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 $\text{Pt}|{\mathrm{Fe}}_x{\mathrm{Co}}_{1-x}|\mathrm{Cu}$ 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.

Figure 1

Structure of the investigated multilayer system $\mathrm{Pt}|{\mathrm{Fe}}_x{\mathrm{Co}}_{1-x}|\mathrm{Cu}$ consisting of a stacking of fcc (111) planes along the $\mathrm{z}$ axis. Cu atoms are colored in blue, ${\mathrm{Fe}}_x{\mathrm{Co}}_{1-x}$ sites in red, and Pt atoms are represented in light gray.

Figure 2

Top: the longitudinal components ${\sigma }_{\mathrm{xx}}={\sigma }_{\mathrm{yy}}$ and ${\sigma }_{\mathrm{zz}}$ of the conductivity tensor $\underline{\mathbf{\sigma }}$ of $\mathrm{Pt}|{\mathrm{Fe}}_x{\mathrm{Co}}_{1-x}|\mathrm{Cu}$ as a function of the concentration $x$. Middle: the corresponding anomalous Hall conductivity ${\sigma }_{\mathrm{xy}}=-{\sigma }_{\mathrm{yx}}$. Bottom: the spin Hall conductivity ${\sigma }_{\mathrm{xy}}^{\mathrm{z}}=-{\sigma }_{\mathrm{yx}}^{\mathrm{z}}$. Open symbols represent calculations without vertex corrections (NV) and filled symbols those including vertex corrections (VC). The blue squares correspond to the Fermi sea contribution (sea), the green diamonds represent contributions from the Fermi surface (surf), and red circles give the total result (tot).

Figure 3

Top: the longitudinal component $t_{\mathrm{xx}}=t_{\mathrm{yy}}$ of the SOT depending on the concentration. Bottom: the transverse component $t_{\mathrm{xy}}=-t_{\mathrm{yx}}$ of the SOT depending on the concentration. Use of symbols and colors as in Fig. 2. The black dotted line represents the total spin conductivity ${\sigma }_{\mathrm{zy}}^{\mathrm{x}}$ in ${\left({10}^{-3}\mathrm{\Omega }\mathrm{cm}\right)}^{-1}$ including vertex corrections.