Periodic asymmetric lattices, viewed from one side to the other, their spatial potential energy are different. This difference affects the electronic structure of valence electrons. Our work shows that pronounced even harmonic signals are observed from periodic asymmetric lattices driven by a multi-cycle pulse field. The phases of the odd and even harmonics driven by parallel and anti-parallel laser polarization directions are compared and show different dependence on laser polarization direction. Moreover, it is found each burst in the synthesized attosecond pulse trains in a periodic asymmetric lattice shows the same carrier-envelope phase. We also show that even-order harmonic efficiency in periodic asymmetric lattices can be enhanced (reduced) by using a multi-cycle driving laser in the presence of a weak terahertz pulse field.

High-order harmonic generation (HHG) from periodic asymmetric potentials is investigated by solving the real-space time-dependent Schrödinger equations and semi-conductor Bloch equations. Our results show that the periodic asymmetric potential will cause asymmetric excitation and HHG dynamics in the time domain, leading to the signal of even order harmonics. From the k-space perspective, a k-dependent transition dipole moment phase reflects the real-space asymmetry and causes the symmetry breaking of excitation and HHG dynamics. We also illustrate that the dephasing effect competes with the asymmetry of interband HHG dynamics. Large amplitude of the driving field is beneficial for observing the symmetry breaking of interband HHG. However, it is found symmetry breaking of intraband HHG dynamics is less influenced by dephasing. Moreover, a carrier-envelope phase controlled driving laser field can be utilized as an isolated attosecond pulse gating mechanism for HHG from the periodic asymmetric potentials.