Knee structure in the laser intensity dependence of high-order harmonic generations for graphene

We investigate the high-order harmonic generations (HHGs) of graphene irradiated by linearly polarized lasers with intensities in a wide range from $10^{8}$ W/cm$^2$ to $10^{13}$ W/cm$^2$. We find a striking knee structure in the laser intensity dependence of HHGs, which consists of a linear growth regime, followed by a plateau of the saturated HHGs, and then a transition to a nonlinear growth. The knee structure is rather universal for the varied harmonic orders and has been certificated by the calculations of two-band density-matrix equations as well as the \textit{ab initio} calculations of time-dependent density functional theory. Based on the two-band model, we reveal the underlying mechanisms: The behavior of linear growth can be depicted analytically by the perturbative theory of optical conductivity; While, the plateau of saturated HHGs and the transition to a nonlinear growth are caused by the quantum destructive interference and constructive interference of harmonics generated by the electrons corresponding to the lattice momentums around Dirac points and M points in Brillouin zone, respectively. In particular, we find that tuning Fermi energy can effectively alter the knee structure while the profile of the knee structure is not sensitive to the temperature. Our calculations of the third-order harmonic vs. tuning Fermi energy are compared with recent experiment showing a good agreement. Our predicted knee structure and its associated properties are observable with the current experimental techniques.