Landau quantization of topological surface states in Bi2Se3

Edge states and integer quantum Hall effect in topological insulator thin films

The integer quantum Hall effect is a topological state of quantum matter in two dimensions, and has recently been observed in three-dimensional topological insulator thin films. Here we study the Landau levels and edge states of surface Dirac fermions in topological insulators under strong magnetic field. We examine the formation of the quantum plateaux of the Hall conductance and find two different patterns, in one pattern the filling number covers all integers while only odd integers in the other. We focus on the quantum plateau closest to zero energy and demonstrate the breakdown of the quantum spin Hall effect resulting from structure inversion asymmetry. The phase diagrams of the quantum Hall states are presented as functions of magnetic field, gate voltage and chemical potential. This work establishes an intuitive picture of the edge states to understand the integer quantum Hall effect for Dirac electrons in topological insulator thin films.

Ultrafast electron dynamics at the Dirac node of the topological insulator Sb2Te3

Topological insulators (TIs) are a new quantum state of matter. Their surfaces and interfaces act as a topological boundary to generate massless Dirac fermions with spin-helical textures. Investigation of fermion dynamics near the Dirac point (DP) is crucial for the future development of spintronic devices incorporating topological insulators. However, research so far has been unsatisfactory because of a substantial overlap with the bulk valence band and a lack of a completely unoccupied DP. Here, we explore the surface Dirac fermion dynamics in the TI Sb2Te3 by time- and angle-resolved photoemission spectroscopy (TrARPES). Sb2Te3 has an in-gap DP located completely above the Fermi energy (EF). The excited electrons in the upper Dirac cone stay longer than those below the DP to form an inverted population. This was attributed to a reduced density of states (DOS) near the DP.

A Roadmap for Controlled Production of Topological Insulator Nanostructures and Thin Films

The group V-VI chalcogenide semiconductors (Bi2 Se3 , Bi2 Te3 , and Sb2 Te3 ) have long been known as thermoelectric materials. Recently, they have been once more generating interest because Bi2 Se3 , Bi2 Te3 and Sb2 Te3 have been crowned as 3D topological insulators (TIs), which have insulating bulk gaps and metallic Dirac surface states. One big challenge in the study of TIs is the lack of high-quality materials with few defects and insulating bulk states. To manifest the topological surface states, it is critical to suppress the contribution from the bulk carriers. Controlled production of TI nanostructures that have a large surface-to-volume ratio is an efficient way to reduce the bulk conductance and to significantly enhance the topological surface conduction. In this review article, the recent progress on the preparation of TI nanostructures is highlighted. Basic production methods for TI nanostructures are introduced in detail. Furthermore, several specific production approaches to reduce the residual bulk carriers from defects are summarized. Finally, the progress and the prospects of the production of TI-based heterostructures, which hold promise in both fundamental study and novel applications are discussed.

Probing Dirac fermion dynamics in topological insulator Bi2Se3 films with a scanning tunneling microscope

Scanning tunneling microscopy and spectroscopy have been used to investigate the femtosecond dynamics of Dirac fermions in the topological insulator Bi2Se3 ultrathin films. At the two-dimensional limit, bulk electrons become quantized and the quantization can be controlled by the film thickness at a single quintuple layer level. By studying the spatial decay of standing waves (quasiparticle interference patterns) off steps, we measure directly the energy and film thickness dependence of the phase relaxation length lϕ and inelastic scattering lifetime τ of topological surface-state electrons. We find that τ exhibits a remarkable (E - EF)(-2) energy dependence and increases with film thickness. We show that the features revealed are typical for electron-electron scattering between surface and bulk states.

Nanoscale determination of the mass enhancement factor in the lightly doped bulk insulator lead selenide

Bismuth chalcogenides and lead telluride/selenide alloys exhibit exceptional thermoelectric properties that could be harnessed for power generation and device applications. Since phonons play a significant role in achieving these desired properties, quantifying the interaction between phonons and electrons, which is encoded in the Eliashberg function of a material, is of immense importance. However, its precise extraction has in part been limited due to the lack of local experimental probes. Here we construct a method to directly extract the Eliashberg function using Landau level spectroscopy, and demonstrate its applicability to lightly doped thermoelectric bulk insulator PbSe. In addition to its high energy resolution only limited by thermal broadening, this novel experimental method could be used to detect variations in mass enhancement factor at the nanoscale level. This opens up a new pathway for investigating the local effects of doping and strain on the mass enhancement factor.

Electron–phonon coupling influences the thermal and electronic properties of many solid materials. Zeljkovic et al. now combine Landau level spectroscopy and scanning tunnelling microscopy to extract quantitative information on electron–phonon coupling in the insulator lead selenide.

Substitution-induced spin-splitted surface states in topological insulator (Bi 1-x Sbx)2Te3

We present a study on surface states of topological insulator (Bi 1-x Sbx)2Te3 by imaging quasiparticle interference patterns (QPI) using low temperature scanning tunneling microscope. Besides the topological Dirac state, we observed another surface state with chiral spin texture within the conduction band range. The quasiparticle scattering in this state is selectively suppressed. Combined with first-principles calculations, we attribute this state to a spin-splitted band induced by the substitution of Bi with Sb atoms. Our results demonstrate that the coexistence of topological order and alloying may open wider tunability in quantum materials.

Strong magneto-optical effects due to surface states in three-dimensional topological insulators

We show that a thin film of a three-dimensional topological insulator such as Bi(2)Se(3)or Bi(2)Te(3) can exhibit strong linear and nonlinear magneto-optical effects in a transverse magnetic field. In particular, one can achieve an almost complete circular polarization of an incident terahertz or mid-infrared radiation and an efficient four-wave mixing.

Topological properties determined by atomic buckling in self-assembled ultrathin Bi(110)

Topological insulators (TIs) are a new type of electronic materials in which the nontrivial insulating bulk band topology governs conducting boundary states with embedded spin-momentum locking. Such edge states are more robust in a two-dimensional (2D) TI against scattering by nonmagnetic impurities than in its three-dimensional (3D) variant, because in 2D the two helical edge states are protected from the only possible backscattering. This makes the 2D TI family a better candidate for coherent spin transport and related applications. While several 3D TIs are already synthesized experimentally, physical realization of 2D TI is so far limited to hybrid quantum wells with a tiny bandgap that does not survive temperatures above 10 K. Here, combining first-principles calculations and scanning tunneling microscopy/spectroscopy (STM/STS) experimental studies, we report nontrivial 2D TI phases in 2-monolayer (2-ML) and 4-ML Bi(110) films with large and tunable bandgaps determined by atomic buckling of Bi(110) films. The gapless edge states are experimentally detected within the insulating bulk gap at 77 K. The band topology of ultrathin Bi(110) films is sensitive to atomic buckling. Such buckling is sensitive to charge doping and could be controlled by choosing different substrates on which Bi(110) films are grown.

Optical conductivity of topological insulator thin films in a quantizing magnetic field

We determine the optical response of topological insulator thin films in the presence of a quantizing, external magnetic field. We explicitly take into account hybridization between the states of top and bottom surface. The interplay between hybridization and Zeeman energies gives rise to topological and normal insulator phases and phase transitions between them. The optical response in the two phases and at the phase transition point is investigated. We show that the difference in magneto-optical response can be used to distinguish the topological phase from the normal phase of the system. Further, the optical response also allows us to determine the gap generated by hybridization between top and bottom surface states of topological insulator thin films.

Topological crystalline insulator nanostructures

Topological crystalline insulators are topological insulators whose surface states are protected by the crystalline symmetry, instead of the time reversal symmetry. Similar to the first generation of three-dimensional topological insulators such as Bi₂Se₃ and Bi₂Te₃, topological crystalline insulators also possess surface states with exotic electronic properties such as spin-momentum locking and Dirac dispersion. Experimentally verified topological crystalline insulators to date are SnTe, Pb₁-xSnxSe, and Pb₁-xSnxTe. Because topological protection comes from the crystal symmetry, magnetic impurities or in-plane magnetic fields are not expected to open a gap in the surface states in topological crystalline insulators. Additionally, because they have a cubic structure instead of a layered structure, branched structures or strong coupling with other materials for large proximity effects are possible, which are difficult with layered Bi₂Se₃ and Bi₂Te₃. Thus, additional fundamental phenomena inaccessible in three-dimensional topological insulators can be pursued. In this review, topological crystalline insulator SnTe nanostructures will be discussed. For comparison, experimental results based on SnTe thin films will be covered. Surface state properties of topological crystalline insulators will be discussed briefly.

Scanning tunneling microscopy studies of topological insulators

Scanning tunneling microscopy (STM), with surface sensitivity, is an ideal tool to probe the intriguing properties of the surface state of topological insulators (TIs) and topological crystalline insulators (TCIs). We summarize the recent progress on those topological phases revealed by STM studies. STM observations have directly confirmed the existence of the topological surface states and clearly revealed their novel properties. We also discuss STM work on magnetic doped TIs, topological superconductors and crystalline symmetry-protected surface states in TCIs. The studies have greatly promoted our understanding of the exotic properties of the new topological phases, as well as put forward new challenges. STM will continue to play an important role in this rapidly growing field from the point view of both fundamental physics and applications.

Stability of Z2 topological order in the presence of vacancy-induced impurity band

Although topological insulators (TIs) are known to be robust against non-magnetic perturbations and exhibit edge or surface states as their distinct feature, experimentally it is known that vacancies often occur in these materials and impose strong perturbations. Here, we investigate effects of vacancies on the stability of Z2 topological order using the Kane-Mele (KM) model as a prototype of topological insulators. It is shown that even though a vacancy is not classified as a topological defect in the KM model, it generally induces a pair of degenerate mid-gap bound states only in the TI phase. Hence mid-gap bound states due to vacancies arise from the same Z2 classification of topological insulators. Furthermore, we show that in the presence of many vacancies, an impurity band is induced and coexists with edge states until a phase transition occurs when the spectral weights of Dirac cones near Dirac points are depleted. Our analyses indicate that the same scenario holds for point vacancies or lines of vacancies in 3D TIs as well.

Atomic and electronic structure of an alloyed topological insulator, Bi1.5Sb0.5Te1.7Se1.3

Bi2-xSbxTe3-ySey has been argued to exhibit both topological surface states and insulating bulk states, but has not yet been studied with local probes on the atomic scale. Here we report on the atomic and electronic structures of Bi1.5Sb0.5Te1.7Se1.3 studied using scanning tunnelling microscopy (STM) and spectroscopy (STS). Although there is significant surface disorder due to alloying of constituent atoms, cleaved surfaces of the crystals present a well-ordered hexagonal lattice with 10 Å high quintuple layer steps. STS results reflect the band structure and indicate that the surface state and Fermi energy are both located inside the energy gap. In particular, quasi-particle interference patterns from electron scattering demonstrate that the surface states possess linear dispersion and chirality from spin texture, thus verifying its topological nature. This finding demonstrates that alloying is a promising route to achieve full suppression of bulk conduction in topological insulators whilst keeping the topological surface state intact.

Surface effects on electronic transport of 2D chalcogenide thin films and nanostructures

The renewed interest in two-dimensional materials, particularly transition metal dichalcogenides, has been explosive, evident in a number of review and perspective articles on the topic. Our ability to synthesize and study these 2D materials down to a single layer and to stack them to form van der Waals heterostructures opens up a wide range of possibilities from fundamental studies of nanoscale effects to future electronic and optoelectronic applications. Bottom-up and top-down synthesis and basic electronic properties of 2D chalcogenide materials have been covered in great detail elsewhere. Here, we bring attention to more subtle effects: how the environmental, surface, and crystal defects modify the electronic band structure and transport properties of 2D chalcogenide nanomaterials. Surface effects such as surface oxidation and substrate influence may dominate the overall transport properties, particularly in single layer chalcogenide devices. Thus, understanding such effects is critical for successful applications based on these materials. In this review, we discuss two classes of chalcogenides - Bi-based and Mo-based chalcogenides. The first are topological insulators with unique surface electronic properties and the second are promising for flexible optoelectronic applications as well as hydrogen evolution catalytic reactions.

The d-p band-inversion topological insulator in bismuth-based skutterudites

Skutterudites, a class of materials with cage-like crystal structure which have received considerable research interest in recent years, are the breeding ground of several unusual phenomena such as heavy fermion superconductivity, exciton-mediated superconducting state and Weyl fermions. Here, we predict a new topological insulator in bismuth-based skutterudites, in which the bands involved in the topological band-inversion process are d- and p-orbitals, which is distinctive with usual topological insulators, for instance in Bi2Se3 and BiTeI the bands involved in the topological band-inversion process are only p-orbitals. Due to the present of large d-electronic states, the electronic interaction in this topological insulator is much stronger than that in other conventional topological insulators. The stability of the new material is verified by binding energy calculation, phonon modes analysis, and the finite temperature molecular dynamics simulations. This new material can provide nearly zero-resistivity signal current for devices and is expected to be applied in spintronics devices.

Identification of helicity-dependent photocurrents from topological surface states in Bi2Se3 gated by ionic liquid

Dirac-like surface states on surfaces of topological insulators have a chiral spin structure with spin locked to momentum, which is interesting in physics and may also have important applications in spintronics. In this work, by measuring the tunable helicity-dependent photocurrent (HDP), we present an identification of the HDP from the Dirac-like surface states at room temperature. It turns out that the total HDP has two components, one from the Dirac-like surface states, and the other from the surface accumulation layer. These two components have opposite directions. The clear gate tuning of the electron density as well as the HDP signal indicates that the surface band bending and resulted surface accumulation are successfully modulated by the applied ionic liquid gate, which provides a promising way to the study of the Dirac-like surface states and also potential applications in spintronic devices.

Thermoelectric properties of an ultra-thin topological insulator

Thermoelectric coefficients of an ultra-thin topological insulator are presented here. The hybridization between top and bottom surface states of a topological insulator plays a significant role. In the absence of a magnetic field, the thermopower increases and thermal conductivity decreases with an increase in the hybridization energy. In the presence of a magnetic field perpendicular to the ultra-thin topological insulator, thermoelectric coefficients exhibit quantum oscillations with inverse magnetic field, whose frequency is strongly modified by the Zeeman energy and whose phase factor is governed by the product of the Landé g-factor and the hybridization energy. In addition to the numerical results, the low-temperature approximate analytical results for the thermoelectric coefficients are also provided. It is also observed that for a given magnetic field these transport coefficients oscillate with hybridization energy, at a frequency that depends on the Landé g-factor.

Warping effect-induced optical absorbance increment of topological insulator films for THz photodetectors with high signal-to-noise ratio

Strong optical absorbance makes topological insulator (TI) surfaces a promising high-performance photodetector in the terahertz (THz) to infrared frequency range. Here, we study the optical absorbance of more realistic TI films with hexagonal warping effect using the Fermi's golden rules. It was found that when the warping term is λ ≠ 0, the absorbance is no longer a universal value as that of graphene or ideal Dirac cone, but increases monotonously with the photon energy. The increment is positively correlated with the parameter λ/vF(3) where vF is the Fermi velocity. The relative signal-to-noise ratio (SNR) of the TI film working as a photoresistor-type photodetector is significantly enhanced by the warping effect-induced absorbance increment. These investigations provide useful information for developing TI-based photodetectors with high SNR in the range of THz to infrared frequency.

A quantum dot in topological insulator nanofilm

We introduce a quantum dot in topological insulator nanofilm as a bump at the surface of the nanofilm. Such a quantum dot can localize an electron if the size of the dot is large enough, ≳5 nm. The quantum dot in topological insulator nanofilm has states of two types, which belong to two ('conduction' and 'valence') bands of the topological insulator nanofilm. We study the energy spectra of such defined quantum dots. We also consider intraband and interband optical transitions within the dot. The optical transitions of the two types have the same selection rules. While the interband absorption spectra have multi-peak structure, each of the intraband spectra has one strong peak and a few weak high frequency satellites.

Topological Defects in Topological Insulators and Bound States at Topological Superconductor Vortices

The scattering of Dirac electrons by topological defects could be one of the most relevant sources of resistance in graphene and at the boundary surfaces of a three-dimensional topological insulator (3D TI). In the long wavelength, continuous limit of the Dirac equation, the topological defect can be described as a distortion of the metric in curved space, which can be accounted for by a rotation of the Gamma matrices and by a spin connection inherited with the curvature. These features modify the scattering properties of the carriers. We discuss the self-energy of defect formation with this approach and the electron cross-section for intra-valley scattering at an edge dislocation in graphene, including corrections coming from the local stress. The cross-section contribution to the resistivity, ρ, is derived within the Boltzmann theory of transport. On the same lines, we discuss the scattering of a screw dislocation in a two-band 3D TI, like Bi_1-xSb_x, and we present the analytical simplified form of the wavefunction for gapless helical states bound at the defect. When a 3D TI is sandwiched between two even-parity superconductors, Dirac boundary states acquire superconductive correlations by proximity. In the presence of a magnetic vortex piercing the heterostructure, two Majorana states are localized at the two interfaces and bound to the vortex core. They have a half integer total angular momentum each, to match with the unitary orbital angular momentum of the vortex charge.

Dirac electron states formed at the heterointerface between a topological insulator and a conventional semiconductor

Topological insulators are a class of semiconductor exhibiting charge-gapped insulating behaviour in the bulk, but hosting a spin-polarized massless Dirac electron state at the surface. The presence of a topologically protected helical edge channel has been verified for the vacuum-facing surface of several topological insulators by means of angle-resolved photoemission spectroscopy and scanning tunnelling microscopy. By performing tunnelling spectroscopy on heterojunction devices composed of p-type topological insulator (Bi1−xSbx)2Te3 and n-type conventional semiconductor InP, we report the observation of such states at the solid-state interface. Under an applied magnetic field, we observe a resonance in the tunnelling conductance through the heterojunction due to the formation of Landau levels of two-dimensional Dirac electrons at the interface. Moreover, resonant tunnelling spectroscopy reveals a systematic dependence of the Fermi velocity and Dirac point energy on the composition x. The successful formation of robust non-trivial edge channels at a solid-state interface is an essential step towards functional junctions based on topological insulators.

Identifying magnetic anisotropy of the topological surface state of Cr(0.05)Sb(1.95)Te(3) with spin-polarized STM

The surface magnetic property plays a key role in determining magnetic related quantum phenomena of magnetic topological insulators. Using spin-polarized scanning tunneling microscopy, we investigate the surface magnetism and anisotropy of a Cr doped topological insulator: Cr(0.05)Sb(1.95)Te(3). It is found that the topological surface state of Cr(0.05)Sb(1.95)Te(3) is spin polarized in the surface plane while the bulk shows a ferromagnetism with an out-of-plane easy axis. The upper and lower branch of the helical Dirac cone harbors the opposite spin polarization and the polarization at the Dirac point is zero. Our results show the complexity of surface magnetism of magnetic doped topological insulators.

Landau level spectroscopy of Dirac electrons in a polar semiconductor with giant Rashba spin splitting

Optical excitations of BiTeI with large Rashba spin splitting have been studied in an external magnetic field (B) applied parallel to the polar axis. A sequence of transitions between the Landau levels (LLs), whose energies are in proportion to √B were observed, being characteristic of massless Dirac electrons. The large separation energy between the LLs makes it possible to detect the strongest cyclotron resonance even at room temperature in moderate fields. Unlike in 2D Dirac systems, the magnetic field induced rearrangement of the conductivity spectrum is directly observed.

Memory effect in a topological surface state of Bi2Te2Se

We demonstrate the controllable local manipulation of the Dirac surface state in a topological insulator, Bi2Te2Se, which has suppressed bulk carrier density. Using scanning tunneling microscopy/spectroscopy under magnetic fields, we observe Landau levels of the Dirac surface state in the conductance spectra. The Landau levels start to shift in their energy once the bias voltage between the tip and the sample exceeds a threshold value. The amount of shift depends on the history of bias ramping. As a result, conductance spectra show noticeable hysteresis, giving rise to a memory effect. The conductance images exhibit spatially inhomogeneous patterns which can also be controlled by the bias voltage in a reproducible way. On the basis of these observations, we argue that the memory effect is associated with the tip-induced local charging effect which is pinned by the defect-generated random potential. Our study opens up a new avenue to controlling the topological surface state.

Metal-insulator transition in variably doped (Bi(1-x)Sb(x))2Se3 nanosheets

Topological insulators are novel quantum materials with metallic surface transport but insulating bulk behavior. Often, topological insulators are dominated by bulk contributions due to defect induced bulk carriers, making it difficult to isolate the more interesting surface transport characteristics. Here, we report the synthesis and characterization of nanosheets of a topological insulator Bi2Se3 with variable Sb-doping levels to control the electron carrier density and surface transport behavior. (Bi(1-x)Sb(x))2Se3 thin films of thickness less than 10 nm are prepared by epitaxial growth on mica substrates in a vapor transport setup. The introduction of Sb in Bi2Se3 effectively suppresses the room temperature electron density from ∼4 × 10(13) cm(-2) in pure Bi2Se3 (x = 0) to ∼2 × 10(12) cm(-2) in (Bi(1-x)Sb(x))2Se3 at x ∼ 0.15, while maintaining the metallic transport behavior. At x ≳ ∼0.20, a metal-insulator transition (MIT) is observed, indicating that the system has transformed into an insulator in which the metallic surface conduction is blocked. In agreement with the observed MIT, Raman spectroscopy reveals the emergence of vibrational modes arising from Sb-Sb and Sb-Se bonds at high Sb concentrations, confirming the appearance of the Sb2Se3 crystal structure in the sample. These results suggest that nanostructured chalcogenide films with controlled doping can be a tunable platform for fundamental studies and electronic applications of topological insulator systems.

Electric-double-layer field-effect transistors with ionic liquids

Charge carrier control is a key issue in the development of electronic functions of semiconductive materials. Beyond the simple enhancement of conductivity, high charge carrier accumulation can realize various phenomena, such as chemical reaction, phase transition, magnetic ordering, and superconductivity. Electric double layers (EDLs), formed at solid-electrolyte interfaces, induce extremely large electric fields. This results in a high charge carrier accumulation in the solid, much more effectively than solid dielectric materials. In the present review, we describe recent developments in the field-effect transistors (FETs) with gate dielectrics of ionic liquids, which have attracted much attention due to their wide electrochemical windows, low vapor pressures, and high chemical and physical stability. We explain the capacitance effects of ionic liquids, and describe the various combinations of ionic liquids and organic and inorganic semiconductors that are used to achieve such effects as high transistor performance, insulator-metal transitions, superconductivity, and ferromagnetism, in addition to the applications of the ionic-liquid EDL-FETs in logic devices. We discuss the factors controlling the mobility and threshold voltage in these types of FETs, and show the ionic liquid dependence of the transistor performance.

Quasiballistic transport of Dirac fermions in a Bi2Se3 nanowire

Quantum coherent transport of surface states in a mesoscopic nanowire of the three-dimensional topological insulator Bi(2}Se(3) is studied in the weak-disorder limit. At very low temperatures, many harmonics are evidenced in the Fourier transform of Aharonov-Bohm oscillations, revealing the long phase coherence length of spin-chiral Dirac fermions. Remarkably, from their exponential temperature dependence, we infer an unusual 1/T power law for the phase coherence length L(φ)(T). This decoherence is typical for quasiballistic fermions weakly coupled to their environment.

Charging Dirac states at antiphase domain boundaries in the three-dimensional topological insulator Bi2Se3

Using scanning tunneling microscopy and transmission electron microscopy, we demonstrate the existence of antiphase boundaries between neighboring grains shifted by a fraction of a quintuple layer in epitaxial (0001) films of the three-dimensional topological insulator Bi(2)Se(3). Scanning tunneling spectroscopy and first-principles calculations reveal that these antiphase boundaries provide electrostatic fields on the order of 10(8) V/m that locally charge the Dirac states, modulating the carrier density, and shift the Dirac point by up to 120 meV. This intrinsic electric field effect, demonstrated here near interfaces between Bi(2)Se(3) grains, provides direct experimental evidence at the atomic scale that the Dirac states are indeed robust against extended structural defects and tunable by electric field. These results also shed light on the recent observation of coexistence of Dirac states and two-dimensional electron gas on Bi(2)Se(3) (0001) after adsorption of metal atoms and gas molecules.

Quantum interference and Aharonov-Bohm oscillations in topological insulators

Topological insulators (TIs) have an insulating bulk but a metallic surface. In the simplest case, the surface electronic structure of a three-dimensional (3D) TI is described by a single two-dimensional (2D) Dirac cone. A single 2D Dirac fermion cannot be realized in an isolated 2D system with time-reversal symmetry, but rather owes its existence to the topological properties of the 3D bulk wavefunctions. The transport properties of such a surface state are of considerable current interest; they have some similarities with graphene, which also realizes Dirac fermions, but have several unique features in their response to magnetic fields. In this review we give an overview of some of the main quantum transport properties of TI surfaces. We focus on the efforts to use quantum interference phenomena, such as weak anti-localization and the Aharonov-Bohm effect, to verify in a transport experiment the Dirac nature of the surface state and its defining properties. In addition to explaining the basic ideas and predictions of the theory, we provide a survey of recent experimental work.

Anisotropic topological surface states on high-index Bi2Se3 films

A high-index topological insulator thin film, Bi2 Se3 (221), is grown on a faceted InP(001) substrate by molecular-beam epitaxy (see model in figure (a)). Angle-resolved photoemission spectroscopy measurement reveals the Dirac cone structure of the surface states on such a surface (figure (b)). The Fermi surface is elliptical (figure (c)), suggesting an anisotropy along different crystallographic directions. Transport studies also reveal a strong anisotropy in Hall conductance.

Generation of entangled photons in graphene in a strong magnetic field

Entangled photon states attract tremendous interest as the most vivid manifestation of nonlocality of quantum mechanics and also for emerging applications in quantum information. Here we propose a mechanism of generation of polarization-entangled photons, which is based on the nonlinear optical interaction (four-wave mixing) in graphene placed in a magnetic field. Unique properties of quantized electron states in a magnetized graphene and optical selection rules near the Dirac point give rise to a giant optical nonlinearity and a high rate of photon production in the mid- or far-infrared range. A similar mechanism of photon entanglement may exist in topological insulators where the surface states have a Dirac-cone dispersion and demonstrate similar properties of magneto-optical absorption.

Quantum capacitance of an ultrathin topological insulator film in a magnetic field

We present a theoretical study of the quantum magnetocapacitance of an ultrathin topological insulator film in an external magnetic field. The study is undertaken to investigate the interplay of the Zeeman interaction with the hybridization between the upper and lower surfaces of the thin film. Determining the density of states, we find that the electron-hole symmetry is broken when the Zeeman and hybridization energies are varied relative to each other. This leads to a change in the character of the magnetocapacitance at the charge neutrality point. We further show that in the presence of both Zeeman interaction and hybridization the magnetocapacitance exhibits beating at low and splitting of the Shubnikov de Haas oscillations at high perpendicular magnetic field. In addition, we address the crossover from perpendicular to parallel magnetic field and find consistency with recent experimental data.

Three-dimensional models of topological insulators: engineering of Dirac cones and robustness of the spin texture

We design three-dimensional models of topological insulator thin films, showing a tunability of the odd number of Dirac cones driven by the atomic-scale geometry at the boundaries. A single Dirac cone at the Γ-point can be obtained as well as full suppression of quantum tunneling between Dirac states at geometrically differentiated surfaces. The spin texture of surface states changes from a spin-momentum-locking symmetry to a surface spin randomization upon the introduction of bulk disorder. These findings illustrate the richness of the Dirac physics emerging in thin films of topological insulators and may prove utile for engineering Dirac cones and for quantifying bulk disorder in materials with ultraclean surfaces.

Quantum oscillations in magnetothermopower measurements of the topological insulator Bi2Te3

We report the magnetothermopower measurements of the nonmetallic topological insulator Bi(2)Te(3) in magnetic fields up to 35 T. Quantum oscillations arising from surface states are observed in both thermoelectric and conductivity tensors. The inferred surface thermopower has a peak magnitude ~1 mV/K possibly as a result of surface electron and bulk phonon interaction. At the n = 1 Landau level, we resolve additional quantum oscillations signaling Landau sublevels.

Scanning tunneling microscopy of interface properties of Bi2Se3 on FeSe

We investigate the heteroepitaxial growth of Bi(2)Se(3) films on FeSe substrates by low-temperature scanning tunneling microscopy/spectroscopy. The growth of Bi(2)Se(3) on FeSe proceeds via van der Waals epitaxy with atomically flat morphology. A striped moiré pattern originating from the lattice mismatch between Bi(2)Se(3) and FeSe is observed. Tunneling spectra reveal the spatially inhomogeneous electronic structure of the Bi(2)Se(3) thin films, which can be ascribed to the charge transfer at the interface.

Topological surface states with persistent high spin polarization across the Dirac point in Bi2Te2Se and Bi2Se2Te

Helical spin textures with marked spin polarizations of topological surface states have been unveiled for the first time by state-of-the-art spin- and angle-resolved photoemission spectroscopy for two promising topological insulators, Bi(2)Te(2)Se and Bi(2)Se(2)Te. Their highly spin-polarized natures are found to be persistent across the Dirac point in both compounds. This novel finding paves a pathway to extending the utilization of topological surface states of these compounds for future spintronic applications.

Visualizing Landau levels of Dirac electrons in a one-dimensional potential

Using scanning tunneling spectroscopy, we study a 3D topological insulator Bi(2)Te(3) with a periodic structural deformation (buckling). The buckled surface allows us to measure the response of Dirac electrons in a magnetic field to the presence of a well-defined potential variation. We find that while the n=0 Landau level exhibits a 12 meV energy shift across the buckled structure at 7 T, the amplitude of this shift changes with the Landau level index. Modeling these effects reveals that the Landau level behavior encodes information on the spatial extent of their wave functions. Our findings have important implications for transport and magnetoresistance measurements in Dirac materials with engineered potential landscapes.

Quantum capacitance in topological insulators

Topological insulators show unique properties resulting from massless, Dirac-like surface states that are protected by time-reversal symmetry. Theory predicts that the surface states exhibit a quantum spin Hall effect with counter-propagating electrons carrying opposite spins in the absence of an external magnetic field. However, to date, the revelation of these states through conventional transport measurements remains a significant challenge owing to the predominance of bulk carriers. Here, we report on an experimental observation of Shubnikov-de Haas oscillations in quantum capacitance measurements, which originate from topological helical states. Unlike the traditional transport approach, the quantum capacitance measurements are remarkably alleviated from bulk interference at high excitation frequencies, thus enabling a distinction between the surface and bulk. We also demonstrate easy access to the surface states at relatively high temperatures up to 60 K. Our approach may eventually facilitate an exciting exploration of exotic topological properties at room temperature.

Robust nodal structure of Landau level wave functions revealed by Fourier transform scanning tunneling spectroscopy

Scanning tunneling spectroscopy is used to study the real-space local density of states of a two-dimensional electron system in a magnetic field, in particular within higher Landau levels. By Fourier transforming the local density of states, we find a set of n radial minima at fixed momenta for the nth Landau levels. The momenta of the minima depend only on the inverse magnetic length. By comparison with analytical theory and numerical simulations, we attribute the minima to the nodes of the quantum cyclotron orbits, which decouple in a Fourier representation from the random guiding center motion due to disorder. Adequate Fourier filtering reveals the nodal structure in real space in some areas of the sample with relatively smooth potential disorder.

Thickness-independent transport channels in topological insulator Bi(2)Se(3) thin films

With high quality topological insulator Bi(2)Se(3) thin films, we report thickness-independent transport properties over wide thickness ranges. Conductance remained nominally constant as the sample thickness changed from 256 to ∼8  QL (where QL refers to quintuple layer, 1  QL≈1  nm). Two surface channels of very different behaviors were identified. The sheet carrier density of one channel remained constant at ∼3.0×10(13)  cm(-2) down to 2 QL, while the other, which exhibited quantum oscillations, remained constant at ∼8×10(12)  cm(-2) only down to ∼8  QL. The weak antilocalization parameters also exhibited similar thickness independence. These two channels are most consistent with the topological surface states and the surface accumulation layers, respectively.

Manifestation of topological protection in transport properties of epitaxial Bi2Se3 thin films

The massless Dirac fermions residing on the surface of three-dimensional topological insulators are protected from backscattering and cannot be localized by disorder, but such protection can be lifted in ultrathin films when the three-dimensionality is lost. By measuring the Shubnikov-de Haas oscillations in a series of high-quality Bi2Se3 thin films, we revealed a systematic evolution of the surface conductance as a function of thickness and found a striking manifestation of the topological protection: The metallic surface transport abruptly diminishes below the critical thickness of ~6 nm, at which an energy gap opens in the surface state and the Dirac fermions become massive. At the same time, the weak antilocalization behavior is found to weaken in the gapped phase due to the loss of π Berry phase.

Spatial and energy distribution of topological edge states in single Bi(111) bilayer

By combining scanning tunneling microscopy and spectroscopy, angle-resolved photoemission spectroscopy, and density functional theory band calculations, we directly observe and resolve the one-dimensional edge states of single bilayer (BL) Bi(111) islands on clean Bi(2)Te(3) and Bi(111)-covered Bi(2)Te(3) substrates. The edge states are localized in the vicinity of step edges having an ∼2  nm wide spatial distribution in real space and reside in the energy gap of the Bi(111) BL. Our results demonstrate the existence of nontrivial topological edge states of single Bi(111) bilayer as a two-dimensional topological insulator.

In-plane magnetic anisotropy of Fe atoms on Bi2Se3(111)

The robustness of the gapless topological surface state hosted by a 3D topological insulator against perturbations of magnetic origin has been the focus of recent investigations. We present a comprehensive study of the magnetic properties of Fe impurities on the prototypical 3D topological insulator Bi(2)Se(3) using local low-temperature scanning tunneling spectroscopy and integral x-ray magnetic circular dichroism techniques. Single Fe adatoms on the Bi(2)Se(3) surface, in the coverage range ≈ 1% of a monolayer, are heavily relaxed into the surface and exhibit a magnetic easy axis within the surface plane, contrary to what was assumed in recent investigations on the supposed opening of a gap. Using ab initio approaches, we demonstrate that an in-plane easy axis arises from the combination of the crystal field and dynamic hybridization effects.