Spin orbit torque on a curved surface

We provide a general formulation of the spin-orbit coupling on a 2D curved surface. Considering the wide applicability of spin-orbit effect in spinor-based condensed matter physics, a general spin-orbit formulation could aid the study of spintronics, Dirac graphene, topological systems, and quantum information on curved surfaces. Particular attention is then devoted to the development of an important spin-orbit quantity known as the spin-orbit torque. As devices trend smaller in dimension, the physics of local geometries on spin-orbit torque, hence spin and magnetic dynamics shall not be neglected. We derived the general expression of a spin-orbit anisotropy field for the curved surfaces and provided explicit solutions in the special contexts of the spherical, cylindrical and flat coordinates. Our expressions allow spin-orbit anisotropy fields and hence spin-orbit torque to be computed over the entire surfaces of devices of any geometry.