Dipolar field and energy anisotropy in magnetic thin films

a

PHYSICAL REVIEW B 68, 184410 ~2003!

Dipolar field and energy anisotropy in magnetic thin films

E. Estevez-Rams,1,2,* J. Martinez-Garcia,3 J. Martinez-Garcia,1 J. Hiram-Espina,1 and H. Fuess21Instituto de Materiales y Reactivos, Universidad de la Habana (IMRE), San Lazaro y L., Caixa Postal 10400, Habana, Cub

2University of Technology Darmstadt, Institute for Materials Science, Petersenstrasse 23, D-64287 Darmstadt, Germany3Facultad de Fisica-IMRE, Universidad de La Habana, San Lazaro y L., Caixa Postal 10400, Habana, Cuba

~Received 17 April 2003; published 11 November 2003!

The formalism for dipolar field calculations in thin films is addressed. It is shown that in the limit of verythin films the separation of the dipolar field contribution perpendicular to the film plane cannot be split into ashape contribution and a lattice contribution. The failure in recognizing this leads to the wrong interpretationof magnetic-susceptibility measurements. It is shown that lattice summation can be carried out generalizing theEwald-Born procedure to the case when the magnetization and the point where the field is being calculated arenot constrained to the dipolar planes. This procedure avoids the direct summation of lattice dipoles and the splitof the summation into a discrete part and a continuum integration. The formalism is applied to the 2H and 3Rclose-packed stacking arrangements.

DOI: 10.1103/PhysRevB.68.184410 PACS number~s!: 75.70.Ak, 75.30.Gw, 45.10.Na

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I. INTRODUCTION

The nanoscale is increasingly becoming the focusmuch attention in magnetic research. The emergence of sing new phenomena at the nanoscale has drawn particattention. Especially attractive is the potential realizationmaterials with new properties, or a combination of propertnot found at larger scales.

The orientation of the magnetization in a magnetizbody is determined by the competing effect of the exchaand dipolar energies. The exchange energy tends, in theof a ferromagnet, to align the dipole moments of the atoparallel to each other. The dipolar energy, on the other hafavors the compensation of the dipolar contribution by tryito lower the occurrence of free poles at the body surfaceattempting to reduce the internal self-magnetic-field.

It has been argued that the continuum approximationthe dipolar field and dipolar energy calculations in magnebodies breakdown for the nanoscale sizes.1–4 The discussioninvolves the validity of the demagnetizing tensor formalisfor dipolar calculations in nanoscale particles and ultratfilms.5 The demagnetizing tensor is introduced in the cotinuum theory to solve the Maxwell equations inside tmagnetized body, while the field outside the materials isessentially affected by this consideration. The magnetic fiHmed inside the uniformly magnetized body is related to tapplied fieldHappl and the magnetizationM through the re-lation

Hmed~r!5Happl2D~r!•M, ~1!

whereD is the so-called demagnetizing tensor,D tensor orshape tensor.6,7 The D tensor is in general a function of thposition for an arbitrarily shaped body, but can be constanthe case of ellipsoidal bodies, where it can be analyticacalculated.8 The D tensor, as defined through Eq.~1!, doesnot depend on the size of the magnetized body but onlythe geometry of the body and the position at which the fiis being calculated.6

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In this paper we address the mathematical formalismdipolar calculations in thin films and its physical implications. The reason for doing so is, on one hand, the wealtinteresting magnetic behaviors discovered in such systwhich has made them a very attractive research area~see, forexample, Refs. 9,10, and references cited therein!. On theother hand, there seems to be a sort of confusion indefinition and use of the tensorial magnitudes involved indipolar calculations of these nanoscale objects.4,5,9,11–14

The most striking behavior in some magnetic thin films,perhaps, the occurrence of out-of-the-plane magnetizafor certain thickness range.4,15This surprising result has beeexplained by the change of magnetocrystalline anisotrwith film thickness.15,16 Other properties, independent of thdimensions for ‘‘bulk’’ materials, can also show thicknedependence in thin films, such is the case of the Cutemperature.14

While nanoparticles have reduced their size to the naneter range in all three dimensions, thin films can be consered objects where the nanoscale is realized in only onemension perpendicular to the film plane. In general, oneconsider nanoscale thin films as being built by the arranment of one to several atomic layers. In this limit, somauthors have argued that, contrary to the continuum approgiven by Eq.~1!, the discrete nature of the lattice must btaken into account in any dipolar calculation.1–4 Based onsusceptibility measurements, other authors have questiothe approximation of discrete dipoles.9,12Vedmedenkoet al.5

have analyzed the discrepancy between the continuumproximation and the direct discrete sum of dipole momein platelet shaped nanoparticles. To account for the thickndependence of the demagnetizing dipole field, they introdan additional factor~thickness dependent!, multiplying theDtensor in Eq.~1!. Then theD tensor is taken as a functioonly of the sample geometry.

Even in the case of bulk objects, the local field at a powithin an object has to go beyond Eq.~1! and consider thediscrete nature of the atoms.6 The calculation of the locafield then reduces to considering the contribution arisfrom each atom carrying dipolar moment and the contrib

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E. ESTEVEZ-RAMSet al. PHYSICAL REVIEW B 68, 184410 ~2003!

tion of the applied field. In the case of crystalline materiathe periodic arrangement of the dipole moments allowsto transform the otherwise slower convergent dipole suma faster convergent series. This is done by different maematical procedures, perhaps the Ewald-Born techniqueing the more commonly applied.17

In the case of thin films, dipolar sums have been generdone by direct summation. The field at a point is foundselecting a disk of dipoles around the point and performthe discrete sum over them, while the contribution fromrest of the body is found by the integration ofcontinuum.2–4 The reason for not performing the EwaldBorn summation in the general case, when the magneticments have components outside the film plane and the pwhere the field is being calculated can be anywhere in spis the complication arising from the fact that the pointr isnot in the film plane. We will show that in spite of this, aEwald-Born procedure can be developed for thin films, evin the case of magnetic dipole moments not constrainethe film plane. The use of such a formalism is not only coputationally more efficient, but also physically more soun

The developed formalism will be applied to the caseclose-packed structures with a 3R and 2H stacking order.While it is generally believed that dipolar interaction favothe alignment of the dipole moments in the plane of the fiDraaisma and Jonge2 have shown for a tetragonal structuthat for decreasing interlayer/intralayer dipole distance rathe dipolar interaction could favor magnetization out of tfilm plane. It will be shown that this will also be the case fclose-packed structures.

The paper is organized as follows. In Sec. II the dipofield formalism in the discrete three-dimensional lattice wbe reviewed to introduce the definition of dipole sum tensLorentz tensor, andD tensor, all central to the formalismlater developed. The section will also help us in the discsion about the validity of the use of theD tensor in thecontinuum approach. In Sec. III a similar formalism as in tpreceding section will be developed for the case of tfilms. The total magnetic susceptibility for the case of thfilms will be discussed in Sec. IV. In Sec. V an Ewald-Bortype procedure will be presented for the lattice sum of dipmoments in a two-dimensional lattice. The obtained eqtions will be valid for any crystal system. The case of clospacked 3R and 2H thin films will be discussed in Sec. VIThe results will be summarized in Sec. VII.

II. THE DIPOLE SUM TENSOR OF A MAGNETIZEDPARTICLE

The dipolar field at a pointr due to a single dipolem inthe origin can be written as

H51

4pm•,H ,S 1

r D J , ~2!

whereH is the local dipolar field andr is the absolute valueof the position vectorr. If every lattice point in a crystacarries the same dipolar moment, then the dipolar local fiin a point r due to the assembly of dipoles will be given b

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H loc~r!5C~r!•M, ~3!

whereM is the magnetization or magnetic moment per uvolume andC is the so-called dipole sum tensor,6

C~r!5Vdip

4p (¯

t,H ,S 1

ur2tu D J . ~4!

The bar over the sum indicates that the finite sum is mover all the lattice pointst inside the crystal, avoiding thesingularity if r happens to be a lattice node.Vdip is the vol-ume associated with each dipole (M5m/Vdip).

According to definition~4!, the dipole sum tensor is symmetric, and being real it is always possible to write it dowin diagonal form. It is also possible to show that in aorthonormal base the dipole sum tensor is traceless.

The dipolar energy density for an ellipsoidal shaped bowhereC attains the same value at every lattice point, wthen be

UD52 12 m0M•C•M. ~5!

In order to decompose the contributions of the latticeone side and the geometry of the magnetized body onother, the concept of Lorentz field is introduced followinthe transformation6

(¯

t,H ,S 1

ur2tu D J 2¹H ¹S 1

VdipE

V

d3t

ur2tu D J5 (¯

t¹H ¹H 1

ur2tu2

1

VdipE

Vdip(t)

d3t

ur2tu D J'(

t

8

¹H ¹S 1

ur2tu2

1

VdipE

Vdip(t)

d3t

ur2tu D J , ~6!

and the last sum is now made over the infinite lattice. Tprime over the summation symbol indicates that ifr happensto be a lattice vector, the singularity is avoided in the sumation. The last step where a sum made over the latnodes interior to the finite object is extended to a sum othe whole infinite lattice is a valid approximation as longthe pointr under consideration is far from the surface of tbody. In such a case the term$1/ur2tu21/Vdip*Vdip(t)d

3t/ur2tu% will tend to zero fort values far fromr.

We now define the Lorentz tensor as

L~r!5Vdip

4p,$,w~r!%

5Vdip

4p (t

8

,H ,S 1

ur2tu2

1

VdipE

Vdip(t)

d3t

ur2tu D J , ~7!

wherew is the so-called Lorentz potential. From the defintion of the Lorentz tensor and Eq.~4!, the dipolar sum tensocan be written as

C~r!5L~r!2D~r!, ~8!

D is again theD tensor.

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DIPOLAR FIELD AND ENERGY ANISOTROPY IN . . . PHYSICAL REVIEW B68, 184410 ~2003!

D~r!521

4p,H ,S E

V

d3t

ur2tu D J . ~9!

In an orthonormal base theD tensor and the Lorentz tensor will have a trace equal to one.

The local dipolar field can now be divided into a contbution due to the lattice and a contribution due to the geoetry of the body:

H loc~r!5L~r!•M2D~r!•M. ~10!

For an ellipsoidal body whereD(r)5D, the correspond-ing dipolar energy density will be given by

UD52 12 m0M•L•M1 1

2 m0M•D•M, ~11!

where the Lorentz tensor is evaluated at a lattice point.The extension to a lattice with several atoms associate

each lattice node is straightforward. For each atom carrydipole moment within the unit cell, we can write an expresion similar to Eq.~4!. If every atom carries the same dipolmoment, the field at a pointr can be written as in Eq.~3!, butnow the effective dipole sum tensor will be given by the suof the dipole sum tensors corresponding to each atom.

Ce f f~r!5(rp

C~r2rp!, ~12!

where rp is the position vector of thep atom carrying adipole moment in the unit cell.

The dipolar energy density for an ellipsoidal shaped bocan also be formally written as in Eq.~5!; in this case theenergy dipole sum tensor will be given by

C5(rp

Ce f f~rp!. ~13!

Ce f f andC will also be traceless tensors in an orthonormbase.

In terms of the Lorentz and demagnetizing tensors,expressions for the effective tensors will be

Le f f~r!5(rp

L~r2rp!,

De f f~r!5(rp

D~r2rp!, ~14!

which can be used for calculating the Lorentz and demagtizing field at a pointr. For the energy density calculation thcorresponding expression can be written as

L5(rp

Le f f~rp!, ~15!

D5(rp

De f f~rp!5P2D, ~16!

whereP is the total number of dipoles in the unit cell and tlast expression in the definition ofD is valid for an ellipsoi-dal shaped body.

18441

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The first term in Eq.~11! is usually taken into consideration as part of the magnetocrystalline anisotropy andsecond term is the one usually referred to as the shapeisotropy energy. According to Eq.~9!, the D tensor is sym-metric, andC also being symmetric the Lorentz tensor wbe symmetric. Both theD tensor and the Lorentz tensor cabe written in diagonal form. The connection between tcontinuum equations for the field in the magnetized mediand the microscopic discrete dipolar sum defining the lofield H loc can be made through Eqs.~10! and ~1!.

III. THE DIPOLE SUM TENSOR IN THE CASE OFULTRATHIN FILMS

There is a subtle approximation in the expression fordipolar energy~11!. This definition comes from the definitionof the Lorentz and theD tensor and, therefore, will be validas long as the contribution resulting from the dipoles neathe surface is negligible, compared to the dipoles whereproximation ~6! of the Lorentz field is valid~Lorentz di-poles!.

When the size of the magnetized body is reduced tonanoscale, and the number of surface dipoles are of the oof the Lorentz dipoles, the split of the dipolar energy into ttwo contributions, one to the magnetocrystalline energy athe other to the shape anisotropy energy, will no longervalid. The approximation made in Eq.~6! will be valid for ~atmost! as many dipoles as those where it is not valid. In sua case the definition for aD tensor and the Lorentz tensoloses physical significance, while the definition for theCtensor will still remain valid.

In the continuum approximation the crystal nature of tmagnetized body is completely hidden. The anisotropy risfrom the ~lack of! symmetry of the crystal is taken into account through the empirically determined magnetic anisropy energy term, usually written as a Taylor expansion18

Therefore, the lattice dipolar energy term given by the fiterm in Eq.~11! is contained in the Taylor expansion coefcientski for the magnetocrystalline anisotropy energy. In tcase of a nanosized body this is no longer valid. This inaity to split the lattice term from the shape contribution terand absorb the former into the magnetocrystalline anisotrenergy, is the main reason for the failure of the continuapproach for calculating the dipolar energy.

The above discussion for the general nanosized bodythus be followed to the case of magnetic films with onseveral monatomic layers of thickness. One could first csider the contribution of one monatomic layer and write

2C~r!5Vdip

4p (t

,H ,S 1

ur2tu D J , ~17!

where the sum in this case is over the layer plane andpoint r is not necessarily contained in the layer. We can fmally proceed as before and try to split the2C into twocomponents by making use of a similar transformation asEq. ~6!, but then being the integral over the layer plane. Tresults in a similar expression as Eq.~8! but the Lorentztensor 2L and theD tensor 2D are now given by

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2L~r!5Vdip

4p,$,w2~r!%

5Vdip

4p (t

8

,H ,S 1

ur2tu2

1

AdipE

Adip(t)

d2t

ur2tu D J~18!

2D~r!521

4p,H ,S Vdip

AdipE

A

d2t

ur2tu D J . ~19!

2L and 2D are traceless tensors in an orthonormal baThe dipolar field due to the contribution of all the laye

in the film can be calculated as

H~r!52Le f f~r!•M22De f f•M, ~20!

where

2Le f f~r!5 (w50

nl21

2L~r2Rw! ~21!

and

2De f f~r!5 (w50

nl21

2D~r2Rw!, ~22!

nl being the number of layers in the film andRw the positionof the w layer with respect to the origin.

The dipolar energy will then be given by

UD52 12 m0M•2L•M1 1

2 m0M•2D•M, ~23!

where we have written

2D5(t

2De f f~ t!, ~24!

2L5(t

2Le f f~ t!, ~25!

and the sum is carried out over every node in the film caing dipolar moment.

If the layers in the thin film can be considered effectiveinfinite, then the2D tensor given by Eq.~19! will vanish. Itwill also follow that for each layer,Le f f(t) will take the samevalue at each lattice node, and Eq.~25! reduces to

2L51

nl(w50

nl21

(j 50

nl21

2L~Rj2Rw!. ~26!

Formally, Eq.~23! is equivalent to Eq.~11! but, strictlyspeaking,2L is not a pure Lorentz tensor. In the definitioof 2Le f f there is a demagnetizing contribution which arisfrom the sum along the layers stacking direction. In the sasense, the definition of the demagnetizing tensor givenEq. ~22! does not contain the information on the film thicness. Contrary to the bulk case,2Le f f and 2De f f are tracelesstensors in a Cartesian base. In accordance with the discus

18441

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at the beginning of the section, no split of the dipole sutensor can be made along the directions where the nanosis realized.

We can split the different contributions of the layers2L into two types, those layers which can be consideinterior to the film and those which can be consideredlonging to the surface

2L5S 12ns

nlDL int1

ns

nlLsur f, ~27!

wherens is the number of layers which can be considerbelonging to the surface.

The difference in dipolar energy between the configution with magnetization perpendicular to the film surfacand that within the film surface, will determine the energecally favorable configuration. Let us call such a differenDUD . A negative value ofDUD means that the perpendicular configuration is favored, while a positive value is an idication that the dipolar energy is lower with the magnetiztion lying in the plane of the film.

When studying the dipolar interaction in nanosized thfilms, the demagnetizing tensor has been sometimes usethe dipole sum tensor. This leads to several errors, in socases it is assumed that the trace of the calculated teshould be one in a Cartesian system, when in such coordisystem, the dipole sum tensor is traceless. The relatioincorrectly used to calculate only one value of the diagoelement and deduce the others from the calculated onesymmetry of the problem, and the assumed trace.12 In otherapproaches such as that of Vedmedenkoet al.,5 the confusionbetween the dipole sum tensor and the demagnetizing teleads us to introduce an artificial termX. The new term isused to account for the thickness dependence of the dipsum tensor in order to fulfill the trace relation for the demanetizing tensor. In reality, they are trying to force the diposum tensor to behave as a demagnetizing tensor. Thereneed for introducing such additional term which is not jusfied from physical grounds.

IV. MAGNETIC SUSCEPTIBILITY

Different susceptibilities can be defined in the hystereloop of a magnetic material. For the discussion that followthe total susceptibilityxa defined throughM5xa•Ha will besufficient.Ha will be the applied field andM the measuredmagnetization. In general,xa is a function of the magnetization.

We could also define a susceptibility response functxm for the individual layers in the thin film,M5xm•H loc .Now H loc is the local field, given by Eq.~20!, sensed at thelayer. The measured susceptibilityxa can be related to thesusceptibility of each layerxm by

xa5H nl I2(m

xm•2Ce f fmJ 21

•H(m

xmJ , ~28!

I is the identity tensor,nl is again the number of layers, an2Ce f fm

52Le f fm22De f fm

. If we assume, for the sake of simplicity, that all terms in Eq.~28! are scalars, then

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xa5

(m

xm

nl2(m

xm 2Ce f fm

. ~29!

In general, the magnetization in all layers need not besame. If there is a strong exchange coupling betweenmagnetic moments of neighboring layers, then the magnzation at each layer in a thin film can be assumed to behcollectively, and the productxm 2Ce f fm

will have the samevalue independent ofm. Even in the case where the couplinbetween magnetic moments of neighboring layers is nostrong, for the qualitative discussion that will follow we catake an average value of the productxm 2Ce f fm

and use it foreach layer independent ofm. In both cases we can write

xa5S 1

nlD (

mxm

12x0 2Ce f f0

. ~30!

where the subscript 0 refers to the value taken at any laFrom the constant character ofxm 2Ce f fm

, we can write for

the mth layer

xm5x0

2Ce f f0

2Ce f fm

and Eq.~30! can then be written as

xa51

nl

x0 2Ce f f0

12x0 2Ce f f0(m

1

2Ce f fm

5x0 2Ce f f0

12x0 2Ce f f0K 1

2Ce f fL . ~31!

where

K 1

2Ce f fL 5

1

nl(m

1

2Ce f fm

.

Let us take 2Ce f f05^1/2Ce f f&

21[2Cav while keeping

xav 2Cav5x0 2Ce f f0to the correct value, then we can wri

for Eq. ~31!,

xa5xav

12xav 2Cav, ~32!

if the susceptibility at the layers tends to infinity,xav willalso tend to infinity, then the limit for the measured susctibility will be

limxo→`

xa521

2Cav. ~33!

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For a given applied field direction and resulting magnezation, we can derive from Eq.~32! and~33! several differentcases, given as follows.

~1! xav 2Cav,0: The measured susceptibility is smallthan the average internal susceptibility. This comes as asult of the demagnetizing field prevailing over the Lorenfield. The dipoles sense a smaller field than the applied oAt zero applied field, the dipoles sense a ‘‘negative’’ fiefrom the rest of the dipole moments. The applied field hasovercome such a field for the dipoles to sense a positive fiin the direction of the applied one. The magnitude of tinternal demagnetizing field that needs to be overcomeproportional to2Cav . In the limit of an infinite internal sus-ceptibility, the measured susceptibilityxa cannot be greatethan Eq.~33!.

~2! 2Cav50: This very special case corresponds to tshape of the sample tuned to cancel the Lorentz field.measured susceptibility and the internal average suscepity are the same.

~3! 0,xav 2Cav,1: The measured susceptibility is largthan the internal average susceptibility. The Lorentz fiprevails over the demagnetizing field. The dipole momesense a larger field than the applied one.

~4! 1,xav 2Cav : The measured susceptibility is negativThis does not make physical sense and therefore, whenLorentz field prevails over the demagnetizing field, the intnal average susceptibility cannot become infinity and expsion ~33! loses its physical meaning.

The Lorentz field is fixed by the lattice of the layers. Thdemagnetizing field, on the other hand, is a function ofsample geometry in the plane of the film. This geometfactor is determined by the size and shape of the areas ofilm where the layers can be considered periodic. When this a zero demagnetizing term2Dav , corresponding to aneffective infinite layer, it will correspond to case~IV ! and theinternal average susceptibilityxav cannot have a valuegreater than 1/2Cav51/2Lav .

Contrary to the continuum approach, the demagnetizterm 2D alone does not impose an upper limit to the susctibility, but the contribution of the whole dipolar sum tenshas to be taken into account instead. It should be noticedthis is also valid for the tridimensional bulk case. Failingconsider the whole contribution of the dipolar sum tensor clead to an apparent contradiction between the measmaximum values of susceptibility, and the calculated demnetizing factors.12,9 The contradiction is removed if propeaccount is taken of the whole dipolar sum tensor.

V. DIPOLAR FIELD OF THE ATOMIC LAYER

As explained in the Introduction, Lorentz dipolar sumgiven by Eq.~7! for the tridimensional case, are not usuacarried out by direct sum. The direct sum of terms in Eq.~7!is slowly convergent, and suitable transformations of theries can yield more rapidly convergent series.

In the case of two-dimensional layers, Colpa6,7 has shownan Ewald-Born-type calculation when the magnetization vtor is contained in the plane of the layers and the fieldbeing calculated at a point within the layer. In this section

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will generalize the approach used by Colpa, lifting the costraint over the magnetization direction and the pointwhich the field is being calculated. In this case, althoughsum in Eq.~18! remains two dimensional, the fact that thpoint at which the field is calculated is anywhere in sparenders the problem a tridimensional one. The procedurefollows will be valid for any crystal system.

Consider a two-dimensional lattice where each noderies a dipolar moment. We intend to calculate the Loretensor given by Eq.~18! at a pointr not necessarily contained in the layer plane. The vectorr can be decomposed athe sum of two vectors, one lying in the plane of the layrab , and the other one perpendicular to the layer planerc~Fig. 1!.

Making use of the identity

1

uzu5

2

ApE

0

`

e2z2a2dazÞ0, zPR,

the expression for thew2 potential defined in Eq.~18! canbe written as

w2~r!52

ApE

0

`

daH e2r c2a2

(t

8

e2ut2rabu2a2

21

Adip(

t

8 EAdip(t)

e2ur2r8u2a2J , r c[urcu.

~34!

The term

T1~rab ,a!52

Ap(

t

8

e2ut2rabu2a2~35!

FIG. 1. Diagram of an atomic layer and the decompositionthe vectorr, pointing to where the field is calculated, into twvectors.rab is contained in the plane of the layer, whiler c is per-pendicular to the layer plane.

18441

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r-z

,

will be periodic over the two-dimensional lattice and, therfore, can be written as a Fourier series,

T1~rab ,a!5(g

F~g,a!e2p ig•rab, ~36!

whereg is the reciprocal vector dual to the two-dimensionlattice of the layer.

The Fourier coefficients will be given by

F~g,a!51

AdipE

Adip

T1~rab ,a!e22p ig.rabd2rab ~37!

by substituting Eq.~35! into Eq.~37!, and using the fact thaexp@2pig•t#51 we obtain

F~g,a!52Ap

Adip

e24p2g2/(4a2)

a2; g[ugu ~38!

for the Fourier coefficients and the expression forT1 can bewritten as

T1~rab ,a!52Ap

Adip(

g

e24p2g2/(4a2)

a2e2p ig•rab. ~39!

The Ewald-Born summation technique17,10 consists insplitting the integration interval overa in Eq. ~34!, into twointervals, one from 0 to§, and the other one from§ to `, theresulting two integrals can be called2w2(r) and `w2(r),respectively.

The term `w2(r) can be directly integrated

`w2~r!52

Ap(

t

8 E§

`

daH e2ut2ru2a2

21

AdipE

Adip(t)e2ur2r8u2a2

d2r2J5(

t

8 erfc~§ur2tu!ur2tu

22Ap

§Adip,

where erfc(x) is the complementary error function whicdecays exponentially for large values ofx. `w2(r) will rep-resent the contribution of the ‘‘short-range’’ term to the dpolar sum.

The 0w2(r) can be evaluated making use of thT1(rab ,a) Fourier expansion in reciprocal space~39!. Inte-gration of this term leads to

f

0w2~r!52Ap

Adip

e2§2r c2211§r cAperf~§r c!

§1

1

2Adip(

g

8 e22pgrcerfc~pg/§2§r c!1e2pgrcerfc~pg/§1§r c!

g.

The prime over the summation symbol again indicates that singularities in the summations are avoided.As § can take any arbitrary value, in practice it is tuned to guarantee a rapid convergence of0w2(r).The final expression forw2(r) is then

0-6


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Adip

e2§2r c2211§r cAperf~§r c!

§1

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8 erfc~§ur2tu!ur2tu

22Ap

§Adip. ~40!

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Contrary to a previous work,7 the above deduction doenot make any particular assumption of the direction ofmagnetization vector. It can be seen that Eq.~40! is reduced,for r c50, to the expression reported by Colpa.7

The Lorentz tensor will now be given by

2L~r!5Vdip

4p,$,w2~r!%. ~41!

If r is a lattice point, an additional term 4/(3Ap)§3G,whereG is the metric tensor used to describe the layer lattmust be added.

The use of the Ewald-Born expansion~40! avoids theslow direct summation over the thin-film lattice dipolewhere the film is split into a discrete part aroundr, and acontinuum contribution far from the point where the fieldbeing calculated.2

FIG. 2. ~a! Normalized dipolar field perpendicular to the filmplane, sensed at each layer in a 15 layer thick, 3R arrangement. Thefield is given for different (c/a) values.~b! Same as~a! but for a2H arrangement. With increasing (c/a) the difference between thfield at the outer layers and at the inner ones decreases signific

18441

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,

VI. CLOSE-PACKED 3 R AND 2H ARRANGEMENTS

The previous two sections have laid down the geneformalism to study the dipole field in thin films with homogeneous magnetization. In this section this formalism willused to study the magnetic dipole energy for two clopacked arrangements, the 2H and the 3R.

Close-packed structures are formed by stacking layeratoms forming a compact hexagonal net in the plane. Tdistance between atoms in the layer, which is denoted ba,can be considered to be equal to the diameter of the atoClose-packed layers can be stacked one over the othethree different positions, usually labeled asA, B, andC. Eachletter corresponds to a different lateral displacement~a pos-sible letter assignment and lateral displacementrs could beA→rs50, B→rs51/3a21/3b, and C→rs521/3a11/3b). The distance between consecutive layers inideal case follows the relation (c/a)252/3, where we havecalled c the interlayer distance. In real structures thereusually a departure from the idealc/a ratio.

tly.

FIG. 3. Normalized energy difference between the magnettion perpendicular to the film plane and parallel to the film planplotted against the inverse of the number of layers. Each cucorresponds to a differentc/a value. ~a! 3R arrangement, and~b!2H arrangement. The change of slope for each curve indicchange from surface to bulk layer.

0-7

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FIG. 4. ~a! Slope~m! and ~b! intercept (DU`) of the linear region ofDUD plotted againstc/a values. For smallc/a values, the slopeof the 3R and the 2H arrangement differs significantly. With increasingc/a values, the discrete nature of the layers is lost and both cutend to the same values. NegativeDU` , showing energy preference for the out-of-the-plane magnetizations, can be reached for smc/avalues. Again, above a certain interlayer-to-intralayer distance ratio, the behaviors of the 3R and the 2H arrangement are identical.

c

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A number of thin films stack according to the sequenABCABC. . . which leads to a rhombohedral 3R crystalsystem~the face-centered cubic could be considered a pticular case of such an arrangement, with the stacking ocring along the cubic 111& direction!. Another common ar-rangement is the 2H hexagonal compact structurecharacterized by a stacking sequenceABABAB. . . alongthe @0001# direction.

Figure 2~a! showsHloc at each layer normalized by thmagnetization value, in a 15-layer 3R thin film, for differentc/a ratio. The thin film is considered effectively infinite in itlateral dimensions. The magnetization is taken as a veperpendicular to the film surface and the normalized fivalue corresponds to the same direction.

For (c/a)251/36, the dipoles between layers are cloenough to sense the discrete nature of the layers abovebelow them. The field at the surface layers is quite intereing, while the absolute value of the dipolar field is smalfor most external layers compared with the layer at the cter, the field at the second and third layers on each sidactually larger than the field at the center layer. Two regioare clearly distinguishable, the first four layers on each ssense a different field value among them and with the celayer, these layers can be considered belonging to theface. The following seven layers including the center oneall under the same dipole field and can be considered beling to the bulk of the material.

Already at (c/a)251/6 the number of surface layers hreduced to three on each surface side, and the field asurface is smaller than at the bulk. For (c/a)251/3 the fieldfelt at the bulk layers is greater than in (c/a)251/6 case,dropping again for the ideal ratio of (c/a)252/3. When(c/a)252/3 only the most external layers on each surfa

18441

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side sense a different dipole field than the rest. At lar(c/a)2 ratios the layers are so far apart that a distinctbetween surface and bulk layers can not be made.

Figure 2~b! is similar to Fig. 2~a! but now for the 2Harrangement. The distinctive feature of the 2H arrangementwith (c/a)251/36 is the field positive sign for the bulk layers. The surface layers sense a smaller field than that obulk. For (c/a)251/6 the field has changed to negative vaues and, again, the surface layers sense a smaller field.increasing (c/a)2 ratio, the field behavior at each layer resembles that of the 3R structure. The layers do not see thdiscrete nature of the neighboring dipole layers and thereare insensitive to the particular layer arrangement.

Figure 3~a! plots DUD /(m0M2/2) against the inverse othe film thickness for different values ofc/a in a 3R arrange-ment. A linear relation can be found for the whole 1/nl rangefor (c/a)2.1/3. For (c/a)251/12,1/6,1/3 the difference inenergy shows a linear behavior above a certain numbelayers. If we refer to Eq.~27!, we can understand that thlinear region corresponds to film thickness where everyditional layer ‘‘added’’ will be a bulk layer, and the termLint andLsur f remain, together withns , constant. The onlythickness dependence will be given by the 1/nl term in Eq.~27!. In the case of (c/a)251/12,1/6,1/3 for thickness belownl54,3,2, there are no bulk layers in the film, each layadded will change the value ofLsur f in Eq. ~27!, as well aschangingnl and ns . The linear dependence on (1/nl) istherefore lost. The value of 1/nl at which the linear dependence starts can be used to determine the number of lawhich contributes to the surface.

Figure 3~b! is equivalent to Fig. 3~a! but now for the 2Harrangement. With an increase in thec/a value, a point is

0-8

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reached where the dipolar field is insensitive to the acstacking arrangement of the layers and the values of theference in energy for the 2H and 3R arrangements are thsame.

The slope and intercept for the linear part of tDUD /(m0M2/2) versus 1/nl curves are shown in Fig. 4. Thslope is seen to decrease with increasingc/a value, indicat-ing that the difference in energy becomes more insensitivthe film thickness.

The intercept of the curve, on the other hand, will give tdifference in energy for the case of an ‘‘infinite’’ thick crystal. The values obtained are in agreement with the onesculated for a bulk crystal using usual infinite crystprocedures.6 This can be seen as a test of the validity of tsummation procedure described in Sec. V. Above a cerc/a ratio, the behaviors of the 3R and 2H arrangement areidentical. The interlayer/intralayer dipole distance ratiosuch that the discrete nature of the layers is lost fromlayer to the other.

For small values ofc/a the sign of the difference in energy interceptDU` becomes negative and the dipolar enerbecomes an out-of-plane magnetization. The interlayertance has become so close compared to the distance betdipoles in a layer that alignment of the dipoles perpendicuto the layer planes is now favored. Again, above a cerc/a ratio, the behaviors of the 3R and 2H arrangement areidentical.

VII. CONCLUSIONS

In the analysis of demagnetizing dipolar interactionnanoparticles or in thin films, it is often not realized that tdipolar contribution to the magnetic field cannot be split in

*Electronic address: [email protected]. Benson and D.L. Mills, Phys. Rev.178, 839 ~1969!.2H.J.G. Draaisma and W.J.M. de Jonge, J. Appl. Phys.64, 3610

~1988!.3B. Heinrich, S.T. Purcell, J. Dutcher, K.B. Urquhart, J.F. Cochr

and A.S. Arrot, Phys. Rev. B38, 12 879~1988!.4M. Farle, A. Berghaus, and K. Baberschke, Phys. Rev. B39, 4838

~1989!.5E.Y. Vedmedenko, H.P. Oepen, and J. Kirschner, J. Magn. Ma

Mater.256, 237 ~2003!.6J.H.P. Colpa, Physica~Amsterdam! 56, 185 ~1971!.7J.H.P. Colpa, Physica~Amsterdam! 56, 205 ~1971!.8J.A. Osborn, Phys. Rev.67, 351 ~1945!.9M. Farle, Rep. Prog. Phys.61, 755 ~1998!.

10K. de Bell, A.B. MacIsaac, and J.P. Whitehead, Rev. Mod. Ph72, 225 ~2000!.

18441

alif-

to

al-

in

e

ys-eenr

in

a geometrical factor and a lattice-dependent factor alongdimensions where the nanoscale is realized.

We have shown how the dipolar sum formalism canextended to the case of ultrathin films. The full accountthe dipolar sum tensor solves the apparent contradictiontween the experimental susceptibility measurements anddemagnetizing factor calculations found in previous worIn the same sense, it is shown that there is no need for inducing an additional thickness-dependent factor in the tenexpressions.

We have extended the Ewald-Born procedure to the cof a thin film where the magnetization direction is not rstricted to the film plane.

The developed tools were applied to the case of thin filwith 3R and 2H arrangement. In the limit of very thickfilms, the results agree with bulk calculations, confirming tvalidity of the approach. With decreasing interlayer distana limit is reached below which dipolar interactions favor tout-of-the-plane magnetization. The obtained (c/a)2 valuewhere this threshold is found is far below the ideal 2/3 vafor compact structure of only one atomic element. It shobe noticed that in the case of more than one atomic elemthe threshold value for the out-of-the-plane magnetizatcan be reached for larger interlayer distances. As a resulcritical magnitude is the interlayer/intralayer dipole distanratio, as has already been shown in the sections above.

ACKNOWLEDGMENTS

One of the authors~E.E.R.! acknowledges financial support from the Humboldt Foundation. Part of this project wcarried out under TWAS Research Grant No. 99-082 REPHYS/LA and under an Alma Mater Grant from the Univesity of Havana.

,

n.

.

11C.D.H. Williams, D. Evans, and J.S. Thorp, J. Magn. MagMater.73, 123 ~1988!.

12A. Aspelmeier, M. Tischer, M. Farle, M. Russo, K. Baberschkand D. Arvanitis, J. Magn. Magn. Mater.146, 256 ~1995!.

13U. Stetter, M. Farle, K. Baberschke, and W.G. Clark, Phys. Rev45, 503 ~1992!.

14M. Farle, K. Baberschke, U. Stetter, A. Aspelmeier, and F. Ghardter, Phys. Rev. B47, 11 571~1993!.

15G. Andre, A. Aspelmeier, B. Schulz, M. Farle, and K. Babeschke, Surf. Sci.326, 275 ~1995!.

16B. Schulz, A. Aspelmeier, and K. Baberschke, Vacuum46, 1189~1995!.

17P.P. Ewald, Ann. Phys.~Leipzig! 49, 117 ~1916!.18D. Craik, Magnetism, Principles and Applications~J Wiley,

Chichester, 1995!.

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Dipolar field and energy anisotropy in magnetic thin films