Observation of exciton-phonon coupling in ${\mathrm{MoSe}}_2$ monolayers

We study experimentally and theoretically the exciton-phonon interaction in ${\mathrm{MoSe}}_2$ monolayers encapsulated in hexagonal BN, which has an important impact on both optical absorption and emission processes. The exciton transition linewidth down to 1 meV at low temperatures makes it possible to observe high-energy tails in absorption and emission extending over several meV, not masked by inhomogeneous broadening. We develop an analytical theory of the exciton-phonon interaction accounting for the deformation potential induced by the longitudinal acoustic phonons, which plays an important role in exciton formation. The theory allows fitting absorption and emission spectra and permits estimating the deformation potential in ${\mathrm{MoSe}}_2$ monolayers. We underline the reasons why exciton-phonon coupling is much stronger in two-dimensional transition metal dichalcogenides as compared to conventional quantum well structures. The importance of exciton-phonon interactions is further highlighted by the observation of a multitude of Raman features in the photoluminescence excitation experiments.

Figure 1

Optical spectroscopy results. ML ${\mathrm{MoSe}}_2$ encapsulated in hBN, sample 1, (a) typical PL emission spectrum at $T=4$ K. Emission peaks at energy 1.613 and 1.637 eV are due to the charged ($T$) and the neutral exciton ${\mathrm{X}}^0$. [Inset of (a)] Schematics of sample structure: ${\mathrm{MoSe}}_2$ ML encapsulated in hBN, the bottom hBN layer is ($140\pm 5$ nm) thick as determined by AFM, the ${\mathrm{SiO}}_2$ layer is 80 nm thick. (b) PLE experiments: integrated ${\mathrm{X}}^0$ PL intensity plotted in terms of excess energy (bottom axis), defined as the difference between the laser energy (top axis) and the ${\mathrm{X}}^0$ resonance. The average oscillation period corresponds to the $M$-point longitudinal acoustic phonon LA(M). [Inset of (b)] Schematics of 2D Brillouin zone with $K, M$, and $\mathrm{\Gamma }$ points (c) Integrated trion PL intensity as a function of excitation energy scanning laser across ${\mathrm{X}}^0$ resonance and (d) contour plot of trion PL intensity scanning laser across ${\mathrm{X}}^0$ resonance at $T=4$ K. (e) Temperature dependent PL emission at ${\mathrm{X}}^0$ resonance. For better comparison of emission shape, the ${\mathrm{X}}^0$ PL intensity is normalized to the maximum value and the energy shifted to compensate the temperature evolution of the band gap.

Figure 2

Sample 2. (a) PLE spectra of ${\mathrm{MoSe}}_2$ ML plotted in terms of excess energy (bottom axis), defined as the difference between the laser energy (top axis) and the $X^0$ resonance. (b) Contour plot of the same data, where sharp Raman features are superimposed on broad exciton absorption features. Sample 3. (c) Same as (a) but for different sample. (d) Same as (b) but for different sample. Phonons are labeled according to Soubelet et al. [40]. In samples with more disorder and stronger localization, Raman features are more prominent as momentum conservation is relaxed [40], which leads to rich features in (d). In addition to PL emission, the integrated signal in (a) and (c) also contains contributions from Raman scattered photons if the laser energy is a multiple of a prominent phonon energy above the ${\mathrm{X}}^0$ transitions. In (b) and (d), the Raman features are spectrally very narrow due to the narrow linewidth of the excitation laser used.

Figure 3

(a) Absorbance at the A:1s exciton in ${\mathrm{MoSe}}_2$ monolayer. Experimental data same as Fig. 1—red squares, the data at a lower laser excitation power are shown as small blue squares, the full theory by a solid green curve, Eqs. (20) and (21), the black dashed curve shows the result in the absence of phonon-assisted transitions. ${\mathrm{\Gamma }}_0=\mathrm{\Gamma }=0.47$ meV, $|D_c-D_v|=11$ eV, the other parameters are from Table 1, in the calculation $T=4$ K. (b) Normalized to the value at $T=70$ K with the $T\to 0$ offset subtracted intensity of PL emission measured (symbols) and calculated after Eq. (24) (solid lines) as a function of temperature at a fixed detuning of $\mathrm{\Delta }=3$ (red), 4 (black), and 6 meV (blue). The parameters of the calculation are the same as in (a). (c) Schematics illustrating exciton and phonon dispersion, the intersection gives the energy scale $2M{s}^2$, for $E_k\gg M{s}^2$, the exciton-phonon interaction is quasielastic. Steps (1), (2), and (3) describe, respectively, photon absorption with exciton formation in a virtual state, exciton-phonon scattering resulting in a real final state for exciton (on the exciton dispersion, 2), and relaxation toward the radiative states inside the light cone (3). (d) Sketch of the photon absorption accompanied by the phonon emission. Exciton and phonon wave vectors have same absolute value but opposite directions.

Figure 4

Exciton linewidth ($2\times$ half width at half maximum at the low-energy side) as a function of temperature extracted from the PL data in Fig. 1 (points). Fit 1 (red solid curve) is performed by Eq. (A1) with the parameter set $\gamma \left(0\right)=1.52$ meV, $c_1=52\mu \mathrm{eV}/\mathrm{K}, c_2=11.52$ meV, and $\hslash {\mathrm{\Omega }}_0=13.4$ meV. Fit 2 (blue dashed curve) is performed by Eq. (A1) with the parameter set $\gamma \left(0\right)=1.49$ meV, $c_1=88\mu \mathrm{eV}/\mathrm{K}, c_2=14.9$ meV, and $\hslash {\mathrm{\Omega }}_0=15.5$ meV.

Figure 5

Diagrams describing the photon absorption (top) and exciton dressed Greens function (bottom).