Nonlinear optical conductivity resulting from the local energy spectrum at the $M$ point in graphene

Based on the tight-binding model we construct a nonlinear Hamiltonian to describe the effective electric system around the $M$ point for the single layer graphene. The local energy spectrum at the $M$ point is approximated by the perfect hyperbolic geometry, and the modification by the screening effect from the substrate is taken into account. With the method based on the concept of the Floquet states and quasienergies (FSQ) we investigate the third order nonlinear conductivity ${\sigma }_3\left(\omega \right),{\sigma }_3\left(3\omega \right)$ for the different frequency ranges, respectively, in which only the $\pi -{\pi }^{*}$ bands are involved. A positive cusplike peak arises at $\hslash \omega ={\varepsilon }_{\mathrm{gap}}\left(\text{M}\right)/3$ for ${\sigma }_3\left(3\omega \right)$ which originates from the three-photon processes. Also, there is a peak at ${\varepsilon }_{\mathrm{gap}}\left(\text{M}\right)/2$ for ${\sigma }_3\left(\omega \right)$ resulting from two-photon-resonant processes. The analysis of the pole processes indicates that a self-energy-like term transition process plays a role in the nonlinear optical response, and the different transition processes interact with each other during the response to the external field. These interactions can be influenced by the polarization of the external field.

Figure 1

The scheme of equienergy contours for the single graphene. (a) Three classes of configurations (A), (B), (C) between the external field and the local geometry of energy spectrum. (b) Constant-${\tilde{\varepsilon }}^2$ contours for Eq. (3) (red dashed lines) and for Eq. (4) (blue lines).

Figure 2

The linear optical conductivity ${\sigma }_{yy}\left(\omega \right)$ vs frequency calculated with the FSQ method.

Figure 3

The resonance structure for ${\sigma }_3\left(\omega \right)$ and ${\sigma }_3\left(3\omega \right)$. The two-photon-resonant process for the case ${\sigma }_3\left(\omega \right)$ in the vertical section across $\mathrm{\Gamma }-M$ (a) and the vertical section across K-$M$ (b). (c) The third-harmonic generation for ${\sigma }_3\left(3\omega \right)$ in the vertical section across $\mathrm{\Gamma }-M$. Red dashed line: virtual level.

Figure 4

Third nonlinear conductivity with single frequency ${\sigma }_3\left(\omega \right)/{\sigma }_0^{\left(3\right)}={\tilde{\sigma }}_3$, where ${\sigma }_0^{\left(3\right)}=\frac{e^2{\hslash }^2{\left(v_F^R\right)}^2}{\left(t_h^R\right)4}{\sigma }_0$ and ${\sigma }_0=\frac{e^2}{4\hslash }$ (a) The real (solid line) and imaginary (dashed line) part of ${\sigma }_3\left(\omega \right)$ vs frequency (b) The modulus (solid line) and the phase (dashed line) ${\sigma }_3\left(\omega \right)=\left|{\sigma }_3\left(\omega \right)\right|e^{i\varphi }$ vs frequency.

Figure 5

Third nonlinear conductivity with single frequency ${\sigma }_3\left(3\omega \right)/{\sigma }_0^{\left(3\right)}={\tilde{\sigma }}_3$. (a) The real (solid line) and imaginary part (dashed line) of ${\sigma }_3\left(3\omega \right)$ vs frequency. (b) The modulus (solid line) and the phase (dashed line) ${\sigma }_3\left(3\omega \right)=\left|{\sigma }_3\left(3\omega \right)\right|e^{i\varphi }$ vs frequency.

Figure 6

The third-harmonic generation conductivity for different effective field polarization ${\sigma }_3^{\mathrm{A}}\left(\tilde{\omega }\right),{\sigma }_3^{\mathrm{B}}\left(\tilde{\omega }\right)$. Inset: The distribution of the group velocity around the $M$ point.