Deduction of the Tight-Binding Hamiltonian Matrix using the discretization of the Schrödinger Equation

Abstract In this work, we set up a Hamiltonian matrix for a non-interacting two-dimensional electron gas. These systems can be formed at the heterostructures interface of GaAs-AlGaAs and, in the presence of confinement potentials, be used as quantum transistors. Starting from the Schrödinger equation in the approximation of the effective mass, we deduce the Hamiltonians 1D and 2D on the basis of Tight-Binding sites. These Hamiltonians were obtained through the procedure of discretization of the Schrödinger equation. For the one-dimensional case, the result found was the well-known tridiagonal matrix and, for the two-dimensional case, a block tridiagonal matrix. The discretization performed allowed the deduction of the values of the site and hopping energies of the studied system. These results demonstrate the direct link between the Schrödinger equation and the Tight-Binding method, and such results are very useful in the realization of numerical methods, which are not addressed in the basic literature of Solid State Physics.