Theory of plasmonic effects in nonlinear optics: The case of graphene

We develop a microscopic large-$N$ theory of electron-electron interaction corrections to multilegged Feynman diagrams describing second- and third-order non-linear-response functions. Our theory, which reduces to the well-known random-phase approximation in the linear-response limit, is completely general and is useful to understand all second- and third-order nonlinear effects, including harmonic generation, wave mixing, and photon drag. We apply our theoretical framework to the case of graphene, by carrying out microscopic calculations of the second- and third-order non-linear-response functions of an interacting two-dimensional (2D) gas of massless Dirac fermions. We compare our results with recent measurements, where all-optical launching of graphene plasmons has been achieved by virtue of the finiteness of the quasihomogeneous second-order nonlinear response of this inversion-symmetric 2D material.

Figure 1

Large-$N$ theory for the second-order density response function, ${\chi }^{\left(2\right)}$. (a) The Feynman diagram for the noninteracting second-order density response function, ${\chi }_0^{\left(2\right)}$. (b)–(d) Feynman diagrams for the second-order density response function in the large-$N$ approximation. (e) Infinite series of Feynman diagrams for the linear density response function: Dyson equation in the large-$N$ limit. (f) Example of a Feynman diagram for the second-order response function, which is excluded from the large-$N$ approximation. Solid lines stand for noninteracting Green's functions, solid circles represent density vertices, and wavy lines denote the electron-electron interaction $v_{\mathit{q}}$.

Figure 2

First family of large-$N$ diagrams for the third-order density response function. These are obtained by renormalizing the vertices of the noninteracting third-order density response function, shown in panel (a). (b)–(e) Feynman diagrams for the third-order density response function in the large-$N$ approximation.

Figure 3

Second family of Feynman diagrams for the third-order response function. These are obtained by gluing together two second-order diagrams via an electron-electron interaction line. The RPA triangle diagram, which is present in panels (a), (b), and (c), is defined as the sum of the diagrams in panels (a)–(d) in Fig. 1.

Figure 4

Illustrative plots of $|d_{xxxx}^{\left(2\right)}(-2\omega ;\omega ,\omega )|,|d_{xyxy}^{\left(2\right)}(-2\omega ;\omega ,\omega )|$, and $|d_{xyyx}^{\left(2\right)}(-2\omega ;\omega ,\omega )|$—in units of $d_0^{\left(2\right)}=d_0/{\left|E_{\mathrm{F}}\right|}^3$, where $d_0$ has been introduced in Eq. (46)—for the case of clean graphene, at zero temperature. The quantities $d_{xxxx}^{\left(2\right)}$ and $d_{xyxy}^{\left(2\right)}$ (solid black and dotted red lines) show two sharp resonances at $\hslash \omega =|E_{\mathrm{F}}|$ and $2|E_{\mathrm{F}}|$, while $d_{xyyx}^{\left(2\right)}$ (dashed blue line) shows only one resonance at $\hslash \omega =2|E_{\mathrm{F}}|$.

Figure 5

Dimensionless efficiencies for harmonic generation in graphene are shown as functions of the wave vector $q$ (in units of $k_{\mathrm{F}}$) and $\omega$ (in units of $E_{\mathrm{F}}/\hslash$). These plots have been made by taking $I_{\mathrm{in}}=1{\mathrm{GW}/\mathrm{cm}}^2$. Panel (a) illustrates the SHG efficiency ${\mathrm{{\rm Y}}}_{2\omega }$, as dressed by plasmons, i.e., Eq. (51). Panel (b) illustrates the THG efficiency ${\mathrm{{\rm Y}}}_{3\omega }$, in the case in which one sets ${\tilde{\sigma }}_{xxxx}^{\left(3\right)}=0$ in Eq. (52). Panel (c) illustrates the same quantity as in panel (b), but for ${\tilde{\sigma }}_{xxxx}^{\left(3\right)}\ne 0$.

Figure 6

Color maps of the bare sum-frequency and difference-frequency wave mixing conversion efficiencies in graphene. The quantities ${\mathrm{{\rm Y}}}_{\mathrm{SF}}$—panel (a)—and ${\mathrm{{\rm Y}}}_{\mathrm{DF}}$—panel (b)—are plotted as functions of ${\omega }_2$ and ${\omega }_1$. In making this plot we have neglected the effect of electron-electron interactions by setting ${\mathcal{R}}_2=1$. Also, we have set $I_1=I_2=1\mathrm{GW}/{\mathrm{cm}}^2,\theta =0$, and $q_i={\omega }_i/c$. For the sake of simplicity, we have neglected the perpendicular component of all wave vectors in the previous relation. In each panel, we have also reported a one-dimensional cut of the 2D color map, taken along the dashed green line at $\hslash {\omega }_2=\left|E_{\mathrm{F}}\right|$.

Figure 7

Color maps of the bare sum-frequency and difference-frequency wave mixing conversion efficiencies in graphene. The quantities ${\mathrm{{\rm Y}}}_{\mathrm{SF}}$—panel (a)—and ${\mathrm{{\rm Y}}}_{\mathrm{DF}}$—panel (b)—are plotted as functions of the relative angle $\theta$ and relative frequency $({\omega }_2-{\omega }_1)/{\omega }_1$. In making this plot we have neglected the effect of electron-electron interactions by setting ${\mathcal{R}}_2=1$. Also, we have set $I_1=I_2=1\mathrm{GW}/{\mathrm{cm}}^2,\hslash {\omega }_1=\left|E_{\mathrm{F}}\right|$, and $q_i={\omega }_i/c$. For the sake of simplicity, we have neglected the perpendicular component of all wave vectors in the previous relation. In each panel, we have also reported a one-dimensional cut of the 2D color map, taken along the dashed green line at $\theta =0.7\pi$.

Figure 8

(a) Color map of the dimensionless quantity $|{\mathcal{R}}_2{|}^{-1}$ for the case of difference-frequency wave mixing. Here, $\hslash {\omega }_1=\left|E_{\mathrm{F}}\right|$ and $q_i={\omega }_i/c$. We clearly see two sharp plasmon resonances. Also, we notice that $|{\mathcal{R}}_2{|}^{-1}$ drops quickly to zero away from these resonances due to the low-frequency Drude peak in the first-order conductivity. (b) Difference-frequency wave mixing conversion efficiency with the inclusion of electron-electron interactions. The reader is invited to compare this panel with panel (b) in Fig. 7.

Figure 9

(a) Theoretical predictions for the DF wave mixing conversion efficiency for two different laser configurations, as in Ref. [34]. Solid lines: ${\vartheta }_{\mathrm{probe}}={70}^{\circ }$ and ${\vartheta }_{\mathrm{pump}}={50}^{\circ }$. Dashed lines: ${\vartheta }_{\mathrm{probe}}={125}^{\circ }$ and ${\vartheta }_{\mathrm{pump}}={15}^{\circ }$. Different colors stand for different values of the Fermi energy $E_{\mathrm{F}}$: black, $E_{\mathrm{F}}=200\mathrm{meV}$; red, $E_{\mathrm{F}}=300\mathrm{meV}$; and blue, $E_{\mathrm{F}}=400\mathrm{meV}$. The pump beam wavelength is changed in the range ${\lambda }_{\mathrm{pump}}=540$–$620\mathrm{nm}$ in order to observe the plasmonic resonance for the different-frequency wave mixing signal at a frequency $\mathrm{\Delta }\nu =c|{\lambda }_{\mathrm{pump}}^{-1}-{\lambda }_{\mathrm{probe}}^{-1}|$. (b) Our result for the dressed second-order susceptibility ${\mathcal{X}}_{\mathrm{L}}^{\left(2\right),\mathrm{ee}}$ for the DF wave mixing process—Eq. (63)—is plotted as a function of the pump wavelength and for the same laser configurations as in panel (a). The red dashed horizontal line indicates the value reported in Ref. [34]. Other parameters [34]: $I_{\mathrm{pump}}=100\mathrm{GW}/{\mathrm{cm}}^2,I_{\mathrm{probe}}=2\mathrm{GW}/{\mathrm{cm}}^2$. The probe wavelength is fixed at ${\lambda }_{\mathrm{probe}}=615\mathrm{nm}$.