Light–matter interactions in two-dimensional layered WSe2 for gauging evolution of phonon dynamics

Phonon dynamics is explored in mechanically exfoliated two-dimensional WSe2 using temperature-dependent and laser-power-dependent Raman and photoluminescence (PL) spectroscopy. From this analysis, phonon lifetime in the Raman active modes and phonon concentration, as correlated to the energy parameter E0, were calculated as a function of the laser power, P, and substrate temperature, T. For monolayer WSe2, from the power dependence it was determined that the phonon lifetime for the in-plane vibrational mode was twice that of the out-of-plane vibrational mode for P in the range from 0.308 mW up to 3.35 mW. On the other hand, the corresponding relationship for the temperature analysis showed that the phonon lifetime for the in-plane vibrational mode lies within 1.42× to 1.90× that of the out-of-plane vibrational mode over T = 79 K up to 523 K. To provide energy from external stimuli, as T and P were increased, peak broadening in the PL spectra of the A-exciton was observed. From this, a phonon concentration was tabulated using the Urbach formulism, which increased with increasing T and P; consequently, the phonon lifetime was found to decrease. Although phonon lifetime decreased with increasing temperature for all thicknesses, the decay rate in the phonon lifetime in the monolayer (1L) material was found to be 2× lower compared to the bulk. We invoke a harmonic oscillator model to explain the damping mechanism in WSe2. From this it was determined that the damping coefficient increases with the number of layers. The work reported here sheds fundamental insights into the evolution of phonon dynamics in WSe2 and should help pave the way for designing high-performance electronic, optoelectronic and thermoelectric devices in the future.


Calculation of laser spot size
The laser spot size is primarily defined by the laser wavelength and microscope objective using Rayleigh's criterion. The laser spot diameter R is calculated using the following:  (1) where λ is the wavelength of the Raman laser and NA is the numerical aperture of the microscope objective. For our study, the wavelength of the laser was 532 nm and for a 10× microscope objective, the NA was 0.25. Hence, using Equation 1 the laser spot diameter was calculated to be ≈2.6 µm. A slightly modified equation yields the theoretical diffraction limited spatial resolution which is achievable using an optical microscope, and is given by the following formula: = 0.61λ (2) After putting the respective values of wavelength and NA in Equation 2, the spatial resolution was found to be ≈1.3 µm. However, in Raman spectroscopy, scattering of the laser photons and interaction with the interfaces in the sample can reduce the resolution. Thus, typical spatial resolution in the Raman process is more than the theoretical calculated value.

Calculation of error bars for Raman & PL measurements
The Raman and PL measurements were done over a wide range of temperatures on WSe2 comprised of multiple layer thicknesses. Making multiple measurements on the same spot of the sample to extract reproducibility data and generate error bars over the course of the temperature and sample thickness experiments was deemed practically challenging and perhaps not yielding the true impact of the external temperature on the Raman spectra. The laser-sample interactions could unduly heat the local area of the molecular membranes during repeated and successive measurements which could potentially induce defects and affect the structural morphology of the WSe2 nanosheets, convoluting the Raman spectra. Therefore, in order to extract error-bars in our S3 data, we choose three spatially uniform regions marked as a, b, and c on the sample in each of the 1L, ML, and bulk WSe2 sections, as shown in Figure 1a of the manuscript. For Raman and PL spectroscopy measurements, the mean value for measurements conducted in a, b and c were calculated from which the standard deviation (1) were determined in order to extract the error bars.
The Raman and PL characteristics in 1L, ML and bulk WSe2 at a fixed temperature (T = 298 K) and laser power (P = 3.35 mW) are shown in Figure 1e Figure S1a. Similarly, Figure S1b shows the error bars calculated for A-and I-peak in PL characteristics from Figure 1f.

S6
The Raman and PL characteristics in 1L WSe2 at a fixed laser power (P = 3.35 mW) and varying temperature (T = 79 K to T = 523 K) are shown in Figure 3 in the manuscript. Similar to the analysis of the laser power variation study as discussed above, the error bars for the Raman shifts and FWHM of 2g 1 and 1g modes calculated from Figure 3a of the manuscript are shown in Figure   S2d and S2e respectively. The error bars for the PL A-peak shifts and 1/E0 are calculated from Figure 3e of the manuscript and are plotted in Figure S2f.

Optimization of laser power used in Raman and PL measurements
In our study, the laser power P was tuned using a ND filter in LabRAM HR Evolution (Horiba Scientific) spectrometer. The calculated laser powers corresponding to the tunable ND filters are given in Table S1 from which it can be inferred that the minimum achievable P was 0.00325 mW when the laser was tuned with 0.01% of the ND filter. The Raman and PL signals were measured at the minimum P = 0.00325 mW in monolayer (1L), multilayer (ML) and bulk samples. In case of 1L samples a decent signal to noise ratio (SNR) was observed; however, the SNR was found to be very poor when bulk WSe2 was measured under such low laser power. For example, Figure S3 shows below the PL spectra of 1L and bulk WSe2 nanosheets when the ND filter was tuned at 0.01% which reveals that the SNR is very low in case of bulk WSe2 while it is decent in 1L nanosheets. It was found that SNR only improves in bulk WSe2 from P = 0.308 mW (ND filter = 1%) and that is why it was the minimum P chosen throughout the study as we found it was beneficial to use a common minimum laser power to compare and analyze different parameters in the 1L, ML and bulk nanosheets.

Calculation of instrumental broadening and its effect on phonon lifetime analysis
In our study, phonon lifetimes  of the Raman active 2g 1 and 1g modes in 1L, ML and bulk WSe2 nanosheets are calculated from the full-width-half-maximum (FWHM), i.e., phonon linewidth Γ of 2g 1 and 1g modes by using Equation 3 described in the manuscript. The measured Raman linewidths are a convolution of effects of both the actual Lorentzian vibrational distribution of the phonons and the instrument-induced broadening which is typically assumed to have a Gaussian response. The convolution of the phonon line shape and the spectrometer function is known as the Voigt profile [1]. The true phonon linewidth Γ was therefore determined from the Voigt profile fitted to the experimental data illustrated for 1L WSe2 at T = 298 K in Figure S4. The instrumental broadening was calculated to be 0.613 cm −1 which is far lower compared to Γ in 2g 1 and 1g modes and it is discussed in more detail below.

S9
The calculation of instrumental broadening is described below: It is understood that the condition for constructive interference and obtaining a primary maximum for a diffraction grating Raman spectrometer used in our study is given by, where λ is the wavelength (532 nm) of the incident Raman laser, m is the order, d is the slit spacing, θ and θ are the angle of incidence and reflection respectively. Here, we assume that θ = θ = θ, order m = 1, and sin ≈ for very small value of  Hence, Equation 1 can be rewritten as, ∆λ ≈ ∆θ, where ∆λ is the instrument broadening, i.e., change in λ for a small change in or ∆θ.
A slit width b will introduce a ∆θ = , where f is the focal length of the spectrometer; for our spectrometer f is 800 mm. This results in, ∆λ = (4) There is a lower limit that can be achieved by narrowing the slit width, In the case of the PL measurements, we found that the FWHM is 60 nm for 1L WSe2 while ∆λ was found to be 0.017 nm from Equation 4, as described above. Therefore, similar to the Raman measurements, the effect of instrumental broadening on the FWHM of the PL peak was also minimal in our study.

Calculation of hysteresis during cooling down and warming up processes
The Raman shift of the Raman active 2g 1 and 1g modes in 1L WSe2 during cooling to T = 79 K and then warming to T = 298 K are shown in Figure S5a and S5b respectively.

Temperature-dependent Raman spectra in bulk WSe2
The T-dependent Raman spectra for 2g 1 and 1g modes in bulk WSe2 samples is shown in Figure   S6a. The intensity of the 2g 1 mode in bulk WSe2 also remained less sensitive with T similar to that in 1L WSe2 which is shown in Figure 3a. The intensity ratio of the Raman active 1g mode to that of 2g 1 mode is plotted for a wide temperature range for 1L, ML and bulk WSe2 in Figure S6b which shows that the intensity ratio is higher in ML and bulk WSe2 compared to that in 1L WSe2.

AFM, Raman and PL characterization of 1L and bulk WSe2
The optical images of an ultrathin WSe2 mechanically exfoliated on SiO2/Si, is depicted in Figure   S7a in which the 1L WSe2 is highlighted by the color contrast. The AFM characterization was done using Bruker Multimode 8 AFM with Nanoscope V controller at University of North Texas, Denton. Figure S7b depicts the AFM image of the ultrathin WSe2 which shows a thickness from the scan direction of A to B of ∼ 0.77 nm which is consistent with other reported values of thickness in single layer 2D TMDCs [2][3][4]. Similarly, Figure S7c and d show the optical and AFM images of a bulk WSe2 from which the step height in the direction of C to D was calculated to be ≈6 nm. The Raman and PL characterization of these samples are shown in Figure S7e and f.