  A calculation of the high-harmonic generation (HHG) in $\alpha$-quartz using
the time-dependent density functional theory is reported. The inter-band
process is attributed to the dominant in HHG above the band gap. The photon
energy is set to 1.55 eV, and the cutoff energy of the plateau region is found
to be limited at the 19th harmonic (30 eV). The dependence of the HHG
efficiency at the cutoff energy region on laser intensity is consistent with
that of the hole density in the lowest-lying valence band. Numerical results
indicate that electron-hole recombination plays a crucial role in HHG in
$\alpha$-quartz. It is found that a 200 attosecond pulse train is produced
using HHG around the plateau cutoff energy.
