We derive the spin-wave dynamics of a magnetic material from the
time-dependent spin density functional theory in the linear response regime.
The equation of motion for the magnetization includes, besides the static spin
stiffness, a "Berry curvature" correction and a damping term. A gradient
expansion scheme based on the homogeneous spin-polarized electron gas is
proposed for the latter two quantities, and the first few coefficients of the
expansion are calculated to second order in the Coulomb interaction.Comment: 8 pages, no figure

Qian, Zhixin

Vignale, Giovanni

English

arXiv

Zhixin Qian

Giovanni Vignale

Crossref

Physical Review Letters

Spin Dynamics from Time-Dependent Spin-Density-Functional Theory

URL:http://link.aps.org/doi/10.1103/PhysRevLett.88.056404
DOI:10.1103/PhysRevLett.88.056404We derive the spin-wave dynamics of a magnetic material from the time-dependent spin-density-functional theory in the linear response regime. The equation of motion for the magnetization includes, besides the static spin stiffness, a “Berry curvature” correction and a damping term. A gradient expansion scheme based on the homogeneous spin-polarized electron gas is proposed for the latter two quantities, and the first few coefficients of the expansion are calculated to second order in the Coulomb interaction.We gratefully acknowledge support from NSF
Grant No. DMR-0074959 and Research Board Grant
No. URB-00-029 at the University of Missouri

Vignale, Giovanni, 1957-

University of Missouri: MOspace

VOLUME 88, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 4 FEBRUARY 2002056404-1Spin Dynamics from Time-Dependent Spin-Density-Functional TheoryZhixin Qian and Giovanni VignaleDepartment of Physics, University of Missouri, Columbia, Missouri 65211(Received 23 July 2001; published 17 January 2002)We derive the spin-wave dynamics of a magnetic material from the time-dependent spin-density-functional theory in the linear response regime. The equation of motion for the magnetization includes,besides the static spin stiffness, a “Berry curvature” correction and a damping term. A gradient expansionscheme based on the homogeneous spin-polarized electron gas is proposed for the latter two quantities,and the first few coefficients of the expansion are calculated to second order in the Coulomb interaction.DOI: 10.1103/PhysRevLett.88.056404 PACS numbers: 71.15.Mb, 75.30.Ds, 76.50.+gThe study of the ground-state properties of magneticmaterials within the framework of spin-density-functionaltheory (SDFT) is by now a mature field [1–3]. By com-parison, the study of excited-state properties is still in itsinfancy. There has been great interest in recent yearsin deriving a closed equation of motion for the magne-tization starting from a first-principles description of theelectrons as itinerant particles [4–8], rather than from thetime-honored Heisenberg model of interacting local mo-ments [9–11]. An alternative approach is to calculate thespectrum of spin excitations from the imaginary part ofthe linear spin-spin response function [12]. Our objectivein this Letter is to unify the two approaches within theframework of the time-dependent SDFT. We emphasizenew aspects of the physics beyond the adiabatic approxi-mation (namely, dissipation), as well as a practically work-able computational scheme.Elementary spin excitations in itinerant-electron mag-nets fall into two groups (i) Stoner excitations —inwhich a single electron quasiparticle is spin reversedand (ii) spin waves. Both types of excitations can becomputed from the linear spin-spin response functionxijr, r0;v  ib2R`0 dt eivtSˆir, t, Sˆjr0 [Sˆir isthe i component of the spin-density operator, with b ge2mc and h¯  1], which determines the magneti-zation mir,v  2bSir,v induced by an externalmagnetic field Br0,v at a frequency v:mir,v XjZxijr, r0;vBjr0;v dr0. (1)Stoner excitations are distributed along branch cuts of thisresponse function, while collective modes show up as iso-lated poles in the complex frequency plane.A time-dependent SDFT is ideally suited for calculat-ing xij [13]. As a first step in this direction one solves thestatic Kohn-Sham equation [14], whose eigenfunctions andeigenvalues determine the exact equilibrium density andmagnetization. One then constructs the linear spin-spinresponse function xKS,ijr, r0;v of the Kohn-Sham (KS)system. Because this is a stationary noninteracting sys-tem, the calculation can be carried out exactly [15]. Fi-nally, the response of the physical system is calculatedas the response of the KS system to an effective time-0031-90070288(5)056404(4)$20.00dependent field Beff which includes, in addition to the ex-ternal field B, a many-body “exchange-correlation” fieldBxc. When these ideas are cast in formulas one obtainsthe well-known mathematical relation between the matrixinverses of the exact response function and the KS responsefunction, namely,x21ijr, r0;v  x21KSijr, r0;v 2 fxc,ijr, r0;v ,(2)where fxc,ijr, r0;v is the tensor that connects Bxc to theinduced magnetizationBxc,ir,v XjZfxc,ijr, r0;vmj r0;v dr0. (3)Collective spin excitations can then be obtained from thesolution of the eigenvalue problemXjZx21ijr, r0;vmj r0,vdr0  0 , (4)where mjr,v is the magnetization profile in the spinwave. The problem is to find the frequencies v for whichthis equation has nonvanishing solutions. Because x hasboth real and imaginary parts these eigenfrequencies willbe complex in general and will determine both the disper-sion Rev and the linewidth Imv of spin waves.To appreciate the power of Eqs. (2) and (4) we now usethem to derive both the adiabatic spin dynamics [4–6] andthe Landau-Lifshitz (LL) equation [16] in the linear re-sponse regime. First of all, we choose to focus on thetransverse part of the response function, namely, the partthat describes the response to a magnetic field perpendicu-lar to the ground-state magnetization. It turns out that in acollinear magnet the transverse response function is rigor-ously decoupled from the longitudinal one in the absenceof spin-orbit interactions. In general, this decoupling isjustified by the difference between the time scales of thelongitudinal and transverse spin dynamics. Our key as-sumption is that both x21KS and fxc can be Taylor expanded,at low frequency, in powers ofv. Keeping only the first or-der term in the low-frequency expansion of x21 we cometox21ijr, r0;v  aijr, r0 1 ivV˜ij r,r0 , (5)© 2002 The American Physical Society 056404-1VOLUME 88, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 4 FEBRUARY 2002where aijr, r0  x21ijr,r0;v  0 is the (symmet-ric) spin stiffness tensor, given by the second derivative ofthe ground-state energy with respect to the magnetization,andV˜ijr, r0  limv!0≠ Imx21ijr, r0;v≠v, (6)with the derivative taken along the real frequency axis.(V˜ij is purely real because the derivative of Rex vanishesat v  0.) Notice that, by definition,Zaijr, r0mjr0 dr0 dEmdmir, (7)where Em is the ground-state energy regarded as a func-tional of mr [17]. The tensor V˜ij can be split into anti-symmetric and symmetric components as follows:V˜ijr, r0  Vijr, r0 1 gijr, r0 , (8)where Vijr, r0  2Vjir0, r and gijr, r0  gjir0, r.Therefore, substituting (5) into (4), and switching to a realtime representation with the substitution2iv ! ≠≠t, weobtain the equation of motionXjZdr0 Vijr, r0 1 gijr, r0≠mjr0, t≠tdEmdmir, t. (9)This equation reduces to the Niu-Kleinman adiabaticequation of motion [4] if the symmetric tensor gij is ne-glected and the antisymmetric tensor Vij is identified withthe “Berry curvature.” Indeed, after a lengthy but straight-forward calculation we can show thatVijr, r0  22 Imø≠cm≠mirÇ≠cm≠mjr0¿, (10)where cm is the ground-state wave function regarded asa functional of m [17,18]. Thus, the antisymmetric part ofEq. (6) is equivalent to Eq. (10): the former is, however,more amenable to approximation and computation.The symmetric part of Eq. (6), gij , is responsible fordissipation as one can immediately verify by calculatingthe rate of entropy production at temperature T :TdSdt 2Z dEmdmr, t?≠mr, t≠tdr 2XijZdrZdr0≠mir, t≠tgijr, r0≠mjr0, t≠t.(11)Not surprisingly, this term is absent in a purely adiabatictheory such as that of Ref. [4].We now turn to the task of approximating the right-hand side of Eq. (6). A classic approximation scheme isprovided by the gradient expansion [19]. In this schemeone assumes that the two-point function V˜ijr, r0 is ashort ranged function of the distance jr 2 r0j. It is then056404-2permissible, if the density and magnetization are slowlyvarying, to expand V˜ asV˜ijr, r0  V˜0,ijnr,mrdr 2 r01 V˜2,ijnr,mr=rdr 2 r0 ? =r0 1 . . . ,(12)where V˜0,ijn,m and V˜2,ijn,m are the coefficientsof q0 and q2, respectively, in the small-q expansionof V˜homij q RV˜homij r 2 r0e2iq?r2r0 dr in a ho-mogeneous electron gas of density n and magnetiza-tion m.We are now in a position to prove that the standard LLequation [16] is simply the zero-order approximation (i.e.,the local density approximation) in the gradient expan-sion for the Berry curvature. To this end, we considera homogeneous spin-polarized electron gas with the sameground-state density n and magnetization m0 as those ofthe real system at point r. The homogeneous magnetiza-tion is maintained by an external fictitious magnetic fieldB0 parallel to m0,B0 µ≠en,m≠m∂mm0, (13)where en,m is the energy density of the homogeneouselectron gas of density n and magnetization m [20]. Thetransverse spin-spin response function of this system atq  0 isx21ijq  0,v v0dij 1 iveijbm0, (14)where v0  bB0 and eij is the two-dimensional Levi-Civita tensor with the Cartesian indices i, j being orthogo-nal to the direction of m0. Thus, for the homogeneouselectron gas, V˜ijq  0  eijbm0 [see Eq. (6)], andthen, from Eqs. (8) and (12), we see that the local densityapproximation takes the formVijr, r0 	 eij1bm0rdr 2 r0 . (15)With this approximation Eq. (9) reduces to the linearizedLL equation,≠mir, t≠t 2Xjbm0eij≠Em≠mjr. (16)Notice that the dissipative gij is exactly zero at this orderof approximation. Thus, the gradient expansion for gijbegins with a second-order termgijr, r0  g2,ijr=rdr 2 r0 ? =r 0 1 . . . . (17)This makes physical sense because a global rotation ofthe spins must be rigorously undamped in the absence ofspin-orbit interactions.The remainder of this Letter is devoted to the calcu-lation of the leading gradient corrections to the LL equa-tion. After lengthy calculations, which will be described in056404-2VOLUME 88, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 4 FEBRUARY 2002detail elsewhere, we obtain the small-q expansion of the transverse spin-spin response function of the homogeneousspin-polarized electron gas:x21ijq,v  x21ij 0,v 1nq24mm20dij 1q22mm023∑4mTˆ# 2 Tˆ"3Vv0dij 2 iveijv2 2 v2012pn22g0 2 13ma0µdij 2 ieijv 2 v021dij 1 ieijv 1 v02∂1F21v dij 2 ieijv 2 v021F12v dij 1 ieijv 1 v02∏, (18)where V is the volume and a0 is the Bohr radius, Tˆ" andTˆ# are the kinetic energy operators associated with up-spinand down-spin electrons, respectively, the angular brack-ets denote the ground-state or thermal ensemble average,g0 is the pair correlation function at zero separation, andF12v is a four-point response function, defined asF12v  F212v13V3Xk,k0ykyk0k ? k03 Sˆ12krˆk; Sˆ2k0rˆ2k0v . (19)Sˆ6k  Sˆxk 6 iSˆyk are spin-density fluctuation op-erators, yk  4pe2k2, and rˆk is the density fluc-tuation operator. The “Zubarev product” is defined asAˆ; Bˆv  2iR`0 dt eivtAˆt, Bˆ. Taking the small vlimit of this expression we obtain the coefficients of thegradient expansion for V and g as follows:V2,ij  eij1mm0v02∑mT" 2 T#3V12pn22g0 2 13ma0v01ReF120v0212≠ReF12v≠vÇv0∏(20)andg2,ij  2dij2mm0v02limv!0ImF12vv. (21)Since the long wavelength spin-wave frequency in a ferro-magnet is proportional to q2, the above results indicate thatthe gradient corrections to V and g will affect the disper-sion and damping of ferromagnetic spin waves beginningat order q4.Equations (20) and (21) contain both ground-state (ther-mal ensemble) averages, such as T" 2 T# and g0, andthe dynamical response function F12v, which dependson the spectrum of excited states. The former can be cal-culated to a high degree of accuracy by variational anddiffusion Monte Carlo techniques [21]; the major chal-lenge lies therefore in the calculation of F12v. Theform of Eq. (19) suggests that we evaluate F12v to sec-ond order in the Coulomb interaction: this is accomplishedby substituting the noninteracting expression for the fourpoint response function Sˆ12krˆk; Sˆ2k0rˆ2k0v .Even such an approximate calculation turns out to be verydifficult, but we have been able to establish analytically the056404-3limiting forms of the imaginary part of F12v for highand low frequency. At low frequency (v ø EF" 2 EF#,where EF" is the larger of the two Fermi energies) we findImF12v  2m2Gvv2 1 4p2k2BT236p3kF"a20, (22)where kB is the Boltzmann constant and T is the tempera-ture. The dimensionless coefficient G is the sum of “di-rect” and “exchange” terms: G  GD 1 GE which wereport separately for reasons that will become clear in thefollowing:GD 2l1 2 l21u3l 2 1 3l 2 12l1 2 l,GE 12ln1 1 l1 2 l2u3l 2 12lln 2l1 2 l,(23)where l  1 2 z 131 1 z 13, and z  n" 2 n#nis the degree of spin polarization, and u is the step func-tion. Equation (22) yields the damping tensor accordingto Eq. (21):g2,ij  dijG2pkF"3m0v0a02kBT2. (24)Note that the dissipation vanishes as T 2 for T ! 0 [22].To calculate the correction to the Berry curvature wealso need the real part of F12v. To this end, we makeuse of the Kramers-Krönig dispersion relationReF12v 1pZ `2`dv0ImF12v0v0 2 v(25)and use for ImF12v at finite frequency the mode-decoupling approximation of Ref. [23]:1V 2ImS12krk;S2k0r2k0v	 2gxdk,k0pZ v0Imxnnk,v 2 v03 Imx122k,v0dv0, (26)where xnnk,v  V21rk;r2kv is the density-density response function, and the factor gx  GD 1GEGD is used to include the exchange contributionand assure the correct behavior in the most important low-frequency limit.Carrying out the calculations, we find that the last twoterms on the right-hand side of Eq. (20) are056404-3VOLUME 88, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 4 FEBRUARY 2002ReF120  2gxk 3F" EF"3p4a20Pz  , (27)and≠ReF12v≠vÇv0 2gxk 3F"3p4a20Qz  . (28)Pz  and Qz  can be very accurately parametrized asPz   1.9606 2 3.5z 2 1.4z2 lnz 1 2.08z2, andQz   1.18z 2 0.186z2 2 0.842z3 2 0.045z 21.49z 2 lnz .Finally, we need practical approximate expressionsfor the ground-state averages appearing in Eq. (20). Thezero-separation pair-correlation function g0 of a spin-polarized electron gas has recently been calculated byGori-Giorgi and Perdew, and an explicit expression isgiven in Eqs. (47) and (22)–(30) of their paper [24].As for T" 2 T#, in the absence of detailed calcula-tions, we propose a linear interpolation for the correlationcontribution:T" 2 T#  T" 2 T#0 2 Ndrsecdrsz  1z , (29)where ec is the correlation energy per particle, rs is theWigner radius, and T" 2 T#0  k 5F" 2 k 5F# 20p2mis the difference of kinetic energies of the noninteractingsystems.The spin dynamics equation (9) is the central result ofthis Letter. It includes both the adiabatic spin dynamicsand the conventional Landau-Lifshitz equation as specialcases. It reduces to the adiabatic spin dynamics if thedamping tensor g is neglected. It further reduces to a con-ventional Landau-Lifshitz equation if the gradient correc-tions to V are neglected. In addition, we have developeda density-functional scheme for the systematic calculationof V and g. The analytical expressions for the dampingtensor and the Berry curvature make the application of thenew equation of motion to spin dynamics of ferromagneticand antiferromagnetic systems quite promising. If greateraccuracy is required, one can revert to the full-fledged lin-ear response formalism, in which x21KS is treated exactlyand only fxc is approximated.We gratefully acknowledge support from NSFGrant No. DMR-0074959 and Research Board GrantNo. URB-00-029 at the University of Missouri. We thankJohn Perdew for his comments on the manuscript.[1] U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972).056404-4[2] A. K. Rajagopal, Adv. Chem. Phys. 41, 59 (1980), andreferences therein.[3] L. M. Sandratskii, Adv. Phys. 47, 91 (1998); J. Sticht, K.-H.Höck, and J. Kübler, J. Phys. Condens. Matter 1, 8155(1989).[4] Q. Niu and L. Kleinman, Phys. Rev. Lett. 80, 2205 (1998).[5] Q. Niu, X. Wang, L. Kleinman, W. M. Liu, D. M. C. Nichol-son, and G. M. Stocks, Phys. Rev. Lett. 83, 207 (1999).[6] R. Gebauer and S. Baroni, Phys. Rev. B 61, R6459 (2000).[7] V. P. Antropov, M. I. Katsnelson, B. N. Harmon, M. vanSchilfgaarde, and D. Kusnezov, Phys. Rev. B 54, 1019(1996).[8] D. M. Bylander, Q. Niu, and L. Kleinman, Phys. Rev. B61, R11 875 (2000).[9] C. Herring, in Magnetism, edited by G. T. Rado and H.Suhl (Academic Press, New York, 1966), Vol. IV.[10] T. Moriya, Spin Fluctuations in Itinerant Electron Mag-netism, Springer Series in Solid State Sciences Vol. 56(Springer-Verlag, Berlin, 1985).[11] A. Aharoni, Introduction to the Theory of Ferromagnetism(Oxford University Press, New York, 1996).[12] S. Y. Savrasov, Phys. Rev. Lett. 81, 2570 (1998).[13] For an introduction to time-dependent DFT, see E. K. U.Gross, J. F. Dobson, and M. Petersilka, in Density Func-tional Theory II, Topics in Current Chemistry Vol. 181,edited by R. F. Nalewajski (Springer, Berlin, 1996).[14] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).[15] E. K. U. Gross and W. Kohn, Phys. Rev. Lett. 55, 2850(1985).[16] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, inCourse of Theoretical Physics Part II, Vol. 9, edited byL. D. Landau and E. M. Lifshitz (Reed Publishing, Oxford,1980).[17] The precise definition is Em  minc!mrcjHˆjc,where Hˆ is the Hamiltonian and the minimum is soughtamong the wave functions that yield the magnetizationmr. This constrained search also yields the ground-statewave function jcm as a functional of mr.[18] A general expression for the Berry curvature in terms ofa linear response function is also derived in S. R. Jain andA. K. Pati, Phys. Rev. Lett. 80, 650 (1998).[19] K. Burke, J. Perdew, and Y. Wang, in Electronic DensityFunctional Theory, Recent Progress and New Directions,edited by J. Dobson, G. Vignale, and M. P. Das (Plenum,New York, 1998), p. 81.[20] Thus, we do not consider the possibility of spontaneousferromagnetism in the homogeneous electron gas. This ispredicted to occur at extremely low densities [21] and isof no interest here.[21] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566(1980); G. Ortiz and P. Ballone, Phys. Rev. B 50, 1391(1994); G. Ortiz, M. Harris, and P. Ballone, Phys. Rev.Lett. 82, 5317 (1999).[22] B. I. Halperin and P. C. Hohenberg, Phys. Rev. 188, 898(1969). Note that gijr, r0 in Eq. (17) is negative definitewhen g2,ij . 0.[23] R. Nifosì, S. Conti, and M. P. Tosi, Phys. Rev. B 58, 12 758(1998).[24] P. Gori-Giorgi and J. P. Perdew, Phys. Rev. B 64, 155102(2001).056404-4

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Spin dynamics from time-dependent spin density-functional theory

Spin dynamics from time-dependent spin density-functional theory

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