Overlap functions in planar crystals

The overlap function F² = |∫ u^*_k(r)u_k'(r)dr|² is evaluated for several values of k and a wide mesh of points k', where u_k is the modulating part of the Bloch wave function for a π-electron in the valence or conduction band, in planar boron nitride and graphite. It is found that for boron nitride, if u_k and u_k, are in the same band and |k - k'| is less than a quarter of the radius of the Brillouin zone, then F² usually lies between 0.9 and 1. In graphite, however, F² behaves more erratically because the top of the valence band is degenerate with the bottom of the conduction band; a theoretical analysis of this behaviour is also presented.