Physical origin of Davydov splitting and resonant Raman spectroscopy of Davydov components in multilayer ${\mathrm{MoTe}}_2$

We systematically study the high-resolution and polarized Raman spectra of multilayer (ML) ${\mathrm{MoTe}}_2$. The layer-breathing (LB) and shear (C) modes are observed in the ultralow-frequency region, which are used to quantitatively evaluate the interlayer coupling in ML ${\mathrm{MoTe}}_2$ based on the linear chain model, in which only the nearest interlayer coupling is considered. The Raman spectra on three different substrates verify the negligible substrate effect on the phonon frequencies of ML ${\mathrm{MoTe}}_2$. Ten excitation energies are used to measure the high-frequency modes of $N$-layer ${\mathrm{MoTe}}_2$ ($N\mathrm{L} {\mathrm{MoTe}}_2$; $N$ is an integer). Under the resonant excitation condition, we observe $N$–dependent Davydov components in ML ${\mathrm{MoTe}}_2$, originating from the Raman-active $A_1^{\prime }\left(A_{1g}^2\right)$ modes at ∼$172\mathrm{c}{\mathrm{m}}^{-1}$. More than two Davydov components are observed in $N\mathrm{L} {\mathrm{MoTe}}_2$ for $N>4$ by Raman spectroscopy. The $N$-dependent Davydov components are further investigated based on the symmetry analysis. A van der Waals model only considering the nearest interlayer coupling has been proposed to well understand the Davydov splitting of high-frequency $A_1^{\prime }\left(A_{1g}^2\right)$ modes. The different resonant profiles for the two Davydov components in 3L ${\mathrm{MoTe}}_2$ indicate that proper excitation energy of $\sim 1.8-2.2$ eV must be chosen to observe the Davydov splitting in ML ${\mathrm{MoTe}}_2$. Our work presents a simple way to identify layer number of ultrathin ${\mathrm{MoTe}}_2$ flakes by the corresponding number and peak position of Davydov components. Our work also provides a direct evidence from Raman spectroscopy of how the nearest van der Waals interactions significantly affect the frequency of the high-frequency intralayer phonon modes in multilayer ${\mathrm{MoTe}}_2$ and expands the understanding on the lattice vibrations and interlayer coupling of transition metal dichalcogenides and other two-dimensional materials.

Figure 1

(a) Optical microscope image of 1L–6L and bulk ${\mathrm{MoTe}}_2$ flakes on 300-nm ${\mathrm{SiO}}_2/\mathrm{Si}$ substrates. (b) Sample heights along the red line in the inset. Inset: the AFM image of the measured region. (c) PL spectra for 1L and 3L ${\mathrm{MoTe}}_2$ on a 300-nm ${\mathrm{SiO}}_2/\mathrm{Si}$ substrate under the 1.96-eV laser excitation.

Figure 2

(a) The Raman spectra of 1L–6L ${\mathrm{MoTe}}_2$ in the ultralow-frequency region. (b) The Raman spectra of 2L–6L ${\mathrm{MoTe}}_2$ under the parallel (red solid line) and perpendicular (blue solid line) polarization configurations. Blue and red dashed lines are used to link the LB and C modes, respectively. (c,d) The experimental frequencies and the calculated ones of the LB and C modes in 2L–6L ${\mathrm{MoTe}}_2$ based on LCM.

Figure 3

The Raman spectra of 6L ${\mathrm{MoTe}}_2$ with red, green, and blue lines for samples on ${\mathrm{SiO}}_2/\mathrm{Si}$, quartz, and sapphire substrates, respectively. The spectra are normalized to the strongest LB mode located at $\sim 10\mathrm{c}{\mathrm{m}}^{-1}$.

Figure 4

(a,b) Raman spectra of 1L–6L ${\mathrm{MoTe}}_2$ and bulk one on ${\mathrm{SiO}}_2/\mathrm{Si}$ in the high-frequency region excited by 2.28 and 1.96 eV, respectively. The corresponding irreducible representation is labeled for each mode. (c) Raman spectra of 1L–4L ${\mathrm{MoTe}}_2$ in the high-frequency region in parallel and perpendicular polarization configurations excited by 1.96 eV.

Figure 5

(a) The $A_1^{\prime }\left(A_{1g}^2\right)$ modes in 1L–6L ${\mathrm{MoTe}}_2$ excited by 1.96 eV. Raman spectra are normalized to the strongest peak and are offset for clarity. (b) The experimental result of the frequency evolution of the $A_1^{\prime }\left(A_{1g}^2\right)$ modes with layer number. (c) Normal atomic displacements for all the high-frequency modes 2L–6L ${\mathrm{MoTe}}_2$ which are derived from the $A_1^{\prime }$ mode in 1L ${\mathrm{MoTe}}_2$. The relative motion of Te atoms is schematically drawn by the left atoms with purple arrows. The corresponding irreducible is also listed for each mode, where the ${\mathrm{R}}_j$ and $\mathrm{I}{\mathrm{R}}_j$ ($j=1,2$, or 3) in the brackets are used to identify the Raman-active and infrared-active modes for each $N\mathrm{L} {\mathrm{MoTe}}_2$. All the modes are arranged in frequency from low to high from left to right.

Figure 6

(a) Schematic diagram of the vdW model for Davydov splitting in 4L ${\mathrm{MoTe}}_2$. Four Davydov components (two $A_{1g}^2$ and two $A_{2u}$ modes, bottom panel) derived from the $A_1^{\prime }$ mode in 1L ${\mathrm{MoTe}}_2$ and the corresponding four coupling modes (three LB modes and one LA mode, top panel) between four coupled ${\mathrm{MoTe}}_2$ layers are shown. The coupling frequency for the three LB modes is denoted as $\mathrm{\Delta }{\omega }_1, \mathrm{\Delta }{\omega }_2$, and $\mathrm{\Delta }{\omega }_3$, respectively. The $A_{2u}\left(\mathrm{I}{\mathrm{R}}_2\right)$ mode is the uncoupled entities with a frequency of ${\omega }_0$, and the other three modes are the coupled entities with the frequencies of ${{\omega }_c}_1,{{\omega }_c}_2$, and ${{\omega }_c}_3$, respectively. The (i) and (o) in the each atomic displacement denote in-phase and out-of-phase vibrations of Te atoms in adjacent layers, respectively. The number of out-of-phase vibrations of Te atoms in adjacent layers for each $A_{1g}^2$ or $A_{2u}$ mode is the same as that of the corresponding coupling mode. (b) The calculated frequency (open diamonds and squares) of each Davydov component of the $A_1^{\prime }\left(A_{1g}^2\right)$ modes in 3-6L ${\mathrm{MoTe}}_2$ based on the experimental (Exp.) value (gray solid circles) of the Davydov component with highest frequency and the vdW model of ${\omega }_{cj}^2-\mathrm{\Delta }{\omega }_j^2={\omega }_0^2(j=1,2,3)$, and the corresponding experimental (gray solid diamonds and squares) frequency of each Davydov component of the $A_1^{\prime }\left(A_{1g}^2\right)$ modes in 3-6L ${\mathrm{MoTe}}_2$. The experimental frequency of the $A_1^{\prime }\left(A_{1g}^2\right)$ modes in 1-2L ${\mathrm{MoTe}}_2$ is also included.

Figure 7

(a) Raman spectra of Davydov doublets of the $A_1^{\prime }$ modes in 3L ${\mathrm{MoTe}}_2$ excited by ten laser excitation energies, where the Raman intensity is normalized to the $A_3$ mode in quartz at about $465\mathrm{c}{\mathrm{m}}^{-1}$. (b) The intensity of the $A_1^{\prime }\left({\mathrm{R}}_1\right)$ (blue open diamonds) and $A_1^{\prime }\left({\mathrm{R}}_2\right)$ (red solid circles) as a function of the excitation energy. The dashed gray line is the reflectance contrast spectrum (Δ$R/R$) of 3L ${\mathrm{MoTe}}_2$ in the visible range.