Topological Insulators

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THE NEW STATE OF MATTER - TOPOLOGICAL INSULATORS AND QUANTUM PHASE TRANSITION

THE NEW STATE OF MATTER - TOPOLOGICAL INSULATORS AND QUANTUM PHASE TRANSITION

The subject of topological insulators and topological superconductors is now one of the most active fields of research in condensed matter physics, developing at a rapid pace. Theorists have systematically classified topological states in all dimensions. For future progress on the theoretical side, the most important outstanding problems include interaction and disorder effects, realistic predictions for topological Mott insulator materials, a deeper understanding of fractional topological insulators and realistic predictions for materials realizations of such states, the effective field theory description of the topological superconducting state, and realistic materials predictions for topological superconductors. On the experimental part, the most important task is to grow materials with sufficient purity so that the bulk insulating behavior can be reached, and to tune the Fermi level close to the Dirac point of the surface state.
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Scanning Tunnelling Spectroscopic Studies of Dirac Fermions in Graphene and Topological Insulators

Scanning Tunnelling Spectroscopic Studies of Dirac Fermions in Graphene and Topological Insulators

An exciting new development in condensed matter physics over the last decade is the beautiful realization of topological field theories [18] in strongly correlated electronic systems, where topological field theories are shown to provide a classification of order due to macroscopic entanglement that is independently of symmetry breaking [19]. The FQH state is the first known example of such a quantum state that exhibits no spontaneous broken symmetry, and its properties depend only on its topology rather than geometry, which is topologically distinct from all other quantum states classified by broken symmetry. Recently, the quantum spin Hall (QSH) states and the topological insulators (TI) have emerged as a new class of topological states and have stimulated intense research activities [2]. As described in Section 1, one of the novel properties associated with the TI is the presence of a Dirac spectrum of chiral low-energy excitations, which is a salient feature of the Dirac materials. In the case of 3D-STI, an odd number of massless Dirac cones in their SS are ensured by the Z 2 topological invariant of the fully gapped bulk
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Electronic States in Disordered Topological Insulators

Electronic States in Disordered Topological Insulators

In connection to topological insulator possessing chiral edge states or surface states without backscatterings, it is one of the central questions that how these topologically protected states behave in the presence of non-magnetic and magnetic impurities. Moreover, as 3d topological insulators may contain unexpected bulk modes [53] from doping to shift the Fermi energy, how the conventional picture of the Anderson localization applies to the system with helical surface states as well as bulk modes and impurities is a question awaiting resolution. Lastly, physicists numerically found that a trivial insulator can undergo a phase transition into a topological insulator by introducing impurities that renormalize the chemical potential as well as the mass term of the Hamiltonian. The Landauer-Buttiker type conductance calculation shows a well-quantized chiral edge transport in 2d system [54], and for the 3d case by Guo, Resenberg et al.[55] the appearance of a strong topological insulating phase is verified in terms of quantized conductance and the Witten effect.
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Ferromagnetism of magnetically doped topological insulators in CrxBi2− xTe3 thin films

Ferromagnetism of magnetically doped topological insulators in CrxBi2− xTe3 thin films

Topological insulators (TIs) with dispersionless surface state attract attentions both for fundamental science and potential applications. The reason of this nontrivial surface state is their bulk band inversion and massless Dirac-cone- like surface state arising from the strong spin-orbit cou- pling. 1 This surface state is protected by the time reversal symmetry which prohibits the backscattering on non- magnetic impurities and induces a weak antilocalization of Dirac fermions. 2

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Quantum effects in electrical transport properties of Bismuth chalcogenides Topological Insulators

Quantum effects in electrical transport properties of Bismuth chalcogenides Topological Insulators

Topological insulators (TIs) are celebrated novel materials with a peculiar band structure that features a gap for bulk states but may also host spin-polarised surface conducting states, topo- logically protected from scattering mechanisms that do not break time-inversion symmetry [1]. A representative family of such materials are bismuth chalcogenides, that have attracted a lot of interest also for their good properties as thermoelectric compounds [2].

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Acoustic realization of quadrupole topological insulators

Acoustic realization of quadrupole topological insulators

states of matter in classical wave systems [1-5]. The robustness of the topological systems against disorders and the associated one-way edge states provide new opportunities for manipulating classical waves. Besides the analogues of conventional topological insulators [6-24] and topological semimetals [25-34], the recent interest on higher-order topological insulators (HOTIs) [35,36] has opened a new direction of topological phases in classical systems [37-53]. Different from the conventional topological insulators, in which the D-dimensional bulk is gapped and the topological invariant counts the number of gapless modes hosted on the (D-1)-dimensional boundaries of the sample, the HOTIs are a new family of topological phases of matter that goes beyond the conventional bulk-boundary correspondence principle. For example, unlike conventional two-dimensional (2D) topological insulators, a 2D second-order HOTI does not exhibit gapless one-dimensional (1D) topological edge states, but instead has topologically protected zero-dimensional (0D) corner states. Due to the flexibility in sample design, HOTIs have been soon implemented in mechanical [37,45], photonic [38,40-44], electrical circuits [39,46,47] and acoustic [48-53] systems.
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Disorder Effects on Electron Transport in Nanocrystal Assemblies and Topological Insulators

Disorder Effects on Electron Transport in Nanocrystal Assemblies and Topological Insulators

The discovery and classification of distinctive electronic phases of matter has always been an important topic in condensed matter physics. As we know, the behavior of electrons in different materials varies dramatically. The electrical insulator, for instance, is one of the most basic electronic phases of matter, characterized by an energy gap for electronic excitations. Recent work has, however, now uncovered a new class of materials termed topological insulators (TIs) [12, 13, 14, 15, 16]. The most distinguishing feature of these insulators is that they can insulate on the inside but conduct on the outside - analogous to a block of wood covered with a layer of copper, except that the material is actually the same throughout. Due to strong spin-orbit coupling, electrons that move along the surface have their spin locked perpendicular to their momentum (spin- momentum locking) (see Fig. 1.2(a) for the band structure of a typical 3D TI [17]). Furthermore, these gapless surface states are topologically protected against disruptions such as defects, chemical passivation, and thermal fluctuations. First predicted and then discovered experimentally in semiconducting alloy Bi 1 − x S x , 3D TI has attracted
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Theoretical study of topological insulators : models and materials

Theoretical study of topological insulators : models and materials

distinguish between the two types of topological insulators. The experi­ ment involves placing m agnetic im purities along one of the edges of the two- dimensional sample, for instance, by using the tip of a scamiing tunneling microscope. The im p iu ities’ spin will, in general, be aligned in arb itrary directions. The illum ination with low-frequency polarized infrared light can however induce their alignment. This has l)een dem onstrated, for instance, for Mil im purities in CdTe [58]. The infrared pulse im parts a m om entum to align th e im purity s])ins. which subsequently relax back to th eir random orientations. Tlie Andreev reflection coefficient can then be m easured as a function of tim e, and this can l)c related to the inclination angle 0 of the im purity spin S. The dependence of R-^ on 0 is shown in Fig. 4.5. As the si)in rotates tow ards th e 2 : direction. R ‘^ retu rn s back to the perfectly
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Topological Insulators: Tight Binding Models and Surface States

Topological Insulators: Tight Binding Models and Surface States

Many solid materials are crystalline in nature, which means that their constituent atoms are ordered in regular and repeating patterns at the microscopic scale. These patterns are often arranged periodically, and hence form a so-called crystal structure. If one wants to describe a sample of a crystalline solid of macroscopic size, it is a powerful idealization to assume that this crystal extends infinitely far in all directions so that the system acquires a certain translational symmetry. Macroscopic samples typically consist of 10 20 − 10 24 atoms, which means that for an electron in the bulk of the sample, this is a very good approximation. On the other hand, if one is interested in effects taking place at the edge of the sample, one has to take a different approach. The electronic structure inside the bulk and at the boundary are not entirely unrelated however, as there is an important theorem for topological insulators known as the bulk-boundary correspondence. This result roughly states that the Z 2 -index determines the phenomena that take place at the surface, and it
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Scanning Tunneling Microscopy Studies of Topological Insulators Grown by Molecular Beam Epitaxy

Scanning Tunneling Microscopy Studies of Topological Insulators Grown by Molecular Beam Epitaxy

STM is a surface sensitive technique based on the concept of quantum tunneling. The atomically resolved structure of a surface can be obtained by monitoring the tunneling current in an STM. Besides its unprecedented capability to study surface topography, information on electronic structure at a given location of a sample can also be obtained. This type of measurement is called scanning tunneling spectroscopy (STS) and typically results in a plot of the local density of states (LDOS) as a function of energy of the sample. It has been widely used in the characterization of metals, semiconductors and superconductors. Applying it to study of topological surface states, many intriguing properties have recently been unveiled. In this paper, we give an overview on the recent progress of STM study of topological insulators.
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Coupling Organic Molecules to Topological Insulators.

Coupling Organic Molecules to Topological Insulators.

Topological insulators (TIs) are a prime candidate for studying interface effects due to their spin- textured surface states on a bulk insulating material.[4, 133] These topological surface states (TSS) are protected by time-reversal symmetry making them robust against defects and perturbations that preserve this symmetry.[134] Studies of the organic-metal interface have produced exciting and unexpected results, such as beating the Stoner criterion[135] and the Kondo effect[136]. Modification of the magnetic and electronic properties of TIs has mainly focused on the technique of bulk or surface doping leaving room for organic molecules to control TIs.[42, 111, 137] The organic-TI interface is a unique opportunity to probe the coupling of organic molecules to a bulk insulating material with a spin-momentum locked surface state. Transition-metal phthalocyanine molecules (MPcs) are promising molecular candidates considering their success in creating interesting spin-interfaces with metals.[138-143] The ability to tune the central metal atom offers the potential to selectively couple MPcs to the orbitals of the substrate.[54, 144, 145]
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Andreev reflection in two dimensional topological insulators with either conserved or broken time reversal symmetry

Andreev reflection in two dimensional topological insulators with either conserved or broken time reversal symmetry

edge and two right movers at the other, exactly as the integer quantum Hall states. We find that for both these cases, the edge modes lead to a perfect Andreev reflection for electrons with energy smaller than the superconducting gap. In fact, in both cases the edge modes are perfectly Andreev reflected. Normal reflection, where an incident particle is reflected back without being converted into its antiparticle, is completely suppressed for the edge states as long as the Fermi energy lies in the bulk gap, as we have verified numerically. These findings are consistent with recent theoretical and experimental studies for time-reversal symmetric topological insulators. 22,23 Note that
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Statistical topological insulators

Statistical topological insulators

These two examples share one common trait. In order for the surface to avoid localization, the disordered ensemble must be invariant under a certain symmetry: translation for a weak TI or time-reversal for a strong TI with a random magnetic field. We show that this property defines a broad class of systems, which we call statistical topological insulators (STI). An STI is an ensemble of disordered systems belonging to the same symmetry class. This ensemble, as a whole, also has to be invariant under an extra symmetry, which we call statistical symmetry since it is not respected by single ensemble elements. These elements have surfaces pinned to the middle of a topological phase transition and protected from localization due to the combined presence of the statistical symmetry and the symmetry of each element, if any. For example, for a weak TI the statistical symmetry is translation, while the symmetry of each element is time reversal.
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Electronic Transport Properties of Two-Dimensional Semiconductors and Topological Insulators with Device Applications.

Electronic Transport Properties of Two-Dimensional Semiconductors and Topological Insulators with Device Applications.

Topological insulators (TI), a new quantum state of matter, have been identified as promising ma- terials for a new platform for coherent spin-polarized electronics. Three dimensional topological insulators have been theoretically predicted [ FK07; MB07; HK10; QZ11 ] and experimentally ob- served [ Hsi08; Xia09; Hsi09 ] . As a consequence of strong spin orbital coupling and band inversion, TI possess unique gapless surface states with linear energy-momentum dispersion characteristics that coexist with gapped bulk states. The new conducting surface states can be well described by Dirac equation. The spin momentum locking in such system enables the spin polarized current, which gives TIs the potential for applications in the fabrication of fault tolerant and spintronic devices. Moreover, the left handed helical spin texture of the surface states which generates a π - Berry phase when an electronic wave function completes one rotation adiabatically around a scattering center. Due to the spin momentum locking and π - Berry phase transition of reversed trajectory, electron backscattering by nonmagnetic impurity is suppressed. The basic unit for fabrication of logic gates for quantum computing, the Majorana particle, is predicted to be observable at the interface of a TI and a superconductor[FK08; SL13].
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Theory of non-Abelian Fabry-Perot interferometry in topological insulators

Theory of non-Abelian Fabry-Perot interferometry in topological insulators

共 Received 31 December 2009; revised manuscript received 20 April 2010; published 11 May 2010 兲 Interferometry of non-Abelian edge excitations is a useful tool in topological quantum computing. In this paper we present a theory of a non-Abelian edge-state interferometer in a three-dimensional topological insulator brought in proximity to an s-wave superconductor. The non-Abelian edge excitations in this system have the same statistics as in the previously studied 5/2 fractional quantum-Hall 共 FQH 兲 effect and chiral p-wave superconductors. There are however crucial differences between the setup we consider and these systems, like the need for a converter between charged and neutral excitations and the neutrality of the non-Abelian excitations. These differences manifest themselves in a temperature scaling exponent of −7 / 4 for the conductance instead of −3 / 2 as in the 5/2 FQH effect.
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Random-matrix theory and stroboscopic models of topological insulators and superconductors

Random-matrix theory and stroboscopic models of topological insulators and superconductors

A fundamental difference between the quantum Hall and the quan- tum spin Hall effect is that the topological invariant is restricted to the values 0 and 1 in the latter case, meaning either a pair of topologically protected edge states or none at all. The emphasize is here on the topo- logical protection - further edge states may exist but would not be stable. In contrast the quantum Hall effect can realize any integer number of topologically protected edge modes, but they all propagate in the same direction. This ensures the topological protection: if there was a channel in the opposite direction, left and right moving states could scatter into each other and would hybridize, leading to a gap in the spectrum of edge states. In the quantum spin Hall effect this is prevented for a sin- gle pair of time reversed modes since the so-called Kramers degeneracy of the crossing point in the Brillouin zone (see lower panel of Fig. 1.4) is protected by time-reversal symmetry.
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Topological Insulators and Superconductors

Topological Insulators and Superconductors

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共 k , r 兲 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 Majo- rana 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.
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Universal probes of two-dimensional topological insulators: Dislocation and pi flux

Universal probes of two-dimensional topological insulators: Dislocation and pi flux

The topological band insulators (TBIs) in two and three dimensions have attracted a great deal of interest in both theoretical and experimental condensed matter physics [1,2]. An extensive classification of topological phases in free fermion systems with a bulk gap, based on time- reversal symmetry (TRS) and particle-hole symmetry (PHS), has been provided [3], and it defines the so-called tenfold way. This however relies on the spatial continuum limit while TBIs actually need a crystal lattice which, in turn, breaks the translational symmetry. Therefore, the question arises whether the crystal lattice may give rise to an additional subclass of topological phases. The simple M - B model introduced for the two-dimensional (2D) HgTe quantum spin Hall insulator [4] gives away a generic wisdom in this regard. Depending on its parameters, this model describes topological phases which are in a thermo- dynamic sense distinguishable: their topological nature is characterized by a Berry phase Skyrmion lattice in the extended Brillouin zone (BZ), where the sites of this lattice coincide with the reciprocal lattice vectors (‘‘ phase’’) or with the TRS points ( , ) (‘‘ M phase’’).
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Spin pumping and inelastic electron tunneling spectroscopy in topological insulators

Spin pumping and inelastic electron tunneling spectroscopy in topological insulators

cartoons describe a spin-flip scattering event. A right-going electron with up-spin direction (left panel) is inelastically backscattered by the magnetic impurity. In the process both the electron and the impurity spins are reversed (right panel). Note that, given the topological nature of the ribbon, spin flip forbids electron transmission as the edge presenting a right-going spin-up state does not possess a right-going spin-down one.

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Interfaces of topological insulators and superconductors

Interfaces of topological insulators and superconductors

In Section 2.1, we will first discuss the rather simple resistively shunted junction model that has been used to model S/N/S junction. This model does not take MAR into account. The current through a one dimentional S/N/S junction as a result of MAR was first modelled by Averin- Bardas in 1995. Their model is discussed in Section 2.2. As a next step, we are interested in a junction with a topological insulator (TI) in the middle. When a TI is brought into contact with a superconductor, its surface becomes superconducting as well via the proximity effect. This combination of spin-momentum locking and superconductivity allows symmetry protected surface states to host Majorana particles; which has been a hot topic for the past decades. The one dimensional S/TI/S junction and its relation to Majorana particles were considered by Badiane, Houzet and Meyer in 2011. A brief review of their work is given in Section 2.3.
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