High-order harmonic generation from a two-dimensional band structure

In the past few years, harmonic generation in solids has attracted tremendous attention. Recently, some experiments of two-dimensional (2D) monolayer or few-layer materials have been carried out. These studies demonstrated that harmonic generation in the 2D case shows a strong dependence on the laser's orientation and ellipticity, which calls for a quantitative theoretical interpretation. In this work, we carry out a systematic study on the harmonic generation from a 2D band structure based on a numerical solution to the time-dependent Schrödinger equation. By comparing with the 1D case, we find that the generation dynamics can have a significant difference due to the existence of many crossing points in the 2D band structure. In particular, the higher conduction bands can be excited step by step via these crossing points and the total contribution of the harmonic is given by the mixing of transitions between different clusters of conduction bands to the valence band. We also present the orientation dependence of the harmonic yield on the laser polarization direction.

Figure 1

Configuration of the laser-solid interaction. The laser is linearly polarized with an angle $\theta$ with respect to the lattice side. The generated harmonics can be detected behind the 2D material.

Figure 2

The band structures for the 2D Kronig-Penney well. (a) is the full 3D representation for the first 3 energy bands. (b) shows the first 50 bands about the high-symmetry points indicated in the horizontal axis. $\mathrm{\Gamma }$ is the (0,0) point, X is the (1,0) point, M is the (1,1) point. The band clusters (C1, C2,...) have been marked in (b).

Figure 3

Band structures for (a) the 1D model and for (b) the 2D model along the $\mathrm{\Gamma }-\mathrm{X}$ direction. The corresponding harmonic spectrum is shown in (c) for the 1D and in (d) for the 2D model, generated by the same laser pulse of $\lambda =3200$ nm and $I_0=0.436\mathrm{TW}/{\mathrm{cm}}^2$.

Figure 4

HHG by different wavelengths of the laser pulse and by using different number of basis states for the 1200-nm case. Figures (a) and (b) respectively show the spectra for $\lambda =1200$ and $\lambda =3200$ nm at the same peak value of the vector potential. Figure (c) shows the band structure in the $\mathrm{\Gamma }-\mathrm{X}$ direction. The two vertical lines indicate the two crossing points. Figures (d)–(f) show the resultant spectra for the case of $\lambda =1200$ nm by using different number of basis states in the expansion (9).

Figure 5

Harmonic spectra as a function of electric field strength at different laser wavelengths: (a) $\lambda =800$, (b) $\lambda =1200$, and (c) $\lambda =3200$ nm. (d) Vector potential of the electric field for the case in (c), in which the horizontal green dashed line represents the value of crystal momentum (0.365 a.u.) at the boundary of Brillouin zone.

Figure 6

Time-frequency analysis of the HHG and the orientation dependence of the HHG. (a) and (b) show the time-frequency spectrum at different angles $\theta =0^{\circ }$ and   $\theta ={45}^{\circ }$. Figure (c) shows a yield comparison at these two angles and (d) shows the orientation dependence from $0^{\circ }$ to ${180}^{\circ }$.