# Enhancement of the magnetic anisotropy of nanometer-sized 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).

**0**Bookmarks

**·**

**80**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**Low temperature measurements of specific heat and magnetic properties were performed on granular Cu90Co10 ribbons prepared by melt spinning technique. The thermal and magnetic behavior of an as-quenched sample is compared to that of an annealed sample at 500 °C for 1 h. It was found that the electronic specific heat γ decreases about 50% and the magnetic properties change from spin-glass to superparamagnetic with the annealing process. These results are interpreted and discussed considering the role of isolated Co atoms in the RKKY interaction among small nanoparticles.Journal of Magnetism and Magnetic Materials 12/2014; 370:116–121. · 2.00 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Here, we report a phase-solution annealing-induced structural transition of 7 nm-Co nanocrystals from the fcc polycrystalline phase to the hcp single-crystalline phase. For any annealing temperature, contrary to what was down in our previous paper ( Langmuir 2011, 27, 5014), the same solvent (octyl ether) is used preventing any change in adsorbates related to various solvents on the nanocrystal surface. A careful transmission electron microscopy study, combined with the electron diffraction, confirms the nanocrystal recrystallization mechanism. The annealing process results in neither coalescence nor oxidation. The converted nanocrystals can be easily manipulated and due to their low size dispersion self-organize on an amorphous-carbon-coated grid. Magnetic property investigations, keeping the same nanocrystal environment, show that the structural transition is accompanied by a significant increase in both the blocking temperature (to a near room-temperature value) and the coercivity.The Journal of Physical Chemistry C 07/2012; 116(29):15723–15730. · 4.84 Impact Factor - SourceAvailable from: Venugopal Reddy Paduru[Show abstract] [Hide abstract]

**ABSTRACT:**With a view to investigate the influence of nanometric size on the structural, surface, and magnetic properties of nanocrystalline Ti 0.95 Co 0.05 O 2 -diluted mag-netic semiconductors, prepared by a novel simple con-trollable peroxide-assisted reflux chemical route followed by annealing at different temperatures, a systematic inves-tigation has been undertaken. Structural characterizations such as X-ray diffraction followed by Rietveld refinement, electron diffraction pattern, Fourier transform infrared, Raman scattering, and X-ray photoelectron spectroscopy (XPS) measurements have shown anatase phase formation in nanocrystalline Ti 0.95 Co 0.05 O 2 without any additional impurity phases. The modified reflux chemical route was effective in obtaining pure phase Ti 0.95 Co 0.05 O 2 nanoparti-cles. Surface morphological investigations by using trans-mission electron microscopy and atomic force microscopy measurements showed the predominant effect of random distribution of nanoparticles on the aggregation behavior and local microstructural changes. The deconvoluted XPS core level Co 2p spectral study manifested the oxidation state of Co as +2 and is found to be stable with varying particle size and annealing temperature. The ferromagnetic behavior was investigated by vibrating sample magnetome-ter, magnetic force microscopy, and electron spin resonance measurements. These magnetization studies showed all the samples are ferromagnetic at room temperature without any magnetic clusters. The correlation between structure, surface condition of the nanoparticles and local electronic interactions, and magnetization of the samples was analyzed and explored the origin of ferromagnetism.07/2014;

Page 1

arXiv:cond-mat/0108304v1 [cond-mat.mtrl-sci] 20 Aug 2001

Enhancement of the magnetic anisotropy of nanometer-sized Co clusters: influence of

the surface and of the inter-particle 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, CSIC-Universidad 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 spin-glass 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

intra-particle 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, zero-field cooled (ZFC) and field-cooled (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 dipole-dipole 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 energy-dispersive X-ray 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 non-aggregated 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 energy-loss spectroscopy (EELS), X-ray photoelectron spectroscopy and X-ray absorption

spectroscopy. The atomic structure of the clusters was determined by extended X-ray absorption fine structure

(EXAFS) spectroscopy and high resolution TEM.14It was found that the particles bond poorly to the alumina

2

Page 3

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 quasi-ordered 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

bi-layers 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 ZFC-FC 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 Curie-Weiss 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 Curie-Weiss 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 attempt-frequency, 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

3

Page 4

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 low-T 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 Stoner-Wolhfarth 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 so-called 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 co-workers.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 low-T paramagnetic

tail from the total Ms. In this way, xparais also estimated. We list the results in Table I. The low-T 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

4

Page 5

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 co-deposition 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

5

Page 6

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 non-interacting 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

6

Page 7

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 X-ray magnetic dichroism experiments performed on Au/Co/Au layers35and

more recently also on Co disk-like 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 time-dependent

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: dipole-dipole interaction between the particles

There has been some debate during the last years about the effect that the dipole-dipole interaction between

magnetic nano-particles 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 self-organized 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 Curie-Weiss law, with identical values of

C, which confirms that ?D? and σ are practically the same. By contrast, the Curie-Weiss temperature θ is about 2

times smaller for the monolayer, indicating that the average inter-particle 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 Stoner-Wolhfarth 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 so-called 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 ZFC-FC 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 time-dependent 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. E-mail 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. 104-107, 1577 (1992).

3B. V. Reddy, S. N. Khanna, and B. I. Dunlap, Phys. Rev. Lett. 70, 3323 (1993).

4G. M. Pastor, J. Dorantes-D´ 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. 157-158, 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).

36Ding-sheng 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. 157-158, 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 ZFC-FC magnetization curves

measured with two different magnetic fields are also shown for comparison.

Figure 8. Size-dependence 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. Time-dependent 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