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

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_ec²|, Δ≈0.001m_ec². 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.