Double resonance Raman modes in monolayer and few-layer ${\mathrm{MoTe}}_2$

We study the second-order Raman process of mono- and few-layer ${\mathrm{MoTe}}_2$, by combining ab initio density functional perturbation calculations with experimental Raman spectroscopy using 532, 633, and 785 nm excitation lasers. The calculated electronic band structure and the density of states show that the resonance Raman process occurs at the $M$ point in the Brillouin zone, where a strong optical absorption occurs due to a logarithmic Van Hove singularity of the electronic density of states. The double resonance Raman process with intervalley electron-phonon coupling connects two of the three inequivalent $M$ points in the Brillouin zone, giving rise to second-order Raman peaks due to the $M$-point phonons. The calculated vibrational frequencies of the second-order Raman spectra agree with the observed laser-energy-dependent Raman shifts in the experiment.

Figure 1

(a) Raman spectra of mono-, bi-, and trilayer ${\mathrm{MoTe}}_2$ under 2.33 eV (532 nm) laser excitation energy, showing the small intensity peaks ${\omega }_i$ ($i=1,2,...,6$) and an additional calibration peak from Si besides the typical Raman peaks of $A_{1g}, E_{2g}^1$, and $B_{2g}^1$ that are frequently observed in ${\mathrm{MoTe}}_2$ [5, 16]. (b) Raman spectra of monolayer ${\mathrm{MoTe}}_2$ under 2.33 eV (532 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) laser excitation energies.

Figure 2

(a) Electronic energy band structure $E\left(k\right)$ and the density of states (DOS), (b) the real (${\epsilon }^{\prime }$) and imaginary (${\epsilon }^{\prime \prime }$) parts of the dielectric function and the optical absorption coefficient $\alpha$ as a function of photon energy $E_{\gamma }$, and (c) an equienergy contour plot of the two conduction bands [(c1) and (c2)] with the same spin for monolayer ${\mathrm{MoTe}}_2$. Energy bands are split by spin-orbit interaction; the conduction bands (c1) and (c2) and the valence bands which have the same spin [6, 55] are highlighted in red solid lines in (a). When we consider the energy separation between two Van Hove singularities (VHSs) of the DOS, the peak at 1.80 eV in the absorption spectrum corresponds to the optical transition at the $M$ point [black double-headed arrow in (a)]. (c) The equienergy contour plot for VHS $E_{\mathrm{band}}^{c1}=1.52$ and 1.57 eV (left) and $E_{\mathrm{band}}^{c2}=1.72$ eV (right). All equienergy contours appear around the $M$ point. The $E_{\mathrm{band}}$ energies of 1.52, 1.57, and 1.72 eV [horizontal lines in (a)] correspond, respectively, to the experimental laser energies (1.58, 2.33, and 1.96 eV) in the blue, pink, and orange colors.

Figure 3

(a) In the double resonance process, photon-excited electrons are scattered inelastically by emitting phonons and make intervalley (i.e., $M⟷M^{\prime }, M^{\prime }⟷M^{\prime \prime }$, and $M⟷M^{\prime \prime }$) transitions, as shown by the solid double-end arrows in the first BZ. The related phonon vectors $\vec{q}$ measured from the $\Gamma$ point are also given by the six dashed arrows. (b) The schematics of the double resonance process (electron-photon and electron-phonon processes). This process generates two phonons, ($\hslash {\omega }_{p1},-\vec{q}$) and ($\hslash {\omega }_{p2},\vec{q}$).

Figure 4

(a) All possible phonon vectors $q_{{MM}^{\prime }}$ coming from the double resonance Raman process due to the laser energies of 2.33, 1.96, and 1.58 eV. The ${\vec{q}}_{M{M}^{\prime }}$ is equal to either $\overline{\Gamma M}$ (for the laser energy $E_{\gamma }=2.33$ and 1.96 eV), $\sim \frac{1}{3}\overline{MK}$ (for $E_{\gamma }=1.96$ eV), or $\sim 10$%$\overline{{MM}^{\prime }}$ (for $E_{\gamma }=1.58$ eV), as indicated by an arrow away from the $\Gamma$ point. (b) The phonon dispersion relations of a monolayer ${\mathrm{MoTe}}_2$. The second-order phonon modes are analyzed at the $M^{\prime }$ point for the laser energy $E_{\gamma }=2.33$ eV, and then at two $k$ points marked by two vertical dashed lines which are shifted leftward and rightward from the $M$ point for the laser energy $E_{\gamma }=1.58$ eV and 1.96 eV, respectively. The red, green, blue, purple, magenta, and cyan vertical arrows are used for modes ${\omega }_1, {\omega }_2, {\omega }_3, {\omega }_4, {\omega }_5$, and ${\omega }_6$, respectively. The superscripts $+, -$, and 2 for $\omega$ refer to possible “combination modes” [i.e., ${\omega }_4^{+}=A_{1g}\left(M\right)+\mathrm{LA}\left(M\right), {\omega }_6^{+}=E_{2g}^1\left(M\right)+\mathrm{LA}\left(M\right)$], “difference modes” [i.e., ${\omega }_1^{-}=E_{2g}^1\left(M\right)-\mathrm{LA}\left(M\right), {\omega }_2^{-}=E_{2g}^1\left(M\right)-\mathrm{TA}\left(M\right)$], and “overtones” [i.e., ${\omega }_1^2=2\mathrm{TA}\left(M\right), {\omega }_3^2=2\mathrm{LA}\left(M\right)$], respectively.

Figure 5

Visualization of the possible Raman-active modes at the $M$ point for the second-order process suggested in Fig. 4. Both the top and side views are given; red arrows represent the vibrational directions and amplitudes of the motion of individual atoms.