Introduction

The transition metal dichalcogenides (TMD) have recently become the focus of fundamental research and technological applications due to their unique crystal structures, a wide range of chemical compositions, novel and intriguing properties with potential applications in field effect transistors, optoelectronic devices, topological insulators, electrocatalysts and so on1. Among them, the MX2-type transition metal dichalcogenides, such as TaSe2, TaS2, IrTe2, WTe2, MoS2 and MoSe2, have attracted great attention due to their rich physical properties like charge-density wave2,3,4, Mott-insulator to metal transition4, superconductivity5,6,7, catalysis of chemical reactions8, extremely large magneto-resistance9 and potential technological applications9,10,11. A thorough investigation of the electronic structures of these compounds is important for understanding their physical properties and exploring for new phenomena.

Recently, it has been discovered that the MX2-type PdTe2 and its Cu-intercalated counterpart CuxPdTe2 exhibit superconductivity with transition temperature (Tc) of about 1.7 K12. The following angle-resolved photoemission spectroscpoy results and theoretical calculations have revealed the existence of topologically nontrivial surface state with Dirac cone in PdTe213, which offers a new material base for understanding the superconducting mechanism in the transition metal dichalcogenide compounds. More importantly, the coexistence of topological surface state with bulk superconductivity may offer an important material candidate for investigating possible topological superconductivity14,15, which is of particular importance both in fundamental physics such as the searching of long-sought yet elusive Majorana Fermions and in practical applications in the next-generation spintronics technologies.

The electronic structure of the PdTe2 compound has been investigated both theoretically and experimentally16,17,18,19. It turns out that the PdTe2 exhibits a complex Fermi surface topology with a three-fold symmetry. A small electron-like spot appears around the Γ point, which is surrounded by six small electron-like spots locating near the midpoint between Γ and K. The complicated multiband Fermi surface makes it rather difficult in the understanding of the transport properties and possible topological characters. In this paper, we report the de Haas-van Alphen effect (dHvA effect) and magnetoresistance (MR)results of the PdTe2 single crystal. The dHvA results clearly suggest that only one kind of carrier with certain relaxation rate dominates the electronic structure of the PdTe2 compound, which makes the situation more simple. The dominance of one kind of carrier is further supported by the MR results, where the Kohler’s law is well obeyed. The present observation could be important in further investigation of the electronic structure and transport behaviors of the PdTe2-related compound.

Figure 1 gives the powder x-ray diffraction pattern and the single crystal x-ray diffraction pattern of the PdTe2 sample. The powder x-ray diffraction pattern suggests that the PdTe2 sample is well crystallized in the CdI2-type structure with the (No. 164) space group. The refined lattice parameters are a = b = 4.112 Å and c = 5.233 Å, which are consistent with previous reports20. The as-grown samples exhibit shining silvery surface with typical dimensions of 4 × 4 × 0.5 mm3. A picture of the cleaved single crystal is shown in the upper right corner of Fig. 1. The single crystal x-ray diffraction pattern is performed on the cleaved shining surface of the as-grown PdTe2 single crystal. The observed peaks can be indexed into (0 0 n) (with n being integers), indicating that the naturally cleaved surface is the basal ab plane. In the inset of Fig. 1 we show the temperature dependence of in-plane resistivity of the PdTe2 sample. A metallic behavior is observed at all temperature range. Below Tc ~ 2 K, the superconducting transition occurs. These results suggest that the obtained sample is high-quality PdTe2 single crystal sample.

Figure 1
figure 1

The powder and single-crystal x-ray diffraction pattern of the PdTe2 sample.

The inset shows the temperature dependence of in-plane resistivity of the sample. The superconducting transition occurs with Tc = 2.0 K.

Full size image

Figure 2(a) shows the quantum oscillation data of the PdTe2 sample using the de Haas-van Alphen effect (dHvA effect) at T = 0.36 K. The inmagnetization quantum oscillations are probed via highly sensitive torque magnetometry methods which measure the magnetic susceptibility anisotropy of the sample. The experimental setup is schematically depicted in the inset of Fig. 2(a). The sample is mounted on the sample holder with the basal ab plane parallel to the top surface of the sample holder. The magnetic field is applied to the crystal with a tilt angle relative to the crystalline c axis. We find that at θ = 60°, the dHvA effect has the strongest oscillation amplitude comparing to other angles (the angel dependence of dHvA effect will be discussed later). The polynomial background has been subtracted and the dHvA effect exhibits good oscillation feature. The oscillatory pattern in δC/C0 is periodic in 1/μ0H and the oscillation arise from the successive empty of Landau levels when the field is increased (Fig. 2(b)). In the inset of Fig. 2(b) we plot the fast Fourier transformation (FFT) spectra of the oscillatory torque after subtracting the polynomial background. One can clearly see three peaks in the FFT spectra, which are labeled as F1, F2 and F3, respectively. We carefully check the position of these peaks and find that F1 = 121.5 T, F2 = 239 T, F3 = 283.2 T. Since F2 ≈ 2F1, the F2 can be considered as the multiple frequency signal of F1. Besides F1, there is a weak peak locating at F3 = 283.2 T, which is consistent with the multi-band feature of the PdTe2 compound16,17,18,19. However, the FFT peaks of F3 as well as other possible peaks derived from the multi-band feature of PdTe2 compound are all at least one order of magnitude less than the F1 peak. The dominant oscillation frequency is F1 = 121.5 T, which is in agreement with the early dHvA results by A. E. Dunsworth16. According to the Onsager relation , we can get the cross-sectional area of the Fermi surface normal to the field, that is A = 1.16 × 10−2 Å−2. The corresponding Fermi momentum is estimated to be kF = 0.061 Å−1.

Figure 2
figure 2

(a) The de Hass-van Alphen oscillations in the torque data with subtracting the polynomial background at 0.36 K. A schematic of the experimental setup is shown in upper left corner, where the magnetic field is applied to the crystal with a tilt angle relative to the crystalline c axis. (b) The de Hass-van Alphen oscillations ploted against 1/μ0H. The inset is a fast Fourier transformation (FFT) of the oscillatory torque after subtracting the polynomial background.

Full size image

In order to obtain the effective mass and Dingle temperature of the charge carriers, we perform the dHvA measurements at various temperatures. The results are shown in Fig. 3(a), where the magnetic torque is plotted as a function of 1/μ0H. The dHvA oscillation can be described by the Lifshitz-Kosevich formula21,22,23,

Figure 3
figure 3

(a) The de Hass-van Alphen oscillations ploted against 1/μ0H at various temperatures. (b) The temperature dependence of the thermal damping factor of dHvA oscillation. The blue curve is a fit to the Lifshitz-Kosevich formula, from which we can extract the cyclotron effective mass m* ≈ 0.13 m0 and Fermi velocity νF ≈ 5.6 × 10−5 m/s. The inset of (b) shows the Dingle plot at different magnetic fields. The red line is a fit using the Lifshitz-Kosevich formula, which gives TD = 30.4 ± 1.4 K.

Full size image

where RT and RD are the thermal damping factor and the Dingle damping factor, respectively, γ is a phase shift. The thermal damping factor is RT = X/sin h(X), where . And the Dingle damping factor is , where m* is the effective mass and TD is the Dingle temperature. At a fixed magnetic field, the amplitude of the the quantum oscillation is given by A ∝ RT · RD. Figure 3(b) shows the amplitude versus temperature at the fixed magnetic field of H = 12.4 T. We use the thermal damping factor RT to fit these experimental data points and yields the cyclotron effective mass m* ≈ 0.13 me. Based on the effective mass m* and the Fermi momentum kF, we obtain the related Fermi velocity νF ≈ 5.6 × 105 m/s by using the formula νF =  kF/m.

Dingle temperature TD is related to the cyclotron relaxation rate with the formula 24. We get the Dingle temperature by fitting the Dingle plot at different magnetic fields. The experimental data and the fitting line are shown in the inset of Fig. 3(b). We find that the fitting line can well-reproduce the experimental data. The fit yields the Dingle temperature TD = 30.4 ± 1.4 K. The cyclotron relaxation rate is estimated to be τq = 4 × 10−14 s.

From above de Hass-van Alphen oscillation data we notice that a predominant oscillatory frequency which is at least one order of magnitude larger than other oscillation frequencies. In order to clarify which band is corresponded to this oscillatory, we perform the dHvA effect experiments at different tilting angle relative to the crystalline c axis at the fixed temperature T = 1.5 K and compare the results with previous theoretical calculations and experimental data. The experimental setup is the same as that depicted in the inset of Fig. 2(a). It is found that the quantum oscillation has the strongest magnitude at around 60°. The magnitude of dHvA oscillation gradually decreases when the tilting angle is go away from 60°. Particularly, when the tilting angle is less than 16°, the oscillation signal is comparable with the noise signal. We make the fast Fourier transformation on each curve and get the dominant oscillation frequencies for every angle, which are given as the blue circles in Fig. 4(b). We use the two-dimensional (2D) Fermi surface model and three-dimensional (3D) ellipsoid Fermi surface model to fit the experimental data and We find that both the 2D Fermi surface and the 3D ellipsoid Fermi surface cannot well-reproduce the experimental data. The angle dependent dHvA results suggest a complicated Fermi surface topology of the PdTe2 compound, which is consistent with previous de Hass-van Alphen results16. We carefully compare the angle-dependent oscillatory frequency presented in Fig. 4(b) with previous band structure calculations and experimental band structure studies16,17. We find that both the high frequency branches and the medium frequency branches are not detected in the present de Hass-van Alphen results. At low frequency region, there are five branches: ε1, ε2, ε3, ζ and o16. The branch ζ is a long ellipsoid or cylinder lying along c. The branches ε1, ε2 and ε3 are related and can be described by six long ellipsoids or cylinders which are 60° apart from each other and 40° tilted from c. And the branch o could be another orbit around the ε’s or a new piece of Fermi surface, which lies slightly above ε1. We compare the angle dependence of dHvA frequencies and the effective masses of our experimental data with those of the ε1, ε2, ε3, ζ and o branches16,17. It is found that though the ζ branch has the same effective mass with the present data, its angle dependence of dHvA frequency is significant larger than the data given in Fig. 4(b). On the other hand, both the angle dependence of dHvA frequencies and the effective masses in our dHvA oscillations are comparable with those of the ε branches. Since the branches ε1, ε2 and ε3 are related with each other and are described by six long ellipsoids or cylinders which are 60° apart from each other, these branches cannot be distinguished in dHvA measurement. Thus it can be concluded that the predominant oscillatory frequency in the present de Hass-van Alphen data is related to the ε branches. This could probably explain the fact that the de Hass-van Alphen oscillations have the strongest magnitude at around 60°. That is, at 60°, each ellipsoid/cylinder contributes equally to the dHvA oscillations and the sum of the dHvA oscillations has the largest magnitude.

Figure 4
figure 4

(a) The de Hass-van Alphen oscillations measured under different tilt angles. (b) The angle dependence of the dominant oscillation frequency at the temperature of 1.5 K. The solid lines are the fits to the Fermi surfaces using a two-dimensional Fermi surface and a three-dimensional ellipsoid Fermi surface model, respectively.

Full size image

The present de Hass-van Alphen oscillation results suggest that the electronic structure in PdTe2 system could be simplified in a single-band model. In order to further investigate the magnetotransport behaviors of the PdTe2 compound, we perform the magnetoresistance measurements on the PdTe2 single crystal sample at different temperature. The results are shown in Fig. 5(a). At all temperatures, it is found that the resistivity increases with increasing the applied magnetic field, suggesting a positive magnetoresistance. The magnetoresistance decreases with increasing temperature. It is known that the Kohlers Rule is a suitable criterion for judging whether the transport properties of a given material is dominated by a single band or multi-band feature. According to the Kohlers Rule25, for a given metal with only one relaxation rate τ, the increase in resistivity in a magnetic field H relative to the zero-field value ρ0 is a universal function of H/ρ0, Δρ/ρ0 = f(H/ρ0), at all temperatures T and fields H. Here f(H/ρ0) is a universal function for a certain material, independent of temperature or impurity content. Note that this rule is only applicable to single-band metals. The Kohler plots for the PdTe2 single crystal sample are shown in Fig. 5(b) for temperatures between 0.36 K to 70 K and magnetic fields from 0 to 30 T. It is obvious that the magnetoresistance properties of the PdTe2 sample follow the Kohlers law well at all temperature. These facts suggest that despite of the multiband character in PdTe2, only one kind of charge carrier dominates the transport behavior.

Figure 5
figure 5

(a) The magnetic field dependence of in-plane resistance of the PdTe2 sample at different temperatures. (b) Kohler plot for the PdTe2 sample at different temperatures. It can be found that the Kohlers rule is obeyed at all temperatures. The inset plots the variation of Δρ/ρ0 against H2, where the Δρ/ρ0 ∝ H2 criterion is also obeyed when the magnetic field is less than 3 T.

Full size image

We also find that the magnetoresistance of PdTe2 obeys the Δρ/ρ0 ∝ H2 criterion at low magnetic field (H ≤ 3 T), which is another evidence of the predominantly single-band transport behavior in this compound (see the inset in Fig. 5(b)). Interestingly, the magnetoresistance Δρ/ρ0 is approximately linear dependent on H2 under high magnetic field. The magnetoresistance does not saturate even when the applied magnetic field is as high as 30 Tesla. The Δρ/ρ0 value at 30 T is about 900% at 0.36 K. The large unsaturated magnetoresistance phenomenon is recently found in many topologically-related compounds9,26,27, probably meaning a common origin of the unsaturated magnetoresistance in these systems. This result seems to be consistent with the observation of topological surface state by recent angle-resolved photoemission spectroscopy experiments13.

The above de Hass-van Alphen and magnetoresistance results both point to a predominantly single-band feature in the PdTe2 compound. However, due to the lack of Shubnikov-de Hass results on the PdTe2 sample, we are not able to accurately determine which band dominates the transport properties of the present compound. In order to thoroughly understand the electronic structure of the PdTe2 compound, a systematic investigation on the magnetotransport properties is needed.

The identification of the topological superconducting state has become a big challenge in condensed matter physics and materials communities. The superconductivity induced by atomic intercalation in pristine topological insulator Bi2Se3 has attracted great attention in the investigation of possible topological superconductivity14,15,28,29. However, these intercalated materials usually suffer from serious chemical inhomogeneity. And there is no consensus on whether or not these materials are topological superconductors. The PdTe2, on the other hand, is much simple in chemical composition and lattice structure, making this compound a promising candidate for investigating possible topological superconductivity. The existence of topological surface state has been observed in PdTe2 by a recent angle-resolved photoemission spectroscopy study13. The next step would be the study of its superconducitng pairing symmetry and the searching of the Majorona Fermions at the vortex cores of the PdTe2 compound. Combining the simplicity in chemical composition and lattice structure, as well as the predominately single-band electronic structure, the PdTe2 compound could be served as an ideal material base in further investigation of the possible topological superconductivity.

In conclusion, by using de Hass-van Alphen effects and megnetoresistance we study the electronic structure and transport behavior of the two-dimensional TMD material PdTe2. Though the multi-band feature is well constructed in this material, we find the existence of a predominant oscillation frequency in the fast Fourier transformation spectra. Thus it is likely that a predominant single band electronic structure dominates the transport behavior of the PdTe2 compound, with other bands contribute very little. This result will be helpful in understanding the topological superconducting properties of this material and also helps to realize the transport properties of related two-dimensional TMDs.

Methods

Single crystal of PdTe2 was grown by melting stoichiometric mixtures of high-purity elements Pd (99.999%) and Te (99.997%) in a sealed evacuated quartz tube at 780 °C for 48 h, followed by a slow cooling to 500 °C at a rate of 3 °C/h. After that, the crystals were cooled with furnace to room temperature. The crystals can be easily cleaved into sheets with shiny mirror-like surfaces. The structure was checked by the Rigaku- TTR3 x-ray diffractometer using high-intensity graphite monochromatized Cu Kα radiation at room temperature. The orientation of c axis was perpendicular to the shining surface of the as-cleaved single crystal samples, which was confirmed by the single crystal x-ray diffraction measurements on the Rigaku- TTR3 x-ray diffractometer. The b-axis was determined by a back-reflection Laue x-ray photograph measurement.

The magnetoresistance measurements were performed using the standard four-probe technique. Quantum oscillations inmagnetization were probed via highly sensitive torque magnetometry methods which measure the magnetic susceptibility anisotropy of sample. With the tilted magnetic field H confined to the x-z plane and M in the same plane, the torque is . In paramagnetic state, it can be written as τ = χzHzHx − χxHxHz = ΔχH2sin θ cos θ, where Δχ = χz − χx is the magnetic susceptibility anisotropy and θ is the tilt angle of H away from z. Throughout this paper, θ is defined as the angle between the magnetic field direction and the crystallographic c axis. Both magnetoresistance and torque measurements were performed on the Cell5 Water-Cooling Magnet of High Magnetic Field Laboratory of Chinese Academy of Sciences.

Additional Information

How to cite this article: Wang, Y. et al. De Hass-van Alphen and magnetoresistance reveal predominantly single-band transport behavior in PdTe2. Sci. Rep. 6, 31554; doi: 10.1038/srep31554 (2016).