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 $\vec{s}(\vec{r},t)$ for the case of moving electrons is the expectation value $\mathbf{(}\vec{\mathcal{T}}(\vec{r},t),{\mathcal{T}}_4(\vec{r},t)\mathbf{)}$ of the four-component Bargmann-Wigner polarization operator $T_{\mu }=(\vec{T},T_4)$ [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.