Retrieval of fundamental material parameters of monolayer transition metal dichalcogenides from experimental exciton energies: An analytical approach

TL;DR

This work addresses the measurement challenge of extracting intrinsic parameters (E_g, r0, mu, kappa) for monolayer TMDCs from exciton spectra by developing an analytical framework based on the Rytova-Keldysh potential. It provides closed-form relations to obtain E_g and r0 (and kappa once mu is known) from zero-field s-state energies, and introduces a one-parameter, analytic method to determine mu from magnetoexciton energies via an E(B, mu) relation with accurate Padé-like corrections. The study applies the method to WSe2, WS2, MoSe2, and MoS2 in hBN, yielding mu ≈ 0.19 m_e and kappa ≈ 4–4.5, with E_g and r0 values that align with or improve upon prior data, and it delivers analytical diamagnetic constants and exciton radii for cross-checks. Overall, the approach reduces computational cost, clarifies the underlying physics, and provides a practical tool for extracting fundamental material properties from exciton spectroscopy in TMDCs.

Abstract

We propose a straightforward and highly accurate method for extracting material parameters such as screening length, bandgap energy, exciton reduced mass, and the dielectric constant of the surrounding medium from experimental magnetoexciton energies available for monolayer transition metal dichalcogenides (TMDCs). Our approach relies on analytical formulations that allow us to calculate the screening length $r_0$ and bandgap energy $E_g$ directly from the experimental $s$-state exciton energies $E_{1s}$, $E_{2s}$, and $E_{3s}$. We also establish a relationship between the surrounding dielectric constant $κ$ and the exciton reduced mass $μ$. This relationship simplifies the Schr{ö}dinger equation for a magnetoexciton in a TMDC monolayer, transforming it into a one-parameter equation that depends solely on the single material parameter $μ$. Furthermore, we develop an analytical formula with high accuracy for magnetoexciton energies as a function of the exciton reduced mass: $E(B,μ)$. Then, the inverse of this formula allows us to calculate the exciton reduced mass from experimental data on magnetoexciton energies. By applying this method, we extract key material parameters, $E_g$, $r_0$, $μ$, and $κ$, from the magnetoexciton energies of monolayer TMDCs, including WSe$_2$, WS$_2$, MoSe$_2$, and MoS$_2$, encapsulated by hexagonal boron nitride (hBN) slabs in various current experiments. The material properties we retrieve complement and correct existing experimental and theoretical data. Additionally, we develop an analytical method for calculating diamagnetic coefficients and exciton radii with high accuracy compared to numerical calculations. Based on this method, we provide diamagnetic coefficients and exciton radii computed using the extracted material parameters.

Figures (5)

• Figure 1: Schematic flowchart of extracting the material parameters (the bandgap energy $E_g$, screening length $r_0$, exciton reduced mass $\mu$, and surrounding dielectric constant $\kappa$) from experimental exciton energies. Here, for the exciton reduced mass in the last step, we use the analytical formulation \ref{['eq17']} and \ref{['eq18']} for $\mu$ from the experimental data.
• Figure 2: Ratio $\delta(\xi)$ by Formula \ref{['eq5']} as a function of variable $\xi$ (black line) plotted with some experimental data Stier2018Liu2019Chennano2019Molas2019takahashi2024NAT2019 of $\delta_{\text{exp}} = (E_{2s} - E_{1s}) / (E_{3s} - E_{2s})$ marked by different symbols. Solution $\xi (\delta)$\ref{['eq7']} also plotted for comparison (red dots).
• Figure 3: Magnetoexciton energies in monolayer WSe$_2$ encapsulated by hBN slabs, calculated by the highly accurate numerical method (black lines) and by the analytical method (formulas \ref{['eq1']} and \ref{['eq15']}) (red lines) compared with the experimental data (blue triangles) of Refs. Stier2018 (a), Liu2019 (b), and Chennano2019 (c). Material parameters used in the calculations are retrieved from the experimental data using the formulas \ref{['eq9']} (for the band gap energy $E_g$), \ref{['eq10']} (for the screening length $r_0$), \ref{['eq11']} (for the dielectric constant $\kappa$), and \ref{['eq17']} (for the exciton reduced mass $\mu$).
• Figure 4: Magnetoexciton energies of monolayer WS$_2$ (a), MoSe$_2$ (b), and MoS$_2$ (c), calculated numerically (black lines) and analytically (red lines) with the retrieved material parameters and compared with the experimental data (blue triangles) in Ref. NAT2019. Material parameters used in the calculations are retrieved from the experimental data using the formulas \ref{['eq9']} (for the band gap energy $E_g$), \ref{['eq10']} (for the screening length $r_0$), \ref{['eq11']} (for the dielectric constant $\kappa$), and \ref{['eq17']} (for the exciton reduced mass $\mu$).
• Figure 5: Exciton binding energies calculated by numerical method with the retrieved material parameters for monolayer TMDCs in Tables \ref{['tab2']} and \ref{['tab3']} and compared with the experimental in Refs. Stier2018Liu2019Chennano2019Molas2019takahashi2024NAT2019.