Comparison of Numerical and Analytical Methods for Energy Band of the One-Dimensional Dirac Comb Model

In this paper, we discuss the formation mechanism of energy band using the empty lattice model first. Then, we derive the energy dispersion of one-dimensional Dirac comb model in real space, and systematically compare and analyze the main conceptual characteristics of the energy bands by combining numerical simulation results and approximate method formulas. We clarify in more details about the physical significance of the nearly-free electron approximation and tight-binding approximation for band gap and bandwidth. Using the simplest models, we systematically give a visual image of energy band. Finally, a convergence analysis of the plane-wave method is given for further explanation.