Journal of Luminescence 138 (2013) 48–52

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Journal of Luminescence

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Influence of external field and consequent impurity breathing on excitationprofile of doped quantum dots

Suvajit Pal a, Manas Ghosh b,n

a Department of Chemistry, Hetampur Raj High School, Hetampur, Birbhum 731124, West Bengal, Indiab Department of Chemistry, Physical Chemistry Section, Visva-Bharati University, Santiniketan, Birbhum 731235, West Bengal, India

a r t i c l e i n f o

Article history:

Received 27 September 2012

Received in revised form

9 December 2012

Accepted 25 January 2013Available online 1 February 2013

Keywords:

Quantum dot

External field

Impurity domain

Impurity coordinate

Relative oscillation frequency

Excitation rate

13/$ - see front matter & 2013 Elsevier B.V. A

x.doi.org/10.1016/j.jlumin.2013.01.029

esponding author. Tel.: þ91 3463 2615

1 3463 262672.

ail addresses: mgphyschem@gmail.com,

@rediffmail.com (M. Ghosh).

a b s t r a c t

Excitation in quantum dots is an important phenomenon. Realizing the importance we investigate the

excitation behavior of a repulsive impurity doped quantum dot induced by an external oscillatory field.

As an obvious consequence the simultaneous oscillation of spatial stretch of impurity domain has also

been taken into account. The impurity potential has been assumed to have a Gaussian nature. The ratio

of two oscillations (Z) has been exploited to understand the nature of excitation. Indeed it has been

found that the said ratio could orchestrate the excitation in a truly elegant way. Apart from the ratio,

the dopant location also plays some meaningful role towards modulating the excitation rate. The

present study also indicates the attainment of stabilization in the excitation rate as soon as Z surpasses

a threshold value irrespective of the dopant location. Moreover, prior to the onset of stabilization we

also envisage minimization in the excitation rate at some typical Z values depending on the dopant

location. The critical analysis of pertinent impurity parameters provides important perception about

the physics behind the excitation process.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Over the couple of decades we have witnessed an exponentialgrowth in theoretical and experimental researches on impuritystates of low-dimensional heterostructures [1,2]. The quantizedproperties of these doped systems have made them perfectmaterials for scientific study and technological applications. Inview of this, researches on opto-electronic properties of a widerange of semiconductor devices containing impurity have nowbecome worldwide with the excitement of understanding newphysics and potential hope for technological impact [3–15].

Miniaturization of semiconductor devices reaches its limitwith the advent of quantum dots (QDs). With QD, the subtleinterplay between new confinement sources and impurity relatedpotentials has opened up new windows of research in this field[16]. Such confinement, coupled with the dopant location, candramatically alter the electronic and optical properties of thesystem [4]. For this reason there are a seemingly large number oftheoretical studies on impurity states [17–26]. Added to this,there are also some excellent experimental works which includethe mechanism and control of dopant incorporation [27,28].

ll rights reserved.

26/þ91 3463 262751 6x467;

The emergence of novel experimental and theoretical techni-ques together with the improvement of traditional ones havemade the research on carrier dynamics in nanodevices a ubiqui-tous one [29,30]. The time-dependent aspects in nanodevicesnaturally become a hot topic which largely comprises ofresearches on internal transitions between impurity inducedstates in a QD [31,32]. These transitions depend on the spatialrestriction imposed by the impurity. A minute survey of thedynamical features directs us to explore excitation of electronsstrongly confined in QD’s. Detailed analysis on this aspect deemsimportance because such excitation provides us with systems foruse in opto-electronic devices and as lasers. Within the purviewof technological applications such excitation further involvesoptical encoding, multiplexing, photovoltaic and light emittingdevices. The phenomenon also plays some promising role in theeventual population transfer among the exciton states in QD[33,34].

In connection with the dynamical aspects mentioned abovethere are some important works which study the effect of appliedelectric field on doped quantum wells and dots [6,35,36]. Theaspects discussed above impel us to thoroughly investigate theexcitation in doped quantum dots propelled by external oscilla-tory field. As a result, off late, we have made some investigationson the excitation profiles of the doped quantum dots exposed tooscillatory external field [37]. A deeper physical insight in ourproblem hitherto outlined reveals some deficiency. The deficiencyarises as we have so far considered that the spatial stretch of

S. Pal, M. Ghosh / Journal of Luminescence 138 (2013) 48–52 49

impurity (g�1 in this paper) remains incurious to this externalelectric field. In practice this spatial stretch depends heavily onmagnitude of dot–impurity interaction and any perturbation thataffects the said interaction would undoubtedly have tangibleinfluence on the stretch. The energy delivered by an oscillatoryexternal field to the system causes a change in the extent of dot–impurity interaction. And this makes our earlier consideration ofuninterrupted impurity spread too stringent to realize. As a resultit becomes absolutely essential to assimilate the time-variation ofg in the present study. Of course, this time-dependent alterationin the spatial spread of dopant should not be arbitrary but mustbe guided by the oscillation frequency of external field. Thus,notwithstanding the tiresome elevation in the mathematical rigorwe also incorporate the effect of time-dependent variation of g inour calculation. We conceive such a breathing impurity with aview of making the problem more realistic and reasonable.It needs to be mentioned now that in the present study we haveconsidered that external oscillatory field affects the impuritywidth only. The other important impurity parameters beingimpurity location (r0) and impurity strength (V0). We felt that itis the impurity width which is most susceptible to a changeowing to the external field. V0 is the inherent strength of impuritypotential and is thus expected not to undergo any alteration dueto the external field. On the other hand, impurity location may bechanged due to the external field but it would require largeamount of energy input from the external field. The spatialextension of impurity appears to be the most prominent impurityparameter whose variation due to the external field can bephysically realized.

Recently we have done some works on this kind of time-dependent impurity spread in different perspectives [38,39].Thus, in the present enquiry we have tried to decipher thecombined role of oscillatory external field and impurity stretchon the excitation profile. To be precise, in this work we havemonitored the ratio of above two oscillation frequencies (termedas relative oscillation frequency, ROF and denoted by Z wheren1 ¼ Zn2, n1 and n2 being the oscillation frequencies of externalfield and g, respectively) [38,39] in connection with determiningthe time-average excitation rate for different dopant locations.Such a response of linear type (i.e. output frequency proportionalto input frequency) has been considered for mathematical andcomputational convenience. Also, such a linear relation can bemost easily envisaged and could represent the real situation tosome extent. Since the two frequencies are proportional to eachother it suggests that there is a kind of linear dependencebetween them. However, the ratio Z simply behaves as a propor-tionality constant which has been varied arbitrarily in order tomonitor the influence of relative strengths of the two frequencieson the excitation kinetics.

A consequent follow up of the dynamics of the doped dot bysolving the time-dependent Schrodinger equation containing thetime-dependent potential becomes the obvious task to tackle theproblem.

2. Method

The model considers an electron subject to a harmonic con-finement potential Vðx,yÞ and a perpendicular magnetic field B.The confinement potential assumes the form Vðx,yÞ ¼ 1

2 mn

o20ðx

2þy2Þ, where o0 is the harmonic confinement frequency,oc ¼ eB=mnc being the cyclotron frequency (a measure of mag-netic confinement offered by B). In the present work a magneticfield of miliTesla (mT) order has been employed. mn is theeffective electronic mass within the lattice of the material to beused. We have taken mn ¼ 0:5m0 and set _¼ e¼m0 ¼ a0 ¼ 1. This

value of mn closely resembles Ge quantum dots (mn ¼ 0:55 a.u.).We have used Landau gauge ½A¼ ðBy,0,0Þ� where A stands for thevector potential. The Hamiltonian in our problem reads

H00 ¼�_2

2mn

@2

@x2þ@2

@y2

� �þ

1

2mno2

0x2þ1

2mnðo2

0þo2c Þy

2�i_ocy@

@x:

ð1Þ

Define O2¼o2

0þo2c as the effective frequency in the y-direction.

The model Hamiltonian [cf. Eq. (1)] sensibly represents a 2-dquantum dot with a single carrier electron [40,41]. The form ofthe confinement potential conforms to a kind of lateral electro-static confinement (parabolic) of the electrons in the x2y plane[9,17,24,42].

Following earlier works on the effects of a repulsive scattererin multi-carrier dots in the presence of magnetic field [43,44],here we have considered that the QD is doped with a repulsiveGaussian impurity. Now, as the impurity perturbation is attachedto the Hamiltonian [cf. Eq. (1)] it transforms to

H0ðx,y,oc ,o0Þ ¼H00ðx,y,oc ,o0ÞþVimpðx0,y0Þ, ð2Þ

where Vimpðx0,y0Þ ¼ Vimpð0Þ ¼ V0e�g0½ðx�x0Þ2þðy�y0Þ

2� with g040 and

V040 for repulsive impurity, and ðx0,y0Þ denotes the coordinateof the impurity center. V0 is a measure of the strength of impuritypotential whereas g0 determines the spatial stretch of the impur-ity potential. A large value of g0 indicates a highly quenchedspatial extension of impurity potential whereas a small g0

accounts for the spatially dispersed one. Thus, a change in g0 inturn causes a change in the extent of dot–impurity overlap thataffects the excitation pattern noticeably [37–39]. The parameterg0 in the impurity potential is equivalent to 1=d2, where d isproportional to the width of the impurity potential [43,44]. Thevalue of g0 is taken to be 0.001 a.u. which corresponds to anextension of the impurity domain up to 1.41 nm. The dopantstrength (V0) assumes a maximum value of � 10�4 a.u. or2.72 meV. The presence of repulsive scatterer simulates dopantwith excess electrons. The use of such Gaussian impurity poten-tial is quite well-known [45–47]. In view of the ongoing discus-sion the work of Gharaati et al. [48] merits mention. Theyintroduced a new confinement potential for the spherical QD’scalled modified Gaussian potential, MGP and showed that thispotential can predict the spectral energy and wave functions ofa spherical quantum dot.

We write the trial wave function cðx,yÞ as a superposition ofthe product of harmonic oscillator eigenfunctions fnðaxÞ andfmðbyÞ, respectively, as follows [37–39]:

cðx,yÞ ¼Xn,m

Cn,mfnðaxÞfmðbyÞ, ð3Þ

where Cn,m are the variational parameters and a¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimno0=_

pand

b¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimnO=_

p. In the linear variational calculation, we have used

an appreciably large number of basis functions [cf. Eq. (3)] withn,m¼ 0220 for each of the directions (x,y). This direct productbasis spans a space of ð21� 21Þ dimension. It has been verifiedthat the basis of such size scans the 2-d space effectivelycompletely as long as monitoring the observables under investi-gation is concerned. A convergence test run by us with stillgreater number of basis functions confirmed our observation.

The general expressions for the matrix elements of H00 and Vimp

are given in Refs. [38,39]. The pth eigenstate of the system in thisrepresentation can be written as

cpðx,yÞ ¼X

ij

Cij,pffiðaxÞfjðbyÞg, ð4Þ

where i, j are the appropriate quantum numbers, respectively, andðijÞ are composite indices specifying the direct product basis.

S. Pal, M. Ghosh / Journal of Luminescence 138 (2013) 48–5250

The external electric field V1ðtÞ is now switched on with

V1ðtÞ ¼ ex � x � sinðn1tÞþey � y � sinðn1tÞ, ð5Þ

where ex and ey are the field intensities along x and y directionsand n1 being the oscillation frequency. Now the time-dependentHamiltonian reads

HðtÞ ¼H0þV1ðtÞ: ð6Þ

The matrix element involving any two arbitrary eigenstates p andq of H0 due to V1ðtÞ reads

Vextp,qðtÞ ¼/cp9V1ðtÞ9cqS

¼ ex sinðgxtÞX

ij

Xkl

Cn

ij,pCkl,q/fiðaxÞ9x9fkðaxÞSdjl

þey sinðgytÞX

ij

Xkl

Cn

ij,pCkl,q/fjðbyÞ9y9flðbyÞSdik, ð7Þ

where

/fiðaxÞ9x9fkðaxÞS¼1

a

ffiffiffiffiffiffiffiffiffiffiffikþ1

2

rdk,iþ

ffiffiffik

2

rdk�1,i

" #ð8Þ

and

/fjðbyÞ9y9flðbyÞS¼1

b

ffiffiffiffiffiffiffiffiffilþ1

2

rdl,jþ

ffiffiffil

2

rdl�1,j

" #: ð9Þ

We can now introduce the time-dependence into the impurityspread so that g0-gðtÞ. Now the time-dependent Hamiltonian reads

HðtÞ ¼ ½H0�Vimpð0Þ�þV1ðtÞþV2ðtÞ, ð10Þ

where

V2ðtÞ ¼ V0 e�gðtÞ½ðx�x0Þ2þðy�y0Þ

2�: ð11Þ

The matrix element involving any two arbitrary eigenstates p and q

of H0 due to V2ðtÞ reads (Details of derivation and various relevantquantities are given in Refs. [38,39].)

Vimpp,q ðtÞ ¼/cpðx,yÞ9V1ðtÞ9cqðx,yÞS

¼Xnm

Xn0m0

Cn

nm,pCn0m0 ,q/fnðaxÞfmðbyÞ9V1ðtÞ9fn0 ðaxÞfm0 ðbyÞS

¼ V0ðtÞX16

j ¼ 1

Vjp,qðtÞ: ð12Þ

Now, we consider the periodic fluctuation of g as a result of variabledot–impurity interaction arising out of energy transfer from theexternal field. The oscillation frequency of g (viz. n2) has beenconsidered to be multiple and submultiple of n1 and can berepresented as n1 ¼ Zn2, Z is the integers and fractions. The quantityZ could be termed as the relative oscillation frequency (ROF). Thus,the quantity Z actually links the two oscillations and eventuallyturns out to be a key factor in controlling the excitation rate. Thetime-variation of g can be written as

gðtÞ ¼ g0 cosðn2tÞ: ð13Þ

Under the perturbation, the evolving wave function is described by alinear combination of the eigenstates of H0 which is diagonal in thefcg basis

cðx,y,tÞ ¼X

q

aqðtÞcq: ð14Þ

In order to determine the time-dependent superposition coefficientswe need to solve the time-dependent Schrodinger equation (TDSE).

i_@c@t¼Hc or equivalently

i_ _aq ðtÞ ¼HaqðtÞ, ð15Þ

with the initial conditions apð0Þ ¼ 1, aqð0Þ ¼ 0, for all qap, where p

may be the ground or any other excited states of H0. The TDSE in thedirect product basis [cf. Eq. (15)] has been integrated by the sixth

order Runge–Kutta–Fehlberg method with a time step sizeDt¼ 0:01 a.u. and the numerical stability of the integrator has beenchecked. The quantity PkðtÞ ¼ 9akðtÞ9

2indicates the population of kth

state of H0 at time t. There occurs a continuous growth and decay inthe ground state population [P0ðtÞ] during the time evolution.Naturally the quantity Q ðtÞ ¼ 1�P0ðtÞ serves as a measure ofexcitation. In consequence, the quantity RexðtÞ ¼ dQ=dt serves asthe time-dependent rate of excitation. We have calculated the time-average rate of excitation ½/RexS¼ ð1=TÞ

R T0 RexðtÞ dt� with T being

the total time of dynamic evolution as a function of ROF (Z) fordifferent dopant locations (r0).

3. Results and discussion

Our present investigation has made it quite perspicuous thatthe excitation rate is controlled by the interplay between severalfactors of different characteristics. As the dopant is introduced ata greater distance from the dot confinement center (0, 0) theconfines of electric (o0) and magnetic (oc) origins naturallybecome weak and favor excitation. However, the said shift alsosimultaneously decreases the extent of repulsive interactionbetween the dot and the impurity which has a negative impacton the degree of excitation. In order to visualize the role played bythe spatial stretch of impurity potential more comprehensively,the plot of the matrix elements /c09V imp9c0S as a function ofradial position of impurity (r0) assumes importance (see Fig. 1 ofRef. [39]). Also, the functioning of spatial stretch of impurity (g�1)is no longer straightforward and requires deeper realization.An increase in g reduces the spatial stretch of the impuritypotential which in turn reduces the extent of overlap betweenthe dot and the impurity. Such a decrease in the said overlap has atwo-folded role, primarily it reduces the dot–impurity interactionand consequently the strength of dot confinement therebyimpeding and promoting the excitation concurrently. In thiscontext the work of Xie seems pertinent which showed thatabsorptions in a spherical QD containing Gaussian impuritydepend on electron–impurity overlap [49]. Thus, it is the subtlenuances of several impurity parameters that ultimately shape theexcitation process.

Let us now have a close look at the plot that portrays the time-average excitation rate (/RexS) as a function of ROF (Z) for threedifferent dopant locations namely on-center (r0 ¼ 0:0 a.u.), nearoff-center (r0 ¼ 28:28 a.u.), and far off-center (r0 ¼ 70:71 a.u.)(Fig. 1). From the plot it is quite discernible that at all dopantlocations the excitation rate culminates in a saturation at large Zvalues indicating sort of negotiation between several factors thattune excitation. The diversities in the nature of excitation profilesare most prominent in low and medium Z values.

For an on-center dopant it is evident that excitation rate startsfrom a rather high value when Z is extremely small. However, therate falls noticeably to a minimum at around Z� 0:52. As Z isfurther increased /RexS again undergoes an enhancement andfinally settles to a steady value as ZZ2:18. A low to low-mediumZ regime is furnished with a strong g oscillation which in thepresent case seems to undergo a kind of skirmish with oscillatoryexternal field. Thus, instead of promoting the excitation rate –which would have been possible if the two oscillations reinforceeach other – we observe a depletion in the said quantity. Withinthe domain 0:52rZr2:18 the rise in /RexS indicates that thetwo oscillations mutually underpin one another. As Z is increasedfurther (i.e. ZZ2:18) the g oscillation becomes less expressive(but yet it would be imprudent to ignore it because of on-centerlocation of dopant) and invites some semblance of balancebetween the oscillations of its own and the external field leadingto a saturation.

0 1 2 3 4 50.2

0.4

0.6

0.8

1.0

(iii)

(ii)

(i)

<Rex

> x

106

R O F

Fig. 1. Plot of /RexS vs Z for different dopant locations (r0) with (i) r0 ¼ 0:0 a.u.,

(ii) r0 ¼ 28:28 a.u., and (iii) r0 ¼ 70:71 a.u.

0 10 20 30 40 50 60 70

0.2

0.4

0.6

0.8

1.0

1.2

<Rex

> x

106

r0 (a. u.)

Fig. 2. Plot of /RexS vs r0 at Z¼ 5:0.

S. Pal, M. Ghosh / Journal of Luminescence 138 (2013) 48–52 51

At near off-center dopant location the dot–impurity overlapgets somewhat reduced. At this dopant location also we envisagesimilar behavior of /RexS as a function of Z as in on-centerlocation. However, the minimization now occurs at Z� 0:68 andsaturation begins from a Z value of 1.38. Thus, the region aroundthe minima now comes out to be much slender with respect tothe previous case. Also, the drop in the excitation rate around theminima is much shallower in its extent in comparison to the on-center location. The observations indicate that although thenature of mutual adjustment of the two oscillations remain thesame as before, its intensity is much abridged. The obvious reasonbehind this depleted interplay between the two oscillations is theshifted placement of the dopant. It is because of the placement forwhich the dot–impurity overlap is much weaker in comparison tothe on-center counterpart. Accordingly, over the entire range of Z,the influence of g oscillation is not so much pronounced as it is incase of an on-center dopant.

For a far off-center dopant the /RexS vs Z profile is rathermonotonous as the rate declines from the very beginning until itgets saturated. The saturation commences from Z¼ 3:01 onwards.At a far off-center dopant location the dot–impurity overlap getshighly quenched so that g oscillation has little impact in compar-ison to near off-center location. An increase in Z value at thislocation makes g oscillation further downgraded. Thus, althoughthe external field continues to deliver energy it becomes moreand more unable to promote excitation with increase in Z owingto serious crunch in the support from g oscillation. A persistentdecrease in the excitation rate becomes manifestly obvious.

In the present inspection the importance of dopant coordinateon excitation profile becomes quite significant. In this connectionthe notable works of Baskoutas et al. [12], Karabulut et al. [50],and Gomez et al. [51] and others [8,16,25] related to dopantlocated at off-center position are worth-mentioning. Particularly,Karabulut, Baskoutas and their coworkers studied the off-centerimpurities invoking an accurate numerical method (PMM, poten-

tial morphing method). We therefore now present the solitary roleof dopant location on the excitation pattern for a more compre-hensive description of our investigation. Since heretofore we havefound that beyond ZZ3:0 there occurs some kind of stabilization

in /RexS regardless of dopant location, we explore the positiondependence of /RexS keeping Z fixed at 5.0 (Fig. 2). The plotreveals that the excitation rate decreases monotonically as thedopant is progressively shifted to more and more off-centerexcept at r0 � 14:5 a.u. where it exhibits a faint maxima. As wehave pointed out earlier, the position dependent maximization of

the excitation rate occurs when the forces that resist excitationsupersede the forces that have the reverse impact with progres-sive change in the dopant location. However, the decrease in dot–impurity overlap associated with the above dopant shift playshere the decisive role such that the steadfast decrease in /RexSbecomes conspicuous. The faint maxima simply hints about ameager win of forces that favor excitation only at some inter-mediate dopant location. Thus, in keeping with the works citedabove, present investigation also divulges that a change in dopantlocation could bring about new hallmarks of excitation pattern.

4. Conclusions

The excitation profile of repulsive impurity doped quantumdots triggered by oscillatory external field and concomitantoscillation of impurity spread reveals noteworthy features. Theratio of two oscillations (Z) has been found to play governing rolein influencing the excitation rate. Coupled to this, the dopantcoordinate also modulates the excitation pattern to a consider-able extent. We have found a kind of stabilization in the excita-tion rate when Z exceeds some threshold value at all dopantlocations. However, before that threshold value the excitationrates exhibit different trends depending on the dopant coordi-nate. A critical analysis of Z domain when the excitation rateattains the steady behavior unveils the exclusive role played bythe dopant coordinate towards excitation. The analysis evincedminimization in the excitation rate with variation of Z in someparticular dopant coordinate. Whereas the minimization occursdue to a change in the relative preponderance of various factorsthat assist or resist excitation, the observed stabilization can beexplained by arguing with a kind of compromise between theaforesaid factors. The said change in the relative dominance of therelevant factors as well as the compromise between them in turnstems from interplay between two oscillations. The results arethus quite interesting and expected to convey important insightsin various applications of quantum dot nanomaterials.

Acknowledgments

The authors S.P. and M.G. thank D.S.T.-F.I.S.T. (Govt. of India)and U.G.C.-S.A.P (Govt. of India) for partial financial support.Thanks are also due to Mr. Nirmal Kr Datta for his cooperation.

S. Pal, M. Ghosh / Journal of Luminescence 138 (2013) 48–5252

References

[1] P.M. Koenraad, M.E. Flatte, Nat. Mater. 10 (2011) 91.[2] R.N. Bhargava, J. Lumin. 72–74 (1997) 46.[3] C.P. Poole Jr., F.J. Owens, Introduction to Nanotechnology, Wiley, New York,

2003.[4] M.J. Kelly, Low-dimensional Semiconductors, Oxford University Press, Oxford,

1995.[5] W. Xie, Physica B 403 (2008) 4319.[6] W. Xie, Physica B 405 (2010) 3436.[7] W. Xie, Phys. Lett. A 372 (2008) 5498.[8] W. Xie, Superlattices Microstructures 48 (2010) 239.[9] A.J. Peter, Phys. Lett. A 355 (2006) 59.

[10] K.M. Kumar, A.J. Peter, C.W. Lee, Superlattices Microstructures 51 (2012) 184.[11] R. Khordad, Eur. Phys. J. B 78 (2010) 399.[12] S. Baskoutas, E. Paspalakis, A.F. Terzis, J. Phys. 19 (2007) 395024. (9 pp.).[13] I. Karabulut, S. Baskoutas, J. Appl. Phys. 103 (2008) 073512. (5 pp.).[14] F. Ungan, J. Lumin. 131 (2011) 2237.[15] F. Ungan, U. Yesilgul, E. Kasapoglu, H. Sari, I. Sokmen, J. Lumin. 132 (2012)

1627.[16] J.L. Movilla, J. Planelles, Phys. Rev. B 71 (2005) 075319. (7 pp.).[17] B. Gulveren, U. Atav, M. Sahin, M. Tomak, Physica E 30 (2005) 143.[18] B. Gulveren, U. Atav, M. Tomak, Physica E 28 (2005) 482.[19] E. Rasanen, J. Konemann, R.J. Puska, M.J. Haug, R.M. Nieminen, Phys. Rev. B 70

(2004) 115308. (6 pp.).[20] M. Aichinger, S.A. Chin, E. Krotscheck, E. Rasanen, Phys. Rev. B 73 (2006)

195310. (8 pp.).[21] M. Sahin, M. Tomak, Physica E 28 (2005) 247.[22] F.J. Betancur, J. Sierra-Ortega, R.A. Escorcia, J.D. Gonzalez, I.D. Mikhailov,

Physica E 23 (2004) 102.[23] F.J. Betancur, I.D. Mikhailov, L.E. Oliveira, J. Phys. D 31 (1998) 3391.[24] W. Xie, Physica B 358 (2005) 109.[25] W. Xie, Physica B 403 (2008) 2828.[26] E. Kasapoglu, H. Sari, I. Sokmen, Physica B 392 (2007) 213.[27] S.V. Nistor, M. Stefan, L.C. Nistor, E. Goovaerts, G. VanTendeloo, Phys. Rev. B

81 (2010) 035336. (6 pp.).

[28] S.V. Nistor, L.C. Nistor, M. Stefan, C.D. Mateescua, R. Birjega, N. Solovieva,M. Nikl, Superlattices Microstructures 46 (2009) 306.

[29] E. Paspalakis, C. Simserides, S. Baskoutas, A.F. Terzis, Physica E 40 (2008)1301.

[30] A. Ozmen, Y. Yakar, B. C- akir, U. Atav, Opt. Commun. 282 (2009) 3999.[31] C.A. Duque, N. Porras-Montenegro, Z. Barticevic, M. Pacheco, L.E. Oliveira,

Microelectron. J. 36 (2005) 231.[32] C.A. Duque, N. Porras-Montenegro, Z. Barticevic, M. Pacheco, L.E. Oliveira, J.

Phys. 18 (2006) 1877.[33] E. Paspalakis, A.F. Terzis, in Proceedings of the 5th WSEAS International

Conference on Microelectronics, Nanoelectronics, Optoelectronics, Prague,Czech Republic, March 12–14, 2006, pp. 44–49.

[34] A. Fountoulakis, A.F. Terzis, E. Paspalakis, J. Appl. Phys. 106 (2009) 074305. (8pp.).

[35] S. Baskoutas, A.F. Terzis, Physica E 40 (2008) 1367.[36] W. Xie, Q. Xie, Physica B 404 (2009) 1625.[37] N.K. Datta, D. Konar, M. Ghosh, Microelectron. Eng. 88 (2011) 3306.[38] N.K. Datta, S. Pal, M. Ghosh, Chem. Phys. 400 (2012) 44.[39] N.K. Datta, S. Pal, M. Ghosh, J. Appl. Phys. 112 (2012) 014324. (8 pp.).[40] L. Jacak, P. Hawrylak, A. Wojos, Quantum Dots, Springer-Verlag, Berlin, 1998.[41] T. Chakraborty, Quantum Dots—A Survey of the Properties of Artificial

Atoms, Elsevier, Amsterdam, 1999.[42] S. Baskoutas, A.F. Terzis, E. Voutsinas, J. Comput. Theor. Nanosci. 1 (2004)

317.[43] V. Halonen, P. Hyvonen, P. Pietilainen, T. Chakraborty, Phys. Rev. B 53 (1996)

6971.[44] V. Halonen, P. Pietilinen, T. Chakraborty, Europhys. Lett. 33 (1996) 337.[45] J. Adamowski, A. Kwasniowski, B. Szafran, J. Phys. 17 (2005) 4489.[46] S. Bednarek, B. Szafran, K. Lis, J. Adamowski, Phys. Rev. B 68 (2003) 155333.

(9 pp.).[47] B. Szafran, S. Bednarek, J. Adamowski, Phys. Rev. B 64 (2001) 125301. (10

pp.).[48] A. Gharaati, R. Khordad, Superlattices Microstructures 48 (2010) 276.[49] W. Xie, Opt. Commun. 284 (2011) 5730.[50] I. Karabulut, S. Baskoutas, J. Comput. Theor. Nanosci. 6 (2009) 153.[51] S.S. Gomez, R.H. Romero, Physica E 42 (2010) 1563.