JOURNAL OF APPLIED PHYSICS 108, 123920 共2010兲
A collective dynamics description of dipolar interactions and the coercive
field of magnetic nanoparticles
R. K. Das,1 S. Rawal,2 D. Norton,2 and A. F. Hebard1,a兲
1
Department of Physics, University of Florida, Gainesville, Florida 32611, USA
Department of Material Science and Engineering, University of Florida, Gainesville, Florida 32611, USA
2
共Received 11 August 2010; accepted 1 November 2010; published online 30 December 2010兲
The effect of dipolar interactions on the coercive field of Ni nanoparticles embedded as layers in a
Al2O3 host matrix is discussed. The results for two sets of 5 layer samples with different interlayer
spacings and a set of single layer samples are compared for samples with particle size varying from
3 nm 共single domain兲 to 60 nm 共multiple domain兲. The dipolar interactions are strongest in the
samples with shorter interlayer distances and weakest for the single layer samples. Our observation
that dipolar interactions increase the coercive field and decrease the critical diameter separating
single domain from multiple domain behavior reinforces a description based on collective
dynamics. © 2010 American Institute of Physics. 关doi:10.1063/1.3524277兴
I. INTRODUCTION
The origin of the coercive field Hc for coherently rotating ferromagnetic nanoparticles1 is remarkably different than
for bulk, where Hc is dominated by irreversible domain wall
motion.2 For nanoparticles with size smaller than a critical
size dc, the most favorable energy state is described by coherently rotating single domain 共SD兲 particles. When Hc is
plotted as a function of particle diameter d, there is a well
defined peak at dc. Particles with d ⬍ dc are SD and particles
with d ⬎ dc are multidomain 共MD兲.3–8 The formation of multiple domains within each nanoparticle in the region 共d
⬎ dc兲 reduces the magnetostatic energy, and hence Hc, by
simultaneously lowering the energy barriers to domain reversal.
Kittel9,10 has shown that for a spherical particle, dc is
given by
dc = 72冑AK/␮0M s2 ,
共1兲
where A is the exchange stiffness, K is the anisotropy constant, ␮0 is the free space permeability, and M s is the saturation magnetization. In SD particles there are no domain
walls, and the origin of Hc is the finite time required to
reverse the magnetization direction over the magnetic field
dependent anisotropy energy.1 The time required to reverse
the direction of the magnetization of a coherently rotating
SD particle in zero field is given by5,11–13
␶ = ␶0 exp共KV/kBT兲,
共2兲
where ␶0 is the inverse of the attempt frequency to overcome
the energy barrier, V is the volume of the particle, kB is the
Boltzmann constant, and T is the temperature. For ␶0
⬃ 10−9 s and typical measurement times ␶m ⬃ 100 s, Eq. 共2兲
defines a blocking temperature, T = TB = KV / kB ln共␶m / ␶0兲
⬃ KV / 25kB, below which there is hysteresis 共Hc ⬎ 0兲 and
above which there is full reversibility 共Hc = 0兲.
a兲
Electronic mail: afh@ufl.edu.
0021-8979/2010/108共12兲/123920/4/$30.00
Stoner and Wohlfarth have calculated Hc共T兲 of SD particles in the simple case when particles are coherently rotating and the applied magnetic field is along the easy axis of
magnetization of the particles. For H ⬎ 0 the energy barrier
to magnetization reversal decreases according to the relation,
KV共1 − H / Hc0兲2, where Hc0 = 2K / M s and M s is the saturated
magnetization. The coercive field HcSW共T兲 for Stoner–
Wohlfarth particles is,1
HcSW共T兲 =
=
再 冋 册冎
冋 冉 冊册
2K
ln共␶m/␶0兲
1−
Ms
ln共␶/␶0兲
2K
T
1−
Ms
TB
1/2
1/2
.
共3兲
This simple Stoner–Wohlfarth model thus shows that Hc for
the nanoparticle can depend on many different factors: Hc
increases as ␶m and T decrease and as ␶, K, and V increase. In
the presence of dipolar interactions, a widely accepted modification to the above equation is achieved by assuming that
the dipolar interactions give rise to an effective anisotropy
energy.14–24 Thus if dipolar interactions give rise to an increase 共decrease兲 in K, then ␶, according to Eq. 共2兲, will also
increase 共decrease兲 and as a net result Hc will increase 共decrease兲.
The effect of dipolar interaction on Hc has been investigated extensively. Early theoretical work by Néel25 showed
that Hc decreases with an increase in the packing fraction ⑀
according to the relation, Hc = Hc⬁共1 − ⑀兲. The interaction effect is incorporated in the parameter ⑀ and can be considered
as an “interaction field” that lowers the anisotropy energy. In
a later paper, Wohlfarth14 showed that the effect of interactions can either increase or decrease Hc depending on particle orientation, since the dipolar interaction is direction dependent. All of those results have been derived assuming that
K either increases or decreases due to dipolar interactions.
Previous theoretical and experimental works have reported
interaction-induced increases or decreases in Hc and explained the results in terms of a corresponding increase or
decrease in the anisotropy energy.14–24,26
108, 123920-1
© 2010 American Institute of Physics
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123920-2
Das et al.
FIG. 1. 共Color online兲 共a兲 TEM image of a single layer sample with average
particle diameter of ⬃24 nm. Particles are well defined with interparticle
distance of around 4 nm. 共b兲 Schematic of the single layer sample. A 40 nm
thick buffer layer of Al2O3 is first grown on top of the substrate. The Ni
nanoparticles are then grown on top of the buffer layer. Finally a capping
layer of Al2O3 is deposited for protection against oxidation. 共c兲 Schematic of
a 5-layer Ni nanoparticle sample.
In the experimental results reported here, we have found
that an increase in the dipolar interaction increases Hc of
coherently rotating SD particles. In our previous work on the
same nanoparticle system, we have found that interactions
decrease dc.4 Herein lies a dilemma! Equation 共1兲 suggests
that an interaction-induced decrease in dc gives rise to a decrease in K. But a decrease in K will also decrease ␶ 关Eq. 共2兲兴
and thus will decrease Hc 关Eq. 共3兲兴, which is in contradiction
to our results presented below. Thus, a mechanism in which
interactions solely give rise to a change in K cannot be applied to our system of magnetic nanoparticles. Any interaction induced change in K will simultaneously change both Hc
and dc in the same direction. Below, we show qualitatively
that our interaction-induced increase in Hc together with our
previously observed interaction-induced decrease in dc 共Ref.
4兲 can be realized using theory which finds that in the presence of dipolar interactions the dynamics of magnetization
are collective in nature.27,28
J. Appl. Phys. 108, 123920 共2010兲
共see Fig. 1兲. The top layer of Al2O3 acts as a capping layer
that prevents oxidation of the nanoparticles.29 Three different
sets of samples are grown. Set 1 and set 2 samples comprise
five layers of Ni nanoparticles separated by Al2O3 layers. For
set 1 and set 2 the Al2O3 separations are 3 nm and 40 nm,
respectively. Set 3 samples comprise a single layer of Ni
nanoparticles in the Al2O3 matrix. The dipolar interactions
are strongest in set 1, since the particles are close to each
other and the number of nearest neighbors are larger compared to the other sets. The dipolar interaction is moderate in
set 2 and weakest in the single-layer set 3, where there is no
interlayer coupling. Each of the three sample sets comprise
five to six different samples with particle size varying from
3–60 nm.
Figure 1共a兲 shows a cross-sectional transmission electron
microscopy 共TEM兲 image of a single layer Ni nanoparticle
sample 共set 3兲 with average particle diameter of ⬃24 nm.
Schematics of the single and multilayer samples are shown
in Figs. 1共b兲 and 1共c兲. Typical magnetization loops at three
different temperatures are shown in Fig. 2共a兲 for the sample
with a 3 nm thick Al2O3 spacer layer 共set 1兲 and 6 nm diameter nanoparticles. The coercive field Hc共T兲 is determined
from the loop as shown by the arrow for the data at 10 K.
This procedure to determine Hc共T兲 is repeated for all samples
belonging to all three sets. At temperatures above the blocking temperatures 共T ⬎ TB兲, the SD samples behave as superparamagnetic particles. Figure 2共b兲 shows the superparamagnetic behavior of a sample with 6 nm diameter nanoparticles
from set 1. We note that the magnetization data collapse onto
a single curve when plotted as a function of H / T as would be
expected for superparamagnetic coherently rotating SD particles.
III. RESULTS AND DISCUSSION
II. EXPERIMENTAL DETAILS
Samples were grown using the pulsed laser deposition
technique.4 The base pressure of the growth chamber was on
the order of 10−7 Torr and the growth temperature was near
550 ° C. Multilayer structures of Al2O3 and Ni nanoparticles
were grown without breaking the vacuum of the chamber.
First a 40 nm thick buffer layer of Al2O3 is grown on top of
the substrate. This buffer layer provides a substrate independent growth surface for the Ni particles. Ni nanoparticles and
Al2O3 are then sequentially deposited onto this buffer layer
Figure 3 shows Hc as a function of d for each of the
sample sets identified in the legend. The peaks in Hc define a
critical diameter dc labeled by arrows that separates SD 共d
⬍ dc兲 from MD 共d ⬎ dc兲 particles.3–7,10 The data clearly show
that dc decreases with increasing dipolar interactions 共dc1
⬍ dc2 ⬍ dc3兲. Hc on the other hand increases with increasing
dipolar interactions 共vertical dotted arrow兲 in the SD region.
These two results cannot be explained in terms of the commonly reported change in K due to dipolar interactions.14–24
In earlier work, we observed a shift in dc to higher grain size
FIG. 2. 共Color online兲 共a兲 Magnetization loops at the temperatures indicated in the legend of a sample from set 1 having an average particle diameter of
⬃6 nm. The coercive fields Hc共T兲 are determined from the loops as shown by the arrow for the data at T = 10 K. Hc共T兲 decreases with increasing temperature
and goes to zero above the blocking temperature. 共b兲 Magnetization loops above the blocking temperatures for the same sample. Magnetization is plotted as
a function of H / T to show the data collapse onto a single curve as expected for superparamagnetic SD particles above the blocking temperature.
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123920-3
J. Appl. Phys. 108, 123920 共2010兲
Das et al.
␶⫾ = ␶0 exp关K共eff⫾兲V/kBT兴,
FIG. 3. 共Color online兲 The coercive field Hc at T = 10 K is plotted as a
function of the particle diameter d. The peak for each curve separates the SD
from the MD particles. Particles with diameter higher 共smaller兲 than the
peak diameter 共dc兲 are MD 共SD兲. Data for the three different sample sets are
shown and indicated in the legend. The critical diameters dc1, dc2, and dc3
for each sample set are identified with arrows. In the SD region 共i.e., d
⬍ dc兲 the coercivity increases with increasing dipolar interactions as shown
by the vertical dashed arrow.
with increasing temperature in accord with theory that takes
into account the decreasing influence of dipolar magnetic
interactions from thermally induced random orientations of
neighboring grains.4 This interpretation does not rely on interaction induced changes in K but rather on the effect of
fluctuating T-dependent dipolar magnetic fields arising from
fluctuations in the magnetization of nearest neighbor particles. In this present study, the collective dynamics of the
particle magnetizations due to dipolar interactions is found to
be responsible for the increase in Hc. These observations are
shown in Fig. 3 and summarized in Fig. 4.
The effect of dipolar interactions on Hc are usually
treated by including an additive energy term, ⫾Edip,14–24 to
the anisotropy energy KV which appears in the expression
for ␶ given in Eq. 共4兲.26 The resulting expression,
␶⫾ = ␶0 exp关共KV ⫾ Edip兲/kBT兴,
共4兲
can be rewritten as,
共5兲
where K共eff⫾兲 is an effective anisotropy constant that is renormalized by interactions either upwards or downwards depending on the sign of Edip.
Since Hc and hence ␶ is experimentally observed to increase with increasing interactions, Eq. 共5兲 shows that a
larger value for K共eff⫾兲 corresponding to +Edip must be chosen. These correlations of ␶ and Hc with a larger K are consistent with Eqs. 共2兲 and 共3兲. However, according to Eq. 共1兲,
a larger K associated with dipolar interactions should also
result in an increase of dc. The assumption that dipolar interactions can be solely accounted for by changes in K imply
that Hc and dc should simultaneously change in the same
way, either increasing or decreasing. Experimentally, as seen
in Fig. 4, this is not the case; dipolar interactions cause Hc to
increase and dc to decrease. This contradiction strongly suggests an alternative approach to the problem, namely, collective dynamics.27,28
IV. COLLECTIVE DYNAMICS
Magnetization dynamics has been recognized to be collective in nature in the presence of dipolar interactions between magnetic nanoparticles. The relaxation time ␶ⴱ in this
case is given by,27
␶ⴱ = ␶共T/Tg − 1兲−z,
T ⬎ Tg ,
共6兲
where ␶ is the relaxation time of single noninteracting particles as expressed in Eq. 共2兲 and Tg = ␮0M 2 / 4␲kBr3 is a
critical temperature that depends on interparticle distance r,
the magnetic moment M per particle and a critical exponent
z. Interactions give rise to the power law dependence on
共T / Tg − 1兲−z shown in Eq. 共6兲. With decreasing temperature
the presence of this term causes ␶ⴱ to diverge more rapidly
than ␶. Experimental studies of Fe–C monodisperse nanoparticles confirm that dipole-dipole interactions increase the relaxation time at all temperatures studied.27 Accordingly, the
relaxation time will be larger in the presence of dipolar interactions and thus Hc 关Eq. 共3兲兴 will be larger, in agreement
with our experimental observations 共Fig. 4兲. Note that with
this description, K is not affected by dipolar interactions and
the increase in relaxation time is due to collective magnetic
dynamics.27,28
V. CONCLUSIONS
FIG. 4. 共Color online兲 The critical diameter dc 共left axis兲 and coercive field
Hc 共right axis兲 plotted as a function of increasing dipolar interaction. Hc 共dc兲
increases 共decreases兲 with increasing dipolar interaction. The opposite behavior of Hc and dc suggests that the collective dynamics is responsible for
the increase in Hc due to dipolar interactions.
A study of the effect of dipolar interactions on the coercive fields of Ni nanoparticle systems is presented for both
single and multilayer thin-film structures. The coercive field
increases with increasing dipolar interactions and can be understood qualitatively in terms of collective dynamics. Three
sets of samples are investigated. Each set comprises samples
having particle size varying from 3–60 nm in diameter. Dipolar interactions are strongest in set 1 and diminish progressively for sets 2 and 3. The coercive field increases and the
critical SD radius decreases with increasing dipolar interactions. These two behaviors together cannot be modeled by
interaction-induced changes in the anisotropy constant and
are better described by reversals of magnetization that are
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123920-4
collective in nature. To our knowledge, the effect of collective dynamics on the coercive field of nanoparticle systems
has not been previously reported.
This work is supported by the National Science Foundation 共NSF兲 under Contract No. 0704240 共A.F.H.兲.
1
J. Appl. Phys. 108, 123920 共2010兲
Das et al.
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