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Physica E 15 (2002) 99 – 106 www.elsevier.com/locate/physe Binding energy of excitons in inhomogeneous quantum dots under uniform electric eld J. El Khamkhami a , E. Feddi b , E. Assaid c , F. Dujardin d ; , B. St eb e d , J. Diouri e a Facult e des Sciences et Techniques, B.P. 416, Tanger, Maroc b Ecole Normale Sup erieure, B.P. 209, Martil, Tetouan, Maroc c Facult e des Sciences, B.P. 20, El Jadida, Maroc d Institut de Physique et d’Electronique, Laboratoire de Th eorie de la Mati ere Condens ee, 1 Boulevard Arago, 57078 Metz, Cedex 3, France e Facult e des Sciences, B.P. 2121, Tetouan, Maroc Received 13 July 2001; received in revised form 6 December 2001; accepted 22 December 2001 Abstract Excitons in inhomogenous quantum nanospheres have been theoretically studied within the eective mass approximation. An innite deep potential has been used to describe the eects of quantum connement. The binding energy with or without an applied electric eld is determined by the Ritz variational method taking into account the correlation between the electron and the hole in the trial wave function. It appears that the binding energy strongly depends on the core and shell radii. The existence of a radius ratio critical value has been shown: it may be used to distinguish between tridimensional and spherical surface connement. The inuence of a uniform electric eld is analyzed. It has been found that the Stark eect appears even for very small sizes and that the energy shift is more signicant when the exciton is near the spherical surface. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 73.21.La; 71.35.y Keywords: Quantum dots; Excitons; Stark eect 1. Introduction Recent progress in crystal growth and process tech- niques has made it possible to realize zero-dimensional systems such as clusters and nano-crystallites. In these systems, ultimate quantum connement eects restrict the motions of the optically excited electrons and holes in the three spatial directions. As a consequence, free particle energy levels are quantized and coulombic Corresponding author. Tel.: +33-3-87-31-58-78; fax: +33-3- 87-31-58-01. E-mail address: [email protected] (F. Dujardin). correlation eects as well as optical absorption oscil- lator strength are enhanced [1]. For the last ten years, it has been possible to process a new class of spherical quantum dots called quantum dot-quantum well or in- homogeneous quantum dots (IQDs) composed of two semiconductor materials. One of them, that with the smaller bulk band gap, is embedded between a core and outer shell of the material with the larger band gap (Fig. 1). For more details on the chemical fabrication of these new articial structures we refer the readers to the Refs. [2– 4]. The experimental investigations in “coated” nano-spheres: ZnS(core,shell)= CdSe(well) [2], CdS= PbS [3], CdS= HgS [4], CdS= AgI, CdS= TiO 2 [5], 1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII:S1386-9477(02)00448-4

Binding energy of excitons in inhomogeneous quantum dots under uniform electric field

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Page 1: Binding energy of excitons in inhomogeneous quantum dots under uniform electric field

Physica E 15 (2002) 99–106www.elsevier.com/locate/physe

Binding energy of excitons in inhomogeneous quantum dotsunder uniform electric %eld

J. El Khamkhamia, E. Feddib, E. Assaidc, F. Dujardind ; ∗, B. St0eb0ed, J. Diourie

aFacult�e des Sciences et Techniques, B.P. 416, Tanger, MarocbEcole Normale Sup�erieure, B.P. 209, Martil, Tetouan, Maroc

cFacult�e des Sciences, B.P. 20, El Jadida, MarocdInstitut de Physique et d’Electronique, Laboratoire de Th�eorie de la Mati)ere Condens�ee, 1 Boulevard Arago,

57078 Metz, Cedex 3, FranceeFacult�e des Sciences, B.P. 2121, Tetouan, Maroc

Received 13 July 2001; received in revised form 6 December 2001; accepted 22 December 2001

Abstract

Excitons in inhomogenous quantum nanospheres have been theoretically studied within the e4ective mass approximation.An in%nite deep potential has been used to describe the e4ects of quantum con%nement. The binding energy with or withoutan applied electric %eld is determined by the Ritz variational method taking into account the correlation between the electronand the hole in the trial wave function. It appears that the binding energy strongly depends on the core and shell radii. Theexistence of a radius ratio critical value has been shown: it may be used to distinguish between tridimensional and sphericalsurface con%nement. The in9uence of a uniform electric %eld is analyzed. It has been found that the Stark e4ect appearseven for very small sizes and that the energy shift is more signi%cant when the exciton is near the spherical surface. ? 2002Elsevier Science B.V. All rights reserved.

PACS: 73.21.La; 71.35.−y

Keywords: Quantum dots; Excitons; Stark e4ect

1. Introduction

Recent progress in crystal growth and process tech-niques has made it possible to realize zero-dimensionalsystems such as clusters and nano-crystallites. In thesesystems, ultimate quantum con%nement e4ects restrictthe motions of the optically excited electrons and holesin the three spatial directions. As a consequence, freeparticle energy levels are quantized and coulombic

∗ Corresponding author. Tel.: +33-3-87-31-58-78; fax: +33-3-87-31-58-01.

E-mail address: [email protected](F. Dujardin).

correlation e4ects as well as optical absorption oscil-lator strength are enhanced [1]. For the last ten years,it has been possible to process a new class of sphericalquantum dots called quantum dot-quantum well or in-homogeneous quantum dots (IQDs) composed of twosemiconductor materials. One of them, that with thesmaller bulk band gap, is embedded between a coreand outer shell of the material with the larger band gap(Fig. 1). For more details on the chemical fabricationof these new arti%cial structures we refer the readersto the Refs. [2–4].The experimental investigations in “coated”

nano-spheres: ZnS(core,shell)=CdSe(well) [2],CdS=PbS [3], CdS=HgS [4], CdS=AgI, CdS=TiO2 [5],

1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.PII: S 1386 -9477(02)00448 -4

Page 2: Binding energy of excitons in inhomogeneous quantum dots under uniform electric field

100 J. El Khamkhami et al. / Physica E 15 (2002) 99–106

b

a

a b r

Ene

rgy

Vwe

Vwh

Shell

Well

Core

Fig. 1. Schematic description of a spherical IQD and correspondingpotential energies.

ZnSe=CdSe [6] have shown that these structures canexhibit some remarkable and interesting phenom-ena associated with the redistribution of the electronand hole wave function, such as an increase in theband-edge absorption when the shell material has asmall band gap and a rapid change in the lumines-cence eLciency. The original characteristics of thesestructures are that their physical properties can becontrolled and can be adjusted by changing the corediameter, the thickness of the well and the size of theoutermost shell.The %rst theoretical studies have shown the e4ect of

the internal well on single particle (electron or hole)energies and pair overlaps in IQDs and have deter-mined the ground state of an uncorrelated electron–hole pair. Indeed, Kortan et al. [2] have establishedthe theory of electronic structure in layered crystallitesusing the same assumptions and approximations pre-viously used for a homogeneous quantum dot (HQD)[7]. Haus et al. [8] have provided a recursive methodto %nd the wave functions and energies of free carriersin a multiple shell structure. Schooss et al. [9] havegiven a theoretical approach for the calculation of

1s–1s electronic transition by the inclusion of thecoulombic interaction and by the introduction of a %-nite potential at the spherical boundaries in CdS=HgS=CdS structures.Up to now, only two theoretical studies have

been devoted to the behavior of excitons in IQDs.Bryant [10] has extended the large scale con%gura-tion interaction calculation previously used to studymulti-electrons and excitons in HQDs to sphericalCdS=HgS IQDs. He has shown that the internal wella4ects pair correlations in IQDs, and that the modelwith screened pair interaction quantitatively givesexciton ground state energy. Ferreyra et al. [11] havestudied the exciton binding energy in IQDs, takinginto account induced polarization charges, in theapproximation of an in%nite potential barrier and con-sidering the coulombic interaction as a perturbation.They have shown that the exciton binding energy de-creases by increasing the size of the core region andbecomes as pronounced as in homogeneous quan-tum dots when the e4ect of the induced polarizationcharges is taken into account. Nevertheless, they haveneglected the coupling between the electron and thehole in the trial wave function. Such a model o4ers asimple description to investigate the exciton behaviorin an IQD but the approximation used is only validfor the strong con%nement limit. We shall discuss thispoint in the results section.The Stark e4ect in HQDs has been the subject of

numerous experimental and theoretical investigations[12–14]. All these studies have demonstrated that inthe presence of a uniform electric %eld, energy levelsare shifted to low energies and the exciton recombi-nation lifetime increases. To date there has been nostudy concerning the in9uence of a uniform electric%eld on excitons in IQDs.In the present paper we have performed a fully vari-

ational determination of the exciton binding energy inan IQD in the case of strong and moderate con%ne-ment regimes. We have chosen a trial wave functiontaking into account the coulombic interaction betweenthe electron and the hole. First, we have determinedthe exciton binding energy without electric %eld andwe have compared our results with those existing inthe literature. Then, we have investigated the in9u-ence of a uniform electric %eld on the exciton be-havior in an IQD. The paper is organized as follows:in Section 2 we describe the essential features of the

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J. El Khamkhami et al. / Physica E 15 (2002) 99–106 101

theory and some details of variational calculation, theresults are presented in Section 3 and the conclusionsin Section 4.

2. Fundamental state and binding energy

To simplify, we consider a structure composed bythree spherical layers in which the inner and outerlayers correspond to the same semiconductor material.For example, we take [CdS (core)= HgS (well)=CdS(shell)] in water-surrounded medium [9]. The bottomof CdS conduction band is 1:35 eV above the bot-tom of HgS conduction band . The top of the CdSvalence band is situated at 0:65 eV below the top ofthe HgS valence band. Because of these band o4sets,we assume that the electron and hole are completelycon%ned in HgS well by an in%nite potential. Un-der these conditions we can neglect the polarizationcharges because the internal well inhibits charge local-ization at the surfaces [12]. Under the e4ective massapproximation and assuming isotropic, parabolic andnon-degenerated bands, the Hamiltonian of an excitonin the presence of a uniform electric %eld writes:

H =− �e

1 + �− ��h

1 + �− 2reh

+ Vw + f(zh − ze): (1)

We have used as unit of length the 3D exciton e4ec-tive Bohr radius: a∗ = �˜2= e2 and as unit of energy:R∗= e4=2�2˜2 which represents the absolute value ofthe 3D exciton ground state energy. � is the dielectricconstant of the well. = m∗

em∗h =(m

∗e + m∗

h) is the re-duced mass of the exciton. � = m∗

e =m∗h is the electron

to hole e4ective mass ratio. We have introduced thedimensionless parameter f = ea∗F=R∗ which charac-terizes the electric %eld strength. For the ground stateit is suLcient to consider a wave function dependingon the distance re; rh ; reh ; ze and zh :Within these coor-dinates, the Laplacian operators read:

�i =929r2i

+2ri

99ri

+r2i − r2j + r2eh

rireh

929ri9reh

+929r2eh

+2reh

99reh

+929z2i

+2ziri

929zi9ri

+ 2(zi − zj)reh

929zi9reh

; (2)

where i; j = e; h (i �= j): Assuming an in%nitely deepwell, the total con%nement potential Vw is written asfollows:

Vw = V ew + V h

w ; (3)

V iw =

{0; a¡ ri ¡b∞; ri ¡a and ri ¿b

(4)

a and b, respectively, denotes the inner and outerradius of the IQD (Fig. 1). The energy and the en-velope wave function are solutions of the e4ectiveSchrQodinger equation:

H �(re; rh ; reh ; ze; zh) = E�(re; rh ; reh ; ze; zh): (5)

This equation cannot be solved analytically, so wehave to determine its ground state solutions using theRitz variational principle. We have chosen the follow-ing trial wave function which takes into account theelectron–hole correlation:

�(re; rh ; reh ; ze; zh)

=’(re)’(rh) [1 + f�(ze − zh)]exp(−�reh); (6)

where ’(re) and ’(rh) are, respectively, the wavefunctions of the electron and the hole. We recallthat the more general form of the radial part of thesingle-particle wave function in the IQD is given bythe linear combination [11,15]:

’n; l (knlri) = Ajl(knlri) + Byl(knlri); (7)

where jl and yl are the spherical Bessel functions ofthe %rst and second kind. The constants A and B haveto be determined from the boundary conditions at theHgS well edges ri = a and ri = b (i = e; h) whileknl =

√Enl; where Enl denotes the eigenvalue in the

state (n,l). For the particular case of l =m=0 states,Eq. (7) yields an explicit solution. Taking into accountthat J0(r)=sin(r)=r and Y0(r)=cos(r)=r and limitingour study to 1s states we can easily obtain

’(ri) = ’1;0(kri) = Csin k(ri − a)

ri; (8)

where k = k10 = $=(b − a) and C is a normalizationconstant.That is worth mentioning that in the in%nite well

approximation and when a tends to b, each single

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102 J. El Khamkhami et al. / Physica E 15 (2002) 99–106

particle wave function tends to a Dirac function cen-tered on the middle of the interval [a : : : b]:

The exponential factor exp(−�reh) describes thecoulombic spatial correlation between the electron andthe hole. The factor 1 + f�(ze − zh) is introduced inorder to describe the distortion caused by the electric%eld. The variational parameters � and � are deter-mined by the minimization of the mean value of thetotal energy:

E(�; �; f) = 〈�|H |�〉=〈� |�〉: (9)

After integration over the coordinates ze and zh, thetotal energy can be expressed as

E(�; �; f) = k2 − �2 + 2�I2(�; �; f)I1(�; �; f)

− 11 + �

[I3(�; �; f) + �I4(�; �; f)]I1(�; �; f)

+2�I5(�; �; f)I1(�; �; f)

− 2I2(�; �; f)I1(�; �; f)

+I6(�; �; f)I1(�; �; f)

; (10)

where the integrals Ii (�; �; f) are de%ned by

Ii(�; �; f)

=∫ b

adre

∫ b

adrh

∫ re+rh

|re−rh|dreh Fi(re; rh ; reh) (11)

and the functions Fi(re; rh ; reh) by

F1(re; rh ; reh) = rerhreh(�2fr2eh + 3)

×[’(re)’(rh)exp− �reh]2; (12)

F2(re; rh ; reh) =1reh

F1(re; rh ; reh); (13)

F3(re; rh ; reh)

=rh(r2h − r2e − r2eh)[�2(�r2eh − reh)f2 + 3�]

×’(re)9’(re)9e[’(rh)exp− �reh]2; (14)

F4(re; rh ; reh)

=re(r2e − r2h − r2eh)[�2f2(�r2eh − reh) + 3�]

×’(rh)9’(rh)9rh[’(re)exp− �reh]2; (15)

F5(re; rh ; reh)

=�2f2rerhr2eh[’(re)’(rh)exp− �reh]2; (16)

F6(re; rh ; reh)

=− 2�f2rerhr3eh[’(re)’(rh)exp− �reh]2: (17)

3. Results and discussions

In order to investigate the in9uence of the reductionof the HgS thickness on the electron–hole correlation,we have de%ned the exciton binding energy as follows:

Eb(f) = [Ee(f) + Eh(f)]− E(�; �; f)min ; (18)

where Ee(f)+Eh(f) denotes the energy of an uncor-related electron–hole pair in presence of the externalelectric %eld.Firstly, we have studied the particular case of an

exciton in the absence of the electric %eld (f=0). Inthis case, one can easily verify that the kinetic energiesof the electron and the hole are, respectively: Ee(f=0) = k2=(1 + �) and Eh(f = 0) = �k2=(1 + �). Inthese conditions, the exciton binding energy may berewritten:

Eb(f = 0)

=[�2 − 2�

I2(�; �)I1(�; �)

+I3(�; �)I1(�; �)

+ 2I2(�; �)I1(�; �)

]min

:

(19)

The %rst three terms correspond to the kinetic contri-bution and the last one to the coulombic energy.Fig. 2a–c shows the variation of the excitonic bind-

ing energy as a function of the ratio a=b for di4erentrepresentative values of the outer radius. The ratio a=bvaries between 0 and 1 and we begin by analyzing thetwo particular cases a=b= 0 and 1. The %rst situation(a=b=0) corresponds to a HQD. We note that all thecurves tend to the values which are in good agree-ment with the results obtained by Kayanuma [16]. The

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J. El Khamkhami et al. / Physica E 15 (2002) 99–106 103

0.0 0.2 0.4 0.6 0.8 1.020

30

40

50

60

b = 0.1

b = 0.08

b = 0.06

a/b

Bin

ding

Ene

rgy

(R* )

0.0 0.2 0.4 0.6 0.8 1.04

6

8

10

12

14

16

18

b = 0.5

b = 0.4

b = 0.3

b = 0.2

Bin

ding

Ene

rgy

(R* )

a/b

0.0 0.2 0.4 0.6 0.8 1.01.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

b = 4

b = 2

b = 1

b = 0.8

Bin

ding

Ene

rgy

(R* )

a/b

(a) (b)

(c)

Fig. 2. (a) Excitonic binding energy versus the ratio a=b in strong con%nement regime (b = 0:06; 0:08 and 0:1a∗). The full curvescorrespond to our variational results and the dashed curves to those obtained by the perturbation method [11]. (b) Excitonic binding energydrawn against the ratio a=b for b = 0:2–0:5a∗ (full curves: variational; dashed curves: perturbation). (c) Excitonic binding energy versusthe ratio a=b for b = 0:8–4a∗ (full curves: variational; dashed curves: perturbation).

Page 6: Binding energy of excitons in inhomogeneous quantum dots under uniform electric field

104 J. El Khamkhami et al. / Physica E 15 (2002) 99–106

second one (a=b = 1), corresponds to a very narrowwidth HgS well and the problem is reduced to an exci-ton con%ned on a spherical surface which is a curvedspace generally considered as 2D system mainly forlarge b. As shown in Table 1, we notice that for thespherical surfaces with radius b6 0:14a∗, the exci-tonic binding energies approach values of the order of2=b limit which constitutes the absolute value of thecoulombic energy obtained by the %rst order pertur-bation in a spherical surface case [17].Fig. 2a shows that for values of the outer radius such

as b¡ 0:1a∗, the binding energy decreases monoton-ically to the 2=b limit when a=b increases. When theCdS core increases, the excitonic energy moves fromthe HQD situation where the kinetic energy is pre-dominant to the small spherical surface case wherethe coulombic energy gives the leading contribution tothe binding energy. Indeed, for b¡ 0:1a∗, the excitonloses its mobility on the surface and can be considereda %xed dipole. That is why the perturbation approachused by Ferreyra and Proetto [11] (dashed lines) whichconsiders the binding energy as the coulombic poten-tial mean value gives results close to our variationalcalculation (full lines) and the small deviation betweenthe two methods with increasing a=b is not signi%ca-tive.For values of the outer radius between 0:1 and

0:5a∗, Fig. 2b shows the same trend but the di4erencebetween the two methods appears more and more ob-vious particularly when a=b → 1 where the variationalresults move away from the limit value 2=b. Indeedwhen the shell size increases the mobility of the bound(e–h) pair is enhanced which leads to an increase ofits kinetic energy to the detriment of the coulombicinteraction. This fact limits the validity of the pertur-bation approach which considers that the binding en-ergy is given by the coulombic interaction mean value.In these cases, the best results are given by the varia-tional method.Fig. 2c shows the variations of the excitonic binding

energy versus a=b for b radii signi%cantly larger than0:5a∗. The full lines represent our variational resultswhich tend to the well-known two-dimensional exci-ton binding energy limit of 4R∗ [18] when a=b tendsto unity. We notice that for this set of radii, the kineticenergy becomes relatively more important due to thefree rotation of the electron and hole on the spher-ical surface. On the other hand, the dashed curves

represent the binding energy calculated using theperturbation method described by Ferreyra andProetto [11]. First of all, we notice that thesescurves do not recover the 2D limit value of 4R∗

when a=b tends to unity. In consequence, the2=b limit value obtained by the perturbation ap-proach is no longer valid for the spherical sur-faces with radii signi%cantly larger than 0:5a∗.Furthermore, for a given value of b larger than0:5a∗, our excitonic binding energy exhibits a min-imum when a=b increases. Indeed, when a in-creases the tridimensional con%nement decreases,so the kinetic energy becomes less important com-pared with the coulombic energy, thus we ex-pect a diminution of the binding energy. For acritical value (a=b)crit the binding energy is min-imal. This minimum moves toward the smallvalues of a=b when b increases and vanishes for thelarge values of b: When a=b varies from (a=b)crit tounity, the HgS well tends to become a spherical sur-face. Thus, the electron and hole constitute a free rotat-ing pair, so the kinetic energy becomes more and moreimportant compared with the coulombic energy whichleads to an increasing of the excitonic binding energy.Finally, we assume that the knowledge of the criticalvalue (a=b)crit allows to delimit between geometricaltridimensional and spherical surface con%nement.In order to estimate the validity of the perturbation

method, the di4erence between the binding energiesobtained by the two methods is divided by the varia-tional binding energy to obtain the relative deviation.Fig. 3 shows the relative deviation for di4erent valuesof the outer radius. According to this %gure, we thinkthat the perturbation method is only valid for b smallerthan 0:3a∗ which induce a di4erence around 8:5%.Now, we focus on the in9uence of a uniform elec-

tric %eld on the con%ned exciton in an IQD. Generallyspeaking, it is known that in the HQD, the e4ect of theelectric %eld is negligible in the case of strong con-%nement, for radius¡ 1a∗ [12–14]. This is due to thefact that the excitonic orbital is localized at the centerbecause the con%nement e4ect is predominant, whichreduces the excitonic polarizability [14]. In IQDs, ourpresent study shows that the Stark shift exists even forsmall sizes. In order to show that, we have calculatedthe excitonic binding energy in presence of electric%eld (Eq. (18)). We recall that Ee(f) and Eh(f) arecalculated using a variational method.

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J. El Khamkhami et al. / Physica E 15 (2002) 99–106 105

Table 1Coulombic and kinetic energy contributions to the binding energy (of the exciton on a spherical surface) for di4erent outer radii b6 0:14a∗

b (a∗) Coulombic contribution Kinetic contribution Eb (Eb)lim = 2=b [17]

0.02 −100:913 0.461 100.453 1000.06 −34:301 0.497 33.804 33.3330.08 −25:996 0.517 25.479 250.10 −21:027 0.538 20.488 200.12 −17:727 0.562 17.165 16.6670.14 −15:383 0.587 14.796 14.286

The last two columns are, respectively, the binding energies obtained in the present study and in Ref. [17]. All energies aregiven in units of R∗.

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

(a)

(b)

(i)

(g)

(f)

(e)

(d)

(c)

(h)

(a) : b = 0.06(b) : b = 0.1(c) : b = 0.2(d) : b = 0.3(e) : b = 0.4(f) : b = 0.5(g) : b = 0.6(h) : b = 0.8(i ) : b = 1

Rel

ativ

e D

evia

tion

a/b

Fig. 3. Relative deviation between variational and perturbationresults versus the ratio a=b for di4erent values of the outer radius.

Fig. 4 shows the plots of the binding energyEb(f) in the strong and moderate con%nements,b=0:8; 1 and 2a∗, as function of the ratio a=b for twovalues of the electric %eld strength f=0:6 and 1. Weremark that for b¡ 1a∗ and for a small core radius a,the 3D con%nement e4ect is still predominant due toa strong localization of charge density near the centerand the exciton is less sensitive to the electric %eld

0.0 0.2 0.4 0.6 0.8 1.01.2

1.6

2.0

2.4

2.8

3.2

3.6

4.0

4.4

4.8

b = 2.0

b = 1.0

b = 0.8

f = 0.0 f = 0.6 f = 1.0

Bin

ding

Ene

rgy

(R* )

a/b

Fig. 4. Binding energy of the ground state exciton versus the ratioa=b for three representative values of the outer radius (b= 0:8, 1and 2a∗) and for two values of the electric %eld strength (f=0:6and 1). The full curves correspond to the binding energy withoutelectric %eld.

e4ect. Thus, the curves corresponding to di4erent val-ues of f coincide and the binding energy decreaseswith increasing a=b. When the ratio a=b tends to unity,the electron and hole are pushed to the surface wherethey take a rotational motion, the kinetic energy is

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106 J. El Khamkhami et al. / Physica E 15 (2002) 99–106

more important and the electron and hole becomesensitive to the electric %eld. This explains that theenergy shift is more signi%cant when the carriers areplaced in both cases: strong and moderate con%nementregime.To complete the discussion, the in%nite potential ap-

proximation may be questionable. In the present study,the quantum con%nement in the well is described byan in%nitely deep potential so the exciton is supposedto be completely con%ned in the well region andthe probability to %nd the charge carriers in the coreregion is equal to zero. This approximation is valid aslong as the tunnel e4ect is negligible. For a %nite po-tential well, it is known that the leakage of the singleparticle wave function toward the core region may beimportant for very narrow wells but the single particleenergy level remains in the well. On the other hand,for an in%nite con%nement potential the single parti-cle energy tends to in%nity when the well width tendsto zero. This discrepancy is not so important since wefocus on the exciton binding energy and not on the to-tal energy. For a %nite potential well, it is predictablethat when the ratio a=b tends to unity, the excitonbinding energy will approach the CdS (core=shell)value instead of the 4R∗ limit. Some preliminary cal-culations with a %nite potential show that the presentresults remain valid for a=b about 0.9 and we plan toinvestigate and to detail this point in a future paper.

4. Conclusion

In conclusion, we have studied the excitonic bind-ing energy in IQDs. The calculations have been per-formed within the e4ective mass approximation byusing the Ritz variational method. An in%nite deeppotential describes the e4ect of the quantum con%ne-ment well. The same formalism is extended to studythe in9uence of a uniform electric %eld on the exci-tonic binding energy in IQDs. The results show thatin both cases, the binding energy strongly dependson the core and the shell radii. We have shown theexistence of a critical value (a=b)crit which presents amajor interest in the micro-fabrication technique be-cause it may be used to distinguish tridimensional

IQD con%nement from spherical surface con%nement.In addition, we have found that the Stark e4ect appearseven for strong con%nement and that the energy shiftis more signi%cant when the exciton is pushed to thesurface. These results may be useful in technologicalapplications by controlling energy levels and selectingdensity distribution by adjusting the ratio a=b and byapplying an electric %eld.We think that the present study will allow a bet-

ter understanding of the behavior of these new com-posite materials. However, the present model can beimproved by including other relevant e4ects such as%nite band o4sets and induced polarization chargeswhich will be treated in a future paper.

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