A-geometrical approach to Topological Insulators with defects

# A-geometrical approach to Topological Insulators with defects

D. Schmeltzer Physics Department, City College of the City University of New York
New York, New York 10031
###### Abstract

The study of the propagation of electrons with a varying spinor orientability is performed using the coordinate transformation method.

Topological Insulators are characterized by an odd number of changes of the orientability in the Brillouin zone. For defects the change in orientability takes place for closed orbits in real space. Both cases are characterized by nontrivial spin connections.

Using this method , we derive the form of the spin connections for topological defects in three dimensional Topological Insulators.

On the surface of a Topological Insulator, the presence an edge dislocation gives rise to a spin connection controlled by torsion. We find that electrons propagate along two dimensional regions and confined circular contours. We compute for the edge dislocations the tunneling density of states. The edge dislocations violates parity symmetry resulting in a current measured by the in-plane component of the spin on the surface.

I Introduction

The propagation of electrons in solids is characterized by the topological properties of the the electronic band spinors. Topological Insulators Konig (); Volkov (); Gotelman (); Kreutz (); Mele (); Kane (); More (); Essin (); BernewigZhang (); Ludwig (); davidtop (); ZhangField (); Zhangnew () can be identified by an odd number of changes of the davidtop () of the spinors in the Brillouin zone. As a results non trivial spin connections with a non- zero curvature characterized by the Chern numbers can be identified. In time reversal invariant systems one finds that for Kramer’s states the time reversal operator obeys and one thus the second Chern number for four dimensional space is given by , where is an odd number of orientability changes Nakahara () .

Real materials are imperfect and contain topological defects such as dislocations Ran (); Sinova (),disclinations alberto (); Vozmediano () and gauge fields induced by strain in graphene Baruch (); Guinea () ;therefore, a natural question is to formulate the physics of Topological Insulators in the presence of such defects davidtop (). These topological defects can be analyzed using the coordinate transformation method given in ref.kleinert () which modifies the Hamiltonian for a Topological Insulator with a defect by the metric tensor and the spin connection Pnueli (); Green (); Birrell (); Randono (); Ryu ().

The effect of strain fields dislocations and disclinations plays an important role in material science and can be study using Scanning Tunneling Microscopy () and Transmission Electron Spectroscopy ( ). Therefore we expect that the chiral metallic boundary Wu () will be sensitive to such defects.

In this paper we will introduce the tangent space approach used in differential geometry Nakahara (); Randono (); Ryu () to study propagation of electrons for a space dependent coordinate kleinert (). We find that the continuum representation of the edge dislocation kleinert () generates a spin connection Pnueli (); Green (); Birrell () which is controlled by the vector.

Using this formulation we obtain the form of the topological insulator in three dimensions which simplifies for the surface Hamiltonian (on the boundary). For the surface Hamiltonian we find that the electronic excitations are confined to a two-dimensional region and to a set of circular contours of radius .

The contents of this paper is as follows: In chapter we introduce the gemetrical method. In section we present the geometrical method for the edge dislocations and strain fields. In section we consider the effects of the strain fields on the three- dimensional Topological Insulator (). The Chiral model for the boundary surface is presented in section . Section is devoted to the derivation of the metric tensor and spin connection for an edge dislocation kleinert (). In section we identify the stable solutions. Section is devoted to the stable two dimensional solutions and section is devoted to the stable solution for circular contours . Chapter is devoted to the computation of the tunneling density of states. In section we present results for the two dimensional region . Section is devoted to a large number of dislocations. In section we compute the tunneling density of states for the circular contours . In chapter we consider the current which is given by the in-plane spin component. In section we show that this current is zero for a . In section we show that in the presence of an edge dislocation the parity symmetry is violated, and current, representing the in-plane spin component, is generated. Chapter is devoted to conclusions.

II-The Geometrical method for dislocations and strain fields

A-General Considerations

A perfect crystal is described by the lattice coordinates . For a crystal with a deformation , the coordinates are replaced by where is the local lattice deformation and , is the local coordinate which changes when we move from one point to another.

In a deformed crystal we introduced a set of local vectors which are orthogonal to each other and local coordinates , . The unit vector can be represented in terms of a Cartesian fixed frame space with the coordinate basis ,. In the fixed Cartesian frame the coordinates are given by . Using the Cartesian basis we expand the deformed medium in terms of the local tangent vector : (for the particular case where vectors are given by , the transformation between the two basis is ). Any vector (in the deformed space) can be represented in terms of the unit vectors or the (the tangent vectors in the Cartesian fixed coordinates space). The vector can be represented in two different ways, (when an index appear twice is understood as a summation, ). The dual vector is a and can be expanded in terms of the one forms . We have: , where represents the matrix transformation . The scalar product of the components , defines the metric tensors, (in the Cartesian frame ) and in the local medium frame.

B-Application to the Topological insulators in three dimensions

The three dimensional electronic bands for and can be represented using four projected states Chao (), (the Pauli matrix describes the orbital states and the Pauli matrix describes the spin). The effective Hamiltonian in the first quantized form is given by:

 h3D=ℏv0[ky(σ1⊗τ1)−kx(σ2⊗τ1)+ϵkz(σ3⊗τ1)+M(→k)(I⊗τ3)] (1)

The parameter determines if the insulator is trivial or topological. For and the gap is inverted, namely with and therefore topological ZhangField (); Chao (); Zhangnew ().

Using the metric tensor given by the coordinate transformation ( the transformation between the two sets of coordinates - the one without the dislocation and the second with the dislocation ) , defines the Jacobian where . We find that the derivative for a spinor component , is replaced by the derivative Green ():

 ∇μΨ(α)(→r)=∂μΨ(α)(→r)+18ω(a,b)μ[^Γa,^Γb]αβΨ(β)(→r) (2)

where , are the matrixes: ; ; ; ;.

The determines the covariant derivative Green () is given in terms of the tangent vectors : ;     ;     .

 ωa,bμ=12eν,a(∂μebν−∂νebμ)−12eν,b(∂μeaν−∂νeaμ) −12eρ,aeσ,b(∂ρeσ,c−∂σeρ,c)ecμ (3)

We notice the asymmetry between and : and . As a result the Hamiltonian in eq. in the second quantized form is replaced by:

 H(3D)=ℏv0∫d3r√G[Ψ†(→r)[eμa^Γa(−i∇μ)−EF(I⊗I)+^Γ5(−M0)]Ψ(→r) +B1gμ,ν(∇μΨ†1(→r))(∇νΨ1(→r))−B1gμ,ν(∇μΨ†2(→r)∇νΨ2)]

where , , and is the covariant derivative given in terms of the spin connection given in equation :

C-The Mechanical strain effect on

From the work of young () we learn that the effect of the strained field is different on than on . In the strain decreases the Coulombic gap while increasing the inverted gap strength induced by the spin-orbit interaction. We will use the result in equation to analyze the effect of strain. The strain field (symmetric in ) is related to the stress field and elastic stiffness constant and : . Applying a constant stress one can determine the value of the constant strain field which is related to the tangent vectors . In the present case the spin connection and the Christofel tensor vanish. The metric tensor is given by :. Using this formulation we can investigate the effect of the stress on the at the point . The TI Hamiltonian given in eq. , with the inverted case Zhangnew () . The Hamiltonian in eq. is replaced by:

 H(3D−strain)=ℏv0∫d3r√G[Ψ†(→r)[^Γa(δμ,a+ϵμ,a)(−i∂μ))+^Γ4(−M0)+B(1−2ϵμ,ν)∂μ^Γ4∂ν]Ψ(→r) ≈ℏv0∫d3r√G[Ψ†(→r)[^Γa[(δμ,a(1+<ϵ>)(−i∂μ))]+^Γ4(−M0)+B(1−2<ϵ>δμ,ν)∂μ^Γ4∂ν]Ψ(→r)

In equation we have used the average strain field , . We replace the spinor field by . As a result we obtain:

 H(3D−strain)≈ℏv0∫d3r√G[^Ψ†(→r)^Γμ(−i∂μ)+^Γ4(−M0)(1+<ϵ>)+B(1−2<ϵ>)1+<ϵ>∂μ^Γ4∂μ]^Ψ(→r)

For the compressive case is negative, . As a result we observe that the inverted gap is enhanced .

In the same way we can show that the Coulomb interaction is reduced: We introduce the Hubbard Stratonovici field to describe the Coulomb interactions.

 He−e=∫d3r√G[I(−e)⋅a0Ψ†(→r)Ψ(→r)+(1−2ϵμ,ν)2a0∂μ∂νa0] ≈∫d3r√G[I(−e)⋅a0Ψ†(→r)Ψ(→r)+(1−2<ϵ>)2a0∂μ∂νa0]

Next we rescale and obtain:

 He−e≈∫d3r√G[I(−e)√1−2<ϵ>A0Ψ†(→r)Ψ(→r)+A0∂μ∂μA0] (8)

We observe that for the compressive case the effective charge is reduced and therefore the Coulomb gap decreases, while at the same time the inverted gap increases, in qualitative agreement with young ().

III-The chiral metal with an edge dislocation

A-Description of the Chiral model

The low energy Hamiltonian for the bulk in the family was shown to behave on the boundary surface (the - plane) as a two dimensional chiral metal nature () .

 H=∫d2rΨ†(→r)[hT.I−μ]Ψ(→r)]≡ℏvF∫d2rΨ†(→r)[iσ1∂y−iσ2∂x−μ]Ψ(→r)

is the chiral Dirac Hamiltonian in the first quantized language. is the Fermi velocity, is the Pauli matrix describing the electron spin and is the chemical potential measured relative to the Dirac point. The Hamiltonian for the two dimensional surface describes well the excitations smaller than the bulk gap of the at . Moving away from the point, the Fermi velocity becomes momentum dependent; therefore, we will introduce a momentum cut off to restrict the validity of the Dirac model. The chiral Dirac model in the Bloch representation takes the form: The eigen-spinors for this Hamiltonian are : where is the spinor phase and is the eigenvalue for particles . For holes we have the eigenvalue and eigenvectors . The chirality operator is defined in terms of the chiral phase :

 (→σ×→K|→K|)⋅^z≡sin[χ(kx,ky)]σ1−cos[χ(kx,ky)]σ2 (10)

The chirality operator takes the eigenvalue (counter-clockwise) for particles and (clockwise) for holes .

B-The effect of edge dislocation on a two dimensional chiral surface Hamiltonian

We use the notation , and , to describe the media with dislocations. For an edge dislocation in the direction the vector is in the direction . The value of the burger vector is given by the shortest translation lattice vector in the direction. (For the the length of the vector is times the inter atomic distance ). Following kleinert () we introduce the coordinate transformation for an edge dislocation: with the core of the dislocation centered at . The matrix elements fields for the edge dislocation is given by :

 eaμ=∂μXa(→r);a=1,2;μ=x,y (11)

We express the Burger vector in terms of the the partial derivatives with respect the coordinates in the dislocation frame and for the fixed Cartesian frame kleinert ():

 ∂xe2y−∂ye2x=B(2)δ2(→r) (12)

Using Stokes theorem, we replace the line integral by the surface integral . For a system with zero and non zero we find that the surface torsion tensor integral is equal to , and therefore both integrals are equal to the Burger vector.

 ∮dxμe2μ(→r)=∫∫dxμdxν[∂μe2ν−∂νe2μ]=B(2); ∫∫dxμdxνT(2)μ,ν=∫∫dxμdxν[∂μe2ν−∂νe2μ]=B(2);

where represents the surface element. The tangent components can be expressed in terms of the Burger vector density kleinert () :

 e1x=1;e1y=0 (14)

Using the tangent components, we obtain the metric tensor .

 eaμeaν≡e1μe1ν+e2μe2ν=gμ,ν(→r);eaμebμ≡eaxebx+eayeby=δa,b (15)

The inverse of the metric tensor is the tensor defined trough the equation . Using the tangent vectors, we find in the Burger vector the metric tensor and the Jacobian transformation :

 (16)

The inverse tensor is given by:, , . Using the inverse tensor we obtain the inverse matrix which is given by:

 eμa=ea,νgν,μ=(δa,bebν)gν,μ=eaνgν,μ (17)

Using the components we compute the the transformed Pauli matrices. The Hamiltonian in the absence of the edge dislocation is given by where the Pauli matrices are given by , and . (We will use the convention that when an index appears twice we perform a summation over this index.) In the presence of the edge dislocation, the term must be expressed in terms of the Cartesian fixed coordinates . As a result, the spinor transforms accordingly to the transformation . If is the spinor for the deformed lattice, it can be related with the help of an transformation to the spinor in the undeformed lattice: . Where is the rotation angle between the two set of coordinates: . Using the relation between the coordinates , and with the singularity at gives us that the derivative of the phase which is a delta function, . Combining the transformation of the derivative with the rotation in the plane, we obtain the form of the chiral Dirac equation in the Cartesian space (see Appendix A) given in terms of the Nakahara ():

 iγa∂a˜Ψ(→R)=iδa,bγb∂a˜Ψ(→R)=iγaeμa[∂μ+14[γb,γc]ωbcμ]Ψ(→r) (18)

The Hamiltonian is transformed to the dislocation edge Hamiltonian with the explicit form given by:

 hedge=iσ1∂2−iσ2∂1=iσ1eμ2[∂μ+18[σ1,σ2]ω1,2μ]−iσ2eμ1[∂μ+18[σ1,σ2]ω1,2μ] =i(σ1eμ2−σ2eμ1)(∂μ+18[σ1,σ2]ω1,2μ)

To first order in the Burger vector we find : and , see eqs. in Appendix A.

 hedge≈iσ1(∂y−i2σ3B(2)δ2(→r))−iσ2∂x (20)

In the second quantized form the chiral Dirac Hamiltonian in the presence of an edge dislocations is given by :

 Hedge≈∫d2r√GΨ†(→r)[hedge−μ]Ψ(→r) ≡ℏvF∫d2r√GΨ†(→r)[iσ1(∂y−i2σ3B(2)δ2(→r))−iσ2∂x−μ]Ψ(→r)

is the Hamiltonian in the first quantized language, is the chemical potential and is the two component spinor field.

C- The Identification of the physical contours for the edge Hamiltonian

In order to identify the solutions, we will use the complex representation. The coordinates in the complex representation are given by, ,    ,     ,     . In this representation the two dimensional delta function is given by Conformal (); Nair (). We will use the edge Hamiltonian and will compute the eigenfunctions and . The eigenvalue equation is given by:

 ϵUϵ↑(z,¯¯¯z)=−[∂z+(B(2)√2π)∂z(1¯¯¯z)]Uϵ↓(z,¯¯¯z) ϵUϵ↓(z,¯¯¯z)=[∂¯z+(B(2)√2π)∂¯z(1z)]Uϵ↑(z,¯¯¯z)

The eigenfunctions and can be written with the help of a singular matrix Ezawa () :

 uϵ(z,¯¯¯z)=M(z,¯¯¯z)^Fϵ(z,¯¯¯z)≡⎡⎢⎣e−B(2)2π(1z)00e−B(2)2π(1¯z)⎤⎥⎦(Fϵ↑(z,¯¯¯z)Fϵ↓(z,¯¯¯z))

( and are the transformed eigenfunctions for and respectively .) In terms of the transformed spinors the eigenvalue equation and becomes:

 ϵ(Fϵ↑(z,¯¯¯z)Fϵ↓(z,¯¯¯z))=[rrrI(z,¯¯¯z)00(I(z,¯¯¯z)∗][−∂z00∂¯z](Fϵ↑(z¯¯¯z)Fϵ↓,(z,¯¯¯z))

where , , . We search for zero modes and find :

 ∂zFϵ↓(z,¯¯¯z)=0∂¯zFϵ↑(z,¯¯¯z)=0 (23)

The solutions are given by the holomorphic representation and the anti-holomorphic function . The zero mode eigenfunctions are given by :

 uϵ=0,↑(z)=e−B(2)2π(1z)f↑(z),uϵ=0,↓(¯¯¯z)=e−B(2)2π(1¯z)f↓(¯¯¯z) (24)

Due to the presence of the essential singularity at it is not possible to find analytic functions and which vanish fast enough around such that . Therefore, we conclude that zero mode solution does not exists. The only way to remedy the problem is to allow for states with finite energy.

In the next step we look for finite energy states. We perform a coordinate transformation :

 z→W[z,¯¯¯z];¯¯¯z→¯W[z,¯¯¯z] (25)

We demand that the transformation is conformal and preserve the orientation. This restricts the transformations to holomorphic and anti-holomorphic functions Conformal (). This means that we have the conditions and . As a result we obtain and