Velocity Acquired by an Electron in a Finite Electric Field in a Polar Crystal

Abstract

The expectation value of the steady-state velocity acquired by an electron interacting with the longitudinal, optical phonons of a polar crystal in a finite electric field is analyzed quantum mechanically for arbitrary coupling strength, field strength, and temperature. The rate of loss of momentum by an electron drifting through the crystal in the applied field is expressed in a form in which the lattice coordinates (the phonons) have been eliminated exactly by path-integral methods. This expression is then evaluated by a path-integral approach similar to that used to calculate the impedance of electrons in polar crystals. We present numerical calculations of field (loss of energy per unit distance) versus velocity for three coupling strengths using the Fröhlich polaron model. In a single curve, all the expected phenomena appear, including a threshold field for producing hot electrons and a decreasing rate of energy loss with velocity for very fast electrons. Using only the experimentally measured values of the reststrahlen energy and the static and optical dielectric constants, we find an energy loss of 0.025 eV/Å for electrons near the threshold in ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$, which compares favorably with the experimental value of about 0.03 eV/Å. We conclude that optical-phonon scattering can indeed produce the high rate of energy loss that is present in tunnel-cathode structures.