Transparency of graphene and other direct-gap two-dimensional materials

Graphene and other two-dimensional materials display remarkable optical properties, including a simple transparency of $T\approx 1-\pi \alpha$ for visible light. Most theoretical rationalizations of this “universal” opacity employ a model coupling light to the electron's crystal momentum and put emphasis on the linear dispersion of the graphene bands. However, such a formulation of interband absorption is not allowable within band structure theory, because it conflates the crystal momentum label with the canonical momentum operator. We show that the physical origin of the optical behavior of graphene can be explained within a straightforward picture with the correct use of canonical momentum coupling. Its essence lies in the two-dimensional character of the density of states rather than in the precise dispersion relation, and therefore the discussion is applicable to other systems such as semiconductor membranes. At higher energies the calculation predicts a peak corresponding to a van Hove singularity as well as a specific asymmetry in the absorption spectrum of graphene, in agreement with previous results.

Figure 1

(a) Lattice geometry of graphene and (b) band structure in the tight-binding approximation, with conical dispersion near the Dirac points (circled).

Figure 2

Norm of the momentum matrix element $\langle +,\mathbf{k}\left|\widehat{\mathbf{p}}\right|-,\mathbf{k}\rangle$ as a function of $\mathbf{k}$ in the BZ. The black dots are the Dirac points ${\mathbf{K}}_D$. Lighter (darker) shades indicates higher (lower) values, going to zero at $\mathbf{k}=\mathbf{0}$.

Figure 3

Opacity of graphene calculated with $\mathbf{p}$ substitution (solid red curve, lower panel) and $\mathbf{k}$ substitution (dotted blue curve) as a function of photon energy, and the calculated joint density of states (JDOS) for valence-conductance band transitions as a function of energy difference. For comparison, the measured opacity of graphene from Ref. [4] is also shown with brown squares.