We have measured magnetoresistance of type-II Weyl semimetal WTe_{2} in high magnetic fields of *B*//*c* up to *B* = 56 T using a high quality single crystal with the residual resistivity ratio 1330. A Kohler plot demonstrates the existence of a deviation from the *B*^{2} dependence. Significant deviation from Kohler’s rule appears with increasing temperature, indicating wave-vector ** k** dependent anisotropic carrier scattering, which develops with increasing temperature. The fast-Fourier-transform spectra of the observed large-amplitude Shubnikov–de Haas oscillations confirm the existence of four fundamental and one additional (cyclotron frequency of 255 T) peaks. The amplitude of the latter one increases anomalously with increasing field, confirming this oscillation being due to magnetic breakdown (or magnetic breakthrough) between the two cyclotron orbits of the adjacent hole pockets.

## INTRODUCTION

Quantum topological materials, including graphene, Dirac- and Weyl-semimetals^{1–4} have been attracting great interests recent years due to novel physical properties. In Weyl semimetals, Weyl fermions with different chiralities of left or right handed are expected to play an essential role in the transport properties. Large magnetoresistance, which manifests itself in some of Weyl semimetals, is considered to be caused by high carrier mobility of Weyl fermions. Layered transition-metal dichalcogenide WTe_{2} is a type-II Weyl semimetal candidate.^{5} Angle-resolved photoemission spectroscopy measurements have confirmed the existence of Fermi arcs connecting Weyl points of opposite chirality.^{6–8} Fourier transform spectra of Shubnikov-de Haas (SdH) oscillations show that WTe_{2} has four Fermi pockets, i.e., two sets of concentric electron- and hole-Fermi pockets, being consistent with band structure calculations.^{9–12} Extremely large non-saturating magnetoresistance with ∼*B*^{2} dependence suggests nearly perfect compensation of electrons and holes.^{13} However, the features of the electronic transport properties and Fermi-surface topology have not been investigated yet in high fields above 35 T. In this paper, we report magnetoresistance measurements in high fields up to 56 T using a high quality single crystal with the residual resistivity ratio 1330.

## EXPERIMENTAL DETAILS

High quality single crystals of WTe_{2} were grown by Te flux method. Tungsten powder (99.99%) were mixed with excessive amount of Tellurium (99.999%) and loaded into an alumina crucible. The crucible was then placed into a quartz tube and sealed under vacuum. The sealed quartz ampoule was heated in a furnace up to 1050 °C with a rate of 100 °C/h, kept at 1050 °C for 10 h, cooled down to 750 °C at a rate of 1 °C/h, and furnace-cooled down to RT. The excess flux was removed at 750 °C by a centrifuge. X-ray powder diffraction and single crystal structural analysis confirmed that WTe_{2} crystallizes in the orthorhombic noncentrosymmetric structure with the space group *Pmn*2_{1} (#31). The residual resistance ratio *RRR* = *ρ*(300 K)/*ρ*(*T*→0 K) = 1330 is in the highest class in the literature.^{6,10–12,14–16}

We used a four-terminal method to measure the electrical resistivity *ρ* of a WTe_{2} single crystal with a size of 1.175 × 0.289 × 0.128 mm^{3}. The electrical contacts were made by spot welding Au wires (ϕ50 μm) to the sample. The magnetic field dependence of *ρ*(*B*) in pulsed magnetic fields was measured up to 56 T using a nondestructive pulsed magnet at the International MegaGauss Science Laboratory in the Institute for Solid State Physics (University of Tokyo). The duration of the pulsed magnetic field was about 36 ms. The resistivity was measured by a lock-in technique at 100 kHz with ac excitation of 1 - 10 mA. The measurement was made in the transverse magnetic field configuration with the current *I*//*a* in the applied magnetic fields *B*//*c*. The measurements were made in the temperature range of 1.4 - 80 K. We have also measured *ρ* in static magnetic fields up to 9 T in the same configuration using a Quantum Design (QD) Physical Property Measurement System (PPMS) to compare with the pulse-field data.

## RESULTS AND DISCUSSION

Figure 1(a) displays the field dependence of electrical resistivity *ρ*(*B*) of a WTe_{2} single crystal up to 56 T for selected temperatures between 1.4 and 80 K. With increasing field, *ρ*(*B*) increases without any saturation tendency. Clear Shubnikov-de Hass (SdH) quantum oscillations appear above ∼4 T. The amplitude of the SdH oscillations develops with increasing field as well as with decreasing temperature (analyses will be given below).

A log-log plot of the field dependence of magnetoresistance *MR*≡Δ*ρ*(*B*)/*ρ*(0)≡(*ρ*(*B*)-*ρ*(0))/*ρ*_{0}, where *ρ*_{0}≡*ρ*(*B*=0), is shown in Fig. 1(b). At 1.4 K, *MR* reaches 1.5 × 10^{7}% at 56 T, indicating roughly that *ω*_{c}*τ* reaches of the order of 10^{5}, where *ω*_{c} and *τ* represents the cyclotron frequency and the relaxation time of carriers, respectively. This high value of *ω*_{c}*τ* reflects the high quality of the single crystal. In the measured field range, overall behavior of *MR* seems to show ∼*B*^{2} dependence, which is expected for a metal with the carrier compensation.^{17} However, there appears slight deviation above 10 T. This feature can be more clearly seen in the so-called Kohler plot, Δ*ρ*(*B*)/*ρ*_{0} vs [*B*/*ρ*_{0}]^{2}, which is shown in Fig. 1(c). The data show slight deviation from the *B*^{2} dependence with a concave-downward behavior. Figure 1(d) shows a Δ*ρ*(*B*)/*ρ*_{0}/[*B*/*ρ*_{0}]^{2} = Δ*ρ*(*B*)*ρ*_{0}/*B*^{2} vs *B* plot. If Δ*ρ*(*B*)/*ρ*_{0} ∝ *B*^{2}, this quantity becomes a *B*-independent constant. The value of Δ*ρ*(*B*)*ρ*_{0}/*B*^{2} decreases by approximately 30% up to 56 T, clearly demonstrating significant deviation from the *B*^{2} dependence.

Such deviation from the *B*^{2} dependence can be caused by (1) slight carrier uncompensation, i.e., the electron and hole carrier densities are not equal (*n*_{e} ≠ *n*_{h}), and (2) inter-Fermi-pocket scattering as discussed for Cd metal.^{18} In the former case, however, in a simple two carrier model, a deviation from the *B*^{2} dependence is expected to be enhanced nonlinearly with increasing field, which is different from our observation, i.e., Δ*ρ*(*B*)*ρ*_{0}/*B*^{2} seems to decrease almost linearly up to 40 T and plateaus in *B* > 40 T. In the latter case, quantitative analysis is not easy. The reason for the deviation from the *B*^{2} dependence remains unsolved and needs to be studied further.

Another notable feature in Fig. 1(d) is the upward shift of Δ*ρ*(*B*)*ρ*_{0}/*B*^{2} with increasing temperature; it increases by about 50% from 1.4 to 10 K. This feature is often called “deviation from the Kohler’s rule”. In many conventional metals, Kohler’s rule holds at low enough temperatures, i.e., all Δ*ρ*(*B*)/*ρ*_{0}-vs-*B*/*ρ*_{0} curves from different temperatures fall on top of each other. This is because Δ*ρ*(*B*)/*ρ*_{0} depends only on the product of *ω*_{c} ∝ *B* and *τ*(*T*) ∝ 1/*ρ*_{0}(*T*), resulting in the scaling law Δ*ρ*(*B*)/*ρ*_{0} = *f*(*B*/*ρ*_{0}), where *f* is a material-dependent function;^{17} note that *τ* is expected to be the same for all carriers when impurity (or defect) scattering dominates at low temperatures.

The observed deviation from Kohler’s rule in WTe_{2} suggests that different types of scattering mechanisms with different effects on different types of carriers are playing an essential role in the measured *T* range. We expect that this is caused by small-angle electron-phonon scattering (wave-vector ** k** dependent electron scattering by phonons), which becomes more effective with increasing

*T*. In general, around the temperature region with

*T*/

*Θ*

_{D}∼ 0.1, where

*Θ*

_{D}represents the Debye temperature, the carrier scattering is dominated by small wave-vector phonons. In this temperature region, scattering probability becomes largely anisotropic in the reciprocal space. Because of this, the relaxation time becomes anisotropic and

*MR*should increase.

^{17}If such anisotropy is caused by electron-phonon scattering, the maximum in Δ

*ρ*(

*B*)

*ρ*

_{0}/

*B*

^{2}is expected to appear around

*T*/

*Θ*

_{D}∼ 0.1. Since

*Θ*

_{D}=134 K for WTe

_{2},

^{19}the increase in Δ

*ρ*(

*B*)

*ρ*

_{0}/

*B*

^{2}with increasing temperature in 1.4 <

*T*< 10 K is reasonable.

SdH quantum oscillations appear below 10 K as shown in Fig. 1. The SdH oscillatory contribution Δ*ρ*^{SdH} is obtained by a fitting to the Δ*ρ*(*B*) data using Δ*ρ*(*B*) = Δ*ρ*^{SdH} + *aB*^{n}, where *a* and *n* are free fitting parameters. Figure 2(a) displays Δ*ρ*^{SdH} as a function of 1/*B*. The amplitude of the oscillations decreases as the temperature increases. Since the signals comprise a superposition of several oscillatory components with different frequencies, the data are most easily visualized by using a fast Fourier transform (FFT), as shown in Fig. 2(b).

There appear four main fundamental peaks with the frequencies of *F*^{1} = 93.0 ± 0.4 T, *F*^{2} = 126.2 ± 1.4 T, *F*^{3} = 141.0 ± 0.9 T, and *F*^{4} = 162.3 ± 0.6 T. Comparison with band structure calculations and ARPES measurement suggests that *F*^{1} and *F*^{4} (*F*^{2} and *F*^{3}) correspond to two hole (electron) pockets.^{9–11} In addition, there appears a peak at the frequency of *F*^{1} + *F*^{4} = 254.5 ± 0.3 T, whose intensity develops markedly with decreasing temperature. This peak is considered to be a magnetic breakdown (or magnetic breakthrough) corresponding to a high-field orbit enclosing two adjacent hole orbits.^{9} This hypothesis is consistent with our observation, i.e., the ratio of the peak intensities between *F*^{1} + *F*^{4} and the fundamental *F*^{i} increases with increasing field (not shown).

Figure 3 shows the cyclotron frequencies as a function of temperature. No temperature-dependent shift is observed below 6 K, indicating no shift of the Fermi level in this temperature range. Recent ARPES measurements^{8} indicate that the bands of WTe_{2} show an anomalous temperature-dependent shift. With increasing temperature, the bands corresponding to the hole pockets move down in energy and finally they sink completely under the Fermi level leading to a Lifshitz transition around 160 K. The present observation indicates that this phenomenon is frozen below 6 K, even if it exists.

The cyclotron mass (*m*_{c}) for each orbit was determined by fitting the temperature-dependent amplitude of the oscillations to the Lifshitz-Kosevich (LK) formula.^{20} Using the FFT spectra shown in Fig. 2(b), the values of *m*_{c}/*m*_{0} (*m*_{0}: the bare electron mass) are obtained to be 0.19, 0.11, 0.15, and 0.23 for *F*^{1}, *F*^{2}, *F*^{3}, and *F*^{4}, respectively. These values are expected to be underestimated by a factor of a few because of the rather large fitting field range (3.5-56 T), which is necessary to increase the frequency resolution in the FFT spectra for the separation of the four peaks of *F*^{1}∼*F*^{4} located closely each other. Although these values are actually smaller than those obtained in *B*<35 T,^{11} the relation *m*_{c}(*F*^{4})>*m*_{c}(*F*^{1})>*m*_{c}(*F*^{3})>*m*_{c}(*F*^{2}) is consistently observed in both the measurements, confirming that the hole pockets (*F*^{4} and *F*^{1}) have heavier cyclotron mass than the electron pockets (*F*^{3} and *F*^{2}). The value of *m*_{c} for the *F*^{1} + *F*^{4} peak is larger than those for the *F*^{1} and *F*^{4} peaks. This feature is in line with the interpretation that the *F*^{1} + *F*^{4} peak is caused by magnetic breakdown.

## CONCLUSIONS

In summary, we have measured magnetoresistance of type-II Weyl semimetal WTe_{2} in high magnetic fields of *B*//*c* up to *B* = 56 T using a high quality single crystal with the residual resistivity ratio 1330. A Kohler plot demonstrates the existence of a deviation from the *B*^{2} dependence. Significant deviation from Kohler’s rule appears with increasing temperature, indicating wave-vector ** k** dependent anisotropic carrier scattering, which develops with increasing temperature. The fast-Fourier-transform spectra of the observed large-amplitude Shubnikov–de Haas oscillations confirm the existence of four fundamental and one additional (cyclotron frequency of 255 T) peaks. The amplitude of the latter one increases anomalously with increasing field, confirming this oscillation being due to magnetic breakdown between the cyclotron orbits of two adjacent hole pockets.

## ACKNOWLEDGMENTS

We appreciate Prof. Hidetoshi Fukuyama and Prof. Hideyuki Sato for fruitful discussions. This work was supported by JSPS KAKENHI Grant nos. JP15H03693, JP15H05884, JP16F16028 and JP16K05454.