Quantum oscillations of robust topological surface states up to 50 K in thick bulkinsulating topological insulator
As personal electronic devices increasingly rely on cloud computing for energyintensive calculations, the power consumption associated with the information revolution is rapidly becoming an important environmental issue. Several approaches have been proposed to construct electronic devices with lowenergy consumption. Among these, the lowdissipation surface states of topological insulators (TIs) are widely employed. To develop TIbased devices, a key factor is the maximum temperature at which the Dirac surface states dominate the transport behavior. Here, we employ Shubnikovde Haas oscillations (SdH) as a means to study the surface state survival temperature in a highquality vanadium doped Bi_{1.08}Sn_{0.02}Sb_{0.9}Te_{2}S single crystal system. The temperature and angle dependence of the SdH show that: (1) crystals with different vanadium (V) doping levels are insulating in the 3–300 K region; (2) the SdH oscillations show twodimensional behavior, indicating that the oscillations arise from the pure surface states; and (3) at 50 K, the V_{0.04}single crystals (V_{x}:Bi_{1.08x}Sn_{0.02}Sb_{0.9}Te_{2}S, where x = 0.04) still show clear sign of SdH oscillations, which demonstrate that the surface dominant transport behavior can survive above 50 K. The robust surface states in our V doped single crystal systems provide an ideal platform to study the Dirac fermions and their interaction with other materials above 50 K.
Introduction
Since the concept of topology has been introduced into condensed matter physics, an exotic form of quantum matter, namely, the topological insulator (TI), has attracted much attention, in which there is no transport of electrons in the bulk of the material, while the edges/surface can support metallic electronic states protected by timereversal symmetry.^{1}Theoretical predictions have revealed spinmoment locking, and lineardispersion edge states can be addressed in a twodimensional (2D) topological insulator.^{2,3} The experimental breakthrough leading to the 2D system was achieved in a strong spinorbital coupled CdTe/HgTe quantum well, which gave rise to a quantum spin Hall state.^{4} Recently, 1 T′WTe_{2}monolayer^{5,6,7} has been proved to be a new quantum spin Hall insulator, with a conducting edge state that can survive to 100 K^{8,9}, thereby offering a potential application for a 2D TI in ultralowenergy electronics.
In threedimensional (3D) topological insulators, the conducting edge changes to a metallic surface state that emerges due to the nontrivial Z_{2} topology of the bulk band structure.^{10,11,12,13} The unique conducting surface states offer a new playground for studying the physics of quasiparticles with unusual dispersions, such as Dirac or Majorana fermions, as well as showing promising capabilities for spintronics (e.g., efficient spintorque transfer). Experimentally, ever since the novel 3D TI family, including Bi_{2}Se_{3}, Bi_{2}Te_{3}, and Sb_{2}Te_{3}, was developed, the transport and optical properties related to topological surface states have been intensively studied.^{11,12,13,14,15,16,17,18,19} In particular, a quantized anomalous Hall effect (QAHE) was found in a magneticiondoped 3D TI thin film that presented a transverse current with extremely low dispersion.^{20,21} To achieve the QAHE, both the linearly dispersed topological surface states and ferromagnetic ordering are needed.^{21,22,23} More recently, Xray magnetic circular dichroism results^{24} have demonstrated that the ferromagnetic ordering of spinspin related surface states survives better than in the bulk states, which might imply that the critical temperature of the surface states is the key to QAHE.
Since the surface states of a TI are very important, it is a great challenge to develop a system in which surfacedominated transport survives at high temperature. In fact, most of the known TI materials are not bulkinsulating, which hinders study of the transport properties of the surface states. Therefore, bulkinsulating TIs with high resistivity are still required to extend the topological states to the high temperature region. A good example of a widegap TI system is Bi_{2x}Sb_{x}Te_{3y}Se_{y}(BSTS)^{10,11,25,26,27,28}, which shows surface states that dominate the transport behavior at and below 30 K^{10}. Another important bulkinsulating TI compounds are Tl_{1x}Bi_{1+x}Se_{2}system, which shows robust surface oscillations at 50 K.^{29} However, the Tl’s toxicity limits their application in electronic devices. Since in bulkinsulating TIs, the Fermi surface is formed by pure Dirac dispersed surface states, the Shubnikovde Haas (SdH) oscillations are strong evidence that can be used to evaluate the contribution of surface states. Tracing the SdH oscillations is an effective method to study the surface states in bulkinsulating 3D TIs. For example, the angular dependence of the SdH oscillations is key evidence that can be used to isolate surfacefeatures. The oscillation magnitudes also provides insight into other properties of the electronic structure such as the effective mass and quantum mobility. In realistic systems, the effective mass of a Dirac carrier is always nonzero instead of the ideal zero mass, and this causes temperatureinduced damping of the amplitudes of SdH oscillations. One strategy that improves on BSTS, for which the surface band is not ideally linear, is to utilize a different TI: Sndoped Bi_{2x}Sb_{x}Te_{2}S, which has a wide bulk band gap^{30,31,32,33}. In the latter compound, 2% tin is introduced to stabilize the crystal structure, and the corresponding material shows surface states with a linear dispersion over a large energy range.^{30} We therefore focus on further optimization of this particular system by adding V at the bismuth sites to tune the bulk band gap. In this work, we found that the SdH oscillations could survive up to 50 K in all three samples, especially clear in the V_{0.04} sample (V_{x}:Bi_{1.08x}Sn_{0.02}Sb_{0.9}Te_{2}S, where x = 0.04), which allows 3D TIbased study of this substrate above the 50 K region. A special feature of this material is that the SdH oscillations are found in 1 mmthick bulk crystals, which present useable surface states, making these crystals suitable for the fabrication of further van der Waals heterostructures.
Results and discussion
A single crystal exfoliated from the asgrown V_{0.04} ingot is shown in Fig. 1a. The V_{x}:Bi_{1.08x}Sn_{0.02}Sb_{0.9}Te_{2}S (V:BSSTS) single crystal shares the same crystal symmetry as its parent compounds Sb_{2}Te_{3} and Bi_{2}Te_{3}, which belong to the R3m space group, with quintuple layers piled up along the hexagonal caxis (Fig. 1b). Traditionally, Se is widely used as a dopant to tune the band structure, which results in the (Bi,Sb)_{2}(Te,Se)_{3} (BSTS) formula for the most popular topological insulator. Nevertheless, the defect chemistry limits our ability to grow large highquality bulk single crystals of BSTS. Ideally, the new lowenergy electronics industry requires wafersize thick topological insulator materials with stable surface states and a large bulk band gap. Previous work by Kushwaha et al.^{30}demonstrated that Sndoped Bi_{1.1}Sb_{0.9}Te_{2}S single crystal has a low carrier concentration with clean surface states at low temperatures. To further optimize this material, we have employed V as a Bisite dopant, which, we discovered, has the effect of making the bulk states more insulating (as sketched in Fig. 1d), whilst having minimal effects on the surface states. This is a major step forward towards realizing an ideal TI material for the electronics.
Plots of the temperature dependence of bulk resistivity are presented in Fig. 2a for single crystal samples ~1 mm in thickness with different Vdoping levels. The resistivity curves for all of the vanadium doped samples show a steep upturn as the sample is cooled, indicating that the bulk V:BSSTS samples have become highly insulating. The residual resistance ratios (RRR), defined by RRR = ρ(3 K)/ρ(300 K) for the samples, where ρ is the resistivity, were 2.1, 9.4, and 127 for V_{0.02}, V_{0.04}, and V_{0.08}, respectively. As the Vdoping level increases, the resistivity and the RRRincrease dramatically, showing better insulating behavior, e.g., the maximum resistivity of the V_{0.02} sample is near 0.1 Ω·cm, but it can reach about 0.7 and 10 Ω·cm in the V_{0.04} and V_{0.08} samples, respectively. In the low temperature region below 10 K, the resistivity curves of the V_{0.02} and V_{0.04}samples show an additional minor feature and a slight upturn, although this is negligible compared to the major transition at ~100 K. In the more highly insulating V_{0.08} sample, the low temperature resistivity slightly drops with cooling. The temperature at which the surface states finally dominate the bulk resistivity depends on intrinsic factors, but also varies with V doping. The lnρ versus T^{−1} plots (shown in Fig. 2b) exhibit activated behavior, with the activation energy for transport at 110, 240, and 340 meV in the V_{0.02}, V_{0.04}, and V_{0.08} samples, respectively.
We conducted measurements to determine the magnetoresistance (MR) ratio, MR = (R(B)−R(0))/R(0), for all three Vdoped samples in the low temperature region in order to understand the magnetotransport properties. In the V_{0.02} sample, the MR curves first decrease in the low magnetic field region, and then increase with increasing magnetic field, up to about 16% at 3 K and 14 T. The maximum MR value at 14 T increases upon heating, so that it reaches 25% at 50 K. In the V_{0.04} and V_{0.08} samples, the MR values are always positive at all temperatures and under all magnetic field conditions, but they show temperaturedamping behavior, with maximum MR of 50% and 110% at 3 K, respectively. Although the single crystals are similar in size and shape, the MR curves for the three samples are quite different from one another. In the V_{0.02} crystal, the MR curves show a nonmonotonic relationship with increasing magnetic field. Between 0 and 1 T, the 3 K MR curves slightly decrease to about −0.7% due to paramagnetic V ions (see Supplementary Fig. 5 for magnetic properties), and they then show a weak field dependence until about 5 T, after which, obvious oscillation patterns are displayed in the quasiparabolic curves. The aforementioned oscillations, denoted as Shubnikovde Haas (SdH) oscillations, result from Landau quantization. Upon cooling to different temperatures, the phenomenon of lowfield MR decrease shows a similar tendency, but with a larger change in the relative drop, e.g., the minimum MR value (appearing at 1 T) at 5 K is about 1%, but it is near 1.6% at 8–50 K. To explain the negative MR behavior, we employ the magnetization vs. temperature and magnetization vs. field measurements for three samples, shown in Supplementary Fig. 5. At low temperature region, the V doping introduce paramagnetic behavior. The paramagnetic V dopants order with the increasing field, thus reduce the electronic scattering, leading to the negative MR behavior. Unlike the V_{0.02} sample, the V_{0.04} sample becomes more resistive with increasing field. Starting in the lowfield region, the MR curves show a rapid increase at 3 K. Note that the total behavior of the V_{0.04}sample is linearlike with high field SdH oscillations, but it shows a fasterthanlinear increase in applied fields below 0.5 T fields. This faster rate of increase is attributed to the contribution of surface conduction, denoted as the weak antilocalization (WAL) effect, which is analyzed in Supplementary Fig. 6.^{27}During heating, the lowfield behavior remains, but it becomes less significant and vanishes at 50 K. In the V_{0.08} sample, the MRshows similar WAL effect in small field region, after which, the SdH oscillations occur and the MR values increase more slowly than the linear tendency before.
Let’s then focus on the SdH oscillations in the MR curves, which are obtained via subtracting the smooth background (as shown in Supplementary Fig. 3) and are plotted in Fig. 2d, f, h. Since the bulk states are strongly insulating, the oscillation related Fermi surface is due to the contributions of the pure surface states. The oscillation patterns of each sample are almost the same, but with different amplitudes, which are damped during heating. Note that, in all three samples, the SdH oscillations survive at 50 K. Particularly, in the V_{0.04} sample, the SdH oscillations are still clear at 50 K, which means that the surfaces states are quite mobile at this temperature and might still exist at even higher temperatures in the thick crystals. Note that, similar SdH oscillations can also be found in thin flake of V_{0.08} crystal as a representative, which is shown in Supplementary Fig. 7. Previous study in TlBiSe_{2} also present the similar robust surface states dominant SdH oscillations at 50 K with a frequency near 210 T.^{27} Here in V:BSSTS system, the Fermi surface is smaller, which offers a greater opportunity to tune the Dirac point to into Fermi surface.
The SdH oscillations for a TI can be described by the LifshitzKosevich (LK) formula, with a Berry phase being taken into account for the topological system:
$$\frac{\Delta \rho}{\rho (0)} = \frac{5}{2}\left( \frac{B}{2F} \right)^{\frac{1}{2}}R_{T}R_{D}R_{S}\cos \left( 2\pi \left( \frac{F}{B} + \gamma – \delta \right) \right)$$
(1)
Where R_{T} = αTν/Bsinh(αTν/B), R_{D} = exp(−αT_{D}ν/B), and R_{S} = cos(αgν/2). Here, ν = m*/m_{0} is the ratio of the effective cyclotron mass m* to the free electron mass m_{0}; g is the gfactor; T_{D} is the Dingle temperature; and α = (2π^{2}k_{B}m_{0})/ћe, where k_{B} is Boltzmann’s constant, ℏ is the reduced Planck’s constant, and e is the elementary charge. The oscillation of Δρ is described by the cosine term with a phase factor γ−δ, in which γ = 1/2 − Φ_{B}/2, where Φ_{B} is the Berry phase. From the LK formula, the effective mass of carriers contributing to the SdH effect can be obtained through fitting the temperature dependence of the oscillation amplitude to the thermal damping factor R_{T}. From the temperaturedamping relationship, we obtain the Dingle temperatures of the three samples: 4.7, 6.5, and 9.4 K for V_{0.02}, V_{0.04}, and V_{0.08}, respectively. The effective masses for these crystals are 0.16 m_{0}, 0.15 m_{0}, and 0.13 m_{0}, respectively, as shown in Fig. 3c. During the effective masses fitting, we employ the normalized amplitudes of the last peak towards high magnetic fields. We further conduct the fitting to other peaks, and summarize the errors below. The quantum relaxation time and quantum mobility can also be obtained by τ = ħ/2πk_{B}T_{D} and μ = eτ/m∗, respectively. According to the OnsagerLifshitz equation, the frequency of quantum oscillation, F = (φ_{0}/2π^{2})A_{F}, where A_{F} is the extremal area of the crosssection of the Fermi surface perpendicular to the magnetic field, and φ_{0} is the magnetic flux quantum. The crosssections related to the 32, 30, and 25 T pockets are 0.34, 0.31, and 0.26 × 10^{−3} Å^{−2}, respectively. The obtained parameters are summarized in Table 1 below. The Berry phase can be obtained via the Landau fan diagram, which is shown in Fig. 3a. Since ρ_{xx} > ρ_{xy}, we assign the maxima of the SdH oscillation to integer (n) Landau levels, to linearly fit the n versus 1/B curve. As we can see in Fig. 3a, the intercept has a value of around 0.5 for all three samples, which implies that these oscillations are contributed by the topological surface states.
The SdH oscillation frequency is proportional to the area of the Fermi surface. We analyzed the oscillations of the three samples at 3 K and found that the SdH oscillation frequencies decreased with increasing vanadium concentration. We deduced that the vanadium doping changes the chemical potential of the samples, thus causing the Fermi level to shift downwards slightly towards the valence band, although remaining near the Dirac point in all cases. We further measured the Hall effect for all three samples (shown in Supplementary Fig. 4), from which we calculated the Hall coefficient, as shown in Fig. 3c, where it is plotted versus temperature. In the lowfield region, the Hall resistivity curves increase/decrease linearly with the external magnetic field, where the Hall coefficient is positive or negative, respectively. The positive or negative value of the Hall coefficient depends on the carrier type in the samples. Interestingly, the carrier type and density in V:BSSTS changes with doping: in the V_{0.02}and V_{0.04} samples, it is electrons that contribute to the transport behavior, as in pure BSSTS; in contrast, the V_{0.08} sample has ptype carriers. From the Hall coefficient, we can calculate that the carrier density is quite small in all three samples (10^{15−16} cm^{−3}). Via V doping, we observe that the carrier type and density show large changes, which results in the different transport behavior.
To further understand the Fermi surface geometry, we performed angledependent SdH oscillation measurements. A good example of the angular dependence of the MRproperties is shown in Fig. 4a, which is for the V_{0.04} crystal. Note that, during rotation, the magnetic field is always perpendicular to the current, to make it possible to ignore the influence of the angle between the magnetic field and the current (as sketched in Figs 1cand 4a). The rotation parameter θ is the angle between the caxis and the direction of the magnetic field. At higher tilt angles, the MRvalue drops significantly, as along with the SdH oscillation amplitude. The oscillation patterns obtained from the MR curves at different orientations of the magnetic field are shown in Fig. 4b–d with the smooth background subtracted. One can see that the quantum oscillations become weaker in the rotation process and almost vanish at about 60 deg. We calculated each oscillation frequency, which are summarized in Fig. 4eand roughly fitted by the 1/cosθ relationship. The 2Dlike relationship is strong evidence that the quantum oscillations are contributed by the topological surface states.
We employed a simple meltingslowcooling method to grow single crystals of V, Sndoped Bi_{1.1}Sb_{0.9}Te_{2}S single crystals. The RT and Hall measurements show that all three samples are insulating in the 3–300 K region, with low carrier concentrations. SdH oscillations can be detected in the low temperature region, which implies that our insulating single crystals possess Fermi surfaces. Furthermore, we find strong 2Dlike behavior and a π Berry phase in all three samples, which provide compelling evidence that the bulks of these crystals are good insulators. Moreover, V dopant appears to tune both the type and the concentration of carriers in these TI systems, which provide us with an ideal platform to study their physical properties, as well as offering potential for device fabrication related to their surface states.
Methods
To obtain bulkinsulating TIs, defect control is one of the most important factors in the crystal growth process. Here, we employ a simple meltingcooling method in a uniformtemperature vertical furnace to spontaneously crystallize the raw elements into a tetradymite structure (V_{x}:Bi_{1.08x}Sn_{0.02}Sb_{0.9}Te_{2}S, x is the nominated ratio).^{34,35} Briefly, highpurity stoichiometric amounts (∼10 g) of V, Bi, Sn, Sb, Te, and S powders were mixed via ball milling and sealed in a quartz tube as starting materials. The crystal growth was carried out using the following procedures: (i) Heating the mixed powders to completely melt them; (ii) Maintaining this temperature for 24 h to ensure that the melt is uniform; and iii) Slowly cooling down to 500 °C to crystallize the sample. After growth, single crystal flakes with a typical size of 5 × 5 × 1 mm^{3} could be easily exfoliated mechanically from the ingot. Naturally, the single crystals prefer to cleave along the [001] direction, resulting in the normal direction of these flakes being [001]. See Supplementary Figs 1 and 2 for Xray differaction and scanning electron microscopy results.
The electronic transport properties were measured using a physical properties measurement system (PPMS14T, Quantum Design). Hallbar contact measurements were performed on a freshly cleaved c plane using silver paste cured at room temperature. The electric current was parallel to the c plane while the magnetic field was perpendicular to the c plane. The angle dependence of the MRwas also measured using a standard horizontal rotational rig mounted on the PPMS.
Data availability
The data sets generated and/or analyzed in this study are available from the corresponding author upon request.
References
 1.Moore, J. Topological insulators: the next generation. Nat. Phys. 5, 378–380 (2009).
 2.Moore, J. E. The birth of topological insulators. Nature 464, 194–198 (2010).
 3.Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045 (2010).
 4.Bernevig, B. A., Hughes, T. L. & Zhang, S.C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).
 5.Shi, Y. et al. Imaging quantum spin Hall edges in monolayer WTe2. Sci. Adv. 5, eaat8799 (2019).
 6.Tang, S. et al. Quantum spin Hall state in monolayer 1T’WTe2. Nat. Phys. 13, 683–687 (2017).
 7.Zheng, F. et al. On the quantum spin hall gap of monolayer 1T′‐WTe2. Adv. Mater. 28, 4845–4851 (2016).
 8.Fei, Z. et al. Edge conduction in monolayer WTe2. Nat. Phys. 13, 677–682 (2017).
 9.Ok, S. et al. Custodial glide symmetry of quantum spin Hall edge modes in monolayer WTe2. Phys. Rev. B 99(R), 121105 (2019).
 10.Taskin, A., Ren, Z., Sasaki, S., Segawa, K. & Ando, Y. Observation of Dirac holes and electrons in a topological insulator. Phys. Rev. Lett. 107, 016801 (2011).
 11.Ren, Z., Taskin, A., Sasaki, S., Segawa, K. & Ando, Y. Large bulk resistivity and surface quantum oscillations in the topological insulator Bi2Te2Se. Phys. Rev. B 82, 241306 (2010).
 12.Analytis, J. G. et al. Twodimensional surface state in the quantum limit of a topological insulator. Nat. Phys. 6, 960–964 (2010).
 13.Taskin, A. & Ando, Y. Quantum oscillations in a topological insulator Bi1−xSbx. Phys. Rev. B80, 085303 (2009).
 14.Wang, X., Du, Y., Dou, S. & Zhang, C. Room temperature giant and linear magnetoresistance in topological insulator Bi2Te3 nanosheets. Phys. Rev. Lett. 108, 266806 (2012).
 15.Yue, Z., Cai, B., Wang, L., Wang, X. & Gu, M. Intrinsically coreshell plasmonic dielectric nanostructures with ultrahigh refractive index. Sci. Adv. 2, e1501536 (2016).
 16.Qu, D.X., Hor, Y. S., Xiong, J., Cava, R. J. & Ong, N. P. Quantum oscillations and Hall anomaly of surface states in the topological insulator Bi2Te3. Science 329, 821–824 (2010).
 17.Cai, S. et al. Independence of topological surface state and bulk conductance in threedimensional topological insulators. Npj Quantum. Materials 3, 62 (2018).
 18.Lupke, F. et al. In situ disentangling surface state transport channels of a topological insulator thin film by gating. Npj Quantum. Materials 3, 46 (2018).
 19.Zhao, X. W. et al. Reversible and nonvolatile manipulation of the electronic transport properties of topological insulators by ferroelectric polarization switching. npj Quant. Mater. 3, 52 (2018).
 20.Laughlin, R. B. Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395 (1983).
 21.Chang, C. Z. et al. Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).
 22.Wang, X. L. Dirac spingapless semiconductors: promising platforms for massless and dissipationless spintronics and new (quantum) anomalous spin Hall effects. Natl. Sci. Rev. 4, 252–257 (2017).
 23.Wang, X. L. Proposal for a new class of materials: spin gapless semiconductors. Phys. Rev. Lett. 100, 156404 (2008).
 24.Liu, W. et al. Experimental observation of dual magnetic states in topological insulators. Sci. Adv. 5, eaav2088 (2019).
 25.Bagchi, M. et al. Large positive magnetoconductivity at microwave frequencies in the compensated topological insulator BiSbTeSe2. Physcial Rev. B 99(R), 161121 (2019).
 26.Ren, Z., Taskin, A., Sasaki, S., Segawa, K. & Ando, Y. Optimizing Bi2− xSbxTe3− ySey solid solutions to approach the intrinsic topological insulator regime. Phys. Rev. B 84, 165311 (2011).
 27.Bao, L. et al. Weak antilocalization and quantum oscillations of surface states in topological insulator Bi2Se2Te. Sci. Rep. 2, 726 (2012).
 28.Xiong, J. et al. Highfield Shubnikov–de Haas oscillations in the topological insulator Bi2Te2Se. Phys. Rev. B 86, 045314 (2012).
 29.Eguchi, G., Kuroda, K., Shirai, K., Kimura, A. & Shiraishi, M. Surface Shubnikov–de Haas oscillations and nonzero Berry phases of the topological hole conduction in Tl1−xBi1+xSe2. Phys. Rev. B 90, 201307 (2014).
 30.Kushwaha, S. et al. Sndoped Bi1.1Sb0.9Te2S bulk crystal topological insulator with excellent properties. Nat. Commun. 7, 11456 (2016).
 31.Liu, Z. et al. Pointcontact tunneling spectroscopy between a Nb tip and an ideal topological insulator Sndoped Bi1.1Sb0.9Te2S. Sci. China Phys. Mech. Astronomy 62, 997411 (2019).
 32.Cheng, B., Wu, L., Kushwaha, S., Cava, R. J. & Armitage, N. Measurement of the topological surface state optical conductance in bulkinsulating Sndoped Sndoped Bi1.1Sb0.9Te2S single crystals. Phys. Rev. B94, 201117 (2016).
 33.Cai, S. et al. Universal superconductivity phase diagram for pressurized tetradymite topological insulators. Physical Review. Materials 2, 114203 (2018).
 34.Zhao, W. et al. Quantum oscillations in irondoped single crystals of the topological insulator Sb2Te3. Phys. Rev. B 99, 165133 (2019).
 35.Xiang, F.X., Wang, X.L., Veldhorst, M., Dou, S.X. & Fuhrer, M. S. Observation of topological transition of Fermi surface from a spindle torus to a torus in bulk Rashba spinsplit BiTeCl. Phys. Rev. B 92, 035123 (2015).
Acknowledgements
We acknowledge support from the ARC Professional Future Fellowship (FT130100778), DP130102956, DP170104116, DP170101467 and ARC Centre of Excellence in Future LowEnergy Electronics Technologies.
Author information
Affiliations

Institute for Superconducting and Electronic Materials, Australian Institute for Innovative Materials, University of Wollongong, Wollongong, NSW, 2500, Australia
 Weiyao Zhao
 , Lei Chen
 , Zengji Yue
 , Zhi Li
 , David Cortie
 & Xiaolin Wang

ARC Centre of Excellence in Future LowEnergy Electronics Technologies FLEET, University of Wollongong, Wollongong, NSW, 2500, Australia
 Weiyao Zhao
 , Zengji Yue
 , Zhi Li
 , David Cortie
 & Xiaolin Wang

School of Physics & Astronomy, and ARC Centre of Excellence in Future LowEnergy Electronics Technologies FLEET, Monash University, Clayton, VIC, 3800, Australia
 Michael Fuhrer
Contributions
X.W. supervised the study. W.Z. conducted the experiments. W.Z., X.W. and L.C. wrote the manuscript. D.C., M.F., Z.Y. and Z.L. joined the discussion.
Corresponding author
Correspondence to Xiaolin Wang.