Energy Spectrum of Two-Dimensional Excitons in a Nonuniform Dielectric Medium

We demonstrate that, in monolayers (MLs) of semiconducting transition metal dichalcogenides, the $s$-type Rydberg series of excitonic states follows a simple energy ladder: ${\epsilon }_n=-{\mathrm{Ry}}^{*}/(n+\delta {)}^2$, $n=1,2,\dots$, in which ${\mathrm{Ry}}^{*}$ 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 accounted for only by $\delta$. This is justified by the analysis of experimental data on excitonic resonances, as extracted from magneto-optical measurements of a high-quality ${\mathrm{WSe}}_2$ ML encapsulated in hexagonal boron nitride (hBN), and well reproduced with an analytically solvable Schrödinger equation when approximating the electron-hole potential in the form of a modified Kratzer potential. Applying our convention to other ${\mathrm{MoSe}}_2$, ${\mathrm{WS}}_2$, ${\mathrm{MoS}}_2$ 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 ${\mathrm{WSe}}_2$ as well as for ${\mathrm{MoS}}_2$ 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.

Figure 1

(a) Helicity-resolved (${\sigma }^{\pm }$) PL spectra of ${\mathrm{WSe}}_2$ ML at selected magnetic fields. The separate parts of the spectra are normalized to the intensity of the $1s$, $2s$, and $3s$ lines. (b) False-color map of the corresponding PL spectra from 0 to 14 T. (c) Obtained excitonic energies for ${\sigma }^{\pm }$ components as a function of the magnetic field. Mean energies of the ${\sigma }^{\pm }$ components of excitonic resonances measured on ${\mathrm{WSe}}_2$ ML as a function of (d) $B$ and (e) $B^2$. The black lines are obtained by fitting the presented data with (d) $E_{5s}(B)=A+9/2\hslash {\omega }_c^{*}$ ($A$ is a fitting parameter) and (e) $E_{ns}(B)=E_{ns}(B=0)+\sigma B^2$.

Figure 2

Experimentally obtained transition energies for the exciton states as a function $1/(n+\delta {)}^2$ for $\delta =-0.083$. The black line shows a fit of the data to the model described by Eq. (1). The gray lines denote the band gap ($E_g$) and excitonic binding ($E_b$) energies.

Figure 3

Rytova-Keldysh (purple curve), Coulomb (blue curve), and Kratzer potential with $g^2=0.21$ (red curve), as a function of dimensionless parameter $r/r_0^{*}$. The energy scale is measured in units of $U_0=e^2/r_0$. The gray rectangular depicts the region of distances smaller than the lattice constant $a=3.28\AA$ of ${\mathrm{WSe}}_2$ MLs encapsulated in hBN ($r_0^{*}=10\AA$).

Figure 4

Low temperature PL spectra of S-TMD MLs at $T=5\mathrm{K}$. The pink vertical arrows denote the estimated band-gap energies $E_g$. The chosen spectral regions are scaled for clarity. Typically for S-TMDs monolayers, the most pronounced emission feature seen in our spectra is due to the $1s$ excitonic resonance accompanied by low energy peaks commonly assigned to different excitonic complexes [33, 40, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53].