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Keywords:

  • Josephson effect;
  • proximity effect;
  • quantum spin Hall effect;
  • topological insulators

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Topological insulators in two dimensions: Quantum spin Hall insulators
  5. 3 Spin-helical transport in 3D topological insulators
  6. 4 Quantum Hall effect on topological surfaces
  7. 5 Superconducting Klein tunneling on topological surfaces
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

In a topological insulator (TI) the character of electron transport varies from insulating in the interior of the material to metallic near its surface. Unlike, however, ordinary metals, conducting surface states in TIs are topologically protected and characterized by spin helicity whereby the direction of the electron spin is locked to the momentum direction. In this paper we review selected topics regarding recent theoretical and experimental work on electron transport and related phenomena in two-dimensional (2D) and three-dimensional (3D) TIs.

The review provides a focused introductory discussion of the quantum spin Hall effect in HgTe quantum wells as well as transport properties of 3DTIs such as surface weak antilocalization, the half-integer quantum Hall effect, s + p-wave induced superconductivity, superconducting Klein tunneling, topological Andreev bound states and related Majorana midgap states. These properties of TIs are of practical interest, guiding the search for the routes towards topological spin electronics.


1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Topological insulators in two dimensions: Quantum spin Hall insulators
  5. 3 Spin-helical transport in 3D topological insulators
  6. 4 Quantum Hall effect on topological surfaces
  7. 5 Superconducting Klein tunneling on topological surfaces
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

The situation when a material behaves as a metal in terms of its electric conductivity and, at the same time, as an insulator in terms of its band structure is extremely unconventional from the viewpoint of the standard classification of solids. That is why the recent discovery of a class of such materials – topological insulators (TIs) – has generated much interest (see e.g., reviews 1, 2). The dual properties of the TIs are especially well pronounced in two-dimensional (2D) systems which are also known as quantum spin-Hall insulators (QSHIs) 3–7. In QSHIs, the metallic electric conduction is associated with propagating states that occur only near the sample edges, while the conduction in the interior is suppressed by a band gap like in ordinary band insulators. These edge states originate from intrinsic spin–orbit (SO) coupling and are profoundly different from those appearing in quantum Hall systems in a strong perpendicular magnetic field 8, 9. The key distinction lies in the role of the time reversal symmetry. In the QSHIs the SO coupling preserves the time-reversal symmetry, resulting in a pair of counter-propagating channels on the same edge as opposed to one-way directed (chiral) edge states in quantum Hall systems. Remarkably, the spin and momentum directions of the two QSHI edge channels are locked in the opposite ways so that these states are characterized by opposite spin helicities and orthogonal to each other. As a result, such helical edge states have a nodal band dispersion (see also Fig. 1) which is topologically protected against any structural or sample imperfections that do not cause spin scattering.

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Figure 1. (online color at: www.pss-b.com) Schematic of band dispersion in 2D HgTe quantum wells: (a) ordinary band insulator with normal gap equation image versus (b) topological quantum spin-Hall insulator (QSHI) with inverted gap equation image. In the latter case, two crossing spectral branches correspond to a pair of edge states with opposite spin helicities equation image. We use Hamiltonian (4) with equation image, equation image, equation image and boundary condition (6).

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The QSHIs have been experimentally realized in HgTe/CdTe quantum wells with inverted band structure and strong intrinsic SO splitting coming from the atomically heavy mercury 5–7. In these structures the helical edge states generate nonlocal transport effects in zero magnetic field that have no analogs in the conventional 2D semiconductors, allowing one to detect the helical edge states in appropriately designed Hall bar devices 7. A variety of other properties of the QSHIs and, specifically, HgTe quantum wells have also got in the focus of recent theoretical 10–65 and experimental 66–75 research.

The higher dimensional analogs of the QSHIs are the three-dimensional (3D) TIs 76–82. In 3DTIs the topologically protected electronic states appear on the surface of a bulk material. These surface states have a nodal band dispersion in the form of a Dirac-like cone reflecting a continuum of momentum directions on the surface (see also Fig. 2). The family of materials and heterostructures which can host surface states is pretty large. They were first predicted in inverted semiconductor contacts 83, 84. The coexistence of the metallic surface states with the bulk gapped band structure, i.e., the 3DTI phase, has been established theoretically for the semiconducting alloy Bi1−xSbx 77, strained 3D layers of α-Sn and HgTe 77, the tetradymite semiconductors Bi2Se3, Bi2Te3, and Sb2Te3 82, thallium-based ternary chalcogenides TlBiTe2 and TlBiSe2 85–87 as well as Pb-based layered chalcogenides 88, 89. Experimentally, topological surface states have been observed by means of angle-resolved photo-emission spectroscopy (ARPES) in Bi1−xSbx 80, 81, Bi2Se3 90, Bi2Te3 91, TlBiSe2 92–94, TlBiTe2 94, Pb(Bi1−xSbx)2Te4 95, and PbBi2Te4 96.

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Figure 2. (online color at: www.pss-b.com) Energy bands (in meV) of a surface state in a 3DTI versus in-plane wave-numbers kx and ky (in nm−1) from effective Hamiltonian (34). The Fermi level lies in the conduction band at E = 0. We chose equation image meV nm, W = 50 meV nm3 and equation image (adapted from 157).

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Beside the ARPES and band structure calculations, there has been a growing number of experiments 97–124 and theoretical studies 125–187 devoted to helical transport in 3DTI materials. Like in the QSHIs, the surface charge carriers in 3DTIs are characterized by a well-defined spin helicity, i.e., the locking of the spin and momentum directions. This has several implications for electron transport which will be the subject of the present review. First, the helicity conservation along closed electron trajectories prevents ordinary electron localization on the surface. Instead, experiments 97, 104–109, 112, 120–122 and theoretical calculations 157–160, 180, 181 have indicated that the surface quantum transport in the 3DTIs should exhibit weak antilocalization with a positive magnetoresistance. In thin TI films with surface and bulk states, both weak antilocaliation and localization regimes have been predicted 160, 181. Second, in a strong perpendicular magnetic field the unconventional half-integer quantum Hall plateaus are expected for a single spin-helical surface 77, 126, 136, 139, 164, as has been found in graphene 188 and in zero-gap HgTe/CdTe quantum wells 69. Experimentally, unusual odd and even quantum Hall plateaus have been observed in epitaxially strained 3DTI HgTe 111, which may indicate the contribution of two spin-helical surfaces, one at the top and one at bottom of the sample. Beside the magnetotransport, the quantum Hall dynamics in 3DTIs manifests itself in the Faraday effect at THz frequencies 124. Another interesting implication of the spin helicity is the possibility of unconventional surface superconductivity. It is expected to occur in proximity of a singlet s-wave superconductor (e.g., Nb or Al) which induces both singlet s-wave and triplet p-wave correlations, as a result of the broken spin-rotation symmetry 125, 132, 134, 151, 152, 172. As a signature of the p-wave correlations, topological Majorana midgap states have been predicted in superconductor (S)/3DTI junctions (see e.g., 2, 125, 171, 172). The theoretical search for unconventional superconductivity in 3DTIs goes in parallel with experimental progress in fabrication and characterization of such hybrid superconducting systems 114–118, 123.

As transport in TIs is a multifaceted problem, this review aims to discuss it from different angles, starting from the QSH effect and related phenomena in 2D HgTe quantum wells and continuing to the surface weak antilocalization, half-integer quantum Hall effect and s + p-wave induced superconductivity in 3DTIs. Regarding the superconducting properties, we introduce a generalized Fu-Kane model correctly accounting for the energy-dependence of the proximity effect in 3DTIs. We then analyze the superconducting Klein tunneling, topological Andreev bound states and related Majorana midgap states as well as the fractional AC Josephson effect 189, 190 in such proximity S/TI/S junctions.

2 Topological insulators in two dimensions: Quantum spin Hall insulators

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Topological insulators in two dimensions: Quantum spin Hall insulators
  5. 3 Spin-helical transport in 3D topological insulators
  6. 4 Quantum Hall effect on topological surfaces
  7. 5 Superconducting Klein tunneling on topological surfaces
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

2.1 Effective Hamiltonian for HgTe QWs

We will discuss the QSHIs in the context of their experimental realization in HgTe/CdTe quantum wells (QWs) 5–7. HgTe is a zinc-blende-type semiconductor. Due to large relativistic corrections, it has an inverted band structure where a metallic s-type electron band (usually acting as the conduction band) has a lower energy than p-type hole bands 191. Consequently, in HgTe QWs with large thicknesses d the energy subbands are also inverted (see Fig. 3a). In Fig. 3, subbands H1, H2, … originate from the heavy-hole band, whereas the electron-like subbands are denoted by E1, E2, …. With the decreasing QW thickness d, the energies of the E subbands increase as a result of quantum confinement, whereas those of the H subbands decrease (see Fig. 3a). This results in the normal band sequence in narrow QWs. The different d-dependences of E1 and H1 subbands imply a critical thickness, dc, at which the band gap is closed. For d ≈ dc and near the Γ point, an effective four-band model involving double (Kramers) degenerate E1 and H1 subbands can be derived from the eight-band Kane model 4, 26. Introducing basis states equation image, equation image, equation image, and equation image (where ± denotes Kramers partners), one can write the effective four-band Hamiltonian for the system as follows 4, 26:

  • equation image((1))
  • equation image((2))
  • equation image((3))

The two diagonal blocks of HHgTe (1) describe pairs of states related to each other by time reversal symmetry (Kramers partners). Each of the blocks has a matrix 2 × 2 structure with Pauli matrices σx,y,z and unit matrix σ0 representing the two lowest-energy subbands E1 and H1. The linear terms in Eq. (2) (proportional to constant equation image and in-plane wave-vectors kx,y) describe the E1–H1 hybridization, while equation image yields the band gap equation image at the Γ (equation image) point of the Brillouin zone. The positive quadratic terms equation image and equation image take into account the details of the band curvature in HgTe QWs 4. The Hamiltonian (1) can be extended to include the spin–orbit coupling between the Kramers partners 26, 6.

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Figure 3. (online color at: www.pss-b.com) Band structure of a HgTe/Hg0.3Cd0.7Te quantum well: (a) electron E1, E2, … and heavy-hole H1, H2, equation image subband energies versus well thickness d, (b) in plane dispersion at the critical thickness dc ≈ 6.3 nm, and (c) a 3D plot of the Dirac-like low-energy spectrum for d = dc near the Γ point of the Brillouin zone (adapted from 69).

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Using unitary transformation equation image with equation image we can cast the Hamiltonian (1) into a Dirac-like form

  • equation image((4))

where Pauli matrix τz and unit matrix τ0 act on the Kramers partners. We note that despite the effective mass term equation image the Hamiltonian (4) is invariant under time reversal, i.e., equation image, where equation image is the time-reversal operator, with equation image denoting complex conjugation.

2.2 Simple analytic model of the QSH insulator

The QSHI state is realized when a QW with an inverted gap equation image is sided by ordinary band insulators with equation image (see also Fig. 1b). The system boundaries play a role of topological defects – “mass” domain walls – that bind electronic states near the edge so that they decay on both sides of the boundary. In this subsection we discuss specific properties of such edge states:

  • the edge-state spectrum is gapless and merges into the bulk spectrum above the band gap;
  • the edge-states are the orthogonal eigenstates of the helicity operator equation image, where equation image is the unit vector in the direction of the edge-state momentum. For this reason, the QSH edge states are called helical;
  • local static perturbation V preserving time-reversal symmetry does not couple the QSH edge states.

In order to illustrate these properties we will make two simplifications. First, we will omit all the terms equation image in Hamiltonian (4). This is justified since the edge states occur in the vicinity of the equation image point. Hamiltonian (4) takes, then, the form equation image which in position representation corresponds to the following equation for the four-component wave function equation image:

  • equation image((5))

Second, we will assume that our system is confined by a normal band insulator with the infinite mass, equation image. It is known 192 that such infinite mass confinement can be modeled by an effective local boundary condition which in our geometry (e.g., at y = 0) reads

  • equation image((6))

This boundary condition is specific to Dirac fermions with linear spectrum. It ensures vanishing of the normal component of the particle current without putting equation image to zero at the boundary. The use of the infinite mass confinement (6) is complementary to the tight-binding calculations (see e.g., Refs. 6, 18) and other continuum models of the helical edge states, which include the quadratic (equation image) terms and employ the hard wall boundary conditions (see e.g., Refs. 11, 13).

We seek solutions to Eq. (5) in form of the two eigenstates, equation image, of diagonal matrix equation image propagating along the edge (in the x-direction) and decaying exponentially away from it (in the y-direction):

  • equation image((7))
  • equation image((8))

with a real positive decay length equation image. The symbol equation image denotes a tensor product of an eigenstate of equation image (first column) and the wave function in σ space (second column). The conditions for the nontrivial solutions for the coefficients equation image and equation image follow from Eqs. (5) and (6), yielding two equations for λ and ε:

  • equation image((9))
  • equation image((10))

We notice that the left-hand-side of Eq. (10) does not contain index τ, whereas the right-hand-side does. This can only be true if both sides of Eq. (10) [and those Eq. (9)] vanish independently, which yields a solution with a gapless linear dispersion and a real decay length:

  • equation image((11))

Since λ must be positive, the edge states exists only in a system with the inverted negative gap, disappearing when equation image turns positive. Their propagation velocity equation image coincides with that of the bulk states above the gap (see also Fig. 1). In a narrow QSHI the overlap of the edge states from the opposite sides results in a gapped edge-state dispersion 11.

The edge-state wave functions normalized to half-space equation image are given by

  • equation image((12))
  • equation image((13))

The key feature of the edge states (12) and (13) is that they are orthogonal eigenstates of the helicity operator equation image:

  • equation image((14))

The helicity Σ is defined as the projection of vector equation image on the direction of the edge-state momentum equation image. Since the matrix structure of Σ derives from the SO-split energy bands, it is also called the spin helicity. Equation (14) is a manifestation of the time-reversal symmetry and the fact that the QSH state is generally characterized by a equation image topological invariant 3. It is easy to see that one helical channel can be obtained from the other by simply applying the time-reversal operator equation image:

  • equation image((15))

The helicity [see Eq. (14)] protects the edge states from local perturbations that does not break the time-reversal symmetry. Concretely, let us consider a perturbation V which is (up to a phase) invariant under time reversal:

  • equation image((16))

Using Eqs. (15) and (16) we can transform the matrix element equation image as follows

  • equation image((17))
  • equation image((18))

i.e., for a hermitian equation image its matrix element is zero:

  • equation image((19))

In particular, spin-independent potential disorder cannot cause scattering between the helical edge states and their localization. The physics of localization in the QSHIs has been studied by using both field-theoretical 20, 24 and numerical (see e.g., 18, 35, 37, 52, 51) techniques. In particular, Ref. 18 has found that the edge backscattering and magnetoresistance can occur as a combined effect of bulk-inversion asymmetry, sufficiently strong potential disorder and an external magnetic field.

Another question regarding the role of the time-reversal symmetry is what happens to the QSHIs in a strong quantizing magnetic field? In this case the helical edge states appear within the gap between the highest hole (equation image) and lowest particle (equation image) Landau levels 19. The edge-state dispersion is still nodal, but nonlinear, i.e., the two crossing spectral branches have now different group velocities. The latter circumstance makes the helical edge states prone to get coupled by spin-flip scattering, which also generates the edge magnetoresistance 19, 50.

We should emphasize that Eqs. (17)–(19) do not generally hold for interacting helical channels where the electron backscattering may occur provided that the axial spin rotation symmetry of Hamiltonian (1) is broken, e.g. by the Bychkov–Rashba spin–orbit coupling 20, 47, 60. The spatially random Bychkov–Rashba coupling has been predicted to cause localization of the edge states in the presence of weakly screened electron–electron interaction 20. Ref. 60 has shown that weak electron–electron interaction in the absence of the axial spin rotation symmetry allows for inelastic backscattering of a single electron, accompanied by forward scattering of another. The inelastic phonon-induced backscattering in the helical liquid has been considered in Ref. 47.

2.3 Helical versus chiral edge transport. Nonlocal detection of the QSHI state

The helical edge transport in a QSHI differs markedly from the chiral edge transport in QH systems. In order to illustrate this we follow Ref. 7 and consider a four-terminal device shown in Fig. 4. Using the ballistic Landauer–Bütikker approach, we express the current, Ii, injected through contact i in terms of voltages Vj induced on all contacts as

  • equation image((20))

where equation image is the transmission probability from contact i to contact j. For a chiral QH edge channel, equation image connects the neighboring contacts only in one propagation direction (see also Fig. 4a), such that

  • equation image((21))

where N is the number of the terminals (e.g., N = 4 in Fig. 4), with the convention that equation image describes the transmission from terminal N to terminal 1. Assuming, for concreteness, that the current flows from terminal 1 to terminal 4, while leads 2 and 3 are used as voltage probes, we have

  • equation image((22))

This yields a finite two-terminal resistance equation image and zero four-terminal (nonlocal) resistances equation image, equation image, and equation image.

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Figure 4. (online color at: www.pss-b.com) Schematic of edge transport in four-terminal device: (a) chiral edge states in a quantum Hall (QH) system versus (b) helical edge states in the quantum spin Hall (QSH) insulator.

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In contrast, in a QSHI the helical edge channels connect the neighboring contacts in both propagation directions (see also Fig. 4b), such that

  • equation image((23))

with the conventions equation image and equation image. Consequently, for a current flowing from 1 to 4, we find

  • equation image((24))
  • equation image((25))

which yields the two-terminal resistance 7

  • equation image((26))

and the four-terminal resistances 7

  • equation image((27))
  • equation image((28))

The nonzero non-local resistances (27) and (28) are unique to the QSHI state, allowing its unambiguous experimental detection 7. It should be emphasized that the universality of the non-local resistances (27) and (28) is just the consequence of the time-reversal symmetry and, therefore, is expected also for other proposed realizations of the QSHIs, e.g. in inverted InAs/GaSb quantum wells 58. Equations (26)–(28) for the quantized resistances are valid in the zero-temperature limit when the inelastic backscattering processes are negligible 3.

2.4 Helical carriers and weak antilocalization in n-type HgTe quantum wells

The absence of the edge localization in the QSHI regime (i.e., when the Fermi level lies in the band gap) is closely related to the weak antilocalization (WAL) effect observed when the Fermi level is pushed above the gap into the bulk conduction band. The latter case corresponds to n-type HgTe quantum wells in which charge carriers behave as a 2D helical metal. Assuming that it is described by the same Hamiltonian (4), we can calculate the disorder-induced quantum-interference correction equation image to the classical Drude conductivity (for more details see Section 3). In the leading logarithmic order equation image is given by 157

  • equation image((29))
  • equation image((30))
  • equation image((31))

where τ is the elastic scattering time and equation image is the dephasing time. The third new time-scale equation image appears due to the lack of topological protection of the gapped spectrum: its geometrical Berry phase equation image (31) no longer coincides with the universal value π characteristic of a massless Dirac cone 193. The presence of the gap equation image enables scattering between opposite-momenta states equation image and equation image (backscattering) on the 2D Fermi surface (see also Ref. 30). Thus, equation image is the rate of such backscattering. It is an interesting band-structure effect (see also Fig. 5): for the normal band structure with equation image the rate equation image is finite at any carrier density n, whereas for the inverted band structure with equation image the backscattering rate equation image is strongly nonmonotonic with a zero at a specific carrier density equation image.

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Figure 5. (online color at: www.pss-b.com) Backscattering rate equation image [in units of elastic scatering rate equation image, see Eq. (30)] versus carrier density n; equation image and equation image (adapted from 157).

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The role of the geometrical phases in transport quantum-interference effects in HgTe quantum wells has recently been studied numerically in Ref. 62. In Ref. 63 analytical results for the WAL in HgTe quantum wells have been obtained with the account of both bulk-inversion asymmetry and Rashba SO coupling. Experimentally, the WAL in HgTe has been observed in 2D quantum wells 67, 75 and in strained 3D layers 112. Also, experiments have revealed cyclotron resonance phenomena for the helical carriers in HgTe quantum wells 70, 72.

3 Spin-helical transport in 3D topological insulators

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Topological insulators in two dimensions: Quantum spin Hall insulators
  5. 3 Spin-helical transport in 3D topological insulators
  6. 4 Quantum Hall effect on topological surfaces
  7. 5 Superconducting Klein tunneling on topological surfaces
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

The 1D edge states of HgTe/CdTe quantum wells discussed above have higher-dimensional analogues, such as gapless states on a 2D interface between two bulk semiconductors with normal and inverted gaps 83, 84. Generally, a 3D system with gapless surface states occuring inside the bulk band gap realizes a topogically nontrivial insulating state, the 3DTI 1, 2, 76–79, 82. Since the gapless states in 3DTIs appear only on the surface, they are exempt from the fermion doubling theorem 194. For this reason, the topological surface state consists of an odd number of Weyl-like fermions, each described by a two-component spinor wave function. In Bi2Se3 and Bi2Te3 there is a single Weyl-like fermion species 82, 90, 91) which can be described by an effective 2D Hamiltonian 195, 196:

  • equation image((32))
  • equation image((33))

where equation image is the wave vector on the surface and equation image, equation image, and equation image are band structure parameters. Upon unitary transformation equation image with equation image Hamiltonian (32) takes the form:

  • equation image((34))

which looks similar to the HgTe quantum well Hamiltonian (4). There are, however, two important distinctions between these Hamiltonians. First, here the basis functions correspond to equation image and equation image electron spin projections, i.e., Pauli (equation image) and unit (equation image) matrices act on real spin indices. Second, the equation image-term in Eq. (34) is cubic (odd) in momentum equation image, causing no gap at equation image. This term does not break equation image invariance, which is the real time-reversal symmetry in this case. Instead, it causes hexagonal warping 197, 198 of the surface spectrum (see, also Fig. 2):

  • equation image((35))

where the angle equation image indicates the momentum direction.

3.1 Persistence of surface spin-momentum locking in the presence of disorder

In Eqs. (32) and (34) the first linear term is analogous to the Bychkov–Rashba spin–orbit interaction that occurs in a conventional 2D electronic system confined in an asymmetric quantum well 199. The key difference here is the strength of the coupling constant equation image. For the topological surface states, the experimentally determined coupling constants typically are equation image meV nm for Bi2Se3 198 and equation image meV nm for Bi2Te3 91, 197. These values are really giant compared to the Bychkov–Rashba coupling equation image meV nm in conventional GaAs-based 2D electronic systems (see e.g., Ref. 200). This large quantitative difference has two important physical consequences. The first, as mentioned above, is the formation of the conical band dispersion (see also Fig. 2). The second is the robustness of the surface spin-momentum locking against potential impurity scattering (and other spin-independent scattering) as long as the energy separation between the valence and conduction bands [see Eq. (35)] is larger than the spectrum broadening equation image:

  • equation image((36))

where equation image is the elastic life-time, kF and EF is the Fermi momentum and energy (the Fermi level is, for concreteness, located in the conduction band). Due to the large constant equation image the condition (36) can be met already for the Fermi energy values, EF, of several tens of meV. Moreover, the condition (36) simultaneously implies a diffusive metallic transport regime in which

  • equation image((37))

because for a conical dispersion the transport momentum relaxation time equation image 193 (vF is the Fermi velocity and equation image is the transport mean-free path). Thus, the large values of the coupling constant equation image [see Eq. (36)] enable the diffusive metallic regime (37) without significant spin relaxation, i.e., the spin-helicity of the carriers is preserved during their diffusion on a disordered TI surface. For Hamiltonian (34) the conservation of the spin-helicity can be expressed as

  • equation image((38))

where the constant h takes the values equation image for the conduction and valence bands, respectively. The absence of the spin relaxation under condition (38) defines a distinct transport regime – the 2D helical metal – opening hitherto unexplored routes for theoretical investigations of disordered systems (e.g., Refs. 24, 142, 154, 165, 174).

3.2 Surface weak antilocalization in 3D TIs

One important example of observable transport phenomena in disordered systems is the weak localization (WL) effect 201–203. It is associated with closed classical electron trajectories that self intersect with opposite electron momenta, k and −k (see Fig. 6a). On such trajectories, two states which are related to each other by time reversal interfere constructively, yielding, in the absence of the spin–orbit coupling, a negative WL correction δσ to the classical Drude conductivity equation image. The relative change δσ/σD is the measure of the classical return probability integrated from the shortest diffusion time-scale τ to the largest one, given by the dephasing time equation image:

  • equation image((39))

The negative sign here is the consequence of the fact that the particle returns to its original location with the opposite momentum −k (i.e., the Kubo formula for the conductivity correction contains the product of the two opposite-sign velocities). The return probability equation image is proportional to a space element of the trajectory, equation image, divided by the typical area equation image enclosed by the particle trajectory at time t (equation image is the Fermi wavelength and D is the diffusion constant). The conductivity correction (39) is logarithmically divergent and has a universal prefactor 201–203:

  • equation image((40))

where the factor of 2 in the numerator is due to the spin degeneracy.

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Figure 6. (online color at: www.pss-b.com) (a) Schematic of closed electron trajectories giving rise to weak localization [see also Eq. (39) and text]. Shaded areas indicate a space element of the trajectory, equation image, and the area enclosed by the trajectory, Dt, which we use to estimate the return probabilty equation image in Eq. (39). (b) Same trajectories on the surface of a 3D TI involve opposite momenta, k and −k, and opposite spins, σ and −σ, as a result of the conservation of the spin-helicity [see also Eq. (41) and text]. This leads to the weak antilocalization conductivity (42).

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In contrast to Eq. (40), according to experiments 97, 104–109, 112, 120–122 and theoretical calculations 157–160, 180, 181, the topological surface states in 3D TIs exhibit the weak antilocalization (WAL) effect characterized by a positive conductivity correction equation image. The change of the sign can be easily explained by the fact that the classical motion along the loop is now subject to the conservation of the spin-helicity [see Eq. (38)]. Indeed, upon returning to its original location the particle with momentum −k must have the same helicity as it had initially with momentum k. This requires the change of the direction of the electron spin from σ to −σ (see also Fig. 6b) so that we have the identity

  • equation image((41))

The spin rotation equation image yields another “minus” sign in Eq. (39):

  • equation image((42))

This can also be viewed as the result of the π Berry phase accumulated along the trajectory loop 193, 204.

The explicit calculation of equation image with the account of the warping equation image under condition (36) was carried out in Refs. 157, 180. We note that the conductivity correction equation image (42) has the same form as for a conventional 2D electron system with spin–orbit impurity scattering 205 or with Bychkov–Rashba and Dresselhaus spin–orbit interactions (see e.g., Refs. 206, 207). The reason is that the surface states of 3DTIs with hexagonal warping and conventional 2D electron systems with spin–orbit impurity scattering, Bychkov-Rashba or Dresselhaus spin–orbit interactions belong to the same – symplectic – universality class of disordered systems.

In contradiction with the aforementioned symmetry argument, Ref. 162 has found a nonuniversal prefactor in Eq. (42) which depends on the hexagonal warping strength. It is therefore worthwhile to briefly review the calculation idea and show that different warping terms cancel each other, yielding a universal prefactor equation image in Eq. (42). We take the standard formulas for the quantum-interference conductivity correction, which are expressed diagrammatically in Fig. 7 (see e.g., Refs. 203, 208), treating the warping (33) as weak perturbation onto the isotropic spectrum under condition

  • equation image((43))

The first diagram in Fig. 7a – the standard bare Hikami box – gives the following result:

  • equation image((44))

The nonuniversal prefactor τ/τ0 reflects the difference between the elastic life-time τ0 and transport relaxation time τ for helical carriers 157:

  • equation image((45))

formally described by the vertex renormalization in Fig. 7c. In Eq. (45) the bar denotes averaging over the directions of the momentum, equation image, on the Fermi surface. Since equation image, two other diagrams in Fig. 7a – dressed Hikami boxes – also need to be taken into account. Each of them gives the following correction:

  • equation image((46))

For the unwarped Dirac cone (equation image), equation image, in agreement with the calculations of the WL in graphene 208. Thus, the net result is

  • equation image
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Figure 7. (online color at: www.pss-b.com) Diagrammatic representations for (a) bare and dressed Hikami boxes for the correction to Drude conductivity, (b) Bethe-Salpeter equation for the Cooperon, and (c) equation for the renormalized current vertex in the ladder approximation. Thick lines denote disorder-averaged Green's functions in self-consistent Born approximation, dashed lines – the potential (spin-independent) disorder correlation functions.

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3.3 Detecting surface states by WAL magnetotransport

So far we have treated the surface states in 3D TIs as purely two-dimensional. There is, however, a finite length λ of order of a few nm over which they penetrate into the bulk of the material. Since λ is quite small, the surface WAL conductivity is sensitive to the orientation of an external magnetic field with respect to the surface of the material, which could be used in practice to detect the surface states (see e.g., recent experiment 105). Defining the magnetoconductivity as equation image, one can obtain the following B-field dependences for perpendicular (⟂) and parallel (equation image) field orientations:

  • equation image((47))
  • equation image((48))

These equations contain the same phase-coherence length equation image. However, the magnetic-field scales, equation image and equation image, on which the magnetoconductivity decreases, are distinctly different. The field equation image corresponds to the Aharonov–Bohm magnetic flux of order of the quantum equation image through a typical area equation image enclosed by the interfering trajectories 201, while equation image corresponds to the same flux h/e, but through a significantly smaller area equation image which is proportional to the surface-state penetration length in the bulk 209.

In order to identify the surface state one should extract its penetration length λ from Eqs. (47) and (48). Excluding equation image from Eqs. (47) and (48) we can express λ in terms of two parameters, equation image and equation image, which can be obtained independently from fitting the corresponding experimental data 157:

  • equation image((49))

The knowledge of this penetration length also helps to estimate the critical thickness of the sample at which the two surface states start to overlap and the TI state disappears.

To conclude this section, we should mention that the WAL in 3DTIs is suppressed and may even turn into the WL if an energy gap opens in the surface spectrum, as a result of time-reversal symmetry breaking 159. This should apply to 3DTIs with magnetically doped surfaces (see e.g., Ref. 105). The WL behavior has also been found in ultrathin TI films in which the lowest bulk-state subbands are described by a massive 2D Dirac model 160. Besides, quantum transport in electrically gated TI films may depend sensitively on the coupling between surface and bulk states 181. Such a coupling introduces new time scales on which the crossover from WAL to WL may occur 181.

4 Quantum Hall effect on topological surfaces

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Topological insulators in two dimensions: Quantum spin Hall insulators
  5. 3 Spin-helical transport in 3D topological insulators
  6. 4 Quantum Hall effect on topological surfaces
  7. 5 Superconducting Klein tunneling on topological surfaces
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

Another manifestation of the surface spin-helicity in 3DTIs is the unconventional half-integer quantum Hall effect (QHE; see e.g., Refs. 77, 126, 136, 139, 164). Such unconventional QHE was first observed in single atomic layers of carbon – graphene – 188, and has been regarded as a purely 2D phenomenon. The search for the half-integer Hall quantization in 3DTI systems is therefore a new challenging task (see e.g., Refs. 1, 111, 136, 139, 164). Our discussion of the QHE in 3DTIs follows closely Refs. 139, 164.

4.1 Landau quantization of surface states and conductivity tensor

Using a linear Hamiltonian equation image for a single topological surface state in a perpendicular magnetic field (equation image is the vector potential), we arrive at the following eigenvalue problem:

  • equation image((50))

where equation image denotes an eigenspinor, equation image and a are the raising and lowering operators of a harmonic oscillator, respectively, and equation image is the characteristic Landau level (LL) spacing depending on the magnetic length equation image. The solutions for the LLs and eigenspinors are given by

  • equation image((51))
  • equation image((52))
  • equation image

In the LL basis the Kubo conductivity tensor is

  • equation image((53))

where equation image is the equation image-component of the surface current operator (equation image), and fn is the Fermi occupation number of the nth LL. Introducing operators equation image we find that the current matrix elements equation image and equation image obey the dipole selection rules

  • equation image((54))

allowing only transitions between LLs with equation image. Below we illustrate a link between selection rules (54) and the half-integer quantization of the Hall conductivity.

4.2 Half-integer-quantized Hall conductivity equation image. Role of particle-hole symmetry

The unconventional half-integer-quantized Hall conductivity of the helical carriers is linked to their Berry phases. One way to see this is to calculate the topological Chern number associated with the Berry flux in the Brillouin zone, as discussed in detail, e.g., in Refs. 1, 2, 126. Here, we intend to obtain this result directly from Kubo formula (53), as this allows us to address a more general situation in which particle-hole asymmetry and finite-frequency effects also play a role.

We are interested in the QHE regime that requires strong magnetic fields such that

  • equation image((55))

The realization of this regime depends also on the classical surface conductivity equation image and surface carrier density ns (equation image and equation image are the resistance and magnetic flux quanta). Additionally, we consider the zero temperature limit of conductivity (53) in which all LLs below the Fermi level EF are occupied,

  • equation image((56))

while those above EF are empty (see also Fig. 8):

  • equation image((57))

where N is the index of the highest occupied LL given by

  • equation image((58))

where equation image denotes the integer part. In Eqs. (56) and (57) we introduce the number of the hole LLs in the valence band, Nh, and the number of the particle LLs in the valence band, Np, both counted from the charge neutrality point. Using Eqs. (56), (57) and selection rules (54) we can write the Hall conductivity (53) as

  • equation image((59))
  • equation image((60))

In this equation the first term corresponds to an intraband transition between the LLs N and N + 1, both in the conduction band, whereas the second term (59) corresponds to an interband transition from the occupied valence-band LL −N to the empty LL N + 1 in the conduction band (see also Fig. 8). The third term (60) involves interband transitions from the rest of the occupied (hole) LLs equation image to empty (particle) LLs equation image. Since the summand in Eq. (60) is odd under exchange equation image, this term vanishes for the particle-hole symmetric spectrum with equation image. This means that the interband transitions induced by the spin-raising (equation image) and -lowering (equation image) operators between the particle-hole symmetric LLs equation image and equation image exactly compensate each other. Therefore, in the QHE regime (55) the dc conductivity equation image is

  • equation image((61))

Thus, the half-integer Hall conductivity (61) reflects the particle-hole symmetry in two ways. First, it involves the transitions from the symmetric particle and hole LLs, N and −N, to the lowest empty LL N + 1, and, second, interband transitions from deep LLs equation image to higher empty LLs equation image do not affect equation image provided that the conduction and valence bands contain equal LL numbers, equation image. In Fig. 9 we plot the dc Hall conductivity equation image in the whole magnetic field region and at finite temperature T, using Eq. (53) with equation image.

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Figure 8. (online color at: www.pss-b.com) Schematic of surface Landau level spectrum (51). For particle-hole symmetric case, only two dipole transitions equation image (intraband) and equation image (interband) contribute to Hall conductivity, yielding the half-integer quantized equation image (61); N is the index of the highest occupied Landau level (58), EF is the Fermi level.

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Figure 9. (online color at: www.pss-b.com) DC Hall conductivity equation image [given by Eq. (53) for equation image] in units of equation image versus magnetic field B in Tesla. Inset: same dependence in high-field region; surface state parameters are equation image, equation image, and equation image, and temperature T = 2 K.

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4.3 Particle-hole asymmetry Np ≠ Nh. Accuracy of the half-integer quantization

We now proceed by discussing the Hall conductivity for the particle-hole asymmetric LL spectrum with equation image. In this case equation image acquires an additional contribution due to the interband transitions in Eq. (60),

  • equation image((62))
  • equation image((63))

The last term contains an additional allowed transition from the deepest hole LL equation image to the empty particle state equation image (see also Fig. 10a) with matrix element equation image. Therefore, for equation image we have

  • equation image((64))

In the similar way we obtain equation image for the particle-hole asymmetric LL spectrum with equation image:

  • equation image((65))

In this case there is an additional allowed transition from the hole LL equation image to the highest empty particle state Np induced by the spin-lowering operator equation image (see also Fig. 10b), due to which the correction to the half-integer equation image is negative. To summarize, for large equation image the Hall conductivity is given by

  • equation image((66))

where correction δ determines the accuracy of the half-integer quantization.

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Figure 10. (online color at: www.pss-b.com) Particle–hole asymmetric Landau spectrum with (a) equation image and (b) equation image. Additional allowed interband transitions equation image (a) and equation image (b) give rise to non-universal corrections to the half-integer quantized Hall conductivity [see also Eqs. (64)–(66)].

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4.4 Two-surface model

Since in a 3DTI two surface states, one at the top and one at the bottom of the sample, can contribute to transport 111, it is worthwhile to briefly discuss the behavior of the net Hall conductivity:

  • equation image((67))

where the conductivities of the top and bottom surfaces, equation image, are both given by Eq. (53) with, generally, different classical conductivities equation image and carrier densities equation image. Such difference may arise when one surface faces a substrate, while the other – vaccuum 111. Although, separately, each surface exhibits the half-integer Hall quantization, the plateaus in the net conductivity deviate from equation image and are irregular, e.g., Fig. 11 shows an unusual plateau sequence equation image The QHE with unusual odd and even plateaus has been observed in the 3DTI HgTe where a bulk energy gap is induced by epitaxial strain 111. The bulk energy gap is a necessary prerequisite for the observation of the surface contribution in transport which is, otherwise, dominated by bulk carriers 98–101.

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Figure 11. (online color at: www.pss-b.com) Two-surface DC Hall conductivity equation image [see Eqs. (53) and (67) for equation image] in units of equation image versus magnetic field B in Tesla. Inset: same dependence in high-field region; top (t) and bottom (b) surface state parameters are equation image, equation image, and equation image, equation image. For both surfaces equation image and T = 2 K.

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4.5 AC conductivities. Classical cyclotron resonance

Here we briefly discuss the AC Hall equation image and diagonal equation image conductivities. At zero temperature T = 0 for the particle-hole symmetric LL spectrum, they can be obtained from the Kubo formula (53) as

  • equation image((68))
  • equation image((69))

In Eq. (69) the first and second terms arise from the intraband (equation image) and interband (equation image) transitions (see also Fig. 8), while the third term accounts for the interband transitions from the hole LLs equation image to the particle LLs equation image. The latter transitions contribute to equation image regardless of the presence (or absence) of the particle-hole symmetry.

For weak magnetic fields when

  • equation image((70))

the main contribution to Eqs. (68) and (69) comes from the intraband equation image transition, leading to the classical AC conductivities

  • equation image((71))
  • equation image((72))

where equation image is the cyclotron frequency corresponding to a resonant transition between the highest occupied (N) and lowest unoccupied (equation image) LLs (see also Fig. 8):

  • equation image((73))

Unlike the quadratic-dispersion case, equation image depends on the surface carrier density, equation image, through the Fermi wave-vector equation image, while v is the ns-independent band structure parameter.

Complemented by the Maxwell equations, the magnetotransport theory discussed in this section lays the basis for magneto-optical spectroscopy of TIs, which is currently in the focus of both experimental 71, 102, 103, 113, 119, 124 and theoretical 126, 131, 136, 139, 140, 156, 164, 176 research. In particular, the Faraday and Kerr effects have been observed in the TI materials 71, 103, 113, 119, 124. In high-mobility strained 3DTI HgTe, the Faraday effect reveals the QHE oscillations with low LL indices equation image 124. The broken time-reversal symmetry of the surface QH states has been predicted to give rise to rich magnetoelectric phenomena specific to axion electrodynamics 18, 126, 131, 136, 156.

5 Superconducting Klein tunneling on topological surfaces

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Topological insulators in two dimensions: Quantum spin Hall insulators
  5. 3 Spin-helical transport in 3D topological insulators
  6. 4 Quantum Hall effect on topological surfaces
  7. 5 Superconducting Klein tunneling on topological surfaces
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

Recently, experiments on the TIs have advanced to the investigation of superconducting junctions involving 3DTIs as a weak link between conventional (singlet s-wave) superconductors (Ss) 114–118, 123 (see also Fig. 12). The novelty of such S/TI/S junctions is intimately related to the electron spin helicity: since for the helical states the spin-rotation symmetry is broken, a conventional singlet s-wave S (e.g., Nb or Al) is expected to induce not only the singlet s-wave pairing, but also the unconventional triplet p-wave pairing on the surface underneath the superconductor 125, 132, 134, 151, 152, 172.

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Figure 12. (online color at: www.pss-b.com) (a) S/TI/S Josephson junction and (b) schematic energy diagram of the junction, showing Andreev bound states (ABSs) within induced superconducting gap Δin on TI surface (see also text).

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The key question that arises at this stage is what are the potentially observable phenomena that could serve as a smoking gun for the p-wave superconductivity in S/TI/S systems? One of such phenomena, widely discussed in literature, could be the formation of Majorana bound states, topological midgap states that are known to appear on edges (or in a vertex core) of a p-wave S (see e.g., Refs. 2, 171, 172, 189, 210–214). The Majorana bound states are expected to give rise to unconventional Josephson effects 149, 189, 190, 215, dc superconducting tunneling and current noise 40. For the detection of the Majorana bound states it is essential, however, to avoid hybridization between different Majorana states. In Josephson junctions this requirement is not easy to fulfill because when two Ss are brought in contact their midgap edge states would normally hybridize to become a pair of Andreev bound states (ABSs) with finite energies depending on the phase difference between the Ss (see e.g., Ref. 190 for the case of intrinsic p- and d-wave Ss). In such a situation another closely related phenomenon – the superconducting Klein tunneling – can serve as a smoking gun for the p-wave superconductivity in S/TI/S junctions. Below we discuss this in more details.

5.1 Generalized Fu-Kane model of superconducting proximity effect in 3DTIs

We begin by reviewing the superconducting proximity effect for a single lateral contact between a conventional singlet s-wave S and the surface of a 3DTI (see also Fig. 12). Such a hybrid system was first considered by Fu and Kane 125. In their approach the proximity effect on the surface is described by a singlet pairing potential treated phenomenologically as an energy-independent constant. On the other hand, microscopic approaches (e.g., McMillan's model 216) allow for a more general energy-dependent description of the proximity effect in terms of the Green's functions of the S. We will therefore follow McMillan's model 216 and its adaptations to low-dimensional systems (see e.g., 217–220). In this model, the coupling between the systems is described by a tunneling Hamiltonian, allowing one to calculate the Green's function of the normal system, equation image, by summing up relevant Feynman diagrams generated by the tunneling Hamiltonian. The superconducting proximity is accounted for by a tunneling self-energy equation image in the equation for equation image:

  • equation image((74))

where equation image is the Hamiltonian of the surface state in equation image Nambu (particle-hole) representation, with equation image being itself a equation image matrix in spin space. The self-energy equation image is also a matrix in Nambu space with the following structure 216:

  • equation image((75))

Its off-diagonal elements yield the induced singlet pairing potential (χ is the phase of the superconducting order parameter), while the diagonal elements in Eq. (75) account for the spectrum shift due to the tunneling:

  • equation image((76))
  • equation image((77))

Here the tunneling energy scale is given by equation image [it determines the normal-state escape rate into the S, t is the tunneling coupling strength, and equation image is the normal-state density of states at the Fermi level in S]. equation image and equation image are, respectively, the condensate and single-particle quasi-classical Green's functions of the S [equation image is the gap energy in S]. From Eqs. (74) and (75) we explicitly find the effective surface-state Hamiltonian with induced pairing:

  • equation image((78))

Consequently, one can calculate all the Nambu matrix elements of the surface-state Green's function:

  • equation image((79))

where equation image and equation image are the particle (hole) and condensate Green's functions, respectively. We will briefly discuss the spin structure of the condensate function equation image as it reveals induced mixed s + p-wave superconducting correlations:

  • equation image((80))
  • equation image((81))

Here we compare the general spin structure of equation image (80) involving the ground-state expectation values of all time-ordered pairs of the creation operators equation image with the explicit solution (81) obtained from Eq. (74). Note that in addition to the singlet component equation image, the spin-helicity equation image generates the spin-triplet p-wave component equation image with the same strength equation image [equation image is the unit vector in the momentum direction on the Fermi surface]. Additionally, from the denominator of Eq. (81) we see that an isotropic energy gap Δin is induced in the surface spectrum (see also Fig. 12):

  • equation image((82))

The origin of the mixed s- and p-wave superconducting correlations in Eq. (81) is the broken spin-rotation symmetry of the helical surface state. The situation reminds the mixed singlet-triplet intrinsic superconductivity predicted in Ref. 223 for systems without inversion symmetry. Unlike intrinsic p-wave superconductors, the conservation of the spin helicity in TIs guaranties the robustness the s + p – wave proximity effect against potential impurity scattering 151. We also note that Eq. (81) retains the mixed s + p – wave structure in the limit equation image, which corresponds to the case of Ref. 125.

5.2 Superconducting Klein tunneling and topological Andreev bound states

We consider now a short weak link with length equation image, choosing the superconductor phases on the left and right as equation image and equation image. At the simplest level the junction can be described by the following real-space equations:

  • equation image((83))
  • equation image((84))

for the Nambu spinor equation image which combines the particle equation image and hole equation image spinors. The transverse periodic boundary conditions (84) can be realized in a surface array of Josephson junctions. In order to model the interface scattering inside the weak link we introduce the potential equation image (see Ref. 183 for the scattering-free case). The solution of Eq. (83) can be sought as the sum of independent channels equation image, with equation image and equation image. Searching for ABSs with equation image we obtain the eingevalue equation:

  • equation image((85))

where Tn is the n's channel normal-state transparency:

  • equation image((86))

with equation image, determining the number of the open channels in the junction, equation image. Eq. (85) accounts for the energy dependence of both equation image and equation image.

Superconducting Klein tunneling and topological ABSs occur in the n = 0 channel, as it propagates perpendicularly to the junction barrier and is protected against backscattering (equation image, see also Fig. 13). We obtain the ABS spectrum from Eq. (85) using expansion in small parameter equation image:

  • equation image((87))

These states have orthogonal wave functions. Their spinor structure is simplest at zero energy:

  • equation image((88))
  • equation image((89))

where equation image is the healing length. The orthogonality of (88) and (89) prevents mixing of the counter-propagating modes, yielding the gapless 4π-periodic ABS spectrum (87). Moreover, the particle (88) and hole (89) midgap states are constructed of the same two spinors equation image and equation image describing two degenerate orthogonal Majorana modes. Similar midgap states were found in the core of a vertex on the 3DTI surface 125 (see also reviews 1, 2, 171, 172 for more details).

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Figure 13. (online color at: www.pss-b.com) Topological gapless ABSs equation image (87). They propagate perpendicular to junction barrier without backscattering. These states are orthogonal and 4π periodic (see also text).

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We should emphasize the difference between the topological ABSs (87) and those appearing in p-wave Josephson junctions (see e.g., Ref. 190). First, because of the superconducting Klein tunneling the ABSs (87) are completely independent of the details of the junction barrier. Second, the energy dependence of the proximity effect leads to the cubic and, in general, to higher odd-power terms in the spectrum. Additionally, due to the energy dependence of the proximity effect Eq. (85) has high-energy solutions appearing just below the superconductor gap equation image (see thin blue curves in Figs. 13 and 14; these solutions will be discussed in detail elsewhere).

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Figure 14. (online color at: www.pss-b.com) Nontopological ABSs equation image (90) correspond to oblique incidence at junction barrier. Scattering from barrier results in gapped 2π-periodic spectrum (see also text).

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The transport channels with equation image support nontopological ABSs because, at oblique incidence, scattering from the junction barrier generates an energy gap in the spectrum, making it 2π-periodic (see also Fig. 14):

  • equation image((90))

The topological (87) and usual (90) ABSs give rise to a peculiar AC Josephson current discussed below.

5.3 Fractional AC Josephson effect

The AC Josephson effect is observed in voltage-biased junctions where the supercurrent oscillates with the frequency equation image proportional to the bias voltage V and Cooper pair charge 2e 221, 222. In weak links between intrinsic p-wave superconductors, the 4π-periodic ABSs are expected to give rise to an unconventional AC Josephson effect at the fractional frequency equation image (see e.g., Refs. 189, 190). Leaving aside calculations (see Ref. 190 for details) we present the result for the AC Josephson current in our proximity S/TI/S system:

  • equation image((91))

Here the function equation image is the conventional AC Josephson current carried by the 2π-periodic ABSs (90), while the other two terms arise from the topological 4π-periodic ABSs (87). Unlike the p-wave junctions 189, 190, not only the main fractional frequency equation image, but also equation image and, in general, higher odd harmonics appear due to the energy dependence of the proximity-induced equation image (76) and equation image (77). The estimates of J0, J2, and J2 are given by

  • equation image((92))
  • equation image((93))

We can conclude that favourable conditions for the observation of the topological ABSs exist in narrow weak links with a small number of open channels equation image.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Topological insulators in two dimensions: Quantum spin Hall insulators
  5. 3 Spin-helical transport in 3D topological insulators
  6. 4 Quantum Hall effect on topological surfaces
  7. 5 Superconducting Klein tunneling on topological surfaces
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

We thank L. W. Molenkamp, S.-C. Zhang, A. H. MacDonald, H. Buhmann, C. Brüne, J. Oostinga, B. Trauzettel, P. Recher, E. G. Novik, P. Virtanen, A. Pimenov, G. V. Astakhov, K. Richter, P. M. Ostrovsky, A. D. Mirlin, C.-X. Liu, M. Guigou, P. Michetti and J. Budich for many valuable discussions. This work was financially supported by the German research foundation DFG [Grants No. HA 5893/3-1, No. TK 60/1-1].

Biographical Information

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Topological insulators in two dimensions: Quantum spin Hall insulators
  5. 3 Spin-helical transport in 3D topological insulators
  6. 4 Quantum Hall effect on topological surfaces
  7. 5 Superconducting Klein tunneling on topological surfaces
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

Grigory Tkachov obtained his PhD in 1999 at the I.M. Lifshitz Theoretical Physics Chair of Kharkiv National University, Ukraine. He has worked at Lancaster University (UK) as a post-doctoral fellow sponsored by the Royal Society of the UK and, then, as a research scientist at universities in Germany. His main research interests are focused on the theory of novel “Dirac” materials – topological insulators and graphene – and their possible applications for spintronics and quantum information processing. He is an author of over 35 publications in refereed physics journals.

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Biographical Information

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Topological insulators in two dimensions: Quantum spin Hall insulators
  5. 3 Spin-helical transport in 3D topological insulators
  6. 4 Quantum Hall effect on topological surfaces
  7. 5 Superconducting Klein tunneling on topological surfaces
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

Ewelina Hankiewicz obtained her Ph.D. in 2001 from the Institute of Physics, Polish Academy of Sciences, in Warsaw, Poland. She was Research Associate from 2002 till 2007 at several universities in the US. From 2007 till 2008 she was Assistant Professor at Fordham University in New York. She moved to Wuerzburg University in 2008 as Assistant Professor. Since 2011 she has been Associate Professor at Wuerzburg University. She is an author of over 45 publications. She has been working on problems of theoretical condensed matter physics including spintronics, topological phases of matter, and strongly correlated electron systems.

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