Spatially correlated disorder in self-organized precursor magnetic nanostructures
Marcel Porta,
1,2
Teresa Castán,
1,2
Pol Lloveras,
1,2
Antoni Planes,
1,2
and Avadh Saxena
2,3
1
Departament d’Estructura i Constituents de la Matèria, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Catalonia, Spain
2
Institut de Nanociència i Nanotecnologia de la Universitat de Barcelona, 08028 Barcelona, Catalonia, Spain
3
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Received 4 December 2006; revised manuscript received 22 June 2007; published 20 August 2007
We study the scaling behavior of the characteristic length of precursor magnetic nanostructures above the
Curie temperature with the correlation length of quenched-in disorder. We found that the modulation length of
the nanostructures follows the scaling law
D
¯
, where D
¯
is the average size of the magnetized regions in
the material. The scaling behavior of the average size of these regions, D
¯
, with the correlation length of the
disorder, , depends on the properties of the disorder. For Gaussian disorder, we find that D
¯
scales with the
disorder correlation length as D
¯
a/2
, where a is the exponent of the leading term of the pair correlation
function of the disorder in the limit r →0, r 1- 1/ ar /
a
. These results are quite general and
applicable to other systems, e.g., ferroelectric precursors, independent of the nature of the long-range dipolar
forces.
DOI: 10.1103/PhysRevB.76.054432 PACS numbers: 75.30.Kz, 75.40.Mg, 61.43.-j
I. INTRODUCTION
Self-organized nanostructures are of considerable interest
because of their potential importance in engineering func-
tional materials.
1
A peculiar situation is that of modulated
nanoscale textures which originate as precursors to phase
transitions in multiferroic materials. Such textures have been
revealed by high-resolution imaging techniques well above
the phase transition.
2
This type of precursor
3
was first ob-
served in the case of ferroelastic structural transitions with
modulations in strain. The corresponding pattern exhibits an-
isotropic cross-hatched correlations that in real-space strain-
contrast images resemble the tweed textile.
4
More recently,
5
precursor structures with modulations in the magnetization
giving rise to stripelike patterns have been observed in the
Co
0.38
Ni
0.33
Al
0.29
magnetic alloy above the Curie point. This
led to the suggestion
6
that the tweed concept is not just struc-
tural but applicable to a much broader class of materials with
modulations in other physical variables strain, magnetiza-
tion, polarization. In addition, it was shown
6,7
that the origin
of tweed lies in very general requirements, likely to be sat-
isfied in quite different systems undergoing phase transitions.
In short, the tweedlike modulations occurring above a phase
transition are the natural global response of anisotropic long-
range dipolar forces elastic, magnetic, electric to local per-
turbations arising from quenched-in disorder coupling to the
order parameter strain, magnetization, polarization.
The natural source for disorder in magnetic alloys is
statistical compositional fluctuations that are quenched in
during the alloying process. Thus, the alloy composition is
an inhomogeneous quantity that is spatially correlated in or-
der to avoid drastic variations from one point to another. In
general one does not expect such correlations to be long
ranged. Nevertheless, the spatial inhomogeneities might in-
duce long-range interactions e.g., elastic that mediate the
correlations.
Here we shall give special attention to precursor modula-
tions in magnetic alloys. The modulations in the magnetiza-
tion above the Curie point
5
occur at a scale 100 nm and as
mentioned above the pattern is stripelike with stripes being
either vertical or horizontal due to magnetic anisotropy. In a
previous work,
7
we demonstrated that the intermediate
tweedlike regime does exist and corresponds to a paramag-
netic textured phase with modulations occurring at a length
scale smaller than the magnetic domains observed below the
Curie point. In the dipolar or ferromagnetic phase, the size
of the stripes i.e., the modulation follows the standard
8
scaling law given by
L, where L denotes the crystal
size. In the intermediate precursor phase the modulations of
the magnetization are localized in certain regions of the ma-
terial of size D L. Then the modulation length inside
these regions should depend on D, which in turn depends on
the correlation length of the disorder, . In the present paper
we use Gaussian disorder and study the scaling behavior of
both D and for different functional forms of the pair cor-
relation function of the disorder r.
We shall first focus on the fundamental aspects of the
problem and later discuss potential practical implications.
The paper is organized as follows. In the next section we
present a model for a two-dimensional paramagnetic to fer-
romagnetic transition with magnetic tweed present above the
Curie temperature in the presence of quenched-in disorder
and long-range magnetic dipolar interaction. Section III con-
tains the numerical results obtained for the scaling behavior
with of both the size of the magnetized regions, D, and the
stripe modulation length . This is done for different
stretched exponential r functions. In Sec. IV we focus
on the statistical properties of the disorder itself and provide
a theoretical analysis to link such results to the problem of
interest here. In the next section Sec. V we also discuss the
scaling behavior of the stripe modulation length in the dipo-
lar and paramagnetic textured phases. Finally, in Sec. VI, we
summarize our main findings and discuss their relevance in
the context of engineering nanoscale functional materials.
Some technical details pertinent to the pair correlation func-
tion r are relegated to Appendixes A and B.
PHYSICAL REVIEW B 76, 054432 2007
1098-0121/2007/765/0544327 ©2007 The American Physical Society 054432-1