quantum dot using the fem simulation

8/11/2019 Quantum Dot Using the FEM Simulation

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The combined effect of pressure and temperature

on the impurity binding energy in a cubic

quantum dot using the FEM simulation

A. Sali a,, H. Satori b,c

a E.N.S de Fs, Dpartement de Physique, B.P. 5206 Bensouda, Fs, Moroccob Department of Mathematics and Computer Science, Mohammed Premier University, Faculty Pluridisciplinary, 300 Selouane, 62700

Nador, Moroccoc Department of Mathematics and Computer Science, Sidi Mohamed Ben Abdellah University, Faculty of Science, B.P. 1796 Dhar El

Mehraz, Fez, Morocco

a r t i c l e i n f o

Article history:

Received 22 December 2013Received in revised form 9 January 2014Accepted 20 January 2014Available online 6 February 2014

Keywords:

Binding energyDonor impurityQuantum dotHydrostatic pressureTemperatureFinite barrier confining potentialFinite element method

a b s t r a c t

Using the finite element method, the binding energies of a hydro-

genic shallow donor impurity are investigated in a cubic GaAs/Ga1xAlxAs quantum dot structure with realistic potential barrierheight under the combined effect of hydrostatic pressure andtemperature in the framework of the effective mass approxima-tion. In the calculations, we have taken into account the elec-tronic effective mass, dielectric constant, and conduction bandoffset between the dot and barriers varying with pressure andtemperature. The results we have obtained show that the donorbinding energy varies with the dot size, the confining potential,the position of impurity, the hydrostatic pressure and tempera-ture. It is also found that the donor binding energy increases lin-early with the pressure in direct gap regime and its variation is

larger for narrower dots only and drops slightly with the temper-ature. A good agreement is obtained with the existing literaturevalues.

2014 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.spmi.2014.01.0110749-6036/2014 Elsevier Ltd. All rights reserved.

Corresponding author. Tel.: +212 650775673.

E-mail address: [email protected](A. Sali).

Superlattices and Microstructures 69 (2014) 3852

Contents lists available at ScienceDirect

Superlattices and Microstructures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / s u p e r l a t t i c e s
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1. Introduction

In last past decades, great developments in nano-fabrication technology have made it now pos-sible to fabricate low-dimensional semiconductor structures whose dimensions are comparable withinter-atomic distances in solids such as quantum dots (QDs). The movements of charge carriers inthese structures are constrained by potential barriers which confine the carriers in three dimensions.This results in the restriction of the degrees of freedom for motion to zero. The three-dimensionalquantum confinement of carriers in these structures, has led to the formation of atomic-like discreteenergy levels (subbands) which can be tailored to a specific need by changing the size, shape andmaterials of the dot. Such systems have a small scale in zero spatial dimensions that their electronicand optical properties[1]are significantly different from the same material in bulk form. The quan-tum dot systems have many applications in transistors, solar cells, LEDs, and diode lasers, medicalimaging etc.

In the last ten years, a profusion investigation of hydrogenic impurities states in quantum confinedzero-dimensional electron systems has been witnessed due to rapid developments in their synthesis.Shallow hydrogenic impurities increase the conductivity of a semiconductor by several orders of mag-

nitude and play a very important role in its revolution. Without impurities, there would be no elec-tronic components such as diode, transistor, or any semiconductor science and technology.Understanding the effects of impurities on electronic states is a fundamental question in semiconduc-tor physics because their presence can dramatically alter the performance of a quantum device, suchas quantum transport and optical properties. It is has been found that when the dimensions of the sys-tem are reduced, the quantum size effect becomes prominent. This is due to the fact that an electronmoves only in a smaller space and spends most of its time close to the impurity ion. Thus, the effectivestrength of the electronimpurity ion Coulomb interaction increases which lead to an enhancement ofthe binding energy in lower dimensions. As a consequence, the electronic and optical properties ofquantum dot structures such as binding energy, photoionization cross-section[2], absorption spectraand other optical properties[3]can be drastically changed.

The study of shallow hydrogenic impurity states in a semiconductor quantum well was initiated inthe early 1980s through the Bastards pioneering work[4]. Since then, a number of theoretical inves-tigations with various methods of donor impurity states in low dimensional semiconductors and par-ticularly in quantum dots have received considerable attention. The first studies of the confinementeffects on the impurity states in these structures have been made by Porras-Montenegro and Perez-Merchancano[5]and Zhu et al. [6]who calculated the binding energies for the ground and excitedstates as a function of dot size and impurity position.

The eigenstates of quantum dots are obtained by solving time-independent Schrdinger equation,but, in many cases, because of complex geometry, it is very complicated to solve the equation usinganalytical method due to the existence of the three-dimensional confinement and Coulomb potential.In order to solve this problem, many authors adopted several numerical methods. The most of them

have employed the variational approach[715], a technique in which the trial wave function takesinto account the confinement of the carrier in the dot and the influence of the electronimpurityion Coulomb interaction or alternatively, perturbation methods limited to the strong confinement re-gime[1618], an analytical transfer matrix method [19], a potential morphing method[20], exactsolution which has been obtained only for centered charged impurities[21]or impurities[6], Quan-tum Genetic Algorithm and HartreeFock Roothaan method[22], exact diagonalization method[23],the finite difference method[24,25], plane wave method[26]and so on[2729].

In the above works, due to the presence of electronimpurity Coulomb potential and the complex-ity of the configuration, appropriate basis functions are difficult to obtain or are not available in closedform for most QDs with complex geometries. We therefore present in this paper a numerical varia-tional methodology of the finite element method (FEM) which provides a more flexible method to

approximate differential equations since it can easily be extended to higher order approximationsand can be used for complex geometries.The FEM became a dominant method in applied mathematics for numerical modeling of physical

systems in many engineering and scientific disciplines, e.g. electromagnetic, solid or fluid dynamics as

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well as civil and aeronautical engineering due to the intensive computationally of modern supercom-puters with large amounts of memory and higher frequency of CPU processing.

The finite element method is a popular numerical technique and a typical real-space method forsolving problems which are described by partial differential equations (PDE) or can be formulatedas functional minimization and is one such method that transforms the PDE into matrix operations.Unlike conventional variational schemes, the FEM does not require predetermined globally definedtrial functions. Rather, the entire domain of interest is discretized into smaller regions called elementsand a simpler solution is assumed by incorporating a local polynomial representation within each ele-ment. This flexible approach yields well converged eigenvalues.

In our previous work[30], we have calculated the binding energy of an impurity confined in aspherical quantum dot by using the FEM choosing a one dimensional physical domain and quadraticLagrange as basis functions for each finite element. In this paper, we calculate the binding energy ofa hydrogenic donor impurity in a cubic GaAs/Ga1xAlxAs quantum dot structure by considering athree dimensional physical region in which elements are tetrahedral Since accuracy of the resultsand computational time depend heavily besides the number of elements and the size of the localelement, on the dimension of discretization and on the type of the interpolation functions in each

element.Understanding the effects of the quantum confinement on the impurity states in the low dimen-

sional systems and especially in quantum dot structures is important in physics. In addition, uniaxialstress and external perturbations such as the hydrostatic pressure and temperature on the electronicand optical properties of doped nano-structures have been of great interest to researchers from bothexperimental and theoretical points of view. Lefebvre et al. [31,32]presented theoretical and exper-imental calculations on the pressure coefficients of carrier effective mass and the well-width andbarrier-height dependent excitonic transitions in quantum well systems. Their results indicated thatthe increase of the effectives masses and the decrease of the barrier heights were the main reasonfor the decrease of the pressure coefficients with reduced well widths. An experimental investigationof the effect of an in-plane uniaxial stress on the characteristic transition energies of both the heavy-

hole and the light-hole excitons confined in GaAsGa1xAlxAs quantum wells have been carried out[33].

As it is well known, the hydrostatic pressure has an influence on the semiconductor parameterssuch as energy gap, carrier effective mass, dielectric constant, lattice vibration, volume of the hetero-structure and so on and an increase in the pressure results in a nearly linear increase in donor bindingenergy without altering the symmetry of the heterostructure system in direct-gap regime. The effectof hydrostatic pressure on the optical and electrical properties in quantum dot systems has beeninvestigated by several authors[3444].

By means of the LuttingerKohn effective mass equation and the direct diagonalization method,Rezaei and Doostimotlagh[34]have recently studied the combined effects of hydrostatic pressure,electric field, and conduction band non-parabolicity on the binding energies and the diamagnetic

susceptibility of an impurity in a typicalGaAsGa1xAlxAsspherical QD. Dane et al. [35]have reportedcalculations of the normalized ground state binding energy of a hydrogenic donor impurity located atthe center of a GaAs/GaAlAs spherical QD under the influence of hydrostatic pressure and electricfield using a variational procedure. The influence of the hydrostatic pressure on the binding energiesof the ground and the few excited states along with diamagnetic susceptibility of an on-centerhydrogenic impurity confined in typical GaAsGa1xAlxAs spherical QDs is theoretically investigatedusing the matrix diagonalization method and the conduction band non-parabolicity effect [36].Rajashabala and Kannan[37]have investigated the simultaneous effects of hydrostatic pressure andgeometry on the ionization energies of a hydrogenic donor and the metal-insulator transition in aGaAsGa1xAlxAs cubical quantum dot system. Perez-Merchancano et al. [38] have calculated thebinding energies of shallow donors and acceptors in a spherical GaAsGa1xAlxAsquantum dot under

isotropic hydrostatic pressure for both a finite and an infinite high barrier by using a variationalapproach within the effective mass approximation. The binding energy of a shallow hydrogenicimpurity in a spherical quantum dot under hydrostatic pressure with square well potential iscalculated using a variational approach and the non-parabolicity effect [39]. Perez-Merchancanoet al. [40] have studied the influences of hydrostatic-pressure on the donor binding energy in

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GaAs(Ga,Al)As, quantum dots. Within the framework of effective-mass approximation, Xia et al.[41]have investigated the hydrostatic pressure effects on the donor binding energy of a hydrogenic impu-rity inInAs/GaAsself-assembled QD by means of a variational method. John Peter[42]has calculatedthe binding energy of shallow hydrogenic impurities as a function of the dot size inGaAsGa1xAlxAsspherical QDs in the influence of pressure using a variational approach within the effective massapproximation. Correa et al.[43]have calculated the effects of hydrostatic pressure on the bindingenergy and photoionization cross-section in spherical QDs for different dimensions of the structureand radial impurity position. Oyoko et al.[44]have studied the effect of an unaxial stress on the bind-ing energy of shallow impurities in parallelepiped-shaped GaAs/GaAlAsQDs. In all the above works, itis have been found that the pressure increases almost linearly the donor binding energy in the regimeof low hydrostatic pressure and diminishes with the size of the structure.

The effects of temperature on the impurity states of confined charge carriers in a nanostructurequantum dot have been carried out by some authors[4547]. The binding energy of the ground stateand the low-lying excited state of an impurity atom in a GaAsparabolic quantum dot have been re-ported as a function of the temperature and the effective confinement length of the QD by the sec-ond-order RayleighSchrodinger perturbation theory [45]. Elabsy has studied the effect of

temperature on the binding energy of the donor impurity in an artificial semiconductor atom [46]and spherical quantum dot system[47].

Several works on the combined effects of hydrostatic pressure and temperature on the physicalproperties of semiconductor quantum dot structures have drawn the attention of many scientistsrecently[4854].More recently, Sivakami and Gayathri [48]have studied the simultaneous effectsof pressure and temperature with dielectric mismatch effect of an on-center hydrogenic impuritybinding energy in a GaAs spherical QD, using the variational approach within the effective massapproximation. Also, in a more recent work, Kirak et al. [49] have calculated the effects of thehydrostatic pressure and temperature on the electronic and the linear and nonlinear optical prop-erties in a spherical QD in the presence of the electric field. Vaseghi and Sajadi[50]have examinedthe binding energies and diamagnetic susceptibility of an impurity in a spherical GaAsQD under the

simultaneous influence of static pressure, temperature and laser radiation. Based on the effective-mass approximation within a matrix diagonalization scheme, simultaneous effects of external elec-tric field, hydrostatic pressure and temperature on the binding energy of an off-center hydrogenicdonor confined by a spherical Gaussian potential have been calculated[51]. Liang and Xie[52]haveinvestigated the combined effects of the hydrostatic pressure and temperature on the binding en-ergy, the oscillator strength and the third-order susceptibility of third harmonic generation of ahydrogenic impurity in a spherical QD, in the presence of the external electric field, by means ofthe perturbation approach. The combined effects of hydrostatic pressure and temperature on theground state binding energy of two electrons in a GaAs SQD have been studied by Sivakami andMahendran[53]using a perturbation approach within the effective-mass approximation. Their re-sults show that an increment in temperature results in a decrease of the correlation energy while

an increment in the pressure for the same temperature increases the correlation energy at aparticular dot size. Yesilgu et al.[54]have reported a calculation of the binding energy and the pho-toionization cross-section of a shallow hydrogenic impurity in quantum dots under hydrostaticpressure and temperature as a function of the dot sizes, for incident light polarized along the axisof the dots.

The present research is concerned with a theoretical study of the combined effects of hydrostaticpressure and temperature on the binding energy of the ground state of a hydrogenic shallow donorimpurity in a cubic GaAs/Ga1xAlxAsQD with finite confinement potential barriers using the FEM. Re-sults are calculated for different dot sizes, shallow donor impurity positions, hydrostatic pressure andtemperature. The difference of the electron effective masses and dielectric constants between thequantum dot region and barriers have been considered within the effective mass and parabolic band

approximations and by restricting ourselves to range of pressure where there is noC

Xcrossovereffect.The paper is organized as follows: in Section2we describe the theoretical framework, Section3is

dedicated to the results and discussion, and finally, our conclusions are given in Section 4.



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2. Basic theory

Within the effective mass and parabolic band approximations, the hydrogenic donor impurity lo-cated at the position (x0,y0,z0) and confined in a cubic QD of width L, embedded in a dielectric matrix,under the combined effects of hydrostatic pressure (P) and temperature (T) can be described by thestationary Schrdinger equation,

HEw 0 1

where

H me

md;bP; Tr2 2

0d;bP; T

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixx0

2 yy0

2 zz0

2q Vx;y;z;P; T

R

264

375 2

Edenotes a discrete eigenvalue of the Hamiltonian Handw is the eigenfunction expressed as the sumover the elemental wave function:

wx;y;z XNEa

wax;y;z 3

where NEis the number of element andwais the wave function of element(a) which can be expressedas a linear combination of the interpolation polynomials basis functions:

wax;y;z Xni1

Cai uaix;y;z 4

whereCai are unknown coefficients that represent the amplitude of the wave function at a particularnode,uai x;y;z are the basis functions in each element and the summation upper limit is the numberof basis functions per element. In our case, the elements are three dimensional and have tetrahedronshape with four interpolation functions (n= 4).

By inserting Eqs.(3) and (4)in Eq.(1), we obtained a general matrix eigenvalue problem:

HESC 0 5

The elements of the Hamiltonian matrix Hand the overlap matrix Sare given by:

Hij hui;Huji ZX

uiHujdX 6

andS hui;uji ZX

uiujdX 7

in which Xis the entire physical domain of interest.Because the wave function vanishes at the infinity, w(x,y,z)? 0 as (x,y, z)?1 and hence there is

no surface contribution, and because the FEM being a numerical approximation, we therefore reducethe upper limit of integration (1) to a finite value, (x,y,z) = Lc[55]. As a consequence, the entire phys-ical domain is segmented in two finite domains Xdand Xbwhere

Xd

jxj LP=2

jyj LP=2

jzj LP=2

8>: 8

is the domain related to the GaAsquantum dot material and,

XbLP=2< jxj 6 LcLP=2< jyj 6 Lc

LP=2< jzj 6 Lc

8>: 9



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is theGa1xAlxAs barrier domain as seen inFig. 1.HereL(P) is the cubic quantum dot side width depending on the pressure.The Hamiltonian Eq.(2)is written in reduced atomic units which correspond to a length unit of an

effective Bohr radius,a 0h2=mee

2, and an effective Rydberg,R mee4=20h

2. The two subscripts dand b refer to the quantum dot and the barrier layer materials, respectively.

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi

x x0 2

y y0 2

zz0 2

q is the distance between the electron and impurity site.me and

0are, respectively, the static dielectric constant and the electron effective mass of theGaAsquantumdot at zero pressure. The application of hydrostatic pressure modifies the lattice constants, dot size,barrier height, effective masses and dielectric constants. mb;dP; T are the pressure and temperature

dependent effective masses for the electron of the quantum dot and barrier layer, respectively. Fora GaAs quantum dot, the parabolic conduction effective mass is determined from the expression[5658]

mdP; T 1 ECpP; T

2

ECgP; T

1

ECgP; T D0

!" #1m0 10

wherem0is the free electron mass, ECp P; T 7:51 eV is the energy related to the momentum matrix

element, D0= 0.341 eV is the spin-orbit splitting.ECgP; T is the pressure and low temperature depen-

dent energy gap for the GaAsQD semiconductor at the C-point in units of eV taken from Ref.[59].

ECgP;T ECg0;0 5:405 10

4 K1 T2

T 204K 107 10

3 GPa1P 3:77 103 GPa2P2

" #eV:

11

ECg0; 0 1:519 eV is the energy gap forGaAsquantum dot at the C-point and at temperature T= 0and pressureP= 0.

The barrier materials parabolic conduction effective masse as a function of pressure and tempera-ture is determined from the expression[5658]

mbP; T mdP; T 0:083xm0; 12

wherex is the Al mole fraction of Aluminum in the Ga1xAlxAs layer.In the above expression Eq.(1),d;bP; T are the pressure and temperature dependent static dielec-

tric constant of both materials respectively. In the GaAs quantum dot region, dP; T is given by [60,61]

d

b

O

AL/2

x

y

z

Lc

Fig. 1. A 3D-cubic quantum dot with center in origin. OA is the diagonal of the cube.

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dP; T 12:74exp1:67 102 GPa1Pexp9:4 105T 75:6T6 200 K

13:18exp1:73 102 GPa1Pexp20:4 105T 300T>200 K

( 13

The static dielectric constant of material barrier is obtained from a linear interpolation of thedielectric constants ofGaAsandGa1xAlxAsis given by[58]

bx;P; T dP; T 3:12x: 14

V(x,y,z,P, T) in Eq. (2) is the confinement potential energy which confines the donor electron and isgiven by

Vx;y;z;P; T 0;jxj;jyj;jzj 6 LP=2

V0P; T;jxj;jyj;jzj> LP=2;

15

where L(P) is the pressure dependent of the cubic quantum dot side length. V0(P, T) is the barrierheight expressed as a function of the applied pressure P, at temperature Tand is is given by the expres-sion[60,61]

V0P; T 0:658D

E

C

gx;P; T: 16DECgx; P; T stands for the difference in the band gap energy ofGaAsandGa1-xAlxAsat the C point as afunction of temperatureTand pressurePand is expressed as[5658]

DECgx;P; T DECgx CxPDxT: 17

The function DECgx is the variation of the energy gap difference in the absence of pressure andtemperature and is given by

DECgx 1:155x 0:37x2eV: 18

The quantities C(x) and D(x) are usually defined as the pressure and temperature coefficients,

respectively of the band gap difference and are given byCx 1:3 10

2xGPa1 eV; 19

Dx 1:15 104xK1 eV: 20

The hydrostatic pressure effects on the geometric dimensions of the zinc blende low-dimensionalstructures are obtained from the Murnaghan relation[62,63]

aP

a0 1 P

B00B0

13B0

021

The change in lattice constant is related to the change in volume by [63]

DV=V0 3Da=a0; 22

where DVVP V0 and Da aP a0 23

HereVis the volume of the dot versus the pressure, V0is the original volume at atmospheric hydro-static pressure,a(P) is the lattice constant as a function of pressure and a0is the static lattice constant.B0= (C11+ 2C12)/3 is the bulk modulus[64],B

00 is the pressure derivative of the bulk modulus and C11

andC12are the elastic constants.By developing the terma(P) as a function ofP, the variation of the cubic quantum dot side width

L(P) with the pressure is therefore

LP L0 1 P=B0P2

=6B2

0 13

; 24

which can be reduced by omitting the P2 term to

LP L0 1 3S11 2S12P 13; 25



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where L0is the cubic GaAsdot size at atmospheric hydrostatic pressure,S11and S12are the complianceconstants, which can be calculated using elastic constant C11andC12ofGaAs[65]

S11 C11C12=C11C12C11 2C12; 26

S12C12=C11C12C11 2C12; 27whereS11= 11.6 10

3 GPa1 andS12= 3.7 103 GPa1 andL0 is the well width at atmospheric

hydrostatic pressure.In Eq. (1), the hydrostatic pressure and temperature dependence of the ground state binding energy

Eb(P, T) can be given as follows,

EbP; T EsbP; T EiP; T; 28

where Esb(P, T) is the eigenvalue of Hamiltonian in Eq. (2) without the impurity potential term and Ei(P,T) is the eigenvalue with the impurity potential term.

3. Results and discussions

We have calculated the donor binding energy by means of the FEM technique using the equidistantdiscretization and considering tetrahedral elements and taking the critical cubic size value Lcto be tentimes the effective Bohr radius (Lc= 10a

).In our theoretical study, we have considered a cubic QD made ofGaAs andGa1xAlxAs material

where x is the aluminum concentration since it is the most widely investigated and in which allthe material properties are well known. For numerical calculations, in all what follows, we expressedthe binding energies in effective electron Rydberg units R = 5.302 m eV and the sizes of the cubicquantum dot in effective electron Bohr radius a 103; 56 Awhich correspond to an electron effectivemass me 0:067m0 and0 12; 74 without applied hydrostatic pressure and in the regime of lowtemperature[60,61].

We have calculated the donor binding energy of the shallow hydrogenic impurity as functions ofthe size of the cubic QD, the Al concentration x, the impurity position, the hydrostatic pressure Pand the temperature Tin Zinc BlendGaAs/Ga1xAlxAs QD. We limited our calculations in the regimeof small values of the pressure [04 GPa] where the GaAsQD has a direct band gap since the transitionfrom direct to indirect band gap for GaAs occurs at higher pressures (larger than 4 GPa) where thegamma-X crossover effect cannot be neglected[66].

To understand clearly the dot size width effect on the impurity electronic state, we have plotted inFig. 2 the binding energy of an on-center shallow donor impurity as a function of the half quantum dotside width L/2 for an Al concentration x= 0.15 and for five different hydrostatic pressure valuesP= 0, P= 1,P= 2,P= 3 andP= 4 GPa withT= 0 K. The behavior of the binding energies without pres-sure (P= 0) is similar to the previous results found in Ref.[67]using the plane wave basis method. For

each value of the pressure, we observe that the binding energy increases from its bulk value in GaAsasthe dot side width is reduced, reaches a maximum value, and then drops to the bulk value character-istic of the barrier material as the cubic dot side length goes to zero. For large dot side widths, thebinding energy is small and nearly independent of dot size. Since, the electron is away from the impu-rity ion and it behaves as if it is in bulk material, resulting in bulk binding energies. While for verysmall dots, the tunneling effects become dominant and most of the wave function of electron pene-trates into the barrier region, which results in a rapidly decreasing binding energies. For small dots,the maximum of the binding energy is due to a more localized wave function in the dot region causedby the combined effect of confinement and Coulombic attraction potentials.

Note that the binding energy increases with the hydrostatic pressure for any dot side width, reflect-ing the additional confinement due to the pressure; i.e. when the hydrostatic pressure is increased, the

donor impurity becomes more confined and the binding energy increases. Also we observe that thepressure effect is more noticeable for narrow dots, and the maximum position goes slightly to smalldot sizes when the pressure increases. This is due to the increment of the dot effective masses as wellas to the decrease of dielectric constant and the barrier height with the pressure [39,48]. On theother hand, the shrinkage of the dot size with the increment of the pressure [39,48] results in

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shortening of the effective electronimpurity distance which also leads to an increase in thebinding energies. This behavior is clearly shown inFig. 3 for the impurity fractional energy shiftDEb= (Eb(P, L) Eb(P= 0, L0))/Eb(P= 0, L0) as a function of the half quantum dot side width L/2 fordifferent values of the hydrostatic pressures with an Al concentration x = 0, 15 atT= 0 K.

Fig. 4shows the dependence of the binding energy of an on-center hydrogenic donor impurity on

the hydrostatic pressure Pfor two different values of the half quantum dot sizes L/2 = 0.5a

and 1a

with a fixed temperature value T= 200 K and an Al concentrationx = 0.30. By keeping fixe the valueof the temperature and barrier potential confinement and for a given value of the dot size, the bindingenergy shows an approximately linear increase with the applied pressure. These results are inaccordance with those obtained previously in quantum dots[48,51], quantum wires[68]and quan-tum wells[69,70]. Additionally, it is clear that as the half size of the cubic QD structure increases,

Fig. 2. Binding energyEbof an on-center shallow donor impurity in a cubic GaAs/Ga1xAlxAsquantum dot as a function of thehalf quantum dot side width L/2 for five hydrostatic pressure values P= 0, P= 1, P= 2, P=3 and P=4 GPa with an Alconcentration x= 0, 15 and forT= 0 K.

Fig. 3. The impurity fractional energy shift, DEb= (Eb(P,L) Eb(P= 0,L0))/Eb(P= 0, L0) as a function of the half quantum dot sidewidthL/2 for different values of the hydrostatic pressures with an Al concentration x= 0, 15 and forT= 0 K.



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the binding energy decreases, reflecting the lower confinement potential. This is because the wavefunction does not fell the small compression in the structure when the size of the structure is verylarge. Fig. 4reveals also that the difference in binding energy between the two curves L/2 = 0.5a

and 1a increases fromP= 0 toP= 4 GPa with increasing pressure because of the combined effect ofthe QD size and the hydrostatic pressure. As a consequence the application of hydrostatic pressure

is leading to a more confinement of the impurity electron for small dot dimensions, in agreement withan observation made earlier inFig. 2.

The Variation of the binding energy of an on-center hydrogenic donor impurity in a cubic GaAs/Ga1xAlxAsquantum dot as a function of the hydrostatic pressure Pfor three values of the Al concen-trationsx= 0.15,x= 0.30 andx= 0.45 is given inFig. 5for a fixed temperature valueT= 200 K and the

Fig. 4. The dependence of the binding energy of an on-center hydrogenic donor impurity on the hydrostatic pressurePfor twovalues of the half quantum dot sizes L/2 = 0.5a and 1a with a fixed temperature value T= 200 K and an Al concentration

x= 0.30.

Fig. 5. Variation of the binding energy of an on-center hydrogenic donor impurity in a cubicGaAs/Ga1xAlxAsquantum dot as afunction of the hydrostatic pressure P for three values of the Al concentrations x= 0.15, x= 0.30 and x= 0.45 for a fixedtemperature valueT= 200 K and the half quantum dot sizeL/ 2 = 1a.

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half quantum dot sizeL/ 2 = 1a. It can be more clearly seen from this figure that the binding energy isan increasing function of the Al concentrationxfor a fixed dot size, pressure and temperature. Since asthe Al concentrationx increases, the height of the barrier potential increases and the wave functionbecomes more localized inside the dot, which leads to more probability of finding the electron insidethe dot and the electron is better pushed toward the impurity center. The effective Coulomb interac-tion is therefore more enhanced and as a consequence the binding energy increases. In contrast toFig. 4, the three curves of the figure relative to x = 0.15, x= 0.30 andx = 0.45 are almost, parallel, inthe calculated pressure range, from 0 to 4 GPa. This means that the difference in the binding energybetween any two values of the Al concentration x remains nearly the same for all the range of thepressure.

To study the effect of the barrier height on the donor binding energy, we present in Fig. 6, the bind-ing energy Ebof a shallow donor impurity located at the center of a cubic GaAs/Ga1xAlxAsquantum dotas a function of the half dot side widthL/2 for two different Al concentrations x = 0.15 (dashed line)andx = 0.30 (solid line) and three hydrostatic pressure valuesP= 0,P= 1 andP= 2GPa withT= 0 K.As seen from this figure, the binding energy of the shallow hydrogenic impurity depends highly onthe height of the confinement potential since high (low) Aluminum concentration implies a high

(small) potential barrier. The raise of the aluminum content x in the Ga1xAlxAs material increasesthe binding energy for a fixed value of the pressure, reflecting the higher confinement potential. Thisincrement in the binding energy is due to the fact that the excess of aluminum content expands theenergy gap between the two materials which in turn increases the potential barrier height leading tomore confinement of the donor electron inside the QD, and consequently greater binding energy. Wealso note that the increasing of mole fractions of aluminum shifts the maximum of the binding energyto lower dot side width. These results agree with those obtained by Li and Xia[26]who introduced auniform method to calculate the binding energy of the ground state as a function of the cubic QD sidewidth using the plane wave basis in the absence of the applied hydrostatic pressure. For QD with largedot sizes, the effects of the hydrostatic pressure and the potential barrier height on the binding energyis negligible, since the impurity wave function is more spread because the potential barriers are far

away and the impurity feels a bulk like environment for these large dimensions of the QD.In Fig. 7, we display the variation of the binding energy Ebof a hydrogenic donor impurity in a cubic

GaAs/Ga1xAlxAsQD of half sideL/ 2 = 1a as a function of the impurity positionx0inside the dot, along

the cube diagonal (see Fig. 1) withx0=y0=z0wherex0varying from 0 to L/2 at three hydrostatic pres-sure valuesP= 0,P= 1 andP= 2 GPa and for two values of the Al concentrationsx = 0.15 (solid line)

Fig. 6. Binding energy Ebof a shallow donor impurity located at the center of a cubic GaAs/Ga1xAlxAsquantum dot as a functionof the half quantum dot side width L/2 for two different Al concentrations x= 0.15 (dashed line) andx= 0.30 (solid line) andthree hydrostatic pressure valuesP= 0,P= 1 andP= 2 GPa withT= 0 K .



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andx= 0.30 (dashed line). According to this figure, the maximum binding energy is obtained for impu-rity located at the center of the cubic QD for the large band offset (x= 0.30) and P= 2 GPa pressure va-lue. For a specific value of the pressure and the Al concentration x, the binding energy decreases fromits maximum as the impurity moves towards the corner of the cube (L/2). This is in agreement with

the results obtained previously by Mendoza et al.[71]in a cubic QD with infinite potential barrier.We also show that the binding energy increases with increasing applied pressure and Al concen-trationx, depending on the impurity position inside the dot. We have also observed that the variationof the binding energy as a function of the pressure and the Al concentration x for the shallow donorimpurity in different positions is not homogeneous. As an example, for impurity positions closed tothe barriers, the binding energy is nearly insensitive to the hydrostatic pressure and the potential bar-rier height effects due to the small Coulomb interaction between the electron and the impurity and thepotential barrier repulsion. These results are in agreement with those obtained by Perez-Merchancanoet al.[40],who showed that the pressure effects are less pronounced for impurities on the edge.

The binding energy for an on-center shallow donor impurity as a function of the half quantum dotside widthL/2 for an Al concentrationx= 0.15 is shown inFig. 8for five different temperature valuesT

= 0,T

= 200 GPa,T

= 400 GPa,T

= 600 GPa andT

= 800 GPa and forP

= 0. It is clear from Fig. 8 that, at afixed temperature the binding energy increases until it reaches a maximum value at a specific QD sizeand then decreases as the half quantum dot side width increases. In contrast to the hydrostatic pres-sure effect shown in Fig. 2,the binding energy decreases with a raise in the temperature for all the dotsizes with a constant applied pressure. For very large QD side width, all curves approach the bulk limitvalue ofGaAs. These behaviors are the same as of Refs.[5,46]. The increment of temperature leads toan increase of the effective Bohr radius a edh

2=mde

2 due to the decreasing effective mass andincreasing the dielectric constant [43]. As a result, the value of the effective Rydberg constant,R = e2/(2eda

) reduces, which in turns decreases the potential barrier height and as a consequencethe binding energy decreases.

The dependencies of the donor binding energy as a function of the temperature in the cubic GaAs/Ga1xAlxAs quantum dot is shown inFig. 9at three different values of the Al concentration x = 0.15,

x= 0.30 andx = 0.45 with a fixed value of the QD half width, L/2 = 0.5a and an applied hydrostaticpressure P= 2 GPa. The impurity is considered at the center of the dot. As can be seen from this figure,for a given constant applied pressure and at a specified dot size, as the temperature is increased, theground state binding energy of a donor impurity decreases. This effect leads to the weakening of

Fig. 7. Binding energy Ebof a hydrogenic donor impurity in a cubic GaAs/Ga1xAlxAsQD of half sideL/ 2 = 1a as a function of the

impurity positionx0inside the dot along the cube diagonal withx0=y0=z0wherex0varying from 0 to L/2 for three hydrostaticpressure valuesP= 0,P= 1 andP= 2 GPa and for two values of the Al concentrationsx= 0.15 (solid line) andx= 0.30 (dashedline).

http://-/?-


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electron localization near the impurity with the enhancement of temperature. We also note that forT6 200 K, the binding energy decreases more slowly than for T> 200 K. Similar results have been alsoseen in quantum dots [46,48] and quantum wells [72]. This behavior is caused by the differencebetween the temperature coefficients in the dielectric constant for the two ranges of temperature thatwe have considered here. This result indicates that the temperature effect is quite significant in smallquantum dots only.

4. Conclusions

In conclusion, by using the FEM, we have studied and computed the binding energy of a donorimpurity in a cubic GaAs/Ga1xAlxAs quantum dot with the realistic potential barrier height under

Fig. 8. The binding energy for an on-center shallow donor impurity as a function of the half quantum dot side widthL/2 for Alconcentration x= 0.15 and for five different temperature valuesT= 0,T= 200 GPa,T= 400 GPa,T= 600 GPa andT= 800 GPa atzero pressure.

Fig. 9. Variation of an on-center donor binding energy versus the temperature in a cubic GaAs/Ga1xAlxAsquantum dot at threedifferent values of the Al concentrationx= 0.15,x= 0.30 andx= 0.45 with a fixed value of the QD half width,L/2= 0.5a and anapplied hydrostatic pressureP= 2 GPa and impurity position (x0,y0,z0) = (0, 0, 0).



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the simultaneous effects of hydrostatic pressure and temperature. The calculations have been made inthe effective mass and parabolic band approximations. The study considers also variation in the impu-rity position and in the dimensions of the quantum dot. The main findings can be summarized asfollows:

- The hydrostatic pressure increases almost linearly the binding energy and its effect is more notice-able for narrow dots.

- The binding energy is an increasing function of the Al concentrationxfor a fixed dot size, pressureand temperature and the effect of the potential barrier height is more significant at smaller dots.

- The binding energy has a maximum at the center of the QD and decreases as the impurity movestowards the corner of the cube (L/2).

- The increment of temperature leads to a reduction of the donor ground state binding energy and itseffect is quite significant in small quantum dots only.

- We note also that for T6 200 K, the binding energy decreases more slowly than forT> 200 K.- Our results are in a agreement with other previous works.

References

[1]G. Wang, Phys. Rev. B 72 (2005) 155329.[2]A. Sali, H. Satori, M. Fliyou, H. Loumrhari, Phys. Status Solidi B 232 (2002) 209. and references therein.[3]M. Sahin, Phys. Rev. B 77 (2008) 045317.[4]G. Bastard, Surf. Sci. 113 (1982) 165.[5]N. Porras-Montenegro, S.T. Perez-Merchancano, Phys. Rev. B 46 (1992) 9780 .[6]J.L. Zhu, J.J. Xiong, B.L. Gu, Phys. Rev. B 41 (1990) 6001.[7]F. Jiang, C. Xia, S. Wei, Phys. B 403 (2008) 165.[8]A. John Peter, K. Navaneethakrishnan, Physica E 40 (2008) 2747.[9]S.T.P. Merchancano, H.P. Gutierrez, J.S. Valencia, J. Phys.: Condens. Matter 19 (2007) 026225.

[10] C. Bose, K. Midya, M.K. Bose, Physica E 33 (2006) 116 .

[11] R.S.D. Bella, K. Navaneethakrishnan, Solid State Commun. 130 (2004) 773.[12] H. Satori, A. Sali, K. Satori, Physica E 14 (2002) 184.[13] H. Satori, M. Fliyou, A. Sali, A. Nougaoui, L. Tayebi, Phys. Low. Dim. Struct. 1 (2) (2001) 73.[14] Y.P. Varshni, Superlattices Microstruct. 29 (2001) 233.[15] F.J. Ribeiro, A. Latg, Phys. Rev. B 50 (1994) 4913.[16] Y. Yakar, B. akr, A. zmen, Superlattices Microstruct. 60 (2013) 389.[17] J.-H. Yuan, C. Liu, Physica E 41 (2008) 41.[18] C. Bose, Physica E 4 (1990) 180.[19] T. Xu, L. Yuan, J. Fang, Physica B 404 (2009) 3445.[20] A.F. Terzis, S. Baskoutas, J. Phys: Conf. Ser. 10 (2005) 77.[21] W. Xie, Phys. Lett. A 263 (1999) 127.[22] Y. Yakar, B. akr, A. zmen, J. Lumin. 134 (2013) 778.[23] W. Xie, Physica B 403 (2008) 2828.[24] L. Gong, Y. Shu, J. Xu, Q.Z. Wang, Superlattices Microstruct. 60 (2013) 311.

[25] C.S. Yang, Microelectron. J. 39 (2008) 1469.[26] S.-S. Li, J.-B. Xia, Phys. Lett. A 366 (2007) 120.[27] C. Gonzlez-Santander, T. Apostolova, F. Domnguez-Adame, J. Phys.: Condens. Matter 25 (2013) 335802.[28] E. Sadeghi, A. Avazpour, Physica B 406 (2011) 241.[29] A. Gharaati, R. Khordad, Superlattices Microstruct. 48 (2010) 276.[30] H. Satori, A. Sali, Physica E 48 (2013) 171.[31] P. Lefebvre, B. Gil, H. Mathieu, Phys. Rev. B 35 (1987) 5630 .[32] P. Lefebvre, B. Gil, J. Allegre, H. Mathieu, Y. Chen, C. Raisin, Phys. Rev. B 35 (1987) 1230.[33] B. Gil, P. Lefebvre, H. Mathieu, G. Platero, M. Altarelli, T. Fukunaga, H. Nakashima, Phys. Rev. B 38 (1988) 1215 .[34] G. Rezaei, N.A. Doostimotlagh, Physica E 44 (2012) 833.[35] C. Dane, H. Akbas, A. Guleroglu, S. Minez, K. Kasapoglu, Physica E 44 (2011) 186.[36] G. Rezaei, N.A. Doostimotlagh, B. Vaseghi, Commun. Theor. Phys. 56 (2011) 377.[37] S. Rajashabala, R. Kannan, J. Nano-Electron. Phys. 3 (2011) 1041.[38] S.T. Perez-Merchancano, R. Franco, J. Silva-Valencia Microelectron. J. 39 (2008) 383.[39] A. Sivakami, M. Mahendran, Physica B 405 (2010) 1403.

[40] S.T. Perez-Merchancano, H. Paredes-Gutierrez, J. Silva-Valencia, J. Phys.: Condens. Matter 19 (2007) 026225.[41] C. Xia, Y. Lui, S. Wei, Appl. Surf. Sci. 254 (2008) 3479.[42] A. John Peter, Physica E 28 (2005) 225.[43] J.D. Correa, N. Porras-Montenegro, C.A. Duque, Phys. Status Solidi B 241 (2004) 2440 .[44] H.O. Oyoko, C.A. Duque, N. Porras-Montenegro, J. Appl. Phys. 90 (2001) 819 .[45] S.-H. Chen, J.-L. Xiao, Physica B 393 (2007) 213.

http://refhub.elsevier.com/S0749-6036(14)00020-2/h0005http://refhub.elsevier.com/S0749-6036(14)00020-2/h0010http://refhub.elsevier.com/S0749-6036(14)00020-2/h0015http://refhub.elsevier.com/S0749-6036(14)00020-2/h0020http://refhub.elsevier.com/S0749-6036(14)00020-2/h0025http://refhub.elsevier.com/S0749-6036(14)00020-2/h0030http://refhub.elsevier.com/S0749-6036(14)00020-2/h0035http://refhub.elsevier.com/S0749-6036(14)00020-2/h0040http://refhub.elsevier.com/S0749-6036(14)00020-2/h0045http://refhub.elsevier.com/S0749-6036(14)00020-2/h0045http://refhub.elsevier.com/S0749-6036(14)00020-2/h0050http://refhub.elsevier.com/S0749-6036(14)00020-2/h0055http://refhub.elsevier.com/S0749-6036(14)00020-2/h0060http://refhub.elsevier.com/S0749-6036(14)00020-2/h0065http://refhub.elsevier.com/S0749-6036(14)00020-2/h0065http://refhub.elsevier.com/S0749-6036(14)00020-2/h0070http://refhub.elsevier.com/S0749-6036(14)00020-2/h0070http://refhub.elsevier.com/S0749-6036(14)00020-2/h0075http://refhub.elsevier.com/S0749-6036(14)00020-2/h0080http://refhub.elsevier.com/S0749-6036(14)00020-2/h0085http://refhub.elsevier.com/S0749-6036(14)00020-2/h0090http://refhub.elsevier.com/S0749-6036(14)00020-2/h0095http://refhub.elsevier.com/S0749-6036(14)00020-2/h0100http://refhub.elsevier.com/S0749-6036(14)00020-2/h0105http://refhub.elsevier.com/S0749-6036(14)00020-2/h0110http://refhub.elsevier.com/S0749-6036(14)00020-2/h0110http://refhub.elsevier.com/S0749-6036(14)00020-2/h0115http://refhub.elsevier.com/S0749-6036(14)00020-2/h0120http://refhub.elsevier.com/S0749-6036(14)00020-2/h0120http://refhub.elsevier.com/S0749-6036(14)00020-2/h0125http://refhub.elsevier.com/S0749-6036(14)00020-2/h0130http://refhub.elsevier.com/S0749-6036(14)00020-2/h0135http://refhub.elsevier.com/S0749-6036(14)00020-2/h0140http://refhub.elsevier.com/S0749-6036(14)00020-2/h0145http://refhub.elsevier.com/S0749-6036(14)00020-2/h0150http://refhub.elsevier.com/S0749-6036(14)00020-2/h0150http://refhub.elsevier.com/S0749-6036(14)00020-2/h0155http://refhub.elsevier.com/S0749-6036(14)00020-2/h0160http://refhub.elsevier.com/S0749-6036(14)00020-2/h0165http://refhub.elsevier.com/S0749-6036(14)00020-2/h0170http://refhub.elsevier.com/S0749-6036(14)00020-2/h0175http://refhub.elsevier.com/S0749-6036(14)00020-2/h0180http://refhub.elsevier.com/S0749-6036(14)00020-2/h0185http://refhub.elsevier.com/S0749-6036(14)00020-2/h0190http://refhub.elsevier.com/S0749-6036(14)00020-2/h0195http://refhub.elsevier.com/S0749-6036(14)00020-2/h0200http://refhub.elsevier.com/S0749-6036(14)00020-2/h0200http://refhub.elsevier.com/S0749-6036(14)00020-2/h0205http://refhub.elsevier.com/S0749-6036(14)00020-2/h0210http://refhub.elsevier.com/S0749-6036(14)00020-2/h0215http://refhub.elsevier.com/S0749-6036(14)00020-2/h0220http://refhub.elsevier.com/S0749-6036(14)00020-2/h0220http://refhub.elsevier.com/S0749-6036(14)00020-2/h0225http://refhub.elsevier.com/S0749-6036(14)00020-2/h0225http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0220http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0215http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0210http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0205http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0200http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0195http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0190http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0185http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0180http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0175http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0170http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0165http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0160http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0155http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0150http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0145http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0140http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0135http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0130http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0125http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0120http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0115http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0110http://refhub.elsevier.com/S0749-6036(14)00020-2/h0110http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0105http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0100http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0095http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0090http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0085http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0080http://refhub.elsevier.com/S0749-6036(14)00020-2/h0080http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0075http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0070http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0065http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0060http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0055http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0050http://-/?-http://refhub.elsevier.com/S0749-6036(14)00020-2/h0045http://refhub.elsevier.com/S0749-6036(14)00020-2/h0040http://refhub.elsevier.com/S0749-6036(14)00020-2/h0035http://refhub.elsevier.com/S0749-6036(14)00020-2/h0030http://refhub.elsevier.com/S0749-6036(14)00020-2/h0025http://refhub.elsevier.com/S0749-6036(14)00020-2/h0020http://refhub.elsevier.com/S0749-6036(14)00020-2/h0015http://refhub.elsevier.com/S0749-6036(14)00020-2/h0010http://refhub.elsevier.com/S0749-6036(14)00020-2/h0005http://-/?-


15/15

[46] A.M. Elabsy, Egypt. J. Sol. 23 (2000) 267.[47] A.M. Elabsy, Phys. Scr. 59 (1999) 328.[48] A. Sivakami, V. Gayathri, Superlatt. Microstruct. 58 (2013) 218.[49] M. Kirak, Y. Altinok, S. Yilmaz, J. Lumin. 136 (2013) 415.[50] B. Vaseghi, T. Sajadi, Physica B 407 (2012) 2790.[51] G. Rezaei, S.F. Taghizadeh, A.A. Enshaeian, Physica E 44 (2012) 1562.

[52] S.J. Liang, W.F. Xie, Eur. Phys. J. B 81 (2011) 79.[53] A. Sivakami, M. Mahendran, Superlatt. Microstruct. 47 (2010) 530.[54] U. Yesilgu, E. Kasapoglu, H. Sari, I. Sokmen, Superlatt. Microstruct. 48 (2010) 509.[55] L.R. Ram-Mohan, Finite Element and Boundary Element Applications in Quantum Mechanics, Oxford University Press,

Oxford, UK, 2002.[56] B. Welber, M. Cardona, C.K. Kim, S. Rodriquez, Phys. Rev. B 12 (1975) 5729.[57] D.E. Aspnes, Phys. Rev. B 14 (1976) 5331.[58] S. Adachi, J. Appl. Phys. 58 (1985) R1.[59] M.E. Mora-Ramos, S.Y. Lpez, C.A. Duque, Eur. Phys. J. B 62 (2008) 257.[60] R.F. Kopf, M.H. Herman, M. Lamont Schnoes, A.P. Perley, G. Livescu, M. Ohring, J. Appl. Phys 71 (1992) 5004.[61] G.A. Samara, Phys. Rev. B 27 (1983) 3494.[62] F.P. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 244.[63] B. Rockwell, H.R. Chandrasekhar, M. Chandrasekhar, A.K. Ramdas, M. Kobayashi, R.L. Gunshor, Phys. Rev. B 44 (1991)

11307.[64] J.A. Tuchman, I.P. Herman, Phys. Rev. 45 (1992) 11929.

[65] P.Y. Yu, M. Cardona, Fundamentals of Semiconductors, Springer, Berlin, 1998 .[66] D.J. Wolford, J.A. Bradly, Solid State Commun. 53 (1985) 1069.[67] S.-S. Li, J.-B. Xia, J. Appl. Phys. 101 (2007) 093716 .[68] G. Rezaei, S. Mousavi, E. Sadeghi, Physica B 407 (2012) 2637.[69] G.J. Zhao, X.X. Liang, S.L. Ban, Phys. Lett. A 319 (2003) 191.[70] A.L. Morales, A. Montes, S.Y. Lpez, C.A. Duque, J. Phys. Condens. Matter 14 (2002) 987.[71] C.I. Mendoza, G.J. Vazquez, M. del Castillo-Mussot, H. Spector, Phys. Rev. B 71 (2005) 075330 .[72] A. Hakimyfard, M.G. Barseghyan, C.A. Duque, A.A. Kirakosyan, Physica B 404 (2009) 5159.

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quantum dot using the fem simulation

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