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We present a detailed study of a generalised one-dimensional Kronig-Penneymodel using $\delta\text{-}\delta'$ potentials. We analyse the band structureand the density of states in two situations. In the first case we consider aninfinite array formed by identical $\delta\text{-}\delta'$ potentials standingat the linear lattice nodes. This case will be known throughout the paper asthe one-species hybrid Dirac comb. We investigate the consequences of addingthe $\delta'$ interaction to the Dirac comb by comparing the band spectra andthe density of states of pure Dirac-$\delta$ combs and one-species hybrid Diraccombs. Secondly we study the quantum system that arises when the periodicpotential is the one obtained from the superposition of two one-species hybridDirac combs displaced one with respect to the other and with differentcouplings. The latter will be known as the two-species hybrid Dirac comb. Oneof the most remarkable results is the appearance of a curvature change in theband spectrum when the $\delta'$ couplings are above a critical value.

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