Modulating optical rectification, second and third harmonic generation of doped quantum dots: Interplay between hydrostatic pressure, temperature and noise

Superlattices and Microstructures 98 (2016) 385e399

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Modulating optical rectification, second and third harmonic generation of doped quantum dots: Interplay between hydrostatic pressure, temperature and noise Jayanta Ganguly a, Surajit Saha b, Aindrila Bera c, Manas Ghosh c, * a

Department of Chemistry, Brahmankhanda Basapara High School, Basapara, Birbhum 731215, West Bengal, India Department of Chemistry, Bishnupur Ramananda College, Bishnupur, Bankura 722122, West Bengal, India c Department of Chemistry, Physical Chemistry Section, Visva Bharati University, Santiniketan, Birbhum 731235, 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 13 July 2016 Received in revised form 24 August 2016 Accepted 28 August 2016 Available online 30 August 2016

We examine the profiles of optical rectification (OR), second harmonic generation (SHG) and third harmonic generation (THG) of impurity doped QDs under the combined influence of hydrostatic pressure (HP) and temperature (T) in presence and absence of Gaussian white noise. Noise has been incorporated to the system additively and multiplicatively. In order to study the above nonlinear optical (NLO) properties the doped dot has been subjected to a polarized monochromatic electromagnetic field. Effect of application of noise is nicely reflected through alteration of peak shift (blue/red) and variation of peak height (increase/decrease) of above NLO properties as temperature and pressure are varied. All such changes again sensitively depends on mode of application (additive/multiplicative) of noise. The remarkable influence of interplay between noise strength and its mode of application on the said profiles has also been addressed. The findings illuminate fascinating role played by noise in tuning above NLO properties of doped QD system under the active presence of both hydrostatic pressure and temperature. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Quantum dot Impurity Nonlinear optical properties Hydrostatic pressure Temperature Gaussian white noise

1. Introduction Low-dimensional semiconductor systems (LDSS) e.g. quantum wells (QWLs), quantum wires (QWRs) and quantum dots (QDs) are renowned for their extensive applications in the field of applied physics. The strong confinement existing in LDSS compared with their bulk neighbors has insisted enhanced research activities in studying the electronic, magnetic, and optical properties of them, both experimentally and theoretically [1]. Impurity states in LDSS are extremely significant as their presence dramatically alters the optical and transport properties of LDSS [2e4]. This necessitates a deep realization of the effects of shallow impurities on electronic states of LDSS. The opportunities of tremendous technological applications in electronic and optoelectronic devices have fomented experimental and theoretical studies pertinent to deciphering the physical properties of impurities in LDSS [5e13]. External perturbations, such as electric field, magnetic field, hydrostatic pressure (HP), and temperature provide valuable information about LDSS [1,2,14e22]. The physical properties of LDSS can be manipulated by altering the strength of external

* Corresponding author. E-mail address: [email protected] (M. Ghosh). http://dx.doi.org/10.1016/j.spmi.2016.08.052 0749-6036/© 2016 Elsevier Ltd. All rights reserved.

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perturbations without hampering the physical size of the structure [3]. Therefore, it has become a regular practice to fabricate the electronic structure of LDSS by means of external perturbations. In view of device applications, such fabrication paves the way of tailoring the energy spectrum of LDSS to produce desirable optical effects. Moreover, controlled variation of strength of external perturbations can regulate the performance of optoelectronic devices. Thus, external perturbations are marked as important candidates for studying the linear and nonlinear optical (NLO) properties of LDSS [23]. High pressure investigations of LDSS have emerged as a crucial topic in condensed matter physics and materials sciences because of their immense influence on the tunable optical properties relevant to applications in optoelectronics, QD lasers, high-density memory, bio-engineering etc. [24e26]. Application of pressure into LDSS generally causes enhancement of effective mass and reduction of dielectric constant of the system. In consequence, the band structure of LDSS is also affected, which, in turn, affects the transition between different energy levels of confined particles [26,27]. Besides the said modification of band gap, applied HP can also perturb the potential barriers, band-offset, lattice constant, and even the dimension of LDSS which are related to the fractional change in volume. Above discussions nicely highlight the importance of HP as a powerful means of modulating the electron-related NLO properties of LDSS. Naturally we find a plethora of notable studies on LDSS involving HP [28e45]. Apart from HP, temperature effects also tailor the electronic structure of LDSS [27] and consequently the NLO properties which are very much linked with the electron-impurity interaction [29,37,38,43,44,46e50]. Optical rectification (OR) and second harmonic generation (SHG) are the two simplest second-order nonlinear processes having magnitudes usually greater than those of higher-order ones. These two properties are prominently manifested whenever LDSS possess noticeable asymmetry [51e57]. The third-order NLO properties assume importance in LDSS having inversion symmetry. In this case, while the second-order susceptibilities become insignificant because of the inversion symmetry, the third-order one exhibits huge enhancement compared with the bulk material. NLO materials with large thirdorder nonlinear susceptibilities c3 are regularly utilized as important components to manufacture all-optical switching, modulating and computing devices [58,59]. Recently higher harmonic generation in AC electric field driven by the gate electrodes and not light field is observed in carbon nanotubes [60]. As a natural consequence, we can find lots of noticeable works on OR, SHG and third harmonic generation (THG) under the influence of HP and temperature [30,32,61e67]. Recently we have explored the role of noise in regulating the above NLO properties of doped QD [68]. In the present manuscript we have inspected the influence of Gaussian white noise on OR, SHG and THG of doped QD under the combined influence of hydrostatic pressure (HP) and temperature. The system under study is a 2-d QD (GaAs) consisting of single carrier electron under parabolic confinement in the xey plane. The QD is doped with an impurity modeled by a Gaussian potential in the presence of a perpendicular magnetic field which provides an additional confinement. Gaussian white noise has been incorporated to the doped QD via two different pathways i.e. additive and multiplicative [68]. The investigation delineates subtle interplay between noise (which manifestly depends on its mode of application), HP and temperature that finally settles OR, SHG and THG of doped QD. The findings seem to carry practical relevance. 2. Method The impurity doped QD Hamiltonian, subject to external static electric field (F) applied along x and y-directions and spatially d-correlated Gaussian white noise (additive/multiplicative) can be written as



H0 ¼ H00 þ Vimp þ
e
Fðx þ yÞ þ Vnoise :

(1)

Under effective mass approximation, H00 represents the impurity-free 2-d quantum dot containing single carrier electron under lateral parabolic confinement in the xey plane and in presence of a perpendicular magnetic field. Vðx; yÞ ¼ 12m u20 ðx2 þ y2 Þ is the confinement potential with u0 as the harmonic confinement frequency. H00 is therefore given by

H00 ¼

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

(2)

m* represents the effective mass of the electron inside the QD material. Using Landau gauge [A ¼ (By,0,0), where A is the vector potential and B is the magnetic field strength], H00 reads

H00

Z2 ¼  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

(3)

eB being the cyclotron frequency. U2 ¼ u2 þ u2 can be viewed as the effective confinement frequency in the y-direction. uc ¼ m  c 0 The Hamiltonian [cf. eqn. (2)] represents a 2-d quantum dot with a single carrier electron [69,70]. The form of the confinement potential conforms to kind of lateral electrostatic confinement (parabolic) of the electrons in the xey plane. In real QDs the electrons are confined in 3-dimensions i.e. the carriers effectively possess a quasi-zero dimensional domain. The confinement length scales R1, R2, and R3 can, in general, be different in three spatial directions, but usually R3 ≪ R1 x R2. Whenever such QDs are modeled R3 is often taken to be strictly zero and the confinement in the other two directions is described by a potential V with V(x) / ∞ for jxj/∞, x ¼ (x1,x2)2R2. x1 and x2 represent the coordinates in x and y directions,

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respectively. It means that, since R3 is almost zero; the potential in the other two relevant directions extends up to a length which is of the order of confinement length scales in these directions. Thus, our present doped QD assumes a disk shape. In presence of HP and temperature, the effective mass becomes pressure and temperature-dependent and is given by (for GaAs) [1].

" 

m ðP; TÞ ¼ m0

(

1 þ EG P

2 1 þ EgG ðP; TÞ EgG ðP; TÞ þ D0

)#1 ;

(4)

where m0 is the single electron bare mass. EPG ¼ 7:51 eV is the energy related to momentum matrix element. D0 ¼ 0.341 eV is the spin-orbit splitting of the valence band (VB) for GaAs. The pressure and temperature-dependent energy gap for GaAs QD at G point in units of eV is given by

EgG ðP; TÞ ¼ EgG ð0; TÞ þ 1:26  102 P  3:77  105 P 2 ; in the above expression P is in Kbar unit and the factors 1.26  102 and 3.77  105 have units eV/Kbar and eV/Kbar2, respectively. EgG ð0; TÞ is the energy gap at zero pressure and is given by

EgG ð0; TÞ ¼ 1:519 

5:405  104 T 2 : T þ 204

The Pressure and temperature-dependent dielectric constant (for GaAs) is given by Ref. [1].

h i h i εðP; TÞ ¼ 12:74 exp  1:73  103 P :exp 9:4  105 ðT  75:6Þ ; for T  200K;

(5)

h i h i εðP; TÞ ¼ 13:18 exp  1:73  103 P :exp 20:4  105 ðT  300Þ ; for T > 200K:

(6)

and

2

2

Vimp is the impurity (dopant) potential formulated by a Gaussian function [68] viz. Vimp ¼ V0 eg½ðxx0 Þ þðyy0 Þ  . (x0,y0) is the site of dopant inclusion, V0 is the strength of the dopant potential, and g1/2 represents the spatial region over which the influence of impurity potential is dispersed. A large value of g indicates that the spatial extension of impurity potential is highly quenched whereas a small g accounts for spatially diffused one. g can be written as g ¼ kε where ε is the dielectric constant of the medium and k is a constant. The use of repulsive Gaussian potential as a modification of the confinement potential was first introduced by Szafran et al. in the context of exciton spectrum of a quantum ring [71]. The repulsive nature of the potential simulates dopant with excess electrons. Such Gaussian impurity has also drawn the attention of several other researchers. Adamowski studied the screening effect of the LO phonons of the D center confined by a Gaussian potential QD, V(r) ¼ V0 exp[(r/R)2] [72]. Gaussian potential is a smooth potential and therefore is a good approximation to the impurity potential in electrostatic quantum dots [73], in which the spatial restriction originates from an inhomogeneous electric field. In self-assembled quantum dots with a composition modulation [74], the impurity potential can also be represented by the Gaussian potential [75]. The term Vnoise represents the noise contribution to the Hamiltonian H0. It consists of a spatially d-correlated Gaussian white noise [f(x,y)] which assumes a Gaussian distribution (generated by Box-Muller algorithm) having strength z and is described by the set of conditions [68]:

〈f ðx; yÞ〉 ¼ 0;

(7)

the zero average condition, and

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

(8)

the spatial d-correlation condition. The Gaussian white noise can be applied to the system via two different modes (pathways) i.e. additive and multiplicative [68]. In case of additive white noise Vnoise becomes

Vnoise ¼ l1 f ðx; yÞ:

(9)

And with multiplicative noise we can write

Vnoise ¼ l2 f ðx; yÞðx þ yÞ:

(10)

The parameters l1 and l2 absorb in them all the neighboring influences in case of additive and multiplicative noise, respectively. In reality, there exist a variety of physical situations in which external noise can be realized and bears interest. In these situations one deals with system which experiences fluctuations which are not self-originating. These fluctuations can

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be due to a fluctuating environment or can be consequence of an externally applied random force. Whereas additive noise does not interfere with the system coordinate the multiplicative analogue depends on the instantaneous value of the variables of the system. It does not scale with system size and is not necessarily small [76,77]. €dinger equation we have generated the sparse Hamiltonian matrix (H0) In order to solve the time-independent Schro where the matrix elements involve the function j(x,y), constructed as a superposition of the products of harmonic oscillator eigenfunctions. In this context requisite number of basis functions (20) for each of the directions (x,y) have been used after performing the convergence test. Thus, the direct product basis spans a space of (20  20) dimension. And H0 is diagonalized in the direct product basis of harmonic oscillator eigenfunctions to obtain the energy levels and wave functions. We now consider interaction between a polarized monochromatic electromagnetic field of angular frequency n with an ensemble of QDs. If the wavelength of progressive electromagnetic wave is greater than the QD dimension, the amplitude of the wave may be regarded constant throughout QD and the aforesaid interaction can be realized under electric dipole approximation. Under density matrix approach and iterative procedure the expression of OR coefficient (for a two-level quantum system) is given by Ref. [53].





c0ð2Þ

¼

4e3 ss ε0 Z2

Mij2 dij 

DEij2 ð1 þ G2 =G1 Þ þ n2 þ G22 ðG2 =G1  1Þ h

2

DEij  n

þ G22

ih

2

DEij þ n

þ G22

i

;

(11)

where e is the absolute value of electron charge, ε0 being the vacuum permittivity, ss is
the carrier density, ji(jj) are the

b b
x þ y eigenstates and DEij ¼ (EiEj) is the energy difference between these states. M ¼ e〈 j j 〉; ði; j ¼ 0; 1Þ is the matrix

ij i j

elements of the dipole moment, dij ¼
Mii  Mjj
, Gk ¼ 1/Tk with k ¼ (1,2) are damping terms associated with the lifetime (longitudinal and transverse, respectively) of the electrons involved in the transitions. For n z DE01, the peak value of ð2Þ ð2Þ 2 3 2 cð2Þ 0 ð≡c0;max Þ is estimated by the expression c0;max ¼ 2e T1 T2 ss M01 d01 =ðε0 Z Þ [55]. The geometric factor (GF) for the OR co2 d . efficient is given by M01 01 The SHG susceptibility per unit volume under two-photon resonance condition (i.e. Zn ¼ Zn10 ¼ Zn21 ) is given by (for a threelevel quantum system) [78].

c2ð2Þ n ¼

e3 ss

jM01 j:jM12 j:jM20 j ; ε0 Z2 ðn  n10 þ iG10 Þ:ð2n  n20 þ iG20 Þ

(12)

where, nij ¼ ðEi  Ej Þ=Z is the transition frequency. G ¼ G10 ¼ G20 is the off-diagonal relaxation rate. The SHG susceptibility has a resonant peak in the energy position of two-photon resonance, (i.e. Zn ¼ Zn10 ¼ Zn21 ¼ Zn20 =2) obtained by Ref. [52].

c2ð2Þ n;max ¼

e3 ss jM01 j:jM12 j:jM20 j : ε0 ðZGÞ2

(13)

It is obvious that the maximum SHG coefficient is proportional to the geometric factor (GF) jM01 j:jM12 j:jM20 j of the quantum confined system. Therefore, in order to obtain large SHG coefficient, apart from small relaxation rate G, a large GF is also required. Under triple resonance conditions the THG susceptibility per unit volume is given by Ref. [79].

c3ð3Þ n ¼

e4 ss

M01 :M12 :M23 :M30 ; ε0 Z3 ðn  n10 þ iG10 Þ:ð2n  n20 þ iG20 Þ:ð3n  n30 þ iG30 Þ

(14)

where Gij(i s j) ¼ G2 ¼ 1/T2 is the off-diagonal relaxation rate with transverse relaxation time T2. The THG susceptibility has a resonant peak in the energy position of triple resonance, i.e. Zn ¼ Zn10 ¼ Zn21 ¼ Zn32 given by Ref. [79].

c3ð3Þ n;max ¼

e4 ss jM01 j:jM12 j:jM23 j:jM30 j ; ε0 iðZG2 Þ3

(15)

where the off-diagonal relaxation rate G2 ¼ G10 ¼ G20 ¼ G30. The geometric factor (GF) (jM01 j:jM12 j:jM23 j:jM30 j) gives the ð3Þ maximum THG susceptibility (c3n;max ) at resonance peaks [79]. 3. Results and discussion The calculations are performed using the following parameters: ε ¼ 12.4 (without considering pressure and temperature dependence), m* ¼ 0.067 m0 (without considering pressure and temperature dependence), where m0 is the free electron mass. ε0 ¼ 8.8542  1012 Fm1, t ¼ 0.14 ps, s s¼ 5.0  1024 m3, Zu0 ¼ 250:0 meV, F ¼ 100 KV/cm, B ¼ 20.0 T, z ¼ 1.0  104, V0 ¼ 280.0 meV and r0 ¼ 0.0 nm. The parameters are suitable for GaAs QDs. Moreover, when pressure variation is carried out, temperature is kept fixed at T ¼ 100 K and during temperature variation, pressure is kept fixed at P ¼ 50 Kbar.

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3.1. Optical rectification (OR) Fig. 1aec show the pattern of variations of OR against hn at P ¼ 0 Kbar, 25 Kbar, 50 Kbar, 100 Kbar and 200 Kbar in absence of noise, in presence of additive noise and in presence of multiplicative noise, respectively. In absence of noise, as HP increases, OR peaks have been found to undergo red-shift [30,61e63,65] and peak height [63,65] [Fig. 1a]. The red
2
decreases

shift takes place because of drop in the value of DE01 gap as P increases. The quantities
M01
and d01 are independent of HP and therefore the OR peak height is directly related to DE01 value. Thus, the decrease in the peak height also stems from the said drop in the DE01 value. The general pattern of OR profile remains the same even in presence of additive [Fig. 1b] and multiplicative [Fig. 1c] noise. However, with respect to noise-free condition, the shape of the OR profile gets to some extent distorted and the OR values get enhanced indicating greater overlap between the eigenstates concerned. Moreover, the said enhancement appears to be much more prominent in presence of additive noise than its multiplicative counterpart. 2 d ) with pressure at hn ¼ 200 meV, (i) in Fig. 2a and b represent the variation of OR and geometric factor (GF ¼ M01 01 absence of noise (ii) in presence of additive noise and (iii) in presence of multiplicative noise, respectively. In agreement with earlier findings, under all conditions, OR decreases monotonically with increase in HP. Moreover, additive noise causes more enhancement of OR value than multiplicative noise (with respect to noise-free condition) [Fig. 2a]. GF actually stands as a measure of the extent of overlap between the eigenstates involved and explains the variation of OR peak intensities with P. The GF plots [Fig. 2b] further corroborate our earlier observations as under all conditions we find a steady fall of GF with increase in HP. The magnitude of GF maintains the same sequence in absence and in presence of noise (including its mode of application) as found earlier. Fig. 3aec depict the pattern of variations of OR against hn at T ¼ 0 K, 200 K and 500 K, in absence of noise, in presence of additive noise and in presence of multiplicative noise, respectively. In absence of noise, as T increases, OR peaks undergo blueshift [62,65] and peak height increases [62,63,65] [Fig. 3a]. The blue-shift takes place because of rise in the value of DE01 gap as T increases. And an increase in temperature increases the extended area of wave function and consequently OR peak height increases. Increase of OR peak height with temperature has also been found in presence of additive [Fig. 3b] and multiplicative [Fig. 3c] noise. However, whereas in presence of multiplicative noise the OR peaks exhibit blue-shift with increase in temperature (similar to noise-free condition), in presence of additive noise the OR peaks remain nearly unshifted. It can therefore be inferred that the DE01 gap remains almost unchanged with temperature in presence of additive noise. Furthermore, quite similar to what has been observed previously, again we find noise-induced enhancement of OR value which is even more prominent in case of additive noise than its multiplicative analogue.

Fig. 1. Plots of OR vs hn at (i) P ¼ 0 Kbar, (ii) P ¼ 25 Kbar, (iii) P ¼ 50 Kbar, (iv) P ¼ 100 Kbar and (v) P ¼ 200 Kbar: (a) under noise-free condition, (b) in presence of additive noise and (c) in presence of multiplicative noise.

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Fig. 2. Plots of (a) OR vs P and (b) M201 d01 vs P: (i) under noise-free condition, (ii) in presence of additive noise and (iii) in presence of multiplicative noise.

Fig. 3. Plots of OR vs hn at (i) T ¼ 0 K, (ii) T ¼ 200 K and (iii) T ¼ 500 K: (a) under noise-free condition, (b) in presence of additive noise and (c) in presence of multiplicative noise.

2 d ) with temperature at hn ¼ 200 meV, (i) in Fig. 4a and b represent the variation of OR and geometric factor (GF ¼ M01 01 absence of noise (ii) in presence of additive noise and (iii) in presence of multiplicative noise, respectively. In conformity with previous observations, under all conditions, OR increases persistently with increase in T. Moreover, additive noise causes greater enhancement of OR value than multiplicative noise (with respect to noise-free condition) [Fig. 4a]. The GF plots [Fig. 4b] further affirm our earlier observations as under all conditions we find a steady enhancement of GF with increase in temperature. The magnitude of GF exhibits the same sequence in absence and in presence of noise (including its mode of application) as found erstwhile. At this point of discussion it needs to be noted that both during pressure and temperature variations OR values maintain a definite sequence viz. OR (in presence of additive noise) > OR (in presence of multiplicative noise) > OR (in absence of noise). At a first glance it appears that for a fixed extent of disorder introduced into the system, additive noise enhances the extent of overlap between the relevant eigenstates (over that of noise-free condition) much more than multiplicative one. Propelled by aforesaid outcome we now concentrate on the important aspect of how OR changes as noise strength (z) is varied over a range under a given temperature and pressure. Fig. 5a delineates the variation of OR with log(z) in presence of additive [Fig. 5a(i)]

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Fig. 4. Plots of (a) OR vs T and (b) M201 d01 vs T: (i) under noise-free condition, (ii) in presence of additive noise and (iii) in presence of multiplicative noise.

Fig. 5. Plots of (a) OR vs log(z) and (b) DEr vs log(z) at T ¼ 100 K and P ¼ 50 Kbar: (i) in presence of additive noise and (ii) in presence of multiplicative noise.

and multiplicative [Fig. 5a(ii)] noise, respectively, at T ¼ 100 K and pressure P ¼ 50 Kbar. We have varied noise strength over a range from z ¼ 1.0  1018 to z ¼ 1.0  102 and the OR values at the lowest segment of the range nearly correspond to those under noise-free condition. We have chosen above range because of the fact that at the two extremes of it the OR values exhibit prominent saturation with no sign of change even if the said range is broadened further. It has been found that in case of additive noise OR decreases persistently with increase in z at a given temperature and pressure. On the other hand, in presence of multiplicative noise the variation of OR is not at all monotonic; rather it exhibits a maximization at z ~ 108. It appears interesting to note that the OR values become higher in magnitude in presence of multiplicative noise than its addditive counterpart up to z ~ 106. A marked crossover of OR values takes place at z ~ 106 whence OR in presence of additive noise supersedes those in presence of multiplicative one. It can therefore be argued that presence of noise enhances OR value over that of noise-free case regardless of its mode of application. However, up to z ~ 106 multiplicative noise would cause more efficient overlap between the relevant eigenstates than its additive neighbor and the OR values will show a sequence just reverse to what we have found previously. Moreover, multiplicative noise would cause maximum amount of said overlap around z ~ 108. As noise strength is increased beyond z ~ 106 (the “crossover” zone) additive noise begins to cause more fruitful overlap between the pertinent eigenstates than its multiplicative analogue which is reflected through OR values displaying the same sequence as we have previously observed. Such observations nicely highlight a delicate interplay between noise strength and mode of application of noise in shaping magnitude of OR under a given temperature and pressure. It can therefore be inferred that gradual enhancement in the extent of noise applied to the system may deplete or amplify the effective overlap between the concerned eigenstates depending on the mode of application of noise. The “crossover” point prescribes the preferred mode of application of noise in view of achieving higher magnitude of NLO properties. These two different modes, in effect, differ in the extent to which noise perturbs the system. Apparently, multiplicative noise affects the system much more than its additive analogue because of its close linkage with the system coordinates. Additive noise, on the other hand, makes a rather weak contact with the system. Present findings suggest that, before the “crossover” point, multiplicative noise of low strength gets moderately involved with the system and thus, it does not feel the complete influence of system confinement. As a result it overcomes the system confinement much more smoothly than additive one and noticeably amplifies the NLO properties. As soon as noise strength exceeds the “crossover” value, the multiplicative mode gets more involved with the system and comes heavily under strong confinement. It therefore becomes unable to amplify the NLO properties to the same extent as before. Making full use of this opportunity, additive noise now enhances the NLO properties much more prominently than its multiplicative neighbor.

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Above observation can be put on a more rational basis if we compute the energy fluctuations corresponding to various values of noise strength (z) and also analyze how large these fluctuations are in comparison with the confinement energy of electron in QD. For this purpose we have determined the energy separations between the ground state and the first excited state (DE01 ¼ E1  E0) in absence (DE01) and presence (DE01 0) of noise at a given value of confinement energy (Zu0 ¼ 250:0 meV). The ratio of these two values (DEr ¼ DE010 /DE01) gives an estimate of extent of fluctuation because of presence of noise. Fig. 5b depicts the plot of DEr against log(z) in presence of additive [Fig. 5b(i)] and multiplicative [Fig. 5b(ii)] noise, respectively. The plot gives an overview of energy fluctuation as a function of noise strength at a given confinement energy which discernibly depends upon the mode of application of noise. The plot reveals a steady increase of DEr with z for both additive and multiplicative noise indicating larger energy fluctuation with increase in noise strength. The plot also shows greater energy fluctuation in presence of multiplicative noise than its additive counterpart for low values of noise strength. However, as z > ~106 (the “crossover” value) the extent of energy fluctuation exhibits a reverse behavior. 3.2. Second harmonic generation (SHG) Fig. 6aec exhibit the pattern of variations of SHG against hn at P ¼ 0 Kbar, 25 Kbar, 50 Kbar, 100 Kbar and 200 Kbar in absence of noise, in presence of additive noise and in presence of multiplicative noise, respectively. In absence of noise, as HP increases, SHG peaks have been found to undergo red-shift [32] and peak height decreases [32] [Fig. 6a]. The red-shift indicates a drop in the energy level separation as P increases and the drop in the peak height can be attributed to a fall in the asymmetric character of the system with increase of pressure. Moreover, in absence of noise, out of the two sets of SHG peaks the left and the right ones correspond to DE02/2 and DE01 separations, respectively, and the left peaks come out to be much more intense than the right ones [64]. However, emergence of dual SHG peaks completely disappear at P ¼ 0 Kbar. SHG peak height continues to exhibit a fall with increase in pressure even in presence of additive [Fig. 6b] and multiplicative [Fig. 6c] noise. However, unlike noise-free condition, we observe blue-shifted SHG peaks in presence of additive noise [Fig. 6b]. Thus, an increase in pressure enhances the energy level separations in presence of additive noise. In presence of multiplicative noise, on the other hand, SHG peaks remain nearly unshifted with increase in pressure at low to moderate pressure regime (P < ~50 Kbar) indicating almost unchanged energy level separation [Fig. 6c]. However, with further increase in HP (P > ~50 Kbar) the SHG peaks undergo red-shift reflecting depletion of energy level separation as HP increases. Presence of noise enhances SHG peak height over noise-free situation regardless of its mode of application indicating greater asymmetric character of the system. The asymmetric nature becomes even greater in presence of additive noise. This is reflected through

Fig. 6. Plots of SHG vs hn at (i) P ¼ 0 Kbar, (ii) P ¼ 25 Kbar, (iii) P ¼ 50 Kbar, (iv) P ¼ 100 Kbar and (v) P ¼ 200 Kbar: (a) under noise-free condition, (b) in presence of additive noise and (c) in presence of multiplicative noise.

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more intense SHG peaks in presence of additive noise than its multiplicative analogue. It can be further stated that the minor SHG peaks (corresponding to DE01 separations) almost disappear in presence of additive noise and become mildly visible in presence of its multiplicative counterpart. Fig. 7a and b represent the variation of SHG and geometric factor (GF ¼ jM01 j:jM12 j:jM20 j) with pressure at hn ¼ 80 meV, (i) in absence of noise (ii) in presence of additive noise and (iii) in presence of multiplicative noise, respectively. In agreement with earlier findings, under all conditions, SHG decreases steadily with increase in HP. Moreover, additive noise causes greater enhancement of SHG value than multiplicative noise (with respect to noise-free condition) [Fig. 7a]. The GF plots [Fig. 7b] nicely conform to our earlier observations as under all conditions we find a steady fall of GF with increase in HP. The magnitude of GF maintains the same sequence in absence and in presence of noise (including its mode of application) as found earlier. Fig. 8aec depict the pattern of variations of SHG against hn at T ¼ 0 K, 200 K and 500 K, in absence of noise, in presence of additive noise and in presence of multiplicative noise, respectively. In absence of noise, as T increases, SHG peaks remain almost unshifted and peak height increases [Fig. 8a]. The observation suggests nearly unchanged energy level separation with increase in temperature. And increase in SHG peak height indicates increase in asymmetric nature of the system as temperature increases. Furthermore, in absence of noise, out of the two sets of SHG peaks the left and the right ones correspond to DE02/2 and DE01 separations, respectively, and, in contrary to what has been observed during pressure variation [cf (Fig. 6a).], the left peaks come out to be much less intense than the right ones. In presence of both additive and multiplicative noise SHG peak height decreases with increase in temperature indicating fall of asymmetric character and the peaks remain almost unshifted just like noise-free condition [Fig. 8bec]. It can also be noticed that the emergence of dual peak vanishes in presence of additive noise but becomes moderately visible in presence of its multiplicative analogue. Interestingly, unlike noise-free condition, in presence of multiplicative noise, the left peaks (corresponding to DE02/2 separations) becomes more prominent than the other set of peaks. The magnitude of SHG in presence and absence of noise maintains the same sequence as often found earlier. Fig. 9a and b represent the variation of SHG and geometric factor (GF ¼ jM01 j:jM12 j:jM20 j) with temperature at hn ¼ 80 meV, (i) in absence of noise (ii) in presence of additive noise and (iii) in presence of multiplicative noise, respectively. In compliance with previous observations, both SHG and its GF regularly increase with temperature in absence of noise [Fig. 9a] and show exactly the reverse trend in presence of [Fig. 9b] noise. Furthermore, additive noise causes greater enhancement of SHG and its GF than multiplicative noise (with respect to noise-free condition). The plot representing variation of SHG with noise strength appear quite similar to that of OR [cf (Fig. 5).] and therefore not given. 3.3. Third harmonic generation (THG) Fig. 10aec display the pattern of variations of THG against hn at P ¼ 0 Kbar, 25 Kbar, 50 Kbar, 100 Kbar and 200 Kbar in absence of noise, in presence of additive noise and in presence of multiplicative noise, respectively. In absence of noise, as HP increases, THG peaks have been found to undergo red-shift [66] and peak height decreases [66,67] [Fig. 10a]. The red-shift indicates a drop in the energy level separation as P increases and the drop in the peak height can be attributed to a fall in the extent of overlap between the concerned eigenstates with increase in pressure. Moreover, in absence of noise, out of three sets of THG peaks, the left, middle and the right ones correspond to DE02/2, DE01 and DE03/3, separations, respectively. Among them the left set of peaks are most prominent, middle peaks are somewhat less prominent and the right peaks are faintly noticeable. However, emergence of triple THG peaks completely disappear at P ¼ 0 Kbar. THG peak height continues to evince a fall with increase in pressure even in presence of additive [Fig. 10b] and multiplicative [Fig. 10c] noise. However, unlike noise-free condition, we observe blue-shifted THG peaks in presence of noise [Fig. 10bec]. Thus, an increase in pressure enlarges the energy level separations in presence of noise. Presence of noise enhances THG peak height over noise-free

Fig. 7. Plots of (a) SHG vs P and (b) jM01 j:jM12 j:jM20 j vs P: (i) under noise-free condition, (ii) in presence of additive noise and (iii) in presence of multiplicative noise.

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Fig. 8. Plots of SHG vs hn at (i) T ¼ 0 K, (ii) T ¼ 200 K and (iii) T¼500 K: (a) under noise-free condition, (b) in presence of additive noise and (c) in presence of multiplicative noise.

Fig. 9. Plots of (a) SHG vs T and (b) jM01 j:jM12 j:jM20 j vs T: (i) under noise-free condition, (ii) in presence of additive noise and (iii) in presence of multiplicative noise.

situation regardless of its mode of application indicating greater overlap between relevant wave functions. The said overlap becomes even stronger in presence of additive noise. This is revealed through more intense THG peaks in presence of additive noise than its multiplicative analogue. It can be further stated that the minor THG peaks (corresponding to DE01 and DE03/3 separations) almost disappear in presence of additive noise and become mildly visible in presence of its multiplicative relative. Fig. 11a and b represent the variation of THG and geometric factor (GF ¼ jM01 j:jM12 j:jM23 j:jM30 j) with pressure at hn ¼ 80 meV, (i) in absence of noise (ii) in presence of additive noise and (iii) in presence of multiplicative noise, respectively. In conformity with earlier findings, under all conditions, THG decreases consistently with increase in HP. Moreover, additive noise causes greater enhancement of THG value than multiplicative noise (with respect to noise-free condition) [Fig. 11a]. The GF plots [Fig. 11b] also show steady fall with increase in HP under all conditions. The magnitude of GF maintains the same sequence in absence and in presence of noise (depending on its mode of application) as found earlier.

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Fig. 10. Plots of THG vs hn at (i) P ¼ 0 Kbar, (ii) P ¼ 25 Kbar, (iii) P ¼ 50 Kbar, (iv) P¼100 Kbar and (v) P ¼ 200 Kbar: (a) under noise-free condition, (b) in presence of additive noise and (c) in presence of multiplicative noise.

Fig. 11. Plots of (a) THG vs P and (b) jM01 j:jM12 j:jM23 j:jM30 j vs P: (i) under noise-free condition, (ii) in presence of additive noise and (iii) in presence of multiplicative noise.

Fig. 12aec depict the pattern of variations of THG against hn at T ¼ 0 K, 200 K and 500 K, in absence of noise, in presence of additive noise and in presence of multiplicative noise, respectively. In absence of noise, as T increases, THG peaks undergo blue-shift [66] and peak height increases [66,67] [Fig. 12a]. The observation suggests enhanced energy level separation and more efficient overlap between pertinent wave functions with increase in temperature. In contrary to noise-free situation, both in presence of additive and multiplicative noise, THG peak height decreases with increase in temperature indicating noise-induced reduction in the extent of overlap between eigenstates involved [Fig. 12(bec)]. However, similar blue-shift of THG peaks, as observed under noise-free condition, is again obtained as temperature increases. Furthermore, whereas THG peaks corresponding to DE01 and DE03/3 separations are moderately observed in absence of noise and in presence of multiplicative noise, they are nearly absent in presence of additive noise. Fig. 13a and b represent the variation of THG and geometric factor (GF ¼ jM01 j:jM12 j:jM23 j:jM30 j) with temperature at hn ¼ 80 meV, (i) in absence of noise (ii) in presence of additive noise and (iii) in presence of multiplicative noise, respectively. In agreement with previous observations, both THG and its GF steadily increase with temperature in absence of noise

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Fig. 12. Plots of THG vs hn at (i) T ¼ 0 K, (ii) T ¼ 200 K and (iii) T ¼ 500 K: (a) under noise-free condition, (b) in presence of additive noise and (c) in presence of multiplicative noise.

Fig. 13. Plots of (a) THG vs T and (b) GF ¼ jM01 j:jM12 j:jM23 j:jM30 j vs T: (i) under noise-free condition, (ii) in presence of additive noise and (iii) in presence of multiplicative noise.

[Fig. 13a] and show exactly the reverse trend in presence of [Fig. 13b] noise. Furthermore, additive noise causes greater amplification of THG and its GF than multiplicative noise (with respect to noise-free condition). The plot depicting variation of THG with noise strength appear quite similar to that of OR [cf (Fig. 5).] and therefore we refrain from presenting the figure for the brevity of the manuscript. It needs to be mentioned that we have considered a rather simplified model of QD utilizing the effective mass approximation in order to calculate the eigenfunctions of the localized electrons. In view of calculating the OR, SHG and THG coefficients, along with determination of ground state, calculation of the energy gaps between the excited states and matrix elements between the ground and excited states are also very much crucial. This necessitates some quantitative comparison of calculated values of nonlinear coefficients with their experimental counterparts in GaAs based QD structures. Brunhes et al. have experimentally studied the spectral dependence of SHG in InAs/GaAs self-assembled QDs and found a magnitude ~107 (m/V) which is very close to our result [80]. Sauvage et al. experimentally calculated the THG of InAs/GaAs self-

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assembled QDs and found a magnitude ~1014 (m/V)2 [81]. However, in the present work we have found THG having magnitude ~1012 (m/V)2. Our work gives OR coefficients having magnitude ~105 (m/V). Baskoutas et al. have theoretically determined the OR coefficient of semiparabolic QDs and found its magnitude [55] quite close to our results. They have exploited Hartree-Fock approach realizing that the said approach has been successfully used in explaining the experimental results in QD structures. Thus, our model gives good results for OR and SHG coefficients and shows some departure from the experimental values for THG coefficients. It appears that, since determination of THG involves DE03 energy separations and related matrix elements, the said departure takes place. Thus, the present model has its limitation. It comes out that the model is fine for calculating OR and SHG whereas needs some improvement for determination of THG. At this point of discussion we would like to mention that in our calculations for the temperature dependencies of nonlinear coefficients, we have set T ¼ 500 K as upper limit. This value is, however, very close to growth temperature of GaAs crystal at which a real QD dissolves. Moreover, we have kept the value of magnetic field fixed at B ¼ 20 T, which is also close to a limit for available superconducting magnetic setups. However, these facts have not been considered while setting the parameter values. We have just attempted to investigate the influence of temperature on these NLO properties to a fairly large value and that is why the temperature has been increased up to 500 K. In fact, there are some works which involve fairly large value of temperature (T ¼ 400 K [65,66] and T ¼ 500 K [62]) while calculating the NLO properties. Similarly, there are works which involve very large value of magnetic field (up to 50 T [21]) while computing some NLO properties. The present manuscript aims at inquiring the exclusive role of HP and T on some NLO properties under the aegis of noise. The effect of variation of magnetic field strength on OR, SHG and THG has been explored in some of our previous works [82e84] where magnetic field has been varied from a very low value to B ~ 15 T. 4. Conclusion The profiles of OR, SHG and THG of impurity doped QDs under the combined influence of HP and T have been examined in presence and absence of Gaussian white noise. A variation of temperature and pressure often causes shift (blue/red) of the peaks of above NLO properties and also changes the peak heights (increase/decrease) when the doped QD is exposed to a polarized monochromatic electromagnetic field of angular frequency n. Application of noise, quite conspicuously, changes the nature of aforesaid peak shift and variation of peak height. 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