Role of anisotropy, spatially-varying effective mass, and dielectric constant on self-polarization effect of doped quantum dots in presence of noise

Accepted Manuscript Role of anisotropy, spatially-varying effective mass, and dielectric constant on selfpolarization effect of doped quantum dots in presence of noise Anuja Ghosh, Manas Ghosh PII:

S0749-6036(17)30426-3

DOI:

10.1016/j.spmi.2017.02.053

Reference:

YSPMI 4865

To appear in:

Superlattices and Microstructures

Received Date: 20 February 2017 Accepted Date: 28 February 2017

Please cite this article as: A. Ghosh, M. Ghosh, Role of anisotropy, spatially-varying effective mass, and dielectric constant on self-polarization effect of doped quantum dots in presence of noise, Superlattices and Microstructures (2017), doi: 10.1016/j.spmi.2017.02.053. 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 proof before it is published in its final 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.

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60.0 noise-free

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additive noise

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Role of anisotropy, spatially-varying effective mass, and dielectric constant on self-polarization effect of doped quantum dots in presence of noise Anuja Ghosh and Manas Ghosh

∗ †‡

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

Abstract

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The profiles of self-polarization effect (SPE) of impurity doped GaAs quantum dot (QD) have been investigated under the governance of variable effective mass, variable dielectric constant and

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anisotropy of the system. Presence of noise has also been considered to inspect how it interplays with above parameters in modulating SPE. Noise term possesses a Gaussian white character and it has been introduced to the system via two different pathways; additive and multiplicative. The spatially-varying effective mass and spatially-varying dielectric constant mainly affect SPE quantitatively in comparison with the fixed ones. A changing anisotropy of the system also affects

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SPE. However, the extent to which SPE is being affected evidently depends on presence/absence of noise and also on the pathway through which noise has been applied. The findings of the study reveal authentic routes to tailor the SPE of doped QD system through the interplay between noise, anisotropy and spatially-varying effective mass and dielectric constant of the system. quantum dot; position-dependent effective mass; position-dependent dielectric constant;

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Keywords:

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anisotropy; self-polarization effect; Gaussian white noise

∗ † ‡

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

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ACCEPTED MANUSCRIPT I.

INTRODUCTION

Low-dimensional semiconductor systems (LDSS) such as quantum wells (QWLs), quantum wires (QWRs) and quantum dots (QDs) are subjected to tremendous quantum confinement much greater than their bulk relatives. Such strong confinement becomes responsible for the occurrence of partial or complete quantization of electrons in LDSS to a discrete

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spectrum of energy levels. As a result of such stringent confinement LDSS are often exploited as indispensable candidates for the manufacture of high-performance optoelectronic devices with improved optical and electrical characteristics. Apart from its technological importance, study of LDSS also serves the academic need of realizing and appreciating many

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fundamental physical concepts. Such splendid properties of LDSS have accentuated a large number of studies on the nonlinear optical (NLO) properties of LDSS.

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Presence of impurity in LDSS is a common fact. These impurities heavily perturb the probability distribution of electrons and quite obviously the spatial distribution of energy levels. The quenched spatial freedom in LDSS causes a somewhat lengthy residence of electrons close to impurity. Such close residence gives rise to significant mutual interaction between them revealed through amplified binding energy (BE) of the system. Thus, addition of impurity can cause perceptible changes in the thermal, transport, optical and electrical

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properties of LDSS, specifically at low temperatures. These interesting features have induced a large number of studies on various properties of LDSS containing impurity [1–25]. Applied electric field marks its signature on LDSS through polarization of carrier distri-

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bution, energy shift of quantum states and modification of effective well width. Following the change in the well width, effective confinement of the system also changes giving rise to changed energy spectrum. Such modified energy spectrum of the confined states of the

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carriers perturbs the electronic and optical properties of LDSS with ample scope of designing novel optoelectronic devices with desired NLO properties. Position-dependent effective mass (PDEM) of LDSS is a topic that bears enough importance. The importance arises out of the fact that a spatially-varying effective mass can discernibly change the BE of the doped system (with respect to the fixed effective mass) and thus remarkably influences the NLO properties. We, therefore, can find a good number of studies that deal with PDEM in LDSS [26–35]. Similar to PDEM, position-dependent dielectric screening function (PDDSF) also carries

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ACCEPTED MANUSCRIPT substantial importance in harnessing the energy levels and consequently the NLO properties of LDSS. PDDSF plays prominent role in changing the BE of LDSS, in the manufacture of various optoelectronic devices, in the exploration of magnetic properties of LDSS and in depicting various screened interactions of LDSS. Thus, we can envisage a few inspections on LDSS that involve PDDSF [26, 29, 36–39].

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Anisotropy in LDSS is an extremely common phenomenon. It, therefore, appears logical to enquire how anisotropy affects various properties of LDSS. Experimentally, anisotropy can be understood by chemically regulating the nanostructure aspect ratio. Thus, study of anisotropic systems has gathered enough reputation in view of construction of novel

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optoelectronic devices. As a result, we can find a few works concerned with the influence of geometrical anisotropy on the NLO properties of LDSS [40–44].

Exploration of polarizability and self-polarization effect (SPE) of LDSS is undoubtedly

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important as it elucidates the carrier dynamics and NLO properties of them. Thus, there exists plenty of studies related to polarizability [45–61] and SPE [62–67] of LDSS. SPE is defined as the influence of well-potential on impurity. In this case the electronic wave function is perturbed by a combination of impurity and well potential. The well potential (confinement potential) shifts the electronic probability distribution with respect to impurity

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location. The resulting polarization is called SPE.

Introduction of noise to LDSS can noticeably alter its performance. Noise may originate externally, or it may be intrinsic, emerging from the changes in the structure of QD lattice in the vicinity of impurity. It is therefore quite relevant to explore how presence of noise

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perturbs various properties of impurity doped LDSS. In the present work we examine the SPE of GaAs QD doped with impurity in presence of noise with special focus on PDEM,

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PDDSF and geometrical anisotropy. In view of a comprehensive enquiry the results of PDEM and PDDSF are compared with those of fixed effective mass (FEM) and static dielectric constant (SDC), respectively. The comparison highlights the exclusive role played by the spatial variation of effective mass and dielectric constant of the system in modulating SPE. Despite a rigorous literature survey we could not find such type of study in presence of noise. Present study considers a 2-d QD (GaAs) carrying a single electron in presence of a static electric field. The confinement is parabolic in the x − y plane. An orthogonal magnetic field is present too as an additional confinement. Impurity, modeled by a Gaussian potential, has been doped into the QD system. Gaussian white noise has been externally applied to 3

ACCEPTED MANUSCRIPT the system which initiates substantial disorder. There are two different pathways (modes) through which such introduction of disorder can be achieved. These two modes are additive and multiplicative which differ from one another by the extent of interaction with the system. The investigation elucidates how delicately noise (which evidently depends on its mode of application) modulates the SPE of doped QD when the QD is anisotropic and when the

II.

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effective mass and dielectric constant of the system are spatially-varying.

METHOD

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The system Hamiltonian with impurity (H0 ) contains four terms and reads as H0 = H00 + Vimp + |e|F (x + y) + Vnoise .

(1)

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In the above expression, the first, second, third and the fourth terms on the right hand side of the equation stand for impurity-free system containing single carrier electron, the impurity potential, the externally applied electric field having field strength F and noise contribution, respectively. The static electric field has been applied along x and y-directions. |e| is the absolute value of electron charge. The noise term characterizes zero mean and spatially δ-correlated Gaussian white noise (additive/multiplicative).

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In view of a lateral parabolic confinement in the x − y plane and presence of a perpendicular magnetic field, H00 , under effective mass approximation, can be written as e i2 1 ∗ 2 2 1 h −i~∇ + A + m ω0 (x + y 2 ), H00 = 2m∗ c 2

(2)

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where m∗ is the effective mass of the electron in QD and ω0 is the harmonic confinement frequency. A is the vector potential which in Landau gauge becomes A = (By, 0, 0), where

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B is the magnetic field strength. In this gauge H00 can be further written as  2  ~2 ∂ ∂2 1 1 ∂ 0 H0 = − ∗ + 2 + m∗ ω02 x2 + m∗ Ω2 y 2 − i~ωc y , (3) 2 2m ∂x ∂y 2 2 ∂x p where the quantity Ω(= ω02 + ωc2 ) represents the effective confinement frequency in the y-direction, ωc (=

eB ) m∗ c

being the cyclotron frequency. Following Xie, the ratio η =

Ω ω0

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be defined as the anisotropy parameter [40, 41]. Vimp represents the Gaussian impurity (dopant) potential and can be expressed as 2 2 Vimp = V0 e−γ [(x−x0 ) +(y−y0 ) ] . The relevant parameters belonging to this dopant potential are (x0 , y0 ), V0 and γ −1/2 . They represent the site of dopant incorporation, magnitude 4

ACCEPTED MANUSCRIPT of the dopant potential, and the spatial region over which the impurity potential is dispersed, respectively. γ can be given by γ = kε, where k is a constant and ε is the static dielectric constant of the medium. The noise term of eqn.(1) can be generated by Box-Muller algorithm with necessary characteristics as mentioned before. The interaction of noise with system can be tuned in two distinct modes (pathways); additive and multiplicative. These

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two modes actually signify varied extents of system-noise interaction. The time-independent Schr¨odinger equation has been solved numerically by diagonalizing the Hamiltonian matrix (H0 ). The said matrix has been generated by the direct product basis of the harmonic oscillator eigenfunctions. The necessary convergence test has been performed and finally we

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have obtained the energy levels and wave functions.

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The dopant location-dependent effective mass i.e. PDEM; m∗ (r0 ), is given by [26, 29]   1 1 1 (4) = ∗ + 1 − ∗ exp(−βr0 ). m∗ (r0 ) m m p In the above expression r0 = x20 + y02 is the dopant location and β is a constant having value 0.01 a.u. The form of PDEM indicates that the dopant resides in the close proximity of dot confinement center as r0 → 0 i.e. for on-center dopants while m∗ (r0 ) reflects prominent contribution as r0 → ∞ i.e. for far off-center dopants.

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The Hermanson’s impurity position-dependent dielectric constant/dielectric screening function (PDDSF) is given by [26, 29, 36–39]   1 1 1 = + 1− exp(−r0 /α), ε(r0 ) ε ε

(5)

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where ε is the SDC and α = 1.1 a.u. is the screening constant. The concept of SDC appears meaningful only for distances substantially away from the impurity center. Above form of

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PDDSF suggests that ε(r0 ) → 1 as r0 → 0 i.e. for on-center dopants and tends to ε as r0 → ∞ i.e. for far off-center dopants. Such large r0 can be conceived as the screening radius [36].

SPE of impurity doped QD can be given by P = −hψ| (x − x0 ) |ψi + hψ 0 | (x − x0 ) |ψ 0 i − hψ| (y − y0 ) |ψi + hψ 0 | (y − y0 ) |ψ 0 i, e

(6)

where ψ is the wave function describing the system and ψ 0 is the wave function in absence of confinement effects.

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ACCEPTED MANUSCRIPT III.

RESULTS AND DISCUSSION

The calculations are performed with static dielectric constant (SDC) ε = 12.4 and the fixed effective mass (FEM) m∗ = 0.067m0 (m0 is the free electron mass). The values of the other parameters are: ~ω0 = 250.0 meV, F = 100 KV/cm, B = 20.0 T, V0 = 280.0 meV

A.

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and ζ = 1.0 × 10−2 . The parameters are suitable for GaAs QDs.

Role of position-dependent effective mass (PDEM):

Fig. 1 depicts the profile of SPE as a function of dopant location (r0 ) without presence

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of noise [fig. 1(i) and 1(iv), for FEM and PDEM, respectively], in presence of additive noise [fig. 1(ii) and (v) for FEM and PDEM, respectively] and in presence of multiplicative noise

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[fig. 1(iii) and (vi) for FEM and PDEM, respectively]. The ’solid lines’ and the ’dashed lines’ stand for PDEM and FEM. With FEM, under all conditions, SPE increases steadily with 70.0

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FIG. 1: Plot of SPE vs r0 : (i) using FEM without noise; (ii) using FEM and in presence of additive

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noise; (iii) using FEM and in presence of multiplicative noise; (iv) using PDEM without noise; (v) using PDEM and in presence of additive noise; (vi) using PDEM and in presence of multiplicative noise.

r0 [63] up to r0 ∼ 4.0 nm and culminates in saturation thereafter. The observed behavior can be realized since a gradual shift of dopant from on-center to off-center locations reduces the confinement and enhances the spatial elongation of wave function. After the typical off-center location of r0 ∼ 4.0 nm the elongation ceases and SPE stabilizes. Introduction of noise causes noticeable reduction in the SPE value from that of noise-free condition. In 6

ACCEPTED MANUSCRIPT other words, incorporation of noise induces substantial quenching of spatial elongation of wave function. Moreover, the said quenching comes out to be more with multiplicative noise than its additive analogue. Thus, the mode of application of noise also appears important in harnessing the SPE of the system. This is because of the fact that, multiplicative noise - owing to its mode of application - maintains closer connection with the system than its

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additive counterpart leading to greater quenching of spatial freedom of wave function. Use of PDEM leaves the SPE profiles nearly unchanged from that of FEM both in presence and absence of noise. A quantitative difference, however, in the magnitude of SPE can be observed using FEM and PDEM. Regardless of presence of noise, SPE registers a lower value

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using PDEM than using FEM suggesting a drop in the spatial stretch of wave function in case of PDEM. Diminish in the SPE value in presence of noise (from noise-free value) and

as what have been found with FEM.

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the effect of mode of application of noise on SPE value appear to be similar using PDEM

Role of position-dependent dielectric screening function (PDDSF):

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FIG. 2: Plot of SPE vs r0 : (i) using SDC without noise; (ii) using SDC and in presence of additive noise; (iii) using SDC and in presence of multiplicative noise; (iv) using PDDSF without noise; (v) using PDDSF and in presence of additive noise; (vi) using PDDSF and in presence of multiplicative noise.

Fig. 2 depicts the profile of SPE as a function of dopant location (r0 ) without presence of noise [fig. 2(i) and 2(iv), for SDC and PDDSF, respectively], in presence of additive noise [fig. 2(ii) and (v) for SDC and PDDSF, respectively] and in presence of multiplicative noise 7

ACCEPTED MANUSCRIPT [fig. 2(iii) and (vi) for SDC and PDDSF, respectively]. The ’solid lines’ and the ’dashed lines’ represent PDDSF and SDC. It goes without saying that the plots corresponding to SDC are exactly the same as those for FEM. Thus, the features of SPE with SDC are not discussed to avoid repetition. Analogous to what has been found in the previous section, use of PDDSF keeps the SPE

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profiles nearly unchanged from that of SDC under all conditions. However, using PDDSF, the onset of saturation in the SPE values becomes delayed as it occurs at r0 ∼ 7.5 nm (similar saturation occurs at r0 ∼ 4.0 nm using SDC). Thus, using PDDSF, the spatial dispersion of wave function ceases to enhance any further at larger dopant location in comparison with

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SDC. Furthermore, irrespective of presence of noise, SPE exhibits lower values using PDDSF than SDC reflecting a drop in the spatial extension of wave function in case of the former. The role of application of noise (including its mode of application) are exactly the same as

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what has been observed in case of PDEM.

Role of anisotropy (η):

Fig. 3 depicts the profile of SPE as a function of anisotropy parameter (η) without presence of noise [fig. 3(i)] and when noise is introduced through additive [fig. 3(ii)] and

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multiplicative [fig. 3a(iii)] mode, respectively. SPE profiles exhibit persistently decreasing 60.0

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FIG. 3: Plot of SPE vs η: (i) without noise; (ii) in presence of additive noise and (iii) in presence of multiplicative noise.

trend with increase in η under all conditions. However, in absence of noise and in presence

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ACCEPTED MANUSCRIPT of additive noise the decrease is mild. And the decrease becomes severe in presence of multiplicative noise. The findings indicate that a gradual increase in anisotropy of the system sluggishly reduces the spatial elongation of the wave function in absence of noise and in presence of additive noise. The said reduction, however, becomes prominently noticeable in presence of multiplicative noise. As before, at a given anisotropy, ingression of noise lowers

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the SPE value from that of noise-free situation and lowering becomes manifestly large in presence of multiplicative noise.

IV.

CONCLUSION

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The profiles of SPE of doped GaAs QD have been explored under the supervision of variable effective mass, variable dielectric constant and anisotropy of the system. Presence of

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noise has also been taken into account to explore how it interplays with above parameters in modulating SPE. We have found that spatially-varying effective mass and dielectric constant do not produce any new qualitative feature in the SPE profile (from that of fixed effective mass and dielectric constant). However, a quantitative change can be envisaged since use of variable effective mass and dielectric constant causes a drop in the SPE value from the fixed ones. Introduction of noise causes reduction in the magnitude of SPE from that of noise-free

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situation and the reduction is more with multiplicative noise. In absence of noise and in presence of additive noise, SPE displays a modest decrease with increase in anisotropy of the system. And the said decrease becomes severe in presence of multiplicative noise. The

ACKNOWLEDGEMENTS

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V.

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results seem to possess considerable relevance in the related field of research.

The authors A. 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 support.

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Research Highlights

Self-polarization effect (SPE) of doped quantum dot is studied.

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The dot is subjected to Gaussian white noise.

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Special emphasis is put on role of anisotropy.

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Spatial variations of effective mass and dielectric constant have also been considered.

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