Three-dimensionality of the bulk electronic structure in WTe2

TL;DR

This study combines resistivity and ARPES measurements to reveal the three-dimensional bulk electronic structure of WTe2, showing multiple electron and hole pockets and demonstrating strong 3D characteristics.

Contribution

It provides a detailed, experimentally verified picture of WTe2's 3D electronic structure using tunable ARPES and quantum oscillation data, highlighting the material's three-dimensionality.

Findings

Bulk Fermi surface includes three pairs of electron pockets and two pairs of hole pockets.

Photon energy tuning reveals additional tiny electron pockets, indicating 3D electronic structure.

Resistivity and ARPES data are consistent in describing the electronic structure.

Abstract

We use temperature- and field-dependent resistivity measurements [Shubnikov--de Haas (SdH) quantum oscillations] and ultrahigh resolution, tunable, vacuum ultraviolet (VUV) laser-based angle-resolved photoemission spectroscopy (ARPES) to study the three-dimensionality (3D) of the bulk electronic structure in WTe2, a type-II Weyl semimetal. The bulk Fermi surface (FS) consists of two pairs of electron pockets and two pairs of hole pockets along the X-Gamma-X direction as detected by using an incident photon energy of 6.7 eV, which is consistent with the previously reported data. However, if using an incident photon energy of 6.36 eV, another pair of tiny electron pockets is detected on both sides of the Gamma point, which is in agreement with the small quantum oscillation frequency peak observed in the magnetoresistance. Therefore, the bulk, 3D FS consists of three pairs of electron pockets and two pairs of hole pockets in total. With the ability of fine tuning the incident photon energy, we demonstrate the strong three-dimensionality of the bulk electronic structure in WTe2. The combination of resistivity and ARPES measurements reveal the complete, and consistent, picture of the bulk electronic structure of this material.

Equations

$$A_t\propto B^{1/2}{\left|\frac{{\partial }^2{S}_t}{\partial k_H^2}\right|}^{-1/2}{R}_T{R}_D{R}_S$$

$$R_T=\frac{\alpha m^{\ast }T/B}{sinh(\alpha m^{\ast }T/B)}$$

Full text

I Introduction

Extremely large magnetoresistance, i.e. dramatic increase in the resistance of a material upon applied magnetic fields, has recently attracted great interest Mun et al. (2012); Ali et al. (2014); Liang et al. (2015). Materials with this type of property can be potentially very useful for applications such as magnetic field sensors, data storage and processing etc. Interestingly, some of these extremely large magnetoresistive materials are hosts for other exotic properties. Among the first few extremely large magnetoresistive materials, PtSn4 Mun et al. (2012) has been reported to host unusual Dirac node arc structure, that is the Dirac dispersion extending in momentum space in one dimension and gapped out at both ends Wu et al. (2016a). Another material with extremely large magnetoresistance, Cd3As2 Liang et al. (2015) was shown to be one of the first three-dimensional Dirac semimetals with linear dispersion along all three momentum directions Wang et al. (2013); Neupane et al. (2014); Liu et al. (2014); Borisenko et al. (2014); Narayanan et al. (2015). Extremely large magnetoresistive material, WTe2 Ali et al. (2014) has been reported to exhibit pressure-induced superconductivity Kang et al. (2015); Pan et al. (2015), and a pressure-induced Lifshitz phase transition was proposed to explain the emergence of the superconductivity Kang et al. (2015). Surprisingly, a temperature-induced Lifshitz transition was recently reported in WTe2. The significant shift of the chemical potential with moderate temperature change is caused by the close proximity of electron and hole band extrema to the chemical potential Wu et al. (2015). More interestingly, WTe2 was the first material proposed to be a type-II Weyl semimetal Soluyanov et al. (2015). Unlike the type-I Weyl semimetals Huang et al. (2015); Weng et al. (2015); Xu et al. (2015a); Yang et al. (2015); Lv et al. (2015a, b); Xu et al. (2015b), such materials have the Weyl points emerging at the touching points of the electron and hole pockets Soluyanov et al. (2015). Recently, multiple ARPES measurements reported the presence of the Fermi arc surface states in these compounds Belopolski et al. (2016); Huang et al. (2016); Tamai et al. (2016); Deng et al. (2016); Jiang et al. (2016); Liang et al. (2016); Xu et al. (2016); Bruno et al. (2016); Wang et al. (2016); Wu et al. (2016b); Feng et al. (2016). Photon energy dependence measurements have been used to demonstrate the two-dimensionality (surface origin) of the Fermi arc in WTe2 Bruno et al. (2016); Wu et al. (2016b). However, detailed measurements of three-dimensional bulk electronic structures are still lacking.

ARPES has been known as the most direct technique for probing the electronic structures of materials Damascelli et al. (2003); Campuzano et al. (2004). Early ARPES and density-functional based augmented spherical wave calculations have revealed the semimetallic nature of WTe2 Augustin et al. (2000). However, no details close to the Fermi level were clearly resolved. More recent, high resolution ARPES data has revealed one pair of electron pockets and one pair of hole pockets of similar size, supporting the electron-hole carrier compensation theory as the primary origin of the extremely large magnetoresistance Pletikosic et al. (2014). By varying the incident photon energies in the 40–70 eV range, the ${k}_{z}$ dispersion of the states was mapped out with some bands showing low dispersion and some showing variations in intensity, but no solid conclusion can be drawn from these data Pletikosic et al. (2014). Another study reported presence of nine Fermi pockets. However, no significant photon energy dependence along the out of plane direction was observed Jiang et al. (2015). On the other hand, magnetoresistance measurements with varying magnetic field applied at an angle with respect to the $c$ axis of the sample have led to the conclusion of three-dimensional electronic structure in WTe2 Thoutam et al. (2015). Furthermore, the results from quantum oscillations–another technique to probe the Fermi surface structure–have come to similar conclusions. Angle resolved quantum oscillation measurements implied strong three-dimensionality of the band structure in this material Zhu et al. (2015a). The analysis of quantum oscillations is a powerful technique to measure the extrema of the Fermi surface topology. However, one should be cautious when assigning oscillation frequencies to particular FS pockets, e.g. in measurements performed under applied pressure, one group assigned the peaks that survived at high pressure to a particular pair of electron and hole pockets Cai et al. (2015), however, the Hall effect measurements from another group found that only electron carriers were present under high pressure Kang et al. (2015), a result that is consistent with our report of temperature-induced Lifshitz transition in WTe2 Wu et al. (2015). Thus, in order to demonstrate the three-dimensionality of the electronic structure in WTe2, ultrahigh resolution ARPES measurements with fine tunable incident photon energies have real experimental advantage.

Here, we use temperature- and field-dependent resistivity measurements and ultrahigh resolution, tunable VUV laser-based ARPES to probe the three-dimensionality of the bulk electronic structure in WTe2. In the SdH oscillations, a low frequency peak was observed and can be explained by our photon energy dependent ARPES measurements. With the ability of fine tuning of the incident photon energy from 5.77 to 6.7 eV, we have determined Fermi surface and band structure with very high precision. At the incident photon energy of 6.7 eV, we can detect a bulk FS that consists of two pairs of electron pockets and two pairs of hole pockets, while the top of another band is located just below the Fermi level at the $\Gamma$ point. When decreasing the incident photon energy to 6.36 eV, another pair of tiny electron pockets is detected close to the $\Gamma$ point, which corresponds to the so far unaccounted for, low oscillation frequency observed in the quantum oscillation measurements Cai et al. (2015); Wu et al. (2015). Further decreasing the incident photon energy, we observe the disappearance of the tiny electron pockets, thus the bulk FS has only two pairs of electron and hole pockets left for this range of ${k}_{z}$ momenta. Detailed band dispersion along several cuts for different incident photon energies are presented, demonstrating the strong three-dimensionality of the bulk electronic structure in WTe2. These results are consistent with the band structure calculations and quantum oscillations Ali et al. (2014); Cai et al. (2015); Wu et al. (2015). Our photon energy dependent ARPES measurements have solved the mystery of the low frequency peak reported by several quantum oscillation measurements Cai et al. (2015); Wu et al. (2015).

II Experimental details

Single crystals of WTe2 were grown by solution method Canfield and Fisk (1992); Canfield et al. (2016), following the procedure describe in Ref. Wu et al., 2015. The resulting crystals were blade or ribbon like in morphology with typical dimensions of 3 $\times$ 0.5 $\times$ 0.01 mm with the crystallographic $c$ axis being perpendicular to the crystal surface; the crystals are readily cleaved along this crystal surface.

Magnetic field dependent electrical transport measurements were carried out in a Quantum Design Physical Property Measurement System (PPMS) for $H\leq 14$ T. Samples for standard four-probe resistivity measurement were prepared by attaching four Pt wires using Epotek-H20E silver epoxy. The field was applied parallel to the crystallographic $c$ axis, and the current was along the crystallographic $a$ axis. Magnetoresistance was measured at 1.8, 2.5, 4, 6, 8, and 10 K.

Samples used for ARPES measurements were cleaved in situ at 40 K under ultrahigh vacuum (UHV). The data were acquired using a tunable VUV laser ARPES system, that consists of a Scienta R8000 electron analyzer, picosecond Ti:Sapphire oscillator and fourth harmonic generator Jiang et al. (2014). Data were collected with a tunable photon energies from 5.3 to 6.7 eV. Momentum and energy resolutions were set at $\sim$ 0.005 Å*-1* and 2 meV. The size of the photon beam on the sample was $\sim$30 $\mu$m.

III Experimental results

Magnetoresistance (MR) shows parabolic behavior without any saturation at high field as shown in Fig. 1(a). In order to analyze the quantum oscillation spectra, we subtracted the background using a second order polynomial function to fit the background MR in the range of 6 $\leq H\leq$ 14 T for all temperatures. The oscillations show periodic behavior in 1/$H$ as shown in Fig. 1(b).

The frequencies of the oscillation were obtained by FFT analysis shown in Fig. 1(c). Five frequencies, including ${F}^{1}=92~{}\textrm{T}$, ${F}^{2}=132~{}\textrm{T}$, ${F}^{3}=152~{}\textrm{T}$, ${F}^{4}=172~{}\textrm{T}$ and ${F}^{5}=264~{}\textrm{T}$, are similar to published results Cai et al. (2015); Zhu et al. (2015a); Wu et al. (2015). Interestingly, a new low frequency peak, ${F}^{*}=10~{}\textrm{T}$, is also clearly observed in our data. The amplitude of the ${F}^{*}$ decreases with increasing temperature as shown in Fig. 1(d). Note that for the sake of consistency the exactly same data acquisition and processing protocol was followed for all temperatures. The observed decrease of the oscillation amplitude is due to the temperature induced scattering of electrons, described by the Lifshitz-Kosevich formula Shoenberg (1984):

[TABLE]

where the factor ${R}_{T}$ is related to thermal damping, ${R}_{D}$ is related to impurities and ${R}_{S}$ is related to spin Zeeman splitting and superposition of spin-up and spin-down oscillations. The thermal damping part ${R}_{T}$ is defined as

[TABLE]

where $\alpha=2{\pi}^{2}c{k}_{B}/e\hbar$. Using this equation, we calculated the effective mass of the carriers linked to the oscillation frequency ${F}^{*}$, ${m}^{*}_{{F}^{*}}=0.29\pm 0.01~{}{m}_{e}$.

To match this small frequency observed in quantum oscillation to a specific Fermi surface, we carried out photon energy dependent ARPES measurements. By varying the incident photon energies, we are able to map out the band dispersion along the out of plane, ${k}_{z}$ direction Damascelli et al. (2003); Campuzano et al. (2004). Synchrotron radiation based ARPES systems are often used for ${k}_{z}$ dispersion mapping due to the large tunable range of the incident photon energies. However, tuning photon energies with usually utilized coarse steps $\geq$ 1 eV can result in some important details being missed along very key ${k}_{z}$ direction Pletikosic et al. (2014); Jiang et al. (2015). By using tunable VUV laser ARPES with very fine energy steps, we mapped out the ${k}_{z}$ dispersion of WTe2 in great detail. Fig. 2 shows the FS and band dispersion measured using incident photon energies of 6.70, 6.36, 6.05, and 5.77 eV, as indicated at the top center of each plot. In Figs. 2(a)-(d), we can see that the FS of WTe2 measured using different photon energies look similar, with two pairs of electron pockets and two pairs of hole pockets in the first BZ. However, a significant variation between the data sets is also observed. The hole band at the $\Gamma$ point has different intensities and curvatures, although none of them crosses the Fermi level as shown in Figs. 2(e)-(h). Furthermore, the FS close to the $\Gamma$ point in Fig. 2(b) shows a dumb-bell like structure, whereas the other three FSs show only a single hole band at the $\Gamma$ point. The band dispersion along the black dashed line in Fig. 2(b) is shown in panel (f). On either side of the $\Gamma$ point, a tiny electron pocket is visible [marked by the red dashed lines in panel (f)], which is different from the band dispersion observed using other photon energies.

To directly compare the quantum oscillation results with the ARPES measurements, we have plotted the extremal orbits with the areas determined from quantum oscillation measurements on top of the Fermi surface plot as shown in Fig. 2(b). The corresponding extremal areas of the Fermi surface were calculated using the Onsager relation, ${F}^{i}=\frac{\hbar c}{2\pi e}{S}^{i}$ [Ref. Shoenberg, 1984] with ${S}_{{F}^{1}}=0.00874~{}{{\textrm{\AA}}^{-2}}$, ${S}_{{F}^{2}}=0.01262~{}{{\textrm{\AA}}^{-2}}$, ${S}_{{F}^{3}}=0.01456~{}{{\textrm{\AA}}^{-2}}$, ${S}_{{F}^{4}}=0.0165~{}{{\textrm{\AA}}^{-2}}$, and ${S}_{{F}^{*}}=0.000971~{}{{\textrm{\AA}}^{-2}}$. For simplicity, we assume that the areas obtained from quantum oscillations are from simple circle/ellipse orbit extrema and the corresponding shapes are plotted in Fig. 2(b). We can clearly see a good match between the quantum oscillation results and ARPES measurements of the Fermi surface. Furthermore, the electronic structure calculations that take into account the spin-orbit coupling in WTe2 bring extra tiny electron pockets close to the $\Gamma$ point Ali et al. (2014). Thus, our ARPES results have solved the mystery of this unaccounted for, low frequency quantum oscillation peaks. We should note that as the electronic structure of WTe2 is very sensitive to pressure/strain Kang et al. (2015); Pan et al. (2015); Soluyanov et al. (2015); Wu et al. (2016b), it is possible that these tiny electron pockets might be suppressed in some of the quantum oscillation or ARPES measurements Zhu et al. (2015b); Wu et al. (2016b)

Figure 3 shows the detailed band dispersion measured using various photon energies. Panels (a)–(d), (f)–(i), and (k)–(n) present the band dispersions measured using photon energy of 6.7, 6.36, 6.05 and 5.77 eV, respectively, and correspond to the cuts as marked #1, 2, and 3 in Fig. 2(a). In panels (a)–(d), only minor intensity differences can be seen between the four measurements. At the photon energy of 6.05 and 5.77 eV, the electron pockets are clear and symmetric. On the other hand, the electron pockets measured at the photon energy of 6.7 and 6.36 eV are not symmetric in intensity, probably due to the matrix elements effect. To quantify the electron pocket sizes in panels (a)–(d), we have plotted the momentum distribution curves (MDCs) at the Fermi level ${E}_{F}$ in Fig. 3(e). The peak locations of the MDCs show clear differences across these four photon energies (peak locations of the left branches are aligned for easy comparison). Panels (f)–(i) present the band dispersion from cut #2, which clearly shows that the hole pocket measured with 5.77 eV photons is significantly smaller than the other three. Panel (j) shows the MDCs at the ${E}_{F}$ from panels (f)–(i), illustrating different hole pocket sizes [also left aligned as in (e)]. Panels (k)–(n) show the photon energy dependence of the hole bands at the $\Gamma$ point [cut #3 in Fig. 2(a)]. Two hole bands can be clearly seen at the $\Gamma$ point with different separations between them for different photon energies. Panel (o) shows the energy distribution curves (EDCs) from panels (k)–(n), where the black arrows point to the peak locations in the lower hole bands. The upper hole bands sit at roughly the same binding energy for these photon energies, but none of them crosses the Fermi level. On the other hand, the distance between the upper and lower hole bands is very different across these photon energies, with 5.77 eV showing the maximum separation. By fitting two Lorentzian functions to the MDCs in panels (e) and (j), we calculate the electron/hole pocket sizes and summarize the results in panel (p). With the decreasing incident photon energy, the size of the electron pocket decreases and then increases. On the other hand, the size of the hole pocket increases and then decreases. This trend (strong three-dimensionality along ${k}_{z}$ direction) is consistent with the band structure calculations shown in Ref. Ali et al., 2014, where the hole pockets have a concave shape and the electron pockets have a convex shape along ${k}_{z}$ direction toward the center of the zone.

IV Conclusion

In conclusion, we used temperature- and field-dependent resistivity measurements in tandem with ultrahigh resolution laser ARPES to investigate the detailed electronic structure of WTe2. The photon energy dependence measurements with relatively fine energy steps have revealed the three-dimensional character of the electron and hole pockets along the $\Gamma-Z$ direction. With the increase of the incident photon energy from 5.77 to 6.70 eV (i.e., probing along ${k}_{z}$ direction), we have observed that the hole pocket expands and then shrinks, while the electron pocket displays opposite behavior. Strong photon energy dependence is also observed in the hole bands at the $\Gamma$ point. Furthermore, at the photon energy of 6.36 eV we have revealed a pair of tiny electron pockets sitting at the opposite side of the $\Gamma$ point, providing strong support for the low quantum oscillation frequency peak that was not accounted for in the previous studies Wu et al. (2015); Cai et al. (2015).

Acknowledgements.

Research was supported by the US Department of Energy, Office of Basic Energy Sciences, Devision of Materials Science and Engineering. Ames Laboratory is operated for the US Department of Energy by the Iowa State University under Contract No. DE-AC02-07CH11358. Y.W. (Analysis of ARPES data) was supported by Ames Laboratory’s Laboratory-Directed Research and Development (LDRD) funding. N.H.J. was supported by the Gordon and Betty Moore Foundation EPiQS Initiative (Grant No. GBMF4411). L.H. was supported by CEM, a NSF MRSEC, under Grant No. DMR-1420451.

Raw data for this manuscript is available at http://lib.dr.iastate.edu/ameslab_datasets/.