Theory of graphene saturable absorption

Saturable absorption is a nonperturbative nonlinear optical phenomenon that plays a pivotal role in the generation of ultrafast light pulses. Here we show that this effect emerges in graphene at unprecedentedly low light intensities, thus opening avenues to new nonlinear physics and applications in optical technology. Specifically, we theoretically investigate saturable absorption in extended graphene by developing a semianalytical nonperturbative single-particle approach, describing electron dynamics in the atomically-thin material using the two-dimensional Dirac equation for massless Dirac fermions, which is recast in the form of generalized Bloch equations. By solving the electron dynamics nonperturbatively, we account for both interband and intraband contributions to the intensity-dependent saturated conductivity and conclude that the former dominates regardless of the intrinsic doping state of the material. We obtain results in qualitative agreement with atomistic quantum-mechanical simulations of graphene nanoribbons including electron-electron interactions, finite-size, and higher-band effects. Remarkably, such effects are found to affect mainly the linear absorption, while the predicted saturation intensities are in good quantitative agreement in the limit of extended graphene. Additionally, we find that the modulation depth of saturable absorption in graphene can be electrically manipulated through an externally applied gate voltage. Our results are relevant for the development of graphene-based optoelectronic devices, as well as for applications in mode-locking and random lasers.

Figure 1

(a) Schematic representation of a light wave (angular frequency $\omega$, amplitude ${\mathbf{E}}_0$, and intensity $I_0$) normally impinging on a self-standing graphene sheet. (b) Illustration of the temporal displacement of the Dirac cone along the $k_x$ direction due to the oscillatory optical field. The central cone represents the unperturbed conical dispersion of graphene (${\varepsilon }_{\pm }$), while left and right cones indicate the maximum achievable displacement.

Figure 2

Intraband surface current density $J_{\mathrm{intra}}\left(t\right)$ (lower panel) induced in extended graphene by a harmonic external electric field (upper panel) with maximum amplitude $2E_0=\sqrt{8\pi I_0/c}$ and optical frequency $\omega =2\pi c/\lambda$, where $\lambda =1550\mathrm{nm}$ is the optical excitation wavelength. In the lower panel we provide results for incident intensities $I_0=10$, 100, and $1000\mathrm{GW}/{\mathrm{cm}}^2$, compared with the maximum achievable current density $J_{\max }\left(t\right)$ in the $I_0\to \infty$ limit (dashed curve). We assume a high Fermi energy $E_{\mathrm{F}}=1$ eV and zero temperature in order to illustrate the intraband contribution to the current when it is dominant over the interband one ($2E_{\mathrm{F}}/\hslash \omega =2.5$).

Figure 3

Intraband absorption coefficient ${\alpha }_{\mathrm{intra}}$ as a function of incident light intensity $I_0$ for (a) several Fermi energies ($E_{\mathrm{F}}/\hslash \omega =0.25,0.5,0.75,1$) at zero temperature and (b) several electron temperatures ($k_{\mathrm{B}}T/\mu =0.22,0.43,0.65,0.86,1.08$) and fixed chemical potential $\mu =0.4\mathrm{eV}$. The light wavelength is $1550\mathrm{nm}$.

Figure 4

(a) Out-of-equilibrium carrier occupation distribution $f_{\mathrm{c}}\left(\mathbf{k}\right)$ in 2D $\mathbf{k}$ space at $T=300\mathrm{K}$ for a chemical potential $\mu =0.2\mathrm{eV}(\mu /\hslash \omega =0.25$). The electron wave number components $k_x$ and $k_y$ are normalized to $k_{\mathrm{F}}=\mu /\hslash v_{\mathrm{F}}$. (b) Cut along the $k_y$ axis of (a) for several chemical potentials $\mu =0.1\text{–}0.4\mathrm{eV}(\mu /\hslash \omega =0.125,0.250,0.375,0.500$), with $k_y$ scaled to $k_{\mathrm{S}}=\omega /2v_{\mathrm{F}}$. The light intensity and wavelength are $I_0=10\mathrm{GW}/{\mathrm{cm}}^2$ and $\lambda =1550\mathrm{nm}$ in both plots. We assume a phenomenological relaxation time $\tau =22\mathrm{fs}$.

Figure 5

Interband absorption coefficient ${\alpha }_{\mathrm{inter}}$ as a function of incident light intensity $I_0$ for a wavelength of $1550\mathrm{nm}$ for (a) several Fermi energies ($E_{\mathrm{F}}/\hslash \omega =0,0.250,0.375,0.500,0.625$) at zero temperature and (b) several electron temperatures ($k_{\mathrm{B}}T/\mu =0.13,0.43,0.86,1.29,1.72,2.15$) and fixed chemical potential $\mu =0.2$ eV. The inelastic relaxation time is set to $\tau =22\mathrm{fs}$ in all plots.

Figure 6

Dependence of the interband absorption coefficient ${\alpha }_{\mathrm{inter}}$ of undoped graphene on incident light intensity $I_0$ and inelastic relaxation time $\tau$ at zero temperature. The white dashed curve represents the saturation intensity $I_{\mathrm{S}}^{\mathrm{inter}}\left(\tau \right)$ for which ${\alpha }_{\mathrm{inter}}\left(I_{\mathrm{S}}^{\mathrm{inter}}\right)={\alpha }_{\mathrm{inter}}\left(0\right)/2$. The light wavelength is $1550\mathrm{nm}$.

Figure 7

Predicted interband saturation intensity $I_{\mathrm{S}}^{\mathrm{inter}}$ (left scale) as a function of optical wavelength $\lambda$ (horizontal axis) and relaxation time $\tau$ (color scale) at zero temperature and vanishing Fermi energy $E_{\mathrm{F}}=0$. Available experimental results for the saturation intensity are indicated by symbols [10, 11, 12, 13, 14, 15, 16, 17, 18] (see legend).

Figure 8

Comparison between numerical atomistic simulations of a ribbon with zigzag edges and width $W=20$ nm (solid curves) and the semianalytical MDF model for the total absorption coefficient ($\alpha ={\alpha }_{\mathrm{intra}}+{\alpha }_{\mathrm{inter}}$, dashed curves) as a function of incident light intensity $I_0$ for a wavelength of $1550\mathrm{nm}$ and several Fermi energies ($E_{\mathrm{F}}/\hslash \omega =0.250,0.375,0.500,0.625$) in the vanishing temperature limit. In the atomistic calculations the incident light is polarized across the ribbon.

Figure 9

Results from atomistic simulations of graphene ribbons depicting the dependence of the absorption coefficient $\alpha$ either (a) on ribbon width $W$ [for undoped graphene ($E_{\mathrm{F}}=0$) and both zigzag and armchair edges] and (b) on impinging light with intensity $I_0$ (polarized across the ribbon) for a fixed Fermi energy $E_{\mathrm{F}}=0.2$ eV and several values of $W$. The dashed line in (a) indicates the universal absorption limit $\alpha \approx \pi /137$.

Figure 10

Fitted analytical expression for the intensity-dependent absorption coefficient (curves) $\alpha \left(I_0\right)={\alpha }_{\mathrm{ns}}+{\alpha }_0/\sqrt{(1+3I_0/I_{\mathrm{S}})}$ of undoped graphene ($E_{\mathrm{F}}=0$) compared with experiments (symbols) at two different optical wavelengths: $\lambda =800\mathrm{nm}$ (red curves and symbols, ${\alpha }_{\mathrm{ns}}=0.008$) and $\lambda =1550\mathrm{nm}$ (blue curves and symbols, ${\alpha }_{\mathrm{ns}}=0.004$). Experimental results are taken from Refs. [11] (rhombic markers) and [18] (circle markers).