1 INTRODUCTION

A large number of different topological materials have been discovered in recent years, including topological insulators and topological semimetals [1–6]. Considerable attention is drawn to topological semimetals that can be divided into the following three main groups: Weyl semimetals, Dirac semimetals, and topological nodal line semimetals. In Dirac and Weyl semimetals, two double degenerate bands or two nondegenerate bands intersect with each other at special points or nodes near the Fermi level, thereby forming four-fold degenerate Dirac points or two-fold degenerate Weyl points, respectively, and disperse linearly in all three directions of momentum. The corresponding low-energy excitations behave similarly to Dirac and Weyl fermions in high-energy physics.

It is well known that Weyl fermions can occur in systems with broken inversion or time-reversal symmetry [3–6]. In this regard, noncentrosymmetric semimetals, such as TaAs, and magnetic semimetals, such as some Heusler alloys, are candidates for Weyl semimetals. The first experimental confirmation of the existence of the Weyl semimetal phase was obtained on single crystals of the TaAs family (TaAs, TaP, NbAs, and NbP) in 2015 [7]. Moreover, the authors of [8] predicted a special type of band intersection with a strongly tilted Weyl cone along a certain direction in the momentum space, the so-called “type-II Weyl semimetals.” The existence of the phase of the type-II Weyl semimetal was predicted and experimentally confirmed in the following layered transition metal dichalcogenides: WTe2 [8, 9], MoTe2, and ternary compound MoxW1 – xTe2 [10].

The peculiarities of the electronic structure of topological materials are reflected in their electronic properties and lead to a number of unusual effects, such as extremely large magnetoresistance without a tendency to saturation, high mobility and low effective mass of current carriers, nontrivial Berry phase, chiral anomaly and anomalous Hall effect, and linear behavior of optical conductivity [5, 6].

Among such unusual effects are a quadratic temperature dependence of the electrical resistivity of single crystals of WTe2 [11] and MoTe2 [12] in a very wide temperature range from 2 to 70 K and 50 K, respectively. It can be expected that a quadratic temperature dependence of their resistivity should be observed in the presence of a magnetic field as well. It is also worth noting that either single-band [13] or two-band models are normally used when analyzing data on the Hall effect with subsequent calculation of the concentration and mobility of current carriers [14]. At the same time, it is not entirely clear how correct a particular model is.

This study is devoted to the analysis of kinetic properties (electrical resistivity, magnetoresistivity, the Hall effect) of a WTe2 single crystal to establish the type of temperature dependence of resistivity (conductivity) in a magnetic field, and to the use of single-band and two-band models for the analysis of galvanomagnetic properties.

2 MATERIALS AND TECHNIQUES

2.1 Single Crystal Growth and Structural Analysis

WTe2 single crystals were grown using the method of chemical vapor transport [15]. The synthesis scheme is shown in Fig. 1. Tungsten and tellurium in a stoichiometric ratio were placed in a quartz ampoule with a length of 24 cm and a diameter of 1.5 cm. The ampoule was evacuated to a residual pressure of about 10–4 atm and then placed in a horizontal tube furnace with a linear temperature gradient. The hot zone had a temperature of 850°C, and the cold zone, i.e., the crystal growth zone, had a temperature of 770°C. The single crystal growth process lasted 500 h. The resulting crystals have a needle shape with lengths of 3–5 mm, widths of 0.2–1.0 mm, and thicknesses of 50–400 μm.

Fig. 1.
figure 1

The scheme of growing the WTe2 single crystals by the chemical vapor transport method with Br2 as a transport agent.

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A fragment of the diffraction pattern taken from the surface of the WTe2 sample is shown in Fig. 2. All peaks can be indexed as (00l). Hence, the surface of a single crystal of WTe2 coincides with the (001) plane. It is established that compound WTe2 crystallizes in the orthorhombic structure (space group Pmn21) with lattice parameters a = 3.435(8) Å, b = 6.312(7) Å, and c = 14.070(4) Å.

Fig. 2.
figure 2

A fragment of a diffraction pattern (CrKα) taken from the surface of a WTe2 single crystal.

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The surface microstructure and the chemical composition of the crystals were analyzed on an FEI Quanta 200 Pegasus scanning electron microscope equipped with an EDAX attachment for X-ray energy dispersive microanalysis at the Collaborative Access Center “Testing Center of Nanotechnology and Advanced Materials” (TC NTAM), Institute of Metal Physics, Ural Branch, Russian Academy of Sciences. Images of the (001) surface and the lateral surface the WTe2 single crystal are shown in Fig. 3. As can be seen from Fig. 3, the resulting single crystal has a layered structure.

Fig. 3.
figure 3

The microstructure of the surface of a WTe2 single crystal: (a) the surface of the (001) type, (b) the lateral surface of the sample. The region in which the chemical composition of the sample was investigated is highlighted in Fig. 3a.

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The results of X-ray energy dispersive microanalysis of the WTe2 single crystal are shown in Fig. 4. The contents of W and Te in the composition are 33.17 and 66.83 at %, respectively. Thus, the chemical composition of the single crystal corresponds to the stoichiometric ratio WTe2.

Fig. 4.
figure 4

Analysis of the chemical composition of a WTe2 single crystal in the region highlighted in Fig. 3a. The contents of W and Te is are 33.17 and 66.83 at %, respectively.

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2.2 Techniques for Measuring Kinetic Properties

The resistivity and the Hall effect were measured using the four-pin contact method in the temperature range from 12 to 200 K and in magnetic fields up to 9 T on a PPMS-9 universal system for measuring physical properties (Quantum Design, USA) at the TC NTAM of the Institute of Metal Physics, Ural Branch, Russian Academy of Sciences. The electrical contacts were prepared using thin copper wire and silver paste. Measurements were performed with an electric current that flowed in the (001) plane and the magnetic field direction perpendicular to this plane. The ratio of resistivities of the WTe2 single crystal under study at room and helium temperatures is ρ300 K/ρ4.2 K ≈ 55, which indicates its high electrical purity.

In this study, the electrical resistivity in the absence of a magnetic field is denoted by ρ or ρ(0), the magnetoresistivity is denoted as Δρxx = ρ(B) – ρ(0) (where ρ(B) is the resistivity in magnetic field B), and the Hall resistivity is denoted by ρH. For ease of interpretation and presentation of experimental results, some of them are given in the form of magnetoconductivity \({{{{\sigma }}}_{{xx}}} = {{\Delta {{{{\rho }}}_{{xx}}}} \mathord{\left/ {\vphantom {{\Delta {{{{\rho }}}_{{xx}}}} {\left( {\Delta {{\rho }}_{{xx}}^{{\text{2}}} + {{\rho }}_{{\text{H}}}^{{\text{2}}}} \right)}}} \right. \kern-0em} {\left( {\Delta {{\rho }}_{{xx}}^{{\text{2}}} + {{\rho }}_{{\text{H}}}^{{\text{2}}}} \right)}}.\)

3 RESULTS AND DISCUSSION

3.1 Electrical Resistivity

The temperature dependence ρ(T) of the electrical resistivity of the WTe2 single crystal is shown in Fig. 5. In the temperature range from 12 to about 70 K, this dependence can be expressed as follows:

$$\rho = {{\rho }_{0}} + A{{T}^{2}}.$$
(1)
Fig. 5.
figure 5

The temperature dependence ρ(T) of the electrical resistivity of WTe2 in the temperature range from 12 to 200 K. The inset shows dependence ρ = f(T 2) in the temperatures from 12 to 100 K.

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The quadratic temperature dependence of the electrical resistivity was observed in pure metals [16]. As a rule, the contribution ~T2 is associated with electron–electron scattering that is usually observed at temperatures below 10–15 K [16, 17]. At higher temperatures, electron–phonon scattering mechanism should prevail, which leads to the dependence ρ(T) ~ T  5 at T ⪡ ΘD (ΘD is the Debye temperature) and to the linear dependence ρ(T) at temperatures comparable to ΘD. The Debye temperature for WTe2 is 133.8 ± 0.06 K [18]. In our case, the resistivity contribution proportional to T5 is not observed at low temperatures T ⪡ ΘD, i.e., the Bloch–Grüneisen law is not followed.

The quadratic behavior of the ρ(T) dependence at temperatures between 12 K and around 70 K can be explained as follows. According to Drude formula, the conductivity can be expressed as

$$~\sigma = \frac{{n{{e}^{2}}\tau }}{m} = \frac{{n{{e}^{2}}l}}{{mv}},$$
(2)

where n is the concentration of current carriers; e is the elementary charge; τ is the relaxation time; m is the electron mass; and l and v are the mean free path and the velocity of conduction electrons, respectively. In [19], the l value was estimated for our crystal, and it was demonstrated that the mean free path length is l = const + CT–2 in the temperature region 24–55 K, which is consistent with the quadratic behavior of the dependence of the electrical resistivity at temperatures up to 70 K. It can be assumed that contributions from various scattering mechanisms lead to the quadratic nature of the ρ(T) dependence due to the peculiarities of the electronic structure of WTe2 at T ≤ 70 K. This should also manifest itself in the resistivity (conductivity) value measured in the presence of a magnetic field.

3.2 Magnetoresistivity

Field dependence Δρxx = ρ(B) – ρ(0) of the magnetoresistivity of the WTe2 single crystal at a temperature of T = 12 K is shown in Fig. 6a. As can be seen from Fig. 6a, magnetoresistivity Δρxx varies with variation of the field in accordance with a law close to quadratic law Δρxx ~ Bn, where n ≈ 1.93 ± 0.01. Such behavior is typical for compensated conductors with a closed Fermi surface in the region of high effective magnetic fields (ωcτ ⪢ 1, where ωc is the cyclotron frequency) [17].

Fig. 6.
figure 6

(a) The field dependence Δρxx(B) of the magnetoresistivity of the WTe2 single crystal at T = 12 K. (b) Temperature dependence ρ(T) of the resistivity of the WTe2 single crystal in a magnetic field of 9 T in the temperature range from 12 to 100 K. The inset shows conductivity σxx = f(T2) in a magnetic field of 9 T in the temperature range from 12 to 65 K.

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Temperature dependence ρ(T) of the resistivity of the WTe2 single crystal in a magnetic field of 9 T is shown in Fig. 6b. There is a minimum in the ρ(T) curve. A similar dependence was observed, for example, in tungsten single crystals [20], in which the presence of a minimum is explained by the transition from high effective magnetic fields to low ones. According to [17], the conductivity of a compensated metal with a closed Fermi surface in the region of high effective magnetic fields (ωcτ ⪢ 1) is determined by the contributions from various scattering mechanisms. Therefore, it is more convenient to carry out further analysis on the basis of the dependence of conductivity σxx in a magnetic field. To simplify the calculations, the formula of σxx for the case of an isotropic crystal is further used, in which σxx is related to components Δρxx and ρH (Hall resistivity) of the resistivity tensor as \({{{{\sigma }}}_{{xx}}} = {{\Delta {{{{\rho }}}_{{xx}}}} \mathord{\left/ {\vphantom {{\Delta {{{{\rho }}}_{{xx}}}} {\left( {\Delta {{\rho }}_{{xx}}^{2} + {{\rho }}_{{\text{H}}}^{2}} \right)}}} \right. \kern-0em} {\left( {\Delta {{\rho }}_{{xx}}^{2} + {{\rho }}_{{\text{H}}}^{2}} \right)}}.\) The dependence of σxx on T2 is shown in the inset of Fig. 6b, from which one can see that conductivity σxx in a magnetic field also changes with temperature according to a quadratic law, but already in a narrower temperature range from 12 to about 55 K when compared to electrical resistivity. Thus, quadratic temperature dependences are observed for both the electrical resistivity in the absence of a magnetic field and the conductivity in a magnetic field, which is apparently associated with contributions from various scattering mechanisms.

3.3 The Hall Effect

Figure 7a shows the temperature dependences of the Hall coefficient RH, and the concentration n and mobility μ of the main charge carriers of the WTe2 single crystal that were obtained within of a single-band model by formulas

$${{R}_{{\text{H}}}} = \frac{{{{\rho }_{{\text{H}}}}}}{B};$$
(3)
$$n = \frac{1}{{{{R}_{{\text{H}}}}e}};$$
(4)
$$\mu = \frac{{{{R}_{{\text{H}}}}}}{\rho }.$$
(5)
Fig. 7.
figure 7

(a) The temperature dependences of Hall coefficient RH, concentration n, and mobility μ of current carriers in WTe2 according to a single-band zone model in a magnetic field of B = 9 T. (b) Field dependences of Hall resistivity ρH(B) and resistivity ρ(B) in the magnetic field for WTe2 at T = 12 K: open circles show the experimental data; the solid red lines are curves according to a two-band model and calculated with a computer software [26].

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Given that RH < 0, the main charge carriers are electrons with a concentration of n ≈ 5.3 × 1019 cm–3 and a mobility of μ ≈ 5.9 × 103 cm2/(V s) at T = 12 K. The n value determined by formula (4) changes weakly with temperature, which is typical for a number of compensated conductors with a closed Fermi surface [21, 22]. At the same time, the mobility μ calculated by formula (5) strongly decreases with temperature, which can be explained by an increase in the scattering efficiency of current carriers.

As was shown in [19], Hall resistivity ρH in WTe2 is nonlinearly dependent on the magnetic field B. It is assumed that such behavior of the ρH(B) dependence may be related to the mechanism of scattering of conduction electrons on the sample surface. This was observed in [23, 24] when compensated metals with a closed Fermi surface were investigated under conditions of a static skin effect. A strongly nonlinear dependence of ρH on the field in WTe2 was also observed at low temperatures in [14, 25], which was explained by the presence of current carriers of the electron and hole types. In systems containing electron and hole charge carriers, a two-band model is typically used to analyze the field dependences of resistivity ρ in a magnetic field and Hall resistivity ρH. The expressions for ρ and ρH are written in the following form given in [14]:

$$\begin{array}{*{20}{c}} {\rho = \frac{1}{e}\frac{{\left( {{{n}_{h}}{{\mu }_{h}} + {{n}_{e}}{{\mu }_{e}}} \right) + \left( {{{n}_{h}}{{\mu }_{e}} + {{n}_{e}}{{\mu }_{h}}} \right){{\mu }_{h}}{{\mu }_{e}}{{B}^{2}}}}{{{{{\left( {{{n}_{h}}{{\mu }_{h}} + {{n}_{e}}{{\mu }_{e}}} \right)}}^{2}} + {{{\left( {{{n}_{h}} - {{n}_{e}}} \right)}}^{2}}\mu _{h}^{2}\mu _{e}^{2}{{B}^{2}}}},} \end{array}$$
(6)
$$\begin{array}{*{20}{c}} {{{\rho }_{{\text{H}}}} = \frac{B}{e}\frac{{\left( {{{n}_{h}}\mu _{h}^{2} - {{n}_{e}}\mu _{e}^{2}} \right) + \left( {{{n}_{h}} - {{n}_{e}}} \right)\mu _{h}^{2}\mu _{e}^{2}{{B}^{2}}}}{{{{{\left( {{{n}_{h}}{{\mu }_{h}} + {{n}_{e}}{{\mu }_{e}}} \right)}}^{2}} + {{{\left( {{{n}_{h}} - {{n}_{e}}} \right)}}^{2}}\mu _{h}^{2}\mu _{e}^{2}{{B}^{2}}}},} \end{array}$$
(7)

where ne (μe) and nh (μh) are the concentration (mobilities) of electrons and holes, respectively. As shown in Fig. 7b, the field dependences of the resistivity (ρ(B)) in the presence of a magnetic field and the Hall resistivity (ρH(B)) for the WTe2 single crystal at 12 K were described using the two-band model by formulas (6) and (7), respectively. The following values of the concentrations and mobilities of electrons and holes were obtained: ne = (3.14 ± 0.01) × 1019 cm–3, nh = (2.78 ± 0.01) × 1019 cm–3, μe = (4.77 ± 0.02) × 103 cm2/(V s), and μh = (3.42 ± 0.01) × 103 cm2/(V s). Proportion ne ≈ nh points to electron–hole compensation in WTe2.

Thus, estimates of the concentrations and mobilities of current carriers within both single-band and two-band models are in good agreement with each other. This also applies to the values of the Hall coefficient as well, which equal RH = –1.170 × 10–1 cm3/C in the two-band model and RH = –1.168 × 10–1 cm3/C in the single-band model.

4 CONCLUSIONS

Studies of the kinetic properties of the topological semimetal WTe2 single crystal have shown that both electrical resistivity in the absence of a magnetic field and conductivity in the presence of a field depend on the temperature according to the quadratic law in a wide temperature range from 12 K to about 70 and 55 K, respectively. This appears to be associated with the contributions from various scattering mechanisms.

As a result of the analysis of experimental data on the Hall effect and the resistivity in the presence of a magnetic field, the concentrations and mobilities of current carriers in WTe2 are estimated using both single-band and two-band models. These results are in good correlation and point to electron–hole compensation with a slight predominance of electron charge carriers.