Phonon-Coupled Electron Tunneling in Two and Three-Dimensional Tunneling Configurations

TL;DR

This paper investigates how acoustical phonons influence electron tunneling in 2D and 3D systems, revealing that deformation potential coupling causes minor corrections, while piezoelectric coupling significantly alters transmission probabilities.

Contribution

It provides a detailed analysis of phonon-electron interactions in higher-dimensional tunneling, highlighting the different effects of deformation potential and piezoelectric couplings.

Findings

Deformation potential coupling results in small corrections to tunneling transmission.

Piezoelectric coupling significantly modifies the transmission coefficient.

Second order perturbation theory suffices for deformation potential, but not for piezoelectric interactions.

Abstract

We treat a tunneling electron coupled to acoustical phonons through a realistic electron phonon interaction: deformation potential and piezoelectric, in two or three-dimensional tunneling configurations. Making use of 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.