Abstract
A topological insulator protected by timereversal symmetry is realized via spin–orbit interactiondriven band inversion. The topological phase in the Bi_{1−x}Sb_{x} system is due to an odd number of band inversions. A related spin–orbit system, the Pb_{1−x}Sn_{x}Te, has long been known to contain an even number of inversions based on band theory. Here we experimentally investigate the possibility of a mirror symmetryprotected topological crystalline insulator phase in the Pb_{1−x}Sn_{x}Te class of materials that has been theoretically predicted to exist in its end compound SnTe. Our experimental results show that at a finite Pb composition above the topological inversion phase transition, the surface exhibits even number of spinpolarized Dirac cone states revealing mirrorprotected topological order distinct from that observed in Bi_{1−x}Sb_{x}. Our observation of the spinpolarized Dirac surface states in the inverted Pb_{1−x}Sn_{x}Te and their absence in the noninverted compounds related via a topological phase transition provide the experimental groundwork for opening the research on novel topological order in quantum devices.
Introduction
Topological insulators are band insulators are featuring uniquely spinpolarized surface states, which is believed to be a consequence of odd number of inversions of bulk bands driven by spin–orbit interaction alone or in combination with crystal lattice modulations^{1,2,3,4,5}. The odd number of band inversion transitions in the bulk demarcate a Z_{2} topological insulator (TI) phase with spinpolarized surface states from that of a conventional band insulator phase, as previously demonstrated in TI Bi_{1−x}Sb_{x} and the Bi_{2}Se_{3} series^{1,2,3,4,5}. Within band theoretical calculations, lead tin telluride Pb_{1−x}Sn_{x}Te, which is a narrow band gap semiconductor widely used for infrared optoelectronic and thermoelectric devices^{6,7}, has long been known to contain an even number of band inversions as electronic structure is tuned via the Sn/Pb ratio^{8,9,10,11,12,13}. Theoretical calculations have also predicted the occurrence of surface or interface states within the inverted band gap upon the band inversion transition^{11,12,13} but no experimental evidence of the surface states has been reported so far. Recent theoretical works^{14,15} have further stimulated the experimental search for surface states in Pb_{1−x}Sn_{x}Te. In a recent calculation^{15}, Hsieh et al. predicted that the topological surface states in SnTe^{15} are protected by crystalline space group symmetries, in contrast to the timereversal symmetry protection in the wellknown Z_{2} TIs.
The crystal group symmetry of Pb_{1−x}Sn_{x}Te, critical for the realization of the topological crystalline insulator (TCI) phase^{14}, is based on the sodium chloride structure (space group Fmm (225)). In this structure, each of the two atom types (Pb/Sn, or Te) forms a separate facecentred cubic lattice, with the two lattices interpenetrating so as to form a threedimensional checkerboard pattern. The first Brillouin zone (BZ) of the crystal structure is a truncated octahedron with six square faces and eight hexagonal faces. The band gap of Pb_{1−x}Sn_{x}Te is found to be a direct gap located at the L points in the BZ^{8}, which are the centres of the eight hexagonal faces of the BZ. Owing to the inversion symmetry of the crystal, each L point and its diametrically opposite partner on the BZ are completely equivalent. Thus, there are four distinct L point momenta. It is well established that the band inversion transitions in the Pb_{1−x}Sn_{x}Te take place at these four L points of the BZ^{9,10,11,12,13}. As a result, even number of (four) inversions exclude the system from being a Kane–Mele Z_{2} TI^{2}. However, it is interesting to note that the momentum space locations of the band inversions are coincident with the momentum space mirror plane within the BZ. This fact provides a clue that band inversions in Pb_{1−x}Sn_{x}Te may lead to a distinct topologically nontrivial phase that is irrelevant to the timereversal symmetry but may be the consequence of the spatial mirror symmetries of the crystal, which has been theoretically shown in the inverted end compound SnTe^{15}. However, it is experimentally known that SnTe crystals actually are subjected to a rhombohedral distortion^{16}, which, strictly speaking, breaks the crystal mirror symmetries and hence excludes a gapless TCI phase^{15}. More importantly, SnTe crystals are typically heavily ptype^{17} due to the fact that Sn vacancies are thermodynamically stable^{17}, which makes the chemical potential of SnTe to lie below the bulk valence band maximum (VBM), consequently not cutting across the surface states. The surface states, which may exist within the band gap, are thus unoccupied and cannot be experimentally studied by photoemission experiments^{18}. On the other hand, the Pbrich samples are reported to possess the ideal (averaged) sodium chloride structure without global rhombohedral distortion^{16}, which thus preserve the proper crystal symmetry required for the predicted TCI phase^{15}. Moreover, it is possible to grow Pbrich crystals in which the chemical potential can be brought up above the bulk VBM (ingap or even ntype)^{19}, so as to lie near the surface state Dirac point.
Therefore, to experimentally investigate the possibility of a topological phase transition, and thus rigorously proving the existence of a TCI phase in the Pb_{1−x}Sn_{x}Te system, we hereby utilize angleresolved photoemission spectroscopy (ARPES) and spinresolved ARPES to study the lowenergy electronic structure below and above the band inversion topological transition in the Pbrich compositional range of the system, through which the observation of spinpolarized surface states in the inverted regime and their absence in the noninverted regime is demonstrated, correlating the observed spinpolarized Dirac surface states with the expected band inversion topological phase transition. We further map out the critically important spin structure of the surface states protected by mirror symmetries in the inverted topological composition. These results show that the topological order observed in Pb_{1−x}Sn_{x}Te is distinct from that previously discovered in Bi_{1−x}Sb_{x} or Bi_{2}Se_{3} (refs 1,2,3,4,5). Our spinresolved study of Pb_{1−x}Sn_{x}Te paves the way for further investigation of many novel properties of topological crystalline order^{14,20,21,22} in real materials and devices.
Results
Comparison of noninverted and inverted compositions
To capture the electronic structure both below and above the band inversion transition (theoretically predicted to be around x1/3 (refs 9, 10), we choose two representative compositions, namely x=0.2 and x=0.4, for detailed systematic studies. Figure 1c shows the momentumintegrated core level photoemission spectra for both compositions, where intensity peaks corresponding to tellurium 4d, tin 4d and lead 5d orbitals are observed. The energy splitting of the Pb orbital is observed to be larger than that of the Sn orbital, which is consistent with the stronger spin–orbit coupling of the heavier Pb nuclei. In addition, the spectral intensity contribution of the Sn peaks in the x=0.4 sample (red) is found to be higher than that of in the x=0.2 sample (blue), which is also consistent with the larger Sn concentration in the x=0.4 samples. We perform systematic lowenergy electronic structure studies on these two representative compositions. As the lowenergy physics of the system is dominated by the band inversion at L points (L points projected onto points on the (001) surface), we present ARPES measurements with the momentum space window centred at the point, which is the midpoint of the surface BZ edge (see Fig. 1b). Figures 2a show the ARPES Fermi surface and dispersion mappings of the Pb_{0.8}Sn_{0.2}Te sample (x=0.2). The system at x=0.2 is observed to be gapped: No band is observed to cross the Fermi level in the Fermi surface maps (Fig. 2a). The dispersion measurements (Fig. 2b) reveal a single holelike band below the Fermi level. This single holelike band is observed to show strong dependence with respect to the incident photon energy (see Fig. 2b, and also detailed in Supplementary Figs S1–S3 and Supplementary Methods in the Supplementary Information), which reflects its threedimensionally dispersive bulk valence band origin. As a qualitative guide to the ARPES measurements on x=0.2, we present firstprinciplesbased electronic structure calculation on the noninverted end compound PbTe (Fig. 2c). Our calculations confirm that PbTe is a conventional band insulator, whose electronic structure can be described as a single holelike bulk valence band in the vicinity of each point, which is consistent with our ARPES results on Pbrich Pb_{0.8}Sn_{0.2}Te. The threedimensional nature of the calculated bulk valence band is revealed by its k_{z} evolution in Fig. 2c, which is in qualitative agreement with the incident photon energy dependence of our ARPES measurements shown in Fig. 2b.
Now we present comparative ARPES measurements under identical experimental conditions and setups on the Pb_{0.6}Sn_{0.4}Te (x=0.4) sample. In contrast to the conventional band insulator (insulating) behaviour in the x=0.2 sample, the Fermi surface mapping (Fig. 2d) on the x=0.4 sample shows two unconnected metallic Fermi pockets (dots) on the opposite sides of the point. Such twopockets Fermi surface topology cannot be easily interpreted as the bulk valence band as the lowenergy bulk valence band of the Pb_{1−x}Sn_{x}Te system is a single holelike band, unless certain surfaceumklapp processes of the bulk states are considered, which usually have only weak crosssection in ARPES measurements. The assignment of the observed twoFermipockets as surfaceumklapp processes or other bulkrelated origins can be further ruled out by incident photon energy dependence (k_{z} dispersion) measurements and our spin polarization studies of these metallic states (see Supplementary Figs S1–S5 in the Supplementary Information). The dispersion measurements on the x=0.4 sample are shown in Fig. 2e. The single holelike bulk valence band, which is similar to that in the x=0.2 sample, is also observed below the Fermi level. More importantly, a pair of metallic states crossing the Fermi level on the opposite sides of the point is observed along the mirror line momentum space direction. These states are found to show no observable dispersion upon varying the incident photon energy (further details in Supplementary Fig. S3), reflecting its twodimensional character. On the other hand, the single holelike band is observed to disperse strongly upon varying the incident photon energy, suggesting its threedimensional character. At a set of different photon energy (k_{z}) values, the bulk valence band intensity overlaps (intermixing) with different parts of the surface states in energy and momentum space. At a photon energy of 18 eV, the intermixing (intensity overlap) is strong, and the inner two branches of the surface states are masked by the bulk intensity. At photon energies of 10 and 24 eV, the surface states are found to be relatively better isolated. These ARPES measurements suggest that the x=0.4 sample lies on the inverted composition regime and that the observed surface states are related to the band inversion transition in Pb_{1−x}Sn_{x}Te as predicted theoretically^{11,12,13,15}. As a qualitative guide, we present firstprinciplesbased electronic structure calculation on the inverted end compound SnTe (Fig. 2f). The calculated electronic structure of SnTe is found to be a superposition of two k_{z} nondispersive metallic surface states and a single holelike k_{z} dispersive bulk valence band in the vicinity of the point, which is in qualitative agreement with the ARPES results on Pb_{0.6}Sn_{0.4}Te.
Surface state topology of Pb_{0.6}Sn_{0.4}Te
We now perform systematic measurements on the surface electronic structure of the Pb_{0.6}Sn_{0.4}Te. Figure 3b shows the wide range Fermi surface mapping covering the first surface BZ. The surface states are observed to be present, and only present, along the mirror line () directions. No other states are found along any other momentum directions on the Fermi level. In close vicinity to each point, a pair of surface states are observed along the mirror line direction. One lies inside the first surface BZ whereas the other is located outside. Therefore, in total four surface states are observed within the first surface BZ, in agreement with the fact that there are four band inversions in Pb_{1−x}Sn_{x}Te. The mapping zoomedin near the point (Fig. 3c) reveals two unconnected small pockets (dots). The momentum space distance from the centre of each pocket to the point is about 0.09 Å^{−1}. Dispersion measurements (E_{B} vs k) are performed along three important momentum space cuts, namely cuts 1, 2 and 3 defined in Fig. 3c, to further reveal the electronic structure of the surface states. Metallic surface states crossing the Fermi level are observed in both cuts 1 and 2, whereas cut 3 is found to be fully gapped, which is consistent with the theoretically calculated surface states electronic structure shown in Fig. 1e. In cut 1 (Fig. 3d), which is the mirror line () direction, a pair of surface states are observed on the Fermi level. The surface states in our Pb_{0.6}Sn_{0.4}Te samples are found to have a relatively broad spectrum, which can be possibly understood by the strong scattering in the disordered alloy system similar to the broad spectrum of the topological surface states in the Bi_{1−x}Sb_{x} alloy^{23}. In addition to the scattering broadening, the surface states are also observed to tail on the very strong main valence band emission (for example, the white region of intensity distribution in Fig. 3d). In the case of Fig. 2e (photon energy of 24 eV), the bulk VBM locates outside the first surface BZ. The surface state inside the first surface BZ is relatively better isolated from the bulk bands as compared with the one outside the first surface BZ. We thus study the dispersion along cut 2 (Fig. 3d), which only cuts across the surface states inside the first surface BZ. Both the dispersion maps and the momentum distribution curves in cut 2 reveal that the surface states along cut 2 are nearly Diraclike (linearly dispersive) close to the Fermi level. In many TIs, the surface states deviate from ideal linearity^{24}. Fitting of the momentum distribution curves of cut 2 (see Supplementary Fig. S6 for data analysis) shows that the experimental chemical potential (E_{F}) lies roughly at (or just below) the Dirac point energy (E_{D}), E_{F}=E_{D}±0.02 eV. The surface states' velocity is obtained to be 2.8±0.1 eV Å (4.2±0.2 × 10^{5} m s^{−1}) along cut 2, and 1.1±0.3 eV Å (1.7±0.4 × 10^{5} m s^{−1}) for the two outer branches along cut 1, respectively.
Spin polarization measurements of the surface states
We study the spin polarization of the lowenergy states of the Pb_{0.6}Sn_{0.4}Te samples, which are highly dominated by the surface states near the Fermi level. We further compare and contrast their spin behaviour with that of the states at high binding energies away from the Fermi level, which are highly dominated by the bulk valence band in the same sample. Spinresolved (SR) measurements are performed in the spinresolved momentum distribution curve mode^{25,26}, which measures the spinresolved intensity and net spin polarization at a fixed binding energy along a certain momentum space cut direction (detailed in the Methods section). As shown in Fig. 4a, our spinresolved measurements are performed along the mirror line () direction, as the electronic and spin structure along this direction is most critically relevant to the predicted TCI phase^{15}. Considering that the natural Fermi level of our x=0.4 samples are very close to the Dirac point (which is spin degenerate), the spin polarization of the surface states are measured at 60 meV below the Fermi level to gain proper contrast, namely SRCut 1 in Fig. 4a. As shown in the net spin polarization measurement of SRCut 1 in Fig. 4c, in total four spins pointing in the (±) inplane tangential direction are revealed for the surface states along the mirror line direction. This is well consistent with the observed two surface state cones (four branches in total) near an point along the mirror line direction. To compare and contrast the spin polarization behaviour of the surface states (SRCut 1) with that of the bulk states, we perform spinresolved measurement SRCut 2 at E_{B}=0.70 eV, where the bulk valence bands are prominently dominated. Indeed, in contrast to SRCut 1 reflecting the surface states’ spin polarization, no significant net spin polarization is observed for SRCut2, which is expected for the bulk valence bands of the inversion symmetric (centrosymmetric) Pb_{1−x}Sn_{x}Te system. Our observed spin polarization configuration of the surface states is also in qualitative agreement with the firstprinciples calculation spin texture of the SnTe TCI surface states (see Supplementary Figs S7–S9 for texture calculation); and experimental derived topological invariant (Mirror Chern number^{27,28}) n_{M}=−2 also agrees with theoretical prediction for SnTe^{15}.
Discussion
We discuss the possibility that the observed surface states in our data are the signature of the theoretically predicted TCI phase. The TCI phase was predicted to be observed in the band inverted side and absent in the noninverted side^{15}. In our data of the Pb_{0.6}Sn_{0.4}Te samples, which lie on the inverted side, surface states are observed, and only observed, along the two independent mirror line () directions. Within the first surface BZ, two surface states are observed on each mirror line, which are found to locate in vicinity of the points. Dispersion measurements (for example, Cut 2 in Fig. 3) reveal a nearly linear dispersion. Spin polarization measurements show that these surface states are spinpolarized and their spin polarization direction is locked with their momentum (spin–momentum locking). The overall electronic structure observed for our surface states are in qualitative agreement with the theoretically predicted surface states of the TCI phase in SnTe^{15}. It is important to note that in an alloy, the space group symmetry is only of the averaged type. The spatial mirror symmetry required for the topological protection is only preserved in a globally averaged sense. To this date, it is not even theoretically known whether only averaged mirror symmetry is sufficient for the predicted TCI phase. In the absence of any theoretical work for the alloy, the experimental proof of mirror protection in the alloy Pb_{1−x}Sn_{x}Te(Se) system perhaps requires the observation of strictly gapless surface states. The current ARPES works^{29,30} have not demonstrated resolution better than 5 meV, and thus fitting of the energy–momentum distribution curves lacks the resolving power to exclude gap values <5 meV (see Supplementary Fig. S10 and Supplementary Discussion). On the other hand, a proof of strictly gapless surface states in fact requires a surface and spinsensitive experimental setup with energy resolution better than 1 meV, which can even exclude possible small gap values of ≤1 meV at the Dirac point. Even a gap size of even <1 meV can lead to observable change in spintransport experiments, thus destroying the delicate TCI phase. Therefore, although the existence of surface states correlated with band inversion transition is established here in our work, their gapped or gaplessness nature requires surface transport experiments on samples with improved quality, which is currently not possible^{29}.
Our observations of spin–momentum locked surface states in the inverted composition and their absence in the noninverted composition suggest that Pb_{1−x}Sn_{x}Te system is a tunable spin–orbit insulator analogous to the BiTl(S_{1−δ}Se_{δ})_{2} system^{5}, which features a topological phase transition across the band inversion transition. However, the key difference is that odd number of band inversions in the BiTl(S_{1−δ}Se_{δ})_{2} system leads to odd number of surface states, whereas even number of band inversions in Pb_{1−x}Sn_{x}Te system leads to even number of surface states per surface BZ. We further present a comparison of the Pb_{0.6}Sn_{0.4}Te and a single Dirac cone Z_{2} TI GeBi_{2}Te_{4}^{31,32,33}. As shown in Figs 5a–c, for GeBi_{2}Te_{4}, a single surface Dirac cone is observed enclosing the timereversal invariant (Kramers’) momentum in both ARPES and calculation results, demonstrating its Z_{2} TI state and the timereversal symmetry protection of its single Dirac cone surface states. On the other hand, for the Pb_{0.6}Sn_{0.4}Te samples (Figs 5d–f), none of the surface states is observed to enclose any of the timereversal invariant momentum, suggesting their irrelevance to the timereversal symmetryrelated protection. With future ultrahighresolution experimental studies to prove the strict gapless nature of the Pb_{0.6}Sn_{0.4}Te surface states and therefore the topological protection by the crystalline mirror symmetries, it is then possible to realize magnetic yet topologically protected surface states in the Pb_{1−x}Sn_{x}Te system due to its irrelevance to the timereversal symmetryrelated protection, which is fundamentally not possible in the Z_{2} TI systems. Considering the wide applications of magnetic materials in modern electronics, such topologically protected surface states compatible with magnetism will be of great interest in terms of integrating TI materials into future electronic devices.
We note that one advantage of the Pb_{1−x}Sn_{x}Te system is that it can be easily doped with manganese, thallium, or indium to achieve bulk magnetic or superconducting states^{34,35,36}. The symmetry in the Pb_{1−x}Sn_{x}Te system (its nonmagnetic character) can be broken by magnetic or superconducting doping into the bulk or the surface. In future experiments, it would be interesting to explore the modification of our observed surface states brought out by magnetic and superconducting correlations, to search for exotic magnetic and superconducting orders on the surface. Such magnetic and superconducting orders on the surface states in Pb_{1−x}Sn_{x}Te can be different from those recently observed in the Z_{2} TIs^{37,38} because of its very distinct topology of surface electronic structure. The novel magnetic and superconducting states to be realized with this novel topology are not strongly related to the question of gapless or gapped nature of the TCI phase. Therefore, our observation of the spinpolarized surface states presented here provides the much desired platform for realizing unusual surface magnetic and superconducting states in future experiments.
Methods
Electronic structure measurements
Spinintegrated ARPES measurements were performed with incident photon energies of 8–30 eV at beamline 5–4 at the Stanford Synchrotron Radiation Lightsource (SSRL) in the Stanford Linear Accelerator Center, and with 28–90 eV photon energies at beamlines 4.0.3, 10.0.1 and 12.0.1 at the Advanced Light Source (ALS) in the Lawrence Berkeley National Laboratory (LBNL). Samples were cleaved in situ between 10 and 20 K at chamber pressure better than 5 × 10^{−11} torr at both the SSRL and the ALS, resulting in shiny surfaces. Energy resolution was better than 15 meV and momentum resolution was better than 1% of the surface BZ. To crosscheck the cleavage surface orientation in ARPES measurements, the Laue backreflection Xray measurements were performed at the ALS using a Realtime Laue delayline detector from the Multiwire Laboratories (Ithaca, NY), and diffraction spot indexing was performed using the MWL NorthStar orientation package. The Xray Laue measurements along with the analysis results are presented in the Supplementary Fig. S11.
SD measurements
SR ARPES measurements were performed on the SIS beamline at the Swiss Light Source using the COPHEE spectrometer with two 40 kV classical Mott detectors and a photon energy of 24 eV, which systematically measures all three components of the spin of the electron (P_{x}, P_{y} and P_{z}) as a function of its energy and momentum^{25,26}. Energy resolution was bettern than 60 meV and momentum resolution was better than 3% of the surface BZ for the spinresolved measurements. Samples were cleaved in situ at 20 K at a chamber pressure <2 × 10^{−10} torr. Typical electron counts on the detector reach 5 × 10^{5}, which places an error bar of approximately ±0.01 for each point on our measured polarization curves.
Sample growth
Single crystals of Pb_{1−x}Sn_{x}Te were grown by the standard growth method (described in ref. 39 for the end compound PbTe). High purity (99.9999%) elements of lead, tin and tellurium were mixed in the right ratio. Lead was etched by CH_{3}COOH+H_{2}O_{2} (4:1) to remove the surface oxide layer before mixing. The mixture was heated in a clean evacuated quartz tube to 924 °C, where it was held for two days. Afterwards, it was cooled slowly, at a rate of 1.5 °C h^{−1} in the vicinity of the melting point, from the high temperature zone towards room temperature. The nominal concentrations were estimated by the Pb/Sn mixture weight ratio before the growth, in which the two representative compositions presented in the paper were estimated to be x_{nom}=0.20 and x_{nom}=0.36. The concentration values were reexamined on the single crystal samples after growth using highresolution Xray diffraction measurements by tracking the change of the diffraction angle (the 2θ angle) of the sharpest Bragg peak, from which we obtained x=0.22 and x=0.39 (see Supplementary Fig. S12 for diffraction data). In the paper we note the chemical compositions as Pb_{0.8}Sn_{0.2}Te and Pb_{0.6}Sn_{0.4}Te.
Firstprinciples calculation methods
We perform firstprinciples calculations to extract both the electronic structure and the spin texture of the SnTe surface states. Our firstprinciples calculations are within the framework of the density functional theory using fullpotential projected augmented wave method^{40}, as implemented in the VASP package^{41}. The generalized gradient approximation^{42} is used to model exchangecorrelation effects. The spin–orbital coupling is included in the selfconsistent cycles. The surfaces are modelled by periodically repeated slabs of 33atomiclayer thickness, separated by 13Å wide vacuum regions, using a 12 × 12 × 1 MonkhorstPack k point mesh over the BZ with 208 eV cutoff energy. The room temperature crystal structures of SnTe in ideal sodium chloride are used to construct the slab to have the required crystal structure and mirror symmetries required for the predicted TCI phase^{15} (the rhombohedral distortion is found to only occur in the Snrich compositions (x>0.5) and only at low temperature^{16}). The experimental lattice constants of SnTe with the value of 6.327 Å are used^{43}. The experimental lattice constant of PbTe with the value of 6.46 Å^{43} in ideal sodium chloride lattice are used in the PbTe band structure calculation.
Additional information
How to cite this article: Xu, S.Y. et al. Observation of a TCI phase and topological phase transition in Pb_{1−x}Sn_{x}Te. Nat. Commun. 3:1192 doi: 10.1038/ncomms2191 (2012).
Change history
04 August 2016
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Acknowledgements
We thank L. Fu and A. Vishwanath for theoretical discussions on TCI. The work at Princeton and Princetonled synchrotron xraybased measurements and the related theory at Northeastern University are supported by the Office of Basic Energy Sciences, US Department of Energy (grants DEFG0205ER46200, AC0376SF00098 and DEFG0207ER46352). M.Z.H. acknowledges visitingscientist support from the Lawrence Berkeley National Laboratory and additional support from the A. P. Sloan Foundation. The spinresolved and spinintegrated photoemission measurements using synchrotron Xray facilities are supported by the Advanced Light Source, the Stanford Synchrotron Radiation Lightsource, and the Swiss Light Source, by the Basic Energy Sciences of the US Department of Energy and the Swiss National Science Foundation. Theoretical computations benefited from the allocation of supercomputer time at NERSC and Northeastern University’s Advanced Scientific Computation Center. Sample growth and characterization are partially supported by DMR0819860, DMR1006492 and by the National Science Council (NSC)Taiwan under project number NSC1002119M002021. We gratefully acknowledge N. Lavdovsky for assistance with sample preparation. We also thank M. Hashimoto and S.K. Mo for beamline assistance at the Stanford Synchrotron Radiation Lightsource and the Advanced Light Source in Berkeley.
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S.Y.X. and C.L. performed the experiments with assistance from N.A., M.N., D.Q., I.B., L.A.W. and M.Z.H. A.M., E.M., Q.G., R.S., F.C.C. and R.J.C. provided samples; G.L., B.S., J.H.D. and J.D.D. provided beamline assistance; Y.J.W., H.L. and A.B. carried out the theoretical calculations; M.Z.H. was responsible for the conception and the overall direction, planning and integration among different research units.
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Supplementary Figures S1S12, Supplementary Discussion, Supplementary Methods and Supplementary References (PDF 4277 kb)
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Xu, SY., Liu, C., Alidoust, N. et al. Observation of a topological crystalline insulator phase and topological phase transition in Pb_{1−x}Sn_{x}Te. Nat Commun 3, 1192 (2012). https://doi.org/10.1038/ncomms2191
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DOI: https://doi.org/10.1038/ncomms2191
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