Exciton Polariton-Polariton Interactions in Transition-Metal Dichalcogenides

Abstract

Microscopic insights into nonlinear interactions are essential for advancing polaritonic devices. Existing studies often rely on phenomenological models that overlook important many-body processes. Based on a material-specific and predictive approach, we investigate monolayer and homobilayer MoS$_2$ embedded in a Fabry-Pérot cavity to characterize the exchange, saturation, and dipole-dipole contributions to polariton-polariton interactions in these technologically promising materials. A key finding is that the exchange interaction induces asymmetric energy shifts of the lower and upper polariton branches in a detuned cavity, a behavior driven by the difference in their excitonic character. Furthermore, we demonstrate that temperature and electron-photon coupling determine the energy renormalization through the equilibrium polariton distribution. In homobilayers, the dipole-dipole interaction is mediated by the interlayer character, enabling electrical control and facilitating the electric-field-induced closing of anti-crossings due to dipolar-interaction shifts. The gained insights on polariton-polariton interactions are important for the development of ultra-compact polaritonic circuitry.

Figures (5)

• Figure 1: (a) Schematic of a TMD monolayer in a Fabry-Pérot microcavity and (b) polariton-polariton interactions derived from the constituent exciton interactions: (1) Exchange interactions between identical fermions and (2) saturation due to the Pauli exclusion principle. (c) The TMD homobilayer case with: (1) dipole-dipole repulsion between interlayer excitons and (2) intralayer exciton saturation.
• Figure 2: (a) Lower (LP) and upper (UP) polariton dispersions as a function of in-plane momentum $Q$ in the low-density limit at zero detuning ($\Delta=0$). The exciton/photon character is given by the Hopfield coefficients (color gradient). Gray and purple dashed lines show the bare exciton and cavity photon dispersions, respectively. The orange line denotes the lightcone edge. (b) Schematic of different LP and UP energy shift contributions due to nonlinear interactions for a zero- (top) and red- (bottom) detuned cavity. (c) The polariton energy shift $\Delta E$ is plotted against polariton density $n$ in monolayer MoS$_2$ at room temperature and zero detuning. The dashed line indicates the shift solely from exchange interaction, which is identical for the LP and UP at $\Delta=0$. (d) LP and (e) UP polariton energy shift versus detuning at a fixed polariton density of $n=10^{12}\,\mathrm{cm}^{-2}$ shown for cryogenic (blue) and room (red) temperatures. The horizontal gray line marks the temperature-independent shift in the bare exciton limit, which is purely due to the exchange interaction. (f) The resulting LP-UP energy splitting at the two considered temperatures. While the low-density limit (black line) exhibits a minimum at zero detuning, the inclusion of nonlinear interactions at an elevated density shifts the minimum to positive detunings, corresponding to an effective interaction-induced exciton detuning.
• Figure 3: (a) Polariton energy landscape in a 2H-stacked MoS$_2$ homobilayer as a function of an applied out-of-plane electric field $E_\text{z}$ for a detuning of $\Delta=-50\,\mathrm{meV}$. The solid lines denote the four lowest-energy polariton branches, $P_1$ to $P_4$, with the color gradient showing the excitonic/photonic character. Gray and purple dashed lines indicate the hybrid exciton branches and the cavity photon energy, respectively. (b) Hybrid exciton energies from (a) with the color gradient showing their intra- vs. interlayer character. The energy shifts are driven by the interlayer component, which couples to the electric field due to the out-of-plane dipole moment.
• Figure 4: (a) Impact of interaction-induced shifts on the polariton energy landscape in a 2H-stacked MoS$_2$ homobilayer under an applied out-of-plane electric field $E_\text{z}$. Results are shown for the four lowest polariton branches at a detuning of $\Delta=-50\,\mathrm{meV}$, a temperature of $T=300\,\mathrm{K}$, and a polariton density of $n=10^{12}\,\mathrm{cm^{-2}}$. The colormap shows the polariton absorption. The low-density limit, where the interaction-induced shifts are zero, is denoted by the black dashed lines. (b)-(c) Field-dependent nonlinear energy shifts resolved for different polariton branches at (b) 300 K and (c) 10 K. The gray dashed line illustrates the shift of the predominantly interlayer-like hybrid exciton.
• Figure 5: Nonlinear polariton energy shifts in monolayer MoS$_2$ embedded within a FP cavity at (a) $T=10\,K$ and (b) $T=300\,K$ for zero detuning as a function of the electron-photon coupling strength $g_\text{e}$. The dashed line denotes the shift due to just exchange interaction, which is symmetric for the LP and UP branches. The bare exciton energy shift is marked by the horizontal solid gray line. LP occupation vs. $g_\text{e}$ and momentum $Q$ at (c) $T=10\,K$ and (d) $T=300\,K$. The horizontal line indicates the edge of the lightcone. At low temperatures and coupling strengths of above 15 meV most of the polaritons are concentrated within the lightcone, leading to the sharp reduction in the exchange interaction shift observed in (a).