  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.
