Wannier-Bloch approach to localization in high harmonics generation in solids

Emission of high-order harmonics from solids provides a new avenue in attosecond science. On one hand, it allows to investigate fundamental processes of the non-linear response of electrons driven by a strong laser pulse in a periodic crystal lattice. On the other hand, it opens new paths toward efficient attosecond pulse generation, novel imaging of electronic wave functions, and enhancement of high-order harmonic generation (HHG) intensity. A key feature of HHG in a solid (as compared to the well-understood phenomena of HHG in an atomic gas) is the delocalization of the process, whereby an electron ionized from one site in the periodic lattice may recombine with any other. Here, we develop an analytic model, based on the localized Wannier wave functions in the valence band and delocalized Bloch functions in the conduction band. This Wannier-Bloch approach assesses the contributions of individual lattice sites to the HHG process, and hence addresses precisely the question of localization of harmonic emission in solids. We apply this model to investigate HHG in a ZnO crystal for two different orientations, corresponding to wider and narrower valence and conduction bands, respectively. Interestingly, for narrower bands, the HHG process shows significant localization, similar to harmonic generation in atoms. For all cases, the delocalized contributions to HHG emission are highest near the band-gap energy. Our results pave the way to controlling localized contributions to HHG in a solid crystal, with hard to overestimate implications for the emerging area of atto-nanoscience.