Session Z31: Topological Insulators: Junctions and Interfaces

Topological p-n Junction

(Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ is an ideal topological insulator with truly insulating bulk and tunable surface state across the Dirac point. We consider a junction between surface p-type and surface n-type on these ideal topological insulators in which carrier type and density in two adjacent regions are locally controlled by electrostatic gating or planar grade doping. Such junction setting on topological insulators are fundamental to device development. We find a gapless chiral edge state localized at the p-n interface arises when applying a magnetic field, which can be detected by scanning tunneling microscopy. The two-terminal conductance of such p-n junction in the quantum Hall regime will be 1/4 times the quantum of conductance e$^2$/h, which signifies the half-quantum Hall effect of a topological insulator surface.   

Andreev Reflection in two-dimensional Topological Insulators

A metal-superconductor interface may reflect an incident electron from the metal as a positively charged hole with opposite spin, while a cooper pair is formed in the superconductor. This electron-hole conversion is Andreev reflection (AR) and has served as a useful probe for spin-polarized currents. In this work we study AR at topological insulator-superconductor interface, for both time-reversal symmetric ($Z_2$) and time-reversal broken ({\it Chern}) cases. We model $Z_2$ insulators using the proposal of Kane and Mele, while for {\it Chern} insulator we use a spinful version of the Haldane model. By employing Landauer-B\"{u}ttiker scheme we find for both cases perfect AR, which is highly robust to disorder and persists as long as the edge states are present. Further, we propose an experiment to distinguish between the two types of topological insulators. The proposal involves a local doping with magnetic impurities at one of the edges of the two-dimesional material. This suppresses one of the channels for reflection and the AR coefficient drops by a factor of two. No such suppression is seen for the {\it Chern} insulator.   

Theory of Odd-Parity Superconductivity: from Gap Function to Topological Invariants and Surface Andreev Bound States

Three-dimensional superconductors with a nodeless pairing gap can be classified by an integer topological invariant, and gapless Andreev bound states exist on the surface when the topological invariant is nontrivial. In this letter we give a general criterion for determining the topological invariant for centrosymmetric superconductors with \emph{odd-parity} pairing. We show that in a general system with spin-orbit coupling, the superconducting gap function can be expressed by a pseudospin d-vector, and the topological invariant can be determined from the total winding number of the pseudospin d-vector on the Fermi surfaces. We also discuss Andreev surface states, and we found gapless surfaces states when the topological invariant in the bulk is topologically nontrivial.   

Current-phase relation for Josephson effect through helical metal

We compute the current-phase relation of Josephson junctions fabricated on the surface of three-dimensional topological insulators. The Josephson coupling between two superconductors is mediated by the two-dimensional helical metal. It gives rise to the so-called fractional Josephson effect. A short junction is previously known to be a quantum wire of Majorana fermions. We discuss the dependence of the current-phase relation on the length of the junction, the chemical potential of the helical metal, and temperature.   

Interface engineering of quantum Hall effects in digital transition-metal oxide heterostructures

Based on tight-binding modeling and first-principles calculations, we investigate possible quantum Hall effects in transition-metal oxide heterostructures. Bilayers of perovskite-type transition-metal oxides grown along the [111] crystallographic axis are found to be potential candidates for two-dimensional topological insulators. The topological band structure of these materials can be tune-tuned by changing dopant ions, substrates, and external gate voltages. We predict that LaAuO$_3$ bilayers have a topologically-nontrivial energy gap of about 0.15 eV, which is sufficiently large to realize the quantum spin-Hall effect at room temperature. We also discuss intriguing phenomena associated with the nearly flat topologically-nontrivial bands found in eg systems, such as fractional quantum Hall effect.   

Proximity effects at semiconductor/topological insulator interfaces

Using phenomenological model Hamitonians, we study the spatial distribution of topological surface states (TSS) in semiconductor/topological insulator (TI) heterostructures. Due to proximity effects induced by the TI substrate, the location of the TSS can be shifted perpendicularly to the interface. We show that both the direction and magnitude of the shift can be tuned by the cooperative effects of the spin-orbit coupling within the hybrid system, the bandgap of the overlayer, and the thickness of the overlayer. Potential technological applications of these salient properties of the TSS will also be discussed.   

Abrikosov Vortex Lattice in 3D Topological Insulator -- Superconductor Heterostructures

Majorana fermions have been predicted to exist on the surface of the three-dimensional (3D) topological insulator/s-wave superconductor heterostructures by proximity effects [Phys. Rev. Lett. {\bf 100}, 096407 (2008)]. In the diffuse vortex limit, the physics of these non-abelian anyons is theoretically well-understood c.f. Phys. Rev. B {\bf 84}, 144507 (2011). However, the dilute vortex limit is unlikely to be available in experimental systems. In this work, we study the dense vortex limit in 3D topological insulator/s-wave superconductor heterostructures using the self-consistent Bogoliubov-de Gennes (BdG) equations under the application of a uniform magnetic flux. We find that as we approach the dense limit of vortices on the surface, that the hybridization between the vortices leads to the formation of a ``Majorana bandstructure'' which exists within the superconducting gap.We describe the physics of the system as we move from the dilute limit to the the dense limit as we vary the surface chemical potentials and the magnetic field magnitudes.   

Quantum Hall Super uids in Topological Insulator Thin Films

Three-dimensional topological insulators have protected Dirac-cone surface states. In this work we argue that gapped excitonic superfluids with spontaneous coherence between top and bottom surfaces can occur in the TI-thin-film quantum-Hall regime. We find that the large dielectric constants of TI materials increase the layer separation range over which coherence survives and decrease the superfluid sound velocity, but have little influence on the superfluid density or on the charge gap. The coherent state at total Landau-level filling factor $\nu_T = 0$ is predicted to be free of edge modes, qualitatively altering its transport phenomenology compared to the widely studied case of $\nu_T = 1$ in GaAs double quantum wells.   

Junction between surfaces of two topological insulators

We study scattering from a line junction which separates the surfaces of two three-dimensional topological insulators; some aspects of this problem were recently studied in Takahashi and Murakami, Phys. Rev. Lett. 107, 166805 (2011). The velocities of the Dirac electrons on the two surfaces may be unequal and may even have opposite signs; in the latter case, we find that the electrons must, in general, go into the two-dimensional interface separating the two topological insulators. We also study what happens if the two surfaces are at an angle $\phi$ with respect to each other. We find in this case that there are bound states which propagate along the line junction with a velocity and direction of spin which depend on the bending angle $\phi$.   

Dirac cones in the gapless interface states between two topological insulators

When two topological insulators are attached together, the states on the interface become gapped due to the hybridization between the surface states. We have shown that if the two topological insulators have the opposite signs for the Dirac velocities, there exist gapless interface states [1]. In the last March meeting we showed a general proof for the existence of the gapless states using the mirror Chern number, which fixes the chirality of the surface states. In this presentation, we report the dispersions of these gapless interface states. They are in general a collection of Dirac cones. For example, if the system has threefold rotational symmetry, the interface states have six Dirac cones. By using the Fu-Kane-Mele model, which is the tight-binding model on the diamond lattice with the spin-orbit interaction, we calculate the dispersion of this gapless interface states, and discuss the relationship with the mirror Chern number.\\[4pt] [1] R. Takahashi, S. Murakami, Phys. Rev. Lett. 107,166805 (2011).   

Conductance through step junctions in 3D topological insulators

An effective continuous model for low-energy surface states of a 3D topological insulator was presented by Zhang {\it et al.}, {\it Nat. Phys. }{\bf 5}, 438 (2009). We present a general solution for this 3D model in a surface different from the standard (111)-surface. In our solution, surface states consist of a single Dirac cone with a Fermi velocity different from the one in (111)-surfaces, and the energy has an elliptical dispersion in $k$-space. We then study transport through a step junction composed of a (111)-surface -- side-surface -- (111)-surface and predict that the conductance saturates at 2/3 G$_0$, independent of eccentricity and velocity mismatch at the interfaces. We compare our model with a junction in a plane with only (111)-states where conductance saturation does depend on velocity mismatch. We also analyze the Fano factor and highlight experimentally relevant situations where our predictions could be tested.   

Inter-band Tunneling between Doped Topological Insulator Surface States

Thin films of 3D topological insulators (TIs) have been experimentally synthesized recently. Impurity doping of the TI surface has been reported to modify the position of the Fermi level and generate Rashba-like splitting of the surface band structure. Our research uses a Non-Equilibrium Green's Function (NEGF) method to simulate inter-surface transport properties for TI thin films. A tight-binding model is established by discretizing a 4 x 4 k.p Hamiltonian for 3D TIs. Because of confinement, thin TI slabs of several nanometers allows inter-surface tunneling at quantum number matching states. The tunneling intensity can be tuned by surface Coulomb impurity doping or applying an external bias. Unlike regular topologically trivial surface states, inter-band tunneling between TI surfaces presents a conduction minimum when the dispersions of the two surfaces align perfectly over each other. The suppression of transport originates from the momentum coupling with time reversal symmetry, leading to significant non-linear I-V properties for the P-N tunneling at forward bias. This leads to a NDR current minimum when an external bias completely compensates the built-in potential. The study on inter-surface tunneling in TI thin films benefits the understanding of the transport behavior of TI surface states, which calls for further experimental investigations in the future.   

Spin Texture on the Fermi Surface of Strained HgTe

We present \emph{ab initio} and ${\bf k\cdot p}$ calculations of the Fermi surface of strained HgTe obtained by stretching the Zinc-Blende lattice along the (111) axis. Near the Fermi level, strained HgTe exhibits point-like accidental degeneracies between a two-fold degenerate and two non-degenerate bands along the (111) axis. The three bands disperse linearly in all directions about the degenerate points and their low energy physics is described by an effective four band ${\bf k\cdot p}$ Hamiltonian. The Fermi surface consists of two ellipsoids which contact only at the point where the Fermi level crosses the two-fold degenerate band along the (111) axis. The spin expectation value on both ellipsoids is constrained to vanish along the (111) axis due to mirror symmetry about a plane that contains that axis. Furthermore the winding number of spins around the two ellipsoids changes from one end to the other indicating the existence of singular points in the spin texture. Indeed, the \emph{ab initio} and ${\bf k\cdot p}$ calculations confirm the existence of such spin singularities on the Fermi ellipsoids. We show that doping HgTe with Zinc atoms chemically strains the HgTe Zinc-Blende lattice and present \emph{ab initio} calculations on HgZnTe that confirm the above results.   

Spin-transfer torque and spin-polarization in topological-insulator-based magnetic tunnel junctions

We derive a nonequilibrium Green function-based formula for spin-transfer torque (STT) exerted by the conduction electrons on the magnetization of a free ferromagnetic (F) layer where {\em strong} spin-orbit coupling (SOC) is present either in the bulk or at the interface of the F layer. This nonequilibrium Born-Oppenheimer approximation-type formula is employed to predict unconventional STT in N$|$TI$|$F semi- magnetic tunnel junction (MTJ) containing a three-dimensional topological insulator (TI). The STT is driven by the SOC on the surface of TI, as well as by the charge current becoming spin-polarized in the direction of transport as it flows from the normal metal (N) through the bulk of the TI layer. The in- plane and perpendicular STT components in N$|$TI$|$F semi-MTJ are an order of magnitude larger than in conventional F$^\prime|$I$|$F MTJ, or N$|$I$|$F semi-MTJ with the strong Rashba SOC at the I$|$F interface, assuming comparable resistance of all three junctions.   

Time reversal symmetric Kitaev model and topological superconductor

We study a time reversal symmetric quantum spin model in two dimensions that is introduced as a higher spin extension of the Kitaev model and is exactly solvable [1]. The ground state of the topological phase of this model can be viewed as a time reversal symmetric topological superconductor in two dimensions. The helical Majorana edge modes which appear in time reversal pair in the topological phase are explained by topological argument. The correlation functions along the edge are derived from the gapless edge theory.\\[4pt] [1] R. Nakai, S. Ryu, and A. Furusaki, arXiv:1111:1230.