Exciton-exciton interaction in transition metal dichalcogenide monolayers and van der Waals heterostructures

Due to a strong Coulomb interaction, excitons dominate the excitation kinetics in two-dimensional (2D) materials. While Coulomb scattering between electrons has been well studied, the interaction of excitons is more challenging and remains to be explored. As neutral composite bosons consisting of electrons and holes, excitons show nontrivial scattering dynamics. Here, we study exciton-exciton interaction in transition-metal dichalcogenides and related van der Waals heterostructures on microscopic footing. We demonstrate that the crucial criterion for efficient scattering is a large electron/hole mass asymmetry, giving rise to internal charge inhomogeneities of excitons and emphasizing their cobosonic substructure. Furthermore, both exchange and direct exciton-exciton interactions are boosted by enhanced exciton Bohr radii. We also predict an unexpected temperature dependence that is usually associated with phonon-driven scattering, and we reveal an orders of magnitude stronger interaction of interlayer excitons due to their permanent dipole moment. The developed approach can be generalized to arbitrary material systems and will help to study strongly correlated exciton systems, such as moire super lattices.

Figure 1

(a) Schematic illustration of exciton-exciton scattering in a TMD monolayer (left) and a van der Waals heterostructure (right). While the intralayer interaction is determined by the wave function overlap, the interlayer coupling resembles a dipole-dipole interaction. (b) Exemplary scattering channels in monolayer ${\mathrm{WSe}}_2$ including intravalley (I, blue) and intervalley processes (II, orange).

Figure 2

(a) Schematic illustration of the electronic band structure of monolayer ${\mathrm{WSe}}_2$ in the vicinity of the high symmmetry K, K', and $\mathrm{\Lambda }$ points. In this work, we focus on spin-allowed transitions (yellow lines). (b) Excitonic band structures displaying the energetic ordering of exciton states as a function of the center-of-mass momentum. Note that the momentum-dark KK' and $\mathrm{K}\mathrm{\Lambda }$ states are energetically lower than the bright KK exciton.

Figure 3

Momentum and real space representations of the excitonic Coulomb matrix element in the ${\mathrm{WSe}}_2$ monolayer. We distinguish between direct interactions $\mathcal{V}$, illustrated in (a)-(b), and exchange interactions $\mathcal{U}$, shown in (c)-(d). We consider intravalley and intervalley exciton-exciton scattering [i.e., $\mu =\mathrm{KK}$ and $\nu =$ KK, $\mathrm{K}\mathrm{\Lambda }$, KK', respectively, cf. also Fig. 1] in ${\mathrm{WSe}}_2$. The inset in (d) displays the screening dependence of the exchange interaction element with $\mu =\nu$=KK.

Figure 4

Momentum and real space representations of the excitonic Coulomb matrix element in the ${\mathrm{MoSe}}_2\text{−}{\mathrm{WSe}}_2$ heterostructure. We consider scattering of KK interlayer excitons with other bright interlayer excitons (IX-IX) as well as with KK intralayer excitons (IX-X) and pure intralayer scattering between KK excitons (X-X). Note that the latter is strongly suppressed. The intralayer excitons we consider here reside in the ${\mathrm{WSe}}_2$ layer, and the interlayer excitons are composed from holes from the ${\mathrm{MoSe}}_2$ layer and electrons from the ${\mathrm{WSe}}_2$ layer. The inset in (b) illustrates the screening dependence of the matrix elements.

Figure 5

Temperature and density dependence of excitation-induced dephasing in monolayer ${\mathrm{WSe}}_2$. (a) EID as a function of exciton density $n$ for two different temperatures. (b) EID as a function of temperature, showing single scattering contributions including intravalley scattering (KK) as well as intervalley scattering ($\mathrm{K}\mathrm{\Lambda }$, KK') at fixed exciton density $n={10}^{11}{\mathrm{cm}}^{-2}$. (c) Temperature-dependent density $n\left(T\right)$ of KK, $\mathrm{K}\mathrm{\Lambda }$ and KK' excitons.

Figure 6

Excitation-induced dephasing of interlayer (IX) and intralayer KK excitons (X) in the ${\mathrm{MoSe}}_2\text{−}{\mathrm{WSe}}_2$ heterostructure. (a) Density dependence of intralayer and interlayer EID at room temperature. (b) Temperature-dependent EID at fixed density $n={10}^{11}{\mathrm{cm}}^{-2}$ revealing ${\gamma }^{\mathrm{I}/\mathrm{IX}}\left(T\right)\propto T^{-\frac{1}{2}}$.

Figure 7

Screening dependence of excitation-induced dephasing for the (a) ${\mathrm{WSe}}_2$ monolayer and the (b) ${\mathrm{MoSe}}_2\text{−}{\mathrm{WSe}}_2$ heterostructure for $T=50\mathrm{K}$ and $T=300\mathrm{K}$ and at the fixed exciton density $n={10}^{11}{\mathrm{cm}}^{-2}$. Two of the most common substrates, hBN and $\mathrm{Si}{\mathrm{O}}_2$, have been indicated as vertical dashed lines.