[Back]

A. Vernes, B.L. Gyorffy, P. Weinberger:
"Spin currents, spin-transfer torque, and spin-Hall effects in relativistic quantum mechanics\";
Physical Review B, 76 (2007), 0124081 - 0124084.

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)T4(r,t))of the four-component Bargmann-Wigner polarization
operator T =(T,T4) 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.\