Local Density of States and Level Width for Wannier-Stark Ladders

TL;DR

This paper calculates the local density of states for a Bloch electron in an electric field, revealing how Wannier-Stark ladders and state localizations depend on system size, with level widths diminishing as system size grows.

Contribution

It introduces a model combining delta potential barriers and electric potential to analyze Wannier-Stark ladders and demonstrates how level widths decrease with increasing system size.

Findings

Level widths shrink to zero as system size approaches infinity.

Multiple Wannier-Stark ladder sequences observed depending on system size.

Level widths follow an inverse power law decay.

Abstract

The local density of states \rho(x,E) is calculated for a Bloch electron in an electric field. Depending on the system size, we can see one or more sequences of Wannier-Stark ladders in \rho(x,E), with Lorentz type level widths and apparent spatial localization of the states. Our model is a chain of delta function potential barriers plus a step-like electric potential, with open boundary condition at both ends of the system. Using a wave tunneling picture, we find that the level widths shrink to zero as an inverse power as the system size approaches infinity, confirming an earlier result.