Quantum mechanism of nonlocal Gilbert damping in magnetic trilayers

A fully quantum-mechanical calculation of the Gilbert damping constant $\alpha$ in magnetic trilayers is done by employing the torque-correlation formula within a realistic tight-binding model. A remarkable enhancement of $\alpha$ in Co/${\mathrm{NM}}_1$/${\mathrm{NM}}_2$ trilayers is obtained due to adding the caps ${\mathrm{NM}}_2$=Pd, Pt, and it decays with the thickness of the spacers ${\mathrm{NM}}_1$=Cu, Ag, Au in agreement with experiment. Nonlocal origin of the Gilbert damping is visualized with its atomic layer contributions. It is shown that magnetization in Co is damped remotely by strong spin-orbit coupling in ${\mathrm{NM}}_2$ via quantum states with large amplitude in both Co and ${\mathrm{NM}}_2$.

Figure 1

The Gilbert damping constant $\alpha$ in an Co(6 ML) film, Co(6 ML)/${\mathrm{NM}}_1$ bilayers (${\mathrm{NM}}_1$=Cu, Ag and Au), and Co(6 ML)/${\mathrm{NM}}_1$/${\mathrm{NM}}_2$(4 ML) trilayers (${\mathrm{NM}}_2$=Pd, Pt) vs ${\mathrm{NM}}_1$ thickness; the scattering rate $\mathrm{\Gamma }=0.01$ eV.

Figure 2

Gilbert damping constant $\alpha$ in Co(6 ML)/Cu($N$ ML)/ Pt(4 ML) trilayers against the scattering rate $\mathrm{\Gamma }$ for different Cu spacer thicknesses $N$.

Figure 3

Layer contributions to the damping constant $\alpha$ in Co/${\mathrm{NM}}_1$/Pt trilayers (${\mathrm{NM}}_1$=Cu, Au) with different ${\mathrm{NM}}_1$ thicknesses; $\mathrm{\Gamma }=0.01$ eV.

Figure 4

(a) Intraband and interband terms of the damping constant $\alpha$ in the Co(4 ML)/Cu($N$ ML)/Pt(4 ML) trilayers with $\mathrm{\Gamma }=0.01$ eV vs the Cu spacer thickness $N$; (b)–(d) contributions to the intraband term of $\alpha$ in the trilayers with $N=3,9,30$ ML from quantum states $|n\mathit{k}\rangle$ with various fractions in the Co film and the Pt cap. The inclined yellow lines correspond to states with a fixed fraction in the Cu spacer.