Electric field-induced spin-orbit torques are an exchange of angular momentum between the lattice and magnetization, mediated by spin-orbit coupling. Spin-orbit torque-based devices are typically comprised of heavy metal/ferromagnet bilayers. Compared to traditional spin-transfer torque-based magnetic tunneling junctions, these devices possess advantages for applications such as magnetic random access memory.  The inversion symmetry breaking at the interface between the layers leads to dampinglike torques which can deterministically switch the magnetization between in-plane configurations. 

Reducing the device symmetry leads to additional vector components of the spin-orbit torque, which could enable the switching of perpendicularly magnetized layers.   This was demonstrated in recent observations of unconventional out-of-plane dampinglike torque in WTe2/Permalloy bilayer systems [1].  Motivated by these experiments, we calculate the spin-orbit torque generated in a WTe2-Co bilayer using first-principles methods.  Our calculations show both conventional in-plane and unconventional out-of-plane dampinglike torque components.  We find the dampinglike torque is approximately equal to the spin current flux between the WTe2 layers and Co layers, indicating the conventional spin Hall effect plus spin-transfer torque applies to this system.  Finally, we present a method for computing the layer-resolved spin Hall conductivity in bulk systems with non-symmorphic symmetry, such as 1T’-WTe2, where some vector components of spin current are staggered and average to zero.  The bulk spin current in WTe2 is approximately equal to the observed and computed spin-orbit torque in heterostructures.  We also propose a new material candidate, orthorhombic PbTe, in which both uniform and staggered spin current conductivities are one order larger than in WTe2. 

[1] MacNeill, D., Stiehl, G., Guimaraes, M. et al. Control of spin–orbit torques through crystal symmetry in WTe2/ferromagnet bilayers. Nature Phys 13, 300–305 (2017).
[2] Xue, Fei, et al. Unconventional spin-orbit torque in transition metal dichalcogenide–ferromagnet bilayers from first-principles calculations. Physical Review B 102, 014401 (2020).