Excitons in van der Waals heterostructures: The important role of dielectric screening

The existence of strongly bound excitons is one of the hallmarks of the newly discovered atomically thin semiconductors. While it is understood that the large binding energy is mainly due to the weak dielectric screening in two dimensions, a systematic investigation of the role of screening on two-dimensional (2D) excitons is still lacking. Here we provide a critical assessment of a widely used 2D hydrogenic exciton model, which assumes a dielectric function of the form $\epsilon \left(q\right)=1+2\pi \alpha q$, and we develop a quasi-2D model with a much broader applicability. Within the quasi-2D picture, electrons and holes are described as in-plane point charges with a finite extension in the perpendicular direction, and their interaction is screened by a dielectric function with a nonlinear $q$ dependence which is computed ab initio. The screened interaction is used in a generalized Mott-Wannier model to calculate exciton binding energies in both isolated and supported 2D materials. For isolated 2D materials, the quasi-2D treatment yields results almost identical to those of the strict 2D model, and both are in good agreement with ab initio many-body calculations. On the other hand, for more complex structures such as supported layers or layers embedded in a van der Waals heterostructure, the size of the exciton in reciprocal space extends well beyond the linear regime of the dielectric function, and a quasi-2D description has to replace the 2D one. Our methodology has the merit of providing a seamless connection between the strict 2D limit of isolated monolayer materials and the more bulk-like screening characteristics of supported 2D materials or van der Waals heterostructures.

Figure 1

Sketch of the (a) pure 2D and (b) quasi-2D Coulomb interaction. In the latter case the point charges can be thought of as lines of charge extending along the thickness of the material.

Figure 2

Macroscopic dielectric functions for (a) hBN and (b) ${\mathrm{MoS}}_2$. The bulk(black), along with the Q2D (green) and 2D (blue) static dielectric functions are illustrated, the latter corresponding to the linear fits in the small $q_{\parallel }$ region. For these calculations the $q_{\parallel }$ values are taken along the $\mathrm{\Gamma }-K$ direction, but the homogeneity of the materials has been numerically verified. The parameters used in the linear response ab initio calculation are discussed in Sec. 5.

Figure 3

$z$ dependence of the total potentials (solid lines) coming from external perturbations (dashed lines) at different in-plane wave vectors in the case of hBN [(a) and (c)] and ${\mathrm{MoS}}_2$ [(b) and (d)]. Left panels: The external perturbation is generated by a step function density distribution (insets). Right panels: The external perturbation is generated by the actual hole out-of-plane density distribution (insets), which is calculated as ${\rho }_h\left(z\right)={\int }_{A}d{\mathbf{r}}_{\parallel }{\left|{\psi }_{vK}({\mathbf{r}}_{\parallel },z)\right|}^2$, with $v$ indicating the valence band and $K$ the high symmetry point of the first Brillouin zone. In all cases the density distributions are normalized to 1.

Figure 4

Valence (red) and conduction (green) band densities for (a) hBN and (b) ${\mathrm{MoS}}_2$ calculated at the $K$ point.

Figure 5

Macroscopic dielectric functions for (a) hBN and (b) ${\mathrm{MoS}}_2$. The different dielectric functions are calculated with the three different approaches explained in the text: dielectric function from actual electron and hole distributions (magenta), dielectric function from step function distributions (cyan), and Q2D dielectric function (green). The vertical line represent the radius of the exciton in reciprocal space.

Figure 6

Screened Q2D and 2D interaction energies for (a) hBN and (b) ${\mathrm{MoS}}_2$. The interactions are calculated numerically starting from the macroscopic dielectric functions in Fig. 2 and using Eqs. (17) and (18) respectively. The bare Q2D curves are calculated using the first equation but neglecting the screening.

Figure 7

Variation of the macroscopic dielectric function and effective interaction energy in ${\mathrm{MoS}}_2$ due to the change in the thickness $d$ of the averaging region in the Q2D model. The continuous black lines are relative to $d=6.29\AA$ (the interlayer distance in the bulk), while the dashed lines delimiting the shaded region are calculated with a variation of $\pm 10\%$ in $d$.

Figure 8

Convergence plot for the binding energy obtained from the BSE solution against the inverse of the $k$-points density. Extrapolation to infinite $k$-point sampling is shown. In the BSE the exchange contribution is left out. The horizontal dashed lines show the results given by the Q2D model.

Figure 9

LDA band structure and exciton weights for (a) hBN and (b) ${\mathrm{MoS}}_2$. In both materials the exciton is well localized at the $K$ point. The excitonic weights are calculated as the absolute value squared of the eigenvector of the two-particle BSE Hamiltonian associated with the lowest bound exciton. In red the parabolic bands used in the Mott-Wannier model. The values for the electron and hole masses are $0.93$ a.u. and $0.62$ a.u. for hBN and $0.61$ a.u. and $0.49$ a.u. for ${\mathrm{MoS}}_2$.

Figure 10

Left panels: The on-top (a) and sandwich (c) arrangements of the ${\mathrm{MoS}}_2$/hBN heterostructures. Right panels: Effective dielectric function (full line) for the on-top (b) and sandwich (d) configurations. The linear approximations to the dielectric function is shown by dashed lines. The shaded regions in (b) and (d) represent the range of inverse exciton radii found for the considered structures. The $q$ values below these regions are relevant for screening the electron-hole interaction, and for the thicker structures this region extends beyond the linear regime of $\epsilon \left(q\right)$.

Figure 11

Energy and radius of the lowest bound exciton for the [(a) and (c)] on-top and [(b) and (d)] sandwich configuration as function of the number of hBN layers obtained from the Q2D (green) and 2D (blue) approaches.

Figure 12

(a) Effective dielectric function (full line) and (b) energy of the lowest bound exciton for the on-top ${\mathrm{MoS}}_2$-hBN configuration, calculated with the QEH model based on a 2D description of the layers. The linear approximations to the dielectric function is shown by dashed lines in panel (a), along with the range of inverse exciton radii found for the considered structures represented by the shaded region.

Figure 13

Energy of the lowest bound exciton for ${\mathrm{MoS}}_2$ incapsulated in ${\mathrm{MoS}}_2$ layers as function of the total number of ${\mathrm{MoS}}_2$ layers obtained from the Q2D (green) and 2D (blue) approaches. With the Q2D approach we can clearly see the transition from the monolayer exciton to the bulk one.