Surface Andreev bound states of superfluid (3)He and Majorana fermions

Superfluid (3)He is an intensively investigated and well characterized p-wave superfluid. In the bulk Balian-Werthamer state, which is commonly called the (3)He B phase, the superfluid gap is opened isotropically but near a flat boundary such as a wall of a container it can harbor interesting quasi-particle states inside the gap. These states are called surface Andreev bound states, and have not been experimentally explored in detail. Transverse acoustic impedance measurement has revealed their existence and provided spectroscopic details of the dispersion of the bound states. Recent theoretical arguments claim that the surface Andreev bound states of the superfluid (3)He B phase can be recognized as the edge states of the topological superfluid and be regarded as a Majorana fermion, a fancy particle which has not been confirmed in elementary particle physics. In this review, we present up-to-date knowledge on the surface Andreev bound states of the (3)He B phase revealed by acoustic spectroscopy and the possible realization of a Majorana fermion, along with related studies on this topic.     

A brief review on Majorana bound states in topological superconductors

Topological superconductivity has drawn much attention recently, and most interests are focused on the Majorana bound states existing at the edges of one-dimensional topological superconductors. These Majorana bound states are ideal platform for studying non-Abelian statistics. Meanwhile, they are proposed to be useful in quantum computation. In this review, we introduce the basic concepts and models in this area. We begin from the Kitaev model, which is the most concise model for one-dimensional topological superconductivity. Then, we discuss how to realize this model with spin-orbit coupling in realistic materials. Finally, we show some simple methods to detect the Majorana bound states and study their novel properties with the help of adjacent quantum dots.     

Topological Superconductivity in Cu(x)Bi(2)Se(3)

A topological superconductor (TSC) is characterized by the topologically protected gapless surface state that is essentially an Andreev bound state consisting of Majorana fermions. While a TSC has not yet been discovered, the doped topological insulator Cu(x)Bi(2)Se(3), which superconducts below ～3 K, has been predicted to possess a topological superconducting state. We report that the point-contact spectra on the cleaved surface of superconducting Cu(x)Bi(2)Se(3) present a zero-bias conductance peak (ZBCP) which signifies unconventional superconductivity. Theoretical considerations of all possible superconducting states help us conclude that this ZBCP is due to Majorana fermions and gives evidence for a topological superconductivity in Cu(x)Bi(2)Se(3). In addition, we found an unusual pseudogap that develops below ～20 K and coexists with the topological superconducting state.     

Fragility of surface states and robustness of topological order in Bi2Se3 against oxidation

Topological surface states are protected against local perturbations, but this protection does not extend to chemical reaction over the whole surface, as demonstrated by theoretical studies of the oxidation of Bi(2)Se(3) and its effects on the surface spin polarization and current. While chemisorption of O(2) largely preserves the topological surface states, reaction with atomic O removes the original surface states and yields two new sets of surface states. One set forms a regular Dirac cone but is topologically trivial. The other set, while topologically relevant, forms an unusual rounded Dirac cone. The details are governed by the hybridization interaction at the interface.     

Effect of surface Andreev bound states on the Bean-Livingston barrier in d-wave superconductors

We study the influence of surface Andreev bound states in d-wave superconductors on the Bean-Livingston surface barrier for entry of a vortex line into a strongly type-II superconductor. Starting from Eilenberger theory, we derive a generalization of London theory to incorporate the anomalous surface currents arising from the Andreev bound states. This allows us to find an analytical expression for the modification of the Bean-Livingston barrier in terms of a single parameter describing the influence of the Andreev bound states. We find that the field of first vortex entry is significantly enhanced. Also, the depinning field for vortices near the surface is renormalized. Both effects are temperature dependent and depend on the orientation of the surface relative to the d-wave gap.     

Evidence for surface Andreev bound states in cuprate superconductors from penetration depth measurements

Tunneling and theoretical studies have suggested that Andreev bound states form at certain surfaces of unconventional superconductors. Through studies of the temperature and field dependence of the in-plane magnetic penetration depth lambda(ab) at low temperature, we have found strong evidence for the presence of these states in clean single crystal YBCO and BSCCO. Crystals cut to expose (110) surfaces show a strong upturn in lambda(ab) at around 7 K, when the field is oriented along the c axis. In YBCO this upturn is completely suppressed by a field of approximately 0.1 T.     

Spin polarization and transport of surface states in the topological insulators Bi2Se3 and Bi2Te3 from first principles

We investigate the band dispersion and the spin texture of topologically protected surface states in the bulk topological insulators Bi2Se3 and Bi2Te3 by first-principles methods. Strong spin-orbit entanglement in these materials reduces the spin polarization of the surface states to ～50% in both cases; this reduction is absent in simple models but of important implications to essentially any spintronic application. We propose a way of controlling the magnitude of spin polarization associated with a charge current in thin films of topological insulators by means of an external electric field. The proposed dual-gate device configuration provides new possibilities for electrical control of spin.     

Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3

We report the THz response of thin films of the topological insulator Bi2Se3. At low frequencies, transport is essentially thickness independent showing the dominant contribution of the surface electrons. Despite their extended exposure to ambient conditions, these surfaces exhibit robust properties including narrow, almost thickness-independent Drude peaks, and an unprecedentedly large polarization rotation of linearly polarized light reflected in an applied magnetic field. This Kerr rotation can be as large as 65° and can be explained by a cyclotron resonance effect of the surface states.     

Quasiparticle bound states around impurities in d(x(2)-y(2))-wave superconductors

Competition between the two alternative positions (shuffle and glide 111 plane subsets) for the core of a 30 degrees partial dislocation in Si is examined. Using a combination of ab initio total energy calculations with finite temperature free-energy calculations based on an interatomic potential, we obtained free energies for the relevant vacancy-type core defects. Generally, the free energy of vacancy formation in the core of a 30 degrees glide partial dislocation is considerably lower (by more than 1 eV) than in the bulk. However, even at high temperatures, the predicted thermal concentration of the shuffle segments comprised of a row of vacancies in the core is low, placing the 30 degrees partial dislocation in the glide subset position.     

Chiral orbital-angular momentum in the surface states of Bi2Se3

We performed angle-resolved photoemission (ARPES) experiments with circularly polarized light and first-principles density functional calculation with spin-orbit coupling to study surface states of a topological insulator Bi2Se3. We observed circular dichroism (CD) as large as 30% in the ARPES data with upper and lower Dirac cones showing opposite signs in CD. The observed CD is attributed to the existence of local orbital-angular momentum (OAM). First-principles calculation shows that OAM in the surface states is significant and is locked to the electron momentum in the opposite direction to the spin, forming chiral OAM states. Our finding opens a new possibility for strong light-induced spin-polarized current in surface states. We also provide a proof for local OAM origin of the CD in ARPES.     

Surface scattering via bulk continuum states in the 3D topological insulator Bi2Se3

We have performed scanning tunneling microscopy and differential tunneling conductance (dI/dV) mapping for the surface of the three-dimensional topological insulator Bi(2)Se(3). The fast Fourier transformation applied to the dI/dV image shows an electron interference pattern near Dirac node despite the general belief that the backscattering is well suppressed in the bulk energy gap region. The comparison of the present experimental result with theoretical surface and bulk band structures shows that the electron interference occurs through the scattering between the surface states near the Dirac node and the bulk continuum states.     

Superconductivity in the doped topological insulator Cu{x}Bi{2}Se{3} under high pressure

We report a high-pressure single crystal study of the topological superconductor Cu{x}Bi{2}Se{3}. Resistivity measurements under pressure show superconductivity is depressed smoothly. At the same time the metallic behavior is gradually lost. The upper-critical field data B{c2}(T) under pressure collapse onto a universal curve. The absence of Pauli limiting and the comparison of B{c2}(T) to a polar-state function point to spin-triplet superconductivity, but an anisotropic spin-singlet state cannot be discarded completely.     

Theoretical study of influencing factors on the dispersion of bulk band-gap edges and the surface states in topological insulators Bi2Te3 and Bi2Se3

The dispersion of the band-gap edge states in bulk topological insulators Bi₂Te₃ and Bi₂Se₃ is considered within density functional theory. The dependences of this dispersion both on the approximation used for an exchange-correlation functional at fixed unit cell parameters and atomic positions and on these parameters and positions that are obtained upon structural relaxation performed using a certain approximated functional are analyzed. The relative position of the Dirac point of topologically protected surface states and the valence band maximum in the surface electronic structure of the topological insulators is discussed.     

Carrier transport in Bi2Se3 topological insulator slab

Electron transport in Bi₂Se₃ topological insulator slabs is investigated in the thermal activation regime (>50 K) both in the absence (ballistic) and presence of weak and strong acoustic phonon scattering using the non-equilibrium Green function approach. Resistance of the slab is simulated as a function of temperature for a range of slab thicknesses and effective doping in order to gain a handle on how various factors interact and compete to determine the overall resistance of the slab. If the Bi₂Se₃ slab is biased at the Dirac point, resistance is found to display an insulating trend even for strong electron–phonon coupling strength. However, when the Fermi-level lies close to the bulk conduction band (heavy electron doping), phonon scattering can dominate and result in a metallic behavior, although the insulating trend is retained in the limit of ballistic transport. Depending on values of the operating parameters, the temperature dependence of the slab is found to exhibit a remarkably complex behavior, which ranges from insulating to metallic, and includes cases where the resistance exhibits a local maximum, much like the contradictory behaviors seen experimentally in various experiments.     

Common origin of the circular-dichroism pattern in angle-resolved photoemission spectroscopy of SrTiO3 and Cu(x)Bi2Se3

Circular dichroism in the angular distribution of photoelectrons from SrTiO(3):Nb and Cu(x)Bi(2)Se(3) is investigated by 7-eV laser angle-resolved photoemission spectroscopy. In addition to the well-known node that occurs in the circular dichroism pattern when the incidence plane matches the mirror plane of the crystal, we show that another type of node occurs when the mirror plane of the crystal is vertical to the incidence plane and the electronic state is two-dimensional. The flower-shaped circular dichroism patterns in the angular distribution occurring around the Fermi level of SrTiO(3):Nb and around the Dirac point of Cu(x)Bi(2)Se(3) are explained on equal footings. We point out that the penetration depth of the topological states of Cu(x)Bi(2)Se(3) depends on momentum.     

Existence of Majorana fermions and topological order in nodal superconductors with spin-orbit interactions in external magnetic fields

We demonstrate that Majorana fermions exist in edges of systems and in a vortex core even for superconductors with nodal excitations such as the d-wave pairing state under a particular but realistic condition in the case with an antisymmetric spin-orbit interaction and a nonzero magnetic field below the upper critical field. We clarify that the Majorana fermion state is topologically protected in spite of the presence of bulk gapless nodal excitations, because of the existence of a nontrivial topological number. Our finding drastically enlarges target systems where we can explore the Majorana fermion state.     

Rashba spin-splitting control at the surface of the topological insulator Bi2Se3

The electronic structure of Bi(2)Se(3) is studied by angle-resolved photoemission and density functional theory. We show that the instability of the surface electronic properties, observed even in ultrahigh-vacuum conditions, can be overcome via in situ potassium deposition. In addition to accurately setting the carrier concentration, new Rashba-like spin-polarized states are induced, with a tunable, reversible, and highly stable spin splitting. Ab initio slab calculations reveal that these Rashba states are derived from 5-quintuple-layer quantum-well states. While the K-induced potential gradient enhances the spin splitting, this may be present on pristine surfaces due to the symmetry breaking of the vacuum-solid interface.     

Anomalous proximity effect in gold coated (110) YBa2Cu3O7-delta films: penetration of the Andreev bound states

Scanning tunneling spectroscopy of (110)YBa(2)Cu(3)O(7-delta)/Au bilayers reveal a proximity effect markedly different from the conventional one. While proximity-induced mini-gaps rarely appear in the Au layer, the Andreev bound states clearly penetrate into the metal. Zero bias conductance peaks are measured on Au layers thinner than 7 nm with a magnitude similar to those detected on the bare superconductor films. The peaks then decay abruptly with Au thickness and disappear above 10 nm. This length is shorter than the normal coherence length and corresponds to the (ballistic) mean free path.     

Andreev reflection and bound states in topological insulator based planar and step Josephson junctions

A superconductor-topological insulator-superconductor (S/TI/S) junction having normal region at angle θ is studied theoretically to investigate the junction angle dependency of the Andreev reflection and the formation of the Andreev bound states in the step and planar S/TI/S structures. It is found that the Andreev reflection becomes θ dependent only in the presence of the potential barrier at the TI/S interface. In particular, the step and planar TI/S junction have totally different conductive behavior with bias voltage and potential barrier in the regime of retro and specular Andreev reflection. Interestingly, we find that the elliptical cross section of Dirac cone, an important feature of topological insulator with step surface defect, affects the Fabry-Perot resonance of the Andreev reflection induced Andreev bound states (which become Majorana zero energy states at low chemical potential) in the step S/TI/S structure. Unlike the usual planar S/TI/S structures, we find these ellipticity affected Andreev bound states lead to non-monotonic Josephson super-current in the step S/TI/S structure whose non-monotonicity can be controlled with the use of the potential barrier, which may find applications in nanoelectronics.     

Interaction of phonons and dirac fermions on the surface of Bi2Se3: a strong Kohn anomaly

We report the first measurements of phonon dispersion curves on the (001) surface of the strong three-dimensional topological insulator Bi2Se3. The surface phonon measurements were carried out with the aid of coherent helium beam surface scattering techniques. The results reveal a prominent signature of the exotic metallic Dirac fermion quasiparticles, including a strong Kohn anomaly. The signature is manifest in a low energy isotropic convex dispersive surface phonon branch with a frequency maximum of 1.8 THz and having a V-shaped minimum at approximately 2kF that defines the Kohn anomaly. Theoretical analysis attributes this dispersive profile to the renormalization of the surface phonon excitations by the surface Dirac fermions. The contribution of the Dirac fermions to this renormalization is derived in terms of a Coulomb-type perturbation model.