Spin currents, spin-transfer torque, and spin-Hall effects in relativistic quantum mechanics

It is shown that a useful relativistic generalization of the conventional spin density s(r,t) for the case of moving electrons is the expectation value (T(r,t),T-4(r,t)) of the four-component Bargmann-Wigner polarization operator T-mu=(T,T-4) [Proc. Natl. Acad. Sci. U.S.A. 34, 211 (1948)] with respect to the four components of the wave function. An exact equation of motion for this quantity is derived using the one-particle Dirac equation, and the relativistic analogs of the nonrelativistic concepts of spin currents and spin-transfer torques are identified. Using this theoretical framework, in the classical limit the Bargmann-Michel-Telegdi equation [Phys. Rev. Lett. 2, 435 (1959)] for a relativistic wave packet crossing a nonmagnetic/ferromagnetic interface is derived, to lowest order in the spin-orbit coupling it is shown that a contribution to the polarization current occurs with spin-Hall symmetry, and the spin-transfer torque for the simple perfect spin filter model is calculated and discussed.