First-principles quantum transport modeling of spin-transfer and spin-orbit torques in magnetic multilayers.

We review a unified approach for computing both spin-transfer torque (STT) in spin-valves and magnetic tunnel junctions (MTJs), where charge current is injected perpendicularly to interfaces, and spin-orbit torque (SOT) in ferromagnet/spin-orbit-coupled-material (where experimentally explored spin-orbit-coupled-materials include $5d$ heavy metals, topological insulators, Weyl semimetals and monolayers of transition metal dichalcogenides) with charge current injected parallel to the interface. This approach requires to construct torque operator for a given Hamiltonian of the junction whose trace with current-driven density matrix, split into four components and expressed in terms of nonequilibrium Green functions, automatically generates field-like and damping-like components of STT or SOT vector. We provide illustrative examples for its usage by computing STT in one-dimensional toy model of MTJ and Co/Cu/Co spin valve, described by noncollinear density function theory (ncDFT) Hamiltonian, as well as SOT in a ferromagnetic layer, described by tight-binding Hamiltonian, with spin-orbit proximity effect within its monolayers assumed to be generated by the adjacent spin-orbit-coupled-material. Using these examples, we also discuss the difference generated by using self-consistent ncDFT Hamiltonian vs. collinear DFT Hamiltonian with rigidly rotated magnetic moments to create noncollinear magnetization configurations required for spin torque phenomenon, as well as how spin-orbit proximity effect, quantified by using ncDFT-computed spectral functions and spin textures on monolayers of realistic ferromagnetic material like Co, can be tailored to enhance the magnitude of SOT.