Numerical investigation of electrostatically confined excitons in monolayer $\text{MoSe}_2$

TL;DR

The study develops an exact-diagonalization framework for excitons in a MoSe$_2$ monolayer confined along one dimension by a gate-induced in-plane field from a p-i-n junction. By combining a center-of-mass/relative-coordinate formulation with the Rytova–Keldysh screened interaction and a confinement potential derived from electrostatic simulations, the authors map the problem onto a finite grid and solve for the full confined spectrum, including excited states. They identify bright (even parity) and dark (odd parity) states, finding that dark states have oscillator strengths reduced by at least an order of magnitude and can vanish in symmetric confinement; the bright spectrum matches recent experiments, while large gate biases lead to potential saturation due to screening, limiting changes to the spectrum. The results provide a theoretical basis for designing confinement schemes and suggest routes to access previously undetected dark states via controlled asymmetry and electric manipulation of the exciton dipole. Overall, the work advances understanding of non-hydrogenic, device-controlled exciton spectra in 2D TMDs and informs future optoelectronic confinement architectures.

Abstract

We investigate exciton confinement to a quantum wire in monolayer $\text{MoSe}_2$ where the confinement is achieved by a p-i-n junction. We employ an effective-mass exciton model and solve the problem numerically, reflecting device geometries found in experimental state-of-the-art set up. Our method allows us to investigate the entire spectrum of confined states. We show the emergence of quantum confinement and study the dependence of the confined states as a function of electrical gate voltages, which are experimentally tunable parameters. We find that the confined states can be divided into bright and dark states with the dark states having small but finite oscillator strengths. Their oscillator strengths are low enough that they have not yet been detected in experiments, whereas the spectrum of the bright exciton states reproduces recent experimental measurements. Our results provide insight into the theoretical background of confined exciton states beyond the ground state and pave the way for the development of new confinement schemes as well as avenues to access the previously not detected dark states.

Figures (9)

• Figure 1: Schematic representation of the simulated experimental set-up. The TMD monolayer is sandwiched between two layers of h-BN and electrical gates on top (TG) and bottom (BG). Different bias voltages $V_{\text{TG}},V_{\text{BG}}$ applied to the gates combined with the asymmetric spatial extent of the electric gates creates an electrical potential difference in x-direction.
• Figure 2: Subset of in-plane electrostatic potentials $V_{\text{es}}$ obtained through COMSOL simulations of the setup shown in Fig. \ref{['fig:set_up']}, with an hBN thickness of 30nm per slab. For convenience the potential is set to $V_\text{es}(x_0)=0~\eV$, where $x_0$ is the midpoint of the step.
• Figure 3: a) Oscillator strength and corresponding energy of the confined states relative to the unconfined 2D state for a $V_\text{TG}$ range of $[-32\V,0\V]$. The four emerging discrete lines are the bright states with even $n$. The energy levels of the states converge to constant values for large $V_\text{TG}$, due to the minimal changes in the confining potential in the large voltage regime, as seen in Fig. \ref{['fig:potential']}. b) Zoom into the region of $V_\text{TG} \in [–2\V,-7\V]$. We observe the emergence of the odd states with faint oscillator strength. The linewidths in these plots have no physical meaning and are purely for the purpose of visibility, while each horizontal line represents a single data point in our parameter space their length is also arbitrarily chosen for visibility.
• Figure 4: Relative oscillator strength (logarithmic scale) for the bright (dashed) and dark states (dotted). The bright states have oscillator strengths that are an order of magnitude or more stronger than the dark states. The significant difference is explained by the parity of the corresponding wavefunctions. The dip at $V_\text{TG}=8.62$V is explained by the symmetry of the confining potential around the mid-point of the step, which corresponds to $F_n=0$ for odd n. The symmetric confining potential (shown in Fig. \ref{['fig:potential']}) is caused by equivalent doping densities in the p-doped and n-doped region.
• Figure 5: a)-c): The wavefunctions $\phi_n(R_x, \mathbf{r}=0)$ for selected confining potentials. These functions also act as the integrand in Eq. \ref{['relative']}. Parts with opposite sign in the wavefunction reduce the oscillator strength. In the case of odd $n$ this leads to a significant reduction in oscillator strength leading to the dark states. For $V_\text{TG} = -8.62V$, $\phi_n(R_x, \mathbf{r}=0)$ possesses perfect odd parity and leads to the dip in oscillator strength shown in Fig \ref{['fig:KN']}.