Semiclassical Equations of Motion for Bloch Electrons in External Fields, Including Spin-Orbit Interaction.

This talk considers an electron moving in a periodic potential with spin-orbit interaction and perturbed by external slowly varying electric field and uniform magnetic field. Superposition of the time-independent Bloch spinor states of the unperturbed Hamiltonian gives a wave packet state with both wavevector space and spin-orientation amplitude factors. The time-dependent variational principle produces equations of motion for the centers of the wave packet in both configuration and wavevector space and for the spin-orientation factors. For spinless electrons this procedure yields the familiar semiclassical equations of motion augmented by an orbital magnetic moment contribution to the Bloch band energy and an ``anomalous velocity'' proportional to a Berry curvature.\footnote{M. C. Chang \& Q. Niu, Phys. Rev. B {\bf{53}}, 7010 (1996)} The inclusion of spin-orbit interaction gives additional contributions to the velocity involving different Berry curvatures. One is a spin-dependent contribution to the magnetic moment, and another is an electric-dipole-like contribution that is also proportional to the spin operator.