Relativistic treatment of spin-currents and spin-transfer torque

Abstract

It is shown that a useful relativistic generalization of the conventional spin density for the case of moving electrons is the expectation value of the four-component Bargmann-Wigner polarization operator. An exact equation of motion for this quantity is derived, using the one-particle Dirac equation, and the relativistic analogues of the non-relativistic concepts of spin-currents and spin-transfer torques are identified. In the classical limit the time evolution is governed by the equation of motion first proposed by Bargmann, Michel and Telegdi generalized to the case of inhomogeneous systems. In the non-relativistic limit it is found that the spin-current has an intrinsic Hall contribution and to order 1/c^2 a spin-orbit coupling related torque appears in the equation of motion. The relevance of these results to the theory of the intrinsic spin Hall effect and current-induced switching are briefly discussed.