Noninvasive control of excitons in two-dimensional materials

We investigate how external screening shapes excitons in two-dimensional semiconductors embedded in laterally structured dielectric environments. An atomic scale view of these elementary excitations is developed using models which apply to a variety of materials including transition metal dichalcogenides. We find that structured dielectrics imprint a peculiar potential energy landscape on excitons in these systems: While the ground-state exciton is least influenced, higher excitations are attracted towards regions with high dielectric constant of the environment. This landscape is “inverted” in the sense that low energy excitons are less strongly affected than their higher energy counterparts. Corresponding energy variations emerge on length scales of the order of a few unit cells. This opens the prospect of trapping and guiding of higher excitons by means of tailor-made dielectric substrates on ultimately small spatial scales.

Figure 1

Sketches of the electronic quasiparticle and excitonic energies in a homogeneous 2D semiconducting monolayer (blue) embedded in different dielectric environments. (a) A homogeneous environment with macroscopic dielectric constant $\varepsilon$. (b) A laterally structured dielectric environment with different dielectric constants ${\varepsilon }_1$ and ${\varepsilon }_2$. The heterogeneous environment sketched in (b) imprints a potential energy landscape as calculated in Ref. [32] on electronic quasiparticles (c) and electron-hole excitations/excitons (d). Higher energy excitons are attracted towards regions with higher dielectric constant. The lowest-energy excitons are affected more weakly and can be either attracted towards or expelled from regions with higher dielectric constant. Therefore, nanostructured dielectric substrates allow for the realization of potential energy landscape for excitons as shown in (e). Under optically pumping, these can yield “inverted” regions where higher energy excitonic states have a higher occupation than the ones at lower energies.

Figure 2

Schematic illustration of competing contributions to the excitonic gap $E_g^{\text{exc}}$. The single-particle band gap $E_g^0$ is widened to the dressed quasiparticle band gap $E_g^{\text{HF}}$ due to the exchange self-energy ${\mathrm{\Sigma }}^{\text{HF}}$ which results from the ground-state electron-electron interaction. The optically measurable excitonic gap is reduced as compared to $E_g^{\text{HF}}$ due to the electron-hole binding energy $E_{\text{B}}$.

Figure 3

Linear optical absorption spectra and total excitonic density of states for different dielectric constants of the environment of the monolayer for the localized (a) and delocalized (b) hexagonal bilayer (left panels) and honeycomb lattice (right panels). In the insets of the two-particle DOS the lowest-energy excitations are shown in more detail. High $\varepsilon$ correspond to small Coulomb interactions leading in our model systems to the description of independent particles for $\varepsilon =\infty$.

Figure 4

Local two-particle DOS for $\varepsilon =2$ for different models and in-plane hopping parameters $t$. A smaller $t$ describes more strongly localized electrons. The quasiparticle (QP) limit corresponds to the case of vanishing electron-hole interaction ($V_{ijkl}^{ehhe}=0$) but electron-electron interaction being included via the Hartree-Fock self-energy. In addition, the noninteracting (NI) limit $\varepsilon \to \infty$ is depicted. As the QP and the NI limit show no spatial dependence, the corresponding two-particle DOS are spatially averaged. Note that in the “localized” panels a broadening of $\mathrm{\Gamma }=0.01$ eV and in the “delocalized” panels $\mathrm{\Gamma }=0.1$ eV was used.

Figure 5

(a) Energetic shift $\delta E=E_g^{\text{exc}}-E_g^0$ of the lowest-energy exciton $E_g^{\text{exc}}$ with respect to the noninteracting band gap $E_g^0$ as a function of the inverse environmental dielectric constant $1/\varepsilon$. (b) The Bohr radius $a$ of the lowest-energy exciton as a function of $1/\varepsilon$. The gray line marks the lattice constant.

Figure 6

One-particle local DOS for different unit cells along a line perpendicular to the dielectric interface of the environment located in the unit cell 5 (left ${\varepsilon }_1=5$ and right ${\varepsilon }_2=2$).

Figure 7

Local two-particle DOS for a heterogeneous dielectric environment with ${\varepsilon }_1=5$ and ${\varepsilon }_2=2$. The $x$-axis marks the used hole positions whereas the red dot in the picture reveals the chosen electron positions for each subfigure. The quasiparticle limit (leftmost panel) shows all possible electron-hole excitations without electron-hole interaction.

Figure 8

The 2D semiconductors are modeled using two different tight-binding models labeled “hexagonal bilayer” and “honeycomb lattice”, where the schematic side and top views of the model illustrate the hopping matrix elements.