Intrinsic damping phenomena from quantum to classical magnets: An ab initio study of Gilbert damping in a Pt/Co bilayer

A fully quantum-mechanical description of the precessional damping of a Pt/Co bilayer is presented in the framework of the Keldysh Green's function approach using ab initio electronic structure calculations. In contrast to previous calculations of classical Gilbert damping (${\alpha }_{\mathrm{GD}}$), we demonstrate that ${\alpha }_{\mathrm{GD}}$ in the quantum case does not diverge in the ballistic regime due to the finite size of the total spin $S$. In the limit of $S\to \infty$ we show that the formalism recovers the torque correlation expression for ${\alpha }_{\mathrm{GD}}$ which we decompose into spin-pumping and spin-orbital torque correlation contributions. The formalism is generalized to take into account a self-consistently determined dephasing mechanism which preserves the conservation laws and allows the investigation of the effect of disorder. The dependence of ${\alpha }_{\mathrm{GD}}$ on Pt thickness and disorder strength is calculated, and the spin-diffusion length of Pt and the spin mixing conductance of the bilayer are determined and compared with experiments.

Figure 1

Schematic of the combined FM-Bloch electron system. The horizontal planes denote the eigenstates $|S,m\rangle$ of the total ${\mathit{S}}_z$ of the nano-FM with eigenvalues $m=-S,-S+1,...,+S$. For more details see Fig. 2 in Ref. [19].

Figure 2

Gilbert damping versus precessional cone angle calculated from Eq. (5) for a Pt(1-ML)/Co(6-ML) bilayer system for $S=60$ and $\eta =1\mathrm{meV}$, respectively.

Figure 3

Gilbert damping versus broadening parameter for the Pt(1-ML)/Co(6-ML) bilayer for different values of the total spin $S$ of the FM nanocluster. Equations (5) and (6) were used to calculate the Gilbert damping for finite and infinite $S$, respectively.

Figure 4

Gilbert damping of the Pt(1-ML)/Co(6-ML) bilayer versus the broadening parameter $\eta$ (lower abscissa) and interaction strength ${\lambda }_{\mathrm{int}}$ (upper abscissa) using the TC, SOTC, and conserving TC expressions given by Eqs. (6), (10), and (14), respectively.

Figure 5

Ab initio values (circles) of Gilbert damping versus Pt thickness for the $\mathrm{Pt}\left(m\text{−}\mathrm{ML}\right)/\mathrm{Co}\left(6\text{−}\mathrm{ML}\right)$ bilayer where $m$ ranges between 0 and 6 and ${\lambda }_{\mathrm{int}}=1\mathrm{eV}$. The dashed curve is the fit of the Gilbert damping values to Eq. (22). The inset: spin-diffusion length (left ordinate) and effective spin mixing conductance $g_{\uparrow \downarrow }^{\mathrm{eff}}$, (right ordinate) versus interaction strength. The error bar for $g_{\uparrow \downarrow }^{\mathrm{eff}}$ is equal to the root-mean-square deviation of the damping data from the fitted curve.