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.