Determining the electronic states that contribute most to solid-state high-order harmonic radiation

Utilizing realistic simulations of high-order harmonic generation (HHG) in several materials, we study how different regions of the Brillouin zone contribute to the nonlinear response. It is often assumed that the electronic trajectories that start in the vicinity of the Γ point are predominantly responsible for the HHG spectrum, but it is shown here that such an approximation is generally inaccurate. While examples can be identified where merely 0.4% of the Brillouin zone produces semiquantitatively accurate HHG spectra, in most situations one must include at least 30%-50% of the Brillouin-zone volume to obtain accurate above-the-gap harmonics. For the harmonic peaks below the band-gap energy, the current-density responses from the entire Brillouin zone must always be integrated. We also identify the minimal set of electronic bands necessary for the construction of reduced but still realistic HHG models. The results should be useful for a number of HHG applications, including all-optical reconstructions of the band structure and light-matter couplings or considerations involving semiclassical approaches to solid-state high-order harmonic radiation.