REVISTA MEXICANA DE FISICA S53 (7) 124–127 DICIEMBRE 2007

Electron energy loss by 2D quantum dot arraysJ.F. Nossa and A.S. Camacho

Departamento de Fısica, Universidad de los Andes,A.A. 4976, Bogota D.C., Colombia.

J.L. CarrilloDepartament de Fısica Fonamental, Facultat de Fısica, Universitat de Barcelona,

Diagonal 647, 08028 Barcelona, Spain.

Recibido el 30 de noviembre de 2006; aceptado el 8 de octubre de 2007

Dipolar transitions in bidimensional arrays of quantum dots are investigated by means of the simulation of electron energy loss experiments.We assume bidimensional distributions of independent quantum dots, whose dipolar transitions are induced by the interaction with an electronbeam traveling parallel to the plane of the array. We calculate the electron energy loss function corresponding to different distributions andseveral confinement potentials of quantum dots. We found that by using this procedure it would be possible to explore the dielectric andother physical properties of these arrays.Keywords: Composite materials; quantum dots; EELS.

Transiciones dipolares en arreglos bidimensionales de puntos cuanticos son calculados por medio de la simulacion de experimentos deperdida de energıa de electrones. Suponemos distribuciones 2D de puntos cuanticos independientes, en las cuales las transiciones dipolaresson inducidas por la interaccion con rayo de electrones que viajan paralelos al plano del arreglo. Calculamos la funcion de probabilidad deperdida de energıa de los electrones, correspondiente a diferentes distribuciones y varios potenciales de confinamiento de los puntos cuanticos.De esta forma serıa posible investigar, adicionalmente a las propiedades dielectricas, las paraelectricas de este tipo de distribuciones de puntoscuanticos.Descriptores: Materiales compositos; puntos cuanticos; EELS.

PACS: 77.84.Lf; 73.21.La; 79.20.Uv

1. Introduction

One of the most versatile experimental techniques for thestudy of the electronic structure and physical properties ofhomogeneous systems is Electron Energy Loss Spectroscopy(EELS). In EELS a material is exposed to an electron beamwhich places the energy in an appropriate range to interactwith the elementary excitations of the array; some of the elec-trons in the beam lose energy due to this inelastic scattering.Besides energy loss, this inelastic dispersion produces a mo-mentum exchange.

Presently, one of the most interesting applications ofEELS is the study of nanostructured materials. Particularly,the so called granular nanocomposites have attracted the at-tention of researchers in the last few years. The energy rangesof the incident electron beam required to study this kind ofsystems are fully accessible.

The electron energy loss spectra of these materials de-pend on both the electronic properties of the matrix materialand the electronic properties of the inclusions of the com-posed material.

The energy loss spectrum has been calculated by meansof a classical dielectric theory for crystalline and for homo-geneous systems. R. Barrera and R. Fuchs first developed anenergy loss theory of electrons traveling in a non homoge-neous material containing a system of randomly distributedspheres [1]. Two of the central difficulties in developing thistheory were: first, the incorporation of the many body in-teractions among the inclusions and second, the procedureto make the proper average on the position of the inclusions

for a given non homogenous distribution. A version of thistheory was developed to describe the energy loss for distribu-tions of hard spheres in terms of spectral representations ofthe response function in composed media (granular compos-ites) [2]. Also, on this basis, the theory to calculate the energyloss spectra of semi-infinite granular composites was devel-oped by means of the evaluation of the surface response func-tion of the composite, considering that this response is causedby an electron beam traveling parallel to the surface [3].

If one assumes that the inclusions in a nano composite arequantum dots, the theoretical description requires the knowl-edge of the discrete, as well as the continuous part of thespectrum of that quantum dot distribution and the evaluationof the respective effective dispersion section. Consequently,knowledge of the state of polarization of the quantum dot dis-tribution is required. In this work we first develop a theoret-ical procedure to investigate the dielectric properties of twodimensional distributions of quantum dots by calculating theenergy loss spectrum of an electron beam traveling parallel tothe array of non interacting quantum dots. Next, we calculatethe dielectric function and the energy loss function for sev-eral bidimensional distributions of quantum dots consideringdifferent confinement potentials and non homogeneous twodimensional distributions.

2. Energy loss function

The energy loss spectrum in composites has been calculatedfrom the effective inverse longitudinal dielectric response

J.F. NOSSA, A.S. CAMACHO, AND J.L. CARRILLO 125

ε−1(k, ω). Here the effective term comes from the averageof all particles. The main idea of the method is the follow-ing [1]: An external potentialV ext(k, ω) is applied to thesystem. The following is the relationship among the total, theexternal and the induced potentialsV (k, ω) = V ext(k, ω) +V int(k, ω) ; thus, the effective longitudinal dielectric func-tion is given by the relationε−1 = 〈V (k, ω)〉/V ext(k, ω).

If the energy loss of the incident electron isEi − Ef =E = ~ω and the momentum transfer isKi −Kf = k, theprobability per unit time of scattering of a given electron withan energy loss in the interval(E, E+δE) and final wave vec-tor Kf in the solid angledΩ is given by

dP (E, Ω)dt

=m

(2π~2)2Kf

∣∣∣∣4πe2

k2

∣∣∣∣2

S(k, ω)dEdΩ (1)

where

S(k, ω) = − ~k2

4π2e2Im

∣∣ε−1(k, ω)∣∣

is the dynamic structure factor.The spectral representation of the dielectric function can

be calculated by means of the following expression [1]

ε−1(k, ω) = (ε2)−1

1 + f

[Cb

u− 1+

∑s

Cs

u− ns

](2)

wheref is the volume fraction of the spherical inclusions,uis the spectral variable,ε2 is the matrix dielectric function,Cb is the strength of the bulk mode of the spheres,Cs is thestrength of the interfacial modes, andns are the depolariza-tion factors.

To study the energy loss phenomena from electrons trav-eling parallel to the free surface of a semi-infinite composedmedia, it is necessary to calculate the surface response func-tion g(Q,ω) [3]. This quantity depends on the wave vectorparallel to surfaceQ. A schematic description of the proce-dure is the following: We consider a system of spherical in-clusions randomly located in the half-spacez < 0. One takesan external potential produced by an external charge infinitelydistant from the interface in the regionz > 0 with a singlewave-vector componentQ = Qi. A simple form to calculatethe surface response function is to use the model of specu-lar reflection or semiclassical infinite barrier (SCIB) [4]. Themain idea of this model is to extend the bounded system ofinclusions to an unbounded system, whose response functionis given byε(k, ω). In this way, to simulate the presenceof the surface atz = 0, a fictitious external charge of theform ρ(r) = AeiQxδ(z) is introduced, whereA is a constant.Imposing the boundary conditions that the potential and thenormal component of the displacement field are continuousthrough interface atz = 0 [3], one obtains [5]

g(Q.ω) =ε(Q,ω)− 1ε(Q,ω) + 1

(3)

where

1ε(Q,ω)

=Q

π

∫ +∞

−∞

dkz

(Q2 + k2z)εM (k, ω)

(4)

From the surface response function we find a quantityS(Q,ω) = Img(Q,ω), which is used to calculate the energyloss probability of an electron traveling parallel to the inter-face of the system, with an impact parameterz0 and a speedvl in they direction. The probability per unit path length andper unit energy of scattering with energy lossE is given by

F (E) ≡ d2P

dl dE=

1a0El

Ξ(E) (5)

wherea0 is the Bohr radius,El is the kinetic energy of theincident electronE = ~ω and

Ξ(E) =1π

∫ ∞

0

e−2Qz0

QS(Q, ω)dkx, (6)

hereQ =√

k2z + ω2/v2

l .In the local limit εM (k, ω)=εM (k=0, ω)≡εM (ω), one

gets1/ε(Q,ω)=1/εM (ω). Thus,

S(ω) = Im[(εM (ω)− 1)(εM (ω) + 1)

],

by substitutingS(ω) into Eqs. 3 and 5, and by performingthe integration, one gets the following local result, which isvalid in the mean field approximation

F (E) =1

a0El

1π

K0

(2z0vl

ω

)Im

[εM (ω)− 1εM (ω) + 1

], (7)

whereK0 is the modified Bessel function of zero order,a0

is the Bohr radius,El andvl refer to the energy and veloc-ity of the incident electron, respectively, andz0 is the impactparameter.

3. Quantum dots dielectric response

On the basis of the inverse dielectric function, scattering phe-nomena occurring in bulk materials are described. However,in dealing with nanostructured materials, there are some par-ticularities that must be taken into account [6]; a detailed dis-cussion of these points can be read in [7]. On the other hand,it has been theoretically and experimentally shown that thebulk value of the average dielectric constant fits well to mostquantum dot systems [8,9].

The intrinsic physical properties of quantum dots origi-nate a great diversity of unanswered relevant questions forapplied and basic science. This is due to the fact that theycan be conceived as artificial atoms; they can also be manip-ulated by using specific control parameters. Therefore, thecollective response of the system can be tuned by varyingthose parameters [10].

The effective response of these materials is determinedby the dielectric function of the inclusions and the dielec-tric function of the matrix under interaction with an elec-tron beam traveling parallel to the array [2]. Here we shall

Rev. Mex. Fıs. S53 (7) (2007) 124–127

126 J.F. NOSSA, A.S. CAMACHO, AND J.L. CARRILLO

consider quantum dots in which the values of the dielectricfunction, as well as the allowed energies, are discrete and thetransitions among their different quantum states are due to theinteraction with the electrons. The excited states decay aftera certain relaxation timeγ−1 = (2.96253 ∗ 104)−1nm. Wecalculate the dielectric function by numerically solving thetime dependent Schrodinger equation for several geometriesof the quantum dots. In this way, we obtain the respectivepolarization state which allows us to construct the dielectricfunction, which for quantum dots is given by [12].

ε(ω) = 1−4π

~∑

j

∣∣〈φj |x|φ0〉∣∣2

×[

1ω + iγ − ωj

+1

ω + iγ + ωj

], (8)

where 〈φj |x|φ0〉 is the dipolar moment,φ0 is the groundstate wavefunction,φj is the final state wavefunction afterthe dipolar transition,Ej = ~ωj is the energy difference bet-ween the final and ground state, andω is the incident electronfrequency.

From eq. (8) we can notice that the imaginary part isgiven by

ε′′(ω) = 4π~∑

j

∣∣〈φj |x|φ0〉∣∣2 γ

(~ω−Ej)2+~2γ2

and the real part is

ε′(ω) = 1− 4π∑

j

∣∣〈φj |x|φ0〉∣∣2 ~ω−Ej

(~ω−Ej)2+~2γ2 ,

The eigenfunctions, eigenvalues, and polarization statesof the quantum dots for various geometries are numericallyfound using a finite element package COMSOL [13].

4. Results

We consider three geometries of GaAs quantum dots em-bebed in anAlxGal−xAs matrix: a) spherical quantum dotswith radii of 4nm, b) conical quantum dots with lower baseradii of 4nm, upper base radii of2, 134nm and height of4nm, c) cylindrical quantum dots with radii of4nm andheight of4nm. For spherical quantum dots, we obtain fourconfined energies, whereas for cylindrical and conical quan-tum dots we obtain only three confined energies. The ex-perimental parameters used are shown in Table I. The dipolevalues and energies obtained in our calculations are shown inTable II.

Since we suppose independent homogeneous quantumdots we may introduce a linear density of these to directlyobtain the total energy loss function per unit length and unittime. Now, we consider an electron beam with speedvl,whose energy isEl = 100keV , and it is traveling parallelto the surface with an impact parameterz0. The graph of theenergy loss function for the different geometries for two im-pact parameters are shown in Fig. 1. A peak appears in the

TABLE I. Experimental parameters used in our calculations [11]

V (meV) m∗ matrix m∗ dot

500 0.0962me 0.063me

FIGURE 1. Energy loss probability, per unit of time and unit length,F (E), for several values of the impact parametersz0 = 0.1 andz0 = 0.9 and several shapes of quantum dots.

TABLE II. a) Dipole andb) energy values for several quantum dotsgeometries.〈φi|x|φ0〉 are innm and the energies are inmeV .

Geometries 〈φ1|x|φ0〉 〈φ2|x|φ0〉 〈φ3|x|φ0〉a)

Spherical QD 1.19 1.18 0.35

Cylindrical QD 0.479 1.772

Conical QD 1.250 0.846

b)

Geometries E1 E2 E3

Spherical QD 195.084 195.106 195.149

Cylindrical QD 163.675 163.838

Conical QD 199.716 199.721

energy loss probability around the average energy of the tran-sition between the initial and final states in the quantum dots(see Table IIb)). We observe an energy shift of the max-imum probability corresponding to the cylindrical quantumdots. This is related to the fact that the average energy of thetransition in cylindrical quantum dots is lower that the othergeometries. Notice that when the impact parameter increases,the probability peaks move into lower values.

Now we consider inhomogeneous distributions by mixingquantum dots of two different geometries. The correspondinggraphics are shown in Fig. 2, where we used the followingdistribution

F (E) = (1− x)Fa(E) + xFb(E)

with a andb denoting two different quantum dot geometries.One may observe that there are noticeable differences in

the energy loss probability for the various confinement poten-

Rev. Mex. Fıs. S53 (7) (2007) 124–127

J.F. NOSSA, A.S. CAMACHO, AND J.L. CARRILLO 127

FIGURE 2. Energy loss probability:a) cylindrical and conical quantum dot distributions,x = 0 → cylindrical quantum dots,x = 1 →conical quantum dots.b) cylindrical and spherical quantum dots distributions,x = 0 → cylindrical quantum dots,x = 1 → sphericalquantum dots.c) spherical and conical quantum dot distributions,x = 0 → spherical quantum dots,x = 1 → conical quantum dots.

tials we used in our calculations. In principle, these differ-ences could be sharply detected by real EELS experiments.

From the discussion above we may conclude that bymeans of the simulation of EELS experiments, for electronstraveling parallel to the surface that contains the independentquantum dot distribution, it is possible to detect in the effec-tive dielectric response of the material, the resonances cor-responding to the discrete spectra of the quantum dots. Thiswould provide a tool to test the electronic properties of tai-lored quantum dot distributions.

We have calculated the energy loss function for bidimen-sional distributions of independent spherical, cylindrical andconical quantum dots. According to our results, EELS ex-periments would be able to detect the difference in the di-electric response for these geometries and also in the non ho-mogeneous distributions that can be constructed by combin-ing them. Additionally, we have calculated the conditions inwhich it would be more probable to produce transitions in thepolarization states of quantum dots. This opens the possibil-

ity to investigate diverse relaxation mechanisms of the quan-tum dot energetic states and consequently, it would allow usto explore other physical characteristics of the quantum dotdistributions, for instance, the paraelectric properties.

Acknowledgments

The partial financial support by the following institutions isacknowledged: COLCIENCIAS, Colombia, grant 1204-05-46852. Proyectos de investigacion Facultad de Ciencias-Universidad de los Andes. CONACyT, Mexico, grant 44296.Universidad Autonoma de Puebla, Grant Not 05/EXC/06-I.J.L.C. is on sabbatical leave from the Instituto de Fısica dela Universidad Autonoma de Puebla, Mexico. We would liketo thank the Secretarıa de Estado de Universidades e Inves-tigacion of Spain for their partial financial support throughgrant SAB2005-0063, and we would also like to thank J.M.Rubı for his hospitality at the Universitat de Barcelona.

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