Coercivity of a percolative magnetic system
Jose
´
Mejı
´
a-Lo
´
pez,
1,2
Ricardo Ramı
´
rez,
1
Miguel Kiwi,
1
Michael J. Pechan,
3
J. Zachary Hilt,
3
S. Kim,
4
Harry Suhl,
4
and Ivan K. Schuller
4
1
Facultad de Fı ´sica, Pontificia Universidad Cato ´lica, Casilla 306, Santiago, Chile 6904411
2
Facultad de Ciencias, Escuela Superior Polite ´cnica de Chimborazo, Panamericana Sur, Km 1, Riobamba, Ecuador
3
Department of Physics, Miami University, Oxford, Ohio 45056
4
Physics Department, University of California-San Diego, La Jolla, California 92093-0319
Received 9 June 2000; published 27 December 2000
A model that describes the behavior of the coercivity H
c
, of coalescing Ni films, as a percolation phenom-
enon is presented. We show that there is a nonmonotonic dependence, of the coercive field as a function of the
occupation probability p. This behavior is a function of cluster size and topology, and is independent of the
growth direction of the thin film. Based on these ideas we predict theoretically, and confirm experimentally, the
dependence of H
c
as a function of external magnetic field orientation. For a 111 oriented fcc film, if p is less
than the percolation threshold p
cr
, a sixfold symmetry is observed. However, if p p
cr
the percolation is
isotropic.
DOI: 10.1103/PhysRevB.63.060401 PACS numbers: 75.70.Cn, 75.70.Kw, 64.60.Ak
The concept of percolation was introduced in the realm of
science by Broadbent and Hammersley
1
in 1957. Since then,
it has been applied to many problems in completely different
fields, ranging from natural sciences to sociological
phenomena.
2
In physics, percolation has been useful in the
study of microscopic
3
as well as macroscopic phenomena.
4,5
In particular, granular systems inherently prone to percola-
tion, have experimentally shown magnetic anomalies when
the grains are magnetic. Panina et al.
6
found that the effec-
tive magnetic permeability of composite materials containing
small Fe particles of 1–2 m size tends to zero near the
percolation threshold. Xiao and Chien
7
observed that across
the whole volume fraction range, in granular Fe-(SiO
2
) sol-
ids, magnetic coercivity experiences dramatic variations due
to the change of grain size and percolation effects. In thin
films the relation between magnetism and percolation also
has been investigated: for example, the transition from su-
perparamagnetism to ferromagnetism,
8
while recently
Gor’kov and Kresin
9
studied the transition from paramag-
netic to conducting ferromagnetic phases of manganites by
percolation theory. Moreover, Choi et al.
10
observed that the
coercivity H
c
of Ni films, whose magnetic and structural
properties are found in the literature,
11
exhibits a nonmono-
tonic behavior, related to topological morphology changes,
which coincides with the onset of percolation. However, no
theoretical ideas have been advanced which directly connect
coercivity and percolation.
In order to study changes in H
c
as a function of morphol-
ogy in Ni, Choi et al.
10
performed a well controlled experi-
ment evaporating Ni films, by molecular beam epitaxy
MBE, on top of various thickness t
Cu
copper buffer layers.
The morphology of the Cu buffer layer changes with t
Cu
.
First isolated copper clusters form, but as t
Cu
increases pairs
of clusters coalesce. As more Cu is deposited a critical thick-
ness is reached and the Ni covered Cu clusters percolate. In
order to assure that all changes in the coercivity of Ni are due
only to changes in morphology, all other structural param-
eters of the epitaxial Ni were kept constant. These param-
eters include: the thickness 100 Å, the level of contamina-
tion, the growth parameters, the capping layer width 150 Å
of Al, the 111 crystalline orientation perpendicular to the
substrate, and the mosaic spread. All these structural and
chemical properties were checked using in situ Auger elec-
tron spectroscopy AES, reflection high energy electron dif-
fraction RHEED, low energy electron diffraction LEED
and ex situ x-ray diffraction XRD, scanning electron mi-
croscopy SEM, and atomic force microscopy AFM.
Two major variations of the coercivity H
c
become appar-
ent as the morphology reaches the percolation threshold; the
magnitude of H
c
changes in a nonmonotonic fashion and the
in-plane angular anisotropy changes as well. Both of them
display a distinct feature, which coincides with the crossing
of the percolation threshold. Our theoretical model explains
these two distinct features, that is: i the magnitude of H
c
as
a function of the copper substrate thickness; and ii the de-
pendence of H
c
on the direction of the applied field H. The
model shows that H
c
, as a function of site occupation prob-
ability p, has a nonmonotonic dependence due to the frac-
tional variation of the cluster sizes and their topology, as
they form during the film deposition process, and it is also
shows that this behavior is independent of the growth direc-
tion of the magnetic film. Moreover, also the symmetry of
H
c
, when p p
cr
, is in quantitative agreement with experi-
ment.
Our model focuses upon the morphological changes that
take place during the growth of a thin film, which in its early
stage is characterized by a homogeneous distribution of the
small islands. When additional material is deposited these
islands grow in size, due to the large mobility of the small
droplets, until coalescence sufficient for percolation is
achieved. Based on this the coercivity H
c
( p ), as a function
of occupation probability p, can be understood assuming that
the magnetization reversal mechanism depends on cluster
size and topology: i if the particles are smaller than a cer-
tain critical size z
cr
, of the order of a domain wall, H
c
is
large, due to the fact that for such particles wall nucleation is
energetically unfavorable. Consequently, the magnetization
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