Wave mixing and high harmonic generation at two-color multiphoton excitation in two-dimensional hexagonal nanostructures

The high-order wave mixing by two-color multiphoton excitation in nonlinear regime along with the harmonics generation in two-dimensional (2D) nanostructures such as graphene, silicene, germanene, and stanene is investigated. We consider the coherent part of the spectra corresponding to high harmonic generation by two-color intense laser pulses. The developed theory of interaction of free carriers with the driving effective field of two-color waves covers the full Brillouin zone of 2D hexagonal nanostructures. We also consider the case when one of the driving waves is in one-photon resonance with the van Hove singularity at the $M$ point of the Brillouin zone. Obtained results show that novel 2D nanostructures can serve as an effective medium for high-order wave mixing and harmonic generation at the two-color multiphoton excitation with strong laser pulses.

Figure 1

Radiation spectrum from interband (blue-solid line) and intraband (green-dashed line) currents for graphene, at ${\chi }_1={10}^{-2}$ and ${\chi }_2=0.5$. We assume $\hslash {\omega }_1=1.7\mathrm{eV}$ and $\hslash {\omega }_2=0.1\mathrm{eV}$.

Figure 2

Total radiation spectra for graphene, silicene, germanene, and stanene at $\hslash {\omega }_1=1.7\mathrm{eV}$ and $\hslash {\omega }_2=0.1\mathrm{eV}$. The dimensionless interaction parameters calculated for graphene are ${\chi }_1={10}^{-2}$ and ${\chi }_2=0.5$. These correspond to fields' strengths $E_{01}=6.9\times {10}^5\mathrm{V}/\mathrm{cm}$ and $E_{02}=2\times {10}^6\mathrm{V}/\mathrm{cm}$.

Figure 3

Density plot of the interband polarization absolute value $\left|\mathcal{P}\left(1,\mathbf{k},t_i\right)\right|$ (in arbitrary units) at the instant $t_i={\tau }_{1,2}/2$, when fields reach peak values, as a function of scaled dimensionless momentum components $\left(k_x/k_b,k_y/k_b\right)$ for silicene. The waves' field strengths (frequency) are $E_{01}=6.9\times {10}^5\mathrm{V}/\mathrm{cm}\left(\hslash {\omega }_1=1.7\mathrm{eV}\right)$ and $E_{02}=2\times {10}^6\mathrm{V}/\mathrm{cm}\left(\hslash {\omega }_2=0.1\mathrm{eV}\right)$.

Figure 4

Density plot of the electron-hole energy ${\mathcal{E}}_{eh}\left(1,t_i,\mathbf{k}\right)$ (in eV) at the instant $t_i={\tau }_{1,2}/2$, when fields reach peak values, as a function of scaled dimensionless momentum components $\left(k_x/k_b,k_y/k_b\right)$ for silicene. The interaction parameters are the same as in Fig. 3. The black isoline corresponds to energy 1.7 eV, while white isoline corresponds to energy 3.0 eV.

Figure 5

Total radiation spectra for graphene, silicene, germanene, and stanene at $\hslash {\omega }_1=1\mathrm{eV}$ and $\hslash {\omega }_2=0.05\mathrm{eV}$. The dimensionless interaction parameters calculated for graphene are ${\chi }_1={10}^{-2}$ and ${\chi }_2=0.5$. These correspond to fields' strengths $E_{01}=4\times {10}^5\mathrm{V}/\mathrm{cm}$ and $E_{02}={10}^6\mathrm{V}/\mathrm{cm}$.

Figure 6

Total radiation spectra for graphene, silicene, germanene, and stanene at $\hslash {\omega }_1=1\mathrm{eV}$ and $\hslash {\omega }_2=0.03\mathrm{eV}$. The dimensionless interaction parameters calculated for graphene are ${\chi }_1={10}^{-2}$ and ${\chi }_2=0.5$. These correspond to fields' strengths $E_{01}=4\times {10}^5\mathrm{V}/\mathrm{cm}$ and $E_{02}=6\times {10}^5\mathrm{V}/\mathrm{cm}$.

Figure 7

Density plot of the particle distribution function $N_c(1,\mathbf{k},t_i)$ (in arbitrary units) at the instant $t_i={\tau }_{1,2}/2$, when fields reach peak values, as a function of scaled dimensionless momentum components for graphene. The high-frequency wave is in resonance with the van Hove singularity at the $M$ point of Brillouin zone $\hslash {\omega }_1=5.4\mathrm{eV}$. The dimensionless interaction parameters calculated for graphene are ${\chi }_1={10}^{-2}$ and ${\chi }_2=0.5\left(\hslash {\omega }_2=0.1\mathrm{eV}\right)$.

Figure 8

Radiation spectrum at $\hslash {\omega }_1=5.4\mathrm{eV}$ (red-solid line) and $\hslash {\omega }_1=3.4\mathrm{eV}$ (green-dashed line) for graphene. We have taken ${\chi }_1={10}^{-2},{\chi }_2=0.5$, and $\hslash {\omega }_2=0.1\mathrm{eV}$.

Figure 9

Radiation spectrum at $\hslash {\omega }_1=2.174\mathrm{eV}$ (red-solid line) and $\hslash {\omega }_1=1.7\mathrm{eV}$ (green-dashed line) for silicene. We have taken ${\chi }_1={10}^{-2},{\chi }_2=0.5$, and $\hslash {\omega }_2=0.1\mathrm{eV}$.