Even-order high-harmonic generation from solids in velocity gauge

We present a velocity-gauge model for the generation of even-order high harmonics, and reveal that the even-order harmonics originate from the multiple-step transitions among the energy bands in momentum space, while the odd-order harmonics are mainly from direct transitions. The lower valence band is found vital for the generation of even harmonics. Relative intensity of even-order harmonics versus the odd orders is calculated and shows a growing trend as the laser field amplitude increases.