A Kubo-Greenwood-like equation for the Gilbert damping parameter $\alpha$ is
presented that is based on the linear response formalism. Its implementation
using the fully relativistic Korringa-Kohn-Rostoker (KKR) band structure method
in combination with Coherent Potential Approximation (CPA) alloy theory allows
it to be applied to a wide range of situations. This is demonstrated with
results obtained for the bcc alloy system Fe$_x$Co$_{1-x}$ as well as for a
series of alloys of permalloy with 5d transition metals.
  To account for the thermal displacements of atoms as a scattering mechanism,
an alloy-analogy model is introduced. The corresponding calculations for Ni
correctly describe the rapid change of $\alpha$ when small amounts of
substitutional Cu are introduced

D. Ködderitzsch

H. Ebert

M. E. Rose

P. J. Kelly

S. Mankovsky

English

arXiv

A Kubo-Greenwood-like equation for the Gilbert damping parameter α is presented that is based on the linear response formalism. Its implementation using the fully relativistic Korringa-Kohn-Rostoker band structure method in combination with coherent potential approximation alloy theory allows it to be applied to a wide range of situations. This is demonstrated with results obtained for the bcc alloy system Fe1-xCox as well as for a series of alloys of Permalloy with 5d transition metals. To account for the thermal displacements of atoms as a scattering mechanism, an alloy-analogy model is introduced. The corresponding calculations for Ni correctly describe the rapid change of α when small amounts of substitutional Cu are introduced

Ebert, Hubert

Mankovsky, Sergiy

Koedderitzsch, Diemo

Kelly, P. J.

University of Regensburg Publication Server

Ab Initio Calculation of the Gilbert Damping Parameter via the Linear Response FormalismH. Ebert, S. Mankovsky, and D. Ko¨dderitzschUniversity of Munich, Department of Chemistry, Butenandtstrasse 5-13, D-81377 Munich, GermanyP. J. KellyFaculty of Science and Technology and MESAþ Institute for Nanotechnology,University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands(Received 1 March 2011; published 2 August 2011)A Kubo-Greenwood-like equation for the Gilbert damping parameter  is presented that is based on thelinear response formalism. Its implementation using the fully relativistic Korringa-Kohn-Rostoker bandstructure method in combination with coherent potential approximation alloy theory allows it to beapplied to a wide range of situations. This is demonstrated with results obtained for the bcc alloy systemFe1xCox as well as for a series of alloys of Permalloy with 5d transition metals. To account for thethermal displacements of atoms as a scattering mechanism, an alloy-analogy model is introduced. Thecorresponding calculations for Ni correctly describe the rapid change of  when small amounts ofsubstitutional Cu are introduced.DOI: 10.1103/PhysRevLett.107.066603 PACS numbers: 72.25.Rb, 71.20.Be, 71.70.Ej, 75.78.nThe magnetization dynamics that is relevant for theperformance of any type of magnetic device is in generalgoverned by damping. In most cases the magnetizationdynamics can be modeled successfully by means of theLandau-Lifshitz-Gilbert (LLG) equation [1] that accountsfor damping in a phenomenological way. The possibility tocalculate the corresponding damping parameter from firstprinciples would open the perspective of optimizing mate-rials for devices and has therefore motivated extensivetheoretical work in the past. This led among others toKambersky’s breathing Fermi surface (BFS) [2] andtorque-correlation models (TCM) [3], that in principleprovide a firm basis for numerical investigations basedon electronic structure calculations [4,5]. The spin-orbitcoupling that is seen as a key factor in transferring energyfrom the magnetization to the electronic degrees of free-dom is explicitly included in these models. Most ab initioresults have been obtained for the BFS model though thetorque-correlation model makes fewer approximations[4,6]. In particular, it in principle describes the physicalprocesses responsible for Gilbert damping over a widerange of temperatures as well as chemical (alloy) disorder.However, in practice, like many other models it depends ona relaxation time parameter  that describes the rate oftransfer due to the various types of possible scatteringmechanisms. This weak point could be removed recentlyby Brataas et al. [7] who described the Gilbert damping bymeans of scattering theory. This development suppliedthe formal basis for the first parameter-free investigationson disordered alloys for which the dominant scatteringmechanism is potential scattering caused by chemical dis-order [8] or temperature induced structure disorder [9].As pointed out by Brataas et al. [7], their approach iscompletely equivalent to a formulation in terms of thelinear response or Kubo formalism. The latter route istaken in this communication that presents a Kubo-Greenwood-like expression for the Gilbert damping pa-rameter. Application of the scheme to disordered alloysdemonstrates that this approach is indeed fully equivalentto the scattering theory formulation of Brataas et al. [7]. Inaddition a scheme is introduced to deal with the tempera-ture dependence of the Gilbert damping parameter.Following Brataas et al. [7], the starting point of ourscheme is the Landau-Lifshitz-Gilbert (LLG) equation forthe time derivative of the magnetization ~M:1d ~Md¼  ~M ~Heff þ ~M ~Gð ~MÞ2M2sd ~Md; (1)where Ms is the saturation magnetization,  the gyromag-netic ratio, and ~G the Gilbert damping tensor. Accordingly,the time derivative of the magnetic energy is given by_Emag ¼ ~Heff  d~Md¼ 12_~m½ ~Gð ~mÞ _~m (2)in terms of the normalized magnetization ~m ¼ ~M=Ms. Onthe other hand, the energy dissipation of the electronicsystem _Edis ¼ hdH^d i is determined by the underlyingHamiltonian H^ðÞ. Expanding the normalized magnetiza-tion ~mðÞ, that determines the time dependence of H^ðÞabout its equilibrium value, ~mðÞ ¼ ~m0 þ ~uðÞ, one hasH^ ¼ H^0ð ~m0Þ þX~u@@ ~uH^ð ~m0Þ: (3)Using the linear response formalism, _Edis can be writtenas [7]PRL 107, 066603 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending5 AUGUST 20110031-9007=11=107(6)=066603(4) 066603-1  2011 American Physical Society_Edis ¼ @Xii0X_u _uc i~~~~~~~~@H^@u~~~~~~~~c i0c i0~~~~~~~~@H^@u~~~~~~~~c i ðEF  EiÞðEF  Ei0 Þ; (4)where EF is the Fermi energy and the sums run over alleigenstates  of the system. Identifying _Emag ¼ _Edis, onegets an explicit expression for the Gilbert damping tensor~G or equivalently for the damping parameter  ¼~G=ðMsÞ [7]. In full analogy to electric transport [10],the sum over eigenstates jc ii may be expressed in termsof the retarded single-particle Green’s functionImGþðEFÞ ¼ Pijc iihc ijðEF  EiÞ. This leads forthe parameter  to a Kubo-Greenwood-like equation¼ @MsTrace@H^@uImGþðEFÞ @H^@u ImGþðEFÞc(5)with h. . .ic indicating a configurational average in case of adisordered system (see below). Identifying T ¼ @H^=@uwith the component of the magnetic torque operator ~^Ialong the direction ~n, such that I^ ~n ¼ @H^=@ ~uð ~n ~uÞ ¼@H^=@uð ~n ~uÞ this expression obviously gives the pa-rameter  in terms of a torque-torque correlation function.However, in contrast to the conventional TCM the elec-tronic structure is not represented in terms of Bloch statesbut using the retarded electronic Green’s function givingthe present approach much more flexibility.The most reliable way to account for spin-orbit couplingas the source of Gilbert damping is to evaluate Eq. (5)using a fully relativistic Hamiltonian within the frameworkof local spin density formalism (LSDA) [11]:H^ ¼ c ~ ~pþmc2 þ Vð~rÞ þ  ~	 ~mBð~rÞ: (6)Here i and  are the standard Dirac matrices and ~p is therelativistic momentum operator [12]. The functions V andB are the spin-averaged and spin-dependent parts, respec-tively, of the LSDA potential. Equation (6) implies for theT operator occurring in Eq. (5) the expressionT ¼ @@u H^ ¼ B	: (7)The Green’s function Gþ in Eq. (5) can be obtained in avery efficient way by using the spin-polarized relativisticversion of multiple scattering theory [11] that allows us totreat magnetic solids:Gþð ~rn; ~r0m; EÞ ¼X0Znð ~rn; EÞnm0 ðEÞZm0 ð~r0m; EÞXZnð~r<; EÞJn0 ð~r>; EÞnm: (8)Here coordinates ~rn referred to the center of cell nhave been used with j~r<j ¼ minðj~rnj; j~r0njÞ and j~r>j ¼maxðj~rnj; j~r0njÞ. The four-component wave functionsZnð ~r; EÞ ðJnð ~r; EÞÞ are regular (irregular) solutions to thesingle-site Dirac equation for site n and nm0 ðEÞ is the so-called scattering path operator that transfers an electronicwave coming in at site m into a wave going out from site nwith all possible intermediate scattering events accountedfor coherently.Using matrix notation, this leads to the following ex-pression for the damping parameter: ¼ gtotXnTracehT0~0nTn~n0ic (9)with the g factor 2ð1þorb=spinÞ in terms of the spinand orbital moments, spin and orb, respectively, thetotal magnetic moment tot ¼ spin þorb, and ~0n0 ¼12i ð0n0  0n0Þ and with the energy argument EF omitted.The matrix elements of the torque operator, Tn, areidentical to those occurring in the context of exchangecoupling [13] and can be expressed in terms of the spin-dependent part B of the electronic potential with matrixelements:Tn0 ¼Zd3rZn0 ð ~rÞ½	Bxcð~rÞZnð ~rÞ: (10)As indicated above, the expressions in Eqs. (5)–(10) canbe applied straightforwardly to disordered alloys. In thiscase the brackets h. . .ic indicate the necessary configura-tional average. This can be done by describing in a first stepthe underlying electronic structure (for T ¼ 0 K) on thebasis of the coherent potential approximation (CPA) alloytheory. In the next step the configurational average inEq. (5) is taken following the scheme worked out byButler [10] when dealing with the electrical conductivityat T ¼ 0 K or residual resistivity, respectively, of disor-dered alloys. This implies, in particular, that so-calledvertex corrections of the type hTImGþTImGþic hTImGþichTImGþic that account for scattering-inprocesses in the language of the Boltzmann transportformalism are properly accounted for.Thermal vibrations as a source of electron scatteringcan in principle be accounted for by a generalization ofEqs. (5)–(10) to finite temperatures and by including theelectron-phonon self-energy el-ph when calculating theGreen’s function Gþ. Here we restrict ourselves to elasticscattering processes by using a quasistatic representationof the thermal displacements of the atoms from theirequilibrium positions. We introduce an alloy-analogymodel to average over a discrete set of displacementsthat is chosen to reproduce the thermal root mean squareaverage displacementffiffiffiffiffiffiffiffiffiffiffihu2iTp for a given temperature T.This was chosen according to hu2iT¼ 14 3h22mkD½ðD=TÞD=T þ 14with ðD=TÞ the Debye function, h the Planck constant,k the Boltzmann constant, and D the Debye temperature[14]. Ignoring the zero temperature term 1=4 and assuminga frozen potential for the atoms, the situation can be dealtPRL 107, 066603 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending5 AUGUST 2011066603-2with in full analogy to the treatment of disordered alloysdescribed above.The approach described above has been applied to theferromagnetic 3d-transition metal alloy systems bccFe1xCox, fcc Fe1xNix, and fcc Co1xNix. Figure 1 showsas an example results for bcc Fe1xCox for x  0:7. Thecalculated damping parameter ðxÞ for T ¼ 0 K is foundto be in very good agreement with the results based on thescattering theory approach [8] demonstrating numericallythe equivalence of the two approaches. An indispensablerequirement to achieving this agreement is to include thevertex corrections mentioned above. In fact, ignoring themleads in some cases to completely unphysical results. Tocheck the reliability of the standard CPA, that implies asingle-site approximation when performing the configura-tional average, we performed calculations on the basis ofthe nonlocal CPA [15]. Using a four-atom cluster led topractically the same results as the CPA except for the verydilute case. As found before for fcc Fe1xNix [8] thetheoretical results for  reproduce the concentration de-pendence of the experimental data quite well but are foundto be too low (see below). As suggested by Eq. (9) thevariation of ðxÞ with concentration x may reflect to someextent the variation of the average magnetic moment of thealloy, tot. Because the moments and spin-orbit couplingstrength do not differ very much for Fe and Co, thevariation of ðxÞ should be determined in the concentratedregime primarily by the electronic structure at the Fermienergy EF. As Fig. 1 shows, there is indeed a close corre-lation with the density of states nðEFÞ that may be seen as ameasure for the number of available relaxation channels.While the scattering and linear response approach arecompletely equivalent when dealing with bulk alloys thelatter allows us to perform the necessary configurationaveraging in a much more efficient way. This allows usto study with moderate effort the influence of varyingthe alloy composition on the damping parameter .Corresponding work has been done, in particular, usingPermalloy as a starting material and adding transitionmetals (TM) [16] or rare earth metals [17]. If we use thepresent scheme to study the effect of substituting Fe and Niatoms in Permalloy with a 5d TM, we find an increase of nearly linear with the 5d TM content, just as in experiment[16]. The total damping for 10% 5d TM content shown inFig. 2 (top) varies roughly parabolically over the 5d TMseries. In contrast to the Fe1xCox alloys considered above,there is now an S-like variation of the moments 5dspin overthe series (Fig. 2, bottom), characteristic of 5d impuritiesin the pure hosts Fe and Ni [18,19]. In spite of this behaviorof 5dspin the variation of ðxÞ seems again to be correlatedwith the density of states n5dðEFÞ (Fig. 2 bottom). Againthe trend of the experimental data is well reproduced by thecalculated values that are, however, somewhat too low.One possible reason for the discrepancy between thetheoretical and experimental results shown in Figs. 1 and2 might be the neglect of the influence of finite tempera-tures. This can be included as indicated above to accountfor the thermal displacement of the atoms in a quasistateway by performing a configurational average over thedisplacements using the CPA. This leads even for puresystems to a scattering mechanism and this way to a finitevalue for . Corresponding results for pure Ni are given inFig. 3 that show in full accordance with experiment a rapiddecrease of  with increasing temperature until a regimewith a weak variation of with T is reached. This behavioris commonly interpreted as a transition from conductivity-like to resistivitylike behavior reflecting the dominance ofintra- and interband transition, respectively [4], that isrelated to the increase of the broadening of electron energybands caused by the increase of scattering events withtemperature. Adding even less than 1 at.% Cu to Nistrongly reduces the conductivitylike behavior at low tem-peratures while leaving the high-temperature behavior es-sentially unchanged. A further increase of the Cu contentleads to the impurity-scattering processes responsible for0 0.1 0.2 0.3 0.4 0.5 0.6 0.7concentration xCo0123456α(x)x10-3ExptTheory (CPA), bccTheory (NL CPA)Fe-Con(EF)n(EF) (sts./Ry)1020304050600FIG. 1 (color online). Gilbert damping parameter for bccFe1xCox as a function of Co concentration: full circles—thepresent results within CPA; empty circles—within nonlocalCPA (NL CPA); and full diamonds—experimental data byOogane [20].02468α x10-2Ta W Re Os Ir Pt Au-0.300.30.6mspin5d (µB) n5d(EF)5d spin moment061218n5d(EF) (sts./Ry)FIG. 2 (color online). Top: Gilbert damping parameter  forPy/5d TM systems with 10% 5d TM content in comparison withexperiment [16]; bottom: spin magnetic moment m5dspin anddensity of states nðEFÞ at the Fermi energy of the 5d componentin Py/5d TM systems with 10% 5d TM content.PRL 107, 066603 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending5 AUGUST 2011066603-3band broadening dominating . This effect completelysuppresses the conductivitylike behavior in the low-temperature regime because of the increase of scatteringevents due to chemical disorder. Again this is fully in linewith the experimental data, providing a straightforwardexplanation for their peculiar variation with temperatureand composition.From the results obtained for Ni one may conclude thatthermal lattice displacements are only partly responsiblefor the finding that the damping parameters obtained for Pydoped with the 5d TM series, and Fe1xCox are somewhatlow compared with experiment. This indicates that addi-tional relaxation mechanisms like magnon scattering con-tribute. Again, these can be included at least in a quasistaticway by adopting the point of view of a disordered localmoment picture. This implies scattering due to randomtemperature-dependent fluctuations of the spin momentsthat can also be dealt with using the CPA.In summary, a formulation for the Gilbert dampingparameter  in terms of a torque-torque-correlation func-tion was derived that led to a Kubo-Greenwood-likeequation. The scheme was implemented using the fullyrelativistic Korringa-Kohn- Rostoker band structuremethod in combination with the CPA alloy theory. Thisallows us to account for various types of scattering mecha-nisms in a parameter-free way. Corresponding applicationsto disordered transition metal alloys led to very goodagreement with results based on the scattering theoryapproach of Brataas et al. demonstrating the equivalenceof both approaches. The flexibility and numericalefficiency of the present scheme was demonstrated by astudy on a series of Permalloy-5d TM systems. To inves-tigate the influence of finite temperatures on , a so-calledalloy-analogy model was introduced that deals with thethermal displacement of atoms in a quasistatic manner.Applications to pure Ni gave results in very good agree-ment with experiment and, in particular, reproduced thedramatic change of  when Cu is added to Ni.The authors would like to thank the DFG for financialsupport within the SFB 689 ‘‘Spinpha¨nomene in redu-zierten Dimensionen’’ and within project Eb154/23 forfinancial support. P. J. K acknowledges support by EUFP7 ICT Grant No. 251759 MACALO.[1] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).[2] V. Kambersky, Can. J. Phys. 48, 2906 (1970).[3] V. Kambersky, Czech. J. Phys. 26, 1366 (1976).[4] K. Gilmore, Y. U. Idzerda, and M.D. Stiles, Phys. Rev.Lett. 99, 027204 (2007).[5] M. Fa¨hnle and D. Steiauf, Phys. Rev. B 73, 184427 (2006).[6] V. Kambersky, Phys. Rev. B 76, 134416 (2007).[7] A. Brataas, Y. Tserkovnyak, and G. E.W. Bauer, Phys.Rev. Lett. 101, 037207 (2008).[8] A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak, andG. E.W. Bauer, Phys. Rev. Lett. 105, 236601 (2010).[9] Y. Liu, A.A. Starikov, Z. Yuan, and P. J. Kelly, Phys. Rev.B 84, 014412 (2011).[10] W.H. Butler, Phys. Rev. B 31, 3260 (1985).[11] H. Ebert, in Electronic Structure and Physical Propertiesof Solids, edited by H. Dreysse´, Lecture Notes in PhysicsVol. 535 (Springer, Berlin, 2000), p. 191.[12] M. E. Rose, Relativistic Electron Theory (Wiley,New York, 1961).[13] H. Ebert and S. Mankovsky, Phys. Rev. B 79, 045209(2009).[14] E.M. Gololobov, E. L. Mager, Z. V. Mezhevich, and L.K.Pan, Phys. Status Solidi (b) 119, K139 (1983).[15] D. Ko¨dderitzsch, H. Ebert, D.A. Rowlands, and A. Ernst,New J. Phys. 9, 81 (2007).[16] J. O. Rantschler, R. D. McMichael, A. Castillo, A. J.Shapiro, W. F. Egelhoff, B. B. Maranville, D. Pulugurtha,A. P. Chen, and L.M. Connors, J. Appl. Phys. 101, 033911(2007).[17] G. Woltersdorf, M. Kiessling, G. Meyer, J.-U. Thiele, andC.H. Back, Phys. Rev. Lett. 102, 257602 (2009).[18] B. Drittler, N. Stefanou, S. Blu¨gel, R. Zeller, and P. H.Dederichs, Phys. Rev. B 40, 8203 (1989).[19] N. Stefanou, A. Oswald, R. Zeller, and P. H. Dederichs,Phys. Rev. B 35, 6911 (1987).[20] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando, A.Sakuma, and T. Miyazaki, Jpn. J. Appl. Phys. 45, 3889(2006).[21] S.M. Bhagat and P. Lubitz, Phys. Rev. B 10, 179(1974).00.050.10.15α(T) expt:    pure Nitheory: pure Ni00.050.10.15α(T) expt:    Ni + 0.17 wt.%Cutheory: Ni + 0.2 at.%Cu0 100 200 300 400 500Temperature (K)00.050.10.15α(T) expt: Ni + 5 wt.%Cutheory: Ni + 5 at.%CuFIG. 3 (color online). Temperature variation of Gilbert damp-ing of pure Ni and Ni with Cu impurities: present theoreticalresults vs experiment [21].PRL 107, 066603 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending5 AUGUST 2011066603-4

Ab Initio Calculation of the Gilbert Damping Parameter via the Linear Response Formalism

A Kubo-Greenwood-like equation for the Gilbert damping parameter α is presented that is based on the linear response formalism. Its implementation using the fully relativistic Korringa-Kohn-Rostoker band structure method in combination with coherent potential approximation alloy theory allows it to be applied to a wide range of situations. This is demonstrated with results obtained for the bcc alloy system Fe1−xCox as well as for a series of alloys of Permalloy with 5d transition metals. To account for the thermal displacements of atoms as a scattering mechanism, an alloy-analogy model is introduced. The corresponding calculations for Ni correctly describe the rapid change of α when small amounts of substitutional Cu are introduced

Ebert, H.

Mankovsky, S.

Ködderitzsch, D.

Kelly, Paul J.

NARCIS 

Ab initio calculation of the Gilbert damping parameter via the linear response formalism

Crossref

Physical Review Letters

<i>Ab Initio</i>Calculation of the Gilbert Damping Parameter via the Linear Response Formalism

University of Twente Research Information

Ab Initio Calculation of the Gilbert Damping Parameter via the Linear Response FormalismH. Ebert, S. Mankovsky, and D. KödderitzschUniversity of Munich, Department of Chemistry, Butenandtstrasse 5-13, D-81377 Munich, GermanyP. J. KellyFaculty of Science and Technology and MESAþ Institute for Nanotechnology,University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands(Received 1 March 2011; published 2 August 2011)A Kubo-Greenwood-like equation for the Gilbert damping parameter  is presented that is based on thelinear response formalism. Its implementation using the fully relativistic Korringa-Kohn-Rostoker bandstructure method in combination with coherent potential approximation alloy theory allows it to beapplied to a wide range of situations. This is demonstrated with results obtained for the bcc alloy systemFe1xCox as well as for a series of alloys of Permalloy with 5d transition metals. To account for thethermal displacements of atoms as a scattering mechanism, an alloy-analogy model is introduced. Thecorresponding calculations for Ni correctly describe the rapid change of  when small amounts ofsubstitutional Cu are introduced.DOI: 10.1103/PhysRevLett.107.066603 PACS numbers: 72.25.Rb, 71.20.Be, 71.70.Ej, 75.78.nThe magnetization dynamics that is relevant for theperformance of any type of magnetic device is in generalgoverned by damping. In most cases the magnetizationdynamics can be modeled successfully by means of theLandau-Lifshitz-Gilbert (LLG) equation [1] that accountsfor damping in a phenomenological way. The possibility tocalculate the corresponding damping parameter from firstprinciples would open the perspective of optimizing mate-rials for devices and has therefore motivated extensivetheoretical work in the past. This led among others toKambersky’s breathing Fermi surface (BFS) [2] andtorque-correlation models (TCM) [3], that in principleprovide a firm basis for numerical investigations basedon electronic structure calculations [4,5]. The spin-orbitcoupling that is seen as a key factor in transferring energyfrom the magnetization to the electronic degrees of free-dom is explicitly included in these models. Most ab initioresults have been obtained for the BFS model though thetorque-correlation model makes fewer approximations[4,6]. In particular, it in principle describes the physicalprocesses responsible for Gilbert damping over a widerange of temperatures as well as chemical (alloy) disorder.However, in practice, like many other models it depends ona relaxation time parameter  that describes the rate oftransfer due to the various types of possible scatteringmechanisms. This weak point could be removed recentlyby Brataas et al. [7] who described the Gilbert damping bymeans of scattering theory. This development suppliedthe formal basis for the first parameter-free investigationson disordered alloys for which the dominant scatteringmechanism is potential scattering caused by chemical dis-order [8] or temperature induced structure disorder [9].As pointed out by Brataas et al. [7], their approach iscompletely equivalent to a formulation in terms of thelinear response or Kubo formalism. The latter route istaken in this communication that presents a Kubo-Greenwood-like expression for the Gilbert damping pa-rameter. Application of the scheme to disordered alloysdemonstrates that this approach is indeed fully equivalentto the scattering theory formulation of Brataas et al. [7]. Inaddition a scheme is introduced to deal with the tempera-ture dependence of the Gilbert damping parameter.Following Brataas et al. [7], the starting point of ourscheme is the Landau-Lifshitz-Gilbert (LLG) equation forthe time derivative of the magnetization ~M:1d ~Md¼  ~M ~Heff þ ~M ~Gð ~MÞ2M2sd ~Md; (1)where Ms is the saturation magnetization,  the gyromag-netic ratio, and ~G the Gilbert damping tensor. Accordingly,the time derivative of the magnetic energy is given by_Emag ¼ ~Heff  d~Md¼ 12_~m½ ~Gð ~mÞ _~m (2)in terms of the normalized magnetization ~m ¼ ~M=Ms. Onthe other hand, the energy dissipation of the electronicsystem _Edis ¼ hdĤd i is determined by the underlyingHamiltonian ĤðÞ. Expanding the normalized magnetiza-tion ~mðÞ, that determines the time dependence of ĤðÞabout its equilibrium value, ~mðÞ ¼ ~m0 þ ~uðÞ, one hasĤ ¼ Ĥ0ð ~m0Þ þX~u@@ ~uĤð ~m0Þ: (3)Using the linear response formalism, _Edis can be writtenas [7]PRL 107, 066603 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending5 AUGUST 20110031-9007=11=107(6)=066603(4) 066603-1  2011 American Physical Society_Edis ¼ @Xii0X_u _uc i~~~~~~~~@Ĥ@u~~~~~~~~c i0c i0~~~~~~~~@Ĥ@u~~~~~~~~c i ðEF  EiÞðEF  Ei0 Þ; (4)where EF is the Fermi energy and the sums run over alleigenstates  of the system. Identifying _Emag ¼ _Edis, onegets an explicit expression for the Gilbert damping tensor~G or equivalently for the damping parameter  ¼~G=ðMsÞ [7]. In full analogy to electric transport [10],the sum over eigenstates jc ii may be expressed in termsof the retarded single-particle Green’s functionImGþðEFÞ ¼ Pijc iihc ijðEF  EiÞ. This leads forthe parameter  to a Kubo-Greenwood-like equation¼ @MsTrace@Ĥ@uImGþðEFÞ @Ĥ@u ImGþðEFÞc(5)with h. . .ic indicating a configurational average in case of adisordered system (see below). Identifying T ¼ @Ĥ=@uwith the component of the magnetic torque operator ~̂Ialong the direction ~n, such that Î ~n ¼ @Ĥ=@ ~uð ~n ~uÞ ¼@Ĥ=@uð ~n ~uÞ this expression obviously gives the pa-rameter  in terms of a torque-torque correlation function.However, in contrast to the conventional TCM the elec-tronic structure is not represented in terms of Bloch statesbut using the retarded electronic Green’s function givingthe present approach much more flexibility.The most reliable way to account for spin-orbit couplingas the source of Gilbert damping is to evaluate Eq. (5)using a fully relativistic Hamiltonian within the frameworkof local spin density formalism (LSDA) [11]:Ĥ ¼ c ~ ~pþmc2 þ Vð~rÞ þ  ~	 ~mBð~rÞ: (6)Here i and  are the standard Dirac matrices and ~p is therelativistic momentum operator [12]. The functions V andB are the spin-averaged and spin-dependent parts, respec-tively, of the LSDA potential. Equation (6) implies for theT operator occurring in Eq. (5) the expressionT ¼ @@u Ĥ ¼ B	: (7)The Green’s function Gþ in Eq. (5) can be obtained in avery efficient way by using the spin-polarized relativisticversion of multiple scattering theory [11] that allows us totreat magnetic solids:Gþð ~rn; ~r0m; EÞ ¼X0Znð ~rn; EÞnm0 ðEÞZm0 ð~r0m; EÞXZnð~r<; EÞJn0 ð~r>; EÞnm: (8)Here coordinates ~rn referred to the center of cell nhave been used with j~r<j ¼ minðj~rnj; j~r0njÞ and j~r>j ¼maxðj~rnj; j~r0njÞ. The four-component wave functionsZnð ~r; EÞ ðJnð ~r; EÞÞ are regular (irregular) solutions to thesingle-site Dirac equation for site n and nm0 ðEÞ is the so-called scattering path operator that transfers an electronicwave coming in at site m into a wave going out from site nwith all possible intermediate scattering events accountedfor coherently.Using matrix notation, this leads to the following ex-pression for the damping parameter: ¼ gtotXnTracehT0~0nTn~n0ic (9)with the g factor 2ð1þorb=spinÞ in terms of the spinand orbital moments, spin and orb, respectively, thetotal magnetic moment tot ¼ spin þorb, and ~0n0 ¼12i ð0n0  0n0Þ and with the energy argument EF omitted.The matrix elements of the torque operator, Tn, areidentical to those occurring in the context of exchangecoupling [13] and can be expressed in terms of the spin-dependent part B of the electronic potential with matrixelements:Tn0 ¼Zd3rZn0 ð ~rÞ½	Bxcð~rÞZnð ~rÞ: (10)As indicated above, the expressions in Eqs. (5)–(10) canbe applied straightforwardly to disordered alloys. In thiscase the brackets h. . .ic indicate the necessary configura-tional average. This can be done by describing in a first stepthe underlying electronic structure (for T ¼ 0 K) on thebasis of the coherent potential approximation (CPA) alloytheory. In the next step the configurational average inEq. (5) is taken following the scheme worked out byButler [10] when dealing with the electrical conductivityat T ¼ 0 K or residual resistivity, respectively, of disor-dered alloys. This implies, in particular, that so-calledvertex corrections of the type hTImGþTImGþic hTImGþichTImGþic that account for scattering-inprocesses in the language of the Boltzmann transportformalism are properly accounted for.Thermal vibrations as a source of electron scatteringcan in principle be accounted for by a generalization ofEqs. (5)–(10) to finite temperatures and by including theelectron-phonon self-energy el-ph when calculating theGreen’s function Gþ. Here we restrict ourselves to elasticscattering processes by using a quasistatic representationof the thermal displacements of the atoms from theirequilibrium positions. We introduce an alloy-analogymodel to average over a discrete set of displacementsthat is chosen to reproduce the thermal root mean squareaverage displacementffiffiffiffiffiffiffiffiffiffiffihu2iTp for a given temperature T.This was chosen according to hu2iT ¼ 14 3h22mkD½ðD=TÞD=T þ 14with ðD=TÞ the Debye function, h the Planck constant,k the Boltzmann constant, and D the Debye temperature[14]. Ignoring the zero temperature term 1=4 and assuminga frozen potential for the atoms, the situation can be dealtPRL 107, 066603 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending5 AUGUST 2011066603-2with in full analogy to the treatment of disordered alloysdescribed above.The approach described above has been applied to theferromagnetic 3d-transition metal alloy systems bccFe1xCox, fcc Fe1xNix, and fcc Co1xNix. Figure 1 showsas an example results for bcc Fe1xCox for x  0:7. Thecalculated damping parameter ðxÞ for T ¼ 0 K is foundto be in very good agreement with the results based on thescattering theory approach [8] demonstrating numericallythe equivalence of the two approaches. An indispensablerequirement to achieving this agreement is to include thevertex corrections mentioned above. In fact, ignoring themleads in some cases to completely unphysical results. Tocheck the reliability of the standard CPA, that implies asingle-site approximation when performing the configura-tional average, we performed calculations on the basis ofthe nonlocal CPA [15]. Using a four-atom cluster led topractically the same results as the CPA except for the verydilute case. As found before for fcc Fe1xNix [8] thetheoretical results for  reproduce the concentration de-pendence of the experimental data quite well but are foundto be too low (see below). As suggested by Eq. (9) thevariation of ðxÞ with concentration x may reflect to someextent the variation of the average magnetic moment of thealloy, tot. Because the moments and spin-orbit couplingstrength do not differ very much for Fe and Co, thevariation of ðxÞ should be determined in the concentratedregime primarily by the electronic structure at the Fermienergy EF. As Fig. 1 shows, there is indeed a close corre-lation with the density of states nðEFÞ that may be seen as ameasure for the number of available relaxation channels.While the scattering and linear response approach arecompletely equivalent when dealing with bulk alloys thelatter allows us to perform the necessary configurationaveraging in a much more efficient way. This allows usto study with moderate effort the influence of varyingthe alloy composition on the damping parameter .Corresponding work has been done, in particular, usingPermalloy as a starting material and adding transitionmetals (TM) [16] or rare earth metals [17]. If we use thepresent scheme to study the effect of substituting Fe and Niatoms in Permalloy with a 5d TM, we find an increase of nearly linear with the 5d TM content, just as in experiment[16]. The total damping for 10% 5d TM content shown inFig. 2 (top) varies roughly parabolically over the 5d TMseries. In contrast to the Fe1xCox alloys considered above,there is now an S-like variation of the moments 5dspin overthe series (Fig. 2, bottom), characteristic of 5d impuritiesin the pure hosts Fe and Ni [18,19]. In spite of this behaviorof 5dspin the variation of ðxÞ seems again to be correlatedwith the density of states n5dðEFÞ (Fig. 2 bottom). Againthe trend of the experimental data is well reproduced by thecalculated values that are, however, somewhat too low.One possible reason for the discrepancy between thetheoretical and experimental results shown in Figs. 1 and2 might be the neglect of the influence of finite tempera-tures. This can be included as indicated above to accountfor the thermal displacement of the atoms in a quasistateway by performing a configurational average over thedisplacements using the CPA. This leads even for puresystems to a scattering mechanism and this way to a finitevalue for . Corresponding results for pure Ni are given inFig. 3 that show in full accordance with experiment a rapiddecrease of  with increasing temperature until a regimewith a weak variation of with T is reached. This behavioris commonly interpreted as a transition from conductivity-like to resistivitylike behavior reflecting the dominance ofintra- and interband transition, respectively [4], that isrelated to the increase of the broadening of electron energybands caused by the increase of scattering events withtemperature. Adding even less than 1 at.% Cu to Nistrongly reduces the conductivitylike behavior at low tem-peratures while leaving the high-temperature behavior es-sentially unchanged. A further increase of the Cu contentleads to the impurity-scattering processes responsible for0 0.1 0.2 0.3 0.4 0.5 0.6 0.7concentration xCo0123456α(x)x10-3ExptTheory (CPA), bccTheory (NL CPA)Fe-Con(EF)n(EF) (sts./Ry)1020304050600FIG. 1 (color online). Gilbert damping parameter for bccFe1xCox as a function of Co concentration: full circles—thepresent results within CPA; empty circles—within nonlocalCPA (NL CPA); and full diamonds—experimental data byOogane [20].02468α x10-2Ta W Re Os Ir Pt Au-0.300.30.6mspin5d (µ B) n5d(EF)5d spin moment061218n5d (EF) (sts./Ry)FIG. 2 (color online). Top: Gilbert damping parameter  forPy/5d TM systems with 10% 5d TM content in comparison withexperiment [16]; bottom: spin magnetic moment m5dspin anddensity of states nðEFÞ at the Fermi energy of the 5d componentin Py/5d TM systems with 10% 5d TM content.PRL 107, 066603 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending5 AUGUST 2011066603-3band broadening dominating . This effect completelysuppresses the conductivitylike behavior in the low-temperature regime because of the increase of scatteringevents due to chemical disorder. Again this is fully in linewith the experimental data, providing a straightforwardexplanation for their peculiar variation with temperatureand composition.From the results obtained for Ni one may conclude thatthermal lattice displacements are only partly responsiblefor the finding that the damping parameters obtained for Pydoped with the 5d TM series, and Fe1xCox are somewhatlow compared with experiment. This indicates that addi-tional relaxation mechanisms like magnon scattering con-tribute. Again, these can be included at least in a quasistaticway by adopting the point of view of a disordered localmoment picture. This implies scattering due to randomtemperature-dependent fluctuations of the spin momentsthat can also be dealt with using the CPA.In summary, a formulation for the Gilbert dampingparameter  in terms of a torque-torque-correlation func-tion was derived that led to a Kubo-Greenwood-likeequation. The scheme was implemented using the fullyrelativistic Korringa-Kohn- Rostoker band structuremethod in combination with the CPA alloy theory. Thisallows us to account for various types of scattering mecha-nisms in a parameter-free way. Corresponding applicationsto disordered transition metal alloys led to very goodagreement with results based on the scattering theoryapproach of Brataas et al. demonstrating the equivalenceof both approaches. The flexibility and numericalefficiency of the present scheme was demonstrated by astudy on a series of Permalloy-5d TM systems. To inves-tigate the influence of finite temperatures on , a so-calledalloy-analogy model was introduced that deals with thethermal displacement of atoms in a quasistatic manner.Applications to pure Ni gave results in very good agree-ment with experiment and, in particular, reproduced thedramatic change of  when Cu is added to Ni.The authors would like to thank the DFG for financialsupport within the SFB 689 ‘‘Spinphänomene in redu-zierten Dimensionen’’ and within project Eb154/23 forfinancial support. P. J. K acknowledges support by EUFP7 ICT Grant No. 251759 MACALO.[1] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).[2] V. Kambersky, Can. J. Phys. 48, 2906 (1970).[3] V. Kambersky, Czech. J. Phys. 26, 1366 (1976).[4] K. Gilmore, Y. U. Idzerda, and M.D. Stiles, Phys. Rev.Lett. 99, 027204 (2007).[5] M. Fähnle and D. Steiauf, Phys. Rev. B 73, 184427 (2006).[6] V. Kambersky, Phys. Rev. B 76, 134416 (2007).[7] A. Brataas, Y. Tserkovnyak, and G. E.W. Bauer, Phys.Rev. 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Back, Phys. Rev. Lett. 102, 257602 (2009).[18] B. Drittler, N. Stefanou, S. Blügel, R. Zeller, and P. H.Dederichs, Phys. Rev. B 40, 8203 (1989).[19] N. Stefanou, A. Oswald, R. Zeller, and P. H. Dederichs,Phys. Rev. B 35, 6911 (1987).[20] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando, A.Sakuma, and T. Miyazaki, Jpn. J. Appl. Phys. 45, 3889(2006).[21] S.M. Bhagat and P. Lubitz, Phys. Rev. B 10, 179(1974).00.050.10.15α(T)expt:    pure Nitheory: pure Ni00.050.10.15α(T)expt:    Ni + 0.17 wt.%Cutheory: Ni + 0.2 at.%Cu0 100 200 300 400 500Temperature (K)00.050.10.15α(T) expt: Ni + 5 wt.%Cutheory: Ni + 5 at.%CuFIG. 3 (color online). Temperature variation of Gilbert damp-ing of pure Ni and Ni with Cu impurities: present theoreticalresults vs experiment [21].PRL 107, 066603 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending5 AUGUST 2011066603-4

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