Analytical expression for exciton energies in monolayer transition metal dichalcogenides

We have developed an analytical expression for $s$-state exciton energies in monolayer transition metal dichalcogenides (TMDCs) as ${E}_ns=\ensuremath{-}{\text{Ry}}^*\ifmmode\times\else\texttimes\fi{}{P}_n/{(n\ensuremath{-}0.5+0.479\phantom{\rule{0.16em}{0ex}}{r}₀^*/\ensuremath{\kappa})}², n=1,2,3,...$ within the perturbation theory. Here, ${r}₀^*$ and $\ensuremath{\kappa}$ represent the dimensionless screening length and the surrounding medium's dielectric constant, respectively; ${\text{Ry}}^*$ is an effective Rydberg energy scaled by the dielectric constant and exciton reduced mass; ${P}_n({r}₀^*/\ensuremath{\kappa})$ is a function of variables $n$ and ${r}₀^*/\ensuremath{\kappa}$ with values around 1.0, so it is considered a term that corrects the Rydberg energy. Despite the simple form, the suggested formula provides exciton energies with high precision compared to the exact numerical solutions and accurately describes recent experimental data for TMDC materials such as ${\mathrm{WSe}}₂, {\mathrm{WS}}₂, {\mathrm{MoSe}}₂, {\mathrm{MoS}}₂$, and ${\mathrm{MoTe}}₂$ encapsulated by hexagonal boron nitride. To achieve these results, we have developed a regulated perturbation theory by combining the conventional perturbation method with several elements of the Feranchuk-Komarov operator method. These include the Levi-Civita transformation, the algebraic calculation technique using the annihilation and creation operators, and the introduction of a free parameter to optimize the convergence rate of the perturbation series. This analytical form of exciton energies, working well in the range $1.45>{r}₀^*/\ensuremath{\kappa}>0.65$, covers a large class of TMDC monolayers and could be beneficial in various physical analyses, including retrieval of the material parameters such as reduced exciton mass and screening length from the available measured exciton energies.