Dirac equation and energy levels of electrons in one-dimensional wells: Plane wave expansion method

J. D. Valenzuela-Sau, Rafael A. Méndez-Sánchez*, R. Aceves, Raúl García-Llamas

*Autor correspondiente de este trabajo

Producción científica: Contribución a una revistaArtículorevisión exhaustiva


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 Va. 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 |Va|≤0.01er, where er 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 ap∕w for a fixed depth, where ap 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 Va, it was studied and found that the width of this gap is reduced when the magnitude of the potential is increased, i.e., for Va≈|±2mec2|, Δ≈0.001mec2. It was also studied a structure with local periodicity al embedded in a periodic one. This crystal allows to study both, localized and surfaces states. In the first case, the deepness Vc 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 ws of the lateral wells is varied and two-degenerated surface modes appear.

Idioma originalInglés
Número de artículo114298
PublicaciónPhysica E: Low-Dimensional Systems and Nanostructures
EstadoPublicada - oct. 2020

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© 2020 Elsevier B.V.


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