Hysteresis curve of magnetic nanocrystals

JOURNAL OF APPLIED PHYSICS

VOLUME 93, NUMBER 12

15 JUNE 2003

Hysteresis curve of magnetic nanocrystals monolayers: Influence of the structure V. Russier, C. Petit, and M. P. Pilenia) Laboratoire LM2N, Universite´ Pierre et Marie Curie, BP 52, 4 Place Jussieu, 75252 Paris Cedex 05, France

共Received 23 January 2003; accepted 18 March 2003兲 We calculate the magnetization curve at vanishing temperature of a monolayer of spherical single domain magnetic nanocrystals in terms of the structure of the monolayer. The magnetization curve of a square lattice of particles is compared to those of disordered monolayers. The particles on the disordered monolayers are either distributed isotropically on the surface or organized in chains, which are either linear and or totally flexible. A strong effect of the structure is found only in the case of linear chains and when the magnetization is measured along the chains direction. In the experimental part a monolayer of cobalt nanoparticles organized in a chainlike structure is elaborated by applying a magnetic field during the evaporation of a ferrofluid on a substrate. The change of the magnetization curve due to the chainlike structure is compared to that of the model. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1573343兴 I. INTRODUCTION

Due to their size range in between the atom and the bulk, nanometric sized magnetic particles present peculiar magnetic properties which make them of great interest both for themselves and for their potential applications 共see Ref. 1 for a review on magnetic nanoparticles兲. Numerous kind of assemblies of such particles can now be elaborated, differing by the nature and the shape of the particles, their geometrical arrangement or the dimensionality of the assembly. For instance, spherical2–10 or elongated particles11 of Co or Ni,11,12 spherical,1,13–18 and elongated ferrite particles19 can be synthesized and two dimensional 共2D兲2– 4,7–11,13,20,21 structures, quasi 3D13,19,22,23 films or 3D7,10,24 structures of such particles can be elaborated. In the case of close packed structures, the mutual interactions between particles play an important role on their magnetic properties and are widely and studied both experimentally1,11,13,14 –16,20,21,25–27 14,28 –31,11,32,20,21 Among the various systems theoretically. made of magnetic nanoparticles the case of well coated spherical particles deposited as monolayers on a plane substrate is of great importance since then only the dipolar interaction plays a role. Moreover such systems are now experimentally available.2– 4,8,9,13 In these systems the geometric and structural parameters influence the magnetic properties only through the interaction between particles. When no exchange interaction between particles is involved 共i.e., when the particles are well coated in order to avoid a direct contact兲, the importance of the interaction is totally determined by a coupling constant ␣ d , which is nothing but the ratio of the dipolar interaction energy for particles separated by the mean value of the first neighbor distance to the magnetic anisotropy energy of the particle.31,32 The influence of the interaction between particles on the magnetization curve at vanishing temperature has been cala兲

Author to whom correspondence should be addressed; electronic mail: [email protected]

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culated for well ordered monolayers of either square or hexagonal lattices in the case of spherical particles with uniaxial anisotropy.21,32 It has been found that for sufficiently small values of the coupling constant, corresponding typically to what can be reached experimentally with cobalt particles, a mean field type of approximation, where only the demagnetizing field effect is taken into account, leads to a satisfactory approximation of the magnetization curve.20,21 On the other hand in the strong coupling case ( ␣ d ⬎0.20), the correlations lead to a nontrivial orientational structure.32 Since in the mean field regime the local structure of the monolayer is not invoked, one can conclude that in the weak coupling case the magnetic properties do not depend on the precise structure of the monolayer, at least for ordered structures. Indeed, these calculations were performed only for well ordered structures and a more general investigation is necessary to conclude on this point. In the present work, we investigate the influence of the structure in the monolayer in a quite general way. We calculate the magnetization curve at T⫽0 of a monolayer of spherical uniaxial particles whose locations are fixed on the surface and correspond to different structures, well ordered 共square lattice兲 or disordered. In the later case, we consider either an isotropic disorder 共2D hard-sphere like distribution of particles兲 or chainlike structures. In order to estimate the influence of the structure on the magnetization curve, we determine an effective coupling constant, ␣ eff d , which takes into account on the one hand the mean density of particles in the layer and on the other hand the kind of structure considered 共isotropic or chainlike兲. Then, the comparison of the magnetization curve in terms of the applied field H, M o ( ␣ eff d ,H), of a square lattice characterized by the coupling constant ␣ eff d with that of the actual system, M (H) permits us to evaluate the influence of the details of the structure in the layer on the magnetization curve of the system. As Co nanoparticles could present a cubic magnetocrystalline anisotropy,33,34 the case of spheri-

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J. Appl. Phys., Vol. 93, No. 12, 15 June 2003

Russier, Petit, and Pileni

cal particles with a cubic anisotropy is also briefly considered. II. MODEL AND COMPUTATIONAL DETAILS A. Structure of the monolayer

Concerning the structure in the layer we consider the location of n s particles in a square box of length l, and then periodic conditions are used in order to represent a real infinite system. Different situations are considered. The simplest one corresponds to a perfect square lattice, where the particles occupy all the sites of a lattice of lattice spacing d, with l 2 /d 2 ⫽n s . In the second situation, referred to in the following as the isotropic disorder, the particles are randomly distributed on the surface with the only constraint that they do not overlap. Such a distribution corresponds to a hard-sphere like configuration for the particles, which is frozen before the minimization of the total energy in terms of the moment orientations is performed. These configurations are obtained from a Monte Carlo scheme with a hard-sphere interaction potential between the particles. Finally, we have considered chainlike structures, where all the particles belong to chains constructed according to a self avoiding walk procedure in the following way. We first chose at random a point in the box as the starting point of the chain; then the (i⫹1)th particle in the chain is located at contact with the ith one, and the orientation of the bond, rˆ i,i⫹1 , defined by its polar angle ␸ i , is deduced from the preceding one by

␸ i ⫽ ␸ i⫺1 ⫹2 ␲ a ␩ ,

共1兲

where ␩ is a random number in between 0 and 1 and a is an amplitude allowing to control the flexibility of the chain (0 ⭐a⭐1). The first bond angle, ␸ 0 , can take either a random value or a conveniently chosen initial value. When the (i ⫹1)th element of a chain overlap with another particle already put on the surface, a new trial value for the bond angle is chosen, and the chain is stopped when a maximum number of trials, n t , has been reached 共typically, we use n t ⫽20). The building of the chains in the box is stopped when the total number of particles reaches the value n s . The parameters a and ␸ 0 permit to control the characteristics of the chains. Obviously, a nonvanishing value of a leads to nonlinear chains, whose twisting degree increases with a(a ⭐1). In the following we use a⫽1 and ␸ 0 ⫽2 ␲ ␩ or a⫽0 and ␸ 0 ⫽0 in order to get either strongly twisted or linear chains, respectively. The twisted chains corresponding to a ⫽1 will be referred to simply as twisted chains. Examples of the configurations of particles we have obtained are shown in Fig. 1. B. Model

We consider a monodisperse assembly of spherical monodomain magnetic particles, with uniaxial anisotropy. The particles are distributed on a plane surface with fixed positions according to the different structures described above, and the easy axes orientations, 兵 nˆ k 其 , are randomly distributed. This corresponds to the experimental situation where the ferrofluid is evaporated at room temperature and the magnetization measurement is performed subsequently at

FIG. 1. Configurations of the particles in the disordered systems. The length of the arrow is proportional to the projection of the moment in the (xˆ ,yˆ ) plane; stared particles correspond to m z ⬍0: 共top兲 isotropic case, ␾ ⫽0.4948, 共center兲 twisted chain structure ␾ ⫽0.0785, and 共bottom兲 linear chain structure ␾ ⫽0.0785.

very small temperature (T⯝3 K兲.2,3 The normal to the surface defines the zˆ direction. The particles are characterized by an anisotropy constant K, a saturation magnetization M s , and a volume v 0 ⫽( ␲ /6)D 3 , where D is the diameter. The magnetic moment of the particles is ␮ ⫽M s v 0 . Let us denote




by d the first neighbor distance. In the case of a perfect square lattice d coincides with the lattice spacing, d o . The total energy of the system is given by

d o ⫽D

冉冊 ␲ 4␾

10003

1/2

with E⫽⫺ 共 K v 0 兲

兺

k⫽1,n s

ˆ k兲2 共 nˆ k ␮

1 ␮ 2 d3

兺 i⫽ j

␮ˆ i ␮ˆ j ⫺3 共 ␮ˆ i rˆ i j 兲共 ␮ˆ j rˆ i j 兲

⫺H a ␮

兺

hˆ ␮ˆ k .

2

⫺

␾ ⫽n s ␲

1 2

兺

k⫽1,n s

␣ eff d ⫽

共2兲

Here and in the following, hatted symbols denote unit vectors; Ha ⫽H a hˆ is the external field. In the following we deal with the reduced energy, E * ⫽E/(2K v 0 ). We have E * ⫽⫺

ˆ k兲2 共 nˆ k ␮

兺

兺

k⫽1,n s

hˆ ␮ˆ k ,

a⫽ 共3兲

where h a ⫽H a /(2K/M s ) is the reduced applied field and ␣ d is the dipolar coupling constant

␣ d⫽

冉冊

␲ M s2 D 12K d

3

.

兺k 共 ␮ˆ k• hˆ 兲 .

M s2

In order to compare the magnetization obtained with the different structures considered, we introduce an effective coupling constant, ␣ eff d , and this is done by using the expression of the total dipolar energy. We first consider the case of the isotropic disorder. In this case, we take into account only the mean density of particles in the layer and thus we represent the actual system by a perfect square lattice the lattice spacing of which d o is related to the surface occupation fraction ␾ of the layer

3

⫽ ␣ dM

␾ ␲ /4

3/2

共7兲

,

␲ D. 4␾

Because of the anisotropy of the dipole–dipole interaction, Eq. 共2兲, we estimate the total dipolar interaction energy from the sum of the dipole–dipole energies in the two opposite cases where all the particles have their moment oriented in the xˆ or in the zˆ directions, respectively. The corresponding reduced sums depend only on t⫽a/D and are given by m⬘

S x共 t 兲 ⫽

兺

1⫺3 共 xˆ i j 兲 2 共 r i j /D 兲

3

m⬘

S z共 t 兲 ⫽

兺

1 共 r i j /D 兲 3

,

共8兲

where 兺 m ⬘ denotes the sum on the sites of the model lattice excluding r i j ⫽0. Obviously, the case t⫽1 corresponds to the square lattice. We can write the total dipolar energy for the model lattice in the case ␮ˆ k ⫽xˆ or ␮ˆ k ⫽zˆ , for all k in the form

* m ⫽ ␣ dM E dx

冉冊

* m ⫽ ␣ dM E dx

冉冊

and

共5兲

C. Effective coupling constant

冉冊冉冊

D 12␲ K d o

共4兲

Since we deal with the vanishing temperature limit, we determine the configuration of the moments 兵 ␮ˆ k 其 from the minimization of the total energy. The calculation is performed on a two-dimensional square box of finite size, l, including n s particles. We minimize the total energy with respect to the 2n s polar angles 兵 ␪ k , ␸ k 其 from the conjugate gradient method. At each iteration of the numerical scheme, the energy is minimized along the calculated descent direction by a Newton Raphson scheme. Because of the long range of the dipolar interaction, we consider periodic boundary conditions in the two directions and we use the Ewald summation procedure.35,36 Then we calculate the reduced magnetization in the direction of the applied field 1 M 共 ha兲⫽ ns

共6兲

.

where ␣ dM ⫽( ␲ M s2 )/(12K) is the maximum value of the coupling constant defined in Eq. 共4兲. In the case of chainlike structures, we first consider, as an intermediate step, a periodic arrangement of infinite parallel lines of particles at contact, parallel to the xˆ direction. This structure is considered as a model lattice. The distance between the lines of particles is

␮ˆ i ␮ˆ j ⫺3 共 ␮ˆ i rˆ i j 兲共 ␮ˆ j rˆ i j 兲 1 ⫺ ␣d 2 i⫽ j 共 r i j /d 兲 3 ⫺h a

4l 2

We then get

共 r i j /d 兲 3

k⫽1,n s

D2

S x共 t 兲 S ox

S z共 t 兲 S zo

S ox

S zo ,

o where S x,z are relative to a perfect square lattice (t⫽1). We can then define an orientation dependent effective coupling eff in such a way that constant, ␣ dx,z eff * m ⫽E dx,z * o 共 ␣ dx,z E dx,z 兲,

namely eff ␣ dx,z ⫽ ␣ dM

S x,z 共 t 兲 o S x,z

⬅ ␣ dM

S x,z 共 t 兲 . S x,z 共 t⫽1 兲

ˆ Finally we define ␣ eff d as a mean value over the orientations x and zˆ :


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冉

1 M S x共 t 兲 S z共 t 兲 ␣ eff ⫹ o d ⫽ ␣d 2 S ox Sz


冊

with t⫽

␲ a ⫽ . D 4␾

共9兲

It is important to notice that when ␾ →0, in opposite to the isotropic case, Eq. 共7兲, ␣ eff d does not vanish for the chainlike structures since in this latter case, whatever the value of ␾, all the particles have at least two neighbors at contact. We get

lim ␾ →0

冉冊 ␣ eff d ␣ dM

⫽0.6658.

III. RESULTS AND DISCUSSION A. Model calculations

We have performed calculations for both types of structures, isotropic and chainlike, and the applied field is either normal (hˆ ⫽zˆ ) or parallel (hˆ ⫽xˆ ) to the surface. In each case we consider the magnetization curve of the perfect square eff lattice characterized by ␣ d ⫽ ␣ eff d , M o (h a ; ␣ d ), as a reference magnetization curve. Then the deviation of the actual magnetization curve, M (h a ; ␣ dM , ␾ ) from M o (h a ; ␣ eff d ) measures the direct influence of the structure in the layer on the magnetization since obviously all the details of the structure are removed in the reference system. We emphasize that conversely to what is done in Ref. 31, where the magnetic properties 共remanence and coercive field兲 are investigated in terms of a growing volume fraction in particles, x v , here we compare the magnetic properties of ordered and disordered structure at constant surface occupation fraction. We have chosen two particular values of the effective coupling constant ␣ eff d as typical examples for the system under study, eff namely ␣ eff d ⫽0.125 and ␣ d ⫽0.260. The former on the one hand should be reachable experimentally for instance with nanoparticles of cobalt and on the other hand goes slightly beyond the limit of validity of a simple mean-field approximation;23 the latter is far beyond the range of validity of the mean-field approximation and therefore the correlations between particles play an important role in this case.32 We first consider three different possibilities to get the eff same value of ␣ eff d ( ␣ d ⫽0.125), with different values of ␾ M and ␣ d in the isotropic case. The corresponding magnetization curves are shown and compared to the reference one for an applied field normal to the surface (hˆ ⫽zˆ ), and in the case ␣ dM ⫽0.250 and ␾ ⫽0.4849 for the applied field parallel to the surface, hˆ ⫽xˆ , in Fig. 2. We see that when ␣ eff d ⫽0.125, the different magnetization curves are nearly similar, and moreover they are close to the corresponding reference one, even when the applied field is parallel to the surface when the dipolar interaction enhances the applied field. We can conclude that even in the case of an applied field parallel to the surface, the magnetization curve is very well reproduced

FIG. 2. Magnetization in the case of the isotropic disorder: ␣ eff d ⫽0.125. Applied field in the direction hˆ ⫽zˆ : ␾ ⫽0.5741 and ␣ dM ⫽0.200 共solid circles兲; ␾ ⫽0.4948 and ␣ dM ⫽0.250 共solid squares兲; ␾ ⫽0.3775 and ␣ dM ⫽0.375 共solid triangles兲; solid line: perfect square lattice with ␾ ⫽ ␲ /4 and ␣ dM ⫽0.125 共reference system兲. Applied field in the direction hˆ ⫽xˆ : ␾ ⫽0.4948 and ␣ dM ⫽0.250 共open circles兲; dashed line: perfect square lattice with ␾ ⫽ ␲ /4 and ␣ dM ⫽0.125 共reference system兲.

by that of the perfect square lattice characterized by ␣ d eff ⫽ ␣ eff d in the case of the isotropic disorder at least for ␣ d ⭐0.125. Then, still for ␣ eff d ⫽0.125, we consider one example of twisted chains for the two orientations of the applied field. The result is shown and compared to the reference one in Fig. 3. As is the case for the isotropic disorder, the reference magnetization curve reproduces with a good accuracy the magnetization curve of the actual system, although the accuracy is slightly worse when the applied field is parallel to the surface (hˆ ⫽xˆ ). On the other hand, when the particles are arranged in linear and parallel chains along the xˆ axis, the magnetization curve differs notably from that of the reference system, when hˆ ⫽xˆ , as can be seen in Fig. 3. Finally, we consider the cases corresponding to ␣ eff d ⫽0.260. When the particles present an isotropic structure, see Fig. 4, the same kind of agreement with the reference system as we have got for ␣ eff d ⫽0.125 is obtained, except in the very vicinity of the coercive field when hˆ ⫽xˆ . The results for the twisted and linear chains are shown and compared to the corresponding reference magnetization curves for hˆ ⫽zˆ and xˆ in Figs. 5 and 6, respectively. These results show that even in this strong coupling limit, the reference system, reproduces quite well the magnetization of the actual system when hˆ ⫽zˆ . Conversely, when hˆ ⫽xˆ there is no agreement at all between the actual and the reference magnetization curves. In particular, the coercive field is much larger for the chain than for the reference square lattice. For comparison, we also give in Fig. 6 the magnetization curve of linear chains parallel to the yˆ axis, and hˆ ⫽xˆ . Clearly, due to the



FIG. 3. Magnetization for particles arranged in chains: ␣ eff d ⫽0.125, ␾ ⫽0.0785, and ␣ dM ⫽0.1875. 共Solid circles兲 twisted chains, hˆ ⫽zˆ . 共Solid squares兲 twisted chains, hˆ ⫽xˆ . 共Open squares兲 linear chains in the xˆ direction, hˆ ⫽xˆ . Solid and dotted lines: perfect square lattice, with ␣ d ⫽0.125 and hˆ ⫽zˆ and xˆ , respectively.

high value of the dipolar coupling, the linear chains behave roughly as homogeneous wires with an effective easy axis in the direction of the chain although the individual particles have randomly distributed easy axes. Then the difference between the two hysteresis curves corresponding to chains parallel to the xˆ and yˆ axes in Fig. 6 corresponds in a first approximation to the different orientations of the effective

FIG. 4. Magnetization in the case of the isotropic disorder. ␣ eff d ⫽0.260. ␾ ⫽0.558 and ␣ dM ⫽0.4333: 共solid circles兲 hˆ ⫽xˆ , 共solid squares兲 hˆ ⫽zˆ , 共dotted and solid lines兲 perfect square lattice with ␣ d ⫽0.125 and hˆ ⫽xˆ and zˆ , respectively.


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ˆ FIG. 5. Magnetization for particles arranged in chains: ␣ eff d ⫽0.260 and h ⫽zˆ . 共Top curve兲 ␣ dM ⫽0.3750 and ␾ ⫽0.2380; 共bottom curve兲 ␣ dM ⫽0.3916 and ␾ ⫽0.0785. The arrows indicate the corresponding M /M s scale; 共solid line兲 perfect square lattice with ␣ d ⫽0.260, 共open circles兲 twisted chains; and 共solid squares兲 linear chains.

easy axis of the wire with respect to the applied field. This explains also the high value of the coercive field and the high squareness of the hysteresis curve when the chains are parallel to the applied field. The increase in the squareness of the hysteresis curve when going from twisted to linear chains ˆ ˆ in the case ␣ eff d ⫽0.125 and h ⫽x 共see Fig. 3兲 may thus be interpreted as the onset of such a behavior.

ˆ ˆ FIG. 6. Magnetization for particles arranged in chains: ␣ eff d ⫽0.260, h ⫽x , ␣ dM ⫽0.3916, and ␾ ⫽0.0785; 共solid line兲 perfect square lattice with ␣ d ⫽0.260, 共solid triangles兲 twisted chains, 共solid squares兲 linear chains parallel to the xˆ direction, and 共open squares兲 linear chains parallel to the yˆ direction.


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We can conclude that the structure in the layer does not play any role on the magnetization when the particles are distributed isotropically in the surface or when the applied field is normal to the surface. Hence, in order to get the influence of the structure on the magnetization one has to investigate the magnetization of thin layers of nanocrystals arranged in aligned chains with different orientations of the applied field in the surface plane and normal to the surface plane. This is precisely what has been done in the experimental part of this work. However, as we shall see below, in order to get particles deposited on the substrate with a chainlike structure, we impose a magnetic field, H d , during the solvent evaporation phase as was done in Refs. 9 and 19. As a result, we expect the easy axes of particles to be preferentially oriented in the direction of the field.19 In the experiments described below, a constant field of 1 T was necessary in order to obtain a sufficiently well pronounced structure. For such a value of the field H d and at room temperature, we deduce from a Langevin type of calculation, that the magnetization of the system takes a value of M L (T⫽T RT , H d ⫽1T)/M s ⯝0.9 for cobalt particles of D⯝8 nm. Therefore the particles moments are strongly aligned along the direction of the field Hd. Because of the magnetic anisotropy energy term, we can assume that during the evaporation process the particles rotate in the field in such a way that their easy axes orient themselves parallel to the moments which are pinned by the field. The resulting degree of orientation of the easy axes is very difficult to estimate, and this is beyond the scope of the present work. Nevertheless, in order to see whether it is possible to discriminate the effect of the chainlike structure from that of the easy axes distribution, we have calculated the magnetization of a monolayer of particles arranged in a perfect square lattice characterized by ␣ d and the easy axes of which are preferentially oriented in the xˆ direction. The polar axis for the distribution is thus the xˆ axis, the azimuthal angle ␾ is randomly distributed and the polar angle ␪ is distributed according to the following probability law: P共 ␪ 兲⫽

sin共 ␪ 兲 C 共 ␴ 兲 exp关 ⫺ 共兩 cos共 ␪ 兲兩 ⫺1 兲 2 / ␴ 2 兴 , 2

共10兲

where sin(␪)/2 corresponds to the random distribution, and C( ␴ ) is the normalization constant. We have determined the value of ␴ in such a way that the remanence magnetization coincides with that of the monolayer presenting a chainlike structure along the xˆ direction, and characterized by ␣ eff d ⫽ ␣ d . The results for ␣ eff d ⫽0.125 for which we get ␴ ⫽0.40 is shown in Fig. 7. The nearly perfect coincidence between the two curves show that it is not possible to discriminate between the two effects; in any case they add up but it is a priori difficult to determine which one is the most important. It has been found experimentally that Co spherical nanocrystals present a dominating uniaxial anisotropy.33,34 However, this results from crystallization defects 共stacking faults and twin boundaries兲 since Co nanoparticles are generally found to crystallize in the fcc structure and accordingly should present a cubic anisotropy. As a matter of fact some Co nanoparticles sufficiently ‘‘perfect’’ in order to

FIG. 7. Magnetization curve for particles arranged in linear chain along the xˆ direction with randomly distributed easy axes 共solid circles兲; or particles located on a perfect square lattice with easy axes preferentially oriented in the xˆ direction 共open square兲; ␣ eff d ⫽0.125, ␴ ⫽0.40.

present at least two easy axes33,34 may be found and therefore, one cannot rule out the case of the cubic anisotropy. Accordingly, we have considered the case of spherical particles presenting a cubic magnetic anisotropy with K⬎0, where the particles have three easy magnetization axes. Only the case of particles arranged in linear chains parallel to a fixed direction (xˆ ), where we expect the most important effect, has been considered. Although we have not performed extensive calculations we can conclude that the influence of the structure in position of the particles in the monolayer is similar to that obtained for particles with uniaxial anisotropy. Indeed, we find that when hˆ ⫽zˆ , the magnetization curve of the real system is very close to that of the reference system, while a significant deviation is obtained when hˆ ⫽xˆ . This is displayed in Fig. 8 where we compare the magnetization curves of monolayers arranged in linear chains parallel to the xˆ direction to those of the corresponding reference system for the two directions of the measuring field, namely hˆ ⫽zˆ and hˆ ⫽xˆ in the case ␣ eff d ⫽0.125. Because of the similarity with the case of uniaxial particles, 共see Fig. 3兲, we expect that the magnetization curve for particles isotropically distributed in the monolayer are very close to the corresponding reference system ones. B. Experimental samples

Now, we compare the results of the models with experiments. Such a comparison is only qualitative, given on the first hand the simplicity of the model, which cannot reproduce the details of the experimental sample, and on the other hand the difficulty to get the relevant parameters concerning the structure in the experimental sample. The synthesis of the cobalt nanocrystals assemblies was reported previously.3,37 Immersing a transmission electron




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FIG. 8. Magnetization curves for spherical particles with cubic anisotropy, arranged in linear chains along the xˆ axis and comparison with the reference system 共square lattice兲: ␣ eff d ⫽0.125. Solid and open symbols correspond to the chainlike structure and the reference system, respectively: 共circles兲 hˆ ⫽zˆ and 共squares兲 hˆ ⫽xˆ .

microscope carbon grid in a low concentration solution of nanocrystals allows the cobalt to self organize in 2D monolayers, during the evaporation process 关see Figs. 9共C兲 and 9共D兲兴. This self-assembling results from a balance between hard sphere repulsion and van der Waals attraction forces. These forces are isotropic and impose a close-packing structure: a hexagonal network of nanocrystals forming the monolayer.38,39 The interparticle spacing, ⌬, is found to be 1.7⫾0.2 nm for lauric acid coated cobalt nanocrystals. The length of a fully extended lauric acid molecule is 1.6 nm.40 This implies that the interdigitating process is total between

FIG. 10. TEM patterns of cobalt nanocrystals of 5.8 nm in diameter organized in linear structures by applying a field of 1 T parallel to the substrate during the evaporation process.

FIG. 9. Hysteresis curve measured at 3 K of the 2D isotropic monolayers of cobalt nanocrystals for the two orientations of the applied field: 共A兲 5.8 nm and 共B兲 8.0 nm nanocrystals. 共Solid and dashed curves兲 field parallel and perpendicular to the substrate surface, respectively. Corresponding TEM patterns at long range for the 5.8 nm 共C兲 and 8 nm 共D兲 nanocrystals samples, respectively.

the alkyl surrounding the nanocrystals. This favors the selforganization and prevents the coalescence or direct contact between adjacent nanocrystals. Applying a magnetic field of 1 T, parallel to the substrate during the evaporation process, drastically changes the morphology of the film made of uncoalesced cobalt nanocrystals. Figures 10 and 11 show linear chains made of uncoalesced cobalt nanocrystals in the direction of the applied field. In the case of 5.8 nm cobalt nanocrystals 共see Fig. 10兲, the lines appear more disordered than in the case of the 8 nm 共Fig. 11兲 cobalt nanocrystals, which have the hardest magnetic properties. In the first case, the average width is 250⫾100 nm for a length larger than 10 ␮m in average. The interchain distance 共center to center兲 is about 400 nm. For the largest nanocrystals, the average width is 300⫾50 nm and the inter-


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Russier, Petit, and Pileni TABLE I. Structural and magnetic properties of the various cobalt nanocrystals: D is the mean value of the diameter, ␴ the polydispersity, M s the magnetization at saturation, H c the coercive field. M r /M s and (M r /M s ) 储 are the reduced remanences of nanocrystals dispersed in solution and deposited on the surface with the applied field parallel to the substrate surface, respectively.

FIG. 11. Same as Fig. 10 for the 8 nm cobalt nanocrystals.

chain distance is 600 nm. Close view of the chains indicate that they are formed by uncoalesced cobalt nanocrystals in mono- or bilayers. Notice that the same kind of structures has been obtained in Ref. 9 using a similar method; however the magnetic properties in terms of the direction of the field with respect to the direction of the chains in the substrate plane were not measured. Obviously these organizations are not the ideal 1D chains as we have considered in the model. Nevertheless their magnetic properties strongly differ from those of the isotropic monolayers, as predicted theoretically. C. Experimental magnetic properties 1. Isotropic samples

Figure 9 shows the hysteresis loop of the cobalt nanocrystals monolayer on cleaved graphite.41 During the mea-

D 共nm兲

␴

M s 共emu/g兲

H c 共T兲

5.8 8.0

18% 18%

110⫾10 120⫾5

0.12⫾0.02 0.12⫾0.02

M r /M s (M r /M s ) 储 K a (erg/cm3) 0.25 0.45

0.29 0.52

2.6⫻106 1.5⫻106

surements, at 3 K, the sample is kept parallel to the applied field. Whatever the size the hysteresis loop is squarer than that of the same nanocrystals dispersed in solution. Saturation is reached at 2 T for 5.8 nm nanocrystals and at 1.5 T for 8 nm nanocrystals while it is not reached at 2 T for the nanocrystals dispersed in solution. On the other hand, the reduced remanence increases 共Table I兲. The bigger the nanocrystals, the larger is the change in the magnetic properties of the monolayer. This effect is due to the long-range dipolar interaction and can be checked by measuring the hysteresis loop with an applied field either parallel or perpendicular to the sample surface 共Fig. 9兲. This can be quantified by the value of ␥ ⫽(M r /M s )⬜ /(M r /M s ) 储 , the ratio of the reduced remanences in perpendicular and parallel geometries, respectively. The coercive field remains unchanged wile the remanence is decreased when the applied field is perpendicular instead of parallel to the substrate plane, and this is expected from the model presented above.22,23 From the magnetic characteristics of the cobalt nanocrystals the coupling constant ␣ d is evaluated to 0.04 and 0.06 for the 5.8 and the 8 nm nanocrystals, respectively. The presence of vacancies and defects in the monolayer induces a decrease of this coupling constant to 0.03 and 0.05, respectively.22,23 The theoretical value of the ratio ␥ has been calculated with the results: ␥ th⫽0.70 for the 5.8 nm nanocrystals and 0.66 for the 8 nm nanocrystals, while the corresponding experimental values are ␥ exp⫽0.75 and 0.65, respectively. Thus the good agreement between the theoretical and experimental values of ␥ confirms that the collective effects on the magnetic properties of the 2D monolayers made of cobalt nanocrystals are mainly due to the dipolar interactions between adjacent nanocrystals. Comparison of the data for the two nanocrystal samples differing only by their size confirms the validity of the model and shows the strong influence of the size on the dipolar interactions. As expected theoretically, the structural order does not play an important role as the real monolayers are rather disordered 共Fig. 9兲 compared to a perfect square lattice. 2. Samples with chainlike structure

Now let us consider the case of the particles arranged in chains. When the measuring field is parallel to the chain and thus also to the substrate plane, the hysteresis curve at 3 K appears squarer than that of the isotropic 2D monolayer measured with a field parallel to the substrate 共see Fig. 12兲. For cobalt nanocrystals of 5.8 nm in diameter, which have the lowest coupling constant the reduced remanence slightly increases from 0.29 to 0.31 关Fig. 12共A兲兴. However the satura-




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TABLE II. Magnetic properties of the cobalt nanocrystals assemblies arranged in linear chains, for the two orientations of the applied magnetic field: parallel 共储兲 or perpendicular 共⬜兲 to the chains direction.

FIG. 12. Comparison of the hysteresis curves measured at 3 K for linear chains with the field parallel 共open squares兲 or perpendicular 共open circles兲 to the chains to that of the isotropic disordered monolayer with the field parallel to the substrate 共solid squares兲: 共A兲 5.8 nm nanocrystals and 共B兲 8 nm nanocrystals.

tion is reached at 1.5 T instead of 2 T and the coercivity remains unchanged at 0.13 T 关see inset in Fig. 12共A兲兴. For the nanocrystals with a higher coupling constant 共diameter of 8 nm兲 the influence of the structure is more pronounced 关Fig. 12共B兲兴. When the measuring field is parallel to the chains direction the reduced remanence increases to 0.60 instead of 0.52 in the case of the isotropic monolayers. The coercivity increases slightly from 0.13 to 0.14. The saturation is reached at 1 T instead of 1.5 T. Thus in both cases the experimental magnetization curves show first an increase in the remanence and in the squareness, when going from the isolated nanocrystals to the 2D isotropic monolayer with the applied field in the surface. In this case the coercive field is nearly unchanged and the local order does not play a significant role. Then when we compare to the case of nanocrystals arranged in linear chains, the squareness is notably increased when the field is parallel to the chains. This is consistent with the theoretical hysteresis loop presented above, where an increase in the squareness has been obtained when the applied field is in the chains direction, compared to the isotropic case. For 5.8 nm cobalt nanocrystals turning the sample of 90° indeed induces a smoother hysteresis loop, as presented above. The saturation is not reached, even at 2.5 T. The reduced remanence decreases to 0.19 and the coercive field decreases to 0.10 T. For 8 nm nanocrystals the rotation of the sample induces a stronger effect as is expected due to the

D 共nm兲

(M r /M s ) 储

H c (T)

(M r /M s )⬜

H⬜c (T)

5.8 8.0

0.31 0.60

0.13 0.14

0.19 0.25

0.10 0.10

储

larger value of the coupling constant. Again, the saturation is not reached at 2.5 T. The reduced remanence decreases to 0.25 and the coercive field decreases to 0.10 T. In order to quantify this structural effect, we introduce the ratio ␥ ⬘ ⫽(M r /M s )⬜ /(M r /M s ) 储 where here ⬜ and 储 refer to the orientation of the applied field relative to the chains direction. ⬘ and theoretical ␥ ⬘th values are the folThe experimental ␥ exp ⬘ ⫽0.61 and ␥ th ⬘ ⫽0.83 for the 5.8 nm nanocrystals lowing: ␥ exp ⬘ ⫽0.42 and ␥ ⬘th⫽0.77 for the 8 nm nanocrystals. The ex␥ exp perimental results and the calculated results from the model are thus in qualitative agreement. Therefore, the deviation of ␥⬘ from ␥ ⬘ ⫽1 can be attributed to the modification of the structure. The experimental samples do not present the perfect chainlike structure introduced in the model; the experimental chains are larger and present locally some bilayers 共Figs. 10 and 11兲. This probably explains the differences between the experiments and the theory. On an other hand, an influence of the alignment of the easy axes during the deposition under the field cannot be ruled out and can explain in part the deviation we get between the calculated and the experimental results for ␥ ⬘ . The results reported here are nevertheless likely to be mainly structural effects on the collective magnetic properties of the monolayers made of uncoalesced cobalt nanocrystals either distributed isotropically or arranged in linear chains. Our results seem to show that when arranged in linear chains, the nanocrystals behave roughly as an assembly of wires, with an effective easy axis in the wire direction. Turning the substrate of 90°, and keeping the substrate parallel to the applied field, does not change the magnetic properties of the isotropic 2D monolayers. However in the case of the 1D chains, a smoother hysteresis curve is obtained with a marked decrease of the reduced remanence as the applied field is perpendicular to the effective easy axes of the chains 共Table II兲. Thus the experimental magnetization curves show an increase in the remanence and in the squareness, when going from the 2D isotropic monolayer with the applied field in the surface to the case of the linear parallel chains with a field parallel to the chains. In conclusion, when we compare the experimental magnetization curves corresponding to the 2D isotropic distribution and to parallel chains, respectively, with an applied field parallel to the chains, we see that both the squareness and the coercive field are notably increased. This is in agreement, at a qualitative level, with the results of the model as can be seen from Figs. 2 共open circles兲 and 3 共open squares兲. Although one cannot known if the effective coupling constant of the experimental samples takes the same value, for the isotropic and the chainlike organization, the difference between the corresponding magnetization curves of these two samples is likely to be due to the structure in the layer, since


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the coercive field is only very slightly modified by the interactions when the structure in the layer is not changed, at least in the experimental coupling range ( ␣ d ⭐0.10). 23,32 However we cannot rule out a combined effect of both the structure in the layer and a possible partial orientation of the easy axes in the direction of the magnetic field H d used to make the chainlike structure. ACKNOWLEDGMENTS

The authors gratefully thank Dr. A. T. Ngo, Dr. I. Lisiecki, and Dr. L. Motte from the LM2N, Universite´ P. et M. Curie, for fruitful discussions. Thanks are due to Dr. G. Lebras and Dr. E. Vincent 共SPEC/DRECAM/CEA-Saclay兲 for the use of their SQUID apparatus. 1

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