Theory of Spin Loss at Metallic Interfaces

Interfacial spin-flip scattering plays an important role in magnetoelectronic devices. Spin loss at metallic interfaces is usually quantified by matching the magnetoresistance data for multilayers to the Valet-Fert model, while treating each interface as a fictitious bulk layer whose thickness is $\delta$ times the spin-diffusion length. By employing the properly generalized circuit theory and the scattering matrix approaches, we derive the relation of the parameter $\delta$ to the spin-flip transmission and reflection probabilities at an individual interface. It is found that $\delta$ is proportional to the square root of the probability of spin-flip scattering. We calculate the spin-flip scattering probabilities for flat and rough $\mathrm{Cu}/\mathrm{Pd}$ interfaces using the Landauer-Büttiker method based on the first-principles electronic structure and find $\delta$ to be in reasonable agreement with experiment.

Figure 1

$\mathbf{k}$-resolved spin-flip reflectance $R_{\uparrow \downarrow }$ for the test Pd system, in which SOC is gradually suppressed away from a (111) plane. The spin quantization axis points up, parallel to the interface.

Figure 2

$\mathbf{k}$-resolved transmittances $T_{\sigma {\sigma }^{\prime }}$ and reflectances $R_{\sigma {\sigma }^{\prime }}^s$ for the $\mathrm{Cu}/\mathrm{Pd}$ (111) interface. (a) $T_{\uparrow \uparrow }$. (b) $T_{\uparrow \downarrow }$. (c) $R_{\uparrow \downarrow }^{\mathrm{Cu}}$. (d) $R_{\uparrow \downarrow }^{\mathrm{Pd}}$. The spin quantization axis points up, parallel to the interface.