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

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.

Figure 1

The Gilbert damping parameter (a) for bcc Fe${}_{1-x}$V${}_x$ ($T=0$ K) as a function of V concentration and (b) for bcc-Fe as a function of temperature. Solid (open) symbols give results with (without) the vertex corrections.

Figure 2

The Gilbert damping parameter for Py+15$\%$Os as a function of the scaling parameter of spin-orbit coupling applied to all atoms contained in the alloy. The red dashed line in the plot denotes linear fit. The values 0 and 1 for the SOC scaling parameter correspond to the scalar-relativistic Schrödinger-like and fully relativistic Dirac equations, respectively.

Figure 3

Temperature variation of the Gilbert damping parameter of pure systems. Comparison of theoretical results with experiment: (a) bcc-Fe: circles and squares denote the results for ideal bcc Fe for two lattice parameters, $a=5.42$ a.u. and $a=5.45$ a.u.; stars denote 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 denote theoretical results for ideal hcp Co; stars denote results for Co with 0.03$\%$ of vacancies; and pluses denote results for Co with 0.1$\%$ of vacancies (Expt. Ref. 45); and (c) fcc-Ni (Expt. Ref. 45).

Figure 4

Temperature variation of the ${\alpha }_{xx}$ and ${\alpha }_{yy}$ components of the Gilbert damping tensor of bcc Fe with the magnetization direction taken along different crystallographic directions: $\mathbf{M}=\widehat{z}\left|\mathbf{M}\right|\parallel \langle 001\rangle$ (circles), $\mathbf{M}\parallel \langle 011\rangle$ (squares), $\mathbf{M}\parallel \langle 111\rangle$ (diamonds).

Figure 5

(a) Theoretical results for the Gilbert damping parameter of bcc Fe${}_{1-x}$Co${}_x$ as a function of Co concentration: CPA results for the bcc structure (solid circles) describing the random alloy system, and results for the partially ordered system (open square) for $x=0.5$ [i.e., for Fe${}_{1-x}$Co${}_x$ alloy with CsCl structure and alloy components randomly distributed in two sublattices in the following proportions: (Fe${}_{0.9}$Co${}_{0.1}$)(Fe${}_{0.1}$Co${}_{0.9}$), the NL-CPA results for random alloy with bcc structure (open circles) and the NL-CPA results for the the system with short-range order within the first-neighbor shell (open diamonds)]. The dashed line represents the DOS at the Fermi energy, $E_F$, as a function of Co concentration. (b) Spin-resolved DOS for bcc Fe${}_{1-x}$Co${}_x$ for $x=0.01$ (dashed line) and $x=0.5$ (solid line).

Figure 6

The Gilbert damping parameter for Fe${}_{1-x}$Co${}_x$ (a), Ni${}_{1-x}$Co${}_x$ (b), and Ni${}_{1-x}$Fe${}_x$ (c) as a function of Co and Fe concentration, respectively: present results within CPA (solid circles) and experimental data by Oogane[52] (solid diamonds). (d) Results for bcc Fe${}_{1-x}$V${}_x$ as a function of V concentration: $T=0$ K (solid circles) and $T=300$ K (open circles). Squares: experimental data.[53] Open circles: theoretical results by Starikov et al.[5]

Figure 7

Gilbert damping parameter for bcc Fe with $1\%$ (squares) and $5\%$ (circles) of $5d$ impurities calculated for $T=0$ K (solid symbols) and $T=300$ K (open sysmbols).

Figure 8

Gilbert damping parameter for bcc Fe${}_{1-x}$$M_x$ with $M=$ Pt (circles) and $M=$ Os (squares) impurities as a function of temperature for 1$\%$ (a) and 5$\%$ (b) of the impurities.

Figure 9

DOS for Pt in Fe${}_{1-x}$Pt${}_x$ (solid line) and Os in Fe${}_{1-x}$Os${}_x$ (dashed line) for $x=0.01$.

Figure 10

(a) Gilbert damping parameter $\alpha$ for Py/5d TM systems with 10$\%$ 5d TM content in comparison with experiment;[54] (b) spin magnetic moment $m_{\mathrm{spin}}^{5d}$ and density of states $n\left(E_F\right)$ at the Fermi energy of the $5d$ component in Py/5d TM systems with 10$\%$ 5d TM content.