Electron tunneling in the presence of phonons: Classical versus quantum oscillating thin potential barrier

The transmission coefficient for electrons incident on an oscillating thin potential barrier is calculated for the case of a classical barrier and for the case where phonons are present (quantum barrier). In the classical case we find a feedback mechanism between elastic and inelastic tunnel current similar to that found by Gelfand, Schmitt-Rink, and Levi [Phys. Rev. Lett. 62, 1683 (1989)] in the quantum barrier case.