## Resumen

The band structure of an electron in one-dimensional crystals is obtained using the Dirac equation. At low energies, the Dirac equation solved for a periodic 1D potential corresponds to the obtention of the band structure of a 1D graphene periodic superlattice. The plane wave expansion method was used to obtain the theoretical solution to the problem as an eigenvalues equation, which is solved with a standard matrix diagonalization numerical method. Results are presented for the case of the Dirac–Kronig–Penney model for a rectangular potential of width w and depth V_{a}. It is firstly shown that the bands structures, calculated in the first Brillouin zone, using the Schrödinger and Dirac equations, give practically the same results for |V_{a}|≤0.01e_{r}, where e_{r} is the electron's rest energy. Then, a comparison between the limits of the two lowest bands obtained with Schrödinger and Dirac formalisms is presented as a function of the ratio a_{p}∕w for a fixed depth, where a_{p} is the potential's period. Later, since the energies given by the Dirac equation form two subsets separated by a gap Δ for any value of the potential amplitude V_{a}, it was studied and found that the width of this gap is reduced when the magnitude of the potential is increased, i.e., for V_{a}≈|±2m_{e}c^{2}|, Δ≈0.001m_{e}c^{2}. It was also studied a structure with local periodicity a_{l} embedded in a periodic one. This crystal allows to study both, localized and surfaces states. In the first case, the deepness V_{c} of the central potential is varied and two localized states appear at the forbidden gap between the two lower bands. In the second one, the width w_{s} of the lateral wells is varied and two-degenerated surface modes appear.

Idioma original | Inglés |
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Número de artículo | 114298 |

Publicación | Physica E: Low-Dimensional Systems and Nanostructures |

Volumen | 124 |

DOI | |

Estado | Publicada - oct. 2020 |

### Nota bibliográfica

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