Exciton Binding Energy and Nonhydrogenic Rydberg Series in Monolayer ${\mathrm{WS}}_2$

We have experimentally determined the energies of the ground and first four excited excitonic states of the fundamental optical transition in monolayer ${\mathrm{WS}}_2$, a model system for the growing class of atomically thin two-dimensional semiconductor crystals. From the spectra, we establish a large exciton binding energy of 0.32 eV and a pronounced deviation from the usual hydrogenic Rydberg series of energy levels of the excitonic states. We explain both of these results using a microscopic theory in which the nonlocal nature of the effective dielectric screening modifies the functional form of the Coulomb interaction. These strong but unconventional electron-hole interactions are expected to be ubiquitous in atomically thin materials.

Figure 1

(a) Real-space representation of electrons and holes bound into excitons for the three-dimensional bulk and a quasi-two-dimensional monolayer. The changes in the dielectric environment are indicated schematically by different dielectric constants ${ϵ}_{3\mathrm{D}}$ and ${ϵ}_{2\mathrm{D}}$ and by the vacuum permittivity ${ϵ}_0$. (b) Impact of the dimensionality on the electronic and excitonic properties, schematically represented by optical absorption. The transition from 3D to 2D is expected to lead to an increase of both the band gap and the exciton binding energy (indicated by the dashed red line). The excited excitonic states and Coulomb correction for the continuum absorption have been omitted for clarity.

Figure 2

The derivative of the reflectance contrast spectrum $(d/dE)(\mathrm{\Delta }R/R)$ of the ${\mathrm{WS}}_2$ monolayer. The exciton ground state and the higher excited states are labeled by their respective quantum numbers (schematically shown at the bottom right). The spectral region around the $1s$ transition ($AX$) and the trion peak ($A{X}_T$) of the $A$ exciton is scaled by a factor of 0.03 for clarity. The inset shows the as-measured reflectance contrast $\mathrm{\Delta }R/R$ for comparison, allowing for the identification of the $A$, $B$, and $C$ transitions.

Figure 3

(a) Experimentally and theoretically obtained transition energies for the exciton states as a function of the quantum number $n$. The fit of the $n=3,4,5$ data to the 2D hydrogen model for Wannier excitons is shown for comparison. Gray bands represent uncertainty in the quasiparticle band gap from the fitting procedure. Corresponding effective dielectric constants are shown in the inset. (b) Screened 2D interaction Eq. (2) used in the model Hamiltonian (black lines) compared to the 2D hydrogen interaction $1/r$ (red lines); a semilogarithmic plot is given in the inset. Also shown are the corresponding energy levels and radial wave functions up to $n=3$. (c) Schematic representation of electron-hole pairs forming $1s$ and $2s$ excitonic states in a nonuniform dielectric environment.

Figure 4

(a) The derivative of the reflectance contrast spectra for the ${\mathrm{WS}}_2$ 1L, 2L, 4L, and bulk. The positions of the $2s$ exciton resonance are indicated by dotted arrows. (b) Energies of the $1s$ and $2s$ states for various layer thicknesses. Band gaps of the bulk and the monolayer are represented by the dashed lines.