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 Z2 topological invariants, which characterize the ground state. In two dimensions, there is a single Z2 invariant that distinguishes the ordinary insulator from the quantum spin-Hall phase. In three dimensions, there are four Z2 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 threedimensional 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 Z2 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 Bi1−xSbx as well as α-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. Disciplines Physical Sciences and Mathematics | Physics Comments Suggested Citation: L. Fu and C.L. Kane (2007) Topological insulators with inversion symmetry. Physical Review B 76, 045302. © 2007 The American Physical Society http://dx.doi.org/10.1103/PhysRevB.76.045302 This journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/150 Topological insulators with inversion symmetry Liang Fu and C. L. Kane Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA Received 14 November 2006; revised manuscript received 17 January 2007; published 2 July 2007 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 Z2 topological invariants, which characterize the ground state. In two dimensions, there is a single Z2 invariant that distinguishes the ordinary insulator from the quantum spin-Hall phase. In three dimensions, there are four Z2 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 threedimensional 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 Z2 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 Bi1−xSbx as well as -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. DOI: 10.1103/PhysRevB.76.045302 PACS number s : 73.43. f, 72.25.Hg, 73.20. r, 85.75. d