Doping-Induced Second-Harmonic Generation in Centrosymmetric Graphene from Quadrupole Response

For centrosymmetric materials such as monolayer graphene, no optical second-harmonic generation (SHG) is generally expected, because it is forbidden under the electric-dipole approximation. Yet we observe a strong, doping-induced SHG from graphene, with its highest strength comparable to the electric-dipole-allowed SHG in noncentrosymmetric 2D materials. This novel SHG has the nature of an electric-quadrupole response, arising from the effective breaking of inversion symmetry by optical dressing with an in-plane photon wave vector. More remarkably, the SHG is widely tuned by carrier doping or chemical potential, being sharply enhanced at Fermi-edge resonances but vanishing at the charge neutral point that manifests the electron-hole symmetry of massless Dirac fermions. This striking behavior in graphene, which should also arise in graphenelike Dirac materials, expands the scope of nonlinear optics and holds the promise of novel optoelectronic and photonic applications.

Figure 1

SHG from ion-gel gated graphene with oblique incident excitation. (a) Experimental geometry at an oblique incidence of 45°. The graphene supported on a fused silica substrate is covered with ion gel for gating, through which the chemical potential $\mu$ could be tuned from $-0.9$ to 0.5 eV. The laser excitation and signal collection are done through microscope objectives with numerical apertures of 0.35 and 0.40, respectively. The incident photon energy was 0.95 eV, unless otherwise noted. (b) Specularly reflected spectra with $s$-polarized excitation as a function of $2\mu$. (c) Selected spectra in (b) at different chemical potentials $\mu$. The spectra are offset for clarity. (d),(e) Corresponding spectra with the $p$-polarized excitation. The spectra from bare fused silica covered with ion gel at corresponding gate voltages are also shown as black lines in (c) and (e) for comparison. The broadband background at low doping is from the ultrafast nonlinear photoluminescence, which could be switched off at high doping [32, 33]. The incident laser fluence was $2.4{\mathrm{J}/\mathrm{m}}^2$. (f) Incident fluence dependence of SHG (red) and THG (gray) intensities at a heavily doped level ($\mu \approx -0.88\mathrm{eV}$) on a double-logarithmic scale. Solid lines are fits with slopes of $1.85\pm 0.02$ and $2.95\pm 0.06$, respectively.

Figure 2

Comparison of SHG between normal and oblique incidences. (a) Experimental geometry at normal incidence. The excitation and collection are through the same objective with a numerical aperture as high as 0.95. (b) SHG spectra from heavily doped graphene ($\mu =-0.8\mathrm{eV}$) at the same incident fluence of $2.4{\mathrm{J}/\mathrm{m}}^2$. The red and blue data represent the cases of oblique ($s$-polarized) and normal incidences, respectively. The spectrum at normal incidence is magnified by 100 times.

Figure 3

Polarization characteristics of the SHG at an oblique incidence of 45°. (a),(b) Symmetry properties of the overall system defined by both the input electric field and wave vector. For both $s$- and $p$-polarized excitations, ${\mathbf{P}}_{\mathrm{EQ}}(2\omega )$ is confined in the mirror plane defined by the incident plane, while ${\mathbf{P}}_{\mathrm{ED}}(3\omega )$ follows the in-plane projection of $\mathbf{E}(\omega )$. (c)–(e) The SHG (red dots) intensity versus the angle $\theta$ between the beam incident plane and analyzer polarization direction, under at $p$-, $s$-, and 45°-polarized excitations, respectively. The THG (gray dots) data are also shown for comparison. Solid lines are fits using equations in Supplemental Material [29]. The graphene was heavily doped at $\mu =-0.88\mathrm{eV}$.

Figure 4

Chemical potential dependence of the oblique SHG. (a) The extracted second-order nonlinear susceptibility $|{\chi }_{\mathrm{EQ}}^{(3)}{q}_x|$ of the SHG under $s$- (red dots) and $p$-polarized (black dots) light excitation versus chemical potential $2|\mu |$. The data were extracted from Figs. 1 and 1 for negative chemical potentials. (b) Transition diagrams for one- and two-photon Fermi-edge resonances, respectively. (c) Theoretically calculated $|{\chi }_{\mathrm{EQ},xxyy}^{(3)}{q}_x|$ (red) and $|{\chi }_{\mathrm{EQ},xxxx}^{(3)}{q}_x|$ (black) for $s$- and $p$-polarized excitations, respectively. The dashed and solid lines are from the theories by Wang, Tokman, and Belyanin [22] and Cheng, Vermeulen, and Sipe [23], respectively. In the calculation, the temperature $T=300\mathrm{K}$ (ignoring the effect of high electron temperature for $2|\mu |\le \hslash \omega$) and the resonant damping factor $\mathrm{\Gamma }=0.2|\mu |\mathrm{eV}$ are used [14]. (d),(e) Real (yellow) and imaginary (blue) parts of ${\chi }_{\mathrm{EQ},xxxx}^{(3)}$ and ${\chi }_{\mathrm{EQ},xxyy}^{(3)}$, corresponding to EQ SHG under $p$- and $s$-polarized excitations, respectively. On the right vertical axis, the effective second-order nonlinear susceptibilities ${\chi }_{\mathrm{EQ},xxxx}^{(3)}{q}_x$ and ${\chi }_{\mathrm{EQ},xxyy}^{(3)}{q}_x$ are also shown. (f) Real (yellow) and imaginary (blue) parts of ${\chi }_{\mathrm{ED},xxxx}^{(3)}$, corresponding to ED THG. The calculations were followed from the theory by Cheng, Vermeulen, and Sipe [23, 31]. (g) Transition schematics under the operation of $\mu \to -\mu$ and $\mathbf{k}\to -\mathbf{k}$ with the electron-hole symmetry and time-reversal symmetry for $\mathbf{q}\to -\mathbf{q}$, where $\mathbf{k}$ is the electron wave vector and $\mathbf{q}$ the photon wave vector.