Optical evidence of type-II Weyl semimetals MoTe$_2$ and WTe$_2$

TL;DR

This study provides optical evidence supporting the classification of MoTe$_2$ and WTe$_2$ as type-II Weyl semimetals by analyzing their temperature-dependent carrier dynamics and electronic structures.

Contribution

It offers the first optical spectroscopy evidence of type-II Weyl semimetals, revealing two types of carriers and their temperature-dependent behaviors in MoTe$_2$ and WTe$_2$.

Findings

Observation of two Drude peaks indicating different carrier types.

Temperature dependence of narrow Drude peak suggests Weyl band Dirac carriers.

Optical spectra align with band structure calculations including electron correlation effects.

Abstract

The carrier dynamics and electronic structures of type-II Weyl semimetal candidates MoTe$_2$ and WTe$_2$ have been investigated by using temperature-dependent optical conductivity [$\sigma(\omega)$] spectra. Two kinds of Drude peaks (narrow and broad) have been separately observed. The width of the broad Drude peak increases with elevating temperature above the Debye temperature of about 130 K in the same way as those of normal metals, on the other hand, the narrow Drude peak becomes visible below 80 K and the width is rapidly suppressed with decreasing temperature. Because the temperature dependence of the narrow Drude peak is similar to that of a type-I Weyl semimetal TaAs, it was concluded to originate from Dirac carriers of Weyl bands. The result suggests that the conductance has the contribution of two kinds of carriers, normal semimetallic and Dirac carriers, and this observation is an evidence of type-II Weyl semimetals of MoTe$_2$ and WTe$_2$. The obtained $\sigma(\omega)$ spectra in the interband transition region can be explained by band structure calculations with a band renormalization owing to electron correlation.

Equations

$$\sigma (\omega )=\sum _{i=1}^2\frac{{\sigma }_i(0)}{{\omega }^2+{\gamma }_i^2},{\sigma }_i(0)=\frac{N_i^{\ast }{e}^2{\gamma }_i}{m}$$

Full text

I Introduction

Materials with linear band dispersions at the Fermi level ($E_{\rm F}$) such as surface states of topological insulators and Dirac semimetals have been recently attracting attention because the very high mobility as well as high speed devices are expected thanks to the very low effective carrier mass of the linear band dispersion. In the case of the space- or time-reversal symmetry breaking of Dirac semimetals, the spin degeneracy of the Dirac band is released in the momentum space, resulting in forming two Dirac bands with different spin directions at the symmetry points with respect to high symmetry axes. These materials are named as Weyl semimetals.

Weyl semimetals are classed in two groups by the shape of the Fermi surface. One named type-I exhibits point-like Fermi surfaces owing to Lorentz invariant band structures, and another named type-II has tilted Weyl cones appearing between electron and hole Fermi surfaces. $1T^{\prime}$-MoTe2 and $1T^{\prime}$-WTe2 of transition metal dichalcogenides (TMD) are candidates of the type-II Weyl semimetals Soluyanov2015 ; Sum2015 . TMD forms four crystal structures, namely $2H$, $1T$, $1T^{\prime}$, and $T_{d}$. Theoretical predictions are that MoTe2 is a semiconductor in $2H$ and $1T$ forms and the electronic structure has been revealed by optical measurements Frindt1963 ; Grasso1972 ; Davey1972 , on the other hand, that in $1T^{\prime}$ and $T_{d}$ forms is predicted as a semimetal Gao2015 .

MoTe2 with monoclinic $\beta$ structure ($P2_{1}m$, No. 11) at room temperature is regarded as a trivial metal, however, it has a phase transition at about 240 K with structure change to orthorhombic $1T^{\prime}$ structure ($Pmn2_{1}$, No. 31). The $1T^{\prime}$-MoTe2 (this is denoted as MoTe2 hereafter) is believed to be a putative Weyl semimetal Berger2018 . MoTe2 is attracting attention owing to the appearance of superconductivity at the temperatures of 0.1 K at ambient pressure and of 8.2 K at 11.7 GPa Qi2016 . Mo1-xNbxTe2 has anomalous enhancement of thermopower near the critical region between the polar and nonpolar metallic phases Sakai2016 . It is under debate whether these properties are related to the Weyl semimetallic band structure or not.

WTe2 also has orthorhombic crystal structure ($Pmn2_{1}$) below room temperature. This material has a large positive magnetoresistance Xiong2014 , and the electron and hole Fermi surfaces are the same size, which is consistent with other normal semimetals Zhu2015 . In addition to the quadratic magnetoresistance, a field-linear Nernst response distinguishes WTe2 from other dilute metals Zhu2015 .

Since both of MoTe2 and WTe2 have been theoretically predicted to be type-II Weyl semimetals Soluyanov2015 ; ZWang2016 , the investigation of the relation of the electronic structures to interesting physical properties of these materials is important and has been performed so far by using angle-resolved photoemission spectroscopy (ARPES) Jiang2017 ; Rhodes2017 ; Huang2016 ; Deng2016 ; Sakano2017 ; Tamai2016 ; Pletikosic2014 ; CWang2016 ; Sante2017 ; YWu2016 ; Wu2017 ; Bruno2016 ; Feng2016 ; Matsumoto2018 and quantum oscillation measurements Yang2017 ; Berger2018 ; Luo2016 ; Ali2014 ; Li2017 ; Zhu2015 . All measurements pointed out that electron and hole bands commonly exist on $E_{\rm F}$. By surface-sensitive ARPES measurements, Fermi arc surface electronic states have been also observed. The origin of the Fermi arcs has been concluded to be the existence of Weyl points in many experiments, but another opinion is also present Bruno2016 , so the origin is still under debate.

The evidence of the existence of Weyl points can be observed in optical conductivity [$\sigma(\omega)$] spectra. For instance, in the $\sigma(\omega)$ spectra of TaAs, which is one of type-I Weyl semimetals, the rapid decrease of the Drude width and the decrease of the Drude weight with decreasing temperature have been observed Kimura2017 ; Xu2016 . In normal semimetals, because carriers are mainly scattered by lattice vibrations, the electrical resistivity as well as the scattering rate $\gamma$ linearly increases with increasing temperature above the Debye temperature AM . On the other hand, carriers at Weyl points have different temperature-dependent $\gamma$ from normal metals owing to the increase of mobility $\mu$ ($\gamma\propto 1/\mu$) at low temperatures Shekhar2015 ; Kimura2017 . Therefore, the temperature dependence of $\gamma$ is one of important indices to judge the existence of Dirac electrons of Weyl semimallic bands.

So far, temperature-dependent $\sigma(\omega)$ spectra of WTe2 have been reported by Homes et al. Homes2015 . In the paper, owing to the model of existence of two kinds of carriers assumed as electrons and holes included in the Drude analysis, one of $\gamma$, which is equal to inverse relaxation time, becomes very small at low temperatures. They have suggested that the carriers are related to the extremely large nonsaturating magnetoresistance, but they have not discussed the existence of Weyl electrons. The anisotropic $\sigma(\omega)$ spectra of WTe2 in the $ab$ plane have been reported, and the interband spectra can be well explained by ab initio band calculations Frenzel2017 . On the other hand, even though the optical spectra of $2H$-MoTe2 have been reported previously Frindt1963 ; Grasso1972 ; Davey1972 , the optical spectra of $1T^{\prime}$-MoTe2 have never been reported with our full knowledge.

In this paper, we report optical spectra of MoTe2 and WTe2, in which the evidence of Weyl electrons commonly appears in both materials. As a result, the character of Weyl electrons can be identified by the analysis with the combination of two Drude curves, namely narrow and broad Drude components. The $\gamma$ value of the narrow Drude component is strongly suppressed at low temperatures, which is similar to that of a type-I Weyl semimetal TaAs, suggesting the existence of Weyl electrons. In addition, the electron correlation is discussed using the comparison of the experimentally obtained $\sigma(\omega)$ spectra with band calculations and calculated $\sigma(\omega)$ spectra.

II Experimental

High-quality single-crystalline MoTe2 and WTe2 samples with the size of about $2\times 0.5\times 0.1$ mm3 were synthesized by a self-flux method. One of our samples, WTe2, shows a large residual resistivity ratio of 1330 and very large magneto-resistance with obvious Shubnikov-de Haas oscillation Jha2018 ; Onishi2018 . As-grown surfaces were measured for the optical reflectivity [$R(\omega)$] spectra. Near-normal incident $R(\omega)$ spectra were acquired in a wide photon-energy range of 3 meV–30 eV to ensure accurate Kramers-Kronig analysis (KKA) Kimura2013 . Michelson-type and Martin-Puplett-type rapid-scan Fourier spectrometers were used at the photon energy $\hbar\omega$ regions of 8 meV–1.5 eV and 3–20 meV, respectively, with a feed-back positioning system to maintain the overall uncertainty level less than $\pm$0.5 % in the temperature range of 10–300 K Kimura2008 . To obtain the absolute $R(\omega)$ values, in-situ evaporation method was adopted. The obtained temperature-dependent $R(\omega)$ spectra of MoTe2 and WTe2 are shown in Fig. 1. We could not recognize the structural phase transition of MoTe2 at about 250 K by the $R(\omega)$ spectra. This suggests that the change of the electronic structure is very small. In the photon energy range of 1.2–30 eV, the $R(\omega)$ spectrum was only measured at room temperature by using synchrotron radiation, and connected to the spectra for $\hbar\omega\leq 1.5$ eV for the KKA. All measurements were performed along the $ab$-plane. In order to obtain $\sigma(\omega)$ spectra via KKA of $R(\omega)$ spectra, the $R(\omega)$ spectra were extrapolated below 3 meV with a Hagen-Rubens function, and above 30 eV with a free-electron approximation $R(\omega)\propto\omega^{-4}$ DG . The obtained $\sigma(\omega)$ spectra were compared with the spectra derived from LDA band structure calculations including spin-orbit coupling using the Wien2k code Wien2k . Lattice parameters of MoTe2 ZWang2016 and WTe2 Mar1992 with $1T^{\prime}$ crystal structure were adopted.

III Results and Discussion

III.1 Comparison with band calculations

Figure 2 indicates the band dispersions and densities of states (DOS) of MoTe2 and WTe2. Both materials are semimetals because electron and hole bands cross $E_{\rm F}$ at around the $\Gamma$ point. This result is consistent with the previously reported band calculations ZWang2016 ; Dawson1987 . Weyl points are located near the symmetry line of $\Gamma-X$ (not shown in the figure). In comparison between MoTe2 and WTe2, the dispersion curves are similar to each other, but both occupied and unoccupied states of MoTe2 are closer to $E_{\rm F}$ than those of WTe2, which can be also seen in DOS. A peak at about 0.4 eV from $E_{\rm F}$ of WTe2 shifts to about 0.2 eV in MoTe2. This causes an increase in DOS for MoTe2 at around $E_{\rm F}$, i.e., the carrier density of MoTe2 is expected to be higher than that of WTe2.

Calculated $\sigma(\omega)$ spectra obtained from band calculations in Fig. 2a in comparison with experimental ones are indicated in Fig. 3. The experimental $\sigma(\omega)$ spectra in the $ab$-plane have been obtained using non-polarized light, so the spectra are compared with the average spectra [$(\sigma_{xx}+\sigma_{yy})/2$] of $\sigma_{xx}$ and $\sigma_{yy}$ along the $a$- and $b$-axes, respectively. It should be noted that Drude structures appear in the experimental spectra in the photon energy region below 50 meV due to the existence of carriers, but it cannot be reproduced by the band calculations owing to no information on scattering rate. In WTe2, the experimental spectrum below the photon energy of 500 meV has been compared with the DFT-calculated spectrum by Homes et al.Homes2015 , and also with the DFT$+U$ calculation involving electron correlation ($U=2$ eV) Sante2017 . The latter provides better agreement with the experimental spectrum.

In the present data, even though the calculation without electron correlation has been performed, the spectrum below 500 meV, especially peaks at around 100 meV, seems to reproduce the experimental spectrum. However, a peak at $\hbar\omega$ = 550 meV in MoTe2 and two peaks at 300 and 600 meV in WTe2 in the experimental spectra cannot be reproduced by our calculation. There are peaks at 750 meV in MoTe2 and at 400 and 800 meV in WTe2 in the calculated spectra. If these peaks in the calculated spectra are regarded to move to the experimental peaks by a self-energy effect owing to an electron correlation Hewson1993 , the renormalization factor $z$ can be evaluated as $\sim 0.73$ in MoTe2 and $\sim 0.75$ in WTe2, which are almost consistent with each other. These values are larger (this means smaller electron correlation) than those of normal heavy fermion compounds Kimura2009 , but are similar to that of a valence fluctuation material YbAl2 (the Kondo temperature $T_{\rm K}\sim 2000$ K). This is consistent with a relatively smaller correlation energy $U$ of 2.0 eV Sante2017 than those of other strongly correlated electron systems Imada1998 . On the other hand, Frenzel et al. pointed out that the experimental $\sigma(\omega)$ spectra above the wavenumber of 5000 cm*-1* ($\sim 0.6$ eV) in both $a$ and $b$ axes can be explained by the DFT calculations well Frenzel2017 . All of these results suggest the weak electron correlation intensity in MoTe2 and WTe2.

The overall intensity of the experimental $\sigma(\omega)$ spectrum of MoTe2 is larger than that of WTe2. This implies that the DOS of MoTe2 near $E_{\rm F}$ is higher than that of WTe2. This is consistent with the calculated DOS shown in Fig. 2b. In addition, the experimental peaks at 100 and 50 meV in MoTe2 can be reproduced in the calculation. These peaks can be regarded to originate from flat structures in the bands near the $\Gamma$ point.

III.2 Drude analysis

Figure 4 denotes $\sigma(\omega)$ spectra below the photon energy of 500 meV and the direct current [$\sigma(0)$] at the corresponding temperatures. In both of MoTe2 and WTe2, $\sigma(0)$ values are located at the extrapolation to the lower energy side of the $\sigma(\omega)$ spectra at temperatures above 130 K, however, $\sigma(0)$ values below 40 K become one or more orders higher than the extrapolation of the $\sigma(\omega)$ spectra. It should be noted that $\sigma(0)$ at 10 K (higher than $10^{6}~{}\Omega^{-1}$cm*-1* in both materials Jha2018 ) is located out of range. However, in our accessible lowest energy range above 3 meV, upturn structures in $\sigma(\omega)$ spectra with decreasing photon energy have been observed. This suggests that the upturns are tails of the Drude curves connecting to the $\sigma(0)$ values, i.e., new conducting carriers emerge in addition to the conduction manifesting at high temperatures.

Considering these features, the $\sigma(\omega)$ spectra have been fitted by using the combination of two Drude functions as follows;

[TABLE]

Here, $m$ denotes the electron rest mass, and $\sigma_{i}(0)$, $\gamma_{i}$, and $N^{*}_{i}$ are the direct current value, the damping constant of carriers, and the effective electron number of each component, respectively.

The fitting results are shown in Fig. 5. In the fitting, the value of $\sigma_{1}(0)+\sigma_{2}(0)$ has been assumed to be equal to $\sigma(0)$. In MoTe2, at temperatures higher than 130 K, because the $\sigma(\omega)$ spectra are almost flat, a single broad Drude function (namely D2) can be applied. Below 80 K, however, another Drude component (namely D1) with very narrow peak appears below the photon energy of 10 meV in addition to the D2 component. D1 component becomes narrower with decreasing temperature. In WTe2, the temperature dependence is similar to that of MoTe2, but narrow D1 component appears even at 130–300 K, which is the same result as shown by Homes et al. Homes2015

Drude fitting results are shown in Fig. 6. In the two Drude components, firstly, let us discuss the behavior of the broad Drude component D2. In MoTe2, $N^{*}$ shown in Fig. 6a slightly decreases but is almost constant with decreasing temperature, but in WTe2, shown in Fig. 6c, $N^{*}$ below 20 K becomes about three times smaller than that above 200 K. The temperature dependence of $N^{*}$ originates from the shape of DOS near $E_{\rm F}$, i.e., WTe2 has a large change in DOS near $E_{\rm F}$ in comparison with MoTe2 as seen in Fig. 2b. Therefore, the thermal excitation of electrons is considered to be the origin of the large temperature dependence of $\sigma(\omega)$ of WTe2. Even though the calculated temperature dependence of the carrier density using DOS in Fig. 2b has similar temperature dependence to the experimental $N^{*}$, however, the calculated carrier density increases only about 1.5 times with increasing temperature from 10 to 300 K (not shown here). This inconsistency suggests that the electron-hole asymmetry in the DOS should be more accentuated than the calculation shown in Fig. 2b.

The damping constant $\gamma$ of D2 in both materials is almost flat at temperatures below 80 K and slightly increases with increasing temperature above 130 K. Since the $\gamma$ value originates from the scattering of carriers and the boundary temperature $T^{*}$ of about 100 K is consistent with the Debye temperature $\theta_{D}=133.8\pm 0.6$ K of WTe2 Callanan1992 , the temperature dependence of $\gamma$ of D2 originates from the scattering by phonons. Below $T^{*}$, $\gamma$ becomes almost flat, which suggests the existence of constant scattering by impurities and/or defects. This result is consistent with the behavior of other TMDs, i.e., the mobility ($\mu\propto 1/\gamma$) that has a tender slope below about 100 K changes to a rapid drop above the temperature ZWu2016 . Therefore, D2 component is concluded to originate from the semimetallic bands. It should be noted that $N^{*}$ at 10 K evaluated as $1.5\times 10^{20}$ cm*-3* is consistent with the sum of the electron and hole carriers obtained from the size of Fermi surfaces ($\sim 1.35\times 10^{20}$ cm*-3*) Zhu2015 . This also suggests that D2 originates from semimetallic electron and hole bands and, in addition, the effective mass of carriers in the semimetallic bands is almost equal to the rest mass of electrons.

It should be noted that the carrier densities evaluated from the DOS of MoTe2 and WTe2 in Fig. 2b are $1.3\times 10^{22}$ cm*-3* and $4.3\times 10^{21}$ cm*-3*, respectively, which are one order of magnitude larger than the experimentally obtained values. According to the previous quantum oscillation and ARPES data, however, the experimental Fermi surface size is much smaller than the calculated one Sante2017 ; Zhu2015 ; CWang2016 ; Pletikosic2014 ; YWu2016 ; Bruno2016 . Therefore, the evaluated carrier density here is not inconsistent with other experiments.

Next, the narrow Drude component D1 is discussed. In both materials, $N^{*}$ rapidly increases with decreasing temperature below about 100 K. This change cannot be expected from the shape of DOS. In addition, as pointed out by Homes et al. Homes2015 , one of two semimetallic bands of the compensated metals is also a candidate. However, since the sizes of the Fermi surfaces of electron and hole bands are the same as each other Zhu2015 and the dispersions of the bands are also similar to each other Pletikosic2014 , the Drude parameters of the electron and hole bands should be similar to each other. Therefore, it is hard to believe that these two show different temperature dependences. While, $\gamma$ of D1 rapidly decreases with decreasing temperature below 80 K in contrast to the flat of D2. The temperature dependence of $\gamma$ is very similar to that of type-I Weyl semimetals, TaAs and TaP Kimura2017 . Therefore, D1 is considered to reflect the character of Dirac electrons at Weyl points of MoTe2 and WTe2. From this point of view, the rapid increase of $N^{*}$ with decreasing temperature suggests that the rapid decrease of the effective mass $m^{*}$ because of $m^{*}/m=N_{0}/N^{*}$, where $N_{0}$ is the actual carrier density. This suggests the presence of zero mass Dirac electrons.

The temperature dependence of the $\gamma$ of WTe2 follows $\gamma\propto T^{1.6\pm 0.05}$ below the temperature of 80 K as shown in Fig. 6b. The extrapolation of the $\gamma$ to zero temperature is expected to be zero, in other words, the mobility becomes infinity at 0 K. This suggests that the Weyl points are located just at $E_{\rm F}$. It should be noted that the relation of $\gamma\propto T^{1.6\pm 0.05}$ is close to the $T^{2}$-law of Weyl semimetals predicted by Hosur et al. Hosur2012 . On the other hand, the $\gamma$ of MoTe2 has the temperature dependence of $\gamma\propto T^{1.5\pm 0.05}$ at temperatures higher than 40 K, which is similar to that of WTe2 below 80 K. The temperature dependence of $\gamma$ cannot be explained by the Fermi liquid theory but is similar to those of non-Fermi liquid materials Degiorgi1999 ; Kimura2006 . The temperature order of $\gamma$ of MoTe2 decreases at temperatures lower than 40 K, which has the relation of $\gamma\propto T^{0.8\pm 0.1}$. This suggests that the Weyl points of MoTe2 are located a little away (about 40 K $\sim$ 3 meV) from $E_{\rm F}$ because of a residual $\gamma$ value due to the existence of massive electrons and/or the electron correlation intensity increases with approaching $E_{\rm F}$. An excitonic correlation might be effected to the origin Matsumoto2018 . The low-energy electron correlation near the Weyl points might effect to the physical properties such as superconductivity and high thermopower of MoTe2.

The evidence for Dirac/Weyl semimetal phase in optical sepctra also appears in the power-law dependence of the interband optical conductivity$\sigma(\omega)$ spectra Carbotte2016 ; Shao2019 . The power-law dependence must appear in the energy region of Dirac linear bands. In the case of MoTe2 and WTe2, it must appear below 50 meV with reference to band calculations. However, since Drude component is dominant in the energy region, the power-law dependence may not be visible. In other Weyl semimetals, TaAs and TaP, such power-law behaviors can be recognized because of weak Drude components.

IV Conclusion

To summarize, the electronic structures of type-II Weyl semimetal candidates MoTe2 and WTe2 have been investigated by optical conductivity $\sigma(\omega)$ spectra and LDA band calculations. Obtained $\sigma(\omega)$ spectra can be explained by the band calculations with the renormalization factor of $0.74\pm 0.01$. Two kinds of Drude components have been observed below the photon energy of 40 meV. One of the two components originates from the carriers of semimetallic electron and hole bands, the other from Dirac carriers at the Weyl points located at $E_{\rm F}$ that emerge only at low temperatures and the width of the Drude component become sharper continuously with decreasing temperature. Therefore, MoTe2 and WTe2 have two kinds of carriers and are concluded to be type-II Weyl semimetals.

Acknowledgments

Part of this work was performed under the Use-of-UVSOR Facility Program (BL7B, 2016–2017) of the Institute for Molecular Science. This work was partly supported by JSPS KAKENHI (Grant No. 15H03676).