# Energy spectrum of two-dimensional excitons in a non-uniform dielectric medium

## TL;DR

This paper reveals that excitonic energy levels in monolayer transition metal dichalcogenides follow a simple Rydberg-like energy ladder, with the energy spectrum accurately modeled by an analytical Schrödinger equation using a modified potential.

## Contribution

It introduces a simple analytical model for excitonic energy levels in 2D materials, validated by experimental data, and estimates the quantum defect parameter for various monolayers.

## Key findings

- Excitonic states follow a Rydberg-like energy ladder with a small quantum defect.
- The model accurately reproduces experimental magneto-optical data.
- The quantum defect parameter is close to zero for WSe2 and MoS2 monolayers.

## Abstract

We demonstrate that, in monolayers (MLs) of semiconducting transition metal dichalcogenides, the $s$-type Rydberg series of excitonic states follows a simple energy ladder: $\epsilon_n=-Ry^*/(n+\delta)^2$, $n$=1,2,\ldots, in which $Ry^*$ is very close to the Rydberg energy scaled by the dielectric constant of the medium surrounding the ML and by the reduced effective electron-hole mass, whereas the ML polarizability is only accounted for by $\delta$. This is justified by the analysis of experimental data on excitonic resonances, as extracted from magneto-optical measurements of a high-quality WSe$_2$ ML encapsulated in hexagonal boron nitride (hBN), and well reproduced with an analytically solvable Schr\"odinger equation when approximating the electron-hole potential in the form of a modified Kratzer potential. Applying our convention to other, MoSe$_2$, WS$_2$, MoS$_2$ MLs encapsulated in hBN, we estimate an apparent magnitude of $\delta$ for each of the studied structures. Intriguingly, $\delta$ is found to be close to zero for WSe$_2$ as well as for MoS$_2$ monolayers, what implies that the energy ladder of excitonic states in these two-dimensional structures resembles that of Rydberg states of a three-dimensional hydrogen atom.