Yet another position-dependent mass quantum model

Ann. Phys. (Berlin) 523, No. 11, 898 – 909 (2011) / DOI 10.1002/andp.201000158

Electron on a cylinder with topological defectsin a homogeneous magnetic field

Cleverson Filgueiras1,∗ and Breno F. de Oliveira 2

1 Unidade Acadêmica de Física, Universidade Federal de Campina Grande, POB 10071 Campina Grande,PB 58109-970, Brazil

2 Departamento de Física, Universidade Federal da Paraíba, POB 5008 João Pessoa, PB 58051-970, Brazil

Received 30 November 2010, revised 16 August 2011, accepted 17 August 2011 by B. KramerPublished online 20 September 2011

Key words Bound states, curved structures, topological defects, two-dimensional systems.

In this work we study the effects of the geometry and topology of a cylinder on the energy levels of anelectron moving in a homogeneous magnetic field. We consider the existence of topological defects as ascrew dislocation and a disclination. When we take the region of movement as the full cylindrical surface,we find that, by increasing the strength of the screw dislocation, the dispersion on the electronic energylevels is affected and monotonically increasing. For an electron moving in an almost flat region we showthat the dispersion on the Landau levels decrease monotonically as we increase the strength of the screwdislocation. The lowest Landau level can reach a zero value, leaving the energy of the system solely givenby the geometry of the cylinder, which does not depend on the magnetic field. In both situations, as wechange the deficit angle of the disclination, we observe that the energy levels are shifted and the magnitudeof such shift depends on the magnetic field. The Landau levels for a flat sample are recovered in the limit ofan infinite cylinder radius.

c© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Geometry and topology show up in different fields of Physics. It is well known the role of them in gravita-tion and cosmology. In soft condensed matter they both have revealed important features about the natureof liquid crystals [1], physics of membranes [2], foams [3] and so on. In condensed matter physics theRiemann-Cartan geometry provides the interpretation of the curvature and torsion tensors as the surfacedensities of the Frank and Burges vectors, respectively [4]. Such feature leaves to many applications as theinvestigation of the fluid flow and the formation of viscous fingering patterns on a two-dimensional conicalbackground space [5], the determination of the self-energy of a single charge in the presence of either acontinuous distribution of disclinations or a continuous distribution of dislocations [6], the quantum scat-tering of an electron by a topological defect called dispiration [7], the effects of torsion on electromagneticfields [8], the Berry quantum phase [9] and geometric phases in graphitic cones [10], etc. Due to tech-nological progress, the physics of curved two-dimensional quantum systems is of great interest too, boththeoretically and experimentally [11].

In this work we investigate the energy levels of an electron on a cylinder with topological defects suchas screw dislocation and disclination. The cylindrical background is the simplest model to investigate theinfluence of geometry in physical systems [12,13]. We consider here such defects to see also the influenceof topology. We consider an electron moving in an infinite cylinder of radius R, with a disclination ofdeficit angle 2π(1 − C) and screw dislocation of Burges vector �k = kz [4, 13]. Both defects have their

∗ Corresponding author E-mail: [email protected], Phone: +55 83 3310 1195, Fax: +55 83 3310 1005

c© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Ann. Phys. (Berlin) 523, No. 11 (2011) 899

axes coinciding with the cylinder axis. Such surface is characterized by the line element

ds2 =(C2R2 + k2

)dφ2 + 2kdφdz + dz2 , (1)

in cylindrical coordinates. The cylinder geometry is obtained for C = 1 and k = 0.The problem addressed here can find applications in the study of a two-dimensional electron gas (2DEG)

in a homogeneous magnetic field perpendicular to it. This is a subject of great interest since importantphysical properties concerned about magnetic, transport and optic effects can be explored [14]. Since onlythe component of the magnetic field pointing along the surface normal governs the Lorentz force and theelectronic transport in 2DEGs, we also consider the presence of a uniform magnetic field crossing thecylinder transversely, with the vector potential in the symmetric gauge [15]

Az = BR sin φz . (2)

Making the transformation

φ ≡ θ

C;

z ≡ Z −(

k

C

)θ , (3)

the line element takes the form

ds2 = R2dθ2 + dZ2 . (4)

Fig. 1 The cylinder in a homogeneous magnetic field: �B is perpendicularto the cylinder axis at φ = 0.

This work is organized as follows: in Sect. 2 we derive the Schrödinger equation using the metric (4)and the gauge (2). We find the energy levels for the system shown in the Fig. 1 and we study their behaviordue to the magnetic field B, the screw dislocation k, the disclination C and the radius R of the cylinder.Section 3 deals with the electron moving in an almost flat region, that is, the so called parabolic model. Inthis case we analyze the changes in the Landau levels due to C and k. We define a dimensionless quantityin order to appreciate the influence of B and R on them. In the last section we have the concluding remarks.

2 The Schrödinger equation and the energy levels for an electronon the surface of a cylinder with defects

In this section we derive the Schrödinger equation for an electron in the background space given by themetric (4). In a general case, the Hamiltonian for a spinless particle in a two-dimensional surface is givenby [16]

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900 C. Filgueiras and B. F. de Oliveira: Electron on a cylinder with topological defects

H =1

2m∗

[

− �2

√g

∂μ

(√ggμν∂ν

)+

iQ�√g

∂μ

(√ggμνAν

)+ 2iQ�gμνAμ∂ν + Q2gμνAμAν

]

+

− �2

2m∗(M2 − K

), (5)

where gμν is the contravariant component of the metric tensor of the manifold, g = det gμν and Aμ is thecomponent of the vector potential of the magnetic field �B. The last term in (5) is the surface potential [17]:M = (κ1 + κ2) /2 is the mean curvature and K = κ1κ2 is the Gaussian curvature of the surface; κ1 andκ2 are the principal curvatures of the surface. For an electron, m∗ is the effective electron mass and Q = e.Using the metric (4), it yields

H = − �2

2m∗R2

∂2

∂θ2+

12m∗

(−i�

∂

∂Z− eAZ

)2

− �2

8m∗R2, (6)

where we have used κ1 = 1R and κ2 = 0, for the cylinder. Note that because of (3) the derivative in Z is

related to the derivatives in z and in φ by the expression

∂

∂Z=

∂

∂z+

k

C

∂

∂φ. (7)

So, this means that the momentum PZ = −i� ∂∂Z , along the Z-direction, is related to the momentum

Pz = −i� ∂∂z and to the angular momentum Lz = −i� ∂

∂φ , both along the z-direction. This relation is

PZ = Pz +k

CLz . (8)

In classical mechanics, for a free particle moving in a space with a screw dislocation, the quantities Pz

and Lz are no longer conserved quantities of motion. But it has been shown in [18] that Pz + kC Lz is

conserved, so we will consider the eigenvalues of PZ as good quantum numbers for our problem suchthat the Schrödinger equation can be written as H [Ψ(θ) exp (iKZZ)] = E [Ψ(θ) exp (iKZZ)], with KZ

being the wavevector in the Z-direction. Saying that, we take into account (8), and then we have the gaugeAZ = Az = BR sin (θ/C). Finally, the Schrödinger equation to be studied is

− �2

2m∗R2

∂2Ψ(θ)∂θ2

+m∗ω2

c

2

[R sin (θ/C) − �KZ

eB

]2

Ψ(θ) − �2

8m∗R2Ψ(θ) = EΨ(θ) , (9)

where ωc = eB/m∗. Going back to the variable φ, we rewrite (9) as

− �2

2m∗R2

∂2Ψ∂φ2

+m∗C2ω2

c

2

[R sinφ − �KZ

eB

]2

Ψ = ΣΨ , (10)

where Ψ ≡ Ψ(φ) and

Σ = C2

(E +

�2

8m∗R2

). (11)

Equation (10) can be rewritten as

− �2

2m∗R2

∂2Ψ∂φ2

+

[m∗C2ω2

c

2

(�KZ

eB

)2

+m∗C2ω2

cR2

4

− m∗C2ω2cR2

4cos 2φ − m∗ω2

c

�KZ

eBC2R sin φ

]Ψ = ΣΨ . (12)

c© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org


Next, we define the following dimensionless parameters,

χ ≡√

2πeB

�R (13)

and

γ ≡ eBR

�KZ. (14)

Using these definitions together with the coordinate transformation φ = 2α − π/2, we arrive at

∂2Ψ∂α2

+ [λ + 2a cos 2α + 2b cos 4α] Ψ = 0 , (15)

where λ = − 1π2

χ4C2

γ2 − 12π2 χ4C2 + E, E = 8m∗R2

�2 Σ, a = − 1π2

χ4C2

γ and b = − 14π2 χ4C2. Equation

(15) is a periodic differential equation. It is a special case of the differential equation

d2F

dx2+ [λ + V (x)] F = 0 , (16)

which is called Hill’s equation [19–21]. In (16), V (x) is an even periodic function, with period π, expandedas a Fourier series

V (x) =∞∑

κ=1

2uκ cos (2κx) . (17)

In our case we have u1, u2 �= 0 and uκ ≡ 0, for κ ≥ 3.The formal solutions of (15) are obtainable as a trigonometric series which can be either 2π− or π−

periodic. Note that our original potential in (10) is 2π−periodic. To match this condition we must considerjust the π−periodic solutions of (15). So, the even π−periodic solutions are given by

Φeven(α) =∞∑

n=0

A2n cos 2nα , (18)

while the odd π−periodic solutions are

Φodd(α) =∞∑

n=0

A2n+2 sin(2n + 2)α . (19)

Substituting (18) in (15) yields the relations

2λnA0 + 2aA2 + 2bA4 = 0 ,

2aA0 + [λn + b − 4]A2 + aA4 + bA6 = 0 ,

2bA0 + aA2 + [λn − 16]A4 + aA6 + bA8 = 0 ,

bA2n−2 + aA2n +[λn − 4(n + 1)2

]A2n+2 + aA2n+4 + bA2n+6 = 0 , (n ≥ 2) . (20)

From them, we build the pentadiagonal matrix⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 2a 2b 02a b − 4 a b

2b a −16 a. . .

0 b a −36. . .

. . .. . .

. . .

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

.



The eigenvalues λ in the relations (20) are obtained by diagonalizing this matrix. Similarly, for the solution(19), we have

[λn − b − 4]A2 + aA4 + bA6 = 0 ,

aA2 + [λn − 16]A4 + aA6 + bA8 = 0 ,

bA2n−2 + aA2n +[λn − 4(n + 1)2

]A2n+22 + aA2n+4 + bA2n+6 = 0 , (n ≥ 2) (21)

and⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

−b − 4 a b 0a −16 a b

b a −36 a. . .

0 b a −64. . .

. . .. . .

. . .

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

The matrix diagonalization procedure is done numerically using the GNU Scientific Library and thematrices above were truncated at n = 20. In the Figs. 2a, 2b and 3a we plot the highest four energies forthe first matrix as function of the magnetic field B, the screw dislocation k and the radius of the cylinderR, respectively. The Figs. 4a, 4b and 5 are plotted using the second matrix above.

(a)0 2 4 6 8 10 12

520

550

580

610

640

670

700

B/T

E2m

∗ R2/h

2

λ

1

λ2

λ3

λ4

(b)0 1 2 3 4 5

−30

−20

−10

0

10

20

30

40

50

60

k

E2m

∗ R2/h

2

λ

1

λ2

λ3

λ4

Fig. 2 (a) Plot of the energies versus the magnetic field B; here, KZ = 40(e/�)cm−1, R = 3 × 10−3cm andC = 1.2. (b) Plot of the energies versus the screw dislocation parameter k; here, B = 2T, Kz = lφ = 2(e/�)cm−1,R = 3×10−3cm and C = 1.2. As we increase the screw dislocation k, the dispersion is affected and them the energylevels increase. The eigenvalues come from the diagonalization of the first matrix showed in the text (even solutions).

From all these graphics we can see that each energy level has an asymptotic behavior as the radius Rincreases. In fact, in the limit R → ∞ they become the energy levels of a charged particle moving in aplane with a perpendicular constant magnetic field, that is, the usual Landau levels. On the other hand, aswe increase the screw dislocation parameter k, the energy levels increase according to the Figs. 2b and 4b.To see this effect, we have considered, from relation (8), that

KZ = Kz +k

Clφ , (22)

where Kz and lφ are, respectively, the eigenvalues of the wavevector and the angular momentum, both inthe z direction. The disclination parameter C shifts the energy curves, a fact which is showed in Fig. 3b,for an even solution. We note that such shifts are less pronounced for magnetic fields ranging from 1 to 3Tand are greater otherwise.



(a)1 1.5 2 2.5 3

x 10−3

3.5

4

4.5

5

5.5

6

6.5

7x 10

7

R/cm

E2m

∗ /h

2

λ1

λ2

λ3

λ4

(b)0 2 4 6 8 10 12

510

520

530

540

550

560

570

580

B/T

E2m

∗ R2/h

2

C= 0.8 C= 1.0 C= 1.2

Fig. 3 (a) Plot of the energies versus the radius R of the cylinder; here, B = 2T, KZ = 40(e/�)cm−1 and C = 1.2.The energy levels have an asymptotic behavior as the radius R increases. In the limit R → ∞ they become the LandauLevels for an electron in a flat region of movement. (b) Plot of the third highest energy level for different values ofdisclination parameter C; here, KZ = 40(e/�)cm−1, R = 3 × 10−3cm and B = 2T. The eigenvalues come fromthe diagonalization of the first matrix showed in the text (even solutions).

(a)0 2 4 6 8 10 12

520

540

560

580

600

620

640

660

B/T

E2m

∗ R2/h

2

λ

1

λ2

λ3

λ4

(b)0 1 2 3 4 5

−50

−30

−10

10

30

50

k

E2m

∗ R2/h

2

λ

1

λ2

λ3

λ4

Fig. 4 (a) Plot of the energies versus the magnetic field B; here, KZ = 40(e/�)cm−1, R = 3 × 10−3cm andC = 1.2. (b) Plot of the energies versus the screw dislocation parameter k; here, B = 2T, Kz = lφ = 2(e/�)cm−1,R = 3 × 10−3cm and C = 1.2. Again, as we increase the screw dislocation parameter k, the dispersion is affectedand them the energy levels increase. The eigenvalues come from the diagonalization of the second matrix showed inthe text (odd solutions).

3 The parabolic model

In this section we analyze the harmonic approximation for the problem described above. To study this case,we rewrite the Schrödinger equation (10) as

− �2

2m∗R2

∂2Ψ∂φ2

+�

2

2m∗ K2ZC2 [γ sin(φ) − 1]2 Ψ = ΣΨ . (23)

with γ given by (14). So, we can define the effective potential as

V (φ)K2

Z

= C2 [γ sin(φ) − 1]2 . (24)



1 1.5 2 2.5 3

x 10−3

1.5

2.5

3.5

4.5

5.5

6.5x 10

7

R/cm

E2m

∗ /h

2

λ1

λ2

λ3

λ4

Fig. 5 Plot of the energies versus the radius R of the cylin-der; here, B = 2T, KZ = 40(e/�)cm−1, R = 3 × 10−3cmand C = 1.2. The energy levels have an asymptotic behavioras the radius R increases. In the limit R → ∞ they becomethe Landau Levels for an electron in a flat region of movement.The eigenvalues come from the diagonalization of the secondmatrix showed in the text (odd solutions).

The Figs. 6a and 6b show the profile of (24), as a function of φ for some values for the dimensionlessparameter γ that gives the strength of the interaction between the charge and the field, for a given KZ . Notethat the existence of a disclination (C �= 1) does not change the location of the minima of the potential,that is, they are in the same positions as in the case of a cylinder without defects (C = 1 and k = 0). Thiscan be seen in Fig. 6b where we plot V (φ)/K2

Z versus φ for different values of C (the effect due to thescrew dislocation k shows up in γ).

(a)0 pi/2 pi 3pi/2 2pi

0

10

20

30

40

φ

V(φ

)/K

2 Z

γ = 1γ = 3γ = 5

(b)0 pi/2 pi 3pi/2 2pi

0

10

20

30

40

50

60

φ

V(φ

)/K

2 Z

C = 0.8C = 1.0C = 1.2

Fig. 6 (a) Plot of the potential versus φ for some values of the dimensionless parameter γ (strength of the interactionbetween the charge and the field); γ affects the minima location. Moreover, for small γ we have one minimum whilethat for high γ two minima can show up. (b) Plot of the potential with γ = 5 versus φ. The existence of a disclination(C �= 1) does not change the location of the minima of the potential in comparison with the case of a cylinder withoutdefects (C = 1 and k = 0).

Taking γ ≈ 1 as a reference, we can see that the effective potential has one shallow minimum atφ = π/2 for small γ while it has two minima for greater γ. In the first case, the carriers tend to localize ina quasi-one-dimensional channel on one side of the cylinder, where the component of the field normal tothe surface is minimum. The relative sign of KZ and the field decide in which side of the cylinder is a suchminimum. In the former case, the two minima are located at φ = 0 and φ = π, that is, above and below thecylinder. The carriers form two quasi-one-dimensional channels located where the component of the fieldnormal to the cylinder is maximum, and the field is either entering or exiting the surface [15].

Now we define the new coordinate x = Rφ and write (23) as

− �2

2m∗∂2Ψ∂x2

+m∗ω2

cC2

2

[R sin(x/R) − �KZ

eB

]2

Ψ = ΣΨ . (25)



The minimum of the potential present in (25) occurs at xm = R sin−1(

�KZ

eBR

). Expanding it around xm

up to the harmonic term, we arrive at

− �2

2m∗∂2Ψ∂x2

+m∗ω2

2(x − xm)2 Ψ = ΣΨ , (26)

where

ω = Cωc cos[sin−1

(�KZ

eBR

)]= Cωc

√

1 −(

�KZ

eBR

)2

. (27)

These results are approximate and valid for a sufficiently strong normal to the surface component of themagnetic field and for electrons with the Landau oscillator suspension center far enough from the sideedges of the cylindrical strip [22].

Defining the dimensionless coordinate ξ ≡√

m∗ωc

�x and the dimensionless parameter ξm ≡

√m∗ωc

�xm, the Schrödinger equation (26) can be put in the following way:

− ∂2Ψ∂ξ2

+ C2 cos(

ξm

R′

)(ξ − ξm)2 Ψ =

2Σ�ωc

Ψ , (28)

where R′ = R/√

�/m∗ωc. The wavefunctions are given, as we know, in terms of the Hermite polynomialsH(x). But now we have the effect of the dispersion on them. The wavefunctions are

Ψn(ξ) = AnHn (ξ − ξm) e−(ξ−ξm)2 , (29)

where An is a constant of normalization. The Figs. 7a and 7b show how the dispersion given in terms ofξm affects, for C = 1, the first two wavefunctions (29) (n = 0 and n = 1, respectively). So, we see from

(a) flat xm = 0.5 xm = 3

2 m *wcZ x

K4 K2 0 2 4 6

y0

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

(b) flat xm = 0.5 xm = 3

2 m *wcZ x

K4 K2 0 2 4 6

y1

K0,8

K0,6

K0,4

K0,2

0,2

0,4

0,6

0,8

Fig. 7 (a) Plot of Ψ0 versus the dimensionless coordinate ξ for some values of the dimensionless parameter ξm; here,C = 1 and R′ = 100 (dimensionless). We also show Ψ0 for an electron in a plane with a perpendicular homogeneousmagnetic field. (b) Plot of Ψ1 versus the dimensionless coordinate ξ for some values of the dimensionless parameterξm; here, C = 1 and R′ = 100 (dimensionless). We also show Ψ1 for an electron in a plane with a perpendicularhomogeneous magnetic field. We can see that the electronic states tend to localize around the minimum ξm, that is,manipulating B, R or k we dislocate the wavefunctions around the cylinder. The wavefunctions were normalized inthe region of movement.



them that the electronic states tend to localize around the minimum ξm, that is, manipulating B, R or k wedislocate the wavefunctions around the cylinder.

In the Figs. 8a and 8b we plot the n = 0 and n = 1 wavefunctions for a fixed ξm and for some differentvalues of C. In this case the disclination C does not change the minimum location, as we noted in theanalysis of the potential above. Just the height of the wavefunctions are affected, which means that theapex of the density of probability changes as we modify C.

(a) C = 1 C = 0.8 C = 1.4

2 m *wcZ

x

K6 K4 K2 0 2 4 6

y0

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

(b) C = 1 C = 0.8 C = 1.4

2 m *wcZ x

K4 K2 0 2 4 6

y1

K0,6

K0,4

K0,2

0,2

0,4

0,6

Fig. 8 (a) Plot of Ψ0 versus the dimensionless coordinate ξ for ξm = 1 (dimensionless); here, R′ = 100 (di-mensionless). (b) Plot of Ψ1 versus the dimensionless coordinate ξ for ξm = 1 (dimensionless); here, R′ = 100(dimensionless). The disclination C does not change the minimum location. The wavefunctions were normalized inthe region of movement.

The eigenvalues associated with (29) are the Landau levels which are given by

Σn =(

n +12

)�Cωc

√

1 −(

�KZ

eBR

)2

, (30)

with n = 0, 1, 2, 3, .... So, the energy levels are

En =(

n +12

)�ωc

C

√

1 −(

�KZ

eBR

)2

− �2

8m∗R2. (31)

Taking into account the dimensionless parameter (13), we put (31) into the following form:

E =(

n +12

)χ2

C

√

1 − 4π2j2

χ4, (32)

with E =(En + �

2

8m∗R2

)/

(�2

2m∗πR2

)and j = RKZ . These energy levels have dispersions as function

of KZ which can be decomposed as Kz + kC lφ. Taking into account just the parameter C (that is, making

k = 0), we first investigate the effect of the disclination on the energy levels E. In Fig. 9a we plot theenergy as function of j, for different values of C and also for fixed B and R (that is, for fixed χ). Wecan see that for C < 1 the semi-elliptical dispersion curves shift upward(the maximum energies increase)while that for C > 1 they shift downward (the maximum energies decrease). Since the condition x Rmeans that the electrons are in an almost flat region, we plot in the same figure the Landau levels for the



−16 −12 −8 −4 0 4 8 12 160

50

100

150

200

250

300

350

RKZ

E2m

∗ R2/h

2

flatC = 0.8C = 1.0C = 1.2

n = 2

n = 1

n = 0

2 4 6 8 100

50

100

150

200

χ

E2m

∗ R2/h

2

C = 0.8C = 1.0C = 1.2

n = 1

n = 0

(a) (b)

Fig. 9 (a) Plot of energy E versus RKZ , that is, the dispersion of Landau levels with different indices n (n = 0, 1, 2)

in a cylindrical sample for different values of C and k = 0; we have χ ≡√

2πeB�

R = 10.0. The disclinationparameter C affects the maximum of each level. The Landau levels for a flat sample are also showed for this threefirst indices n (they are dispersionless). (b) Plot of energy E versus χ, for RKZ = 1.0 and n = 0 (lower curves) andn = 1 (upper curves).

plane case. Their values correspond to the maximum of the energies (31) with C = 1, k = 0 and R → ∞.In Fig. 9b we draw the energies, for n = 0, 1, versus χ. The influence of C on them can be noted too.

In order to appreciate the influence of the screw dislocation k in the energy levels, we draw Fig. 10,where we make RKZ = RKz + R k

C lφ and put i = RKz and l = Rlφ. It is shown on it the energy levelsas function of k, for C = 1 and for fixed i and l. We note that the energies decrease as we increase thescrew dislocation parameter k, reaching the zero value for some k. So, the lowest Landau Level now canbe zero for a specific k. Note that the energy of the ground state is simply given by − �

2

8m∗R2 , which doesnot depend on the magnetic field. Actually, for any screw dislocation parameter k, we can pick values forKz and lφ which cause this effect. This also happens for k = 0, but now the energy levels do not dependon the angular momentum lφ anymore.

0 1 2 3 4 5 6 7 80

50

100

150

200

250

k

E2m

∗ R2/h

2

n = 0n = 1n = 2

Fig. 10 Plot of energy E versus k (screw dislocation). Here,i ≡ RKz and l ≡ Rlφ. The graphic was made by taking i =1.9, l = 2.0 and χ = 10.0. We can see that increasing the screwdislocation parameter k diminishes the energy for fixed wavevectors Kz and angular momentum lφ.



4 Concluding remarks

In this work we studied how the geometry and topology of a cylinder with both screw a dislocation and adisclination affect the energy levels of an electron in the presence of a homogeneous magnetic field. Twocases were taken into account: first, we considered the electron moving in the full cylindrical surface andin the second case we dealt with the harmonic approach, that is, the almost flat case. In the first case weshowed that the wavefunctions are the solutions of the Hill’s differential equation. Solutions in terms ofFourier series were provided which allowed us to compute the energy levels numerically. We saw that theenergies increase as we increase the screw dislocation parameter k while the disclination parameter C justshifts them. The influence due to the radius R of the cylinder on the energies was drawn too, so they havean asymptotic behavior. This is because in the limit R → ∞ they became the well known Landau Levelsfor a charged particle moving in a plane in a perpendicular homogeneous magnetic field. In the formerone we considered the limit case for a sufficiently strong normal to the surface component of the magneticfield and for electrons with the Landau oscillator suspension center far enough from the side edges of acylindrical strip. This is an almost flat problem and what we have seen was that the Landau levels showdispersion behavior. The screw dislocation parameter k, the radius R and the magnetic field B affect sucha dispersion relation. We have shown the influence of all these parameters in the energies. Again, C justshifts such energy curves, while k makes them to decrease. In fact, for a given k, we always can choosespecific values for both wavevectors and angular momentum in the z-direction in order to have a state withenergy given solely by − �

2

8m∗R2 , which is of geometric nature and does not depend on the applied magneticfield.

The study of bound states are the building blocks for many applications in condensed matter. The studycarried out in this paper could lead to the investigation not only of the role of geometry but also of theinfluence of topology in electronics as, for example, the study of magnetotransport [23]. Knowing theeffect of geometry and the topology in the Landau levels could lead, for example, to the investigation ofthe Quantum Hall Effect [24] in curved substrates with defects. An experimental setup to study curvedtwo-dimensional electron gases is depicted in [25]. The new feature could be to introduce the topologicaldefects on the experiment. The influence of topology on the magnetic moment of an electron gas [26] couldalso be investigated with the results shown here.

Acknowledgements This work was partially supported by CAPES(NANOBIOTEC).

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Yet another position-dependent mass quantum model

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