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

The electronic self-energies in degenerate semiconductors due to interactions of electrons and holes with optical and local-mode phonons (of energy $\hslash {\omega }_0$) are evaluated using second-order perturbation theory. Both (screened) polar and deformation-potential interactions are considered, as are the effects of optical-phonon dispersion: $\hslash \omega \left(\mathbf{q}\right)=\hslash {\omega }_0-\alpha q^2$. 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 $\mathrm{eV}\cong \pm \hslash {\omega }_0$ 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.