Higher Donor Excited States for Prolate-Spheroid Conduction Bands: A Reevaluation of Silicon and Germanium

Abstract

An extension of the Kohn-Luttinger method for the energy levels of the effective-mass Hamiltonian $H=(\frac{P_{x}^{2}}{2{m}_{\ensuremath{\perp}}}+\frac{P_{y}^{2}}{2{m}_{\ensuremath{\perp}}}+\frac{P_{z}^{2}}{2{m}_{\ensuremath{\parallel}}})\ensuremath{-}\frac{{e}^{2}}{\mathrm{Kr}}$ is made via the Rayleigh-Ritz approach. The method is capable of indefinite extension provided one is prepared to deal with large matrices. The first nine $S$-like energy levels and the first eighteen $P$-like energy levels are presented here as a function of the mass ratio $\ensuremath{\gamma}=\frac{{m}_{\ensuremath{\perp}}}{{m}_{\ensuremath{\parallel}}}$ for $0<\ensuremath{\gamma}\ensuremath{\le}1$. The experimental results for the $P$-like excited states of silicon and germanium can be fitted to within experimental error if one takes the low-temperature static dielectric constant of silicon to be 11.40\ifmmode\pm\else\textpm\fi{}0.05, and that of germanium to be 15.36\ifmmode\pm\else\textpm\fi{}0.05. The situation concerning donor levels in GaP is discussed briefly.