Physica E 42 (2010) 2436–2440

Contents lists available at ScienceDirect

Physica E

1386-94

doi:10.1

n Tel.:

E-m

journal homepage: www.elsevier.com/locate/physe

Donor in cylindrical quantum well wire under the action of an appliedmagnetic field

Pablo Villamil n

Departamento de Matematicas y Fısica, Universidad de Sucre, A.A. 406, Sincelejo, Colombia

a r t i c l e i n f o

Article history:

Received 14 April 2010

Received in revised form

26 May 2010

Accepted 1 June 2010Available online 4 June 2010

Keywords:

Quantum well wires

Hydrogenic donor impurity

Binding energy

77/$ - see front matter & 2010 Elsevier B.V. A

016/j.physe.2010.06.001

+57 3116652503.

ail address: pabloe1954@yahoo.es

a b s t r a c t

In this paper we present a calculation of the binding energy of a donor impurity in a cylindrical GaAs-

Ga0.6Al0.4As quantum well wire (QWW) as a function of an applied magnetic field for different wire radii

and different positions of the donor. The model considers an infinite length QWW with a carrier that is

confined laterally by a parabolic potential and the presence of a uniform magnetic field applied parallel

to the wire axis. The 1s-like impurity state is considered using the effective-mass approximation within

the variational approach. We found that for a given wire radius the binding energy of the impurity

increases with the magnetic field and that for a given magnetic field value it decreases with wire radius.

Our results are in good agreement with previous theoretical reports.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

The great progress in the last years of modern technologies incrystal growth have allowed the production of high quality ofvarious low-dimensional systems such as quantum wells, QWs(2 D), quantum well wires, QWWs (1 D), and quantum dots, QDs(0 D), where the quantum mechanical nature of the carriers playan important role, and it has motivated an increasing interest inthe studies of their optical and electrical properties [1–7].

Great deal of theoretical and experimental investigations onthese systems in the presence of shallow impurities has beenpublished [8–25]. Elagoz et al. [22] described the calculation ofhydrogenic impurity binding energies in cylindrical GaAs-Ga0.6

Al0.4As quantum well wires with lateral parabolic confinement inthe presence of an axial magnetic field. They observed sharpchanges in binding energy for critical spatial confinement radiusand magnetic field values. In this work we consider a shallowimpurity inside an infinite length QWWs with lateral parabolicpotential and in the presence of a uniform magnetic field appliedin the axial direction. We use the effective-mass approximationwithin the variational approach to calculate the binding energyfor the 1s-like state of a donor impurity in a GaAs-Ga0.6Al0.4AsQWWs as a function of the magnetic field and the impurityposition inside the wire. In Section 2 we present the theoryfollowed for this calculation. Our results and discussions arepresented in Section 3, and conclusions in Section 4.

ll rights reserved.

2. Theory

The Hamiltonian of a donor impurity in a cylindrical GaAs-Ga0.6Al0.4As QWW with radius R, parabolic confinement potentialand in the presence of an applied magnetic field, in the effective-mass approximation, can be written as

H¼1

2m*½PþeA�2�

e2

e9r�r09þVðrÞ, ð1Þ

where 9r�r09¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr�r0Þ

2þz2

q, r0 is the impurity position, mea-

sured from the center of wire, the z coordinate is the relativeseparation of the electron from the impurity ion along the wireaxis, e is the dielectric constant of GaAs semiconductor inside thewire, m* is the effective electron mass, and A(r) is the vectorpotential of the magnetic field. For a uniform magnetic fieldapplied in the wire axis direction, the vector potential can bewritten as A(r)¼1/2(B� r), with B¼ Bz, and in cylindricalcoordinates becomes Ar¼Az¼0, Aj¼1/2(Br). The lateral para-bolic potential V(r) is defined as

VðrÞ ¼ V0r2

R2, rrR

V0, r4R:

8><>: ð2Þ

The Hamiltonian of the system in cylindrical coordinates andin effective Rydbergs, R*, becomes

H ¼�r 2�ig @

@j

� �þg2r2

4�

2

rþVðrÞ: ð3Þ

The term g¼ e_B=2m*R*, is the electron energy in the firstLandau level (n¼0) due to the action of the magnetic field. For

0

10

20

30

40

first level of the conduction subband

R = 2 a*

R = 1.5 a*

R = 1 a*

E 10 (

R*

)

Magnetic Field ( T )

R = 0.5 a*

10 20 30 40 50

Fig. 1. E10, first energy level of an electron in the conduction subband, as a

function of the magnetic field for different values of the quantum wire radius.

P. Villamil / Physica E 42 (2010) 2436–2440 2437

donor impurities in GaAs, m*¼0.065, e¼12.58, a*

ffi100 A andR*¼5.83 meV.We use the variational method in order to calculate the

expected value of the Hamiltonian in Eq. (3). We assume suitablevariational wave functions for the 1s-like state of impurity, as theproduct of a hydrogenic part G1s, and the appropriate confluenthypergeometric functions. The trial wave functions for the 1s-likestate of impurity is written as

C1sðrÞ ¼

N1sExp �xi

2

� �1F1ða01,1,xiÞG1sðr,l1sÞ, rrR

N1sExp �xi

2

� �1F1ða01,1,xiRÞ

1F1ðau01,1,xeRÞ1F1ðau01,1,xeÞG1sðr,l1sÞ, r4R:

8>>><>>>:

ð4Þ

In Eq. (4), N1s is the normalization constant of the 1s-like state,

1F1(a01, 1, xi), and 1F1ðau01,1,xeÞ are the confluent hypergeometricfunctions, which are the corresponding solutions for the case ofparabolic confinement potential in the presence of a uniformmagnetic field parallel to the wire axis, where

xi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4V0þðRgÞ2

q=2R

� �r2, xe ¼ ðg=2Þr2, and xiR and xeR are the

xi and xe variables evaluated in r¼R. a01 and au01 are theparameters of the confluent hypergeometric functions forthe 1s-like state of the problem inside and outside the wire,respectively, which are calculated numerically by means of the

boundary conditions @Cin@r 9r ¼ R ¼

@Cout@r 9r ¼ R. G1s is the hydrogenic

wave functions [21] corresponding to the 1s-like state and l1s isthe variational parameter.

The binding energy of a hydrogenic impurity is the energynecessary to move the electron from the donor level to the firstlevel of the conduction subband and is calculated by mean of theequation (see Appendix).

Eb,1s ¼ E10�minl1sC1s9H9C1s

� �

E10 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4V0þðRgÞ2

qR

ð1�2a01Þ: ð5Þ

E10 is the first level of the conduction subband in the absenceof the Coulomb term. Eq. (5) shows that when V0¼0, inside thequantum wire, the energy of the first level of the square well isreproduced, E10¼g(1–2a01). The term minl1s

C1s9H9C1s

� �means

that the Hamiltonian expected value is minimized with respect tol1s. The expression of the binding energy, for the 1s-like state, isas follows

Eb,1s ¼ l21sþ

4ðl1s�1ÞðAþBÞdðAþBÞ

dl1s

�2l1s WðC�2a01DÞþðWE�2gau01FÞ

dðAþBÞ

dl1s

þl1sr0

WðG�2a01HÞþðWI�2gau01JÞ

dðAþBÞdl1s

, ð6Þ

where,

W ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4V0þðRgÞ2

q:

In addition, we transform the integrals in a dimensionless formby letting r¼Rt.

A¼ R2

Z1

0

tI0ð2lRtoÞK0ð2lRt4 Þe�RWt2=21F1ða01,1,xiÞ2 dt

B¼ R2 1F1ða01,1,xiRÞ2

1F1ðau01,1,xeRÞ2

Z11

tI0ð2lRtoÞK0ð2lRt4Þe�ðRWt2=2Þ1F1ðau01,1,xeÞ2 dt

C ¼ R3

Z1

0

t3I0ð2lRto ÞK0ð2lRt4 Þe�ðRWt2=2Þ1F1ða01,1,xiÞ2 dt

D¼ R3

Z1

0

t3I0ð2lRto ÞK0ð2lRt4Þe�ðRWt2=2Þ1F1ða01,1,xiÞ1F1ð1þa01,2,xiÞdt

E¼ R3 1F1ða01,1,xiRÞ2

1F1ðau01,1,xeRÞ2

Z11

t3I0ð2lRto ÞK0ð2lRt4 Þe�ðRWt2=2Þ1F1ðau01,1,xeÞ2 dt

F ¼ R4 1F1ða01,1,xiRÞ2

1F1ðau01,1,xeRÞ2

Z11

t3I0ð2lRto ÞK0ð2lRt4Þ

�e�ðRWt2=2Þ1F1ðau01,1,xeÞ1F1ð1þau01,2,xeÞdt

G¼ R2

Z1

0

t2I1ð2lRto ÞK1ð2lRt4Þe�ðRWt2=2Þ1F1ða01,1,xiÞ2 dt

H¼ R2

Z1

0

t2I1ð2lRtoÞK1ð2lRt4Þe�ðRWt2=2Þ1F1ða01,1,xiÞ1F1ð1þa01,2,xiÞdt

I¼ R2 1F1ða01,1,xiRÞ2

1F1ðau01,1,xeRÞ2

Z11

t2I1ð2lRto ÞK1ð2lRt4 Þ

�e�ðRWt2=2Þ1F1ðau01,1,xeÞ2 dt

J¼ R3 1F1ða01,1,xiRÞ2

1F1ðau01,1,xeRÞ2

Z11

t2I1ð2lRto ÞK1ð2lRt4Þ

�e�ðRWt2=2Þ1F1ðau01,1,xeÞ1F1ð1þau01,2,xeÞdt

where to(t4) is the lesser (greater) of r and r0. Furthermore,I0ðxÞ, I1ðxÞ, K0ðxÞ, K1ðxÞ are modified Bessel functions of integer order.

3. Results and discussions

In our calculations we have used the Al concentration, x¼0.4,which corresponds to a barrier height of 316 meV. Also, as it hasbeen mentioned, the magnetic field is applied parallel to the wireaxis. In Fig. 1 we present E10, the first energy level of the

00.4

0.6

0.8

R = 2 a*

R = 0.5 a*

R = 1 a*

R = 1.5 a*

< r >

( a*

)

Magnetic Field ( T )

1s - like Stateρi = 0

10 20 30 40 50

Fig. 3. /rS, expected value of impurity radius as a function of the magnetic field

of a donor impurity located at the center ri¼0 of a cylindrical GaAs-Ga0.6Al0.4As

quantum wire and for different values of the quantum wire radius.

4

5

6E b (

R*

)

1s - like State R = 0.5 a*

ρi = 0

ρi = 0.4 R

ρi = 0.8 R

P. Villamil / Physica E 42 (2010) 2436–24402438

conduction subband GaAs quantum wire vs. magnetic field, B, fordifferent quantum wire radii. Please note that if B¼0, the energylevels decrease as the quantum wire radius increases. For a givenradius quantum wire, the energy level increases with themagnetic field. For small radii of the wire, the energy level has asmooth variation with the magnetic field.

Fig. 2 shows the binding energy, Eb, vs. magnetic field for 4values of the quantum wire radius: 0.5a*, 1a*, 1.5a*, and 2a*, withimpurity at the center quantum wire (ri¼0). If B¼0T, the bindingenergy decreases as the quantum wire radius increases; thisbehavior is due only to the lateral parabolic confinement(geometric confinement). In black circles appear the results,obtained by Branis et al. [24], of the binding energy of the groundstate of a donor in a quantum wire in the presence of a uniformmagnetic field applied parallel to the wire axis. The impurity ion islocated at the axis of the wire. The Al concentration is 0.4. Theyconsidered finite confinement potential (finite square well). Weobserve that for B¼0T and for a given value of quantum well wireradius the Eb is lesser in the finite square well than in the lateralparabolic potential. In addition, in the lateral parabolic potentialwe observe that all the curves have a monotonous increase in thebinding energy with magnetic field. This growth is greater forlarger values of the quantum wire radius. This is equivalent to saythat the expected value of impurity radius /rS, when B¼0T,increases with the quantum wire radius, as shown in Fig. 3. In thisfigure, for a given value of quantum wire radius the expectedvalue of impurity radius /rS, decreases monotonically with themagnetic field except for R¼0.5a* where /rS has a little variationwith the applied magnetic field.

The binding energy of an impurity in a cylindrical GaAs-Ga0.6Al0.4As QWW as a function of the magnetic field, for a wirewith R¼0.5a*, and different positions of the impurity within thequantum wire are presented in Fig. 4. When B¼0T, the bindingenergy decreases when the impurity position varies fromthe center of the wire (ri¼0) towards the frontier of quantumwire (ri¼0.8R). This is due to the expected value of impurityradius /rS increases as the impurity is close to the edge quantumwire, this is, the potential makes the probability amplitude ofthe bound electron to describe a deformed ellipsoidal chargedistribution. The binding energy for each position of the impurityconsidered increases very slowly with the magnetic field.

01

2

3

4

5

6 Lateral Parabolic Potential Finite Square Well

R = 1 a*

R = 1.5 a*

R = 2 a*

R = 0.5 a*

R = 2 a*

R = 1 a*

R = 0.5 a*

E b (

R*

)

Magnetic Field ( T )

1s - like State ρi = 0

10 20 30 40 50

Fig. 2. Binding energy of the 1s-like state of a donor impurity located at the center

ri¼0 of a cylindrical GaAs-Ga0.6Al0.4As quantum wire, as a function of the

magnetic field and for different values of the quantum wire radius.

03

Magnetic Field ( T )10 20 30 40 50

Fig. 4. Binding energy of an impurity in a cylindrical GaAs-Ga0.6Al0.4As QWW as a

function of the magnetic field, for a wire with R¼0.5a*, and different positions of

the impurity within the quantum wire.

4. Conclusions

In this work using the effective mass-approximation within thevariational approach, we have calculated the binding energy of the1s-like state of a hydrogenic donor impurity in a cylindrical GaAs-Ga0.6Al0.4As QWW under the action of a magnetic field applied inthe axial direction, and the carrier is confined laterally by aparabolic potential. Our results are in good agreement withprevious theoretical reports, but with higher binding energies thanin those works which use finite confinement potential, as expected.We have found that the binding energy for the impurity groundstate increases with the magnetic field and it diminishes when theimpurity position varies from the center to the border barrier.According to our results, the impurity position inside the QWW iscrucial in understanding the optical responses of these systemsassociated with impurity states. The electron–phonon interactionwas neglected. This is important when high magnetic fields are

P. Villamil / Physica E 42 (2010) 2436–2440 2439

present as the ones examined. Resonant polaron interaction ispossible in such high magnetic fields. For future work will take intoaccount polaron correction and band nonparabolicity in transitionenergies. We believe the present calculation will be of importancein understanding future experimental work in this subject.

Acknowledgments

The author gratefully acknowledges Universidad de Sucre forthe total financial support to develop this work. Also, the authorthanks Dr. Jose Sierra and Dr. Carlos Beltran for useful discussionsand critical reading of the manuscript.

Appendix

We consider a carrier of charge q in a cylindrical quantum wireof infinite length, radius R and dielectric constant e. The carrier isconfined laterally by a parabolic potential, V(r), inside thequantum wire. If the carrier is a hole, q¼ +e, and if it is anelectron, q¼�e. A uniform magnetic field is applied in the axialdirection of the quantum wire. In the effective mass approxima-tion, the Hamiltonian of the carrier is expressed as

H¼1

2m*½P�qA�2þVðrÞ, ðA:1Þ

where the first term of Eq. (A.1) is the carrier’s kinetic energy in amagnetic field and the last term is the parabolic lateralconfinement potential in the radial direction, modeled by

VðrÞ ¼ V0r2

R2, rrR

V0, r4R

8><>: ðA:2Þ

The vector potential is written as A(r)¼1/2(B� r) with B¼Bz .In cylindrical coordinates, its components are: Ar¼Az¼0, Aj¼

1/2(Br). When the carrier is an electron and on expanding theparenthesis of Eq. (A.1), the Hamiltonian takes the form

H¼1

2m*P2þeBLzþe2 B2r2

4

!" #þVðrÞ ðA:3Þ

where Lz ¼�i_ð@=@jÞ in Eq. (A.3) and the Hamiltonian is writtenas

H¼�_2

2m*r2�

ie_B

2m*

@

@jþ

e2B2r2

8m*þVðrÞ ðA:4Þ

In order to express the Hamiltonian in atomic units of length,a*2 ¼ _2=2m*R*, and in atomic units of energy, R* ¼ _2=2m*a*2, wedefine the lengths r¼ra* and z¼za*. In these units and incylindrical coordinates the electron’s Hamiltonian is

H ¼H

R*¼�r

2�ig @

@j þg2r2

4þV ðrÞ ðA:5Þ

In this equation, g is the energy of the electron in a magneticfield in the first Landau level (n¼0), and is given by

g¼ e_B

2m*R*ðA:6Þ

Expressing the Laplacian in cylindrical coordinates, Eq. (A.5) iswritten in the following way:

H ¼�1

r@

@r r @

@r

!�

1

r2

@2

@j2�@2

@z2�ig @

@j þg2r2

4þV ðrÞ

ðA:7Þ

The time-independent Schrodinger equation is HC¼EC,whose solutions can be written in a product form ascðr,j,zÞ ¼ Rðr,jÞZðzÞ. In the time-independent Schrodinger

equation, we substitute C by RZ and then we divide by RZ, giving

�1

R

@2

@r2�

1

R

1

r@R

@r �1

R

1

r2

@2R

@j2�

igR

@R

@j þg2r2

4þVðrÞ�E ¼

1

Z

d2Z

dz2

ðA:8Þ

The function Z(z) is governed by the differential equation

1

Z

@2Z

@z2¼�k2

z ðA:9Þ

where �k2z is a constant. The solution of Eq. (A.9) is

ZðzÞ ¼ Exp½ikzz� ðA:10Þ

there pz ¼ _kz. The differential equation for R(r,j) is

�1

R

@2R

@r2�

1

R

1

r@R

@r �1

R

1

r2

@2R

@j2�

igR

@R

@jþg2r2

4þVðrÞ�EðrÞ

!¼ 0,

ðA:11Þ

With EðrÞ ¼ E�k2z . Substituting in this equation R(r, j)¼r(r)F(j)

and dividing by R(r,j), we obtain the following equation:

�1

r

@2R

@r2�

1

R

1

r@R

@r �1

F1

r2

@2F@j2�

igF@F@j þ

g2r2

4þVðrÞ�EðrÞ ¼ 0,

ðA:12Þ

we have supposed that F(j)¼eimj, m being the eigenvalue of Lz

in units of _; when you replace F, the lateral confinementpotential inside the quantum wire, V(r), and, moreover, ismultiplied by r Eq. (A.12), we obtain the radial part of thedifferential equation

@2r

@r2þ

1

r@r

@r�

m2

r2þmgþ

g2r2

4þ

V0

R2r2�EðrÞ

" #r¼ 0: ðA:13Þ

With the purpose of expressing this differential equation in awell-known form, we make the changes of variables

x¼ r2, dx¼ 2rdr,@r

@r ¼ 2ffiffiffixp @r

@x,

@2r

@r2¼ 2

@r

@xþ4x

@2r

@x2ðA:14Þ

and the differential equation is obtained as

4x@2r

@x2þ4

@r

@x�

m2

xþmgþg

2x

4þ

V0

R2x�EðxÞ

� �r¼ 0: ðA:15Þ

We suppose that the solution of r(x) is of form

rðxÞ ¼ e�bxx9m9=2wðxÞ ðA:16Þ

Then of some simplifications we obtain the following differ-ential equation:

4x@2w@x2þ4ð1þ9m9�2bxÞ

@w@xþ EðxÞ�4bþ4b2x�mg�g

2x

4�4b9m9�

V0

R2x

� �¼ 0

ðA:17Þ

in order to determinate b we choose

4b2x�g2x

4�

V0

R2x¼ 0, ) b¼

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4V0þðRgÞ2

q2R

: ðA:18Þ

With this done, and with the change of variable

y¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4V0þðRgÞ2

q2R

x, ðA:19Þ

we obtain the differential equation whose solutions are theconfluent hypergeometric functions F(ajmjj,b,y).

y@2F

@y2þð1þ9m9�yÞ

@F

@y�

1

2ð1þ9m9Þþ

ðmgþEðyÞÞR

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4V0þðRgÞ2

q264

375FðyÞ ¼ 0

ðA:20Þ

P. Villamil / Physica E 42 (2010) 2436–24402440

Making

b¼ 1þ9m9, a9m9j ¼b

2þ½mgþEðyÞ�R

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4V0þðRgÞ2

q ðA:21Þ

Eq. (A.20) takes the form

y@2F

@y2þðb�yÞ

@F

@y�a9m9jFðyÞ ¼ 0 ðA:22Þ

If the electron is inside the quantum wire, the solution ofEq. (A.22) is the confluent hypergeometric function 1F1(ajmjj,b,y),where ajmjj corresponds to the state with magnetic quantumnumber jmj. For the first level of energy, m¼0, then ajmjj¼a01.Eq. (A.21) shows that

a01 ¼1

2�

E01R

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4V0þðRgÞ2

q : ðA:23Þ

The energy of an electron, in the first level of the well, withmagnetic field is

E10 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4V0þðRgÞ2

qR

ð1�2a01Þ: ðA:24Þ

Otherwise, if the electron is outside of the quantum wire, thesolution of radial equation is the confluent hypergeometricfunction 1F1(a0jmjj,b,ye) o U(a0jmjj,b,ye). The relationship betweenthe parameters a01 and au01 is

au01 ¼1

21þ

V0

g

� ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4V0þðRgÞ2

q2Rg ð1�2a01Þ: ðA:25Þ

The eigenvalues of the electron energy are determined by thecontinuity of the wave functions and of its first derivative at the

boundar

Cintðr ,j,zÞ9r ¼ R ¼Cextðr,j,zÞ9r ¼ R

@Cintðr,j,zÞ

@r

����r ¼ R

¼@Cextðr,j,zÞ

@r

����r ¼ R

: ðA:26Þ

References

[1] H. Huang Michael, et al., Science 292 (2001) 1897.[2] A.C. Graham, et al., Solid State Commun. 131 (2004) 591.[3] C. Thelander, et al., Solid State Commun. 131 (2004) 573.[4] D.F. Urban, et al., Solid State Commun. 131 (2004) 609.[5] M. Bayer, et al., Physica E 7 (2000) 475.[6] B. Huard, et al., Solid State Commun. 131 (2004) 599.[7] Sassetti Maura, et al., Solid State Commu. 131 (2004) 647.[8] I.D. Mikhailov, et al., Phys. Status Solidi (b) 234 (2) (2002) 590.[9] J.M. Shi, F.M. Peeters, J.T. Devreese, Phys. Rev. B 51 (1995) 7714.

[10] F.M. Peeters, V.A. Schweigert, Phys. Rev. B 53 (1996) 1468.[11] C. Riva, V.A. Schweigert, F.M. Peeters, Phys. Rev. B 57 (1998) 15392.[12] G.J. Zaratiegui, et al., Phys. Rev. B 66 (2002) 195324.[13] I.D. Mikhailov, et al., Phys. Rev. B 67 (2003) 115317.[14] S. Holmes, et al., Phys. Rev. Lett. 69 (1992) 2571.[15] P. Villamil, N. Porras-Montenegro, J. Phys.: Condens. Matter 10 (10) (1998)

599.[16] P. Villamil, N. Porras-Montenegro, Phys. Rev. B 59 (1) (1999) 605.[17] C. Riva, et al., Physica E 22 (2004) 550.[18] V. Lien Nguyen, et al., Physica B 292 (2000) 153.[19] F.M. Peeters, Phys. Rev. B 42 (1990) R1486.[20] J.M. Shi, et al., Phys. Rev. B 44 (1991) 5692.[21] A. Latge, et al., Phys. Rev. B 53 (10) (1996) 160.[22] S. Elagoz, R. Amca, S. Kutlu, I. Sokmen, Superlattices and Microstruct. 44

(2008) 802.[23] A.J. Peter, Solid State Commun. 147 (2008) 296.[24] S. Branis, G. Li, K.K. Bajaj, Phys. Rev. B 47 (1993) 1316.[25] L.J. Slater, in: Confluent Hypergeometric Functions, Cambridge University

Press, London, 1960.