Deviation from the Kohler’s rule and Shubnikov–de Haas oscillations in type-II Weyl semimetal WTe₂: High magnetic field study up to 56 T

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 WTe₂ 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 WTe₂ 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 ∼B² 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.

High quality single crystals of WTe₂ 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 WTe₂ crystallizes in the orthorhombic noncentrosymmetric structure with the space group Pmn2₁ (#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 WTe₂ single crystal with a size of 1.175 × 0.289 × 0.128 mm³. 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.

Figure 1(a) displays the field dependence of electrical resistivity ρ(B) of a WTe₂ 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≡Δρ(B)/ρ(0)≡(ρ(B)-ρ(0))/ρ₀, where ρ₀≡ρ(B=0), is shown in Fig. 1(b). At 1.4 K, MR reaches 1.5 × 10⁷% at 56 T, indicating roughly that ω_cτ reaches of the order of 10⁵, where ω_c and τ represents the cyclotron frequency and the relaxation time of carriers, respectively. This high value of ω_cτ reflects the high quality of the single crystal. In the measured field range, overall behavior of MR seems to show ∼B² 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, Δρ(B)/ρ₀ vs [B/ρ₀]², which is shown in Fig. 1(c). The data show slight deviation from the B² dependence with a concave-downward behavior. Figure 1(d) shows a Δρ(B)/ρ₀/[B/ρ₀]² = Δρ(B)ρ₀/B² vs B plot. If Δρ(B)/ρ₀ ∝ B², this quantity becomes a B-independent constant. The value of Δρ(B)ρ₀/B² decreases by approximately 30% up to 56 T, clearly demonstrating significant deviation from the B² dependence.

Such deviation from the B² 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 B² dependence is expected to be enhanced nonlinearly with increasing field, which is different from our observation, i.e., Δρ(B)ρ₀/B² 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 B² dependence remains unsolved and needs to be studied further.

Another notable feature in Fig. 1(d) is the upward shift of Δρ(B)ρ₀/B² 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 Δρ(B)/ρ₀-vs-B/ρ₀ curves from different temperatures fall on top of each other. This is because Δρ(B)/ρ₀ depends only on the product of ω_c ∝ B and τ(T) ∝ 1/ρ₀(T), resulting in the scaling law Δρ(B)/ρ₀ = f(B/ρ₀), 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 WTe₂ 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 Δρ(B)ρ₀/B² is expected to appear around T/Θ_D ∼ 0.1. Since Θ_D=134 K for WTe₂,¹⁹ the increase in Δρ(B)ρ₀/B² 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 Δρ^SdH is obtained by a fitting to the Δρ(B) data using Δρ(B) = Δρ^SdH + aBⁿ, where a and n are free fitting parameters. Figure 2(a) displays Δρ^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 F¹ = 93.0 ± 0.4 T, F² = 126.2 ± 1.4 T, F³ = 141.0 ± 0.9 T, and F⁴ = 162.3 ± 0.6 T. Comparison with band structure calculations and ARPES measurement suggests that F¹ and F⁴ (F² and F³) correspond to two hole (electron) pockets.^9–11 In addition, there appears a peak at the frequency of F¹ + F⁴ = 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 F¹ + F⁴ and the fundamental Fⁱ 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 measurements⁸ indicate that the bands of WTe₂ 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/m₀ (m₀: the bare electron mass) are obtained to be 0.19, 0.11, 0.15, and 0.23 for F¹, F², F³, and F⁴, 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 F¹∼F⁴ located closely each other. Although these values are actually smaller than those obtained in B<35 T,¹¹ the relation m_c(F⁴)>m_c(F¹)>m_c(F³)>m_c(F²) is consistently observed in both the measurements, confirming that the hole pockets (F⁴ and F¹) have heavier cyclotron mass than the electron pockets (F³ and F²). The value of m_c for the F¹ + F⁴ peak is larger than those for the F¹ and F⁴ peaks. This feature is in line with the interpretation that the F¹ + F⁴ peak is caused by magnetic breakdown.

In summary, we have measured magnetoresistance of type-II Weyl semimetal WTe₂ 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 B² 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.

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.