# Raman Spectra of ZrS\({}_{2}\) and ZrSe\({}_{2}\) from Bulk to Atomically Thin Layers
## Appendix C Raman Spectra
Broad range spectra as well as zooms in the region of the A\({}_{\rm 1g}\) and E\({}_{\rm g}\) modes with their corresponding fitting are presented. Fitting results are given in Table 13.  
The Raman spectra measured at three different wavelengths (532 nm, 635 nm, and 785 nm) for two thick flakes of ZrS\({}_{2}\) and ZrSe\({}_{2}\) are shown in Figure 6. Comparing the different spectra for each material (and, in particular, focusing on the intensity of the A\({}_{\rm{1g}}\) mode), we can conclude that an optimized signal can be observed at 532 nm for ZrS\({}_{2}\) and at 638 nm for ZrSe\({}_{2}\). Figure 6: Raman spectra measured at different wavelengths (\(\lambda\) = 532 nm, 638 nm, and 785 nm) for (**a**) ZrS\({}_{2}\) and (**b**) ZrSe\({}_{2}\). The intensity has been normalized to the height of the silicon phonon peak. The Raman modes A\({}_{\rm{1g}}\) and E\({}_{\rm{g}}\) are labeled. For ZrS\({}_{2}\), the shoulder marked with ‘**’ is attributed to a broadening of the A\({}_{\rm{1g}}\) peak due to non-harmonic effects. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Thickness (\(\pm\)1 Layer)**} & \multicolumn{3}{c}{E\({}_{\rm{g}}\)} & \multicolumn{3}{c}{**A\({}_{\rm{1g}}\)**} \\ \cline{2-7}  & **Position (cm\({}^{-1}\))** & **Intensity (Counts/s)** & **FWHM (cm\({}^{-1}\))** & **Position (cm\({}^{-1}\))** & **Intensity (Counts/s)** & **FWHM (cm\({}^{-1}\))** \\ \hline
48 L & 145.80 \(\pm\) 0.19 & 5.4 & 3.31 \(\pm\) 1.16 & 195.36 \(\pm\) 0.02 & 291.9 & 3.59 \(\pm\) 0.08 \\
14 L & - & - & - & 195.67 \(\pm\) 0.02 & 123.8 & 3.66 \(\pm\) 0.09 \\
9 L & - & - & - & 195.68 \(\pm\) 0.09 & 3.8 & 3.1 \(\pm\) 0.5 \\
5 L & - & - & - & 195.20 \(\pm\) 0.07 & 2.5 & 2.7 \(\pm\) 0.4 \\ \hline \end{tabular}
\end{table} 
Table 3: Thickness dependence of the fitted parameters (position, intensity, and FWHM) of the E\({}_{\rm{g}}\) and A\({}_{\rm{1g}}\) modes obtained from the Raman spectra for ZrSe\({}_{2}\) on a Si/SiO\({}_{2}\) substrate (\(\lambda\) = 638 nm) presented in Figure 5.  
In order to compare crystals degradation under ambient conditions, the Raman spectra for bulk crystals of freshly cleaved specimens and uncleaved ones are shown in Figure 17. The intensity has been normalized to the height of the A\({}_{\mathrm{1g}}\) peak. Appendix D E\({}_{\mathrm{g}}\) and A\({}_{\mathrm{1g}}\) Raman Modes in ZrX\({}_{\mathrm{2}}\)  
The primitive cell of the crystal has three atoms (one formula unit) and thus nine vibrational modes: three of them are acoustic and the other six are optical modes. The decomposition by group theory (at the Gamma point) is (assuming the Wyckoff positions \(a\) and \(d\) for the Zr and the S, respectively):  
\[\Gamma=\,\mathrm{A_{1g}}\,+\,\mathrm{E_{g}}\,+\,2\mathrm{A_{2u}}\,+\,2\mathrm{ E_{u}}. \tag{1}\]  
Half of the modes are even (gerade) and the other half are odd (ungerade) because the space group has a center of symmetry. The A modes are one-dimensional (the atoms move in the z direction), while the E modes are two-dimensional (the displacement can be, equivalently, in the x or y direction). Since the Raman modes are even, the only option is that the Zr atom remains at rest, while the two S atoms move in opposite directions. There is no dipole moment, but a change in the Raman polarizability. In the case of the A\({}_{\mathrm{1g}}\) mode, there is a plane of S atoms against the other (like a stretching mode) while in the case of the E\({}_{\mathrm{g}}\) mode, the S planes slide one against the other (see Figure 1c). Thus, the first mode must have a substantially higher frequency as compared to the second mode. In fact, as already stated in the literature [33], the E\({}_{\mathrm{g}}\) mode has a frequency of 251 cm\({}^{-1}\), while A\({}_{\mathrm{1g}}\) has a frequency of 333 cm\({}^{-1}\). Now, following a simple rule, the frequency can be taken as:  
\[\omega\,=\,(\mathrm{k/M})^{1/2}, \tag{2}\]  
where k is the spring constant [N/m].