Theory of Phonon-Assisted Tunneling in Semiconductors

Abstract

We calculate the phonon-assisted tunneling current for a model $p-n$ junction (as opposed to the homogeneous-electric-field model) due to two mechanisms. A first-order mechanism in which an electron on, say, the $p$ side scatters to a state on the $n$ side with the emission of a phonon yields results similar to those calculated by other workers for the homogeneous-electric-field model and is about three orders of magnitude too small to account for the experimentally observed current. A second-order process in which an electron on the $p$ side tunnels to an intermediate state in a higher band on the $n$ side via the interband term in the Hamiltonian and then scatters with the emission of a phonon to a final state on the $n$ side yields a current equal in magnitude to the experimentally observed current. This mechanism also succeeds, where the first one fails, in accounting for the magnitude of an differences between the experimentally measured pressure coefficients $\pi _{\mathrm{LA}}^{}{}_{}{}^{+}$, $\pi _{\mathrm{LA}}^{}{}_{}{}^{-}$, $\pi _{\mathrm{TA}}^{}{}_{}{}^{+}$, $\pi _{\mathrm{TA}}^{}{}_{}{}^{-}$ where $\pi =\frac{J^{-1\mathrm{}}\mathrm{dJ}}{\mathrm{dP}}$, the superscripts identify the direction of current flow, and the subscripts, the branch of the phonon involved in the tunneling process (LA = longitudinal acoustic, TA = transverse acoustic).