Kane, Charles L.
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Publication Josephson Current and Noise at a Superconductor/Quantum-Spin-Hall-Insulator/Superconductor Junction(2009-04-28) Fu, Liang; Kane, Charles LWe study junctions between superconductors mediated by the edge states of a quantum-spin-Hall insulator. We show that such junctions exhibit a fractional Josephson effect, in which the current phase relation has a 4π rather than a 2π periodicity. This effect is a consequence of the conservation of fermion parity—the number of electron mod 2—in a superconducting junction and is closely related to the Z2 topological structure of the quantum-spin-Hall insulator. Inelastic processes, which violate the conservation of fermion parity, lead to telegraph noise in the equilibrium supercurrent. We predict that the low-frequency noise due these processes diverges exponentially with temperature T as T→0. Possible experiments on HgCdTe quantum wells will be discussed.Publication Topological Defects and Gapless Modes in Insulators and Superconductors(2010-09-22) Teo, Jeffrey C.Y.; Kane, Charles L.We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians.We consider Hamiltonians H(k,r) that vary slowly with adiabatic parameters r surrounding the defect and belong to any of the ten symmetry classes defined by time-reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three-dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions, and helical Majorana fermions, as well as zero-dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes.Publication Theoretical Investigation of the Evolution of the Topological Phase of Bi2Se3 under Mechanical Strain(2011-08-19) Young, Steve M.; Chowdhury, Sugata; Mele, Eugene J.; Kane, Charles L; Rappe, Andrew M; Walter, Eric J.The topological insulating phase results from inversion of the band gap due to spin-orbit coupling at an odd number of time-reversal symmetric points. In Bi2Se3, this inversion occurs at the Γ point. For bulk Bi2Se3, we have analyzed the effect of arbitrary strain on the Γ point band gap using density functional theory. By computing the band structure both with and without spin-orbit interactions, we consider the effects of strain on the gap via Coulombic interaction and spin-orbit interaction separately. While compressive strain acts to decrease the Coulombic gap, it also increases the strength of the spin-orbit interaction, increasing the inverted gap. Comparison with Bi2Te3 supports the conclusion that effects on both Coulombic and spin-orbit interactions are critical to understanding the behavior of topological insulators under strain, and we propose that the topological insulating phase can be effectively manipulated by inducing strain through chemical substitution.Publication Topological Insulators with Inversion Symmetry(2007-07-07) Fu, Liang; Kane, Charles LTopological 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.Publication Topology, Delocalization via Average Symmetry and the Symplectic Anderson Transition(2012-12-14) Fu, Liang; Kane, Charles LA field theory of the Anderson transition in two-dimensional disordered systems with spin-orbit interactions and time-reversal symmetry is developed, in which the proliferation of vortexlike topological defects is essential for localization. The sign of vortex fugacity determines the Z2 topological class of the localized phase. There are two distinct fixed points with the same critical exponents, corresponding to transitions from a metal to an insulator and a topological insulator, respectively. The critical conductivity and correlation length exponent of these transitions are computed in an N = 1 - ε expansion in the number of replicas, where for small epsilon; the critical points are perturbatively connected to the Kosterlitz-Thouless critical point. Delocalized states, which arise at the surface of weak topological insulators and topological crystalline insulators, occur because vortex proliferation is forbidden due to the presence of symmetries that are violated by disorder, but are restored by disorder averaging.Publication Topological Insulators in Three Dimensions(2007-03-07) Kane, Charles L; Fu, Liang; Mele, Eugene JWe study three-dimensional generalizations of the quantum spin Hall (QSH) effect. Unlike two dimensions, where a single Z2 topological invariant governs the effect, in three dimensions there are 4 invariants distinguishing 16 phases with two general classes: weak (WTI) and strong (STI) topological insulators. The WTI are like layered 2D QSH states, but are destroyed by disorder. The STI are robust and lead to novel ‘‘topological metal’’ surface states. We introduce a tight binding model which realizes the WTI and STI phases, and we discuss its relevance to real materials, including bismuth.Publication Interface Between Topological and Superconducting Qubits(2011-03-28) Jiang, Liang; Kane, Charles L; Preskill, JohnWe propose and analyze an interface between a topological qubit and a superconducting flux qubit. In our scheme, the interaction between Majorana fermions in a topological insulator is coherently controlled by a superconducting phase that depends on the quantum state of the flux qubit. A controlled-phase gate, achieved by pulsing this interaction on and off, can transfer quantum information between the topological qubit and the superconducting qubit.Publication Surface State Magnetization and Chiral Edge States on Topological Insulators(2013-01-25) Kane, Charles L; Zhang, Fan; Mele, Eugene J.We study the interaction between a ferromagnetically ordered medium and the surface states of a topological insulator with a general surface termination that were identified recently [F. Zhang et al.Phys. Rev. B 86 081303(R) (2012)]. This interaction is strongly crystal face dependent and can generate chiral states along edges between crystal facets even for a uniform magnetization. While magnetization parallel to quintuple layers shifts the momentum of the Dirac point, perpendicular magnetization lifts the Kramers degeneracy at any Dirac points except on the side face, where the spectrum remains gapless and the Hall conductivity switches sign. Chiral states can be found at any edge that reverses the projection of the surface normal to the stacking direction of quintuple layers. Magnetization also weakly hybridizes noncleavage surfaces.Publication One-dimensional Diffusion-limited Relaxation of Photoexcitations in Suspensions of Single-walled Carbon Nanotubes(2006-07-26) Mele, Eugene J; Kane, Charles L; Russo, Richard M; Therien, Michael; Luzzi, David E; Rubtsov, I. VWe report pump-probe transient absorption spectroscopy on carbon nanotubes with a high initial excitation density. We find that the recovery of the ground state optical absorption is well described by a 1/√t relaxation, indicating that the long time population relaxation is controlled by one-dimensional diffusion limited two body recombination.Publication Colloquium: Topological Insulators(2010-11-08) Hasan, M. Zahid; Kane, Charles L.Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator but have protected conducted states on their edge or surface. These states are possible due to the combination of spin-orbit interactions and time-reversal symmetry. The two-dimensional (2D) topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A three-dimensional (3D) topological insulator supports novel spin-polarized 2D Dirac fermions on its surface. In this Colloquium the theoretical foundation for topological insulators and superconductors is reviewed and recent experiments are described in which the signatures of topological insulators have been observed. Transport experiments on HgTe/CdTe quantum wells are described that demonstrate the existence of the edge states predicted for teh quantum spin hall insulator. Experiments on Bi1-xSbx, Bi<2Se3, Bi2Te3 and Sb2Te3 are then discussed that establish these materials as 3D topological insulators and directly probe the topology of their surface states. Exotic states are described that can occur at the surface of a 3D topological insulator due to an induced energy gap. A magnetic gap leads to a novel quantum Hall state that gives rise to a topological magnetoelectric effect. A superconducting energy gap leads to a state that supports Majorana fermions and may provide a new venue for realizing proposals for topological quantum computation. Prospects for observing these exotic states are also discussed, as well as other potential device applications of topological insulators.