Large magnetoresistance and quantum oscillations in Sn_0.05Pb_0.95Te

Topological phases of materials have attracted enormous attention in recent years because of the novel properties of their surface states [1–4]. A topological insulator (TI) has an insulating bulk and highly conducting surface states arising from the non-trivial topology of the bulk states. TIs that have been predicted are experimentally verified using various techniques such as angle-resolved photoemission spectroscopy (ARPES) [5–8], quantum oscillations [9–11], weak antilocalization (WAL) [12–16], etc. ARPES is a commonly used technique for studying topological materials as it can directly

visualize the Dirac dispersion of surface states. Magnetoresistance (MR)/magnetization under high magnetic fields show quantum oscillations known as Shubnikov de Haas (SdH)/de Haas–van Alphen oscillations [17–19]. Surface electrons are Dirac fermions and have a linear dispersion in the band structure. The two-dimensional Fermi surface of surface states can be identified by carrying out measurements of the angle dependence of quantum oscillations and Berry phase calculations [3, 16, 20]. The origin of surface states in TIs is both fascinating and pivotal for understanding fundamental physics, and surface states have potential applications in modern electronic and computer devices. However, only a few TIs are known to exist in the real world, including bismuth- and thallium-based TIs (also known as Z₂-TIs) [20, 21]. This is due to the requirement of strong spin–orbit interaction and time-reversal symmetry for Z₂-TIs. Also, most of the Z₂-TIs studied thus far do not have an insulating bulk due to crystal defects (antisite or point). The bulk conduction channel interferes with the surface conduction channel, which makes the transport characterization of surface states very challenging, although not impossible [22, 23].

In the search for other topological compounds, a new system, known as a topological crystalline insulator (TCI), has been discovered [24]. A TCI has a simple rock salt crystal structure and includes elements from groups IV–VI, namely Sn_x Pb_1−x Se/Te [25–28]. Like the Z₂-TIs, these materials possess semiconducting bulk states and metallic gapless surface states. However, there are some fundamental differences between the surface states in the two types of TIs. First, surface states in TCIs are protected by the mirror symmetry or reflection symmetry present in the crystal, whereas those in Z₂-TIs are protected by the time-reversal symmetry [29, 30]. Second, there is an even number of Dirac cones on the surface terminations in TCIs, and they are oriented perpendicular to the mirror symmetry planes, whereas Z₂-TIs possess an odd number of Dirac cones [21, 31].

Recent studies have shown that the topological properties of Sn_x Pb_1−x Se/Te can be tuned either via doping [32, 33] or external pressure [34], making it an ideal system for investigating the topological phase transition. With increasing Sn content, Sn_x Pb_1−x Se/Te shows a band inversion at critical doping of x_c ∼ 0.35 for Sn_x Pb_1−x Te [27] and ∼0.17 for Sn_x Pb_1−x Se [35]. This means that the system transforms from a topologically trivial to non-trivial phase at x_c. The existence of surface states and topological phase transition in TCIs has been evidenced using methods such as ARPES [25, 27, 36], magneto-optical absorption [37], WAL [38–40], etc. However, there have been limited magnetotransport studies [41–45] of TCIs, especially of their quantum oscillations, although this is one of the most powerful techniques for studying topological surface states. One of the reasons could be the higher magnetic field range needed to observe the Landau-level quantization, typically above 9 T, which is not easily attainable in most labs. By analyzing the quantum oscillations, we can determine the Fermi surface topology in addition to its shape, size, and dimensionality. Therefore, it would be interesting to explore how the Fermi surface topology changes with x in Sn_x Pb_1−x Se/Te.

Here, we have synthesized high-quality single crystals of Sn_x Pb_1−x Te, and we present the results of detailed magnetotransport studies of one of the samples, Sn_0.05Pb_0.95Te. The sample shows a large positive MR without showing any sign of saturation. At higher fields, it exhibits clear SdH oscillations. We have analyzed these oscillations and we discuss the Fermi surface topology.

Single crystals of Sn_x Pb_1−x Te were grown by the chemical vapor transport method. All of the elements used in the crystal growth, lead, tin, and tellurium, were of high purity (99.9999%). First, high-purity powder samples of PbTe (62 at.% Pb and 39 at.% Te) and SnTe (48 at.% Sn and 52 at.% Te) were prepared by solid-state reaction. Powder mixtures with nominal concentrations estimated by the Pb/Sn mixture weight ratio were each heated in a clean evacuated quartz tube maintained at 900°C over 48 h prior to crystal growth. Afterward, each powder mixture was sealed in an evacuated quartz tube of dimensions 2.2 × 2 × 40 cm³ along with a transporting agent of ∼150 mg iodine (I₂). The quartz tube was placed in a horizontal 2 zone furnace maintained for 10 d at 900°C–800°C, before being further cooled down to room temperature.

Longitudinal and Hall resistance under applied magnetic fields up to 13 T were measured in an Oxford cryostat. A special homemade probe was built for rotating the sample in the magnetic field. A linear resistance bridge (LR-700) and a Lake Shore (LS336) temperature controller were used for measuring the resistance and temperature, respectively, of the sample.

Figures 1(a)–(c) shows the x-ray diffraction (XRD) analysis of Sn-doped PbTe single-crystal powder along with the calculated pattern at selected Sn doping (x). It is well known that PbTe crystallizes in the cubic structure; when Sn is doped at the Pb site of PbTe, the lattice parameter decreases systematically. The figure 1(d) inset shows the lattice parameter (a) determined from the Rietveld refinement at different x values. A systematic decrease of the lattice parameter with x indicates that a periodic doping was successfully implemented. The temperature dependence of the longitudinal resistance (R_xx ) for Sn_x Pb_1−x Te (x = 0.05) is displayed in figure 1(d). The resistance decreases with decreasing temperature, showing typical metallic behavior.

Zoom In Zoom Out Reset image size

To investigate the magnetotransport properties of this system, we have measured MR in one of the samples, Sn_0.05Pb_0.95Te. The tilt angle (θ) is defined as the angle between H and the normal to the [001] surface. Figure 2 shows the longitudinal (R_xx ) and Hall resistance (R_xy ) at θ = 0°. MR is expressed as a percentage, i.e. MR (%) = $\frac{{R}_{xx}\left(H\right)-{R}_{xx}\left(0\right)}{{R}_{xx}\left(0\right)}\enspace {\times}$ 100%, where R_xx (0) and R_xx (H) are the resistance values at 0 and H applied fields, respectively. As shown in figure 2(a), MR is positive and increases almost linearly with H. It reaches 310% at 13 T without showing any sign of saturation. Such unsaturated large positive MR in Sn_x Pb_1−x Te has also been reported in previous magnetotransport studies [41, 44, 46]. Roychowdhury et al [46] observed a systematic decrease in the MR value with increasing x. Here, we have estimated MR ∼ 210% at H = 9 T for x = 0.05, which is slightly higher than the previously reported MR ∼ 190% for x = 0.4. Therefore, our MR data lies within the same doping dependence trend reported by Roychowdhury et al [46]. The Hall resistance, R_xy , also increases linearly with H, as shown in figure 2(b). The positive slope of the R_xy vs H plot confirms the presence of the hole-like bulk charge carriers in Sn_0.05Pb_0.95Te. It should be noted that both longitudinal and Hall resistance show a signature of SdH oscillations at higher fields, and these oscillations can be clearly resolved in the polynomial background-subtracted data, as shown in the insets to figures 2(a) and (b), respectively. The oscillations are periodic and well-defined, and appear to have a single frequency. The frequency of the oscillations can be determined by obtaining the fast Fourier transform (FFT).

Zoom In Zoom Out Reset image size

Figure 3 shows the frequencies of the SdH oscillations at θ = 0°. Both the longitudinal and transverse oscillations have a single frequency located at f_α = 57 T. As the frequency of SdH oscillations is directly proportional to the cross-section of the Fermi surface, the single frequency indicates the presence of one Fermi surface in Sn_0.05Pb_0.95Te. Using the Onsager's relation [17–19, 47], we can determine the Fermi-wave vector as

$$\begin{equation}f=\left(\frac{\hslash }{2e}\right){K}_{\text{F}}^{2},\end{equation} \tag{ 1 }$$

where ℏ = $\frac{h}{2\pi }$, h is Planck's constant, and K_F is the Fermi wave vector. Using equation (1), we have estimated that K_F = 4.12 × 10⁶ cm⁻¹.

Zoom In Zoom Out Reset image size

In order to map the Fermi surface, we have measured the angle dependence of the SdH oscillations. Figures 4(a) and (b) show the SdH oscillations after polynomial background subtraction for longitudinal and Hall resistance, respectively, at different θ values. The oscillations at θ = 0° appear smooth and periodic, and they exhibit a single frequency of 57 T (see figure 3). At higher θ values, the oscillations in the longitudinal resistance (figure 4(a)) are similar to those at θ = 0°, but they are aperiodic, possibly due to the superposition of more than one frequency. Similar aperiodic oscillations are also seen in the Hall component (figure 4(b)). The superposition of multiple frequencies in the oscillations can be separated by obtaining the Fourier transform.

Zoom In Zoom Out Reset image size

Figure 5 shows the frequency plots at different θ values for both longitudinal and Hall oscillations. The frequency values for the longitudinal and Hall components are comparable to one another. At θ = 0°, there exists a single frequency, f_α = 57 T, and it decreases to 45 T as θ is increased to 15° (figure 5(b)). It should be noted that an additional frequency, f_β = 69 T, appears at θ = 15°. A third frequency of 118 T is seen at θ = 45° (figure 4(d)) and is nearly the sum of f_α and f_β , i.e. (f_α + f_β ). Although there exists a single frequency at θ = 0°, we have observed two distinct frequencies (f_α and f_β ) at higher θ values. The observation of two frequencies suggests that there exist two distinct Fermi wave vectors representing two different sections (α and β pockets) of the Fermi surface.

Zoom In Zoom Out Reset image size

Previous magnetotransport studies [41, 44] have also reported two distinct frequencies in PbTe, although the frequency values that were found are slightly different than those observed here. Ma et al [41] reported two frequencies, one ≈ 39 T and another that appears as a broad shoulder ≈70 T. Furthermore, Akiba et al [44] found two frequencies at 22.6 T and 45.2 T, although they claimed that the second frequency is the second harmonic of the first. Since the frequency of the SdH oscillations in PbTe depends on the bulk carrier concentration [48], the slight difference in the frequency values in our Sn_0.05Pb_0.95Te sample could be due to a different bulk carrier concentration arising from the small amount of Sn doping. However, detailed studies at higher fields are required to determine unambiguously both the number of frequencies and their values.

The angle dependence of f_α and f_β is shown in the inset to figure 6(a). With increasing θ, f_α decreases gradually, whereas f_β first increases and then decreases slightly. In order to obtain information on the Fermi surface topology, we have calculated the Berry phase (δ). We chose the oscillation data at θ = 0° for this calculation to avoid interference from the second frequency. Figure 6(a) shows the SdH oscillations in conductance, Δσ_xx , which were obtained after subtracting the polynomial background. Here, σ_xx = $\frac{{\rho }_{xx}}{\left({\rho }_{xx}^{2}+{\rho }_{xy}^{2}\right)}$, where ρ_xx and ρ_xy represent the longitudinal and Hall resistivity, respectively. Minima and maxima positions of the oscillations were assigned as integer (N) and half-integer (N + $\frac{1}{2}$) values, respectively, and the constructed Landau-level fan diagram is shown in figure 6(b). From the linear extrapolation of the data at the limit of $\frac{1}{H}\enspace \to$ 0, we have obtained δ = 0.11 ± 0.07 and f_α = (56.8 ± 0.6) T. This value of δ is very close to 0, which is expected for a topologically trivial system [9–11, 49]. The frequency obtained from the linear extrapolation, 56.8 T, is also consistent with the f_α = 57 T obtained by the FFT of the SdH oscillations (figure 3). It should be noted that our Sn_0.05Pb_0.95Te sample has Sn content, x = 0.05, that is below x_c = 0.35. Therefore, the trivial topology of the α pocket determined from the analyses of SdH oscillations is consistent with the results from other measurement techniques [27, 28, 33, 38]. The topology of the α pocket is well understood based on the current data. However, it will be necessary to acquire more SdH oscillations data above the current maximum field of 13 T to calculate δ for, and determine the topology of, the β pocket.

Zoom In Zoom Out Reset image size

We have synthesized high-quality single crystals of Sn_x Pb_1−x Te, and studied the Fermi surface topology of one of the samples, Sn_0.05Pb_0.95Te. The sample exhibits a large positive MR that increases almost linearly with increasing applied field and does not show any sign of saturation. At higher fields, SdH oscillations appear for both longitudinal and Hall resistance. At θ = 0°, the SdH oscillations are smooth and periodic, and they consist of a single frequency, f_α = 57 T. At higher θ values, the SdH oscillations are complex and consist of two frequencies, f_α and f_β . The presence of two frequencies suggests that two Fermi surface pockets (α and β pockets) are present in the Sn_0.05Pb_0.95Te sample. For the α pocket, we have estimated the Berry phase (δ) to be ∼0.1, which is very close to the expected value of 0 for a topologically trivial system. In order to construct the Landau-level plot for the β pocket and hence determine its topology, studies will need to be conducted above the current maximum field of 13 T. Although our motivation is to explore the topological phase transition in Sn_x Pb_1−x Te, this work only presents the Fermi surface topology of the sample below the critical doping level (x_c ∼ 0.35). Detailed magnetotransport studies at higher fields and of samples with higher Sn content (particularly above x_c) are underway, and the obtained results will be reported elsewhere.

We are grateful to Dr. F C Chou for providing high-quality single crystals of Sn_x Pb_1−x Te. This work is supported in part by the US Air Force Office of Scientific Research Grant FA9550-15-1-0236, the TLL Temple Foundation, the John J and Rebecca Moores Endowment, and the State of Texas through the Texas Center for Superconductivity at the University of Houston. The research at West Texas A&M University is supported by a start-up Grant from the Paul Engler College of Agriculture and Natural Sciences and by the Welch Foundation (Grant No. AE-025).

The data that support the findings of this study are available upon reasonable request from the authors.