Determination of the thickness and orientation of few-layer tungsten ditelluride using polarized Raman spectroscopy

Figure 1(a) shows the crystal structure of WTe₂ with orthorhombic, distorted-1T structure with space group Pmn2₁ [2, 21–23, 26, 28]. The a-axis is defined as the zigzag direction of the W–W chains. Figure 1(b) is a low magnification transmission electron microscope (TEM) image of a thick (bulk) WTe₂ sample on TEM grid, and figure 1(c) is a corresponding transmission electron diffraction (TED) pattern. The a-axis direction determined from the TED pattern is indicated in figure 1(b). Unlike the case of black phosphorus, another anisotropic two-dimensional material, long straight edges do not necessarily correspond to a principal axis (supplementary figure 1).

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Figure 2(a) shows polarized Raman spectra of this sample in parallel polarization, using the excitation wavelength of 632.8 nm. The polarization angles are measured with respect to the a-axis direction determined from the TED measurement. The spectrum varies significantly depending on the polarization, and so the spectra taken in three representative polarizations are shown. For convenience, those peaks that are clearly resolved are numbered from the lowest frequency peak (P1). The optical phonon modes of space group Pmn2₁ are A₁, A₂, B₁, and B₂ [2, 21–23, 26, 28, 29], but B₁ and B₂ modes are forbidden in the back-scattering geometry. The complex Raman tensors, which take absorption into account [30–34], of A₁ and A₂ modes can be written as

$$\begin{eqnarray}{A}_{1}=\left(\begin{array}{ccc}| a| {{\rm{e}}}^{{\rm{i}}{\phi }_{a}} & 0 & 0\\ 0 & | b| {{\rm{e}}}^{{\rm{i}}{\phi }_{b}} & 0\\ 0 & 0 & | c| {{\rm{e}}}^{{\rm{i}}{\phi }_{c}}\end{array}\right),\\ {A}_{2}=\left(\begin{array}{ccc}0 & | d| {{\rm{e}}}^{{\rm{i}}{\phi }_{d}} & 0\\ | d| {{\rm{e}}}^{{\rm{i}}{\phi }_{d}} & 0 & 0\\ 0 & 0 & 0\end{array}\right).\end{eqnarray} \tag{ 1 }$$

The Raman intensity in parallel polarization is given by

$$\begin{eqnarray}&&I({A}_{1})\propto {| a| }^{2}{{\rm{\cos }}}^{4}\theta +{| b| }^{2}{{\rm{\sin }}}^{4}\theta \\ &&\quad +2| a| | b| {{\rm{\cos }}}^{2}\theta {{\rm{\sin }}}^{2}\theta \,{\rm{\cos }}\,{\phi }_{ab}\end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray}&&I({A}_{2})\propto {| d| }^{2}{{\rm{\cos }}}^{2}\theta {{\rm{\sin }}}^{2}\theta ,\end{eqnarray} \tag{ 3 }$$

where ${\phi }_{ab}={\phi }_{a}-{\phi }_{b},$ and the angle $\theta$ is measured with respect to the a-axis. By comparing with equations (2) and (3), we can assign P1, P2, P5, P6, P7, P8, P10, P11, and P12 as A₁ modes, and P3, P4, and P9 as A₂ modes (supplementary figure 2), which is consistent with theoretical predictions [2, 21–23, 26]. The atomic displacements of some of these modes can be found in [21, 23, 26]. Figure 2(b) shows that the polarization dependence of the A₁ modes can be used to determine the crystallographic orientation of the sample: the polarization direction in which the intensity of P2 is maximum corresponds to the b-axis.

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Figure 3(a) shows the Raman spectra as a function of thickness. For the low-frequency range (<35 cm⁻¹), the Raman spectra for the polarization angle of 0° are shown because the low-frequency peaks have the maximum intensity at this polarization angle (supplementary figure 3). For the higher frequency range (>70 cm⁻¹), the spectra for 45° polarization are shown because some closely spaced peaks (P9 and P10 or P11 and P12) are best resolved at this polarization angle. (See supplementary figures 4–6 for spectra in different polarizations.) The low-frequency region of the spectrum shows the most pronounced thickness dependence. Both shear and breathing modes are observed. The breathing mode is stronger for the 632.8 nm excitation, but the shear mode is only slightly stronger for the 514.5 nm excitation. Both modes have the same polarization dependence (supplementary figure 3). The sharp peak at ∼9 cm⁻¹ for bulk WTe₂ seems to be an overlap of the breathing mode and a shear mode, but the sharpness of the peak seems to indicate that the contribution from the shear mode is stronger. In the higher frequency range, several peaks show clear shift with thickness. There have been several attempts to correlate the Raman spectrum with the thickness of WTe₂ [8, 22, 23, 26, 27]. The difference between the two Raman peaks positioned at ∼160 cm⁻¹ (P10) and ∼210 cm⁻¹ (P11 and P12) has been proposed as an indicator of the thickness of WTe₂. However, because the Raman peak near ∼210 cm⁻¹ is in fact a superposition of two separate Raman peaks, P11 and P12, with strong polarization dependence (supplementary figure 4), its position is not well defined unless the polarization is specified. The significant scatter of the reported separation between the peak at ∼210 cm⁻¹ and the peak at ∼160 cm⁻¹ (supplementary figure 7) can be thus explained. We find that the separation between P2 and P3 also depends on the thickness. Figures 3(b) and (c) summarize the positions of P2, P3, P10, P11, and P12 as a function of thickness. Figure 3(d) compares the difference between these peaks also as a function of thickness. It is clear that all three differences, P3–P2, P11–P10, and P12–P10, correlate well with thickness. However, since one has to deconvolute the peak at ∼210 cm⁻¹ to determine the positions of P11 and P12, their positions are less reliable. A small difference between the data from 632.8 and 514.5 nm excitations is a result of this limitation. For this reason, we propose that P3–P2 is a more reliable measure of the thickness, because these peaks are not convolution of multiple peaks. Also, because P3 is very weak for 514.5 nm but strong and clearly resolved for 632.8 nm, we suggest that the excitation wavelength of 632.8 nm is more preferable (supplementary figure 5).

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Figure 4(a) shows the polarization behaviors of some representative peaks measured with the 632.8 nm excitation. We take the a-axis direction (0°) of few-layer WTe₂ so that the polarization dependences of the Raman peaks coincide with those of bulk WTe₂. The polarization behaviors are essentially identical for all thicknesses, which give credence to the above assignment of the a-axis direction. On the other hand, figure 4(b) shows that for 514.5 nm excitation, there is a strong thickness dependence of the polarization behavior. For example, the polarization behavior of P2 is similar to that for 632.8 nm excitation for all thicknesses except for mono-layer WTe₂, which shows an opposite polarization dependence. In the case of P11, the polarization behavior varies gradually from mono-layer to bulk WTe₂. (See supplementary figures 6, 8, and 9 for complete data.) Figure 4(c) compares the polarization behaviors of some of the Raman peaks from mono-layer WTe₂, measured with excitation wavelengths of 632.8 and 514.5 nm.

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Similar anomalous polarization behavior that depends on the excitation wavelength or the sample thickness was observed in black phosphorus. The anisotropic electron–phonon interaction as well as the interference effect combined with birefringence and dichroism due to anisotropy could partly explain this anomalous behavior [31–33, 35]. Unlike black phosphorus, WTe₂ has only weak birefringence (supplementary figure 10), and so its contribution to the anomalous polarization behavior, if any, would not be large. We propose that dichroism is the major factor that contributes to this anomalous behavior. If we compare the polarization dependence of the Raman intensities with equation (2), we find that the variation of the polarization behaviors with the thickness or the excitation wavelength can be accounted for by varying the phase factor, ${\phi }_{ab}={\phi }_{a}-{\phi }_{b}.$ The phase factors in the Raman tensor elements are related to the imaginary part of the dielectric function, which is directly correlated with optical absorption. Because the band structure determines the optical absorption through the joint density of states, the phase factors reflect the differences in the band structure. The band structure of WTe₂ depends sensitively on the number of layers [4, 22], the observed thickness dependence can be explained in terms of the thickness-dependent dielectric function. The excitation wavelength dependence is understood in the same way because the anisotropy in the dielectric function would depend on the wavelength for a given thickness. Supplementary figures 8 and 9 summarize the polarization dependence of all the peaks. The A₂ modes do not show any dependence on the excitation wavelength or the thickness because the phase factor does not affect the intensity in equation (3). The polarization behavior of each A₁ mode except P5 does not vary much with thickness for the 632.8 nm excitation but shows significant variation for the 514.5 nm excitation. This is somewhat different from the case of black phosphorus, in which a shorter-wavelength excitation showed less variation with thickness [32]. An exception is P9, which appears to be an A₂ mode for 632.8 nm but an A₁ mode for 514.5 nm. Because P9 is relatively weak for the 514.5 nm excitation, its deconvolution from P10 is less reliable than for the 632.8 nm excitation. We tentatively assign P9 as an A₂ mode which is also consistent with theoretical calculations [2, 21, 22, 26].

WTe₂ samples were mechanically exfoliated on p-type silicon substrates covered with 90 nm thick silicon dioxide. Because WTe₂ can get degraded in air [8, 21, 23] (supplementary figure 11), we placed the samples in an optical chamber with ultrapure N₂ environment immediately after exfoliation. After suitable thin WTe₂ samples were identified using an optical microscope, the substrate with WTe₂ samples was transferred to a micro vacuum chamber for Raman measurements. All Raman measurements were carried out in a vacuum of 10⁻⁵ Torr in order to minimize degradation. A long-working-distance objective (40×, N.A. 0.6) with a correction ring was used to focus the laser beam of 514.5 and 632.8 nm wavelengths to a spot size of about 1 μm. The Raman signal was dispersed by a spectrometer with a focal length of 550 mm with a grating with 2400 grooves mm⁻¹ and detected using a charge-coupled device. For low frequency Raman measurements in the range below 100 cm⁻¹, volume holographic filters were used to clean the laser lines and to eliminate the Rayleigh-scattered light (supplementary figure 12). A half-wave plate and a polarizer were used to control the polarization of the excitation laser and another half-wave plate was used to compensate for the polarization dependent throughput of the spectrometer (supplementary figure 12). Raman scattered light with the same polarization as the excitation laser (parallel polarization) was measured. The laser power was kept at 300 μW to avoid laser-induced damages to the sample. The slight day-to-day fluctuations in the optical alignment, etc, are corrected for by normalizing against the intensity of the Raman peak from a piece of Si, by measuring the same piece in the same polarization. The number of layers of WTe₂ was determined by comparing the optical contrast and atomic force microscopy (AFM) measurements after all the Raman measurements were completed (supplementary figure 13). The inter-layer spacing of bulk WTe₂ is about 0.7 nm, but the height of monolayer WTe₂ measured by AFM is about 1 nm, similar to other TMD materials.

For TED measurements, exfoliated WTe₂ flakes on the SiO₂/Si substrate were transferred onto a TEM grid by a wet direct transfer method. In detail, for comparison between Raman and TED data, TED analysis should be performed at the particular flakes characterized by Raman spectroscopy. The wet direct transfer method without polymer coating is adopted for deterministic transfer because this method is capable of fixing the grid on a particular flake through optical microscopy. To bond WTe₂ flake onto the amorphous carbon film on TEM grids, isopropyl alcohol (IPA) was dropped on top of the grid. As IPA evaporates, surface tension induces tight bonding between amorphous carbon and the WTe₂ flake [36]. Then, for detaching the grid bonded with the flake from the SiO₂/Si substrate, a KOH solution was used for etching away SiO₂. In order to minimize damages on WTe₂, we soaked the grid/flake/SiO₂/Si stack in the KOH solution to etch the surface of SiO₂ rapidly. After a few seconds, the grid with the WTe₂ flake was detached from the SiO₂/Si substrate. Rinsing the KOH residue was the final stage for the preparation of TEM specimens. Electron diffraction pattern analysis was carried out using a TEM, FEI Titan³ G2 60-300, operated at 80 kV acceleration voltage.