Wannier-Stark ladders and the energy spectrum of an electron in a finite one dimensional crystal

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Abstract
The one-band hamiltonian for an electron in an open finite tight-binding chain in a uniform external electric field is exactly diagonalized. The exact energy eigenvector components are given in terms of the Lommel polynomials and the energy spectrum, for a chain of N atoms, corresponds to the zeros of the Lommel polynomial of degree N. From this general eigenvalue problem solution, limiting cases of the model are investigated. In the infinite N limit, with fixed field strength and hopping parameter, for all eigenvalues bounded with respect to the centre of the spectrum, the components of the eigenvectors converge to Bessel functions whose order is determined by the energy. An asymptotic analysis shows that these bounded eigenvalues converge quite rapidly to a Wannier-Stark ladder when N to infinity . The results can be applied to the study of certain new one-dimensional solids.

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