Inﬂuence of topology in a quantum ring

In this Letter we study the quantum rings in the presence of a topological defect. We use geometric theory of defects to describe one and two-dimensional quantum rings in the presence of a single screw dislocation. In addition we consider some potential in a two dimensional ring and calculate their energy spectrum. It is shown that the energy spectrum depend on the parabolic way on the burgers vectors of the screw dislocation. We also show that the presence of a topological defect introduces a new contribution for the Aharonov–Bohm effect in the quantum ring. ©


Introduction
In recent years, a number of papers analyzed the quantum effects in mesoscopic structure.For example, observations on the quantum nature of electrons in the interference of electron in Aharonov-Bohm rings has been explored by some authors and their results has been explored in various physical experiments [1,2].In addition, the study of rings of mesoscopic dimension in the presence of external fields exhibits a series of interesting phenomena in physics such as Aharonov-Bohm [3,4], quantum hall [5] and spin orbit [6] effects, persistent currents [7,8] and Berry's phase [9].Furthermore, the simplest model for a ring-like device is a one-dimensional ring in which the applied magnetic field can be represented by an Aharonov-Bohm flux.Such as one-dimensional model has played an important role in the understanding of the quantum interference effect in rings [6,9,10] and the recent progress in nanotechnology has made possible the production of curved two-dimensional layers and nanometer-size objects of several forms, geometry and controlled shapes [10].
The model for a two-dimensional ring has been studied by Tan and Inkson [11,12].These authors proposed a simple model of confinement potential for a two-dimensional quantum ring and have obtained a series of physical properties of this system.Recently, the effect of the surface curvature on the spectral, magnetic and transport properties of nanostructures has attracted substantial interest [13][14][15][16][17] in the physical properties of quantum ring.A few years ago, Bulaev et al. [18] have also investigated the persistent current and magnetic moment in quantum rings in a negative curvature surface (hyperbolic space).
The theory of defects in solids is viewed as the analogous of three-dimensional gravity in the approach of Katanaev and Volovich [19].The theory, in the continuum limit, describes the solid by a Riemann-Cartan manifold where curvature and torsion are associated to disclinations and dislocations, respectively, in the medium.The Burgers vector of a dislocation is associated to torsion, and the Frank angle of a disclination to curvature.In this approach, the elastic deformations introduced in the medium by defects are incorporated in the metric of the manifold.The quantum and classical problems in the Riemann-Cartan manifold representing a crystal with a topological defect have been extensively analyzed in Refs.[20][21][22].It is worth mentioning that two decades ago, Kawamura [23] and Bausch and coworkers [24] investigated the scattering of a single particle in dislocated media by a different approach and demonstrated that the equation that governs the scattering is of Aharonov-Bohm type.Recently, Aurell [25] has investigated the influence of the torsion in the motion of electron in quantum dots.Also in Ref. [26] the influence of the two-dimensional quantum dot in conical geometry was investigated.
In this contribution we investigated the influence of a topological defect in the center of quantum ring by using the geometric description to introduce the topological defect.In addition, we analyze the dynamics of a charged particle constrained to move in a quantum ring of radius r 0 in the presence of topological defect.We study the one-dimensional and twodimensional quantum ring using a hard wall potential model to confine the particle in a finite width region.The choice of this geometry is due to its simplicity, and to the possibility of experimental test.

Cylindrical wire
In this section we will investigate the dynamics of particles in a medium with a screw dislocation along the z direction.The metric describing this medium is (1) We are interested in studying a quantum ring with a constant radius.Considering ρ = R in Eq. ( 1), we obtain the following metric (2) Then, we can obtain the general Hamiltonian in terms of the metric, which is given by The Hamiltonian the describe an electron in quantum ring for the metric 2 is given by ( 4) We use the following ansatz to the solution of Schrödinger equation ( 5) Ψ (ϕ, z) = e imϕ e ikz , and we found the following expression for the energy eigenvalues The first term in the expression ( 6) represents the translational kinetic energy due to the fact that the particle has a free motion along the axis of the defect, a consequence of translational invariance.The second term in (6) represents the rotational kinetic energy (around the wire's center) of the particle.Besides the second term in Eq. ( 6) is different from the case without defects.Notice that the case with topological defect is obtained if we made the following identification , in the Euclidean results.Besides, the quantum number m is shifted by an amount that depends on β which is related to the Burgers vector.This is a manifestation of the Aharonov-Bohm effect of the similar to the case of a onedimensional quantum ring piercewised by a magnetic flux [27].Note that the energy spectrum depends on the Burgers vector b z = 2πβ as well as on R 0 , that is related to the geometry of medium.Now, we will investigate the introduction of an Aharonov-Bohm flux Φ AB 2πR 0 in the center of the one-dimensional quantum ring.In this case, the Hamiltonian is given by ( 7) Using the ansatz Eq. ( 5), we found with m ≡ m − Φ AB Φ 0 and Φ 0 ≡ hc/|e|.Notice that, the quantum number m is shifted in the presence of an external magnetic field.Note that in the present case the we have a one-dimensional quantum ring in the presence of a screw dislocation and Aharonov-Bohm flux.The energy spectrum exhibit a parabolic dependence on the burgers vector, similarly to the dependence in the magnetic flux.The presence of the defect introduce a topological term that produce an Aharonov-Bohm contribution on the energy levels.Analyzing our results we conclude that the particle in a space with a topological defect has behavior like a particle in an Euclidean space in the presence of a effective magnetic flux crossing the ring.This effective flux has two contribution, the first one has topological nature due to topological defect, the other is due to the magnetic flux Φ AB .We can use the magnetic flux as external fine tuning to compensate the topological contribution introduced by the defect.In this way, no Aharonov-Bohm effect in the ring is observed when the term of magnetic flux compensate the topological contribution, the energy spectrum is similar to that one concerning a particle moving in quantum ring in a space without topological defects.

The confinement potential in a two-dimensional ring
In this section we introduce a more realistic model to the quantum ring.We consider a two-dimensional ring in the presence of a topological defect and a confinement potential.In this section, we consider a quantum ring with a screw dislocation along the z direction and a hardwall-like potential.In this way, we impose the particle to move inside a cylindrical shell between the radii a and b, a < b.Therefore, the regions ρ < a and ρ > b are forbidden.This last fact is introduced in the following boundary conditions Ψ (ρ = a) = Ψ (ρ = b) = 0. We introduce a finite width of the quantum ring.In this model the particle moves constrained to a circular strip but is free to move in the perpendicular direction.
We consider the geometric theory of defects to describe the medium with dislocation and this is represented by the metric (1).The Hamiltonian of the particle in the present case can be written using Eq. ( 3) and the time-independent Schrödinger equation obtained has the following form in the Schrödinger equation, we obtain Therefore, the solution of the radial equation is given by ( 12) were α ≡ m − kβ and γ 2 ≡ 2μE h2 − k 2 .Now, we use the boundary conditions Ψ (ρ = a) = Ψ (ρ = b) = 0, to confine the particle in a quantum ring with inner radius a and outer radius b, we have obtained The non-trivial solution for Eq. ( 14) is obtained if the matrix with Bessel functions has null determinant.In this way, we have following condition Therefore, the radial solution becomes with γ α,n being the nth root for Eq. ( 15).The eigenvalues are Obviously, the energy levels depend on the parameter α, because Ψ (ρ, ϕ, z) depends on these parameters.In order to obtain the energy spectrum explicitly we will consider γ a 1 and γ b 1.Then, using Hankel's asymptotic expansion when ν is fixed, we get and similar expressions for J |ν| (γ b) and N |ν| (γ b) with a interchanged for b.Putting Eqs. ( 17) and ( 18) and the expressions for J |α| (γ b) and N |α| , (γ b) in the conditions given by Eq. ( 14), we obtain the following result The quantum number n corresponds to the oscillating modes in the interval [a, b].Then, using the definition of k 2 , we obtain the energy levels given by with n is integer number.Note in this expression the energy of the particle in quantum ring is characterized by appearance of subbands identified by m.The energy depends on the parameter β that characterize the defect.The flux in the center of ring can be added in the expression (20) using the fact m → m − Φ AB Φ 0 .Considering the presence of an Aharonov-Bohm flux in the expression (20), the energy levels are given by Notice that the magnetic field can be used to compensate the influence of topological defect, we can use the Aharonov-Bohm flux as a external fine tuning to compensate the influence of the screw dislocation.From the expression for the energy (20) we see that when b → a, E → ∞, so that in order to get the limit E → const as b → a we have to introduce an attractive potential in the region a ρ b in order to compensate the increasing of the energy of the radial modes in this limit.Doing this we get Analyzing the expression (22) we can see that the first term is a contribution due the free movement of the particle in zdirection.The second term is due the confinement of the particle in the ring geometry in the presence of a screw dislocation.The third term in Eq. ( 22) corresponds to the contribution due the dynamics in a two-dimensional surface inside a three-dimensional space [28].This additional term is like a geometrical confinement potential of a constrained particle in a cylindrical surface immersed in a three-dimensional space.Note that for β = 0, the energy expression is formally equal to the Aharonov-Bohm effect for bound states [29,30].

Concluding remarks
In this contribution we have analyzed the quantum dynamics of an electron in one an two-dimensional ring with a screw dislocation and AB flux in the center.We have investigated the one-dimensional ring geometry and have found the eigenvalues and eigenvector in this geometry.We notice that in this case the AB flux applied in opposed direction of defect can be used to compensate the quantum effects produced by screw dislocations.We have observed that the presence of a topological defect introduces a quantum effect similar to Aharonov-Bohn effect in a quantum ring.We also consider a more realistic model where the electron is confined in the internal region between the radii a and b, a < b.The exact solutions for the eigenvalues and eigen wavefunctions was obtained in this two-dimensional case.A geometrical contribution is obtained in this case due to confinement of the particle in two-dimensional surface immersed in a three-dimensional space.We have demonstrated that all systems investigated exhibit an Aharonov-Bohm effect that combine topological and electromagnetic contribution.