Optical spectroscopy of excited exciton states in ${\mathrm{MoS}}_2$ monolayers in van der Waals heterostructures

The optical properties of ${\mathrm{MoS}}_2$ monolayers are dominated by excitons, but for spectrally broad optical transitions in monolayers exfoliated directly onto ${\mathrm{SiO}}_2$ substrates detailed information on excited exciton states is inaccessible. Encapsulation in hexagonal boron nitride (hBN) allows approaching the homogenous exciton linewidth, but interferences in the van der Waals heterostructures make direct comparison between transitions in optical spectra with different oscillator strength more challenging. Here we reveal in reflectivity and in photoluminescence excitation spectroscopy the presence of excited states of the $A$ exciton in ${\mathrm{MoS}}_2$ monolayers encapsulated in hBN layers of calibrated thickness, allowing us to extrapolate an exciton binding energy of $\approx 220$ meV. We theoretically reproduce the energy separations and oscillator strengths measured in reflectivity by combining the exciton resonances calculated for a screened two-dimensional Coulomb potential with transfer matrix calculations of the reflectivity for the van der Waals structure. Our analysis shows a very different evolution of the exciton oscillator strength with principal quantum number for the screened Coulomb potential as compared to the ideal two-dimensional hydrogen model.

Figure 1

Optical spectroscopy results. Sample 1. (a) Optical microscope image of an ${\mathrm{MoS}}_2$ flake encapsulated in hBN. The scale bar is $20\mu \mathrm{m}$. (b) Schematic of sample structure, the top (bottom) hBN layers are $7\pm 3$ nm ($130\pm 5$ nm) thick as determined by AFM, the ${\mathrm{SiO}}_2$ layer is 83 nm thick. (c) Differential reflectivity measurement $(R_{ML}-R_{\mathrm{sub}})/R_{\mathrm{sub}}$ performed with a power-stabilized white halogen lamp. The strong transition at lower energy is due to the $A$-exciton ground state absorption. At higher energies, three more transitions are clearly visible; the broader one is due to the $B$-exciton ground state while the other two are tentatively ascribed to be the first two excited states of the $A$ exciton: $A:2s$ and $A:3s$. Model simulations using the hBN thicknesses as determined by AFM are shown by the bold, red curve. Additional simulations (fine magenta, green, and blue) for different hBN and ${\mathrm{SiO}}_2$ thicknesses show how the depth and shape of the exciton resonances depends on the individual layer thicknesses. The exciton resonance parameters are as follows: The energy positions of $A:1s$ exciton were tuned to the observed $A:1s$ peak; energies of excited states are calculated and shown in Fig. 3; radiative damping for the ground states ${\mathrm{\Gamma }}_{0,\mathrm{A}:1\mathrm{s}}={\mathrm{\Gamma }}_{0,\mathrm{B}:1\mathrm{s}}=1$ meV, for excited states found from calculation, Fig. 3; nonradiative damping ${\mathrm{\Gamma }}_{\mathrm{A}}=2.5$ meV, ${\mathrm{\Gamma }}_{\mathrm{B}}=25$ meV. Refractive indices: $n_{\mathrm{hBN}}=2.2$, $n_{{\mathrm{SiO}}_2}=1.46$, $n_{\mathrm{Si}}=3.5$.

Figure 2

Sample 2. (a) Polarization-resolved photoluminescence at $T=4$ K following linear (top) and circular (bottom) excitation at 2.1 eV, exhibiting efficient valley coherence and valley initialization, respectively. (b) PLE measurements. The integrated $X^0$ (i.e., transition $A:1s$) intensity as a function of excitation energy is shown (black circles), as well as the linear (resp. circular) polarization obtained under linear (resp. circular) excitation (red triangles, open squares). (c) Schematic single particle band structure of ML ${\mathrm{MoS}}_2$, for simplicity the small spin splitting in the conduction band (CB) is neglected.

Figure 3

Results of calculations. (a) Exciton binding energy for screened $r_0=2.95$ nm, Eq. (1) and unscreened 2D Coulomb potential as a function of principal quantum number $n$ for ML ${\mathrm{MoSe}}_2$. (b) Same as (a), but for the relative oscillator strength normalized at the $A:1s$ exciton oscillator strength. (c) Comparison of the bound states in the screened and unscreened Coulomb potential, see Supplemental Material for more details [64].