Optoelectronic properties of graphene in the presence of optical phonon scattering

We study in detail the optoelectronic properties of graphene. Considering the electron interactions with photons and phonons, we employ the mass- and energy-balance equations to self-consistently evaluate the photoinduced carrier densities, the optical conductance, and the transmission coefficient in the presence of a linearly polarized radiation field. We demonstrate that the photoinduced carrier densities increase around the electron-photon-phonon resonant transition. They depend strongly on the radiation intensity and frequency, temperature, and dark carrier density. For short-wavelength radiation $\left(\mathcal{L}<3\mu \text{m}\right)$, we obtain the universal optical conductance ${\sigma }_0=e^2/\left(4\hslash \right)$. Importantly, there exists an optical-absorption window in the radiation wavelength range $4–100\mu \text{m}$, which is induced by different transition energies required for interband and intraband optical absorption. The position and width of this window depend sensitively on the temperature and the carrier density of the system. These theoretical results are in line with recent experimental findings and indicate that graphene exhibits important features not only in the visible regime but also in the midinfrared bandwidth.

Figure 1

Photoexcited electron density $\Delta n_e$ as a function of the radiation wavelength, at temperature $T=150\text{K}$ and a fixed dark electron density $n_0=5\times {10}^{11}{\text{cm}}^{-2}$, for different strengths of the radiation field $F_0$. The inset shows $\Delta n_e$ as a function of $F_0$ at a fixed radiation wavelength $\mathcal{L}=6.0\mu \text{m}$. Note that $\Delta n_e$ is equal to $n_h$, the photoinduced hole density.

Figure 2

Photoexcited electron density $\Delta n_e$ as a function of the radiation wavelength, for fixed dark electron density $n_0=1\times {10}^{12}{\text{cm}}^{-2}$ and radiation intensity $F_0=500\text{V}/\text{cm}$, at different temperatures.

Figure 3

$\Delta n_e$ as a function of the radiation wavelength at a temperature $T=150\text{K}$ and fixed strength of the radiation field $F_0=500\text{V}/\text{cm}$, for different dark electron densities.

Figure 4

Contribution to the total optical conductance (solid line) from different transition channels at room temperature for a fixed dark electron density $n_0=1\times {10}^{12}{\text{cm}}^{-2}$. Here ${\sigma }_0=e^2/\left(4\hslash \right)$ and ${\sigma }_{-+}$ (dashed curve) is for transition from valence band $(-)$ to conduction band $(+)$. The curves for intraband transitions, ${\sigma }_{++}$ and ${\sigma }_{--}$, coincide roughly.

Figure 5

(a) A graphene system in the absence of the radiation field $\left(F_0=0\right)$. The conducting carriers are electrons with a Fermi energy ${\mu }_e^{\ast }$ in the conduction band. The hatched area shows the occupied states. (b) Optical-absorption channels in the presence of a radiation field $\left(F_0\ne 0\right)$. Here ${\mu }_e^{\ast }$ and ${\mu }_h^{\ast }$ are the quasi-Fermi energies for, respectively, electrons and holes and there are three optical-absorption channels: ${\alpha }_{-+}$, ${\alpha }_{++}$, and ${\alpha }_{--}$. $\hslash {\omega }_0$ is the energy of an emitted optical phonon.

Figure 6

Optical conductance and transmission coefficient (inset) as a function of the radiation wavelength, at a fixed dark electron density $n_0=1\times {10}^{12}{\text{cm}}^{-2}$, for different temperatures $T=10\text{K}$ (solid curve), 77 K (dashed curve), 150 K (dotted curve), and 300 K (dotted-dashed curve).

Figure 7

Optical conductance and transmission coefficient (inset) as a function of the radiation wavelength, at room temperature, for different dark electron densities $n_0=5\times {10}^{11}{\text{cm}}^{-2}$ (solid curve), $8\times {10}^{11}{\text{cm}}^{-2}$ (dashed curve), and $1\times {10}^{12}{\text{cm}}^{-2}$ (dotted curve).