High-order harmonic generation from gapped graphene: Perturbative response and transition to nonperturbative regime

We consider the interaction of gapped graphene in the two-band approximation using an explicit time-dependent approach. In addition to the full high-order harmonic generation (HHG) spectrum, we also obtain the perturbative harmonic response using the time-dependent method at photon energies covering all the significant features in the responses. The transition from the perturbative to the fully nonperturbative regime of HHG at these photon energies is studied in detail.

Figure 1

The structure of graphene and gapped graphene in position space. The elementary lattice vectors, ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$, as well as the atomic sites A and B (the two inequivalent sublattices) are shown. The yellow dashed lines denote the connections of A with its nearest neighbors. The $x$ and $y$ axes are also indicated.

Figure 2

The absolute value of the (a) first harmonic (linear response) and (b) second harmonic, for 1-eV gapped graphene and ${\tau }_1={\tau }_2=5$ fs, obtained using the time-dependent calculation for $M=48$ cycles (black curves) and the frequency-dependent method for infinitely periodic pulses.

Figure 3

The absolute value of the (a) first harmonic (linear response) and (b) second harmonic, for 1-eV gapped graphene and ${\tau }_1={\tau }_2=5\mathrm{fs}$, obtained using the condition (19) and compared to the perturbative result. The numbers in the legends in panels (a) and (b) denote the peak field strength (in a.u.) of the pulses used in the calculation. The curve labeled “num pert” denotes the numerical result that compares best with the perturbation theory at peak fields (a) ${10}^{-4}$ a.u., and (b) ${10}^{-5}$ a.u.; see the text.

Figure 4

The absolute value of the (a) first harmonic (linear response) and (b) second harmonic, for 1-eV gapped graphene and ${\tau }_1={\tau }_2=5\mathrm{fs}$, obtained with full calculation and compared to the perturbative result. The numbers in the legends in panels (a) and (b) denote the peak field strength (in a.u.) of the pulses used in the calculation.

Figure 5

The absolute value of the third harmonic for 1-eV gapped graphene, ${\tau }_1={\tau }_2=5$ fs, obtained at different peak field strengths and compared to the perturbative result. The numbers in the legend denote the peak field strength (in a.u.) of the $M=48$ cycle pulses used in the calculation.

Figure 6

Harmonic spectra (divided by peak field strength squared) for different peak fields (given in the legend in atomic units) at (a) $\hslash \omega =1.5$ eV, (b) $\hslash \omega =1$ eV, and (c) $\hslash \omega =0.5$ eV photon energy. The yield is proportional to ${\left|j\left(\mathrm{\Omega }\right)\right|}^2/F_0^2$.

Figure 7

Harmonic spectra (divided by peak field strength squared) at different photon energies of the driving field on color logarithmic scale in arbitrary units: (a) spectra for a peak field strength of $2\times {10}^{-5}$ a.u. (close to the perturbation regime), (b) spectra for a peak field strength of $2\times {10}^{-4}$ a.u., and (c) spectra for a peak field strength of $2\times {10}^{-3}$ a.u. (deeply in the nonperturbative regime). The lines in the color plots denote the borders defining how many harmonics fit (in order from left to right on the figure) at the 1-eV gap (K point), at the 6.2-eV gap at van Hove singularity ($M$ point) and at the maximum gap of 18.03 eV ($\mathrm{\Gamma }$ point).

Figure 8

First harmonic at the photon energy equal to the gap ($\hslash \omega =\mathrm{\Delta }$). The peak field strengths (scaled to the gap) are given in a.u. in the legend. To obtain the actual field strength in a.u. used for a given gap, $\mathrm{\Delta }$ in the legend should be given in eV.

Figure 9

Second harmonic at the photon energy equal to the gap ($\hslash \omega =\mathrm{\Delta }$) and at photon energy equal to the half of the gap ($\hslash \omega =\mathrm{\Delta }/2$), obtained using the frequency-domain method of Ref. [26], ${\tau }_1={\tau }_2=5$ fs, and $6000\times 6000$ grid in $\mathbf{k}$ space.