Dynamic screening and electron–electron scattering - Donostia ...

Surface Science 601 (2007) 4546–4552
www.elsevier.com/locate/susc
Dynamic screening and electron–electron scattering
in low-dimensional metallic systems
V.M. Silkin a , M. Quijada a,b ,R.Díez Muiño a,c , E.V. Chulkov a,b,c , P.M. Echenique a,b,c, *
a Donostia International Physics Center (DIPC), P. de Manuel Lardizabal 4, 20018 San Sebastián, Spain
b Departamento de Física de Materiales, Facultad de Ciencias Químicas UPV/EHU, Apartado 1072, 20080 San Sebastián, Spain
c Unidad de Física de Materiales, Centro Mixto CSIC-UPV/EHU, P. de Manuel Lardizabal 3, 20018 San Sebastián, Spain
Available online 1 May 2007
Abstract
The modification of dynamic screening in the electron–electron interaction in systems with reduced dimensionality and tunable oneparticle
electronic structure is studied. Two examples of such systems are considered, namely, the adsorbate-induced quantum well states
at the Na adlayer covered Cu(111) surface, and metal clusters of sizes up to few nanometers. The dependence of the electron–electron
decay rates on the Na coverage in the former case and on the cluster size in the latter is investigated. The role played by the dynamical
screened interaction in such processes is addressed as well.
Ó 2007 Elsevier B.V. All rights reserved.
Keywords: Many-body and quasiparticle theories; Surface electronic phenomena; Noble metals; Adsorption; Quantum well states; Clusters
1. Introduction
* Corresponding author. Address: Departamento de Física de Materiales,
Facultad de Ciencias Químicas UPV/EHU, Apartado 1072, 20080
San Sebastián, Spain. Tel.: +34 943 015963; fax: +34 943 015600.
E-mail address: wapetlap@sc.ehu.es (P.M. Echenique).
Electronic excitations in metallic media can decay
through various mechanisms, the most important of them
being electron–electron (e–e) and electron–phonon (e–ph)
scattering. Over the last years, a large amount of theoretical
and experimental work has been devoted to understand
and predict the lifetime of these excitations in solids [1–5]
and at surfaces [6–10]. Focusing on the e–e scattering process,
one of the key quantities that determine the rate at
which the electronic excitations can decay is the dynamic
screening among the medium electrons. However, the
screening properties of metallic systems are profoundly affected
by their dimensionality. Confinement and surface effects
are, among others, two features that can drastically
change the electronic screening and thus the lifetime of
electronic excitations as compared with the bulk case.
The modification at wish of the electronic properties of metal
systems and the subsequent change in the lifetime of
electronic excitations is not a question of purely academic
interest, but has important implications for many technological
applications.
In present work, we analyze two cases in which the
reduction of dimensionality leads to significant modifications
in the electron–electron interaction. First, we study
the case of quantum well states formed under the deposition
of alkali atoms on metal surfaces [11,12]. Whereas
the number of adsorbed alkali atoms is small, and they
can be considered as independent, the picture of almost total
transition of s valence electron from the alkali atom to
the substrate is appropriate. However, when the coverage
reaches rates of around 1 monolayer (ML) [13] an adsorbate-induced
electron band which forms a quasi twodimensional
electron gas confined between the vacuum
barrier and the metal substrate [14,15] (called a quantum
well state (QWS)), unoccupied for low coverage, starts to
fill [16]. A prominent example of this is the case of Na
adsorption on Cu(111), for that the properties of this
QWS have been intensively studied. Thus, the dynamics
of holes created at wave vector k k = 0 has been studied
for this system, both experimentally and theoretically
0039-6028/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.susc.2007.04.232

V.M. Silkin et al. / Surface Science 601 (2007) 4546–4552 4547
[17]. In this study it was proven that e–e and the e–ph
mechanisms are equally important for the hole decay for
1 ML coverage. Very recently a joint experimental and theoretical
investigation of electron dynamics for Cs and Na
adlayers on Cu(111) was performed as well [18]. Here,
we investigate the energy dependence of the e–e contribution
for several positions of the QWS band relative to the
Fermi level E F , which, in turn, depends on the Na coverage.
For the sake of comparison with the experiment, a reasonable
assumption could be that the e–ph contribution
should not change significantly for such small variation
of the QWS energy.
As a second case study, we analyze dynamic screening
properties in metal clusters. In clusters and nanoparticles,
chemical, optical, and electronic properties can be tuned
by varying the size and/or the shape of the system. Besides
its fundamental interest, tailoring the electronic properties
can be of vast importance for many processes of technological
interest, such as photochemical reactivity. Over the last
years, the advent of femtosecond lasers has fueled the
experimental study of electron relaxation processes. Particular
attention has been paid to the dependence on size of
the electron–electron interaction processes [19–21]. We will
show that the interplay between screening effects and space
localization of the initial excitation makes the lifetime of
the excitation vary with respect to the bulk equivalent.
2. Theoretical methods
In very recent years, improvements in experimental
methods, in particular in the field of ultrafast laser spectroscopies,
are making it possible to study electronic excitations
at time scales below the femtosecond [22,23]. In this
time scale, the screening of excited electrons in condensed
matter can be incomplete and the description of the electronic
excitations as quasiparticles is questionable [24].
However, most standard experimental techniques provide
information on time scales for which the screening of excited
electrons can be approximated as instantaneous in
practice. In this case, the description of the many-body
electronic excitation can be simplified by means of the quasiparticle
picture. In the following, we focus into this
situation.
The scattering rate of an excited electron or hole due to
inelastic e–e scattering can be evaluated by means of
many-body calculations based on the electronic self-energy
[25,26]. In this approach the lifetime broadening of a quasiparticle
in a quantum state characterized by an energy E 0
and wave function w 0 is obtained as the projection of the
imaginary part of the self-energy R(r,r 0 ,E 0 ) onto the state
itself (atomic units are used throughout, i.e., h = e 2 =
m e = 1, unless otherwise is stated)
Z Z
C e–e ¼ s 1
e–e ¼ 2 drdr 0 w 0 ðrÞImRðr; r0 ; E 0 Þw 0 ðr 0 Þ: ð1Þ
Frequently the self-energy R is represented by the so-called
GW approximation [25], which is the first term in a series
expansion of R in terms of the screened Coulomb interaction
W. Usually, the non-interacting Green function G 0 is
used to replace the full one-electron Green function G. In
the following we provide a brief description of the numerical
methods used in the calculation of scattering rates in
the two examples considered in this work, namely, quantum
well states at metal surfaces and metal clusters. A more
detailed account can be found elsewhere [26,27].
2.1. Quantum well states
In the case of Na adlayers on the Cu(111) surface, we
assume that the charge density and the one-electron potential
only vary along the z-direction perpendicular to the
surface and are constant in the (x,y) plane parallel to the
surface. This assumption is valid for Na induced QWS’s
at coverage rates close to 1 ML, since the wave functions
lie mainly in the Na layer and in the vacuum side, i.e., in
a region with little potential variation in the direction parallel
to the surface [28]. This leads to significant simplification
of Eq. (1), which now takes a form
C e–e ¼ s 1
e–e ¼ 2 X 0
f
Z
Z Z dqk
ð2pÞ 2
dzdz 0 / 0 ðzÞ/ f ðz0 Þ
Im½ W ðz; z 0 ; q k ; E 0 E f ÞŠ/ 0 ðz 0 Þ/ f ðzÞ; ð2Þ
where the summation is performed over all available final
states f, W(z,z 0 ,q k ,x) is the two-dimensional Fourier transform
of the screened Coulomb interaction, and / n (z) is the
z-dependent component of the wave function w n (r):
w n ðrÞ ¼p 1 ffiffiffi e iq kr k
/ n ðzÞ
ð3Þ
S
with S being a normalization area. To describe the Cu(111)
surface covered by Na adlayers, we employ a slab containing
31 atomic layers of Cu together with a region corresponding
to Na and a vacuum region corresponding to 20
Cu interlayer spacings. For the description of the surface
electronic structure, the model potential of Ref. [17], based
on the model potential of the bare Cu(111) surface [29], has
been employed. This potential reproduces the QWS energy
position of E QWS ¼ 0:127 eV at the surface Brillouin zone
C
center for 1 ML Na coverage [30]. To investigate the linewidth
dependence on the QWS energy position relative to
the Fermi level and demonstrate the dramatic change of
the screening upon the QWS band shrink below E F , we evaluate
C e–e for the cases with QWS energies E QWS ¼ 0eV 1 and
E QWS
C
¼ 0:042 eV. We assume that the small variation of
the QWS binding energy at the scale of 0.1 eV does not
significantly affect the QWS wave function, and therefore
use the same set of wave functions obtained for the
E QWS
C
¼ 0:127 eV case, just shifting the QWS band to the
corresponding energy positions.We also assume here that
1 As the QWS energy position at the C point is gradually moved down
with increasing Na coverage [16] this energy position should roughly
correspond to h 0.9.
C

4548 V.M. Silkin et al. / Surface Science 601 (2007) 4546–4552
the QWS wave functions do not change along the QWS
energy band, although this can have a significant effect on
the excited electron linewidth in some cases [31,32]. For
the QWS energy dispersion curve vs the two-dimensional
wave vector k k , we use a parabolic form
E QWS ðk k Þ¼E QWS þ k 2 C k =2mQWS , taking as effective mass the
experimental value m QWS = 0.7 [30]. Further calculation
details on the evaluation of the screened Coulomb interaction
W(z,z 0 ,q k ,x) can be found in Refs. [33,34].
2.2. Metal clusters
The spherical jellium model [35,36] is used to represent
metal clusters of variable size. The average one-electron
radius r s in the cluster can be defined from r s = R/N 1/3 ,
where R is the radius of the cluster and N the number of
electrons in it. If the cluster is neutral, the equality
1=n þ 0 ¼ 4pr3 s =3 holds as well, with nþ 0
being the constant
background positive charge of the cluster.
We restrict our discussion to neutral clusters in closedshell
configurations. The ground state properties of the
cluster are described using density functional theory
(DFT) and Kohn–Sham (KS) equations [37]. The exchange-correlation
potential is calculated in the local density
approximation (LDA) with the parametrization of
Ref. [38]. The KS wave functions w(r) are calculated
numerically after expansion in the spherical harmonic basis
set Y lm (X r ). The set of KS equations are solved self-consistently
using an iterative procedure.
In real space, when the exact Green function G is replaced
by the independent-electrons Green function G 0 in
Eq. (1), one can show that
C e–e ¼ 2 X Z
0
drdr 0 w 0 ðrÞw f ðr0 Þ
f
ImW ðr; r 0 ; xÞw 0 ðr 0 Þw f ðrÞ;
where w 0 (r) is a KS wave function with eigenvalue E 0 ,
x = E 0 E f , and the sum over f runs over all unoccupied
KS states of eigenvalues E f < E 0 . Hence we are approximating
the energy levels and wave functions in the decay
by the eigenvalues and wave functions of unoccupied KS
levels. This is a reasonable first-order approximation [39].
In practice, the spherical symmetry of the problem simplifies
the calculation [27]. All quantities in Eq. (4) are expanded
in the spherical harmonics basis set. The screened
interaction W(r,r 0 ,x) is calculated from the response function
of the system v(r,r 0 ,x) obtained in the random phase
approximation (RPA) [40,41]. The latter is built in a similar
way to that of Ref. [42]: one-electron Green functions are
built from the KS wave functions for every x. These Green
functions are used to calculate the multipole components
of the independent-electrons response function v 0 (r,r 0 ,x),
and the RPA integral equation is solved in real space after
matrix inversion for every polar component. Finally, the
decay rate C e–e is calculated as a sum over the independent
contributions of each of the available final states [27].
ð4Þ
3. Calculation results and discussion
3.1. Quantum well states
In Fig. 1, the calculated dependence of the linewidth C e–e
on the excited energy is presented for three Na QWS energy
positions. A large variation of the linewidth as a function of
coverage is observed. A particularly strong enhancement of
C e–e for small excitation energies and for the two systems
with partially occupied QWS band is observed, with respect
to the case of unoccupied QWS. Additionally, one can note
the different behavior between the two occupied QWS cases.
For small energies, C e–e for E QWS ¼ 0:042 eV is larger than
C
C e–e for E QWS ¼ 0:127 eV. Indeed, the two curves cross
C
around E = 0.25 eV, and the linewidth for E QWS ¼ 0:127
C
starts to be the largest for higher energies. This finding is
rather difficult to understand in simple terms such as phase
space available for the quasiparticle decay, since this phase
space is the same for all three systems as is seen in Fig. 2.
Note that these intraband transitions within the QWS band
itself are the main source for excited electron decay in all
systems under study. Thus, for small energies this contribution
accounts for almost 100% of the total decay rate. With
the increase of the excitation energy the contribution from
interband transitions to the empty bulk electronic states increases
faster than that from intraband transitions. Nevertheless,
even for energies around 2 eV the intraband
contribution accounts for more than 90% of the total
C e–e . Hence, the two-dimensional effects in Na/Cu(111) systems
are much stronger than in the case of surface states at
clean metal surfaces [8,43]. This fact can be explained by the
larger decoupling of the QWS state wave functions from the
bulk electron system [17].
To understand the behavior of C e–e as a function of the
QWS energy position let us see Eq. (2), where the double
integration in z and z 0 is performed. As mentioned above,
Linewidth (meV)
400
300
200
100
0
0 0.5 1 1.5 2
Energy (eV)
Fig. 1. Calculated e–e contribution, C e–e , to the linewidth of a quantum
well state in Na/Cu(111) for QWS energy position E QWS ¼ 0:127 eV
C
(solid line), E QWS ¼ 0:042 eV (dashed line), and E QWS ¼ 0 (dotted line).
C
C

V.M. Silkin et al. / Surface Science 601 (2007) 4546–4552 4549
Fig. 2. Quantum well states energy dispersion for E QWS ¼ 0:127 eV
C
(solid line), E QWS ¼ 0:042 eV (dashed line), and E QWS ¼ 0 (dotted line).
C
C
Grey area shows the regions of projected bulk electronic structure of bare
Cu(111) surface being most efficient decay channel in Na/Cu(111)
systems as well [18]. The two main decay channels, namely intraband and
interband transitions, are shown schematically by arrows.
the QWS wave functions / 0 (z) and the bulk electronic structure
for all the three different QWS energy positions have
been hold the same in the present calculation. In addition
to the above-mentioned similarity of the available phase
space for decay, the remaining difference between these calculations
is the screened Coulomb interaction W(z,z 0 ,q k ,x)
originated from different QWS energy positions. We perform
calculation of W(z,z 0 ,q k ,x) for all the three systems
and analyze them below. First of all, we note a strong
variation of ImW as a function of all its variables z, q k ,
and x. To give a general idea of it in Fig. 3, we present
ImW(z,z,q k ,x) as a function of z and x for the E QWS ¼
C
0:127 eV and E QWS ¼ 0:042 eV cases for two-dimensional
momentum value q k = 0.05 a.u.
1 . The use of
C
ImW(z,z,q k ,x) as an example is reasonable as inside the
solid and near the surface ImW(z,z 0 ,q k ,x) has the maximum
magnitude at z = z 0 and only far from the surface into
the vacuum the maximum of this quantity occurs at z 5 z 0
[44]. InFig. 3 one can observe a strong enhancement of
ImW(z,z,q k ,x) for z P 0 and energies x 6 0.25 eV and
x 6 0.18 eV, respectively. Comparing both figures one
can see that the amplitude of ImW in low left corner for
the E QWS ¼ 0:042 eV case is much larger than for the
C
E QWS ¼ 0:127 eV case. Therefore, although the phase
C
space in the former case is smaller as one can appreciate
in Fig. 5, this ImW enhancement leads to a larger linewidth
for the E QWS ¼ 0:042 eV case for small energies. When the
C
energy increases, the relative role of this enhanced region
of ImW decreases leading to a smaller linewidth for
the E QWS ¼ 0:042 eV case, as compared to the E QWS ¼
C
C
0:127 eV one. In Fig. 4, ImW(z,z,q k ,x) for E QWS ¼
C
0:127 eV and E QWS ¼ 0 are presented for two values of
C
q k , demonstrating that, at the surface, the amplitude of
Fig. 3. Imaginary part of the screened interaction, ImW(z,z,q k ,x), for the
E QWS ¼ 0:127 eV (upper panel) and E QWS ¼ 0:042 eV (lower panel)
C
C
cases as a function of distance z and energy x for the two-dimensional
momentum q k = 0.05 a.u.
1 . Na adlayer is located at z = 0 and negative
(positive) z values correspond to the solid (vacuum) side.
ImW is two orders of magnitude larger than in the former
case, whereas the latter is very similar to the clean Cu(111)
case [44].
Furthermore, in Fig. 3, one can see additional peaks at
the maximum energy for QWS intraband e–h excitations
which correspond to the collective electronic excitations
similar to the acoustic surface plasmon (ASP) at the clean
metal surfaces [33,34]. Their dispersions closely follow the
upper edge of regions for e–h excitations within the QWS
bands and are shown in Fig. 5. The origin of this collective
mode is the coexistence at the surface of both the QWS and
bulk electrons. More detailed analysis of its properties will
be given elsewhere. From Fig. 5, it is clear that the ASP
contributes to the total linewidth C e–e . Nevertheless, it is
difficult to separate this contribution from the very efficient
decay channel due to intraband QWS e–h excitations.
3.2. Metal clusters
One of the most appealing features of low-dimensional
systems such as clusters is that their properties can often

4550 V.M. Silkin et al. / Surface Science 601 (2007) 4546–4552
Fig. 5. Phase space for intraband electron–hole excitations in the QWS
band. Grey colored area shows the region for the E QWS ¼ 0:127 eV case
C
and hatched area stands for E QWS ¼ 0:042 eV. Intraband bulk and
C
interband QWS-bulk electron–hole excitations are allowed everywhere.
Thick solid and dashed lines on the left sides of each area show the
dispersion of corresponding acoustic surface plasmons. As an example,
thin dashed (dotted) line delimits the phase space region available for
intraband transitions for an excited electron of energy of 0.3 eV above E F
for the E QWS ¼ 0:127 eV ðE QWS ¼ 0:042 eVÞ case.
C
C
Fig. 4. Imaginary part of the screened interaction, ImW(z,z,q k ,x), as a
function of z for q k = 0.05 a.u.
1 (upper panel) and q k = 0.10 a.u.
1 (lower
panel). Thick (thin) lines stand for the E QWS ¼ 0:127ðE QWS ¼ 0Þ case.
C
C
Solid, long dashed, dashed, dashed-dotted, and dotted lines show ImW
for x = 0.06 eV, 0.12 eV, 0.18 eV, 0.24 eV, and 0.30 eV, respectively.
be tuned by varying the size and thus they can be different
from the bulk case. For this reason and in the following, we
focus into clusters of variable size and fixed average electronic
density (r s = 4, that represents for instance sodium
clusters). In this respect, the difference between the dynamics
of hot electrons in clusters and in bulk has been discussed
bringing up two different effects. First, the electron
lifetime can be enhanced in the cluster because of the discretization
of levels that reduces the number of final states
to which the electronic excitation can decay. Then again,
the electron lifetime can decrease in the cluster because
of the reduction of dynamic screening in the proximity of
the cluster surface. We show in the following that the
screening in clusters is not only modified due to the presence
of the surface but also due to the different mobility
of the cluster electrons with respect to those in bulk. The
dynamically screened interaction between the cluster electrons
is affected by this factor, in particular for low excitation
energies.
In Fig. 6, we show the linewidth C e–e of electron excitations
with energy E 0 1 eV measured from the Fermi level
of the cluster. Due to the discretization of levels in the cluster,
the initial energy of the excitation cannot be fixed exactly
to 1 eV, but the eigenvalue closest in energy is
chosen. In any case, the variation with respect to the reference
value is never larger than 10%. We come back to this
Fig. 6. Linewidth (in meV) of electron excitations with energy E 0 1eV
above the Fermi level of the cluster, as a function of the cluster radius (in
a.u.). All systems have r s = 4. The value of C e–e obtained from an RPA
calculation in an homogeneous electron gas with the same parameter r s is
also shown.

V.M. Silkin et al. / Surface Science 601 (2007) 4546–4552 4551
point below. C e–e is plotted as a function of the cluster radius
R. For small cluster sizes, the discretization of levels is
clearly observable in the large oscillations found. For clusters
of nanometer size, however, these oscillations are
damped and the value of the linewidth is C e–e 150 meV.
Let us also mention that an RPA calculation of the electron
lifetime in a homogeneous electron gas, with the same
parameters that we fix in our calculation (E 0 = 1 eV and
r s = 4) gives a value C e–e = 60 meV [45], smaller than the
ones obtained for these big clusters.
One of the reasons not often discussed in the literature
for the difference between the value of C e–e in bulk and in
nanometer-sized clusters is the difference in the electron–
electron screened interaction W(r,r 0 ,x) between an infinite
electron gas and a finite jellium system, even for values of r
and r 0 far from the cluster surface. In Fig. 7, we quantify
this statement by showing the imaginary part of
ImW(r,r 0 ,x) for a value of r nearby the cluster center
(r = 0.31) and as a function of r 0 . The dimension of the
chosen is sufficiently big (R = 38.8 a.u.) to consider that
no surface effects are relevant. ImW is plotted for two different
values of the energy, namely x = 0.815 meV and
x = 54.4 eV. The former is representative of actual values
of x that enter the calculation of C e–e in clusters. Fig. 7
shows that ImW(0, r 0 ,x = 0.815 eV) is clearly different in
a cluster as compared to the bulk counterpart. The dynamic
screening between electrons is thus modified due to
the different mobility of electrons in a confined system.
However, for higher values of x, the response of the system
is similar to that calculated in the independentelectron
approximation v 0 and finite-size effects are not
relevant.
Confinement effects are also relevant in the choice of the
initial state in the decay process. The discretization of levels
in the cluster makes it impossible to exactly define the same
initial energy E 0 for all the cluster sizes considered. An estimate
of the variation that this feature can introduce in the
linewidth value is also shown in Fig. 6. For a cluster size of
R = 31 a.u., the value of C e–e for two initial states very
close in energy is plotted. One of them is the first l =13
state with E 0 = 1 eV, and the second one is a l = 4 state
with E 0 = 0.99 eV. In spite of the proximity of the two energy
levels, the value of C e–e varies as much as 35%. The
main reason for that is the different coupling between the
initial state wave function and the available final states
for the decay. In Fig. 8, we show the normalized radial distribution
of the KS wave functions rR l;E0 ðrÞ, where
w 0 ðrÞ ¼R l;E0 ðrÞY lm ðX r Þ, for the two cases considered. The
integrated radial distributions
QðrÞ ¼
Z r
0
dr 0 r 02 R l;E0 ðr 0 2
Þ
ð5Þ
are plotted in Fig. 8 as well. While the l = 13 wave function
does not have any nodes, oscillations in the l = 4 wave
function are conspicuous. A proper definition of the initial
excitation in the decay process can thus be crucial when
calculating the linewidth.
Fig. 7. Imaginary part of the self-energy ImW(r 0 ,r 0 ,x) (in eV) for a
cluster of N = 912 and r s =4(R = 38.8 a.u.), as a function of the radial
coordinate r 0 (in a.u.), for two different values of the energy x = 0.815 eV
and 54.4 eV. The coordinate r 0 is fixed to 0.31 a.u. The corresponding
values for ImW in an homogeneous electron gas obtained from Mermin
response functions are also plotted for comparison.
Fig. 8. Normalized radial distribution of wave functions with l =4, 13
and E 0 = 0.99, 1 eV, respectively, corresponding to a cluster with N = 440
and r s =4 (R = 30.42 a.u.). E 0 is measured from the Fermi level of the
cluster. The integrated value of the wave functions is plotted in the upper
panel.

4552 V.M. Silkin et al. / Surface Science 601 (2007) 4546–4552
4. Summary and conclusions
Low-dimensional systems bear electronic properties that
can be very different from those well studied in bulk systems.
Here, we have shown that the modification of the dynamic
screening in the electron–electron interaction and
the change in the structure of energy levels in these systems
can combine to drastically modify the linewidth of electronic
excitations as compared to the bulk reference. In
particular, we have analyzed the screened interaction between
electrons and the phase space availability for the decay
in two different cases: quantum well states in overlayers
of Na over Cu(111) and gas-phase metal clusters of sizes
up to few nanometers. In the former case, the amount of
alkali atoms coverage on the metal surface leads to a shift
of the QWS energy band, that produces an appreciable
change in the linewidth. In the latter case, the variation
in size of the cluster is responsible for modifications in
the decay rate. Our detailed analysis of these properties
shows that a proper choice of the system characteristics
should allow to tune the excitation linewidths due to electron–electron
scattering processes.
Acknowledgements
Financial support by the Basque Departamento de Educación,
Universidades e Investigación, the University of
the Basque Country UPV/EHU (Grant No. 9/UPV
00206.215-13639/2001), the Spanish MEC (Grant No.
FIS2004-06490-C03-00), and the EU Network of Excellence
NANOQUANTA (Grant No. NMP4-CT-2004-
500198) is acknowledged.
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