The self-consistent calculation of a spherical quantum dot: A quantum genetic algorithm study

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doi:10.1016/j.ph

�Correspondifax: +90332 24

E-mail addre

m.sahin@lycos.

Physica E 28 (2005) 247–256

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The self-consistent calculation of a spherical quantum dot:A quantum genetic algorithm study

Mehmet S-ahina,�, Mehmet Tomakb

aSelcuk University, Faculty of Sciences and Arts, Physics Department, Kampus, 42031 Konya, TurkeybMiddle East Technical University, Physics Department, Inonu Bulvarı 06531 Ankara, Turkey

Received 7 March 2005; accepted 21 March 2005

Available online 31 May 2005

Abstract

In this study, we have calculated the subband energy level, potential profile, and the corresponding wavefunction and

chemical potential for different temperatures and donor concentrations in a spherical quantum dot self-consistently. We

have also investigated the effect of exchange-correlation potential on the energy levels. In addition, we have checked the

applicability of quantum genetic algorithm to a realistic self-consistent quantum dot problem. In all computations, the

penetration of wavefunction to the barrier region is taken into account.

r 2005 Elsevier B.V. All rights reserved.

PACS: 02.60.Pn; 73.20.Dx; 73.40.Kp

Keywords: Quantum dot; Self-consistent calculation; Genetic algorithm

1. Introduction

Semiconductor quantum nanostructures (quan-tum wells, wires or dots) have found variousapplication areas especially as electronic devicessuch as single electron transistor, quantum welland quantum dot infrared photodetector (QWIPand QDIP) [1–3]. Therefore, these structures have

e front matter r 2005 Elsevier B.V. All rights reserve

yse.2005.03.010

ng author. Tel.: +90332 223 2598;

1 0106.

sses: [email protected],

com (M. S-ahin).

been intensively studied both theoretically andexperimentally in condensed matter physics [4–7].Many analytical and numerical studies on energylevels and other physical properties of quantumdots (QDs) have been reported [7–9]. Differenttechniques and approximations had been used forthis purpose such as variational method, perturba-tion method, matrix diagonalization, Monte Car-lo, etc. Each of these techniques has someadvantages and disadvantages. For example, inthe traditional variational method, trial wavefunc-tion chosen must be well suited to the systemconsidered. If the wavefunction is not properly

d.

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Rdot

Rdot

Vb

ED

GaAs

AlGaAs 0

Fig. 1. Schematic representation of a quantum dot.

M. S- ahin, M. Tomak / Physica E 28 (2005) 247–256248

chosen, the results may be far from the exact ones.Most of the structures considered include a fewelectrons system [10]. However, recently, theinvestigation of the doped QD or box problem isdone by many researchers [11–16]. An extensivereview on electronic structure of quantum dots isreported by Reimann and Manninen [17]. In mostof the technological application of QD, such asQDIP, modulation-doped QD heterostructureshave been used [18]. Therefore, that this kind ofheterostructure is investigated theoretically isvery important for the device application of QD.The self-consistent effects of the accumulatedelectronic charge of bound states in a quantumbox have been investigated by Kumar et al. [16].Todorovic et al. [11] determined electronic struc-ture of spherical semiconductor QD self-consis-tently using Morse-type parametrized potential.They have not considered wavefunction penetra-tion to the barrier region and the exchange-correlation potential. In this study, we haveconsidered the penetration of the wavefunctionto the barrier region and also inspected effect ofthe exchange-correlation potential on the energylevels.Earlier, we have investigated the electronic

properties of 2DEG in a heterojunction by thequantum genetic algorithm [19]. In addition, wehave discussed this method, which is basedessentially on energy minimization procedure,and compared it with the standard variationaltechniques and demonstrated that it has importantdifferences, and advantages over the standardvariational method in a hydrogenic impurityproblem in a spherical QD [20]. In this paper, wepresent a calculation of ground state energy levelof a spherical QD by using quantum geneticalgorithm (QGA). Here, we achieved self-consis-tency in contrast to our earlier work [20].QGA method has been used in many different

fields in science and engineering [21,22], andefficiency and superiority of GA method has beenestablished [20,21] and so, the details of the QGAmethod are not discussed in this paper. In spite ofthe fact that QGA method is based on energyminimization, it exhibits some important differ-ences from the variational techniques and thesedifferences were presented in Ref. [20].

To the best of our knowledge, QGA method hasnot been applied to any self-consistent QDproblem.The main purpose of this study is to show the

applicability of QGA on self-consistent electronicstructure calculations of a donor-doped sphericalQD and investigate the ground state energy level,potential profile, the corresponding envelopefunction, chemical potential and their dependenceon the temperature and donor doping concentra-tion using the Hartree approximation.This paper is organized as follows: The next

section presents the theory and formulation. InSection 3, a brief description of QGA method ispresented. Results and discussion are given inSection 4. In the last section, we present theconclusion.

2. Theory and formulation

Let us consider a spherically symmetric quan-tum dot with radius Rdot which is embedded in abulk semiconductor as seen in Fig. 1. The coupledPoisson and Schrodinger equations have to besolved self-consistently in the effective massapproximation and Ben Daniel–Duke boundaryconditions for the energy levels and the chargetransfer in a quantum dot

�_2

2

qqr

1

mðrÞ

qqr

þ2

mðrÞr

qqr

�‘ð‘ þ 1Þ

mðrÞr2

� �Ri;‘ðrÞ

þ ½VbðrÞ � eV scðrÞ�Ri;‘ðrÞ ¼ Ei;‘Ri;‘ðrÞ, ð1Þ

and

d2

dr2þ2

r

d

dr

� �VHðrÞ ¼

4pe

kðrÞ½nðrÞ � Nþ

DðrÞ�, (2)

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M. S- ahin, M. Tomak / Physica E 28 (2005) 247–256 249

where VHðrÞ is the Hartree potential, mðrÞ is theposition-dependent effective mass, nðrÞ is theelectron density in the QD, Nþ

D is the ionizeddonor concentration, kðrÞ is the position-depen-dent dielectric constant, RiðrÞ is ith subband radialwavefunction, ‘ is orbital quantum number, VbðrÞ

is the barrier potential, _ is the reduced Planckconstant, e is the electronic charge, and V scðrÞ isthe self-consistent potential, defined as

V sc ¼ VHðrÞ þ VxcðrÞ, (3)

where V xcðrÞ is the exchange-correlation potential.The charge density in the QD is given by

nðrÞ ¼1

4p

X‘¼0

2ð2‘ þ 1ÞXi¼1

jRi;‘j2f FDðEi;‘Þ, (4)

where the second summation term is over thenumber of bound states for any particular value of‘; 2ð2‘ þ 1Þ expression is for spin and magneticquantum number degeneracy and f FDðEiÞ is theFermi–Dirac distribution function given as

f FDðEi;‘Þ ¼1

1þ exp½ðEi;‘ � mÞ=kBT �, (5)

where m is the chemical potential, kB is theBoltzmann constant and T is the temperature.The largest contribution to the charge density inthe QD is originated from the ‘ ¼ 0 state. Thispoint is also elaborated by Todorovic et al. [11].The ionized donor concentration is determined bymeans of the following relation:

NþD ¼

ND

1þ 2 exp ðm�ðVbðrÞ�EDÞ�eVHðrÞÞkBT

h i , (6)

where ND is the doped donor concentration andED is the donor binding energy at the barrierregion. As for the Poisson equation Eq. (2), theheterostructure is required to be in electricalequilibrium, namelyZ 1

0

½nðrÞ � NþDðrÞ�r

2 dr ¼ 0. (7)

It is also required for thermodynamical equili-brium, that the chemical potential m is a constantand equal in both the QD and barrier region.The many-body effects are taken into account.

By using the Hedin–Lundqvist [23] expression

used for the exchange-correlation potential,

Vxc ¼ �2

kðrÞbðrsÞ

3nðrÞ

8p

� �1=3

,

bðrsÞ ¼ 1þ 0:0368rs ln 1þ21:0

rs

� �,

rs ¼3

4pnðrÞ

� �1=3

. ð8Þ

Eq. (2) is solved by the finite difference methodto calculate the Hartree potential VHðrÞ. We useQGA to solve the coupled Poisson and Schrodin-ger equations self-consistently.

3. Genetic process

Genetic algorithms (GA) are general search andnumerical optimization methods inspired by bothnatural selection and genetics [24]. GA is firstlyproposed by Holland [25]. Recently, this methodhas appeared to be used more frequently in theoptimization and minimization problems for thequantum mechanical systems [19–21]. GA processis based on three basic operations: reproduction(or copy), crossover and mutation.In this study, we have used the Gaussian-type

wavefunction, which includes penetration into thebarrier region,

RkðrÞ ¼ Ar‘ expð�akrMk Þ, (9)

where A is normalization constant, ‘ is the orbitalquantum number, ak is a random number selectedat the interval ð0;RmaxÞ, Mk is a random integer,which is 2 or 3, Rdot is the QD radius and Rmax isthe distance from center of QD to the edge of thedoped barrier region. The functional form given inEq. (9) is used to generate only the initialpopulation then the population is let free toevolve. The functional form given in Eq. (9) ischosen so as to satisfy some general requirementssuch as integrability of the wavefunction andcontinuity of the wavefunction and its derivatives,etc. These requirements are valid for one, two orthree dimensional problems [21].

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The normalization constant A is determinednumerically by the normalization condition,Z 1

0

jRkðrÞj2r2 dr ¼ 1. (10)

Initial population has been created numerically fromEq. (9) for random values of ak and Mk ðk ¼

1 . . . npopÞ and assigned to the two-dimensional vectorarrays. Here, the population number ðnpopÞ is selectedto be 100. The randomly created population has beennormalized using Eq. (10). Thus, a normalizedrandom population of wavefunction (or individuals)is obtained as an initial generation. Expectationvalues of energy are determined for each individualfrom this generation by means of

Ek ¼ hRkðrÞjHjRkðrÞi. (11)

These energy eigenvalues are used for determining thefitness values. For this reason, we use the followingexpression:

Fitness ½Rk� ¼ exp½�sðEk � EavÞ�, (12)

where s is a constant and Eav is the average of theenergy eigenvalues. By using this fitness value, aroulette wheel [26] is constituted and a selectionprocedure has been performed. In this selectionprocedure, usually, better individuals are selected;however, sometimes less fit individuals can also beselected and a new generation is created from this setof chosen individuals. This process is usually knownas reproduction or copy.The crossover is realized over two randomly

chosen individuals or wavefunctions. For thispurpose, we take two randomly chosen individualsand thus two new wavefunctions are produced byusing them as

R01ðrÞ ¼ R1ðrÞcrðrÞ þ R2ðrÞ½1� crðrÞ�

R02ðrÞ ¼ R2ðrÞcrðrÞ þ R1ðrÞ½1� crðrÞ�, ð13Þ

where crðrÞ is a random number, its value israndomly selected from a uniform distributionbetween (0,1). The crossover probability is takenas 0.35.In the mutation procedure, a Gaussian-like

function is used. A random mutation functionRM is constructed by using initial populationfunction through Eq. (9) and then added to arandomly chosen wavefunction Rk for creating a

new one as

R0kðrÞ ¼ RkðrÞ þ DBRMðrÞ, (14)

where B is the random number in (0,1) and D is avariable mutation intensity [27].

D ¼ ð1� Cð1�iter=maxitÞd Þ, (15)

where C is a number in (0,1), iter is the currentiteration number, maxit is the maximum iterationnumber and d is a system-dependent parameter.As seen that when the iter goes to maxit, D is goingto zero. We have selected the mutation probabilityas 0.01.Here, we have performed linear scaling over the

individuals for efficiency of GA. Details of thesegenetic processes (copy, crossover, mutation,elitism, scaling, etc.) are explained in Refs. [24,26].Through the whole GA iteration, copy or

reproduction, crossover and mutation operationswere randomly performed over the individuals.After the application of the genetic operations, thenew obtained populations were renormalized.In this study, all calculations are performed

numerically. The derivatives are calculated overthe five mesh points. In computation of theintegrals, Simpson’s method was used.Our algorithm may be summarized briefly as

follows:

(i)
Initial population is created and normalized. (ii) The expectation values of energy are deter-
mined for each individual in the barrierpotential. Hartree potential initially is takenas zero.

(iii)
Fitness values are computed with these energyeigenvalues and the best fitness is determined.
(iv)
A new generation is created from the old onewith genetic operations (copy or reproduction,crossover and mutation operations) and thenis normalized.
(v)
Poisson’s equation is solved and Hartreepotential is determined for each of theindividual using the new generations.
(vi)
Hartree potential calculated is added to thebarrier potential and returned back to step (ii).
This process is repeated until the best convergenceis obtained.

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4. Results and discussion

In most of theoretical studies on sphericalquantum dots, authors use the material para-meters of GaAs for well region and that ofAlGaAs for barrier region [11,28–31], since theirbasic physical properties such as band mismatch,effective masses and dielectric constants are wellknown [32]. Hence, we use these material para-meters and give detailed results for the GaAs/AlGaAs quantum dot.We have used atomic units through all

calculations, where Planck constant _ ¼ 1, theelectronic charge e ¼ 1 and the electron massm ¼ 1. Effective Bohr radius is a

0 ¼ 100 (Aand effective Rydberg energy is R

y ¼ 5:5meV.We have taken mGaAs ¼ 0:067m0; mAlGaAs ¼

0:092m0; ED ¼ 60meV; Vb ¼ 228meV; kGaAs ¼

13:1; kAlGaAs ¼ 12:2 as material parameters. Wehave taken the effective masses of electronsinside GaAs and AlGaAs as m1 and m2, andsimilarly dielectric constants as k1 and k2,respectively. The position-dependent effectivemass and dielectric constant can be defined as

Number o0 5000 10000 15000

Ene

rgy

(eV

)

0.0670

0.0675

0.0680

0.0685

0.0690

0.0695

T=300 KND=5.1015cm-3

Best fitness not kept

25600

Ene

rgy(

eV)

0.06736

0.06738

0.06740

0.06742

0.06744

0.06746

Fig. 2. The evolution of the energy eigenvalue with the number of ite

follows [29]:

mðrÞ ¼

1; roRdot

m2

m1; r4Rdot

8<: ,

kðrÞ ¼1; roRdot

k2k1

; r4Rdot

8<: . ð16Þ

The traditional variational calculations for thefinite well heterostructures, the widely employedapproach is to select two-parts wavefunction tospecify the system, i.e. one-part related to the innerregion of the well and the other corresponds to theouter region. But, in this study, we have used awavefunction for describing both parts of thestructure, and based on this function, we havecreated the initial population numerically. Thistype of wavefunction may not be appropriate forother variational calculations.In Fig. 2, the evolution of energy eigenvalue

with the number of iterations is plotted for keepingand not keeping of the best fitness. As seen fromthis figure, if the best fitness is not kept, the

f Generations20000 25000 30000 35000

Best fitness kept

Number of Generations25620 25640 25660 25680 25700

Best fitness not kept

rations. The inset shows small oscillation in an enlarge manner.

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r (a0*)0 1 2 3 4 5

Ene

rgy

(meV

)

0

50

100

150

200

250

300

R10

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Best fitness not keepingBest fitness keeping

E10

µ

T=300 KND=5.1015cm-3

Fig. 3. The calculated self-consistent potential, the ground subband energy, chemical potential and the corresponding envelope

function at T ¼ 300K.


variation of the energy eigenvalue with an iterationshows an oscillatory behavior during wholegenerations. However, evolution of the energydecreases with increasing number of iterations.Nearly, after 5000 iterations, the energy eigenvalueconverges and oscillations are in the very narrowrange of �0:03meV.In Fig. 3, the calculated self-consistent poten-

tials V sc, the single particle energy level and thechemical potential are shown in both situations forRdot ¼ 1:0a

0;T ¼ 300K and ND ¼ 5� 1015 cm�3.In this calculation, the exchange and correlationenergies have not been considered. The groundsubband wavefunction, determined by QGA, isalso shown in this figure. As seen from this figure,no difference is observed between energy levels,chemical potentials and self-consistent potentialprofiles in the keeping or not keeping of the bestfitness. Nevertheless, very little difference is seenbetween wavefunctions especially in ro0:8a

0.As it is seen from Figs. 2 and 3, in both

situations, it really does not change energy muchwhether one keeps the best fitness or not and thesystem converges to the same stable solution.

Fig. 4 shows the ground subband energy leveland chemical potential for different dot radii atT ¼ 300K and ND ¼ 5� 1015 cm�3. The groundsubband energy values are dependent on dot radiias expected. However, the chemical potential doesnot seem to depend on the dot radius. A smallincrease is seen only for ro0:8a

0.In Fig. 5, variation of energy level and chemical

potential with the donor doping concentrations isseen for Rdot ¼ 1:0a

0 at 300K. While the energylevel does not depend strongly on the donordoping concentration, the chemical potentialdepends strongly on the donor concentration.Similar results for the energy level have beenreported in Ref. [11]. At the low doping donorconcentration, the ionized donor concentration israther small. So, the accumulating electron con-centration is small in the dot region and, conse-quently, the chemical potential is lower. However,while the doping concentration increases, thechemical potential shows sudden increase, andcontinues to increase linearly.The occupation probability is shown in Fig. 6.

As seen from the figure, while the occupation

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Rdot(a0*)0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Ene

rgy

(meV

)

0.0

30.0

60.0

90.0

120.0

150.0210.0

212.5

215.0

Empty symbols refer to chemical potentialFilled symbols refer to single particle energy level

T=300 KND=5.1015cm-3

Fig. 4. Variation of the subband energy and chemical potential with the QD radius.

ND (x1015 cm-3)

1 100

Ene

rgy

(meV

)

40

80

120

160

200

240

Single particle energyChemical potential

T=300 KRdot=1.0 a0*

10

Fig. 5. Variation of ground subband energy and chemical potential with the donor doping concentration.


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probability is nearly 0.5 at low doping concentra-tions, in the region of sudden jump, this valueapproaches to 1. Consequently, although theground subband is not fully filled at the low donorconcentrations, the ground level is filled and

ND(x1015cm-3)

1 10 100

f FD

0.4

0.5

0.6

0.7

0.8

0.9

1.0

T=300 KRdot=1.0 a0*

Fig. 6. Occupation probability of the ground subband with the

donor doping concentration.

T 50 100 150 200 2

Ene

rgy

(meV

)

65

70

75160

180

200

220

240

ND=1.1016cm-3

ND=5.1016cm-3

Empty symbols refer to chemical potFilled symbols refer to single particle

R

Fig. 7. Variation of ground energy level and

furthermore the upper energy levels begin to befilled when the doping concentration is increased.Fig. 7 shows temperature effect on ground

subband energy level and chemical potentialbetween 100 and 400K for two different dopingconcentrations, 1� 1016 and 5� 1016 cm�3. Asseen from the figure, ground energy level is notdependent strongly on the temperature especiallyfor higher doping concentration. The chemicalpotential is not varying with temperature at lowerdoping concentration, but it increases linearly withtemperature at higher concentrations. This in-crease indicates that the ionized donor concentra-tion is increasing with temperature at higherconcentrations while it is not changed at lowerconcentrations.As seen from Figs. 4, 5 and 7, the chemical

potential strongly depends on the temperature anddonor concentration. However, in contrast tochemical potential, ground subband energy showsstronger dependency on QD radius.In Fig. 8, variation of occupation probability of

the ground subband level is seen for different

(K)50 300 350 400 450

ential energy level

dot=1.0 a0*

chemical potential with temperature.

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doping concentrations of 1� 1016 and5� 1016 cm�3. In lower doping, while the occupa-tion probability is 1.0 at T ¼ 100K, this value isdecreasing to about 0.93 at 400K. The ionizeddonor concentration does not increase withtemperature at lower doping levels as seen clearlyin Fig. 7. The excitations from ground to upper

T (K)50 100 150 200 250 300 350 400 450

f FD

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1.00

ND=1.1016cm-3

ND=5.1016cm-3

Rdot=1.0 a0*

Fig. 8. Occupation probability of the ground subband with

temperature.

r (a0*0 1 2

Ene

rgy

(meV

)

0

50

100

150

200

250

300

E10

µ

Fig. 9. Effect of the exchange-correlation potential on the ground

levels increase with temperature. However, theseexcitations are not excessive and therefore, thelargest contribution to the charge density is stilloriginated from the ground subband. At higherdoping concentrations, number of electrons in thedot is much more, so upper levels may be filledand, so the transition from ground to upper levelsis smaller.In Fig. 9, the effect of the exchange-correlation

potential is investigated on the energy levels andwavefunction for ND ¼ 5� 1015 cm�3 at T ¼

300K and compared with no exchange-correlationcase. As can be seen from the figure, the groundsubband energy decreases by about 5meV, but thechemical potential does not change significantly inthe presence of the exchange-correlation potential.The amplitude of the wavefunction is a bit biggerwith many-body effect in dot region, especiallyaround the center of the dot. But, no significantvariation of the potential profile is observed withthe exchange-correlation effect.In this study, we have neglected the image term

in the calculations, the effect of this term isapparently extremely small for these materials anddiscussed in detail by Stern and Das Sarma [33].

)3 4 5

R10

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

No exc-corr. potentialWith exc-corr. potential

T=300 KND=5.1015cm-3

subband energy, chemical potential and the potential profile.

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5. Conclusion

We have calculated the electronic structure of amodulation-doped spherical quantum dot as afunction of the dot radius, doping concentrationand temperature. In the calculations, we usequantum genetic algorithm in Hartree approxima-tion and solve Poisson and Schrodinger equationsself-consistently. The effect of the exchange-correla-tion potential is also investigated. However, none ofthese effects seem to affect dramatically the energylevels and potential profile of the problem. At thesame time, applicability of QGA is checked. Wehave shown that the QGA method is quite efficientfor the realistic self-consistent quantum dot problem.Therefore, this method can be utilized in thecalculation of the electronic structure of the quan-tum nanostructures. Keeping or not keeping of thebest fitness is not important for solution of theproblem. The application of this technique is rathereasy in comparison to other stochastic techniques,and run time is surprisingly shorter than the others.

Acknowledgements

This study is partially supported by SelcukUniversity under grant No. BAP 2001/112.

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Documents

The self-consistent calculation of a spherical quantum dot: A quantum genetic algorithm study