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Direct and inverse spin-orbit torques

In collinear magnets lacking inversion symmetry application of electric currents induces torques on the magnetization and conversely magnetization dynamics induces electric currents. The two effects, which both rely on spin-orbit interaction (SOI), are reciprocal to each other and denoted direct spin-orbit torque (SOT) and inverse spin-orbit torque (ISOT), respectively. We derive expressions for SOT and ISOT within the Kubo linear response formalism. We show that expressions suitable for density-functional theory calculations can be derived either starting from a Kohn-Sham Hamiltonian with time-dependent exchange field or by expressing general susceptibilities in terms of the Kohn-Sham susceptibilities. For the case of magnetic bilayer systems we derive the general form of the ISOT current induced under ferromagnetic resonance. Using \textit{ab initio} calculations within density-functional theory we investigate SOT and ISOT in Co/Pt(111) magnetic bilayers. We determine the spatial distribution of spin and charge currents as well as torques in order to expose the mechanisms underlying SOT and ISOT and to highlight their reciprocity on the microscopic level. We find that the spin Hall effect is position-dependent close to interfaces.