Effective anisotropies and energy barriers of magnetic nanoparticles with Neel surface

anisotropy

R. Yanes and O. Chubykalo-FesenkoInstituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain

H. KachkachiGroupe d’Etude de la Matiere Condensee, Universite de Versailles St. Quentin,

CNRS UMR8635, 45 av. des Etats-Unis, 78035 Versailles, France

D. A. GaraninDepartment of Physics and Astronomy, Lehman College, City University of New York,

250 Bedford Park Boulevard West, Bronx, New York 10468-1589, U.S.A.

R. Evans and R. W. ChantrellDepartment of Physics, University of York, Heslington, York YO10 5DD, UK

Magnetic nanoparticles with Neel surface anisotropy, different internal structures, surface arrange-ments and elongation are modelled as many-spin systems. The results suggest that the energy ofmany-spin nanoparticles cut from cubic lattices can be represented by an effective one-spin potentialcontaining uniaxial and cubic anisotropies. It is shown that the values and signs of the correspondingconstants depend strongly on the particle’s surface arrangement, internal structure and shape. Par-ticles cut from a simple cubic lattice have the opposite sign of the effective cubic term, as comparedto particles cut from the face-centered cubic lattice. Furthermore, other remarkable phenomena areobserved in nanoparticles with relatively strong surface effects: (i) In elongated particles surfaceeffects can change the sign of the uniaxial anisotropy. (ii) In symmetric particles (spherical andtruncated octahedral) with cubic core anisotropy surface effects can change the sing of the latter.We also show that the competition between the core and surface anisotropies leads to a new energythat contributes to both the 2nd

− and 4th−order effective anisotropies.

We evaluate energy barriers ∆E as functions of the strength of the surface anisotropy and theparticle size. The results are analyzed with the help of the effective one-spin potential, which allowsus to assess the consistency of the widely used formula ∆E/V = K∞ + 6Ks/D, where K∞ is thecore anisotropy constant, Ks is a phenomenological constant related to surface anisotropy, and Dthe particle’s diameter. We show that the energy barriers are consistent with this formula only forelongated particles for which the surface contribution to the effective uniaxial anisotropy scales withthe surface and is linear in the constant of the Neel surface anisotropy.

PACS numbers: 75.75.+a, 75.10.HK

I. INTRODUCTION

Understanding the thermally-activated switching ofmagnetic nanoparticles is crucial to many technologicalapplications and is challenging from the point of viewof basic research. Surface effects have a strong bearingon the behavior of magnetic nanoparticles. They existalmost inevitably due to many physical and chemical ef-fects including crystallographic arrangement on the sur-face, oxidation, broken surface bonds, existence of sur-factants, etc. Magnetic particles are often embedded innon-magnetic matrices which alter the magnetic proper-ties of the surface in many ways such as additional surfacetension and possible charge transfer.

Consequently, the properties of magnetic particles arenot the same as those of the bulk material, and manyexperiments have shown an increase in the effective mag-netic anisotropy due to surface effects1–4. Quantum ab-

initio studies have revealed different anisotropy and mag-

netic moment at the surface of magnetic clusters em-bedded in matrices5. Synchrotron radiation studies haveconfirmed that both spin and orbital moments at thesurface differ significantly from their bulk counterparts6.Recent µ-SQUID experiments on isolated clusters1,2 pro-duced more reliable estimations of surface anisotropy.

Thermal measurements have become an importantpart of the characterization of systems of magneticnanoparticles. Often, these measurements include a com-plex influence of inter-particle interactions. However, inother cases, measurements on dilute systems can provideinformation on individual particles. The results showthat even in these cases the extracted information is notalways consistent with the approximation picturing theparticle as a macroscopic magnetic moment, and this isusually attributed to surface effects.

Experimentally, the enhancement of the anisotropy atthe surface often leads to an increase in the blocking tem-perature of single-domain particles from which the valuesof the energy barriers ∆E may be extracted. The influ-

2

ence of the surface manifests itself in the fact that thevalues ∆E/V are different from that of the bulk, i.e.,there is an effective anisotropy Keff that is not exactlyproportional to the particle’s volume V .

One can expect that the effect of the surface reduceswhen the particle size increases. In Ref. 3 it was sug-gested that the effective anisotropy scales as the inverseof the particle’s diameter D according to

∆E/V = Keff = K∞ +6Ks

D. (1)

Here K∞ is the anisotropy constant for D → ∞, pre-sumably equal to the bulk anisotropy, and Ks is the “ef-fective” surface anisotropy. We would like to emphasizethat this formula has been introduced in an ad hoc man-ner, and it is far from evident that the surface shouldcontribute into an effective uniaxial anisotropy (relatedto the barrier ∆E) in a simple additive manner. Ac-tually one cannot expect Ks to coincide with the atom-istic single-site anisotropy, especially when strong devia-tions from non-collinearities leading to “hedgehog-like”structures appear7. The effective anisotropy Keff ap-pears in the literature in relation to the measurementsof energy barriers, extracted from the magnetic viscos-ity or dynamic susceptibility measurements. The surfaceanisotropy should affect both the minima and the saddlepoint of the energy landscape in this case. The origin ofKeff may be expected to be different from that obtainedfrom, e. g., magnetic resonance measurements. In thelatter case the magnetization dynamics depends on thestiffness of the energy minima modified by the surfaceeffects. Despite its ad hoc character, Eq. (1) has becomethe basis of many experimental studies with the aim toextract the surface anisotropy from thermal magnetiza-tion measurements (see, e.g., Refs. 4,10,11) because ofits mere simplicity. Up to now there were no attemptsto assess the validity of Eq. (1) starting from an atom-istic surface anisotropy model such as the Neel model forsurface anisotropy.

The aim of the present paper is to understand theinfluence of the surface anisotropy on the behavior ofmagnetic particles with different surface arrangements,shapes and sizes. Although various crystal structureshave been investigated, the overall particle propertieswere kept close to Co. With the help of numerical mod-elling of magnetic particles as many-spin systems, weshow that the energy of the many-spin particle can beeffectively represented by that of an effective one-spinparticle with both uniaxial and cubic anisotropy terms.The effective anisotropy constants depend on the surfacearrangement of the particle, crystal structure, and shape.We numerically evaluate the energy barriers of many-spinparticles and show that they can also be understood interms of the effective one-spin approximation. Inciden-tally, this allows us to establish the conditions for thevalidity of Eq. (1) which turns out to correctly describethe magnetic behavior of elongated particles only.

II. FROM THE ATOMISTIC TO THE

EFFECTIVE ENERGY

A. The atomistic model

We consider the atomistic model of a magneticnanoparticle consisting of N classical spins si (with|si| = 1) taking account of its lattice structure, shape,and size7,12,14–21. The magnetic properties of the parti-cle can be described by the anisotropic Heisenberg model

H = −1

2J

∑

ij

si · sj + Hanis, (2)

where J is the nearest-neighbor exchange interaction andHanis contains core and surface anisotropies.

The surface anisotropy is often thought to favor thespin orientation normal to the surface. This is the so-called transverse anisotropy model (TSA). However, amore solid basis for understanding surface effects is pro-vided by the Neel surface anisotropy (NSA),1,8,9,18 whichtakes into account the symmetry of local crystal environ-ment at the surface. The simplest expression for the NSAthat will be used below is, as the exchange energy, a dou-ble lattice sum over nearest neighbors i and j,

HNSAanis =

Ks

2

∑

ij

(si · uij)2, (3)

where uij are unit vectors connecting neigboring sites.One can see that for perfect lattices the contributionsof the bulk spins to HNSA

anis yield an irrelevant constant,and the anisotropy arising for surface spins only becauseof the local symmetry breaking is, in general, biaxial.For the simple cubic (sc) lattice and the surface parallelto any of crystallographic planes, an effective transversesurface anisotropy arises for Ks > 0. In all other casesNSA cannot be reduced to TSA. In fact, Eq. (3) canbe generalized so that it describes both surface and bulkanisotropy. It is sufficient to use different constants Ks

for different bond directions uij to obtain a second-ordervolume anisotropy as well. However, to obtain a cubicvolume anisotropy that is fourth order in spin compo-nents, a more serious modification of Eq. (3) is required.Thus, for simplicity, we will simply use Eq. (3) to de-scribe the surface anisotropy and add different kinds ofanisotropy in the core.

For the core spins, i.e., those spins with full coordi-nation, the anisotropy energy Hanis is taken either asuniaxial with easy axis along z and a constant Kc (peratom), that is

Hunianis = −Kc

∑

i

s2i,z, (4)

or cubic,

Hcubanis =

1

2Kc

∑

i

(

s4i,x + s4i,y + s4i,z)

. (5)

3

Dipolar interactions are known to produce an ad-ditional “shape” anisotropy. However, in the atom-istic description, their role in describing the spin non-collinearities is negligible as compared to that of all othercontributions. In order to compare particles with thesame strength of anisotropy in the core, we assume thatthe shape anisotropy is included in the core uniaxialanisotropy contribution. We also assume that in theellipsoidal particles the magnetocrystalline easy axis isparallel to the elongation direction.

In magnetic nanoparticles or nanoclusters, the num-ber of surface spins Ns is comparable to or even largerthan the number of core spins Nc. In addition, the sur-face anisotropy has a much greater strength than thecore anisotropy because of the local symmetry break-ing. We use the core anisotropy value typical for Cobalt,Kc ' 3.2 × 10−24 Joule/atom. Numerous experimen-tal results4,10,11,13 show that the value of the surfaceanisotropy in Co particles embedded in different matricessuch as alumina, Ag or Au, as well as in thin films andmultilayers, could vary from Ks ' 10−4J to Ks ' J. Inthe present study we will consider the surface anisotropyconstant Ks as a variable parameter. For illustration wewill choose values of Ks 10-50 times larger than that ofthe core anisotropy, in accordance with those reportedin Ref. 4 for Co particles embedded into alumina matri-ces. Such values of Ks are also compatible with thosereported in Refs. 29–31. All physical constants will bemeasured with respect to the exchange coupling J , so wedefine the following reduced anisotropy constants

kc ≡ Kc/J, ks ≡ Ks/J. (6)

The core anisotropy constant will be taken as kc '0.01 and kc ' 0.0025. On the other hand, the sur-face anisotropy constant ks will be varied. Note thatwe are using atomistic temperature-independent con-stants for anisotropies that should not be confused withtemperature-dependent micromagnetic anisotropy con-stants K(T ). The relation between atomistic and micro-magnetic uniaxial and cubic anisotropy constants withinthe mean-field approximation can be found in Ref. 32.

B. The effective energy

Investigation of the magnetization switching of a parti-cle consisting of many atomic spins is challenging becauseof the multidimensionality of the underlying potentialthat can have a lot of minima of different topologies, con-nected by sophisticated paths. Examples are the “hedge-hog” structures realized in the case of a strong enoughsurface anisotropy that causes strong noncollinearity ofthe spins. Obviously in this case Eq. (1), relevant in theone-spin description of the problem, cannot be a goodapproximation. Thus an important question that ariseshere is whether it is possible to map the behavior of amany-spin particle onto that of a simpler model systemsuch as one effective magnetic moment. Analysis of the

energy potential is unavoidable since it is a crucial step incalculating relaxation rates and thereby in the study ofthe magnetization stability against thermally-activatedreversal.

In the practically important case of dominating ex-change interaction, or equivalently, only a small non-collinearity of the spins, the problem dramatically sim-plifies, so that the initial many-spin problem can be re-duced to an effective one-spin problem (EOSP). In thefirst approximation, one can consider spins as collinearand calculate the contribution of the surface anisotropyto the energy of the system depending on the orientationof its global magnetization m, (|m| = 1). The resultingenergy scales as the number of surface spins, is linear inKs and depends on the crystal structure and shape. Thiswe refer to as the first-order surface-anisotropy energy E1.Together with the core anisotropy energy (per spin)

Ec =Nc

NKc

{

−m2z, uniaxial

12

(

m4x +m4

y +m4z

)

, cubic(7)

E1 can lead to Eq. (1). However, for crystal shapes suchas spheres or cubes E1 vanishes by symmetry. In Ref.18

it was shown that for an ellipsoid of revolution with axesa and b = a(1 + ε), ε� 1, cut out of an sc lattice so thatthe ellipsoid’s axes are parallel to the crystallographicdirections, the first-order anisotropy is given by

E1 = −Kuam2z, Kua ∼ −

Ns

NKsε. (8)

where the z axis is assumed to be parallel to the crystal-lographic axis b. That is, E1 scales with the particle sizeas ∼ 1/N 1/3 ∼ 1/D. One can see that, for the uniax-ial core anisotropy along the elongation direction of theellipsoid, Ks < 0 and ε > 0, Eq. (1) follows. On thecontrary, in other cases, as for example, ε > 0,Ks > 0Eq. (1) does not apply.

If one takes into account the noncollinearity of thespins that results from the competition of the exchangeinteraction and surface anisotropy and is described by theangles of order δψ ∼ N 1/3Ks/J, a contribution of secondorder in Ks arises in the particle effective energy.18 Thespin noncollinearity depends on the orientation of m andresults in the effective cubic anisotropy

E2 = Kca

(

m4x +m4

y +m4z

)

, Kca ∼ κK2

s

zJ, (9)

where z is the number of nearest neighbors and for thesc lattice one has κ ' 0.53466. This equation was ob-tained analytically for Ks � J in the range of particlesizes large enough (N � 1) but small enough so that δψremains small. Numerical calculations yield Kca slightlydependent on the size since the applicability conditionsfor Eq. (9) are usually not fully satisfied. The ratio ofthe second- to first-order surface contributions is

E2

E1

∼Ks

J

N 1/3

ε. (10)

4

It can be significant even for Ks � J due to the com-bined influence of the large particle size and small de-viation from symmetry, ε � 1. Since Kca is nearly sizeindependent (i.e., the whole energy of the particle scaleswith the volume), it is difficult to experimentally distin-guish between the core cubic anisotropy and that dueto the second-order surface contribution (see discussionlater on). The reason for the size independence of Kca

is the deep penetration of spin noncollinearities into thecore of the particle. This means that the angular depen-dence of the noncollinear state also contributes to theeffective anisotropy. Interestingly this implies that theinfluence of the surface anisotropy on the overall effec-tive anisotropy is not an isolated surface phenomena andis dependent on the magnetic state of the particle. Wenote that this effect is quenched by the presence of thecore anisotropy which could screen the effect at a distanceof the order of domain wall width from the surface.

Taking into account the core anisotropy analytically todescribe corrections to Eq. (9) due to the screening ofspin noncollinearities in the general case is difficult. How-ever, one can consider this effect perturbatively, at leastto clarify the validity limits of Eq. (9). One obtains27 anadditional mixed contribution that is second order in Ks

and first order in Kc

E21 = Kcsm g(m), Kcsm ∼ κNsKcK

2s

J2(11)

where g(m) is a function of mα which comprises, amongother contributions, both the 2nd- and 4th-order contribu-tions in spin components27. For example for an sc lattice,g(θ, ϕ = 0) = − cos2 θ+3 cos4 θ−2 cos6 θ, which is shownlater to give agreement with numerical simulations. Thismixed contribution, called here the core-surface mixing

(CSM) contribution, should satisfy Kcsm . Kca whichrequires

NsKc/J . 1. (12)

This is exactly the condition that the screening length(i.e., the domain-wall width) is still much greater than

the linear size of the particle, δ ∼√

J/Kc & D ∼ N 1/3.For too large sizes the perturbative treatment becomesinvalid.

Thus we have seen that in most cases the effectiveanisotropy of a magnetic particle, considered as a sin-gle magnetic moment, can be approximately described asa combination of uniaxial and cubic anisotropies18,21,27.Consequently, collecting all these contributions anddefining the EOSP energy as

EEOSP =1

J(Ec + E1 + E2 + E21) (13)

one can model the energy of a many-spin particle as

EEOSP = −keffuam

2z −

1

2keffca

∑

α=x,y,z

m4α. (14)

The subscripts ua/ca stand for uniaxial/cubic anisotropy,respectively. The effective anisotropy constants are nor-malized to the exchange constant J , according to thedefinition Eq. (6). Note that we have changed the signof the cubic anisotropy constant to be consistent withmore customary notations.

We would like to remark here that the effective energypotential (14) is an approximation to the full energy of amany-spin particle, especially with respect to CSM con-tribution (11). More precisely, the function g(m) in (11)contains terms of various orders (see the expression in thetext et seq.), and thus fitting the many-spin particle’s en-ergy to the effective potential (14) amounts to truncatingthe function g(m) to a 4th−order polynomial. As such,when the orders of the anisotropy contributions of the ini-tial many-particle match those of the effective potential(14), they get renormalized by the corresponding contri-butions from (11). Indeed, in Ref. 21, where the energyof a many-spin spherical particle with uniaxial anisotropyin the core and TSA on the surface was computed usingthe Lagrange-multiplier technique (see below), it turnedout that the core anisotropy is modified. However, whenthere is no uniaxial anisotropy in the initial many-spinparticle, i.e., when the core anisotropy is cubic and theparticle is perfectly symmetrical (no elongation), trun-cating the function g(m) to the 4th−order generates anartificial uniaxial anisotropy, though very small. On theother hand, even if the core anisotropy is not uniaxialbut the particle presents some elongation, the effectiveenergy does exhibit a relatively large uniaxial contribu-tion induced by the surface due to the term in Eq. (8).Therefore, the 2nd-order term with the coefficient keff

ua inEq. (14) stems, in general, from the two contributions(8) and (11). Similarly, the 4th-order term with the co-efficient keff

ca comprises the contribution (9) from the sur-face, and part of the CSM contribution in Eq. (11), andalso a contribution from the core if the latter has a cubicanisotropy.

We now are going to numerically calculate the effectiveenergy of spherical, ellipsoidal and truncated octahedralmagnetic particles cut from a lattice with sc, fcc, andhcp structures. We will plot this energy as a functionof the polar angles of the net magnetic moment m ofthe particle and fit it to Eq. (14). From these fits weextract the effective anisotropy constants keff

ua and keffca

and compare their behavior with those predicted by theanalytical formulas discussed above in the case of the sclattice. We will investigate the differences between theresults for different lattice structures and particle shapes.

III. NUMERICAL METHOD AND RESULTS

A. Computing method

As mentioned above, the problem of studying the mul-tidimensional energy landscape of the multispin particleis, in general, very difficult. However, if the exchange

5

J is dominant over the anisotropy and the spin non-collinearity is small, one can, to a good approximation,describe the particle by its net or global magnetization m

(|m| = 1) as a slow master variable. All other variablessuch as spin noncollinearities quickly adjust themselvesto the instantaneous direction of m. Thus one can treatthe energy of the particle as a function of m only. Tech-nically this can be done with the help of the Lagrangemultiplier technique18 by considering the augmented en-ergy F = H−Nλ·(ν−m), where ν ≡

∑

i si/|∑

i si| andH is the atomistic energy of the particle, Eq. (2). TheLagrange-multiplier term produces an additional torqueon the atomic spins that forces the microscopic net mag-netization ν to coincide with m. The equilibrium stateof the spin system is determined by solving the Landau-Lifshitz equation (without the precession term and withthe damping coefficient α = 1) and an additional equa-tion for λ

dsi

dt= − [ si × [si × Fi]] ,

dλ

dt= −N (ν − m), (15)

where the effective field Fi = −∂F/∂si depends on λ.It is worth noting that the stationary points of F foundwith this method are also stationary points of the actualHamiltonian H. Indeed, for these orientations of m noadditional torque is needed to support this state, thusthe solution of our equations yields [λ × m] = 0. For allother directions, the unphysical Lagrange-multiplier fieldintroduces distortions of the microscopic spin configura-tion. Nevertheless, if the exchange is dominant, these dis-tortions remain small. A further advantage of this tech-nique is that it can produce highly non-collinear multi-dimensional stationary points18,21,22 in the case of strongsurface anisotropy. Here, again, these stationary pointsare true points because of the condition [λ × m] = 0. Inorder to check the correct loci of the saddle points, wecomputed the eigenvalues and gradients of the Hessianmatrix associated with the Hamiltonian of Eq. (2), withthe results that are exactly the same as those obtainedby the much simpler Lagrange-multiplier method.

In this work we show that the magnetic behavior ofsmall particles is very sensitive to the surface arrange-ment, shape of the particles and underlying crystallo-graphic structure. To investigate the various tenden-cies, we have considered particles cut from lattices withthe simple cubic (sc), body-centered cubic (bcc), face-centered cubic (fcc), hexagon closed-packed (hcp). Al-though experimental studies providing transmission elec-tron microscopy images show particles resembling trun-cated octahedra1,2, making realistic particle shapes andsurface arrangements proves to be rather complex. Trun-cated octahedra have been included in our studies as anideal case for fcc crystals. The reality is somewhat morecomplicated, though. In Ref. 2, in order to interpretthe experimental results of the 3D-dimensional switch-ing field curve, the so-called Stoner-Wohlfarth astroid,it was assumed that a few outer layers in the truncatedoctahedral particle were magnetically “dead”, leading to

FIG. 1: Two particles cut from fcc structure: spherical (left)and truncated octahedron (right).

an effective elongation and thereby to a non-perfect oc-tahedron. Producing such a faceted elongated particleby somehow cutting the latter is an arbitrary procedure.In order to minimize the changes in the surface structurecaused by elongation, we assumed a spherical particle orintroduced elliptical elongation along the easy axis. Thiskind of structure has been the basis of many theoreticalstudies using the Heisenberg Hamiltonian (see, e.g., Refs.7,14,17–21,23).

Regarding the arrangement at the particle surface, anappropriate approach would be to use molecular-dynamictechniques24,25 based on the empirical potentials for spe-cific materials. This would produce more realistic non-perfect surface structures, more representative of what itis hinted at by experiments. However, these potentialsexist only for some specific materials and do not fullyinclude the complex character of the surface. Moreover,the particles thus obtained (see, e.g., Ref. 26), may havenon-symmetric structures, and may present some dislo-cations. All these phenomena lead to a different behaviorof differently prepared particles which will be studied ina separate publication.

In the present work, in order to illustrate the generaltendency of the magnetic behavior, we mostly presentresults for particles with “pure” non modified surfaces,namely spheres, ellipsoids and truncated octahedra cutfrom regular lattices. Even in this case, the surface ar-rangement may appear to be very different (see Fig. 1)leading to a rich magnetic behavior.

B. Spherical particles

We compute the 3D energy potential as a function ofthe spherical coordinates (θ, ϕ) of the net magnetizationm of a many-spin particle. We do this for a spherical par-ticle with uniaxial anisotropy in the core and NSA, cutfrom sc, fcc and hcp lattices, and for different values ofthe surface anisotropy constant, ks. Fig. 2 shows energylandscapes for spherical particles cut from an sc lattice.One can see that as ks increases the global minima moveaway from those defined by the core uniaxial anisotropy,i.e., at θ = 0, π and any ϕ, and become maxima, whilenew minima and saddle points develop which are remi-

6

FIG. 2: Energy potentials of a spherical many-spin particle of N = 1736 spins on an sc lattice with uniaxial anisotropy in thecore (kc = 0.0025) and NSA with constant (a) ks = 0.005, (b) ks = 0.112, (c) ks = 0.2, (d) ks = 0.5.

niscent of cubic anisotropy. Now, in Fig. 3 we present thecorresponding 2D energy potential (ϕ = 0). From thisgraph we see that the energy of the many-spin particleis well reproduceed by Eq. (14) when ks is small, whichshows that such a many-spin particle can be treated asan EOSP with an energy that contains uniaxial and cu-bic anisotropies. However, as was shown in Ref. 21, whenthe surface anisotropy increases to larger values this map-ping of the many-spin particle onto an effective one-spinparticle fails.

Repeating this fitting procedure for other values of ks

we obtain the plots of keffua and keff

ca in Fig. 4. We firstsee that these effective constants are quadratic in ks, inaccordance with Eqs. (9) and (11). In addition, the ploton the right shows an agreement between the constantkeffua , excluding the core uniaxial contribution in Eq. (7).

These results confirm those of Refs. 21,27,28 that the coreanisotropy is modified by the surface anisotropy, thoughonly slightly in the present case.

Comparing the energy potential in Fig. 2 for the sclattice and that in Fig. 5 for the fcc lattice one re-alizes that, because of the fact that different underly-ing structure produces different surface spin arrange-ments, the corresponding energy potentials exhibit dif-ferent topologies. For instance, it can be seen that thepoint θ = π/2, ϕ = π/4 is a saddle in particles cut froman sc lattice and a maximum in those cut from the fcc lat-tice. Spherical particles cut from the sc lattice exhibit aneffective four-fold anisotropy with keff

ca < 0 [see Fig. 4 andEq. (14)]. As such, the contribution of the latter to the

effective energy is positive, and this is compatible withthe sign of Kca in Eq. (9). In Figs. 6, 7 we plot the 2Denergy potential and the effective anisotropy constants,respectively, for a spherical particle with fcc structure.For a spherical fcc particle, the effective cubic constantkeffca is positive [see Fig. 7], and as for the sc lattice, it is

quadratic in ks. As mentioned earlier, the coefficient κin (9) depends on the lattice structure and for fcc it maybecome negative. To check this one first has to find ananalytical expression for the spin density on the fcc lat-tice, in the same way the sc lattice density was obtainedin Ref. 18 [see Eq. (6) therein]. The corresponding de-velopments are somewhat cumbersome and are now inprogress. Likewise, the coefficient κ in Eq. (11) shouldchange on the fcc lattice, thus changing the uniaxial andcubic contributions as well.

Finally, in particles with the hcp lattice and large sur-face anisotropy, we have found that the effective energypotential is six-fold, owing to the six-fold symmetry in-herent to the hcp crystal structure. The global magneti-zation minimum is also shifted away from the core easydirection.

C. Ellipsoidal particles and effect of elongation

Now we investigate the effect of elongation. As dis-cussed earlier, due to the contribution in Eq. (8), even asmall elongation may have a strong effect on the energybarrier of the many-spin particle, and in particular on

7

FIG. 3: (Shifted) 2D energy potentials of a spherical many-spin particle of N = 1736 spins on an sc lattice with uniaxialanisotropy in the core (kc = 0.01) and NSA with constant ks = 0.1 (left), 0.5 (right). The solid lines are plots of Eq. (14).

FIG. 4: Effective anisotropy constants against ks for a spherical many-spin particle of N = 1736 spins cut from an sc lattice.The panel on the right shows the CSM contribution obtained numerically as keff

ua , excluding the core uniaxial contribution fromEq. (7). The thick solid lines are plots of Eqs. (9) and (11) with ϕ = 0.

the effective uniaxial constant keffua , as will be seen below.

Fig. 8 shows the energy potential of an ellipsoidal many-spin particle with aspect ratio 2:3, cut from an fcc lat-tice. Unlike the energy potentials of spherical particles,the result here shows that for large surface anisotropythe energy minimum corresponds to θ = π/2. Indeed,due to a large number of local easy axes on the surfacepointing perpendicular to the core easy axis, the totaleffect is to change this point from a saddle for small ks

to a minimum when ks assumes large values. The effec-tive uniaxial and cubic anisotropy constants are shownin Fig. 9. As expected the effective uniaxial constantshows a strong linear variation and even changes sign atsome value of ks, as opposed to the case of a sphericalmany-spin particle. On the other hand, as for the lattercase, the constant keff

ca retains its behavior as a quadraticfunction of ks. Again, in the case of an sc lattice keff

ca < 0and on an fcc lattice keff

ca > 0.

D. Truncated octahedral particles

Here we consider the so-called truncated octahedralparticles as an example of a minimum close-packed clus-ter structure. Real particles are often reported as havingthis structure with fcc or bcc underlying lattice [see, e.g.Co or Co/Ag particles in Ref. 1,2,4].

Regular truncated octahedrons having six squares andeight hexagons on the surface have been constructed cut-ting an ideal fcc lattice in an octahedral (two equal mutu-ally perpendicular pyramids with square bases parallel tothe XY plane) and subsequent truncation. Equal surfacedensities in all hexagons and squares can be obtained ifthe fcc lattice is initially rotated by 45 degrees in the XYplane (the X-axis is taken parallel to the (1,1,0) direc-tion and the Z-axis to the (0,0,1) direction). We performthe same calculations as before for a many-spin particlecut from an fcc lattice, but now with cubic single-site

8

FIG. 5: Energy potentials of a spherical many-spin particle with uniaxial anisotropy in the core (kc = 0.0025) and NSA withconstant (a) ks = 0.005, (b)ks = 0.1, (c) ks = 0.175, (d) ks = 0.375. The particle contains N = 1264 spins on an fcc lattice.

FIG. 6: The same as in Fig. 3 but here for the fcc lattice and ks = 0.025, 0.375.

anisotropy in the core and NSA. The results presented inFig.10 show vanishing uniaxial contribution and the mod-ification of the cubic anisotropy by surface effects. It canbe seen that, similarly to the results discussed above, theeffective cubic constant is again proportional to k2

s forsmall ks. This is mainly due to the two contributions,one coming from the initial core cubic anisotropy and theother from the surface contribution as in Eq. (9). It is in-teresting to note that the surface contribution can changethe sign of the initially negative cubic core anisotropyconstant. We can also observe an asymmetric behavior

of the effective anisotropy constants with respect to thechange of sign of the surface anisotropy which we found,in general, in all particles with fcc underlying lattice.

If the fcc lattice is initially oriented with crystallo-graphic lattice axes parallel to those of the system ofcoordinates, then different atomic densities are createdon different surfaces. This way the surface density alongthe XY circumference is different from that along theXZ one. In Fig. 11 we plot the dependence of the effec-tive cubic constant keff

ca as a function of ks for kc > 0 andkc < 0. The surface contribution can again change the

9

FIG. 7: Effective anisotropy constants against ks for a spher-ical particle of N = 1264 spins cut from an fcc lattice withuniaxial core anisotropy kc = 0.0025. The lines are guides tothe eye.

sign of the initially negative cubic core anisotropy con-stant. Besides, we clearly see that the many-spin particledevelops a negative uniaxial anisotropy contribution, in-duced by the surface in the presence of core anisotropy.This constitutes a clear example of importance of thesurface arrangement for global magnetic properties of in-dividual particles.

IV. ENERGY BARRIERS

A. Dependence of the energy barrier on ks

Now we evaluate the energy barriers of many-spin par-ticles by numerically computing the difference betweenthe energy at the saddle point and at the minimum, us-ing the Lagrangian multiplier technique described ear-lier. On the other hand, the EOSP energy potential (14)can be used to analytically evaluate such energy barriersand to compare them with their numerical counterparts.Namely, we have investigated minima, maxima and sad-dle points of the effective potential (14) for different val-ues and signs of the parameters keff

ua and keffca and calcu-

lated analytically the energy barrier in each case. Theresults are presented in Table I. The energy barriers forthe case keff

ua > 0 are plotted in Fig. 12. We remark thatfor large surface anisotropy |ζ| >> 1, where ζ ≡ keff

ca /keffua ,

all energy barriers are simple linear combinations of thetwo effective anisotropy constants.

Note that in a wide range of the parameters severalenergy barriers (corresponding to different paths of mag-netization rotation) coexist in the system in accordancewith the complex character of the effective potential withtwo competing anisotropies [see Figs. 2(c), 8(c)]. Be-cause of this competition the symmetry of the anisotropycan be changed leading to relevant energy barriers in

the θ or ϕ direction, the former case is illustrated inFig. 8(a,b,c), where switching occurs between global min-ima at θ = 0 and θ = π and the latter is shown inFig. 8(d) where the large surface anisotropy has givenrise to an easy plane with the stable states correspond-ing to (θ = π/2, ϕ = n × π/2) where n is an integer. Insome cases there are multiple energy barriers, but herewe consider only the relevant energy barrier for switching,corresponding to the lowest energy path between globalminima.

We have seen that in the case of a spherical particlecut from an sc lattice, and in accordance with the EOSPenergy potential (14), keff

ua > 0 and keffca < 0 [see Fig. 4].

For a spherical particle with an fcc lattice, keffua > 0, keff

ca >0, as can be seen in Fig. 7. Finally, for ellipsoidal andtruncated octahedral many-spin particles, the results inFigs. 9, 11 show that keff

ua may become negative at somevalue of ks, since then the contributions similar to (8)and (11) become important.

Fig. 13 shows the energy barrier of a spherical particlecut from an sc lattice as a function of ks. The non-monotonic behavior of the energy barrier with ks followsquantitatively that of the EOSP potential (14). Indeed,the solid line in this plot is the analytical results (2) and(1.1) from Table I, using analytical expressions of Eq. (9)together with the pure core anisotropy contribution (7).The discrepancy at the relatively large ks is due to thefact that the analytical expressions are valid only if thecondition (12) is fulfilled; the CSM contribution has notbeen taken into account.

Fig. 14 represents the energy barriers against ks forparticles with different shapes and internal structures.First of all, one can see a different dependence on ks ascompared to particles with the sc lattice. In the presentcase, i.e., keff

ua > 0, keffca > 0, the energy barriers are given

by Eqs. (3) and (4.1) in Table I. Consequently, for smallvalues of ks with |ζ| < 1 and neglecting the CSM term,the energy barrier is independent of ks. Accordingly, thenearly constant value of the energy barrier, coincidingwith that of the core, is observed for particles in a largerange of ks. For larger ks, the energy barrier increases,since keff

ua > 0 for particles cut from an fcc lattice. At verylarge values of ks, i.e., ks & 100kc the energy barriersstrongly increase with ks and reach values larger thanthat inferred from the pure core anisotropy.

The energy barriers for ellipsoidal particles are shownin Fig. 15. Note that in this case the effective uniaxialanisotropy constant keff

ua is a linear function of ks, ac-cording to the analytical result (8) and the numericalresults presented in Fig. 9. Here, the energy barriers arenot symmetric with respect to the change of sign of ks.This is due to the fact that for ks < 0 the effective uni-axial constant is a sum of the core anisotropy and thefirst-order contribution owing to elongation. On the con-trary, when ks > 0, the “effective core anisotropy” keff

ua

is smaller than the pure core anisotropy Ec in Eq. (7).This means that at some ks k

effua may change sign. At

the same time the effective cubic anisotropy keffca remains

10

FIG. 8: Energy potentials of an ellipsoidal particle cut from an fcc lattice and with uniaxial anisotropy in the core (kc = 0.0025)and NSA with constant (a) ks = 0.0125, (b)ks = 0.075, (c) ks = 0.1, (d) ks = 0.175.

TABLE I: Energy barriers for the effective one-spin particle. The critical angle θc(ϕ) is defined by cos(θc(ϕ))2 = (keffua +

keffua (sin(ϕ)4 + cos(ϕ)4))/(keff

ca (1 + sin(ϕ)4 + cos(ϕ)4))

keffua > 0

ζ = keffca /keff

ua Minima(θ, ϕ) Saddle points(θ, ϕ) Energy barriers, ∆EEOSP

−∞ < ζ < −1 θc(π/4); π/4 π/2; π/4 keffua/3 − keff

ca /12 − (keffua )2/3keff

ca (1.1)

θc(π/4); π/4 θc(π/2); π/2 −keffua/6 − keff

ca /12 − (keffua )2/12keff

ca (1.2)

−1 < ζ < 0 0; 0 π/2; 0 keffua + keff

ca /4 (2)

0 < ζ < 1 0; π/2 π/2; π/2 keffua (3)

1 < ζ < 2 θc(0); π/2 θc(π/2); π/2 keffua/2 + keff

ca /4 + (keffua )2/4keff

ca (4.1)

π/2; π/2 θc(π/2); π/2 −keffua/2 + keff

ca /4 + (keffua )2/4keff

ca (4.2)

2 < ζ < ∞ 0; π/2 θc(π/2); π/2 keffua/2 + keff

ca /4 + (keffua )2/4keff

ca (5.1)

π/2; π/2 π/2; π/4 keffca /4 (5.2)

π/2; π/2 θc(π/2); π/2 −keffua/2 + keff

ca /4 + (keffua )2/4keff

ca (5.3)

keffua < 0

−∞ < ζ < −1 π/2; π/2 θc(π/2); π/2 −keffua/2 + keff

ca /4 + (keffua )2/4keff

ca (6.1)

π/2; 0 π/2; π/4 keffca /4 (6.2)

0; π/2 θc(π/2); π/4 keffua/2 + keff

ca /4 + (keffua )2/4keff

ca (6.3)

−1 < ζ < 0 π/2; 0 π/2; π/4 keffca /4 (7)

0 < ζ < 1 π/2; π/4 π/2; π/2 keffca /4 (8)

1 < ζ < 2 π/2; π/4 θc(π/2); π/2 −keffua/2 + (keff

ua )2/4keffca (9)

2 < ζ < ∞ θc(π/4); π/4 π/2; π/4 −keffua/6 − keff

ca /12 − (keffua )2/3keff

ca (10.1)

θc(π/4); π/4 θc(π/2); π/2 keffua/3 − keff

ca /12 − (keffua )2/12keff

ca (10.2)

positive and is proportional to k2s . Accordingly, at the

vicinity of the point at which keffua ≈ 0 (see Fig. 9), rapid

changes of the character of the energy landscape occur.The analysis, based on the EOSP potential shows thatwhen keff

ua > 0 the energy barriers of ellipsoidal particleswith sc lattice are defined by Eqs. (2) and (1.1) in Table

I and for negative keffua < 0 these are given by Eq. (7) and

(6.1) in Table I. Note that a regime of linear behaviorin ks exists for both ks < 0 and ks > 0 (see Fig. 15),specially when ks . 0.1(ζ � 1). In some region of theeffective anisotropy constants, e.g., keff

ua > 0, |ζ| < 1, theenergy barrier ∆EEOSP ≈ keff

ua , i.e., it is independent of

11

FIG. 9: Effective anisotropy constants against ks for an ellip-soidal particle of N = 2044 spins on sc and fcc lattices, withuniaxial core anisotropy kc = 0.0025. The lines are guides tothe eye.

FIG. 10: Effective anisotropy constants against ks for a reg-ular truncated octahedral particle of N = 1289 spins, fccstructure and cubic anisotropy in the core with kc > 0 andkc < 0.

the cubic contribution (ignoring the CSM term). Theinterval of these parameters is especially large in fcc par-ticles with ks < 0, for which keff

ua does not change signand the energy barriers are exactly defined by Eq. (3) inTable I.

B. Dependence of the energy barrier on the system

size

As N → ∞, the influence of the surface should be-come weaker and the energy barriers should recover thefull value KcN . Fig. 16 shows energy barriers against the

FIG. 11: Effective anisotropy constants against ks for a trun-cated octahedral particle with different atomic densities onsurfaces. The particle has N = 1080 spins, fcc structure andcubic anisotropy in the core with kc > 0 and kc < 0.

FIG. 12: Analytical energy barriers of the EOSP with thepotential (14) as functions of keff

ca /keffua = ζ with keff

ua > 0.Analytical formulas in Table I are labelled.

total number of spins N in particles of spherical shapecut from an sc lattice and with two values of ks > 0.First of all, we note that in this case the main contri-bution to the effective anisotropy consists of two terms:the core anisotropy and the surface second-order contri-bution (9). In agreement with this all energy barriers ofthese particles are always smaller than KcN since, as weshowed previously for the sc lattice, keff

ca is negative andthe energy barriers in this case are defined by Eq. (2) inTable I.

Both uniaxial core anisotropy (7) and the main con-tribution to the effective cubic anisotropy (9) scale withN . As N → ∞, the core anisotropy contribution slowly

12

FIG. 13: Energy barrier as a function of ks for a sphericalparticle cut from an sc lattice. The particle contains N =20479 spins and has the uniaxial core anisotropy kc = 0.0025.The solid line is a plot of the analytical expressions (1.1) and(2) from Table 1, using Eqs. (7) and (9).

FIG. 14: Energy barriers against ks for truncated octahedralparticles cut from an fcc lattice (N = 1688) and sphericalparticles cut from the fcc (N = 1289) and hcp (N = 1261)lattices. Uniaxial core anisotropy with kc = 0.0025 is as-sumed.

recovers its full value, i.e., Ec/(KcN ) → 1. However,from the analytical expressions Eqs. (7) and (9), evenwhen N → ∞, when neglecting the CSM contribution,∆E/(KcN ) should approach the value 1 − κk2

s/12kc,which is independent of the system size. Hence, we mayconclude that it is the CSM contribution (11) that is re-sponsible for the recovery of the full one-spin uniaxialpotential. Being very small, this contribution producesa very slow increase of the energy barrier with the sys-tem size. In fact, we have estimated that even sphericalparticles of diameter D = 20 nm (an estimation basedon the atomic distance of 4 A) would have an effectiveanisotropy ∆E/(KcN ) that is 13% smaller than that of

FIG. 15: Energy barriers versus ks of ellipsoidal particles withdifferent aspect ratio (a/c = 0.6667,N = 21121, and a/c =0.81,N = 21171, with uniaxial core anisotropy kc = 0.0025.The solid lines are linear fits.

FIG. 16: Energy barrier as a function of the total number ofspins N for two different values of the surface anisotropy,spherical particles cut from the sc lattice with uniaxialanisotropy kc = 0.0025 in the core. The inset shows a slowdependence of the difference between these results and theuniaxial one-particle energy barrier KcN in the logarithmicscale. The lines in the inset are the analytical expressions inEqs. (7) and (9).

the bulk.

Truncated octahedra [see Fig. 17] show a behavior sim-ilar to that of the spherical particles. The energy barriersin this case behave very irregularly due to the rough vari-ation of the number of atoms on the surface. The sameeffect was observed in other particles of small sizes. Fortruncated octahedra this effect arises as a consequence ofnon-monotonic variation of the number of spins on thesurface for particles cut from regular lattices. The ef-fective anisotropy of truncated octahedra particles withlarge ks > 0 is larger than the core anisotropy in accor-dance with the fact that keff

ca is positive for fcc structures

13

FIG. 17: Energy barriers versus N for truncated octahedrawith internal fcc structure and uniaxial core anisotropy withkc = 0.0025.

FIG. 18: Energy barriers as a function of the particle sizefor ellipsoidal particles with internal sc structure, uniaxialanisotropy kc = 0.0025 and different values of ks.

and the energy barriers are defined by Eqs. (4.1) and(5.1) in Table I.

Finally, in Fig. 18 we present the energy barriers of el-lipsoidal particles with different values of ks and internalsc structure. The energy barriers in this case are definedby formulas (1.1) and (2) in Table I. The main contri-bution comes from the effective uniaxial anisotropy. Thecorrection to it due to elongation is positive when ks < 0and negative in the opposite case. Consequently, parti-cles with ks < 0 have energy barriers larger than that in-ferred from the core anisotropy, and for those with ks > 0the energy barriers are smaller. In this case, the energybarrier approximately scales with the number of surfacespins Ns (see Fig. 19), in agreement with the first-ordercontribution from elongation (11).

C. Applicability of the formula Keff = K∞ + 6Ks/D

The results presented above show that in the most gen-eral case, studied here, of a many-spin particle with NSA,this formula is not applicable, for the following reasons:(i) It assumes that the overall anisotropy of the parti-cle remains uniaxial. However, we have shown that thesurface anisotropy induces an additional cubic contribu-tion. (ii) It assumes that the surface anisotropy alwaysenhances that in the core. In the previous section wesaw that both situations can arise. (iii) It is implic-itly based on the hypothesis that the core and surfaceanisotropies are additive contributions. As we have seenabove for large ks (Table I) the energy barrier indeed canbe represented as a sum of the effective cubic and uniaxialanisotropies. However, the cubic anisotropy term is pro-portional to k2

s , which is inconsistent with formula (1).(iv) It assumes a linear dependence of energy barriers onthe parameter 1/D, or equivalently Ns/N .

Consequently, spherical or octahedral particles cannotbe described by formula (1), since in this case (i) No termlinear in ks is obtained. (ii) No term scales as the ratio ofthe surface-to-volume number of spins Ns/N . However,in the case of elongated particles with a not too largesurface anisotropy, (i.e., |ζ| < 1 for fcc lattice or |ζ| �1 for sc lattice), the energy barriers are independent ofthe effective cubic anisotropy. In this case, for weaklyellipsoidal particles, for example, we may write

∆EEOSP = keffua ≈ kcNc/N +A|ks|/N

1/3 (16)

where A is a parameter that depends on the particle elon-gation and surface arrangement, and which is positive forks < 0 and negative in the opposite case. Hence, the be-havior is as predicted by formula (1). An approximatelylinear behavior in Ns/N was also observed in the caseof large surface anisotropy ζ >> 1 (see Fig. 12). How-ever, when N → ∞, the “uniaxial anisotropy term” K∞

is modified by the effective cubic anisotropy keffca ∼ k2

s .In Fig. 19 we plot the energy barriers of small ellip-soidal particles with sc structure, aspect ratio 2:3, andks < 0 from Fig. 18. For such particles, formula (1)should be modified as Keff = K∞ + |Ks|Ns/N . Accord-ingly, in Fig. 19 we plot the energy barrier against Ns/N .These data are highly linear, especially when small par-ticle sizes are removed, as shown by the fit in Fig.19. Wenote that in the case of relatively small surface anisotropyks = −0.041 (though 17 times larger than in the core),the full core anisotropy K∞ = Kc/v ( v is the atomicvolume) can be extracted. However, for the larger sur-face anisotropy ks = −0.1125, K∞ is renormalized bythe surface contribution (9). On the other hand, it is notpossible to extract the value of ks, since the exact pro-portionality coefficient of Eq. (11) is dependent on theparticles surface arrangement and elongation. The effec-tive anisotropy constant Ks obtained from this fit is muchsmaller than the input value, namely, for ks/kc = 45 weobtain from the fit (ks/kc)eff = 4.3.

14

FIG. 19: Linear fit of energy barriers, versus the surface-to-total number of spins NS/N , of ellipsoidal particles withaspect ratio 2:3 with ks = −0.041 (circles) and ks = −0.1125(squares).

V. CONCLUSIONS

We have studied in detail effective anisotropies and en-ergy barriers of small magnetic particles within the Neel’ssurface anisotropy model. The present calculations havebeen performed in a many-spin approach allowing devi-ations of the spins from the collinear state. They showthat the magnetic behavior of small particles is rich andstrongly dependent on the particle surface arrangement.The particular structure of each particle and the strengthof its surface anisotropy makes each particle unique andits magnetic properties different in each case.

Our calculations show that the magnetic behavior ofnanoparticles with Neel surface anisotropy and underly-ing cubic lattices is consistent with the effective modelof one-spin particle with uniaxial and cubic anisotropies.The strength of this additional cubic anisotropy is depen-dent on many parameters, including the shape and elon-gation of the particles, and the underlying crystal struc-ture which produces a different surface arrangement. Theanalytical results have made it possible to classify the var-ious surface contributions and their effects as: (i) Firstorder contribution from elongation (8), which producesan additional uniaxial anisotropy, is proportional to ks

and scales with the number of surface atoms. (ii) Sur-face second-order contribution (9) which is cubic in thenet magnetization components, proportional to k2

s andscales with the particle’s volume; (iii) core-surface mix-ing contribution (11), which is smaller than the other twocontributions and scales with both surface and volume ofthe particle.

For particles with sc lattices we compared analyticaland numerical calculations for many-spin particles ob-taining a very good agreement. Numerical modelling ofparticles with other structures has confirmed the general

character of these effects. The possibility to describe avariety of many-spin particles by a macroscopic magneticmoment with effective uniaxial and cubic anisotropiesconstant opens a unique possibility to model a collectionof small particles in a multiscale manner, taking into ac-count surface effects through the effective potential (14).

Several very interesting effects were observed in par-ticles with strong surface anisotropy. Particles withmagneto-crystalline uniaxial anisotropy develop cubicanisotropy. At the same time the uniaxial anisotropyis also modified by the surface anisotropy. In ellip-soidal particles, surface anisotropy can change the signof the effective uniaxial anisotropy. Some signaturesof these behaviors can be found in the literature. Forexample in Ref. 1, the magnetic behavior of Co parti-cles with fcc structure and, presumably, cubic magneto-crystalline anisotropy, have demonstrated the effect ofuniaxial anisotropy.

The energy barriers of many-spin particles have beenevaluated using the Lagrangian-multiplier technique.Their behavior could be well understood with the helpof the effective one-spin potential (14). The energy bar-riers larger than KcN have been obtained for all parti-cles with very large surface anisotropy, ks & 100kc, orfor elongated particles with ks < 0. This confirms a well-known fact that the surface anisotropy may contribute tothe enhancement of the thermal stability of the particle.However, in the case ks > 0, the strength of the surfaceanisotropy has to be very strong.

We have found that the effective anisotropy extrapo-lated from the energy barriers measurements is consis-tent with formula (1) only for elongated particles. In thecase of relatively weak surface anisotropy ks, the value ofthe core anisotropy could be correctly recovered. How-ever, for larger ks this effective uniaxial anisotropy K∞ isrenormalized by the surface. The applicability conditionsof formula (1) are never fulfilled in spherical or truncatedoctahedral particles. The control of the parameters gov-erning effective anisotropies does not seem to be possiblein real experimental situations and therefore, the extrac-tion of the parameters based on formula (1) in some casesmay be unreliable.

On the other hand, we should note here that the con-clusions of our work have been drawn on the basis of theNeel anisotropy model [see also Ref. 21 for the case oftransverse surface anisotropy]. We would like to empha-size that the model itself has been based on microscopicconsiderations. Although later there appeared some at-tempts to justify the model (see e.g. Ref. 9), in mostof cases it remains ad-hoc and not based on the realspin-orbit coupling considerations. A more adequate ap-proach to model magnetic properties of nanoparticlesshould involve first principle calculations of the mag-netic moments and MAE with atomic resolution, like inCo/Cu5,33. However at the present state of the art thistask remains difficult. Moreover, it is not clear how suchmodels could be used to calculate, for example, thermalproperties of small particles. The multiscale hierarchical

15

approach34 proposes to incorporate the ab-initio calcula-tions into classical spin models. We conclude that morework on theory is necessary to understand surface mag-netism, especially in relation with small particles.

Acknowledgments

This work was supported by a joint travel grant of

the Royal Society (UK) and CSIC (Spain) and by theIntegrated Action Project between France (Picaso) andSpain. The work of R.Yanes and O.Chubykalo-Fesenkohas been also supported by the project NAN-2004-09125-C07-06 from the Spanish Ministry of Science and Edu-cation and by the project NANOMAGNET from Comu-nidad de Madrid. D. A. Garanin is a Cottrell Scholar ofResearch Corporation.

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