Micromagnetic analysis of exchange-coupled hard-soft planar nanocomposites

PHYSICAL REVIEW B 69, 174401 ~2004!

Micromagnetic analysis of exchange-coupled hard-soft planar nanocomposites

Giovanni Asti, Massimo Solzi, Massimo Ghidini, and Franco M. NeriINFM and Dipartimento di Fisica dell’Universita`, Parco Area delle Scienze 7/A, I-43100 Parma, Italy~Received 5 November 2003; revised manuscript received 13 February 2004; published 4 May 2004!

A complete magnetic phase diagram for exchange-coupled planar hard-soft nanocomposites has been ob-tained in the frame of a one-dimensional micromagnetic model describing the dependence of the propertiesalong the growth direction. The phase diagram in terms of layer thicknesses provides information on the typeof demagnetization processes and the critical fields at which nucleation and reversal take place. The basiccriterion to this purpose is the analytical expression we have obtained of the critical susceptibility at thenucleation field. The phase diagram is divided into three regions: the exchange-spring magnet~ES!, the rigidcomposite magnet~RM!, and the decoupled magnet~DM!. The main boundary line is an U-shaped linecorresponding to divergence of the critical susceptibility. The diagram also reports the isocritical field linesboth for the nucleation and the reversal field. These lines bifurcate along the RM boundary line. The essentialcharacteristics of the phase diagram are directly connected with the intrinsic properties of the chosen soft andhard materials. With increasing ratio between the anisotropy constants of soft to hard phases the ES region isreduced until it disappears at a critical value. The model includes as limiting cases the classical problems of theplanar soft inclusion in a bulk magnet and of the domain-wall depinning at the hard-soft interface.

DOI: 10.1103/PhysRevB.69.174401 PACS number~s!: 75.50.Ww, 75.60.Jk, 75.60.Ej, 75.70.Cn

teit

mnS

ha

termmedson

an-

etoein

hcusicc

aeea

vio

therenteti-ys-h ases.rop-theytly

an-theed

etic

I. INTRODUCTION

In the present technological scenario the microsystechnologies are becoming more and more important wparticular focus on microelectromechanical syste~MEMS! and micromachines. Typical applications are sesors, actuators, accelerometers, relays, bioMEMS, rfMEMand microfluidic and microwave devices. In this context tdevelopment of MEMS based on high performance permnent magnets is very promising. This goal has stimularesearch towards the development of new concepts in penent magnetism, without any constraints concerning theterial cost. In recent years some attention has been devotcomposite magnets combining the best properties of aand a hard magnetic material via exchange coupling onanometric scale~exchange-spring magnet,1 see Fig. 1 be-low!. In particular it has been shown that oriented plancomposite systems in the form of multilayers offer in priciple the possibility of achieving the 1 MJ/m3 theoreticallimit for the energy density.2,3 The planar magnetic nano-composites are technologically relevant, at least in persptive. As an example, the millisize engines and microactuahave great potentialities in fields like bioengineering. Othapplications could concern high-density magnetic recordand microwave integrated devices.

A multilayer is an intrinsically simple ideal system witan easily controllable and reproducible structure. In partilar, thin-film growth allows the control of layer thicknesand, in principle, it provides a mean for crystallographalignment of the hard-phase easy axis. The dimensional sparameter is the ferromagnetic exchange lengthl ex, whichcan be considered as roughly proportional to the domain w~DW! average thickness. In the case of very hard magnthis quantity is only a few nanometers, while it is of the ordof several tenths of nm for soft magnetic materials, sucha-Fe. In order to predict the macroscopic magnetic beha

0163-1829/2004/69~17!/174401~19!/$22.50 69 1744

mhs-,

e-da-

a-tofta

r

c-rsrg

-

ale

llts,rsr

of the exchange-spring magnet, Knelleret al.1 proposed aqualitative one-dimensional model that takes into accountmagnetostatic and exchange interactions among diffecrystallographically coherent grains. However, the magnzation reversal and coercivity mechanisms of real bulk stems certainly depend on the microstructural details, sucthe three-dimensional spatial distribution of the two phasOn the other hand, the layered structures can be more perly modeled as one-dimensional structures, becauseshow variations in the magnetic properties predominanalong the growth direction. Therefore, an additional advtage of these systems is to allow a realistic estimate ofultimate gain in performance that may potentially be realiz

FIG. 1. Basic scheme for the one-dimensional micromagncontinuum model of the planar hard-soft nanocomposite.

©2004 The American Physical Society01-1

-ne

l

iona

capttitedi

edierstrisuoneto

foyves.

rins,-s

ehoe

lnaginfe

heg

o.pe

inr,t

o

ize

in-m-

teoft-

ple

ed

nthe

eticr ofsys-

allajor

inbe

-

tiont

isiumndlongse-eticns,

ewn

nlem,

on-rd

par-ft-het-an-in

ASTI, SOLZI, GHIDINI, AND NERI PHYSICAL REVIEW B 69, 174401 ~2004!

in exchange-spring bulk magnets.Recent studies on hard/soft bilayers and multilayers~pla-

nar exchange-spring magnet! have shown that the flux reversal occurs in general in two stages beginning at well deficritical fields: the nucleation fieldHc1 , at which the mag-netic moments start to deviate reversibly and nonuniformfrom the easy direction; the reversal fieldHc2 that promotesthe irreversible rotation of the whole system.

In order to gain a deep insight into the magnetizatreversal process of the hard-soft composite magnet, we hdeveloped a one-dimensional micromagnetic model andculated the analytical expression of the differential suscebility at the nucleation field. The dependence of this quanon the structural parameters allows to define the magnphase diagram of the planar hard-soft composite. In thisgram, boundary critical lines separate the different regimcorresponding to distinct reversal mechanisms. The pretions of the model are here compared with existing expmental data and with the results of other theoretical analyreported in the literature. The model, far from the nanomescale, also includes important aspects of classic systemsas soft inclusions in a hard bulk. Finally some consideratiare given concerning the best conditions for obtaining pmanent magnets with the highest energy density. Beforepresentation of our model, in Sec. II we give an overviewthe main endeavors to realize permanent magnets in theof planar magnetic nanocomposite systems, followed bdetailed analysis of the theoretical models especially deoped to describe the magnetic behavior of these system

II. INTRODUCTORY OVERVIEW

The first attempts to realize a planar exchange-spmagnet focused on isotropic systems, either bilayer4,5

trilayers,6,7 or multilayers,8–10 constituted by Sm-Co or NdFeB as the hard phase and Fe, NiFe, and Co as thephase. Shi-Shen Yanet al.11,12 studied extensively theNi80Fe20/Sm40Fe60 system, by varying over a wide rangboth the hard- and the soft-layer thicknesses. Other autfollowed a different approach, involving the growth of thnanocomposite thin films Co/Sm-Co~Ref. 13!, Co/Pr-Co~Ref. 14!, and Fe/FePt~Ref. 15!, obtained by rapid thermaannealing of multilayered structures. In this case, the fitwo-phase nanostructured film shows the exchanhardening behavior although keeping no trace of the origlayered structure. These first endeavors have shown thesibility of a multilayer nanocomposite magnet, but on tother hand, they have also stressed the difficulties existinthe way of achieving the ideal 1 MJ/m3 magnet,2,3 e.g., theinterfacial roughness, which can influence the exchange cpling between layers6 and the alignment of their easy axes

An attempt to overcome the latter problem was the etaxial growth of Co/Sm-Co and Fe/Sm-Co bilayers obtainby sputtering deposition.16 In principle, the stacking of bilay-ers of this type should produce a soft/hard multilayer havessentially the same properties of the component bilayethe reversal process occurs homogeneously. However,growth of multilayers of this kind seems now not s

17440

d

y

vel-i-ytica-sc-i-escchsr-hefrmal-

g

oft

rs

le-ala-

on

u-

i-d

gifhe

straightforward,17 although there are some attempts to realCo/Sm-Co superlattices.10,17 From a different point of view,other authors explored the possibility of modulating thetensity of interfacial exchange, by exploiting the phenoenon of interdiffusion between Co/CoPt bilayers.18,19 Fol-lowing a different approach, thin-film nanocomposimagnets were obtained by randomly dispersing sinclusions in an oriented hard matrix.20,21

Starting from a different viewpoint, Astiet al.22–24 real-ized a model system constituted by alternate layers of simmaterials~elemental Fe and Co! but with special care for theinterface condition. For this purpose, they have employultra high-vacuum deposition techniques~electron-beamevaporation! with very low deposition rates to obtain cleaand sharp interfaces. Furthermore, they have investigatedeffect of the layer thickness on the structural and magnproperties of Fe/Co multilayers having a constant numbelayers and a given Fe/Co thickness ratio. The obtainedtems show a single-phase-like behavior~almost bistable!with a steep magnetic reversal. The recoil susceptibility ofthe studied multilayers approaches the slope of the mloop at remanence, as in a classical permanent magnet.

III. THEORETICAL MODELS

The literature on the theory of magnetization reversalmultilayered exchange-coupled thin film structures canclassified by the following scheme:~i! continuum-approachmicromagnetic models,25–27which allow one to deduce analytic expressions of relevant parameters,~ii ! discretemodels16,28,29that are useful for very thin films,~iii ! compu-tational approaches based on solving the dynamic equaof motion,30,31 and~iv! first-principles calculations, aimed adetermining the band-energy structure.32

The inhomogeneous distribution of the magnetizationgenerally to be expected in such a manner that at equilibrthe magnetization is always parallel to the film plane athus the problem can be considered as one dimensional aan axis perpendicular to the multilayer plane. As a conquence of the chosen in-plane distribution of the magnmoments, corresponding to Bloch-wall-type configuratiothe stray fields are vanishing.

A. Continuum models

The use of the continuum approximation for thmultilayer problem is justified down to thicknesses of a featomic layers.25,27 Within this approach, models have beemainly utilized to deduce analytically the instability field isome special cases, such as the two-coupled-layers probfirst addressed by Gotoet al.25 In their model, the spin con-figuration inside the isotropic soft layer corresponds to a ctinuous rotation, as in Bloch walls. The spins in the hasubstrate with infinite anisotropy are assumed to remainallel, therefore imposing the boundary condition to the solayer magnetization. Through the analytical solution of tEuler equation, Gotoet al.25 deduce the shape of the theoreical magnetization curve. In particular, they express thegular distribution of magnetization vector in the soft layer

1-2

a

ft ttoc

film

thin

t-t

ucarg

thg

toal

aaial. Abcen

befoghin-

ucduel

e

ntwasndh

ee

hesoic

lellesid-y, in

resisntsalandase

edandone-

n

o-

--wot-ri-ey-

ns,ob-th-l ofess

ne-eenof

tic’ oran-yersr. In

ringnit

tis,ingthen-

micmr-

ns

MICROMAGNETIC ANALYSIS OF EXCHANGE-COUPLED . . . PHYSICAL REVIEW B69, 174401 ~2004!

terms of Jacobi elliptic functions. They individuate a criticvalue of the field ~instability field, usually called theexchange-bias field! corresponding to the initial deviation othe spins in the soft layer from the parallelism with respecthe spins of hard layer. This field is inversely proportionalthe squared soft-layer thickness. The same authors alsosidered the more general case of two exchange-coupledwith different uniaxial anisotropy,25 but did not explicitlyreduce the problem to a set of elliptic integrals, due tocomplexity of mathematical expressions. They deducedstead approximate solutions in special cases,33,34such as, thelimit of vanishing hard-layer thickness with infinite anisoropy ~degenerate hard film!, with the condition of a constanthickness-anisotropy product.25 Hagedorn26 further devel-oped the above model by investigating the stability of eqlibrium magnetic configurations by deducing an analytiexpression for the second variation of the magnetic enewhich he evaluated through numerical calculations.

The background of the above-described approach isproblem of calculating the nucleation field of a ferromanetic crystal with inclusions having lower uniaxial magnecrystalline anisotropy.35,36 For example, a one-dimensionmodel was developed by Aharoni,35 in which a crystal ex-tending infinitely in all directions is considered, includingzero-anisotropy slab of finite width. The procedure to obtan analytical expression for the nucleation field essentirelies on the solution of the linearized Brown’s equationsfurther evolution of these models was proposedAbraham,37 where a bulk crystal is covered by a surfalayer having zero anisotropy, but different exchange amagnetization.

In recent years Leineweber and Kronmu¨ller27,38examinedthe case of a soft layer with finite thickness sandwichedtween two infinite hard layers. They derived expressionsthe angular orientation of the magnetization vector throuout the layers, both with and without an applied magnetizfield, as a function of the intrinsic~exchange, magnetocrystalline anisotropy! and extrinsic~thickness! characteristics ofthe layers. On the basis of these results, the authors dedthe remanence, reversal field and maximum energy proof the triple-layer system. They calculated the reversal fi~they called this field ‘‘nucleation field’’Hn) as a function ofthe soft-layer half-thickness~therein denotedds) and de-duced that this critical field coincides with the hard-layanisotropy field, if 2ds is below thehard Bloch wall thick-nesspdh5pAAh /Kh. In this region, the value ofHn doesnot depend on the particulards value, and the system cadevelop a complete exchange coupling between thephases. Furthermore the hysteresis loop is rectangularthe case of a pure hard bulk, with only one critical field, athe magnetization reversal is only due to a rotation mecnism. A different behavior is instead deduced if 2ds is of theorder of one or a fewpdh , whenHn turns out to decreassteeply with increasingds . This situation corresponds to thso-called exchange-coupled phase,38 in which an inhomoge-neous ~irreversible! rotation mechanism predominates tmagnetization reversal, following a reversible process cauby the exchange coupling. The hysteresis loop is no mrectangular in shape and one can distinguish a first crit

17440

l

o

on-s

e-

i-ly,

e--

nly

y

d

-r-

g

edctd

r

oin

a-

edreal

field, at which the collinearity is broken and the reversibprocess starts~the exchange-bias field!, and a second criticafield corresponding to the irreversible switch of the whosystem. The hysteresis behavior of the system can be conered as a single-phase magnetic behavior. On the contrarthe extreme case, corresponding to 2ds@pdh , the hard andsoft phase can be considered as decoupled and the hystecycle is typical of a two-phase system, with two differereversal fields. In this region, the magnetization revertakes place by rotation of the moments independentlyirreversibly in the hard and soft phases and the hard-phreversal field decreases with an inverse power ofds .

A remarkably different analytical approach is that bason the analogy between quantum mechanicsmicromagnetics.39,40 In essence, the micromagnetic equatiwhich is obtained by the minimization of free energy corrsponds to the Schro¨dinger’s equation for a particle moving ia three-dimensional potential of the form 2K1(r )/m0M (r ).In particular the treatment of the multilayer limit is analgous to the periodic multiple quantum well problem~onedimensional!, leading to the implicit equation for the nucleation field @see Eq.~A11!# as a function of hard and softlayer thicknesses and of intrinsic parameters of the tphases.39 At this place we should also cite an earlier treament of the problem of inhomogeneous nucleation in peodic hard-soft multilayers based on the Kronig-Pennmodel,41 allowing the derivation of the linearized micromagnetic equation as a one-dimensional Schro¨dinger’s equation.

B. Other models

The micromagnetic approach suffers of some limitatiopartly because the complete solution of the variational prlem is possible only by approximation and numerical meods, and partly because it is based on a continuum modematter, which becomes inaccurate when the layers thicknis very small. The discrete models assume the odimensional character of the reversal mode and have bapplied to explain the observed magnetic behaviorexchange-spring bilayers,16,17,29,42 triple-layers,43 andmultilayers.29,42 The basic assumption is that the magneproperties are considered to be constant within a ‘‘sheet’‘‘layer’’ parallel to the interfaces and to depend only onplane indexi, which corresponds to the coordinate perpedicular to the planes. Most authors assume that such laare separated by a distance equal to the lattice parametethese models the coupling only occurs between neighbo‘‘layers’’ and may be described, e.g., by an energy per usurface area.28 On this basis Amatoet al.29 demonstrated thaon increasing the nanostructuration of the multilayer, thaton increasing the number of alternate layers while keepconstant the total thickness and the hard/soft ratio, bothcoercivity field and the maximum energy product are ehanced.

A different approach is based on the magnetodynaLandau-Lifshitz equation for distributed ferromagnetic filsystems.30,31The explicit purpose of the authors was to ovecome the complications of the quasistatic description~mainlyconcerning the verification of the stability of the solutio

1-3

aan

chsu-ag.

fesd

gra

rol p

rinpe

roheeerr,

efticnioaces

r

r

n

k

en-

ac-ithueof

ti-

opy

nne-

be-. In

pyththe

-ro-ge-te

onion.

iallytheree-

alpe-

ann7.

er

e


obtained from energy minimization! in the case of distributedsystems such as films with laminar magnetization. Thethors claim the dynamic solutions to be less complicatemore convenient, even including the problem of stability.

IV. THE ADOPTED MODEL

Following the guidelines of the micromagnetic approawe handled the problem of the physical existence of thelutions arising from variational calculus by utilizing an heristic viewpoint, essentially based on the study of the mnetic susceptibility in correspondence to the critical fields44

We calculated indeed the analytical expression of the difential susceptibility (xc) at the critical field that correspondto the start of flux reversal~hereafter called nucleation fielHc1 , as usual in the micromagnetic treatment45!, on the basisof a one-dimensional micromagnetic model of the exchanspring multilayer. The value ofxc is an important parametecharacterizing the type of reversal process and it is of pticular interest the evaluation of its dependence on the perties of the soft and the hard phases and on structurarameters.

The adopted model considers the exchange-spmultilayer as composed by alternate soft and hard layerspendicular to thex axis ~see the scheme in Fig. 1!. The twocomponent layers are supposed to have uniaxial anisotwith parallel easy axes both lying in the film plane along tz axis ~Fig. 1!. The search for solutions is limited to thoswhich are periodical, so that the median planes of the layare symmetry planes for the magnetic structure. The pposed approach is~i! based on a continuum model of matte~ii ! it does not take explicitly into account temperaturefects, except for the fact they are implicit in the magneparameters normally utilized in a micromagnetic treatme~iii ! it also neglects the time dependence of magnetizatthus considering a quasistatic magnetization reversal,~iv! it does not consider the possible contribution of surfaand interface anisotropy. In the case of an applied magnfield H opposite to thez axis, the expression of the Gibbfree energy can be written as

G5 (i 51,2

~21! iEx0

xi FAi S dq

dx D 2

1m0MiH cosq

1Ki sin2 qGdx, ~1!

where indexes 1 and 2 refer to the soft and hard layer,spectively, andx0 , x1 , and x2 denote the position of theinterface and of two contiguous median planes~Fig. 1!, q(x)is the angle between thez axis and the magnetization vectoM , andAi is the exchange stiffness constant of layeri. Themagnetocrystalline anisotropy term is limited to the secoorder. We also define the quantitiest15(x02x1), t25(x22x0), which represent half of the respective layer thicnesses.

17440

u-d

,o-

-

r-

e-

r-p-a-

gr-

py

rso-

-

t,n,ndetic

e-

d

-

The magnetization of both layers is supposed to lie gerally in the film plane. Hence in Eq.~1! no demagnetizingfield contribution to the free energy has been taken intocount. In the case of an exchange-coupled multilayer wperpendicular anisotropy, a further energy contribution dto stray fields should be introduced into the model in forma demagnetizing anisotropy constant for each layer (Ki

dm

5m0Mi2/2). In this case the magnetic field is applied an

parallel to thex axis and the intrinsic anisotropy constantsKishould be substituted by the corresponding total anisotrconstantsLi5Ki2m0Mi

2/2, where a positiveKi indicatesthat thex axis is an easy axis. A further assumption is thatM iin both layers lies in thexy plane and the angleq is theorientation angle ofM i with respect to the positive directioof x axis. This means that the problem has again a odimensional character with one degree of freedom~y com-ponent ofM i) and the mathematical treatment presentedlow also applies to the case of perpendicular anisotropyparticular, in the limitK15K2 , M15M2 , the system re-duces to a classic infinite slab with perpendicular anisotro~see Ref. 45, p. 194!. Our assumptions are consistent withis case characterized by a spin reversal triggered bynucleation of a uniform rotation mode at a critical fieldHc5(2K2m0M2)/m0M ~Ref. 46!. So we expect that the onedimensional approach we utilize in the model is also apppriate for describing the nucleation process in the exchancoupled multilayer with remanent saturated staperpendicular to the film plane. However this descripticould be no more adequate in the region far from saturatIn particular the second critical fieldHc2 , at which the com-plete reversal of the whole system occurs, could substantdiffer from the values predicted by the model becausesystem could take advantage of the additional degree of fdom ~z component ofM i).

The determination of equilibrium state is a variationproblem. In this particular case, one has to consider theriodic boundary conditions (dq/dx)50 for x5x1 and x5x2 and A1(dq/dx)ux5x025A2(dq/dx)ux5x01 for x5x0 .The last condition corresponds to the Weierstrass-Erdmlaw along the surface normal, which is also cited in Ref. 2The absence of an explicit dependence onx of the integrandallows to directly obtain from the Euler equation a first ordintegral, which reads

m0MiH~cosq2cosq i !1Ki sin2 q2Ai S dq

dx D 2

2Ki sin2 q i

50, ~2!

from which

dq

dx5Aa i~cosq2cosq i !1b i~sin2 q2sin2 q i !, ~3!

where i 51 for x1,x,x0 and i 52 for x0,x,x2 and a i5m0MiH/Ai , b i5Ki /Ai . The boundary angles areq i forx5xi and q0 for x5x0 . Because we are interested in th

1-4

ig

te

th

ilio

x-qua-

is,

d

e

al

n the


behavior for small angular deviations around the fieldHc1 ,Eqs. ~3! are then expanded up to the fourth order inq andbecome

dq

Auq22q i2u

F11pi

2~q21q i

2!G5dxAUb i2a i

2 U, ~4!

where

p15a128b1

12~2b12a1!,

p25a228b2

12~a222b2!. ~5!

In Eqs. ~4!, the argument of the modulus has negative sfor the soft layer (q,q1) and positive sign for the hardlayer (q.q2). The obtained equations are then integrabetween the extrema, which areq5q1 and q5q0 for thesoft layer, q5q0 and q5q2 for the hard layer, and thecorresponding values for variablex ~see Ref. 47!.

Then we obtain two linear equations for the soft andhard layer in terms of higher order infinitesimal variablesh j( j 50,1,2) that define the deviations of anglesq j from theirunperturbed values corresponding to the onset of instabat H5Hc1 . The magnetic field itself is expressed in termsan infinitesimal reduced variablet5(H2Hc1)/Hc1 , thatvanishes whenH5Hc1 ~see Appendix A!. A third equation isobtained by considering the boundary condition atx5x0 ,

s

riemb

esl

17440

n

d

e

tyf

which corresponds to the equilibrium condition for the echange forces. Thus we obtain a system of three linear etions in the anglesh j . At H5Hc1 ~i.e., t50) it becomes asystem of homogeneous linear equations: the conditionthat the determinantD of the coefficient matrix is vanishingleading to an implicit equation for the critical fieldHc1 ~seeAppendix A!. For tÞ0 a mathematical relation is obtainebetween the infinitesimal fieldt and anglesq j @see Eq.~A10!#, which in turn allows to obtain an expression of thslope ofM (H) at the nucleation fieldHc1 .

Appendix B reports the calculation of the infinitesimdecrease of the reduced magnetization,dm5dM /Ms (Ms isthe saturation magnetization of the whole system! which isproportional toq0

2 through a factorG, which is a function ofthe layer thicknesses and the magnetic parameters. Thereduced susceptibility is@Appendix B, Eq.~B5!#

x5dm

t5

q02G

t. ~6!

Equation~A10! of Appendix A yields the ratioq02/t and thus

the volume susceptibility at the critical fieldHc1 ~in SI units!turns out to be

xc5dM

dH5

dmMs

~H2Hc1!5

q02GMs

Hc1t5x

Ms

Hc1~7!

or, equivalently,44

xc5H M1F t1

cos2~ t1g1!1

tan~ t1g1!

g1G1M2F t2

cosh2 ~ t2g2!1

tanh~ t2g2!

g2G

3p1

cos2~ t1g1! F 2t1g1

sin~2t1g1!11G2

3p2

cosh2~ t2g2! F 2t2g2

sinh~2t2g2!11G12~p12p2!

J3H a1F 1

g12 1

2t1

g1 sin~2t1g1!G1a2F 1

g22 1

2t2

g2 sinh~2t2g2!G J •F 1

4Hc1~ t11t2!G , ~8!

ursof

Inrvef

whereg1 andg2 are defined in Appendix A as functions ofaandb.

V. THE MAGNETIC PHASE DIAGRAM

The dependence ofxc on the structural parameters allowto define a phase diagram in the (t1 ,t2) plane. Figure 2 rep-resents the phase diagram of a typical exchange-spmultilayer, calculated for the case of a Fe/Sm-Co systwith the same layer intrinsic parameters already utilizedFullerton et al.16 (M151.7 MA/m, M250.55 MA/m, K15102 J/m3, K255 MJ/m3, A152.8310211 J/m, and A251.2310211 J/m). Starting from the thick layers region, thcritical susceptibility is positive and increases continuou

ng,

y

y

when reducing the half-layer thicknesst1 and t2 , until xc

diverges along a critical line (xc→`). This line correspondsto the onset of instability and, in the region below it,xc

becomes negative, indicating that the flux reversal occirreversibly. This critical condition separates the regimethe so-called exchange-spring magnet~ES,xc.0) from thatof the magnetically rigid composite magnet~RM, xc,0),which is very similar to a conventional oriented magnet.the latter, the reversible portion of the demagnetization cuon the left of Hc1 ~usually interpreted to be indicative o‘‘exchange-spring behavior’’! is absent.

The first critical fieldHc1 is the solution of the implicitequation~A11! of Appendix A. Givent1 and t2 , this is anequation inH throughg1 andg2 , and it is usually solved byusing the simple method of bisection~see Ref. 48, p. 353!. In

1-5

ticlu

.e

bln

S

eshesfeye

d

sibhag.2

io

nue

thek-

ions.Mriti-s-

ndenteeldt ofin-oft

hei-redmp-

d--

ckr,

d

scal

o4

-

tne


this way theHc1 lines have been traced on the magnephase diagram. The curves corresponding to constant vaof the critical susceptibilityxc in the (t1 ,t2) plane are foundby a similar method: givent1 , we find the correspondingpoint of the iso-xc curve with the bisection, by lettingt2vary. Since expression~8! for xc also depends onHc1 , wehave to solve equation~A11! for each step of the bisectionIn order to trace theHc2 curves, we first have to calculate thdemagnetization curves~see Appendix C!. The condition todetermine this critical field is the divergence of the reversisusceptibility, a criterion discussed in detail by Hubert aRave.49

The lines of constant critical fieldsHc1 andHc2 are pre-sented in Fig. 2 over the half-layer thicknessest1 and t2 ,together with the line of constant critical susceptibilityxc→` (x` line!, which separates the RM regime from the Eregime. In the RM regime the isocritical field lines forHc1andHc2 coincide, while there is a bifurcation of these linon thex` line and a separation in the ES region. With tmaterial parameters used, the RM region correspondrather small thicknesses for both layers, of the order of anm. In general, by setting a specific value of the soft-lahalf-thicknesst1 , Hc1 becomes larger by increasingt2 .However, due to the fact that theHc1 lines show a verticalasymptote, by increasing the hard-layer thickness beyoncertain value,Hc1 changes no more.

Another system that has been proposed as a posexchange-coupled planar composite is Fe/NdFeB. The pdiagram for a Fe/NdFeB infinite multilayer is shown in Fi3~a!, based on the material parameters given in Ref.(M151.7 MA/m, M251.28 MA/m, K154.33104 J/m3,K254.3 MJ/m3, A152.5310211 J/m, A257.7310212 J/m). If we extend the phase diagram to the regof large soft half-layer thicknesses@Fig. 3~a!#, we observethat thex` line assumes a typical U shape. This line presetwo vertical asymptotes, corresponding to particular val

FIG. 2. Magnetic phase diagram in the plane of half-layer thinessest1 ~soft! andt2 ~hard! for the case of a Fe/Sm-Co multilayewith parameters M151.7 MA/m, M250.55 MA/m, K1

5102 J/m3, K255 MJ/m3, A152.8310211 J/m, A251.2310211 J/m ~from Ref. 16!. The figure reports the criticalx` line(xc→`, dotted line! together with the lines of constant critical fielHc1 ~solid! andHc2 ~dashed!.

17440

es

ed

towr

a

lese

7

n

tss

of the soft-layer thickness. As a consequence, if we focusattention to the case of relatively large hard-layer thicnesses, the phase diagram is subdivided into three reg~1! On the left of first asymptote, the system is in the Rregime dominated by the hard phase, that is, with large ccal fields.~2! On the right of the second asymptote, the sytem is in a decoupled regime@decoupled magnet~DM!#,where the hard and soft phases behave as almost indepecomponents.~3! The intermediate region pertains to thusual ES regime. In the decoupled region the nucleation fiHc1 does not mark the starting of a reversible detachmenthe magnetic moments from saturated state: it indicatesstead the occurrence of the irreversible switching of the sphase, which is followed by a similar process involving thard phase at the fieldHc2 . The corresponding demagnetzation curve is typical of a two-phase system, as prefiguin Refs. 1 and 38. For Fe/Sm-Co system, the second asytote occurs at very larget1 values~of the order of 200 nm!.

Figure 3~a! also reports the iso-critical lines corresponing to the reversal fieldHc2 . Note that those lines corre

-

FIG. 3. ~a! Fe/NdFeB composite system~data from Ref. 27!:magnetic phase diagram in the plane of half-layer thicknesset1

~soft! andt2 ~hard!. The U-shaped dotted line represents the critix` line (xc→`). The figure also reports theHc1 ~solid! and Hc2

lines~dashed!, which coincide in the RM region~left-bottom side ofthe diagram!. ~b! Same as~a! but with an enlarged scale, in order tgive evidence to theHc2* line corresponding to the value 542.7340kA/m: this line discriminates theHc2 lines having a vertical asymptote from those having horizontal asymptote~two examples aretraced on the diagram, see text!. The figure also shows the differenregions in which the U-shaped critical line and the bifurcation lisubdivide the phase diagram. The bifurcation line~dotted line! de-notes the boundary between coupled and decoupled systems.

1-6

ue-k

ck

lairiee

dia

toe

iob

alp:seemealtlythhi

attedthis

urve

m-infi-h-r ine

oseckos-es,p-

cir-ty

ternh is

ersot-

n

ft/epses

w

es

e

nts


sponding to fields above a specific valueHc2* show a verticalasymptote, while those referring to a field lower thanHc2*present a horizontal asymptote. This critical field val(Hc2* 5542.73404 kA/m with the utilized material parameters! can be obtained by imposing that both layer thicnesses are infinite: the corresponding iso-Hc2 curve appearsto have an oblique asymptote in the region of large thinesses. It is worth noting@Fig. 3~b!# that a very small changeof this critical field valueHc2* in either directions, even onthe eighth digit (Hc2* 60.04 A/m), gives rise to isocriticafield curves that are practically coincident up to a certpoint, where they bifurcate taking either a vertical or an hozontal asymptote. Figure 4 reports the calculated thicknpositions of the horizontal and vertical asymptotes, resptively, as a function of the critical fieldHc2 . The divergenceof both curves occurs at the critical field valueHc2* .

Another peculiar characteristic of the obtained phasegram, as can be seen from Fig. 3~a!, is that the bifurcation ofthe Hc1 and Hc2 lines occurs exactly in correspondencethe critical line x` only for fields above a specific valu~about 210 kA/m, with the utilized material parameters!. Forfields lower than this value, the bifurcation occurs alongline that branches out of the critical linex` . This ‘‘detach-ment’’ of the bifurcation from thex` critical line can bevisualized by plotting on the phase diagram the bifurcatpoints corresponding to different field values, as it candeduced from Fig. 3~a!. One obtains thus a critical [email protected]~b!#, called the bifurcation line, which is nearly horizontand denotes the boundary between coupled and decousystems. Below this line the system is exchange coupledparticular it is a rigid magnet dominated by the soft phathat is, with small critical fields. Above the bifurcation thhard and soft phases behave as nearly independent conents~DM regime!. The bifurcation line converges into thleft-hand side of the U-shaped critical line to a tri-criticpoint G. The definition of the decoupled magnet is inherensubject to certain arbitrariness. We have considered inpresent analysis a decoupled system as the one for wuHc1u<uHc2u while xc,0. Another possible definition could

FIG. 4. Fe/NdFeB composite system~data from Ref. 27!: hard-layer half-thickness positiont2 of the horizontal asymptotes for thHc2 lines, as a function of theHc2 value. On the same figure it ialso reported the soft-layer half-thickness positiont1 of the verticalasymptotes for theHc2 lines. The discontinuity corresponds to thcritical field Hc2* 5542.7340384 kA/m~see Fig. 3!.

17440

-

-

n-ssc-

-

a

ne

ledin,

po-

ech

be connected with the introduction of a third critical fieldwhich the irreversible transition occurs when the saturaand the intermediate states have equal free energy. Incase we could have again a two-step demagnetization cin the region whereuHc1u.uHc2u and the new bifurcationline would be shifted to lowert2 values while converging onthe same tricritical pointG.

Figure 5 shows the isocritical susceptibility lines superiposed to the magnetic phase diagram of the Fe/NdFeBnite multilayer, with the hard-layer thickness on a logaritmic scale, in order to put in evidence the peculiar behaviothe limit of very large hard-layer thickness. In this limit, thiso-xc lines are still open lines and tend to concentrate clto the U-shaped critical line. This means that for very thihard-layer thicknesses, the critical susceptibility of compite system is practically zero for all soft-layer thicknessexcept in a very narrow region around the vertical asymtotes, where it diverges. This fact reflects the obviouscumstance thatxc is a weighted average of the susceptibilicontribution of both components~hard and soft!. With in-creasing hard-layer thickness the demagnetization pattends to be invariant and involves the same volume, whica decreasing fraction of the whole system.

VI. DEPENDENCE OF THE MAGNETIC PHASEDIAGRAM ON MATERIAL PARAMETERS

Let us examine first the effect of an increasing soft-layanisotropy on the phase diagram. Starting from the aniropy ratior5K1* /K2 as in the case of Fig. 2~Fe/Sm-Co!, thephase diagram modifies as shown in Fig. 6~a!. The two as-ymptotes of the critical linex` approach one another oincreasingr, until they meet for a particular valuer8 ~forsoft/Sm-Co it is about 0.06, while in the case of the soNdFeB multilayer it is 0.034!, so that the region of existencof the ES magnet vanishes. So the U-shaped curve collain a half-line starting from the tricritical pointG, where itlinks up with the bifurcation line thus forming a unique ne

FIG. 5. Fe/NdFeB composite system~data from Ref. 27!: mag-netic phase diagram in the plane of half-layer thicknessest1 ~soft!and t2 ~hard!. On the diagram the U-shaped dotted line represethe x` line, and the other lines~solid! are isocritical susceptibilitylines. The hard-layer thickness scale is logarithmic.

1-7

ursth

h

ha

heilehis

i-of

he

ete

bery

p ofniol-

of

otes a

f

-r a

m

r

al

the


boundary line separating the RM and DM regions. With fther increase ofr.r8 the new boundary line moves upwardto the right in the phase diagram, progressively reducingDM region @Fig. 6~c!#. At a second special value~see Sec.IX !

r95J1

J2

HA1*

HA25

J1

J2

1

~112j12Aj!, ~9!

wherej5A1J1 /A2J2 , the region at infinite thickness of botphases (t1 ,t2→`) becomes RM, so that forr.r9 the DMregion either disappears or becomes an island in the p

FIG. 6. Soft/Sm-Co composite system~starting data from Ref.16!: ~a! modification of thex` line in the magnetic phase diagraon varying ther5K1* /K2 ratio, with fixedK2 . ~b! Soft-phase half-thickness position of first and second asymptotes of thex` line as afunction of r. The positiont151.23 nm represents the limit foK1→0. The two asymptotes coalesce at a critical ratior5r850.06, corresponding tot152.8 nm: for higher values ofr the ESphase disappears.~c! Phase diagram forr.r8 ~broken lines repre-sent the new boundary lines separating RM and DM phases!. Forcomparison we have also included the phase diagram for a vr50.05,r8, with thex` line ~solid! and the bifurcation line~bro-ken!.

17440

-

e

se

diagram. The numerical values ofr9 are 0.096 for soft/NdFeB and 0.148 for soft/Sm-Co.

The variation of the anisotropy ratio has little effect on tsoft half-layer thickness position of the first asymptote, whstrongly influences the position of second asymptote. Tfact is summarized in Fig. 6~b!, where the asymptote’s postions in the phase diagram are reported as a functionK1 /K2 ratio for the case of the soft/Sm-Co system. In tlimit of vanishing soft phase anisotropy, the positiont1 offirst asymptote tends to a value~1.23 nm!, which is close tothe exchange lengthd25AA2 /K2 of the hard Sm-Co phas~about 1.55 nm!, while the position of the second asymptotends to infinite thickness.

In the particular caseK150, K25K, A15A25A, J15J2 , the position of the first asymptote turns out tot1AK/A50.985013, in good agreement with the boundathickness between the continuous and discontinuous jumthe magnetization at nucleation as reported by Aharo35

(t1AK/A50.984). The positions of the two asymptotes clapse, as already stressed, forK1 /K250.06, in correspon-dence to a soft half-layer thickness of 2.8 nm for the casethe soft/Sm-Co system@Fig. 6~b!#.

We also analyzed how the position of the first asymptis influenced by the variation of exchange constants. Afirst attempt, we fixedK1 andK2 to the initial values of Ref.16 and changed eitherA1 or A2 . The results for the case othe soft/Sm-Co system are reported in Figs. 7~a! and 7~b!,from which it can be inferred thatA1 determines the strongest variation of the asymptote position: about 70% fochange of a factor 10 of theA1 /A2 ratio.

ue

FIG. 7. Soft/Sm-Co composite system~starting data from Ref.16!: ~a! soft-phase half-thickness position of first asymptote ofx` line as a function of theA1 /A2 ratio, with A2 fixed. ~b! Soft-phase half-thickness position of first asymptote of thex` line as afunction of theA2 /A1 ratio, with A1 fixed. The dots on the linesrepresent the starting values ofA1 andA2 taken from Ref. 16.

1-8

thr

dem

,n-th

nc

me

rre

n

th

tize

at

rue

nomalltio

k-

ask-0, touesI

tour-al-w-turnsisachew

-C

gs

-essia-

iteyer-

Ref.

half-


VII. THE CALCULATED DEMAGNETIZATION CURVES

The procedure for tracing by numerical calculationsdemagnetization curves of an exchange-coupled multilayereported in Appendix C. Figure 8 reports the calculatedmagnetization curves for a Fe/Sm-Co system, with the saintrinsic parameters utilized by Fullertonet al.16 In this casethe curves refer to the same Sm-Co layer thickness~20 nm!and different Fe layer thicknesses in the range 1–20 nmin Fig. 10~a! of Ref. 16. The calculation was performed cosidering a tilting angle of 3° between the easy axis andfield direction, as in Fullertonet al.16 The comparison ofcalculated curves with those reported in the cited referegives a good agreement concerning the shape and theHc1andHc2 values down to a soft-layer thickness of about 5 nthat is, for systems lying in the ES region. However, wobserve an increasing discrepancy for the calculatedHc2value on reducing the 2t1 value below 5 nm.

The values ofHc1 andHc2 as functions of the soft-layethickness are reported in Fig. 9, together with the corsponding critical field values calculated in Fullertonet al.16

There is good agreement forHc2 , especially at large thick-nesses, while ourHc1 values seem to be slightly larger thathose obtained in Fullertonet al.16 The transition to the RMregime appears in both calculations to occur at roughlysame thickness 2t1'2 nm. Amatoet al.,29 on the basis of adiscrete micromagnetic model, calculated the demagnetion curves for the Fe/Sm-Co exchange-spring multilayutilizing the same data given by Fullertonet al.16 The com-parison of the results from our model and those of Amet al.29 is difficult because they consider afinite multilayersystem, with constant total thickness and different nanostturation degree, that is, different number of layers. Furthmore, because the systems of Amatoet al.29 start and finishwith a hard layer, they show a symmetry that we canexactly reproduce: this is true in particular for the systewith a small number of layers. Nevertheless, we numericcalculated, on the basis of our model, the demagnetiza

FIG. 8. Demagnetization curves for the case of a Fe/Smmultilayer, with parametersM151.7 MA/m, M250.55 MA/m,K15102 J/m3, K255 MJ/m3, A152.8310211 J/m, A251.2310211 J/m ~data from Ref. 16!. The hard-layer thickness (2t2) isfixed at 20 nm and those of the soft layer (2t1) are indicated~innm! on the curves. The calculation was performed considerintilting angle of 3° between the easy axis and the field direction, aRef. 16.

17440

eis-e

as

e

e

,

-

e

a-r,

o

c-r-

tsyn

curves for infinite multilayers having the same layer thicnesses of the finite systems considered by Amatoet al.29 Theonly exception is the case of trilayer (n53) for which wehave also calculated the curve considering the systemcomposed by two symmetrical bilayers having half thicnesses 10 nm/10 nm. These curves are reported in Fig. 1be compared with Fig. 2 of Ref. 29 and the deduced valof the critical fieldsHc1 and Hc2 are compared in Tablewith the corresponding values of Amatoet al.29 Our criticalfields Hc1 andHc2 are larger than those deduced by Amaet al.,29 independently of the nanostructuration degree. Fthermore, our system enters the RM regime for smaller vues of the nanostructuration degree of the multilayer. Hoever, the general features of the demagnetization curvesout to be similar. In general, from this comparative analywe can say that the continuous micromagnetic approgives reliable results down to layer thicknesses of a fatomic layers.

o

ain

FIG. 9. Comparison between the calculated critical fieldsHc1

~continuous line! andHc2 ~dashed line! and the corresponding values obtained in Ref. 16, as functions of the soft-layer thickn(2t1), having fixed the hard-layer thickness at 20 nm. Open dmonds,Hc1 ; open dots,Hc2 .

FIG. 10. Series of calculated demagnetization curves for infinFe/Sm-Co exchange-spring multilayer having the same lathicknesses of the finite systems of Ref. 29. Then value reported onthe curves represents the nanodispersion index, as defined in29, that is,n53 corresponds to a trilayer,n55 to a pentalayer, andso on. The numbers in parentheses represent the soft and hardlayer thickness expressed in nm.

1-9

othnontycooo

orre

ase. Iaip

Ane

sea

as

aea

w

insw

s at

-ick-

ls ofck-in

in

ft-rial

ick-ent

blegetheh

mt isthe

ntal-

ns.in a

iza-

s are

ate


VIII. COMPARISON WITH EXPERIMENTAL DATA

The magnetic behavior of a planar magnetic nanocompite described in the previous paragraphs, in particulardemagnetization processes, can in principle be conveniecompared with experimental results on real systems. Amthe observed phenomena we believe that the nucleation acritical field Hc1 in the ES region is properly described bthe present model because the system undergoes a seorder transition to a reversible state, so that it is in a favable condition to be insensitive to localized defectsinhomogeneities.50 These are instead known to be of majimportance in the coercivity mechanisms involving an irversible transition~Brown paradox!. As a matter of fact thevalues of theHc2 field that the model predicts are based oncondition of instability of inhomogeneous rotation procesand are strictly valid for a perfectly homogeneous systemother words our model does not include inverse domnucleation and propagation, at least for the case of a proexchange-coupled system, i.e., in the ES and RM states.consequence the obtainedHc2 values are to be intended as aupper limit for the real reversal field, in a way similar to thswitching field in the Stoner-Wohlfarth model.

We made an attempt to compare the hysteresis cyclethe Ni80Fe20/Sm40Fe60 system reported in Ref. 11 with thosdeduced from our model, on the basis of the following mterial parameters: the saturation magnetization of both ph(M15860 kA/m, M25286 kA/m, from Ref. 12! and the an-isotropy constant~assumed to be zero! of Ni80Fe20. Theother quantities~exchange constants of both phases andisotropy constant of the hard phase! were considered as freparameters in a best-fit procedure concerning the soft-hard-layer thickness dependence ofHc1 , as deduced fromFigs. 3 and 4 of Ref. 11. The results of the best fit are shoin Figs. 11~a! and 11~b!, corresponding to a given 2t1 value~57.6 nm! and to a given 2t2 value ~100 nm!, respectively.The obtained values of the free parameters areK252.23104 J/m3 andA15A256.5310212 J/m.

For the above explained reasons, it has to be underlthat we only took into account the experimental hysterecycles that manifestly are in the ES regime and for which

TABLE I. The calculated critical fieldsHc1 andHc2 comparedwith the corresponding values obtained in Ref. 29. Then valuerepresents the nanodispersion index, as defined in Ref. 29, thn53 corresponds to a trilayer,n55 to a pentalayer, and so on. Thsoft and hard half-layer thickness is expressed in nm.

H~MA/m!

Nanodispersion index

n511 n59 n57 n55 n53 n53soft/hard layer thickness

2/1.7 2.5/2 3.3/2.5 5/3.4 10/10 10/5

Hc1a 1.79 1.42 0.99 0.60 0.21 0.21

Hc1b 2.24 1.90 1.49 0.93 0.36 0.36

Hc2a 1.77 1.54 1.31 1.18 1.07 1.07

Hc2b 2.24 1.90 1.54 1.21 1.13 1.04

aRef. 27.bThis work.

17440

s-etlyg

the

nd-r-r

-

snners a

of

-es

n-

nd

n

edise

identify the nucleation field with the critical fieldHc1 of ourmodel. In these cycles it turns out that the reversal occurfields substantially lower than the theoreticalHc2 values@Figs. 11~a! and 11~b!#. It is worth noting that this discrepancy is progressively reduced on increasing both layer thnesses, i.e., approaching the DM regime@see Fig. 11~b!#.With 2t1.60 nm and 2t25100 nm we have a substantiaagreement between experimental and theoretical valueHc2 , which appear to be independent of the soft-layer thiness. This limit ofHc2 corresponds amazingly to the domawall depinning field at the hard-soft interface as describedthe next section.

An example of demagnetization curves for different solayer thicknesses, calculated by using the obtained mateparameters, is shown in Fig. 12, where the hard-layer thness is fixed at 100 nm. They are in qualitative agreemwith the experimental curves particularly on the reversiportion where there is a slight difference in the averaslope. Moreover Fig. 13 reports the phase diagram ofNi80Fe20/Sm40Fe60 system obtained from our theory, witthe critical linex` , which is compared with the critical linereported in Fig. 7 of Ref. 11, defining the transition frosingle-switching process to exchange-spring process. Ievident that there is a pronounced discrepancy between

FIG. 11. Comparison between calculated and experime~from Ref. 11! critical fieldsHc1 andHc2 for a series of exchangespring bilayers and trilayers based on the Ni80Fe20/Sm40Fe60 sys-tem:~a! in the case of a fixed soft-layer thickness 2t1557.6 nm,~b!in the case of a fixed hard-layer thickness 2t25100 nm. The opensymbols refer toHc1 , while the filled symbols toHc2 ; trianglesrefer to data taken from Ref. 11 and circles refer to our calculatioThe material parameters were considered as free parametersbest-fit procedure, with the exception of the saturation magnettion of both layers and the anisotropy constant of Ni80Fe20, as-sumed to be zero. The obtained values of the free parameterK252.23104 J/m3 and A15A256.5310212 J/m. The lines areguides for the eyes.

is,

1-10

era

yionino

eldio

ed-a

atis

nveg-ld,

in

elu-on

ardto

d-le,our

k-three

oft-er

e of-

llreionr ofSr the

er

o

d

t

hase


two curves, particularly in the low soft-layer thickness rgion. We explain this difference by admitting that in genefor the experimental hysteresis loops~Fig. 4 of Ref. 11! ex-hibiting a single switching the reversal occurs before the stem could reach the instability condition at the nucleatfield. This irreversible switching could be ascribed to anhomogeneous reversal caused by extrinsic factors, whilemodel gives, as an upper limit, an irreversible switching fiHc2 only based on rotation processes. Domain observathave been indeed recently performed on Ni80Fe20/Sm40Fe60exchange-spring films using Kerr microscopy in applimagnetic fields.51 On the other hand the critical curve reported in Fig. 7 of Ref. 11 can be considered at most asempirical fitting curve of the experimental results. As a mter of fact the theoretical interpretation given in Ref. 11inconsistent because it is based on an inverse power law@Eq.~1!# with exponent 1.75. The point is that this power depedence, as explained in Ref. 27, does not refer to the contional definition of the nucleation field given in micromanetism, but rather to the irreversible switching fie

FIG. 12. Series of calculated demagnetization curvesNi80Fe20/Sm40Fe60 multilayers ~data from Ref. 11! in the ES re-gime, in the case of a fixed hard-layer thickness 2t25100 nm fordifferent soft-layer thicknesses 2t1 .

FIG. 13. Phase diagram of the Ni80Fe20/Sm40Fe60 system~datafrom Ref. 11!, with the critical linex` ~triangles and dash-dotteline! compared with the critical line~solid line! for the transitionfrom single-switching process to exchange-spring process,reported in Fig. 7 of Ref. 11. The material parameters utilizedtrace the phase diagram were deduced from a best-fit procedure~seeFig. 11!.

17440

-l

s-

-ur

ns

n-

-n-

according to the meaning given to the term ‘‘nucleation’’Ref. 27~see also next section!.

IX. THE LIMIT OF A HARD BULK WITH A SOFTPLANAR INCLUSION

As outlined in Sec. III, the problem of reduction of thnucleation field in a hard bulk containing a soft planar incsion was first discussed by Aharoni and, later, Abrahamthe basis of a micromagnetic one-dimensional model,35,37

which is rather similar to that adopted by Leineweberet al.27

for the hard-soft exchange-coupled triple layer~where theinfinite extension of the hard layers makes the system a hbulk!. Moreover, the same model was also utilized by Goet al.25 with the only difference that the soft layer is consiered coupled to a hard bulk only on one side. In principthe results of all these models should be obtained frommodel in the limit case of a very thick~ideally infinite! hardlayer. Note that both Aharoni35 and Gotoet al.25 assumed azero-anisotropy soft layer.

As already outlined in Sec. V, for large hard-layer thicnesses the phase diagram appears to be subdivided intoregions by the vertical asymptotes of the critical linex` : thehard-dominated RM region, the ES region, and the sdominated DM region. We report in Fig. 14 the soft-laythickness dependence of the critical fieldsHc1 andHc2 in thecase of a Fe/NdFeB composite, having set a large valuthe hard-layer thickness (t25100 nm). The two vertical asymptotes correspond in this case tot15t1A52.15 nm'(p/2)d2 , wherepd2 represents the hard-layer Bloch wawidth, and t15t1B516.3 nm: the asymptote positions aindicated in Fig. 14 by dashed lines. Inside the RM regthe critical fields coincide, as expected, and the characteHc1 in the DM region is modified with respect to the Eregion, because it represents here a true reversal field fosoft layer, that is, the starting of an irreversible process.

For the case of a NdFeB/Fe/NdFeB trilayer, Leineweb

f

aso

FIG. 14. Fe/NdFeB composite system~data from Ref. 27!: softhalf-layer thickness dependence of critical fieldsHc1 and Hc2 , inthe case of a thick hard layer (t25100 nm). The critical fields arenormalized to the hard-layer anisotropy field, whilet1 is normalizedto the hard-layer DW half width (p/2)d2 . The vertical dashed linesrepresent the boundaries of different regions in the magnetic pdiagram~see Fig. 3!.

1-11

th

ae

vin

-

ulthois-e

ca-rda

ecth

nee

n--

stheopy

5

I,

dnd-con-ve-

ults

ed5,

cedrs.a-

dicein-

ss:

theer-an-isof

rowallnt

-lu-

tersely

0


et al. calculated the soft-layer thickness dependence offield at which the irreversible inversion occurs~Fig. 4 of Ref.27!. Our analysis shows the following~see Fig. 14!. ~i! Fort1,(p/2)d2 , the nucleation fieldHc1 is rather different fromthe anisotropy field of the hard layerHA252K2 /J2 , exceptthatHc1 reaches the value ofHA2 only whent1→0. ~ii ! It isreduced by a factor 3 whent15t1A . This means that evenvery thin soft layer coating on a hard bulk dramatically rduces the nucleation~or reversal! field in the RM regime.~iii ! The nucleation fieldHc1 asymptotically tends to thevalue of the soft-layer anisotropy fieldHA152K1 /J1 , as ex-pected. These results are in contradiction to the behashown in Fig. 4 of Ref. 27, where the reversal field is costant for t1,(p/2)d2 , while for t1.10(p/2)d2 we cannotobserve neither forHc1 nor for Hc2 the inverse power-lawbehavior}(t1)21.75, as reported in Ref. 27.

The limit of Hc2 for t1→` corresponds to the domainwall depinning field at the hard-soft interface,52 which is animportant coercivity mechanism in the case of a hard benclosing a soft inclusion. In bulk permanent magnets,depinning of a domain wall may be the rate limiting stepflux reversal, thus representing a more realistic mechanas compared to intrinsic nucleation~characterized by the anisotropy field!.53 Nucleation centers exist indeed on surfacand along hard-soft grain boundaries~in the very commoncase of presence of soft phase inclusions!, which obviate theonset of the intrinsic process.

In this particular case it is possible to obtain an analytiexpression of the reversal fieldHc2 . To this purpose we assume botht1 andt2→`, a situation corresponding to a habulk exchange-coupled to a soft bulk at the interface. Wein general in the case of a system in the DM regime~exceptwhen K1 /K2 is above a certain value, as explained in SVI !, and we are considering the intermediate state afterreversal of the soft phase, so that we assumeq15p andq250. The system of equations is given by Eq.~3! plus Eq.~A7!. From this an equation inH andq0 is obtained:

F~H,q0!5@A2J2~cosq021!2A1J1~cosq02cosq1!#H

2~A1K12A2K2!sin2 q01A1K1 sin2 q150.

~10!

The condition for Hc2 is dH/dq050 that implies]F(H,q0)/]q050. Elimination of q0 between the latterequation and Eq.~10! allows one to obtain the followingequation forH:

~A2J22A1J1!2H224~A2K22A1K1!~A2J21A1J1!H

14~A2K22A1K1!250. ~11!

The only solution having physical meaning is the lower oIt provides the analytical expression of the domain-wall dpinning fieldHDW ~Ref. 52!:

17440

e

-

or-

kefm

s

l

re

.e

.-

HDW52A2K22A1K1

A2J21A1J112AA1A2J1J2

. ~12!

In the case of a Fe/NdFeB system, Eq.~12! gives HDW50.543 MA/m, which is about 10% of the hard-phase aisotropy fieldHA2 . Table II reports the calculated DW depinning field HDW for different hard-soft composites: it iworth noting that the obtained values are in all cases oforder of 10–20% with respect to the hard-phase anisotrfield.

The reversal field calculated by Aharoni in Ref. 3~therein termed ‘‘coercive force’’! for the caseK150, K25K, A15A25A, J15J2 , as already mentioned in Sec. Vcan be considered as a particular case of Eq.~12!, whichgivesHDW50.25HA2 , in agreement with the value reportein Ref. 35. It should be emphasized that the periodic bouary conditions at the basis of the present treatment aresistent with the boundary conditions assumed in the abomentioned work.

To summarize, we were not able to reproduce the resof the analysis of Leineweberet al. as a limit of our modelfor very larget2 values. Nevertheless, the results reproducin Fig. 14 compare well with those of Fig. 1 of Ref. 3where the nucleation field and the coercive~reversal! fieldare reported as functions of the soft-layer size in reduunits. The agreement is even better for very thin soft laye

There is however, in principle, a possible physical sitution that is not included in the abovementioned perioboundary conditions for the composite multilayer. As whave seen these conditions imply a reversal mechanismvolving the whole system in a cooperative rotation procethe displacement of a domain wall isa priori excluded for aRM and an ES state, while a DW depinning process athard-soft interface occurs in the case of DM phase. A diffent situation can in principle be envisaged by assumingtiparallel domains at infinity as a boundary condition: in thcase the periodicity of the system is in fact lost. The studythe multilayer under these new circumstances could thnew light on the coercivity mechanism of the domain wpinning. A similar situation was considered in the treatmeof Friedberg and Paul54 concerning domain wall pinning process in a bulk uniaxial magnet, having a planar soft inc

TABLE II. The calculated domain wall depinning fieldHDW fordifferent hard-soft composites. The intrinsic magnetic parameare taken from Refs. 16 and 27. Co-soft represents an extremfine-grained Co with anisotropy constant reduced by a factor 13.

compositeHDW

~MA/m!HA2

~MA/m!HDW /HA2

~%!

Fe/NdFeBa 0.540 5.347 10.10Co/NdFeBa,b 0.669 5.347 12.52Fe/Sm-Cob 2.011 14.469 13.90

Co-soft/Sm-Cob 3.021 14.469 20.88Co/Sm-Cob 2.859 14.469 19.76

aParameters taken from Ref. 27.bParameters taken from Ref. 16.

1-12

hisonteu

cky’etain

deo

at

ce

ou

tiormosth

thto

tintosiada

-Cmye

om-

ion

ng

eron,for

am

erior,ers,re-t

toof

ise

theno-l-

ac-

’’

i.e.,


sion, to be compared with the classical nucleation mecnism in the bulk.35 In the case of a composite multilayer, thcondition of antiparallel domains should correspond to csider the propagation of a domain wall throughout the sysand eventually to observe an intrinsic type of coercivity, dto the matching between the DW width and the layer thinesses. The analogy is with the proper ‘‘intrinsic coercivitconnected with the DW pinning in a homogeneous magncrystal having very high anisotropy and thus narrow domwalls. The intrinsic coercivity originates from the modulatioof the DW energy by the lattice periodicity.55

Having introduced the depinning field we can reconsihere the modifications of the magnetic phase diagramvarying the anisotropy ratior, as explained in Sec. VI. Inparticular, there is a critical valuer5r9 for which the DMphase disappears at infinite thickness of both phases@see Eq.~9!#. In order to obtain this critical value the condition is ththe depinning field equals the critical fieldHc1 , which in thislimit coincides with the soft phase anisotropy field. Henwe have the equation

2A2K22A1K1

A2J21A1J112AA1A2J1J2

52K1

J1~13!

which yields

HA1

HA25

1

~112j12Aj!, ~14!

wherej5A1J1 /A2J2 . Expression~14! provides the criticalvaluer9 given by Eq.~9!.

As a final remark we have verified the agreement ofmodel with the one of Gotoet al.25 In that model theexchange-bias field, which corresponds to the nucleafield Hc1 of our treatment, results to be inversely propotional to the squared soft-layer thickness. In order to copare the two models, we fixed the soft-layer anisotropy tvery low value (1026), while the hard-layer anisotropy wachosen equal to that of Sm-Co in Ref. 10. Furthermore,thickness ratiot2 /t1 was kept equal to 103. We obtained aperfect agreement between the results of our treatment inlimit and the power law predicted by the model of Goet al., concerning the soft-layer thickness dependenceHc1 .

X. THE MAXIMUM ENERGY PRODUCT

Although the exchange-spring shows such an interesreversibility property it is not perhaps the best conditionachieve the maximum energy density, which is a prerequifor a high-performance permanent magnet. It is likely ththe best composite permanent magnet should be realizethe RM state because the magnetization remains at its mmum value in the presence of a reverse field.

A. Optimum „BH …max : a convenient criterion

The analysis of the phase diagram for the Fe/Smmultilayer ~Fig. 2! allows us to deduce that an RM systewith significant technical parameters is expected for la

17440

a-

-me-’icn

rn

r

n--a

e

is

of

g

tetin

xi-

o

r

thicknesses of about 8 and 3 nm, for the soft and hard cponents respectively.44 In this case,Hc1 would be of theorder of 1 MA/m, while the average remanent magnetizat

Mr5~M1t11M2t2!/~ t11t2! ~15!

turns out to be close to 1.4 MA/m, with a correspondi(BH)max of about 0.6 MJ/m3.

In general, if we are interested in obtaining a multilaywith the largest possible layer thicknesses in the RM regithen an optimum point in the phase diagram should existthe maximum energy product. This point must be on thex`

critical line and is determined from the conditionHc15Mr /2. As an example we have utilized the phase diagrcalculated for the Fe/NdFeB system~Fig. 3! to obtain theoptimum point corresponding to (BH)max: the deduced val-ues are t154.4 nm, t251.8 nm, Hc150.8 MA/m, and(BH)max50.8 MJ/m3. For real samples, one has to considthe influence of extrinsic factors on the coercive behavsuch as, the imperfect moment orientation of the hard laywhich could reduce considerably the coercive field withspect to the critical fieldHc1 , and then the energy produc~see below!.

B. The limit of extreme nanostructuration

A further possibility offered by the phase diagram isanalyze the region of very thin layers, in the neighborhoodthe origin, corresponding to the limit of smallt1 and t2 , ascompared to (a1/2)21/2 and (b2)21/2, respectively~see Sec.IV !. In such conditions the magnetic moments rotationalmost uniform andHc1 approaches the value of an effectivanisotropy field

HA52~K1t11K2t2!/m0~M1t11M2t2!, ~16!

that is the weighted average of the anisotropy fields oftwo phases. This condition is equivalent to an extreme nastructuration of the multilayer, which, in principle, could alow us to achieve very high-energy products.16,29In this case,it could be of interest to evaluate the soft-phase volume frtion l5t1 /(t11t2) for an optimum (BH)max, in correspon-dence to a given hard material. To define the ‘‘optimumenergy product, we refer again to the criterionHc15Hc1*

5Mr /2, in this case withHc1'HA . By substituting theabove conditions in Eq.~15! we obtain an equation inl ~Ref.44!:

~M12M2!2l212FM2~M12M2!12S K2

m02

K1

m0D Gl1M2

2

24K2

m050. ~17!

If we assume a negligible anisotropy for the soft phase,K150, and that the hard-phase anisotropy constantK2 is

1-13

a

plti

ldrgryth

tio

eracy

hetex

iothWth

ial

Set,slylly

fterhe

eticrticu-eref ainthefirstnalngramrfulIna-

andtivensial

we

weing

thr

sy-es


very large with respect tom0M12, the solution of Eq.~17!

simply reduces tol'12m0M12/4K2 . It turns out that the

soft phase volume ratio is independent of the hard-phmagnetization. In other words, for a very hard phase,l ap-proaches unity and the relevance ofM2 in the overall mag-netization is accordingly reduced. Therefore, in the caseextreme nanostructuration, it would be possible in princito utilize hard phases having low or virtually zero magnezation, i.e., ferrimagnets or even antiferromagnets, to buiplanar nanocomposite magnet with the optimum eneproduct. In this case, one might think to exploit the velarge crystal anisotropy of intermetallic phases such asheavy rare-earth–transition metal (TbCo5) or the Lavesphases. This principle has recently found practical realizain the anti-ferromagnetically coupled DyFe2-YFe2superlattices,56 grown by molecular beam epitaxy. Moreovthe high degree of nanostructuration implies a large surfto volume ratio that makes surface or interface anisotropfurther element to come into play.

C. Effect of misalignment

As a further aspect, the orientation of the hard phasean important role in the overall performance of real magnFrom this viewpoint, we have analyzed the effect of theistence of a nonzero angle« ~deg.! between the soft- andhard-layer easy axis and the field direction. This situatresembles that of a real system, in which a distribution ofhard-layer easy axis orientations is likely to be realized.have observed two distinct behaviors of the systems in

FIG. 15. Series of calculated demagnetization curves forcase of a Fe/Sm-Co multilayer, with the material parametersported in Ref. 16, with different angles« ~deg.! between the mag-netic field direction and the easy-axis direction:~a! in the caset1

5t252 nm ~RM region!; ~b! in the caset1510 nm andt255 nm~ES region!.

17440

se

ofe-ay

e

n

ea

ass.-

neee

RM and ES regime. If we consider a multilayer with materparameters as in Ref. 16, the reversal fieldHc2 decreaseswith increasing the angle« if the system belongs to the RMregime @Fig. 15~a!#, while the opposite occurs for the Eregime @Fig. 15~b!#. The singularity associated with thnucleation fieldHc1 is only present for a perfect alignmeni.e., «50°. A finite angle«, whatever small, actually ruleout the singularity and the kink point is progressiverounded off. It is a case of a broken symmetry and actuaall the magnetic moments start to rotate immediately athe application of an infinitesimal field. Figure 16 shows teffect of a small misorientation on the critical fieldHc2 , inthe case of systems in the RM and ES regimes.

XI. CONCLUSIONS

The magnetic composite can present peculiar magnproperties that are absent in homogeneous phases, in palar depending on the microstructural parameters. In fact this the need of a basic phenomenological description ophysical system of this kind. A magnetic phase diagramterms of important microstructural parameters such aslayer thicknesses, in the case of a planar composite, is astep in this direction. We have developed a one-dimensiomicromagnetic model of the multilayer exchange-sprimagnet, which leads to a complete magnetic phase diagfor the planar hard-soft nanocomposite, providing a powetool for a general overview of its magnetic properties.particular it gives information on the type of demagnetiztion processes and the critical fields at which nucleationreversal take place. The diagram has in principle a predicpotential on the behavior of particular configuratioand therefore is useful for the tailoring of these artificmaterials.

A key point to this purpose is the analytical expressionhave obtained in Eq.~8! for the critical susceptibility at thenucleation field. Depending on the sign of this quantity,have a reversible or an irreversible switching, correspondrespectively to the exchange-spring magnet~ES! or to therigid composite magnet~RM! regime, for very small soft-

ee-

FIG. 16. Critical fieldHc2 as a function of the misorientationangle« ~deg.! between the magnetic field direction and the eaaxis direction for the case of a Fe/Sm-Co multilayer. The two lincorrespond to the cases considered in Fig. 15. TheHc2 values arenormalized to that corresponding to«50°.

1-14

sitynoThpeityterdrannd

ree

obuM

-luthcle

ithmth

roa

th

edean

do

rteiss

tourt

lsanpl

ecn

ce

andeshere

eto

za-

e inzedob-

thee,

tiveruc-

Wure

jectedn a

o

d


layer thicknesses. On the side of large soft-layer thickneswe find again a negative value of the critical susceptibilbut it corresponds now to the condition of decoupled mag~DM!, in which the irreversible switching occurs in twsteps, i.e., almost independently in the two phases.boundary line between the different regimes is an U-shaline corresponding to divergence of the critical susceptibilThen for large values of the soft-layer thickness the sysgoes directly from RM to DM phase on increasing the halayer thickness, without crossing the ES region. The diagalso reports the iso-critical lines both for the nucleation athe reversal field. These lines bifurcate along the RM bouary line.

Noticeably, for a given hard material the ES phase is pdicted to occur only below a threshold of the ratio betwethe anisotropy constants of the two phasesK1 /K2 . For ex-ample, the threshold is of the order of a few % for the caseSm-Co and NdFeB hard phases. Well above this limit,remarkably still far from unity, it happens that even the Ddisappears.

The limit of infinite hard-layer thickness gives information on the behavior of bulk magnets with planar soft incsions. It is worth noting that even a soft layer as thin ashard-phase Bloch wall is enough to cause a fall of the nuation field to 1/3 of the hard-phase anisotropy field. The limof both infinite hard-and soft-layer thicknesses providesanalytical expression of the critical field pertinent to an iportant coercivity mechanism of permanent magnets:domain-wall depinning.

In general, the adopted model is limited to rotational pcesses: in fact, the deduced values of the reversal fieldssystematically larger than the experimental ones. Thisclearly due to the fact that the model does not considerrole of domains and DW nucleation and pinning,57 except inthe abovementioned case of the decoupled system. Hencobtained values of the reversal field should be considerean upper limit, in analogy with the switching field of thStoner-Wohlfarth model with respect to the coercivity of rebulk magnets. Moreover, because of the chosen odimensional ansatz, we excludea priori the possible role oftopological singularities.57

Another intrinsic limitation of our model is that it is baseon the continuum approximation. However, the compariswith the results of discrete one-dimensional models repoin the literature shows that the micromagnetic approachgood representation of the real system, down to thicknesa few atomic layers.

A characteristic, which is common to our model andpractically all the other treatments reported in the literatand discussed in the present work, is the assumption ofmagnetization lying in the film plane. However we have aconsidered the implications of admitting a perpendicularisotropy and we have given some indications how to apour analysis to this case.

Furthermore, we have addressed the problem of the tnical performance of a planar composite permanent magThe maximum energy density requires the magnetizationbe as high as possible, so that best performance is expein the RM state. If one wants to conveniently utilize th

17440

es,et

ed.m-md-

-n

ft

-e-

te-e

-re

ise

theas

le-

ndaof

eheo-y

h-et.toted

largest layer thicknesses, the optimum (BH)max should besearched along the RM-ES boundary line. On the other hit turns out that, in the limit of vanishing layer thicknessand of very large hard-phase anisotropy, an even hig(BH)max is achievable for an optimum soft-phase volumfraction ~close to unity! depending only on the soft phasmagnetization. This means that it is possible in principleemploy hard phases having low or virtually zero magnetition, i.e., ferrimagnets or even antiferromagnets.

The orientation of the hard phase has an important rolthe overall performance of real systems. We have analythe effect of a small misalignment of the easy axis andserved that the reversal field decreases with increasingmisorientation angle if the system belongs to the RM regimwhile the opposite occurs for the ES regime.

The planar nanocomposite magnets have in perspecother reasons of interests, such as the role of the microstture ~multilayers vs hard-granular layered systems!, the DWpinning process, and the rediscovery of the intrinsic Dpinning on a mesoscopic scale, by matching microstructlength scale with domain wall width.

ACKNOWLEDGMENTS

The present work has been supported by a FIRB Proof the Italian Ministry of Education and Research, entitl‘‘Microsystems based on magnetic materials structured onanoscopic scale.’’

APPENDIX A: FOURTH-ORDER EXPANSION

We start from Eq.~4!, which is the power expansion up tthe fourth order of Eq.~3!. After integration between theextrema (q1 ,q0) for the soft layer and (q0 ,q2) for the hardlayer we obtain

q02p1

4

1

S 113

4p1q1

2D q0~q122q0

2!

5q1 cosF ~x02x1!

S 113

4p1q1

2D Aa1

22b1G , ~A1!

q01p2

4

1

S 113

4p2q2

2D q0~q022q2

2!

5q2 coshF ~x02x2!

S 113

4p2q2

2D Ab22a2

2 G . ~A2!

We perform now the substitutionsq j→q j1h j ( j 50,1,2),where theh j are higher order infinitesimal quantities, anneglect in the calculations the terms of order higher thanq3,

1-15

sr,

one

-

lect-

ing


such as,hq2 or h2q. Then, we obtain two linear equationin the variablesh j , which are, in the case of the soft laye

h02h1 cosF ~x02x1!Aa1

22b1G

52q01p1

4q0~q1

22q02!1q1H cosF ~x02x1!

3Aa1

22b1G1q1

2~x02x1!3p1

4Aa1

22b1

3sinF ~x02x1!Aa1

22b1G J ~A3!

and for the hard layer

h02h2 coshF ~x02x2!Ab22a2

2 G52q01

p2

4q0~q0

22q22!1q2

3H coshF ~x02x2!Ab22a2

2 G2q22~x02x2!

33p2

4Ab22

a2

2sinhF ~x02x2!Ab22

a2

2 G J . ~A4!

The next step consists of substituting, in the above equatithe reduced variablesa i ( i 51,2), which are related to th

-he

th

17440

s,

applied fieldH, with their expression in terms of the infinitesimal fieldt. a i→a icr(11t), wherea icr is the value ofa iat H5Hc1 . Furthermore, we introduce the variablesg1cr5Aa1cr/22b1 andg2cr5Ab22a2cr/2. Similarly, the criticalvariablespicr correspond to the expressions forpi with a i5a icr . As a consequence of these substitutions and neging higher order terms, we transform Eqs.~A3! and ~A4! in

h02h1 cos@~x02x1!g1cr#

52q1t~x02x1!a1cr

4g1crsin@~x02x1!g1cr#1q1

3g1cr

3~x02x1!3p1cr

4sin@~x02x1!g1cr#1

p1cr

4q0~q1

22q02!

~A5!

for the soft layer and

h02h2 cosh@~x02x2!g2cr#

52q2t~x02x2!a2cr

4g2crsinh@~x02x2!g2cr#

2q23g2cr~x02x2!

3p2cr

4sinh@~x02x2!g2cr#

2p2cr

4q0~q0

22q22! ~A6!

for the hard layer. A third equation is obtained by considerthe boundary condition atx5x0 , which corresponds to theequilibrium condition for the exchange forces

S A1

dq

dx Ux5x02

5 D A1Aa1~cosq02cosq1!1b1~sin2 q02sin2 q1!

5S A2

dq

dx Ux5x01

5 D A2Aa2~cosq02cosq2!1b2~sin2 q02sin2 q2!. ~A7!

By performing the substitutionsq j→q j1h j ( j 50,1,2),and a i→a icr(11t) ( i 51,2), and considering that the expansion of Eq.~A7! truncated to the second order gives tadditional condition

A12~q1

22q02!g1

252A22~q2

22q02!g2

2, ~A8!

we obtain finally a system of three linear equations inanglesh j :

e

h02h1 cos@~x02x1!g1cr#

5~x02x1!g1crsin@~x02x1!g1cr#

3F2ta1crq1

2~a1cr22b1!1

3p1cr

4q1

3G1

p1cr

4q0~q1

22q02!,

1-16

sh

ohf

re

oe--

ies

of

rder.

ion

s-


h02h2 cosh@~x02x2!g2cr#

5~x22x0!g2crsinh@~x02x2!g2cr#

3F2ta2crq2

2~2b22a2cr!2

3p2cr

4q2

3G2

p2cr

4q0~q0

22q22!,

q0h0~g1cr2 A1

21g2cr2 A2

2!2g1cr2 q1h1A1

22g2cr2 q2h2A2

2

5t

4@~a2cr

2 A222a1cr

2 A12!q0

21a1crA12q1

22a2crA22q2

2#

11

6A2

2~q042q2

4!S a2cr

82b2D1

1

6A1

2~q142q0

4!

3S a1cr

82b1D . ~A9!

SinceD50 it is necessary that the augmented matrix~ob-tained by adding the column of the right-hand side term!has the same rank as the matrix of the coefficients. Tcondition can be written in the formD850 whereD8 is thedeterminant of the square matrix obtained by substitutionthe column of the right-hand side terms in place of whicever column ofD. Then we substitute the first column odeterminantD with the right-hand side terms of Eqs.~A9!.The condition of vanishing determinant then reads

2tS a1cr

g1cr2 1

a2cr

g2cr2 1

t1a1cr

g1crsin~ t1g1cr!cos~ t1gcr!

1t2a2cr

g2crsinh~ t2g2cr!cosh~ t2g2cr!D

1q02S 3p1crt1g1cr

sin~ t1g1cr!cos3~ t1g1cr!2

3p2crt2g2cr

sinh~ t2g2cr!cosh3~ t2g2cr!D

1q02S 3p1cr

cos2~ t1g1cr!2

3p2cr

cosh2~ t2g2cr!12p1cr22p2crD50,

~A10!

where t15(x02x1) and t25(x22x0) represent half of thelayer thicknesses.

If we truncate the power expansion of Eq.~3! to the sec-ond order, we obtain, after integration, a system of thhomogeneous linear equations in the anglesq j . It is worthnoting that the obtained system has the same matrix of cficients of system~A9!. The condition of vanishing determinantD at H5Hc1 leads to an implicit equation for the critical field

A1g1 tan~ t1g1cr!5A2g2 tanh~ t2g2cr!. ~A11!

Equation~A11! assumes the same form as that obtainedRefs. 2, 26, 37. Moreover, from this expansion to the loworder, we can deduce the relationsq0 /q15cos(t1g1cr) andq0 /q25cosh(t2g2cr).

17440

is

f-

e

f-

nt

APPENDIX B: INFINITESIMAL VARIATION OFMAGNETIZATION

From the definitions of the layer magnetizations

Mz5M1*x1

x0dx cosq1M2*x0

x2dx cosq

~x22x1!5

M1I 11M2I 2

~x22x1!,

~B1!

whereM1 andM2 represent the saturation magnetizationthe soft and hard layer, respectively, and with

I 15Ex1

x0dx cosq>2E

x1

x0cosq

dq

g1Aq122q2

,

I 25Ex0

x2dx cosq>2E

x0

x2cosq

dq

g2Aq22q22

~B2!

considering a power expansion truncated to the second oThe above integrals result

I 15S 12q0

2

4 cos2@g1~x02x1!#D ~x02x1!

2q0

2

4g1tan@g1~x02x1!#,

I 25S 12q0

2

4 cosh2@g2~x22x0!#D ~x22x0!

2q0

2

4g2tanh@g2~x22x0!# ~B3!

and therefore the infinitesimal variation of the magnetizatturns out to be

dM5Ms2Mz5M1~x02x1!1M2~x22x0!

~x22x1!2Mz

5q0

2

4~x22x1! H M1~x02x1!

cos2@g1~x02x1!#1

M1

g1tan@g1~x02x1!#J

1q0

2

4~x22x1! H M2~x22x0!

cosh2@g2~x22x0!#

1M2

g2tanh@g2~x22x0!#J , ~B4!

whereMs is the saturation magnetization of the whole sytem. The reduced infinitesimal variation ofM is then

dm5dM

Ms5

dM ~x22x1!

M1~x02x1!1M2~x22x0!

5q02 1

4~M1t11M2t2!3H M1t1

cos2 @g1t1#1

M1

g1tan@g1t1#

1M2t2

cosh2@g2t2#1

M2

g2tanh@g2t2#J 5q0

2G. ~B5!

1-17

ie

a

-

-

rt,to-heed

e.

et

dnfield

or

s-

nne of

de,


APPENDIX C: GENERAL PROCEDURE FOR TRACINGTHE DEMAGNETIZATION CURVES

The problem of tracing the demagnetization curve implthe integration of Eq.~3! with the boundary conditions(dq/dx)50 andq5q i for x5xi ( i 51,2) andA1(dq/dx)5A2(dq/dx) and q5q0 for x5x0 . This calculation in-volves elliptical integrals and is performed by numericmethods.

For a specific multilayer system, given a particular fieldHwe have to find the anglesq0 , q1 , q2 by solving the systemgiven by the three equations

F1~q0 ,q1 ,q2 ,H !

5Eq0

q1 dq

Aa1~cosq2cosq1!1b1~sen2 q2sen2 q1!2t1

50,

F2~q0 ,q1 ,q2 ,H !

5Eq2

q0 dq

Aa2~cosq2cosq2!1b2~sen2 q2sen2 q2!

2t250, ~C1!

F3~q0 ,q1 ,q2 ,H !

5a1~cosq02cosq1!1b1~sen2 q02sen2 q1!

2S A2

A1D 2

@a2~cosq02cosq2!

1b2~sen2 q02sen2 q2!#50.

Using vector symbols we writeF(Q)50, with F5(F1 ,F2 ,F3) and Q5(q0 ,q1 ,q2). We use the Newton-Raphson method, as described in the Numerical Recipes~seeRef. 48, p. 379!, to find this solution. We start from a tentative valueQstart. In the neighborhood of any nonsingularQthe functionF(Q) can be expanded in Taylor series

F~Q1dQ!5F~Q!1J•dQ1O~dQ2!, ~C2!

where J5](F1 ,F2 ,F3)/](q0 ,q1 ,q2) is the Jacobian matrix of the system. By neglecting terms of orderdQ2 andhigher and by settingF(Q1dQ)50 we obtain a set of lineaequations for the correctionsdQ that move each componenfunction of the vectorF closer to zero simultaneouslynamely, dQ52J21

•F. The corrections are then addedthe starting valueQnew5Qstart1dQ and the procedure is iterated untiluFu is as small as we want, thus obtaining tdesired value ofQ. With this value we can compute thmagnetizationMz from Eq. ~B1!, once we have obtaineq(x) after integration of Eq.~3!. We generally start from a

17440

s

l

value of the fieldH0 very close to the nucleation fieldHc1and from a tentative value of the anglesQtent given by thenucleation field equations@see Appendixes A, B and Eq.~7!#:

q05Axc~H02Hc1!

GMs,

q15q0

cos~ t1g1!, ~C3!

q25q0

cosh~ t2g2!

then we compute the right value of the anglesQ0 ~and themagnetizationM0) with the above-described procedurThen we increase the field toH1 , take Q0 as the startingvalue and computeQ1 ~andM1), and so on. Each step we sHn5Hn211step and computeQn usingQn21 as the startingvalue. This way we are able to findM for each value ofH, upto the reversal fieldHc2 . In correspondence of the seconcritical field Hc2 an irreversible jump of the magnetizatiotakes place. This means that a very small increase of theH leads to very large changes in the values of the anglesq0 ,q1 , q2 , that is, the first derivativesdq0 /dH, dq1 /dH,dq2 /dH diverge. The derivatives are given by the vectequation

S ]q0

]H

]q1

]H

]q2

]H

D 52S ]~F1 ,F2 ,F3!

]~q0 ,q1 ,q2! D21 ]~F1 ,F2 ,F3!

]H

52J21S ]F1

]H

]F2

]H

]F3

]H

D , ~C4!

where as usualJ represents the Jacobian matrix of the sytem. Thus in order to getdq i /dH→`, we must seek theconditionJ50. The step of the fieldH has to be repeatedlydecreased while approachingHc2 ~since the demagnetizatiocurve becomes very steep! and we stop the procedure whethe step has become small enough not to affect the valuthe field within the set precision of 1027– 1028. By thispoint J has usually decreased by 3 or 4 orders of magnituand we consider this to be a convergence to zero.

1E. F. Kneller and R. Hawig, IEEE Trans. Magn.27, 3588~1991!.2R. Skomski, J. Appl. Phys.76, 7059~1994!.3R. Skomski and J. M. D. Coey, IEEE Trans. Magn.29, 2860

~1993!.

4K. Mibu, T. Nagahama, T. Shinjo, and T. Ono, Phys. Rev. B58,6442 ~1998!.

5B. Z. Cui and M. J. O’Shea, J. Magn. Magn. Mater.256, 348~2003!.

1-18

n

J

.

G

A

s

J

. D

gn

J.

n.

d

u

n.

J.

n

pl

ta,

, B.

nys-

ichfa-

.

.

d,B


6S. Parhofer, J. Wecker, C. Kuhrt, and G. Gieres, IEEE TraMagn.32, 4437~1996!.

7C. J. Yang and S. W. Kim, J. Appl. Phys.87, 6134~2000!.8M. Shindo, M. Ishizone, A. Sakuma, H. Kato, and T. Miyazaki,

Appl. Phys.81, 4444~1997!.9I. A. Al-Omari and D. J. Sellmyer, Phys. Rev. B52, 3441~1995!.

10E. E. Fullerton, J. S. Jiang, C. H. Sowers, J. E. Pearson, and SBader, Appl. Phys. Lett.72, 380 ~1998!.

11Shi-shen Yan, J. A. Barnard, Feng-ting Xu, J. L. Weston, andZangari, Phys. Rev. B64, 184403~2001!.

12Shi-shen Yan, W. J. Liu, J. L. Weston, G. Zangari, and J.Barnard, Phys. Rev. B63, 174415~2001!.

13J. P. Liu, R. Skomski, Y. Liu, and D. J. Sellmyer, J. Appl. Phy87, 6740~2000!.

14J. P. Liu, Y. Liu, and D. J. Sellmyer, J. Appl. Phys.83, 6608~1998!.

15J. P. Liu, Y. Liu, C. P. Luo, Z. S. Shan, and D. J. Sellmyer,Appl. Phys.81, 5644~1997!.

16E. E. Fullerton, J. S. Jiang, M. Grimditch, C. H. Sowers, and SBader, Phys. Rev. B58, 12193~1998!.

17E. E. Fullerton, J. S. Jiang, and S. D. Bader, J. Magn. MaMater.200, 392 ~1999!.

18J. Kim, K. Barmak, M. De Graef, L. H. Lewis, and D. C. Crew,Appl. Phys.87, 6140~2000!.

19D. C. Crew, J. Kim, L. H. Lewis, and K. Barmak, J. Magn. MagMater.233, 257 ~2001!.

20H. Kato, T. Nomura, M. Ishizone, H. Kubota, T. Miyazaki, anM. Motokawa, J. Appl. Phys.87, 6125~2000!.

21Jai-Lin Tsai, Tsung-Shune Chin, Jhy-Chau Shih, and Shi-KChen, IEEE Trans. Magn.35, 3337~1999!.

22G. Asti, M. Carbucicchio, M. Rateo, and M. Solzi, J. MagMagn. Mater.196–197, 59 ~1999!.

23G. Asti, M. Carbucicchio, M. Ghidini, M. Rateo, and M. Solzi,Appl. Phys.87, 6689~2000!.

24M. Solzi, M. Ghidini, and G. Asti, inMagnetic Nanostructures,1st ed., edited by H. S. Nalwa~American Scientific, StevensoRanch, CA, 2002!, Vol. 1, Chap. 4, p. 124.

25E. Goto, N. Hayashi, T. Miyashita, and K. Nakagawa, J. ApPhys.36, 2951~1965!.

26F. B. Hagedorn, J. Appl. Phys.41, 2491~1970!.27T. Leineweber and H. Kronmu¨ller, J. Magn. Magn. Mater.176,

145 ~1997!.28W. Andra, IEEE Trans. Magn.2, 560 ~1966!.29M. Amato, M. G. Pini, and A. Rettori, Phys. Rev. B60, 3414

~1999!.30H. Chang, Y. S. Lin, and A. Priver, J. Appl. Phys.38, 2294

~1967!.

17440

s.

.

D.

.

.

.

.

.

.

n

.

31Y. S. Lin and H. Chang, J. Appl. Phys.40, 604 ~1969!.32R. F. Sabiryanov and S. S. Jaswal, Phys. Rev. B58, 12071~1998!.33N. Hayashi and E. Goto, J. Appl. Phys.37, 3715~1966!.34E. Goto, N. Hayashi, N. Honma, R. Kuroda, and T. Miyashi

Jpn. J. Appl. Phys., Part 14, 712 ~1965!.35A. Aharoni, Phys. Rev.119, 127 ~1960!.36C. Abraham and A. Aharoni, Phys. Rev.120, 1576~1960!.37C. Abraham, Phys. Rev.135, A1269 ~1964!.38T. Leineweber and H. Kronmu¨ller, Phys. Status Solidi B201, 291

~1997!.39R. Skomski and J. M. D. Coey, Phys. Rev. B48, 15812~1993!.40R. Skomski, J. Appl. Phys.83, 6503~1998!.41S. Nieber and H. Kronmu¨ller, Phys. Status Solidi B153, 367

~1989!.42G. J. Bowden, J. M. L. Beaujour, S. Gordeev, P. A. J. de Groot

D. Rainford, and M. Sawicki, J. Phys.: Condens. Matter12,9335 ~2000!.

43S. Wuchner, J. C. Toussaint, and J. Voiron, Phys. Rev. B55,11576~1997!.

44G. Asti, M. Solzi, and M. Ghidini, J. Magn. Magn. Mater.226,1464 ~2001!.

45A. Aharoni, Introduction to the Theory of Ferromagnetism, Ith ed.~Clarendon Press, Oxford, 1996!, Chap. 9.

46This is the mode if the limit of infinite slab is reached from aasymmetric configuration. In case of a rotation-symmetric stem the reversal mode is thecurling with the same nucleationfield ~see Ref. 45, p. 197!.

47In the calculation we assume solutions withq1 andq2 having thesame sign. In principle there should be also solutions in whthe two angles have opposite sign. These are certainly lessvored configurations, with higher energy or even unstable.

48Numerical Recipes, URL http://www.library.cornell.edu/nr/bookcpdf.html.

49A. Hubert and W. Rave, Phys. Status Solidi B211, 815 ~1999!.50W. F. Brown, Phys. Rev.124, 1348~1961!.51D. Chumakov, R. Scha¨fer, D. Elefant, D. Eckert, L. Schultz, S. S

Yan, and J. A. Barnard, Phys. Rev. B66, 134409~2002!.52G. Asti, M. Ghidini, F. M. Neri, and M. Solzi, J. Magn. Magn

Mater.272-276, 650 ~2004!.53S. T. Chui and Y. Yu, Phys. Rev. B63, 140419~R! ~2001!.54R. Friedberg and D. I. Paul, Phys. Rev. Lett.34, 1234~1975!.55H. Zijlstra, IEEE Trans. Magn.6, 179 ~1970!.56M. Sawicki, G. J. Bowden, P. A. J. de Groot, B. D. Rainfor

J.-M. L. Beaujour, R. C. C. Ward, and M. R. Wells, Phys. Rev.62, 5817~2000!.

57A. Hubert and R. Scha¨fer, Magnetic Domains~Springer-Verlag,Berlin, 1998!.

1-19

Micromagnetic analysis of exchange-coupled hard-soft planar nanocomposites

Documents

Transcript of Micromagnetic analysis of exchange-coupled hard-soft planar nanocomposites