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

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.