Complex carrier-envelope-phase effect of solid harmonics under nonadiabatic conditions

We theoretically investigate the carrier-envelope-phase (CEP) effect of laser pulses in the generation of solid high-order harmonics under nonadiabatic conditions using the time-dependent density functional theory. It is found that the CEP dependence of solid harmonics is still complex even if the Keldysh parameter $\ensuremath{\gamma}$ remains unchanged. Further analysis shows that the multiphoton ionization still contributes to the generation of high-order harmonics for solids with inversion symmetry when the minimum band gap between the highest valence band and the lowest conduction band is equal to even multiples of the laser frequency. The relative contribution of tunneling ionization and multiphoton ionization to the harmonic generation is responsible for this complex CEP dependence.