Interface Enhancement of Gilbert Damping from First Principles

The enhancement of Gilbert damping observed for ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$ (Py) films in contact with the nonmagnetic metals Cu, Pd, Ta, and Pt is quantitatively reproduced using first-principles scattering calculations. The “spin-pumping” theory that qualitatively explains its dependence on the Py thickness is generalized to include a number of extra factors known to be important for spin transport through interfaces. Determining the parameters in this theory from first principles shows that interface spin flipping makes an essential contribution to the damping enhancement. Without it, a much shorter spin-flip diffusion length for Pt would be needed than the value we calculate independently.

Figure 1

Calculated (solid lines) Gilbert damping of $\mathrm{NM}|\mathrm{Py}|\mathrm{NM}$ ($\mathrm{NM}=\mathrm{Cu}$, Pd, Ta, and Pt) compared to experimental measurements (dotted lines) [4] as a function of the Py thickness $d$. Inset: sketch of the structure used in the calculations. The dashed frame denotes one structural unit consisting of a Py film between two NM films.

Figure 2

Total damping calculated for $\mathrm{Pt}|\mathrm{Py}|\mathrm{Pt}$ and $\mathrm{Cu}|\mathrm{Py}|\mathrm{Cu}$ as a function of the Py thickness. The open symbols correspond to the case without backflow while the full symbols are the results shown in Fig. 1 where backflow was included. The lines are linear fits to the symbols. The asterisks on the $y$ axis are the values of ${\overline{G}}^{\text{mix}}$ calculated without SOC using Eq. (2).

Figure 3

$\alpha$ as a function of the Pt thickness $l$ calculated for $\mathrm{Pt}(l)|\mathrm{Py}(d)|\mathrm{Pt}(l)$. The dashed and solid lines are the curves obtained by fitting without and with interface spin memory loss, respectively. Inset: fractional spin conductances $G^{\uparrow \uparrow }/G^{\uparrow }$ and $G^{\uparrow \downarrow }/G^{\uparrow }$ when a fully polarized up-spin current is injected into bulk Pt at room temperature. $G^{\sigma {\sigma }^{\prime }}$ is ($e^2/h$ times) the transmission probability of a spin $\sigma$ from the left-hand lead into a spin ${\sigma }^{\prime }$ in the right-hand lead and $G^{\uparrow }=G^{\uparrow \uparrow }+G^{\uparrow \downarrow }$. The value of the spin-flip diffusion length for a single spin channel obtained by fitting is $l^{\sigma }=7.8\pm 0.3\mathrm{nm}$.

Figure 4

(a) $l_{\text{sf}}$ for diffusive Pt as a function of its conductivity $\sigma$ (solid black circles) calculated by injecting a fully polarized current into Pt. The solid black line illustrates the linear dependence. Bulk values extracted from experiment that are either not sensitive to interface spin flipping [37] or take it into account [17, 35, 38] are plotted (squares) for comparison. The empty circles are values of $l_{\text{sf}}$ determined from the interface-enhanced damping using Eq. (1) with $\delta =0$. (b) Fit values of $R^{*}/\delta$ (squares) and $\delta$ (diamonds) as a function of the conductivity of Pt obtained using Eq. (1). The solid red line is the average value (for different values of $\sigma$) of $\delta =3.7$.