Quantum scattering by a magnetic flux screw dislocation

15 October 2001 Physics Letters A 289 (2001) 160–166 www.elsevier.com/locate/pla Quantum scattering by a magnetic flux screw dislocation Cláudio Furtado a , V.B. Bezerra b , Fernando Moraes a,∗ a Laboratório de Física Teórica e Computacional, Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil b Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa, PB, Brazil Received 13 June 2001; accepted 27 August 2001 Communicated by V.M. Agranovich Abstract We investigate the quantum scattering of one electron by a screw dislocation with an internal magnetic flux. The Aharonov– Bohm effect for bound states is analyzed and we demonstrate that the wave function and the energy spectra associated with the particle depend on the Burgers vectors of dislocation and the magnetic flux. We also calculate Berry’s phase associated to the dynamics of the electrons in this background. For some specific values of the magnetic flux there is a matching of the effects produced by the flux and by the dislocation in such a way that there is neither scattering nor Berry’s phase.  2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.Bz; 03.65.Nk; 02.40.Ky 1. Introduction The theory of defects in solids is viewed as the analogous of three-dimensional gravity in the geometric approach of Katanaev and Volovich [1]. In this formalism the boundary conditions imposed by defects in elastic media are accounted for by non-Euclidean metrics. The theory, in the continuum limit, describes the solid by a Riemann–Cartan manifold where curvature and torsion are associated to disclinations and dislocations in the medium, respectively. The Burgers vector of a dislocation is associated to torsion, and the Frank angle of a disclination to curvature. In this theory, the elastic deformations introduced in the medium by defects are incorporated into the metric of the manifold. * Corresponding author. E-mail address: [email protected] (F. Moraes). The quantum and classical problems in the Riemann–Cartan manifold representing a crystal with a topological defect have been extensively analyzed in recent years [2–4]. Also, Kawamura [5] and Bausch and coworkers [6] investigated the scattering of a single particle in dislocated media by different approaches and demonstrated that the equation that governs the scattering is of Aharonov–Bohm type. Recently, alternative approaches using gauges fields have been proposed to study this kind of problem [7,8]. The appearance of topological phases in the quantum dynamics of a single particle moving freely in multiply connected space–times has been studied in a variety of physical systems. The prototype phase being the electromagnetic Aharonov–Bohm one [9], which appears as a phase factor in the wave function of an electron which moves around a magnetic flux line. The gravitational analogue of this effect has also been investigated and discussed [10]. 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 6 1 5 - 6 C. Furtado et al. / Physics Letters A 289 (2001) 160–166 The quantum phase holonomy [11] has purely geometrical origin and plays an important and fundamental role in various areas of physics. In the early eighties Berry discovered [12] that a slowly evolving (adiabatic) quantum system retains information of its evolution when returned to its original physical state. This information corresponds to what is termed Berry’s phase. The appearance of this phase has been generalized to the case of non-adiabatic [13] evolution of a quantum system. In any case the phase depends only on the geometrical nature of the pathway along which the system evolves. This phenomenon has been investigated in several areas of physics. Among them we mention, for example, the investigation of the Berry’s phase acquired by a particle in a gravitational background [14]. There are various experiments which have been reported concerning the appearance of the adiabatic and non-adiabatic geometric phases, including observations on photons [15], neutrons [16] and nuclear spins [17]. The manifestations of Berry’s quantum phase in high-energy electron diffraction in a deformed crystal, which contains a screw dislocation, has been observed [18]. In this case the equation that governs the high-energy electron diffraction is shown to be equivalent to a Schrödinger equation with a time-dependent Hamiltonian. In a recent work [4] we demonstrated the acquisition of Berry’s quantum phase by an electron in the presence of a screw dislocation. In this Letter we study the influence of a combined defect that consists in a screw dislocation with Burgers vector in the z-direction with an internal magnetic field in the same direction but pointing toward the negative z-axis. This defect combines the electromagnetic Aharonov–Bohm effect [9] produced by an internal magnetic flux with an elastic Aharonov–Bohm effect produced by a screw dislocation. We have demonstrated [4] that the torsion of topological defects plays a role of magnetic field in the dynamics of a quantum particle. Here, we analyze the role of the combined defect in the dynamics, and observe that the electromagnetic Aharonov–Bohm effect can compensate the elastic Aharonov–Bohm effect for specific values of the magnetic field. The aim of this Letter is to study the quantum scattering of an electron by such a combined defect, that we will call magnetic flux dislocation. The Aharonov– Bohm effect for bound states is analyzed in the pres- 161 ence of this defect. The geometric phase for this problem is analyzed with emphasis into the role played by these topological defects in this phase. This Letter is organized as follows: In Section 2 we introduce the topological defect and study the scattering problem. In Section 3, we study the Aharonov– Bohm effect for bound states. In Section 4, the Berry’s quantum phase is determined and finally in Section 5 the conclusions are presented. 2. Scattering of a electron by a magnetic flux dislocation In the Katanaev–Volovich theory a dislocated medium that contains one screw dislocation is described by the geometry with the following line element [3]: ds 2 = gij dx i dx j = (dz + β dϕ)2 + dρ 2 + ρ 2 dϕ 2 , (1) where ρ, ϕ and z are cylindrical coordinates and β is a parameter related to the Burgers vector b by β = b/2π . In gravitation theory this metric represents the spatial section of the metric of a chiral cosmic string [19] whose angular momentum vanishes. This defect contains an internal flux whose associated vector potential is −Φ (2) 2πρ which is generated by an internal magnetic field Bz = −Φδ(ρ). This topological defect carries torsion but not curvature. The torsion associated to this defect corresponds to a conical singularity at the origin. The only non-zero component of the torsion tensor in this case is given by the two-form Aφ = T 1 = 2πβδ 2 (ρ) dρ ∧ dϕ, δ 2 (ρ) (3) where is the two-dimensional delta function in flat space. The three-dimensional geometry of the medium, in this case, is characterized by non-trivial torsion, which is identified with the surface density of the Burgers vector in the classical theory of elasticity. In this way, the Burgers vector can be viewed as a flux of torsion given by 
1 T = e1 = 2πβ = b, (4) Σ S 162 C. Furtado et al. / Physics Letters A 289 (2001) 160–166 where we have chosen the following triad representation (one-form basis) for metric (1): e1 = dz + β dϕ, (5) 2 (6) 3 (7) e = dρ, e = ρ dϕ. The torsion two-form is related with the triad by T = de + Ŵ (L) ∧ e, (8) (L) where Ŵ is the Lorentz connection, which is zero for this geometry since there is no curvature involved. This equation leads to result (3) when we substitute (5)–(7) into it. For future comparison with the electromagnetic field strength Fµν we write the torsion in tensor notation as a Tµν = ∂µ eνa − ∂ν eµa , (9) a dx µ ∧ dx ν where the two-form component T a = Tµν a a µ and the triad component e = eµ dx . The Hamiltonian of the charged particle in the presence of a vector potential and in some geometry described by the metric tensor gij is given by     1 q q √ ij A A − − gg H= p p √ i i j . (10) j 2m g c c The Hamiltonian for this problem in the geometry given by (1) is   1 −h̄2 1 ∂ρ (ρ∂ρ ) + 2 (∂ϕ − β∂z + iα)2 + ∂z2 , Ĥ = 2m ρ ρ (11) where α = eΦ/2π and e the electric unit charge. Using (11) the time-independent Schrödinger equation reads   1 h̄2 2 1 2 ∂ + ∂ρ (ρ∂ρ ) + 2 (∂ϕ − β∂z + iα) Ψ − 2m z ρ ρ = EΨ (ρ, ϕ, z). (12) The solution of this equation can be obtained using the ansatz Ψ = Ceikz z φ(ρ, ϕ), (13) where C is a normalization constant and kz is a constant. The equation for ρ and ϕ can be written as   h̄2 1 1 2 2 − ∂ρ (ρ∂ρ ) + 2 (∂ϕ − iβkz + iα) + k 2m ρ ρ × φ(ρ, ϕ) = 0, (14) where k 2 = 2mE/h̄2 − kz2 . We adopt the Dirac phase factor method [20] to solve the scattering problem. A solution of the Schrödinger’s equation for a particle in the presence of this topological defect can be obtained from the wave function corresponding to β = 0 and Φ = 0 using Dirac prescription of multiplication by a phase factor. Therefore we can write φ(ρ, ϕ) as φ(ρ, ϕ) = e{i ∞ 0 (βkz −α) dϕ ′ } φ0 (ρ, ϕ), where φ0 (ρ, ϕ) is the solution of the equation   1 2 2 ∂ρ (ρ∂ρ ) + ∂ϕ + k φ0 (ρ, ϕ) = 0, ρ (15) (16) given by  φ0 (ρ, ϕ) = eiℓφ CJ|ℓ| (kρ) + BN|ℓ| (kρ) , (17) φ ρ = exp −ikρ cos φ + i(βkz − α)φ . (18) where J|ℓ| and N|ℓ| are the Bessel and Neumann functions, respectively. We observe that the solution obtained by Dirac’s prescription is multivalued. But if it really represents the state of the system, then the combined topological defect would have no effect on the probability density, which is obviously not the case. We use the Dirac prescription to show that we can obtain the exact wave function in the presence of this defect. Let us consider now the quantum scattering of a particle by a magnetic flux dislocation. The incident beam has wave function The multivaluedness is now explicit in (18), because φ changes by a factor exp(2πi(α − βkz )) along a closed path around the defect. The procedure to be adopted consists in decomposing φ0 into an infinite number of components (whirling waves), which is multivalued and then apply (15) to each whirling wave to get the exact single-valued wave function. The decomposition of φ0 in partial waves is given by exp(−ikρ cos ϕ) = ∞ (−i)|ℓ| J|ℓ| (kρ) exp(iϕ). ℓ=−∞ (19) Using Poisson summation formula ∞ ℓ=−∞ F (ℓ) = ∞
∞ m=−∞ −∞ dλ F (λ) exp(2πimϕ), (20) C. Furtado et al. / Physics Letters A 289 (2001) 160–166 in (18) we get φ= ∞ (21) Tm (ρ, ϕ), m=−∞ where
∞   −iπλ J|λ| (kρ) dλ exp − 2 −∞  × exp iλ(ϕ + 2πm) . Tm (ρ, ϕ) = (22) In fact, Tm is not single valued, but sum (21) is single-valued. Observe that (22) satisfies Tm (ρ, ϕ) = Tm+1 (ρ, ϕ), which proves that Tm is not single-valued. Therefore, Dirac’s prescription can be applied to a whirling wave satisfying Schrödinger equation in the presence of the defect. The phase is given by (15), but instead of ϕ we must write ϕ + 2πm which is the total angle turned through. The whirling waves for this problem are therefore  Tm (ρ, ϕ) = Tm (ρ, ϕ) exp i(βkz − α)(ϕ + 2πm) . (23) Performing the sum over m, we get φ(ρ, ϕ) =
∞ ∞ m=−∞−∞   −iπ|λ| exp J|λ| (kρ) 2 On defining λ + (βkz − α) as a new variable this can be reverse-Poisson-transformed to give ∞ (−i)|ℓ+(α−βkz )| ℓ=−∞ × J|ℓ+(α−βkz )| (kρ) exp(iℓϕ). (25) From the asymptotic form of the Bessel function it follows that φ(ρ, ϕ) → ∞ and the scattering amplitude associated to this problem is   1 2πδ(ϕ − π) 1 − cos π(α − βkz ) f (ϕ) = √ 2iπk   e−i(ϕ/2) − ieiN(ϕ−π) sin π(α − βkz ) , cos(ϕ/2) (28) where N is the largest integer less than or equal to α − βkz . The expression for the scattering amplitude is analogous to the expression for the Aharonov– Bohm effect [4,9,21–23] for β = 0, which is the result for one particle scattering by a magnetic flux. This result agrees with results of Kawamura [5] and Bausch et al. [6] for Φ = 0, which correspond to the results for a screw dislocation. Also, similar results have been found for the spinning cone [24]. Note that for α = βkz (29) this implies that δl = 0 and f (φ) = 0. Therefore, we have no scattering. In this way for values of the magnetic field such that Φ = h̄bkz /e the scattering does not take place. 3. Aharonov–Bohm effect for bound states   × exp i λ + (βkz − α) [ϕ + 2πm] . (24) φ(ρ, ϕ) = 163 e(−i/2)|ℓ−(α−βkz)| ℓ=−∞   π |ℓ − (α − βkz )|π − × cos kρ − . 2 4 (26) The phase shift associated to the scattering of a single particle by a magnetic flux dislocation is given by  π π δℓ = − ℓ(α − βkz ) + |ℓ|, (27) 2 2 Now, we assume that the scalar particle is restricted to move in a region bounded by the cylindrical surfaces ρ = a and ρ = b, where b > a. Considering the boundaries of this region as impenetrable, we require the wave function ψ(ρ, ϕ, z) to vanish on the boundaries and outside them. It is interesting to examine the quantum particle restricted to move between two cylinders because this permits us to determine the energy spectrum of the particle from the boundary conditions φ(a) = φ(b) = 0. (30) Eq. (30) yields the following result: J|µ| (ka)N|µ| (kb) − J|µ| (kb)N|µ| (ka) = 0, (31) where µ = ℓ + (α − βkz ). Obviously, the energy levels depend on the parameters β and α, because φ(ρ, ϕ, z) depends on these parameters. In order to obtain the energy spectrum explicitly we will consider the situation in which ka ≫ 1 and kb ≫ 1. Then, using Hankel’s 164 C. Furtado et al. / Physics Letters A 289 (2001) 160–166 asymptotic expansion when µ is fixed, we get     µ 2 π cos ka − π − J|µ| (ka) ∼ πka 2 4   2 4µ − 1 µπ π − sin ka − − , 8ka 2 4 N|µ| (ka) ∼ (32)     π 2 µπ − sin ka − πka 2 4   2 µπ π 4µ − 1 cos ka − − + , (33) 8ka 2 4 and similar expressions for J|µ| (kb) and N|µ| (kb) with a interchanged for b. Putting Eqs. (32) and (33) and similar expressions for J|µ| (λb ) and N|µ| (λb ) in the conditions given by Eq. (31), we obtain the following result:   nπ 2 [ℓ + (α − βkz )]2 2 . + k ≈ (34) b−a 4ab The quantum number n corresponds to the oscillating modes in the interval [a, b]. Then, using the definition of k 2 , we finally get that the energy levels are h̄2 kz2 h̄2 (ℓ + (α − βkz ))2 + 2m 2mab 4abh̄2 (nπ)2 − (b − a)2 . + 8mab(b − a)2 Eℓkz Φ = + (35) From the expression for the energy 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 h̄2 kz2 h̄2 (ℓ + (α − βkz ))2 h̄2 + − . (36) 2m 2m a2 8ma 2 Note that for β = 0, the energy expression is formally equal to the Aharonov–Bohm case [25] and for α = 0 this result is the elastic Aharonov–Bohm effect for bound states for an electron in the presence of screw dislocation. From the above equations we see that the wave equation and the energy depend on the Burgers vector and the magnetic flux. Then we conclude that the wave function and therefore the probability of finding a particle around the defect, as well as the energy Eℓkz = are affected by the global (topological) properties of this magnetic flux dislocation. This corresponds to the phenomenon called Aharonov–Bohm effect for bound states [25]. Observe from (36) that for α = βkz the Aharonov–Bohm effect for bound state disappears. In this case, the energy levels (36) do not depend on the parameters α and β, and the spectrum of energy is the same as the one of a quantum particle confined in a cylindrical box. 4. Berry’s quantum phase Now, let us consider the calculation of Berry’s phase [12] in a dislocated medium which contains one screw dislocation with an internal magnetic flux. The quantum evolution of a single particle is given by a time-dependent Schrödinger equation    d  ψ(t) = H Ri (t) ψ(t) , (37) dt where H (Ri (t)) is the system Hamiltonian which depends on a set of externally controllable parameters R(t) = Ri . The solution of the Schrödinger equation in the adiabatic approximation is given by 
t      ψ(t) = exp −i E(t) dt exp iγ (C) ψ R(t) , i 0 (38) where |ψ(R(t)) are instantaneous eigenstates of the Hamiltonian with eigenvalues E(t). The first phase factor is the usual dynamical one. The extra phase factor exp(γ (C)) becomes physically important when the parameters are changed, over some time T , along a path C in parameter space, such that R(T ) = R(0). The non-trivial phase factor is Berry’s phase for the path C, which is given by     γn (C) = i ψ(R)∇R ψ(R) , (39) C where the term    AInJ = ψ(R)∇R ψ(R) (40) is denominated Berry connection. Let us calculate Berry’s phase for the electron in the presence of a magnetic flux screw dislocation. In order to do this we adopt the Dirac phase factor method to describe the C. Furtado et al. / Physics Letters A 289 (2001) 160–166 solution of the Schrödinger equation in the presence of defects in terms of the solution of the defect free case. The Schrödinger equation for one electron in the presence of the screw dislocation is given by i h̄ dψ(x, t) dt   h̄2 1 1 2 2 = ∂z + (∂ρ ρ∂ρ ) + 2 (∂ϕ + iα) 2m ρ ρ  β2 2β − 2 ∂z (∂ϕ + iα) + 2 ∂z2 ψ(x, t), ρ ρ (41) whose solutions can be written as ψ(x, t) = e−iEt /h̄ eikz z φ(ρ, ϕ). (42) In this way the Schrödinger equation is given by i ϕ ϕ0 (kz β−α) dϕ φ0 (ρ, ϕ), is given by Eq. (44). Let the box be transported around a path C threaded by the defect. Due to the degeneracy of the energy eigenvalues, in order to compute Berry’s geometric phase, it is necessary to use the non-Abelian version of the corresponding connection [26,27] given by    AInJ = ψnI (R − x)∇R ψnJ (R − x) , (45) where I and J stand for possible degeneracy labels. The inner product in (45) may be evaluated using the Dirac phase factor,   I  ψn (Ri − xi )∇R ψnJ (Ri − xi )
= i dS ψn∗I (Ri − xi ) Σ Eφ(ρ, ϕ)  1 1 = ∂ρ (ρ∂ρ ) + 2 (∂ϕ + iα)2 ρ ρ    β2 2 2iβkz k φ(ρ, ϕ). (∂ + iα) − 1 − − ϕ ρ2 ρ2 z (43) The solution for this equation can be written in the Dirac phase factor method as φ(ρ, ϕ) = e 165 (44) where φ0 (ρ, ϕ) is the solution for the case where β = 0 (defect free). In this form the solution of the Schrödinger equation in this background is given by ψ(t, ρ, ϕ, z) = eiEt /h̄ eikz φ(ρ, ϕ), where φ(ρ, ϕ) is given by (44). In what follows, we will adopt the analogous treatment of the Aharonov–Bohm effect used by Berry [12] taking into account the degeneracy of the energy. In order to calculate Berry’s phase let us consider a quantum particle inside a perfectly reflecting box, which means that the particle is described by a wave packet that is a linear combination of different eigenfunctions of the Hamiltonian. These eigenmodes we will labeled by n. The vector that localizes the box in relation to the defect is called R. This vector is oriented from the origin of the coordinate system (localized on the defect) to the center of the box. In the absence of defects the wave function corresponding to mode n is given by ψn (R − x), where x represents the coordinates of the particle centered at R. When we consider the defect, the wave function in the interior of the box is obtained by using the Dirac phase factor and  × (kn β − α)ψnJ (Ri − xi ) + ∇R ψnJ (Ri − xi ) . (46) The integrand is calculated and the result is   I  ψn (Ri − xi )∇R ψnJ (Ri − xi ) = i(kn β − α)δI J . (47) This result leads to the following expression for Berry’s phase γn (C) = 2π(α − βkn ), (48) where the labels I , J and δI J have been omitted for convenience. The effect is observed as an interference between the particle in the transported box and one in a box which was not transported around the path. This result demonstrates the existence of Berry’s quantum phase in media with dislocations by use of the geometric theory of defects of Katanaev and Volovich. Note that geometric phase disappears for α = βkn . For β = 0 we obtain Berry’s result for the Aharonov– Bohm effect [12]. For α = 0 we obtain our previous results for a screw dislocation [4]. 5. Conclusions In this Letter we analyzed the quantum dynamics of a particle in the presence of a magnetic screw dislocation. In a recent article we have demonstrated that a single particle in the presence of screw dislocation has a behavior similar to that of a quantum particle in the presence of an Aharonov–Bohm flux. In this way 166 C. Furtado et al. / Physics Letters A 289 (2001) 160–166 the study of a single particle in the presence of this combined defect demonstrates that an internal magnetic flux can compensate the quantum effects produced by the dislocation. We observe that the scattering is of Aharonov–Bohm type and combines the electromagnetic Aharonov–Bohm effect with the elastic Aharonov–Bohm effect due to the flux and dislocation, respectively. In the limit of null magnetic flux we get the result for the elastic Aharonov–Bohm effect. For β = 0 we obtain the well know result for the Aharonov–Bohm effect. For α = βkz the particle does not notice the defect. This a manifestation of the well known Ramsauer–Tounsend effect of quantum mechanics. The analysis of the dynamics of a quantum particle confined between two cylindrical surfaces shows that the energy levels depend on the global features of the background under consideration through parameters α and β which are responsible for the manifestation of the Aharonov–Bohm effect. Observe that for the special case in which α = βKz the energy is the same of a particle in the absence of the defect. 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