A fully relativistic description of spin-orbit torques by means of linear response theory

TL;DR

This paper develops a fully relativistic linear response theory for spin-orbit torques, incorporating symmetry constraints and first-principles calculations to analyze material-specific effects in alloy systems.

Contribution

It introduces a comprehensive relativistic framework for SOT using the Kubo formalism, including symmetry restrictions and disorder effects via CPA, applied to alloy systems.

Findings

Material-specific torkance tensor elements calculated for Pt|FeCo|Cu alloys.

Symmetry constraints on response tensors derived and discussed.

Analysis of contributions to SOT in various alloy compositions.

Abstract

Symmetry and magnitude of spin-orbit torques (SOT), i.e., current-induced torques on the magnetization of systems lacking inversion symmetry, are investigated in a fully relativistic linear response framework based on the Kubo formalism. By applying all space-time symmetry operations contained in the magnetic point group of a solid to the relevant response coefficient, the torkance expressed as torque-current correlation function, restrictions to the shape of the direct and inverse response tensors are obtained. These are shown to apply to the corresponding thermal analogues as well, namely the direct and inverse thermal SOT in response to a temperature gradient or heat current. Using an implementation of the Kubo-Bastin formula for the torkance into a first-principles multiple-scattering Green's function framework and accounting for disorder effects via the so-called coherent potential approximation (CPA), all contributions to the SOT in pure systems, dilute as well as concentrated alloys can be treated on equal footing. This way, material specific values for all torkance tensor elements in the fcc (111) trilayer alloy system Pt | Fe$_x$Co$_{1-x}$ | Cu are obtained over a wide concentration range and discussed in comparison to results for electrical and spin conductivity, as well as to previous work - in particular concerning symmetry w.r.t. magnetization reversal and the nature of the various contributions.