Wannier interpolation study of the Elliot-Yafet spin relaxation in metals

Energy states of nonmagnetic metals may be chosen to be purely spin up and down in the absence of spin-orbit coupling. Spin-orbit coupling mixes the two states by a small amount $b^2$. A spin-conserving interaction (e.g. electron-phonon) causes transitions between the two states, and flips the electron's spin. Some insight into this Elliot-Yafet spin relaxation mechanism can be obtained by averaging $b^2$ over the Fermi surface. In trivalent metals, such as aluminum, $b^2\ll 1$ almost everywhere on the Fermi surface, except at small ``hot spot'' regions. \footnote{J. Fabian and S. Das Sarma, Phys. Rev. Lett. {\bf 81}, 5624 (1998).} Although the small regions of large $b^2$ dominate the spin relaxation process, they are difficult to capture numerically. We describe a Wannier interpolation strategy \footnote{X. Wang, J. Yates, I. Souza, and D. Vanderbilt, Phys.\ Rev.\ B, in press (cond-mat/0608257).} to compute $\langle b^2\rangle$. We validate it by performing {\it ab initio} calculations on aluminum, finding good agreement with previous results.$^1$ We also discuss interpolating {\it ab initio} electron-phonon matrix elements to compute the spin relaxation rate.