Exploring electro-optic effect of impurity doped quantum dots in presence of Gaussian white noise

Author’s Accepted Manuscript Exploring electro-optic effect of impurity doped quantum dots in presence of Gaussian white noise Suvajit Pal, Jayanta Ganguly, Surajit Saha, Manas Ghosh www.elsevier.com/locate/jpcs

PII: DOI: Reference:

S0022-3697(15)30070-6 http://dx.doi.org/10.1016/j.jpcs.2015.10.002 PCS7642

To appear in: Journal of Physical and Chemistry of Solids Received date: 14 August 2015 Revised date: 16 September 2015 Accepted date: 4 October 2015 Cite this article as: Suvajit Pal, Jayanta Ganguly, Surajit Saha and Manas Ghosh, Exploring electro-optic effect of impurity doped quantum dots in presence of Gaussian white noise, Journal of Physical and Chemistry of Solids, http://dx.doi.org/10.1016/j.jpcs.2015.10.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Exploring electro-optic effect of impurity doped quantum dots in presence of Gaussian white noise Suvajit Pala , Jayanta Gangulyb , Surajit Sahac and Manas Ghoshd a

∗†‡

Department of Chemistry, Hetampur Raj High School, Hetampur, Birbhum 731124, West Bengal, India.

b

Department of Chemistry, Brahmankhanda Basapara High School, Basapara, Birbhum 731215, West Bengal, India. c

Department of Chemistry, Bishnupur Ramananda College, Bishnupur, Bankura 722122, West Bengal, India. d

Department of Chemistry, Physical Chemistry Section, Visva Bharati University, Santiniketan, Birbhum 731 235, West Bengal, India.

∗ † ‡

e-mail address: [email protected] Phone:(+91)(3463)261526, (3463)262751-6 (Ext.467) Fax : +91 3463 262672

1

Abstract We explore the profiles of electro-optic effect (EOE) of impurity doped quantum dots (QDs) in presence and absence of noise. We have invoked Gaussian white noise in the present study. The quantum dot is doped with Gaussian impurity. Noise has been administered to the system additively and multiplicatively. A perpendicular magnetic field acts as a confinement source and a static external electric field has been applied. The EOE profiles have been followed as a function of incident photon energy when several important parameters such as electric field strength, magnetic field strength, confinement energy, dopant location, relaxation time, Al concentration, dopant potential, and noise strength possess different values. In addition, the role of mode of application of noise (additive/multiplicative) on the EOE profiles has also been scrutinized. The EOE profiles are found to be adorned with interesting observations such as shift of peak position and maximization/minimization of peak intensity. However, the presence of noise and also the pathway of its application bring about rich variety in the features of EOE profiles through some noticeable manifestations. The observations indicate possibilities of harnessing the EOE susceptibility of doped QD systems in presence of noise. Keywords: A. electronic materials, A. nanostructures, A. quantum wells, D. defects, D. optical properties

2

I.

INTRODUCTION

The fascinating role played by impurity in designing the electronic and optical properties of low-dimensional semiconductor devices has garnered widespread recognition. Investigations on impurity states have been further augmented because of their importance in physics and technological applications of quantum dots (QDs). Impurity causes substantial change in the spatial disposition of the energy levels of doped QD system and helps attainment of desirable optical transitions. A well-harnessed optical transition is an integral part of designing optoelectronic devices with tunable emission or transmission properties and ultranarrow spectral linewidths. Furthermore, the adjacency of the optical transition energy and the confinement strength (or the quantum size) allows fine-tuning of the resonance frequency. As a natural follow-up, optical properties of doped QDs and other low-dimensional systems have visualized broad research activities [1–34]. Nonlinear optical (NLO) properties of semiconductor QDs and quantum wells (QWLs) possess profound capacity to be utilized as a probe for the electronic structure of mesoscopic media. Far-infrared spectroscopy of these systems provides the opportunity to study their internal excitation [35]. Moreover, NLO properties of low-dimensional materials appear prolific for application in electronic and optoelectronic devices in the infra-red region of the electromagnetic spectrum [36, 37]. The NLO properties connected with intersubband transitions in low-dimensional systems illuminate important fundamental physics. The illumination occurs owing to the profound escalation of the nonlinear effects in these lowdimensional quantum systems over those in bulk materials exploiting quantum confinement effect [38]. The said confinement favors small energy separation between the subband levels, large value of electric dipole matrix elements and greater scope for the establishment of resonance conditions. Production of large optical nonlinearities accompanying the intersubband transitions of QD appears highly pertinent in the domain of integrated optics and optical communications [39, 40]. In what follows, these nonlinear properties have become the cornerstone of fabricating many optoelectronic devices such as far-IR laser amplifiers, photo-detectors, and high-speed electro-optical modulators [41–43]. Among the NLO properties much attention has been devoted to the second-order nonlinear processes e.g. nonlinear optical rectification (NOR), second harmonic generation (SHG), and electro-optical effect (EOE) over that of other NLO properties. This is because the

3

second-order nonlinear processes are the simplest and the lowest-order nonlinear processes with enhanced magnitudes compared with the higher-order ones if the quantum systems are enriched with significant asymmetry [44, 45]. Indeed, the second-order nonlinear susceptibility gets completely eliminated in symmetric systems as because optical transitions between the electronic states with the same parity are forbidden [37]. In general, even-order susceptibilities disappear in a symmetric quantum well structure and only a small contribution from bulk susceptibility persists. Therefore, in order to generate a strong second-order optical nonlinearity, the inversion symmetry of the quantum systems should be annihilated [37, 44, 46, 47]. In general, these asymmetries in the confinement potential can be obtained in two ways, one is by using the advanced material growing technology such as molecular beam epitaxy (MBE) and metal-organic chemical vapor deposition (MOCVD), and the other is through the introduction of an electric field to the system [44]. The second-order NLO effects actually undergoes pronounced enhancement with increase in the magnitude of the electric field. In recent times we have come across some important studies on second-order EOE by Guo et. al. [35], Zhang and Xie [48], and Yu and Guo [49]. Guo et. al. also put special emphasis on the role played by the applied magnetic field in modulating EOE [35]. The emphasis originates because of the fact that the NLO properties of a material can be modified in presence of a magnetic field. The changes in refractive index as a functions of the applied magnetic field are responsible for many electro-optic effects. These effects can be conceived as nonlinear optical mixing effects in the limit where one of the electromagnetic field components is having either zero or vanishingly small frequency. Physically, electro-optic effects result from both ionic or molecular movement and deformation of electronic cloud induced by the applied magnetic field. Thus, EOEs have been widely used as optical modulators [35]. In some of our recent works we have made thorough discussions on how noise modifies the optical properties of QD devices [50–52]. In these works the role of Gaussian white noise on the polarizabilities of doped QDs has been rigorously explored. In the current manuscript we make an exhaustive analysis of the influence of Gaussian white noise on the second-order electro-optic effect (EOE) susceptibility of doped QD. The system under investigation is a 2-d QD (GaAs) consisting of single carrier electron under parabolic confinement in the x − y plane. The QD is doped with an impurity represented by a Gaussian potential in the presence 4

of a perpendicular magnetic field which provides additional confinement. An external static electric field has also been applied to the system. Gaussian white noise has been administered to the doped QD via two different pathways i.e. additive and multiplicative [50–52]. It needs to be mentioned that the static electric and/or magnetic fields, and possible impurities are important means for fine-tuning the electronic and optical properties in semiconductor QDs. Recently, Zeng et. al. have systematically addressed the impurity-related properties under the combined effects of the static fields [53, 54]. In the present work the profiles of EOE coefficients are meticulously monitored with variations of confinement frequency (ω0 ), electric field strength (F ), dopant location (r0 ), magnetic field strength (B), impurity potential (V0 ), relaxation time (τ ), noise strength (ζ), and the mode of application of noise (additive/multiplicative). In addition, Alx Ga1−x As QD has also been explored in order to inspect the role played by Al concentration (x) on the profiles of EOE coefficients in presence and absence of noise.

II.

METHOD

The impurity doped QD Hamiltonian, exposed to external static electric field (F ) applied along x and y-directions and spatially δ-correlated Gaussian white noise (additive/multiplicative) can be written as H0 = H00 + Vimp + |e|F (x + y) + Vnoise .

(1)

Under effective mass approximation, H00 represents the impurity-free 2-d quantum dot with single carrier electron arrested by lateral parabolic confinement in the x − y plane and in presence of a perpendicular magnetic field. V (x, y) = 21 m∗ ω02 (x2 + y 2 ) is the confinement potential with ω0 as the harmonic confinement frequency. H00 is therefore given by H00 =

1 h e i2 1 ∗ 2 2 −i~∇ + A + m ω0 (x + y 2 ). 2m∗ c 2

(2)

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

~2 =− ∗ 2m



∂2 ∂2 + ∂x2 ∂y 2



1 ∂ 1 + m∗ ω02 x2 + m∗ (ω02 + ωc2 )y 2 − i~ωc y , 2 2 ∂x 5

(3)

ωc =

eB m∗

being the cyclotron frequency. Ω2 = ω02 + ωc2 can be envisaged as the effective

confinement frequency in the y-direction. Vimp is the impurity (dopant) potential formulated by a Gaussian function [50–52] viz. 2 2 Vimp = V0 e−γ [(x−x0 ) +(y−y0 ) ] . Positive values for γ and V0 represent repulsive impurity. (x0 , y0 ) is the site of dopant inclusion, V0 is the strength of the dopant potential, and γ −1 represents the spatial region over which the influence of impurity potential is disseminated. γ here acts equivalent to that of static dielectric constant (ε) of the medium and can be written as γ = kε, where k is a constant. At this point of discussion we would like to mention that Khordad and his co-workers used a new type of confinement potential for spherical QD’s called Modified Gaussian Potential, MGP [55, 56]. It needs to be mentioned here that we have neglected the dielectric mismatch effect as an approximation. However, if one desires to use the results presented herein to have an approximate idea about EOE in some other material systems (e.g. colloidal quantum dot-matrix systems where high dielectric mismatch exists), the local field effect will be very important and should be carefully taken into account [57]. The term Vnoise represents the noise contribution to the Hamiltonian H0 . It comprises of a spatially δ-correlated Gaussian white noise [f (x, y)] which assumes a Gaussian distribution (generated by Box-Muller algorithm) having strength ζ and is described by the set of conditions [50–52]: hf (x, y)i = 0,

(4)

hf (x, y)f (x0 , y 0 )i = 2ζδ ((x, y) − (x0 , y 0 )) ,

(5)

the zero average condition, and

the spatial δ-correlation condition. The Gaussian white noise can be applied to the system by means of two different modes (pathways) i.e. additive and multiplicative [50–52]. These two different modes actually modulate the extent of system-noise interaction. In case of additive white noise Vnoise becomes Vnoise = λ1 f (x, y).

(6)

And with multiplicative noise we can write Vnoise = λ2 f (x, y)(x + y). 6

(7)

The parameters λ1 and λ2 absorb in them all the neighboring influences in case of additive and multiplicative noise, respectively. At a first glance, it appears that the presence of multiplicative noise would bring about greater deviation of the optical properties from that of noise-free condition than due to the presence of additive noise. This is because of greater involvement of noise with system coordinates in case of multiplicative noise than the additive counterpart. We have invoked a variational procedure to solve the time-independent Schr¨odinger equation with trial function ψ(x, y), constructed as a superposition of the products of harmonic oscillator eigenfunctions φn (px) and φm (qy), respectively, as [50–52] X ψ(x, y) = Cn,m φn (px)φm (qy),

(8)

n,m

where Cn,m are the variational parameters and p =

q

m∗ ω0 ~

and q =

q

m∗ Ω . ~

In this context

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. The matrix elements (|k, li) corresponding to the first three terms of H0 [cf. eqn(1)] can be obtained using routine procedure. However, the matrix element for the noise term has been computed numerically using the relation p p exp [−(x2 + y 2 + x02 + y 02 )/2] √ δ( x2 + y 2 − x02 + y 02 ) = π p p ∞ 2 X Hk ( x + y 2 )Hk ( x02 + y 02 ) , × k k! 2 k=0 where Hk (x) stands for the Hermite polynomial of k th order. In order to calculate the optical EOE coefficient we envisage interaction between a polarized monochromatic electromagnetic field of angular frequency ν 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. Now, the electric field of incident optical wave can be expressed as [14]  h i  ˆ ˜ iνt + E˜ ∗ e−iνt k. E(t) = E(t)kˆ = 2E˜ cos (νt) kˆ = Ee

(9)

The electronic polarization P (t) induced by the incident field E(t) is given by (limited up to first two orders): (2) ˜ 2 2iνt ˜ iνt + ε0 χ(2) ˜2 P (t) = ε0 χ(1) + c.c. 0 E + ε0 χ2ν E e ν Ee

7

(10)

(1)

(2)

(2)

χν , χ0 , χ2ν , are known as linear, nonlinear optical rectification (NOR), and second harmonic generation (SHG) susceptibilities, respectively. ε0 being the vacuum permittivity. (2)

χ2ν = 0 for spherically symmetric systems and thus these kind of systems become unable to generate second-order optical interactions. It appears only in non-centrosymmetric structures [41, 45]. EOE is related to NOR through the relation [35] χEOE =

1 n4r ε20

(2)

χ0 .

(11)

By means of density matrix approach and iterative procedure, under two-level system approximation, the expression of EOE coefficient is given by [48] χEOE =

8e3 σs 2 ν 2 Γ4     M δ . ij n4r ε30 ~2 ij (ωij − ν)2 + Γ2 . (ωij + ν)2 + Γ2

(12)

where e is the absolute value of electron charge, σs is the carrier density, Mij = ehψi |ˆ x+ yˆ|ψj i, (i, j = 1, 2) is the matrix elements of the dipole moment, δij = |Mii − Mjj |, ψi (ψj ) are the eigenstates and ωij = (Ei − Ej )/~ is the transition frequency, Γ = 1/τ is the relaxation rate with τ as the relaxation time.

III.

RESULTS AND DISCUSSION

The calculations are performed using the following parameters: ε = 12.4, m∗ = 0.067m0 , where m0 is the free electron mass, ε0 = 8.8542×10−12 F m−1 , τ = 0.14ps, and nr = 3.2. The parameters are suitable for GaAs QDs. Regarding other parameters; if some of them are not varying with reference to a particular study their values are kept fixed as follows: ~ω0 = 2.72meV , F = 100KV /cm, B = 1.0T, ζ = 1.0 × 10−8 , V0 = 272meV , σs = 5.0 × 1024 m−3 , and r0 = 0.0nm.

A.

Role of electric field (F ):

Fig. 1a shows the pattern of variations of EOE coefficient with incident photon energy ~ν for three different values of F (25.0KV /cm, 50.0KV /cm, and 100.0KV /cm) in absence of noise. Increase in electric field strength amplifies the maximum value of EOE susceptibility and causes red-shift of EOE peaks [48]. Such enhancement of EOE coefficient originates from the increase in the asymmetry of the system induced by the electric field. And the 8

shift of the EOE coefficient maximum can be connected with the decrease of ∆E01 gap with increase in F . b) (i) (ii) (iii)

(iii)

(i)

2.0

 x 10 m/V

2.0 (ii)

(i) (ii) (iii) (ii)

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 x 10 m/V

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1.5 (i)

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(i)

M01  x 10 (m )

c) 3.0

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(iii)

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F (KV/cm)

FIG. 1: Plots of χEOE vs ~ν at three different F values: (i) F = 25.0KV /cm, (ii) F = 50.0KV /cm, and (iii) F = 100.0KV /cm. (a) Under noise-free condition, (b) in presence of additive noise, and 2 δ (c) in presence of multiplicative noise. (d) Plot of M01 01 vs F : (i) Under noise-free condition, (ii)

in presence of additive noise, and (iii) in presence of multiplicative noise.

Figs. 1b and 1c show the similar profiles in presence of additive and multiplicative noise, respectively. In presence of additive noise, an enhancement in F alters the scenario in an opposite manner. Now we find a blue-shift of EOE peaks and monotonic drop of peak heights as F increases [fig. 1b]. Presence of additive noise thus enhances the ∆E01 energy separation and lowers the extent of asymmetry as F increases. In presence of multiplicative noise the EOE peak position remains nearly unshifted within the range 25.0KV /cm ≤ F ≤ 50.0KV /cm after which it exhibits a blue-shift with further increase in 9

F up to 100KV /cm. The change in EOE peak height also lacks monotonicity and displays minimization at F = 50KV /cm [fig. 1c]. It thus appears that in presence of multiplicative noise the ∆E01 gap remains nearly unchanged up to a threshold value of electric field strength (i.e. F = Fth = 50KV /cm) beyond which the said gap increases noticeably as F increases. Moreover, multiplicative noise reduces the extent of asymmetry to a minimum at the same threshold value of F . 2 Fig. 1d displays the variations of geometric factor (M01 δ01 ) as a function of F for above

three cases. The geometric factor (GF) actually stands as a measure of the extent of asymmetry of the system and acts as an indicator of how the EOE peak intensities change with variation of different relevant parameters. In absence of noise GF increases steadily with F [fig. 1d(i)] [48], decreases persistently with F in presence of additive noise [fig. 1d(ii)], and exhibits minimization at F = 50KV /cm in presence of multiplicative noise [fig. 1d(iii)]. Thus, the GF profiles support our previous findings. A perfectly symmetric condition arises in absence of noise if F = 0 and if the dopant is on-center. Fig. 1d(i) also shows that under 2 these typical sets of conditions the factor M01 δ01 becomes zero and so also EOE. This is

obvious as the system becomes symmetric and EOE disappears.

B.

Role of magnetic field (B):

Fig. 2a delineates the profiles of EOE coefficient with incident photon energy ~ν for three different values of B (1T , 5T , and 10T ) in absence of noise. As B increases, EOE peaks undergo a blue-shift and the peak value increases [35]. The observation suggests that the effective confinement of electron and accordingly the energy eigenvalues and energy intervals are enhanced as B increases. We could also infer that an increase in the magnetic field strength promotes the asymmetric character of the system leading to enhancement of EOE peak height. This feature makes doped QDs very promising candidates as NLO materials. Figs. 2b and 2c evince the similar profiles in presence of additive and multiplicative noise, respectively. In presence of additive noise we observe a non-monotonic shift of EOE peaks as B increases. The peaks undergo a blue-shift as B increases from 1.0T to 5.0T followed by a red-shift as B increases further [fig. 2b]. Thus, additive noise enhances ∆E01 interval up to B = 5.0T but diminishes the same with further increase in magnetic field strength. The 10

EOE peak heights, however, display steady increase with increase in B indicating persistent enhancement of asymmetric character of the system. In presence of multiplicative noise, the EOE peaks depict prominent red-shift as B increases from 1.0T to 5.0T after which the peak positions become nearly static as B increases [fig. 2c]. The observation reflects that multiplicative noise reduces the ∆E01 interval between magnetic field strengths of 1.0T to 5.0T but maintains a fixed separation thereafter. The variation of peak height exhibits a non-uniform behavior in presence of multiplicative noise. Within a magnetic field strength of 1.0T to 5.0T a feeble drop in the peak height is envisaged indicating small decline of asymmetry. Interestingly, as B is increased further to 10T we observe a sharp rise of EOE peak intensity which even prominently surpasses the corresponding values in absence of noise or in presence of additive noise [fig. 2c] indicating pronounced enhancement of asymmetry. 2 δ01 ) as a function of B for above Fig. 2d displays the variations of geometric factor (M01

three cases. In absence of noise GF rises steadily with B [fig. 2d(i)]. This occurs as B not only provides extra binding to the electron, but also enhances the electron-photon coupling [35]. In presence of additive noise GF increases steadily with B up to B = 5T where we find onset of saturation [fig. 2d(ii)]. In presence of multiplicative noise GF displays minimization at B ∼ 5T followed by a sharp rise [fig. 2d(iii)]. The observations appear in good agreement with our prior findings.

C.

Role of confinement frequency (ω0 ):

Fig. 3a displays the variation of EOE susceptibility with incident photon energy ~ν for three different values of ~ω0 (1.36meV , 2.72meV , and 5.44meV ) in absence of noise. Dependence of EOE on ~ν reveals a single peak [48]. From the plot it becomes discernible that as ω0 increases, the EOE peaks undergo blue-shift with steady decrease in the peak values [48, 49]. The said shift occurs due to enhancement of the transition energy between the ground state and the first excited state as ω0 increases. Moreover, the enhanced confinement resulting from increase in ω0 causes a reduction in the overlap between |ψ0 i and |ψ1 i states thereby weakening M01 . Consequently, the EOE peak intensities are decreased. The observations thus suggest that attainment of considerable EOE coefficient requires low value of ω0 . Figs. 3b and 3c exhibit the similar profiles in presence of additive and multiplicative 11

b) (iii)

(i) (ii) (iii)

6.0

-7

(iii)

 x 10 m/V

a)

(ii)

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(i) (ii) (iii)

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(m )

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M01  x 10

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B (T)

FIG. 2: Plots of χEOE vs ~ν at three different B values: (i) B = 1.0T , (ii) B = 5.0T , and (iii) B = 10.0T . (a) Under noise-free condition, (b) in presence of additive noise, and (c) in presence 2 δ of multiplicative noise. (d) Plot of M01 01 vs B: (i) Under noise-free condition, (ii) in presence of

additive noise, and (iii) in presence of multiplicative noise.

noise, respectively. In presence of additive noise the EOE profiles looks quite similar to their noise-free relatives, at least qualitatively, both with respect to nature of peak shift and variation of peak intensity as ω0 increases [fig. 3b]. Additive noise, therefore, does not qualitatively rearrange the energy level distribution which is prevalent under noise-free condition. Application of multiplicative noise brings about large change in the scenario. Now, the EOE peaks undergo red-shift as ω0 increases and the peak intensity passes through a distinct maximum at ω0 = 2.72meV [fig. 3c]. The observations indicate a drop in the energy level separation and maximization of asymmetric character (at ω0 = 2.72meV ) induced by multiplicative noise with increase in confinement potential. 12

a)

1.8

(ii)

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EOE x 10 m/V

(iii)

(i) (ii) (iii)

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FIG. 3: Plots of χEOE vs ~ν at three different ~ω0 values: (i) ~ω0 = 1.36meV , (ii) ~ω0 = 2.72meV , and (iii) ~ω0 = 5.44meV . (a) Under noise-free condition, (b) in presence of additive noise, and (c) 2 δ in presence of multiplicative noise. (d) Plot of M01 01 vs ~ω0 : (i) Under noise-free condition, (ii)

in presence of additive noise, and (iii) in presence of multiplicative noise. 2 Fig. 3d displays the variations of geometric factor (M01 δ01 ) as a function of ~ω0 in absence

of noise and in presence of additive and multiplicative noise. In absence of noise and in presence of additive noise GF manifests nearly similar behavior and reveals a steady decline with increase in ω0 [figs. 3d(i) and (ii), respectively]. However, in presence of multiplicative noise, GF shows maximization at ~ω0 ∼ 2.7meV [fig. 3d(iii)] suggesting emergence of maximum asymmetric character. Thus, the GF profiles conform to our previous findings.

D.

Role of dopant potential (V0 ):

Fig. 4a exhibits the variations of EOE coefficient with incident photon energy ~ν for three different values of V0 (0.0meV , 136.0meV , and 272.0meV ) in absence of noise. The 13

plots deem special physical significance as they divulge how the EOE coefficient can be tuned as the QD composition is gradually changed from an impurity free state (V0 = 0.0 meV) to a state with a given impurity potential (V0 = 272.0 meV). The plot reveals that as V0 increases, the EOE peaks undergo a blue-shift and the peak heights steadily fall. The blueshift takes place because of increase in the energy interval as V0 increases owing to increased confinement. The said confinement exerted on the carrier motion reduces the extended area of the wave function. In consequence, the magnitude of the EOE peak decreases following an increase in V0 . Fig. 4b and 4c describe the analogous profiles in presence of additive and multiplicative noise, respectively. In both the cases EOE peaks remain unshifted even for different values of V0 indicating noise-induced one photon resonance enhancement. Thus, in presence of 2.0

a) 1.5

(i) (ii) (iii)

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V0 (meV)

h (meV)

FIG. 4: Plots of χEOE vs ~ν at three different V0 values: (i) V0 = 0.0meV , (ii) V0 = 136.0meV , and (iii) V0 = 272.0meV . (a) Under noise-free condition, (b) in presence of additive noise, and (c) 2 δ in presence of multiplicative noise. (d) Plot of M01 01 vs V0 : (i) Under noise-free condition, (ii) in

presence of additive noise, and (iii) in presence of multiplicative noise.

14

noise, a variation of dopant potential does not change the energy level separations. In presence of additive noise, the variation of peak intensity becomes non-uniform and exhibits minimization at V0 = 136.0meV [fig. 4b] indicating severe fall in the asymmetric character of the system. With the application of multiplicative noise, however, the peak intensity increases manifestly with increase in V0 confirming persistent enhancement of asymmetric character [fig. 4c]. 2 Fig. 4d delineates the variations of geometric factor (M01 δ01 ) as a function of V0 in

absence of noise and in presence of additive and multiplicative noise. In absence of noise GF falls steadily with increase in V0 [fig. 4d (i)]. In presence of additive noise GF displays minimization at V0 = 136.0meV [fig. 4d (ii)]. On the other hand, GF displays steady enhancement with increase in V0 in presence of multiplicative noise [fig. 4d (iii)]. The GF plots, therefore, run in agreement with earlier observations.

E.

Role of dopant location (r0 ):

Fig. 5a evinces the variation of EOE coefficient with incident photon energy ~ν for three different values of dopant locations i.e. on-center (r0 = 0.0nm), near off-center (r0 = 0.4nm), and far off-center (r0 = 0.75nm) in absence of noise. The plot reveals that as the dopant is gradually shifted from on-center to off-center locations, the EOE peaks undergo a red-shift with prominent increase in the peak height. The said red-shift takes place as an increase in r0 reduces the ∆E01 gap. The peak intensity steadily increases because of increasing asymmetry of the system associated with the dopant shift from on-center to more and more off-center locations. Fig. 5b and 5c display the similar profiles in presence of additive and multiplicative noise, respectively. In both the cases we find unshifted EOE peaks for different dopant sites proclaiming emergence of noise-induced one photon resonance enhancement. Thus, in presence of noise, a shift of dopant coordinate refrains from changing the energy interval. The EOE peak exhibits minimization at r0 = 0.4nm in presence of additive noise [fig. 5b] whereas a modest maximization of EOE peaks are found at the same r0 in presence of the multiplicative analogue [fig. 5c]. Thus, a change in the mode of application of noise (from additive to multiplicative) modifies the asymmetric character of the system from a minimum value to a maximum value in case of a near off-center impurity. 15

a)

b) 1.5

(i) (ii) (iii)

(iii)

(i) (ii) (iii)

(i)

 x 10 m/V

(ii)

-7

1.5

-7

 x 10 m/V

2.0

(i)

1.0

1.3 (iii)

1.0 0.8 0.5 (ii)

0.5 0.3 0.0

0.0 75

150

225

300

375

450

75

150

h (meV) d) 0.12

-27

1.0 (iii)

450

0.09 (ii)

0.06

2

0.8

375

0.5

(i) (ii) (iii)

(iii)

3

(i)

(m )

(i) (ii) (iii)

(ii)

1.3

300

h (meV)

M01  x 10

-7

 x 10 m/V

c) 1.5

225

(i)

0.03

0.3 0.0 75

150

225

300

375

0.0

450

0.2

0.4

0.6

0.8

r0 (nm)

h (meV)

FIG. 5: Plots of χEOE vs ~ν at three different r0 values: (i) r0 = 0.0nm, (ii) r0 = 0.4nm, and (iii) r0 = 0.75nm. (a) Under noise-free condition, (b) in presence of additive noise, and (c) in presence 2 δ of multiplicative noise. (d) Plot of M01 01 vs r0 : (i) Under noise-free condition, (ii) in presence of

additive noise, and (iii) in presence of multiplicative noise. 2 Fig. 5d demonstrates the variations of geometric factor (M01 δ01 ) as a function of r0 in

absence of noise and in presence of additive and multiplicative noise. In absence of noise GF rises steadily with increase in r0 [fig. 5d (i)]. In presence of additive and multiplicative noise GF exhibits minimization and a feeble maximization at r0 = 0.4nm [fig. 5d(ii-iii)], respectively. The GF plots therefore support the earlier observations.

F.

Role of relaxation time (τ ):

Fig. 6a exhibits how EOE susceptibilities vary with incident photon energy ~ν for three different values of τ (0.14ps, 0.20ps, and 0.26ps) in absence of noise. The plot reveals that EOE peak value increases with increase in τ [49]. It needs to be mentioned that τ is related 16

not only to the quantum dot material and confining potential, but also to other factors, such as temperature and boundary conditions. a) 1.75 (iii)

1.25 (ii)

-7

1.00 0.75

1.8 1.5

 x 10 m/V

-7

 x 10 m/V

1.50

b)

(i) (ii) (iii)

(i)

0.50

(i) (ii) (iii)

(i)

1.3 (ii)

1.0 0.8

(iii)

0.5 0.3

0.25

0.0

0.00 75

150

225

300

375

75

450

150

-7

375

450

1.8 1.5

 x 10 m/V

300

h (meV)

h (meV) c)

225

(i) (ii) (iii)

(i)

1.3 (ii)

1.0 0.8

(iii)

0.5 0.3 0.0 75

150

225

300

375

450

h (meV)

FIG. 6: Plots of χEOE vs ~ν at three different τ values: (i) τ = 0.14ps, (ii) τ = 0.20ps, and (iii) τ = 0.26ps. (a) Under noise-free condition, (b) in presence of additive noise, and (c) in presence of multiplicative noise.

Fig. 6b and 6c demonstrate the similar profiles in presence of additive and multiplicative noise, respectively. Application of noise completely reverses the nature of τ -dependence of χEOE . In both the cases the EOE peak intensity has been found to diminish with increase in τ . These observations bear important experimental significance.

G.

Role of Al concentration (x):

In order to inspect the role played by aluminium concentration we now consider Alx Ga1−x As QD with effective mass given as m∗ = (0.067 + 0.083x) m0 , where x is the Al concentration [14]. Fig. 7a exhibits how EOE coefficient vary with incident photon en17

ergy ~ν for three different values of x (0.0, 0.2, and 0.5) in absence of noise. From the plot 1.8

 x 10 m/V

1.5

b)

(i) (ii) (iii)

(i)

1.3

-7

-7

1.0 0.8

1.8 1.5

 x 10 m/V

a)

(iii)

0.5

(i) (ii) (iii)

(i)

1.3 1.0 0.8 (ii)

0.5 (iii)

0.3

0.3 (ii)

0.0

0.0 75

150

225

300

375

450

75

150

225

h (meV) d) 0.12

1.5

450

(i) (ii) (iii)

3

(i)

(m )

(i) (ii) (iii)

0.10

-27

M01  x 10

1.0

-7

 x 10 m/V

375

h (meV)

c) 1.3

300

0.08

(iii)

0.06

(ii)

2

0.8 (ii)

0.5 (iii)

0.3

0.04 0.02

(i)

0.0 75

150

225

300

375

0.0

450

0.1

0.2

0.3

0.4

0.5

x

h (meV)

FIG. 7: Plots of χEOE vs ~ν at three different x: (i) x = 0.0, (ii) x = 0.2, and (iii) x = 0.5. (a) Under noise-free condition, (b) in presence of additive noise, and (c) in presence of multiplicative 2 δ noise. (d) Plot of M01 01 vs x: (i) Under noise-free condition, (ii) in presence of additive noise,

and (iii) in presence of multiplicative noise.

it becomes conspicuous that as x increases the EOE peaks undergo a weak red-shift indicating minor reduction in the energy interval. Moreover, the peak height undergoes prominent minimization at some intermediate x = 0.2 reflecting minimization of asymmetric character. Fig. 7b and 7c show the similar profiles in presence of additive and multiplicative noise, respectively. In both the cases we find qualitatively similar behavior of EOE peaks whence they undergo red-shift as x increases. Thus, application of noise via both the pathways also causes a decline of energy intervals. However, it is the behavior of peak height which shows departure from noise-free case. Now, the peak height persistently decreases with increase in x and indicates steady fall of asymmetric character.

18

2 δ01 ) as a function of x in absence of Fig. 7d depicts the variations of geometric factor (M01

noise and in presence of additive and multiplicative noise. In agreement with our erstwhile findings, the GF plot exhibits minimization at x ∼ 0.25 in absence of noise [fig. 7d(i)], and declines steadily with x reflecting monotonic fall of asymmetric nature in presence of noise applied through both the pathways [fig. 7d(ii) & (iii)].

H.

Role of noise strength (ζ):

Finally we explore the EOE profiles in view of the most significant parameter of the present study i.e. the noise strength. Figs. 8a and 8b exhibit the variations of EOE a)

2.5

(ii)

1.5 1.0

(i)

0.5 0.0

(i) (ii) (iii)

(ii)

1.5

-7

-7

(i)

 x 10 m/V

2.0

 x 10 m/V

b)

(i) (ii) (iii)

(iii)

1.0 (iii)

0.5

0.0 75

150

225

300

375

450

75

150

h (meV)

225

300

375

450

h (meV)

(i) (ii)

(i)

0.2

2

M01  x 10

-27

3

(m )

c) 0.3

0.1

(ii)

0

20

40

60

 x 10

80

100

8

FIG. 8: Plots of χEOE vs ~ν at three different ζ: (i) ζ = 1.0 × 10−10 , (ii) ζ = 1.0 × 10−8 , and (iii) ζ = 1.0 × 10−6 . (a) in presence of additive noise, (b) in presence of multiplicative noise. (c) Plot 2 δ of M01 01 vs ζ: (i) in presence of additive noise and (ii) in presence of multiplicative noise.

coefficient with incident photon energy ~ν for three different values of ζ (1.0 × 10−10 , 1.0 × 10−8 , and 1.0 × 10−6 ) with additive and multiplicative noise, respectively. In both the 19

cases the EOE peaks emerge at the same position even for different values of ζ indicating noise-induced one photon resonance enhancement. The observation further proclaims that a variation of noise strength becomes unable to alter the energy intervals. However, the variation of peak intensity with ζ depends on the mode of application of noise. In presence of additive noise, the EOE peak height increases persistently with increase in ζ [fig. 8a]. On the other hand, the situation becomes exactly opposite in presence of multiplicative noise [fig. 8b]. Thus, the change in the extent of asymmetry occurs in contrasting ways if we change the pathway of application of noise. 2 Fig. 8c depicts the variations of geometric factor (M01 δ01 ) as a function of ζ in presence

of additive and multiplicative noise. In case of additive noise GF increases consistently with ζ [fig. 8c(i)], whereas, exactly opposite behavior is noticed in case of multiplicative analogue [fig. 8c(ii)]. The outcomes nicely corroborate our earlier findings.

IV.

CONCLUSION

The electro-optic effect (EOE) of impurity doped QD has been investigated in minute detail in presence and absence of noise. The EOE profiles have been analysed as a function of incident photon energy when several pertinent parameters such as electric field strength, magnetic field strength, confinement energy, dopant location, relaxation time, aluminium concentration, dopant potential, and noise strength assume different values. Furthermore, the role of mode of application of noise (additive/multiplicative) on the EOE profiles has also been rigorously analyzed. In most of the cases the EOE profiles are enriched with shift of peak position and variation of peak intensity as several parameters are varied over a range. Application of noise affects the EOE profiles with great subtlety and they display noticeable deviation from their behavior under noise-free condition. Quite often, in presence of noise, the variation of peak intensities lacks monotonicity and exhibit maximization and minimization. Moreover, presence of noise sometimes promotes emergence of one photon resonance enhancement. In addition, a change in the mode of application of noise (additive/multiplicative) frequently influences the characteristics of EOE profiles in diverse and often conspicuously contrasting ways. The findings illuminate promising scope of tailoring the EOE coefficient of doped QD systems by exploiting noise.

20

V.

ACKNOWLEDGEMENTS

The authors S. P., J. G., S. S. 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.

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25

Figure Captions Fig. 1: Plots of χEOE vs ~ν at three different F values: (i) F = 25.0KV /cm, (ii) F = 50.0KV /cm, and (iii) F = 100.0KV /cm. (a) Under noise-free condition, (b) in presence 2 of additive noise, and (c) in presence of multiplicative noise. (d) Plot of M01 δ01 vs F :

(i) Under noise-free condition, (ii) in presence of additive noise, and (iii) in presence of multiplicative noise. Fig. 2: Plots of χEOE vs ~ν at three different B values: (i) B = 1.0T , (ii) B = 5.0T , and (iii) B = 10.0T . (a) Under noise-free condition, (b) in presence of additive noise, and (c) in 2 δ01 vs B: (i) Under noise-free condition, (ii) presence of multiplicative noise. (d) Plot of M01

in presence of additive noise, and (iii) in presence of multiplicative noise. Fig. 3: Plots of χEOE vs ~ν at three different ~ω0 values: (i) ~ω0 = 1.36meV , (ii) ~ω0 = 2.72meV , and (iii) ~ω0 = 5.44meV . (a) Under noise-free condition, (b) in presence 2 of additive noise, and (c) in presence of multiplicative noise. (d) Plot of M01 δ01 vs ~ω0 :

(i) Under noise-free condition, (ii) in presence of additive noise, and (iii) in presence of multiplicative noise. Fig. 4: Plots of χEOE vs ~ν at three different V0 values: (i) V0 = 0.0meV , (ii) V0 = 136.0meV , and (iii) V0 = 272.0meV . (a) Under noise-free condition, (b) in presence of 2 additive noise, and (c) in presence of multiplicative noise. (d) Plot of M01 δ01 vs V0 : (i) Under

noise-free condition, (ii) in presence of additive noise, and (iii) in presence of multiplicative noise. Fig. 5: Plots of χEOE vs ~ν at three different r0 values: (i) r0 = 0.0nm, (ii) r0 = 0.4nm, and (iii) r0 = 0.75nm. (a) Under noise-free condition, (b) in presence of additive noise, 2 δ01 vs r0 : (i) Under noise-free and (c) in presence of multiplicative noise. (d) Plot of M01

condition, (ii) in presence of additive noise, and (iii) in presence of multiplicative noise. Fig. 6: Plots of χEOE vs ~ν at three different τ values: (i) τ = 0.14ps, (ii) τ = 0.20ps, and (iii) τ = 0.26ps. (a) Under noise-free condition, (b) in presence of additive noise, and (c) in presence of multiplicative noise. Fig.

7: Plots of χEOE vs ~ν at three different x: (i) x = 0.0, (ii) x = 0.2, and

(iii) x = 0.5. (a) Under noise-free condition, (b) in presence of additive noise, and (c) in 2 presence of multiplicative noise. (d) Plot of M01 δ01 vs x: (i) Under noise-free condition, (ii)

in presence of additive noise, and (iii) in presence of multiplicative noise.

26

Fig. 8: Plots of χEOE vs ~ν at three different ζ: (i) ζ = 1.0 × 10−10 , (ii) ζ = 1.0 × 10−8 , and (iii) ζ = 1.0 × 10−6 . (a) in presence of additive noise, (b) in presence of multiplicative 2 noise. (c) Plot of M01 δ01 vs ζ: (i) in presence of additive noise and (ii) in presence of

multiplicative noise.

27

-7

 x 10 m/V

a) 2.5

(i) (ii) (iii)

(iii)

2.0 (ii)

1.5 (i)

1.0 0.5 0.0 100

200

300

h (meV)

400

500

b) (i)

(ii)

1.5

(iii)

-7

 x 10 m/V

2.0

(i) (ii) (iii)

1.0

0.5

0.0 0

100

200

h (meV)

300

400

500

c) 3.0

(i) (ii) (iii)

(i)

2.0

-7

 x 10 m/V

2.5

1.5 (iii)

1.0 (ii)

0.5 0.0 0

100

200

h (meV)

300

400

500

d)

(i) (ii) (iii)

3.0 (i)

2.0

2

-27

3

M01  x 10 (m )

4.0

(iii)

1.0 (ii)

0.0 0

20

40

60

F (KV/cm)

80

100

a)

(iii)

(i) (ii) (iii)

-7

 x 10 m/V

6.0 (ii)

4.0 (i)

2.0

0.0 100

200

h (meV)

300

400

500

b) (i) (ii) (iii)

2.0

1.5

-7

 x 10 m/V

(iii)

(ii)

1.0 (i)

0.5

0.0 100

200

300

h (meV)

400

500

c) (iii)

 x 10 m/V

6.0

-7

(i) (ii) (iii)

4.0

2.0 (i) (ii)

0.0 100

200

300

h (meV)

400

500

(i) (ii) (iii)

1.5

(iii)

(ii)

1.0

2

M01  x 10

-27

3

(m )

d)

0.5

(i)

0.0 2

4

6

8

B (T)

10

12

14

16

a)

1.8 (i) (ii) (iii)

(i)

1.5

-7

EOE x 10 m/V

(ii)

1.3 1.0

(iii)

0.8 0.5 0.3 0.0 0

272

544

h (meV)

817

1089

(i) (ii) (iii)

(i)

(ii)

1.5

-7

EOE x 10 m/V

b) 2.0

1.0 (iii)

0.5

0.0 0

272

544

h (meV)

817

1089

c) 1.5

-7

 x 10 m/V

1.3

(i) (ii) (iii)

(ii)

1.0 0.8 (iii)

0.5

(i)

0.3 0.0 0

272

544

h (meV)

817

1089

d)

1.3 (ii)

0.8 (i)

(iii)

0.5

2

M01  x 10

-27

3

(m )

1.0

(i) (ii) (iii)

0.3 0.0 0

1

2

3

h (meV)

4

5

a) 1.5

(i) (ii) (iii)

(i) (ii)

(iii)

1.0

-7

 x 10 m/V

1.3

0.8 0.5 0.3 0.0 100

200

300

h (meV)

400

500

2.0 (i) (ii) (iii)

(i)

1.5 (iii)

-7

 x 10 m/V

b)

1.0 (ii)

0.5

0.0 75

150

225

300

h (meV)

375

450

1.5

(i) (ii) (iii)

c) (iii)

-7

 x 10 m/V

1.3 1.0 0.8

(ii)

0.5 0.3

(i)

0.0 75

150

225

300

h (meV)

375

450

(i) (ii) (iii)

0.10

(iii)

0.08

(ii)

0.06 (i)

2

M01  x 10

-27

3

(m )

d) 0.12

0.04 0.02

0

50

100

150

V0 (meV)

200

250

300

a) (iii) (ii)

1.5

-7

 x 10 m/V

2.0

(i) (ii) (iii)

(i)

1.0

0.5

0.0 75

150

225

300

h (meV)

375

450

b) 1.5

(i) (ii) (iii)

-7

 x 10 m/V

(i)

1.3 (iii)

1.0 0.8 0.5 (ii)

0.3 0.0 75

150

225

300

h (meV)

375

450

-7

 x 10 m/V

c) 1.5

(i) (ii) (iii)

(ii) (i)

1.3 1.0

(iii)

0.8 0.5 0.3 0.0 75

150

225

300

h (meV)

375

450

d) 0.12

(m )

(iii)

3

0.09 (ii)

0.06

2

M01  x 10

-27

(i) (ii) (iii)

(i)

0.03

0.0

0.2

0.4

r0 (nm)

0.6

0.8

a) 1.75

-7

 x 10 m/V

1.50

(i) (ii) (iii)

(iii)

1.25 (ii)

1.00 0.75

(i)

0.50 0.25 0.00 75

150

225

300

h (meV)

375

450

b)

1.8

-7

 x 10 m/V

1.5

(i) (ii) (iii)

(i)

1.3 (ii)

1.0 0.8

(iii)

0.5 0.3 0.0 75

150

225

300

h (meV)

375

450

c)

1.8

-7

 x 10 m/V

1.5

(i) (ii) (iii)

(i)

1.3 (ii)

1.0 0.8

(iii)

0.5 0.3 0.0 75

150

225

300

h (meV)

375

450

a)

1.8

-7

 x 10 m/V

1.5

(i) (ii) (iii)

(i)

1.3 1.0 0.8

(iii)

0.5 0.3 (ii)

0.0 75

150

225

300

h (meV)

375

450

b)

1.8

-7

 x 10 m/V

1.5

(i) (ii) (iii)

(i)

1.3 1.0 0.8 (ii)

0.5 (iii)

0.3 0.0 75

150

225

300

h (meV)

375

450

1.5 (i) (ii) (iii)

c) (i)

1.0

-7

 x 10 m/V

1.3

0.8 (ii)

0.5 (iii)

0.3 0.0 75

150

225

300

h (meV)

375

450

a)

2.5

(i) (ii) (iii)

(iii)

-7

 x 10 m/V

2.0 (ii)

1.5 1.0

(i)

0.5 0.0 75

150

225

300

h (meV)

375

450

b) (i) (ii)

 x 10 m/V

1.5

-7

(i) (ii) (iii)

1.0 (iii)

0.5

0.0 75

150

225

300

h (meV)

375

450

0.3 (i) (ii)

(i)

0.2

2

M01  x 10

-27

3

(m )

c)

0.1

(ii)

0

20

40

60

 x 10

80 8

100