Gilbert damping in magnetic layered systems

The Gilbert damping constant present in the phenomenological Landau-Lifshitz-Gilbert equation describing the dynamics of magnetization is calculated for ferromagnetic metallic films as well as Co/nonmagnet (NM) bilayers. The calculations are done within a realistic nine-orbital tight-binding model including spin-orbit coupling. The convergence of the damping constant expressed as a sum over the Brillouin zone is remarkably improved by introducing finite temperature into the electronic occupation factors and subsequent summation over the Matsubara frequencies. We investigate how the Gilbert damping constant depends on the ferromagnetic film thickness as well as on the thickness of the nonmagnetic cap in Co/NM bilayers $(\text{NM}=\text{Cu}$, Pd, Ag, Pt, and Au). The obtained theoretical dependence of the damping constant on the electron-scattering rate, describing the average lifetime of electronic states, varies substantially with the ferromagnetic film thickness and it differs significantly from the dependence for bulk ferromagnetic metals. The presence of nonmagnetic caps is found to largely enhance the magnetic damping in Co/NM bilayers in accordance with experimental data. Unlike Cu, Ag, and Au a particularly strong enhancement is obtained for Pd and Pt caps. This is attributed to the combined effect of the large spin-orbit couplings of Pd and Pt and the simultaneous presence of $d$ states at the Fermi level in these two metals. The calculated Gilbert damping constant also shows an oscillatory dependence on the thicknesses of both ferromagnetic and nonmagnetic parts of the investigated systems which is attributed to quantum-well states. Finally, the expression for contributions to the damping constant from individual atomic layers is derived. The obtained distribution of layer contributions in Co/Pt and Co/Pd bilayers proves that the enhanced damping which affects the dynamics of the magnetization in the Co film originates mainly from a region within the nonmagnetic part of the bilayer. Such a nonlocal damping mechanism, related to spin pumping, is almost absent in other investigated bilayers: Co/Cu, Co/Ag, and Co/Au.

Figure 1

Gilbert damping constant $\alpha$ (solid lines) and its intraband (dotted line) and interband (dashed line) terms vs scattering rate $\Gamma$ for bulk Fe and Co found with (solid symbols) and without (open squares) the SO interaction $H_{\mathrm{SO}}$ in the band structure calculation.

Figure 2

Convergence of the Gilbert damping constant $\alpha$ with the number ${(2N_{\mathrm{k}}+1)}^3$ $\mathit{k}$ points in the 3D BZ for bulk Fe and Co at $T=0$ (dashed lines) and $T=300$ K (solid lines), for different scattering rates $\Gamma$.

Figure 3

Convergence of the Gilbert damping constant $\alpha$ with the number ${(2N_{\mathrm{k}}+1)}^2$ $\mathit{k}$ points in the 2D BZ for (001) fcc Co(5 ML) and Co(20 ML) films at $T=0$ (open symbols, dashed lines) and $T=300$ K (solid symbols and lines), for different scattering rates: $\Gamma =0.001$ eV (diamonds), $\Gamma =0.01$ eV (circles), and $\Gamma =0.1$ eV (triangles).

Figure 4

Gilbert damping constant $\alpha$ vs the scattering rate $\Gamma$ in (001) fcc Co(5 ML) and Co(10 ML) films at $T=0$ and $T=300$ K. Open symbols show damping at $T=0$ and the solid ones stand for $T=300$ K. The error bars shown for $\Gamma =0.002$ eV at $T=0$ reflect the not fully converged summation over the BZ with $N_{\mathrm{k}}=1000$ for Co(5 ML) and $N_{\mathrm{k}}=700$ for Co(10 ML).

Figure 5

Gilbert damping constant $\alpha$ vs scattering rate $\Gamma$ in bulk ferromagnets (bcc Fe, fcc Co, fcc Ni) and (001) ferromagnetic films with different thicknesses $N$.

Figure 6

Gilbert damping constant $\alpha$ vs film thickness for (001) bcc Fe, fcc Co, and fcc Ni films, calculated with the scattering rates $\Gamma =0.01$ eV and $\Gamma =0.1$ eV, $T=300$ K. The horizontal lines mark the bulk values of $\alpha$ for $\Gamma =0.1$ eV (dashed line) and $\Gamma =0.01$ eV (solid line).

Figure 7

Gilbert damping constant $\alpha$ vs film thickness at $T=0$ and $T=300$ K in (001) fcc Co films; $\Gamma =0.01$ eV.

Figure 8

Gilbert damping constant vs Co thickness in (001) fcc Co/NM(6 ML) bilayers; $\Gamma =0.01$ eV.

Figure 9

Inverse Co thickness dependence of the additional damping in (001) fcc Co/NM(6 ML) bilayers due to adding the NM caps; $\Gamma =0.01$ eV.

Figure 10

Gilbert damping constant vs NM cap thickness in (001) fcc Co(6 ML)/NM bilayers; $\Gamma =0.01$ eV.

Figure 11

Gilbert damping constant vs NM cap thickness in (001) fcc Co(6 ML)/NM bilayer $(\text{NM}=\text{Pt}$ and Au), in the presence (solid circles) and the absence (open circles) of the SO coupling in NM; $\Gamma =0.01$ eV.

Figure 12

Layer contributions to the Gilbert damping constant in Fe, Co, and Ni films of 18-ML thickness; $\Gamma =0.01$ eV.

Figure 13

Layer contributions to the Gilbert damping constant in Co(6 ML) film as well as in Co(6 ML)/Pd(12 ML), and Co(6 ML)/Pt(12 ML) bilayers; $\Gamma =0.01$ eV.

Figure 14

Layer contributions to the Gilbert damping constant in Co(6 ML)/NM(12 ML) bilayers $(\text{NM}=\text{Cu}$, Au); $\Gamma =0.01$ eV.

Figure 15

Layer contributions to the Gilbert damping constant in Co(6 ML)/NM(42 ML) bilayers $(\text{NM}=\text{Pd}$, Pt); $\Gamma =0.01$ eV. The dashed line marks a fit with the function ${\alpha }_{7}exp(-z_l/\lambda )$ with $\lambda =0.45$nm inside the NM (see text).