Universal infrared absorbance of two-dimensional honeycomb group-IV crystals

We show that the low-frequency absorbance of undoped graphene, silicene, and germanene has a universal value, only determined by the Sommerfeld fine-structure constant. This result is derived by means of ab initio calculations of the complex dielectric function for optical interband transitions applied to two-dimensional (2D) crystals with honeycomb geometry. The assumption of chiral massless Dirac fermions is not necessary. The low-frequency absorbance does not depend on the group-IV atom, neither on the sheet buckling nor on the orbital hybridization. We explain these findings via an analytical derivation of the relationship between absorbance and fine-structure constant for 2D Bloch electrons. The effect of deviations of the electronic bands from linearity is also discussed.

Figure 1

Interband transition energies along high-symmetry lines in the BZ for graphene (a), silicene (b), and germanene (c). The red horizontal lines indicate energies of van Hove singularities which give peak structures in the absorbance in Figs. 2 and 5. The resulting joint densities of states $D\left(\omega \right)$ are displayed in addition [in units of (eV)${}^{-1}$].

Figure 2

Ab initio calculated optical absorbance of graphene (black solid line), silicene (red dashed line), and germanene (blue dotted line) $vs$ photon energy. The absorbance is normalized to $\pi \alpha$. The reduced infrared absorbance is depicted in the inset vs photon-energy square.

Figure 3

(a) Transition matrix elements of the pure $\pi -{\pi }^{*}$ transitions along high-symmetry lines in the BZ for graphene (black solid line), silicene (red dashed line), and germanene (blue dotted line). The longitudinal representation (2) has been used in the numerical calculations. (b) For illustration, the $\pi$ and ${\pi }^{*}$ bands that are involved in the optical transitions are also shown, indicated by vertical arrows.

Figure 4

Wave-function squares in silicene for the highest occupied $\pi$ state (a) and the lowest unoccupied ${\pi }^{*}$ state (b) at $K$. The atomic positions in the isolated Si sheet indicate the buckled honeycomb geometry.

Figure 5

Spectral absorbance (in units of $\pi \alpha$) for graphene (black solid line), silicene (red dashed line), and germanene (blue dotted line).

Figure 6

The frequency-dependent absorbance for (a) graphene, (b) silicene, and (c) germanene. Besides the longitudinal gauge (2) (black solid line) also the transversal gauge (3) (red solid line) has been used.