Measurement of the Optical Conductivity of Graphene

Optical reflectivity and transmission measurements over photon energies between 0.2 and 1.2 eV were performed on single-crystal graphene samples on a ${\mathrm{SiO}}_2$ substrate. For photon energies above 0.5 eV, graphene yielded a spectrally flat optical absorbance of $(2.3\pm 0.2)\%$. This result is in agreement with a constant absorbance of $\pi \alpha$, or a sheet conductivity of $\pi e^2/2h$, predicted within a model of noninteracting massless Dirac fermions. This simple result breaks down at lower photon energies, where both spectral and sample-to-sample variations were observed. This “nonuniversal” behavior is explained by including the effects of doping and finite temperature, as well as contributions from intraband transitions.

Figure 1

Experimental data for the reflection from a graphene monolayer on the ${\mathrm{SiO}}_2$ substrate for photon energies in the range of 0.5–1.2 eV. The curves associated with the left vertical axis are the measured reflection spectra of the substrate alone (lower curve) and of the graphene monolayer on the substrate (upper curve). The curve associated with the right vertical axis represents the fractional change in reflectance from the graphene monolayer.

Figure 2

Absorption spectra for three different samples of graphene over the range of photon energies between 0.5 and 1.2 eV. The left scale gives the absorbance in units of $\pi \alpha$, while the right scale gives the corresponding optical sheet conductivity in units of $\frac{\pi }{4}{G}_0=6.08\times {10}^{-5}\mathrm{S}$. The black horizontal line corresponds to the universal result of an absorbance of $\pi \alpha =2.293\%$ with a range indicated of $\pm 0.1$ $\pi \alpha$ or approximately $\pm 0.2\%$.

Figure 3

Calculated graphene absorption spectra from 0 to 1.2 eV, including both the inter- and intraband contributions at a temperature of 300 K. The calculation is based on a model of noninteracting massless fermions. Results are shown for different values of the chemical potential $\mu$ and scattering rate $\Gamma$.

Figure 4

Measured graphene absorption spectra of samples 1 and 2 over a range of photon energies between 0.25 and 0.8 eV. The smooth lines are based on the theory shown in Fig. 3, with $\mu =200$ and 100 meV for samples 1 and 2, respectively.