Abstract
Topological insulators constitute a new phase of matter protected by symmetries. Timereversal symmetry protects strong topological insulators of the Z_{2} class, which possess an odd number of metallic surface states with dispersion of a Dirac cone. Topological crystalline insulators are merely protected by individual crystal symmetries and exist for an even number of Dirac cones. Here, we demonstrate that Bidoping of Pb_{1−x }Sn_{ x }Se (111) epilayers induces a quantum phase transition from a topological crystalline insulator to a Z_{2} topological insulator. This occurs because Bidoping lifts the fourfold valley degeneracy and induces a gap at \(\bar \Gamma \), while the three Dirac cones at the \({\bar{\rm M}}\) points of the surface Brillouin zone remain intact. We interpret this new phase transition as caused by a lattice distortion. Our findings extend the topological phase diagram enormously and make strong topological insulators switchable by distortions or electric fields.
Introduction
Topological insulators are bulk insulators with a metallic surface state^{1}. Provided that the system stays in the same symmetry class of the Hamiltonian^{2}, it is fundamentally impossible to follow a path from this topologically distinct phase of matter to a trivial phase without closing the insulating bulk band gap. For a strong topological insulator, metallic surface states are necessarily present at the boundary to a trivial insulator as, for example, air or vacuum. These topological surface states are protected by timereversal symmetry and their energy vs. momentum dispersion mimics quasirelativistic, massless particles, with the shape of a Dirac cone and a peculiar helical spin texture^{1,2,3,4}. The topological classification is given by the socalled Z_{2} invariant ν _{0}, which for odd number of Dirac cones is ν _{0} = 1, giving rise to strong (Z_{2}) topological insulators, but for even number of cones is zero, characterizing weak topological or trivial insulators^{5, 6}.
It is possible to transform a strong (Z_{2}) topological insulator to a trivial insulator by alloying as has been shown for Bi_{2}Se_{3}:In^{7, 8} and BiTl(S_{1−x }Se_{ x })_{2} ^{9}. The fundamental principle of bulkboundary correspondence dictates again that this topological phase transition proceeds through a transition point where the bulk band gap closes. In this picture, it is supposed that the crystal symmetry is maintained through the phase transition. However, the crystal symmetry itself can protect topologically distinct phases as well, termed topological crystalline insulators (TCIs)^{5, 10,11,12,13,14}. For TCIs, the decisive role of the crystal symmetry renders the topological protection dependent on the specific crystal face^{10}. The topological invariants allow for an even number of Dirac cones, which are, however, not robust against disorder^{5}. Pb_{1−x }Sn_{ x }Se and Pb_{1−x }Sn_{ x }Te represent such mirrorsymmetry protected TCIs with fourfold valley degeneracy^{5, 11} in which the trivialtoTCI phase transition is reached for sufficiently large Sn contents^{10, 15}. Upon cooling, the lattice contracts and the enhanced orbital overlap leads to an inverted (i.e., negative) bulk band gap, which, via bulkboundary correspondence, gives rise to Dirac cone surface states. This has impressively been shown by temperaturedependent angleresolved photoemission (ARPES)^{15}.
On the other hand, there are crystals that are not necessarily topological, but change their symmetry and their electronic properties with temperature. Such phasechange materials have been studied intensively for nonvolatile data storage because their properties can be altered dramatically at the structural phase transition^{16}. GeTe is such a prototypical material^{16, 17} that transforms upon cooling from the cubic rock salt to a rhombohedral structure characterized by a large relative sublattice displacement. This gives rise to pronounced ferroelectricity of GeTe^{18} and has recently been found to allow for an electrical switching of electronic properties^{19}. Symmetry changes are potentially very interesting also for topological insulators. In fact, it has been shown for the TCI Pb_{1−x }Sn_{ x }Se that by breaking of mirror symmetries two out of the four Dirac cones at its (100) surface can be gapped^{11, 20,21,22}. In that case, the topological phase remains, however, unchanged by the symmetry breaking and the Z_{2} invariant stays even. It is crucial for the present work to note that in principle the topological phase does not need to be preserved by a distortion. Indeed, topological phase transitions have recently been predicted for twodimensional TlSe^{23} and threedimensional SnTe by distortions^{24} and by finitesize effects^{25}.
Here, we investigate the band topology of the (111) surface of Pb_{1−x }Sn_{ x }Se by cooling through the complete trivial to topological phase transition. We demonstrate a new type of phase transition from crystalsymmetryprotected to timereversal symmetryprotected topology controlled by Bi incorporation. We show that when Bi is introduced in the bulk making the system ntype, a gap is opened up at the one Dirac cone at the center of the surface Brillouin zone at \(\bar \Gamma \) _{,} while the three Dirac cones located at the \({\bar{\rm M}}\) points at the zone boundaries behave as in pure Pb_{1−x }Sn_{ x }Se, that is, are gapless at low temperature. Our findings provide the first experimental evidence for a topological phase transition from a TCI with an even number of Dirac cones to a Z_{2} timereversal symmetryprotected strong topological insulator where the number is odd (three). Following the recent prediction^{24}, the origin of the novel topological phase transition is interpreted as due to a sublattice shift and rhombohedral distortion along the [111] direction, which lifts the bulk band inversion only at the Zpoint (Lpoint in the undistorted phase) projected onto \(\bar \Gamma \). At the same time, we do not find any evidence for a bulk band gap closing across the phase transition, most likely, because the rhombohedral distortion does not leave the system in the same symmetry class where the TCI is defined.
Results
Effect of temperature and Sn concentration
We have grown both undoped and Bidoped epitaxial (111) Pb_{1−x }Sn_{ x }Se films of high quality by molecular beam epitaxy (see Supplementary Fig. 1–6 and Supplementary Notes 1–4 for details). The samples were capped insitu by a thin Se layer to protect the surface during transport to the ARPES setup, where the cap was desorbed by annealing (see also “Methods” section and Supplementary Note 5). ARPES measurements were performed using linearly polarized light incident on the sample under the geometry shown in Fig. 1a, which also depicts the bulk and (111) surface Brillouin zones of rock salt Pb_{1−x }Sn_{ x }Se. In contrast to the natural (100) cleavage plane of bulk crystals previously studied^{12, 13, 15, 26}, for the (111) orientation, the four bulk Lpoints project on the following four timereversal invariant surface momenta: \(\bar \Gamma \) and three equivalent \({\bar{\rm M}}\) points^{27}. This is seen in the ARPES data shown in Fig. 1b, where the intensity from the Dirac cones at the three \({\bar{\rm M}}\) points is enhanced by a photoemission finalstate effect. Due to the sensitive dependence of the bulk band inversion on the lattice constant, the trivial to topological phase transition can be monitored during cooling by tracing the evolution of the bulk band gap^{28} or by observation of the appearance of the Dirac cones in ARPES^{15, 26}. Indeed, as seen in Fig. 1c–q, for our (111) films with low Sn concentrations and without Bi doping (x _{Sn} = 10%, Fig. 1c, f, i, and l), the Dirac cone at the \(\bar \Gamma \) point does not form down to 22 K, while for x _{Sn} = 20% (Fig. 1d, g, j, and m) and x _{Sn} = 28% (Fig. 1e, h, k, and n), the gapless Dirac cone appears at around 90 and 130 K, respectively.
Impact of Bi incorporation in the bulk
Figure 2 shows the effect of bulk Bi doping on the \(\bar \Gamma \) Dirac cone of Pb_{0.72}Sn_{0.28}Se. When substitutionally incorporated at cation (Pb, Sn—group IV) lattice sites, the group V element Bi acts as electron donor due to its excess valence electron^{29}. Increasing the Bi concentration n _{Bi} thus leads to a strong upward shift of the Fermi level by 200 meV for n _{Bi} = 2.2% (Fig. 2a–e). Remarkably, while for undoped Pb_{0.72}Sn_{0.28}Se, an intact Dirac cone is seen at 30 K, a surface gap as large as ~100 meV opens up at the \(\bar \Gamma \) Dirac cone upon Bi doping and increases strongly with increasing Bi content (see Fig. 2f, g, Supplementary Fig. 7, and Supplementary Note 6). This leads to the general conclusion that for Bi concentrations >~0.6%, the gap at \(\bar \Gamma \)does not close. Figure 2h–n demonstrates that within the experimental error bars, this gapped surface state does not disperse with the photon energy, i.e., momentum perpendicular to the surface plane, pinpointing its twodimensional nature (see also Supplementary Fig. 8 and discussion in Supplementary Note 7). This observation is important to rule out that the probed state corresponds to the gapped bulk states.
The effect of Bi on the Dirac cones at the \(\bar \Gamma \) and \({\bar{\rm M}}\) points was evaluated in dependence of the Bi and Sn contents as well as of temperature. On the topologically trivial side (x _{Sn} < 16%), for Bi doping n _{Bi} < 0.6%, no difference in the gaps Δ at \(\bar \Gamma \) and \({\bar{\rm M}}\) is found, which remain open at all temperatures (Supplementary Figs. 9, 10, and Supplementary Note 8). On the topologically nontrivial side (x _{Sn} > 16%), the Dirac cones at \(\bar \Gamma \) (Fig. 3a–d) and \({\bar{\rm M}}\) (Fig. 3e–h) close synchronously as a function of temperature for low Bi concentrations. This is shown by Fig. 3a, b, e, and f, where corresponding ARPES dispersions at 305 and 50 K are presented. In contrast, for high Bi concentrations (Fig. 3c, d, g, and h), the fourfold valley degeneracy is completely lifted such that the gap is closed to zero at all three \({\bar{\rm M}}\) points (Fig. 3h–k), but opens as wide as 100 meV at the \(\bar \Gamma \) point at 30 K (Fig. 3d). Thus, we conclude that of the even numbered Dirac cones per surface Brillouin zone, characteristic of a TCI, only three remain, qualifying Pb_{1−x }Sn_{ x }Se:Bi as a strong Z_{2} topological insulator already for moderate Bi doping. This is the central result of the present work. While a TCI is protected by mirror symmetry, the odd numbered Dirac cones of strong topological insulators are protected by timereversal symmetry and robust against disorder. Figure 3i shows that the new Z_{2} phase, which has never been observed before for this material class, exists in a wide temperature range from ~150 K down to the lowest temperature probed in our experiments.
Discussion
Due to the bulkboundary correspondence, the formation of the Dirac cones at the surface indicates a bulk band inversion at the parent momenta in the Brillouin zone as indicated by Fig. 1a. The band structures and shape of the Dirac cones corresponding to the different topological phases are illustrated schematically in Fig. 4a–c, where the open and closed Dirac cones are colored such as to indicate the changes of the predominant charge density at the anion (orange) and cation sites (blue) occurring during the band inversion (see Supplementary Fig. 11 and Supplementary Note 9). The topological phase transition so far reported for Pb_{1−x }Sn_{ x }Se^{15, 26} occurs between trivial (Fig. 4a) and TCI phase (Fig. 4b) where all bulk band inversions behave equally. For the new topological phase transition into the Z_{2} topological insulator phase (Fig. 4c), the bulk band inversion along the momenta normal to the surface does not occur, triggered by the Bi doping. This means the discovery of two new topological phase transitions in Pb_{1−x }Sn_{ x }Se: (i) from trivial to Z_{2} topological by cooling and (ii) from topological crystalline to Z_{2} topological by adding Bi. The resulting topological phase diagram (Fig. 4d) derived from our data shows the interdependence of these two pathways.
The observed behavior resembles the predictions by Plekhanov et al.^{24} for SnTe. SnTe is known for its ferroelectric phase transition in which the anion and cation sublattices are shifted against each other along the [111] direction^{17}, connected to a transverse optical phonon softening^{30}. Plekhanov et al.^{24} tested small displacements and theoretically predicted that for a certain range of displacements a Z_{2} topological insulator can exist.
In general, all IV–VI compounds are close to such a structural phase transition due to their mixed covalentionic bonding. They belong to the family of 10 electron systems and crystallize either in the cubic rock salt, rhombohedral, or orthorhombic structure. As calculated by Littlewood^{31, 32}, the type of structure in which a IV–VI compound crystallizes critically depends on the values of two bond orbital coordinates—one is a measure of the ionicity and the other of covalency or s–p hybridization, and based on these a phase diagram has been established^{31, 32}. Due to the not fully saturated pbonds, the rock salt structure is inherently unstable against rhombohedral distortions^{31,32,33}, which reduces the six nearest neighbors to three. The cubic/rhombohedral phase boundary is determined by the electronegativity difference between the constituting elements and as a physical explanation of the instability the resonating bond model was invoked^{34}. For the six nearest neighbors, the number of available pelectrons (six per atom pair) is not sufficient to stabilize the cubic bonds^{35}. Thus, SnTe and GeTe assume a rhombohedral structure and even for PbTe minute addition of <0.5% of Ge suffices to drive Pb_{1x }Ge_{ x }Te into the rhombohedral phase, rendering it ferroelectric^{36, 37}. Thus, very small changes on the group IV lattice sites give rise to structural phase transitions. Indeed, for our Bidoped films, we find a small Biinduced rhombohedral distortion along the [111] direction using xray diffraction (Supplementary Figs. 4–6 and Supplementary Note 4). This indicates a ferroelectric inversion symmetry breaking as the underlying physical mechanism, which has been suggested to be much enhanced at the surface^{21, 38}. Because of the (111) orientation of our films, this symmetry breaking lifts the even number degeneracy of the Dirac cones such that a gap is opened only at the \(\bar \Gamma \) point. This leaves an odd number of Dirac cones intact at the three \({\bar {\rm M}}\) points, causing the topological phase transition.
Finally, we would like to address the role of Bi as decisive ingredient for symmetry breaking even at small concentrations. Bi is another 10 electron system^{33}, covalently bonded and crystallizing in a rhombohedral structure. Accordingly, when incorporated at group IV lattice sites in Pb_{1−x }Sn_{ x }Se, it shifts the alloy toward a more covalently bonded structure in the phase diagram^{31, 32, 39}. Moreover, on such lattice sites, Bi also reduces the cation vacancy concentration and the latter strongly enhances the ferroelectric Curie temperature T _{C} of the cubictorhombohedral phase transition^{30, 35, 38, 40}. Indeed, in SnTe thin films with reduced vacancy concentration, ferroelectricity up to room temperature was recently reported^{38}. Thus, both effects of the Bi, higher covalency, and lower vacancy concentration, contribute to structural symmetry breaking and trigger the novel TCItoZ_{2} topological phase transition discovered in the present work.
Methods
Sample growth and characterization
Epitaxial growth of (111) Pb_{1−x }Sn_{ x }Se films on BaF_{2} substrates was performed using molecular beam epitaxy (MBE) in ultrahigh vacuum conditions better than 5 × 10^{−10} mbar at a substrate temperature of 380 °C. Effusion cells filled with stoichiometric PbSe and SnSe were used as source materials, as well as a ternary Pb_{1−x }Sn_{ x }Se source with x _{Sn} = 25%. Bidoping was realized using a Bi_{2}Se_{3} effusion cell. The chemical composition of the layers was varied over a wide range from x _{Sn} = 0 to 40% by control of the SnSe/PbSe beam flux ratio, and a twodimensional growth was observed by in situ reflection highenergy electron diffraction (Supplementary Fig. 1). The film thickness was in the range of 1–3 µm.
Highresolution ARPES
For the ARPES measurements, the films were capped in situ in the MBE chamber with a 200 nm thick amorphous Se layer at room temperature to protect the surface against oxidation during transport to the BESSY II synchrotron radiation source in Berlin, Germany. There the Se cap was completely desorbed in the ARPES preparation chamber by annealing at about 230 °C for 15 min in 3 × 10^{−10} mbar. ARPES measurements were performed at the UE112PGM2a beamline of BESSY II at pressures better than 1 × 10^{−10} mbar using linearly polarized p + s photons incident on the sample under an angle ϕ = 45°. We used photon energies between 18 and 23 eV for the temperaturedependent ARPES measurements of the band dispersions, and a photon energy of 90 eV for the core levels (see Supplementary Fig. 12 and Supplementary Note 10). Emitted photoelectrons were detected with a Scienta R8000 electron energy analyzer at the ARPES 1^{2} endstation. Overall resolutions of the ARPES measurements were 5 meV (energy) and 0.3° (angular).
Composition and structural characterization
The composition of the epilayers was determined using highresolution xray diffraction and the Vegard’s law (Supplementary Fig. 2 and Supplementary Note 2). We employed a Seifert diffractometer equipped with primary and secondary monochromator crystals. Upon Bi incorporation, the rhombohedral lattice distortion was determined from reciprocal space maps recorded around the (513) reflection (see Supplementary Figs. 4–6 and Supplementary Note 4).
Electrical characterization
The dopant concentration and transport properties were assessed by Hall effect measurements at 77 K (Supplementary Fig. 3 and Supplementary Note 3), evidencing carrier mobilities as high as 10^{4} cm^{2} V^{−1} s^{−1} in dependence of the carrier concentration.
Data availability
The authors declare that all data supporting the findings of this study are available within the paper and its Supplementary Information files.
References
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Ezawa, M., Tanaka, Y. & Nagaosa, N. Topological phase transition without gap closing. Sci. Rep. 3, 2790 (2013).
Qi, X.L. & Zhang, S.C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
Bernevig, B. A. & Hughes, T. L. Topological Insulators and Topological Superconductors (Princeton University Press, Princeton, 2013).
Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).
Fu, L., Kane, C. L. & Mele, E. J. Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007).
Brahlek, M. et al. Topologicalmetal to bandinsulator transition in (Bi_{1−x }In_{ x })_{2}Se_{3} thin Films. Phys. Rev. Lett. 109, 186403 (2012).
Wu, L. et al. A sudden collapse in the transport lifetime across the topological phase transition in (Bi_{1−x }In_{ x })_{2}Se_{3}. Nat. Phys. 9, 410–414 (2013).
Xu, S.Y. et al. Topological phase transition and texture inversion in a tunable topological insulator. Science 332, 560–564 (2011).
Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011).
Hsieh, T. H. et al. Topological crystalline insulators in the SnTe material class. Nat. Commun. 3, 982 (2012).
Tanaka, Y. et al. Experimental realization of a topological crystalline insulator in SnTe. Nat. Phys. 8, 800–803 (2012).
Xu, S.Y. et al. Observation of a topological crystalline insulator phase and topological phase transition in Pb_{1x }Sn_{ x }Te. Nat. Commun. 3, 1192 (2012).
Ando, Y. & Fu, L. Topological crystalline insulators and topological superconductors: from concepts to materials. Annu. Rev. Condens. Matter Phys. 6, 361–381 (2015).
Dziawa, P. et al. Topological crystalline insulator states in Pb_{1x }Sn_{ x }Se. Nat. Mater. 11, 1023–1027 (2012).
Wuttig, M. & Yamada, N. Phasechange materials for rewriteable data storage. Nat. Mater. 6, 824–832 (2007).
Schmitte, F. J. in Physics of NonTetrahedrally Bonded Binary Compounds II (ed. Madelung, O.) LandoltBörnstein, New Series III/17F (Springer, Berlin, 1983).
Chattopadhyay, T., Boucherle, J. X. & von Schnering, H. G. Neutron diffraction study on the structural phase transition in GeTe. J. Phys. C 20, 1431 (1987).
Di Sante, D., Barone, P., Bertacco, R. & Picozzi, S. Electric control of the giant Rashba effect in bulk GeTe. Adv. Mater. 25, 509–513 (2013).
Okada, Y. et al. Observation of Dirac node formation and mass acquisition in a topological crystalline insulator. Science 341, 1496–1499 (2013).
Zeljkovic, I. et al. Dirac mass generation from crystal symmetry breaking on the surfaces of topological crystalline insulators. Nat. Mater. 14, 318–324 (2015).
Wojek, B. M. et al. Direct observation and temperature control of the surface Dirac gap in a topological crystalline insulator. Nat. Commun. 6, 8463 (2015).
Niu, C. et al. Twodimensional topological crystalline insulator and topological phase transition in TlSe and TlS monolayers. Nano. Lett. 15, 6071–6075 (2015).
Plekhanov, E., Barone, P., Di Sante, D. & Picozzi, S. Engineering relativistic effects in ferroelectric SnTe. Phys. Rev. B 90, 161108(R) (2014).
Safaei, S., Galicka, M., Kacman, P. & Buczko, R. Quantum spin hall effect in IVVI topological crystalline insulators. New J. Phys. 17, 063041 (2015).
Neupane, M. et al. Topological phase diagram and saddle point singularity in a tunable topological crystalline insulator. Phys. Rev. B 92, 075131 (2015).
Polley, C. M. et al. Observation of topological crystalline insulator surface states on (111)oriented Pb_{1−x }Sn_{ x }Se films. Phys. Rev. B 89, 075317 (2014).
Strauss, A. J. Inversion of conduction and valence bands in Pb_{1x }Sn_{ x }Se alloys. Phys. Rev. 157, 608–611 (1967).
Zykov, V. A., Gavrikova, T. A., Il’in, V. I., Nemov, S. A. & Savintsev, P. V. Effect of bismuth impurity on carrier density in PbSe:Bi:Se epitaxial layers. Semiconductors 35, 1254–1258 (2001).
Sugai, S. et al. Carrier density dependence of soft TOphonon in SnTe by Raman scattering. Solid State Commun. 24, 407–409 (1977).
Littlewood, P. B. The crystal structure of IVVI compounds. I. Classification and description. J. Phys. C Solid State Phys. 13, 4855–4873 (1980).
Littlewood, P. B. The crystal structure of IVVI compounds. II. A microscopic model for cubic/rhombohedral materials. J. Phys. C Solid State Phys. 13, 4875–4892 (1980).
Littlewood, P. B. Structure and bonding in narrow gap semiconductors. Crit. Rev. Solid State Mater. Sci. 11, 229–285 (1983).
Nimtz, G. & Schlicht, B. in Springer Tracts in Modern Physics 1–117, Vol. 98, Narrowgap semiconductors (Springer, Berlin, 1983).
Jantsch, W., BussmannHolder, A., Bilz, H. & Vogel, P. in Springer Tracts in Modern Physics 198, Vol. 99, Dynamical properties of IVVI compounds (Springer, Berlin, 1983).
Bangert, E., Bauer, G., Fantner, E. J. & Pascher, H. Magnetooptical investigations of phasetransitioninduced bandstructure changes of Pb_{1x }Ge_{ x }Te. Phys. Rev. B 31, 7958 (1985).
Lebedev, A. I. & Sluchinskaya, I. A. Influence of random fields on the ferroelectric phase transition in IV–VI semiconductors. Ferroelectrics 169, 293–301 (1995).
Chang, K. et al. Discovery of robust inplane ferroelectricity in atomicthick SnTe. Science 353, 274–278 (2016).
Kool, B. J. & Noheda, B. Ferroelectric chalcogenides–materials at the edge. Science 353, 221–222 (2016).
Iizumi, M., Hamaguchi, Y., Komatsubara, K. F. & Kato, Y. Phase transition in SnTe with low carrier concentration. J. Phys. Soc. Jpn. 38, 443–449 (1975).
Acknowledgements
This work was supported by SPP 1666 of DFG and the SFB025 IRON of the Austrian Science Funds. P.S.M. is supported by Helmholtz Virtual Institute “New states of matter and their excitations.” We thank Andrzej Szczerbakow for providing the MBE source material.
Author information
Authors and Affiliations
Contributions
P.S.M., G.S., V.V.V., O.C., and J.S.B. performed photoemission experiments with support from A.V. and E.G.; G.S. prepared and characterized the samples; G.S., V.V.V., and O.C. performed xray diffraction experiments; J.S.B. and P.S.M. performed data analysis, figure and draft planning; J.S.B., O.R., G.S., and G.B. coordinated the project and wrote the manuscript with input from all the coauthors.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Mandal, P.S., Springholz, G., Volobuev, V.V. et al. Topological quantum phase transition from mirror to time reversal symmetry protected topological insulator. Nat Commun 8, 968 (2017). https://doi.org/10.1038/s41467017012040
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467017012040
This article is cited by

Energy gap of topological surface states in proximity to a magnetic insulator
Communications Physics (2023)

Emerging group11 monochalcogenide MX (M = Cu, Ag, Au; X = S, Se, Te) monolayers: Twodimensional polarized metals and visiblelightdriven photocatalysts
Science China Materials (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.