Universal Optical Conductance of Graphite

We find experimentally that the optical sheet conductance of graphite per graphene layer is very close to $(\pi /2)e^2/h$, which is the theoretically expected value of dynamical conductance of isolated monolayer graphene. Our calculations within the Slonczewski-Weiss-McClure model explain well why the interplane hopping leaves the conductance of graphene sheets in graphite almost unchanged for photon energies between 0.1 and 0.6 eV, even though it significantly affects the band structure on the same energy scale. The $f$-sum rule analysis shows that the large increase of the Drude spectral weight as a function of temperature is at the expense of the removed low-energy optical spectral weight of transitions between hole and electron bands.

Figure 1

The real part of the optical sheet conductance of graphite per layer [(a) experiment, (b) calculation)] as well as the calculated conductance of isolated undoped graphene (c). The error bar indicates the inaccuracy of the absolute value due to systematic experimental uncertainties; the relative inaccuracy of the temperature dependence is much smaller. In (b) the dashed lines indicate the interband conductance only, while the solid line represents the total conductance with a Drude peak added (the scattering rate is 5 meV). In (c) the thermally activated Drude peak is not shown. The arrows in (b) correspond to optical transitions indicated in Fig. 3. A certain noise in (b) and (c) is of numerical origin. The inset of (c) depicts the optical transitions between hole and electron bands in monolayer graphene.

Figure 2

In-plane infrared reflectivity spectra of highly ordered pyrolytic graphite. The inset shows the real and the imaginary parts of the dielectric function obtained by ellipsometry at higher energies.

Figure 3

The calculated dispersion of $p_z$ bands in graphite along the $K\mathrm{\text{−}}H$ line (along the $K’\mathrm{\text{−}}H’$ line the dispersion is the same). Different bands are shown with different colors. The arrows indicate interband transitions which form the correspondingly marked peaks in Fig. 1b.

Figure 4

The integrated conductance of graphite (symbols, experiment; solid lines, calculation) and of monolayer undoped graphene (dashed lines, calculation).