AP Distinguished Seminar - Uncovering the Fourier Structure of Wavefunctions in Semiconductors

Symmetry dictates the physical properties of materials. In the free electron picture, the symmetry of the Bravais lattice defines a set of points, lines, and planes over which sets of planewaves are degenerate. In a real solid, point-group symmetry determines the interaction potentials in the reciprocal space which may lift such degeneracies. This results in wavefunctions which are largely single planewaves throughout the Brillouin zone (BZ), except in the vicinity of the removed degeneracies. As optical transitions between any two planewaves are forbidden, only regions of the BZ near these lifted degeneracies contribute to optical properties. Application to optical response of Si and other semiconductors reveals that a single band transition, with only two planewaves, well describes their electric susceptibility χ⁽¹⁾. Further, it provides a framework to understand non-linear optical response which to date is largely intractable. We show that χ⁽²⁾ arises mostly from higher order degeneracy only existing along high symmetry lines/points of the BZ.