Finite diffraction models for electron bandstructures

Truncations of the infinite determinant resulting from the plane wave expansion method for an electron in a periodic potential are analysed to determine how well they can reproduce the low-lying eigenvalues or energy bands. The availability of rigorous expansions for the solutions of the Mathieu equation, essentially the Schrödinger equation for an electron in a one-dimensional cosine potential, suggests that problem for definitive comparisons. Since the only models which can reproduce the fundamental quadratic behaviour of bands at the zone centre have symmetric (odd) sets of reciprocal lattice vectors, the lowest order candidate has an odd determinant of size 3 × 3 through the reciprocal lattice vectors which define the 1st. Brillouin zone of the one-dimensional lattice. As zone centred determinants are not symmetric about zone boundaries they will not give a vanishing group velocity at that point and the effects of truncation will be at their worst. The 3 × 3 or cubic model factorises at the zone centre and a detailed analysis in closed form is straightforward. Agreement with the rigorous results is better than 1% everywhere. The next largest model, with a 5 × 5 determinant, gives errors which are several orders of magnitude smaller throughout the 1st. Brillouin zone. In addition it is found that even the 3 × 3 model gives much better results for the lowest eigenvalue than does second order perturbation theory. Numerical comparisons are also made for the group velocity and (inverse) effective mass using a Hellman-Feynmann approach to calculate the derivatives. Extensions to complex periodic potentials and higher dimensions are briefly discussed.