Topologically Nontrivial Interband Plasmons in Type-II Weyl Semimetal MoTe$_2$

In many realistic topological materials, more than one kind of fermions contribute to the electronic bands crossing the Fermi level, leading to various novel phenomena. Here, using momentum-resolved inelastic electron scattering, we investigate the plasmons and their evolution across the phase transition in a type-II Weyl Semimetal MoTe$_2$, in which both Weyl fermions and trivial nonrelativistic fermions contribute to the Fermi surface in the Td phase. One plasmon mode in the 1T' phase at high temperature and two plasmon modes in the topological T$_d$ phase at low temperature are observed. Combining with first-priciples calculations, we show that all the plasmon modes are dominated by the interband correlations between the inverted bands of MoTe$_2$. Especially in the T$_d$ phase, since the electronic bands split due to inversion symmetry breaking and spin-orbit coupling, the plasmon modes manifest the interband correlation between the topological Weyl fermions and the trivial nonrelativistic electrons. Our work emphasizes the significance of the interplay between different kinds of carriers in plasmons of topological materials.

Most of these fascinating plasmons in topological materials can be well described within a single band picture with only one kind of carriers involved. This is similar to the cases in normal metals, in which the intraband correlation dominates [34]. Consequently, the origin of these plasmons can be directly understood in terms of the simple Fermi surfaces. However, in many topological materials, such as NbP and (W, Mo)Te2, the topologically nontrivial bands and the trivial bands coexist around the Fermi level, and thus more than one kind of fermions contribute to the Fermi surface [35][36][37][38][39][40][41][42][43][44]. As the result, the plasmon modes in these systems may be contributed from multiple kinds of Fermions. Especially, when the topologically nontrivial and trivial bands highly mix with each other, their couplings could significantly affect the properties of the plasmons, which is beyond the scope of the single band picture. Therefore, the exotic plasmon properties based on the assumption of pure topological bands may not be applicable in real topological materials.
In this paper, using high-resolution electron energy loss spectroscopy (HREELS) [45], we studied the plasmons in MoTe2, which exhibits the structural phase transition from the high temperature monoclinic 1T' phase to the orthorhombic Td phase with the critical temperature of ~250 K. We observe one plasmon mode in the 1T' phase and two plasmon modes in the Td phase.
The energies of these plasmon modes are almost dispersionless and exhibit distinct nonlinear temperature dependences. Combining with first-principles calculations, we reveal that unlike the conventional plasmons dominated by the intraband correlations, all these plasmon modes in MoTe2 mainly originate from the interband correlations. Especially in the Td phase, the two plasmon modes are dominated by the correlations between the nontrivial Weyl Fermion bands and the trivial nonrelativistic electronic bands. Our work clearly demonstrates that the interplay of nonrelativistic fermions and Weyl fermions plays a crucial role in plasmons of realistic topological materials.

A. Crystal Preparation
Single crystal of 1T'-MoTe2 were grown by using Te as flux. Starting materials Mo (Column, 99.9999%) and Te (Lump, 99.9999%) were mixed in an Ar-filled glove box at a molar ratio of Mo : Te = 1 : 20. The mixture was placed in an alumina crucible, which was then sealed in an evacuated quartz tube. The tube was heated to 1100 °C over 20 hours and dwelt for 10 hours.
Then, the tube was slowly cooled down to 950 °C at a rate of 1 °C/h followed by separating the crystals from the Te flux by centrifuging. Shiny crystals with the size of 1×5 mm 2 were obtained on the bottom of the crucible.
The good crystal quality is characterized ex situ by X-ray diffraction (XRD) and the surface of the cleaved sample is checked by X-ray photoelectron spectroscopy (XPS), which can exclude the possible surface contamination or oxidation of them. The details of the sample characterizations are shown in Appendix A.

B. HREELS Measurement
As a surface sensitive technique, HREELS is an ideal candidate to explore the low-energy collective excitations of MoTe2. Compared with conventional HREELS, our recently developed two-dimensional (2D)-HREELS can directly obtain a 2D energy-momentum mapping in a very large momentum scale without rotating sample, monochromator, or analyzer [46].
The energy and momentum of the collective excitations (either plasmon or phonon) are obtained using the conservation of energy and momentum for incident and scattered electrons.
As given by ℏ ‖ = ℏ( sin − sin ) (where and are the incident and scattering angles, respectively), the parallel momentum ‖ depends on incident energy , energy loss , and according to In this study, all the HREELS measurements were performed in situ within ~10 hours after fresh cleavage in ultra-high vacuum (~1×10 -10 Torr). We obtained the information around the first Brillouin zone center  with the incident energy ranging from 15 eV to 110 eV at room temperature. And the temperature-dependent measurements were obtained with the incident energy of 110 eV only.

C. Details of First-principles Calculations
The first-principles calculations are performed by using Vienna ab initio simulation package

D. Calculations of Dynamical Dielectric Function
Within the random phase approximation (RPA), the dynamical dielectric function RPA ( , ) q  can be calculated from the formula: where the first term is the intraband term with a broadening parameter η1 and the second term is the interband term with a broadening parameter η2.

A. Crystal Structures of MoTe2
MoTe2 is a layered van der Waals material with two different structures at 300 K: hexagonal (2H) phase or monoclinic (1T') phase, due to different growth conditions. In this study we focus on the 1T' phase (a trivial metal/semimetal), which exhibits a structural phase transition to the Td phase at ~ 250 K [51]. Although these two phases share the same in-plane structure, the Td phase share the same in-plane structure. The angle-resolved inelastic electron scattering was performed by the HREELS with the capability of 2D energy-momentum mapping [46]. The results in both the 1T' and Td phases at different temperatures were obtained.

B. Plasmons from HREELS Measurements
Figures 2(a) and 2(b) display the 2D energy-momentum mappings obtained from the HREELS measurements along the X  direction with the incident beam energy of 110 eV for the Td phase at 44 K and for the 1T' phase at 294 K, respectively. In the Td phase, the energy distribution curve (EDC) integrated over momentum [red curve in Fig. 2(a)] shows two distinct energy-loss peaks around 90 and 170 meV, which are labeled as α and β, respectively. In contract, the integrated EDC for the 1T' phase only shows one energy-loss peak around 220 meV, which is labeled as γ. The energies of all these peaks are much higher than the highest phonon energy (around 35 meV) in MoTe2 [52], indicating that they are not phonons. Through XPS measurement to test the 3d binding energy of Te and Mo, and the comparison of the HREELS spectra between clean and exposed surfaces, we show that these loss peaks are not vibrations of possibly adsorbed molecules (e.g., H2O) or the Te-O bond due to possible surface oxidation.
Instead, they are most likely to be plasmons.
Then the dispersions of these modes are checked from q-dependent EDCs extracted from the 2D mapping, as shown in Figs. 2(c) and 2(d). Due to the semimetallic nature of MoTe2, there exists a strong Drude background in the energy loss EDCs. We employ a background subtraction method based on a polynomial fitting of the baseline, which has been used in graphene [53] and graphite [54], to extract the information of the exact energy loss peaks, including energy, full width at half maximum (FWHM), and intensity (see details in Appendix B). The obtained experimental energy-momentum points of these modes are plotted in Figs. 2(e-h) (solid dots).
All of the three modes are almost dispersionless, and their intensities show quick damping with the increasing momentum q, becoming invisible when q > 0.05 Å -1 . In addition, their energies are nonzero at q = 0, evidencing that they are plasmons originating from bulk bands. In the Td phase, the average ratio of energy between α and β is ~ 0.56 ± 0.04, significantly deviating from the conventional ratio between the energies of bulk plasmon (bp) and its corresponding surface plasmon (sp) from the same electronic band: Details of the calculation methods and analyses can be found in Appendix D.

C. Temperature-dependence of Plasmons
To gain more insights into the behaviors of plasmons accompanied with the structural phase transition, we performed temperature-dependent HREELS measurements from 294 K to 44 K. Figure 3(a) shows the stacked EDCs at  point at several temperatures (a complete set of temperature-dependent measurements is provided in Appendix C). With the temperature decreasing, the γ mode gradually evolves into the β mode, while the α mode gradually appears.
The energies and intensities of these three modes as a function of temperature are extracted by using the fitting method mentioned above, with the results plotted in Figs. 3(b) and (c). When the temperature is above ~260 K, the data can be only well fitted with one peak. And when the temperature is below ~200 K, the data can be only well fitted with two peaks. It should be noted that, in the temperature range from 200 K to 260 K where the structural phase transition temperature TC locates, the data can be fitted equally well either with one peak or with two peaks.
This temperature range was marked by a gray rectangle in Figs  temperature-dependent behaviors. In the Td phase below TC, the energy of the β mode shows a very weak temperature-dependence, while the α mode shows a very strong nonmonotonic temperature-dependence. In the 1T' phase above TC, the energy of the γ mode shows very slight increase with increasing temperature. It has been theoretically predicted that the energy of the intrinsic plasmon in type-I Weyl/Dirac semimetals has a non-linear temperature-dependence when only a single Weyl/Dirac cone contributes to the Fermi surface [13,14], similar to the temperature dependence of the α mode in our experiment. However, since the electronic structure of MoTe2 is more complicated than the single Weyl/Dirac cone model, and all of the three plasmon modes mainly originate from the interband correlations. As a result, the different temperature dependences of energies of these modes should be hard to be faithfully captured by the theoretical calculations based on the simple ideal model.  [9]. Accompanied with the structural phase transition from the 1T' phase to the Td phase, the bands a, b, c split into a1/a2, b1/b2, and c1/c2 due to the inversion symmetry breaking, forming eight Weyl points at the crossing points of the bands a1/a2 and b1/b2. Accordingly, the γ mode of the 1T' phase splits into two plasmon modes, α and β. The interband correlations between a2 and c1 contribute to the α mode while the correlations between a1 and c2 contribute to the β mode [shown in Fig. 4(d)].
These two plasmon modes in the Td phase are dominated by the interband correlations between the topologically nontrivial bands and trivial bands.

IV. Conclusions
In conclusion, we have systematically investigated the plasmon modes of MoTe2 at different temperatures. Our HREELS experiments indicate the presence of two plasmons (α and β) in the low temperature Td phase and only one mode (γ) in the high temperature 1T' phase. Combining with first-principles calculations, we find that all the modes are dominated by interband correlations between the inverted bands of MoTe2. Especially, the modes in the topological Td phase, dominated by the interband correlations between the topologically nontrivial bands and trivial bands, are attributed to the band splitting due to inversion symmetry breaking. This work reveals the limitation of the single band picture and significantly broadens the understanding of plasmon modes in realistic topological materials, in which band mixing usually exists. Figure 5 shows the X-ray diffraction (XRD) pattern measured from the (001) surface of MoTe2 single crystal that we used in our HREELS study. The XRD pattern shows a series of (00l) peaks, indicating good crystalline quality. By comparing with the reported XRD data in the existing studies of MoTe2 topological properties [39,57,58], the crystalline quality in our study is better than most of the reported samples.

Appendix B: Fitting of the HREELS spectra
The typical 2D-HREELS data set is a mapping like Fig. 2(a) or Fig. 2(b). We can extract EDCs at different momentum and fit the curves. Figure. 7 shows a typical fitting case. respectively. In particular, the normalized height of these energy loss peaks relative to the elastic peak is extremely low (~10 -4 ), which needs a rather critical analysis to obtain the intrinsic information. As MoTe2 is a semimetal, there's a huge Drude background, which can be described as an analytical form where the spectrum intensity () fx is a function of x (energy loss) and the coefficients 0 x , A, B, C and D to extrapolate the background. This background form was used in fitting the plasmons in graphite [54] and graphene [53]. Fitting the subtracted spectra with Lorentz lines shape can obtain the exact information of loss peaks, including energy, full width at half maximum (FWHM), height and area/intensity. Typical subtracted spectra and the fittings are illustrated in Fig. 7(b).   Fig. 9, the HREELS spectra along these three directions are illustrated, indicating the plasmon modes are isotropic.
All other data presented in the manuscript are those collected along the X  direction, except that the data in Fig. 11 is from the Y  direction. Moreover, the intensity ratio between the surface plasmon and the corresponding bulk plasmon should increase with the increasing incident angle (defined as the angle between the incident electron beam and the surface normal) [59], since the electrons penetrate deeper at smaller incident angle. Here, the intensity ratio of α and β modes is weakly dependent on the incident angle, which is demonstrated in Fig. 11. These analyses suggest that β and α cannot be the conventional bulk plasmon and its corresponding surface plasmon. But it may not be rigorous to verify the nature of α and β only based on the energy ratio, since the energy ratio between the surface and bulk plasmons may become complicated when band structure effects are operative. Yet considering there is only one mode (γ) observed above 250 K, the same measurement at lower temperature observing two modes should be the result of the phase transition instead of a surface effect.

HREELS intensity check with different incident energies
We performed the HREELS measurements with the incident energy ranging from 15 eV to 110 eV at room temperature. The results are plotted in Fig. 12. The worse signal-to-noise ratio of the 50 and 80 eV data is due to the shorter data acquisition time used than the other incident energies. The peak position and intensity of the plasmon peak shows very weak dependence on the incident energy. We chose 110 eV to perform most of measurements in this study, since 110 eV is the mostly used incident energy in our facility and thus can generate the most stable beam intensity during the measurements.
The independence of the loss peaks on the incident energy can also provide auxiliary evidence that the measured modes are bulk plasmons. The mean-free path of incident electrons is strongly dependent on the incident electron energy. If the loss peak were from surface plasmon, the intensity would be strongly dependent on the incident energy.

Additional temperature-dependent HREELS data
In addition to the data shown in Fig. 3(a) in the main manuscript, here in Fig. 13 we also provide the temperature-dependent data with more dense temperature points, for the sake of completeness.   Fig. 4 in the main text.