Effect of Spin-Orbit Coupling and Other Relativistic Corrections on Donor States in Ge and Si

Four corrections to the effective-mass approximation are considered with first-order perturbation theory; namely, the deviation of the total perturbing potential $U$ from the potential $U_0=-\frac{e^2}{\kappa r}$ ($\kappa =\mathrm{static}\mathrm{dielectric}\mathrm{constant}$) and the three relativistic corrections: (a) spin-orbit coupling, (b) $s$-shift correction, and (c) massvelocity correction. The number of independent matrix elements is determined for each perturbation by means of the selection-rule theorem. There is no effect of spin-orbit coupling on the effective-mass ground state; the corresponding effect on the exact eigenstates of the nonrelativistic Hamiltonian appears to be small for donors in Si, and does not cause a splitting of the sextet state (with spin) in Ge or Si. Both of the other two relativistic corrections give rise to a shift and a splitting of the degenerate effective-mass ground state, as does the perturbation $U-U_0$. The magnitude of the relativistic corrections and their relative contribution to the observed splitting of the effective-mass ground state are discussed briefly.