dipolar transformations of two-dimensional quantum dots arrays
TRANSCRIPT
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
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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)
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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)
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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)
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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)
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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)
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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.
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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)
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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.
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