Memory in nanomagnetic systems: Superparamagnetism versus Spinglass behavior

Abstract

The slow dynamics and concomitant memory (aging) effects seen in nanomagnetic systems are analyzed on the basis of two separate paradigms : superparamagnets and spinglasses. It is argued that in a large class of aging phenomena it suffices to invoke superparamagnetic relaxation of individual single domain particles but with a distribution of their sizes. Cases in which interactions and randomness are important in view of distinctive experimental signatures, are also discussed. Comment: 11 pages and 19 figures

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arXiv:cond-mat/0606233v1 [cond-mat.stat-mech] 9 Jun 2006
Memory in nanomagnetic systems: Superparamagnetism versus Spinglass behavior
Malay Bandyopadhyay1and Sushanta Dattagupta1,2
1S.N. Bose National Centre for Basic Sciences,JD Block, Sector III, Salt Lake, Kolkata 700098, India.
2Jawaharlal Neheru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India.
(Dated: February 6, 2008)
The slow dynamics and concomitant memory (aging) effects seen in nanomagnetic systems are
analyzed on the basis of two separate paradigms : superparamagnets and spinglasses. It is argued
that in a large class of aging phenomena it suffices to invoke superparamagnetic relaxation of indi-
vidual single domain particles but with a distribution of their sizes. Cases in which interactions and
randomness are important in view of distinctive experimental signatures, are also discussed.
PACS numbers: 75.75.+a,75.50.Lk,75.50.Tt,75.47.Lx
I.INTRODUCTION
The subjects of both superparamagnetism and spin-
glasses are quite old and well studied [1, 2, 3, 4, 5, 6, 7].
Yet they have been rejuvenated in recent years in the
context of fascinating memory and aging properties of
nanomagnets. These properties, which are believed to be
of great practical usages, have been recently investigated
in a large number of experiments on magnetic nanopar-
ticles [8, 9, 10, 11, 12, 13, 14, 15]. The observed slow dy-
namical behavior has been variously interpreted, based
on the paradigm of either superparamagnet or spinglass,
sometimes even obscuring the difference between the two
distinct physical phenomena. The purpose of this paper
is to reexamine some of the data, others’ as well as our
own, and critically assess the applicability of the physics
of either superparamagnets or spinglasses and occasion-
ally, even a juxtaposition of the two. Our main point is,
spinglasses are marked by Complexity, arising out of two
separate attributes —– Frustration and Disorder. While
the manifested properties, such as stretched exponential
relaxation and concomitant aging effects, can also occur
due to ‘freezing’ of superparamagnetism, especially in a
polydisperse sample, the physics of spinglasses is natu-
rally much richer than that of superparamagnets. A dis-
cernible experimental signature of superparamagnetism
versus spinglass behavior seems to be the magnitude of
the field-cooled (FC) magnetization memory effect that
is significantly larger for the interacting glassy systems
than the one in non-interacting superparamagnetic par-
ticles [16]. Therefore, invoking spinglass physics in in-
terpreting data on the slow dynamics of nanomagnets
can sometimes be like ‘killing a fly with a sledge ham-
mer’, especially if a simpler interpretation on the basis of
superparamagnetism is available. We explore such situ-
ations in this paper.
Superparamagnetism was discussed quite early by
Frenkel and Dorfman and later by Kittel, as a property
arising out of single-domain behavior when a bulk ferro-
magnetic or an antiferromagnetic specimen is reduced to
a size below about 50 nm [2]. For such a small particle-
size the domination of surface to bulk interactions yields
a mono-domain particle inside which nearly 105magnetic
moments are coherently locked together in a given direc-
tion, thus yielding a giant or a supermoment. Clearly, for
this to happen, the ambient temperature must be much
less than the bulk ordering temperature, so that the in-
tegrity of the super moment is maintained. However, as
Neel pointed out, in the context of magnetic properties
of rocks in Geomagnetism, the direction of the supermo-
ment is not fixed in time [1]. Indeed, because of thermal
fluctuations, this direction can undergo rotational relax-
ations across an energy-barrier due to the anisotropy of
the single-domain particle, governed by the Neel relax-
ation time:
τ = τ0exp
?KV
kBT
?
.(1)
In Eq. (1), the preexponential factor τ0 is of the order
of 10−9sec, V is the volume of the particle, and K is the
anisotropy energy, the origin of which lies in the details
of all the microscopic interactions. For our purpose K
would be treated as a parameter whose typical value is
about 10−1Joule/cm3. Therefore, at room temperature,
τ can be as small as 10−1sec for a particle of diame-
ter 11.5 nm but can be astoundingly as large as 109sec
for a particle of diameter just about 15 nm. Thus, a
slight polydispersity (i.e., a distribution in the volume
V), can yield a plethora of time scales, giving rise to in-
teresting slow dynamics. For instance, if τ < τE, where
τEis a typical measurement time in a given experiment,
the supermoment would have undergone many rotations
within the ‘time-window’ of the experiment, thereby av-
eraging out to zero the net magnetic moment. One then
has superparamagnetism. On the other hand, if τ > τE,
the supermoment hardly has time to rotate within the
time-window, thus yielding a ‘Frozen-moment’ behavior.
The consequent nonequilibrium features have led to the
phrase: “Magnetic Viscosity” while depicting the time-
dependent freezing of moments [3, 4, 17, 18]. Further, the
transition from superparamagnetism to frozen-moment
behavior occurs at a temperature, referred to in the lit-
erature as the blocking temperature Tb, defined by
τE= τ0exp
?KV
kBTb
?
.(2)
When the measurement temperature T is less than
Tb the magnetic particles are blocked whereas in the

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other extreme they display facile response to applied
fields.Therefore, we emphasize that even within a
single particle picture, sans any form of inter-particle
interactions, such as in a dilute nanomagnetic specimen,
one can obtain apparently intriguing effects such as
‘stretched exponential’ relaxation simply because of
size distributions. The latter will be shown to be
responsible for much of the data on slow relaxations in
nanomagnets.
Turning now to spinglasses, historically the phe-
nomenon was first observed in dilute alloys such as
Au1−xFex (or Cu1−xMnx) in which magnetic im-
purities Fe (or Mn) in very low concentrations were
“quenched-in” from a solid solution with a host metallic
system of Au (or Cu) [19]. The localized spin is cou-
pled with the s-electron of the host metal which itself
interacts with the other conduction electrons via what
is called the Ruderman-Kittel-Kasuya-Yoshida (RKKY)
Hamiltonian, thereby setting up an indirect exchange
interaction between the localized moments.
the coupling constant of the exchange interaction, in
view of the RKKY coupling, alternates in sign (between
ferro and antiferromagnetic bonds),
‘frustrated’. Thus the ground state is highly degenerate
yielding a zero-temperature entropy.
effect is due to disorder. Because the dilute magnetic
moments are quenched-in at random sites, the exchange
coupling-strengths are randomly distributed. The dual
occurrence of frustration and disorder has led to novel
concepts in the Statistical Mechanics of spinglasses
such as configuration-averaging, replica-techniques (for
computing the free energy), broken-ergodicity, etc.
[20]. Experimentally, spinglasses are characterized by
a ‘cusp’ in the susceptibility and stretched exponential
relaxation of time-dependent correlation functions [19].
It is no wonder then that spinglasses also exhibit slow
dynamics with associated memory and aging effects,
albeit the root causes are much more complex than a
system of polydisperse, noninteracting single-domain
nanomagnetic particles, discussed earlier. Indeed spin-
glasses, because of their complexity, have been employed
as paradigms for studying real structural glasses, an
unresolved problem of modern condensed matter physics
[21].
Given this background on two distinct physical
phenomena (and yet manifestly similar properties) of
superparamagnets and spinglasses, a natural question
to ask is: can there be spinglass-like physics emanat-
ing from a collection of single-domain nanomagnetic
particles embedded in a non-metallic, non-magnetic
host? The answer is clearly an YES when the system
is no longer a diluted one such that the supermoments
start interacting via dipole-dipole coupling. Because the
dipolar interaction (like the RKKY-mediated exchange
interaction) is also endowed with competing ferro and
antiferromagnetic bonds [22], as well as randomness due
to random locations of the magnetic particles, all the
attributes of spinglasses can be simulated in interacting
Because
the system is
An additional
single-domain particles. This will be analyzed below.
With the preceding discussion the plan of the paper
is as follows.In Sec. II we discuss the relaxation
of non-interacting single-domain particles based on a
rudimentary rate theory. The results of this rate theory,
coupled with polydispersity of the particles, are applied
in Sec. III to a large body of recently published data
on the slow dynamics of nanomagnets.
deals with a different set of experiments that necessarily
requires incorporation of interactions between
nanoparticles and hence spinglass-like physics.
Section IV
the
II.RATE THEORY OF RELAXATION OF
SINGLE-DOMAIN PARTICLES
The Neel formula (Eq. (1)) of relaxation time is the
outcome of a generic class of ‘Escape over Barriers’ prob-
lems studied by Kramers [23].
a single-domain magnetic particle, due to spontaneous
thermal fluctuations, is envisaged to undergo rotational
Brownian motion across an anisotropy barrier. The lat-
ter, in a large class of systems characterized by uniaxial
anisotropy, can be described by the energy
The supermoment of
E(θ) = KV sin2Θ,(3)
where K and V have been introduced earlier, and Θ is
the angle between the anisotropy axis, chosen as Z (de-
termined by the host crystal) and the direction of the
supermoment. Thermal fluctuations of the system can
be studied in terms of the Fokker-Planck equation for
P(Θ,t), which defines the probability that the supermo-
ment makes an angle Θ with the anisotropy axis, at a
time t [5, 18]:
∂
∂tP(Θ,t) = d
1
sinΘ
+∂P(Θ,t)
∂Θ
∂
∂Θ
?
?
sinΘ
?
1
kBT
∂E(Θ)
∂Θ
P(Θ,t)
,0 ≤ Θ ≤ π,(4)
where d (having the dimension of frequency) is the rota-
tional diffusion constant. Application of Kramers’ anal-
ysis to Eq. (4) not only yields the Neel relation (Eq. (1))
but also a set of rate equations in the high barrier / weak
noise limit, i.e. KV >> kBT. In this limit the dynam-
ics is basically restricted to Θ = 0 and Θ = π regions
and consequently, the Fokker-Planck equation reduces to
a set of two-state rate equations:
d
dtn0(t) = −d
dtnπ(t) = −λ0→πn0(t) + λπ→0nπ(t), (5)
where the subscripts on n indicate the two allowed values
of Θ and the rate constants are as follows:
λ0→π= λ0exp[−V (K + Msh)
kBT
], (6)
λπ→0being obtained by switching the sign of h. Here Ms
is the saturation magnetization per unit volume. Note

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that Eq. (6) is a generalization of Eq. (1) in order to take
cognizance of an external magnetic field h. As discussed
in Ref. [9] the rate equations can be solved analytically
for any temperature and field protocol represented by
T(t) and h(t), from a given initial condition. For the
sake of completeness we rewrite the main result for the
time dependent magnetization M(t):
M(t) = M(t = 0)exp(−¯λt) + µV N∆λ
¯λ
[1 − exp(−¯λt)],
where
¯λ = λ0→π+ λπ→0;∆λ = λπ→0− λ0→π. (7)
The observed magnetization of the system is obtained by
averaging over a volume distribution
¯
M(t) =
?
dV P(V )M(t,V ). (8)
The superposition of relaxation rates, caused by the vol-
ume distribution P(V), can alter the exponential relax-
ation indicated in Eq. (7) into a variety of forms, e.g.
stretched exponential or logarithmic [17, 18].
models of P(V) are extant in the literature, all leading
to aging effects. Examples are bimodal distribution [9], a
flat distribution bounded by two volumes Vminand Vmax
[17] or a log-normal distribution (assumed below)
Several
P(V ) =exp[−ln(V2)/(2γ2)]
(γV√2π)
, (9)
γ being a fitting parameter.
Until now we have discussed the relaxation effects of
isolated (i.e. non-interacting) single-domain nanomag-
netic particles. The question we would like to next ad-
dress is : what happens when these particles are brought
closer and the dipolar interaction between their magnetic
moments starts becoming non-negligible? Recall that the
interaction between two dipole moments ? miand ? mj, lo-
cated at the sites i and j at a distance |? rij| apart is given
by [24]
Hd−d=
?
ij
γiγj{3? mi· ? mj− (? mi· ˆ rij)(? mj· ˆ rij)}
|? rij|3
. (10)
Here γi and γj are the gyromagnetic ratios of the ith
and jthparticles respectively, ? rij is the vector distance
between the ‘sites’ at which the two magnetic particles
are located, and ˆ rij is the corresponding unit vector. It
is well known that dipolar couplings, being long-ranged,
anisotropic and alternating in the sign of interaction,
can indeed lead to frustration and very complex mag-
netic order of the ground state, depending on the crystal
structure [22]. Incorporation of the dipolar interaction
into the dynamics is a further complication involving a
multi-particle Fokker-Planck equation in which the ‘drift
term’, proportional to E(Θ) in Eq. (4), would have to
be replaced by Eq. (10). The underlying theory is quite
daunting and is not attempted here, as a simpler treat-
ment is possible for nanoparticles with large anisotropy,
as is sketched below.
Recall that the rate equation abstraction of the Fokker-
Planck equation is itself a discrete version of a continuous
stochastic process, applicable in the high barrier/weak
noise limit when the basins of dynamical attractors are
restricted to the Θ = 0 and Θ = π regions. In the context
of the dipolar coupling, which is after all an anisotropic
Heisenberg interaction, this approximation implies that
we are in the so-called Ising limit. In this, only the Z-
components of the magnetic moments are relevant. The
dipolar interaction can now be described by its truncated
form [24]:
Hd−d= µ2V2?
ij
γiγj?2(1 − 3cos2θij)
| ? rij|3
cosΘicosΘj,
(11)
where θijis the angle between ? rijand the anisotropy axis
and ? mi is replaced by µV cosΘi, µ being the magnetic
moment per unit volume and Θi defined after Eq. (3).
The Fokker-Planck dynamics including Eq. (11) is still
very formidable. For our purpose we invoke a mean field
theory in which the ithnanoparticle say, is envisaged to
be embedded in an effective medium that creates a local
mean field (MF) at the site i which is proportional to
the average magnetization itself. Therefore, Hd−din Eq.
(11) may be replaced by its MF form
HMF
d−d= γ?µ2V2cosΘ
?
j
γj?(1 − 3cos2θij)
| ? rij|3
?cosΘj?,
(12)
wherein the angular brackets ?...? represent thermal av-
eraging. In addition, and in conformity with our stated
assumption about the largeness of the anisotropy energy,
cosΘ can be replaced by the two-state Ising variable σ :
HMF
d−d= γµ2V2σ
?
j
γj?(1 − 3cos2θij)
| ? rij|3
?σj?.(13)
If the nanoparticles are located at random sites of the
host matrix, such as grown by the sol-gel technique (for
instance NiFe2O4magnetic particles in a SiO2host [9]),
the interaction in Eq. (13) is random because of random
values of |? rij| and is also alternating in sign due to differ-
ent allowed values of θij. Within the spirit of the mean
field theory the local field H is to be derived self consis-
tently from the following expression:
H = µΛV tanh(µV H
kBT),(14)
where Λ is a random variable [9]. Since the local field can
point either along Θ = 0 or Θ = π direction, within the
two-state model, Eq. (14) naturally admits both positive
and negative solutions for H.
Summarizing, the effect of interaction within the sim-
plified mean field approximation, enumerated above, is

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4
to modify the rate theory in which the rate constant in
Eq. (6) is replaced by
λ0→π= λ0exp[−V (K + Msh + H)
kBT
].(15)
Clearly Eq.
incorporate dipole-dipole interaction into the dynamics
of nanomagnetic particles, and is therefore, expected to
have limited validity. The actual spinglass dynamics is
a much more complex subject that requires application
of sophisticated theoretical tools [25]. Yet we find that
the simple-minded extension of two-state rate theory, as
encapsulated in Eq. (15), is adequate to interpret aging
data in interacting systems (Sec. IV).
(15) is an extremely crude attempt to
III.SUPERPARAMAGNETIC SLOW
DYNAMICS
Recently Sun et al have made a series of measure-
ments on a permalloy (Ni81Fe19) nanoparticle sample
which demonstrate striking memory effects in the dc
magnetization [8]. These involve field-cooled (FC) and
zero-field cooled (ZFC) relaxation measurements under
the influence of temperature and field changes. We have
also observed very similar memory effects in NiFe2O4
magnetic particles in a SiO2 host [9].
Sasaki et al [10] and Tsoi et al [11] have reported similar
results for the noninteracting (or weakly interacting)
superparamagnetic system of γ − Fe2O3 nanoparticles
and ferritin (Fe-N) nanoparticles respectively. Further,
to understand the mechanisms of the experimental
approach of Sun et al, Zheng et al [12] replicated the
experiments on a dilute magnetic fluid with Co particles
and observed similar phenomena.
present a comparison of simulated results with all the
above mentioned experimental observations on the basis
of our simple two-state noninteracting model plus a
log-normal distribution of particle size, described in Sec.
II.
More recently
In this section we
050 100150200250300
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ZFC
FC
H=50 Oe
M (10
-5 emu)
T (K)
-3 -2 -1
H (kOe)
0123
-10
-5
0
5
10
10 K
150 K
M (10
-5 emu)
FIG. 1: (color online) Temperature dependence of the dc mag-
netization for the FC and ZFC processes. Inset shows the M-H
curves below and above Tb. (Sun et al Phys. Rev. Lett. 91,
167206 (2003)).
We begin our discussion from the most basic and well
known protocol, viz. the measurement of the zero field-
cooled magnetization (ZFCM) and the field-cooled mag-
netization (FCM). Figure 1 shows the ZFCM and FCM
curves in a 50 Oe field for Ni81Fe19. The ZFCM has a
peak at Tmax= 78K, which corresponds to the blocking
temperature Tb. The magnetization of the FC curve con-
tinues to increase with decreasing temperature as would
be expected for a system in thermal equilibrium. The two
curves depart from one another at a temperature higher
than Tmax. The inset shows the M-H curve below and
above the blocking temperature. Figure 2 and Figure 3
show the simulated FC-ZFC curves and the M-H curve
respectively. Our simulations, based on the two-state
noninteracting model, match well with the experimental
results.
0
50
100
150
T
200
250
300
0
0.01
0.02
0.03
0.04
0.05
0.06
FC
ZFC
M (arb. unit)
FIG. 2: (color online) Numerically calculated dc magnetiza-
tion for the FC and ZFC process.
-10
-5
0
5
10
-2
-1
0
1
2
3
-10
-5
0
5
10
-0.2
-0.1
0
0.1
0.2
T = 10 K
T = 150 K
M (arb. unit)
M (arb. unit)
H (arb. unit)
H (arb. unit)
FIG. 3: (color online) Numerically calculated M Versus H
curve below and above Tb.
The most striking experimental observation of Sun et
al is the memory effect in the dc magnetization (Fig.
4) obtained from the following procedure. The sample
is cooled in 50 Oe field at a constant cooling rate of
2K/min from 200K (TH) to 10K (Tbase). After reaching
Tbase, the sample is heated continuously at the same rate
to TH. The obtained M(T) curve is the normal FC curve
which is referred to as the reference curve. Then the
sample is cooled again at the same rate, but the cooling
is arrested three times (at T = 70, 50 and 30K) below
Tbwith a wait of tw= 4h at each stop. During tw, the
applied field is also turned off to let the magnetization

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5
decay. After each stop and wait period, the 50 Oe field is
reapplied and cooling is resumed. The cooling procedure
produces a step like M(T) curve.
base temperature, the sample is warmed continuously at
the same rate to TH in the continual presence of the 50
Oe field. Surprisingly, the M(T) curve obtained in this
way also shows the step like behavior. Similar memory
effects, following the same protocol were seen by us in
NiFe2O4 sample in which the magnetic particles were
embedded in a host SiO2matrix [9]. The effects can be
explained in terms of a bimodal distribution of particle
size (i.e. P(V) a sum of two delta functions at volumes
V1and V2) [9]. Our simulated results based on the two
state non-interacting model but accompanied by a log
normal distribution, are represented in Fig.
indicate satisfactory agreement with experiments.
After reaching the
5, which
0 10 20 30 40 50 60 70 80 90 100
2.0
2.1
2.2
2.3
2.4
2.5
H=50 Oe
decreasing T
increasing T
reference
M (10
-5 emu)
T (K)
FIG. 4: (color online) “Memory effect” observed in the dc
magnetization measurements in Ni81Fe19. (Sun et al Phys.
Rev. Lett. 91, 167206 (2003)).
0 30
60
90
0.15
0.16
0.17
M (arb. unit)
T (K)
Decreasing T
Increasing T
(b)
FIG. 5: (color online) Numerically simulated memory effect
observed in dc magnetization curves.
We further discuss the memory effect observed in
ZFC response measurements (Fig.
experiment, the sample is cooled down to T0 = 30K
6).In the ZFC
in zero field. Then a field of 50 Oe is applied and the
magnetization is recorded as a function of time. After a
time t1, the sample is quenched to a lower temperature,
T0− ∆T = 22K, and the magnetization is recorded
for time t2.Finally the temperature is turned back
to T0 and the magnetization is recorded for another
period t3.The field of 50 Oe is kept on during the
entire aging process.When the field is first turned
on, a slow logarithmic relaxation takes place following
an immediate jump.During the temporary cooling
the relaxation is rather weak. When the temperature
returns back to T0, viz. 30K, the magnetization comes
back to the level it reached before temporary cooling.
Moreover it is found that the relaxation curve during t3
is a continuation of the curve during t1.
2.1
(b)
05000100001500020000
0.9
1.0
1.1
1.2
1.3
(a)
30 K
22 K
30 K
M (10
-5 emu)
time (second)
1.0
1.2
04000800012000
t3
t1
100010000
FIG. 6: (color online) Magnetic relaxation measurements in
Ni81Fe19 with temporary cooling for the ZFC method. Inset
shows the same data vs the total time spent at 30 K for both
normal and logarithmic time scales.(Sun et al Phys.
Lett. 91, 167206 (2003)).
In Fig. 7 we show the numerically calculated results
of ZFC relaxation, again on the basis of the two-state
non-interacting model, which are qualitatively similar
to those in experimental measurements (Fig. 6). When
a field of 50 Oe is applied, the magnetization reaches a
certain value which is determined by the particles with
Tb≤ 30K. Then a logarithmic relaxation begins which
is due to those particles whose Tb are higher than 30
K [26].The sudden increase in magnetization during
t2 is due to the particles with Tb ≤ 22K which had
flipped during t1in order to reach their new equilibrium
state at T = 22K. On the other hand the particles
with Tb > 30K are not in equilibrium state and relax
extremely slowly at 22K to yield an almost constant
curve during t2. Finally, when the sample is heated
back to 30K, the particles with Tb ≤ 30K and those
flipped during t1+ t2, return back to the pre-quenching
equilibrium state and therefore the relaxation during t3
is the continuation of the curve in t1.
In Fig.8 we show the relaxation measurements of
Sun et al in the FC method with temporary cooling, in
which the sample is cooled to T0= 30K in a 50 Oe field
and the relaxation is measured for a time t1 after the
field is cut-off. The sample is quenched to T = 22K, and
the magnetization is recorded for time t2. Finally, the
Rev.