Topological insulators with inversion symmetry
Open accessAbstract
Topological insulators are materials with a bulk excitation gap generated by the spin-orbit interaction that are different from conventional insulators. This distinction is characterized by ${Z}_{2}$ topological invariants, which characterize the ground state. In two dimensions, there is a single ${Z}_{2}$ invariant that distinguishes the ordinary insulator from the quantum spin-Hall phase. In three dimensions, there are four ${Z}_{2}$ invariants that distinguish the ordinary insulator from ``weak'' and ``strong'' topological insulators. These phases are characterized by the presence of gapless surface (or edge) states. In the two-dimensional quantum spin-Hall phase and the three-dimensional strong topological insulator, these states are robust and are insensitive to weak disorder and interactions. In this paper, we show that the presence of inversion symmetry greatly simplifies the problem of evaluating the ${Z}_{2}$ invariants. We show that the invariants can be determined from the knowledge of the parity of the occupied Bloch wave functions at the time-reversal invariant points in the Brillouin zone. Using this approach, we predict a number of specific materials that are strong topological insulators, including the semiconducting alloy ${\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x}$ as well as $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{Sn}$ and HgTe under uniaxial strain. This paper also includes an expanded discussion of our formulation of the topological insulators in both two and three dimensions, as well as implications for experiments.
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If the chemical potential lies inside the bandgap, as it should in purely insulating Bi2Se3, its surface must carry a global π mod 2π Berry phase.
“Therefore, if the chemical potential (Fermi level) lies inside the bandgap, as it should in purely insulating Bi 2 Se 3 , its surface must carry a global π mod 2π Berry phase.”
From Observation of a large-gap topological-insulator class with a single Dirac cone on the surface (2009) · cites this paper as [7]
The “Strong topological insulator” section says that an electron circling a surface Fermi arc acquires a quantized Berry phase when the arc encloses a single Dirac point (page unavailable in our copy). It does not establish this for Bi₂Se₃ or say that placing the chemical potential in a bulk gap alone guarantees it; the result is conditional on the strong topological-insulator phase. (LLM, conf=0.88)
Experimental — accuracy not yet measuredThis provides evidence that insulating Bi2Se3 belongs to the Z2 = −1 topological class in the Fu-Kane-Mele topological classification scheme for band insulators.
“This provides evidence that insulating Bi 2 Se 3 belongs to the Z 2 = -1 topological class in the Fu-Kane-Mele topological classification scheme for band insulators.”
From Observation of a large-gap topological-insulator class with a single Dirac cone on the surface (2009) · cites this paper as [7]
Section II describes a Z₂ classification of band insulators, supporting the attribution of the classification scheme to this paper. But the paper’s materials discussion identifies Bi₁₋ₓSbₓ, strained α-Sn and HgTe—not Bi₂Se₃—as strong topological insulators. It does not provide evidence for the claimed classification of insulating Bi₂Se₃. Page numbers were unavailable in our copy. (LLM, conf=0.86)
Experimental — accuracy not yet measuredThis calculated topology is consistent with the Z2 = −1 class in the Fu-Kane-Mele classification scheme.
“This topology is consistent with the Z 2 = -1 class in the Fu-Kane-Mele classification scheme 7 .”
From Observation of a large-gap topological-insulator class with a single Dirac cone on the surface (2009) · cites this paper as [7]
Constraints from theory that make surface-band superconductivity unconventional will set important features of the pairing wavefunction.
“and that theoretical constraints ensuring unconventional superconductivity induced in the topological surface band will define key properties of the pairing wavefunction 3,12,28 .”
From Observation of topological order in a superconducting doped topological insulator (2010) · cites this paper as [3]
The bulk band-structure topology fully determines the topological surface states.
“Topological surface states are completely determined by the bulk band-structure topology 3,29”
From Observation of topological order in a superconducting doped topological insulator (2010) · cites this paper as [3]
Simpler forms of these invariants in inversion-symmetric materials were found independently and used to predict the first topological insulator to be discovered, Bi_x Sb_{1-x}.
“Simpler forms of these invariants in inversion-symmetric materials were found independently (6) and used to predict the first topological insulator to be discovered, Bi x Sb 1 -x (7).”
From Three-Dimensional Topological Insulators (2011) · cites this paper as [7]
In the case of inversion symmetry, a useful trick allows the integrals in the approach to be expressed simply in terms of the parity eigenvalues at the time-reversal-invariant points.
“In the case of inversion symmetry, a useful trick (7, 2) allows the integrals in the following approach to be expressed simply in terms of the parity eigenvalues at the time-reversal-invariant points.”
From Three-Dimensional Topological Insulators (2011) · cites this paper as [7]
Bi2Se3 and related materials have been proposed as three-dimensional topological insulators with a single Dirac cone on the surface and verified by angle-resolved photoemission spectroscopy experiments.
“3D topological insulators with the simplest possible surface states consisting of a single Dirac cone, on the other hand, have been proposed theoretically [10] in stoichiometric compounds Bi 2 Se 3 , Bi 2 Te 3 and Sb 2 Te 3 , and topological surface states have been observed independently by angle-resolved photoemission spectroscopy (ARPES) experiments in Bi 2 Se 3 [11].”
From Aharonov–Bohm interference in topological insulator nanoribbons (2010) · cites this paper as [10]
Neither bulk-related effect could result in B-periodicity over the large field range in this experiment; therefore the AB effect in Bi2Se3 nanoribbons indisputably manifests the existence of conducting surface states.
“Nevertheless, neither of these bulk-related effects could result in B-periodicity over the large field range in this experiment. Therefore, the AB effect in Bi 2 Se 3 nanoribbons indisputably manifests the existence of the conducting surface states”
From Aharonov–Bohm interference in topological insulator nanoribbons (2010) · cites this paper as [10]
Bi2Se3 has a simple band structure with a single Dirac cone on the surface and a large non-trivial bulk gap of 0.3 eV.
“Among these materials, Bi 2 Se 3 , a narrow gap semiconductor for IR detectors and thermoelectric applications [13], has a simple band structure with a single Dirac cone on the surface and a large non-trivial bulk gap of 0.3eV [10, 11].”
From Aharonov–Bohm interference in topological insulator nanoribbons (2010) · cites this paper as [10]
The recent proposals that many Heusler compounds may be topological insulators, together with the antiferromagnetic order in GdBiPt below 9 K, suggest a possible candidate for the state proposed here.
“The recent proposals that many Heusler compounds may be topological insulators, 20,21 together with the antiferromagnetic order in GdBiPt below 9 K, 19 suggest a possible candidate for the state proposed here.”
From Antiferromagnetic topological insulators (2010) · cites this paper as [20]
Transport measurements show a gapless edge channel, demonstrating that the HgTe/CdTe quantum well is a two-dimensional topological insulator with the quantum spin Hall effect.
“Transport measurements[6, 7] show the existence of the gapless edge channel, which demonstrates that HgTe/CdTe quantum well is a two-dimensional (2D) TI with quantum spin Hall effect.”
From Model Hamiltonian for topological insulators (2010) · cites this paper as [7]
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Cites
Two-dimensional gas of massless Dirac fermions in graphene
Novoselov, K. S., Geim, A. K., Morozov, S. V., Jiang, D., Katsnelson, M. I., Grigorieva, I. V., Dubonos, S. V., & Firsov, A. A. (2005) · Nature · doi:10.1038/nature04233 · Cited by 21,611 · see citing works
Experimental observation of the quantum Hall effect and Berry's phase in graphene
Zhang, Y., Tan, Y.-W., Stormer, H. L., & Kim, P. (2005) · Nature · doi:10.1038/nature04235 · Cited by 13,435 · see citing works
Quantum Spin Hall Effect in Graphene
Kane, C. L., & Mele, E. J. (2005) · Physical Review Letters · doi:10.1103/physrevlett.95.226801 · Cited by 8,197 · see citing works
Self-citationQuantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
Bernevig, B. A., Hughes, T. L., & Zhang, S.-C. (2006) · Science · doi:10.1126/science.1133734 · Cited by 7,090 · see citing works
Quantized Hall Conductance in a Two-Dimensional Periodic Potential
Thouless, D. J., Kohmoto, M., Nightingale, M. P., & den Nijs, M. (1982) · Physical Review Letters · doi:10.1103/physrevlett.49.405 · Cited by 6,973 · see citing works
Su, W. P., Schrieffer, J. R., & Heeger, A. J. (1979) · Physical Review Letters · doi:10.1103/physrevlett.42.1698 · Cited by 6,783 · see citing works
Kane, C. L., & Mele, E. J. (2005) · Physical Review Letters · doi:10.1103/physrevlett.95.146802 · Cited by 6,460 · see citing works
Self-citationHaldane, F. D. M. (1988) · Physical Review Letters · doi:10.1103/physrevlett.61.2015 · Cited by 6,301 · see citing works
Topological Insulators in Three Dimensions
Fu, L., Kane, C. L., & Mele, E. J. (2007) · Physical Review Letters · doi:10.1103/physrevlett.98.106803 · Cited by 4,651 · see citing works
Self-citationTheory of polarization of crystalline solids
King-Smith, R. D., & Vanderbilt, D. (1993) · Physical Review B · doi:10.1103/physrevb.47.1651 · Cited by 4,472 · see citing works
Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System
Hikami, S., Larkin, A. I., & Nagaoka, Y. (1980) · Progress of Theoretical Physics · doi:10.1143/ptp.63.707 · Cited by 2,774 · see citing works
Observation of the Spin Hall Effect in Semiconductors
Kato, Y. K., Myers, R. C., Gossard, A. C., & Awschalom, D. D. (2004) · Science · doi:10.1126/science.1105514 · Cited by 2,636 · see citing works
Macroscopic polarization in crystalline dielectrics: the geometric phase approach
Resta, R. (1994) · Reviews of Modern Physics · doi:10.1103/revmodphys.66.899 · Cited by 2,567 · see citing works
Halperin, B. I. (1982) · Physical Review B · doi:10.1103/physrevb.25.2185 · Cited by 2,547 · see citing works
Quantized Hall conductivity in two dimensions
Laughlin, R. B. (1981) · Physical review. B, Condensed matter · doi:10.1103/physrevb.23.5632 · Cited by 2,478 · see citing works
Topological invariants of time-reversal-invariant band structures
Moore, J. E., & Balents, L. (2007) · Physical Review B · doi:10.1103/physrevb.75.121306 · Cited by 2,350 · see citing works
Bernevig, B. A., & Zhang, S.-C. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.106802 · Cited by 2,164 · see citing works
Berry’s phase for energy bands in solids
Zak, J. (1989) · Physical Review Letters · doi:10.1103/physrevlett.62.2747 · Cited by 1,950 · see citing works
Dissipationless Quantum Spin Current at Room Temperature
Murakami, S., Nagaosa, N., & Zhang, S. (2003) · Science · doi:10.1126/science.1087128 · Cited by 1,947 · see citing works
Condensed-Matter Simulation of a Three-Dimensional Anomaly
Semenoff, G. W. (1984) · Physical Review Letters · doi:10.1103/physrevlett.53.2449 · Cited by 1,832 · see citing works
The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal
Nielsen, H. B., & Ninomiya, M. (1983) · Physics Letters B · doi:10.1016/0370-2693(83)91529-0 · Cited by 1,708 · see citing works
Chern number and edge states in the integer quantum Hall effect
Hatsugai, Y. (1993) · Physical Review Letters · doi:10.1103/physrevlett.71.3697 · Cited by 1,499 · see citing works
Wunderlich, J., Kaestner, B., Sinova, J., & Jungwirth, T. (2005) · Physical Review Letters · doi:10.1103/physrevlett.94.047204 · Cited by 1,494 · see citing works
A method for simulating chiral fermions on the lattice
Kaplan, D. B. (1992) · Physics Letters B · doi:10.1016/0370-2693(92)91112-m · Cited by 1,196 · see citing works
Halperin, B. I. (1982) · Perspectives in condensed matter physics · doi:10.1007/978-94-010-9709-3_15 · Cited by 1,185 · see citing works
Quantized Hall conductance as a topological invariant
Niu, Q., Thouless, D. J., & Wu, Y.-S. (1985) · Physical Review B · doi:10.1103/physrevb.31.3372 · Cited by 1,136 · see citing works
Topological invariant and the quantization of the Hall conductance
Kohmoto, M. (1985) · Annals of Physics · doi:10.1016/0003-4916(85)90148-4 · Cited by 1,090 · see citing works
Theory of the Insulating State
Kohn, W. (1964) · Physical Review · doi:10.1103/physrev.133.a171 · Cited by 998 · see citing works
Fu, L., & Kane, C. L. (2006) · Physical Review B · doi:10.1103/physrevb.74.195312 · Cited by 905 · see citing works
Self-citationBerry Curvature on the Fermi Surface: Anomalous Hall Effect as a Topological Fermi-Liquid Property
Haldane, F. D. M. (2004) · Physical Review Letters · doi:10.1103/physrevlett.93.206602 · Cited by 760 · see citing works
Helical Liquid and the Edge of Quantum Spin Hall Systems
Wu, C., Bernevig, B. A., & Zhang, S.-C. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.106401 · Cited by 759 · see citing works
A method for simulating chiral fermions on the latticePreprint
Kaplan, D. B. (1992) · Cited by 741 · see citing works
Quantum Spin-Hall Effect and Topologically Invariant Chern Numbers
Sheng, D. N., Weng, Z.-Y., Sheng, L., & Haldane, F. D. M. (2006) · Physical Review Letters · doi:10.1103/physrevlett.97.036808 · Cited by 653 · see citing works
Quantum Spin Hall Effect and Enhanced Magnetic Response by Spin-Orbit Coupling
Murakami, S. (2006) · Physical Review Letters · doi:10.1103/physrevlett.97.236805 · Cited by 639 · see citing works
Mean-field theory of spin-liquid states with finite energy gap and topological orders
Wen, X.-G. (1991) · Physical review. B, Condensed matter · doi:10.1103/physrevb.44.2664 · Cited by 627 · see citing works
Xu, C., & Moore, J. E. (2006) · Physical Review B · doi:10.1103/physrevb.73.045322 · Cited by 571 · see citing works
Electronic structure and thermoelectric properties of bismuth telluride and bismuth selenide
Mishra, S. K., Satpathy, S., & Jepsen, O. (1997) · Journal of Physics: Condensed Matter · doi:10.1088/0953-8984/9/2/014 · Cited by 540 · see citing works
Dornhaus, R., Nimtz, G., & Schlicht, B. (1983) · Springer tracts in modern physics · doi:10.1007/bfb0044919 · Cited by 521 · see citing works
Crossover from Symplectic to Orthogonal Class in a Two-Dimensional Honeycomb Lattice
Suzuura, H., & Ando, T. (2002) · Physical Review Letters · doi:10.1103/physrevlett.89.266603 · Cited by 502 · see citing works
Dimmock, J. O., Melngailis, I., & Strauss, A. J. (1966) · Physical Review Letters · doi:10.1103/physrevlett.16.1193 · Cited by 461 · see citing works
Blount, E. I. (1962) · Solid state physics · doi:10.1016/s0081-1947(08)60459-2 · Cited by 448 · see citing works
Fractional charge and zero modes for planar systems in a magnetic field
Jackiw, R. (1984) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.29.2375 · Cited by 394 · see citing works
Electronic structure of the semimetals Bi and Sb
Liu, Y., & Allen, R. E. (1995) · Physical Review B · doi:10.1103/physrevb.52.1566 · Cited by 375 · see citing works
Murakami, S., Nagaosa, N., & Zhang, S.-C. (2004) · Physical Review Letters · doi:10.1103/physrevlett.93.156804 · Cited by 346 · see citing works
SU ( 2 ) non-Abelian holonomy and dissipationless spin current in semiconductors
Murakami, S., Nagosa, N., & Zhang, S. (2004) · Physical Review B · doi:10.1103/physrevb.69.235206 · Cited by 342 · see citing works
Groves, S., & Paul, W. M. (1963) · Physical Review Letters · doi:10.1103/physrevlett.11.194 · Cited by 329 · see citing works
Transport properties of Bi-RICH Bi-Sb alloys
Lenoir, B., Cassart, M., Michenaud, J.-P., Scherrer, H., & Scherrer, S. (1996) · Journal of Physics and Chemistry of Solids · doi:10.1016/0022-3697(95)00148-4 · Cited by 281 · see citing works
Relativistic Band Structure and Electronic Properties of SnTe, GeTe, and PbTe
Tung, Y., & Cohen, M. L. (1969) · Physical Review · doi:10.1103/physrev.180.823 · Cited by 250 · see citing works
Fermi Surface of Bi(111) Measured by Photoemission Spectroscopy
Ast, C. R., & Höchst, H. (2001) · Physical Review Letters · doi:10.1103/physrevlett.87.177602 · Cited by 230 · see citing works
Band Structure of Bismuth: Pseudopotential Approach
Golin, S. (1968) · Physical Review · doi:10.1103/physrev.166.643 · Cited by 212 · see citing works
Quasiparticle localization in superconductors with spin-orbit scattering
Senthil, T., & Fisher, M. P. A. (2000) · Physical review. B, Condensed matter · doi:10.1103/physrevb.61.9690 · Cited by 200 · see citing works
Sugawara, K., Sato, T., Souma, S., Takahashi, T., Arai, M., & Sasaki, T. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.046411 · Cited by 158 · see citing works
Physical Realization of the Parity Anomaly in Condensed Matter Physics
Fradkin, E., Dagotto, E. R., & Boyanovsky, D. (1986) · Physical Review Letters · doi:10.1103/physrevlett.57.2967 · Cited by 154 · see citing works
Diophantine equation for the three-dimensional quantum Hall effect
Kohmoto, M., Halperin, B. I., & Wu, Y.-S. (1992) · Physical review. B, Condensed matter · doi:10.1103/physrevb.45.13488 · Cited by 151 · see citing works
Localization in a Quantum Spin Hall System
Onoda, M., Avishai, Y., & Nagaosa, N. (2007) · Physical Review Letters · doi:10.1103/physrevlett.98.076802 · Cited by 143 · see citing works
Electronic Band Structure of Arsenic. I. Pseudopotential Approach
Falicov, L. M., & Golin, S. (1965) · Physical Review · doi:10.1103/physrev.137.a871 · Cited by 138 · see citing works
Electron Spectrum of Gapless Semiconductors
Tsidilkovski, I. M. (1997) · Springer series in solid-state sciences · doi:10.1007/978-3-642-60403-4 · Cited by 93 · see citing works
Stress-Induced Band Gap and Related Phenomena in Gray Tin
Roman, B. J., & Ewald, A. W. (1972) · Physical review. B, Solid state · doi:10.1103/physrevb.5.3914 · Cited by 91 · see citing works
Band Model for Bismuth-Antimony Alloys
Golin, S. (1968) · Physical Review · doi:10.1103/physrev.176.830 · Cited by 91 · see citing works
Energy-Band Structure and Optical Spectrum of Grey Tin
Pollak, F. H., Cardona, M., Higginbotham, C. W., et al. (1970) · Physical review. B, Solid state · doi:10.1103/physrevb.2.352 · Cited by 81 · see citing works
Strong Energy Dependence of the Electron-Phonon Coupling Strength on Bi(100)
Gayone, J. E., Hoffmann, S. V., Li, Z., & Hofmann, P. (2003) · Physical Review Letters · doi:10.1103/physrevlett.91.127601 · Cited by 66 · see citing works
Transport properties of Bi1−xSbx alloy thin films grown on CdTe(111)B
Cho, S., DiVenere, A., Wong, G. K., et al. (1999) · Physical review. B, Condensed matter · doi:10.1103/physrevb.59.10691 · Cited by 63 · see citing works
Electronic structure and Fermi surface of Bi(100)
Hofmann, P., Gayone, J. E., Bihlmayer, G., Koroteev, Y. M., & Chulkov, E. V. (2005) · Physical Review B · doi:10.1103/physrevb.71.195413 · Cited by 62 · see citing works
Moon, C.-Y., & Wei, S. (2006) · Physical Review B · doi:10.1103/physrevb.74.045205 · Cited by 60 · see citing works
Lerner, L. S., Cuff, K. F., & Williams, L. R. (1968) · Reviews of Modern Physics · doi:10.1103/revmodphys.40.770 · Cited by 57 · see citing works
Transport properties of Bi1−xSbx alloy nanowires synthesized by pressure injection
Lin, Y.-M., Cronin, S. B., Rabin, O., et al. (2001) · Applied Physics Letters · doi:10.1063/1.1385800 · Cited by 55 · see citing works
Anomalous currents, induced charge and bound states on a domain wall of a semiconductor
Boyanovsky, D., Dagotto, E. R., & Fradkin, E. (1987) · Nuclear Physics B · doi:10.1016/0550-3213(87)90343-9 · Cited by 54 · see citing works
First-principles calculations of the II-VI semiconductor β-HgS: Metal or semiconductor
Delin, A. (2002) · Physical review. B, Condensed matter · doi:10.1103/physrevb.65.153205 · Cited by 42 · see citing works
Galvanomagnetic properties of single-crystal bismuth-antimony thin films
Morelli, D. T., Partin, D. L., & Heremans, J. P. (1990) · Semiconductor Science and Technology · doi:10.1088/0268-1242/5/3s/056 · Cited by 36 · see citing works
Demler, E., Nayak, C., Kee, H., et al. (2002) · Physical review. B, Condensed matter · doi:10.1103/physrevb.65.155103 · Cited by 31 · see citing works
X. Speculations on the Energy Band Structure of Zinc-Blende-Type Crystals∗
Herman, F. (1955) · Journal of Electronics and Control · doi:10.1080/00207215508961393 · Cited by 28 · see citing works
On the $Z_2$ classification of Quantum Spin Hall ModelsPreprint
Roy, R. (2006) · arXiv (Cornell University) · doi:10.48550/arxiv.cond-mat/0604211
Flagged first, then by citations
Research works printed but not linked: 4
- [14]14. R. Roy, arXiv:cond-mat/0607531͑unpublished͒. · arXiv
- [21]21. T. Fukui and Y. Hatsugai, arXiv:cond-mat/0607484 ͑unpub- lished͒. · arXiv
- [44]44. B. Lenoir, A. Dauscher, X. Devaux, R. Martin-Lopez, Yu. I. Ravich, H. Scher- rer, and S. Scherrer, Proceedings of 15th International Confer- ence on Thermoelectrics, 1996 ͑unpublished͒, pp. 1-13.
- [58]58. G. Nimtz and B. Schicht, Narrow Gap Semiconductors, Springer Tracts in Modern Physics Vol. 98 ͑Springer-Verlag, Berlin, 1983͒.
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<i>Colloquium</i> : Topological insulators
Hasan, M. Z., & Kane, C. L. (2010) · Reviews of Modern Physics · doi:10.1103/revmodphys.82.3045 · Cited by 19,930 · see citing works
Self-citationTopological insulators and superconductors
Qi, X.-L., & Zhang, S.-C. (2011) · Reviews of Modern Physics · doi:10.1103/revmodphys.83.1057 · Cited by 14,394 · see citing works
Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface
Zhang, H., Liu, C.-X., Qi, X.-L., Dai, X., Fang, Z., & Zhang, S.-C. (2009) · Nature Physics · doi:10.1038/nphys1270 · Cited by 6,359 · see citing works
Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator
Fu, L., & Kane, C. L. (2008) · Physical Review Letters · doi:10.1103/physrevlett.100.096407 · Cited by 4,868 · see citing works
Self-citationWan, X., Turner, A. M., Vishwanath, A., & Savrasov, S. Y. (2011) · Physical Review B · doi:10.1103/physrevb.83.205101 · Cited by 4,718 · see citing works
Weyl and Dirac semimetals in three-dimensional solids
Armitage, N. P., Mele, E. J., & Vishwanath, A. (2018) · Reviews of Modern Physics · doi:10.1103/revmodphys.90.015001 · Cited by 4,528 · see citing works
“It is also beneficial (but not required) to have direct gaps at an odd number of points in the Brillouin zone, such as at the L points in Bi1−xSbx (Fu and Kane, 2007), or at the Γ point (particularly in inversion symmetric systems).”
BackgroundInfluentialObservation of a large-gap topological-insulator class with a single Dirac cone on the surface
Xia, Y., Qian, D., Hsieh, D., Wray, L., Pal, A., Lin, H., Bansil, A., Grauer, D., Hor, Y. S., Cava, R. J., et al. (2009) · Nature Physics · doi:10.1038/nphys1274 · Cited by 3,765 · see citing works
“Topological quantum properties are directly probed from the nature of the electronic states on the surface by studying the way surface bands connect the material’s bulk valence and conduction bands in momentum space [5, 6, 7].”
BackgroundResultInfluentialChen, Y. L., Analytis, J. G., Chu, J.-H., Liu, Z. K., Mo, S.-K., Qi, X. L., Zhang, H. J., Lu, D. H., Dai, X., Fang, Z., et al. (2009) · Science · doi:10.1126/science.1173034 · Cited by 3,585 · see citing works
A topological Dirac insulator in a quantum spin Hall phase
Hsieh, D., Qian, D., Wray, L., Xia, Y., Hor, Y. S., Cava, R. J., & Hasan, M. Z. (2008) · Nature · doi:10.1038/nature06843 · Cited by 3,523 · see citing works
“c, Schematic variation of bulk band energies of Bi1−xSbx as a function of x (based on band calculations and on [7, 17]).”
BackgroundInfluentialClassification of topological insulators and superconductors in three spatial dimensions
Schnyder, A. P., Ryu, S., Furusaki, A., & Ludwig, A. W. W. (2008) · Physical Review B · doi:10.1103/physrevb.78.195125 · Cited by 3,322 · see citing works
Quantum spin Hall effect in two-dimensional transition metal dichalcogenides
Qian, X., Liu, J., Fu, L., & Li, J. (2014) · Science · doi:10.1126/science.1256815 · Cited by 2,053 · see citing works
Self-citationTopological Insulator Materials
Ando, Y. (2013) · Journal of the Physical Society of Japan · doi:10.7566/jpsj.82.102001 · Cited by 1,789 · see citing works
<i>Colloquium</i> : Topological band theory
Bansil, A., Lin, H., & Das, T. (2016) · Reviews of Modern Physics · doi:10.1103/revmodphys.88.021004 · Cited by 1,587 · see citing works
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Hsieh, T. H., Lin, H., Liu, J., Duan, W., Bansil, A., & Fu, L. (2012) · Nature Communications · doi:10.1038/ncomms1969 · Cited by 1,449 · see citing works
Self-citationBenalcazar, W. A., Bernevig, B. A., & Hughes, T. L. (2017) · Physical Review B · doi:10.1103/physrevb.96.245115 · Cited by 1,389 · see citing works
Observation of Unconventional Quantum Spin Textures in Topological Insulators
Hsieh, D., Xia, Y., Wray, L., Qian, D., Pal, A., Dil, J. H., Osterwalder, J., Meier, F., Bihlmayer, G., Kane, C. L., et al. (2009) · Science · doi:10.1126/science.1167733 · Cited by 1,239 · see citing works
Self-citationMurakami, S. (2007) · New Journal of Physics · doi:10.1088/1367-2630/9/9/356 · Cited by 1,176 · see citing works
The quantum spin Hall effect and topological insulators
Qi, X.-L., & Zhang, S.-C. (2010) · Physics Today · doi:10.1063/1.3293411 · Cited by 1,283 · see citing works
Topological surface states protected from backscattering by chiral spin texture
Roushan, P., Seo, J., Parker, C. V., Hor, Y. S., Hsieh, D., Qian, D., Richardella, A., Hasan, M. Z., Cava, R. J., & Yazdani, A. (2009) · Nature · doi:10.1038/nature08308 · Cited by 1,052 · see citing works
Hor, Y. S., Williams, A. J., Checkelsky, J. G., Roushan, P., Seo, J., Xu, Q., Zandbergen, H. W., Yazdani, A., Ong, N. P., & Cava, R. J. (2010) · Physical Review Letters · doi:10.1103/physrevlett.104.057001 · Cited by 1,028 · see citing works
Topological bands for ultracold atoms
Cooper, N. R., Dalibard, J., & Spielman, I. B. (2019) · Reviews of Modern Physics · doi:10.1103/revmodphys.91.015005 · Cited by 940 · see citing works
From DFT to machine learning: recent approaches to materials science–a review
Schleder, G. R., Padilha, A. C. M., Acosta, C. M., Costa, M., & Fazzio, A. (2019) · Journal of Physics: Materials · doi:10.1088/2515-7639/ab084b · Cited by 852 · see citing works
Classification of stable three-dimensional Dirac semimetals with nontrivial topology
Yang, B.-J., & Nagaosa, N. (2014) · Nature Communications · doi:10.1038/ncomms5898 · Cited by 864 · see citing works
Quantum spin Hall state in monolayer 1T'-WTe2
Tang, S., Zhang, C., Wong, D., Pedramrazi, Z., Tsai, H.-Z., Jia, C., Moritz, B., Claassen, M., Ryu, H., Kahn, S., et al. (2017) · Nature Physics · doi:10.1038/nphys4174 · Cited by 846 · see citing works
Dirac Line Nodes in Inversion-Symmetric Crystals
Kim, Y., Wieder, B. J., Kane, C. L., & Rappe, A. M. (2015) · Physical Review Letters · doi:10.1103/physrevlett.115.036806 · Cited by 832 · see citing works
Self-citationDzero, M., Sun, K., Galitski, V., & Coleman, P. (2010) · Physical Review Letters · doi:10.1103/physrevlett.104.106408 · Cited by 826 · see citing works
Topological crystalline insulator states in Pb1−xSnxSe
Dziawa, P., Kowalski, B. J., Dybko, K., Buczko, R., Szczerbakow, A., Szot, M., Łusakowska, E., Balasubramanian, T., Wojek, B. M., Berntsen, M. H., et al. (2012) · Nature Materials · doi:10.1038/nmat3449 · Cited by 820 · see citing works
Aharonov–Bohm interference in topological insulator nanoribbons
Peng, H., Lai, K., Kong, D., Meister, S., Chen, Y., Qi, X.-L., Zhang, S.-C., Shen, Z.-X., & Cui, Y. (2010) · Nature Materials · doi:10.1038/nmat2609 · Cited by 780 · see citing works
Teo, J. C. Y., Fu, L., & Kane, C. L. (2008) · Physical Review B · doi:10.1103/physrevb.78.045426 · Cited by 761 · see citing works
Self-citationRecent developments in transport phenomena in Weyl semimetals
Hosur, P., & Qi, X. (2013) · Comptes Rendus. Physique · doi:10.1016/j.crhy.2013.10.010 · Cited by 743 · see citing works
Lin, H., Wray, L. A., Xia, Y., Xu, S., Jia, S., Cava, R. J., Bansil, A., & Hasan, M. Z. (2010) · Nature Materials · doi:10.1038/nmat2771 · Cited by 736 · see citing works
Antiferromagnetic topological insulators
Mong, R. S. K., Essin, A. M., & Moore, J. E. (2010) · Physical Review B · doi:10.1103/physrevb.81.245209 · Cited by 663 · see citing works
Hor, Y. S., Richardella, A., Roushan, P., Xia, Y., Checkelsky, J. G., Yazdani, A., Hasan, M. Z., Ong, N. P., & Cava, R. J. (2009) · Physical Review B · doi:10.1103/physrevb.79.195208 · Cited by 647 · see citing works
Sato, M., & Fujimoto, S. (2009) · Physical Review B · doi:10.1103/physrevb.79.094504 · Cited by 597 · see citing works
Probing Neutral Majorana Fermion Edge Modes with Charge Transport
Fu, L., & Kane, C. L. (2009) · Physical Review Letters · doi:10.1103/physrevlett.102.216403 · Cited by 566 · see citing works
Self-citationQuantum oscillations from surface Fermi arcs in Weyl and Dirac semimetals
Potter, A. C., Kimchi, I., & Vishwanath, A. (2014) · Nature Communications · doi:10.1038/ncomms6161 · Cited by 558 · see citing works
Exploring topological phases with quantum walks
Kitagawa, T., Rudner, M. S., Berg, E., & Demler, E. (2010) · Physical Review A · doi:10.1103/physreva.82.033429 · Cited by 534 · see citing works
Alpichshev, Z., Analytis, J. G., Chu, J.-H., Fisher, I. R., Chen, Y. L., Shen, Z. X., Fang, A., & Kapitulnik, A. (2010) · Physical Review Letters · doi:10.1103/physrevlett.104.016401 · Cited by 539 · see citing works
Quantum anomalous Hall effect and related topological electronic states
Weng, H., Yu, R., Hu, X., Dai, X., & Fang, Z. (2015) · Advances in Physics · doi:10.1080/00018732.2015.1068524 · Cited by 506 · see citing works
Observation of topological order in a superconducting doped topological insulator
Wray, L. A., Xu, S.-Y., Xia, Y., Hor, Y. S., Qian, D., Fedorov, A. V., Lin, H., Bansil, A., Cava, R. J., & Hasan, M. Z. (2010) · Nature Physics · doi:10.1038/nphys1762 · Cited by 464 · see citing works
Xu, Y., Miotkowski, I., Liu, C., Tian, J., Nam, H., Alidoust, N., Hu, J., Shih, C.-K., Hasan, M. Z., & Chen, Y. P. (2014) · Nature Physics · doi:10.1038/nphys3140 · Cited by 430 · see citing works
Kim, H.-J., Kim, K.-S., Wang, J.-F., Sasaki, M., Satoh, N., Ohnishi, A., Kitaura, M., Yang, M., & Li, L. (2013) · Physical Review Letters · doi:10.1103/physrevlett.111.246603 · Cited by 425 · see citing works
Hedgehog spin texture and Berry’s phase tuning in a magnetic topological insulator
Xu, S.-Y., Neupane, M., Liu, C., Zhang, D., Richardella, A., Andrew Wray, L., Alidoust, N., Leandersson, M., Balasubramanian, T., Sánchez-Barriga, J., et al. (2012) · Nature Physics · doi:10.1038/nphys2351 · Cited by 388 · see citing works
Classification of reflection-symmetry-protected topological semimetals and nodal superconductors
Chiu, C.-K., & Schnyder, A. P. (2014) · Physical Review B · doi:10.1103/physrevb.90.205136 · Cited by 339 · see citing works
Topological Semimetals from First Principles
Gao, H., Venderbos, J. W. F., Kim, Y., & Rappe, A. M. (2019) · Annual Review of Materials Research · doi:10.1146/annurev-matsci-070218-010049 · Cited by 263 · see citing works
Non-Fermi-liquid and topological states with strong spin-orbit coupling.
Moon, E.-G., Xu, C., Kim, Y. B., & Balents, L. (2013) · Physical Review Letters · doi:10.1103/physrevlett.111.206401 · Cited by 260 · see citing works
Sato, T., Segawa, K., Kosaka, K., Souma, S., Nakayama, K., Eto, K., Minami, T., Ando, Y., & Takahashi, T. (2011) · Nature Physics · doi:10.1038/nphys2058 · Cited by 235 · see citing works
Schumann, T., Galletti, L., Kealhofer, D. A., Kim, H., Goyal, M., & Stemmer, S. (2018) · Physical Review Letters · doi:10.1103/physrevlett.120.016801 · Cited by 185 · see citing works
Recent Progress in the Study of Topological Semimetals
Bernevig, A., Weng, H., Fang, Z., & Dai, X. (2018) · Journal of the Physical Society of Japan · doi:10.7566/jpsj.87.041001 · Cited by 176 · see citing works
Are the surface Fermi arcs in Dirac semimetals topologically protected?
Kargarian, M., Randeria, M., & Lu, Y.-M. (2016) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1524787113 · Cited by 165 · see citing works
Lattice generalization of the Dirac equation to general spin and the role of the flat band
Dóra, B., Kailasvuori, J., & Moessner, R. (2011) · Physical Review B · doi:10.1103/physrevb.84.195422 · Cited by 159 · see citing works
Bulmash, D., Liu, C.-X., & Qi, X.-L. (2014) · Physical Review B · doi:10.1103/physrevb.89.081106 · Cited by 144 · see citing works
Fidkowski, L., Jackson, T. S., & Klich, I. (2011) · Physical Review Letters · doi:10.1103/physrevlett.107.036601 · Cited by 123 · see citing works
Topological Nodal Cooper Pairing in Doped Weyl Metals.
Li, Y., & Haldane, F. D. M. (2018) · Physical Review Letters · doi:10.1103/physrevlett.120.067003 · Cited by 113 · see citing works
Topological insulators beyond the Brillouin zone via Chern parity
Essin, A. M., & Moore, J. E. (2007) · Physical Review B · doi:10.1103/physrevb.76.165307 · Cited by 98 · see citing works
Weng, H., Dai, X., & Fang, Z. (2014) · MRS Bulletin · doi:10.1557/mrs.2014.216 · Cited by 91 · see citing works
Topological states of quantum matter
Zhang, S.-C. (2008) · Physics · doi:10.1103/physics.1.6 · Cited by 86 · see citing works
Wang, H.-X., Chen, Y., Hang, Z. H., Kee, H.-Y., & Jiang, J.-H. (2017) · npj Quantum Materials · doi:10.1038/s41535-017-0058-z · Cited by 70 · see citing works
Topological materials discovery using electron filling constraints
Chen, R., Po, H. C., Neaton, J. B., & Vishwanath, A. (2018) · Nature Physics · doi:10.1038/nphys4277 · Cited by 47 · see citing works
Liu, X., Wang, Z., Xie, X. C., & Yu, Y. (2011) · Physical Review B · doi:10.1103/physrevb.83.125105 · Cited by 13 · see citing works
Weyl and Dirac Semimetals in Three Dimensional Solids
Armitage, N. P., Mele, E. J., & Vishwanath, A. (2017) · arXiv
“It is also beneficial (but not required) to have direct gaps at an odd number of points in the Brillouin zone, such as at the L points in Bi1−xSbx (Fu and Kane, 2007), or at the Γ point (particularly in inversion symmetric systems).”
BackgroundInfluential
- Authors
- Fu, L., & Kane, C. L.
- Affiliations
- University of Pennsylvania (US)
- ORCID
- Venue
- Physical Review B
- Publisher
- American Physical Society (APS)
- Published
- 2007-07-02
- Volume
- 76
- Issue
- 4
- ISSN
- 1098-0121 (ISSN-L)
- 1550-235X
- OpenAlex
- W2031028572
- Cited by
- 4,252
- References
- 72
- Citations by year
- 2026: 150 · 2025: 219 · 2024: 229 · 2023: 240 · 2022: 267 · 2021: 315 · 2020: 340 · 2019: 284 · 2018: 296 · 2017: 231 · 2016: 291 · 2015: 256 · 2014: 245 · 2013: 260 · 2012: 238
- FWCI
- 40.80
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Physical Review B (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- ScholarlyCommons (University of Pennsylvania) (submittedVersion)
- Scholarly Commons (University of Pennsylvania) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (0605066)
- American Chemical Society Petroleum Research Fund
- Division of Materials Research
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Advanced Condensed Matter Physics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Magnetic properties of thin films
- Keywords
- topological insulators, inversion symmetry, Z2 topological invariant, quantum spin Hall phase, strong topological insulator, weak topological insulator, gapless surface states, Bi1-xSbx, alpha-Sn, HgTe, uniaxial strain, spin-orbit coupling
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