Physics Letters A 306 (2002) 21–24

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Charged particle with magnetic moment in a spacewith a screw dislocation

Sérgio Azevedo

Departamento de Física, Universidade Estadual de Feira de Santana, Km-03, BR116-Norte, 44031-460 Feira de Santana, BA, Brazil

Received 11 February 2002; accepted 3 April 2002

Communicated by A.R. Bishop

Abstract

In this work we study the interaction of a charged quantum mechanical particle with magnetic moment in a space with ascrew dislocation. We focus on the influence of the screw dislocation on the energy spectrum and eigenfunctions of the electronwith spin-1/2 in a uniform magnetic field. We use the approach of the continuum theory of defects that is isomorphic tothree-dimensional gravity. 2002 Elsevier Science B.V. All rights reserved.

PACS: 61.72.Lk; 41.20.Cv; 41.20.-q

1. Introduction

The influence of line defects on the electronic prop-erties of a crystal is an old issue in condensed matterphysics [1]. Recently, alternative approaches to studythis problem have been proposed, which use eithera gauge field [2,3] or a gravity-like approach [4,5]. Themodeling of the influence of the defects on the quan-tum motion of electrons becomes reasonably simplein the latter approach, where the boundary conditionsimposed by the defects are incorporated into a metric.This geometric approach is based on the isomorphismthat exists between the theory of defects in solids andthree-dimensional gravity [6].

E-mail address: sazevedo@uefs.br (S. Azevedo).

A topological defect consists of a core region char-acterized by absence of order and a smooth far fieldregion. In the approximation that we use, the core isshrunk until a singularity. Although not very realistic,this model is very useful to show the appearance ofglobal phenomena, related rather to topology than tolocal geometry induced by the defect. These defects,although formed during phase transitions involvingsymmetry breaking, can be conceptually generated asa “cut-and-glue” process, known in the literature as theVolterra process [7]. Among the variety of line defectsgenerated, the more commons are disclination and dis-location. Dislocations, which are much more realis-tic line defects in a three-dimensional crystal, appearfrequently in semiconductor [8,9]. Recently, Tamuraet al. [10] and Ihara et al. [11] showed that screw dislo-cation can be considered in graphite. The one are alsoknown to alter the energy spectrum of electrons mov-

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9601(02)00381-X

22 S. Azevedo / Physics Letters A 306 (2002) 21–24

ing in a uniform magnetic field, as reported, for exam-ple, by Kaner and Feldman [12] and Kosevich [13].

In a recent work, we [14] have demonstrated thatthe change in topology caused by the defect produceson the spectrum of a free particle an analogue ofthe bound-state Aharonov–Bohm effect in condensedmatter. In another paper [15] we study a charged par-ticle with spin-1/2 in the field of a magnetic string ina space of a disclination. In early 1999, Furtado andMoraes [16] study electron moving in a magnetic fieldin the presence of a screw dislocation. In this Letter weintend to extend this result for a particle with spin-1/2.Namely, we will consider a charged quantum mechan-ical particle with spin-1/2 in the presence of a uniformmagnetic field, parallel to the defect, in a space witha screw dislocation. We study the effects of this defecton the energy spectrum and eigenfunction of this par-ticle. The only approximation that we make is to workin the continuum elastic medium, where the geometricapproach makes sense.

In general, the defects correspond to singular curva-ture or torsion (or both) along of the defect line [6]. Weconsider an infinitely long linear screw dislocation ori-ented along thez-axis. The three-dimensional geom-etry of the medium, in this case, is characterized bynon-trivial torsion which is identified with the surfacedensity of the Burgers vector in the classical theory ofelasticity. This way, the Burger vector can be viewedas flux of torsion. The screw dislocation is describedby the following metric in cylindrical coordinates [17]:

(1)ds2 = (dz + β dθ)2 + dρ2 + ρ2dθ2,

whereβ is a parameter related to the Burger vectorb

by β = b/2π . This topological defect carries torsionbut no curvature. The torsion associated to this defectcorresponds to a conical singularity at the origin.

2. Eigenvalues and eigenfunctions

In this section, we are interested in the study ofbound states and eigenfunctions of a particle withmagnetic moment in a space with a screw dislocationsubmitted to potential of a uniform magnetic field.

The magnetic field, in this case, is given by

(2)�H(ρ) = Φ

πR2Θ(R − ρ)z;

namely, the magnetic field is uniform, in thez-direc-tion, inside a cylinder of radiusR. Note that magneticfield is not influenced by the topological defect.

The contribution due to magnetic field–magneticmoment interaction is given by

(3)H1 = µH(ρ).

In the expression above

(4)µ = gµB

hs,

whereµB = eh/2mc andg is its gyromagnetic ratio.Due to conservation spin, the magnetic interaction

can be replaced by

(5)±1

2gµBH(ρ),

wherein± corresponds to spin projection on the fluxline. From here, we have interested ourselves to theminus sign in which the magnetic moment leads toa binding force. Furthermore, we chooseΦ > 0; forΦ < 0 the sign direction must be reversed.

The vector potential associated to the magneticfield (2) is given by

(6)�A = Φ

2πρ

(ρ2

RΘ(R − ρ)+Θ(ρ −R)

)θ .

In order to compute the modification on the energyspectrum and eigenvalues due to screw dislocation,one needs to write the time-independent Schrödingerequation in the space described by metric (1). In thisspace, the one is

(7)

(1

2m

(h

i�∇β − e

c�A)2

− gµBH(ρ)

2

)Ψ = EΨ,

where H(ρ) and �A(ρ) are given by (2) and (6),respectively. The second term in the above expressionis due to the magnetic interaction, and is given by (5).

The Laplace–Beltrami operator�∇2β , described by

metric (1), is given by

�∇2βΨ = ∂2Ψ

∂z2 + 1

ρ2

(∂

∂θ− β

∂

∂z

)2

Ψ

(8)+ 1

ρ

∂

∂ρ

(ρ

∂

∂ρ

)Ψ.

Using the ansatz

(9)Ψ (ρ, θ, z) = eikzei!θR(ρ),

S. Azevedo / Physics Letters A 306 (2002) 21–24 23

and substituting in (7), using (8) we obtain

1

ρ

d

dρ

(ρ

d

dρ

)R(ρ) − 1

ρ2

[(!′ − δ

ρ2

R2

)]2

R(ρ)

(10)+ g

2δ

2

R2R(ρ) −K2R(ρ) = 0,

whereδ = Φe/2πhc. The constants!′, K are given by

(11)!′2 = (!− βk)2 and K2 = −2mE

h2+ k2.

Note that expression (7) is obtained usingρ < R in (3)since we are interested in the region of uniform field.

We can see that Eq. (7) is the same as that obtainedin Ref. [18]. Therefore the solution of this equation iswell known and its solution is given by

R(ρ) =(δρ2

R2

)|!−βk|1F1

(a,1+ |!− βk|; δ ρ

2

R2

)

(12)× e−δρ2/2R2,

where

a = 2− g

4+ |!− βk| − (!− βk)

2+ k2R2

4δ.

In the expression above we used the change!′ =(!− βk).

From (12) and the expression fora above, we candetermine the condition of normalization of the wave-function as

(13)a = −n.

Substituting Eq. (11) in (13), we obtain the eigen-values

En,l = 2h2

mR2 δ

(n + |!− βk| − (!− βk)

2+ 2− g

4

)

(14)+ h2k2

2m.

Note thatR → 0, with all other parameters fixed;namely the uniform magnetic field coinciding with thedefect, the bound-state tends to be infinite as can beseen from (14). This indicates that it is not a physi-cal limit, at least in the model used here. From (12)and (14), we can see that both eigenfunctions andeigenvalues are affected by the magnetic moment andtopological defect simultaneously.

The ground state,n, l = 0 in (14), is given by

(15)E0,0 = 2h2

mR2δ

(βk + 2− g

4

)+ h2k2

2m.

Note that becauseE depends on theβ , the groundstate is influenced by the topological defect, even thespace is locally flat. Namely, the particle depends onthe global aspect features of this space.

From (15), we can see thatg = 2(2βk + 1) doesnot correspond to bound state; on the other hand, ifg < 2(2βk + 1) we obtain scattering. Finally,g >

2(2βk + 1) corresponds to bound state. Whenβ → 0,space without defect, these results are the same as thatobtained in Ref. [18]. Therefore, we can conclude that,in condensed matter, an electron in a solid with a screwdislocation as topological defect is influenced by theglobal aspects of these defects, affecting in this waysome properties of the solid.

3. Conclusions

In this Letter we used the geometric approach tothe particle charged with magnetic moment in a spacewith screw dislocation, which it is in a region of uni-form field magnetic. This approach leads straightfor-ward solution of the Schrödinger equation, giving ex-act expressions for the energy eigenvalues and eigen-functions. It was shown that a quantum particle withmagnetic moment in space around a topological de-fect like screw dislocation is influenced by the globalaspects of this space. It is important to remark that theregion outside the defect has curvature zero, while thedefect region concentrates all curvature.

We have demonstrated that the binding energy of aparticle is affected by the existence of a screw dislo-cation. Besides, it was investigated the case where themagnetic flux coincide with defect(R → 0), the wellknown Aharonov–Bohm effect, the bound state ener-gies tend to be infinite, so that we conclude that thislimit is not physical. We showed, in the casen, ! = 0,that only exists bound state forg > 2(2βk + 1), andthat this case depends on the both magnetic momentand topological defect, different from the result ob-tained in the space without defect [18]. A detailedstudy on the this dependence can be seen in Ref. [18].In this Letter we intended to show the modification dueto screw dislocation.

24 S. Azevedo / Physics Letters A 306 (2002) 21–24

Acknowledgements

The author acknowledges partial financial supportfrom Programa de Apoio à Instalação de Doutores noEstado da Bahia (PRODOC).

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