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

Abstract

Abstract 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 Schrodinger and Dirac equations, give practically the same results for | V a | ≤ 0 . 01 e r , where e r is the electron’s rest energy. Then, a comparison between the limits of the two lowest bands obtained with Schrodinger 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 ≈ | ± 2 m e c 2 | , Δ ≈ 0 . 001 m 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.