Exploring optical refractive index change of impurity doped quantum dots driven by white noise

Superlattices and Microstructures 88 (2015) 620e633

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Exploring optical refractive index change of impurity doped quantum dots driven by white noise Surajit Saha a, Suvajit Pal b, Jayanta Ganguly c, Manas Ghosh d, * a

Department of Chemistry, Bishnupur Ramananda College, Bishnupur, Bankura, 722122 West Bengal, India Department of Chemistry, Hetampur Raj High School, Hetampur, Birbhum, 731124 West Bengal, India c Department of Chemistry, Brahmankhanda Basapara High School, Basapara, Birbhum, 731215 West Bengal, India d Department of Chemistry, Physical Chemistry Section, Visva Bharati University, Santiniketan, Birbhum, 731 235 West Bengal, India b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 30 June 2015 Received in revised form 10 October 2015 Accepted 15 October 2015 Available online 26 October 2015

We make an extensive exploration of total refractive index (RI) change of impurity doped quantum dots (QDs) in presence and absence of noise. The noise invoked in the present study is a Gaussian white noise. The quantum dot is doped with repulsive Gaussian impurity. Noise has been incorporated to the system additively and multiplicatively. A perpendicular magnetic field acts as a source of confinement and a static external electric field has been applied. The total RI change profiles have been studied as a function of incident photon energy when several important parameters such as optical intensity, electric field strength, magnetic field strength, confinement energy, dopant location, relaxation time, Al concentration, dopant potential, and noise strength assume different values. Additionally, the role of mode of application of noise (additive/multiplicative) on the total RI change profiles has also been examined minutely. The total RI change profiles mainly comprise of interesting observations such as shift of total RI change peak position and maximization/minimization of peak intensity. However, presence of noise conspicuously alters the features of total RI change profiles through some interesting manifestations. Moreover, the mode of application of noise (additive/multiplicative) also governs the total RI change profiles in diverse as well as often contrasting manners. The observations indicate possibilities of tailoring the linear and nonlinear optical properties of doped QD systems in presence of noise. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Quantum dot Impurity Gaussian white noise Total refractive index change Noise strength

1. Introduction Over the last few decades we have observed a plethora of investigations on nonlinear optical (NLO) properties bearing formal proximity with the intersubband transitions in the low-dimensional semiconductor systems. The said systems include quantum wells (QWLs), quantum wires (QWRs) and quantum dots (QDs). The findings of the research have enriched our understanding of fundamental physics and simultaneously indicated potential hope for promising applications in electronic and optoelectronic devices. These low-dimensional systems reveal more encouraging optical properties than their bulk neighbors. The small energy separations between the subband levels and the large values of electric dipole matrix elements are the main factors behind the prominent manifestation of the said optical effects. The above factors also promote

* Corresponding author. E-mail address: [email protected] (M. Ghosh). http://dx.doi.org/10.1016/j.spmi.2015.10.021 0749-6036/© 2015 Elsevier Ltd. All rights reserved.

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occurrence of resonance conditions. The large value of dipole matrix element favors optical transition between the QD subbands. In consequence, photons, whose energies are of the order of intersubband transition energies, can lead to a huge change in the (complex) dielectric constant of host materials, which, in turn, changes the index of refraction and absorption coefficient [1]. Moreover, the nonlinear parts offer greater contribution to the dielectric constants and other important optical properties [1]. The various optical properties have a wide range of applications in high-speed electro-optical modulators, far IR photodetectors, left-handed materials, semiconductor optical amplifiers etc. QD, the 3-d confined low-dimensional system, displays prominent intersubband optical transitions. Research in this area has been further bolstered by rapid progress in semiconductor growth techniques such as molecular beam epitaxy (MBE) and metal organic chemical vapor deposition (MOCVD). Impurity plays vital role in semiconductor devices as it modulates the electronic properties of quantum nanostructures [2]. Shallow impurity augments the conductivity of semiconductors by several orders of magnitude. Impurity effects in QD also appear to be quite evident in the emerging field of nanaoelectronics, an area that has drawn huge attention as it opens up immense possibilities in the domain of applied physics. The performance of a majority of optical devices involving both bulk and nanoscale semiconductors depends heavily on the presence of impurity. In QDs, presence of impurity strongly affects the basic physics and technological applications thereby necessitating detailed investigations on impurity states. The strong quantum confinement in QD has made their size and shape very much controllable which in effect modulates their electronic structure. In presence of impurities the modulation becomes much more delicate and ultimately modifies the energy spectrum of doped QD system to obtain adjustable optical transitions. A controlled optical transition is an essential ingredient to fabricate optoelectronic devices with tunable emission or transmission properties and ultranarrow spectral linewidths. Moreover, the intimacy between optical transition energy and the confinement strength (or the quantum size) favors finetuning of resonance frequency. In consequence, NLO properties of doped QDs have become a subject of full-fledged experimental and theoretical research [1,3e26]. Electric field is a major external perturbation that elucidates the important aspects of confined impurities [27e38]. The interplay between impurity and the applied field modulates the optical properties and turns out to be useful in view of fundamental physics and device applications. The electric field modifies the electronic structure and thus provides a feasible way of tailoring the energy spectrum to attain desirable optical transitions. It deserves to be mentioned now that the NLO properties of semiconductor QD is governed by the asymmetry of the confining potential. Controllable asymmetry of confinement potential turns out to be conducive for generation of NLO properties which usually vanish in a symmetric quantum structure. Among the NLO properties the second order ones are found in noncentrosymmetric structures whereas the third order nonlinearities become evident in both centrosymmetric and noncentrosymmetric structures. In this context application of electric field bears immense significance owing to its capacity of producing such asymmetry in confinement potential. Hence, both from theoretical and experimental perspectives, the involvement of an applied electric field in the study of physical properties of low-dimensional systems have become a widely followed research topic looking at the development of innovative semiconductor devices. An electric field, applied in the growth direction of heterostructures, causes an energy shift of quantum states and polarization of the spatial disposition of the carriers. These effects are utilized to harness and modulate the intensity output of optoelectronic devices [34]. Additionally, enhanced influence of external electric field has been found for dopants introduced into far off-center locations [35]. The magnetic field is also an important additional perturbation. The applied magnetic field modifies the symmetry of the impurity states and hence the nature of the wave functions. Owing to above modification subtle changes are observed in the binding energy and consequently in other auxiliary properties of doped QD systems [39e41]. Magnetic field can be applied very much under experimental control and thus offers a suitable means of altering the electronic structure. Magnetic field applied perpendicular to the QD plane has been found to be much more effective in tailoring the energy spectrum than a parallel one. This fact has a straightforward impact on the nature of electronic and optical properties of these systems [42e44]. Naturally we find an abundance of notable investigations on NLO properties of QD in presence of a magnetic field [45e51]. Of late, we have made detailed discussions on the importance of noise in governing the performances of QD devices [52e55]. In these works the influence of Gaussian white noise on the polarizability profiles of doped QDs has been rigorously studied. In the present study we make a detailed analysis of the influence of Gaussian white noise on the total change of optical refractive index (RI) (which is a combination of linear and the third-order nonlinear changes of RI) of doped QD relevant to transition between jj0〉 and jj1〉 states. We have considered a 2-d QD (GaAs) containing single carrier electron laterally arrested (parabolic) in the xey plane. In connection with the model described here it needs to be mentioned that QDs are created mainly by imposing a lateral confinement to electrons in a very narrow quantum well. QDs fabricated in this way usually have the shape of a flat disk with lateral dimension considerably exceeding their thickness. The energy of single electron excitation can be considered as strictly 2-d. In most studies a harmonic oscillator potential was used to describe the lateral confinement of electrons. Thus, optical properties of impurity in a 2-d disc-like QD with parabolic confinement potential in an applied electric field is a problem which is very much in vogue. In the present study the QD is doped with a repulsive Gaussian impurity in the presence of a perpendicular magnetic field. And the doped system is subjected to an external static electric field. Gaussian white noise has been incorporated to the system additively and multiplicatively [52e55]. Thus, the full system is under the influence of external probes: crossed axially directed homogeneous magnetic (perpendicular to the quantum disk plane) and in-plane electric fields. In view of a comprehensive investigation the RI profiles are monitored as several pertinent parameters viz. the confinement frequency (u0), electric field strength (F), dopant

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location (r0), magnetic field (B), optical intensity (I), impurity potential (V0), noise strength (z), and the mode of application of noise (additive/multiplicative) are varied over a range. Furthermore, AlxGa1xAs QD has also been invoked to analyze the role played by Al concentration (x) on RI in presence of noise. 2. Method Under effective mass approximation, the Hamiltonian representing a 2-d quantum dot with single carrier electron under lateral parabolic confinement in the xey plane with simultaneous presence of a perpendicular magnetic field (B) is given by

H00 ¼

  1 h e i2 1  iZV þ A þ m u20 x2 þ y2 ; 2m c 2

(1)

where Vðx; yÞ ¼ 12m u20 ðx2 þ y2 Þ is the confinement potential with u0 as the harmonic confinement frequency. m* represents the effective mass of the electron inside the QD material. In Landau gauge [A ¼ (By,0,0),A being the vector potential] the Hamiltonian reduces to

H00 ¼ 

Z2 2m

v2 v2 þ 2 2 vx vy

!

 1 1  v þ m u20 x2 þ m u20 þ u2c y2  iZuc y ; 2 2 vx

(2)

uc ¼ eB/m*c being the cyclotron frequency. U2 ¼ u20 þ u2c can be viewed as the effective confinement frequency in the ydirection. In this context it needs to be mentioned that Landau gauge yields different effective confinement frequencies in x and y-directions. This fact makes analysis of anisotropy effect combined with noise on the optical properties of one-electron parabolic quantum disk quite interesting. Recently, Correa et al. have studied the dimensionality effect on two-electron energy spectrum using a fractional-dimension-based formulation in which the effective dimension of the space has been modified by changing the confinement frequencies [56]. 2

2

The impurity (dopant) inside QD can be represented by a Gaussian potential given by Vimp ¼ V0 eg½ðxx0 Þ þðyy0 Þ . Positive values for g and V0 indicate repulsive impurity. Other impurity parameters include (x0,y0) (the dopant coordinate), V0 (the strength of impurity potential), and g1 (the spatial stretch of impurity potential). g here plays the role similar to that of static dielectric constant (ε) of the medium and thus we consider g ¼ kε, where k is a constant. Recently Khordad and his coworkers used a new type of confinement potential for spherical QD's called Modified Gaussian Potential, MGP [57,58]. The Hamiltonian representing the doped system reads

H0 ¼ H00 þ Vimp :

(3)

When an external static electric field (F) is applied along x and y-directions the Hamiltonian becomes

  H0 ¼ H00 þ Vimp þ eFðx þ yÞ:

(4)

A spatially d-correlated Gaussian white noise [f(x,y)] is now introduced into the system. The noise assumes a Gaussian distribution (produced by BoxeMuller algorithm) having strength z and possesses following characteristics [52e55]:

〈f ðx; yÞ〉 ¼ 0;

(5)

the zero spatial-average condition, and

〈f ðx; yÞf ðx0 ; y0 Þ〉 ¼ 2zd½ðx; yÞ  ðx0 ; y0 Þ;

(6)

the spatial d-correlation condition. Introduction of Gaussian white noise can be done additively as well as multiplicatively [52e55]. With additive white noise the Hamiltonian becomes

  H0 ¼ H00 þ Vimp þ eFðx þ yÞ þ l1 f ðx; yÞ;

(7)

and in case of multiplicative noise H0 reads

  H0 ¼ H00 þ Vimp þ eFðx þ yÞ þ l2 f ðx; yÞðx þ yÞ:

(8)

The parameters l1 and l2 take care of surrounding influences in case of additive and multiplicative noise, respectively. € dinger equation has been solved using variational method with the trial function j(x,y), The time-independent Schro formed by the superposition of the products of harmonic oscillator eigenfunctions [52e55] fn(px) and fm(qy) respectively, as

S. Saha et al. / Superlattices and Microstructures 88 (2015) 620e633

jðx; yÞ ¼

X

623

Cn;m fn ðpxÞfm ðqyÞ;

(9)

n;m

where Cn,m are the variational parameters and p ¼

qffiffiffiffiffiffiffiffiffi m u0 Z

and q ¼

qffiffiffiffiffiffiffiffi m U. In the linear variational calculation the above Z

expansion needs to be truncated at some particular values of n and m. We have used basis functions with n,m ¼ 020 for each of the directions (x,y). The direct product basis spans a space of (20  20) dimension. We have checked that the basis functions span the 2-d space effectively completely, at least with respect to representing the observable under investigation. We have made the convergence test with still greater number of basis functions. For ready comparison, Table 1 shows the energy values of the ground and first excited state for a number of basis functions. The general expressions for the matrix elements of 0 H0 and Vimp in the chosen basis have been derived [52e55]. The matrix element (jk,l〉) due to the electric field reads

jejF〈jk ðx; yÞjx þ yjjl ðx; yÞ〉 ¼ jejF

X X nm n0 m0

 Cnm;k Cn0 m0 ;l 〈fn ðpxÞfm ðqyÞjx þ yjfn0 ðpxÞfm0 ðqyÞ〉

9 2 8rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi0 < n0 þ 1 = 1 n  Cnm;k Cn0 m0 ;l 4 ¼ jejF dn0 þ1;n þ dn0 1;n dm;m0 ; p: 2 2 nm n0 m0 8rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 3 rffiffiffiffiffi0ffi 0 = 1< m þ 1 m þ dm0 þ1;m þ dm0 1;m dn;n0 5: : ; q 2 2 X X

(10)

The matrix elements for the noise terms viz.

 0  0   〈fn ðpxÞfm ðqyÞf ðx; yÞfn ðpxÞfm ðqyÞ〉;

(11)

for additive noise, and

 0  0   〈fn ðpxÞfm ðqyÞf ðx; yÞðx þ yÞfn ðpxÞfm ðqyÞ〉;

(12)

for multiplicative noise, are computed numerically with the help of following relation i.e.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0  ∞ Hk x2 þ y2 Hk x2 þy2 X exp  x2 þ y2 þ x 2 þ y 2 2 02 02 2 2 pffiffiffi  ; d x þy  x þy ¼ p 2k k! k¼0

(13)

where Hk(x) stands for the Hermite polynomial of kth order. In the linear variational calculation, requisite number of basis functions have been used after performing the convergence test. And H0 is diagonalized in the direct product basis of harmonic oscillator eigenfunctions to obtain the energy levels and wave functions. We consider interaction of a polarized monochromatic electromagnetic field of angular frequency n with an ensemble of QDs. The optical RI changes have been determined using standard compact density matrix approach. Assuming optical transition between two states jj0〉 and jj1〉 the linear and the third-order nonlinear changes in RI are given by

 2

  ss Mij  DEij  Zn Dnð1Þ ðnÞ 1 ¼ 2 $



; nr 2nr ε0 DEij  Zn 2 þ ZGij 2

(14)

and

Table 1 The energies of ground and first excited states for different sizes of the basis sets: (all the energies are in meV). Basis space

E0

E1

55 10  10 15  15 20  20 30  30 40  40

122.4513 120.6169 120.60045 111.89599 111.89599 111.89599

184.7533 122.4513 122.45069 122.45069 122.45069 122.45069

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 2  2  

2  ss Mij 

 2

n

Mjj  Mii Dnð3Þ ðnÞ mcI  4 M ¼ $  4 DE  Zn  $ DE  Zn  DEij DEij  Zn  h i

ij ij ij  2

2

2 2 2 nr 4ε0 n3r  DEij þ ZGij DEij  Zn þ ZGij 3

2 o

2  ZGij  ZGij 2DEij  Zn 5: (15)

Finally, the total change in RI becomes

DnðnÞ Dnð1Þ ðnÞ Dnð3Þ ðnÞ ¼ þ : nr nr nr

(16)

In the above equations Gij is the phenomenological relaxation rate, caused by the electron-phonon, electroneelectron and other collision processes. The diagonal matrix element i.e. Gjj gives the relaxation rate of state jj〉 and Gjj ¼ 1/tjj, where tjj is the relaxation time of jj〉-th state. The off-diagonal matrix element i.e. Gij(¼1/tij,isj) gives the relaxation rate of ji〉-th and jj〉-th     states with relaxation time tij. ε0 is the vacuum permittivity. ss is the carrier density, Mij ¼ e〈ji b xþb y jj 〉; ði; j ¼ 0; 1Þ is the matrix elements of the dipole moment, ji(jj) are the eigenstates and DEij ¼ EiEj is the energy difference between these pffiffiffiffi states. nr is the static component of refractive index (RI) in QD (¼ εr , where εr is the static dielectric constant). c is the speed of light in free space, m is the magnetic permeability of the system (¼1/ε0c2), and I is the intensity of the electromagnetic field. 3. Results and discussion The calculations are performed using the following parameters: ε ¼ 12.4, m* ¼ 0.067m0, where m0 is the free electron mass, ε0 ¼ 8.8542  1012 Fm1, t ¼ 0.14 ps, and nr ¼ 3.2. The parameters are suitable for GaAs QDs. Regarding other parameters; if some of them are not varying with reference to a particular observation their values are kept fixed as follows: Zu0 ¼ 8.2 meV, F ¼ 100 KV/cm, B ¼ 1.0 T, z ¼ 1.0  1010, V0 ¼ 272 meV, I ¼ 1.5  1010 W/m2, ss ¼ 5.0  1024 m3, and r0 ¼ 0.0 nm.

Fig. 1. Plots of Dn(n)/nr vs Zn at four different I values: (i) 0.0 W/m2, (ii) 8.6  109 W/m2, (iii) 1.72  1010 W/m2, (iv) 2.58  1010 W/m2: (a) Under noise-free condition, (b) in presence of additive noise, and (c) in presence of multiplicative noise. Plots 1(b) and 1(c) contain two additional intensities viz. (v) 1.0  1011 W/m2 and (vi) 2.5  1011 W/m2.

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A. Role of optical intensity (I): Fig. 1a depicts the variation of total optical RI change i.e. Dn(n)/nr with incident photon energy Zn for four different values of I (0.0 W/m2, 8.6  109 W/m2, 1.72  1010 W/m2, and 2.58  1010 W/m2), in absence of noise. As I increases, total RI change steadily decreases. The reason behind the said decrease lies inside the expressions of linear and nonlinear RI changes [cf. eqn.14 and 15]. With increase in optical intensity, linear RI does not increase much whereas nonlinear RI (intensity-dependent) is highly enhanced (in absolute sense). Consequently, as I increases we visualize a persistent fall in the total RI change (because of high negative contribution from nonlinear RI change term). Thus, nonlinear RI change deserves serious consideration specially at high optical intensity region. Furthermore, a relatively weak optical intensity needs to be employed if attainment of large RI change is desired [36]. It becomes also evident from the figure that as the illumination intensity is increased up to I ¼ 2.58  1010 W/m2, the total RI change displays a new behavior and the profile now consists of maximum and minimum values. It becomes difficult to provide any convincing explanation of appearance of maxima and minima in the profile of total RI change at a typical value of intensity viz. I ¼ 2.58  1010 W/m2. It appears that the frequency of the external field, in conjunction with various dipole matrix elements and energy interval between the concerned eigenstates may be responsible for this. A close look at Eqs. 14e16 suggests that at this typical intensity value the undoubtedly complex combination of external frequency, various dipole matrix elements and the energy interval maximizes the nonlinear RI change contribution at Zn  145:0 meV resulting into minimization of total RI change profile. On the other hand, at the same optical intensity, the said factors minimize the dominance of nonlinear RI change at Zn  170:0 meV giving rise to maximization of total RI change profile. However, optical intensities lower than this typical value refrain from maximizing/minimizing the nonlinear RI change contribution within such a small range of change in the energy of external field (145:0 meV  Zn  170:0 meV) and the profile gets devoid of maximization/minimization. Fig. 1b and c displays the similar profiles in presence of additive and multiplicative noise, respectively. Presence of noise simply reverses the effect of illumination intensity on the profile of total RI change as now the total RI change has been found to increase steadily as I increases. Thus, an increase in optical intensity, in turn, invites more response from noise which eventually reverses the intensity effect. Moreover, the magnitude of total RI change undergoes moderate enhancement and the total RI profiles become bit stretched in presence of noise than in absence of it. It needs to be further noted that the presence of noise eliminates the appearance of maximum and minimum in the total RI change plot at high optical intensity which has been found in the absence of noise. It can be taken into account that greater the optical intensity the higher and deeper will be the maximum and minimum, respectively. However, the noise-induced elimination of the emergence of maximum and minimum in the total RI change profile may raise doubts. We, therefore, have delineated the total RI change profile at two different illumination intensities which are greater than 2.58  1010 W/m2 viz. 1.0  1011 W/m2 and 2.5  1011 W/m2 in presence of both additive and multiplicative noise (Fig. 1b(v/vi) and 1c(v/vi)). However, even with such

Fig. 2. Plots of Dn(n)/nr vs Zn at three different F values: (i) F ¼ 0.0 KV/cm, (ii) F ¼ 50.0 KV/cm, and (iii) F ¼ 100.0 KV/cm. (a) Under noise-free condition, (b) in presence of additive noise, and (c) in presence of multiplicative noise.

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high intensities we do not find any maximum or minimum in the total RI change profile. It can therefore be concluded that presence of noise suppresses the appearance of maximum and minimum in the total RI change profile even at fairly high optical intensities. B. Role of electric field (F): Fig. 2a shows the pattern of variations of Dn(n)/nr with incident photon energy Zn for three different values of F (0.0 KV/cm, 50.0 KV/cm, and 100.0 KV/cm) in absence of noise. It has been found that as the electric field strength increases the total RI change peaks undergo a red-shift [59] but the magnitude of total RI change remains nearly unaltered. Thus, electric field does not substantially affect the peak values of total RI change. It seems that an increase in F diminishes the effective confinement of electron and accordingly lowers the energy eigenvalues. The decrease of E0 being smaller than that of E1 there occurs a net decrease in DE01 gap. However, the said decrease in confinement causes simultaneous enhancement of stretched area of j and therefore the magnitude of dipole moment matrix elements (jM01j). It comes out that the drop in DE01 separation supersedes the influence of jM01j on the optical properties. As a result we envisage the red-shift of total RI change peaks with progressive increase in F. Fig. 2b and c shows the similar profiles in presence of additive and multiplicative noise, respectively. In presence of additive noise, as before, we find red-shift in total RI change peaks with increase in F (Fig. 2b). However, the peak height passes through a maximum at F ¼ 50.0 KV/cm. On the other hand, in presence of multiplicative noise, we observe a blue-shift in total RI change peaks with increase in F (Fig. 2c). Thus, an increase in electric field strength appears to reduce DE01 gap in presence of additive noise whereas the consequence becomes just reverse in presence of the multiplicative counterpart. Moreover, we find a prominent rise in the magnitude of total RI change between F ¼ 0 KV/cm to F ¼ 50 KV/cm followed by saturation with further increase in F. The outcome thus indicates electric field-induced maximization of total RI change peaks in presence of additive and multiplicative noise (obviously to different extents and gaining prominence particularly in case of multiplicative noise). In other words, application of noise causes maximum overlap of eigenstates involved in the optical transition at a particular value of electric field strength. However, the magnitude of total RI change remains more or less same in presence and absence of noise. C. Role of magnetic field (B): Fig. 3a delineates the profiles of Dn(n)/nr with incident photon energy Zn for three different values of B (0T, 10T, and 20T) in absence of noise. As B increases the total RI change peaks move rightwards indicating a strong magnetic field-induced blue-

Fig. 3. Plots of Dn(n)/nr vs Zn at three different B values: (i) B ¼ 0.0T, (ii) B ¼ 10.0T, and (iii) B ¼ 20.0T. (a) Under noise-free condition, (b) in presence of additive noise, and (c) in presence of multiplicative noise.

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shift with more or less constant peak intensities. Thus, magnetic field does not significantly influence the peak values of total RI change [37]. The physical reason behind the blue-shift being the stronger confinement imposed by the magnetic field. Thus, the wave functions shrink and the separation between the energy levels increases with marginal increase in dipolar transition matrix elements [48,49,51,59]. Similar profile in presence of additive noise manifests a red-shift and gradual decrease in the peak height of total RI change as B increases (Fig. 3b). The red-shift also persists in presence of multiplicative noise, although, with maximization of peak height (at B ¼ 10T) as B increases (Fig. 3c). Thus, an increase in magnetic field strength reduces the DE01 energy separation in presence of noise. However, in presence of additive noise, an increase in magnetic field strength happens to steadily diminish the overlap between the eigenstates involved. Multiplicative noise, on the other hand, causes a maximization in the said overlap at B ¼ 10T under similar situation. We further envisage an enhancement in the total RI change (by an order of magnitude) in presence of noise over that of noise-free condition. D. Role of confinement frequency (u0): Fig. 4a displays the variation of Dn(n)/nr with incident photon energy Zn for three different values of Zu0 (0.82 meV, 8.2 meV, and 82.0 meV) in absence of noise. It becomes evident from the plot that as confinement strength Zu0 increases the total RI change peaks move rightwards showing a strong confinement-induced blue-shift. However, the peak height decreases steadily and gets progressively widened. The said blue-shift stems from increase in the energy difference between the states involved with increase in u0. An increase in u0 offers stronger confinement, which, in turn, causes more localization of the wave function thereby lowering the dipole matrix elements. Such lowering reduces the height of total RI change peaks and makes them more broadened with increase in u0. We, therefore, could infer that low u0 should be employed to obtain considerable total RI change. Fig. 4b and c depict the similar profiles in presence of additive and multiplicative noise, respectively. In case of additive noise, as confinement potential increases, there occurs a blue-shift of total RI change peaks and the peak height passes through a maxima at Zu0 ¼ 8:2 meV (Fig. 4b). Thus, in presence of additive noise, there occurs confinement potential-induced maximization of transition dipole matrix elements. With the application of multiplicative noise, on the other hand, we find a red-shift of total RI change peaks as u0 increases and the peak height undergoes a steady increase (Fig. 4c). Thus, multiplicative noise rearranges the energy level separation in a reverse manner (with respect to noise-free case) as u0 increases. Moreover, it promotes enhancement of dipole matrix elements leading to more intense total RI change peaks with increase in u0. Added to this, we observe a general enhancement of total RI change value in presence of noise (by an order of magnitude) over that of noise-free condition.

Fig. 4. Plots of Dn(n)/nr vs Zn at three different Zu0 values: (i) Zu0 ¼ 0:82 meV, (ii) Zu0 ¼ 8:2 meV, and (iii) Zu0 ¼ 82:0 meV. (a) Under noise-free condition, (b) in presence of additive noise, and (c) in presence of multiplicative noise.

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E. Role of dopant location (r0): Fig. 5a evinces the variation of Dn(n)/nr with incident photon energy Zn for three different values of dopant locations i.e. on-center (r0 ¼ 0.0 nm), near off-center (r0 ¼ 0.4 nm), and far off-center (r0 ¼ 0.75 nm) in absence of noise. It becomes conspicuous from the figure that an increase in r0 (i.e. shift of dopant from on to more off-center locations) shifts the total RI change peak positions to smaller frequency (red-shift) because of decrease in the energy level separation. Moreover, the total RI change peak exhibits a position-dependent maximization at r0 ¼ 0.4 nm [23] indicating maximum overlap between the eigenstates concerned. Fig. 5b and c displays the similar profiles in presence of additive and multiplicative noise, respectively. In presence of additive noise the peaks exhibit small blue-shift as the dopant is shifted from on to off-center locations. However, peak positions remain nearly unshifted for near and far off-center dopants (Fig. 5b). Importantly, quite similar to noise-free situation, the peak height exhibits maximization for a near off-center dopant (r0 ¼ 0.4 nm). In presence of multiplicative noise, the total RI change profile qualitatively resembles its additive analogue with minor difference in their magnitudes (Fig. 5c). The observations thus suggest small enhancement in DE01 energy gap under the sway of noise accompanying a dopant shift from on to off-center locations. However, the said enhancement becomes nominal between two different off-center locations. As before, presence of noise amplifies the magnitude of total RI change in comparison with noise-free situation. It becomes further evident from the figures that presence of noise (both additive and multiplicative) inverts the RI peak heights (with respect to noise-free case) for on-center (r0 ¼ 0.0 nm) and far off-center (r0 ¼ 0.75 nm) dopants whereas does not affect the said peak for a near off-center dopant (r0 ¼ 0.40 nm). Moreover, the RI peak position remains almost unshifted in presence of noise for a near off-center dopant w.r.t noise-free situation. However, in presence of noise, the RI peak undergoes red-shift and blue-shift w.r.t. noise-free condition for on-center and far off-center dopants, respectively. These observations provide an overall impression that it is the near off-center dopant location (r0 ¼ 0.40 nm) which remains rather unaffected in presence of noise. The observations thus indicate that the delicate interplay between noise and a near off-center dopant does not alter the energy interval from that of noise-free situation whereas the said interplay decreases (increases) the energy interval for an on-center (far off-center) dopant. Apart from these, the subtlety of noise-dopant interplay reverses the extent of overlap between the relevant eigenstates in presence of noise (w.r.t noise-free condition). Such reversal causes greater overlap between the relevant eigenstates for a far off-center dopant over that of an on-center one. Quite discernibly, the said overlap follows just the opposite sequence in absence of noise.

Fig. 5. Plots of Dn(n)/nr vs Zn at three different r0 values: (i) r0 ¼ 0.0 nm, (ii) r0 ¼ 0.4 nm, and (iii) r0 ¼ 0.75 nm. (a) Under noise-free condition, (b) in presence of additive noise, and (c) in presence of multiplicative noise.

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F. Role of relaxation time (t): Fig. 6a exhibits how Dn(n)/nr vary with incident photon energy Zn for three different values of t (0.14 ps, 0.2 ps, and 0.3 ps) in absence of noise. The total RI change peaks exhibit a feeble blue-shift as t increases indicating nearly unaltered separation between the energy levels. However, the amplitude of total RI change peaks increases prominently with increase in t suggesting enhancement in the extent of overlap between the relevant eigenstates. In practice, t is governed not only by the materials constituting the QD but also by other factors like temperature of the system, boundary conditions, electroneimpurity interaction etc. Hence, in order to obtain large total RI change the influence of these factors should be minimized as much as practicable [44]. Fig. 6b and c demonstrate the similar profiles in presence of additive and multiplicative noise, respectively. In both the cases we find that the total RI change peak remains almost unshifted with increase in t. However, the peak height increases as relaxation time increases. And at a small t value of 0.14 ps the total RI change profile evinces occurrence of maximization and minimization. The total RI change value increases by an order of magnitude only in presence of additive noise (over that of noise-free condition). G. Role of Al concentration (x): In order to explore the role played by aluminum concentration we now consider AlxGa1xAs QD with effective mass given as m* ¼ (0.067 þ 0.083x)m0, where x is the Al concentration [1]. Fig. 7a exhibits how Dn(n)/nr vary with incident photon energy Zn for three different values of x (0.0, 0.3, and 0.5) in absence of noise. It becomes discernible from the figure that as x increases the total RI change peaks undergo a blue-shift but the peak values remain almost unchanged. Thus, increase in aluminum concentration enhances the DE01 gap between the QD levels and maintains a steady overlap between the concerned eigenstates. Fig. 7b and c delineate similar plots in presence of additive and multiplicative noise, respectively. The plots reveal that with increase in Al concentration the total RI change peaks undergo a blue-shift in presence of additive noise, whereas, the shift becomes red in presence of the multiplicative counterpart. However, in both the cases the peak height passes through a minima at x ¼ 0.3 indicating minimum overlap between the related wave functions. Thus, it is the mode of application of noise that modifies the separation between the energy levels in contrasting ways as x increases. As before, the total RI change value increases by an order of magnitude only in presence of additive noise (over that of noise-free condition). H. Role of dopant potential (V0): Fig. 8a exhibits the variations of Dn(n)/nr with incident photon energy Zn for three different values of V0 (0.0 meV, 136.0 meV, and 272.0 meV) in absence of noise. The plots carry important physical significance. It reveals how the total RI

Fig. 6. Plots of Dn(n)/nr vs Zn at three different t: (i) t ¼ 0.14 ps, (ii) t ¼ 0.2 ps, and (iii) t ¼ 0.3 ps. (a) Under noise-free condition, (b) in presence of additive noise, and (c) in presence of multiplicative noise.

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Fig. 7. Plots of Dn(n)/nr vs Zn at three different x: (i) x ¼ 0.0, (ii) x ¼ 0.3, (iii) and x ¼ 0.5. (a) Under noise-free condition, (b) in presence of additive noise, and (c) in presence of multiplicative noise.

Fig. 8. Plots of Dn(n)/nr vs Zn at three different V0 values: (i) V0 ¼ 0.0 meV, (ii) V0 ¼ 136.0 meV, and (iii) V0 ¼ 272.0 meV. (a) Under noise-free condition, (b) in presence of additive noise, and (c) in presence of multiplicative noise.

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change is modulated as the QD composition is progressively changed from an impurity free state (V0 ¼ 0.0 meV) to a state with a given impurity potential (V0 ¼ 272.0 meV). The plot evinces blue-shift of total RI change peaks with increase in V0 accompanied by a steady increase in the peak intensities. The shift occurs because of the enhanced energy separation between the energy levels with increase in V0 in compliance with increased confinement on carrier motion [37]. And the increase in peak height originates from increase in transition dipole matrix elements because of greater dot-impurity interaction as V0 increases [44]. Fig. 8b and c describe the analogous profiles in presence of additive and multiplicative noise, respectively. In both the cases we find blue-shift of total RI change peaks as V0 increases. Thus, presence of noise does not qualitatively alter the spatial arrangement of energy levels as V0 increases. However, it is the peak height which shows contrasting behavior depending on mode of application of noise. In presence of additive noise, the peak height decreases steadily as V0 increases (Fig. 8b). However, exactly opposite behavior has been observed in presence of multiplicative noise (Fig. 8c). Thus, gradual enhancement of dopant potential affects the overlap of wave functions in diverse ways depending on whether noise has been introduced additively or multiplicatively. I. Role of noise strength (z): Finally we pay attention to examining the total RI change profiles in the light of a key parameter of present study i.e. the noise strength. Fig. 9a and b exhibit the variations of Dn(n)/nr with incident photon energy Zn for three different values of z (1.0  1010, 1.0  108, and 1.0  106) with additive and multiplicative noise, respectively. In presence of additive noise the shift of total RI change peak position with change in noise strength is feeble (Fig. 9a) indicating a steady separation between the energy levels. However, the peak height persistently falls with increase in z owing to drop in the extent of overlap between the relevant wave functions. In presence of multiplicative noise sharp total RI change peaks are observed at different values of z (Fig. 9b). Interestingly, a change in noise strength seems to have very little impact on the total RI change peak values as well as the peak positions. In other words, a variation of z, even over a considerably wide range (from ~1010 to ~106) produces rather identical total RI change profiles (and sometimes the profiles nearly overlap making it extremely difficult to distinguish them). It suggests that, in the present case, a variation of noise strength favors a well-settled dot-impurity interaction. The observation apparently contradicts our expectation. Since multiplicative noise is more deeply coupled with the system coordinates than additive noise, a variation in noise strength is expected to affect the optical properties more profoundly in case of the former than the latter. However, in practice, we have found distinct total RI change profiles in case of additive noise and rather overlapping profiles with the multiplicative analogue. In this context, it can be conceived that the strong system-noise coupling in case of multiplicative noise makes the system quite robust with respect to the variation of noise strength. On the other hand, as the system-noise coupling gets much diminished with additive noise, the robustness diminishes at the cost of high sensitivity of total RI change peaks to the noise strength variation. 4. Conclusion The total optical refractive index change of impurity doped QD has been thoroughly investigated in presence and absence of noise. The total RI change profiles have been monitored as a function of incident photon energy when several important parameters such as optical intensity, electric field strength, magnetic field strength, confinement energy, dopant location, relaxation time, aluminum concentration, dopant potential, and noise strength assume different values. An increase in optical intensity has been found to decrease total change in RI because of negative contribution from nonlinear RI change. Moreover, the role of mode of application of noise (additive/multiplicative) on the total RI change profiles has also been critically

Fig. 9. Plots of Dn(n)/nr vs Zn at three different z values: (i) z ¼ 1.0  1010, (ii) z ¼ 1.0  108, and (iii) z ¼ 1.0  106. (a) in presence of additive noise and (b) in presence of multiplicative noise.

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analyzed. In most of the cases we have envisaged shift of total RI change peak position and variation of total RI change peak intensity as several parameters are varied over a range. We have also found that the variation of the parameters mentioned above affects the total RI change both in presence and absence of noise with different extents of delicacy. Moreover, the variation of peak intensities often looses their uniformity in presence of noise and exhibit maximization and minimization. Most often we have also found enhancement of peak intensity in presence of noise over that of noise-free condition. Added to this, a change in the mode of application of noise (additive/multiplicative) can affect the behavior of total RI change peak shift and peak intensity in diverse and often contrasting ways. Interestingly, when noise strength itself is varied over a broad range, the total RI change profiles remain rather indifferent in case of multiplicative noise indicating robustness of the system to noise strength variation. However, in presence of additive noise, the said robustness declines and we observe distinct total RI change peaks. The findings announce the possibility of tuning the linear and nonlinear optical properties of doped QD systems by exploiting noise and bear technological relevance. Acknowledgments The authors S.S., S.P., J.G. 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 computational support. References [1] G. Rezaei, M.R.K. Vahdani, B. Vaseghi, Nonlinear optical properties of a hydrogenic impurity in an ellipsoidal finite potential quantum dot, Curr. Appl. Phys. 11 (2011) 176e181. [2] H. Ta ̧s, M. S¸ahin, The electronic properties of core/shell/well/shell spherical quantum dot with and without a hydrogenic impurity, J. Appl. Phys. 111 (2012), 083702 (8 pages). [3] H. Ta ̧s, M. S¸ahin, The inter-sublevel optical properties of a spherical quantum dot-quantum well with and without a donor impurity, J. Appl. Phys. 112 (2012) 053717. [4] S. Yilmaz, M. S¸ahin, Third-order nonlinear absorption spectra of an impurity in a spherical quantum dot with different confining potential, Phys. Status Solidi B 247 (2010) 371e374. [5] I. Karabulut, Ü. Atav, H. S¸afak, M. Tomak, Linear and nonlinear intersubband optical absorptions in an asymmetric rectangular quantum well, Eur. Phys. J. B 55 (2007) 283e288. [6] Y. Yakar, B. Çakir, Calculation of linear and nonlinear optical absorption coefficients of a spherical quantum dot with parabolic potential, Opt. Commun. 283 (2010) 1795e1800. [7] C.J. Zhang, K.X. Guo, Z.E. Lu, Exciton effects on the optical absorptions in one-dimensional quantum dots, Phys. E 36 (2007) 92e97. [8] W. Xie, Linear and nonlinear optical properties of a hydrogenic donor in spherical quantum dots, Phys. B 403 (2008) 4319e4322. [9] W. Xie, Nonlinear optical properties of a hydrogenic donor quantum dot, Phys. Lett. A 372 (2008) 5498e5500. [10] C.-H. Liu, B.-R. Xu, Theoretical study of the optical absorption and refraction index change in a cylindrical quantum dot, Phys. Lett. A 372 (2008) 888e892. [11] W. Xie, Dipole-allowed optical absorption of an exciton in a spherical parabolic quantum dot, Opt. Commun. 282 (2009) 2604e2607. € [12] A. Ozmen, Y. Yakar, B. Çakir, Ü. Atav, Computation of the oscillator strength and absorption coefficients for the intersubband transitions of the spherical quantum dot, Opt. Commun. 282 (2009) 3999e4004. [13] J. Yuan, W. Xie, L. He, An off-center donor and nonlinear absorption spectra of spherical quantum dots, Phys. E 41 (2009) 779e785. [14] T. Chen, W. Xie, S. Liang, Optical and electronic properties of a two-dimensional quantum dot with an impurity, J. Lumin. 139 (2013) 64e68. € [15] B. Çakir, Y. Yakar, A. Ozmen, Refractive index changes and absorption coefficients in a spherical quantum dot with parabolic potential, J. Lumin. 132 (2012) 2659e2664. € € [16] B. Çakir, Y. Yakar, A. Ozmen, M. Ozgür Sezer, M. S¸ahin, Linear and nonlinear optical absorption coefficients and binding energy of a spherical quantum dot, Superlattices Microstruct. 47 (2010) 556e566. _ [17] S. Ünlu, I. Karabulut, H. S¸afak, Linear and nonlinear intersubband optical absorption coefficients and refractive index changes in a quantum box with finite confining potential, Phys. E 33 (2006) 319e324. [18] 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 (5 pages). [19] M.R.K. Vahdani, G. Rezaei, Linear and nonlinear optical properties of a hydrogenic donor in lens-shaped quantum dots, Phys. Lett. A 373 (2009) 3079e3084. [20] Z. Zeng, C.S. Garoufalis, A.F. Terzis, S. Baskoutas, Linear and nonlinear optical properties of ZnS/ZnO core shell quantum dots: effect of shell thickness, impurity, and dielectric environment, J. Appl. Phys. 114 (2013) 023510. [21] K.M. Kumar, A.J. Peter, C.W. Lee, Optical properties of a hydrogenic impurity in a confined Zn1xCdxSe/ZnSe quantum dot, Superlattices Microstruct. 51 (2012) 184e193. €kmen, C.A. Duque, Electron-related optical responses in triangular [22] A. Tiutiunnyk, V. Tulupenko, M.E. Mora-Ramos, E. Kasapoglu, F. Ungan, H. Sari, I. So quantum dots, Phys. E 60 (2014) 127e132. [23] R. Khordad, H. Bahramiyan, Impurity position effect on optical properties of various quantum dots, Phys. E 66 (2015) 107e115. [24] J.C. Martínez-Orozco, M.E. Mora-Ramos, C.A. Duque, Electron-related linear and nonlinear optical responses in vertically coupled triangular quantum dots, Phys. B 452 (2014) 82e91. [25] G. Rezaei, B. Vaseghi, F. Taghizadeh, M.R.K. Vahdani, M.J. Karimi, Intersubband optical absorption coefficient changes and refractive index changes in a two-dimensional quantum pseudodot system, Superlattices Microstruct. 48 (2010) 450e457. [26] G.H. Wang, K.X. Guo, Interband optical absorptions in a parabolic quantum dot, Phys. E 28 (2005) 14e21. [27] W. Xie, Impurity effects on optical property of a spherical quantum dot in the presence of an electric field, Phys. B 405 (2010) 3436e3440. [28] S. Baskoutas, C.S. Garoufalis, A.F. Terzis, Linear and nonlinear optical absorption coefficients in inverse parabolic quantum wells under static external electric field, Eur. Phys. J. B 84 (2011) 241e247. [29] N.V. Lien, N.M. Trinh, Electric field effects on the binding energy of hydrogen impurities in quantum dots with parabolic confinements, J. Phys. Cond. Mater. 13 (2001) 2563e2571. [30] G. Murillo, N. Porras-Montenegro, Effects of an electric field on the binding energy of a donor impurity in a spherical GaAs(Ga,Al)As quantum dot with parabolic confinement, Phys. Status Solidi B 220 (2000) 187e190. € kmen, M.E. Mora-Ramos, C.A. Duque, Donor impurity states and related optical responses in triangular quantum [31] E. Kasapoglu, F. Ungan, H. Sari, I. So dots under applied electric field, Superlattices Microstruct. 73 (2014) 171e184. [32] E. Sadeghi, Electric field and impurity effects on optical property of a three-dimensional quantum dot: a combinational potential scheme, Superlattices Microstruct. 50 (2011) 331e339.

S. Saha et al. / Superlattices and Microstructures 88 (2015) 620e633

633

[33] W. Xie, Q. Xie, Electric field effects of hydrogenic impurity states in a disc-like quantum dot, Phys. B 404 (2009) 1625e1628. [34] C.A. Duque, A. Montes, A.L. Morales, Binding energy and polarizability in GaAs(Ga,Al)As quantum-well wires, Phys. B 302 (2001) 84e87. [35] S. Baskoutas, E. Paspalakis, A.F. Terzis, Electronic structure and nonlinear optical rectification in a quantum dot: effects of impurities and external electric field, J. Phys. Cond. Mat. 19 (2007), 395024 (9-pages). [36] M. Kirak, S. Yilmaz, M. S¸ahin, M. Gençasian, The electric field effects on the binding energies and the nonlinear optical properties of a donor impurity in a spherical quantum dot, J. Appl. Phys. 109 (2011) 094309. [37] M. Narayanan, A. John Peter, Electric field induced exciton binding energy and its non-linear optical properties in a narrow InSb/InGaxSb1x quantum dot, Superlattices Microstruct. 51 (2012) 486e496. [38] L. He, W. Xie, Effects of an electric field on the confined hydrogen impurity states in a spherical parabolic quantum dot, Superlattices Microstruct. 47 (2010) 266e273. [39] Z. Xiao, J. Zhu, F. He, Magnetic field dependence of the binding energy of a hydrogenic impurity in a spherical quantum dot, J. Appl. Phys. 79 (1996) 9181. [40] Z. Xiao, Exciton binding energy in spherical quantum dots in a magnetic field, J. Appl. Phys. 86 (1999) 4509. ~ o, R. Rosas, J.L. Marín, R. Riera, Hydrogenic impurities in spherical quantum dots in a magnetic field, J. Appl. Phys. 90 (2001) [41] A. Corella-Maduen 2333e2337. €kmen, Shallow donor impurity binding energy in the V-shaped quantum well under the crossed electric and magnetic fields, Phys. E [42] E. Kasapoglu, I. So 27 (2005) 198e203. €kmen, Intersubband optical absorption in a quantum well under a tilted magnetic field, Superlattices Microstruct. 29 (2001) [43] E. Kasapoglu, H. Sari, I. So 25e32. € kmen, Effect of magnetic fields on the linear and nonlinear intersubband optical absorption coefficients and refractive index changes in [44] E. Ozturk, I. So square and graded QWLs, Superlattices Microstruct. 48 (2010) 312e320. [45] M.J. Karimi, G. Rezaei, Effects of external electric and magnetic fields on the linear and nonlinear intersubband optical properties of finite semiparabolic quantum dots, Phys. B 406 (2011) 4423e4428. [46] G. Liu, K. Guo, C. Wang, Linear and nonlinear intersubband optical absorption in a disk-shaped quantum dot with a parabolic potential plus an inverse squared potential in a static magnetic field, Phys. B 407 (2012) 2334e2339. [47] G. Rezaei, B. Vaseghi, F. Taghizadeh, M.R.K. Vahdani, M.J. Karimi, Intersubband optical absorption coefficient changes and refractive index changes in a two-dimensional quantum pseudodot system, Superlattices Microstruct. 48 (2010) 450e457. [48] W. Xie, Nonlinear optical properties of an off-center donor in a quantum dot under applied magnetic field, Solid State Commun. 151 (2011) 545e549. [49] L. Lu, W. Xie, Impurity and exciton effects on the nonlinear optical properties of a disc-like quantum dot under a magnetic field, Superlattices Microstruct. 50 (2011) 40e49. [50] G. Liu, K. Guo, H. Hassanabadi, L. Lu, Linear and nonlinear optical properties in a disk-shaped quantum dot with a parabolic potential plus a hyperbolic potential in a static magnetic field, Phys. B 407 (2012) 3676e3682. [51] Z.H. Zhang, K.X. Guo, B. Chen, R.Z. Wang, M.W. Kang, Theoretical studies on the optical absorption coefficients and refractive index changes in parabolic quantum dots in the presence of electric and magnetic fields, Superlattices Microstruct. 47 (2010) 325e334. [52] J. Ganguly, M. Ghosh, Influence of Gaussian white noise on the frequency-dependent linear polarizability of doped quantum dot, Chem. Phys. 438 (2014) 75e82. [53] J. Ganguly, M. Ghosh, Influence of Gaussian white noise on the frequency-dependent first nonlinear polarizability of doped quantum dot, J. Appl. Phys. 115 (2014), 174313 (10 pages). [54] J. Ganguly, M. Ghosh, Exploring static and frequency-dependent third nonlinear polarizability of doped quantum dots driven by Gaussian white noise, Phys. Status Solidi B 252 (2015) 289e297. [55] S. Pal, J. Ganguly, S. Saha, M. Ghosh, Oscillator strength of impurity doped quantum dots: Influence of Gaussian white noise, Phys. B 474 (2015) 41e46. rrez, I. Mikhailov, M.R. Fulla, J.H. Marín, Dimensionality effect on two-electron energy spectrum: A fractional-dimension-based [56] R. Correa, W. Gutie formulation, Phys. Lett. A 379 (2015) 1457e1463. [57] A. Gharaati, R. Khordad, A new confinement potential in spherical quantum dots: modified Gaussian potential, Superlattices Microstruct. 48 (2010) 276e287. [58] R. Khordad, Use of modified Gaussian potential to study an exciton in a spherical quantum dot, Superlattices Microstruct. 54 (2013) 7e15. €kmen, The linear and nonlinear intersubband optical absorption coefficients and refractive index [59] U. Yesilgul, F. Ungan, E. Kasapoglu, H. Sari, I. So changes in a V-shaped quantum well under the applied electric and magnetic fields, Superlattices Microstruct. 50 (2011) 400e410.