Polariton interactions in microcavities with atomically thin semiconductor layers

We investigate the interactions between exciton-polaritons in $N$ two-dimensional semiconductor layers embedded in a planar microcavity. In the limit of low-energy scattering, where we can ignore the composite nature of the excitons, we obtain exact analytical expressions for the spin-triplet and spin-singlet interaction strengths, which go beyond the Born approximation employed in previous calculations. Crucially, we find that the strong light-matter coupling enhances the strength of polariton-polariton interactions compared to that of the exciton-exciton interactions, due to the Rabi coupling and the small photon-exciton mass ratio. We furthermore obtain the dependence of the polariton interactions on the number of layers $N$, and we highlight the important role played by the optically dark states that exist in multiple layers. In particular, we predict that the singlet interaction strength is stronger than the triplet one for a wide range of parameters in most of the currently used transition metal dichalcogenides. This has consequences for the pursuit of polariton condensation and other interaction-driven phenomena in these materials.

Figure 1

(a) Schematic illustration of the microcavity structure with $N$ embedded monolayers, where $N=4$. A polariton consists of a cavity photon (shaded red) and a superposition of $N$ in-phase 2D excitons (blue ellipses), where the relative phase of each exciton is represented by an in-plane arrow. (b) Example of a scattering process involving intermediate dark states which are uncoupled to light. (c) Polariton dispersion (red) at zero detuning, together with the uncoupled cavity photon and exciton modes (dashed black lines). For high momenta $k>{\epsilon }_X/\hslash c$ (shaded region), where ${\epsilon }_X$ is the exciton energy and $c$ is the speed of light, the exciton is far detuned in energy from the photon and thus essentially uncoupled. We use the parameters for ${\mathrm{MoSe}}_2$ (see Table 1) with ${\epsilon }_X=1.66$ eV [21] and $N=4$ layers.

Figure 2

Polariton and exciton triplet elastic scattering as a function of relative momentum. (a) Real and imaginary parts of the full polariton $T$ matrix, Eq. (13), together with the analytical formula in the small photon mass approximation, Eq. (16). (b) Low-energy exciton triplet $T$ matrix $T_{\sigma \sigma }^X\left(k\right)$, Eq. (17). (c) Polariton and exciton $T$ matrices on a larger momentum scale. We use the ${\mathrm{MoSe}}_2$ parameters with $m_X=1.14m_0$ [58], $m_C={10}^{-5}{m}_0$ [28], $\delta =0$, and $N=1$, giving the inflection wave vector $q_0\simeq 2.29\mu {\mathrm{m}}^{-1}$ [thin vertical line in (a) and (c)].

Figure 3

Polariton interactions in several 2D materials. (a) and (e) Monolayer triplet interactions ${\alpha }_1$ and the corresponding singlet-triplet ratio ${\alpha }_2/{\alpha }_1$ as a function of the exciton-photon detuning $\delta$. (b)–(d) Multilayer triplet interactions as a function of $N$ for different $\delta$, and (f)–(h) the corresponding singlet-triplet ratio. Parameters are taken from Table 1, with $m_0$ the free electron mass.