Residual strain measurements in InGaAs metamorphic buffer layers on GaAs

Abstract

This work deals with the strain relaxation mechanism in InGaAs metamorphic buffers (MBs) grown on GaAs substrates and overgrown by InAs quantum dots (QD). The residual strain is measured by using Raman scattering and X-ray diffraction, both in Reciprocal Space Map and in single ω-2θ scan modes (ω and θ being the incidence angles on the sample surface and on the scattering planes, respectively). By relating the GaAs-like longitudinal optical phonon frequency ω LO of InGaAs MBs to the in-plane residual strain ε measured by means of photoreflectance (PR), the linear ε-vs.-ω LO working curve is obtained. The results of Raman and XRD measurements, as well as those obtained by PR, are in a very satisfactory agreement. The respective advantages of the techniques are discussed. The measurements confirm that strain relaxation depends on the thickness t of the buffer layer following a ~t -1/2 power law, that can be explained by an energy-balance model. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Full text

Eur. Phys. J. B 56, 217–222 (2007)
DOI: 10.1140/epjb/e2007-00105-8
THE EUROPEAN
PHYSICAL JOURNAL B
Residual strain measurements in InGaAs metamorphic buffer
layers on GaAs
V. Bellani
1,a
,C.Bocchi
2
, T. Ciabattoni
1
, S. Franchi
2
,P.Frigeri
2
,P.Galinetto
1
,M.Geddo
3
,F.Germini
2
,
G. Guizzetti
1
,L.Nasi
2
,M.Patrini
1
,L.Seravalli
2
,andG.Trevisi
2
1
Dipartimento di Fisica “A. Volta” and CNISM, Universit`a di Pavia, 27100 Pavia, Italy
2
CNR-IMEM Institute, Parco delle Scienze 37a, 43100 Parma, Italy
3
Dipartimento di Fisica and CNISM, Universit`a di Parma, 43100 Parma, Italy
Received 28 November 2006 / Received in final form 8 March 2007
Published online 13 April 2007 –
c
 EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2007
Abstract. This work deals with the strain relaxation mechanism in InGaAs metamorphic buffers (MBs)
grown on GaAs substrates and overgrown by InAs quantum dots (QD). The residual strain is measured
by using Raman scattering and X-ray diffraction, both in Reciprocal Space Map and in single ω − 2θ
scan modes (ω and θ being the incidence angles on the sample surface and on the scattering planes,
respectively). By relating the GaAs-like longitudinal optical phonon frequency ω
LO
of InGaAs MBs to the
in-plane residual strain ε measured by means of photoreflectance (PR), the linear ε-vs.-ω
LO
working curve
is obtained. The results of Raman and XRD measurements, as well as those obtained by PR, are in a
very satisfactory agreement. The respective advantages of the techniques are discussed. The measurements
confirm that strain relaxation depends on the thickness t of the buffer layer following a ∼t
−1/2
power law,
that can be explained by an energy-balance model.
PACS. 78.30.Fs III-V and II-VI semiconductors – 61.10.-i X-ray diffraction and scattering – 71.70.Fk
Strain-induced splitting
1 Introduction
Lattice strain in semiconducting materials is an effec-
tive tool to modify energy gaps, to shift and reverse the
band edges of heavy-hole and light-hole bands, to remove
band degeneracy at critical points of the Brillouin zone
and to change band curvatures and hence-carrier effective
masses [1,2]. The strain can be induced in a semiconductor
by applying an external pressure or by epitaxially growing
thematerialtobestressedon a lattice-mismatched sub-
strate or layer that behaves as a virtual substrate. While
the first approach is used to study the electronic band
structure and related parameters, only the second one is
suited for exploitation in devices. In this case, the struc-
tures may consist of a substrate, a buffer layer (termed
as metamorphic buffer — MB) and the active part of the
structure; by controlling the lattice parameter of the MB
the mismatch between buffer and the upper part of the
structure is changed and, hence, the strain in the active
layer is affected. The lattice parameter of the MB depends
on its composition and thickness through the mechanism
of strain relaxation, by which — for MB thickness larger
than a critical thickness — the elastic strain is partially
relaxed and the formation of a network of misfit disloca-
tions takes place.
a
e-mail: bellani@unipv.it
The approach of engineering material properties by
means of lattice strain has been used not only for semicon-
ductors with a three-dimensional system of carriers, but
also for quantum wells and superlattices (two-dimensional
systems), quantum wires (one-dimensional systems) and
quantum dots (zero-dimensional systems). Examples of
structures that make use of MBs to improve device perfor-
mances are: (a) high electron mobility transistor (HEMT)
structures, grown on MBs to take advantage of the higher
electron mobility and the better carrier confinement in the
channel region made of In-rich InGaAs alloys [3]; (b) het-
erojuction bipolar transistor (HBT) structures prepared
with base-layers of high In-content so as to have relatively
small band gap and increased mobilities, saturation veloc-
ities, as well as reduced base contact-resistance [4]; and (c)
multi-junction solar cells grown on MBs [5] that utilise a
wider part of the solar spectrum, thus increasing the con-
version efficiency. Also SiGe-based structures have taken
and will take increasing advantages of strain engineering
since the pioneering work of Abstreiter et al. [6] up to the
most recent proposals [7].
In InAs/InGaAs QD structures light emission can
be red-shifted to long-wavelengths (λ  1.3 µm) [8–10]
by QD strain engineering. In particular emission at
1.3–1.4 µm could be obtained at room temperature
(RT) from InAs/InGaAs QDs by using the QD strain
as a tuning parameter, which is controlled by the

218 The European Physical Journal B
thickness-dependent strain relaxation of suitably designed
InGaAs MB [9,11,12]. Moreover, it has been shown
that the QD strain engineering yields two degrees-of-
freedom [13] that can be used not only to red-shift the
emission, but also to enhance the RT emission efficiency,
possibly up to 1.55 µm. Metamorphic buffers may have ei-
ther constant composition or a continuously graded one; in
the second case, in spite of the more sophisticated growth
process that is required, advantage can be taken of the
possibility to control the misfit dislocation distribution
and of confining them close to the substrate-MB heteroin-
terface — far from the active part of the structure — so
that not to spoil its optoelectronic properties [14].
The use of strain to tailor the electronic properties of
materials requires the capability of modelling strain relax-
ation and the availability of techniques to measure it in
structures suited for specific applications, such as light-
emitters.
Raman scattering has been usefully applied to mea-
sure the strain in InGaAs layers grown by different epitax-
ial techniques on GaAs or InP substrates [15–19]. These
works have shown that the strain of an InGaAs layer can
be derived by measuring the frequencies of the GaAs-like
or the InAs-like optical phonon modes of the alloy.
Well-known and established X-ray diffraction meth-
ods [20,21] based on the measurement of asymmetric
Bragg reflections both in linear scan mode and by recipro-
cal space mapping (RSM) allow the determination of the
strain relaxation of mismatched heterostructures. By sin-
gling out the effects due to strain and to composition, the
strain tensor components as well as the alloy composition
can be obtained separately also for the general case of an
arbitrary distortion of the epilayer lattice unit cell [22].
The main advantage of X-ray diffraction methods is the
high accuracy in the measurements of lattice parameter a,
with uncertainties ∆a/a < 1 × 10
−5
.
In this work we present the investigation by micro-
Raman scattering and by X-ray diffraction methods of
strain relaxation in InGaAs metamorphic buffers incor-
porated in InAs/InGaAs QD nanostructures grown by
Molecular Beam Epitaxy (MBE). The results show that
the strain relaxation of MBs can be effectively predicted
by the Mar´ee et al. model [23] and that the strain can
be measured in QD InAs/InGaAs structures for long-
wavelengths operation at RT by means of the above men-
tioned techniques, the respective advantages of which are
discussed. These results, along with those reported in
reference [24], justify the approach of QD strain engineer-
ing [9,12,13] that may open the way to the fabrication of
QD nanostructures for 1.55 µm operation at RT, a result
of huge technological interest for telecom and datacom
applications.
2 Experimental procedures
The structures grown on (100) GaAs substrates consist of:
(i) a 100 nm-thick GaAs buffer layer; (ii) a In
x
Ga
1−x
As
partially-relaxed metamorphic buffer MB (which acts also
as lower confining layer (LCL) for QDs carriers) with
thickness t ranging from 20 to 1000 nm and grown by MBE
at 490
◦
C; (iii) a plane of InAs QDs with 3-monolayer cov-
erage, deposited by Atomic Layer Molecular Beam Epi-
taxy (ALMBE) [25] at 460
◦
C. The structures have been
grown both with and without a 20 nm-thick In
x
Ga
1−x
As
upper confining layer (UCL); such a layer is grown by
ALMBE at lower temperature (360
◦
C) in order to reduce
the interaction among confining layers and QDs. The In
content of LCLs and UCLs is identical and ranges from
x =0.09 to x =0.31. Before and after the deposition
of QDs, the growth has been interrupted for 210 s to
change the substrate temperature. More details on the
growth conditions can be found in reference [9]. The resid-
ual strain of MBs (LCLs) has been measured in structures
with QDs in order to correlate the strain measurements
done in this work to the emission energy studied by PL
and reported in references [9,12,13].
Micro-Raman spectra have been measured with a Dilor
LabRam system. In this set-up the excitation laser beam
is sent to the sample through microscope optics. A HeNe
laser light source (λ = 632.8 nm), with 15 mW laser power
and a 100 magnification objective optics have been used.
The Raman signal has been recorded by a silicon CCD
camera cooled down to 210 K, with spectral resolution of
1cm
−1
. The spectra have been taken using 2 min integra-
tion times to improve the signal to noise ratio.
X-ray diffraction measurements have been performed
by using a Philips high resolution diffractometer equipped
with a four (220) reflections Ge-crystal monochromator
for selecting the CuKα
1
X-ray radiation line. Reciprocal
Space Maps (RSM) of the coherent scattered intensity
have been obtained by using a triple bounce Ge crystal
analyser scanning the reciprocal space with an angular res-
olution of 12 arcsec. In order to evaluate the strain relax-
ation of MB layers, RSMs around the asymmetrical (-224)
and (2-24) have been used. The lattice parameters parallel
and perpendicular to the interfaces and, then, the compo-
sition and the strain components have been obtained from
the position in reciprocal space of the diffraction peaks due
to the MB layer and the substrate [26]. Furthermore, lin-
ear ω −2θ scans (ω and θ being the incidence angles on the
sample surface and on the scattering planes, respectively)
through the asymmetrical 335 and symmetrical 004 re-
flections have been also collected for a selected number
of structures in order to verify the occurrence of differ-
ences in the strain level of UCLs and LCLs. In this case
advantage has been taken of the better signal-to-noise ra-
tio of single scan measurements, as compared to the area
scan (RSM) counterparts, related to the more favourable
counting statistics.
3 Results and discussion
In Figure 1 the Raman spectra of the InAs/In
0.15
Ga
0.85
As
structures with UCL in the frequency region of the
GaAs-like longitudinal (LO) and transversal (TO) optical
phonons are plotted. The structures related to InAs-like
LO phonons are less intense and are not reported here. We
note that the Raman frequency of our structures shifts

V. Bellani et al.: Residual strain measurements in InGaAs metamorphic buffer layers on GaAs 219
Fig. 1. Raman spectra of In
0.15
Ga
0.85
As metamorphic buffers
(MBs) of different thicknesses incorporated in InAs/InGaAs
quantum dot nanostructures with GaAs substrates, in the spec-
tral region of the GaAs-like LO phonon. Spectra are shifted
vertically for clarity.
monotonically to lower values with increasing the LCL
thickness, and the total shift is of ∼5cm
−1
for t ranging
from 20 to 360 nm. In the spectra we can also observe a
less intense structure, which in the sample with t =20nm
is around 265 cm
−1
, due to the TO GaAs-like mode acti-
vated by disorder [16]. This structure shifts to lower fre-
quencies and broadens with increasing t. The spectra have
been carefully analysed in order to derive the peak energy
of the GaAs-like LO mode by best fitting a Lorentzian
lineshape to the experimental data. The resulting uncer-
tainty in the value of the peak energy is 0.2 cm
−1
.
According to references [16,19,27] a linear relation be-
tween the GaAs-like LO phonon frequency and the in-
plane residual strain ε =(a
MB
− a
InGaAs
)/a
InGaAs
of the
InGaAs layer has been assumed. Here a
MB
represents
the lattice parameter of the MB in the growth plane,
while a
InGaAs
is the lattice parameter of free-standing
In
x
Ga
1−x
As with the same x. In pseudomorphic InGaAs
layers grown on GaAs, i.e. with thicknesses smaller than
the critical thickness for plastic relaxation of the strain,
a
MB
coincides with the lattice parameter of free-standing
GaAs.
The phonon frequency shift due to strain in the alloy
layer is given by [19]:
∆ω
LO
=ω
LO
− ω
LO
0
=

S
12
K
LO
11
S
11
+ S
12
+ K
LO
12

ω
LO
0

ε = ξε
(1)
where S
ij
and K
LO
ij
are the elastic compliance and phonon
deformation potential tensors, respectively; ω
LO
and ω
LO
0
are the LO phonon frequency of strained and unstrained
InGaAs, respectively.
In order to find the parameters of the linear depen-
dence of ε on the measured ω
LO
in x =0.15 samples,
instead of relying on literature data for S
ij
, K
LO
ij
and
ω
LO
0
that are somewhat scattered we performed a best-fit
of equation (1) by using the values of ω
LO
from Raman
measurements and of residual strain ε measured by means
of photoreflectance (PR) measurements [11,24,28] on the
same samples; this approach allow us to minimize the ef-
fects of unintentional differences among sample prepared
by different techniques and under different conditions.
Photoreflectance measurements yield the strain-related
splitting between heavy- and light-hole bands at the Γ
point of the Brillouin zone; according to the deformation
potential theory [27,29]; such a splitting is linearly pro-
portional to the in-plane strain.
The best-fit procedure yielded ω
LO
0
= 285 cm
−1
and
ξ = −666.9cm
−1
for x =0.15. The value of ω
LO
0
well
compare with those measured in literature on strain-free
InGaAs alloy layers [16].
We note that previously published PR results [24] for
the same samples (with UCL) and similar samples without
UCL, reported on the negligible variations observed in the
parameter values of the HH and LH transitions, ensuring
that UCLs are pseudomorphic to LCLs. Consequently, in
the following, PR, Raman and XRD strain results will
be equally well compared with theoretical predictions on
strain relaxation.
Then ε values have been calculated by equation (1)
with the above ξ and ω
LO
0
parameters and experimental
ω
LO
values. These values are reported in Figure 2 as a
function of the MB thickness; the figure also shows the ε
values from RSM measurements in the vicinity of (224)
reciprocal lattice nodes for structures with MBs with In
composition in the 0.09–0.31 range, without UCLs and
with QDs.
In Figure 3 we plot the reciprocal space map in
the vicinity of (-224) reciprocal lattice node for the
structure with In composition x =0.09. The con-
tours of constant scattered intensity around a node
have been derived from a series of ω − 2θ scans with
different ω offsets. The conversion of each intensity-
peak position (ω,2θ) in reciprocal space coordinates
(Q
x
and Q
z
parallel to [-110] and [001], respectively)
is given by [26]: Q
x
= R[cos(2ω

) − cos(2ω

− ω)] and
Q
z
= R[sin(ω) − sin(2ω

− ω)]; where R is the Ewald
sphere radius (R = |k
i
| =1/λ)and2ω

=2θ
B
when
the Bragg condition is satisfied.
The broadening of the substrate and MB peaks are
comparable along the reciprocal space [112] direction, thus
demonstrating that strain (and composition) in MBs have
not significant variations in depth. Instead, the much
larger broadening of the MB intensity distribution per-
pendicular to [112] direction as compared to the substrate
one is related to the “mosaicity” of MBs induced by the
misfit dislocations intentionally formed to relax the elas-
tic strain. The separation between the intensity maxima
is directly related to the strain relaxation.
The values of ε in the range of plastic relaxation have
been corrected by the so-called thermal misfit [30] ∆
T
that is due to the different contraction of substrate and
layer during cooling from the growth temperature T
g
to

220 The European Physical Journal B
Fig. 2. Absolute value of residual in-plane strain ε as a func-
tion of the thickness t of In
x
Ga
1−x
As metamorphic buffers ob-
tained by Raman scattering (open squares), reciprocal space
map (RSM, closed circles), and photoreflectance (PR, closed
diamonds) measurements. The continuous line shows the thick-
ness dependence of strain both in the pseudomorphic regime
(horizontal line, x =0.15) and in the partial relaxation one
where ε can be approximated by a t
−1/2
dependence [23]; the
dotted line represents the t
−1
behaviour foreseen by the equi-
librium models [33–35]. Raman and PR measurements refer to
x =0.15 structures, while the MB compositions of structures
for RSM are given in the figure.
the room-temperature, under a condition where the de-
fects can be considered as “frozen”. The thermal misfit
for the InGaAs/GaAs is definitely smaller than the value
∆
T
= −3.76 × 10
−4
for T
g
= 490
◦
C of the InAs/GaAs
counterpart, that is calculated using the thermal expan-
sion coefficients reported in the literature [31].
In Figure 2 we also report ε alues obtained by means
of PR spectroscopy on the same samples [11,24]. We
note that the strain values obtained by Raman scatter-
ing, X-ray diffraction and photoreflectance are in a very
satisfactory agreement.
As for errors in the RMS measurements of the residual
strain, we note that the occurrence of misfit dislocations
at the substrate/buffer interface and the consequent sam-
ple curvature [32] result in the broadening of both sub-
strate and buffer layer peaks (Fig. 3). The peak widening
gives rise to a certain ambiguity in the determination of
the angular position of the intensity maxima. This was
particularly evident for the samples with thinnest MBs.
Notwithstanding, the peak separation between the LCL
and the substrate could be evaluated with an accuracy
of ±30 arcsec, for which an error of ±1.5 × 10
−4
in the
calculated ε values was estimated. Concerning the MB
composition x, the relative difference between the nom-
inal and the values of the MB composition measured by
X-ray diffraction is always less than 5% and the estimated
Fig. 3. Reciprocal space map (RSM) around the -224 asym-
metrical node of a structure consisting of a 1000 nm-thick
In
0.09
Ga
0.91
As metamorphic buffer (MB) and a GaAs sub-
strate (broader and sharper diffraction peaks, respectively).
The [112] crystallographic direction and its perpendicular (the
mosaic broadening direction) are indicated. The relaxation line
joining the reciprocal lattice nodes (black dots) associated to
the full strained and full relaxed MB conditions is also shown.
The open circle represents the position of the maximum of the
buffer diffraction peak. The dashed line crossing the substrate
and the full strained MB nodes is parallel to the [001] direction.
error is ∼±2.5%, except for one case where it was ±4.0%.
The reported composition of samples measured by Ra-
man scattering and PR are the nominal ones, very close
to those measured by XRD techniques.
As regards the Raman measurements, the uncertainty
of 0.2 cm
−1
in the peak energy ω
LO
yields an error in the
determination of ε values that is ±2 × 10
−4
.
Let us compare now our experimental results with
predictions of existing models of strain relaxation. It is
well-known that when the critical thickness is exceeded
the epitaxial growth of a lattice mismatched layer is no
longer pseudomorphic. The lattice mismatch is accommo-
dated partly by elastic strain and partly by the formation
of a misfit dislocation network. In the frame of the contin-
uum elasticity theory the equilibrium between elastic and
plastic accommodation is found by minimizing the total
energy of the system [33], given by the elastic strain en-
ergy and the dislocation energy. The models based on this
assumption, lead to a strain relaxation rate proportional
to t
−1
,wheret is the epilayer thickness [33,35]. Most of
the experimental observations made on different epitax-
ial systems, indicate that while the equilibrium models
quite properly give the critical thickness for the onset of

V. Bellani et al.: Residual strain measurements in InGaAs metamorphic buffer layers on GaAs 221
the formation of misfit dislocations, they cannot explain
the strain relaxation rate as well. If the energy-balance
model [23] is assumed, the relation ε
2
t ∼ const. is ob-
tained, that gives a ∼t
−1/2
dependence for the residual
strain. Figure 2 shows that the experimental values of ε
obtained by both Raman scattering and X-ray diffraction
experiments, as well as by PR, are in better agreement
to the prediction of the model of Mar´ee et al. than with
those of references [33–35], thus producing further sup-
port to former model of strain relaxation in mismatched
materials.
The former model [23] considers the nucleation and
expansion of dissociated half loop dislocations originating
from the free surface of the layer and propagating down to
the interface between layer and substrate. The assumption
is that the energy of the half loop system increases during
the expansion until a critical loop radius is achieved where
the energy is a maximum. Then, the expansion continues
lowering the total energy of the system.
Raman measurements performed on structures with
t = 220 nm and higher composition (not reported here)
have revealed well-behaved LO phonon spectral features
also for x =0.35. This indicates the possibility to opti-
cally determine the strain status even when it cannot be
obtained through PR measurements. Indeed it has been
shown that PR determination of strain stems from the ac-
curate measurement of the splitting between the LH- and
HH-related interband transitions: this procedure may be
hindered by the broadening of optical transitions which
increases with increasing In composition. On the other
hand, Raman measurements of residual strain can be done
provided that the proportionality constants between ω
LO
shift and ε are known for the alloy composition of inter-
est, either from the literature or from calibrations, as it
has been done in the present paper.
It should be noted that the values of ω
LO
of strained
In
x
Ga
1−x
As depend not only on the values of the residual
strain ε but also on the material composition x. Therefore
any uncertainty on x affects the accuracy of the measure-
ment of ε. On the other hand, XRD measurements simul-
taneously and directly give x and ε, without the need of
any calibration with other techniques, but — as a draw-
back — are much more time-consuming than the Raman
characterization of residual strain. It is useful to remind
that results of the PR approach to measure strain are
fairly independent of layer composition, since ε is deduced
from the difference of two quantities (the HH and the LH
energy gaps) that present a similar dependence on x.
From the analysis of the ω − 2θ scans, structures with
and without UCL on top of QDs show no significant dif-
ferences in the strain status. An exception is made for a
structure with x =0.31 and a 19-nm thick LCL, that —
for reasons under investigation — showed an anomalously
high density of threading dislocations in Transmission
Electron Microscopy (TEM) cross-sections. Moreover, in
references [24] and [28] it was shown by PR measurements
of the same structures studied in the present work that
negligible differences exists between the in-plane strain in
structures with and without UCLs, thus confirming that
UCLs are pseudomorphic to LCLs. Hence, the compar-
ison presented in Figure 2 between the results obtained
in structures with and without UCL is justified and the
assumption of the model developed to calculate the light
emission energy from strain-engineered QD InAs/InGaAs
nanostructures [9] is substantiated.
4 Conclusions
In order to study the strain relaxation of constant-
composition metamorphic buffers as a function of their
thickness, we have considered two well known techniques
to determine quantities linearly dependent on the in-
plane residual strain ε of an epitaxial layer grown on a
mismatched substrate. Those are Raman scattering, that
gives the LO phonon frequency ω
LO
of the strained mate-
rial, and X-ray diffraction either in the Reciprocal Space
Map and in the single ω − 2θ scan modes, that directly
yields the values of lattice parameter a
MB
of the metamor-
phic buffer and, then, of ε.Theε-vs.-ω
LO
working curve
for Raman measurements has been obtained by fitting the
Raman shift data to the values of the residual strain de-
duced by photoreflectance on thesamesamples.Wehave
shown that the ε values measured by means of Raman,
XRD and PR are in a very satisfactory agreement and
the respective advantages of the techniques are discussed.
The residual strain values ε versus the thickness t of
metamorphic buffers have been compared to the results
of strain relaxation models, that give different ε(t) depen-
dences; the experimental data confirm the validity of the
energy-balance model [23] that foresees an approximate
t
−1/2
dependence.
These results are of great interest in designing meta-
morphic epitaxial structures where the strain induced by
buffer layers can be used as a tool to modify in a pre-
dictable way the electronic band structure of the upper
layers. An interesting example is the QD strain engineer-
ing [9,12] of InAs/InGaAs QD nanostructures grown on
GaAs substrates, that results in the shift of the RT light
emission wavelength towards the 1.55 µm spectral window
of telecom and datacom applications.
In addition, the knowledge of the mechanism that
determine the strain relaxation may allow the de-
sign of advanced heteroepitaxial structures, where
graded-composition MBs are incorporated to take specific
advantages over the constant-composition counterparts,
as regards the confinement of misfit dislocations far away
from the active region of the structures [14].
The work has been partially supported by the “SANDiE” Net-
work of Excellence of EU, contract no. NMP4-CT-2004-500101
and by the FIRB Project “Nanotecnologie e Nanodispositivi
per la Societ`a dell’Informazione”. V.B. acknowledges support
from Spanish Ministry of Education an Science (FIS2006-
00716).

222 The European Physical Journal B
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