# Effective spin-mixing conductance of topological-insulator/ferromagnet and heavy-metal/ferromagnet spin-orbit-coupled interfaces: A first-principles Floquet-nonequilibrium-Green-function approach

## TL;DR

This paper introduces a first-principles approach combining density functional theory and Green's functions to accurately compute spin-mixing conductance at complex interfaces with strong spin-orbit coupling, crucial for spintronics.

## Contribution

It develops a novel method to directly calculate effective spin-mixing conductance using first-principles Hamiltonians and Floquet-nonequilibrium Green's functions, accounting for strong spin-orbit effects.

## Key findings

- SOC reduces pumped spin current at interfaces.
- The method accurately extracts SMC from spin current dependence.
- Effective SMC is validated against different material interfaces.

## Abstract

The spin mixing conductance (SMC) is a key quantity determining efficiency of spin transport across interfaces. Thus, knowledge of its precise value is required for accurate measurement of parameters quantifying numerous effects in spintronics, such as spin-orbit torque, spin Hall magnetoresistance, spin Hall effect and spin pumping. However, the standard expression for SMC, provided by the scattering theory in terms of the reflection probability amplitudes, is inapplicable when strong spin-orbit coupling (SOC) is present directly at the interface. This is the precisely the case of topological-insulator/ferromagnet and heavy-metal/ferromagnet interfaces of great contemporary interest. We introduce an approach where first-principles Hamiltonian of these interfaces, obtained from noncollinear density functional theory (ncDFT) calculations, is combined with charge conserving Floquet-nonequilibrium-Green-function formalism to compute {\em directly} the pumped spin current $I^{S_z}$ into semi-infinite left lead of two-terminal heterostructures Cu/X/Co/Cu or Y/Co/Cu---where X=Bi$_2$Se$_3$ and Y=Pt or W---due to microwave-driven steadily precessing magnetization of the Co layer. This allows us extract an effective SMC as a prefactor in $I^{S_z}$ vs. precession cone angle $\theta$ dependence, as long as it remains the same, $I^{S_z} \propto \sin^2 \theta$, as in the case where SOC is absent. By comparing calculations where SOC in switched off vs. switched on in ncDFT calculations, we find that SOC consistently reduces the pumped spin current and, therefore, the effective SMC.