Eur. Phys. J. B (2013) 86: 485DOI: 10.1140/epjb/e2013-40160-x

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

Screw dislocation-induced influence of transverse modeson Hall conductivity

Andre G. de Lima1, Armelle Poux2,3, Denise Assafrao1, and Cleverson Filgueiras3,a

1 Departamento de Fısica, Centro de Ciencias Exatas, Universidade Federal do Espırito Santo, 29075-910 Vitoria, ES, Brazil2 UFR Faculte de Sciences et d´ Ingenierie, Universite Toulouse III - Paul Sabatier, 31062 Toulouse, Cedex 9, France3 Unidade Academica de Fısica, Universidade Federal de Campina Grande, POB 10071 Campina Grande, PB 58109-970, Brazil

Received 25 February 2013 / Received in final form 4 July 2013Published online 28 November 2013 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2013

Abstract. The Hall conductivity of an electron gas on an interface showing a topological defect called screwdislocation is investigated. This kind of defect induces a singular torsion on the medium which in turninduces transverse modes in the quantum Hall effect. It is shown that this topology decreases the plateaus’widths and shifts the steps in the Hall conductivity to lower magnetic fields. The Hall conductivity isneither enhanced nor diminished by the presence of this kind of defect alone. We also consider the presenceof two defects on a sample, a screw dislocation together with a disclination. For a specific value of deficitangle, there is a reduction in the Hall conductivity. For an excess of angle, the steps shift to higher magneticfields and the Hall conductivity is enhanced. Our work could be tested only in common semiconductorsbut we think it opens a road to the investigation on how topological defects can influence other classes ofHall effect.

1 Introduction

The integer quantum Hall effect (IQHE) was discoveredin 1980 by von Klitzing et al. [1]. It occurs in two-dimensional electron systems in the limit of strong per-pendicular magnetic fields and sufficiently low tempera-tures. In this regime, plots of the Hall conductivity againstthe magnetic field show plateaus at some field values. Thismeans that Hall conductivity takes quantized values whichare integer multiples of a quantum of conductivity givenby e2/h, where e is the electrical charge and h is thePlanck’s constant. An important aspect of the quantumHall effect is that the Hall conductivity is independentof the material, geometry and microscopic details of thesample. So, the quantized Hall resistance (h/e2) is used asa resistance standard [2,3].

A two-dimensional electron gas (2DEG) built from asemiconductor heterostructure [4] is, in fact, a quasi two-dimensional system. Nowadays, graphene is considered atruly 2DEG and it shows an unusual QHE due to its pe-culiar properties [5,6]. Its honeycomb lattice makes theconducting electrons behave like massless fermions; thusthey are described by the Dirac equation [7].

Over the last few years, technological advanceshave made it possible to investigate non-planar two-dimensional electron gasses (NP2DEG) [8–18]. Bilayergraphene sheets are candidates for the exploration of phe-nomena induced by geometry in a NP2DEG [19–23]. The

a e-mail: cleversonfilgueiras@yahoo.com.br

Hall conductivity (or resistance) is independent of the ge-ometry of a 2DEG but curvature can change the plateaus’widths and shift the steps in the Hall conductivity [8].

It would be interesting to investigate how topologicaldefects can influence the physics of quantum Hall systems.Since this is not a simple task, we think that we must startby considering common semiconductors. The conductingelectrons on tridimensional solids with topological defectsare studied within the geometric theory of defects [24]. Forexample, the Landau levels (LL) in a background spacewith a disclination was investigated in reference [25] andwith a screw dislocation in reference [26]. Since topologi-cal defects introduce singularities in the medium, we musttake into account how these LL are affected by singular-ities [27]. In reference [28], the influence of a disclinationon the IQHE was addressed.

In this work, we will consider a 2DEG with a screw dis-location. We start by considering a tridimensional solid.The electrons are constrained to a thin interface so thequantum Hall effect can manifest. We consider the groundstate at T = 0 K. We do not consider the temperaturesmearing of the steps of the quantized conductivity [8].As stated above, our findings are only applicable to com-mon semiconductors, but we think that understandinghow topological defects affect the QHE in this class of ma-terials is a first step to exploring the influence of topologyon the QHE in other types of materials.

This work is divided as follows: in Section 2, we dis-cuss the Landau levels (LL) around a screw dislocation,

Page 2 of 6 Eur. Phys. J. B (2013) 86: 485

taking into account the singular effects. In Section 3, wecompute the IQHE for some values of the ratio betweenthe Burgers vector and the thickness of a quasi-2DEG. Inorder to have a complete analysis with both disclinationsand screw dislocations, we consider a third type of defectcalled dispiration. This defect shows both curvature andtorsion. In Section 4, we have the concluding remarks.

2 Landau levels

In this paper we study the effect of a screw dislocationon the Hall conductivity of a 2DEG. We consider an in-finitely long linear screw dislocation oriented along thez-axis. The three-dimensional geometry of the medium ischaracterized by torsion which is identified with the sur-face density of the Burgers vector in the classical theoryof elasticity. The metric of the medium with this kind ofdefect is given by (see Ref. [24])

ds2 = (dz + βdφ)2 + dρ2 + ρ2dφ2, (1)

in cylindrical coordinates. In equation (1), β is a parame-ter related to the Burgers vector b by β = b/2π. No cur-vature is induced by this topological defect. Its inducedmetric describes a flat medium with a singularity at theorigin. The only non-zero component of the torsion tensorin this case is given by the two form

T 1 = 2πβδ2(ρ)dρ ∧ dφ, (2)

with δ2(ρ) being the two-dimensional delta function inflat space. We consider a constant magnetic field in thez-direction, B = Bz. We choose the vector potential inthe background space (1) as Aφ = Bρ

2 [25].The Schrodinger equation for a free particle in a back-

ground gμν is given by

−�2

2m1√g∂μ

√ggμν∂νΨ = EΨ, (3)

where g ≡ det gμν . For electrons in the presence of anexternal magnetic field, we have i∂μ → i∂μ − eAμ. Then,the Schrodinger equation for an electron on a medium witha screw dislocation in a homogeneous magnetic field in thez-direction is

− �2

2m∗

[1ρ

∂

∂ρ

(ρ∂

∂ρ

)+

1ρ2

(∂

∂φ− β

∂

∂z

)2]Ψ

+i�eB

2m∗

(∂

∂φ− β

∂

∂z

)Ψ +

e2B2ρ2

8m∗ Ψ

− �2

2m∗∂2Ψ

∂z2= EΨ. (4)

By considering the wavefunction

Ψ(ρ, φ, z) = ei�φeikzzψ(ρ),

(a) (b)

Fig. 1. (a) Cylindrical portion of a 3D solid showing a disloca-tion. (b) Interface on a 3D solid with a screw dislocation. Theline crossing the center of the figure represents the dislocationcore.

with ∈ Z and kz ∈ �, equation (4) yields

− �2

2m∗

[1ρ

d

dρ

(ρd

dρ

)− ν2

ρ2

]ψ − ν�eB

2m∗ ψ

+e2B2ρ2

8m∗ ψ = En,�ψ, (5)

where En,� = E − �2k2

z

2m∗ and ν = − βkz . This equationdescribes the two-dimensional harmonic oscillator in thepresence of a singular delta-like potential. This singularpotential is introduced by a screw dislocation in the prob-lem of a 2DEG in the magnetic field that yields the aboveequation.

Let us point out that equation (5) looks the same asfor a disclination [28]. The only difference is that, in thecase of a screw dislocation, the parameter ν is related tothe Burgers vector while in the case of a disclination itis related to the deficit/excess angle α characterizing thedefect (ν = l/α). Thus, the two problems can be treatedon completely equal footing.

The energy eigenvalues are

En,� = �ωc

(n+

|ν| − ν

2+

12

), (6)

where ωc ≡ eB/m∗ is the cyclotron frequency, n =0, 1, 2, . . . and = 0,±1,±2, . . . From a theoretical pointof view, the electrons on an interface as shown in Figure 1are confined by an infinite square well potential in thez-direction (0 ≤ z ≤ d). This way, we have kz = lπ

d , wherel = 1, 2, 3 . . . Then, the parameter ν in the LL given byequation (6) can be written as

ν = − βlπ

d. (7)

This means that the LL depends on three quantum num-bers: n, and l. Thus, the transverse modes l are goingto have some influence on the Hall conductivity. Actually,we should write En,�,l, but we will keep the notation En,�

since we will consider just the first transverse mode l = 1filled. For a sample without defects, the LL depends onlyon n and . For a 2DEG showing disclinations, there is nodependence on the transverse modes l either.

In both cases, the wavefunction is

ψ(ρ) = C exp(−mωc

4�ρ2

)ρνF

(−n, ν + 1,

mωc

2�ρ2

), (8)

where C is a normalization constant.

Eur. Phys. J. B (2013) 86: 485 Page 3 of 6

As stated above, the screw dislocation induces a sin-gularity in the medium. This means that the electronsare moving in a sample with a singularity. The investiga-tion of quantum systems in this scenario has been carriedout over several years [29–40]. In reference [27], the har-monic oscillator in spaces with singularities was addressedand the effect of such singularities is to restrict the rangeof allowed values for the effective angular momentum ν,that is,

−1 < ν < 1. (9)

This relation holds in order to satisfy the square-integrablecondition of wavefunctions, that is,∫ ∞

0

|ψ|2 ρdρ <∞.

Notice that the degeneracy of the LL is broken due to thetorsion parameter β. In fact, if β ≡ 0,

En,� = �ωc

(n+

|| −

2+

12

). (10)

If > 0, we have En = �ωc (n+ 1/2). On the other hand,if = − ||, then we have the same expression as abovewith n changed by n+ ||, with n and || being non nega-tive integers.

In order to complete our analysis, we will consider athird type of defect called dispiration. This corresponds toa singular curvature and a singular torsion along the defectline. This defect is described by the following metric [41],

ds2 = (dz + βdφ)2 + dρ2 + α2ρ2dφ2. (11)

Note that this only differs from the metric in equation (1)by the inclusion of α, the deficit/excess angle. Then, theparameter ν is now given by

ν =− β lπ

d

α. (12)

If α = 1, which means no curvature, we recoverexpression (7).

In what follows, we are interested in the physical con-sequences due to the presence of a screw dislocation ona 2DEG. We will consider the case α = 1 first. The in-fluence of α was investigated in reference [28] but we willshow here how the Hall effect is influenced by the presenceof both a disclination and a screw dislocation.

3 Hall conductivity

In this section, we first compute the Hall conductivity ona 2DEG showing a screw dislocation characterized by theparameter β. We consider just the integer quantum Halleffect, consisting of a quantization of the conductivity andplateaus at particular values of the external magnetic field.This effect manifests at low temperatures. We considerjust the ground state, T = 0. As we have pointed out

Fig. 2. Landau levels distribution for π βd

< 1. The solid lines

are for En,1 = �ωc

(n + 1

2

)and the dotted lines are for En,0 =

�ωc

(n + π β

d+ 1

2

)(the energy spacing depends on the ratio

π βd).

before, the Hall conductivity is independent of sample ge-ometry. Each LL contributes one quantum of conductivityto the electronic transport. If j LL’s are completely filled,the Hall conductivity will be given by

σH = −jσo, (13)

where σo ≡ e2/h is the quantum of conductivity, e is theelectron charge and h is Planck’s constant. j is the fillingfactor.

Consider that only the fundamental state in the trans-verse direction is occupied, i.e. the l = 1 mode. From thecondition (9), we have

−1 + πβ

d< < 1 + π

β

d. (14)

Taking into account only the case π βd < 1, the possible

values for the angular momentum are = 0, 1. For eachLandau level n, we have

En,1 = �ωc

(n+

12

)(15)

for = 1 and

En,0 = �ωc

(n+ π

β

d+

12

)(16)

for = 0.From these two last equations, we can see how the

degeneracy of the Landau levels is broken, that is, eachLandau level is split into two other levels. In Figure 2 weshow how they are distributed.

In equations (15) and (16), we have n = 0, 1, 2, 3 . . .Performing the transformation n = no − 1, with no =1, 2, 3 . . ., equations (15) and (16) yield

no =[14

+EF

2�ωc

](17)

Page 4 of 6 Eur. Phys. J. B (2013) 86: 485

and

no′ =

[−1

4− πβ

2d+

EF

2�ωc

], (18)

respectively ([x] is the integer part of x). Suppose that,for some value of the external magnetic field, we haveEn,0 = EF (dotted lines in Fig. 2), with n = no−1. As weraise the magnetic field intensity, the next LL which willcoincide with the Fermi energy is En,1, with n = no − 1(solid lines in Fig. 2). This situation corresponds to a Hallconductivity being diminished by one quantum σo. Fromthe equation En,1 = En,0 = EF , we find the plateau widthin this transition to be

ΔB =m∗EF

�e

b2d(

no − 1/2 + b2d

)(no − 1/2)

. (19)

As we continue to raise the external magnetic field inten-sity, one quantum of conductivity σo is diminished againand we have transitions from En+1, 1 = EF (solid lines inFig. 2) to En,0 = EF (dotted lines in Fig. 2). In this case,the plateau width is

ΔB′ =m∗EF

�e

1 − b2d(

no − 1/2 + b2d

)(no + 1/2)

. (20)

We have assumed β = b/2π. The two types of plateaufound above have lower widths in comparison to the flatcase. Notice that if b ≡ 0, ΔB = 0 and ΔB′ coincides withthe plateau width for a flat sample without defects.

For a sample with a dispiration, condition (9) yields

−α+ πβ

d< < α+ π

β

d. (21)

For a deficit angle (α < 1), the LL’s are

En,0;α =�ωc

α

(n+

b

2dα+

12

). (22)

Notice that there is no energy splitting in this case sincecondition (9) allows = 0 only. However, the energy spac-ing is affected and the plateau width is given by

ΔB =αm∗EF

�e

1(no + b

2dα

)2 − 14

. (23)

Without a screw dislocation (b ≡ 0), expression (23) willbe the same as that found in reference [28] for a deficitangle α. In that reference, it was observed that the deficitangle decreases the quantum Hall plateaus’ widths, shift-ing the steps in the Hall conductivity to lower magneticfields and reducing the Hall conductivity. The presenceof a screw dislocation in this case changes only a plateauwidth for a specific value of the deficit angleα. The screwdislocation diminishes the plateau width, shifting it evenmore to lower magnetic fields. There is no extra reductionof the Hall conductivity (see Fig. 3).

In Figure 4 we plot the Hall conductivity for somevalues of the ratio b/d. We also plot σH in the case with-out screw dislocation, which we call the flat case. All the

Fig. 3. Plot of equation (23) for a deficit angle α = 0.7 andno = 15. The screw dislocation diminishes the plateau width,shiftting it to lower values. The Hall conductivity remains thesame, that is, σH = 15σo. The same occurs at any no.

Fig. 4. Hall conductivity versus magnetic field B for differ-ent values of the ratio b/d. The quantum Hall plateau widthsdecrease and the steps in the Hall conductivity are shifted tolower magnetic fields in comparison to the flat case (b/d ≡ 0).Notice that there is no enhancement/reduction of the Hallconductivity.

graphics start with the same filling factor at B = 10 T.This happens because of formula (17) which is the sameas in the flat case. We can see that the steps on the Hallconductivity shift to lower magnetic fields, but there is noenhancement or reduction of the Hall conductivity.

As we raise the ratio b/d, plateaus from equation (19)get higher and the ones given by (20) diminish. On theother hand, as b/d → 0, the influence of both the screwdislocation and sample thickness diminish, returning the

Eur. Phys. J. B (2013) 86: 485 Page 5 of 6

(a)

(b)

Fig. 5. (a) Landau levels on a sample showing a dispirationand (b) Hall conductivity on a 2DEG with a screw dislocation(lower curve) and with a dispiration (upper curve). Notice thatwhile the screw dislocation only shifts the plateaus to lowermagnetic fields (see Fig. 4), the presence of a disclination withan excess angle α shifts the plateaus to higher magnetic fieldsand enhances the Hall conductivity.

Hall conductivity to that of a sample without defects. Foran excess angle, the situation is quite different. Condi-tion (9) yields = −1, 0, 1. This means that each LL withindex no splits as

Eno,1 =�ωc

α

(no − 1

2

),

Eno,0 =�ωc

α

(no +

b

2dα− 1

2

),

Eno,−1 =�ωc

α

(no +

b

2dα+

1α− 1

2

). (24)

Their plot is depicted in Figure 5. Consider EF = Eno,1.As we raise the magnetic field intensity, the next lowest LL

coinciding with EF will be Eno−1,0. Continuing to raisethe magnetic field, a transition goes from this level toEno−1,−1 and the next one goes from this level to Eno−1,1.We then find three different plateaus,

ΔB1 =m∗EF

�e

α− 1 − b2d(

no + 1α + b

2dα − 32

) (no − 1

2

) ,ΔB2 =

m∗EF

�e

1(no + 1

α + b2dα − 3

2

) (no − 3

2 + b2dα

) ,ΔB3 =

m∗EF

�e

b2d(

no − 32

) (no − 3

2 + b2dα

) . (25)

In Figure 5, we see that the presence of an excess anglecauses a shifting of the steps to higher magnetic fieldsand an enhancement of the Hall conductivity. Notice thatat 10 T, the filling factor for the sample is 17 with adisclination and 22 without. This plot was created usingequations (25). For a dispiration, there are three differ-ent expressions for the plateaus. If we consider the ab-sence of screw dislocations, equations (25) will be given byΔB1 = ΔB2 = ΔB3. That is, we have only two types ofplateau, just as was found in reference [28] where just theinfluence of disclinations was investigated. On the otherhand, if we take α = 1 in equations (25), we recover thoseexpressions for a screw dislocation, (19) and (20).

4 Concluding remarks

In conclusion, we have studied the effect of a screw dis-location on the Hall conductivity. It was shown that thethickness of the sample changes the position of the quan-tum Hall plateaus. We have shown that the torsion of themedium, β, decreases the quantum Hall plateau widthsand shifts the steps in the Hall conductivity to lowervalues of magnetic fields. However, there is no enhance-ment/reduction of the Hall conductivity. We noticed theappearance of two types of plateaus in this case. This hashappened because of the splitting of each Landau level intotwo levels for = 0, 1. In order to investigate the presenceof both a screw dislocation and a disclination, we haveconsidered a 2DEG with a dispiration. The steps in theHall conductivity shift to lower values of magnetic fieldswhen the deficit angle α < 1 [28]. The screw dislocationdiminishes the plateau width for a specific deficit anglevalue but does not drop the Hall conductivity. In the caseof an excess angle (α > 1), the presence of a screw dislo-cation induces three different expressions for the plateaussince the Landau levels now split for = −1, 0, 1. In thiscase, there is a shift of the steps to higher magnetic fieldsand an enhancement of the Hall conductivity.

As a final point: if higher transverse modes are excited,we think different plateaus could be observed since the LLwill be split more than twice as addressed in the text forthe first transverse state filled. As we have said, the QHEtranscends common semiconductors, so we think that thestudy of the influence of topology on the Hall conductivityin those materials would lead to its investigation in otherclasses of QHE.

Page 6 of 6 Eur. Phys. J. B (2013) 86: 485

The authors would like to thank the referee for valuable sug-gestions and for critically reading the manuscript. D.A. thanksFAPES for financial support. C.F. thanks CNPq (Universal)and CAPES (Nanobiotec) for financial support.

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