Valence band effective-mass expressions in the ${\mathrm{sp}}^3{d}^5{s}^{*}$ empirical tight-binding model applied to a Si and Ge parametrization

Exact, analytic expressions for the valence band effective masses in the spin-orbit, ${\mathrm{sp}}^3{d}^5{s}^{*}$ empirical tight-binding model are derived. These expressions together with an automated fitting algorithm are used to produce improved parameter sets for Si and Ge at room temperature. Detailed examinations of the analytic effective-mass expressions reveal critical capabilities and limitations of this model in reproducing simultaneously certain gaps and effective masses. The [110] masses are shown to be completely determined by the [100] and [111] masses despite the introduction of $d$ orbitals into the basis.