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

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 analogs 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 function framework and accounting for disorder effects via the so-called coherent potential approximation, 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|FeₗCo₁-ₗ|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 with respect to magnetization reversal and the nature of the various contributions.