Enhancement of the magnetic anisotropy of nanometersized Co clusters: Influence of the surface and of interparticle interactions
ABSTRACT We study the magnetic properties of spherical Co clusters with diameters between 0.8 nm and 5.2 nm (25–7000 atoms) prepared by sequential sputtering of Co and Al2O3. The particle size distribution has been determined from the equilibrium susceptibility and magnetization data and it is compared with previous structural characterizations. The distribution of activation energies has been independently obtained from a scaling plot of the ac susceptibility. Combining these two distributions we have accurately determined the effective anisotropy constant Keff. We find that Keff is enhanced with respect to the bulk value and that it is dominated by a strong anisotropy induced at the surface of the clusters. Interactions between the magnetic moments of adjacent layers are shown to increase the effective activation energy barrier for the reversal of the magnetic moments. Finally, this reversal process is shown to proceed classically down to the lowest temperature investigated (1.8 K).

Article: Luis et al. Reply:
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ABSTRACT: A Reply to the Comment by Mikkel F. Hansen and Steen Mørup.Physical Review Letters 02/2003; 90(5). · 7.73 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: (FexNi1−x)4N (0.5 ≤ x ≤ 0.8) compounds were prepared by combining a simple reduction route with nitridation process under flowing ammonia gas at various temperatures. The microstructure as well as magnetic and microwave absorbing properties of asprepared products was investigated. The reflection loss of (FexNi1−x)4Nparaffin composites with a filler mass fraction of 50% was calculated from permittivity and permeability, and the magnetization hysteresis loops of the composites were measured at 300 K. It has been found that the saturation magnetization value of asprepared products increases as x rises from 0.50 to 0.80. Particularly, the reflection loss of (Fe0.67Ni0.33)4Nparaffin composite (x = 0.67) with a matching thickness of 2 mm is −17.5 dB at 12.0 GHz, lower than that of the composites with x = 0.50, 0.75, and 0.80. Moreover, (Fe0.67Ni0.33)4Nparaffin composite with a matching thickness of 2 mm has the reflection loss exceeding −10 dB in a maximum frequency range of 10.5–14.0 GHz, which is ascribed to its proper match of electromagnetic parameters and large magnetic loss. In one word, (Fe0.67Ni0.33)4N has great potential as a thin broadband microwave absorbent.Journal of Applied Physics 03/2013; 113(11). · 2.21 Impact Factor  SourceAvailable from: Bashar Issa
Article: Magnetic Nanoparticles: Surface Effects and Properties Related to Biomedicine Applications.
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ABSTRACT: Due to finite size effects, such as the high surfacetovolume ratio and different crystal structures, magnetic nanoparticles are found to exhibit interesting and considerably different magnetic properties than those found in their corresponding bulk materials. These nanoparticles can be synthesized in several ways (e.g., chemical and physical) with controllable sizes enabling their comparison to biological organisms from cells (10100 μm), viruses, genes, down to proteins (350 nm). The optimization of the nanoparticles' size, size distribution, agglomeration, coating, and shapes along with their unique magnetic properties prompted the application of nanoparticles of this type in diverse fields. Biomedicine is one of these fields where intensive research is currently being conducted. In this review, we will discuss the magnetic properties of nanoparticles which are directly related to their applications in biomedicine. We will focus mainly on surface effects and ferrite nanoparticles, and on one diagnostic application of magnetic nanoparticles as magnetic resonance imaging contrast agents.International Journal of Molecular Sciences 01/2013; 14(11):2126621305. · 2.46 Impact Factor
Page 1
arXiv:condmat/0108304v1 [condmat.mtrlsci] 20 Aug 2001
Enhancement of the magnetic anisotropy of nanometersized Co clusters: influence of
the surface and of the interparticle interactions
F. Luis1, J.M. Torres1, L.M. Garc´ ıa1, J. Bartolom´ e1, J. Stankiewicz1, F. Petroff2, F. Fettar2†, J.L Maurice2, and
A. Vaur` es2
1Instituto de Ciencia de Materiales de Arag´ on, CSICUniversidad de Zaragoza, 50009 Zaragoza, Spain
2Unit´ e Mixte de Physique CNRS/THALES, UMR 137, Domaine de Corbeville, 91404 Orsay Cedex, France
†Present address: Laboratoire de Nanostructures et Magn´ etisme, DRFML/SP2M, CEA, 38054 Grenoble Cedex 9, France
(February 1, 2008)
We study the magnetic properties of spherical Co clusters with diameters between 0.8 nm and
5.4 nm (25 to 7500 atoms) prepared by sequential sputtering of Co and Al2O3. The particle size
distribution has been determined from the equilibrium susceptibility and magnetization data and
it is compared to previous structural characterizations. The distribution of activation energies was
independently obtained from a scaling plot of the ac susceptibility. Combining these two distribu
tions we have accurately determined the effective anisotropy constant Keff. We find that Keff is
enhanced with respect to the bulk value and that it is dominated by a strong anisotropy induced at
the surface of the clusters. Interactions between the magnetic moments of adjacent layers are shown
to increase the effective activation energy barrier for the reversal of the magnetic moments. Finally,
this reversal is shown to proceed classically down to the lowest temperature investigated (1.8 K).
PACS:75.50.Tt,75.70.i,75.40.Gb,75.30.Pd
1
Page 2
I. INTRODUCTION
Single domain magnetic particles are attractive for applications in data storage. Their properties differ from those
of the bulk magnets1because, as the size of the particles decreases, an increasing fraction of the total magnetic atoms
lies at the surface. The electronic and magnetic structure of these atoms can be modified by the smaller number of
neighbors as compared to the bulk2–4and/or by the interaction with the surrounding atoms of the matrix where the
particles are dispersed. For example, it was shown by Van Leeuwen et al. that the bonding of CO at the surface of
Ni clusters induces quenching of the magnetic moments of those atoms located at the surface.5In some other cases,
the surface layer is oxidized and shows antiferro or spinglass like arrangement of the magnetic moments, which also
leads to a smaller net magnetic moment of the particle.6By contrast, measuring ”bare” particles of Fe, Co and Ni
produced in beams, de Heer and coworkers7found that the net magnetic moment per atom increases as the size of
the cluster decreases, approaching the limiting value for a free atom. In addition, the net anisotropy of the particle
exceeds the bulk value.8,9This excess was recently correlated to the augmentation of the orbital magnetic moment of
the peripheral atoms.10,11
Magnetic nanoparticles are also good candidates for the study of quantum effects in intermediate scales between
the microscopic and the macroscopic classical world.12,13In real systems however, we usually deal with macroscopic
ensembles of particles with different sizes and shapes. The average magnetic properties of these systems come from
intraparticle as well as interparticle phenomena, which are usually difficult to disentangle. Therefore, in this field of
research it is desirable to obtain systems in which each of the parameters, such as the average particle size, the particle
size distribution, the crystalline structure, and the spatial arrangement of the particles, can be varied independently
of each other.
We believe that the work reported here is a step forward in this direction. We present the magnetic characterization
of a new type of systems of Co nanoparticles, embedded in an amorphous matrix of Al2O3, prepared by sequential
deposition of both materials.14–16By varying the deposition time, the diameter of the aggregates can be controlled
between below 1 nm and 7 nm. An important advantage of this preparation method is that it gives a rather ho
mogeneous dispersion of the particles in the matrix. It was also found that, for a range of thicknesses, a relatively
ordered disposition of the particles is obtained, in which they are arranged in layers separated from the adjacent
ones by a controllable distance.16The paper is organized as follows. In the first two sections we briefly describe the
method employed to prepare the samples and their physical characterization. Then, we present our experimental
results. Using the data obtained from ac magnetic susceptibility, zerofield cooled (ZFC) and fieldcooled (FC) mag
netization measurements, and isotherms of magnetization as a function of the field, we have determined the particle
size distribution in samples which have been prepared with different Co deposition times. We compare these results
with available data from a previous structural characterization. This important information is then used to determine
accurately the effective anisotropy constant and its variation with the size of the particles. We have also been able to
separate surface anisotropy effects from the effect of the dipoledipole interaction between the magnetic moments of
the particles. The last section is left for the conclusions.
II. MORPHOLOGY AND STRUCTURE OF THE SAMPLES
Details of the sample preparation and of its structural characterization have already been reported elsewhere.14–16
The Co aggregates were prepared by sputter deposition of Co atoms on a smooth alumina surface. The amount of
deposited Co is given here by the nominal thickness tCothat the deposits would have if they were homogeneous. This
amount was measured by using energydispersive Xray spectroscopy in the transmission electron microscope, and
found to be within less than 5% of the planned dose in all cases. Clusters are formed below the percolation limit
which appears to occur at tCo= 2 nm. On top of each Co layer a new alumina layer of about 3 nm was deposited.
Oxidized Si was used as a substrate. A given sample is usually made by piling up a number N (1 to 100 for the
samples studied here) of these layers.16The deposition rates of both Co and alumina were respectively 0.114˚ A s−1
and 0.43˚ A s−1. It was found that the amount of Co deposited on the surface is larger than the Co mass which forms
clusters visible by transmission electron microscopy (TEM). The relative difference between these quantities increases
as tCodecreases. Therefore, we have in our samples nonaggregated atoms or very small clusters, which contribute to
the magnetic signal of the samples, in addition to Co aggregates. One of the difficulties of the interpretation of the
magnetic data is to separate these two contributions.
Because of the upper alumina layer the aggregatesshow no trace of oxide even after exposure to air. The lack of oxide
was checked by electron energyloss spectroscopy (EELS), Xray photoelectron spectroscopy and Xray absorption
spectroscopy. The atomic structure of the clusters was determined by extended Xray absorption fine structure
(EXAFS) spectroscopy and high resolution TEM.14It was found that the particles bond poorly to the alumina
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matrix, and that the Co crystallizes in the fcc phase for tCo< 1 nm. The presence of a fcc phase in place of the
hcp phase which is the stable phase for bulk Co is not uncommon for small particles. It was theoretically predicted17
and also found experimentally18that the fcc phase becomes more stable below some diameter which depends on the
matrix.
The morphology, size and spatial distribution of the aggregates were also studied using the TEM data. The
aggregates are of approximately spherical shape (at least for tCo< 1 nm). The average diameter ?D? of the particles
increases linearly with tCo. We give in Table I a list of the important parameters obtained from these experiments
for all samples studied. Finally, the TEM pictures reveal a quasiordered arrangement of the Co clusters16that is
induced by the topology of the layers: the clusters of a layer nucleate preferentially in the hollows left by the previous
layer. In each layer, the average distance between the borders of adjacent clusters is of order 2 nm and approximately
independent of tCo.
III. EXPERIMENTAL DETAILS
The magnetic measurements were performed using a commercial SQUID magnetometer. The temperature range
of the measurements was 1.8 K < T < 320 K and magnetic fields up to 5 T could be applied by means of a
superconducting magnet. The ac susceptibility was measured by applying a small ac field (4.5 Oe) to the sample
and using the ac detection option of the same magnetometer. The frequency ω/2π of the ac magnetic field can be
varied continuously between 0.01 Hz and 1.5 kHz. The samples had a rather large diamagnetic signal arising from
the silicon substrate. This contribution was estimated independently by measuring a bare substrate and found to
be linear in field and independent of the temperature. It was subsequently subtracted from all experimental data.
Unless indicated otherwise, the data shown in this paper were measured on samples having more than 20 Co/Al2O3
bilayers in order to maximize their magnetic signals. We have checked for tCo= 0.3 nm and tCo= 0.7 nm that the
variation of the magnetization and of the ac susceptibility with temperature and magnetic field is rather insensitive
to the precise value of N, provided that N is larger than 10 layers.
IV. RESULTS AND DISCUSSION
A. Superparamagnetic blocking
The magnetic dc susceptibility was measured by cooling the samples in zero field (ZFC) or in the presence of the
measuring magnetic field (FC). Typical ZFCFC magnetization curves are plotted in Fig. 1. At high temperatures
the ZFC and FC curves coincide, indicating that the samples behave as superparamagnets. In this region, both curves
follow the CurieWeiss law C/(T −θ). The value of C increases as tCoincreases (see the inset of Fig. 1), as expected
for larger clusters formation as the deposition time of Co increases. The CurieWeiss temperature θ is nearly zero but
for the two samples containing the largest particles. This is brought about by the interaction between the particles,
which we shall consider in a separate section below. At lower temperature, the two curves start to separate. The ZFC
curve shows a maximum at a temperature TBbelow which the magnetic moments are blocked in fixed directions. It is
well known that the phenomenon of blocking is related to the magnetic anisotropy of the particles.19The anisotropy
favors some particular orientations of the magnetic moment, two opposite to each other in the simplest case of uniaxial
anisotropy, which are separated by activation energy barriers U. As the temperature decreases, the number of thermal
phonons of energy equal or larger than U decreases, thus leading to an exponential increase of the time τ needed to
reverse the magnetic moment of a particle19–21
τ = τ0exp(U/kBT)(1)
Here τ0≈ 10−10− 10−13s is an inverse attemptfrequency, which depends on the damping of the magnetic moment
by the phonon or the magnon baths. In this simple picture, the superparamagnetic blocking takes place when τ
equals the measurement time of each experimental point te, thus TB≃ αU/kBln(te/τ0), where α is a constant which
depends on the width of the particle size distribution (more details are given below). We have indeed observed that
TB increases with the Co deposition time, that is, with the average volume of the aggregates. Therefore, we write
U = KeffV , where Keff is an effective anisotropy constant with contributions from the intrinsic magnetocrystalline
anisotropy of the fcc Co and from other sources, such as stress induced anisotropy or surface induced anisotropy. The
dependence of Keff on V will be considered below in section IVC.
As expected, the blocking of the magnetic moments by the anisotropy also leads to a maximum in the temperature
dependence of both the real and the imaginary components of the ac magnetic susceptibility. A typical experimental
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result is shown in Fig. 2. The position of the susceptibility peak shifts towards lower temperatures as the frequency
of the ac magnetic field decreases since teequals 1/ω.
Above TBthe magnetization isotherms are fully reversible because the magnetic moments are in thermal equilibrium.
As shown in Fig. 3, the experimental data measured well above the blocking temperature of each sample collapse
into a single curve when they are plotted as a function of H/T, indicating that the effect of the anisotropy is weak.
Furthermore, pure Langevin curves fit the experimental data reasonably well, which shows that the size distributions
of all these samples are narrow. Below TB, the magnetization shows hysteresis (see Fig. 4) with both the coercive
field Hcand the remanence Mrincreasing as the temperature decreases (see Fig. 5).
We plot in Fig. 6 the lowT values of Mr and of the saturation magnetization Ms as a function of tCo. It is
interesting to note that, for tCo < 0.7 nm, the reduced remanence mr = Mr/Ms is smaller than the value 1/2
predicted by the StonerWolhfarth model.22We attribute the decrease of mrto a paramagnetic contribution, which
adds to that of the blocked particles. A Curie tail shown by the saturation magnetization at the lowest temperatures
(cf Fig. 5) is also related to this extra contribution. The excess paramagnetism arises likely from single atoms or
very small clusters that are formed in the first stages of the preparation process and which do not give rise to further
aggregation.14,15,23It was found that the fraction xparaof Co which is deposited but is not detected by TEM increases
as tCodecreases. Accordingly, mrdecreases as the amount of deposited Co decreases. On the other hand, the sample
with tCo= 1 nm has mr= 0.71, that is, larger than 1/2, likely because of the predominant ferromagnetic coupling
between particles.
It is also remarkable that the average magnetic moment per atom for the whole sample, as obtained from Msof
Fig. 6, is smaller than the value for bulk Co (1.7µB per Co atom) for all samples and that it decreases as tCoand,
thus as the average size of the particles decrease. This dependence is opposite to that observed for free Co clusters in
beams7and also for Co particles of similar size supported in a solid matrix.9,24In those experiments, the measured
magnetic moment per atom exceeded the bulk value and it was found to increase as the diameter of the particles
decreases.
The reduced value of Ms that we measure could be caused by an oxide layer at the surface of the particles,
which orders antiferromagnetically. However, as we mentioned before, we did not find any trace of oxide in EELS
measurements. Moreover, it is known that the exchange interaction between this layer and the magnetic core of the
particles would also induce a net anisotropy on the latter.25This socalled exchange anisotropy leads to a shift of
the hysteresis loops when the sample is cooled down in the presence of a magnetic field. For example, Peng and
coworkers26have recently measured an exchange bias field as large as 10.2 kOe for CoO coated Co clusters having a
diameter of 6 nm and 13 nm. By contrast, as we show in Fig. 4, the hysteresis loops measured after cooling the sample
in zero field or in 5 T from room temperature are nearly identical, thus with no evidence for an antiferromagnetic
order at the surface layer. Thus, we conclude that most of the particles are free from oxidation.
It is however still possible that some of the Co atoms, in close contact with the Al2O3matrix, have a weak chemical
link with it. This chemical bonding can reduce the number of unpaired electrons and then quench the magnetic
moment of the metal atom, as was shown by van Leeuwen and coworkers.5From our data, it is not possible to
determine whether the atoms involved in the reduction of the average magnetic moment are located at the periphery
of the particles or are those atoms which do not form aggregates, because the relative concentration of both increases
as the average size of the clusters decreases.
Therefore, in what follows, we approach the problem in a different way. As a starting point of the analysis we consider
that the spheres have the bulk magnetization: Msb = 1.7µB per Co atom, whereas the missing magnetic moment
is exclusively attributed to the paramagnetic Co fraction. The contribution of the clusters to the net saturation
magnetization of each sample then equals (1 − xpara)Msb. It was obtained by subtracting the lowT paramagnetic
tail from the total Ms. In this way, xparais also estimated. We list the results in Table I. The lowT paramagnetic
magnetization is found to be compatible with a free spin 1/2 for all samples, which indicates that the isolated atoms
have in average only one unpaired electron.
In order to fit the magnetic data, it is necessary to know the fraction xpara of paramagnetic atoms and their
magnetic moment.
B. Determination of the particle size distribution
In this section we will try to determine the particle size distribution from the equilibrium magnetic properties of each
sample and compare it with the results obtained by TEM. In Fig. 3 we have plotted the equilibrium magnetization M
of two different samples having tCo= 0.3 nm and 0.7 nm, respectively. We recall that, for a set of magnetic moments
µ without anisotropy, M(H,T,µ) = MsL(µH/T), where L denotes the Langevin function. If the anisotropy energy is
taken into account, there is no analytical expression for M(H,T,µ) and the shape of the magnetization curve deviates
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from the pure Langevin form when U/kBT is large. However, it can still be evaluated numerically, as was described
in Ref.27. As we said above, the good scaling on a single curve of data measured at different temperatures indicates
therefore that the influence of the anisotropy is not very important.
For a real sample, we have to average M(H,T,µ) over the appropriate distribution of particle’s sizes. Comparing
the calculated magnetization to the experimental data we shall try next to get information about this distribution. In
order to directly compare our results with those obtained previously by TEM, we define g(D) as the distribution of
number of particles having a diameter equal to D. For spherical particles µ = πMsbD3/6, where Msbis, for the reasons
given in the previous section, taken as equal to the saturation magnetization of bulk Co. We fit the experimental
using the following expression:
M(H,T) = xparaµBtanh
?µBH
kBT
?
+ (1 − xpara)Msb
?g(D)V M(H,T,µ)dD
?g(D)V dD
(2)
taking for g a Gaussian distribution.
For each sample, we fit only data measured at temperatures for which the two calculations, with and without
anisotropy, give approximately the same result. We give an example of this method in the lower plot of Fig. 3 for a
multilayer with tCo= 0.3 nm. Above 30 K, the calculations performed with and without anisotropy almost coincide.
For lower temperatures, close to TB= 8.6 K, the experimental magnetization starts to deviate from the pure isotropic
behavior, as happens for the data measured at T = 12 K. Even then, the experimental data are rather well reproduced
by our calculations if we use the value of Keff determined from the blocking of the ac susceptibility (see section IVC
below). A list of the ?D? values obtained from the fit for all samples is given in Table I. We find that ?D? increases
with tCo, as expected. For most of the samples it is however larger by ten to forty percent than the values that were
previously found by TEM. This discrepancy can be ascribed to the fact that the TEM experiments were performed on
single layers deposited on a special carbon substrate, whereas we have measured multilayers prepared on a Si oxide.
However, we have also measured a monolayer with tCo= 0.7 nm and obtained almost the same magnetization results
(see the upper plot of Fig. 3) as for a multilayer. An alternative explanation is that the saturation magnetization
of the smallest particles is enhanced with respect to the bulk, as was found in similar systems of Co clusters,7,9,24.
However, even if we had used the maximum value of 2.3µB per Co atom, which was found by Respaud et al., ?D?
would have decreased by only a 10 percent, that is, within the uncertainty of the fitting procedure. In order to get
the same diameters that were observed by TEM, Msshould be as large as 3µBto 4µBper Co atom for the smallest
clusters.
The fit of the magnetization curves is more sensitive to the value of ?D? than to the width of the distribution σ. In
fact, it is possible to obtain a reasonably good fit by using a single Langevin curve with almost the same value of ?D?
(see Fig. 3, upper plot). In order to get a better estimation of σ, we have also fitted the equilibrium susceptibility, as
obtained from the low field dc or ac measurements well above TB, using the following expression
χeq=
?g(D)V?M2
sbV/3kBT?dD
?g(D)V dD
.(3)
This formula is valid also for particles with uniaxial magnetic anisotropy if, as it is the case for our samples, the
anisotropy axes are not oriented.28It turns out that the equilibrium susceptibility is very sensitive to the presence
of large particles in the distribution and therefore to σ, as it is shown in the inset of Fig. 1. For this reason, the
contribution of the paramagnetic moments to χeq can be neglected in all cases. The values for ?D? and σ that are
given in Table I are those which reproduce best both the equilibrium magnetization isotherms and the equilibrium
susceptibility. The width of the distribution is found to be rather constant and in good agreement with the value
found previously by TEM. The slight increase of σ as the average size of the aggregates decreases was also observed
in the TEM data.14In conclusion, the codeposition of Co and Al2O3 gives us Co clusters of controllable size and
with a narrow and nearly constant distribution of diameters.
C. Magnetic anisotropy
Next, we want to study the dependence of the effective anisotropy on the cluster size. The anisotropy of the particles
can be estimated by comparing the average activation energy ?U? to the average volume π/6?D?3. Usually ?U? is
estimated as ?U? = 25kBTB, where TBis the temperature of the maximum of the ZFC susceptibility. However, this
procedure leads to an overestimation of the anisotropy because TBdepends not only on ?U? but also increases with
σ.29,30
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In order to get more reliable values we need a way to obtain the full distribution of activation energies f(U), and
then to find which value of U corresponds to particles with a diameter equal to ?D?. Fortunately, f(U) can be
directly determined from ac susceptibility data measured near the superparamagnetic blocking temperature TB.30,31
As mentioned above, the blocking occurs when the average relaxation time becomes of the order of 1/ω. It is therefore
clear that the temperature dependence of χ′and χ′′near TBis determined by the distribution of U among the particles.
In order to relate χ′and χ′′to the distribution f(U) it is common to assume that those particles having U > Ubare
fully blocked and the ones that fulfill the opposite condition are in equilibrium. This hypothesis is reasonable because
the relaxation time depends exponentially on U according to Eq. 1. For noninteracting particles, the relation that
we were looking for reads as follows,
χ′≃
?Ub
0
χeq(U,T)f(U)dU +2
3
?∞
Ub
χ⊥(U,T)f(U)dU (4)
χ′′≃π
2kBTχeq(T,Ub)f (Ub), (5)
where Ub= kBT ln(1/ωτ0) is the activation energy of those particles having exactly τ = teat a given temperature.
χeq= M2
susceptibility, respectively.28It follows from Eq. 5 that Uf(U) can be directly determined by plotting χ′′versus the
scaling variable Ub. In Fig. 7 we show the result for a multilayer with tCo= 0.7 nm. Similar results were obtained
for the other samples.
It is important to note here that f(U) is the fraction of the total magnetic volume occupied by particles having
the activation energy U, since the susceptibility is mainly dominated by the contribution of the largest particles.
Contrary, g(D) gives instead the number of particles of a given size. The two distributions are related as follows
sbV/3kBT and χ⊥= M2
sb/2Keff are the equilibrium susceptibility and the reversible (high frequency limit)
f(U) =
V g(D)
(dU/dD)?∞
0V g(D)dD
(6)
For spherical particles U(D) = Keff(π/6)D3. Therefore f(U) is, apart from normalization factors, proportional
to (U/Keff)1/3g?(6U/πKeff)1/3?. Using this relationship and taking as above a gaussian g(D), it is possible to fit
χ′′(Ub). The anisotropy constant is then simply the ratio between U(?D?) and V (?D?). Although the fit is rather
good, we find that the function g that is extracted in this way from f(U) (or χ′′) is systematically narrower than the
size distribution obtained previously using the equilibrium magnetization and magnetic susceptibility. We will discuss
later on the possible physical origin of this discrepancy.
Before we comment on the variation of the anisotropy with the size, we would like to show that the distribution
f(U) can also be obtained by a different method, which makes use of the ZFC and FC dc susceptibility curves
measured at low enough magnetic fields. The difference between the ZFC and FC magnetization curves stems from
the different contribution that the blocked particles make to each of them. Neglecting the weak variation of Msbwith
T, this contribution only depends on T via the critical energy Ubwhich determines the relative number of blocked
and superparamagnetic particles at a given temperature. Using the same approximation which led to Eqs. 4 and 5
for the ac susceptibility, it is possible to show that
∂ (MFC− MZFC)
∂T
= −Mirr(Ub,T,H)f (Ub).(7)
where Mirr= Meq−Mrevand Mrevis the magnetization brought by the reversible rotation of the magnetic moments.
This expression is valid provided that the applied magnetic field is much smaller than the anisotropy field Hk =
2Keff/Msb, as it is actually the case in our experiments. If this condition was not fulfilled, the activation energy
would be a function of the field and of its orientation with respect to the easy axes of the particles. It is also possible
to approximate Mirr(Ub) ≃ χeq(Ub)H. Therefore, Eq. 7 gives an independent method to determine f(U). We plot
in Fig. 6 the results obtained for an applied field of 10 Oe, which are in good agreement with the ac susceptibility
data. In the same figure, data obtained for H = 100 Oe are also shown. In this case the maximum of the distribution
shifts towards lower values of Ub, indicating that the activation energy decreases in a magnetic field. In addition, the
distribution function broadens a bit as a result of the random orientation of the easy axes.
We now come back to our main goal. The anisotropy constant is plotted in Fig. 8 as a function of the average
diameter of the aggregates. It is interesting to compare these experimental data with the constant Keff that is
estimated using only the intrinsic magnetocrystalline anisotropy of bulk Co. For hcp Co, the stable phase for large
particles, Keff equals the intrinsic uniaxial anisotropy constant K = 4.3 × 106erg/cm3. However, the structural
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characterization of all samples studied here shows that they crystallize in the fcc phase. Therefore, we would expect
that the intrinsic anisotropy of the particles in our samples would be smaller than for hcp Co. For cubic anisotropy29,32
Keff = K/4, where K is the second order intrinsic anisotropy constant. Taking K = 2.8 × 106erg/cm3for fcc Co,
this gives Keff = 7 × 105erg/cm3. Therefore, the values that we find for all samples are almost one to two orders
of magnitude larger than expected for magnetocrystalline anisotropy. Furthermore, Keff is observed to increase as
?D? decreases. The size dependence of the effective anisotropy follows approximately the following phenomenological
expression
Keff= K∞+6Ks
?D?
(8)
with K∞ = 5(2) × 105erg/cm3and Ks = 3.3(5) × 10−1erg/cm2. This result is robust in the sense that it does
not change qualitatively if we use the average diameter found by TEM, instead of the values obtained from the
magnetization data. The first term is close to K/4 and can therefore be identified as the contribution of the intrinsic
anisotropy. The second one is proportional to the fraction of atoms located at the periphery of the particles, which
can be more than 80 % of all Co atoms for the smallest clusters studied here. Our experimental results indicate then
that there exists a rather large contribution of the surface of the particles to the net anisotropy.
The enhancement of the magnetic anisotropy of nanometer sized metallic particles with respect to the bulk has
been previously reported by several authors8,9,24. For Co particles with diameters varying between 4.4 and 1.8 nm,
Chen and coworkers9obtained Keff which increases from about 5 × 106erg/cm3to about 3 × 107erg/cm3. These
values are even larger than ours. However, they are of the same order as the values that would have been obtained
if we had used the temperature of the maximum of the ZFC susceptibility, as it was done by the authors. More
recently, Respaud et al.24studied the anisotropy of Co particles of 1.5 and 1.9 nm by fitting the whole ZFC and FC
magnetization curves, a method that can be considered as equivalent to ours. They found Keff≃ 8.3× 106erg/cm3
and Keff ≃ 7.3 × 106erg/cm3, respectively, in reasonably good agreement with our data. The existence of a large
surface anisotropy in metallic particles is thus well established experimentally.
The origin of this extra anisotropy has been related to the modification of the electrostatic and exchange interactions
of the atoms located at the surface,4,33,34which depends largely on whether the surface is oxidized or not. However,
as we argued above, the characterization of our samples by EXAFS, EELS, and XPS does not indicate the presence
of an oxide layer.15The same conclusion is derived from the hysteresis loops measured below TB. Therefore, we
have to consider how the properties of a ”bare” metallic surface are modified with respect to the bulk. The value
of Ks that we have found is actually comparable to the perpendicular anisotropy measured in free Co surfaces.35
It is commonly accepted that this perpendicular anisotropy is related to the appearance of a large orbital magnetic
moment on these atoms.36The 3d electrons become more localized at the surface and the localization gives rise to an
increase of the orbital moment. The same theoretical interpretation can be applied to the atoms at the periphery of
small metallic clusters.4In this case, the enhanced anisotropy at the surface extends to the inner atoms via the strong
exchange interaction with them, which leads to an increase of the average anisotropy of even spherical clusters.37This
interpretation has been confirmed by Xray magnetic dichroism experiments performed on Au/Co/Au layers35and
more recently also on Co disklike aggregates supported on Au surfaces.10. It was found that the orbital component
mL of the total magnetic moment scales with the fraction of atoms located at the surface of the aggregates. For
spherical clusters, as the ones studied here, we expect then that mL ∝ 1/?D?, dependence that we have indeed
observed for Keff. We therefore conclude that the observed increase of Keffis likely due to the increasingly localized
character of the 3d electrons of the atoms located at the surface.
Once the particle size distribution and the anisotropy are known, it is possible to predict the timedependent
magnetic response of the samples and compare it to the experiment. Examples of these calculations are compared to
the experiments in Figs. 1, 3 and 5. The calculations account very well for the experimental data measured above TB,
as expected. They also reproduce in Fig. 1 the deviation of the FC susceptibility from the equilibrium susceptibility
that takes place below 5 K. However, they reproduce neither the position nor the shape of the maximum of the ZFC
susceptibility. Another example of this discrepancy is shown in Fig. 9, where we plot the experimental χ′for a
multilayer with tCo= 0.3 nm and the values calculated (dotted line) with Eq. 4. Again, the width of the blocking
transition is clearly overestimated by the calculations.
We recall here that we have found that the activation barriers distribution is systematically narrower than the
size distribution for all samples. As an example, in Fig. 10 we plot the size distribution g(D) of a multilayer with
tCo= 0.7 nm extracted from χ′′and directly observed by TEM. The horizontal scale for the former distribution is
(6Ub/πKeff)1/3, with Keff= 107erg/cm3. It is tempting now to attribute the ”narrowing” of the blocking transition
to the effect of the surface anisotropy. When Ks/D ≫ K∞ then U ≈ KsS, where S = πD2is the surface of the
particle. It follows then from Eq. 6 that f(U) ∝ D2g(D) and the width of the distribution of activation energies must
then be smaller than when U ∝ V . Figure 10 shows indeed that when the same susceptibility data are represented
7
Page 8
versus the variable (Ub/πKs)1/2the ensuing size distribution is in better agreement with what it is found by TEM or
from the equilibrium magnetization and susceptibility. In this way, we also obtain Kswhich turns out to be between
0.2 and 0.3 erg/cm2for all samples. This value can be then used to recalculate the ac susceptibility and the ZFC
magnetization. We find that the calculations performed with the same parameters σ and ?D? as before (see Table I)
but taking U ∝ D2are in much better agreement with the experiment (see Figs. 1 and 9). Although the width of
the of size distribution is not always accurately determined, it seems that the influence of the surface anisotropy also
modifies the shape of the susceptibility peak at the blocking. We conclude that the dynamical response of very small
particles is therefore determined by the special physical properties of the atoms which are located at their surface.
D. Influence of the number of layers: dipoledipole interaction between the particles
There has been some debate during the last years about the effect that the dipoledipole interaction between
magnetic nanoparticles has on their relaxation times. Shtrikman and Wolfarth38and later Dormann et al.39predicted
that the effective activation energy increases by an amount that depends on the number and spatial arrangement of
the neighbor particles. By contrast, in the model proposed by S. Mørup and E. Tronc40the interaction between the
particles leads to a lower U. The experimental validation of one of these two models is complicated because, for some
preparation methods, it is difficult to vary the density of particles in the sample without modifying the distribution
of particle’s sizes.39A different approach is to dissolve the particles in a fluid and to change the concentration by
varying the amount of solvent. However, it is possible that the particles agglomerate in the fluid because of their
mutual interaction, so that the interaction with the nearest neighbors is not greatly affected.41
The preparation method of our samples presents a number of advantages. We have seen that the average size can
be controlled by changing the deposition time, but also the packing of the particles can be controlled. The TEM
images show that the clusters in a layer do not agglomerate and, furthermore that the deposition of several layers of
Co and Al2O3leads to a selforganized spatial arrangement of the particles (see ref.16). For a multilayer each cluster
has, in average, six nearest neighbors in the same plane, three above and another three below it. For tCo= 0.7 nm,
the average distance between nearest Co clusters in the same layer is Λ?≃ 5.4 nm, whereas the distance to nearest
neighbors in adjacent layers is λ = 4.5 nm.16In this section, we compare the relaxation rate of two samples having
both tCo= 0.7 nm (?D? ≃ 3 nm), but very different number of layers, namely 30 and only one. By going from a
monolayer to a multilayer we certainly expect that the average energy of interaction of a particle with the others
changes. The interaction energy between particles in adjacent layers is the largest and of the order of µ2/λ3≈ 40 K.
By contrast, in a sample with a single layer, each particles has, in average, only six neighbors coupled by a weaker
interaction (µ2/Λ3
?≈ 20 K).
In order to attribute any difference between the two samples to the effect of the interparticle interactions, it is very
important to check beforehand that the sizes of the aggregates are the same in both. We showed in Fig. 3 that the
equilibrium magnetization curves of the two samples are almost identical, and we compare in Fig. 11 the inverse
of their ac susceptibility curves. Above TB, the susceptibility follows the CurieWeiss law, with identical values of
C, which confirms that ?D? and σ are practically the same. By contrast, the CurieWeiss temperature θ is about 2
times smaller for the monolayer, indicating that the average interparticle interaction is notably reduced. It is also
apparent that the blocking temperature of the monolayer is smaller than that of the multilayer. As we have done
before, the activation energy of the two samples can be compared by plotting χ′′measured at different frequencies
as a function of the scaling variable Ub. This comparison is shown in Fig. 12. The maximum of the curve for the
monolayer is clearly shifted towards lower values of Ubwith respect to the maximum obtained for the multilayer. Our
data give strong evidence that the interaction between the aggregate layers tends to increase the activation energy
of each particle, by an amount of about 200 K. This difference is of the same order of magnitude as the interaction
energy with the six nearest neighbors in the multilayer. We also find that the relative width of Uf(U) has the same
value for the two samples, which confirms again that the distribution of particle’s sizes is the same.
E. Magnetic relaxation at low temperatures
In the previous sections, the reversal of the magnetic moments has been treated as a classical process assisted
by the interaction with a thermal bath. However, taken as a quantum variable, the spin of a magnetic particle
S = MsV/gµBcan in principle flip also by quantum tunneling across the barrier if the effective Hamiltonian contains
terms which deviate from the uniaxial symmetry.13This possibility is very attractive because it would show the
existence of quantum effects at the intermediate scale between the microscopic and the macroscopic worlds. Quantum
relaxation can dominate over the thermal activation at very low temperatures, when the thermal population of the
8
Page 9
first excited state doublet ±(S − 1) becomes negligible, and should lead to a saturation of the relaxation rate to a
nearly temperature independent value.42Such a saturation has indeed been observed in some systems of single domain
particles in the past.12,13,26,43
In this section, we present measurements of the relaxation of the remanent magnetization of an initially saturated
sample. We have chosen the sample with the smallest Co clusters for two reasons; first, because the rate for quantum
relaxation must be the largest for these clusters of only about 50−100 atoms; and second, because this sample shows
the strongest anisotropy. The separation of the two lowest lying state doublets, which is roughly given by Ω0≈ gµBHk
is then about 3 K, thus larger than the lowest temperature that our magnetometer can reach (Tmin= 1.7 K). We
have measured the decay of the magnetization of the sample that takes place after a magnetic field of 5 T is switched
off at different temperatures. The decay of Mr is approximately logarithmic in time. An important advantage of
recording the relaxation at zero field is that it can then be easily calculated using our knowledge of the activation
energies distribution. At zero field, the equilibrium magnetization is zero for all particles. Therefore, using the same
approximation as before, the time dependent magnetization is given by
M(t,T) =Ms
2
?∞
Ub
f(U)d(U) (9)
where we have made the reasonable approximation that the magnetic moments of the particles are initially saturated
by the magnetic field. The factor 1/2 arises from the reversible rotation of the magnetic moments for a random
orientation of the easy axes, as in the StonerWolhfarth model.22As pointed out by Labarta et al.,44if the magnetic
moments flip by a thermally activated process the relaxation curves measured at different temperatures should scale
when plotted as a function of Ub. This plot also gives a picture of the relaxation at very long times, which are not
experimentally accessible. Our experimental data, which we plot in Fig. 13 do indeed show a rather good scaling for
the same τ0= 10−13s that was obtained from the shift of the maximum of χ′′with frequency. The full line in the
same figure was calculated with Eq. 9 using the distribution f(U) that we determined with the method described
in section IVC. The scaling of the data confirms that the relaxation mechanism is classical (not tunneling) down to
T = 1.7 K.
In the inset of the same Fig. 13 we show the temperature dependence of the socalled magnetic viscosity Sr,
determined as the slope of the Mr vs ln(t) curves. Below about 2.5 K, Sr does not vary much with T. We note
however that, according to Eq. 9, the magnetic viscosity is just
Sr≡
∂M
∂ lnt= −kBTMs
2f (Ub)(10)
and it is therefore proportional to f(U). The apparent saturation of S measured between 1.7 K and 2.5 K just reflects
the shape of the distribution f(U), and it is indeed rather well described by the ”classical” calculation. These data
give an example of how important it is to have information about f(U) in order to adequately interpret the relaxation
data.45
V. FINAL REMARKS AND CONCLUSIONS
We have presented a detailed and extensive study of the magnetic properties of Co aggregates prepared by sequential
deposition of Co and Al2O3. This preparation method enables us to control both the average size and the number
of layers independently. We have shown that the distribution of activation energies can be accurately determined
from ac susceptibility and ZFCFC magnetization measurements. We have investigated the variation of the effective
anisotropy as the size of the aggregates decreases from about 5 nm to below 1 nm. We find that Keffscales with the
fraction of atoms located at the periphery of the aggregates. The strength of the surface anisotropy is of the same order
of what is found for free Co surfaces and we therefore attribute it to the increase of the orbital magnetic moment of
these atoms. Furthermore, the activation energies distribution resembles the distribution of particle’s surfaces rather
than the volume distribution. For these small clusters, it is therefore more appropriate to write U = KsS than the
”traditional” U = KeffV . Using the distributions of sizes and of activation energies that we have determined, we are
able to give a quantitative account of all the equilibrium and timedependent experimental quantities. We have also
shown that the activation energy increases when the average number of nearest neighbors per particle increases, in
agreement with the model of Dormann et al. Finally, the decay of the remanent magnetization of clusters containing
only about 50 to 100 atoms is shown to proceed via a thermally activated mechanism down to the lowest temperatures
investigated.
9
Page 10
ACKNOWLEDGMENTS We would like to thank Dr C. Paulsen and Dr. J. Carrey for assistance with some of
the experiments reported in this work. This work has been partly funded by Spanish Grant MAT 99/1142 and the
European ESPRIT contract ”MASSDOTS”.
∗To whom all correspondence should be addressed. Email address: barto@posta.unizar.es
1J. L. Dormann, D. Fiorani, and E. Tronc, Adv. Chem. Phys. 98, 283 (1997).
2P. V. Hendricksen, S. Linderoth, and P.A. Lindgard, J. Mag. Mag. Mater. 104107, 1577 (1992).
3B. V. Reddy, S. N. Khanna, and B. I. Dunlap, Phys. Rev. Lett. 70, 3323 (1993).
4G. M. Pastor, J. DorantesD´ avila, S. Pick, and H. Dreyss´ e, Phys. Rev. Lett. 75, 326 (1995).
5D. A. van Leeuwen, J. M. van Ruitenbeek, L. J. de Jongh, A. Ceriotti, G. Pacchioni, O. D. H¨ uberlen, and N. R¨ osch, Phys.
Rev. Lett. 73, 326 (1995).
6R. H. Kodama, A. E. Berkowitz, E. J. McNiff, Jr., and S. Foner, Phys. Rev. Lett. 77, 394 (1996).
7M. L. Billas, A. Chatelain, and W. A. de Heer, Science 265, 1682 (1994).
8F. Bødker, S. Mørup, and S. Linderoth, Phys. Rev. Lett. 72, 282 (1994).
9J. P. Chen, C. M. Sorensen, K. J. Klabunde, and G. C. Hadjipanayis, Phys. Rev. B, 51, 11527 (1995).
10H. A. D¨ urr et al., Phys. Rev. B 59, R701 (1999).
11K. W. Edmons et al., Phys. Rev. B 60, 472 (1999).
12Quantum Tunneling of Magnetization, edited by L. Gunther and B. Barbara (Kluwer, Dordrecht, 1995).
13E. M. Chudnovsky and J. Tejada, Macroscopic Quantum Tunneling of the Magnetic Moment. (Cambridge University Press,
1998).
14J. L. Maurice, J. Bri´ atico, J. Carrey, F. Petroff, L. F. Schelp, and A. Vaur` es, Phil. Magazine A 79, 2921 (1999).
15J. Bri´ atico, J.L. Maurice, J. Carrey, D. Imhoff, F. Petroff, and A. Vaur` es, Eur. Phys. J. D 9, 517 (1999).
16D. Babonneau, F. Petroff, J.L. Maurice, F. Fettar, and A. Vaur´ es, Appl. Phys. Lett. 76, 2892 (2000).
17O. Kitakami, H. Sato, Y. Shimada, F. Sato, and M. Tanaka, Phys. Rev. B 56, 13849 (1997).
18C. G. Granquist and R. Buhrman, J. Apll. Phys. 47, 2200 (1976).; W. Gong, H. Li, Z. Zhao, and J. Chen, J. Appl. Phys.
69, 5119 (1991); M. E. Mohenry, S. A. Majetich, J. O. Artman, M. DeGraef, and S. W. Staley, Phys. Rev. B 49, 11358
(1994).
19L. N´ eel, C.R. Acad. Sci. Paris 228, 664 (1949).
20W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963).
21W. T. Coffey, D. S. F. Crothers, Yu P. Kalmykov, E. S. Massawe, and J. T. Waldron, Phys. Rev. E 49, 1869 (1994).
22E. C. Stoner and E. P. Wohlfarth, Phil. Trans. Roy. Soc. A 240, 599 (1948).
23J. Carrey, J.L. Maurice, P. Jensen, and A. Vaur` es, Appl. Surf. Sci. 164, 48 (2000).
24M. Respaud et al., Phys. Rev. B 57, 2925 (1998).
25A. E. Berkowitz and K. Takano, J. Mag. Mag. Mater. 200, 552 (2000).
26D. L. Peng, K. Sumiyama, T. Hihara, and S. Yamamuro, Appl. Phys. Lett. 75, 3856 (1999); D. L. Peng, K. Sumiyama, T.
Hihara, S. Yamamuro, and T. J. Konno, Phys. Rev. B 61, 3103 (2000).
27M. Hanson, C. Johanson, and S. Mørup, J. Phys.: Condens. Matter 5, 725 (1993).
28M. I. Shliomis and V. I. Stepanov, Adv. Chem. Phys. 87, 1 (1994).
29J. I. Gittleman, B. Abeles, and S. Bosowski, Phys. Rev. B 9, 3891 (1974).
30F. Luis, E. del Barco, J. M. Hern´ andez, E. Remiro, J. Bartolom´ e, and J. Tejada, Phys. Rev. B 59, 11837 (1999).
31F. Luis, J. Bartolom´ e, J. Tejada, and E. Mart´ ınez, J. Mag. Mag. Mater. 157158, 266 (1996).
32C. P. Bean and J. D. Livingston, J. Appl. Phys. 30, 120 S (1959).
33P. Bruno, Phys. Rev. B 39, R865 (1989).
34R. H. Kodama and A. E. Berkowitz, Phys. Rev. B 59, 6321 (1999).
35D. Weller et al., Phys. Rev. Lett. 75, 3752 (1995).
36Dingsheng Wang, R. Wu, and A. J. Freeman, Phys. Rev. Lett. 70, 869 (1993).
37D. A. Dimitrov and G. M. Wysin, Phys. Rev. B 50, 3077 (1994); ibid 51, 11947 (1995).
38S. Shtrikman and E. P. Wholfarth, Phys. Lett. A 85, 467 (1981).
39J. L. Dormann, L. Bessais, and D. Fiorani, J. Phys. C 21, 2015 (1988).
40S. Mørup and E. Tronc, Phys. Rev. Lett. 72, 3278 (1994).
41R. W. Chantrell, G. N. Coverdale, M. El Hilo, and K. O’Grady, J. Mag. Mag. Mater. 157158, 250 (1996).
42N. V. Prokof’ev and P. C. E. Stamp, J. Low Temp. Phys. 104 143 (1996).
43Ll. Balcells, J.L. Tholence, S. Linderoth, B. Barbara, and J. Tejada, Z. Phys. B 89, 209 (1992); C. Paulsen et al., Europhys.
Let. 19, 643 (1992); J. Tejada et al., J. Appl. Phys. 73, 6952 (1993); X. X. Zhang and J. Tejada, J. Appl. Phys. 75, 5637
(1994); R. H. Kodama, C. L. Seaman, A. E. Berkowitz, and B. M. Maple, J. Appl. Phys. 75, 5639 (1994); M. M. Ibrahim, S.
10
Page 11
Darwish, and M. S. Seehra, Phys. Rev. B 51, 2955 (1995); X. X. Zhang, J. M. Hern´ andez, J. Tejada, and R. F. Ziolo, Phys.
Rev. B 54, 4101 (1996).
44A. Labarta, O. Iglesias, Ll. Balcells, and F. Bad´ ıa, Phys. Rev. B 48, 10240 (1993).
45B. Barbara and L. Gunther, J. Mag. Mag. Mater. 128, 35 (1993).
11
Page 12
(a)(b)
tCo (nm)
0.1
0.2
0.3
0.4
0.7
1
?D?a(nm)
σa
?D?b(nm)
0.8(1)
1.3(1)
1.4(1)
2.2(1)
3.1(3)
5.2(3)
σb
xpara
0.7
0.7
0.22
0.24
0.25
0.13
0.35(5)
0.3(1)
0.32(5)
0.2(1)
0.2(1)
0.25(5)
0.83(20)
1.4(3)
1.4(3)
2.9(6)
4.2(8)
0.3
0.3
0.22
0.23
0.27
TABLE I. Parameters of the gaussian distribution of particle’s sizes obtained by TEM (a) and from the fit of the magneti
zation data (b). The width σ of the distribution is given in units of the average diameter. The last column gives the estimated
fraction of Co atoms which do not aggregate in particles.
12
Page 13
Figure captions
Figure 1. dc susceptibility of a multilayer with tCo= 0.1 nm and N = 100 measured with a field of 100 Oe; open
symbols, FC; closed symbols, ZFC. The lines represent the results of calculations performed with the parameters of the
size distribution given in Table I: dashed line, equilibrium susceptibility; dotted line, ZFC susceptibility calculated
taking U = KeffV and Keff = 2.4 × 107erg/cm3; full lines, ZFC and FC susceptibilities calculated for surface
anisotropy with Ks= 0.3 erg/cm2. Inset: Inverse suceptibility of three multilayers: (a), tCo= 0.1 nm, N = 100; (b),
tCo= 0.3 nm, N = 40; (c), tCo= 0.7 nm, N = 30. The lines represent the equilibrium susceptibility of (b) calculated
for ?D? = 1.4 nm and three values of σ: 0.25 (upper curve), 0.3 (medium curve), and 0.35 (lower curve).
Figure 2. Real and imaginary parts of the ac susceptibility of a multilayer with tCo= 0.3 nm and N = 40.
Figure 3. Equilibrium magnetization of multilayers with tCo= 0.7 nm (a) and tCo= 0.3 nm (b), measured at
different temperatures. The lines represent the calculated results. (a): dotted line, pure Langevin curve for D = 3.1
nm; full lines, results calculated averaging a Langevin curve over a Gaussian distribution of sizes (see Eq. 2) with
σ = 0.2 and three different values of the average diameter. (b): full line, as in the upper picture for σ = 0.32 and
?D? = 1.4 nm; dotted lines, equilibrium magnetization calculated for T = 12 K and T = 30 K with the same size
distribution but for uniaxial anisotropy with U = πKsD2and Ks= 0.2 erg/cm2.
Figure 4. Hysteresis loop of a multilayer with tCo= 0.1 nm measured at T = 2 K after cooling the sample in zero
field or in 5 T from room temperature.
Figure 5. Left axis: Temperature dependence of the remanent magnetization and of the saturation magnetization
(measured with H = 50 kOe) of a multilayer. The lines are calculated with Eqs. 2 and 9, respectively using the
parameters given in Table I and the distribution f(U) estimated from the blocking of the ac susceptibility. Right
axis: Temperature dependence of the coercive field of the same sample.
Figure 6. Variation of the reduced remanent magnetization mr (open symbols, right axis) and of the low tem
perature saturation magnetization Ms (closed symbols, left axis) with the amount of deposited Co for all samples
studied.
Figure 7. Imaginary part of the susceptibility of a multilayer with tCo= 0.7 nm and N = 30 plotted as a function
of the scaling variable UB/kB = T ln(1/ωτ0), with τ0 = 10−13seconds. The full line is a fit according to Eq. 5
taking a Gaussian for g(D). Results obtained as explained in the text (cf Eq. 7) from ZFCFC magnetization curves
measured with two different magnetic fields are also shown for comparison.
Figure 8. Sizedependence of the effective anisotropy constant for all samples investigated. The full line is a best
squares fit of the data to Eq. 8.
Figure 9. Real part of the susceptibility of a Co multilayer with tCo = 0.3 nm and N = 40 measured for two
different frequencies. The dotted line is calculated for ω/2π = 0.1 Hz with Eq. 4 using the parameters of Table I and
taking U = πKeffD3/6, with Keff = 1.15 × 107erg/cm3. The full lines are calculated taking U = πKsD2, with
Ks= 2 × 10−1erg/cm2.
Figure 10. The size distribution determined by TEM is compared to the distributions obtained from χ′′for
two limiting cases where the anisotropy is either dominated by the intrinsic (volume) contribution (full line) or by
the surface anisotropy (dotted line). The scaling in the horizontal axis gives respectively Keff = 107erg/cm3and
Ks= 0.33 erg/cm2.
Figure 11. Inverse ac susceptibility of two samples with the same tCo= 0.7 nm but different number of layers:
open symbols, N = 30; full symbols, N = 1.
Figure 12. Scaling plot of χ′′for two samples with tCo = 0.7 nm but different number of layers. For both
τ0= 10−13seconds.
Figure 13. Timedependent remanent magnetization of a (Co 0.1 nm Al2O3 3 nm)100 multilayer plotted as a
function of the scaling variable UB/kB = T ln(te/τ0) with τ0 = 10−13seconds. The inset shows the temperature
dependence of the magnetic viscosity. The full lines are calculated according to Eqs. 9 and 10.
13
Page 14
010 2030
0.0
0.1
0.2
0.3
0.4
0 204060 80100 120 140
0
2
4
6
8
10
12
χ(emu/cm
3 Co)
T(K)
PRB F. Luis et al. Fig. 1
c
b
a
1/χ (cm3 Co/emu)
T(K)
Page 15
0 1020 30
0.0
0.4
0.8
1.2
1.6
T(K)
χ"
χ'
0.1 Hz
1 Hz
10 Hz
100 Hz
χ(emu/cm
3 Co)
PRB F. Luis et al. Fig. 2