Energy-Band Structure of Germanium and Silicon: The k·p Method

Abstract

The energy bands of germanium and silicon, throughout the entire Brillouin zone, have been obtained by diagonalizing a k\ifmmode\cdot\else\textperiodcentered\fi{}p Hamiltonian referred to 15 basis states at k=0. The basis states of the k\ifmmode\cdot\else\textperiodcentered\fi{}p Hamiltonian correspond to plane-wave states of wave vector (in units of $\frac{2\ensuremath{\pi}}{a}$) [000], [111], and [200]. For matrix elements and energy gaps we have used, when available, experimental data from cyclotron resonance and optical measurements. The parameters not available from experimental information have been adjusted until the calculated energy bands agree with ultraviolet reflection data. The energies of the ${\ensuremath{\Lambda}}_{3}\ensuremath{-}{\ensuremath{\Lambda}}_{1}$ transition for germanium and the ${\ensuremath{\Sigma}}_{4}\ensuremath{-}{\ensuremath{\Sigma}}_{1}$ transition for germanium and silicon, which were not explicitly fitted, are in good agreement with experimental data. The eigenvectors of the k\ifmmode\cdot\else\textperiodcentered\fi{}p matrix provide an expansion of the wave function for any value of k in terms of the k=0 basis states. These eigenvectors have been used (1) to calculate effective masses of the lowest conduction bands in germanium (${L}_{1}$) and silicon (${\ensuremath{\Delta}}_{1}$), which are in good agreement with experiment, and (2) to calculate the effects of the spin-orbit interaction on the band structure of germanium. The extension of the k\ifmmode\cdot\else\textperiodcentered\fi{}p method to calculate the band structure of zinc-blende-and wurtzite-type materials from that of the isoelectronic group IV material is discussed briefly.