Spin dependent non-tunneling transport in granular ferromagnet
Soumik Mukhopadhyay and I. Das
We have shown that, at low temperature, the effect of localization of charge carriers in granular metals cannot be ignored. The electron localization in granular ferromagnetic metals can give rise to non-linear transport in the moderate electric field regime, which is dominated by spin dependent inter-grain non-tunneling emission. In contrast to the general opinion that spin dependent tunneling gives rise to low field magnetoresistance in granular ferromagnetic metals we observe that spin dependent non-tunneling transport is primarily responsible for the sharp magnetic field dependence of resistance, at least in the low temperature and moderate electric field regime.
Mukhopadhyay_pr.tex
\documentclass[prl,twocolumn,showpacs]{revtex4}
%\documentclass[twocolumn,aps]{revtex4}
\usepackage{graphics}
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\begin{document}
\title{Spin dependent non-tunneling transport in granular ferromagnet}
\author{Soumik Mukhopadhyay and I. Das}
\affiliation{Saha Institute of Nuclear Physics, 1/AF, Bidhannangar, 
Kolkata 700064, India}
\begin{abstract}
We have shown that, at low temperature, the effect of localization
of charge carriers in granular metals cannot be ignored. 
The electron localization in granular ferromagnetic metals can give rise to 
non-linear transport in the moderate electric field regime, 
which is dominated by spin dependent inter-grain non-tunneling emission.
In contrast to the general opinion that
spin dependent tunneling gives rise to low field magnetoresistance
in granular ferromagnetic metals we observe that spin dependent non-tunneling
transport is primarily responsible for the sharp magnetic field dependence of 
resistance, at least in the low temperature and moderate electric field regime. 
\end{abstract}
\pacs{72.25.-b, 73.63.-b, 85.75.-d}
\maketitle
\indent
Recent emergence of spin based electronics or spintronics has initiated 
extensive studies on spin polarized transport in magnetic tunnel junctions, 
granular ferromagnetic films and nanocomposites. The low field 
magnetoresistance in magnetic tunnel junctions (MTJ), and granular 
ferromagnetic systems is attributed to spin dependent 
tunneling~\cite{mtj1,abeles,intergrain}. However, it has been shown
recently that the superposition of tunneling conduction and ballistic
channels can lead to inversion of magnetoresistance~\cite{soumik} in MTJs.
This letter explores the possibility of whether there
exists any transport mechanism other than spin dependent tunneling 
which might affect low field magnetoresistance in granular ferromagnets.
Although the study of granular ferromagnetic metals spans more than three
decades~\cite{abeles}, recently, there is a renewed interest in
this field with the granular systems based on ``half metallic'' 
materials like CrO$_{2}$ and La$_{0.67}$Sr$_{0.33}$MnO$_{3}$ 
showing enhanced low field magnetoresistance~\cite{intergrain} leading to the 
possibility of their application as magnetic field sensors. 
We will analyze the effect of electron localization
in granular ferromagnetic systems at sufficiently low temperature. 
The consequent dominance of spin dependent inter-grain non-tunneling
emission over inter-grain tunneling in the moderate
electric field regime will be discussed. 

\indent 
For the preparation of granular film with almost uniform grain size 
distribution and quasi-homogeneous distribution of conducting and 
non-conducting regions, we have utilized the well studied strain relaxation 
properties of manganite films. Manganites like La$_{0.67}$Sr$_{0.33}$MnO$_{3}$ 
(LSMO) can be grown epitaxially on single crystal insulating substrate 
LaAlO$_{3}$ (LAO). However, due to finite lattice mismatch 
($\{a_{LSMO}-a_{LAO}\}/a_{LAO} \sim +2\%$) with the 
substrate, thin films of La$_{0.67}$Sr$_{0.33}$MnO$_{3}$ undergoes a biaxial 
in plane compressive stress leading to higher out of plane lattice parameter 
compared to the substrate~\cite{strain1,strain2,strain3}. Relatively thick 
films (usually above $50$ nm) become strain relaxed forming granular 
nanostructures and the grain size depends on the thickness of the film.
In granular manganite systems, it is an established fact that 
the core of grain is ferro-magnetically ordered but the grain surface is
spin-disordered and insulating~\cite{balcells,barnali}. 

\indent
Thick films of LSMO, having thickness $\sim3000\AA$,
were grown on LaAlO$_{3}$($100$), using pulsed laser deposition. 
The surface morphology of the film was inspected, using
atomic force microscopy which shows orderly arrangement of connected
grains with average size $80$ nm (Fig:~\ref{fig:afm}).
In order to prove that this is not a mere surface corrugation,
the x-ray diffraction $\theta-2\theta$ scan around $(002)$
peaks of LSMO/LAO are shown in Fig:~\ref{fig:xrd}A and the
magnetic measurements in Fig:~\ref{fig:mht}. The Bragg peak corresponding to 
that portion of the film which matches with the in-plane lattice parameters
of the substrate leading to higher c-axis parameter is indicated in the
figure as $(1)$ which gradually relaxes to the bulk lattice parameter value
shown by the higher angle peak among the twin peaks separated by a plateau 
indicated as $(2)$. The intensity of Bragg peak which corresponds to strain 
relaxed part of the film is much higher compared to the strained part 
indicating that the granularity extends deep inside the film and
hence bound to influence the transport and magnetic properties of the film.
The magnetic characterization was done using a Quantum Design SQUID
magnetometer which shows the ferromagnetic transition temperature 
$T_{c}$ at around $360$ K.
The large irreversibility in the temperature dependence of zero field cooled
(ZFC) and field cooled (FC) magnetization $(M)$ together with the hysteric 
$M-H$ loop, where the magnetization does not saturate even at $2$ kOe, 
are typical of granular ferromagnets (Fig:~\ref{fig:mht}A,B). 
The virgin $M-H$ curve shows (Fig:~\ref{fig:mht}) an `S'-like shape with 
the point of inflexion around $100$ Oe indicating non-ferromagnetic 
inter-grain coupling. The coercivity of the film has roughly the same value.
\begin{figure}
\resizebox{7cm}{3cm}
{\includegraphics{Mukhopadhyay_Fig1.ps}}
\caption{AFM picture of the strain relaxed film over $2\times2 \mu m^2$ area
showing the granular surface morphology along with a schematic cross-sectional
blown-up view with strain-relaxed granular part and a thin strained bottom
part, the grains (the ferromagnetically ordered core part) being assimilated
to localized sites at low temperature, leading to the dominance of inter-grain
non-tunneling transport.}
\label{fig:afm}
\end{figure}
\begin{figure}
\resizebox{4.5cm}{3.8cm}
{\includegraphics{Mukhopadhyay_Fig2.EPS}}
\resizebox{3.4cm}{3.8cm}
{\includegraphics{Mukhopadhyay_Fig3.ps}}
\caption{{\bf A:}
X-ray diffraction pattern of the LSMO film on LAO substrate. The c-axis 
parameter of the film relaxes gradually to the bulk value of LSMO. The broad 
hump at the lower angle $(1)$ indicates the strained part whereas the two 
broad peaks $(2)$ and the plateau in between corresponds to the gradual 
relaxation of the strain. {\bf B:} The conduction is due to the combined 
effect of tunneling through the barrier and thermal activation over the 
barrier. The effective barrier height depends on the relative spin 
orientation of adjacent sites. The conduction is in the moderate electric 
field regime where $R_{j}-{R_{1}}^{M}<<R_{j}-R_{i}$}
\label{fig:xrd}
\end{figure}

\indent
The transport properties were studied using four probe technique with the 
magnetic field being in-plane to the electric field. The granular film 
exhibits a sharp rise in resistance (Fig:~\ref{fig:res}B) with lowering of 
temperature below around $6$ K in the
absence of external magnetic field. At higher bias current, the 
rise in resistance reduces drastically suggesting non-ohmic transport at low 
temperature. A significant reduction of percentage rise in resistance 
on application of magnetic field is observed. Hence Ziese's model of 
quantum interference 
effect~\cite{ziese} is not applicable here. Below $6$ K, the strain 
relaxed film exhibits low field magnetoresistance which is extremely 
sensitive to the applied bias (Fig:~\ref{fig:res}C). 

\indent
The electric field dependence of conductivity at $2$ K, plotted in 
Fig:~\ref{fig:res}B, shows that the logarithm of conductivity is linearly 
proportional to square root of the electric field $(E)$. At $5$ K, 
very close to the point of upturn in resistance with 
lowering of temperature, the transport becomes nearly ohmic 
(Fig:~\ref{fig:res}D). Such electric field 
dependence is strikingly similar to the 
charge transport mechanism called Poole-Frenkel emission (PFE)~\cite{frenkel}. 
The PFE, usually observed in disordered semiconducting systems,
is a non-tunneling transport phenomenon where the
charge carriers are thermally activated over the potential barrier
from localized charge traps in the presence of electric field.
The reduction in barrier height $\phi_{PF}$ is 
proportional to the square root of applied electric field
and the conductivity is given by 
$G=G_{0}\exp\left(\beta_{PF}E^{1/2}/KT\right)$, where $\beta_{PF}$
is a constant.

\indent
However, in this case what is fundamentally different from traditionally
observed PFE is the tunability of the phenomenon
by application of even a small amount of magnetic field. 
It is evident from Fig:~\ref{fig:film}A that the threshold electric field
for the onset of charge emission $E_{th}$ (indicated by the vertical 
arrow) and the slope of the curve are higher at $100$ Oe magnetic field
compared to that at $1$ kOe. The origin of PF type emission in granular 
metal can be explained following Mostefa's model~\cite{mostefa}.
This model considers a granular metallic system with a narrow grain
size distribution with almost equal Fermi levels for all grains and
as a result the activation energy is entirely due to the potential
barrier between two adjacent grains. In each grain, electrons can be 
considered as being localized at sufficiently low temperature. 
At low temperature, the leakage of electron wave function
outside the grain is minimal, which is particularly true for insulating grain
surfaces, as in this case~\cite{balcells, barnali}. This justifies the 
assumption of grains as localized sites at sufficiently low temperature. 
In the weak field regime, the transport is purely thermally activated.
The electric field will affect two important parameters of the
system if it is above the weak field regime.

\indent
Firstly, \emph{the shape of the potential barriers and hence the
decay of the wave function associated with the electronic states}. 
The tunneling probability $P(R_{ij})$ between two grains assumed to be located
at sites $\vec{R_{i}}$ and $\vec{R_{j}}$ separated by a distance
$R_{ij}=|\vec{R_{j}}-\vec{R_{i}}|$ can be written, using WKB
approximation,
\[P(R_{ij})\simeq \exp\left[-\frac{2}{\hbar}\int_{\vec{R_{i}}}^{\vec{R_{j}}}\{2m[V(x)-\varepsilon_{i}]\}^{1/2}\,dx\right]\] 
Here $V(x)$ is barrier height, $x$ is the position of the electron
on the path $\vec{R_{ij}}$ and $\varepsilon_{i}$ is the electron energy
in the grain located at $i$th site. It is assumed that the 
potential barrier is in the moderate field regime where the 
electron is emitted as a free particle into the insulator over a 
path much shorter than the total distance between the grains.
The effective barrier height is $\phi_{i}(x)=V(x)-\varepsilon_{i}$.
Taking these factors into consideration and assuming directed flow of
electrons we get,
$P(R_{ij})\simeq \exp\left\{-\frac{A}{E}\right\}$, where $A$ is a constant.
%where $\alpha_{0}=\left(\frac{2m\phi(0)}{\hbar^{2}}\right)^{1/2}$ 
\begin{figure}
\resizebox{6.5cm}{6.5cm}
{\includegraphics{Mukhopadhyay_Fig4.EPS}}
\caption{{\bf A:} The zero field cooled (ZFC) (open symbols) and
field cooled (FC) (filled symbols) M vs. T curves at magnetic fields
$H=20$ Oe and $H=100$ Oe showing irreversibility between the ZFC and FC curves
which is an evidence for the granularity of the film.
{\bf B:} Virgin M vs. H curve for the strain relaxed film at $2$ K, exhibiting
inflexion at $100$ Oe shown by the slope of the curve plotted as continuous
line. {\bf C:} M vs. H hysteresis loops at $2$ K, showing coercivity of
about $100$ Oe.}
\label{fig:mht}
\end{figure}
\begin{figure}
\resizebox{4.8cm}{4cm}
{\includegraphics{Mukhopadhyay_Fig5.EPS}}
\resizebox{3.7cm}{4cm}
{\includegraphics{Mukhopadhyay_Fig6.EPS}}
\caption{{\bf A}: Rise in resistance with lowering of
temperature below $6$ K at different
bias currents, suggesting non-ohmic transport below $6$ K.
Inset: Magnetic field dependence of MR at $3$ K at different bias currents.
{\bf B}: Plot of logarithm of dc conductivity vs. $E^{1/2}$ in absence of
magnetic field at $2$ K and $5$ K for the same sample. The linear region in
the curve (for $2$ K) indicates thermally assisted non-tunneling emission
due to localization of charge carriers at low temperature.
The non-tunneling emission is almost suppressed at $5$ K.}
\label{fig:res}
\end{figure}

\indent
Secondly, \emph{the activation energy $E_{a}$ required
for one electron to hop from a neutral grain to another neutral grain}.
In the non-negligible field regime, the activation energy required to transfer
an electron from site $i$ to site $j$ is, ${E_{a}}^{ij}=\varepsilon_{ij}-e\vec{E}.\vec{R_{ij}}$, where, $\varepsilon_{ij}=\varepsilon_{i}-\varepsilon_{j}$.
Hence the activation probability,
\[P({E_{a}}^{ij})\simeq\exp\left\{{\frac{-\varepsilon_{ij}}{KT}}+\frac{eER_{ij}\cos\theta}{KT}\right\}\]
The electrical conductance $G_{ij}$ between sites $i$ and $j$ 
can be calculated from the \emph{total transport probability $P_{ij}$, which 
is given by the product of the tunneling probability 
across the barrier and the activation probability over the barrier}.
Assuming that the temperature is low enough to help the hopping transport, 
the conductance can be calculated employing percolative treatment.
\begin{figure}
\resizebox{7.5cm}{9.5cm}
{\includegraphics{Mukhopadhyay_Fig7.EPS}}
\caption{Sensitivity of the PF type emission to magnetic field.
Plot of logarithm of dc conductivity vs. $E^{1/2}$
in the antiparallel and parallel magnetic configuration with the
corresponding magnetic field dependence of magnetoresistance at
the same temperature in the inset for the strain relaxed
film (A) and (LSMO)$_{1-x}$(ALO)$_{x}$ composite with $x=0.5$ (B).
The threshold electric field $E_{th}$ is indicated by the vertical arrow.
The additional inset (II) in (B) describes the temperature dependence of
resistance of the same sample with a rapid increase in slope below $10$ K
shown by the continuous line.}
\label{fig:film}
\end{figure}

\indent
However, Mostefa's model alone cannot explain the sharp magnetic field
dependence discussed earlier. Therefore, we incorporate an additional 
inter-grain magnetic exchange energy ($E_{m}$) term in the model. 
It arises due to the relative spin orientation of the two adjacent localized 
sites when the spin is conserved in hopping from one site to another. 
The value of $E_{m}$ depends on the overlap of 
electronic wave function across the grains. If the spin orientation
is anti-parallel an additional energy $+E_{m}$ is necessary for charge carrier
emission while less energy $-E_{m}$ is required if the orientation is parallel.
Obviously, the value of $E_{m}$ can easily be manipulated by applying
even a tiny magnetic field $H$.
Hence the prescription is to simply replace the effective 
potential barrier $\phi_{i}(x)$ by 
$\phi_{i}^{m}(x)=\phi_{i}(x)\pm E_{m}^{ij}(H)$ and the activation
energy ${E_{a}}^{ij}$ by 
${E_{a}}^{ij,m}=\varepsilon_{ij}-e\vec{E}.\vec{R_{ij}}\pm E_{m}^{ij}(H)$.
Now we employ the percolation criterion $G_{ij}\geq G_{C}$, where $G_C$ is the
critical conductance for percolation. This sets a limit on the
energy and the length scales ($\varepsilon_{ij}\leq \varepsilon_{M}$ and
$R_{ij}\leq R_{M}$) involved and one arrives at,
$\frac{\varepsilon_{ij}}{\varepsilon_{M}}-\frac{R_{ij}}{R_{M}}\leq 1$
~\cite{mostefa}
where, $\varepsilon_{M}=\left(\frac{KT}{1\pm\xi_{m}}\right)Y^{\prime}$,
$S_{M}=\left(\frac{KT}{eE}\right)Y^{\prime}$ and 
$Y^{\prime}=\ln\frac{G_{0}^\prime}{G_{C}}$
Here the magnetic exchange energy has been expressed in the unit 
of $\varepsilon_{ij}$ as $E_m=\xi_{m}\varepsilon_{ij}$, where $\xi_{m}$ is a
dimensionless quantity. Then following Mostefa's 
derivation one obtains,
\[\log\left(\frac{G_{C}}{G_{0}^{\prime}}\right)=B(H) \frac{E^{1/2}}{KT}\]
where $B_{\uparrow\downarrow,\uparrow\uparrow} \propto \sqrt{1\pm\xi_{m}}$ 
for a given system, with $\uparrow\downarrow$ and
$\uparrow\uparrow$ indicating the antiparallel and parallel magnetic
configuration between nearest neighboring grains, respectively.
Evidently, the value of $B$ is higher in the antiparallel configuration
compared to the parallel one.

\indent
It is interesting to compare the magnetoresistance 
(Inset, Fig:~\ref{fig:film}A) with the conductivity curves at different 
magnetic fields in Fig:~\ref{fig:film}A at low temperature. The  
magnetoresistance is positive at $100$ Oe, which is equal to the coercivity 
and the inflexion point in the virgin $M-H$ curve 
(Fig:~\ref{fig:mht}). This suggests that at this point, the spin orientation 
between the adjacent grains becomes anti-parallel, 
resulting in increase of exchange energy. The increase in the slope of the 
logarithmic conductivity vs. $E^{1/2}$ curve at $H=100$ Oe in 
Fig:~\ref{fig:film} compared to that at $H=1$ kOe is 
consistent with our model. The value of exchange energy can be estimated
from the slope of the curves at $100$ Oe (antiparallel configuration) 
and $1$ kOe (parallel configuration) using the relation 
$\xi_{m}=({{B_{\uparrow\downarrow}}^2-{B_{\uparrow\uparrow}}^2})/({{B_{\uparrow\downarrow}}^2+{B_{\uparrow\uparrow}}^2})$ which gives $\xi_{m}=0.79$
and therefore $E_{m}=0.79\varepsilon_{ij}$. 
The validity of the non-tunneling emission being limited to the moderate
electric field regime (as defined in Fig:~\ref{fig:xrd}B) is determined partly
by the potential barrier height. Therefore,
the shift in the threshold electric field ($E_{th}$) for the onset of PF type
emission should be magnetic field dependent since magnetic exchange energy
contributes substantially to the total barrier height. 
In the antiparallel configuration, the value of $E_{th}$ should be higher
in comparison with the parallel configuration, which is consistent with
our observation. A rough estimation of the average total activation energy 
(including the magnetic contribution) is possible from the fitting of 
temperature dependence of resistance (at low temperature and bias) in 
absence of magnetic field by an activation term with a metallic contribution in
parallel. The metallic channel includes whatever the effect of the 
thin continuous strained part
might have. Consequently, the magnetic exchange energy turns out
to be $E_{m}\sim 0.24$ K. Similar value ($\sim 0.23$ K) can be extracted 
from the difference in the total activation energy values in the absence and 
in presence of magnetic field, respectively. 

\indent
In order to find out whether the same phenomenon is operative in
a different micro-structure, we have studied magnetotransport properties 
in a nanocrystalline LSMO/ 
$Al_{2}O_{3}$(ALO) granular composite with different weight concentrations 
$x$ of $Al_{2}O_{3}$ (where $x = 0 - 0.5$). The average grain size
of the sol-gel prepared LSMO was measured, using Transmission Electron 
Microscope (TEM) and Scanning electron 
microscope (SEM), which turns out to be about $50$ nm. 
Fig:~\ref{fig:film}B shows similar spin dependent non-tunneling
emission in LSMO/ALO composite with $x=0.5$ at $3$ K.
The temperature dependence of resistance shows a low temperature upturn 
with a rapid increase in the slope below $10$ K (Inset II, 
Fig:~\ref{fig:film}B). 
All other samples with different values of $x$, exhibit similar behavior. 
This suggests that the effect
of localization at low temperature and the consequent dominance
of non-tunneling effect might be generic to a large variety of granular 
ferromagnetic metals. Employing the same method for estimation of magnetic 
exchange energy described in the previous paragraph we obtain $\xi_{m}=0.04$ 
only. The temperature dependence of resistance can be fitted
using the parallel channel conduction model of ref.~\cite{andres}, which gives
a rough estimate of activation energy to be $\sim 0.8$ K.
The reduction in the slope of the curve as well as $E_{th}$
with applied magnetic field is not as pronounced
as in the case of strain-relaxed film indicating that the value of exchange
energy should be comparatively smaller. The small value of exchange energy
in this case can be understood considering the larger average inter-particle 
separation in the composite. Thus the sensitivity of the slope of the 
logarithmic conductance vs. $E^{1/2}$ curves in the non-tunneling regime 
as well as that of $E_{th}$ to the magnetic field depends on the relative 
contribution of the magnetic exchange to the total activation energy. 
This factor is determined by the microstructure of the granular system along 
with the spin polarization of the ferromagnetic grains. 

\indent
To conclude, we have observed spin dependent non-tunneling transport 
in LSMO granular film and LSMO/ALO metal-insulator nanocomposites, a 
phenomenon which appears to be generic to ferromagnetic granular metallic
systems, particularly at low temperature. In contrast to the usual 
Poole-Frenkel emission, the electric field dependence
of conductivity is sensitive to external magnetic field. 
The observed phenomenon has been analyzed considering the
dominance of thermally assisted inter-grain non-tunneling 
transport (with the grains acting as localized sites) in the moderate 
electric field regime and the existence of an additional inter-grain 
magnetic exchange barrier. 

\indent
The authors acknowledge Prof. A. K. Raychaudhuri and Dr. Barnali Ghosh 
of SNBNCBS, Kolkata for providing some experimental facilities.
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\end{thebibliography}


\end{document}



cond-mat0506739.extracted_bib
\begin{thebibliography}{99}

\bibitem{mtj1}
J. S. Moodera, Lisa R. Kinder, Terrilyn M. Wong, R. Meservey,
Phys. Rev. Lett., {\bf 74}, 3273 (1995)

\bibitem{abeles} Helman J. S. and Abeles B.,
%Tunneling of spin polarized
%electrons and Magnetoresistance in granular Ni films
Phys. Rev. Lett., {\bf 37}, 1429 (1976)

\bibitem{intergrain} H. Y. Hwang, S-W. Cheong, N. P. Ong, and B. Batlogg
Phys. Rev. Lett. {\bf 77}, 2041 (1996)

\bibitem{soumik} Soumik Mukhopadhyay, I. Das, Phys. Rev. Lett, {\bf 96},
026601 (2006)

\bibitem{strain1} S. I. Khartsev, P. Johnson, and A. M. Grishin, 
J. Appl. Phys. {\bf 87}, 2394 (2000)

\bibitem{strain2} Mandar Paranjape, A. K. Raychaudhuri, N. D. Mathur, and
M. G. Blamire, Phys. Rev. B {\bf 67}, 214415 (2003)

%\bibitem{strain3} J. Z. Sun, D. W. Abraham, R. A. Rao, C. B. Eom, 
%Appl. Phys. Lett. {\bf 74}, 3017 (1999) 

\bibitem{strain3} M. Ziese, H. C. Semelhack, K. H. Han, S. P. Sena, and 
H. J. Blythe, J. Appl. Phys. {\bf 91}, 9930 (2002)

\bibitem{balcells} L. Balcells, J. Fontcuberta, B. Martínez, and X. Obradors
Phys. Rev. B {\bf 58}, R14697 (1998)

\bibitem{barnali}
Barnali Ghosh, Sohini Kar, Loveleen K. Brar, A. K. Raychaudhuri, J. Appl. Phys.
{\bf 98}, 094302 (2005)

\bibitem{ziese} M. Ziese, Phys. Rev. B, {\bf 68}, 132411 (2003)

\bibitem{frenkel} Frenkel J., 
%On pre-breakdown phenomena in insulators
%and electronic semiconductors 
Phys. Rev., {\bf 54}, 647 (1938)

%\bibitem{amor} Ben-Chorin M., Moller F., and Koch F., 
%Nonlinear electrical
%transport in porous silicon 
%Phys. Rev. B, {\bf 49}, R2981 (1994)

\bibitem{mostefa} M. Mostefa, D. Bourbie and G. Olivier, 
Physica B, {\bf 160}, 186 (1989)

\bibitem{andres} A. de Andres, M. Garcia-Hernandez, 
and J. L. Martinez, Phys. Rev. B {\textbf 60}, 7328 (1999)

\end{thebibliography}