Ab Initio Calculation of the Gilbert Damping Parameter via the Linear Response Formalism H. Ebert, S. Mankovsky, and D. Ko ¨dderitzsch University of Munich, Department of Chemistry, Butenandtstrasse 5-13, D-81377 Munich, Germany P. J. Kelly Faculty of Science and Technology and MESA þ 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 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 Fe 1x Co 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  when small amounts of substitutional Cu are introduced. DOI: 10.1103/PhysRevLett.107.066603 PACS numbers: 72.25.Rb, 71.20.Be, 71.70.Ej, 75.78.n The magnetization dynamics that is relevant for the performance of any type of magnetic device is in general governed by damping. In most cases the magnetization dynamics can be modeled successfully by means of the Landau-Lifshitz-Gilbert (LLG) equation [1] that accounts for damping in a phenomenological way. The possibility to calculate the corresponding damping parameter from first principles would open the perspective of optimizing mate- rials for devices and has therefore motivated extensive theoretical work in the past. This led among others to Kambersky’s breathing Fermi surface (BFS) [2] and torque-correlation models (TCM) [3], that in principle provide a firm basis for numerical investigations based on electronic structure calculations [4,5]. The spin-orbit coupling that is seen as a key factor in transferring energy from the magnetization to the electronic degrees of free- dom is explicitly included in these models. Most ab initio results have been obtained for the BFS model though the torque-correlation model makes fewer approximations [4,6]. In particular, it in principle describes the physical processes responsible for Gilbert damping over a wide range of temperatures as well as chemical (alloy) disorder. However, in practice, like many other models it depends on a relaxation time parameter  that describes the rate of transfer due to the various types of possible scattering mechanisms. This weak point could be removed recently by Brataas et al. [7] who described the Gilbert damping by means of scattering theory. This development supplied the formal basis for the first parameter-free investigations on disordered alloys for which the dominant scattering mechanism 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 is completely equivalent to a formulation in terms of the linear response or Kubo formalism. The latter route is taken in this communication that presents a Kubo- Greenwood-like expression for the Gilbert damping pa- rameter. Application of the scheme to disordered alloys demonstrates that this approach is indeed fully equivalent to the scattering theory formulation of Brataas et al. [7]. In addition 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 our scheme is the Landau-Lifshitz-Gilbert (LLG) equation for the time derivative of the magnetization ~ M: 1  d ~ M d ¼ ~ M  ~ H eff þ ~ M   ~ Gð ~ MÞ  2 M 2 s d ~ M d  ; (1) where M s 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 _ E mag ¼ ~ H eff  d ~ M d ¼ 1  2 _ ~ m½ ~ Gð ~ mÞ _ ~ m (2) in terms of the normalized magnetization ~ m ¼ ~ M=M s . On the other hand, the energy dissipation of the electronic system _ E dis ¼h d ^ H d i is determined by the underlying Hamiltonian ^ HðÞ. Expanding the normalized magnetiza- tion ~ mðÞ, that determines the time dependence of ^ HðÞ about its equilibrium value, ~ mðÞ¼ ~ m 0 þ ~ uðÞ, one has ^ H ¼ ^ H 0 ð ~ m 0 Þþ X  ~ u  @ @~ u  ^ Hð ~ m 0 Þ: (3) Using the linear response formalism, _ E dis can be written as [7] PRL 107, 066603 (2011) PHYSICAL REVIEW LETTERS week ending 5 AUGUST 2011 0031-9007= 11=107(6)=066603(4) 066603-1 Ó 2011 American Physical Society