Phonon-coupled electron tunneling in two- and three-dimensional tunneling configurations

We treat a tunneling electron coupled to acoustical phonons through a realistic electron-phonon interaction: deformation potential and piezoelectric coupling, in two- or three-dimensional tunneling configurations. Making use of the slowness of the phonon system compared to electron tunneling, and using a Green function method for imaginary time, we are able to calculate the change in the transition probability due to the coupling to phonons. It is shown using standard renormalization procedure that, contrary to the one-dimensional case, second-order perturbation theory is sufficient in order to treat the deformation potential coupling, which leads to a small correction to the transmission coefficient prefactor. In the case of piezoelectric coupling, which is found to be closely related to the piezoelectric polaron problem, vertex corrections need to be considered. Summing leading logarithmic terms, we show that the piezoelectric coupling leads to a significant change of the transmission coefficient.