1 INTRODUCTION

Layered tungsten ditelluride (WTe2) crystals are semimetals with the Td structure (orthorhombic space group Pmn21) [1], which remains stable at a change in temperature [2, 3]. Recently, it was found that resistance of WTe2 (and some other compounds with metallic conductivity) increases strongly in constant magnetic fields [4–7]. The magnetoresistance (MR), which is defined as a ratio of a change in the resistance to the resistance at a zero field, was approximately proportional to the squared magnetic-field induction. A specific feature of MR in WTe2 is the absence of saturation in the fields of up 60 T [4]. Shubnikov–de Haas (SdH) oscillations were also observed in the field dependence of MR [8–11]. The oscillation spectrum yields direct information about the Fermi energy of electron and hole pockets in WTe2. These data are of particular interest in view of the fact that, based on the theoretical predictions and experimental results, obtained by different methods (see, e.g., [12–14] and references therein), tungsten ditelluride has been considered as a possible Weyl semimetal. Measurement of MR and SdH oscillations at different temperatures makes it possible to study the temperature evolution of the electron and hole pockets, which is important at analysis of the transformation of the electronic band topology [15]. The existing data on the frequencies of quantum oscillations, their temperature dependences, and MR values are not always in agreement, which can be caused by different electron properties of the crystals, grown under different conditions and using different methods. The purpose of this study was to investigate MR and SdH oscillations in a WTe2 single crystal, grown at the Institute of Metal Physics (Ural Branch of the Russian Academy of Sciences, Yekaterinburg).

2 EXPERIMENTAL

A single crystal of tungsten ditelluride WTe2 was grown by the chemical vapor transport method. Bromine was used as a transport agent. The crystal growth in an evacuated and sealed quartz-glass ampoule lasted three weeks. The sample for studies in the form of a thin plate with a thickness of about 0.2 mm was cleaved from the grown crystal perpendicular to the c axis. The crystal structure, single crystallinity, and directions of the crystallographic axes were controlled using the X-ray diffraction (XRD) analysis. Triple atomic layers Te–W–Te (W–Te bond is covalent) form planes, perpendicular to the c axes, and coupled by weak van der Waals interaction. Tungsten atoms form chains within layers, oriented along the crystallographic a axis.

The sample resistance in the direction of the a crystalline axis was measured with a PPMS-16 system (Quantum Design) using the built-in four-probe technique. The temperature dependence of the resistance in a zero magnetic field was obtained in the range from 2 to 300 K. The dependence of resistance on magnetic field up to 14 T was measured at temperatures of 2 and 5 K. The magnetic field was directed along the c crystalline axis. According to [9], the largest MR values are observed in this geometry.

3 RESULTS AND DISCUSSION

The temperature dependence of resistance in a zero magnetic field is shown in Fig. 1. An increase in the resistance with increasing temperature exhibits metallic character. In the range from 90 to 200 K, the sample resistance depends linearly on temperature. Despite the strong increase in resistance with temperature, the RRR ratio of the resistance values at 300 and 2 K is 31, which is much lower than the values, obtained in [4, 8–10]. Apparently, it is due to somewhat larger number of defects in the sample under study.

Fig. 1.
figure 1

Temperature dependence of resistance in a zero external magnetic field. Straight line indicates a linear dependence.

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The MR curves, measured at temperatures of 2 and 5 K, are shown in Fig. 2. The MR values (in %) were calculated according to the formula MR = \([R(B) - R(0)]{\text{/}}R(0)\) × 100, where R(B) and R(0) are the sample resistances in field B and in the zero field, respectively. At 14 T, MR reaches the values of 3.45 × 103 and 3.31 × 103% at temperatures of 2 and 5 K, respectively. The obtained MR values are smaller than those, obtained in [4, 8–10], which is in agreement with smaller RRR ratio. As in the earlier studies of MR in WTe2 semimetal, no saturation was observed in the field dependence of MR. For both temperatures, a monotonic rise in MR at increasing magnetic field (with the oscillations disregarded) is described in the entire field range by a power-law close-to-quadratic function \({\text{MR}} \propto {{B}^{{1.78}}}\). The quadratic dependence at weak magnetic fields was predicted theoretically for metals, semimetals, and semiconductors. It was also observed for compensated semiconductors with closed Fermi surface at strong effective magnetic fields [16]. Absence of MR saturation indicates compensation of electrons and holes [4, 17]. Within the two-band model [18], in which one electron and one hole bands are considered, the longitudinal resistivity in a magnetic field is described by the following relation:

$$\rho = \frac{1}{e}\frac{{({{n}_{e}}{{\mu }_{e}} + {{n}_{h}}{{\mu }_{h}}) + ({{n}_{e}}{{\mu }_{e}}\mu _{h}^{2} + {{n}_{h}}{{\mu }_{h}}\mu _{e}^{2}){{B}^{2}}}}{{{{{({{n}_{e}}{{\mu }_{e}} + {{n}_{h}}{{\mu }_{h}})}}^{2}} + {{{({{n}_{h}} - {{n}_{e}})}}^{2}}\mu _{e}^{2}\mu _{h}^{2}{{B}^{2}}}},$$
(1)

in which \({{n}_{e}}\) and \({{n}_{h}}\) are the electron and hole concentrations, respectively, and \({{\mu }_{e}}\) and \({{\mu }_{h}}\) are the corresponding mobilities. At the compensation condition \({{n}_{e}} = {{n}_{h}}\), having introduced average mobility \(\bar {\mu }\), we obtain for MR

$${\text{MR}} = {{(\bar {\mu }B)}^{2}}.$$
(2)
Fig. 2.
figure 2

Dependences of magnetoresistance on an external magnetic field at temperatures of 2 and 5 K.

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Strong MR in bismuth and graphite was explained using the two-band model; however, saturation in them was observed even in the fields of about several teslas, which indicates incomplete compensation of electrons and holes [19, 20].

Based on the obtained data, we can find the average mobility \(\bar {\mu }\). At 2 K, the average mobility is 4.2 × 103 cm2 V–1 s–1, which is close to the mobility value, obtained in [21]. Thus, within the two-band model, larger MR values correspond to higher electron and hole mobilities under condition of their equal concentrations.

Note that extreme MR was also interpreted using the models, which take into account the specific feature of the electronic band structure in the topologic materials, limiting carrier backscattering [6, 22–25]. However, additional studies of WTe2 (a possible Weyl semimetal) favor the validity of compensation of electrons and holes [26].

One can see quantum SdH oscillations in the MR curves, recorded at 2 and 5 K (Fig. 2). As one would expect, SdH oscillations weaken at an increase in temperature. Quantum oscillations of MR were analyzed using the discrete Fourier transform. To select oscillations against the background of the MR dependence on magnetic field at both temperatures, a monotonic increase in MR was approximated by a fourth-order polynomial and was subtracted from the experimental curves. Figure 3 shows the SdH oscillations in dependence on the reciprocal magnetic-field induction. Figures 4 and 5 show the Fourier transforms, obtained at temperatures of 2 and 5 K, respectively. Along with the calculation results, the contributions of individual spectral components and their sum are shown. Five frequencies F, which correspond to individual components, and their intensities are listed in Table 1. According to [9, 10], the calculated first four frequencies should be related to electron and hole pockets, split due to the spin–orbit coupling. These frequencies in the SdH oscillation spectrum correspond to indices h1, e1, e2, and h2 in Table 1, where h and e indicate hole and electron pockets, respectively. Numbers 1 and 2 correspond to two possible spin orientations. Apparently, the fifth frequency, which is indicated by sym-bol *, is the second harmonic of the lowest frequency. The Onsager relation

$$F = ({{\Phi }_{0}}{\text{/}}2{{\pi }^{2}})A,$$
(3)

makes it possible to estimate size A of the maximum cross section of the Fermi surface of electron and hole pockets in the direction, perpendicular to the magnetic field, based on the calculated quantum oscillation frequencies. In (3), \({{\Phi }_{0}}\) is a magnetic-flux quantum. The cross-sectional areas are also given in Table 1. The sum of cross sections for electron pockets differs from the sum of cross sections for hole pockets by as little as 3%, which indicates that the electron and hole concentrations in the crystal under study are close.

Fig. 3.
figure 3

Shubnikov–de Haas oscillations in dependence of the reciprocal field induction at temperatures of 2 and 5 K.

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Fig. 4.
figure 4

Oscillation spectra at a temperature of 2 K. Points, thin solid curves, and thick solid curves correspond to the Fourier-transform results, calculated individual spectral components, and the total contribution of the spectral components, respectively.

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Fig. 5.
figure 5

Oscillation spectra at a temperature of 5 K. Points, thin solid curves, and thick solid curves correspond to the Fourier-transform results, calculated individual spectral components, and the total contribution of the spectral components, respectively.

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Table 1. Frequencies of spectral components F, their relative intensities Irel, and cross-sectional areas A of the Fermi surface of electron and hole pockets, calculated for temperatures of 2 and 5 K
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The SdH oscillation frequencies, obtained in this study, can be compared to the quantum oscillation frequencies, which were previously found for the WTe2 crystal [8–10, 21, 27, 28]. In most studies, except for [21], four fundamental quantum oscillation frequencies were found, the spread of their values overlaps the difference with the frequencies, obtained by us (Table 1). Thus, the presence of defects in the crystal under investigation, which decreases RRR, barely affects the cross-sectional area of the Fermi surface of electron and hole pockets.

SdH oscillations can be interpreted using the Lifshitz–Kosevich theory [10]. According to this theory, the temperature dependence of the oscillation intensity makes it possible to estimate effective electron mass m*. However, in our case of measurements only at two temperatures the estimate for the effective mass has low accuracy.

4 CONCLUSIONS

The investigated single-crystal sample of WTe2 semimetal is characterized by the ratio RRR = 31, which is indicative of the presence of a large number of defects. MR at temperatures of 2 and 5 K reaches the values, exceeding 3 × 103%. The spectrum of SdH oscillations shows four fundamental frequencies, consistent with the previously obtained results for MR in tungsten ditelluride. Thus, the crystal-structure defects, which decrease MR, barely affect the electronic band topology in WTe2.