Microscopic Origin of Spin-Orbit Torque in Ferromagnetic Heterostructures: A First Principles Approach

We present an {\it ab initio}-based theoretical framework which elucidates the origin of the spin-orbit torque (SOT) in Normal-Metal(NM)/Ferromagnet(FM) heterostructures. The SOT is decomposed into two contributions, namely, {\it spin-Hall} and the {\it spin-orbital} components. We find that {\it (i)} the Field-Like (FL) SOT is dominated by the spin-orbital component and {\it (ii)} both components contribute to the damping-like torque with comparable magnitude in the limit of thick Pt film. The contribution of the spin-orbital component to the DL-SOT is present only for NMs with strong SOC coupling strength. We demonstrate that the FL-SOT can be expressed in terms of the non-equilibrium spin-resolved orbital moment accumulation. The calculations reveal that the experimentally reported oxygen-induced sign-reversal of the FL-SOT in Pt/Co bilayers is due to the significant reduction of the majority-spin orbital moment accumulation on the interfacial NM atoms.

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