Band spectra of periodic hybrid $\delta$-${\delta }^{\mathrm{\prime }}$ structures

We present a detailed study of a generalised one-dimensional Kronig-Penney model using $\delta \text{-}{\delta }^{\mathrm{\prime }}$ potentials. We analyse the band structure and the density of states in two situations. In the first case we consider an infinite array formed by identical $\delta \text{-}{\delta }^{\mathrm{\prime }}$ potentials standing at the linear lattice nodes. This case will be known throughout the paper as the one-species hybrid Dirac comb. We investigate the consequences of adding the ${\delta }^{\mathrm{\prime }}$ interaction to the Dirac comb by comparing the band spectra and the density of states of pure Dirac-$\delta$ combs and one-species hybrid Dirac combs. Secondly we study the quantum system that arises when the periodic potential is the one obtained from the superposition of two one-species hybrid Dirac combs displaced one with respect to the other and with different couplings. The latter will be known as the two-species hybrid Dirac comb. One of the most remarkable results is the appearance of a curvature change in the band spectrum when the ${\delta }^{\mathrm{\prime }}$ couplings are above a critical value.