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Based on the multiband semiconductor Bloch equations a microscopic approachto high-harmonic generation in crystalline solids which is able to properlydescribe degenerate bands and band crossings is presented and analyzed. It iswell-known that numerical band structure calculations typically provideelectronic wave functions with an undetermined k-dependent phase which resultsin matrix elements which contain arbitrary k-dependent phases. In addition,such approaches usually mix degenerate bands and bands with an energydifference smaller than the numerical precision in an arbitrary way for eachpoint in k-space. These ambiguities are problematic if one considers thedynamics induced by electric fields since the matrix elements of the positionoperator involve a derivative of the wave functions with respect to k. When thelight-matter interaction is described in the length gauge, the problem ofarbitrary phases and degenerate subspace mixing of Bloch states is solved byadopting a smooth gauge transformation along the field direction. The resultsobtained within this method are validated by comparing with calculations in thevelocity gauge. Although we obtain in both gauges the same overall result, thelength gauge is advantageous since it converges with a smaller number of bandsand thus requires significantly less numerical effort than the velocity gauge.Also an unique distinction between inter- and intraband contributions and thusan instructive physical interpretation is possible in the length gauge whereasin the velocity gauge this is unclear. The computedpolarization-direction-dependent high-harmonic spectra agree well withexperimental data reported for GaAs. Furthermore, it is demonstrated that,under proper conditions, the Berry curvature is largely responsible for theeven-order harmonics which are polarized perpendicular to the driving field.

Microscopic analysis of high harmonic generation in semiconductors with degenerate bands