Linear Scaling of the Exciton Binding Energy versus the Band Gap of Two-Dimensional Materials

The exciton is one of the most crucial physical entities in the performance of optoelectronic and photonic devices, and widely varying exciton binding energies have been reported in different classes of materials. Using first-principles calculations within the $GW$–Bethe-Salpeter equation approach, here we investigate the excitonic properties of two recently discovered layered materials: phosphorene and graphene fluoride. We first confirm large exciton binding energies of, respectively, 0.85 and 2.03 eV in these systems. Next, by comparing these systems with several other representative two-dimensional materials, we discover a striking linear relationship between the exciton binding energy and the band gap and interpret the existence of the linear scaling law within a simple hydrogenic picture. The broad applicability of this novel scaling law is further demonstrated by using strained graphene fluoride. These findings are expected to stimulate related studies in higher and lower dimensions, potentially resulting in a deeper understanding of excitonic effects in materials of all dimensionalities.

Figure 1

Side (upper panel) and top view (lower panel) of (a) phosphorene and (b) graphene fluoride. Each unit cell is indicated by the solid lines. (c),(d) The QP band gap as a function of the inverse spatial separation $1/L_z$ for phosphorene and graphene fluoride, respectively. In (c) and (d), the dots indicate the results of the $G_0{W}_0$, $G_2{W}_0$, and BSE calculations, while the dashed lines represent the corresponding extrapolations.

Figure 2

(a) Calculated QP band structure and (b) optical absorption spectrum of phosphorene. The absorption spectrum for incident light is along the armchair direction. In (a), the zero energy is set to the valence band maximum. In (b), the positions of the first absorption peak and the QP band gap are denoted by the dotted line and the arrow, respectively.

Figure 3

(a) Calculated QP band structure and (b) optical absorption spectrum of graphene fluoride.

Figure 4

The exciton binding energy ($E_b$) versus the QP band gap ($E_g$) for various representative 2D materials. The dashed line represents the fitted linear relation in the form of $E_b=\alpha E_g+\beta$, with $\alpha =0.21$ and $\beta =0.40$. The fitted data are denoted by the filled symbols, while the unfilled data point for strained graphene fluoride helps to demonstrate the broad applicability of the scaling relationship.