Phonon-assisted hopping of an electron on a Wannier-Stark ladder in a strong electric field

Abstract

With the application of a spatially constant electric field, the degeneracy of electronic energy levels of geometrically equivalent sites of a crystal is generally lifted. As a result, the electric field causes the electronic eigenstates of a one-dimensional periodic chain to become localized. In particular, they are Wannier-Stark states. With sufficiently large electric-field strengths these states become sufficiently well localized that it becomes appropriate to consider electronic transport to occur via a succession of phonon-assisted hops between the localized Wannier-Stark states. In this paper, we present calculations of the drift velocity arising from acoustic- and optical-phonon-assisted hopping motion between Wannier-Stark states. When the intersite electronic transfer energy is sufficiently small so that the Wannier-Stark states are essentially each confined to a single atomic site, the transport reduces to that of a small polaron. In this regime, while the drift velocity initially rises with increasing electric field strength, the drift velocity ultimately falls with increasing electric-field strength at extremely large electric fields. More generally, for common values of the electronic bandwidth and electric field strength, the Wannier-Stark states span many sites. At sufficiently large electric fields, the energy separation between Wannier-Stark states exceeds the energy uncertainty associated with the carrier's interaction with phonons. Then, it is appropriate to treat the electronic transport in terms of phonon-assisted hopping between Wannier-Stark states. The resulting high-field drift velocity falls with increasing field strength in a series of steps. Thus, we find a structured negative differential mobility at large electric fields. Such a situation may be experimentally realized in superlattice structures at low temperatures.