summationdisplay j=0δjζj, γ(ζ) =1 ζ/bracketleftbig 1−(1−ζ)k/bracketrightbig =k−1/summationdisplay i=0γiζi. We lettn=nτ, n= 0,...,N,be a uniform partition of the interval [0 ,¯t] with time stepτ=¯t/N.For thek-step method we require kstarting values mifor i= 0,...,k−1. Forn/greaterorequalslantk, we determine the approximation mntom(tn) as follows. We first extrapolate the known values mn−k,...,mn−1to a preliminary normalized approximation /hatwidermnattn, (2.1) /hatwidermn:=k−1/summationdisplay j=0γjmn−j−1/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay j=0γjmn−j−1/vextendsingle/vextendsingle/vextendsingle. To avoid potentially undefined quantities, we define /hatwidermnto be an arbitrary fixed unit vector if the denominator in the above formula is zero.6 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH The derivative approximation ˙mnand the solution approximation mnare related by the backward difference formula (2.2) ˙mn=1 τk/summationdisplay j=0δjmn−j,i.e.,mn=/parenleftig −k/summationdisplay j=1δjmn−j+τ˙mn/parenrightig /δ0. We determine mnby requiring that for all ϕ∈T(/hatwidermn)∩H1(Ω)3, (2.3)α/parenleftbig ˙mn,ϕ/parenrightbig +/parenleftbig /hatwidermn×˙mn,ϕ/parenrightbig +/parenleftbig ∇mn,∇ϕ/parenrightbig =/parenleftbig H(tn),ϕ/parenrightbig ˙mn∈T(/hatwidermn),i.e.,/hatwidermn·˙mn= 0. Here we note that on inserting the formula in (2.2) for mnin the third term of (2.3), we obtain a linear constrained elliptic equation for ˙mn∈T(/hatwidermn)∩H1(Ω)3of the form α/parenleftbig ˙mn,ϕ/parenrightbig +/parenleftbig /hatwidermn×˙mn,ϕ/parenrightbig +τ δ0/parenleftbig ∇˙mn,∇ϕ/parenrightbig =/parenleftbig fn,ϕ/parenrightbig ∀ϕ∈T(/hatwidermn)∩H1(Ω)3, wherefnconsistsofknownterms. Thebilinearformontheleft-handsideis H1(Ω)3- coercive on T(/hatwidermn)∩H1(Ω)3, and hence the above linear equation has a unique solution ˙mn∈T(/hatwidermn)∩H1(Ω)3by the Lax–Milgram lemma. Once this elliptic equation is solved for ˙mn, we obtain the approximation mn∈H1(Ω)3tom(tn) from the second formula in (2.2). 2.2.Full discretization by BDF and higher-order finite elements .For a familyofregularandquasi-uniformfiniteelement triangulationsof Ωwithmaximum meshwidth h >0 we form the Lagrange finite element spaces Vh⊂H1(Ω) with piecewise polynomials of degree r/greaterorequalslant1. We denote the L2-orthogonal projections onto the finite element space by Πh:L2(Ω)→VhandΠh=I⊗Πh:L2(Ω)3→V3 h. With a function m∈H1(Ω)3that vanishes nowhere on Ω, we associate the discrete tangent space (2.4)Th(m) ={ϕh∈V3 h: (m·ϕh,vh) = 0∀vh∈Vh} ={ϕh∈V3 h:Πh(m·ϕh) = 0}. This space is different from the discrete tangent space used in [4, 5 ], where the orthogonality constraint m·ϕh= 0 is required to hold pointwise at the finite element nodes. Here, the constraint is enforced weakly on the finit e element space, as is done in various saddle point problems for partial differential equ ations, for example forthedivergence-free constraint inthe Stokes problem [14, 25]. Incontrast to that example, here the bilinear form associated with the linear con straint, i.e., b(m;ϕh,vh) = (m·ϕh,vh), depends on the state m. This dependence substantially affects both the implementation and the error analysis. Following the general approach of [4, 5] with this modified discrete ta ngent space, we discretize (1.5) in space by determining the time derivative ∂tmh(t)∈Th(mh(t)) such that (omitting the argument t) (2.5)α/parenleftbig ∂tmh,ϕh/parenrightbig +/parenleftbig mh×∂tmh,ϕh/parenrightbig +/parenleftbig ∇mh,∇ϕh/parenrightbig =/parenleftbig H,ϕh/parenrightbig ∀ϕh∈Th(mh), where the brackets ( ·,·) denote again the L2inner product over the domain Ω.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 7 The full discretization with the linearly implicit BDF method is then readily obtained from (2.3): determine ˙mn h∈Th(/hatwidermn h) such that (2.6)α/parenleftbig˙mn h,ϕh/parenrightbig +/parenleftbig/hatwidermn h×˙mn h,ϕh/parenrightbig +/parenleftbig ∇mn h,∇ϕh/parenrightbig =/parenleftbig Hn,ϕh/parenrightbig ∀ϕh∈Th(/hatwidermn h), where/hatwidermn hand˙mn hare related to mn−j hforj= 0,...,kin the same way as in (2.1) and (2.2) above with mn−j hin place ofmn−j, viz., (2.7) ˙mn h=1 τk/summationdisplay j=0δjmn−j h,/hatwidermn h=k−1/summationdisplay j=0γjmn−j−1 h/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay j=0γjmn−j−1 h/vextendsingle/vextendsingle/vextendsingle. To avoid potentially undefined quantities, we define /hatwidermn hto be an arbitrary fixed unit vector if the denominator in the above formula is zero. (We will, ho wever, show that this does not occur in the situation of sufficient regularity.) To implement the discrete tangent space Th(/hatwidermn h), there are at least two options: using the constraints Πh(m·ϕh) = 0 or constructing a local basis of Th(m). (a)Constraints : Letφifori= 1,...,N:= dimVhdenote the nodal basis of Vhand denote the basis functions of V3 hbyφi=ek⊗φifori= (i,k), where ekfork= 1,2,3 are the standard unit vectors of R3. We denote by MandA the usual mass and stiffness matrices, respectively, with entries mij= (φi,φj)L2(Ω) andaij= (∇φi,∇φj)L2(Ω)3. We further introduce the sparse skew-symmetric matrix Sn= (sn i,j)∈R3N×3Nwithentries sn i,j= (/hatwidermn h×φi,φj)L2(Ω)3andthesparseconstraint matrixCn= (cn i,j)∈R3N×Nbycn i,j= (/hatwidermn h·φi,φj)L2(Ω). Finally, we denote the matrix of the unconstrained time-discrete problem as Kn=αI⊗M+τ δ0I⊗A+Sn. Let ˙mn∈R3Ndenote the nodal vector of ˙mn h∈Th(/hatwidermn h). In this setting, (2.6) yields a system of linear equations of saddle point type Kn˙mn+(Cn)Tλn=fn, Cn˙mn= 0, whereλn∈RNis the unknown vector of Lagrange multipliers and fn∈R3Nis a known right-hand side. (b)Local basis : It is possible to compute a local basis of Th(m) by solving small local problems. To see that, let ω⊂Ωdenote a collection of elements of the mesh and letω⊃ωdenote the same set plus the layer of elements touching ω(the patch ofω). A sufficient (and necessary) condition for ϕh∈V3 hwith supp(ϕh)⊆ωto belong toTh(m) is (2.8) ( m·ϕh,ψh) = 0 for all ψh∈Vhwith supp(ψh)⊆ω. If we denote by # ωthe number of generalized hat functions of Vhsupported in ω, the space of functions in V3 hwith support in ωis 3#ω-dimensional. On the other hand, the space of test functions in (2.8) is # ω-dimensional. We may choose ω sufficiently large (depending only on shape regularity) such that 3# ω >#ωand hence (2.8) has at least one solution which is then a local basis functio n ofTh(m). Choosing different ωto coverΩyields a full basis of Th(m).8 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH Let us denote the so obtained basis of Th(/hatwidermn h) by (ψn ℓ), given via ψn ℓ=/summationtext iφibn iℓ, and the sparse basis matrix by Bn= (bn iℓ). Then, the nodal vector ˙ mn=Bnxnis obtained by solving the linear system (Bn)TKnBnxn= (Bn)Tfn. An advantage of this approach is that the dimension is roughly halved compared to the formulation with constraints. However, the efficiency of one approach versus the other depends heavily on the numerical linear algebra used. Suc h comparisons are outside the scope of this paper. Remark 2.1. The algorithm described above does not enforce the norm constra int |m|= 1 at the nodes. The user might add a normalization step in the definit ion ofmnin (2.2). However, here we do not consider this normalized variant of the method, whose convergence properties are not obvious to derive . Remark 2.2. Differently to [4], we do not use the pointwise discrete tangent space Tpw h(m) ={ϕh∈V3 h:m·ϕ= 0 in every node } ={ϕh∈V3 h:Ih(m·ϕh) = 0}=IhP(m)V3 h, whereIh:C(¯Ω)→Vhdenotesfiniteelementinterpolationand Ih=I⊗Ih:C(¯Ω)3→ V3 h. It is already reported in [4, Section 4] that an improvement of the o rder with higher-degree finite elements could not be observed in numerical ex periments when using the pointwise tangent spaces in the discretization (2.5). Our a nalysis shows a lack of consistency of optimal order in the discretization with Tpw h(m), which originates from the fact that IhP(m) is not self-adjoint. The order reduction can, however, be cured by adding a correction term: in the nth time step, determine ˙mn h∈Tpw h(/hatwidermn h) such that for all ϕh∈Tpw h(/hatwidermn h), (2.9)α/parenleftbig˙mn h,ϕh/parenrightbig +/parenleftbig/hatwidermn h×˙mn h,ϕh/parenrightbig +/parenleftbig ∇mn h,∇ϕh/parenrightbig −/parenleftbig ∇/hatwidermn h,∇(I−P(/hatwidermn h))ϕh/parenrightbig =/parenleftbig P(/hatwidermn h)H(tn),ϕh/parenrightbig , with notation /hatwidermn hand˙mn has in (2.7). With the techniques of the present paper, it can be shown that like (2.6), also this discretization converges with o ptimal order in theH1norm under sufficient regularity conditions. Since this paper is alread y rather long, we do not include the proof of this result. In contrast to (2.6) for the first- and second-order BDF methods, the method (2.9) does not admit anh- and τ-independent bound of the energy that is irrespective of the smoo thness of the solution. 3.Main results 3.1.Error bound and energy inequality for BDF of orders 1 and 2. For the full discretization with first- and second-order BDF methods a nd finite elements of arbitrary polynomial degree r/greaterorequalslant1 we will prove the following optimal-order error bound in Sections 5 to 7. Theorem 3.1 (Error bound for orders k= 1,2).Consider the full discretization (2.6)of the LLG equation (1.4)by the linearly implicit k-step BDF time discretiza- tion fork/lessorequalslant2and finite elements of polynomial degree r/greaterorequalslant1from a family ofHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 9 regular and quasi-uniform triangulations of Ω. Suppose that the solution mof the LLG equation is sufficiently regular. Then, there exist ¯τ >0and¯h >0such that for numerical solutions obtained with step sizes τ/lessorequalslant¯τand meshwidths h/lessorequalslant¯h, which are restricted by the very mild CFL-type condition τk/lessorequalslant¯ch1/2 with a sufficiently small constant ¯c(independent of handτ), the errors are bounded by (3.1) /ba∇dblmn h−m(tn)/ba∇dblH1(Ω)3/lessorequalslantC(τk+hr)fortn=nτ/lessorequalslant¯t, whereCis independent of h,τandn(but depends on αand exponentially on ¯t), provided that the errors of the starting values also satisfy such a bound. The precise regularity requirements are as follows: (3.2)m∈Ck+1([0,¯t],L∞(Ω)3)∩C1([0,¯t],Wr+1,∞(Ω)3), ∆m+H∈C([0,¯t],Wr+1,∞(Ω)3). Remark 3.1 (Discrepancy from normality ).Sincem(x,tn) are unit vectors, an immediate consequence of the error estimate (3.1) is that (3.3) /ba∇dbl1−|mn h|/ba∇dblL2(Ω)/lessorequalslantC(τk+hr) fortn=nτ/lessorequalslant¯t, with a constant Cindependent of n,τandh. The proof of Theorem 3.1 also shows that the denominator in the definition of the normalized extrapolate d value/hatwidermn h satisfies /vextenddouble/vextenddouble/vextenddouble1−/vextendsingle/vextendsinglek−1/summationdisplay j=0γjmn−j−1 h/vextendsingle/vextendsingle/vextenddouble/vextenddouble/vextenddouble L∞(Ω)/lessorequalslantCh−1/2(τk+hr)/lessorequalslant1 2fortn=nτ/lessorequalslant¯t, which in particular ensures that /hatwidermn his unambiguously defined. Testing with ϕ=∂tm∈T(m) in (1.5), we obtain (only formally, if ∂tmis not inH1(Ω)3) α(∂tm,∂tm)+(∇m,∂t∇m) = (H,∂tm), which, by integration in time and the Cauchy–Schwarz and Young ineq ualities, im- plies the energy inequality /ba∇dbl∇m(t)/ba∇dbl2 L2+1 2α/integraldisplayt 0/ba∇dbl∂tm(s)/ba∇dbl2 L2ds/lessorequalslant/ba∇dbl∇m(0)/ba∇dbl2 L2+1 2α/integraldisplayt 0/ba∇dblH(s)/ba∇dbl2 L2ds. Similarly, wetestwith ϕh=˙mn h∈Th(/hatwidermn h)in(2.6). Thenwecanprovethefollow- ing discrete energy inequality, which holds under very weak regularit y assumptions on the data. Proposition 3.1 (Energy inequality for orders k= 1,2).Consider the full dis- cretization (2.6)of the LLG equation (1.4)by the linearly implicit k-step BDF time discretization for k/lessorequalslant2and finite elements of polynomial degree r/greaterorequalslant1. Then, the10 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH numerical solution satisfies the following discrete energy inequality :forn/greaterorequalslantkwith nτ/lessorequalslant¯t, γ− k/ba∇dbl∇mn h/ba∇dbl2 L2+1 2ατn/summationdisplay j=k/ba∇dbl˙mj h/ba∇dbl2 L2/lessorequalslantγ+ kk−1/summationdisplay i=0/ba∇dbl∇mi h/ba∇dbl2 L2+τ 2αn/summationdisplay j=k/ba∇dblH(tj)/ba∇dbl2 L2, whereγ± 1= 1andγ± 2= (3±2√ 2)/4. This energy inequality is an important robustness indicator of the nu merical method. In [5, 11], such energy inequalitys are used to prove conve rgence with- out rates (for a subsequence τn→0 andhn→0) to a weak solution of the LLG equation for the numerical schemes considered there (which have γ±= 1, but this is inessential in the proofs). As the proof of Proposition 3.1 is short, we give it here. Proof.TheproofreliesontheA-stabilityofthefirst-andsecond-orderB DFmethods via Dahlquist’s G-stabilitytheoryasexpressed inLemma 10.1ofthe Ap pendix, used withδ(ζ) =/summationtextk ℓ=1(1−ζ)ℓ/ℓandµ(ζ) = 1. The positive definite symmetric matrices G= (gij)k i,j=1are known to be G= 1 fork= 1 and (see [27, p.309]) G=1 4/parenleftbigg 1−2 −2 5/parenrightbigg fork= 2, which has the eigenvalues γ±= (3±2√ 2)/4. We test with ϕh=˙mn h∈Th(/hatwidermn h) in (2.6) and note/parenleftbig /hatwidermn h×˙mn h,˙mn h/parenrightbig = 0, so that α/ba∇dbl˙mn h/ba∇dbl2 L2+(∇mn h,∇˙mn h) = (Hn,˙mn h). The right-hand side is bounded by (Hn,˙mn h)/lessorequalslantα 2/ba∇dbl˙mn h/ba∇dbl2 L2+1 2α/ba∇dblHn/ba∇dbl2 L2. Recalling the definition of ˙mn h, we have by Lemma 10.1 (∇mn h,∇˙mn h)/greaterorequalslant1 τk/summationdisplay i,j=1gij(∇mn−i+1 h,∇mn−j+1 h)−1 τk/summationdisplay i,j=1gij(∇mn−i h,∇mn−j h). We fix ¯nwithk/lessorequalslant¯n/lessorequalslant¯t/τand sum from n=kto ¯nto obtain k/summationdisplay i,j=1gij(∇m¯n−i+1 h,∇m¯n−j+1 h)+1 2ατ¯n/summationdisplay n=k/ba∇dbl˙mn h/ba∇dbl2 L2 /lessorequalslantk/summationdisplay i,j=1gij(∇mk−i h,∇mk−j h)+τ 2α¯n/summationdisplay n=k/ba∇dblHn/ba∇dbl2 L2. Noting that γ−/ba∇dbl∇m¯n h/ba∇dbl2 L2/lessorequalslantk/summationdisplay i,j=1gij(∇m¯n−i+1 h,∇m¯n−j+1 h),HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 11 k/summationdisplay i,j=1gij(∇mk−i h,∇mk−j h)/lessorequalslantγ+k−1/summationdisplay i=0/ba∇dbl∇mi h/ba∇dbl2 L2, we obtain the stated result. /square 3.2.Error bound for BDF of orders 3to5.For the BDF methods of orders 3 to 5 we prove the following result in Section 8. Here we require a stro nger, but still moderate stepsize restriction in terms of the meshwidth. More importantly, we must impose a positive lower bound on the damping parameter αof (1.1). Theorem 3.2 (Error bound for orders k= 3,4,5).Consider the full discretization (2.6)of the LLG equation (1.4)by the linearly implicit k-step BDF time discretiza- tion for3/lessorequalslantk/lessorequalslant5and finite elements of polynomial degree r/greaterorequalslant2from a family of regular and quasi-uniform triangulations of Ω. Suppose that the solution mof the LLG equation has the regularity (3.2), and that the damping parameter αsatisfies (3.4)α>α kwith αk= 0.0913,0.4041,4.4348,fork= 3,4,5,respectively. Then, for an arbitrary constant ¯C >0, there exist ¯τ >0and¯h >0such that for numerical solutions obtained with step sizes τ/lessorequalslant¯τand meshwidths h/lessorequalslant¯hthat are restricted by (3.5) τ/lessorequalslant¯Ch, the errors are bounded by /ba∇dblmn h−m(tn)/ba∇dblH1(Ω)3/lessorequalslantC(τk+hr)fortn=nτ/lessorequalslant¯t, whereCis independent of h,τandn(but depends on αand exponentially on ¯C¯t), provided that the errors of the starting values also satisfy such a bound. Theorem 3.2 limits the use of the BDF methods of orders higher than 2 (and more severely for orders higher than 3) to applications with a large dampin g parameter α, such as cases described in [24, 39]. We remark, however, that in man y situations αis of magnitude 10−2or even smaller [10]. A very small damping parameter α affects not only the methods considered here. To our knowledge, t he error analysis of any numerical method proposed in the literature breaks down as α→0, as does the energy inequality. It is not surprising that a positive lower bound on αarises for the methods of ordersk/greaterorequalslant3, since they are not A-stable and a lower bound on αis required also for the simplified linear problem ( α+i)∂tu=∆u, which arises from (1.4) by freezing m in the termm×∂tmand diagonalizing this skew-symmetric linear operator (with eigenvalues ±i and 0) and by omitting the projection P(m) on the right-hand side of (1.4). The proof of Theorem 3.2 uses a variant of the Nevanlinna–Odeh mult iplier tech- nique [34], which is described in the Appendix for the convenience of th e reader. While for sufficiently large αwe have an optimal-order error bound in the case of a smooth solution, there is apparently no discrete energy inequality under weak regularity assumptions similar to Proposition 3.1 for the BDF methods of orders 3 to 5.12 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH As in Remark 3.1, the error bounds also allow us to bound the discrepa ncy from normality. 4.A continuous perturbation result In this section we present a perturbation result for the continuou s problem, be- cause we will later transfer the arguments of its proof to the discr etizations to prove stability and convergence of the numerical methods. Letm(t) be a solution of (1.4) for 0 /lessorequalslantt/lessorequalslant¯t, and letm⋆(t), also of unit length, solve the same equation up to a defect d(t) for 0/lessorequalslantt/lessorequalslant¯t: (4.1)α∂tm⋆+m⋆×∂tm⋆=P(m⋆)(∆m⋆+H)+d =P(m)(∆m⋆+H)+r, with r=−/parenleftbig P(m)−P(m⋆)/parenrightbig (∆m⋆+H)+d. Then,m⋆also solves the perturbed weak formulation α(∂tm⋆,ϕ)+(m⋆×∂tm⋆,ϕ)+(∇m⋆,∇ϕ) = (r,ϕ)∀ϕ∈T(m)∩H1(Ω)3, and the error e=m−m⋆satisfies the error equation (4.2)α(∂te,ϕ)+(e×∂tm⋆,ϕ)+(m×∂te,ϕ)+(∇e,∇ϕ) =−(r,ϕ) ∀ϕ∈T(m)∩H1(Ω)3. Before we turn to the perturbation result, we need Lipschitz-typ e bounds for the orthogonal projection P(m) =I−mmTapplied to sufficiently regular functions. Lemma 4.1. The projection P(·)satisfies the following estimates, for functions m,m⋆,v:Ω→R3, wheremandm⋆take values on the unit sphere and m⋆∈ W1,∞(Ω)3: /ba∇dbl(P(m)−P(m⋆))v/ba∇dblL2(Ω)3/lessorequalslant2/ba∇dblv/ba∇dblL∞(Ω)3/ba∇dblm−m⋆/ba∇dblL2(Ω)3,/vextenddouble/vextenddouble∇/parenleftbig (P(m)−P(m⋆))v/parenrightbig/vextenddouble/vextenddouble L2(Ω)3×3/lessorequalslant2/ba∇dblm⋆/ba∇dblW1,∞(Ω)3/ba∇dblv/ba∇dblW1,∞(Ω)3/ba∇dblm−m⋆/ba∇dblL2(Ω)3 +6/ba∇dblv/ba∇dblL∞(Ω)3/ba∇dbl∇(m−m⋆)/ba∇dblL2(Ω)3×3. Proof.Settinge=m−m⋆, we start by rewriting (P(m)−P(m⋆))v=−(mmT−m⋆mT ⋆)v=−(meT+emT ⋆)v. The first inequality then follows immediately by taking the L2norm of both sides of the above equality, using the fact that mandm⋆are of unit length. The second inequality is proved similarly, using the product rule ∂i(P(m)−P(m⋆))v=−∂i(eeT+m⋆eT+emT ⋆)v =−(∂ieeT+e∂ieT+∂im⋆eT+m⋆∂ieT+∂iemT ⋆+e∂imT ⋆)v +(meT+emT ⋆)∂iv, theL∞bound of∂im⋆, and the fact that /ba∇dble/ba∇dblL∞/lessorequalslant/ba∇dblm/ba∇dblL∞+/ba∇dblm⋆/ba∇dblL∞/lessorequalslant2./square We have the following perturbation result.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 13 Lemma 4.2. Letm(t)andm⋆(t)be solutions of unit length of (1.5)and(4.1), respectively, and suppose that, for 0/lessorequalslantt/lessorequalslant¯t, we have (4.3)/ba∇dblm⋆(t)/ba∇dblW1,∞(Ω)3+/ba∇dbl∂tm⋆(t)/ba∇dblW1,∞(Ω)3/lessorequalslantR and/ba∇dbl∆m⋆(t)+H(t)/ba∇dblL∞(Ω)3/lessorequalslantK. Then, the error e(t) =m(t)−m⋆(t)satisfies, for 0/lessorequalslantt/lessorequalslant¯t, (4.4) /ba∇dble(t)/ba∇dbl2 H1(Ω)3/lessorequalslantC/parenleftig /ba∇dble(0)/ba∇dbl2 H1(Ω)3+/integraldisplayt 0/ba∇dbld(s)/ba∇dbl2 L2(Ω)3ds/parenrightig , where the constant Cdepends only on α,R,K, and¯t. Proof.Let us first assume that ∂tm(t)∈H1(Ω)3for allt. Following [21], we test in the error equation (4.2) with ϕ=P(m)∂te∈T(m). By the following argument, this test function is then indeed in H1(Ω)3and can be viewed as a perturbation of∂te: ϕ=P(m)∂te=P(m)∂tm−P(m)∂tm⋆ =P(m)∂tm−P(m⋆)∂tm⋆−(P(m)−P(m⋆))∂tm⋆ =∂tm−∂tm⋆−(P(m)−P(m⋆))∂tm⋆, and so we have (4.5)ϕ=P(m)∂te=∂te+qwithq=−(P(m)−P(m⋆))∂tm⋆. By Lemma 4.1 and using (4.3) we have (4.6) /ba∇dblq/ba∇dblL2/lessorequalslant2R/ba∇dble/ba∇dblL2and/ba∇dbl∇q/ba∇dblL2/lessorequalslantCR/ba∇dble/ba∇dblH1. Testing the error equation (4.2) with ϕ=∂te+q, we obtain α(∂te,∂te+q)+(e×∂tm⋆,∂te+q)+(m×∂te,∂te+q) +(∇e,∇(∂te+q)) =−(r,∂te+q), where, by (4.1) and Lemma 4.1 with (4.3), ris bounded as (4.7)/ba∇dblr/ba∇dblL2/lessorequalslant/ba∇dbl/parenleftbig P(m)−P(m⋆)/parenrightbig (∆m⋆+H)/ba∇dblL2+/ba∇dbld/ba∇dblL2 /lessorequalslant2K/ba∇dble/ba∇dblL2+/ba∇dbld/ba∇dblL2. By collecting terms, and using the fact that ( m×∂te,∂te) vanishes, we altogether obtain α/ba∇dbl∂te/ba∇dbl2 L2+1 2d dt/ba∇dbl∇e/ba∇dbl2 L2=−α(∂te,q)−(e×∂tm⋆,∂te+q)−(m×∂te,q) −(∇e,∇q)−(r,∂te+q). For the right-hand side, the Cauchy–Schwarz inequality and /ba∇dblm/ba∇dblL∞= 1 yield α/ba∇dbl∂te/ba∇dbl2 L2+1 2d dt/ba∇dbl∇e/ba∇dbl2 L2/lessorequalslantα/ba∇dbl∂te/ba∇dblL2/ba∇dblq/ba∇dblL2+R/ba∇dble/ba∇dblL2(/ba∇dbl∂te/ba∇dblL2+/ba∇dblq/ba∇dblL2) +/ba∇dbl∂te/ba∇dblL2/ba∇dblq/ba∇dblL2+/ba∇dbl∇e/ba∇dblL2/ba∇dbl∇q/ba∇dblL2+/ba∇dblr/ba∇dblL2(/ba∇dbl∂te/ba∇dblL2+/ba∇dblq/ba∇dblL2).14 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH Young’s inequality and absorptions, together with the bounds in (4.6 ) and (4.7), yield α1 2/ba∇dbl∂te/ba∇dbl2 L2+1 2d dt/ba∇dbl∇e/ba∇dbl2 L2/lessorequalslantc/ba∇dble/ba∇dbl2 H1+c/ba∇dbld/ba∇dbl2 L2. Here, we note that 1 2d dt/ba∇dble/ba∇dbl2 L2= (∂te,e)/lessorequalslant1 2/ba∇dbl∂te/ba∇dbl2 L2+1 2/ba∇dble/ba∇dbl2 L2,so that /ba∇dbl∂te/ba∇dbl2 L2/greaterorequalslantd dt/ba∇dble/ba∇dbl2 L2−/ba∇dble/ba∇dbl2 L2. Combining these inequalities and integrating in time, we obtain /ba∇dble(t)/ba∇dbl2 H1/lessorequalslantc/ba∇dble(0)/ba∇dbl2 H1+c/integraldisplayt 0/ba∇dble(s)/ba∇dbl2 H1ds+c/integraldisplayt 0/ba∇dbld(s)/ba∇dbl2 L2ds. By Gronwall’s inequality, we then obtain the stated error bound. Finally, if∂tm(t) is not inH1(Ω)3for somet, then a regularization and density argument, which we do not present here, yields the result, since th e error bound does not depend on the H1norm of∂tm. /square 5.Orthogonal projection onto the discrete tangent space For consistency and stability of the full discretization, we need to s tudy properties of theL2(Ω)-orthogonal projection onto the discrete tangent space Th(m), which we denote by Ph(m):V3 h→Th(m). We do not have an explicit expression for this projection, but the pr operties stated in Lemmas 5.1 to 5.3 will be used for proving consistency and stability. W e recall that we consider a quasi-uniform, shape-regular family Thof triangulations with Lagrange finite elements of polynomial degree r. The first lemma states that the projection Ph(m) approximates the orthogonal projection P(m) =I−mmTonto the tangent space T(m) with optimal order. It will be used in the consistency error analysis of Section 6. Lemma 5.1. Form∈Wr+1,∞(Ω)3with|m|= 1almost everywhere we have /ba∇dbl(Ph(m)−P(m))v/ba∇dblL2(Ω)3/lessorequalslantChr+1/ba∇dblv/ba∇dblHr+1(Ω)3, /ba∇dbl(Ph(m)−P(m))v/ba∇dblH1(Ω)3/lessorequalslantChr/ba∇dblv/ba∇dblHr+1(Ω)3, for allv∈Hr+1(Ω)3, whereCdepends on a bound of /ba∇dblm/ba∇dblWr+1,∞(Ω)3. The second lemma states that the projection Ph(m) has Lipschitz bounds of the same type as those of the orthogonal projection P(m) given in Lemma 4.1. It will be used in the stability analysis of Sections 7 and 8. Lemma 5.2. Letm∈W1,∞(Ω)3and/tildewiderm∈H1(Ω)3with|m|=|/tildewiderm|= 1almost everywhere and /ba∇dblm/ba∇dblW1,∞/lessorequalslantR. There exist CR>0andhR>0such that for h/lessorequalslanthR, for allvh∈V3 h, (i)/ba∇dbl(Ph(m)−Ph(/tildewiderm))vh/ba∇dblL2(Ω)3/lessorequalslantCR/ba∇dblm−/tildewiderm/ba∇dblLp(Ω)3/ba∇dblvh/ba∇dblLq(Ω)3, for(p,q)∈ {(2,∞),(∞,2)}, and (ii)/ba∇dbl(Ph(m)−Ph(/tildewiderm))vh/ba∇dblH1(Ω)3/lessorequalslantCR/ba∇dblm−/tildewiderm/ba∇dblH1(Ω)3/ba∇dblvh/ba∇dblL∞(Ω)3HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 15 +CR/ba∇dblm−/tildewiderm/ba∇dblL2(Ω)3/ba∇dblvh/ba∇dblW1,∞(Ω)3. The next lemma shows the Ws,p-stability of the projection. It is actually used for p= 2 in the proof of Lemmas 5.1 and 5.2 and will be used for p= 2 in Section 6 and forp=∞in Sections 7 and 8. Lemma 5.3. There exists a constant depending only on p∈[1,∞]and the shape regularity of the mesh such that for all m∈W1,∞(Ω)3with|m|= 1almost every- where, /ba∇dblPh(m)vh/ba∇dblWs,p(Ω)3/lessorequalslantC/ba∇dblm/ba∇dbl2 W1,∞(Ω)3/ba∇dblvh/ba∇dblWs,p(Ω)3 for allvh∈V3 hands∈ {−1,0,1}. These three lemmas will be proved in the course of this section, in whic h we formulate also three more lemmas that are of independent interest but will not be used in the following sections. In the following, we use the dual norms /ba∇dblv/ba∇dblW−1,q:= sup w∈W1,p(v,w) /ba∇dblw/ba∇dblW1,pfor 1/p+1/q= 1. The space W−1,1(Ω) is not the dual space of W1,∞(Ω) but rather defined as the closure ofL2(Ω) withrespect to thenorm /ba∇dbl·/ba∇dblW−1,1. Wealso recall that Πh:Ws,p(Ω) →Ws,p(Ω) is uniformly bounded for s∈ {0,1}andp∈[1,∞] (see, e.g., [20] for proofs in a much more general setting). By duality, we also obta in uniform boundedness for s=−1 andp∈[1,∞]. A useful consequence is that for vh∈Vh, /ba∇dblvh/ba∇dblW−1,q= sup w∈W1,p(vh,Πhw) /ba∇dblw/ba∇dblW1,p /lessorequalslantsup w∈W1,p(vh,Πhw) /ba∇dblΠhw/ba∇dblW1,psup w∈W1,p/ba∇dblΠhw/ba∇dblW1,p /ba∇dblw/ba∇dblW1,p/lessorsimilarsup wh∈Vh(vh,wh) /ba∇dblwh/ba∇dblW1,p. Lemma 5.4. There holds /ba∇dblv/ba∇dblWs,p(Ω)≃supw∈W−s,q(Ω)(v,w) /bardblw/bardblW−s,q(Ω)with1/p+1/q= 1 forp∈[1,∞]ands∈ {−1,0,1}. Proof.The interesting case is ( s,p) = (1,∞) since all other cases follow by duality. Forv∈W1,∞(Ω), thereexists asequence offunctions qn∈C∞ 0(Ω)3with/ba∇dblqn/ba∇dblL1= 1 such that /ba∇dbl∇v/ba∇dblL∞= lim n→∞(∇v,qn) = lim n→∞−(v,divqn)/lessorequalslantsup q∈W1,1(v,divq) /ba∇dblq/ba∇dblL1. Moreover, there holds /ba∇dbldivq/ba∇dblW−1,1/lessorequalslantsup w∈W1,∞(q,∇w) /ba∇dbl∇w/ba∇dblL∞/lessorequalslant/ba∇dblq/ba∇dblL1. Combining the last two estimates shows /ba∇dbl∇v/ba∇dblL∞/lessorequalslantsup w∈W−1,1(v,w) /ba∇dblw/ba∇dblW−1,1.16 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH Since /ba∇dblv/ba∇dblL∞= sup w∈L1(v,w) /ba∇dblw/ba∇dblL1/lessorequalslantsup w∈W−1,1(v,w) /ba∇dblw/ba∇dblW−1,1, we conclude the proof. /square Letthediscretenormalspace Nh(m) :=V3 h⊖Th(m)begivenasthe L2-orthogonal complement of Th(m) inV3 h. We note that (5.1) Nh(m) ={Πh(mψh) :ψh∈Vh} by the definition of Th(m). The functions in the discrete normal space are bounded from below as follows. Lemma 5.5. For everyR >0, there exist hR>0andc >0such that for all m∈W1,∞(Ω)3with|m|= 1almost everywhere and /ba∇dblm/ba∇dblW1,∞(Ω)/lessorequalslantRand for all h/lessorequalslanthR, /ba∇dblΠh(mψh)/ba∇dblWs,p(Ω)3/greaterorequalslantc/ba∇dblψh/ba∇dblWs,p(Ω) for allψh∈Vhand(s,p)∈ {−1,0,1}×[1,∞]. Proof.(a) We first prove the result for s∈ {−1,0}. LetIh:C(Ω)→V3 hdenote the nodal interpolation operator and define mh:=Ihm∈V3 h. There holds /ba∇dblΠh(mhψh)/ba∇dblLp/greaterorequalslant/ba∇dblmhψh/ba∇dblLp−/ba∇dbl(I−Πh)(mhψh)/ba∇dblLp. Moreover, stability of ΠhinLp(Ω)3, for 1/lessorequalslantp/lessorequalslant∞, see [20], implies the estimate /ba∇dbl(I−Πh)(mhψh)/ba∇dblLp/lessorequalslant(1+C) inf vh∈V3 h/ba∇dblmhψh−vh/ba∇dblLp. In turn, this implies /ba∇dbl(I−Πh)(mhψh)/ba∇dblLp/lessorsimilar/ba∇dbl(I−Ih)(mhψh)/ba∇dblLp =/parenleftig/summationdisplay T∈Th/ba∇dbl(I−Ih)(mhψh)/ba∇dblp Lp(T)3/parenrightig1/p . For each element, the approximation properties of Ihshow /ba∇dbl(I−Ih)(mhψh)/ba∇dblLp(T)3/lessorsimilarhr+1/ba∇dbl∇r+1(mhψh)/ba∇dblLp(T)3 /lessorequalslanthr+1/summationdisplay i+j=r+1/ba∇dbl∇min{i,r}mh/ba∇dblL∞(T)3/ba∇dbl∇min{j,r}ψh/ba∇dblLp(T)3. Thus, multiple inverse estimates yield /ba∇dbl(I−Ih)(mhψh)/ba∇dblLp(T)3/lessorsimilarh/ba∇dblmh/ba∇dblW1,∞/ba∇dblψh/ba∇dblLp(T)3. Moreover, we have /ba∇dblmhψh/ba∇dblLp/greaterorequalslant/ba∇dblmψh/ba∇dblLp−/ba∇dbl(m−mh)ψh/ba∇dblLp/greaterorequalslant1 2/ba∇dblψh/ba∇dblLp provided that /ba∇dblm−mh/ba∇dblL∞/lessorequalslant1 2, which in view of /ba∇dblm−mh/ba∇dblL∞=/ba∇dbl(I−Ih)m/ba∇dblL∞/lessorsimilarh/ba∇dbl∇m/ba∇dblL∞ is satisfied for h/lessorequalslanthRwith a sufficiently small hR>0 that depends only on R. Altogether, this shows /ba∇dblΠh(mhψh)/ba∇dblLp/greaterorsimilar/ba∇dblψh/ba∇dblLpHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 17 forh/lessorequalslanthR. Similarly we estimate /ba∇dblΠh((m−mh)ψh)/ba∇dblLp/lessorsimilar/ba∇dblm−mh/ba∇dblL∞/ba∇dblψh/ba∇dblLp/lessorsimilarh/ba∇dbl∇m/ba∇dblL∞/ba∇dblψh/ba∇dblLp. Altogether, we obtain /ba∇dblΠh(mψh)/ba∇dblLp/greaterorsimilar/ba∇dblΠh(mhψh)/ba∇dblLp−/ba∇dblΠh((mh−m)ψh)/ba∇dblLp/greaterorsimilar/ba∇dblψh/ba∇dblLp forh/lessorequalslanthR. This concludes the proof for s= 0. Finally, for s=−1 we note that by using the result for s= 0 and an inverse inequality, /ba∇dbl(I−Πh)(mψh)/ba∇dblW−1,p/lessorsimilarh/ba∇dblψh/ba∇dblLp /lessorsimilarh/ba∇dblΠh(mψh)/ba∇dblLp/lessorsimilar/ba∇dblΠh(mψh)/ba∇dblW−1,p. Since/ba∇dblmψh/ba∇dblW−1,p/greaterorsimilar/ba∇dblm/ba∇dbl−1 W1,∞/ba∇dblψh/ba∇dblW−1,p, this concludes the proof for s∈ {−1,0}. (b) It remains to prove the result for s= 1. Note that the result follows from duality if we show (5.2) /ba∇dblΠh(m·wh)/ba∇dblW−1,q/greaterorsimilar/ba∇dblwh/ba∇dblW−1,q for allwh∈Nh(m). To see this, note that (5.2) implies /ba∇dblΠh(mψh)/ba∇dblW1,p/greaterorequalslantsup wh∈Nh(m)(ψh,Πh(m·wh)) /ba∇dblwh/ba∇dblW−1,q /greaterorsimilarsup wh∈Nh(m)(ψh,Πh(m·wh)) /ba∇dblΠh(m·wh)/ba∇dblW−1,q= sup ωh∈Vh(ψh,ωh) /ba∇dblωh/ba∇dblW−1,q≃ /ba∇dblψh/ba∇dblW1,p, whereweusedinthesecondtolastequalitythatpart(a)for s= 0alreadyshowsthat dim(Nh(m)) = dim(Vh) and since (5.2) implies that the map Nh(m)→Vh,wh/ma√sto→ Πh(m·wh) is injective, it is already bijective. It remains to prove (5.2). To tha t end, we first show for wh=Πh(mωh)∈Nh(m) for someωh∈Vh, using the reverse triangle inequality, that /ba∇dblm·wh/ba∇dblW−1,q/greaterorequalslant/ba∇dblωh/ba∇dblW−1,q−/ba∇dblm·(I−Πh)(mωh)/ba∇dblW−1,q /greaterorsimilar/ba∇dblm/ba∇dbl−1 W1,∞/ba∇dblwh/ba∇dblW−1,q−/ba∇dblm·(I−Πh)(mωh)/ba∇dblW−1,q. Withmh:=Ih(m)∈V3 h, the last term satisfies /ba∇dblm·(I−Πh)(mωh)/ba∇dblW−1,q/lessorsimilarh/ba∇dblm/ba∇dblW1,∞/ba∇dbl(I−Πh)(mωh)/ba∇dblLq /lessorsimilarh/ba∇dblm/ba∇dblW1,∞(/ba∇dblm−mh/ba∇dblL∞/ba∇dblωh/ba∇dblLq+h/ba∇dblmh/ba∇dblW1,∞/ba∇dblωh/ba∇dblLq), where we used the same arguments as in the proof of part (a) to ge t the estimate /ba∇dbl(I−Πh)(mhωh)/ba∇dblLq/lessorsimilarh/ba∇dblmh/ba∇dblW1,∞/ba∇dblωh/ba∇dblLq. The fact /ba∇dblmh/ba∇dblW1,∞/lessorsimilar/ba∇dblm/ba∇dblW1,∞, the approximation property /ba∇dblm−mh/ba∇dblL∞/lessorsimilarh/ba∇dblm/ba∇dblW1,∞, and an inverse inequality con- clude (5.3) /ba∇dblm·wh/ba∇dblW−1,q/greaterorsimilar/ba∇dblwh/ba∇dblW−1,q with (hidden) constants depending only on /ba∇dblm/ba∇dblW1,∞and shape regularity of the mesh.18 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH Toprove(5.2), itremainstoboundtheleft-handsideaboveby /ba∇dblΠh(m·wh)/ba∇dblW−1,q. To that end, we note /ba∇dbl(I−Πh)(m·wh)/ba∇dblW−1,q/lessorsimilarh/ba∇dblwh/ba∇dblLq=hsup v∈Lp(wh,v) /ba∇dblv/ba∇dblLp /lessorsimilarhsup v∈Nh(m)(wh,v) /ba∇dblv/ba∇dblLp=hsup v∈Vh(Πh(m·wh),v) /ba∇dblΠh(mv)/ba∇dblLp/lessorsimilarh/ba∇dblΠh(m·wh)/ba∇dblLq, where we used part (a) for s= 0 for the last inequality. An inverse inequality and the combination with (5.3) imply (5.2) for h >0 sufficiently small in terms of /ba∇dblm/ba∇dbl−1 W1,∞. This concludes the proof. /square Lemma 5.6. Define the matrix M∈RN×N, whereNdenotes the dimension of Vh, byMij:=h−3(Πh(mφj),Πh(mφi)). Under the assumptions of Lemma 5.5, there existsC >0such that for h/lessorequalslanthR, /ba∇dblM/ba∇dblp+/ba∇dblM−1/ba∇dblp/lessorequalslantCfor1/lessorequalslantp/lessorequalslant∞, whereCdepends only on the shape regularity. Proof.Lemma 5.5 shows for x∈RN (5.4) Mx·x=h−3/ba∇dblΠh(mN/summationdisplay i=1xiφi)/ba∇dbl2 L2/greaterorsimilarh−3/ba∇dblN/summationdisplay i=1xiφi/ba∇dblL2≃ |x|2, where|·|denotes the Euclidean norm on RN. Letd(i,j) := dist(zi,zj)h−3denote the metric which (approximately) measures the number of elements between the supports of φiandφj, corresponding to the nodes ziandzj, and letBd(z) denote the corresponding ball. In the following, we use a localization propert y of theL2- projection, i.e., there exist a,b>0 such that for all ℓ∈N, (5.5) /ba∇dblΠh(mφi)/ba∇dblL2(Ω\Bℓ(zi))3/lessorequalslantae−bℓ/ba∇dblmφi/ba∇dblL2. The proof of this bound is essentially contained in the proof of [9, Le mma 3.1]. Since we use the very same arguments below, we briefly recall the st rategy: First, one observes that the mass matrix /tildewiderM∈RN×Nwith entries /tildewiderMij:=h−3(φj,φi) is bandedinthesense that d(i,j)/greaterorsimilar1implies /tildewiderMij= 0, anditsatisfies /tildewiderMx·x/greaterorsimilar|x|2. As shown below, this implies that the inverse matrix /tildewiderM−1satisfies|(/tildewiderM−1)ij|/lessorsimilare−bd(i,j) for someb >0 independent of h >0. Note that each entry of the vector field Πh(mφi)∈V3 hcan be represented by/summationtextN j=1xk,jφj,k= 1,2,3,and is computed by solving/tildewiderMxk=gk∈RNwithm= (m1,m2,m3)Tandgk,j:= (mkφi,φj). Hence, the exponential decay of /tildewiderM−1directly implies (5.5). From the decay property (5.5), we immediately obtain |Mij|/lessorequalslant/tildewideae−/tildewidebd(i,j) for all 1/lessorequalslanti,j/lessorequalslantNand some /tildewidea,/tildewideb>0. This already proves /ba∇dblM/ba∇dblp/lessorequalslantC. We follow the arguments from [28] to show that also M−1decays exponentially. To that end,HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 19 note that (5.4) implies the existence of c >0 such that /ba∇dblI−cM/ba∇dbl2=:q <1 and hence (5.6) M−1=c(I−(I−cM))−1=c∞/summationdisplay k=0(I−cM)k. Clearly,I−cMinherits the decay properties from Mand therefore |((I−cM)k+1)ij|/lessorequalslant/tildewideak+1N/summationdisplay r1,...,rk=1e−/tildewideb(d(i,r1)+···+d(rk,j)) /lessorequalslant/tildewideak+1/parenleftig max s=1,...,NN/summationdisplay r=1e−/tildewidebd(s,r)/2/parenrightigk e−/tildewidebd(i,j)/2. The value of max s=1,...,N/summationtextN r=1e−/tildewidebd(s,r)/2depends only on the shape regularity of the triangulation and on /tildewideb, but is independent of h(it just depends on the number of elements contained in an annulus of thickness ≈h). This implies the existence of /tildewidec/greaterorequalslant1 such that |((I−cM)k+1)ij|/lessorequalslantmin{qk+1,/tildewideck+1e−/tildewidebd(i,j)/2}. Thus, for /tildewideck+1/lessorequalslante/tildewidebd(i,j)/4, we have |((I−cM)k+1)ij|/lessorequalslante−/tildewidebd(i,j)/4, whereas for /tildewideck+1> e/tildewidebd(i,j)/4, we have |((I−cM)k+1)ij|/lessorequalslantqk+1<q/tildewidebd(i,j)/(4log(/tildewidec)). Altogether, we find some /tildewideb>0 (we reuse the symbol), independent of hsuch that |((I−cM)k+1)ij|/lessorequalslantq(k+1)/2|((I−cM)k+1)ij|1/2/lessorsimilarq(k+1)/2e−/tildewidebd(i,j). Plugging this into (5.6), we obtain |(M−1)ij|/lessorsimilar∞/summationdisplay k=0q(k+1)/2e−/tildewidebd(i,j)/lessorsimilare−/tildewidebd(i,j). This yields the stated result. /square We are now in a position to prove Lemma 5.3. Proof of Lemma 5.3. (a) We first consider the case s= 0. In view of (5.1), we write (I−Ph(m))vh∈Nh(m) as (I−Ph(m))vh=h−3/2N/summationdisplay i=1xiΠh(mφi) for some coefficient vector x∈RNand letbi:=h−3/2(vh,mφi) fori= 1,...,N. Then, there holds Mx=bwith the matrix Mfrom Lemma 5.6. This lemma and theLp-stability of the L2-orthogonal projection Π h[20] imply that for p∈[1,∞], /ba∇dbl(I−Ph(m))vh/ba∇dblLp=/ba∇dblΠhh−3/2N/summationdisplay i=1ximφi/ba∇dblLp/lessorsimilar/ba∇dblh−3/2N/summationdisplay i=1ximφi/ba∇dblLp /lessorsimilarh−3/2/parenleftigN/summationdisplay i=1h3|xi|p/parenrightig1/p =h3/p−3/2|x|p=h3/p−3/2|M−1b|p/lessorsimilarh3/p−3/2|b|p.20 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH With|bi|/lessorequalslanth−3/2/ba∇dblvh/ba∇dblLp(supp(φi))3h3(1−1/p)=/ba∇dblvh/ba∇dblLp(supp(φi))3h3/2−3/p, this shows /ba∇dblPh(m)vh/ba∇dblLp/lessorsimilar/ba∇dblvh/ba∇dblLp. (b) We now turn to the cases s=±1. Define the operator /tildewideP⊥ h(m)vh:=Πh(mΠh(m·vh)) andnotethat /tildewideP⊥ h(m)vh∈Nh(m)aswellasker /tildewideP⊥ h(m) =Th(m)(duetoLemma5.5). However, /tildewideP⊥ h(m) is no projection. We observe for vh=Πh(mψh)∈Nh(m) that /ba∇dbl(I−/tildewideP⊥ h(m))vh/ba∇dblW−1,p=/ba∇dblΠhmψh−Πh(mΠh(m·Πh(mψh)))/ba∇dblW−1,p /lessorsimilar/ba∇dblm/ba∇dblW1,∞/ba∇dblψh−m·Πh(mψh)/ba∇dblW−1,p =/ba∇dblm/ba∇dbl2 W1,∞/ba∇dbl(I−Πh)(mψh)/ba∇dblW−1,p /lessorsimilar/ba∇dblm/ba∇dbl2 W1,∞h/ba∇dblψh/ba∇dblLp. With Lemma 5.5 we conclude /ba∇dbl(I−/tildewideP⊥ h(m))vh/ba∇dblW−1,p/lessorsimilar/ba∇dblm/ba∇dbl2 W1,∞h/ba∇dblvh/ba∇dblLp. Since/tildewideP⊥ h(m)Ph(m) = 0 by definition of Th(m), we obtain with part (a) and an inverse inequality that for all vh∈V3 h, /ba∇dbl(I−Ph(m)−/tildewideP⊥ h(m))vh/ba∇dblW−1,p=/ba∇dbl(I−/tildewideP⊥ h(m))(I−Ph(m))vh/ba∇dblW−1,p /lessorsimilar/ba∇dblm/ba∇dbl2 W1,∞h/ba∇dbl(I−Ph(m))vh/ba∇dblLp /lessorsimilar/ba∇dblm/ba∇dbl2 W1,∞h/ba∇dblvh/ba∇dblLp /lessorsimilar/ba∇dblm/ba∇dbl2 W1,∞/ba∇dblvh/ba∇dblW−1,p. TheW−1,p(Ω)-stability of Πhimplies/ba∇dbl/tildewideP⊥ h(m)vh/ba∇dblW−1,p/lessorsimilar/ba∇dblm/ba∇dbl2 W1,∞/ba∇dblvh/ba∇dblW−1,pand the triangle inequality concludes the proof for s=−1. The case s= 1 follows by duality. /square Proof of Lemma 5.2. (a) (s= 0) The projection vh:=Ph(m)vis given by the equation (vh,ϕh) = (v,ϕh)∀ϕh∈Th(m), which in view of the definition of Th(m) is equivalent to the solution of the saddle point problem (with the Lagrange multiplier λh∈Vh) (vh,wh)+(m·wh,λh) = (v,wh)∀wh∈V3 h, (m·vh,µh) = 0 ∀µh∈Vh. By the first equation, we also obtain the identity Πh(mλh) = (I−Ph(m))vh, which will be used below. Furthermore, /tildewidevh:=Ph(/tildewiderm)vis given by the same system with /tildewidermin place ofm, yielding a corresponding Lagrange multiplier /tildewideλh. Hence, the differenceseh:=vh−/tildewidevhandδh:=λh−/tildewideλhsatisfy (eh,wh)+(m·wh,δh) =−(wh,(m−/tildewiderm)/tildewideλh)∀wh∈V3 h, (m·eh,µh) = −((m−/tildewiderm)·/tildewidevh,µh)∀µh∈Vh.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 21 The classical results on saddle-point problems (see [13, Proposition 2.1]) require two inf-sup conditions to be satisfied. First, inf qh∈Vhsup vh∈V3 h(m·vh,qh) /ba∇dblvh/ba∇dblHs/ba∇dblqh/ba∇dblH−s>0 holds uniformly in hdue to Lemma 5.5. Second, inf wh∈Th(m)sup vh∈Th(m)(vh,wh) /ba∇dblvh/ba∇dblHs/ba∇dblwh/ba∇dblH−s>0 holdsuniformlyin hduetothestabilityestimatesfromLemma5.3(notingthat vh= Ph(m)vhandwh=Ph(m)whforvh,wh∈Th(m)). For the above saddle-point problems, these bounds for s= 0 give us an L2bound foreh=Ph(m)v−Ph(/tildewiderm)v: From [13] we obtain /ba∇dbl/tildewidevh/ba∇dblL2+/ba∇dbl/tildewideλh/ba∇dblL2/lessorsimilar/ba∇dblv/ba∇dblL2 and /ba∇dbleh/ba∇dblL2+/ba∇dblδh/ba∇dblL2/lessorsimilar/ba∇dbl(m−/tildewiderm)/tildewideλh/ba∇dblL2+/ba∇dbl(m−/tildewiderm)·/tildewidevh/ba∇dblL2. With the stability from Lemma 5.3 and Lemma 5.5, we also obtain /ba∇dbl/tildewidevh/ba∇dblL∞+/ba∇dbl/tildewideλh/ba∇dblL∞/lessorsimilar/ba∇dblPh(/tildewiderm)v/ba∇dblL∞+/ba∇dbl(I−Ph(/tildewiderm))v/ba∇dblL∞/lessorsimilar/ba∇dblv/ba∇dblL∞. Altogether, this implies /ba∇dbleh/ba∇dblL2+/ba∇dblδh/ba∇dblL2/lessorsimilar/ba∇dblm−/tildewiderm/ba∇dblLp/ba∇dblv/ba∇dblLq for (p,q)∈ {(2,∞),(∞,2)}. (b) (s= 1) For the H1(Ω)-estimate, we introduce the Riesz mapping Jhbetween Vh⊂H1(Ω) and its dual Vh⊂H1(Ω)′, i.e., the isometry defined by (vh,Jhψh)H1=/a\}b∇acketle{tvh,ψh/a\}b∇acket∇i}ht ∀vh∈Vh, ψh∈Vh. ByJh:=I⊗Jhwe denote the corresponding vector-valued mapping on V3 h. We consider the bilinear form on V3 h×V3 hdefined by ah(vh,wh) =/a\}b∇acketle{tvh,J−1 hwh/a\}b∇acket∇i}ht,vh,wh∈V3 h, and reformulate the saddle-point problem for ( vh,λh)∈V3 h×Vh⊂H1(Ω)3×H1(Ω)′ as ah(vh,wh)+/a\}b∇acketle{tm·J−1 hwh,λh/a\}b∇acket∇i}ht=a(v,wh)∀wh∈V3 h, /a\}b∇acketle{tm·vh,J−1 hµh/a\}b∇acket∇i}ht = 0 ∀µh∈Vh. As in the case s= 0 (algebraically it is the same system), we have vh=Ph(m)v andΠh(mλh) = (I−Ph(m))v. The system for eh=vh−/tildewidevhandδh=λh−/tildewideλh reads ah(eh,wh)+/a\}b∇acketle{tm·J−1 hwh,δh/a\}b∇acket∇i}ht=−/a\}b∇acketle{t(m−/tildewiderm)·J−1 hwh,/tildewideλh/a\}b∇acket∇i}ht ∀wh∈V3 h, /a\}b∇acketle{tm·eh,J−1 hµh/a\}b∇acket∇i}ht =−/a\}b∇acketle{t(m−/tildewiderm)·/tildewidevh,J−1 hµh/a\}b∇acket∇i}ht ∀µh∈Vh. The above inf-sup bounds for s= 1 ands=−1 are precisely the inf-sup condi- tions that need to be satisfied for these generalized saddle-point p roblems (see [15, Theorem 2.1]), whose right-hand sides are bounded by |ah(v,wh)|/lessorequalslant/ba∇dblv/ba∇dblH1/ba∇dblJ−1 hwh/ba∇dblH−1≃ /ba∇dblv/ba∇dblH1/ba∇dblwh/ba∇dblH122 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH and |/a\}b∇acketle{t(m−/tildewiderm)·J−1 hwh,/tildewideλh/a\}b∇acket∇i}ht|/lessorsimilar/ba∇dbl(m−/tildewiderm)/tildewideλh/ba∇dblH1/ba∇dblwh/ba∇dblH1, |/a\}b∇acketle{t(m−/tildewiderm)·/tildewidevh,J−1 hµh/a\}b∇acket∇i}ht|/lessorequalslant/ba∇dbl(m−/tildewiderm)·/tildewidevh/ba∇dblH1/ba∇dblµh/ba∇dblH1. As in the case s= 0, we obtain from Lemma 5.3 and Lemma 5.5 that /ba∇dbl/tildewidevh/ba∇dblW1,∞+/ba∇dbl/tildewideλh/ba∇dblW1,∞/lessorsimilar/ba∇dblPh(/tildewiderm)v/ba∇dblW1,∞+/ba∇dbl(I−Ph(/tildewiderm))v/ba∇dblW1,∞ /lessorsimilar/ba∇dblv/ba∇dblW1,∞. Hence, we obtain from [15, Theorem 2.1], for ( p,q)∈ {(2,∞),(∞,2)}, /ba∇dbleh/ba∇dblH1/lessorsimilar/ba∇dbl(m−/tildewiderm)/tildewideλh/ba∇dblH1+/ba∇dbl(m−/tildewiderm)·/tildewidevh/ba∇dblH1 /lessorsimilar1/summationdisplay s′=0/parenleftig /ba∇dblm−/tildewiderm/ba∇dblH1/ba∇dbl/tildewideλh/ba∇dblW1−s′,q+/ba∇dblm−/tildewiderm/ba∇dblWs′,p/ba∇dbl/tildewidevh/ba∇dblW1−s′,q/parenrightig /lessorsimilar1/summationdisplay s′=0/ba∇dblm−/tildewiderm/ba∇dblWs′,p/ba∇dblv/ba∇dblW1−s′,q. This implies the H1(Ω)3estimate and hence concludes the proof. /square Proof of Lemma 5.1. SincePh(m)vis the Galerkin approximation of the saddle point problem for P(m)v(as in the previous proof), the C´ ea lemma for saddle- point problems (see [13, Theorem 2.1]) shows in L2 /ba∇dbl(Ph(m)−P(m))v/ba∇dblL2 /lessorsimilarinf (wh,µh)∈V3 h×Vh/parenleftig /ba∇dblP(m)v−wh/ba∇dblL2+/ba∇dblm·v−µh/ba∇dblL2/parenrightig /lessorsimilarhr+1/ba∇dblm/ba∇dblWr+1,∞/ba∇dblv/ba∇dblHr+1 and similarly in H1, using [15, Theorem 2.1], /ba∇dbl(Ph(m)−P(m))v/ba∇dblH1 /lessorsimilarinf (wh,µh)∈V3 h×Vh/parenleftig /ba∇dblP(m)v−wh/ba∇dblH1+/ba∇dblm·v−µh/ba∇dblH1/parenrightig /lessorsimilarhr/ba∇dblm/ba∇dblWr+1,∞/ba∇dblv/ba∇dblHr+1. This concludes the proof. /square 6.Consistency error and error equation To study the consistency errors, we find it instructive to separat e the issues of consistency for the time and space discretizations. Therefore, w e first show defect estimates for the semidiscretization in time, and then turn to the fu ll discretization.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 23 6.1.Consistency error of the semi-discretization in time. The order of both the fully implicit k-step BDF method, described by the coefficients δ0,...,δ kand 1, andtheexplicit k-step BDFmethod, thatis themethoddescribed by thecoefficients δ0,...,δ kandγ0,...,γ k−1,isk,i.e., (6.1)k/summationdisplay i=0(k−i)ℓδi=ℓkℓ−1=ℓk−1/summationdisplay i=0(k−i−1)ℓ−1γi, ℓ= 0,1,...,k. We first rewrite the linearly implicit k-step BDF method (2.3) in strong form, (6.2) α˙mn+/hatwidermn×˙mn=P(/hatwidermn)(∆mn+Hn), with Neumann boundary conditions. The consistency error dnof the linearly implicit k-step BDF method (6.2) for the solutionmis the defect by which the exact solution misses satisfying (6.2), and is given by (6.3) dn=α˙mn ⋆+/hatwidermn ⋆×˙mn ⋆−P(/hatwidermn ⋆)(∆mn ⋆+Hn) forn=k,...,N, where we use the notation mn ⋆=m(tn) and (6.4)/hatwidermn ⋆=k−1/summationdisplay j=0γjmn−j−1 ⋆/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay j=0γjmn−j−1 ⋆/vextendsingle/vextendsingle/vextendsingle, ˙mn ⋆=P(/hatwidermn ⋆)1 τk/summationdisplay j=0δjmn−j ⋆∈T(/hatwidermn ⋆). Notethat thedefinition of ˙mn ⋆contains theprojection P(/hatwidermn ⋆), while ˙mnwasdefined without a projection (see the first formula in (2.2)), since ˙mn=P(/hatwidermn)˙mnis automatically satisfied due to the constraint in (2.3). The consistency error is bounded as follows. Lemma 6.1. If the solution of the LLG equation (1.4)has the regularity m∈Ck+1([0,¯t],L2(Ω)3)∩C1([0,¯t],L∞(Ω)3)and∆m+H∈C([0,¯t],L∞(Ω)3), then the consistency error (6.3)is bounded by /ba∇dbldn/ba∇dblL2(Ω)3/lessorequalslantCτk forn=k,...,N. Proof.We begin by rewriting the equation for the defect as (6.5)dn=α˙mn ⋆+/hatwidermn ⋆×˙mn ⋆−P(mn ⋆)(∆mn ⋆+Hn) −/parenleftbig P(/hatwidermn ⋆)−P(mn ⋆)/parenrightbig (∆mn ⋆+Hn). In view of (1.4), we have P(mn ⋆)(∆mn ⋆+Hn) =α∂tm(tn)+mn ⋆×∂tm(tn), and can rewrite (6.5) as dn=α/parenleftbig ˙mn ⋆−∂tm(tn)/parenrightbig +/parenleftbig /hatwidermn ⋆×˙mn ⋆−mn ⋆×∂tm(tn)/parenrightbig −/parenleftbig P(/hatwidermn ⋆)−P(mn ⋆)/parenrightbig (∆mn ⋆+Hn),24 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH i.e., dn=α/parenleftbig˙mn ⋆−∂tm(tn)/parenrightbig +(/hatwidermn ⋆−mn ⋆)×˙mn ⋆+mn ⋆×/parenleftbig˙mn ⋆−∂tm(tn)/parenrightbig −/parenleftbig P(/hatwidermn ⋆)−P(mn ⋆)/parenrightbig (∆mn ⋆+Hn). Therefore, (6.6)dn=α˙dn+/hatwidedn×˙mn ⋆+mn ⋆×˙dn−/parenleftbig P(/hatwidermn ⋆)−P(mn ⋆)/parenrightbig (∆mn ⋆+Hn), with (6.7) ˙dn:=˙mn ⋆−∂tm(tn),/hatwidedn:=/hatwidermn ⋆−mn ⋆. Now, in view of the first estimate in Lemma 4.1, we have /ba∇dbl/parenleftbig P(/hatwidermn ⋆)−P(mn ⋆)/parenrightbig (∆mn ⋆+Hn)/ba∇dblL2/lessorequalslantC/ba∇dbl/hatwidermn ⋆−mn ⋆/ba∇dblL2, i.e., (6.8) /ba∇dbl/parenleftbig P(/hatwidermn ⋆)−P(mn ⋆)/parenrightbig (∆mn ⋆+Hn)/ba∇dblL2/lessorequalslantC/ba∇dbl/hatwidedn/ba∇dblL2. Therefore, it suffices to estimate ˙dnand/hatwidedn. To estimate /hatwidedn, we shall proceed in two steps. First we shall estimate the extrap- olation error (6.9)k−1/summationdisplay j=0γjmn−j−1 ⋆−mn ⋆ and then /hatwidedn. By Taylor expanding about tn−k,the leading terms of order up to k−1 cancel, due to the second equality in (6.1), and we obtain (6.10)k−1/summationdisplay i=0γimn−i−1 ⋆−mn ⋆=1 (k−1)!/bracketleftiggk−1/summationdisplay j=0γj/integraldisplaytn−j−1 tn−k(tn−j−1−s)k−1m(k)(s)ds −/integraldisplaytn tn−k(tn−s)k−1m(k)(s)ds/bracketrightigg , withm(ℓ):=∂ℓm ∂tℓ,whence (6.11)/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay i=0γimn−i−1 ⋆−mn ⋆/vextenddouble/vextenddouble/vextenddouble L2/lessorequalslantCτk. Now, for a normalized vector aand a non-zero vector b,we have a−b |b|= (a−b)+1 |b|(|b|−|a|)b, whence/vextendsingle/vextendsinglea−b |b|/vextendsingle/vextendsingle/lessorequalslant2|a−b|. Therefore, (6.11) yields (6.12) /ba∇dbl/hatwidedn/ba∇dblL2/lessorequalslantCτk.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 25 To bound ˙dn,we use the fact that P(m(tn))∂tm(tn) =∂tm(tn)∈T(m(tn)), so that we have ˙dn=P(/hatwidermn ⋆)1 τk/summationdisplay j=0δjm(tn−j)−∂tm(tn) =P(/hatwidermn ⋆)/parenleftig1 τk/summationdisplay j=0δjm(tn−j)−∂tm(tn)/parenrightig +/parenleftbig P(/hatwidermn ⋆)−P(m(tn))/parenrightbig ∂tm(tn). By Lemma 4.1 and (6.12), we have for the last term /ba∇dbl/parenleftbig P(/hatwidermn ⋆)−P(m(tn))/parenrightbig ∂tm(tn)/ba∇dblL2/lessorequalslantCτk. By Taylor expanding the first term about tn−k,we see that, due to the order condi- tions of the implicit BDF method, i.e., the first equality in (6.1), the leadin g terms of order up to k−1 cancel, and we obtain (6.13)1 τk/summationdisplay j=0δjm(tn−j)−∂tm(tn) =1 k!/bracketleftigg 1 τk/summationdisplay j=0δj/integraldisplaytn−j tn−k(tn−j−s)km(k+1)(s)ds −k/integraldisplaytn tn−k(tn−s)k−1m(k+1)(s)ds/bracketrightigg , whence (6.14) /ba∇dbl˙dn/ba∇dblL2/lessorequalslantCτk, provided the solution mis sufficiently regular. Now, (6.6), (6.8), (6.14), and (6.12) yield (6.15) /ba∇dbldn/ba∇dblL2/lessorequalslantCτk. This isthe desired consistency estimate, which isvalidfor BDFmethod s of arbitrary orderk. /square 6.2.Consistency error of the full discretization. Wedefine theRitzprojection Rh:H1(Ω)→Vhcorresponding to the Poisson–Neumann problem via/parenleftbig ∇Rhϕ,∇ψ/parenrightbig +/parenleftbig Rhϕ,1/parenrightbig/parenleftbig ψ,1/parenrightbig =/parenleftbig ∇ϕ,∇ψ/parenrightbig +/parenleftbig ϕ,1/parenrightbig/parenleftbig ψ,1/parenrightbig for allψ∈Vh, and we denote Rh=I⊗Rh:H1(Ω)3→V3 h. We denote again theL2-orthogonal projections onto the finite element space by Πh:L2(Ω)→Vh andΠh=I⊗Πh:L2(Ω)3→V3 h. As in the previous section, we write Ph(m) for theL2-orthogonal projection onto the discrete tangent space at m. We insert the following quantities, which are related to the exact solution, mn ⋆,h=Rhm(tn), /hatwidermn ⋆,h=k−1/summationdisplay j=0γjmn−j−1 ⋆,h/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay j=0γjmn−j−1 ⋆,h/vextendsingle/vextendsingle/vextendsingle, (6.16) ˙mn ⋆,h=Ph(/hatwidermn ⋆,h)1 τk/summationdisplay j=0δjmn−j ⋆,h∈Th(/hatwidermn ⋆,h),26 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH intothelinearlyimplicit k-stepBDFmethod(2.6)andobtainadefect dn h∈Th(/hatwidermn ⋆,h) from (6.17)α/parenleftbig ˙mn ⋆,h,ϕh/parenrightbig +/parenleftbig /hatwidermn ⋆,h×˙mn ⋆,h,ϕh/parenrightbig =−/parenleftbig ∇mn ⋆,h,∇ϕh/parenrightbig +/parenleftbig Hn,ϕh/parenrightbig +/parenleftbig dn h,ϕh/parenrightbig for allϕh∈Th(/hatwidermn ⋆,h). By definition, there holds ( Rhϕ,1) = (ϕ,1) (this can be seen by testing with ψ= 1) and hence/parenleftbig ∇mn ⋆,h,∇ϕ/parenrightbig =/parenleftbig ∇m(tn),∇ϕ/parenrightbig =−/parenleftbig ∆m(tn),ϕ/parenrightbig . Thus, we obtain the consistency error for the full discretization b y (6.18)dn h=Ph(/hatwidermn ⋆,h)Dn hwithDn h=α˙mn ⋆,h+/hatwidermn ⋆,h×˙mn ⋆,h−∆m(tn)−H(tn) forn=k,...,N. The consistency error is bounded as follows. Lemma 6.2. If the solution of the LLG equation (1.4)has the regularity m∈Ck+1([0,¯t],L2(Ω)3)∩C1([0,¯t],Wr+1,∞(Ω)3)and ∆m+H∈C([0,¯t],Wr+1,∞(Ω)3), then the consistency error (6.18)is bounded by /ba∇dbldn h/ba∇dblL2(Ω)3/lessorequalslantC(τk+hr) fornwithkτ/lessorequalslantnτ/lessorequalslant¯t. Proof.We begin by defining Dn:=α∂tm(tn)+m(tn)×∂tm(tn)−∆m(tn)−H(tn) and note that P(mn ⋆)Dn= 0. Here we denote again mn ⋆=m(tn) and in the following we use also the notations ˙mn ⋆and/hatwidermn ⋆as defined in (6.4). With this, we rewrite the equation for the defect as dn h=Ph(/hatwidermn ⋆,h)Dn h−P(mn ⋆)Dn =Ph(/hatwidermn ⋆,h)/parenleftbig Dn h−Dn/parenrightbig +/parenleftbig Ph(/hatwidermn ⋆,h)−Ph(/hatwidermn ⋆)/parenrightbig Dn +/parenleftbig Ph(/hatwidermn ⋆)−P(/hatwidermn ⋆)/parenrightbig Dn+/parenleftbig P(/hatwidermn ⋆)−P(mn ⋆)/parenrightbig Dn ≡I+II+III+IV. For the term IVwe have by Lemma 4.1 /ba∇dblIV/ba∇dblL2/lessorequalslant2/ba∇dbl/hatwidermn ⋆−mn ⋆/ba∇dblL2/ba∇dblDn/ba∇dblL∞, where the last term /hatwidermn ⋆−mn ⋆has been bounded in the L2norm byCτkin the proof of Lemma 6.1. The term IIIis estimated using the first bound from Lemma 5.1, under our regularity assumptions, as /ba∇dblIII/ba∇dblL2/lessorequalslantChr. For the bound on IIwe use Lemma 5.2 ( i) (withp= 2 andq=∞), to obtain /ba∇dblII/ba∇dblL2/lessorequalslantCR/ba∇dbl/hatwidermn ⋆,h−/hatwidermn ⋆/ba∇dblL2/ba∇dblDn/ba∇dblL∞, where, using (7.11), we obtain /ba∇dbl/hatwidermn ⋆,h−/hatwidermn ⋆/ba∇dblL2/lessorequalslant2/ba∇dbl/summationtextk i=1γi(Rh−I)mn−i ∗/ba∇dblL2 min/vextendsingle/vextendsingle/summationtextk i=1γimn−i ∗/vextendsingle/vextendsingle/lessorequalslantChr.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 27 The denominator is bounded from below by 1 −Cτk, because |mn ∗|= 1 and |/summationtextk i=1γimn−i ∗−mn ∗|/lessorequalslantCτk. For the first term we have /ba∇dblI/ba∇dblL2/lessorequalslant/ba∇dblDn−Dn h/ba∇dblL2 /lessorequalslantα/ba∇dbl∂tm(tn)−˙mn ⋆,h/ba∇dblL2+/ba∇dblm(tn)×∂tm(tn)−/hatwidermn ⋆,h×˙mn ⋆,h/ba∇dblL2. The terms /ba∇dbl∂tm(tn)−˙mn ⋆/ba∇dblL2and/ba∇dblmn ⋆×∂tm(tn)−/hatwidermn ⋆×˙mn ⋆/ba∇dblL2can be handled as in the proof of Lemma 6.1. Standard error estimates for the Ritz projectionRh (we do not exploit the Aubin–Nitsche duality here) imply /ba∇dbl(I−Rh)˙mn ⋆/ba∇dblL2/lessorequalslantchr/ba∇dbl˙mn ⋆/ba∇dblHr+1. Together this yields, under the stated regularity assumption, /ba∇dblI/ba∇dblL2/lessorequalslantC(τk+hr), and the result follows. /square 6.3.Error equation. We recall, from (2.6), the fully discrete problem with the linearly implicit BDF method: find ˙mn h∈Th(/hatwidermn h) such that for all ϕh∈Th(/hatwidermn h), (6.19) α(˙mn h,ϕh)+(/hatwidermn h×˙mn h,ϕh)+(∇mn h,∇ϕh) = (H(tn),ϕh). Then, similarly as we have done in Section 4, we first rewrite (6.17): fo r all ϕh∈Th(/hatwidermn h), (6.20) α(˙mn ⋆,h,ϕh)+(/hatwidermn ⋆,h×˙mn ⋆,h,ϕh)+(∇mn ⋆,h,∇ϕh) = (rn h,ϕh) with (6.21) rn h=−(Ph(/hatwidermn h)−Ph(/hatwidermn ⋆,h))(∆m⋆(tn)+H(tn))+dn h. The erroren h=mn h−mn ⋆,hsatisfies the error equation that is obtained by sub- tracting (6.20) from (6.19). We use the notations /hatwideen h=/hatwidermn h−/hatwidermn ⋆,h, (6.22) ˙en h=˙mn h−˙mn ⋆,h=1 τk/summationdisplay j=0δjen−j h+sn h, (6.23) withsn h= (I−Ph(/hatwidermn ⋆,h))1 τk/summationdisplay j=0δjmn−j ⋆,h. We have the following bound for sn h. Lemma 6.3. Under the regularity assumptions of Lemma 6.2, we have (6.24) /ba∇dblsn h/ba∇dblH1(Ω)3/lessorequalslantC(τk+hr). Proof.We use Lemmas 5.1 and 5.3, and the bounds in the proof of Lemma 6.2. We start by subtracting ( I−P(/hatwidermn ⋆,h))∂tmn ⋆= 0, and obtain (with ∂τmn ⋆,h:= 1 τ/summationtextk j=0δjmn−j ⋆,h) sn h= (I−Ph(/hatwidermn ⋆,h))∂τmn ⋆,h−(I−P(/hatwidermn ⋆,h))∂tmn ⋆ = (∂τmn ⋆,h−∂tmn ⋆)−/parenleftbig Ph(/hatwidermn ⋆,h)∂τmn ⋆,h−P(/hatwidermn ⋆,h)∂tmn ⋆/parenrightbig .28 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH The first term above is bounded as O(τk+hr) via the techniques of the consistency proofs, Lemma 6.1 and 6.2. For the second term we have Ph(/hatwidermn ⋆,h)∂τmn ⋆,h−P(/hatwidermn ⋆,h)∂tmn ⋆ =Ph(/hatwidermn ⋆,h)(∂τmn ⋆,h−∂tmn ⋆)+/parenleftbig Ph(/hatwidermn ⋆,h)−P(/hatwidermn ⋆,h)/parenrightbig ∂tmn ⋆, where the first term is bounded as O(τk+hr), using Lemma 5.3 and the previous estimate, while the second term is bounded as O(hr) by theH1estimate from Lemma 5.1. Altogether, we obtain the stated H1bound forsn h. /square We then have the error equation (6.25)α(˙en h,ϕh)+(/hatwideen h×˙mn ⋆,h,ϕh)+(/hatwidermn h×˙en h,ϕh)+(∇en h,∇ϕh) =−(rn h,ϕh), for allϕh∈Th(/hatwidermn h), which is to be taken together with (6.21)–(6.23). 7.Stability of the full discretization for BDF of orders 1 and 2 For the A-stable BDF methods (those of orders 1 and 2) we obtain t he follow- ing stability estimate, which is analogous to the continuous perturba tion result Lemma 4.2. Lemma 7.1 (Stability for orders k= 1,2).Consider the linearly implicit k-step BDF discretization (2.6)fork/lessorequalslant2with finite elements of polynomial degree r/greaterorequalslant1. Letmn handmn ⋆,h=Rhm(tn)satisfy equations (2.6)and(6.17), respectively, and suppose that the exact solution m(t)is bounded by (4.3)and/ba∇dblH(t)/ba∇dblL∞/lessorequalslantMfor 0/lessorequalslantt/lessorequalslant¯t. Then, for sufficiently small h/lessorequalslant¯handτ/lessorequalslant¯τ, the erroren h=mn h−mn ⋆,h satisfies the following bound, for kτ/lessorequalslantnτ/lessorequalslant¯t, (7.1)/ba∇dblen h/ba∇dbl2 H1(Ω)3/lessorequalslantC/parenleftigk−1/summationdisplay i=0/ba∇dblei h/ba∇dbl2 H1(Ω)3+τn/summationdisplay j=k/ba∇dbldj h/ba∇dbl2 L2(Ω)3+τn/summationdisplay j=k/ba∇dblsj h/ba∇dbl2 H1(Ω)3/parenrightig , where the constant Cis independent of h,τandn, but depends on α,R,K,M , and¯t. This estimate holds under the smallness condition that the r ight-hand side is bounded byˆchwith a sufficiently small constant ˆc(note that the right-hand side is of size O((τk+hr)2)in the case of a sufficiently regular solution ). Combining Lemmas 7.1, 6.2 and 6.3 yields the proof of Theorem 3.1 : These lem- mas imply the estimate /ba∇dblen h/ba∇dblH1(Ω)3/lessorequalslant/tildewideC(τk+hr) in the case of a sufficiently regular solution. Since then /ba∇dblRhm(tn)−m(tn)/ba∇dblH1(Ω)3/lessorequalslant Chrand because of mn h−m(tn) =en h+(Rhm(tn)−m(tn)), this implies the error bound (3.1). The smallness condition imposed in Lemma 7.1 is satisfied under the very mild CFL condition, for a sufficiently small ¯ c>0 (independent of h,τandn), τk/lessorequalslant¯ch1/2. Taken together, this proves Theorem 3.1.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 29 Proof.(a)Preparations. The proof of this lemma transfers the arguments of the proof of Lemma 4.2 to the fully discrete situation, using energy estim ates obtained by testing with (essentially) the discrete time derivative of the erro r, as presented in the Appendix, which is based on Dahlquist’s G-stability theory. However, testing the error equation (6.25) directly with ˙en his not possible, since ˙en his not in the tangent space Th(/hatwidermn h). Therefore, as in the proof of Lemma 4.2, we again start by showing that the test function ϕh=Ph(/hatwidermn h)˙en h∈Th(/hatwidermn h)∩H1(Ω)3 is a perturbation of ˙en hitself: ϕh=Ph(/hatwidermn h)˙en h=Ph(/hatwidermn h)˙mn h−Ph(/hatwidermn h)˙mn ⋆,h =Ph(/hatwidermn h)˙mn h−Ph(/hatwidermn ⋆,h)˙mn ⋆,h+(Ph(/hatwidermn ⋆,h)−Ph(/hatwidermn h))˙mn ⋆,h. Here we note that Ph(/hatwidermn h)˙mn h=˙mn h∈Th(/hatwidermn h) by construction of the method(2.6), andPh(/hatwidermn ⋆,h)˙mn ⋆,h=˙mn ⋆,h∈Th(/hatwidermn ⋆,h) by the definition of ˙mn ⋆,hin (6.4). So we have ϕh=˙mn h−˙mn ⋆,h−(Ph(/hatwidermn h)−P(/hatwidermn ⋆,h))˙mn ⋆,h, and hence (7.2)ϕh=Ph(/hatwidermn h)˙en=˙en h+qn hwithqn h=−(Ph(/hatwidermn h)−P(/hatwidermn ⋆,h))˙mn ⋆,h. Theproofnowtransferstheproofofthecontinuousperturbat ionresultLemma4.2 to the discrete situation with some notable differences, which are em phasized here: (i) Instead of using the continuous quantities it uses their spatially d iscrete coun- terparts, in particular the discrete projections Ph(/hatwidermn h) andPh(/hatwidermn ⋆,h), defined and studied in Section 5. In view of the definition (2.1) and (6.16) of /hatwidermn hand/hatwidermn ⋆,h, re- spectively, thisrequiresthat/summationtextk−1 j=0γjmn−j−1 h(x)and/summationtextk−1 j=0γjmn−j−1 ⋆,h(x)arebounded away from zero uniformly for all x∈Ω. (ii) Instead of Lemma 4.1 we use Lemma 5.2 (with /hatwidermn hand/hatwidermn ⋆,hin the role of /tildewiderm andm, respectively) to bound the quantity qn h. This requires that /hatwidermn ⋆,hand˙mn ⋆,h are bounded in W1,∞independently of h. Ad(i): In order to show that |/summationtextk−1 j=0γjmn−j−1 h(x)|stays close to 1 for all x∈Ω, we need to establish an L∞bound for the errors en−j−1 h=mn−j−1 h−mn−j−1 ⋆,h. We use an induction argument and assume that for some time step nu mber ¯n with ¯nτ/lessorequalslant¯twe have (7.3) /ba∇dblen h/ba∇dblL∞/lessorequalslantρ,for 0/lessorequalslantn<¯n, where we choose ρsufficiently small independent of handτ. (In this proof it suffices to chooseρ/lessorequalslant1/(4Cγ), whereCγ=/summationtextk−1 j=0|γj|= 2k−1.) Note that the smallness condition of the lemma implies that (7.3) is satis fied for ¯n=k, because for the L∞errors of the starting values we have by an inverse inequality, for i= 0,...,k−1, /ba∇dblei h/ba∇dblL∞/lessorequalslantCh−1/2/ba∇dblei h/ba∇dblH1/lessorequalslantCh−1/2(ˆch)1/2=Cˆc1/2/lessorequalslantρ, provided that ˆ cis sufficiently small (independent of τandh), as is assumed. We will show in part (b) of the proof that with the induction hypothes is (7.3) we obtain also /ba∇dble¯n h/ba∇dblL∞/lessorequalslantρso that finally we obtain (7.3) for all¯nwith ¯nτ/lessorequalslant¯t.30 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH Using reverse and ordinary triangle inequalities, the error bound of [12, Corol- lary 8.1.12] (noting that m(t)∈W2,∞(Ω) under our assumptions) and the L∞ boundedness of ∂tm, and the bound (7.3), we estimate (7.4)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay j=0γjmn−j−1 h/vextendsingle/vextendsingle/vextendsingle−1/vextenddouble/vextenddouble/vextenddouble/vextenddouble L∞=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay j=0γjmn−j−1 h/vextendsingle/vextendsingle/vextendsingle−|mn ⋆|/vextenddouble/vextenddouble/vextenddouble/vextenddouble L∞/lessorequalslant/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay j=0γjmn−j−1 h−mn ⋆/vextenddouble/vextenddouble/vextenddouble/vextenddouble L∞ /lessorequalslant/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay j=0γjen−j−1 h/vextenddouble/vextenddouble/vextenddouble L∞+/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay j=0γj(Rhmn−j−1 ⋆−mn−j−1 ⋆)/vextenddouble/vextenddouble/vextenddouble/vextenddouble L∞+/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay j=0γj(mn−j−1 ⋆−mn ⋆)/vextenddouble/vextenddouble/vextenddouble/vextenddouble L∞ /lessorequalslant/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay j=0γjen−j−1 h/vextenddouble/vextenddouble/vextenddouble L∞+Ch+Cτ/lessorequalslantk−1/summationdisplay j=0|γj| ·ρ+Ch+Cτ/lessorequalslant1 2, provided that handτare sufficiently small. The same argument also yields that/vextenddouble/vextenddouble|/summationtextk−1 j=0γjmn−j−1 ⋆,h|−1/vextenddouble/vextenddouble L∞/lessorequalslant1 2, and so we have (7.5)1 2/lessorequalslant/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay j=0γjmn−j−1 h(x)/vextendsingle/vextendsingle/vextendsingle/lessorequalslant3 2and1 2/lessorequalslant/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay j=0γjmn−j−1 ⋆,h(x)/vextendsingle/vextendsingle/vextendsingle/lessorequalslant3 2 for allx∈Ω. In particular, it follows that /hatwidermn hand/hatwidermn ⋆,hare unambiguously defined. Ad(ii): The required W1,∞bound formn ⋆,h=Rhm(tn) follows from the W1,∞- stability of the Ritz projection: by [12, Theorem 8.1.11] and by the as sumedW1,∞ bound (4.3) for m(t), (7.6) /ba∇dblmn ⋆,h/ba∇dblW1,∞/lessorequalslantC/ba∇dblm(tn)/ba∇dblW1,∞/lessorequalslantCR. The bounds (7.5) and (7.6) for n/lessorequalslant¯nimply that also (7.7) /ba∇dbl/hatwidermn ⋆,h/ba∇dblW1,∞/lessorequalslantCR forn/lessorequalslant¯n(with a different constant C). Using this bound in Lemma 5.3 and the assumedW1,∞bound (4.3) for ∂tm(t), we obtain with δ(ζ)/(1−ζ) =/summationtextk ℓ=1(1− ζ)ℓ−1/ℓ=:/summationtextk−1 j=0µjζjthat /ba∇dbl˙mn ⋆,h/ba∇dblW1,∞=/ba∇dblPh(/hatwidermn ⋆,h)1 τk/summationdisplay j=0δjmn−j ⋆/ba∇dblW1,∞ =/ba∇dblPh(/hatwidermn ⋆,h)k−1/summationdisplay j=0µj1 τ(mn−j ⋆−mn−j−1 ⋆)/ba∇dblW1,∞ =/ba∇dblPh(/hatwidermn ⋆,h)k−1/summationdisplay j=0µj1 τ/integraldisplaytn−j tn−j−1∂tm(t)dt/ba∇dblW1,∞ /lessorequalslantCR/ba∇dblk−1/summationdisplay j=0µj1 τ/integraldisplaytn−j tn−j−1∂tm(t)dt/ba∇dblW1,∞HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 31 /lessorequalslantCRk−1/summationdisplay j=0|µj|R. We can now establish a bound for qn has defined in (7.2), using Lemma 5.2 together with the above W1,∞bounds for /hatwidermn ⋆,hand˙mn ⋆,hto obtain (7.8) /ba∇dblqn h/ba∇dblL2/lessorequalslantc/ba∇dbl/hatwideen h/ba∇dblL2and/ba∇dbl∇qn h/ba∇dblL2/lessorequalslantc/ba∇dbl/hatwideen h/ba∇dblH1. With theW1,∞bound of /hatwidermn ⋆,hwe also obtain a bound of rn hdefined in (6.21). Using Lemma 5.2 ( i) and recalling the L∞bound of∆m+Hof (4.3), we find that rn his bounded by (7.9)/ba∇dblrn h/ba∇dblL2/lessorequalslant/ba∇dbl(Ph(/hatwidermn h)−Ph(/hatwidermn ⋆,h))(∆mn ⋆+Hn)/ba∇dblL2+/ba∇dbldn h/ba∇dblL2 /lessorequalslantc/ba∇dbl/hatwideen h/ba∇dblL2+/ba∇dbldn h/ba∇dblL2. (b)Energy estimates. Forn/lessorequalslant¯nwith ¯nof (7.3), we test the error equation (6.25) withϕh=˙en h+qn hand obtain α(˙en h,˙en h+qn h)+(/hatwideen h×˙mn ⋆,h,˙en h+qn h)+(/hatwidermn h×˙en h,˙en h+qn h) +(∇en h,∇(˙en h+qn h)) =−(rn h,˙en h+qn h). By collecting the terms, and using the fact that ( /hatwidermn h×˙en h,˙en h) = 0, we altogether obtain α/ba∇dbl˙en h/ba∇dbl2 L2+(∇en h,∇˙en h) =−α(˙en h,qn h)−(/hatwideen h×˙mn ⋆,h,˙en h+qn h) −(/hatwidermn h×˙en h,qn h)−(∇en h,∇qn h)−(rn h,˙en h+qn h). We now estimate the term ( ∇en h,∇˙en h) on the left-hand side from below using Dahlquist’s Lemma 10.1, so that the ensuing relation (10.2) yields (∇en h,∇˙en h)/greaterorequalslant1 τ/parenleftig /ba∇dbl∇En h/ba∇dbl2 G−/ba∇dbl∇En−1 h/ba∇dbl2 G/parenrightig +(∇en h,∇sn h), whereEn h= (en−k+1 h,...,en h) and theG-weighted semi-norm is given by /ba∇dbl∇En h/ba∇dbl2 G=k/summationdisplay i,j=1gij(∇en−k+i h,∇en−k+j h). This semi-norm satisfies the relation (7.10) γ−k/summationdisplay j=1/ba∇dbl∇en−k+j h/ba∇dbl2 L2/lessorequalslant/ba∇dbl∇En h/ba∇dbl2 G/lessorequalslantγ+k/summationdisplay j=1/ba∇dbl∇en−k+j h/ba∇dbl2 L2, whereγ−andγ+are the smallest and largest eigenvalues of the positive definite symmetric matrix G= (gij) from Lemma 10.1. The remaining terms are estimated using the Cauchy–Schwarz inequ ality and /ba∇dbl/hatwidermn h/ba∇dblL∞= 1; we altogether obtain α/ba∇dbl˙en h/ba∇dbl2 L2+1 τ/parenleftig /ba∇dbl∇En h/ba∇dbl2 G−/ba∇dbl∇En−1 h/ba∇dbl2 G/parenrightig /lessorequalslantα/ba∇dbl˙en h/ba∇dblL2/ba∇dblqn h/ba∇dblL2+/ba∇dbl/hatwideen h/ba∇dblL2(/ba∇dbl˙en h/ba∇dblL2+/ba∇dblqn h/ba∇dblL2) +/ba∇dbl˙en h/ba∇dblL2/ba∇dblqn h/ba∇dblL2+/ba∇dbl∇en h/ba∇dblL2(/ba∇dbl∇qn/ba∇dblL2+/ba∇dbl∇sn h/ba∇dblL2)+/ba∇dblrn h/ba∇dblL2(/ba∇dbl˙en h/ba∇dblL2+/ba∇dblqn h/ba∇dblL2).32 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH We now show an L2error bound for /hatwideen hin terms of (en−j−1 h)k−1 j=0. Using the fact that fora,b∈R3\{0}, (7.11)/vextendsingle/vextendsingle/vextendsingle/vextendsinglea |a|−b |b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(|b|−|a|)a+|a|(a−b) |a| |b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant2|a−b| |b|, and the lower bounds in (7.5) for both |/summationtextk−1 j=0γjmn−j−1 h|and|/summationtextk−1 j=0γjmn−j−1 ⋆,h|, we can estimate (7.12)/ba∇dbl/hatwideen h/ba∇dblL2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationtextk−1 j=0γjmn−j−1 h/vextendsingle/vextendsingle/vextendsingle/summationtextk−1 j=0γjmn−j−1 h/vextendsingle/vextendsingle/vextendsingle−/summationtextk−1 j=0γjmn−j−1 ⋆,h/vextendsingle/vextendsingle/vextendsingle/summationtextk−1 j=0γjmn−j−1 ⋆,h/vextendsingle/vextendsingle/vextendsingle/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2/lessorequalslantCk−1/summationdisplay j=0/ba∇dblen−j−1 h/ba∇dbl2 L2. To show a similar bound for /ba∇dbl∇/hatwideen h/ba∇dblL2we need the following two observations: First, theW1,∞bounds formn−j−1 ⋆,hfrom (7.6) imply W1,∞boundedness for /hatwidermn ⋆,hby /vextendsingle/vextendsingle/vextendsingle/vextendsingle∂j/parenleftbiggb |b|/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂jb |b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingleb(∂jb,b) |b|3/vextendsingle/vextendsingle/vextendsingle/vextendsingle. Second, similarly we have/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂j/parenleftbigga |a|−b |b|/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ja |a|−∂jb |b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsinglea(∂ja,a)|b|3−b(∂jb,b)|a|3 |a|3|b|3/vextendsingle/vextendsingle/vextendsingle/vextendsingle /lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ja |a|−∂jb |b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+||a|3−|b|3||∂jb| |a|3|b|+|a(∂ja,a)−b(∂jb,b)| |b|3 /lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ja |a|−∂jb |b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+|a−b|(|b|2+|b||a|+|a|2)|∂jb| |a|3|b| +|a|2|∂ja−∂jb| |b|3+|a||∂jb||a−b| |b|3+|a−b||∂jb| |b|2. Combining these two observations, again with mhandm⋆,hin the role of aandb, respectively, and the upper and lower bounds from (7.5) altogethe r yield (7.13) /ba∇dbl∇/hatwideen h/ba∇dbl2 L2/lessorequalslantCk−1/summationdisplay j=0/ba∇dblen−j−1 h/ba∇dbl2 H1. We estimate further using Young’s inequality and absorptions into th e term /ba∇dbl˙en/ba∇dbl2 L2, together with the bounds in (7.8) and (7.9), to obtain α1 2/ba∇dbl˙en h/ba∇dbl2 L2+1 τ/parenleftig /ba∇dbl∇En h/ba∇dbl2 G−/ba∇dbl∇En−1 h/ba∇dbl2 G/parenrightig /lessorequalslantck/summationdisplay j=0/ba∇dblen−j h/ba∇dbl2 H1+c/ba∇dbldn h/ba∇dbl2 L2+c/ba∇dbl∇sn h/ba∇dbl2 L2. Multiplying both sides by τ, summing up from kton/lessorequalslant¯n, and using an absorption yield α1 2τn/summationdisplay j=k/ba∇dbl˙ej h/ba∇dbl2 L2+/ba∇dbl∇En h/ba∇dbl2 G /lessorequalslant/ba∇dbl∇Ek−1 h/ba∇dbl2 G+cτn/summationdisplay j=k/ba∇dblej h/ba∇dbl2 H1+cτn/summationdisplay j=k/parenleftbig /ba∇dbldj h/ba∇dbl2 L2+/ba∇dblsj h/ba∇dbl2 H1/parenrightbig +ck−1/summationdisplay i=0/ba∇dblei h/ba∇dbl2 L2.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 33 We then arrive, using (7.10), at (7.14)α1 2τn/summationdisplay j=k/ba∇dbl˙ej h/ba∇dbl2 L2+/ba∇dbl∇en h/ba∇dbl2 L2/lessorequalslantcτn/summationdisplay j=k/ba∇dblej h/ba∇dbl2 H1+cτn/summationdisplay j=k/parenleftbig /ba∇dbldj h/ba∇dbl2 L2+/ba∇dblsj h/ba∇dbl2 H1/parenrightbig +ck−1/summationdisplay i=0/ba∇dblei h/ba∇dbl2 L2, withcdepending on α. Similarly as in the time continuous case in the proof of Lemma 4.2, we con nect /ba∇dblen h/ba∇dbl2 L2andτ/summationtextn j=k/ba∇dbl˙ej h/ba∇dbl2 L2. We rewrite the identity 1 τk/summationdisplay j=0δjen−j h=˙en h−sn h, n/greaterorequalslantk, as 1 τn/summationdisplay j=kδn−jej h=˙ehn−sn h−gn h, n/greaterorequalslantk, withδℓ= 0 forℓ>kand where gn h:=1 τk−1/summationdisplay i=0δn−iei h depends only on the starting errors and satisfies gn h= 0 forn/greaterorequalslant2k. With the inverse power series of δ(ζ), κ(ζ) =∞/summationdisplay n=0κnζn:=1 δ(ζ), we then have, for n/greaterorequalslantk, en h=τn/summationdisplay j=kκn−j(˙ej h−sj h−gj h). By the zero-stability of the BDF method of order k/lessorequalslant6, the coefficients κnare uniformly bounded: |κn|/lessorequalslantcfor alln/greaterorequalslant0. Therefore we obtain via the Cauchy– Schwarz inequality /ba∇dblen h/ba∇dbl2 L2/lessorequalslant2τ2/vextenddouble/vextenddouble/vextenddoublen/summationdisplay j=kκn−j(˙ehj−sj h)/vextenddouble/vextenddouble/vextenddouble2 L2+2τ2/vextenddouble/vextenddouble/vextenddouble2k−1/summationdisplay j=kκn−jgj h/vextenddouble/vextenddouble/vextenddouble2 L2 /lessorequalslant(2nτ)τc2n/summationdisplay j=k/ba∇dbl˙ehj−sj h/ba∇dbl2 L2+2τ2c2k2k−1/summationdisplay j=k/ba∇dblgj h/ba∇dbl2 L2 /lessorequalslantCτn/summationdisplay j=k/ba∇dbl˙ej h/ba∇dbl2 L2+Cτn/summationdisplay j=k/ba∇dblsj h/ba∇dbl2 L2+Ck/summationdisplay i=0/ba∇dblei h/ba∇dbl2 L2.34 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH Inserting this bound into (7.14) then yields α/ba∇dblen h/ba∇dbl2 L2+/ba∇dbl∇en h/ba∇dbl2 L2/lessorequalslantcτn/summationdisplay j=k/ba∇dblej h/ba∇dbl2 H1+cτn/summationdisplay j=k/parenleftbig /ba∇dbldj h/ba∇dbl2 L2+/ba∇dblsj h/ba∇dbl2 H1/parenrightbig +ck−1/summationdisplay i=0/ba∇dblei h/ba∇dbl2 L2, and a discrete Gronwall inequality implies the stated stability result fo rn/lessorequalslant¯n. It then follows from this stability bound, the smallness condition of the le mma and the inverse estimate from H1toL∞that (7.3) is satisfied also for ¯ n+1. This completes the induction step for (7.3) and proves the stated error bound. /square 8.Stability of the full discretization for BDF of orders 3 to 5 Stability for full discretizations using the BDF methods of orders 3 t o 5 can be shown under additional conditions on the damping parameter αand the stepsize τ. Lemma 8.1 (Stability for orders k= 3,4,5).Consider the linearly implicit k-step BDF discretization (2.6)for3/lessorequalslantk/lessorequalslant5with finite elements of polynomial degree r/greaterorequalslant2. Letmn handmn ⋆,hsatisfy(2.6)and(6.17), respectively, and suppose that the regularity assumptions of Lemma 7.1 hold. Furthermore, ass ume that the damping parameterαsatisfies (8.1) α>α k:=ηk 1−ηk with the multiplier ηkof Lemma 10.2, and that τandhsatisfy the mild CFL-type condition, for some ¯c>0, (8.2) τ/lessorequalslant¯ch. Then, for sufficiently small h/lessorequalslant¯handτ/lessorequalslant¯τ, the erroren h=mn h−mn ⋆,hsatisfies the following bound, for kτ/lessorequalslantnτ/lessorequalslant¯t, (8.3)/ba∇dblen h/ba∇dbl2 H1(Ω)3/lessorequalslantC/parenleftigk−1/summationdisplay i=0/ba∇dblei h/ba∇dbl2 H1(Ω)3+τn/summationdisplay j=k/ba∇dbldj h/ba∇dbl2 L2(Ω)3+τn/summationdisplay j=k/ba∇dblsj h/ba∇dbl2 H1(Ω)3/parenrightig , where the constant Cis independent of τ,handn, but depends on α,R,K,M , and exponentially on ¯c¯t. This estimate holds under the smallness condition that the right-hand side is bounded by ˆch3with a constant ˆc(note that the right-hand side is of sizeO((τk+hr)2)in the case of a sufficiently regular solution ). Together with the defect bounds of Section 6, this stability lemma pr oves Theo- rem 3.2. We remark that the thresholds αk>0 defined here are the same as those appearing in Theorem 3.2. Proof.The proof of this lemma combines the arguments of the proof of Lem ma 7.1 with a nonstandard variant of the multiplier technique of Nevanlinna a nd Odeh, as outlined in the Appendix. Since the size of the parameter αdetermines which BDF methods satisfy the stability estimate, the dependence on αwill be carefully traced all along the proof.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 35 (a)Preparations. As in the previous proof, we make again the induction hypoth- esis (7.3) for some ¯ nwith ¯nτ/lessorequalslant¯t, but this time with ρ=c0hfor some positive constantc0: (8.4) /ba∇dblen h/ba∇dblL∞/lessorequalslantc0h, n< ¯n. By an inverse inequality, this implies that /ba∇dblen h/ba∇dblW1,∞has anh- andτ-independent bound, and hence also /ba∇dblmn h/ba∇dblW1,∞forn<¯n. Together with (7.5), this implies (8.5) /ba∇dbl/hatwidermn h/ba∇dblW1,∞/lessorequalslantC and further (8.6) /ba∇dbl/hatwideen h/ba∇dblL∞/lessorequalslantCh. As in the Appendix, we aim to subtract ηktimes the error equation for time stepn−1 from the error equation for time step n, and then to test with ϕh= Ph(/hatwidermn h)˙en h∈Th(/hatwidermn h) (similarly as in the proof of Lemma 7.1). However, this is not possible directly due to the different test spaces at different time st eps: α(˙en h,ϕh)+(/hatwideen h×˙mn ⋆,h,ϕh) +(/hatwidermn h×˙en h,ϕh)+(∇en h,∇ϕh) =−(rn h,ϕh),(8.7a) for allϕh∈Th(/hatwidermn h), and α(˙en−1 h,ψh)+(/hatwideen−1 h×˙mn−1 ⋆,h,ψh) +(/hatwidermn−1 h×˙en−1 h,ψh)+(∇en−1 h,∇ψh) =−(rn−1 h,ψh),(8.7b) for allψh∈Th(/hatwidermn−1 h). As in (7.2), we have (8.8)ϕh=Ph(/hatwidermn h)˙en h=˙en h+qn h,withqn h=−(Ph(/hatwidermn h)−Ph(/hatwidermn ⋆,h))˙mn ⋆,h, whereqn his bounded by (7.8). In turn, the test function ψh=Ph(/hatwidermn−1 h)˙en h∈Th(/hatwidermn−1 h) is a perturbation of ϕh=˙en h+qn h, since using (8.8) we obtain ψh=Ph(/hatwidermn−1 h)˙en h =Ph(/hatwidermn h)˙en h−(Ph(/hatwidermn h)−Ph(/hatwidermn−1 h))˙en h =˙en h+qn h+pn hwithpn h=−(Ph(/hatwidermn h)−Ph(/hatwidermn−1 h))˙en h. The perturbation pn his estimated using the second bound in Lemma 5.2 ( i) with p=∞,q= 2, and noting (8.5). We obtain /ba∇dblpn h/ba∇dblL2/lessorequalslant/ba∇dbl(Ph(/hatwidermn h)−Ph(/hatwidermn−1 h))˙en h/ba∇dblL2 /lessorequalslantc/ba∇dbl˙en h/ba∇dblL2/ba∇dbl/hatwidermn h−/hatwidermn−1 h/ba∇dblL∞ /lessorequalslantc/ba∇dbl˙en h/ba∇dblL2/parenleftig /ba∇dbl/hatwideen h/ba∇dblL∞+/ba∇dbl/hatwidermn ⋆,h−/hatwidermn−1 ⋆,h/ba∇dblL∞+/ba∇dbl/hatwideen−1 h/ba∇dblL∞/parenrightig /lessorequalslantc/ba∇dbl˙en h/ba∇dblL2/parenleftig /ba∇dbl/hatwideen h/ba∇dblL∞+k−1/summationdisplay j=0|γj|/integraldisplaytn−j−1 tn−j−2/ba∇dblRh∂tm(t)/ba∇dblL∞dt+/ba∇dbl/hatwideen−1 h/ba∇dblL∞/parenrightig .36 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH We have /ba∇dblRh∂tm(t)/ba∇dblL∞/lessorequalslantc/ba∇dbl∂tm(t)/ba∇dblW1,∞by [12, Theorem 8.1.11]. In view of (8.6) we obtain, for τ/lessorequalslant¯Ch, (8.9) /ba∇dblpn h/ba∇dblL2/lessorequalslantCh/ba∇dbl˙en h/ba∇dblL2, and by an inverse estimate, (8.10) /ba∇dbl∇pn h/ba∇dblL2/lessorequalslantC/ba∇dbl˙en h/ba∇dblL2. We also recall the bound (7.9) for /ba∇dblrn h/ba∇dblL2. (b)Energy estimates. By subtracting (8.7a) −ηk(8.7b) with the above choice of test functions, we obtain (8.11)α(˙en h−ηk˙en−1 h,˙en h+qn h)+(/hatwideen h×˙mn ⋆,h−ηk/hatwideen−1 h×˙mn−1 ⋆,h,˙en h+qn h) +(/hatwidermn h×˙en h−ηk/hatwidermn−1 h×˙en−1 h,˙en h+qn h)+(∇en h−ηk∇en−1 h,∇(˙en h+qn h)) −ηk/bracketleftbig α(˙en−1 h,pn h)+(/hatwideen−1 h×˙mn−1 ⋆,h,pn h) +(/hatwidermn−1 h×˙en−1 h,pn h)+(∇en−1 h,∇pn h)/bracketrightbig =−(rn h−ηkrn−1 h,˙en h+qn h)−ηk(rn−1 h,pn h). We estimate the terms of the error equation (8.11) separately and track carefully the dependence on ηkandα. The termα(˙en h−ηk˙en−1 h,˙en h) is bounded from below, using Young’s inequality and absorptions, by α(˙en h−ηk˙en−1 h,˙en h)/greaterorequalslantα/parenleftbig 1−1 2ηk/parenrightbig /ba∇dbl˙en h/ba∇dbl2 L2−α 2ηk/ba∇dbl˙en−1 h/ba∇dbl2 L2, while the term ( ∇en h−ηk∇en−1 h,∇˙en h) is bounded from below, via the relation (10.2) and (6.23), by (∇en h−ηk∇en−1 h,∇˙en h)/greaterorequalslant1 τ/parenleftig /ba∇dbl∇En h/ba∇dbl2 G−/ba∇dbl∇En−1 h/ba∇dbl2 G/parenrightig +(∇en h−ηk∇en−1 h,∇sn h), withEn h= (en−k+1 h,...,en h), and where the G-weighted semi-norm is generated by the matrix G= (gij) from Lemma 10.1 for the rational function δ(ζ)/(1−ηkζ). The remaining terms outside the rectangular bracket are estimate d using the Cauchy–Schwarz and Young inequalities (the latter often with a suffi ciently small but fixedh- andτ-independent weighting factor µ >0) and/ba∇dbl/hatwidermn h/ba∇dblL∞= 1 and orthogonality. We obtain, with varying constants c(which depend on αand are inversely proportional to µ) α(˙en h−ηk˙en−1 h,qn h)+(/hatwideen h×˙mn ⋆,h−ηk/hatwideen−1 h×˙mn−1 ⋆,h,˙en h+qn h) +(/hatwidermn h×˙en h−ηk/hatwidermn−1 h×˙en−1 h,˙en h+qn h)+(∇en−ηk∇en−1 h,∇qn h) /lessorequalslant/parenleftbig αµ+µ+1 2ηk/parenrightbig /ba∇dbl˙en h/ba∇dbl2 L2+/parenleftbig αµηk+1 2ηk/parenrightbig /ba∇dbl˙en−1 h/ba∇dbl2 L2 +c/parenleftbig /ba∇dblqn h/ba∇dblL2+/ba∇dbl/hatwideen h/ba∇dbl2 L2+/ba∇dbl/hatwideen−1 h/ba∇dbl2 L2/parenrightbig +1 2/parenleftbig /ba∇dbl∇en h/ba∇dbl2 L2+η2 k/ba∇dbl∇en−1 h/ba∇dbl2 L2+/ba∇dbl∇qn h/ba∇dblL2/parenrightbig /lessorequalslant/parenleftbig αµ+µ+1 2ηk/parenrightbig /ba∇dbl˙en h/ba∇dbl2 L2+/parenleftbig αµηk+1 2ηk/parenrightbig /ba∇dbl˙en−1 h/ba∇dbl2 L2+ck/summationdisplay j=0/ba∇dblen−j−1 h/ba∇dbl2 H1, where in the last inequality we used (7.12) and (7.13) to estimate /hatwideen h.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 37 The terms inside the rectangular bracket are bounded similarly, usin g (8.9) and (8.10) and the condition τ/lessorequalslant¯Ch, by α(˙en−1 h,pn h)+(/hatwideen−1 h×˙mn−1 ⋆,h,pn h)+(/hatwidermn−1 h×˙en−1 h,pn h)+(∇en−1 h,∇pn h) /lessorequalslantµ/ba∇dbl˙en h/ba∇dbl2 L2+ch/ba∇dbl˙en−1 h/ba∇dbl2 L2+c/parenleftbig /ba∇dbl/hatwideen−1 h/ba∇dbl2 L2+/ba∇dbl∇en−1 h/ba∇dbl2 L2/parenrightbig /lessorequalslantµ/ba∇dbl˙en h/ba∇dbl2 L2+ck/summationdisplay j=0/ba∇dblen−j−1 h/ba∇dbl2 H1. Hereµis an arbitrarily small positive constant (independent of τandh), andc depends on the choice of µ. In view of (7.9), the terms with the defects rn hare bounded by −(rn h−ηkrn−1 h,˙en h+qn h)−ηk(rn−1 h,pn h) /lessorequalslantµ/ba∇dbl˙en h/ba∇dbl2 L2+c/parenleftbig /ba∇dblrn h/ba∇dbl2 L2+/ba∇dblrn−1 h/ba∇dbl2 L2+/ba∇dblqn h/ba∇dbl2 L2/parenrightbig /lessorequalslantµ/ba∇dbl˙en h/ba∇dbl2 L2+ck/summationdisplay j=0/ba∇dblen−j−1 h/ba∇dbl2 L2+c1/summationdisplay j=0/ba∇dbldn−j h/ba∇dbl2 L2. Combination of these inequalities yields /parenleftig α(1−1 2ηk)−1 2ηk−µ/parenrightig /ba∇dbl˙en h/ba∇dbl2 L2−/parenleftig α 2ηk+1 2ηk+µαηk/parenrightig /ba∇dbl˙en−1 h/ba∇dbl2 L2 +1 τ/parenleftig /ba∇dbl∇En h/ba∇dbl2 G−/ba∇dbl∇En−1 h/ba∇dbl2 G/parenrightig /lessorequalslantck/summationdisplay j=0/ba∇dblen−j−1 h/ba∇dbl2 H1+c1/summationdisplay j=0/ba∇dbldn−j h/ba∇dbl2 L2+c/ba∇dbl∇sn h/ba∇dbl2 L2. Under condition (8.1) we have ω:=α(1−ηk)−ηk>0. Multiplying both sides by τand summing up from ktonwithn/lessorequalslant¯nyields, for sufficiently small µ, 1 2ωτn/summationdisplay j=k/ba∇dbl˙ej h/ba∇dbl2 L2+/ba∇dbl∇En h/ba∇dbl2 G /lessorequalslantcτ/ba∇dbl˙ek−1 h/ba∇dbl2 L2+/ba∇dbl∇Ek−1 h/ba∇dbl2 G+cτn−1/summationdisplay j=0/ba∇dblej h/ba∇dbl2 H1+cτn/summationdisplay j=k/ba∇dbldj h/ba∇dbl2 L2+cτn/summationdisplay j=k/ba∇dbl∇sj h/ba∇dbl2 L2. The proof is then completed using exactly the same arguments as in t he last part of theproofofLemma7.1,byestablishinganestimatebetween /ba∇dblen h/ba∇dbl2 L2andτ/summationtextn j=k/ba∇dbl˙ej h/ba∇dbl2 L2 and using a discrete Gronwall inequality, and completing the induction step for (8.4). /square 9.Numerical experiments To obtain significant numerical results, we prescribe the exact solu tionmon given three-dimensional domains Ω:= [0,1]×[0,1]×[0,L] withL∈ {1/100,1/4}.38 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH The discretizations of these domains will consist of a few layers of ele ments inz- direction (one layer for L= 1/100 and ten layers for L= 1/4) and a later specified number of elements in xandydirections. This mimics the common case of thin film alloys as for example in the standard problems of the Micromagnet ic Modeling ActivityGroupatNISTCenterforTheoreticalandComputational MaterialsScience (ctcms.nist.gov ). Moreover, this mesh structure helps to keep the computationa l requirements reasonable and allow us to compute the experiments o n a desktop PC. We are aware that these experiments are only of preliminary nat ure and are just supposed to confirm the theoretical results. A more thorou gh investigation of the numerical properties of the developed method is needed. Th is will require us to incorporate preconditioning, parallelization of the computatio ns, as well as lower order energy contributions in the effective field (1.3) to be able to compare to benchmark results from computational physics. This, however, is beyond the scope of this paper, and will be the topic of a subsequent work. We consider the time interval [0 ,¯t] with¯t= 0.2 and define two different exact solutions. Since within our computational budget either the time disc retization error or the space discretization error dominates, we construct the solutions such that the first oneis harder to approximate inspace, while thesecon d oneis harder to approximate in time. Both solutions are constant in z-direction as is often observed in thin-film applications. 9.1.Implementation. The numerical experiments were conducted using the finite element package FEniCS ( www.fenicsproject.org ) on a desktop computer. As al- readydiscussed inSection2.2, thereareseveral ways toimplement thetangent space restriction. We decided to solve a saddle point problem (variant (a) in Section 2.2) for simplicity of implementation. For preconditioning, we used the blac k-box AMG preconditioner that comes with FEniCS. Although this might not be th e optimal solution, it keeps the number of necessary iterative solver steps w ithin reasonable bounds. Assuming perfect preconditioning, the cost per time-ste p is then propor- tional to the number of mesh-elements. We observed this behavior approximately, although further research beyond the scope of this work is requir ed to give a definite conclusion. 9.2.Exact solutions. We choose the damping parameter α= 0.2 and define g(t) := (¯t+0.1)/(¯t+0.1−t) as well as d(x) := (x1−1/2)2+(x2−1/2)2, which is the squared distance of the projection of xto [0,1]×[0,1] and the point (1 /2,1/2). For some constant C= 400 (a choice made to have pronounced effects), define (9.1)m(x,t) := Ce−g(t) 1/4−d(x)(x1−1/2) Ce−g(t) 1/4−d(x)(x2−1/2)/radicalig 1−C2e−2g(t) 1/4−d(x)d(x) ifd(x)/lessorequalslant1 4andm(x,t) := 0 0 1 else. It iseasy to check that |m(x,t)|= 1for all ( x,t)∈Ω×[0,¯t]. Moreover, ∂nm(x,t) = 0 for allx∈∂Ω. We may calculate the time derivative of min a straightforwardHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 39 fashion, i.e., ∂tm(x,t) = 0 ford(x)>1/4 and ∂tm(x,t) = −g′(t) 1/4−d(x)Ce−g(t) 1/4−d(x)(x1−1/2) −g′(t) 1/4−d(x)Ce−g(t) 1/4−d(x)(x2−1/2) g′(t) 1/4−d(x)C2e−2g(t) 1/4−d(x)d(x) m3(x,t) ifd(x)/lessorequalslant1 4. Here,m3denotes the third component of mas defined above. The second exact solution is defined via (9.2) /tildewiderm(x,t) := −(x3 1−3x2 1/2+1/4)sin(3πt/¯t)/radicalbig 1−(x3 1−3x2 1/2+1/4)2 −(x3 1−3x2 1/2+1/4)cos(3πt/¯t) . Due to the polynomial nature in the first and the third component, a nd the well- behaved square-root, the space approximation error does not d ominate the time approximation. 9.3.The experiments. We now may compute the corresponding forcings Hresp. /tildewiderHto obtain the prescribed solutions by inserting into (1.4), i.e., H=α∂tm+m×∂tm−∆m. (Note that we may disregard the projection P(m) from (1.4) since we solve in the tangent space anyway.) We compute Hnumerically by first interpolating m and∂tmand then computing the derivatives. This introduces an additional e rror which is not accounted for in the theoretical analysis. However, th e examples below confirm the expected convergence rates and hence conclude tha t this additional perturbation is negligible. Figure 9.1 shows slices of the exact solution at different time steps. Figure 9.2 shows the convergence with respect to the t ime step size τ, while Figure 9.3 shows convergence with respect to the spatial mesh sizeh. All the experiments confirm the expected rates for smooth solutions. Finally, we consider an example with nonsmooth initial data and consta nt right- hand side. The initial data are given by (9.3)m0(x) := x1−1/2 x2−1/2/radicalbig 1−d(x) ifd(x)/lessorequalslant1 4andm0(x) := 0 0 1 else. With the constant forcing field H:= (0,1,1)Twe compute a numerical approxima- tion to the unknown exact solution. Note that we do not expect any smoothness of the solution (even the initial data is not smooth). Figure 9.4 neverth eless shows a physically consistent decay of the energy /ba∇dbl∇m(t)/ba∇dblL2(Ω)3over time as well as a good agreement between different orders of approximation. Moreover , the computed ap- proximation shows little deviation from unit length as would be expecte d for smooth solutions.40 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH Figure 9.1. Thefirstrowshowstheexactsolution m(x,t)from(9.1) forx∈[0,1]×[0,1]× {0}andt∈ {0,0.05,¯t}(from left to right), whereas the second row shows the exact solution /tildewiderm(x,t) from (9.2) forx∈[0,1]×[0,1]×{0}andt∈ {0,0.2/6,0.2/3}(from left to right). While the problems are three-dimensional, the solutions are constan t inz-direction and we only show one slice of the solution.PSfrag replacements 10−810−610−410−2100 10−310−210−1k= 1 k= 2 k= 3 k= 4 timestep τPSfrag replacements 10−610−510−410−310−210−1100101 10−310−210−1k= 1 k= 2 k= 3 k= 4 timestep τ Figure 9.2. The plots show the error between computed solutions and exact solution /tildewidermfor a given time stepsize with a spatial poly- nomial degree of r= 2 and a spatial mesh size 1 /40 which results in≈6·104degrees of freedom per time step in the left plot. In the right plot we use a thicker domain D= [0,1]×[0,1]×[0,1/4] with 10 elements in z-direction. This results in ≈4·105degrees of freedom per timestep. We use the k-step methods of order k∈ {1,2,3,4}and observe the expected rates O(τk) indicated by the dashed lines. The coarse levels of the higher order methods are missing because the kth step is already beyond the final time ¯t.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 41 PSfrag replacements r= 1 r= 2 r= 3 r= 4 meshsize h10−410−310−210−1100 10−1 Figure 9.3. The plot shows convergence in meshsize hwith respect to the exact solution mfrom (9.1) on the domain D= [0,1]×[0,1]× [0,1/100]withonelayerofelementsin z-direction. Weusedthesecond order BDF method with τ= 10−3and spatial polynomial degrees r∈ {1,2,3,4}. The mesh sizes range from 1 /2 to 1/32. We observe the expected rates O(hr) indicated by the dashed lines. The finest mesh-size for r= 4 does reach the expected error level. This is due to the fact that the time-discretization errors start to dominate in t hat region. 10.Appendix: Energy estimates for backward difference formula e The stability proofs of this paper rely on energy estimates, that is, on the use of positive definite bilinear forms to bound the error ein terms of the defect d. This is, of course, a basic technique for studying the time-continuo us problem and also for backward Euler and Crank–Nicolson time discretizations (se e, e.g., Thom´ ee [38]), but energy estimates still appear to be not well known for bac kward difference formula (BDF) time discretizations of order up to 5, which are widely u sed for solving stiff ordinary differential equations. To illustrate the basic me chanism, we here just consider the prototypical linear parabolic evolution equa tion in its weak formulation, given by two positive definite symmetric bilinear forms ( ·,·) anda(·,·) on Hilbert spaces HandVwith induced norms |·|and/ba∇dbl·/ba∇dbl, respectively, and with Vdensely and continuously embedded in H. The problem then is to find u(t)∈V such that (10.1) ( ∂tu,v)+a(u,v) = (f,v)∀v∈V, with initial condition u(0) =u0. Ifu⋆is a function that satisfies the equation up to a defectd, that is, (∂tu⋆,v)+a(u⋆,v) = (f,v)+(d,v)∀v∈V,42 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICHPSfrag replacements 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.050 0.05 0.1 0.15 0.2 timer=k= 1 r=k= 2 r=k= 3 r=k= 4r=k= 1 r=k= 2 r=k= 3 r=k= 4PSfrag replacements10−1 10−2 10−3 0 0 .05 0.1 0.15 0.2 time Figure 9.4. Left plot: Decay of energies /ba∇dbl∇m(t)/ba∇dblL2(Ω)3for the ap- proximations to the unknown solution with m0andHgiven in (9.3) and one line after (9.3). We plot four approximations of the k-step method with polynomial degree rforr=k∈ {1,2,3,4}. The spa- tial mesh-size is 1 /40 and the size of the timesteps is 10−3(blue) and 10−2(red). Right plot: Deviation from unit length /ba∇dbl1−|m(t)|2/ba∇dblL∞(Ω) plotted over time for step sizes τ= 10−2(blue),τ= 10−3(red), and τ= 10−4(green). The solid lines indicate k= 1, whereas the dashed lines indicate k= 2. The spatial mesh-size is 1 /40 withr= 1. then the error e=u−u⋆satisfies, in this linear case, an equation of the same form, (∂te,v)+a(e,v) = (d,v)∀v∈V, with initial value e0=u0−u⋆ 0. Testing with v=eyields 1 2d dt|e|2+/ba∇dble/ba∇dbl2= (d,e). Estimating the right-hand side by ( d,e)/lessorequalslant/ba∇dbld/ba∇dbl⋆/ba∇dble/ba∇dbl/lessorequalslant1 2/ba∇dbld/ba∇dbl2 ⋆+1 2/ba∇dble/ba∇dbl2, with the dual norm/ba∇dbl·/ba∇dbl⋆, and integrating from time 0 to tresults in the error bound |e(t)|2/lessorequalslant|e(0)|2+/integraldisplayt 0/ba∇dbld(s)/ba∇dbl2 ⋆ds. On the other hand, testing with v=∂teyields |∂te|2+1 2d dt/ba∇dble/ba∇dbl2= (d,∂te), which leads similarly to the error bound /ba∇dble(t)/ba∇dbl2/lessorequalslant/ba∇dble(0)/ba∇dbl2+/integraldisplayt 0|d(s)|2ds. This procedure is all-familiar, but it is not obvious how to extend it to tim e dis- cretizations beyond the backward Euler and Crank–Nicolson metho ds. The use of energy estimates for BDF methods relies on the following remarkable results. Lemma 10.1. (Dahlquist [18]; see also [8] and [27, Section V.6]) Letδ(ζ) =δkζk+ ···+δ0andµ(ζ) =µkζk+···+µ0be polynomials of degree at most k(and at leastHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 43 one of them of degree k)that have no common divisor. Let (·,·)be an inner product with associated norm |·|.If Reδ(ζ) µ(ζ)>0for|ζ|<1, then there exists a positive definite symmetric matrix G= (gij)∈Rk×ksuch that forv0,...,v kin the real inner product space, /parenleftigk/summationdisplay i=0δivk−i,k/summationdisplay j=0µjvk−j/parenrightig /greaterorequalslantk/summationdisplay i,j=1gij(vi,vj)−k/summationdisplay i,j=1gij(vi−1,vj−1). In combination with the preceding result for the multiplier µ(ζ) = 1−ηkζ,the following property of BDF methods up to order 5 becomes important . Lemma 10.2. (Nevanlinna & Odeh [34]) Fork/lessorequalslant5,there exists 0/lessorequalslantηk<1such that forδ(ζ) =/summationtextk ℓ=11 ℓ(1−ζ)ℓ, Reδ(ζ) 1−ηkζ>0for|ζ|<1. The smallest possible values of ηkare η1=η2= 0, η3= 0.0836, η4= 0.2878, η5= 0.8160. Precise expressions for the optimal multipliers for the BDF methods of orders 3,4 and 5 are given by Akrivis & Katsoprinakis [1]. An immediate consequence of Lemma 10.2 and Lemma 10.1 is the relation (10.2)/parenleftigk/summationdisplay i=0δivk−i,vk−ηkvk−1/parenrightig /greaterorequalslantk/summationdisplay i,j=1gij(vi,vj)−k/summationdisplay i,j=1gij(vi−1,vj−1) with a positive definite symmetric matrix G= (gij)∈Rk×k; it is this inequality that plays a crucial role in our energy estimates, and the same inequality f or the inner producta(·,·). The errorequationfortheBDFtimediscretization ofthelinear para bolicproblem (10.1) reads (˙en,v)+a(en,v) = (dn,v)∀v∈V,where ˙en=1 τk/summationdisplay j=0δjen−j, with starting errors e0,...,ek−1. When we test with v=en−ηken−1, the first term can be estimated from below by (10.2), the second term is bounded f rom below by (1−1 2ηk)/ba∇dblen/ba∇dbl2−1 2ηk/ba∇dblen−1/ba∇dbl2, and the right-hand term is estimated from above by the Cauchy-Schwarz inequality. Summing up from ktonthen yields the error bound (10.3) |en|2+τn/summationdisplay j=k/ba∇dblej/ba∇dbl2/lessorequalslantCk/parenleftigk−1/summationdisplay i=0/parenleftbig |ei|2+τ/ba∇dblei/ba∇dbl2/parenrightbig +τn/summationdisplay j=k/ba∇dbldj/ba∇dbl2 ⋆/parenrightig , whereCkdepends only on the order kof the method. This kind of estimate for the BDF error has recently been used for a variety of linear and non linear parabolic problems [33, 3, 2, 30].44 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH On the other hand, when we first subtract ηktimes the error equation for n−1 from the error equation with nand then test with ˙ en, we obtain (˙en−ηk˙en−1,˙en)+a(en−ηken−1,˙en) = (dn−ηkdn−1,˙en). Here, thesecond termis boundedfrombelow by (10.2)withthe a(·,·)inner product, the first term is bounded from below by (1 −1 2ηk)|˙en|2−1 2ηk|˙en−1|2, and the right- hand term is estimated from above by the Cauchy–Schwarz inequalit y. Summing up fromktonthen yields the error bound (10.4) /ba∇dblen/ba∇dbl2+τn/summationdisplay j=k|˙ej|2/lessorequalslantCk/parenleftigk−1/summationdisplay i=0/ba∇dblei/ba∇dbl2+τn/summationdisplay j=k|dj|2/parenrightig . It is this type of estimate that we use in the present paper for the n onlinear problem considered here. It has previously been used in [29]. Acknowledgment. The work of Michael Feischl, Bal´ azs Kov´ acs and Christian Lu- bichissupportedbyDeutscheForschungsgemeinschaft –projec t-id 258734477–SFB 1173. References 1. G. Akrivis and E. Katsoprinakis, Backward difference formulae :new multipliers and stability properties for parabolic equations , Math. Comp. 85(2016) 2195–2216. 2. G. Akrivis, B. Li, and C. 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Teubner, Stuttgart, 1997. 33. C. Lubich, D. Mansour, and C. Venkataraman, Backward difference time discretisation of parabolic differential equations on evolving surfaces , IMA J. Numer. Anal. 33(2013) 1365– 1385. 34. O. Nevanlinna and F. Odeh, Multiplier techniques for linear multistep methods , Numer. Funct. Anal. Optim. 3(1981) 377–423. 35. D.Praetorius,M.Ruggeri,andB.Stiftner, Convergence of an implicit-explicit midpoint scheme for computational micromagnetics , Comput. Math. Appl. 75(2018) 1719–1738. 36. A. Prohl, Computational Micromagnetism , Teubner, Stuttgart, 2001. 37. D. Suess, V. Tsiantos, T. Schrefl, J. Fidler, W. Scholz, H. Fors ter, R. Dittrich, and J. J. Miles, Time resolved micromagnetics using a preconditioned time i ntegration method , J. Magn. Magn. Mater.248(2002) 298–311. 38. V.Thom´ ee, Galerkin Finite Element Methods for Parabolic Problems , 2nded., Springer–Verlag, Berlin, 2006. 39. D. Thonig and J. Henk, Gilbert damping tensor within the breathing Fermi surface m odel: anisotropy and non-locality , New J. Phys. 16(2014) 013032.46 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH Department of Computer Science & Engineering, University o f Ioannina, 45110 Ioannina, Greece, and Institute of Applied and Computation al Mathematics, FORTH, 70013 Heraklion, Crete, Greece E-mail address :akrivis@cse.uoi.gr Institute for Analysis and Scientific Computing (E 101), Te chnical University Wien, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria E-mail address :michael.feischl @kit.edu E-mail address :michael.feischl @tuwien.ac.at Mathematisches Institut, Universit ¨at T¨ubingen, Auf der Morgenstelle, D-72076 T¨ubingen, Germany E-mail address :kovacs@na.uni-tuebingen.de Mathematisches Institut, Universit ¨at T¨ubingen, Auf der Morgenstelle, D-72076 T¨ubingen, Germany E-mail address :lubich@na.uni-tuebingen.de

2019-03-13

For the Landau--Lifshitz--Gilbert (LLG) equation of micromagnetics we study linearly implicit backward difference formula (BDF) time discretizations up to order $5$ combined with higher-order non-conforming finite element space discretizations, which are based on the weak formulation due to Alouges but use approximate tangent spaces that are defined by $L^2$-averaged instead of nodal orthogonality constraints. We prove stability and optimal-order error bounds in the situation of a sufficiently regular solution. For the BDF methods of orders $3$ to~$5$, this requires %a mild time step restriction $\tau \leqslant ch$ and that the damping parameter in the LLG equations be above a positive threshold; this condition is not needed for the A-stable methods of orders $1$ and $2$, for which furthermore a discrete energy inequality irrespective of solution regularity is proved.

Higher-order linearly implicit full discretization of the Landau--Lifshitz--Gilbert equation

1903.05415v2

Gilbert damping in metallic ferromagnets from Schwinger-Keldysh field theory: Intrinsically nonlocal and nonuniform, and made anisotropic by spin-orbit coupling Felipe Reyes-Osorio and Branislav K. Nikoli´ c∗ Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA (Dated: March 1, 2024) Understanding the origin of damping mechanisms in magnetization dynamics of metallic ferro- magnets is a fundamental problem for nonequilibrium many-body physics of systems where quantum conduction electrons interact with localized spins assumed to be governed by the classical Landau- Lifshitz-Gilbert (LLG) equation. It is also of critical importance for applications as damping affects energy consumption and speed of spintronic and magnonic devices. Since the 1970s, a variety of linear-response and scattering theory approaches have been developed to produce widely used for- mulas for computation of spatially-independent Gilbert scalar parameter as the magnitude of the Gilbert damping term in the LLG equation. The largely unexploited for this purpose Schwinger- Keldysh field theory (SKFT) offers additional possibilities, such as to rigorously derive an extended LLG equation by integrating quantum electrons out. Here we derive such equation whose Gilbert damping for metallic ferromagnets is nonlocal —i.e., dependent on all localized spins at a given time—and nonuniform , even if all localized spins are collinear and spin-orbit coupling (SOC) is absent. This is in sharp contrast to standard lore, where nonlocal damping is considered to emerge only if localized spins are noncollinear—for such situations, direct comparison on the example of magnetic domain wall shows that SKFT-derived nonlocal damping is an order of magnitude larger than the previously considered one. Switching on SOC makes such nonlocal damping anisotropic , in contrast to standard lore where SOC is usually necessary to obtain nonzero Gilbert damping scalar parameter. Our analytical formulas, with their nonlocality being more prominent in low spatial dimensions, are fully corroborated by numerically exact quantum-classical simulations. I. INTRODUCTION The celebrated Landau-Lifshitz equation [1] is the foundation of standard frameworks, such as classical mi- cromagnetics [2, 3] and atomistic spin dynamics [4], for modelling the dynamics of local magnetization within magnetic materials driven by external fields or currents in spintronics [2] and magnonics [3]. It considers localized spins as classical vectors M(r) of fixed length normalized to unity whose rotation around the effective magnetic fieldBeffis governed by ∂tM=−M×Beff+M×(D ·∂tM), (1) where ∂t≡∂/∂t. Although spin is a genuine quan- tum degree of freedom, such phenomenological equation can be fully microscopically justified from open quantum many-body system dynamics where M(r) tracks the tra- jectories of quantum-mechanical expectation value of lo- calized spin operators [5] in ferromagnets, as well as in antiferromagnets as long as the spin value is sufficiently large S >1. The presence of a dissipative environment in such justification invariably introduces damping mecha- nisms, which were conjectured phenomenologically in the earliest formulation [1], as well as in later renderings us- ing the so-called Gilbert form of damping [6, 7] written as the second term on the right-hand side (RHS) of Eq. (1). The Gilbert damping Dwas originally considered as a spatially uniform scalar D ≡αG, or possibly tensor [8, 9], ∗bnikolic@udel.edudependent on the intrinsic properties of a material. Its typical values are αG∼0.01 in standard ferromagnetic metals [10], or as low as αG∼10−4in carefully designed magnetic insulators [11] and metals [12]. Furthermore, recent extensions [13–21] of the Landau-Lifshitz-Gilbert (LLG) Eq. (1) for the dynamics of noncollinear magneti- zation textures find Dto be a spatially nonuniform and nonlocal tensor Dαβ=αGδαβ+ηX β′(M×∂β′M)α(M×∂β′M)β,(2) where ∂β′≡∂/∂β′, and α, β, β′∈ {x, y, z}. It is generally believed that αGisnonzero only when SOC [22, 23] or magnetic disorder (or both) are present [15, 24, 25]. For example, αGhas been ex- tracted from a nonrelativistic expansion of the Dirac equation [22, 23], and spin-orbit coupling (SOC) is vir- tually always invoked in analytical (conducted for sim- plistic model Hamiltonians) [26–28] or first-principles calculations [24, 25, 29–33] of αGvia Kubo linear- response [9, 30, 34–36] or scattering [8] theory-based for- mulas. The second term on the RHS of Eq. (2) is the particular form [13] of the so-called nonlocal (i.e., magnetization-texture-dependent) and spatially nonuni- form (i.e., position-dependent) damping [13–21, 37]. The search for a proper form of nonlocal damping has a long history [19, 37]. Its importance has been revealed by ex- periments [10] extracting very different Gilbert damping for the same material by using its uniformly precessing localized spins versus dynamics of its magnetic domain walls, as well as in experiments observing wavevector- dependent damping of spin waves [38]. Its particulararXiv:2306.13013v4 [cond-mat.mes-hall] 29 Feb 20242 B L lead R leadxyz MnJsd e ee (a) (b) (c) FIG. 1. Schematic view of (a) classical localized spins, mod- eled by unit vectors Mn(red arrows), within an infinite metal- lic ferromagnet defined on a cubic lattice in 1D–3D (1D is used in this illustration); or (b) finite-size metallic ferromag- net (central region) attached to semi-infinite NM leads termi- nating in macroscopic reservoirs, whose difference in electro- chemical potentials inject charge current as commonly done in spintronics. The localized spins interact with conduction electron spin ⟨ˆs⟩(green arrow) via sd-exchange of strength Jsd, while both subsystems can experience external magnetic fieldB(blue arrow). (c) Nonlocal damping λD nn′[Eq. (10)] obtained from SKFT vs. distance |rn−rn′|between two sites nandn′of the lattice for different dimensionality Dof space. form [13] in Eq. (2) requires only noncollinear and non- coplanar textures of localized spins, so it can be nonzero even in the absence of SOC, but its presence can greatly enhance its magnitude [18] (without SOC, the nonlocal damping in Eq. (2) is estimated [18] to be relevant only for small size ≲1 nm noncollinear magnetic textures). However, recent quantum-classical and numerically ex- act simulations [39, 40] have revealed that αGcan be nonzero even in the absence of SOC simply because ex- pectation value of conduction electron spin ⟨ˆs⟩(r) isal- ways somewhat behind M(r). Such retarded response of electronic spins with respect to motion of classical lo- calized spins, also invoked when postulating extended LLG equation with phenomenological time-retarded ker- nel [41], generates spin torque ∝ ⟨ˆs⟩(r)×M(r) [42] and, thereby, effective Gilbert-like damping [39–41] that is nonzero in the absence of SOC and operative even if M(r) at different positions rarecollinear [40]. Including SOC in such simulations simply increases [43] the an- gle between ⟨ˆs⟩(r) and M(r) and, therefore, the effective damping. To deepen understanding of the origin of these phe- nomena observed in numerical simulations, which are analogous to nonadiabatic effects discussed in diversefields where fast quantum degrees of freedom interact with slow classical ones [44–47], requires deriving an an- alytical expression for Gilbert damping due to interac- tion between fast conduction electrons and slow local- ized spins. A rigorous path for such derivation is offered by the Schwinger-Keldysh nonequilibrium field theory (SKFT) [48] which, however, remains largely unexplored for this problem. We note that a handful of studies have employed SKFT to study small systems of one or two localized spins [49–54] as they interact with conduction electrons. While some of these studies [49, 53, 54] also arrive at extended LLG equation with nonlocal damp- ing, they are only directly applicable to small magnetic molecules rather than macroscopic ferromagnets in the focus of our study. It is also worth mentioning that an early work [55] did apply SKFT to the same model we are using—electrons whose spins interact via sdexchange interaction with many Heisenberg-exchange-coupled lo- calized spins representing metallic ferromagnet in self- consistent manner—but they did not obtain damping term in their extended Landau-Lifshitz equation, and in- stead focused on fluctuations in the magnitude of Mn. In contrast, the vectors Mnare of fixed length in classical micromagnetics [2, 3] and atomistic spin dynamics [4], as well as in our SKFT-derived extended LLG Eq. (9) and all other SKFT-based analyses of one or two localized spin problems [49–54]. In this study we consider either an infinite [Fig. 1(a)], or finite [Fig. 1(b)] but sandwiched between two semi- infinite normal metal (NM) leads terminating in macro- scopic electronic reservoirs [8, 52, 53], metallic magnet whose localized spins are coupled by ferromagnetic ex- change in equilibrium. The setups in Fig. 1 are of di- rect relevance to experiments [10, 38] on external field [Fig. 1(a)] or current-driven dynamics [Fig. 1(b)] of lo- calized spins in spintronics and magnonics. Our princi- pal result is encapsulated by Fig. 1(c)—Gilbert damping, due to conduction electron spins not being able to instan- taneously follow changes in the orientation of classical localized spins, is always nonlocal and inhomogeneous, with such features becoming more prominent in low- dimensional ferromagnets. This result is independently confirmed [Fig. 2] by numerically exact simulations (in one dimension) based on time-dependent nonequilibrium Green’s function combined with LLG equation (TD- NEGF+LLG) scheme [40, 43, 56, 57]. We note that conventional linear-response formulas [9, 30, 34–36] produce unphysical divergent Gilbert damp- ing [33] in a perfectly crystalline magnet at zero tempera- ture. In contrast to previously proposed solutions to this problem—which require [58–60] going beyond the stan- dard picture of electrons that do not interact with each other, while interacting with classical localized spins— our formulas are finite in the clean limit, as well as in the absence of SOC. The scattering theory [8] yields a formula for αGwhich is also always finite (in the absence of SOC, it is finite due to spin pumping [61]). However, that result can only be viewed as a spatial average of our3 nonlocal damping which cannot produce proper LLG dy- namics of local magnetization [Fig. 3]. The paper is organized as follows. In Sec. II we for- mulate the SKFT approach to the dynamics of local- ized spins interacting with conduction electrons within a metallic ferromagnet. Sections III A and III B show how this approach leads to nonlocal and isotropic, or nonlocal and anisotropic, damping in the presence or ab- sence of SOC, respectively. The SKFT-derived analyt- ical results are corroborated by numerically exact TD- NEGF+LLG simulations [40, 43, 56, 57] in Sec. III C. Then, in Secs. III D and III E we compare SKFT-derived formulas with widely used scattering theory of conven- tional scalar Gilbert damping [8, 61, 62] or spin-motive force (SMF) theory [13, 19] of nonlocal damping, respec- tively. Finally, in Sec. III F, we discuss how to com- bine our SKFT-derived formulas to first-principles calcu- lations on realistic materials via density functional theory (DFT). We conclude in Sec. IV. II. SCHWINGER-KELDYSH FIELD THEORY FOR METALLIC FERROMAGNETS The starting point of SKFT is the action [48] of metal- lic ferromagnet, S=SM+Se, SM=Z CdtX nh ∂tMn(t)·An− H[Mn(t)]i ,(3a) Se=Z CdtX nn′h ¯ψn(t) i∂t−γnn′
ψn′(t) (3b) −δnn′JsdMn(t)·sn′(t)i , where SMis contribution from localized spins and Seis contribution from conduction electrons. The integrationR Cis along the Keldysh closed contour C[48]. Here the subscript nlabels the site of a D-dimensional cubic lat- tice;∂tMn·Anis the Berry phase term [63, 64]; H[Mn] is the Hamiltonian of localized spins; ψn= (ψ↑ n, ψ↓ n)T is the Grassmann spinor [48] for an electron at site n;γnn′=−γis the nearest-neighbor (NN) hopping; sn=¯ψnσψnis the electronic spin density, where σis the vector of the Pauli matrices; and Jsdis the magni- tude of sdexchange interaction between flowing spins of conduction electrons and localized spins. For simplicity, we use ℏ= 1. The Keldysh contour C, as well as all functions defined on it, can be split into forward (+) and backward ( −) segments [48]. These functions can, in turn, be rewritten asM± n=Mn,c±1 2Mn,qfor the real-valued localized spins field, and ψ± n=1√ 2(ψ1,n±ψ2,n) and ¯ψ± n=1√ 2(¯ψ2,n± ¯ψ1,n) for the Grassmann-valued fermion fields ψnand¯ψn. The subscripts candqrefer to the classical and quantum components of time evolution. This rewriting yields thefollowing expressions for the two actions SM=Z dtX nMα nq ϵαβγ∂tMβ n,cMγ nc+Bα eff[Mn,c]
,(4a) Se=Z dtdt′X nn′¯ψσ nˇG−1 nn′δσσ′−JsdˇMα nn′σα σσ′
ψσ′ n′,(4b) where subscript σ=↑,↓is for spin; summation over repeated Greek indices is implied; ψ≡(ψ1, ψ2)T; Beff=−δH/δMis the effective magnetic field; ϵαβγis the Levi-Civita symbol; and ˇOare 2×2 matrices in the Keldysh space, such as ˇGnn′= GRGK 0GA nn′,ˇMα nn′= McMq 2Mq 2Mcα nδnn′. (5) Here GR/A/K nn′(t, t′) are electronic re- tarded/advanced/Keldysh Green’s functions (GFs) [48] in the real-space representation of sites n. The electrons can be integrated out [49] up to the sec- ond order in Jsdcoupling, thereby yielding an effective action for localized spins only Seff M=Z dtX nMα n,qh ϵαβγ∂tMβ n,cMγ n,c+Bα eff[Mn,c] +Z dt′X n′Mα n′,c(t′)ηnn′(t, t′)i , (6) where ηnn′(t, t′) = iJ2 sd GR nn′(t, t′)GK nn′(t′, t) +GK nn′(t, t′)GA nn′(t′, t) , (7) is the non-Markovian time-retarded kernel. Note that terms that are second order in the quantum fluctuations Mn,qare neglected [48] in order to write Eq. (6). The magnetization damping can be explicitly extracted by analyzing the kernel, as demonstrated for different ferro- magnetic setups in Secs. III A and III B. III. RESULTS AND DISCUSSION A. Nonlocality of Gilbert damping in metallic ferromagnets in the absence of SOC Since ηnn′(t−t′) depends only on the difference t−t′, it can be Fourier transformed to energy ε. Thus, the kernel can be written down explicitly for low energies as ηnn′(ε) =J2 sdiε 2πX k,qeik·(rn−rn′)eiq·(rn−rn′)Ak(µ)Aq(µ), (8) where Ak(µ)≡i[GR k(µ)−GA k(µ)] is the spectral func- tion [52] evaluated at chemical potential µ;kis a4 wavevector; and rnandrn′are the position vectors of sites nandn′. Equation (8) remains finite in the clean limit and for low temperatures, so it evades unphysical divergences in the linear-response approaches [58–60]. By transforming it back into the time domain, we minimize the effective action in Eq. (6) with respect to the quan- tum fluctuations to obtain semiclassical equations of mo- tion for classical localized spins. This procedure is equiv- alent to the so-called large spin approximation [65, 66] or a one loop truncation of the effective action. The higher order terms neglected in Eq. (6) contribute a stochas- tic noise that vanishes in the low temperature and large spin limit. Although the fluctuating effect of this noise can modify the exact dynamics [54, 65], the determinis- tic regime suffices for a qualitative understanding and is often the main focus of interest [66, 67]. Thus, we arrive at the following extended LLG equa- tion ∂tMn=−Mn×Beff,n+Mn×X n′λD nn′∂tMn′,(9) where the conventional αGMn×∂tMnGilbert term is replaced by the second term on the RHS exhibit- ing nonlocal damping λD nn′instead of Gilbert damping scalar parameter αG. A closed expression for λD nn′can be obtained for one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) metallic ferromagnets by considering quadratic energy-momentum dispersion of their conduction electrons λD nn′= 2J2 sd πv2 Fcos2(kF|rn−rn′|) 1D , k2 FJ2 sd 2πv2 FJ2 0(kF|rn−rn′|) 2D , k2 FJ2 sd 2πv2 Fsin2(kF|rn−rn′|) |rn−rn′|2 3D.(10) Here kFis the Fermi wavevector of electrons, vFis their Fermi velocity, and J0(x) is the 0-th Bessel function of the first kind. B. Nonlocality and anisotropy of Gilbert damping in metallic ferromagnets in the presence of SOC Taking into account that previous analytical calcu- lations [26–28] of conventional Gilbert damping scalar parameter always include SOC, often of the Rashba type [68], in this section we show how to generalize Eq. (8) and nonlocal damping extracted in the presence of SOC. For this purpose, we employ the Rashba Hamil- tonian in 1D, with its diagonal representation given by, ˆH=P kσεkσˆc† kσˆckσ, where ˆ c† kσ/ˆckσcreates/annihilates an electron with wavenumber kand spin σoriented along they-axis, εkσ=−2γcosk+ 2σγSOsinkis the Rashba spin-split energy-momentum dispersion, and γSOis the strength of the Rashba SOC coupling. By switching from second-quantized operators ˆ c† kσ/ˆckσto Grassmann- valued two-component fields [64] ¯cσ n/cσ n, where cσ n= FIG. 2. (a) Time evolution of two localized spins Mn, lo- cated at sites n= 1 and n′= 3 within a chain of 19 sites in the setup of Fig. 1(b), computed numerically by TD- NEGF+LLG scheme [40, 43, 56, 57]. The two spins are collinear at t= 0 and point along the x-axis, while mag- netic field is applied along the z-axis. (b) The same infor- mation as in panel (a), but for two noncollinear spins with angle ∈ {0,45,90,135,180}between them. (c) and (d) Ef- fective damping extracted from TDNEGF+LLG simulations (red dashed line) vs. the one from SKFT [black solid line plots 1D case in Eq. (10)] as a function of the site n′of the second spin. The two spins are initially parallel in (c), or antiparallel in (d). The Fermi wavevector of conduction electrons is cho- sen as kF=π/2a, where ais the lattice spacing. (cσ 1,n, cσ 2,n)T, we obtain for the electronic action Se=Z dtdt′X nn′¯cσ n (ˇGσ nn′)−1δσσ′−JsdˇMα nn′σβ σσ′ cσ′ n′. (11) Here ˇGσ nn′is diagonal, but it depends on spin through εkσ. In addition, ˇMx,y,z nn′, as the matrix which couples to the same σx,y,zPauli matrix in electronic action without SOC [Eq. (3b)], is coupled in Eq. (11) to a different Pauli matrix σy,z,x. By integrating electrons out up to the second order in Jsd, and by repeating steps analogous to those of Sec. II while carefully differentiating the spin-split bands, we find that nonlocal damping becomes anisotropic λ1D nn′= α⊥ nn′0 0 0α∥ nn′0 0 0 α⊥ nn′. . (12)5 where α⊥ nn′=J2 sd πcos2(k↑ F|rn−rn′|) v↑ F2+cos2(k↓ F|rn−rn′|) v↓ F2 , (13a) α∥ nn′=J2 sd π|v↑ Fv↓ F| cos (k↑ F+k↓ F)|rn−rn′| (13b) + cos (k↑ F−k↓ F)|rn−rn′| , andk↑/↓ Fandv↑/↓ Fare the Fermi wavevectors and veloc- ities, respectively, of the Rashba spin-split bands. This means that the damping term in Eq. (9) is now given by Mn×P n′λ1D nn′·∂tMn′. We note that previous experimental [69], numeri- cal [9, 70], and analytical [26–28] studies have also found SOC-induced anisotropy of Gilbert damping scalar pa- rameter. However, our results [Eqs. (12) and (13)] ex- hibit additional feature of nonlocality (i.e., damping at sitendepends on spin at site n′) and nonuniformity (i.e., dependence on |rn−rn′|). As expected from Sec. III A, nonlocality persists for γSO= 0, i.e., k↑ F=k↓ F=kF, with λ1D nn′properly reducing to contain αnn′three di- agonal elements. Additionally, the damping component α∥ nn′given by Eq. (13b) can take negative values, re- vealing the driving capability of the conduction electrons (see Sec. III C). However, for realistic small values of γSO, the driving contribution of nearby localized spins is like- wise small. Furthermore, the decay of nonlocal damping with increasing distance observed in 2D and 3D, together with the presence of intrinsic local damping from other sources, ensures that the system tends towards equilib- rium. C. Comparison of SKFT-derived formulas with numerically exact TDNEGF+LLG simulations An analytical solution to Eq. (9) can be obtained in few special cases, such as for two exchange-uncoupled lo- calized spins at sites n= 1 and n′̸= 1 within 1D wire placed in an external magnetic field Bext=Bextez, on the proviso that the two spins are collinear at t= 0. The same system can be simulated by TDNEGF+LLG scheme, so that comparing analytical to such numeri- cally exact solution for trajectories Mn(t) makes it pos- sible to investigate accuracy of our derivation and ap- proximations involved in it, such as: truncation to J2 sd order; keeping quantum fluctuations Mn,qto first order; and low-energy approximation used in Eq. (8). While such a toy model is employed to verify the SKFT-based derivation, we note that two uncoupled localized spins can also be interpreted as macrospins of two distant ferro- magnetic layers within a spin valve for which oscillatory Gilbert damping as a function of distance between the layers was observed experimentally [71]. Note that semi- infinite NM leads from the setup in Fig. 1(b), always usedin TDNEGF+LLG simulations to ensure continuous en- ergy spectrum of the whole system [40, 56], can also be included in SKFT-based derivation by using self-energy ΣR/A k(ε) [52, 72] which modifies the GFs of the central magnetic region in Fig. 1(b), GR/A k= (ε−εk−ΣR/A k)−1, where εk=−2γcosk. The TDNEGF+LLG-computed trajectory M1(t) of lo- calized spin at site n= 1 is shown in Figs. 2(a) and 2(b) using two localized spins which are initially collinear or noncollinear, respectively. For the initially parallel [Fig. 2(a)] or antiparallel localized spins, we can ex- tract Gilbert damping from such trajectories because Mz 1(t) = tanh¯λ1D nn′Bextt/(1 + ( ¯λ1D nn′)2)
[4, 40], where the effective damping is given by ¯λ1D nn′=λ1D 00±λ1D nn′ (+ for parallel and −for antiparallel initial condition). The nonlocality of such effective damping in Figs. 2(c) and 2(d) manifests as its oscillation with increasing sep- aration of the two localized spins. The same result is predicted by the SKFT-derived formula [1D case in Eq. (10)], which remarkably closely traces the numeri- cally extracted ¯λ1D nn′despite approximations involved in SKFT-based analytical derivation. Note also that the two localized spins remain collinear at all times t, but damping remains nonlocal. The feature missed by the SKFT-based formula is the decay of ¯λ1D nn′with increasing |rn−rn′|, which is present in numerically-extracted effec- tive damping in Figs. 2(c) and 2(d). Note that effective drastically reduced for antiparallel initial conditions, due to the driving capabilities of the conduction electrons, in addition to their dissipative nature. For noncollinear ini- tial conditions, TDNEGF+LLG-computed trajectories become more complicated [Fig. 2(b)], so that we can- not extract the effective damping λ1D nn′akin to Figs. 2(c) and 2(d) for the collinear initial conditions. D. Comparison of SKFT-derived formulas with the scattering theory [8] of uniform local Gilbert damping The scattering theory of Gilbert damping αGwas formulated by studying a single domain ferromagnet in contact with a thermal bath [8]. In such a setup, energy [8] and spin [61] pumped out of the system by time-dependent magnetization contain information about spin-relaxation-induced bulk [8, 62] and interfa- cial [61] separable contributions to αG, expressible in terms of the scattering matrix of a ferromagnetic layer attached to two semi-infinite NM leads. For collinear lo- calized spins of the ferromagnet, precessing together as a macrospin, scattering theory-derived αGis a spatially- uniform scalar which can be anisotropic [62]. Its expres- sion is equivalent [62] to Kubo-type formulas [9, 34–36] in the linear response limit, while offering an efficient al- gorithm for numerical first-principles calculations [24, 25] that can include disorder and SOC on an equal footing. On the other hand, even if all localized spins are ini- tially collinear, SKFT-derived extended LLG Eq. (9) pre-6 FIG. 3. (a) Comparison of trajectories of localized spins Mz n(t), in the setup of Fig. 1(b) whose central region is 1D metallic ferromagnet composed of 5 sites, using LLG Eq. (9) with SKFT-derived nonlocal damping (solid red lines) vs. LLG equation with conventional spatially-independent αG= 0.016 (black dashed line). This value of αGis ob- tained by averaging nonlocal damping over the whole ferro- magnet. The dynamics of Mn(t) is initiated by an external magnetic field along the z-axis, while all five localized spins point along the x-axis at t= 0. (b) Comparison of spin cur- rentISz R(t) pumped [56, 57, 61] by the dynamics of Mn(t) for the two cases [i.e., nonuniform Mn(t) for nonlocal vs. uniform Mn(t) for conventional damping] from panel (a). The Fermi wavevector of conduction electrons is chosen as kF=π/2a. dicts that due to nonlocal damping each localized spin will acquire a distinct Mn(t) trajectory, as demonstrated by solid red lines in Fig. 3(a). By feeding these trajec- tories, which are affected by nonlocal damping [1D case in Eq. (10)] into TDNEGF+LLG simulations, we can compute spin current ISz R(t) pumped [56, 57] into the right semi-infinite lead of the setup in Fig. 1(b) by the dynamics of Mn(t). A very similar result for pumped spin current is obtained [Fig. 3(b)] if we feed identical Mn(t) trajectories [black dashed line in Fig. 3(a)] from conventional LLG equation with Gilbert damping scalar parameter, αG, whose value is obtained by averaging the SKFT-derived nonlocal damping over the whole ferro- magnet. This means that scattering theory of Gilbert damping [8], which in this example is purely due to inter- facial spin pumping [61] because of lack of SOC and dis- order (i.e., absence of spin relaxation in the bulk), would predict a constant αGthat can only be viewed as the spatial average of SKFT-derived nonlocal and nonuni- form λ1D nn′. In other words, Fig. 3 reveals that different types of microscopic magnetization dynamics Mn(t) can yield the same total spin angular momentum loss into the external circuit, which is, therefore, insufficient on its own to decipher details (i.e., the proper form of ex- tended LLG equation) of microscopic dynamics of local magnetization. 1.5 2.0 2.5 3.0 w/a024vDW(aJ/¯ h)×10−2 αG= 0.1 Eq.(9) Ref.[13] withη= 0.05 Ref.[19] withη= 0.05 Eq.(1) withη= 0(a) 0 25 50 75 Site i−1.0−0.50.00.51.0Mα(t)t = 410 ¯ h/J α=x,y,z(b) 0 25 50 75 Site i−2−101(Mn×/summationtext n/primeλd nn/prime·∂tMn/prime)α×10−2 (c) 0 25 50 75 Site i−2−101(M×D·∂tM)α×10−3 (d)FIG. 4. (a) Comparison of magnetic DW velocity vDWvs. DW width wextracted from numerical simulations using: ex- tended LLG Eq. (9) with SKFT-derived nonlocal damping [Eq. (10), red line]; extended LLG Eq. (1) with SMF-derived in Ref. [13] nonlocal damping [Eq. (2), blue line] or SMF- derived nonlocal damping (green line) in Ref. [19] [with ad- ditional term when compared to Ref. [13], see Eq. (14)]; and conventional LLG Eq. (1) with local Gilbert damping [i.e., η= 0 in Eq. (2), black line]. (b) Spatial profile of DW within quasi-1D ferromagnetic wire at time t= 410 ℏ/J, where Jis exchange coupling between Mnat NN sites, as obtained from SKFT-derived extended LLG Eq. (9) with nonlocal damping λ2D nn′[Eq. (10)]. Panels (c) and (d) plot the corresponding spa- tial profile of nonlocal damping across the DW in (b) using SKFT-derived expression [Eqs. (9) and Eq. (10)] vs. SMF- derived [13] expression [second term on the RHS of Eq. (2)], respectively. E. Comparison of SKFT-derived formulas with spin motive force theory [13] and [19] of nonlocal damping The dynamics of noncollinear and noncoplanar magne- tization textures, such as magnetic DWs and skyrmions, leads to pumping of charge and spin currents assumed to be captured by the spin motive force (SMF) the- ory [16, 73, 74]. The excess angular momentum of dy- namical localized spins carried away by pumped spin cur- rent of electrons appears then as backaction torque [57] exerted by nonequilibrium electrons onto localized spins or, equivalently, nonlocal damping [13, 17–19]. From this viewpoint, i.e., by using expressions for pumped spin cur- rent [13, 17–19], a particular form for nonlocal damp- ing [second term on the RHS of Eq. (2)] was derived in Ref. [13] from the SMF theory, as well as extended in Ref. [19] with an additional term, while also invoking a number of intuitively-justified but uncontrolled approxi- mations. In this Section, we employ an example of a magnetic field-driven DW [Fig. 4(b)] of width wwithin a quasi-7 1D ferromagnetic wire to compare its dynamics obtained by solving extended LLG Eq. (1), which includes non- local damping tensor [Eq. (2)] of Ref. [13], with the dynamics obtained by solving SKFT-derived extended LLG Eq. (9) whose nonlocal damping is different from Ref. [13]. By neglecting nonlocal damping in Eq. (2), the ferromagnetic domain wall (DW) velocity vDWis found [75] to be directly proportional to Gilbert damping αG,vDW∝ −BextwαG, assuming high external magnetic fieldBextand sufficiently small αG. Thus, the value of αG can be extracted by measuring the DW velocity. How- ever, experiments find that αGdetermined in this fashion can be up to three times larger than αGextracted from ferromagnetic resonance linewidth measurement scheme applied to the same material with uniform dynamical magnetization [10]. This is considered as a strong evi- dence for the importance of nonlocal damping in systems hosting noncollinear magnetization textures. In order to properly compare the effect of two different expressions for the nonlocal damping, we use αG= 0.1 in Eq. (1) and we add the same standard local Gilbert damping term, αGMn×∂tMn, into SKFT-derived ex- tended LLG Eq. (9). In addition, we set λ2D 00=ηin Eq. (10), so that we can vary the same parameter ηin all versions of extended LLG Eqs. (1), and (9). Note that we use λ2D nn′in order to include realistic decay of nonlo- cal damping with increasing distance |rn−rn′|, thereby assuming quasi-1D wire. By changing the width of the DW, the effective damping can be extracted from the DW velocity [Fig. 4(a)]. Figure 4(a) shows that vDW∝wre- gardless of the specific version of nonlocal damping em- ployed, and it increases in its presence—compare red, blue, and green data points with the black ones obtained in the absence of nonlocal damping. Nevertheless, the clear distinction between red, and blue or green data points signifies that our SKFT-derived nonlocal damping can be quite different from previously discussed SMF- derived nonlocal damping [13, 19], which are compara- ble regardless of the inclusion of the nonadiabatic terms. For example, the effective damping extracted from blue or green data points is D= 0.17 or D= 0.15, respec- tively, while λ2D nn′= 0.48. This distinction is further clar- ified by comparing spatial profiles of SKFT-derived and SMF-derived nonlocal damping in Figs. 4(c) and 4(d), respectively, at the instant of time used in Fig. 4(b). In particular, the profiles differ substantially in the out- of-DW-plane or y-component, which is, together with thex-component, an order of magnitude greater in the case of SKFT-derived nonlocal damping. In addition, the SKFT-derived nonlocal damping is nonzero across the whole wire , while the nonlocal damping in Eq. (2) is nonzero only within the DW width, where Mnvec- tors are noncollinear [as obvious from the presence of the spatial derivative in the second term on the RHS of Eq. (2)]. Thus, the spatial profile of SKFT-derived nonlocal damping in Fig. 4(c) illustrates how its nonzero value in the region outside the DW width does not re- quire noncollinearity of Mnvectors.Since SKFT-derived formulas are independently con- firmed via numerically exact TDNEGF+LLG simula- tions in Figs. 2(c) and 2(d), we conclude that previously derived [13] type of nonlocal damping [second term on the RHS of Eq. (2)] does not fully capture backaction of nonequilibrium conduction electrons onto localized spins. This could be due to nonadiabatic corrections [16, 19, 74] to spin current pumped by dynamical noncollinear mag- netization textures, which are present even in the ab- sence of disorder and SOC [43]. One such correction was derived in Ref. [19], also from spin current pumping ap- proach, thereby adding a second nonlocal damping term ηX β′h (M·∂β′∂tM)M×∂β′M−M×∂2 β′∂tMi ,(14) into the extended LLG Eq. (1). However, combined us- age [green line in Fig. 4(a)] of both this term and the one in Eq. (2) as nonlocal damping still does not match the effect of SKFT-derived nonlocal damping [compare with red line in Fig. 4(a)] on magnetic DW. As it has been demonstrated already in Fig. 3, the knowledge of total spin angular momentum loss carried away by pumped spin current [Fig. 3(b)], as the key input in the deriva- tions of Refs. [13, 19], is in general insufficient to decipher details of microscopic dynamics and dissipation exhibited by localized spins [Fig. 3(a)] that pump such current. F. Combining SKFT-derived nonlocal damping with first-principles calculations Obtaining the closed form expressions for the nonlocal damping tensor λnn′in Secs. III A and III B was made possible by using simplistic model Hamiltonians and ge- ometries. For realistic materials and more complicated geometries, we provide in this Section general formulas which can be combined with DFT quantities and evalu- ated numerically. Notably, the time-retarded dissipation kernel in Eq. (7), from which λnn′is extracted, depends on the Keldysh GFs. The same GFs are also commonly used in first-principles calculations of conventional Gilbert damping scalar parameter via Kubo-type formulas [29– 33]. Specifically, the retarded/advanced GFs are ob- tained from first-principles Hamiltonians ˆHDFTDFT as ˆGR/A(ε) = ε−ˆHDFT+ˆΣR/A(ϵ)−1. Here, ˆΣR/A(ε) are the retarded/advanced self-energies [52, 72] describing es- cape rate of electrons into NM leads, allowing for open- system setups akin to the scattering theory-derived for- mula for Gilbert damping [8, 62] and its computational implementation with DFT Hamiltonians [24, 25]. Since escape rates are encoded by imaginary part of the self- energy, such calculations do not require iηimaginary pa- rameter introduced by hand when using Kubo-type for- mulas [29–33] (where η→0 leads to unphysical divergent results [58–60]). Therefore, ˆHDFTcan be used as an input to compute the nonlocal damping tensor, via the8 calculation of the GFs ˆGR/A(ε) and the spectral function ˆA(ε) =iˆGR(ε)−ˆGA(ε) . For these purposes, it is convenient to separate the nonlocal damping tensor into its symmetric and anti- symmetric components, λαβ nn′=λ(αβ) nn′+λ[αβ] nn′, where the parenthesis (brackets) indicate that surrounded indices have been (anti)symmetrized. They are given by λ(αβ) nn′=−J2 sd 2πZ dε∂f ∂εTrspin σαAnn′σβAn′n , (15a) λ[αβ] nn′=−2J2 sd πZ dε∂f ∂εTrspin σαReˆGR nn′σβAn′n −σαAnn′σβReˆGR n′n +J2 sd 2πZ dε(1−2f) ×Trspin σαReˆGR nn′σβ∂An′n ∂ε−σα∂Ann′ ∂εσβReˆGR n′n , (15b) where f(ε) is the Fermi function, and the trace is taken in the spin space. The antisymmetric component either vanishes in the presence of inversion symmetry, or is of- ten orders of magnitude smaller than the symmetric one. Therefore, it is absent in our results for simple models on hypercubic lattices. As such, the nonlocal damping tensors in Eqs. (10) and (13), are fully symmetric and special case of Eq. (15a) when considering specific energy- momentum dispersions and assuming zero temperature. IV. CONCLUSIONS AND OUTLOOK In conclusion, we derived a novel formula, displayed as Eq. (15), for magnetization damping of a metallic fer- romagnet via unexploited for this purpose rigorous ap- proach offered by the Schwinger-Keldysh nonequilibrium field theory [48]. Our formulas could open a new route for calculations of Gilbert damping of realistic materials by employing first-principles Hamiltonian ˆHDFTfrom den- sity functional theory (DFT) as an input, as discussed in Sec. III F. Although a thorough numerical exploration of a small two-spin system based on SKFT was recently pursued in Ref. [54], our Eqs. (15) are not only applica- ble for large systems of many localized spins, but are also refined into readily computable expressions that depend on accessible quantities.While traditional, Kubo linear-response [9, 30, 34– 36] or scattering theory [8] based derivations produce spatially uniform scalar αG, SKFT-derived damping in Eqs. (15) is intrinsically nonlocal and nonuniform as it depends on the coordinates of local magnetization at two points in space rnandrn′. In the cases of model Hamil- tonians in 1D–3D, we reduced Eqs. (15) to analytical ex- pressions for magnetization damping [Eq. (10)], thereby making it possible to understand the consequences of such fundamental nonlocality and nonuniformity on lo- cal magnetization dynamics, such as: ( i) damping in Eq. (10) osc illates with the distance between xandx′ where the period of such oscillation is governed by the Fermi wavevector kF[Figs. 1(c), 2(c), and 2(d)]; ( ii) it always leads to nonuniform local magnetization dy- namics [Fig. 3(a)], even though spin pumping from it can appear [Fig. 3(b)] as if it is driven by usually an- alyzed [8, 61] uniform local magnetization (or, equiv- alently, macrospin); ( iii) when applied to noncollinear magnetic textures, such as DWs, it produces an order of magnitude larger damping and, therefore, DW wall velocity, than predicted by previously derived [13] non- local damping [second term on the RHS of Eq. (2)]. Remarkably, solutions of SKFT-based extended LLG Eq. (9) are fully corroborated by numerically exact TD- NEGF+LLG simulations [40, 43, 56, 57] in 1D, despite the fact that several approximations are employed in SKFT-based derivations. 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2023-06-22

Understanding the origin of damping mechanisms in magnetization dynamics of metallic ferromagnets is a fundamental problem for nonequilibrium many-body physics of systems where quantum conduction electrons interact with localized spins assumed to be governed by the classical Landau-Lifshitz-Gilbert (LLG) equation. It is also of critical importance for applications, as damping affects energy consumption and speed of spintronic and magnonic devices. Since the 1970s, a variety of linear-response and scattering theory approaches have been developed to produce widely used formulas for computation of spatially-independent Gilbert scalar parameter as the magnitude of the Gilbert damping term in the LLG equation. The largely unexploited for this purpose Schwinger-Keldysh field theory (SKFT) offers additional possibilities, such as to rigorously derive an extended LLG equation by integrating quantum electrons out. Here we derive such equation whose Gilbert damping for metallic ferromagnets is nonlocal, i.e., dependent on all localized spins at a given time, and nonuniform, even if all localized spins are collinear and spin-orbit coupling (SOC) is absent. This is in sharp contrast to standard lore, where nonlocal damping is considered to emerge only if localized spins are noncollinear; for such situations, direct comparison on the example of magnetic domain wall shows that SKFT-derived nonlocal damping is an order of magnitude larger than the previously considered one. Switching on SOC makes such nonlocal damping anisotropic, in contrast to standard lore where SOC is usually necessary to obtain nonzero Gilbert damping scalar parameter. Our analytical formulas, with their nonlocality being more prominent in low spatial dimensions, are fully corroborated by numerically exact quantum-classical simulations.

Gilbert damping in metallic ferromagnets from Schwinger-Keldysh field theory: Intrinsically nonlocal and nonuniform, and made anisotropic by spin-orbit coupling

2306.13013v4

Damped bead on a rotating circular hoop - a bifurcation zoo Shovan Dutta Department of Electronics and Telecommunication Engineering, Jadavpur University, Calcutta 700 032, India. Subhankar Ray Department of Physics, Jadavpur University, Calcutta 700 032, India. The evergreen problem of a bead on a rotating hoop shows a multitude of bifurca- tions when the bead moves with friction. This motion is studied for dierent values of the damping coecient and rotational speeds of the hoop. Phase portraits and trajectories corresponding to all dierent modes of motion of the bead are presented. They illustrate the rich dynamics associated with this simple system. For some range of values of the damping coecient and rotational speeds of the hoop, linear stability analysis of the equilibrium points is inadequate to classify their nature. A technique involving transformation of coordinates and order of magnitude arguments is pre- sented to examine such cases. This may provide a general framework to investigate other complex systems.arXiv:1201.1218v1 [physics.class-ph] 5 Jan 20122 I. INTRODUCTION The motion of a bead on a rotating circular hoop1shows several classes of xed points and bifurcations2{4. It also exhibits reversibility, symmetry breaking, critical slowing down, homoclinic and heteroclinic orbits and trapping regions. It has been shown to provide a mechanical analogue of phase transitions5. It can also operate as a one-dimensional pon- deromotive particle trap6. The rigid pendulum, with many applications, can be considered a special case of this system7,8. In this article we examine the motion of a damped bead on a rotating circular hoop. Damping alters the nature of the xed points of the system, showing rich nonlinear fea- tures. The overdamped case of this model3,9and a variant involving dry friction10has been previously studied. For certain values of the damping coecient and the rotational speed of the hoop, linear stability analysis predicts a line of xed points and some of the xed points appear as degenerate nodes. However, such xed points are borderline cases, sensitive to nonlinear terms. By transforming to polar coordinates and employing order of magnitude arguments we analyze these borderline cases to determine the exact nature of these xed points. To our knowledge, such analytical treatment does not appear in literature. The basic equations obtained for this system are quite generic and arise in other systems (e.g. electrical systems) as well. Hence, our technique may serve as a framework for investigating other more complex nonlinear systems. II. THE PHYSICAL SYSTEM A bead of mass m, moves on a circular hoop of radius a. The hoop rotates about its vertical diameter with a constant angular velocity !. The position of the bead on the hoop is given by angle , measured from the vertically downward direction ( zaxis), andis the angular displacement of the hoop from its initial position on the x-axis (Figure 1). The Lagrangian of the system with no damping is, L(;_) =ma2 2(_2+!2sin2) +mga cos;where!=_is a constant : (1) Using the Euler-Lagrange equation, the equation of motion is obtained as, = sin(!2cosg=a): (2)3 Φ ΘΩ xz y FIG. 1. Schematic gure of a bead sliding on a rotating hoop showing the angles and. To include friction, a term b_is introduced in (2) as, = sin(!2cosg=a)b_; (3) wherebis the damping coecient. We identify !2 c=g=aas the critical speed of rotation of the hoop, and write k=!2=!2 c,=b=!c. Dening=!ct, (3) may be made dimensionless by changing from tto, 00= sin(kcos1)0;whered2 d2=00: (4) For phase plane analysis, we dene a new variable 1=0, and write (4) as, 0=1 (5) 10=sin(1kcos)1: (6) The parameter kcan take only positive values whereas may be either positive or negative. Due to the symmetry of the hoop about its vertical axis, (5) and (6) remain invariant under the transformations !;  1!1. This implies that alternate quadrants of the 1plane have similar trajectories. Similarly, it is easily veried that if ( (t);1(t)) is a solution for positive damping ( >0), then for negative damping ( <0), ((t);1(t)) and ((t);1(t)) are two solutions. The phase portrait of the system for negative damping will just be the re ection of the positive damping phase portrait with the arrows reversed. Hence we conne our attention to 2[0;] and>0.4 When there is no damping11, the xed points are at (0 ;0) and (;0) for 0k1, whereas for k > 1, an additional xed point appears at ( 1= cos1(1=k);0). Damping changes the nature of xed points and not their number or location. III. NATURE OF THE FIXED POINT (0;0) The Jacobian matrix at (0 ;0) is obtained by Taylor expanding (5) and (6) about (0 ;0) and retaining the linear terms. J(0;0) =0 @0 1 k11 A: (7) Let and
denote the trace and determinant of the above matrix. 1. Whenk>1, both and
are negative. The xed point is a saddle with eigenvalues and eigenvectors given by, 1;2=1 2;v1;2=0 @1 (1)=21 A; (8) where1=p 2+ 4(k1). Saddles are robust and do not get perturbed by non- linearities. Thus, (0 ;0) will remain a saddle even if nonlinear terms are taken into account (see Figures 8, 9 and 16). For= 0, both1and2equalp k1. Ask!1+,1!0+and2!, which means that the saddle will start looking like a line of xed points along the direction ofv1with solutions decaying along v2. 2. For 0k <1, =is negative, whereas
= 1 kis positive. When there is no damping, the point (0 ;0) is a center. As is increased, the center transforms into a stable spiral for <2p 1kas shown in Figure 2(a). The frequency of spiralling is p 1k2=4. As!2p 1k,!0+. For > 2p 1k, the xed point (0;0) transforms to a stable node (Figure 2(b)). For = 2p 1k, it is a degenerate node. However, degenerate nodes are borderline cases and are sensitive to nonlinear terms.5 3. Fork= 1, (5) and (6) simplify to, 0=1 (9) 0 1=sin(1cos)1: (10) In the linearized dynamics, 1decays exponentially as et. In the phase plane, all trajectories move along a straight line with slope and stop on reaching the axis. However, the inclusion of nonlinear terms changes this situation. (a) stable spiral, <2p 1k (b) stable node, >2p 1k FIG. 2. Phase trajectories around (0 ;0) fork= 0:91 showing (a) stable spiral for = 0:3 and (b) stable node for = 1. A. Nature of (0;0)with nonlinearities 1. 0k<1and<2p 1k: To include the eect of nonlinear terms, let us dene two new variables, =rcos;  1=rsin (11) Equations (5) and (6) then may be written as, r0=r[cossin2p 1ksin2]sinsin(rcos)[1kcos(rcos)] (12) 0=p 1ksin(2)sin2cos rsin(rcos)[1kcos(rcos)] (13) We wish to examine the xed point(s) in the rplane corresponding to (0 ;0) in the1 plane, to determine their true nature. Strictly speaking, (12) and (13) are meaningful only6 whenr >0. Neither nor0have any meaning when r= 0. Hence, we may assign any arbitrary function f() to0atr= 0 without altering physical predictions. However, (13) describes accurately the approach to r= 0 (if any) in the rplane at arbitrarily small scales. We therefore set f() equal to the limiting value of 0asr!0. f() = lim r!00=p 1ksin(2) +k 2cos(2) 1k 2
(14) Equations (12) and (13) are periodic in with period . Hence, the phase portrait in the rplane is periodic along the -axis with period . This means that in the 1plane, the phase space is symmetric about (0 ;0). Using the identity (p 1k)2+ (k=2)2= (1k=2)2, one can write f() in the form, f() = 1k 2 [cos(2+)1]; (15) where= 2 tan1p 1k,n= 0;1;2;:::. Therefore, xed points (0 ;) in ther plane, where f() = 0, are given by  n=ntan1p 1kwithn= 0;1;2;:::. These correspond to the point (0 ;0) in the1plane. The xed points (0 ; n) in therplane are separated by n(wherenis any integer). Hence, in the 1plane, there are no trajectories that can approach (0 ;0) along two independent directions. So we can say that (0 ;0) in the1plane cannot be a stable node. In the rplane, close to some xed point (0 ; n), if there exist trajectories that approach this point and stop there, the corresponding xed point (0 ;0) in the1plane cannot be a stable spiral. For a spiral, !1 asr!0. As (12) and (13) are periodic inwith period , the nature of all xed points on the -axis separated by is identical. So we may choose to investigate (0 ;=tan1p 1k). Linearization about this point incorrectly predicts the whole axis to be a line of xed points. So, we must include the eects of the nonlinear terms. Let =. For small r, we may write (12) and (13) as, r0=rhp 1k 1k 2
sin(2)i +O(r3) (16) 0= 1k 2 [1cos(2)] +O(r2) (17) Consider an initial condition, r(= 0) =r0and(= 0) =0, where 0 <  0< tan1(p 1k=2). For any nite positive value of 1 k,0is nite and <1=2. There- fore,r0may be chosen suciently small, 0 <r 00, so as to make all terms of O(r2) and7 higher, negligible compared to the leading terms in (16) and (17). When these terms are neglected, (16) and (17) may be solved to yield, () = tan1h1 (2k)+ cot0i (18) r() =r0sin0p [(2k)+ cot0]2+ 1 exp(p 1k) (19) According to this solution, the trajectory monotonically approaches the point (0 ;0) in the rplane as!1 . This behaviour will hold even with inclusion of nonlinear terms, provided, the terms independent of rand ofO(r) remain dominant over the entire trajectory. The following arguments establish that it is indeed so. First, the trajectory cannot reach the axis at a positive value of . This is because on theaxis,r0= 0 and0=f()<0 in between two xed points. Thus, there is already a trajectory running along the axis directed towards = 0. For the initial point ( r0;0), with the choice 0 <r 00<1=2, bothrandwill start to decrease as per (16) and (17). Hence r2will become more negligible compared to r. Also, for all 00, theindependent term, namely, p 1kr, in (16), will be dominant. However, if decays more rapidly, such that at some stage r, then theO(r2) term will contribute on the same scale as the rst term in (17), which is O(2). Similarly, the O(r3) term will contribute on the same scale as the 2nd term in (16) if in the course of decay, at some point r2. However, such situations will never arise as is shown below. Let us assume that r0=20 01 and that decreases very rapidly, such that, at some instance,r=2. Asrhas decreased monotonically from its initial value, we must have =r1=2< r1=2 0. However, r1=2 0=10 01. Hence, along the entire trajectory, up to this instance, terms of O(r2) and higher are negligible compared to the leading terms in (16) and (17). Thus the solutions (18) and (19) are valid and give the correct orders of magnitudes of the dynamical quantities. As 0<  < 10 01 and 0<  0<1=2, we may write, tan and tan00. Hence, 0<tan <tan100, which implies tan  <(p 1k=2)101. Combining this with (18) and using the fact that (2 k)1, we get&(cot0)10, a very large quantity. Meanwhile (18) and (19) together imply, r 2r0sin0h3 exp(p 1k)i (20) Bothr0sin0and the quantity within brackets are 1, which implies that r2. This is8 in contradiction to the initial assumption that r=2. Therefore, we conclude that starting with the prescribed initial condition, rwill never become equal to 2. This ensures that along the entire trajectory, the terms independent of rand ofO(r) remain the dominant terms in (16) and (17). Both randwill decrease monotonously toward their respective zero values. Neither can the trajectory cross the curve r=2nor reach the axis before  becomes zero. Thus, the trajectories in the rplane must approach (0 ;0) tangential to theaxis and slow to a halt there. In the1plane, the above arguments imply that, trajectories exist which start at a nite distance from (0 ;0) and reach this point along a line of slope p 1k. Also, no other such line with a dierent slope exists. These facts clearly establish that (0 ;0) is a stable degenerate node (Figure 3). (a) stable degenerate node, = 0:6 (b) central region magnied FIG. 3. Phase trajectories around (0 ;0) fork= 0:91 and= 2p 1k. 2. k =1: Proceeding as before, (9) and (10) may be written in terms of randas, r0=r[cossinsin2]sinsin(rcos)[1cos(rcos)] (21) 0=cossinsin2cos rsin(rcos)[1cos(rcos)] (22) f() is dened as, f() = lim r!00=cossinsin2 (23) The phase portrait is periodic in with period . The xed points in the rplane of the form (0, ) are given by,  1=n and 2=ntan1 n= 0;1;2;:::: (24)9 Fortan1<< 0,f()>0 whereas for 0 <<tan1,f()<0. The positive and negative nature of f() repeats periodically along the axis. The Jacobian at the points (0 ;ntan1) given by, J(0;2) =0 @0 01 A (25) is traceless and has a negative determinant
= 2. So, this family of xed points are saddles having stable manifold along raxis and unstable manifold along axis. Linear analysis of the family of xed points (0 ;n), incorrectly predicts =nto be lines of xed points. Let us examine the point (0 ;0) in therplane for simplicity. Consider the condition, 0jjrminf1;g (26) If (26) holds, then neglecting terms of O(r3) and smaller in (21) and (22), we may write, r0=r+1 (27) 0=r2 22+2 (28) where12rand23or2r2, whichever is larger. Note that as long as (26) is satised, both 1and2can at most be of the order of r3. Let us take the initial condition 0= 0 and 0< r 01. Then,0(t= 0) =r2 0=2 and r0(t= 0) = 0. Hence, will start decreasing and become negative. As a result, r0will become negative and remain so until orrvanishes, provided (26) remains true. It is seen that as long as the trajectory is above the curve =r2=2, (26) is satised and both r0 and0are negative. Therefore, the trajectory approaches and eventually crosses this curve, where0is still negative, being of the order of 2. Let us consider the `trapping region' in Figure 4, which shows the phase ow on the curves=rand=r2=2. At any point on the line =rfor whichrminf1;g, 0 r0= r+O(1) Asr,j0=r0j1. Thus the phase ow is almost vertically upward, as shown in Figure 4. Everywhere inside the region, (26) is satised and hence r0<0 and nite. Consequently, after entering the region at point P, the trajectory must constantly move towards left. Again, it cannot penetrate the curves APorAB, because other trajectories are actually10 FIG. 4. Trapping Region. owing inward across them. Hence, we have trapped it. Upon arrival at any point on the arcAP, a trajectory must inevitably land up at (0 ;0). Note that at any point on the line =mr, (0<m1), 0 r0=m(m2+ 1=2)r+O(m3r2) mr+O(m2r2): For any nite value of m, this approaches 1 asr!0, meaning that the phase ow is almost vertically upward on any line of non-zero slope near (0 ;0). Therefore, the trajectory must reach (0 ;0) along the raxis. Thus, the xed points (0 ;n) in therplane are stable nodes having slow eigenvector alongraxis and fast eigenvector along axis. In between these, lie the saddle points (0;ntan1) (Figure 5). (a) stable node and saddle points (b) region near (0 ;0) magnied FIG. 5. Phase portrait in the rplane. These results from the rplane mean that in the 1plane, two trajectories exist11 which reach (0 ;0) along the line of slope and all other neighbouring trajectories reach it along theaxis. In other words, (0 ;0) is a stable node here. (a) stable node (b) central region magnied FIG. 6. Phase trajectories about (0 ;0) in the1plane fork= 1 and= 0:5. Figure 7a shows the nature of the xed point at (0 ;0) over the entire parameter space. (a) (0;0), curve is = 2p 1k (b) ( 1;0), curve is = 2p k1=k FIG. 7. Nature of xed points (0 ;0) and ( 1;0) over the kplane IV. NATURE OF THE FIXED POINT ( 1;0) This xed point exists when k1. Linearization of (5) and (6) gives us the Jacobian at ( 1;0) as, J( 1;0) =0 @0 1 1 kk
1 A (29)12 For 0<2q k1 k, ( 1;0) is a stable spiral with eigenvalues given by, = 2ir (k1 k)2 4 Trajectories spiral in with an angular frequency p (k1=k)2=4, while their radial distance decreases as et=2. As!0, this decay rate vanishes and ( 1;0) turns into a center (Figure 8). Also, vanishes as !2p k1=k, representing a smooth transition to a stable node, similar to the behaviour of the xed point (0 ;0). When > 2q k1 k, (a) center,= 0 (b) stable spiral, = 0:1 FIG. 8. Phase trajectories around (0 ;0) fork= 1:1 and 0<2q k1 k. 24
>0, hence ( 1;0) is a stable node (Figure 9), with eigenvalues and eigenvectors given by, 1;2=3 2;v1;2=0 @1 (3)=21 A; where3=p 24(k1=k). Both1and2approach the value p k1=kas! 2p k1=k+, indicating a stable degenerate node. For = 2q k1 k, 24
= 0. In the linear stability analysis, ( 1;0) is a stable degenerate node with a single eigenvector, v=0 @1 p k1=k1 A; (30) corresponding to the eigenvalue =p k1=k. However, degenerate nodes can be trans- formed into stable nodes or stable spirals due to perturbation introduced by nonlinear terms.13 (a) stable node, = 1 (b) stable node, = 2:5 FIG. 9. Phase trajectories around (0 ;0) fork= 1:1 and>2q k1 k. A. Nature of ( 1;0)with nonlinearities As discussed in subsection III A, (5) and (6) may be transformed to equations in rand andwith the substitutions,  1=rcosand1=rsin. Dene, f() = lim r!00=r k1 ksin(2)1 2(k1 k1) cos(2)1 2(k1 k+ 1) (31) f() is negative at all points on the axis except at the xed points given by (0 ;) with =n, where= tan1p k1=k, (n= 0;1;2;:::), where it is zero. These xed points are separated by n. Then, by the same reasoning as used in III A, we can argue that (0;0) cannot be a stable node. The remaining possibilities are a spiral or a degenerate node. Let us consider the xed point (0 ;), and let=+. Then we may expand r0and 0uptoO(r), r0=1 2 k1 k+ 1 sin(2)sin(2)
r+O(r2) (32) 0= k1 k+ 1 sin23 2rr 11 r2cos3 9 4r 11 k2cossin(2)r+O(r2) (33) We choose an initial point, ( r0;0), and 0< r 00< minf1;g. This choice ensures that bothrandwill start decreasing. In the course of this monotonic decay, rcannot reach zero before becomes zero, as there is a straight line trajectory moving downward14 along theaxis. From (32) and (33) we note that r0and0each contains an independent term ofO(r). Therefore, as randdecrease toward their respective zero values, the 1st order approximation gets even better. However, if at some stage, r, then the O(r2) terms would contribute on the same scale as some of the terms of O(r) in (32) and (33). But the following argument rules out such a possibility. Let us consider a specic case and choose r0=20 0. Ifis to become O(r), at some stage, we must have r=2. However, for r=2, we have r0=r k1 k2+O(3) 0=2 4(k1 k+ 1) +3 2q 11 k2 (k1 k+ 1)3 23 52+O(3) whereas along the curve r=2,dr=d = 2. Therefore, for a given value of k >1, we can always select an 0, suciently small, for which, at all points on the curve r=2contained between= 0 and=0, the ratior0=0>dr=d . This would guarantee that the trajectory cannot penetrate down this curve, which means that cannot reach zero before rdoes. Thus, for a suitable choice of initial conditions, the trajectory must slow to a halt at ( r= 0;= 0). In the1plane, this means that there exist trajectories which start at a nite distance from ( 1;0) and reach it along the line of slope . Also, there is no other such line with a dierent slope. Hence, ( 1;0) is a stable degenerate node (Figure 10). Figure 7b gives the (a) stable degenerate node (b) region around ( 1;0) magnied FIG. 10. Phase trajectories for k= 1:1 and= 2q k1 k. nature of the xed point at (  1;0) in dierent parts of the parameter space.15 V. NATURE OF THE FIXED POINT (;0) The Jacobian matrix at ( ,0) is given as, J(;0) =0 @0 1 k+ 11 A: (34) The xed point is a saddle for all values of kand remains so even with the inclusion of nonlinear terms. The eigenvalues and corresponding eigenvectors are given by, 1;2=2 2;v1;2=0 @1 (2)=21 A; where2=p 2+ 4(k+ 1). A. Trajectories Damping of the bead leads to some qualitatively dierent trajectories in addition to those observed for the frictionless case11. These are mainly the dierent kinds of damped oscillations (underdamped, critically damped, overdamped) about the stable equilibrium points. Some of these are illustrated with the following numerical plots. (a)k= 0:75,= 0:05,0(0) = 0 (b)k= 4,= 0:5,0(0) = 0 FIG. 11. Underdamped oscillation about (a) = 0 and (b) = 1. VI. PHASE PORTRAITS AND BIFURCATION For 0k <1, the xed point at (0 ;0) transforms its nature as the damping coecient is varied. It is a center at = 0, asincreases, it becomes a stable spiral. At = 2p 1k,16 (a)k= 0:75,= 1:5,0(0) = 0 (b)k= 4,= 4,0(0) = 0 FIG. 12. Overdamped oscillation about (a) = 0 (b)= 1. it turns into a stable degenerate node. It makes a smooth transition to a stable node as damping is increased further. Thus, a spiral-node bifurcation takes place at this critical condition (Figures 2 and 3). Physically, as damping is gradually increased from 0, the system undergoes a continuous transition from undamped oscillations of the bead about = 0 (center), to underdamped oscillations (stable spiral). At = 2p 1k, the system is critically damped (degenerate node) and becomes overdamped (stable node) as is increased further. (a) unstable spiral, =1 (b) center, = 0 (c) stable spiral, = 1 FIG. 13. Phase portraits for k= 0 showing degenerate Hopf bifurcation. For negative damping, (0 ;0) becomes an unstable spiral and changes to an unstable node asis made more negative. Consequently, as one crosses = 0, the xed point (0 ;0), undergoes a degenerate Hopf bifurcation (Figure 13). With increase in the angular speed of the hoop (i.e., k), the stability of the origin degrades continuously. When k= 1, (0;0) is a weak center. A special case of Hopf bifurcation occurs, whenis swept from negative to positive values acroos 0, keeping kxed at 1 (Figure 14).17 Askis increased beyond 1, (0 ;0) transforms from a stable (  >0) or unstable (  <0) node to a saddle. Two new stable nodes appear at  1=cos1(1=k) and branch out in opposite directions. Thus, a supercritical pitchfork bifurcation occurs at fk= 1g(Figure 15a). (a) unstable node, =0:2 (b) stable node, = 0:2 FIG. 14. Phase portraits for k= 1. (a) Supercritical bifurcation at k= 1,= 0 (b) Supercritical bifurcation at k= 1,=p 2 FIG. 15. Section of the bifurcation diagram for (a) = 0 and (b) >0. In (a) solid curve represents center, dashed curve represents saddle. In (b) solid curve represents stable node, densely dashed curve represents saddle and sparsely dashed curve represents stable spiral. Supercritical pitchfork bifurcation occurs at k= 1 and spiral-node bifurcation occurs at k= 0:5 andk= 1:28. Askincreases from 1 to 1, 1varies from 0 to =2. The xed point ( 1;0), is a center for zero damping, a stable spiral in the region 0 <  < 2p k1=k(underdamped oscillation), and becomes a stable degenerate node at critical damping = 2p k1=k. For the overdamped condition >2p k1=k, it is a stable node.18 For negative damping, we get just the unstable counterparts. Accordingly, a spiral-node bifurcation is observed at =2p k1=k(Figures 8, 9, 10 and 15)b. A degenerate Hopf bifurcation is observed for k>1 and= 0 (Figure 16). The xed points ( ;0) are saddles (a) unstable spiral, =1 (b) center, = 0 (c) stable spiral, = 1 FIG. 16. Phase portraits for k= 4 showing degenerate Hopf bifurcation. for all values of k. They have saddle connections between them at = 0, which break in opposite directions for positive and negative damping. The above observations are summarized in Figure 17 and Table I below. FIG. 17. The bifurcation diagram In Fig 17, red denotes stable node, green denotes stable spiral, blue denotes unstable spiral, yellow denotes unstable node, brown denotes saddle, pink denotes center, gray denotes unstable degenerate node, and black denotes stable degenerate node. Up to now, we have limited attention to those bifurcations resulting from a variation of k or a variation of . From Figure 17, we see that the curves 2= 2(1k),2= 2(k1=k), andk= 1 divide the parameter space into 8 distinct regions of dierent dynamics. All these19 TABLE I. Bifurcation Table. Points in parameter space Bifurcation along k Bifurcation along  Figure references 1) (k;0) ;k6= 1 { degenerate Hopf Figures 13, 16 2) (k;2p 1k) ; 0k<1 spiral-node spiral-node Figures 2-3, 15b 3) (1;0) supercritical pitchfork Hopf Figures 13b, 8a, 14 , 15a 4) (1;) ;6= 0 supercritical pitchfork { Figures 2b, 9a, 14b, 15b 5) (k;2q k1 k) ;k>1 spiral-node spiral-node Figures 8-10, 15b regions meet at the point fk= 1;= 0g. Traversing suitable curves in kspace, one can move from any one region to another, yielding new kinds of bifurcation. Following such a curve amounts to keeping a certain function (k;) constant, while varying some other function(k;). Mathematically, the possibilities are rich. But whether it is possible to actually implement this in the bead-hoop system is subject to further inquiry. However, this would attain physical signicance if there exists another system where andthemselves are the control parameters. CONCLUDING REMARKS The simple introduction of damping to the bead-hoop system enriches its dynamics and leads to various new modes of motion and dierent classes of bifurcations. We have studied this system over the entire parameter space and presented phase portraits and trajectories. This serves to illustrate the qualitative changes in the system's dynamics across dierent bi- furcation curves. We have presented exact analytical treatment of the borderline cases where linearization fails, for which no general methods are available in the literature. The method of transforming to polar coordinates and using order of magnitude arguments, employed in this article, can serve as a useful technique for other dynamical systems as well.20 REFERENCES sray.ju@gmail.com 1Goldstein H 1980-07 Classical Mechanics (Addison-Wesley, Cambridge, MA) 2Jordan D W and Smith P 1999 Nonlinear Ordinary Dierential Equations : An Introduction to Dynamical Systems (Oxford University Press, New York) 3Strogatz S 2001 Nonlinear Dynamics And Chaos: Applications To Physics, Biology, Chemistry, And Engineering (Addison-Wesley, Reading, MA) 4Marsden J E and Ratiu T S 1999 Introduction to Mechanics and Symmetry (Springer-Verlag, New York) 5Fletcher G 1997 A mechanical analogue of rst- and second-order phase transitions Am. J. Phys. 6574 6Johnson A K and Rabchuk J A 2009 A bead on a hoop rotating about a horizontal axis : A one-dimensional ponderomotive trap Am. J. Phys. 771039 7Butikov E I 1999 The rigid pendulum - an antique but evergreen physical model Eur. J. Phys. 20429 8Butikov E I 2007 Extraordinary oscillations of an ordinary forced pendulum Eur. J. Phys. 29 215 9Mancuso R V 1999 A working model for rst- and second-order phase transitions and the cusp catastrophe Am. J. Phys. 68271 10Burov A A 2009 On bifurcations of relative equilibria of a heavy bead sliding with dry friction on a rotating circle Acta Mechanica 212349 11Dutta S and Ray S 2011 Bead on a rotating circular hoop: a simple yet feature-rich dynamical system arXiv:1112.4697v1

2012-01-05

The evergreen problem of a bead on a rotating hoop shows a multitude of bifurcations when the bead moves with friction. This motion is studied for different values of the damping coefficient and rotational speeds of the hoop. Phase portraits and trajectories corresponding to all different modes of motion of the bead are presented. They illustrate the rich dynamics associated with this simple system. For some range of values of the damping coefficient and rotational speeds of the hoop, linear stability analysis of the equilibrium points is inadequate to classify their nature. A technique involving transformation of coordinates and order of magnitude arguments is presented to examine such cases. This may provide a general framework to investigate other complex systems.

Damped bead on a rotating circular hoop - a bifurcation zoo

1201.1218v1

The eects of time-dependent dissipation on the basins of attraction for the pendulum with oscillating support James A. Wright1, Michele Bartuccelli1, Guido Gentile2 1Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK 2Dipartimento di Matematica e Fisica, Universit a di Roma Tre, 00146 Roma, Italy Abstract We consider a pendulum with vertically oscillating support and time-dependent damping coecient which varies until reaching a nite nal value. Although it is the nal value which determines which attractors eventually exist, however the sizes of the corresponding basins of attraction are found to depend strongly on the full evolution of the dissipation. In particular we investigate numerically how dissipation monotonically varying in time changes the sizes of the basins of attraction. It turns out that, in order to predict the behaviour of the system, it is essential to understand how the sizes of the basins of attraction for constant dissipation depend on the damping coecient. For values of the parameters where the systems can be considered as a perturbation of the simple pendulum, which is integrable, we characterise analytically the conditions under which the attractors exist and study numerically how the sizes of their basins of attraction depend on the damping coecient. Away from the perturbation regime, a numerical study of the attractors and the corresponding basins of attraction for dierent constant values of the damping coecient produces a much more involved scenario: changing the magnitude of the dissipation causes some attractors to disappear either leaving no trace or producing new attractors by bifurcation, such as period doubling and saddle-node bifurcation. Finally we pass to the case of an initially non-constant damping coecient, both increasing and decreasing to some nite nal value, and we numerically observe the resulting eects on the sizes of the basins of attraction: when the damping coecient varies slowly from a nite initial value to a dierent nal value, without changing the set of attractors, the slower the variation the closer the sizes of the basins of attraction are to those they have for constant damping coecient xed at the initial value. Furthermore, if during the variation of the damping coecient attractors appear or disappear, remarkable additional phenomena may occur. For instance it can happen that, in the limit of very large variation time, a xed point asymptotically attracts the entire phase space, up to a zero measure set, even though no attractor with such a property exists for any value of the damping coecient between the extreme values. Keywords: action-angle variables, attractors, basins of attraction, dissipative systems, non-constant dissipation, periodic motions, simple pendulum. Mathematical Subject Classication (2000) 34C60, 34C25, 37C60, 58F12, 70K40, 70K50. 1 Introduction Consider the ordinary dierential equations x+G(x;t) + _x= 0; +F(;t) + _= 0; (1.1) 1arXiv:1403.4996v1 [math.DS] 19 Mar 2014where (x;_x)2R2and (;_)2TR, with T=R=2Z. The functions FandGare smooth and 2-periodic in time t(Fis also 2-periodic in ); the dots denote derivatives with respect to time. Equations to describe the motion of one-dimensional physical systems are often of this form, in which case the functions G(x;t) andF(;t) can be considered as an external driving force and the parameter represents the damping coecient, which we shall assume positive. First, for convenience, let us summarise some of the already known ideas regarding systems of the form (1.1), which can be found in the literature [1, 3, 5]. When is xed at zero, the system is Hamiltonian and no attractors are present. For >0 numerical experiments show that a nite set of attractors exist: this is consistent with Palis' conjecture [31, 16, 33]. The number of attractors present and the percentage of phase space covered by their basins of attraction depend upon the chosen values of the parameters (perturbation parameter "and damping coecient ), but, for all values of the parameters, the union of the corresponding basins of attraction completely ll the phase space, up to a set of zero measure. Moreover if the system is a perturbation of an integrable system (perturbation regime), all attractors found numerically turn out to be either xed points or periodic solutions with periods that are rational multiples of the forcing period (subharmonic solutions); we cannot exclude the presence of chaotic attractors [20, 15], but apparently they either do not arise or seem to be irrelevant. Generally in the literature the damping coecient is taken as constant, but in many physical systems it changes non-periodically over time. This can be due to several factors, such as the heating or cooling of a mechanical system and the wear out or rust on mechanical parts. Despite this, usually models and numerical simulations of such systems only take the nal value of dissipation into account when calculating basins of attraction. The recent paper [3] puts forward the idea that, although the nal value of dissipation determines which attractors exist, the relative sizes of their basins of attraction depend on the evolution of the dissipation. In particular the eect of dissipation increasing to some constant value over a given time span induces a signicant change to the sizes of the basins of attraction in comparison to those when dissipation is constant. Let us illustrate in more detail the phenomenology. Suppose that for two values 0and 1of the damping coecient, with 06= 1, the same set of attractors exists. Provided the dierence between the two values is suciently large, the relative sizes of the basins of attraction under the two coecients will in general be appreciably dierent. If we allow the damping coecient to depend on time, = (t), and vary from 0to 1over an initial period of time T0, after which it remains constant at the value 1, then the sizes of the basins of attraction will be dierent from those where the system has constant coecient 1throughout. Moreover if T0is taken larger, the sizes of the basins of attraction tend towards those for the system under constant = 0: this re ects the fact that the damping coecient remains close to 0for longer periods of time. Now consider two values 0and 1of the damping coecient for which the corresponding sets of attractorsA0andA1are not the same. As a system evolving under dissipation is expected to have only nitely many attractors, there can only be a nite number of attractors which exist for one of the two values and not for the other one. What happens is that, by varying (t) from 0to 1, an attractor can either appear or disappear, and in the latter case it can disappear either without leaving any trace or being replaced by a new attractor by bifurcation. Suppose, for instance, that the only dierence between A0andA1is that the attractor a02A 0simply disappears, that is A0nA1=fa0g; then, if the time T0over which (t) varies is large, each remaining attractor tends to have a basin of attraction not smaller than that it has for xed at 0: the reason being, again, that the damping coecient remains close to 0for a long time and, moreover, the trajectories which would be attracted by a0at = 0will move towards some other attractor when a0disappears. If, instead, the only dierence between the sets of attractors A0andA1is that the attractor a02A 0 is replaced by an attractor a1, say by period doubling bifurcation, then, letting (t) vary from 0to 1over a suciently large time T0causes the size of the basin of attraction of a1to tend towards that of the basin of attraction that a0has for = 0. We summarise our results by the following statements. 1. IfA0, the set of attractors at = 0, is a subset ofA1, the set of attractors which exist at = 1, that isA0A 1, then, as the time T0over which (t) is varied from 0to 1is taken larger, the basins of attraction tend towards those when is kept constant at = 0. In 2particular, if an attractor belongs to A1nA0, then the larger T0the more negligible is the corresponding basin of attraction. 2. If the set of attractors at xed = 1is a proper subset of those which exist at = 0, that isA1A 0, then, asT0is taken larger, the basins of attraction for the attractors which exist at both 0and 1change so that for (t) varying from 0to 1they tend to become greater than or equal to those for constant = 0. 3. If an attractor a0exists for = 0but is destroyed as (t) tends towards 1, and a new attractora1is created from it by bifurcation (we will explicitly investigate the case of saddle- node or period doubling bifurcations), then the size of the basin of attraction of a1, asT0is taken larger, tends towards that of a0at constant = 0. 4. IfA01is the set of attractors which exist at both = 0and = 1, that isA01=A0\A 1, and none of the elements in A0nA01are linked by bifurcation to elements in A1nA01, then, as T0is taken larger, the phase space covered by the basins of attraction of the attractors which belong toA01tends towards 100%. Moreover, all such attractors have a basin of attraction larger than or equal to that they have when the coecient of dissipation is xed at = 0. The main model used in [3] to convey some of the ideas above is a version of the forced cubic oscillator, which is of the form of the rst equation in (1.1), with G(x;t) = (1 +"cost)x3. This system, considered in the perturbation regime (both "and small), apart from the xed point and as far as the numerics fortells, exhibits only oscillatory attractors with dierent periods depending on the parameter values. Also discussed in [3] is the relevance to the spin-orbit problem, describing an asymmetric ellipsoidal satellite moving in a Keplerian elliptic orbit around a planet [28]: the corresponding equations of motion are of the form of the second equation in (1.1), with the tidal friction term (t) (_1) instead of _, with (t) slowly increasing in time because of the the cooling of the satellite. In the present paper we wish to extend the discussion to the pendulum with periodically oscil- lating support [25, 32]. The latter is a system which has been already extensively studied in the literature (we refer to [5] for a list of references): it oers a wide variety of dynamics and, because of the separatrix of the unperturbed system, in the perturbation regime, unlike the cubic oscillator, also includes rotatory attractors in addition to the oscillatory attractors. An important dierence with respect to the results in [1] is the following. In [1], if an attractor exists for some value of , it is found to exist for smaller values of too. This is not always true for the pendulum considered in the present paper, where we will see that, at least for some values of the parameters, both increas- ing and decreasing can destroy attractors as well as create new ones. However, this occurs away from the perturbation regime, where the system can no longer be considered as a perturbation of an integrable one: the appearance and disappearance of attractors would occur also in the case of the cubic oscillator for larger values of the forcing. In addition to the case of increasing dissipation studied in [3], here we also include the case where the damping coecient decreases to a constant value, which is appropriate for physical systems where joints are initially tight and require time to loosen. In this case similar phenomena are expected. For instance, as the value of variation time T0is taken larger, the amount of phase space covered by each of the basins of attraction should tend towards that corresponding to original value 0of , providing the set of attractors remains the same. The non-linear pendulum with vertically oscillating support is described by +f(t) sin+ _= 0; f (t) =g `b!2 `f0(!t) ; (1.2) wheref0is a smooth 2 -periodic function and the parameters `,b,!andgrepresent the length, amplitude and frequency of the oscillations of the support and the gravitational acceleration, re- spectively, all of which remain constant; for the sake of simplicity we shall take f0(!t) = cos(!t) in (1.2), as in [5, 6, 7]. As mentioned above, the parameter represents the damping coecient, 3which, for analysis where it remains constant, we shall model as =Cn"n, where"is small and n is an integer. We shall consider (1.2) as a pair of coupled rst order non-autonomous dierential equations by letting x=andy= _x, such that the phase space is TRand the system can be written as _x=y; _y=g `b!2 `cos (!t) sinx y: The system described by (1.2) can be non-dimensionalised by taking =g `!2;  =b `;  =!t; so that it becomes 00+f() sin+ 0= 0; f () = (cos); (1.3) or, written as a system of rst order dierential equations, x0=y; y0=f() sinx y; (1.4) where the dashes represent dierentiation with respect to the new time and has been normalised so as not to contain the frequency. Linearisation of the system about either xed point results in a system of the form of Mathieu's equation, see for instance [27]. When the downwards xed point is linearly stable, it is possible, for certain parameter values, to prove analytically the conditions for which the xed point attracts a full measure set of initial conditions; see Appendix A. In the Sections that follow we shall use the non-dimensionalised version of the system (1.3), preferable for numerical implementation as it reduces the number of parameters in the system. In Section 2 we detail the calculations of the threshold values for the attractors, that is the values of constant below which periodic attractors exist in the perturbation regime (small ). As we shall see, because of the presence of the separatrix for the unperturbed pendulum, this will be of limited avail for practical purposes: the persisting periodic solutions found to rst order are in general too close to the separatrix for the perturbation theory to converge. In Section 3 we present numerical results, in the case of both constant and non-constant (either increasing or decreasing) dissipation, for values of the parameters in the perturbation regime. Since for such values the downwards position turns out to be stable, we shall refer to this case as the downwards pendulum. Next, in Section 4 we perform the numerical analysis for values of the parameters for which the upwards position is stable (hence such a case will be referred to as the inverted pendulum). Of course such parameter values are far away from the perturbation regime: as a consequence additional phenomena occur, including period doubling and saddle-node bifurcations. In Section 5 we include a discussion of numerical methods used. Finally in Section 6 we draw our conclusions and brie y discuss some open problems and possible directions for future investigation. 2 Thresholds values for the attractors The method used below to calculate the threshold values of below which given attractors exist follows that described in [1, 3], where it was applied to the damped quartic oscillator and the spin- orbit model. We consider the system (1.4), with ="and =C1", where";C1>0. This approach is well suited to compute the leading order of the threshold values. In general, it would be preferable to write as a function of "of the form =C1"+C2"2+:::(bifurcation curve), and x the constants Ckby imposing formal solubility of the equations to any perturbation order, see [18]; however this only produces higher order corrections to the leading order value. For"= 0 the system reduces to the simple pendulum 00+sin= 0, which admits periodic solutions inside the separatrix (librations or oscillations) and outside the separatrix (rotations). In terms of the variables ( x;y) the equations (1.4) become x0=y,y0=sinx: the librations are described by( xosc() = 2 arcsin [ k1sn (p(0);k1)]; yosc() = 2k1pcn (p(0);k1);k1<1; (2.1) 4while the rotations are described by ( xrot() = 2 arcsin [sn (p(0)=k2;k2)]; yrot() = 2k1 2pdn (p(0)=k2:k2);k2<1; (2.2) where cn (
;k), sn(
;k) and dn(
;k) are the Jacobi elliptic functions with elliptic modulus k[8, 11, 26, 35], and k1andk2are such that k2 1= (E+)=2andk2 2= 1=k2 1, withEbeing the energy of the pendulum. From (2.1) and (2.2) it can be seen that the solutions are functions of ( 0), so that the phase of a solution depends on the initial conditions. We can x the phase of the solution to zero without loss of generality by instead writing f() in equation (1.4) as f(0). This moves the freedom of choice in the initial condition to the phase of the forcing. The dynamics of the simple pendulum can be conveniently written in terms of action-angle variables (I;'), for which we obtain two sets of variables: for the librations inside the separatrix one expresses the action as I=8 ph (k2 11)K(k1) +E(k1)i ; (2.3) where K(k) and E(k) are the complete elliptic integrals of the rst and second kind, respectively, and writes x= 2 arcsin k1sn2K(k1) ';k1 ; y = 2k1pcn2K(k1) ';k1 ; (2.4) withk1obtained by inverting (2.3), while for the rotations outside the separatrix one expresses the actions as I=4 k2pE(k2); (2.5) and writes x= 2 arcsin snK(k2) ';k2 ; y =2 k2pdnK(k2) ';k2 ; (2.6) withk2obtained by inverting (2.5); further details can be found in Appendix B. For"small, in order to compute the thresholds values, we rst write the equations of motion for the perturbed system in terms of the action-angle coordinates ( I;') of the simple pendulum, then we look for solutions in the form of power series expansions in ", I() =1X n=0"nI(n)(); ' () =1X n=0"n'(n)(); (2.7) whereI(0)() and'(0)() are the solutions to the unperturbed system, that is, see Appendix B, (I(0)();'(0)()) = (Iosc;'osc()) and (I(0)();'(0)()) = (Irot;'rot()), in the case of oscillations and rotations, respectively, with Iosc=8 ph (k2 11)K(k1) +E(k1)i ; ' osc() = 2K(k1)p; Irot=4 k2pE(k2); ' rot() = K(k2)p k2;(2.8) and with given k1=k(0) 1andk2=k(0) 2. As the solution (2.7) is found using perturbation theory, its validity is restricted to the system where"is comparatively small. In particular this limitation has the result that the calculations of the threshold values are not valid for the inverted pendulum, where large "is required to stabilise the system. On the other hand the regime of small "has the advantage that we can characterise analytically the attractors and hence allows a better understanding of the dynamics with respect to the case of large ", where only numerical results are available. 52.1 Librations We rst write the equations of motion (1.4) in action-angle variables, see Appendix D, as '0=p 2K(k1)" 2K(k1)p sn2(
) +k2 1sn2(
) cn2(
) 1k2 1Z(
) sn(
) cn(
) dn(
) 1k2 1 cos(0) +C1"cn(
) 2K(k1)sn(
) dn(
)+k2 1sn(
) cn2(
) (1k2 1) dn(
)Z(
) cn(
) 1k2 1 ; I0=8"k2 1K(k1) cos(0) sn(
) cn(
) dn(
)8C1"k2 1pK(k1) cn2(
);(2.9) where Z(
) is the Jacobi zeta function, see [26]. Here and throughout Section 2.1 to save clutter we dene (
) = 2K(k1) ';k1 . Note that in (2.9), the dependence on Iof the vector eld is through the variable k1, according to (2.3). The coordinates for the unperturbed system ( "= 0) satisfy '0=dE dI:= (I) =p 2K(k1); I0= 0: (2.10) Linearising around ( '(0)();I(0)()) = ( (I(0));I(0)), we have '0=@ @I(I(0))I; I0= 0; (2.11) where, see Appendix B, (I) :=@ @I(I) =2 16k2 1K3(k1)E(k1) 1k2 1K(k1) : (2.12) SinceI=I(k1), that is the action is a function of k1, settingI=I(0)xesk1=k(0) 1, yielding (I(0)) =(0), with(0)given by (2.12) with k1=k(0) 1. The linearised system (2.11) can by written in compact form as  '0 I0 = 0(0) 0 0 ' I : (2.13) The Wronskian matrix W() is dened as the solution of the unperturbed linear system W0() = 0(0) 0 0 W(); W (0) =I; where Iis the 22 identity matrix. Hence W() = 1(0) 0 1 ; (2.14) with (1;0) and ((0);1) two linearly independent solutions to (2.13). We now look for periodic solutions ( '();I()) to (2.9) with period T= 2q= 4K(k1)p=p, with p=q2Q, of the form (2.7); see also [18, 19] for a more general discussion. A solution of this kind will be referred to as a p:qresonance. The functions ( '(n)();I(n)()) are formally obtained by introducing the expansions (2.7) into the equations (2.9) and equating the coecients of order n. This leads to the equations ('(n))0 (I(n))0 = (0)I(n) 0 + F(n) 1() F(n) 2()! (2.15) 6withF(n) 1() andF(n) 2() given by F(n) 1() =p 2K(k1)(0)I(n) +"  2K(k1)p sn2(
) +k2 1 1k2 1sn2(
) cn2(
) Z(
) 1k2 1sn(
) cn(
) dn(
) cos(0) +C1cn(
) 2K(k1)sn(
) dn(
)+k2 1sn(
) cn2(
) (1k2 1) dn(
)Z(
) cn(
) 1k2 1#(n1) ; F(n) 2() =" 8k2 1K(k1) cos(0) sn(
) cn(
) dn(
)8C1k2 1pK(k1) cn2(
)#(n1) : The notation [ :::](n)means that one has to take all terms of order nin"of the function inside [:::]. By construction, F(n) 1() andF(n) 2() depend only on the coecients '(p)() andI(p)(), with p<n , so that (2.15) can be solved recursively. Then, by using the Wronskian matrix (2.14), see [30], one can integrate (2.15) so as to obtain '(n)() I(n)() =W()'(n) I(n) +W()Z 0dW1() F(n) 1() F(n) 2()! (2.16) where '(n)and I(n)are thenthorder in the "-expansion of the initial conditions for 'andI, respectively. In the last term of (2.16) we have W()Z 0dW1() F(n) 1() F(n) 2()! =Z 0dW() F(n) 1() F(n) 2()! : This yields '(n)() = '(n)+(0)I(n)+Z 0dF(n) 1() +(0)Z 0dZ 0d0F(n) 2(0); I(n)() =I(n)() +Z 0dF(n) 2():(2.17) For a periodic function g, let us denote the average of gwithhgiand the zero-average function ghgiwith g. Suppose that hF(n) 2i:=1 TZT 0dF(n) 2() = 0; (2.18) whereT= 4K(k1)p; we will check later on the validity of (2.18). Then we may write F(n) 1() =Z 0dF(n) 1() =hF(n) 1i+Z 0dF(n) 1(); F(n) 2() =Z 0F(n) 2() d=Z 0dF(n) 2(); and subsequently rewrite (2.17) as '(n)() = '(n)+(0)I(n)+hF(n) 1i+Z 0dF(n) 1() +(0)hF(n) 2i+(0)Z 0dF(n) 2(); I(n)() =I(n)+Z 0dF(n) 2(); in which all the terms which are not linear in are periodic. If we choose our initial conditions I(n) such that they satisfy I(n)=1 (0)hF(n) 1ihF(n) 2i; 7the above reduces to '(n)() = '(n)() +Z 0dF(n) 1() +(0)Z 0dF(n) 2(); I(n)() =I(n)+Z 0dF(n) 2(~); so that both '(n)() andI(n)() are periodic functions with period T, provided (2.18) holds. Lemma 1 Consider the series (2.7) . If p=q= 1=2m,m2NandC1is small enough, then it is possible to x the initial conditions ( '(n);I(n))in such a way that (2.18) holds for all n1. If p=q6= 1=2mfor allm2N, then (2.18) can be satised only for C1= 0. Proof Forn= 1 we have F(1) 2() =8k2 1K(k1) cos(0) sn(p;k 1) cn(p;k 1) dn(p;k 1) 8C1k2 1pK(k1) cn2(p;k 1); withk1=k(0) 1here and henceforth. Moreover set, see Appendix E,
:=p 4K(k1)Z4K(k1)=p 0dcn2(p;k 1) =1 2K(k1)Z2K(k1) 0dcn2(;k1) =1 k2 11 2K(k1)E(2K(k1);k1)(1k2 1) ;(2.19) where E(u;k) is the incomplete elliptic integral of the second kind, and 1(0;p;q) := sin(0)G1(p;q), with G1(p;q) =1 TZT 0sn(p;k 1) cn(p;k 1) dn(p;k 1) sin() =1 4K(k1)pZ4K(k1)p 0dsn(;k1) cn(;k1) dn(;k1) sin(=p):(2.20) Under the resonance condition =2K(k1) = p=q, one has sin(=p) = sin 2K(k1)q p ; where pand qare relatively prime. By expanding the Jacobi elliptic functions in Fourier series, see Appendix E, we nd that p;qmust also satisfy the condition p (2m11)(2m21)2m3 q= 0 forG1(p;q) to be non-zero. Thus q= 2mp,m2N, that is p= 1 and q22N, andhF(1) 2i= 0 providedC1and0satisfy C1=sin(0)p
G1(p;q): (2.21) Note that the existence of a value of 0satisfying (2.21) is possible only if jC1jC1(p=q) :=1p
G1(p;q): Some values of the constants C1(p=q) for= 0:5 are listed in Table 1. 8qk1G1(1=q)
C1(1=q) 2 0.885201568846 0.172135 0.407121 0.597944 4 0.998888384493 0.077675 0.224342 0.489649 6 0.999986981343 0.051734 0.150043 0.487616 8 0.999999846887 0.038800 0.112539 0.487578 10 0.999999998199 0.031040 0.090032 0.487577 12 0.999999999979 0.025867 0.075026 0.487577 Table 1: Constants for the oscillating attractors with = 0:5. For alln2 we can write F(n) 2() as F(n) 2() =8k2 1K(k1) @ @' cos(0) sn(
) cn(
) dn(
)pC1cn2(
)
 '='(0)'(n1)+R(n)(); whereR(n)() is a suitable function which does not depend on  '(n1). It can be seen that hF(n) 2i= 0 if and only if hR(n)i=8k2 1K(k1)  1 TZT 0d2K(k1)p@ @ sn(p) cn(p) dn(p) cos(0) pC1 TZT 0d2K(k1)p@ @ cn2(p)! '(n1): This can be rewritten as hR(n)i=16k2 1K2(k1)p2cos(0)G1(p;q) '(n1): We refer the reader to Appendix E for more details on the evaluation of the integrals. The coecient of '(n1)is non-vanishing for 0chosen such that (2.21) is satised. Therefore it is possible to x the initial conditions  '(n1)in such a way that one has hF(n) 2i= 0 at all orders, thus completing the proof of the lemma. 2 Lemma 1 implies that the threshold values of the p:qresonances are (p=q) =C1(p=q)"forp= 1 and qeven, with the constants C1(p=q) in Table 1, while the threshold values of the other resonances are at least O("2). 2.2 Rotations Similarly for the rotating scenario, again further details can be found in Appendix D, the perturbed system can be written as '0=p k2K(k2)+"k2pK(k2)k2 2sn2(
) cn2(
) 1k2 2Z(
) sn(
) cn(
) dn(
) 1k2 2 cos(0) C1" K(k2)k2 2sn(
) cn(
) dn(
) 1k2 2Z(
) dn2(
) 1k2 2 ; I0=4"K(k2) cos(0) sn(
) cn(
) dn(
)4C1"pK(k2) k2dn2(
):(2.22) Here and throughout this subsection we set (
) = K(k2) ';k2 . In this scenario, the Wronskian matrixW() can be written as in (2.14), with (0)given by, see Appendix B, (I) =2p 4K3(k2)E(k2) 1k2 2K(k2) ; (2.23) 9fork2=k(0) 2. We again look for solutions ( '();I()) with period T= 2q= 4k2K(k1)p=p corresponding to a resonance p:q, of the form (2.7), the functions '(n)() andI(n)() being dened as in (2.16), with F(n) 1() andF(n) 2() dened as F(n) 1() =p k2K(k2)(0)I(n) +" k2pK(k2)k2 2sn2(
) cn2(
) 1k2 2 Z(
) sn(
) cn(
) dn(
) 1k2 2 cos(0) C1 K(k2)k2 2sn(
) cn(
) 1k2 2Z(
) dn(
) 1k2 2#(n1) ; F(n) 2() =4K(k2) cos(0) sn(
) cn(
) dn(
)4C1pK(k2) k2dn2(
)(n1) : The theory goes through exactly as previously shown for the case of libration and we must show thathF(n) 2i= 0. Lemma 2 Consider the series (2.7) . If p=q= 1=2m,m2N, andC1is small enough, then it is possible to x the initial conditions ( '(n);I(n))in such a way that hF(n) 2i= 0 for alln1. If p=q6= 1=2mfor allm2NthenhF(n) 2i= 0only whenC1= 0. Proof One has F(1) 2() =4K(k2) cos (0) snp k2;k2 cnp k2;k2 dnp k2;k2 4C1pK(k2) k2dn2p k2;k2 ; withk2=k(0) 2here and henceforth. Dene, see Appendix E,
:=p 2k2K(k2)Z2k2K(k2)=p 0ddn2p k2;k2 =1 2K(k2)E 2K(k2);k2 : and 1(0;p;q) := sin(0)G1(p;q), where G1(p;q) =1 TZT 0dsnp k2;k2 cnp k2;k2 dnp k2;k2 sin() =1 4K(k2)pZ4K(k2)p 0dsn(;k2) cn(;k2) dn(;k2) sin(k2=p); then use the resonance condition to set sink2p = sin 2K(k2)q p : On inspection of the above we see that 1(0;p;q) can be calculated similarly to the case inside the separatrix. It follows that the same applies and p=q= 1=2mform2N. ThenhF(1) 2i= 0 if C1=k2sin(0)p
G1(p;q); (2.24) which requires jC1jC1(p=q) :=k2p
G1(p;q): 10Some values of the constants C1(p=q) for= 0:5 are listed in Table 2. qk2G1(1=q)
C1(1=q) 2 0.924397052341 0.156774 0.474414 0.432005 4 0.998899257272 0.077612 0.225808 0.485542 6 0.999986983601 0.051734 0.150063 0.487439 8 0.999999846887 0.038800 0.112540 0.487577 10 0.999999998199 0.031040 0.090032 0.487577 12 0.999999999978 0.025867 0.075026 0.487577 Table 2: Constants for the rotating attractors with = 0:5. Forn2 one has F(n) 2() =4K(k2) @ @' cos(0) sn(
) cn(
) dn(
)pC1dn2(
)
 '='(0)'(n1)+R(n)(); where again R(n)() will be a suitable function which does not depend on  '(n1). Similarly to the case of libration, hF(n) 2i= 0 if and only if hR(n)i=4K(k2)  1 TZT 0k2K(k1)p@ @ sn(p) cn(p) dn(p) cos(0) d pC1 TZT 0k2K(k1)p@ @ dn2(p) d! '(n1)=4k2K2(k2)p2cos(0)G1(p;q) '(n1); so that the coecient of  '(n1)turns out to be non-vanishing for 0chosen such that equation (2.24) is satised. Therefore it is possible to x the initial conditions  '(n1)in such a way that one has hF(n) 2()i= 0 at all orders, thus completing the proof. 2 Lemma 2 implies that the threshold values of the p:qresonances are (p=q) =C1(p=q)"forp= 1 and qeven, with the constants C1(p=q) in Table 2, while the threshold values of the other resonances are at least O("2). Note, in Tables 1 and 2 it is apparent that, for = 0:5, increasing qcauses the value of C1(1=q) to converge to 0.487577 in both cases. However this does not mean that for <0:487577"there are innitely many attracting solutions with increasing period: this would be a counter-example to Palis' conjecture! The explanation for this seeming paradox is as follows: As qincreases the solutions move closer and closer to the separatrix (this can be seen by the corresponding values of k1andk2), where the power series expansions (2.7) for the solutions I() and'() which were constructed with perturbation theory converge only for very small values of ": the larger q, the smaller must be ". In particular, for any xed "there is only a nite number of periodic solutions which can be studied by perturbation theory. In particular, for the chosen parameters the only periodic solution corresponds to the resonance 1:2 inside the separatrix. We also note that the above analysis applies only to periodic attractors with p= 1 and qeven. However we shall see that the numerical simulations provide also rotating attractors with period 2 , that is the same period as the forcing: we expect that continuing the analysis to second order and writing =C2"2, see [1], would give the threshold values for these periodic attractors. 3 Numerics for the downwards pendulum We shall investigate the system (1.3) in the same region of phase space used in [5], namely 2[;], 02[4;4] and calculate the relative areas of the basins of attraction, that is the percentage of phase space they cover relative to this region. 11Throughout this Section we x the parameters = 0:5 and= 0:1. These parameter values, also investigated in [5], correspond to a stable region of the stability tongues for the system linearised around= 0, see [22], so that the downwards conguration is stable. The chosen values for the damping coecient span values between = 0:002 and 0:06, of which only = 0:03 was previously investigated in [5]. For some values of , the system exhibits three non-xed-point attractors, examples of which are shown in Figure 1, as well as the downwards xed point attractor. Here and henceforth, for brevity, we shall say that a solution has period nif it comes back to its initial value after nperiods of the forcing. Of course the upwards xed point also exists as a solution to the system, however it is unstable and thus does not attract any non-zero measure subset of phase space. It can also be seen from Figure 1 that the attractive solutions are near the separatrix of the unperturbed system: this is evident as the curves described by the two rotating attractors are close to that of the oscillating attractor and the separatrix lies between them. This observation conrms the reasoning as to why the computation of the threshold values can only produce valid results for the period 2 oscillating attractor (see the conclusive remarks in Section 2). Figure 1: Attracting solutions for the system (1.3) with = 0:5,= 0:1 and = 0:02, namely two period 1 rotations and one period 2 solution which oscillates about the downwards xed point. Periods can be deduced by circles corresponding to the Poincar e map. For <  0, for a suitable  02(0:4;0:5), the system has three periodic attractors, in addition to the downwards xed point: one oscillating and two rotating attractors. For  0the two rotating attractors no longer exist, leaving just the oscillatory attractor and the xed point. The basins of attraction for = 0:02, 0:03, 0:04 and 0:05 are shown in Figure 2, from which we can see that the entire phase space is covered: this suggests that no other attractors exist, at least for the values of the parameters considered. The corresponding relative areas, as estimated by the numerical simulations, are given in Table 3 and plotted in Figure 3. The relative areas of the basins of attraction for positive and negative rotations have been listed in the same column: numerical simulations found a dierence in size no greater than 102% and, due to the symmetries of the system, it is expected that this dierence is numerically induced by the selection of initial conditions The basins of attraction were estimated using numerical simulations with 600 000 random initial conditions in phase space. More notes on the numerics used can be found in Section 5. It can be seen that the results in Table 3 are in agreement with the calculations for the threshold value for the period 2 oscillatory attractor. The calculations in Appendix A predict that, for the chosen values = 0:5 and= 0:1, taking >  10:1021 ensures for the origin to capture a full measure set of initial conditions. From Table 3 a stronger result emerges numerically: the xed point attracts the full phase space, up to a zero-measure set, for  20:06. Upon comparing results in Table 3, we see that, essentially, the basin of attraction for the xed point becomes smaller with an increase in , up to approximately 0 :035, after which it grows again. Similarly, by increasing the value of , the basins of attraction of the oscillating and rotating solutions attractors increase initially, up to some value (about 0 :025 and 0:035, respectively), after which they become smaller. 12Furthermore, the variations of the relative areas of the basins of attraction are never monotonic, as one observes slight oscillations for small variations of . These features seem contrary to systems such as the cubic oscillator, where decreasing dissipation seems to cause the relative area of the basin of attraction of the xed point to decrease monotonically, while the basins of attraction of the periodic attractors reach a maximum value, after which their relative areas slightly decrease, see for instance Table III in [3]. We note, however, that a more detailed investigation shows that oscillations occur also in the case of the cubic oscillator. This was already observed for some values of the parameters (see Table IX in [3]), but the phenomenon can also be observed for the parameter values of Table III, simply by considering smaller changes of the value of with respect to the values in [3]. For instance, by varying slightly around 0:0005 (see Table III in [3] for notations), one nds for the main attractors the relative areas in Table 4. (a) (b) (c) (d) Figure 2: Basins of attraction for the system (1.3) with = 0:5,= 0:1 and (a) = 0:02, (b) = 0:03, (c) = 0:04 and (d) = 0:05. The xed point (FP) is shown in blue, the positive and negative rotating solutions (PR and NR) are shown in red and yellow, respectively. and the oscillating solution (OSC) in green. In conclusion, for the pendulum, apart from small oscillations, by decreasing the value of from 0:06 to 0:002, the basins of attraction of the periodic attractors, after reaching a maximum value, becomes smaller. A similar phenomenon occurs also in the cubic oscillator (albeit less pronounced). However, a new feature of the pendulum, with respect to the cubic oscillator, is that the basin of attraction of the origin after reaching a minimum value increases again: the increase seems to be too large to be ascribed simply to an oscillation, even though this cannot be excluded. In the case of the cubic oscillator the slight decrease of the sizes of the basins of attraction of the periodic attractors was due essentially to the appearance of new attractors and their corresponding basins 13of attraction. It would be interesting to investigate further, in the case of the pendulum, how the basins of attractions, in particular that of the xed point, change by taking smaller and smaller values of . We intend to come back to this in the future [36]. Basin of attraction % FP PR/NR OSC 0.0020 84.57 3.35 8.73 0.0050 79.91 3.88 12.32 0.0100 72.24 4.60 18.57 0.0200 71.95 4.57 18.90 0.0230 70.73 5.18 18.90 0.0250 69.28 5.19 20.35 0.0300 69.94 4.42 21.23 0.0330 68.92 3.75 23.59 0.0350 68.77 3.16 24.90 0.0400 73.84 1.42 23.32 0.0500 85.61 0.00 14.39 0.0590 96.96 0.00 3.04 0.0597 98.59 0.00 1.41 0.0600 100:00 0.00 0.00 Table 3: Relative areas of the basins of attraction with = 0:5,= 0:1 and con- stant . Figure 3: Plot of the relative areas of the basins of attraction with constant as per Table 3. Basin of attraction % 0 1/2 1/4 1a 1b 1/6 3a 3b 0.00052 39.03 41.73 14.72 1.22 1.22 1.59 0.25 0.25 0.00051 39.73 41.70 13.88 1.24 1.24 1.66 0.27 0.27 0.00050 38.72 41.85 14.65 1.28 1.28 1.65 0.29 0.29 0.00049 39.26 41.96 13.81 1.29 1.29 1.77 0.27 0.32 0.00048 38.48 41.87 14.60 1.30 1.30 1.75 0.34 0.34 Table 4: Relative areas of the basins of attraction of the main attractors for the system x+ (1 +"cost)x3+ _x= 0, with"= 0:1 and around 0:0005. The basins of attraction were estimated using numerical simulations with 1000000 random initial conditions in the square [ 1;1][1;1] in phase space. 3.1 Increasing dissipation In this section we shall investigate the case where dissipation increases with time, up to a time T0, after which it remains constant. We will consider a linear increase in dissipation from a value 0at timet= 0 up to 1at timeT0, that is (see Figure 4) = (t) =( 0+ ( 1 0) T0;0 <T 0; 1; T 0:(3.1) Although this is a greatly simplied model of what might take place in reality, it serves the purpose of demonstrating the signicant eects of initially non-constant dissipation on the nal basins of attraction. Below we will consider explicitly the cases 0= 0:2 and 1= 0:3, 0:4 and 0:5. As previously mentioned, we expect that increasing the value of T0results in the relative areas of the basins of attraction moving along the curves plotted for constant . The movement along these curves starts at 1and goes towards 0. In particular, any values of the relative area of a basin of attraction for constant values of the damping coecient between 1and 0are traced as the value ofT0varies for time-dependent dissipation. This movement along the curve is not linear with 14the value of T0but asymptotic, with the relative area of the basin of attraction tending towards the value at = 0asT0!1 , providing the attractors existing at = 0also persist at 1. When the attractors which persist at = 1are a proper subset of those which exist at 0, we expect the persisting attractors to absorb the remaining phase space left by the attractors which have disappeared: thus their basins of attraction should be greater than or equal to those at = 0. For the values of the parameters in the chosen range, only these two cases may occur as the solutions which exist for = 1also exist at = 0, see Table 3. Figure 4: Plot of equation (3.1) with 0= 0:1, 1= 0:2 and varying T0. The results in Tables 5, 6 and 7 are in agreement with the expectations above. It can be seen from Tables 5 and 6 that the relative areas of the basins of attraction trace those of constant . In particular, the relative area of the basins of attraction of the rotating attractors tends towards that at = 0from above, despite having a smaller basin of attraction for the chosen values of 1. More precisely, the longer T0, the closer is the relative area of the basin of attraction to the value it has at = 0. However, the convergence to the asymptotic value is rather slow: for instance in Table 5, evenT0= 2000 is not enough to reach the values corresponding to = 0:02. The simulations for time varying dissipation have in general taken 300 000 or 400 000 initial conditions in phase space. In some cases more points were used for additional accuracy. Basin of Attraction % FP PR/NR OSC T00 69.94 4.42 21.23 25 69.80 4.42 21.36 50 69.57 4.45 21.52 75 69.40 4.64 21.33 100 68.84 4.85 21.47 200 68.82 5.10 20.99 500 69.86 5.17 19.80 1000 70.65 5.18 18.99 2000 71.17 5.11 18.61 Table 5: Relative areas of the basins of attraction with 0= 0:02, 1= 0:03 and T0varying. Figure 5: Plot of the relative areas of the basins of attraction as per Table 5. In Table 7, we see that for 0= 0:02 and 1= 0:05 only two attractors are present: indeed the oscillating attractors no longer exist for = 0:05. Hence, when (t) increases and crosses the value at which those attractors disappear, all the trajectories that up to this time were converging to them, will fall into the basins of attraction of the persisting attractors, that is the xed point and the oscillating solution. In particular the corresponding basins of attraction will acquire relative areas larger than those they have at constant = 0, because of the absorption of all these trajectories. It is dicult to predict how such trajectories are distributed among the persisting attractors. In 15the case of Table 7 they seem to be attracted slightly more by the xed point, even though the percentage increase is larger for the oscillating solution. Basin of Attraction % FP PR/NR OSC T00 73.84 1.42 23.32 25 73.66 1.44 23.45 50 73.37 1.50 23.63 75 72.15 2.22 23.41 100 68.69 3.50 24.31 200 67.46 4.76 23.03 500 69.05 5.02 20.92 1000 69.85 5.17 19.81 2000 70.63 5.18 19.02 Table 6: Relative areas of the basins of attraction with 0= 0:02, 1= 0:04 and T0varying. Figure 6: Plot of the relative areas of the basins of attraction as per Table 6. Basin of Attraction % FP OSC T00 85.61 14.39 25 86.01 13.99 50 86.18 13.82 75 84.19 15.87 100 80.42 19.58 200 75.47 24.53 500 77.95 22.06 1000 77.55 22.45 1500 78.03 21.97 Table 7: Relative areas of the basins of attraction with 0= 0:02, 1= 0:05 andT0varying. Figure 7: Plot of the relative areas of the basins of attraction as per Table 7. 3.2 Decreasing dissipation In this section we conversely look at the damping coecient decreasing from some value 0> 1, with dierent rates of decrease, see Figure 8. We will consider the cases 0= 0:04 and 1= 0:02, 0= 0:04 and 1= 0:03, 0= 0:05 and 1= 0:02. Figure 8: Plot of equation (3.1) with 0= 0:23, 1= 0:2 and varying T0. 16In this situation it is possible that more attractors exist at 1than at 0, see Table 3. We again expect that increasing T0causes the relative areas of the basins of attraction to tend towards those at 0. The result of this is that solutions which do not exist at 0will attract less and less of the phase space as T0increases, and for T0large enough their basins of attraction will tend to zero. Basin of Attraction % FP PR/NR OSC T00 71.95 4.57 18.90 25 71.85 4.60 18.94 50 72.36 4.48 18.69 75 73.64 4.43 17.51 100 74.10 4.32 17.27 200 72.31 2.99 21.71 500 71.51 2.09 24.31 1000 72.61 1.79 23.81 2000 73.11 1.63 23.64 Table 8: Relative areas of the basins of attraction with 0= 0:04, 1= 0:02 and T0varying. Figure 9: Plot of the relative areas of the basins of attraction as per Table 8. Basin of Attraction % FP PR/NR OSC T00 69.94 4.42 21.23 25 69.73 4.50 21.28 50 70.78 4.23 20.77 75 72.03 3.45 21.07 100 71.77 2.95 22.33 500 72.61 1.79 23.81 1000 73.11 1.63 23.64 Table 9: Relative areas of the basins of attraction with 0= 0:04, 1= 0:03 and T0varying. Figure 10: Plot of the relative areas of the basins of attraction as per Table 9. Basin of Attraction % FP PR/NR OSC T00 71.95 4.57 18.90 25 72.13 4.58 18.72 50 72.97 4.24 18.55 75 76.14 3.15 17.56 100 77.02 2.18 18.62 200 77.71 0.31 21.67 500 81.94 0.00 18.06 Table 10: Relative areas of the basins of attraction with 0= 0:05, 1= 0:02 andT0 varying. Figure 11: Plot of the relative areas of the basins of attraction as per Table 10. 17Tables 8 and 9 illustrate cases in which the system admits the same set of attractors for both values 0and 1of the damping coecient. An example of what happens when an attractor exists at 1but not at 0can be seen in the results of Table 10, where (t) varies from 0 :05 to 0:02. As the damping coecient starts o at a larger value, then decreases to some smaller value, we also expect the change in the basins of attraction to happen over shorter values of T0. The reasoning for this is simply that larger values of dissipation cause trajectories to move onto attractors in less time. IncreasingT0results in the system remaining at higher values of dissipation for more time and thus trajectories land on the attractors in less time. 4 Numerics for the inverted pendulum The upwards xed point of the inverted pendulum can be made stable for large values of , i.e when the amplitude of the oscillations is large relative to the length of the pendulum. In this section we numerically investigate the system (1.3) for parameter values for which this happens. For simplicity, as mentioned in the introduction, we refer to this case as the inverted pendulum. It can be more convenient to set x=+, so as to centre the origin at the upwards position of the pendulum. Then the equations of motion become 00+f() sin+ 0= 0; f () =(+cos); (4.1) =g `!2;  =b `;  =!t: The dierence between equations (1.3) and (4.1) is that here the parameter has changed sign. (a) (b) (c) (d) Figure 12: Attracting solutions for the system (4.1) with = 0:1 and= 0:545. Figure (a) shows an example of the positive and negative rotating attractors with period 2, taken for = 0:05. Figure (b) shows the positive and negative attractors with period 1 when = 0:23. Figures (c) and (d) show the oscillatory attractors with periods 2 and 4 respectively when is taken equal to 0.2725. These solutions oscillate about the downwards xed point =and the axis has been shifted to show a connected curve in phase space. The period of each solution can be deduced from the circles corresponding to the Poincar e map. 18The stability of the upwards xed point creates interesting dynamics to study numerically, how- ever it means that the system is no longer a perturbation of the simple pendulum system. This in turn has the result that the analysis in Section 2 to compute the thresholds of friction cannot be applied. However, we shall see that the very idea that attractors have a threshold value below which they always exist does not apply to the inverted pendulum: both increasing and decreasing the damping coecient can create and destroy solutions. For numerical simulations of the inverted pendulum throughout we shall take parameters = 0:1 and= 0:545, which are within the stable regime for the upwards position. For these parameter values the function f() changes sign. As such, the analysis in Appendix A cannot be applied. Again these particular parameter values were also investigated in [5], but with a small value for the damping coecient, that is = 0:08, where only three attractors appeared in the system: the upwards xed point and the left and right rotating solutions. We have opted to focus on larger dissipation because the range of values considered for allows us to incorporate already a a wide variety of dynamics, in which remarkable phenomena occur, and, at the same time, larger values of are better suited to numerical simulation because of the shorter integration times. We note that, for the values of the parameters chosen, no strange attractors arise: numerically, besides the xed points, only periodic attractors are found. For constant dissipation we provide results for 2[0:05;0:6]. These values of are considered to correspond to large dissipation, however non-xed-point solutions still persist due to the large coecientof the forcing term. Some examples of the persisting non-xed-point solutions can be seen in Figure 12; of course, the exact form of the curves depends on the particular choices of . For varying in the range considered the following attractors arise (we follow the same convention as in Section 3 when saying that a solution has period n): the upwards xed point (FP), the downwards xed point (DFP), a positively rotating period 1 solution (PR), a negatively rotating period 1 solution (NR), a positively rotating period 2 solution (PR2), a negatively rotating period 2 solution (NR2), an oscillating period 2 solution (DO2) and an oscillating period 4 solution (DO4). However, as we will see, the solution DO2 deserves a separate, more detailed discussion. (a) (b) (c) Figure 13: Basins of attraction for constant dissipation with = 0:2, 0:23 and 0:2725 from left to right respectively. The xed point (FP) is shown in blue, the positively rotating solution (PR) in red, the negatively rotating solution (NR) in yellow, the downwards oscillation with period 2 (DO2) in green and nally the downwards oscillation with period 4 (DO4) in orange. The basins of attraction corresponding to the values = 0:2, 0:23 and 0:2725 are shown in Figure 13. The relative areas of the basins of attraction for 2[0:05;0:6] are listed in Table 11. Again the positive and negative rotations have been listed together as any dierence in the size of their basins of attraction is expected to be due to numerical inaccuracies. In Figure 14, there is a large jump in the relative area of the basin of attraction for the upwards xed point (FP) between the values of = 0:22 and 0:225, approximately: this is due to the appearance of the oscillatory solution which oscillates about the downwards pointing xed point (=). For values of slightly larger than 0 :22 large amounts of phase space move close to the solution DO2, where they remain for long periods of time; however they do not land on the solution 19and are eventually attracted to FP. The percentage of phase space which does this is marked in Figure 14 by a dotted line, which becomes solid when the trajectories remain on the solution for all time (however, see comments below). Basin of Attraction % FP DFP PR/NR PR2/NR2 DO2 DO4 0.0500 4.30 0.00 0.00 47.85 0.00 0.00 0.0750 5.08 0.00 0.00 47.46 0.00 0.00 0.0900 7.41 0.00 0.00 46.30 0.00 0.00 0.1000 8.51 0.00 45.74 0.00 0.00 0.00 0.1700 49.65 0.00 25.17 0.00 0.00 0.00 0.2000 64.31 0.00 17.84 0.00 0.00 0.00 0.2230 72.09 0.00 13.95 0.00 0.00 0.00 0.2250 27.60 0.00 13.59 0.00 45.22 0.00 0.2300 25.00 0.00 12.68 0.00 49.61 0.00 0.2500 15.87 0.00 8.49 0.00 67.16 0.00 0.2690 16.50 0.00 2.13 0.00 79.25 0.00 0.2694 17.26 0.00 0.00 0.00 82.74 0.00 0.2700 17.28 0.00 0.00 0.00 82.73 0.00 0.2725 17.21 0.00 0.00 0.00 79.44 3.35 0.2800 17.30 0.00 0.00 0.00 82.70 0.00 0.2900 17.30 0.00 0.00 0.00 82.70 0.00 0.3000 16.97 0.00 0.00 0.00 83.03 0.00 0.4600 9.61 0.00 0.00 0.00 90.39 0.00 0.4700 9.80 90.20 0.00 0.00 0.00 0.00 0.5000 10.06 89.94 0.00 0.00 0.00 0.00 0.5500 10.32 89.68 0.00 0.00 0.00 0.00 0.6000 8.79 91.21 0.00 0.00 0.00 0.00 Table 11: Relative areas of the basins of attraction with constant damping coecient . The solutions are named as per Figures 12 and 13 with the addition of the downwards xed point (DFP) and the rotating period 2 solutions (PR2/NR2). For details on the DO2 solution we refer to the text. Figure 14: Relative sizes of basins of attraction with constant as per Table 11. The lines are labeled as in Table 11 and regions in which a bifurcation takes place are marked with a dot. The basin of attraction for the oscillatory solution with period 4 (DO4), has not been included due to its small size and the solutions low range of persistence with respect to . The broken lines for FP and DO2 represent areas of transition just before DO2 (and the solutions created by the period doubling bifurcation) becomes stable, see text. 20The solution DO4 listed in Table 11 is found to persist only in the interval [0 :272;0:27422], where it only attracts a small amount of the phase space. As such it has not been included in Figure 14. (a) (b) (c) (d) Figure 15: The transition from the period 2 rotating solutions to the period 1 rotating solutions. = 0:09, 0.094, 0.096 and 0.098 from (a) to (d) respectively. Numerical simulations can be used to nd estimates for the values of at which solutions appear/disappear. This is done by starting with initial conditions on a solution, then allowing the parameter to be varied to see for which value that solution vanishes. In Figure 15 the transition from the period 2 rotating solution to the period 1 rotating solution can be observed: by moving towards smaller values of , this corresponds to a period doubling bifurcation [21, 13] (period halving, if we think of as increasing). Similarly, starting on the period 1 rotating attractor and increasing further , the solution disappears at 0:2694. The same analysis can be done for the oscillatory solutions. We nd that the downwards oscilla- tory solution labeled DO2 persists for in the interval [0 :224;0:46], approximately. However such a solution is really a period 2 solution only for greater than 0:24. In the interval [0 :224;0:24] the trajectory is \thick", see Figure 16: only due to its similarity to the solution DO2 and to prevent Table 11 having yet more columns, the basin of attraction of these solutions in that range has also been listed under that of DO2. Nevertheless, by moving backwards starting from 0 :24 we have a sequence of period doubling bifurcations, corresponding to values of closer and closer to each other. A period doubling cascade is expected to lead to a chaotic attractor, which, however, may survive only for a tiny window of values of (at = 0:223 it has already denitely disappeared) and has a very small basin of attraction (for getting closer to the value 0.223 its relative area goes to zero). The appearance of chaotic attractors for small sets of parameters and with small basins of attraction has been observed in similar contexts of multistable dissipative systems close to the conservative limit [15]. For the value = 0:223 numerical simulations nd that trajectories remain in the region of phase space occupied by DO2 for a long time, before eventually moving onto the xed point. As increases further towards 0:46, the amplitude of the period 2 oscillatory solution gradually decreases and taking larger causes a slow spiral into the downwards xed point, which now becomes stable. (a) (b) (c) (d) Figure 16: The solution DO2 for dierent values of time independent . As the damping coecient is increased, the solution becomes more clearly dened: this is due to a period halving bifurcation which stops when the period becomes 2. The damping coecient is = 0:23, 0.235, 0.239 and 0.24 from (a) to (d), respectively. The position of the trajectory at every 2 , i.e the period of the forcing, is shown by circles. 214.1 Increasing dissipation As mentioned in Section 3.1, as increases from 0to 1it is expected that taking T0larger causes the sizes of the basins of attraction to tend towards the sizes of the corresponding basins when = 0, when the set of attractors remains the same for all values in between. If an attractor is replaced by a new attractor (by bifurcation), then the new attractor inherits the basin of attraction of the old one. We shall begin by xing 1= 0:2 and 02[0:05;0:2], as for = 1the basins of attraction are not so sensitive to initial conditions, see Figure 13(a), and for in that range the set of attractors consists only of the the upwards xed point and two rotating solutions; moreover the proles of the corresponding relative areas plotted in Figure 14 are rather smooth and do not present any sharp jumps. Basin of Attraction % FP PR/NR T00 64.31 17.84 25 51.41 24.30 50 32.09 33.95 75 23.48 38.26 100 17.09 41.45 200 6.85 46.58 500 4.35 47.82 1000 4.15 47.92 1500 4.09 47.96 Table 12: Relative areas of the basins of attraction with 0= 0:05, 1= 0:2 andT0 varying. Figure 17: Plot of the relative areas of the basins of attraction as per Table 12. Basin of Attraction % FP PR/NR T00 64.31 17.84 25 49.06 25.47 50 38.54 30.73 75 31.25 34.37 100 25.67 37.16 150 18.12 40.94 200 14.14 42.93 500 9.81 45.09 1000 9.45 45.27 Table 13: Relative areas of the basins of attraction with 0= 0:1, 1= 0:2 andT0 varying. Figure 18: Plot of the relative areas of the basins of attraction as per Table 13. Tables 12, 13 and 14 show the relative area of each basin of attraction as increases from 0.05, 0.1 and 0.17, respectively, to 0.2 with varying T0. It can be seen from the results in Tables 13 to 14 that the numerical simulations are in agreement with the above expectation. With the exception of Table 12 the relative areas of the basins of attraction tend towards those when is kept constant at 0. The exception of the case of Table 12 is due to the fact that the set of attractors has changed as passes from 0 :05 to 0:2: the period 2 rotating solutions have been destroyed and replaced by the period 1 rotating solutions. However, when the transition occurs, the new attractors are located in 22phase space very close to the previous ones and we nd that the initial conditions which were heading towards or had indeed landed on the period 2 rotating solutions move onto the now present period 1 rotating solutions. On the other hand, when the damping coecient crosses the value 0:1, the attractor undergoes topological changes, but, apart from that, the transition is rather smooth: the location in phase space and the basin of attraction change continuously. In conclusion, we nd that the relative areas of the basins of attraction for the two period 1 rotating attractors (PR/NR) tend towards those the now destroyed period 2 rotating attractors (PR2/NR2) had at = 0. As in Section 3 we expect that the sizes of the basin of attraction at = 0are recovered asymptotically asT0!1 . Nevertheless, once more, the larger T0the smaller is the variation in the relative area: for instance in Table 12 for T0= 100 the relative area of the basin of attraction of the xed point has become nearly 1 =4 of the value for = 0:2 constant, while in order to have a further reduction by a factor of 4 one has to take T0= 1000. Basin of Attraction % FP PR/NR T00 64.31 17.84 25 58.54 20.73 50 56.78 21.62 100 55.41 22.29 200 53.69 23.15 500 51.93 24.04 1000 50.80 24.60 1500 50.62 24.69 Table 14: Relative areas of the basins of attraction with 0= 0:17, 1= 0:2 andT0 varying. Figure 19: Plot of the relative areas of the basins of attraction as per Table 14. Basin of Attraction % FP PR/NR DO2 T00 25.00 12.68 49.61 10 24.86 13.81 47.52 15 24.34 14.84 45.98 20 24.43 15.57 44.43 25 24.77 16.04 43.15 50 27.60 16.98 38.45 75 30.12 17.28 35.32 100 33.08 17.42 32.08 150 38.29 17.56 26.58 200 42.60 17.66 22.08 300 49.36 17.73 15.18 400 54.20 17.75 10.30 500 57.37 17.77 7.08 1000 63.39 17.78 1.06 1500 64.26 17.79 0.16 2000 64.38 17.80 0.03 Table 15: Relative areas of the basins of attraction with 0= 0:2, 1= 0:23 and T0varying. Figure 20: Plot of the relative areas of the basins of attraction as per Table 15. 23We now consider the case where either 0= 0:2 and 1= 0:23 or 0:2725 or 0= 0:23 and 1= 0:2725. Such values for oer more complexities as not only are there more attractors to consider, but one may have attractors (PR and NR) that are destroyed without leaving any trace. When this happens it is not obvious which persisting attractor will inherit their basins of attraction. The result of this could cause the nal basins of attraction to be drastically dierent from those for constant and even not monotonically increasing or decreasing as the value of T0is increased. In Table 15 we see that initially, for values of T0not too large, the basin of attraction of FP slightly reduces in size, while those of the rotating solutions PR/NR increase substantially. Instead, for larger values of T0, the basin of attraction of FP increases appreciably, while those of PR/NR increase very slowly. Apparently, the rotating solutions react more quickly as is varied, attracting phase space faster, so that the relative areas of their basins of attraction tend towards the values at = 0for shorter initial times T0. It would be interesting to study further this phenomenon. When (t) varies from = 0:2 to = 0:2725 and from = 0:23 to = 0:2725, the rotating solutions PR/NR disappear, so that their basins of attractions are absorbed by the persisting at- tractors. In Table 16 one sees a very slow movement towards global attraction of the upwards xed point, which is the only attractor persisting for both 0and 1. However even taking T0= 5000 is not enough for the asymptotic behaviour to be approached. The results in Table 17 show that, by takingT0larger and larger, the relative areas of the basins of attraction of FP and DO4 both tend to the values corresponding to 0= 0:23 (in particular the basin of attraction of DO4 becomes negligible). Nearly all trajectories which were converging towards the rotating solutions before the latter disappeared are attracted by the period two oscillations. This could be due to the fact that DO2 is the closest attractor in phase space which persists at both 0and 1. Basin of Attraction % FP DO2 DO4 T00 17.21 79.44 3.35 25 18.63 78.33 3.04 50 20.32 79.29 0.39 75 23.38 76.52 0.12 100 25.71 74.27 0.02 150 28.45 71.55 0.01 200 30.92 69.08 0.00 300 35.70 64.30 0.00 400 39.82 60.18 0.00 500 42.76 57.24 0.00 980 54.19 45.81 0.00 990 54.39 45.81 0.00 995 54.30 45.70 0.00 1000 90.11 9.89 0.00 1005 54.58 45.43 0.00 1010 54.52 45.48 0.00 1020 90.34 9.66 0.00 1030 54.81 45.19 0.00 1050 55.13 44.87 0.00 1500 59.78 40.22 0.00 2000 62.12 37.88 0.00 3000 63.89 36.11 0.00 5000 64.29 35.71 0.00 Table 16: Relative areas of the basins of attraction with 0= 0:2, 1= 0:2725 and T0varying. Figure 21: Plot of the relative areas of the basins of attraction as per Table 16. However, the more striking feature of Figures 21 and 22 are the jumps corresponding T0= 1000 24in the prior and T0= 100 and T0= 500 in the latter. Moreover such jumps are very localised: for instance in Figure 22 for T0= 99 andT0= 100 the basins of attraction of FP and DO2 are found to be about 44% and 56%, respectively, whereas by slightly increasing or decreasing T0they settle around 20% and 80%. The quantity of phase space exchanged in these instances is roughly equal to that attracted to the rotating solutions for 0. For particular values of T0when the rotating attractors disappear their trajectories move to the upwards xed point rather than the period 2 oscillations. The reason for this to happen is not clear. Moreover, note that in principle there could be other jumps, corresponding to values of T0which have not been investigated: however, it seems hard to make any prediction as far as it remains unclear how the disappearing basins of attractions are absorbed by the persisting ones. Basin of Attraction % FP DO2 DO4 T00 17.21 79.44 3.35 25 17.96 79.45 2.60 50 16.04 83.36 0.60 75 18.39 80.27 1.34 90 19.56 80.38 0.05 95 20.05 79.91 0.04 97 20.08 79.89 0.04 98 20.01 79.96 0.03 99 44.14 55.83 0.03 100 43.96 56.01 0.03 101 20.54 79.42 0.03 105 20.40 79.57 0.03 110 20.79 79.19 0.02 125 21.55 78.45 0.01 150 22.46 77.54 0.00 200 23.33 76.67 0.00 300 24.06 75.94 0.00 400 24.36 75.64 0.00 490 24.54 75.46 0.00 500 49.45 50.55 0.00 510 24.42 75.58 0.00 1000 24.81 75.19 0.00 Table 17: Relative areas of the basins of attraction with 0= 0:23, 1= 0:2725 and T0varying. Figure 22: Plot of the relative areas of the basins of attraction as per Table 17. 4.2 Decreasing dissipation Tables 18, 19, 20 and 21 and the corresponding Figures 23, 24, 25 and 26 illustrate the cases when dissipation decreases over an initial period of time T0. We have considered the cases with 0= 0:23, 0:02725 and 1= 0:2, with 0= 0:2725 and 1= 0:23 and with 0= 0:3 and 1= 0:2725. In particular they show that if the set of attractors at = 1is a proper subset of the set of attractors which exist at = 0, then, asT0!1 , the basin of attraction of each attractor which exists at 1turns out to have a relative area which tend to be greater than or equal to that found for = 0. In Table 18 we consider the situation in which the attractor DO2, which has a large basin of attraction for 0= 0:23, is no longer present when (t) has reached the nal value 1= 0:2: as a consequence the trajectories which would be attracted by DO2 at = 0end up onto the other attractors: in fact most of them are attracted by the xed point. 25Basin of Attraction % FP PR/NR T00 64.31 17.84 25 70.69 14.65 50 72.57 13.72 75 73.24 13.38 100 73.57 13.22 200 74.10 12.95 500 74.42 12.79 Table 18: Relative areas of the basins of attraction with 0= 0:23, 1= 0:2 andT0 varying. Figure 23: Plot of the relative areas of the basins of attraction as per Table 18. Basin of Attraction % FP PR/NR T00 64.31 17.84 5 65.80 17.10 10 69.52 15.24 15 74.40 12.80 20 77.90 11.05 25 80.38 9.81 50 86.22 6.89 75 88.64 5.68 100 90.07 4.97 200 92.79 3.61 500 95.92 2.04 1000 99.34 0.33 1500 99.35 0.32 Table 19: Relative areas of the basins of attraction with 0= 0:2725, 1= 0:2 and T0varying. Figure 24: Plot of the relative areas of the basins of attraction as per Table 19. Basin of Attraction % FP PR/NR DO2 T00 25.00 12.68 49.61 25 24.73 7.44 60.40 50 19.67 5.35 69.63 75 17.13 4.44 73.99 100 16.42 3.88 75.83 200 16.29 2.72 78.28 500 17.03 0.76 81.45 Table 20: Relative areas of the basins of attraction with 0= 0:2725, 1= 0:23 and T0varying. Figure 25: Plot of the relative areas of the basins of attraction as per Table 20. We also notice the interesting features in Table 19: as the xed point is the only attractor which 26exists for both 0and 1, we nd that as T0increases its basin of attraction tends towards 100%, which corresponds to attraction of the entire phase space, up to a zero-measure set. This happens despite the fact that (t) does not pass through any value for which global attraction to the xed point is satised. It also suggests that it is possible to provide conditions on the intersection of the two setsA0andA1of the attractors corresponding to 0and 1, respectively, in order to obtain that all trajectories move towards the same attractor when the time T0over which (t) is varied is suciently large. In particular, it is remarkable that it is possible to create an attractor for almost all trajectories by suitably tuning the damping coecient as a function of time. In Table 20, the relative areas of the rotating solutions, which are absent at = 0, tend to become negligible when T0is large. Similarly, in Table 21, the basin of attraction of the period 4 oscillating attractor, which exists only for the nal value 1of the damping coecient, tends to disappear when T0is taken large enough. This conrms the general expectation: the basin of attraction of the disappearing attractor is absorbed by the closer attractor, that is the solution DO2 in this case. Basin of Attraction % FP DO2 DO4 T00 17.21 79.44 3.35 25 17.15 80.99 1.86 50 16.83 83.03 0.14 75 16.75 83.24 0.01 100 16.79 83.21 0.00 200 16.81 83.19 0.00 500 16.80 83.20 0.00 Table 21: Relative areas of the basins of attraction with 0= 0:3, 1= 0:2725 and T0varying. Figure 26: Plot of the relative areas of the basins of attraction as per Table 21. 5 Numerical Methods The two main numerical methods implemented for the simulations throughout were a variable order Adams-Bashforth-Moulton method and the method of analytic continuation [10, 23, 12, 17, 34, 36]; the latter consists in a numerical implementation of the Frobenius method. Also used to check the results was a Runge-Kutta method. The Adams-Bashforth-Moulton integration scheme used is the built in integrator found in matlab , ODE113, whereas the programs based on the method of analytic continuation and Runge-Kutta scheme were written in C. Of the three methods, the slowest was the Adams-Bashforth-Moulton method, however it was found that the method worked well for the system with the chosen parameter values and the results produced were reliable. Both the Adams-Bashforth-Moulton and Runge-Kutta methods are standard methods for solving ODEs of this type. When implementing these two integrators to calculate the basins of attraction, two dierent methods for both choosing initial conditions in phase space and classifying attractors were used. The rst method for picking initial conditions was to take a mesh of equally spaced points in the phase space: this method ensures uniform coverage of the phase space. When taking this approach a mesh of either 321 141 or 503 289 points was used depending on the desired accuracy. The second method was to take random initial conditions: this can be done by choosing initial conditions from a stream of random points, which allows the user to use the same random initial conditions in each simulation if required. This method is often preferred as the accuracy of the estimates of the relative areas of the basins of attraction compared to the number of initial conditions used can be calculated [29]; also it is easier to run additional simulations for extra random points to improve estimates later on, when needed. When using random points, the number of points used to calculate the basins of attraction was 300 000 or 400 000 depending on the expected complexity 27of the system under given parameters. In some cases where extra accuracy was required due to some attractors having particularly small basins of attraction, additional 200 000 or 300 000 random points were used. This allowed us to obtain an error less than 0.20 on the relative areas of the basins of attraction. In the most delicate cases, where more precise estimates were needed to distinguish between values very close to each other (for instance in Table 4), the error was made smaller by increasing the number of points. We decided to express in all cases the relative areas up to the second decimal digit because often further increasing the number of points did not alter appreciably that digit. To detect and classify solutions, two methods can be used. The rst method consists in nding all attractors as a rst step, before computing the corresponding basins of attraction: this required a complete characterisation of both the period and the location in phase space of the attractors. In principle, this works very well, but has the downside of having to nd initially all the attractors, and dierentiate between those that occupy the same region in phase space. The second method for classifying the solutions was to create a library of solutions. This was created as the program ran and built up as new solutions were found. The solution of each integration was then checked against the library and, if not already known, was added. In this way, the program nds solutions as it goes, so has the advantage of the user not having to know the existing solutions in the system prior to calculating the basins of attraction. The method of analytic continuation was implemented and produced results very similar to those of the Adams-Bashforth-Moulton method, but in general was much quicker to run. When imple- menting this method of integration, we only used random initial conditions in the phase space and the library method for classifying solutions. The reason for using the Adams-Bashforth-Moulton method, despite being the slowest of the three integrators, was for comparison with the method of analytic continuation. Analytic continuation has not previously been used for numerically integrat- ing an ODE of the form (1.2), that is an equation with an innite polynomial nonlinearity that satises an addition formula, thus it was good to have a reliable method to check results with. The method for using analytic continuation for integrating ODE's that have innite polynomial nonlin- earity that satisfy an addition formula will be described more extensively in [36]. The similarity in results of these completely dierent methods for integration, initial condition selection and solution classication provides reassurance and condence that the results produced are accurate. 6 Concluding remarks In this work we have numerically shown the importance of not only the nal value of dissipation but its entire time evolution, for understanding the long time behaviour of the pendulum with oscillating support. This extends the work done in [3] to a system which, even for values of the parameters in the perturbation regime, exhibits richer and more varied dynamics, due to the presence of the separatrix in the phase space of the unperturbed system. In addition we have considered also values of the parameters beyond the perturbation regime (the inverted pendulum), where the system cannot be considered a perturbation of an integrable one. In particular this results in a more complicated scenario, with bifurcation phenomena and the appearance of attractors which exist only for values of the damping coecient in nite intervals away from zero, say 2[ 1; 2], with 1>0. We have preliminarily studied the behaviour of the system in the case of constant dissipation. Firstly, in the perturbation regime, we have analytically computed to rst order the threshold values below which the periodic attractors exist. We have also discussed why this approach fails due to intrinsic perturbation theory limitations, in particular why the method cannot be applied to stable cases of the upwards conguration or to solutions too close to the unperturbed separatrix. Next, we have studied numerically the dependence of the sizes of the basins of attraction on the damping coecient. Then we have explicitly considered the case of damping coecient varying monotonically between two values and outlined a few expectations for the way in which the basins of attraction accordingly change with respect to the case of constant dissipation. These expectations were later illustrated and backed up with numerical simulations: in particular the relevance of the study of the dynamics 28at constant dissipation was argued at length. While the expectations account for many features observed numerically, there are still some facts which are dicult to explain, even at an heuristic level, and which would deserve further investigation, such as the relationship between the xed point solution and the oscillating attractors, to better understand why in some cases they exchange large areas of their basins of attraction when the damping coecient varies in time. More generally, an in-depth numerical study of the system with constant dissipation, also for other parameter values, would be worthwhile. In particular it would be interesting to perform a more detailed bifurcation analysis with respect to the parameter and to study the system for very small values of the damping coecient (which relates to the spin-orbit model in celestial mechanics), both in the perturbation regime and for large values of the forcing amplitude. We think that, in order to study cases with very small dissipation, the method of analytical continuation brie y described in Section 5 could be particularly fruitful. Some interesting features appeared in our analysis which would deserve further consideration are: the increase in the basin of attraction of the xed point observed in Figure 3 when the damping coecient becomes small enough; the appearance of the period 4 solution for a thin interval of values of the damping coecient, as emerges in Table 11; the rate at which the values of the relative areas of the basins attraction corresponding to the initial value 0of (t) are approached as the variation time T0increases; the way in which the basins of attraction of the disappearing attractors distribute among the persisting ones; the oscillations through which the relative areas of the basins of attraction approach the asymptotic value when taking larger and larger values of the variation time T0, as observed for instance in Tables 16 and 17; the jumps corresponding to T0= 1000 in Figure 21 and T0= 100,T0= 500 in Figure 22; the computation of the threshold values to second order, so as to include the rotating solutions found numerically in the perturbation regime investigated in Section 3. Finally, investigating analogous systems such as the pendulum with periodically varying length to see if similar dynamics occur would also be fascinating in its own right. Another interesting model to investigate further, especially in the case of very small dissipation, is the spin-orbit model already considered in [3], which is expected to be of relevance to understand the locking into the resonance 3 : 2 of the system Mercury-Sun. Acknowledgements The Adams-Bashforth-Moulton method used was MATLAB 's ODE113. We thank Jonathan Deane for helpful conversations on analytic continuation and support with coding in C. This research was completed as part of an EPSRC funded PhD. A Global attraction to the two xed points To compute the conditions for attraction to the origin we use the method outlined in [4]; see also [2]. We dene f() as in (1.3) and require f()>0: the consequences of this restriction are that the method can only be applied to the downwards pointing pendulum when > . Then we apply the Liouville transformation ~=Z 0p f(s)ds (A.1) 29and write our equation (1.3) in terms of the new time ~ as ~~+0 @~f(~)~ 2~f(~)+ q ~f(~)1 A~+ sin= 0; (A.2) where the subscript ~ represents derivative with respect to the new time ~ and ~f(~) :=f(). This can be represented as the two-dimensional system on TR, by setting x(~) =(~) and writing x~=y; y ~=yq ~f0 @~f~ 2q ~f+ 1 Asinx; (A.3) for which we have the energy E(x;y) = 1cosx+y2=2. By setting H(~) =E(x(~);y(~)), one nds H~=y2 q ~f0 @~f~ 2q ~f+ 1 A; (A.4) thusH~0, i.ex,yare bounded given that satises >min ~0~f~ 2q ~f=min 0f0 2f: (A.5) Moreover we have that for all ~  >0 H(~) +Z~ 0y2 q ~f0 @~f0 2q ~f+ 1 Ads=H(0); (A.6) so that, as ~ !1 , using the properties above we can arrive at min s02 41q ~f0 @~f~ 2q ~f+ 1 A3 5Z1 0y2(s)ds<1: (A.7) Hencey!0 as time tends to innity. There are two regions of phase space to consider. Any level curve ofHstrictly inside the separatrix of the unperturbed pendulum is the boundary of a positively invariant set Dcontaining the origin: since S=f(x(~);y(~)) :H~= 0g[Dconsists purely of the origin, we can apply the local Barbashin-Krasovsky-La Salle theorem [24] to conclude that every trajectory that begins strictly inside the separatrix will converge to the origin as ~ !+1. Outside of the separatrix we may use equation (A.4), which shows the energy to be strictly decreasing while y6= 0, provided is chosen large enough, coupled with y!0 as time tends to innity. The result is that all trajectories tend to the invariant points on the x-axis as time tends to innity. One of two cases must occur: either the trajectory moves inside the separatrix or it does not. In the rst instance we have already shown that the limiting solution is the origin. In the latter there is only one possibility. As all points on the x-axis are contained within the separatrix other than the unstable xed point, the trajectory must move onto such a xed point and hence belongs to its stable manifold, which is a zero-measure set. Therefore we conclude that a full measure set of initial conditions are attracted by the origin. Reverting back to the original system with time , we conclude that for that system too the basin of attraction of the origin has full measure, provided  < and satises (A.5). 30B Action-angle variables In this section we detail the calculation of the action-angle variables for the simple pendulum in time. More details on calculating action-angle variables can be found in [14, 32, 9]. The simple pendulum has equation of motion given by 00+sin= 0; (B.1) where the dashes represent derivative with respect to the scaled time . The Hamiltonian for the simple pendulum in this notation is E=H(;0) =1 2(0)2cos (B.2) or, in terms of the usual notation for Hamiltonian dynamics, E=H(p;q) =1 2p2cosq; (B.3) whereq=andp=q0=0. Rearranging this for pwe obtainp=p(E;q), with p(E;q) =p 2(E+cosq) =p 2(E0+ cosq); (B.4) whereE0=E=. It is clear that there are two types of dynamics, oscillatory dynamics when E0<1 and rotational dynamics when E0>1, separated at a separatrix when E0= 1, for which no action-angle variables exist. B.1 Librations We rst consider the case E0<1. The action variable is I=1 2I pdq=2 p 2Zq1 0p E0+ cosqdq=8 ph (k2 11)K(k1) +E(k1)i ; (B.5) wherek2 1= (E0+1)=2 andq1= arccos(E0). The functions K(k) and E(k) are the complete elliptic integrals of the rst and second kinds respectively. The angle variable 'can be found as follows '0=@H @I=dE dI=dI dE1 ; (B.6) so that dI dE=d dE2 Zq1 0p E+cosqdq=2 pK(k1): (B.7) Hence we have '() = 2K(k1)p(0): (B.8) Takes= sin (q=2); then using equation (B.2) it is easy to show that (s0)2=g l(1s2) k2 1s2
: (B.9) Integrating using the Jacobi elliptic functions s() =k1snrg l(0);k1 ; (B.10) the expresssion can then be rearranged to achieve the following result: q= 2 arcsin k1sn2K(k1) ';k1 ; p = 2k1pcn2K(k1) ';k1 ; (B.11) which coincide with equations (2.4). By using (E.3) in Appendix E, one obtains from (B.5) @I @k1=8 k1K(k1)p; (B.12) a relation which has been used to derive (2.12). 31B.2 Rotations In the case of rotational dynamics we have I=1 2Z2 0pdq=1 2pZ2 0p E0+ cosqdq=4 k2pE(k2); (B.13) where this time we let k2 2= 2=(E0+ 1) = 1=k2 1. The angle variable 'can similarly be found using (B.6), where d I=dEcan be similarly calculated as dI dE=d dE1 2Z2 0p E+cosqdq=k2 pK(k2); (B.14) which hence gives '() = K(k2)p(0) k2: (B.15) Using (B.9) and the denition of k2, for the rotating solutions we nd that s() = snp(0) k2;k2 ; (B.16) and similarly, by simple rearrangement, we nd that q= 2 arcsin snK(k2) ';k2 ; p =2 k2pdnK(k2) ';k2 ; (B.17) which again yields equations (2.6) By using (E.3) in Appendix E, one obtains from (B.13) @I @k2=4 k2 2K(k2)p: (B.18) which has been used to derive (2.23). C Jacobian determinant Here we compute the entries of the Jacobian matrix Jof the transformation to action-angle variables, which will be used in the next Appendix. As a by-product we check that Jdeterminant equal to 1, that is@q @'@p @I@q @I@p @'= 1: (C.1) For further details on the proof of (C.1) we refer the reader to [9], where the calculations are given in great detail. The derivative with respect to 'is straightforward in both the libration and rotation case, however the dependence of pandqon the action Iis less obvious. That said, the dependence of pandqonk1in the oscillating case and k2in the rotating case is clear and we know the relationship betweenIandkin both cases, hence the derivative of the Jacobi elliptic functions can be calculated by using that @ @I=@k @I@ @k+@u @I@ @u=@k @I@ @k+@u @k@ @u ; (C.2) whereuis the rst argument of the functions, i.e sn( u;k), etc. Then for the oscillations we have @q @I= 4k1K(k1)psn(
) dn(
)+2E(k1)'cn(
) k02 1+k2 1sn(
) cn2(
) k02 1dn(
)E(
) cn(
) k02 1 ; @p @I= 4k1K(k1) cn(
)2E(k1)'sn(
) dn(
) k02 1k2 1sn2(
) cn(
) k02 1+E(
) sn(
) dn(
) k02 1 ; @q @'=4k1K(k1) cn(
) ; @p @'=p4k1K(k1) sn(
) dn(
) ;(C.3) 32where (
) = 2K(k1)' ;k1 andk0 1=p 1k2 1. From the above it is easy to check that equation (C.1) is satised. Similarly, for the rotations we have @q @I=k2 2 2K(k2)p'E(k2) dn(
) k2k02 2+k2sn(
) cn(
) k02 2E(
) dn(
) k2k02 2 ; @p @I=k2 2 2K(k2)dn(
) k2 2+'E(k2) sn(
) cn(
) k02 2+sn2(
) dn(
) k02 2E(
) sn(
) cn(
) k02 2 ; @q @'=2K(k2) dn(
) ; @p @'=p2k2K(k2) sn(
) cn(
) ;(C.4) where (
) = K(k2) ';k2 andk0 2=p 1k2 2. It is once again easily checked from the above that (C.1) is satised. D Equations of motion for the perturbed system By (C.1) one has@'=@q @'=@p @I=@q @I=@p =@p=@I@q=@I @p=@' @q=@': (D.1) We rewrite the equation (1.3) in the action-angle coordinates introduced in Appendix B as follows. D.1 Librations In this section we want to write (1.3) in terms of the action-angle introduced in Appendix B. By taking into account the forcing term coscosin (1.3) one nds I0=@I @qq0+@I @pp0=@p @'q0+@q @'p0 =8k2 1K(k1) cos(0) sn(
) cn(
) dn(
); '0=@' @qq0+@' @pp0=@p @Iq0@q @Ip0 =p 2K(k1) 2K(k1)p sn2(
) +2E(k1)'sn(
) cn(
) dn(
) k02 1 +k2 1sn2(
) cn2(
) 1k2 1E(
) sn(
) cn(
) dn(
) 1k2 1 cos(0);(D.2) where we have used the properties of the Jacobi elliptic functions in Appendix E. As in Appendix C, we are shortening (
) = 2K(k1)' ;k1 . We then wish to add the dissipative term 0. This results in the following equations: I0=8k2 1K(k1) cos(0) sn(
) cn(
) dn(
)8 k2 1pK(k1) cn2(
); '0=p 2K(k1) 2K(k1)p sn2(
) +2E(k1)'sn(
) cn(
) dn(
) (1k2 1) +k2 1sn2(
) cn2(
) 1k2 1E(
) sn(
) cn(
) dn(
) 1k2 1 cos(0) + cn(
) 2K(k1)sn(
) dn(
)+2E(k1)'cn(
) (1k2 1)+k2 1sn(
) cn2(
) (1k2 1) dn(
)E(
) cn(
) 1k2 1 :(D.3) Using the property that, see [26], E(u;k) =E(k)u=K(k) +Z(u;k) we arrive at equations (2.9). The function Z(u;k) is the Jacobi zeta function, which is periodic with period 2 K(k) inu. 33D.2 Rotations The presence of the forcing term leads to te equations I0=@p @'q0+@q @'p0=4K(k2) cos(0) sn(
) cn(
) dn(
); '0=@p @I_q@q @I_p=p k2K(k2)+k2pK(k2)E(k2)'sn(
) cn(
) dn(
) (1k2 2) +k2 2sn2(
) cn2(
) 1k2 2E(
) sn(
) cn(
) dn(
) 1k2 2 cos(0):(D.4) Again, if we wish to add a dissipative term, we arrive at the equations I0=4K(k2) cos(0) sn(
) cn(
) dn(
)4 pK(k2) k2dn2(
); '0=p k2K(k2)+k2pK(k2)E(k2)'sn(
) cn(
) dn(
) (1k2 2) +k2 2sn2(
) cn2(
) 1k2 2E(
) sn(
) cn(
) dn(
) 1k2 2 cos(0)  K(k2)E(k2)'dn2(
) (1k2 2)+k2 2sn(
) cn(
) dn(
) 1k2 2E(
) dn2(
) 1k2 2 :(D.5) Again using that E(u;k) =E(k)u=K(k) +Z(u;k) we arrive at the equations (2.22). E Useful properties of the elliptic functions The complete integrals of the rst and second kind are, respectively, K(k) =Z=2 0d p 1k2sin2 ; E(k) =Z=2 0d q 1k2sin2 ; (E.1) whereas the incomplete elliptic integral of the second kind is E(u;k) =Zsn(u;k) 0dxp 1k2x2 p 1x2: (E.2) One has @K(k) @k=1 kE(k) 1k2K(k) ;@E(k) @k=1 k(E(k)K(k)): (E.3) The following properties of the Jacobi elliptic functions have been used in the previous sections. The derivatives with respect to the rst arguments are @ @usn(u;k) = cn(u;k) dn(u;k); @ @ucn(u;k) =sn(u;k) dn(u;k); @ @udn(u;k) =k2sn(u;k) cn(u;k);(E.4) while the derivatives with respect to the elliptic modulus are @ @ksn(u;k) =u kcn(u;k) dn(u;k) +k k02sn(u;k) cn2(u;k)1 kk02E(u;k) cn(u;k) dn(u;k); @ @kcn(u;k) =u ksn(u;k) dn(u;k)k k02sn2(u;k) cn(u;k) +1 kk02E(u;k) sn(u;k) dn(u;k); @ @kdn(u;k) =kusn(u;k) cn(u;k)k k02sn2(u;k) dn(u;k) +k k02E(u;k) sn(u;k) cn(u;k);(E.5) 34wherek02= 1k2. Finding the value of
for rotations in Section 2 requires use of Zx1 0dn2(x;k) dx=Zsn(x1;k) 0p 1k2^x2 p 1^x2d^x=E(x1;k): (E.6) In the case of librations we also require the relation k2cn2(
) + (1k2) = dn2(
). The integral for 1(0;p;q) in equation (2.20) is found by 1(0;p;q) =1 TZT 0sn(p) cn(p) dn(p) cos(0) d =cos(0) TZT 0sn(p) cn(p) dn(p) cos() d +sin(0) TZT 0sn(p) cn(p) dn(p) sin() d =sin(0) TZT 0sn(p) cn(p) dn(p) sin() d;(E.7) whereT= 2q= 4K(k1)p. The Jacobi elliptic functions can be expanded in a Fourier series as sn(u;k) =2 kK(k)1X n=1qn1=2 1q2n1sin(2n1)u 2K(k) ; cn(u;k) =2 kK(k)1X n=1qn1=2 1 +q2n1cos(2n1)u 2K(k) ; dn(u;k) = 2K(k)+2 K(k)1X n=1qn 1q2ncos2nu 2K(k) ;(E.8) where qis the nome, dened as q= exp K(k0) K(k) ; withk0=p 1k2. In the calculation of hR(n)iforn2, when the pendulum is in libration, we require the evaluation of the integrals 1 TZT 02K(k1)p@ @ cn2(p) d= 0; (E.9) and 1 TZT 02K(k1)p@ @ sn(p) cn(p) dn(p) cos(0) d =1 TZT 02K(k1)psn(p) cn(p) dn(p) sin(0) d =cos(0) TZT 02K(k1)psn(p) cn(p) dn(p) sin() d sin(0) TZT 02K(k1)psn(p) cn(p) dn(p) cos() d;(E.10) whereT= 2q= 4K(k1)p. The integral multiplying sin( 0) vanishes due to parity and hence 2K(k1)pTZT 0@ @ sn(p) cn(p) dn(p) cos(0) d=2K(k1)pcos(0)G1(p;q):(E.11) 35References [1] M.V. Bartuccelli, A. Berretti, J.H.B. Deane, G. Gentile, S.A. Gourley, Selection rules for periodic orbits and scaling laws for a driven damped quartic oscillator , Nonlinear Anal. Real World Appl. 9(2008), no. 5, 1966{1988. [2] M.V. Bartuccelli, J.H.B. Deane, G. Gentile, Globally and locally attractive solutions for quasi- periodically forced systems , J. Math. Anal. Appl. 328(2007), no. 1, 699{714. [3] M.V. Bartuccelli, J.H.B. Deane, G. Gentile, Attractiveness of periodic orbits in parametrically forced systems with time-increasing friction , J. Math. Phys. 53(2012), no.10, 102703, 27 pp. [4] M.V. Bartuccelli, J.H.B. Deane, G. Gentile, S.A. Gourley, Global attraction to the origin in a parametrically-driven nonlinear oscillator , Appl. Math. Comput. 153(2004), no. 1, 1{11. [5] M.V. Bartuccelli, G. Gentile, K.V. Georgiou, On the dynamics of a vertically driven damped planar pendulum , Proc. R. Soc. Lond. A 457(2001), no. 2016, 3007{3022. [6] M.V. Bartuccelli, G. Gentile, K.V. Georgiou, On the stability of the upside-down pendulum with damp- ing, Proc. R. Soc. Lond. A 458(2002), no. 2018, 255{269. [7] M.V. Bartuccelli, G. Gentile, K.V. Georgiou, KAM theory, Lindstedt series and the stability of the upside-down pendulum , Discrete Contin. Dyn. Syst. 9(2003), no. 2, 413{426. [8] F. Bowman, Introduction to elliptic functions with applications , English Universities Press, London, 1953. [9] A.J. Brizard, Jacobi zeta function and action-angle coordinates for the pendulum , Commun. Nonlinear Sci. Numer. Simul. 18(2013), 511{518. [10] K. Brown, Topics in calculus and probability , Self-published on Lulu (2012), 64{77. Also available at: www.mathpages.com. [11] P.F. 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2014-03-19

We consider a pendulum with vertically oscillating support and time-dependent damping coefficient which varies until reaching a finite final value. The sizes of the corresponding basins of attraction are found to depend strongly on the full evolution of the dissipation. In order to predict the behaviour of the system, it is essential to understand how the sizes of the basins of attraction for constant dissipation depend on the damping coefficient. For values of the parameters in the perturbation regime, we characterise analytically the conditions under which the attractors exist and study numerically how the sizes of their basins of attraction depend on the damping coefficient. Away from the perturbation regime, a numerical study of the attractors and the corresponding basins of attraction for different constant values of the damping coefficient produces a much more involved scenario: changing the magnitude of the dissipation causes some attractors to disappear either leaving no trace or producing new attractors by bifurcation, such as period doubling and saddle-node bifurcation. For an initially non-constant damping coefficient, both increasing and decreasing to some finite final value, we numerically observe that, when the damping coefficient varies slowly from a finite initial value to a different final value, without changing the set of attractors, the slower the variation the closer the sizes of the basins of attraction are to those they have for constant damping coefficient fixed at the initial value. If during the variation of the damping coefficient attractors appear or disappear, remarkable additional phenomena may occur. For instance, a fixed point asymptotically may attract the entire phase space, up to a zero measure set, even though no attractor with such a property exists for any value of the damping coefficient between the extreme values.

The effects of time-dependent dissipation on the basins of attraction for the pendulum with oscillating support

1403.4996v1

arXiv:1012.4455v1 [math.AP] 20 Dec 2010GLOBAL ATTRACTORS FOR THE ONE DIMENSIONAL WAVE EQUATION WITH DISPLACEMENT DEPENDENT DAMPING A.KH.KHANMAMEDOV Abstract. We study the long-time behavior of solutions of the one dimen sional wave equation with nonlinear damping coefficient. We prove that if the dampi ng coefficient function is strictly positive near the origin then this equation possesses a glob al attractor. 1.INTRODUCTION In this paper, we consider the following Cauchy problem: utt+σ(u)ut−uxx+λu+f(u) =g(x),(t,x)∈(0,∞)×R, (1.1) u(0,x) =u0(x), u t(0,x) =u1(x), x ∈R, (1.2) whereλis a positive constant, g∈L1(R)+L2(R) and nonlinear functions f(·) andσ(·) satisfy the following conditions: f∈C1(R),f(u)u≥0,/notturnstileu∈R, (1.3) σ∈C(R),σ(0)>0,σ(u)≥0,/notturnstileu∈R. (1.4) Applying standard Galerkin’s method and using techniques of [6, Pro position 2.2], it is easy to prove the following existence and uniqueness theorem: Theorem 1. Assume that the conditions (1.3)-(1.4) hold. Then for any T >0and(u0,u1)∈ H:=H1(R)×L2(R)the problem (1.1)-(1.2) has a unique weak solution u∈C([0,T];H1(R))∩ C1([0,T];L2(R))∩C2([0,T];H−1(R))on[0,T]×Rsuch that /bardbl(u(t),ut(t))/bardblH≤c(/bardbl(u0,u1)/bardblH),/notturnstilet≥0, wherec:R+→R+is a nondecreasing function. Moreover if v∈C([0,T];H1(R))∩C1([0,T];L2(R))∩ C2([0,T];H−1(R))is also weak solution to (1.1)-(1.2) with initial data (v0,v1)∈ H, then /bardblu(t)−v(t)/bardblL2(R)+/bardblut(t)−vt(t)/bardblH−1(R)≤ ≤/tildewidec(T,/tildewideR)/parenleftBig /bardblu0−v0/bardblL2(R)+/bardblu1−v1/bardblH−1(R)/parenrightBig ,/notturnstilet∈[0,T], where/tildewidec:R+×R+→R+is a nondecreasing function with respect to each variable an d/tildewideR= max{/bardbl(u0,u1)/bardblH,/bardbl(v0,v1)/bardblH}. Thus, by the formula ( u(t),ut(t)) =S(t)(u0,u1), the problem (1.1)-(1.2) generates a weak con- tinuous semigroup {S(t)}t≥0inH, whereu(t,x) is a weak solution of (1.1)-(1.2), determined by Theorem 1.1, with initial data ( u0,u1). The attractors for equation (1.1) in the finite interval were studie d in [2], assuming positivity of σ(·). For two dimensional case, the attractors for the wave equatio n with displacement dependent damping were investigated in [7] under conditions σ∈C1(R), 0< σ0≤σ(u)≤c(1+|u|q),/notturnstileu∈R, 0≤q <∞, and |σ′(u)| ≤c[σ(u)]1−ε,/notturnstileu∈R, 0< ε <1, (1.5) on the damping coefficient. Recently, in [3], condition (1.5) has been im proved as |σ′(u)| ≤cσ(u),/notturnstileu∈R. 2000Mathematics Subject Classification. 35B41, 35L05. Key words and phrases. Attractors, wave equations. 12 A.KH.KHANMAMEDOV For the three dimensional bounded domain case, the existence of a global attractor for the wave equation with displacement dependent damping was proved in [6] when σ(·) is a strictly positive and globally bounded function. In this case, when σ(·) is not globally bounded, but is equal to a positive constant in a large enough interval, the existence of a globa l attractor has been established in [4]. In the articles mentioned above, the existence of global attracto rs was proved under positivity or strict positivity condition on the damping coefficient function σ(·). In this paper, we study a global attractor for (1.1)-(1.2) under weaker conditions on σ(·) and prove the following theorem: Theorem 2. Under conditions (1.3)-(1.4) a semigroup {S(t)}t≥0generated by (1.1)-(1.2) possesses a global attractor in H. 2.Proof of Theorem 1.2 To prove this theorem we need the following lemma: Lemma 1. Let conditions (1.3)-(1.4) hold and let Bbe a bounded subset of H. Then for any ε >0 there exist T0=T0(ε,B)>0andr0=r0(ε,B)>0such that /bardblS(t)ϕ/bardblH1(R\(−r0,r0))×L2(R\(−r0,r0))< ε,/notturnstilet≥T0,/notturnstileϕ∈B. (2.1) Proof.Let (u0,u1)∈BandS(t)(u0,u1) = (u(t),ut(t)). Multiplying (1.1) by utand integrating over (0,t)×Rwe obtain /bardblut(t)/bardbl2 L2(R)+/bardblu(t)/bardbl2 H1(R)+t/integraldisplay 0/integraldisplay Rσ(u(τ,x))u2 t(τ,x)dxdτ≤c1,/notturnstilet≥0.(2.2) Letη∈C1(R), 0≤η(x)≤1,η(x) =/braceleftbigg0,|x| ≤1 1,|x| ≥2,ηr(x) =η(x r) and Σ( u) =u/integraltext 0σ(s)ds. Multiplying (1.1) by η2 rΣ(u), integrating over (0 ,t)×Rand taking into account (2.2) we have t/integraldisplay 0/integraldisplay Rη2 r(x)σ(u(τ,x))u2 x(τ,x)dxdτ+λt/integraldisplay 0/integraldisplay Rη2 r(x)Σ(u(τ,x))u(τ,x)dxdτ≤ ≤c2(1+√ t+t r+t/bardblg/bardblL1(R\(−r,r))+L2(R\(−r,r))),/notturnstilet≥0,/notturnstiler >0. (2.3) By (1.4), there exists l >0, such that σ(0) 2≤σ(s)≤2σ(0),/notturnstiles∈[−l,l]. (2.4) Using embedding H1 2+ε(R)⊂L∞(R) and taking into account (2.2) and (2.4) we find t/integraldisplay 0/integraldisplay Rη2 r(x)u2(τ,x)dxdτ≤2 σ(0)t/integraldisplay 0/integraldisplay {x:|u(τ,x)|≤l}η2 r(x)Σ(u(τ,x))u(τ,x)dxdτ+ +c3t/integraldisplay 0/integraldisplay {x:|u(τ,x)|>l}η2 r(x)|u(τ,x)|dxdτ≤2 σ(0)t/integraldisplay 0/integraldisplay {x:|u(τ,x)|≤l}η2 r(x)Σ(u(τ,x))u(τ,x)dxdτ+ +2c3 σ(0)lt/integraldisplay 0/integraldisplay {x:|u(τ,x)|>l}η2 r(x)Σ(u(τ,x))u(τ,x)dxdτ and consequently t/integraldisplay 0/bardblηru(τ)/bardbl5 L∞(R)dτ≤c4t/integraldisplay 0/bardblηru(τ)/bardbl2 L2(R)dτ≤c5t/integraldisplay 0/integraldisplay Rη2 r(x)Σ(u(τ,x))u(τ,x)dxdτ, (2.5)GLOBAL ATTRACTORS 3 forr≥1. So by (2.2), (2.3) and (2.5), we get t/integraldisplay 0/bracketleftbigg/vextenddouble/vextenddouble/vextenddoubleη2rσ1 2(u(τ))ut(τ)/vextenddouble/vextenddouble/vextenddouble2 L2(R)+/vextenddouble/vextenddouble/vextenddoubleη2rσ1 2(u(τ))ux(τ)/vextenddouble/vextenddouble/vextenddouble2 L2(R)+λ/vextenddouble/vextenddouble/vextenddoubleη2rσ1 2(u(τ))u(τ)/vextenddouble/vextenddouble/vextenddouble2 L2(R)+ +/bardblηru(τ)/bardbl5 L∞(R)/bracketrightBig dτ≤c6(1+√ t+t r+t/bardblg/bardblL1(R\(−r,r))+L2(R\(−r,r))),/notturnstilet≥0,/notturnstiler≥1.(2.6) Now denote Φ r(u(t)) :=1 2/bardblηrut(t)/bardbl2 L2(R)+1 2/bardblηrux(t)/bardbl2 L2(R)+µ/an}bracketle{tηrut(t), ηru(t)/an}bracketri}ht+λ 2/bardblηru(t)/bardbl2 L2(R)+ /an}bracketle{tηrF(u(t)), ηr/an}bracketri}ht+/an}bracketle{tηrg, ηru(t)/an}bracketri}ht, where µ= min/braceleftbigg/radicalBig λ 2,σ(0) 5,λ 2σ(0)/bracerightbigg ,/an}bracketle{tu, v/an}bracketri}ht=/integraltext Ru(x)v(x)dxand F(u) =u/integraltext 0f(s)ds.By (2.4) and (2.6), it follows that for any δ >0 there exist Tδ=Tδ(B)>0, r1,δ=r1,δ(B)>1 and for any r≥r1,δthere exists t∗ δ,r∈[0,Tδ] such that Φr(u(t∗ δ,r))< δ,/notturnstiler≥r1,δ. (2.7) Again by (2.2), we have /bardblηru(t)/bardblL2(R)≤/vextenddouble/vextenddoubleηru(t∗ δ,r)/vextenddouble/vextenddouble L2(R)+t/integraldisplay t∗ δ,r/bardblηrut(s)/bardblL2(R)ds≤/vextenddouble/vextenddoubleηru(t∗ δ,r)/vextenddouble/vextenddouble L2(R)+c7(t−t∗ δ,r) and consequently /bardblηru(t)/bardbl3 L∞(R)≤c8/bardblηru(t)/bardblL2(R)≤c9(Φ1 2r(u(t∗ δ,r))+/bardblg/bardbl1 2 L1(R\(−r,r))+L2(R\(−r,r))+t−t∗ δ,r)< < c9(δ1 2+/bardblg/bardbl1 2 L1(R\(−r,r))+L2(R\(−r,r))+t−t∗ δ,r),/notturnstilet≥t∗ δ,r,/notturnstiler≥r1,δ. Denoting T∗ δ,r=t∗ δ,r+l3 3c9and choosing δ∈(0,l6 9c2 9), by the last inequality, we can say that there existsr2,δ≥2r1,δsuch that /bardblu(t)/bardblL∞(R\(−r2,δ,r2,δ))< l,/notturnstilet∈[t∗ δ,r,T∗ δ,r]. (2.8) Now multiplying (1.1) by η2 r(ut+µu), integrating over Rand taking into account (2.4) and (2.8) we obtain d dtΦr(u(t))+c10Φr(u(t))≤c11(1 r+/bardblg/bardblL1(R\(−r,r))+L2(R\(−r,r))),/notturnstilet∈[t∗ δ,r,T∗ δ,r], and consequently Φr(u(t))≤Φr(u(t∗ δ,r))e−c10(t−t∗ δ,r)+c11(1 r+/bardblg/bardblL1(R\(−r,r))+L2(R\(−r,r)))1−e−c10(t−t∗ δ,r) c10,(2.9) forr≥r2,δ. By (2.7) and (2.9), there exists r3,δ≥r2,δsuch that Φr(u(t))< δ,/notturnstiler≥r3,δ,/notturnstilet∈[t∗ δ,r,T∗ δ,r]. Hence denoting by nδthe smallest integer number which is not less than3c9Tδ l3and applying above procedure at most nδtime, we find Φr(u(Tδ))< δ,/notturnstiler≥r4,δ, for some r4,δ≥2nδr1,δ. From the last inequality it follows that for any ε >0 there exist /hatwideTε= /hatwideTε(B)>0 and/hatwiderε=/hatwiderε(B)>0 such that/vextenddouble/vextenddouble/vextenddoubleS(/hatwideTε)ϕ/vextenddouble/vextenddouble/vextenddouble H1(R\(−/hatwiderε,/hatwiderε))×L2(R\(−/hatwiderε,/hatwiderε))< ε,/notturnstileϕ∈B. Since, by (2.2), B0=∪ t≥0S(t)Bis a bounded subset of H, for any ε >0 there exist T0=T0(ε,B)>0 andr0=r0(ε,B)>0 such that /bardblS(T0)ϕ/bardblH1(R\(−r0,r0))×L2(R\(−r0,r0))< ε,/notturnstileϕ∈B0. Taking into account positively invariance of B0, from the last inequality we obtain (2.1). /square4 A.KH.KHANMAMEDOV By (2.1) and (2.4), for any bounded subset BofHthere exist /hatwideT0=/hatwideT0(B)>0 and/hatwider0=/hatwider0(B)>0 such that σ(u(t,x))≥σ(0) 2,/notturnstilet≥/hatwideT0,/notturnstile|x| ≥/hatwider0. (2.10) Hence using techniques of [5] one can prove the asymptotic compac tness of the semigroup {S(t)}t≥0, which is included in the following lemma: Lemma 2. Assume that conditions (1.3)-(1.4) hold and Bis bounded subset of H. Then every sequence of the form {S(tn)ϕn}∞ n=1,{ϕn}∞ n=1⊂B,tn→ ∞, has a convergent subsequence in H. By (2.10) and the unique continuation result of [8], it is easy to see tha t problem (1.1)-(1.2) has a strict Lyapunov function (see [1] for definition). Thus according t o [1, Corollary 2.29] the semigroup {S(t)}t≥0possesses a global attractor. Remark 1. We note that, for the problem considered in [2], from compact embedding H1 0(0,π)⊂ C[0,π], it immediately follows that σ(u(t,x))≥σ(0) 2,/notturnstilet≥0,/notturnstilex∈[0,ε]∪[π−ε,π]for some ε∈(0,π). So a global attractor still exists if one replaces the posit ivity condition on σ(·)by the σ(0)>0. References [1] I. Chueshov, I. Lasiecka, Long-time behavior of second o rder evolution equations with nonlinear damping, Memoirs of AMS, 195 (2008). [2] S. Gatti, V. Pata, A one-dimensional wave equation with n onlinear damping, Glasg. Math. J. , 48 (2006), 419–430. [3] A. Kh. Khanmamedov, Global attractors for 2-D wave equat ions with displacement dependent damping, Math. Methods Appl. Sci. , 33 (2010) 177-187. [4] A. Kh. Khanmamedov, A strong global attractor for 3-D wav e equations with displacement dependent damping, Appl. Math. Letters , 23 (2010) 928-934. [5] A. Kh. Khanmamedov, Global attractors for the plate equa tion with a localized damping and a critical exponent in an unbounded domain, J. Diff. Eqs., 225 (2006) 528-548. [6] V. Pata, S. Zelik, Global and exponential attractors for 3-D wave equations with displacement dependent damping, Math. Methods Appl. Sci. , 29 (2006), 1291–1306. [7] V. Pata, S. Zelik, Attractors and their regularity for 2- D wave equations with nonlinear damping, Adv. Math. Sci. Appl., 17 (2007), 225–237. [8] A. Ruiz, Unique continuation for weak solutions of the wa ve equation plus a potential, J. Math. Pures Appl. , 71 (5) (1992) 455–467. Department of Mathematics, Faculty of Science, Hacettepe U niversity, Beytepe 06800 , Ankara, Turkey E-mail address :azer@hacettepe.edu.tr

2010-12-20

We study the long-time behavior of solutions of the one dimensional wave equation with nonlinear damping coefficient. We prove that if the damping coefficient function is strictly positive near the origin then this equation possesses a global attractor.

Global attractors for the one dimensional wave equation with displacement dependent damping

1012.4455v1

arXiv:1402.1787v1 [math.DS] 7 Feb 2014One-Dimensional Random Attractor and Rotation Number of the Stochastic Damped Sine-Gordon Equation Zhongwei Shena,1, Shengfan Zhoua,1,∗, Wenxian Shenb,2 aDepartment of Applied Mathematics, Shanghai Normal Univer sity, Shanghai 200234, PR China bDepartment of Mathematics and Statistics, Auburn Universi ty, Auburn 36849, USA Abstract : This paper is devoted to the study of the asymptotic dynamic s of the stochastic damped sine-Gordon equation with homogen eous Neumann boundary condition. It is shown that for any positive dampin g and diffusion coefficients, the equation possesses a random attractor, and when the damping and diffusion coefficients are sufficiently large, the random att ractor is a one- dimensionalrandomhorizontal curveregardlessofthestre ngthofnoise. Hence its dynamics is not chaotic. It is also shown that the equatio n has a rotation number provided that the damping and diffusion coefficients are sufficiently large, whichimpliesthatthesolutionstendtooscillatewi ththesamefrequency eventually and the so called frequency locking is successfu l. Keywords : Stochastic damped sine-Gordon equation; random horizont al curve; one-dimensional random attractor; rotation number ; frequency lock- ing AMS Subject Classification : 60H10, 34F05, 37H10. 1 Introduction Let (Ω,F,P) be a probability space , where Ω ={ω= (ω1,ω2,...,ω m)∈C(R,Rm) :ω(0) = 0}, the Borel σ-algebra Fon Ω is generated by the compact open topology (see [1]), and Pis the corresponding Wiener measure on F. Define ( θt)t∈Ron Ω via θtω(·) =ω(·+t)−ω(t), t∈R. Thus, (Ω ,F,P,(θt)t∈R) is an ergodic metric dynamical system. Consider the following stochastic damped sine-Gordon equa tion with additive noise: dut+αdu+(−K∆u+sinu)dt=fdt+m/summationdisplay j=1hjdWjinU×R+(1.1) 1The first two authors are supported by National Natural Scien ce Foundation of China under Grant 10771139, and the Innovation Program of Shanghai Municipal Education Commission under Grant 08ZZ70 2The third author is partially supported by NSF grant DMS-090 7752 ∗Corresponding Author: zhoushengfan@yahoo.com 1complemented with the homogeneous Neumann boundary condit ion ∂u ∂n= 0 on ∂U×R+, (1.2) whereU⊂Rnis a bounded open set with a smooth boundary ∂U,u=u(x,t) is a real function ofx∈Uandt≥0,α, K >0 are damping and diffusion coefficients, respectively, f∈H1(U), hj∈H2(U) with∂hj ∂n= 0 on∂U,j= 1,...,m, and{Wj}m j=1are independent two-sided real- valued Wiener processes on (Ω ,F,P). We identify ω(t) with (W1(t),W2(t),...,W m(t)), i.e., ω(t) = (W1(t),W2(t),...,W m(t)), t∈R. Sine-Gordon equations describe the dynamics of continuous Josephosn junctions (see [18]) and have been widely studied (see [3], [4], [5], [11], [13], [ 14], [15], [17], [18], [19], [25], [26], [29], [30], [31], etc.). Various interesting dynamical scenario s such as subharmonic bifurcation and chaotic behavior are observed in damped and driven sine-Gor don equations (see [3], [4], [19], etc.). Note that interesting dynamics of a dissipative syst em occurs in its global attractor (if it exists). It is therefore of great importance to study the exi stence and the structure/dimension of a global attractor of a damped sine-Gordon equation. As it is known, under various boundary conditions, a determi nistic damped sine-Gordon equation possesses a finite dimensional global attractor (s ee [15, 16, 27, 29, 30, 31]). Moreover, some upperboundsof thedimension of the attractor were obta ined in [15, 29, 30, 31]. In[26, 27], the authors proved that under Neumann boundary condition, w hen the damping is sufficiently large, the dimension of the global attractor is one, which ju stifies the folklore that there is no chaotic dynamics in a strongly damped sine-Gordon equation . Recently, the existence of attractors of stochastic damped sine-Gordon equations has been studied by several authors (see [5], [13], [14]). For exampl e, for the equation (1.1) with Dirichlet boundary condition considered in [13], the author proved th e existence of a finite-dimensional attractor in the random sense. However, the existing works o n stochastic damped sine-Gordon equations deal with Dirichlet boundary conditions only. Th e case of a Neumann boundary condition is of great physical interest. It is therefore imp ortant to investigate both the existence andstructureofattractors ofstochasticdampedsine-Gord onequationswithNeumannboundary conditions. Observe that there is no bounded attracting set s in such case in the original phase space due to the uncontrolled space average of the solutions , which leads to nontrivial dynamics and also some additional difficulties. Nevertheless, it is st ill expected that (1.1)-(1.2) possesses an attractor in the original phase space in proper sense. The objective of the current paper is to provide a study on the existence and structure of random attractors (see Definition 2.2 for the definition of random attractor) of stochastic damped sine-Gordon equations with Neumann boundary condit ions, i.e. (1.1)-(1.2). We will do so in terms of the random dynamical system generated by (1.1) -(1.2) (see Definition 2.1 for the definition of random dynamical system). The following are the main results of this paper. (1) For any α >0 andK >0, (1.1)-(1.2) possesses a random attractor (see Theorem 4. 1 and Corollary 4.2). (2) When Kandαaresufficientlylarge, therandomattractorof (1.1)-(1.2)i saone-dimensional random horizontal curve (and hence is one dimensional) (see Theorem 5.3 and Corollary 5.4). 2(3) When Kandαare sufficiently large, the rotation number of (1.1) exists (S ee Theorem 6.4 and Corollary 6.5). The above results make an important contribution to the unde rstanding of the nonlinear dynamics of stochastic damped sine-Gordon equations with N eumann boundary conditions. Property (1) extends the existence result of random attract or in the Dirichlet boundary case to the Neumann boundary case and shows that system (1.1)-(1. 2) is dissipative. By property (2), the asymptotic dynamics of (1.1)-(1.2) with sufficientl y largeαandKis one dimensional regardless of the strength of noise and hence is not chaotic. Observe that ρ∈Ris called therotation number of (1.1)-(1.2) (see Definition 6.1 for detail) if for any solu tionu(t,x) of (1.1)-(1.2) and any x∈U, the limit lim t→∞u(t,x) texists almost surely and lim t→∞u(t,x) t=ρfora.e. ω∈Ω. Property (3) then shows that all the solutions of (1.1)-(1.2 ) tend to oscillate with the same fre- quency eventually almost surely and hence frequency lockin g is successful in (1.1)-(1.2) provided thatαandKare sufficiently large. We remark that the results in the current paper also hold for s tochastic damped sine-Gordon equations with periodic boundary conditions. It should be pointed out that the dynamical behavior of varie ty of systems of the form (1.1) have been studied in [22, 23, 24, 25] for ordinary differential equations, [26, 27] for partial differential equations and [6, 21, 28] for stochastic (random ) ordinary differential equations. In above literatures, two main aspects considered are the st ructure of the attractor and the phenomenon of frequency locking. For example, in [28], the a uthors studied a class of nonlinear noisy oscillators. They proved the existence of a random att ractor which is a family of horizontal curves and the existence of a rotation number which implies t he frequency locking. The rest of the paper is organized as follows. In section 2, we present some basic concepts and properties for general random dynamical systems. In sec tion 3, we provide some basic settings about (1.1)-(1.2) and show that it generates a rand om dynamical system in proper function space. We prove in section 4 the existence of a uniqu e random attractor of the random dynamical system φgenerated by (1.1)-(1.2) for any α,K >0. We show in section 5 that the random attractor of φis a random horizontal curve provided that αandKare sufficiently large. In section 6, we prove the existence of a rotation number of (1 .1)-(1.2) provided that αandK are sufficiently large. 2 General Random Dynamical Systems In this section, we collect some basic knowledge about gener al random dynamical systems (see [1, 8] for details). Let ( X,/ba∇dbl ·/ba∇dblX) be a separable Hilbert space with Borel σ-algebra B(X) and (Ω,F,P,(θt)t∈R) be the ergodic metric dynamical system mentioned in sectio n 1. Definition 2.1. A continuous random dynamical system over (Ω,F,P,(θt)t∈R)is a(B(R+)× F ×B(X),B(X))-measurable mapping ϕ:R+×Ω×X→X,(t,ω,u)/ma√sto→ϕ(t,ω,u) such that the following properties hold 3(1)ϕ(0,ω,u) =ufor allω∈Ωandu∈X; (2)ϕ(t+s,ω,·) =ϕ(t,θsω,ϕ(s,ω,·))for alls,t≥0andω∈Ω; (3)ϕis continuous in tandu. For given u∈XandE,F⊂X, we define d(u,E) = inf v∈E/ba∇dblu−v/ba∇dblX and dH(E,F) = sup u∈Ed(u,F). dH(E,F) is called the Hausdorff semi-distance fromEtoF. Definition 2.2. (1) A set-valued mapping ω/ma√sto→D(ω) : Ω→2Xis said to be a random set if the mapping ω/ma√sto→d(u,D(ω))is measurable for any u∈X. IfD(ω)is also closed (compact) for each ω∈Ω, the mapping ω/ma√sto→D(ω)is called a random closed (compact) set. A random set ω/ma√sto→D(ω)is said to be bounded if there exist u0∈Xand a random variable R(ω)>0such that D(ω)⊂ {u∈X:/ba∇dblu−u0/ba∇dblX≤R(ω)}for allω∈Ω. (2) A random set ω/ma√sto→D(ω)is called tempered provided for P-a.s.ω∈Ω, lim t→∞e−βtsup{/ba∇dblb/ba∇dblX:b∈D(θ−tω)}= 0for allβ >0. (3) A random set ω/ma√sto→B(ω)is said to be a random absorbing set if for any tempered random setω/ma√sto→D(ω), there exists t0(ω)such that ϕ(t,θ−tω,D(θ−tω))⊂B(ω)for allt≥t0(ω), ω∈Ω. (4) A random set ω/ma√sto→B1(ω)is said to be a random attracting set if for any tempered random setω/ma√sto→D(ω), we have lim t→∞dH(ϕ(t,θ−tω,D(θ−tω),B1(ω)) = 0for allω∈Ω. (5) A random compact set ω/ma√sto→A(ω)is said to be a random attractor if it is an random attracting set and ϕ(t,ω,A(ω)) =A(θtω)for allω∈Ωandt≥0. Theorem 2.3. Letϕbe a continuous random dynamical system over (Ω,F,P,(θt)t∈R). If there is a tempered random compact attracting set ω/ma√sto→B1(ω)ofϕ, thenω/ma√sto→A(ω)is a random attractor of ϕ, where A(ω) =/intersectiondisplay t>0/uniondisplay τ≥tϕ(τ,θ−τω,B1(θ−τω)), ω∈Ω. Moreover, ω/ma√sto→A(ω)is the unique random attractor of φ. Proof.See [8, Theorem 1.8.1]. 43 Basic Settings In this section, we give some basic settings about (1.1)-(1. 2) and show that it generates a random dynamical system. Define an unbounded operator A:D(A)≡/braceleftBig u∈H2(U) :∂u ∂n/vextendsingle/vextendsingle/vextendsingle ∂U= 0/bracerightBig →L2(U), u/ma√sto→ −K∆u. (3.1) Clearly,Ais nonnegative definite and self-adjoint. Its spectral set c onsists of only nonnegative eigenvalues, denoted by λi, satisfying 0 =λ0< λ1≤λ2≤ ··· ≤λi≤ ···,(λi→+∞asi→ ∞). (3.2) It is well known that −Agenerates an analytic semigroup of bounded linear operator s{e−At}t≥0 onL2(U) (andH1(U)). LetE=H1(U)×L2(U), endowed with the usual norm /ba∇dblY/ba∇dblH1×L2=/parenleftbig /ba∇dbl∇u/ba∇dbl2+/ba∇dblu/ba∇dbl2+/ba∇dblv/ba∇dbl2/parenrightbig1 2forY= (u,v)⊤, (3.3) where/ba∇dbl·/ba∇dbldenotes the usual norm in L2(U) and⊤stands for the transposition. The existence of solutions to problem (1.1)-(1.2) follows f rom [10]. We next transform the problem (1.1)-(1.2) to a deterministic system with a random parameter, and then show that it generates a random dynamical system. Let(Ω,F,P,(θt)t∈R)betheergodicmetricdynamicalsysteminsection1. For j∈ {1,2,...,m}, consider the one-dimensional Ornstein-Uhlenbeck equatio n dzj+zjdt=dWj(t). Its unique stationary solution is given by zj(θtωj) =/integraldisplay0 −∞es(θtωj)(s)ds=−/integraldisplay0 −∞esωj(s+t)ds+ωj(t), t∈R. Note that the random variable |zj(ωj)|is tempered and the mapping t/ma√sto→zj(θtωj) isP-a.s. continuous (see [2, 12]). More precisely, there is a θt-invariant Ω 0⊂Ω withP(Ω0) = 1 such that t/ma√sto→zj(θtωj) is continuous for ω∈Ω0andj= 1,2,···,m. Putting z(θtω) =/summationtextm j=1hjzj(θtωj), which solves dz+zdt=/summationtextm j=1hjdWj. Now, let v=ut−z(θtω) and take the functional space Einto consideration, we obtain the equivalent system of (1.1)-(1.2), /braceleftBigg ˙u=v+z(θtω), ˙v=−Au−αv−sinu+f+(1−α)z(θtω).(3.4) LetY= (u,v)⊤,C=/parenleftbigg0I −A−αI/parenrightbigg ,F(θtω,Y) = (z(θtω),−sinu+f+(1−α)z(θtω))⊤, problem (3.4) has the following simple matrix form ˙Y=CY+F(θtω,Y). (3.5) We will consider (3.4) or (3.5) for ω∈Ω0and write Ω 0as Ω from now on. 5Clearly,Cis an unbounded closed operator on Ewith domain D(C) =D(A)×H1(U). It is not difficult to check that the spectral set of Cconsists of only following points [27] µ± i=−α±√ α2−4λi 2, i= 0,1,2,... andCgenerates a C0-semigroup of bounded linear operators {eCt}t≥0onE. Furthermore, let Fω(t,Y) :=F(θtω,Y), it is easy to see that Fω(·,·) :R+×E→Eis continuous in tand globally Lipschitz continuous in Yfor each ω∈Ω. By the classical theory concerning the existence and uniqueness of the solutions, we obtain (see [20, 29]) Theorem 3.1. Consider (3.5). For each ω∈Ωand each Y0∈E, there exists a unique function Y(·,ω,Y0)∈C([0,+∞);E)such that Y(0,ω,Y0) =Y0andY(t,ω,Y0)satisfies the integral equation Y(t,ω,Y0) =eCtY0+/integraldisplayt 0eC(t−s)F(θsω,Y(s,ω,Y0))ds. (3.6) Furthermore, if Y0∈D(C), there exists Y(·,ω,Y0)∈C([0,+∞);D(C))∩C1((R+,+∞);E) which satisfies (3.6)andY(t,ω,Y0)is jointly continuous in t,Y0, and is measurable in ω. Then,Y:R+×Ω×E→E(orR+×Ω×D(C)→D(C))is a continuous random dynamical system. We now define a mapping φ:R+×Ω×E→E(orR+×Ω×D(C)→D(C)) by φ(t,ω,φ0) =Y(t,ω,Y0(ω))+(0,z(θtω))⊤, (3.7) whereφ0= (u0,u1)⊤andY0(ω) = (u0,u1−z(ω))⊤. It is easy to see that φis a continuous random dynamical system associated with the problem (1.1)- (1.2) on E(orD(C)). We next show a useful property of just defined random dynamical syste ms. Lemma 3.2. Suppose that p0= (2π,0)⊤. The random dynamical system Ydefined in (3.6)is p0-translation invariant in the sense that Y(t,ω,Y0+p0) =Y(t,ω,Y0)+p0, t≥0, ω∈Ω, Y0∈E. Proof.SinceCp0= 0 and F(t,ω,Y) isp0-periodic in Y,Y(t,ω,Y0) +p0is a solution of (3.5) with initial data Y0+p0. Thus,Y(t,ω,Y0)+p0=Y(t,ω,Y0+p0). Note that µ+ 1→0 asα→+∞, which will cause some difficulty. In order to overcome it, we introduce a new norm which is equivalent to the usual norm /ba∇dbl·/ba∇dblH1×L2onEin (3.3). Here, we only collect some results about the new norm (see [27] for det ails). Since Chas at least two real eigenvalues 0 and −αwith correspondingeigenvectors η0= (1,0)⊤andη−1= (1,−α)⊤, letE1= span{η0},E−1= span{η−1}andE11=E1+E−1. For any u∈L2(U), define ¯ u=1 |U|/integraltext Uu(x)dx, i.e., the spatial average of u, let˙L2(U) ={u∈L2(U) : ¯u= 0},˙H1(U) =H1(U)∩˙L2(U) and E22=˙H1(U)×˙L2(U). It’s easy to see that E=E11⊕E22andE1is invariant under C. We now define two bilinear forms on E11andE22respectively. For Yi= (ui,vi)⊤∈E11,i= 1,2, let /an}b∇acketle{tY1,Y2/an}b∇acket∇i}htE11=α2 4/an}b∇acketle{tu1,u2/an}b∇acket∇i}ht+/an}b∇acketle{tα 2u1+v1,α 2u2+v2/an}b∇acket∇i}ht, (3.8) 6where/an}b∇acketle{t·,·/an}b∇acket∇i}htdenotes the inner product on L2(U), and for Yi= (ui,vi)⊤∈E22, i= 1,2, let /an}b∇acketle{tY1,Y2/an}b∇acket∇i}htE22=/an}b∇acketle{tA1 2u1,A1 2u2/an}b∇acket∇i}ht+(α2 4−δλ1)/an}b∇acketle{tu1,u2/an}b∇acket∇i}ht+/an}b∇acketle{tα 2u1+v1,α 2u2+v2/an}b∇acket∇i}ht,(3.9) whereA1 2=√ K∇(see (3.1) for the definition of A) andδ∈(0,1]. By the Poincar´ e inequality /ba∇dblA1 2u/ba∇dbl2≥λ1/ba∇dblu/ba∇dbl2,∀u∈˙H1(U), (3.9) is then positive definite. Note that for any Y∈E,¯Y=/integraltext UY(x)dx∈E11andY−¯Y∈E22, thus we define /an}b∇acketle{tY1,Y2/an}b∇acket∇i}htE=/an}b∇acketle{t¯Y1,¯Y2/an}b∇acket∇i}htE11+/an}b∇acketle{tY1−¯Y1,Y2−¯Y2/an}b∇acket∇i}htE22forY1,Y2∈E. (3.10) Lemma 3.3 ([27]).(1)(3.8)and(3.9)define inner products on E11andE22, respectively. (2)(3.10)defines an inner product on E, and the corresponding norm /ba∇dbl· /ba∇dblEis equivalent to the usual norm /ba∇dbl·/ba∇dblH1×L2in(3.3), where /ba∇dblY/ba∇dblE=/parenleftBigα2 4/ba∇dblu/ba∇dbl2+/ba∇dblα 2u+v/ba∇dbl2+/ba∇dblA1 2(u−¯u)/ba∇dbl2−δλ1/ba∇dblu−¯u/ba∇dbl2/parenrightBig1 2 =/parenleftBigα2 4/ba∇dblu/ba∇dbl2+/ba∇dblα 2u+v/ba∇dbl2+/ba∇dblA1 2u/ba∇dbl2−δλ1/ba∇dblu−¯u/ba∇dbl2/parenrightBig1 2(3.11) forY= (u,v)⊤∈E. (3) In terms of the inner product /an}b∇acketle{t·,·/an}b∇acket∇i}htE,E1andE11are orthogonal to E−1andE22, respec- tively. (4) In terms of the norm /ba∇dbl·/ba∇dblE, the Lipschitz constant LFofFin(3.5)satisfies LF≤2 α. (3.12) Now let E2=E−1⊕E22, thenE2is orthogonal to E1andE=E1⊕E2. Thus,E2is also invariant under C. Denote by PandQ(=I−P) the projections from EintoE1andE2, respectively. Lemma 3.4. (1) For any Y∈D(C)∩E2,/an}b∇acketle{tCY,Y/an}b∇acket∇i}htE≤ −a/ba∇dblY/ba∇dbl2 E, where a=α 2−/vextendsingle/vextendsingle/vextendsingleα 2−δλ1 α/vextendsingle/vextendsingle/vextendsingle. (3.13) (2)/ba∇dbleCtQ/ba∇dbl ≤e−atfort≥0. (3)eCtPY=PYforY∈E,t≥0. Proof.See Lemma 3.3 and Corollary 3.3.1 in [27] for (1) and (2). We no w show (3). For Y∈ D(C)∩E1, sinced dteCtY=eCtCY= 0, we have eCtY=eC0Y=Y. Then, by approximation, eCtY=Yforu∈E1, t≥0, sinceD(A)∩E1is dense in E1. Thus,eCtPY=PYforY∈E, t≥0. 7We will need the following lemma and its corollaries. Lemma 3.5. For any ǫ >0, there is a tempered random variable r: Ω/ma√sto→R+such that /ba∇dblz(θtω)/ba∇dbl ≤eǫ|t|r(ω)for allt∈R, ω∈Ω, (3.14) wherer(ω),ω∈Ωsatisfies e−ǫ|t|r(ω)≤r(θtω)≤eǫ|t|r(ω), t∈R, ω∈Ω. (3.15) Proof.Forj∈ {1,2,...,m}, since|zj(ωj)|is a tempered random variable and the mapping t/ma√sto→ln|zj(θtωj)|isP-a.s. continuous, it follows from Proposition 4.3.3 in [1] t hat for any ǫj>0 there is a tempered random variable rj(ωj)>0 such that 1 rj(ωj)≤ |zj(ωj)| ≤rj(ωj), whererj(ωj) satisfies, for P-a.s.ω∈Ω, e−ǫj|t|rj(ωj)≤rj(θtωj)≤eǫj|t|rj(ωj), t∈R. (3.16) Takingǫ1=ǫ2=···=ǫm=ǫ, then we have /ba∇dblz(θtω)/ba∇dbl ≤m/summationdisplay j=1|zj(θtωj)|·/ba∇dblhj/ba∇dbl ≤m/summationdisplay j=1rj(θtωj)/ba∇dblhj/ba∇dbl ≤eǫ|t|m/summationdisplay j=1rj(ωj)/ba∇dblhj/ba∇dbl. Letr(ω) =/summationtextm j=1rj(ωj)/ba∇dblhj/ba∇dbl, (3.14) is satisfied and (3.15) is trivial from (3.16). Corollary 3.6. For any ǫ >0, there is a tempered random variable r′: Ω/ma√sto→R+such that /ba∇dblA1 2z(θtω)/ba∇dbl ≤eǫ|t|r′(ω)for allt∈R, ω∈Ω, wherer′(ω) =/summationtextm j=1rj(ωj)/ba∇dblA1 2hj/ba∇dblsatisfies e−ǫ|t|r′(ω)≤r′(θtω)≤eǫ|t|r′(ω), t∈R, ω∈Ω. Corollary 3.7. For any ǫ >0, there is a tempered random variable r′′: Ω/ma√sto→R+such that /ba∇dblAz(θtω)/ba∇dbl ≤eǫ|t|r′′(ω)for allt∈R, ω∈Ω wherer′′(ω) =/summationtextm j=1rj(ωj)/ba∇dblAhj/ba∇dblsatisfies e−ǫ|t|r′′(ω)≤r′′(θtω)≤eǫ|t|r′′(ω), t∈R, ω∈Ω. 84 Existence of Random Attractor In this section, we study the existence of a random attractor . Throughout this section we assume that p0= 2πη0= (2π,0)⊤∈E1andδ∈(0,1] is such that a >0, where ais as in (3.13). We remark in the end of this section that such δalways exists. The space D(C) can be endowed with the graph norm, /ba∇dblY/ba∇dbl˜E=/ba∇dblY/ba∇dblE+/ba∇dblCY/ba∇dblEforY∈D(C). SinceCis a closed operator, D(C) is a Banach space under the graph norm. We denote (D(C),/ba∇dbl·/ba∇dbl˜E) by˜Eand let˜E1=˜E∩E1,˜E2=˜E∩E2. By Lemma 3.2 and the fact that operator Chas a zero eigenvalue, we will define a random dynamical system Ydefined on torus induced from Y. Then by properties of Yrestricted on E2, we can prove the existence of a random attractor of Y. Thus, we can say that Yhas a unbounded random attractor. Now, we define Y. LetT1=E1/p0ZandE=T1×E2. ForY0∈E, letY0:=Y0(modp0) =Y0+p0Z⊂E denotes the equivalence class of Y0, which is an element of E. And the norm on Eis denoted by /ba∇dblY0/ba∇dblE= inf y∈p0Z/ba∇dblY0+y/ba∇dblE. Note that, by Lemma 3.2, Y(t,ω,Y0+kp0) =Y(t,ω,Y0) +kp0,∀k∈Zfort≥0,ω∈Ω and Y0∈E. With this, we define Y:R+×Ω×E→Eby setting Y(t,ω,Y0) =Y(t,ω,Y0) (modp0), (4.1) whereY0=Y0(modp0). It is easy to see that Y:R+×Ω×E→Eis a random dynamical system. Similarly, the random dynamical system φdefined in (3.7) also induces a random dynamical systemΦonE. By (3.7) and (4.1), Φis defined by Φ(t,ω,Φ0) =Y(t,ω,Y0)+ ˜z(θtω) (modp0), (4.2) whereΦ0=φ0(modp0), ˜z(θtω) = (0,z(θtω))⊤andY0=Φ0−˜z(ω) (modp0). The main result of this section can now be stated as follows. Theorem 4.1. The random dynamical system Ydefined in (4.1)has a unique random attractor ω/ma√sto→A0(ω), where A0(ω) =/intersectiondisplay t>0/uniondisplay τ≥tY(τ,θ−τω,B1(θ−τω)), ω∈Ω, in which ω/ma√sto→B1(ω)is a tempered random compact attracting set for Y. Corollary 4.2. The induced random dynamical system Φdefined in (4.2)has a random attrac- torω/ma√sto→A(ω), whereA(ω) =A0(ω)+ ˜z(ω) (modp0)for allω∈Ω. Proof.It follows from (4.2) and Theorem 4.1. To prove Theorem 4.1, we first introduce the concept of random pseudo-balls and prove a lemma on the existence of a pseudo-tempered random absorbin g pseudo-ball. 9Definition 4.3. LetR: Ω→R+be a random variable. A random pseudo-ball ω∈Ω/ma√sto→B(ω)⊂ Ewith random radius ω/ma√sto→R(ω)is a set of the form ω/ma√sto→B(ω) ={b(ω)∈E:/ba∇dblQb(ω)/ba∇dblE≤R(ω)}. Furthermore, a random set ω/ma√sto→B(ω)⊂Eis called pseudo-tempered provided ω/ma√sto→QB(ω)is a tempered random set in E, i.e., for P-a.s.ω∈Ω, lim t→∞e−βtsup{/ba∇dblQb/ba∇dblE:b∈B(θ−tω)}= 0for allβ >0. Notice that any random pseudo-ball ω/ma√sto→B(ω) inEhas the form ω/ma√sto→E1×QB(ω), where ω/ma√sto→QB(ω) is a random ball in E2, which implies the measurability of ω/ma√sto→B(ω). By Definition 4.3, if ω/ma√sto→B(ω) is a random pseudo-ball in E, thenω/ma√sto→B(ω) (modp0) is random bounded set in E. And if ω/ma√sto→B(ω) is a pseudo-tempered random set in E, then ω/ma√sto→B(ω) (modp0) is tempered random set in E. Lemma 4.4. Leta >0. Then there exists a tempered random set ω/ma√sto→B0(ω) :=B0(ω) (modp0) inEsuch that, for any tempered random set ω/ma√sto→B(ω) :=B(ω) (modp0)inE, there is a TB(ω)>0such that Y(t,θ−tω,B(θ−tω))⊂B0(ω)for allt≥TB(ω), ω∈Ω, whereω/ma√sto→B0(ω)is a random pseudo-ball in Ewith random radius ω/ma√sto→R0(ω)andω/ma√sto→B(ω) is any pseudo-tempered random set in E. Proof.Forω∈Ω, we obtain from (3.6) that Y(t,ω,Y0(ω)) =eCtY0(ω)+/integraldisplayt 0eC(t−s)F(θsω,Y(s,ω,Y0(ω)))ds. (4.3) The projection of (4.3) to E2is QY(t,ω,Y0(ω)) =eCtQY0(ω)+/integraldisplayt 0eC(t−s)QF(θsω,Y(s,ω,Y0(ω)))ds. (4.4) By replacing ωbyθ−tω, it follows from (4.4) that QY(t,θ−tω,Y0(θ−tω)) =eCtQY0(θ−tω)+/integraldisplayt 0eC(t−s)QF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))ds, and it then follows from Lemma 3.4 and Q2=Qthat /ba∇dblQY(t,θ−tω,Y0(θ−tω))/ba∇dblE ≤e−at/ba∇dblQY0(θ−tω)/ba∇dblE+/integraldisplayt 0e−a(t−s)/ba∇dblF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/ba∇dblEds.(4.5) 10By (3.11), Lemma 3.5 and Corollary 3.6 with ǫ=a 2, /ba∇dblF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/ba∇dblE =/parenleftBigα2 4/ba∇dblz(θs−tω)/ba∇dbl2+/ba∇dbl(1−α 2)z(θs−tω)−sin(Yu)+f/ba∇dbl2+/ba∇dblA1 2z(θs−tω)/ba∇dbl2 −δλ1/ba∇dblz(θs−tω)−z(θs−tω)/ba∇dbl2/parenrightBig1 2 ≤/parenleftBig (α2−3α+3)/ba∇dblz(θs−tω)/ba∇dbl2+3/ba∇dblsin(Yu)/ba∇dbl2+3/ba∇dblf/ba∇dbl2+/ba∇dblA1 2z(θs−tω)/ba∇dbl2/parenrightBig1 2 ≤/parenleftBig (α2−3α+3)ea(t−s)(r(ω))2+ea(t−s)(r′(ω))2+3|U|+3/ba∇dblf/ba∇dbl2/parenrightBig1 2 ≤a1ea 2(t−s)r(ω)+ea 2(t−s)r′(ω)+a2, whereYusatisfies Y(s,θ−tω,Y0(θ−tω)) = (Yu,Yv)⊤,a1=√ α2−3α+3,a2=/radicalbig 3|U|+3/ba∇dblf/ba∇dbl2 and|U|is the Lebesgue measure of U. We find from (4.5) that /ba∇dblQY(t,θ−tω,Y0(θ−tω))/ba∇dblE≤e−at/ba∇dblQY0(θ−tω)/ba∇dblE+2 a(1−e−a 2t)(a1r(ω)+r′(ω))+a2 a(1−e−at). Now for ω∈Ω, define R0(ω) =4 a(a1r(ω)+r′(ω))+2a2 a. Then, for any pseudo-tempered random set ω/ma√sto→B(ω) inEand any Y0(θ−tω)∈B(θ−tω), there is aTB(ω)>0 such that for t≥TB(ω), /ba∇dblQY(t,θ−tω,Y0(θ−tω))/ba∇dblE≤R0(ω), ω∈Ω, which implies Y(t,θ−tω,B(θ−tω))⊂B0(ω) for all t≥TB(ω), ω∈Ω, whereω/ma√sto→B0(ω) is the random pseudo-ball centered at origin with random ra diusω/ma√sto→R0(ω). In fact,ω/ma√sto→R0(ω) is a tempered random variable since ω/ma√sto→r(ω) andω/ma√sto→r′(ω) are tempered random variables. Then the measurability of random pseudo- tempered ball ω/ma√sto→B0(ω) is obtained from Definition 4.3 and ω/ma√sto→B0(ω) is a random pseudo-ball. Hence, ω/ma√sto→B0(ω) := B0(ω) (modp0) is a tempered random ball in E. It then follows from the definition of Ythat Y(t,θ−tω,B(θ−tω))⊂B0(ω) for all t≥TB(ω), ω∈Ω, whereTB(ω) =TB(ω) forω∈Ω. This complete the proof. We now prove Theorem 4.1. Proof of Theorem 4.1. By Theorem 2.3, it suffices to prove the existence of a random at tracting set which restricted on E2is tempered and compact, i.e., there exists a random set ω/ma√sto→B1(ω) such that ω/ma√sto→QB1(ω) is tempered and compact in E2and for any pseudo-tempered random setω/ma√sto→B(ω) inE, dH(Y(t,θ−tω,B(θ−tω)),B1(ω))→0 ast→ ∞, ω∈Ω, 11wheredHis the Hausdorff semi-distance. Since pseudo-tempered rand om sets in Eare absorbed by the random absorbing set ω/ma√sto→B0(ω), it suffices to prove that dH(Y(t,θ−tω,B0(θ−tω)),B1(ω))→0 ast→ ∞, ω∈Ω. (4.6) Clearly, if such a ω/ma√sto→B1(ω) exists, then ω/ma√sto→B1(ω) :=B1(ω) (modp0) is a tempered random compact attracting set for Y. We next show that (4.6) holds. By the superposition principle, (3.5) with initial data Y0(ω) can be decomposed into ˙Y1=CY1+F(θtω,Y(t,ω,Y0(ω))), Y10(ω) = 0 (4.7) and ˙Y2=CY2, Y20(ω) =Y0(ω), (4.8) whereY(t,ω,Y0(ω)) is the solution of (3.5) with initial data Y0(ω)∈B0(ω). LetY1(t,ω,Y10(ω)) andY2(t,Y20(ω)) be solutions of (4.7) and (4.8), respectively. We now give some estimations of Y1(t,ω,Y10(ω)) andY2(t,Y20(ω)), which ensure the existence of a random attracting set whi ch restricted on E2is tempered and compact. We first estimate Y2(t,Y20(ω)). Clearly, (4.8) is a linear problem. It is easy to see that Y2(t,Y20(ω)) =eCtY20(ω), which implies (with ωbeing replaced by θ−tω) that /ba∇dblQY2(t,Y20(θ−tω))/ba∇dblE≤ /ba∇dbleCtQ/ba∇dbl·/ba∇dblQY20(θ−tω)/ba∇dblE≤e−atR0(θ−tω)→0 ast→ ∞.(4.9) ForY1(t,ω,Y10(ω)), we show that it is bounded by a tempered random bounded clo sed set in˜E, which then is compact in Esince˜Eis compactly imbedded in E. Note that Y1(t,ω,Y10(ω)) =/integraldisplayt 0eC(t−s)F(θsω,Y(s,ω,Y0(ω)))ds, (4.10) it then follows that /ba∇dblQY1(t,θ−tω,Y10(θ−tω))/ba∇dblE≤2 a(1−e−a 2t)(a1r(ω)+r′(ω))+a2 a(1−e−at),(4.11) wherea1=√ α2−3α+3 and a2=/radicalbig 3|U|+3/ba∇dblf/ba∇dbl2are the same as in the proof of Lemma 4.4, |U|denotes the Lebbesgue measure of U. We next estimate CQY1(t,θ−tω,Y10(θ−tω)). We find from (4.10) that CQY1(t,θ−tω,Y10(θ−tω)) =/integraldisplayt 0eC(t−s)CQF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))ds =/integraldisplayt 0eC(t−s)CF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))ds. Then, /ba∇dblCQY1(t,θ−tω,Y10(θ−tω))/ba∇dblE≤/integraldisplayt 0e−a(t−s)/ba∇dblCF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/ba∇dblEds.(4.12) 12Obviously, CF(θs−tω,Y(s,θ−tω,Y0(θ−tω))) =/parenleftbigg−sin(Yu)+f+(1−α)z(θs−tω) αsin(Yu)−αf−α(1−α)−Az(θs−tω)/parenrightbigg , whereYusatisfiesY(s,θ−tω,Y0(θ−tω)) = (Yu,Yv)⊤. By (3.11), Lemma 3.5, Corollary 3.6 and Corollary 3.7 with ǫ=a 2, /ba∇dblCF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/ba∇dbl2 E ≤7 4α2/ba∇dblsin(Yu)/ba∇dbl2+7 4α2/ba∇dblf/ba∇dbl2+7 4α2(1−α)2/ba∇dblz(θs−tω)/ba∇dbl2+4/ba∇dblAz(θs−tω)/ba∇dbl2 +3/ba∇dblA1 2sin(Yu)/ba∇dbl2+3/ba∇dblA1 2f/ba∇dbl2+3(1−α)2/ba∇dblA1 2z(θs−tω)/ba∇dbl2 ≤a2 3+7 4α2(1−α)2ea(t−s)(r(ω))2+3(1−α)2ea(t−s)(r′(ω))2 +4ea(t−s)(r′′(ω))2+3/ba∇dblA1 2sin(Yu)/ba∇dbl2 ≤/parenleftBig a3+√ 7 2α|1−α|ea 2(t−s)r(ω)+√ 3|1−α|ea 2(t−s)r′(ω) +2ea 2(t−s)r′′(ω)+√ 3/ba∇dblA1 2sin(Yu)/ba∇dbl/parenrightBig2 , wherea3=/radicalBig 7 4α2|U|+7 4α2/ba∇dblf/ba∇dbl2+3/ba∇dblA1 2f/ba∇dbl2. Then, (4.12) implies /ba∇dblCQY1(t,θ−tω,Y10(θ−tω))/ba∇dblE ≤a3 a(1−e−at)+√ 3/integraldisplayt 0e−a(t−s)/ba∇dblA1 2sin(Yu)/ba∇dblds +2 a/parenleftBig√ 7 2α|1−α|r(ω)+√ 3|1−α|r′(ω)+2r′′(ω)/parenrightBig (1−e−a 2t).(4.13) For the integral on the right-hand side of (4.13), we note tha t /ba∇dblA1 2sin(Yu)/ba∇dbl ≤ /ba∇dblA1 2Yu/ba∇dbl ≤a4/ba∇dblQY(s,θ−tω,Y0(θ−tω))/ba∇dblE, wherea4=/radicalbig 2/(2−δ). Since /ba∇dblQY(s,θ−tω,Y0(θ−tω))/ba∇dblE ≤e−as/ba∇dblQY0(θ−tω)/ba∇dblE+/integraldisplays 0e−a(s−τ)/ba∇dblF(θτ−tω,Y(τ,θ−tω,Y0(θ−tω)))/ba∇dblEdτ, 13we find that /integraldisplayt 0e−a(t−s)/ba∇dblA1 2sin(Yu)/ba∇dblds ≤a4te−at/ba∇dblQY0(θ−tω)/ba∇dblE+a4/integraldisplayt 0/integraldisplays 0e−a(t−τ)/ba∇dblF(θτ−tω,Y(τ,θ−tω,Y0(θ−tω)))/ba∇dblEdτds ≤a4te−at/ba∇dblQY0(θ−tω)/ba∇dblE +a4/integraldisplayt 0/parenleftBigg 2 a/parenleftBig a1r(ω)+r′(ω)/parenrightBig (e−a 2(t−s)−e−a 2t)+a2 a(e−a(t−s)−e−at)/parenrightBigg ds =a4te−at/ba∇dblQY0(θ−tω)/ba∇dblE+2a4 a/parenleftBig a1r(ω)+r′(ω)/parenrightBig/parenleftBig2 a(1−e−a 2t)−te−a 2t/parenrightBig +a2a4 a/parenleftBig1 a(1−e−at)−te−at/parenrightBig . (4.14) Combining (4.11), (4.13) and (4.14), there is a T(ω)>0 such that for all t≥T(ω), /ba∇dblQY1(t,θ−tω,Y10(θ−tω))/ba∇dbl˜E =/ba∇dblQY1(t,θ−tω,Y10(θ−tω))/ba∇dblE+/ba∇dblCQY1(t,θ−tω,Y10(θ−tω))/ba∇dblE ≤R1(ω), (4.15) whereR1(ω) =a5r(ω) +a6r′(ω) +8 ar′′(ω) +a7is a tempered random variable, in which a5=4a1+2√ 7α|1−α| a+8√ 3a1a4 a2,a6=4+4√ 3|1−α| a+8√ 3a4 a2anda7=2a2+2a3 a+2√ 3a2a4 a2. Now, let ω/ma√sto→B1(ω) be the random pseudo-ball in ˜Ecentered at origin with random radius ω/ma√sto→R1(ω), thenω/ma√sto→B1(ω) is tempered and measurable. By (4.9), (4.15) and QY(t,θ−tω,φ0(θ−tω)) =QY1(t,θ−tω,Y10(θ−tω))+QY2(t,Y20(θ−tω)), we have for ω∈Ω, dH(Y(t,θ−tω,B0(θ−tω)),B1(ω))→0 ast→ ∞. Then by the compact embedding of ˜EintoE,ω/ma√sto→QB1(ω) is compact in E2, which implies thatω/ma√sto→B1(ω) :=B1(ω) (modp0) is a tempered random compact attracting set for Y. Thus by Theorem 2.3, Yhas a unique random attractor ω/ma√sto→A0(ω), where A0(ω) =/intersectiondisplay t>0/uniondisplay τ≥tY(τ,θ−τω,B1(θ−τω)), ω∈Ω. This completes the proof. Remark 4.5. (1) For any α >0andλ1=K˜λ1>0(see(3.2)), there is a δ∈(0,1]such that a >0holds, where ais as in(3.13)and˜λ1is the smallest positive eigenvalue of −△and a constant. (2) We can say that the random dynamical Y(orφ) has a unique random attractor in the sense that the induced random dynamical system Y(orΦ) has a unique random attractor, 14and we will say that Y(orφ) has a unique random attractor directly in the sequel. We denote the random attractor of Yandφbyω/ma√sto→A0(ω)andω/ma√sto→A(ω)respectively. Indeed, ω/ma√sto→A0(ω)andω/ma√sto→A(ω)satisfy A0(ω) =A0(ω) (modp0),A(ω) =A(ω) (modp0), ω∈Ω. (3) For the deterministic damped sine-Gordon equation with homogeneous Neumann boundary condition, the authors proved in [27] that the random attrac tor is a horizontal curve pro- vided that αandKare sufficiently large. Similarly, we expect that the random a ttractor ω/ma√sto→A(ω)ofφhas the similar property, i.e., A(ω)is a horizontal curve for each ω∈Ω provided that αandKare sufficiently large. We prove that this is true in next secti on. (4) By (2), system (1.1)-(1.2)is dissipative (i.e. it possesses a random attractor). In sec tion 6, we will show that (1.1)-(1.2)with sufficiently large αandKalso has a rotation number and hence all the solutions tend to oscillate with the same fr equency eventually. 5 One-dimensional Random Attractor In this section, we apply the theory established in [7] to sho w that the random attractor ofY(orφ) is one-dimensional provided that αandKare sufficiently large. This method has been used by Chow, Shen and Zhou [6] to systems of coupled nois y oscillators. Throughout this section we assume that p0= 2πη0= (2π,0)⊤∈E1anda >4LF(see (3.13) for the definition ofaand see (3.12) for the upper bound of LF). We remark in the end of this section that this condition can be satisfied provided that αandKare sufficiently large. Definition 5.1. Suppose {Φω}ω∈Ωis a family of maps from E1toE2andn∈N. A family of graphsω/ma√sto→ℓ(ω)≡ {(p,Φω(p)) :p∈E1}is said to be a random np0-periodic horizontal curve if ω/ma√sto→ℓ(ω)is a random set and {Φω}ω∈Ωsatisfy the Lipshcitz condition /ba∇dblΦω(p1)−Φω(p2)/ba∇dblE≤ /ba∇dblp1−p2/ba∇dblEfor allp1,p2∈E1, ω∈Ω and the periodic condition Φω(p+np0) = Φω(p)for allp∈E1, ω∈Ω. Clearly, for any ω∈Ω,ℓ(ω) is a deterministic np0-periodic horizontal curve. When n= 1, we simply call it a horizontal curve. Lemma 5.2. Leta >4LF. Suppose that ω/ma√sto→ℓ(ω)is a random np0-periodic horizontal curve inE. Then,ω/ma√sto→Y(t,ω,ℓ(ω))is also a random np0-periodic horizontal curve in Efor allt >0. Moreover, ω/ma√sto→Y(t,θ−tω,ℓ(θ−tω))is a random np0-periodic horizontal curve for all t >0. Proof.First, since Yis a random dynamical system and ω/ma√sto→ℓ(ω) is a random set in E, ω/ma√sto→Y(t,ω,ℓ(ω)) andω/ma√sto→Y(t,θ−tω,ℓ(θ−tω)) are random sets in Efor allt >0. We next show the Lipschitz condition and periodic condition. It is sufficient to prove the Lipschitz condition and periodic condition valid for ω/ma√sto→ℓ(ω) in D(C) sinceD(C) is dense in E. Clearly, for ω∈Ω andt >0, Y(t,ω,ℓ(ω)) ={(PY(t,ω,p+Φω(p)),QY(t,ω,p+Φω(p))) :p∈E1∩D(C)}. 15Forp1,p2∈E1∩D(C),p1/ne}ationslash=p2, letYi(t,ω) =Y(t,ω,pi+Φω(pi)),i= 1,2,p(t,ω) =P(Y1(t,ω)− Y2(t,ω)) andq(t,ω) =Q(Y1(t,ω)−Y2(t,ω)), where P,Qare defined as in section 1. We have by Lemma 3.4 PYi(t,ω) =eCtP(pi+Φω(pi))+/integraldisplayt 0eC(t−s)PF(θsω,Yi(s,ω))ds =P(pi+Φω(pi))+/integraldisplayt 0PF(θsω,Yi(s,ω))ds, i= 1,2, and then,d dtPYi(t,ω) =PF(θtω,Yi(t,ω)), i= 1,2, it then follows that d dtp(t,ω) =d dtP(Y1(t,ω)−Y2(t,ω)) =P(F(θtω,Y1(t,ω))−F(θtω,Y2(t,ω))).(5.1) Sincep(t,ω)+q(t,ω) =Y1(t,ω)−Y2(t,ω), d dt(p(t,ω)+q(t,ω)) =d dt(Y1(t,ω)−Y2(t,ω)) =C(Y1(t,ω)−Y2(t,ω))+F(θtω,Y1(t,ω))−F(θtω,Y2(t,ω)), then, by the orthogonal decomposition, d dtq(t,ω) =C(Y1(t,ω)−Y2(t,ω))+Q(F(θtω,Y1(t,ω))−F(θtω,Y2(t,ω))) =Cq(t,ω)+Q(F(θtω,Y1(t,ω))−F(θtω,Y2(t,ω))).(5.2) We find from (5.1) that d dt/ba∇dblp(t,ω)/ba∇dbl2 E= 2/angbracketleftbig p(t,ω),d dtp(t,ω)/angbracketrightbig E ≥ −2/ba∇dblp(t,ω)/ba∇dblE·/ba∇dblP(F(θtω,Y1(t,ω))−F(θtω,Y2(t,ω)))/ba∇dblE ≥ −2LF(/ba∇dblp(t,ω)/ba∇dbl2 E+/ba∇dblp(t,ω)/ba∇dblE/ba∇dblq(t,ω)/ba∇dblE). Similarly, by (5.2) and Lemma 3.4, d dt/ba∇dblq(t,ω)/ba∇dbl2 E≤ −2a/ba∇dblq(t,ω)/ba∇dbl2 E+2LF(/ba∇dblp(t,ω)/ba∇dblE/ba∇dblq(t,ω)/ba∇dblE+/ba∇dblq(t,ω)/ba∇dbl2 E). Becausea >4LF, ifthereisa t0≥0suchthat /ba∇dblq(t0,ω)/ba∇dblE=/ba∇dblp(t0,ω)/ba∇dblEandsince /ba∇dblp(t,ω)/ba∇dblE/ne}ationslash= 0 fort≥0, then d dt/vextendsingle/vextendsingle/vextendsingle t=t0/parenleftBig /ba∇dblq(t,ω)/ba∇dbl2 E−/ba∇dblp(t,ω)/ba∇dbl2 E/parenrightBig ≤(8LF−2a)/ba∇dblq(t0,ω)/ba∇dbl2 E<0, which means that there is a ¯t0> t0such that for t∈(t0,¯t0), /ba∇dblq(t,ω)/ba∇dbl2 E−/ba∇dblp(t,ω)/ba∇dbl2 E</ba∇dblq(0,ω)/ba∇dbl2 E−/ba∇dblp(0,ω)/ba∇dbl2 E =/ba∇dblΦω(p1)−Φω(p2)/ba∇dbl2 E−/ba∇dblp1−p2/ba∇dbl2 E ≤0, 16namely,/ba∇dblq(t,ω)/ba∇dblE</ba∇dblp(t,ω)/ba∇dblEfort∈(t0,¯t0). If there is another t1≥¯t0such that /ba∇dblq(t1,ω)/ba∇dblE=/ba∇dblp(t1,ω)/ba∇dblE, then d dt/vextendsingle/vextendsingle/vextendsingle t=t1/parenleftBig /ba∇dblq(t,ω)/ba∇dbl2 E−/ba∇dblp(t,ω)/ba∇dbl2 E/parenrightBig ≤(8LF−2a)/ba∇dblq(t1,ω)/ba∇dbl2 E<0, which means that there is a ¯t1> t1such that for t∈(t1,¯t1),/ba∇dblq(t,ω)/ba∇dblE</ba∇dblp(t,ω)/ba∇dblE. Continue this process, we have for all t≥0,/ba∇dblq(t,ω)/ba∇dblE≤ /ba∇dblp(t,ω)/ba∇dblE, i.e., /ba∇dblQ(Y1(t,ω)−Y2(t,ω))/ba∇dblE≤ /ba∇dblP(Y1(t,ω)−Y2(t,ω))/ba∇dblE, which shows that ω/ma√sto→Y(t,ω,ℓ(ω)) satisfies the Lipschitz condition in Definition 5.1. We next show the periodic condition. We find from Lemma 3.2 tha t Y(t,ω,p+Φω(p))+np0=Y(t,ω,p+np0+Φω(p)).. Since Φω(p) = Φω(p+np0),Y(t,ω,p+Φω(p))+np0=Y(t,ω,p+np0+Φω(p+np0)). It follows that QY(t,ω,p+Φω(p)) =QY(t,ω,p+np0+Φω(p+np0)). Consequently, ω/ma√sto→Y(t,ω,ℓ(ω)) is a random np0-periodic horizontal curve for all t >0. Moreover, for any fixed ω∈Ω andt >0, ¯ω=θ−tω∈Ω is fixed. Then, Y(t,¯ω,ℓ(¯ω)) is a deterministic np0-periodic horizontal curve, which yields the assertion. Chooseγ∈(0,a 2) such that 2 α/parenleftBigg 1 γ+1 a−2γ/parenrightBigg <1, (5.3) where2 αis the upper bound of the Lipschitz constant of F(see (3.12)). We remark in the end of this section that such a γexists provided that αandKare sufficiently large. We next show the main result in this section. Theorem 5.3. Assume that a >4LFand that there is a γ∈(0,a 2)such that (5.3)holds. Then the random attractor ω/ma√sto→A0(ω)of the random dynamical system Yis a random horizontal curve. Proof.By the equivalent relation between φandY, we mainly focus on equation (3.5), which can be viewed as a deterministic system with a random paramet erω∈Ω. We write it here as (3.5)ωfor some fixed ω∈Ω. Observe that the linear part of (3.5) ω, i.e. ˙Y=CY (5.4) has a one-dimensional center space Ec= span{(1,0)}=E1and a one co-dimensional stable spaceEs=E2. We first show that (3.5) ωhas a one-dimensional invariant manifold, denoted by W(ω), and will show later that W(ω) exponentially attracts all the solutions of (3.5) ω. LetFω(t,Y) =F(θtω,Y),ω∈Ω. For fixed ω∈Ω, consider the following integral equation ˜Y(t) =eCtξ+/integraldisplayt 0eC(t−s)PFω(s,˜Y(s))ds+/integraldisplayt −∞eC(t−s)QFω(s,˜Y(s))ds, t≤0,(5.5) 17whereξ=P˜Y(0)∈E1. Forg: (−∞,0]→Esuch that supt≤0/ba∇dbleγtg(t)/ba∇dblE<∞, define (Lg)(t) =/integraldisplayt 0eC(t−s)Pg(s)ds+/integraldisplayt −∞eC(t−s)Qg(s)ds, t≤0. It is easy to see that sup t≤0/ba∇dbleγt(Lg)(t)/ba∇dblE≤/parenleftBigg 1 γ+1 a−γ/parenrightBigg sup t≤0/ba∇dbleγtg(t)/ba∇dblE≤/parenleftBigg 1 γ+1 a−2γ/parenrightBigg sup t≤0/ba∇dbleγtg(t)/ba∇dblE, which means that /ba∇dblL/ba∇dbl ≤1 γ+1 a−2γ. Then, Theorem 3.3 in [7] shows that for any ξ∈E1, equation (5.5) has a unique solution ˜Yω(t,ξ) satisfying supt≤0/ba∇dbleγt˜Yω(t,ξ)/ba∇dblE<∞. Let h(ω,ξ) =Q˜Yω(0,ξ) =/integraldisplay0 −∞e−CsQFω(s,˜Yω(s,ξ))ds, ω∈Ω. Let W(ω) ={ξ+h(ω,ξ) :ξ∈E1}, ω∈Ω. For anyǫ∈(0,γ) in Lemma 3.5 and Corollary 3.6, we have /ba∇dblh(θ−tω,ξ)/ba∇dblE≤1 a−ǫ(a1r(ω)+r′(ω))eǫt+a2 a, t≥0. (5.6) Observe that ˜Yω(t,ξ) =eCtξ+/integraldisplayt 0eC(t−s)PFω(s,˜Yω(s,ξ))ds+/integraldisplayt −∞eC(t−s)QFω(s,˜Y(s,ω,ξ))ds =eCt(ξ+h(ω,ξ))+/integraldisplayt 0eC(t−s)Fω(s,˜Yω(s,ξ))ds, i.e.,˜Yω(t,ξ) is the solution of (3.5) with initial data ξ+h(ω,ξ) fort≤0. Thus, for Y0(ω) = ξ+h(ω,ξ)∈W(ω), there is a negative continuation of Y(t,ω,Y0(ω)), i.e., Y(t,ω,Y0(ω)) =˜Yω(t,ξ), t≤0. (5.7) Moreover, for t≥0, we obtain from (3.6) and (5.7) that Y(t,ω,Y0(ω)) =eCt(ξ+h(ω,ξ))+/integraldisplayt 0eC(t−s)F(θsω,Y(s,ω,Y0(ω)))ds =eCtξ+/integraldisplayt 0eC(t−s)F(θsω,Y(s,ω,Y0(ω)))ds+/integraldisplay0 −∞eC(t−s)QFω(s,˜Yω(s,ξ))ds =eCtξ+/integraldisplayt 0eC(t−s)F(θsω,Y(s,ω,Y0(ω)))ds+/integraldisplay0 −∞eC(t−s)QF(θsω,Y(s,ω,Y0(ω)))ds =eCtξ+/integraldisplayt 0eC(t−s)PF(θsω,Y(s,ω,Y0(ω)))ds+/integraldisplayt −∞eC(t−s)QF(θsω,Y(s,ω,Y0(ω)))ds =eCt/parenleftBig ξ+/integraldisplayt 0e−CsPF(θsω,Y(s,ω,Y0(ω)))ds/parenrightBig +/integraldisplay0 −∞e−CsQF(θt+sω,Y(t+s,ω,Y0(ω)))ds. 18Then by the uniqueness of solution of (5.5) for fixed ω∈Ω, we have h/parenleftBig θtω,eCt/parenleftBig ξ+/integraldisplayt 0e−CsPF(θsω,Y(s,ω,Y0(ω)))ds/parenrightBig/parenrightBig =/integraldisplay0 −∞e−CsQF(θt+sω,Y(t+s,ω,Y0(ω)))ds, and then for t≥0, Y(t,ω,W(ω)) =W(θtω). (5.8) By (5.7) and (5.8), W(ω) is an invariant manifold of (3.5) ω. Next, we show that W(ω) attracts the solutions of (3.5) ω, more precisely, for the given ω∈Ω, we prove the existence of a stable foliation {Ws(ω,Y0) :Y0∈W(ω)}of the invariant manifold W(ω) of (3.5) ω. Consider the following integral equation ˆY(t) =eCtη+/integraldisplayt 0eC(t−s)Q/parenleftBig Fω(s,ˆY(s)+Yω(s,ξ+h(ω,ξ))) −Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig ds +/integraldisplayt ∞eC(t−s)P/parenleftBig Fω(s,ˆY(s)+Yω(s,ξ+h(ω,ξ))) −Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig ds, t≥0,(5.9) whereξ+h(ω,ξ)∈W(ω),η=QˆY(0)∈E2andYω(t,ξ+h(ω,ξ)) :=Y(t,ω,ξ+h(ω,ξ)),t≥0 is the solution of (3.5) with initial data ξ+h(ω,ξ) for fixed ω∈Ω. Theorem 3.4 in [7] shows that for any ξ∈E1andη∈E2, equation (5.9) has a unique solution ˆYω(t,ξ,η) satisfying supt≥0/ba∇dbleγtˆYω(t,ξ,η)/ba∇dblE<∞and for any ξ∈E1,η1, η2∈E2, sup t≥0eγt/ba∇dblˆYω(t,ξ,η1)−ˆYω(t,ξ,η2)/ba∇dblE≤M/ba∇dblη1−η2/ba∇dblE. (5.10) whereM=1 1−2 α/parenleftbig 1 γ+1 a−2γ/parenrightbig. Let ˆh(ω,ξ,η) =ξ+PˆYω(0,ξ,η) =ξ+/integraldisplay0 ∞e−CsP/parenleftBig Fω(s,ˆYω(s,ξ,η)+Yω(s,ξ+h(ω,ξ))) −Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig ds. Then,Ws(ω,ξ+h(ω,ξ)) ={η+h(ω,ξ)+ˆh(ω,ξ,η) :η∈E2}is the stable foliation of W(ω) at ξ+h(ω,ξ). Observe that ˆYω(t,ξ,η)+Yω(t,ξ+h(ω,ξ))−Yω(t,ξ+h(ω,ξ)) =ˆYω(t,ξ,η) =eCt(η+h(ω,ξ)+ˆh(ω,ξ,η)−ξ−h(ω,ξ)) +/integraldisplayt 0eC(t−s)/parenleftBig Fω(s,ˆYω(s,ξ,η)+Yω(s,ξ+h(ω,ξ))) −Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig ds(5.11) 19and Yω(t,η+h(ω,ξ)+ˆh(ω,ξ,η))−Yω(t,ξ+h(ω,ξ)) =eCt(η+h(ω,ξ)+ˆh(ω,ξ,η)−ξ−h(ω,ξ)) +/integraldisplayt 0eC(t−s)/parenleftBig Fω(s,Yω(s,η+h(ω,ξ)+ˆh(ω,ξ,η))) −Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig ds.(5.12) Comparing (5.11) with (5.12), we find that ˆYω(t,ξ,η) =Yω(t,η+h(ω,ξ)+ˆh(ω,ξ,η))−Yω(t,ξ+h(ω,ξ)), t≥0.(5.13) In addition, if η= 0, then by the uniqueness of solution of (5.9), ˆYω(t,ξ,0)≡0 fort≥0, which associates with (5.10) and (5.13) show that sup t≥0eγt/ba∇dblYω(t,η+h(ω,ξ)+ˆh(ω,ξ,η))−Yω(t,ξ+h(ω,ξ))/ba∇dblE≤M/ba∇dblη/ba∇dblE(5.14) for anyξ∈E1andη∈E2. We now claim that ω/ma√sto→W(ω) is the random attractor of Y. Letω/ma√sto→B(ω) be any pseudo- tempered random set in E. For any ω/ma√sto→Y0(ω)∈ω/ma√sto→B(ω), there is ω/ma√sto→ξ(ω)∈E1such that Y0(θ−tω)∈Ws(θ−tω,ξ(θ−tω)+h(θ−tω,ξ(θ−tω))). Letη(θ−tω) =QY0(θ−tω)−h(θ−tω,ξ(θ−tω)). By (5.6), it is easy to see that sup Y0(θ−tω)∈B(θ−tω)/ba∇dblη(θ−tω)/ba∇dbl ≤ sup Y0(θ−tω)∈B(θ−tω)/ba∇dblQY0(θ−tω)/ba∇dbl+1 a−ǫ(a1r(ω)+r′(ω))eǫt+a2 a. It then follows from (5.14) and the fact that ω/ma√sto→QB(ω) is tempered that sup Y0(θ−tω)∈B(θ−tω)/ba∇dblY(t,θ−tω,Y0(θ−tω))−Y(t,θ−tω,ξ(θ−tω)+h(θ−tω,ξ(θ−tω)))/ba∇dblE ≤Me−γtsup Y0(θ−tω)∈B(θ−tω)/ba∇dblη(θ−tω)/ba∇dblE ≤Me−γtsup Y0(θ−tω)∈B(θ−tω)/ba∇dblQY0(θ−tω)/ba∇dbl+M a−ǫ(a1r(ω)+r′(ω))e(ǫ−γ)t+a2M ae−γt →0 ast→ ∞, which associates with (5.8) lead to dH(Y(t,θ−tω,B(θ−tω)),W(ω))→0 ast→ ∞. Therefore, A0(ω) =W(ω) forω∈Ω. Next, we show that ω/ma√sto→A0(ω) is a random horizontal curve. Infact, forsomerandomhorizontalcurve ω/ma√sto→ℓ(ω)inE, forexample, ℓ(ω)≡ {(p,Φω(p)) : 20Φω(p) =c,p∈E1},ω∈Ω, where c∈E2is constant, it must be contained in some pseudo- tempered random set, for example ω/ma√sto→B2/bardblc/bardblE(ω), where B2/bardblc/bardblE(ω) is a pseudo-ball with radius 2/ba∇dblc/ba∇dblE. Then, for ω∈Ω, dH(Y(t,θ−tω,ℓ(θ−tω)),A0(ω))→0 ast→ ∞, which means that lim t→∞Y(t,θ−tω,ℓ(θ−tω))⊂A0(ω). SinceA0(ω) is one-dimensional, we have forω∈Ω, A0(ω) = lim t→∞Y(t,θ−tω,ℓ(θ−tω)). It then follows from Lemma 5.2 that ω/ma√sto→A0(ω) is a random horizontal curve. Corollary 5.4. Assume that a >4LFand that there is a γ∈(0,a 2)such that (5.3)holds. Then the random attractor ω/ma√sto→A(ω)of the random dynamical system φis a random horizontal curve. Proof.It follows from Corollary 4.2, Remark 4.5 and Theorem 5.3. Remark 5.5. At the beginning of this section, we assume that a >4LF. Sincea=α 2−|α 2−δλ1 α| andLF≤2 α, we can take α,λ1satisfyingα 2−/vextendsingle/vextendsingle/vextendsingleα 2−δλ1 α/vextendsingle/vextendsingle/vextendsingle>8 α, whereλ1is the smallest positive eigenvalue of Aand its value is determined by the diffusion coefficient K. On the other hand, we need some γ∈(0,a 2)such that (5.3)holds. Note that min γ∈(0,a 2)/parenleftBigg 1 γ+1 a−2γ/parenrightBigg =/parenleftBigg 1 γ+1 a−2γ/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle γ=(2−√ 2)a 2=√ 2 (3√ 2−4)a, which implies that there exist α,λ1satisfying α 2−/vextendsingle/vextendsingle/vextendsingleα 2−δλ1 α/vextendsingle/vextendsingle/vextendsingle>2√ 2 (3√ 2−4)α>8 α. (5.15) Indeed, let c=2√ 2 3√ 2−4, then for any α >√ 2candλ1> c, there is a δ >0satisfying c λ1< δ <min/braceleftBigα2−c λ1,1/bracerightBig such that (5.15)holds. 6 Rotation Number In this section, we study the phenomenon of frequency lockin g, i.e., the existence of a rotation number of the stochastic damped sine-Gordon equat ion (1.1)-(1.2), which characterizes the speed that the solution of (1.1)-(1.2) moves around the o ne-dimensional random attractor. 21Definition 6.1. The stochastic damped sine-Gordon equation (1.1)with boundary condition (1.2)is said to have a rotation number ρ∈Rif, forP-a.e.ω∈Ωand each φ0= (u0,u1)⊤∈E, the limit limt→∞Pφ(t,ω,φ0) texists and lim t→∞Pφ(t,ω,φ0) t=ρη0, whereη0= (1,0)⊤is the basis of E1. We remark that the rotation number of (1.1)-(1.2) (if exists ) is unique. In fact, assume that ρ1andρ2are rotation numbers of (1.1)-(1.2). Then there is ω∈Ω such that for any φ0∈E, ρ1η0= lim t→∞Pφ(t,ω,φ0) t=ρ2η0. Therefore, ρ1=ρ2and then the rotation number of (1.1)-(1.2) (if exists) is un ique. From (3.7), we have Pφ(t,ω,φ0) t=PY(t,ω,Y0(ω)) t+P(0,z(θtω))⊤ t, (6.1) whereφ0= (u0,u1)⊤andY0(ω) = (u0,u1−z(ω))⊤. By Lemma 2.1 in [12], it is easy to prove that lim t→∞P(0,z(θtω))⊤ t= (0,0)⊤. Thus, it sufficient to prove the existence of the rotation number of the random system (3.5). By the random dynamical system Ydefined in (4.1), we define the corresponding skew- product semiflow Θt: Ω×E→Ω×Efort≥0 by setting Θt(ω,Y0) = (θtω,Y(t,ω,Y0)). Obviously, (Ω ×E,F ×B,(Θt)t≥0) is a measurable dynamical system, where B=B(E) is the Borelσ-algebra of E. Lemma 6.2. There is a measure µonΩ×Esuch that (Ω×E,F ×B, µ,(Θt)t≥0)becomes an ergodic metric dynamical system. Proof.LetPrΩ(E) be the set of all random probability measures on EandPrP(Ω×E) be the set of all probability measures on Ω ×Ewith marginal P. We know from Proposition 3.3 and Proposition 3.6 in [9] that PrΩ(E) andPrP(Ω×E) are isomorphism. Moreover, both PrΩ(E) andPrP(Ω×E) are convex, and the convex structure is preserved by this is omorphism. Let Γ = {ω/ma√sto→µω∈PrΩ(E) :P-a.s.µω(A0(ω)) = 1, ω/ma√sto→µωis invariant for Y}. Clearly, Γ is convex. Since ω/ma√sto→A0(ω) is the random attractor of Y, we obtain from Corollary 6.13 in [9] that Γ/ne}ationslash=∅. Letω/ma√sto→µωbe an extremal point of Γ. Then, by the isomorphism between PrΩ(E) andPrP(Ω×E) and Lemma 6.19 in [9], the corresponding measure µon Ω×Eofω/ma√sto→µωis (Θt)t≥0-invariant and ergodic. Thus, (Ω ×E,F×B, µ,(Θt)t≥0) is an ergodic metric dynamical system. We next show a simple lemma which will be used. For any pi= (si,0)⊤∈E1,i= 1,2, we define p1≤p2ifs1≤s2. Then we have 22Lemma 6.3. Suppose that a >4LF. Letℓbe any deterministic np0-periodic horizontal curve (ℓsatisfies the Lipschitz and periodic condition in Definition 5 .1). For any Y1, Y2∈ℓwith PY1≤PY2, there holds PY(t,ω,Y1)≤PY(t,ω,Y2)fort >0, ω∈Ω. (6.2) Proof.Clearly, if PY1=PY2, then (6.2) holds. We now prove that (6.2) holds for PY1< PY2. If not, then by the continuity of Ywith respect to t, there is a t0>0 such that PY(t0,ω,Y1) = PY(t0,ω,Y2), which implies that Y(t0,ω,Y1) =Y(t0,ω,Y2) sinceY(t0,ω,Y1) andY(t0,ω,Y2) belong to the same deterministic np0-periodic horizontal curve Y(t0,ω,ℓ), which leads to a contradiction. The lemma is thus proved. We now show the main result in this section. Theorem 6.4. Assume that a >4LF. Then the rotation number of (3.5)exists. Proof.Note that PY(t,ω,Y0) t=PY0 t+1 t/integraldisplayt 0PF(θsω,Y(s,ω,Y0))ds. SinceF(θsω,Y(s,ω,Y0)+kp0) =F(θsω,Y(s,ω,Y0)),∀k∈Z, wecanidentify F(θsω,Y(s,ω,Y0)) withF(θsω,Y(s,ω,Y0)). Precisely, define h:E→ E,Y/ma√sto→ {Y}, whereEis the collection of all singleton sets of E, i.e.E={{Y}:Y∈E}(see Remark 6.6 for more details of the space E). Clearly,his a homeomorphism from EtoE. Then, F(θsω,Y(s,ω,Y0)) =h−1(F(θsω,Y(s,ω,Y0))). Thus, PY(t,ω,Y0) t=PY0 t+1 t/integraldisplayt 0Ph−1(F(θsω,Y(s,ω,Y0)))ds =PY0 t+1 t/integraldisplayt 0F(Θs(ω,Y0))ds.(6.3) whereF=P◦h−1◦F∈L1(Ω×E,F ×B, µ). Lett→ ∞in (6.3), lim t→∞PY0 t= (0,0)⊤and by Lemma 5.2 and Ergodic Theorems in [1], there exist a consta ntρ∈Rsuch that lim t→∞1 t/integraldisplayt 0F(Θs(ω,Y0))ds=ρη0, which means lim t→∞PY(t,ω,Y0) t=ρη0 forµ-a.e.(ω,Y0)∈Ω×E. Thus, there is Ω∗⊂Ω withP(Ω∗) = 1 such that for any ω∈Ω∗, there isY∗ 0(ω)∈Esuch that lim t→∞PY(t,ω,Y∗ 0(ω)) t=ρη0. By Lemma 3.2, we have that for any n∈Nandω∈Ω∗, lim t→∞PY(t,ω,Y∗ 0(ω)±np0) t= lim t→∞PY(t,ω,Y∗ 0(ω))±np0 t=ρη0. (6.4) 23Now for any ω∈Ω∗and any Y0∈E, there is n0(ω)∈Nsuch that PY∗ 0(ω)−n0(ω)p0≤PY0≤PY∗ 0(ω)+n0(ω)p0 and there is a n0(ω)p0-periodic horizontal curve l0(ω) such that Y∗ 0(ω)−n0(ω)p0,Y0,Y∗ 0(ω)+ n0(ω)p0∈l0(ω). Then by Lemma 6.3, we have PY(t,ω,Y∗ 0(ω)−n0(ω)p0)≤PY(t,ω,Y0)≤PY(t,ω,Y∗ 0(ω)+n0(ω)p0), which together with (6.4) implies that for any ω∈Ω∗and any Y0∈E, lim t→∞PY(t,ω,Y0) t=ρη0. Consequently, for any a.e. ω∈Ω and any Y0∈E, lim t→∞PY(t,ω,Y0) t=ρη0. The theorem is thus proved. Corollary 6.5. Assume that a >4LF. Then the rotation number of the stochastic damped sine-Gordon equation (1.1)with the boundary condition (1.2)exists. Proof.It follows from (6.1) and Theorem 6.4. Remark 6.6. We first note that the space E={{Y}:Y∈E}in the proof of Theorem 6.4 is a linear space according to the linear structure defined by α{X}+β{Y}={αX+βY},forα,β∈R,{X},{Y} ∈ E. Also, for {X},{Y} ∈ E, we define /an}b∇acketle{t{X},{Y}/an}b∇acket∇i}htE=/an}b∇acketle{tX,Y/an}b∇acket∇i}htE. (6.5) It is easy to verify that the functional /an}b∇acketle{t·,·/an}b∇acket∇i}htE:E ×E → Rdefined by (6.5)is bilinear, symmetric and positive, thus defining the scalar product in EoverR. Moreover, the completeness of Eis from the completeness of E. Hence, Eis a Hilbert space. Remark 6.7. In the proof of Theorem 6.4, we used an ergodic invariant measur eµof(Ω× E,F ×B, µ,(Θt)t≥0). It should be pointed out that the measure µonΩ×Ethat makes (Ω× E,F ×B, µ,(Θt)t≥0)becomes an ergodic metric dynamical system may not be unique , because the convex set Γin the proof of Lemma 6.2 may have more than one extremal point s. However, as mentioned above, the rotation number in Theorem 6.4 and Coro llary 6.5 are independent of µand are unique. Acknowledgement . We would like to thank the referee for carefully reading the manuscript and making very useful suggestions. 24References [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. [2] P. W. Bates, K. Lu and B. Wang, Random attractors for stoch astic reaction-diffusion equations on unbounded domains, J. Differential Equations, 2 46(2009), 845-869. [3] A. R. Bishop, K. Fesser, P. S. Lomdahl, and S. E. Trullinge r, Influence of solitons in the initial state on chaos in the driven damped sine-Gordon syst em, Phys. D 7 (1983), 259–279. [4] A. R. Bishop and P. S. 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2014-02-07

This paper is devoted to the study of the asymptotic dynamics of the stochastic damped sine-Gordon equation with homogeneous Neumann boundary condition. It is shown that for any positive damping and diffusion coefficients, the equation possesses a random attractor, and when the damping and diffusion coefficients are sufficiently large, the random attractor is a one-dimensional random horizontal curve regardless of the strength of noise. Hence its dynamics is not chaotic. It is also shown that the equation has a rotation number provided that the damping and diffusion coefficients are sufficiently large, which implies that the solutions tend to oscillate with the same frequency eventually and the so called frequency locking is successful.

One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation

1402.1787v1

arXiv:1608.00984v2 [cond-mat.mes-hall] 13 Aug 2016Ferromagnetic Damping/Anti-damping in a Periodic 2D Helic al surface; A Non-Equilibrium Keldysh Green Function Approach Farzad Mahfouzi1,∗and Nicholas Kioussis1 1Department of Physics, California State University, North ridge, California 91330-8268, USA In this paper, we investigate theoretically the spin-orbit torque as well as the Gilbert damping for a two band model of a 2D helical surface state with a Ferromagn etic (FM) exchange coupling. We decompose the density matrix into the Fermi sea and Fermi sur face components and obtain their contributions to the electronic transport as well as the spi n-orbit torque (SOT). Furthermore, we obtain the expression for the Gilbert damping due to the surf ace state of a 3D Topological Insulator (TI) and predicted its dependence on the direction of the mag netization precession axis. PACS numbers: 72.25.Dc, 75.70.Tj, 85.75.-d, 72.10.Bg I. INTRODUCTION The spin-transfer torque (STT) is a phenomenon in which spin current of large enough density injected into a ferromagnetic layer switches its magnetization from one static configuration to another [1]. The origin of STT is absorption of itinerant flow of angular momen- tum components normal to the magnetization direc- tion. It represents one of the central phenomena of the second-generation spintronics, focused on manipulation of coherent spin states, since reduction of current den- sities (currently of the order 106-108A/cm2) required for STT-based magnetization switching is expected to bring commercially viable magnetic random access mem- ory (MRAM) [2]. The rich nonequilibrium physics [3] arising in the interplay of spin currents carried by fast conduction electrons and collective magnetization dy- namics, viewed as the slow classical degree of freedom, is of great fundamental interest. Very recent experiments [4, 5] and theoretical stud- ies[6] havesoughtSTT innontraditionalsetupswhich do not involvetheusual two(spin-polarizingandfree) F lay- ers with noncollinear magnetizations [3], but rely instead on the spin-orbit coupling (SOC) effects in structures lacking inversion symmetry. Such “SO torques” [7] have been detected [4] in Pt/Co/AlO xlateral devices where current flows in the plane of Co layer. Concurrently, the recent discovery [8] of three-dimensional (3D) topologi- cal insulators (TIs), which possess a usual band gap in the bulk while hosting metallic surfaces whose massless Dirac electrons have spins locked with their momenta due to the strong Rashba-type SOC, has led to theoreti- cal proposals to employ these exotic states of matter for spintronics [9] and STT in particular [10]. For example, magnetizationofa ferromagneticfilm with perpendicular anisotropy deposited on the TI surface could be switched by interfacial quantum Hall current [10]. In this paper, we investigate the dynamical properties of a FM/3DTI heterostructure, where the F overlayer ∗farzad.mahfouzi@gmail.comcovers a TI surface and the device is periodic along in- planex−ydirections. The effect of the F overlayer is a proximityinduced exchangefield −∆surf/vector m·/vectorσ/2superim- posed on the Dirac cone dispersion. For a partially cov- ered FM/TI heterostructure, the spin-momentum-locked Dirac electrons flip their spin upon entering into the in- terface region, thereby inducing a large antidamping-like SOT on the FM [15–17]. The antidamping-like SOT driven by this mechanism which is unique to the sur- face of TIs has been predicted in Ref. [17], where a time- dependent nonequilibrium Green function [18] (NEGF)- based framework was developed. The formalism made it possible to separate different torque components in the presenceofarbitraryspin-flipprocesseswithinthedevice. Similaranti-dampingtorqueshasalsobeen predicted[19] to exist due to the Berry phase in periodic structures where the device is considered infinite in in-plane direc- tions and a Kubo formula was used to describe the SOT as a linear response to homoginiuos electric field at the interface. However, the connection between the two ap- proaches is not clear and one of the goals of the current paper is to address the similarities and the differences between the two. In the following we present the theo- retical formalism of the SOT and damping in the regime ofslowlyvaryingparametersofaperiodicsysteminspace and time. Generally, in a quantum system with slowly varying parameters in space and/or time, the system stays close toits equilibrium state ( i.e.adiabaticregime)and the ef- fects of the nonadiabaticity is taken into account pertur- batively using adiabatic expansion. Conventionally, this expansion is performed using Wigner representation [20] after the separation of the fast and slow variations in space and/or time. [21] The slow variation implies that the NEGFs vary slowly with the central space ( time ), /vector xc= (/vector x+/vector x′)/2(tc= (t+t′)/2 ), while they changefast with the relative space (time), /vector xr=/vector x−/vector x′(tr=t−t′). Here we use an alternative approach, where we consider (x,t) and (/vector xr,tr) as the natural variables to describe the close to adiabatic apace-time evolution of NEGFs and then perform the following Fourier transform ˇG(/vector xt;/vector x′t′) =/integraldisplaydE 2πd/vectork ΩkeiE(t−t′)+i/vectork·(/vector x−/vector x′)ˇG/vectorkE(/vector xt).(1)2 where, Ω kis the volume of the phase space that the /vectork-integration is being performed. The standard Dyson equation of motion for ˇG(/vector xt;/vector x′t′) is cumbersome to ma- nipulate[22,23]orsolvenumerically,[24]sotheyareusu- ally transformed to some other representation.[11] Gen- eralizing the equation to take into account slowly varying time and spatial dependence of the Hamiltonian we ob- tain, ˇG=/parenleftbigg GrG< 0Ga/parenrightbigg , (2) =/parenleftbigg Gr,−1 ad−iDxtΣ< 0 Ga,−1 ad−iDxt/parenrightbigg−1 , where, Gr,−1 ad= (E−iη)1−H(/vectork,t)−µ(/vector x),(3a) Σ<=−2iηf(E−i∂ ∂t−µ(/vector x)), (3b) Dxt=∂ ∂t+∂H ∂/vectork·/vector∇, (3c) and,η=/planckover2pi1/2τis the phenomenological broadening pa- rameter, where τis the relaxation time. It is worth mentioning that for a finite ηthe number of particles is not conserved, and a more accurate interpretation of the introduced broadening might be to consider it as an energy-independent scape rate of electrons to fictitious reservoirs attached to the positions /vector x. Consequently, a finite broadening could be interpreted as the existence of an interface in the model between each atom in the system and the reservoir that is spread homogeneously along the infinite periodic system. Eq. (2) shows that the effect of the space/time varia- tion is to replace E→E−i∂/∂tand/vectork→/vectork−i/vector∇in the equation of motion for the GFs in stationary state. To the lowest order with respect to the derivatives we can write, ˇG=ˇGad−i∂ˇGad ∂E∂ˇG−1 ad ∂tˇGad−i∂ˇGad ∂/vectork·/vector∇ˇG−1 adˇGad, (4) where, ˇG−1 ad=/parenleftbigg Gr,−1 ad−2iηf(E−µ(/vector x)) 0 Ga,−1 ad/parenrightbigg .(5) For the density matrix of the system, ρ(t) =1 iG<(t,t), we obtain, ρneq /vectork,t≈ −/integraldisplaydE 2πℜ/parenleftbigg [D(Gr ad),Gr ad]f+2iηD(Gr ad)Ga ad∂f ∂E/parenrightbigg (6) whereD=∂ ∂t−/vector∇µ·∂ ∂/vectorkis the differential operator act- ing on the slowly varying parameters in space and time.The details of the derivation is presented in Appendix.A. The density matrix in Eq. (6) is the central formula of the paper and consists of two terms; the first term con- tains the equilibrium Fermi distribution function from the electrons bellow the Fermi surface occupying a slowly (linearly) varying single particle states that has only in- terband contributions and can as well be formulated in terms of the Berry phase as we will show the following sections, and; the second term corresponds to the elec- trons with Fermi energy (at zero temperature we have, ∂f/∂E=δ(E−EF)) which are the only electrons al- lowed to get excited in the presence of the slowly varying perturbations. The fact that the first term originates from the assumption that the electric field is constant inside the metallic FM suggests that this term might dis- appearoncethescreeningeffect isincluded. Onthe other hand, duetothe factthat thesecondtermcorrespondsto the nonequilibrium electrons injected from the fictitious reservoirs attached to the device through the scape rate η, it might capture the possible physical processes that occur at the contact region and makes it more suitable for the calculation of the relevant physical observables in such systems. Using the expression for the nonequilibrium density matrix the local spin density can be obtained from, /vectorSneq(t) =/angb∇acketleft/vector σ/angb∇acket∇ightneq=1 4π2/integraldisplay d2/vectorkTr[ρneq /vectork,t/vector σ],(7) where/angb∇acketleft.../angb∇acket∇ightneqrefers to the ensemble average over many- body states out of equilibrium demonstrated by the nonequilibrium density matrix of the electrons and, Tr refers to the trace. In this case the time derivative in the differential operator Dleads to the damping of the dy- namics of the ferromagnet while the momentum deriva- tive leads to either damping or anti-damping of the FM dynamics depending on the direction of the applied elec- tric field. In the followingsection we apply the formalism to a two band helical surface state model attached to a FM. II. SOT AND DAMPING OF A HELICAL 2D SURFACE A two band Hamiltonian model for the system can be generally written as, H(/vectork,t) =ε0(/vectork)1+/vectorh(/vectork,t)·/vectorσ (8) where,/vectorh=/vectorhso(/vectork)+∆xc(/vectork) 2/vector m(t), with/vectorhso(/vectork) =−/vectorhso(−/vectork) and ∆ xc(/vectork) = ∆ xc(−/vectork) being spin-orbit and magnetic exchange coupling terms respectively. In particular in the case of Rashba type helical states we have /vectorhso= αsoˆez×/vectork. In this case for the adiabatic single particle GF we have, Gr ad(E,t) =(E−ε0−iη)1+/vectorh·/vectorσ (E−ε0−iη)2−|/vectorh|2(9)3 From Eq. (7) for the local spin density, we obtain (See Appendix B for details), /vectorSneq(t) =/integraldisplayd2/vectork 4π2/parenleftBigg/vectorh×D/vectorh 2|/vectorh|3(f1−f2)−(/vector∇µ·/vector v0)/vectorh 2η|/vectorh|(f′ 1−f′ 2) +(/vectorh×D/vectorh 2|/vectorh|2+ηD/vectorh−1 η(/vectorh·D/vectorh)/vectorh 2|/vectorh|2)(f′ 1+f′ 2)/parenrightBigg (10) where,f1,2=f(ε0± |/vectorh|) and/vector v0=∂ε0/∂/vectorkis the group velocity of electrons in the absence of the SOI. Here, we assumeη≪ |/vectorh|which corresponds to a system close to the ballistic regime. In this expression we kept the ηD/vectorh because of its unique vector orientation characteristics. As it becomes clear in the following, the first term in Eq. (10) is a topological quantity which in the presence of an electric field becomes dissipative and leads to an anti-damping torque. The second term in this expression leads to the Rashba-Edelstein field-like torque which is a nondissipative observable. The third term has the exact formasthe firstterm with the difference that it is strictly a Fermi surface quantity. The fourth term, also leads to a field like torque that as we will see in the following has similar features as the Rashba-Edelstein effect. It is im- portant to pay attention that unlike the first term, the rest of the terms in Eq. (10) are solely due to the flow of the non-equilibrium electrons on the Fermi surface. Furthermore, we notice that the terms that lead to dissi- pation in the presence of an electric field ( D ≡/vector∇µ·∂ ∂/vectork) become nondissipative when we consider D ≡∂/∂tand vice versa. A. Surface State of a 3D-TI In the case of the surface state of a 3D-TI, as an ap- proximation we can ignore ε0(/vectork) and consider the helical term as the only kinetic term of the Hamiltonian. In this casethelocalchargecurrentandthenonequilibriumlocal spin density share a similar expression, /vectorI=/angb∇acketleft∂(/vectorh·/vector σ)/∂/vectork/angb∇acket∇ight. For the conductivity, analogous to Eq. (10), we obtain, σij=e/integraldisplayd2/vectork 4π2 /vectorh·∂/vectorh ∂ki×∂/vectorh ∂kj 2|/vectorh|2/parenleftBigg f1−f2 |/vectorh|+f′ 1+f′ 2/parenrightBigg δi/negationslash=j +−η|∂/vectorh ∂ki|2+1 η(∂|/vectorh|2 ∂ki)2 2|/vectorh|2(f′ 1+f′ 2)δij (11) This shows that the Fermi sea component of the density matrixcontributesonlytotheanomalousHallconductiv- ity which is in terms of a winding number. On the other hand, the second term is finite only for the longitudinal components of the conductivity and can be rewritten in terms of the group velocity of the electrons in the system which leads to the Drude-like formula.Should the linear dispersion approximation for the ki- netic term in the Hamiltonian be valid in the range of the energy scale corresponding to the magnetic exchange coupling ∆ xc(i.e. when vF≫∆xc), the effect of the in- plane component of the magnetic exchange coupling is to shift the Dirac point (i.e. center of the k-space integra- tion) which does not affect the result ofthe k-integration. In this case after performing the partial time-momentum derivatives, ( D(/vectorh) =∆xc 2∂/vector m ∂t−vFˆez×/vector∇µ), we use /vectorh(/vectork,t) =vFˆez×/vectork+∆xc 2mz(t)ˆez, to obtain, /vectorSneq(t) =/integraldisplaykdk 4π|/vectorh|2/parenleftBigg /vectorS1f1−f2 |/vectorh|+(/vectorS1+/vectorS2)(f′ 1+f′ 2)/parenrightBigg , (12) where, /vectorS1(/vectork,t) =∆2 xc 4mz(t)ˆez×∂/vector m ∂t+∆xcvF 2mz(t)/vector∇µ(13) /vectorS2(/vectork,t) =∆xc 4η(2η2−v2 F|k|2)(∂mx ∂tˆex+∂my ∂tˆey) +∆xc 4η(2η2−∆2 xcm2 z 2)∂mz ∂tˆez −vF η(η2−v2 F|k|2 2)ˆez×/vector∇µ (14) The dynamics of the FM obeys the LLG equation where the conductions electrons insert torque on the FM mo- ments through the magnetic exchange coupling, ∂/vector m ∂t=/vector m× γ/vectorBext+∆xc 2/vectorSneq(t)−/summationdisplay ijαij 0∂mi ∂tˆej (15) where,αij 0=αji 0, withi,j=x,y,z, is the intrinsic Gilbert damping tensor of the FM in the absence of the TI surface state and /vectorBextis the total magnetic field applied on the FM aside from the contribution of the nonequilibrium electrons. While the terms that consist of /vector∇µare called SOT, the ones that contain∂/vector m ∂tare generally responsible for the damping of the FM dynamics. However, we no- tice that ˆ ez×∂/vector m ∂tterm in Eq. (13) which arises from the Berry curvature, becomes mz∂/vector m ∂tin the LLG equa- tion that does not contribute to the damping and only renormalizes the coefficient of the left hand side of the Eq. (15). The second term in the Eq. (13), is the anti-damping SOT pointing along ( ez×/vector∇µ)-axis. The cone angle dependence of the anti-damping term can be checked by assuming an electric field along the x- axis when the FM precesses around the y-axis, (i.e. /vector m(t) = cos(θ)ˆey+sin(θ)cos(ωt)ˆex+sin(θ)sin(ωt)ˆez). In this case the average of the SOT along the y-axis in one period of the precession leads to the average of the an- tidamping SOT that shows a sin2(θ) dependence, which is typical for the damping-like torques. Keeping in mind4 that in this section we consider vF≫∆xc, the first and second terms in Eq. (14) show that the Gilbert damp- ing increases as the precession axis goes from in-plane ( x ory) to out of plane ( z) direction. Furthermore, when the precession axis is in-plane (e.g. along y-axis), the damping rate due to the oscillation of the out of plane component of the magnetization ( ∂mz/∂t) has a sin4(θ) dependence that can be ignored for low power measure- ment of the Gilbert damping θ≪1. This leaves us with the contribution from the in-plane magnetization oscilla- tion (∂mx∂t) only. Therefore, the Gilbert damping for in-plane magnetization becomes half of the case when magnetization is out-of-plane. The anisotropic depen- dence of the Gilbert damping can be used to verify the existence of the surface state of the 3DTI as well as the proximity induce magnetization at the interface between a FM and a 3DTI. Finally, the third term in Eq. (14) demonstrates a field like SOT with the same vector field characteristics as the Rashba-Edelstein effect. III. CONCLUSION In conclusion, we have developed a linear response NEGF framework which provides unified treatment of both spin torque and damping due to SOC at interfaces. We obtained the expressions for both damping and anti- damping torques in the presence of a linear gradiance of the electric field and adiabatic time dependence of the magnetization dynamics for a helical state correspond- ing to the surface state of a 3D topological insulator. We present the exact expressions for the damping/anti- damping SOT as well as the field like torques and showed that, (i); Both Fermi surface and Fermi sea contribute similarly to the anti-damping SOT as well as the Hall conductivity and, ( ii); The Gilbert damping due to the surface state of a 3D TI when the magnetization is in- plane is less than the Gilbert damping when it is in the out-of-plane direction. This dependence can be used as a unique signature of the helicity of the surface states of the 3DTIs and the presence of the proximity induced magnetic exchange from the FM overlayer. ACKNOWLEDGMENTS We thank Branislav K. Nikoli´ c for the fruitful discus- sions. F. M. and N. K. were supported by NSF PREM Grant No. 1205734.Appendix A: Derivation of the Density Matrix Using Eqs. (2) and . (4) it is straightforwardto obtain, G<=(Gr ad−Ga ad)f−2ηf′∇µ·∂Gr ad ∂kGa ad +i∂G< ad ∂E∂H ∂tGa ad+i∂Gr ad ∂E∂H ∂tG< ad +i∂G< ad ∂k·∇HGa ad+i∂Gr ad ∂k·∇HG< ad(A1) We plug in the expression for the adiabatic lesser GF in equilibrium, G< ad= 2iηfGr adGa ad= (Gr ad−Ga ad)f, and obtain, G<= (Gr ad−Ga ad)f−2ηf′∇µ·∂Gr ad ∂kGa ad +if∂(Gr ad−Ga ad) ∂E∂H ∂tGa ad+if∂Gr ad ∂E∂H ∂t(Gr ad−Ga ad) +if′(Gr ad−Ga ad)∂H ∂tGa ad+if∂(Gr ad−Ga ad) ∂k·∇HGa ad +if∂Gr ad ∂k·∇H(Gr ad−Ga ad). (A2) Expanding the terms, leads to, G<=/parenleftbigg Gr ad−Ga ad+iGa ad∂H(t) ∂t∂Ga ad ∂E −iGr ad∂H ∂k·∇µ(x)∂Gr ad ∂E−i∂Ga ad ∂E∂H(t) ∂tGa ad +i∂Gr ad ∂E∂H ∂k·∇µ(x)Gr ad/parenrightbigg f +if′Gr ad∇µ·∂H ∂k(Gr ad−Ga ad) +if′(Gr ad−Ga ad)∂H ∂tGa ad, (A3) where,forthefirstandthirdlineswehaveusedtheiranti- Hermitian forms instead. Since to calculate the density matrix we integrate G<over energy, we can use integra- tion by parts and obtain, G<=/parenleftbigg Gr ad−Ga ad+2iGa ad∂H(t) ∂t∂Ga ad ∂E −2iGr ad∂H ∂k·∇µ(x)∂Gr ad ∂E/parenrightbigg f −if′/parenleftbigg Gr ad∇µ·∂H ∂kGa ad−Gr ad∂H ∂tGa ad/parenrightbigg = (Gr ad−Ga ad)f +i/parenleftbigg 2Gr adD(H)∂Gr ad ∂Ef+Gr adD(H)Ga adf′/parenrightbigg .(A4) where we define, D=∂ ∂t− ∇µ·∂ ∂k. Finally, using the identity, 2Gr adD(H)∂Gr ad ∂E=Gr adD(H)∂Gr ad ∂E−∂Gr ad ∂ED(H)Gr ad +∂(Gr adD(H)Gr ad) ∂E, (A5)5 and performing the differential over the energy, we arrive at Eq. (6). Appendix B: Derivation of the Local Spin Density From Eqs. (6) and (7), the local spin density can be written as, /vectorSneq=/vectorS(1) neq+/vectorS(2) neq, where /vectorS(1) neqis due to the nonequilibrium electrons at the Fermi surface and for a two band mo del can be calculated as the following, /vectorS(1) neq≈−2ηℑ/integraldisplaydE 2πTr /parenleftBig D(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig/parenleftBig (E−ε0+iη)1+/vectorh·/vectorσ/parenrightBig ((E−ε0−iη)2−|/vectorh|2)((E−ε0+iη)2−|/vectorh|2)/vectorσ +D(1 (E−ε0−iη)2−|/vectorh|2)((E−ε0+iη)1+/vectorh·/vectorσ)/vectorσ((E−ε0−iη)1+/vectorh·/vectorσ) (E−ε0+iη)2−|/vectorh|2/bracketrightBigg f′(E) (B1) Preforming the trace over the Pauli matrix, we obtain, /vectorS(1) neq≈ −4η/integraldisplaydE 2π/parenleftBigg ηD/vectorh+D(/vectorh)×/vectorh ((E−ε0−iη)2−|/vectorh|2)(E−ε0+iη)2−|/vectorh|2) +4ℑ(/parenleftBig /vectorh·D(/vectorh)+(E−ε0−iη)(/vector∇µ·/vector v0)/parenrightBig (E−ε0)/vectorh ((E−ε0−iη)2−|/vectorh|2)2((E−ε0+iη)2−|/vectorh|2)) f′(E) (B2) In the limit of small broadening, η≪ |/vectorh|, we obtain, /vectorS(1) neq≈−ηD/vectorh−/vectorh×D/vectorh+1 η/vectorh·D(/vectorh)/vectorh 2|/vectorh|2/parenleftBig f′(ε0+|/vectorh|)+f′(ε0−|/vectorh|)/parenrightBig −1 η(/vector∇µ·/vector v0)/vectorh 2|/vectorh|/parenleftBig f′(ε0+|/vectorh|)−f′(ε0−|/vectorh|)/parenrightBig (B3) For the Fermi sea contribution to the nonequilibrium local spin densit y we have, /vectorS(2) neq≈ℜ/integraldisplaydE 2πTr/bracketleftBigg ∂ ∂E/parenleftBigg (E−ε0−iη)1+/vectorh·/vectorσ (E−ε0−iη)2−|/vectorh|2/parenrightBigg/parenleftBig D(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig(E−ε0−iη)1+/vectorh·/vectorσ (E−ε0−iη)2−|/vectorh|2/vectorσ −(E−ε0−iη)1+/vectorh·/vectorσ (E−ε0−iη)2−|/vectorh|2/parenleftBig D(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig∂ ∂E/parenleftBigg (E−ε0−iη)1+/vectorh·/vectorσ (E−ε0−iη)2−|/vectorh|2/parenrightBigg /vectorσ/bracketrightBigg f(E). (B4) Similarly, we obtain, /vectorS(2) neq≈ℜ/integraldisplaydE 2πTr /bracketleftBig D(/vectorh·/vectorσ)+/vector∇µ·/vector v01,(E−ε0−iη)1+/vectorh·/vectorσ/bracketrightBig ((E−ε0−iη)2−|/vectorh|2)2/vectorσ f(E) ≈2ℑ/integraldisplaydE πD(/vectorh)×/vectorh ((E−ε0−iη)2−|/vectorh|2)2f(E) ≈−1 2D(/vectorh)×/vectorh |/vectorh|3/parenleftBig f(ε0+|/vectorh|)−f(ε0−|/vectorh|)/parenrightBig (B5) [1] D. Ralph and M. Stiles, Journal of Magnetism and Mag- netic Materials 320, 1190 (2008).[2] J. Katine and E. E. Fullerton, Journal of Magnetism and Magnetic Materials 320, 1217 (2008).6 [3] C. Wang et al., Nature Physics 7, 496 (2011). [4] I. M. Miron et al., Nature Materials 9, 230 (2010). [5] L. Liu et al., Phys. Rev. Lett. 109, 096602 (2012). [6] A.Manchon andS. Zhang, Physical ReviewB 78, 212405 (2008). [7] P. Gambardella and I. M. Miron, Philos. Transact. A Math. Phys. Eng. Sci. 369, 3175 (2011). [8] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [9] D. Pesin and A. H. MacDonald, Nature Mater. 11, 409 (2012). [10] I. Garate and M. Franz, Phys. Rev. Lett. 104, 146802 (2010). [11] F. Mahfouzi, J. Fabian, N. Nagaosa, and B. K. Nikoli´ c, Phys. Rev. B 85, 054406 (2012). [12] A. Manchon, Phys. Rev. B 83, 172403 (2011).[13] E. Zhao, C. Zhang, and M. Lababidi, Phys. Rev. B 82, 205331 (2010). [14] N. P. Butch et al., Phys. Rev. B 81, 241301 (2010). [15] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Nat Phys 4, 273 (2008). [16] F. Mahfouzi, B. K.Nikoli´ c, S.-H.Chen, andC.-R.Chang , Phys. Rev. B 82, 195440 (2010). [17] F. Mahfouzi, B. K. Nikoli´ c, and N. Kioussis, Phys. Rev. B93, 115419 (2016). [18] L. Keldysh, Sov. Phys. JETP 20, 1018 (1965). [19] H. K. et al, Nature Nanotechnology 9, 211 (2014). [20] H. Haug and A.-P. Jauho, Quantum kinetics in transport and optics of semiconductors (Springer-Verlag, Berlin, ADDRESS, 2008). [21] N. Bode et al., Phys. Rev. B 85, 115440 (2012). [22] L. Arrachea, Phys. Rev. B 75, 035319 (2007). [23] A.-P. Jauho, N. S. 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2016-08-02

In this paper, we investigate theoretically the spin-orbit torque as well as the Gilbert damping for a two band model of a 2D helical surface state with a Ferromagnetic (FM) exchange coupling. We decompose the density matrix into the Fermi sea and Fermi surface components and obtain their contributions to the electronic transport as well as the spin-orbit torque (SOT). Furthermore, we obtain the expression for the Gilbert damping due to the surface state of a 3D Topological Insulator (TI) and predicted its dependence on the direction of the magnetization precession axis.

Ferromagnetic Damping/Anti-damping in a Periodic 2D Helical surface; A Non-Equilibrium Keldysh Green Function Approach

1608.00984v2

arXiv:2309.08281v1 [math.AP] 15 Sep 2023ON THE FORMATION OF SINGULARITIES FOR THE SLIGHTLY SUPERCRITICAL NLS EQUATION WITH NONLINEAR DAMPING PAOLO ANTONELLI AND BORIS SHAKAROV Abstract. We consider the focusing, mass-supercritical NLS equation augmented with a nonlin- ear damping term. We provide sufficient conditions on the nonl inearity exponents and damping coefficients for finite-time blow-up. In particular, singula rities are formed for focusing and dissipa- tive nonlinearities of the same power, provided that the dam ping coefficient is sufficiently small. Our result thus rigorously proves the non-regularizing effe ct of nonlinear damping in the mass- supercritical case, which was suggested by previous numeri cal and formal results. We show that, under our assumption, the damping term may be co ntrolled in such a way that the self-similar blow-up structure for the focusing NLS is a pproximately retained even within the dissipative evolution. The nonlinear damping contributes as a forcing term in the equation for the perturbation around the self-similar profile, that may p roduce a growth over finite time inter- vals. We estimate the error terms through a modulation analy sis and a careful control of the time evolution of total momentum and energy functionals. 1.Introduction In this work, we consider the NLS equation with nonlinear damping (1)/braceleftBigg i∂tψ+∆ψ+|ψ|2σ1ψ+iη|ψ|2σ2ψ= 0, ψ(0) =ψ0∈H1(Rd), whereψ:R+×Rd→Candη>0 is the damping coefficient. More precisely, our goal is to investi- gate the formation of singularities in finite time. Equation (1) arises a s an effective model in various contexts, see for instance [13, 7, 6]. The nonlinear damping appearing in (1) is usually introduced as a regula rizing term of the singu- lar dynamics provided by the focusing NLS equation. It is well known t hat the undamped NLS equation (1) with η= 0 andσ1≥2 dmay experience the formation of singularity in finite time, see [21, 24]. From the modeling point of view, this means that the NLS effec tive description fails to be accurate close to the blow-up time and further effects, that were neglected in the derivation of the NLS equation, become relevant near the singularity. Several phen omena may be taken into account within the NLS description, see for instance [12] for a quite general overview. In this work, we focus on nonlinear damping terms as in (1). A relevant question in this perspective is to determine whether the n onlinear damping truly acts as a regularization in the vanishing dissipation regime. More precisely, giv en the singular dynamics for η= 0, we are interested in determining whether the regularized equat ion (1) has a global solution for anyη>0, no matter how small. The eventual (weak) limit as η→0 of such solutions (if it exists) may be seen as a possible criterion to continue the solution beyond the singularity, in the same spirit of v anishing viscosity limits for conservation laws [8]. To our knowledge, the only related rigorous re sults available in the literature are due to Merle [30, 29], where Hamiltonian-type regularization is ado pted, see also [28] for a re- lated result for the generalized KdV equation. On the other hand, s everal numerical simulations 12 P. ANTONELLI AND B. SHAKAROV were performed to investigate this issue, see for instance [39, 14, 15, 17] and the references therein. The regularizing property of nonlinear damping is already established in some cases. For2 d≤σ1< σ2≤2 (d−2)+[4, 3] and for2 d=σ1=σ2[11], the Cauchy problem (1) is globally well-posed in H1 for anyη >0. On the other hand, there are also other cases where the dampin g does not act as a regularization and the dynamics remains singular for sufficiently small values ofη >0. This is the case for instance of a linear damping σ2= 0 [40, 37]. Moreover, in [11] it is also shown that, for 0≤σ2< σ1=2 d, it is possible to determine an open set of initial data that develop a sin gularity in finite time. This is achieved by adapting the analysis developed in [33, 3 2, 31, 34, 38] by Merle and Rapha¨ el for the mass-critical NLS, where they study the st ability of blow-up in a self-similar regime. In the physically relevant case2 d< σ1=σ2≤2 (d−2)+, this question remains unanswered in the general case. In [3] the authors prove global well-posedness of ( 1) inH1only for sufficiently large η, namely by requiring η≥min(σ1,√σ1). It is not clearwhether this result is sharp, howevernumerical simulations [39, 14, 15] suggest that finite time blow-up still occurs f or small values of η. This work provides a rigorous answer to this question in the slightly su percritical case, confirming the numerical findings of [15]. In particular, we prove that for2 d<σ1=σ2<2 d+δ∗, whereδ∗>0 is sufficiently small, it is possible to provide an open set of initial data tha t develop a singularity in finite time in a self-similar regime. Our result shows that the self-similar blow-up regime studied in [36] remains unaltered even under the action of the dissipative effec ts encoded in (1). In fact, we are going to prove a more general theorem on finite time blow-up for a lar ger class of nonlinear damping terms, see Theorem 1.1 below. We first introduce the following notations. Let sc=sc(σ1) be the Sobolev critical exponent associ- ated withσ1, (2) sc=d 2−1 σ1, namelyscdetermines the critical regularity ˙Hscfor the well-posedness of equation (1) with η= 0. We also define (3) σ∗=2sc d−2sc=scσ1, σ∗=2 d−2−σ1. Moreover, we also set (4) σ2,max=/braceleftbiggσ1ford≤3, σ∗ford≥4. Our main result is stated as follows. Theorem 1.1. There exists σcrit>2 d, such that for any2 d< σ1< σcrit,σ∗< σ2≤σ2,maxthe following holds true. There exists η∗=η∗(σ1)>0such that for any 0< η≤η∗, there exists an open set O ⊂H1(Rd)such that if ψ0∈ Othen the corresponding solution ψ∈C([0,Tmax),H1(Rd)) to(1)develops a singularity in finite time, that is Tmax<∞and lim t→Tmax/ba∇dbl∇ψ(t)/ba∇dblL2=∞. A more precise statement of our blow-up result is provided later in Th eorem 1.2. In particular, the explicit blow-up rate is given there. As previously said, our proof follows the strategy developed in [36] for the Hamiltonian dynamics, given by (1) with η= 0, which in turn exploited previous results by the same authors related to the formation of singularitie s in the mass-critical case [31, 32]. More precisely, we construct a set of initial data whose evolution is a lmost self-similar. By a fine control of the modulation parameters entering the description of the self-similar regime, it is thenBLOW-UP OF THE DAMPED NLS EQUATION 3 possible to show the occurrence of finite-time blow-up. Let us emphasize that the introduction of a dissipative term in the dy namics introduces further mathematicaldifficulties. First ofall, the self-similarprofileweconside rin ouranalysisisdetermined by the undamped equations, see also (6) below, for instance. At pr esent, it is not even clear whether it would be possible to determine a profile that takes into account also the dissipative term. A generalized notion of dissipative solitons is present in the literature [ 1, 2, 23], where the profiles are determined not only by the balance between dispersion and focusing effects but also between gain and loss terms. In (1) the sole presence of a nonlinear damping cann ot be balanced by other effects. The fact that the self-similar profile is determined by the Hamiltonian p art of the equation, yields a non-trivial forcing term in the equation for the perturbation, giv en by the nonlinear damping itself. Through a careful modulation analysis, we determine conditio ns under which this forcing can be controlled, so that the perturbation is shown to be sufficiently sm all with respect to the self- similar profile. In particular, in the case σ1=σ2, the control is determined by imposing a smallness condition on η>0, as stated in the main theorem above. Moreover,asecondmaindifficultyisthatinourcasethefunctionalsr elatedtothephysicalquantities such as total mass, momentum and energy are - straightforward ly - not conserved in time anymore. We thus need a suitable control on the time evolution of these quant ities that will in turn provide the necessary bounds on the perturbation and the modulation par ameters. Ford≥4, the restriction σ2≤σ∗<σ1prevents us to consider the case σ2=σ1. This is a technical condition, needed to ensure the validity of the Sobolev embedding H1(Rd)֒→L2(σ1+σ2+1)(Rd) in (17), (18) below, see also Section 5. On the other hand, the condit ionσ∗<σ2is motivated by the fact that we cannot control Sobolev norms ˙Hsnorms with s < scin the self-similar regime, see (103). We remark that for the undamped dynamics (1) with η= 0, it was proved in [35] that all radially symmetric blowing-up solutions leave the critical space ˙Hscat blow-up time. Let us further remark that the smallness condition η≤η∗is only necessary when σ2=σ1. In the caseσ1> σ2, it is also possible to show that for any η, there exists a set of initial conditions depending on η,σ1andσ2whose corresponding solutions blow-up in finite time. We will further discuss this point in Section 6 and throughout this work. Moreover, as it will be clear in our analysis, the smallness of η∗is determined in terms of the smallness of sc, which is related to σ1through identity (2). Indeed, we will see that η∗∼s3 c. For this reason, with some abuse of notation, in what follows we often wr iteη∗(sc) instead of η∗(σ1), wherescandσ1are related by identity (2). Finally, ourresultprovidesadditionalevidencethatthemasssuper criticalself-similarcollapseisvery different from the one occurring in the mass critical case. In the lat ter case, indeed, any damping coefficientη>0 regularizes the dynamics and prevents the formation of finite time blow-up [11, 4]. In Section 6 we will provide an argument explaining why the solution esc apes the self-similar regime in the case σ1=σ2=2 dfor anyη>0. We will now present a road map of how Theorem 1.2 will be proved. 1.1.Singularity formation. We start with an initial condition which can be decomposed as a soliton profile and a small perturbation (5) ψ0(x) =λ−1 σ1 0/parenleftbigg Qb0/parenleftbiggx−x0 λ0/parenrightbigg +ξ0/parenleftbiggx−x0 λ0/parenrightbigg/parenrightbigg eiγ0, where 0< b0,λ0≪1 are small parameters, and Qb0is roughly a localized radial solution to the stationary equation (6) ∆ Qb0−Qb0+ib0/parenleftbigg1 σ1Qb0+x·∇Qb0/parenrightbigg +|Qb0|2σ1Qb0= 0.4 P. ANTONELLI AND B. SHAKAROV It is known (see, for instance, [39, 36]) that self-similar blowing-up s olutions of the supercritical NLS focus as a zero energy solution to (6) plus a non-focusing radiation . For any initial value Qb0(0) and anyb0∈R, nontrivial solutions to (6) exist [41, 9], but any zero energy solutio n does not belong toL2(Rd) [25] and thus we employ a suitable localization in space. Moreover, th e parameter b0is chosen to be close to a value b∗=b∗(sc)>0. By continuity, we may find a time interval where the solution can be decomposed as (7) ψ(t,x) =1 λ1 σ1(t)/parenleftbigg Qb(t)/parenleftbiggx−x(t) λ(t)/parenrightbigg +ξ/parenleftbigg t,x−x(t) λ(t)/parenrightbigg/parenrightbigg eiγ(t) where the parameter bis still close enough to b∗andλandξare still small enough. By perturbation techniques, we choose the parameters b,λ,x,γ so thatξsatisfies four suitable orthogonality condi- tions. Next, we will use a local virial law and a suitable Lyapunov functional t o prove that the parame- terb(t) is trapped around the value b∗for all times. This yields the following law for the scaling parameter λ(t)∼/radicalBig −2b∗t+λ2 0. In particular, there exists a time Tmax(sc,λ0)>0 such that λ(t)→0 ast→Tmax. Consequently, the kinetic energy of the solution diverges lim t→Tmax/ba∇dbl∇ψ(t)/ba∇dblL2=∞. We will use the coercivity property stated in Proposition2.12 below to provethe dynamical trapping of the parameter b. In order to control the six negative directions on the right-hand side of (37), we will use four orthogonality conditions implied by the selection of the pa rametersb,λ,x,γ and two almost orthogonal conditions which come from suitable bounds on th e energy and the momentum of the solution. In ourcase, therearetwomoredifficulties thatdo notarisewhen η= 0. First, since Qbapproximates a solution ofthe undamped NLS, one issue will be to show that the con tribution ofthe dampingterm in (1) can be considered of a smaller order with respect to the rest o f the dynamics. in particular, the damping term generates a forcing in the equation of the remaind er, which will be controlled using the smallness of ηandλ. Second, the presence of the damping implies that the energy E[ψ(t)] =/integraldisplay1 2|∇ψ(t)|2−1 2σ+2|ψ(t)|2σ+2dx and the momentum P[ψ(t)] =/parenleftbig ∇ψ(t),iψ(t)/parenrightbig are not conserved. But to show the dynamical trapping of the par ameterbaroundb∗, we needE andPto remain small enough to control two negative directions in the coe rcivity (37) below. In comparison with the undamped case, this is not a trivial consequenc e implied by the smallness of the energy and momentum of the initial condition. Thus, we will study the ir time evolution and show that they remain sufficiently small until the blow-up time under the as sumptiond≤3,σ2≤σ1and η≤η∗(sc) ord≥4 andσ2<σ∗. Finally, we recall that the localized profile Qbis close inH1(Rd) to the ground state of the undamped NLS which is the unique real-va lued solution Q∈H1(Rd) [27, 18] to the elliptic equation (8) ∆ Q−Q+|Q|2σ1Q= 0. In light of the discussion above, Theorem 1.1 will be the consequence of the following result.BLOW-UP OF THE DAMPED NLS EQUATION 5 Theorem 1.2. There exists s∗ c>0such that for any 0<sc<s∗ candσ∗<σ2≤σ1whend≤3or σ∗<σ2<σ2,maxwhend≥4, there exists η∗(sc)>0such that if η≤η∗, then there exists an open setO ⊂H1(Rd)such that if ψ0∈ O, then for any t∈[0,Tmax)the corresponding maximal solution to(1)can be written as ψ(t,x) =1 λ1 σ1(t)/parenleftbigg Q/parenleftbiggx−x(t) λ(t)/parenrightbigg +ζ/parenleftbigg t,x−x(t) λ(t)/parenrightbigg/parenrightbigg eiγ(t), whereλ,γ∈C1([0,Tmax);R),x∈C1([0,Tmax),Rd), lim t→Tmaxλ(t) = 0,lim t→Tmaxx(t) =x∞∈Rd, there exists 0<δ(sc) =δ≪1such that /ba∇dbl∇ζ/ba∇dblL∞([0,Tmax),L2)≤δ, and the blow-up rate is given by /ba∇dbl∇ψ(t)/ba∇dbl2 L2∼(Tmax−t)−(1−sc). More precisely, we will show that if we define (9) b∗=−π(1+δ) ln(sc) then the law of the scaling parameter will satisfy the following bounds (10) λ2(0)−2(1+δ)b∗t≤λ2(t)≤λ2(0)−2(1−δ)b∗t. Remark 1.3.Recently, in [5], the authors provided a complete description of a ze ro energy solution to (6) forσ1∈/parenleftbig2 d,2 d+ε/parenrightbig andεsmall enough. 2.Preliminaries We start this section with a list of notations that we are going to use t hroughout this work. We use the symbol A∼B, to indicate the fact that there exist two constants C1,C2>0 such that C1B≤A≤C2B. For anyf∈H1(Rd), we define the operators (11) Λ f=1 σ1f+x·∇f, Df=d 2f+x·∇f. We notice that (12) Λf=Df−scf. Moreover, by integrating by parts we have (13) ( f,Λg) =−2sc(f,g)−(g,Λf) and (14) ∆Λ f= 2∆f+Λ∆f. We use the following notation 1 (d−2)+=/braceleftBigg ∞,ford≤2, 1 (d−2),ford≥3. Next, we recall some preliminary results.6 P. ANTONELLI AND B. SHAKAROV Definition 2.1. We say that a pair ( q,r) is admissible if 2 ≤q,r≤ ∞, (q,r,d)/\e}atio\slash= (2,∞,2) and 2 q=d/parenleftbigg1 2−1 r/parenrightbigg We will exploit the following Strichartz estimates [20, 26]. Theorem 2.2. For everyφ∈L2(Rd)and every (q,r)admissible, there exists a constant C >0 such that for any t>0, /ba∇dbleit∆φ/ba∇dblLq((0,t),Lr)≤C/ba∇dblφ/ba∇dblL2. Moreover, if f∈Lγ′((s,t),Lρ′(Rd))where(γ,ρ)is an admissible pair, and N(f) =/integraldisplayt sei(t−τ)∆f(τ)dτ, then there exists a constant C >0such that for any (q,r)admissible, we have /ba∇dblN(f)/ba∇dblLq((s,t),Lr)≤C/ba∇dblf/ba∇dblLγ′((s,t),Lρ′). Byusingthe Strichartzestimates, itis possibletoprovethe localex istenceofsolutionstoequation (1) (see [3, Proposition 2 .3]). Theorem 2.3. Letη∈R,σ1,σ2<2/(d−2)ifd≥3. Then for any ψ0∈H1(Rd)there exists Tmax>0and a unique solution ψto(1)such that for any (q,r)admissible, we have (15) ψ,∇ψ∈C/parenleftbig [0,Tmax),L2(Rd)/parenrightbig ∩Lq loc/parenleftbig (0,Tmax),Lr(Rd)/parenrightbig . Moreover, either Tmax=∞orTmax<∞and lim t→Tmax/ba∇dbl∇ψ(t)/ba∇dblL2=∞. The usual physical quantities (the total mass, the total energy and the momentum) are governed by the following time-dependent functions [10]. Theorem 2.4. Letψ0∈H1(Rd)andψ∈C/parenleftbig [0,Tmax),H1(Rd)/parenrightbig the corresponding solution to (1). Then the total mass, the total energy and the momentum satisf y (16) M[ψ(t)] =/integraldisplay |ψ(t)|2dx=M[ψ0]−2η/integraldisplayt 0/ba∇dblψ(s)/ba∇dbl2σ2+2 L2σ2+2ds, (17)E[ψ(t)] =/integraldisplay1 2|ψ(t)|2−1 2σ1+2|ψ(t)|2σ1+2dx =E[ψ0]+η/integraldisplayt 0/integraldisplay |ψ|2σ1+2σ2+2−|ψ|2σ2|∇ψ|2 −2σ2|ψ|2σ2−2Re/parenleftbig¯ψ∇ψ/parenrightbig2dxds, and (18) P[ψ(t)] =/parenleftbig ∇ψ(t),iψ(t)/parenrightbig =P[ψ0]−2η/integraldisplayt 0/integraldisplay |ψ|2σ2Im(¯ψ∇ψ)dxds, respectively. Remark 2.5.The dissipative terms appearing in (16), (17) and (18) are always we ll defined because of Strichartz estimates in Theorem 2.3, see (15).BLOW-UP OF THE DAMPED NLS EQUATION 7 We also notice that if σ2<σ1<1 d−2, then it follows 4σ2+2≤2(σ1+σ2+1)≤4σ1+2<2d (d−2)+. Thus, by the Sobolev embedding theorem, the positive term on the r ight-hand side of (17) can be bounded by/integraldisplayt s/integraldisplay |ψ|2(σ1+σ2+1)dxdτ≤(t−s)/ba∇dblψ/ba∇dbl2(σ1+σ2+1) L∞((s,t),H1). Analogously, we can also estimate the term on the right-hand side of (18) as /integraldisplayt s/integraldisplay |ψ|2σ2Im(¯ψ∇ψ)dxdτ/lessorsimilar(t−s)/ba∇dblψ/ba∇dbl2σ2+1 L4σ2+2/ba∇dbl∇ψ/ba∇dblL2/lessorsimilar(t−s)/ba∇dblψ/ba∇dbl2σ2+2 H1. These computations will be exploited in Section 5. Indeed, they will be crucial to control uniformly in time the right-hand side of (17) and (18). 2.1.Construction of the Approximated Soliton Core. In this subsection, we are going to constructasuitablelocalizedsolutiontothe stationaryequation(6 ), thatwillconstitutethe blowing- up soliton core in the self-similar regime. This construction already ap peared in [36, Section 2], however for the sake of completeness we are going to present the main related results and properties here. Let us notice that the damping term does not enter into the constr uction of the approximated blow- up core. Letρ∈(0,1),b>0, we define the following radii Rb=2 b/radicalbig 1−ρ, R− b=/radicalbig 1−ρRb. Letφb∈C∞(Rd) be a radially symmetric cut-off function such that (19) φb(x) =/braceleftBigg 0,for|x| ≥Rb, 1,for|x| ≤R− b,0≤φb(x)≤1. We also denote the open ball in Rdof radiusRband centered at the origin by B(0,Rb) ={x∈Rd:|x|<Rb}. We recall that scdenotes the Sobolev critical exponent defined in (2). We start with the following lemma. Lemma 2.6. There exists σ(1) c>2 dandρ(1)>0, such that for any2 d< σ1< σ(1) c,ρ∈(0,ρ(1)), there exists b(1)(ρ)>0, such that for any 0< b≤b(1), there exists a unique radial solution Pb∈H1 0(B(0,Rb))to the elliptic equation /parenleftbigg −∆+1−b2|x|2 4/parenrightbigg Pb−P2σ1+1 b= 0, withPb>0inB(0,Rb). Moreover Pb∈C3(B(0,Rb))and /ba∇dblPb−Q/ba∇dblC3→0 asb→0, whereQthe unique positive solution to (8). Finallyb(1)(ρ)→0asρ→0.8 P. ANTONELLI AND B. SHAKAROV This lemma was stated in [36, Proposition 2 .1] and its proof is a straightforward adaptation of that given for [31, Proposition 1] in the mass-critical case. To prov e the existence, one shows that Pbis a suitably scaled minimizer of the functional F(u) =/integraldisplay B(0,Rb)|∇u|2+/parenleftbigg 1−b2|x|2 4/parenrightbigg |u|2dx in the set U=/braceleftbig u∈H1 0,rad(B(0,Rb)) :/ba∇dblu/ba∇dblL2σ1+2= 1/bracerightbig , whereH1 0,rad(B(0,Rb))isthesubsetof H1 0(B(0,Rb))containingradiallysymmetricfunctions. Notice that the functional Fis bounded from below in B(0,Rb) and−∆+1−b2|x|2 4is an uniformly elliptic operator. In particular, both the maximum principle and standard r egularity results for elliptic equations are satisfied, see [19] for instance. Next, we define the function ˆQb=φbe−ib|x|2 4Pb, in the whole Rdspace, where the cut-off φbis defined in (19). Notice that the profile ˆQb∈H1(Rd) satisfies the equation (20) ∆ ˆQb−ˆQb+ib/parenleftbiggd 2ˆQb+x·∇ˆQb/parenrightbigg +|ˆQb|2σ1ˆQb=−˜Ψb, where the remainder term ˜Ψbcomes from the localization in space and is defined by (21) ˜Ψb=/parenleftbig 2∇φb·Pb+Pb(∆φb)+(φ2σ1+1 b−φb)P2σ1+1 b/parenrightbig e−ib|x|2 4. We recall some properties of the profile ˆQbwhich were shown in [36, Proposition 2 .1]. Proposition 2.7. For any polynomial pand anyk= 0,1, there exists a constant C >0such that (22)/vextenddouble/vextenddouble/vextenddouble/vextenddoublepdk dxk˜Ψb/vextenddouble/vextenddouble/vextenddouble/vextenddouble L∞≤e−C |b|. Moreover, the function ˆQbsatisfies P[ˆQb] =/parenleftBig ∇ˆQb,iˆQb/parenrightBig = 0,/parenleftBig ΛˆQb,iˆQb/parenrightBig =/parenleftBig x·∇ˆQb,iˆQb/parenrightBig =−b 2/ba∇dblxˆQb/ba∇dbl2 L2, whereΛis defined in (11)and d db2/parenleftbigg/integraldisplay |ˆQb|2dx/parenrightbigg |b=0=C(σ1), withC(σ1)→C >0asσ1→2 d. Notice that heuristically ˆQbprovides an approximating solution to (6) in the ball B(0,Rb). In the complementary region, Rd\B(0,Rb) nonlinear effects become negligible, hence it is sufficient to study the outgoing radiation defined in the next lemma, that first appeared in [32, Lemma 15]. The results in the lemma below will be exploited, in particular, to estimat e the mass flux leaving the collapsing core in Lemma 4.6 below. Lemma 2.8. There exists ρ2>0such that for any 0<ρ<ρ 2there exists b2(ρ)>0such that for any0<b<b 2, there exists a unique radial solution ζb∈˙H1(Rd)to (23) ∆ ζb−ζb+ib/parenleftbiggd 2ζb+x·∇ζb/parenrightbigg =˜Ψb,BLOW-UP OF THE DAMPED NLS EQUATION 9 where˜Ψbis defined in (21). Moreover there exists C >0such that (24) Γ b= lim |x|→+∞|x|d|ζb(x)|2/lessorsimilare−C b, and there exists c>0such that for |x| ≥R2 b (25) e−(1+cρ)π b≤4 5Γb≤ |x|d|ζb(x)|2≤e−(1−cρ)π b. Furthermore, the following estimates hold (26) /ba∇dbl∇ζb/ba∇dbl2 L2≤Γ1−cρ b, /vextenddouble/vextenddouble/vextenddouble|y|d 2(|ζb|+|y||∇ζb|)/vextenddouble/vextenddouble/vextenddouble L∞(|y|≥Rb)≤Γ1 2−cρ b. Finally, we have that/vextenddouble/vextenddouble/vextenddouble|ζb|e−|y|/vextenddouble/vextenddouble/vextenddouble C2(|y|≤Rb)≤Γ3 5 b,/ba∇dbl∂bζb/ba∇dblC1≤Γ1 2−cρ b. Remark 2.9.We observe that the profile ζb∈˙H1 rad(Rd) is not inL2(Rd) because of the logarithmic divergence implied by (24), (25). We now want to suitably modify the profile ˆQbto obtain an approximating solution to (27) i ∂tQb+∆Qb−Qb+ib/parenleftbigg1 σ1Qb+x·∇Qb/parenrightbigg +|Qb|2σ1Qb= 0, whereb=b(t) is a function depending on time. Notice that if we suppose that ˙b(t) = 0 anddσ1= 2, thenˆQbis already an approximating solution to (27), see (20) and (22). In o ur case, we suppose that there exists β(b)>0 such that ˙b(t) =βsc. In other words, we search for localized solutions to the following equation (28) i scβ∂bQb+∆Qb−Qb+ib/parenleftbigg1 σ1Qb+x·∇Qb/parenrightbigg +|Qb|2σ1Qb= 0. We find a solution to this equation as a suitable perturbation of the pr ofileˆQb. We make the following ansatz on Qb, Qb=ˆQb+scTb, by requiring that it solves equation (28) inside the ball |x| ≤R− b, up to an error of order s1+C cfor someC >0. Such a solution was already studied in [36, Proposition 2 .6]. Proposition 2.10. There exists ˜σc>2 dandρ3>0such that for any2 d< σ1<˜σc,ρ∈(0,ρ3) there exists b3(ρ)>0such that for any 0<b<b 3there exists a radial function Tb∈C3(Rd)and a constantβ >0such thatQb=e−ib|y|2 4(Pbφb+scTb)satisfies (29) iscβ∂bQb+∆Qb−Qb+ibΛQb+|Qb|2σ1Qb=−Ψb where (30) Ψb=˜Ψb+Φb, and we have (31)/vextenddouble/vextenddouble/vextenddouble/vextenddoubledk dxkΦb/vextenddouble/vextenddouble/vextenddouble/vextenddouble L∞(|x|≤R− b)/lessorsimilars1+C c, /vextenddouble/vextenddouble/vextenddouble/vextenddoubledk dxkΦb/vextenddouble/vextenddouble/vextenddouble/vextenddouble L∞(|x|≥R− b)/lessorsimilarsc,10 P. ANTONELLI AND B. SHAKAROV fork= 0,1andC >0. Moreover, Qbsatisfies (32)|E[Qb]|/lessorsimilarΓ1−cρ b+sc, P[Qb] = (∇Qb,iQb) = 0, (ΛQb,iQb) = (x·∇Qb,iQb) =−b 2/ba∇dblxQ/ba∇dbl2 L2(1+δ1(sc,b)), /ba∇dblQb/ba∇dbl2 L2=/ba∇dblQ/ba∇dbl2 L2+O(sc)+O(b2), whereδ1(sc,b)→0as asb,sc→0andcρ≪1is defined in (26)andQis a solution to (8). The profileQbsatisfies also the uniform estimate (33)/vextendsingle/vextendsingle/vextendsingle/vextendsingleP(x)dk dxkQb(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−(1+c)|x| for any polynomial P(x). Finally, we have that (34)/vextenddouble/vextenddouble/vextenddoublee(1−ρ)π 4|x|Tb/vextenddouble/vextenddouble/vextenddouble C3+/vextenddouble/vextenddouble/vextenddoublee(1−ρ)π 4|x|∂bTb/vextenddouble/vextenddouble/vextenddouble C2+|∂bβ(b)|/lessorsimilar1. We will now show a useful Pohozaev-type estimate for the profile Qb. Proposition 2.11. LetQbbe a solution to (29). Then we have (35) 2 E[Qb]−(iscβ∂bQb+Ψb,ΛQb) =sc/parenleftbig 2E[Qb]+/ba∇dblQb/ba∇dbl2 L2/parenrightbig . Proof.We take the scalar product of equation (29) with Λ Qb 0 = (∆Qb−Qb+ibΛQb+|Qb|2σ1Qb,ΛQb)+(iscβ∂bQb+Ψb,ΛQb). Now by integration by parts, we observe that (∆Qb+|Qb|2σ1Qb,1 σ1Qb+x·∇Qb) = 2(sc−1)E[Qb], and −(Qb,ΛQb) =sc/ba∇dblQb/ba∇dbl2 L2. as a consequence, we obtain that 0 = 2(sc−1)E[Qb]+sc/ba∇dblQb/ba∇dbl2 L2+(iscβ∂bQb+Ψb,ΛQb) which is equivalent to (35). /square Finally, by applying the operator Λ to (29), we see that Λ Qbsatisfies the following equation (36)iβscΛQb+Λ∆Qb−ΛQb+ibΛ(ΛQb)+|Qb|2σ1ΛQb +2σ1Re(¯Qbx·∇Qb)|Qb|2σ1−2Qb=−ΛΨb, that will be used in the Appendix. 2.2.Coercivity Property. We conclude the section by discussing the coercivity properties of t he linearized operator around the ground state in the mass-critical c ase. Since our analysis deals with the slightly mass-supercritical case, we derive a similar property by perturbative arguments. Let us first define the following quadratic form associated with the linearize d operator around the ground state in the mass-critical case, (37)H(f,f) =/ba∇dbl∇f/ba∇dbl2 L2+2 d/parenleftbigg 1+4 d/parenrightbigg/integraldisplay Q4 d−1 c(y·∇Qc)Re(f)2dy +2 d/integraldisplay Q4 d−1 c(y·∇Qc)Im(f)2dy.BLOW-UP OF THE DAMPED NLS EQUATION 11 whereQcis the ground state profile for the mass-critical NLS, namely is the u nique positive solution to (38) ∆ Qc−Qc+|Qc|4 dQc= 0. We recall that DQc=2 dQc+y·∇Qc. We statethefollowingcoercivitypropertyforthe quadraticform H. Letusremarkthat, eventhough we present it as a proposition, the following result was proved rigoro usly only in the one-dimensional case [33], while for the multi-dimensional case d≤10 it was proved only numerically [16, 42]. Proposition 2.12. For anyf∈H1(Rd)there exists c>0such that (39)H(f,f)≥c/parenleftbigg /ba∇dbl∇f/ba∇dbl2 L2+/integraldisplay |f|2e−|y|dy/parenrightbigg −c/parenleftbigg (f,Qc)2+(f,|y|2Qc)2+(f,yQc)2+(f,iDQc)2 +(f,iD(DQc))2+(f,i∇Qc)2/parenrightbigg . 3.Setting up the Bootstrap In this section, we are going to define the set of initial conditions tha t we consider for our analysis. These data are suitable perturbations of the scaled self-similar solit on profile defined in Proposition 2.10 and are going to form singularities in finite time. In what follows, we also discuss how to estimate the solution in such a way that it retains its self-similar struc ture along the evolution. Finally, we are going to show that there exists a finite time when the sc aling parameter λ(t), related to the size of the solution, goes to zero, thus exhibiting the format ion of a singularity. The idea is to find a set that is almost invariant under the dynamics of equation ( 1) and to use a bootstrap argument to show that the solution remains inside this set until the s caling parameter becomes zero and consequently the solution blows up. We use the quantity ρ3and the profile Qbdefined in Proposition 2.10, and the constant Γ bdefined in (24). Moreover, we denote by ˜sc=d 2−1 ˜σc, where ˜σcis determined in Proposition 2.10. In the next definition and subseque nt propositions, it will be useful to write the assumptions on σ1by exploiting identity (2). This means that every time a condition on scis found, such as for instance 0 < sc< Scfor some constant Sc, it should be interpreted as a condition on the exponent σ1<σ1(Sc) whereσ1(Sc) is given by σ1(Sc) =2 d−2Sc. Definition 3.1. Let 0<sc<˜scand 0< ρ<ρ 3. We define the set O ⊂H1(Rd) as the family of all functions φ∈H1such that there exist λ0,b0>0,x0∈Rd,γ0∈Randξ0∈H1(Rd) such that (40) φ(x) =λ−1 σ1 0/parenleftbigg Qb0/parenleftbiggx−x0 λ0/parenrightbigg +ξ0/parenleftbiggx−x0 λ0/parenrightbigg/parenrightbigg eiγ0, with the following conditions: there exists 0 <ν(ρ)≪1 such that (41) Γ1+ν10 b0≤sc≤Γ1−ν10 b0,12 P. ANTONELLI AND B. SHAKAROV the scaling parameter satisfies (42) 0<λ0<Γ100 b0, the initial momentum and energy are bounded as (43) λ2−2sc|E[φ]|+λ1−2sc|P[φ]|<Γ50 b0, there exists s=s(σ1,σ2) in the interval (44) sc<s<min/parenleftbiggdσ1 2σ1+2,dσ2 2σ2+2,1 2/parenrightbigg such that the following inequality is verified (45)/integraldisplay ||∇|sξ0|2+|∇ξ0|2+|ξ0|2e−|y|dy<Γ1−ν b0, where y=x−x0 λ. Finally, the remainder ξ0satisfies also the orthogonality conditions (46) ( ξ0,|y|2Qb0) = (ξ0,yQb0) = (ξ0,iΛ(ΛQb0)) = (ξ0,iΛQb0) = 0. In this definition, the constant ν=ν(ρ) is chosen so that (47) Γ1−cρ b0≤Γ1−ν50 b0, wherec>0 is the constant in (25). Note that from the definition of Γ bin (25) and from condition (41), it follows that sc∼Γb∼e−(1+cρ)π b that is b0∼b∗=−π(1+δ) ln(sc) namelyb0is chosen to be close to the value b∗(sc)>0 defined in (9). The constant s=s(σ1,σ2) is chosen so that the L2σ1+2andL2σ2+2norms ofξmay be controlled by interpolating between ˙Hs and˙H1, see (78) and (103) below. Finally, we recall that the set Ois non-empty, see [36, Remark 2.10]. We remark that this is the same set defined in [36], whose evolution s develop a singularity in finite time. In this work, we are going to prove that the same initial da ta produce singular solutions also under the dissipative dynamics (1). Thus, our result implies that the nonlinear damping is not able to regularize the dynamics and to prevent the formation of sing ularities, in the cases under our consideration. This is not in contradiction with the global well-pos edness result proven in [3], where the condition η≥min(σ1,√σ1) was needed. In fact, here we prove our main result under a smallness assumption on η, namely we require (48) η≤s3 c. Moreover, in Section 6 we are going to provide an alternative definitio n of the set O, in the case σ2< σ1. In particular, hypothesis (48) will no longer be needed, even thou gh the set will depend onσ1,σ2andη. Now, let us consider an initial datum ψ0∈ Oand letψ∈C([0,Tmax),H1(Rd)) be the corresponding solution to (1), where Tmax≤ ∞is its maximal time of existence. By continuity of the solution, there exists a time interval [0 ,T1), withT1≤Tmax, where the inequalities in Definition 3.1 are still valid but with slightly larger bounds. Moreover, by standard per turbation techniques, we can also preserve the orthogonal conditions in (46) for any t∈[0,T1) by modulational analysis, see forBLOW-UP OF THE DAMPED NLS EQUATION 13 instance [31, Lemma 2] where this was proved for the Hamiltonian dyn amics. The proof of a similar property for solutions to the dissipative dynamics (1) follows straig htforwardly. Proposition 3.2. There exists 0< T1≤Tmax,b,λ,γ∈C1([0,T1),R),x∈C1([0,T1),Rd)and ξ∈C([0,T1),H1(Rd))such that for all t∈[0,T1), the solution ψto(1)may be decomposed as (49) ψ(t,x) =λ(t)−1 σ1/parenleftbig Qb(t)(y)+ξ(t,y)/parenrightbig eiγ(t), with (50) y=x−x(t) λ(t), whereξsatisfies the following orthogonal conditions: (51) ( ξ(t),|y|2Qb(t)) = (ξ(t),yQb(t)) = (ξ(t),iΛ(ΛQb(t))) = (ξ(t),iΛQb(t)) = 0, and we have Γ1+ν2 b(t)≤sc≤Γ1−ν2 b(t), (52) 0≤λ(t)≤Γ10 b(t), (53) λ2−2sc|E[ψ(t)]|+λ1−2sc(t)|P[ψ(t)]| ≤Γ2 b(t), (54) /integraldisplay ||∇|sξ(t)|2dy≤Γ1−50ν b(t), (55) /integraldisplay |∇ξ(t)|2+|ξ(t)|2e−|y|dy≤Γ1−20ν b(t). (56) We notice that (52) and the smallness of νandscimply that Γ b(t)<1 for anyt∈[0,T1). Our main goal is to prove that the solution remains in this self-similar regime untilTmax, that is the decomposition of ψ, along with the properties of the modulation parameters as stated in Proposition 3.2 are valid until the maximal time of existence. This is achieved by usin g a bootstrap argument. We show that the bounds in Proposition 3.2 can be improved and thus t he self-similar regime may be extended in time. This amounts to finding a dynamical trapping of t he parameter bto improve (52) and (56), to find a suitable differential equation for λto improve (53), and to prove that the energy and the momentum of the solution remain small enough to impr ove (54). The trapping of the control parameter bwill be achieved in Section 4 by finding a lower bound for ˙busing a virial-type argument, and an upper bound with a monotonicity formula. The equ ation satisfied by the scaling parameterλwill be a direct consequence of the dynamical trapping of b(t) aroundb∗, see Section 5. Finally, the controls on EandPwill be obtained in Section 5 as a consequence of the choice of the damping parameter ηin (48). In fact, we will show the following bootstrap result. Proposition 3.3. There exists s∗ c>0, such that for any sc< s∗ c, there exist smax(sc)> sc, and ν∗(sc)>0such that for any sc<s<s max(sc)and0<ν <ν∗and for any t∈[0,T1),the following inequalities are true: Γ1+ν4 b(t)≤sc≤Γ1−ν4 b(t), (57) 0≤λ(t)≤Γ20 b(t), (58) λ2−2sc|E[ψ(t)]|+λ1−2sc|P[ψ(t)]| ≤Γ3−10ν b(t), (59) /integraldisplay ||∇|sξ(t)|2≤Γ1−45ν b(t), (60) /integraldisplay |∇ξ(t)|2+|ξ(t)|2e−|y|dy≤Γ1−10ν b(t). (61)14 P. ANTONELLI AND B. SHAKAROV As a consequence of this proposition, we see that the solution satis fies improved bounds with respect to those stated in Proposition 3.2. A standard continuity a rgument then implies that the same bounds are satisfied in the whole interval [0 ,Tmax). Let us remark that the same dynamical trapping argument to show self-similar blow-up was already exploited in [36]. In particular, this implies that a similar argument to [36] works also in our case for the damped NLS, under the assumptions of Theorem 1.2. On the other hand, dealing with the dissipative dynamics (1) entails new mathematical difficulties with resp ect to [36]. First of all, we notice that the self-similar profile Qb, given by equation (29), does not determine an approximate solution to (1), not even close to the collapsing core. In particular, this implies that the equation for the perturbation ξbears a forcing term of order one, depending on Qb. In the case under our consideration here, we can see that the forcing term produces an error that becomes non-negligible on a range of times of order/parenleftBig ηλ2−2σ2 σ1/parenrightBig−1 . We overcome this difficulty by choosing the parameters (and in partic ular, the damping coefficient in the case σ1=σ2) in such a way that Tmax≪/parenleftBig ηλ2−2σ2 σ1/parenrightBig−1 . The second mathematical difficulty induced by the dissipative dynamic s is that the global quantities, such as total momentum and total energy, are not conserved alo ng the flow of (1), see (18) and (17), respectively. Consequently, while the bootstrap condition (54) is s traightforwardly satisfied in the Hamiltonian case, a fundamental step in our analysis would be to cont rol the time evolution of these quantities. In Section 6 we will also show that in the case σ2< σ1, it is possible to deal with these issues by choosing the scaling parameter λsmall enough and depending on σ1,σ2andηso that the initial condition is very close to the blow-up point and the damping term is not strong enough to force the solution out of the self-similar regime before the blow-up time. As the last step, we will show that inside the self-similar regime, the sc aling parameter λ(t) goes to zero. In particular, the following Corollary can be inferred from Propositio n 3.2. Corollary 3.4. For anysc<s∗ cwheres∗ cis defined in Proposition 3.3, the bounds in Proposition 3.2 are valid for any t∈[0,Tmax). Moreover, there exists a constant 0< C=C(ν,sc)≪1such that for any t∈[0,Tmax), (62) λ2 0−2(1+C)t≤λ2(t)≤λ2 0−2(1−C)t. As a consequence of the corollary above, there exists Tmax(ν,sc,λ0)<∞such thatλ→0 as t→Tmax(ν,sc,λ0) andλbehaves like λ2(t)∼Tmax−t. Moreover, the estimate (62) also provides a blow-up rate for the L2-norm of the gradient of the solution, as we have /vextenddouble/vextenddouble∇ψ(t)/vextenddouble/vextenddouble2 L2=λ2(sc−1)(t)/vextenddouble/vextenddouble∇(Qb(t)+ξ(t))/vextenddouble/vextenddouble2 L2∼(Tmax−t)−(1−sc). 4.Estimates on the Modulation Parameters In this section, we start our analysis of solutions emanating from init ial conditions in the set O, defined in Definition 3.1. Let ψ0∈ O, we denote by ψ∈C([0,Tmax),H1(Rd)) its corresponding maximal solution, with Tmax≤ ∞. Then we decompose the solution as the sum of a soliton core and remainder as in (49). By Proposition 3.2, the bounds (52) - (56) are satisfied for all t∈[0,T1], together with the four orthogonal conditions in (51). The purpose of this section is to obtain preliminary estimates on the p arameters that are requiredBLOW-UP OF THE DAMPED NLS EQUATION 15 to prove the dynamical trapping of bin the next section. A crucial step will be to use the coercivity property stated in Proposition 2.12. Even though this proposition is related to the mass critical case, we will use the smallness of scand property (12) to obtain a similar coercivity property in the slightly super-critical case. We will then show that the negative par t of the right-hand side of (39) is controlled by the positive part. When computing the quadratic for m (39) on the perturbation ξ, we see that four of the negative terms appearing in are small enoug h because of the orthogonality conditions (51), the smallness of scand property (12). We now show how to control the two remaining negative terms. By plugging the decomposition (49) into equation (1), we obtain the f ollowing equation for the perturbation ξ, i∂τξ+Lξ+i(˙b−βsc)∂bQb−i/parenleftBigg˙λ λ+b/parenrightBigg Λ(Qb+ξ)−i˙x λ·∇(Qb+ξ) −(˙γ−1)(Qb+ξ)+iηλ2−2σ2 σ1|Qb+ξ|2σ2(Qb+ξ)+R(ξ)−Ψb= 0,(63) where the scaled time τis defined according to (64) τ(t) =/integraldisplayt 01 λ2(v)dv, and we recall that Λf=1 σ1f+y·∇f. In (63) the dot denotes the derivative with respect to τand the space derivatives are intended to be performed with respect to the scaled space variable ywhere (65) y=x−x(t) λ(t). Moreover, we used the notation Lfor the linearized operator around Qb, Lξ= ∆ξ−ξ+ibΛξ+2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb+|Qb|2σ1ξ, (66) andRcontains all nonlinear terms in ξ, (67)R(ξ) =|Qb+ξ|2σ1(Qb+ξ)−|Qb|2σ1Qb−2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb−|Qb|2σ1ξ, whereas Ψ bis defined in (30). We notice that the damping term yields a forcing in the equation for ξ, given by (68) i ηλ2−2σ2 σ1|Qb|2σ2Qb depending on Qb,ηandλ. One of the main goals in our subsequent analysis is to exploit the smallness of the product ηλ2−2σ2 σ1in order to obtain a suitable control on the nonlinear damping term so that the self-similar regime is maintained along the evolution. T his task is achieved by the selection of ηfor instance as in (48) or by the smallness of λforσ2<σ1, see Section 6. Let us notice that the assumption σ∗<σ2≤σ2,max, see (3) and (4), implies that 0≤2−2σ2 σ1<2(1−sc) ford≤3, while 0<4(d−2)σ1−1 (d−2)σ1≤2−2σ2 σ1<2(1−sc), ford≥4. In what follows, we will use the following estimates on scalar products .16 P. ANTONELLI AND B. SHAKAROV Lemma 4.1. For anyt∈[0,T1), any polynomial P(y)and any integers k∈ {0,1,2,3}andn∈ {0,1}, we have (69)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg ξ,Pdk dykQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/parenleftbigg/integraldisplay |ξ|2e−|y|dy/parenrightbigg1 2 , (70)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg ξ,Pdk dyk∂bQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/parenleftbigg/integraldisplay |∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1 2 , (71)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbiggdk dykQb,Pdn dynΨb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg ξ,Pdn dynΨb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓ1−ν b, (72)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg ∂bQb,Pdk dykQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg |y|2Qb,Pdk dykQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar1. These inequalities have been proven in [31, Lemma 4]. They follow from t he Cauchy-Schwarz inequalityand theuniform estimate(33). In the next twoLemmas4.2 and4.3weprovethe smallness of the scalar products ( ξ(t),Qb(t)) and (ξ,i∇Qb(t)), so to conclude our control of the negative terms appearing in (39) up to additional terms controlled by the smallness o fsc. This will be achieved by exploiting the balance laws satisfied by the total momentum and ener gy (18) and (17), respectively, together with the bounds (54). Lemma 4.2. There exists ν1>0such that for any ν <ν1andt∈[0,T1), we have (73) |(ξ(·,t),Qb(t))|/lessorsimilar/integraldisplay |ξ|2e−|y|+|∇ξ|2dy+Γ1−2ν b(t). Let us remark that, by combining (73) with the bound (56), we would obtain the following estimate (74) |(ξ(·,t),Qb(t))|/lessorsimilarΓ1−20ν b(t). Although this rougher bound is sufficient to show the coercivity of th e linearized operator, the estimate (73) will be crucial to prove Proposition 3.3. Proof.We plug the decomposition of the solution (49) into the energy equat ion to obtain (75)2λ2−2scE[ψ] = 2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2 L2−2(∆Qb,ξ) −1 σ1+1/integraldisplay |Qb+ξ|2σ1+2−|Qb|2σ1+2dy. By using equation (29) satisfied by Qbin the expression above, we get 2λ2−2scE[ψ] = 2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2 L2 +2(iscβ∂bQb−Qb+ibΛQb+Ψb,ξ)−/integraldisplay R(2)(ξ)dy, whereR(2)(ξ) is defined by (76) R(2)(ξ) =1 σ1+1/parenleftbig |Qb+ξ|2σ1+2−|Qb|2σ1+2−(2σ1+2)|Qb|2σ1Re(Qb¯ξ)/parenrightbig . Equivalently, we write the identity above as 2(Qb,ξ) =−2λ2−2scE[ψ]+2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2 L2 +2(iscβ∂bQb+ibΛQb+Ψb,ξ)−/integraldisplay R(2)(ξ)dy.(77)BLOW-UP OF THE DAMPED NLS EQUATION 17 Now we bound the terms on the right-hand side of (77). For the firs t term, we use the control on the energy (54)/vextendsingle/vextendsingle−2λ2−2scE[ψ]/vextendsingle/vextendsingle≤2Γ2 b. For the second term, we use the properties of Qblisted in (32), inequality (47) and the control (52) onscto obtain that/vextendsingle/vextendsingleE[Qb]/vextendsingle/vextendsingle≤Γ1−cρ b+sc≤Γ1−ν50 b+Γ1−ν2 b. For the fourth term, we use again the control (52) on sc, estimates (70), (71) and (56) to get |sc(iβ∂bQb,ξ)+(Ψ b,ξ)|/lessorsimilar/parenleftBigg sc/parenleftbigg/integraldisplay |∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1 2 +Γ1−ν b/parenrightBigg /lessorsimilar/parenleftBig Γ1−ν2 bΓ1 2−10ν b+Γ1−ν b/parenrightBig . Moreover, we also observe that (i bΛQb,ξ) = 0 from (51). Finally, we use estimates (69) and (56) to control the remainder term R(2)(ξ) defined in (76) as follows, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay R(2)(ξ)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/integraldisplay |∇ξ|2+|ξ|2e−|y|dy+/ba∇dblξ/ba∇dbl2σ1+2 L2σ1+2. In order to bound the L2σ1+2-norm ofξ, we observe that since sc<s<d 2−d 2σ1+2=dσ1 2σ1+2=s(σ1) then by Sobolev embedding and subsequent interpolation between H1(Rd) andHs(Rd), we have (78) /ba∇dblξ/ba∇dbl2(σ1+1) L2(σ1+1)/lessorsimilar/ba∇dbl|∇|s(σ1)ξ/ba∇dbl2(σ1+1) L2≤ /ba∇dblξ/ba∇dbl2θ(σ1+1) ˙H1/ba∇dblξ/ba∇dbl2(1−θ)(σ1+1) ˙Hs, for someθ=θ(s)∈(0,1), wheresis determined in Definition 3.1, see (44). Now by using the controls (56) and (55) we obtain /ba∇dbl∇ξ/ba∇dbl2θ(σ1+1) ˙H1/ba∇dbl∇ξ/ba∇dbl2(1−θ)(σ1+1) ˙Hs ≤Γ(1−20ν)θ(σ1+1) bΓ(1−50ν)(1−θ)(σ1+1) b ≤Γ(1−50ν)(σ1+1)+30 νθ(1+σ1) b. By collecting everything together, we obtain that |(ξ,Qb)|/lessorsimilarΓ2 b+Γ1−ν50 b+Γ1−ν2 b+/integraldisplay |∇ξ|2+|ξ|2e−|y|dy +Γ1−ν2 bΓ1 2−10ν b+Γ1−ν b+Γ(1−50ν)(σ1+1)+30 νθ(1+σ1) b. By choosing νsmall enough, we have thus obtained estimate (73). /square We also exploit the equation of the momentum (18) to derive a bound o n (ξ(·,t),i∇Qb(t)), as in the following lemma. Lemma 4.3. There exists ν(1)>0such that for any 0<ν <ν(1)and anyt∈[0,T1), we have (79) |(ξ(·,t),i∇Qb(t))| ≤Γ1−50ν b(t). Proof.We plug decomposition (49) into the momentum (18) to obtain 2(ξ,i∇Qb) =−λ1−2scP[ψ]+P[Qb]+P[ξ]. The first term on the right-hand side is bounded by (54), /vextendsingle/vextendsingle−λ1−2scP[ψ]/vextendsingle/vextendsingle≤Γ2 b.18 P. ANTONELLI AND B. SHAKAROV Next, we observe that by the properties of Qb, we haveP[Qb] = 0, see (32). For the last term, since s<1 2, we use (44) to estimate |P[ξ]|/lessorsimilar/ba∇dblξ/ba∇dbl2 ˙H1 2/lessorsimilar/ba∇dblξ/ba∇dbl2θ ˙H1/ba∇dblξ/ba∇dbl2(1−θ) ˙Hs where θ=2−4s 2−s>0. Now we use estimates (56) and (55) to obtain /ba∇dblξ/ba∇dbl2θ ˙H1/ba∇dblξ/ba∇dbl2(1−θ) ˙Hs≤Γ(1−20ν)θ bΓ(1−50ν)(1−θ) b= Γ(1−50ν)+30νθ b. By combining all previous estimates, we get |(ξ,i∇Qb)|/lessorsimilarΓ2 b+Γ(1−50ν)+30νθ b. Inequality (79) thus follows by choosing νsufficiently small. /square Our next step is to obtain suitable estimates on the modulational par ameters defined in the decomposition (49). This is accomplished by exploiting the equation (6 3) satisfied by the remainder ξ, and the orthogonality conditions listed in (51). Lemma 4.4. For anyt∈[0,T1), we have (80)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle˙λ(τ) λ(τ)+b(τ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+|˙b(τ)|/lessorsimilarΓ1−20ν b(τ), and/vextendsingle/vextendsingle/vextendsingle/vextendsingle(˙γ(τ)−1)−1 /ba∇dblΛQb(τ)/ba∇dbl2 L2(ξ(τ),LΛ(ΛQb(τ))/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle˙x(τ) λ(τ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle /lessorsimilarδ2/parenleftbigg/integraldisplay |∇ξ(τ)|2+|ξ(τ)|2e−|y|dy/parenrightbigg1 2 +Γ1−20ν b(τ),(81) whereδ2=δ2(sc)>0. Moreover, δ2(sc)→0, assc→0. Proof.The lemma is proved by taking the scalar product of equation (63) wit h suitable terms that allow us to exploit the orthogonality conditions (51). The same appro ach was already used in [36], see Lemma 3.1 and Proposition 3.3 therein, see also [38, Appendix A], to study the undamped dynamics. For this reason we write equation (63) as (82) U(ξ)+iηλ2−2σ2 σ1|Qb+ξ|2σ2(Qb+ξ) = 0, so that, in what follows, we exploit the analysis already developed in [36 ]. Let us first consider the bound on |˙b|. We take the scalar product of (82) with Λ Qb. Following the computations in Appendix A, we use estimates (69), (74), (79), (5 5) and (56) to obtain |˙b|/lessorsimilar/integraldisplay |∇ξ|2+|ξ|2e−|y|dy+Γ1−11ν b+ηλ2−2σ2 σ1/vextendsingle/vextendsingle/parenleftbig i|Qb+ξ|2σ2(Qb+ξ),ΛQb/parenrightbig/vextendsingle/vextendsingle. Similarly, we obtain the estimate for other parameters /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle˙λ λ+b/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/integraldisplay |∇ξ|2+|ξ|2e−|y|dy+Γ1−11ν b+ηλ2−2σ2 σ1/vextendsingle/vextendsingle/parenleftbig |Qb+ξ|2σ2(Qb+ξ),|y|2Qb/parenrightbig/vextendsingle/vextendsingle.BLOW-UP OF THE DAMPED NLS EQUATION 19 and /vextendsingle/vextendsingle/vextendsingle/vextendsingle(˙γ−1)−1 /ba∇dblΛQb/ba∇dbl2 L2(ξ,LΛ(ΛQb)/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle˙x λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarδ2/parenleftbigg/integraldisplay |∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1 2 +/integraldisplay |∇ξ|2dy+Γ1−11ν b +ηλ2−2σ2 σ1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbig |Qb+ξ|2σ2(Qb+ξ),yQb/parenrightbig +/parenleftbig i|Qb+ξ|2σ2(Qb+ξ),Λ(ΛQb)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle. Let us now control the contributions coming from the damping term . We write |Qb+ξ|2σ2(Qb+ξ) =|Qb|2σ2Qb+R(1)(ξ). We use (72) to obtain that (83)/vextendsingle/vextendsingle/parenleftbig i|Qb|2σ2Qb,ΛQb+Λ(ΛQb)+iyQb+i|y|2Qb/parenrightbig/vextendsingle/vextendsingle/lessorsimilar1. On the other hand, by using (69) and (56), we also have that /vextendsingle/vextendsingle/vextendsingle/parenleftBig R(1)(ξ),ΛQb+Λ(ΛQb)+iyQb+i|y|2Qb/parenrightBig/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/parenleftbigg/integraldisplay |∇ξ|2dy+/integraldisplay |ξ|2e−|y|dy/parenrightbigg1 2 . Thus it follows that (84)ηλ2−2σ2 σ1/vextendsingle/vextendsingle/parenleftbig i|Qb+ξ|2σ2(Qb+ξ),ΛQb+Λ(ΛQb)+iyQb+i|y|2Qb/parenrightbig/vextendsingle/vextendsingle /lessorsimilarηλ2−2σ2 σ1/parenleftbigg 1+/integraldisplay |∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1 2 /lessorsimilarΓ3−3ν2 bΓ20−20σ2 σ1 b≤Γ2 b, where we used the hypothesis on η(48), (52) and (53). /square One can see that the estimates (81) and (80) are the same as thos e in [36, Lemma 3 .1] for the undamped case η= 0. This is because our choice of ηandσ2≤σ1imply that the damping term is of lower order with respect to other terms in equation (63). in part icular, the scalar products with the forcing term defined in (68) are well-controlled by the smallness o fηλ2−2σ2 σ1. 4.1.Local Virial Law. We will now derive suitable inequalities on ˙bto prove that bis trapped around the value b∗defined in (9). We first obtain a lower bound, see (85) below. This est imate is connected with the local virial law for the remainder ξ. For details, see [33, Section 3] and [31, Section 4]. Lemma 4.5. There exists s(2) c>0such that for any sc<s(2) cand for any t∈[0,T1), there exists C >0such that (85) ˙b(t)≥C/parenleftbigg sc+/integraldisplay |∇ξ(t)|2+|ξ(t)|2e−|y|dy−Γ1−ν6 b(t)/parenrightbigg .20 P. ANTONELLI AND B. SHAKAROV Proof.By taking the scalar product of (63) with Λ Qbwe obtain 0 = (i∂τξ,ΛQb)+˙b(i∂bQb,ΛQb)−(iβsc∂bQb+Ψb,ΛQb)+(Lξ+R(ξ),ΛQb) −/parenleftBigg i/parenleftBigg˙λ λ+b/parenrightBigg ΛQb+i˙x λ·∇Qb+(˙γ−1)Qb,ΛQb/parenrightBigg −/parenleftBigg i/parenleftBigg˙λ λ+b/parenrightBigg Λξ+(˙γ−1)ξ+i˙x λ·∇ξ,ΛQb/parenrightBigg +ηλ2−2σ2 σ1(i|Qb+ξ|2σ2(Qb+ξ),ΛQb). Notice that only the last term in this equation depends on the damping . The computation involving the other terms will be shown in Appendix A following the exposition of [ 36, Proposition 3 .3]. These computations yield the following inequality ˙b≥C/parenleftbigg sc+/integraldisplay |∇ξ|2dy+/integraldisplay |ξ|2e−|y|dy−Γ1−ν6 b/parenrightbigg −ηλ2−2σ2 σ1/vextendsingle/vextendsingle(i|Qb+ξ|2σ2(Qb+ξ),ΛQb)/vextendsingle/vextendsingle, for someC >0. On the other hand, the contribution of the damping term is negligib le as we control it with the calculations in the previous lemma (see (84)) ηλ2−2σ2 σ1/vextendsingle/vextendsingle(i|Qb+ξ|2σ2(Qb+ξ),ΛQb)/vextendsingle/vextendsingle/lessorsimilarΓ2 b. /square 4.2.Refined virial estimate. In this subsection, we derive an upper bound for ˙b. We will study the mass flux escaping the self-similar soliton core region. The outgo ing radiation ζbdefined in Lemma 2.8 will play a central role in this task. Let φ∈C∞ c(Rd) be a radial cut-off defined by φ(r)∈[0,1], φ(r) =/braceleftBigg 1 forr∈[0,1), 0 forr≥2, We also define φA(r) =φ/parenleftBigr A/parenrightBig whereA=A(t) is determined by (86) A(t) = Γ−a b(t) anda=a(ν)>0 will be chosen later. We denote by ˜ζb=φAζbthe localization of the outgoing radiation. We notice that the equation for the outgoing radiation (2 3) implies that ˜ζbsatisfies the equation (87) ∆ ˜ζb−˜ζb+ib/parenleftbiggd 2+y·∇/parenrightbigg ˜ζb=φA˜Ψb+F=˜Ψb+F where (88) F= (∆φA)ζb+2∇φA·∇ζb+iby·∇φAζb, and˜Ψbwas defined in (21). In particular, by recalling that suppΨ b⊂B(0,Rb)⊂B(0,2 b) and by the definition of A(t), we have that φA˜Ψb=˜Ψb. Moreover, from the properties of ζbestablished in Lemma 2.8, we infer that ˜ζb∈H1 radand, by suitably choosing a=a(ν)>0, we also have (89) /ba∇dbl˜ζb/ba∇dbl2 H1≤Γ1−cρ b.BLOW-UP OF THE DAMPED NLS EQUATION 21 We define the following refined soliton core and remainder, ˜Qb=Qb+˜ζband˜ξ=ξ−˜ζb, so that we decompose the solution as (90) ψ(t,x) =λ(t)−1 σ1/parenleftbigg ˜Qb(t)/parenleftbiggx−x(t) λ(t)/parenrightbigg +˜ξ/parenleftbigg t,x−x(t) λ(t)/parenrightbigg/parenrightbigg eiγ(t). We now claim that also the new soliton core ˜Qbis an approximating solution to (28). By using (87) and (29), we obtain iβsc∂b˜Qb+∆˜Qb−˜Qb+ibΛ˜Qb+|˜Qb|2σ1˜Qb= iβsc∂bQb+∆Qb−Qb +ibΛQb+|Qb|2σ1Qb +∆˜ζb−˜ζb+i/parenleftbiggd 2˜ζb+y·∇˜ζb/parenrightbigg −iscb˜ζb −iscb˜ζb+iscβ∂b˜ζb +|Qb+˜ζb|2σ1(Qb+˜ζb)−|Qb|2σ1Qb =−Ψb+˜Ψb+F−iscb˜ζb+iscβ∂b˜ζb +|Qb+˜ζb|2σ1(Qb+˜ζb)−|Qb|2σ1Qb. We observe that by using (30), we get −Ψb+φA˜Ψb= Φb. Thus the new profile ˜Qbsatisfies (91) i βsc∂b˜Qb+∆˜Qb−˜Qb+ibΛ˜Qb+|˜Qb|2σ1˜Qb=−˜Φb+F where (92) ˜Φb=−Φb−iscb˜ζb+iβ∂b˜ζb+|Qb+˜ζb|2σ1(Qb+˜ζb)−|Qb|2σ1Qb, andFis defined in (88). By using the equation (91) for ˜Qband decomposition (90), we derive the equation for ˜ξ, i∂τ˜ξ+˜L˜ξ+i(˙b−βsc)∂b˜Qb−i/parenleftBigg˙λ λ+b/parenrightBigg Λ(˜Qb+˜ξ)−i˙x λ·∇(˜Qb+˜ξ) −(˙γ−1)(˜Qb+˜ξ)+iηλ2−2σ2 σ1|˜Qb+˜ξ|2σ2(˜Qb+˜ξ)+˜R(˜ξ)−˜Φb+F= 0,(93) where ˜L˜ξ= ∆˜ξ−˜ξ+ibΛ˜ξ+2σ1Re(˜ξ˜Qb)|˜Qb|2σ1−2˜Qb+|˜Qb|2σ1˜ξ, and ˜R(˜ξ) =|˜Qb+˜ξ|2σ1(˜Qb+˜ξ)−|˜Qb|2σ1˜Qb−2σ1Re(˜ξ˜Qb)|˜Qb|2σ1−2˜Qb−|˜Qb|2σ1˜ξ. We recall again that the scaled time and space denoted by τandyare defined in (64) and (65), respectively. With some abuse of notations, in what follows we often explicit the τ−dependence of functions. For instance, we denote ˜ξ(τ), to actually mean ˜ξ(t−1(τ)). Byexploitingagainthe equation(93) forthe remainder ˜ξ, the controls(81) and(80)and estimates in Proposition 3.2, we obtain the following bounds. Lemma 4.6. There exists C >0such that any t∈[0,T1), there exists C >0such that C2/parenleftbigg/integraldisplay |∇˜ξ(τ)|2+|˜ξ(τ)|2e−|y|dy+Γb(τ)/parenrightbigg ≤C/parenleftbiggd dτf(τ)+sc/parenrightbigg +/integraldisplay {A(τ)≤|y|≤2A(τ)}|ξ(τ)|2dy,(94)22 P. ANTONELLI AND B. SHAKAROV where (95) f(τ) =−1 2(i˜Qb(τ),y·∇˜Qb(τ))−(i˜ξ(τ),Λ˜Qb(τ)). Proof.As for Lemmas 4.4 and 4.5, we conveniently write equation (93) as U(1)(˜ξ)+iηλ2−2σ2 σ1|˜Qb+˜ξ|2σ2(˜Qb+˜ξ) = 0, so that forU(1)((˜ξ) we again exploit the analysis already presented in [36, Lemma 3 .5], [34, Lemma 6] and reported in Appendix B . By taking the scalar product of equa tion (93) with Λ ˜Qb, we obtain (96) ( P(˜ξ),Λ˜Qb)+(iηλ2−2σ2 σ1|˜Qb+˜ξ|2σ2(˜Qb+˜ξ),Λ˜Qb) = 0. Exploiting the computations in Appendix B yields the following inequality d dτf≥C/parenleftbigg/integraldisplay |∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg −1 C/parenleftBigg sc+/integraldisplay2A A|ξ|2dy/parenrightBigg −ηλ2−2σ2 σ1/vextendsingle/vextendsingle/vextendsingle(i|˜Qb+˜ξ|2σ2(˜Qb+˜ξ),Λ˜Qb)/vextendsingle/vextendsingle/vextendsingle. We will now show that in this inequality, the contribution of the damping term can be considered negligible. From (89) and (72) it follows that (97)/vextendsingle/vextendsingle/vextendsingle/parenleftBig i|˜Qb|2σ2˜Qb,Λ˜Qb/parenrightBig/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/parenleftBig i|Qb+˜ζb|2σ2(Qb+˜ζb),Λ(Qb+˜ζb)/parenrightBig/vextendsingle/vextendsingle/vextendsingle ≤/vextendsingle/vextendsingle/parenleftbig i|Qb|2σ2Qb,ΛQb/parenrightbig/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/parenleftBig i|Qb|2σ2Qb,Λ˜ζb/parenrightBig/vextendsingle/vextendsingle/vextendsingle +/vextendsingle/vextendsingle/vextendsingle/parenleftBig i|Qb+˜ζb|2σ2(Qb+˜ζb)−i|Qb|2σ2Qb,Λ(Qb+˜ζb)/parenrightBig/vextendsingle/vextendsingle/vextendsingle. We already know from (83) that/vextendsingle/vextendsingle/parenleftbig i|Qb|2σ2Qb,ΛQb/parenrightbig/vextendsingle/vextendsingle/lessorsimilar1. For the other terms in (97), we use (89) and (47) to obtain /vextendsingle/vextendsingle/vextendsingle/parenleftBig i|Qb|2σ2Qb,˜ζb/parenrightBig/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/parenleftBig i|Qb+˜ζb|2σ2(Qb+˜ζb)−i|Qb|2σ2Qb,Λ(Qb+˜ζb)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓ1−ν b. Thus we obtain that ηλ2−2σ2 σ1/vextendsingle/vextendsingle/vextendsingle(i|˜Qb+˜ξ|2σ2(˜Qb+˜ξ),Λ˜Qb)/vextendsingle/vextendsingle/vextendsingle/lessorsimilarηλ2−2σ2 σ1≤Γ2 b where we used again (48), (52) and (53). Inequality (94) follows st raightforwardly from (166) and (52). /square Notice that the control (94) is the same as that in [36, Lemma 3 .5]. This is again because the contribution of the damping term is negligible in the self-similar regime du e to the choice of ηand the fact that σ2≤σ1. We proceed by finding a suitable bound for the mass flux term/integraldisplay {A(τ)≤|y|≤2A(τ)}|ξ|2dy that appears on the right-hand side of (94). To do this, we introdu ce a further radial, smooth cut-off χ∈[0,1] withχ(r) = 0 forr≤1 2, andχ(r) = 1 forr≥3, withχ′≥0 andχ′(r)∈/bracketleftbig1 4,1 2/bracketrightbig for 1≤r≤2. Let (98) χA(r) =χ/parenleftBigr A/parenrightBigBLOW-UP OF THE DAMPED NLS EQUATION 23 whereA=A(t) is defined in (86). In the following lemma, we will choose s>0 depending also on σ2in order to be able to control the L2σ2+2-norm of the remainder ξ, see (103) below. Lemma 4.7. For anyt∈[0,T1), we have (99)b(τ)/integraldisplay {A(τ)≤|y|≤2A(τ)}|ξ(τ)|2dy/lessorsimilarλ−2sc(τ)d dτ/parenleftbigg λ2sc(τ)/integraldisplay χA|ξ(τ)|2dy/parenrightbigg +Γ3 2−10ν b(τ)+Γ2a b(τ)/ba∇dbl∇ξ(τ)/ba∇dbl2 L2. Proof.We take the scalar product of equation (1) with i χA/parenleftBig x−x(t) λ(t)/parenrightBig ψ(t,x) and obtain 1 2d dt(ψ,χAψ)−(ψ,(∂tχA)ψ)−(∇xψ,i(∇xχA)ψ)+η(|ψ|2σ2ψ,χAψ) = 0. By using decomposition (49) and the scaled space and time variables, we rewrite the equation above as (100)0 =1 2λ2scd dτ/parenleftbig λ2sc(ξ,χAξ)/parenrightbig −(ξ,(∂τχA)ξ)−(∇ξ,i(∇χA)ξ) +ηλ2−2σ2 σ1(|ξ|2σ2ξ,χAξ)+R(1)(Qb,ξ), where the gradient and the scalar products are taken with respec t to the variable y(τ) =x−x(τ) λ(τ) and R(1)(Qb,ξ) =1 2λ2scd dτ/parenleftbig λ2sc((Qb,χAQb)+2(Qb,χAξ))/parenrightbig −(Qb,(∂τχA)Qb) −2(Qb,(∂τχA)ξ)−(∇Qb,i(∇χA)Qb)−2(∇Qb,i(∇χA)ξ) +ηλ2−2σ2 σ1/parenleftbig (|Qb+ξ|2σ2(Qb+ξ),χA(Qb+ξ))−(|ξ|2σ2ξ,χAξ)/parenrightbig . We now estimate the reminder term R(1)(Qb,ξ). Let us recall that for |y| ≥2 bwe haveQb=scTb, whereTbis exponentially decreasing, see Proposition 2.10. We claim that, for t he terms depending only onQb, we have /vextendsingle/vextendsingle/vextendsingleR(2)(Qb)/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 2λ2scd dτ/parenleftbig λ2sc(Qb,χAQb)/parenrightbig −(Qb,(∂τχA)Qb)−(∇Qb,i(∇χA)Qb) +ηλ2−2σ2 σ1/parenleftbig |Qb|2σ2Qb,χAQb)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle /lessorsimilars2 c+ηλ2−2σ2 σ1s2σ2+2 c. Here we used the bound d dτ/parenleftbig λ2sc(Qb,χAQb)/parenrightbig =s2 cd dτ/parenleftbigg λ2sc/integraldisplay χA|Tb|2dy/parenrightbigg /lessorsimilars3 cλ2sc/parenleftBigg˙λ λ−b+b/parenrightBigg /ba∇dblQb/ba∇dbl2 L2+s2 cλ2sc˙b∂b/ba∇dblQb/ba∇dbl2 L2 /lessorsimilars3 cλ2sc/parenleftbigg Γ1 2−10ν b−c lnsc/parenrightbigg +s2 cλ2scΓ1 2−10ν b/lessorsimilars2 c.24 P. ANTONELLI AND B. SHAKAROV that follows from the properties of Qb(32). All other terms may be estimated by /vextendsingle/vextendsingle/vextendsingleR(1)(Qb,ξ)−R(2)(Qb)/vextendsingle/vextendsingle/vextendsingle/lessorsimilarsc/parenleftbigg/integraldisplay |∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1 2 . To prove the two claims above, we use the following property |(Tb,ξ)| ≤/parenleftbigg/integraldisplay |∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1 2 , that can be proven similarly to (69) and (70). Thus by using (52), (5 3) and (48), we obtain that /vextendsingle/vextendsingle/vextendsingleR(1)(Qb,ξ)/vextendsingle/vextendsingle/vextendsingle/lessorsimilars2 c+ηλ2−2σ2 σ1s2σ2+2 c+sc/parenleftbigg/integraldisplay |∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1 2 /lessorsimilarΓ3 2−10ν b. In particular, from (100) we get the following inequality 1 2λ2scd dτ/parenleftbig λ2sc(ξ,χAξ)/parenrightbig ≥(ξ,(∂τχA)ξ)+(∇ξ,i(∇χA)ξ) −ηλ2−2σ2 σ1(|ξ|2σ2ξ,χAξ)−Γ3 2−10ν b. From straightforward computations, we see that ∂τχA(τ)/parenleftbiggx−x(τ) λ(τ)/parenrightbigg =−1 A/parenleftBigg ˙x λ+/parenleftBigg˙λ λ+˙A A/parenrightBigg y/parenrightBigg ·∇χ/parenleftbiggx−x(τ) λ(τ)A(τ)/parenrightbigg . and consequently, we can rewrite the inequality above as (101)1 2λ2scd dτ/parenleftbig λ2sc(ξ,χAξ)/parenrightbig ≥b(ξ,y·∇χAξ) −/parenleftBigg ξ,1 A/parenleftBigg ˙x λ+/parenleftBigg˙λ λ+b/parenrightBigg y+˙A Ay/parenrightBigg ·(∇χ)ξ/parenrightBigg +(∇ξ,i(∇χA)ξ)−ηλ2−2σ2 σ1/integraldisplay χA|ξ|2σ2+2dy−Γ3 2−10ν b. The definition of χA(98) implies the following chain of estimates (102)1 8/integraldisplay {A≤|y|≤2A}|ξ|2dy≤1 2/integraldisplay {A≤|y|≤2A}χ′(|y| A)|ξ|2dy≤1 2/integraldisplay {A≤|y|≤2A}|y| Aχ′(|y| A)|ξ|2dy =1 2/integraldisplay {A≤|y|≤2A}y A·∇χ(|y| A)|ξ|2dy. We will now exploit it to bound the terms in inequality (101). For the firs t term, we can easily infer the following bound b/integraldisplay y·∇χA|ξ|2dy≥b 8/integraldisplay {A≤|y|≤2A}|ξ|2dy. For the second term, the definition of A(t) (86) and estimate (25) for Γ ballow us to infer ˙A A=−ac˙b b2, which yields 1 A/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBigg˙λ λ+b+˙A A/parenrightBigg/integraldisplay y·∇χ|ξ|2dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓa bΓ1−20ν b/integraldisplay A≤|y|≤2A|ξ|2dy,BLOW-UP OF THE DAMPED NLS EQUATION 25 where we have used (80) and (102). By using (81) we may analogous ly estimate 1 A/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay˙x λ·∇χ|ξ|2dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓa b/parenleftbigg (/integraldisplay |∇ξ|2+|ξ|2e−|y|dy)1/2+Γ1−20ν b/parenrightbigg/integraldisplay A≤|y|≤2A|ξ|2dy /lessorsimilarΓa bΓ1 2−10ν b/integraldisplay A≤|y|≤2A|ξ|2dy, where we used (56) in the last inequality. For the third term on the right-hand side of (101) we use Young’s ine quality and get 1 A|(∇ξ,i(∇χ)ξ)| ≤1 A/ba∇dbl∇ξ/ba∇dblL2/parenleftBigg/integraldisplay A≤|y|≤2A|ξ|2dy/parenrightBigg1/2 ≤40 bA2/ba∇dbl∇ξ/ba∇dbl2 L2 +b 40/integraldisplay A≤|y|≤2A|ξ|2dy. Finally, we consider the contribution coming from the nonlinear dampin g. By recalling that σ2> σ∗= 2sc/(d−2sc), we have sc<d 2−d 2σ2+2=dσ2 2σ2+2. This implies that we can choose ssuch that sc<s<dσ2 2σ2+2=s(σ2). Consequently, we can interpolate the space ˙Hs(σ2)(Rd) between ˙H1(Rd) and˙Hs(Rd) and obtain that (103) /ba∇dblξ/ba∇dbl2σ2+2 L2σ2+2/lessorsimilar/ba∇dbl|∇|s(σ2)ξ/ba∇dbl2σ2+2 L2/lessorsimilar/ba∇dblξ/ba∇dblθ(2σ2+2) ˙H1/ba∇dblξ/ba∇dbl(1−θ)(2σ2+2) ˙Hs for someθ(s)∈(0,1). Now from (56) and (55) it follows that /ba∇dblξ/ba∇dblθ(2σ2+2) ˙H1/ba∇dblξ/ba∇dbl(1−θ)(2σ2+2) ˙Hs ≤Γ(1−50ν)(σ2+1)+30 νθ(1+σ2) b. Thus, by collecting everything, we have that (104)1 2λ2scd dτ/parenleftbig λ2sc(ξ,χAξ)/parenrightbig ≥/parenleftbiggb 8−cΓa bΓ1 2−10ν b−b 40/parenrightbigg/integraldisplay {A≤|y|≤2A}|ξ|2dy −Γ2a b/ba∇dbl∇ξ/ba∇dbl2 L2−Γ3 2−10ν b −ηλ2−2σ2 σ1Γ(1−50ν)(σ2+1)+30 νθ(1+σ2) b. Inequality (99) is a simple consequence of the choice of η(48). /square Let us emphasize that the assumption σ2> σ∗, see (3), is required to obtain the bound (103). As already remarked, this assumption is related to the fact that it is not possible to control Sobolev norms ofξrougher than the critical norm ˙Hsc. Consequently, our argument cannot be applied for instance to the linearly damped NLS equation (equation (1) with σ2= 0) since we would need to estimate the term ηλ2/integraldisplay χA|ξ|2dy. By using estimates (94) and (99), it is possible to define a Lyapunov f unctional that provides an upper bound for ˙b. In the next lemma, it will be fundamental to exploit the decay of the total mass. We notice that this fact suggests that a different regularization of the focusing NLS dynamics could26 P. ANTONELLI AND B. SHAKAROV not yield the same result. In what follows we define (105)J(τ) =/integraldisplay (1−χA(τ))|ξ(τ)|2dy+/ba∇dblQb(τ)/ba∇dbl2 L2−/ba∇dblQ/ba∇dbl2 L2+2(ξ(τ),Qb(τ)) −b(τ)f(τ) +/integraldisplayb(τ) 0f(v)dv, wherefis defined in (95) and Qis the unique positive solution to (8). Lemma 4.8. There exist a1=a1(ν)>0such that for any a < a1and anyt∈[0,T1), there exist c>0such that (106)d dτJ(τ)/lessorsimilarb(τ)sc+Γ2a b(τ)/ba∇dbl∇ξ(τ)/ba∇dbl2 L2 −b(τ)/parenleftbigg Γb(τ)+/integraldisplay |∇˜ξ(τ)|2+|˜ξ(τ)|2e−|y|dy/parenrightbigg . Proof.By multiplying inequality (94) by b, we obtain (107) C2b/parenleftbigg/integraldisplay |∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg ≤Cb/parenleftbiggd dτf+sc/parenrightbigg +b/integraldisplay {A≤|y|≤2A}|ξ|2dy. The last term on the right-hand side of (107) may be estimated by (9 9), so that (108) b/integraldisplay {A(τ)≤|y|≤2A(τ)}|ξ|2dy/lessorsimilarλ−2scd dτ/parenleftbigg λ2sc/integraldisplay χA|ξ|2dy/parenrightbigg +Γ3 2−10ν b+Γ2a b/ba∇dbl∇ξ/ba∇dbl2 L2. In order to find a satisfactory bound on the first term on the right -hand side of the inequality above, we exploit the fact that the total mass is non-increasing. By writing d dτ/ba∇dblψ/ba∇dbl2 L2=d dτ/parenleftbigg λ2sc/integraldisplay (χA+(1−χA))|ξ|2+|Qb|2dy+2λ2sc(ξ,Qb)/parenrightbigg ≤0, we have that (109)λ−2scd dτ/parenleftbigg λ2sc/integraldisplay χA|ξ|2dy/parenrightbigg ≤ −d dτ/parenleftbigg/integraldisplay (1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg −2sc˙λ λ/parenleftbigg/integraldisplay (1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg . From the definition of χAgiven in (98) and by Hardy’s inequality, we have /integraldisplay (1−χA)|ξ|2≤/integraldisplay |y|≤3A|y|2 |y|2|ξ|2dy/lessorsimilarA2/ba∇dbl∇ξ/ba∇dbl2 L2. Similarly, we can obtain a comparable bound in dimensions one and two, s ee [34, Appendix C]. In particular, for any dimension we conclude that (110)/integraldisplay (1−χA)|ξ|2/lessorsimilarA2/integraldisplay |∇ξ|2+|ξ|e−|y|dy.BLOW-UP OF THE DAMPED NLS EQUATION 27 Thus, by using estimates (80), (74) and (56) we can bound the sec ond term on the right-hand side of (109) by/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesc˙λ λ/parenleftbigg/integraldisplay (1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesc/parenleftBigg˙λ λ+b/parenrightBigg/parenleftbigg/integraldisplay (1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle +/vextendsingle/vextendsingle/vextendsingle/vextendsinglescb/parenleftbigg/integraldisplay (1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle /lessorsimilarsc/parenleftbig Γ1−20ν b+b/parenrightbig/parenleftbigg (1+A2)/integraldisplay |∇ξ|2+|ξ|e−|y|dy+/ba∇dblQb/ba∇dbl2 L2/parenrightbigg . Now we want that (111) A2/integraldisplay |∇ξ|2+|ξ|e−|y|dy≤A2Γ1−20ν b= Γ−2a+1−20ν b/lessorsimilar1, where we used the definition of Ain (86) and (56). That is amust satisfy the following inequality a≤1−20ν 2. This implies that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesc˙λ λ/parenleftbigg/integraldisplay (1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarscb. By combining (108) with (109), we obtain b/integraldisplay {A≤|y|≤2A}|ξ|2dy/lessorsimilar−d dτ/parenleftbigg/integraldisplay (1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg +scb+Γ3 2−10ν b+Γ2a b/ba∇dbl∇ξ/ba∇dbl2 L2. By plugging the above inequality into (107), we infer (112)b/parenleftbigg/integraldisplay |∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg /lessorsimilarbd dτf −d dτ/parenleftbigg/integraldisplay (1−χA)|ξ|2dy+/ba∇dblQb/ba∇dbl2 L2+2(ξ,Qb)/parenrightbigg +scb+Γ3 2−10ν b+Γ2a b/ba∇dbl∇ξ/ba∇dbl2 L2. Finally, let us consider f=f(τ) as defined in (95). By the monotonicity property of b, see (85) for instance, we denote - by some abuse of notation - f=f(b(τ)). In this way, we may write bd dτf=d dτ(bf)−d dτ/integraldisplayb 0f(v)dv. Let us now recall the definition of J(τ) given in (105), we have J(τ) =/integraldisplay (1−χA(τ))|ξ(τ)|2dy+/ba∇dblQb(τ)/ba∇dbl2 L2−/ba∇dblQ/ba∇dbl2 L2+2(ξ(τ),Qb(τ)) −b(τ)f(τ)+/integraldisplayb(τ) 0f(v)dv.28 P. ANTONELLI AND B. SHAKAROV By using the previous identities and (52), we see that (112) implies th e following estimate (113) b/parenleftbigg/integraldisplay |∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg /lessorsimilar−d dτJ+bsc+Γ2a b/ba∇dbl∇ξ/ba∇dbl2 L2, which readily gives (106). /square Let us now discuss how the functional Jis related to the control parameter b. First, we define K(τ) as (114) K(τ) =/ba∇dblQb(τ)/ba∇dbl2 L2−/ba∇dblQ/ba∇dbl2 L2−b(τ)f(τ) +/integraldisplayb(τ) 0f(v)dv whereQis the ground state profile to (1), see (8). In this way, we obtain th at J(τ)−K(τ) =/integraldisplay (1−χA(τ))|ξ(τ)|2dy+2(ξ(τ),Qb(τ)). Thus, by using (110) and (73) we get (115) J(τ)−K(τ)/lessorsimilar(1+A2)/integraldisplay |ξ|2e−|y|+|∇ξ|2dy+Γ1−ν b. We stress that to provide a satisfactory upper bound on the differ enceJ−K, it is not sufficient to consider the weaker bound (74) in the inequality above. Indeed, in t he proof of the bootstrap in the next section, we will first show that the control (56) is not satisfa ctory to obtain the better bound (61). To obtain a lower bound we observe that, by using (77), we have J−K=/integraldisplay (1−χA)|ξ|2dy−2λ2−2scE[ψ]+2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2 L2 +2(iscβ∂bQb+ibΛQb+Ψb,ξ)−/integraldisplay R(2)(ξ)dy whereR(2)(ξ) is defined in (76). We rewrite the equation above as (116)J−K=−2λ2−2scE[ψ]+2E[Qb]+2(iscβ∂bQb+Ψb,ξ) +(L(1)ξ,ξ)−/integraldisplay χA|ξ|2dy−1 σ1+1/integraldisplay R(3)(ξ)dy where (117) L(1)ξ=−∆ξ+ξ−2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb−|Qb|2σ1ξ and R(3)(ξ) =|Qb+ξ|2σ1+2−|Qb|2σ1+2−(2σ1+2)|Qb|2σ1Re(Qb¯ξ) −(2σ1+2)|Qb|2σ1−2/parenleftbig |Qb|2|ξ|2+2σ1Re(Qb¯ξ)2/parenrightbig . We recall the coercivity property (L(1)ξ,ξ)−/integraldisplay χA|ξ|2dy≥C/integraldisplay |ξ|2e−|y|+|∇ξ|2dy−Γ3 2−10ν b which was proved in [34, Appendix D]. Then in (116) by further using (54), (32), (70) and (71) to bound the rests as /vextendsingle/vextendsingle/vextendsingle/vextendsingle2(iscβ∂bQb+Ψb,ξ)−1 σ1+1/integraldisplay R(3)(ξ)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓbBLOW-UP OF THE DAMPED NLS EQUATION 29 we obtain (118) J−K/greaterorsimilar/integraldisplay |ξ|2e−|y|+|∇ξ|2dy−Γb−sc. By combining (118) and (115), we obtain that (119)/integraldisplay |ξ|2e−|y|+|∇ξ|2dy−Γb−sc/lessorsimilarJ−K/lessorsimilar(1+A2)/integraldisplay |ξ|2e−|y|+|∇ξ|2dy+Γ1−ν b. Now we want to prove that K(τ) is a small perturbation of b2(τ). By slightly abusing notations again, we write K(τ) =K(b(τ)). The following lemma can be found in [36, Section 4]. Lemma 4.9. There exists b(1)>0such that for any 0<b<b 1<b(1), we have (120) K(b)−K(b1)/lessorsimilarsc and (121) b2−sc/lessorsimilarK(b)/lessorsimilarb2+sc. The proofof this lemma follows from the properties of Qbstated in (32), the definition of fin (95) the decomposition (90), and the smallness of the outgoing radiation (89). Observe that by collecting together (119) and (121), we have that J(τ) is close to b2(τ) up to smaller order corrections and up to the term A2/integraltext |ξ|2e−|y|+|∇ξ|2dy. Consequently, if aandbare small enough, we obtain the following inequality (122)/vextendsingle/vextendsingleJ(τ)−db2(τ)/vextendsingle/vextendsingle/lessorsimilarΓ1−20ν b(τ)+A2/integraldisplay |ξ|2e−|y|+|∇ξ|2dy≤νb2(τ). 5.Proof of the Bootstrap In this section, we are going to prove Proposition 3.3. We proceed in t he following order. (1) Using the monotonicity properties (85) and (106), we refine th e control over the remainder ξas in (61). (2) Inequalities (85) and (106) also imply the dynamical trapping of b(57). In particular, bis almost constant and close to the value b∗(sc)>0 defined in (9). (3) From (80), we obtain the equation for the scaling parameter ˙λ λ∼ −b+Γb. The previous points yield (58) and a precise law for λ. (4) By finding suitable bounds on the time derivatives of the energy a nd the momentum, we will prove (59). (5) Finally, we will deduce the ˙Hs-norm control of ξin (60). We shall stress that the first three points of our scheme are cons equences of the local virial law (85) and the monotonicity formula (106) and their proofs are very similar to those in [36]. On the other hand, in the undamped case, the fourth point comes natura lly from the conservation of the energy and the momentum. In our case, we will show that the growt h of the time-dependent energy and momentum (see equations (18) and (17)) can be controlled unt il the blow-up time by our choice ofη. Finally, with respect to the proof in [36], the fifth point requires an a dditional change to treat the newdissipative term. Forthe convenienceofthe reader, we will restatethe bootstrapproposition below.30 P. ANTONELLI AND B. SHAKAROV Proposition 5.1. There exists s∗ c>0,s∗>s∗ c,ν∗>0anda∗(ν∗)>a∗(ν∗)>0, such that for any sc<s∗ c,sc<s<s∗,ν <ν∗anda∗<a<a∗and for any t∈[0,T1),the following inequalities are true: Γ1+ν4 b(t)≤sc≤Γ1−ν4 b(t), (123) 0≤λ(t)≤Γ20 b(t), (124) λ2−2sc|E[ψ(t)]|+λ1−2sc|P[ψ(t)]| ≤Γ3−10ν b(t), (125) /integraldisplay ||∇|sξ(t)|2≤Γ1−45ν b(t), (126) /integraldisplay |∇ξ(t)|2+|ξ(t)|2e−|y|dy≤Γ1−10ν b(t). (127) Proof.We chooses∗ cto be the minimum of all the conditions for scfound in previous sections. Then we fixsc<s∗ cand we choose ssuch that sc<s<min/parenleftbiggdσ1 2σ1+2,dσ2 2σ2+2,1 2/parenrightbigg such that (103), (78) and (79) are true. Now we choose an initial c onditionψ0∈ O, where the set O is defined in 3.1. Consequently, there exists a time interval [0 ,T1] where the estimates in Proposition 3.2 are true. In particular, by choosing scsmall enough we obtain that b(t) is small for any t∈[0,T1] and also that Γ b(t)<1. We start by proving inequality (127). We proceed by contradiction. First, suppose that there exists τ0∈[0,τ−1(T1)) such that /integraldisplay |∇ξ(τ0)|2+|ξ(τ0)|2e−|y|dy>Γ1−10ν b(τ0). Notice that the initial condition satisfies (45), and in particular, sinc e Γb<1, for anyt∈[0,T1), see (52), we have/integraldisplay |∇ξ(0)|2+|ξ(0)|2e−|y|dy<Γ1−7ν b(0), Then by continuity of ξ,band Γb, there exists a time interval [ τ1,τ2]⊂[0,τ0) such that (128)/integraldisplay |∇ξ(τ1)|2+|ξ(τ1)|2e−|y|dy= Γ1−7ν b(τ1), /integraldisplay |∇ξ(τ2)|2+|ξ(τ2)|2e−|y|dy= Γ1−10ν b(τ2), and for any τ∈[τ1,τ2] (129)/integraldisplay |∇ξ(τ)|2+|ξ(τ)|2e−|y|dy≥Γ1−7ν b(τ). The virial estimate (85), controls (52), (53) and (54) imply that fo r anyτ∈[τ1,τ2], we have ˙b/greaterorsimilarsc+/integraldisplay |∇ξ(τ)|2+|ξ(τ)|2e−|y|dy−Γ1−ν6 b/greaterorsimilarΓ1+ν2 b+Γ1−7ν b−Γ1−ν6 b>0, forνsmall enough and hence (130) b(τ2)≥b(τ1). We notice that the definition of Γ b(25) implies also that (131) Γ b(τ2)≥Γb(τ1).BLOW-UP OF THE DAMPED NLS EQUATION 31 On the other hand, from the smallness of ˜ζb(89), we obtain the inequality (132)/integraldisplay |∇˜ξ|2+|˜ξ|2e−|y|dy≥1 2/integraldisplay |∇ξ|2+|ξ|2e−|y|dy−Γ1−ν b which combined with the second monotonicity estimate (106) and (52 ) yields d dτJ/lessorsimilarbsc+Γ2a b/ba∇dbl∇ξ/ba∇dbl2 L2−bΓb−b/integraldisplay |∇˜ξ(τ)|2+|˜ξ(τ)|2e−|y|dy /lessorsimilarb/parenleftbigg sc−Γb−1 2/integraldisplay |∇ξ|2+|ξ|2e−|y|dy+Γ1−ν b+Γ2a b b/ba∇dbl∇ξ/ba∇dbl2 L2/parenrightbigg /lessorsimilarb/parenleftBig Γ1−ν2 b−Γb−Γ1−7ν b+Γ1−ν b/parenrightBig ≤0, forνsmall enough where we have chosen a>0 so that Γ2a b b≤1 4. Thus we obtain that (133) J(τ2)≤J(τ1). Next, using (119) and (133) we get K(τ2)+/integraldisplay |ξ(τ2)|2e−|y|+|∇ξ(τ2)|2dy−Γb(τ2)−sc/lessorsimilarJ(τ2)≤J(τ1) /lessorsimilarK(τ1)+(1+A2)/integraldisplay |ξ(τ1)|2e−|y|+|∇ξ(τ1)|2dy+Γ1−ν b(τ1). Equivalently, by using (128) we get /integraldisplay |ξ(τ2)|2e−|y|+|∇ξ(τ2)|2dy= Γ1−10ν b(τ2) /lessorsimilarK(τ1)−K(τ2)+sc+Γb(τ2)+(1+A2)Γ1−7ν b(τ1)+Γ1−ν b(τ1). On the right-hand side of the equation above, we use (120) to get |K(τ1)−K(τ2)|/lessorsimilarsc. Thus using the inequality sc≤Γ1−ν2 b(τ1)(52) and the definition of A(86), we obtain Γ1−10ν b(τ2)/lessorsimilarΓ1−ν2 b(τ1)+Γb(τ2)+Γ1−7ν−2a b(τ1)+Γ1−ν b(τ1). Now we suppose that a≤ν 2. This implies that there exists C >0 such that Γ1−10ν b(τ2)≤CΓ1−8ν b(τ1). By exploiting (131), and for bsmall enough, this implies that Γ1−10ν b(τ2)≤CΓ1−8ν b(τ1)≤CΓ1−8ν b(τ2)=CΓν b(τ2)Γ1−9ν b(τ2)≤Γ1−9ν b(τ2), which is a contradiction because Γ b(τ2)<1. As our next step, we will prove (123). Assume by contradiction tha t there exists τ0∈[0,τ−1(T1)] such thatsc>Γ1−ν4 b(τ0).By continuity, this implies that there exists τ1<τ0such that sc= Γ1−ν5 b(τ1) and for any τ∈[τ1,τ0], sc>Γ1−ν5 b(τ).32 P. ANTONELLI AND B. SHAKAROV In particular, we have that d dτΓb(τ1)<0 which is equivalent to ˙b(τ1)<0 from the definition of Γ bin (25). On the other hand, the local virial inequality (85) implies that ˙b(τ1)/greaterorsimilarsc−Γ1−ν6 b(s6)/greaterorsimilarΓ1−ν5 b(s6)−Γ1−ν6 b(s6)>0, which is a contradiction. Similarly, suppose by contradiction that there exists τ0∈[0,τ−1(T1)) such that sc<Γ1+ν4 b(τ0).Then, by using (122), we get that sc≤Γ1+ν5 /radicalBig J(τ0) d Now letτ1∈[0,τ0] be the largest time such that sc= Γ1+ν6 /radicalBig J(τ1) d Thus, by definition of τ1and Γb(25) we obtain that d dτJ(τ1)≥0, while from the monotonicity formula (106) and (132) we obtain d dτJ(τ1)/lessorsimilarb(τ1)/parenleftBigg sc−Γb(τ1)−1 2/integraldisplay |∇ξ(τ1)|2+|ξ(τ1)|2e−|y|dy+Γ1−ν b(τ1)+Γ2a b(τ1) b(τ1)/ba∇dbl∇ξ(τ1)/ba∇dbl2 L2/parenrightBigg /lessorsimilarb(τ1)/parenleftBigg Γ1+ν6 /radicalBig J(τ1) d−Γ1+ν2 /radicalBig J(τ1) d/parenrightBigg ≤0, which is a contradiction. The next step will be to proveinequality (124). From the control (1 23), it followsthat the parameter bis dynamically trapped around b0, that is, for any t∈[0,T1), we have the following upper and lower bounds 1−ν4 1+ν10b0≤b(t)≤1+ν4 1−ν10b0. Thus, we notice that the control on the parameter (80), and ineq uality (61) imply that for any t∈[0,T1) 0<1−2ν4 1+ν10b0≤ −˙λ λ=−1 2d dtλ2≤1+2ν4 1−ν10b0, where we used that d dτλ=˙λ=λ2d dtλ from (64). Equivalently, we obtain the law of the parameter λ (134) λ2 0−2/parenleftbigg1+2ν4 1−ν10/parenrightbigg b0t≤λ2(t)≤λ2 0−2/parenleftbigg1−2ν4 1+ν10/parenrightbigg b0t. in particular, λ(t) is a non-increasing function of time. This estimate and the dynamica l trapping of the parameter b(123) imply inequality (124).BLOW-UP OF THE DAMPED NLS EQUATION 33 The next step is to obtain the bound on the total energy Eand momentum P(125). We start by deriving a suitable bound for the energy functional. We use (17) to o btain that E[ψ(t)] =E[ψ0]+η/integraldisplayt 0/integraldisplay |ψ|2(σ1+σ2+1)−|ψ|2σ2|∇ψ|2−2σ2|ψ|2σ2−2Re/parenleftbig¯ψ∇ψ/parenrightbig2dxdv ≤E[ψ0]+η/integraldisplayt 0λ2sc−2σ2 σ1−2/integraldisplay |Qb+ξ|2(σ1+σ2+1)dydv. We notice that if d≤3 andσ2≤σ1<1 (d−2)+or ifd≥4, andσ2< σ∗, thenH1(Rd)֒→ L2(σ1+σ2+1)(Rd). Thus, for scsmall enough we can use the Jensen inequality, (61), (55) and inter - polation /ba∇dblξ/ba∇dbl2(σ1+σ2+1) L2(σ1+σ2+1)/lessorsimilar/ba∇dblξ/ba∇dbl2θ(σ1+σ2+1) ˙H1 /ba∇dblξ/ba∇dbl2(1−θ)(σ1+σ2+1) ˙Hs /lessorsimilar1 for someθ(σ1,σ2,s)∈(0,1), to obtain /ba∇dblQb+ξ/ba∇dbl2(σ1+σ2+1) L2(σ1+σ2+1)≤C/parenleftBig /ba∇dblQb/ba∇dbl2(σ1+σ2+1) L2(σ1+σ2+1)+/ba∇dblξ/ba∇dbl2(σ1+σ2+1) L2(σ1+σ2+1)/parenrightBig ≤C whereC=C(σ1,σ2,b∗)>0. If follows that λ2−2scE[ψ(t)]≤λ2−2scE[ψ0]+Cηλ2−2sc/integraldisplayt 0λ2sc−2σ2 σ1−2dv. (135) Now we observe that, from (134), there exists a constant 0 < c=c(ν,b0)≪1 such that, for any t∈[0,T1), λ2 0−2(1+c)t≤λ2(t)≤λ2 0−2(1−c)t. Integrating λin time, it is straightforward to see that for any α∈R, we have (136)/integraldisplayt 0λ(τ)αdv/lessorsimilarλ(t)α+2. Thus, from (135), it follows that (137) λ2−2scE[ψ(t)]≤λ2−2scE[ψ0]+Cηλ2−2σ2 σ1. From (58) and (43) we have λ2−2sc|E[ψ0]| ≤Γ50 b. Moreover, we bound the last term using the smallness of η(48), (57) and (57) Cηλ2−2σ2 σ1≤Cs3 c≤Γ3−9ν b forνsmall enough. This implies the bootstrapped control on the energy λ2−2sc|E[ψ(t)]| ≤Γ3−10ν b. We use the same procedure to bound the momentum functional P. From (18), we have P[ψ(t)] =P[ψ0]+2η/integraldisplayt 0/integraldisplay |ψ|2σ2Im(¯ψ∇ψ)dxdτ =P[ψ0]+2η/integraldisplayt 0λ−2σ2 σ1+2sc−1/ba∇dblQb+ξ/ba∇dbl2σ2+1 L4σ2+2/ba∇dbl∇(Qb+ξ)/ba∇dblL2dv. Again, ifd≤3,scis small enough and σ2≤σ1or ifd≥4, andσ2<σ∗, then we use use the Jensen inequality, (61), (55), interpolation and (136) to prove that ther e existsC=C(σ2,b∗)>0 such that (138) λ1−2scP[ψ(t)]≤λ1−2scP[ψ0]+Cηλ2−2σ2 σ1.34 P. ANTONELLI AND B. SHAKAROV Consequently, (43), (48) and (57) imply that λ1−2scP[ψ]≤Γ3−10ν b, forνsmall enough. This concludes the proof of (59). The last step is to obtain the ˙Hs-norm control (126). We define ˆQ(t,x) =λ−1 σ1Qb/parenleftbiggx−x(t) λ/parenrightbigg eiγ, ˆξ(t,x) =λ−1 σ1ξ/parenleftbigg t,x−x(t) λ/parenrightbigg eiγ, we decompose the solution as ψ=ˆQ+ˆξ. From (1), we see that the function ˆξsatisfies the equation (139) i ˆξt+∆ˆξ=−E(ˆQ)−N1(ˆξ)−iηN2(ˆξ), whereEdoes not depend on ˆξ E(ˆQ) = i∂tˆQ+∆ˆQ+|ˆQ|2σ1ˆQ+iη|ˆQ|2σ2˜Q and N1(ˆξ) =|ˆQ+ˆξ|2σ1(ˆQ+ˆξ)−|ˆQ|2σ1ˆQ, N2(ˆξ) =|ˆQ+ˆξ|2σ2(ˆQ+ˆξ)−|ˆQ|2σ2ˆQ. Using the equation satisfied by Qb(29), we obtain that E(ˆQ) =1 λ2+1 σ1eiγ/parenleftBigg −Ψb+i(˙b−βsc)∂bQb−i/parenleftBigg˙λ λ+b/parenrightBigg ΛQb−i˙x λ·∇Qb −(˙γ−1)Qb+ iηλ2−2σ2 σ1|Qb|2σ2Qb/parenrightBig . We notice that inequality (126) is equivalent to (140)/integraldisplay ||∇|sˆξ|2dx≤λ2(sc−s)Γ1−45ν b since /integraldisplay ||∇|sˆξ|2dx=λ2(sc−s)/integraldisplay ||∇|sξ|2dy, and thus we will now prove (140). We define for j= 1,2 where (141) rj=d(2σj+2) d+2sσj, γj=4(σj+1) σj(d−2s),2 γj=d 2−d rj. We notice that ( γj,rj) are Strichartz admissible pairs (see Definition 2.1). Now we write equ ation (139) in the integral form ˆξ(t) =ei∆tˆξ(0)+/integraldisplayt 0ei(t−τ)∆/parenleftBig E(Qb)+N1(ˆξ)+iηN2(ˆξ)/parenrightBig dτ. and we use the Strichartz estimates in Theorem 2.2 to obtain (142)/ba∇dbl|∇|sˆξ/ba∇dblL∞([0,T1],L2)/lessorsimilar/ba∇dbl|∇|sˆξ0/ba∇dblL2+/ba∇dbl|∇|sE/ba∇dblL1([0,T1],L2) +/ba∇dbl|∇|sN1(ˆξ)/ba∇dblLγ1[0,T1],Lr1+η/ba∇dbl|∇|sN2(ˆξ)/ba∇dblLγ2([0,t],Lr2).BLOW-UP OF THE DAMPED NLS EQUATION 35 From the initial bound (45), we have λ2(s−sc)/integraldisplay ||∇|sˆξ(0)|2dx=/integraldisplay ||∇|sξ0|2dy≤Γ1−ν b0, and thus the bootstrapped controls on b, see (57) and λ, see (58) imply (143) /ba∇dbl|∇|sˆξ0/ba∇dbl2 L2≤Γ1−ν b0λ2(sc−s). We claim that the remaining terms in (142) are bounded as (144) /ba∇dbl|∇|sE/ba∇dbl2 L1([0,T1],L2)≤Γ1−15ν bλ2(sc−s), and (145) /ba∇dbl|∇|sN1(ˆξ)/ba∇dblLγ1([0,T1],Lr1)+η/ba∇dbl|∇|sN2(ˆξ)/ba∇dblLγ2([0,T1],Lr2)≤Γ1 2(1−41ν) bλsc−s. Notice that (144) and the bound on N1(ˆξ) in (145) has been already proven in [36, Section 4], up to the term coming from the damping in E. In particular, from the estimates on the parameters (81), (80), the bound on Ψ b(31), and the bootstrap bounds for b,λandξ, see (57), (58) and (61) respectively, and from the smallness condition on ηin (48) there holds for any t∈[0,T1) /ba∇dbl|∇|sE(ˆQ)/ba∇dblL2/lessorsimilar1 λ2+s−sc/parenleftbigg/integraldisplay |∇ξ|2dy+/integraldisplay |ξ|2e−|y|dy+Γ1−11ν b+ηλ2−2σ2/σ1/parenrightbigg1 2 /lessorsimilarΓ1 2(1−12ν) b λ2+s−sc.(146) We use inequality (136) to integrate (146) in time which yields /integraldisplayt 0/ba∇dbl|∇|sE(ˆQ)/ba∇dblL2dτ/lessorsimilar/integraldisplayt 0Γ1 2(1−12ν) b0 λ2+(s−sc)dτ/lessorsimilarΓ1 2(1−12ν) b0 λs−sc. Lastly, we will show inequality (145). The main ingredient is the inequalit y (147) /ba∇dbl|∇|sNj(ˆξ)/ba∇dblLr′ j/lessorsimilarλ(2σj+1)(sc−s+2/γj)/ba∇dbl|∇|s+2/γjξ/ba∇dblL2 for anyj= 1,2 which has been proven in [36, Appendix] and will be shown in Appendix C. Notice that since s+2/γj=s+(d−2sc)σj 2σj+2, for anyj= 1,2, then if we choose sto be close enough to sc, we haves<s+2/γj<1. In particular, we can interpolate and use (61) and (55) to obtain /ba∇dbl|∇|s+2/γjξ/ba∇dbl2 L2/lessorsimilar/ba∇dbl|∇|sξ/ba∇dbl2θj L2/ba∇dbl∇ξ/ba∇dbl2(1−θj) L2≤Γ1−(10+40θj)ν b where θj=1−s−2/γj 1−s. Forj= 1, we choose sc≪1 small enough and sclose enough to scto have /ba∇dbl|∇|s+2/γjξ/ba∇dblL2/lessorsimilarΓ1 2(1−40ν) b0. It follows that /ba∇dbl|∇|sN1(ˆξ)/ba∇dblLγ1[0,T1],Lr1/lessorsimilarΓ1 2(1−40ν) b0/parenleftbigg/integraldisplay λγ′ 1(2σ1+1)(sc−s+2/γj)/parenrightbigg1 γ′ 1. Since γ′ 1(2σ1+1)(s+2/γ1) = 2+γ′ 1(s−sc),36 P. ANTONELLI AND B. SHAKAROV we can use (136) to integrate in time and obtain the first inequality in ( 145) /ba∇dbl|∇|sN1(ˆξ)/ba∇dblLγ1[0,T1],Lr1/lessorsimilarΓ1 2(1−41ν) b0 λ(s−sc). Moreover, we make the same computations for j= 2 and use the control on η(48) and (57) to conclude that (148) η/ba∇dbl|∇|sN2(ˆξ)/ba∇dblLγ2([0,T1],Lr2)≤ηΓ1−(10+40θ2)ν bλsc−s≤Γ2−50ν bλsc−s. /square Thus Proposition 3.3 has been proven for any t∈[0,T1). Consequently, we can extend by continuity the time interval of the self-similar regime, that is the time interval where the bounds in Proposition 3.2 are true. Recursively, we extend the self-similar reg ime to the whole time interval [0,Tmax) whereTmax(ψ0) is the maximal time of existence of the solution stemming from ψ0. Now we show that ψexperiences a collapse in finite time. Corollary 5.2. There exists a time Tmax<∞such that lim t→Tmaxλ(t) = 0. Moreover, there exists x∞∈Rdsuch that lim t→Tmaxx(t)→x∞, wherex(t)is defined in (50). Proof.Letψ0∈ OwhereOis defined in 3.1 and let Tmax≤ ∞be the maximal time of existence of the corresponding solution to (1). Then Proposition 3.3 implies tha t for anyt∈[0,Tmax), the bounds on the scaling parameter in (134) are true, namely, there e xists 0< c=c(ν,b0)≪1 such that (149) λ2 0−2(1+c)t≤λ2(t)≤λ2 0−2(1−c)t. Consequently, there exists a time 0 <T=T(λ0,ν,b0)<∞such that lim t→Tλ(t) = 0. Furthermore, by using decomposition (49) we also obtain that lim t→T/ba∇dbl∇ψ(t)/ba∇dbl2 L2= lim t→Tλ2(sc−1)(t)/ba∇dbl∇(Qb(t)+ξ(t))/ba∇dbl2 L2=∞. Indeed, in the limit above, the exponent 2( sc−1) is negative because sc<1, and for any t∈[0,T], Qb(t)∈H1(Rd),/vextenddouble/vextenddouble∇Qb(t)/vextenddouble/vextenddouble L2≥C >0, while (123) and (127) imply that /vextenddouble/vextenddouble∇ξ(t)/vextenddouble/vextenddouble2 L2≤Γ1−10ν b/lessorsimilars1−10ν 1+ν4 c. in particular, the maximal time of existence is given by T <∞. Finally, we prove the convergence to a blow-up point. By exploiting (81) and (149), we get /vextendsingle/vextendsinglex(T)−x(t)/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT tdx dsds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplayT t1 λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 λd dτx/vextendsingle/vextendsingle/vextendsingle/vextendsingleds /lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg/integraldisplay |∇ξ(t)|2+|ξ(t)|2e−|y|dy/parenrightbigg1 2 +Γ1−20ν b(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L∞([0,T])/integraldisplayT t/parenleftbig λ2 0−2(1+c)s/parenrightbig−1 2ds.BLOW-UP OF THE DAMPED NLS EQUATION 37 We use (57) and (61) to obtain /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg/integraldisplay |∇ξ(t)|2+|ξ(t)|2e−|y|dy/parenrightbigg1 2 +Γ1−20ν b(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L∞([0,T])/lessorsimilar1. Moreover, we compute the integral as follows /integraldisplayT t/parenleftbig λ2 0−2(1+c)s/parenrightbig−1 2ds=−2 2+c(λ2 0−2(1+c)s)1 2|T t. This implies that lim t→T/vextendsingle/vextendsinglex(T)−x(t)/vextendsingle/vextendsingle/lessorsimilarlim t→T(λ2 0−2(1+c)s)1 2|T t= 0. /square 6.The case σ2<σ1 In this section, we discuss the case where the exponent of the dam ping term is strictly smaller than that of the power-type nonlinearity. First, we observe that the condition which we have chosen for the d amping parameter η≤s3 cis not strictly necessary. In fact, this condition was used to prove that the damping term is of smaller order with respect to other terms while the solution is in the self-similar regim e. In particular, it was used for instance in (104), (84), (146), (148), (137) and (138), whe re the damping term contributions can be always bounded by ηλ2−2σ2 σ1(t)/parenleftBigg C(Qb(t))+c/parenleftbigg/integraldisplay |∇ξ(t)|2+|ξ(t)|2e−|y|dy/parenrightbigg1 2/parenrightBigg , whereC(Qb)>0 is a constant depending only on Qb. We know that in the self-similar regime, from (56), we have that /parenleftbigg/integraldisplay |∇ξ(t)|2+|ξ(t)|2e−|y|dy/parenrightbigg1 2 /lessorsimilarΓ1 2−10ν b(t)/lessorsimilarC(Qb(t)) forb(t) small enough. Thus, in order to prove that the damping term is sma ller than the leading order terms in the equation referred to above, which are of the siz e Γb, we shall have for example that ηλ2−2σ2 σ1(t)≤Γ2 b(t) for anyt∈[0,T1). This is equivalent to asking that the scaling parameter satisfies th e inequality λ(t)<η−1Γσ1 σ1−σ2 b(t). Notice that the right-hand side of this inequality degenerates for σ2→σ1, and consequently, for σ2=σ1, the supposition on the smallness of ηbecomes necessary to carry on with the bootstrap. On the other hand, for σ2<σ1, we can use this condition to proceed with the bootstrap instead of that in (53). We observe that in this case, the set of initial condition s would depend on η,σ1and σ2. For instance, it would be necessary to choose the initial condition s atisfying, say λ0<η−1Γ10σ1 σ1−σ2 b(t) instead of the weaker bound (42). In particular, the bigger ηis and the closer σ2is toσ1, the more focused the initial condition needs to be to experience a collapse in fin ite time. This implies the following.38 P. ANTONELLI AND B. SHAKAROV Theorem 6.1. There exists s∗ c>0such that for any 0< sc< s∗ c, anyσ∗< σ2< σ1whend≤3 orσ∗< σ2< σ∗whend≥4and anyη >0there exists a set Oη,σ1,σ2⊂H1(Rd)such that if ψ0∈ Oη,σ1,σ2then the corresponding solution ψ∈C([0,Tmax),H1(Rd))to(1)develops a singularity in finite time. We emphasize againthat ourargumentworksonlywhen σ1>σ2. Whenσ1=σ2, it is crucialthat σ1>2 dandηto be small enough. On the other hand, it is already known that when σ1=σ2=2 d, solutions are global in time. This means that a solution escapes the se lf-similar regime no matter how smallηandλ0are. Heuristically, this is explained by the fact that, in the self-similar regime, whenσ1=2 d, the control parameter converges to zero in time, that is b(t)→0 fort→Tmax(see [34] for example). Consequently, for σ2=σ1=2 d, there would exist a time T=T(η,b0,λ0)>0 such that for some t>T, the critical inequality η/lessorsimilarΓb(t)is not true. Thus the damping term would become of the leading order inside the self-similar dynamics and interr upt it. As our final remark, we notice that if σ2> σ1, then heuristically the self-similar regime would be interrupted. Indeed the convergence of λto zero implies that contributions of damping terms will eventually be of the leading order in the dynamics because λ2−2σ2 σ1(t)→ ∞. 7.Appendix 7.1.Appendix A. In this appendix we will report the proof of [36, Proposition 3 .3] that contains computations used in Lemma 4.5. We state the result below for the re ader’s convenience. Lemma 7.1. Suppose that η= 0. Then there exists s(1) c>0such that for any sc< s(1) cand for anyt∈[0,T1), there exists C >0such that (150) ˙b≥C/parenleftbigg sc+/integraldisplay |∇ξ|2+|ξ|2e−|y|dy−Γ1−ν6 b/parenrightbigg . We now present some preliminary computations. We start with the fo llowing. Lemma 7.2. LetQbbe a solution to (29). Then (151) (i ∂bQb,ΛQb) =−1 4/ba∇dblxQ/ba∇dbl2 L2(1+δ1(sc,b))+sc(iQb,∂bQb), whereδ1(sc,b)→0assc,b→0andQis the unique positive solution to (8). Proof.We recall the third property in (32) (152) (iQb,x·∇Qb) = (iQb,ΛQb) =−b 2/ba∇dblxQ/ba∇dbl2 L2(1+δ1(sc,b)) whereδ1(sc,b)→0 assc,b→0. By taking the derivative of (152) with respect to bwe obtain d db(iQb,ΛQb) = (i∂bQb,ΛQb)+(iQb,Λ∂bQb) =−1 2/ba∇dblxQ/ba∇dbl2 L2(1+δ1(sc,b)). Let us also recall identity (14), (iQb,Λ∂bQb) =−2sc(iQb,∂bQb)+(i∂bQb,ΛQb). By plugging it into the previous formula, we obtain (i∂bQb,ΛQb) =−1 4/ba∇dblxQ/ba∇dbl2 L2(1+δ1(sc,b))+sc(iQb,∂bQb). /squareBLOW-UP OF THE DAMPED NLS EQUATION 39 Next, we recall the definition of the linearized operator around Qb, see (66) (153) Lξ= ∆ξ−ξ+ibΛξ+2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb+|Qb|2σ1ξ, and the nonlinear remainder term in the equation (63) for the pertu rbation, namely R(ξ) =|Qb+ξ|2σ1(Qb+ξ)−|Qb|2σ1Qb−2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb−|Qb|2σ1ξ. Lemma 7.3. We have that (154)(Lξ+R(ξ),ΛQb) =−2scb(iξ,ΛQb)−βsc(ξ,iΛ∂bQb)−(ξ,ΛΨb) −2λ2−2scE[ψ]+2E[Qb]+H(ξ,ξ)+E(ξ,ξ) +1 σ1+1/integraldisplay R(3)(ξ)dy+(R3(ξ),ΛQb). whereHis the quadratic form defined in (37),E(ξ,ξ)satisfies the following bound |E(ξ,ξ)| ≤δ2(sc)/integraldisplay |ξ|2e−|y|+|∇ξ|2dy, withδ2(sc)→0assc→0and the remainder terms R(3)(ξ),R3(ξ), are given by (155)R(3)(ξ) =|Qb+ξ|2σ1+2−|Qb|2σ1+2−(2σ1+2)|Qb|2σ1Re(Qb¯ξ) −(2σ1+2)|Qb|2σ1−2/parenleftbig |Qb|2|ξ|2+2σ1Re(Qb¯ξ)2/parenrightbig , and (156) R3(ξ) =R(ξ)−2σ1|Qb|2σ1−2/parenleftbigg Re(Qb¯ξ)ξ+(2σ1−1)|Qb|−2Re(Qb¯ξ)2Qb+Qb|ξ|2/parenrightbigg . Proof.We observe that by using properties (13), (14) we have (∆ξ,ΛQb) = 2(ξ,∆Qb)+(ξ,Λ∆Qb) and (ibΛξ,ΛQb) =−2scb(iξ,ΛQb)+(ξ,ibΛ(ΛQb)). Consequently, by using the equation (36) satisfied by Λ Qband the definition (153), we obtain that (Lξ,ΛQb) = 2(ξ,∆Qb)−2scb(iξ,ΛQb) +(ξ,Λ∆Qb−ΛQb+ibΛ(ΛQb)+|Qb|2σ1ΛQb+2σ1Re(¯QbΛQb)|Qb|2σ1−2Qb) = 2(ξ,∆Qb)−2scb(iξ,ΛQb)+2σ1(ξ,Re(¯Qb,1 σ1Qb)|Qb|2σ1−2Qb) +(ξ,Λ∆Qb−ΛQb+ibΛ(ΛQb)+|Qb|2σ1ΛQb+2σ1Re(¯Qby·∇Qb)|Qb|2σ1−2Qb) = 2(ξ,∆Qb+|Qb|2σ1Qb)−2scb(iξ,ΛQb)−βsc(ξ,iΛ∂bQb)−(ξ,ΛΨb). Now for the first term on the right-hand side of the equation above we notice that (75) implies 2(∆Qb+|Qb|2σ1Qb,ξ) =−2λ2−2scE[ψ]+2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2 L2−/integraldisplay R(2)(ξ)dy, where R(2)(ξ) =1 σ1+1/parenleftbig |Qb+ξ|2σ1+2−|Qb|2σ1+2−(2σ1+2)|Qb|2σ1Re(Qb¯ξ)/parenrightbig . Thus we arrive to the preliminary equation (Lξ+R(ξ),ΛQb) =−2scb(iξ,ΛQb)−βsc(ξ,iΛ∂bQb)−(ξ,ΛΨb) −2λ2−2scE[ψ]+2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2 L2−/integraldisplay R(2)(ξ)dy+(R(ξ),ΛQb).40 P. ANTONELLI AND B. SHAKAROV It remains to extract the quadratic terms in −/integraltext R(2)(ξ)dy+(R(ξ),ΛQb). We write /integraldisplay R(2)(ξ)dy= 2/integraldisplay |Qb|2σ1−2/parenleftbig |Qb|2|ξ|2+2σ1Re(Qb¯ξ)2/parenrightbig +1 σ1+1/integraldisplay R(3)(ξ)dy whereR(3)(ξ) is the rest defined in (155). Then we also write that (R(ξ),ΛQb) = (R3(ξ),ΛQb) +2σ1/parenleftbigg |Qb|2σ1−2/parenleftbigg Re(Qb¯ξ)ξ+(2σ1−1)|Qb|−2Re(Qb¯ξ)2Qb+Qb|ξ|2/parenrightbigg , 1 σ1Qb+y·∇Qb/parenrightbigg whereR3(ξ) is defined in (156). In the equation above, we notice that 2σ1/parenleftbigg |Qb|2σ1−2/parenleftbigg Re(Qb¯ξ)ξ+(2σ1−1)|Qb|−2Re(Qb¯ξ)2Qb+Qb|ξ|2/parenrightbigg ,1 σ1Qb/parenrightbigg = 2/integraldisplay |Qb|2σ1−2/parenleftbig |Qb|2|ξ|2+2σ1Re(Qb¯ξ)2/parenrightbig dy that is we can write that −/integraldisplay R(2)(ξ)dy+(R(ξ),ΛQb) =−1 σ1+1/integraldisplay R(3)(ξ)dy+(R3(ξ),ΛQb) +2σ1/parenleftbigg |Qb|2σ1−2/parenleftbigg Re(Qb¯ξ)ξ+(2σ1−1)|Qb|−2Re(Qb¯ξ)2Qb +Qb|ξ|2/parenrightbigg ,y·∇Qb/parenrightbigg . Consequently, by replacing Qbwith the ground state Qgenerating an error which we denote by E(ξ,ξ), one obtains that /ba∇dbl∇ξ/ba∇dbl2 L2−/integraldisplay R(2)(ξ)dy+(R(ξ),ΛQb) =H(ξ,ξ)+E(ξ,ξ) −1 σ1+1/integraldisplay R(3)(ξ)dy+(R3(ξ),ΛQb) whereH(ξ,ξ) is the quadratic form defined in (37). For scsmall enough, we also obtain that E(ξ,ξ) can be bounded by (157) |E(ξ,ξ)| ≤δ2(sc)/integraldisplay |ξ|2e−|y|+|∇ξ|2dy, whereδ2(sc)→0 assc→0. /square For details of the proof above, we refer to [36, Proposition 3 .3]. We are now ready to prove the local virial property 4.5 when η= 0.BLOW-UP OF THE DAMPED NLS EQUATION 41 Proof of Lemma 7.1. By taking the scalar product of equation (63) with Λ Qb, we obtain (158)0 = (i∂τξ,ΛQb)+˙b(i∂bQb,ΛQb)+(2E[Qb]−(iβsc∂bQb+Ψb,ΛQb)) +(Lξ+R(ξ),ΛQb)−2E[Qb] −/parenleftBigg i/parenleftBigg˙λ λ+b/parenrightBigg Λ(Qb+ξ)+i˙x λ·∇Qb+(˙γ−1)Qb,ΛQb/parenrightBigg −/parenleftBigg i/parenleftBigg˙λ λ+b/parenrightBigg Λξ+(˙γ−1)ξ+i˙x λ·∇ξ,ΛQb/parenrightBigg . We will study the contributions of the terms in (158) separately. Fo r the first term, we observe that (159) (i ∂τξ,ΛQb) =d dτ(iξ,ΛQb)−(iξ,∂τΛQb) =d dτ(iξ,ΛQb)−˙b(iξ,Λ∂bQb). For the second term, we use (151) to get ˙b(i∂bQb,ΛQb) =−˙b1 4/ba∇dblxQ/ba∇dbl2 L2(1+δ1(sc,b))+sc(iQb,∂bQb). Moreover, from (70) and (56) , we bound the second term on the r ight-hand side of (159) as |(iξ,Λ∂bQb)|/lessorsimilar/parenleftbigg/integraldisplay |ξ|2e−|y|+|∇ξ|2dy/parenrightbigg1 2 ≤Γ1 2−10ν b, which implies that there exists c>0 such that ˙b((i∂bQb,ΛQb)−(iξ,Λ∂bQb)) =˙bsc(∂bQb,iQb)−˙b/parenleftbigg/ba∇dblyQ/ba∇dbl2 L2 4(1+δ1(sc,b))−cΓ1 2−10ν b/parenrightbigg . For the third term in (158), we use the Pohozaev-type estimate (3 5) to obtain 2E[Qb]−(iβsc∂bQb+Ψb,ΛQb) =sc(2E[Qb]+/ba∇dblQb/ba∇dbl2 L2). For the fourth term, we use (154) to obtain that (Lξ+R(ξ),ΛQb)−2E[Qb] =−2scb(iξ,ΛQb)−βsc(ξ,iΛ∂bQb)−(ξ,ΛΨb) −2λ2−2scE[ψ]+H(ξ,ξ)+E(ξ,ξ) −1 σ1+1/integraldisplay R(3)(ξ)dy+(R3(ξ),ΛQb) whereR(3)is defined in (155). For the fifth term in (158), by straightforward computations we get −/parenleftBigg i/parenleftBigg˙λ λ+b/parenrightBigg ΛQb+i˙x λ·∇Qb+(˙γ−1)Qb,ΛQb/parenrightBigg =sc(˙γ−1)/ba∇dblQb/ba∇dbl2 L2. By combining the previous computations, we have (160)−˙b((i∂bQb,ΛQb)−(iξ,Λ∂bQb)) =d dτ(iξ,ΛQb) +sc/parenleftbig −2b(iξ,ΛQb)−β(ξ,iΛ∂bQb)+2E[Qb]+/ba∇dblQb/ba∇dbl2 L2+(˙γ−1)/ba∇dblQb/ba∇dbl2 L2/parenrightbig −(ξ,ΛΨb)−2λ2−2scE[ψ]+H(ξ,ξ)+E(ξ,ξ)+1 σ1+1/integraldisplay R(3)(ξ)dy −/parenleftBigg i/parenleftBigg˙λ λ+b/parenrightBigg Λξ+(˙γ−1)ξ+i˙x λ·∇ξ,ΛQb/parenrightBigg ,42 P. ANTONELLI AND B. SHAKAROV or equivalently (161)˙b/parenleftbigg/ba∇dblyQ/ba∇dbl2 L2 4(1+δ1(sc,b))−cΓ1 2−10ν b/parenrightbigg ≥d dτ(iξ,ΛQb) +sc/parenleftBig ˙b(∂bQb,iQb)−2b(iξ,ΛQb)−β(ξ,iΛ∂bQb)+2E[Qb]+/ba∇dblQb/ba∇dbl2 L2+(˙γ−1)/ba∇dblQb/ba∇dbl2 L2/parenrightBig −(ξ,ΛΨb)−2λ2−2scE[ψ]+H(ξ,ξ)+E(ξ,ξ)+1 σ1+1/integraldisplay R(3)(ξ)dy −/parenleftBigg i/parenleftBigg˙λ λ+b/parenrightBigg Λξ+(˙γ−1)ξ+i˙x λ·∇ξ,ΛQb/parenrightBigg . We will now study term by term the right-hand side of (161). The firs t term vanishes for any t∈[0,T1) because of one orthogonality condition in (51). Furthermore, by choosingscsmall enough, we notice that /parenleftbigg/ba∇dblyQ/ba∇dbl2 L2 4(1+δ1(sc,b))−cΓ1 2−10ν b/parenrightbigg ≥/ba∇dblyQ/ba∇dbl2 L2 8. For the second term, we observe that by using estimates (52), (5 6) and (80), we obtain that sc/parenleftBig ˙b(∂bQb,iQb)−2b(iξ,ΛQb)−β(ξ,iΛ∂bQb)+2E[Qb]+/ba∇dblQb/ba∇dbl2 L2+(˙γ−1)/ba∇dblQb/ba∇dbl2 L2/parenrightBig ≥sc/ba∇dblQb/ba∇dbl2 L2 2, forbsmall enough. Furthermore, we use (70) to bound the term |(ξ,ΛΨb)|/lessorsimilarΓ1−ν b, and the control on the energy (54) to bound /vextendsingle/vextendsingle2λ2−2scE[ψ]/vextendsingle/vextendsingle/lessorsimilarΓ2 b. All the rests are bounded using (157) and (69) /vextendsingle/vextendsingle/vextendsingle/vextendsingleE(ξ,ξ)−1 σ1+1/integraldisplay R(3)(ξ)dy+(R3(ξ),ΛQb)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δ4(sc)/integraldisplay |ξ|2e−|y|+|∇ξ|2dy for someδ4(sc)>0 withδ4(sc)→0 assc→0. For the last term on the right-hand side of (160), we notice that using (13) and two of the orthogonality conditions in ( 51), we have (iΛξ,ΛQb) =−(iξ,Λ(ΛQb))−2sc(iξ,ΛQb) = 0. Moreover, by using (81), Cauchy-Schwartz inequality and (56), w e obtain /vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg i˙x λ·∇ξ,ΛQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/ba∇dbl∇ξ/ba∇dblL2/parenleftBigg δ2(sc)/parenleftbigg/integraldisplay |∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1 2 +/integraldisplay |∇ξ|2dy+Γ1−11ν b/parenrightBigg /lessorsimilarδ2(sc)Γ1−20ν b+Γ3 2−31ν b. In the same way, we observe that −((˙γ−1)ξ,ΛQb) =−/parenleftbigg/parenleftbigg (˙γ−1)−1 /ba∇dblΛQb/ba∇dbl2 L2(ξ,LΛ(ΛQb)/parenrightbigg ξ,ΛQb/parenrightbigg −/parenleftbigg/parenleftbigg1 /ba∇dblΛQb/ba∇dbl2 L2(ξ,LΛ(ΛQb)/parenrightbigg ξ,ΛQb/parenrightbiggBLOW-UP OF THE DAMPED NLS EQUATION 43 and we use again (81) to obtain that /vextendsingle/vextendsingle/vextendsingle/vextendsingle−/parenleftbigg/parenleftbigg (˙γ−1)−1 /ba∇dblΛQb/ba∇dbl2 L2(ξ,LΛ(ΛQb)/parenrightbigg ξ,ΛQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓb. Now we use the coercivity property in Proposition 2.12 to obtain that there exists c>0 such that H(ξ,ξ)−((˙γ−1)ξ,ΛQb)≥c/integraldisplay |ξ|2e−|y|+|∇ξ|2dy −c/parenleftbig (ξ,Q)2+(ξ,|y|2Q)2+(ξ,yQ)2 +(ξ,iDQ)2+(ξ,iD(DQ))2+(ξ,i∇Q)2/parenrightbig , whereDξis defined in (11). By using property (12), (56) and the closeness o fQbwithQ, we have that there exists a constant δ5(sc)>0 such that (ξ,Q)2+(ξ,|y|2Q)2+(ξ,yQ)2+(ξ,iDQ)2+(ξ,iD(DQ))2+(ξ,i∇Q)2 ≥(ξ,Qb)2+(ξ,|y|2Qb)2+(ξ,yQb)2+(ξ,iΛQb)2+(ξ,iΛ(ΛQb))2+(ξ,i∇Qb)2 −δ5(sc)(sc+Γb) whereδ5(sc)→0 assc→0. Finally, we use the orthogonality conditions (51), inequalities (74) and (79) to obtain /vextendsingle/vextendsingle(ξ,Qb)2+(ξ,|y|2Qb)2+(ξ,yQb)2+(ξ,iΛQb)2+(ξ,iΛ(ΛQb))2+(ξ,i∇Qb)2/vextendsingle/vextendsingle/lessorsimilarΓ3 2−20ν b. Thus (160) implies (150). /square 7.2.Appendix B. In this appendix we report the computations needed in Lemma 4.6, ba sed on [36, Section 3 .3] and [34, Lemma 6] and stated in Lemma 7.4 below. We recall that the remainder term˜ξin (90) satisfies i∂τ˜ξ+˜L˜ξ+i(˙b−βsc)∂b˜Qb−i/parenleftBigg˙λ λ+b/parenrightBigg Λ(˜Qb+˜ξ)−i˙x λ·∇(˜Qb+˜ξ) −(˙γ−1)(˜Qb+˜ξ)+iηλ2−2σ2 σ1|˜Qb+˜ξ|2σ2(˜Qb+˜ξ)+˜R(˜ξ)−˜Φb+F= 0,(162) where ˜L˜ξ= ∆˜ξ−˜ξ+ibΛ˜ξ+2σ1Re(˜ξ˜Qb)|˜Qb|2σ1−2˜Qb+|˜Qb|2σ1˜ξ, ˜R(˜ξ) =|˜Qb+˜ξ|2σ1(˜Qb+˜ξ)−|˜Qb|2σ1˜Qb−2σ1Re(˜ξ˜Qb)|˜Qb|2σ1−2˜Qb−|˜Qb|2σ1˜ξ. (163) F= (∆φA)ζb+2∇φA·∇ζb+iby·∇φAζb, and (164) ˜Φb=−Φb−iscb˜ζb+iβ∂b˜ζb+|Qb+˜ζb|2σ1(Qb+˜ζb)−|Qb|2σ1Qb. Here Φ bis defined in (30), ζbis the outgoing radiation of Lemma 2.8, ˜ζb=φAζbis the localized outgoing radiation where φAis a smooth cut-off defined in the beginning of Subsection 4.2. By construction, ˜ζbsatisfies the following inequality (165)/vextenddouble/vextenddouble/vextenddouble(1+|y|)10(|˜ζb|+|∇˜ζb|2/vextenddouble/vextenddouble/vextenddouble L2+/vextenddouble/vextenddouble/vextenddouble(1+|y|)10(|∂b˜ζb|+|∇∂b˜ζb|2/vextenddouble/vextenddouble/vextenddouble2 L2≤Γ1−cρ b, wherecρ≪1 is defined in Lemma 2.8. By exploiting equation (162) and reproducing the steps in Lemma 7.1, we obtain the following.44 P. ANTONELLI AND B. SHAKAROV Lemma 7.4. Letη= 0. Then for any t∈[0,T1), there exists C >0such that d dτf(τ)≥C/parenleftbigg/integraldisplay |∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg −1 C/parenleftBigg sc+/integraldisplay2A A|ξ|2dy/parenrightBigg , (166) where (167) f=−1 2Im/parenleftbigg/integraldisplay y·∇˜Qb˜Qbdy/parenrightbigg −/parenleftBig ˜ζb,iΛ˜Qb/parenrightBig +(ξ,iΛ˜ζ). Proof.By taking the scalar product of equation (162) with Λ ˜Qb, we obtain that (168)0 = (i∂τ˜ξ,Λ˜Qb)+˙b(i∂b˜Qb,Λ˜Qb)+/parenleftBig 2E[˜Qb]−/parenleftBig iβsc∂b˜Qb+˜Φb−F,Λ˜Qb/parenrightBig/parenrightBig +(˜L˜ξ+˜R(˜ξ),Λ˜Qb)−2E[˜Qb] −/parenleftBigg i/parenleftBigg˙λ λ+b/parenrightBigg Λ(˜Qb+˜ξ)+i˙x λ·∇˜Qb+(˙γ−1)˜Qb,Λ˜Qb/parenrightBigg −/parenleftBigg i/parenleftBigg˙λ λ+b/parenrightBigg Λ˜ξ+(˙γ−1)˜ξ+i˙x λ·∇˜ξ,Λ˜Qb/parenrightBigg . Now we repeat the steps in Lemma 7.1, and arrive to the preliminary es timate (169)−˙b/parenleftBig (i∂b˜Qb,Λ˜Qb)−(i˜ξ,Λ∂b˜Qb)/parenrightBig −d dτ(i˜ξ,Λ˜Qb) =sc/parenleftBig −2b(i˜ξ,Λ˜Qb)−β(˜ξ,iΛ∂b˜Qb)+2E[˜Qb]+/ba∇dbl˜Qb/ba∇dbl2 L2+(˙γ−1)/ba∇dbl˜Qb/ba∇dbl2 L2/parenrightBig −(˜ξ,Λ(˜Φb−F))−2λ2−2scE[ψ]+H(˜ξ,˜ξ)+E(˜ξ,˜ξ)+1 σ1+1/integraldisplay R(3)(˜ξ)dy −/parenleftBigg i/parenleftBigg˙λ λ+b/parenrightBigg Λ˜ξ+(˙γ−1)˜ξ+i˙x λ·∇˜ξ,Λ˜Qb/parenrightBigg . The most important difference between (169) and (160) is the leadin g order term ( ˜ξ,Λ(˜Φb−F)) instead of ( ˜ξ,ΛΨb). First, by integration by parts and by using (13), we have that (i∂b˜Qb,Λ˜Qb) =∂b(i˜Qb,Λ˜Qb)−(i˜Qb,Λ∂b˜Qb) =∂b(i˜Qb,y·∇˜Qb)+(iΛ˜Qb,∂b˜Qb)+2sc(i˜Qb,∂b˜Qb), which equivalently implies that −˙b(i∂b˜Qb,Λ˜Qb) =−1 2d dτ(i˜Qb,y·∇˜Qb)−˙bsc(i˜Qb,∂b˜Qb). Thus we can write the left-hand side of (169) as −˙b/parenleftBig (i∂b˜Qb,Λ˜Qb)−(i˜ξ,Λ∂b˜Qb)/parenrightBig −d dτ(i˜ξ,Λ˜Qb) =d dτf(τ)+˙b(i˜ξ,Λ∂b˜Qb)−˙bsc(i˜Qb,∂b˜Qb) where f(τ) =−1 2(i˜Qb,y·∇˜Qb)−(i˜ξ,Λ˜Qb).BLOW-UP OF THE DAMPED NLS EQUATION 45 We notice that by using the estimate on |˙b|(80), the smallness of ˜ζb(165), (69) and the controls (56), (52) we obtain −˙b(i˜ξ,Λ∂b˜Qb) =−˙b(i(ξ−˜ζb),Λ∂b(Qb+˜ζb)) /greaterorsimilarΓ1−20ν b/parenleftBigg/parenleftbigg/integraldisplay |∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1 2 +Γ1−cρ b/parenrightBigg /greaterorsimilarΓ3 2−31ν b. Next, all the terms on the right-hand side of (169) are bounded in t he same way as in Lemma 7.1 except for the scalar product ( ˜ξ,Λ(˜Φb−F)). In this way, we obtain that there exists a constant C >0 such that (170)d dτf≥C/integraldisplay |˜ξ|2e−|y|+|∇˜ξ|2dy−C(sc+Γ3 2−50ν b)−(˜ξ,Λ(˜Φb−F)). and Φ bis defined in (29) and Fin (88). We notice that from the definition of ˜Φbin (92) and the bound on Φ bin (31) and the control on the outgoing radiation (165) we get /vextenddouble/vextenddouble/vextenddouble(1+|y|10)(|˜Φb|+|∇˜Φb|)/vextenddouble/vextenddouble/vextenddouble2 L2≤Γ1+ν4 b+sc, which implies that/vextendsingle/vextendsingle/vextendsingle(˜ξ,Λ˜Φb)/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓ1+ν4 b. Finally, it remains to study the last term ( ˜ξ,ΛF) in (170). We observe that (˜ξ,ΛF) = (ξ−˜ζb,ΛF) = (ξ,ΛF)−(˜ζb,ΛF). For the second term on the right-hand side of the equation above, we notice that −(˜ζb,ΛF) =−(˜ζb,DF)+sc(˜ζb,F) and from (165)/vextendsingle/vextendsingle/vextendsinglesc(˜ζb,F)/vextendsingle/vextendsingle/vextendsingle≤scΓ1−cρ≤Γ3 2, while using (25) −(˜ζb,DF)≥C1Γb for someC1>0. Finally, we use the Young inequality and (165) to obtain |(ξ,DF)|/lessorsimilar/parenleftBigg/integraldisplay2A A|ξ|2dy/parenrightBigg1 2/parenleftBigg/integraldisplay2A A|F|2/parenrightBigg1 2 ≤1 C2/integraldisplay2A A|ξ|2dy+C1 2Γb. for someC2>0. Collecting everything we see that from (170) we obtain d dτf≥C/integraldisplay |˜ξ|2e−|y|+|∇˜ξ|2dy−C(sc+Γ1+ν2 b)−(˜ξ,Λ(˜Φb−F)) ≥C/integraldisplay |˜ξ|2e−|y|+|∇˜ξ|2dy+C1Γb−C1 2Γb−C3/parenleftBigg sc+/integraldisplay2A A|ξ|2dy/parenrightBigg for someC3>0. This is equivalent to (166) /square46 P. ANTONELLI AND B. SHAKAROV 7.3.Appendix C. In this appendix, we will report the proof inequality (147) which has a lready been done in the appendix of [36]. Proof.We give a proof only for the term N1(ξ), since that for N2(ξ) is equivalent. We define the functionF:C→Cas F(z) =|z|2σ1z. From the definition of r1(141) it follows that /ba∇dbl|∇|sN1(ˆξ)/ba∇dblLr′/lessorsimilar1 λ(2σ1+1)(˜s−sc)/ba∇dbl|∇|s(F(Qb+ξ)−F(Qb))/ba∇dblLr′ where (171) ˜ s=s+d 2−d r. We claim the following estimate: /ba∇dbl|∇|s(F(Qb+ξ)−F(Qb))/ba∇dblLr′/lessorsimilar/ba∇dbl|∇|˜sξ/ba∇dblL2. Indeed observe that (172) F(Qb+ξ)−F(Qb) =/parenleftbigg/integraldisplay1 0∂zF(Qb+τξ)dτ/parenrightbigg ξ+/parenleftbigg/integraldisplay1 0∂¯zF(Qb+τξ)dτ/parenrightbigg ¯ξ. Both terms on the right-hand side of (147) are treated in the same way. We define q∈Rsuch that (173)1 q=1 r′−1 r. For any function h, by the definition of r(141), ˜s(171) and by Sobolev embedding we have (174) /ba∇dblh/ba∇dblL2σ1q/lessorsimilar/ba∇dbl|∇|sh/ba∇dblLr/lessorsimilar/ba∇dbl|∇|˜sh/ba∇dblL2. By the fractional Leibniz rule (see for example [22]), it follows that /vextenddouble/vextenddouble/vextenddouble/vextenddouble|∇|s/parenleftbigg ξ/integraldisplay1 0∂zF(Qb+τξ)dτ/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble Lr′/lessorsimilar/ba∇dbl|∇|sξ/ba∇dblL2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay1 0∂zF(Qb+τξ)dτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble Lq(175) +/ba∇dblξ/ba∇dblL2σ1q/vextenddouble/vextenddouble/vextenddouble/vextenddouble|∇|s/integraldisplay1 0∂zF(Qb+τξ)dτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble Lu, where (176)1 u=1 r′−1 2σ1q. Then, from (174), we have /vextenddouble/vextenddouble/vextenddouble/vextenddouble|∇|s/parenleftbigg ξ/integraldisplay1 0∂zF(Qb+τξ)dτ/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble Lr′/lessorsimilar/ba∇dbl|∇|˜sξ/ba∇dblL2/parenleftbigg/integraldisplay1 0/ba∇dbl∂zF(Qb+τξ)/ba∇dblLqdτ (177) +/integraldisplay1 0/ba∇dbl|∇|s∂zF(Qb+τξ)/ba∇dblLudτ/parenrightbigg , Now it remains to prove that (178)/parenleftbigg/integraldisplay1 0/ba∇dbl∂zF(Qb+τξ)/ba∇dblLqdτ+/integraldisplay1 0/ba∇dbl|∇|s∂zF(Qb+τξ)/ba∇dblLudτ/parenrightbigg /lessorsimilar1. By homogeneity ∀τ∈[0,1],|∂zF(Qb+τξ)|/lessorsimilar|Qb|2σ1+|ξ|2σ1,BLOW-UP OF THE DAMPED NLS EQUATION 47 and so /integraldisplay1 0/ba∇dbl∂zF(Qb+τξ)/ba∇dblLqdτ/lessorsimilar/integraldisplay1 0(|Qb|2σ1+|ξ|2σ1 L2σ1q)dτ/lessorsimilar/integraldisplay1 0(|1+|ξ|2σ1 L2)dτ/lessorsimilar1. Moreover, recall [22] the equivalent definition of the homogeneous Besov norm: ∀0<˜s<1,/ba∇dblu/ba∇dbl2 ˙B˜s q,2∼/integraldisplay∞ 0/parenleftbig R−˜ssup |y|≤R|u(.−y)−u(.)|Lq/parenrightbig21 RdR. Recall also that /ba∇dbl|∇|˜sψ/ba∇dblLq/lessorsimilar/ba∇dblψ/ba∇dbl˙B˜s q,2. Observe that, for 1 ≤d≤3, 2σ1>2 and it follows by homogeneity, |∂zF(u)−∂zF(v)|/lessorsimilar|u−v|(|u|2σ1−1+|v|2σ1−1). We define hτ=Qb+τξ,0≤τ≤1. We estimate from H¨ older and (174) |∂zF(hτ)(.−y)−∂zF(hτ)(.)/ba∇dblLu/lessorsimilar/vextenddouble/vextenddouble((hτ)(.−y)−(hτ)(.)|)(|hτ(.−y)|2σ1−1+|hτ(.)|2σ1+1)/vextenddouble/vextenddouble Lu /lessorsimilar/ba∇dblhτ(.−y)−hτ(.)/ba∇dblLr/ba∇dblhτ/ba∇dbl2σ1−1 L2σ1q /lessorsimilar/ba∇dblhτ(.−y)−hτ(.)/ba∇dblLr/ba∇dbl|∇|˜shτ/ba∇dbl2σ1−1 L2, and so we have that /ba∇dbl|∇|s∂zF(hτ)/ba∇dblLu/lessorsimilar/integraldisplay∞ 0/parenleftbig R−˜ssup |y|≤R|∂zF(hτ)(.−y)−∂zF(hτ)(.)|Lu/parenrightbig21 RdR /lessorsimilar/ba∇dbl|∇|˜shτ/ba∇dbl2σ1−1 L2/ba∇dbl|∇|shτ/ba∇dblLr/lessorsimilar/ba∇dbl|∇|˜shτ/ba∇dbl2σ1 L2, and this concludes the proof of (147). /square References [1] V.V. Afanasjev, N. Akhmediev, and J.M. Soto-Crespo. Thr ee forms of localized solutions of the quintic complex Ginzburg-Landau equation. Phys. Rev. E , 53(2):1931, 1996. [2] N. Akhmediev and A. Ankiewicz. Solitons of the complex Gi nzburg-Landau equation. In Spatial solitons , pages 311–341. Springer, 2001. [3] P. Antonelli, R. 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Blow-up dynamics and s pectral property in the L2-critical nonlinear Schr¨ odinger equation in high dimensions. Nonlinearity , 31(9):4354–4392, 2018. Gran Sasso Science Institute, viale Francesco Crispi, 7, 67 100 L’Aquila, Italy Email address :paolo.antonelli@gssi.it Gran Sasso Science Institute, viale Francesco Crispi, 7, 67 100 L’Aquila, Italy Email address :boris.shakarov@gssi.it

2023-09-15

We consider the focusing, mass-supercritical NLS equation augmented with a nonlinear damping term. We provide sufficient conditions on the nonlinearity exponents and damping coefficients for finite-time blow-up. In particular, singularities are formed for focusing and dissipative nonlinearities of the same power, provided that the damping coefficient is sufficiently small. Our result thus rigorously proves the non-regularizing effect of nonlinear damping in the mass-supercritical case, which was suggested by previous numerical and formal results. We show that, under our assumption, the damping term may be controlled in such a way that the self-similar blow-up structure for the focusing NLS is approximately retained even within the dissipative evolution. The nonlinear damping contributes as a forcing term in the equation for the perturbation around the self-similar profile, that may produce a growth over finite time intervals. We estimate the error terms through a modulation analysis and a careful control of the time evolution of total momentum and energy functionals.

On the formation of singularities for the slightly supercritical NLS equation with nonlinear damping

2309.08281v1

9-11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 Comparison Between Damping Coefficients of Measured Perforated Structures and Compact Models T. Veijola1, G. De Pasquale2, and A. Somá2 1 Department of Radio Science and Engineering, Helsinky University of Technology P.O. Box 3000, 02015 TKK, Finland. 2 Department of Mechanics, Polytechnic of Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Abstract - Measured damping coefficients of six different perforated micromechanical test structures are compared with damping coefficients given by published compact models. The motion of the perforated plates is almost translational, the surface shape is rectangular, and the perforation is uniform validating the assumptions made for compact models. In the structures, the perforation ratio varies from 24% - 59%. The study of the structure shows that the compressibility and inertia do not contribute to the damping at the frequencies used (130kHz - 220kHz). The damping coefficients given by all four compact models underestimate the measured damping coefficient by approximately 20%. The reasons for this underestimation are discussed by studying the various flow components in the models. I. I NTRODUCTION Perforations are used in micromechanical squeeze-film dampers for several reasons. The main purpose is to reduce the damping and spring forces of oscillating structures due to the gas flow in small air gaps. Generally, the modeling problem is quite complicated, since the damping force acting on the moving structure depends on the 3D fluid flow in the perforations, in the air gap, and also around the structure. Compact models have been published in the literature, but their verification is generally questionable. Verification methods used are FEM solutions of the Navier-Stokes equations of the fluid volume and measurements [1]. In this paper, responses of four compact models are compared with measured responses of test structures. Six different perforated plates (figures 1 and 2) with different topologies have been measured at their first out-of-plane resonant frequencies, and the damping coefficients have been calculated from the quality factors (Q values) and effective masses. The measurement setup, the testing procedure and specimens characteristics are presented in [2]; here it is observed that dynamic parameters of the microsystem characterizing the fluidic and structural coupling can be extracted from the experimental frequency response function (FRF). The dynamic performance of microstructures are discussed based on the analytical solutions to perforated parallel-plate problems in [3], [4] and [5]. Since the perforation is uniform, the motion is almost translational, and also since the shape of the surface is rectangular, analytic damping models are applicable. First, the oscillating flow is analyzed using several characteristic numbers, the applicable modeling method is then chosen, the damping coefficients are calculated and compared with the measured ones. Finally, the results are discussed. II. T EST STRUCTURES AND MEASUREMENTS Figures 1 and 2 show the structures of the test specimen. The height of the plate hc = 15µm, the air gap height h = 1.6µm. Table I shows the other dimensions of the measured devices. In the table, q is the perforation ratio in percent, in this case q = MNs 02/(LW), M and N are the number of holes in the length and width directions, respectively. Fig. 1. Geometrical shape and dimensions of the vibrating structures. The measurements are made using the interferometric microscope Fogale ZoomSurf 3D (figure 3), with 20x objective magnification factor, 0.1nm of vertical resolution and 0.6µm of lateral resolution. The Frequency Shift technique is used, consisting of the excitation of the structure by an alternate voltage, the frequency of which is progressively increased by discrete steps. For each level of actuation frequency, the corresponding amplitude of vibration is stored and the experimental FRF is plotted for the detection of the resonance peak (figure 4). The first detection is made across a wide 9-11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 frequency range (0-500kHz) in order to roughly locate the resonance; five successive identical detections are then performed across a more precise and narrow range. These are statistically treated to extract the values of resonance reported in Table II. TABLE I DIMENSIONS OF MEASURED TOPOLOGIES type L [µm] W [µm] M×N L:W s0 [µm] s1 [µm] q % A 372.4 66.4 36x6 6:1 5.0 5.2 24 B 363.9 63.9 36x6 6:1 6.1 3.9 37 C 373.8 64.8 36x6 6:1 7.3 3.0 50 D 369.5 64.5 36x6 6:1 7.9 2.3 59 E 363.8 123.8 36x12 3:1 6.2 3.8 38 F 363.8 243.8 36x24 3:2 6.2 3.8 38 Fig. 2. Microscope image of specimen F. The measurement technique described uses a red monochromatic light source for the interferometric fringes detection. The vibration amplitude is detected optically in correspondence of a selectable region ( detection window ) of the specimen, located at the center of the suspended plate. The output value of the oscillation amplitude is averaged between the values captured by each pixel of the CCD camera inside the active window. Fig. 3. Interferometric microscope Fogale ZoomSurf 3D (a) and the 20x Nikon objective (b). 020406080100120140 180 190 200 210 220 230 240 Frequency (kHz)Amplitude (nm) Fig. 4. Displacement vs. frequency diagram of specimen C. The quality factor is extracted from the experimental curve, that was previously interpolated by a 6-order polynomial; the damping coefficient is finally calculated from the quality factor, resonant frequency and the effective mass by means of the method of the half power bandwidth . These are shown in Table II. The effective mass is calculated from FEM eigenmode analysis. The mass ratio α is the ratio between the modal mass and total mass. TABLE II MEASURED DAMPING COEFFICIENTS AND RESONANT FREQUENCIES OF SIX DIFFERENT TEST STRUCTURES type cm measured [10-6Ns/m] f0 measured [kHz] mass ratio α A 47.38 201.637 0.918 B 19.46 204.329 0.893 C 9.863 211.011 0.885 D 7.609 222.282 0.856 E 38.22 173.904 0.946 F 67.44 138.564 0.974 It is ensured that the amplitude of the oscillation is small compared with the gap height, and the static bias voltage caused by the excitation signal and DC bias voltage does not deflect the plate changing the air gap height. III. M ODELING OF THE DAMPING COEFFICIENT A. Analysis of the fluid flow The actual mass is supported with thin beams in such a way that the movement of the mass is approximately translational. This justifies starting with an analysis where the velocity of the plate surface is constant. Flow patterns In perforated dampers in perpendicular motion, two different flow patterns can be distinguished. The first is the “closed borders” pattern, where the fluid flows only through the holes. The second pattern, the “closed holes” pattern considers only the flow from the damper borders. In practical dampers both patterns exist simultaneously. The perforation ratios q (the area of the surface without holes divided by the area of the holes) are here considerably high, ranging from 24% - 59%. Also, the holes are relatively wide compared to the air gap height. It is then evident that the first “closed borders” flow pattern is strong here. Here compact models that consider both flow patterns are selected; the contribution the flow patterns in the measured cases will be discussed later in this paper. Rarefied gas Air at standard atmospheric conditions is used in the measurements. The pressure P A = 101kPa, the density ρ = 1.155 kg/m3, the viscosity coefficient µ = 18.5 ⋅10-6 Ns/m2, and the mean free path λ = 65nm. The air gap height h = 1.6µm, which makes the Knudsen number of the air gap flow Kch = λ/h to be 0.04, and the smallest hole diameter is 5µm making the Knudsen number of the perforation Ktb = λ/s0 to be 0.013. The estimated contributions to the damping coefficient are 1/(1 + 6 K ch) and 1/(1 + 7.567 Ktb), that is -24% 9-11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 and -9.8%, respectively. The assumption of a slightly rarefied gas is justified, and the slip velocity model is sufficient. Compressibility Next, the contribution of compressibility is analyzed. This can be made by studying the squeeze number. For a rigid surface, the squeeze number is 2212 hPW Aω ησ= (1) W is here the smallest, “dominating” characteristic dimension. For example, for case A without holes ω σ6108.3−⋅ = (2) At 200kHz σ = 4.8. This means that without perforation, compressibility should be considered (when σ ∼ 20, the viscous and spring forces are equal). When the surface is perforated, the situation changes completely. The characteristic dimension can now be estimated to be the space between the holes. In case A this is 5.2µm. The squeeze number now becomes σ = 0.03 at 200kHz. This is an overestimate of the real situation, since accounting for the damping in the holes will make the effective squeeze number considerably smaller. According to the squeeze number analysis, the spring forces are negligible compared with the damping forces, and noncompressible gas can be assumed without loss of any accuracy. The damping coefficient can be considered constant up to several MHz. Also, it is expected that the spring force due to the gas is much smaller than the force due to the effective spring of the mechanical structure. The frequency shift due to gas compressibility is expected to be very small. The spring force in the system is only due the effective spring of the mass-spring system. The spring coefficient can also be considered constant at least up to several MHz. Gas Inertia One can suspect that the inertia of the gas may contribute to the damping coefficient. The place where the inertia is the largest is the “widest” flow channel, that is the perforation holes. The contribution of inertia is characterized by the Reynolds number R e, specified for a circular channel as [6]: µωρ2 erR= (3) The “worst” case, where the inertial is the largest, is case D where the hole diameter is the largest. If the square channel is approximated with a circular channel having a radius of r = 4µm, that’s half of the hole side, gives R e = 0.998·10-6ω, and at 200 kHz R e = 1.255. The real and imaginary parts of the impedance are equal when Re ~ 6. The additional imaginary part does not directly influence the damping coefficient, but the change is due to the frequency dependent real part. At R e ~ 6, the change in the real part in only 3.2% [6]. This Reynolds number study shows that the inertia needs not to be considered even in the accurate analysis. B. Compact models A model for the noncompressible perforation cell is sufficient, as indicated in the study above. Four models that consider both “closed holes” and “closed borders” flow patterns are selected to be compared. The first one, M1, is a model by Bao [3] for a rectangular damper that is much longer than wide. In this model the air gap regime flow resistance, the flow resistance of a circular perforation, and its constant elongation are included. Continuum flow conditions are assumed. The 2 nd model M2 has been also presented in [3], but now an arbitrary rectangular surface is assumed. Next, the model M3 in [4] is used. The air gap flow resistance model, the circular perforation flow channel model, and four different elongations of the flow channels, that vary depending on the ratios of the cell dimensions, are included in the model. Slip velocity conditions are used for the air gap and the perforations. The 4th model, M4, is made especially for square holes [5]. It includes similar components as the previous model and accounts also for the rarefied gas in the slip flow regime. Model M4 for square perforations accepts directly the dimensions given in Table I (size of the perforation cell s x = s0 + s1). To apply the other models, an effective radius of the circular perforation r0 and the cell rx need to be specified first. Matching the areas of the actual cell sx2, and the equivalent circular cell πr02 gives πx xsr= (4) The radius r0 is determined by requiring the acoustic impedances of square and rectangular channels to match. For relatively small Knudsen numbers this leads to approximately [3], [4] 00 0 547.02096.1ssr ≈ = (5) The Appendix shows all equations needed in computing the damping coefficients using models M1…M4. IV. S IMULATION RESULTS AND DISCUSSION The results of the comparison are shown in Table III. ∆i is the relative error of the simulated damping coefficient cs of model M i compared to the measured damping coefficients. TABLE III RELATIVE ERRORS OF THE COMPACT MODELS type ∆1 [%] M1 ∆2 [%] M2 ∆3 [%] M3 ∆4 [%] M4 A -23.53 -25.74 -33.51 -33.27 B -16.36 -18.06 -21.02 -21.96 C -5.21 -6.59 -4.11 -6.65 D -14.66 -15.72 -12.46 -15.29 E -17.27 -18.94 -19.03 -20.14 F -4.77 -6.70 -5.19 -6.52 The results of models M3 and M4 are quite close to each other, the largest error between them is only 2.8%-points, showing that the effective radius approach is sufficient. Continuum conditions were assumed for M1 and M2, giving approximately 10% larger values than with slip velocity conditions. If these models are corrected to account for the 9-11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 rarefied gas, the errors become approximately 10% worse, than those shown in Table III. The drag on the sidewalls is expected to increase the damping coefficient since the structure is relatively high. The models do not account for this drag force. The moving supporting beams will have also an additional contribution to the damping. The length of the supporting beams in all cases is about L b = 122µm, and their widths are about Wb = 4µm. A rough approximation for the contribution of the supporting beams is ) 61()3.1 ( 34 ch33 b b bK hh WLc ++=µ (6) This gives a damping coefficient of cb = 0.16 ⋅10-6 Ns/m. This approximation shows that the damping due to the beams is very small. The responses of models M1 and M2 are quite close. One could expect that the error of M1 would be larger in case E and especially in case F, since the length to width ratio is quite small in these cases. The explanation for this can be found by studying the contribution of the different flow patterns. This can be easily done using the perforation cell model that assumes the “closed borders” flow pattern, where the pressure distribution is independent of the shape of the damper. The flow resistances R P for a perforation cell in [4] and [5] are derived using this assumption. The damping coefficient becomes simply c P = NMR P (MN is the number of holes). Table IV shows the errors of the damping coefficients c P compared with the measured values using the models for circular cells, M5, and rectangular cells, M6, as presented in [4] and [5], respectively. TABLE IV RELATIVE ERROR OF “PERFORATION CELL” COMPACT MODELS type ∆5 [%] M5 ∆6 [%] M6 A -17.25 -16.92 B -7.81 -9.00 C 7.38 4.36 D -3.55 -6.83 E -11.45 -12.73 F 0.37 -1.08 The results in Table IV show that the “closed borders” flow pattern is the dominant one. The contribution of the “closed holes” flow is only 6% - 16% of the damping coefficient. This explains why models M1 and M2 differ only slightly in this case: the contribution of the shape-dependent damping is quite small. TABLE V RELATIVE CONTRIBUTIONS OF THE FLOW RESISTANCES OF MODEL M5 type Rs [%] Ris [%] R ib [%] Ric [%] Rc [%] Re [%] A 8.15 9.78 0.78 5.63 68.01 7.65 B 7.62 12.94 1.87 5.13 64.05 8.40 C 6.48 15.30 3.50 4.55 61.19 8.98 D 4.51 14.49 5.03 4.06 62.59 9.31 E 7.51 13.18 2.00 5.07 63.80 8.45 F 7.51 13.18 2.00 5.07 63.80 8.45 To study further the sources of damping, Table V shows the contributions of the flow resistance components in M5 [4]. The flow resistance of the perforations R S is the most significant source of damping; its contribution is approximately 65%. The 2nd important contribution comes from the intermediate region resistances RIS, RIB, and RIC: 15 - 20%. Next important is the elongation at the perforation outlet R E, about 8%. V. C ONCLUSIONS Measured damping coefficients have been compared to those obtained with four different compact models for perforated dampers. After analyzing the oscillating flow with several characteristic numbers, sufficient models were selected. Only translational motion was assumed. The results of all models were quite close to each other, a systematic underestimate of the damping coefficient was about -20%. The reasons for this were discussed and the contribution of various flow components were presented. For a more accurate analysis, the realistic modes of the plates should be considered. It is expected that the “closed holes” flow pattern will become relatively stronger in this case. The comparison showed also how a model for circular perforations can be used to model square holes. REFERENCES [1] E.S. Kim, Y.H. Cho and M.U. Kim, “Effect of Holes and Edges on the Squeeze Film Damping of Perforated Micromechanical Structures” Proceedings of IEEE Micro Electro Mechanical Systems Conference , pp. 296-301, 1999. [2] A. Somà and G. De Pasquale, “Identification of Test Structures for Reduced Order Modeling of the Squeeze Film Damping in MEMS”, Proc. DTIP Symposium on Design, Test, Integration and Packaging of MEMS & MOEMS, pp. 230-239, 2007. [3] M. Bao, H. Yang, Y. Sun and P.J. French, “Modified Reynolds’ equation and analytical analysis of perforated structures”, J. Micromech. Microeng. , vol. 13, pp. 795-800, 2003. [4] T. Veijola, “Analytic Damping Model for an MEM Perforation Cell”, Microfluidics and Nanofluidics, vol. 2, pp. 249-260, 2006. [5] T. Veijola, “Analytic Damping Model for a Square Perforation Cell”, Proc. of the 9th International Conference on Modeling and Simulation of Microsystems , pp. 554-557, 2006. [6] C. J. Morris and F. K. Forster, “Oscillatory Flow in Microchannels”, Experiments in Fluids , vol. 36, pp. 924-937, 2004. 9-11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 APPENDIX This Appendix contains equations for four compact models M1…M4. The dimensions and symbols in Fig. 1 are used: the length and width of the perforated plate are L and W. The side lengths of the square holes and the square perforation cells are s 0 and sX = s 0 + s 1, respectively. A. Model M1 equations The equations for a narrow hole plate (L>>W) are given in [3]. Note, in the following equations a = W/2 and b = L/2. The equivalent radii for the circular cell and hole are given in Eqs. (4) and (5). The damping coefficient c is − + =la al hhKr rhaL c CCtanh 1 16)( 318234 0 2 02β βµ where () () ()() 831631323 ln4 4 0 C eff03 C4 02 02eff34 2 rh HrrhhKrrHhlK C πβββηββηβ β β β + ==+==− − − = B. Model M2 equations The equations for an arbitrary shaped rectangular plate are also included in [3]. Also, in the following equations a = W/2 and b = L/2. The equivalent radii for the circular cell and hole are given in Eqs. (4) and (5). The damping coefficient c is () () 332 2 hb acµγ−= where () () () ()[]∑∞ =++− − = ,...5,3,123 2 22 232 3 2 2/ 12/ 1tanh24/2 sinh/1 sinh6 3 nn nn παακπα πκαααα α γ al ba= = α κ , Above, l is the same as used in M1 equations. C. Model M3 equations A model for a circular perforation cell is derived in [4], and the damping coefficient of a rectangular perforated plate is given in the paper. Note, in the following equations a = W and b = L. The equivalent radii for the circular cell and hole are given in Eqs. (4) and (5). The damping coefficient c is ())1(/1 ,1 ,...5,3,1 ,... 5,3,1 , eff eff ,CR ba Gc mn nm nm∑∑∞ =∞ = += Where the effective surface dimensions are () () hK b bhK a a ch effch eff 3.313.13.313.1 + +=+ += and () 422P ,ch3622 22 22 , 64ab 768, πµπ nmMNRRQh nm bn amba G nmnm = + = The flow resistance of a single perforation cell is () () ∆+ = +∆ =∆ =∆−= − +− =+ + + + + = 0E tbC E CC0 ICB0 IBS 2 022 02 X IS4 X4 0 2 X2 0 0X 3 ch4 X SE C IC 4 04 X IB IS S P 88868 283ln21 12 rQhR Rr Rr RhrrrRrr rr rr hQrRR R R rrR R R R πµπµπµπµπµ where the elongations are ++ − =∆++ − =∆ hh hrfKK rrKrr rr C 0 B chtb 2 X2 0 Bch2 X2 0 X0 S ,1732.01812.0133.15.2186.0 32.0 56.0 () () − +×+ ⋅=∆ − − +=∆ hrf rr rrKrr rrK 0 E 4 X4 0 2 X2 0tb E2 X2 0 X0 tb C 754.0 2.0116216.013 944.025.0 41.0 66.0 61 π where the functions are ()() ()()ch5.3 E334 B 5.17117811 4311.71 , Kxxfyyxyxf ++=++= The flow rate coefficients and Knudsen numbers for the air gap and the holes are 9-11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 0tb tb tbch ch ch , 41, 61 rK K QhK K Q λλ = +== += D. Model M4 equations A model for a rectangular perforation cell has been given in [5]. Note, in the following equations a = W and b = L. The damping coefficient c is given by (C1), where RP for a rectangular hole is () () ∆+ = +∆ ==∆−= − +− =+ + + + + = 0E sqC E CC0 ICIBS 2 022 02 X IS4 X4 E0 2 X2 E0 E0X 3 ch4 X SE C IC 4 04 X IB IS S P 454.28454.28038 283ln21 12 sQhR Rs RRhss sRrr rr rr hQrRR R R ssR R R R µµµπµ where the elongations are () () () + − + =∆=∆− + =∆ 83.2 0 4 sq EC2 S 019.01 1 41242.0302.08.3 5.61122.0 hsK ξξ ξ where X0 ss=ξ The equation for ∆E includes a misprint in [5]. The corrected equation is shown above. The flow rate coefficients and Knudsen numbers for the square hole are 0sq sq sq , 567.71sK K Qλ= += The effective radius is 4 20 E0008.0 02108.0158076.0 ξ ξ+ +=sr

2008-05-07

Measured damping coefficients of six different perforated micromechanical test structures are compared with damping coefficients given by published compact models. The motion of the perforated plates is almost translational, the surface shape is rectangular, and the perforation is uniform validating the assumptions made for compact models. In the structures, the perforation ratio varies from 24% - 59%. The study of the structure shows that the compressibility and inertia do not contribute to the damping at the frequencies used (130kHz - 220kHz). The damping coefficients given by all four compact models underestimate the measured damping coefficient by approximately 20%. The reasons for this underestimation are discussed by studying the various flow components in the models.

Comparison Between Damping Coefficients of Measured Perforated Micromechanical Test Structures and Compact Models

0805.0893v1

1 Growth parameters of Bi 0.1Y2.9Fe5O12 thin films for high frequency applications Ganesh Gurjar1,4, Vinay Sharma2, S. Patnaik1,*, Bijoy K. Kuanr3 1School of Physical S ciences, Jawaharlal Neh ru University, New Delhi, INDIA 110067 2Department of Physics, Morgan State University, Baltimore, MD, USA 21251 3Special C entre for Nanosciences, Jawaharlal Nehru University, New Delhi , INDIA 110067 4Shaheed Rajguru College of Applied Sciences for Women, University of Delhi, INDIA 110096 Abst ract The growth and characterization of Bismuth (Bi) substituted YIG ( Bi-YIG, Bi0.1Y2.9Fe5O12) thin films are reported. Pulsed laser deposited (PLD) films with thicknesses ranging from 20 to 150 nm were grown o n Gadolinium Gallium Garnet substrates . Two substrate orientations of (100) and (111) were considered . The enhanced distribution of Bi3+ ions at dodecahedral site along (111) is observed to lead to an increment in lattice constant from 12.379 Å in (1 00) to 12.415 Å in (1 11) orient ed films. Atomic force microscopy images show ed decreasing roughness with increasing film thickness. Compared to (100) grown films, (111) orient ed films showed an increase in ferromagnetic resonance linewid th and consequent increase in Gilbert dampin g. The lowest Gilbert damping values are found to be (1.06±0. 12) × 10-4 for (100) and (2.30±0. 36) × 10-4 for (111) oriented films with thickness of ≈150 nm . The observed value s of extrinsic linewidth, effective magnetization , and anisotropic field are related to thickness of the films and substrate orientation. In addition, the in-plane angular variation establishe d four-fold symmetry for the (100) deposited films unlike the case of (111) deposited films. This study prescribes growth condition s for PLD grow n single-crystalline Bi -YIG films towards desired high frequency and magneto -optic al device applications. Keyword s: Bi-Yttrium iron oxide; Thin film; Lattice mismatch; Pulsed Laser Deposition; Ferromagnetic resonance; Gilbert damping; Inhomogeneous br oadening . Corresponding authors: spatnaik@mail.jnu.ac.in 2 1.1 Introduction One of the most important magnetic materials for studying high frequency magnetization dynamics is the Yttrium Iron Garnet (YIG, Y 3Fe5O12). Thin film form of YIG have attracted a huge attention in the field of spintronic devices due to its large spin -wave propagation length , high Curie temperature T c ≈ 560 K [1], lowest Gilbert damping and strong magneto -crystalline anisotropy [2-7]. Due to these merits of YIG, it finds several ap plications such as in magneto - optical (MO) devices, spin-caloritronics [8,9] , and microwave resonators and filters [10-14]. The crystal structure of YIG is body centered cubic under Ia3̅d space group . In Wyckoff notation, t he yttrium (Y) ions are located at the dodecahedral 24c sites, whereas the Fe ions are located at two distinct sites ; octahedral 16a and tetrahedral 24d . The oxygen ions are located in the 96h sites [7]. The ferrimagnetism of YIG is induced via a super -exchange interaction at the ‘d’ and ‘a’ site between the non-equivalent Fe3+ ions. It has already been observed that substituting Bi/Ce for Y in YIG improves magneto -optical responsiv ity [13,15 -21]. In addition, Bi substitution in YIG (Bi -YIG) is known to generate growth -induced anisotropy, therefore, perpendicular magnetic anisotropy (PMA) can be achieved in Bi doped YIG, which is beneficial in applications like magnetic memory and logic devices [7,22,23] . Due to its u sage in magnon -spintronics and related disciplines such as caloritronics, the study of fundamental characteristics of Bi -YIG materials is of major current interest due to their high uniaxial anisotropy and F araday rotation [17, 24-27]. Variations in the concentration of Bi3+ in YIG, as well as substrate orientation and film thickness, can improve strain tuned structural properties and magneto -optic characteristics . As a result, selecting the appropriate substrate orientation and film thickness is important for identifying the growth of Bi-YIG thin films. 3 The structural and magnetic characteristics of Bi -YIG [Bi 0.1Y2.9Fe5O12] thin film have been studied in the current study. Gadolinium Gallium Garnet (GGG) substrates with orientations of (100) and (111) were used to grow thin films . The Bi-YIG films of four different thickness (≈20 nm, 50 nm, 100 nm and 150 nm ) were deposited in -situ by pulsed laser deposit ion (PLD) method [19,2 8] over single -crystalline GGG substrates . Along with structural characterization of PLD grown films , magnetic properties were ascertained by using vibrating sample magnetometer (VSM) in conjunction with ferromagnetic resonance (FMR) techniques. FMR is a highly effective tool for studying magnetization dynamics. The FMR response not only provides information about the magnetization dynamic s of the material such as Gilbert damping and anisotropic field, but also about the static magnetic properties such as saturation magnetization and anisotropy field. 1.2 Experiment Polycrystalline YIG an d Bi -YIG targets were synthesized via the solid -state reaction method. Briefly, yttrium oxide (Y 2O3) and iron oxide (Fe 2O3) powders from Sigma -Aldrich were grounded for ≈14 hours before calcination at 1100 ℃. The calcined powders were pressed into pellets of one inch and sintered at 1300 ℃. Using these polycrystalline YIG and Bi -YIG targets, thin films of four thicknesses ( ≈20 nm, 50 nm, 100 nm, and 150 nm) were synthesized in -situ on (100) - and (111) -oriented GGG substrates using the PLD method. The samples are labelled in the text as 20 nm (100), 20 nm (111) , 50 nm (100) , 50 nm (111), 100 nm (100) , 100 nm (111) , 150 nm (100), and 150 nm (111) . Before deposition, GGG substrates were cleaned in an ultrasonic bath with acetone and isopropanol for 30 minute s. The deposition chamber was cleaned and evacuated to 5.3×10-7 mbar. For PLD growth, a 248 nm KrF excimer laser (Laser fluence (2.3 J cm-2) with 10 Hz pulse rate was used to ablate the material from the target . Oxygen pressure, target -to-4 substrate distance, and substrate temperature were maintained at 0.15 mbar, 5.0 cm, and 825 oC, respectively. Growth rate of deposited films were 6 nm/min . The as -grown films were annealed in-situ for 2 hours at 825 oC in the presence of oxygen (0.15 mbar). The structural characterization of thin films were ascertained using X -ray diffraction (XRD) with Cu-Kα radiation (1.5406 Å). We have performed the XRD me asurement at room temperature in -2 geometry and incidence angle are 20 degrees. The film's surface morphology and thickness were estimated using atomic force microscopy (AFM) (WITec GmbH , Germany ). The magnetic properties were studied using a vibrating sample magnetomet ry (VSM) in Cryogenic 14 Tesla Physical Property Measurement System (PPMS). FMR measurements were done on a coplanar waveguide (CPW) in a flip -chip arrangement with a dc magnetic field applied perpendicular to the high -frequency magnetic field (hRF). A Keysight Vector Network Analyzer was used for this purpose. The CPW was rotated in the film plane from 0º to 360º for in -plane () measurement s and from 0º to 18 0º for out of plane (θ) measurement. In this study, the thickness of Bi -YIG was determined by employing methods such as laser lithography and AFM. We have calibrated the thickness of thin films with PLD laser shots. Photoresist by spin coating is applied to a silicon substrate, and then straight-line patterns were drawn on the photoresist coated substrates using laser photolithography. The PLD technique was used to deposit thin films of the required material onto a pattern -drawn substrate. It is then necessary to wet etch the PLD grown thin fi lm in order to remove the photoresist coating. Then, AFM tip is scanned over the line pattern region in order to estimate the thickness of the grown samples from the AFM profile image. 5 1.3 Results and Discussion 1.3.1 Structural properties Figure 1 (a)-(d) show the XRD pattern of (100)- and (111)-oriented Bi-YIG grown thin films with thickness ≈20-150 nm (Insets depict the zoomed image of XRD patterns) . XRD data indicate single -crystalline growth of Bi -YIG thin films . Figures 1 (e) and 1 (f) show the l attice constant and lattice mismatch (with respect to substrate) determined from XRD data, respectively . The cubic lattice constant 𝒂 is calculated using the formula , 𝒂=𝜆√ℎ2+𝑘2+𝑙2 2sin𝜃 (1) where the wavelength of Cu -Kα radiation is represented by 𝜆, diffraction angle by 𝜃, and the Miller indices of the corresponding XRD peak by [h, k, l] . Further, the l attice mismatch parameter (𝛥𝑎 𝑎) is calculated using the equation , 𝛥𝑎 𝑎=(𝑎𝑓𝑖𝑙𝑚 − 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 ) 𝑎𝑓𝑖𝑙𝑚 100 (2) Here lattice constant of film and substrate are represented by 𝑎𝑓𝑖𝑙𝑚 and 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 , respectively . The reported lattice constant values are consistent with prior findings [15,17,21]. Lattice constant slightly increases with the increase in thickness of the film in the case of (111) as compared to (100) . Since the distribution of Bi3+ in the dodecahedral sit e is dependent on the substrate orientation [7,23,2 9], the (111) oriented films show an increase in the lattice constant . In Bi -YIG films, this slight increase in the lattice constant (in the 111 direction) leads to a compar atively larger lattice mismatch as seen in Fig. 1 (f). For 50 nm (111) Bi -YIG film, we achieved a lattice mismatch of ~0.47 , which is close to what has been reported earlier [30,31]. 6 Smaller value of lattice mismatch can reduc e the damping constant of the film [31]. We want to underline the importance of lattice plane dependen t growth in conjunction with film thickness in indicating structural and magnetic property changes. 1.3.2 Surface morphology Figure 2 (a) -(h) shows room temperature AFM images with root mean square (RMS) roughness. Roughness is essential from an application standpoint because the roughness directly impacts the inhomogeneous linewidth broadening which leads to increase in the Gil bert damping. We have observed RMS roughness around 0.5 nm or less for all grown Bi -YIG films which are comparable to previous reported YIG films [32,33]. We have observed that RMS roughness decreases with increase in thickness of the film. With (100) and (111) orientations, there is no discernible difference in roughness. Furthermore, roughness would be more affected by changes in growth factors and by substrate orientation [7,33,34]. 1.3.3 Static magnetization study The room temperature ( ≈296 K ) VSM magnetization measurements were carried out with applied magnetic field parallel to the film plane (in-plane) . The paramagnetic contribution s from the GGG substrate were carefully subtracted. F igure 3 (a)-(h) show s the magnetization plot s of Bi- YIG thin films of thickness ≈20-150 nm . Inset of Fig. 3 (i) shows the measured saturation magnetization ( µ0MS) data of as-grown (100) and (111) -oriented Bi -YIG films which are consistent with the previous reports [6,17,22,3 5,36]. Figure 3 ( i) shows plot of µ0Ms × t Vs. t, where ‘t’ is film thickness . This is done t o determine thickness of dead layer via linear 7 extrapolati on plot to the x -axis. The obtaine d magnetic dead -layer for (100) and (111) -oriented GGG substrates are 2.88 nm and 5.41 nm , which are comparable to previous reports [37-39]. The saturation magnetization of Bi -YIG films increases as the thickness of the films increases . The increase in saturation magnetization with increase in thickness can be understood by the following ways . Firstly, ferromagnetic thin films are generally deposited with a thin magnetically dead layer over the interface with the substrate. This magnetic dead layer effect is larger in thinner films that leads to the decrease in net magnetization with the decrease in thickness [40,41]. Figure 3 (i) shows the effect of magnetic dead laye r region near to the substrate. Secondly, t hicker films exhibit the bulk effect of YIG which, in turn, results in increas ed magnetization. 1.3.4 Ferromagnetic r esonance study Figure 4 (a) -(d) shows the FMR absorption spectra of (100) and (111) -oriented films that are labeled with open circle ( Ο) and open triangle ( Δ) respectively . FMR experiment s were carried out at room temperature. In -plane dc magnetic field was a pplied parallel to film surface . To find the effective magnetization and Gilbert damping, the FMR linewidth (∆H) and resonance magnetic field (H r) are calculated using a Lorentzian fit of the FMR absorption spectra measured at 𝑓 = 1 GHz to 12 GHz. Effective magnetization field ( 0𝑀𝑒𝑓𝑓) were obtained from the fitting of Kittel's in-plane equation (Eq. 3) [42]. 𝑓=𝛾 2𝜋0√(𝐻𝑟)(𝐻𝑟+𝑀𝑒𝑓𝑓) (3), Here, 0𝑀𝑒𝑓𝑓=0(𝑀𝑠−𝐻𝑎𝑛𝑖), anisotropy field 𝐻𝑎𝑛𝑖=2𝐾1 0𝑀𝑠, and 𝛾 being the gyromagnetic ratio. Further, the dependence of FMR linewidth on microwave frequency shows a linear variation (Eq. 4) [42] from which the Gilbert damping parameter (α) and FMR linewidth broad ening ( 𝛥𝐻 0) were obtained: µ0𝛥𝐻=µ0𝛥𝐻 0+4𝜋𝛼 𝛾𝑓 (4) 8 where, 𝛥𝐻 0 is the inhomogeneous broadening linewidth and α is the Gilbert damping. Figures 4 (e) and 4 (f) show Kittel and linewidth fitted graphs, respectively . Figure 5 (a) -(d) shows the derived parameters acquired from the FMR study. The estimated Gilbert damping is con sistent with data reported for sp in-wave propagation [3,22] . The value of α decreases as the thickness of the film increases ( Fig. 5 (c)). Howe ver, in the instance of Bi -YIG with (111) orientation, there is a substantial increase. This might be attributed qualitatively to the presence of Bi3+ ions, which cause strong spin-orbit coupling [43-45] as well as electron scattering inside the lattice when the lattice mismatch (or strain) increases [46]. Our earlier study [7] revealed a clear distribution of Bi3+ ions along (11 1) planes, as well as slightly larger lattice mismatch in Bi -YIG (111). These results explain the larger values of Gilbe rt damping, 0𝑀𝑒𝑓𝑓, and ΔH 0 values in Bi -YIG (111) (Fig. 5). The change in 0𝑀𝑒𝑓𝑓 is due to u niaxial in -plane magne tic anisotropy and it is observed from magnetization measurements using 0𝑀𝑒𝑓𝑓=0(𝑀𝑠−𝐻𝑎𝑛𝑖) [36,47,48]. The enhanced anisotropy field in the lower thic kness of Bi -YIG ( Fig. 5 (d) ) signifies the effect of dead magnetic layer at the interface. The lattice mismatch between films and GGG substrates induces uniaxial in - plane magnetic anisotropy [36,47]. ΔH 0 has a magnitude that is similar to previously published values for the same substrate orientation [7,47]. In conclusion, Bi -YIG with (100) orientation produces the lowest Gilbert damping facto r and inhomogeneous broadening linewidth . These are the required optimal parameters for spintronics based devices. Figure 6 (a) shows the variation of resonance field with polar angle ( ) for the grown 20 nm-150 nm films , H is the angle measured between applied magnetic field and surface of film (shown in inset of Fig. 4 (a)). The FMR linewidth (ΔH) were extracted fr om fitting of FMR spectra with L orentzian absorption functions. From Fig. 6 (a), we observe change in H r value for 50 nm Bi-YIG film as 0.22 T and 0.27 T for (100) and (111) orientation respectively. Similarly, 0.21 T and 0.31 T change is observed in (100) and (111) orientation respectively for 100 nm Bi -YIG film . We see that H r increases slightly in case o f (111) oriented film by changing the di rection of H from 0º to 90º with regard to sample surface (inset of Fig. 4 (a)). The change in H r decreases with increase in film thickness in cas e of (100) while it is reversed in case of (111 ). Figure 6 (b) shows 9 the variation of FMR linewidth with polar angle for 150 nm Bi -YIG film . Maximum FMR linewidth is observed at 90º and it is slightly more as compared with (100) orientation. The enhanced variation of FMR linewidth in (111) oriented samples is generated due to the higher contribution of two -magnon scatte ring in perpendicular geometry [49]. This can be understood due to the higher anisotropy field in (111) oriented samples ( Fig. 5 (d)). Figure 6 (c) & (e) shows the azimuthal angle ( ) variation of H r. Frequency of 5 GHz is used in the measurement . From variation data (by changing the direction of H from 0 º to 360 with regard to sample surface (inset of Fig. 4 (a)). We can see clearly in-plane anisotropy of four- fold in Bi-YIG (100) (Fig. 6 (c)) unlike in Bi-YIG (111) (Fig. 6 (e)). According to crystalline surface symmetry there would be six -fold in -plane anisotropy in case of (111) orientation but we have not observe d it, based on previous reports, it can be superseded by a mis cut-induced uniaxial anisotropy [33,50]. This reinforces our grown films' single -crystalline nature . The observed change in H r (H=0 to 45) is 6 .6 mT in 50 nm (100 ), 0.17 mT for 50 nm (111) , 6.2 mT in 100 nm (100) , 0.17 mT for 100 nm (111) ) and 5.1 mT in 150 nm (100 ). As a result, during in -plane rotation, the higher FMR field change observed along the (100) orientation. The dependent FMR field data shown in figure 6 (c) were fitted using the following Kittel relation [50] 𝑓=𝛾 2𝜋0√([𝐻𝑟cos(𝐻−𝑀)+𝐻𝑐cos4(𝑀−𝐶)+𝐻𝑢cos2(𝑀−𝑢)])× (𝐻𝑟cos(𝐻−𝑀)+𝑀𝑒𝑓𝑓+1 4𝐻𝑐(3+cos4(𝑀−𝐶))+𝐻𝑢𝑐𝑜𝑠2(𝑀−𝑢)) (5) With respect to the [100] direction of the GGG substrate, in -plane directions of the magnetic field, magnetization, uniaxial, and cubic anisotropies are given by H, M, u and c, respectively. 𝐻𝑢=2𝐾𝑢 µ0𝑀𝑠 and 𝐻𝑐=2𝐾𝑐 µ0𝑀𝑠 correspond to the uniaxial and cubic anisotropy fields, respectively, with 𝐾𝑢 and 𝐾𝑐 being the uniaxial and cubic magnetic anisotropy constants, respectively. 10 Figure 6 (d) shows t he obtained uniaxial anisotropy field, cubic anisotropy field and saturation magnetization field for (100) orientation. The obtained saturation magnetization field follows the same pattern as we have obtained from the VSM measurements. The cubic anisotropy field increases and then saturates with the thickness of the film. A large drop in the uniaxial anisotropy field is observed with the thickness of the grown films. We have not got the in -plane angular variation data for the 20 nm thick Bi -YIG sample and m ay be due to the low thickness of the Bi - YIG, it is not detected by our FMR setup. 1.4 Conclusion In conclusion, we compare the properties of high-quality Bi -YIG thin films of four distinct thicknesses (20 nm, 50 nm, 100 nm, and 150 nm) grown on GGG substrates with orientations of (100) and (111). Pulsed laser deposition was used to synthesize the se films. AFM and XRD characterizations reveal th at the deposited thin films have smooth surfaces and are phase pure. According to FMR data, t he Gilbert damping value decreases with increase in film thickness . This is explained i n the context of a dead m agnetic layer . The (100) orientation has a lower va lue of Gilbert damping, indicating that it is the preferable substrate for doped YIG thin films for high frequency application . Bi-YIG on (111) orientation , on the other hand, exhibits anisotropic dominance, which is necessary for magneto -optic devices. Th e spin -orbit coupled Bi3+ ions are responsible for the enhanced Gilbert damping in (111). We have also correlated ∆H 0, anisotropic field, and effective magnetization to the variations in film thickness and substrate ori entation . In (100) oriented films, there is unambiguous observation of four-fold in -plane anisotropy. In particular, Bi-YIG grown on (111) GGG substrates yields best result for optim al magnetization dynamics. This is linked to an enhanced magnetic anisotropy. Therefore, proper substrate 11 orientation and thickness are found to be important parameters for growth of Bi-YIG thin film towards high frequency applications. Acknowledgments This work is supported by the MHRD -IMPRINT grant, DST (SERB, AMT , and PURSE - II) gran t of Govt. of India. Ganesh Gurjar acknowledges CSIR, New Delhi for financial support . 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Akansel, A. Kumar, N. Behera, S. Husain, R. Brucas, S. Chaudhary, and P. Svedlindh, Thickness -dependent enhancement of damping in Co 2FeAl/β -Ta thin films , Physical Review B 97.13 (2018): 134421. https://doi.org/10 .1103/PhysRevB.97.134421 20 List of f igure caption s Figure 1: (a)-(d) X -ray diffraction (XRD) patterns of 20 nm -150 nm Bi -substituted YIG films in (100) and (111) orientations. Insets in (a) -(d) depict the zoomed image of XRD patterns. Variation of lattice constant (e) and (f) lattice mismatch with thickness are shown . Figure 2: (a)-(h) A tomic force microscopy images of 20 nm -150 nm Bi -YIG film in (100) and (111) orientations are shown . Figure 3: (a)-(h) Static magnetization graph of 2 0 nm -150 nm Bi-substituted YIG (Bi-YIG) films in (100) and (111) orientations. ( i) Graph to determine the magnetic dead -layer thickness of Bi - YIG films on (100) and (111) -oriented GGG substrates is depicted (inset shows the variation of saturation magnetiz ation value with the film thickness). Figure 4: (a)-(d) Ferromagnetic resonance ( FMR ) absorption spectra of 20 nm -150 nm Bi- substituted YIG films with (100) and (111) orientations. Inset in (a) shows the geometry of an applied field angle measured from the sample surface. (e) shows frequency -dependent FMR magnetic field data fitted with Kittel Eq. 3 . (f) shows frequency -dependent FMR linewidth data fitted with Eq. 4 . Figure 5: Variation s of (a) extrinsic linewidth, (b) effective magnetization, (c) Gilbert damping, and (d) magnetic anisotropy with thickness for (100) and (111) oriented Bi-substituted YIG films are depicted . 21 Figure 6: (a) Angular variation of Ferromagnetic resonance (FMR) magnetic field for 20 nm -150 nm Bi -substituted YIG (Bi -YIG) film with (100) and (111) orientations is shown. (b) Angular variation of FMR linewidth of 150 nm thick Bi -YIG film with (100) and (111) orientation is shown. Variations of FMR magnetic field as a function of azimuthal angle ( ) for (c) 50 nm, 100 nm and 150 nm Bi -YIG film with (100) orientation is depicted (d) obtained uniaxial anisotropy field, cubic anisotropy field and saturation magnetization field for (100) orientation. (e) dependent FMR fi eld data for 50 nm and 100 nm Bi -YIG film with (111) orientation is depicted. 22 Figure 1 23 Figure 2 24 Figure 3 25 Figure 4 26 Figure 5 27 Figure 6

2022-09-01

The growth and characterization of Bismuth (Bi) substituted YIG (Bi-YIG, Bi0.1Y2.9Fe5O12) thin films are reported. Pulsed laser deposited (PLD) films with thicknesses ranging from 20 to 150 nm were grown on Gadolinium Gallium Garnet substrates. Two substrate orientations of (100) and (111) were considered. The enhanced distribution of Bi3+ ions at dodecahedral site along (111) is observed to lead to an increment in lattice constant from 12.379 angstrom in (100) to 12.415 angstrom in (111) oriented films. Atomic force microscopy images showed decreasing roughness with increasing film thickness. Compared to (100) grown films, (111) oriented films showed an increase in ferromagnetic resonance linewidth and consequent increase in Gilbert damping. The lowest Gilbert damping values are found to be (1.06) * 10E-4 for (100) and (2.30) * 10E-4 for (111) oriented films with thickness of 150 nm. The observed values of extrinsic linewidth, effective magnetization, and anisotropic field are related to thickness of the films and substrate orientation. In addition, the in-plane angular variation established four-fold symmetry for the (100) deposited films unlike the case of (111) deposited films. This study prescribes growth conditions for PLD grown single-crystalline Bi-YIG films towards desired high frequency and magneto-optical device applications.

Growth parameters of Bi0.1Y2.9Fe5O12 thin films for high frequency applications

2209.00558v1

arXiv:1212.1772v3 [math.AP] 14 Mar 2013A NOTE ON THE LIFESPAN OF SOLUTIONS TO THE SEMILINEAR DAMPED WAVE EQUATION MASAHIRO IKEDA AND YUTA WAKASUGI Abstract. This paper concerns estimates of the lifespan of solutions t o the semilinear damped wave equation /squareu+Φ(t,x)ut=|u|pin (t,x)∈[0,∞)×Rn, where the coefficient of the damping term is Φ( t,x) =/angbracketleftx/angbracketright−α(1 +t)−βwith α∈[0,1), β∈(−1,1) andαβ= 0. Our novelty is to prove an upper bound of the lifespan of solutions in subcritical cases 1 < p <2/(n−α). 1.Introduction We consider the semilinear damped wave equation (1.1) utt−∆u+Φ(t,x)ut=|u|p,(t,x)∈[0,∞)×Rn, with the initial condition (1.2) ( u,ut)(0,x) =ε(u0,u1)(x), x∈Rn, whereu=u(t,x) is a real-valued unknown function of ( t,x), 1< p, (u0,u1)∈ H1(Rn)×L2(Rn) andεis a positive small parameter. The coefficient of the damping term is given by Φ(t,x) =/an}b∇acketle{tx/an}b∇acket∇i}ht−α(1+t)−β withα∈[0,1), β∈(−1,1) andαβ= 0. Here /an}b∇acketle{tx/an}b∇acket∇i}htdenotes/radicalbig 1+|x|2. Our aim is to obtain an upper bound of the lifespan of solutions to (1.1) . We recall some previous results for (1.1). There are many results a bout global existenceofsolutionsfor(1.1) andmanyauthorshavetried todet ermine the critical exponent(see [3, 4, 6,8, 10,11, 12, 13, 16,18, 20, 23, 24]and t he referencestherein). Here “critical” means that if pc<p, all small data solutions of (1.1) are global; if 1<p≤pc, the local solution cannot be extended globally even for small data. In the constant coefficient case α=β= 0, Todorova and Yordanov [18] and Zhang [23] determined the critical exponent of (1.1) with compactly supported data as pc=pF= 1+2 n. Thisisalsothe criticalexponentofthe correspondingheatequatio n−∆v+vt=|v|p and called the Fujita exponent (see [2]). On the other hand, there are few results about upper estimates o f the lifespan for (1.1). When n= 1,2, Li and Zhou [10] obtained the sharp upper bound: (1.3) Tε≤/braceleftbiggexp(Cε−2/n),ifp= 1+2/n, Cε−1/κ,if 1<p<1+2/n, 2000Mathematics Subject Classification. 35L71. Key words and phrases. semilinear damped wave equation; lifespan, upper bound. 12 M. IKEDA AND Y. WAKASUGI whereC=C(n,p,u0,u1)>0andκ= 1/(p−1)−n/2forthe data u0,u1∈C∞ 0(Rn) satisfying/integraltext (u0+u1)dx>0. Nishihara [14] extended this result to n= 3 by using the explicit formula of the solution to the linear part of (1.1) with initial data (0,u1): u(t,x) =e−t/2W(t)u1+J0(t)u1. HereW(t)u1is the solution of the wave equation /squareu= 0 with initial data (0 ,u1) andJ0(t)u1behaves like a solution of the heat equation −∆v+vt= 0. However, both the methods of [10] and [14] do not work in higher dimensional ca sesn≥4, because they used the positivity of W(t), which is valid only in the case n≤3. In this paper we shall extend both of the results to n≥4 in subcritical cases 1<p<1+2/n. Next, we recall some results of variable coefficient in cases α/ne}ationslash= 0 orβ/ne}ationslash= 0. There are many results on asymptotic behavior of solutions in conne ction with the diffusion phenomenon, Here the diffusion phenomenon means that so lution of the damped wave equation behaves like a solution for the corresponding heat equation ast→+∞. For more details about the diffusion phenomenon, see, for example [19, 21, 22]. For the case α∈[0,1),β= 0, Ikehata, Todorova and Yordanov [8] determined the critical exponent for (1.1) as pc= 1+ 2/(n−α), which also agrees with that of the corresponding heat equation −∆v+/an}b∇acketle{tx/an}b∇acket∇i}ht−αvt=|v|p. Here we emphasize that in this case there are no results about upper estimates for the lifes pan. It will be given in this paper. Next, for the case β∈(−1,1),α= 0, Nishihara [15] and Lin, Nishihara and Zhai [11] proved pc= 1 + 2/n, which is also same as that of the heat equation −∆v+(1+t)−βvt=|v|p. On the other hand, upper estimates of the lifespan have not been well studied. Recently, Nishihara [15] obtained a similar resu lt of [10, 14]: letn≥1,β≥0and(u0,u1)satisfy/integraltext Rnui(x)dx≥0(i= 0,1),/integraltext Rn(u0+u1)(x)dx> 0. Then there exists a constant C >0 such that Tε≤/braceleftbigg eCε−(1+β)/n,ifp= 1+(1+β)/n, Cε−1/ˆκ,if 1+2β/n≤p<1+(1+β)/n, where ˆκ= (1 +β)/(p−1)−n. We note that the rate ˆ κis not optimal, because it is not same as that of the corresponding heat equation. Moreove r, there are no results for 1+(1+ β)/n<p≤1+2/n. We note that the proof by Todorova and Yordanov [18] also gives the same upper bound in the case β= 0,1<p<1+1/n. In this paper we will improve the above result for all 1 <p<1+2/nand give the sharp upper estimate. Finally, we mention that our method is not applicable to αβ/ne}ationslash= 0. On the other hand, the second author [20] proved a small data global exis tence result for (1.1) withα,β≥0, α+β≤1, whenp>1+2/(n−α). This also agrees with the criticalexponent of the correspondingheat equation −∆v+/an}b∇acketle{tx/an}b∇acket∇i}ht−α(1+t)−βvt=|v|p. Therefore,it isexpectedthat when1 <p≤1+2/(n−α),thereisablow-upsolution for (1.1) in this case. 2.Main Result First,wedefinethesolutionof(1.1). Wesaythat u∈X(T) :=C([0,T);H1(Rn))∩ C1([0,T);L2(Rn)) is a solution of (1.1) with initial data (1.2) on the interval [0 ,T)LIFESPAN OF SOLUTIONS TO THE SEMILINEAR DAMPED WAVE EQUATIO N 3 if the identity/integraldisplay [0,T)×Rnu(t,x)(∂2 tψ(t,x)−∆ψ(t,x)−∂t(Φ(t,x)ψ(t,x)))dxdt =ε/integraldisplay Rn{(Φ(0,x)u0(x)+u1(x))ψ(0,x)−u0(x)∂tψ(0,x)}dx (2.1) +/integraldisplay [0,T)×Rn|u(t,x)|pψ(t,x)dxdt holds for any ψ∈C∞ 0([0,T)×Rn). We also define the lifespan for the local solution of (1.1)-(1.2) by Tε:= sup{T∈(0,∞];there exists a unique solution u∈X(T) of (1.1)-(1.2) }. We first describe the local existence result. Proposition 2.1. Letα≥0,β∈R,1<p≤n/(n−2) (n≥3),1<p<∞(n= 1,2),ε >0and(u0,u1)∈H1(Rn)×L2(Rn). ThenTε>0, that is, there exists a unique solution u∈X(Tε)to(1.1)-(1.2). Moreover, if Tε<+∞, then it follows that lim t→Tε−0/ba∇dbl(u,ut)(t,·)/ba∇dblH1×L2= +∞. For the proof, see, for example [7]. Next, we give an alomost optimal lower estimate of Tε. Proposition 2.2. Let(u0,u1)∈H1(Rn)×L2(Rn)be compactly supported and δ any positive number. We assume that α∈[0,1),β∈(−1,1),αβ≥0andα+β <1. Then there exists a constant C=C(δ,n,p,α,β,u 0,u1)>0such that for any ε>0, we have Cε−1/κ+δ≤Tε, where κ=2(1+β) 2−α/parenleftbigg1 p−1−n−α 2/parenrightbigg . The proof of this proposition follows from the a priori estimate for t he energy of solutions. For the proof, see [11, 8, 20]. We note that the above proposition is valid even for the case αβ/ne}ationslash= 0. Next, we state our main result, which gives an upper bound of Tε. Theorem 2.3. Letα∈[0,1),β∈(−1,1),αβ= 0and let1<p <1+2/(n−α). We assume that the initial data (u0,u1)∈H1(Rn)×L2(Rn)satisfy (2.2) /an}b∇acketle{tx/an}b∇acket∇i}ht−αBu0+u1∈L1(Rn)and/integraldisplay Rn(/an}b∇acketle{tx/an}b∇acket∇i}ht−αBu0(x)+u1(x))dx>0, where B=/parenleftbigg/integraldisplay∞ 0e−/integraltextt 0(1+s)−βdsdt/parenrightbigg−1 . Then there exists C >0depending only on n,p,α,β and(u0,u1)such thatTεis estimated as Tε≤C ε−1/κif1+α/(n−α)<p<1+2/(n−α), ε−(p−1)(log(ε−1))p−1ifα>0, p= 1+α/(n−α), ε−(p−1)ifα>0,1<p<1+α/(n−α)4 M. IKEDA AND Y. WAKASUGI for anyε∈(0,1], where κ=2(1+β) 2−α/parenleftbigg1 p−1−n−α 2/parenrightbigg . Remark 2.1. The results of Theorem 2.3 and Proposition 2.2 can be express ed by the following table: α= 0 β= 0 pc 1+2 n1+2 n−α Tε/lessorsimilar ε−1/κ ε−1/κ,/parenleftbigg 1+α n−α<p<1+2 n−α/parenrightbigg ε−(p−1)(log(ε−1))p−1,/parenleftbigg p= 1+α n−α/parenrightbigg ε−(p−1),/parenleftbigg 1<p<1+α n−α/parenrightbigg Tε/greaterorsimilarε−1/κ+δε−1/κ+δ κ(1+β)/parenleftbigg1 p−1−n 2/parenrightbigg2 2−α/parenleftbigg1 p−1−n−α 2/parenrightbigg Remark 2.2. It is expected that the rate κin Theorems 2.3 is sharp except for the caseα>0,1<p≤1+α/(n−α)from Proposition 2.2. Remark 2.3. The explicit form of Φ =/an}b∇acketle{tx/an}b∇acket∇i}ht−α(1+t)−βis not necessary. Indeed, we can treat more general coefficients, for example, Φ(t,x) =a(x)satisfyinga∈ C(Rn)and0≤a(x)/lessorsimilar/an}b∇acketle{tx/an}b∇acket∇i}ht−α, orΦ(t,x) =b(t)satisfyingb∈C1([0,∞))and b(t)∼(1+t)−β. Remark 2.4. The same conclusion of Theorem 2.3 is valid for the correspon ding heat equation −∆v+Φ(t,x)vt=|v|pin the same manner as our proof. Our proof is based on a test function method. Zhang [23] also used a similar way to determine the critical exponent for the case α=β= 0. By using his method, many blow-up results were obtained for variable coefficient cases (s ee [1, 8, 11]). However, the method of [23] was based on a contradiction argumen t and so upper estimates of the lifespan cannot be obtained. To avoid the contrad iction argument, we use an idea by Kuiper [9]. He obtained an upper bound of the lifespan for some parabolic equations (see also [5, 17]). We note that to treat the time -dependent damping case, we also use a transformation of equation by Lin, Nishih ara and Zhai [11] (see also [1]). At the end of this section, we explain some notation and terminology u sed throughout this paper. We put /ba∇dblf/ba∇dblLp(Rn):=/parenleftbigg/integraldisplay Rn|f(x)|pdx/parenrightbigg1/p . We denote the usual Sobolev space by H1(Rn). For an interval Iand a Banach spaceX, we define Cr(I;X) as the Banach space whose element is an r-timesLIFESPAN OF SOLUTIONS TO THE SEMILINEAR DAMPED WAVE EQUATIO N 5 continuously differentiable mapping from ItoXwith respect to the topology in X. The letter Cindicates the generic constant, which may change from line to line. We also use the symbols /lessorsimilarand∼. The relation f/lessorsimilargmeansf≤Cgwith some constantC >0 andf∼gmeansf/lessorsimilargandg/lessorsimilarf. 3.Proof of Theorem 2.3 We first note that if Tε≤C, whereCis a positive constant depending only on n,p,α,β,u 0,u1, then it is obvious that Tε≤Cε−1/κfor anyκ >0 andε∈(0,1]. Therefore, once a constant C=C(n,p,α,β,u 0,u1) is given, we may assume that Tε>C. In the case β/ne}ationslash= 0, (1.1) is not divergence form and so we cannot apply the test function method. Therefore, we need to transform the equ ation (1.1) into divergence form. The following idea was introduced by Lin, Nishihara a nd Zhai [11]. Letg(t) be the solution of the ordinary differential equation /braceleftBigg −g′(t)+(1+t)−βg(t) = 1, g(0) =B−1. The solution g(t) is explicitly given by g(t) =e/integraltextt 0(1+s)−βds/parenleftbigg B−1−/integraldisplayt 0e−/integraltextτ 0(1+s)−βdsdτ/parenrightbigg . By the de l’Hˆ opital theorem, we have lim t→∞(1+t)−βg(t) = 1 and sog(t)∼(1+t)β. We note that B= 1 andg(t)≡1 ifβ= 0. By the definition ofg(t), we also have |g′(t)|/lessorsimilar|(1+t)−βg(t)−1|/lessorsimilar1. Multiplying the equation (1.1) byg(t), we obtain the divergence form (3.1) ( gu)tt−∆(gu)−((g′−1)/an}b∇acketle{tx/an}b∇acket∇i}ht−αu)t=g|u|p, here we note that αβ= 0. Therefore, we can apply the test function method to (3.1). We introduce the following test functions: φ(x) :=/braceleftBigg exp(−1/(1−|x|2)) (|x|<1), 0 ( |x| ≥1), η(t) := exp(−1/(1−t2)) exp(−1/(t2−1/4))+exp( −1/(1−t2))(1/2<t<1), 1 (0≤t≤1/2), 0 (t≥1). It is obvious that φ∈C∞ 0(Rn),η∈C∞ 0([0,∞)) and there exists a constant C >0 such that for all |x|<1 we have |∇φ(x)|2 φ(x)≤C. Using this estimate, we can prove that there exists a constant C >0 such that the estimate (3.2) |∆φ(x)| ≤Cφ(x)1/p6 M. IKEDA AND Y. WAKASUGI is true for all |x|<1. Indeed, putting ϕ:=φ1/qwithq=p/(p−1), we have |∆φ(x)|=|∆(ϕ(x)q)|/lessorsimilar|∆ϕ(x)|ϕ(x)q+|∇ϕ(x)|2ϕ(x)q−2/lessorsimilarϕ(x)q−1=φ(x)1/p. In the same way, we can also prove that (3.3) |η′(t)| ≤Cη(t)1/p,|η′′(t)| ≤Cη(t)1/p fort∈[0,1). Letube a solution on [0 ,Tε) andτ∈(τ0,Tε),R≥R0parameters, where τ0∈ [1,Tε),R0>0 are defined later. We define ψτ,R(t,x) :=ητ(t)φR(x) :=η(t/τ)φ(x/R) and Iτ,R:=/integraldisplay [0,τ)×BRg(t)|u(t,x)|pψτ,R(t,x)dxdt, JR:=ε/integraldisplay BR(/an}b∇acketle{tx/an}b∇acket∇i}ht−αBu0(x)+u1(x))φR(x)dx, whereBR={|x|< R}. Sinceψτ,R∈C∞ 0([0,Tε)×Rn) anduis a solution on [0,Tε), we have Iτ,R+JR=/integraldisplay [0,τ)×BRg(t)u∂2 tψτ,Rdxdt−/integraldisplay [0,τ)×BRg(t)u∆ψτ,Rdxdt +/integraldisplay [0,τ)×BR(g′(t)−1)/an}b∇acketle{tx/an}b∇acket∇i}ht−αu∂tψτ,Rdxdt =:K1+K2+K3. Here we have used the property ∂tψ(0,x) = 0 and substituted the test function g(t)ψ(t,x) into the definition of solution (2.1). We note that for the correspo nding heat equation, we have the same decomposition without the term K1and so we can obtain the same conclusion (see Remark 2.4). We first estimate K1. By the H¨ older inequality and (3.3), we have K1≤τ−2/integraldisplay [0,τ)×BRg(t)|u||η′′(t/τ)|φR(x)dxdt (3.4) ≤Cτ−2/integraldisplay [τ/2,τ)×BRg(t)|u|η(t)1/pφR(x)dxdt ≤τ−2I1/p τ,R/parenleftBigg/integraldisplayτ τ/2g(t)dt·/integraldisplay BRφR(x)dx/parenrightBigg1/q ≤Cτ−2+1/q(1+τ)β/qRn/qI1/p τ,R.LIFESPAN OF SOLUTIONS TO THE SEMILINEAR DAMPED WAVE EQUATIO N 7 Using (3.2) and a similar calculation, we obtain K2≤R−2/integraldisplay [0,τ)×BRg(t)|u||∆φ(x/R)|η(t/τ)dxdt (3.5) ≤CR−2/integraldisplay [0,τ)×BRg(t)|u||φ(x/R)|1/pη(t/τ)dxdt ≤CR−2I1/p τ,R/parenleftbigg/integraldisplayτ 0g(t)η(t/τ)dt·/integraldisplay BR1dx/parenrightbigg1/q ≤C(1+τ)(1+β)/qR−2+n/qI1/p τ,R. ForK3, using (3.3) and |g′(t)−1|/lessorsimilarC, we have K3≤τ−1/integraldisplay [0,τ)×BR/an}b∇acketle{tx/an}b∇acket∇i}ht−α|u||η′(t/τ)|φR(x)dxdt (3.6) ≤τ−1I1/p τ,R/parenleftBigg/integraldisplayτ τ/2g(t)−q/pdt·/integraldisplay BR/an}b∇acketle{tx/an}b∇acket∇i}ht−αqφR(x)dx/parenrightBigg1/q ≤Cτ−1+1/q(1+τ)−β/pFp,α(R)I1/p τ,R, where Fp,α(R) = R−α+n/q(αq<n), (log(1+R))1/q(αq=n), 1 ( αq>n). Thus, putting D(τ,R) :=τ−(1+β)/p(τ−1+βRq/n+τ1+βR−2+q/n+Fp,α(R)) and combining this with the estimates (3.4)-(3.6), we have (3.7) JR≤CD(τ,R)I1/p τ,R−Iτ,R. Now we use a fact that the inequality acb−c≤(1−b)bb/(1−b)a1/(1−b) holds for all a >0,0< b <1,c≥0. We can immediately prove it by considering the maximal value of the function f(c) =acb−c. From this and (3.7), we obtain (3.8) JR≤CD(τ,R)q. On the other hand, by the assumption on the data and the Lebesgu e dominated convergence theorem, there exist C >0 andR0such thatJR≥Cεholds for all R>R 0. Combining this with (3.8), we have (3.9) ε≤CD(τ,R)q for allτ∈(τ0,Tε) andR>R 0. Now we difine τ0:= max{1,R(2−α)/(1+β) 0 }, and we substitute (3.10) R=/braceleftBigg τ(1+β)/(2−α)(αq<n), τ (αq≥n)8 M. IKEDA AND Y. WAKASUGI into (3.9). Here we note that R>R 0ifRis given by (3.10). As was mentioned at the beginning of this section, we may assume that τ0<Tε. Finally, we have ε≤C τ−κ(αq<n), τ−1/(p−1)log(1+τ) (αq=n), τ−1/(p−1)(αq>n), with κ=2(1+β) 2−α/parenleftbigg1 p−1−n−α 2/parenrightbigg . We can rewrite this relation as τ≤C ε−1/κif 1+α/(n−α)<p<1+2/(n−α), ε−(p−1)(log(ε−1))p−1ifα>0, p= 1+α/(n−α), ε−(p−1)ifα>0,1<p<1+α/(n−α). Here we note that κ >0 if and only if 1 < p <1+2/(n−α) and that αq=nis equivalent to p= 1+α/(n−α). Sinceτis arbitrary in ( τ0,Tε), we can obtain the conclusion of the theorem. Acknowledgement The authorsaredeeply gratefultoProfessorTatsuoNishitani. H e gaveus helpful advice. Theywouldalsoliketoexpresstheirdeepgratitudetoanano nymousreferee for many useful suggestions and comments. References [1]M. D’abbicco, S. Lucente ,A modified test function method for damped wave equations , arXiv:1211.0453v1. [2]H. Fujita ,On the blowing up of solutions of the Cauchy problem for ut= ∆u+u1+α, J. Fac. Sci. Univ. Tokyo Sect. I, 13(1966), 109-124. [3]N. Hayashi, E. I. Kaikina, P. I. Naumkin ,Damped wave equation with super critical non- linearities , Differential Integral Equations, 17(2004), 637-652. [4]T. Hosono, T. Ogawa ,Large time behavior and Lp-Lqestimate of solutions of 2-dimensional nonlinear damped wave equations , J. Differential Equations, 203(2004) 82-118. [5]M. Ikeda ,Lifespan of solutions for the nonlinear Schr¨ odinger equat ion without gauge invari- ance, arXiv:1211.6928v1. [6]R. Ikehata, Y. Miyaoka, T. Nakatake ,Decay estimates of solutions for dissipative wave equations in RNwith lower power nonlinearities , J. Math. Soc. Japan, 56(2004), 365-373. 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Yordanov ,Weighted L2-estimates for dissipative wave equations with variable coefficients , J. Differential Equations, 246(2009), 4497-4518. [20]Y. Wakasugi ,Small data global existence for the semilinear wave equatio n with space-time dependent damping , J. Math. Anal. Appl., 393(2012) 66-79. [21]J. Wirth ,Wave equations with time-dependent dissipation I. Non-effe ctive dissipation , J. Differential Equations, 222(2006), 487-514. [22]J. Wirth ,Wave equations with time-dependent dissipation II. Effecti ve dissipation , J. Dif- ferential Equations 232(2007), 74-103. [23]Qi S. Zhang ,A blow-up result for a nonlinear wave equation with damping: the critical case , C. R. Acad. Sci. Paris S´ er. I Math., 333(2001), 109-114. [24]Y. Zhou ,A blow-up result for a nonlinear wave equation with damping a nd vanishing initial energy in RN, Appl. Math. Lett., 18(2005), 281-286. E-mail address :m-ikeda@cr.math.sci.osaka-u.ac.jp E-mail address :y-wakasugi@cr.math.sci.osaka-u.ac.jp Department of Mathematics, Graduate School of Science, Osa ka University, Toy- onaka, Osaka, 560-0043, Japan

2012-12-08

This paper concerns estimates of the lifespan of solutions to the semilinear damped wave equation. We give upper estimates of the lifespan for the semilinear damped wave equation with variable coefficients in all space dimensions.

A note on the lifespan of solutions to the semilinear damped wave equation

1212.1772v3

Calculation of Gilbert damping and magnetic moment of inertia using torque-torque correlation model within ab initio Wannier framework Robin Bajaj1, Seung-Cheol Lee2, H. R. Krishnamurthy1, Satadeep Bhattacharjee2,∗and Manish Jain1† 1Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012, India 2Indo-Korea Science and Technology Center, Bangalore 560065, India (Dated: January 2, 2024) 1arXiv:2401.00714v1 [cond-mat.mtrl-sci] 1 Jan 2024Abstract Magnetization dynamics in magnetic materials are well described by the modified semiclassical Landau-Lifshitz-Gilbert (LLG) equation, which includes the magnetic damping ˆαand the magnetic moment of inertia ˆItensors as key parameters. Both parameters are material-specific and physi- cally represent the time scales of damping of precession and nutation in magnetization dynamics. ˆαandˆIcan be calculated quantum mechanically within the framework of the torque-torque corre- lation model. The quantities required for the calculation are torque matrix elements, the real and imaginary parts of the Green’s function and its derivatives. Here, we calculate these parameters for the elemental magnets such as Fe, Co and Ni in an ab initio framework using density functional theory and Wannier functions. We also propose a method to calculate the torque matrix elements within the Wannier framework. We demonstrate the effectiveness of the method by comparing it with the experiments and the previous ab initio and empirical studies and show its potential to improve our understanding of spin dynamics and to facilitate the design of spintronic devices. I. INTRODUCTION In recent years, the study of spin dynamics[1–5] in magnetic materials has garnered significant attention due to its potential applications in spintronic devices and magnetic storage technologies[6–9]. Understanding the behaviour of magnetic moments and their interactions with external perturbations is crucial for the development of efficient and reliable spin-based devices. Among the various parameters characterizing this dynamics, Gilbert damping[10] and magnetic moment of inertia play pivotal roles. The fundamental semi- classical equation describing the magnetisation dynamics using these two crucial parameters is the Landau-Lifshitz-Gilbert (LLG) equation[11, 12], given by: ∂M ∂t=M× −γH+ˆα M∂M ∂t+ˆI M∂2M ∂t2! (1) where Mis the magnetisation, His the effective magnetic field including both external and internal fields, ˆαandˆIare the Gilbert damping and moment of inertia tensors with the tensor components defined as αµνand Iµν, respectively, and γis the gyromagnetic ∗s.bhattacharjee@ikst.res.in †mjain@iisc.ac.in 2ratio. Gilbert damping, ˆαis a fundamental parameter that describes the dissipation of energy during the precession of magnetic moments in response to the external magnetic field. Accurate determination of Gilbert damping is essential for predicting the dynamic behaviour of magnetic materials and optimizing their performance in spintronic devices. On the other hand, the magnetic moment of inertia, ˆIrepresents the resistance of a magnetic moment to changes in its orientation. It governs the response time of magnetic moments to external stimuli and influences their ability to store and transfer information. The moment of inertia[13] is the magnetic equivalent of the inertia in classical mechanics[14, 15] and acts as the magnetic inertial mass in the LLG equation. Experimental investigations of Gilbert damping[16–23] and moment of inertia involve various techniques, such as ferromagnetic resonance (FMR) spectroscopy[24, 25], spin-torque ferromagnetic resonance (ST-FMR), and time-resolved magneto-optical Kerr effect (TR- MOKE)[26, 27]. Interpreting the results obtained from these techniques in terms of the LLG equation provide insights into the dynamical behaviour of magnetic materials and can be used to extract the damping and moment of inertia parameters. In order to explain the experimental observations in terms of more macroscopic theoretical description, various studies[28–34] based on linear response theory and Kambersky theory have been carried out. Linear response theory-based studies of Gilbert damping and moment of inertia involve perturbing the system and calculating the response of the magnetization to the pertur- bation. By analyzing the response, one can extract the damping parameter. Ab initio calculations based on linear response theory[33] can provide valuable insights into the mi- croscopic mechanisms responsible for the damping process. While formal expression for the moment of inertia in terms of Green’s functions have been derived within the Linear response framework[11], to the best of our knowledge, there hasn’t been any first principle electronic structure-based calculation for the moment of inertia within this formalism. Kambersky’s theory[35–37] describes the damping phenomena using a breathing Fermi surface [38] and torque-torque correlation model [39], wherein the spin-orbit coupling acts as the perturbation and describes the change in the non-equilibrium population of electronic states with the change in the magnetic moment direction. Gilmore et al.[32, 34] have reported the damping for ferromagnets like Fe, Ni and Co using Kambersky’s theory in the projector augmented wave method[40]. The damping and magnetic inertia have been derived within the torque-torque correlation 3model by expanding the effective dissipation field in the first and second-time derivatives of magnetisation[29–31]. In this work, the damping and inertia were calculated using the combination of first-principles fully relativistic multiple scattering Korringa–Kohn–Rostoker (KKR) method and the tight-binding model for parameterisation[41]. However, there hasn’t been any full ab intio implementation using density functional theory (DFT) and Wannier functions to study these magnetic parameters. The expressions for the damping and inertia involves integration over crystal momentum kin the first Brillouin zone. Accurate evaluation of the integrals involved required a dense k- point mesh of the order of 106−108points for obtaining converged values. Calculating these quantities using full ab initio DFT is hence time-consuming. To overcome this problem, here we propose an alternative. To begin with, the first principles calculations are done on a coarse kmesh instead of dense kmesh. We then utilize the maximally localised Wannier functions (MLWFs)[42] for obtaining the interpolated integrands required for the denser k meshes. In this method, the gauge freedom of Bloch wavefunctions is utilised to transform them into a basis of smooth, highly localised Wannier wavefunctions. The required real space quantities like the Hamiltonian and the torque-matrix elements are calculated in the Wannier basis using Fourier transforms. The integrands of integrals can then be interpolated on the fine kmesh by an inverse Fourier transform of the maximally localised quantities, thereby enabling the accurate calculations of the damping and inertia. The rest of this paper is organized as follows: In Sec. II, we introduce the expressions for the damping and the inertia. We describe the formalism to calculate the two key quantities: Green’s function and torque matrix elements, using the Wannier interpolation. In Sec. III, we describe the computational details and workflow. In Sec. IV, we discuss the results for ferromagnets like Fe, Co and Ni, and discuss the agreement with the experimental values and the previous studies. In Sec. V, we conclude with a short summary and the future prospects for the methods we have developed. II. THEORETICAL FORMALISM First, we describe the expressions for Gilbert damping and moment of inertia within the torque-torque correlation model. Then, we provide a brief description of the MLWFs and the corresponding Wannier formalism for the calculation of torque matrix elements and the 4Green’s function. A. Gilbert damping and Moment of inertia within torque-torque correlation model If we consider the case when there is no external magnetic field, the electronic structure of the system can be described by the Hamiltonian, H=H0+Hexc+Hso=Hsp+Hso (2) The paramagnetic band structure is described by H0andHexcdescribes the effective lo- cal electron-electron interaction, treated within a spin-polarised (sp) local Kohn-Sham exchange-correlation (exc) functional approach, which gives rise to the ferromagnetism. Hso is the spin-orbit Hamiltonian. As we are dealing with ferromagnetic materials only, we can club the first two terms as Hsp=H0+Hexc. During magnetization dynamics, (when the magnetization precesses), only the spin-orbit energy of a Bloch state |ψnk⟩is affected, where nis the band index of the state. The magnetization precesses around an effective field Heff=Hint+Hdamp+HI, where Hintis the internal field due to the magnetic anisotropy and exchange energies, Hdampis the damping field, and HIis the inertial field, respectively. From Eqn. (1), we can see that the damping field Hdamp =α Mγ∂M ∂t, while HI=I Mγ∂2M ∂t2. Equating these damping and inertial fields to the effective field corresponding to the change in band energies as magnetization processes, we obtain the mathematical description of the Gilbert damping and inertia. It was proposed by Kambersky [39] that the change of the band energies∂εnk ∂θµ(θ=θˆndefines the vector for the rotation) can be related to torque operator or matrix depending on how the Hamiltonian is being viewed Γµ= [σµ,Hso], where σµare the Pauli matrices. Eventually, within the so-called torque-torque correlation model, the Gilbert damping tensor can be expressed as follows: αµν=g msπZ Z −d f(ϵ) dϵ Tr[Γµ(IG)(Γν)†(IG)]d3k (2π)3dϵ (3) Here trace Tris over the band indices, and msis the magnetic moment. Recently, Thonig et al [30] have extended such an approach to the case of moment of inertia also, where they 5deduced the moment of inertia tensor components to be, Iνµ=gℏ msπZ Z f(ϵ)Tr[Γν(IG)(Γµ)†∂2 ∂ϵ2(RG) +Γν∂2 ∂ϵ2(RG)(Γµ)†(IG)]d3k (2π)3dϵ (4) Here f(ϵ) is the Fermi function, ( RG) and ( IG) are the real and imaginary parts of Green’s function G=(ϵ+ιη− H)−1with ηas a broadening parameter, mis the magnetization in units of the Bohr magneton, Γµ= [σµ,Hso] is the µthcomponent of torque matrix element operator or matrix, µ=x, y, z .αis a dimensionless parameter, and I has units of time, usually of the order of femtoseconds. To obtain the Gilbert damping and moment of inertia tensors from the above two k- integrals calculated as sums of discrete k-meshes, we need a large number of k-points, such as around 106, and more than 107, for the converged values of αand I respectively. The reason for the large k-point sampling in the first Brillouin zone (BZ) is because Green’s function becomes more sharply peaked at its poles at the smaller broadening, ηwhich need to be used. For I, the number of k-points needed is more than what is needed for αbecause ∂2RG/∂ϵ2has cubic powers in RGand/or IGas: ∂2RG ∂ϵ2= 2 (RG)3−RG(IG)2−IGRGIG−(IG)2RG (5) We note that to carry out energy integration in αit is sufficient to consider a limited number of energy points within a narrow range ( ∼kBT) around the Fermi level. This is mainly due to the exponential decay of the derivative of the Fermi function away from the Fermi level. However, the integral for I involves the Fermi function itself and not its derivative. Consequently, while the Gilbert damping is associated with the Fermi surface, the moment of inertia is associated with the entire Fermi sea. Therefore, in order to adequately capture both aspects, it is necessary to include energy points between the bottom of the valence band and the Fermi level. 6B. Wannier Interpolation 1.Maximally Localised Wannier Functions (MLWFs) The real-space Wannier functions are written as the Fourier transform of Bloch wave- functions, |wnR⟩=v0 (2π)3Z BZdke−ιk.R|ψnk⟩ (6) where |ψnk⟩are the Bloch wavefunctions obtained by the diagonalisation of the Hamiltonian at each kpoint using plane-wave DFT calculations. v0is the volume of the unit cell in the real space. In general, the Wannier functions obtained by Eqn. (6) are not localised. Usually, the Fourier transforms of smooth functions result in localised functions. But there exists a phase arbitrariness of eιϕnkin the Bloch functions because of independent diagonalisation at each k, which messes up the localisation of the Wannier functions in real space. To mitigate this problem, we use the Marzari-Vanderbilt (MV) localisation procedure[42– 44] to construct the MLWFs, which are given by, |wnR⟩=1 NX qNqX m=1e−ιq.RUq mn|ψmq⟩ (7) where Uq mnis a (Nq×N) dimensional matrix chosen by disentanglement and Wannierisation procedure. Nare the number of target Wannier functions, and Nqare the original Bloch states at each qon the coarse mesh, from which Nsmooth Bloch states on the fine k-mesh are extracted requiring Nq>Nfor all q,Nis the number of uniformly distributed qpoints in the BZ. The interpolated wavefunctions on a dense k-mesh, therefore, are given via inverse Fourier transform as: |ψnk⟩=X Reιk.R|wnR⟩ (8) Throughout the manuscript, we use qandkfor coarse and fine meshes in the BZ, respec- tively. 7FIG. 1: The figure shows the schematic of the localisation of the Wannier functions on a Rgrid. The matrix elements of the quantities like Hamiltonian on the Rgrid are exponentially decaying. Therefore, most elements on the Rgrid are zero (shown in blue). We can hence do the summation till a cutoff Rcut(shown in red) to interpolate the quantities on a fine kgrid. 2.Torque Matrix elements As described in the expressions of αµνand Iνµin Eqns. (3) and (4), the µthcomponent of the torque matrix is given by the commutator of µthcomponent of Pauli matrices and spin- orbit coupling matrix i.e.Γµ= [σµ,Hso]. Physically, we define the spin-orbit coupling (SOC) and spin-orbit torque (SOT) as the dot and cross products of orbital angular momentum and spin angular momentum operator, respectively, such that Hso=ξℓ.where ξis the coupling amplitude. Using this definition of Hso, one can show easily that −ι[σ,Hso] = 2ξℓ×which represents the torque. There have been several studies on how to calculate the spin-orbit coupling using ab initio numerical approach. Shubhayan et al. [45] describe the method to obtain SOC matrix elements in the Wannier basis calculated without SO interaction, using an approximation of weak SOC in the organic semiconductors considered in their work. Their method involves DFT in the atomic orbital basis, wherein the SOC in the Bloch basis can be related to the SOC in the atomic basis. Then, by the basis transformation, they get the SOC in the Wannier basis calculated in the absence of SO interaction. Farzad et al. [46] calculate the SOC by extracting the coupling amplitude from the Hamiltonian in the Wannier basis, treating the Wannier functions as atomic-like orbitals. 8We present a different approach wherein we can do the DFT calculation in any basis (plane wave or atomic orbital). Unlike the previous approaches, we perform two DFT calculations and two Wannierisations: one is with spin-orbit interaction and finite magnetisation (SO) and the other is spin-polarised without spin-orbit coupling (SP). The spin-orbit Hamiltonian, Hsocan then be obtained by subtracting the spin-polarized Hamiltonian, Hspfrom the full Hamiltonian, HasHso=H−H sp. This, however, can only be done if both the Hamiltonians, HandHsp, are written in the same basis. We choose to use the corresponding Wannier functions as a basis. It should however be noted that, when one Wannierises the SO and SP wavefunctions, one will get two different Wannier bases. As a result, we can not directly subtract the HandHspin these close but different Wannier bases. In order to do the subtraction, we find the transformation between two Wannier bases , i.e. express one set of Wannier functions in terms of the other. Subsequently, we can express the matrix elements ofHandHspin the same basis and hence calculate Hso. In the equations below, the Wannier functions, the Bloch wavefunctions and the operators defined in the corresponding bases in SP and SO calculations are represented with and without the tilde ( ∼) symbol, respectively. TheNSO Wannier functions are given by: |wnR⟩=1 NX qNqX m=1e−ιq.RUq mn|ψmq⟩ (9) where Uq mnis a (Nq× N) dimensional matrix. The wavefunctions and Wannier functions from the SO calculation for a particular qandRare a mixture of up and down spin states and are represented as spinors: |ψnq⟩= |ψ↑ nq⟩ |ψ↓ nq⟩ |wnR⟩= |w↑ nR⟩ |w↓ nR⟩ (10) The ˜NsSP Wannier functions are given by: |˜ws nR⟩=1 NX q˜Ns qX m=1e−ιq.R˜Uqs mn|˜ψs mq⟩ (11) where s=↑,↓.˜Uqs mnis a ( ˜Ns qטNs) dimensional matrix. Since the spinor Hamiltonian doesn’t have off-diagonal terms corresponding to opposite spins in the absence of SOC, the wavefunctions will be |˜ψs nq⟩=|˜ψnq⟩⊗|s⟩. The combined expression for ˜Uqfor˜N↑+˜N↓=˜N 9FIG. 2: This figure shows the implementation flow chart of the theoretical formalism described in Sec. II SP Wannier functions is: ˜Uq= ˜Uq↑0 0˜Uq↓ (12) where ˜Uqis˜NqטNdimensional matrix with ˜Nq=˜N↑ q+˜N↓ q. Dropping the sindex for SP kets results in the following expression for ˜NSP Wannier functions: |˜wnR⟩=1 NX q˜NqX m=1e−ιq.R˜Uq mn|˜ψmq⟩ (13) 10We now define the matrix of the transformation between SO and SP Wannier bases as: TRR′ mn=⟨˜wmR|wnR′⟩ =1 N2X qq′˜Nq,Nq′X p,l=1eι(q.R−q′.R′)˜Uq† mp⟨˜ψpq|ψlq′⟩Uq′ ln =1 N2X qq′eι(q.R−q′.R′)[˜Uq†Vqq′Uq′]mn (14) HereVqq′ pl=⟨˜ψpq|ψlq′⟩. Eqn. (14) is the most general expression to get the transformation matrix. We can reduce this quantity to a much simpler one using the orthogonality of wavefunctions of different q. Eqn. (14) hence becomes, TRR′ mn=1 N2X qeιq.(R−R′)[˜Uq†(NVq)Uq]mn =1 NX qeιq.(R−R′)[˜Uq†VqUq]mn (15) where Vq pl=⟨˜ψpq|ψlq⟩. Using this transformation, we write SP Hamiltonian in SO Wannier bases as: (Hsp)RR′ mn=⟨wmR|Hsp|wnR′⟩ =X plR′′R′′′⟨wmR|˜wpR′′⟩ ⟨˜wpR′′|Hsp|˜wlR′′′⟩⟨˜wlR′′′|wnR′⟩ =X plR′′R′′′(T†)RR′′ mp(˜Hsp)R′′R′′′ pl TR′′′R′ ln (16) Since Wannier functions are maximally localised and generally atomic-like, the major con- tribution to the overlap TRR′ mnis for R=R′. Therefore, we can write TRR mn=T0 mn. The reason is that it depends on relative R−R′, we can just consider overlaps at R=0. Eqn. 16 becomes, (Hsp)RR′ mn=X pl(T†)0 mp(˜Hsp)RR′ plT0 ln (17) Therefore, we write the Hsoin Wannier basis as, (Hso)RR′ mn=HRR′ mn−(Hsp)RR′ mn (18) 11The torque matrix elements in SO Wannier bases are given by, (Γµ)RR′ mn= (σµHso)RR′ mn−(Hsoσµ)RR′ mn (19) Consider ( σµHso)RR′ mnand insert the completeness relation of the Wannier functions, and also neglecting SO matrix elements between the Wannier functions at different sites because of their being atomic-like. (σµHso)RR′ mn=P pR′′(σµ)RR′′ mp(Hso)R′′R′ pn = (σµ)RR′ mp(Hso)0 pn (20) (σµ)RR′ mpis calculated by the Fourier transform of the spin operator written in the Bloch basis, just like the Hamiltonian. (σµ)RR′ mp=1 NX qe−ιq.(R′−R) Uq†(σµ)qUq mp(21) We interpolate the SOT matrix elements on a fine k-mesh as follows: (Γµ)k mn=X R′−Reιk.(R′−R)(σµ)RR′ mn (22) This yields the torque matrix elements in the Wannier basis. In the subsequent expres- sions, WandHsubscripts represent the Wannier and Hamiltonian basis, respectively. In order to rotate to the Hamiltonian gauge, which diagonalises the Hamiltonian interpolated on the fine kmesh using its matrix elements in the Wannier basis. (HW)k mn=X R′−Reιk.(R′−R)HRR′ mn (23) (HH)k mn= (Uk)†(HW)kUk mn(24) Here Uk(not to be confused with Uq) are the matrices with columns as the eigenvectors of (HW)k, and ( HH)k mn=ϵmkδmn. We use these matrices to rotate the SOT matrix elements in Eqn. (22) to the Hamiltonian basis as: (Γµ H)k mn= (Uk)†(Γµ W)kUk mn(25) 123.Green’s functions The Green’s function at an arbitrary kandϵon a fine k-mesh in the Hamiltonian basis is given by: Gk H(ϵ+ιη) = (ϵ+ιη−(HH)k)−1(26) where ηis a broadening factor and is caused by electron-phonon coupling and is generally of the order 5 −10 meV. Gk H(ϵ+ιη) is aN × N dimensional matrix. Therefore, we can calculate RG,IGand∂2RG/∂ϵ2as defined in Eqn. (5) and hence, α and I. III. COMPUTATIONAL DETAILS Plane-wave pseudopotential calculations were carried out for the bulk ferromagnetic tran- sition metals bcc Fe, hcp Co and fcc Ni using Quantum Espresso package[47, 48]. The conventional unit cell lattice constants ( a) used for bcc Fe and fcc Ni were 5.424 and 6.670 bohrs, respectively and for hcp Co, a=4.738 bohr and c/a=1.623 were used. The non- collinear spin-orbit and spin-polarised calculations were performed using fully relativistic norm-conserving pseudopotentials. The kinetic energy cutoff was set to 80 Ry. Exchange- correlation effects were treated within the PBE-GGA approximation. The self-consistent calculations were carried out on 16 ×16×16 Monkhorst-Pack Grid using Fermi smearing of 0.02 Ry. Non-self-consistent calculations were carried out using the calculated charge den- sities on Γ-centered 10 ×10×10 coarse k-point grid. For bcc Fe and fcc Ni, 64 bands were calculated and hcp Co 96 bands were calculated (because there are two atoms per unit cell for Co). We define a set of 18 trial orbitals sp3d2,dxy,dxz, and dyzfor Fe, 18 orbitals per atom s,panddfor Co and Ni, to generate 18 disentangled spinor maximally-localized Wannier functions per atom using Wannier90 package [43]. From the Wannier90 calculations, we get the checkpoint file .chk, which contains all the information about disentanglement and gauge matrices. We use .spnand.eigfiles generated bypw2wannier90 to get the spin operator and the Hamiltonian in the Wannier basis. We evaluate the SOT matrix elements in the Wannier Basis. We get αby simply summing up on a fine- kgrid with appropriate weights for the k- integration, and we use the trapezoidal rule in the range [-8 δ,8δ] for energy integration 1310−610−410−2100 10−610−410−210010−610−410−210010 10 10 10 10 101 0 1−2 −2 −310−410−310−210−1100101 10−410−310−210−1100101 Fe Ni Co(a) (b) (c) FIG. 3: (a), (b) and (c) shows the αvsηfor Fe, Ni and Co, respectively. Damping constants calculated using the Wannier implementation are shown in blue. Damping calculated using the tight binding method based on Lorentzian broadening and Green’s function by Thonig et al[29] are shown in brown and green, respectively. Comparison with damping constants calculated by Gilmore et al[32] using local spin density approximation (LSDA) are shown in red. The dotted lines are guides to the eye. around the Fermi level where δis the width of the derivative of Fermi function ∼kBT. We consider 34 energy points in this energy range. We perform the calculation for T= 300K. For the calculation of I, we use a very fine grid of 400 ×400×400k-points. For η >0.1, we use 320 energy points between VBM and Fermi energy. For 0 .01< η < 0.1, we use 3200-6400 energy points for the energy integration. TABLE I Material η(meV) -I (fs) α(×10−3) τ(ps) Fe 6 0.210 3.14 0.42 8 0.114 2.77 0.26 10 0.069 2.51 0.17 Ni 10 2.655 34.2 0.48 Co 10 0.061 1.9 0.21 14FIG. 4: Schematic explaining the dependence of intraband and interband contribution in αwith η. IV. RESULTS AND DISCUSSION A. Damping constant In this section, we report the damping constants calculated for the bulk iron, cobalt and nickel. The magnetic moments are oriented in the z-direction. For reference direction z, the damping tensor is diagonal resulting in the effective damping constant α=αxx+αyy. In Fig. 3, we report the damping constants calculated by the Wannier implementation as a function of broadening, ηknown to be caused by electron-phonon scattering and scattering with impurities. We consider the values of ηranging from 10−6to 2 eV to understand the role of intraband and interband transitions as reported in the previous studies[29, 32]. We note that the experimental range is for the broadening is expected to be much smaller with η∼5−10 meV. The results are found to be in very good agreement with the ones calculated using local spin density approximation (LSDA)[32] and tight binding paramterisation[29]. The expression for Gilbert damping[3] is written in terms of the imaginary part of Green’s functions. Using the spectral representation of Green’s function, Ank(ω), we can rewrite Eqn. 15(3) as: αµν=gπ msX nmZ Tµ nm(k)T∗ν nm(k)Snmdk (27) where Snm=R η(ϵ)Ank(ϵ)Amk(ϵ)dϵis the spectral overlap. Although we are working in the basis where the Hamiltonian is diagonal, the non-zero off-diagonal elements in the torque matrix lead to both intraband ( m=n) and interband ( m̸=n) contributions. For the sake of simple physical understanding, we consider the contribution of the spectral overlaps at the Fermi level for both intraband and interband transitions in Fig. 4. But in the numerical calculation temperature broadening has also been considered. For the smaller η, the contribution of intraband transitions decreases almost linearly with the increase in ηbecause the overlaps become less peaked. Above a certain η, the interband transitions become dominant and the contribution due to the overlap of two spectral functions at different band indices m and n becomes more pronounced at the Fermi level. Above the minimum, the interband contribution increases till η∼1 eV. Because of the finite Wannier orbitals basis, we have the accurate description of energy bands only within the approximate range of ( ϵF−10, ϵF+ 5) eV for the ferromagnets in consideration. The decreasing trend after η∼1 eV is, therefore, an artifact. 10−2 0.000.050.100.150.20 10−1100-0.001 0.000 0.001 0.002 10−1100 FIG. 5: Plot showing moment of inertia, −I versus broadening, η. The moment of inertia in the range 0 .03−3.0 eV is shown as an inset. The values using the Wannier implementation and the tight binding method[30] are shown in blue and green, respectively. 160.020.040.060.08 Co(a) 10−210−11000.0010−1100−0.0075−0.0050−0.0050−0.0025 0.000 10−210−1100− 0.00.51.0 1.01.5 Ni(b) 0.000.010.02 100101 FIG. 6: (a) and (b) show negative of the moment of inertia, −I versus broadening, ηfor Co and Ni, respectively. The values using the Wannier implementation and the tight binding method are shown in magenta and cyan, respectively. The moment of inertia is shown as an inset in the range 0 .03−2.0 eV and 0 .03−3.0 eV for Co and Ni, respectively. B. Moment of inertia In Fig. 5, we report the values for the moment of inertia calculated for bulk Fe, Co and Ni. Analogous to the damping, the inertia tensor is diagonal, resulting in the effective moment of inertia I = Ixx+ Iyy. The behaviour for I vs ηis similar to that of the damping, with smaller and larger ηtrends arising because of intraband and interband contributions, respectively. The overlap term in the moment of inertia is between the ∂2RG/∂ϵ2andIGunlike just IGin the damping. In Ref. [30], the moment of inertia is defined in terms of torque matrix elements and the overlap matrix as: Iµν=−gℏ msX nmZ Tµ nm(k)T∗ν nm(k)Vnmdk (28) where Vnmis an overlap function, given byR f(ϵ)(Ank(ϵ)Bmk(ϵ) +Bnk(ϵ)Amk(ϵ))dϵand Bmk(ϵ) is given by 2( ϵ−ϵmk)((ϵ−ϵmk)2−3η2)/((ϵ−ϵmk)2+η2)3. There are other notable features different from the damping. In the limit η→0, the overlap Vmnreduces to 2 /(ϵmk− ϵnk)3. For intraband transitions ( m=n), this leads to I → −∞ . In the limit η→ ∞ , Vmn≈1/η5which leads to I →0. The behaviour at these two limits is evident from Fig. 5. The large τ(small η) behaviour is consistent with the expression I = −α.τ/2πderived by 17104 −10−310−2−10−1100 10−2−10−110−1100101102 FIG. 7: The damping, magnetic moment of inertia and relaxation rate are shown as a function of broadening, ηin blue, green and red, respectively. The grey-shaded region shows the observed experimental relaxation rate, τ, ranging from 0 .12 to 0 .47 ps. The corresponding range of ηis shown in purple and is 6 −12 meV. This agrees with the experimental broadening in the range of 5 −10 meV, arising from electron-phonon coupling. The numbers are tabulated in Table I F¨ahnle et al. [49]. Here τis the Bloch relaxation lifetime. The behaviour of τas a function ofηusing the above expression in the low ηlimit is shown in Fig. 7. Apart from these limits, the sign change has been observed in a certain range of ηfor Fe and Co. This change in sign can be explained by the Eqn. (5). In the regime of intraband contribution, at a certain η, the negative and positive terms integrated over ϵandkbecome the same, leading to zero inertia. Above that η, the contribution due to the negative terms decreases until the interband contribution plays a major role leading to maxima in I (minima in −I). Interband contribution leads to the sign change from + to - and eventually zero at larger η. The expression I = −α.τ/2πderived from the Kambersky model is valid for η <10 meV, which indicates that damping and moment of inertia have opposite signs. By analyzing the rate of change of magnetic energy, Ref. [11] shows that Gilbert damping and the moment of inertia have opposite signs when magnetization dynamics are sufficiently slow (compared toτ). Experimentally, the stiffening of FMR frequency is caused by negative inertia. The softening caused by positive inertia is not observed experimentally. This is because the experimentally realized broadening, ηcaused by electron-phonon scattering and scattering with impurities, is of the order of 5 −10 meV. The values of Bloch relaxation lifetime, τ 18measured at the room temperature with the FMR in the high-frequency regime for Ni 79Fe21 and Co films of different thickness, range from 0 .12−0.47 ps. The theoretically calculated values for Fe,Ni and Co using the Wannier implementation for the ηranging from 5 −10 meV are reported in Table. I and lies roughly in the above-mentioned experimental range for the ferromagnetic films. V. CONCLUSIONS In summary, this paper presents a numerical method to obtain the Gilbert damping and moment of inertia based on the torque-torque correlation model within an ab initio Wannier framework. We have also described a technique to calculate the spin-orbit coupling matrix elements via the transformation between the spin-orbit and spin-polarised basis. The damping and inertia calculated using this method for the transition metals like Fe, Co and Ni are in good agreement with the previous studies based on tight binding method[29, 30] and local spin density approximation[32]. We have calculated the Bloch relaxation time for the approximate physical range of broadening caused by electron-phonon coupling and lattice defects. The Bloch relaxation time is in good agreement with experimentally reported values using FMR[27]. The calculated damping and moment of inertia can be used to study the magnetisation dynamics in the sub-ps regime. In future studies, we plan to use the Wannier implementation to study the contribution of spin pumping terms, arising from the spin currents at the interface of ferromagnetic-normal metal bilayer systems due to the spin-orbit coupling and inversion symmetry breaking to the damping. We also plan to study the magnetic damping and anisotropy in experimentally reported 2D ferromagnetic materials[50] like CrGeTe 3,CrTe ,Cr3Te4etc. The increasing interest in investigating the magnetic properties in 2D ferromagnets is due to magnetic anisotropy, which stabilises the long-range ferromagnetic order in such materials. Moreover, the reduction in dimensionality from bulk to 2D leads to intriguingly distinct magnetic properties compared to the bulk. VI. ACKNOWLEDGEMENTS This work has been supported by a financial grant through the Indo-Korea Science and Technology Center (IKST). 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2024-01-01

Magnetization dynamics in magnetic materials are well described by the modified semiclassical Landau-Lifshitz-Gilbert (LLG) equation, which includes the magnetic damping $\alpha$ and the magnetic moment of inertia $\mathrm{I}$ tensors as key parameters. Both parameters are material-specific and physically represent the time scales of damping of precession and nutation in magnetization dynamics. $\alpha$ and $\mathrm{I}$ can be calculated quantum mechanically within the framework of the torque-torque correlation model. The quantities required for the calculation are torque matrix elements, the real and imaginary parts of the Green's function and its derivatives. Here, we calculate these parameters for the elemental magnets such as Fe, Co and Ni in an ab initio framework using density functional theory and Wannier functions. We also propose a method to calculate the torque matrix elements within the Wannier framework. We demonstrate the effectiveness of the method by comparing it with the experiments and the previous ab initio and empirical studies and show its potential to improve our understanding of spin dynamics and to facilitate the design of spintronic devices.

Calculation of Gilbert damping and magnetic moment of inertia using torque-torque correlation model within ab initio Wannier framework

2401.00714v1

arXiv:1608.04071v1 [physics.class-ph] 14 Aug 2016Mechanical energy and mean equivalent viscous damping for SDOF fractional oscillators Jian Yuan1,∗, Bao Shi, Mingjiu Gai, Shujie Yang Institute of System Science and Mathematics, Naval Aeronau tical and Astronautical University, Yantai 264001, P.R.China Abstract Thispaperaddressesthetotalmechanical energyofasingledegr eeoffreedom fractional oscillator. Based on the energy storage and dissipation properties of the Caputo fractional derivatives, the expression for total m echanical en- ergy in the single degree of freedom fractional oscillator is firstly pr esented. The energy regeneration due to the external exciting force and t he energy loss due to the fractional damping force during the vibratory motio n are an- alyzed. Furthermore, based on the mean energy dissipation of the fractional damping element in steady-state vibration, a new concept of mean e quiva- lent viscous damping is suggested and the value of the damping coeffic ient is evaluated. Keywords: Fractional oscillators, linear viscoelasticity, fractional constitutive relations, mechanical energy, mean equivalent viscou s damping 1. Introduction Viscoelasticmaterialsanddampingtreatmenttechniqueshavebeen widely applied in structural vibration control engineering, such as aeros pace indus- try, military industry, mechanical engineering, civil and architectu ral engi- neering[1]. Describingtheconstitutive relationsforviscoelastic mat erialsisa top priority to seek for the dynamics of the viscoelastically damped s tructure and to design vibration control systems. ∗Corresponding author. Tel.: +8613589862375. Email address: yuanjianscar@gmail.com (Jian Yuan) Preprint submitted to Elsevier September 19, 2018Recently, theconstitutiverelationsemploying fractionalderivativ es which relatestressandstraininmaterials, alsotermedasfractionalvisc oelasticcon- stitutive relations, have witnessed rapid development. They may be viewed asanaturalgeneralizationoftheconventional constitutiverelat ionsinvolving integer order derivatives or integrals, and have been proven to be a power- ful tool of describing the mechanical properties of the materials. Over the conventional integer order constitutive models, the fractional o nes have vast superiority. The first attractive feature is that they are capable of fitting ex- perimental results perfectly and describing mechanical propertie s accurately in both the frequency and time domain with only three to five empirical pa- rameters[2]. Thesecondisthattheyarenotonlyconsistent withth ephysical principles involved [3] and the molecule theory [4], but also represent t he fad- ing memory effect [2] and high energy dissipation capacity [5]. Finally, fr om mathematical perspectives the fractional constitutive equation s and the re- sulting fractional differential equations of vibratory motion are co mpact and analytic [6]. Nowadays many types of fractional order constitutive relations h ave been established via a large number of experiments. The most frequently used models include the fractional Kelvin-Voigt model with three paramet ers [2]: σ(t) =b0ε(t)+b1Dαε(t), the fractional Zener model with four parameters [3]:σ(t)+aDασ(t) =b0ε(t)+b1Dαε(t), and the fractional Pritz model with five parameters [7]: σ(t)+aDασ(t) =b0ε+b1Dα1ε(t)+b2Dα2ε(t). Fractionaloscillators, orfractionallydampedstructures, aresy stemswhere the viscoelastic damping forces in governing equations of motion are de- scribed by constitutive relations involving fractional order derivat ives [8]. The differential equations of motion for the fractional oscillators a re frac- tional differential equations. Researches on fractional oscillator s are mainly concentratedontheoreticalandnumericalanalysisofthevibrat ionresponses. Investigations on dynamical responses of SDOF linear and nonlinear frac- tional oscillators, MDOF fractional oscillators and infinite-DOF frac tional oscillators have been reviewed in [8]. Asymptotically steady state beh avior of fractional oscillators have been studied in [9, 10]. Based on the fu nc- tional analytic approach, the criteria for the existence and the be havior of solutions have been obtained in [11-13], and particularly in which the im- pulsive response function for the linear SDOF fractional oscillator is derived. The asymptotically steady state response of fractional oscillator s with more than one fractional derivatives have been analyzed in [14]. Consider ing the memory effect and prehistory of fractional oscillators, the histor y effect or 2initialization problems for fractionally damped vibration equations has been proposed by Fukunaga, M. [15-17] and Hartley, T.T., and Lorenzo, C.F. [18, 19]. Stability synthesis for nonlinear fractional differential equations h ave re- ceived extensive attention in the last five years. Mittag-Leffler sta bility theo- rems [20, 21]andtheindirect Lyapunov approach[22] based onthe frequency distributed model are two main techniques to analyze the stability of non- linear systems, though there is controversy between the above t wo theories due to state space description and initial conditions for fractional systems [23]. In spite of the increasing interest in stability of fractional differ ential equations, there’s little results on the stability of fractionally dampe d sys- tems. For the reasons that Lyapunov functions are required to c orrespond to physical energy and that there exist fractional derivatives in the differential equations of motion for fractionally damped systems, it is a primary t ask to define the energies stored in fractional operators. Fractional energy storage and dissipation properties of Riemann- Liouville fractional integrals is defined [24, 25] utilizing the infinite state appr oach. Based on the fractional energies, Lyapunov functions are propo sed and sta- bility conditions of fractional systems involving implicit fractional der iva- tives are derived respectively by the dissipation function [24, 25] an d the energy balance approach [26, 27]. The energy storage properties of frac- tional integrator and differentiator in fractional circuit systems h ave been investigated in [28-30]. Particularly in [29], the fractional energy for mula- tion by the infinite-state approach has been validated and the conv entional pseudo-energy formulations based on pseudo state variables has been inval- idated. Moreover, energy aspects of fractional damping forces described by the fractional derivative of displacement in mechanical elements ha ve been considered in [31, 32], in which the effect on the energy input and ener gy return, as well as the history or initialization effect on energy respo nse has been presented. On the basis of the recently established fractional energy definitio ns for fractional operators, our main objective in this paper is to deal wit h the total mechanical energy of a single degree of freedom fractional oscillator. To this end, we firstly present the mechanical model and the differe ntial equation of motion for the fractional oscillator. Then based on the energy storage and dissipation in fractional operators, we provide the ex pression of total mechanical energy in the single degree of freedom fractiona l oscillator. Furthermore, we analyze the energy regeneration due to the ext ernal exciting 3force and the energy loss due to the fractional damping force in th e vibration processes. Finally, based on the mean energy dissipation of the fra ctional dampingelementinsteady-statevibration, weproposeanewconce ptofmean equivalent viscous damping and determine the expression of the dam ping coefficient. The rest of the paper is organized as follows: Section 2 retrospect some basic definitions and lemmas about fractional calculus. Section 3 intr oduces the mechanical model and establishes the differential equation of m otion for the single degree of freedom fractional oscillator. Section 4 provid es the expression of total mechanical energy for the SDOF fractional o scillator and analyzes the energy regeneration and dissipation in the vibration pr ocesses. Section 5 suggests a new concept of mean equivalent viscous dampin g and evaluatesthevalueofthedamping coefficient. Finally, thepaperisco ncluded in section 6 with perspectives. 2. Preliminaries Definition 1. The Riemann-Liouville fractional integral for the function f(t) is defined as aIα tf(t) =1 Γ(α)/integraldisplayt a(t−τ)α−1f(τ)dτ, (1) whereα∈R+is an non-integer order of the factional integral, the subscripts aandtare lower and upper terminals respectively. Definition 2. The Caputo definition of fractional derivatives is aDα tf(t) =1 Γ(n−α)/integraldisplayt af(n)(τ)dτ (t−τ)α−n+1,n−1< α < n. (2) Lemma 1. The frequency distributed model for the fractional integra tor [33- 35]Theinput ofthe Riemann-Liouvilleintegralis denotedb yv(t)andoutput x(t), thenaIα tv(t)is equivalent to /braceleftBigg ∂z(ω,t) ∂t=−ωz(ω,t)+v(t), x(t) =aIα tv(t) =/integraltext+∞ 0µα(ω)z(ω,t)dω,(3) withµα(ω) =sin(απ) πω−α. 4System (3) is the frequency distributed model for fractiona l integrator, which is also named as the diffusive representation. Lemma 2. The following relation holds[26] /integraldisplay∞ 0ωµα(ω) ω2+Ω2dω=sinαπ 2Ωαsinαπ 2. (4) 3. Differential equation of motion for the fractional oscill ator This section will establish the differential equation of motion for a sin- gle degree of freedom fractional oscillator, which consists of a mas s and a spring with one end fixed and the other side attached to the mass, d epicted in Fig.1. The spring is a solid rod made of some viscoelastic material with the cross-sectional area Aand length L, and provides stiffness and damping for the oscillator. m Figure 1: Mechanical model for the SDOF fractional oscillator. In accordance with Newton’s second law, the dynamical equation fo r the SDOF fractional oscillator is m¨x(t)+fd(t) =f(t),fd(t) =Aσ(t), (5) wherefd(t) is the forceprovided by the viscoelastic rodandcan be separated into two parts: the resilience and the damping force. f(t) is the vibration exciting force acted on the mass. The kinematic relation is ε(t) =x(t) L. (6) 5As for the constitutive equation of viscoelastic material, the followin g frac- tional Kelvin-Voigt model (7) with three parameters will be adopted σ(t) =b0ε(t)+b1Dα tε(t), (7) whereα∈(0,1) is the order of fractional derivative, b0andb1are positive constant coefficients. The above three relations (5) (6) and (7) form the following differen tial equa- tion of motion for the single degree of freedom fractional oscillator m¨x(t)+cDαx(t)+kx(t) =f(t), (8) wherec=Ab1 L,k=Ab0 L. For the reason that the Caputo derivative is fully compatible with the clas- sical theory of viscoelasticity on the basis of integral and different ial consti- tutive equations [36], the adoption of the Caputo derivative appear s to be the most suitable choice in the fractional oscillators. For the simplific ation of the notation, the Caputo fractional-order derivativeC 0Dα tis denoted as Dα in this paper. Comparing the forms of differential equations for the fractional o scillator (8) with the following classical ones m¨x(t)+c˙x(t)+kx(t) =f(t), (9) onecansee thatthefractionalone(8)isthegeneralizationofthe classical one (9)byreplacing thefirst orderderivative ˙ xwiththefractionalorderderivative Dαx. However, the generalization induces the following essential differe nces between them. •In view of the formalization of the mechanical model, the classical os - cillator is composed of a mass, a spring and a dashpot, where kis the stiffness coefficient of the spring offering restoring force kxandcis the damping coefficient of the dashpot offering the damping force c˙x. The fractional oscillator is formed by a mass and an viscoelastic rod. The rod offers not only resilience but also damping force. In fraction al differential equation(8), the coefficient candkare determined by both the constitute equation (7) for the viscoelastic material and the g eo- metrical parameters for the rod, which can be interpreted respe ctively as the fractional damping coefficient and the stiffness coefficient. A s a 6result, the physical meaning of candkin the fractional oscillator (8) andtheclassical one(9) aredifferent. Thefractional damping for cecan be viewed as a parallel of a spring component kx(t) and a springpot component cDαx(t) which is termed in [37] and illustrated in Fig.2. The hysteresis loop of the fractional damping force is dipicted in Fig.3 . mkx cD xD Figure 2: Abstract mechanical model for the SDOF fractional osc illator. −2−1.5 −1−0.5 00.5 11.5−4−3−2−101234 x(t)Dalpha x(t) Figure 3: Hysteresis loop of the fractional damping force. •Fractional operators are characterized by non-locality and memo ry properties, so fractional oscillators (8) also exhibit memory effect and the vibration response is influenced by prehistory. While the classica l one (9) has no memory effect and the vibration response is irrelevan t with prehistory. •In the aspect of mechanical energy, the fractional term Dαxin (8) not only stores potential energy but also consumes energy due to the fact 7that fractional operators exhibit energy storage and dissipation simul- taneously [24]. As a result, the total mechanical energy in fraction al oscillator consists of three parts: the kinetic energy1 2m˙x2stored in the mass, the potential energy corresponding to the spring element1 2kx2, and the potential energy e(t) stored in the fractional derivative. How- ever, in [38] the potential energy e(t) stored in the fractional term Dαxhas been neglected and the expression1 2m˙x2+1 2kx2for the total mechanical energy is incomplete. 4. The total mechanical energy Given the above considerations, we present the total mechanical energy of the SDOF fractional oscillator (8)in this section. The fractional system is assumed to be at rest before exposed to the external excitation . We firstly analyze the energy stored in the Caputo derivative, based on which the ex- pression for total mechanical energy is derived. Then we obtain th e energy regeneration due to external excitation and the energy dissipatio n due to the fractional viscoelastic damping. Bydefinitions(1)and(2),theCaputoderivativeiscomposedofone Riemann- Liouville fractional order integral and one integer order derivative , Dαx(t) =I1−α˙x(t). In view of Lemma 1, the frequency distributed model for the Caput o deriva- tive is /braceleftBigg ∂z(ω,t) ∂t=−ωz(ω,t)+ ˙x, Dαx(t) =/integraltext∞ 0µ1−α(ω)z(ω,t)dω.(10) In terms of the fractional potential energy expression for the f ractional inte- gral operator in [24], the stored energy in the Caputo derivative is e(t) =1 2/integraldisplay∞ 0µ1−α(ω)z2(ω,t)dω. (11) The total mechanical energy of the SDOF fractional oscillator is th e sum of the kinetic energy of the mass1 2m˙x2, the potential energy corresponding to the spring element1 2kx2, and the potential energy stored in the fractional derivative ce(t) E(t) =1 2m˙x2+1 2kx2+c 2/integraldisplay∞ 0µ1−α(ω)z2(ω,t)dω. (12) 8To analyze the energy consumption in the fractional viscoelastic os cillator, taking the first order time derivative of E(t),one derives dE(t) dt=m˙x¨x+kx˙x+c/integraldisplay∞ 0µ1−α(ω)z(ω,t)∂z(ω,t) ∂tdω.(13) Substituting the first equation in the frequency distributed model (10) into the third term of the above equation (13), one derives dE(t) dt=m˙x¨x+kx˙x+c/integraldisplay∞ 0µ1−α(ω)z(ω,t)[−ωz(ω,t)+ ˙x]dω =m˙x¨x+kx˙x+c˙x/integraldisplay∞ 0µ1−α(ω)z(ω,t)dω −c/integraldisplay∞ 0ωµ1−α(ω)z2(ω,t)dω. (14) Substituting the second equation in the frequency distributed mod el (10) into the second term of the above equation (14), one derives dE(t) dt=m˙x¨x+kx˙x+c˙xDαx−c/integraldisplay∞ 0ωµ1−α(ω)z2(ω,t)dω = ˙x[m¨x+cDαx+kx]−c/integraldisplay∞ 0ωµ1−α(ω)z2(ω,t)dω. (15) Substituting the differential equation of motion (8) for the fractio nal oscilla- tor into the first term of the above equation (15), one derives dE(t) dt=f(t) ˙x(t)−c/integraldisplay∞ 0ωµ1−α(ω)z2(ω,t)dω. (16) From Eq. (16) it is clear that the energy regeneration in the fractio nal oscillator due to the work done by the external excitation in unit time is P(t) =f(t) ˙x(t). (17) On the other hand, the energy consumption or the Joule losses due to the fractional viscoelastic damping is J(t) =c/integraldisplay∞ 0ωµ1−α(ω)z2(ω,t)dω. (18) 9The mechanical energy changes in the vibration process can be obs erved through the following numerical simulations. Parameters in the frac tional oscillator (8) are taken respectively as m= 1,c= 0.4,k= 2,α= 0.56, the external force are assumed to be f(t) = 30cos6 t. Fig.4 shows the fractional potential energy ce(t); Fig.5 shows comparison between the fractional energy ce(t)and the total mechanical energy E(t); Fig.6 illustrates the mechanical energy consumption J(t). 0 5 10 15 20 25 3000.050.10.150.20.250.30.350.4 Time(sec)Fractional Energy Ec(t) Figure 4: Fractional energy of the SDOF fractional oscillator. 0 5 10 15 20 25 30051015202530354045 Time(sec)Energy Figure 5: Comparison between the fractional energy and the tota l mechanical energy. 100 5 10 15 20 25 3000.511.522.533.54 Time(sec)Energy Consumption J (t) Figure 6: Mechanical energy consumption in the SDOF fractional os cillator. Remark 1. IfthefollowingmodifiedfractionalKelvin-Voigtconstituteequa- tion (19)which is proposed in [39] is taken to describe the viscoelastic stress- strain relation σ(t) =b0ε(t)+b1Dα1ε(t)+b2Dα2ε(t), (19) withα1,α2∈(0,1), the differential equation of motion for the SDOF frac- tional oscillator is m¨x(t)+c1Dα1x(t)+c2Dα2x(t)+kx(t) =f(t), (20) wherec1=Ab1 L,c2=Ab2 L,k=Ab0 L. In view of the following equivalences (21) and (22) between the Capu to derivatives and the frequency distributed models Dα1x(t) =I1−α1˙x(t)⇔/braceleftBigg ∂z1(ω,t) ∂t=−ωz1(ω,t)+ ˙x(t) Dα1x(t) =/integraltext∞ 0µ1−α1(ω)z1(ω,t)dω(21) and Dα2x(t) =I1−α2˙x(t)⇔/braceleftBigg ∂z2(ω,t) ∂t=−ωz21(ω,t)+ ˙x(t) Dα2x(t) =/integraltext∞ 0µ1−α2(ω)z2(ω,t)dω(22) 11the total mechanical energy of the fractional oscillator (20) is ex pressed as E(t) =1 2m˙x2+1 2kx2+c1 2/integraldisplay∞ 0µ1−α1(ω)z2 1(ω,t)dω +c2 2/integraldisplay∞ 0µ1−α2(ω)z2 2(ω,t)dω. (23) In the above expression (23) for the total mechanical energy, P1(t) =c1 2/integraldisplay∞ 0µ1−α1(ω)z2 1(ω,t)dω represents the potential energy stored in Dα1x(t), whereas P2(t) =c2 2/integraldisplay∞ 0µ1−α2(ω)z2 2(ω,t)dω represents thepotential energystored in Dα2x(t). Taking the first order time derivative of E(t) in Eq.(23), one derives ˙E(t) =f(t) ˙x(t)−c1/integraldisplay∞ 0ωµ1−α1(ω)z2 1(ω,t)dω −c2/integraldisplay∞ 0ωµ1−α2(ω)z2 2(ω,t)dω. It is clear that the energy dissipation due to the fractional viscoela stic damp- ingc1Dα1xis Jα1(t) =c1/integraldisplay∞ 0ωµ1−α1(ω)z2 1(ω,t)dω, (24) and the energy dissipation due to the fractional viscoelastic dampin gc2Dα2x is Jα2(t) =c2/integraldisplay∞ 0ωµ1−α2(ω)z2 2(ω,t)dω. (25) 5. The mean equivalent viscous damping The resulting differential equations of motion for structures incor porating fractional viscoelastic constitutive relations to dampen vibratory motion are fractional differential equations, which are strange and intricate ly to tackled withfor engineers. In engineering, complex descriptions fordampin g areusu- allyapproximately represented by equivalent viscous damping to simp lify the 12theoretical analysis. Inspired by this idea, we suggest a new conce pt of mean equivalent viscous damping based on the expression of fractional e nergy (18). Using thismethod, fractionaldifferential equations aretransfor med into clas- sical ordinary differential equations by replacing the fractional da mping with the mean equivalent viscous damping. The principle for the equivalenc y is that the mean energy dissipation due to the desired equivalent damp ing and the fractional viscoelastic damping are identical. To begin with, some comparisons of the energy dissipation between t he frac- tional oscillator (8) and the classical one (9) are made in the following .In view of the concept of work and energy in classical physics, the wor k done by any type of damping force is expressed as W(t) =/integraldisplayt 0fc(τ)dx(τ), (26) wherefc(t) is some type of damping force, x(t) is the displacement of the mass. In the classical oscillators, the viscous damping force is fc1(t) =c˙x(t). The work done by the viscous damping force is W1(t) =/integraldisplayt 0c˙x(τ)dx(τ) =/integraldisplayt 0c˙x2(τ)dτ. (27) It is well known that the energy consumption in unit time is J1(t) =c˙x2(t), (28) which is equal to the rate of the work done by the viscous damping fo rce J1(t) =dW1(t) dt. Obviously, the entire work done by the viscous damping force is conv erted to heat energy. However, the case in the fractional oscillators is different. As a mat ter fact, the fractional damping force is fc2(t) =cDαx(t). 13The work done by the fractional damping force is W2(t) =/integraldisplayt 0cDαx(τ)dx(τ). Due to the property of energy storage and dissipation in fractiona l deriva- tives, the entire work done by the fractional damping force W2is converted to two types of energy: one of which is the heat energy J(t) =c/integraldisplay∞ 0ωµ1−α(ω)z2(ω,t)dω, and the other is the fractional potential energy P(t) =c 2/integraldisplay∞ 0µ1−α(ω)z2(ω,t)dω. However, in [40] the equivalent viscous damping coefficient was obtain ed by the equivalency /contintegraldisplay cDαx(τ)dx(τ) =/contintegraldisplay ceq˙x(τ)dx(τ) Bythisequivalencythepropertiesoffractionalderivativehavebe enneglected and the work done by the fractional damping force is considered to be con- vertedintotheheatentirely. Asaresult, theaboveequivalencyisp roblematic and the value of the derived equivalent viscous damping coefficient is la rger than the actual value. In terms of the energy consumption (18), (24) and (25) due to th e fractional damping force, we suggest a new the concept of mean equivalent vis cous damping and evaluate the expression of the damping coefficient. Assuming the steady-state response of the fractional oscillator (8) is x(t) =XejΩt, whereXis the amplitude and Ω is the vibration frequency. Step1. We firstly need to calculate the mean energy consumption due to the fractional viscoelastic damping element, i.e. Jα(t) =c/integraldisplay∞ 0ωµ1−α(ω)z(ω,t)2dω. (29) 14To this end, we evaluate the mean square of z(ω,t), i.e.z(ω,t)2. In terms of the first equation in the diffusive representation of Cap uto deriva- tive (10) ˙z(ω,t) =−ωz(ω,t)+ ˙x(t), we get z(ω,t) =˙x(t) ω+jΩ=jΩxejΩt √ ω2+Ω2ejθ, whereθ= arctanΩ ω. Furthermore we get z(ω,t)2=1 2z(ω,t)z(ω,t)∗=1 2Ω2x2 ω2+Ω2, (30) wherez(ω,t)∗is the complex conjugate of z(ω,t). Substituting Eq. (30) into Eq.(29), one derives Jα(t) =c/integraldisplay∞ 0ωµ1−α(ω)z(ω,t)2dω =c 2Ω2X2/integraldisplay∞ 0ωµ1−α(ω) ω2+Ω2dω. (31) Applying the relation (4) in Lemma 2 ,one derives /integraldisplay∞ 0ωµ1−α(ω) ω2+Ω2dω=sin(1−α)π 2Ωαsin/parenleftbig1−α 2/parenrightbig π. (32) Substituting Eq.(32) into Eq. (31) one derives Jα(t) =c 4Ω1+αX2sin(1−α)π sin/parenleftbig1−α 2/parenrightbig π. (33) Step2. Now we calculate the mean energy loss due to the viscous damping force in the classical oscillator. From the relation(28), we have J(t) =cmeq˙x2(t), wherecmeqis denoted as the mean equivalent viscous damping coefficient for the fractional viscoelastic damping. 15Then the mean of the energy loss is derived as J(t) =cmeq˙x2=1 2cmeq˙x˙x∗=1 2cmeqΩ2X2. (34) Step3. Letting Jα(t) =J(t) and from the relations (33) and (34) one derives c 4Ω1+αX21 2sin(1−α)π sin/parenleftbig1−α 2/parenrightbig π=1 2cmeqΩ2X2. Consequently, we obtain the mean equivalent viscous damping coeffic ient for the fractional viscoelastic damping cmeq=c 2Ωα−1sin(1−α)π sin/parenleftbig1−α 2/parenrightbig π. (35) It is clear from (35) that the mean equivalent viscous damping coeffic ient for the fractional viscoelastic damping is a function of the vibration fre quency Ω and the order αof the fractional derivative. To this point, the fractional differential equations for the SDOF fractional oscillator (8) is appr oximately simplified to the following classical ordinary differential equation m¨x(t)+cmeq˙x(t)+kx(t) =f(t). (36) With the aid of numerical simulations, we compare the vibration respo nses of the approximate integer-order oscillator (36) with the fraction al one (8). The coefficients are respectively taken as m= 1,c= 0.4,k= 2,α= 0.56, the external force is taken asthe form f=FcosΩt, whereF= 30, Ω = 6. In terms of Eq.(35), we derive the mean equivalent viscous damping coe fficient cmeq= 0.14. 160510152025303540−2−1.5−1−0.500.511.5 Time(sec)x(t) Figure 7: The mean equilvalent damping coefficient for the SDOF fract ional oscillator of Kelvin-Voigt type. Remark 2. By the above procedure, we can furthermore evaluate the mean equivalent viscous damping coefficient for the SDOF fractional oscilla tor (20) containing two fractional viscoelastic damping elements. Letting Jα1(t) + Jα2(t) =J(t) and from the relations (24) (25) and (34), we get c1 4Ωα1+1X2sin(1−α1)π sin/parenleftbig1−α1 2/parenrightbig π+c2 4Ωα2+1X2sin(1−α2)π sin/parenleftbig1−α2 2/parenrightbig π =1 2c(α1,α2,Ω)Ω2X2.(37) From (37) we obtain the mean equivalent viscous damping coefficient cmeq=c1 2Ωα1−1sin(1−α1)π sin/parenleftbig1−α1 2/parenrightbig π+c2 2Ωα2−1sin(1−α2)π sin/parenleftbig1−α2 2/parenrightbig π.(38) With the aid of numerical simulations, we compare the vibration respo nses of the approximate integer-order oscillator (36) with the fraction al one(20). The coefficients are respectively taken as m= 1,c1= 0.4,c2= 0.2,k= 2, 17α1= 0.56,α2= 0.2, the external force is taken as the form f=FcosΩt, whereF= 30, Ω = 6. In terms of Eq.(38), we derive the mean equivalent viscous damping coefficient cmeq= 0.56. 0510152025303540−2−1.5−1−0.500.511.5 Time(sec)x(t) Figure 8: The mean equilvalent damping coefficient for the SDOF fract ional oscillator of modified Kelvin-Voigt type. 6. Discussion The total mechanical energy in single degree of freedom fractiona l oscil- lators has been dealt with in this paper. Based on the energy storag e and dissipation properties of the Caputo fractional derivative, the to tal mechani- cal energy is expressed asthe sum ofthe kinetic energy of themas s1 2m˙x2, the potential energy corresponding to the spring element1 2kx2, and the potential energy stored in the fractional derivative e(t) =1 2/integraltext∞ 0µ1−α(ω)z2(ω,t)dω. The energy regeneration and loss in vibratory motion have been ana lyzed by means of the total mechanical energy. Furthermore, based on t he mean en- ergy dissipation of the fractional damping element in steady-state vibration, a new concept of mean equivalent viscous damping has been suggest ed and the expression of the damping coefficient has been evaluated. 18By virtue of the total mechanical energy in SDOF fractional oscillat ors, it becomes possible to formulate Lyapunov functions for stability an alysis and control design for fractionally damped systems as well as othe r types of fractional dynamic systems. As for the future perspectives, our research efforts will be focused on fractional control design for fractiona lly damped oscillators and structures. Acknowledgements The author Yuan Jian expresses his thanks to Prof. Dong Kehai fr om Naval Aeronautical and Astronautical University, and Prof. Jian g Jianping from national University of Defense technology. All the authors a cknowledge the valuable suggestions from the peer reviewers. This work was su pported by the Natural Science Foundation of the Province Shandong of Ch ina ti- tled Controls for fractional systems with applications to hyperson ic vehicles (Grant Nos. ZR2014AM006). References References [1] IBRAHIMRA.Recentadvancesinnonlinearpassivevibrationisolat ors. Journal of Sound and Vibration, 2008, 314(3-5): 371-452 [2] BAGLEY R L. Applications of Generalized Derivatives to Viscoelastic - ity. Air Force Institute of technology,1979 [3] BAGLEY R L, TORVIK J. Fractional calculus-a different approach to the analysis of viscoelastically damped structures. AIAA journal, 1 983, 21(5): 741-748 [4] BAGLEY R L, TORVIK P. A theoretical basis for the application of fractionalcalculus toviscoelasticity. JournalofRheology(1978- present), 1983, 27(3): 201-210 [5] XU Z-D, XU C, HU J. Equivalent fractional Kelvin model and experimental study on viscoelastic damper. J Vib Control, 2013, 1077546313513604 [6] MOREAU X, RAMUS-SERMENT C, OUSTALOUP A. 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2016-08-14

This paper addresses the total mechanical energy of a single degree of freedom fractional oscillator. Based on the energy storage and dissipation properties of the Caputo fractional derivatives, the expression for total mechanical energy in the single degree of freedom fractional oscillator is firstly presented. The energy regeneration due to the external exciting force and the energy loss due to the fractional damping force during the vibratory motion are analyzed. Furthermore, based on the mean energy dissipation of the fractional damping element in steady-state vibration, a new concept of mean equivalent viscous damping is suggested and the value of the damping coefficient is evaluated.

Mechanical energy and mean equivalent viscous damping for SDOF fractional oscillators

1608.04071v1

arXiv:2307.05981v1 [math.AP] 12 Jul 2023PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM VALENTIN LEMARIÉ Abstract. We study on the whole space Rdthe compressible Euler system with damping coupled to the Poisson equation when the damping coefficient tends towards i nfinity. We first prove a result of global existence for the Euler-Poisson system in the case where the damping is large enough, then, in a second step, we rigorously justify the passage to the limit to the parabol ic-elliptic Keller-Segel after performing a diffusive rescaling, and get an explicit convergence rate. The overal l study is carried out in ‘critical’ Besov spaces, in the spirit of the recent survey [16] by R. Danchin devoted to p artially dissipative systems. 1.Introduction In this article, we will focus on two systems : Euler-Poisson and parabolic-elliptic Keller-Segel system. Let us first present these systems and motivate their study. The Euler-Poisson system with damping, set on the whole spac eRd(whered≥2) reads: ∂tρε+div(ρεvε) = 0, ∂t(ρεvε)+div(ρεvε⊗vε)+∇(P(ρε))+ε−1ρεvε=−ρε∇Vε, −∆Vε=ρε−ρ(1.1) whereε >0,ρε=ρε(t,x)∈R+represents the density of the gas (with ρ >0a constant state), vε= vε(t,x)∈Rdthe velocity, the pressure P(z) =Azγwithγ >1andA >0, as well as Vε=Vε(t,x)the potential. This is the classical Euler compressible system with dampin g, to which we added a coupling with the Poisson equation. Without this coupling, System (1.1) reduces to the compress ible Euler system with the damping coeffi- cientε−1: /braceleftbigg ∂tρε+div(ρεvε) = 0, ∂t(ρεvε)+div(ρεvε⊗vε)+∇(P(ρε))+ε−1ρεvε= 0. This system was recently studied by Crin-Barat and Danchin in [11] where they established a result of existence and uniqueness of global solutions for sufficientl y small data, and obtained optimal time decay estimates for these solutions. In parallel, they studied th e singular limit of the system when the damping coefficient tends towards infinity. They then obtained the equ ation of porous media. In this article, we will draw freely on this study and the method used to obtain a prior i estimates. The system that we will study is usually used to describe the t ransport of charge carriers (electrons and ions) in semiconductor devices or plasmas. The system co nsists of conservation laws for mass density and current density for carriers, with a Poisson equation fo r electrostatic potential. This system is also of interest in other fields like e.g. in chemotaxis: then, ρεrepresents cell density and Vε,the concentration of chemoattractant secreted by cells. For recent results on th e Euler-Poisson system: well-posedness, existence of global solutions, study of long-time behavior or other si ngular limits, the reader can refer to [1], [2], [3], [9], [18],[27], [31], [39] and [41]. As in [16], in order to investigate the asymptotics of soluti ons of (1.1) when εgoes to0,we perform the following so-called diffusive change of variable: (˜̺ε,˜vε)(τ,x) = (̺ε,ε−1vε)(ε−1τ,x) (1.2) so that we have ∂t˜̺ε+div(˜̺ε˜vε) = 0 ε2(∂t˜vε+ ˜vε·∇˜vε)+∇(P(˜̺ε)) ˜̺ε+ ˜vε+∇(−∆)−1(˜̺ε−̺) = 0.(1.3) 12 VALENTIN LEMARIÉ We then define the damped mode: ˜Wε:=∇(P(˜̺ε)) ˜̺ε+ ˜vε+∇(−∆)−1(˜̺ε−̺). (1.4) As the first equation of (1.3) can be rewritten as ∂t˜̺ε−∆(P(˜̺ε))−div/parenleftbig ˜̺ε∇(−∆)−1(˜̺ε−̺)/parenrightbig = div(˜̺ε˜Wε), we expect the limit density Nto satisfy the following parabolic-elliptic Keller-Segel system : /braceleftbigg ∂tN−∆(P(N)) = div( N∇V) −∆V=N−̺(1.5) supplemented with the initial data lim ε→0˜̺ε 0. Our second aim is to justify the passage to the limit when ε→0of the Euler-Poisson system towards the parabolic-elliptic Keller-Segel system. Recall that (1.5) is a model for describing the evolution of d ensityN=N(t,x)∈R+of a biological population under the influence of a chemical agent with conce ntration V=V(t,x)∈Rd. Chemotaxis are an important means of cell communication. How cells are arrang ed and organized is determined by communi- cation by chemical signals. Studying such a biological proc ess is important because it has repercussions in many branches of medicine such as cancer [8], [32], embryoni c development [30] or vascular networks [7], [20]. The previous system is famous in biology and comes from E.F Ke ller and L.A Segel in [26]. This basic model was used to describe the collective movement of bacteria pos sibly leading to cell aggregation by chemotactic effect. We refer to the articles [25] and [21] for more details and information about the different Keller-Segel models studied since the 1970s. Our aim here is to demonstrate that (1.5) may be obtained from the Euler-Poisson system with damping when the parameter εtends to 0.This question has been addressed in [28] on the torus case and Sobolev spaces in a situation where the potential satisfies a less sin gular equation : the author justifies the passage to the limit for regular periodic solutions. A lot of article s justify another limit: the passage from the parabolic-parabolic Keller-Segel system to the parabolic -elliptic Keller-Segel system (see e.g. the paper [29] by P-G. Lemarié-Rieusset for the case of Morrey spaces). In the same spirit as this article, T. Crin-Barat, Q. He and L. S hou in [13] justified the high relaxation asymptotics for the (less singular) parabolic-parabolic K eller-Segel system (the potential satisfies the equation −∆V+bV=aNwitha,b >0) : this other system comes from the system (HPC) (hyperbolic -parabolic- chemotaxis) which is a damped isentropic compressible Eule r system with a potential satisfying an elliptical equation. In comparison with what is done here, T. Crin-Barat et alused a parabolic approach to justify their passage to the limit. Here, we have to handle the more si ngular case where the limit system is parabolic- elliptic. 2.Main results and sketch of the proof In this section, we will first present and motivate the functi onal spaces used. Secondly we will state the results and the sketch of the proofs about the well-posed ness behavior of Euler-Poisson system and the justification of the passage to the limit to parabolic-ellip tic Keller-Segel system. 2.1.Functional spaces. Before describing the main results of this article, we introd uce the different notations and definitions used throughout this document. We will designate by C >0an independent constant of εand time, and f/lessorsimilarg will mean f≤Cg. For any Banach space Xand all functions f,g∈X, we denote /ba∇dbl(f,g)/ba∇dblX:=/ba∇dblf/ba∇dblX+/ba∇dblg/ba∇dblX. We designate by L2(R+;X)the set of measurable functions f: [0,+∞[→Xsuch that t/mapsto→ /ba∇dblf(t)/ba∇dblXbelongs toL2(R+)and write /ba∇dbl·/ba∇dblL2(R+;X):=/ba∇dbl·/ba∇dblL2(X). In this article we will use a decomposition in Fourier space, called the homogeneous Littlewood-Paley decomposition . To this end, we introduce a regular non-negative function ϕonRdwith support in thePARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 3 annulus{ξ∈Rd,3/4≤ |ξ| ≤8/3}and satisfying /summationdisplay j∈Zϕ(2−jξ) = 1, ξ/ne}ationslash= 0. For allj∈Z, the dyadic homogeneous blocks ˙∆jand the low frequency truncation operator ˙Sjare defined by ˙∆j:=F−1(φ(2−j·)Fu),˙Sju:=F−1(χ(2−j·)Fu), whereFandF−1designate respectively the Fourier transform and its inver se. From now on, we will use the following shorter notation : uj:=˙∆ju. LetS′ hthe set of tempered distributions uonRdsuch that lim j→−∞/ba∇dbl˙Sju/ba∇dblL∞= 0. We have then : u=/summationdisplay j∈Zuj∈ S′,˙Sju=/summationdisplay j′≤j−1uj′,∀u∈ S′ h. Homogeneous Besov spaces ˙Bs p,rfor allp,r∈[1,+∞]ands∈Rare defined by: ˙Bs p,r:=/braceleftBig u∈ S′ h/vextendsingle/vextendsingle/vextendsingle/ba∇dblu/ba∇dbl˙Bsp,r:=/ba∇dbl{2js/ba∇dbluj/ba∇dblLp}j∈Z/ba∇dbllr<∞/bracerightBig · In this article, we will only consider Besov spaces of indices p= 2andr= 1. As we will need to restrict our Besov norms in specific regions o f low and high frequencies, we introduce the following notations : /ba∇dblu/ba∇dblh ˙Bs 2,1:=/summationdisplay j≥−12js/ba∇dbluj/ba∇dblL2,/ba∇dblu/ba∇dbll ˙Bs 2,1:=/summationdisplay j≤−12js/ba∇dbluj/ba∇dblL2,/ba∇dblu/ba∇dbll−,ε ˙Bs 2,1:=/summationdisplay j≤−1 2j≤ε2js/ba∇dbluj/ba∇dblL2, /ba∇dblu/ba∇dbll+,ε ˙Bs 2,1:=/summationdisplay j≤−1 2j≥ε2js/ba∇dbluj/ba∇dblL2. We put in the appendix several results about Besov spaces: the reader may refer to Chapter 2 of [4] for more information on this topic. 2.2.Main results, sketch of the proofs and article organization . The starting point is that, formally, (1.1) rewrites : /braceleftbigg∂tρε+div(ρεvε) = 0, ∂t(ρεvε)+div(ρεvε⊗vε)+∇(P(ρε))+ε−1ρεvε=−ρε∇(−∆)−1(ρε−ρ).(2.1) By the variable change (ρ,v) = (ρε,vε)(εt,εx), (2.2) we get the following system: : /braceleftbigg∂tρ+div(ρv) = 0, ∂t(ρv)+div(ρv⊗v)+∇(P(ρ))+ρv=−ε2ρ∇(−∆)−1(ρ−ρ).(2.3) For the study of this system, the key is to obtain suitable glo bal-in-time a priori estimates. Then, very classical arguments lead to existence and uniqueness of glo bal solutions (see Theorem 2.1 below). Obtaining estimates will be strongly inspired by the work do ne by Crin-Barat et al in [11] where we consider the classic compressible Euler system. As the syst em we are studying is very close to a partially dissipative system, we will follow [16] so as to obtain a prio ri estimates : the standard energy method is not enough to conclude because we do not obtain all the informati on through this (mainly at the low frequencies) on the dissipated part. Hence, we must better use the couplin g, exhibiting a combination of unknowns (the "purely" damped mode) that will allow us to recover the whole dissipation. For the estimates, therefore, we follow the strategy propos ed by Danchin in [16]: first (second part of the third section), we analyze how to obtain the estimates fo r the linear system in Besov space ˙Bs 2,1where4 VALENTIN LEMARIÉ s∈Ris any at the moment. For high frequencies, we follow step by s tep the approach of [16], by putting the negligible term containing ε2∇∆−1̺, in order to obtain exponential decay. For low frequencies, the task is slightly more complicated because the latter term is no mo re negligible : we lose the symmetry condition and we can not apply Danchin’s method. But by looking at the sys tem differently (essentially by changing vto another variable dependent on ε), we obtain a symmetrical system for which we find the associa ted estimates (go back to the initial unknowns to get the estimat es). We then see the appearance of two regimes within the low frequencies that will be named respectively v ery low frequencies (frequencies below ε) and medium frequencies (frequencies of magnitude between εand1). To recover the full dissipative properties of the system, we introduce the damped mode W:=−∂tv.Then, from the estimate satisfied by W,we will improve the estimate for the low frequency part of v. Let us next explain how to choose the indices of regularity fo r the solution. Since our system is very similar to Euler’s without coupling, we make the same choice as in [11] : s=d 2for the medium frequencies ands=d 2+1for the high frequencies. We taked 2−1for the very low frequencies of the density in view of the estimate obtained for the linear system. For proving a priori estimates for (2.3), we have to take into account non-linear terms now. To do this, for high frequencies, we again follow the method described by [1 6] with a precise analysis of the system using commutators. Concerning the low frequencies, we need some information on the damped mode W=−∂tv, before studying the estimates of the density and velocity. By combin ing the obtained inequalities,we deduce the desired a priori estimates. Before stating our main results, we provide the reader with th e following diagram so as to clarify the notations of the theorem: 1• ε−1• |ξ|l l+,1, ε−1h, ε−1 l−, ε−1 Our first result states the global well-posedness of the Eule r Poisson system with high relaxation. We point out an explicit dependence of the estimates with respe ct to the damping parameter, that we believe to be optimal: Theorem 2.1. Letε >0. There exists a positive constante αsuch that for all ε≤1/2and initial data Zε 0= (̺ε 0−̺,vε 0)∈/parenleftbigg ˙Bd 2−1 2,1∩˙Bd 2+1 2,1/parenrightbigg ×/parenleftbigg ˙Bd 2 2,1∩˙Bd 2+1 2,1/parenrightbigg satisfying : Zε 0:=/ba∇dbl̺ε 0−̺/ba∇dbll ˙Bd 2−1 2,1+/ba∇dbl̺ε 0−̺/ba∇dbll+,1, ε−1 ˙Bd 2 2,1+/ba∇dblvε 0/ba∇dbll−, ε−1 ˙Bd 2 2,1+ε/ba∇dbl(̺ε 0−̺,vε 0)/ba∇dblh, ε−1 ˙Bd 2+1 2,1≤α, System (1.1)supplemented with the initial data (̺ε 0−¯ρ,vε 0)admits a unique global-in-time solution Zε= (̺ε−¯ρ,vε)in the set E:=/braceleftbigg (̺ε−̺,vε)/vextendsingle/vextendsingle/vextendsingle/vextendsingle(̺ε−̺)l∈ Cb(R+:˙Bd 2−1 2,1), ε(̺ε−̺)l∈L1(R+;˙Bd 2−1 2,1), (̺ε−̺)l+,1, ε−1∈ Cb(R+:˙Bd 2 2,1), ε(̺ε−̺)l+,1, ε−1∈L1(R+;˙Bd 2+1 2,1), (vε)l−, ε−1∈ Cb(R+;˙Bd 2 2,1),(vε)l∈L1(R+;˙Bd 2 2,1),(vε)l+,1, ε−1∈L1(R+;˙Bd 2+1 2,1), (̺ε−̺,vε)h,ε−1∈ Cb(R+;˙Bd 2−1 2,1)∩L1(R+;˙Bd 2+1 2,1), wε∈ Cb(R+;˙Bd 2 2,1)∩L1(R+;˙Bd 2 2,1)/bracerightbigg where we have denoted wε:=ε∇(P(̺ε)) ̺ε+vε+ε∇(−∆)−1(̺ε−̺).PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 5 Moreover, we have the following inequality : Zε(t)≤CZε 0 where Zε(t):=/ba∇dbl̺ε−̺/ba∇dbll L∞/parenleftbig ˙Bd 2−1 2,1/parenrightbig+/ba∇dbl̺ε−̺/ba∇dbll+,1, ε−1 L∞/parenleftbig ˙Bd 2 2,1/parenrightbig+/ba∇dblvε/ba∇dbll−, ε−1 L∞/parenleftbig ˙Bd 2 2,1/parenrightbig+ε/ba∇dbl(̺ε−̺,v)/ba∇dblh, ε−1 L∞/parenleftbig ˙Bd 2+1 2,1/parenrightbig +/ba∇dblε(̺ε−̺)/ba∇dbll L1/parenleftbig ˙Bd 2−1 2,1/parenrightbig+/ba∇dblvε/ba∇dbll L1/parenleftbig ˙Bd 2 2,1/parenrightbig+ε/ba∇dbl̺ε−̺/ba∇dbll+,1, ε−1 L1/parenleftbig ˙Bd 2+2 2,1/parenrightbig+/ba∇dblvε/ba∇dbll+,1, ε−1 L1/parenleftbig ˙Bd 2+1 2,1/parenrightbig +/ba∇dbl(̺ε−̺,v)/ba∇dblh,ε−1 L1/parenleftbig ˙Bd 2+1 2,1/parenrightbig+/ba∇dblwε/ba∇dbl L∞/parenleftbig ˙Bd 2 2,1/parenrightbig+ε−1/ba∇dblwε/ba∇dbl L1/parenleftbig ˙Bd 2 2,1/parenrightbig. Remark 2.1. In summary, for large enough damping and small enough initia l data, we get a global solution to the Euler-Poisson system. The demonstration will allow u s to understand the choice of regularity indices for the different frequency groups. In addition, the control of the damped mode wεin the above theorem which enable us to obtain on the one hand the a priori estimate of the theorem and on the other hand the following theorem on the singular limit of (1.1)when the damping coefficient ε−1tends to infinity. By the change of variable (1.2) and the existence theorem on Eu ler-Poisson, we then have ˜Wε=O(ε) inL1(R+;˙Bd 2 2,1)where˜Wεis defined by (1.4) and we need to look at the solutions of (1.3) in the following functional space : ˜E:=/braceleftbigg (̺ε−̺,vε)/vextendsingle/vextendsingle/vextendsingle/vextendsingle(̺ε−̺)l∈ Cb(R+:˙Bd 2−1 2,1), ε(̺ε−̺)l∈L1(R+;˙Bd 2−1 2,1), (̺ε−̺)l+,1, ε−1∈ Cb(R+:˙Bd 2 2,1), ε(̺ε−̺)l∈L1(R+;˙Bd 2+1 2,1),(̺ε−̺)l+,1, ε−1∈L1(R+;˙Bd 2+1 2,1), ε(vε)l−, ε−1∈ Cb(R+;˙Bd 2 2,1),(vε)l∈L1(R+;˙Bd 2 2,1),(vε)l+,1, ε−1∈L1(R+;˙Bd 2+1 2,1), (̺ε−̺,vε)h,ε−1∈ Cb(R+;˙Bd 2−1 2,1)∩L1(R+;˙Bd 2+1 2,1), wε∈ Cb(R+;˙Bd 2 2,1)∩L1(R+;˙Bd 2 2,1)/bracerightbigg · By studying the system satisfied by the difference between the s olution of Euler-Poisson and that of Keller- Segel parabolic-elliptic and thanks to the previous theore m, we manage to justify that the solutions of the Euler-Poisson system that has been scaled back will converg e to the solutions of the Keller-Segel system towards the following theorem : Theorem 2.2. We consider (1.3)forε >0small enough. Then, there exists a positive constant α(inde- pendent of ε) such that for all initial data N0∈˙Bd 2−1 2,1∩˙Bd 2 2,1for(1.5)and˜Zε 0∈˜Efor(1.3)satisfying /ba∇dblN0/ba∇dbl˙Bd 2 2,1∩˙Bd 2−1 2,1≤α, (2.4) ˜Zε 0:=/ba∇dbl˜̺ε 0−̺/ba∇dbll ˙Bd 2−1 2,1+/ba∇dbl˜̺ε 0−̺/ba∇dbll+,1, ε−1 ˙Bd 2 2,1+ε/ba∇dbl˜vε 0/ba∇dbll, ε−1 ˙Bd 2 2,1+ε/ba∇dbl/parenleftbig ˜̺ε 0−̺,ε˜vε 0/parenrightbig /ba∇dblh, ε−1 ˙Bd 2+1 2,1≤α, the system (1.5)admits an unique solution Nin the space Cb/parenleftbig R+;˙Bd 2−2 2,1∩˙Bd 2 2,1/parenrightbig ∩L1/parenleftbig R+;˙Bd 2−2 2,1∩˙Bd 2 2,1/parenrightbig , satisfying for all t≥0, /ba∇dblN(t)−̺/ba∇dbl˙Bd 2−1 2,1∩˙Bd 2 2,1+/integraldisplayt 0/ba∇dblN−̺/ba∇dbl˙Bd 2+2 2,1∩˙Bd 2+1 2,1dτ≤C/ba∇dblN0−̺/ba∇dbl˙Bd 2−1 2,1∩˙Bd 2 2,1, (2.5) and the system (1.3)has an unique global-in-time solution ˜Zεin˜Esuch that ˜Z(t)≤C˜Z06 VALENTIN LEMARIÉ where (2.6)˜Zε(t):=/ba∇dbl˜̺ε−̺/ba∇dbll L∞/parenleftbig ˙Bd 2−1 2,1/parenrightbig+/ba∇dbl˜̺ε−̺/ba∇dbll+,1, ε−1 L∞/parenleftbig ˙Bd 2 2,1/parenrightbig+ε/ba∇dbl˜vε/ba∇dbll, ε−1 L∞/parenleftbig ˙Bd 2 2,1/parenrightbig+ε/ba∇dbl(˜̺ε−̺,ε˜vε)/ba∇dblh, ε−1 L∞/parenleftbig ˙Bd 2+1 2,1/parenrightbig +/ba∇dbl˜̺ε−̺/ba∇dbll L1/parenleftbig ˙Bd 2−1 2,1/parenrightbig+/ba∇dbl˜vε/ba∇dbll L1/parenleftbig ˙Bd 2 2,1/parenrightbig+/ba∇dbl˜̺ε−̺/ba∇dbll+,1, ε−1 L1/parenleftbig ˙Bd 2+2 2,1/parenrightbig+/ba∇dbl˜vε/ba∇dbll+,1, ε−1 L1/parenleftbig ˙Bd 2+1 2,1/parenrightbig +ε−1/ba∇dbl(˜̺ε−̺,ε˜vε)/ba∇dblh,ε−1 L1/parenleftbig ˙Bd 2+1 2,1/parenrightbig+ε/ba∇dbl˜Wε/ba∇dbl L∞/parenleftbig ˙Bd 2 2,1/parenrightbig+ε−1/ba∇dbl˜Wε/ba∇dbl L1/parenleftbig ˙Bd 2 2,1/parenrightbig where˜Wεhas been defined in (1.4). Moreover, if, /ba∇dblN0−˜̺ε 0/ba∇dbl˙Bd 2−1 2,1≤Cε, then we have /ba∇dblN−˜̺ε/ba∇dbl L∞/parenleftbig R+;˙Bd 2−1 2,1/parenrightbig+/ba∇dblN−˜̺ε/ba∇dblh L1/parenleftbig R+;˙Bd 2+1 2,1/parenrightbig+/ba∇dblN−˜̺ε/ba∇dbll L1/parenleftbig R+;˙Bd 2 2,1/parenrightbig≤Cε. Remark 2.2. This theorem ensures that the solution densities of the Eule r-Poisson system converge (in the space highlighted in the theorem) towards the only solution of the Keller-Segel system. For the velocity limit, we can take the limit in (1.4). 3.Study of the Euler-Poisson system with damping This section is devoted to the proof of theorem 2.1. 3.1.Study of the linearized system. Linearizing (2.3) around (̺,0)yields the following system : /braceleftbigg∂t̺+divv= 0 ∂tv+P′(ρ)∇̺+v=−ε2∇(−∆)−1̺. Performing the change of unknown ˜ρ:=̺(t,P′(ρ)x)reduces the study to the case P′(ρ) = 1 (after changing εintoε′:=ε/radicalbig P′(ρ)). Hence we focus on the following linear system : /braceleftbigg∂t̺+divv= 0, ∂tv+∇̺+v=−ε2∇(−∆)−1̺.(3.1) By the Fourier transform, we get : d dt/parenleftbigg /hatwide̺ /hatwidev/parenrightbigg +/parenleftBigg0 iξ i/parenleftBig 1+ε2|ξ|−2/parenrightBig ξtId/parenrightBigg/parenleftbigg /hatwide̺ /hatwidev/parenrightbigg = 0. The eigenvalues of the matrix of this system are: •1is with multiplicity d−1, • λ±(ξ) =1 2(1±/radicalbig 1−4(|ξ|2+ε2))if|ξ|2+ε2<1 4 λ±(ξ) =1 2(1±i/radicalbig 4(|ξ|2+ε2)−1)elseare the two remaining eigenvalues. For the high frequencies, we thus have : Re/parenleftbig λ±(ξ)/parenrightbig = 1. As for partially dissipative hyperbolic systems, we expect exponential decay for high frequencies. For the low frequencies, in the case ε <1/2,two regimes have to be considered: a very low frequency regime (i.e for |ξ| ≤/radicalbig 1/4−ε2) and another for the medium frequencies (for |ξ| ≥/radicalbig 1/4−ε2). In what follows, we prove a priori estimates for a smooth enou gh solution (̺,v)of (3.1).PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 7 3.1.1. High frequency behavior: case |ξ| ≥1/2. Proposition 3.1. LetZ= (̺,v)and/ba∇dblZ/ba∇dblh ˙Bs 2,1:=/summationtext j≥−12js/ba∇dbl˙∆jZ/ba∇dblL2. Then we have : /ba∇dblZ(t)/ba∇dblh ˙Bs 2,1+/integraldisplayt 0/ba∇dblZ/ba∇dblh ˙Bs 2,1dτ/lessorsimilar/ba∇dblZ0/ba∇dblh ˙Bs 2,1. Proof. Since the classical energy method does not provide enough in formation, we consider, as in [16], the evolution equation of ∇̺·v, namely ∂t(∇̺·v) =∇∂t̺·v+∇̺·∂tv=−∇divv·v−∇̺·/parenleftbig ∇̺+v+ε2∇(−∆)−1̺/parenrightbig · We then integrate on Rdand by integration by parts :/integraldisplay Rd∂t(∇̺·v)dx+/integraldisplay Rd∇̺·vdx=/ba∇dbldivv/ba∇dbl2 L2−/ba∇dbl∇̺/ba∇dbl2 L2−ε2/ba∇dbl̺/ba∇dbl2 L2. By taking the gradient in (3.1) and taking the scalar product w ith∇̺(respectively ∇v), we get : 1 2d dt/ba∇dbl∇̺/ba∇dbl2 L2−/integraldisplay Rd∇∇̺·∇vdx= 0 1 2d dt/ba∇dbl∇v/ba∇dbl2 L2+/integraldisplay Rd∇∇̺·∇vdx+/ba∇dbl∇v/ba∇dblL2=−ε2/integraldisplayt 0∇∇(−∆)−1̺·∇vdx. These identities are also true for (̺j,vj)(where̺j=˙∆j̺andvj=˙∆jv) since the studied system is linear with constant coefficients. To study high frequencies, we wil l further assume that j∈N. We then set Lj:=1 2/ba∇dbl∇̺j/ba∇dbl2 L2+1 2/ba∇dbl∇vj/ba∇dbl2 L2+1 4/integraldisplay Rd∇̺j·vjdxwhich verifies : 3 8/parenleftbig /ba∇dbl∇̺j/ba∇dbl2 L2+/ba∇dbl∇vj/ba∇dbl2 L2/parenrightbig ≤ Lj≤5 8/parenleftbig /ba∇dbl∇̺j/ba∇dbl2 L2+/ba∇dbl∇vj/ba∇dbl2 L2/parenrightbig because for j≥0,/ba∇dblvj/ba∇dblL2≤1 2/ba∇dbl∇vj/ba∇dblL2by Bernstein’s inequality. Then we have : d dtLj+/ba∇dbl∇vj/ba∇dbl2 L2+1 4/integraldisplay Rd∇̺j·vjdx−1 4/ba∇dbldivv/ba∇dbl2 L2+1 4/ba∇dbl∇̺j/ba∇dbl2 L2≤ε2/ba∇dbl̺j/ba∇dblL2/ba∇dbl∇vj/ba∇dblL2+ε2 4/ba∇dbl̺j/ba∇dbl2 L2 ≤2ε2 4/ba∇dbl∇̺j/ba∇dbl2 L2+2ε2 4/ba∇dbl∇vj/ba∇dbl2 L2. By the inequalities of Cauchy-Schwarz and Bernstein, we have : /ba∇dbl∇vj/ba∇dbl2 L2+1 4/integraldisplay Rd∇̺j·vjdx−1 4/ba∇dbldivv/ba∇dbl2 L2+1 4/ba∇dbl∇̺j/ba∇dbl2 L2−2ε2 4/ba∇dbl∇̺j/ba∇dbl2 L2+2ε2 4/ba∇dbl∇vj/ba∇dbl2 L2 ≥ −1 4/ba∇dbl∇̺j/ba∇dblL2/ba∇dblvj/ba∇dblL2+3−2ε2 4/ba∇dbl∇vj/ba∇dbl2 L2+1−2ε2 4/ba∇dbl∇̺j/ba∇dbl2 L2 ≥4−4ε2 8/ba∇dbl∇vj/ba∇dbl2 L2+1−4ε2 8/ba∇dbl∇̺j/ba∇dbl2 L2 ≥1−4ε2 5Lj. By the inequality on Lj, we thus get : d dtLj+1−4ε2 5Lj≤0. Forε≤1 4, so we get : /ba∇dbl∇(̺j,vj)(t)/ba∇dblL2+3 20/integraldisplayt 0/ba∇dbl∇(̺j,vj)/ba∇dblL2dτ≤ /ba∇dbl∇(̺j,vj)(0)/ba∇dblL2. We multiply by 2j(s−1)and we sum up on j∈N, we get then the inequality announced in the proposition (the case j=−1that presents no particular difficulty can be studied separat ely). /square8 VALENTIN LEMARIÉ 3.1.2. Low frequency behavior. We have to proceed differently since the term −ε2∇(−∆)−1is of order −1. The goal will be here to understand his role. Let us use the Helmholtz decomposition to highlight the two b ehaviors corresponding respectively to the solenoid and the irrotational part of v: v=Pv+Qv wherePandQverifyP= Id+∇(−∆)−1divandQ=−∇(−∆)−1div. First, let us look at the equation verified by Pv. By applying Pto the second equation of the system (3.1) and using that P∇= 0, we get : ∂tPv+Pv= 0. By applying ˙∆jand taking the scalar product with Pvj, we have: d dt/ba∇dblPvj/ba∇dbl2 L2+/ba∇dblPvj/ba∇dbl2 L2= 0. Then we have by (A.1): /ba∇dblPvj(t)/ba∇dblL2+/integraldisplayt 0/ba∇dblPvj/ba∇dblL2dτ≤ /ba∇dblPv0,j/ba∇dblL2. (3.2) Now let us look at the system satisfied by the divergence of vand̺. We then set u= divv. By taking the divergence in the second equation of (3.1), we get the follow ing2×2system : /braceleftbigg ∂t̺+u= 0, ∂tu+∆̺+u=ε2̺. In Fourier variables, it becomes : /braceleftbigg ∂t/hatwide̺+/hatwideu= 0, ∂t/hatwideu−(|ξ|2+ε2)/hatwide̺+/hatwideu= 0. Setting/hatwidew:=1/radicalbig |ξ|2+ε2/hatwideuyields : ∂t˜Z+A(D)˜Z+B(D)˜Z= 0 (3.3) where ˜Z=/parenleftbigg ̺ w/parenrightbigg , A(ξ) =/parenleftbigg 0/radicalbig |ξ|2+ε2 −/radicalbig |ξ|2+ε20/parenrightbigg , B(ξ) =/parenleftbigg 0 0 0 1/parenrightbigg · Let us build by hand a Lyapunov functional, allowing us to rec over the dissipative properties of the system on˜Z. By taking the scalar product with ˜Zin (3.3) and looking at the time derivative of Re(/hatwide̺·/hatwidew), we have: 1 2d dt|/hatwide̺|2+/radicalbig |ξ|2+ε2Re(/hatwide̺·/hatwidew) = 0, 1 2d dt|/hatwidew|2−/radicalbig |ξ|2+ε2Re(/hatwide̺·/hatwidew)+|/hatwidew|2= 0, d dtRe(/hatwide̺·/hatwidew) =−/radicalbig |ξ|2+ε2|/hatwidew|2+/radicalbig |ξ|2+ε2|/hatwide̺|2−Re(/hatwide̺·/hatwidew). With these equations, we can easily deduce the Lyapunov func tional and the equation it verifies: d dt/parenleftBig |/hatwide̺|2+|/hatwidew|2−/radicalbig |ξ|2+ε2Re(/hatwide̺·/hatwidew)/parenrightBig +(2−(|ξ|2+ε2))|/hatwidew|2+(|ξ|2+ε2)|/hatwide̺|2−/radicalbig |ξ|2+ε2Re(/hatwide̺·/hatwidew) = 0.PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 9 Yet /radicalbig |ξ|2+ε2Re(/hatwide̺·/hatwidew)≤|ξ|2+ε2 2|/hatwide̺|2+|/hatwidew|2 2· Thus for |ξ|2+ε2≤1, we have: d dt/parenleftBig |/hatwide̺|2+|/hatwidew|2−/radicalbig |ξ|2+ε2Re(/hatwide̺·/hatwidew)/parenrightBig +|ξ|2+ε2 2|/hatwidew|2+|ξ|2+ε2 2|/hatwide̺|2≤0. We have then: |(/hatwide̺,/hatwidew)(t,ξ)| ≤2e−1 8(|ξ|2+ε2)t|(/hatwide̺0,/hatwidew0)(ξ)|. (3.4) So we have after spectral localization of the system by means of˙∆jwithj≤ −1and forεsmall enough: /ba∇dbl(̺j,wj)(t)/ba∇dblL2+1 8(22j+ε2)/integraldisplayt 0/ba∇dbl(̺j,wj)/ba∇dblL2dτ≤2/ba∇dbl(̺0,j,w0,j)/ba∇dblL2. (3.5) We have a priori estimate on ˜Zj. A similar estimate has yet to be obtained for (̺j,uj). By definition of wand by multiplying (3.5) by/radicalbig 22j+ε2, we have as an estimate : /ba∇dbl(/radicalbig 22j+ε2̺j,uj)(t)/ba∇dblL2+/parenleftbig 22j+ε2/parenrightbig/integraldisplayt 0/ba∇dbl(/radicalbig 22j+ε2̺j,uj)/ba∇dblL2dτ/lessorsimilar/ba∇dbl(/radicalbig 22j+ε2̺0,j,u0,j)/ba∇dblL2. These estimates reveal two distinct regimes within the low f requencies : |ξ| ≤ε("very low frequencies") and|ξ| ≥ε("medium frequencies"). For very low frequencies, we have 22j+ε2≃ε2, and thus /ba∇dbl(ε˜̺j,uj)(t)/ba∇dblL2+ε2/integraldisplayt 0/ba∇dbl(ε˜̺j,uj)/ba∇dblL2dτ/lessorsimilar/ba∇dbl(ε˜̺0,j,u0,j)/ba∇dblL2withu= divv, and for the medium frequencies, since 22j+ε2≃22j, /ba∇dbl(˜̺j,vj)(t)/ba∇dblL2+22j/integraldisplayt 0/ba∇dbl(˜̺j,vj)/ba∇dblL2dτ/lessorsimilar/ba∇dbl(˜̺0,j,v0,j)/ba∇dblL2. Consequently, if we denote : /ba∇dblZ/ba∇dbll−,ε ˙Bs 2,1:=/summationdisplay j≤−1 2j≤ε2js/ba∇dblZj/ba∇dblL2et/ba∇dblZ/ba∇dbll+,ε ˙Bs 2,1:=/summationdisplay j≤−1 2j≥ε2js/ba∇dblZj/ba∇dblL2. (3.6) then, we obtain: /ba∇dbl(ε˜̺,div(v))(t)/ba∇dbll−,ε ˙Bs 2,1+ε2/integraldisplayt 0/ba∇dbl(ε˜̺,div(v))/ba∇dbll−,ε ˙Bs 2,1dτ/lessorsimilar/ba∇dbl(ε˜̺0,div(v0))/ba∇dbll−,ε ˙Bs 2,1 /ba∇dbl(˜̺,v)(t)/ba∇dbll+,ε ˙Bs 2,1+/integraldisplayt 0/ba∇dbl(˜̺,v)/ba∇dbll+,ε ˙Bs+2 2,1dτ/lessorsimilar/ba∇dbl(˜̺0,v0)/ba∇dbll+,ε ˙Bs 2,1, /ba∇dblPv(t)/ba∇dbll ˙Bs 2,1+/integraldisplayt 0/ba∇dblPv/ba∇dbll ˙Bs 2,1dτ/lessorsimilar/ba∇dblPv0/ba∇dbll ˙Bs 2,1(incompressible part) . 3.1.3. Damped mode and improvement of estimates for v. Like in [16], we consider the damped mode : ˜W:=−∂tv=∇̺+ε2∇(−∆)−1̺+v. We have : ∂t˜W+˜W=−(∇+ε2∇(−∆)−1)divv.10 VALENTIN LEMARIÉ By applying the localization operator ˙∆j, taking the scalar product with ˜Wj, multiplying by 2js, summing up onjcorresponding to very low and medium frequencies and applyi ng the lemma A.1, we get : /ba∇dbl˜W(t)/ba∇dbll−,ε ˙Bs 2,1+/integraldisplayt 0/ba∇dbl˜W/ba∇dbll−,ε ˙Bs 2,1dτ/lessorsimilar/ba∇dbl˜W0/ba∇dbll−,ε ˙Bs 2,1+/integraldisplayt 0ε2/ba∇dblv/ba∇dbll−,ε ˙Bs 2,1dτ /ba∇dbl˜W(t)/ba∇dbll+,ε ˙Bs 2,1+/integraldisplayt 0/ba∇dbl˜W/ba∇dbll+,ε ˙Bs 2,1dτ/lessorsimilar/ba∇dbl˜W0/ba∇dbll+,ε ˙Bs 2,1+/integraldisplayt 0/ba∇dblv/ba∇dbll+,ε ˙Bs+2 2,1dτ. In particular, we have : /ba∇dbl˜W0/ba∇dbll−,ε ˙Bs 2,1/lessorsimilarε2/ba∇dbl̺0/ba∇dbll−,ε ˙Bs−1 2,1+/ba∇dblv0/ba∇dbll−,ε ˙Bs 2,1 /ba∇dbl˜W0/ba∇dbll+,ε ˙Bs 2,1/lessorsimilar/ba∇dbl̺0/ba∇dbll+,ε ˙Bs+1 2,1+/ba∇dblv0/ba∇dbll+,ε ˙Bs 2,1. By using the fact v=˜W−∇̺−ε2∇(−∆)−1̺and the estimates on the low frequencies obtained previousl y, we have : /ba∇dblv(t)/ba∇dbll−,ε ˙Bs 2,1≤ /ba∇dbl˜W(t)/ba∇dbll−,ε ˙Bs 2,1+ε2/ba∇dbl̺(t)/ba∇dbll−,ε ˙Bs−1 2,1/lessorsimilarε2/ba∇dbl̺0/ba∇dbll−,ε ˙Bs−1 2,1+/ba∇dblv0/ba∇dbll−,ε ˙Bs 2,1+/integraldisplayt 0ε2/ba∇dblv/ba∇dbll−,ε ˙Bs 2,1dτ /integraldisplayt 0ε/ba∇dblv/ba∇dbll−,ε ˙Bs 2,1dτ≤/integraldisplayt 0ε/ba∇dbl˜W/ba∇dbll−,ε ˙Bs 2,1dτ+/integraldisplayt 0ε3/ba∇dbl̺/ba∇dbll−,ε ˙Bs−1 2,1dτ/lessorsimilarε/ba∇dbl̺0/ba∇dbll−,ε ˙Bs−1 2,1+/ba∇dblv0/ba∇dbll−,ε ˙Bs 2,1+/integraldisplayt 0ε2/ba∇dblv/ba∇dbll−,ε ˙Bs 2,1dτ /ba∇dblv(t)/ba∇dbll+,ε ˙Bs 2,1/lessorsimilar/ba∇dbl˜W(t)/ba∇dbll+,ε ˙Bs 2,1+/ba∇dbl̺(t)/ba∇dbll+,ε ˙Bs+1 2,1/lessorsimilar/ba∇dbl̺0/ba∇dbll+,ε ˙Bs+1 2,1+/ba∇dblv0/ba∇dbll+,ε ˙Bs 2,1+/integraldisplayt 0/ba∇dblv/ba∇dbll+,ε ˙Bs+2 2,1dτ /integraldisplayt 0/ba∇dblv/ba∇dbll+,ε ˙Bs+1 2,1dτ≤/integraldisplayt 0/ba∇dbl˜W/ba∇dbll+,ε ˙Bs+1 2,1dτ+/integraldisplayt 0/ba∇dbl̺/ba∇dbll+,ε ˙Bs+2 2,1dτ/lessorsimilar/ba∇dbl̺0/ba∇dbll+,ε ˙Bs 2,1+/ba∇dblv0/ba∇dbll+,ε ˙Bs 2,1+/integraldisplayt 0/ba∇dblv/ba∇dbll+,ε ˙Bs+2 2,1dτ By summing up these inequalities and noticing that some terms in the right-hand side are negligible compared to those of the left-hand side, we get : /ba∇dblv(t)/ba∇dbll−,ε ˙Bs 2,1+/integraldisplayt 0ε/ba∇dblv/ba∇dbll−,ε ˙Bs 2,1dτ/lessorsimilarε/ba∇dbl̺0/ba∇dbll−,ε ˙Bs−1 2,1+/ba∇dblv0/ba∇dbll−,ε ˙Bs 2,1 /ba∇dblv(t)/ba∇dbll+,ε ˙Bs 2,1+/integraldisplayt 0/ba∇dblv/ba∇dbll+,ε ˙Bs+1 2,1dτ/lessorsimilar/ba∇dbl̺0/ba∇dbll+,ε ˙Bs 2,1+/ba∇dblv0/ba∇dbll+,ε ˙Bs 2,1 3.2.A priori estimates for the non-linear system. Let us now prove similar estimates for the non-linear system . To do this, we use Makino symmetrization, which consists in setting c:=(γA)1 2 ˜γ̺˜γwith˜γ=γ−1 2· (3.7) After this change of unknown, we obtain: ∂tc+v·∇c+ ˜γcdiv(v) = 0 ∂tv+v·∇v+ ˜γc∇c+v=−ε2∇(−∆)−1/parenleftBigg/parenleftbigg ˜γ (γA)1 2/parenrightbigg1 ˜γ c1 ˜γ−̺/parenrightBigg .(3.8) We setf(x):=/parenleftBigg ˜γ (γA)1 2/parenrightBigg1 ˜γ x1 ˜γ.So we have by Taylor’s formula with integral rest : /parenleftBigg ˜γ (γA)1 2/parenrightBigg1 ˜γ/parenleftBig c1 ˜γ−c1 ˜γ/parenrightBig =f(˜c+c)−f(c) = ˜cf′(c)+/integraldisplayc c(c−y)f′′(y)dy. Let us set F(˜c) =/integraldisplayc c(c−y)f′′(y)dyandG(˜c) = ˜cf′(c) +/integraldisplayc c(c−y)f′′(y)dywhich vanishes at 0 where c:=(γA)1 2 ˜γ̺˜γand˜c=c−c.PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 11 Then we get: /braceleftbigg∂t˜c+v·∇˜c+ ˜γcdiv(v)+ ˜γ˜cdiv(v) = 0, ∂tv+v·∇v+ ˜γc∇c+ ˜γ˜c∇c+v=−ε2∇(−∆)−1G(˜c).(3.9) After changing v(t,x)andc(t,x)intov(t,˜γcx)andc(t,˜γcx), respectively, we can look at the following system (keeping the previous notations) : ∂t˜c+1 ˜γcv·∇˜c+div(v)+˜c cdiv(v) = 0, ∂tv+1 ˜γcv·∇v+∇c+˜c c∇c+v=−ε2∇(−∆)−1G(˜c).(3.10) To simplify the presentation, suppose from now on that ˜γc= 1: the general case works the same way. We’re going to assume throughout this section that: /ba∇dbl(˜c,v)/ba∇dbl˙Bd 2 2,1≪1. (3.11) In view of the linear analysis, we will start the following st udy with the choice of index: •d 2−1for very low frequencies, •d 2for medium frequencies, •d 2+1for high frequencies. This choice of index is strongly inspired by the results of [1 5] where to study the relaxation limit, a similar choice is taken. Let us pose then : ˜L(t):=/ba∇dbl(ε˜c,divv)(t)/ba∇dbll−,ε ˙Bd 2−1 2,1+/ba∇dbl(˜c,v)(t)/ba∇dbll+,ε ˙Bd 2 2,1+/ba∇dblPv(t)/ba∇dbll ˙Bd 2 2,1+/ba∇dbl(˜c,v)(t)/ba∇dblh ˙Bd 2+1 2,1 and ˜H(t):=ε2/ba∇dbl(ε˜c,divv)(t)/ba∇dbll−,ε ˙Bd 2−1 2,1+/ba∇dbl(˜c,v)(t)/ba∇dbll+,ε ˙Bd 2+2 2,1+/ba∇dblPv(t)/ba∇dbll ˙Bd 2 2,1+/ba∇dbl(˜c,v)(t)/ba∇dblh ˙Bd 2+1 2,1. Lemma 3.2. We have the following inequalities : /ba∇dbl(˜c,v)(t)/ba∇dbll ˙Bd 2 2,1+/ba∇dbl(˜c,v)(t)/ba∇dblh ˙Bd 2+1 2,1/lessorsimilar˜L(t), /ba∇dbl(˜c,v)(t)/ba∇dbll ˙Bd 2+2 2,1+/ba∇dbl(˜c,v)(t)/ba∇dblh ˙Bd 2+1 2,1/lessorsimilarε2/ba∇dbl(˜c,v)(t)/ba∇dbll−,ε ˙Bd 2 2,1+/ba∇dbl(˜c,v)(t)/ba∇dbll+,ε ˙Bd 2+2 2,1+/ba∇dbl(˜c,v)(t)/ba∇dblh ˙Bd 2+1 2,1/lessorsimilar˜H(t), ε2/ba∇dbl(˜c,v)(t)/ba∇dbl2 ˙Bd 2 2,1+/ba∇dbl(˜c,v)(t)/ba∇dbl2 ˙Bd 2+1 2,1/lessorsimilar˜L(t)˜H(t). Proof. The first two inequalities are easily deduced from the definit ion (3.6). Let us set Z= (˜c,v). We have by definition and the second inequality: ε2/ba∇dblZ/ba∇dbl2 ˙Bd 2 2,1=/parenleftBigg ε2/ba∇dblZ/ba∇dbll−,ε ˙Bd 2 2,1+ε2/ba∇dblZ/ba∇dbll+,ε ˙Bd 2 2,1+ε2/ba∇dblZ/ba∇dblh ˙Bd 2 2,1/parenrightBigg /ba∇dblZ/ba∇dbl˙Bd 2 2,1 /lessorsimilar/parenleftBigg ε2/ba∇dblZ/ba∇dbll−,ε ˙Bd 2 2,1+/ba∇dblZ/ba∇dbll+,ε ˙Bd 2+2 2,1+ε2/ba∇dblZ/ba∇dblh ˙Bd 2+1 2,1/parenrightBigg /ba∇dblZ/ba∇dbl˙Bd 2 2,1 /lessorsimilar˜L˜H.12 VALENTIN LEMARIÉ We have also : /ba∇dblZ/ba∇dbl2 ˙Bd 2+1 2,1/lessorsimilar/parenleftBigg /ba∇dblZ/ba∇dbll−,ε ˙Bd 2+1 2,1/parenrightBigg2 +/parenleftBigg /ba∇dblZ/ba∇dbll+,ε ˙Bd 2+1 2,1/parenrightBigg2 +/parenleftBigg /ba∇dblZ/ba∇dblh ˙Bd 2+1 2,1/parenrightBigg2 /lessorsimilarε2/ba∇dblZ/ba∇dbll−,ε ˙Bd 2 2,1+/ba∇dblZ/ba∇dbll+,ε ˙Bd 2 2,1/ba∇dblZ/ba∇dbll+,ε ˙Bd 2+2 2,1+˜L˜H /lessorsimilar˜L˜H. /square 3.2.1. High Frequency Estimates. We rely on the high-frequency analysis carried out in [16]. Proposition 3.3. For high frequencies, we have the estimate: /ba∇dblZ(t)/ba∇dblh ˙Bd 2+1 2,1+/integraldisplayt 0/ba∇dblZ/ba∇dblh ˙Bd 2+1 2,1dτ≤ /ba∇dblZ0/ba∇dblh ˙Bd 2+1 2,1+C/integraldisplayt 0˜L(τ)˜H(τ)dτ. Proof. Let us denote Lj:=1 2/ba∇dblZj/ba∇dbl2 L2+2−2j 4/integraldisplay Rd∇˜cj·vjdx. The following lemma will enable us to handle the first term of Lj: Lemma 3.4. We have the following inequality with Z=/parenleftbigg ˜c v/parenrightbigg andε′=/radicalbig f(c)ε: 1 2d dt/ba∇dblZj/ba∇dbl2 L2+/ba∇dblvj/ba∇dbl2 L2+/integraldisplayt 0ε′2∇(−∆)−1˜cj·vjdx≤Caj2−j(d 2+1)/parenleftbig /ba∇dbl∇Z/ba∇dbl˙Bd 2 2,1/ba∇dblZ/ba∇dbl˙Bd 2+1 2,1+ε2/ba∇dbl˜c/ba∇dbl2 ˙Bd 2 2,1/parenrightbig /ba∇dblZj/ba∇dblL2 with(aj)verifying /summationdisplay j∈Zaj= 1. (3.12) Proof. Let us apply ˙∆jto the system (3.10), then we get: ∂tcj+v·∇˜cj+c cdiv(vj)+/integraldisplayt 0ε′2∇(−∆)−1˜cj·vjdx= [v·∇,˙∆j]˜c+[˜c c,˙∆j]div(v), ∂tvj+v·∇vj+c c∇cj+vj+ε′2∇(−∆)−1˜cj=−ε2˙∆j∇(−∆)−1F(˜c)+[v·∇,˙∆j]v+[˜c c,˙∆j]∇˜c. By performing the scalar product with Zj:=/parenleftbigg ˜cj vj/parenrightbigg inL2(Rd;Rn), we get: 1 2d dt/ba∇dblZj/ba∇dbl2 L2+/ba∇dblvj/ba∇dbl2 L2=−/integraldisplay Rd/parenleftBig v·∇˜cj+c cdiv(vj)/parenrightBig ˜cj+/integraldisplay Rd[v·∇,˙∆j]˜c˜cjdx +/integraldisplay Rd[˜c c,˙∆j]div(v)˜cjdx−/integraldisplay Rd/parenleftBig v·∇vj+c c∇cj/parenrightBig vjdx +/integraldisplay Rd/bracketleftBig v·∇,˙∆j/bracketrightBig vvjdx+/integraldisplay Rd[˜c c,˙∆j]∇˜c·vjdx −ε2/integraldisplay Rd˙∆j∇(−∆)−1F(˜c)·vjdx. In order to bound the right-hand side, we use the following fa cts: • /ba∇dblε2∇(−∆)−1F(˜c)/ba∇dbl˙Bd 2+1 2,1≤ε2/ba∇dblF(˜c)/ba∇dbl˙Bd 2 2,1≤ε2C(/ba∇dbl˜c/ba∇dblL∞)/ba∇dbl˜c/ba∇dbl2 ˙Bd 2 2,1/lessorsimilarε2/ba∇dbl˜c/ba∇dbl2 ˙Bd 2 2,1.PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 13 •The following commutator estimates with s′=d 2+1: /ba∇dbl[v·∇,˙∆j]Z/ba∇dblL2≤Caj2−js′/ba∇dbl∇Z/ba∇dbl˙Bd 2 2,1/ba∇dblZ/ba∇dbl˙Bs′ 2,1where/summationtext j∈Zaj= 1, /ba∇dbl[˜c c,˙∆j]div(v)/ba∇dblL2≤Caj2−js′/ba∇dbl∇Z/ba∇dbl˙Bd 2 2,1/ba∇dblZ/ba∇dbl˙Bs′ 2,1, /ba∇dbl[˜c c,˙∆j]∇˜c/ba∇dblL2≤Caj2−js′/ba∇dbl∇Z/ba∇dbl˙Bd 2 2,1/ba∇dblZ/ba∇dbl˙Bs′ 2,1. •The integration by parts: /integraldisplay Rdv·∇˜cj˜cjdx=−/integraldisplay Rd1 2div(v)|˜cj|2dx. •/integraldisplay Rdv·∇˜vj˜vjdx=−/integraldisplay Rd1 2div(v)|˜vj|2dx, •/integraldisplay Rdc c(div(vj)˜cj+∇˜cjvj)dx=−1 c/integraldisplay Rdvj∇(c˜cj)dx+1 c/integraldisplay Rdc∇˜cjvjdx=−/integraldisplay Rdvj˜cj∇c cdx, hence using the injection ˙Bd 2 2,1(Rd)֒→ Cb(Rd) /integraldisplay Rdv·∇˜cj˜cjdx+/integraldisplay Rdv·∇˜vj˜vjdx+/integraldisplay Rdc c(div(vj)˜cj+∇˜cjvj)dx ≤Caj2−js′/ba∇dbl∇Z/ba∇dbl˙Bd 2 2,1/ba∇dblZ/ba∇dbl˙Bs′ 2,1/ba∇dblZj/ba∇dblL2. Whence the result by putting together all the above inequali ties. /square For the other term of Lj, look at (3.10) as: ∂t˜c+div(v) =−v·∇c−˜c cdiv(v) ∂tv+∇˜c+ε′2∇(−∆)−1˜c+v=−v·∇v−˜c c∇˜c−ε2∇(−∆)−1F(˜c) whereε′=/radicalbig f(c)ε. (3.13) Let us denote S1:=−v·∇c−˜c cdiv(v)andS2:=−v·∇v−˜c c∇˜c−ε2∇(−∆)−1F(˜c)as well as S:= (S1,S2). Analogously to linear analysis, we obtain: d dt/parenleftbigg/integraldisplay Rd∇˜cj·vjdx/parenrightbigg +/integraldisplay Rd∇˜cj·vjdx−/ba∇dbldivvj/ba∇dbl2 L2+/ba∇dbl∇˜cj/ba∇dbl2 L2+ε′2/ba∇dbl˜cj/ba∇dblL2=/integraldisplay RdRe(˙∆jS1·v)dx +/integraldisplay RdRe(∇cj·˙∆jS2)dx. By Cauchy-Schwarz and Bernstein inequalities, we see that Lj:=1 2/ba∇dblZj/ba∇dbl2 L2+2−2j 4/integraldisplay Rd∇˜cj·vjdxverifies : (3.14)3 8/ba∇dblZj/ba∇dbl2 L2≤ Lj≤1 2/ba∇dblZj/ba∇dbl2 L2+2−2j 8/ba∇dbl∇˜cj/ba∇dbl2 L2+1 8/ba∇dblvj/ba∇dbl2 L2≤5 8/ba∇dblZj/ba∇dbl2 L2. We then summarize the previous inequality with the inequali ty of the lemma 3.4: d dtLj+/ba∇dblvj/ba∇dbl2 L2+/integraldisplayt 0ε′2∇(−∆)−1˜cj·vjdx−2−2j 4/ba∇dbldivvj/ba∇dbl2 L2+2−2j 4/ba∇dbl∇cj/ba∇dbl2 L2+ε′22−2j 4/ba∇dbl˜cj/ba∇dbl2 L2+2−2j 4/integraldisplay Rd∇cj·vjdx ≤Caj2−js′/parenleftBigg /ba∇dbl∇Z/ba∇dbl˙Bd 2 2,1/ba∇dblZ/ba∇dbl˙Bs′ 2,1+ε2/ba∇dbl˜c/ba∇dbl2 ˙Bd 2 2,1/parenrightBigg /ba∇dblZj/ba∇dblL2+2−2j 4(/ba∇dbl˙∆jS1/ba∇dblL2/ba∇dblvj/ba∇dblL2+/ba∇dbl˙∆jS2/ba∇dblL2/ba∇dbl˜cj/ba∇dblL2).14 VALENTIN LEMARIÉ But owing to the Cauchy-Schwarz and Bernstein inequalities, w e have /integraldisplayt 0ε′2∇(−∆)−1˜cj·vjdx+2−2j 4/integraldisplay Rd∇˜cj·vjdx≥ −1−2ε′2 8/ba∇dbl˜cj/ba∇dbl2 L2−1−2ε′2 8/ba∇dblvj/ba∇dbl2 L2 which allows us to obtain, thanks to (3.14), /ba∇dblvj/ba∇dblL2+/integraldisplayt 0ε′2∇(−∆)−1˜cj·vjdx−2−2j 4/ba∇dbldivvj/ba∇dbl2 L2+1 4/ba∇dblcj/ba∇dbl2 L2+ε′22−2j 4/ba∇dbl˜cj/ba∇dbl2 L2+2−2j 4/integraldisplay Rd∇cj·vjdx ≥5−2ε′2 8/ba∇dblvj/ba∇dbl2 L2+1−2ε′2 8/ba∇dbl˜cj/ba∇dbl2 L2 ≥1−2ε′2 5Lj. We then get for ε′≤1/2: d dtLj+Lj/lessorsimilar/parenleftBigg 2−j/ba∇dbl˙∆jS/ba∇dblL2+aj2−js′(/ba∇dblZ/ba∇dbl2 ˙Bs′ 2,1+ε2/ba∇dbl˜c/ba∇dbl2 ˙Bd 2 2,1)/parenrightBigg /ba∇dblZj/ba∇dblL2. (3.15) By decomposing Z=Zl+Zhwhere/hatwiderZl=/hatwideZ1|ξ|≤1and/hatwiderZh=/hatwideZ1|ξ|≥1as well as using the lemmas 3.2 and A.3, we have : /ba∇dblZ∇Z/ba∇dblh ˙Bd 2 2,1≤ /ba∇dblZ∇Zl/ba∇dbl˙Bd 2+1 2,1+/ba∇dblZ∇Zh/ba∇dbl˙Bd 2 2,1/lessorsimilar/ba∇dblZ/ba∇dbl2 ˙Bd 2+1 2,1+/ba∇dblZ/ba∇dbl˙Bd 2 2,1/ba∇dblZ/ba∇dbll ˙Bd 2+2 2,1+/ba∇dblZ/ba∇dbl˙Bd 2 2,1/ba∇dblZ/ba∇dblh ˙Bd 2+1 2,1/lessorsimilar˜L˜H /ba∇dbl−ε2∇(−∆)−1F(˜c)/ba∇dblh ˙Bd 2 2,1/lessorsimilarε2/ba∇dblF(˜c)/ba∇dblh ˙Bd 2−1 2,1≤ε2/ba∇dblF(˜c)/ba∇dbl˙Bd 2 2,1/lessorsimilarε2/ba∇dblZ/ba∇dbl2 ˙Bd 2 2,1/lessorsimilar˜L˜H. Consequently, we have : 2js′/radicalBig Lj(t)+c2js′/integraldisplayt 0/radicalbig Ljdτ≤2js′/radicalBig Lj(0)+C/integraldisplayt 0aj˜L(τ)˜H(τ)dτ,withs′=d 2+1 We deduce, by summing on j∈N, the estimate for the high frequencies of the proposition. /square 3.2.2. Damped mode. As explained in [16], at the low frequency level, we have a los s of information on the part undergoing dissipation (here v). To overcome this loss, Danchin highlighted "the damped mo de" to recover the missing information: we then pose W:=−∂tv. However here we cannot proceed as in linear analysis: some no n-linear terms do not allow us to conclude. So we are going to proceed differently here: we will first study the damped mode, which will allow us to have very precise estimates. In a second step, we will study t he equation on cusing this damped mode and finally we will get the information on vthat we need using the estimates on Wandc. From now on, let us take the constants (other than ε) appearing in our system equal to 1to simplify the presentation: they play no role in what will follow. Let us first look at this damped mode at the high frequencies. Lemma 3.5. We have the following estimate for the damped mode: /ba∇dblW(t)/ba∇dblh ˙Bd 2 2,1+/integraldisplayt 0/ba∇dblW/ba∇dblh ˙Bd 2 2,1dτ/lessorsimilar˜L(t)+/integraldisplayt 0˜H(τ)dτ+/integraldisplayt 0˜L(τ)˜H(τ)dτ. Proof. By definition of W, we have : W=v+∇c+ε2∇(−∆)1˜c+v·v+˜c∇˜c+ε2∇(−∆)−1F(˜c). We then have (by using the fact that ˙Bd 2 2,1is a multiplicative algebra) : /ba∇dblW(t)/ba∇dblh ˙Bd 2 2,1/lessorsimilar/ba∇dblv(t)/ba∇dblh ˙Bd 2 2,1+/ba∇dbl˜c(t)/ba∇dblh ˙Bd 2+1 2,1+ε2/ba∇dbl˜c(t)/ba∇dblh ˙Bd 2−1 2,1+/ba∇dblv(t)/ba∇dblh ˙Bd 2 2,1/ba∇dblv(t)/ba∇dblh ˙Bd 2+1 2,1+/ba∇dbl˜c(t)/ba∇dblh ˙Bd 2 2,1/ba∇dbl˜c(t)/ba∇dblh ˙Bd 2+1 2,1 +ε2/ba∇dbl˜c(t)/ba∇dblh ˙Bd 2 2,1/ba∇dbl˜c(t)/ba∇dblh ˙Bd 2−1 2,1.PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 15 By the lemma 3.2, we have : /ba∇dblW(t)/ba∇dblh ˙Bd 2 2,1/lessorsimilar˜L(t)and/integraldisplayt 0/ba∇dblW(τ)/ba∇dblh ˙Bd 2 2,1dτ/lessorsimilar/integraldisplayt 0˜H(τ)dτ+/integraldisplayt 0˜L(τ)˜H(τ)dτ. /square As∂tv+W= 0, thus we have : ∂tW+W=∂tv+W+∂tv·∇v+v·∇(∂tv)+∇(∂tc)+(∂tc)∇c+˜c∇(∂tc)+ε2∇(−∆)−1∂tc +ε2∇(−∆)−1F′(˜c)∂tc =−W·∇v−v·∇W−∇(v·∇˜c)−∇divv−∇(˜cdivv)−v·∇˜c∇˜c−divv∇˜c−˜cdivv∇˜c −˜c∇(v·∇˜c)−˜c(∇divv)−˜c∇(divv∇˜c)−ε2∇(−∆)−1(v·∇˜c)−ε2∇(−∆)−1(divv) −ε2∇(−∆)−1(˜cdivv). Lemma 3.6. The following estimate holds true: (3.16)/ba∇dblW(t)/ba∇dbll ˙Bd 2 2,1+/integraldisplayt 0/ba∇dblW/ba∇dbll ˙Bd 2 2,1dτ/lessorsimilar/ba∇dblW0/ba∇dbll ˙B2,1+/integraldisplayt 0˜L(τ)˜H(τ)dτ+/integraldisplayt 0/ba∇dblv/ba∇dbll ˙Bd 2+2 2,1dτ+ε2/integraldisplayt 0/ba∇dbldivv/ba∇dbll ˙Bd 2−1 2,1dτ. Proof. By applying ˙∆jto the previous equation verified by W, taking the scalar product with Wj, multiplying by2jd 2, summing up on j∈Z−and using the lemma A.1, we get owing to the product laws of the lemma A.3 (each non-linear term in the right-hand side appears in t he same order as the following estimate): /ba∇dblW(t)/ba∇dbll ˙Bd 2 2,1+/integraldisplayt 0/ba∇dblW/ba∇dbll ˙Bd 2 2,1dτ/lessorsimilar/ba∇dblW0/ba∇dbll ˙Bd 2 2,1+/integraldisplayt 0/ba∇dblW/ba∇dbl˙Bd 2 2,1/ba∇dblv/ba∇dbl˙Bd 2+1 2,1dτ+/integraldisplayt 0/ba∇dblv·W/ba∇dbll ˙Bd 2−1 2,1dτ +/integraldisplayt 0/parenleftBigg /ba∇dbl∇v·∇˜c/ba∇dbl˙Bd 2 2,1+/ba∇dblv·∇∇c/ba∇dbll ˙Bd 2 2,1/parenrightBigg dτ+/integraldisplayt 0/ba∇dblv/ba∇dbll ˙Bd 2+2 2,1dτ +/integraldisplayt 0/parenleftBigg /ba∇dbl∇˜cdivv/ba∇dbl˙Bd 2 2,1+/ba∇dbl˜c∇divv/ba∇dbl˙Bd 2 2,1/parenrightBigg dτ+/integraldisplayt 0/ba∇dblv/ba∇dbl˙Bd 2 2,1/ba∇dbl˜c/ba∇dbl2 ˙Bd 2+1 2,1dτ +/integraldisplayt 0/ba∇dblv/ba∇dbl˙Bd 2+1 2,1/ba∇dbl˜c/ba∇dbl˙Bd 2+1 2,1dτ+/integraldisplayt 0/ba∇dbl˜c/ba∇dbl˙Bd 2+1 2,1/ba∇dblv/ba∇dbl˙Bd 2+1 2,1/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1dτ +/integraldisplayt 0/ba∇dbl˜c∇(v·∇˜c)/ba∇dbll ˙Bd 2 2,1dτ+/integraldisplayt 0/ba∇dbl˜c(∇divv)/ba∇dbll ˙Bd 2 2,1dτ+/integraldisplayt 0/ba∇dbl˜c∇(divv∇˜c)/ba∇dbll ˙Bd 2 2,1dτ +ε2/integraldisplayt 0/ba∇dblv/ba∇dbl˙Bd 2 2,1/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1dτ+/integraldisplayt 0ε2/ba∇dbldivv/ba∇dbll ˙Bd 2−1 2,1dτ+ε2/integraldisplayt 0/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1/ba∇dblv/ba∇dbl˙Bd 2 2,1dτ. Let us estimate one by one the terms appearing in the right-ha nd side: •/integraldisplayt 0/ba∇dblW/ba∇dbll ˙Bd 2 2,1/ba∇dblv/ba∇dbl˙Bd 2+1 2,1dτcan be absorbed by the left-hand side as we know that /ba∇dblZ/ba∇dbl L∞(˙Bd 2 2,1)is small; •/integraldisplayt 0/ba∇dblW/ba∇dblh ˙Bd 2 2,1/ba∇dblv/ba∇dbl˙Bd 2+1 2,1dτ/lessorsimilar/integraldisplayt 0˜L(τ)˜H(τ)dτby the lemma 3.5; •/integraldisplayt 0/ba∇dblv·W/ba∇dbll ˙Bd 2−1 2,1dτ/lessorsimilar/integraldisplayt 0/ba∇dblv/ba∇dbl˙Bd 2 2,1(/ba∇dblW/ba∇dbll ˙Bd 2 2,1+/ba∇dblW/ba∇dblh ˙Bd 2 2,1)dτhas the low frequency part absorbed by the left-hand side and the other part below/integraldisplayt 0˜L(τ)˜H(τ)dτby the lemma 3.5; •By using the lemma 3.2 and the fact that /ba∇dblZ/ba∇dbl L∞(˙Bd 2 2,1)is small, we have : /integraldisplayt 0/parenleftbigg /ba∇dbl∇v·∇c/ba∇dbl˙Bd 2 2,1+/ba∇dbl∇˜c·divv/ba∇dbl˙Bd 2 2,1+/ba∇dblv/ba∇dbl˙Bd 2 2,1/ba∇dbl˜c/ba∇dbl2 ˙Bd 2+1 2,1+/ba∇dblv/ba∇dbl˙Bd 2+1 2,1/ba∇dblc/ba∇dbl˙Bd 2+1 2,1+/ba∇dbl˜c/ba∇dbl˙Bd 2+1 2,1/ba∇dblv/ba∇dbl˙Bd 2+1 2,1/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1/parenrightbigg dτ /lessorsimilar/integraldisplayt 0/ba∇dblZ/ba∇dbl2 ˙Bd 2+1 2,1dτ/lessorsimilar/integraldisplayt 0˜L(τ)˜H(τ)dτ;16 VALENTIN LEMARIÉ •/integraldisplayt 0ε2/ba∇dblv/ba∇dbl˙Bd 2 2,1/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1dτ+ε2/integraldisplayt 0/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1/ba∇dblv/ba∇dbl˙Bd 2 2,1/lessorsimilar/integraldisplayt 0˜L(τ)˜H(τ)dτby using the lemma 3.2; • /ba∇dblv·∇∇c/ba∇dbll ˙Bd 2 2,1/lessorsimilar/ba∇dblv·∇∇cl/ba∇dbl˙Bd 2 2,1+/ba∇dblv·∇∇ch/ba∇dbl˙Bd 2 2,1/lessorsimilar/ba∇dblv/ba∇dbl˙Bd 2 2,1/ba∇dbl˜c/ba∇dbll ˙Bd 2+2 2,1+/ba∇dblv·∇∇ch/ba∇dbll ˙Bd 2−1 2,1 /lessorsimilar˜L˜H+/ba∇dblv/ba∇dbl˙Bd 2 2,1/ba∇dbl˜c/ba∇dblh ˙Bd 2+1 2,1/lessorsimilar˜L˜Hby the lemmas 3.2 and A.3; •By proceeding in the same way as previously, we discover that : /ba∇dbl˜c∇divv/ba∇dbll ˙Bd 2 2,1/lessorsimilar/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1/parenleftBigg /ba∇dblv/ba∇dbll ˙Bd 2+2 2,1+/ba∇dblv/ba∇dblh ˙Bd 2+1 2,1/parenrightBigg /lessorsimilar˜L˜H; • /ba∇dbl˜c∇(v·∇˜c)/ba∇dbll ˙Bd 2 2,1≤ /ba∇dbl˜cdivv∇˜c/ba∇dbl˙Bd 2 2,1+/ba∇dbl˜cv·∇∇˜c/ba∇dbl˙Bd 2 2,1. For the first term, the lemma 3.2 is used and for the second, we d o the same as for the previous two points. We find that this term is less than ˜L˜H. So we have inequality (3.16). /square 3.2.3. Study of the low frequencies of ˜c. We have : ∂t˜c+divv=−˜cdivv−v·∇˜c. However , divv= divW−∆c+ε2˜c+ε2F(˜c)−div(˜c∇˜c)−div(v·∇v). So by rewriting the equation of ˜c, we get ∂t˜c−∆˜c+ε2c=−divW−ε2F(˜c)+div(˜c∇˜c)+div(v·∇v)−˜cdivv−v·∇˜c After spectral localization by means of ˙∆j, taking the scalar product with ˙∆j˜c, multiplying by ε2j(d 2−1) (respectively 2jd 2) and, finally, summing up on 2j≤ε(respectively ε≤2j≤1), we get Lemma 3.7. The following estimates are satisfied by ˜c: /ba∇dblε˜c(t)/ba∇dbll−,ε ˙Bd 2−1 2,1+ε2/integraldisplayt 0/ba∇dblε˜c(t)/ba∇dbll−,ε ˙Bd 2−1 2,1dτ/lessorsimilar/ba∇dblε˜c0/ba∇dbll−,ε ˙Bd 2−1 2,1+ε/integraldisplayt 0/ba∇dblW/ba∇dbll−,ε ˙Bd 2 2,1dτ+/integraldisplayt 0C(τ)˜C(τ)dτ +ε/integraldisplayt 0/ba∇dblv·∇v/ba∇dbll−,ε ˙Bd 2 2,1+ε/integraldisplayt 0/ba∇dbl˜cdivv/ba∇dbl˙Bd 2−1 2,1dτ+ε/integraldisplayt 0/ba∇dblv·∇˜c/ba∇dbl˙Bd 2−1 2,1dτ, /ba∇dbl˜c(t)/ba∇dbll+,ε ˙Bd 2 2,1+/integraldisplayt 0/ba∇dbl˜c/ba∇dbll+,ε ˙Bd 2+2 2,1dτ/lessorsimilar/ba∇dbl˜c0/ba∇dbll+,ε ˙Bd 2 2,1+/integraldisplayt 0/ba∇dblW/ba∇dbll+,ε ˙Bd 2+1 2,1dτ+/integraldisplayt 0C(τ)˜C(τ)dτ+/integraldisplayt 0/ba∇dbldiv(v·∇v)/ba∇dbll+,ε ˙Bd 2 2,1dτ +/integraldisplayt 0/ba∇dbl˜cdivv/ba∇dbl˙Bd 2 2,1dτ+/integraldisplayt 0/ba∇dblv·∇˜c/ba∇dbl˙Bd 2 2,1dτ where, in view of the linear estimates, we set C(t):=/ba∇dblε˜c(t)/ba∇dbll−,ε ˙Bd 2−1 2,1+/ba∇dbl˜c(t)/ba∇dbll+,ε ˙Bd 2 2,1+/ba∇dbl˜c(t)/ba∇dblh ˙Bd 2+1 2,1and˜C(t):= ε2/ba∇dblε˜c(t)/ba∇dbll−,ε ˙Bd 2−1 2,1+/ba∇dbl˜c(t)/ba∇dbll+,ε ˙Bd 2+2 2,1+/ba∇dbl˜c(t)/ba∇dblh ˙Bd 2+1 2,1. 3.2.4. Study of the low frequencies of v. It is now possible to deduce optimal bounds for vfrom the ones we have just derived for Wand˜c. We need to decompose vby using the damped mode as follows : v=W−∇c+ε2∇(−∆)−1˜c+v·∇v+˜c∇c+ε2∇(−∆)−1F(˜c) (3.17) Lemma 3.8. Based on the following estimates, we will set: V(t):=/ba∇dblv(t)/ba∇dbll ˙Bd 2 2,1+/ba∇dblv(t)/ba∇dblh ˙Bd 2+1 2,1and˜V(t):=ε/ba∇dblv(t)/ba∇dbll−,ε ˙Bd 2 2,1+/ba∇dblv(t)/ba∇dbll+,ε ˙Bd 2+1 2,1+/ba∇dblv(t)/ba∇dblh ˙Bd 2+1 2,1.PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 17 We then obtain the following estimates: /ba∇dblv(t)/ba∇dbll−,ε ˙Bd 2 2,1+ε/integraldisplayt 0/ba∇dblv/ba∇dbll−,ε ˙Bd 2 2,1dτ/lessorsimilar/ba∇dblW(t)/ba∇dbll−,ε ˙Bd 2 2,1+(ε2+/ba∇dblZ(t)/ba∇dbl˙Bd 2+1 2,1+εC(t))C(t)+V(t)/ba∇dblZ(t)/ba∇dbl˙Bd 2+1 2,1 +/integraldisplayt 0ε/ba∇dblW/ba∇dbll−,ε ˙Bd 2 2,1dτ+/integraldisplayt 0˜C(τ)dτ+/integraldisplayt 0C(τ)˜C(τ)dτ+ε/integraldisplayt 0V(τ)˜V(τ)dτ. /ba∇dblv(t)/ba∇dbll+,ε ˙Bd 2 2,1+/integraldisplayt 0/ba∇dblv(t)/ba∇dbll+,ε ˙Bd 2+1 2,1dτ/lessorsimilar/ba∇dblW(t)/ba∇dbll+,ε ˙Bd 2 2,1+/ba∇dbl˜c(t)/ba∇dbll+,ε ˙Bd 2+1 2,1+(V(t)+C(t))/ba∇dblZ(t)/ba∇dbl˙Bd 2+1 2,1+ε(C(t))2 +/integraldisplayt 0/ba∇dblW/ba∇dbll+,ε ˙Bd 2+1 2,1dτ+/integraldisplayt 0˜C(τ)dτ+/integraldisplayt 0V(τ)˜V(τ)dτ+/integraldisplayt 0C(τ)˜C(τ)dτ. Proof. By lemmas A.3, A.5 and (3.17), we obtain : /ba∇dblv(t)/ba∇dbll−,ε ˙Bd 2 2,1/lessorsimilar/ba∇dblW/ba∇dbll−,ε ˙Bd 2 2,1+ε2/ba∇dbl˜c/ba∇dbll−,ε ˙Bd 2−1 2,1+/ba∇dblv/ba∇dbl˙Bd 2 2,1/ba∇dblv/ba∇dbl˙Bd 2+1 2,1+/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1/ba∇dbl˜c/ba∇dbl˙Bd 2+1 2,1+ε2/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1/ba∇dbl˜c/ba∇dbl˙Bd 2−1 2,1. We deduce : /ba∇dblv(t)/ba∇dbll−,ε ˙Bd 2 2,1/lessorsimilar/ba∇dblW(t)/ba∇dbll−,ε ˙Bd 2 2,1+(ε+/ba∇dblZ(t)/ba∇dbl˙Bd 2+1 2,1+εC(t))C(t) +V(t)/ba∇dblv(t)/ba∇dbl˙Bd 2+1 2,1 ε/integraldisplayt 0/ba∇dblv/ba∇dbll−,ε ˙Bd 2 2,1dτ/lessorsimilar/integraldisplayt 0ε/ba∇dblW/ba∇dbll−,ε ˙Bd 2 2,1dτ+/integraldisplayt 0˜C(τ)dτ+/integraldisplayt 0C(τ)˜C(τ)dτ+ε/integraldisplayt 0V(τ)˜V(τ)dτ. Similarly, we get : /ba∇dblv(t)/ba∇dbll+,ε ˙Bd 2 2,1/lessorsimilar/ba∇dblW(t)/ba∇dbll+,ε ˙Bd 2 2,1+/ba∇dbl˜c(t)/ba∇dbll+,ε ˙Bd 2+1 2,1+(V(t)+C(t))/ba∇dblZ(t)/ba∇dbl˙Bd 2+1 2,1+ε(C(t))2. By lemmas A.3 and A.5 and inequality /ba∇dblZ/ba∇dbll+,ε ˙Bd 2+1 2,1≤ /ba∇dblZ/ba∇dbll+,ε ˙Bd 2 2,1, we have also : /ba∇dblv/ba∇dbll+,ε ˙Bd 2+1 2,1/lessorsimilar/ba∇dblW/ba∇dbll+,ε ˙Bd 2+1 2,1+/ba∇dbl˜c/ba∇dbll+,ε ˙Bd 2+2 2,1+/ba∇dblv/ba∇dbl˙Bd 2 2,1/ba∇dblv/ba∇dbl˙Bd 2+1 2,1+/ba∇dbl˜c·∇c/ba∇dbll+,ε ˙Bd 2+1 2,1+ε2/ba∇dbl˜c/ba∇dbl2 ˙Bd 2 2,1. Like the proof of lemma 3.2, we have that ε2/ba∇dbl˜c/ba∇dbl2 ˙Bd 2 2,1/lessorsimilarC˜C. Moreover, /ba∇dbl˜c·∇c/ba∇dbll+,ε ˙Bd 2+1 2,1/lessorsimilar/ba∇dbl˜c·∇cl/ba∇dbll ˙Bd 2+1 2,1+/ba∇dbl˜c·∇ch/ba∇dbll ˙Bd 2+1 2,1 /lessorsimilar/ba∇dbl˜c/ba∇dbl˙Bd 2+1 2,1/ba∇dbl˜c/ba∇dbll ˙Bd 2+1 2,1+/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1/ba∇dbl˜c/ba∇dbll ˙Bd 2+2 2,1+/ba∇dbl˜c·∇ch/ba∇dbll ˙Bd 2 2,1 /lessorsimilarC˜C+/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1/ba∇dbl˜c/ba∇dbl˙Bd 2+1 2,1/lessorsimilarC˜C. We deduce : /integraldisplayt 0/ba∇dblv/ba∇dbll+,ε ˙Bd 2 2,1dτ/lessorsimilar/integraldisplayt 0/ba∇dblW/ba∇dbll+,ε ˙Bd 2+1 2,1dτ+/integraldisplayt 0˜C(τ)dτ+/integraldisplayt 0V(τ)˜V(τ)dτ+/integraldisplayt 0C(τ)˜C(τ)dτ. /square 3.2.5. Final a priori estimates. Let us denote (3.18) L(t):=C(t)+V(t)+/ba∇dblW(t)/ba∇dbll ˙Bd 2 2,1andH(t):=˜C(t)+˜V(t)+/ba∇dblW(t)/ba∇dbll ˙Bd 2 2,1. We notice that : ˜L(t)≤ L(t)and˜H(t)≤ H(t).18 VALENTIN LEMARIÉ Proposition 3.9. We have the following estimate : L(t)+/integraldisplayt 0H(τ)dτ/lessorsimilarL(0)+/integraldisplayt 0L(τ)H(τ)dτ. If we take L(0)sufficiently small, thus we obtain the final a priori estimate : L(t)+/integraldisplayt 0H(τ)dτ/lessorsimilarL(0). Proof. First, we note that by summing up the previous inequalities ( lemmas 3.6, 3.7, 3.8), terms in the right-hand side can be absorbed by those of the left-hand sid e. Indeed : •In (3.16), we have that the term/integraldisplayt 0/ba∇dblv/ba∇dbll ˙Bd 2+2 2,1dτ+ε2/integraldisplayt 0/ba∇dbldivv/ba∇dbl˙Bd 2−1 2,1dτis negligible compared to /integraldisplayt 0˜V(τ)dτ(so also to/integraldisplayt 0H(τ)dτ). •In the estimates of the lemma 3.7, we have that/integraldisplayt 0ε/ba∇dblW/ba∇dbll−,ε ˙Bd 2 2,1dτ,/integraldisplayt 0/ba∇dblW/ba∇dbll+,ε ˙Bd 2+1 2,1dτare negligible compared to/integraldisplayt 0/ba∇dblW/ba∇dbll ˙Bd 2 2,1dτ. By using that /ba∇dblZ/ba∇dbl˙Bd 2 2,1is small and the lemma A.3, we also have that termsε/integraldisplayt 0/ba∇dblv·∇v/ba∇dbll−,ε ˙Bd 2 2,1dτand/integraldisplayt 0/ba∇dbldiv(c·∇v)/ba∇dbll+,ε ˙Bd 2 2,1dτare negligible compared to/integraldisplayt 0˜V(τ)dτ. •(ε2+εC(t))C(t)and/ba∇dbl˜c(t)/ba∇dbll+,ε ˙Bd 2+1 2,1are negligible compared to L(t). Using the definitions of the various introduced norms and L,Hand the lemma A.3, we have : •/integraldisplayt 0˜L˜Hdτ/lessorsimilar/integraldisplayt 0LHdτ, •/integraldisplayt 0C(τ)˜C(τ)dτ+/integraldisplayt 0V(τ)˜V(τ)dτ/lessorsimilar/integraldisplayt 0L(τ)H(τ)dτ, •/integraldisplayt 0ε/ba∇dbl˜cdivv/ba∇dbll−,ε ˙Bd 2−1 2,1dτ/lessorsimilar/integraldisplayt 0/ba∇dblε˜c/ba∇dbl˙Bd 2−1 2,1/ba∇dblv/ba∇dbl˙Bd 2+1 2,1dτ/lessorsimilar/integraldisplayt 0L(τ)H(τ)dτ, •/integraldisplayt 0/ba∇dbl˜cdivv/ba∇dbll−,ε ˙Bd 2 2,1dτ/lessorsimilar/integraldisplayt 0/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1/ba∇dblv/ba∇dbl˙Bd 2+1 2,1dτ/lessorsimilar/integraldisplayt 0L(τ)H(τ)dτ, •ε/ba∇dblv·∇c/ba∇dbll−,ε ˙Bd 2−1 2,1dτ/lessorsimilar/integraldisplayt 0/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1/ba∇dblεv/ba∇dbl˙Bd 2 2,1dτ/lessorsimilar/integraldisplayt 0L(τ)H(τ)dτ. Now, if we sum up the previous inequalities by using what we ju st did before and by removing "negligible terms compared to the right term", we get: L(t)+/integraldisplayt 0H(τ)dτ/lessorsimilarL(0)+/integraldisplayt 0L(τ)H(τ)dτ+/integraldisplayt 0/ba∇dblv·∇c/ba∇dbll+,ε ˙Bd 2 2,1dτ. To handle the last term, let us use the fact that vandware interrelated as follows: v=W−∇c+ε2∇(−∆)−1˜c+v·∇v+˜c∇c+ε2∇(−∆)−1F(˜c). By the lemma A.3, we have also : /ba∇dblv·∇c/ba∇dbl˙Bd 2 2,1/lessorsimilar/ba∇dblW/ba∇dbl˙Bd 2 2,1/ba∇dbl˜c/ba∇dbl˙Bd 2+1 2,1+/ba∇dblε˜c/ba∇dbl˙Bd 2−1 2,1/ba∇dbl˜c/ba∇dbl˙Bd 2+1 2,1+/ba∇dbl˜c/ba∇dbl2 ˙Bd 2+1 2,1+/ba∇dblv/ba∇dbl˙Bd 2 2,1/ba∇dblv/ba∇dbl˙Bd 2+1 2,1/ba∇dbl˜c/ba∇dbl˙Bd 2+1 2,1 +/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1/ba∇dbl˜c/ba∇dbl2 ˙Bd 2+1 2,1+/ba∇dbl˜c/ba∇dbl˙Bd 2 2,1/ba∇dblε˜c/ba∇dbl˙Bd 2−1 2,1/ba∇dblε˜c/ba∇dbl˙Bd 2+1 2,1. We then have by lemma 3.2 and definition of L,H: /ba∇dblv·∇c/ba∇dbl˙Bd 2 2,1/lessorsimilar˜L˜H+LH/lessorsimilarLH,PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 19 hence the result. We have the second inequality of the propos ition by using lemma A.2. /square 3.3.A global well-posedness theorem. Here is the theorem that we will prove in the rest of this secti on: Theorem 3.10. We assume ε′≤1/2withε′defined in (3.13). Then, there exists a positive constant αsuch that for all Zε 0= (˜c0,v0)∈˙Bd 2 2,1∩˙Bd 2+1 2,1satisfying Zε 0:=/ba∇dblε˜c0/ba∇dbll−,ε′ ˙Bd 2−1 2,1+/ba∇dbl˜c0/ba∇dbll+,ε′ ˙Bd 2 2,1+/ba∇dblv0/ba∇dbll ˙Bd 2 2,1+/ba∇dbl(˜c0,v0)/ba∇dblh ˙Bd 2+1 2,1≤α, the system (3.8)with the initial data (c0,v0)admits a unique global-in-time solution Z= (˜c,v)in the set E:=/braceleftbigg (˜c,v)/vextendsingle/vextendsingle/vextendsingle/vextendsingleε˜cl−,ε′∈ Cb(R+:˙Bd 2−1 2,1), ε3˜cl−,ε′∈L1(R+;˙Bd 2−1 2,1),˜cl+,ε′∈ Cb(R+:˙Bd 2 2,1), ˜cl+,ε′∈L1(R+;˙Bd 2+2 2,1), vl∈ Cb(R+;˙Bd 2 2,1),εvl−,ε′∈L1(R+;˙Bd 2 2,1),vl+,ε′∈L1(R+;˙Bd 2+1 2,1), (˜c,v)h∈ Cb(R+;˙Bd 2+1 2,1)∩L1(R+,˙Bd 2+1 2,1), Wl∈ Cb(R+;˙Bd 2 2,1)∩L1(R+;˙Bd 2 2,1)/bracerightbigg where we denote W:=−∂tv. Moreover, we have the following inequality : Z(t)≤CZε′ 0 where Z(t):=/ba∇dblε˜c/ba∇dbll−,ε′ L∞/parenleftbig ˙Bd 2−1 2,1/parenrightbig+/ba∇dbl˜c/ba∇dbll+,ε′ L∞/parenleftbig ˙Bd 2 2,1/parenrightbig+/ba∇dblv/ba∇dbll L∞/parenleftbig ˙Bd 2 2,1/parenrightbig+/ba∇dbl(˜c,v)/ba∇dblh L∞/parenleftbig ˙Bd 2+1 2,1/parenrightbig+ε2/ba∇dblε˜c/ba∇dbll−,ε′ L1/parenleftbig ˙Bd 2−1 2,1/parenrightbig+ε/ba∇dblv/ba∇dbll−,ε′ L1/parenleftbig ˙Bd 2 2,1/parenrightbig +/ba∇dbl˜c/ba∇dbll+,ε′ L1/parenleftbig ˙Bd 2+2 2,1/parenrightbig+/ba∇dblv/ba∇dbll+,ε′ L1/parenleftbig ˙Bd 2+1 2,1/parenrightbig+/ba∇dbl(˜c,v)/ba∇dblh L1/parenleftbig ˙Bd 2+1 2,1/parenrightbig+/ba∇dblW/ba∇dbl L∞/parenleftbig ˙Bd 2 2,1/parenrightbig+/ba∇dblW/ba∇dbl L1/parenleftbig ˙Bd 2 2,1/parenrightbig. The first step is to approximate (3.9). (1)Approximate systems Let us take Jnthe spectral truncation operator on {ξ∈Rd, n−1≤ |ξ| ≤n}. We consider the following system : d dt/parenleftbigg ˜c v/parenrightbigg +/parenleftbigg Jn(Jn(v)·(∇Jn(c)))+ ˜γJn(Jn(c)div(Jn(v))) Jn(Jn(v)·∇(Jn(v)))+ ˜γJn(Jn(c)∇(Jn(c)))+Jn(v)/parenrightbigg =/parenleftbigg 0 −ε2∇(−∆)−1Jn(G(Jn(˜c)))/parenrightbigg . •By the Cauchy-Lipschitz theorem, we have (using the spectral truncation operator) that this system admits a maximal solution (cn,vn)∈ C1([0,Tn[:L2)with initial data ( Jnc0,Jnv0)for alln∈N. •We have Jncn=cnandJn(vn) =vn(by using the uniqueness in the previous system) and thus: /braceleftbigg∂tcn+Jn(vn·∇cn)+ ˜γJn(cndiv(vn)) = 0, ∂tvn+Jn(vn·∇vn)+ ˜γJn(cn∇cn)+vn=−ε2∇(−∆)−1Jn(G(˜cn)). •From the lemma 3.9, we deduce (the nindex corresponding to the sequence (cn,vn)): Ln(t)+/integraldisplayt 0Hn(τ)dτ/lessorsimilarLn(0)≤ L(0). In particular (by argument of extension of the maximal solut ion), we have that Tn= +∞. (2)Convergence of the sequence The previous estimates guarantee that (˜cn,vn)n∈Nis a bounded sequence of E,whereEis the functional space described in the theorem.20 VALENTIN LEMARIÉ In particular, (˜cn,vn)n∈Nis bounded in L∞(R+;˙Bd 2 2,1)∩L1(R+;˙Bd 2+2 2,1)and inL∞(R+;˙Bd 2+1 2,1)∩L1(R+;˙Bd 2+1 2,1) at low and respectively high frequencies level, so bounded ( by interpolation) in L2/parenleftbigg ˙Bd 2+1 2,1/parenrightbigg . We know that ˙Bd 2 2,1is included continuously in L∞, hence˙Bd 2+1 2,1is locally compact in L2. We can therefore apply the Ascoli theorem and after diagonal extraction, we gather that, up to subse- quence,(cn,vn)n∈Nconverges to some (c,v)inC([0,T[;S′(Rd)). By classical arguments of weak compactness, one can conclude as in e.g [4] that (c,v)belongs to Eand that(c,v)is a solution of the initial system. 3.4.Proof of uniqueness. LetZ1= (c1,v1)andZ2= (c2,v2)be two solutions. We denote δc:=c1−c2, δv:=v1−v2andδZ:=Z1−Z2. In particular, we have : /braceleftBigg ∂tδc+v2·∇δc+ ˜γc2div(δv) =−δv∇c1−˜γδcdiv(v1) ∂tδv+v2·∇δv+ ˜γc2∇δc+ε′2∇(−∆)−1δc+δv=−δv·∇v1−˜γδc∇c1−ε2∇(−∆)−1(F(˜c1)−F(˜c2)). Lemma 3.11. We have the inequality : /ba∇dblδZ(t)/ba∇dblh ˙Bd 2 2,1+/ba∇dblδv(t)/ba∇dbll ˙Bd 2 2,1+/ba∇dblδc(t)/ba∇dbll+,ε ˙Bd 2 2,1+ε/ba∇dblδc(t)/ba∇dbll−,ε ˙Bd 2−1 2,1/lessorsimilar/integraldisplayt 0(L1+L2+H1+H2+1)(τ)/parenleftbigg /ba∇dblδZ(t)/ba∇dblh ˙Bd 2 2,1 +/ba∇dblδv(t)/ba∇dbll ˙Bd 2 2,1+/ba∇dblδc(t)/ba∇dbll+,ε ˙Bd 2 2,1+ε/ba∇dblδc(t)/ba∇dbll−,ε ˙Bd 2−1 2,1/parenrightbigg dτ whereLi,Hifori∈ {1,2}correspond to LandHin(3.18) forZi. Once this lemma is proven, it is easy to conclude the uniquene ss by Grönwall’s lemma. Proof. (1) Estimate for high frequencies : Let us start by proving an estimate for high frequencies. By ap plying the localization operation ˙∆j, we get : ∂tδcj+v2·∇δcj+ ˜γc2div(δvj) =/bracketleftBig v2·∇,˙∆j/bracketrightBig δc+/bracketleftBig ˜γc2,˙∆j/bracketrightBig div(δc)−˙∆j(δv·∇c1+ ˜γδcdiv(v1)) ∂tδvj+v2·∇δvj+ ˜γc2∇δcj+ε′2∇(−∆)−1δcj+δvj=/bracketleftBig v2·∇,˙∆j/bracketrightBig δv+/bracketleftBig ˜γc2∇,˙∆j/bracketrightBig δc −˙∆j/parenleftbig δv·∇v1+ ˜γδc∇c1+ε2∇(−∆)−1(F(˜c1)−F(˜c2))/parenrightbig On the one hand, owing to commutator estimates (see e.g [4]), we have: /ba∇dbl/bracketleftBig v2·∇,˙∆j/bracketrightBig δc+/bracketleftBig ˜γc2,˙∆j/bracketrightBig div(δc) +/bracketleftBig v2·∇,˙∆j/bracketrightBig δv+/bracketleftBig ˜γc2∇,˙∆j/bracketrightBig δc/ba∇dblh ˙Bd 2 2,1/lessorsimilar/ba∇dblZ2/ba∇dbl˙Bd 2+1 2,1/ba∇dblδZ/ba∇dblh ˙Bd 2 2,1 /lessorsimilarH2(τ)/ba∇dblδZ/ba∇dblh ˙Bd 2 2,1. On the other hand, we have by integration by parts: • −/integraldisplay Rdv2·∇δcjδcjdx=1 2/integraldisplay Rddiv(v2)|δcj|2dx, • −/integraldisplay Rdv2·∇δvj·δvjdx=1 2/integraldisplay Rddiv(v2)|δvj|2dx, • −/integraldisplay Rd˜γ˜c2div(δvj)δcjdx−/integraldisplay Rd˜γ˜c2∇δcj·δvjdx=−/integraldisplay Rd˜γ˜c2div(δcjδvj)dx=/integraldisplay Rd˜γ∇c2·(δcjδvj)dx.PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 21 Then we have for all j≤ −1: −/integraldisplay Rdv2·∇δcjδcjdx−/integraldisplay Rdv2·∇δvj·δvjdx−/integraldisplay Rd˜γ˜c2div(δvj)δcjdx−/integraldisplay Rd˜γ˜c2∇δcj·δvjdx =1 2/integraldisplay Rddiv(v2)|δZj|2dx+/integraldisplay Rd˜γ∇c2·(δcjδvj)dx /lessorsimilaraj2−jd 2/ba∇dblZ2/ba∇dbl˙Bd 2+1 2,1/ba∇dblδZ/ba∇dbl˙Bd 2 2,1/ba∇dblδZ/ba∇dbl˙Bd 2 2,1 /lessorsimilaraj2−jd 2H2(t)/ba∇dblδZ/ba∇dbl˙Bd 2 2,1/ba∇dblδZ/ba∇dbl˙Bd 2 2,1. We have also : /ba∇dbl−˙∆j(δv·∇c1+ ˜γδcdiv(v1))−˙∆j/parenleftbig δv·∇v1+ ˜γδc∇c1+ε2∇(−∆)−1(F(˜c1)−F(˜c2))/parenrightbig /ba∇dblL2 /lessorsimilaraj2−jd 2/ba∇dblδZ/ba∇dbl˙Bd 2 2,1/ba∇dblZ1/ba∇dbl˙Bd 2+1 2,1 /lessorsimilaraj2−jd 2/ba∇dblδZ/ba∇dbl˙Bd 2 2,1H1. We then have (taking the scalar product with δZjin the previous system) the following estimate: /ba∇dblδZ/ba∇dblh ˙Bd 2 2,1+/integraldisplayt 0/ba∇dblδZ/ba∇dblh ˙Bd 2 2,1dτ/lessorsimilar/integraldisplayt 0(H1+H2)(τ)/ba∇dblδZ/ba∇dblh ˙Bd 2 2,1dτ. (2) Estimates for low frequencies : As for the study of the Euler-Poisson system, we will look at δvin˙Bd 2 2,1, the very low frequencies ofεδcin˙Bd 2−1 2,1and the medium ones of δcin˙Bd 2 2,1. For the above system, we obtain the following estimates: /ba∇dblεδc(t)/ba∇dbll−,ε ˙Bd 2−1 2,1/lessorsimilar/integraldisplayt 0ε/bracketleftBigg /ba∇dblδv/ba∇dbll−,ε ˙Bd 2 2,1+/ba∇dbl(Z1,Z2)/ba∇dbl˙Bd 2 2,1/ba∇dblδZ/ba∇dbl˙Bd 2 2,1/bracketrightBigg dτ /ba∇dblδc(t)/ba∇dbll+,ε ˙Bd 2 2,1/lessorsimilar/integraldisplayt 0/bracketleftBigg /ba∇dblδv/ba∇dbll+,ε ˙Bd 2+1 2,1+/ba∇dblZ2/ba∇dbl˙Bd 2 2,1/parenleftBigg /ba∇dblδZ/ba∇dbll ˙Bd 2+1 2,1+/ba∇dblδZ/ba∇dblh ˙Bd 2 2,1/parenrightBigg +/ba∇dblδZ/ba∇dbl˙Bd 2 2,1/ba∇dblZ1/ba∇dbl˙Bd 2+1 2,1/bracketrightBigg dτ /ba∇dblδv/ba∇dbll ˙Bd 2+1 2,1+/integraldisplayt 0/ba∇dblδv/ba∇dbll ˙Bd 2+1 2,1dτ/lessorsimilar/integraldisplayt 0/bracketleftbigg ε2/ba∇dblδc/ba∇dbll−,ε ˙Bd 2−1 2,1+/ba∇dblδc/ba∇dbll+,ε ˙Bd 2+1 2,1+/ba∇dblZ2/ba∇dbl˙Bd 2 2,1/parenleftBigg /ba∇dblδZ/ba∇dbll ˙Bd 2+1 2,1+/ba∇dblδZ/ba∇dblh ˙Bd 2 2,1/parenrightBigg +/ba∇dblδZ/ba∇dbl˙Bd 2 2,1/ba∇dblZ1/ba∇dbl˙Bd 2+1 2,1+ε2/parenleftbig /ba∇dblδc/ba∇dbl˙Bd 2 2,1/ba∇dbl(c1,c2)/ba∇dbl˙Bd 2−1 2,1 +/ba∇dblδc/ba∇dbl˙Bd 2−1 2,1/ba∇dbl(c1,c2)/ba∇dbl˙Bd 2 2,1/parenrightbig/bracketrightbigg dτ. (3) Final estimate : We now put together all the estimates and observe that some t erms in the right-hand side are negligible compared to the left-hand side (thanks i n particular to our lemma about 3.9 a priori estimates) and we obtain the final result. /square We deduce by change of variable (2.2) and lemma A.4 the statem ent of the theorem 2.1. 4.Keller-Segel parabolic/elliptical system The goal of this section is to justify the convergence of the d ensity solution of the first equation of (1.3) to the unique solution of (1.5) when εtends to 0. We deduce from the theorem 2.1 the following theorem which wi ll allow us to study the singular limit of the Euler-Poisson system:22 VALENTIN LEMARIÉ Theorem 4.1. Let beε >0. Letε′be defined as (3.13). There exists a positive constant αsuch that for all ε′≤1 2and data Zε 0= (̺ε 0−̺,vε 0)/parenleftbigg ˙Bd 2−1 2,1∩˙Bd 2+1 2,1/parenrightbigg ×/parenleftbigg ˙Bd 2 2,1∩˙Bd 2+1 2,1/parenrightbigg satisfying : Zε 0:=/ba∇dbl̺ε 0−̺/ba∇dbll−, ε′ε−1 ˙Bd 2−1 2,1+/ba∇dbl̺ε 0−̺/ba∇dbll+, ε′ε−1, ε ˙Bd 2 2,1+ε/ba∇dblvε 0/ba∇dbll, ε−1 ˙Bd 2 2,1+ε/ba∇dbl(̺ε 0−̺,εvε 0)/ba∇dblh, ε−1 ˙Bd 2+1 2,1≤α, the system (1.1)with the initial data (̺ε 0,vε 0)admits a unique global-in-time solution Zε= (̺ε−¯ρ,vε)in the set ˜E:=/braceleftbigg (̺ε−̺,vε)/vextendsingle/vextendsingle/vextendsingle/vextendsingle(̺ε−̺)l−, ε′ε−1∈ Cb(R+:˙Bd 2−1 2,1), ε(̺ε−̺)l−, ε′ε−1∈L1(R+;˙Bd 2−1 2,1), (̺ε−̺)l+, ε′ε−1, ε−1∈ Cb(R+:˙Bd 2 2,1), ε(̺ε−̺)l+, ε′ε−1∈L1(R+;˙Bd 2+1 2,1),(̺ε−̺)l+, ε′ε−1, ε−1∈L1(R+;˙Bd 2+1 2,1), ε(vε)l, ε−1∈ Cb(R+;˙Bd 2 2,1),(vε)l−, ε′ε−1∈L1(R+;˙Bd 2 2,1),(vε)l+, ε′ε−1∈L1(R+;˙Bd 2+1 2,1), (̺ε−̺,vε)h,ε−1∈ Cb(R+;˙Bd 2−1 2,1)∩L1(R+;˙Bd 2+1 2,1), wε∈ Cb(R+;˙Bd 2 2,1)∩L1(R+;˙Bd 2 2,1)/bracerightbigg where we have denoted wε:=ε∇(P(̺ε)) ̺ε+vε+ε∇(−∆)−1(̺ε−̺). Moreover, we have the following inequality : Zε(t)≤CZε 0 where Zε(t):=/ba∇dbl̺ε−̺/ba∇dbll−, ε′ε−1 L∞/parenleftbigg ˙Bd 2−1 2,1/parenrightbigg+/ba∇dbl̺ε−̺/ba∇dbll+, ε′ε−1, ε−1 L∞/parenleftbigg ˙Bd 2 2,1/parenrightbigg+ε/ba∇dblvε/ba∇dbll, ε−1 L∞/parenleftbigg ˙Bd 2 2,1/parenrightbigg+ε/ba∇dbl(̺ε−̺,εvε)/ba∇dblh, ε−1 L∞/parenleftbigg ˙Bd 2+1 2,1/parenrightbigg +/ba∇dbl̺ε−̺/ba∇dbll−, ε′ε−1 L1/parenleftbigg ˙Bd 2−1 2,1/parenrightbigg+/ba∇dblvε/ba∇dbll−, ε′ε−1 L1/parenleftbigg ˙Bd 2 2,1/parenrightbigg+/ba∇dbl̺ε−̺/ba∇dbll+, ε L1/parenleftbigg ˙Bd 2+2 2,1/parenrightbigg+/ba∇dblvε/ba∇dbll+, ε L1/parenleftbigg ˙Bd 2+1 2,1/parenrightbigg +ε−1/ba∇dbl(̺ε−̺,εvε)/ba∇dblh,ε−1 L1/parenleftbigg ˙Bd 2+1 2,1/parenrightbigg+/ba∇dblwε/ba∇dbl L∞/parenleftbigg ˙Bd 2 2,1/parenrightbigg+ε−2/ba∇dblwε/ba∇dbl L1/parenleftbigg ˙Bd 2 2,1/parenrightbigg. We notice with theorem 4.1 ensures that ˜Wε=O(ε)inL1(R+;˙Bd 2 2,1). As the first equation of (1.3) can be rewritten as ∂t˜̺ε−∆(P(˜̺ε))−div/parenleftbig ˜̺ε∇(−∆)−1(˜̺ε−̺)/parenrightbig = div(˜̺ε˜Wε), we suspect that the density will tend to satisfy the paraboli c-elliptic Keller-Segel system (1.5) supplemented with the initial data lim ε→0˜̺ε 0. Let us now rigorously prove the theorem 2.2 : Proof. Let us justify quickly that for all N0satisfying (2.4), there exists a unique global-in-time sol utionN of (1.5) in Cb/parenleftbigg R+;˙Bd 2−1 2,1∩˙Bd 2 2,1/parenrightbigg ∩L1/parenleftbigg R+;˙Bd 2+2 2,1∩˙Bd 2−1 2,1/parenrightbigg satisfying (2.5). In terms of ˜N:=N−̺,the equation (1.5) is rewritten : ∂t˜N−∆(P(N))−div/parenleftBig N∇(−∆)−1˜N/parenrightBig = 0. By the Taylor-Lagrange formula, we notice that : P(N)−P(̺) =˜NP′(̺)+g(˜N) wheregis a smooth function vanishing at 0(and also its first derivative). (1) Low frequency analysis We can rewrite this system as : ∂t˜N+̺˜N= ∆/parenleftbig˜NP′(̺)+g(˜N)/parenrightbig +div/parenleftBig ˜N∇(−∆)−1˜N/parenrightBig ·PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 23 We then obtain the following estimates: /ba∇dbl˜N(t)/ba∇dbll ˙Bd 2−1 2,1+/integraldisplayt 0/ba∇dbl˜N/ba∇dbll ˙Bd 2−1 2,1dτ/lessorsimilar/ba∇dbl˜N0/ba∇dbll ˙Bd 2−1 2,1+/integraldisplayt 0/ba∇dblg(˜N)/ba∇dbl˙Bd 2+1 2,1dτ+/integraldisplayt 0/ba∇dbl˜N/ba∇dbll ˙Bd 2+1 2,1dτ+/integraldisplayt 0/ba∇dbl˜N/ba∇dbl˙Bd 2 2,1/ba∇dbl˜N/ba∇dbl˙Bd 2−1 2,1dτ, We note that the term/integraldisplayt 0/ba∇dbl˜N/ba∇dbll ˙Bd 2+1 2,1dτis negligible compared to/integraldisplayt 0/ba∇dbl˜N/ba∇dbll ˙Bd 2−1 2,1dτ. We deduce that : /ba∇dbl˜N(t)/ba∇dbll ˙Bd 2−1 2,1+/integraldisplayt 0/ba∇dbl˜N/ba∇dbll ˙Bd 2−1 2,1dτ/lessorsimilar/ba∇dbl˜N0/ba∇dbll ˙Bd 2−1 2,1+/integraldisplayt 0/ba∇dbl˜N/ba∇dbl˙Bd 2−1 2,1∩˙Bd 2 2,1/ba∇dbl˜N/ba∇dbl˙Bd 2+2 2,1∩˙Bd 2−1 2,1dτ. (2) High frequency analysis We can rewrite the system as : ∂t˜N−P′(̺)∆˜N= ∆(g(˜N))−div(N∇(−∆)−1˜N). Then, we get : /ba∇dbl˜N(t)/ba∇dblh ˙Bd 2 2,1+/integraldisplayt 0/ba∇dbl˜N/ba∇dblh ˙Bd 2+2 2,1dτ/lessorsimilar/ba∇dbl˜N0/ba∇dblh ˙Bd 2 2,1+/integraldisplayt 0/ba∇dbl˜N/ba∇dbl˙Bd 2+2 2,1/ba∇dbl˜N/ba∇dbl˙Bd 2 2,1dτ+/integraldisplayt 0/ba∇dbl˜N/ba∇dblh ˙Bd 2 2,1dτ +/integraldisplayt 0/parenleftBigg /ba∇dbl˜N/ba∇dbl˙Bd 2+1 2,1/ba∇dbl˜N/ba∇dbl˙Bd 2−1 2,1+/ba∇dbl˜N/ba∇dbl2 ˙Bd 2 2,1/parenrightBigg dτ. Thus, we have : /ba∇dbl˜N(t)/ba∇dblh ˙Bd 2 2,1+/integraldisplayt 0/ba∇dbl˜N/ba∇dblh ˙Bd 2+2 2,1dτ/lessorsimilar/ba∇dbl˜N0/ba∇dblh ˙Bd 2 2,1+/integraldisplayt 0/parenleftBigg /ba∇dbl˜N/ba∇dbl˙Bd 2+2 2,1/ba∇dbl˜N/ba∇dbl˙Bd 2 2,1+/ba∇dbl˜N/ba∇dbl˙Bd 2−1 2,1/ba∇dbl˜N/ba∇dbl˙Bd 2+1 2,1/parenrightBigg dτ. (3) A priori estimate By gathering the previous information, we get: /ba∇dbl˜N(t)/ba∇dbl˙Bd 2−1 2,1∩˙Bd 2 2,1+/integraldisplayt 0/ba∇dbl˜N/ba∇dbl˙Bd 2+2 2,1∩˙Bd 2−1 2,1dτ/lessorsimilar/ba∇dbl˜N0/ba∇dbl˙Bd 2−1 2,1∩˙Bd 2 2,1+/integraldisplayt 0/ba∇dbl˜N/ba∇dbl˙Bd 2−1 2,1∩˙Bd 2 2,1/ba∇dbl˜N/ba∇dbl˙Bd 2+2 2,1∩˙Bd 2−1 2,1dτ. Then, we have: /ba∇dbl˜N(t)/ba∇dbl˙Bd 2−1 2,1∩˙Bd 2 2,1+/integraldisplayt 0/ba∇dbl˜N/ba∇dbl˙Bd 2+2 2,1∩˙Bd 2−1 2,1dτ/lessorsimilar/ba∇dbl˜N0/ba∇dbl˙Bd 2−1 2,1∩˙Bd 2 2,1. Hence (2.5). By taking advantage of the Picard fixed point theorem in the fun ctional framework given by the inequalities above, we obtain an unique global-in-time sol ution˜Nof (1.5) in Cb/parenleftBig R+;˙Bd 2−1 2,1∩˙Bd 2 2,1/parenrightBig ∩ L1/parenleftBig R+;˙Bd 2+2 2,1∩˙Bd 2−1 2,1/parenrightBig satisfying (2.5). In order to prove the last part of this theorem, let us observe that(N,˜̺ε)satisfies : /braceleftBigg ∂t˜̺ε−∆(P(˜̺ε))−div/parenleftbig ˜̺ε∇(−∆)−1(˜̺ε−̺/parenrightbig = div/parenleftBig ˜̺ε˜Wε/parenrightBig , ∂tN−∆(P(N))−div/parenleftbig N∇(−∆)−1(N−̺)/parenrightbig = 0. Let us denote δN=N−˜̺ε. We obtain : ∂tδN+∆(P(˜̺ε)−∆(P(N))+̺ δN−div/parenleftbig δN∇(−∆)−1(N−̺)/parenrightbig −div/parenleftbig (˜̺ε−̺)∇(−∆)−1δN/parenrightbig =−div/parenleftBig ˜̺ε˜Wε/parenrightBig · Let us look estimate δNat the level of regularity ˙Bd 2−1 2,1. We have : /integraldisplayt 0/ba∇dbldiv(˜̺ε˜Wε)/ba∇dbl˙Bd 2−1 2,1dτ/lessorsimilar/integraldisplayt 0/ba∇dbl˜̺ε˜Wε/ba∇dbl˙Bd 2 2,1dτ/lessorsimilar/ba∇dbl˜̺ε/ba∇dbl L∞ t/parenleftbigg ˙Bd 2 2,1/parenrightbigg/integraldisplayt 0/ba∇dbl˜Wε/ba∇dbl˙Bd 2 2,1dτ/lessorsimilarαε,24 VALENTIN LEMARIÉ /integraldisplayt 0/ba∇dbldiv/parenleftbig δN∇(−∆)−1(N−̺/parenrightbig /ba∇dbl˙Bd 2−1 2,1dτ/lessorsimilar/integraldisplayt 0/ba∇dblδN∇(−∆)−1(N−̺)/ba∇dbl˙Bd 2 2,1dτ /lessorsimilar/integraldisplayt 0/ba∇dblδN/ba∇dbl˙Bd 2 2,1/ba∇dbl(N−̺)/ba∇dbl˙Bd 2−1 2,1dτ /lessorsimilarα/integraldisplayt 0/ba∇dblδN/ba∇dbl˙Bd 2 2,1dτ, /integraldisplayt 0/ba∇dbldiv/parenleftbig (˜̺ε−̺)∇(−∆)−1δN/parenrightbig /ba∇dbl˙Bd 2−1 2,1dτ/lessorsimilar/integraldisplayt 0/ba∇dbl(˜̺ε−̺)∇(−∆)−1δN/ba∇dbl˙Bd 2 2,1dτ /lessorsimilar/ba∇dbl˜̺ε−̺/ba∇dbl L∞(˙Bd 2 2,1)/integraldisplayt 0/ba∇dblδN/ba∇dbl˙Bd 2−1 2,1dτ /lessorsimilarα/integraldisplayt 0/ba∇dblδN/ba∇dbl˙Bd 2−1 2,1dτ. In order to study ∆(P(˜̺ε)−P(N)), we use the identity : ∆(P(φ)) = ∆φP′(φ)+|∇φ|2P′′(φ). Hence, we have : ∆(P(˜̺ε)−P(N)) =P′(̺)∆(δN)+(P′(N)−P′(̺))∆(δN) +∆˜̺ε/parenleftbig P′(N)−P′(˜̺ε)/parenrightbig +/parenleftbig |∇N|2−|∇˜̺ε|2/parenrightbig P′′(N)+|∇˜̺ε|2/parenleftbig P′′(N)−P′′(˜̺ε)/parenrightbig · Let bound the r.h.s in L1(R+;˙Bd 2−1 2,1)(in particular, the estimate of Theorem 4.1 will be used but a lso the different lemmas in the appendix) : /integraldisplayt 0/ba∇dbl∆(δN)/parenleftbig P′(N)−P′(̺)/parenrightbig /ba∇dbl˙Bd 2−1 2,1dτ/lessorsimilar/integraldisplayt 0/ba∇dblδN/ba∇dbl˙Bd 2+1 2,1/ba∇dblN−̺/ba∇dbl˙Bd 2 2,1dτ/lessorsimilarα/integraldisplayt 0/ba∇dblδN/ba∇dbl˙Bd 2+1 2,1dτ, /integraldisplayt 0/ba∇dbl∆˜̺ε/parenleftbig P′(N)−P′(˜̺ε)/parenrightbig /ba∇dbl˙Bd 2−1 2,1dτ/lessorsimilar/integraldisplayt 0/ba∇dbl˜̺ε−̺/ba∇dbl˙Bd 2+1 2,1/ba∇dblδN/ba∇dbl˙Bd 2 2,1/ba∇dbl(N,˜̺ε)/ba∇dbl˙Bd 2 2,1dτ /lessorsimilar(α2+α)/ba∇dblδN/ba∇dbl L∞(˙Bd 2 2,1), /integraldisplayt 0/ba∇dbl|∇˜̺ε|2/parenleftbig P′′(N)−P′′(̺)/parenrightbig /ba∇dbl˙Bd 2−1 2,1dτ/lessorsimilar/integraldisplayt 0/ba∇dbl˜̺ε/ba∇dbl˙Bd 2+1 2,1/ba∇dbl˜̺ε/ba∇dbl˙Bd 2 2,1/ba∇dblδN/ba∇dbl˙Bd 2 2,1/ba∇dbl(˜N,˜̺ε)/ba∇dbl˙Bd 2 2,1dτ /lessorsimilar(α3+α2)/integraldisplayt 0/ba∇dblδN/ba∇dbl˙Bd 2 2,1dτ, /integraldisplayt 0/ba∇dbl/parenleftbig |∇N|2−|∇˜̺ε|2/parenrightbig P′′(N)/ba∇dbl˙Bd 2−1 2,1dτ/lessorsimilar/integraldisplayt 0/ba∇dbl∇N−∇˜̺ε/ba∇dbl˙Bd 2 2,1/ba∇dbl(∇N,∇˜̺ε)/ba∇dbl˙Bd 2−1 2,1/ba∇dblN/ba∇dbl˙Bd 2 2,1dτ /lessorsimilar(α2+α)/integraldisplayt 0/ba∇dblδN/ba∇dbl˙Bd 2+1 2,1dτ. In particular, δNsatisfies : ∂tδN−P′(̺)∆δN+̺δN=−div/parenleftBig ˜̺ε˜Wε/parenrightBig +div/parenleftbig δN∇(−∆)−1(N−̺/parenrightbig +div/parenleftbig (˜̺ε−̺)∇(−∆)−1δN/parenrightbig −∆(δN)(P′(N)−P′(̺))−∆˜̺ε/parenleftbig P′(N)−P′(˜̺ε)/parenrightbig −|∇˜̺ε|2/parenleftbig P′′(N)−P′′(̺)/parenrightbig −/parenleftbig |∇N|2−|∇˜̺ε|2/parenrightbig P′′(N).PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 25 By using all previous inequalities, we get : /ba∇dblδN/ba∇dbl L∞ t(R+;˙Bd 2−1 2,1)∩L1(R+;˙Bd 2−1 2,1)∩L1(R+;˙Bd 2+1 2,1)/lessorsimilar/ba∇dblδN(0)/ba∇dbl˙Bd 2−2 2,1+αε +(α+α2+α3)/ba∇dblδN/ba∇dbl L∞ t(R+;˙Bd 2−1 2,1)∩L1(R+;˙Bd 2−1 2,1)∩L1(R+;˙Bd 2+1 2,1). Therefore, for αsmall enough, we get : /ba∇dblδN/ba∇dbl L∞(R+;˙Bd 2−1 2,1)∩L1(R+;˙Bd 2−1 2,1∩˙Bd 2+1 2,1)/lessorsimilar/ba∇dblδN(0)/ba∇dbl˙Bd 2−1 2,1+αε which completes the proof of the theorem. /square Appendix A. Here we recall classic lemmas involving differential inequa lities and some basic properties on Besov spaces and product estimates have been be used repeatedly in the art icle. Lemma A.1. LetX: [0,T]→R+be a continuous function such that X2is differentiable. Assume that there exist a constant c≥0and a measurable function A: [0,T]→R+such that 1 2d dtX2+cX2≤AXa.e. on[0,T]. Then, for all t∈[0,T], we have: X(t)+c/integraldisplayt 0X(τ)dτ≤X0+/integraldisplayt 0A(τ)dτ. This classical lemma can be found for instance in [16] : Lemma A.2. LetT >0. LetL: [0,T]→RandH: [0,T]→Rtwo continous positive functions on [0,T] such that L(t)+c/integraldisplayt 0H(τ)dτ≤ L(0)+C/integraldisplayt 0L(τ)H(τ)dτ, andL(0)≤α <<1then for all t∈[0,T], we have : L(t)+c 2/integraldisplayt 0H(τ)dτ≤ L(0). Proof. Letα∈]0,c 2C[. We set T0= sup T1∈[0,T]/braceleftBigg sup t∈[0,T1]L(t)≤α/bracerightBigg . Thissupexists because the previous set is non empty ( 0belongs to this set) and since Lis continuous, T0>0. In time t=T0, we get : L(T0)+c/integraldisplayT0 0H(τ)dτ≤ L(0)+C/integraldisplayT0 0H(τ)L(τ)dτ≤ L(0)+αC/integraldisplayT0 0H(τ)dτ. Hence we have : L(T0)+c 2/integraldisplayT0 0H(τ)dτ≤ L(0). AsL(t)≤ L(T0)for allt∈[0,T0]and/integraldisplayt 0H(τ)dτ≤/integraldisplayT0 0H(τ)dτ, we obtain with the previous inequality : L(t)≤α∀t∈[0,T0]. With the continuity of L, we must have T0=T, whence the result. /square The following lemmas are classic results on Besov spaces ( see e.g. [4]).26 VALENTIN LEMARIÉ Lemma A.3. For alld≥2, the pointwise product extends in a continuous application of˙Bd 2−1 2,1(Rd)×˙Bd 2 2,1(Rd) to˙Bd 2−1 2,1(Rd)and˙Bd 2 2,1is a multiplicative algebra for all d≥1. For alld≥1, we have for (u,v)∈˙Bd 2 2,1∩˙Bd 2+1 2,1thatuv∈˙Bd 2+1 2,1and the following inequality : /ba∇dbluv/ba∇dbl˙Bd 2+1 2,1/lessorsimilar/ba∇dblu/ba∇dbl˙Bd 2 2,1/ba∇dblv/ba∇dbl˙Bd 2+1 2,1+/ba∇dblu/ba∇dbl˙Bd 2+1 2,1/ba∇dblv/ba∇dbl˙Bd 2 2,1. The following lemma comes from the proof in [4] of the followi ng well-known property on Besov spaces : /ba∇dblz(α·)/ba∇dbl˙Bs 2,1≃αs−d 2/ba∇dblz/ba∇dbl˙Bs 2,1for allα >0. Lemma A.4. Lets∈Randz∈˙Bs 2,1.We have that : /ba∇dblz(α·)/ba∇dbll−,ε ˙Bs 2,1≃αs−d 2/ba∇dblz/ba∇dbll−,εα−1 ˙Bs 2,1;/ba∇dblz(α·)/ba∇dbll+,ε ˙Bs 2,1≃αs−d 2/ba∇dblz/ba∇dbll+,εα−1, α−1 ˙Bs 2,1;/ba∇dblz(α·)/ba∇dblh ˙Bs 2,1≃αs−d 2/ba∇dblz/ba∇dblh, εα−1 ˙Bs 2,1 where we denote /ba∇dbl·/ba∇dbll+, α, β ˙Bs 2,1:=/summationdisplay j∈Z α≤2j≤β 42js/ba∇dbl·/ba∇dblL2and/ba∇dbl·/ba∇dblh,α ˙Bs 2,1:=/summationdisplay j≥−2 2j≥α2js/ba∇dbl·/ba∇dblL2. Lemma A.5. LetF:Rd→Rpa smooth function with F(0) = 0 . Then for all (p,r)∈[1,∞]2,s >0and u∈˙Bs p,r∩L∞, we have F(u)∈˙Bs p,r∩L∞and /ba∇dblF(u)/ba∇dbl˙Bsp,r≤C/ba∇dblu/ba∇dbl˙Bsp,r, withCdepending only on /ba∇dblu/ba∇dblL∞,F′(and higher order derivatives), s,pandd. Lemma A.6. Letg∈C∞(R)such that g′(0) = 0 . Then, for all u,v∈˙Bs 2,1∩L∞withs >0, /ba∇dblg(v)−g(u)/ba∇dbl˙Bs 2,1≤C/parenleftBig /ba∇dblv−u/ba∇dblL∞/ba∇dbl(u,v)/ba∇dbl˙Bs 2,1+/ba∇dblv−u/ba∇dbl˙Bs 2,1/ba∇dbl(u,v)/ba∇dblL∞/parenrightBig · References [1] G. Ali: Global existence of smooth solutions of the N-dim ensional Euler-Poisson model, SIAM J. Math. Anal., 35 (2003 ) 389–422. [2] G. Ali, L. Chen, A. Jüngel and Y. J. Peng: The zero-electro n-mass limit in the hydrodynamic model for plasmas, Nonline ar Anal., 72 (2010) 4415–4427. [3] G. Ali and A. Jüngel: Global smooth solutions to the multi -dimensional hydrodynamic model for two-carrier plasma, J . Diff. Equations, 190 (2003) 663–685. [4] H. Bahouri, J.-Y. Chemin and R. Danchin: Fourier Analysis and Nonlinear Partial Differential Equati ons,Grundlehren der mathematischen Wissenschaften, 343, Springer, 2011. [5] K. Beauchard and E. Zuazua: Large time asymptotics for pa rtially dissipative hyperbolic systems, Arch. Rational Mech. Anal,199, 177–227, 2011. [6] S. Benzoni-Gavage and D. Serre: Multi-dimensional Hyperbolic Partial Differential Equati ons : First-order Systems and Applications. Oxford Science Publications, New-York, 2007. [7] P. Carmeliet: Mechanisms of angiogenesis and arterioge nesis, Nature Medicine 6 (2000) 389-395. [8] Condeelis, R. H. Singer and J. E. Segall: The great escape : When cancer cells hijack the genes for chemotaxis and motil ity, Ann. Rev. Cell Dev. Biol. 21 (2005): 695-718. [9] S. Cordier and E. Grenier: Quasineutral limit of an Euler -Poisson system arising from plasma physics, Comm. Part. Di ff. Equations, 25 (2000) 1099–1113. [10] T. Crin-Barat and R. Danchin: Partially dissipative on e-dimensional hyperbolic systems in the critical regulari ty setting, and applications, Pure and Applied Mathematics ,4(1), 85-–125, (2022). [11] T. Crin-Barat and R. 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Danchin: Partially dissipative hyperbolic systems in the critical r egularity setting ,EMS Survey , EMS Surv. Math. Sci. 9, no. 1, 135–192 (2022). [17] R. Danchin and J. Xu: Optimal time-decay estimates for t he compressible Navier-Stokes equations in the critical Lp framework, Arch. Rational Mech. Anal. ,224, 53–90, 2017. [18] D. Y. Fang, J. Xu and T. Zhang: Global exponential stabil ity of classical solutions to the hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci. 17 (2007) 1 507–1530. [19] S. K. Godunov: An interesting class of quasi-linear sys tems,Dokl. Akad. Nauk SSSR ,139, 521–523, (1961) (Russian). [20] G. Helmlinger, M. Endo, N. Ferrara, L. Hlatky and R. Jain : Formation of endothelial cell networks, Nature 405 (2000) : 139-141. [21] D. Horstmann: From 1970 until present: the Keller–Sege l model in chemotaxis and its consequences I.Jahresbericht e DMV 105(3), 103–165 (2003) [22] S. Kawashima: Systems of a hyperbolic-parabolic composite type, with app lications to the equations of magneto- hydrodynamics. Thesis, Kyoto University, 1984. [23] S. Kawashima and W.-A. Yong: Dissipative structure and entropy for hyperbolic systems of balance laws, Arch. Rational Mech. Anal., 174, 345–364, 2004. [24] S. Kawashima and W.-A. Yong: Decay estimates for hyperb olic balance laws, Journal for Analysis and its Applications, 28, 1–33, 2009. [25] E. Keller, L. Segel, Initiation of slime mold aggregati on viewed as an instability, J. Theoret. Biol. 26 (1970) 399- 415. [26] E. Keller and L. Segel: Model for chemotaxis. Journal of theoretical biology, 30(2):225–234, 1971. [27] C. Lattanzio: On the 3-D bipolar isentropic Euler-Pois son model for semiconductors and the drift-diffusion limit, Math. Models Methods Appl. Sci., 10 (2000) 351–360. [28] C. Lattanzio and A. E. Tzavaras: From gas dynamics with l arge friction to gradient flows describing diffusion theorie s, Communications in Partial Differential Equations 42 (2017) 261-290. [29] P.G. Lemarié-Rieusset: Small data in an optimal Banach space for the parabolic-parabolic and parabolic-elliptic Keller- Segel equations in the whole space. Adv. Differential Equati ons, 18:1189–1208, 2013. [30] S. G. Li and K. Muneoka: Cell migration and chick limb dev elopment: chemotactic action of FGF-4 and the AER, Dev. Biol. 211 (1999) 335-347. [31] Y.P.Li and X.F.Yang: Global existence and asymptotic b ehavior of the solutions to the three-dimensional bipolar E uler- Poisson systems, J. Diff. Equations, 252 (2012) 768–791. [32] J. D. Murray: Mathematical biology, II: spatial models and biomedical application, vol. 18 of Interdisciplinary A pplied Mathematics, Springer-Verlag, New York, third ed., 2003. [33] Y.-J. Peng and V. 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2023-07-12

We study on the whole space R d the compressible Euler system with damping coupled to the Poisson equation when the damping coefficient tends towards infinity. We first prove a result of global existence for the Euler-Poisson system in the case where the damping is large enough, then, in a second step, we rigorously justify the passage to the limit to the parabolic-elliptic Keller-Segel after performing a diffusive rescaling, and get an explicit convergence rate. The overall study is carried out in 'critical' Besov spaces, in the spirit of the recent survey [16] by R. Danchin devoted to partially dissipative systems.

Parabolic-elliptic Keller-Segel's system

2307.05981v1

1 Determination of intrinsic damping of perpendicularly magnetized ultrathin films from time resolved precessional magnetization measurements Amir Capua1,*, See -hun Yang1, Timothy Phung1, Stuart S. P. Parkin1,2 1 IBM Research Division, Almaden Research Center, 650 Harry Rd., San Jose, California 95120, USA 2 Max Planck Institute for Microstructure Physics, Halle (Saale), D -06120, Germany *e-mail: acapua@us.ibm.com PACS number(s) : 75.78. -n Abstract: Magnetization dynamics are strongly influenced by damping, namely the loss of spin angular momentum from the magnetic system to the lattice. An “effective” damping constant αeff is often determined experimentally from the spectral linewidth of the free induction decay of the magnetization after the system is excited to its non -equilibrium state . Such an αeff, however, reflects both intrinsic damping as well as inhomogeneous broadening that arises , for example, from spatial variations of the anisotropy field. In this paper we compare measurements of the m agnetization dynamics in ultrathin non -epitaxial films having perpendicular magnetic anisotropy using two different techniques, time- resolved magneto optical Kerr effect (TRMOKE ) and hybrid optical -electrical ferromagnetic resonance (OFMR) . By using a n external magnetic field that is applied at very small angles to the film plane in the TRMOKE studies , we develop an explicit closed - form analytical expression for the TRMOKE spectral linewidth and show how this can be used to reliably extract the intrinsic Gilbert damping constant. The damping constant determined in this way is in exc ellent agreement with that determined from the OFMR method on the same samples. Our studies indicate that the asymptotic high -field approach that is often used in the TRMOKE method to distinguish the intrinsic damping from the 2 effective damping may result in significant error , because such high external magnetic fields are required to make this approach valid that they are out of reach . The error becomes larger the lower is the intrinsic damping constant, and thus may account for the anomalously high damping constants that are often reported in TRMOKE studies . In conventional ferromagnetic resonance ( FMR ) studies , inhomogeneous contributions can be readily distinguished from intrinsic damping contributions from the magnetic field dependence of the FMR linewidth. Using the analogous approach, w e show how reliable values of the intrinsic damping can be extracted from TRMOKE in two distinct magnetic systems with significant perpendicular magnetic anisotropy: ultrathin CoFeB layers and Co/Ni/Co trilayers. 3 I. Introduction Spintronic nano -devices have been identified in recent years as one of the most promisin g emerging technologies for future low power microelectronic circuits1, 2. In the heart of the dynamical spin -state transi tion stands the energy loss parameter of the Gilbert damping . Its accurate dete rmination is of paramount importance as it determines the performance of key building blocks required for spin manipulation such as t he switching current threshold of the spin transfer torq ue magnetic tunnel junction (MTJ) used in magnetic random access memory (MRAM) as w ell as the skyrmion velocities and the domain wall motion current threshold . Up-scaling for high logic and data capacities while obtaining stability with high retention energies require in addition that large magnetic anisotropies be ind uced. T hese cannot be achieved simply by engineering the geometrical asymmetries in the nanometer -scale range , but rather require harnessing the induced spin- orbit interaction a t the interface of the ferromagnet ic film to obtain perpendicular magnetic anisotropy (PMA)2. Hence an increasing effort is invested in the quest for perpendicular magnetized materials having large anisotropies with low Gilbert damping3-11. Two distinct families of experimental methods are typically used for measurement of Gilbert damping , namely, time-resolved pump -probe and continuous microwave stimulated ferromagnetic resonance ( FMR ), either of which can be implemented using optical and/or electrical methods . While in some cases good agreement between these distinct techniques have been reported12, 13, there is often significant disagreement between the methods14, 15. 4 When the time resolved pump -probe method is implemented using the magneto optical Kerr effect ( TRMOKE), a clear advantage over the FMR method is gained in the ability to operate at very high fields and frequencies16, 17. On the other hand , the FMR method a llows operation over a wide r range of geometrical configurations . The fundamental geometrical restriction of the TRMOKE comes from the fact that the magnetization precession s are initiated from the perturbation of the effective anisotropy field by the pump pulse , by momentarily increasing the lattice temperature18, 19. In cases where the torque exerted by the effective anisotropy field is n egligible , the pump pulse cannot sufficiently perturb the magnetization . Such a case occurs for example whenever the magnetization lays in the plane of the sample in uniaxial thin films having perpendicular magneti c anisotropy . Similar limitations exist if the magnetic field i s applied perpendicular to the film . Hence in TRMOKE experiments , the external field is usually applied at angles typically not smaller than about 10 from either the film plane or its normal. This fact has however the consequence that the steady state magnetization orientation , determined by the bal ancing condition for the torques , cannot be described using a n explicit -form algebraic expression , but rather a numerical approach should be taken5. Alternatively , the dynamics can be described using an effective damping from which the intrinsic damping , or at least an upper bound o f its value , is estimated at the high magnetic field limit with the limit being undetermined . These approaches are hence less intuitive while the latter does not indicate directly on the energy losses but rather on the combination of the energy loss rate , coherence time of the spin ensemble and geometry of the measurement . 5 In this paper , we present an approach where the TRMOKE system is operated while applying the magnetic field at very sm all angles with respect to the sample plane. This enables us to use explicit closed -form analytical expressions derived for a perfectly in - plane external magnetic field as an approximate solution. Hence , extraction of the intrinsic Gilbert damping using an analytical model becomes possible without the need to drive the system to the high magnetic field limit providing at the same time an intuitive understanding of the measured responses. The validity of t he method is verified using a highly sensitive hybrid optical -electrical FMR system (OFMR) capable of operating with a perfectly in -plane magnetic field where the analytical expressions hold. In particular, we bring to test the high -field asymptotic approa ch used for evaluation of the intrinsic damping from the effective damping and show that in order for it to truly indicate the intrinsic damping, extremely high fields need to be applied. Our analysis reveals the resonance frequency dispersion relation as well as the inhomogeneous broadening to be the source of this requirement which becomes more difficult to fulfill the smaller the intrinsic damping is. The presented method is applied on two distinct families of technologically relevant perpendicularly mag netized systems; CoFeB4, 6 and Co/Ni/Co20-23. Interestingly, the results indicate that the Ta seed layer thickness used in CoFeB films strongly affects the intrinsic damping , while t he static characteristics of the films remain intact . In the Co/Ni/Co trilayer system which has in contrast a large effective anisotropy field, unexpected ly large spectral linewidth s are measured when the external magnetic field is comparable to the effective anisotropy field, which cannot be explained by the conventional model of no n-interacting spins describing the inhomogeneous broadening . This suggest s that under the low stiffness 6 conditions associated with such bias fi elds, cooperative exchange interactions, as two magnon scattering, become relevant8, 24. II. EXPERIMENT The experiments present ed were carried out on three PMA samples: two samples consist ed of Co36Fe44B20 which differed by the thickness of the underlayer and a third sample consisting of Co/Ni/Co trilayer . The CoFeB samples were characterized by low effective anisotropy (Hkeff) values as well as by small distribution of its value in contrast to the Co /Ni/Co trilayer system . We define here Hkeff as 2Ku/Ms-4πMs where Ku is the anisotropy energy constant and Ms being the saturation magnetization. The structure s of the two CoFeB samples were 50Ta|11CoFeB |11MgO |30Ta, and 100Ta|11CoFeB |11MgO |30Ta (units are in Å) and had similar Ms value s of 1200 emu/cc and Hkeff of 1400 Oe and 1350 Oe respectively. The t hird system studied was 100AlO x|20TaN |15Pt|8Pt 75Bi25|3Co|7Ni|1.5Co |50TaN with Ms of 600 emu/cc and Hkeff value of about 4200 Oe . All samples were grown on oxidized Si substrates using DC magnetron sputtering and exhibited sharp perpendicular switching characteristics . The samples consisting of CoFeB were annealed for 30 min at 275 C in contrast to the Co/Ni/Co which was measured as deposited. Since the resultant film has a polycrystalline texture , the in -plane anisotropy is averaged out and the films are regarded as uniaxial crystals with the symmetry axis being perpendicular to the film plane. 7 The t wo configurations of the experimental setup were driven by a Ti:Sapphire laser emitting 70 fs pulses at 800 nm having energy of 6 nJ. In the first configuration a standard polar pump -probe TRMOKE was implemented with the probe pulse being a ttenuated by 15 dB compared to the pump pulse. Both beams were focused on the sample to an estimated spot size of 10.5 m defined by the full width at half maximum (FWHM) . In the hybrid optical -electrical OFMR system , the Ti:Sapphire laser served to pro be the magnetization state via the magneto -optical Kerr effect after being attenuated to pulse energies of about 200 pJ and was phase -locked with a microwave oscillator in a similar configuration to the one reported in Ref. [ 25]. For this measurement , the film was patterned into a 20 m x 20 m square island with a Au wire deposited in proximity to it, which was driven by the microwave signal. Prior to reaching the sample, the probing laser beam traversed the optical delay line that enabled mapping of the time axis and in particular the out of plane - mz component of the magnetization as in the polar TRMOKE experiment . With this configuration the OFMR realizes a conventional FMR system where the magnetization state is read in the time-domain using the magneto optical Kerr effect and hence its high sensitivity . The OFMR system therefore enables operation even when the external field is applied fully in the sample plane. III. RESULTS AND DISCUSSION A. TRMOKE measurements on 50 Å-Ta CoFeB film The first experiments we present were performed on the 50 Å-Ta CoFeB system which is similar to the one studied in Ref. [4]. The TRMOKE measurement was carried 8 out at two angles of applied magnetic field, H, of 4 and 1 measured from the surface plane as indicated in Fig. 1. We de fine here in addition the comple mentary angle measured from the surface normal, 2HH . Having i ts origin in the effective anisotropy, the torque generated by the optical pump is proportional to cos( )sins keffMH with θ being the angle of the magnetization relative to the normal of the sample plane. Hence, f or 1H , the angle θ becomes close to /2 , and the resultant torque generated by the optical pump is not strong enough to initiate reasonable precessions . For the same reason, the maximum field measureable for the 1H case is significantly lower than for the 4H case. This is clearly demonstrated in the m easured MOKE signals for the two H angles in Fig. 2 (a). While for 4H the precessional motion is clearly seen even at a bias field of 12 kOe, with 1H the precessions are hardly observable already at a bias field of 5.5 kOe. Additionally, it is also possible that the lower signal to noise ratio observed for 1H may be due to a breakdown into domains with the almost in -plane applied magnetic field26. After reduction of the background signal, the measured data can be fitted to a decaying sinusoidal response from which t he frequency and decay time can be extracted in the usual manner 6 (Fig. 2(b)) . The measured precession frequency as a function of the applied external field , H0, is plotted in Fig. 3(a). Significant differences near Hkeff are observed for merely a change of three degrees in the angle of the applied magnetic field . In particular, the trace for 1H exhibits a minimum point at approximately Hkeff in contrast to the monotonic behavior of the 4H case. The theoretical dependence of the resonance 9 frequency on the magnetic bias field expressed in normalized units, /keffH , with being the resonance angular frequency and the gyromagnetic ratio, is presented in Fig. 3(b) for several representative angles of the applied field. The resonance frequency at the vicinity of Hkeff is very sensitive to slight changes in the angle of the applied field as observed also in the experiment . Actually the derivative of the resonance frequency with respect to the applied field at the vicinity of Hkeff is even more sensitive where it diverges for 90 but reaches a value of zero for the slightest angle divergence. A discrepancy between the measurement and the theoretical solution exists however. At field values much higher than Hkeff the precession frequenc y should be identical for all angles (Fig. 3(b)) but in practice the resonance frequency measured for H of 4 is consistently higher by nearly 2 GHz than at 1 . The t heory also predicts that for the case of 4 , the resonance frequencies should exhibit a minimum point as well which is not observed in the measurement . The origin of the difference is not clear and may be related to the inhomogeneities in the local fields or to the higher orders of the interface induced anisotropy which were neglected in the theoretical calculation . In Fig. 3(c), we plot the effective Lorentzian resonance linewidth in the frequency domain , eff , defined by 2/eff eff with eff being the measured decay time extracted from the measured responses. Decompos ing the measured linewidth t o an intrinsic contribution that represent s the energy loss es upon precession and an extrinsic contribution which represent s the inhomogeneities in the local fields and is not related to energy loss of 10 the spin system , we express the linewidth as : int eff IH . int is given by the Smit -Suhl formula27, 28 and equals 2/ with denoting the intrinsic spin precession decay time where as IH represents the dispersion in the resonance frequencies due to the inhomogeneities. If the variations in the resonance frequency are assumed to be primarily caused by variations in the local effective anisotrop y field keffH , IH may be given by : /IH keff keff d dH H . For the case of /2H or 0H , eff has a closed mathematical form. In PMA films with bias field applied in the sample plane , the expression for eff becomes : 0 002 00 0 0022 002 for H 2 2 for Heff keff keff keff keff keff keff eff keff keff keffkeffHH H H H H H H HH HH H HHH HH , (1) with denot ing the Gilbert damping . The first term s in Eq. (1) stem from the intrinsic damping , while the second term s stem from the inhomogeneous broadening . Eq. (1) shows that while the contribution of the intrinsic part to the total spectral linewidth is finite, as the external field approaches Hkeff either from higher or lower field values, the inhomogeneous contribution diverges. Equation (1) further shows that for H0 >> Hkeff , the slope of eff becomes 2 with a constant offset given by /2keffH . Although Eq. (1) is valid only for /2H , it is still instructi ve to apply it on the measured linewidth for the 4H case. 11 The theoretical intrinsic linewidth for /2H , inhomogeneous contribution and the sum of the two a fter fitting and keffH in the range H 0 > 5000 Oe are plotted in Fig. 3(c). The resul tant fitting values were 0.023 ±0.002 for the Gilbert damping and 175 Oe for keffH . At external fields comparable to Hkeff the theoretical expression derived for the inhomogeneous broadening for a perfectly in -plane field does not describe properly the experiment . In the theoretical analysis , at fields comparable to Hkeff, the derivative 0/d dH diverges and therefore also the derivative /keff d dH as understood from Fig. 3(b). In the experiment however , /2H and the actual derivative /keff d dH approache s zero. Hence any variation in Hkeff result s in minor variation of the frequency . This mean s that the contribution of the inhomogeneous broadening to the total linewidth is suppressed near Hkeff in the experiment as opposed to being expanded in the theoretical calculation which was carried out for /2H . The result is an overestimate d theoretical linewidth near Hkeff. After reduction of the inhomogeneous broadening , the extracted intrinsic measured linewidth is presented in Fig. 3(c) as well showing the deviation from the theor etical intrinsic contribution as the field approaches Hkeff. To further investigate the e ffect of tilt ing the magnetic field , we study the TRMOKE responses for the 1H case. The measured linewidth for this case is presented in Fig. 3(d). In contrast to the 4H case, the measured linewidth now increases at fields near Hkeff as expected theoretically . Furthermore, the measured linewidth for the 1H case is 12 well describe d by Eq. (1) even in the vicinity of Hkeff as well as for bias fields smaller than Hkeff. The fitting result s in the same damping value of 0.023 ±0.0015 as with the 4H case, and a variation in keffH of 155 Oe, which is 20 Oe smaller than the value fitted for the 4H case. We next turn to examine the G ilbert damping. In the absence of the demagnetization and crystalline anisotrop y fields, the expression for the intrinsic Gilbert damping is given by: 1 . (2) Once the anisotropy and the demagnetization field s are included , the expression for the intrinsic Gilbert damping becomes : 0 0 0 0 001 for 21 for 2keff keff keff keffdHHHd dHHHd H H H H , (3) and is valid only for 2H and for crystals having uniaxial symmetry. At oth er angles a numerical method5 should be used to relate the precession decay time to the Gilbert damping. Eq. (3) is merely the intrinsic contribution in Eq. (1) written in the form resembling Eq. (2) . At high fields both Eq s. (2) and (3) converge to the same result since 13 1 0dH d. As seen in Fig. 3(b), at bias field s comparable to Hkeff the additional derivative term of Eq. (3) becomes very significant . When substituting the measured decay time, eff , for , Eq. (2) gives what is often interpreted as the “effective ” damping , αeff, from which the intrinsic damping is measured by evaluating it at high fields when the damping becomes asymptotically field independent. Additionally, t he asymptotic limit should be reached with respect to the inhomogeneous contribution of Eq. (1). In Fig. 3(e), we plot the effective damping using eff and Eq. (2) . We further show the intrinsic damping value after extracting the intrinsic linewidth and using Eq. (3). Examining first the effective damping values, we see that for the two angles , the values are distinctively different at low fields but converge at approximately 41 00 Oe (Beyond 5500 Oe the data for the 1H case could not be measured). In fact , the behavior of the effective damping seems to be related to the dependence of the resonance frequency on H0 (Fig. 3(a)) in which for the 1H case reaches an extremum while the 4H case exhibits a monotonic behavior . Since Eq. (2) lacks the derivative term 0/dH d , near Hkeff the effective damping is related to the Gilbert damping by the relation: 01 effd dH for H0 > Hkeff. Furthermore, since does not depend on the magnetic field to the first order, the dependence of the effective damping, eff , on the bias field stems from the derivative term 0 d dH which becomes larger and eventually diverges to infinity when the magnetic field reaches Hkeff as can be inferred from Fig. 3(b) for the case of 0H for which Eq. (3) was derived . Hence the increase in eff at bias 14 fields near Hkeff. The same considerations apply also for H0 < Hkeff. As the angle H increases , this analysis becomes valid only for bias fields which are large enough or small enough relative to Hkeff. When examined separately, each effective damping trace may give the impression that at the higher fields it has become bias field independent and reached its asymptotic value from which two very distinct Gilbert damping values of ~0.027 and ~0.039 are extracted at field values of 12 kOe and 5.5 kOe for the 4H and 1H measurements , respectively . These values are also rather different from the intrinsic damping value of 0.023 extracted using the analytical model . In contrast to the effective damping , the intrinsic damping obtained from the analytical model reveal s a constant and continuous behavior which is field and angle independent. The presumably negative values measure d for the 4H case stem of course from the fac t that the expressions in Eqs. (1) and (3) are derived for the 2H case. The error in using the effective damping in conjunction with the asymptotic approximation compared to using the analytical model is therefore 17% and 70% for the 4H and 1H measurements respectively. It is important in addition to understand th e conse quence of using Eq. (2) rathe r than Eq. (3) . In Fig. 3(f) we present the error in the damping value after accounting for the inhomogeneous broadening using Eq. (2) instead of the complete expression of Eq. (3) . As expected , the error increases as the applied field approaches Hkeff. For the measurement taken with 4H the error is significantly smaller due to the smaller value of the derivative 0/d dH . 15 As mentioned previously, i n order to evaluate the intrinsic damping from the total measured linewidth , the asymptotic limit should be reached with respect to the inhomogeneous broadening as well (Eq. (1) ). In Fig s. 3(c) and 3(d) we see that this is not the case where the contribution of the inhomogeneous linewidth is still large compared to the intrinsic l inewidth . Examining Figs. 3(d) and 3(f) for the case of 1H , we see that the overall error of 70% resulting in the asymptotic evaluation stems from both the contribution of inhomogeneous broadening as well as from the use of Eq. ( 2) rather than Eq. (3) while for 4H (Figs. 3(c) and 3(f)) the error of 17% is solely due to contribution of the inhomogeneous broadening which was not as negligible as conceived when applying the asymptotic approximation . B. Comparison of TRMOKE and OFMR measurements in 100 Å-Ta CoFeB film We next turn to study the magnetization dynamics using the OFMR system where the precession s are driven with the microwave signal . Hence, the external magnetic field can be applied perfectly in the sample plane. The 100 Å-Ta CoFeB sample was used for this experiment. Before patterning the film for the OFMR measurement, a TRMOKE measurement was carried out at 4H which exhibited a similar behavior to that observed with the sample having 50 Å Ta as a seeding layer . The dependence of the resonance frequency on the magnetic field as well as the measured linewidth and its different contributions are presented in Figs. 4(a) and 4(b). Before reduction of the inhomogeneous 16 broadening the asymptotic effective damping was measured to be ~0.0168 while after extraction of the intrinsic damping a value of 0.0109 ±0.0015 was measured marking a difference of 54% (Fig. 4(c)). The fitted keffH was 205 Oe. Fig. 4(b) shows that the origin of the error stems from significan t contribution of the inhomogeneous broadening compared to the intrinsi c contribution which plays a mor e significant role when the damping is low. By us ing the criteria for the minimum field that results in 10IH eff to estimat e the point where the asymptotic approximation would be valid , we arrive to a value of at least 4.6 T which is rather impractic al. The threshold of this minimal f ield is highly dependent on the damping so that for a lower damping an even higher field would be required. An example of a measured trace using the OFMR system at a low microwave frequency of 2.5 GHz is presented in Fig. 4(d). The square root of the magn etization amplitude (out of plane mz component) while preserving its sign is plotted to show detail . The high sensitivity of the OFMR system enable s operation at very low frequencies and bias fields. For every frequency and DC magnetic field value , several cycles of the magnetization precession were recorded by scanning the optical delay line. The magnetic field was then swept to fully capture the resonance . The trace should be examined separately in two sections, be low Hkeff and above Hkeff (marked in the figure by black dashed line). For frequencies of up to keffH two resonances are crossed as indicated by the guiding red dashed line which represents the out -of-phase component of the magnetization, namely the imaginary part of the magnetic susceptibility . Hence the cross section along this line 17 gives the field dependent absorption spectrum from which the resonance frequency and linewi dth can be identified. This spectrum is show n in Fig. 4( e) together with the fitted lorentzian lineshapes for bias fields below and above Hkeff. The resultant resonance frequencies of all measurements are plotted in addition in Fig. 4(a). The resonance linewidth s extracted for bias fields larger than Hkeff, are presented in Fig. 4(f). Here the effective magnetic field linewidth , ΔHeff, that includes the contribution of the inhomogeneous broadening derived from the same principles that led to Eq. (1) with /2H is given by: 02 2 0 00 2 2 0211 for 2 4 2 with keff eff keff keff keff keff keff eff keff keff keffHH H H H H HH HHHH H H HH 0 for keff HH (5). The second terms in Eq. (5) denote the contribution of the inhomogeneous broadening , IHH , and are frequency dependent as opposed to the case where the field is applied out of the sample plane9. The dispersion in the effective anisotropy , keffH , and the intrinsic Gilbert damping were found by fitting the linewidth in the seemingly linear range at frequencies larger th an 7.5 GHz . The contribution s of the intrinsic and inhomogeneous parts and the ir sum are presented as well in Fig. 4(f). 18 It is apparent that the measured linewidth at the lower frequencies is much broader than the theoretical one. The reason for that lies in the fact that in practice the bias field is not applied perfectly in the sample plane as well as in the fact that there migh t be locally different orientation s of the polycrystalline grains due to the natural imperfections of the interfaces that further result in angle distribution of H . Since the measured field linewidth is a projection of the spectral linewidth into the magnetic field domain, the relation between the frequency and the field intrinsic linewidth s is given by: 1 int int 0dHdH . The intrinsic linewidth , int , in the frequency domain near Hkeff is finite , as easily seen from Eq. (1) while the derivative term near Hkeff is zero for even the slightest angle misalignment as already seen. H ence the field-domain linewidth diverg es to infinity as observed experimentally. The inhomogeneous broadening component does not diverge in that manner but is rather suppressed . To show that the excessive linewidth at low field s is indeed related to the derivative of 0/d dH we empirically multiply the total theoretical linewidth by the factor 0 / ( )d d H which turns out to fit the data surprisingly well (Fig. 4(f)). This is merely a phenomenological qualitative description, and a rigorous description should still be derived. The fitted linewidth of Fig. 4(f) results in the intrinsic damping value of 0.011 ±0.0005 and is identical to the value obtained by the TRMOKE method . Often concerns regarding the differences between the TRMOKE and FMR measurements such as spin wave emission away from the pump laser spot in the TRMOKE29, increase of 19 damping due to thermal heating by the pump pulse as well as differences in the nature of the inhomogeneous broadening are raised. Such effects do not seem to be significant here . Additionally , it is worth noting that s ince the linewidth seems to reach a linear dependence with respect to the field at high fields , it may be naively fitted using a constant frequency - independent inhomogeneous broadening factor . In that case an underestimated value of ~0.0096 would have been obtained . The origin of this misinterpretation is seen clearly by examining the inhomogeneous broadening contribu tion in Fig. 4(f) that show s it as well to exhibit a seemingly linear dependence at the high fields. Regarding the inhomogeneous broadening , the anisotropy field dispersion, effKH , obtained with the TRMOKE was 205 Oe while the value obtained from the OFMR system was 169 Oe . Although these values are of the same order of magnitude , the difference is rather significant. It is possible that the discrepancy is related to the differences in the measurement techniques. For instance, the fact that both the pump and probe beams have the same spot size may cause an uneven excitation across the probed region in the case of the TRMOKE measurement while in the case of the OFMR measurement the amplitude of the microwave field decays at increasin g distances away from the microwire. These effects may be reflect ed in the measurements as inhomogeneous broadening. Nevertheless , the measured intrinsic damping values are similar. Finally , we compare the effective damping of the OFMR and the TRMOKE measu rements without correcting for the inhomogeneous broadening in Fig. 4(g). The 20 figure shows a deviation in the low field values which is by now understood to be unrelated to the energy losses of the system . Furthermore, we observe that the thickness of the Ta underlayer affects the damping. The comparison of the 50 Å -Ta CoFeB and the 100 Å -Ta CoFeB samples shows that the increase by merely 50 Å of Ta, reduced significantly the damping while leaving the anisot ropy field unaffected. C. TRMOKE and OFMR measurements in Co /Ni/Co film In the last set of measurements we study the Co /Ni/Co film which has distinctively different static properties compared to the CoFeB samples . The sample was studied using the TRMOKE setup at two H angles of 1 and 4 and using the OFMR system at 0H . The resultant resonance frequency traces are depicted in Fig. 5(a). The spectral linewidth measure d for 4H using the TRMOKE setup is presented in Fig . 5(b). A linear fit at the quasi linear high field range results in a large damping value of 0.081 ±0.015 and in a very large effKH of 630 Oe . The large damping is attributed to the efficient spin pumping into PtBi30 layer having large spin -orbit coupling . When the angle of the applied magnetic f ield is reduced to 1H a clearer picture of the contribution of the inhomogeneous broadening to the total linewidth is obtained (Fig. 5(c)) revealing that it cannot explain solely the measured spectral linewidth s. While the theoretical model predicts that the increase in bandwidth spans a relatively narrow field range around Hkeff, the measurement shows an increase over a much larger range around Hkeff. The linewidth broadening originating from 21 the anisotropy dispersion was theoretically calculated under the assumption of a small perturbation of the resonance frequency. A large keffH value was measured however from the TRMOKE measurement taken at 4H . Calculating numerically the exact variation of the resonance frequency improved slightly the fit but definitely did not resolve the discrepancy (not presented) . From this fact we understand that there should be an additional source contributing to the line broadening at least near Hkeff. A possible explanation may be related to the low stiffness27 associated with the 0 keff HH conditions . Under such conditions weaker torques which are usually neglected may become relevant24, 31. These torques could possibly originate from dipolar or exchange coupling resulting in two magnon scattering processes or even in a breakdown into magnetic domains as described by Grolier et al.26. From the limited data range at this angle, the damping could not be measured. The OFMR system enabl ed a wider range of fields and frequencies than the ones measured with the TRMOKE for 1H (Fig. 5(a)). Fig. 5(d) presents t he measured OFMR linewidth . The quasi -linear regime of the linewidth seems to be reached at frequencies of 12 GHz corresponding to bias field values which are larger than 7500 Oe . The resultant intrinsic damping after fitting to this range was 0.09±0.005 with a keffH of 660 Oe which differ by approximately 10% from the values obtained from the TRMOKE measurement . The effective measured damping is plotted in Fig. 5(e). The asymptotic damping value , though not fully reached for this high damping sample , would be about 0.1. 22 This represents an error of about 10% which is smaller compared to the errors of 17% and 54% encountered in the CoFeB samples because of the larger damping of the Co /Ni/Co sample. D. Considerations of two -magnon scattering In general, two -magnon spin wave scattering by impurities may exist in our measurements at all field ranges32, 33, not only near Hkeff as suggested in the discussion of the previous section32, 33. The resultant additional linewidth broadening would then be regarded as an extr insic contribution to the damping34-36. While in isotropic films which exhibit low crystalline anisotropy or in films having in -plane crystalline anisotropy, two - magnon scattering is maximized when the external field is applied in the film plane, in PMA films this is not necessarily the case and the highest efficiency of two -magnon scattering may be obtained at some oblique angle35. In films where two -magnon scatt ering is significant , the measured linewidth should exhibit an additional nonlinear dependence on the external field which cannot be accounted for by the present model . In such case, a s trong dependence on the external field would be observed for fields below Hkeff due to the variation in the orientation of the magnet ization with the external magnetic field. At higher fields the dependence on the external field is expected to be moderate35. While at bias field values below Hkeff our data is relatively limited, at external magnetic fields that are larger than Hkeff, the observed linewidth seems to be described well 23 by our model resulting in a field independent Gilbert damping coefficient . This seems to support our model that the scattering of spin waves does not have a prominent effect. It is possible however that a moderate dependence on the bias field, especially at high field values, may have been “linearized” and classified as intrinsic damping. IV. CONCLUSION In conclusion, in this paper we studied the time domain magnetization dynamics in non-epitaxial thin films having perpendicular magnetic anisotropy using the TRMOKE and OFMR system s. The analytical model used to interpret the magnetization dynamics from the TRMOKE responses indicated that the asymptotic high -field approach often used to distinguish the intrinsic damping from the effective damping may result in significant error that increases the lower the damping is . Two sources for the error were identified while validity of th e asymptotic approach was shown to require very high magnetic fields. Additionally, the effective damping was shown to be highly affected by the derivative of the resonance frequency with respect to the magnetic field 0/d dH . The analytical approach developed here was verified by use of the OFMR measurement showing excellent agreement whenever the intrinsic damping was compared and ruled out the possibility of thermal heating by the laser or emission of spin waves away from the probed area. 24 As to the systems studied, a large impact of the seed layer on the intrinsic damping with minor effect on the static characteristics of the CoFeB system was observed and may greatly aid in engineering the proper materials for the MTJ. Interestingly, the use of the analytical model enabled identification of an additional exchange torque when low stiffness conditions prevailed. While effort still remains to understand th e limits on the angle of the applied magnetic field to which the analytical solution is valid , the approach presented is believed to accelerate the discovery of novel materials for new applications . 25 Acknowledgments: A.C. thanks the Viterbi foundation and the Feder Family foundation for supporting this research. References: 1 Z. Yue, Z. Weisheng, J. O. Klein, K. Wang, D. Querlioz, Z. Youguang, D. Ravelosona, and C. Chappert, in Design, Automation and Test in Europe Conference and Exhibition (DATE), 2014 , p. 1. 2 A. D. Kent and D. C. Worledge, Nat Nano 10, 187 (2015). 3 M. Shigemi, Z. Xianmin, K. Takahide, N. Hiroshi, O. Mikihiko, A. Yasuo, and M. Terunobu, Applied Physics Express 4, 013005 (2011). 4 S. Iihama, S. Mizukami, H. Nagan uma, M. Oogane, Y. Ando, and T. Miyazaki, Physical Review B 89, 174416 (2014). 5 S. Mizukami, Journal of the Magnetics Society of Japan 39, 1 (2015). 6 G. Malinowski, K. C. Kuiper, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans, Applied Physics Letters 94, 102501 (2009). 7 A. J. Schellekens, L. Deen, D. Wang, J. T. Kohlhepp, H. J. M. Swagten, and B. Koopmans, Applied Physics Letters 102, 082405 (2013). 8 J. M. Beaujour, D. Ravelosona, I. Tudosa, E. E. Fullerton, and A. D. Kent, Physical Review B 80, 180415 (2009). 9 J. M. Shaw, H. T. Nembach, and T. J. Silva, Applied Physics Letters 105, 062406 (2014). 10 T. Kato, Y. Matsumoto, S. Okamoto, N. Kikuchi, O. Kitakami, N. Nishizawa, S. Tsunashima, and S. Iwata, Magnetics, IEEE Transactions on 47, 3036 (2011). 11 H.-S. Song, K. -D. Lee, J. -W. Sohn, S. -H. Yang, S. S. P. Parkin, C. -Y. You, and S. -C. Shin, Applied Physics Letters 103, 022406 (2013). 12 S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, and J. P. Nibarger, Journ al of Applied Physics 99, 093909 (2006). 13 M. Shigemi, A. Hiroyuki, W. Daisuke, O. Mikihiko, A. Yasuo, and M. Terunobu, Applied Physics Express 1, 121301 (2008). 14 I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev, D. C. Ralph, and R. A. Buhrman , Science 307, 228 (2005). 15 I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno, G. Gubbiotti, and C. H. Back, Journal of Magnetism and Magnetic Materials 307, 148 (2006). 16 A. Mekonnen, M. Cormier, A. V. Kimel, A. Kirilyuk, A. Hrabec, L. Ranno, and T. Rasing, Physical Review Letters 107, 117202 (2011). 17 S. Mizukami, et al., Physical Review Letters 106, 117201 (2011). 18 J.-Y. Bigot, M. Vomir, and E. Beaurepaire, Nat Phys 5, 515 (2009). 19 B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. Fahnle, T. Roth, M. Cinchetti, and M. Aeschlimann, Nat Mater 9, 259 (2010). 20 K.-S. Ryu, L. Thomas, S. -H. Yang, and S. Parkin, Nat Nano 8, 527 (2013). 21 S. Parkin and S. -H. Yang, Nat Nano 10, 195 (2015). 22 K.-S. Ryu, S. -H. Yang, L. Thomas, and S. S. P. Parkin, Nat Commun 5 (2014). 23 S.-H. Yang, K. -S. Ryu, and S. Parkin, Nat Nano 10, 221 (2015). 24 E. Schlomann, Journal of Physics and Chemistry of Solids 6, 242 (1958). 26 25 I. Neudecker, K. Perzlma ier, F. Hoffmann, G. Woltersdorf, M. Buess, D. Weiss, and C. H. Back, Physical Review B 73, 134426 (2006). 26 V. Grolier, J. Ferré, A. Maziewski, E. Stefanowicz, and D. Renard, Journal of Applied Physics 73, 5939 (1993). 27 J. Smit and H. G. Beljers, phili ps research reports 10, 113 (1955). 28 H. Suhl, Physical Review 97, 555 (1955). 29 Y. Au, M. Dvornik, T. Davison, E. Ahmad, P. S. Keatley, A. Vansteenkiste, B. Van Waeyenberge, and V. V. Kruglyak, Physical Review Letters 110, 097201 (2013). 30 Y. Tserkovny ak, A. Brataas, and G. E. W. Bauer, Physical Review Letters 88, 117601 (2002). 31 C. E. Patton, Magnetics, IEEE Transactions on 8, 433 (1972). 32 K. Zakeri, et al., Physical Review B 76, 104416 (2007). 33 J. Lindner, K. Lenz, E. Kosubek, K. Baberschke, D. Spoddig, R. Meckenstock, J. Pelzl, Z. Frait, and D. L. Mills, Physical Review B 68, 060102 (2003). 34 H. Suhl, Magnetics, IEEE Transactions on 34, 1834 (1998). 35 M. J. Hurben and C. E. Patton, Journal of Applied Physics 83, 4344 (1998). 36 D. L. Mills and S. M. Rezende, in Spin Dynamics in Confined Magnetic Structures II , edited by B. Hillebrands and K. Ounadjela (Springer Berlin Heidelberg, 2003). 27 Figure 1 FIG 1. Illustration of the angles H , H and . M and H0 vectors denote the magnetization and external magnetic field , respectively. 28 Figure 2 FIG. 2. Measured TRMOKE responses at H angles of 4 and 1 . (a) TRMOKE signal at low and high external magnetic field values. Traces are shifted for clarity. (b) Measured magnetization responses after reduction of background signal (open circles) 29 superimposed with the fitted decaying sine wave (solid lines). Traces are shifted and normalized to have the same peak amplitude. Data presented for low and high external magnetic field values. 30 Figure 3 FIG. 3. TRMOKE measurements at 4H and 1H . (a) Measured resonance frequency versus magnetic field. (b) Theoretical dependence of resonance frequency on magnetic field presented in normalized units . (c) & (d) Measured linewidth (blue) , fitted theoretical con tributions to l inewidth (green, cyan, magenta) and extracted intrinsic linewidth from measurement (red) for 4H and 1H , respectively. (e) Intrinsic and effective damping. (f) Error in damping value when using Eq. (2) instead of Eq. (3 ). 31 Figure 4 FIG. 4. TRMOKE and OFMR measurements at 4H and 0H , respectively. (a) Measured resonance frequency versus magnetic field. (b) Measured linewidth (blue), fitted theoretical contributions to linewidth (green, cyan, magenta) and extracted intrinsic 32 linewidth from measurement (red) using the TRMOKE with 4H . (c) Intrinsic and effective damping using TRMOKE . (d) Representative OFMR trace at 2.5 GHz. The function sign( mz)(mz)1/2 is plotted. (e) Field dependent absorption spectrum (blue) extracted from the cross section along the red dashed lined of (d) together with fitted lorentzian lineshapes (red). (f) Measured linewidth (blue), fitted theoretical contributions to linewidth (green, cyan, black) and empirical fit that describes the angle misalignment (magenta) using the OFMR with 0H . (g) Effective damping using the OFMR and TRMOKE . 33 Figure 5 FIG. 5. TRMOKE at 4H and 1H and OFMR measurement at 0H for Co/Ni/Co sample . (a) Measured resonance frequency versus magnetic field. (b ) Measured linewidth (blue), fitted theoretical contributions to linewidth (green, cyan, magenta) and extracted intrinsic linewidth from measurement (red) using the TRMOKE with 4H . (c) Measured linewidth (blue), fitted theoretical c ontributions to linewidth (green, cyan, magenta) using the TRMOKE with 1H . (d) Measured linewidth (blue), fitted theoretical contributions to linewidth (green, cyan, black) using the OFMR with 0H . 34 (e) Effecti ve (blue) and intrinsic (black ) damping using the TRMOKE at 4H and effective damping measured with the OFMR at 0H (red).

2015-11-16

Magnetization dynamics are strongly influenced by damping. An effective damping constant {\alpha}eff is often determined experimentally from the spectral linewidth of the free induction decay of the magnetization after the system is excited to its non-equilibrium state. Such an {\alpha}eff, however, reflects both intrinsic damping as well as inhomogeneous broadening. In this paper we compare measurements of the magnetization dynamics in ultrathin non-epitaxial films having perpendicular magnetic anisotropy using two different techniques, time-resolved magneto optical Kerr effect (TRMOKE) and hybrid optical-electrical ferromagnetic resonance (OFMR). By using an external magnetic field that is applied at very small angles to the film plane in the TRMOKE studies, we develop an explicit closed-form analytical expression for the TRMOKE spectral linewidth and show how this can be used to reliably extract the intrinsic Gilbert damping constant. The damping constant determined in this way is in excellent agreement with that determined from the OFMR method on the same samples. Our studies indicate that the asymptotic high-field approach that is often used in the TRMOKE method to distinguish the intrinsic damping from the effective damping may result in significant error, because such high external magnetic fields are required to make this approach valid that they are out of reach. The error becomes larger the lower is the intrinsic damping constant, and thus may account for the anomalously high damping constants that are often reported in TRMOKE studies. In conventional ferromagnetic resonance (FMR) studies, inhomogeneous contributions can be readily distinguished from intrinsic damping contributions from the magnetic field dependence of the FMR linewidth. Using the analogous approach, we show how reliable values of the intrinsic damping can be extracted from TRMOKE.

Determination of intrinsic damping of perpendicularly magnetized ultrathin films from time resolved precessional magnetization measurements

1511.04802v1

arXiv:1604.04688v1 [cond-mat.mtrl-sci] 16 Apr 2016A broadband Ferromagnetic Resonance dipper probe for magne tic damping measurements from 4.2 K to 300 K Shikun Hea)and Christos Panagopoulosb) Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological Univers ity, Singapore 637371 Adipper probefor broadband FerromagneticResonance (FMR)op erating from4.2K to room temperature is described. The apparatus is based on a 2-p ort transmitted microwave signal measurement with a grounded coplanar waveguide . The waveguide generates a microwave field and records the sample response. A 3- stagedipper design is adopted for fast and stable temperature control. The tempera ture variation due to FMR is in the milli-Kelvin range at liquid helium temperature. We also desig ned a novel FMR probe head with a spring-loaded sample holder. Improve d signal-to- noise ratio and stability compared to a common FMR head are achieved . Using a superconducting vector magnet we demonstrate Gilbert damping m easurements on two thin film samples using a vector network analyzer with frequency up to 26GHz: 1) A Permalloy film of 5 nm thickness and 2) a CoFeB film of 1.5nm thicknes s. Ex- periments were performed with the applied magnetic field parallel and perpendicular to the film plane. a)Electronic mail: skhe@ntu.edu.sg b)Electronic mail: christos@ntu.edu.sg 1I. INTRODUCTION In recent years, the switching of a nanomagnet by spin transfer t orque (STT) using a spin polarized current has been realized and intensively studied.1–3This provides avenues to new types of magnetic memory and devices, reviving the interest on mag netization dynamics in ultrathinfilms.4,5Highfrequencytechniques playanimportantroleinthisresearchdir ection. Amongthem, FerromagneticResonance(FMR)isapowerfultool. M ostFMRmeasurements have been performed using commercially available systems, such as e lectron paramagnetic resonance (EPR) or electron spin resonance (ESR).6These techniques take advantage of the high Q-factor of a microwave cavity, where the field modulation a pproach allows for the utilization of a lock-in amplifier.7The high signal-to-noise ratio enables the measurement of evensub-nanometerthickmagneticfilms. However, theoperating frequencyofametalcavity is defined by its geometry and thus is fixed. To determine the damping of magnetization precession, whichisinprincipleanisotropic, several cavitiesarereq uiredtostudytherelation between the linewidth and microwave frequency at a given magnetiza tion direction.8–10The apparent disadvantage isthat changing cavities canbetedious and prolongthe measurement time. Recently, an alternative FMR spectrometer has attracted consid erable attention.11–17 The technique is based on a state of the art vector network analyz er (VNA) and a coplanar waveguide (CPW). Both VNA and CPW can operate in a wide frequency range hence this technique is also referred to as broadband FMR or VNA-FMR. The br oadband FMR tech- nique offers several advantages. First, it is rather straightforw ard to measure FMR over a wide frequency range. Second, one may fix the applied magnetic field and acquire spectra with sweeping frequency in a matter of minutes.17Furthermore, a CPW fabricated on a chip using standard photolithography enables FMR measurements on pa tterned films as well as on a single device.18In brief, it is a versatile tool suitable for the characterization of ma g- netic anisotropy, investigation of magnetization dynamics and the s tudy of high frequency response of materials requiring a fixed field essential to avoid any ph ase changes caused by sweeping the applied field. Although homebuilt VNA-FMRs are designed mainly for room temperat ure measure- ments, a setup with variable temperature capability is of great inter est both for fundamental studies and applications. Denysenkov et al. designed a probe with va riable sample temper- ature, namely, 4-420K,19however, the spectrometer only operates in reflection mode. In a more recent effort, Harward et al. developed a system operating at frequencies up to 70GHz.12However the lower bound temperature of the apparatus is limited to 27K. Here we present a 2-port broadband FMR apparatus based on a superc onducting magnet. A 3-stage dipper probe has been developed which allows us to work in th e temperature range 4.2- 300K. Taking advantage of a superconducting vector magnet , measurements can be performed with the magnetic moment saturated either parallel or p erpendicular to the film plane. Wealsodesigned aspring-loadedsample holderforfastandre liablesample mounting, quicktemperatureresponseandimprovedstability. Thissetupallow sforswiftchangesofthe FMR probe heads and requires little effort for the measurement of d evices. To demonstrate the capability of this FMR apparatus we measured the temperature dependence of magne- 2FIG. 1. View of the FMR dipper probe. Top panel: The schematic of the entire design with a straight type FMR head. All RF connectors are 2.4mm. The vacu um cap mounted on the 4K stage, using In seal, and the radiation shield mounted on the second stage are not shown for clarity. Bottom panel: photograph of the components inside the vacuu m cap. tization dynamics of thin film samples of Permalloy (Py) and CoFeB in diffe rent applied magnetic field configurations. II. APPARATUS A. Cryostat and superconducting magnet system Our customized cryogenic system was developed by Janis Research Company Inc. and includes a superconducting vector magnet manufactured by Cryo magnetics Inc. A vertical field up to 9T is generated by a superconducting solenoid. The field ho mogeneity is ±0.1% over a 10mm region. A horizontal split pair superconducting magnet provides a field up to 4T with uniformity ±0.5% over a 10mm region. The vector magnet is controlled by a Model 4G-Dual power supply. Although the power supply gives field r eadings according to the initial calibration, to avoid the influence of remnant field we employ an additional Hall sensor. The cryostat has a 50mm vertical bore to accommodate v ariable temperature inserts and dipper probes. Our dipper probe described below is confi gured for this cryostat, however, the principle can be applied also to other comme rcially available superconducting magnets and cryostats. 3B. Dipper probe Fig. 1shows a schematic of our dipper probe assembly and a photograph o f the com- ponents inside the vacuum cap. The dipper probe is 1.2m long and is mou nted to the cryogenic system via a KF50 flange. The sliding seal allows a slow insert ion of the dipper probe directly into the liquid helium space. Supporting arms (not show n) lock the probe and minimize vibration, with the sample aligned to the field center. The c onnector box on top has vacuum tight Lemo and Amphenol connectors for 18 DC sign al feedthrough. Two 2.4mm RF connection ports allow for frequencies up to 50GHz . A vacu um pump port can be shut by a Swagelok valve. We adopted a three stage design as sho wn in the photograph ofFig. 1. The 4K stage and the vacuum cap immersed in the He bath provide co oling power for the probe. The intermediate second stage acts as an isolator o f heat flow and as thermal sink for the RF cables, providing improved temperature control. Fu rthermore, it allows one to change probe heads conveniently as we discuss later. A separat e temperature sensor on the second stage is used for monitoring purpose. The third stage, namely, the FMR probe head with the spring loaded sample holder, is attached to the lower en d of the intermediate stage using stainless steel rods. Apairof0.086”stainlesssteelSemi-RigidRFcablesrunfromthetopo ftheconnectorbox to the non-magnetic bulkhead connector (KEYCOM Corp.) mounted on the second stage. BeCu non-magnetic Semi-Rigid cables (GGB Industries, Inc.) are use d for the connection between the second stage and the probe head. The cables are car efully bent to minimize losses. The rods connecting the stages are locked by set screws. Loosening the set screw allows the rod length to be adjusted to match the length of the RF ca bles. Reflection coefficient (S 11) and transmission coefficient (S 21) can be recorded simultaneously with this 2-port design. The leads for the temperature sensors, heater, Hall sensor and for optional transport measurements are wrapped around Cu heat-sinks at t he 4K stage before being soldered to the connection pins. C. Probe head with spring-loaded sample holder The key part of the dipper probe, namely, the FMR probe head is sch ematically shown in Fig. 2. Theassemblyisplacedinaradiationshieldtubewithaninnerdiametero f32mm. To maximize thermal conduction between parts, homebuilt component s are machined from Au plated Cu. The 1” long customized grounded coplanar waveguide (GC PW) has a nominal impedanceof50Ohm. Thestraight-lineshapeGCPWwasmadeonduro idR/circlecopyrtR6010(Rogers) board, with a thickness of 254 µm and dielectric constant 10.2. The width of the center conductor is 117 µm and the gap between the latter and the ground planes is 76 µm. For the connection, first the GCPW is soldered to the probe head, and s ubsequently the center pin of the flange connector (Southwest Microwave) is soldered to t he center conductor of the GCPW. The response of the dipper with the straight-line shape GCPW installed is shown inFig. 3. The relatively large insertion loss (-16.9dB at 26GHZ) is due to a tota l cable length of more than 3m and multiple connectors. The high frequency current flowing in the CPW generates a magnetic field of the same frequency. This RF field d rives the precession 4FIG. 2. Schematic of the spring-loaded FMR probe head with st raight shape grounded coplanar waveguide (GCPW). 1 Au plated Cu housing; 2 straight shape GC PW; 3 flange connector; 4 strain gauge thin film heater; 5 CernoxTMtemperature sensor; 6 Hall-sensor housing; 7 housing for 4- pin Dip socket or pingo pin; 8 sample; 9 sample holder; 10 Cu sprin g; 11 spring housing; 12 sample holder handle nut. of the magnetic material placed on top of the signal line, and gives ris e to a change in the system’s impedance, which in turn alters the transmitted and reflec ted signals. A spring-loaded sample holder depicted in Fig. 2by items 9 to 12 is designed to mount the sample. The procedure for loading a sample is as follows: 1) Pull up the handle nut and apply a thin layer of grease (Apiezon N type) to the sample holder ; 2) Place the sample at the center of the sample holder; 3) Mount the spring-loaded sam ple holder to the FMR head; 4) Release the handle nut gradually so that the spring pushes the sample towards the waveguide. The mounting-hole of the spring-housing is slightly lar ger than the outer diameterofthespring. Thisallowsthesampleholdertomatchthesur faceoftheGCPWself- adaptively. With the spring-loaded FMR head design, the sample moun ting is simple and leaves no residue from the commonly used tapes. It maximizes the sig nal by minimizing the gap between waveguide and sample, and enhances the stability. Fur thermore, it is suitable for variable temperature measurements due to the enhanced the rmal coupling between the sample, cold head and sensors ( items 9 to 12 in Fig. 2.). The temperature sensor is mounted at the backside of the probe h ead. Due to limited space, the heater consists of three parallel connected strain ga uges with a resistance of 120 Ohm. TheHallsensor canbemountedaccordingtotherequiredmea surement configuration. The position of the Hall sensor shown in Fig. 2is an example for measurements in the presence of a magnetic field applied parallel to the sample surface. D. Probe-head using end-launch connector Although the probe head with straight-line CPW works well in our expe riment, the necessary replacement of CPW due to unavoidable performance fa tigue over time, or for testing new CPW designs can be time consuming. In response, end-la unch connectors (ELC) utilizing a clamping mechanism allow for a smooth transition from R F cables to CPW. Soldering the launch pin to the center conductor of CPW is optio nal and reduces the effort for modifications. In Fig. 4, we show our design of a FMR probe-head using ELC 50 5 10 15 20 25-30-20-100S (dB) f (GHz) S11 S21 FIG. 3. The reflection (S 11) and transmission (S 21) coefficients of the dipper probe with the straight-line shape GCPW mounted. The measurement was perf ormed at room temperature. form Southwest Microwave, Inc. and a homebuilt U-shape GCPW. Sim ilar to the design of Fig. 2, a Au plated Cu housing is used to mount the GCPW, ELC and the tempe rature and Hall sensors. There are two locations for sample mounting. In posit ion A, the vertical field is used for measurements with the magnetic field applied parallel to th e surface of the thin film sample whereas the horizontal field is used for measurements wit h field perpendicular to the sample surface. On the other hand, measurements for bot h configurations can be accomplishedonlybyusingthehorizontalfieldifthesampleisplacedinp ositionB.Asshown inFig. 4(b) and (c), to change between configurations simply requires rot ating the dipper probeby90degrees. Nevertheless, weprefertoplacethesample inpositionAfortheparallel configurationsincethesolenoidfieldismoreuniform. However, weno tethatthesamedesign with the sample placed in position B is suitable also for an electromagnet . Furthermore, adding a rotary stage to the probe enables angular dependent FMR measurements. III. EXPERIMENTAL TEST In this section, we present data to assess the performance of th e FMR probe head and discuss two sets of magnetic damping measurements, demonstrat ing the capabilities and performance of the appratus. A. Spring-loaded sample holder We tested our setup using a Keysight PNA N5222A vector network a nalyzer with maxi- mum frequency 26.5GHz. The output power of the VNA is always 0dBm in our test. Note that with 2.4mm connectors and customized GCPW, our design can in p rinciple operate up to 50GHz. The performance of the spring-loaded sample holder is first studied at room temperature with a 2nm thick Co 40Fe40B20film. For direct comparison, the FMR spectra are recorded with two sample loading methods: One with a spring-load ed sample holder 6FIG. 4. FMR probe-head with u-shape GCPW and end-launch conn ector. (a) Photograph of the probe-head using end-launch connector and U-shape GCPW. Se nsors are mounted at the backside and at the bottom of the Cu housing. Simplified sketch of the co nfiguration for measuring with an external field generated by the split coils (b) parallel and ( c) perpendicular to the sample plane. Rotating the dipper probe in the horizontal plane changes fr om one configuration to the other. (Fig. 2) and the other using the common method12which only requires Kapton tape. The magnetic field is applied parallel to the plane of the thin film sample. Six se ts of data were obtained by reloading the sample for each measurement. In Fig. 5, we show the amplitude of the power transmission coefficient from Port 1 to Port 2 (S 21) at a frequency of 10GHz and a temperature of 300K. The open circles represent a spectru m for a spring-loaded sam- ple mounting whereas the open squares is the spectrum showing larg est signal for the six flip-sample loadings. The averaged spectra for all six spectra are s hown by solid line and dotted line, for spring and flip-sample loading, respectively. Two obs ervations are evident: First, the best signal we obtained using the flip sample method is appr oximately 20 percent lower compared to the spring-loaded method. Thus the spring-load ed method gives a better signal to noise ratio and sensitivity. Second, for the spring-loaded method, the difference between the averaged spectrum and single spectra is negligibly small. On the other hand the variation between measurements for the flip-sample method can be as large as 20 percent. Hence the spring-loaded method has better stability and is reprodu cible. B. Temperature response As detailed in the previous section, the probe head is made of Au plate d Cu blocks with highinternalthermalconductionandgoodthermalcontact. Con sequently, theresponsetime of the temperature control will be small as the characteristic the rmal relaxation time of a system is C/k, whereCis the heat capacity and kis the overall thermal conduction. Also, the temperature difference between sample and sensor is minimized e ven with the heater turned on. Shown in Fig. 6are the FMR spectra and temperature variation for a CoFeB film of 3nm thickness measured at 4.4K. The external field was swept at a rate of about - 10Oe/s. Forfields close to which FMRpeaks areobserved, we detec ted a temperaturerise of 7FIG. 5. Comparison between S 21signals obtained using spring-loaded sample holder mounti ng and flip-sample mounting at 300K. The sample has a stack of MgO(3n m)—CoFeB(2nm)—MgO(3nm) deposited on silicon substrate. (Numbers in parenthesis of the sample composition represent the thickness of the respective layer.) The frequency is 10 GHz a nd the FMR center field is at 1520 Oe. a few mK. In fact, the field values corresponding to maximum temper atures are about 20Oe lower than the fields satisfying FMR condition, showing that the char acteristic relaxation time between the sample and cold head is approximately 2 seconds. Th e temperature rise of the probe head due to FMR indicates that the magnetic system ab sorbs energy from the microwave and dissipates into the thermal bath. Specifically, at the field satisfying the FMR condition, the damping torque is balanced by the torque gen erated by the RF field. However, the dissipation power of such process is propotiona l to the thickness of the magnetic film hence is very small. The successful detection of a t emperature rise adds credence to the high thermal conduction within the probe head and relative low thermal conduction between different stages. This demonstration shows t hat the probe head is capable of measuring samples with phase transitions in a narrow temp erature range, such as a superconducting/ferromagnetic bilayer system.20 C. Magnetic damping measurements Although the FMR probe can be used to determine the energy anisot ropy of magnetic materials, our primary purpose is to study magnetic damping parame ter. In the following, two examples of such measurements are briefly described. Shown in Fig. 7is FMR response of a Py film of 5nm thickness deposited on a silicon substrate, measur ed at 4.4K. The sweeping external magnetic field is parallel to the sample surface. R eal and Imaginary parts of the spectra obtained at selected frequencies are plotted with o pen circles in Fig. 7(a) and (b), respectively. In FMR measurements, the change in the tr ansmittance, S21, is a direct measure of the field-dependent susceptibility of the magnet ic layer. According to the 8FIG. 6. Sample temperature variation due to FMR at selected f requencies. (upper panel) Ampli- tude of S 21and (lower panel) temperature variation of MgO(3nm)—CoFeB (3nm)—MgO(3nm) at 4.4K measured with external field parallel to the film plane. LandauLifshitzGilbert formalism, the dynamic susceptibility of the ma gnetic material in the configuration where the field is applied parallel to the plane of the thin film be described as:21 χIP=4πMs(H0+Huni+4πMeff+i∆H/2) (H0+Huni)(H0+Huni+4πMeff)−H2 f+i(∆H/2)·[2(H0+Huni)+4πMeff](1) Here, 4πMsis the saturation magnetization, Huniis the in-plane uniaxial anisotropy, 4πMeffis the effective magnetization, Hf= 2πf/γ, and ∆His the linewidth of the spectrum – the last term is of key importance to determine the damping parame ter. As shown by solid lines in Fig. 7(a) and (b), the spectra can be fitted very well by adding a backgr ound, a drift proportional to time, and a phase factor.11,22The field linewidth as a function of frequency – ∆ H(f) is plotted in Fig. 7(c). The data points fall on a straight line. The damping parameter αGL= 0.012±0.001 is therefore determined by the slope through9,23: ∆H=4π γαGLf+∆H0 (2) The error bar here is calculated from the confidence interval of th e fit. We have also tested the setup with a magnetic field applied perpendicu lar to the sample plane. The results for a MgO(3nm)—Co 40Fe40B20(1.5nm)—MgO(3nm) stack deposited on silicon substrate are shown in Fig. 8. Comparing the spectra obtained at different tempera- tures and fixed frequency, two observations are evident. First, the FMR peak position shifts to higher field as the temperature is lowered due to changes in the eff ective magnetization. 9FIG. 7. FMR data of a Py thin film of thickness 5nm measured at 4. 4K with magnetic field applied parallel to the sample plane. (a) Real and (b) Imagin ary parts of transmitted signal S 21 at selected frequencies. The data are normalized and the rel ative strength between the spectra at different frequencies are kept. (c) FMR linewidth as a functio n of frequency. The damping was calculated to be 0.012 ±0.001, using a linear fit. Second, the FMR linewidth increases with decreasing temperature. Although the interfacial anisotropy can be determined by fitting the FMR peak positions to th e Kittel formula,24 here, we are more interested in the damping parameter as a functio n of temperature. The dynamic susceptibility in this configuration is25: χOP=4πMs(H−4πMeff−i∆H/2) (H−4πMeff)2−H2 f+i∆H·(H−4πMeff)(3) Following the same procedure as for Py, the real and imaginary part of the spectra are fitted simultaneously to obtain the linewidth. In Fig. 8(b), we plot the linewidth as a function of frequency at the two boundaries of our measured tem peratures. Although the measured linewidth at lower temperature is larger, the slope of the t wo curves is in good agreement. The additional linewidth at 6K is primarily due to zero freq uency broadening, which quantifies the magnitude of dispersion of the effective magnet ization. The results are summarized in Fig. 8(c). Gilbert damping is essentially independent of temperature although there is a minimum at 40 K. The room temperature value obta ined is in agreement with the value for a thicker CoFeB.21,26On the other hand, the inhomogeneous broaden- ing increases with lowering temperatue. The value at 6K is more than d ouble compared to room temperatue. Notably, neglecting the zero-frequency off set ∆H(0), arising due to inhomogeneity, would give rise to an enhanced effective damping comp ared to the intrinsic contribution. Cavity based, angulardependent FMRmayalso disting uish theGilbertdamp- ing from inhomogeneity effects. A shortcoming however, is the need to take into account the possible contribution of two magnon scattering, which causes in creased complications in the analysis of the data.27,28On the other hand, broadband FMR using a dipper probe with the applied magnetic field in the perpendicular configuration, rule s out two magnon scattering making this technique relatively straightforward to imple ment.29 Thedipperprobediscussedhereisnotlimitedtomeasurementsofth edampingcoefficient. The broadband design is also useful for time-domain measurements .30Furthermore, a spin 10FIG. 8. Temperature dependent FMR measurement for a CoFeB th in film of thickness 1.5nm with the magnetic field applied perpendicular to the film plan e. (a) Transmitted FMR signal at 20GHz obtained at different temperatures. (b) FMR linewidth a s a function of frequency at 6K and 280K. (c) Damping constant and inhomogeneous broadenin g as a function of temperature. The solid lines are the guides for the eye. transfer torque ferromagnetic resonance31measurement on a single device can be performed with variable temperature using bias Tee and a separate sample holde r. IV. CONCLUSION We have developed a variable temperature FMR to measure the magn etic damping pa- rameter in ultra thin films. The 3-stage dipper and FMR head with a spr ing-loaded sample holder design have a temperature stability of milli Kelvin during the FMR measurements. This apparatus demonstrates improved signal stability compared t o traditional flip-sample mounting. The results for Py and CoFeB thin films show that the FMR d ipper can measure the damping parameter of ultra thin films with: Field parallel and perpe ndicular to the film plane in the temperature range 4.2-300K and frequency up to at lea st 26 GHz. ACKNOWLEDGMENTS TheauthorsaregratefultoSzeTerLimatDataStorageInstitut eforpreparingtheCoFeB samples. We acknowledge Singapore Ministry of Education (MOE), Ac ademic Research Fund Tier 2 (Reference No: MOE2014-T2-1-050) and National Res earch Foundation (NRF) of Singapore, NRF-Investigatorship (Reference No: NRF-NRFI2 015-04) for the funding of this research. REFERENCES 1J. C. Slonczewski, Journal of Magnetism and Magnetic Materials 159, L1 (1996) . 2J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ra lph, Physical Review Letters 84, 3149 (2000) . 113S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, a nd E. E. Fullerton, Nat Mater 5, 210 (2006) . 4N. Locatelli, V. Cros, and J. Grollier, Nat Mater 13, 11 (2014) . 5A. Brataas, A. D. Kent, and H. Ohno, Nat Mater 11, 372 (2012) . 6M. Farle, Reports on Progress in Physics 61, 755 (1998) . 7B. Heinrich, Measurement Techniques and Novel Magnetic Properties (Springer Berlin Heidelberg, 2006). 8B. Heinrich, G. Woltersdorf, R. Urban, and E. Simanek, Journal of Applied Physics 93, 7545 (2003) . 9K. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy, and A. J´ an ossy, Physical Review B 73, 144424 (2006) . 10B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y.-Y. S ong, Y. Sun, and M. Wu, Physical Review Letters 107, 066604 (2011) . 11S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P . Kabos, T. J. Silva, and J. P. Nibarger, Journal of Applied Physics 99, 093909 (2006) . 12I. Harward, T. O’Keevan, A. Hutchison, V. Zagorodnii, and Z. Celins ki, Review of Scientific Instruments 82, 095115 (2011) . 13J. Wei, J. Wang, Q. Liu, X. Li, D. Cao, and X. Sun, Review of Scientific Instruments 85, 054705 (2014) . 14E. Montoya, T. McKinnon, A. Zamani, E. Girt, and B. Heinrich, Journal of Magnetism and Magnetic Materials 356, 12 (2014) . 15H. Gowiski, M. Schmidt, I. Gociaska, J.-P. Ansermet, and J. Dubowik , Journal of Applied Physics 116, 053901 (2014) . 16M. 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Vernier, J.-V. Kim, B. Ockert, and D. Ravelosona, Applied Physics Letters 102, 022407 (2013) . 26S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nat Mater 9, 721 (2010) . 27S. Mizukami, Y. Ando, and T. Miyazaki, Physical Review B 66, 104413 (2002) . 1228J.Lindner, I.Barsukov, C.Raeder,C.Hassel, O.Posth, R.Meck enstock, P.Landeros, and D. L. Mills, Physical Review B 80, 224421 (2009) . 29R. Arias and D. L. Mills, Physical Review B 60, 7395 (1999) . 30Y.-T. Cui, G. Finocchio, C. Wang, J. A. Katine, R. A. Buhrman, and D. C. Ralph, Phys. Rev. Lett. 104, 097201 (2010) . 31L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011) . 13

2016-04-16

A dipper probe for broadband Ferromagnetic Resonance (FMR) operating from 4.2 K to room temperature is described. The apparatus is based on a 2-port transmitted microwave signal measurement with a grounded coplanar waveguide. The waveguide generates a microwave field and records the sample response. A 3-stage dipper design is adopted for fast and stable temperature control. The temperature variation due to FMR is in the milli-Kelvin range at liquid helium temperature. We also designed a novel FMR probe head with a spring-loaded sample holder. Improved signal-to-noise ratio and stability compared to a common FMR head are achieved. Using a superconducting vector magnet we demonstrate Gilbert damping measurements on two thin film samples using a vector network analyzer with frequency up to 26 GHz: 1) A Permalloy film of 5 nm thickness and 2) a CoFeB film of 1.5 nm thickness. Experiments were performed with the applied magnetic field parallel and perpendicular to the film plane.

A broadband Ferromagnetic Resonance dipper probe for magnetic damping measurements from 4.2 K to 300 K

1604.04688v1

arXiv:2010.01044v4 [math.NA] 6 Oct 2022Multilevel quasi-Monte Carlo for random elliptic eigenval ue problems I: Regularity and error analysis Alexander D. Gilbert1Robert Scheichl2 October 7, 2022 Abstract Stochastic PDE eigenvalue problems are useful models for quantify ing the uncer- tainty in several applications from the physical sciences and engine ering, e.g., struc- tural vibration analysis, the criticality of a nuclear reactor or phot onic crystal struc- tures. In this paper we present a multilevel quasi-Monte Carlo (MLQ MC) method for approximatingtheexpectationofthe minimaleigenvalueofanelliptic e igenvalueprob- lem with coefficients that are given as a series expansion of countably -many stochastic parameters. The MLQMC algorithm is based on a hierarchy of discret isations of the spatial domain and truncations of the dimension of the stochastic p arameter domain. To approximate the expectations, randomly shifted lattice rules ar e employed. This paper is primarily dedicated to giving a rigorous analysis of the error o f this algo- rithm. A key step in the error analysis requires bounds on the mixed d erivatives of the eigenfunction with respect to both the stochastic and spatial variables simulta- neously. Under stronger smoothness assumptions on the parame tric dependence, our analysis also extends to multilevel higher-order quasi-Monte Carlo r ules. An accom- panying paper [Gilbert and Scheichl, 2022], focusses on practical ex tensions of the MLQMC algorithm to improve efficiency, and presents numerical resu lts. 1 Introduction Consider the following elliptic eigenvalue problem (EVP) −∇·/parenleftbig a(x,y)∇u(x,y)/parenrightbig +b(x,y)u(x,y) =λ(y)c(x,y)u(x,y),forx∈D, u(x,y) = 0 for x∈∂D,(1.1) where the differential operator ∇is with respect to the physical variable x, which belongs to a bounded, convex domain D⊂Rd(d= 1,2,3), and where the stochastic parameter y= (yj)j∈N∈Ω:= [−1 2,1 2]N, is an infinite-dimensional vector of independently and iden tically distributed (i.i.d.) uni- form random variables on [ −1 2,1 2]. The dependence of the coefficients on the stochastic paramete rs carries through to the eigenvalues λ(y), and corresponding eigenfunctions u(y):=u(·,y), and as such, in this 1School of Mathematics and Statistics, University of New Sou th Wales, Sydney NSW 2052, Australia. alexander.gilbert@unsw.edu.au 2Institute for Applied Mathematics & Interdisciplinary Cen tre for Scientific Computing, Universit¨ at Heidelberg, 69120 Heidelberg, Germany and Department of Ma thematical Sciences, University of Bath, Bath BA2 7AY UK. r.scheichl@uni-heidelberg.de 1paper we are interested in computing statistics of the eigen values and of linear functionals of the corresponding eigenfunction. In particular, we woul d like to compute the expecta- tion, with respect to the countable product of uniform densi ties, of the smallest eigenvalue λ, which is an infinite-dimensional integral defined as Ey[λ] =/integraldisplay Ωλ(y) dy:= lim s→∞/integraldisplay [−1 2,1 2]sλ(y1,y2,...,ys,0,0...) dy1dy2···dys. The multilevel Monte Carlo (MLMC) method [22, 30] is a varian ce reduction scheme that has been successfully applied to many stochastic simul ation problems. When applied to stochastic PDE problems (see, e.g., [6, 8]), the MLMC meth od is based on a hierarchy ofL+ 1 increasingly fine finite element meshes {Tℓ}L ℓ=0(corresponding to a decreasing sequenceof meshwidths h0> h1>···> hL>0), andanincreasing sequenceoftruncation dimensions s0< s1<···< sL<∞. Letting the dimension-truncated FE approximation on level ℓbe denoted by λℓ:=λhℓ,sℓ, by linearity, we can write the expectation on the finest level as Ey[λL] =Ey[λ0]+L/summationdisplay ℓ=1Ey[λℓ−λℓ−1]. (1.2) Each expectation Ey[λℓ−λℓ−1] is then approximated by an independent Monte Carlo method. Defining uℓ:=uhℓ,sℓwe can write a similar telescoping sum for Ey[G(uL)] for any linear functional G(u). Quasi-Monte Carlo (QMC) methods are equal-weight quadratu re rules where the sam- ples are deterministically chosen to be well-distributed, see [13]. Multilevel quasi-Monte Carlo (MLQMC) methods, whereby a QMC quadrature rule to appr oximate the expec- tation on each level, were first developed in [23] for path sim ulation with applications in option pricing and then later applied to stochastic PDE prob lems (e.g., [34, 33]). For certain problems, MLQMC methods can be shown to converge fas ter than their Monte Carlo counterpart, and for most problems the gains from usin g multilevel and QMC are complementary. In this paper we present a rigorous analysis of the error of a M LQMC algorithm for approximating the expectation of the smallest eigenvalue o f (1.1) in the case where the coefficients are given by a Karhunen–Lo` eve type series expan sion. The main result proved in this paper is that under some common assumptions on the sum mability of the terms in thecoefficientexpansion, theroot-mean-squareerror(RMSE )ofaMLQMCapproximation ofEy[λ], which on each level ℓ= 0,1,...,Luses a randomly shifted lattice rule with Nℓ points, a FE discretisation with meshwidth hℓ>0 and a fixed truncation dimension, is bounded by RMSE/lessorsimilarh2 L+L/summationdisplay ℓ=0N−1+δ ℓh2 ℓ,forδ >0, (1.3) with a similar result for the eigenfunction (see Theorems 3. 1, 3.2 and Remark 5.1). This error boundis clearly better thanthe correspondingresult fora MLMC method, which has N−1+δ ℓreplaced by N−1/2 ℓ, and in terms of the overall complexity to achieve a RMSE less than some tolerance ε >0 the total cost compared to a single level QMC approximation is reducedbyafactor of ε−1inspatial dimensions d≥2(seeCorollary3.1). Underequivalent assumptions, the convergence rates in (1.3) coincide with t he rates in the corresponding error bound for source problems from [34, 33]. Although it is not unexpected that we are able to obtain the same convergence rates as for source pr oblems, the analysis here is completely new and because of the nonlinear nature of eige nvalue problems presents several added difficulties not encountered previously in the analysis of source problems. 2Indeed, the key intermediate step is an in-depth analysis of the mixed regularity of the eigenfunction, simultaneously in both the spatial and stoc hastic variables. The result, presented in Theorem 4.1, is a collection of explicit bounds on the mixed derivatives of the eigenfunction, where the derivatives are second order with respect to the spatial variable xand arbitrarily high order with respect to the stochastic va riabley. The proof of these bounds forms a substantial proportion of this paper, and req uires a delicate multistage induction argument along with a considerable amount of tech nical analysis (see Section 4 and the Appendix). These bounds significantly extend the pre vious regularity results for stochastic EVPs from [1], which didn’t give any bounds on the derivatives, and [19], which gave bounds that were first-order with respect to xand higher order with respect to y. Furthermore, many other multilevel methods require simil ar mixed regularity bounds for their analysis, e.g., multilevel stochastic collocati on [42]. Hence, the bounds are of independent interest and open the door for further research into methods for uncertainty quantification for stochastic EVPs. In particular, we show h ow the mixed regularity bounds can be immediately applied to extend the analysis als o to multilevel quasi-Monte Carlo methods for EVPs based on higher-order interlaced polynomial lattice rules [9, 24], following the papers [11, 12] for source problems (see Secti on 5.4). The focus of this paper is the theoretical analysis of our MLQ MC algorithm for EVPs. As such numerical results and practical details on how to effic iently implement the algo- rithm will be given in a separate paper [21]. EVPs provide a useful way to model problems from a diverse ran ge of applications, such as structural vibration analysis [43], the nuclear cri ticality problem [15, 31, 44] and photonic crystal structures [14, 18, 32, 36]. More recently , interest in stochastic EVPs has been driven by a desire to quantify the uncertainty in applic ations such as nuclear physics [2, 3, 45, 46], structural analysis [40] and aerospace engin eering [39]. The most widely used numerical methods for stochastic EVPs are Monte Carlo m ethods [40]. More recently stochastic collocation methods [1] and stochastic Galerki n/polynomial chaos methods [17, 45, 46] have been developed. In particular, to deal with the h igh-dimensionality of the parameter space, sparse and low-rank methods have been cons idered, see [1, 16, 26, 28, 29]. Additionally, the present authors (along with colleag ues) have applied quasi-Monte Carlo methods to (1.1) and proved some key properties of the m inimal eigenvalue and its corresponding eigenfunction, see [19, 20]. Although we consider the smallest eigenvalue, the MLQMC met hod and analysis in this paper can easily be extended to any simple eigenvalue th at is well-separated from the rest of the spectrum for all parameters y. If the quantity of interest depends on a cluster of eigenvalues, or on the corresponding subspace of eigenfu nctions, then, in principle, the method in this paper could be used in conjunction with a subsp ace-based eigensolver. Again, one important point for the theory would be that the ei genvalue cluster is well- separated from the rest of the spectrum, uniformly in y. The structure of the paper is as follows. In Section 2 we give a brief summary of the required mathematical material. Then in Section 3 we presen t the MLQMC algorithm along with a cost analysis. Section 4 proves the key regulari ty bounds, which are then required for the error analysis in Section 5. Finally, in the appendix we give the proof of the two key lemmas from Section 5. 2 Mathematical background In this section we briefly summarise the relevant material on variational EVPs, finite element methods and quasi-Monte Carlo methods. For further details we refer the reader to the references indicated throughout, or [19]. 3As a start, we make the following assumptions on the coefficien ts, which will ensure that the problem (1.1) is well-posed and admits fast converg ence rates of our MLQMC algorithm. In particular, we assume that all coefficients are bounded from above and below, independently of xandy. Assumption A1. 1.aandbare of the form a(x,y) =a0(x)+∞/summationdisplay j=1yjaj(x)andb(x,y) =b0(x)+∞/summationdisplay j=1yjbj(x),(2.1) whereaj, bj∈L∞(D), for allj≥0, andc∈L∞(D)depend onxbut noty. 2. There exists amin>0such that a(x,y)≥amin,b(x,y)≥0andc(x)≥amin, for all x∈D,y∈Ω. 3. There exist p,q∈(0,1)such that ∞/summationdisplay j=1max/parenleftbig /⌊a∇d⌊laj/⌊a∇d⌊lL∞,/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/parenrightbigp<∞and∞/summationdisplay j=1/⌊a∇d⌊l∇aj/⌊a∇d⌊lq L∞(D)<∞. For convenience, we define amax<∞so that max/braceleftbig /⌊a∇d⌊la(y)/⌊a∇d⌊lL∞,/⌊a∇d⌊l∇a(y)/⌊a∇d⌊lL∞,/⌊a∇d⌊lb(y)/⌊a∇d⌊lL∞,/⌊a∇d⌊lc/⌊a∇d⌊lL∞/bracerightbig ≤amaxfor ally∈Ω.(2.2) 2.1 Variational eigenvalue problems To introduce the variational form of the PDE (1.1), we let V:=H1 0(D), the first order Sobolev space of functions with vanishing trace, and equip Vwith the norm /⌊a∇d⌊lv/⌊a∇d⌊lV:= /⌊a∇d⌊l∇v/⌊a∇d⌊lL2(D). The space Vtogether with its dual, which we denote by V∗, satisfy the well- known chain of compact embeddings V⊂⊂L2(D)⊂⊂V∗, where the pivot space L2(D) is identified with its own dual. Forv,w∈V, define the inner products A(y;·,·),M(·,·) :V×V→Rby A(y;w,v):=/integraldisplay Da(x,y)∇w(x)·∇v(x)dx+/integraldisplay Db(x,y)w(x)v(x)dx, M(w,v):=/integraldisplay Dc(x)w(x)v(x)dx, and let their respective induced norms be given by /⌊a∇d⌊lv/⌊a∇d⌊lA(y):=/radicalbig A(y;v,v) and/⌊a∇d⌊lv/⌊a∇d⌊lM:=/radicalbig M(v,v). Further, let M(·,·) also denote the duality paring on V×V∗. In the usual way, multiplying (1.1) by v∈Vand performing integration by parts with respect tox, we arrive at the following variational EVP, which is equiva lent to (1.1). Find λ(y)∈R,u(y)∈Vsuch that A(y;u(y),v) =λ(y)M(u(y),v) for all v∈V, (2.3) /⌊a∇d⌊lu(y)/⌊a∇d⌊lM= 1. The classical theory for symmetric EVPs (see, e.g., [5]) ens ures that the variational EVP (2.3) has countably many strictly positive eigenvalues , which, counting multiplicities, we label in ascending order as 0< λ1(y)≤λ2(y)≤ ···. 4The corresponding eigenfunctions, u1(y), u2(y), ..., can be chosen to form a basis of Vthat is orthonormal with respect to the inner product M(·,·), and, by (2.3), also orthogonal with respect to A(y;·,·). Proposition 2.1. The smallest eigenvalue is simple for all y∈Ω. Furthermore, there existsρ >0, independent of y, such that λ2(y)−λ1(y)≥ρfor ally∈Ω. (2.4) Proof.The Krein–Rutmann Theorem and [19, Proposition 2.4]. Henceforth, we will let the smallest eigenvalue and its corr esponding eigenfunction be simply denoted by λ=λ1andu=u1. It is often useful to compare the eigenvalues λkto the eigenvalues of the negative Laplacian on D, also with homogeneous Dirichlet boundary conditions and w ith respect to the standard L2inner product. These are denoted by 0< χ1< χ2≤χ3≤ ···, (2.5) and will often simply be referred to as Laplacian eigenvalue s or eigenvalues of the Lapla- cian, without explicitly stating the domain or boundary con ditions. The following form of the Poincar´ e inequality will also be u seful throughout this paper /⌊a∇d⌊lv/⌊a∇d⌊lL2≤χ−1/2 1/⌊a∇d⌊lv/⌊a∇d⌊lV,forv∈V. (2.6) It follows by the min-max representation for the Laplacian e igenvalue χ1. The upper and lower bounds on the coefficients (2.2), along wit h the Poincar´ e inequal- ity (2.6), ensure that the A(y)- andM-norms are equivalent to the V- andL2-norms, respectively, with √amin/⌊a∇d⌊lv/⌊a∇d⌊lV≤/⌊a∇d⌊lv/⌊a∇d⌊lA(y)≤/radicalBigg amax/parenleftbigg 1+1 χ1/parenrightbigg /⌊a∇d⌊lv/⌊a∇d⌊lV, (2.7) √amin/⌊a∇d⌊lv/⌊a∇d⌊lL2≤/⌊a∇d⌊lv/⌊a∇d⌊lM≤√amax/⌊a∇d⌊lv/⌊a∇d⌊lL2. (2.8) Finally, as is to be expected, for our finite element error ana lysis we require second- order smoothness with respect to the spatial variables, whi ch we characterise by the space Z=H2(D)∩V, equipped with the norm /⌊a∇d⌊lv/⌊a∇d⌊lZ:=/parenleftbig /⌊a∇d⌊lv/⌊a∇d⌊l2 L2+/⌊a∇d⌊l∆v/⌊a∇d⌊l2 L2/parenrightbig1/2. In particular, the eigenfunctions belong to Z, see [19, Proposition 2.1]. 2.2 Stochastic dimension truncation The first type of approximation we make is to truncate the infin ite dimensional stochastic domain to finitely many dimensions, which, for a truncation d imension s∈N, we do by simply setting yj= 0 for all j > s. The result is that the coefficients aandbnow only depend on sterms. We define the following notation: ys= (y1,y2,...,ys), as(x,y):=a0(x)+s/summationdisplay j=1yjaj(x), bs(x,y):=b0(x)+s/summationdisplay j=1yjbj(x), 5and As(y;w,v):=/integraldisplay Das(x,y)∇w(x)·∇v(x)dx+/integraldisplay Dbs(x,y)w(x)v(x)dx. So that the truncated approximations, denoted by ( λs(y),us(y)), satisfy As(y;us(y),v) =λs(y)M(us(y),v) for all v∈V. (2.9) 2.3 Finite element methods for EVPs To begin with, we first describe the finite element (FE) spaces used to discretise the EVP (2.3). Let {Vh}h>0be a family of conforming FE spaces of dimension Mh, where eachVhcorresponds to a shape regular triangulation ThofDand the index parameter h= max{diam(τ) :τ∈Th}is called the meshwidth. Since we have only assumed that the domainDis convex and a∈W1,∞(D), throughoutthis paperweonly consider continuous, piecewise linear FE spaces. However, under stricter condit ions on the smoothness of the domain and the coefficients, one could easily extend our algor ithm to higher-order FE methods. Furthermore, we assume that the number of FE degree s of freedom is of the order of h−d, so that Mh/equalorsimilarh−d. This condition is satisfied by quasi-uniform meshes and also allows for local refinement. Forh >0, eachy∈Ω yields a FE (or discrete) EVP, which is formulated as: Find λh(y)∈R,uh(y)∈Vhsuch that A(y;uh(y),vh) =λh(y)M(uh(y),vh) for all vh∈Vh, (2.10) /⌊a∇d⌊luh(y)/⌊a∇d⌊lM= 1. The discrete EVP (2.10) has Mheigenvalues 0< λ1,h(y)≤λ2,h(y)≤ ··· ≤ λMh,h(y), and corresponding eigenfunctions u1,h(y), u2,h(y), ..., u Mh,h(y), which are known to converge to the first Mheigenvalues and eigenfunctions of (2.3) as h→0, see, e.g., [5] or [19] for the stochastic case. From [19, Theorem 2.6] we have the following bounds on the FE e rror for the minimal eigenpair, which we restate here because they will be used ex tensively in our error analysis in Section 5. Theorem 2.2. Leth >0be sufficiently small and suppose that Assumption A1 holds. Then, for all y∈Ω,λhsatisfies |λ(y)−λh(y)| ≤Cλh2, (2.11) the corresponding eigenfunction uhcan be chosen such that /⌊a∇d⌊lu(y)−uh(y)/⌊a∇d⌊lV≤Cuh, (2.12) and forG ∈H−1+t(D)witht∈[0,1] /vextendsingle/vextendsingleG(u(y))−G(uh(y))/vextendsingle/vextendsingle≤CGh1+t, (2.13) where0< Cλ, Cu, CGare positive constants independent of yandh. 6We have already seen that the minimal eigenvalue of the conti nuous problem (2.3) is simple for all y, and that the spectral gap is bounded independently of y. It turns out that the spectral gap of the FE eigenproblem (2.10) is also bo unded independently of y andh, provided that the FE eigenvalues are sufficiently accurate. Specifically, if h≤h:=/radicalbiggρ 2Cλ, (2.14) then λ2,h(y)−λ1,h(y)≥λ2(y)−λ1(y)−/parenleftbig λ1,h(y)−λ1(y)/parenrightbig ≥ρ−Cλh2≥ρ 2,(2.15) where we have used the FE error estimate (2.11) and that λ1,h(y) converges from above. In fact, it is well known that for conforming methods all of th e FE eigenvalues con- verge from above, so that λk,h(y)≥λk(y). Then, as in [19], we can use the eigenvalues of the Laplacian (or rather their FE approximations) to boun d the FE eigenvalues and eigenfunctions independently of y. Hence, for k= 1,2,...,M hand for ally∈Ω, there existλkanduk, which are independent of both yandh, such that λk:=amin amaxχk≤λk(y)≤λk,h(y)≤amax amin(χk,h+1)≤λk, (2.16) max/braceleftbig /⌊a∇d⌊luk(y)/⌊a∇d⌊lV,/⌊a∇d⌊luk,h(y)/⌊a∇d⌊lV/bracerightbig ≤/radicalbig amax(χk,h+1) amin≤uk,(2.17) whereχk,his the FE approximation of the kth Laplacian eigenvalue χk. In addition to converging from above, for the Laplacian eigenvalues it i s known that χk≤χk,h≤ χk+Ckh2, for some constant that is independent of h(see [7, Theorem 10.4]). As such, forhsufficiently small there exists an upper bound on χk,hthat is independent of h, which in turn allows us to choose the final upper bounds λkandukso that they are independent of bothyandh. To conclude this section we introduce some notation and some properties of Vhthat will be useful later on. First, the spaces Vhsatisfy the best approximation property : inf vh∈Vh/⌊a∇d⌊lw−vh/⌊a∇d⌊lV/lessorsimilarh/⌊a∇d⌊lw/⌊a∇d⌊lZ,for allw∈Z. (2.18) Then, for h >0, letPh(y) :V→Vhdenote the A(y)-orthogonal projection of Vonto Vh, which satisfies A(y;w−Ph(y)w,vh) = 0,for allw∈V, vh∈Vh, (2.19) and hence also /⌊a∇d⌊lw−Ph(y)w/⌊a∇d⌊lA(y)= inf vh∈Vh/⌊a∇d⌊lw−vh/⌊a∇d⌊lA(y). 2.4 Quasi-Monte Carlo integration Quasi-Monte Carlo (QMC) methods are a class of equal-weight quadrature rules that can be used to efficiently approximate an integral over the s-dimensional (translated) unit cube Isf:=/integraldisplay [−1 2,1 2]sf(y)dy. There are several different flavours of QMC rules, however in th is paper we focus on randomly shifted rank-1 lattice rules . In Section 5.4 we will also briefly discuss how to 7extend our method to higher-order interlaced polynomial lattice rules , see [9, 24]. For further details on different QMC methods see, e.g., [13]. A randomly shifted rank-1 lattice rule approximation to IsfusingNpoints is Qs,N(∆)f:=1 NN−1/summationdisplay k=0f(tk−1 2), (2.20) where for a generating vector z∈Nsand a uniformly distributed random shift ∆∈[0,1)s, the pointstkare given by tk=tk(∆) =/braceleftbiggkz N+∆/bracerightbigg fork= 0,1,...,N−1. Here{·}denotes taking the fractional part of each component of a vec tor and1 2:= (1 2,1 2,...1 2). The standard spaces for analysing randomly shifted lattice s rules are the so-called weighted Sobolev spaces that were introduced in [41]. Here the term “w eighted” is used to indicate that the space depends on a collection of positiv e numbers called “weights” that model the importance of different subsets of variables an d enter the space through its norm. To be more explicit, given a collection of weights γ:={γu>0 :u⊆ {1,2,...,s}}, letWs,γbe thes-dimensional weighted Sobolev space of functions with squa re-integrable mixed first derivatives, equipped with the (unanchored) nor m /⌊a∇d⌊lf/⌊a∇d⌊l2 Ws,γ=/summationdisplay u⊆{1:s}1 γu/integraldisplay [−1 2,1 2]|u|/parenleftbigg/integraldisplay [−1 2,1 2]s−|u|∂|u| ∂yuf(y) dy−u/parenrightbigg2 dyu.(2.21) Hereyu:= (yj)j∈uandy−u:= (yj)j∈{1:s}\u. Note also that we have used here setnotation to denote the mixed first derivatives, as this is the conventi on in the QMC literature. However, when we later give results for higher-order mixed d erivatives we will switch to multi-index notation. A generating vector that leads to a good randomly shifted lat tice rule in practice can be constructed using the component-by-component (CBC) algorithm, or the more efficient fast CBC construction [37, 38]. In particular, it can be shown (see, e .g., [13, Theorem 5.10]) that the root-mean-square (RMS) error of a randomly s hifted lattice rule using a generating vector constructed by the CBC algorithm satisfie s /radicalBig E∆/bracketleftbig |Isf−Qs,Nf|2/bracketrightbig ≤/parenleftBigg 1 ϕ(N)/summationdisplay ∅/\e}atio\slash=u⊆{1:s}γξ u/parenleftbigg2ζ(2ξ) (2π2)ξ/parenrightbigg|u|/parenrightBigg1/2ξ /⌊a∇d⌊lf/⌊a∇d⌊lWs,γfor allξ∈(1 2,1].(2.22) Hereϕis the Euler totient function, ζis the Riemann zeta function and E∆denotes the expectation with respect to the random shift ∆. ForNprime one has ϕ(N) =N−1 or forNa power of 2 one has ϕ(N) =N/2, and so in both cases taking ξclose to 1 /2 in (2.22) results in the RMS error converging close to O(N−1). In practice, it is beneficial to perform several independent QMC approximations corre- spondingto a small number of independent random shifts, and then take the final approxi- mation to betheaverage over thedifferent shifts. In particul ar, let∆(1),∆(2),...,∆(R)be Rindependent uniform random shifts, and let the average over the QMC approximations with random shift ∆(r)be denoted by /hatwideQs,N,Rf:=1 RR/summationdisplay r=1Qs,N(∆(r))f. 8Then, the sample variance, /hatwideV[/hatwideQs,N,R]:=1 R(R−1)R/summationdisplay r=1/bracketleftbig/hatwideQs,N,Rf−Qs,N(∆(r))f/bracketrightbig2, (2.23) can be used as an estimate of the mean-square error of /hatwideQs,N,Rf. 3 MLQMC for random EVPs Applying a QMC rule to each term in the telescoping sum (1.2), using a different number Nℓof samples on each level, a simple MLQMC approximation of Ey[λ] is given by QML L(∆)λ:=L/summationdisplay ℓ=0Qℓ(∆ℓ)/parenleftbig λℓ−λℓ−1/parenrightbig . (3.1) Here, we define Qℓ(∆ℓ):=Qsℓ,Nℓ(∆ℓ) (see (2.20)) and we treat the L+ 1 independent random shifts, ∆ℓ∈[0,1)sℓ, as a single vector of dimension/summationtextL ℓ=0sℓ, denoted by ∆= (∆0,∆1,...,∆L). Recall also that λℓ=λhℓ,sℓforℓ= 0,1,...,L, and for simplicity denoteλ−1= 0. By using a different random shift for each level, the approx imations across different levels will be statistically independent. F or a linear functional G ∈V∗, the MLQMC approximation to Ey[G(u)] is defined in a similar fashion. As for single level QMC rules, it is beneficial to use multiple random shifts, so that we can estimate the variance on each level. Letting ∆(1),∆(2),...,∆(R)beRindependent random shifts of dimension/summationtextL ℓ=0sℓ, the shift-averaged MLQMC approximation is /hatwideQML L,Rλ:=L/summationdisplay ℓ=01 RR/summationdisplay r=1Qℓ(∆(r) ℓ)/parenleftbig λℓ−λℓ−1/parenrightbig . (3.2) Ifin practice theparameters arenot specified beforehand, t henwe set hℓ/equalorsimilar2−ℓ,sℓ/equalorsimilar2ℓ and use the adaptive algorithm from [23] to choose the number of QMC points Nℓ. The mean-square error (with respect to the random shift(s) ∆) of the MLQMC esti- mator can be written as the sum of the bias and the total varian ce as follows E∆/bracketleftbig |Ey[λ]−/hatwideQL(∆)λ|2/bracketrightbig =|Ey[λ−λL]|2+L/summationdisplay ℓ=0V∆[Qℓ(λℓ−λℓ−1)].(3.3) In the equation above, we have simplified the first term (corre sponding to the bias) by the telescoping property, and the variance on each level is defin ed by V∆[Qℓ(λℓ−λℓ−1)]:=E∆/bracketleftbig |Ey[λℓ−λℓ−1]−Qℓ(∆ℓ)(λℓ−λℓ−1)/vextendsingle/vextendsingle2/bracketrightbig , where the cross-terms have vanished because randomly shift ed QMC rules are unbiased. By the linearity of G ∈V∗, the error for the eigenfunction approximation can be decom - posed in the same way. Assuming that the total bias and the variance on each level de cay at some given rates, then the decomposition of the mean-square error (3.3 ) gives the following abstract complexity theorems (oneeach, fortheeigenvalue andforfu nctionals oftheeigenfunction). As is usual with the analysis of multilevel algorithms, the d ifficult part is to verify the assumptions on the decay of the variance and to determine the corresponding parameters. This analysis will be performed in Section 5. 9Theorem 3.1 (Eigenvalues) .Suppose that E∆[Qℓ(λℓ−λℓ−1)] =Ey[λℓ−λℓ−1], and that there exist positive constants αλ,α′,βλ,β′,ηsuch that M1.|Ey[λ−λL]|/lessorsimilarhαλ L+s−α′ L, and M2.V∆[Qℓ(λℓ−λℓ−1)]/lessorsimilarR−1N−η ℓ/parenleftBig hβλ ℓ−1+s−β′ ℓ−1/parenrightBig , for allℓ= 0,1,2,...,L. Then E∆/bracketleftBig/vextendsingle/vextendsingleEy[λ]−/hatwideQML L,R(λ)/vextendsingle/vextendsingle2/bracketrightBig /lessorsimilarhαλ L+sα′ L+1 RL/summationdisplay ℓ=01 Nη ℓ/parenleftBig hβλ ℓ−1+s−β′ ℓ−1/parenrightBig . Theorem 3.2 (Functionals) .ForG ∈V∗, suppose E∆[G(uℓ−uℓ−1)] =Ey[G(uℓ−uℓ−1)], and that there exist positive constants αG,α′,βG,β′,ηsuch that M1.|Ey[G(u−uL)]|/lessorsimilarhαG L+s−α′ L, and M2.V∆[Qℓ(G(uℓ−uℓ−1))]/lessorsimilarR−1N−η ℓ/parenleftBig hβG ℓ−1+s−β′ ℓ−1/parenrightBig , for allℓ= 0,1,2,...,L. Then E∆/bracketleftBig/vextendsingle/vextendsingleEy[G(u)]−/hatwideQML L,R(G(u))/vextendsingle/vextendsingle2/bracketrightBig /lessorsimilarhαG L+sα′ L+1 RL/summationdisplay ℓ=01 Nη ℓ/parenleftBig hβG ℓ−1+s−β′ ℓ−1/parenrightBig . Remark 3.1. In the case of a single truncation dimension, sℓ=sLfor allℓ= 1,2,...,L, the terms s−β′ ℓ−1can be dropped from the theorems above. In Section 5, we verify that if Assumption A1 on the coefficient s holds, then Assump- tions M1 and M2 above are satisfied, and we give explicit value s of the rates. To better illustrate the power of our MLQMC algorithm, we giv e here the following complexity bound for the special case of geometrically deca ying meshwidths and a fixed truncation dimension. We only give the eigenvalue result, b ut an analogous result holds also for linear functionals G ∈L2(D). For less smooth functionals, G ∈H−1+t(D) for t∈[0,1], similar results hold but with slightly adjusted rates. Corollary 3.1. Let0< ε≤e−1and suppose that Assumption A1 holds with p,q≤2/3. Also, let hℓ/equalorsimilar2−ℓwithh0sufficiently small and let sℓ=sL/equalorsimilarh2p/(2−p) L. Finally, suppose that each Qℓis anNℓ-point lattice rule corresponding to a CBC-constructed gener ating vector. If there exists 0< γ < d+1such that the cost on each level ℓ∈Nsatisfies M3.cost/parenleftbig Qℓ(λℓ−λℓ−1)/parenrightbig /lessorsimilarRNℓ/parenleftbig sℓh−d ℓ+h−γ ℓ/parenrightbig , then,LandNℓ= 2nℓ, fornℓ∈N, can be chosen such that E∆/bracketleftBig/vextendsingle/vextendsingleEy[λ]−/hatwideQML L,R(λ)/vextendsingle/vextendsingle2/bracketrightBig /lessorsimilarε2 and forδ >0 cost/parenleftbig/hatwideQML L,R(λ)/parenrightbig /lessorsimilar ε−1−p/(2−p)−δifd= 1, ε−1−p/(2−p)−δlog2(ε−1)3/2+δifd= 2, ε−d/2−p/(2−p)ifd >2. Proof.In Section 5 (cf., (5.1) and Theorem 5.3) we verify that Assum ptions M1, M2 from Theorem 3.1 hold with αλ= 2,α′= 2/p−1,βλ= 2αλ= 4 and η= 2−δ. The remainder of the proof follows by a standard minimisation argument as i n, e.g., [33, Cor. 2]. Remark 3.2. In [21] we verify that the cost does indeed satisfy Assumptio n M3 with γ≈d, which is the same order cost as the source problem. 104 Stochastic regularity In order for a randomly shifted lattice rule approximation t o achieve the error bound (2.22), we require that the integrand belongs to Ws,γ, which in turn requires bounds on the mixed first derivatives. For the eigenproblem (1.1), thi s means that we need to study the regularity of eigenvalues (and eigenfunctions) with re spect to the stochastic parameter y. In order to bound the variance on each level of our MLQMC esti mator, it is necessary to also study the FE error in Ws,γ(cf. (5.4)), whereas the single level analysis in [19] only required the expected FE error. This analysis of the FE e rror in a stronger norm requires mixed regularity of the solution with respect to bo thxandysimultaneously, which has not been shown previously. The theorem below prese nts the required bounds foru, along with the bounds from [19] with respect to yonly, which are included here for completeness. Analyticity of simple eigenvalues and eigen functions with respect to ywas shown in [1], however, explicit bounds on the derivatives we re not given there and they also did not consider the mixed xandyregularity required for the ML analysis. Although the analysis of randomly shifted lattice rules req uires only the mixed first derivatives (cf., (2.21)), we also give results for arbitra ry higher-order mixed derivatives. We do this because the proof technique is the same, and also si nce these bounds may be useful for the analysis of higher-order methods, e.g., high er-order QMC (see Section 5.4) or sparse grid rules (see, e.g., [25, 47]). As such, to simpli fy notation we will write mixed higher-order derivatives using multi-index notation inst ead of the set notation used in Section 2.4. For a multi-index ν= (νj)j∈Nwithνj∈N∪ {0}and only finitely-many nonzero components, let ∂ν ydenote the mixed partial differential operator where the orde r of derivative with respect to the variable yjisνj. Define|ν|:=/summationtext j≥1νjand denote the set of all admissible multi-indices by F:={ν∈NN:|ν|<∞}. All operations and relations between multi-indices will be performed componentwise, e. g., forν,m∈ Faddition is given byν+m= (νj+mj)j∈N, andν≤mif and only if νj≤mjfor allj∈N. Similarly, forν,m∈ Fand a sequence β∈ℓ∞define the following shorthand for products /parenleftbiggν m/parenrightbigg :=∞/productdisplay j=1/parenleftbiggνj mj/parenrightbigg andβν:=∞/productdisplay j=1βνj j. Note that since ν,m∈ Fhave finite support these products have finitely-many terms. Theorem 4.1. Letν∈ Fbe a multi-index, let ǫ∈(0,1), and suppose that Assumption A1 holds. Also, define the sequences β= (βj)j∈Nandβ= (βj)j∈Nby βj:=Cβmax/parenleftbig /⌊a∇d⌊laj/⌊a∇d⌊lL∞,/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/parenrightbig , (4.1) βj:=Cβmax/parenleftbig /⌊a∇d⌊laj/⌊a∇d⌊lL∞,/⌊a∇d⌊lbj/⌊a∇d⌊lL∞,/⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞/parenrightbig , (4.2) whereCβ≥1, given explicitly below in (4.6), is independent of ybut depends on ǫ. Then, for all y∈Ω, the derivative of the minimal eigenvalue with respect to yis bounded by |∂ν yλ(y)| ≤λ|ν|!1+ǫβν, (4.3) and the derivative of the corresponding eigenfunction sati sfies both /⌊a∇d⌊l∂ν yu(y)/⌊a∇d⌊lV≤u|ν|!1+ǫβν, (4.4) /⌊a∇d⌊l∂ν yu(y)/⌊a∇d⌊lZ≤C|ν|!1+ǫβν, (4.5) whereλ,uare as in (2.16),(2.17), respectively, and Cin(4.5)is independent of ybut depends on ǫ. Moreover, for h >0sufficiently small, the bounds (4.3)and(4.4)are also satisfied by λh(y)anduh(y), respectively. 11Proof.To facilitate the proof with a single constant for both seque ncesβandβwe define Cβ:=2λ2 ρaminλ a2maxλ/parenleftbigg3λ λCǫ+1/parenrightbigg , (4.6) whereCǫfrom [19, Lemma 3.3] is given by Cǫ:=21−ǫ 1−2−ǫ/parenleftbigge2 √ 2π/parenrightbiggǫ , which is independent of yandh. Then clearly it follows that Cβis independent of yand h. Later we will use that 1 /amin≤Cβ/2 and 1/(aminχ1/2)≤Cβ/2, which both follow from the lower bounds Cǫ≥1 for all ǫ∈(0,1) andλ/λ≥a2 max/a2 min(1+1/χ). Theproofforthebounds(4.3) and(4.4)isgiven in[19, Theor em3.4]. If hissufficiently small such that the FE eigenvalues resolve the spectral gap ( i.e., (2.15) holds) then the bounds also hold for λh(y) anduh(y) because Vh⊂V, cf. [19, Rem. 3.2 and 3.5]. For the bound (4.5), we first prove a recursive bound on /⌊a∇d⌊l∂νu(y)/⌊a∇d⌊lZand then use an induction result from [12] to prove the final bound. Consid er the strong form of the eigenproblem (1.1) for the pair ( λ(y),u(y)), which, omitting the xandydependence, is given by −∇·(a∇u)+bu=cλu. Theνth derivative with respect to ycommutes with the spatial derivatives ∇. Thus, using the Leibniz general product rule we have −∇·(a∇∂ν yu)+b∂ν yu+∞/summationdisplay j=1νj/parenleftbig −∇·(aj∇∂ν−ej yu)−bj∂ν−ej yu/parenrightbig =c/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg ∂m yλ ∂ν−m yu, whereejis the multi-index that is 1 in the jth entry and zero elsewhere. Then we can use the identity ∇·(φψ) =φ∇·ψ+∇φ·ψto simplify this to a∆∂ν yu=−∇a·∇∂ν yu+b∂ν yu−c/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg ∂m yλ ∂ν−m yu +∞/summationdisplay j=1νj/parenleftbig −aj∆∂ν−ej yu−∇aj·∇∂ν−e yu+bj∂ν−ej yu/parenrightbig . Sincea≥amin>0;a,aj∈W1,∞andb,bj∈L∞for allj∈N; and∂m yu∈Vfor allm∈ F, it follows by induction on |ν|that ∆∂ν yu∈L2. This allows us to take the L2-norm of both sides, which, after using the triangle inequal ity and the bounds in (2.2), gives the following recursive bound for ∆ ∂ν yu /⌊a∇d⌊l∆∂ν yu/⌊a∇d⌊lL2≤/⌊a∇d⌊l∇a/⌊a∇d⌊lL∞ amin/⌊a∇d⌊l∂ν yu/⌊a∇d⌊lV+/⌊a∇d⌊lb/⌊a∇d⌊lL∞ amin/⌊a∇d⌊l∂ν yu/⌊a∇d⌊lL2 +/⌊a∇d⌊lc/⌊a∇d⌊lL∞ amin/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg |∂m yλ|/⌊a∇d⌊l∂ν−m yu/⌊a∇d⌊lL2+∞/summationdisplay j=1νj/⌊a∇d⌊laj/⌊a∇d⌊lL∞ amin/⌊a∇d⌊l∆∂ν−ej yu/⌊a∇d⌊lL2 +1 amin∞/summationdisplay j=1νj/parenleftbig /⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞/⌊a∇d⌊l∂ν−ej yu/⌊a∇d⌊lV+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/⌊a∇d⌊l∂ν−ej yu/⌊a∇d⌊lL2/parenrightbig . 12Adding/⌊a∇d⌊l∂ν yu/⌊a∇d⌊lL2to both sides and then using the definition of βj, we can write this bound in terms of the Z-norm as /⌊a∇d⌊l∂ν yu/⌊a∇d⌊lZ≤ /⌊a∇d⌊l∂ν yu/⌊a∇d⌊lL2+/⌊a∇d⌊l∆∂ν yu/⌊a∇d⌊lL2≤∞/summationdisplay j=1νjβj/⌊a∇d⌊l∂ν−ej yu/⌊a∇d⌊lZ+Bν, (4.7) where we used that 1 /amin≤Cβ, and then defined Bν:=/⌊a∇d⌊l∇a/⌊a∇d⌊lL∞ amin/⌊a∇d⌊l∂ν yu/⌊a∇d⌊lV+/⌊a∇d⌊lc/⌊a∇d⌊lL∞ amin/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg |∂m yλ|/⌊a∇d⌊l∂ν−m yu/⌊a∇d⌊lL2 /parenleftbigg/⌊a∇d⌊lb/⌊a∇d⌊lL∞ amin+1/parenrightbigg /⌊a∇d⌊l∂ν yu/⌊a∇d⌊lL2+1 amin∞/summationdisplay j=1νj/parenleftbig /⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞/⌊a∇d⌊l∂ν−ej yu/⌊a∇d⌊lV+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/⌊a∇d⌊l∂ν−ej yu/⌊a∇d⌊lL2/parenrightbig . Now, the sum on the right of (4.7) only involves lower-order v ersions of the object we are interested in bounding (namely, /⌊a∇d⌊l∂ν yu/⌊a∇d⌊lZ), whereas the terms in Bνonly involve derivatives that can be bounded using one of (4.3) or (4.4). We bound the remaining L2-norms in Bνby the Poincar´ e inequality (2.6) to give Bν≤/parenleftbigg/⌊a∇d⌊l∇a/⌊a∇d⌊lL∞ amin+/⌊a∇d⌊lb/⌊a∇d⌊lL∞+amin amin√χ1/parenrightbigg /⌊a∇d⌊l∂ν yu/⌊a∇d⌊lV+/⌊a∇d⌊lc/⌊a∇d⌊lL∞ amin√χ1/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg |∂m yλ|/⌊a∇d⌊l∂ν−m yu/⌊a∇d⌊lV +1 amin∞/summationdisplay j=1νj/parenleftbigg /⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞ √χ1/parenrightbigg /⌊a∇d⌊l∂ν−ej yu/⌊a∇d⌊lV ≤amax amin/parenleftbigg 1+2√χ1/parenrightbigg/parenleftBigg /⌊a∇d⌊l∂ν yu/⌊a∇d⌊lV+/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg |∂m yλ|/⌊a∇d⌊l∂ν−m yu/⌊a∇d⌊lV/parenrightBigg +1 amin∞/summationdisplay j=1νj/parenleftbigg /⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞ √χ1/parenrightbigg /⌊a∇d⌊l∂ν−ej yu/⌊a∇d⌊lV, where in the last inequality we have boundedthe L∞-norms on the second line using (2.2), and then simplified. Then, substituting in the bounds (4.3) a nd (4.4) gives Bν≤amax amin/parenleftbigg 1+2√χ1/parenrightbigg/parenleftBigg u|ν|!1+ǫβν+/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg λ|m|!1+ǫβm·u|ν−m|!1+ǫβν−m/parenrightBigg +1 amin∞/summationdisplay j=1νj/parenleftbigg /⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞√χ1/parenrightbigg u(|ν|−1)!1+ǫβν−ej =uamax amin/parenleftbigg 1+2√χ1/parenrightbigg βν/parenleftBigg |ν|!1+ǫ+λ/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg |m|!1+ǫ|ν−m|!1+ǫ/parenrightBigg +u amin(|ν|−1)!1+ǫ∞/summationdisplay j=1νj/parenleftbigg /⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞ √χ1/parenrightbigg βν−ej. 13Using the fact that (1+ χ−1/2)/amin≤Cβand also that clearly βj≤βj, we have Bν≤uamax amin/parenleftbigg 1+2√χ1/parenrightbigg βν/parenleftBigg |ν|!1+ǫ+λ/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg |m|!1+ǫ|ν−m|!1+ǫ/parenrightBigg +u amin(|ν|−1)!1+ǫ∞/summationdisplay j=1νj/parenleftbig 1+χ−1/2/parenrightbig max/parenleftbig /⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞,/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/parenrightbig βν−ej ≤uamax amin/parenleftbigg 1+2√χ1/parenrightbigg βν/parenleftBigg |ν|!1+ǫ+λ/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg |m|!1+ǫ|ν−m|!1+ǫ/parenrightBigg +u|ν|!1+ǫβν. The sum that remains can be bounded using the same strategy as in the proof of [19, Lemma 3.4], as follows /summationdisplay m≤ν/parenleftbiggν m/parenrightbigg |m|!1+ǫ|ν−m|!1+ǫ=2|ν|!1+ǫ+|ν|−1/summationdisplay k=1k!1+ǫ(|ν|−k)!1+ǫ/summationdisplay m≤ν,|m|=k/parenleftbiggν m/parenrightbigg =2|ν|!1+ǫ+|ν|−1/summationdisplay k=1k!1+ǫ(|ν|−k)!1+ǫ/parenleftbigg|ν| k/parenrightbigg =|ν|!1+ǫ/parenleftbigg 2+|ν|−1/summationdisplay k=1/parenleftbigg|ν| k/parenrightbigg−ǫ/parenrightbigg ≤|ν|!1+ǫ/parenleftBigg 2+21−ǫ 1−2−ǫ/parenleftbigge2 √ 2π/parenrightbiggǫ /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright Cǫ/parenrightBigg , (4.8) where for the inequality on the last line we have used [19, Lem ma 3.3]. Hence,Bνis bounded above by Bν≤CB|ν|!1+ǫβν, where CB:=u/bracketleftbiggamax amin/parenleftbigg 1+2√χ1/parenrightbigg/parenleftbig 1+λ(2+Cǫ)/parenrightbig +1/bracketrightbigg <∞ is clearly independent of yandν. Now we can bound the recursive formula (4.7) using the bound a bove on Bν, which gives /⌊a∇d⌊l∂ν yu/⌊a∇d⌊lZ≤∞/summationdisplay j=1νjβj/⌊a∇d⌊l∂ν−ej yu/⌊a∇d⌊lZ+CB|ν|!1+ǫβν. Finally, by [12, Lemma 4] we can bound this above by /⌊a∇d⌊l∂ν yu/⌊a∇d⌊lZ≤/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg |m|!βmCB|ν−m|!1+ǫβν−m =CBβν/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg |m|!|ν−m|!1+ǫ≤CB(2+Cǫ)|ν|!1+ǫβν, where to obtain the final result we have again used (4.8). 145 Error analysis We now provide a rigorous analysis of the error for (3.1), whi ch we do by verifying the assumptions from Theorems 3.1 and 3.2. Recall that we use the shorthand λℓ:=λhℓ,sℓfor the dimension-truncated FE approx- imation of the minimal eigenvalue on level ℓ, whereas λsdenotes the minimal eigenvalue of the dimension-truncated version of the continuous EVP (2 .9). The bias (the first term) in (3.3) can be bounded by the triangle inequality to give |Ey[λ−λhL,sL]| ≤ |Ey[λ−λsL]|+|Ey[λsL−λhL,sL]|, and similarly for the eigenfunction. Now, both terms on the r ight can be bounded above using the results from the single level algorithm. Explicit ly, forG ∈H−1+t(D) with t∈[0,1] using Theorem 4.1 from [19] and then Theorem 2.2 gives the b ounds |Ey[λ−λL]|/lessorsimilars−2/p+1 L+h2 L, (5.1) |Ey[G(u−uL)]|/lessorsimilars−2/p+1 L+h1+t L, (5.2) with constants independent of sLandhL. That is, we have verified Assumptions M1 from both Theorems 3.1 and 3.2 with αλ= 2,αG= 1+tandα′= 2/p−1. For the variance terms on each level in (3.3) (alternatively to verify Assumption M2), we must study the QMC error of the differences λℓ−λℓ−1. Sinceλℓ−λℓ−1∈ Wsℓ,γfor all ℓ= 0,1,2,...,Land each QMC rule Qℓuses CBC-constructed generating vector zℓ, by (2.22) we have the upper bound V∆[Qℓ(λℓ−λℓ−1)]≤C2 ξ,ℓ ϕ(Nℓ)1/ξ/⌊a∇d⌊lλℓ−λℓ−1/⌊a∇d⌊l2 Wsℓ,γfor allξ∈(1/2,1],(5.3) whereCξ,ℓis the constant from (2.22) with s=sℓ. Thus, in Assumption M2 we can take η= 1/ξ∈[1,2) and for the other parameters we must study the norm of the di fference on each level. By the triangle inequality, we can separate truncation and F E components of the error /⌊a∇d⌊lλℓ−λℓ−1/⌊a∇d⌊lWsℓ,γ ≤ /⌊a∇d⌊lλsℓ−λsℓ−1/⌊a∇d⌊lWsℓ,γ+/⌊a∇d⌊lλsℓ−λhℓ,sℓ/⌊a∇d⌊lWsℓ,γ+/⌊a∇d⌊lλsℓ−1−λhℓ−1,sℓ−1/⌊a∇d⌊lWsℓ−1,γ.(5.4) In contrast to the single level setting [19], here we need to s tudy the truncation and FE errors in the weighted QMC norm (2.21) instead of simply the e xpected truncation and FE errors. Each term will be handled separately in the subsec tions that follow. Thekey ingredient intheerroranalysisaretheboundsofthe derivatives oftheminimal eigenvalue and its eigenfunction that were given in Section 4. 5.1 Estimating the FE error As a first step towards bounding the FE errors in the Ws,γ-norm, we bound their deriva- tives with respect to y, which are given below in Theorem 5.1. The bulk of the work to bound the FE error in Ws,γis dedicated to proving these regularity bounds. As in Theorem 4.1 we also present bounds on higher-order mixed der ivatives instead of simply the mixed first derivatives required in the Ws,γnorm. The strategy for proving these bounds is similar to the proof [19, Lemma 3.4], except in the current multilevel setting we need to bound the deriva tives of the FE errors of the eigenvalue and eigenfunction, in addition to the deriva tives of the eigenvalue and 15eigenfunction themselves. First, we differentiate variatio nal equations involving the errors to obtain a recursive formula for each of the eigenvalue and e igenfunction errors, and then prove the bounds by induction on the cardinality of |ν|. Once we have proved the bound for the eigenfunction in (5.11), the result for any function alG(u(y)) in (5.12) follows by a duality argument. Throughout the proofs in this section we will omit the xandy dependence. Note also that throughout we must explicitly tr ack the constants to ensure that they are independent of yandh, but also to make sure that in both of the inductive steps the constants are not growing, since this could interf ere with the summability of /hatwideβ. Also, the results in this section are all shown for hsufficiently small, wherehere sufficiently smallmeans that the FE eigenvalues resolve the spectral gap. Expl icitly, we assume that h≤h(see (2.14) and (2.15)) for some h >0 that is independent of y. This ensures that the condition that his sufficiently small (i.e., h≤h) is also independent of y. In the following key lemma, we bound the derivative of the diffe rence between the eigenfunctionanditsprojection Phu(y)ontoVh, whichisnotequaltotheFEeigenfunction uh(y), but is easier to handle. The proof relies on the new mixed re gularity estimate (4.5). Lemma 5.1. Letν∈ Fbe a multi-index, let h >0be sufficiently small and suppose that Assumption A1 holds. Then /⌊a∇d⌊l∂ν yu−Ph∂ν yu/⌊a∇d⌊lV≤CPh|ν|!1+ǫβν, (5.5) whereβis as defined in (4.2), andCPis independent of y,handν. Proof.Using the equivalence of the V-norm and the induced A-norm in (2.7), along with theA-orthogonality of the projection and the best approximatio n property (2.18), we get /⌊a∇d⌊l∂ν yu−Ph∂ν yu/⌊a∇d⌊lV≤/radicalBigg amax amin/parenleftbigg 1+1 χ1/parenrightbigg inf vh∈Vh/⌊a∇d⌊l∂ν yu−vh/⌊a∇d⌊lV ≤/radicalBigg amax amin/parenleftbigg 1+1 χ1/parenrightbigg Ch/⌊a∇d⌊l∂ν yu/⌊a∇d⌊lZ≤CPh|ν|!1+ǫβν, where for the last inequality we have used the bound (4.5). Th e final constant CPis independent of y,hand alsoν. The three recursive formulae presented in the next two lemma s are the key to the induction proof to bound the derivatives of the FE error. The general strategy is to differentiate variational equations involving the FE errors . However, the proofs are quite long and technical, and as such are deferred to the Appendix. Lemma 5.2. Letν∈ Fbe a multi-index, let h >0be sufficiently small and suppose that Assumption A1 holds. Then, for all y∈Ω, the following two recursive bounds hold |∂ν y(λ−λh)| ≤CI/parenleftBigg h|ν|!βν+∞/summationdisplay j=1νjβj/⌊a∇d⌊l∂ν−ej y(u−uh)/⌊a∇d⌊lV +/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg |m|!1+ǫβm/bracketleftBig /⌊a∇d⌊l∂ν−m y(u−uh)/⌊a∇d⌊lV+|∂ν−m y(λ−λh)|/bracketrightBig/parenrightBigg (5.6) 16and |∂ν y(λ−λh)| ≤CII/parenleftBigg/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg /⌊a∇d⌊l∂ν−m y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂m y(u−uh)/⌊a∇d⌊lV (5.7) +∞/summationdisplay j=1/summationdisplay m≤ν−ejνj/parenleftbiggν−ej m/parenrightbigg βj/⌊a∇d⌊l∂ν−ej−m y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂m y(u−uh)/⌊a∇d⌊lV +/summationdisplay m≤ν/summationdisplay k≤m/parenleftbiggν m/parenrightbigg/parenleftbiggm k/parenrightbigg |ν−m|!1+ǫβν−m/⌊a∇d⌊l∂m−k y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂k y(u−uh))/⌊a∇d⌊lV/parenrightBigg , whereβ,βare defined in (4.1),(4.2), respectively, and CI, CIIare independent of y,h andν. Lemma 5.3. Letν∈ Fbe a multi-index, let h >0be sufficiently small and suppose that Assumption A1 holds. Then, for all y∈Ω, /⌊a∇d⌊l∂ν y(u−uh)/⌊a∇d⌊lV≤CIII/parenleftBigg h|ν|!1+ǫβν+∞/summationdisplay j=1νjβj/⌊a∇d⌊l∂ν−ej y(u−uh)/⌊a∇d⌊lV +/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg |m|!1+ǫβm/bracketleftBig /⌊a∇d⌊l∂ν−m y(u−uh)/⌊a∇d⌊lV+|∂ν−m y(λ−λh)|/bracketrightBig/parenrightBigg ,(5.8) whereβ,βare defined in (4.1),(4.2), respectively, and CIIIis independent of y,handν. The astute reader may now ask, why do we need both the bounds (5 .6) and (5.7) on the derivative of the eigenvalue error? The reason is that the upper bound in (5.6) depends only on derivatives with order strictly less than ν, whereas the bound in (5.7) depends on ∂ν y(u−uh). Hence the inductive step for the eigenfunction (see (5.11) below) only works with (5.6). On the other hand, (5.6) cannot be used for the inductive step for theeigenvalue result (see (5.10) below), because it will only result in a bo und of order O(h). Hence, the second bound (5.7) is required to maintain the o ptimal rate of O(h2) for the eigenvalue error. We now have the necessary ingredients to prove thefollowing boundson thederivatives of the FE error. Theorem 5.1. Letν∈ Fbe a multi-index, let h >0be sufficiently small and suppose that Assumption A1 holds. Define the sequence /hatwideβ= (/hatwideβj)j∈Nby /hatwideβj:=/hatwideCβmax/parenleftbig /⌊a∇d⌊laj/⌊a∇d⌊lL∞,/⌊a∇d⌊lbj/⌊a∇d⌊lL∞,/⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞/parenrightbig , (5.9) where/hatwideCβ, given explicitly below in (5.14), is independent of y,handj. Then /vextendsingle/vextendsingle∂ν y/bracketleftbig λ(y)−λh(y)/bracketrightbig/vextendsingle/vextendsingle≤C1|ν|!1+ǫ/hatwideβνh2, (5.10) /⌊a∇d⌊l∂ν y/bracketleftbig u(y)−uh(y)/bracketrightbig /⌊a∇d⌊lV≤C2|ν|!1+ǫ/hatwideβνh, (5.11) and forG ∈H−1+t(D) /vextendsingle/vextendsingle∂ν yG(u(y)−uh(y))/vextendsingle/vextendsingle≤C3|ν|!1+ǫ/hatwideβνh1+t, (5.12) withC1,C2,C3all independent of y,handν. 17Proof.Throughoutwe usethe convention that 0! = 1. Then, dueto thee rror bound(2.11) for the FE eigenvalue error, the base case of the induction ( ν=0) for the eigenvalue result (5.10) holds provided C1≥Cλ. Thus, let C1:= max/braceleftbig Cλ, CIIC2 u(2+Cǫ)(4+Cǫ)/bracerightbig . (5.13) Similarly, defining C2=Cuthe base case of the induction for (5.11) also clearly holds d ue to (2.12). For the inductive step, let νbe such that |ν| ≥1 and assume that (5.10) and (5.11) hold for allmwith|m|<|ν|. Now, since the recursive bound for the eigenvalue (5.7) still depends on a term of order ν, whereas the recursive bound for the eigenfunction (5.8) only depends on strictly lower order terms, for our inductiv e step to work we first prove the result (5.11) for the eigenfunction, before proving the result (5.10) for the eigenvalue. Substituting the induction assumptions (5.10) and (5.11) f or|m|<|ν|into (5.8) gives /⌊a∇d⌊l∂ν y(u−uh)/⌊a∇d⌊lV≤CIII/parenleftBigg |ν|!1+ǫβνh+∞/summationdisplay j=1νjβjC2(|ν|−1)!1+ǫ/hatwideβν−ejh +/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg |m|!1+ǫβm(C2+C1h)|ν−m|!1+ǫ/hatwideβν−mh/parenrightBigg ≤/hatwideβνhCIII/parenleftBigg/bracketleftbigg/parenleftbiggCβ /hatwideCβ/parenrightbigg|ν| +C2Cβ /hatwideCβ/bracketrightbigg |ν|!1+ǫ +/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg |ν−m|!1+ǫ|m|!1+ǫ(C1h+C2)/parenleftbiggCβ /hatwideCβ/parenrightbigg|m|/parenrightBigg where we have rescaled each product by using the definitions o fβ,βand/hatwideβin (4.1), (4.2) and (5.9). The constants can be simplified by defining /hatwideCβ:=1 C2Cβmax(1,CIII)/bracketleftbig 1+C2+Cǫ(C1h+C2)/bracketrightbig , (5.14) which is independent of y,handν. This guarantees that Cβ//hatwideCβ≤1, and thus since |m|,|ν−m| ≥1, we have the bound /vextenddouble/vextenddouble∂ν y(u−uh)/vextenddouble/vextenddouble V ≤/hatwideβνhCIIICβ /hatwideCβ/parenleftBigg (1+C2)|ν|!1+ǫ+ (C1h+C2)/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg |ν−m|!1+ǫ|m|!1+ǫ/parenrightBigg ≤CIIICβ /hatwideCβ/bracketleftbig 1+C2+Cǫ(C1h+C2)/bracketrightbig |ν|!1+ǫ/hatwideβνh≤C2|ν|!1+ǫ/hatwideβνh, where we have used [19, Lemma 3.3] to bound the sum from above b yCǫ|ν|!1+ǫ(see also (4.8)), as well as (5.14) to give the final result. For the inductive step for the eigenvalue, we substitute the result (5.11), which has just been shown to hold for all multi-indices of order up to an d including |ν|, into (5.7) 18and then simplify, to give |∂ν y(λ−λh)| ≤CII(C2)2/hatwideβνh2/parenleftBigg/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg |ν−m|!1+ǫ|m|!1+ǫ +∞/summationdisplay j=1νj/summationdisplay m≤ν−ej/parenleftbiggν−ej m/parenrightbigg |ν−ej−m|!1+ǫ|m|!1+ǫ +/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg |ν−m|!1+ǫ/summationdisplay k≤m/parenleftbiggm k/parenrightbigg |m−k|!1+ǫ|k|!1+ǫ/parenrightBigg , where we have used the fact that βj≤/hatwideβj. The sums can again be bounded using (4.8) (using it twice for the double sum on the last line), to give |∂ν y(λ−λh)| ≤CII(C2)2/hatwideβνh2(2+Cǫ)(4+Cǫ)|ν|!1+ǫ, which, with C1as defined in (5.13) and C2=Cu, gives our desired result (5.10). The final result for the derivative of the error of the linear f unctional G(u) (5.12) follows by considering the same dual problem (A.16) as in [19 ]. But instead, here we let w=∂ν y(u−uh) and then use the upper bound (5.11) in the last step. We can now simply substitute these bounds on the derivatives of the FE error into (5.4), in order to bound the FE component of the error. For the second and third term in (5.4), in the case of the eigenvalue, this gives /⌊a∇d⌊lλs(y)−λh,s(y)/⌊a∇d⌊lWs,γ≤h2/parenleftBigg C2 1/summationdisplay u⊆{1:s}|u|!2(1+ǫ) γu/productdisplay j∈u/hatwideβ2 j/parenrightBigg1/2 . (5.15) Similar results hold for u(y) andG(u(y)). To ensure that the constant on the RHS of (5.15), and the const ants in the bounds that follow, are independent of the dimension, for ξ∈(1 2,1] to be specified later, we will choose the weights γby γj= max/parenleftbig/hatwideβj,βp/q j/parenrightbig , γ u=/parenleftBigg (|u|+3)!2(1+ǫ)/productdisplay j∈u(2π2)ξ 2ζ(2ξ)γ2 j/parenrightBigg1/(1+ξ) ,(5.16) wherep,qarethesummability parametersfromAssumptionA1.3, sotha t(γj)j∈N∈ℓq(R). 5.2 Estimating the truncation error It remains to estimate the first term in (5.4) — the truncation error. Theorem 5.2. Suppose that Assumption A1 holds and let s,/tildewides∈Nwiths >/tildewides. Addition- ally, suppose that the weights γare given by (5.16), then /⌊a∇d⌊lλs−λ/tildewides/⌊a∇d⌊lWs,γ/lessorsimilar/tildewides−1/p+1/q/parenleftBigg/summationdisplay u⊆{1:/tildewides}(|u|+3)!2(1+ǫ) γu/productdisplay j∈uβ2 j/parenrightBigg1/2 ,(5.17) with the constant independent of /tildewidesands. 19Proof.Sinceλsis analytic we can expand it as a Taylor series about 0in the variables {y/tildewides+1,...,ys}: λs(ys) =λs(y/tildewides;0)+s/summationdisplay i=/tildewides+1yi/integraldisplay1 0∂ ∂yiλs(y/tildewides;ty{/tildewides+1:s}) dt, where we use the notation ys= (y1,y2,...,ys), (y/tildewides;0) = (y1,y2,...,y/tildewides,0,...0) and (y/tildewides;ty{/tildewides+1:s}) = (y1,y2,...,y/tildewides,ty/tildewides+1,ty/tildewides+2,...,ty s). Sinceλ/tildewides(y/tildewides) =λs(y/tildewides;0) (this is simply different notation for the same object), this can be rearranged to give λs(ys)−λ/tildewides(y/tildewides) =s/summationdisplay i=/tildewides+1yi/integraldisplay1 0∂ ∂yi(y/tildewides;ty{/tildewides+1:s}) dt. (5.18) Letu⊆ {1,2...,/tildewides}, then differentiating (5.18) with respect to yugives ∂|u| ∂yu/parenleftbig λs(ys)−λ/tildewides(y/tildewides)/parenrightbig =s/summationdisplay i=/tildewides+1yi/integraldisplay1 0∂|u|+1 ∂yu∪{i}λs(y/tildewides;ty{/tildewides+1:s}) dt. Taking the absolute value, using the triangle inequality an d the fact that |yj| ≤1/2, we have the upper bound /vextendsingle/vextendsingle/vextendsingle/vextendsingle∂|u| ∂yu/parenleftbig λs(ys)−λ/tildewides(y/tildewides)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1 2s/summationdisplay i=/tildewides+1/integraldisplay1 0/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂|u|+1 ∂yu∪{i}λs(y/tildewides;ty{/tildewides+1:s})/vextendsingle/vextendsingle/vextendsingle/vextendsingledt. Now, substituting in the upper bound on the derivative of λsfrom [19, Lemma 3.4, equa- tion (3.6)] gives /vextendsingle/vextendsingle/vextendsingle/vextendsingle∂|u| ∂yu/parenleftbig λs(ys)−λ/tildewides(y/tildewides)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤λ 2s/summationdisplay i=/tildewides+1(|u|+1)!1+ǫβi/productdisplay j∈uβj =λ 2/parenleftBiggs/summationdisplay i=/tildewides+1βi/parenrightBigg (|u|+1)!1+ǫ/productdisplay j∈uβj, (5.19) withβjas in (4.1). Lettingu⊆ {1,2,...,s}withu∩ {/tildewides+ 1,/tildewides+ 2,...,s} /\e}atio\slash=∅, the derivative λs−λ/tildewidesis simply/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂|u| ∂yu/parenleftbig λs(ys)−λ/tildewides(y/tildewides)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂|u| ∂yuλs(ys)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤λ 2|u|!1+ǫ/productdisplay j∈uβj, (5.20) where we have again used the upper bound [19, equation (3.6)] . We now boundthe norm(2.21) of λs−λ/tildewidesinWs,γ. Splitting thesumover u⊆ {1,2,...} by whether ucontains any of {/tildewides+1,/tildewides+2,...,s}, we can write /⌊a∇d⌊lλs−λ/tildewides/⌊a∇d⌊l2 Ws,γ=/summationdisplay u⊆{1:/tildewides}1 γu/integraldisplay [−1 2,1 2]|u|/parenleftbigg/integraldisplay [−1 2,1 2]s−|u|∂|u| ∂yu/bracketleftbig λs(ys)−λ/tildewides(y/tildewides)/bracketrightbig dy−u/parenrightbigg2 dyu +/summationdisplay u⊆{1:s} u∩{/tildewides+1:s}/\e}atio\slash=∅1 γu/integraldisplay [−1 2,1 2]|u|/parenleftbigg/integraldisplay [−1 2,1 2]s−|u|∂|u| ∂yu/bracketleftbig λs(ys)−λ/tildewides(y/tildewides)/bracketrightbig dy−u/parenrightbigg2 dyu. 20Substituting in the bounds (5.19) and (5.20) then yields /⌊a∇d⌊lλs−λ/tildewides/⌊a∇d⌊lWs,γ≤/bracketleftBigg/parenleftBiggs/summationdisplay i=/tildewides+1βi/parenrightBigg2/summationdisplay u⊆{1:/tildewides}(|u|+1)!2(1+ǫ) γu/productdisplay j∈uβ2 j +/summationdisplay u⊆{1:s} u∩{/tildewides+1:s}/\e}atio\slash=∅|u|!2(1+ǫ) γu/productdisplay j∈uβ2 j/bracketrightBigg1/2 . (5.21) Next, we can bound the sum over iin (5.21) by the whole tail of the sum, which can then be bounded using [19, eq. (4.7)], to give s/summationdisplay i=/tildewides+1βi≤∞/summationdisplay i=/tildewides+1βi≤min/parenleftbiggp 1−p,1/parenrightbigg /⌊a∇d⌊lβ/⌊a∇d⌊lℓp/tildewides−(1/p−1). (5.22) Then for weights given by (5.16), following the proof of [34, Theorem 11] we can bound /summationdisplay u⊆{1:s} u∩{/tildewides+1:s}/\e}atio\slash=∅|u|!2(1+ǫ) γu/productdisplay j∈uβ2 j/lessorsimilar/tildewides−2(1/p−1/q)/summationdisplay u⊆{1:/tildewides}(|u|+3)!2(1+ǫ) γu/productdisplay j∈uβ2 j,(5.23) with a constant that is independent of /tildewidesands. Since 1/q >1, after substituting the bounds (5.22) and (5.23) into (5.2 1) we obtain the final result (5.17). 5.3 Final error bound In the previous two sections we have successfully bounded th e FE and truncation error in theWs,γnorm, now these bounds can simply be substituted into (5.4) t o bound the variance on each level. Theorem 5.3. LetL∈N, let1 =h−1> h0> h1>···> hL>0withh0sufficiently small, let 1 =s−1=s0≤s1··· ≤ ··· sL, and suppose that Assumption A1 holds with p < q. Also, let each Qℓbe a lattice rule using Nℓ= 2nℓ,nℓ∈N, points corresponding to a CBC-constructed generating vector with weights γgiven by (5.16). Then, for all ℓ= 0,1,2,...L, V∆[Qℓ(λℓ−λℓ−1)]≤C1N−η ℓ/parenleftbig h4 ℓ−1+s−2(1/p−1/q) ℓ−1/parenrightbig , (5.24) and, for G ∈H−1+t(D)with0≤t≤1, V∆[Qℓ(G(uℓ)−G(uℓ−1))]≤C2N−η ℓ/parenleftbig h2(1+t) ℓ−1+s−2(1/p−1/q) ℓ−1/parenrightbig , (5.25) where, for 0< δ <1, η= 2−δifq∈(0,2 3], 2 q−1ifq∈(2 3,1). The second term in (5.24)and(5.25)can be dropped if sℓ=sL, forℓ= 1,2,...L. Proof.We prove the result for the eigenvalue, since the eigenfunct ion result follows anal- ogously. For ℓ≥1, substituting the bounds (5.15) and (5.17) into (5.4) give s /⌊a∇d⌊lλℓ−λℓ−1/⌊a∇d⌊lWs,γ≤C/parenleftBigg/summationdisplay u⊆{1:sℓ}(|u|+3)!2(1+ǫ) γu/productdisplay j∈u/hatwideβ2 j/parenrightBigg1/2/parenleftbig h2 ℓ−1+s−(1/p−1/q) ℓ−1/parenrightbig , 21where we have simplified by using that hℓ< hℓ−1,sℓ−1< sℓ,βj≤/hatwideβj, and also merged all constants into a generic constant C, which may depend on ǫ. Substituting the bound above into (5.3), then using that Nℓ= 2nℓand thus ϕ(Nℓ) = Nℓ/2, the variance on level ℓcan be bounded by V∆[Qℓ(λℓ−λℓ−1)]≤Cℓ,γ,ξN−1/ξ ℓ/parenleftbig s−2(1/p−1/q) ℓ−1+h4 ℓ−1/parenrightbig . (5.26) The constant is given by Cℓ,γ,ξ:=C221/ξ/parenleftBigg/summationdisplay u⊆{1:sℓ}(|u|+3)!2(1+ǫ) γu/productdisplay j∈u/hatwideβ2 j/parenrightBigg/parenleftBigg/summationdisplay ∅/\e}atio\slash=u⊆{1:sℓ}γξ u/parenleftbigg2ζ(2ξ) (2π2)ξ/parenrightbigg|u|/parenrightBigg1/ξ . Forℓ= 0 we can similarly substitute (4.3) into the CBC bound (2.22 ), and then since h−1=s−1= 1 and βj≤/hatwideβjit follows that (5.26) also holds for ℓ= 0 with the constant C0,γ,ξas above. All that remains to be shown is that this constant can be bound ed independently of sℓ−1andsℓ. To this end, substituting the formula (5.16) for γuand using the fact that /hatwideβj≤γjthen simplifying, we can bound Cℓ,γ,ξabove by Cℓ,γ,ξ≤C/parenleftBigg/summationdisplay |u|<∞(|u|+3)!2ξ(1+ǫ) 1+ξ/productdisplay j∈uγ2ξ 1+ξ j/parenleftbigg2ζ(2ξ) (2π2)ξ/parenrightbigg1 1+ξ /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright Sξ/parenrightBigg1+ξ ξ , where again Cis a generic constant, which may depend on ǫ. We now choose the exponents ξandǫso that the sum Sξis finite. For 0 < δ <1, let ξ= 1 2−δifq∈(0,2 3], q 2−qifq∈(2 3,1),andǫ=1−ξ 4ξ>0. (5.27) With this choice of ξwe have (4 ξ−q−3qξ)/(1−ξ)≥qfor anyq∈(0,1), and so ∞/summationdisplay j=1γ4ξ−q−3qξ 1−ξ j <∞. (5.28) Then define the sequence αj=/parenleftBigg 1+∞/summationdisplay i=1γq i/parenrightBigg−1 γq i,so that∞/summationdisplay j=1αj<1. (5.29) Substituting in our choice (5.27) for ǫ, and multiplying and dividing each term by the product of α(1+3ξ)/(2(1+ξ)) j , we can write Sξ=/summationdisplay |u|<∞(|u|+3)!1+3ξ 2(1+ξ)/parenleftBigg/productdisplay j∈uα1+3ξ 2(1+ξ) j/parenrightBigg/parenleftBigg/productdisplay j∈uγ2ξ 1+ξ jα−1+3ξ 2(1+ξ) j/parenleftbigg2ζ(2ξ) (2π2)ξ/parenrightbigg1 1+ξ/parenrightBigg . 22Applying H¨ older’s inequality with exponents 2(1+ ξ)/(1+3ξ)>1 and 2(1+ ξ)/(1−ξ)>1 gives Sξ≤/parenleftBigg/summationdisplay |u|<∞(|u|+3)!/productdisplay j∈uαj/parenrightBigg1+3ξ 2(1+ξ)/parenleftBigg/summationdisplay |u|<∞/productdisplay j∈uγ4ξ 1−ξ jα−1+3ξ 1−ξ j/parenleftbigg2ζ(2ξ) (2π2)ξ/parenrightbigg2 1−ξ/parenrightBigg2−ξ 2(1+ξ) ≤/bracketleftBigg 6/parenleftbigg 1−∞/summationdisplay j=1αj/parenrightbigg−4/bracketrightBigg1+3ξ 2(1+ξ) ·exp/bracketleftBigg 2−ξ 2(1+ξ)/parenleftbigg2ζ(2ξ) (2π2)ξ/parenrightbigg2 1−ξ/parenleftbigg 1+∞/summationdisplay j=1γq j/parenrightbigg1+3ξ 1−ξ∞/summationdisplay j=1γ4ξ−q−3qξ 1−ξ j/bracketrightBigg , where we have used [35, Lemma 6.3]. From (5.28) and (5.29) it f ollows that Sξ<∞, and soCℓ,γ,ξcan be bounded independently of sℓ. Finally, letting η= 1/ξforξas in (5.27) gives the desired result with a constant independent of sℓ. Remark 5.1. Hence, we have verified that Assumptions M2 from Theorems 3.1 and 3.2 hold with βλ= 2αλ= 4,βG= 2αG= 2(1+t),β′= 1/p+1/q, andηas given above. The upper bounds in Theorem 5.1, (5.15) and Theorem 5.2 are th e same as the corre- sponding bounds from the MLQMC analysis for the source probl em (see [34, Theorems 7, 8, 11]), the only differences are in the values of the constants and in the extra 1+ ǫfactor in the exponent of |u|!. As such the final variance bounds in Theorem 5.3 also coinci de with the bounds for the source problem from [34] for all q <1. The only difference is that our result does not hold for q= 1, whereas the results for the source problem do. 5.4 Extension to higher-order QMC As mentioned earlier, the bounds on the higher-order deriva tives that we proved in Sec- tion 4 imply higher order methods can also be used for the quad rature component of our ML algorithm, which will provide a faster convergence rate i nNℓ. We now provide a brief discussionofhowtoextendourMLalgorithm, andtheerroran alysis, to higher-order QMC (HOQMC) rules. From an algorithm point of view, one can simply use HOQ MC points instead of lattice rules for the quadrature rules Qℓin (3.1). We denote this ML-HOQMC approximation by QMLHO L. To extend the error analysis to HOQMC we can again use a general framework as in Theorems 3.1 and 3.2. We stress that t he difficult part is to verify the assumptions, and in particular to show the required mixe d higher-order derivative bounds that we have already proved in Theorem 4.1. The remain der of the analysis then follows the same steps as in the previous sections with only s light modifications to handle the higher-order norm as in [12], where ML-HOQMC methods wer e applied to PDE source problems. As such, we don’t present the full details here but only an outline. A HOQMC rule is an equal-weight quadrature rule of the form (2 .20) that can achieve faster than 1 /Nconvergence for sufficiently smooth integrands. A popular cl ass of de- terministic HOQMC rules are interlaced polynomial lattice rules , see [9, 24] and [11, 12] for their application to PDE source problems. Loosely speak ing, a polynomial lattice rule is a QMC rule similar to a lattice rule, except the points are g enerated by a vector of polynomials instead of integers, the number of points Nis a prime power and the points are not randomly shifted. Higher order convergence in sdimensions is then achieved by taking a polynomial lattice rule in a higher dimension, ν·sforν∈N, and cleverly in- terlacing the digits across the dimensions of each ( νs)-dimensional point to produce an s-dimensional point. The factor ν∈Nis called the interlacing order and it determines 23the convergence rate. Good interlaced polynomial lattice r ules can also be constructed by a CBC algorithm. See [24] for the full details. Following [12], for ν∈Nand 1≤r≤ ∞we introduce the Banach space Wν,r s,γ, which is a higher-order analogue of the first-order space Ws,γ, with the norm /⌊a∇d⌊lf/⌊a∇d⌊lWν,r s,γ= max u⊆{1:s}1 γu/parenleftBigg/summationdisplay v⊆u/summationdisplay τu\v∈{1:ν}|u\v| /integraldisplay [−1 2,1 2]|v|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay [−1 2,1 2]s−|v|∂(νv,τu\v,0) y f(y) dy−v/vextendsingle/vextendsingle/vextendsingle/vextendsingler dyv/parenrightBigg1/r .(5.30) Here (νv,τu\v,0)∈ Fis the multi-index with jth entry given by νifj∈v,τjifj∈u\v and0otherwise. For f∈ Wν,r s,γ, anorder νinterlaced polynomiallattice ruleusing Npoints insdimensions can be constructed using a CBC algorithm such tha t the (deterministic) error converges at a rate N−ηfor 1≤η < ν(see [11, Theorem 3.10]). Letνp=⌊1/p⌋+ 1 forp <1 as in Assumption A1 and 1 ≤r≤ ∞, then it follows from (4.3) that λs∈ Wνp,r s,γfor alls. Hence, the error of a single level QMC approximation ofEy[λs] using an order νpinterlaced polynomial lattice rule will converge as N−1/p. Similarly, the ML analysis can be extended to show that a ML-H OQMC method achieves higher order convergence in Nℓ, where in this case we choose the interlacing factor to be νq=⌊1/q⌋+ 1 forq <1 as in Assumption A1. Indeed, (5.10) implies that the bound (5.15) can easily beextended to Wνq,r s,γand(4.3) implies that (5.17) can also beextended to Wνq,r s,γfor alls. In both cases, the sums over uon the right hand sides need to be updated to account for the form of (5.30), but the exponents of handsremain the same. Hence, by following the proof of Theorem 5.3 it can be shown that the f ollowing deterministic analogue of the variance bound (5.3) holds for interlaced po lynomial lattice rules. Theorem 5.4. Suppose that Assumption A1 holds with p < q <1. Forℓ∈N, letQHO ℓbe an interlaced polynomial lattice rule, constructed using a CBC algorithm with Nℓa prime power number of points and interlacing factor νq=⌊1/q⌋+1. ThenQHO ℓsatisfies /vextendsingle/vextendsingleQHO ℓ(λℓ−λℓ−1)/vextendsingle/vextendsingle/lessorsimilarN−1/q ℓ/parenleftbig h2 ℓ+s−1/p+1/q ℓ/parenrightbig , (5.31) where the implied constant is independent of hℓ,sℓandNℓ. The fact that the implied constant in (5.31) is independent o fsℓcan be shown by following similar arguments as in [12] using a special form o fγucalledsmoothness-driven, product and order-dependent (SPOD) weights, as introduced in [11, eq. (3.17)]. Thus the following deterministic version of Theorem 3.1 holds fo r the error of the ML-HOQMC approximation. Theorem 5.5. Suppose that Assumption A1 holds with p < q < 1, letL∈Nand for ℓ= 0,1,...,LletQHO ℓbe an interlaced polynomial lattice rule as in Theorem 5.4. The n the multilevel HOQMC approximation QMLHO Lwith quadrature rule QHO ℓon each level satisfies /vextendsingle/vextendsingleEy[λ]−QMLHO L(λ)/vextendsingle/vextendsingle/lessorsimilarh2 L+s−2/p+1 L+L/summationdisplay ℓ=0N−1/q ℓ/parenleftbig h2 ℓ+s−1/p+1/q ℓ/parenrightbig , where the implied constant is independent of hℓ,sℓandNℓfor allℓ= 0,1,...,L. Similar arguments can also be used to obtain an error bound wi th the same conver- gence rates for QMLHO L(G(u)), i.e., for the approximation of the expected value of smoo th functionals of the eigenfunction. 246 Conclusion We have presented a MLQMC algorithm for approximating the ex pectation of the eigen- value of a random elliptic EVP, and then performed a rigorous analysis of the error. The theoretical results clearly show that for this problem the M LQMC method exhibits better complexity than both single level MC/QMC and MLMC. In the com panion paper [21], we will present numerical results that also verify this supe rior performance of MLQMC in practice. In that paper, we will in addition present novel ideas on how to efficiently implement the MLQMC algorithm for EVPs. Other interesting avenues for future research would be to co nsider non-self adjoint EVPs, e.g., convection-diffusion problems, or to use the mult i-index MC framework from, e.g., [10, 27] to separate theFE anddimension truncation ap proximations on each level. In principle, the algorithm studied in this paper can also be ap plied to the lognormal setting as in, e.g., [33], i.e., whereeach coefficient is theexponent ial of a Gaussian randomfield, by using QMC rules for integrals on unbounded domains. However , in this case, the difficulty for both single level and multilevel QMC is that the coefficien ts are no longer uniformly boundedfromabove and below. As such, it is possiblethat the spectral gap, λ2(y)−λ1(y), becomes arbitrarily small for certain parameter values. Si nce all aspects of the method (the stochastic derivative bounds, the FE error, the perfor mance of the eigenvalue solver etc.) depend inversely on the spectral gap, then both the met hod and the theory fail if the gap becomes arbitrarily small. Thetechnique for boundingt he spectral gap in [19, 20] fails in this case because the stochastic parameters belong to an u nbounded domain. On the other hand, we conjecture that the spectral gap only becomes small with low probability, and so probabilistic arguments may be able to be used to bound the gap from below. This is again another example of the differences between stochasti c EVPs and source problems, and such analysis would make for interesting future work. Acknowledgements. 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[45] M. M. R. Williams. A method for solving stochastic eigen value problems. Appl. Math. Comput. , 215:4729––4744, 2010. [46] M. M. R. Williams. A method for solving stochastic eigen value problems II. Appl. Math. Comput. , 219:4729––4744, 2013. [47] J. Zech, D. Dung, and Ch. Schwab. Multilevel approximat ion of parametric and stochastic PDEs. Math. Models Methods Appl. Sci. , 29, 2019. A Proofs of recursive bounds on derivatives of the FE error Here, we give the proofs of the recursive bounds on the deriva tives of the FE error from Section 5.1 (Lemmas 5.2 and 5.3), which were key to the induct ive steps in the proofs of the explicit bounds in Theorem 5.1. Throughout we omit the xandydependence. Proof of Lemma 5.2 (eigenvalue bounds). Letv=vh∈Vhin thevariational eigenproblem (2.3), andthensubtracttheFEeigenproblem(2.10), withth esamevh,togivethefollowing variational relationship between the two FE errors A(u−uh,vh) =λM(u−uh,vh)+(λ−λh)M(uh,vh), (A.1) which holds for all vh∈Vh. Differentiating (A.1) using the Leibniz general product rule , gives the following recur- sive formula for the νth derivatives of the eigenvalue and eigenfunction errors 0 =A(∂ν y(u−uh),vh)−λM(∂ν y(u−uh),vh)−(λ−λh)M(∂ν yuh,vh) +∞/summationdisplay j=1νj/parenleftbigg/integraldisplay Daj∇/bracketleftbig ∂ν−ejy(u−uh)/bracketrightbig ·∇vh+/integraldisplay Dbj/bracketleftbig ∂ν−ejy(u−uh)/bracketrightbig vh/parenrightbigg −/summationdisplay m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg/bracketleftbig ∂ν−m yλM(∂m y(u−uh),vh)+∂ν−m y(λ−λh)M(∂m yuh,vh)/bracketrightbig . 28Adding extra terms and usingthe A-orthogonality of Ph, we can write this in the following more convenient form 0 =A(Ph∂ν y(u−uh),vh)−λhM(Ph∂ν y(u−uh),vh) −(λ−λh)M(∂ν yu,vh)−λhM(∂ν yu−Ph∂ν yu,vh) +∞/summationdisplay j=1νj/parenleftbigg/integraldisplay Daj∇/bracketleftbig ∂ν−ej y(u−uh)/bracketrightbig ·∇vh+/integraldisplay Dbj/bracketleftbig ∂ν−ej y(u−uh)/bracketrightbig vh/parenrightbigg −/summationdisplay m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg/bracketleftbig ∂ν−m yλM(∂m y(u−uh),vh)+∂ν−m y(λ−λh)M(∂m yuh,vh)/bracketrightbig .(A.2) Lettingvh=uhin (A.2) and separating out the m=0term, we obtain the following formula for the derivative of the eigenvalue error ∂ν y(λ−λh) = (λh−λ)M(∂ν yu,uh)−λhM(∂ν yu−Ph∂ν yu,uh)−(∂ν yλ)M(u−uh,uh) +∞/summationdisplay j=1νj/parenleftbigg/integraldisplay Daj∇/bracketleftbig ∂ν−ej y(u−uh)/bracketrightbig ·∇uh+/integraldisplay Dbj/bracketleftbig ∂ν−ej y(u−uh)/bracketrightbig uh/parenrightbigg −/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg/bracketleftbig ∂ν−m yλM(∂m y(u−uh),uh)+∂ν−m y(λ−λh)M(∂m yuh,uh)/bracketrightbig , where we have used the fact that uhis normalised. Also the first two terms in (A.2) cancel because the bilinear form is symmetric and ( λh,uh) satisfy the FE eigenvalue problem (2.10) with Ph∂ν y(u−uh)∈Vhas a test function. Taking the absolute value, then using the triangle and Cauch y–Schwarz inequalities gives the upper bound |∂ν y(λ−λh)| ≤ |λ−λh|/⌊a∇d⌊l∂ν yu/⌊a∇d⌊lM+λh/⌊a∇d⌊l∂ν yu−Ph∂ν yu/⌊a∇d⌊lM+|∂ν yλ|/⌊a∇d⌊lu−uh/⌊a∇d⌊lM +∞/summationdisplay j=1νj/bracketleftbig /⌊a∇d⌊laj/⌊a∇d⌊lL∞/⌊a∇d⌊l∂ν−ej y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊luh/⌊a∇d⌊lV+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/⌊a∇d⌊l∂ν−ej y(u−uh)/⌊a∇d⌊lL2/⌊a∇d⌊luh/⌊a∇d⌊lL2/bracketrightbig +/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg/bracketleftbig |∂ν−m yλ|/⌊a∇d⌊l∂m y(u−uh)/⌊a∇d⌊lM+|∂ν−m y(λ−λh)|/⌊a∇d⌊l∂m yuh/⌊a∇d⌊lM/bracketrightbig , where we have again simplified by using /⌊a∇d⌊luh/⌊a∇d⌊lM= 1. Then, using the equivalence of norms (2.8) and the Poincar´ e inequality (2.6), we can bound theM- andL2-norms by the corresponding V-norms, to give |∂ν y(λ−λh)| ≤/radicalbiggamax χ1/bracketleftBig |λ−λh|/⌊a∇d⌊l∂ν yu/⌊a∇d⌊lV+λ/⌊a∇d⌊l∂ν yu−Ph∂ν yu/⌊a∇d⌊lV+|∂ν yλ|/⌊a∇d⌊lu−uh/⌊a∇d⌊lV/bracketrightBig +u/parenleftbigg 1+1 χ1/parenrightbigg∞/summationdisplay j=1νjβj Cβ/⌊a∇d⌊l∂ν−ej y(u−uh)/⌊a∇d⌊lV +/radicalbiggamax χ1/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg/bracketleftbig |∂ν−m yλ|/⌊a∇d⌊l∂m y(u−uh)/⌊a∇d⌊lV+|∂ν−m y(λ−λh)|/⌊a∇d⌊l∂m yuh/⌊a∇d⌊lV/bracketrightbig , where we have also used the upper bounds (2.16) and (2.17), an d the definition of βj(4.1). 29Substituting in the upper bounds on the derivatives (4.3) an d (4.4), the bound on the projection error (5.5), and then the bounds on the FE errors ( 2.11) and (2.12), we have the upper bound |∂ν y(λ−λh)| ≤/radicalbiggamax χ1/bracketleftbig Cλhuβν+λCPβν+λCuβν/bracketrightBig h|ν|!1+ǫ +u Cβ/parenleftbigg 1+1 χ1/parenrightbigg∞/summationdisplay j=1νjβj/⌊a∇d⌊l∂ν−ej y(u−uh)/⌊a∇d⌊lV+/radicalbiggamax χ1/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg ·/bracketleftBig λ|ν−m|!1+ǫβν−m/⌊a∇d⌊l∂m y(u−uh)/⌊a∇d⌊lV+u|m|!1+ǫβm|∂ν−m y(λ−λh)|/bracketrightBig . Note that we can simplify the sum on the last line using the sym metry of the binomial coefficient,/parenleftbign k/parenrightbig =/parenleftbign n−k/parenrightbig , as follows. First, we separate it into two sums /summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg/bracketleftBig λ|ν−m|!1+ǫβν−m/⌊a∇d⌊l∂m y(u−uh)/⌊a∇d⌊lV+u|m|!1+ǫβm|∂ν−m y(λ−λh)|/bracketrightBig =λ/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg |ν−m|!1+ǫβν−m/⌊a∇d⌊l∂m y(u−uh)/⌊a∇d⌊lV +u/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg |m|!1+ǫβm|∂ν−m y(λ−λh)|/bracketrightBig =(λ+u)/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg |m|!1+ǫβm/bracketleftBig /⌊a∇d⌊l∂ν−m y(u−uh)/⌊a∇d⌊lV+|∂ν−m y(λ−λh)|/bracketrightBig , (A.3) where to obtain the last equality we have simply relabelled t he indices in the first sum. Then, since βj≤βjandhis sufficiently small (i.e., h≤hwithhas in (2.14)), the result (5.6) holds. The constant is given by CI:= max/braceleftbigg/radicalbiggamax χ1/bracketleftbig uhCλ+λ(CP+Cu)/bracketrightbig ,u Cβ/parenleftbigg 1+1 χ1/parenrightbigg ,/radicalbiggamax χ1(λ+u)/bracerightbigg , which is independent of y,handν. For the second result (5.7), using [4, Lemma 3.1] the eigenva lue error can also be written as λ−λh=−A(u−uh,u−uh)+λM(u−uh,u−uh), which after taking the νth derivative becomes ∂ν y(λ−λh) =−/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg A(∂ν−m y(u−uh),∂m y(u−uh)) −∞/summationdisplay j=1/summationdisplay m≤ν−ejνj/parenleftbiggν−ej m/parenrightbigg/bracketleftbigg/integraldisplay Daj∇∂ν−ej−m y(u−uh)·∇∂m y(u−uh) +/integraldisplay Dbj∂ν−ej−m y(u−uh)∂m y(u−uh)/bracketrightbigg +/summationdisplay m≤ν/summationdisplay k≤m/parenleftbiggν m/parenrightbigg/parenleftbiggm k/parenrightbigg ∂ν−m yλM(∂m−k y(u−uh),∂k y(u−uh)). 30Taking the absolute value, then usingthe triangle, Cauchy– Schwarz and Poincar´ e (2.6) inequalities, along with the norm equivalences (2.7), (2.8 ), gives |∂ν y(λ−λh)| ≤amax/parenleftbigg 1+1 χ1/parenrightbigg/summationdisplay m≤ν/parenleftbiggν m/parenrightbigg /⌊a∇d⌊l∂ν−m y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂m y(u−uh)/⌊a∇d⌊lV +/parenleftbigg 1+1 χ1/parenrightbigg∞/summationdisplay j=1/summationdisplay m≤ν−ejνj/parenleftbiggν−ej m/parenrightbiggβj Cβ/⌊a∇d⌊l∂ν−ej−m y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂m y(u−uh)/⌊a∇d⌊lV +amax χ1/summationdisplay m≤ν/summationdisplay k≤m/parenleftbiggν m/parenrightbigg/parenleftbiggm k/parenrightbigg |∂ν−m yλ|/⌊a∇d⌊l∂m−k y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂k y(u−uh))/⌊a∇d⌊lV. Finally, substituting in the upper bound (4.3) on the deriva tive ofλgives the desired result (5.7). The constant is given by CII:=/bracketleftbig 1/Cβ+amax(1+λ)/bracketrightbig/parenleftbigg 1+1 χ1/parenrightbigg , which is independent of h,yandν. Proof of Lemma 5.3 (eigenfunction bound). Wedealwiththeeigenfunctionerrorprojected ontoVh, as opposed to ∂ν y(u−uh), because the latter belongs to Vbut not to Vh. As such, we first separate the error as /⌊a∇d⌊l∂ν y(u−uh)/⌊a∇d⌊lV≤ /⌊a∇d⌊lPh∂ν y(u−uh)/⌊a∇d⌊lV+/⌊a∇d⌊l∂ν yu−Ph∂ν yu/⌊a∇d⌊lV ≤ /⌊a∇d⌊lPh∂ν y(u−uh)/⌊a∇d⌊lV+CPh|ν|!1+ǫβν, (A.4) where in the second inequality we have used the bound (5.5). Similar to the proof of [19, Lemma 3.4], the bilinear form tha t acts on Ph∂ν y(u−uh) (namely, A −λhM) is only coercive on the orthogonal complement of the eigens pace corresponding to λh, which we denote by E(λh)⊥. Hence, to obtain the recursive for- mula for the derivative of the eigenfunction error, we first m ake the following orthogonal decomposition. The FE eigenfunctions form an orthogonal ba sis forVh, and so we have Ph∂ν y(u−uh) =M(Ph∂ν y(u−uh),uh)uh+ϕh, (A.5) whereϕh∈E(λh)⊥. Then we can bound the norm by /⌊a∇d⌊lPh∂ν y(u−uh)/⌊a∇d⌊lV≤ |M(Ph∂ν y(u−uh),uh)|/⌊a∇d⌊luh/⌊a∇d⌊lV+/⌊a∇d⌊lϕh/⌊a∇d⌊lV. (A.6) To bound the first term in this decomposition (A.6), first obse rve that we can write |M(Ph∂ν y(u−uh),uh)| ≤|M(∂ν yu,u)−M(∂ν yuh,uh)| +|M(∂ν yu,u−uh)|+|M(∂ν yu−Ph∂ν yu,uh)|.(A.7) Thefirstterm ontheright in(A.7) can beboundedby differentia ting thenormalisation equations /⌊a∇d⌊lu/⌊a∇d⌊lM= 1 and /⌊a∇d⌊luh/⌊a∇d⌊lM= 1 (see [19, eq. (3.15)]) to give M(∂ν yu,u)−M(∂ν yuh,uh) =−1 2/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg/bracketleftbig M(∂ν−m yu,∂m yu)−M(∂ν−m yuh,∂m yuh)/bracketrightbig =−1 2/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg/bracketleftbig M(∂ν−m y(u−uh),∂m yu)+M(∂ν−m yuh,∂m y(u−uh))/bracketrightbig . 31Then, using the triangle inequality, the Cauchy–Schwarz in equality, the equivalence of norms (2.8) and the Poincar´ e inequality (2.6), gives the up per bound |M(∂ν yu,u)−M(∂ν yuh,uh)| ≤amax 2χ1/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg/bracketleftbig /⌊a∇d⌊l∂ν−m y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂m yu/⌊a∇d⌊lV+/⌊a∇d⌊l∂ν−m yuh/⌊a∇d⌊lV/⌊a∇d⌊l∂m y(u−uh)/⌊a∇d⌊lV/bracketrightbig ≤uamax 2χ1/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg/bracketleftbig |m|!1+ǫβm/⌊a∇d⌊l∂ν−m y(u−uh)/⌊a∇d⌊lV +|ν−m|!1+ǫβν−m/⌊a∇d⌊l∂m y(u−uh)/⌊a∇d⌊lV/bracketrightbig =uamax χ1/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg |m|!1+ǫβm/⌊a∇d⌊l∂ν−m y(u−uh)/⌊a∇d⌊lV, where for the second last inequality we have used the upper bo und(4.4) and the analogous bound for uh. For the equality on the last line, we have simplified the sum u sing the symmetry of the binomial coefficient as in (A.3). To bound the second and third terms in (A.7) we use the Cauchy– Schwarz inequality, the equivalence of norms (2.8), and the Poincar´ e inequalit y (2.6), followed by the bound on the projection error (5.5) and the bound on the FE error (2. 12), which gives |M(∂ν yu,u−uh)|+|M(∂ν yu−Ph∂ν yu,uh)| ≤ /⌊a∇d⌊l∂ν yu/⌊a∇d⌊lM/⌊a∇d⌊lu−uh/⌊a∇d⌊lM+/⌊a∇d⌊l∂ν yu−Ph∂ν yu/⌊a∇d⌊lM ≤amax χ1/⌊a∇d⌊l∂ν yu/⌊a∇d⌊lV/⌊a∇d⌊lu−uh/⌊a∇d⌊lV+/radicalbiggamax χ1/⌊a∇d⌊l∂ν yu−Ph∂ν yu/⌊a∇d⌊lV ≤amax χ1Cuh|ν|!1+ǫβν+/radicalbiggamax χ1CPh|ν|!1+ǫβν ≤/parenleftbiggamax χ1Cu+/radicalbiggamax χ1CP/parenrightbigg h|ν|!1+ǫβν, (A.8) where in the last inequality we have used that βj≤βj. Substitutingthese twoboundsinto(A.7)then multiplyingb y/⌊a∇d⌊luh/⌊a∇d⌊lVgives thefollowing upper bound on the first term of the decomposition (A.6) |M(Ph∂ν y(u−uh),uh)|/⌊a∇d⌊luh/⌊a∇d⌊lV≤u/parenleftbiggamax χ1Cu+/radicalbiggamax χ1CP/parenrightbigg h|ν|!1+ǫβν +u2amax χ1/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν|m|!1+ǫβm/⌊a∇d⌊l∂ν−m y(u−uh)/⌊a∇d⌊lV.(A.9) Note that we have also used (2.17). Next, to bound the norm of ϕh(the second term in the decomposition (A.6)), we let 32vh=ϕhin (A.2) and then rearrange the terms to give A(Ph∂ν y(u−uh),ϕh)−λhM(Ph∂ν y(u−uh),ϕh) = (λ−λh)M(∂ν yu,ϕh) +λhM(∂ν yu−Ph∂ν yu,ϕh)+∂ν yλM(u−uh,ϕh)+∂ν y(λ−λh)M(uh,ϕh) −∞/summationdisplay j=1νj/parenleftbigg/integraldisplay Daj∇/bracketleftbig ∂ν−ej y(u−uh)/bracketrightbig ·∇ϕh+/integraldisplay Dbj/bracketleftbig ∂ν−ej y(u−uh)/bracketrightbig ϕh/parenrightbigg (A.10) +/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg/bracketleftbig ∂ν−m yλM(∂m y(u−uh),ϕh)+∂ν−m y(λ−λh)M(∂m yuh,ϕh)/bracketrightbig . Again using the decomposition (A.5) and the fact that uhsatisfies the eigenproblem (2.10) with ϕh∈Vhas atest function, the lefthandsideof (A.10) simplifiesto A(ϕh,ϕh)− λhM(ϕh,ϕh). Since ϕh∈E(λh)⊥, we can use the FE version of the coercivity estimate [19, Lemma 3.1] (see also Remark 3.2 that follows), to bound t his from below by A(ϕh,ϕh)−λhM(ϕh,ϕh)≥amin/parenleftbiggλ2,h−λh λ2,h/parenrightbigg /⌊a∇d⌊lϕh/⌊a∇d⌊l2 V≥aminρ 2λ2/⌊a∇d⌊lϕh/⌊a∇d⌊l2 V,(A.11) where in the last inequality we have used the upper bound (2.1 6), along the lower bound (2.15) on the FE spectral gap, which is applicable for hsufficiently small. Taking the absolute value, the right hand side of (A.10) can b e bounded using the triangle inequality, the Cauchy–Schwarz inequality, the e quivalence of norms (2.8), and the Poincar´ e inequality (2.6), which, combined with the lo wer bound (A.11), gives aminρ 2λ2/⌊a∇d⌊lϕh/⌊a∇d⌊l2 V≤amax χ1/parenleftBig |λ−λh|/⌊a∇d⌊l∂ν yu/⌊a∇d⌊lV+λ/⌊a∇d⌊l∂ν yu−Ph∂ν yu/⌊a∇d⌊lV +|∂ν yλ|/⌊a∇d⌊lu−uh/⌊a∇d⌊lV+|∂ν y(λ−λh)|/⌊a∇d⌊luh/⌊a∇d⌊lV/parenrightBig /⌊a∇d⌊lϕh/⌊a∇d⌊lV +∞/summationdisplay j=1νj/parenleftbigg /⌊a∇d⌊laj/⌊a∇d⌊lL∞+1 χ1/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/parenrightbigg /⌊a∇d⌊l∂ν−ej y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊lϕh/⌊a∇d⌊lV +amax χ1/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg/parenleftbig |∂ν−m yλ|/⌊a∇d⌊l∂m y(u−uh)/⌊a∇d⌊lV+|∂ν−m y(λ−λh)|/⌊a∇d⌊l∂m yuh/⌊a∇d⌊lV/parenrightbig /⌊a∇d⌊lϕh/⌊a∇d⌊lV. Dividing through by aminρ/(2λ2)/⌊a∇d⌊lϕh/⌊a∇d⌊lV, then using the bounds (2.11), (4.3), (4.4), (5.5), along with the fact that that βj≤βjfor allj∈Nandh≤h, we have that the norm of ϕhis bounded by /⌊a∇d⌊lϕh/⌊a∇d⌊lV≤amax amin2λ2 ρχ1/bracketleftBig uhCλ+λCP+λCu/bracketrightBig h|ν|!1+ǫβν+amax amin2uλ2 ρχ1|∂ν y(λ−λh)|(A.12) +2λ2 aminρCβ/parenleftbigg 1+1 χ1/parenrightbigg∞/summationdisplay j=1νjβj/⌊a∇d⌊l∂ν−ej y(u−uh)/⌊a∇d⌊lV +amax amin2λ2 ρχ1(λ+u)/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg |m|!1+ǫβm/bracketleftBig /⌊a∇d⌊l∂ν−m y(u−uh)/⌊a∇d⌊lV+/⌊a∇d⌊l∂ν−m y(λ−λh)|/bracketrightBig . Note that to get the sum on the last line we have again simplifie d similarly to (A.3). 33Substituting the bounds (A.6), followed by (A.9) and (A.12) , into the decomposition (A.4) gives the following recursive bound on the derivative of the eigenfunction error /⌊a∇d⌊l∂ν y(u−uh)/⌊a∇d⌊lV ≤/bracketleftbiggamax amin2λ2 ρχ1/parenleftbig uhCλ+λCP+λCu/parenrightbig +u/parenleftbiggamax χ1Cu+/radicalbiggamax χ1CP/parenrightbigg +CP/bracketrightbigg h|ν|!1+ǫβν +amax amin2uλ2 ρχ1|∂ν y(λ−λh)|+2λ2 aminρCβ/parenleftbigg 1+1 χ1/parenrightbigg∞/summationdisplay j=1νjβj/⌊a∇d⌊l∂ν−ej y(u−uh)/⌊a∇d⌊lV +amax amin2λ2 ρχ1(λ+u)/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg |m|!1+ǫβm/bracketleftBig /⌊a∇d⌊l∂ν−m y(u−uh)/⌊a∇d⌊lV+/⌊a∇d⌊l∂ν−m y(λ−λh)|/bracketrightBig +u2amax χ1/summationdisplay 0/\e}atio\slash=m≤ν m/\e}atio\slash=ν/parenleftbiggν m/parenrightbigg |m|!1+ǫβm/⌊a∇d⌊l∂ν−m y(u−uh)/⌊a∇d⌊lV. Observe that all of the constant terms are independent of y,handν. To obtain the final result with a right handside that does not d ependon any derivative of orderν, we now substitute the recursive formula (5.6) for ∂ν y(λ−λh). After grouping the similar terms and collecting all of the constants into CIIIwe have the final result. SinceCIfrom (5.6) and all of the constants above are independent of y,h, andν, the final constant CIIIis as well. 34

2020-10-02

Stochastic PDE eigenvalue problems are useful models for quantifying the uncertainty in several applications from the physical sciences and engineering, e.g., structural vibration analysis, the criticality of a nuclear reactor or photonic crystal structures. In this paper we present a multilevel quasi-Monte Carlo (MLQMC) method for approximating the expectation of the minimal eigenvalue of an elliptic eigenvalue problem with coefficients that are given as a series expansion of countably-many stochastic parameters. The MLQMC algorithm is based on a hierarchy of discretisations of the spatial domain and truncations of the dimension of the stochastic parameter domain. To approximate the expectations, randomly shifted lattice rules are employed. This paper is primarily dedicated to giving a rigorous analysis of the error of this algorithm. A key step in the error analysis requires bounds on the mixed derivatives of the eigenfunction with respect to both the stochastic and spatial variables simultaneously. Under stronger smoothness assumptions on the parametric dependence, our analysis also extends to multilevel higher-order quasi-Monte Carlo rules. An accompanying paper [Gilbert and Scheichl, 2022], focusses on practical extensions of the MLQMC algorithm to improve efficiency, and presents numerical results.

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: Regularity and error analysis

2010.01044v4

Stoner-Wohlfarth switching of the condensate magnetization in a dipolar spinor gas and the metrology of excitation damping Seong-Ho Shinn,1Daniel Braun,2and Uwe R. Fischer1 1Department of Physics and Astronomy, Seoul National University, 08826 Seoul, Korea 2Eberhard-Karls-Universit at T ubingen, Institut f ur Theoretische Physik, 72076 T ubingen, Germany (Dated: May 13, 2020) We consider quasi-one-dimensional dipolar spinor Bose-Einstein condensates in the homogeneous- local-spin-orientation approximation, that is with unidirectional local magnetization. By analyti- cally calculating the exact eective dipole-dipole interaction, we derive a Landau-Lifshitz-Gilbert equation for the dissipative condensate magnetization dynamics, and show how it leads to the Stoner- Wohlfarth model of a uni-axial ferro-magnetic particle, where the latter model determines the stable magnetization patterns and hysteresis curves for switching between them. For an external magnetic eld pointing along the axial, long direction, we analytically solve the Landau-Lifshitz-Gilbert equa- tion. The solution explicitly demonstrates that the magnetic dipole-dipole interaction accelerates the dissipative dynamics of the magnetic moment distribution and the associated dephasing of the magnetic moment direction. Under suitable conditions, dephasing of the magnetization direction due to dipole-dipole interactions occurs within time scales up to two orders of magnitude smaller than the lifetime of currently experimentally realized dipolar spinor condensates, e.g., produced with the large magnetic-dipole-moment atoms166Er. This enables experimental access to the dissipation parameter in the Gross-Pitaevski  mean-eld equation, for a system currently lacking a complete quantum kinetic treatment of dissipative processes and, in particular, an experimental check of the commonly used assumption that is a single scalar independent of spin indices. I. INTRODUCTION Ever since a phenomenological theory to describe the behavior of super uid helium II near the point has been developed by Pitaevski  [1], the dynamics of Bose- Einstein condensates (BEC) under dissipation has been intensely studied, see, e.g., [2{8]. Experimentally, the im- pact of Bose-Einstein condensation on excitation damp- ing and its temperature dependence has for example been demonstrated in [9{12]. Dissipation in the form of condensate loss is dened by a dimensionless damping rate entering the left- hand side of the Gross-Pitaevski  equation, replacing the time derivative as i@t!(i)@t. While a micro- scopic theory of condensate damping is comparatively well established in the contact-interaction case, using var- ious approaches, cf., e.g., [5, 13{15], we emphasize the absence of a microscopic theory of damping in dipolar spinor gases. While for scalar dipolar condensates, par- tial answers as to the degree and origin of condensate- excitation damping have been found see, e.g., Refs. [16{ 19], in spinor or multicomponent gases the interplay of anisotropic long-range interactions and internal spinor or multicomponent degrees of freedom leads to a highly in- tricate and dicult-to-disentangle many-body behavior of condensate-excitation damping. In this paper, we propose a method to experimen- tally access in a dipolar spinor condensate by using the dynamics of the unidirectional local magnetization in a quasi-one-dimensional (quasi-1D) dipolar spinor BEC in the presence of an external magnetic eld. To this end, we rst derive an equation of motion for the mag- netization of the BEC that has the form of a Landau- Lifshitz-Gilbert (LLG) equation [20{22], with an addi-tional term due to the dipole-dipole interaction between the atoms. The LLG equation is ubiquitous in nano- magnetism, where it describes the creation and dynam- ics of magnetization. The static limit of this equation is, in the limit of homogeneous local spin-orientation, de- scribed by the well-known Stoner-Wolfarth (SW) model [23{25] of a small magnetic particle with an easy axis of magnetization. We then investigate the magnetization switching after ipping the sign of the external magnetic eld, and demonstrate the detailed dependence of the switching dynamics on the dissipative parameter . For a quasi-2D spinor BEC with inhomogeneous local magnetization, Ref. [26] has studied the magnetic domain wall formation process by deriving a LLG type equa- tion. Here, we derive the LLG equation in a quasi-1D spinor BEC with unidirectional local magnetization, in order to establish a most direct connection to the orig- inal SW model. In distinction to [27], which studied the eective quasi-1D dipole-dipole interaction resulting from integrating out the two transverse directions within a simple approximation, we employ below an exact ana- lytic form of the dipole-dipole interaction. In Section II, we establish the quasi-1D spinor Gross-Pitaevski  (GP) equation with dissipation, and equations of motion for the magnetization direction (unit vector) M. Section V shows how the LLG equation and the SW model result, and Section VI derives analytical solutions to the equa- tions of motion for Mwhen the external magnetic eld points along the long, zaxis. We summarize our results in section VII. We defer two longer derivations to Appendices. The analytical form of the eective dipole-dipole interaction energy is deduced in Appendix A, and the quasi-1D GP mean-eld equation with dissipation is described in detailarXiv:2002.08723v2 [cond-mat.quant-gas] 12 May 20202 in Appendix B. Finally, in Appendix C, we brie y discuss to which extent relaxing the usual simplifying assumption that dissipation even in the spinor case is described by a single scalar changes the LLG equation, and whether this aects the SW model and its predictions. II. GENERAL DESCRIPTION OF DAMPING IN BECS The standard derivation of the quantum kinetics of Bose-Einstein condensate damping [5] starts from the microscopic Heisenberg equation of motion for the quan- tum eld operator ^ (r;t), for a scalar (single compo- nent) BEC in the s-wave scattering limit. Using their results, [28] obtained a mean-eld equation to describe the dissipation of scalar BEC, whose form is (i)~@ @t=H (1) where is the (in the large Nlimit) dominant mean-eld part upon expanding the full bosonic eld operator ^ . In Ref. [1], Pitaevski  obtained a similar but slightly dierent form of the dissipative mean-eld equation based on phenomenological considerations, i~@ @t= (1i)H , by parametrizing the deviation from exact continuity for the condensate fraction while minimizing the energy [1]. The latter deviation is assumed to be small, which is equivalent to assuming that remains small. This provides a clear physical interpretation of the damping mechanism, namely one based on particle loss from the condensate fraction. The version of Pitaevski  can be written as (i)~@ @t= 1 + 2
H : (2) It can thus be simply obtained by rescaling time with a factor 1 + 2compared to (1). Hence, as long as one does not predict precisely , the two dissipative equa- tions (1) and (2) cannot be distinguished experimentally from the dynamics they induce. From the data of [11], [4] estimated typical values of '0:03 for a scalar BEC of23Na atoms (see also [12]), which shows that to distin- guish between (1) and (2) experimentally the theoretical predictions of would need to be precise to the order of 104. How eqs.(1) and (2) can be generalized to the dipolar spinor gases is comparatively little investigated. Using a symmetry-breaking mean-eld approach by writing the quantum eld operator as ^ (r;t) as ^ (r;t) = (r;t) + ^ (r;t), with (r;t) =h^ (r;t)iandh^ (r;t)i= 0, [5] and [28] showed that is derived from the three-eld correlation function h^ y(r;t)^ (r;t)^ (r;t)iin a ba- sis whereh^ (r;t)^ (r;t)i= 0. From this microscopicorigin, based on correlation functions, it is clear that in principle might depend on the spin indices in a spinor BECs and hence become a tensor (see Appendix C for a corresponding phenomenological generalization). Nev- ertheless, it is commonly assumed cf.,e.g., [26, 29], that does not depend on spin indices, and the scalar value found specically in [4] for a scalar BEC of23Na atoms is commonly used, while a clear justication of this as- sumption is missing. Extending the microscopic derivations in [5] and [28] to the spinor case would be theoretically interesting, but is beyond the scope of the present paper. Here, we instead focus on the question whether the standard assumption that the damping of each spinor component can be de- scribed by the mean-eld equation [28] leads to exper- imentally falsiable dynamical signatures. It will turn out that this assumption introduces an additional strong dephasing in the spin-degrees of freedom, amplied by the dipolar interaction. Hence, even on time scales on which the decay of the condensate fraction according to (1) can be neglected, the relaxation of the magnetization of the BEC potentially oers valuable insights whether the scalar- assumption is justied. Indeed, in [30] it was shown experimentally that on the time scale of the switching dynamics of the magnetization the number of particles in the condensates remains approximately con- stant. One might wonder, then, which dissipative mech- anism is left. However, as we will show, by assuming the same GP equation for each component of the spinor as for scalar bosons, additional dephasing occurs that is in fact much more rapid than the decay of condensate den- sity due to dephasing accelerated by the dipole-dipole interaction. III. MEAN-FIELD DYNAMICS OF DAMPING IN DIPOLAR SPINOR BECS For a spinor BEC, linear and quadratic Zeeman inter- actions are commonly included in the Hamiltonian. The quadratic Zeeman interaction is related to a second-order perturbation term in the total energy that can be induced by the interaction with an external magnetic eld ( qB) as well as with the interaction with a microwave eld (qMW) [31]. Specically, by applying a linearly polarized microwave eld, one can change qMWwithout changing qB[32, 33]. Hence, we will assume that the quadratic Zeeman term can be rendered zero by suitably changing qMW. Following [26], we thus assert that for a dipolar spinor BEC without quadratic Zeeman term, the mean-eld equation can be written as3 (i)~@ (r;t) @t= ~2 2mr2+Vtr(r) +c0j (r;t)j2~fbbdd(r;t)g
^f (r;t) +SX k=1c2kX 1;2;


;k=x;y;zF1;2;


;k(r;t)^f1^f2


^fk (r;t): (3) where (r;t) is a vector quantity whose -th component in the spinor basis is (r;t) (spin-space indices from the beginning of the Greek alphabet such as ;; ;::: are integers running from StoS). In this expres- sion, ~^fis the spin- Soperator where the spin ladder is dened by ^fzji=jiandhji=;, while F1;2;


;k(r;t):= y(r;t)^f1^f2


^fk (r;t) are the components of the expectation value of ^f1^f2


^fk. The Larmor frequency vector reads b=gFBB=~ (with Land e g-factor gF, Bohr magneton B, and the external magnetic induction B),~bdd(r;t)
e= cddR d3r0P 0=x;y;zQ;0(rr0)F0(r0;t):Here,cdd= 0(gFB)2=(4) andeis a unit vector along the  axis [31] (by convention, indices from the middle of the Greek alphabet such as ;;;;::: =x;y;z denote spa- tial indices), and Q;0is the spin-space tensor dened in Eq. (A2) of Appendix A. Finally, mis the boson mass, c0the density-density interaction coecient, and c2kthe interaction coecient parametrizing the spin-spin inter- actions, where kis an positive integer running from 1 toS[26]. For example, c2is the spin-spin interaction coecient of a spin-1 gas ( S= 1). To develop a simple and intuitive physical approach, we consider a quasi-1D gas for which one can perform analytical calculations. We set the trap potential as Vtr(x;y;z ) =1 2m!2 ? x2+y2
+V(z); (4) so that the long axis of our gas is directed along the z axis and the gas is strongly conned perpendicularly. For a harmonic trap along all directions, i.e. when V(z) =m!2 zz2=2, we set!?!z. For a box trap along z, i.e. when V(z) = 0 forjzj LzandV(z) =1 FIG. 1. Schematic of the considered geometry in a quasi-1D gas (shaded ellipsoid). The length of the red magnetization arrows, all pointing in the same direction (homogeneous local- spin-orientation limit), represents jd(z;t)j.forjzj> Lz, our gas will be strongly conned along z as long as the quasi-1D condition is satised, where we will discuss below whether the condition is satised, in section VI A. Single-domain spinor BECs have been already real- ized, for example, using spin-187Rb [34]. This single- domain approximation is common in nanomagnetism, see for example [24], by assuming magnetic particles much smaller than the typical width of a domain wall. The local magnetization is related to the expectation value ~F(r;t)~ y(r;t)^f (r;t) of the spatial spin density operator byd(r;t) =gFBF(r;t). An unidirectional local magnetization d(z;t) is then given by dx(z;t) =d(z;t) sin(t) cos(t); dy(z;t) =d(z;t) sin(t) sin(t); (5) dz(z;t) =d(z;t) cos(t); whered(z;t) =d(z;t)
eis the-th component of d(z;t),d(z;t) =jd(z;t)j,(t) is polar angle of d(z;t), and(t) is azimuthal angle of d(z;t). For an illustra- tion of the geometry considered, see Fig. 1. For a single component dipolar BEC, F(r;t) has a xed direction. To study the relation of the Stoner-Wohlfarth model, in whichF(r;t) changes its direction, with a dipolar BEC, a multi-component dipolar BEC should therefore be em- ployed. In the quasi-1D approximation, the order parameter (r;t) is commonly assumed to be of the form (r;t) =e2=(2l2 ?) l?p (z;t): (6) wherel?is the harmonic oscillator length in the xy plane and=p x2+y2. Assuming our gas is in the homogeneous local spin-orientation limit, we may also apply a single mode approximation in space so that (z;t) = uni(z;t)(t). The time-dependent spinor part is (t) =hjei^fz(t)ei^fy(t)jSi; (7) for spin-Sparticles [26, 31] and the normalization reads j(t)j2:=y(t)(t) = 1. Finally, due to the ( i) factor on the left-hand side of Eq. (3), for the ease of calculation, we may make the following ansatz for the (r;t), cf. Ref. [35], (r;t) =e2=(2l2 ?) l?p (z;t)(t)e(i+)!?t=(1+2): (8)4 From our ans atze in Eq. (7) and (8), one concludes that the expectation value of the (spatial) spin-density oper- ator is ~Fx(r;t) =~Se2=l2 ? l2 ?j (z;t)j2e2!?t=(1+2) sin(t) cos(t); ~Fy(r;t) =~Se2=l2 ? l2 ?j (z;t)j2e2!?t=(1+2) sin(t) sin(t); ~Fz(r;t) =~Se2=l2 ? l2 ?j (z;t)j2e2!?t=(1+2) cos(t): (9) The above equations lead to unidirectional local magneti- zation, which has been assumed in Eqs. (5), in the quasi- 1D limit (after integrating out the strongly conning xandyaxes). Note however that our ansatz in Eq. (8) is sucient, but not necessary for the homogeneous-local- spin-orientation limit, and the homogeneous-local-spin- orientation ansatz is thus designed to render our ap- proach as simple as possible. Because we are not assuming any specic form of (z;t) in our ansatz in Eq. (8),we cover every possible time behavior of j (r;t)j2:= y(t) (t): j (r;t)j2=e2=l2 ? l2 ?j (z;t)j2e2!?t=(1+2):(10) Eq. (10) explicitly shows that Eq. (8) does not imply an exponentially decaying wavefunction with time since j (z;t)j2can be any physical function of time t. How- ever, the ansatz (8) simplies the resulting equation for (z;t), Eq.(11) below. By integrating out the xandydirections, the GP equa- tion for a quasi-1D spin- SBEC can be written as (see for a detailed derivation Appendix B) (i)~@f (z;t)(t)g @t= ~2 2m@2 @z2+V(z) +c0 2l2 ?n(z;t) (z;t)(t) +~[b+SfM(t)3Mz(t)ezgPdd(z;t)]
8 < :SX =S ^f ; (z;t)(t)9 = ; +SX k=1c2k 2l2 ?n(z;t)X 1;2;


;k=x;y;zSM1;2;


;k(t)8 < :SX =S ^f1^f2


^fk ; (z;t)(t)9 = ;; (11) where we dened the two functions M1;2;


;k(t):=1 SSX ;=Sy (t) ^f1^f2


^fk ;(t); (12) Pdd(z;t):=cdd 2~l3 ?Z1 1dz0n(z0;t) Gjzz0j l? 4 3zz0 l? ; (13) with the axial density n(z;t):=R d2j (r;t)j2= j (z;t)j2e2!?t=(1+2), whereR d2 :=R1 1dxR1 1dy. Finally, the function Gappearing inPddis dened as G():=r 2 2+ 1
e2=2Erfcp 2 :(14) We plotG() as a function of in Fig. 2. Eq.(11) rep- resents our starting point for analyzing the dynamics of magnetization. We will now proceed to show how it leads to the LLG equation and the Stoner-Wolfarth model.IV. EFFECTIVE LANGRANGIAN DESCRIPTION To provide a concise phase space picture of the conden- sate magnetization dynamics, we discuss in this section a collective coordinate Lagrangian appropriate to our sys- tem. LetM(t):=d(z;t)=d(z;t) where the magne- tizationd(z;t) is dened in Eq. (5). Explicitly, the local magnetization direction reads M(t) = (sin(t) cos(t);sin(t) sin(t);cos(t)). Then, from Eqs. (9) and (10), F(r;t) =SM(t)j (r;t)j2and one5 1 2 3 4 50.20.40.60.81.01.2 FIG. 2. The function G() dened in Eq. (14). Note that G()'2=3+O 5
for1, soG() is always positive for0. obtains (see for a detailed derivation Appendix B) @M @t=Mfb+S0 dd(t)MzezgM@M @t;(15) where the renormalized interaction function 0 dd(t) reads 0 dd(t) =3 N(t)Z1 1dzn(z;t)Pdd(z;t);(16) andN(t):=R d3rj (r;t)j2=R1 1dz n (z;t). From Eqs. (A9), (A12), and (13), 0 dd(t) is connected to the dipole-dipole interaction contribution Vdd(t) by Vdd(t) =3 2~S2 sin2(t)2 3Z1 1dzn(z;t)Pdd(z;t) =~ 2S2N(t) 0 dd(t)1 3cos2(t) : (17) We note that in order to obtain the eective quasi-1D dipolar interaction (17), we did not use, in distinction to Ref. [27], any simplifying approximation. A detailed derivation is provided in Appendix A. Eq.(15) is the LLG equation with the external mag- netic eld in z-direction modied by the magnetiza- tion inz-direction due to the dipole-dipole interac- tion. The corresponding term in units of magnetic eld, ~S0 dd(t)Mzez=(gFB), can be seen as an additional magnetic eld that is itself proportional to the magne- tization inz-direction, and which leads to an additional nonlinearity in the LLG equation. From Eqs. (13) and (16), to get how 0 dd(t) depends on timet, one has to calculate the double integral Z dzZ dz0n(z;t)n(z0;t) Gjzz0j l? 4 3zz0 l? : (18) To achieve a simple physical picture, we assume that n(z;t) does not depend on time twithin the time range we are interested in. Then we may write 0 dd(t) = 0 dd. The lifetime of a typical dipolar BEC with large atomicmagnetic dipole moments such as164Dy [36],162Dy and 160Dy [37], or166Er [30] is of the order of seconds. Since taking into account the time dependence of n(z;t) gen- erally requires a numerical solution of Eq. (11), we here consider the case where n(z;t) is constant in time tas in [26], to predominantly extract the eect of magnetic dipole-dipole interaction per se . We also neglect the possible eect of magnetostriction. The latter eect, amounting to a distortion of the aspect ratio of the condensate in a harmonic trap as a func- tion of the angle of the external magnetic eld with the symmetry axis of the trap, was measured in a conden- sate of Chromium atoms [38] (with a magnetic moment of 6B). The magnetostriction eect in that experiment was of the order of 10%. For alkali atoms with spin-1 the eect should be a factor 62smaller. In addition, the- oretical analyses in the Thomas-Fermi limit show that magnetostriction in harmonic traps becomes particularly small for very small or very large asymmetries of the trap [39, 40]. More specically, Ref. [41] has shown that magne- tostriction is due to the force induced by the dipole-dipole mean-eld potential  dd(r;t). In Appendix D, we ap- ply the approach of [41] to a dipolar spinor BEC. From Eqs. (16), (17), (A1), and (D5), 0 dd(t) contains  dd(z;t) [the quasi-1D form of  dd(r;t) dened in Eq. (D5)] by S2 13M2 z(t) N(t)~0 dd(t) = 3Z1 1dzn(z;t) dd(z;t):(19) Hence, our LLG-type equation in Eq. (15) eectively con- tains the dipole-dipole mean-eld potential which causes magnetostriction and the form of Eq. (15) itself will not be changed whether the eect of magnetostriction is large or not. Only the value of 0 dd(t) will be changed because magnetostriction changes the integration domain. Fur- thermore, we show in Appendix D that for our quasi-1D system, the eect of magnetostriction is smaller in a box trap than in harmonic trap. In fact, for the box trap, this eect can be neglected if Lz=l?is suciently large. Thus, we may neglect the eect of magnetostriction un- der suitable limits for both box and harmonic traps. To get a simple physical idea of the dynamical behavior of our system, let us, for now, assume that there is no damping, = 0. When the external magnetic eld is chosen to lie in the xzplane,B= (Bx;0;Bz), Eq. (15) becomes d dt=bxsin; d dt=bxcotcosbzS0 ddcos: (20) where we already dened the Larmor frequency vector b=gFBB=~below Eq. (3). By using the Lagrangian formalism introduced in [42],6 the Lagrangian Lof this system then fullls L ~=_cos+bxsincos+bzcos+S 40 ddcos (2); (21) where _=d=dt . The equations of motion are 1 ~@L @=_sin+bxcoscosbzsinS 20 ddsin (2); @L @_= 0;1 ~@L @=bxsinsin;1 ~@L @_= cos:(22) One easily veries that Eq. (21) is indeed the Lagrangian which gives Eqs. (20). Let pbe the conjugate momen- tum of the coordinate . Sincep= 0 andp=~cos (~times thezcomponent of M), the Hamiltonian His given by H=bxq ~2p2 cosbzp+~22p2  4~S0 dd:(23) Note that the energy ~E:=H~S0 dd=4 is conserved. Hence, if we put p= (p)inand==2 at some time t=t0,~E=bz(p)inS0 dd(p)2 in=2~. We can then expressas a function of pas cos=~E+bzp+1 2~S0 ddp2  bxq ~2p2  = (p)inp bz+S0 dd(p)in+p 2~ bxq ~2p2 : (24) The canonical momentum premains the initial ( p)in whenbx= 0, implying that does not change when bx= 0, consistent with Eqs. (20). If jbxjis larger than jbzS0 ddj, we can have p6= (p)inwithjcosj1, which allows for the switching process of the magneti- zation. Below a threshold value of jbxjthat depends on bzandS0 dd,phas to remain constant for Eq. (24) to be satised, which corresponds to simple magnetization precession about the zaxis. Whenpis a function of time, there are two important cases: (a)jbzjS0 dd: cos=bz bx(p)inpq ~2p2 ; (b)jbzjS0 dd: cos=S0 dd 2bx(p)2 inp2  ~q ~2p2 :(25) We plot the corresponding phase diagrams ( vs) in Fig. 3. Let (p)in=~cosin,bx=bsin0, andbz=bcos0. When case (a) holds jbzjS0 dd, one concludes that cos0cos+ sin0sincos= cos0cosin, which is constant. Since d
b=db(cos0cos+ sin0sincos), in case (a) the magnetization dprecesses around the ex- ternal magnetic eld B, as expected. When (b) holds, SW switching can occur, to the description of which we proceed in the following. 0.5 1.0 1.5 2.0 -1.0-0.50.51.0(p)in=~=2 andin==2 0.5 1.0 1.5 2.0 -1.0-0.50.51.0 (p)in=~=2andin==2 FIG. 3.p=~vs=when = 0 (no dissipation), with initial values (p)inandin(initial value of ) as shown. (1) Dashed blue:bz=bx= 0:2 andjbzjS0 dd. (2) Black line: bz=bx= 0:2 andS0 dd=bx= 0:6. (3) Dash-Dotted red: S0 dd=bx= 0:6 andjbzjS0 dd. (4) Dotted orange horizontal line: bx= 0. V. CONNECTION TO STONER-WOHLFARTH MODEL The phenomenological SW model can be directly read o from the equations in the preceding section. From Eq. (23), ~H:=H+~S0 dd=4 is given by ~H ~=bxsincosbzcos+S0 dd 2sin2: (26) Let (b)crbe the value of bat the stability limit where @~H=@ = 0 and@2~H=@2= 0. Then one obtains the critical magnetic elds (bx)crcos=S0 ddsin3;(bz)cr=S0 ddcos3: (27) which satisfy the equation f(bx)crcosg2=3+ (bz)2=3 cr=fS0 ddg2=3: (28) We coin the curve in the ( bx;bz)-plane described by Eq. (28) the switching curve, in accordance with the ter- minology established in [43]. Because changes in time [see Eqs. (20) and Fig. 3], the switching curve depends in general on the timing of the applied external magnetic elds. We note that, for = 0, Eqs. (26) and (28) are identical to the SW energy functional HSW ~=bxsinbzcos+Ksin2 (29)7 and the SW astroid [43], respectively, if we identify K= S0 dd=2. The LLG equation in Eq. (15) has stationary solu- tions withMparallel to the eective magnetic eld ~fb+S0 dd(t)Mzezg=(gFB). Since we set bto lie in thexzplane,will go to zero for suciently large times. Thus Eq. (26) leads to the SW model (29) due to the damping term in (15) if >0. In Appendix C, we demonstrate that a more general tensorial damping coecient introduces additional terms on the right- hand side of the LLG equation (15), which involve time derivatives . While these will thus not aect the SW phe- nomenology, which results from the steady states as func- tion of the applied magnetic elds, and which is thus gov- erned by the vanishing (in the stationary limit) of the rst term on the right-hand side of the LLG equation, they aect the detailed relaxation dynamics of the magneti- zation and its time scales. These deviations can hence can be used to probe deviations from assuming a single scalar . Before we move on to the next section, we show the characteristic behavior of 0 dddened in Eq. (16), for a box-trap scenario dened by n(z;t) =N=(2Lz) for LzzLzandn(z;t) = 0 otherwise ( Nis number of particles). We stress that due to the nite size of the trap along the \long" zdirection, in variance with the Hohenberg- Mermin-Wagner theorem holding for innitely extended systems in the thermodynamic limit, a quasi-1D BEC can exist also at nite temperatures [44]. This remains true up to a ratio of its proper length to the de-Broglie wave- length [45], beyond which strong phase uctuations set in [46]. In fact, these strongly elongated quasi-1D BECs at nite temperature have been rst realized already long ago, cf., e.g. [47]. For the box trap, 0 dd= dd(Lz=l?) where dd() =3Ncdd 2~l3 ?1 (Z2 0dv 1v 2 G(v)2 3) : (30) From Eq. (14), G(v)'2=v3+O v5
forv1, so that dd()'Ncdd 2~l3 ?1 for=Lz l?1: (31) Hence dd() is a slowly decreasing function of the cigar's aspect ratio (keeping everything else xed). We will see below that for the parameters of experiments such as [30], the eective magnetic eld due to dipolar interactions greatly exceeds the externally applied mag- netic elds (in the range relevant for SW switching to be observed) [48]. VI. ANALYTICAL RESULTS FOR AXIALLY DIRECTED EXTERNAL MAGNETIC FIELD Without dissipation, when bx= 0,p=~cos=~Mz is rendered constant; see Eq. (20). However, in the pres-ence of dissipation, Mzchanges in time even if bx= 0. By employing this change, we propose an experimental method to measure . For simplicity, we will assume that the number density is constant in time (also see section IV) and the external magnetic eld points along the zdirection,B=Bzez. Let a critical (see for a detailed discussion below) value of the magnetization be (Mz)cr:=bz S0 dd: (32) Then Eq. (15) can be written as @M @t=S0 ddMezfMz(Mz)crgM@M @t =Mez(bz+S0 ddMz)M@M @t:(33) SinceM
@M @t= 0, by taking the cross product with M on both sides of Eq. (15), one can derive an expression forM@M @t: @Mz @t=S0 dd 1 + 2fMz(Mz)crg M2 z1
= 1 + 2(bz+S0 ddMz) M2 z1
:(34) SinceMis the scaled magnetization, jMj= 1 with a condensate. Hence, 1Mz1. Also, according to the discussion below Eq. (26), the generally positive SW coecient (with units of frequency) KisS0 dd=2. From Eq. (34), for time-independent 0 dd, one con- cludes that there are three time-independent solutions, Mz= (Mz)crandMz=1. For a box-trapped BEC and constant number density, 0 dd= ddwhich is always positive in the quasi-1D limit (cf. Eq. (30) and the discus- sion following it). For some arbitrary physical quasi-1D trap potential, in which the number density is not con- stant in space, from Eqs. (13), (16), and Fig. 2, one can infer that 0 dd>0, due to the fact that the quasi-1D num- ber density n(z;t)>0,n(z;t) has its maximum value nearz= 0 for a symmetric trap centered there, and then G() also has its maximum value near = 0. Then, if j(Mz)crj<1,Mz= (Mz)cris an unstable solution and Mz=1 are stable solutions. When j(Mz)crj<1 and 1< Mz<(Mz)cr,Mzgoes to1. Likewise, Mzgoes to 1 when ( Mz)cr< Mz<1. This bifurcation does not occur ifj(Mz)crj>1. For simplicity, we assume that j(Mz)crj<1. This is the more interesting case due to the possibility of a bifurcation of stable solutions leading to SW switching. Let (Mz)inbe the value of Mzatt= 0. The analytic8 solution of Eq. (34) satises t=1 + 2 S0 dd" 1 f(Mz)crg21ln(Mz)in(Mz)cr Mz(Mz)cr 1 2f1(Mz)crgln1Mz 1(Mz)in +1 2f1 + (Mz)crgln1 + (Mz)in 1 +Mz =1 + 2 " S0 dd b2z(S0 dd)2lnbz+S0 dd(Mz)in bz+S0 ddMz 1 2 (bz+S0 dd)ln1Mz 1(Mz)in 1 2 (bzS0 dd)ln1 + (Mz)in 1 +Mz : (35) The above equation tells us that, if ( Mz)in6= (Mz)crand (Mz)in6=1,Mzgoes to its stable time-independent solution (jMzj= 1) at time t=1. Thus, we dene acritical switching time tcrto be the time when jMzj= 0:99. Also, note that the form of LLG equation (Eq. (33)) does not change whether BEC is conned in a quasi- 1D, quasi-2D, or a three-dimensional geometry. This is because one can nd a connection between 0 ddand the eective dipole-dipole-interaction potential Ve, so one can measure even if the BEC is eectively conned in a space with dimension higher than one, using Eq. (35). We point out, in particular, that tcris inversely pro- portional to 0 dd. Hence, for a constant density quasi- 1D BEC conned between LzzLz, 0 dd= dd(Lz=l?), and thus tcris also inversely proportional to the linear number density along z. This follows from the relation between  dd(Lz=l?) and the linear numberdensity along zdisplayed in Eq. (30). For large dipolar interaction, the asymptotic expres- sion fortcris, assuming 1 tcr'1 S0 ddln" 5p 2(1(Mz)2 in) j(Mz)in(Mz)crj# (36) providedS0 ddjbzj () j (Mz)crj1: The above tcrdiverges at ( Mz)in= (Mz)cror1, as expected, since Mz= (Mz)crandMz=1 are time- independent solutions of the LLG equation. We stress that Eq. (36) clearly shows that the magnetic dipole- dipole interaction accelerates the decay of Mz. Hence, by using a dipolar spinor BEC with large magnetic dipole moment such as produced from164Dy or166Er one may observe the relaxation of Mzto the stable state within the BEC lifetime, enabling the measurement of . Before we show how the critical switching time tcr depends on ( Mz)inand , we will qualitatively discuss when our quasi-1D assumption and homogeneous-local- spin-orientation assumption are valid. Typically, spin- spin-interaction couplings are much smaller than their density-density-interaction counterparts, by two orders of magnitude. For spin 123Na BEC or spin 187Rb BEC, c0'100jc2j[31, 34]. Thus we may neglect to a rst ap- proximation the S2timesc2kterms in Eq. (11) (see the discussion at the end of Appendix D). We also require j(Mz)crj<1. Thus, we may additionally neglect the b term compared to the Pdd(z;t) term since, for b=bzez, S0 dd>jbjshould be satised to make j(Mz)crj<1 (see Eq. (32)) and 0 ddis related to Pdd(z;t) by Eq. (16). When = 0, using our ansatz in Eq. (8) and integrating out thexandydirections, Eq. (D4) can be approximated by the expression (t) (z;t) = ~2 2m@2 @z2+V(z) +c0 2l2 ?j (z;t)j2+ dd(z;t) (z;t); (37) where, from Eqs. (D5), (A1), and (17), the dipole-dipole interaction mean-eld potential reads dd(z;t) =~S2 13M2 z(t) Pdd(z;t) =cdd 2l3 ?S2 13M2 z(t) Z1 1dz0j (z0;t)j2 Gjz0zj l? 4 3z0z l? =cdd 2l2 ?S2 13M2 z(t) Z1 1dzj (z+ zl?;t)j2G(jzj)4 3j (z;t)j2 : (38) From Fig. 2, the function G() is positive and decreases exponentially as increases. Thus, if l?is small enough such thatj (z+ zl?;t)j2does not change within the rangejzj5, one may conclude that dd(z;t)'2 3S2 13M2 z(t) cdd 2l2 ?j (z;t)j2;(39)due to the propertyR1 0dG () = 1. A spinor (S= 6) dipolar BEC has been realized using 166Er [30]. For this BEC, c0= 4~2a=m wherea'67aB (aBis Bohr radius) and 2 S2cdd=3 = 0:4911c0. Due to jMz(t)j1 from the denition of M(t), the maximum value of the chemical potential (t) is achieved when9 Mz(t) = 0, where (t)'V(z) + c0+2 3S2cddn(z;t) 2l2 ?: (40) From above Eq. (40), we may regard the 3D number density as n(z;t)= 2l2 ?
. In [30], N= 1:2105, !?=(2) =p156198 Hz = 175 :75 Hz,!z=(2) = 17:2 Hz,l?= 0:589m, and the measured peak number density npeakis 6:21020m3. Using Eq. (37) and (39), by denoting Lzas the Thomas-Fermi radius along z, (LzzLz) withV(z) =m!2 zz2=2, one derives Lz=( 3 c0+ 2S2cdd=3
N 4m!2zl2 ?)1=3 ; (41) and the mean number density  n= (N=2Lz)= 2l2 ?
= 6:7211020m3as well as chemical potential =(~!?) = m!2 zL2 z=(2~!?) = 23:22. Note that  n'1:1 npeak. Be- causeis not less than ~!?, the experiment [30] is not conducted within the quasi-1D limit. The homogeneous-local-spin-orientation approxima- tion is valid when the system size is on the order of the spin healing length sor less, which has been experimen- tally veried in in [34]. Using c0'100jc2j,s'10d whered=p ~2=(2mc0n) is the density healing length ands=p ~2=(2mjc2jn) is the spin healing length. Thus, ifLzis on the order of 10 d, the homogeneous- local-spin-orientation approximation is justied. Using the S= 6 element166Er, we can provide numerical values which satisfy both the quasi-1D and homogeneous-local-spin-orientation limits, as well as they enable us to explicitly show how tcrdepends on (Mz)inin a concretely realizable setup. We consider be- low two cases: (A) box trap along z[49] and (B) harmonic trap alongz. A. Box traps We setV(z) = 0 forjzj< Lzand1other- wise. Then n(z;t) =N=(2Lz) and we estimate ' c0+ 2S2cdd=3
N= 4l2 ?Lz
from Eq. (40). In this case, 0 dd= dd(Lz=l?) as is calculated in Eq. (30). FixingBz=0:03 mG and N= 100, we con- sider the following two cases: (1) !?=(2) = 2:4 104Hz andLz= 3:125m. Then Lz=l?= 62:03, =(~!?) = 0:1692, and Lz=d= 29:55. Thus, the system is in both the quasi-1D and homogeneous- local-spin-orientation limit. Sdd(Lz=l?) = 4:074 103Hz,~Sdd(Lz=l?)=(gFB) = 0:3969 mG, and cr:= cos1(Mz)cris 85:67. (2)!?=(2) = 1:2104Hz andLz= 6:250m. ThenLz=l?= 87:72,=(~!?) = 0:0846, and Lz=d= 29:55. Thus, again the system is in both the quasi-1D and homogeneous-local-spin- orientation limits. Sdd(Lz=l?) = 1:028103Hz, ~Sdd(Lz=l?)=(gFB) = 0:1002 mG, and cr= 72:57. Fig. 4 shows the relation between tcrand (Mz)in. -1.0 -0.5 0.5 1.00.050.100.15!?=(2) = 2:4104Hz,Lz= 3:125m, andl?= 0:0504m whereN= 4Lzl2 ?
= 10:031020m3((Mz)cr= 0:0756). -1.0 -0.5 0.5 1.00.20.40.6 !?=(2) = 1:2104Hz,Lz= 6:250m, andl?= 0:0712m whereN= 4Lzl2 ?
= 2:5081020m3((Mz)cr= 0:2995). FIG. 4.tcras a function of ( Mz)inwhenB=Bzezwhere Bz=0:03 mG and particle number N= 100. From top to bottom: Red for = 0 :01, black for = 0 :03, and blue for = 0:09. Lines are from exact analytic formula in Eq. (35), and dot-dashed are from asymptotic expression in Eq. (36). Generally,tcrdecreases as increases. Also, note that tcr diverges as ( Mz)in!(Mz)cr. For larger mean number den- sityN= 4Lzl2 ?
(top), the asymptotic expression of tcris essentially indistinguishable from the exact analytic formula oftcr. B. Harmonic traps We setV(z) =m!2 zz2=2. Using the Thomas-Fermi approximation, from Eq. (40), =m!2 zL2 z=2 whereLz is given by Eq. (41). c0+ 2S2cdd=3
n(z;t)= l2 ?
= m!2 z L2 zz2
forjzjLzandn(z;t) = 0 forjzj>Lz. From thisn(z;t), we performed a numerical integration to calculate 0 ddin Eq. (16). Fixing Bz=0:03 mG, we consider the following two cases: (1)N= 240,!?=(2) = 2000 Hz, and !z=(2) = 50 Hz, for which Lz= 5:703m andLz=l?= 32:68. We obtain again the quasi-1D and homogeneous-local- spin-orientation limits since =(~!?) = 0:3337 and Lz=d= 17:85. Furthermore, S0 dd= 1:644103Hz, ~S0 dd=(gFB) = 1:602101mG, andcr= 79:21. (2)N= 340,!?=(2) = 1000 Hz, and !z=(2) = 25 Hz, where Lz= 8:070m andLz=l?= 32:70. Again, we have the quasi-1D and with homogeneous-local-spin- orientation limits fullled due to =(~!?) = 0:3341 and10 -1.0 -0.5 0.5 1.00.10.20.30.4 N= 240,!?=(2) = 2000 Hz, and !z=(2) = 50 Hz. Lz= 5:703m andl?= 0:1745m where N= 4Lzl2 ?
= 1:0101020m3((Mz)cr= 0:1873). -1.0 -0.5 0.5 1.00.20.40.60.8 N= 340,!?=(2) = 1000 Hz, and !z=(2) = 25 Hz. Lz= 8:070m andl?= 0:2468m where N= 4Lzl2 ?
= 0:5501020m3((Mz)cr= 0:3741). FIG. 5.tcras a function of ( Mz)inwhenB=Bzezwhere Bz=0:03 mG, for two particle numbers Nas shown. From top to bottom: Red for = 0 :01, black for = 0 :03, and blue for = 0 :09. Lines are from exact analytic formula in Eq. (35), and dot-dashed are from asymptotic expression in Eq. (36). Generally, tcrdecreases as increases. Also, note thattcrdiverges as ( Mz)in!(Mz)cr. For larger mean number densityN= 4Lzl2 ?
(top), the asymptotic expression of tcris essentially indistinguishable from the exact analytic formula oftcr. Lz=d= 17:87. In addition, S0 dd= 8:230102Hz, ~S0 dd=(gFB) = 8:019102mG, andcr= 68:03. Fig. 5 shows for the harmonic traps the relation be- tweentcrand (Mz)in. C. Measurability of critical switching time Figs. 4 and 5 demonstrate that the critical switching timetcris much smaller than the lifetime of BEC (sev- eral seconds [30]) and thus, by measuring tcrby varying (Mz)in, one will be able to obtain the value of , pro- vided indeed does not depend on spin indices as for example Refs. [26, 29] have assumed. Conversely, if one obtains from the measurements a dierent functional re- lation which does not follow Eq. (35), this implies that may depend on spin indices. Note that both gures, Figs. 4 and 5, show that tcrisinversely proportional to the mean number density N= 4Lzl2 ?
. Eq. (36) states that tcris inversely pro- portional to 0 dd, but except for the box trap case, in which one can analytically calculate 0 dd= dd(Lz=l?) in Eq. (30), the dependence of 0 ddand the mean number densityN= 4Lzl2 ?
is not immediately apparent. Thus, at least for harmonic traps, and in the Thomas-Fermi ap- proximation, one may use the box trap results of Eq. (30) for provide an approximate estimate of the behavior of tcr. VII. CONCLUSION For a quasi-1D dipolar spinor condensate with unidirectional local magnetization (that is in the homogeneous-local-spin-orientation limit), we provided an analytical derivation of the Landau-Lifshitz-Gilbert equation and the Stoner-Wohlfarth model. For an exter- nal magnetic eld along the long axis, we obtained an exact solution of the quasi-1D Landau-Lifshitz-Gilbert equation. Our analytical solution demonstrates that the magnetic dipole-dipole interaction accelerates the relax- ation of the magnetization to stable states and hence strongly facilitates observation of this process within the lifetime of typical dipolar spinor BECs. Employing this solution, we hence propose a method to experimentally access the dissipative parameter(s) . We expect, in particular, that our proposal provides a viable tool to verify in experiment whether is indeed independent of spin indices, as commonly assumed, and does not have to be replaced by a tensorial quantity for spinor gases. We hope that this will stimulate further more detailed investigations of the dissipative mechanism in dipolar BECs with internal degrees of freedom. We considered that the magnetization along z,Mz, has contributions solely from the atoms residing in the con- densate, an approximation valid at suciently low tem- peratures. When the magnetization from noncondensed atoms is not negligible, as considered by Ref. [5] for a contact interacting scalar BEC, correlation terms mix- ing the noncondensed part and the mean eld, such asPS =S  (r;t)h^ (r;t)^ (r;t)iwill appear on the right-hand side of Eq. (3). Here, ^ (r;t) is the-th component of quantum eld excitations above the mean- eld ground state in the spinor basis. Considering the eect of these terms is a subject of future studies. ACKNOWLEDGMENTS The work of SHS was supported by the National Research Foundation of Korea (NRF), Grant No. NRF-2015-033908 (Global PhD Fellowship Program). SHS also acknowledges the hospitality of the Uni- versity of T ubingen during his stay in the summer of 2019. URF has been supported by the NRF11 under Grant No. 2017R1A2A2A05001422 and Grant No. 2020R1A2C2008103. Appendix A: Derivation of the eective potential Ve The dipole-dipole interaction term Vdd(t) in the total energy is given by [31] Vdd(t) =cdd 2Z d3rZ d3r0X ;0=x;y;zF(r;t)Q;0(rr0)F0(r0;t); (A1) wherecddis dipole-dipole interaction coecient, F(r;t) = y(r;t)^f (r;t), andQ;0(r) is dened as the tensor Q;0(r):=r2;03rr0 r5(A2) in spin space, where r=jrjandr=r
e, withebeing the unit vector along the axis. From now on, we dene = (x;y) such that dxdy =d2=d'd where tan'=y=x. Using the convolution theorem, the dipole-dipole interaction term Vdd(t) can be expressed by Vdd(t) =cdd 2(2)D=2Z d3k~n(k;t) ~n(k;t)~Udd(k;t) (A3) with the Fourier transform Udd(;t) =1 n(r;t)n(r0;t)X ;0=x;y;zF(r;t)Q;0()F0(r0;t); (A4) where ~g(k;t) = (2)D=2R drg(r;t)eik
ris the Fourier transform of the function g(r;t) inD-dimensional space r (in our case, D= 3),=rr0, andn(r;t) =j (r;t)j2. By denoting k= (k;kz), wherek= (kx;ky) withk=q k2x+k2yand tan'k=ky=kx, with our mean-eld wavefunction in Eq. (8), one derives ~n(k;t) =1 l2 ?1 (2)3=2Z d2Z1 1dze(=l?)2n(z;t)ei
keikzz=1 2~n(kz;t)ek2 l2 ?=4; (A5) wheren(z;t):=j (z;t)j2e2!?t=(1+2). Note the factor of (2 )1appearing, when compared to Eq. (12) in Ref. [27], which is stemming from our denition of Fourier transform. Denoting=jj, by writingefor the unit vector along , we obtain Udd(;t) =1 3r 6 5hn Y2 2(e)e2i(t)+Y2 2(e)e2i(t)o S2sin2(t) n Y1 2(e)ei(t)Y1 2(e)ei(t)o S2sinf2(t)gi +1 3r 6 5Y0 2(e)r 2 3S2 3 sin2(t)2 ; (A6) whereYm l(e) are the usual spherical harmonics. Its Fourier transform ~Udd(k;t) is ~Udd(k;t) =1 (2)3=24 3S2 13 2sin2(t) 3k2 z k2+k2z1 +1p 2k2  k2+k2zS2sin2(t) cos 2'k2(t) +r 2 kkz k2+k2zS2sinf2(t)gcos 'k(t) :(A7) By plugging Eq. (A5) and Eq. (A7) into Eq. (A3), we nally obtain Vdd(t) as Vdd(t) =cdd 2p 2Z1 1dkz~n(kz;t) ~n(kz;t)2S2 l2 ?p 2 13 2sin2(t) k2 zl2 ?=2
ek2 zl2 ?=2E1 k2 zl2 ?=2
1 3 ;(A8)12 whereE1(x) =R1 xdueu=uis exponential integral. Note that Eq. (A8) can be also written as Vdd(t) =cdd 2p 2Z1 1dkz~n(kz;t) ~n(kz;t)~Ve(kz;t) =cdd 2Z1 1dzZ1 1dz0n(z;t)n(z0;t)Ve(zz0;t):(A9) Due to the fact that ~Ve(kz;t) can be obtained by Eq. (A8), we can get Ve(z;t) by inverse Fourier transform. As a preliminary step, we rst write down some integrals of E1(x) as follows: Z1 1dxex2E1 x2
eikx=Z1 1dxeikxZ1 x2dte(tx2) t= ()3=2ek2=4Erfc (jkj=2): (A10) Dierentiating Eq. (A10) with respect to ktwo times results in Z1 1dxx2ex2E1 x2
eikx=()3=21 2k2 2+ 1 ek2=4Erfc (jkj=2)jkj 2p2p(k) : (A11) Therefore,Ve(z;t) can be calculated as Ve(z;t) =1p 2Z1 1dkz2S2 l2 ?p 2 13 2sin2(t) k2 zl2 ?=2
ek2 zl2 ?=2E1 k2 zl2 ?=2
1 3 eikzz =S2 l3 ?3 2sin2(t)1 G(jzj=l?)4 3(z=l?) ; (A12) whereG(x) is dened in Eq. (14), and (x) is the Dirac delta function. The Fourier transform of Eq. (A12) acquires the form ~Ve(kz;t) =1p 2Z1 1dzV e(z;t)eikzz=r 2 S2 l2 ?3 2sin2(t)1Z1 0dvG (v) cos (kzl?v)2 3 =r 2 S2 l2 ?3 2sin2(t)1Z1 0dunp 2u2+ 1
eu2Erfc (u)2uo cosp 2kzl?u 2 3 :(A13) From [50], the following integral involving the complementary error function is Z1 0dueu2Erfc (u) cos (bu) =1 2peb2=4E1 b2=4
: (A14) By dierentiating Eq. (A14) two times with respect to b, we get Z1 0duu2eu2Erfc (u) cos (bu) =1 2p1 2b2 2+ 1 eb2=4E1 b2=4
1 +2 b2 : (A15) Hence, Eq. (A13) becomes ~Ve(kz;t) =r 2 S2 l2 ?3 2sin2(t)1 1 2 k2 zl2 ?+ 1
ek2 zl2 ?=2E1 k2 zl2 ?=2
1 +1 k2zl2 ? +1 2ek2 zl2 ?=2E1 k2 zl2 ?=2
+1 k2zl2 ?2 3 =2S2 l2 ?p 2 13 2sin2(t) k2 zl2 ?=2
ek2 zl2 ?=2E1 k2 zl2 ?=2
1 3 : (A16) Comparing Eq. (A8) with Eq. (A16), one veries that Eq. (A12) is the correct result for the eective interaction of the quasi-1D dipolar spinor gas.13 Appendix B: Quasi-1D Gross-Pitaevski  equation with dissipation By introducing an identical damping coecient for each component of the spinor, cf., e.g. Refs. [26, 29] (i.e. as if each component eectively behaves as a scalar BEC [28]), and neglecting a possible quadratic Zeeman term, the GP equation for a spin- SBEC can be written as [26] (i)~@ (r;t) @t= ~2 2mr2+Vtr(r) +c0j (r;t)j2 (r;t)~SX =Sfbbdd(r;t)g
 ^f ; (r;t) +SX k=1c2kX 1;2;


;k=x;y;zF1;2;


;k(r;t)SX =S ^f1^f2


^fk ; (r;t); (B1) where (r;t) is the-th component of the mean-eld wavefunction (r;t) (the spin-space index is an integer taking 2S+ 1 values running from SandS),F1;2;


;k(r;t):= y(r;t)^f1^f2


^fk (r;t),~^fis the spin- S operator,b=gFBB=~(gFis the Land e g-factor, Bis the Bohr magneton, and Bthe external magnetic eld). Finally, ~bdd(r;t)
e=cddR d3r0P 0=x;y;zQ;0(rr0)F0(r0;t), whereeis the unit vector along the axis (=x;y;z ) [31]. Applying the formalism of Ref. [1] to a spinor BEC assuming that does not depend on spin indices, one just needs to transform t! 1 + 2
tin Eq. (B1) and (8). We then integrate out the xandydirections in Eq. (B1) to obtain the quasi-1D GP equation. From Eq. (8) in the main text, we have Z d2SX =Se2=(2l2 ?) l?p ~bdd(r)
 ^f ; (r;t) =cdd 2l3 ?Z1 1dz0n(z0;t) Gjzz0j l? 4 3zz0 l? (z;t)ei+ 1+2!?tSfM(t)3Mz(t)ezg
SX =S ^f ;(t); (B2) whereR d2:=R1 1dxR1 1dyandn(z;t):=R d2j (r;t)j2=j (z;t)j2e2!?t=(1+2). For a spin-SBEC, from Eq. (B1), for the trap potential given in Eq. (4) and if we use Eq. (8), by integrating out thexandydirections, one acquires the expression (i)~@f (z;t)(t)g @t= ~2 2m@2 @z2+V(z) +c0 2l2 ?n(z;t) (z;t)(t) + [~b+~SfM(t)3Mz(t)ezgPdd(z;t)]
8 < :SX =S ^f ; (z;t)(t)9 = ; +SX k=1c2k 2l2 ?n(z;t)X 1;2;


;k=x;y;zSM1;2;


;k(t)8 < :SX =S ^f1^f2


^fk ; (z;t)(t)9 = ;; (B3) whereM1;2;


;k(t) is dened in Eq. (12) and Pdd(z;t) =cdd 2~l3 ?Z1 1dz0n(z0;t) Gjzz0j l? 4 3zz0 l? =cdd ~S2 3 sin2(t)2 Z1 1dz0n(z0;t)Ve(zz0;t); (B4) withVedened in (A12). It is already clear from Eq. (B3) that, besides particle loss from the condensate encoded in a decayingj (z;t)j, dissipation also leads to a dephasing , i.e. the decay of (t) due to the term @(t)=@t. From now on, if there is no ambiguity, and for brevity, we drop the arguments such as x;y;z;t from the functions.14 From Eq. (B3), we then get ~@ @t=~ @ @t +i 1 + 2 ~2 2m1 @2 @z2+V+c0 2l2 ?n + +i 1 + 2f~bS(M3Mzez)~Pddg
8 < :SX =S ^f ;9 = ; +i 1 + 2SX k=1c2k 2l2 ?nX 1;2;


;k=x;y;zSM1;2;


;k8 < :SX =S ^f1^f2


^fk ;9 = ;; (B5) Since@jj2 @t= 0 due to the normalization jj2= 1, we then have 0 = 2Re ~ @ @t 1 + 2 ~2 2m1 @2 @z2+V+c0 2l2 ?n +i 1 + 2~2 2m1 @2 @z21 @2  @z2 +2 1 + 2f~bS(M3Mzez)~Pddg
SM2 1 + 2SX k=1c2k 2l2 ?nX 1;2;


;k=x;y;zS2M2 1;2;


;k: (B6) Hence the dynamics of the magnetization direction follows the equation ~S@M @t= 2Re8 < :SX ;=Sy  ^f ; ~@ @t9 = ; =2 1 + 2S2Mf~bS(M3Mzez)~Pddg
M+2 1 + 2MSX k=1c2k 2l2 ?nX 1;2;


;k=x;y;zS3M2 1;2;


;k + 1 + 2X =x;y;zf~bS(M3Mz;z)~PddgSf;+ (2S1)MMg 1 1 + 2X ;=x;y;zf~bS(M3Mz;z)~Pddg;;SM 2Re8 < : +i 1 + 2SX k=1c2k 2l2 ?nX 1;2;


;k=x;y;zSM1;2;


;kSX ;=Sy  ^f^f1^f2


^fk ;9 = ;; (B7) since the scalar product y ^f^f+^f^f =Sf;+ (2S1)MMg[26]. By direct comparison, we can identify Eq. (B8) below as being identical to Eq. (B21) in [26], the only dierence consisting in the denition of M1;2;


;k: We employ a scaled version of M1;2;


;k, which is normalized to Sin [26]. From Eq. (7) in the main text, X 1;2;


;k=x;y;zM1;2;


;kSX ;=Sy  ^f^f1^f2


^fk ;=X 1;2;


;k=x;y;zM2 1;2;


;kS2M; (B8) which is real. Therefore, Eq. (B7) can be written in the following form @M @t= 1 + 2M[MfbS(M3Mzez)Pddg] +1 1 + 2MfbS(M3Mzez)Pddg =1 1 + 2M(b+ 3SPddMzez) 1 + 2M[M(b+ 3SPddMzez)] =M(b+ 3SPddMzez)M@M @t; (B9) sinceM
@M @t= 0 holds. AsPis a function of zandt, butMis independent of z[Mis the scaled local magnetization and our aim is to study a dipolar spinor BEC with unidirectional local magnetization (the homogeneous-local-spin-orientation limit)], by multiplying with n(z;t) both sides of Eq. (B9) and integrating along z, we nally get the LLG equation @M @t=M(b+S0 ddMzez)M@M @t; (B10) where 0 ddis dened in Eq. (16). Note here that 0 ddbecomes  dd(Lz=l?) dened in Eq. (30) when n(z;t) =N=(2Lz) forLzzLzandn(z;t) = 0 otherwise.15 Appendix C: Modication of the LLG equation for a spin-space tensor When depends on spin indices, i.e. is a tensor, Eq. (B3) can be generalized to read SX =S(i;;)~@f (z;t)(t)g @t=SX =SH; (z;t)(t): (C1) The spinor part of the wavefunction is normalized to unity, jj2= 1. Hence, we know that@jj2 @t= 0. Therefore, from Eq. (C1), we derive the expression SX ;=SRe i ;@ @ti ;1 @ @ti1 ~  H;  Re1 @ @t = 0: (C2) This then leads us to @M @t=2 SSX ;; =SRe i  ^f ;; @ @ti  ^f ;;  1 @ @ti1 ~   ^f ;H;   2Re M1 @ @t : (C3) For scalar , ;!;, the equation above becomes Eq. (B7). From Eqs. (C2) and (C3), one concludes that the stationary solution Mof Eq. (C3) is independent of . In other words, whether depends on spin indices or not, the SW model (29) is left unaected, also see the discussion in Section V of the main text. Appendix D: Description of magnetostriction For a dipolar spinor BEC without quadratic Zeeman term, when there is no dissipation ( = 0), the mean-eld equation in Eq. (3) can be written as (t) (r;t) =8 < :~2 2mr2+Vtr(r) +c0SX =Sj (r;t)j29 = ; (r;t)~fbbdd(r;t)g
SX =S ^f ; (r;t) +SX k=1c2kX 1;2;


;k=x;y;zSX 1;1;=S ^f1^f2


^fk 1;1 ^f1^f2


^fk ;  1(r;t) 1(r;t) (r;t): (D1) where we have substituted i~@ (r;t) @t=(t) (r;t). Since we consider the homogeneous-local-spin-orientation limit, we may write (r;t) = uni(r;t)(t). In this limit, we have j (r;t)j2:= y(r;t) (r;t) =SX =S y (r;t) (r;t) =j uni(r;t)j2; (D2) sinceSX =Sj(t)j2= 1 from the denition of (t) in Eq. (7). Thus j uni(r;t)j2is equal to the number density. Then Eq. (D1) can be written as (t)(t) uni(r;t) = ~2 2mr2+Vtr(r) +c0j uni(r;t)j2 (t) uni(r;t) ~fbbdd(r;t)g
SX =S ^f ;(t) uni(r;t) +SSX k=1c2kX 1;2;


;k=x;y;zSX =SM1;2;


;k(t) ^f1^f2


^fk ;(t)j uni(r;t)j2 uni(r;t): (D3)16 Now, we decompose the chemical potential (t) as(t):=SX =S(t)j(t)j2. Then one obtains (t) uni(r;t) =" ~2 2mr2+Vtr(r) +( c0+S2SX k=1c2kX 1;2;


;k=x;y;zM2 1;2;


;k(t)) j uni(r;t)j2# uni(r;t) + [dd(r;t)S~fb
M(t)g] uni(r;t); (D4) where dd(r;t):=S2cdd2 4Z d3r08 < :X ;0=x;y;zM(t)Q;0(rr0)M0(t)9 = ;j uni(r0;t)j23 5; (D5) is the dipole-dipole mean-eld potential [41] following from the denition of bddbelow Eq. (3) in the main text. Due toMx(t) = sin(t) cos(t),My(t) = sin(t) sin(t), andMz(t) = cos(t), from Eqs. (A4) and (A6), we have X ;0=x;y;zM(t)Q;0()M0(t) =1 3r 6 5hn Y2 2(e)e2i(t)+Y2 2(e)e2i(t)o sin2(t) n Y1 2(e)ei(t)Y1 2(e)ei(t)o sinf2(t)gi +1 3r 6 5Y0 2(e)r 2 3 3 sin2(t)2 ; (D6) whereYm l(e) are the usual spherical harmonics. By using Eq. (A2), an alternative form of Eq. (D6) can be obtained: X ;0=x;y;zM(t)Q;0()M0(t) =X ;0=x;y;z2;030 5M(t)M0(t) =2jM(t)j23f
M(t)g2 5 =23f
M(t)g2 5: (D7) Thus, dd(r;t) can be written as dd(r;t) =S2cdd"Z d3r0jrr0j23f(rr0)
M(t)g2 jrr0j5j uni(r0;t)j2# =S2cdd"Z d3r0jrr0j23f(rr0)
M(t)g2 jrr0j5j uni(r0;t)j2# =3 2S2cddsin2(t)Z d3j uni(+r;t)j21 5h 22 z2fxsin(t)ycos(t)g2i 3S2cddsinf2(t)gZ d3j uni(+r;t)j2z 5fxcos(t) + ysin(t)g +1 2S2cdd 13 cos2(t) Z d3j uni(+r;t)j21 5 32 z2
: (D8) where r:=r=LwithLbeing some length which scales r(so that ris a dimensionless vector). For example, in quasi-1D with trap potential being Eq. (4), L=l?. Note that, in the special case where M(t) =Mz(t)ez, the form of Eq. (D8) becomes identical to Eq.(6) in Ref. [40]. Since we concentrate on quasi-1D gases, with trap potential given by Eq. (4) in the main text, we will explicitly compute the form of  dd(r;t) for the quasi-1D setup. By writing j uni(r;t)j2=e2=l2 ? l2 ?j (z;t)j2; (D9)17 -20 -10 10 20 -0.4-0.20.20.40.60.81.0 -60 -40 -20 20 40 60 -0.4-0.20.20.40.60.81.0 FIG. 6. Scaled dipole-dipole mean-eld potential dd(z) as a function of zfor a quasi-1D box trap. (Left) Lz=l?= 10. (Right)Lz=l?= 30. and integrating out xandydirections, one can get the quasi-1D dipole-dipole-interaction mean-eld potential  dd(z;t) as follows (which is in Eq. (38)): dd(z;t) =cdd 2l2 ?S2 13M2 z(t) Z1 1dzj (z+ zl?;t)j2G(jzj)4 3j (z;t)j2 : (D10) Now, let us consider box trap in quasi-1D case, i.e. V(z) = 0 forjzjLzandV(z) =1forjzj>LzwhereV(z) is in Eq. (4). Then we may write j (z;t)j2=2 64N 2LzforjzjLz, 0 forjzj>Lz,(D11) sinceV(z) = 0 forLzzLz. Thus,  dd(z;t) can be written as dd(z;t) =2 666664dd(t)(Z(Lzz)=l? (Lz+z)=l?dzG(jzj)4 3) forjzjLz, dd(t)Z(Lzz)=l? (Lz+z)=l?dzG(jzj) forjzj>Lz,(D12) where dd(t):=NcddS2 13M2 z(t) = 2Lzl2 ?
. dd(z;t) is discontinuous at z=Lzbecause of the sudden change of the density at the boundary ( z=Lz) due to box trap potential. Dening the scaled density-density mean-eld potential dd(z):= dd(z;t)=dd(t), we obtain Fig. 6, for two dierent axial extensions, Lz=l?= 10 and 30. As Fig. 6 clearly illustrates, in a box-trapped quasi-1D gas,  dd(z;t) becomes approximately constant for jzj<LcandLc!LzforLz=l?1. Depending on the value of M(t), dd(r;t) will introduce either a repulsive or an attractive force. This force will however exist only near the boundary for a box trap, where it can lead to a slight modication of the density of atomes. Its relative in uence decreases with increasing extension of the trapped gas along the zaxis, and can therefore be consistently neglected in the approximation of constant particle-density. However, to assess whether signicant magnetostriction occurs, one has to consider, in addition to  dd, the trap potentialVtrand the `quasi' density-density interaction mean eld potential  0dened as 0(r;t):=( c0+S2SX k=1c2kX 1;2;


;k=x;y;zM2 1;2;


;k(t)) j uni(r;t)j2: (D13) We can coin  0(r;t) a `quasi' density-density interaction mean eld potential because only c0is a density-density interaction coecient ( c2kare interaction coecients parametrizing the spin-spin interactions for spin- Sgas wherek is an integer with 1 kS. For example, c2is the spin-spin interaction coecient of a spin-1 gas). In our quasi-1D case, this  0(r;t) potential is  0(z;t) where 0(z;t):=( c0 2l2 ?+S2SX k=1c2k 2l2 ?X 1;2;


;k=x;y;zM2 1;2;


;k(t)) j (z;t)j2: (D14)18 In the main text, we assume that c0S2PS k=1c2kP 1;2;


;k=x;y;zM2 1;2;


;k(t). For spin-123Na or87Rb,S= 1 andc0'100jc2j[31, 34], so this is an appropriate assumption (note thatPS k=1P 1;2;


;k=x;y;zM2 1;2;


;k(t) = 1). The values of the c2kare not yet established for166Er. We therefore tacitly assume in the main text, when calculating concrete numerical examples for166Er, that the above condition also still holds, despite the prefactor S2enhancing the importance of spin-spin interactions in  0(z;t). When this assumption is not applicable, one is required to take into account the time dependence of  0(z;t) due toM(t) together with magnetostriction due to  dd(z;t), which will change the system size Lzas a function of t. This will in turn change the integration domain and quasi-1D density n(z;t) =j (z;t)j2in Eq. (19), and incur also a changed time dependence of 0 dd(t), and the solution of the coupled system of equations (B10) and (D4) needs to be found self-consistently. For a harmonic trap, due to the resulting inhomogeneity of j (z;t)j2, dd(z;t) will have more signicant spatial dependence than its box trap counterpart shown in Fig. 6. Here, we note that Ref. 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Ketterle, \Realization of Bose-Einstein Condensates in Lower Di- mensions," Phys. Rev. Lett. 87, 130402 (2001). [48] This fact is equivalent to the critical dimensionless mag- netization in Eq. (32) being always much less than unity. [49] We note, while box traps so far have been created for scalar BECs with contact interaction only, there is no obstacle in principle to create them as well for dipolar spinor gases (J. Dalibard, private communication). [50] M. Geller, \A table of integrals of the error functions," J. Res. Nat. Bureau Stand., B 73, 1{20 (1969).

2020-02-20

We consider quasi-one-dimensional dipolar spinor Bose-Einstein condensates in the homogeneous-local-spin-orientation approximation, that is with unidirectional local magnetization. By analytically calculating the exact effective dipole-dipole interaction, we derive a Landau-Lifshitz-Gilbert equation for the dissipative condensate magnetization dynamics, and show how it leads to the Stoner-Wohlfarth model of a uni-axial ferro-magnetic particle, where the latter model determines the stable magnetization patterns and hysteresis curves for switching between them. For an external magnetic field pointing along the axial, long direction, we analytically solve the Landau-Lifshitz-Gilbert equation. The solution explicitly demonstrates that the magnetic dipole-dipole interaction {\it accelerates} the dissipative dynamics of the magnetic moment distribution and the associated dephasing of the magnetic moment direction. Under suitable conditions, dephasing of the magnetization direction due to dipole-dipole interactions occurs within time scales up to two orders of magnitude smaller than the lifetime of currently experimentally realized dipolar spinor condensates, e.g., produced with the large magnetic-dipole-moment atoms ${}^{166} \textrm{Er}$. This enables experimental access to the dissipation parameter $\Gamma$ in the Gross-Pitaevski\v\i~mean-field equation, for a system currently lacking a complete quantum kinetic treatment of dissipative processes and, in particular, an experimental check of the commonly used assumption that $\Gamma$ is a single scalar independent of spin indices.

Stoner-Wohlfarth switching of the condensate magnetization in a dipolar spinor gas and the metrology of excitation damping

2002.08723v2

1Thickness-dependentmagneticpropertiesinPt/[Co/Ni]n multilayerswithperpendicularmagneticanisotropy* ChunjieYan(晏春杰)1,LinaChen(陈丽娜)1,2,#,KaiyuanZhou(周恺元)1,Liupeng Yang(杨留鹏)1,QingweiFu(付清为)1,WenqiangWang(王文强)1,Wen-Cheng Yue(岳文诚)3,LikeLiang(梁立克)1,ZuiTao(陶醉)1,JunDu(杜军)1,Yong-Lei Wang(王永磊)3andRonghuaLiu(刘荣华)1* 1NationalLaboratoryofSolidStateMicrostructures,SchoolofPhysicsand CollaborativeInnovationCenterofAdvancedMicrostructures,NanjingUniversity, Nanjing210093,China 2SchoolofScience,NanjingUniversityofPostsandTelecommunications,Nanjing 210023,China. 3SchoolofElectronicsScienceandEngineering,NanjingUniversity,Nanjing210093, China. WesystematicallyinvestigatedtheNiandCothickness-dependentperpendicular magneticanisotropy(PMA)coefficient,magneticdomainstructures,and magnetizationdynamicsofPt(5nm)/[Co(tConm)/Ni(tNinm)]5/Pt(1nm)multilayersby combiningthefourstandardmagneticcharacterizationtechniques.The magnetic-relatedhysteresisloopsobtainedfromthefield-dependentmagnetizationM andanomalousHallresistivity(AHR)xyfoundthatthetwoserialmultilayerswith tCo=0.2and0.3nmhavetheoptimumPMAcoefficientKUwellasthehighest coercivityHCattheNithicknesstNi=0.6nm.Additionally,themagneticdomain structuresobtainedbyMagneto-opticKerreffect(MOKE)microscopyalso significantlydependonthethicknessandKUofthefilms.Furthermore,the thickness-dependentlinewidthofferromagneticresonanceisinverselyproportionalto KUandHC,indicatingthatinhomogeneousmagneticpropertiesdominatethe linewidth.However,theintrinsicGilbertdampingconstantdeterminedbyalinear fittingoffrequency-dependentlinewidthdoesnotdependonNithicknessandKU.OurresultscouldhelppromotethePMA[Co/Ni]multilayerapplicationsinvarious spintronicandspin-orbitronicdevices. Keywords:perpendicularmagneticanisotropy,magneticdomain,damping, multiayers PACS：75.30.Gw,75.70.Kw,75.40.Gb,68.65.Ac1.INTRODUCTION Magneticmultilayerswithstrongperpendicularmagneticanisotropy(PMA)and lowmagneticdampinghaveattractedmuchattentionbecauseoftheirpotential applicationsinhigh-densitymagneticrandomaccessmemories(MRAM)[1-5]andspin torquenano-oscillators[6-9].Comparedtothein-planemagnetizedferromagnets, ferromagneticfilmswithPMAfacilitatetherealizationofnonvolatileMRAMwith lowerpowerandhigherdensitystoragebecausethelatterhaslowercriticalswitching currentandhigherthermalstabilitythantheformerasthecontinuousdownscalingof thecellsizeofdevices[10].Inaddition,PMAcanbeaneffectivemagneticfieldto achievezeroexternalmagneticfieldworkingspin-torquenano-oscillatorswith ferromagnetswithstrongPMAandlowdampingasitsfreelayer.[11]Therefore,the controllabletailoringPMAofmagneticfilmsisanessentialprerequisitefor developinghigh-performancespintronicdevices.Themagneticmultilayers,e.g., [Co/Pd],[Co/Pt],and[Co/Ni],provideanopportunitytotunetheirmagnetic propertiesbychangingthethicknessratiocontrollablyandthenumberofbilayer repeatsthankstotheinterface-inducedPMAduetointerfacialspin-orbitcouplingand interfacialstrainrelevantmagnetoelasticeffects[12-17].AmongthesePMAmultilayers, thePMA[Co/Ni]multilayeralsoexhibitslowdampingconstant[14],whichgetsmuch attention,especiallyforthefieldsofcurrent-drivenauto-oscillationofmagnetization andexcitationandmanipulationofspin-waves[18,19].Furthermore,thePMA[Co/Ni] multilayeralsousefulforspin-orbittorquedevices[20-22].Therefore,[Co/Ni] multilayersareconsideredoneofthemostpromisingPMAferromagnetsinvarious spintronicdevices.Althoughthereareafewstudiesonthemagneticanisotropy, magnetotransport,andmagneticdampingofPt/[Co/Ni]multilayers[6,14,23],the systematicallystudiedevolutionofmagnetostaticproperties,includingthetopography ofmagneticdomainsandmagneticdynamicswiththethicknessratioofCoandNi layersforthismultilayerfilmstillneedstomakeathoroughinvestigationforfacilitatingitbetterusedinfurtherspintronics. Here,wesystematicallyinvestigatehowtocontrolthemagneticfilmPMAby tailoringtheinterfacialeffectbyvaryingthethicknessoftheNilayeranditsimpact onmagneticdomainstructureanddynamicaldampingintwoserialCo/Nimultilayers withtCo=0.2and0.3nm.ThehighestPMAcoefficientKU~3×106ergcm-3and coercivityHC~250OearefoundattheoptimumNithicknesstNi=0.6nmforthe studiedtwoserials.Thenucleationofthemagneticdomainoccursatonlyafew nucleationsitesandgraduallyexpandswithmagneticfieldsforthemultilayerswith theoptimumNithicknesses0.4nm~0.6nm.Finally,theintrinsicGilbertdamping constantαisnotsensitivetothickness-dependentKUanddomainstructureseven thoughthelinewidthofferromagneticresonanceisinverselyproportionaltoKUand HC,whichisdominatedbyinhomogeneousmagneticproperties. 2.EXPERIMENT TwoserialmultilayersofPt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1)and Pt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1),namedasPt/[Co(0.2)/Ni(tNi)]andPt/[Co(0.3)/Ni(tNi)] respectively,weredepositedonSi/SiO2substratesatroomtemperatureby dc-magnetronsputteringwithArpressure3×10-3torr.Theunitinparenthesesisthe thicknessinnm.Thebasepressureofthesputteringdepositionchamberisbelow2× 10−8torr.Thedepositionratewasmonitoredbythequartzcrystalmonitorinsituand calibratedbyspectroscopicellipsometry(SE).Thestaticmagneticpropertieswere characterizedbythevibratingsamplemagnetometer(VSM),theanomalousHall resistivity(AHR)measurement，andtheMagneto-opticKerreffect(MOKE) microscopy,respectively.Thefilms'ferromagneticresonance(FMR)spectra, obtainedbycombiningcoplanarwaveguide(CPW)andlock-intechniques,werealso adoptedtocharacterizetheirdynamicmagneticproperties.Allthesemagnetic characterizationswereperformedatroomtemperature.3.RESULTSANDDISCUSSION 3.1Quasi-staticmagneticproperties TodirectlyobtainthethicknessdependenceofPMApropertiesintheCo/Nifilms, wefirstperformedthemagnetichysteresisloopsofsampleswithdifferentthicknesses usingVSM.Figure1showsthemagnetizationhysteresisloopswiththeout-of-plane andin-planefieldgeometriesforthetwoserialmultilayersofPt/[Co(0.2)/Ni(tNi)]and Pt/[Co(0.3)/Ni(tNi)]samples.Thewell-definedsquareM-Hloopsunderout-of-plane field[Figs.1(a)and(c)]indicatethattwostudiedserialPt/[Co/Ni]multilayersexhibit aperpendicularmagneticanisotropy.Additionally,thesaturationmagnetizationMSof themultilayersdecreaseswithincreasingthethicknessoftheNilayertNi,from673 emucm-3to495emucm-3forPt/[Co(0.2)/Ni(tNi)]and723emucm-3to639emucm-3 forPt/[Co(0.3)/Ni(tNi)],whichagreeswiththemuchlowerMS~484emucm-3ofthe metalnickelcomparedtothatofthecobaltlayerMS~1422emucm-3.Basedonthe out-of-planeandin-planemagnetizationhysteresisloops,theperpendicular anisotropyfieldHKwasdeterminedbyusingthedefinedformulaforthePMA[12,24]: K=2 s0S⊥−0S∥ +4πS.ThecalculatedHK,MSandthe coercivityHC,obtainedfromtheM-Hloops,weresummarizedbelowinFig.5. Fig.1.(a)-(b)MagnetizationloopsofthefilmsPt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1)with out-of-plane(a)andin-plane(b)magneticfield.(c)-(d)Sameas(a)-(b),for Pt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1). ThesestaticmagneticpropertiesofthemetalPt/[Co/Ni]multilayerfilmsalsocan bedeterminedbytheelectrictransportsinmagneticfield,e.g.,anomalousHall resistivity(AHR)andmagnetoresistiveeffect.Comparedtothestandard magnetometerabove,theelectrictransportsinmagneticfieldmeasurementsprovide analternativeapproachand,especially,moreusefulforspintronicnano-devices becausetheycaneasilyaccessthemagneticpropertiesofthemicroscaleand nanoscalesamples[25,26].Therefore,wealsoperformtheout-of-planeandin-plane AHRloopsasafunctionoftheappliedmagneticfieldsforthestudiedtwoserial multilayers,asshowninFig.2.ThecoercivityHCdeterminedfromtheout-of-plane AHRloopsarewellconsistentwiththevaluesobtainedbytheM-Hloops,andare alsosummarizedinthefollowingFig.5.Meanwhile,wecancalculateHKofthe studiedfilmsfromthein-planeAHRloopsbyusingthefollowingrelation[27]:K= ∥∙tanarcsinxy xy(0)+4πS,wherexy0istheAHRvalueatzeroin-planefield. TheevolutionofHKwiththethicknessoftheNilayerisoverallconsistentwiththe resultsdeterminedbytheVSMmeasurement.Furthermore,theAHRmeasurements alsoprovideustheadditionalinformation,whichcannotbeeasilyaccessedbyVSM, aboutthestudiedtwoserialPt/[Co/Ni]multilayers.Forexample,wefindthatthe in-planeAHRnear-zeromagneticfieldismuchsmallerthantheout-of-planeAHR forthesampleswithcertainNithickness,indicatingthatthesesamplesformthe multi-domainstructuresatthelowin-planemagneticfields.Therefore,thevalueof thedifferencebetweenout-of-planeandin-planeAHRatnear-zerofieldshintsthat thedifferentNithicknessfilmsmayexhibitdistinctmagneticdomainstructures[28].Fig.2.(a)-(b)AnomalousHallresistivityasafunctionofout-of-plane(a)and in-planemagneticfieldwith5otiltanglefromthefileplane(b)forthesamples Pt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1).Theinsetshowsthegeometricrelationshipbetween magneticfield,magnetizationandeffectiveanisotropicfieldKeff=K–4πS (c)-(d)Sameas(a)-(b),forPt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1).AllAHRweremeasuredby usingthefilmspatternedintoa0.3×10mmHallcross. 3.2Magneticdomainstructures TodirectlyrevealtheevolutionofmagneticdomainstructurewithNithicknessand thedetailofmagnetizationreversalprocessunderexternalfield,theMOKE microscopyisalsoperformedforPt/[Co(0.2)/Ni(tNi)]serialsamples.Figure3shows MOKEloopsandtherepresentativeMOKEimagesduringscanningout-of-plane magneticfields.Brightanddarkregionsrepresentthedomainswithmagneticmoment point-upandpoint-down,respectively.Exceptforthemultilayerwiththenickel thicknesstNi=0.2nm[Fig.3(a)],allfilmsexhibitawell-definedPMAcorresponding rectangularordumbbell-shapehysteresisloops[Figs.3(b)-(h)].TheMOKEimagesshowthatthesamplewithtNi=0.3nmbeginstonucleatewithnumerousnucleation pointsapproximatelyuniformdispersiononthewholefilmatthefieldof138Oe, thengraduallyexpandwithincreasingfield,andsaturatetotheuniformstateat⊥˃ 300Oe[Fig.3(b)].Combiningwithfield-dependentmagneticsusceptibilityandAHR characterizations,HCandHKofPt/[Co(0.2)/Ni(tNi)]firstenhancewithincreasingtNi, reachthemaximumattNi=0.6nm,andthenreducewithcontinueincreasingtNito0.9 nm.Forthesampleswith0.4nm≤Ni≤0.6nm,theMOKEloopsexhibita well-definedrectangularshape.Meanwhile,incontrasttotNi=0.3nm,onlyafew nucleationpointsappearatthecriticalmagneticfield,andthenbegintoexpandthe magneticdomainwithacontinuouslyincreasingfield.ForincreasingtNitoabove0.7 nm,theMOKEloopsbegintotransferintothedumbbellshapefromtheprior well-definedrectangularshape.ThecorrespondingMOKEimagesshowthatthefilms withtNi≥0.7nmformtree-likedomainwallswithmoreirregularbranchingsas increasingtNiduringthefieldrangenearbelowthesaturationfield[themiddlepicture ofFig.3].Inotherwords,forthefilmswithtNi≥0.7nm,thelengthofthedomainwall increaseswithincreasingtNi,whichisconsistentwiththetrendofdependenceofHK onNithickness.Asweknowthatthetotalenergydensityofthemagneticdomain wallperunitareawisproportionaltomagneticanisotropyenergy,exchangeenergy anddemagnetizationenergybasedonthewidelyrecognizedformula[29]:w=2 + U 2+S2 4µ02 +,whereAistheexchangeconstant,δisthethicknessofthedomain wall,KUisthePMAcoefficient,MSisthesaturationmagnetization,tisthethickness oftheentirefilm,andμ0isthepermeabilityofvacuum.Tominimizethetotalenergy oftheentirefilm,thevolume(orlength)ofthedomainwallneedstoreduceand increasedemagnetizationenergywhenKUincreaseswithincreasingtNiintherangeof 0.4nm~0.6nm.Fig.3.(a)-(h)Magneto-opticKerrhysteresisloopsandmagneticdomainimagesof thefilmsPt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1)withlabeledthicknesstNi=0.2nm(a),0.3nm (b),0.4nm(c),0.5nm(d),0.6nm(e),0.7nm(f),0.8nm(g),0.9nm(h),respectively. Thecorrespondingmagneticdomainimageswiththesizeof100µm×150µmwereobtainedatthelabeledout-of-planemagneticfields(alsomarkedasthereddots onloops). 3.3Magnetizationdynamics TofurtherinvestigatetheNithickness-dependentmagnetizationdynamicsofCo/Ni multilayers,weperformthebroadbandFMRmeasurementwiththeexternalfield perpendiculartothefilmplane.AllFMRmeasurementswerecarriedoutwitha home-madedifferentialFMRmeasurementsystemcombiningLock-intechniqueat roomtemperature.Acontinuous-waveOerstedfieldwithaselectedradiofrequencyis generatedviaconnectingcoplanewaveguide(CPW)withanRFgenerator,which producesamicrowavesignaltoexciteFMRofferromagneticfilm,whichwithfilm surfacewasadheredontheCPW.TheRFpowerusedintheexperimentsis15dBm. Toimprovethesignal-to-noiseratio(SNR),alock-indetectiontechniqueisemployed throughthemodulationofsignals.Themodulationofadirectcurrent(DC)magnetic fieldHisprovidedbyapairofsecondaryHelmholtzcoilspoweredbyanalternating current(AC)sourcewith129.9Hz[seeFig.4(a)][16,30].Thedifferentialabsorption signalismeasuredbysweepingthemagneticfieldwithafixedmicrowavefrequency. TherepresentativeFMRspectrumofPt(5)/[Co(0.2)/Ni(0.3)]5/Pt(1)obtainedat9GHz isshownintheinsetofFig.4(b).ThedifferentialFMRspectrumcanbewellfittedby usingacombinationofsymmetricandantisymmetricLorentzianfunction,asfollows: =s4∆−res 4−res2+∆22+∆2−4−res2 4−res2+∆22, (1) whereVSandVArepresentthesymmetricandantisymmetricfactors,Histhe externalmagneticfield,Hresistheresonancefield,andΔHisthelinewidthofFMR corresponding3timesofthepeak-to-dipwidthintheFMRspectrum.The relationshipbetweenthefrequencyfandtheresonancefieldHresofthetwoseriesof Pt/[Co(0.2)/Ni(tNi)]andPt/[Co(0.3)/Ni(tNi)]samples[Figs.4(b)and(d)]canbewell fittedbytheKittelequation[31]=γ 2πres+eff, (2) whereγ 2π=2.8MHzOe−1isthegyromagneticratio,Heffiseffective demagnetization[32]eff=K−4S.Therefore,themagneticanisotropyfieldHK alsocanbedirectlydeterminedfromthedispersionrelationoffversusHresbyusinga parameterMSobtainedbyVSM.Inaddition,wecanobtaintheintrinsicGilbert dampingαbyfittingtheexperimentaldataoflinewidthΔHversusresonance frequency[Figs.4(c)and(e)]withtheformula:∆=∆0+4πα γ,hereΔH0isan inhomogeneouslinewidthindependentofthefrequency,andthesecondtermisthe intrinsiclinewidthlinearlyproportionaltothefrequency.Theinhomogeneous linewidthofsamplesisderivedfromroughness,defectsandinhomogeneousPMA andmagnetization[33]. Fig.4.(a)DifferentialFMRspectraexperimentalsetup.(b)Dependenceofthe resonancefieldHresonthefrequencyfwiththeout-of-planefieldforthefilms Pt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1).SolidlinesindicatetheKittelfittingcurves.Theinsetis therepresentativeFMRspectrumobtainedat9GHz,whichcanbewellfittedbyEq.(1)(solidredline).(c)Thelinewidthversusfrequency(symbols)forthesamples Pt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1).Thesolidlineisalinearfitting,whichcanextractthe correspondingdampingconstantαbasedonEq.(2).(d)-(e)Sameas(b)-(c),forthe filmsPt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1). Figure5summarizesthedependenceofthedeterminedmaterialparameters:the saturationmagnetizationMS,thecoercivityHC,theanisotropyfieldHK,the inhomogeneouslinewidthΔH0andthemagneticdampingconstantαonNithickness tNiforthestudiedtwoseriesofPt/[Co(0.2)/Ni(tNi)]andPt/[Co(0.3)/Ni(tNi)]samples. ThedeterminedHKbythreeindependentmethodsshowsanoverallconsistent behavior.TheHKbeginstoincreasewithincreasingtNi,andreachesthemaximumat tNi~0.6nm,whereafterreducesagainwithcontinuingtoincreasetNi.Severalreasons accountforthisphenomenon.First,themagneticanisotropyofthestudiedmultilayer ismainlycontributedfromtheinterfacialmagneticanisotropyoftheCo/NiandPt/Co interfaces[34].Second,theCo/Nimultilayers'interfacequalitydependshighlyonthe Nilayer'sthickness.Inotherwords,toothinnickellayermaynotgetagoodCo/Ni interfaceduetoinevitableelementsdiffusionduringsputteringdeposition.However, theHKwilldropduetoreducingtheratiooftheinterfacialanisotropytothevolume anisotropyenergyiftheNilayeristoothick.LikeHK,theHCshowsasimilartrend withvaryingthicknessoftheNilayer.Aswewellknowthatthecoercivitydepends onPMA,aswellasdefects-inducedpinningeffects.But,inourcase,theresultsshow thatthecombinationofPMAandmagnetization-relevantdemagnetizationfield dominatethecoercivity,whichcanbewellexplainedbytheBrownformula[35]:C= 2U S−S,whereKU=(MS*HK)/2andNarethemagneticanisotropyconstantand thedemagnetizationfactorofthefilm,respectively. Figures5(d)and5(i)showtheinhomogeneouslinewidth(H0)ofFMRspectraas afunctionoftNiforPt/[Co(0.2)/Ni(tNi)]andPt/[Co(0.3)/Ni(tNi)],respectively.Forthin thicknessNisamples,islandstructuresaremostlikelyformed.Thisresultsina broadeningoftheresonancelinewidthduetoadistributionofeffectiveinternal anisotropyanddemagnetizationfields[37].OnecanseethattheminimumlinewidthoftwoserialsamplescorrespondstothemaximumPMAfieldHK,suggestingthe inhomogeneousmagneticanisotropy-inducedlinearbroadeningistheminimumatthe optimumPMAcondition[36].Althoughtheintrinsicdampingconstantisalmost independentoftheNithicknessforthestudiedtwoserials,butthePt/[Co(0.2)/Ni(tNi)] filmshavealowerdampingconstantα~0.04thanα~0.07ofPt/[Co(0.3)/Ni(tNi)]. Theobviousdifferenceindampingconstantbetweenthetwoserialmultilayer systemsindicatesthattheformerhasbettermagneticdynamicproperties. Fig.5.(a)-(e)DependenceofthesaturationmagnetizationMS(a),thecoercivityHC(b), theanisotropyfieldHK(c),theinhomogeneouslinewidthΔH0(d)andthemagnetic dampingconstantα(e)ontheNithicknesstNiinthefilms Pt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1).(f)-(j)Sameas(a)-(e)forthesamples Pt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1).MS,HC,andHKweredeterminedfromthepreviousmagnetizationloops,AHRloops,MOKEloops,andtheferromagnetresonance spectra.ThelinewidthwasdeterminedbyfittingtheexperimentalFMRspectrumwith aLorentzianfunctionbasedonEq.(1).Themagneticdampingconstantwasobtained byalinearfittingofΔHvs.fcurvesbasedonEq.(2). 4.CONCLUSION NithicknesseffectonthestaticmagneticpropertiesandmagneticdynamicsofPt(5 nm)/[Co(0.2nmand0.3nm)/Ni(tNinm)]5/Pt(1nm)multilayersdemonstratethatthe twostudiedserialmultilayersystemsexhibittheoptimumPMAcoefficientKUwellas thehighestcoercivityHCattheNithicknesstNi=0.6nm.TheMOKEimagesfurther confirmthatthemaximumKUcorrespondstothemagneticdomainstructurewiththe shortestlengthofdomainwallthroughminimizingthetotalenergy,whichconsistsof magneticanisotropyenergy,exchangeenergy,anddemagnetizationenergy. Furthermore,thefrequency-dependentFMRspectrashowthatthedampingconstant remainsalmostconstantwiththedifferentNithicknessesforbothserials,butthe Pt/[Co(0.2)/Ni(tNi)]multilayerserialhasalowerdampingconstantα~0.04than0.07 ofthePt/[Co(0.3)/Ni(tNi)]serial.Accordingtotheobtainedresults,wefindthatthe optimumPMAcoefficientKU=3.3×106ergcm-3,thehighestcoercivityHC=250 Oe,andaswellasthelowestdampingconstantα=0.04canbeachievedat Pt(5)/[Co(0.2)/Ni(0.6)]5/Pt(1).Ourresultsofoptimizingmagneticpropertiesofthe Pt/[Co/Ni]multilayerbytuningtheratioofCo/Nilayersishelpfultofacilitateits applicationsinvariousspintronicdevices. Acknowledgement：ProjectissupportedbytheNationalNaturalScienceFoundationof China(GrantNos.11774150,12074178,12004171,12074189and51971109),theAppliedBasicResearchProgramsofScienceandTechnologyCommissionFoundationofJiangsuProvince, China(GrantNo.BK20170627),theNationalKeyR&DProgramofChina(GrantNo. 2018YFA0209002),theOpenResearchFundofJiangsuProvincialKeyLaboratoryfor Nanotechnology,andtheScientificFoundationofNanjingUniversityofPostsand Telecommunications(NUPTSF)(GrantNo.NY220164). 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2023-04-19

We systematically investigated the Ni and Co thickness-dependent perpendicular magnetic anisotropy (PMA) coefficient, magnetic domain structures, and magnetization dynamics of Pt(5 nm)/[Co(t_Co nm)/Ni(t_Ni nm)]5/Pt(1 nm) multilayers by combining the four standard magnetic characterization techniques. The magnetic-related hysteresis loops obtained from the field-dependent magnetization M and anomalous Hall resistivity (AHR) \r{ho}_xy found that the two serial multilayers with t_Co = 0.2 and 0.3 nm have the optimum PMA coefficient K_U well as the highest coercivity H_C at the Ni thickness t_Ni = 0.6 nm. Additionally, the magnetic domain structures obtained by Magneto-optic Kerr effect (MOKE) microscopy also significantly depend on the thickness and K_U of the films. Furthermore, the thickness-dependent linewidth of ferromagnetic resonance is inversely proportional to K_U and H_C, indicating that inhomogeneous magnetic properties dominate the linewidth. However, the intrinsic Gilbert damping constant determined by a linear fitting of frequency-dependent linewidth does not depend on Ni thickness and K_U. Our results could help promote the PMA [Co/Ni] multilayer applications in various spintronic and spin-orbitronic devices.

Thickness-dependent magnetic properties in Pt[CoNi]n multilayers with perpendicular magnetic anisotropy

2304.09366v1