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Band spectra of periodic hybrid \(\delta \text {-}\delta '\) structures

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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.

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Notes

  1. It is of note that the critical values \(w_1=\pm 1\) occur when the parameter \(\lambda \) in (8) satisfies \(|\lambda |= \hbar ^2/m=7.62\, \mathrm{eV}{\AA }^2\) for the electron.

  2. This statement excludes the extreme situations in which one recovers the continuum spectrum of the free particle (\(w_1=\pm \infty \) or \(w_1=w_0=0\)).

  3. The behaviour of this system as a conductor or insulator depends on the number of charge carriers in the crystal that together with the band spectrum fixes the position of the Fermi level.

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Acknowledgements

This work was partially supported by the Spanish Junta de Castilla y León and FEDER projects (BU229P18 and VA137G18). L.S.S. is grateful to the Spanish Government for the FPU-fellowships programme (FPU18/00957). The authors acknowledge the fruitful discussions with M. Bordag, K. Kirsten, G. Fucci and C. Romaniega.

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Authors and Affiliations

  1. Departamento de Física Teórica, Atómica y Óptica, Universidad de Valladolid, 47011, Valladolid, Spain

    M. Gadella, J. M. Muñoz-Castañeda, L. M. Nieto & L. Santamaría-Sanz

  2. Departamento de Física Fundamental, IUFFyM, Universidad de Salamanca, Salamanca, Spain

    J. M. Mateos Guilarte

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  1. M. Gadella
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Correspondence to L. M. Nieto.

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Gadella, M., Guilarte, J.M.M., Muñoz-Castañeda, J.M. et al. Band spectra of periodic hybrid \(\delta \text {-}\delta '\) structures. Eur. Phys. J. Plus 135, 786 (2020). https://doi.org/10.1140/epjp/s13360-020-00818-6

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