Local density of states and level width for Wannier-Stark ladders

The local density of states ρ(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 ρ(x,E), with Lorentz-type level widths and apparent spatial localization of the states. Our model is a chain of δ-function potential barriers plus a steplike electric potential, with open boundary conditions at both ends of the system. Using a wave-tunneling picture, we find that the level widths shrink to zero as an inverse power of the system size as the system size approaches infinity, confirming an earlier result. The level width will not approach zero if the δ-function barriers are replaced by the Kronig-Penney potential or smoother ones, as is commonly believed.