Analysis of the Tunneling Measurement of Electronic Self-Energies Due to Interactions of Electrons and Holes with Optical Phonons in Semiconductors

Abstract

The electronic self-energies in degenerate semiconductors due to interactions of electrons and holes with optical and local-mode phonons (of energy $\ensuremath{\hbar}{\ensuremath{\omega}}_{0}$) are evaluated using second-order perturbation theory. Both (screened) polar and deformation-potential interactions are considered, as are the effects of optical-phonon dispersion: $\ensuremath{\hbar}\ensuremath{\omega}(\mathbf{q})=\ensuremath{\hbar}{\ensuremath{\omega}}_{0}\ensuremath{-}\ensuremath{\alpha}{q}^{2}$. One-electron models of tunneling in metal-oxide-semiconductor junctions are constructed. Their consequences are investigated numerically for indium-Si${\mathrm{O}}_{2}$-silicon junctions. The results of these calculations are parametrized by simple models of the barrier penetration factor for use in evaluating fine structure at $\mathrm{eV}\ensuremath{\cong}\ifmmode\pm\else\textpm\fi{}\ensuremath{\hbar}{\ensuremath{\omega}}_{0}$ due to electron-phonon interactions. The transfer-Hamiltonian model is utilized to classify such fine structure as due to either inelastic tunneling processes or (electrode) self-energy effects. The analytical and experimental distinction between these two types of effects is described. The combined model obtained using second-order self-energies characteristic of the semiconductor electrode and simplified approximate barrier penetration factors is utilized to interpret experimental data on indium-Si${\mathrm{O}}_{2}$-silicon and Au-CdS junctions. The satisfactory description of these data suggests that $\frac{{d}^{2}I}{d{V}^{2}}$ measurements on junctions in which one electrode is a very heavily doped semiconductor can provide a direct experimental determination of the energy-shell electronic self-energies in the semiconductor electrode.