A finite element method formulation is given for solving Schrodinger's wave equation for a single electron in a crystal lattice cell subject to a known periodic potential. The formulation has been implemented for a two-dimensional lattice, with an arbitrary potential profile, modelled by quadratic isoparametric elements.
The FEM solver returns a specified number of electronic energy states, En, and nodal values of the complex wavefunction &,. Input data is generated by a standard FEM mesh generator. The postprocessing, given n, for reproducing a full 2-D E-k Brillouin diagram and given k, the electronic distribution, has been implemented. Tests on a 2-D generalized Kronig-Penney energy band model showed excellent agreement between FEM results and analysis. The solver was further satisfactorily checked against published augmented plane wave calculations for a circular potential well within a square lattice. Specimen results are presented for the same circular well but with graded potential distributions and for a rectangular potential barrier set askew in a square lattice. Two-dimensional energy band solvers have application to superlattice nanostructures, whilst a general, full 3-D FEM quantum solver seems feasible.