Role of trajectory symmetry in solid high-order harmonic generation

High-order harmonic generation (HHG) from a ZnO crystal has been investigated theoretically using a two-band model driven by a few-cycle laser pulse. We observe that harmonics in the cut-off region exhibit periodic frequency shifts with changes in the carrier envelope phase (CEP) of the laser field. When the CEP of the laser pulse is an integer multiple of π, the cut-off region is dominated by even-order harmonics rather than odd-order harmonics. To illustrate the physical mechanism behind the even-order harmonics, we track the trajectories of electrons and holes between two successive half-cycles by performing time-frequency analysis and applying the recollision model. The results show that the maximum electron displacement is symmetric between successive half-cycles for odd-order harmonics. In contrast, the half-cycle symmetry of the maximum displacement is broken in the case of even-order harmonics.