Quantum theory of the third-order nonlinear electrodynamic effects of graphene

The linear energy dispersion of graphene electrons leads to a strongly nonlinear electromagnetic response of this material. We develop a general quantum theory of the third-order nonlinear local dynamic conductivity of graphene ${\sigma }_{\alpha \beta \gamma \delta }({\omega }_1,{\omega }_2,{\omega }_3)$, which describes its nonlinear response to a uniform electromagnetic field. The derived analytical formulas describe a large number of different nonlinear phenomena such as the third-harmonic generation, the four-wave mixing, the saturable absorption, the second-harmonic generation stimulated by a dc electric current, etc., which may be used in different terahertz and optoelectronic devices.

Figure 1

Left: the honeycomb lattice of graphene: ${\mathbf{a}}_1=a(1/2,\sqrt{3}/2)$, ${\mathbf{a}}_2=a(-1/2,\sqrt{3}/2)$, $\mathbf{b}=a(0,1/\sqrt{3})$, $a$ is the lattice constant. Right: the Brillouin zone of graphene: ${\mathbf{G}}_1=(2\pi /a)(1,1/\sqrt{3})$, ${\mathbf{G}}_2=(2\pi /a)(1,-1/\sqrt{3})$, $|{\mathbf{G}}_1|=|{\mathbf{G}}_2|=G=4\pi /\left(\sqrt{3}a\right)$; ${\mathbf{K}}_1=-{\mathbf{K}}_4=(2\pi /a)(1/3,1/\sqrt{3})$, ${\mathbf{K}}_2=-{\mathbf{K}}_5=(2\pi /a)(2/3,0)$, ${\mathbf{K}}_3=-{\mathbf{K}}_6=(2\pi /a)(1/3,-1/\sqrt{3})$. Dashed lines show the boundaries of the elementary cell in the direct and reciprocal space.

Figure 2

(a) The intraband (47), (b) interband (48), and (c) total conductivity of a single graphene layer as a function of the frequency $\mathrm{\Omega }=\hslash \omega /E_F$ at $\mathrm{\Gamma }=\hslash \gamma /E_F=0.1$. In (c) the conductivity in the relaxation-free limit ($\mathrm{\Gamma }=0$, thin curves) is also shown.

Figure 3

The relations $\hslash \omega =2E_F=2\hslash v_F\sqrt{\pi n_s}$ (label 2) and $\hslash \omega =2E_F/3$ (label $2/3$) plotted as the frequency $f=\omega /2\pi$ and the wavelength $\lambda =c/f$ versus the electron density $n_s$.

Figure 4

Four contributions to the third-order dimensionless conductivity ${\mathcal{S}}_{xxxx}^{\left(3\right)}(\omega ,\omega ,\omega )$: (a) $(3/0)$ contribution, (b) $(2/1)$ contribution, (c) $(1/2)$ contribution, (d) $(0/3)$ contribution. The effective relaxation rate is $\mathrm{\Gamma }=0.1$.

Figure 5

The total third-order conductivity ${\sigma }_{xxxx}^{\left(3\right)}(\omega ,\omega ,\omega )/{\sigma }_0^{\left(3\right)}$ under the conditions of Fig. 3(b) of Ref. [34]. The parameter $\mathrm{\Gamma }=0.11$ corresponds to the numbers (the relaxation rate 33 meV, the Fermi energy 0.3 eV) used in Ref. [34].

Figure 6

The total third-order conductivity ${\sigma }_{xxxx}^{\left(3\right)}(\omega ,\omega ,\omega )/{\sigma }_0^{\left(3\right)}$ (a) near the triple-photon absorption edge $3\hslash \mathrm{\Omega }\simeq 2E_F$, and (b) near the double-photon absorption edge $2\hslash \mathrm{\Omega }\simeq 2E_F$, in the relaxation-free limit $\mathrm{\Gamma }=0$ and at a finite $\mathrm{\Gamma }=0.01$. In the panels (c) and (d), the exact solution (62) is compared to the approximation (92) at (c) $\mathrm{\Gamma }=0.005$ and $0.001$. The width of the resonance decreases, its height increases, and the accuracy of the approximation (92) improves when $\mathrm{\Gamma }$ tends to zero.

Figure 7

The up-conversion efficiency $I_{3\omega }/I_{\omega }$ [Eq. (91)] as a function of the electron density, in the range of (a) infrared and (b) terahertz frequencies. $\lambda$ is the wavelength of the incident wave. The resonances in (a) correspond to $\mathrm{\Omega }=2$, 1, and $\frac{2}{3}$ (from left to right).

Figure 8

Four contributions to the third-order conductivity ${\sigma }_{xxxx}^{\left(3\right)}(-\omega ,\omega ,\omega )/{\sigma }_0^{\left(3\right)}$: (a) $(3/0)$ contribution, (b) $(2/1)$ contribution, (c) $(1/2)$ contribution, (d) $(0/3)$ contribution. The effective relaxation rate is $\mathrm{\Gamma }=0.1$.

Figure 9

The third-order dimensionless conductivity ${\mathcal{S}}_{xxxx}^{\left(3\right)}(-\mathrm{\Omega },\mathrm{\Omega },\mathrm{\Omega })$ as a function of the frequency $\mathrm{\Omega }$ (a) under the conditions of Fig. 4 from Ref. [34] ($\mathrm{\Gamma }=0.11$) and (b) in the vicinity of the most important feature at $\mathrm{\Omega }\simeq 2$ and $\mathrm{\Gamma }=0.01$. The real (c) and imaginary (d) parts of the function ${\mathcal{S}}_{xxxx}^{\left(3\right)}(-{\mathrm{\Omega }}_1,\mathrm{\Omega },\mathrm{\Omega })$ for ${\mathrm{\Omega }}_1=0.9\mathrm{\Omega }$, $0.95\mathrm{\Omega }$, $1.05\mathrm{\Omega }$, and $1.1\mathrm{\Omega }$. One sees that the large (amplitude $\simeq 4000$) resonant singularity in (a) is a consequence of the crossing and interaction of three smaller (amplitudes $\simeq 100$) resonances: the second-order pole at $2\mathrm{\Omega }-{\mathrm{\Omega }}_1=2$, and two logarithmic singularities at $\mathrm{\Omega }=2$ and ${\mathrm{\Omega }}_1=2$.

Figure 10

The dimensionless absorption coefficient (96) in a single (isolated) graphene layer at (a) low and (b) high frequencies, $\mathrm{\Gamma }=0.1$, and (c) high frequencies at $\mathrm{\Gamma }=0.01$. The curves are labeled by the value of the dimensionless electric field $F=e{E}_0/E_F{k}_F$.