Session Q31: Focus Session: Topological Insulators - Edge States

Electrical control of spin in topological insulators

All-electrical manipulation of electron spin in solids becomes a central issue of quantum information processing and quantum computing. The many previous proposals are based on spin-orbit interactions in semiconductors. Topological insulator, a strong spin-orbit coupling system, make it possible to control the spin transport electrically. Recent calculations proved that external electric fields can drive a HgTe quantum well from normal band insulator phase to topological insulator phase [1]. Since the topological edge states are robust against local perturbation, the controlling of edge states using local fields is a challenging task. We demonstrate that a p-n junction created electrically in HgTe quantum wells with inverted band structure exhibits interesting intraband and interband tunneling processes. We find a perfect intraband transmission for electrons injected perpendicularly to the interface of the p-n junction. The opacity and transparency of electrons through the p-n junction can be tuned by changing the incidence angle, the Fermi energy and the strength of the Rashba spin-orbit interaction (RSOI). The occurrence of a conductance plateau due to the formation of topological edge states in a quasi-one-dimensional p-n junction can be switched on and off by tuning the gate voltage. The spin orientation can be substantially rotated when the samples exhibit a moderately strong RSOI [2]. An electrical switching of the edge-state transport can also be realized using quantum point contacts in quantum spin Hall bars. The switch-on/off of the edge channel is caused by the finite size effect of the quantum point contact and therefore can be manipulated by tuning the voltage applied on the split gate [3,4]. The magnetic ions doped on the surface of 3D TI can be correlated through the helical electrons. The RKKY interaction mediated by the helical Dirac electrons consists of the Heisenberg-like, Ising-like, and Dzyaloshinskii-Moriya (DM)-like terms, which can be tuned by changing the gate voltage. It provides us a new way to control surface magnetism electrically. The gap opened by doped magnetic ions can lead to a short-range Bloembergen-Rowland interaction. The competition among the Heisenberg, Ising, and DM terms leads to rich spin configurations and an anomalous Hall effect on different lattices [4]. There are many proposals for quantum computation scheme are based on the spin in semiconductor quantum dots. Topological insulator quantum dots display a very different behavior with that of conventional semiconductor quantum dots [5]. In sharp contrast to conventional semiconductor quantum dots, the quantum states in the gap of the HgTe QD are fully spin-polarized and show ring-like density distributions near the boundary of the QD and optically dark. The persistent charge currents and magnetic moments, i.e., the Aharonov-Bohm effect, can be observed in such a QD structure. This feature offers us a practical way to detect these exotic ring-like edge states by using the SQUID technique. \\[0pt]Refs: [1] W. Yang, Kai Chang, and S. C. Zhang, Phys. Rev. Lett. 100, 056602 (2008); J. Li and Kai Chang, Appl. Phys. Lett. 95, 222110 (2009). [2] L. B. Zhang, Kai Chang, X. C. Xie, H. Buhmann and L. W. Molenkamp, New J. Phys. 12, 083058 (2010). [3] L. B. Zhang, F. Cheng, F. Zhai and Kai Chang, Phys. Rev. B 83 081402(R) (2011); Z. H. Wu, F. Zhai, F. M. Peeters, H. Q. Xu and Kai Chang, Phys, Rev. Lett. 106, 176802 (2011). [4] J. J. Zhu, D. X. Yao, S. C. Zhang, and Kai Chang, Phys. Rev. Lett. 106, 097201 (2011). [5] Kai Chang, and Wen-Kai Lou, Phys. Rev. Lett. 106, 206802 (2011).   

Conductance of the Quantum Spin Hall Edge in HgTe Quantum Wells

A two-dimensional electron system with band inversion due to spin-orbit interactions can support counterpropagating edge channels, each with one unit of conductance. Unlike the conventional quantum Hall effect, however, these channels can back scatter into each other in the presence of magnetic impurities (or other time-reversal breaking scattering sources). With topgates, we can create 1 um edges of these QSH states and characterize their transmission. In some regions, we are able to see the quantized conductance that is expected of the QSH effect. In other regions, we see a higher resistance corresponding to a nonzero amount of scattering in the channel.   

Inelastic electron backscattering in a generic helical edge channel

We calculate the low-temperature conductance of a generic one-dimensional helical liquid which exists at the edge of a two-dimensional topological insulator (quantum spin Hall insulator). In a generic case, the $S_z$ spin-symmetry is absent, which opens a possibility of single-particle inelastic electron backscattering. We show that although time-reversal invariance is preserved, inelastic backscattering gives rise to a temperature-dependent deviation from the quantized conductance, $\delta G \propto T^4$. In addition, $\delta G$ is sensitive to the position of the Fermi level in the gap of the insulator. We present an effective model for this type of helical liquid and determine its parameters explicitly from numerical solutions of microscopic models for two-dimensional topological insulators in the presence of Rashba spin-orbit coupling.   

Gate controlled Rotating Spin Wave and chiral FFLO Superconducting phases in Quantum Spin Hall edge

We explore the phases exhibited by an interacting quantum spin Hall edge state in the presence of finite chemical potential (applied gate voltage) and spin imbalance (applied magnetic field). We find that the helical nature of the edge state gives rise to orders that are expected to be absent in non-chiral one-dimensional electronic systems. For repulsive interactions, the ordered state has an oscillatory spin texture whose ordering wavevector is controlled by the chemical potential. We analyze the manner in which a magnetic impurity provides signatures of such oscillations. For attractive interactions, finite spin-imbalance, which acts to set up a finite current in unordered QSH edges, results in superconducting order that is characterized by FFLO-type oscillations.   

Phonon induced backscattering in helical edge states

A single pair of helical edge states as realized at the boundary of a quantum spin Hall insulator is known to be robust against elastic single particle backscattering as long as time reversal symmetry is preserved. However, there is no symmetry preventing inelastic backscattering as brought about by phonons in the presence of Rashba spin orbit coupling. In this work, we show that the quantized conductivity of a single channel of helical Dirac electrons is protected even against this inelastic mechanism to leading order. We further demonstrate that this result remains valid even when Coulomb interaction is included in the framework of a helical Tomonaga Luttinger liquid.   

Optically engineering the topological properties of helical edge states

Time-periodic perturbations can be used to engineer topological properties of matter by altering the Floquet band structure. This is demonstrated for the helical edge state of a spin Hall insulator in the presence of monochromatic circularly polarized light. We first demonstrate that the inherent spin structure of the edge state is influenced by the Zeeman coupling and not by the orbital effect. The photocurrent (and the magnetization along the edge) develops a finite, helicity dependent expectation value and turns from dissipationless to dissipative with increasing radiation frequency, signalling a change in the topological properties. The connection with Thouless' charge pumping and non-equilibrium Zitterbewegung is discussed, together with possible experiments. B. Dora, J. Cayssol, F. Simon, and R. Moessner, Optically engineering the topological properties of a spin Hall insulator, arXiv:1105.5963   

Fingerprints of a bidimensional topological insulator in Bismuth nanocontacts

We report on experimental and theoretical work of the electronic transport in Bismuth nanocontacts as created by repeated breaking and indentation using scanning tunneling microscopy techniques. The conductance exhibits a number of unusual features, not shared by normal metals, one of the most striking ones being the presence of plateaus at fractional values of the quantum of conductance at low temperatures. We understand this phenomenon on the basis of the formation of a bilayer of Bismuth (a predicted bidimensional topological insulator) which supports a maximum of a quantum of conductance as expected for its odd number of gapless edge modes. Theoretical transport results based on an atomistic tight-binding model with disorder and spin-orbit coupling permit us associate the fractional-valued plateaus to the final stages of the breaking of the bilayer.   

Spin Josephson effect at the quantum spin Hall edge

We study a spin Josephson effect in a ferromagnetic junction at the quantum spin Hall (QSH) edge. The helical nature of the QSH edge states has striking consequences for the transport properties of such a junction. We derive an expression for the spin current through the junction as a function of the change in magnetization, similar to the current-phase relation of a Josephson junction. We discuss the novel transport properties of the junction that result from fractionally charged excitations hosted by the QSH edge.   

Equality of certain bulk wave functions and edge correlations in $d=2$ and $d=3$

Ground state wavefunctions and gapless edge physics provide two complementary approaches to the study of quantum Hall liquids. Seminal work of Read and Moore establishes a connection between wavefunctions and 1+1 D Conformal Field Theories, which also describe edge states. Here we provide a transparent derivation of the edge correlation-wavefunction {\em equality} for certain topological superconductors - theories with edge states, where charge is not conserved. By studying the 2+1 D $p+ip$ superconductor in some detail, we show that the only necessary ingredient is an approximate Lorentz invariance. We are therefore able to extend the derivation to other dimensions, for example an analogous equality of bulk wavefunctions and edge correlations is derived for superfluid ${}^3He-B$ in $d=3$. A key realization is that ground state wavefunctions can be extracted by considering Euclidean partition functions with a time dependent chemical potential. We also demonstrate that the method works for interacting phases, by studying a ``fractional'' topological superconductor using the parton construction. This connection may help identify novel topological phases in various dimensions.   

Transport on the Surface of Weak Topological Insulators

Weak topological insulators have an even number of Dirac cones in their surface spectrum and are thought to be unstable to disorder, which leads to an insulating surface. Here we argue that the presence of disorder alone will not localize the surface states, rather, the presence of a time-reversal symmetric mass term is required for localization. Through numerical simulations, we show that in the absence of the mass term the surface always flow to a stable metallic phase and the conductivity obeys a one-parameter scaling relation, just as in the case of a strong topological insulator surface. With the inclusion of the mass, the transport properties of the surface of a weak topological insulator follow a two-parameter scaling form.   

Coherent transport of topological insulator surface states

Topological insulators (TIs) are a new state of matter recently predicted theoretically\footnote{C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).}$^,$\footnote{X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78,195424 (2008).} and realized experimentally. In 3D they are characterized by the presence of gapless surface states which exhibit a linear dispersion, typical of Dirac fermions. Moreover, contrary to conventionnal materials, these Dirac cones occur in an odd number of Dirac fermions at the surface: ARPES experiments\footnote{Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature Physics 5, 398 (2009).}$^,$\footnote{Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X.L.Qi,H.J.Zhang,D.H.Lu,X.Dai,Z.Fang,S.C. Zhang, I. R. Fisher, Z. Hussain, and Z.-X. Shen, Science 325, 178 (2009).} have found a single Dirac cone at the surface of Bi2Se3, Bi2Te3. This work focuses on the electronic transport properties calculations in the diffusive limite of a single Dirac cone. Specificities of the TI surface states, like the hexagonal warping coupling are taken into account.   

Spin polarized multi-terminal transport in helical edge states

We propose a three-terminal spin polarized scanning tunneling microscope setup for probing the helical nature of the edge states that appear in the quantum spin Hall system. We show that the three-terminal tunneling conductance depends on the magnetic anisotropy, i.e., the angle between the magnetization of the tip and the local orientation of the electron spin on the edge. We show that chiral injection of an electron into the helical Luttinger liquid is associated with fractionalization of the spin and charge of the injected electron. Finally, we show that the magnetic anisotropy of the probe also leads to Fabry-Perot like two-terminal resonances in charge transport.