Environmentally sensitive theory of electronic and optical transitions in atomically thin semiconductors

We present an electrostatic theory of band-gap renormalization in atomically thin semiconductors that captures the strong sensitivity to the surrounding dielectric environment. In particular, our theory aims to correct known band gaps, such as that of the three-dimensional bulk crystal. Combining our quasiparticle band gaps with an effective-mass theory of excitons yields environmentally sensitive optical gaps as would be observed in absorption or photoluminescence. For an isolated monolayer of ${\mathrm{MoS}}_2$, the presented theory is in good agreement with ab initio results based on the $GW$ approximation and the Bethe-Salpeter equation. We find that changes in the electronic band gap are almost exactly offset by changes in the exciton binding energy such that the energy of the first optical transition is nearly independent of the electrostatic environment, rationalizing experimental observations.

Figure 1

Band structures of bulk and monolayer ${\mathrm{MoS}}_2$ calculated with density functional theory. The direct band gap (at the $K$ point) is 0.09 eV larger for the monolayer than for the bulk due to the carrier confinement effect.

Figure 2

Idealized dielectric slab geometry used to model the electrostatics of atomically thin semiconductors.

Figure 3

The quasiparticle band gap and optical gap (i.e., excitonic transition energy) of monolayer ${\mathrm{MoS}}_2$ as a function of (a) the substrate dielectric constant with vacuum above (${\varepsilon }_3=1$) and (b) the encapsulating dielectric constant (${\varepsilon }_2={\varepsilon }_3$). The bulk band gap, which is a fixed parameter in the theory, is indicated by a dotted gray line. The filled circles at ${\varepsilon }_2=1$ indicate the ab initio $G_0{W}_0$ result (the black circle) and the Bethe-Salpeter equation results (the red and blue circles) for an isolated monolayer from Ref. [20].