Ising-like dynamics and frozen states in systems of ultrafine magnetic
  particles
Ising-like dynamics and frozen states in systems of ultrafine magnetic particles
Stefanie Russ1 and Armin Bunde1
Institut für Theoretische Physik III, Justus-Liebig-Universität Giessen, D-35392 Giessen, Germany
(Dated: February 28, 2022)
We use Monte-Carlo simulations to study aging phenomena and the occurence of spinglass phases
in systems of single-domain ferromagnetic nanoparticles under the combined influence of dipolar
interaction and anisotropy energy, for different combinations of positional and orientational disorder.
We find that the magnetic moments oriente themselves preferably parallel to their anisotropy axes
and changes of the total magnetization are solely achieved by 180 degree flips of the magnetic
moments, as in Ising systems. Since the dipolar interaction favorizes the formation of antiparallel
chain-like structures, antiparallel chain-like patterns are frozen in at low temperatures, leading to
aging phenomena characteristic for spin-glasses. Contrary to the intuition, these aging effects are
more pronounced in ordered than in disordered structures.
PACS numbers: 75.75.+a, 75.40.Mg, 75.50.Lk, 75.50.Tt
INTRODUCTION
In the last decade, systems of ultrafine magnetic
nanoparticles have received considerable interest, due
both to their important technological applications
(mainly in magnetic storage and recordings) and their
rich and often unusual experimental behavior, which is
related to their role as a complex mesoscopic system
[1, 2]. It has been discussed controversially in the past,
under which circumstances these systems are able to
show spin-glass phases. While experiments on disor-
dered magnetic materials present indications of a spin-
glass phase [2, 3, 4] or of a glassy-like random anisotropy
system [5], the situation is less clear on the theoretical
side. Simulations of the zero-field cooling (ZFC) and
field-cooling susceptibility showed no indication of a spin-
glass phase [6, 7]. In contrast, simulations on aging [8]
(on a simplified system, where the dipolar interaction
was only considered up to a cut-off radius) and magnetic
relaxation [9, 10] favorize the spin-glass hypothesis, but
the structure of the frozen history-dependent states as
well as the actual mechanism leading to them has not
yet been clarified.
In this letter, in order to clarify these questions, we
use Monte Carlo simulations [11] to study aging phe-
nomena on a large variety of systems of ultrafine mag-
netic nanoparticles (see Fig. 1). Our simulations do not
only point to the existence of frozen history-dependent
states at low temperatures that are characteristic for spin
glasses, but also yield an insight into the structure of the
frozen states and the underlying dynamics. We find that
under the combined influence of dipolar and anisotropy
energy, the magnetic moments have a tendency to align
in an Ising-like manner either parallel or antiparallel to
their anisotropy axes and change their directions by 180
degree flips as in Ising systems. This way, chain-like
structures are formed where all magnetic moments point
into the same direction and neighboring chains have the
tendency to oriente themselves in an antiparallel way.
(a) (b)
(c) (d)
FIG. 1: Two-dimensional sketches of the geometries consid-
ered in this paper: (a) cubic arrangement of the particles and
all anisotropy axes aligned into the z-direction, (b) liquid-like
arrangement and all axes arranged, (c) cubic arrangement
and all axes randomly oriented and (d) liquid-like arrange-
ment and all axes randomly oriented. In the simulations, the
systems were three-dimensional (64 particles per cube).
These topological chains that freeze in at low tempera-
tures, form simple straight lines, when the particles are
arranged on the sites of a cubic lattice [10] and form com-
plex winded curves, when the arrangement of the parti-
cles is liquid-like. As a consequence, if a small external
magnetic field is applied, the magnetic moments can fol-
low the field more easily in a disordered system than in
the ordered configuration. This leads, contrary to the in-
tuition, to more pronounced aging effects (characteristic
for spin glasses) in ordered than in disordered structures.
http://arxiv.org/abs/0704.0580v1
MODEL SYSTEM AND NUMERICAL
SIMULATIONS
For the numerical calculations, we focus on the same
model as in earlier papers [6, 9], which (i) assumes a
coherent magnetization rotation within the anisotropic
particles, and (ii) takes into account the magnetic dipo-
lar interaction between them. Every particle i of volume
Vi is considered to be a single magnetic domain ~µi with
all its atomic magnetic moments rotating coherently and
the Vi are taken from a Gaussian distribution of width
σV = 0.4 and 〈V 〉 = 1 (see also [6, 9]). This results in
a constant absolute value |µi| = MsVi of the total mag-
netic moment of each particle, whereMs is the saturation
magnetization. The energy of each particle consists of
three contributions: anisotropy energy, dipolar interac-
tion and magnetic energy of an external field. We assume
a temperature independent uniaxial anisotropy energy
A = −KVi((~µi~ni)/|~µi|)
2, where K is the anisotropy
constant and the unit vector ~ni denotes the easy direc-
tions. Eventually, the magnetic moments are coupled to
an external field H leading to the additional field energy
H = −~µi
~H . Finally, the energy of the magnetic dipo-
lar interaction between two particles i and j separated by
~rij is given by E
(i,j)
D = (~µi~µj)/r
ij − 3(~µi~rij)(~µj~rij)/r
Adding up the three energy contributions and summing
over all N particles we obtain the total energy
j 6=i
(i,j)
. (1)
In the Monte Carlo simulations we concentrate on sam-
ples of N = L3 particles placed inside a cube of side
length L = 4 and average over 1000 configurations. Dur-
ing the simulations, both, the positions of the particles
and their easy axes are kept fixed. The unitless concen-
tration c is defined as the ratio between the total vol-
Vi occupied by the particles and the volume
Vs of the sample. Here, we focus on the concentra-
tion c/c0 ≈ 0.3, where c0 = 2K/M
s is a dimensionless
material-dependent constant, c0 ∼ 1.4 for iron nitride
and c0 ∼ 2.1 for maghemite nanoparticles [9]. We also
tested systems with higher concentrations c/c0 ≈ 0.4 and
(the extremely high concentration) c/c0 ≈ 0.6 and found
that the results remain qualitatively unchanged. The
temperature is measured in units of the reduced temper-
ature T̃ ≡ 1/(2βKV ), where 2KV is the height of the
anisotropy barrier and β = 1/(kBT ). Similarly, the mag-
netic field is measured in units of the anisotropy field
Ha = 2K/Ms. The relaxation of the individual magnetic
moments is simulated by the standard Metropolis algo-
rithm [12]. In contrast to [8], where dipole interactions
between the particles were only considered up to a cut-off
radius, we calculate the interaction energies by the Ewald
sum method with periodic boundary conditions in x, y
and z-direction [6, 13] and thus are able to account fully
(a) (b)
(c) (d)
FIG. 2: (Colors online) The magnetization m(τ ) after wait-
ing times tw = 0 (filled symbols) and tw = 10000 Monte
Carlo steps (open symbols) is plotted versus τ (number of
Monte Carlo steps with applied external field) for (a) cubic
lattice and aligned axes, (b) liquid-like system and aligned
axes, (c) cubic system and random axes and (d) liquid-
like systems and random axes for the reduced temperatures
T̃ = kBT/(2KV ) = 5 (black symbols, circles), T̃ = 1/10 (red
symbols, squares) and T̃ = 1/40 (blue symbols, diamonds).
for the long-range character of the dipole forces. The
magnetic moment ~µi is characterized by the spherical
angles θi and ϕi relative to a coordinate frame, where
the z-axis is parallel to the external field [9, 14, 15]. To
study the magnetic relaxation we determine as a function
of time t (number of Monte Carlo steps) for each particle
i the angle θi between the magnetic moment ~µi and the
z-axis, from which we obtain the relevant quantities, as
e.g. the normalized magnetization,
m(t) =
cos θi(t). (2)
To obtain the orientation of ~µi relative to ~ni, we intro-
duce the ”orientational order parameter” Oµ ≡ 〈|~µi~ni|〉,
i.e. the average of the absolute values of the scalar prod-
uct ~µi~ni over all N particles and all configurations. Oµ
does not distinguish between the parallel and the an-
tiparallel alignment. It is equal to zero when all ~µi are
perpendicular to their axes ~ni and equal to 1 if they are
all parallel or antiparallel to them.
To study aging phenomena, we determine the magne-
tization in a ZFC simulation. First, starting in a ran-
dom configuration of the magnetic moments, the system
is cooled down in the absence of an external field, from
T = ∞ to a reduced temperature T̃ with a constant cool-
ing rate of ∆β/∆t = 0.1, corresponding to 400 Monte
Carlo steps for T̃ = 1/40 and 10 steps for T̃ = 1. Sec-
ond, the cooling process is stopped at T̃ and the system is
allowed to relax for a certain waiting time tw. Finally, in
the third step, a small external field h = 0.1Ha is applied
in z-direction. The magnetization m(τ) is determined as
a function of τ ≡ t− tw (number of Monte Carlo steps af-
ter switching on the field). Aging effects are represented
by differences between the m(τ)-curves for different tw
and occur, when many different relaxation rates exist in
the system, so that after a given waiting time tw, the sys-
tem has only partly relaxed towards equilibrium. Experi-
mentally, aging effects have already been found in several
spin-glasses, as e.g. in Permalloy/alumina granular films
[16], rare-earth manganates [17], CuMn spin-glasses [18],
multilayer systems [19] and in Fe3N nanoparticle sys-
tems [20].
NUMERICAL RESULTS
Figure 2 shows m(τ) for the systems of Fig. 1 without
waiting time, tw = 0 (filled symbols), and for tw = 10
(open symbols). The different colors (and symbols) stand
for three different temperatures T̃ = 1/5, 1/10 and 1/40.
Clearly, all curves show aging effects, similar to the ex-
perimental results of Refs. [16, 17, 18]. Systems with no
or only small tw follow the external field faster than the
systems with longer waiting times, indicating that the
longer relaxation leads to more stable chains. The ag-
ing effects are most pronounced for those systems where
all anisotropy axes are oriented into the direction of the
external field (Fig. 2(a,b)) and less pronounced but still
visible for the systems with disordered anisotropy axes
(Fig. 2(c,d)). In these systems with orientational disor-
der, the m(τ) curves coincide for small τ and show aging
effects only after a certain crossover time (close to 102
Monte Carlo steps). This indicates that in these systems
a certain fraction of dipoles does not belong to quasi-
stable chain-like structures and can follow the external
field nearly instantaneously, independentely of the wait-
ing time and thus dominate the short-time behavior. The
aging effects decrease with increasing T̃ , when the order
is destroyed by the thermal fluctuations.
In order to understand the dynamical behavior in a
more microscopic way, we compare m(τ) with the time-
dependence of the corresponding orientational order pa-
rameters Oµ(τ). Figure 3 shows Oµ in the 3rd step of
the aging process for tw = 0 and tw = 10000 (filled and
open symbols, respectively) and for the same geometries
as before (see Fig. 1). The figure shows that quite con-
trary to the expectation, apart from a slight minimum
at intermediate τ , Oµ is constant in time for the systems
of tw = 10000. Without waiting time, the curves start
at much smaller values of Oµ, but increase rapidly un-
til they reach at a crossover time τc of about 10
3 Monte
Carlo steps the common plateau value. In the plateau
regime, the dipolar moments ~µi are either oriented par-
(a) (b)
(c) (d)
FIG. 3: (Colors online) The order parameter Oµ after a wait-
ing time tw = 0 (filled symbols) and tw = 10000 (open sym-
bols) is plotted versus τ (number of Monte Carlo steps) for
the same geometries, temperatures, system parameters and
symbols and colors as in Fig. 2.
(a) (b)
(c) (d)
FIG. 4: (Colors online) The percentage Nup of particles per
system pointing upwards after waiting times tw = 0 (filled
symbols) and tw = 10000 (open symbols) is plotted versus
τ (number of Monte Carlo steps) for the same geometries,
temperatures, system parameters and symbols and colors as
in Fig. 2.
allel or antiparallel to their easy axes ~ni and do therefore
flip only between these two directions. Accordingly, the
value of Oµ does neither depend on the external field nor
on the functional form of m(τ). Since Oµ(τ) stays con-
stant for large tw or τ > τc, while m(τ) increases with
time (see Fig. 2), the ~µi have already reached their par-
FIG. 5: (Colors online) The magnetization m(τ ) after a wait-
ing time tw = 0 (filled symbols) and tw = 10000 (open sym-
bols) for T̃ = 1/10 (red symbols) and T̃ = 1/40 (blue symbols)
of systems with aligned and randomly oriented anisotropy
axes (red circles and diamonds, respectively for T̃ = 1/10
and blue squares and triangles respectively for T̃ = 1/40) are
plotted versus τ (number of Monte Carlo steps) for systems
without dipole-interaction.
allel or antiparallel position and can only perform spin
flips by 180 degrees, thereby increasing m(τ) and leav-
ing Oµ unchanged. To make this point still clearer, we
plot in Fig. 4 the percentage Nup of particles pointing
upwards, i.e. with ϑi < π/2, again for the geometries of
Fig. 1. The similarity between Fig. 4 and Fig. 2 is obvi-
ous, showing that the number of the magnetic moments
oriented upwards determine the shape of m(τ).
We therefore arrive at a remarkably simple Ising-like
dynamics of these ultrafine magnetic particles. The
amount of aging is directly related to the degree of or-
der a system can achieve during tw. In the fully ordered
system of Figs. 1-3(a), after a long waiting time tw, the
~µi prefer to be aligned in stable chains [10] along the z-
direction and thus cannot follow an external field easily.
Single magnetic moments inside a chain will hardly flip to
the other side and flips of whole chains possess extremely
large relaxation times. Without waiting time, on the
other hand, the ~µi are in unstable positions which allows
them to follow the external field quite rapidly, leading to
large aging effects in ordered systems. As Figs. 2(b-d)
show, the situation is different in systems with positional
and/or orientational disorder. The relaxation times for
spin flips decrease with the amount of disorder, in partic-
ular with the amount of orientational disorder. When the
chains are winded and aligned into different directions,
they are less stable and possess a large variety of inter-
mediate positions to flip to the other side. Accordingly,
aging effects become weaker with increasing disorder.
For illustration, we visualize the aging process in Fig. 6
for the system with the highest order and the strongest
aging effects, i.e. for the cubic system with aligned
anisotropy axes. For this visualization, we follow the
definition of the transversal order parameter of Ref. [10]:
each of the L2 sites in the xy plane can be either a + site
or a − site, if a chain has already been formed and all
magnetic moments in the chain point into the positive or
negative z direction, respectively (white sites). If this is
not the case, the site is a 0 site (grey sites). The figure
shows that chains are quite obviously formed in the sec-
ond step of the aging process during the waiting time tw,
as can most easily be seen by comparing Fig. 6(a), where
tw = 10000 with 6(d) where tw = 0. In (a), many chains
are formed during tw that appear to be quite stable in
the following 3rd step of the aging process (Fig. 6(b,c)),
when an external field is applied in the + direction. We
can see that most of the − chains persist in spite of the
external field. The situation is different in Fig. 6(d-f),
where only few chains exist at the end of the 2nd step
of the aging process (Fig. 6(d)). Here, after switching
on the external magnetic field, new chains can be built
from the 0 sites and the system therefore follows the field
much easier than in Fig. 6(a-c).
Recently, it has been argued that also a broad distri-
bution of anisotropy energy barriers might lead to aging
effects in superparamagnetic systems [20]. To show that
these kinds of aging effects are in fact negligible com-
pared with systems where both energy contributions are
present, we have studied systems without dipole interac-
tion (solely anisotropy energy) at temperatures T̃ = 1/10
and 1/40. In this case, the particle positions play no role,
so that the geometry of Fig. 1(a) and (b) as well as (c)
and (d) are physically identical. The results of m(τ) for
these two geometries are shown in Fig. 5 for the same
aging procedure as before. The figure shows that the dif-
ferences between the curves for tw = 0 and tw = 10000
are orders of magnitude smaller than in the systems with
dipolar interaction. It is interesting to note that also for
systems with only dipolar interaction, some kind of ag-
ing can be seen, but orders of magnitude smaller than for
systems with both energy contributions.
In summary, analyzing the microscopic dynamics of ul-
trafine magnetic particles, we found that irrespective of
the strength of the dipolar interaction, the dipoles ori-
ente themselves either parallel or antiparallel to their
anisotropy axes. We therefore arrive at a remarakably
simple picture of the dipole dynamics, where similar to
the Ising model, the ~µi perform ”spin flips” between these
two orientations. Aging effects occur when after a certain
waiting time, the magnetic dipoles have arranged them-
selves in stable configurations and flips of single magnetic
moments are suppressed. These aging effects increase in
a counter-intuitive way with the order of the system and
are thus most pronounced in completely orderded sys-
tems with cubic arrangement of the particles and axes
aligned into the direction of the magnetic field.
(a) (b) (c)
(d) (e) (f)
FIG. 6: Visualization of the chains perpendicular to the xy-
plane in the cubic system with aligned anisotropy axes at
T̃ = 1/5 for one typical system. The complete chains are
indicated by white sites and by + or − signs, depending on
the direction of the chain. Sites, where chains have not (yet)
been built are indicated by the grey shade. (a-c) System with
waiting time tw = 10000, i.e. (a) after the cooling process
and tw = 10000 (b,c) after an external magnetic field in the +
direction has been applied for (b) 1000 and (c) 10000 Monte
Carlo steps. (d-f) System without waiting time (tw = 0),
i.e. (d) after the cooling process (and tw = 0) (e,f) after an
external magnetic field in the + direction has been applied
for (e) 1000 and (f) 10000 Monte Carlo steps.
ACKNOWLEDGEMENTS
We gratefully acknowledge very valuable discussions
with W. Kleemann and financial support from the
Deutsche Forschungsgemeinschaft.
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