Galvanomagnetic properties of the putative type-II Dirac semimetal PtTe2

Platinum ditelluride has recently been characterized, based on angle-resolved photoemission spectroscopy data and electronic band structure calculations, as a possible representative of type-II Dirac semimetals. Here, we report on the magnetotransport behavior (electrical resistivity, Hall effect) in this compound, investigated on high-quality single-crystalline specimens. The magnetoresistance (MR) of PtTe2 is large (over 3000% at T = 1.8 K in B = 9 T) and unsaturated in strong fields in the entire temperature range studied. The MR isotherms obey a Kohler’s type scaling with the exponent m = 1.69, different from the case of ideal electron-hole compensation. In applied magnetic fields, the resistivity shows a low-temperature plateau, characteristic of topological semimetals. In strong fields, well-resolved Shubnikov – de Haas (SdH) oscillations with two principle frequencies were found, and their analysis yielded charge mobilities of the order of 103 cm2 V−1 s−1 and rather small effective masses of charge carriers, 0.11 me and 0.21 me. However, the extracted Berry phases point to trivial character of the electronic bands involved in the SdH oscillations. The Hall effect data corroborated a multi-band character of the electrical conductivity in PtTe2, with moderate charge compensation.

T = 300 K to 0.17 μΩ cm at T = 2 K, yielding the residual resistivity ratio RRR = ρ(300K)/ρ(2K) equal to 142. The magnitudes of both ρ(2K) and RRR indicate high crystallinity of the specimen measured (RRR is approximately 5 times larger than that reported for PtTe 2 in ref. 18 ). As displayed in Fig. 1a (note the red solid line), in the whole temperature range covered, ρ(T) can be very well approximated with the Bloch-Grüneisen (BG) law: where ρ 0 is the residual resistivity, accounting for scattering conduction electrons on crystal imperfections, while the second term represents electron-phonon scattering (Θ D stands for the Debye temperature). The least-squares fitting yielded the parameters: ρ 0 = 0.17 μΩ cm, Θ D = 253 K, A = 11.6 μΩ cm and k = 3.23. The value of k is smaller than k = 5, expected for simple metals, yet similar to k exponents determined for several monopnictides [25][26][27] . The temperature dependencies of the electrical resistivity of PtTe 2 measured in transverse magnetic field (i ⊥ c axis and B c axis) are gathered in Fig. 1b. In non-zero B, ρ(T) is a non-monotonous function of temperature, showing an upturn below a certain slightly field-dependent temperature T*, and then forming a plateau at the lowest temperatures. With increasing magnetic field, the magnitude of the resistivity in the turn-on and plateau regions distinctly increases. Similar behavior was considered as a fingerprint of the presence of nontrivial topology in the electronic structure of several TSs [28][29][30][31] . Another possible explanation of this type of magnetic field governed changes in ρ(T) is a metal-insulator transition 29,32,33 . However, as discussed first for WTe 2 34 , and afterwards, e.g., for rare-earth monopnictides [25][26][27][35][36][37][38][39][40][41] , the magnetic field induced upturn in ρ(T) may appear also in trivial semimetals, which are close to perfect charge carriers compensation. We presume that the latter mechanism is also fully appropriate for the electrical transport behavior in PtTe 2 .
In order to examine the actual nature of the galvanomagnetic behavior observed for PtTe 2 , transverse magnetoresistance, MR = [ρ(B) − ρ(B = 0)]/ρ(B = 0), measurements were performed at several constant temperatures in the configuration i ⊥ c axis and B c axis. As can be inferred from Fig. 2a, in B = 9 T, MR taken at T = 1.8 K achieves a giant value of 3060%, which is an order of magnitude larger than MR determined in the same conditions for the related system PdTe 2 23 , and almost equal to that reported for the type-II Weyl semimetal MoTe 2 42 .  With increasing temperature, MR measured in B = 9 T does not change significantly up to about 10 K (i.e., in the plateau region of ρ(T)), and then decreases rapidly. However, even at T = 150 K, MR remains exceptionally large exceeding 500% in 9 T. At each of the temperatures studied, MR shows no tendency towards saturation in strong fields. MR behavior similar to that of PtTe 2 was established before for several TSs 23,28,30,[42][43][44] . However, unsaturated MR can be also attributed to perfect or almost perfect carrier compensation in a semimetallic material 45 .
Remarkably, as demonstrated in Fig. 2b, the MR isotherms of PtTe 2 obey the Kohler's rule in the entire temperature range studied. This finding rules out the scenario of metal-insulator transition as a possible mechanism of the magnetic field driven changes in the electrical transport of PtTe 2 . The MR data collapse onto a single curve, which can be approximated by the expression: MR ∝ (B/ρ 0 ) m , with the exponent m = 1.69 (note the red solid line in Fig. 2b). While m = 2 is expected for materials with perfect electron-hole balance, and the obtained value is also smaller than m = 1.92 reported for WTe 2 34 , it is similar to those found for some monopnictides, which were reported as trivial semimetals fairly close to charge compensation 26,37,40 . Quantum oscillations. In order to characterize the Fermi surface in PtTe 2 , we investigated quantum oscillations in ρ(B) (Shubnikov-de Haas effect) at a few different temperatures. Figure 3a shows the oscillatory part of the electrical resistivity, Δρ, obtained by subtraction of the second-order polynomial from the experimental data, plotted as a function of reciprocal magnetic field, 1/B. As can be inferred from this figure, the SdH oscillations remain discernible at temperatures up to at least 15 K, however, their amplitudes systematically decrease with increasing temperature. Fast Fourier transform (FFT) analysis, the results of which are presented in Fig. 3b, disclose four features at the oscillations frequencies f i FFT (i represents the Fermi surface pocket label). The most prominent peak occurs at likely arises as the third harmonics of α f FFT . It is worthwhile noting that the FFT spectrum of PtTe 2 is very similar to that reported for PdTe 2 in ref. 23 but differs from the FFT data shown for the same compound in ref. 24 . Using the Onsager relation where S stands for the area of Fermi surface cross-section, one finds for the two pockets in PtTe 2 the values S α = 1.03 × 10 −2 Å −2 and S β = 2.34×10 −2 Å −2 . Assuming circular cross-sections, the corresponding Fermi wave vectors are k F,α = 5.73 ×10 −2 Å −1 and k F,β = 8.63 × 10 −2 Å −1 . Then, if we assume that both Fermi surface pockets are spherical, which is poor approximation, the carrier densities in these two pockets would be equal to n α = 6.35 × 10 18 cm −3 and n β = 2.17 × 10 19 cm −3 .
The electronic structure calculated for PtTe 2 3,18 comprises three bands crossing the Fermi level, one hole-like band, located at the center of the Brillouin zone, and two electron-like bands. The fact that we observed experimentally only two principle frequencies is probably due to very small size of the third Fermi surface pocket or might arise owing to somewhat lower position of the Fermi level in the single crystal studied. From the comparison of the values of k F,i obtained from the FFT analysis and sizes of the calculated electronic bands 3 , one may presume that the α Fermi surface pocket corresponds to one of the electron-like bands and the β Fermi surface pocket represents the hole-like band.
The inset to Fig. 3b displays the temperature variations of the FFT amplitudes, R i (T), corresponding to the α and β Fermi surface pockets in PtTe 2 . The gradual damping of both oscillations with rising temperature can be described by formula 46 : where ⁎ m i is the effective cyclotron mass of charge carriers, B eff = 4.5 T was calculated as B eff = 2/(1/B 1 + 1/B 2 ) (B 1 = 3 T and B 2 = 9 T are the borders of the magnetic field window in which the FFT analysis was performed), β ⁎ m m 0 21 e . It is worth noting that the effective mass determined for the α Fermi surface pocket in PtTe 2 is almost equal to that reported for the Pd-bearing counterpart 23 .
In the next step, we attempted to determine the phase shift, ϕ i , in the SdH oscillations, which is directly related to the Berry phase, ϕ B,i , of the carries involved. There are known a few methods of extracting ϕ i , and the proper interpretation of their values remains debatable [47][48][49][50] . The most reliable approach is direct fitting to the experimental data of the Lifshitz-Kosevich (LK) function 46 : where p i is the harmonic number, f i is the oscillation frequency, and T D,i stands for the Dingle temperature. In Fig. 4a, there is shown the result of fitting the LK function to the oscillating resistivity of PtTe 2 observed at T = 1.8 K (note the red solid line). As discussed above, at this temperature, the FFT spectrum comprises three peaks, and thus the sum in Eq. 3 consists of as many as three contributions. In order to reduce the total number of free parameters in this equation, the effective masses were fixed at the values obtained from Eq. 2 (see above). With this simplification, one obtained the parameters: f α = 108.1 T, T D,α = 9.1 K and ϕ α = 0.65 for the α band, and f β = 246 T, T D,β = 5 K and ϕ β = 0.54 for the β band. Remarkably, the so-obtained values of f i are almost identical to those derived from the afore-described FFT analysis, hence confirming the internal consistency of the approach applied. Using the Dingle temperatures, the quantum relaxation time, τ q,i , could be calculated from the relation τ q,i = ħ/(2πk B T D,i ) to be equal to 1.34 × 10 −13 s and 2.43 × 10 −13 s for the α and β bands, respectively. Then, the quantum mobility of charge carriers, μ q,i , were estimated from the relationship μ to be 2138 cm 2 V −1 s −1 and 2038 cm 2 V −1 s −1 for the α and β bands, respectively.
The phase shift ϕ i in Eq. 3 is generally a sum ϕ i = −1/2 + ϕ B,i + δ i , where δ i represents the dimension-dependent correction to the phase shift 49 . In two-dimensional (2D) case, this parameter amounts zero, while in three-dimensional (3D) case δ i is equal to ±1/8, and its sign depends on type of charge carriers and kind of cross-section extremum. Supposing that the SdH oscillations in PtTe 2 originate from 3D bands with carriers moving on their maximal orbits, one can set δ = −1/8 for electrons and δ = 1/8 for holes. With this assumption, the Berry phases ϕ B,α = 0.55π and ϕ B,β = 1.83π were obtained for the α and β bands, respectively.
To check the reliability of the LK analysis performed, Eq. 3 was also used to describe the experimental data measured at T = 10 K. At this temperature, just one peak in the FFT spectrum is discernible (see Fig. 3b), which corresponds to the α Fermi surface pocket. The result of fitting the LK formula is presented in Fig. 4b (note the red solid line), and the so-derived values of the parameters are: f α = 108.1 T, T D,α = 12.6 K and ϕ B,α = 0.46π. Notably, the agreement between the f α values obtained at T = 10 K and T = 1.8 K is perfect. The value of T D,α implies μ q = 1544 cm 2 V −1 s −1 . Clearly, with increasing temperature, the Dingle temperature increases and consequently the quantum charge carriers mobility becomes smaller, which is probably due to increasing the scattering rate. In turn, the Berry phase of the α band was found almost independent of temperature. All the parameters obtained from the LK approach to the magnetotransport in PtTe 2 are gathered in Table 1.
Another commonly applied technique for Berry phase derivation is using Landau level (LL) fan diagrams. Though in case of multi-frequency oscillations this method is obstructed by possible superposition of the quantum oscillation peaks that hinders precise determination of the Landau level index for a given frequency 51 , we made an attempt to construct the LL fan diagram for the α Fermi pocket in PtTe 2 . As it is apparent from Fig. 3b, the FFT maximum occurring at = α f 108 T FFT is fairly well separated from the other FFT peaks, and hence one could filter this oscillation with reasonably high accuracy. For PtTe 2 one finds ρ ρ > xy (ρ xy is the Hall resistivity discussed in the next section), and therefore the maxima in the oscillatory resistivity measured at T = 1.8 K (see Fig. 4a) were numbered by integers, n, and the minima by half-integers, n + 1/2. The result of this approach is shown in the main panel of Fig. 5. A linear fit of the LL indices (note the solid line) gives an intercept of 0.62, At T = 10 K, the electrical resistivity of PtTe 2 oscillates in the transverse magnetic field with only one frequency (cf. Fig. 3b), so building the LL fan diagram is straightforward. As can be inferred from the inset to Fig. 5, the LL indices plot yields the intercept 0.63, which is almost the same as that obtained at the lower temperature. Also, the slope of the straight line (f α = 108.3 T) is identical with that determined at 1.8 K, and furthermore it is very close to the FFT value (see Table 1). Most importantly, all the parameters extracted from the LL indices plots are in perfect agreement with the quantities obtained for the α Fermi pocket from the LK analysis, which unambiguously corroborates the correctness of both techniques applied for PtTe 2 . T -temperature; f i FFToscillation frequency obtained from the FFT analysis; f i -oscillation frequency obtained from Eq. 3; m *effective mass; T D -Dingle temperature; τ q -quantum relaxation time; μ q -quantum mobility; ϕ B -Berry phase.   Fig. 6a. At T = 2 K, ρ xy (B) behaves in a rather complex manner. In weak magnetic fields, it is negative and exhibits a shallow minimum. Near 2 T, the Hall resistivity changes sign to positive, and then its magnitude increases with increasing B. The ρ xy (B) isotherm measured at 25 K shows a fairly similar field variation, yet the positive contribution in strong fields remains too small to cause sign reversal. At higher temperatures, one observes a gradual straightening of the ρ xy (B) with rising T. The overall behavior of the Hall effect in PtTe 2 confirms the multi-band character of the electrical transport in this material. It is worth recalling that very similar Hall response was observed for the closely related compound PdTe 2 24 .
For quantitative analysis of the experimental data, first a two-bands Drude model was applied. For this purpose, ρ xy (B) measured at T = 2 K was converted to the Hall conductivity σ ρ ρ ρ = + /( ) xy xy xy 2 2 , as displayed in Fig. 6b. Next, σ xy (B) was fitted by the formula: where n e1 and μ e1 , n h and μ h stand for the carrier concentrations and the carrier mobilities of electron-and hole-like bands, respectively. As can be inferred from Fig. 6b, the so-obtained approximation of the measured σ xy (B) data (note the blue dashed line) is not ideal. An obvious reason for the discrepancy between the experiment and the two-band model could be contribution from another band, the presence of which was revealed in the ab-initio calculations of the electronic structure of PtTe 2 3,18 . Therefore, in the next step, the Hall conductivity was analysed in terms of a three-bands model: where n e2 and μ e2 account for the carrier concentration and the carrier mobility, respectively, of another electron-like band in PtBi 2 . The result of fitting Eq. 5 to the experimental σ xy (B) data is shown as red solid line in Fig. 6b. Clearly, the obtained description is much better than that with the two-bands model. The fitting parameters derived in the two approaches are listed in Table 2. Both models yielded large carriers concentrations of the order of 10 20 cm −3 . It is worth noting that very similar charge densities were found in the Dirac semimetal PtBi 2 52 . On the contrary, for the type-II Weyl semimetal WTe 2 the carrier concentrations were reported to be up to two orders of magnitude larger than those in PtTe 2 53 . As regards the level of carrier compensation, the two-bands model yielded considerable charge imbalance given by the ratio n h /n e1 = 1.19, however the three-bands model led to fairly balanced scenario n h /(n e1 + n e2 ) = 1.04. Recently, similar degree of electron-hole compensation was established, e.g., in semimetallic monobismuthides YBi and LuBi 27 . The mobilities of charge carriers in PtTe 2 were found very high, especially that obtained for one of the electron-like Fermi surface pockets (μ ∼ × − − 2 10 cm V s The discrepancy likely arises due to the fact that the quantum mobility is affected by all possible scattering processes, whereas the Hall mobility is sensitive to small-angle scattering only 46 . Angle-dependent magnetoresistance. In order to check whether PtTe 2 demonstrates CMA, angledependent magnetotransport measurements were performed at T = 2 K. In these experiments, electric current was always flowing within the hexagonal a − b plane, while the angle θ between current and magnetic field direction was varied from θ = 90° (B ⊥ i) to θ = 0° (B i). As can be inferred from Fig. 7, the electrical resistivity rapidly decreases on deviating from the transverse configuration, and eventually for the longitudinal geometry ρ measured in B = 9 T is about an order of magnitude smaller than that for B ⊥ i.
Clearly, the longitudinal MR experiments did not provide any evidence for CMA in PtTe 2 . A possible source for that may be large contribution of non-Dirac states to the measured resistivity. At odds with the Drude theory, which predicts zero MR for B i 57 , in several materials sizeable positive longitudinal MR was observed. Among the theories which interpret this phenomenon 58-61 , that accounting for Fermi surface anisotropy 61 seems appropriate for PtTe 2 . Within the latter approach, positive longitudinal MR up to ∼100% can be expected for strongly anisotropic systems. In consequence, CMA would be discernible only if its negative contribution to the longitudinal MR is larger than the positive term due to trivial electronic bands.

Methods
Single crystals of PtTe 2 were grown by flux method. High-purity constituents (Pt 5 N, Te 6 N), taken in atomic ratio 1:20, were placed in an alumina crucible covered by molybdenum foil strainer and capped with another inverted alumina crucible. This set was sealed inside a quartz tube under partial Ar gas atmosphere. The ampoule tube was heated up to 1150 °C, held at this temperature for 24 hours, then quickly cooled down to 850 °C at a rate of 50 °C/h, kept at this temperature for 360 hours, followed by slow cooling down to 550 °C at a rate of 5 °C/h. Subsequently, the tube was quenched in cold water. Upon flux removal by centrifugation, multitude of single crystals with typical dimensions 3 × 2 × 0.4 mm 3 were isolated. Their had metallic luster and were found stable against air and moisture. Chemical composition of the single crystals obtained was checked by energy-dispersive X-ray analysis using a FEI scanning electron microscope equipped with an EDAX Genesis XM4 spectrometer. The average elemental ratio Pt: Te = 35.2(5): 64.8(3) was derived, in accord with the expected stoichiometry. The crystal structure of the single crystals was examined by X-ray diffraction on a KUMA Diffraction KM-4 four-circle diffractometer equipped with a CCD camera, using graphite-monochromatized Cu-Kα1 radiation. The hexagonal CdI 2 -type crystal structure (space group P m 3 1, Wyckoff No. 164) reported in ref. 62 was confirmed, with the lattice parameters very close to the literature values.
Crystallinity and orientation of the crystal used in the electrical transport studies was checked by Laue backscattering technique employing a Proto LAUE-COS system. Due to the layered crystal structure of PtTe 2 it was possible to obtain very thin samples by scotch-tape technique with their surface corresponding to the a − b plane of the hexagonal unit cell of the compound. Rectangular-shaped specimen with dimensions 2.9 × 1.3 × 0.04 mm 3 was cut from the cleaved single crystal using a scalpel. Electrical contacts were made from 50 μm thick silver wires attached to the sample using silver epoxy paste. Electrical transport measurements were carried out within the temperature range 2-300 K and in magnetic field up to 9 μT using a conventional four-point ac technique implemented in a Quantum Design PPMS platform.
Data availability. The datasets analysed during the current study are available from the corresponding author on reasonable request.