Energies of an Electron in a One-Dimensional Lattice Using the Dirac Equation: The Coulomb Potential

The energies of an electron in a one-dimensional crystal are studied with both the Schrödinger and Dirac equations using the plane wave expansion method. The crystalline potential sensed by the electron in a cell was calculated by accounting for the Coulombic (electrostatic) interaction between the electron and the surrounding cores (immobile positive ions at the center of the crystal cells). The energies and wave functions of the electron were calculated as a function of four parameters: the period (Formula presented.) of the lattice, the dimension (Formula presented.) of the matrix in the momentum space, the partition number (Formula presented.) in which the unit cell is divided to calculate the potential and the number of cores (Formula presented.) that affect the electron. It was found that 8000 cores (surrounding the electron) were needed to reach our convergence criterion. An analytical equation that accurately describes the behavior of the energies in function of the cores that affect the electron was also found. As case studies, the energies for pseudo-lithium and pseudo-graphene were obtained as a first approximation for one-dimensional lattices. Subsequently, the energies of an isolated dimer nanoparticle were also calculated using the supercell method.