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Diamagnetic susceptibility of a magneto-donorin Inhomogeneous Quantum Dots

A. Mmadi, K. Rahmani, I. Zorkani ⇑, A. JorioGroupe des Nanomatériaux et Energies Renouvelables, LPS, Faculté des Sciences Dhar Mehraz, BP 1796 Fès, Morocco

a r t i c l e i n f o

Article history:Received 2 September 2012Received in revised form 17 January 2013Accepted 21 January 2013Available online 1 February 2013

Keywords:Inhomogeneous Quantum DotsQuantum dotQuantum wellDiamagnetic susceptibilityMagnetic fieldDonor impurityBinding energy

a b s t r a c t

The binding energy and diamagnetic susceptibility vdia are investi-gated for a shallow donor confined to move in a spherical Inhomo-geneous Quantum Dots ‘‘IQD’’ in the presence of a magnetic field.The calculation was performed with the use of a variational methodin the effective mass approximation. We describe the effect of thequantum confinement by an infinite deep potential. The resultsfor a spherical Inhomogeneous Quantum Dots made out of [Ga1�x

AlxAs (Core)/GaAs (Well)/Ga1�xAlxAs (Shell)] show that the diamag-netic susceptibility and the binding energy increase with the mag-netic field. There are more pronounced for large spherical layer. Thebinding energy and the diamagnetic susceptibility depend stronglyon the donor position. We remark that the diamagnetic susceptibil-ity presents a minimum corresponding to a critical value of the ratio

of the inner radius to the outer radius R1R2

� �crit

, this critical value is

important for nanofabrication techniques.� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In the last decades, the progress in crystal growth of low dimensional systems such as QuantumWells (QWs), Quantum Well Wires (QWWs) and Quantum Dots (QDs) have motivated several studies[1–3]. With the progress in the techniques of nano-material fabrication, it has been possible to processa new class of spherical quantum dots called quantum dot-quantum well or Inhomogeneous QuantumDots (IQDs) composed of two semiconductor materials. One of them, that with the smaller bulk bandgap, is embedded between a core and outer shell of the material with the larger band gap (Fig. 1).Experimental and theoretical studies on these IQD have been reported [4–6]. It is important to

0749-6036/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.spmi.2013.01.006

⇑ Corresponding author.E-mail address: [email protected] (I. Zorkani).

Superlattices and Microstructures 57 (2013) 27–36

Contents lists available at SciVerse ScienceDirect

Superlattices and Microstructures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o ca t e / s u p e r l a t t i c es


understand the electronic and optical properties of impurities in HQD and IQD because these proper-ties are strongly affected by the presence of shallow impurities [7–10]. The systems CdS/HgS/CdS[7,16,17] and Ga1�xAlxAs/GaAs/Ga1�xAlxAs [10,18] are more investigated in the present time.

Understanding the electric and magnetic field effects in these nanostructures is important for tech-nological applications. Zounoubi et al. [11] have studied the influence of magnetic fields on the bind-ing energy and polarizability of a shallow donor impurity placed at the center of a cylindrical quantumdot (CQD). They have demonstrated that the magnetic field increases the binding energy and stronglyreduces the polarizability. For higher field strength and large dot, the magnetic field effects are pre-dominant. Bilekkaya et al. [12] calculated the ground state energies for an electron in QWW with dif-ferent shapes in the presence of an applied electric and magnetic fields using the finite differencemethod. They showed that the binding energy depends strongly on the structural confinement andon the electric and magnetic field. Didi Seddik and Zorkani [13] have studied the absorption spectraassociated with transition between the ground state of a hydrogenic donor impurity to the conductionband in the presence of a magnetic field for spherical CdSe QD. They found that the absorption coef-ficient varies systematically as a function of QDs size. The application of the magnetic field may hinderthe absorption coefficient of an on-center impurity and displace the threshold energy towards highenergy and towards low energy transition. Corella-Maduena et al. [14] have calculated the groundstate and binding energies for a hydrogenic impurity in QD within a uniform magnetic field, theyhad used a trial wave function to treat the cases of on-center, off center or edge impurities. Sadeghiand Rezaie [15] have worked on the effect of magnetic field on the impurity binding energy of theexcited states in QD with a finite barrier potential. El Khamkhamia et al. [16] have investigated abinding energy of excitons in Inhomogeneous Quantum Dots under uniform electric field. Their resultsshow that in both cases, the binding energy strongly depends on the core and the shell radii with orwithout an applied electric field. Chen and Xie [17] have concentrated the nonlinear optical propertiesof nanospherical layer systems in the presence of an electric field. Rahmani and Zorkani [18] havestudied the magnetic and electric field effect on the binding energy of a shallow donor in Ga1�xAlx

As–GaAs IQD. Their results show that the corrections due to the magnetic and electric field are very

Fig. 1. Schematic diagram of Spherical Inhomogeneous Quantum Dots. R1 and R2 are the inner and the outer radius of the IQDrespectively.

28 A. Mmadi et al. / Superlattices and Microstructures 57 (2013) 27–36


important and cannot be neglected or ignored and they show the existence of a critical value whichcan be used to distinguish the three dimensions confinement from the spherical surface confinement.

Otherwise, in parallel with the recent developments of spintronics, several authors have studiedthe magnetic field induced metal–insulator transition [19,20]. Since theoretical [21] and experimental[22] work of diamagnetic susceptibility in doped Si has drawn considerable attention. Peter and Ebe-nezar [23] have computed and compared the susceptibility for a hydrogenic donor in a spherical con-finement, harmonic oscillator-like and rectangular well-like potentials for a finite QD. They observed astrong influence of the shape of confining potential and geometry of the dot on the susceptibility.Khordad and Fathizadeh [24] have reported the effect of temperature and pressure simultaneously,on the diamagnetic susceptibility and binding energy of a hydrogenic on-center donor impurity. Ina recent paper, Rahmani et al. [10] have investigated the diamagnetic susceptibility of a confined do-nor in Ga1�xAlxAs–GaAs IQD. They found that the binding energy and the diamagnetic susceptibilitydepend strongly on the core radius and the shell radius. They have demonstrated that the bindingenergy shows a minimum for a critical value of ratio R1/R2 depending on the value of the outer radiusand shows a maximum when the donor is placed at the center of the spherical layer. Kilicarslan et al.[25] have investigated the effects of the magnetic field and the dielectric screening on the diamagneticsusceptibility of a donor in a QW with anisotropic effective mass. Sharkey et al. [26] have reported astudy on the magnetic field effects on the binding energy and diamagnetic susceptibility of a donor ina GaN/AlGaN quantum dot. They showed that binding energy and diamagnetic susceptibility increaseswith the magnetic field and is pronounced for large dot. Koksal et al. [27] have studied the magneticfield effects on the diamagnetic susceptibility and binding energy of a hydrogenic impurity in a QWWby taking into account spatially dependent screening. They show that the diamagnetic susceptibility ismore important for donors in QWW over a large range of wire dimensions. Kilcarslan et al. [28] havestudied the magnetic field effects on the diamagnetic susceptibility in a QW (GaxIn1�x NyAs1�y/GaAs)and found that the diamagnetic susceptibility and binding energy of the magneto-donor in the(GaxIn1�x NyAs1�y/GaAs) QW increases with nitrogen mole fraction. Peter [29] has studied thepolarizability and diamagnetic susceptibility of shallow donors in finite-barrier (GaAs/Ga1�xAlxAs)of harmonic oscillator nanodots and has seen that there is magnetic field dependence the polarizabil-ity and the diamagnetic susceptibilities for different magnetic field strengths. To the best of ourknowledge, theoretical or experimental studies on the diamagnetic susceptibility of a magneto-donorplaced in a IQD have not been reported yet. In the present work, we use a variational method to cal-culate the hydrogenic donor binding energy and the diamagnetic susceptibility in a GaAs/Ga1�xAlxAsIQD in presence of the magnetic field. We will take account the donor position (see Fig. 1). R1 and R2,respectively, denotes the inner and outer radius of the IQD and Vw is the confinement potential.

This paper is organized as follows: in Section 2 we explain the Hamiltonian of hydrogenic impurityground state in the presence of a magnetic fields; we deduce the expression of the magneto-donorbinding energy and the diamagnetic susceptibility. The numerical results and conclusion arepresented in Section 3 and a close comparison will be made with the Ref. [10].

2. General formalism

We consider a donor impurity located at the position r0 in a spherical Inhomogeneous QuantumDots made out of [Ga1�xAlxAs (Core)/GaAs (Well)/Ga1�xAlxAs (Shell)] in presence of the magnetic fieldB!

. In the effective mass approximation, the Hamiltonian of the system is written as:

H ¼ 12m�

P!� e

cA!� �2

� e2

e0j~r �~r0jþ Vw ð1Þ

where e0 is the dielectric constant, m� is the effective electron mass, A!

is the vector potential of themagnetic field, B

!¼ r!� A!

, r0 is the impurity position. As in our previous papers the case of finite bar-rier will be considered later [30]. For the present work we consider an infinitely deep well [16,17]

Vw ¼0 R1 < r < R2

1 r < R1 and r > R2

�ð2Þ

A. Mmadi et al. / Superlattices and Microstructures 57 (2013) 27–36 29


The effect of hight barrier are in progress. This hypothesis depends on the nature of materialsconsidered and also on the size of the IQD.

For a homogeneous magnetic field B!ð0:0;BÞ along the z-axis, the vector potential is chosen as

A!¼ 1

2 B!� vecr. The Hamiltonian for the ground state, in spherical coordinates, can be expressed as:

H ¼ �r2 � 2j~r �~r0j

þ 14c2r2 sin2 hþ Vw ð3Þ

with

j~r �~r0j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ r2

0 � 2rr0 Coshq

ð4Þ

We use the effective Bohr radius a� ¼ �h2e0m�e2 and the effective Rydberg R� ¼ m�e4

2�h2e20

as the units of lengthand energy (respectively). Furthermore, we introduce the dimensionless parameter c ¼ �hxc

R� character-izing the strength of the magnetic field, xc ¼ eB

m�c is the effective cyclotron frequency and r0 is given by

r0 ¼ R1þR22 .

We use a variational method approach to determine the ground state binding energy; we adopt thewave function given by:

w ¼ sin½Kðr � R1Þ�r

expð�aj~r �~r0jÞ ð5Þ

K ¼ pR2�R1

and a is a variational parameter. The exponential factor exp-aj~r �~r0j describes the Coulombspatial interaction. The corresponding energy is obtained by minimization with respect to the varia-tional parameter a:

E ¼mina

hwjHjwihwjwi

�ð6Þ

The binding energy Eb of the donor impurity is given by:

Eb ¼ ESub � E ð7Þ

where E and ESub represents the state energy of an electron in IQD with and without the impurityrespectively.

The diamagnetic susceptibility vdia of the donor impurity in IQD, in atomic unit (a.u), is given by[31]:

vdia ¼ �e2

6m�e0c2 hð~r �~r0Þ2i ð8Þ

where c is the velocity of light (c = 137 and e = 1, m0 = 1 in a.u.) and hð~r �~r0Þ2i is the mean squaredistance of the electrons from the nucleus.

3. Numerical results and discussion

As in our previous Ref. [10] without a magnetic field; two extreme cases that present themselves:R1 = 0 (i.e. Homogeneous Quantum Dot ‘‘HQD’’ of radius R2) and R1 ? R2 for R2 fixed which corre-sponds to an infinitely thin spherical layer. We will present the results for the cases of a HQD and IQD.

3.1. Quantum dot R1 = 0

We consider first the case of a HQD (i.e. R1 = 0; Vw = 0, r 6 R2 and Vw =1, r > R2). In order to inves-tigate the influence of the magnetic field, we display in Figs. 2 and 3, the donor binding energy Eb andthe diamagnetic susceptibility vdia as a function of the dot radius R2 for different values of magneticfield (c = 0, 0.4 and 0.8). The magneto-donor is placed in the center of the dot. There is a competitionbetween the geometric confinement and the magnetic confinement. For small dot (R2 < 2a�), themagnetic field effect on the binding energy and the diamagnetic susceptibility is not remarkable. Sothe magnetic field effect becomes important for large QD (R2 P 2a�). The binding energy and the



diamagnetic susceptibility increase with the magnetic field. It is important to note that the diamag-netic susceptibility vdia decreases when the dot radius R2 increases. We remark that the susceptibilityvdia in the absence of magnetic field (c = 0) tend the value �1.1 a.u. which correspond to the bulk limitcase (see Refs. [10,26,27]).

In order to investigate the influence of the magnet-donor position, we display in Fig. 4, the diamag-netic susceptibility vdia as a function of the ratio r0/R2 for two different radius of the structure (R2 = 1and 2a�). For small QD R2 = 1a� (strong confinement regime) and for different donor positions, themagnetic effect is not appreciable and the geometric confinement is predominate. While for large

Fig. 2. Variations of the donor binding energy Eb as a function of the dot radius R2 for different values of the magnetic field(c = 0, 0.4 and 0.8).

Fig. 3. Variation of the diamagnetic susceptibility vdia as a function of the dot radius R2 for three values of magnetic field (c = 0,0.4 and 0.8).



QD R2 = 2a� (weak confinement regime), the effect of the magnetic field on the susceptibility is moreappreciable and the susceptibility increases with the magnetic field. We remark that the diamagneticsusceptibility vdia decreases as the ratio r0/R2 increases (i.e. the donor impurity moves from the centerto the surface). This effect is more pronounced when the donor becomes close of the surface (r0/R2 ? 1). These results are in good agreement with those found in Ref. [10] in the absence of magneticfield (c = 0). This could be explained by the fact that for the one-center donor, the orbital electronicwave function has a spherical symmetry and vdia is maximal. While for the off-center donor, thespherical symmetry is broken. We note that the diamagnetic susceptibility is more sensitive for largeQD radius. In Fig. 5, we present the diamagnetic susceptibility as a function of magnetic field for threedifferent values of the radius (R2 = 1a�, 1.25a� and 2a�). We see that for a small dot (R2 = 1a� and1.25a�), the magnetic field effect is not pronounced. For large dot (R2 = 2a�), the diamagnetic suscep-tibility vdia increases when the magnetic field increases and reaches a limit value when the magneticfield becomes very strong.

Fig. 4. Variation of the diamagnetic susceptibility vdia as a function of the ratio r0/R2 for two values radius of the structure(R2 = 1a� and 2a�) and three values of magnetic field (c = 0, 1 and 2).

Fig. 5. Variation of the diamagnetic susceptibility vdia function of magnetic field with three values R2 (R2 = 1a�, 1.25a� and 2a�).



3.2. Inhomogeneous quantum dot

In Figs. 6 and 7, we plot the variation of the binding energy Eb and the diamagnetic susceptibilityvdia as a function of the ratio R1/R2 for two values of the outer radius (R2 = 1a� and 2a�) and for variousvalues of the magnetic field (c = 0, 0.4 and 0.8). The donor is placed in the center of the spherical layerr0 ¼ R1þR2

2

� �. The ratio R1/R2 varies between 0 and 1. In Fig. 6, we observe that the binding energy in-

creases as the magnetic field increases. The magnetic field effect is more pronounced when the ratioR1/R2 go to 1. In the absence of magnetic field (c = 0), the binding energy approaches the value of 4R�

which corresponds to the case of 2D confinement [32]. In Fig. 7, we remark that the magnetic field

Fig. 6. Variations of the binding energy Eb as a function of the ratio R1/R2 for two different values of the outer radius of theQDQW, R2 = 1a� and 2a� and various values of the magnetic field c = 0, 0.4 and 0.8.

Fig. 7. Variation of the diamagnetic susceptibility vdia as a function of the ratio R1/R2 and for different values of the outer radius(R2 = 1a� and R2 = 2a�) and various values of the magnetic field (c = 0, 0.4 and 0.8).



effect on the diamagnetic susceptibility is not pronounced for small spherical layer (R2 = 1a�, strongconfinement regime). For large spherical layer (R2 = 2a�, weak confinement regime), vdia increaseswith the magnetic field. When the ratio R1/R2 tends to 1, the diamagnetic susceptibility vdia tendsto the limit which represents the case of two-dimensional. We remark that the diamagnetic suscep-tibility presents a minimum corresponding to a certain critical value of the ratio R1/R2. From this fig-ure, we can say that the susceptibility vdia decreases as R2 increases in the absence of magnetic fieldcase (c = 0).

In Fig. 8, we present the diamagnetic susceptibility vdia as a function of magnetic field for three dif-ferent values of the outside radius (R2 = 1.125a�, 1.25a� and 2a�). The impurity is located in the centerof the spherical layer. By setting (R1 = 1a�), we see that the diamagnetic susceptibility increases withthe magnetic field. For small spherical layer (R2 = 1.125a�, R2 = 1.25a�), the magnetic field effect is notsignificant and the geometric confinement is dominant. The magnetic field effect on the susceptibility

Fig. 8. Variation of the diamagnetic susceptibility vdia function of magnetic field with three values R2 (R2 = 1.125a�, 1.25a� and2a�), with R1 = 1a�.

Fig. 9. Variation of the diamagnetic susceptibility vdia as a function of the impurity position r0 with three values of magneticfield (c = 0, 0.4 and 0.8).



is more significant for a large spherical layer (R2 = 2a�). We note that vdia increases when the magneticfield increases. We also remark that the susceptibility decreases as the outside radius of the structureincreases. These results are in good agreement with those presented in the case of a homogeneousquantum dot (see Section 3.1).

Fig. 9 presents the diamagnetic susceptibility as a function of the impurity position for three differ-ent values of the magnetic field (c = 0, 0.4 and 0.8) with R1 = 1a� and R2 = 3a�. We note that the diamag-netic susceptibility decreases as the donor moves towards the surface of the spherical well. Theseresults are in good agreement with the case of QW. The susceptibility increases as the magnetic fieldincreases and this magnetic field effect is more pronounced when the donor is placed near the extrem-ities of spherical layer.

4. Conclusion

In this work, the effects of a magnetic field on the diamagnetic susceptibility of a donor placed in-side an Inhomogeneous Quantum Dot are investigated. The calculation has been performed within theeffective mass approximation by using the variationale method. The results show that the magneticeffect on the diamagnetic susceptibility is appreciable especially for large structures. We have shown

the existence of a critical value R1R2

� �crit

which may be important for the nanofabrication techniques.

This critical value can be used to distinguish the three dimensional confinement from the sphericalsurface confinement. We think that the present study will allow a better understanding of the behav-ior of these new structures. We believe that our results of the susceptibility in IQD will be relevant inthe interpretation of experimental results. Unfortunately, we could not compare our results as noexplicit experimental data are available in the literature. The present model can be improved byincluding other relevant effects such as finite band offsets which will be treated in a future work.

Acknowledgements

Two of the authors, Izeddine Zorkani and Ali Mmadi, would like to thank the Abdus Salam Interna-tional Centre for Theoretical Physics (Trieste, Italy, Dr. I.Z. is associate of the ICTP and we have aFederation Scheme with ICTP) for its support and hospitality.

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