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

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 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 ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x$ 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 $\alpha$ when small amounts of substitutional Cu are introduced.

Figure 1

Gilbert damping parameter for bcc ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x$ as a function of Co concentration: full circles—the present results within CPA; empty circles—within nonlocal CPA (NL CPA); and full diamonds—experimental data by Oogane [20].

Figure 2

Top: Gilbert damping parameter $\alpha$ for Py/$5d$ TM systems with 10% $5d$ TM content in comparison with experiment [16]; bottom: spin magnetic moment $m_{\mathrm{spin}}^{5d}$ and density of states $n(E_F)$ at the Fermi energy of the $5d$ component in Py/$5d$ TM systems with 10% $5d$ TM content.

Figure 3

Temperature variation of Gilbert damping of pure Ni and Ni with Cu impurities: present theoretical results vs experiment [21].