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PII: DOI: Reference:

S1386-9477(19)30966-X https://doi.org/10.1016/j.physe.2019.113903 PHYSE 113903

To appear in:

Physica E: Low-dimensional Systems and Nanostructures

Received date : 8 July 2019 Revised date : 12 September 2019 Accepted date : 12 December 2019 Please cite this article as: O. Mommadi, A.E. Moussaouy, M. Chnafi et al., Exciton-phonon properties in cylindrical quantum dot with parabolic confinement potential under electric field, Physica E: Low-dimensional Systems and Nanostructures (2019), doi: https://doi.org/10.1016/j.physe.2019.113903. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2019 Published by Elsevier B.V.

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Exciton-phonon properties in cylindrical quantum dot with parabolic confinement potential under electric field O. Mommadia , A. El Moussaouya,b,∗, M. Chnafia , M. El Hadia , A. Nougaouia , H. Magrezb Laboratory of Dynamics and Optical Materials, Department of Physics, Faculty of Sciences, University Mohamed I, 60000 Oujda, Morocco. b The Regional Centre for the Professions of Education and Training, Oujda, 60000, Morocco.

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Abstract

The mixed confinement potential effect on exciton in cylindrical quantum dot (QD) is calcu-

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lated in the presence of an electric field with and without confined longitudinal optical (LO) phonon mode contribution. The variational approach within the effective mass approximation is used to describe the exciton-phonon interaction with three variational parameters of the trial wave function. The ground state binding energy of exciton has been calculated

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numerically for the parabolic (axial direction) and square (lateral direction) finite confinement potentials under the electric field applied along the growth direction of the QD. The system both with and without LO-phonon contribution is investigated. The combined effects of applied electric field and parabolic confinement on the ground state binding energy, the polaronic correction and the Stark shift are examined in detail. The findings indicate that the electric field strongly reduces the binding energy in both ignoring and considering LO-phonon interaction especially for wider thickness of QD. We have shown that the contribution of LO-phonon leads to an increase in the binding energy and to decrease in

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the Stark shift energy. Furthermore, our numerical results illustrate a strong dependence of Stark shift on the QD height, electric field strength and LO- phonon contribution. Keywords: Parabolic potential; Exciton-LO-phonon; Electric field; Binding energy; Stark shift. ∗

Corresponding author. Tel.: 00212 5 36 50 06 01/02; fax: 00212 5 36 50 06 03. Email address: [email protected] (A. El Moussaouy)

Preprint submitted to Physica E: Low-dimensional Systems and Nanostructures - JournalSeptember 11, 2019

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1. Introduction Due to their improved optoelectronic proprieties, the recent progress of semiconductor quantum well (QW), quantum well wire (QWW) and quantum dot (QD) have have devoted

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numerous theoretical and experimental studies [1–13]. The principal property of the QDs is the possibility to limit the motion of the charges carriers in all three spatial dimensions.

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Their specific characteristics allow to exploit it in different applications[14]. The reduction of the QD size leading to a partial or total quantization of the energy levels permits to these systems to produce a profound improvement of quantum effects in their physical characteristics. They have accelerated the development in microelectronic device technology.

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External effects, such as electric field strength, temperature, hydrostatic pressure, and so on can remarkably control many properties and essential characteristics of these systems [15–19]. As we all know, excitonic aspects in QDs become the main subject of different the-

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oretical studies, because of the high potential applications in fundamental physics. Due to the confining effect of the electron and the hole in the QD structures, the quantized energy levels appeared. The density of states is gama function-like, and the wave function of the particles is localized indoors the QD. Therefore, excitons play a dominant role in improving some physical characteristics of the QD, and their stability is important for possible devices requiring this property. The theoretical works have presented a strong competition between the change in the cross sectional geometry, Coulomb interaction and external perturbations. El-Yadri et al. [20] have examined the optical transition and the binding energy in a quantum disk influenced by the temperature and the hydrostatic pressure. Their results indicate

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that the optical properties and the binding energy are very sensitive to the temperature and stress effects. Kasapoglu et al. [21] have studied the effects of the electric strength and intense laser field on shallow donor binding energies of the ground and excited states, as well as their total optical absorption between the related states. Satori and Sali [22, 23] have implemented the finite element method to investigate the electronic state of a shallow 2

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hydrogenic impurities in cubic and spherical GaAs (Al, As) QDs. Tshipa et al. [24] have investigated the combined effect of the parallel applied magnetic field and the electric confinement in the binding energies of a donor charge assumed to be located exactly at the

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center of the symmetry of two concentric cylindrical quantum wires. They have reported that the binding energy is more important when increasing the radius of the core, confine-

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ment potential and the magnetic field applied parallel to the axis of the wire. Charrour et al. [25] and Zounoubi et al. [26] have investigated the ground state binding energy of a hydrogenic impurity in cylindrical QD under the action of an external strong magnetic field. They have found that the impurity binding energy depends strongly on the magnetic field

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strength. The same behavior was found for an exciton in the case of QWW by Bouhassoune et al. [27]. Minimala et al. [28] have evaluated the electric field influence on the nonlinear optical properties for an exciton in a ZnO/Zn1−x M gx O QD, by considering the spontaneous

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and the piezoelectric polarizations which produce the biaxial strain and the internal electric field. Very recently, Aghoutane et al. [3] have developed the influences of the hydrostatic pressure combined to the QD size effect on the binding energy and the optical absorption coefficient for an exciton in 2D GaN/AlN quantum ultra-thin disk. The action of the electric field strength is a main probe for investigating the physical properties of the nanostructures. The results found by Mora-Ramos et al. [29] and Erdogan et al. [30] have shown that the binding energy of exciton diminishes with increasing the electric field strength and that effect is stronger for a larger dot. Kasapoglu et al. [31] have reported the simultaneous effects of the intense laser and the electric field on donor impurity

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binding energy in cylindrical Gax Ln(1−x) As(1−y) /GaAs QD for different nitrogen and indium concentration. Recently, Vartanian et al. [32] have investigated the effects of image charge and transverse electric field on the binding energy of shallow donor impurity in near-surface QW. In addition, some studies have indicated that the linear and nonlinear optical properties in quantum rings are more sensitive by a donor position and radial applied electric field 3

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[33–37]. Another work was performed by Baskoutas et al. [38], which have shown that the electric field strength and both the position of the impurity influence on the nonlinear optical rectification process. The presence of the electric field strength generated by a dc voltage

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leads to enhance the average value of the electron-hole distance. Up to now, a number of studies have focused on the coupling of electrons, impurities, and excitons with phonons in

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low-dimensional semiconductors. The polaronic process in nanostructures under the influence of the external perturbation has been widely researched by scholars [39–49]. Wei Xiao and Jinglin Xiao [39] have examined the vibration frequency properties and the binding energy of the polaron weakly coupling in the asymmetrical Gaussian QW. Vartanian et al.

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[40] have calculated the influence of the electric field and LO phonon mode contribution on the hydrogenic impurity in a cylindrical QD. Later, M’zerd et al. [41] have investigated the influence of the polaron on linear and nonlinear optical properties in spherical core/shell

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QD. They have found that the electron-LO-phonon interaction causes a red-shift of the peak position of the optical response. El Moussaouy et al. [42–44] have studied the hydrostatic pressure and the temperature effects on the exciton-phonon coupled states and the bound polaron in cylindrical QD. Their findings demonstrate that the binding energy and its polaronic shift increase linearly with enhancing pressure, and these energies decrease when the temperature increases. The dependence of exciton binding energy and the optical susceptibilities of second and third harmonic generation with the LO-phonon coupling in cylindrical QD have been examined in the presence of a magnetic field by Vinolin et al. [45]. The effect of the electric field on the binding energy of electron by the interaction with the confined

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LO and SO phonon in spherical QD has been developed by Huangfu et al. [46]. They have shown that the electric field increases the phonon contributions to the binding energy. More recently, Chen et al. [47, 48] by using the Lee-Low Pines unitary transformation and the variational method of Peker type to study the Rashba and temperature effects on the first excited state and oscillation period of the bound polaron in quantum pseudodot. Wang et 4

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al. [49] have investigated the characteristic of shallow-donor in GaAs semiconductors subjected the intense terahertz laser field modulated magnetopolaron effect. Consequently, it is provided that the contribution of phonon modes influences the binding energy and optical

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properties of nanostructures which will foster radiative and non-radiative transitions. To our Knowledge, till now, any theoretical study for the binding energy of exciton in a cylin-

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drical QD system with the contribution LO phonon mode under the electric field effect by considering the parabolic confinement potential has not been given.

In the present work, we consider the excitonic polarons states in Ga1−x Alx As cylindrical QD under the electric field effect, assuming a parabolic (axial direction) and square (lat-

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eral direction) finite confinement potentials. We use the effective mass approximation in combination with a variational method to investigate the effect of the external electric field on the excitonic binding energy. We chose a trial wave function taking into consideration

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the Coulombic interaction and the electric field applied along the z-direction. The coupling effect of the exciton with confined LO phonon is considered in our calculation. This leads us to discuss the effect of electric field on the binding energy and both their polaronic correction and Stark shift. The results illustrate the competing effects between the geometric confinement, the applied electric field and the exciton coupling with the LO phonons. This paper is structured as follows: In Sec. 2, we present our theoretical model with calculation of exciton ground-state binding energy and its polaronic shift in the presence of electric field. In Sec. 3, we show the numerical findings for the binding energy, the Stark shift and the polaronic correction to the ground state under electric field effect. Finally, in Sec. 4, our

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conclusions are given.

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2. Formalism Let us consider a GaAs semiconductor cylindrical QD of radius R and height H=2d surrounded by Ga1−x Alx As compound. In the presence of applied electric field, the basic

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Hamiltonian of the exciton-phonon system in GaAs/Ga1−x Alx As can be written in the

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effective mass approximation by:

H = Hex + HLO + Hex−LO ,

(1)

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where Hex is the Hamiltonian of exciton which, in the presence of electric field strength and the absence of the confined LO phonons mode, is given by:

~2 2 e2 ~2 2 ∇ − ∇ − 2m∗e e 2m∗h h ε0 | ~re − ~rh |

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Hex (re , rh ) = −

+ Vwe (re ) + Vwh (rh ) ± eF z,

(2)

where m∗i (i = e, h) are the effective masses of the electron and hole. ~re (~rh ) is the elec2

tron (hole) position. − ε0 |~ree −~rh | presents the Coulomb potential between the electron and hole. It is worth mentioning that in problems with two particles, the Hylleraas coordinates (ρe , ρh , ρeh , ze , zh ) are well adapted. In the presence of the phonon modes, the Coulomb potential will be taken as

ε0 ε∞

2 , ρ2eh +(ze −zh )2

√

where ε0 (ε∞ ) is the low (high) frequency dielectric

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constant. The confinement potential for our system is defined as:

Vwi (ρi ) =

Vwi (zi )

=

0 if ρi ≤ R

V0i otherwise

V0i 2 z d2 i

if | zi |≤ d

V0i otherwise 6

(i = e, h),

(i = e, h).

(3)

(4)

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where the expression of the barrier potential height V0i (x) is given by [42]: (5)

Vh (x) = 0.342∆Eg (x),

(6)

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Ve (x) = 0.658∆Eg (x),

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where ∆Eg (x) represents the band gap difference between GaAs and the Ga1−x Alx As materials as a function of Al concentration. At T=0 K, ∆Eg (x) is expressed in units of eV by [44] :

∆Eg (x) = EgGa(1−x)Al(x)As (x) − EgGaAs (x) = 1.155x + 0.37x2 .

(7)

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In Eq.(1), the second term is the Hamiltonian operator for confined LO phonon mode in the cylindrical QD which can be written as [42]: X

~ωLO a+ ln1 aln1 ,

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HLO =

(8)

l,n1

where a+ ln1 (aln1 ) are the creation (annihilation) operators of the (l, n1)th mode for the confined LO phonon, with wave vector (K// =

χn1 , Kz R

=

lπ ) 2d

and frequency ωLO where χn1 is

the n1-th root of the Bessel function of the zero order. The last term in Eq.(1) indicates the exciton-LO-phonon interaction Hamiltonian operator

with

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which is written as [42]:

Hex−LO = He−LO + Hh−LO ,

(9)

Hi−LO = −eφLO (~ri ),

(10)

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where φLO (~ri ) =

X l,n1

~ 8πωLO

1/2

al,n1 + a+ ri ). l,n1 φ± (~

(11)

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X n1

J0

χ

n1

R

ρi

where 1 = V

lπi z 2d i

al,n1 +

a+ l,n1

+ (i = e, h),

al,n1 + a+ l,n1

1 4πe2 ~ωLO 1 2 ε − ε , ∞ 0 χn1 2 2 lπR J1 (χn1 ) 1 + 2dχ R n1

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2 Vl,n1

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l=1,3.. Vl,n1 cos × P l=2,4.. Vl,n1 sin

lπi z 2d i

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Hi−LO = −

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After obtaining the expression of φ± (~ri ) from Ref. [50], we express Hi−LO as:

(12)

(13)

with V = 2πR2 d is the volume of cylindrical QD.

The effective Hamiltonian of exciton-phonon system in the atomic unit system (a∗ex = ∗ excitonic Bohr radius and Rydberg energy Rex =

reads:

2 1 ∂ 1 ∂ ρ2eh + ρ2e − ρ2h ∂ 2 ∂2 = − + + + 1 + σ ∂ρ2e ρe ∂ρe ρe ρeh ∂ρe ∂ρeh ∂ze2 2 ∂ 1 ∂ ρ2eh + ρ2h − ρ2e ∂ 2 ∂2 σ − + + + 1 + σ ∂ρ2h ρh ∂ρh ρh ρeh ∂ρh ∂ρeh ∂zh2 2 ∂ 1 ∂ ε0 2 q − + − 2 ∂ρeh ρeh ∂ρeh ε∞ ρ2eh + (ze − zh )2

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Hef f

µe4 ), 2ε20 (0)~2

ε0 (0)~2 µe2

+ Vwe (ρe , ze ) + Vwh (ρh , zh ) + Ve−LO (ρe , ze )

exc + Vh−LO (ρh , zh ) + Ve−LO−h (ρe , ze , ρh , zh ) ± f (ze − zh ),

(14) 8

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where σ =

m∗e m∗h

is the electron and hole mass ratio. The electric field is supposed oriented in

the Oz direction and we have introduced the reduced electric field f =

F FI

where FI =

∗ Rex (ea∗ex )

corresponds to the ionization field of the exciton. Ve−LO (ρe , ze ) (Vh−LO (ρh , zh )) are the

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effective potentials induced by the interaction between the electron-LO-phonon (hole-LOphonon) coupling, these expressions are determined by using unitary transformation at low

X J02

Vi−LO (ρi , zi ) = −

~ωLO # X X lπ lπ i i 2 2 cos2 sin2 Vln1 zi + Vln1 zi , (i = e, h). 2d 2d l=1,3.. l=2,4.. n1

(15)

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"

χn1 ρ R i

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temperature limit [27].

exc We define Ve−LO−h (ρe , ze , ρh , zh ) as the exchange potential exciton interaction via the LO

phonon, which is expressed as:

X 2J0 ( χn1 ρe )J0 ( χn1 ρh )

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exc Ve−LO−h (ρe , ze , ρh , zh )

=

R

R

~ωLO n1 " # X X lπ lπ lπ lπ e e h h 2 2 Vln1 cos Vln1 sin . ze cos zh + ze sin zh 2d 2d 2d 2d l=1,3.. l=2,4..

(16)

In order to determine the ground state binding energy of the exciton, we select the following

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wave function [51]:

ψex (ρe , ρh , ze , zh ) = Fe (ρe , ze )Fh (ρh , zh )Feh (ρeh , | ze − zh |), (17)

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with Feh (ρeh , | ze − zh |) = exp (−αρeh ) exp (−γ(ze − zh )2 ) exp (−δf (ze − zh )),

(18)

Fi (ρi , zi ) = fi (ρi )gi (zi ), (i = e, h),

(19)

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and

where fi (ρi ) and gi (zi ) are the ground state solution of the Schr¨odinger equation in the plane

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and z-direction, respectively, which are written as:

J0 (θi ρi ) f or ρi ≤ R R fi (ρi ) = A K (β ρ ) f or ρ > R i

0

i i

(i = e, h),

(20)

i

re-

exp−(η 2 zi2 ) f or | zi |≤ d i 2 gi (zi ) = Bi exp(−ki | zi |) f or | zi |> d

(i = e, h).

(21)

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J0 and K0 are the modified Bessel functions of 0th order. βi and ki are expressed as:

2m∗i i βi = (V − Eρi ) ~2 0

where

Eρi

~2 = 2m∗i

12

2m∗i i , ki = (V − Ezi ) ~2 0

12

,

2 ~2 θi , Ezi = (ηi )2 . ∗ R 2mi

(22)

(23)

By applying the boundary conditions at the interfaces ρi = R and | zi |= d, the parameters Ai , θi , Bi and ηi are taken the forms:

d2

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exp−(ηi2 2i ) 2 ki J1 (θi ) K1 (βi R) J0 (θi ) Ai = , θi = βi R , Bi = , ηi = . K0 (βi R) J0 (θi ) K0 (βi R) exp(−ki d) d

(24)

In Eq.(18), the term exp (−αρeh ) exp (−γ(ze − zh )2 ) describes the coulombic correlations effects between the electron and hole. The last term exp (−δf (ze − zh )) represents the effect 10

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of the electric field. α, γ and δ are three variational parameters and the in-plane electron and hole distance ρeh is given by ρ~eh = ρ~e − ρ~h . The binding energy of the exciton system without phonon is defined by:

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The binding energy of the exciton system with phonon is:

(25)

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0 EB0 = Ee0 + Eh0 − EX .

ph , EBph = Eeph + Ehph − EX

(26)

where Ee and Eh are the free electron and hole energies. The EX term is the exciton ground-

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state energy determined by minimizing the following expression:

EX = minhψex | Hef f | ψex i. α,γ,δ

(27)

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The polaronic correction to the exciton binding energy in the presence and absence of the electric field can be defined as the difference between the exciton binding energy with phonon and that without phonon:

ph 0 ∆Eph(0,F ) = EB(0,F ) − EB(0,F ) .

(28)

3. Numerical results and discussions

The above theory is now applied to study the binding energy of the ground state of the exciton coupling with LO phonon mode and its polaronic correction in GaAs cylindrical QD embedded in Ga1−x Alx As material in presence of the external electric field. The Al

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concentration in the barrier material is taken as x=0.3. The physical parameters used in our work are m∗e = 0.067 m0 , m∗h = 0.087 m0 , ε0 = 12.83, ε∞ = 10.9, ~ωLO = 36.25 and the electron-phonon coupling αF = 0.0681. In all what follows, our numerical results are ∗ presented in units of the effective Rydberg energy Rex = 3 meV and the effective Bohr radius

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a∗ex = 18.434 nm. Fig. 1 represents the variation of exciton binding energy with and without LO phonon mode in a cylindrical QD of the radius R = 1 a∗ex as a function of the dot height H for

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different values of the electric field. We can see that with finite barriers, the binding energy increases as the height decreases and reaches a maximum at height critical value Hc,

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and then decreases. At small height values H < 0.4 a∗ex , discrete exciton levels vanish in the parabolic well, and the excitonic wave function penetrates into barrier material. In the strong spatial confinement case (H < 0.6 a∗ex ), the binding energy is relatively insensitive to the electric field effects. For (H > 0.6 a∗ex ), we observe that the binding energy decreases

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when the dot height increases, and this behavior becomes important as the electric field strength increases. Consequently, the average value of the distance | ze − zh | increases and the strength of the coulombic interaction becomes lower. This behavior is in agreement with

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those reported in [29, 51] without phonon. The contribution of LO phonon mode leads to increase the binding energy and it is more influenced for smaller dot height than the larger one due to the geometrical confinement effect, which results in a significant amplification of the coulombic potential. This behavior is analogous to those found by Vinolin et al. [45] and El Moussaouy et al. [42] in the cylindrical QD with a square finite confining potential. The exciton binding energy with and without LO phonon mode consideration for different QD height as a function of the electric field strength applied along the z-direction is presented in Fig. 2, with QD radius R = 1 a∗ex . The binding energy shows a nearly linear decrease with electric field. As the electric field is increased, the charge carriers (electron and hole) are

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pulled toward the opposite side of the QD; as a result, the binding energy decrease and this diminution is more important for wider dots. Thus, the exciton state becomes less stable. Note that the curve slope depends on the presence (absence) of LO phonon mode and the QD height. In this case, the higher value is found with the contribution of LO phonon and for the larger dot size. As shown in the figure, it could be easily seen for a fixed electric 12

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field value, the binding energy enhances significantly with decreasing QD height due to the geometrical confinement effect. Our finding agree with that obtained by Minimala et al. [28] in the case of the different radius values of spherical QD without consideration of the

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LO phonon mode. It is important to notice that in the parabolic confinement model, the contribution of LO phonon leads to enhance the binding energy.

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To clearly understand the effect of LO phonon mode on the binding energy in the presence of electric field, we have plotted in Fig. 3 the variation of the polaronic correction for the binding energy as a function of the QD height with radius R = 1 a∗ex for different electric field values. We remark that the polaronic correction ∆EB has the same behavior as that

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seen in Fig. 1. Indeed, when the well thickness is very small, the correction ∆EB due to LO phonons increases and becomes maximum at H = 0.38 a∗ex , and converges to the quasiQWW value with increasing the QD height. Furthermore, we note that the polaronic shift

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is significantly sensitive to the electric field for (H > 0.6 a∗ex ) and the ∆EB moves toward the small values when the electric field increases. In order to more understand the influence of the LO phonon contribution to the exciton binding energy, we have represented in Fig. 4 the evolution of the polaronic correction as a function of electric field strength for different QD heights (H = 0.5, 1 and 2 a∗ex ) for R = 1 a∗ex . We observe that the polaronic shift decreases as the electric field increases. For larger QD thickness, the polaronic shift is less important because the electric field leads to easier separation of charge carrier, which reduces the overlap of the electron and hole wave functions. We see from the same figure that the polaronic correction takes a maximum value when the

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height of the dot is reduced due to the geometrical confinement effect. These results agree with those illustrated in Fig. 2. To more illustrate the phonon influence on the Stark shift, we have plotted this exciton energy shift with and without phonons as a function of the dot height in Fig. 5. We have considered a QD of radius R = 1 a∗ex and two significative values of electric field strength 13

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(F = 10 and 20 kV /cm). The Stark shift with phonon ∆Esph (without phonon ∆Es0 ) can ph(0)

be defined from the simple expression: ∆Es

ph(0)

= EB

ph(0)

(F ) − EB

(F = 0), where EB (F )

and EB (0) are the excitonic binding energies with and without the presence of electric field,

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respectively. It can be seen that the shift enhances with increasing the QD thickness as well as the electric field strength, and the phonon field polarization leads to diminish the Stark

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effect. Here again, the reason for this can be explained by the fact that the charge carriers (electron and hole) move to the sides of the z-direction, and therefore the electric field has a more obvious influence on the binding energy. It is found that the phonon effect on the Stark shift is more influential in the stronger electric field than the weaker electric field.

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In this case, the phonon contribution cannot be neglected. This behavior agrees with that obtained as a function of the disk radius [52, 53] in the presence of lateral electric field and without phonon coupling.

On the other hand, Fig. 6 describes the variation of the Stark shift versus the electric field

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strength. The curves are obtained with a dot radius R = 1 a∗ex for two height confinement regimes (H = 1 and 2 a∗ex ) with and without LO phonon mode contribution. This figure shows that the Stark shift decreases almost linearly as the field strength increases. Furthermore, we also remark that the energy shift increases with enhancing nanostructure height. It can be explained by that the electric field effect is less important for the small dot sizes due to dominant of quantum confinement, and therefore the Stark shift is more significant for larger dot size. This aspect is similar to that obtained by Minimala et al. [28]. Note that, for a given value of the QD height, the energy shift decreases when including LO phonon

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mode. This curve tells us that a system that operates under electric field by taking into account the LO phonon mode contribution can be exploited to synchronize the output of optoelectronic devices without changing the QD geometry. In order to have a global picture about the exciton binding energy under simultaneous effects of the geometrical confinement (height of the cylinder) and the applied electric field strength 14

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with and without LO phonon mode contribution, we have presented in Fig. 7 the variation of the exciton binding energy in a cylindrical QD of the radius R = 1 a∗ex as a function of both the height H and the electric field F. For each electric field value, the binding energy

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keeps the same variation as a function of the dot height H with and without LO phonon mode. It reaches a maximum and then rapidly decreases when dot height reduced. This

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happens due to the penetration of the excitonic wave function into the barrier material and recovers a higher dimensionality. It is clearly seen that the electric field effect has more importance on the larger dot height than the smaller one. Furthermore, the binding energy increases with the contribution of LO phonon mode. This allows us to claim that the LO

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phonon mode creates a greater electronic confinement in the nanostructure. Let us now discuss the LO phonon mode effect on the binding energy in the presence of electric field. In Fig. 8 we can observe the combined dependence of the polaronic correction

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in a Ga1−x Alx As cylindrical QD as a function of both the height H and electric field F. We see that the dot height and electric field strength display the same effects as exhibited in Fig. 3, Fig. 4 and Fig. 7. Additionally, the polaronic correction of exciton is more pronounced at smaller dot height and is reduced by enhancing the electric field at large thickness of parabolic well. We believe that these properties can be exploited to make and improve new optical devices based on low-dimensional semiconductor structures.

4. Conclusions

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By using the variational approach and the effective mass approximation, we have investigated the ground state binding energy of an exciton confined in a cylindrical QD under electric field effect, taking into account the LO phonon mode. The influence of parabolic confinement as well as the electric field, applying along the QD growth axis (z-direction), on the binding energy and its polaronic shift has been clarified. Numerical results for our parabolic system show that the electric field effect has an important impact on the excitonic 15

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properties in cylindrical QD. So, the binding energy of an exciton in this nanostructure decreases with increasing the electric field. Similarly, the polaronic correction to the binding energy diminishes almost linearly with enhancing the electric field strength. Furthermore, the influence of the electric field on the Stark shift without considering LO phonon mode

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becomes very significant in the case for large QD. We hope that the present study will contribute in goodl understanding of the excitonic properties with consideration of LO phonon dimensional optoelectronic devices.

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mode under the application of the electric field and to potentially exploit them in low-

Solidi B 256 (2019) 1800361.

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[3] N. Aghoutane, M. El-Yadri, A. El Aouami, E. M. Feddi, F. Dujardin, M. El Haouari, Phys. Status

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Figure captions • Fig1 Variation of the exciton binding energy in a GaAs/Ga(1−x) Al(x) As cylindrical QD as a function of the dot height H with and without LO phonon mode for three

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values of electric field strength (F = 0, 10 and 20 kV /cm).

• Fig.2 Variation of the exciton binding energy in a GaAs/Ga(1−x) Al(x) As cylindrical

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QD as a function of the electric field F with and without LO phonon mode for three values of the dot height (H = 0.5, 1 and 2 a∗ex ).

• Fig.3 Polaronic correction to the binding energy of exciton in a GaAs/Ga(1−x) Al(x) As

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cylindrical QD as a function of the cylinder height H for three values of the electric field strength (F = 0, 10 and 20 kV /cm).

• Fig.4 Variation of the polaronic correction to the binding energy as a function of the

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electric field F for three values of the dot height (H = 0.5, 1 and 2 a∗ex ). • Fig.5 The exciton Stark shift in a GaAs/Ga(1−x) Al(x) As cylindrical heterostructure versus the dot height H with and without LO phonon mode for two electric field values (F = 10 and 20 kV /cm).

• Fig.6 Variation of the exciton Stark shift as a function of electric field strength F with and without LO phonon mode for two values of the dot height (H = 1 and 2 a∗ex ). • Fig.7 Variation of the exciton binding energy in a GaAs/Ga(1−x) Al(x) As cylindrical

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QD as a function of the dot height H and electric field F with and without LO phonon mode.

• Fig.8 Variation of the polaronic correction to the exciton binding energy in a cylindrical QD as function of the cylinder height H and electric field strength F .

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Highlights

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The parabolic confinement potential influence on exciton-LO-phonon coupling in quantum dot is studied in the presence of electric field . A variational method within the effective mass approximation have been used. The ground state binding energy and its polaronic correction decrease linearly with the increase of electric field. The stark effect decreases when increasing the confinement effect and considering the LO phonon mode.

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