Probing combinations of acoustic phonons in $\mathrm{Mo}{\mathrm{S}}_2$ by intervalley double-resonance Raman scattering

In this work, we present measurements of the temperature dependence of the resonance Raman spectra of $\mathrm{Mo}{\mathrm{S}}_2$ in two-dimensional and bulk forms, performed to identify the processes related to different combinations of two acoustic phonons involved in the intervalley scattering. The resonance Raman spectra of samples of different thicknesses (single layer, bilayer, trilayer, and bulk) were measured near the resonance with the $A$ excitonic transition and at different temperatures. Measurements of the Raman spectra of bulk $\mathrm{Mo}{\mathrm{S}}_2$ were performed using several laser energies across the resonances with the $A$ and $B$ excitonic transitions. Based on the electronic and vibrational structures of the samples with different thicknesses and the evolution of the bands as a function of the laser excitation energy and temperature, we propose correct assignments to the Raman bands appearing at approximately 380, 395, and $405\mathrm{c}{\mathrm{m}}^{-1}$. According to our measurements and data analysis, the peaks at 380, 395, and $405\mathrm{c}{\mathrm{m}}^{-1}$ correspond to the combinations of 2TA around the K point, LA and TA phonons around the M point, and LA and out-of-plane acoustic (ZA) phonons around the M point, respectively. This work sheds light on the double-resonance processes of $\mathrm{Mo}{\mathrm{S}}_2$ and how it is related to the electronic structure of this material. The results presented here establish the assignment and the scattering mechanism for some two-phonon and double-resonance Raman bands whose origins were still a matter of debate in the literature. It can also be an important basis to explain the double-resonance processes in other transition-metal dichalcogenides since we present the fundamental electron scattering mechanism near the resonance with the $A$ and $B$ excitonic transitions, and how it is affected by thermal effects.

Figure 1

Raman spectra of 1L, 2L, 3L, and bulk $\mathrm{Mo}{\mathrm{S}}_2$ collected with a 1.92-eV laser at the temperatures of (a) 300 and (b) 80 K. The peaks identified by α, β, and γ are discussed in the text. We highlighted the presence of the first-order $E_{2\mathrm{g}}$ and $A_{1\mathrm{g}}$ modes and the α, β, γ, $b$, and 2LA bands. The diagram in (b) illustrates (exaggerated) the intervalley scattering of the electron from the vicinity of the K point to the vicinity of the ${\mathbf{K}}^{\prime }$ point (${\mathbf{KK}}^{\prime }$) and from the vicinity of the K point to the vicinity of the Q point.

Figure 2

(a) Raman spectra of bulk $\mathrm{Mo}{\mathrm{S}}_2$ collected with laser excitation energies in the range of 1.92 to 2.18 eV at 80 K. The intensity is normalized by the intensity of the $E_{2\mathrm{g}}$ mode. The spectrum corresponding to the 1.98-eV laser energy shows the fitting with a sum of Lorentzian and Gaussian peaks, where the γ band was fitted with two Gaussian components. (b) Frequency of the Raman spectral features between 375 and $430\mathrm{c}{\mathrm{m}}^{-1}$ as a function of the laser excitation energy.

Figure 3

Raman spectra of bulk $\mathrm{Mo}{\mathrm{S}}_2$ in the spectral range of 165 to $430\mathrm{c}{\mathrm{m}}^{-1}$ at three different temperatures collected with the laser energies of (a) 1.92, (b) 2.03, and (c) 2.09 eV.

Figure 4

Frequency of the analyzed bands as a function of temperature of bulk $\mathrm{Mo}{\mathrm{S}}_2$ for the (a) 1.92-, (b) 2.03-, and (c) 2.09-eV laser excitation energies. The slopes of the fitting lines are presented in Table 1.

Figure 5

(a) Colored map showing the calculated intensity of the α band (color scale) as a function of the laser excitation energy ($x$ axis) and the temperature of measurement ($y$ axis). (b) The intensity of the α band as a function of the laser excitation energy. (c), (d), and (e) show the intensity of the α band as a function of temperature for the laser excitation energies of 1.92, 2.03, and 2.09 eV, respectively. The intensities of the experimental data are all normalized by the intensity of the $E_{2\mathrm{g}}$ band. The solid lines are a fit with a Gaussian function, while the dashed lines represent the calculated results from (a), at the respective laser excitation energies and temperatures, multiplied by a constant factor to match the intensity observed.