Spectral asymmetry of phonon sideband luminescence in monolayer and bilayer ${\mathrm{WSe}}_2$

We report an experimental study of temperature-dependent spectral line shapes of phonon sideband emission stemming from dark excitons in monolayer and bilayer ${\mathrm{WSe}}_2$. Using photoluminescence spectroscopy in the range from 4 to 100 K, we observe a pronounced asymmetry in the phonon-assisted luminescence from momentum-indirect exciton reservoirs. We demonstrate that the corresponding spectral profiles are distinct from those of bright excitons with direct radiative decay pathways. The line-shape asymmetry reflects thermal distribution of exciton states with finite center-of-mass momenta, characteristic for phonon sideband emission. The extracted temperature of the exciton reservoirs is found to generally follow that of the crystal lattice, with deviations reflecting overheated populations. The latter are most pronounced in the bilayer case and at lowest temperatures. Our results add to the understanding of phonon-assisted recombination of momentum-dark excitons and, more generally, establish means to access the thermal distribution of finite-momentum excitons in atomically thin semiconductors with indirect band gaps.

Figure 1

(a), (b) Logarithmic false-color plots of the PL from samples A and B of ML ${\mathrm{WSe}}_2$ encapsulated in hBN, where the former exhibits features of residual doping and the latter is tuned to charge neutrality. (c) PL of BL ${\mathrm{WSe}}_2$ from sample B tuned close to charge neutrality with a marginal contribution of the negative bilayer trion ${\mathrm{B}}^{-}$. Neutral spin-bright and spin-dark $KK$ exciton peaks are denoted as X and D, and the trion doublet as ${\mathrm{X}}_1^{-}$ and ${\mathrm{X}}_2^{-}$. The remaining ML and BL peaks are labeled by an increasing subscript number $n$ as ${\mathrm{M}}_n$ and ${\mathrm{B}}_n$, respectively.

Figure 2

(a) Left panel: PL spectra of ML in sample B at temperatures from 4 to 120 K (colored from purple to red), normalized to unity and shifted horizontally to the energy $E_X$ of the bright exciton peak X at 4 K. Right panel: High-energy shoulder of the spectra with energy axis normalized by $k_{B}T$. (b) Same for BL in the temperature range from 4 to 80 K with the energy of the peak ${\mathrm{B}}_1$ as reference. (c) Schematic representation of direct and indirect decay processes for excitons. The center-of-mass momenta of $KK$ excitons, such as X or D, with zero-phonon radiative decay are restricted to cold excitons (blue shaded population inside the orange light cone). In contrast, phonon sideband luminescence of momentum-dark excitons such as $KQ$ with energy minimum $E_0=E_{KQ}$ and exciton center-of-mass momentum $Q-K$ is assisted by phonons with momentum $Q$ and thus also includes emission from hot excitons (thermally activated red shaded population). The thermal Boltzmann occupancy of exciton states with finite kinetic energy $E_{\mathrm{kin}}>E_0$ above their respective dispersion minimum is represented by the area of colored shades. (d) Illustration of the phonon sideband asymmetry: The homogeneously broadened Lorentzian spectrum with linewidth $\mathrm{\Gamma }$ is convoluted on the high-energy side with the contribution from excitons with finite kinetic energy $E_{\mathrm{kin}}>0$ according to their thermal occupancy in Boltzmann approximation.

Figure 3

Model fits of asymmetric peaks in (a) and (b) for MLs of sample A and B, respectively, and (c) BL of sample B. The fits (solid lines) to the spectra for temperatures ranging from 4 to 50 K were obtained according to Eq. (1) with exciton temperature, Lorentzian linewidth, peak energy, and intensity as free fit parameters. Note the pronounced asymmetry on the high-energy side of all ML and BL peaks and the good agreement with the model of contributions from thermally activated excitons with finite center-of-mass momenta.

Figure 4

Least-squares parameters obtained from model fits. (a) Evolution of the peak energy with temperature for X in ML as well as for ${\mathrm{B}}_1$ and ${\mathrm{B}}_2$ in BL of sample B. The solid lines show best fits to thermal band-gap renormalization according to Eq. (2) with parameters from Ref. [23]. (b) Lorentzian linewidth of the ML peak ${\mathrm{M}}_2$ in sample A and BL peaks ${\mathrm{B}}_1$ and ${\mathrm{B}}_2$ in sample B. The solid line shows best fit for BL peaks taking into account thermal phonon broadening according to Eq. (3) (c) Effective exciton temperature $T_{\mathrm{exc}}$ obtained from best fits to temperature-dependent spectra with Eq. (1) and plotted versus the nominal sample temperature (the solid grey line shows the lattice temperature; ML and BL peaks from sample A and B, respectively). (d) Blue shoulder of ${\mathrm{B}}_1$ with energy axis normalized by the effective exciton temperature $k_B{T}_{\mathrm{exc}}$ for direct comparison with the right panel of Fig. 2. Note the universality of the high-energy shoulder asymmetry arising from the phonon sideband emission by thermally activated excitons.