Spin Pumping in the Presence of Spin-Orbit Coupling

Spin pumping and related phenomena have been observed recently in heavy metals and topological insulators, where the spin-orbit coupling plays an essential role. We have developed a spin-pumping formalism that explicitly includes the spin-orbit coupling at interfaces and disorder in the layers. Spin pumping across an interface with spin-orbit coupling and the attendant backflow are treated on an equal footing. We resolve some long-standing issues on the conflicting conclusions about the spin-diffusion length for Pt, and the origin of spin-memory loss at interfaces with heavy metals. In addition, we predict some heretofore unanticipated spin-pumping phenomena.

Figure 1

Illustration of spin pumping (a) without and (b) with Rashba SOC at the interface. In both cases, (c) and (d), the spin current decays oscillatorily within the FM layer and exponentially in the NM. The maximum spin current $j_{\max }$ occurs at $z=0^{-}$. The dependence of $j_{\max }$, normalized to its disorder-free value $j_{\max }^0$, on the ferromagnetic exchange constant for several values of the disorder parameter $\mathrm{\Delta }$, (e), where $\mathrm{\Delta }$ is the imaginary part of self-energy. The ratio of the interface spin-current jump $\mathrm{\Delta }{j}_{\mathrm{int}}$ to the spin current relaxed in the NM layer as a function of the SOC strength for several thickness of the NM layer, (f) (in-plane spin polarization).

Figure 2

The enhanced damping parameter and its anisotropy of spin pumping for several different SOC parameters. (a) The precessing axis is parallel to the layers with the backflow fraction at large NM thickness $\epsilon \equiv \xi /(1+\xi )=10\%$, and (b) $\epsilon =40\%$. The dashed lines indicate the effective thickness of the interface. (c) The difference of the enhanced damping parameter for the precessing axis perpendicular and parallel to the layers for $\epsilon =10\%$, and (d) $\epsilon =40\%$. The inset in (a) is from the experimental data in Ref. [8].