Spin-group theory on Edelstein effect and spin-orbit torque in Collinear Ferromagnets

Abstract

Current-induced spin-orbit torques (SOTs) are central to the electrical manipulation of magnetic order in spintronic devices. In transition-metal/collinear ferromagnet bilayers, field-like and damping-like torques have been described only phenomenologically via the spin or orbital Hall effect, lacking a rigorous symmetry-based foundation. The precise role of spin-orbit coupling (SOC) in both the Edelstein effect and SOTs has remained unresolved. Here we develop a spin-group symmetry theory for the Edelstein effect and SOTs in collinear ferromagnets, treating SOC as a symmetry-breaking perturbation. For 4mm (C4v) point group symmetry, we derive the full forms of field-like and damping-like torques, which arise predominantly from first- and second-order SOC. We further show that SOTs in both orbital-Hall-dominated Ti/Ni and spin-Hall-dominated Pt/CoFe bilayers originate at first-order SOC. Taking the 3m (C3v) torque as a paradigmatic example, we elucidate the role of second- and higher-order SOC torques in field-free switching of perpendicular magnetic anisotropy. Remarkably, in PtMnSb, we demonstrate that SOTs under certain point group symmetries deviate from the conventional form: zeroth- and first-order SOC contributions vanish identically, with the leading SOT emerging at second order. All symmetry-based predictions from spin-group theory are in excellent quantitative agreement with first-principles calculations. Our work establishes a unified symmetry framework for the microscopic understanding of the Edelstein effect and current-induced spin torques in ferromagnetic systems.