Accepted Manuscript

Additive Gaussian white noise modulated excitation kinetics of impurity doped

quantum dots: Role of confinement sources

Jayanta Ganguly, Suvajit Pal, Manas Ghosh

PII: S0749-6036(13)00284-X

DOI: http://dx.doi.org/10.1016/j.spmi.2013.09.003

Reference: YSPMI 2995

To appear in: Superlattices and Microstructures

Received Date: 28 July 2013

Revised Date: 28 August 2013

Accepted Date: 4 September 2013

Please cite this article as: J. Ganguly, S. Pal, M. Ghosh, Additive Gaussian white noise modulated excitation kinetics

of impurity doped quantum dots: Role of confinement sources, Superlattices and Microstructures (2013), doi: http://

dx.doi.org/10.1016/j.spmi.2013.09.003

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Additive Gaussian white noise modulated excitation kinetics of

impurity doped quantum dots: Role of confinement sources

Jayanta Gangulya, Suvajit Palb and Manas Ghoshc ∗†‡

a Department of Chemistry, Brahmankhanda Basapara High School,

Basapara, Birbhum 731215, West Bengal, India.

b Department of Chemistry, Hetampur Raj High School,

Hetampur, Birbhum 731124, West Bengal, India.

c Department of Chemistry, Physical Chemistry Section,

Visva Bharati University, Santiniketan,

Birbhum 731 235, West Bengal, India.

Abstract

We investigate the excitation kinetics of a repulsive impurity doped quantum dot initiated by the

application of additive Gaussian white noise. The noise and the dot confinement sources of electric

and magnetic origin have been found to fabricate the said kinetics in a delicate way. In addition

to this the dopant location also plays some prominent role. The present study sheds light on how

the individual or combined variation of different confinement sources could design the excitation

kinetics in presence of noise. The investigation reveals emergence of maximization and saturation

in the excitation kinetics as a result of complex interplay between various parameters that affect

the kinetics. The phase space plots are often invoked and they lend credence to the findings. The

present investigation is believed to provide some useful perceptions of the functioning of mesoscopic

systems where noise plays some profound role.

Keywords: quantum dot, additive Gaussian white noise, confinement potential, dopant location, excitation

rate

∗ e-mail address: pcmg77@rediffmail.com† Phone : (+91)(03463)261526, (03463)262751-6 (Ext. 467)‡ Fax : +91 3463 262672

1

1. INTRODUCTION

Last couple of decades have witnessed a great enthusiasm in theoretical and experi-

mental researches on impurity states of low-dimensional heterostructures [1]. Among these

heterostructures, Quantum dots (QD) have been found to be one of the most prominent

mesoscopic systems. The QD gains importance since the journey towards miniaturization

of semiconductor devices culminates with them. Furthermore, the properties of doped QD’s

have made them suitable candidates for scientific study and technological applications. With

QD, we envisage subtle interplay between new confinement sources and impurity potentials

which has paved the way for new areas of research in this field [2]. Under the confinement,

the dopant location monitors the electronic and optical properties of the system [3]. This

resulted in an abundance of literature comprising of theoretical studies on impurity states

[4–16] in general, and also on their opto-electronic properties, in particular, for a wide range

of semiconductor devices [3, 17–31]. In addition to these there are also some experimental

works which encompass the mechanism and control of dopant incorporation [32–35].The

highly practised research trend not only explores new physics but also indicates profound

technological impact simultaneously. The research trend, as one of its major components,

includes studies on carrier dynamics in nanodevices [36, 37] which largely consists of inter-

nal transitions between impurity induced states in QD [38, 39]. The dynamical aspects also

initiate researches on excitation kinetics of electrons strongly confined in QD’s. Looking

at the possibility of potential usage in opto-electronic devices and as lasers, a scrupulous

analysis on excitation kinetics deems real importance. From the perspective of technological

applications such excitation further involve optical encoding, multiplexing, photovoltaic and

light emitting devices. The phenomenon also plays some important role in the population

transfer among the exciton states in QD [37, 40]. Of late, we have performed some inspec-

tions which deal with excitation in doped QD’s that has been triggered by some external

time-varying field [41, 42].

Noise in mesoscopic systems influences the functioning of the device to a great extent.

This is because of the key role played by the noise as the size of the electronic system reaches

the nanometer scale [43]. In practice, noise appears as the main hurdle in the development

of semiconductor heterostructure devices for various applications and often it also restricts

the device performance [44]. Hastas et al. have also showed the importance of noise in self-

2

assembled InAs QDs embedded in GaAs [45]. The noise can result externally, or it may be

intrinsic. Intrinsic noises generally result as a result of modifications in impurity structures

[46]. However,low frequency noise is one of the prime requirements in QD heterostructures

in view of its potential use in a wide range of optoelectronic devices. These applications

include industrial manufacturing, medicine, remote sensing, space communications, and

military uses [47]. Such noise measurements are also becoming useful as a benign method

for the determination of structural disorders entered during production or operation of the

devices [48, 49]. Since QD suffers from a lot of stress during its growth, lattice defects

are produced and they diffuse inside the QD structure. The said low-frequency noise also

provides a diagnostic tool for defect related properties of materials and structures [50].

Asriyan et al. have rigorously studied the role of low-frequency noise in non-homogeneously

doped semiconductors [48].

In the present manuscript we have investigated the role of Gaussian white noise, applied

to the system additively, on the excitation kinetics of doped QD. In this connection, the

coupling of noise strength with confinement potential and magnetic field is also analyzed.

The confinement potential in quantum dots plays important role in transport processes and

the form of the confinement potential can be experimentally modulated [51]. Magnetic field

has important implications in the research of doped quantum dots [7, 16, 38, 39, 52]. A

variation in magnetic field can help us examine the transitions between the bound states

of the dot. Recently, it has also been shown that some typical orientation of magnetic

field can tune the binding energy of surface impurities in QD’s [53]. Moreover, nowa-

days the importance of dopant location in fabricating dot properties are well recognized

[2, 8, 20, 24, 26, 41, 42]. The distance dependency of dopant unfolds promising oppor-

tunities to engineer QD electron dynamics in doped heterostructures. Particularly, there

are some important works on off-center impurities invoking an accurate numerical method

(PMM, potential morphing method) [24, 26]. We therefore, have determined the time-

average excitation rate (〈Rex〉) as a function of dot confinement potential (ω0) and cyclotron

frequency (ωc, a measure of strength of magnetic field) in the presence of additive noise. To

make the study more self-contained we have also explored the combined role of ω0 and ωc

(also a kind of confinement) on the excitation kinetics in terms of their ratio (ωr = ω0/ωc).

During the study, the delicate role played by the dopant location (r0) has also been critically

analyzed to address the problem with finer details.

3

2. METHOD

The model considers an electron subject to a harmonic confinement potential V (x, y) and

a perpendicular magnetic field B. The confinement potential assumes the form V (x, y) =

12m∗ω2

0(x2 + y2), where ω0 is the harmonic confinement frequency, ωc = eB

m∗cbeing the

cyclotron frequency (a measure of magnetic confinement offered by B). In the present work

a magnetic field of miliTesla (mT) order has been employed. m∗ is the effective electronic

mass within the lattice of the material to be used. We have taken m∗ = 0.5m0 and set

~ = e = m0 = a0 = 1. This value of m∗ closely resembles Ge quantum dots (m∗ = 0.55

a.u.). We have used Landau gauge [A = (By, 0, 0)] where A stands for the vector potential.

The Hamiltonian in our problem reads

H ′0 = − ~2

2m∗ (∂2

∂x2+

∂2

∂y2) +

1

2m∗ω2

0x2 +

1

2m∗(ω2

0 + ω2c )y

2 − i~ωcy∂

∂x. (1)

Define Ω2 = ω20 + ω2

c as the effective frequency in the y-direction. The model Hamiltonian

[cf. eqn(1)] sensibly represents a 2-d quantum dot with a single carrier electron [54, 55].

The form of the confinement potential conforms to kind of lateral electrostatic confinement

(parabolic) of the electrons in the x− y plane [4, 5, 21, 28, 30, 56].

In the present problem we have considered that the QD is doped with a repulsive Gaussian

impurity [57, 58]. Introducing the impurity potential to the Hamiltonian [cf. eqn(1)] it

transforms to:

H0(x, y, ωc, ω0) = H ′0(x, y, ωc, ω0) + Vimp(x0, y0), (2)

where Vimp(x0, y0) = Vimp(0) = V0 e−γ0[(x−x0)2+(y−y0)2] with γ0 > 0 and V0 > 0 for repulsive

impurity, and (x0, y0) denotes the coordinate of the impurity center. V0 is a measure of the

strength of impurity potential whereas γ−10 determines the spatial stretch of the impurity

potential. The presence of repulsive scatterer simulates dopant with excess electrons. The

use of such Gaussian impurity potential is quite well-known [59–61]. Gharaati et al. [62]

introduced a new confinement potential for the spherical QD’s called Modified Gaussian

Potential, MGP and showed that this potential can predict the spectral energy and wave

functions of a spherical quantum dot. The time-independent Schrodinger equation has been

solved using variational method expressing the trial wave function ψ(x, y) as a superposition

of the product of harmonic oscillator eigenfunctions [41, 42]. In the linear variational calcu-

4

lation, an appreciably large number of basis functions have been exploited after making the

required convergence test.

With the application of external additive noise the time-dependent Hamiltonian reads

H(t) = H0 + V (t). (3)

The noise consists of random term (ξ(t)) quite often assumed to be following a Gaussian

distribution and characterized by the equations:

〈ξ(t)〉 = 0, (4)

the zero mean value condition, and

〈ξ(t)ξ(t′)〉 = 2ζδ(t− t′), (5)

the two-time correlation condition with a negligible correlation time, ζ being the noise

strength. The highly fluctuating term ξ(t) is called white noise because of a flat spectrum in

frequency space, like that of white light. The noises strength ζ becomes simply a measure

of intensity of fluctuation. We have invoked Box-Muller algorithm to generate ξ(t). An

additive noise term is a random term that does not depend on coordinates of system so that

V (t) = ξ(t) [63].

The time-dependent Schrodinger equation (TDSE) containing the evolving wave func-

tion has now been solved numerically by 6-th order Runge-Kutta-Fehlberg method. In

the numerical solution appropriate initial conditions have been used and numerical stabil-

ity has been checked. We define the quantity Pk(t) = |ak(t)|2 to indicate the population

of kth state of H0 at time t where ak(t) is the time-dependent superposition coefficient

for the kth eigenstate of the unperturbed Hamiltonian. We observe a continuous variation

in the ground state population [P0(t)] during the time evolution. Naturally the quantity

Q(t) = 1−P0(t) serves as a measure of excitation. In consequence, the quantity Rex(t) = dQdt

serves as the time-dependent rate of excitation from which the time-average rate of excita-

tion [〈Rex〉 = 1T

∫ T

0Rex(t)dt] can be calculated where T is the total time of dynamic evolution

[41, 42].

5

3. RESULTS AND DISCUSSION

3.1. Role of confinement potential (ω0):

We embark on the problem with the plot of 〈Rex〉 as a function of confinement potential

(ω0) at three different dopant locations viz. on-center (r0 = 28.28) a.u., near off-center

(r0 = 28.28 a.u.), and far off-center (r0 = 70.71 a.u.) for fixed value of noise strength

(ζ = 1.0 × 10−6 a.u.) [fig. 1]. The overall pattern of kinetics has been found to be more or

less similar at all dopant locations. The plot reveals that at low value of (ω0) the rate profile

is quite haphazard containing lots of ups and downs. This indicates occurrence of excitation

and deexcitation in quick successions when confinement potential is small. However, the

kinetics begins to be increasingly regular with increase in (ω0) and at large (ω0) the rate

saturates. The saturation zone commences at ω0 = 2.21 × 10−5 a.u., ω0 = 3.38 × 10−5 a.u.,

and ω0 = 4.02 × 10−5 a.u. for on, near off, and far off-center dopants, respectively. The

pattern of variation of 〈Rex〉 quite clearly reflects the conflict between the noise and the

confinement potential in shaping the excitation kinetics. The desultory nature of kinetics

indicates dominance of noise term over the confinement when the latter is small. As ω0 begins

to take intermediate values we envisage a drop in the excitation rate at all dopant locations

as such increase in confinement potential arrests excitation overcoming the influence of noise.

It needs to be realized that the role of ω0 on excitation is no longer straightforward, rather it

is quite subtle. Although, on the basis of its inherent character the confinement potential of

the dot attempts to resist excitation, but, an increase in ω0 in turn causes an enhancement

in the extent of dot-impurity repulsive interaction favoring excitation. The subtlety is nicely

exhibited through the emergence of saturation in the rate profile as ω0 is increased further.

If it would have been a streamline way by which ω0 influences excitation, the saturation

zone would be absent resulting in a persistent fall of excitation rate. The saturation in

kinetics announces kind of compromise between several factors that influence excitation at

large ω0. In the present context the role of dopant location is displayed by a varied rate of

excitation. Fig. 1 depicts a prominently increasing excitation rate as the dopant is shifted

from on-center to more and more off-center locations. Such a shift reduces the strength of

effective dot confinement on the dopant and excitation is favored.

We now take a close look at the phase space plots to understand the interplay between the

6

dopant location, confinement strength and the noise strength in modulating the excitation

kinetics. Figs. 2(a-c) exhibit the 〈px〉 vs 〈x〉 plots for on-center dopants with ω0 equal

to 1.0 × 10−6 a.u., 2.5 × 10−4 a.u., and 1.0 × 10−3 a.u., respectively. The plots are quite

composed and they also display that with increase in the strength of confinement potential

the spreading in x-direction becomes increasingly quenched. The corresponding 〈py〉 vs

〈y〉 plots are quite erratic (figures not shown). The on-center location is characterized by

strong confinement because of proximity of the dopant with the dot confinement center and

effectively passivates the influence of noise. The phase space plots indicate steady transfer of

energy from x to y-direction through the coupling term [cf. eqn(1)] taking advantage of this

weakened influence of noise. An increase in value of ω0 offers more confinement and further

shrinks the spreading in x-direction. A noticeable change in the nature of the phase plots

is envisaged as soon as the dopant is shifted to near off-center location (r0 = 28.28 a.u.).

At this position the dopant is quite away from the confinement center and the weakened

confinement allows noise come into play. The noise, by virtue of its random nature and the

weakened confinement effect causes almost equal sharing of energy in x and y-directions.

Thus, the phase space plots are equivalently composed in these two directions. Figs. 3a

and 3b represent the appropriate phase space plots for x and y-directions, respectively, for

ω0 = 1.0× 10−3 a.u. An alteration in the value of ω0 does not bring about any new feature

in the phase space plots but to alter the spatial extensions in both directions. We, thus,

refrain from showing those figures for the brevity of manuscript. For a dopant situated

at far off-center location (r0 = 70.71 a.u.), the confinement is further weakened and the

influence of noise becomes more conspicuous. The noise, just like near off-center location,

here also causes nearly equal sharing of energy in x and y-directions resulting in similar

phase plots in the two directions. However, because of highly depleted confinement strength

and consequent dominance of noise the phase space plots appear much less composed in

comparison with their near off-center counterparts with severely enhanced spatial extensions.

Fig. 4 delineates the 〈px〉 vs 〈x〉 plot for a far off-center dopant with ω0 = 4.5 × 10−5 a.u.

The phase space plot for the y-direction is almost identical and therefore not shown. Here

also, as usual, a change in value of ω0 causes consequent change in the spatial stretch of the

system.

7

3.2. Role of cyclotron frequency (ωc):

Fig. 5 depicts 〈Rex〉 as a function of (ωc) at three different dopant locations viz. on-center

(r0 = 28.28 a.u.), near off-center (r0 = 28.28 a.u.), and far off-center (r0 = 70.71 a.u.) for

fixed value of noise strength (ζ = 1.0 × 10−6 a.u.). Similar to the previous plot [fig. 1],

here also we envisage a catch-as-catch-can appearance of excitation rate when the magnetic

confinement is extremely small, although its severity is much diminished in comparison with

when ω0 has been varied. A small increase in cyclotron frequency regularizes the kinetics

resulting in a noticeable drop in the excitation rate. With further increase in ωc the rate

terminates into saturation. The behavior thus outlined, seems to be more or less valid

at all dopant locations. The irregularity exhibited by said kinetics at extremely low ωc

can be understood on the basis of low magnetic confinement and consequent surge in the

influence of noise. A close look at eqn(1) shows that whereas ω0 offers confinement in both

x and y-directions, the confinement imposed by ωc is solely restricted in y-direction. The

restriction just mentioned makes the ωc confinement inherently quite strong so that the

noise term becomes unable to produce irregularity in the excitation kinetics to the same

extent as in the case of ω0 variation even when ωc is quite small. When ωc is increased to

moderate values, the influence of noise is further drained out leading to significant drop in

the excitation rate. The steepness of the fall in 〈Rex〉 has been found to depend upon dopant

location. Whereas the steepness is more or less same at on and near off-center locations,

the rate falls somewhat sluggishly for a far off-center dopant. It seems that, since for a

far off-center dopant the magnetic confinement is inherently feeble, the influence of noise

remains more or less significant over the entire range of ωc. The said influence slows down

the drop in 〈Rex〉 at far off-center locations. The role of dopant location is further manifested

by a monotonic increase in the value of 〈Rex〉 associated with a passage from on to more

and more off-center locations. This monotonic increase in 〈Rex〉, as expected, originates

under the sway of increased effective noise strength that ensues the above passage. The

saturation in 〈Rex〉 at high ωc indicates some sort of negotiation between diverse factors

that affect excitation. However, the dopant location again marks its signature as the onset

of saturation gets progressively delayed with the shift of dopant away from dot confinement

center. Although the saturation zone begins almost at ωc ∼ 4.65 × 10−3 a.u. for on and

near off-center dopants, for far off-center dopant it is pushed back to ωc ∼ 5.8 × 10−3 a.u.

8

Since the far off-center location is inherently under the influence of a feeble confinement,

the arrival of saturation zone requires higher ωc value whence a compromise between several

factors that affect excitation can take place.

Figs. 6a and 6b depict the phase space plots in x-direction for an on-center dopant with

ωc = 7.5 × 10−4 a.u. and ωc = 1.0 × 10−3 a.u., respectively. Figs. 6c and 6d represent the

similar plots for y-direction. The plots, quite expectedly, reveal a decline in the spatial spread

in both directions with increase in ωc. Moreover, we also notice a drastic reduction of spatial

spread in y-direction in comparison with x-direction under a given magnetic confinement

(compare fig. 6a with 6c and 6b with 6d). The reason behind such immense depletion of

spatial extension in y-direction is due to the fact that the impact of magnetic confinement

operates almost solely in this direction [cf. eqn(1)]. For off-center dopants the phase space

plots appear quite scattered in both directions over a range of ωc values indicating the

dominance of noise factor in controlling the spatial dissemination of system (figures not

shown).

3.3. Role of confinement ratio (ωr):

In the concluding section we study the role of confinement ratio (ωr = ω0

ωc) on excita-

tion kinetics. In effect, such an investigation reveals the role of relative strengths of two

confinement sources on excitation. The plot of 〈Rex〉 vs ωr displays maximization in the

excitation rate at previously mentioned on-center, near off-center, and far off-center dopant

locations [fig. 7]. The maximization occurs at ωr ∼ 5.0 for on and near off-center dopants

and at ωr ∼ 6.0 for far off-center dopants. The emergence of maximization indicates that

these two confinements do not always act in compliance with one another. In practice, the

two confinement sources are intrinsically different with respect to the directions along which

they are operative. Consequently, their intensities are also becoming expectedly dissimilar.

In the low ωr domain, the y-direction is under strong confinement whereas the confinement

along x-direction is relatively weak. Thus, the noise factor chiefly operates in x mode. A

gradual increase in ωr in this domain causes simultaneous enhancement and decline in the

influences of ω0 and ωc. Since the confinement imposed by ω0 gets distributed along both the

directions, increase of ωr in effect increases the confinement along x-direction whereas the

confinement in the y mode becomes less intense. In the low ωr regime the noise term seems

9

to supersede the overall confinement imposed on the system accompanying the variation of

ωr. This leads to enhancement of 〈Rex〉 until maximization. Beyond maximization, further

increase in ωr enhances the influence of ω0 to a large extent. Thus, along with x mode, y

mode also becomes under stringent confinement overcoming the shortfall because of weak-

ened ωc in this domain. The overall enhancement in the confinement suppresses the noise

parameter resulting in prominent fall of excitation rate. Further increase in ωr invites kind

of balance between the factors that affect excitation and saturation sets in. As explained

earlier, the dopant location reflects its influence through delayed emergence of maxima and

saturation in the kinetics as we move from on to more and more off-center regions.

4. CONCLUSIONS

The excitation kinetics of impurity doped quantum dots insisted by additive Gaussian

white noise reveals some intriguing aspects. The coupling between the noise strength and

the electric and magnetic confinement potentials of dot delicately modulate the kinetics.

The dopant location has been found to affect the said interplay in a subtle way. Whereas

low values of confinement sources lead to erratic kinetics because of noise factor, an increase

in either of the confinement regularizes the kinetics. Such increase suppresses the noise

factor and causes a drop in the excitation rate. Both the confinement sources, varied either

individually or together, result in saturation of kinetics when their values are quite large

because of a balance between diverse parameters that influence excitation. A continuous

variation of confinement ratio leads to maximization in the kinetics owing to a changeover

of relative dominance of two confinement sources mingled with the noise term. The results

are thus quite interesting and expected to convey important discernment in related field of

research.

It needs to be mentioned that the present work and the work published in reference [64]

are widely different and they are in no way close. Both the works focus on the same

phenomenon i.e. the excitation kinetics of doped quantum dot since the phenomenon merits

importance as we have mentioned in the manuscript. And the general methodologies also

exhibit some resemblance. However, the modes of perturbing the doped dot possess hell

and heaven difference in the two works. Consequently, the findings also become perceptibly

different. In the previous paper the authors attempted to investigate the excitation in the

10

doped quantum dot exposed to a pulsed field which is continuous in nature. They have

exploited pulses of different shapes to ascertain if a variation in the shape can modulate the

excitation. The electromagnetic pulsed field has been applied along both x and y directions

to the doped quantum dot. The variation in the pulse shape basically implies a change in

the pattern of energy transfer from the external field to the QD. Alongside pulse shape,

the number of pulses fed into the system from the external field (np) as well as the dopant

location (r0) played some remarkable role so far as excitation is concerned.

The present work stands distinct in itself and bears some innovation. In the manuscript

we amply described the importance of deciphering the excitation kinetics induced by noise.

Noise governs the functioning of mesoscopic systems to a great extent and its interplay with

the impurity potential has been deeply analyzed. In complete contrast to previous pulsed

field [64], the white noise is a random perturbation with special features of zero mean and

negligible correlation time. The present work also critically explores the roles played by the

confinement potential (ω0), the cyclotron frequency (ωc), their ratio (ωr), dopant location

(r0) and becomes a self-contained independent investigation. The observations have also

been amply supported by phase space contours as and when required.

5. ACKNOWLEDGEMENTS

The authors J. G., 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.

11

References

[1] P. M. Koenraad, M. E. Flatte, Single dopants in semiconductors, Nature Materials 10 (2011)

91-100.

[2] J. L. Movilla, J. Planelles, Off-centering of hydrogenic impurities in quantum dots, Phys. Rev.

B 71 (2005) 075319-1-075319-7.

[3] M. J. Kelly, Low-dimensional Semiconductors, Oxford University Press, Oxford, 1995.

[4] S. Akgul, M. Sahin, K. Koksal, A detailed investigation of the electronic properties of a multi-

layer spherical quantum dot with a parabolic confinement, J. Lumin. 132 (2012) 1705-1713.

[5] B. Gulveren, U. Atav, M. Sahin, M. Tomak, A parabolic quantum dot with N electrons and

an impurity, Physica E 30 (2005) 143-149.

[6] E. Rasanen, J. Konemann, R. J. Puska, M. J. Haug, R. M. Nieminen, Impurity effects in

quantum dots: Toward quantitative modeling, Phys. Rev. B 70 (2004) 115308-1-115308-6.

[7] M. Aichinger, S. A. Chin, E. Krotscheck, E. Rasanen, Effects of geometry and impurities on

quantum rings in magnetic fields, Phys. Rev. B 73 (2006) 195310-1-195310-8.

[8] W. Xie, Binding energy of an off-center hydrogenic donor in a spherical Gaussian quantum

dot, Physica B 403 (2008) 2828-2831.

[9] F.J. Betancur, J. Sierra-Ortega, R.A. Escorcia, J. D. Gonzalez, I.D. Mikhailov, Density of

impurity states in doped spherical quantum dots, Physica E 23 (2004) 102-107.

[10] U. Yesilgul, S. Sakiroglu, E. Kasapoglu, H. Sari, I. Sokmen, Hydrogenic impurities in quantum

dots under intense high frequency laser field, Physica B 406 (2011) 1441-1444.

[11] H. Tas, M. Sahin, The electronic properties of core/shell/well/shell spherical quantum dot

with and without a hydrogenic impurity, J. Appl. Phys. 111 (2012) 083702-1-083702-8.

[12] Shu-Shen Li, Jian-Bai Xia, Binding energy of a hydrogenic donor impurity in a rectangular

parallelepiped-shaped quantum dot: Quantum confinement and Stark effects, J. Appl. Phys.

101 (2007) 093716-1-093716-6.

[13] E. Sadeghi, Impurity binding energy of excited states in spherical quantum dot, Physica E 41

(2009) 1319-1322.

[14] S. Elagoz, R. Amca, S. Kutlu, I. Sokmen, Shallow impurity binding energy in lateral parabolic

12

confinement under an external magnetic field, Superlattices and Microstructures 44 (2008)

802-808.

[15] C. Xia, Z. Zeng, S. Wei, Hydrogenic impurity states in zinc−blende InGaN/GaN asymmetric

coupled quantum dots, Superlattices and Microstructures 47 (2010) 624-630.

[16] E. Kasapoglu, H. Sari, I. Sokmen, Density of impurity states of hydrogenic impurities in an

inverse parabolic quantum well under the magnetic field, Physica B 392 (2007) 213-216.

[17] W. Xie, Linear and nonlinear optical properties of a hydrogenic donor in spherical quantum

dots, Physica B 403 (2008) 4319-4322.

[18] W. Xie, Impurity effects on optical property of a spherical quantum dot in the presence of an

electric field, Physica B 405 (2010) 3436-3440.

[19] W. Xie, Nonlinear optical properties of a hydrogenic donor quantum dot, Phys. Lett. A 372

(2008) 5498-5500.

[20] W. Xie, Optical properties of an off-center hydrogenic impurity in a spherical quantum dot

with Gaussian potential, Superlattices and Microstructures 48 (2010) 239-247.

[21] A. J. Peter, Polarizabilities of shallow donors in spherical quantum dots with parabolic con-

finement, Phys. Lett. A 355 (2006) 59-62.

[22] K. M. Kumar, A. J. Peter, C. W. Lee, Optical properties of a hydrogenic impurity in a

confined Zn1−xCdxSe/ZnSe spherical quantum dot, Superlattices and Microstructures 51

(2012) 184-193.

[23] R. Khordad, Diamagnetic susceptibility of hydrogenic donor impurity in a V -groove

GaAs/Ga1−xAlxAs quantum wire, Eur. Phys. J. B 78 (2010) 399-404.

[24] S. Baskoutas, E. Paspalakis, A. F. Terzis, Electronic structure and nonlinear optical rectifica-

tion in a quantum dot: effects of impurities and external electric field, J. Phys:Cond. Mat. 19

(2007) 395024-1-395024-9.

[25] I. Karabulut, S. Baskoutas, Linear and nonlinear optical absorption coefficients and refractive

index changes in spherical quantum dots: Effects of impurities, electric field, size, and optical

intensity, J. Appl. Phys. 103 (2008) 073512-1-073512-5.

[26] I. Karabulut, S. Baskoutas, Second and Third Harmonic Generation Susceptibilities of Spher-

ical Quantum Dots: Effects of Impurities, Electric Field and Size, J. Comput. Theor. Nanosci.

6 (2009) 153-156.

[27] B. Cakir, Y. Yakar, A. Ozmen, M. Ozgur Sezer, M. Sahin, Linear and nonlinear optical

13

absorption coefficients and binding energy of a spherical quantum dot, Superlattices and

Microstructures 47 (2010) 556-566.

[28] Y. Yakar, B. Cakir, A. Ozmen, Calculation of linear and nonlinear optical absorption co-

efficients of a spherical quantum dot with parabolic potential, Optics Commun. 283 (2010)

1795-1800.

[29] C. A. Duque. M. E. Mora-Ramos, E. Kasapoglu, F. Ungan, U. Yesilgul, S. Sakiroglu, H. Sari,

I. Sokmen, Impurity-related linear and nonlinear optical response in quantum-well wires with

triangular cross section, J. Lumin. 143 (2013) 304-313.

[30] F. Ungan, U. Yesilgul, E. Kasapoglu, H. Sari, I. Sokmen, Effects of applied electromagnetic

field on the linear and nonlinear optical properties in an inverse parabolic quantum well, J.

Lumin. 132 (2012) 1627-1631.

[31] S. Baskoutas, E. Paspalakis, A. F. Terzis, Effects of excitons in nonlinear optical rectification

in semiparabolic quantum dots, Phys. Rev. B 74 (2006) 153306.

[32] S. V. Nistor, M. Stefan, L. C. Nistor, E. Goovaerts, G. VanTendeloo, Incorporation and

localization of substitutional Mn2+ ions in cubic ZnS quantum dots. Phys. Rev. B 81 (2010)

035336 (6 pages).

[33] S. V. Nistor, L. C. Nistor, M. Stefan, C. D. Mateescua, R. Birjega, N. Solovieva, M. Nikl,

Synthesis and characterization of Mn2+ doped ZnS nanocrystals self-assembled in a tight

mesoporous structure, Superlattices and Microstructures 46 (2009) 306-311.

[34] M. Stefan, S. V. Nistor, D. Ghica, C. D. Mateescu, M. Nikl, R. Kucerkova, Substitutional and

surface Mn2+ centers in cubic ZnS : Mn nanocrystals. A correlated EPR and photolumines-

cence study, Phys. Rev. B 83 (2011) 045301 (11 pages).

[35] S. V. Nistor, D. Ghica, L. C. Nistor, M. Stefan, C. D. Mateescu, Local structure at Mn2+ ions

in vacuum annealed small cubic ZnS nanocrystals self-assembled into a mesoporous structure,

J. Nanosci. Nanotechnol. 11 (2011) 9296-9303.

[36] E. Paspalakis, C. Simserides, S. Baskoutas, A. F. Terzis, Electromagnetically induced popu-

lation transfer between two quantum well subbands, Physica E 40 (2008) 1301-1304.

[37] E. Paspalakis, A. Kalini, A. F. Terzis, Local field effects in excitonic population transfer in a

driven quantum dot system, Phys. Rev. B 73 (2006) 073305.

[38] C. A. Duque, N. Porras-Montenegro, Z. Barticevic, M. Pacheco, L. E. Oliveira, Electron-hole

transitions in self-assembled InAs/GaAs quantum dots: Effects of applied magnetic fields

14

and hydrostatic pressure, Microelectronic J. 36 (2005) 231-233.

[39] C. A. Duque, N. Porras-Montenegro, Z. Barticevic, M. Pacheco, L. E. Oliveira, Effects of

applied magnetic fields and hydrostatic pressure on the optical transitions in self-assembled

InAs/GaAs quantum dots, J. Phys.:Condens. Matter 18 (2006) 1877-1884.

[40] E. Paspalakis, A. F. Terzis. Proceedings of the 5th WSEAS International Conference on Micro-

electronics, Nanoelectronics, Optoelectronics, Prague, Czech Republic, March 12-14, (2006)

(pp 44-49).

[41] K. Sarkar, N. K. Datta, M. Ghosh, Dynamics of electron impurity doped quantum dots in the

presence of time-varying fields: influence of impurity location, Physica E 43 (2010) 345-353.

[42] S. Pal, M. Ghosh, Influence of external field and consequent impurity breathing on excitation

profile of doped quantum dots, J. Lumin. 138 (2013) 48-52.

[43] H. K. Zhao, Shot noise in the hybrid systems with a quantum dot coupled to normal and

superconducting leads, Phys. Lett. A 299 (2002) 262-270.

[44] N. A. Hastas, C. A. Dimitriadis, L. Dozsa, E. Gombia, S. Amighetti, P. Frigeri, Low frequency

noise of GaAs Schottky diodes with embedded InAs quantum layer and self-assembled quan-

tum dots, J. Appl. Phys. 93 (2003) 3990-3994.

[45] N. A. Hastas, C. A. Dimitriadis, L. Dozsa, E. Gombia, R. Mosca, Investigation of single

electron traps induced by InAs quantum dots embedded in GaAs layer using the low-frequency

noise technique, J. Appl. Phys. 96 (2004) 5735-5739.

[46] M. Pioro-Ladriere, J. H. Davies, A. R. Long, A. S. Sachrajda, L. Gaudreau, P. Zawadzki, J.

Lapointe, J. Gupta, Z. Wasilewski, S. Studenikin, Origin of switching noise in GaAs/AlxGa1−

xAs lateral gated devices, Phys. Rev. B 72 (2005) 115331-1-115331-8.

[47] P. Yuan, O. Baklenov, H. Nie, A. L. Holmes Jr., B. G. Streetman, J. C. Campbell, High-

speed and low-noise avalanche photodiode operating at 1.06/splmu/m”, IEEE J. Select. Top.

Quantum Electron. 6 (2000) 422-425.

[48] H. V. Asriyan, F. V. Gasparyan, V. M. Aroutiounian, S. V. Melkonyan, P. Soukiassian, Low-

frequency noise in non-homogeneously doped semiconductor, Sensors and Actuators A 113

(2004) 338-343.

[49] Z. Chabola, A. Ibrahim, Noise and scanning by local illumination as reliability estimation for

silicon solar cells, Fluctuation and Noise Lett. 1 (2001) L21-L26.

[50] J. I. Lee, H. D. Nom, W. J. Choi, B. Y. Yu, J. D. Song, S. C. Hong, S. K. Noh, A. Chovet,

15

Low frequency noise in GaAs structures with embedded In(Ga)As quantum dots, Current

Appl. Phys. 6 (2006) 1024-1029.

[51] K. Lis, S. Bednarek, B. Szafran, J. Adamowski, Electrostatic quantum dots with designed

shape of confinement potential, Physica E 17 (2003) 494-497.

[52] M. G. Barseghyan, A. A. Kirakosyan, C. A. Duque, Donor-impurity related binding energy and

photoionization cross-section in quantum dots: electric and magnetic fields and hydrostatic

pressure effects, Eur. Phys. J. B 72 (2009) 521-529.

[53] Z. Zeng, C. S. Garoufalis, S. Baskoutas, A. F. Terzis, Tuning the binding energy of surface

impurities in cylindrical GaAs/AlGaAs quantum dots by a tilted magnetic field, J. Appl.

Phys. 112 (2012) 064326-1-064326-5.

[54] L. Jacak, P. Hawrylak, A. Wojos, Quantum Dots, Springer-Verlag, Berlin, 1998.

[55] T. Chakraborty, Quantum Dots-A Survey of the Properties of Artificial Atoms, Elsevier,

Amsterdam, 1999.

[56] S. Baskoutas, A. F. Terzis, E. Voutsinas, Binding Energy of Donor States in a Quantum Dot

with Parabolic Confinement, J. Comput. Theor. Nanosci. 1 (2004) 317-321.

[57] V. Halonen, P. Hyvonen, P. Pietilainen, T. Chakraborty, Effects of scattering centers on the

energy spectrum of a quantum dot, Phys. Rev. B 53 (1996) 6971-6974.

[58] V. Halonen, P. Pietilinen, T. Chakraborty, Optical-absorption spectra of quantum dots and

rings with a repulsive scattering centre, Europhys. Lett. 33 (1996) 337-382.

[59] J. Adamowski, A. Kwasniowski, B. Szafran, LO-phonon-induced screening of electronelectron

interaction in D− centres and quantum dots, J. Phys:Cond. Mat. 17 (2005) 4489-4500.

[60] S. Bednarek, B. Szafran, K. Lis, J. Adamowski, Modeling of electronic properties of electro-

static quantum dots, Phys. Rev. B 68 (2003) 155333-1-155333-9.

[61] B. Szafran, S. Bednarek, J. Adamowski, Parity symmetry and energy spectrum of excitons in

coupled self-assembled quantum dots, Phys. Rev. B 64 (2001) 125301-1-125301-10.

[62] A. Gharaati, R. Khordad, A new confinement potential in spherical quantum dots: Modified

Gaussian potential, Superlattices and Microstructures 48 (2010) 276-287.

[63] J. Garcıa-Ojalvo, J. M. Sancho, Noise in Spatially Extended Syatems, Springer, New York,

USA, 1999.

[64] S. Pal, M. Ghosh, Influence of pulse shape in modulating excitation kinetics of impurity doped

quantum dots, Superlattices and Microstructures, 55 (2013) 118130.

16

Figure Captions

Fig. 1: Plot of 〈Rex〉 vs ω0 with ζ = 1.0 × 10−6 a.u. at three different dopant locations

(r0): (i) 0.0 a.u., (ii) 28.28 a.u., and (iii) 70.71 a.u.

Fig. 2: Plot of 〈px〉 vs 〈x〉 for on-center dopant with ζ = 1.0 × 10−6 a.u. with three

different values of ω0: (a) 1.0× 10−6 a.u., (b) 2.5× 10−4 a.u., and (c) 1.0× 10−3 a.u.

Fig. 3: Phase space plots for near off-center dopant (r0 = 28.28 a.u.) with ζ = 1.0×10−6

a.u. and ω0 = 1.0× 10−3 a.u.: (a) for x-direction and (b) for y-direction.

Fig. 4: Plot of 〈px〉 vs 〈x〉 for far off-center dopant (r0 = 70.71 a.u.) with ζ = 1.0×10−6

a.u. and ω0 = 4.0× 10−5 a.u.

Fig. 5: Plot of 〈Rex〉 vs ωc with ζ = 1.0 × 10−6 a.u. at three different dopant locations

(r0): (i) 0.0 a.u., (ii) 28.28 a.u., and (iii) 70.71 a.u.

Fig. 6a: Plot of 〈px〉 vs 〈x〉 for on-center dopant with ζ = 1.0 × 10−6 a.u. and ωc =

7.5× 10−4 a.u.

Fig. 6b: Plot of 〈px〉 vs 〈x〉 for on-center dopant with ζ = 1.0 × 10−6 a.u. and ωc =

1.0× 10−3 a.u.

Fig. 6c: Plot of 〈py〉 vs 〈y〉 for on-center dopant with ζ = 1.0 × 10−6 a.u. and ωc =

7.5× 10−4 a.u.

Fig. 6d: Plot of 〈py〉 vs 〈y〉 for on-center dopant with ζ = 1.0 × 10−6 a.u. and ωc =

1.0× 10−3 a.u.

Fig. 7: Plot of 〈Rex〉 vs ωr with ζ = 1.0 × 10−6 a.u. at three different dopant locations

(r0): (i) 0.0 a.u., (ii) 28.28 a.u., and (iii) 70.71 a.u.

17