Second-order envelope equation of graphene electrons

A treatment of graphene׳s electronic states based on the tight-binding method is presented. Like Dirac equation, this treatment uses envelope functions to eliminate crystal potential. Besides, a density-functional-theory Kohn–Sham (KS) orbital of an isolated carbon atom is employed. By locally expanding envelope functions into second-order polynomials and by involving up to third-nearest atoms in calculating orbital integrals, the second-order envelope equation is obtained. This equation does not contain any experimental data except graphene׳s crystal structure, and its coefficients are determined through several kinds of integrals of the carbon KS orbital. As an improvement, it leads to more accurate energy dispersion than Dirac equation including the triangular warping effect and asymmetry for electrons and holes, and gives the Fermi velocity which is in good agreement with the experimental value.