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

TL;DR

This paper investigates the band structure and density of states in a generalized one-dimensional Kronig-Penney model with hybrid delta-delta prime potentials, revealing critical coupling effects on the spectrum.

Contribution

It introduces a comprehensive analysis of hybrid delta-delta prime potentials in periodic structures, including new insights into band spectrum changes at critical couplings.

Findings

Addition of delta' interaction alters band spectra compared to pure delta potentials.

Critical delta' coupling causes a curvature change in the band spectrum.

Analysis of both one-species and two-species hybrid Dirac combs.

Abstract

We present a detailed study of a generalised one-dimensional Kronig-Penney model using $\delta\text{-}\delta'$ 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 $\delta\text{-}\delta'$ 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 $\delta'$ 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 $\delta'$ couplings are above a critical value.