Topological insulators with inversion symmetry

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.

[1]  S. Groves,et al.  BAND STRUCTURE OF GRAY TIN , 1963 .

[2]  S. Hoffmann,et al.  Strong energy dependence of the electron-phonon coupling strength on Bi(100). , 2003, Physical review letters.

[3]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[4]  Holger Bech Nielsen,et al.  The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal , 1983 .

[5]  Shuichi Murakami,et al.  Quantum spin Hall effect and enhanced magnetic response by spin-orbit coupling. , 2006, Physical review letters.

[6]  S. Golin.  Band Structure of Bismuth: Pseudopotential Approach , 1968 .

[7]  F. Herman.  X. Speculations on the Energy Band Structure of Zinc-Blende-Type Crystals∗ , 1955 .

[8]  G. Semenoff,et al.  Condensed-Matter Simulation of a Three-Dimensional Anomaly , 1984 .

[9]  P. Kim,et al.  Experimental observation of the quantum Hall effect and Berry's phase in graphene , 2005, Nature.

[10]  Jairo Sinova,et al.  Experimental observation of the spin-Hall effect in a two-dimensional spin-orbit coupled semiconductor system. , 2005 .

[11]  Dagotto,et al.  Physical realization of the parity anomaly in condensed matter physics. , 1986, Physical review letters.

[12]  J. Fleurial.  Proceedings of 15th International Conference on Thermoelectrics , 1996 .

[13]  A. Delin.  First-principles calculations of the II-VI semiconductor β-HgS: Metal or semiconductor , 2002 .

[14]  Liang Fu,et al.  Topological insulators in three dimensions. , 2006, Physical review letters.

[15]  Raffaele Resta,et al.  MACROSCOPIC POLARIZATION IN CRYSTALLINE DIELECTRICS : THE GEOMETRIC PHASE APPROACH , 1994 .

[16]  A. I. Larkin,et al.  Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System , 1980 .

[17]  Robert B. Laughlin,et al.  Quantized Hall conductivity in two-dimensions , 1981 .

[18]  M. E. Ertl,et al.  Narrow gap semiconductors , 1977, Nature.

[19]  I. M. Tsidilkovski.  Electron Spectrum of Gapless Semiconductors , 1997 .

[20]  Haldane,et al.  Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the "parity anomaly" , 1988, Physical review letters.

[21]  D. Morelli,et al.  Galvanomagnetic properties of single-crystal bismuth-antimony thin films , 1990 .

[22]  B. Halperin.  Quantized Hall conductance, current carrying edge states, and the existence of extended states in a two-dimensional disordered potential , 1982 .

[23]  M. Kohmoto,et al.  Topological invariant and the quantization of the Hall conductance , 1985 .

[24]  C. Moon,et al.  Band gap of Hg chalcogenides: Symmetry-reduction-induced band-gap opening of materials with inverted band structures , 2006 .

[25]  D. B. Kaplan.  A Method for simulating chiral fermions on the lattice , 1992 .

[26]  C. Ast,et al.  Fermi surface of Bi(111) measured by photoemission spectroscopy. , 2001, Physical review letters.

[27]  Marvin L. Cohen,et al.  Relativistic Band Structure and Electronic Properties of SnTe, GeTe, and PbTe , 1969 .

[28]  Ove Jepsen,et al.  Electronic structure and thermoelectric properties of bismuth telluride and bismuth selenide , 1997 .

[29]  Y. Hatsugai,et al.  Chern number and edge states in the integer quantum Hall effect. , 1993, Physical review letters.

[30]  L. Sheng,et al.  Quantum spin-Hall effect and topologically invariant Chern numbers. , 2006, Physical review letters.

[31]  I. M. T︠S︡idilʹkovskiĭ.  Electron spectrum of gapless semiconductors , 1996 .

[32]  Shou-Cheng Zhang,et al.  Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells , 2006, Science.

[33]  Wu,et al.  Quantized Hall conductance as a topological invariant. , 1984, Physical review. B, Condensed matter.

[34]  Shuichi Murakami,et al.  Dissipationless Quantum Spin Current at Room Temperature , 2003, Science.

[35]  E. J. Mele,et al.  Z2 topological order and the quantum spin Hall effect. , 2005, Physical review letters.

[36]  Zak,et al.  Berry's phase for energy bands in solids. , 1989, Physical review letters.

[37]  G. Bihlmayer,et al.  Electronic structure and Fermi surface of Bi(100) , 2005 .

[38]  B. J. Roman,et al.  Stress-Induced Band Gap and Related Phenomena in Gray Tin , 1972 .

[39]  E. I. Blount.  Formalisms of Band Theory , 1962 .

[40]  M. Cardona,et al.  Energy-Band Structure and Optical Spectrum of Grey Tin , 1970 .

[41]  C. Kane,et al.  Time Reversal Polarization and a Z 2 Adiabatic Spin Pump , 2006, cond-mat/0606336.

[42]  Wen,et al.  Mean-field theory of spin-liquid states with finite energy gap and topological orders. , 1991, Physical review. B, Condensed matter.

[43]  A. Geim,et al.  Two-dimensional gas of massless Dirac fermions in graphene , 2005, Nature.

[44]  D. Thouless,et al.  Quantized Hall conductance in a two-dimensional periodic potential , 1992 .

[45]  Wu,et al.  Diophantine equation for the three-dimensional quantum Hall effect. , 1992, Physical review. B, Condensed matter.

[46]  Liu,et al.  Electronic structure of the semimetals Bi and Sb. , 1995, Physical review. B, Condensed matter.

[47]  M. Fisher,et al.  Quasiparticle localization in superconductors with spin-orbit scattering , 1999, cond-mat/9906290.

[48]  Spin-Hall insulator. , 2004, Physical review letters.

[49]  L. Balents,et al.  Topological invariants of time-reversal-invariant band structures , 2007 .

[50]  B Andrei Bernevig,et al.  Helical liquid and the edge of quantum spin Hall systems. , 2005, Physical review letters.

[51]  M. Dresselhaus,et al.  Transport properties of Bi1−xSbx alloy nanowires synthesized by pressure injection , 2001 .

[52]  Tsuneya Ando,et al.  Crossover from symplectic to orthogonal class in a two-dimensional honeycomb lattice. , 2002, Physical review letters.

[53]  Alan J. Heeger,et al.  Solitons in polyacetylene , 1979 .

[54]  S. Golin.  Band Model for Bismuth-Antimony Alloys , 1968 .

[55]  L. Falicov,et al.  Electronic Band Structure of Arsenic. I. Pseudopotential Approach , 1965 .

[56]  Jackiw.  Erratum: Fractional charge and zero modes for planar systems in a magnetic field , 1984, Physical review. D, Particles and fields.

[57]  B Andrei Bernevig,et al.  Quantum spin Hall effect. , 2005, Physical review letters.

[58]  Berry curvature on the fermi surface: anomalous Hall effect as a topological fermi-liquid property. , 2004, Physical review letters.

[59]  L. S. Lerner,et al.  Energy-Band Parameters and Relative Band-Edge Motions in the Bi-Sb Alloy System near the Semimetal—Semiconductor Transition , 1968 .

[60]  Walter Kohn,et al.  THEORY OF THE INSULATING STATE , 1964 .

[61]  E. J. Mele,et al.  Quantum spin Hall effect in graphene. , 2004, Physical review letters.

[62]  Minoru Toda,et al.  Springer Series in Solid-State Sciences , 1989 .

[63]  H. Scherrer,et al.  Transport properties of Bi-rich Bi-Sb alloys , 1996 .

[64]  D. Vanderbilt,et al.  Theory of polarization of crystalline solids. , 1993, Physical review. B, Condensed matter.

[65]  J. Dimmock,et al.  Band Structure and Laser Action in Pb x Sn 1-x Te , 1966 .

[66]  E. Dagotto,et al.  Anomalous currents, induced charge and bound states on a domain wall of a semiconductor , 1987 .

[67]  D. D. Awschalom,et al.  Observation of the Spin Hall Effect in Semiconductors , 2004, Science.

[68]  Stability of the quantum spin Hall effect: Effects of interactions, disorder, and Z 2 topology , 2005, cond-mat/0508291.