Scaling Universality between Band Gap and Exciton Binding Energy of Two-Dimensional Semiconductors

Using first-principles $GW$ Bethe-Salpeter equation calculations and the $\mathbf{k}·\mathbf{p}$ theory, we unambiguously show that for two-dimensional (2D) semiconductors, there exists a robust linear scaling law between the quasiparticle band gap ($E_g$) and the exciton binding energy ($E_b$), namely, $E_b\approx E_g/4$, regardless of their lattice configuration, bonding characteristic, as well as the topological property. Such a parameter-free universality is never observed in their three-dimensional counterparts. By deriving a simple expression for the 2D polarizability merely with respect to $E_g$, and adopting the screened hydrogen model for $E_b$, the linear scaling law can be deduced analytically. This work provides an opportunity to better understand the fantastic consequence of the 2D nature for materials, and thus offers valuable guidance for their property modulation and performance control.

Figure 1

Linear relationship between $E_g$ and $E_b$. Calculations cover almost all the popular 2D monolayer semiconductors including TMDs [monolayer ${\mathrm{MoS}}_2$, ${\mathrm{MoSe}}_2$, ${\mathrm{WS}}_2$, ${\mathrm{SnS}}_2$, and ${\mathrm{TiS}}_3$], graphene derivatives [graphane (CH), chlorinated graphene (CCl), fluorinated graphene (CF), and fluorinated graphene under external field with 10% pressure ($\mathrm{CF}-10\%$) or 10% tension ($\mathrm{CF}+10\%$)], IV/III–V compounds [SiC, GaN, AlN, and BN in the honeycomb lattice], black phosphorus, functionalized MXenes [15] [${\mathrm{Ti}}_2{\mathrm{CO}}_2$, ${\mathrm{Zr}}_2{\mathrm{CO}}_2$, and ${\mathrm{Sc}}_2{\mathrm{CF}}_2$] and topological materials Sn [16], SnF [16], PbTe [17], single-quintuple layer ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, and ${\mathrm{GaBiCl}}_2$ [18]. The exciton is determined by the lowest peak in absorption spectrum with the only exception of ${\mathrm{SnS}}_2$ where the first dark exciton is used. The “minimum direct band gap” is used for the indirect gap materials.

Figure 2

Comparison of the results between analytical model and first principles calculations. $E_g$ (left) and $E_b$ (right) as a function of ${\alpha }_{2D}$. The brown lines are from analytical model of Eqs. (12) and (14) while the scattered points are from the $GW$-BSE calculations. They agree rather well when $E_g<4\mathrm{eV}$.

Figure 3

Illustration of the relationship between the energy gap and exciton. For a system with a large $E_g$, the screening is weak and the exciton binds strongly, giving rise to a relatively narrow spatial extension and high $E_b$. In contrast, for a system with a small $E_g$, the screening is strong and the exciton binds loosely, giving rise to a relatively wide spatial extension and a low $E_b$.

Figure 4

Dependence of $E_g$ as the functions of (a) electron effective mass $m_e^{*}$ and (b) reduced mass $\mu$. Only the data of 14 2D materials are presented in (a) due to that the scalar $m_e^{*}$ is ill-defined for the cases with emerging anisotropy and degeneracy at the conduction band minimum. In (b), apart from the well-defined $\mu$ for Sn, PbTe, ${\mathrm{MoS}}_2$, ${\mathrm{WS}}_2$, and ${\mathrm{MoSe}}_2$, the filled and half filled diamonds, respectively, correspond to the $\mu$ calculated from heavy and light holes owing to the twofold degeneracy at the valence band maximum.