Bright and dark singlet excitons via linear and two-photon spectroscopy in monolayer transition-metal dichalcogenides

We discuss the linear and two-photon spectroscopic selection rules for spin-singlet excitons in monolayer transition-metal dichalcogenides. Our microscopic formalism combines a fully $k$-dependent few-orbital band structure with a many-body Bethe-Salpeter equation treatment of the electron-hole interaction, using a model dielectric function. We show analytically and numerically that the single-particle, valley-dependent selection rules are preserved in the presence of excitonic effects. Furthermore, we definitively demonstrate that the bright (one-photon allowed) excitons have $s$-type azimuthal symmetry and that dark $p$-type excitons can be probed via two-photon spectroscopy. The screened Coulomb interaction in these materials substantially deviates from the $1/{\varepsilon }_{0}r$ form; this breaks the “accidental” angular momentum degeneracy in the exciton spectrum, such that the $2p$ exciton has a lower energy than the $2s$ exciton by at least 50 meV. We compare our calculated two-photon absorption spectra to recent experimental measurements.

Figure 1

Single-particle band structure of ${\mathrm{MoS}}_2$ predicted by a linearized two-band model (blue solid) and a nonlinear three-band model (red dashed) compared to first-principles density functional theory with the local density approximation (DFT, solid black).

Figure 2

Valence to lowest conduction band momentum matrix elements for a linearized two-band model (top) and a nonlinear three-band model (bottom). Blue is positive, red is negative, and white is zero. The results are qualitatively very similar in the immediate vicinity of the $K$ and $K^{\prime }$ points, but differ elsewhere in the Brillouin zone.

Figure 3

Valence to lowest conduction band momentum matrix elements squared, for circular polarization, for a linearized two-band model (top) and a nonlinear three-band model (bottom). Black is positive and white is zero. The results are qualitatively very similar in the immediate vicinity of the $K$ and $K^{\prime }$ points, but differ elsewhere in the Brillouin zone.

Figure 4

Imaginary part of the dielectric function for ${\mathrm{MoS}}_2$ calculated in the presence of excitonic effects. The band gap has been rigidly increased to 2.41 eV such that the $1s$ exciton peak occurs near 2.0 eV (spin-orbit splitting into $A$ and $B$ peaks is neglected, as described in the text). A Gaussian broadening of 50 meV (FWHM) has been applied to all peaks.

Figure 5

Reciprocal space plots of the selection-rule-determining product $A_{vc}^X\left(\mathit{k}\right)P_{-}^{vc}\left(k\right)$. In the presence of right-handed circular polarization, it is seen that excitons are only created at the $K$ point, and not at the $K^{\prime }$ point, as was found in Ref. [3] and in Sec. 2c in the absence of exciton effects.

Figure 6

Two-photon absorption (TPA) intensity for monolayer (a) ${\mathrm{WSe}}_2$ and (b) ${\mathrm{WS}}_2$ evaluated numerically with Eq. (38) (blue lines). The spectra have been artificially broadened with a Gaussian linewidth (FWHM) of 80 meV (thicker line) and 20 meV (thinner line). The experimental two-photon photoluminescence excitation (TPPLE) spectrum for ${\mathrm{WSe}}_2$ [9] and ${\mathrm{WS}}_2$ [19] is included for comparison (blue circles). The theoretical linear absorption spectrum from the same model (FWHM of 50 meV) is overlaid for reference (grey lines) along with the experimental result (gray circles) for ${\mathrm{WSe}}_2$ [9] and ${\mathrm{WS}}_2$ [15].