Dipolar transformations of two-dimensional quantum dots arrays proven byelectron energy loss spectroscopyR. E. Moctezuma, J. F. Nossa, A. Camacho, J. L. Carrillo, and J. M. Rubí Citation: J. Appl. Phys. 112, 024105 (2012); doi: 10.1063/1.4737791 View online: http://dx.doi.org/10.1063/1.4737791 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v112/i2 Published by the American Institute of Physics. Related ArticlesElectrical effects of metal nanoparticles embedded in ultra-thin colloidal quantum dot films Appl. Phys. Lett. 101, 041103 (2012) Configuration interaction approach to Fermi liquid–Wigner crystal mixed phases in semiconductornanodumbbells J. Appl. Phys. 112, 024311 (2012) Photocurrent spectroscopy of site-controlled pyramidal quantum dots Appl. Phys. Lett. 101, 031110 (2012) Influence of impurity propagation and concomitant enhancement of impurity spread on excitation profile of dopedquantum dots J. Appl. Phys. 112, 014324 (2012) A single-electron probe for buried optically active quantum dot AIP Advances 2, 032103 (2012) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

Downloaded 31 Jul 2012 to 148.228.150.54. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

Dipolar transformations of two-dimensional quantum dots arrays provenby electron energy loss spectroscopy

R. E. Moctezuma,1 J. F. Nossa,2 A. Camacho,2 J. L. Carrillo,1 and J. M. Rubı31Instituto de Fısica, Universidad Autonoma de Puebla, A.P. J-48 Puebla 72570, Mexico2Departamento de Fısica, Universidad de los Andes, A.A. 4976, Bogota D.C., Colombia3Departament de Fısica Fonamental, Facultat de Fısica, Universitat de Barcelona,C. Martı i Franquez 1, 08028, Barcelona, Spain

(Received 8 June 2011; accepted 2 July 2012; published online 26 July 2012)

Dipolar transformations of two-dimensional arrays of quantum dots are investigated theoretically.

Homogeneous and non-homogeneous two-dimensional distributions are modeled by considering

sections of the quantum dot array with different confinement potential. The dipolar transformations

are tested by means of simulation of the dispersion of electrons of a beam traveling parallel to the

plane of the quantum dot array. We calculate the electron energy loss function as a function of the

temperature for different non-homogeneous distributions, considering several confinement

potentials and coupling potentials among the quantum dots. Our results indicate that it would be

possible by electron energy loss spectroscopy experiments to detect these dipolar transformations.VC 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4737791]

I. INTRODUCTION

Electron scattering due to the interaction with elementary

excitations of many kinds of physical systems, and in a wide

variety of modalities, has been a useful tool for the experi-

mental physics, chemistry, and material science research.1,2

One of the most versatile experimental techniques for the

study of the electronic structure, and other associated physical

properties of non homogeneous systems is electron energy

loss spectroscopy (EELS). In EELS experiments, a material is

exposed to an electron beam which possess an energy in the

appropriate range to interact with the elementary excitations

of that system. Some of the electrons of the beam lose energy

and exchange momentum due to this inelastic scattering. The

analysis of the energy and momentum exchanges provide rich

information about the energy spectra of the elementary excita-

tions and other physical characteristics of the system.3

In the present work, the electron energy loss theory

developed by Fuchs et al.4–6 is adapted to investigate theo-

retically the dipolar properties, dielectric, paraelectric, and

ferroelectric7 of two-dimensional (2D) arrays of self-

assembled quantum dots. A quantum dot (QD) of this kind is

a semiconductor nanostructure which confines in the three

spatial directions one or more electrons.8 The dielectric func-

tion and the energy loss function, namely, the probability of

electron energy loss per unit energy interval, unit time, and

unit length, for several conditions and several 2D distribu-

tions of quantum dots are calculated. Homogeneous as well

as non-homogenous 2D arrays composed by QDs of different

confinement potentials are analyzed.

The model here considered for a 2D granular nano-

composite composed by a AlxGa1-xAs matrix, containing em-

bedded GaAs QDs is the following: the matrix is assumed to

be a continuous semiconductor medium where the electron

has an effective mass me ¼ 0:0962 containing embedded

QDs of three possible geometries (spherical, cylindrical, or

conical), where the electron has an effective mass

me ¼ 0:063. Three kinds of QDs distributions will be consid-

ered here; in the first one, that would correspond to the

dielectric case, it is assumed a distribution of independent

QDs that in the presence of an external electric field acquire

a dipolar moment. In the second kind, paraelectric case, we

assume a distribution of independent QDs with an intrinsic

dipolar moment. In the third one, we assume a distribution of

interactive QDs.

In the ferroelectricity phenomenon, the ordering forces

come from strong long-range electric dipolar interactions. It

has been exhaustively discussed that the mean-field theoreti-

cal approach becomes exact in the long-range limit of equal

interactions between all constituents.9 Assuming that this is

the condition in an infinite 2D QD distribution, we adopt a

mean field approximation to describe a system of interacting

QDs. Furthermore, we consider that additionally to the mean

field interaction, the QDs interact through a wetting layer.10

The coupling among the quantum dots through this wetting

layer yields that the electronic states exhibit a probability of

localization predominantly within the quantum dots, but

there exist also a probability of being located outside them,

i.e., in the region of the wetting layer.

II. ELECTRON ENERGY LOSS DUE TOHOMOGENEOUS QD DISTRIBUTIONS

The procedure by which we address the description of

some ferroic properties of 2D distributions of QDs is the fol-

lowing: the eigenfunctions and eigenvalues of the quantum

dots are found solving numerically the time independent

Schrodinger equation by using a finite element package.11

From the knowledge of the eigenstates, Wi the corresponding

electric dipolar moments ~pi are then obtained by means of

the expression,

~pi ¼ e

ðW�~rWd~r: (1)

0021-8979/2012/112(2)/024105/8/$30.00 VC 2012 American Institute of Physics112, 024105-1

JOURNAL OF APPLIED PHYSICS 112, 024105 (2012)

Downloaded 31 Jul 2012 to 148.228.150.54. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

From here, the electric susceptibility, the dielectric func-

tion, and other quantities can be directly calculated.

If the eigenfunction Wi has a definite parity, the dipolar

moment is null. The three geometries of the confinement

potential mentioned above generate eigenfunctions of defi-

nite parity. The azimuthal symmetry can be broken by add-

ing an external electric potential. This is the physical

situation we will consider to investigate the dielectric proper-

ties of QD distributions. For the discussion of the paraelec-

tric and ferroelectric cases, in addition to this external field,

we will consider another electric field, Eext, that interact with

the dipolar moment of the QDs to align them, counteracting

the disorientation effect of thermal fluctuations.

To study the alignment of the interactive QDs’ dipolar

moments generated by an external field, an internal potential

originated in the QD distribution is assumed.12 This would

be the behavior analogous to a ferroelectric response in a

molecular ferroelectric material. In this way, the Hamilto-

nian of the system can be written in the form

H ¼ �XN

i¼1

pizEext þ1

2

XN

i¼1

XN

j¼1

Jpizpjz; (2)

where piz and pjz are given by expression (1), and Eext is a

constant which in this work has been taken as 1 meV/nm.

In the Ising approximation, this Hamiltonian can be cast

into

H ¼ �XN

i¼1

pizEext þXN

i¼1

�Xk

j¼1

1

2Jpjz

�piz; (3)

with k the coordination number and J the exchange integral.

By defining the internal field in the mean field approxi-

mation by the expression

Eint ¼Xk

j¼1

1

2Jpjz; (4)

we arrive to a Weiss form of the Hamiltonian

H ¼ �XN

i¼1

pizEext �XN

j¼1

pizEint (5)

that can be rewritten in the form

H ¼ �XN

i¼1

piðEext þ EintÞ ¼ �XN

i¼1

pizETot: (6)

The internal constant field, Eint, value can be properly

chosen to describe the main symmetries and the intensity of

the interaction among the nearest neighbors.

By following this model, the numerical solution of the

time independent Schrodinger equation allows us to calcu-

late the linear dielectric response function, and on these basis

to describe the dipolar properties: dielectric, paraelectric,

and within the limitation of a mean field approximation, the

ferroelectric ones of the QD distribution. Furthermore, it has

been shown that by the knowledge of the dielectric function

it is possible to calculate the energy loss function of an elec-

tron beam traveling parallel to the surface that contains the

quantum dots, i.e., the probability that the electron beam loss

energy due to the interaction with the QD distribution per

unit path length and unit energy. Since this quantity is meas-

ured in EELS experiments, it could be the procedure to test

experimentally our theoretical predictions.

First, the dielectric properties of a 2D homogeneous dis-

tributions of QDs are analyzed on the basis of the energy

loss function. We consider an electron beam with speed vl,

energy El ¼ 100 keV, traveling parallel to the surface that

contains the quantum dots.

The probability per unit path length and per unit energy

interval, of scattering with energy loss E, is given by6

FðEÞ ¼ 1

a0El

1

pK0

2z0vl

x

� �Im

eMðxÞ � 1

eMðxÞ þ 1

� �; (7)

where K0 is the modified zeroth-order Bessel function, a0 is

the Bohr radius, El and vl the energy and velocity of the inci-

dent electron beam, respectively, and z0 is the impact param-

eter that typically has the value of some few nm.

We start assuming independent QDs, then, to describe

homogeneous arrays, one may introduce a linear density of

these QDs to obtain directly the total energy loss function

per unit length and unit time. Empirically, we have found

that for QDs of radius r¼ 4 nm, if the density is chosen to be

3:5� 105 QDs per linear centimeter, the electronic density

of QDs do not overlap, behaving the QDs as independent.

We aim to use the theoretical procedure developed by

Fuchs et al.4,5 to analyze the dielectric response of nano-

composites, to investigate the dielectric properties of the QD

distribution. To study these granular nano-composites by

simulating EELS experiments, we consider an electron beam

traveling parallel to the surface with an energy El ¼ 100 keV

and calculate the energy loss function for several physical

situations. Some other proposals to analyze the dielectric

response of independent QDs of different nature have been

reported in the literature.13,14

In Fig. 1, the energy loss probability for the three geo-

metries of the confinement potential mentioned above and

for two different QDs radii are depicted. The behavior of the

energy loss probability exhibits evident differences for QD

of r ¼ 4 nm in comparison to the corresponding energy loss

for the QD of r ¼ 10 nm. For QDs of r ¼ 4 nm there appears

a shift in the energy of the maximum probability belonging

to the distribution of QDs of cylindrical geometry, respect to

the energy of the peak for the distributions with other geome-

tries of the QDs. This shift is related to the fact that the aver-

age energy of the transitions in cylindrical quantum dots is

lower than the corresponding to the other geometries. Notice

that when the radius increases, the probability peaks move

toward lower values. The density considered in these distri-

butions is, for QDs of r¼ 10 nm, 1:5� 105 QDs per linear

centimeter.

In Fig. 1, it is also observed that for the distribution with

QDs of r¼ 4 nm, the probability that the electron beam loses

energy, producing transitions within the QD electronic

024105-2 Moctezuma et al. J. Appl. Phys. 112, 024105 (2012)

Downloaded 31 Jul 2012 to 148.228.150.54. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

system, is smaller in comparison with the case in which the

distribution is conformed by QDs of r¼ 10 nm. However,

the latter distribution exhibits the peak of the resonance at

smaller values of the energy loss.

III. ENERGY LOSS FUNCTION DUE TOINHOMOGENEOUS QD DISTRIBUTIONS

Now we consider inhomogeneous distributions by mix-

ing quantum dots of two different geometries, the graphs of

the corresponding energy loss functions are shown in Fig. 2.

Since we are considering non interacting QDs, the following

relation can be used to calculate the energy loss function:

FðEÞ ¼ ð1� xÞFaðEÞ þ xFbðEÞ; (8)

where a and b stand for the two different QDs geometries.

Fig. 2 shows the graphs of the energy loss function for

inhomogeneous QD distributions of radius r¼ 10 nm. As

in Fig. 1, it is assumed an electron beam of energy El

¼ 100 keV with an impact parameter z0 ¼ 1 nm and a damp-

ing coefficient c ¼ 1� 104 1ns. The first row of graphs corre-

sponds to distributions of independent QDs; in this way, the

analogous to the dielectric properties of the distribution is

explored. The second row shows the loss function for distri-

butions of independent QDs, but these having an intrinsic

dipolar moment. The third row shows the energy loss func-

tion due to distributions conformed by interactive QDs in the

mean field approximation discussed above. The rule of mix-

ing of the QDs in the distribution is given by expression (8).

The first column shows the behavior of the energy loss func-

tion as a function of the energy loss, corresponding to distri-

butions of spherical QDs mixed with cylindrical QDs, the

second column contains the energy loss function calculated

for non homogeneous distributions of spherical QDs mixed

with conical QDs. In the third column, there appears the

results corresponding to inhomogeneous distributions of

cylindrical QDs mixed with conical QDs.

In the first row, corresponding to the dielectric case, we

observe that the behavior of the energy loss function does

not vary drastically for any of the three different distributions

(Figs. 2(a)–2(c)). The peak of the curve is around an energy

loss of 130 meV, indicating that the probe beam is unable to

distinguish between the two geometries conforming these

distributions. A similar behavior of the energy loss function

is shown for the paraelectric case (Figs. 2(d)–2(f)).

In the third row of Fig. 2 which corresponds to distribu-

tions of interactive QDs, namely a sort of ferroelectric

FIG. 1. Energy loss probability, per unit time, unit length, and unit energy

interval, FðEÞ, for an incident electron beam of energy El ¼ 100 keV, an

impact parameter z0 ¼ 1 nm and a characteristic electronic damping constant

c ¼ 1� 104 1ns, for two sizes of quantum dots 4 and 10 nm for the three dif-

ferent confinement potentials.

FIG. 2. Energy loss probability: (a)

spherical and cylindrical independent

QD distribution; (b) spherical and coni-

cal independent QD distributions; (c)

cylindrical and conical independent QD

distribution; (d) distribution conformed

by spherical and cylindrical independent

QDs with intrinsic dipolar moment; (e)

distribution conformed by spherical and

conical independent QD with intrinsic

dipolar moment; (f) distribution con-

formed by cylindrical and conical inde-

pendent QD with intrinsic dipolar

moment. Graphs (g), (h), and (i) show

the corresponding energy loss function

for inhomogeneous distributions of

interactive QDs.

024105-3 Moctezuma et al. J. Appl. Phys. 112, 024105 (2012)

Downloaded 31 Jul 2012 to 148.228.150.54. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

distribution, we may observe that there appear two peaks at

well differentiated values of the energy loss; these peaks

indicate that it is possible for these conditions that the field

generated by the incident beam couples to the dipolar

moment of the distribution, producing energetic transitions

in the distributions. It must be pointed out that the energy

loss function is sensitive to the QD size. Of course, given the

roughness of the mean field approximation for strongly cor-

related systems, one would expect that our estimations about

the ferroelectric behavior of the QD distribution would be

more accurate for physical conditions far from the transition.

IV. DIELECTRIC RESPONSE OF QD DISTRIBUTIONS

QDs can be thought as artificial atoms whose electronic

properties can be, at some extent, tailored. When dealing

with distributions of interactive QDs, it is expected that the

collective response of the system is also sensitive to changes

in the design parameters of the QDs. The effective response

of these materials to a beam of electrons traveling parallel to

the plane that contains the QD distribution, is determined by

the dielectric function of the inclusions and the matrix. In a

distribution of quantum dots the values of the dielectric func-

tion and the allowed energies of the QDs are discrete and the

transitions between different quantum states are induced by

the interaction with the electric field of the electronic beam.

The excited states decay after a certain relaxation time 1c. To

calculate the dielectric function, we solve numerically the

time independent Schrodinger equation for various geome-

tries of QDs. The eigenfunctions in this way obtained, allows

us to calculate the respective QD dipolar moment, and from

here, it is possible to construct the dielectric function. For a

QD distribution, the dielectric function is given by15

�ðxÞ ¼ 1� 4p�h

Xj

jh/jjxj/0ij21

xþ ic� xjþ 1

xþ icþ xj

� �;

(9)

where h/jjxj/0i is the matrix element of the transition

between the zeroth and the jth states, Ej ¼ �hxj is the energy

difference between the final and ground states, and �hx is the

incident electron beam energy.

From Eq. (9), we obtain that the imaginary part is

given by

�00ðxÞ ¼ 4p�hX

j

jh/jjxj/0ij2c

ð�hx� EjÞ2 þ �h2c2; (10)

and the real part is

�0ðxÞ ¼ 1� 4pX

j

jh/jjxj/0ij2�hx� Ej

ð�hx� EjÞ2 þ �h2c2: (11)

From the real and imaginary parts of the dielectric func-

tion, one may obtain some insights on the excitation and

relaxation processes of the distribution collective modes.

A. Imaginary part

The imaginary part of the dielectric function �00 is related

to the decay processes of the electronic states, i.e., the proc-

esses that dissipate energy from the electronic QD system. For

a distribution of spherical QDs with radius 10 nm, and a

damping coefficient c ¼ 1� 104 1ns, �

00 has the behavior shown

in Fig. 3. In the graph (a), it is depicted the imaginary part of

the dielectric function corresponding to a distribution of inde-

pendent QDs, in (b) there appears �00 for a distribution of inde-

pendent QDs with intrinsic dipolar moment and the graph in

(c) for a distribution of interactive QDs. The linear density of

QDs in the distribution is 1:5� 105. In the dielectric and para-

electric cases, the peak of the energy loss function appears

approximately at the energy loss of 200 meV. On the other

hand, for the case of a distribution of interactive QDs, the

peak in the imaginary part of dielectric function is lower in

comparison to the other two distributions and is located about

a loss of energy of 160 meV. As it was expected, there exists a

difference in the behavior of the imaginary part of the dielec-

tric function corresponding to a distribution of independent

QDs respect to the ferroelectric-like behavior. This is because,

in the latter, the imaginary part of the dielectric function is

related to the dissipative processes belonging to relaxation of

collective states of the system.

B. Real part

The imaginary part of the dielectric function for the

three different geometries of the confinement potential and

the three different cases, dielectric, paraelectric, and ferro-

electric, studied here, are shown in Fig. 3. In order to obtain

more illustrative results and trying to compare QDs of simi-

lar volume, for the dielectric and paraelectric cases, the ra-

dius of the considered QDs was chosen to be r¼ 10 nm. For

the ferroelectric case, the radius of the spherical and cylindri-

cal QDs is r¼ 10 nm and for the conical QD is r¼ 15 nm.

This consideration is needed because in the case of inde-

pendent QDs, the electronic distributions of the QDs do not

overlap each other; therefore, it does not matter the volume

of the involved QDs; however, for the ferroelectric case it

certainly does.

FIG. 3. Imaginary part of the dielectric

function in two-dimensional QD distribu-

tions with r¼ 10 nm, an impact parame-

ter z0 ¼ 1 nm and a damping coefficient

c ¼ 1� 104 1ns, and three different geo-

metries of the confinement potential:

(a) Independent quantum dots; (b) inde-

pendent quantum dots with intrinsic

dipolar moment; and (c) interactive

quantum dots.

024105-4 Moctezuma et al. J. Appl. Phys. 112, 024105 (2012)

Downloaded 31 Jul 2012 to 148.228.150.54. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

By comparing these graphs, one may distinguish that for

a distribution of cylindrical QDs we get practically the same

behavior observed for the distribution of spherical ones; how-

ever, the peak of �00 is higher for the case where we have a dis-

tribution of cylindrical QDs. On the other hand, one observes

that for the distribution of conical QDs, a lower peak appears,

shifted to smaller values of the energy loss for the dielectric

and ferroelectric cases, while for the paraelectric case, this

peak appears shifted to the right. Another interesting aspect of

Fig. 3(c) is that for the cylindrical geometry, there are two dif-

ferent values of the loss energy for which, in a predominant

way, the states of the QDs decay.

The behavior shown in Fig. 3 can be understood consider-

ing the characteristics of the possible transitions, consistent with

the corresponding spectra for each one of the geometries of the

confinement potential and their respective dipolar moments.

In the QDs distributions, the real part of the dielectric

function is related to excitation transitions. The transitions

between quantum states of the QDs generated by the interac-

tion with the electrons of the beam lead to quantum states in

which the magnitude of dipolar moment yields some charac-

teristics of the dielectric response; it is the greater the magni-

tude of the dipoles, the greater the value of the dielectric

function. In Fig. 4, we present the real part of the dielectric

FIG. 4. Real part of the dielectric func-

tion in two-dimensional quantum dots dis-

tributions with r¼ 10 nm, an impact

parameter z0 ¼ 1 nm, a damping coeffi-

cient c ¼ 1� 104 1ns, and different geome-

tries of the confinement potential: (a)

Independent quantum dots; (b) independ-

ent quantum dots with intrinsic dipolar

moment; and (c) interactive quantum dots.

FIG. 5. Dielectric function of two-

dimensional distributions of independ-

ent quantum dots with r¼ 10 nm, an

impact parameter z0 ¼ 1 nm, and a

damping coefficient c ¼ 10 1ns. (a) Imagi-

nary part, (b) real part of the dielectric

function for a distribution of spherical

quantum dots. (c) Imaginary part, (d)

real part of the dielectric function for a

distribution of cylindrical quantum dots.

(e) Imaginary part, (f) real part of the

dielectric function for a distribution of

conical quantum dots.

024105-5 Moctezuma et al. J. Appl. Phys. 112, 024105 (2012)

Downloaded 31 Jul 2012 to 148.228.150.54. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

function for the dielectric (Fig. 4(a)), paraelectric (Fig. 4(b)),

and ferroelectric (Fig. 4(c)) cases. It is evident from this fig-

ure that the real part of the dielectric function is sensitive to

the geometry of the quantum dots.

A general an obvious conclusion which can be obtained

from the graphs of Figs. 3 and 4 is that the geometry of the

confinement potential, as well as the polarization condition,

namely, dielectric, paraelectric, or ferroelectric of the distri-

bution, strongly affect the behavior of the dielectric function.

C. Dielectric function for small damping coefficients

When the damping coefficient c decreases, the peaks in

the dielectric function and the energy loss function are better

appreciated. Now, we will discuss that when this coefficient,

c ¼ 10 1ns has a small value, i.e., for greater relaxation times,

more detailed information can be obtained from the curves

of energy loss function and dielectric function. The 2D inho-

mogeneous distributions we analyze here are the following:

spherical and cylindrical QDs of radius 10 nm, and conical

QDs of radius 15 nm.

In Fig. 5, there are depicted graphs corresponding to the

real and imaginary parts of the dielectric function for the distri-

bution of independent QDs. In the first row, the distribution is

formed by spherical QDs, in the second row by cylindrical QDs,

and in the third one conical. In these graphs, one observes that

under these conditions there appear two peaks in the real and

the imaginary parts of dielectric function for the three geome-

tries of the confinement potential. If a relaxation time of ( 110

ns)

is considered, larger and better defined peaks are obtained, in

comparison with the curves obtained with c ¼ 1� 104 1ns.

Comparing the imaginary part of the dielectric function

for the distribution of spherical QDs (Fig. 5(a)), with the dot-

ted curve of Fig. 3(a), we observe that the peak of the latter

appears at a value of the energy loss close to the value where

the two peaks in Fig. 5(a) appears. The same behavior is

found for the real part and the three geometries of the con-

finement potential. From this, one may conclude that when

the damping coefficient is c ¼ 1� 104 1ns, the curves are

envelopes containing the peaks of the curves corresponding

to the damping coefficient c ¼ 10 1ns.

Figs. 6 and 7 show the real and imaginary parts of the

dielectric function for the paraelectric and ferroelectric cases,

respectively. In these graphs, one observes a peak in the real

and imaginary parts of the dielectric function. These peaks are

very sharp and appear close to each other in the paraelectric

case, while in the ferroelectric case, the separation between

them is larger. Comparing these graphs with those for the

FIG. 6. Dielectric function of two-

dimensional distributions of independ-

ent quantum dots with an intrinsic dipo-

lar moment, r¼ 10 nm, an impact

parameter z0 ¼ 1 nm and a damping

coefficient c ¼ 10 1ns. (a) Imaginary part,

(b) real part of the dielectric function for

a distribution of spherical quantum dots.

(c) Imaginary part, (d) real part of the

dielectric function for a distribution of

cylindrical quantum dots. (e) Imaginary

part, (f) real part of the dielectric func-

tion for a distribution of conical quan-

tum dots.

024105-6 Moctezuma et al. J. Appl. Phys. 112, 024105 (2012)

Downloaded 31 Jul 2012 to 148.228.150.54. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

energy loss function with c ¼ 1� 104 1ls, we obtain again an

envelope curve when the damping coefficient is greater.

On the other hand, we also found that when the damping

coefficient has a value of one or two orders of magnitude

less than c ¼ 10 1ns, the magnitude of the peak becomes

higher and in some cases, there appear more peaks on both

sides of the dielectric function. However, when the damping

ratio is three or more orders of magnitude smaller than

c ¼ 10 1ns, the trends are more similar to that shown with

c ¼ 1� 104 1ns, i.e., a curve without structure, in some way

meaning an envelope of the peaks that appear for c ¼ 10 1ns.

Again, in the ferroelectric behavior, our estimations about

the effect of the damping coefficient could be overestimated

by the mean field approximation.

V. CONCLUDING REMARKS

Electric dipolar properties of 2D distributions of QDs of

three different confinement potential were studied theoreti-

cally. The dielectric function and the energy loss function were

calculated for 2D homogeneous and non-homogeneous distri-

butions formed by spherical, cylindrical, and conical QDs. The

results make evident that there are noticeable differences in the

energy loss probability for the various confinement potentials

and for the dielectric, paraelectric, and ferroelectric cases. The

loss energy function is also sensitive to the size of the QDs.

The mean field approximation used here to explore the ferro-

electric response of course is severely limited in particular in

dealing with phase transformations. Nevertheless, these kind

of approximations are capable to predict trends and qualitative

behavior in the ordered phase provided far from the critical

conditions. Therefore, our results indicates that, in principle,

some of the trends here discussed could be detected by real

EELS experiments. From here, one may conclude that by

means of numerically simulated EELS experiments for elec-

trons traveling parallel to the surface of the QD array, it is pos-

sible to detect in the effective dielectric response of the

material, the resonances corresponding to the discrete spectra

of the quantum dots and that corresponding to collective

modes, as well as, the shifts due to the presence of regions

with different geometries of quantum dots. Thus, this theoreti-

cal scheme could provide a tool to design and test the dipolar

properties of tailored quantum dot distributions.

ACKNOWLEDGMENTS

The partial financial support by the following institu-

tions is acknowledged: COLCIENCIAS, Colombia, Grant

FIG. 7. Dielectric function of two-

dimensional distributions of interactive

quantum dots with r¼ 10 nm, an impact

parameter z0 ¼ 1 nm and a damping

coefficient c ¼ 10 1ns. (a) Imaginary part,

(b) real part of the dielectric function for

a distribution of spherical quantum dots.

(c) Imaginary part, (d) real part of the

dielectric function for a distribution of

cylindrical quantum dots. (e) Imaginary

part, (f) real part of the dielectric func-

tion for a distribution of conical quan-

tum dots.

024105-7 Moctezuma et al. J. Appl. Phys. 112, 024105 (2012)

Downloaded 31 Jul 2012 to 148.228.150.54. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

No. 1204-05-46852, Proyectos de investigacion Facultad de

Ciencias-Universidad de los Andes, CONACyT, Mexico,

Grant No. 44296, Universidad Autonoma de Puebla, Grant

No. 05/EXC/06-I. J.L.C. in sabbatical leave from Instituto de

Fısica de la Universidad Autonoma de Puebla, Mexico,

thanks the Secretarıa de Estado de Universidades e Investi-

gacion of Spain for financial support through the Grant

SAB2005-0063 and thanks J. M. Rubı for hospitality at Uni-

versitat de Barcelona.

1J. B. Pendry and L. M. Moreno, Phys. Rev. B 50, 5062 (1994).2J. M. Pitarke, J. B. Pendry, and P. M. Echenique, Phys. Rev. B 55, 9550

(1997).3H. Cohen, B. I. Lembrikov, M. A. Itskovsky, and T. Maniv, Nano Lett. 3,

203 (2003).4R. G. Barrera and R. Fuchs, Phys. Rev. B 52, 3256 (1995).5R. Fuchs, R. G. Barrera, and J. L. Carrillo, Phys. Rev. B 54, 12824

(1996).

6R. Fuchs, C. I. Mendoza, R. G. Barrera, and J. L. Carrillo, Physica A 241,

29 (1997).7H. J. Krenner, M. Sabathil, E. C. Clark, A. Kress, D. Schuh, M. Bichler,

G. Abstreiter, and J. J. Finley, Phys. Rev. Lett. 94, 057402 (2005).8H. Y. Ramirez, A. S. Camacho, and L. C. Lew Yan Voon, Nanotechnology

17, 1286 (2006).9M. E. Lines and A. M. Glass, Principles and Applications of Ferroelec-trics and Related Materials (Oxford University Press, 1977).

10R. V. N. Melnik and K. N. Zotsenko, Lecture Notes in ComputerScience, LNCS-2659, edited by P. M. A. Sloot, D. Abramson, A. V.

Bogdanov, J. J. Dongarra, A. Y. Zomaya, and Y. E. Gorbachev, Part

III, International Conference on Computational Science, Melbourne, 343

(Springer, 2003).11R. V. N. Melnik and M. Willatzen, Nanotechnology 15, 1 (2004).12R. E. Moctezuma, J. L. Carrillo, and L. Meza-Montes, Integr. Ferroelectr.

126, 171 (2011).13X. Cartoixa and L. W. Wang, Phys. Rev. Lett. 94, 236804 (2005).14F. Remacle and R. D. Levine, Int. J. Quantum Chem. 99, 743 (2004).15H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic

Properties of Semiconductors, 3rd ed. (World Scientific, reprinted,

2001).

024105-8 Moctezuma et al. J. Appl. Phys. 112, 024105 (2012)

Downloaded 31 Jul 2012 to 148.228.150.54. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions