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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The scattering of Dirac electrons by topological defects could be one of the most relevant sources of resistance in graphene and at the boundary surfaces of a three-dimensional topological insulator (3D TI). In the long wavelength, continuous limit of the Dirac equation, the topological defect can be described as a distortion of the metric in curved space, which can be accounted for by a rotation of the Gamma matrices and by a spin connection inherited with the curvature. These features modify the scattering properties of the carriers. We discuss the self-energy of defect formation with this approach and the electron cross-section for intra-valley scattering at an edge dislocation in graphene, including corrections coming from the local stress. The cross-section contribution to the resistivity, _{1−}_{x}_{x}

Boundaries in a topological insulator (TI) host Dirac electrons propagating with a linear dispersion in energy [

In the recent past, the charge carrier mobility in a single graphene layer has been extensively investigated [

On the other hand, the analysis of the conductance properties of boundary states at the surface of a three-dimensional (3D) TI, like Bi_{2}Se_{3}, Bi_{2}Te_{3} and Bi_{1−}_{x}_{x}

Dirac electrons at the surface of nanowires have been measured [

In a previous paper [

In the case of graphene, we see that the contribution of a collection of isolated edge dislocation to resistivity in the Boltzmann semiclassical limit is found to be ∝ 1/_{F}_{F}

Similarly, the presence of screw dislocations of Burgers vector
_{ν}_{1−}_{x}_{x}_{2}Se_{3} and Bi_{2}Te_{3} cannot fulfill the constraint, because their 3D Dirac point is located at Γ. We exhibit the wavefunctions of the bound states and their properties in Section 5.

Superconductive proximity induced at an interface between a 3D TI and a conventional superconductor has been attracting a lot of interest recently, due to the expectation that Majorana bound states (MBS) could exist under appropriate conditions [

In Section 2, we sketch our approach by introducing some generalities about the change of coordinates in the presence of defects and curvature, to make the paper self contained [

The long wavelength dynamics of Dirac electrons propagating on a flat two-dimensional boundary at energies close to the neutrality point of the Dirac cone is described by the Dirac equation:

where ^{0} = _{z}, γ^{1} = _{y}^{2} = −_{x}^{a}^{μ}^{a}_{μ}

The inverse of the tetrads, _{a}^{μ}_{a}^{μ}^{a}_{ν}^{μ}_{ν}_{ab}_{µν}^{a}_{μ}_{b}^{μ}^{a}_{b}

The Dirac matrices γ^{μ}_{a}^{μ}^{a}

On formulating the Lorentz covariance of the Dirac equation locally, we replace the derivatives with the covariant derivatives:
_{ab}_{ab}_{a}_{b}^{a}_{μ}^{b}^{a}_{μ}^{b}^{a}_{ν}∂_{μ}e^{b}^{ν}

Making explicit the rotation of the Dirac matrices and the covariant derivatives, the stationary part of Dirac equation on curved space time is:

In full generality, defining the components of the affine connection as Γ_{λη}^{μ}_{λ}e^{a}_{η}_{a}^{μ}

while the Riemann tensor is defined as:

Equation (^{μ}^{a}

An edge dislocation in the graphene sheet, centered on the origin, is obtained by cutting the plane; let us say in correspondence of the negative _{1} half axis and by adding a half line of carbon atoms [

Let
_{2} –axis, orthogonal to the negative _{1}-axis. The Burgers vector defines a singular coordinate transformation:

where the branch cut of the inverse tangent is on the negative _{1}-axis. The tetrads are easily derived from the infinitesimal transformation ^{a}^{a}_{μ}dx^{μ}

Equation (

The Dirac matrices γ^{μ}_{a}^{μ}^{a}

In the case of an edge dislocation, the Riemann tensor generated by these tetrads, _{μνλ}^{κ}

Since the connection vanishes, the spin connection is also zero. Therefore, only the rotation of Dirac matrices appears in the Dirac equation.

Let
_{k}

the solution of the rotated Dirac equation can be written down in full generality as (

Here, the integral is to be performed along the geodesic path, _{r}^{a}_{μ}k_{a}_{μ}_{μ}

This is indeed a solution, because, by substituting the spinor of Equation (

where the equality to zero stems from the definition of Φ_{0} given in Equation (

Equation (

The quantities,

It follows that the integral in Equation (

The edge dislocation produces a vortex-like singularity in the graphene sheet, unless
_{3} rotational symmetry of the graphene lattice requires that

The phase shifts of a particle incoming with momentum

Defining the outgoing wave (_{r}_{k}

the scattering amplitude,

The total cross-section, with

Close to the neutrality point (limit

The resistivity,

where

The relaxation rate, related to the imaginary part of the self-energy, is expressed in terms of the ^{e f f}

(both waves are normalized to the square root of the area, ^{2}). It depends on the energy, _{p}_{m}

Finally:

Now, the sum can be performed. Defining

The prefactor of Equation (

Equation (

When multiplying this result by the number of dislocations, _{d}

The resistivity is proportional to the density of the defects and inversely proportional to the density of carriers
_{F}

The singularity point, which is the origin of the branch cut, due to the edge dislocation, is also the center of a strain texture induced by the defect. In this section, we show that the contribution of the strain to the total cross-section of Equation (_{F}

where _{ij}

Explicitly, the potential arising from the strain due to the edge dislocation, away from the core of the defect (r >

and is limited to an area < ^{2}, where

is scattered by the potential. The Green’s function is required, which solves the Dirac equation inclusive of the A-B flux,

Its spectral representation is given in

We keep just the contribution coming from the pole Equation (

Using the decomposition of a plane wave in angular momentum eigenfunctions, Equation (

The integral giving the scattered part of the wave function is (

where _{in}

These integrals are special cases of the Weber–Schafheitlin integral:

with (

Similarly, the second contribution yields:

where the same integral as Equation (

The dominant contributions to the

Putting aside the consideration that these matrix elements imply incoming waves of relatively high order (

We conclude that their contribution to the cross-section goes as:

which is higher order, when compared with

We now evaluate the formation the self-energy of the dislocation. In the long wavelength limit, we cannot account for the cost of the pentagon-heptagon defect formation, but we can include the strain cost, which is long range, because it decays as 1/

by adding a perturbation potential,

^{λ} is the Green function for our system, with an A-B flux, λ

We have subtracted a reference term in which the dislocation is absent. This is independent of λ and is immaterial, except for the fact that it provides the vanishing of the perturbation, when λ → 0. The prescription, ^{i}^{ωη}^{+} to get the correct ordering in the Green’s functions, which is

Using the Dyson equation

The strain potential of the edge state is taken from Equation (

According to Equation (^{λ}, Equation (

The correct time ordering requires that, before taking the limit, the variables are exchanged: _{r}_{r′}

Now the integral over

Plugging all together, the awkward prefactor, ^{iπλf/}^{2}, disappears, and the final result is (

This is the expected self-energy for a long-range strain potential.

Bismuth-based materials are mostly studied since the prediction that Bi_{(1}_{−x}_{)}Sb_{x}_{2}Se_{3}, is the prototype of a class of 3D TIs. This material was predicted to have boundary states with energy dispersion forming a Dirac cone centered at the Γ point located within the insulating gap [_{z}^{T}_{z}_{z}_{z}_{z}^{T}

where
_{n}_{α}(

the time reversal symmetry

the inversion symmetry

the three-fold rotational symmetry around the

Here,
_{i}_{i}

For states with
_{F}_{0} _{1}_{2} _{1}_{,}_{2} qualifies the insulator as being topologically non-trivial or trivial. _{F}^{i}^{j}^{ij}

Other important symmetries that could be present are the particle-hole symmetry, Ξ, and the chiral symmetry, Γ.
^{T}^{2} = ^{2} = 0, Γ^{2} = 0, it belongs to the class,

In _{1}_{,}_{2} = 0 and that

In a 3D material, a screw dislocation can occur. The Volterra process for the creation of such a defect requires cutting the material along a half plane. Next, the two free surfaces are twisted with a relative displacement along the direction of the plane and glued back, so that the right-hand side is displaced upward and the left-hand side is displaced downward, as shown in

is fulfilled [_{ν}_{1}G_{1} + _{2}G_{2} + _{3}G_{3}))_{1}_{2}_{3}). (_{1}_{2}_{3}) are the “weak topological invariants”, which contribute in classifying the 3D TI [_{2} variables _{i}_{i}_{ν}_{2}Se_{3} discussed above, there is no such non-vanishing M_{ν}_{ν}_{1}_{−x}_{x}_{0} = 1 and (_{1}_{2}_{3}) = (1_{i}_{3}, the condition is fulfilled.

In the case of the screw dislocation, the Burgers vector is oriented along the defect line, at difference with the edge dislocation, which has the Burgers vector orthogonal to the defect axis. The change of coordinates, describing a screw dislocation with

The coordinates with overlined indexes refer to the local inertial set of coordinates, ^{3} coordinate is displaced by

while the inverse tetrads are:

The only non-vanishing component of the torsion is
_{F}

Here,
_{ν}_{1}_{−x}_{x}

is the slowly varying part in cylindrical coordinates, the coordinates

In case

It follows that the full wavefunction
^{3}, acquires an extra phase
^{3} has to be displaced by

We explicitly derive Ψ. According to Equation (

The slowly varying function, Ψ, turns out not to be an eigenstate of the integer angular momentum,

The functions, _{ν}^{−}^{1} to be determined. The only normalizable solution is with

with the eigenvalue:
^{2} = ^{2}, the energy dispersion is gapless and linear:
_{1}^{2}), provided we choose _{1}^{2}

The state, which is time reversed with respect to Equation (

These states have opposite helicity. At zero energy,

where:

In the class, _{2} [_{L}_{R}

A superconductor in close contact with a normal metal induces Cooper pairing in it by proximity. The bulk states located at the Fermi energy in the metal penetrate the superconductor, even when their energy is below the energy of the superconducting gap, thanks to the Andreev reflection mechanism. The matching at the boundary builds up a superposition of particles and holes in the metal, which are quasiparticles nicknamed “bogoliubons”. A pairing order parameter is induced in the normal metal within a distance from the boundary, which depends on whether transport in the metal is diffusive or “clean” [

An undoped 3D TI, being a semiconductor, has a Fermi energy located inside the gap separating the bulk bands. Therefore, in principle, bulk quasiparticle states cannot be involved in the proximity. However, the interface of a TI hosts boundary states, whose energy dispersion is the Dirac cone occupying energies within the band gap. The boundary acts as a semimetal of reduced dimension and proximity can take place. However, the properties of the bogoliubons differ from those of the topologically trivial metal, because orbital and spin degrees of freedom are strongly coupled in the Dirac boundary states.

In the presence of a magnetic field, a vortex can be trapped, piercing the heterostructure in which a 3D TI slab is sandwiched between two conventional superconductors. We assume that the _{2}Se_{3} becomes a topological superconductor, undergoing the superconducting phase transition with an odd-parity order parameter [

Usually, proximity is described in the Nambu basis, of the kind:

Superconductive proximity induced by an even parity singlet pairing requires that .
^{a}^{a}^{a}_{s}_{u↑}ψ_{u↓}_{g↑}ψ_{g↓}_{u↓}ψ_{u↑}_{g↓}ψ_{g↑}

In the odd parity case, the pairing is chosen with zero spin projection along the spin quantization axis, which is pinned to the

where ∆_{p}_{u↑}ψ_{g↓}_{g↑}ψ_{u↓}_{g↓}ψ_{u↑}_{u↓}ψ_{g↑}_{p}

We will choose different representation bases for the two symmetries. They are connected to the Nambu basis by unitary transformations, but differ from it. This is convenient, as it can be shown that the induced even and odd-parity proximities give rise to the same matrix form of the model Hamiltonian, when the two different bases are adopted, each for the two different cases. We consider a vortex line of charge _{vF}

where, outside the vortex core, the Hamiltonian blocks, _{±}

with:

(_{1}_{2}

(^{iqθ}_{s}_{p}

Equation (

while a unitary transformation, which changes the basis to:

transforms the model to describe the odd-parity pairing, provided ∆_{s}_{p}

We now search for zero energy excitations corresponding to quasiparticles bound to the vortex. When proximity induces s-wave , singlet superconductive correlations, Majorana quasiparticles are bound to the vortex. The wavefunction decays exponentially, as exp(_{o}_{F}^{+}^{−}_{o}

with ^{−}^{1} _{o}_{s}_{o}

Let us compare the two Majorana excitations of Equations (_{x}

where _{x}

Now, let us assume the vortex charge to be _{J}_{J}_{J}_{2}RuO_{4} [

States localized at a vortex core occur also when Cooper pairing induced by proximity in the Hamiltonian of Equation (

Outside the vortex core
_{p}

Inside the vortex core
_{p}_{p}|

The partner state to the one given in Equation (_{gσ}

Away from the mid-gap (

There is no possibility for an MBS to exist, when proximity-induced pairing is odd-parity. The reason can be found in the effective parity of the pairing, which is developed in the TI by proximity. It was shown by Fu and Kane [

The bulk of 3D TI exhibits an odd number of time reversal invariant wavevectors in the Brillouin zone, in the vicinity of which the Hamiltonian can be expanded in the
^{′}

In this work, we have shown that an approach similar to the one often used in graphene [

To make a connection with reality, we have interpreted the results as the modeling of one single edge dislocation in graphene for electrons belonging to one valley only, with no inter-valley scattering. In a picture in which defects are considered as non-interacting and very dilute, a Boltzmann approach to conductivity is acceptable, when averaging over their orientation. The contribution to the resistivity is found to be, of course, proportional to the density of defects, but inversely proportional to the density of carriers, in agreement with the measured resistivity. We have also derived the self-energy of the defect connected with the stress involved in the rearrangement of the lattice in Section 3.3. This information could be relevant to estimate the temperature at which the Boltzmann approach breaks down and a defect-mediated phase transition to a disordered phase takes place. As we find a log-dependence of the self-energy on the size of the dislocation, one could surmise that the transition is of Kosterlitz–Thouless type, as in 2D crystal melting. In this case, a temperature scale is determined the stiffness of the lattice, _{ij}_{F}

On the same lines, the corresponding topological defects in a 3D TI are the screw dislocations. The odd Dirac point, which is responsible for the material being topologically non-trivial, is the best candidate for hosting the branch point of a screw dislocation. Emphasis is put in our work on the fact that the defect could harbor gapless helical electron states propagating along the dislocation axis. Again, we approach the description of boundary states at a flat surface of a 3D TI and at the defect in the long wavelength limit. We have shown that an analytic form of the bound state wavefunctions can be given easily, because the dislocation acts as an effective flux in the 3D Dirac Hamiltonian. The gapless states exist, provided the constraint on the Burgers vector of Equation (_{2}Se_{3} and Bi_{2}Te_{3}, which have the 3D Dirac point at Γ, cannot fulfill the constraint, while the alloy, Bi_{1}_{−x}_{x}

When a 3D TI is sandwiched between two even-parity superconductors, a vortex piercing the structure can host a zero energy bound state, which is a real fermion field. This is a Majorana bound state. There is great excitement at present for the possibility of revealing Majorana bound states in Josephson junctions between TI, in proximity with superconductors [

The conserved quantity is total angular momentum along the vortex axis due to SO. The two Majorana at opposite surfaces have a total angular momentum of 1/2 along the vortex axis, and the sum matches the unitary vortex orbital momentum (the “vortex charge”). However, it is remarkable that, because of their opposite chirality, one of them has no orbital angular momentum and spin polarization up, while the other one has spin angular momentum down, to be subtracted from one positive unity of orbital angular momentum. Hence, opposite chiralities imply that the orbital angular momentum of the vortex fixes the spin content of the Majorana fields. It is also interesting that, in the case the induced superconductivity by proximity is of the odd-parity type, the zero energy states are Andreev bound states localized at the free surface, with damped oscillations away from the surface.

An alternative route to realize in a controlled way structures with emerging MBS excitations could be to resort to pertinently designed Josephson junction rings [

Useful discussions with Piet Brouwer, Emmanuele Cappelluti, Procolo Lucignano and Baskaran are gratefully acknowledged. This work was done with financial support from FP7/2007-2013 under the grant no. 264098—MAMA (Multifunctional Advanced Materials and Nanoscale Phenomena), MIUR (Ministero dell’ Istruzione, dell’ Università e della Ricerca)-Italy through the Prin-Project 2009 “Nanowire high critical temperature superconductor field-effect devices” and Futuro In Ricerca (FIRB)/2013-2015. Vincenzo Parente and Francisco Guinea acknowledge financial support from MINECO (Ministerio Economía y Competitividad), Spain, through grant FIS2011-23713, and the European Union, through grant 290846.

The authors declare no conflict of interest.

The scattering of an electron on an edge dislocation is analogous to the scattering off a Aharonov–Bohm flux,

where _{r}_{k}

When fixing the boundary conditions far away from the scattering center, we use the asymptotic expansion of the Bessel functions, _{ν}

The matching requires that:

The outgoing wave is the superposition of the unscattered wave and of the one diffused with scattering amplitude

The scattering amplitude can be identified in terms of the phase shift as:

We first address the Green’s function for Dirac electrons propagating freely in two dimensions (2D). The Green’s function for the complete 2D Dirac operator, which includes the time dependence, satisfies the defining equation:

where
^{0} = −_{z}^{1} = −_{y}^{2} = −_{x}

satisfies:

according to Equation (_{z}

In the plane wave representation and circular coordinates, we obtain:

where
_{ρ}_{k}

The integral over the modulus of

where _{0}(_{1}(

This result is asymptotically sound. The density of state (including spin degeneracy) is:

where

The spectral representation is also useful. Retaining just the half pole contribution, again, we have for

The Bessel functions, being of integer order, are real. However, by keeping the star in the notation, we intend to remind that, when _{r}_{r′}

The wavefunction in the presence of an A-B flux can be expressed through a Fourier series (_{r}−θ_{p}

where the functions, _{m}_{m}_{F}

They are:

Green’s function is (

By imposing scattering causal conditions for ^{out}^{out}_{m}_{+0(1)+}_{α}^{in}^{in}_{m}_{+0(1)}(

Furthermore, in this case, Green’s function allows for the trivial integration of the angle, _{p}_{p}

Again, we can apply the Graf summation theorem with the condition

to get:

where we have used the approximation _{ρ}_{r}_{r′}

For the scattering of a central potential, outgoing spherical waves for

We can exhibit a simple analytical form of the electronic wavefunctions of a two-band TI in the long wavelength limit, which is described by the model Hamiltonian of Equation (_{0}_{1}, C_{2}

The two bands have opposite parity. The toy model Hamiltonian, close to the Γ point, derived from Equation (

Deep in the bulk, for

where

Localized states at the surface could be:

However, so as they stand, the wavefunctions are not continuous at

Therefore ^{†}^{†} is a good Hamiltonian for the other half space,

Apart from a factor ^{ikk·rk}

As the spinor with B is opposite to the one with C, the determinant vanishes, and the solution exists. By inspection, we find:

By exchanging

Note that, at the neutrality point (

They are time reversed states.

_{2}Se

_{3}

_{2}Se

_{3}

_{2}Se

_{3}nanowire

_{x}

_{2}Se

_{3}

_{2}Se

_{3}, Bi

_{2}Te

_{3}and Sb

_{2}Te

_{3}with a single Dirac cone on the surface

_{1−}

_{x}

_{x}

_{M}_{ν}_{ν}_{0} of Equation(58). However, according to [58], the _{2} rotational symmetry only plays a distinctive role when a magnetic field is present and, in the absence of it, the linearized _{0}.

_{x}

_{2}Se

_{3}

_{x}

_{2}Se

_{3}

_{2}RuO

_{4}

_{x}

_{2}Se

_{3}

_{2}Se

_{3}

_{1−}

_{x}

_{x}

_{2}Se

_{3}and (Bi

_{1−}

_{x}

_{x}

_{2}Se

_{3}

_{4}Te

_{7}

Edge dislocation appearing as a pentagon-heptagon pair in the perfect lattice of a graphene monolayer.

Volterra process for the screw dislocation (taken from [

Integration path in the case of an edge dislocation.