Exact band structures for 1D superlattices beyond the tight-binding approximation

TL;DR

This paper derives exact band structures for 1D superlattices without relying on the tight-binding approximation, enabling precise assessment of its validity and revealing differences in band touching behavior.

Contribution

It provides an analytical method to compute exact band structures for 1D superlattices beyond the tight-binding approximation, clarifying its limitations.

Findings

Exact band structures obtained analytically.

Tight-binding approximation's predictions are quantitatively assessed.

Central energy bands do not touch for certain periodicities, contrary to tight-binding predictions.

Abstract

The band structures describing non-interacting particles in one-dimensional superlattices of arbitrary periodicity are obtained by an analytical diagonalization of the Hamiltonian without adopting the popular tight-binding approximation. The results are compared with those of the tight-binding approximation. In this way, a quantitative prediction of the validity and failure of the tight-binding approximation becomes possible. In particular, it is demonstrated that in contrast to the prediction of the tight-binding approximation the central energy bands do not touch for periodicities $\tau$ of the lattice where $\tau=4n$ and $n$ is an integer.