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MAGNETIC HYSTERESIS

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MAGNETIC HYSTERESIS

Edward Della TorreThe George Washington University

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10 9 8 7 6 5 4 3 2

ISBN 0-7803-6041-9IEEE Order Number: PP5766

The Library of Congress has catalogued the hard coveredition of this title as follows:

DellaTorre, Edward 1934-MagneticHysteresis I Edward Della Torre.

p. em.Includes bibliographical references and index.ISBN0-7803-4719-61. Hysteresis. I. Title

QC754.2H9T67 1999538'.3- -dc21 98-46940

CIP

To the memory ofCharles V. Longo

CONTENTS

Preface xi

Acknowledgments xiii

Chapter 1 Physics of Magnetism 11.1 Introduction 11.2 Diamagnetism and Paramagnetism 21.3 Ferro-, Antiferro-, and Ferrimagnetic

Materials 51.4 Micromagnetism 81.5 Domains and Domain Walls 12

1.5.1 Bloch Walls 131.5.2 Neel Walls 151.5.3 Coercivity of a Domain Wall 16

1.6 The Stoner-Wohlfarth Model 171.7 Magnetization Dynamics 26

1.7.1 Gyromagnetic Effects 261.7.2 Eddy Currents 281.7.3 Wall Mobility 28

1.8 Conclusions 29References 30

Chapter 2 The Preisach Model 312.1 Introduction 312.2 Magnetizing Processes 312.3 Preisach Modeling 332.4 The Preisach Differential Equation 40

2.4.1 Gaussian Preisach Function 412.4.2 Increasing Applied Field 43

vii

viii CONTENTS

2.4.3 Decreasing Applied Field 442.5 Model Identification: Interpolation 462.6 Model Identification: Curve Fitting 472.7 The Congruency and the Deletion Properties

492.8 Conclusions 51References 51

Chapter 3 Irreversible and Locally ReversibleMagnetization 533.1 Introduction 533.2 State-Independent Reversible Magnetization

533.3 Magnetization-Dependent Reversible Model

553.4 State-Dependent Reversible Model 583.5 Energy Considerations 62

3.5.1 Hysteron Assemblies 643.6 Identification of Model Parameters 663.7 Apparent Reversible Magnetization 673.8 Crossover Condition 713.9 Conclusions 73References 73

Chapter 4 The Moving Model and the Product Model 754.1 Introduction 754.2 Hard Materials 754.3 Identification of the Moving Model 80

4.3.1 The Symmetry Method 804.3.2 The Method of Tails 84

4.4 The Variable-Variance Model 864.5 Soft Materials 924.6 Henkel Plots 934.7 Congruency Property 95

4.7.1 The Classical Preisach Model 974.7.2 Output-Dependent Models 97

4.8 Deletion Property 1004.8.1 Hysteresis in Intrinsically

Nonhysteretic Materials 1024.8.2 Proof of the Deletion Property 104

4.9 Conclusions 107References 108

CONTENTS ix

Chapter 5 Aftereffect and Accommodation 1115.1 Introduction 1115.2 Aftereffect 1125.3 Preisach Interpretation of Aftereffect 1205.4 Aftereffect Dependence on Magnetization

History 1235.5 Accommodation 1255.6 Identification of Accommodation Parameters

1345.7 Properties of Accommodation Models 137

5.7.1 Types of Accommodation Processes139

5.8 Deletion Property 1435.9 Conclusions 144References 144

Chapter 6 Vector Models 1476.1 Introduction 1476.2 General Properties of Vector Models 1486.3 The Mayergoyz Vector Model 1516.4 Pseudoparticle Models 1526.5 Coupled-Hysteron Models 154

6.5.1 Selection Rules 1546.5.2 The m2 Model 1586.5.3 The Simplified Vector Model or SVM

Model 1596.6 Loss Properties 1646.7 Conclusions 165References 165

Chapter 7 Preisach Applications 1677.1 Introduction 1677.2 Dynamic Effects 1677.3 Eddy Currents 1687.4 Frequency Response of the Recording

Process 1707.5 Pulsed Behavior 172

7.5.1 Dynamic Accommodation Model173

7.5.2 Single-Pulse Simulation 1787.5.3 Double-Pulse Simulation 181

7.6 Noise 1817.6.1 The Magnetization Model 183

x CONTENTS

7.6.1 The Magnetization Model 1837.6.2 The Effectof the Moving Model

1847.6.3 The Effect of the Accommodation

Model 1867.7 Magnetostriction 1887.8 The Inverse Problem 1947.9 Conclusions 195References 195

Appendix A

Appendix B

Appendix C

Index 211

The Playand StopModels 199

The Log-Normal Distribution 203

Definitions 207

About the Author 215

PREFACE

The modeling of magnetic materials can be performed at various levels ofresolution. The highest level of resolution is the atomic level. At this level, one canuse quantum mechanics to understand the basic processes involved. The next stepdown in resolution is the micromagnetic level, where the magnetization is acontinuous function of position. At a still lower level of resolution, one uses thedomain level of modeling, where the material is divided into uniformly magnetizeddomains separated by domain walls of zero thickness. Finally at the lowestresolution, the nonlinear medium level, the magnetization is the average of manydomains, and the physical nature of their formation is ignored. In this last level, themedium is characterized by an input/output relationship.

Preisach modeling is a mathematical tool that has been used principally at thenonlinear medium level, but it can also give some insight at all the levels. Itseffectiveness in describing magnetic materials is due to its ability to have abehavior when the applied field is increasing which is different from its behaviorwhen the applied field is decreasing. It is thus able to describe minor loops andother complex magnetizing processes. The classical Preisach model is limited bythe congruency property and the deletion property, neither of which is possessedby magnetic materials. Although these limitations could be removed using aphenomenological approach, this book relies on physical reasoning as much aspossible to make necessary modifications. This practice usually results in simplermodels that give physical insight into the processes of interest. Although thesemodifications have been shown to be robust, the book uses physical reasoningrather than mathematical rigor to justify its derivations.

In Chapter 1, the physics of magnetization processes is briefly summarized.Chapter 2 summarizes the classical Preisach model, which is the basis for thestatistical analysis used in modeling hysteresis. However, since it cannot describe

xi

xii PREFACE

many of the subtleties in the behavior of magnetic materials, modifications basedupon physical reasoning are presented in the subsequent chapters. In particular, theconcept of reversible magnetization is discussed in Chapter 3. Accurate behaviorof the susceptibility is obtained by a magnetization-dependent reversiblecomponent, called the DOK model. This is further improved by adding a morecomplex state-dependent reversible component, called the CMH model. As shownin Chapter 4, the congruency limitation can be removed by means of an output­dependent model, such as the moving model or the product model. Including eitheraccommodation, aftereffect, or both in the model, as shown in Chapter 5, removesthe deletion property. Even with all these modifications, the resulting model is stilla scalar model, so in Chapter 6, we discuss methods of generalizing it to a vectormodel.

Some applications are discussed in Chapter 7. First, since the model isessentially a magnetostatic model, this chapter presents two brief extensions todynamics. These extensions include the effect of eddy currents on themagnetization, the effect of the accommodation model on the pulsed behavior, andthe effect of the moving model and the accommodation model on noise. Anotherextension is the development ofa magnetostriction model. Finally, the developmentof an inverse model, which would be useful in control applications, is discussed.

I hope that this book is useful in showing how the Preisach model can beextended to describe accurately a wide range of magnetic phenomena. Although thediscussion is limited to magnetic phenomena, it can give deep insight into theanalysis of hysteretic many-body problems. The techniques presented here aregeneral and can be applied to hysteresis problems in disciplines other thanmagnetism.

Edward Della Torre

ACKNOWLEDGMENTS

Ferenc Vajda deserves my special thanks. This book is the result of the manyfruitfulandstimulatingdiscussions thatwehavehad.Manyof thenumerous paperson whichwe hadcollaborated forman important part of this book.I wouldlike tosingle himout for his earlierhelp, insightand encouragement.

I also thankthe following students, whoattended a coursein whichI used themanuscript of this book as a text: Jason Eicke, Luis Lopez-Diaz, Jie Lou, AnnReimers, and PattanaRugkwamsook. I am grateful to Lawrence H. Bennett,whohas beena constantsourceof adviceandencouragement. AlsoMichael Donahue,Robert McMichael, and Lydon Swartzendruber deserve my thanks. My manycolleagues, too numerous to mention, with whomdiscussions resulted in a richexchangeof ideas,also are acknowledged here withthanks.

I alsothankmywife,Sonia,whoreadthismanuscript andmademanyhelpfulsuggestions as it progressed.

Edward Della Torre

xiii

CHAPTER 1

PHYSICS OF MAGNETISM

1.1 INTRODUCTION

The aim of this book is to characterize the magnetization that results in a materialwhen a magnetic field is applied. This magnetization can vary spatially because ofthe geometry of the applied field. The models presented in this book will computethis variation accurately, provided the scale is not too small. In the case ofparticulate media, the computation cells must be large enough to encompass asufficient number of basic magnetic entities to ensure that the deviation from themean number of particles is a small fraction of the number of particles in that cell.In the case of continuous media, the computation cells must be large enough toencompass many inclusions. The study of magnetization on a smaller scale, knownas micromagnetics, is beyond the scope of this book. Nevertheless, we will see thatit is possible to have computation cells as small as the order of micrometers.

This book presents a study of magnetic hysteresis based on physical principles,rather than simply on the mathematical curve-fitting of observed data. It is hopedthat the use of this method will permit the description of the observed data withfewer parameters for the same accuracy, and also perhaps that some physicalinsight into the processes involved will be obtained. This chapter reviews thephysics underlying the magnetic processes that exhibit hysteresis only in sufficientdetail to summarize the theory behind hysteresis modeling; it is not intended as anintroduction to magnetic phenomena.

2 CHAPTER 1 PHYSICS OF MAGNETISM

This chapter's discussion begins at the atomic level, where the behavior of themagnetization is governed by quantum mechanics. This analysis will result in amethodology for computing magnetization patterns called micrornagnetisrn. For amore detailed discussion of the physics involved, the reader is referred to theexcellent books by Morrish [1] and Chikazumi [2].

Since micromagnetic problems involve hysteresis, there are many possiblesolutions for a given applied field. The particular solution that is appropriatedepends on the history of the magnetizing process. We view the magnetizingprocess of hysteretic media as a many-body problem with hysteresis. In thischapter, we start by reviewing some physical principles of magnetic materialbehavior as a basis for developing models for behavior. Special techniques aredevised in future chapters to handle this problem mathematically. The Preisach andPreisach-type models, introduced in the next chapter, form the basic framework forthis mathematics. The discussion presented relies on physical principles, and wewill not discuss the derived equations with mathematical rigor. There are excellentmathematical books addressing this subject, including those by Visintin [3] and byBrokate and Sprekels [4]. In subsequent chapters, when we modify the Preisachmodel so that it can describe accurately phenomena observed in magnetic materials,we will see all these physical insights and techniques.

1.2 DIAMAGNETISM AND PARAMAGNETISM

Both diamagnetic and paramagnetic materials have very weak magnetic propertiesat room temperature; neither kind displays hysteresis. Diamagnetism occurs inmaterials consisting of atoms with no net magnetic moment. The application of amagnetic field induces a moment in the atom that, by Lenz's law, opposes theapplied field. This leads to a relative permeability for the medium that is slightlyless than unity.

Paramagnetic materials, on the other hand, have a relative permeability that isslightly greater than 1. They may be in any material phase, and they consist ofmolecules that have a magnetic moment whose magnitude is constant. In thepresence of an applied field, such a moment will experience a torque tending toalign it with the field. At a temperature of absolute zero, the electrons or atoms witha magnetic moment in assembly would align themselves with the magnetic field.This would produce a net magnetization, or magnetic moment per unit volume,equal to the product of their moment and their density. This is the maximummagnetization that can be achieved with this electron concentration, and thus it willbe called the saturation magnetization Ms. Atoms possess a magnetic moment thatis an integer number of Bohr magnetons. The magnetic moment of an electron, mB,

is one Bohr magneton, which in SI units is 0.9274 x 10-23 A-m2• We note that the

permeability of free space flo, and Boltzmann's constant, k, are in SI units41t x 10-7 and 1.3803 x 10-23J/mole-deg, respectively.

Paramagnetic behavior occurs when these atoms form a reasonably diluteelectron gas. At temperatures above absolute zero, for normal applied field

SECTION 1.2 DIAMAGNETISM AND PARAMAGNETISM 3

strengths, thermal agitation will prevent them from completely aligning with thatfield. Let us define B as the applied magnetic flux density, and T as the absolutetemperature. Then if we define the Langevin function by

1L(~) = coth ~ - ~' (1.1)

then the magnetization is proportional to the Langevin function, so that

M = Ms L(~), (1.2)

where

JlogJmBH = JlomH

kT kT(1.3)

(1.4)

6542

v '\

L(f)

---- L'(~) ----I\. ~

V-:/

/~ - /

/ <,<,

V<,

;--... ..........

"' ..... ----------oo

u>'j 0.8·5~

J1~O.6

~

=o·~O.4

~=.~>~O.2

~~

Here the moment of the atom, m, is the product of g, the gyromagnetic ratio, J theangular momentumquantum number, and ma the Bohr magneton. It can be shownthat thedistributionof magneticmomentsobeysMaxwell-Boltzmann statistics [5].Figure 1.1 shows a plot of the Langevin function and its derivative. It is seen thatfor small ~ the function is linear with slope 1/3 and saturates at unity for large ~.

The susceptibility of the gas, the derivative of the magnetization with respectto the applied field, is given by

dM M~ [ 1 2]x(H) = - = - - - csch ro .dH H ~2

1

Figure 1.1 Langevin function(solidline) and its derivative(dashedline).

4 CHAPTER 1 PHYSICS OF MAGNETISM

(1.6)

For small ~, thequantityin bracketsapproaches 1/3.Thus, whenthe appliedfieldsare small, the susceptibility, Xo is given by

Ms~ llomMsXo = 3H 3kT (1.5)

The small field susceptibility as a functionof temperature is shown in Fig. 1.2.At roomtemperature, theargumentof theLangevin functionis verysmall,and

this effect is very weak; that is, the misalignment due to thermal motionsis muchgreater than the effect of the appliedfield.Thus, thiseffect is not significantin thedescriptionof hysteresis; however, when discussingferromagnetic materials, wewillsee that theirbehaviors abovetheCurietemperature aresimilar,exceptthatthesusceptibility diverges at the Curie temperature rather than at absolute zero.

The previousanalysis did not includequantization effects.Sincethe magneticmomentcan varyonlyin integermultiples, theLangevin functionmustbereplacedby the Brillouin function, BJ..~). The Brillouin function is defined by

21+1 21+1 1 ~B (~) = --coth--~ - -coth-.J 21 21 2J 2J

Thus, the magnetization M(T) at temperature T, is given by

M(T) = NmBgJBJ(~)' (1.7)

where,N is the numberof atoms per unit volume, g is 0.5 for the electron, and J,an integer, is the angular momentumquantumnumber. The Brillouin function iszero if ~ is zero, and approaches one if ~ becomes large, as seen in Fig. 1.3.Therefore, from (1.7) we have

\\\\

0----r--------I----.

Absolute temperature

Figure 1.2 Paramagnetic susceptibility as a function of temperature.

SECTION 1.3 FERRO·, ANTIFERRO- AND FERRIMAGNETIC MATERIALS

l·········_·······················~·---."."...

~/'

/~ /" /~~ /

// ur

/BJ.~) ,

0

~

Figure 1.3 Plot of BJ (~) and the linear function,~kTas a function of ~ for J = 1.

and so

M(1) = Bj.~).M(O)

For small ~, the Brillouin function is given approximately by

Bj.~) = J+l~.3J

1.3 FERRO-, ANTIFERRO- AND FERRIMAGNETIC MATERIALS

(1.8)

(1.9)

(1.10)

5

In accordance with the Pauli exclusion principle, electrons obey Fermi-Diracstatistics; that is, only one electron can occupy a discrete quantum state at a time.When atoms are placed close together as they are in a crystal, the electron wavefunctions of adjacent atoms may overlap. Here, it is found that given a certaindirection ofmagnetization for one atom, the energy of the second atom is higher forone direction ofmagnetization than the other. This difference in energy between thetwo states is called exchange energy. Furthermore, when the parallel magnetizationis the lower energy state, the exchange is said to be ferromagnetic, but when theantiparallel magnetization is the lower energy state, the exchange is said to beantiferromagnetic. In ferromagnetic materials, this energy is very large and causesadjacent atoms to be magnetized in essentially the same direction at normaltemperatures. Pure metal crystals of only three elements, iron, nickel, and cobalt,are ferromagnetic.

6 CHAPTER 1 PHYSICS OF MAGNETISM

Since the electron wave functions are very localized, the overlap of wavefunctions between adjacent atoms decreases very quickly to zero as a function ofthe distance between them. Thus, exchange energy is usually limited to nearestneighbors. Sometimes the intervening atoms in a compound can act as a medium sothat more distant atoms can be exchange coupled. Here, the resulting exchange iscalled superexchange.This, can also be either ferromagnetic or antiferromagnetic.Thus, compounds such as chromium dioxide can also be ferromagnetic.

The effect ofexchange energy can be accounted for by an equivalent exchangefield. Thus, the field, H, that an atomic moment experiences is given by

H = HA + NwM, (1.11)

where HA is the applied field, Nwis the molecular field constant, and NwM is theexchange field. Substituting this into (1.3), one sees that ~ is now given byThe remanence is obtained by setting H equal to zero in this equation and solving

~ogJmB[H+N,.M(1)]~ = -------

kT(1.12)

for Men. Thus,

(1.13)

and we can use (1.8) to write this as follows:

(1.14)~kT.~kT

IL g2m 2J2NN'-0 B W

M(1) = _M(O)

(1.15)J+l = 19.3J

Since this must also be equal to the Brillouin function, we can obtain a graphicalsolution by plotting the two functions on the same graph, as illustrated in Fig. 1.3.For low temperatures, the slope of (1.14) is very small, so the intersection occursat large values of ~, and thus normalized magnetization approaches unity. As thetemperature increases, the slope also increases, and thus, the magnetizationdecreases.

At the Curie temperature, S, the slopes of (1.14) and that of the Brillouinfunction are equal. This intersection occurs at a point where both ~ and themagnetization are zero. The Curie temperature can be computed, since from (1.10),the slope of the Brillouin function is given by

dBJ.~)l~ =0

Thus, the Curie temperature is

SECTION 1.3 FERRO-, ANTIFERRO- AND FERRIMAGNETIC MATERIALS 7

e flog 2m B2J(J +1)NNw

3k(1.16)

Beckerand Doring[6]computed thesaturation magnetization as a functionoftemperature and the totalangularmomentum. A comparison withmeasuredvaluesfor iron and nickel,as shownin Fig. 1.4,appears to be a good fit with theory if Jis taken to beeither0.5 or 1.

AbovetheCurietemperature, thematerial actsasaparamagnetic mediumwiththe susceptibility diverging at a temperature called the Curie-Weiss temperaturerather than at absolute zero degrees. The latter temperature is close to the Curietemperature for mostmaterials. This typeof behavioroccursregardless of whetherthe material is singlecrystalor consistsof manyparticlesor grains that are largerthan a certain critical size. However, for small particlesor grains another effectoccurs.We willshowin Section1.6that if thesegrainsare sufficientlysmall, theymayhave only two stablestates separated by an energybarrier.It is then possiblethat at a temperature smaller than the Curie temperature, called the blockingtemperature, the thermodynamic energykT willbecomecomparable to the barrierenergy, In that case, the particles or grains can spontaneously reverse and thematerial no longer will appear to be ferromagnetic. Above the blockingtemperature, it behaveslikeaparamagnetic material withgrainsthathavemomentsmuch larger than the spin of a single electron. This type of behavior is called

1.0

0.8

f 0.6

,-..0

~ 0.4o Ni

~ x Fe

0.2

0.2 0.4 0.6Tie--+-

0.8 1.0

Figure 1.4 Temperature variation of saturation magnetization for atoms with different total angularmomentum. [After Becker and Doring, 1939.]

8 CHAPTER 1 PHYSICS OF MAGNETISM

superparamagnetism. As the temperature is raised from below, the materialappears to lose its remanence and has a sudden large increase in its susceptibility.For a medium with a distribution of grain sizes, there is a distribution in energybarriers so that the blocking temperature is diffuse.

If the exchange energy is negative, it is convenient to think of the material ascomposed of two sublattices magnetized in the opposite directions. If themagnitude of the magnetization is the same for these two antiparallel sublattices,the net magnetization will be zero, and the material is said to be antiferromagneticand appears to be nonmagnetic. On the other hand, if the magnitude differs, thematerial will have a net magnetization; such a material is said to beferrimagnetic.

Ferrimagnetic materials usually have smaller saturation magnetization valuesthan ferromagnetic materials, because the two sublattices have oppositemagnetization. These materials are important, since they usually occur in ceramicsthat either are insulators or have very high resistivity. Such materials will supportnegligible eddy currents and so will be useful to very high frequencies.

The materials will be ferrimagnetic for all temperatures below a criticaltemperature, known as the Neel temperature. Above that temperature the materialsalso become superparamagnetic, similar to the way ferromagnetic materialsbehave above the Curie temperature. Since the two sublattices may have differenttemperature behavior, it is possible that at a given temperature the two momentsmay be equal but opposite in sign, as illustrated in Fig. 1.5. At this temperature,known as the compensation temperature, the two sublattices have equalmagnetization so that the net magnetization is zero. This magnetization is themagnitude of the difference between the two sublattice magnetizations and will bepositive, since above or below the compensation temperature, the material willbecome magnetized in the direction of an applied field.

The compensation temperature occurs above, below, or at room temperature,depending on the elements in the crystal. Unlike the demagnetized state above theCurie temperature, this state "remembers" its magnetic state, and changing itstemperature from the compensation temperature reproduces the previous magneticstate. This property is useful in magneto-optical disks to render the storedinformation impervious to stray fields. This is done by choosing a compensationtemperature that is close to the storage temperature. For practical devices thestorage temperature is usually room temperature.

1.4 MICROMAGNETISM

In this section we assume that the temperature is fixed so that material parameters,such as saturation magnetization, may be regarded as constants. We then computethe equilibrium magnetization patterns in a ferromagnetic medium. The dynamicsof magnetization are discussed in later sections. Thus, we choose the magnetizationvariation that minimizes the total energy. This total energy is the sum of theexchange energy, the magnetocrystalline anisotropy energy, and the Zeemanenergy.

SECTION 1.4 MICROMAGNETISM 9

-- --- ---- Sublattice 1~

<, -- Sublattice 2<, - - - - - -- Total

" -,--~

~-,

- -

\~... __ ._ ..~\

1.5

oo 0.25 0.5 0.75

Normalized absolute temperature1

Figure 1.5 A ferrimagnetic material with a compensation temperature of approximately65% of its Nee) temperature.

The exchange energy, the source of the ferromagnetism, is given by

Wex = L JSi·Sj ,n.n.

(1.17)

where n.n. denotes that the sum is carried out over all pairs of nearest neighbors,J is the exchange integral, and S is the spin vector. Since the wave functions are notisotropic, the exchange energy is not only a function of the difference in orientationof adjacent spins, but is also a function of the direction of the spins. Since a spininteracts with several nearest neighbors, the orientation energy depends upon thecrystal structure. This variation in the exchange energy with spin orientation iscalled the magnetocrystalline anisotropyenergy.We take it into account by addingan anisotropy energy density term to (1.17). For cubic crystals, the simplest formof this is given by

(1.18)

where the ex'sare the direction cosines with respect to the crystalline axes, and Kis the anisotropy constant. If K is positive, the minimum anisotropy energy densityoccurs along each of the three axes of the crystal. On the other hand, if K isnegative, the minimum anisotropy energy density occurs along the four axes thatmake equal angles with respect to the three crystal axes. Higher order terms maybe added to this in certain cases.

10 CHAPTER 1 PHYSICS OF MAGNETISM

Another type of anisotropy energy density that commonly occurs is theuniaxial anisotropyenergydensity.This is given by

Wu = Kusin20 , (1.19)

where K; is the uniaxial anisotropy constant, e is the angle the magnetizationmakes with respect to the z axis, and z is the easy axis if K; is positive. If K; isnegative,thenz is the hardaxis, and theplaneperpendicularto thez axis is theeasyplane. We will denote the anisotropyenergydensity by Wanis whether it is cubic oruniaxial.

For the present, we willconsideronlyone additionalenergyterm, the Zeemanenergy, which is the energy that a magnetic dipole m has due to a magneticfield.This energy is given by

WZeeman = -m·B, (1.20)

where B is the total magneticfield, which is the sum of the external applied fieldand the demagnetizing fieldof the body.We willdecomposethis terminto the sumof the applied field energy and the demagnetizing energy. The energy of amagnetizedbody in an external field is given by

WH = JB·D dV. (1.21 )v

Since B is Jlo(H + M), and sinceM 2 is constant, by choosing a different referenceenergy, this reduces to

(1.22)

where H is that applied field and V is the volumeof the material. Similarly, theself-demagnetizing energy is given by

where DDis the demagnetizing field.Thus, the total energyof the body is given by

W = Wex + Wanis + WD + WH •

(1.23)

(1.24)

The magnetization pattern is then determined by adjusting the orientation of themagnetization ateachpoint in thematerialto minimize the totalenergy.In principlewe could find theorientationof the magnetization of each atomin the medium, butunless theobject is verysmall,this wouldinvolvetoo manycomputations. Instead,in micromagnetism, we will define a continuous function whose value at eachatomic site is the magnetization of that atom.

SECTION 1.4 MICROMAGNETISM 11

(1.25)

(1.26)

(1.28)

(1.29)

Micromagnetism is the study of magnetization patterns in a material at a levelof resolution at which the discrete atomic structure is blended into a continuum, butthe details are still visible. Thus, the orientation of the magnetization in the mediumis obtained from a continuous function defined over the medium. Summations arereplaced by integrations, and differences by derivatives. In particular, if r is theposition of an atom and a is the relative position of a neighbor, the exchangeenergy density between them is given by

w = -lim 2Js(r)·s(r+a).ex a-'O a 3

Since the magnetization and the spin vector are in the same direction, we canreplace s by sMlMs, where s is the magnitude of the spin vector and Ms is themagnitude of M. Then, if we expand S(r+a) in a Taylor series, we get

s [ aM(r) a 2 a2M(r) 1s(r+al ) = - M(r) +a-- + + ... ,x Ms ax 2 ax 2

where a is the distance to the nearest neighbor atom in the x direction and I, is aunit vector in the x direction. Then

s 2 [ aM(r) a 2 a2M(r) ]s(r)·s(r+al ) = - 1 + aM(r)--- + -M(r)- + .... (1 27)x 2 ax 2 a 2 .Ms x

The first term in the Taylor series is a constant and can be omitted by choosing adifferent energy reference. Since

M_ aM = .!..aM2

ax 2 ax 'and since M 2 is a constant, the second term in (1.27) is zero, If we sum the termsin the y and z directions as well, then for a simple cubic crystal, the total exchangeenergy becomes

oW"x :: _~r MO( a2M + a2

M + a2M ) dV

M;Jv ax2 ay2 az 2

_~r MoV2MdV,M 2JVs

where

Js2

A =-a

(1.30)

Because of the additional atoms in a unit cell, for a body-centered cubic lattice theexchange constant A is twice the value of a simple cubic lattice, and for a face­centered cubic lattice it is four times the value of a simple cubic lattice.

12 CHAPTER 1 PHYSICS OF MAGNETISM

It is noted that (1.29) is approximate in two respects. First, the Taylor seriesis truncated. Thus, the change in magnetization between adjacent atoms is assumedto be small to allow the series to converge rapidly. This assumption is usually valid.The second approximation is more subtle in that we are approximating a discretefunction by a continuous function. Since M 2 is constant, the second derivative ofthe magnetization diverges at the center of a vortex. Thus, (1.29) would calculatean infinite energy, although Js(r)· s(r +al ) remains finite at the center of the

xvortex.

The equilibrium magnetization in a medium is obtained by varying thedirection of the magnetization so as to minimize the total energy. This can be doneby directly minimizing the energy or by solving the Euler-Lagrange partialdifferential equation corresponding to this variational problem. The resultingmagnetization pattern is referred to as the micromagneticsolution. This calculationmust be performed numerically, except for a few cases, two of which are discussedin the next two sections. This introduces an additional discretization error thatcalculates a finite energy at the center of the vortex. This energy is incorrect unlessthe discretization distance is the same as the size of the magnetic unit cell.

If one is interested in the details of the magnetization change when the appliedfield changes, the dynamics of the process must be introduced. Two such effects ­eddy currents, in materials with finite conductivity, and gyromagnetism - arediscussed later.

1.5 DOMAINS AND DOMAIN WALLS

An equilibrium solution to the micromagnetic problem in an infinite medium isuniform magnetization along an easy axis. Then, both the exchange energy and theanisotropy energy are zero. Such a region of uniform magnetization is called adomain. In an infinite medium that is not uniformly magnetized, we will now seethat the equilibrium solution is the division of the medium into many domains thatare separated by domain walls that have essentially a finite thickness. Domain wallsof many types are possible, but in this section we discuss only the two simplesttypes: the Blochwall and the Neel wall.Furthermore, domain walls are classifiedby the difference in the orientations of the domains that they separate, expressedin degrees. For brevity, we limit ourselves to 180 0 walls.

We will consider a domain wall whose center is at x = 0, which divides adomain that is magnetized in the y direction as x goes to infinity and that ismagnetized in the - y direction as x goes to minus infinity. As one goes from onedomain to the other, if the magnetization rotates about the x axis, it remains in theplane of the wall, and the wall is said to be a Blochwall.On the other hand, if themagnetization rotates about the z axis, the wall is said to be a Neel wall.

SECTION 1.5 DOMAINS AND DOMAIN WALLS

1.5.1 Bloch Walls

13

Let us consider a Bloch wall that lies in the yz plane and that separates twodomains: one magnetized in the y direction and the other magnetized in the - ydirection. If the domain magnetized in the y direction lies in the region of positivex, and the domain magnetized in the - y direction lies in the region of negative x,then the magnetization can be written as

M(x) = Ms {cos[e(x)]l, + sin[e(x)]l~}, (1.31)

with the boundary conditions 6( - 00) =0 and 6(00) ='ft. That is, the magnetizationis in the zdirection for large negative values of x and in the - zdirection for largepositive values of x. Differentiating twice with respect to x, we have

a2M = -M( ae)2, (1.32)ax2 dx

so that

(1.33)

If there is no applied field, and since there is no demagnetizing field, the Zeemanenergy is zero. Summing the remaining energies, the anisotropy energy andexchange energy, from (1.29), the energy in a domain wall per unit area is asfollows:

w =i~ [A ~~r + g[6<X)]]dx, (1.34)

where g[6(x)] is the volume density anisotropy energy function.We obtain the domain wall shape by finding the O(x), which minimizes this

integral subject to the constraints that O( - 00) = 0 and 0(00) ='ft. This minimum isfound, using the calculus of variations, by solving the corresponding Lagrangedifferential equation corresponding to the minimization of this integral. In this case,this is given by

dg(6) _ 2A( d26) = o.

de dx?(1.35)

If we integrate this from 0 to 0, since g(O) is zero and since d8Idxlx=_oo is zero,we obtain

14 CHAPTER 1 PHYSICS OF MAGNETISM

or

LCXJd 2e f x d ( de) 2 ( de) 2g(e) = 2A -2de = A - - dx = A - ,o dx -CXJ dx dx dx

(1.36)

(1.37)

For crystals with uniaxial anisotropy, from (1.19),

g(e) = Kucos2e .

Then

x = IT r64!L = lw In( tan'!!') ,~ J(~ Jo sinf 1t 2

where l; is the classical wall width given by

lw = rtJAI«.

(1.38)

(1.39)

(1.40)

For iron, this is approximately42 nm, or roughly 150 atoms wide. Solving for a,one gets

a -I( 1tx) (rtx) 1t= tan exp-c = gd -c -"2' (1.41)

where gd, defined by this equation, is called the Gudermannian. Figure 1.6 plotseas a function of x. It is seen that more than half of the rotation in angle takesplace between ±lw. In fact, in the equal angle approximationall the rotation takesplace between±lw' Since for manymagneticmaterialst; is the order of 0.1 urn, thedomain wall is very localized.Substituting (1.36) into (1.34), we see that the totalenergy density per unit wall area is given by

f rr.l2 dx f1t.2W = 2 g(a)-d8 = 2 JAg(a)de.

-rtl2 de -rtl2

Thus, for uniaxial materials, this becomes

f rt/2

w = 2 VAKusin26d8 = 4JAKu'-n/2

(1.42)

(1.43)

SECTION 1.5 DOMAINS AND DOMAIN WALLS 15

10.5o-0.5-1

I /,I / I

I I I i I I- - -+- - - ~ - - -1- -l- - - -+ --

I I I I II I I I II I I I I

--~--4------~--~--I I I I II I I I II I I I I

--..L--J- -I---L--J---I I 'I I I I

I / I 'I 'II I IO-==~~-_-L-__....L-_---JL....--_--L..__--J

-1.5 1.5

Position in units ofwall width

Figure 1.6 Variationof the magnetization angle for a Blochwall:Dashedline indicates the equalangle approximation to the angle variation.

1.5.2 Neel Walls

For an infinite Neel wall, the magnetization is given by

M = Ms [cos6(x)lx + sin6(x)lzl, (1.44)

where 6 goes from 0 to 1t as x goes from - 00 to 00. The only difference betweenthis and the Bloch wall is that the magnetization now turns so that when x is zeroit points from one domain to the other. In this case, the divergence of M is nolonger zero, and there is a Zeeman term in the total energy. Since the divergenceof B is zero, the divergence of M is the negative of the divergence H. In particular,

aMx dediv M = - = M cos6(x)- = -div H. (1.45)ax s dx

Since H has only an x component, when we integrate this equation and use theboundary conditions that H( - 00) = H(00) = 0, we are led to the conclusion thatH, = - Ms. From (1.23), the demagnetizing energy of the moments in this field isgiven by

w =D(1.46)

Comparison with (1.38) shows that this has the same variation as the uniaxialanisotropy energy. Thus, a Neel wall has the same shape as a Bloch wall whose

16 CHAPTER 1 PHYSICS OF MAGNETISM

anisotropy energy is given by Ku + JlMS2

• Since the wall energy is proportional tothe square root of Kg, it is seen that the Neel wall will have greater energy than aBloch wall. Thus, in infinite media, Bloch walls are energetically preferable to Neelwalls. Furthermore, since the wall width is inversely proportional to the square-rootof Kg, it is seen that the Neel wall will be thinner than a Bloch wall.

We have just discussed domain walls in infinite media. In finite media, thewalls will interact with boundaries. Thus, in thin films, 180 0 walls betweendomains magnetized in the plane of the film tend to be Neel walls, to minimizedemagnetizing fields. Furthermore, at the junction of two walls of oppositerotation, complex wall structures can form, such as cross-tie walls. This subject isbeyond the scope of this chapter.

1.5.3 Coercivity ofaDomain Wall

In the continuous micromagnetic case, the energy is not a function of the positionof the domain wall. Thus the slightest applied field will raise the energy of thedomain on one side of the wall with respect to the other, and there will be nothingto impede its motion, thus predicting zero coercivity. In a real crystal, themagnetization is not continuous because there are preferred positions of the domainwall, so there is a very small coercivity. The sources ofcoercivity in a real materialare the imperfections in the crystal structure. We will briefly discuss imperfectionsof two types: inclusions and dislocations in the crystal lattice.

Inclusions are small "holes" in the medium, usually formed by the entrapmentof bits of foreign matter. The inclusions either are nonmagnetic or have a muchsmaller magnetization than their surroundings. Such an inclusion will havemagnetic poles induced on its surface, which will repel an approaching domainwall, thus impeding its progress. The equilibrium position of this wall in theabsence of an applied field will be between the inclusions. The absence ofexchange and anisotropy energy in the inclusion implies that the domain wall willhave lower energy when it is situated on the inclusion also impeding its progress.

When a field is applied to a material with inclusions, the wall will bend in adirection that increases the volume of the domain that is closer to being parallel tothe applied field. When the field is increased beyond a critical value, the domainwill snap past that inclusion and become attached to another inclusion. We willdenote the applied field behavior of the magnetization of the volume swept out bythis motion as a hysteron. Even if it were possible to sweep that volume back, thefield required to sweep the domain wall back generally would differ from thenegative of the preceding field, which is now being restrained by differentinclusions. Furthermore, these two fields are statistically independent ofeach other.

Dislocations in the crystal lattice also interact with domain walls. In somecases, the easy axes on the two sides of the dislocation may be aligned differently.This permits walls to be noninteger multiples of 90 0

• If the dislocations aresufficiently severe, the exchange interaction between atoms on the two sides of the

SECTION 1.6 STONER-WOHLFARTH MODEL 17

wall may become negligible and a domain wall might not be able to cross theboundary.

A hysteron can switch either by rotation of the magnetization in the domain,as discussed in the next section, or by wall motion. In the latter case, if there is awall, it has to be translated past the inclusions. On the other hand, if the materialhad been saturated, so that all the domain walls were annihilated, a new wall wouldhave to be nucleated, The nucleation of a reversed domain requires a much higherfield than that required to move a wall past each inclusion. Thus, nucleation usuallytakes place only when there are no domain walls anywhere in the crystal. If onemeasures the hysteresis loop of a material by controlling the rate of change ofmagnetization to a very slow rate, the field required for the initial change inmagnetization is found to be larger than that needed for subsequent changes inmagnetization. The resulting loop is said to be reentrant. Such a loop is shown inFig. 1.7. The random variation in width is due to the variation in coercivity frominclusion to inclusion.

1.6 THE STONER-WOHLFARTH MODEL

A magnetic medium consisting of tiny particles can have a much higher coercivitythan a continuous medium with inclusions. A model to analyze this case by meansof an ellipsoidal particle was proposed by Stoner and Wohlfarth [7], who used atheorem, shown by Maxwell, that the demagnetizing field of a uniformlymagnetized ellipsoid is also uniform. Thus, it is possible to have an object in whichthe applied field, the demagnetizing field, and the magnetization are all uniform.This model is called the coherentmagnetization model. Other magnetization modesare possible if the material is large enough, but for bodies whose largest dimensionis smaller than the width of a domain wall, only the uniform magnetization mode

Applied field

Figure 1.7 A typical reentranthysteresis loop.

18 CHAPTER 1 PHYSICS OF MAGNETISM

is possible. In such cases, we say that the particle is a single domain particle. Ofcourse if the particle is too small, thermal energy might be sufficient todemagnetize it, and the particle would become superparamagnetic. That is, itwould behave like a paramagnetic particle with a very large moment.

The Stoner-Wohlfarth model assumes that the particle is an ellipsoid and thatits long (easy) axis is aligned with its magnetocrystalline uniaxial easy axis. It isalso assumed that as the magnetization rotates, its magnitude remains constant.Because we assume that the particle is single domain, that is, it is uniformlymagnetized, its exchange energy is seen to be zero. As the magnetization of theparticle is rotated, the demagnetizing field changes in magnitude, and thus thedemagnetizing energy changes because the demagnetizing factors along thedifferent axes of the particle differ. This energy is referred to as shape anisotropyenergy. Then magnetization will be oriented in such a way that the total energy­the sum of the applied field energy, the demagnetizing energy, and the shapeanisotropy energy - is minimized. The sum of the latter two energies will bereferred to simply as the anisotropy energy.

We will assume that a field is applied horizontally to a particle whose long axismakes an angle p with it, as shown in Fig. 1.8. All angles are measured in thecounterclockwise direction, so that 6, the angle the magnetization makes withrespect to the particle's long axis, as pictured, is negative. We will presently seethat if the applied field is zero, the magnetization will lie along the easy axis of theparticle; however, it could be oriented either way along that axis. Thus, theanisotropy energy will be doubly periodic as the magnetization rotates. We willalso see that the applied field energy is unidirectional and thus is singly periodic.

Maxwell showed that for a uniformly magnetized general ellipsoid, thedemagnetizing field is also uniform, though not antiparallel to it. Thedemagnetizing field can be written as the product of the demagnetization tensor andthe magnetization. The demagnetization tensor is diagonalized if the coordinateaxes are chosen to be the principal axes of the ellipsoid. In that case, the diagonalelements are referred to as the demagnetizing factors, and the demagnetizing field

Figure 1.8 Stoner-Wohlfarth description of a spheroidal particle.

SECTION 1.6 STONER-WOHLFARTH MODEL

Ho is given by

19

HD = D%M%l% + DyMyl y + D%~lZ' (1.47)

whereDx' Dy, Dr. are the demagnetizing factorsalongthe three principalaxes of theellipsoid. Maxwell also showed that

D% + Dy + o, = 1. (1.48)

For a spheroid,an ellipsoidof revolution, if they and z are the twoequal axes, then

I-DD = D = __x (1.49)

y z 2

It is well known that for this spheroid

D" = ULI[VU~-I ~ U+VU2-1) -I], for U> I, (1.50)

and

D = _1_[1 - _11_ sin-1V1- (12] for a < 1x 2 r;-::;. , ,1- a. V1- u2

(1.51)

wherea is the ratio of the lengthof the particlealong thex axis to the lengthof theparticle along other axes (see Bozorth [8]). It can be shown that as ex approachesone for both formulas, the demagnetization factor approaches 1/3, the value for asphere. It can also be shownthat when a =0, then Dx =1, and for large a (1.50)becomes

1D = -(1n2a - 1)% 2 'a.

(1.52)

(1.53)

and thus, goes to zero essentially as l/a2• A graph of D as a function of a,

illustratingthat 0 s D, ~ 1, is shownin Fig. 1.9.Usingthe variablesillustratedinFig. 1.8 and theexpressionfor demagnetizing

energy in (1.23), it is seen that the demagnetizing energy is given by

f.L f.L M2V( I-D)w = ~M·H V = - 0 s D cos28 + --%sin28 .D 2 D 2 x 2

If D, is less than 1/3, then WD is a minimum when6 =o. If the applied field is nownonzero, then we have to add an appliedfield energy,WH, to this, whereaccordingto (1.21),

(1.54)

20 CHAPTER 1 PHYSICS OF MAGNETISM

~ 1.00 _+ __._._1.._ _ 1._ __ _ _

i i·····--····-··-····-···t··········-··..----i·-·-··--··· i - - .

0.00 I ! i

02345Aspect ratio, a

Figure 1.9 Thedemagnetizing factor of a spheroid as a function of its aspectratio.

If the body remains uniformly magnetized, then the exchange energy isconstant. Since the uniaxial magnetocrystalline anisotropy has the same spatialvariation as the demagnetizing field, if their easy axes coincide, the two can becombined into a single term, and the effective demagnetizing factor must beincreased by K; However, if the long particle axis does not line up with themagnetocrystalline axis, an effective easy axis between the two must be computed.A plot of the total energy, the sum of (1.53) and (1.54), is shown in Fig 1.10, forthree applied field values: zero, H/2, and HIc , where HIc = 2KIM is called theanisotropy field. It is seen that for zero applied field, the energy has two equalminima 180 0 apart. Then the magnetization could be oriented along either of thesedirections. As the field is increased, the minimum near 180 0 decreases in energyand moves to the left while the minimum near 00 increases and moves to the right.At the critical field, the minimum near 0 0 disappears, and above that field there isonly a single minimum. When the field is decreased back to zero, the energy barrierbetween the two minima prevents the magnetization from going to the minimumnear 0 0

• Thus, saturating a magnetic material is one method of putting it in a uniquemagnetic state.

In order to solve for the minimum energy, we take the total energy given by

2 [ 1JlMsV I-D

W = -floMsV[H cose+H sine] - 0 D cos2e+-_x sin2e, (1.55)x y 2 x 2

differentiate it with respect to e, and set it equal to zero. Thus, after dividing by floMs ~ we get

SECTION 1.6 STONER-WOHLFARTH MODEL 21

1.5 r-----,....----,....----,.-------r

360-1.5 L---__L---__L----L.. ---'

o 90 180 270e(degrees)

Figure 1.10 Energyas a function of magnetization angle for three applied fields.

1 aw = Hxsin6 - Hycos6 - Csin6cos6 = 0,JloMsV ae (1.56)

where

_ [1 - 3Dx ]C - Ms .2

(1.57)

It is noted that for prolate particles, D, is less than 1/3, so that C will be positive.To determine whether this is a minimum or a maximum, we take the secondderivative of the energy with respect to 6, and obtain

1 a2w= H cos 8 + H sin 8 + C(sin28 - cos28). (1.58)

JloMsV ae2 x y

Since the system seeks an energy minimum, this quantity must be positive at astable equilibrium. To find the critical field, HIc, that is, the value of the field atwhich one of the minima disappears, we solve for the value that makes the secondderivative zero. Thus, we obtain

(1.59)

We can solve for cos eby multiplying (1.56) by sin e, multiplying (1.59) by cos B,and adding the results. Then one obtains

22 CHAPTER 1 PHYSICS OF MAGNETISM

(1.60)

Similarly, we can solve for sin e by multiplying (1.59) by sin e, multiplying(1.56) by -cos 6, and adding the results, yielding

sine = _(Hy/C)1I3 or H

y= -Csirr'fl. (1.61)

Since sin2e + cos2e = 1, we can eliminate e from (1.60) and (1.61). Thus,

Hx'1J3 + H:'3 = e2J3 • (1.62)

The solution to this equation is called the Slonczewski asteroid [9], which isillustrated in Fig. 1.11.

To determine the magnetization and its stability for a Stoner-Wohlfarthparticle, one plots the vector magnetic field from the origin, as shown for two fieldvectors in Fig. 1.11. The direction of the magnetization is obtained by drawing atangent from the asteroid to the tip of the field vector. The magnetization vector isobtained by drawing a vector whose length is given by MsV along that line. It isseen that when HI is applied, the tip of the field vector falls outside the asteroid,and there is a unique state for the magnetization, indicated by M1; however, whenH1 is applied, it falls inside the asteroid, and there are two stable states for themagnetization, both of which are indicated by M2•

Figure 1.11 Slonczewski asteroid used to determine the state of a Stoner-Wohlfarthparticle.

SECTION 1.6 STONER-WOHLFARTH MODEL 23

765432

I~ I I I I~Magnetization switches

~\~-----

0.00

0.75

0.50

0.25

-0.50-I 0

-0.25

Figure 1.12 Variation of awith the applied field for p= 0.5.

The applied field that achieves this magnetization can be obtained by solving(1.56) as

H = Csin(28).2sin(8 + P) (1.64)

The variation of ewith applied field is illustrated in Fig. 1.12. It is seen that forpositive fields, Bapproaches - p monotonically as the magnetization tries to alignitself with the applied field. For negative fields, e increases until it reaches itsmaximum, and then it switches,

We will define the critical angle eM as the angle at which the particle switches.It is obtained by solving for the value ofethat makes (1.58) equal to zero. It is thuspossible to plot m as a function of H by varying abetween - p and aM- That is,one must solve the transcendental equation

Hcos(P + 8M)-Ccos(28M) = o. (1.65)

If we substitute (1.64) into this, and use the tangent trigonometric identities, weobtain

(1.66)

If one plotted the component of the magnetization along the applied field'saxis, that is, Mscos(8 + P), as a function of the applied field, one would obtain thehysteresis loops shown in Fig. 1.13 for three values of~. These loops show that for

24 CHAPTER 1 PHYSICS OF MAGNETISM

.............

Applied field

!~:1I~!~/1 »: ,."

.........~.t., ..·····,··............................... _1-----

II

/I

/,/

---------Figure 1.13 Possible Stoner-Wohlfarth particle hysteresis loops for p= 2°,25°, and 45°.

If one plotted the component of the magnetization along the applied field'saxis, that is, Mscos(6 + P), as a function of the applied field, one would obtain thehysteresis loops shown in Fig. 1.13 for three values of p. These loops show that forparticles in the negative state, when the applied field reaches the critical field Hie'the particle abruptly switches to the positive state. If the magnetization was stillnegative before switching, this field is also the coercivity. On the other hand, if themagnetization was already positive, Hie is larger than that of the coercivity. Thelargest value of pfor which Hie is equal to the coercivity is 45 0

• It is seen that allthe hysteresis loops have two critical fields that are the same in magnitude butopposite in sign.

The critical field of a particle as a function of particle angle pwith respect tothe applied field can then be computed, from (1.62), as

(1.67)c

(cos p2l3 + sin p2l3)3nHk = -------

As shown in Fig. 1.14, this field is a maximum when p =0 or 1t/2. When pincreases from 0, the critical field of the particle decreases until p=1t/4, and thenincreases back to the value it had at p=0 when p= 1t/2.

For values of pbeyond 1t/4, as the field is increased from negative saturation,the magnetization goes through zero before the magnetization switches. Thus, wehave to distinguish between the critical field, the field at which the magnetizationswitches, and the coercivity, the field at which the magnetization is zero. Thecoercivity follows the critical field until1t/4. Beyond that it obeys (1.64) with eset

SECTION 1.6 STONER-WOHLFARTH MODEL 25

<,-,

<,-,

-,\

\O~-----oor--"""""'-----~-------1

0.2

0.8

o 30 60Particle field angle (degrees)

90

Figure 1.14 Coercivity and critical field variation with particle angle.

to the complement of p, as illustrated in Fig. 1.14. So the field at which the lowersection of the curve crosses the H axis is a monotonic decreasing function of p.

For particles that are larger but still single domain, other nonuniform reversalmodes are possible. These modes are characterized by smaller values of Hc and aresometimes referred to as incoherent reversal modes. Although these modes havea different pdependence, they have the same properties as the Stoner-Wohlfarthparticles: two stable states, a monotonic decreasing function of He with p, and amaximum in He when p is 0 or 1t/2.

Real particles are generally ellipsoidal but with "corners." These cornerspermit magnetization reversals to be nucleated with fields considerably smallerthan those necessary to nucleate reversals in ellipsoids. Since the shape of theparticles prevents the existence of analytical solutions for them, reversal modes ofthese types have been studied numerically [10,11]. It was seen that for realparticles, although their specific properties differ in magnitude and in variousdetails, their general properties are the same as those of Stoner-Wohlfarthparticles: that is, they have two stable states for a certain range of particle sizes;their switching field at first decreases with angle and then increases; and theircoercivity is a monotonic decreasing function of angle.

One difference between ellipsoidal and nonellipsoidal particles is that for thelatter there is a nucleation volume that, once reversed, causes the whole particle toreverse. This is also referred to as the activation volume, and it usually has anaspect ratio of unity. It may be thought of as the largest sphere that can be inscribedwithin the particle.

26

1.7 MAGNETIZATION DYNAMICS

CHAPTER 1 PHYSICS OF MAGNETISM

Hysteresis is a rate-independent phenomenon; that is, the final state is the same nomatter how fast the input changes to the final value. In fact, hysteresis is only afunction of the field extrema. Thus, to obtain the possible final states, it isnecessary only to solve the static equilibrium problem. To choose the particularmagnetization pattern that is appropriate for a given input sequence if onlyhysteresis were involved, one would have to be sure only that the energy, in thesequence of magnetization patterns that were traversed by this magnetizing process,was a monotonically decreasing function of time. Other dynamic effects, which wewill now discuss, may alter this sequence of equilibria.

There are two categories of dynamic effects: those that have time constantsmuch slower than the rate of the applied field, and those that are comparable to orfaster than the rate of the applied field. The former type includes magneticaftereffect, which causes the magnetization to drift with time, while the latter typeincludes eddy currents and gyromagnetic effects. A rate-independent effectsometimes confused with these is accommodation. Accommodation is anotherprocess that causes the magnetization to drift; however, this process requires achange in applied field to trigger it. It is observed that repetitive minor loopsapparently drift toward an equilibrium loop. As such, it is a rate-independentprocess and is discussed in Chapter 5.

Aftereffect refers to the slow change in magnetization with time that resultsfrom thermal processes. The magnetization is held in an equilibrium pattern byenergy potential barriers. They may be surmounted by thermal energy according tothe Arrhenius law. When this happens, the magnetization will find another localenergy minimum. The higher the potential barrier, the longer it will take to besurmounted, but given enough time, any barrier may be surmounted. With thisprocess, a magnetization pattern will change from a local energy minimum to aglobal energy minimum. For soft materials, with small energy barriers, this processwill take the order of many minutes, but with harder materials, withcorrespondingly larger energy barriers, it may take centuries. This also is discussedin greater detail in Chapter 5.

1.7.1 Gyromagnetic Effects

We now turn our attention to gyromagnetic effects. When a magnetic field isapplied to an electron, it creates a torque T on its magnetic moment m to align itwith the magnetic field B. That is,

T = rnxB. (1.68)

Since an electron also has an angular momentum, k, we write

SECTION 1.7 MAGNETIZATION DYNAMICS 27

m - gJloe k -yk,2m

(1.69)

where the minus sign is due to the sign of the charge of the electron, elm is the ratioof the charge to the mass of an electron, and g is the gyromagnetic ratio, which isone for orbital motion and two for spin motion. The term y is normally referred toas the gyromagnetic ratio of an electron. Thus, when an electron is subject to anapplied magnetic field, its magnetization is unable to align itself with the field, butinstead its magnetization precesses about the magnetic field. The precessionfrequency Wo is given by

Wo = yB. (1.70)

This rotating magnetic moment radiates energy, thus permitting the electron toeventually align itself with the magnetic field. Therefore, the time rate of changeof angular momentum is given by the Landau-Lifshitz equation

dk- = -ymxB - amx(mxB), (1.71)dt

where a is the damping factor. For small damping factors, the moment will precessmany times about the applied field, but for large damping factors, the moment willmake a small fraction of a revolution about the applied field as it approachesequilibrium.

When an alternating rf magnetic field with frequency wis applied to a materialthat is magnetized by a de field acting along the z-direction, the material appearsto have a nonreciprocal permeability tensor given by

1+Xxx x, 0

[Jll = Jl -Xxy 1+Xxx 0,

o 0

where the reciprocal susceptibility is given by

woyBXxx = --­

W2_W2o

and the nonreciprocal susceptibility is given by

X = jwyBxy W2_W2

o

(1.72)

(1.73)

(1.74)

28 CHAPTER 1 PHYSICS OF MAGNETISM

It is noted that B is the internal field in the material, which in ferromagneticmaterials is given by

(1.75)

where D is the demagnetizing factor along the axis on which the material ismagnetized. The nonreciprocal nature of this permeability permits one to buildnonreciprocal passive devices, such as isolators, circulators, and other similarmicrowave devices.

1.7.2 Eddy Currents

When a field parallel to the magnetization on one side of a domain is applied, thedomain wall experiences a "pressure" in a direction that would make the domainparallel to the applied field grow. In conductors, eddy currents are induced byFaraday's law whenever the applied field changes and consequently themagnetization changes. The eddy current field opposes the applied field andgenerally shields the interior of the material from it. For low frequencies, theapplied field eventually penetrates the entire material. For high frequencies, theinduced currents and the applied fields are limited to a very thin region close to thesurface of the conductor, and so this effect is called the skin effect.

1.7.3 Wall Mobility

We will now address the question of how a domain wall moves in view of theconstraints imposed by the Landau-Lifshitz equation. Consider a t80° Bloch wallbetween two domains magnetized in the +z direction and the -z direction. A z­directed field applies a pressure on the wall tending to move it in a direction suchthat the domain magnetized in the z direction would grow. This field would notapply a torque on the magnetic moments in either domain, since it has nocomponent perpendicular to the magnetization. The atoms in the wall, however,experience a torque and will start to precess about the applied field. If thiscontinues, the Bloch wall will become a Neel wall and will experience ademagnetizing field perpendicular to the applied field. The magnetic moments inthe wall can now precess about this new field, and thus propagate the wall.

The larger the applied field, the faster the atoms in the domain wall willprecess, and the more the Bloch wall will convert into a Neel wall. This willproduce a larger demagnetizing field in the wall, causing it to precess faster, andthus the wall will move faster. Therefore, the wall's velocity will be proportionalto the applied field, and its motion will be characterized by a mobility. This linearvariation of wall velocity with the applied field terminates when the wall hascompletely converted to a Neel wall, and then the wall will have achieved a limitingvelocity, referred to as the Walker velocity. This velocity depends on the material,but for most materials it is of the order of meters per second. The slowness of thismotion was a limiting factor in bubble memories.

SECTION 1.8 CONCLUSIONS

1.8 CONCLUSIONS

29

Modeling magnetic materials can be performed at various levels of detail: theatomic level, the micromagnetic level, the domain level, and finally at the nonlinearlevel. The first of these involves the use of quantum mechanics to compute themagnetization of individual electrons in atoms. The second level smears out theeffect of individual atoms into a continuous function, and one can see the variationof the magnetization in the medium on a greater scale. At the domain level, thedetails of domain walls are invisible, and one sees only uniformly magnetizeddomains separated by domain walls of zero thickness. Finally, at the nonlinearlevel, one averages the magnetization over many thousands of atoms in order toreplace the constituent equations that complete the definition of magnetic fieldsalong with Maxwell's equation.

Preisach modeling, which we will describe in the subsequent chapters, fallsinto the nonlinear level of magnetization detail. This type of modeling describes notonly gross effects, such as the major hysteresis loop, but also the details of minorloops. When coupled with the appropriate equations, it can describe dynamiceffects as well. Finally, it can be coupled with phenomena of other types todescribe hysteresis in such effects as magnetostriction.

The solution for the magnetization involves the calculation of the magneticstate of the system, since the behavior depends upon this. Then one can computethe magnetization of the system under the influence of an applied field when themagnetization is in this state. This type of problem is similar to a many-bodyproblem, except that the system displays hysteresis. Thus, it can be referred to it asthe hysteretic many-body problem.

In modeling coercivity, the quantities of interest are the discrete magnetizationstates and the Barkhausen jumps that occur when going from one state to another.The minimum change of state is the reversal of a single hysteron or magnetic entity.When there are many interacting hysterons, one is solving a hysteric many-bodyproblem. Then one can go to the limit of a continuous density of hysterons.Preisach modeling is one of the mathematical tools for handling such densities.

The definition of the magnetic state will be based on the Preisach definitionof hysteron, that is, a region that switches as a single entity and has two magneticstates. For hard materials, this region might be a single particle in particulate mediaor a single grain in thin-film media; for soft materials, it might be the volumeswitched by a single Barkhausen jump. A discrete entity with more than two statescan be decomposed into several hysterons. Thus, the basic approach is identical forhard and soft materials, but the parameters chosen will differ. The classicalPreisach model, which is discussed in the next chapter, is able to describehysteresis in general, but the details do not accurately describe real-worldphenomena. Subsequent chapters modify this model to correct these errors, usingthe physical principles just discussed.

30 CHAPTER 1 PHYSICS OF MAGNETISM

REFERENCES

[1] A. H. Morrish, The Physical Principles ofMagnetism, Wiley: New York,1965.

[2] S. Chikazumi with S. H. Charap, Physics ofMagnetism, Wiley: New York,1964.

[3] A. Visintin, Differential Models of Hysteresis, Springer-Verlag: Berlin,1994.

[4] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer­Verlag: New York, 1996.

[5] F. W. Sears, Thermodynamics, Addison-Wesley: Reading, MA, 1959.[6] R. Becker and W. Doring, Ferromagnetismus, Springer-Verlag: Berlin,

1939.[7] E. C. Stoner and E. P. Wohlfarth, "A mechanism of magnetic hysteresis in

heterogeneous alloys," Philos. Trans. R. Soc. London, A240, 1948, pp.599-642.

[8] R. M. Bozorth, Ferromagnetism, An IEEE Classic Reissue, IEEE Press:New York, 1994, p. 849.

[9] J. C. Slonczewski, IBM Research Memorandum. No. RM 003.111.224,October 1, 1956.

[10] M. E. Schabes and H. N. Bertram, "Magnetization processes inferromagnetic cubes," J. Appl. Phys., 64, August 1988, pp. 1347-1357.

[11] Y. D. Yan and E. Della Torre, "Modeling of elongated fine ferromagneticparticles," J. Appl. Phys., 66, July 1989, pp. 320-327.

CHAPTER2

THE PREISACH MODEL

2.1 INTRODUCTION

Hysteresis is a rate-independent branching nonlinearity; that is, the slope of theinput-output curvedependsonly upon the signof the rate of changeof the input.Ferenc Preisach [1], developed a model [2] to explainhysteresis in soft magneticmaterials. Although othermodels havebeenused,suchastheplaymodeldiscussedin Appendix A, they cannot give the physical insights into the magnetizationprocess that are possible with this model. The Preisach model is capable ofdescribingminorloops,as wellas the majorhysteresis loop;however, it is limitedin its ability to describe magnetic materials by the congruency property and thedeletion property. For this reason,manymodifications of thePreisachmodelhavebeen suggested. To differentiate between them, we will refer to the originalPreisach model as the classical Preisach model and to the modifications of it asPreisach-type models.

2.2 MAGNETIZING PROCESSES

Beforewebeginthestudyofhysteresis inmagnetic materials, wemustdefinesomeof the common magnetizing processesneededto test the modelswe willdevelop.The first magnetizing process we will use, the major hysteresis loop, starts froma negativefieldlargeenoughto saturatethe material in the negative direction, goesto a positivefield largeenoughtosaturatethematerial in thepositivedirection,andthen goes back.The sectionof the majorhysteresis loop from negativesaturationto positive saturation is called the ascending major curve. The other half of the

31

32 CHAPTER 2 THE PREISACH MODEL

major hysteresis loop is called the descending major curve. A typical major loopis illustrated by the solid line in Fig. 2.1.

The largest achievable magnetization is called the saturation magnetization,Ms. The magnetic field that increases the initial magnetization on the ascendingmajor loop to zero is called the coercivity, Hc. The magnetization whenever theapplied field is reduced to zero is called the remanence, Mrem• The squareness ofthe hysteresis loop, S, is the maximum remanence normalized to the saturationmagnetization.

A hysteresis loop similar to the major loop is called the remanence loop. Fora given applied field, a point on the remanence loop is measured by applying thatfield and reducing it to zero. The resulting remanence is plotted as a function ofthat applied field. Such a loop is illustrated by the dashed line in Fig. 2. I. The lightdashed line indicates the relationship between the major loop and the remanentloop for a typical point. The magnetic field that increases the initial remanence tozero on the ascending remanent loop is called the remanent coercivity, HRC• It isseen from the figure that HRC is always larger than Hc.

The slope of the magnetization curve is called the susceptibility and denotedby X. We will differentiate between the remanent susceptibility Xr , the slope of theremanent curve, and x. We will also define other susceptibilities later as needed.

If at some point on the ascending major loop the field is decreased, or if atsome point on the descending major loop the field is increased, the locus of pointson the magnetization field curve will enter the hysteresis loop. Such points are

32

Major loop

-I 0 1Applied field

-2

-1 -=::;;.__....Io-__~==--_---L .J._______l____.J

-3

I,------,.-----r----r----~~--___...--~

a~~i 0E

"'d

:-!

1-0.5 I---~.-#-__I____+--__+-_II_--t----t---__iz

0.5 ~--_+_---I---__H-_+_--__1f---__#__#__+_--___i

Figure 2.1 Major loop and remanent loop for material with unit coercivity.

SECTION 2.3 PREISACH MODELING 33

(2.1)

called turning points and such traversals are called first-order reversal curves. Afurther reversal from one of these curves would be called a second-order reversalcurve, and so on. A closed loop formed by two higher order reversal curves iscalled a minor loop.

A magnetization curve starting from the demagnetized state - that is, zeromagnetization at zero field - and going to saturation is called a magnetizing curve.Such a curve is not unique but depends on how the material was demagnetized. Wewill reserve the name virgin magnetizing curve for the curve that starts from thestate that was demagnetized by applying an ac field large enough to saturate thematerial and slowly reducing its magnitude to zero. This technique of obtaining ademagnetized state is called ac demagnetization.

2.3 PREISACH MODELING

The Preisach model considers the material to be a collection of square-loophysterons, as shown in Fig. 2.2. The hysteron has a unique normalizedmagnetization, m, equal to one whenever the applied field H is greater than U, anda unique m equal to -1 whenever the applied field H is less than V. If the appliedfield lies between V and V, the magnetization may have either value depending onits history. Whenever the applied field increases beyond U, the magnetization statewill switch to the positive state. Consequently the hysteresis loop is traversed, asshown by the arrows. Since the materials are passive, and since the energy loss isthe area enclosed by the loop going in a counterclockwise direction, U ~ V for allmaterials.

The Preisach function P(V, V), where the up-switching field U and the down­switching field V are the coordinates defining the Preisach plane, is the densityfunction of hysterons. With this definition, if a sufficiently large positive field isapplied to the material, all the hysterons will be switched in the positive direction,and the resulting magnetization will be

J Jp(U,V)dUdV = Ms·U"lV

If we define the Preisach function to be zero when U < V, we can integrate over theentire plane. For a negatively saturated material, the subsequent application of apositive field HI will switch all hysterons that have a U less than HI' We will definethe normalized Preisach function, p(U,V), so that its integral is the normalizedmagnetization, that is, the magnetization divided by its saturation value. Then

00 U 00 U

fdUfdV p(U,V) = fdUfdV P~:V) = 1. (2.2)-00 -00 -00 -00

34 CHAPTER 2 THE PREISACH MODEL

M

H

y u

Figure 2.2 A typicalhysteron in a Preisach model.

Let us consider a magnetizing process that starts from negative saturation followedby an applied field HI. Then, since the change in magnetization when a hysteronswitches from its negative value to its positive value is twice its magnitude, thenormalized magnetization will be given by

H) U

m = : = -1 + 2 JdU JdVp(U,V). (2.3)s -00-00

The magnetization during field traversal to H I will follow the ascending part of themajor loop, that is, the magnetization curve starting from negative saturation andgoing to positive saturation. If HI is not a saturating field, and the applied field isthen decreased to a value H2, the magnetization will follow a first-order reversalcurve from H. to H2• Subsequent traversals in the magnetization after additionalreversals in the applied field are called higherorder reversalcurves.

Since the critical fields of an isolated hysteron, H, and -Hk' must be thenegative of each other, we say that to each of them is added an interaction field Hito form U and V. Thus,

U = H k + Hi and V = -H, + Hi· (2.4)

Since the interaction field varies as the magnetization of the other hysteronschanges, one must be concerned with the stability of the Preisach function. Moreabout this will be said in later chapters.

Starting from negative saturation, we will now obtain the sequence ofmagnetization due to the sequence of fields HI' H2, H3, etc., as shown in Fig. 2.3.We note that the sequence of fields has the property that

Hk > Hk+2 if k is odd, and H, < Hk+2 if k is even. (2.5)

SECTION 2.3 PREISACH MODELING 35

Figure 2.3 Arbitrary magnetizing process.

The normalized remanence after this sequence is applied is givenby

m = !!... = -1 + 2 J Jp(U,V)dVdU, (2.6)Ms U<L

whereL is the lineillustrated inFig.2.4.Thelineisoftendescribed as thestaircasedividingthePreisachplanebetween positively andnegatively magnetized regions,and the cornerson this line are referredto as the stepsof the staircase. For odd k,if HIcwereto be greaterthanHIc-2, the effectof Hie wouldbe deleted. This property

:u

.....-.: Magnetize~negatively :

I •

Magnetizedpositively

H2

•••••••--•••-•••••••••-.---••-.---.---.--.----

Figure 2.4 Division of the Preisachplaneinto a negatively magnetizedregionand a positively magnetized region.

36 CHAPTER 2 THE PREISACH MODEL

of the Preisach model, known as the deletion property, is discussed in the nextsection. Similarly, the effect of a negative extremum is deleted by any subsequentmore negative fields.

A minor loop is a magnetization curve that oscillates between two fields, HIand H2• This curve may be obtained by any history prior to beginning this loop andso may be situated at any elevation inside the major loop. Three such loops areshown in Fig. 2.5. Section 2.7 will show that all these loops must be congruent toeach other, if the process can be modeled by the Preisach model. This is known asthe congruency property.

Mayergoyz has shown [3] that the congruency property and the deletionproperty are the necessary and sufficient conditions for a process to berepresentable by a classical Preisach model. Magnetic materials do not possessthese properties, and to describe these processes accurately the Preisach modelmust be modified. This will be demonstrated later.

We now discuss several standard magnetizing processes that are referred tothroughout this book. A de-magnetizing process is the application ofa dc magneticfield to a material and then its removal, leaving the material in a remanent state.The resulting remanence depends on the magnetic state of the material before theapplication of the field. If the material was saturated in the positive directionfollowed by a negative field, then this negative field is referred to as the bias field.Normally the bias field used is sufficiently negative to saturate the material in orderto achieve a unique state, but other bias values can be used. The resultingremanence is computed from (2.6) where L is the line U = HI.

An anhysteretic magnetizing process is one in which an ac and an offset demagnetic field are simultaneously applied to the magnetic material as shown in Fig.2.6. First the ac field is reduced to zero, and this is followed by the reduction of the

Figure 2.5 Minor loops oscillating between HI and H2o

SECTION 2.3 PREISACH MODELING 37

A A A

nA n ~

AA A fI

11/

1

1/\I~~oc

V v vV v

~ ~v

y

1.5

-o.s

-1o 25 50 75 100

Time (arbitrary units)125

Figure 2.6 An anhysteretic magnetizing processas a functionof time.

de field to zero.We assumethat the bias field is reducedso slowlythat the appliedfield goes through manycycles as the ae field is reduced to zero.

Then the steps in the staircaseon the Preisachdiagrambecomeverysmallandthe staircasemaybe approximated bya straightline.If theae field is largeenough,then the resultingremanence is computedfrom(2.6),wherethe de field is Hdc, andL is the line U = -V + Hde as illustrated in Fig. 2.7.Theellipse labeled "Preisachfunction" indicatesa typicalcross sectionof the Preisach function.

Figure 2.7 Divisionof the plane into positivelyand negatively magnetizedregionsbyan anhysteretic magnetizingprocess.

38 CHAPTER 2 THE PREISACH MODEL

An de-magnetizing process is similar to the anhysteretic magnetizing processexcept that the two fields are simultaneously decreased to zero while the sameproportion of their amplitudes is maintained, as shown in Fig. 2.8. In that case, ifHac is the peak of the ac field and Hdc is the value of the de field, then the resultingremanence is computed from (2.6), where L is the line U = PV, and p is given by

p = Hdc + Hac, (2.7)Hdc - Hac

as shown in Fig. 2.9. The particular ac-magnetizing process where Hdc is zero isknown as ac-demagnetization, since the material will be left demagnetized if thePreisach function is an even function with respect to the line U =- V; that is,P(U, V) =P( - V, -U). It is seen that if Hac and H dc are so large that the staircase

Figure 2.8 An AC magnetizing process.

Figure 2.9 Division of the plane by the field in Fig. 2.8.

SECTION 2.3 PREISACH MODELING 39

(2.8)

divides the entire nonzero portion of the Preisach plane, the resulting magnetizationis only a function of the ratio of Hac to Hdc.

A question sometimes raised is whether minor loops close on themselves. Thisis tested by the repetitive cycling between two applied fields, H. and H 2• Such aprocess is called an appliedfield accommodation process. If the minor loop thustraversed drifts with the cycle number, instead of closing on itself, the material issaid to have accommodation. Other accommodation processes, also involvingrepetiti ve cycling, are defined later.

An important question in Preisach modeling is whether the Preisach functionis stable at all: that is, whether the density of states is constant as the magnetizationvaries. It will be seen that although a constant Preisach function can be used todescribe many observed magnetic hysteresis phenomena, such as finite anhystereticsusceptibility, it is not indeed constant. This instability in the Preisach functionleads to violations of the congruency and deletion properties, as discussed inChapters 4 and 5.

In some cases, the geometric interpretation of the Preisach model iscumbersome, so we now introduce the Preisach statefunction Q to facilitate ourmathematical description. This function of the U and V is 1 if hysterons with theseswitching fields are magnetized in the positive direction and -1 if hysterons withthese switching fields are magnetized in the negative direction. The state functioncan take intermediate values as well. For example, if a material is demagnetized byraising it above the Curie temperature, Q is zero everywhere. Later, we will alsopermit intermediate values of Q in the case of the accommodation and vectormodels. The net magnetization is given by

M = f fQ(U,V)P(U,v)dUdV.

U>V

Using the normalized Preisach function, (2.2), we have

m = : = f f Q(U,V)p(U,V)dUdV.s U>V

(2.9)

The state function Q will change during a magnetizing process, while P willnot. In particular, if a material is saturated in the negative direction, then

Q(U,V) = -1 (2.10)

for all points in the Preisach plane. In this case, the integral in (2.8) will become-Ms. If a positive field H is then applied, Qchanges to

Q(U,V) = sgn(H -lJ), (2.11)

where the sign function sgn(x) is defined to be one if x is positive and minus oneif x is negative. For the anhysteretic magnetizing process, when a large ac field in

40 CHAPTER 2 THE PREISACH MODEL

the presence of a de field, HDC' is reduced to zero, after which the DC field isreduced to zero, we have

Q(U,V) = sgn(Hdc - U - V). (2.12)

(2.13)

We will denote Preisach functions that are zero if U is negative or if V ispositive as single-quadrant Preisaehfunctions, since they are limited to the fourthquadrant of the Preisach plane. On the other hand, Preisach functions that extendinto the first and third quadrants will be called three-quadrant Preisach functions.The hysteresis loops of single-quadrant Preisach functions have zero slope after aturning point. Three-quadrant functions do not decrease horizontally for H> 0, nordo they increase horizontally for H < O. This behavior appears to be reversible butit is not, and therefore, it is referred to as apparent reversible behavior, asdiscussed further in Chapter 3.

2.4 THE PREISACH DIFFERENTIAL EQUATION

An alternative approach to computing the magnetization by integrating over thePreisach plane is the differential equation approach [4]. This method is veryconvenient for computing the magnetization as a function of time in real processes.The magnetic history of the material must be stored for the computation of themagnetization using the Preisach model. This can be done easily using apush-downstaek* for the extrema of the input. At the bottom of the stack is the largestmagnitude applied field, and the successively smaller maxima and minima arestored above it until at the top of the stack is the current applied field. In thefollowing analysis, we will assume that the applied field H is increasing and thestack contains the values Ht, H2, H3, etc. Since the magnetization changes by afactor of 2, in going from negative saturation to positive saturation, as long as Hcontinues to increase and as long as H <H2, the magnetization is computed as thesolution to the following differential equation:

dm = 2(H (H V)dVdH JH) P, ,

where HI is now the largest previous minimum. The upper limit could be set toinfinity for physical Preisach functions, since p(U,V) is zero whenever V is greaterthan U; however, if we use an artificial function for p(U,V) that is nonzero whenV is greater than U, we should leave the upper limit as H. Whenever H =H2, wepop the top two values from the stack; that is, we set HI equal to H3, H2 equal to H4,

and so forth. The popping of the top two values from the stack is identical with the

lieA stack is a programming tool in which data are stored in the order created rather than by position. Apush-down stack is a last-in-first-out (LIFO) stack; that is, data are retrieved in inverse order fromwhich they were stored. Data are said to be "pushed" on the stack when stored and "popped" from thestack when retrieved.

SECTION 2.4 THE PREISACH DIFFERENTIAL EQUATION 41

(2.14)

(2.17)

deletion property of the Preisach model. Then the process is computed by meansof the same differential equation, but with a new lower limit on the integral.

If H starts to decrease, the present value of H is pushed on the stack; that is,we sequentially set HI equal to H, then H2 equal to HI' and so forth. Thus, HI isnow the previous smallest undeleted maximum. Then, as long as H continues todecrease, and as long as H> H2, the magnetization is computed as the solution tothe similar differential equation

dm = 2fHI (U H) su.dH H P ,

In this case, whenever H =H2, we again pop the top two values from the stack, andcontinue.

If we are interested only in the normalized magnetization at the conclusion ofa process, it can beexpressed as an normalized Everett integral. In particular, if theprocess ends in "', H3, H2, HI' then the magnetization is given by

m(-..,H3,H2,Ht) = m(-..,H3,H2) + E(H2,H1) , (2.15)

where E(H2' HI) is the normalized Everett integral. If when changing the field fromHI to H2 no deletions of previous extrema occur, E is given by

H2 U

E(HI'H2) = 2 f fp(U, V)dUdV. (2.16)HI HI

It is noted that the sign of the integral is determined by whether HI is larger orsmaller than H2•

2.4.1 Gaussian Preisach Function

A useful approximation for hard materials is to assume that the Preisach functionis Gaussian, in both the interaction-free critical field Hie of the hysteron, and theinteraction field Hi' Then this integral can be evaluated in closed form. Theinteraction field dependence can be justified on the basis of the central limittheorem of statistical theory, since the interaction field is the sum of the fields dueto all the other hysterons, which are independent and identically distributed. Thecritical field dependence is an approximation to a log-normal dependence for thecase when the mean critical field, hk' is more than twice its standard deviation, Ok'

The relationship between the Gaussian function and the log-normal function isdiscussed in Appendix B. Thus, we will assume that for hard materials the Preisachfunction is given by

{I -2 2] }1 1 (H -h ) Hip(Hk,H;) = exp __ k k + _ ,

21t 0iOk 2 0; 0;

42 CHAPTER 2 THE PREISACH MODEL

where o, and 0; are the standard deviations in the critical field and interaction field,respectively. We will later reserve lowercase h for operative fields, _but sincecritical fields and operative critical fields are the same, we will use hk for theaverage critical field to beconsistent with later treatments. Since the critical fieldsand the interaction fields are independent phenomena, we expect their respectivePreisach functions to have different means and standard deviations. Thus, the jointprobability density will be the product of the individual density functions. It isnoted that this function is valid to better than 0.5% if

(2.18)

(2.19)

since the Preisach function must go to zero when Hie goes to zero. If this is not thecase, one should use, for example, a log-normal function for the H, variation.Alternatively, one can use a truncated Gaussian, but the normalization must bechanged appropriately.

We can express this relationship in the U-V plane by using the inverserelationship of (2.4) between the U and V variables and the H, and H; variables:

U-V U+VH =-- and H. = --.

k 2 '2

Noting that the Jacobian for the change in variables from HIc and Hi to U and V is0.5, the Preisach function in terms of U and V is given by

p(U,V) = 1 exp o;(U-V-2hk): : o~(U+V)21. (2.20)

41t0;Ok 80; Ok

This may be rewritten

where

(2.21)

and

a = Jo; + o~,2 2

A. = (Ok-a;)

20; ok't'=--

o

20~K = -' = 1 - A.

0 2

(2.22)

It is seen from (2.13) and (2.14) that the behavior of a hysteretic material dependson whether the applied field is increasing or decreasing. We will now compute thesusceptibility for these cases separately.

SECTION 2.4 THE PREISACH DIFFERENTIAL EQUATION

2.4.2 Increasing Applied Field

43

When H is increasing, we carry out the V integration in (2.13) and set U equal to Hto obtain the susceptibility X, which is given by

line = tim = _1_ exp[- (H - hi] [_~ V + AH + Kh") t=HdH o{ii 20 2 en~ 't{i =HJ

_1_ exp[- (H-hi][_~ (I+A)H+Kh,,) (2.23)o{ii 20 2 en~ 't{i

- en( HI + 'AH+ KhJ:)],'t{i

where the error function erf(x) is an odd, monotonically increasing function ofx thatapproaches 1 as x approaches infinity, approaches -1 as x approaches minusinfinity, and is defined by

(2.24)

We note that at a reversal point, the upper limit is equal to the lower limit; thus, thesusceptibility is zero. This property is true for any Preisach function.

If (2.18) holds, then the upper limit in (2.23) can be replaced by infinity, andXinc is given by

timline = dH' (2.25)

or

line = _1_ exp[- (H - hi] [1 _ ) HI + 'AH+ KhJ:) ]. (2.26)c {i1t 20 2 IJ.l 't{i.

The error in this is a function of how much larger hIe is compared to aIe'

For the ascending major loop, HI is negative infinity, and the error function is- 1. Thus, in this case, the major loop is an error function and its slope is aGaussian. Then the major loop susceptibility is described by

tim = .! f2 exp[ - (H-hJ2]. (2.27)dB (J~ -; 20 2

Therefore, the ascending curve of the major loop is given by

44

(H -h )

m = erf 01/'

CHAPTER 2 THE PREISACH MODEL

(2.28)

If we are traversing a minor loop, then at corners of the staircase, although m iscontinuous, dm/dH is not.

2.4.3 Decreasing Applied Field

and we can use (2.14) to rewrite (2.23) as

dm 1 [ (H + iik)2] [ (u + 'AH - Kiik) U=H1

X =-=-- exp erfdec dH 0 '2i 202 't- '2

Vkit. v- U=H

(2.29)

(2.30)

(2.32)

(2.33)

or

Xdec

= ~exp[ (8 +hk)2][erf( HI +).,H -Kiik) -erf( (1 +).,)H -Khk)].(2.31)oy21t 202 r:{i ~(i

If (2.18) holds, the lower limit can be replaced by minus infinity so that the seconderror function is - 1, and then (2.31) can be approximated by

1 [ (H + ii/c)2] [ ( HI +AH - Khk) ]Xd :::: -- exp 1 + erf .ec o.fii 202 ~{i

We conclude this section by computing three special cases to illustrate thedependence of first-order reversal curves, starting at -hk from the descendingmajor loop, on the standard deviations. In the first case 0; is equal to Ok' in the nextOk is equal to zero, and in the last 0; is equal to zero. In all the cases, we willassume that the value of (J is the same, but we will vary the ratio of 0; to Ok'

Case I: First we set 0; equal to Ok' Then, A is zero and K is 1. Thus, forincreasing H, (2.26) becomes

dm 1 [ (H-iik)2][ ( H)+hk) 1- = -- exp 1 - erf -- .dH o.fii 202 ~.fi

Case II: In the second case, if we set o, equal to zero, then, ~ is equal to zero,K is 2 and A is -1. Since the argument of the error function now is

SECTION 2.4 THE PREISACH DIFFERENTIAL EQUATION 45

(2.35)

(HI - H + 2hJ/T;Ii, whose magnitude is infinite, the value is either + 1 or -1depending on the sign of the argument. When the error function is positive, thequantity in the square bracket is 2, but otherwise it is zero. Then

1~ [(H-hJ2] if H>HI +2 hidm

- -exp-- = o 1t 20 2

(2.34)dH -

0 if H~ HI +2 hie

Case III: Finally, if we set 0; equal to zero, then t' is again zero, but this timeK is zero and Ais 1. Since the argument of the error function is now (HI + H)/tIi 'the magnitude of the error function is unity, and the sign in front of the errorfunction is the same as the sign of its argument. Thus,

1. r2 exp[ (H-hi)2] if H<-HIdm = o~ -; 20 2

ato

This is the case for no interaction. According to Wohlfarth [5] the slope for thevirgin curve, that is, the magnetization curve starting from the demagnetized state,should be half the slope of the major loop for the same field. In this case, it does notmatter how the material was demagnetized.

A plot of these three special cases is shown in Fig. 2.10 for the case of a first­order reversal curve starting from - h k on the descending major loop, using a valuefor a of 0.35. It is seen that if o,= 0, then m remains zero until it meets the majorloop. If 0; = 0, there is no interaction, and the slope is half that of the major loop,as suggested by Wohlfarth. If o,= a; the magnetization is half at H = h k • All theseloops will then follow the major loop when they eventually encounter it. Finally,minor loops between the same extrema will be congruent, since the Everett integralswiII not be a function of the magnetization.

It is noted from (2.26) that the slope at any given applied field depends uponthe choice of HI. For example, a specimen can be demagnetized in various ways.For an increasing field, if the specimen had been ac demagnetized, then HI is equalto H. If it had been de demagnetized, the process would start from saturation, go tothe opposite coercive field, and then go to zero. Then HI would be either h k or 00,

depending upon whether the process had started from positive saturation or negativesaturation. For other processes, other slopes are possible. The range of slopes isdetermined by 0;, and the range is zero if a; is zero.

For the anhysteretic magnetizing process, it can be shown that themagnetization is given by

46 CHAPTER 2 THE PREISACH MODEL

! If .1'f /Ii

l/' .........

Applied field

0i= 0°i=O,0,=0

Figure 2.10 First-order reversal curvesthat originate fromthe descending majorloopat thecoercivefield. Curves are shown for three pairsof values of 0; and 0v but with the same o.

m = erf ( Hdc

) ,CJ;{i

where Hdc is the previously defined offset field.

2.5 MODEL IDENTIFICATION: INTERPOLATION

(2.36)

To characterize a material by the Preisach model, one must first identify thePreisach function. If the type of function is unknown, the only recourse is toexplore the entire Preisach plane. To beable to use the Preisach model to computethe magnetization, first one must know the saturation magnetization. This can beaccomplished easily by measuring the magnetization in a large field. Then thePreisach function can be normalized by (2.2), so that P(U,V) is equal to Msp(U,V).The identification is then performed by using first-order reversal processes; that is,for various HI and H2, one starts from negative saturation, then applies a field HI'followed by H2, which is less than HI. This magnetization is given by M(H1, H2) .

Thus, for small E'S, we have

p(H1,H2) =

M(H 1+ El'H2+€2)+M(Hl'H 2)-M(H1 + E 1,H2)- M(Hl' H2+ €2) (2.37)

E}€2

In the limit as e goes to zero, this becomes

SECTION 2.6 MODEL IDENTIFICATION: CURVE FiniNG 47

p(U,V) (2.38)

Alternately, wecan expressthePreisachfunctionin termsof Everettintegrals;thatis,

(2.39)

(2.42)

An alternate method for computing the Preisach function [6] utilizes itssymmetry. Since there is no preferred direction of magnetization, for a classicalPreisach model, we must have

p(u,v) = p( -v, -u); (2.40)

that is, the Preisachfunctionmust be symmetrical about the u =-v axis. Consideran ac-demagnetized sample that is then subject to an anhysteretic magnetizingprocess,startingfrom the point U =HI and V = H2' We willdenote the normalizedremanenceat the conclusionof this process by manhys(H.,H2)' In a fashion similarto the derivationof (2.38), it can be shown that

a2manhYs(U,V)p(U V) - (2.41), - au av

The problemwithboth theseapproaches is thatexperimentally one has to takesecond differences of measuredvalues. The error in taking second differences ismuch larger than the error in makingthe measurements. Thus, in the next sectionwe introducea methodthat is much less sensitive to errors.

2.6 MODEL IDENTIFICATION: CURVE FITTING

The preceding method of identifying the Preisach function required taking thesecond partialderivativeof the magnetization resultingfrom a first-orderreversalcurve.This methodis veryproneto experimental errors.Furthermore, if one wantsto obtain the Preisach function for the entire plane, one has to map out the entireplane. An alternatemethodof identifyingthe Preisach function is to assume that,similar to (2.17), it is of the form

M 1 [<H -ii )2 H2

] }P(H ,H.) = S exp _.!. k k + _i .k: I 21t 0.0 2 2 2

I k ~ ~

This functionhas four unknownparameters: Ms, ~, 0/c, ando; If wecan determinetheseparametersdirectly,thenweknowthePreisachfunctionover theentireplane.

48 CHAPTER 2 THE PREISACH MODEL

The first two parameters can be obtained from the major loop: M, is theasymptotic value of the magnetization for large fields, and ~ is the value of theappliedfield that reduces the magnetization to zero.The other twoparameters, at,and 0/, mustbeobtainedin twosteps:first;0 2

, the sumof theirsquares,is obtainedbyfitting the majorloop,and thentheir ratiois obtainedby measuring a first-orderreversal curve. The first step is performed by fitting a Gaussian curve to thederivativeof the major loop. The meanof this Gaussian is anothermeasureof ~and its standarddeviationis a measureof o.

To separate a into its two parts, let us measure the magnetization at theconclusionof the process that starts from positivesaturation, reduces the field to-~, and thenfollowsthe first-ordertransitionbackto ~ [7].At the conclusionofthis process, m is equal to E(-~, ~). Thus,

iile ii: -a,-H,)

m = E( -hk,iik) = f dU f dV p(U, V) = fan, f an, p(Hk,H;). (2.43)-ii le hIe-HIe

Let us define

(2.44)

(2.45)

Substitutingthis and (2.22), into (2.17) gives us the followingexpression for thePreisach function:

2 1(HIc-~)2+p2H;21P(Hk,Hi) = --ex ·

1tot 2a2t 2

If we make the substitutionusing the dummy variables rand e, where

Hk-iik = r cosf and Hi = r sinfl, (2.46)

we obtain

00 1tI4 [2 26 2· 2e 1- 2 fd f de r (cos + p sin )m--- r rexpnor 2a2t 2

o 07tl4

=2p f de =.3.tan-tp .1t 0 cos26 + p2sin26 1t

Thus,

Ok (m1t) (m1t) . (m1t)p=~=tan 2""",0;=0 cos 2"""' and °k=O sm 2"""'

I

(2.47)

(2.48)

SECTION 2.7 THE CONGRUENCY AND THE DELETION PROPERTIES 49

Since m varies between zero and one, both atand o, vary between zero and o. Forthe three cases shown in Fig. 2.10, m at ~ has this property. This identificationmethod does not use any differentiation to obtain the Preisach function andfurthermore can integrate many observations to obtain the parameters, furtherimproving accuracy.

2.7 THE CONGRUENCY AND THE DELETION PROPERTIES

We now show that the congruency property and the deletion property are thenecessary and sufficient conditions for a process to be modeled by a Preisachmodel, as was first shown by Mayergoyz [3]. As stated earlier, one property of theclassical Preisach model is that all minor loops between the same pair of appliedfields are congruent. From Fig. 2.11, it can be seen that cycling between the twoapplied fields HI and H2 divides the Preisach plane into four regions. The regionR2 is always set negative by H2, and the region R3 is always set positive by HI'while the region R1 alternates between positive and negative as the applied field iscycled. Region R4 on the other hand is unaffected by this process. This latter regiondetermines only the position of the minor loop within the major loop. Thus, thecongruency property is a necessary condition for a process to be described by aPreisach model. A consequence of this analysis is that minor loops are alwayscontained within the major loop.

1,-.----1-----

R4------

Ir----------­1==._----11-------It------1.----------

Figure 2.11 Division of the plane to illustrate the congruency property.

50 CHAPTER 2 THE PREISACH MODEL

The deletion property can be understood by means of the process illustratedin Fig. 2.12. In this case we have a staircase line dividing the Preisach plane intotwo regions: the union of R I and R3, a positively magnetized region, and the unionof R2 and R4, a negatively magnetized region. It is assumed that the smallestpositive corner of the staircase is at H2• When a field HI is applied, which is lessthan H2, the region R3 is then switched from negative to positive, but the corner atH2 still maintains its identity. When the applied field is increased to H3, the regionR4 is now switched, so that it becomes part of RJ thereby deleting the effect of H2•

Thus, we have illustrated that the deletion property is a necessary condition for aprocess to be described by a Preisach model.

To show that these are also the sufficient conditions, we will show that theyuniquely determine the Preisach function. That is, a process that possesses thecongruency and deletion properties is capable of defining a unique Preisachfunction. This can be seen because the deletion property ensures that the saturationstate is unique, and the congruency property ensures that the Everett functions areunique. Thus, the Preisach function determined by (2.39) is unique. We have nowshown that these two properties are the necessary and sufficient conditions for aprocess to be described by a classical Preisach model.

In Chapters 4 and 5 we discuss the absence of both congruency and deletionproperties in the magnetic properties of real materials. This does not mean that wecannot use the Preisach framework for describing them; but instead, we show thatthe Preisach model can be used as an element in the description of the entireprocess. The alterations that we will make to the Preisach model will be based onphysical principles.

We call the model without any alterations the classical Preisach model. Theclassical model has some additional properties that are a characteristic of theunaltered model only. First of all, because of the deletion property, minor loopsretrace themselves after the first iteration. Furthermore, whenever the

Figure 2.12 Divisionof the Preisachplane that illustratesthe deletion property.

REFERENCES 51

magnetization changes direction, the susceptibility instantly goes to zero and thenincreases again. The ascending major loop is continuous and has a continuous firstderivative. For single-quadrant media, the magnetizationis constant until the fieldreaches zero, but for three-quadrant media, the magnetization starts changingsooner and has a finite slope at zero field. Also, the small-signal susceptibility isonly a function of the applied field. All these limitationsare violatedto some extentin real media, and these limitations will likewise be corrected.

We havecompletedour discussion of theclassical Preisach model by showingits definition, its derivation, its identification techniques, and its properties. Itworks surprisingly well, considering its limitations. The hysteresis loops that itpredicts, for nonsingular Preisach functions, have unit squareness, and in the nextchapter we add a reversible component to remove this limitation.

2.8 CONCLUSIONS

In this chapter we presented the classical Preisach model. It describes a hysteresisloop with four variables: Ms, hk, ai' ok. The parameter Ms transforms thenormalized Preisach loop into one whose height matches the magnetization of themedium.The parameter hk determines the valueof the coercivityand for Gaussianand other symmetrical functions is equal to the coercivity. The parametero = (07 + 0;)°·5 determines the slope of the hysteresis loop at the coercivity, and theratio OfO/Ok determines the height of minor loops vis-a-vis the major loop.

We have shown explicitly that the Preisach model, for the case of a GaussianPreisach function, computes a different slope and hence a different curve when theinput is increasing from the slope and curve when the input is decreasing.Furthermore, for increasing inputs, the effect of history is contained in the lastundeleted minimum, and for decreasing inputs, the history is contained in the lastundeleted maximum. As each minimum is deleted, the slope is discontinuous.

The classical Preisach modelcreates minorloops that have thecongruency andthe deletion properties. The magnetizationchangescomputed by it are irreversible.To characterize real magnetic materials, in Chapter 3 we will add reversiblemagnetization,in Chapter 4 we will relax the congruencyproperty, in Chapter 5 wewill relax the deletion property, and in Chapter 6 we willdiscuss vector properties.

REFERENCES

[1] F. Vajda and E. Della Torre, "Ferenc Preisach, In Memoriam," IEEE Trans.Magn. MAG·31, March 1995, pp. i-ii.

[2] F. Preisach, "Uber die magnetische Nachwirkung," Z. Phys., 94, 1935, pp.277-302.

[3] I. D.Mayergoyz,Mathematical ModelsofHysteresis, Springer-Verlag:NewYork, 1991.

52 CHAPTER 2 THE PREISACH MODEL

[4] F. Vajda and E. Della Torre, "Efficient numerical implementation ofcomplete-moving-hysteresis models," IEEETrans. Magn.,MAG-29,March1993,pp.1532-1537.

[5] E. P. Wohlfarth, "Relations between different modes of acquisition of theremanent magnetization of ferromagnetic particles," J. Appl. Phys., 29,March 1958, pp. 595-596.

[6] J. G. Woodward and E. Della Torre, "Particle interaction in magneticrecording tapes," J. Appl. Phys., 31, January 1960, pp. 56-62.

[7] E. Della Torre and F. Vajda, "The identification of the switching fielddistribution components," IEEETrans. Magn., MAG-31, September 1995,pp. 2536-2542.

CHAPTER3

IRREVERSIBLE AND LOCALLYREVERSIBLE MAGNETIZATION

3.1 INTRODUCTION

This chapter deals with the first of the corrections to the classical Preisach model,the introduction of reversible magnetization, so that the model can describemagnetization phenomena accurately. Although the classical Preisach model candescribe reversible magnetization, it is limited to a state-independent description.In magnetization-dependent models, the susceptibility can be a function of theapplied field, but is independent of the magnetization. In state-dependent models,the susceptibility can be a function of both the field and the magnetization. Thus,the various models can give increasingly accurate descriptions of the reversiblemagnetization in real media.

3.2 STATE-INDEPENDENT REVERSIBLE MAGNETIZATION

The magnetization changes of a classical Preisach hysteron located in thisphysicallyrealizable regionof the Preisach plane, that is, the region in which u >v, are totally irreversible. By this we mean that the energy transfer associated withthis change is not recoverable. We will refer to the component of the magnetizationassociated with this process as the irreversible magnetization, Mi. Examination ofthe behavior of a single hysteron, such as the Stoner-Wohlfarth particle in Fig.1.11, shows that the magnetization can change reversibly as long as the appropriate

53

54 CHAPTER 3 IRREVERSIBLE AND LOCALLY REVERSIBLE MAGNETIZATION

critical field is notexceeded.This typeof behaviorcan be characterizedbyaddinga reversible componentto the magnetization.

Energy transferred by an applied field to the reversible component is storedand can be totally recovered when the applied field is returned to zero. Thus, thereversible magnetization, M; is given by a single-valuedfunction of the appliedfield,

(3.1)

This single-valued function has the property that F(O) is zero and F((0) is finite.Since there is no preferred direction of magnetization,

F(oo) = -F( -(0). (3.2)

This type of behaviorcould be characterizedby hysteronson the u = v diagonal.The Preisach function necessaryto achieve this magnetization is

P(U,V) = Mr(U) o(U-V) = F(lJ) o(U-V), (3.3)

where (, is the Dirac delta function, which is zero unless its argument is zero butwhose integral is unity; however, we will simply add the function, F(U), to thePreisachintegralto obtain the totalmagnetization. Then the totalmagnetization, orwhat we will simplycall the magnetization, is given by

M = M; + Mr· (3.4)

The remanence,Mrem, is the magnetization whentheappliedfield is zero.Since thereversible magnetization M, is zero, whenH is equal to zero, then Mrem is equal tothe M; at zero field for single-quadrant media.

We define the squareness, S, of a material to be the ratio of the maximumremanence to the maximum magnetization. Then in terms of normalizedmagnetizations, we have

M(H) = Ms[Sm;(H)+(I-S)mr(H)] = Ms[Sm;(H)+(I-S)f(H)]· (3.5)

This functionality is illustrated by the blockdiagramin Fig. 3.1.The normalizedreversiblemagnetization is definedto be 1as the appliedfield

approaches infinity. Furthermore, since the materialdoes not have any preferreddirection of magnetization, the reversible magnetization must be an odd functionof the applied field. Thus,j{O) and all even derivativesof m, at zero applied field,have to be zero, and since the reversible magnetization must saturate, the secondderivativemustdecreasewith increasingH. With this definition,the magnitudeofm is less than one, and the normalized function,J{H), as defined by

_ F(H)f{B) - (l -S)M

s• (3.6)

has the following properties:

SECTION 3.3 MAGNETIZATION-DEPENDENT REVERSIBLE MODEL 55

H m

Figure 3.1 Blockdiagram of a Preisach transducer withstate-independent reversible magnetization.

fix) = -f( -X)t f(0) = O, and f(oo) = 1. (3.7)

It is seen from the Stoner-Wohlfarth model, (Fig. 1.11), that when thehysteron is in its positive state, its reversible susceptibility, dmldll, is amonotonically decreasing function of H, for all H greater than -Hs. Similarly in itsnegative state, dmldll is a monotonically increasing function of H, for all H lessthan Hs. Thus, the reversible magnetization has to be either magnetizationdependent or state dependent, as shown in the following sections.

This type of behavior is state independent, since the reversible componentdepends only upon the applied field. The next section discusses magnetization­dependent and state-dependent models.

3.3 MAGNETIZATION-DEPENDENT REVERSIBLE MODEL

The DOK model [1] t a magnetization-dependent model, assumes that the reversiblemagnetization depends upon the magnetization state of the hysterons. Let thereversible magnetization when the hysteron is in the positive state be f(H). Then,if Q. and Q_ are the fractions of hysterons in the positive and negative states,respectively, the reversible magnetization is given by

m, = a, f(H) - a_.f{ -H). (3.8)

With this definition of a reversible component, we can remove the restriction forlarge negative fields; hence, the function/is restricted only by

j{0) =0 and j{oo) = 1. (3.9)

The decomposition of a hysteron's loop into an irreversible component and amagnetization-dependent reversible component is shown in Fig. 3.2. It is seen thatthe a's are given by

56 CHAPTER 3 IRREVERSIBLE AND LOCALLY REVERSIBLE MAGNETIZATION

Q==-lQ=l

Hysteron Irreversible Reversiblecomponent component

Figure 3.2 Decomposition of hysteron into irreversible and locally reversible components.

and

I-m;a =--

2

(3.10)

(3.11)

Thus,

(3.12)

(3.13)

A block diagram of the resulting model is shown in Fig. 3.3. With this model, thereversible magnetization is now magnetization dependent. This type of reversiblemagnetization changes abruptly when the state of the hysteron changes, so we callit locallyreversible magnetization. In this case, since for large negative fields, mi.approaches -1, consequently a, approaches zero. Thus, there is no restriction onhow f(ll) behaves for large negative values of H. It follows that unlike the case ofthe magnetization-dependent reversible magnetization, f(H) could be amonotonically decreasing function of H, andf'(H) could be negative for all H,since neither contributes to the magnetization for large negative values of H. Theonly restrictions onfi..H) are that it approaches one as H goes to infinity, and thatit is zero when H is zero. Since

dmXr(H,M) =(1 -S)Ms d; =(1 -S)Ms[aJ'(1/) +aJ'( -1/)],

we have

SECTION 3.3 MAGNETIZATION-DEPENDENT REVERSIBLE MODEL 57

H m

Figure 3.3 Preisach model with state-dependent reversible magnetization.

(3.14)

This is independent of m.; which is where the magnetization curve crosses the axis.Since in many materials this property is not present, we will examine this in moredetail in the next section.

For a collection of Stoner-Wohlfarth particles,j(H) should be the normalizedreversible component of the magnetization curve. It is useful to approximate thisfunction by

(3.15)

This approximation is illustrated in Fig. 3.4 as compared to the Stoner-Wohlfarthmodel, and in Fig. 3.5, as compared to y-Fe203 data. Although theStoner-Wohlfarth fits the measurements better than the exponential, the error is notlarge.

Since the susceptibility is given by

x = dM = (1-S) M f'(h) = (1-S) M ~e-~HdH S s'

then

Xo

where Xo is the zero-field susceptibility.

(3.16)

(3.17)

58 CHAPTER 3 IRREVERSIBLE AND LOCALLY REVERSIBLE MAGNETIZATION

1

1_-

8 0.9 ....

1~~ 0.8 ...

0

0.7 :-0 Applied field

Figure 3.4 Comparison of the exponential approximation (circles)with the Stoner-Wohlfarth model (solid line).

• Measurement................ .f(ll)= (l-S)(l-e~

Applied fieldFigure 3.5 Exponential fit to y-Fe20 3 data from the descending major loop.

3.4 STATE-DEPENDENT REVERSIBLE MODEL

Hysteron loops of isolated hysterons have to be symmetrical with respect to theorigin; consequently, we can attribute the asymmetry required in the Preisachmodel to interaction between hysterons. Thus, we say that the field that a hysteronsees is the sum of the applied field and the interaction field. The effect of this field

SECTION 3.4 STATE-DEPENDENT REVERSIBLE MODEL 59

Vi UH

Figure 3.6 Hysteron in the presence of an interaction field.

is to displacethe hysteresis loop horizontally, as shownin Fig. 3.6 for a materialthathaslocallyreversible magnetization. It is seenthatwhen thishappens, notonlydoes U not equal the negative of V, but the positiveremanence, Mrem+, does notequal the negative of the negative remanence, Mrem-. This differenceis taken intoaccountby the eMH model [2], whichis a state-dependent magnetization model.

It is seen that the new values of the remanence are givenby

(3.18)

and

(3.19)

where Hi is the value of the interaction field andf is the same type of functiondiscussedin section3.3. Then, for positivehysterons withan averagesquarenessSA' the magnetization is givenby

(3.20)

Then, summing over all hysterons, we obtain

m = JJQ(H;, Hk)p(Hk,H;){SA + (1- SA )f[Q(Hk,H;)(H+H;)]}dHkdH;, (3.21)HyO

60 CHAPTER 3 IRREVERSIBLE AND LOCALLY REVERSIBLE MAGNETIZATION

where, as before, Q is +1 in the region that is magnetized positively and -1otherwise. The irreversible component m, changesonly whenthere is a changeofstate. The reversible component changes with the applied field in a manner thatdependson the stateand is zerowhenthe appliedfield is zero.Thus, wecan solvefor m, by settingH equal to zero in this equation. Then we get

mi=J JQ(Hk,Hi)p(Hk,Hi){SA +(I-SA)f[Hi Q(Hk.Hi)]} dHkdHi,H/?O

(3.22)

(3.23)

We see that in this model, the remanence is affected by this correction in thesecond term inside the braces. For example, at saturation Q is unityeverywhere,and this reduces to the observedsquareness S, whichis now

S = f fp(Hk,Hi){SA +(l-SA)ft.-Hi)}dHkdHiH/?O

= SA +(l-SA) J fp(Hk,H)fl.-Hi)dHkdHi·Hk>O

Weseethatif theaverage valueoff(-Hi) werezero,theobservedsquareness wouldbe the same as the average squareness; however, for real materials there is acorrectiongiven by the secondterm.

Whentheirreversible magnetization issubtracted fromthetotalmagnetization,(3.21), the remainder is the locallyreversible component

m, = (l-S){ f fp(Hk.HiHft.H+Hi) -ft.Hi)]dHkdH;Q=l

+QLfp(Hk.Hi)fft.H;>-ft.H+Hi)]dHkdH}

We note that if (3.15) is applied, then

f(H + H;) - J(H;) = e -~(H+Hi) - e -~H, = e -QI1[e -~H-1],

so that we can write

where now

a, = f f exp(-~H) p(Hk.H;) dHkdHi,Q=l

and

(3.24)

(3.25)

(3.26)

(3.27)

SECTION 3.4 STATE-DEPENDENT REVERSIBLE MODEL

Q- = f fexp{~H,) p(.HltH,) dH,/lH,.Q=-I

61

(3.28)

This model is now state dependent, since even with the samemagnetization, different valuescan be obtained for a+ and a.. A majordifferencebetween thismodeland thepreceding modelis thata, anda. no longerhaveto addup to one.Thus,thesusceptibility is nowa function of themagnetic state;hencethezero-field susceptibility depends on the magnetization and howthat magnetizationwas achieved.

To illustrate the effect of this model, let us considerthe variationof thesusceptibility along the M axis for a de magnetizing process using a GaussianPreisachfunction. In that case, a; is givenby

Q+ = 1 f feXP(-~H,)exp{-.!.I( HJ:-hJ:]2

+ (H,]2i an, dH"21t0ko, 2 a" a,

Q-I

(3.29)

(3.30)

For a de magnetizing processstartingfrom negative saturation going to a positivefield HI and then returning to zero, we have

eXP(~20:/2)[ (HI-h,,+~a2)]a = l+erf

+ 2 a

and

(3.31)

It is seen that if ~a? is zero, then the sum of the a's is again unity. Since ~a? isalways positive, the two functions overlap, as indicated in Fig. 3.7. The resultingzero-field susceptibility, as shownin the figure, is largestat the coercivefield andapproaches exp(;2a/12) as the magnitude of H increases. Sincethe remanence is asingle-valued function of the applied field, the susceptibility as a function of theremanence has a similarshapethat increases to a maximum at zeroremanence andthendecreases. This is generally similarto theobserved susceptibility in recordingmedia[3].The maindifference between thiscalculation andthe observation is thatthe observed peak in susceptibility does not occur for zero magnetization. Thisdiscrepancy can be explained by the moving model, which is discussed in the nextchapter.

62 CHAPTER 3 IRREVERSIBLE AND LOCALLY REVERSIBLE MAGNETIZATION

~t - -..- ></'------~ / .•..

00 ~~

//

./....... '"

·..·············11_--10

..•.".".

...<; .

Applied field

Figure 3.7 Variationof susceptibility with magnetization for a DC-magnetizing process.

3.5 ENERGY CONSIDERATIONS

It is wellknownthat theenergydensitylost in a closedmagnetization cycleis equalto theareaenclosedbytheM-H loop.FriedmanandMayergoyz [4]havesuggestedthat theenergylost in an openprocesscan be computedfromtheclassicalPreisachmodel. They later extended this analysis to input-dependent Preisach models [5].We now address the question of energy storage and dissipation of the modelsdiscussed in the precedingsections.The irreversiblecomponentdissipatesenergyevery time it changes, and it is incapableof storing energy.The locally reversiblecomponent,on theotherhand,storesan amountof energythatdependson the stateof the system.Since the energystored varies when the systemchanges state, evenif the appliedfield is unchanged, thischangein energymustbe addedor subtractedfrom energy dissipated by the irreversible component. This fact complicates thecomputationof the energy loss for an open cycle.

The energy relations for a single hysteron can be obtained by examining thehysteresis loop for an isolated hysteron,as shown in Fig. 3.8. The magnetizationM, the solidcurve,can be decomposed into the sumof a reversiblecomponent,M"illustratedby the two dashedcurves,and an irreversiblecomponentMit illustratedby the rectangularhysteresis loop. That is,

M(H) = M;(Q) + M,(H,Q), (3.32)

where Q is the state of the hysteron. The functional variation of the reversiblecomponent for the magnetization f(H) is a concave, monotonic, single-valuedfunction of the applied field, which saturates as H approaches infinity.

We willnowcomputew, theenergydissipatedin goingfromzeroappliedfieldto Hk, and back to zero. This is equal to flo times the area between the hysteresisloop and the M axis and can be written

SECTION 3.5 ENERGY CONSIDERATIONS 63

w21-M2___~~_J__

AM(O)

-r----'--.--

Figure 3.8Hysteresis loopof an isolated hysteron.

(3.33)

It is seen that the hatched area of the rectangle at the lower right-hand corner of thehysteresis loop, is given by

(3.34)

and the area of the hatched rectangle at the upper right-hand corner of the hystere­sis loop is given by

(3.35)

The discontinuity in the hysteresis loop when the hysteron changes state, !1M(Hk) ,

can be obtained from the height of the irreversible loop at its center, !1M(O), as

dM(Hk) =dM(O) +MI -M2 • (3.36)

Thus, (3.33) becomes

w=J..lO[dM(Hk) Hk + WI - w3] · (3.37)

The first term is the energy loss corresponding to the discontinuity in themagnetization, while the remaining terms correspond to the change in the energystored in the locally reversible magnetization.

The energy stored in the locally reversible magnetization, w, is

wr = 110 fo MrHdMrlQ=consl. = 110 [H Mr-.foHMrdHIQ=consl} (3.38)

64

Since

CHAPTER 3 IRREVERSIBLE AND LOCALLY REVERSIBLE MAGNETIZATION

M1 = f(HJ and M2 = -f(-HJ, (3.39)

then WJ and W3 can be written as

WI = wRIQ=-I = -lJ.oH"J(-HJ+IJ.O!oHi J(-H) dH, (3.40)

and

(3.41)

(3.43)

Therefore, the energy dissipated in traversing the left half of the hysteresisloop is given by

W = lJ.o{nJ~-J(HJ-J(-HJ]+ foHi [f{H)+J(-H)]dH}. (3.42)

The first term is the magnetization change in the irreversible component and isequal to the product of the coercivity and the size of the Barkhausen jump. Thesecond two terms are due to the magnetization change in the reversible component,and the integral is the change in stored energy in the reversible component.

3.5.1 Hysteron Assemblies

We will now generalize this result for an assembly of hysterons by essentiallysumming this over all the hysterons. The result will be a generalization of (3.42).The energy supplied to a magnetic medium is given by

iM iH tIMW=~o HtIM=Jio H-dH.o 0 dH

If the magnetization M, is the sum of an irreversible component M; and a locallyreversible component Mr, then the rate of change in the magnetization with respectto the applied field is

tIM tIM, aM, dQ eu,-=-+----+--.dH dQ aQ dH en

(3.44)

The first two terms correspond to irreversible changes in the magnetization withrespect to the applied field and, therefore, are a source of dissipation. There is anadditional dissipation term due to the changing ability of the medium to storeenergy in the different irreversible states for the same applied field. The energystored in the reversible component for a given state, Ws, is given by

SECTION 3.5 ENERGY CONSIDERATIONS

w = t " U aM,(U,Q)dUs JloJo au '

65

(3.45)

where U is a dummy variable of integration. Thus, the rate of increase in the energystored in the reversible component is given by

dWs aMrdH =l1oH aH • (3.46)

and the dissipated energy, WD, in the medium is given by

dWD = aws dQ + rHU [dMr + aM,dQ]dU. (3.47)dH aQ dH Jo dU aQ dU

If, furthermore, the reversible component can be factored, as in (3.15), then aspecific formula for the energy dissipated can be derived. The reversible compo­nent of the magnetization at the jth element is then given by

M . = ±g(±H. .) f{±H), (3.48)r,} I,}

where the upper sign is to be used if the hysteron is in the upper magnetizationstate, H is the applied field, and HiJ is the interaction field at the jth hysteron. Therate of increase of this reversible magnetization with the applied field is given by

dMr aM, da, aM, da_ aMr da, da; aM,dH = aa+ dH + aa_ dH + aH =f(H) dH -f(-H) dH + on ' (3.49)

where the first two terms are the change in M, due to a change in state, and the lastterm is the change in M, due to the change in the applied field. The last term isgiven by

aMr _ d.f{H) df{-H)---a --+a--aH + dH - dH '

and the derivatives of the a's are given by

da f.H( H+H)d; = ±(l-S) g ±T P(H,HJdH_.

HI

(3.50)

(3.51 )

where HI is defined in (2.30). It is noted that if g(u, v) is zero outside the fourthquadrant, the derivatives are zero when the magnitude of H is decreasing, and Wsis the recoverable energy.

Thus, the rate of energy dissipation is given by

66 CHAPTER 3 IRREVERSIBLE AND LOCALLY REVERSIBLE MAGNETIZATION

[J:H ]dW Ba; Ba;

dH = ~cIl P(H,HJdH+ + dH j{H) - dH j{-H)

HI LH [ 1aa+ aa+ U - f(U) + -- f(-U) dll .

dH dHo

(3.52)

The first termcorrespondsto the rateof energydissipationdue to thediscontinuouschange in the irreversible magnetization; the second term corresponds to thediscontinuous change in the reversible magnetization when the system changesstate; and the last term is the change in the energy stored in the reversiblemagnetization due to changes in the state of the system.

The energy dissipated by a magnetic materialhas been computed for both asingle hysteron and an assembly of hysterons represented by a state-dependentPreisach model. For an isolated hysteron, the two componentsof the energy lossare due to the sudden change in irreversiblemagnetization and due to the changein the ability of the magnetization to store energy because of the change in state.The former loss is equal to the product of the permeability, the applied field, andthe discontinuous magnetization change. It is noted that the size of thediscontinuouschange in magnetization is generallynot equal to the difference ofthe tworemanentmagnetizations. Theseenergylosscomponentscarryover intothecomputation for an assembly of hysterons. The reversible component is stateindependent,if and only if both the sum of the a, and a. is constant, andf(H) is anodd function: that is, f(H) =-f{-II).

3.6 IDENTIFICATION OF MODEL PARAMETERS

In this section, we will limit our attention to the identificationof the additionalparameters necessary to identify the reversible component of magnetization forsingle-quadrantmedia.In thiscase, the identificationof theirreversiblecomponentof magnetization for anypoint in the M-Hplane can be performedby reducing theapplied field to zero.Then, the reversiblecomponentis reduced to zero, and sincethe mediumis singlequadrant,there is nochangein the irreversiblecomponent.Inthe next section,we willdiscusstheproblemencounteredin three-quadrantmedia.

For all these media,to completelyidentifythe reversiblecomponentone mustmeasure the squareness S and the function f{H). The squareness is simply themeasured ratio of the maximumremanence to the saturation magnetization. Themethod of measuringj{H) depends on the model.

Forstate-independentreversiblemagnetization, onemustfindthefunctionthatdescribes the reversiblecomponentas a function of the applied field. This can bemeasured directly by simply applying the field and reducing it to zero. Theaccuracy of the measurement is determined by the accuracy with which one can set

SECTION 3.7 APPARENT REVERSIBLE MAGNETIZATION 67

(3.53)a

a field and measure the magnetization. If observed variation is approximated by afunction such as a hyperbolic tangent or a Gudermannian, the few parametersassociated with these functions can be obtained by a technique such as curvefitting. The validity of the model could be determined by simply seeing if themeasured value of f(H) is indeed independent of the magnetizing process.

For the magnetization-dependent models, the functionf{H) can be obtained forpositive values of H from the descending major remanence loop directly. Fornegative values ofH, the function can be obtained, oncef(H) has been obtained forpositive H, by measuring the reversible component and substituting into

a+ f(H) - m,f(-H) = ---

since a, and a: can be determined directly from the measurement of mi.The function for the state-dependent reversible component is the most

complicated to obtain, since it depends on the shape of the Preisach function aswell as the magnetizing process. Furthermore, the observed squareness differs fromthe average squareness of the hysterons as computed in (3.23). Although it ispossible to perform the necessary integrations numerically and obtain thefunctional variation directly, it is preferable to approximate the Preisach functionappropriately by a function such as the Gaussian, and the reversible magnetizationby an exponential, as discussed earlier. In that case, the process is described by twoparameters, Sand Xo.

3.7 APPARENT REVERSIBLE MAGNETIZATION

The preceding discussion of reversible magnetization models was limited to single­quadrant media, that is, materials with moments sufficiently smaller than thecoercivity to confine the only significant portion of the Preisach function to thefourth quadrant of the Preisach plane. On the other hand, media with largermoments can have a standard deviation of the interaction field, ai' large enough toallow the Preisach function to spill outside the fourth quadrant. High momentmaterials, such as Co-Cr-Ta, typical of media used in hard drives, under certaincircumstances can have three-quadrant Preisach functions.

For three-quadrant media, one must distinguish between the remanence and theirreversible component of the magnetization. The remanence is the measurablemagnetization when the applied field is removed, whereas the irreversiblecomponent of the magnetization usually is not measurable, but merely a convenientcomponent of the decomposition of the total magnetization. These two quantitiesare identical for single-quadrant media, and the former has been used inexperiments as an estimate for the latter, since the locally reversible component ofthe magnetization is zero in the absence of an applied field. The difference betweenthe remanence and the state-dependent irreversible componentof the magnetizationis called the apparent reversible effect [6].

68 CHAPTER 3 IRREVERSIBLE AND LOCALLY REVERSIBLE MAGNETIZATION

Preisach hysterons that lie in the first or third quadrant are hysteretic,although they have a unique state when the applied field is zero. For example, first­quadrant hysterons have both up- and down-switching fields that are positive, so inthe absence of an applied field they are always magnetized negatively. Themagnetization of such hysterons subtract from the maximum possible positiveremanence. Since they traverse a hysteresis loop, whenever the applied field iscycled between zero and a value larger than its up-switching field, they willdissipate energy. This is not to be confused with the reversible component ofmagnetization, which does not dissipate energy as long as the magnetization statedoes not change. Thus, in the first quadrant of the hysteresis loop, the irreversiblecomponent of magnetization for decreasing applied fields is no longer horizontalfor these materials and is not directly measurable. .

We will now compute the correction to the descending major remanencecurve. For convenience, we will extend the definition of the descending remanencecurve to positive fields by setting it equal to the remanence at zero applied field,when the applied field is positive. Then, the apparent reversible magnetization, mAR'

can be defined for all H as

mAR = m/(H) - m,tm(H) , (3.54)

where mrtm is the remanence. When the remanence is computed after a positive fieldhas been applied, the irreversible component of the magnetization must be reducedby the integral of the Preisach function over the region with the vertical hatching inFig. 3.9. On the other hand, for negative fields, it must be increased by the integralof the Preisach function over the region with the horizontal hatching in that figure.

For the descending major loop, we can get the variation in irreversiblemagnetization by setting HI equal to negative infinity in (2.31). Thus,

X = dmpl) = 1. 12e--C (H+hi] for H>O. (3.55)AR dH o~ -; x, 202

We see that at H equal to zero, the slope of m; is given by

u

Figure 3.9 Regions to be corrected for positive fields (vertical hatching) andnegative fields (horizontal hatching).

SECTION 3.7 APPARENT REVERSIBLE MAGNETIZATION

We see that at H equal to zero, the slope of m, is given by

69

(3.56)

(3.58)

which is not zero as it is in the case of single quadrant media. It was shown ingeneral that mj(H) for the descending major loop is given by

(H +hk )

mpf) = erf ofi · (3.57)

Thus, when there is no reversible magnetization, the squareness due to apparentreversible magnetization of this medium, SA' is given by

SA = erf( o~) ·If the material has in addition a reversible component of magnetization, it must beadded to m.; as before. If the squareness due to m, is called S" then the squarenessof this material is given by

(3.59)

For positi ve applied fields, the descending major remanence loop is a constantgiven by SA. Therefore, the magnetization due to apparent reversible magnetization,mAR' the vertically hatched region of Fig. 3.9, is the difference of between SA andm; (H). That is,

(3.60)

For negative applied fields, the remanence loop is obtained by adding mAR' thecontribution of the horizontally hatched region of Fig. 3.9, to m, (H). We see that

o

mAR = !'XAiH)dH, (3.61)H

where H is a negative number. We can obtain XAR by substituting zero for HI in thenegative of (2.30). Thus,

70 CHAPTER 3 IRREVERSIBLE AND LOCALLY REVERSIBLE MAGNETIZATION

(3.62)

dmARXAR= dB

=_l_exJ (H+iiiIeJ ')..H + tehk) -eJ (1+')..)H+tehk) ] ,

o{ii 'l 202 Hl t{(2) Hl t{(2)

for H < O. It is noted that MR is zero when H is zero.A plot of MR as a function of the applied field is shown in Fig. 3.10. The

applied field is normalized to the coercivity, 0; was taken to be four times thecoercivity, and o, was taken to 0.4 times the coercivity. It is seen that thesusceptibility is always positive.

The effect of apparent susceptibility can be seen by examining Fig. 3.11.Withthis set of parameter values, the apparent squareness SA is 0.2. It is seen that theremanence is constant for positive fields and decreases with decreasing negativefields. Furthermore, the slope of the remanence is positive for small negative fields,an indication ofsubstantial apparent reversible magnetization. Most important, theirreversible magnetization is distinctly different from the remanence. Also, theremanence coercivity, HRC' is not a good measure of ~. For these values ofparameters, the remanent coercivity is only 8% greater than the mean critical field.This slightly complicates the identification problem, as we will see in the nextchapter.

0.2 ,..------r---~----r----r---_r_-~--__r_-___,

-10 0 10Applied field

Figure 3.10 Variation in apparent susceptibility with applied field.

SECTION 3.8 CROSSOVER CONDITION 71

I ,-------.--.,-----r---.--~,__-___._--,.__-__,

10

---_._-~._.__..._---- ----_.-

oApplied field

-10-I '---_-'-_--:'--:--...-..c:::.J...-._-'-__'--_..,-'-:-_--:'-_----'

Figure 3.11Effectof apparent reversible magnetization on remanence.

3.8 CROSSOVER CONDITION

(3.63)

Thereis a limiton thechoiceof parameters thatwillproducea physically realizablemodel. In the case of state-independent reversal models, the only limits on thefunctions are that the Preisachfunction be zero if U < V and that the reversiblecomponent be a monotonic single-valued function so that the material does notviolateconservation of energy. For magnetization-dependent andstate-dependentmodels there is another condition that all physically realizable hysterons mustsatisfy,theso-calledthecrossover condition [7],whichlimitstherangeof possibleparameters permitted wherethe Preisachfunction is nonzero. Let us examineFig.3.12, wherethe valueof the criticalfieldwaschosento be too large for the valuesof the squareness and the zero-field susceptibility. In that case, the part of thehysteresis loopthatis abovethecrossoverpointis traversed in theclockwise sense,violating conservation of energy.

We can write the magnetization as

m = {S+(l-S)f(ffl , if Q=l(If) -S-(l-S)f(-lf), if Q=-l.

In order to avoidcrossovers, we requirethat for all hysterons we have

m(lf)IQ=t ~ m(lf)IQ=_t, VH such that V<H<U. (3.64)

Thus, for a particular hysteron we musthave

72 CHAPTER 3 IRREVERSIBLE AND LOCALLY REVERSIBLE MAGNETIZATION

Figure 3.12 Loopof a hysteron that violates the crossover condition.

or

f(Hk) +f( -Hk) S----- ~ ---

2 1-8

(3.65)

(3.66)

This puts a lower bound on the permissiblesquarenessor an upper bound on thepermissible zero-field susceptibility, depending on the position on the Preisachplane. Note that there is no limit in the case of S =1. We can interpret this as themaximumpermissibleextentof the Preisachfunctionon the Preisachplane, if Sisuniformover the Preisachplane. Alternately, if S is permittedto bea function ofHIc, then (3.65) is theconstraintonpermissible functions. In thatcase, the functionsusedmustalsosatisfytheconditionthattheaverageS mustagreewiththeobservedvalue.This is of course even morecomplicated if there is any apparentreversiblecomponentto the magnetization.

If we use the exponential variation forf{H), as in the precedingsections, then

AHk) +f( -Hk)---- = 1 - cosh(~Hk)' (3.67)

2

and (3.65) becomes

1cosh(~Hk) ~ -.

1-8(3.68)

If we permit either S or ~ to be a functionof Hie' it must satisfy, respectively

S ~ 1 - sech(~Hk)' (3.69)

or

~ ~ _1 COSh(_l).n, I-S(3.70)

REFERENCES 73

Since the hyperbolic secant lies between zero and one, it is seen that this lowerlimit for S also lies betweenzero and one, and approaches one for large Hie.

We haveseen that in order to obtaina realistichysteresis loop, wehaveto adda reversiblecomponentto the Preisachmodel. This component maybe a functionof the appliedfield only,or maydependon the magnetization or the state as well.The consequences of the state dependence on the hysteresis loop was discussed.If the Preisach function is nonzero outside the fourth quadrant, the irreversiblecomponent of the magnetization will be different from the remanence. Thiscomplicates the identification problemand leads to apparentreversiblebehavior.

3.9 CONCLUSIONS

In this chapter we discussed how reversible magnetization can be added to aPreisachmodel. As a result two newparameters wereaddedto the model: Sand x.The first is the fraction of the saturation magnetization due to irreversiblemagnetization, and the second is the saturation reversible susceptibility at zerofield. Three types of reversible magnetization processes were discussed:magnetization-independent, magnetization-dependent and state-dependentreversiblemagnetization. Onlythe first of thesecouldhave beencharacterized bytheclassical Preisach model. Formagnetization-dependent reversibleprocesses, thereversiblesusceptibility is a function of the appliedfieldonly.For a magnetizationdependent process the reversible susceptibility is a function of both the appliedfield and the magnetization; however, at zerofield, it is a constant.Onlyfor state­dependentreversible processesdoes the susceptibility varyat zero field.

It may be said that if one is only interested in computing the remanence, it isnot necessaryto computethe reversible component of the magnetization. This istruefor theclassicalPreisachmodel, butnotfor the moving modeland the productmodeldiscussed the next chapter. We will see that for these two models, even tocomputethe remanence, we mustcomputeboth the reversible and the irreversiblecomponents of the magnetization. Errors can be considerable in these cases,especiallyfor soft magnetic materials, if one neglects the reversible componentordoes not include the correct variation of it.

REFERENCES

[1] E. Della Torre, J. Oti, and G. Kadar, "Preisach modeling and reversiblemagnetization," IEEE Trans. Magn, MAG·26, November 1990, pp.3052-3058.

[2] F. Vajda and E. Della Torre, "Characteristics of magnetic media models,"IEEE Trans. Magn., MAG·28, September 1992, pp. 2611-2613.

[3] F. VajdaandE.DellaTorre,"Reversiblemagnetization modelsfor magneticrecordingmedia,"Physica B, 223, June 1997,pp. 330-336.

[4] I. D. Mayergoyz and G. Friedman, "The Preisach model and hystereticenergylosses," J. Appl. Phys., 61, April 1987,3910-3912.

74 CHAPTER 3 IRREVERSIBLE AND LOCALLY REVERSIBLE MAGNETIZATION

[5] G. Friedman and I. D. Mayergoyz, "Input-dependent Preisach models andhysteretic energy losses," J. Appl. Phys.,69, April 1991, pp. 4611-4613.

[6] O. Benda, "The question of the reversible processes in the Preisach model,"Elect. Engg. J. SlovakAcad. Sci., 6,1991.

[7] F. Vajda and E. Della Torre, "Scalar characterization of magnetic recordingmedia (invited)," Nanophases and nanocrystalline structures, R. D. Shulland J. M. Sanchez, eds. TMS: Warrendale, PA, 1993, pp. 121-133.

CHAPTER4

THE MOVING MODEL AND THEPRODUCT MODEL

4.1 INTRODUCTION

So far, we have assumed that a Preisach function exists for a given magneticmaterial. In this chapter, we address the questions of why it should exist at all,whether it is stable, and what its properties are. We will see that the structure of themodel must be altered in two different ways, depending on whether the material ishard or soft. Models of magnetic phenomena that are based on physical principleswill be more accurate and have fewer parameters. Therefore, the appropriatemodification will be made on the basis of the physical principles that underlie theprocess. This will result in a stable Preisach function that will no longer have thecongruency property. It will still have the deletion property, a subject for the nextchapter.

4.2 HARD MATERIALS

We will view hard materials as consisting of particles or grains that can supportonly a single or at most a few domains. Each domain will be assumed to be a singlehysteron with two stable states. Since an isolated magnetic particle has asymmetrical hysteresis loop, particle interaction is thought to be the cause for theasymmetry of hysterons throughout the Preisach plane. The source of asymmetry

75

76 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

is particle interaction of two types: exchange and magnetostatic. Exchange can atbest be due to nearest neighbors. If it is very strong, two hysterons act as a singlehysteron, thereby reducing the number of independent particles. This has noiseconsiderations as discussed in Chapter 7. If it is weaker, it is either negligible orcan be included in the magnetostatic interaction. In any case, exchange will not beconsidered by itself.

To characterize media, an accurate model for this interaction must bedeveloped [1]. The local field that each particle experiences is the sum of theexternal applied field, the demagnetizing field, and the interaction field. Theinteraction field, that is, the local field in the absence of an applied field, fluctuatesfrom particle to particle. A positive interaction field increases both the positive andnegative values of the critical field, Hie. This makes the hysteresis loop, as viewedby an external field, appear to be asymmetrical.

Thus, we will view each hysteron as having a position on the Preisach planethat is determined by its interaction-free critical field and the interaction field thatit sees. The interaction field at a particle will vary as the medium's magnetizationchanges, so the question of whether the Preisach function has any significance atall may well be asked. There are two aspects to this question. First we must askwhether the function is statistically stable, that is, whether at any region of theplane the hysteron density is constant. Then we must ask whether all the hysteronsin that region have the same magnetization. The answer to the first question,addressedin this chapter, will result in the elimination of the congruency property.The answer to the second question, which will result in the elimination of thedeletion property, is discussed in Chapter 5.

Consider an ensemble of randomly dispersed particles with moment rnJ• It isnoted that if the medium is perfectly aligned, then mJ takes on only the values I,and -1., where 1. is the alignment axis. For more general alignments, in this chapterwe will consider only a scalar model; that is, we will assume that the applied fieldis in a given direction and that we are interested only in the magnetization in thatdirection. The interaction field at a particle may be decomposed into a componentalong the applied field, which adds to it directly, and a component perpendicularto the applied field, which changes the critical field. If the Stoner-Wohlfarth modelis applicable, the largest critical field is a little over twice the smallest one. Thus,for a particle making roughly a 20 0 angle with respect to the applied field, themaximum change in critical field is about ±33%. In the following analysis, we willneglect this variation. Later, we will consider the more general case.

In the scalar case, we will compute the component of the interaction field Hiseen by a particle. This interaction field in general is given by

Hi = L m j • T ij • (4.1);~j

where T ij is the interaction field tensor between the ith and the jth particle, and mj

is the moment of thejth particle. We will assume that the magnetization of eachhysteron is in the x direction, and the only component of the interaction field is in

SECTION 4.2 HARD MATERIALS 77

(4.2)

the x direction. This is consistent with the idea that we are developing a scalarmodel. The relaxation of this condition will be discussed later in connection withvector models.

The tensor T;j is given by

1T .. = V. V.-,I} 'J 41tr..

'}

where the subscripts on the V's indicate differentiation with respect to thosecoordinates. Thus, T ij is independent of the values of the magnetization of thehysteron. For a magnetic medium that consists of a large number of randomlydispersed particles, T is a random variable and under certain conditions is inde­pendent and identically distributed. In particular for perfectly aligned media, (4.1)can be written as

(4.3)

where F is the fraction of the volume taken up by the magnetic material whosesaturation magnetization is Ms» and therefore, MglF is the saturation magnetizationof the hysteron. Then, the central limit theorem applies to each of these sums, andthus the interaction field distribution in such media is expected to beGaussian. Wewill make the assumption that the interaction field is Gaussian and is completelydefined by two numbers: its mean and its variance.

If all subsets of T ij are also independent of the m., then the standard deviationis constant, and the expectation value of the interaction field is given by

- Ms· I:;,..Qj VjT ij (4.4)H.= } =a.mM., F S

Thus, the expectation value of the interaction field is directly proportional to thetotal magnetization, that is, the sum of the irreversible component and thereversible component. We will call the constant of proportionality the movingconstant, «.

The method of the Lorentz cavity can be used to calculate a. In this method,a typical particle is replaced by an empty cavity, and the local field, due to all theother particles, is computed at this location [2]. The average value of the local fieldis computed by replacing all the other particles by a continuum whose averagemagnetization is the same as that of the particles. It is then seen that <T ij> is equalto the negative of the demagnetization tensor of the cavity. Thus for well-alignedhighly acicular particles both <.Tij>,and thus a, are very small; however, aMs maybe substantial.

It is important not to confuse this correction for the local field with that for thedemagnetizing field. Not only do the two corrections usually have opposite signs,

78 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

but the local field correction is a material property and depends only on themagnetization in the immediate area of the calculation, while the demagnetizingfield is a device property and depends on the entire magnetization as well as theshape of the material. However, they would be indistinguishable in an ellipsoidalsample that is uniformly magnetized. In fact, one could be used to balance out theother to simplify the identification process by using an appropriately shapedsample, as was done in [3].

To compute the standard deviation of the interaction field, one could use

(4.5)

(4.6)

The computation of the standard deviation requires knowledge of the correlationbetween interaction tensors and can be performed only in certain special cases. Wewill assume that 0/ is constant for most of the following calculations. Thepossibility that the standard deviation varies with the magnetization is consideredin the variable-variance model [4], and is discussed in Section 4.4. A constantvariance is apparently appropriate for longitudinal media. For particulate media, themoving constant was shown to be equal to the average of the x component of thedemagnetization factors of the particles.

The variation of the critical fields of the hysterons is determined by the size,shape, orientation, and composition of the particles or grains that constitute themedium. We will assume that this distribution is log normal, since the critical fieldsmust be positive. If the standard deviation of these fields is relatively smallcompared with their mean, it is possible to approximate the log-normal distributionby the normal distribution. This is usually the case for hard materials.

Thus, the Preisach function is given by

1 { l!(H1c - h,,)2 (Hi + (XM)2]}p(Hk,H;) = --exp -- + •

21to;o" 2 o~ 0:This is a Gaussian distribution whose peak moves with the magnetization of themedium, hence, this is called the moving model. When a field is applied to themedium, the term aM must be added to the effect of the field.

It is convenient to describe this distribution in the operative plane, which wewill denote as the hihl;-plane, where the operative variables are defined by

(4.7)

In this representation, the Preisach function appears to bestable, and its peak is atthe origin. Figure 4.1 is a block diagram of this moving model. The box "reversiblefield component computer" can contain any of the models for the reversiblecomponent of the magnetization as discussed in Chapter 3. For example, if itcomputes a magnetization-dependent reversible field, this reduces to the

SECTION 4.2 HARD MATERIALS

Reversible M"field componentcomputer

M

Preillchmodel u,

79

Figure 4.1 Blockdiagramof the movingmodel.

magnetization-dependent DOK model [5], but if it computes a state-dependentreversiblefield, then it reduces to the state-dependent eMH model [6].

The effect of the positivefeedback due to the moving constant is to increasethe slope of the hysteresis loop. Thus, for the same distribution, the measuredswitchingfield standarddeviation 0meas decreases as a increases, althoughthe realvalueof 0 stays the same.To see this, we first note that the slopeof the hysteresisloop, the rate of changein magnetization with respectto the appliedfield, may bewritten

dM = dMdhdH dh dH

(4.8)

Then since h is H + ccM, we have

dhdH

Therefore,

dM

dH

dM+ a-.

dH

dMldh

l-a.dMldh'

(4.9)

(4.10)

(4.11)

where dMldh would be the slope of the major loop if there were no positivefeedback. Since both the slope of the major loop and (X are positive,

{

dM . dMdM > - If (X- < 1-= dh dhdH < 0 otherwise.

It can be shown that the sameis true for the remanence loop.

80 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

For a GaussianPreisachmodel, the irreversible magnetization is symmetricalwith respect to the mean operative critical field. To find the relationship of theirreversiblemagnetization with respect to the appliedfield, we subtract ctM fromtheoperativefield.If wetakeequalincrements anddecrements in the appliedfieldfrom the mean critical field ~, the change in the reversible magnetization issmallerfor theincrementdue to saturation. Thus, IM;(H - ~)I is smallerthan IM;(H+ ~)I. Furthermore, the peakslopeof the irreversible magnetization occursabove~.

4.3 IDENTIFICATION OF THE MOVING MODEL

The identification of the sevenmovingmodelparameters involves identifyingthefour classical Preisach model parameters, ~, Ok' 0; , and Ms' the two reversiblefield parametersS and ~, which are model-dependent, and the movingparameterct. We have alreadydiscussed techniques for identifying the first six parameters.Several methods have been proposed for the identification of the movingparameter. Since there is no best methodfor all valuesof parameters, we will nowdiscuss two methods: the symmetrymethodand themethodoftails.Both methodsinvolvechoosing the parameterto obtain the best fit to a desired curve.

4.3.1 The Symmetry Method

The symmetry method utilizes two facts. The first fact is that for a GaussianPreisachfunction, the remanent majorloop has odd symmetry about the remanentoperativecoercivity. This coercivity, for a singlequadrantmedium, is equal to themean critical field ~. The second fact is that the moving model is a classicalPreisach model when its input is the operativefield. In particular, if the PreisachfunctionisGaussian,themajorremanentloopis anerrorfunction. Thus, if Mrem(H)is the remanentmajor loop,

Mrem(H+aM-hk) = -Mrem[-(H+aM-hk) ] · (4.12)

(4.13)

Unless the squareness is 1, a real material does not have this propertybecausethereversiblecomponentof the magnetization is nonlinear. Thus, Mrem(H - 11k) is notthe negativeof -Mrem( -H - hk) and hencethe majorloop is not symmetrical aboutthecoercivity.Thiswastheprincipalcauseof secondharmonic distortionwhendc­bias recording was used. The problem with simply finding the value of a thatminimizes this difference is that we do not know what 11k is. It differs from theremanentcoercivityby aM, and althoughM; is zero there, Mrem is not.

The method, therefore, involves measuring themajorhysteresis loopand thenfinding the valueof a that minimizes the following integral:

rii{- - }2/(ex) = Jo

iMr(R+aM-hk) + Mr[-(R+exM-hk) ] dR.

SECTION 4.3 IDENTIFICATION OF THE MOVING MODEL 81

We can replace hk by the remanent coercivity, HRC plus aM(h k) , which isHRC+ aM,( ~), in evaluating this integral. Differentiating this with respect to a andfinding the value that makes it equal to zero is another method of measuring a.Thus,

(4.14)

(4.16)

The first method involves finding a minimum, while the second method involvesfinding a zero crossing. Thus, the second method is more sensitive. When thesecond method was attempted on several recording media, a precise a was foundthat reduces the value of I by many orders of magnitude and was limited only byexperimental error [7].

The identification of the parameters in the eMH model must be performed ina particular order. The technique we are now presenting applies to single-quadrantmedia. The first step is to measure the major hysteresis loop MJ(H) and the majorremanent loop Mrem(H) as a function of the applied field H. The first twoparameters identified are the saturation magnetization Ms and the squareness S,which are defined by

Ms = MJ(co) and S = Mrem(co)/MS' (4.15)

The operative field h is given by

h = H + aM.

We expect the remanence to be symmetric function of the operative field, withrespect to the remanent operative coercivity. We use this criterion as a method ofchoosing the correct value for a. This method requires a good starting set ofparameters, which we will now obtain.

As shown in Fig. 4.2, we define the coercivity, He to be the field at which themajor loop magnetization is zero, and we define the remanent coercivity, HRC' tobe the field at which the major remanent loop magnetization is zero. On theascending major loop, let

(4.17)

and

(4.18)

We note that Mc is usually negative. Let HI be the field at which the remanentmagnetization is the positive quantity, MRC' That is,

Mrem(H1) = -Me' (4.19)

We will also define

(4.20)

82 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

HIC HI Applied field

Figure 4.2 Definition of terms.

Usually MI and MRC are positive quantities. Then a first approximation for themovingcoefficientis given by

Ht + Hc - 2HRCa = (4.21).2MRC - M t

The effectof thischoiceof a is to makethecurvesymmetrical, as a functionof theoperative field, at three points. This approximation can be quite poor, since in(4.21) 2MRC is only slightlylarger thanMI' Thus, it is recommended to use (4.14)if possible.

In this case, the averageoperativecritical field is given by

(4.22)

For a GaussianPreisachfunction, the remanentcurveas a functionof theoperativefield should be an error functiongiven by

M (h) (h -h) (H-H +a[M(H)-M l)m (h) = rem =Serf __k =Serf RC RC • (4.23)rem Moos

Since the error functionof 0.25 is 0.2763,wecan defineH2 as the field that makesthe normalized remanence 28% of the saturation. Then, an approximation of 0 isgiven by

(4.24)

SECTION 4.3 IDENTIFICATION OF THE MOVING MODEL 83

This valuecan be quite rough,since the approximation for a given by (4.24)was determined by only a few measurements. A better approximation of thestandarddeviationof thecriticalfield0 andthemovingparametera fora GaussianPreisachfunction, is obtainedby fittingMrem(h) to an error function. Then, (4.21)and (4.24) could be used as a startingpoint for a two-variable search algorithm.

To describe completely the irreversible component of the magnetization wedivide the standarddeviation of the switching field into the standarddeviationofthecriticalfield ole andthestandarddeviation of theinteraction fieldOJ. Toperformthis separation, wesaturatethematerial in thepositivedirection, applya field -lik ,

followedby a field hk' and then measure the magnetization MJc , where hk is givenby (4.22). If we define

Mkr =--

SM 's

then

We note that with this definition,222a = 0; + Ok.

(4.25)

(4.26)

(4.27)

(4.28)

To identify the reversible component of the magnetization, we need thesusceptibility at zero field, Xo. We now have the seven parameters of the CMHmodel: Ms, S, ~, ex, o; Ole' and Xa.

For three-quadrant media, the identification is a bitmorecomplicated becausethe remanence is not the sameas the irreversible component of the magnetization.Thus,S is not givenby (4.15); rather, we use a modification of (4.23) withS as anadditional parameterto fit. This modification is

M; (h - iik )- = Serf --,Ms a

where M; is the remanence plus the change in magnetization due to apparentreversalMAR. Hence

(4.29)

Since M; is not observable, we have to compute it from the observableMrem» asdiscussedin Chapter3. Then, in computing Xo' we have to subtract the zero-fieldslope of (4.28) from the remanence susceptibility.

84

4.3.2 The Method ofTails

CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

The method of tails subjects the material to an anhysteretic magnetizing process.If we use the classical Preisach function, the resulting remanence magnetizationcan be computed in closed form and is given by

(4.30)

The anhysteretic susceptibility, the derivative of (4.4) with respect to Hdc is alsoGaussian.

To test the validity of this analysis, a sample of y-Fe20 j magnetic recordingtape was measured. This sample was chosen to avoid the problems of apparentreversible magnetization [8]. The data points in Fig. 4.3 show the result of anormalized anhysteretic susceptibility measurement using a vibrating samplemagnetometer (VSM), and the solid line is a Gaussian fit to these data [9]. It isseen that this fit is good for only small values of Hdc• For large values of Hdc, themeasurements appear to decrease more slowly to zero than the Gaussianapproximation. We will now see that the moving model corrects this apparent error.

Since the Preisach function is constant only in the operative plane, 'this curveshould have been plotted as a function of the operative field. In an experiment, itwas assumed that for small values of Hdc the magnetization is a linear function ofthe applied field, the operative field is directly proportional to the applied field, andno distortion in the Gaussian occurs. As the value of Hdc increases, however, themagnetization saturates and the operative field does not increase as quickly withthe applied field as before. This results in a tail that goes to zero much more slowlywith respect to the applied field than the Gaussian.

1

o-2000 -1000 0

Applied field1000 2000

Figure 4.3 Anhysteretic susceptibility of y-Fe 20 3 recording tape.

SECTION 4.3 IDENTIFICATION OF THE MOVING MODEL 85

The correctwayto obtaina Gaussian anhysteretic susceptibility is to applyananhysteretic sequenceof operative fieldsratherthanordinaryfields.This requiresa priori knowledge of u and the simultaneous measurement of the magnetizationto computethe correctcurrentoperative field. Then the field is slowlyincreaseduntil the desiredoperativefield is reachedwithoutovershooting. This is possibleto do using a computer-controlled VSM.

To correctthesemeasurements, the susceptibility wasplottedas a functionofthe remanent operative field. That is, the remanent magnetization was multipliedbya andaddedtoHdc to approximate theoperative field.ThedatapointsinFig.4.4indicatethe susceptibility measurements as a function of the operativefield. Thesolidline is a fit of thiscurvewitha Gaussianfunction of theoperativefield.Sincethe curve is symmetrical aboutHDC equal to 0, only the values for positivevaluesof HDC areshown.ThisGaussian function hada standard deviation of 2200versus130 for the Gaussian in Fig. 4.4, since it has to compensate for the movingconstant. It is seenthatthisfit stillaccurately describes theregionwherethe valuesof Hdc are small,but now it also fits the tail correctly.

This correction assumes that whenan anhysteretic field processis appliedtothe material, the operative field is also anhysteretic. If the applied field isanhysteretic, the operative field is only approximately anhysteretic; thus, the linedividingthe operative Preisach plane is only approximately straightand does notquite have the correct slope. Furthermore, we have neglected the reversiblecomponent of the magnetization in themagnetizing process. Thesecorrections areexactonlyat remanence; however, theerrorstendtocanceleachotherout.In order

1.0

~0.8

S~ 0

ir~0.6fI}

~.~J0.4

0.2o

oL---------=~~~~'----o 1000 2000 3000 4000 5000 6000

Operative field offset

Figure 4.4 Gaussianfit to the operativesusceptibility.

86 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

to be certainabout the natureof the interactionfield,operative field measurementsmust be performed.

Theapproximation of theinteractionfielddistributionof thePreisachfunctionby a Gaussian function is based on a theoretical applicationof the Central LimitTheorem.Furthermore, the mean valueof this Gaussian is directlyproportionaltothe magnetization, and the standard deviation appears to be constant duringanhysteretic magnetizing processes. Measurements on a sample of y-Fe203

recording tape show that this is an excellentapproximation if plottedas a functionof the operative field.

4.4 THE VARIABLE-VARIANCE MODEL

In longitudinal recording media, a hysteron is surrounded by particles in alldirections; hence, even if they are all magnetized in the same direction, there canbe a considerable variation in the local field. In some thin-film perpendicularmedia,the materialis uniformlyonegrainthick,withall thegrainswellalignedandsimilarlysituated. When the material is saturated, it is expected that the variationin the interactionfield will be small;however, in the demagnetized state, half thegrains are magnetized in one direction and half in the other direction. Thus, eachgrainmaybesurroundedbya substantiallydifferentfield, leadingto a muchhigherstandard deviation in the interactionfield.

In work usingan artificialmagneto-optic medium[10], it has beenshown thatfor perpendicular media, the standard deviation of the interaction field ismagnetization dependent. For this medium [11], the variance dependence onmagnetization appearsto be linearand is smallerin thedemagnetized state.For theCo-Cr medium, it is seen that the variancealso dependson the magnetization, butin this case the demagnetized state has the smallervariance. The basic differenceis that the artificialmediumwas verydilute, and therefore,one considers only thefew nearest neighbors. For this material, in the demagnetized state a particle cansee a wide varietyof configurations, but in the saturatedstate the configuration isveryuniform.The Co-Crmedium, on the other hand, is verydense and the field atany givenparticle is the sum of manyneighbors. Surface roughnesscan cause thisinteraction field to vary considerably from particle to particle, and its standarddeviation is linear with the magnetization. At zero magnetization, there is aminimumvariancein the distribution.

In these perpendicular media, the demagnetizing field shears the hysteresisloop.It has beenshownthat this shearingdependson thedemagnetizing factor, thethickness,and themagneticpropertiesof thefilm [12].In the full descriptionof themodel, these demagnetizing effects will be combined into an effective movingparameter. Paraphrasing our earlier work [4], we now describe this medium interms of a model in which the variancevaries with the magnetization. In additionto the identification of the other parameters, one now must also specify thevariationof the standard deviation of the switching field o. The identification issimplifiedin thismedium, since thereversiblecomponentappearsto be negligible.

SECTION 4.4 THE VARIABLE-VARIANCE MODEL 87

This removes one of the parameters from consideration. Although a generalidentification strategy is not possible, since the Preisach function is not limited tothe fourth quadrant, it is still possible to model accurately the major loop for thismedium.

We now illustrate the effect of the remaining parameters and show how theycan be identified, explaining the nature of the major loop only. The Preisachfunction is described in the operative plane, since it is statistically stable there. Inthe variable-variance model, the operative interaction field h; is obtained bydividing H + aM, the operative interaction field, by the standard deviation in theinteraction field, o; If a; is a constant, it simply acts as a change of scale. In thevariable-variance model, OJ is not a constant, but a function of the magnetization.It is assumed that the Preisach function is Gaussian in both the interaction field andthe critical field in the operative plane. If we define the operative critical field h,by Hla k, then the Preisach function is given by

[2 - 2]hI +(hJ:-hJ (431)

P{h"hJ= A exp - 2' ·

where A is a suitable constant and h k is the operative remanent coercivity. Since thecritical field of a particle is determined by its physical properties only and not themagnetic state of the system, it is reasonable to expect that 0 A: is constant.Therefore, we will assume that only 0; varies and that it is a function only ofM. Wenote that in obtaining the major loop only the switching field variance is required,which is given by

(4.32)

The particular variation that we will assume for a; as a function of themagnetization is given by

a, = 0 10(1 - vJMI~, (4.33)

where v and k are suitably chosen constants. If either V or k is zero, the variance isconstant, and the model reduces to the ordinary state-dependent model. Since themeasurements indicated that the reversible component is negligible, we will assumethat it is zero, causing the model to reduce to the simple moving model. The blockdiagram for this model is shown in Fig. 4.5. The term "modified Preisachtransducer" indicates that it includes reversible components as discussed previously.

The major hysteresis loopof a magnetically uniaxial, rfsputtered Co-Cr filmwith 23% Cr, deposited on a silicon substrate, was measured using a computer­controlled vibrating sample magnetometer [4]. To ensure proper nucleation and toobtain a nearly perpendicular anisotropy over the entire thickness of the film, thefilm was deposited on a germanium seed layer over an Si02 layer on the substrate.The major loops along the film plane and perpendicular to the film plane are shownin Fig. 4.6. It is seen that the in-plane hysteresis curve is much narrower and almost

88 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

H Modified Preisachtranducer

Figure 4.5 Blockdiagramof the variable-variance movingmodelfor media with noreversiblecomponent.

completely reversible. On the other hand, the normal hysteresis curve has anegligible reversible component, and the knee of the ascending major loop occursfor negative values of the applied field. Furthermore, it is observed that theascending major branch is asymmetrical, and its susceptibility below the coercivityis greater that above the coercivity. The measured saturation magnetization is 292kA/m, and the squareness of the loop is 0.28.

Since the knee of the magnetization of the ascending major loop of thismaterial occurs when the applied field is still negative, and since we are assumingthat there is no reversible component, this implies that this increase is due toapparentreversible magnetization [13]. Apparent reversible magnetization is dueto "particles" or hysterons, both of whose switching fields are negative. Thesehysterons are described by points in the third quadrant of the Preisach plane. Thepresence of such particles introduces an error in the routine identification of theirreversible component of the magnetization by measuring the corresponding

Applied field (kOe)

-- Normal M........... In-plane

-4 -3

-------/ -1

2 3 4

Figure 4.6 Normalized majorhysteresis loopsof Co-Cr perpendicularmedia,measurednormalto and in the film plane.

SECTION 4.4 THE VARIABLE-VARIANCE MODEL 89

remanence. In fact, the locationof the kneeof the ascending major loop indicateswhetherthePreisachfunctionspillsover into the thirdquadrant, and by symmetrywhetherit spills into the first quadrant.

The major M-H loop can be computed easily for this model, since themagnetization is an errorfunctionof theoperativefieldh. The appliedfield is thencomputedby

-H = oh - aM. (4.34)

Thus, wecancomputebothMandH parametrically as a functionh. The techniqueunfortunately does not work this simplyfor minorloops.

Thecomputedascending branchof themajorloop,assuming that the varianceis constant (k = 0), is shown in Fig. 4.7. It is seen that as the moving parameterabecomes more negative, the slope of the branchdecreases and the knee moves tothe left. Thus, one effectof the negative ex is to push the Preisachfunctionoutsidethe fourth quadrant as the medium becomes magnetized. In all these cases,however, thecurvesmaintain theirpointsymmetry aboutthecoercivity, a propertyof the state-dependent model when viewedin the operativeplane only.

If k is not zero, the symmetry aboutthecoercivity is lost, as shownin Fig. 4.8.The locationof the kneecontinuesto moveto the left, and the slope of the curvecontinues to decreaseas ex becomes more negative. In this case, since k is 1, thesusceptibility is greater if the applied field is an amount dH greater than thecoercivity, than if it is an amountdH less than the coercivity. As a increases,theslope can even becomenegative.

To illustratetheeffectof theexponentk, thecurvesinFig.4.9 werecomputed.The caseof k=0 is the case in whichthe variance is constant. In the case of k =1,the variancein the interaction fieldvarieslinearly withthemagnetization. It is seenthat the larger the value for k, the more distortion in the symmetry about the

96-3

-3

-2

-1

o

a

-1"- ~....... ~~__~ L..______1

-6 o 3

Applied field

Figure 4.7 Effect of movingparameter, «, if the variance is constant(k = 0).

90 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

I I..... I '

I iI . ;, ;, ;

0.5I

"a If;s If;

I,;.~ _._._._._. -3 (i;

·i 0 - .... ......... ...... -2

~ ----- -1 170/ :i

,I ....,t-0.5

/)I ....//,I

. ,<~.:.... II

-1

-10 -5 0 5 10Applied field

Figure 4.8 The ascending branch of the major hysteresis loop for different a's when k is 1.

coercivity. Although for nonzero values of k the coercivity is larger than the caseof k =0, it is not a strong function of k.

For k = I, as v is varied, we see in Fig. 4. I0 that the type of asymmetrychanges . In particular, for positive values of v, the second derivative of themagnetization at the coercivity is positive, while for negative values of v, thesecond derivative of the magnetization at the coercivity is negative. Theseobservations are useful in fitting the measured curves with this model.

The identification of the parameters in the model has not been solved in

11II

A;/I fIl-

k //If_._.-._._.- 4 /1....................... 2

Ii------ I if"- -- 0 1./

~/

Il.l '~

4/''l

I:l.g ns

.~~e 0."

]§o -o.sZ

-t

-s oApplied field

to

Figure 4.9 Computed ascending major branch for different values of the exponent k:

SECTION 4.4 THE VARIABLE·VARIANCE MODEL 91

i r1. /

1/I

I

V r_._._._.-. 1.5 !f

.................... 0.5----- -0.5 ,-- -1.5 Iii

,Iii

II/ /1/ / 1.

.1 l ot

§.~

.~ 0.5

t 0

]] -0.5

~-I

-10 ~ 0 5

Applied field10

Figure 4.10 Effect of v on the computed ascending branch of the major hysteresis loop.

general; however, for this particular medium a good fit of the major loop wasobtained with the following data: ~=6, 0/ =100 Oe, 0 1 =50 Oe, k =2, 'V =-1.4,and IX =-2.7 . Since 0 is 112 Oe, when M is zero oiik is 672 Oe. The negative 'V

indicates that the variance is larger when the medium is saturated. The resultingsimulation is shown by the solid curve in Fig. 4.11. It is seen that the agreement isvery good between the results of the model, indicated by the solid line, and themeasured values, indicated by the data points. This is all the more remarkable,

//

l/j

- I---.. ..... [7

1.5

I:l 0.5·IS.~ 0lib~-0.5

-1

-1.5

-6 -4 -2 0 2Applied field

4 6

Figure 4.11 Comparison of measurements (dots) of the ascending major loop of the Co-Crsample with the variable-variance model (solid line).

92 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

because the Preisach function in the operative plane is a very smooth andsymmetrical Gaussian curve, whereas, the observed susceptibility is veryasymmetrical.

The state-dependent model is a seven-parameter model that can adequatelydescribe longitudinal media, for which it can be shown that the variance isapproximately constant. This is not true in the case of thin-film Co-Crperpendicular media, which consist essentially of a single layer of particles. Thus,the particles are surrounded by other particles in basically two dimensions, ratherthan three. Therefore, the interaction between any pair of particles is alwaysnegative. The resulting moving parameter, unlike longitudinal media, is negative.This parameter includes the demagnetizing effect of the shape of the medium, sinceon this scale it is difficult to distinguish between the two.

4.5 SOFT MATERIALS

The behavior of soft materials is different as a result of the inherent difference inthe nature of the hysterons. For hard materials the hysteron was a well-definedparticle or grain and always switched the same way. For soft materials, themagnetization changes by the domain wall motion from one pinning configurationto the next. In general, these configurations are not repeatable, especially when thewall moves in the opposite direction. Still, the magnetization changes in quantumjumps, and the overall effects are similar.

The main difference is that the probability of a hysteron switching in onedirection is essentially independent of that switching in the opposite direction. Ifwe then interpret the Preisach function as the probability density function that ahysteron has switching fields U and ~ which are statistically independent, thenthey can be factored into the product of the individual switching fields; that is,

p(U,V) =p(U)p(V). (4.35)

To comply with the symmetry of nature, the individual probabilities must beidentical. Thus, if the distribution is Gaussian, it is described by a single standarddeviation. Furthermore, since p(U) must be zero if U is negative, there can be noapparent reversible magnetization. If the standard deviation becomes comparableor larger than the coercivity, the distribution cannot be Gaussian but is probablylog-normal.

In the case of soft materials, one can ask whether the Preisach function isstable. In most cases, the domains are so shaped that their demagnetization factoris zero. Thus, we would expect a to be zero. On the other hand, the number ofhysterons available for Barkhausen jumps depends on the magnetization. Inparticular, for the last hysteron to switch before saturation is reached, there is onlyone possibility. In the case of a demagnetized specimen, we can have a longdomain wall with many possibilities for hysterons to switch. This leads to theproduct model [14] proposed by Kadar, which assumes that the Preisach functionis the product of a function of magnetization and a function of the switching fields.

SECTION 4.5 SOFT MATERIALS 93

Therefore, the rate of change in the magnetization with respect to an applied fieldis given by

dm fH- = K(lrnl) p(U)dU,dH 0

(4.36)

where K is a function of the magnitude of the magnetization and must be zero whenIml is unity. The simplest such function is 1 - rn2

, which we use in the subsequentexamples. When the material is saturated, the susceptibility is zero. Hence, themagnetization cannot be changed until a reversal has been nucleated, so that mmust be incrementally reduced from unity.

In this model, a magnetization-dependent reversible magnetization can beadded very easily by including an additional single-valued function in (4.36). Thusit is seen that for any choice of X(H), the magnetization cannot exceed saturation,since K(lrnl) will not permit it. The simplest choice for X is a constant.

For example, if K is 1 - m2 and X is a constant, let us apply a positive fieldlarge enough to saturate the sample and reduce it to zero. When a positive field His reapplied, all the changes in magnetization will then be reversible. In that case,(4.36) will reduce to

rm(H) dm = ioHXdU.Jm(O) 1-m 2

On integrating we obtain

m(H) = tanhLxH + tanh-tm(O)].

(4.37)

(4.38)

This is a very reasonable curve for the reversible component when the irreversiblecomponent is saturated.

A block diagram for this model appears in Fig. 4.12. The main differencebetween the product model and the moving model is that the former usesmultiplicative feedback instead of additive feedback. We will see that the productmodel, like the moving model, also violates the congruency property.

4.6 HENKEL PLOTS

Wohlfarth suggested that if there were no interaction between hysterons, the slopeof the major remanent hysteresis loop would be twice the slope of the remanentvirgin curve for any applied field. He did not specify the method of demagnetizingthe material, since if there were no interaction, it wouldn't matter. This suggesteda method of measuring the amount of interaction. If we let m.,,(H) be the virginremanent curve and mlH) be the major remanent curve, a plot of mAH) as afunction of mI.-H), called a Henkel plot, should be a straight line from (0, -1) to(1,2). Any deviation from a straight line would then be due to interaction. Analternate method of measuring interaction would be to plot mtH) - 2 rnv(H), called

94 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

M,

Figure 4.12 Block diagramof the productmodel.

am, as a function of the applied field. This plot would be a horizontal line throughthe origin if there were no interaction.

Measurements of this sort gave curves that fell both above and below thenoninteracting locus. Bertotti et a1. have shown [15] that for the classical Preisachmodel, only curves beneath that locus are predicted, but in the moving model, bothtypes of behavior are possible. Except for the case of noninteracting particles, thevirgin curve is different depending on how the material is demagnetized. In thefollowing analysis, we will assume that ac-demagnetization is used to obtain thevirgin curve.

From (2.27), the major remanence loop, for a equal to zero, that is, theclassical Preisach model, is given by

dmJ = 1. 12 exp (H - hJ2 . (4.39)dB a~ -; 2a2

By substituting -H for HI in (2.23) we obtain the virgin magnetization curve givenby

(4.40)

which reduces to

dmy

dB1 dmJ{ 4a, (H-h')]l--1 + e .2 dB a1 a

(4.41)

(4.42)

Thus, we see that if there is no interaction, (J i is zero. Then the argument of the errorfunction is zero, and thus, the error function itself is zero. Therefore,

dm y ! dmJdB 2 dB

SECTION 4.7 CONGRUENCY PROPERTY 95

This is the Wohlfarthconjecture, whichstates that if there is no interaction, then

mjH) = 2m~H) - 1. (4.43)

To try to quantify"interaction,"researchers usedplotsof two typesto illustratethe deviation from (4.43). In Henkel plots, m; is plotted as a function of m.. Fornoninteracting materials, this shouldbe a straightline from (0, -1) to (1, 1). In ~mcurves, 2mv -1 - mJ is plottedas a function of the appliedfield. For noninteractingmaterials, this should be a horizontal line throughthe origin.

It is seen from (4.40) that

1 dm, -tim y =

>--- if H> hi2dH (4.44)

dH 1 dm, -<--- if H< hi.

2dH

Thus, a dm curvewouldstartat the origin,havea negative slopeuntil hi' and havea positiveslope after that until it returns to zero, as H increases.

As an illustration, am curvesfor a square loop material with a/Ok equal to 0,0.25, 0.50, and 1.0 are shown in Fig. 4.13. The appliedfield is normalized to thecoercivity, and o, is fixed at 0.3. It is seen that if there is no interaction, then OJ isequal to zero, and indeeda horizontal line is the result.As atincreases, so does thedeviationfromthe horizontal line.The slope of the~m curvesis negative up to thecoercivity and positiveafter that.

When a is a positive number, positive feedback is introduced around thetransducer. If there is no reversible magnetization, then

:; = ::: = ::( 1 + U:;). (4.45)

Since a, dm/dH, and dm/dh are all positive, we have

tim = tim/dh > dmdH 1 - u dmldh dh ·

(4.46)

Thus, the effect of the moving constant, a, is to increase the slope of themagnetization curve.Its effecton am, as a function of the appliedfield normalizedto the coercivity, is illustrated in Fig. 4.14, when a; is O.Sat for a square loopmaterial. It is seen that for the classicalPreisachmodel, a is equal to zero, and thedm curve is always negative. When a is greater than zero, the dm curve can bepositiveas well as negative.

4.7 CONGRUENCY PROPERTY

The relationships between the moving model and the product model are exploredby examining both the irreversible and the reversible susceptibility variations

96 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

0.2

0.------------

-0.2e

<I

-0.4

-0.6

a,t"to0.25

0.5

1-0.8 L.-- ~ .....c..

o 0.5 1 1.5 2Normalized applied field

Figure 4.13 dm curve for classicalPreisachmodel.

IE<I

0.2

o

-0.2

-0.4 o0.1

0.2

0.3

0.5 1 1.5 2Normalized appliedfield

-0.6 L..-- ~ ~

o

Figure 4.14 dm curve for variousmovingconstants.

predicted by each model [16]. It is shown that for the moving model thesesusceptibilities are a function of the sum of the applied field and a term proportionalto the total magnetization. For the product model they are the product of a functionof the total magnetization and a function of the applied field. This leads to adifferent variation in the height of minor loops, and thus, a means of differentiatingbetween the models. Measurements reported elsewhere show that for particulatemagnetic recording media, the moving model yields more realistic results. Thereversible magnetization component of the moving model had to be modified bydevising two new models, for the reversible magnetization, compatible with themoving model.

SECTION 4.7 CONGRUENCY PROPERTY 97

We nowdiscusshowthesepropertiesmaybe usedtodifferentiatebetweenthevarious models, considering each model's unique way of circumventing thecongruencyproperty by meansof an examinationof the irreversiblesusceptibilitypredicted by each model.The resultsof experiments[17] indicate that for a partic­ulatey-F~03 medium,the modelmostapplicableappearsto be the movingmodel;the sameworkgivesmeasurements of the variationsin the reversiblesusceptibilityin the interior region of the hysteresis loop. We will discuss the properties ofseveral models for the reversible component in order to compare them withexperiment.

4.7.1 The Classical Preisach Model

The classical Preisach model computes the irreversiblemagnetization using

M j =! !Q(u,v)P(u,v)dudv , (4.47)u>v

whereP(u,v) is the Preisachdensityfunctionof thepositiveand negativeswitchingfields. The function Q is processdependent,and for scalar irreversiblemodelscantake on only the values-lor +1,dependingon the sequenceof applied input fieldextrema.The change in magnetization whenthe appliedfield is increasedfrom HIto H2, can be computed from the Everett integral:

E(H l'H2) = J: H2dvJ: vdu P(u,v). (4.48)H) H)

Therefore, wedefinethe irreversiblesusceptibilityfor theclassicalPreisachmodel'Xci as

.(H) = E(H,H+t1H)XCI dB' (4.49)

where AH is a small incrementin the applied field. This can be interpreted as theratio of the height of an incremental minor loop to its width. It is seen from (4.49)that for the classical Preisach model, the susceptibilityis a function of the appliedfield and the width of the minor loop. Since the susceptibilityvaries with the sizeof l1Hand is in fact zero when l1H is zero, in all subsequentcalculations, we willuse the same value for AH.

To demonstrate the congruencyproperty of the classical Preisach model, wepoint out that the susceptibility is not a function of the magnetization. This isillustratedin Fig.4.15, whichshows the variationof the susceptibilitypredictedbythis model in the interior of the major hysteresis loop.

4.7.2 Output-Dependent Models

The effect of the moving model is to replace the applied field H in the classicalPreisach model with an operative field, h = H + aM, where a, the movingparameter, is a constant for a given medium. Thus, the irreversible susceptibility

98 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

... ~ .

:( [::r-'. \'

: : . ~ :. ~ . .': :

; ,: ~ .;: : r':' :

i .i/· ··

/ ' ' ....

o -r-~J.J.llLLI-l·2

oAppliedfield 2

Figure 4.15 Variation in susceptibility in the classical Preisach model.

of the moving model, 'Xmi' is a function of the magnetization as well as of theapplied field and is given by

I'Xmi(H,M) = 'Xci(H+ aM). (4.50)

The prime on 'X~i indicates that the Preisach function P was modified to P I in themoving model.

The product model [14], on the other hand, is defined by its property of givingits susceptibility directly. If the reversible component is zero, then the irreversiblesusceptibility of the product model, 'Xpi ' is given by

dM H'Xp;(H,M) = dH

i= K(M)!P"(u,H)du = K(M)'X~~(H), (4.51)

where K(M) is the noncongruency function , P"(u,H) is the residual Preisachfunction of the product model, and Z ;;(H) is the modified classical irreversiblesusceptibility. The double prime indicates the product model modification.

From the control point of view, the moving model is a nonlinear feedbackprocess, as shown in the block diagram of Fig. 4.12 [18]. Thus, it is necessary tosolve for the magnetization iteratively. This process is computationally lessefficient than the product model in which the magnetization-dependent and thefield-dependent parts of the modified Preisach function are separated. On the otherhand, the moving model can be directly related to physical material parameters.Therefore, it is desirable to understand the relationship between the two models.The moving model relaxes the congruency limitation of the classical Preisach

SECTION 4.7 CONGRUENCY PROPERTY

.: .

99

x

,: : ::' ....:; :: :

:: ~ )" ~.: : : : :

··••··,····li':"·····. : r: ".

; ....

Applied field 2

......

Figure 4.16 Variation in the susceptibility in the moving model.

model, replacing it with the skew-congruency property. Thus , minor loopsconnected by lines of slope -Va are congruent, as illustrated in Fig. 4.16.

In addition, the product model from a control point of view is a simplePreisach transducer followed by a nonlinearity (18]. This model does not involvefeedback because of the assumption that the magnetization-dependent and the field­dependent parts of the Preisach function can be separated. Thus, the identificationproblem is greatly simplified: K(M) is obtained by measuring the variation in heightof minor loops along the M axis, and the residual Preisach function P"(u,H) can beused to obtain first-order reversal curve information [19]. The product model re­places the congruency limitation of the classical model by the nonlinear con­gruencyproperty, which is equivalent to the existence of the nonlinear functionS(N) . A plot of the variation of the susceptibility for the product model is shownin Fig. 4.17.

For the moving model, from (4.50), it is seen that

Xmi(H,M) = xmi(H+aM,O) . (4.52)

That is, for any line parallel to the H axis, the variation in the susceptibility is givenby the variation along the M axis shifted by the amount aM.Thus, the susceptibilitypeak along a line parallel to the H axis will not occur on the M axis, as shown inFig. 4.18(a) . Also from (4.50) , it is seen that

Xmi(H, M) = Xm{0, M + ~) . (4.53)

That is, for any line parallel to the M axis, the variation in the susceptibility is givenby the variation along the H axis shifted by the amount Ht«. Thus, the

100 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

x

. , .

o

Appliedfield

. . : ~ .

.... ~ , .

-. )

2

Figure 4.17 Varialion of the susceptibility in the product model.

susceptibility peak along a line parallel to the M axis will not occur on the H axis,as shown in Fig. 4.l8(b).

For the product model, on the other hand, from (4.52) it follows that

K(M)XJH,M) =XJH,O) K(O) . (4.54)

Thus, the variation in the susceptibility along any axis parallel to the M axis is thesame. It also follows from (4.50) that

/IXci(H)

XJH,M) =xiO,M)-,-,-. (4.55)Xci(O)

Thus, the variation in the susceptibility along any axis parallel to the H axis is alsothe same. Therefore, if Xd'(H) is symmetrical, then all projections of the suscep­tibility along any axis parallel to the H axis are symmetrical, as shown in Fig.4.19(a). Similarly, since K(M) is symmetrical [19], all projections of the suscepti­bility along any axis parallel to the M axis are symmetrical, as shown in Fig.4.19(b) .

4.8 DELETION PROPERTY

Some interesting properties of the Preisach model obtained in [20] will bedescribed here. The moving model computes the irreversible component of themagnetization, MI, using

M;= ! !Q(w,v)P(w +cxM,v +cxM)dvdw, (4.56)v>w

SECTION 4.8 DELETION PROPERTY 101

-M=--- M>O

(a)Applied field

-H=O---H>O

I

(b)Magnetization

Figure 4.18 Preisach cross sections forthemoving model.

whereQ is a process-dependent function, whichfor scalarprocessesis either +1or -1, P is thePreisachprobability densityfunction, whichis bydefinitiongreaterthanzero,M is the totalmagnetization, andwand v are thePreisachvariables (i.e.,the positiveandnegative switching fields,respectively). The limitof integration isthe entire regionof the Preisachplane wherev> w, that is, the hatchedregion inFig. 4.20.

A line thatconsistsof horizontal and vertical segments only,as shownin Fig.4.20, is the boundary that separates the simply connected region where Q is +1from the simplyconnected region where Q is -1. In the case of an anhystereticmagnetization process, it becomes in the limitof many cyclesa continuous curvewitha negative slope.The sequenceof discontinuities in the boundary in the firstPreisachvariable wk is a monotonically increasing sequencein k corresponding tothe sequenceof successive maxima of the inputvariable, whilethe correspondingsequence in the second variable Vk is a monotonically decreasing sequence in kcorresponding to the successive minima of the input variable. In the case of themoving model, the input variable is the sum of the applied field plus the productof (X and the magnetization.

We define the Everettintegral by:

-M=--M>

H=O

--H>O

102

u;e

J

CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

(a) Appliedfield

(b) Magnetization

Figure 4.19 Projection of the irreversible susceptibility for the productmodelalongan axisparallelto (a) the H axis, and (b) the Maxis.

s v

E(r,s) = f fp(w, v)dwdv.r r

(4.57)

If the applied field is increased from HI to H2 with a corresponding increase inmagnetization fromM1 to M2, and if H2 + aM2 is less than the previousmaximumof H2 + a.M2, the change in magnetization can be computed from the Everettintegral:

(4.58)

The same formula applies if the applied field is decreased from HI to H2 with acorrespondingdecreasein magnetization fromM I toM2, and if H2 + a.M2 is greaterthan the previousminimaof H2 + a.M2• In order to computethe magnetization M2,

(4.58) must be solved implicitly.

4.8.1 Hysteresis inIntrinsically Nonhysteretic Materials

For materialsthat haveno intrinsichysteresis, P is a delta functionin w whoselineof singularity is the line w equal to -v. In this case, the Preisach function can beexpressedas a functionof a singlevariable, theappliedfieldH. The magnetization

SECTION 4.8 DELETION PROPERTY

Figure 4.20 Boundary betweenthe regionsof oppositessignsof Q,

103

is the cumulative distribution of the Preisach function, F(H) , and it increasesmonotonically.

The moving model can introduce hysteresis for these materials if a issufficiently large. Thiscan be seenbyexamining Fig.4.21,whichshowsa typicalplot of F(H) for such a material, and howit is modified by aM to obtainthe curveF(H + aM). It is seen that for IHI < Hie' thereare threepossiblevaluesof M. Thecentralvalueis unstable, but the other two values are stable.At H =Hie' if one ison the lower curve, the magnetization will switch discontinuously to the uppercurve,as indicated bythedashedline,leadingtoextrinsic hysteresis. For IHI > Hie'the curve is single-valued. In materials in which dF/dH is a monotonicallydecreasing function, the condition for hysteresis is a > I/X, whereXis the initialsusceptibility; that is, Xis dF/dH at H = O.

The behavioris morecomplex for materials in whichdF/dH increases at first

F(H+aM)

II,,,'

"j.' I,

I,,I

I,,,II,",f

:," ,.. ,

."

, F(H)

H

Figure 4.21 Effectof (X on magnetization process,

104 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

M

~.

...( ..... (I

.: ' :~

y----

F(H+a.M)--~-- .

~:)!....... F(1l)

.i->

H

Figure 4.22 Effect of a on the magnetizing process when Xmax is at a positive H.

and then decreases monotonically to zero. In this case, there are again two possiblestates at zero H, but four discontinuous regions of operation. For small values ofa, as shown in Fig. 4.22, there is no hysteresis at zero H, but there are two minorhysteresis loops symmetrically displaced from the origin. The condition for thistype of hysteresis to occur is a> l/Xmax' where Xmax is the maximum susceptibility.If a is increased further to a second critical value, the situation pictured in Fig. 4.23is obtained. If one starts with a demagnetized sample, at a certain critical field ajump occurs to the major loop after which it is not possible to demagnetize thesample by any sequence of applied fields. The behavior in this case outwardlyappears to have simple hysteresis.

4.8.2 Proof of The Deletion Property

According to the deletion property, the final state of magnetization is the same ifa local maximum and its subsequent local minimum are deleted whenever they arefollowed by a larger local maximum. This sequence results in a shorter sequenceand guarantees that all minor loops close. The same is true if the roles of maximaand minima are interchanged. This deletion is illustrated in Fig. 4.24 where themaximum labeled a and the subsequent minimum labeled b may be deleted fromthe sequence of extrema that define the magnetic state of the system. The proof ofthis is based on the fact that the magnetic state at point a' is the same as at point a.

The magnetic state is completely defined by the boundary line, shown in Fig.4.20, dividing the Preisach plane into the region where Q is -1 from the regionwhere Q is +1. To show that the moving model has the deletion property, it isnecessary only to show that the same boundary configuration is attained when aminor extremum is encountered and the same applied field is returned to. That is,

SECTION 4.8 DELETION PROPERTY

...~~. :'

,"'"IIII

M4 F(H+a.M)

II

II"I .:'F(H)

'\'/4·····

H

105

Figure 4.23 Similarmagnetizing process as in Fig.4.20t but with largera.

the minor loop in going from H, to He and back to H, is a closed loop, as shown inFig. 4.25.

A rigorous mathematical proof of this property is beyond our scope. Weinstead give a heuristic proof based on the properties of the Everett integral shownin Fig. 4.26. The Everett integral E(r,s) is a monotonically increasing function ofs that saturates if s is large. It is also zero when s is equal to r and has a slope,11 =aElas, that is zero at that point. Furthermore, it is an odd function with aninterchange of its arguments, so that E(r,s) is equal to -E(s, r). Starting from a givenapplied field, Ho, with a corresponding magnetization, Mo, to find the change inmagnetization tiM, it is necessary to find the solution to

b

Time

Figure 4.24 A sequence of applied fields in which extrema a and b aredeleted by maximum c.

106 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

M

MIII H

Figure 4.25 Minor loop predictedby the movingmodel.

~M = E(r,r+~H+a~M), (4.59)

wherer =Ho+ aMoand 4H is the changein appliedfield fromHo. The solutioncan befound graphically by locating the intersection of the Everettintegral curveand the straightline intersecting thes-r axisat I1H withslope l/a. The solutionisunique as long as ex is less than l/11mu. When a is greater than 1/11mu' then forcertainfields therecan be threepossiblesolutions; however, only the lowestoneis physically realizable. In that case, there may be a discontinuity in themagnetization when the applied field is increased to the point that only a singlesolutionexistsagain. Thisis illustrated in Fig.4.27,whichshowshowthe movingmodel transferfunction is constructed fromthe Everettintegral.

The change in magnetization in going from Hb to He is given byE(Hb+ exMb, H e+ exMe). Similarly, in goingfromHe to Hb, thechangein magnetiza­tion is given by E(He+ «Me' Hb .+ aMb,) . Since the properties of the Everett integral

E(r,s)

s

Figure 4.26 Everettintegralas a functionof the differencebetween its arguments.

SECTION 4.9 CONCLUSIONS 107

MMoving modeltransfer function

H

Figure 4.27 Construction of the moving model transfer function from the Everett integral.

lead to a unique solution, we must have M; =Mb" and thus, the minor loop isclosed. Even if a reversible componentof magnetization is added to the irreversiblecomponent computed by the Everett integral, the proof holds provided thereversible component is a function of the applied field and the irreversiblemagnetization only.

A direct consequence of the deletion property is that a process having thisproperty cannot have accommodation, since returning to the same applied fieldmust produce the same final state. Thus, to be able to reproduce accommodation,a further modification of the model must be made. Elsewhere [21] we havesuggested such a modification. The next chapter shows that accommodation modelsdo not have the deletion property.

4.9 CONCLUSIONS

We now summarize the results of the last three chapters. Four models have beenpresented for the irreversible magnetization: the classical Preisach model, themoving model, the product model, and the variable-variance model. In addition, wepresented three models for the locally reversible magnetization: the state­independent model, the magnetization-dependent model, and the state-dependentmodel. Each of these models has its own characteristic, and we may take anyirreversible magnetization model and add it to any locally reversible magnetizationmodel and obtain a new model. These models can be used to describe any materialwith varying degrees of accuracy. If it is not important to characterize all the effectsthat the more accurate models were devised to do, choose the model that is

108 CHAPTER 4 THE MOVING MODEL AND THE PRODUCT MODEL

sufficiently accurate for the desired application but also is most efficientcomputationally.

The concept of an operative field permits one to use the formulation of theclassical Preisach model with either the moving model or the variable-variancemodel. This in effect distorts the field axis so that irreversible susceptibility is nolonger symmetrical about its peak, ~. Furthermore, the peak no longer occurs atthe remanent coercivity, Hso but to the left of it by the amount aM( ~). Since theirreversible component of the magnetization is zero at ~, the total magnetizationM(~) is due purely to the locally reversible magnetization. Thus,

(4.60)

Furthermore, if we are using state-dependent reversible magnetization, Mr(~) isnot uniquely defined unless one knows the history of the magnetizing process.

Although these models affect different portions of the magnetizing curve, andsome of them remove the congruency property, they all possess the deletionproperty. In Chapters 2-4, we have concentrated on hysterons that are uniquely setby the applied field, ignoring the hysterons that are not supposed to be affected byit. In the next chapter, when we examine the behavior of the latter hysterons, wewill find that their effect is to cause minor loops to drift. This in turn serves toremove the deletion property from the resulting model.

REFERENCES

[1] E. Della Torre and F. Vajda, "Effect of apparent reversibility on parameterestimation," IEEE Trans. Magn., MAG·33, March 1997, pp. 1085-1092.

[2] E. Della Torre, "Effect of interaction on the magnetization of single domainparticles," IEEETrans. AudioElectroacoust., AE·14, June 1966, pp. 86-93.

[3] E. Della Torre, "Measurements of interaction in an assembly ofgamma-ironoxide particles," J. Appl. Phys., 36, February 1965, pp. 518-522.

[4] E. Della Torre, F. Vajda, M. Pardavi-Horvath, and C. J. Lodder,"Application of the variable variance model to Co-Cr perpendicularrecording media," J. Magn. Soc. Japan, 18, suppl. SI, 1994, pp. 117-120.

[5] E. Della Torre, J. Oti, and G. Kadar, "Preisach modeling and reversiblemagnetization," IEEE Trans. Magn, MAG·26, November 1990, pp.3052-3058.

[6] F. Vajda and E. Della Torre, "Characteristics of magnetic media models,"IEEE Trans. Magn., MAG-28, September 1992, pp. 2611-2613.

[7] F. Vajda and E. Della Torre, "Measurements of output-dependent Preisachfunction," IEEETrans. Magn.,MAG-27, November 1991, pp. 4757--4762.

[8] E. Della Torre and F. Vajda, "Parameter identification of the complete­moving hysteresis model using major loop data," IEEE Trans Magn., MA G­30, November 1994, pp. 4987-5000.

[9) E. Della Torre and F. Vajda, "Computation and measurement of the

REFERENCES 109

interaction field distribution in recording media," J. Appl. Phys., 81(8), April1997,pp.3815-3817.

[10] M. Pardavi-Horvath and G. Vertesy, "Measurement of switching propertiesof a regular 2-D array ofPreisach particles," IEEE Trans. Magn., MAG·30,January 1994, pp. 124-127.

[11] F. Vajda, E. Della Torre, M. Pardavi-Horvath and G. Vertesy, "A variablevariance Preisach model," IEEE Trans. Magn., MAG·29, November 1993,pp. 3793-3795.

[12] G. J. Gerritsma, M. T. H. C. W. Starn, J. C. Lodder, and Th. J. A. Popma,"Initial slope of the hysteresis curve," J. Phys. Colloq, C8, S12, 49,December 1988, pp. 1997-1998.

[13] O. Benda, "To the question of the reversible processes in the Preisachmodel," Electrotech. Cas., 42, 1991, pp. 186-191.

[14] G. Kadar, "On the Preisach function of ferromagnetic hysteresis," J. Appl.Phys., 61,1987,4013-4015.

[15] V. Basso, M. Lo Bue, and G. Bertotti, "Interpretation of hysteresis curvesand Henkel plots by the Preisach model," J. Appl. Phys., 75(10), May 1994,pp. 5677-5682.

[16] F. Vajda and E. Della Torre, "Minor loops in magnetization-dependentPreisach models," IEEE Trans. Magn., MAG-2S, March 1992, pp.1245-1248.

[17] F. Vajda and E. Della Torre, "Measurements of output-dependent Preisachfunction (Invited)," IEEE Trans. Magn., MAG-27, November 1991, pp.4757--4762.

[18] E. Della Torre, "Existence of magnetization-dependent Preisach models,"IEEE Trans. Magn., MAG·27, July 1991, pp. 3697-3699.

[19] G. Kadar and E. Della Torre, "Hysteresis Modeling I: Noncongruency,"IEEE Trans. Magn., MAG·23, September 1987, pp. 2820-2822.

[20] M. Brokate and E. Della Torre, "The wiping-out property of the movingmodel," IEEE Trans. Magn., MAG·27, September 1991, pp. 3811-3814.

[21] E. Della Torre and G. Kadar, "Hysteresis Modeling II: Accommodation,"IEEE Trans. Magn., MAG-23, September 1987, pp. 2823-2825.

CHAPTER5

AFTEREFFECT ANDACCOMMODATION

5.1 INTRODUCTION

This chapter treats two further corrections to Preisach modeling: aftereffect andaccommodation. Due to these effects minor loops do not in general close onthemselves, so both corrections remove the deletion propertyofthe Preisach model.They do this in different ways: one is time dependent and the other is rateindependent. Both usually involve small drifts of magnetization with time, so theyare easily confused with each other in many cases.

Aftereffect changes the magnetization as a function of time and is mainly dueto thermal effects. A magnetization state is relatively stable if it is surrounded byan energy barrier that is sufficiently high; however, no matter how high that barrieris, the magnetization will eventually revert to the ground state. The higher thebarrier, the longer before reversion to the ground state is completed. In the nextsection, when we discuss the relationship between the height of the barrier and thelength of time needed to revert to the ground state, we will see that changing thephysical size of the hysteron can change that time from a few minutes to manycenturies.

Accommodation, on the other hand, is rate independent and is a direct resultof the hysteretic many-body interpretation of the Preisach model. The drift inmagnetization occurs only when the magnetization is cycled, and this drift is afunction not of time but of the number ofcycles that have elapsed. If one cycles themagnetization at a constant rate, the drift will appear to be a function of time. Both

111

112 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

effects are interpreted here in terms of the Preisach model. The resultingmodifications of the model generally agree with observations.

5.2 AFTEREFFECT

When a magnetic material is subject to a step function in the applied field, itsmagnetization will change very quickly to a new value and then slowly drift to afinal value. The time constant associated with the first change in magnetization isof the order of nanoseconds, while the second is of the order of seconds. The firstchange can be computed with the models already discussed, but the latter must becomputed differently and is the subject of this section. Diffusion aftereffect andthermal aftereffect, the main types identified thus far, are similar in behavior,although they have quite different causes. A history of the research in this area isgiven by Chikazumi [1].

A mechanism for diffusion aftereffect was first proposed by Snook [2]. Itinvolved the diffusion ofcarbon atoms in a-iron as the magnetization rotated. Sincethe carbon atoms occupy interstitial sites in the body-centered cubic that elongatethe lattice, they reduce the magnetocrystalline anisotropy in that direction. Thus,when the magnetization is rotated, if the carbon atoms diffuse to a new position,they can lower the energy of the crystal. When a field is applied, the magnetizationrotates quickly to the new position, but the diffusion is much more gradual, and theenergy approaches the equilibrium value asymptotically. The time constantassociated with the process is

l' = l' e WlkTo ' (5.1)

where W is the barrier energy, and 1'0is an appropriate constant whose dimensionis time. This equation is referred to as the Arrhenius law. Experiments by Tornono[3] have shown that the logarithm of T varies linearly with lIT. The slope that hemeasured for this variation corresponded to a value for W of 0.99 eV for thisprocess.

Thermal aftereffect, on the other hand, involves the reversal of themagnetization of hysterons not the diffusion of atoms. This type of aftereffect,discovered by Preisach [4], is sometimes referred to as magnetic viscosity or astrainage. When a field is applied, all hysterons that have critical fields less than theapplied field will switch very quickly; however, the remaining hysterons that havecritical fields larger than the applied field would not switch at all if the temperaturewere absolute zero. At finite temperatures, this energy barrier can be overcomethermally. Since different hysterons have different barrierenergies, they will switchat different rates. Thus, the aftereffect does not decay exponentially.

Let us assume that the rate of switching is given by (5.1), where W is now theenergy barrier that must be overcome to reverse the magnetization of a hysteron.Then when a step change in the applied field occurs, the aftereffect magnetization,that is, the magnetization after the step change, is given by

SECTION 5.2 AFTEREFFECT

where

m(t) = m(O) + f(t),

113

(5.2)

(5.3)

In (5.3),m(O)is the magnetization just after thestepchange,dm is the totalchangein magnetization due to aftereffect, and Pt(r) is the normalized probability that ahysteronwillswitchwithtimeconstantor. Sinceall magnetizations are normalized,the maximum remanence is unity.

The properchoiceof Pt(r) determines the behaviorof the aftereffect. Severaldistributions have been suggested for it. Chikazumi [1] has suggested a 1/ordependence between t. and t 2, while Aharoni [5] has suggested the r functiondependence, alsowithtwoadjustableparameters,p andto. Neitherdistributionhasanyphysical basisnoranypredictive power.ThePreisach-Arrheniusmodel, on theother hand, links the phenomenon to hysteresis, suggests a distribution with onlyone adjustableparameter, 'to, andcan describethe variation of the aftereffectwiththe applied field.

Korman andMayergoyz [6]and Bertotti[7]suggestedthat the dependenceofthe aftereffecton magnetization historycouldbe describedby the Preisachmodel.The following extensionof their work was recentlyproposed [8]. If aftereffect isto be described in terms of the Preisach model, it is preferable to express theprobability in termsof switching fields.To do this, let us consider the applicationof an operative field h to a material that has been saturated in the negativedirection. For clarity, we will hold h constant throughout this process. If we areusing the movingmodel, then since h dependson the magnetization, the appliedfield would have to be adjusted to keep it constant throughout the process;however, for hard materials, the decay rate is usuallyso small that any change inmagnetization may be neglected for reasonable periodsof time. For the classicalmodel,then,a is zero,andnoadjustment in thefieldis necessary. Hysterons whoseswitching fields are less than h will instantaneously be switched to positivemagnetization, while the remaining ones will remain switched negatively, sincethey are protectedby an energy barrier from switching immediately. If h is largeenough,thermal energywillovercome thisbarrierand the material willeventuallybe saturated. We will discuss what is "large enough" in the next section inconnectionwithmoregeneralmagnetizing processes. The valuesof mGQ and11m forthis process then are

m(O) = r:du p(u) and am = 2fh

oodu p(u), (5.4)

wherep(u) is given by

fOO dm.

p(u) = p(u,v)dv =-' .dh

(5.5)

114 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

(5.6)

(5.7)

The factor of 2 in am comes from the fact the Q in that regionchanges from-1 to+1. We extend the upper limit to infinity in the v integration, since the Preisachfunction is zero for v greater than u. We note that the integrationswould have tobe carriedout in the operativeplane in the case of the movingmodel.Furthermore,if the materialis not square loop, an appropriatereversiblecomponentwouldhaveto be added. The considerationof these effects is beyond the scope of this book.

If we assume that the Preisach function is Gaussian, then

1 [ (u-ii )2]p(u) = ----- exp k,

o{fi 202

where ~ is the average value of the critical field. Note that in the case of single­quadrant media, "" is equal to the remanentcoercivity. It follows that

(h -ii )

m(O) = erf -7 ·and

(h) (h-iik)11m; = erf OJ - erf -0- · (5.8)

In this case, the medium will eventually become saturated as all the hysteronsovercomethe energybarrier.Figure5.1 plots of am;, the change in magnetizationduring the relaxationprocess, for various valuesof 0, when o, is zero and hk is 1.It is seen that the field that maximizes am; is half hk, since this is the difference oftwo error functions, one centered at h, and the other centered at zero. Since themaximumchange in magnetization is limitedto 2, the curve saturatesat that valuefor small 0. The curve is symmetrical with respect to the peak only in this case,since 0; is equal to o when o, is zero. Since 0; is alwaysless than or equal to 0, theslopeat the origin is usuallysteeper thanat hk, and the peakof thiscurve willoccurat a value less than 1/2.

If we neglect the change in the energy stored in the reversiblecomponentofthe magnetization, the energyrequiredto switcha hysteronin a process describedby the Preisach model is given by

W = J,loMV(u -h), (5.9)

where V is the averageactivationvolumeof the hysteron. ThusMV is the magneticmomentof the activationvolumeof the hysteron,h is the operativefield, and u - his the additional field required to switch the hysteron.A micromagnetic study ofrecordingmediashowedthat it is necessaryto switchonlya fractionof the volumeof a hysteron to cause it to reverse [9]. Observations of recording materials [10]have shown that this can range from valuesas small 0.2 of the hysteron's volumeto the entire volume. The lattervalueis validfor verysmallparticles.Thus, V is theminimum volume that has to be switched to nucleate a reversal, and MV is the

SECTION 5.2 AFTEREFFECT 115

olh.

0.2

-----. 0.4............ 0.6

-._._. 0.8

0.8 1.2 1.6 2Operative holding field

0.4OL...-----A----L-----'-~----..-.-~

o

2r----....--".----,r-----r-----------,

81.5 ....-+--I-----.t~'"_:_____+._--t----t

·a.Ju

t.su

X0.5t------+----+-------\--"I-~--+-----i

Figure 5.1 Variation in the total changein magnetization, am;, withnormalized holdingfield,hi It.,for relaxation to the ground state.

(5.10)

minimum moment thatmustbeswitched toreverse theentirehysteron. Thenusing(5.1), this hysteron would have a time constant given by

JllaMV(u - h)]r= '0 ex, kT for ic-h,

or

u = hi IO( ;01 + h for r > '0' (5.11)

where

(5.12)

The parameter hfis referred to as thefluctuationfield, and has the units of magneticfield. It is equal to the field required to make the hysteron's energy barrier equal tothe thermal energy. If this factor is large compared to the switching field, thehysteron will be superparamagnetic. In the study of aftereffect, we are interestedonly in small values of hI' For useful recording media hI is small compared to theswitching field of the hysteron, and therefore, its magnetization is retained for longperiods of time.

We note that

116 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

Then, using (5.6) and (5.11), p[u( r)] is given by

1 { [h - hk + hI In(1:'Ito)]2}p[u(t)] =-- exp .o~ 202

Thus,

(5.13)

(5.14)

(5.15)

(5.16)

It is noted that the lower limit is changed to to because of the limitation imposedby (5.11). Note thatj{O) is approximately one as t approaches zero. If we changethe variable of integration to y =In (t/to), we obtain

hfHL00 { te -Y [h -hk +hf yJ2}j(t) = - - dy exp .a 1t 0 to 202

Using (5.2), we see that the magnetization as a function of time is given by

~mihHLoo {te-Y [h-h +h Y]2}met) = 1 - __I - dy exp -_ k f . (5.17)o 1t 0 to 20 2

This shows that the amount of aftereffect is a function of the applied field.To illustrate the time dependence of (5.17), this expression was integrated

numerically and plotted on a semi-log plot in Fig. 5.2 for two values of hp Theparameter used in the plot, which is reasonable for a recording medium with fairlylarge hysterons, was a =0.6. The value of to used in this simulation was 0.1. Afield equal to the coercivity is applied so that the initial magnetization is zero. Sincethe hysterons that are positively magnetized will remain magnetized because of thisfield, and since the hysterons that are negatively magnetized will eventually alsobecome magnetized, the magnetization will approach saturation.

It is seen that for times somewhat greater than to, the magnetization increaseslinearly with the logarithm of the time. The effect of hf is to change the slope of thelinear portion of the aftereffect on the log-time curve. This linearity can continuefor many decades, as seen from the curve when hll~ is equal to 0.007; however,when the magnetization approaches about half its final value, the curve starts todeviate from the straight line, as seen from the curve when hI I h7c is equal to 0.07.It is characteristic of this process that a small change in hIcan cause a large change

SECTION 5.2 AFTEREFFECT

1 ..---_.

117

(5.18)

(5.19)

",.-

~./"

'-" /E /

S 0.8 /.,d /

.~ /u 0.6 / h,t /

/ 0.007~

0.4 / --- 0.07

~ /~ /0

0.2 /Z //

/00.01 100 106 1010

Time (units of\)

Figure 5.2 Aftereffect as a function of log time for twovaluesof hI"

in the behavior of the aftereffect. These results have been studied for a wide rangeof materials and generally agree with these conclusions [11].

It is noted that as h is increased from zero, the total range of the aftereffectdecreases until when it saturates the medium, the range of the aftereffect is zero.The second effect of h is to change the slope, S, of the aftereffect on the log timecurve in the linear region. To evaluate the slope we first differentiate (5.17) withrespect to time:

dm dm;h/HLoo

[ te-Y (h-hk-h/ y)2]- = -- - exp-y dy.dt toO 'It 0 to 202

We now define the coefficient of magnetic viscosity, S, as discussed byWohlfarth [12]:

s = dM(t) = SM dm(t).d logt s d logt

This is the rate of decay of the magnetization on a logarithmic scale. It has been sodefined because many materials appear to decay linearly on such a scale. We willsee that for "permanent magnet" materials this is the case over a range of times thatare accessible to experimenters. However, when t is very small or very large, logt diverges and the decay is no longer linear.

Since

d

dlogt

d= t-

dt' (5.20)

118 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

If hI is small enough to permit us to neglect the term h, y, this reduces to

S _ 4mihf texp [-(h - hk)2/202]~L00 [ tYr- - - exp -y-- y. (5.22)

SMs TOO 1t 0 To

If we substitute u for e -Y, then this becomes

- 2 2~S hI texp [-(h - h,e> /20] 2 ii_tufT- e °du

SMs too 1t 0(5.23)

hi texp [ -(h - {,kiI2(J2] 12 (1 _e -Iul,o I~.'00 ~1t

Thus, if t is much larger than To, this reduces to

~ = Amjhlexp [ -(h - {,/1202

] r2. (5.24)

SMs (J ~ 1t

It is seen that the slope is independent of both t and To. Furthermore, it isproportional to a Gaussian whose maximum occurs when the applied field is equalto Ii;. and whose standard deviation is o, The maximum slope at h equal to ~ is0.7979 hl/o.

The decay coefficient would have a maximum for h =~ were it not for thevariation in ~m,. Since am, is a decreasing function of h, the location of the peakin S must be located at a value smaller than ~. The amount of decrease in thelocation of the peak depends on the slope of 4m; versus h, which is roughlyinversely proportional to o. This variation in decay coefficient with holding fieldhas a maximum that is inversely proportional to 0, as illustrated in Fig. 5.3 for fourdifferent values of a.

It is noted that at HRC the irreversible component of the magnetization is zero,and thus the peak occurs at HRC - aM,(HRC) ' where Mr(HRC) is the locally reversible(and only) component of the magnetization. It has been shown that a is a positivenumber less than one; thus, ~ is less than HRC' There is a further correction, asdiscussed in Section 3.7, if the material is a three-quadrant material, that is, if 0;

is not negligible compared to the coercivity.The irreversible susceptibility Xi can be computed by substituting (5.6) into

(5.5). Thus,

0.1

--·0.2......... 0.3

_._ .. 0.4

SECTION 5.2 AFTEREFFECT

1.5.----------r_r-----.,r------..,

'0'~ 1.06....Co),Co)

D' 0.5 ~---..-.-.------#-.f__ll-+----....,I-------tu~

0.5 1 1.5 2Operative holding field

Figure 5.3 Variationin decaycoefficientwith holdingfield for variouscritical field distributionsfor negligibleh,.

119

(5.25)

Then (5.22) can be written

S = 2b.m;hfSMsX;. (5.26)

This result is comparable to that obtained by Streetand Woolley [13].Figure5.4 is a plot of (5.21)for 0 =0.6, hll~ = 0.006, and 'to = 1. It is seen

thatfor t less than0.1'to,theslopeisessentially zero.It thenbeginsto rise,reachingabout 64% of its maximum valueat to. By IOto, it is essentiallyat its maximum,and then is essentially constant for many decades. In particular at I08t o itsmagnitude has decreased less than2.5% fromthe peak.If hfliik weredecreased to0.0006, then the decrease wouldbe less than 0.0125%.

The model accurately predictsthataftereffect isessentially linearas afunctionof the logarithm of time. Furthermore, theslopeof thiscurveis a maximum aroundthecoercivity. It assumes thathysterons in a fieldlargeenoughto switchthemwillremain switched, but hysterons that can be in either state will on average bedemagnetized. Theresultspresentedwerebasedontheclassical Preisachmodelforsimplicity, but the corrections for motion and state-dependent reversiblemagnetization mustbe madeif high accuracy is desired. The apparentreversiblemagnetization of three-quadrant mediawillaffectthefieldthatmaximizes theslopeof the aftereffecton the log time curve. Other effects, discussed in the comingsections, also affect these results.

120 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

0.01

~\

if \ hf

·13 \ 0.007IS \

-_. 0.07§ 0.005

~ \~ \

-,<,

<,0 '""""--- .....0.01 100 106 10 10

Time (units of~o)

Figure 5.4 Slopeof the aftereffect on a log timescale.(Notethe decaycoefficientis multiplied by10forhl = .007fordirect comparison.)

This model has three parameters: 0, hI' and 'to. The first is the same standarddeviation of the switching field distribution of the Preisach model and can bedetermined in the same way.The second, the fluctuation field, is defined in termsof physical quantitiesand can be measured,since the modelpredicts that the slopeof the log-timevariationis 1.253olh; The lastparameter,whichis analogousto themean free time between collisions in a paramagnetic gas, can be obtained byseveral methods. One finds that a small change in hi will cause a small change inthe S; since this slope is very small, however,and since the scale is logarithmic,asmallchangecan change the timebymanyordersof magnitude. Thus, doublingthesize of the hysteron will cut hI in half, but maychange the time from the order ofminutes to manycenturies.

5.3 PREISACH INTERPRETATION OF AFTEREFFECT

Since aftereffect can be explainedin termsof thePreisachmodel,we will now usethe Preisach model to calculate aftereffect, and address the question of how torelax the restriction before (5.4) so that will h be "large enough." The precedinganalysisneglects theeffect of the down-switchingfield becauseif the appliedfieldwere large enough, its effect would be negligible. However, the omission leads tothe wrongconclusionabout the groundstate magnetization. The first thing that wenotice is that the aftereffect is time dependent, so a static interpretation of thePreisach diagram will not suffice. Therefore, to include time dependence, we willlet the state variableQbe a functionof time. We will thinkof Q at any point on the

SECTION 5.3 PREISACH INTERPRETATION OF AFTEREFFECT 121

(5.28)

(5.29)

Preisachplane as consisting of a fraction q+ hysterons in the positivestate and afraction 1 - q+ hysterons in the negative state.Then

Q(t) = 2q+(t) - 1. (5.27)

To derive the equations for the magnetization state of each point in thePreisach plane, let us start from negative saturation, as in Section 5.2; thenq+(O) =0, and Q=-1, for all points in the plane. When we applya positivefield,as illustrated in Fig. 5.5, the Preisachplaneis dividedinto threeregions. RegionIhysterons will be switched to the positivestateand regionIII hysterons willbe inthe negative state.Hysterons in regionII couldbe ineitherstate,butstartout in thenegative state.For example, the hysterons in a smallregionabouta point (u,v), asshown,requirea fieldu - h to switchthemto thepositivedirection. Thus,to switchintothepositivedirection theymustovercome anenergybarrierJloMV(u - h).Thentheir magnetization will varyexponentially witha timeconstant

(U -h)r , = "oexp -;;; ·

Similarly, any positive hysterons have to overcome an energybarrierJloMV(h-v)and will do so with a timeconstant

(h-V)"_ = "oexp -,;; ·

Figure 5.S Preisach interpretation of aftereffect.

122 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

The state variable then must obey the following differential equation:

dq+ 1-q+ s.(5.30)

(5.31)

and we can use (5.27), to convert the differential equation for the magnetizationstate at each point in the Preisach plane to

dQ ( 't + + t _) _ r + - r _-+Q -- ---.dt t +t' _ l' +l' _

The initial condition for this differential equation is Q=-1. When the applied fieldis not constant, the rs are functions of position; otherwise this is a first-orderdifferential equation with constant coefficients.

For a constant applied field, it is seen that the steady-state solution of (5.31)is

(5.32)

The time constant to reach this solution is different for each point on the plane. Forexample, from (5.31) at a particular point it is given by (1'+ - r, )/1'+'t., where 1'+ andr. are given by (5.28) and (5.29), respectively. Furthermore, points for whichu - h =h - v have a steady-state value of zero. Starting from a state where Q isdiscontinuous about the magnetization history staircase, as time progresses, thestate becomes continuous over the Preisach plane. In particular, as t becomes large,the magnetization is asymptotic to

mjh) = fftanh( h~~h ) p(u,v)du dv

u>v

foo ( ) [ 2]1 h.-h h.=-- tanh -'- exp -~ dh.,

a{ii hI 20;-00

(5.33)

This function varies between +1 and -1 as h varies from minus infinity to plusinfinity. There are two limiting cases: when hI goes to zero, this approaches theerror function; and when a goes to zero, this approaches the hyperbolic tangent of(h - hi) / hI . In the limit of hI going to zero, Q approaches the sign function at hi=h. This is the same result as obtained by the Korman-Mayergoyz model [6].

Aftereffect can be described by the Preisach model; however, when this isdone, the process is no longer rate independent. The technique for includingaftereffect in the Preisach model is to make the magnetization state a function oftime. In this case, the magnetization state in any region of the Preisach plane is no

SECTION 5.4 AFTEREFFECT DEPENDENCE ON MAGNETIZATION HISTORY 123

longer uniform. In the next section, we will describe accommodation by a similartechniqueand thendiscusshowto computebotheffects whenthe movingconstantis not zero and when a state-dependent magnetization is added. We will see in alater section that both aftereffectand accommodation modifythe deletion propertyof the model.

5.4 AFTEREFFECT DEPENDENCE ON MAGNETIZATION HISTORY

Whena constantfield is appliedto a materialaftera complexmagnetizingprocess,the state of each point in the Preisach plane obeys (5.31), which then may berewrittenas

where

and

1:dQ

+ Q = Qdt -'

't=---1:+ +1:_ 2cosh[(hl - h)/h)

(5.34)

(5.35)

(5.36)

It is noted that Q, Qoo, and t are all functions of u and v. The solution to thisdifferential equation is given by

Q = (Qo - Q.)e-tlt+Q.. (5.37)

Thus, each point in the Preisachplane must approacha differentequilibrium,andeach point approaches that equilibriumat a different rate.

To illustratethiseffect,let usconsiderthefirst-orderreversalprocessthatstartsat a large positive field and then goes to a field HI and finally to a field H2, asillustrated in Fig. 5.6. The dashed line indicates the anhysteretic limit of amagnetizing processascomputedby(5.33).Whentheappliedfieldattainsthe valueHI' the resultingnormalizedmagnetization is m.. As the field is changed to H2, themagnetization follows the minor loop to m2, and finally the aftereffect causes themagnetization todrift to m3• Forexample,ifHI weretheremanencecoercivity,HRC'

ml would be zero. Furthermore, if H2 were zero, m3 would be zero. If the materialwere a single quadrant medium, then m2 would also be zero; however, themagnetization will not remain at zero as the applied field is set at zero, sincedifferent points in the Preisach plane relax at different velocities. Thus, at theinstant the applied field is reduced to zero, the magnetization will indeed be zero,but even though the field is maintained at zero, the magnetization will become

124 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

oAppliedfield

-1

Iio~~--

Figure 5.6 Study of the aftereffect in a first-order reversal process.

positive and then relax back to zero.The descriptionof this process by this model is illustratedin Fig. 5.7. When

HRC is applied,the regionabovethehorizontal lineat HRC willbecomemagnetizednegatively, whilethe rest of the Preisachplane will remainmagnetized positively.After the applied field is reduced to zero, the vertically hatched region willessentiallyrevert to positivemagnetization, while the horizontally hatchedregionwill essentially become magnetized negatively. Since hysterons in the verticallyhatched region have a smallerenergybarrier, they will change morequickly thanhysteronsin thehorizontally hatchedregion.Thus, themagnetization willfirstdriftin a positive direction and then eventuallyrelax back to zero. For materials thathave a smallfluctuation field, and thus wouldmakegood recordingmaterials, thepeakof thedrift wouldoccuraftera verylongtime,and measurements haveshownthat even after days, the magnetization will continue to drift upward.

HRC

Figure S.7 Preisach planeexplanation of first-order aftereffect process.

SECTION 5.5 ACCOMMODATION

3.25r-----.-----r----r----r---~-~-,.______,

2.75L-----I---..a.----"----L----'-----'---~--l5 7 9 11 13

Naturallogarithm of time (seconds)

Figure 5.8 Plotof the aftereffect due to a first-order reversal process.

125

This process can be accelerated by not reducing the field to zero. Then themagnetization can relax to a different value but still change direction in the process.As an illustration, Fig. 5.8 shows the behavior on a log-time scale of themagnetization after a first-order reversal process in which the material started frompositive saturation and then is subject to a field of -1600 Oe, which wasimmediately increased to -1000 Oe, and maintained at that value throughout theremainder of the measurement. The material was assumed to have a coercivity of1140 Oe and a 0 of 970 Oe, which is typical of recording media. The values usedfor to and hI were 10-11 and 14.5 Oe, respectively. This type of behavior wasobserved in spring magnets by LoBue et al. [14]. Further discussion andexperimental verification of these effects can be found elsewhere [15].

5.5 ACCOMMODATION

We now turn to a further statistical modification to the Preisach model to includeaccommodation; we will discuss the properties of such a model and theidentification ofits parameters. When minor hysteresis loops in magnetic media arecycled between two fields, they gradually drift toward an equilibrium loop, asshown in Fig. 5.9. This phenomenon, known as accommodation, requires a changein the applied field for the drift to occur. It is to be distinguished from aftereffect,in which drift takes place even when the applied field is held constant.Accommodation cannot be described by any of the pure Preisach models presentedthus far, since they possess the deletion property[16]. In contrast to purely pheno­menological attempts to explain this effect [17-19], we will describe a statisticalinterpretation of Preisach models that is not limited by the deletion property. Thusthe model naturally exhibits accommodation.

126 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

H

Figure 5.9 Dlustration of an accommodating minor loop.

Although a hysteron has a symmetrical hysteresis loop as a function of its localfield, when this loop is observed as a function of an applied field, it appears to beasymmetrical. The local field is the sum of the external applied field, whichincludes any long-range demagnetizing effects, and the interaction field, which isthe source of the shift in the hysteron's loop. We will assume that this interactionfield is Gaussian, that its standard deviation is constant, and that its mean value isproportional to the local magnetization. We will describe the Preisach functions inthe operative plane, so that the distribution is statistically stable for mostlongitudinal and thick perpendicular media, which means that despite the motionof all the hysterons, the net population density at all points in the operative planeis constant.

We will now compute the irreversible magnetization component by a Preisachtype model by neglecting aftereffect and using

Mj=SMs f f Q(u,v)p(u,v)dudv. (5.38)u>v

For nonaccommodating scalar media, the state variable is +1 in the region that ispositively magnetized and -1 in the region that is negatively magnetized. When anincreasing field h is applied to a magnetic material, the operative plane may bedivided into three regions, as shown in Fig. 5.10. The boundary between region R1

and R2 is a vertical line that intersects the u axis at h.The boundary between R.andR3 is the customary staircase that contains the relevant history of the magnetizingprocess. Region R. is magnetized in the positive direction by the applied field.Although R, is normally positively magnetized and R3 is normally negativelymagnetized, since any hysteron in these regions has a critical field greater than theapplied field, any hysteron that moves into these regions can maintain its originalmagnetization.

When a given hysteron moves in the plane as a result of a change in its localfield, it takes its magnetization with it. If in the new location it experiences a fieldlarge enough to change it, it will reverse its magnetization; however, if in the newlocation the hysteron experiences a field smaller than its critical field, it may notconform to the magnetization of the hysterons in that region. Thus, the magneti-

SECTION 5.5 ACCOMMODATION

______.... u

127

Figure 5.10 Division of the Preisach plane into three regions by an applied field.

zation of the hysteron is determined both by the region it came from and bywhetherin its newregionit experiences a fieldlargeenoughto switchit. Table5.1summarizes the effect of this motion in the operativePreisachplane.The columnlabeled"State" showsthe signof the magnetization of a hysteron thatmovedfromtheregionlabeled"InitialLocation" to theregionlabeled"FinalLocation."On theother hand, the columnlabeled"Other Models"showsthe sign the magnetizationwould have in the nonaccommodating interpretation of the Preisach model. Forexample,it is seenthat if a hysteronoriginally in R1 hadendedin R3, it wouldhavethe "wrong" valueof magnetization.

When a hysteron has the "wrong" value of magnetization, it will dilute thestrength of the magnetization component due to this region.The dilution will beaccounted for by changing the interpretation of the state variable Q(u,v) to theaverageof the statevariables in the region,as in the aftereffectmodel.The change

Table 5.1 Hysteron Motion in the Preisach Plane

Initial Location Final Location State

R, R, +

R, R, +

R, RJ +

R, R, +

R, R, +

R, Ra +

R3 R, +

R] R,

R3 R)

Other Models

+

+

+

+

+

+

128 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

(5.39)

(5.40)

in Q normally may not be the same for all locations of the same region; however,as a first approximation, we will change all values in a given region by the samefactor. Thus, in this model Q is uniform in a given region. In the followinganalysis, we will denote the magnetization associated with the region Rj by Mj , andthe associated state variable by QJ. We will also define the component of thePreisach function in Rj by

r, = f fp(u,v)dudv.Rj

Note that with this definition all the p/s are positive numbers less than one.Furthermore, with this definition, we see that they are normalized, so that

LPj=l,j

and we can compute the irreversible component of the magnetization by means of

Mj =SMs L QjPr (5.4l)j

We assume that whenever the magnetization changes, the hysterons move inthe Preisach plane and carry their magnetization state with them. The followinganalysis is restricted to longitudinal media whose distribution is both stable andGaussian in the operative plane. Furthermore, as stated earlier, we assume that themotion of a given hysteron is not a function of its original position in the plane. Ina special case not considered here, the motion of hysterons is restricted along linesfor which hie = (u + v)/2 is a constant. This is a reasonable approximation for thin,well-aligned films; however, more sophisticated models are possible.

The positions of all the hysterons in the Preisach plane change whenever themagnetization of even a single hysteron changes. The amount of motion of ahysteron depends on its proximity to the hysteron that switched. Since there is nomotion of hysterons in the Preisach plane unless at least one hysteron switches, itis reasonable to assume that the amount of motion is proportional to the change inmagnetization. Furthermore, when a hysteron moves to a new position in thePreisach plane, the probability that its magnetization is of a certain polarity is thesame as the fraction of all the hysterons in the plane that have that polarity.

Consider a magnetizing process consisting ofapplied field extrema, which willbe referred to as events. Let us call the fraction ofhysterons replaced in an elementof the Preisach plane by hysterons coming from other parts of the plane the positivereplacement factor ~, which is less than one. Therefore, at event n + 1 in amagnetizing process, the value of or: at pointj, in terms of the value of or,would be given by

(5.42)

where ~n) is the average Q(n) throughout the plane, and ~ lies between zero and one.Note that in this formulation, the magnitude of Q will always be less than one,

SECTION 5.5 ACCOMMODATION 129

since ~ is less than one. We expect this replacement factor to be proportional to thechange in magnetization, since there will be no replacement unless the state of thesystem changes. Thus,

l:=LIL\M.ICo:» SM "s

(5.43)

where the proportionality constant P is a dimensionless constant for a givenmedium. We note that pmust have a value that allows the magnitude of ~ to be lessthan one. Since the average value of Qat event n is given by

Q(n) =;:; , (5.44)s

we have

n laM.<n)l( M.(n) )dQ.(n) = Q.(n+l) _Q.(n) = _p__, ' __ Q.(n) •

) ) ) SMs

SMs

)(5.45)

This is the amount that Q changes in a given leg of the magnetizing process. In acontinuous process, this difference equation is replaced by the differential equation

dQ = P(M;-SMsQ) dM;

dH S2M2 dHS

(5.46)

This model, like all static Preisach hysteresis models, still is a time­independent process. We can, therefore, fully describe a magnetizing process bygiving only the values of successive extrema of the applied field. The part of theprocess between two successive extrema will be referred to as a leg of themagnetizing process. Since this model does not possess the deletion property, wemust consider all extrema, not only the ones that normally are undeleted.

For simplicity let us consider small hysteresis loops in a medium whosesquareness is unity. We will now consider the cycling of a material with anarbitrary magnetization history between two operative fields: hA and hB, wherehA > hB, and the difference between them is small. Then we can compute themagnetization changes by solving the differential equation (5.46) by Euler'smethod, with one step per leg. We will start the accommodation process from hA

letting the first leg of the process be the transition to hB• The values of the variousquantities during a given leg of the process will be denoted by a superscriptcontaining the leg number in parentheses. Thus, the value of Q in region j at thefirst application of hA will be denoted by Q/l).

When an applied field iterates between the operative fields hA and hB, theregion labeled R1 in Fig. 5.11 is entirely switched. In the classical Preisach modeland in the moving model, the height of a minor loop between these extremities will

130 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

Figure S.ll Description of minorloopbehavior in theoperative Preisach plane.

be equal to MI. This height is independentof M. Furthermore,all the minor loopsbetweenthe fields hA and hB are congruent in the operativeplane, and the loops arestable immediatelyafter a first applicationof hA•

On the other hand, if this model is applied to the process illustrated in Fig.5.11, when the field is hAt then Q1 and Q2will be+1.The valueof Q3' which wouldnormally be +1t in this model will be Jess than 1, since it is diluted by hysteronscoming in from R4and Rs. For similar reasons Q4 and QSt that would normallybe-1 t willhavea valuesomewhatgreater than 1.The differencebetweenthe handlingof QIand Q2 is that the former will oscillate between+1 and -1 and the latter willoscillate between 1and a valueonly somewhatless than 1, whenh =hB• It is alsonoted that only Qs will have a value of +1 at that field.

We will consider only the part of a magnetizingprocess that comes after asuitable history has created the desired staircase on the Preisach plane. The firstapplication of a field hA will be called thefirst iteration,and this iteration numberwillbe indexedafter eachsuccessivelegof the magnetizingprocess.Thus, the firsttime that h equals hAt each of the state variables Ql (1) and Q2(1) will be set equal to+1. The values of the other Q(1)'s have a magnitude less than 1 determined by themagnetizationhistory. It is noted that the value of Q3(I) starts out positive and thatthe values of both Q4(1) and Qs(1) start out negative.

Since taM/")1 is equal top.SMs, wecan use (5.41), to rewrite (5.45) as follows:

~Qt> = PPl[t Qt>Pk -Qt>j. (5.47)

We will use this equation to solve for the Q's at the conclusion of each leg of themagnetizing process. For even indices, the applied field is hB, and we set

SECTION 5.5 ACCOMMODATION

Q(2n) _ Q('2n) - -11 - 5 -

Qj(2n) _Qj(2n-l) = PPl(Pl +P2+Q;2n-l)P3 +Q:2n-l)P4 -PS-Q/2n-l»)

,

for j = 2, 3, and 4.

131

(5.48)

We note that in this calculation we always reset Q2(2n.l) equal to +1. For oddindexes, the appliedfield is hA, and the Q's are givenby

Ql('2n+l) = Qi2n +1) = 1

(5.49)

We note that in thiscalculation wealways reset Qs(2n) equal to -1. It is seen that ifP is zero, then Q/2n+l) =Q/2n) =Q/2n.l), for j =3 and 4, so there is noaccommodation.

We see that the differentregions havedifferentroles in the accommodationprocess.RegionR1 is actively switched as the minorloopsare traversed. ThusPIdrivestheaccommodation processbyforcingthehysterons tomovein thePreisachplane. In alternate half-cycles, regions RI and R, suffer a small amount ofaccommodation, but thenthemagnitude of Qis restoredtounity. Thehistoryof themagnetizing processiscontained inR3 andR4• Duringtheaccommodation process,this historygradually fadesaway. If P3 andP4 are zero,as in the case of the majorloop,thenthereis nohistoryto bedilutedand noaccommodation of theend pointsof minorloopscan takeplace,even if pis not zero.Finally, thereare someminorloops for whichno accommodation takesplace.For example, no accommodationcan takeplacewheno, is zero,sinceif (hA - hB)/2 is greaterthan Fi7c, thenP3 andP4are zero, but if (hA - hs)/2 is smallerthan Fi7c, thenPI is zero.

The equilibrium minorloop, that is, the loop that finally closeson itself, canbe computed by letting Qj('2n+2) =Qj(2n) =Q (even), forj =3 and 4. Thus,

2(P2-Pj) A ( )---PP -P +P -P

Q .(even) = PI +Pz+P5 1 1 2 5 (5.50)

J 2-PPl(Pl +P2+PS)

and by letting Q}2n+l) = Q/2n-l) = Q(odd) ,for j = 3 and 4, we have

(5.51)

The limiting magnetization is obtainedfrom(5.41),so thatat the upperend of thelimiting minor loop we have

132 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

where

(5.52)

(5.53)

while at the lowerend of the limitingminorloop we have

MB =SMs[-PI + QieveO)P2- Ps+ Q(even)(P3 +P4)] ,

where

(5.54)

(5.55)

The height of the minor loop, MA - Ms. will be slightlysmaller than 2SMsPlt thevalueit would haveif therewereno accommodation.

To illustrate the behavior of this model, let us consider the case Pi =0.2,P2 =0.03, P3 =0.3, P4 =0.22, Ps=0.27, and J3 = 0.3. For these specific values, Fig.5.12shows the variation in Q3and Q4as a functionof the numberof timesa minorloop is traversed. It is seen that both curves exponentially approach the samelimiting value asymptotically. Thus, accommodation is caused by the gradualdisappearance of the staircasethatdividesthepartof thePreisachplanethatwould

.. State variable

Q3

------ Q4

~~<,~

~r----------r-----~ ..........------- ------ ------ ------.--,--

~ ............

././

.//

//

/l/

1

0.5

OJ.....c

~.~

0>

~00

-0.5

-Io 5 10 15 20

Minor loop traversal number

2S 30

Figure 5.12 Changein statevariables withnumber of minorloopstraversed.

SECTION 5.5 ACCOMMODATION 133

be unaffected by thisprocessin otherPreisachmodels.Furthermore,in this model,the actual structure of the staircaseis immaterial. Only the values of the integralsof the Preisach function over each of the two areas are used.

The gradual shift in the minor loops can be seen by using (5.52) and (5.54) tocalculate the magnetization at their ends as a function of the number of times aminor loop is traversed. For the same valuesofp and p, the variationin the end ofthe minor loops is shownin Fig. 5.13.Accommodation beginseven at the first leg.Thus, in a nonaccommodating model, starting from negative saturation andapplying a field of ~ woulddemagnetizethe sample. In this model, for the samevalues, the magnetization will have a small positive value.

Whenone startsfromnegativesaturation,the valuefor Q in theentire Preisachplane is equal to -1. When a field h is applied, the part of the Preisach plane to theleft of h has a Q =+1.Thus, if hysteronsfrom the right part of the plane move intothe left part, they will experiencea field sufficient to correcttheir magnetization.On the other hand, the part of the plane to the right of h will have a value greaterthan -1, since hysteronsmovingthere will not have their magnetization corrected.That is, along this leg of the major loop, the valueof Q in that part of the plane isa monotonically increasing function of h, which approaches a limit less than 1.When the appliedfield is largeenoughto saturatethematerial,the entireplane willachieve a value of +1 for Q. Thus, in agreementwith experimentalobservations,there is no accommodation of the ends of the major loop as calculated by this

Accommodation of minorloops

Upperend

Lowerend

.... ... -.... ....... .. _-.. _- --­--------- ----------- ------------ -----------

.. .. .. ..

r-,

~r-------.-1-----'"-..----r----+-----1

~6g.. 0.5 -+-----;----+--~-r_--____r---___r_---~j~

6',=.J 0~-_+__---+---__+---_f__-__+_-__t

t"'d

~ -0.5--t--~--t-----__t_-___t__--_+__----4--~6Z

-1 ---L-.----~ ..L- r; __l. ___L __J

o 5 10 15 20Number of minorlooptraversals

25 30

Figure 5.13 Magnetization accommodation of minorloops.

134 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

(5.56)

model.When one is obtaining the end values of a minor loop for both large changes

in the applied field and large values of p, there will be large errors if a leg of theprocess is traversed in a single step and a low order error method is used to solvethe differential equations. Such a method is Euler's method, used above. Moreaccurate methods, which have higher orders of error, such as the Runge-Kuttamethod and predictor-corrector methods, are discussed in standard numericalmethods books.

An important problem in recording is the gradual decrease in the magnetizationof a recording during successive playbacks. A major cause of this loss is theaccommodation cycle caused by the playback head. A magnetized medium issubject to a demagnetizing field. This field is reduced when the medium is near orin contact with the playback head, since the medium acts as a keeper. Thus, anelement of the medium repeatedly passed in contact with a head is subject to manyminor loop cycles. These cycles range between effective fields that are the productof the element's magnetization and the two demagnetizing factors: one in thepresence and one in the absence of the playback head. The most expedient way toreduce this decay in magnetization is to reduce Pl.

It is noted that in this analysis, unless PI is identically zero, accommodationwill take place. This is an artifact of the approximation ofa discrete particulate tapeby a continuous Preisach function. In a real medium, the smallest entity that can beswitched is the magnetization associated with a hysteron. Thus, if PI is less thanthat due to a single hysteron, it is for all practical purposes zero, and no

accommodation occurs. Furthermore, if only a single hysteron is switched back andforth by this cycling, no accommodation will occur, since the original state of theinteraction field is restored at the conclusion of the cycle. This latter extension canprobably be extended to the switching of a few hysterons.

We note that in the limit as papproaches zero,

Q.(even) =Q~odd) = P2 - Ps .J J

PI +P2 +Ps

These are the equilibrium values of Q that the minor loops try to achieve byaccommodation; however, since there is no accommodation in this case, thesevalues will never be achieved.

5.6 IDENTIFICATION OF ACCOMMODATION PARAMETERS

This model has only one new parameter, p, to be identified. The identification ofthe parameters of the CMH model has been possible from major loop data only[20], since these are not affected by accommodation. A way to identify thisparameter is to measure the drift in a minor loop. To obtain the most accuratemeasure of p, it is necessary to obtain the greatest amount of accommodation. Tomaximize accommodation, one must simultaneously maximizePI (to maximize the

SECTION 5.6 IDENTIFICATION OF ACCOMMODATION PARAMETERS 135

motion of hysterons in the Preisach plane) and maximize either P3 or P4 (tomaximize the magnitude of the magnetization that must be forgotten). Forsymmetrical Preisach functions, this is done by choosing hA to be ~ and hB to be-~ for the extrema of the minor loop. For a nonaccommodating model, these fieldswould be the operative remanent coercivities; because ofaccommodation, however,the magnetization is not zero when the field is hA• In that case, P2 is equal to Ps andPI is equal to P3 + P4' Furthermore, if we start from negative saturation, then P3 iszero. In the subsequent calculations, we will assume that hA =s; =-hB•

The ratio ofPI to P2 is determined by the ratio of o, to 0;; for example if 0; =0,then P2 =0, if 0; =Ok' then PI =P2' and if o, =0, then PI =O. In the followinganalysis we will assume that this is the case. Then, since the piS are normalized, wehave PI =P2 =P4 =P5 =1/4. Since the drift in the minor loops is small, it is possibleto use the Euler method of solution discussed above to describe theaccommodation. The value of pdoes not affect the equilibrium value of the minorloop, but it does affect the rate at which equilibrium is approached. At hB, we use(5.48) to find that for even indices and whenj =2 and 4, the Q's are given by

Qt)_Q?n-1) = i6 [1+Q~2II-1)_4Q?n-l)]. (5.57)

while for odd indices, at hA, whenj = 4 and 5, the Q's are given by

Q.(2n +1) _ Q~2n) =1..[-1 +Q4(2n) _4Q.(2n>].) ) 16 )

For even n, (5.57) reduces to

Q(2n) =Q(2n) =-1 Q(2n) =( 1_~) + 1..Q(2n-1)J 5 '2 16 16 4 '

and

Q(2n) =( 1_3P) Q(2n -1) +1..4 16 4 16 '

and for odd n, (5.58) reduces to

QI(21l+) =Q2(2n +1) =1, Q(2n+l) =( 1 _ 3~) Q(2n)_1..4 16 4 16'

and

Q(2n + 1) _ (-1 3P) PQ (2n)5 - +- +- 4 .

16 16

We note that

Q(I) _ Q(1) - -1 + P4 - 5 - --.

8

(5.58)

(5.59)

(5.60)

(5.61)

(5.62)

(5.63)

Therefore, at the end of the first leg of the magnetizing process, the magnetizationis given by

136 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

PSMsM.=--.I 16 (5.64)

We see that if pwere zero, the magnetization would be zero, and we would be atthe remanentoperativecoercivefield.

The recursion relationsfor Q4(even) can be written

Q(2n+2) =( 1 _ 3P)2Q(2n) + 3p2 (5.65)4 16 4 256'

and for Q4(odd) can be written

Q(2n+l) =( 1- 3P)2Q(2n-I) _ 3p2 .4 16 4 256

(5.66)

Thus, two legs later in the magnetizing process,when the field is hA again, Q4(3) isgiven by

Since Qs(even) is always-1, then Qs(3) is given by

Q(3) =-1 + 3P +1..Q (2) =-1 +~ .5 16 16 5 8

Then the magnetization will be

M; _ 9P I59p2 3p3----+--+-SMs 8 256 128

(5.67)

(5.68)

(5.69)

(5.70)

For small values of p, we can neglect higher powers of pand approximate themagnetization by takingonly the first term.This magnetization is larger than thatof the first leg by approximately PSM/8, and thus the loop does not close. Bycomparingthese two valuesof the magnetization at hA we can obtain an estimatefor p. Thus,

8AMp", 17SM

s'

If this valueof pis too small to be measured accurately, a more appropriateformulacan bederivedusingmorecycles.For smallp, wecan againneglecthigherorder terms, and (5.66) can be written

dQ(2n-l),.., _ 3P Q(2n-l) (5.71)4 ,.., 8 4 •

SECTION 5.7 PROPERTIES OF ACCOMMODATION MODELS

The solution to this equation is

Q~2n+I)~ _( 1-%)( 1- 3:)2n.

137

(5.72)

It is seen that Q4 goes from (-1 + p/8) to zero, as n increases. The approach toequilibrium implied in this equation is the same exponential variation illustrated inFig. 5.13.

It is noted that in these calculations, the operative fields ofhA and hB were keptconstant in the accommodation process and the applied fields were allowed to vary.This can be done on a vibrating sample magnetometer (VSM), especially on aprogrammable one, once the value of ex is known. Bymeasuring the magnetizationas the field is applied, one can iteratively modify the applied field accordingly.Alternately, to keep the applied field limits constant during the accommodationprocess, it is necessary to derive new formulas, since the magnetization changes asthe accommodation process develops.

A statistically derived Preisach model and some of its properties have beenpresented for the accommodation in minor loops. The model has been deri ved froma statistical interpretation of the physical principles underlying the Preisach model.In addition, a measurement technique has been suggested to calculate theparameter, p, introduced by this model. The identification process must beextended to the case where a/ale is not unity, and the method of the identificationof the accommodation parameter must be extended to include accommodationcorrections.

Experiments have yet to be done to determine the applicability of this model.It is believed that this model is appropriate for longitudinal magnetic recordingmedia that can be accurately described by the CMH model. For vertical media, asimilar calculation based on the variable-variance model [21] must be derived. Itis also suggested that a more sophisticated model might be necessary to fitexperimental results. In the more sophisticated model, the state variable, Q, in agiven region is not simply a constant, but a function of the critical field, h/c. Thiscould be the case for a thin film medium that is perfectly aligned. Finally, it ishoped that this model, along with the aftereffect model, might be useful todetermine the archivability of recordings.

5.7 PROPERTIES OF ACCOMMODATION MODELS

We can use these definitions and the notation and method of computing Q givenin the preceding sections to compute the reversible and the irreversible componentsof the magnetization by generalizing the results obtained for the state-dependentreversible magnetization model [20]. A simplification results if we assume that thenormalized reversible function can be factored into the product of a function of theapplied field and a function of the interaction field. For example, if a branch of anisolated hysteresis loop can be written as

138 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

j{H-h)-j{h;) = f(H)g(h)

j{-hi)-j{-H+hi) = j{H)g(h)if Q = 1if Q = -I, (5.73)

then the irreversible component of the media magnetization is given by

Mj = SMsJJmj(hj)p(hj,hk)dhkdhj'-00 0

where

(5.74)

(5.75)

The locally reversible, state-dependent component of magnetization is given by

M,(H) = a , fiH) -a_.f{ -H), (5.76)

where the reversible coefficient is given by

(5.77)

It is seen that for square loop materials, S is 1 and m;(h;) reduces to Q. Fornonaccommodating models, the magnitude of Q is unity and the term (1 + Q)/2 isone in regions that are magnetized positively and zero where they are magnetizednegatively, thus, reducing to the definitions in the eMH model.

We will define the regional reversible coefficients by

aj ± = (l-S)MsJJg(h)p(hj,hk)dhjdhk'RJ

(5.78)

(5.80)

where Qjis the valueof the state variable in region Rjo This definition depends onlyon the shape of the region. Thus, (5.76) still holds with the definition that

5 I±Q.a± = E __Ja

j±,(5.79)

j=1 2

which explicitly illustrates the state variable dependence of the locally reversiblemagnetization. Similarly, to illustrate the state dependence of the irreversiblemagnetization, wecan define regional irreversible coefficients that depend only onthe shape of the region by

Pj = J J[(l-S).f{ -QJh) +S]p(hj,hk)dhkdhj ,

RJ

SECTION 5.7 PROPERTIES OF ACCOMMODATION MODELS 139

Withthis definition, wesee that the sumof thep's is unity; therefore, using(5.74)we mayrewrite(5.75)as follows:

sM;=SMsL Qj Pj·

j=l(5.81)

In this analysis, we willexamine only theend pointsof the minorloopsand studytheir drift. Hence, for a process that oscillates between the same two operativefields, the five regions in Fig. 5.I I are stationary and the integrals in (5.79) and(5.80) are constantat the limitsof the magnetization cycles. Thus, the only driftwill be due to the changing values of the Q.

5.7.1 Types ofAccommodation Processes

Anaccommodation processoccurswhenamagnetic mediumiscycledbetween twovalues of applied field. We will define three types of accommodation process:operative field accommodation (OFA), appliedfieldaccommodation (AFA),anddemagnetizing factor accommodation (DFA).

In OFA,the magnetization is cycledbetween a pairof operativefields;that is,theappliedfieldsarechangedby«LiM whenever themagnetization changesbytheamount LiM. To be able to apply an operative field, one must measure themagnetization as the processproceeds, and iteratively and monotonically correcttheappliedfielduntilthedesiredoperative fieldis attained. It is suggested that thisprocessmaybe usedto identifythe accommodation parameter, since the Preisachfunctions arestablein theoperativeplane.Whendiscussing cyclingbetween fieldshA andhB, we willrefer to thepoint(hA, hB ) in theoperative planeas theoperatingpoint.

In AFA, the magnetization is cycledbetween a pair of appliedfields. This isthe easiest type of accommodation process to performexperimentally, since theappliedfieldjust oscillates between a pair of field extrema. It is moredifficult tointerpretthan the OFA process becauseas the magnetization accommodates, theoperatingpointmoves. Furthermore, the minorhysteresis loopschangeduringtheaccommodation process because of the lack of the congruency property in themedium.

Thefinalaccommodation process, DFA,occurswhenever thegeometry of themagnetic circuitchanges(e.g., upon the application and subsequent removal of akeeper). This process also occurs whenever a recording head passes over arecording medium. In these cases the demagnetization factor changes with thegeometrical changes, thuseffectivelycyclingthedemagnetizing field.Justas in theAFA process, the operating point moves during accommodation, but, since theappliedfield dependson the magnetization, it changes as well.

We will first consideran operativefield accommodation process, since thelimits of the minor loop excursions are constant in the operativeplane and thussimplerto describe. For a givenmedium, the amount of accommodation dependson the values of h, and hb, as wellas the magnetization history. The loop willdrift

140 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

with each subsequent cycle and eventually reach a stable loop, which we will callthe equilibrium loop. Since accommodation wipes out the magnetization history,the equilibrium loop is only a function of hA and hB; however, the way this loop isapproached does depend on the magnetization history. It has been observed thatmajor loops do not accommodate. Since there exist media that do notspontaneously demagnetize, there must be a threshold field below which noaccommodation occurs. We will now discuss these and other properties of theaccommodation model.

The limit fields hA and hB define a point on the Preisach plane that divides thisplane into four regions: RI , R2, R34, and Rs. The region R34 is the combination of R3

and R4• With this division, the regions R1 and R2 are magnetized positively whenthe applied field is hA and the regions R1and Rsare magnetized negatively when theapplied field is hB• The region R34 is unaffected directly by this process; however,the motion of the hysterons in the Preisach plane, causes the magnetic state of thisregion to tend to become homogeneous.

For this accommodation process, the most positive value the magnetization cantake is found when R)4 is initially magnetized positively and the applied field is hA •

At this point the minor hysteresis loop is near the upper branch of the major loopand Q34 is almost 1. During subsequent cycles, Q34 will decrease and the loop willdrift downward. Similarly, the most negative value the magnetization can takeoccurs when initially R34 is magnetized negatively and the applied field is hB. Atthis point the minor hysteresis loop is near the lower branch of the major loop, andQ34 is almost -1. Thus, in this model, a minor loop will always lie inside the majorloop. Furthermore, the maximum accommodation that can take place is at the pointwhere the major loop is widest, and that occurs for loops where hA =-hB. Theseloops will be referred to as symmetricalminor loops.

The size of the first drift in the positive end of a minor loop is proportional tothe product of PI and P34' For symmetrical minor loops, PI is a monotonicincreasing function of hA starting from zero when hA is zero, and P34 is a monotonicdecreasing function of hA that goes to zero for large hA• Therefore, their productstarts at zero and will go through a maximum as hA is increased from zero. It canbe shown that the maximum occurs at the operative remanence coercivity. On theother hand, for major loops, R34 is zero, and there is no accommodation of the endpoints of the loop.

For symmetrical minor loops, the equilibrium loop will also be symmetrical inthe magnetization as well as the operative field. Since the magnetization at the twoends of the minor loop are equal in magnitude but opposite in sign, the minor loopwill be symmetrical with respect to the applied field as well. That is, the center ofthe equilibrium loop will be the origin. Thus, in an ac demagnetization process, itis not necessary to have a field large enough to saturate the sample to delete themagnetization history, but simply to go through a sufficient number of cyclesbefore the applied ac field is reduced to zero. It can be shown that for this model,these two demagnetization processes and the Curie point demagnetization producethe same magnetization sequence for the same applied field sequence.

SECTION 5.7 PROPERTIES OF ACCOMMODATION MODELS 141

It should be pointed out that this model has one other property: allaccommodating minor remanence loops lie within the major remanence loop, andtheir equilibrium position lies at the midpoint of the section of the major loopbetween the two field limits. This can be seen from the fact that m, on theascending major remanence loop at hA is given by

m;asc(hA ) = SMS(P1 +Pz -P34-ps)· (5.82)

For any minor loop, since the magnitudesof all the q's are less than one, it is seenthat m, (hA ) is given by .

m;(hA) = SMS(Pl+P2-Q34P34-PS) > m;asc(hA) · (5.83)

Thus, the right ends of all minor loops lie above the ascending major remanenceloop. Furthermore, the descending major remanence loop magnetization at hA isgiven by

(5.84)

where v, a positive fraction that is less than 1, is the fraction of R, that is stillpositive when the applied field is reduced to HA• Furthermore, for Preisachfunctions that are limited to the fourth quadrant, if HA is positive, then » is 1.Comparing with (5.83), it is seen that this is greater than m;(hA) . Therefore, the rightend of the minor loop also lies below the descending major remanence loop. Sincethe reversible component in the CMH loop is also largest for the major loop, theanalysis above can be extended to the total magnetization.Bysimilar reasoning, itcan be shown that the left end of minor loops lie above the ascending majorremanence loop.

For small pthe equilibrium loop, the state variable Q34 is given by

Q34

= P2- Ps . (5.85)PI+P2+PS

Thus, it can be shown that the average magnetization for the equilibrium loop isgiven by

(5.86)

This magnetization is the averageof the magnetizationof the region that is affectedby the applied fields and generally lies in the center of the major remanence loop.Therefore, minor loops starting at the major loop will accommodateaway from themajor loop. Furthermore, this limiting average magnetization is zero forsymmetrical minor loops. When there is cycling between two applied fields, theoperating point changes with the magnetization, as illustrated in Fig. 5.14. If theprocess observes thecongruencyproperty, the locusof operating points is a straightline with unit slope.

142 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

TheDFAprocess occurswhenever thegeometry changes (e.g.,whena keeperis broughtup to a permanent magnet or whena recorded medium is passednear arecording head). If this activity is repeated cyclically, accommodation can takeplace,and in certaincasesthe medium can become demagnetized. In thiscase, thefield in a magnetized mediumchanges because the geometric demagnetizationfactors change. Thus, if the geometry is cycled, the medium will experience acyclical appliedfield. Thisfieldis similarto theAFAprocess exceptthatbothfieldlimitsare nowof the samesignand as the magnetization accommodates, the fieldlimitschange.

It is seen that in this case, the "applied fieldIt is only of one sign. Thus, atequilibriumthe averagemagnetization will not be zero; however, it can becomeverysmall. It is important that this limitbe calculated, since if it is too small themedium willbe useless for recording. This phenomenon limits the usefulness ofmediafor all wavelengths of recording, but for short wavelength recording thereis anothereffect.

In longitudinal digitalmagnetic recording, ideallythe medium is magnetizedto saturation in alternating directions in regions separated by an abrupt transitionthat is perpendicular to the track. In a real medium this transition occurs over afinitedistance, and a detailed plotof themagnetization alongthe trackis showninFig. 5.15. However, DFA maycause the medium to be slightlydemagnetized asshown, thus resulting in a more gradual transition. This is one of the limitsassociated withrecording density.

Thisaccommodation process[5] isabletodescribe accommodation withonlya single newparameter. It is able to predictwhata stableminorloop wouldbe asa function of the limits of theapplied fieldexcursions. Preliminary measurements[3]haveshownthat it appears to describe thegeneral features of accommodation.

vu

IIIII,IIII

HB -------,

...III:--~

h//~B J \

\

\ Accommodation of theoperating point

Figure 5.14 Motion of theoperating pointduringan AFA process.

SECTION 5.8 DELETION PROPERTY 143

-1

Before

--- After

Position along trackFigure 5.15 Transition broadening in longitudinal digitalmagnetic recording due to

accommodation.

The features of this model are as follows: The major loop does not accommodate.Minor loops always lie inside the major loop. Minor loops accommodate awayfrom the major loop. The magnetization is stable if the applied field does notchange. Accommodation distorts the symmetry of all loops, and if hysteroninteraction decreases, accommodation decreases.

5.8 DELETION PROPERTY

In Chapter 2 we saw that the deletion property of the Preisach model was directlyrelated to the uniqueness of the Everett integral as a description of themagnetization change. The proof of the deletion property was based on theassumption that changes in magnetization are completely determined by this Everettintegral. This is no longer the case when there is accommodation, aftereffect, orboth.

Whenever a field is applied, the Preisach plane is divided into three regions:the two regions where the field determines the magnetic state of the hysterons, anda region where the hysteron can be in either state. It is this latter region-alsocalled the unaffected region, since it would not be affected by the magnetizingprocess in the classical Preisach model-that causes the violation of the deletionproperty. For the state to be determined by the Everett integral, it is necessary forthe state vector to be constant in this region; however, it can be shown from (5.31)and (5.46) that the state vector in this region obeys the following differentialequation:

dQ(u,v)dt

t -t+ -

(5.87)

144 CHAPTER 5 AFTEREFFECT AND ACCOMMODATION

The timederivativeof the magnetization is the sum of the integralof this functionover the unaffected region, plus the magnetization changes for the other regions,as computed in the precedingchapters.

5.9 CONCLUSIONS

The gradualdrift of minorloopscan be rate independentdue to accommodation orrate dependent due to aftereffect. The first will vary with cycle number whenexecuting repeated minor loops, while the second will drift with time even if theapplied field does not change. If one applies a small alternatingapplied field, thetwo methodscan be easily confused.

Both types of processes can be modeled by Preisach models and relax thedeletionpropertybychangingthemagnitudeof thestate variable. A newparameterpmust be introducedto modelaccommodation, and two newparametersT hfmustbe introduced to model accommodation.

REFERENCES

[1] S. Chikazumiwith S. H. Charap,PhysicsofMagnetism , Wiley:New York,1964.

[2] J. L. Snook,"Timeeffectsin magnetization," Physica,S, 1938,pp. 663-688.[3] Y. Tomono, "Magnetic after effect of cold rolled iron, I," J. Phys. Soc.

Japan, 7, 1952,pp. 174-179.[4] F. Preisach, "tiber die magnetische Nachwirkung," Z. Phys., 94, 1935,pp.

277-302.[5] A.Aharoni,Introduction to theTheoryofFerromagnetism, ClarendonPress:

Oxford, 1996.[6] C. Korman and I. D. Mayergoyz, "Preisach model driven by stochastic

inputsasa modelfor aftereffect,"IEEE Trans. Magn., MAG·32, September1996,pp.4204-4209.

[7] G.Bertotti,"Energeticandthermodynamic aspectsof hysteresis,"Phys.Rev.Lett., 76, 1996, pp.1739-1742.

[8] E. Della Torre and L. H. Bennett, "A Preisachmodelfor aftereffect," IEEETrans. Magn., MAG·34, July 1998,pp. 1276-1278.

[9] Y. D. Van and E. Della Torre, "Particle interaction in numerical micro-magneticmodeling,"J. Appl. Phys., 67(9), May 1990,pp. 5370-5372.

[10] G. Bottoni, "Size effect on the time dependence of magnetization of ironoxide particles," IEEE Trans. Magn., MAG·33, September 1997, pp.3049-3051.

[11] G. Bottoni, D. Candolfo,and A. Cecchetti,"Interaction effects of the timedependenceof the magnetization in recordingparticles,"J. Appl. Phys., 81,1997,pp.3809-3811.

[12] E. P. Wohlfarth, "The coefficient of magnetic viscosity," J. Phys. F: Met.Phys., 14, August 1984,L 155-LI59.

REFERENCES 145

[13] R. Street and J. C. Woolley, "A study of magnetic viscosity," Proc. Phys.Soc., A 62, 1949, pp. 562-572.

[14] M. 1.,0Bue, V. Basso, G. Bertotti, and K.-H. Muller, "Magnetic aftereffectin spring magnets and the Preisach model of hysteresis," IEEE Trans.Magn., MAG·33, September 1997, pp. 3862-3864.

[15] (a) E. Della Torre, L. H. Bennett, and L. J. Swartzendruber, "Modelingcomplex aftereffect behavior in recording materials using a Preisach­Arrhenius approach," Mat. Res. Soc. Symp. Proc.• 517, 1998, pp. 291-296.(b)L.J. Swartzendruber, L.H.Bennett, E. Della Torre, H. I. Brown, and I.H. Judy, "Behavior of magnetic aftereffect along a magnetization reversalcurve in a metal particle recording material," Mat. Res. Soc. Symp. Proc.•517,1998, pp. 360-366.

[16] M. Brokate and E. Della Torre, "The wiping-out property of the movingmodel," IEEE Trans. Magn., MAG·27, September 1991, pp. 3811-3814.

[17] E. Della Torre and G. Kadar, "Hysteresis Modeling II: Accommodation,"IEEE Trans. Magn., MAG·23, September 1987, pp. 2823-2825.

[18] O. Benda, "Possibilities and limits of the Preisach model," J. Magn. &Magn. Mater., 112, 1992, pp. 443-446.

[19] I. D. Mayergoyz, Mathematical Models ofHysteresis, New York: Springer­Verlag, 1991, p. 108.

[20] E. Della Torre and F. Vajda, "Parameter identification of the complete­moving hysteresis model using major loop data," IEEE Trans Magn.,MAG·30, November 1994, pp. 4987-5000.

[21] F. Vajda, E. Della Torre, M. Pardavi-Horvath, and G. Vertesy, "A variablevariance Preisach model," IEEE Trans. Magn., MAG·29, November 1993,pp. 3793-3795.

CHAPTER 6

VECTOR MODELS

6.1 INTRODUCTION

So far we have been discussing increasingly accurate scalar models for themagnetizing process. We can think of these as processes in which all the fieldvariations lie along an axis, and we are interested only in the component of themagnetization along that axis. In a real magnetizing process, besides changing itsvalue, the applied field could rotate. Furthermore, especially if the material is notisotropic, the resulting magnetization might not be in the same direction as theapplied field. Thus, it is necessary to characterize material behavior in two or moredimensions. In this chapter we discuss how the work of Chapters 1 to 5 can beextended into two- and three- dimensional situations.

Before we address specific models, we will identify the general properties ofvector models that are physically realizable. Besides the limits imposed on thescalar models, we will add two more properties. The saturation property refers tothe requirement that all magnetizations calculated by the model not exceedsaturation. The loss property refers to the fact that as the size of a rotating fieldincreases, the losses first increase and then decrease. Both properties can beachieved by vector models.

147

148 CHAPTER 6 VECTOR MODELS

We discuss three types of vector models. The Mayergoyz vector model is apurely phenomenological extension of the scalar Preisach models. On the otherhand we can construct pseudoparticle models based on micromagnetic models,such as the Stoner-Wohlfarth model. These models can require substantialcomputation intensity. A middle course is the coupled-hysteron model, whichcouples three scalar models to obtain three-dimensional vectors, and adjusts themso that they satisfy the general requirements for vector models.

6.2 GENERAL PROPERTIES OF VECTOR MODELS

When the magnetization changes in a magnetic material, energy may be dissipatedby various causes. It is convenient to categorize these energy losses as static lossesand dynamic losses. The static losses are those that would occur when themagnetization is cycled arbitrarily slowly; the dynamic losses, some of which arediscussed in Chapter 7, are the additional losses that occur when the magnetizationis cycled more quickly and are a function of how quickly the magnetization varieswith time. Static losses are caused by sudden changes in magnetization, when afield threshold is exceeded, such as those due to Barkhausen jumps.

In earlier chapters we discussed how these effects can be modeled to varyingdegrees of accuracy by various scalar models, for applied fields acting along asingle axis. When the applied field changes its direction as well as its magnitude,the modeling becomes more complicated. Several vector extensions of Preisachmodels have been proposed in the last decade.

One of the properties a vector model should have is the saturationproperty:that is, for a large applied field in any direction, the magnetization should neverexceed saturation. Furthermore, it should be able to achieve saturation, and for anydirection of the applied field, by means of the application of a sufficiently largefield. Then, as long as the field is applied, the magnetization should be in the samedirection as the field. Thus, for a large rotating field, the locus of magnetizationvector tips should trace out a circle. The three types of models discussed in thefollowing sections all have this property.

We now describe some of the energy loss properties vector models shouldpossess, and discuss how these models may be modified to achieve the desired lossvariation with the applied field. We will concentrate on two such vector lossmechanisms in magnetic materials: that associated with anisotropy and thatassociated with wall motion. Other types of rotational loss mechanisms have beenobserved, but these are beyond the scope of this work. Since the models behavedifferently under a rotating field whose magnitude is increasing, this property canbe used to distinguish between the various proposed models.

When an increasing oscillating field is applied to a magnetic material, theenergy loss per cycle due to hysteresis is zero until a threshold field is reached.Then the loss increases until the material is saturated. Any further increase in themagnitude of the field does not increase or decrease the static loss per cycle. For

SECTION 6.2 GENERAL PROPERTIES OF VECTOR MODELS 149

both types of rotational hysteresis loss, the situation is different when the materialis subject to an increasing rotating magnetic field.

The first type of rotational hysteresis, called anisotropy hysteresis, occurs insingle domain particles when the magnetization attempts to follow a rotatingapplied field but is prevented from doing so by either shape or magnetocrystallineanisotropy. This type of hysteresis is characterized by a zero loss for small fields,which first increases and then decreases to ~ero as the applied field is increased.The analytic properties of this loss can be derived by considering theStoner-Wohlfarth model for uniformly magnetized ellipsoidal particles.

The second type of rotational hysteresis, called wall motion hysteresis, occursin materials that are large enough to support multidomains. When two adjacentdomains, separated by a domain wall, have different orientations, then the domainwhose orientation is closer to the applied field will grow at the expense of theother. As the applied field rotates, the direction of wall motion can even change.In these cases, the loss mechanism is due to Barkhausen jumps in wall motion,when domains with lower Zeeman energy grow at the expense of those with higherenergy. Then the hysteresis loss for fields smaller than the minimum required toproduce a Barkhausen jump is zero. As the field increases above this threshold, theloss increases as larger regions of the material are traversed by the domain walls.For fields large enough to saturate the material, the loss again decreases to zero,since all domain walls are eliminated. The range of fields for which hysteresis lossis present is much larger for these effects than for anisotropy hysteresis.

Thus, in both these cases, as the rotating magnetic field increases, the energyloss per cycle due to hysteresis is essentially zero until a threshold field is reached.Then the loss increases until the contribution of the new thresholds is less than thedecreasing effect due to the thresholds that have been previously exceeded. At thatpoint, unlike the case of an oscillating field, the loss starts to decrease to zero as thematerial saturates.

In particular, a model for anisotropy hysteresis is the uniform magnetizationmodel for an isolated spheroidal magnetic particle, proposed by Stoner andWohlfarth and discussed in Chapter 1. When the energy loss is plotted as afunction of the applied rotating field, one obtains a curve as shown in Fig. 6.1. Itis seen that there is no energy loss for applied fields that are less than the thresholdrequired to change the state of the particle, since the process is entirely reversible.When the threshold is exceeded, the loss suddenly increases and thenmonotonically decreases with the applied field until it is reduced to zero. Furtherincreases in the applied field, as is well known, do not produce losses, since forlarge fields, the magnetization is able to follow the applied field.

For an array of particles, although each particle behaves essentially in this way,the threshold field will be different for each particle. Furthermore, particleinteraction may result in different magnitudes for the positive and negativeswitching fields. Nevertheless, as the rotating field is increased in magnitude, theloss will at first increase monotonically. At a critical field, the increase in lossassociated with the switching of additional particles is equal to the decrease in lossof the particles with smaller critical fields. At this point the loss will decrease with

150 CHAPTER 6 VECTOR MODELS

Applied rotating field magnitude

Figure 6.1 Rotational energylossper cyclefor a Stoner-Wohlfarth particle.

increasing field magnitude until all the particles are following the applied field.This is in sharp contrast with the loss associated with an alternating field thatincreases monotonically to saturation withthe applied field.

The variation of the threshold field withtheanglethat it makes with the easyaxis is fairly complicated for anisotropy hysteresis. In particular for aStoner-Wohlfarth particle, the switching field variation with the angle of theappliedfield is an asteroid, discussed in Chapter1.For a realparticle, the angularvariation is muchmorecomplicated. Forwallmotion hysteresis, on theotherhand,the energy that the applied field supplies to the domain wall, to overcome theenergythreshold, is theZeeman energy. Thisenergyvaries as thecosineof a, andthe threshold fieldvaries as its reciprocal; thatis, as thesecantof 6. Thus to makeareasonable model for thevectorinterpretation of thethreshold fieldit is necessaryto know the orientation of the easy axis and the mechanism of hysteresis. Sincedomainpatternsin unsaturated specimens are random, evenif theirmagnetizationhistoryis known, such an analysis mustbe statistical.

Thus,a vectormodel forhysteresis mustbeabletodescribetheseeffects.Thatis, it must reduce to the scalar model under the appropriate conditions, and inaddition must obey the saturation property and the loss property in order to bephysically realizable. Onceit is physically realizable, the model shouldreproduceobservedmeasurements. Oneof theseresultsis the remanence loop, whichis thelocusof points tracedout by the vectorremanence as the direction of the appliedfieldcausingit is rotated. Thisremanence loopformanymaterials is anellipse,andthesematerials arecalledellipsoidally magnetizable. Themajoraxisof thatellipseis the easy axis, and the minor axis is the hard axis. For isotropic media, theremanence loop is a circle.

SECTION 6.3 THE MAYERGOYZ VECTOR MODEL

6.3 THE MAYERGOYZ VECTOR MODEL

151

(6.2)

Mayergoyz proposedbuilding a vectormodel froma continuum of scalarPreisachtransducers [1], each incrementally rotatedfrom its neighbor. The input to eachtransduceris thecomponent of theapplied field in thatdirection, and the outputofeach transducer is a magnetization in that direction. The output of the completemodel is the vector sum of the output of all the transducers. Since his basicbuildingblockis a Preisachtransducer, he quickly showsthat his modelreducesto the scalar Preisachtransducer for processes that have a unique line of action.Furthermore, his model has the generalized congruency property; that is, for allcyclicmagnetizing processes, the magnetization is also cyclicand the loops thusformedare all congruent to each other.

The Mayergoyz vector model computes the irreversible component of themagnetization as

m. = f ffoKp(6, uo,vo)Q(6,uo,vo)d6duodve, (6.1)u>v

whereQ is a unit vectorlyingeitheralongthe Ie or the -Ie direction. For isotropicmedia,the Preisachfunction p(6, Ue, v e), does not varywith6. If a large field isappliedalongthe line6 =n/2, thenQliesalongIe for all 6. Thenit is seen that m,is in the Ie direction, andthecomponents perpendicular to thatdirectioncanceloutin pairs.For anisotropic media, p(6,ue,ve) varieswith6. Then it is seen that if weapply a large field and rotate it, the magnitude of the magnetization will vary.Moreover, its direction will not normally be in the samedirection as the appliedfield, but will always makean acuteanglewithrespectto it.

Forsmallerfields, theirreversiblecomponentof themagnetization willdependon the magnetizing history, sincethemedium is hysteretic. Eachhysteron can havea different history, becauseit experiences a differentsequenceof appliedfields.Thus, a different"staircase"mustbe storedfor each hysteron.

The identification of isotropic media is comparatively simple, since all thehysterons are identical. Then all one has to do is to identify a typical hysteron;when a field is applied to one hysteron, however, the other hysterons mayexperiencedifferent fields. So even if they are identical, they will have differentmagnetization histories. We illustrate the identification process for two­dimensional processes. For simplicity, if one applies a field H in the direction 6=0, which will be taken as the x-axis, then hysterons in the direction e willexperience a field H cos 6. If one considers a first-order reversalprocess startingfrom a large negative value, goingto a field HI and thento a smallerfield H2, theresulting magnetization is givenby

1t H. H.

m. = lxfd6f dvef dUe cos6p(uecos6,vecos 6).o H2

152 CHAPTER 6 VECTOR MODELS

Differentiating with respect to Ueand Vogives

a2m 11

__I = lx!cos6p(uecos6,vacos6).auac3ve 0

(6.3)

Unlike the case of the scalar model, the second partial derivative of themagnetization at the conclusion of a first-order reversal process does not yield thePreisach function directly .

Mayergoyz suggests two methods [1] to obtain the Preisach function from(6.3) . The first method involves the evaluation of polynomial coefficients if (6.3)can be approximated by a polynomial. The second method involves a simpletransformation that converts the integral equation into one of the Abel type. Foranisotropic media, one must measure the magnetization for first-order reversalprocesses at all angles . The Preisach function is then obtained in terms of sphericalharmonics. It is easy to show that this model has the saturation property, since themagnetization that it computes is always bounded . Therefore, if the saturationmagnetization is set to be the least upper bound of these values, one can neverexceed saturation.

6.4 PSEUDOPARTICLE MODELS

The pseudoparticle models approximate a hysteron by a small number of basicparticles that are combined into a so-called pseudoparticle. Two such models havebeen proposed by Oti: one uses the Stoner-Wohlfarth model for the basic particles[2]; the other uses the results of a micromagnetic calculation for the basic particles[3]. Although the models assume that the hysterons are particles, their result caneasily be extended to granular media. To illustrate how they work, let us assumethat the pseudoparticle consists of three identical basic particles, as shown in Fig.

Side particles

x

Figure 6.2 A pseudoparticle consisting of three basic particles.

SECTION 6.4 PSEUDOPARTICLE MODELS 153

6.2. If higher accuracy is desired, one can easily extend this model to include morebasic particles.

We assume that the x axis, also called the PMA (Preisach measurement axis),is the easy axis of the medium and that the size of the moment and the angle madeby the two side particles with the easy axis are the same. We therefore, have threeindependent variables: the moment of the central particle, the moment ofone of theside particles, and the angle of the side particles. We can solve for these variablesby requiring the pseudoparticles to have the same squareness as the medium as awhole, along three directions: the x direction, the y direction, and at an angle, say450

, with respect to these axes.If we call the moment of the central particle ml , and of each of the side

particles m2, and the angle that each of the side particles makes with respect to thex axis 0, then the x squareness S, is given by

m1 + 2m2cos 6s, :: (6.4)

m1+2m2

Similarly, the y squareness S, is given by

2m2sin 6m2 = --­

m1 +2m2(6.5)

If we apply a large field at other angles, we will find that the remanence is not inthe same direction as the applied field. In particular, if e is 45 0 and the appliedfield is also at 45 0

, we can assume that the lower of the two side particles is on theaverage demagnetized. The vector remanence of the pseudoparticle at zero fieldthen is

(6.6)

Thus, by changing a, we can change the magnetization properties at other anglesand thereby the shape of the remanence loop.

Each of the basic particles contains the angular variation of the process;however, each particle also represents a distribution of critical fields. Thus, thestate of a particular basic particle is computed by a Preisach process. The Preisachdistribution can be a normal distribution, and a moving model can be used toaccount for the variation in the local field with magnetization. Aftereffect andaccommodation can also be introduced into this model, as discussed in Chapter 5.The identification of the Preisach parameters for each basic particle can beperformed as a generalization of the identification of the scalar Preisach model. Ifwe assume that the basic particles are identical. then once e is known, we canproject the effect of the two side particles on the PMA and use scalar identificationon the composite particle.

154 CHAPTER 6 VECTOR MODELS

(6.8)

Since the basic particles behave like real particles, in the case of themicromagnetic modelor for smallparticlesusingtheStoner-Wohlfarthmodel, thesystemwillnaturally havethecorrectrotational properties. Inparticular, thesystemwill have the saturation propertyand the loss property. The net magnetization ofthe systemis obtainedby takingthe vectorsum of the magnetization of the basichysterons. It is notedthattheStoner-Wohlfarthmodel naturally computes the totalmagnetization of the hysteron. Hence, it is unnecessary to decompose themagnetization into a reversible and an irreversible component. Although in thismodel, we must maintain the magnetization history of only a few hysterons, incomparison to the many hysterons in the Mayergoyz model, since each basicparticleproducesacorrectspatialfieldvariation, thepremiseof thepseudoparticlemodel may be no less accurate.

6.5 COUPLED-HYSTERON MODELS

Anothercategoryof vectormodels consistsof thecoupled-hysteron models[4]. Inthiscase weplacea Preisachmodelalongtheprincipal axesof the system: twofortwo-dimensional models and three for three-dimensional models. If these modelsarepermittedto be independent, thesaturation propertycaneasilybe violated. Thecouplingis accomplished through a combined Preisachfunction. Forexample, forthree-dimensional models, there are six Preisach variables: the up- and down­switching fields in the x, y, and z directions, respectively. The magnetization iscomputed by meansof

mj =r·"!Q(ux'vx.uy• vy'UZ' vz)p(ux'VX,uy'vy'UZ' vz)duzdvZduydvyduxdvx· (6.7)OR

whereOR is theregionwhereu.>vx' u;> vy and u;> vt " ByrequiringthatQ's be lessthanor equaltoone, weguaranteethatthemagnitude of m, is always less thanone.Wewilldefinethecomposite Preisachvolume as thesix-dimensional hypervolumewhose axes are UX' vx' uy, vy' uz' and vr., • A point in this six-dimensional space willbedenotedsimplyby 0, so that this equationcan be written

ml = !Q(O)p(O)dO.

OR

6.5.1 Selection Rules

Theselectionof Q, thestatevector, isdetermined byselectionrules.Forsimplicity,in this section.we assumethat the x axis is the easy axis and the y and z axes arerelatively harderaxes of the material. Then the appliedfield will be decomposedintothex-direction, they-direction components, andthezcomponents. Modelswillthen be built to computethecorresponding components of the magnetization. Thiswill avoidcross termsin the calculations.

SECTION 6.5 COUPLED·HYSTERON MODELS 155

Thestatevectorat a pointin thePreisach volume represents the average stateof a group of hysterons thathavethesameswitching fieldbut mayhavedifferentorientations, size, shape, etc. Two such hysterons are indicated schematically inFig.6.3.Whena largehorizontal fieldisapplied, thehorizontal component of theirmagnetization will become positive. In that case, their vertical components willcancel. Similarly, a vertical fieldwillmagnetize thehysterons vertically andreducethe horizontal component to zero. This concept is the basis for choosing thefollowing selection rules.

We willassume that if thex component of theapplied fieldis greaterthan U.t'

the y component is between v, and u,and the z component is between Vz and uz•

thenthe hysteron willbe magnetized in thex-direction. Then,for thatpoint in thePreisach plane

(6.9)

where Ux<hx' vy<hy<uy' and vt <ht <ut ' andwhere theh's areoperative fields.This typeof selection rulewillbecalleda simpleselection rule.Therearesixsuchrules for three-dimensional models. They apply to each point in the Preisachhypervolume where onecomponent of theapplied fieldis sufficientto switchthathysteron, but the othercomponents are not. Undertheseconditions, the hysteronis switched into that unique direction.

Whenmorethanonecomponent issufficient toswitch themagnetization, com­poundselection rulesgovern theselection of thestatevector. Theyare required toavoid the indeterminacy of the direction of Q for large fields. When they arenecessary. many selections are possible. For vector models, however, theremanence is notjust a function of theapplied fieldextrema but isa function of thepath taken by the applied field as it is reduced to zero. The choiceof compoundselection rulesmusttakethisadditional dependence intoaccount. A possible rulewould be to choosethe state vectorto be in whichever direction the applied fieldhasthelargestexcess overthatcomponent of theswitching field.Thus,whichevercomponent of the applied field last exceeds the coercivity in that direction willdetermine the direction of the hysteron's magnetization. This would give acontinuous function for Q over the Preisach volume, but the derivative of Q isdiscontinuous whenever twocomponents havethesameexcess overthecoercivity.

(a) (b)

Figure 6.3 When both membersof a pair of hysterons, at the same point In the Preisachvolume,are magnetized horizontally (a), the verticalcomponentof magnetization is zero.When both are magnetized vertically (b), the horizontal componentis zero.

156 CHAPTER 6 VECTOR MODELS

Furthermore, we will see later that for largefields, the irreversible magnetizationdoesnot tendto followtheapplied fieldas itdoesfor thefollowing selection rules.

We now describe a betterchoiceof compound selection rules that meet thedesirablecriteriawe willuse.Whentwoor morecomponents of the appliedfieldexceedthe switching fieldsof the hysteron, we will select the components of thestate vectorto be in the sameratioas the excessof the applied field's componentsover the respective switching field components. This would makeQ a functionwithcontinuous derivatives overthePreisach hypervolume. Toconserve space,wewill summarize these rules for the two-dimensional case only, since thegeneralization to threedimensions is routine. Therulesare summarized in Tables6.1 and 6.2, whichgive the components of Q for thesecompound selection rules

Table 6.1 Values for Q,r

Qx v.>», Vx< h;« u,

vy > hy

hx -vx 0Ihx -v) + Ihy - vyl

vy< hy< Uy -1 Nochange

hy >uy

hx -vx 0Ih;x - v;xl + thy - uyl

Ih -u 1+lh -v Ix x y y

Table 6.2 Values for Q,

Qy v.>». vx<hx<ux hx > u,

vy > h;h

y-vy -1

hy-vy

Ihx- vxl + Ihy- vyl Ih -u 1+lh -v Ix x y y

vy < hy < uy 0 No change 0

h.> uy

hy-uy hy-uyIh - v I + Ih - u I Ih -u 1+lh -u Ix x y y x x y y

SECTION 6.5 COUPLED-HYSTERON MODELS 157

as a function of the operativefield. They apply to every point in the compositePreisach volume. It is noted, however, that when the applied field changes, allpointsare not necessarily affectedandonlythepointsaffectedhaveto bechanged.Furthermore, these rules reduceto the simpleselectionrules when they apply.

If we assumethat all thecouplingbetween the twoaxes is entirelythroughthestate vectors, then the Preisachfunction can be factored as

(6.10)

(6.13)

where

Ox = (ux' vx) ' ely = (uy'vy)' and o, = (uz,vz)· (6.11)

Examination of Tables6.1 and 6.2 shows that as a resultof the application ofthe selectionrules, at any point on the Preisachplane, the sum of the magnitudesof the Cartesiancomponents of the state vector is set equal to 1; that is,

IQ) + IQyl + IQzl :: 1. (6.12)

Let us define the following two integrals:

Ij =fQiO)p(O)dO =fQiOj)piOj)dOj for j = x, y, or z,OR OR

or

(6.14)

where the Q's are computed using the selection rules as above. It is seen from(6.12) that

(6.15)

The equalityin this equationoccursonly whenfor all points at which the Q's arenot zero, all the Qx's in I, are of the same sign, all the Q,'s in I, are of the samesign, and all of the Qz's in I, are of the same sign. For example, if the remanenceis obtainedbyrotatinga largefield, thenequalityoccursfor theentireprocess.Wenote,for example, that if I, is zeroand l, and I, havethe samesign, so that the termI, + I, is equal to one, then

1-; + I: = u, + I y)2 - 21/y = 1 - 211xl-It-Ixl. (6.16)

Thisequationimpliesthat thesumof thesquaresof'theJ's is a functionof lx, henceof the direction of the magnetization. This would be true even for isotropicmaterials under large fields.

158 CHAPTER 6 VECTOR MODELS

Thus, we cannot let

mIx ee Ix, mIy ee Iy and mIz ee I" (6.17)

since the simple application of these selection rules yields neither circularremanence paths for isotropic materials nor ellipsoidal remanence paths foranisotropic materials. As the applied field is rotated from the x direction to the ydirection, the normalized remanent path traces a straight line from the point (1, 0, 0)to the point (0, 1, 0). These results can easily be generalized to three dimensions.A pair of two-dimensional models [5] was proposed to correct for this limitation:the m2 model and the SVM model.

6.5.2 The m2 Model

In a possible coupled-hysteron model, the m2 model, we compute the square of theirreversible components of the magnetization using the appropriate component ofQ. Thus, using (6.13) in two dimensions we have

2 2mix = t, and m ty = Iy ' (6.18)

where

(6.19)

or

(6.20)

where

(6.21)

If we wish the material to be ellipsoidally magnetizable, then the major remanencepath must obey

(6.22)

or

(6.23)

We see that this is indeed the case for large rotating fields, by substituting (6.15)into (6.18) with the equality sign.

The problem with this approach is that (6.18) gives only the magnitude of thecomponents of the remanence and not their sign. The sign must be computedseparately. For example, the sign of m, could be given by a formula such as

SECTION 6.5 COUPLED-HYSTERON MODELS 159

It is noted that in the case of a scalar applied field in the x direction, Qx is one.However, (6.18) computes the square of the magnetization, not the magnetizationdirectly. Thus, the vector Preisach function does not reduce simply to the scalarPreisach function. For example, an attempt to identify the Preisach function bycalculating an x-directed magnetization by one starting from a negative x saturationstate and applying fields only in the x direction, would not yield the same Preisachfunction obtained from a scalar Preisach model.

6.5.3 The Simplified Vector Model orSVM Model

A better way of coupling the two Preisach models is the SVM model [6]. In thismodel, we use a rotational correction R(Ix' Iy' 1

1) to compute the normalized

magnetization, and we compute the components of the normalized irreversiblemagnetization by means of

mIx = R(/xJyJz)/x' ml y = R(/1llyJz)/y' and m Iz = R(/xJy/z)Iz' (6.25)

or

(6.26)

(6.28)

where R(lx' Iy' 11) is the rotational correction. We then compute the magnetizationby substituting these expressions into

~x = MsSxmtx' ~y = MSSymty, and ~z = MsSzmtz' (6.27)

where the S's are the squareness of the material. Then

M1 = u, S ml ,

(6.29)s

where, as a result of the choice of the coordinate system, S is the following matrix:

S1l 0 0

o Sy 0 .o 0 s,

This model is designed to handle anisotropic media by choosing different valuesfor the S's along each of the axes, and different parameters in the basic Preisachmodels. If the parameters are the same along the three axes, the model describesisotropic media, and the major remanent path will be a circle. In addition, if all thebasic Preisach models have the same parameters, for any circular applied field path,all the remanent paths are circles and the model is isotropic. The model can also

160 CHAPTER 6 VECTOR MODELS

describe scalar processesif the applied field is along one of the principal axes. Inthat case, the magnetization will be along that axis. For the material to be

ellipsoidallymagr;;jl:.:h(~Oj:e~r~:) ~a~:~ rotating field mus;:::

or

(6.31)

(6.32)

ToobtainellipsoidaUy magnetized behavior,fora saturatedmedium,anacceptablerotationalcorrectioncouldbe(1; +I: +1%2)-112. However, thisrotationalcorrectiontries to keep the mediumsaturatedas the I's are decreased. To correct for this, wewill use the rotationalcorrection given by

R(I",IyI,,) = IIxl + 11,1 +11,,1.(12

+12+12

):x y %

It can be shownthat for any directionof magnetization the rotationalcorrectionisboundedby

1 s R ~ {i, (6.33)

and if the magnetization lies in a principalplane, the upper limit is {i.From (6.15) it is seen that settinganyone of thers equal to 1 forces the other

I's to O. Thus, if we apply a largeenoughfield along any of the principalaxes, allthe Q's willbedirectedalongthat axis and the I alongthat axis willbe set equal to1. Thus, after applyinga large field in the x direction, for processes in which thefield alwayslies along thex axis, the rotationalcorrectionwill remainat unityandthe process will act like a scalarprocess.Then the irreversible magnetization is

m", = JQx(Ox)piOx)aDx' (6.34)1Iz!>":r

where

Qx(O,,) = Jl x·Q(O)aD, .OR

(6.35)

So, for these processes, the SVM model reduces to the ordinaryscalar processes.Similarly, processes along either the y or z axes also reduce to ordinary scalarprocesses. Thus, like the scalar model, the model can be modified to havenoncongruency and exhibitaftereffect and accommodation. Also, for incrementalchangesin theappliedfieldonlya smallregionof thePreisachvolumewillchange,so the differential equation approach to magnetization changes can be veryeffective. Therefore, the scalar models along the three principal axes can beidentified individually in the same manner as previously described for scalarprocesses.

SECTION 6.5 COUPLED-HYSTERON MODELS 161

(6.37)

If we computethe magnitude of the magnetization for the remanencedue to alarge field in any direction,since

J/xl + J/,I + IIzl = 1, (6.36)

the rotationalcorrectionis(!; +I: + 1;>-112, and we have

(RIx)2 + (Rl,)2 + (RIz)2 = 1.

Thus, we see that

(6.38)

This states that the normalized major remanentpath lies on a sphere,and thus, themajor remanentpath itself lies on an ellipsoidunlessall the S 's are equal. If morecomplexpaths are desired, additional rotationalcorrections can be added.

In particularfor isotropicmedia,for large h the selectionrules require that

hQJ = "j ,where j = x, y, and z. (6.39)

Ihzl + Ih~ + Ihzl

Since the field is large,

~ = JJQJ pJC)df1 = Qj' where j = x, y, and z,

oJl

From (6.27) we see that

- QJmy - , where j = x, y, and z.

IQ1l2 + Q:+ Q;Thus, again

Furthermore, for an appliedfield rotating in the xy plane,

Qx = hz = mix

Qy hy m"

(6.40)

(6.41)

(6.42)

(6.43)

Thus,themagnetization willbealignedwiththeappliedfieldandwillhaveconstantmagnitude.

If the individual scalar process is modeled with accommodation, aftereffect,and state-dependent reversiblemagnetization, and is a movingmodel,the resultingvector model will have all these properties. In this case, for an applied rotatingfield, the magnetization path willbe an ellipticalhelixwhosepitch decreases witheach rotationuntil finally it reaches an elliptical limitcycle, as shown in Fig. 6.4.

162 CHAPTER 6 VECTOR MODELS

Limitcycle

Figure 6.4 Magnetization path of an accommodating anisotropicmedium due to a rotatingappliedfield.

For isotropic media, the Preisachmodelsalong the x and y axes are identical,so onlyone identification is necessary. For anisotropicmedia,theparametersof thethree models will be different, especially the mean critical fields and thesquarenesses. Then,for largefields,theirreversiblecomponentof themagnetizationis in the same direction as the applied field only whenthe applied field lies alongthe principalaxes. In general, the magnetization will lie closer to the easy axis.Forsmaller fields, the magnetization will also lag behind the applied field, and theaspect ratio of these paths can be different from that of the major path.

So far we have computedthe irreversiblecomponentof the magnetization. Ifthej(H) is the samealong the threeprincipalaxes, the reversiblecomponentof themagnetization is also a vector and can be computedby first computing

mR = 8+ j(IB) + 8_ j(-IHD, (6.44)

whereitlHI) has the properties given in (3.9). For the DOK model,1 + m.·l u 1 - m.·lu

8+ = 2 1H and 8_ = 2 1H • (6.45)

(6.46)

where 1" is a unit vector in the H direction. ThenM. = L Ms{l-SJ} mR1J 1J•

j=%~.%

For fields along the principal axes, similar to the irreversible component, thiscomponent also reduces to the reversible component of the scalar OOK model.Therefore, if the magnetization originallyis alongone of the principalaxes and theapplied field is constrainedto that axis, then the magnetization will remain alongthat axis and the model will reduce to the DOK model. We could obtain similarexpressionsfor the a's in CMH model.

It can be shownthat for large fields, ImRI =1. Thus,

SECTION 6.5 COUPLED-HYSTERON MODELS

and

163

(6.47)

(6.48)

(6.49)

Thus, for both isotropicand anisotropic mediain the presenceof large fields, thenormalized reversiblecomponent of magnetization has a constantmagnitude, andthe reversiblemagnetization tracesout an ellipse.The magnitude of the reversiblemagnetization, therefore, tracesoutanellipsewhosemajoraxisis theeasyaxisandwhose minor axis is the hard axis, as shown in Fig. 6.5. It can be shown using(6.37) that

M; + M: + M z2

= (MIx+MRx)2 + (M ly +MRy)2 + (MIl.+MRz)2

=M;[(RI/ + (RI/ + (RIll =M;.

Thus, the magnitude of the magnetization is a constantequalto Ms in the directionof the appliedfield.Sincefor anisotropic mediathe irreversible magnetization liesbetween the applied field and the easy axis, the reversible magnetization liesbetweenthe appliedfield and the hard axis, as shown in Fig. 6.6.

Forthisrotational correction, themodel is, ingeneral, elliptically magnetizableand has the saturation property. Whenmagnetized alongeithertheeasyaxisor thehardaxis,themodelreducesproperlyto thescalarmodel, and thesimplifiedmodelcan be computed directly. This simplifies the identification of the parameters. It isnoted that whenthe appliedfield is not alonga principal axis, noneof the modelsreduce to simple Preisach models because the magnetization is not in the same

Hardaxis

axis--+-+-----+----+-~----....-4---~-!L.-

Figure 6.5 Magnetization loci for a large rotating field.

164

Hard axis

CHAPTER 6 VECTOR MODELS

Appliedfield

Irrevtiblerna etization

Easyaxis

Figure 6.6 Vectordecomposition of magnetization.

direction as the applied field. Since the model involves only the computation ofPreisach models along the principal axes, like the scalar Preisach model, it iscomputationally efficient.

6.6 LOSS PROPERTIES

In the case of the Mayergoyz model and the coupled Preisach model, one iscomputing the irreversible component of the magnetization, while in thepseudoparticle model one computes the total magnetization. Thus, one must add areversible component to the first two categories of models. For isotropic media, allthe models predict that for large applied rotational fields, the computedmagnetization will be in the same direction as the applied field. Any reversiblemagnetization will also be in that direction. The energy the field supplies to amagnetic medium is given by

dw = H. dM.dt dt

(6.50)

Since the magnitude of the magnetization is constant, its time rate of change mustbe perpendicular to it, and thus, no energy will then be supplied to the material. Thestored energy in the reversible component of the magnetization does not change,because the magnitude of the vector remains the same.

For anisotropic media, the total magnetization will still be in the direction ofan applied field if it is large enough; however, the models that compute theirreversible component of the magnetization compute a component that lags behindthe rotating field. Thus, they would compute an energy supplied to the medium. For

REFERENCES 165

the total magnetization to be in phase with the applied field, the irreversiblecomponent must then lead the applied field. The amount of lead depends on theirreversible state; thus, the reversible magnetization must be state dependent.Furthermore, it would compute energy given up by the medium which is equal tothat supplied to the irreversible component of the magnetization. Thus, the netenergy supplied to the medium in this case is also zero. For smaller fields, not onlydoes the irreversible magnetization start lagging behind, but also the lead of thereversible component decreases. Hence, there will be hysteresis loss in the material.

6.7 CONCLUSIONS

Vector hysteresis models must obey all the physical realizability conditions ofscalar models. These limits put certain constraints on.the parameters of a model.These constraints include the conditions that the magnetization cannot exceedsaturation, and that the energy dissipated by the material, for any change in appliedfield, must be positive. The latter constraint includes the crossover condition [7]which prevents minor loops from being traversed in the clockwise direction. Inaddition, vector models should be able to calculate magnetizations that do notexceed saturation and also correctly calculate the energy loss for large rotatingfields. For rotating fields, these losses for most materials must eventually decreaseas the amplitude of an applied rotating field increases, but for oscillating fields, theymust saturate as the amplitude of an applied field increases.

Many vector models have been proposed that have the correct rotationalproperties and reduce to scalar Preisach models under the appropriate conditions.Of these, the m model is the most computationally efficient. It is also the easiest oneto correct for observed deviations from the classical Preisach model, such asaccommodation and aftereffect.

REFERENCES

[1] I. D. Mayergoyz, MathematicalModelsofHysteresis,Springer-Verlag: NewYork, 1991.

[2] J. Oti and E. Della Torre, "A vector moving model of both reversible andirreversible magnetizing processes," J. Appl. Phys., 67(9), May 1990, pp.5364-5366.

[3] J. Oti and E. Della Torre, "A vector moving model of non-aligned particulatemedia," IEEE Trans. Magn., MAG.26, September 1990, pp. 2116-2118.

[4] E. Della Torre and F. Vajda, "Vector hysteresis modeling for anisotropicrecording media," IEEE Trans. Magn.,MAG·32, May 1996, pp. 1116--1119.

[5] F. Vajda and E. Della Torre, "A vector moving hysteresis model withaccommodation," J. Magn. Magn. Mater., 155, 1996, pp. 25-27.E. Della Torre and F. Vajda, "Vector hysteresis modeling for anisotropicrecording media," IEEE Trans. Magn., MAG-32, May 1996, pp.1116-1119.

166 CHAPTER 6 VECTOR MODELS

[6] E. DellaTorre, "A simplified vectorPreisachmodel," IEEE Trans. Magn.,MAG·34, March 1998,pp. 495-501.

[7] F. Vajda and E. Della Torre, "Characteristics of magnetic media models,"IEEE Trans. Magn., MAG·28, September 1992, pp. 2611-2613.

CHAPTER7

PREISACH APPLICATIONS

7.1 INTRODUCTION

This chapter introduces several indirectapplications of the Preisach model. Oneapplication dealswithmodifications to includedynamic effects.Anotherexplainshow magnetostriction can be introduced into the Preisach formalism. Theseapplications involvecouplingto other fields, such as eddy currentfields or stressfields inducedby the material'smagnetization. Theyare presentedto indicatethegenerality of Preisachmodeling.

7.2 DYNAMIC EFFECTS

In Chapter5, we discussed aftereffect, which is principally a long time-constanteffect.Wewillnowdiscussshorttime-constant dynamic effects.The twoprincipalshort time-constant dynamic effects are: eddy currents in conductors and inertialeffects,suchasgyromagnetism. Eddycurrents areinduced inconductors wheneverthefieldchanges. Inthecaseof magnetic materials thechangeinmagnetization canin turn induceeddycurrents. Eddycurrentshavetheeffectof shieldingthe interiorof the material from changes in the appliedfield. Thus, there is a strong spatialinteraction involved in the computation of the material's behavior.

In nonconductors, the principal dynamic effects are gyromagnetic; that is,whena magnetic moment isplacedin a magnetic field,its moment precesses aboutthefieldandeventually alignsitselfwiththefieldby dissipating someof itsenergy,

167

168 CHAPTER 7 PREISACH APPLICATIONS

since the aligned state is lower in energy. Thus, as we saw in Chapter 1, precessioncauses a domain wall to move with finite mobility, and causes the phenomenon offerromagnetic resonance. In othergeometries, many additional complex effects canbe observed, such as nonreciprocity. These effects are beyond the scope of thisbook and are not discussed further. The example of dynamic effects that we willdiscuss in the next sections are associated with eddy currents and reversal times.

7.3 EDDY CURRENTS

To understand the effects of eddy currents in magnetic materials, we consider firsta simplified model, seen in Fig. 7.1, in which a tape of magnetic material is woundinto a thin toroid whose inner diameter is almost equal to its outer diameter. Wefurther wrap a conducting wire around the toroid, carrying a current I, to producean almost uniform field inside the tape. We assume that the tape is made of auniform ferromagnetic material, and is rectangular in cross section, as shown inFig. 7.2. Furthermore, we assume that the coercivity is uniform throughout thematerial , that the tape is uniformly magnetized when saturated, and that themagnetization changes by nucleating a domain wall at each surface that propagatesinward and reverses the magnetization of each region that it passes.

The motion of the wall is retarded by eddy currents . They have the effect ofshielding the interior of the tape from the applied field . For this geometry, we cancalculate all the fields if we neglect the effect of the ends . Then the eddy currentsare uniform in the region between the surface and the domain wall, and zero insidethe domain wall. Their effect is to reduce the applied field to the coercive field atthe domain wall so that the wall can begin to move. The dynamic behavior of themagnetizing process is determined by balancing the wall's velocity with the effectof the eddy currents. If the wall velocity is too large, then the field at the wall willfall below the coercivity and the wall cannot move. If the velocity is too small, then

Figure 7.1 Toroid used to illustrate eddy current effects.

SECTION 7.3 EDDY CURRENTS

Figure 7.2 Crosssection of a tape.

169

there will be insufficient shielding, and the wall will be accelerated.With this geometry the eddy current density is uniform, so the total eddy

current is given by I, the eddy current density by J, and the distance that the wallis from the surface by x. Therefore, the field at the wall Hw, equal to the appliedfield H less the effect of the eddy currents, is given by

H w = H - Jx. (7.1)

The applied field is given by

H = NI ,r

(7.2)

(7.5)

where N is the number of turns in the magnetizing coil and r is the radius of a giventape element. Thus each turn of the tape experiences a slightly different field. Theeddy currents are determined by Ohm's law; that is,

J = oE, (7.3)

where E is the field induced by the eddy currents. This field is the negative of therate of change of magnetic flux divided by the path length. If the tape thickness isS, the rate of change of magnetic flux per unit length is given by

dxE = Ms- for x ~ s/2. (7.4)

dt

The total shielding current is computed by reducing the applied field to thecoercivity at the domain wall. Therefore, we have

dxI = oM x- = H - H .s dt C

This equation could be solved to give us the net magnetization M(s - 2x) as afunction of the applied field.

This model would assume that every time the applied field changes sign, a newdomain wall starts propagating inward from the surface. Unfortunately, thebehavior of a real material is much more complicated. The nucleation of a domainwall requires fields much higher than those required to propagate it. Furthermore,

170 CHAPTER 7 PREISACH APPLICATIONS

(7.6)

(7.7)

the coercive field is a random variable of the position. Thus, the domain wall doesnot propagate inward as a plane, but becomes distorted and may even break up intomany sections.

An alternate approach is a nongeometric one in which average magnetizationis computed without worrying about how it is distributed in the material. Bertottisuggested [1] that each point in the Preisach plane has a state, Q, that variescontinuously between -1 and 1 as a function of time. He then computes themagnetization as a function of time by

M(t) = SMs f fp(u,v)Q(u,v,t)dudv.

u>v

If at a particular point in the Preisach plane, the applied field is greater than u, thenfor that point the state function will vary according to

aQ = {k. [h(t) - u], when h > uat k·[h(t)-v], when h < v,

where k is an unknown parameter. This parameter varies inversely with theconductivity of the material and would be infinite if the material had zeroconductivity. In this case, the state function would change instantaneouslywhenever the field exceeded u, as in the case of the classical Preisach model. Thus,this calculation correctly reduces to the classical model for insulators.

This model predicts a hysteresis loss as a function of magnetizing frequencythat can be described by

(7.8)

where w is the hysteresis loss per cycle, W is the frequency of the applied field, andCI and c2 are monotonic increasing functions of the peak of the applied sine wavemagnetic field. The latter two constants are a function of the material and thegeometry. This is consistent with measurements.

7.4 FREQUENCY RESPONSE OF THE RECORDING PROCESS

The frequency response of the recording process is determined principally bythe recorded wavelength. Thus, if the media speed past the recording head isincreased, the recorded wavelength is increased for a given frequency signal. Thus,neglecting the effect of the circuit parameters in the head, doubling the speedeffectively doubles the frequency response of the process. Of course the reactancesassociated with the head windings and the ability of the media to respond to theapplied fields will ultimately limit the ability of the medium to respond to thesignal.

Two factors control the frequency response of the recording process. The firstis due to the inability to localize the magnetic field. Thus, even if one uses a ring­type recording head with zero gap width, the magnetic field is not very welllocalized, as shown in Fig. 7.3. In particular, one gap length away from the head,

SECTION 7.4 FREQUENCY RESPONSE OF THE RECORDING PROCESS 171

the perpendicular component of the magnetic field eventually decreases as thereciprocal of the distance from the gap, and the longitudinal field eventuallydecreases as the square of the reciprocal of the distance from the gap. It is seen thata vector model is necessary to analyze properly the recording process . Even thoughthe strongest component of the magnetic field is longitudinal, there is a sizableperpendicular component. Furthermore, the perpendicular component eventuallydominates the magnetizing process.

The second factor that controls the recording process is related to thecharacteristics of the recording media. Even if the standard deviation of theswitching field is zero, a transition would have finite width because of the spreadof the head field. We see that a nonzero switching field distribution also affects thefrequency response. This is because it takes the vulnerable part of the medium afinite time to pass the region of the recording head where the field is of the orderof the coercivity .

For a step function in the applied field and zero switching field distribution,the transition occurs at the place where the applied field has decreased to thecoercivity of the material. Therefore, if the amplitude of the applied field ischanged, the location of the transition will move to a position that satisfies thiscriterion. If the switching field distribution is not zero, the transition will have afinite width. The hysterons with the smallest switching fields will be writtenfurthest downstream from the gap, and the hysterons with the largest switching

.__L_L__..-J-Jt l\ +-.-f---.+--+-

.. j ; '1-' ..······1--· ~.. . '--j-'" .. . ·.. 1··_···...L,..·._--+_.-t·---- ----+----,-- --- --- - -"-'-1_.._- .._..._.- -·-r----+----ii i i i i I

, i V \ i ' Jo5 1---1---+-- I . ~. ...-:;--r- --j-.--j--+-----j• ! ' I! !\ .... i i

'C ! I / i \ 1~r-' I

i C--r---4-~:i=::::: +-- =:~::1=:::~ 0 . ._ ) j - 1... - -.1--.-- ; 1 _ - ...1 - ...1 .

! ! I! : .. "·-·,,r-=·:.-t-..- ""'--i-t---f'-f--- Applied field ._-I ,..... i I . L . dinal_..·_..·..r ..·..·..t-·....:.:,. ':~-"'-r---+"'!" -- ongitu I-

I i ····. ! I : ......... Perpendicular I_i i I·····.. i ' I I I I

-0.5 r--r-r I i i iI

-6 -4 -2 o 2 4 6

Distancefromgap (units of gap width )

Figure 7.3 Field due 10 a ring-type head.

172 CHAPTER 7 PREISACH APPLICATIONS

fields, which are affected by this head, will be written closest to the gap. Thiseffect, for ac-bias recording, is discussed elsewhere [2].

7.5 PULSED BEHAVIOR

Accommodation has been observed in particulate recording media [3,4], underpulsed conditions. It appears that the source of this effect may be the statisticalstability of the Preisach model. We now examine how this variation can beexplained by the Preisach accommodation model discussed in Chapter 5. When afield is applied, the magnetization normally will change, causing all the hysteronsto move within the Preisach plane as well. Normally, hysterons whose positiveoperative switching fields are less than an applied de field will switch to theirpositive state. While the field is reduced to zero, for fourth-quadrant media,normally no further switching occurs; however, the final magnetic state of thesystem will be different if the pulses are long enough to nucleate a magnetizationreversal, but not long enough for the reversal to take place.

The experiment described by Flanders et al. [3] involves the change in the finalremanence caused by a pulse when it precedes a larger pulse. There are twosources for this difference in the model described below: the motion of thedistribution as a whole, which is described by the moving model, and the motionof hysterons within the distribution, which is described by the Preisachaccommodation model as the source of the accommodation.

It has been known for some time that in soft materials the field required tonucleate a domain wall is much larger than that required to propagate it. Numericalmicromagnetic studies [5] have shown that for hysterons used in recording media,the field required to nucleate a reversal HN is also much larger than that requiredto propagate the reversal throughout the hysteron.

The simulation of the dynamics in this process is based on two characteristictimes associated with the reversal process: the nucleation time and the actualreversal time. Nucleation time tN is the length of time that the nucleation field mustbe applied in order for the magnetization reversal to nucleate; reversal time tR is thetotal time required to complete that reversal. The nucleation time decreases if thefield applied is increased beyond the minimum field, but even the longestnucleation time is usually much shorter than the reversal time. When a reversal hasbeen nucleated, the applied field may be drastically reduced, sometimes even tonegative values, without affecting the completion of the reversal process.

Although each hysteron in a medium can have a different applied-field­dependent tN and a different applied-field-dependent tR , to simplify the model wewill assume that these times are the same and constant for all hysterons. The fielddependence of tRis not a serious source of error, since in the experiment described,the field will always bezero during reversal. We assume that a hysteron subjectedto a field pulse, whose strength is HNand whose duration is tp , will switch if tp > tN'

and will not switch if tp < tN. Furthermore, if tN< tp <tR, the hysteron will reverse,but it will complete its reversal after the pulse has ended.

SECTION 7.5 PULSED BEHAVIOR

7.5.1 Dynamic Accommodation Model

173

The source of accommodation in the model discussed in Chapter 5 is the motionof hysterons in the Preisach plane whenever the magnetization changes. Unlike thehysterons of the classical Preisach model, in its new position a hysteron mightfind itself with a different magnetization from nearby hysterons - perhaps becauseit acquired its magnetization at its old position, and during its motion in theoperative plane did not experience a field large enough to reverse it. In theaccommodation model, it was assumed that the field was applied until themagnetization had achieved steady state. For short pulses it is probable that ahysteron's position will start to change after the pulse has ended. It will then havea different remanence from that which it would have had if the field had been keptconstant until steady state had been achieved.

The Preisach plane is divided into six regions, as shown in Fig. 7.4, for amedium whose history is suggested by the staircase line. At zero field, themagnetization in region I is always kept positive, and the magnetization in regionVI is always kept negative. In regions II and III the magnetization is essentiallypositive owing to its history, but may become diluted as a result of accommodation.Similarly, regions IV and V are essentially magnetized negatively. Application ofHA nucleates reversals in region V, which starts the motion of the hysterons in thePreisach plane. Subsequent application of a field pulse HA will nucleate anynegative hysterons that have moved into regions II and V, leading to accommo­dation. Table 7.1 compares the magnetization state of a region before theapplication of a field with that computed by Preisach models and with thatcomputed by this model after the application of a pulse.

In Table 7.1, "Same" indicates that the hysteron remains in the state definedin the region it came from, which may be different from that computed by theclassical Preisach model. It is seen that for long pulses, the Preisachaccommodation model dilutes regions III and IV, while short pulses dilute regions

~---..l~---VI------4_____ 8+

Figure 7.4 Division of the Preisachplaneinto six behavioral regions.

174 CHAPTER 7 PREISACH APPLICATIONS

II and V as well. Thus, there is a change in magnetizationat theconclusionof eachpulse. After many pulses, it is expected that the Preisach accommodation modelwill asymptoticallyapproach the same equilibrium magnetizationfor either shortor long pulses, but this magnetizationwill be different from that computed by theclassical Preisach model. It was shown in Chapter 5 that the amount of dilutiondepends on the average magnetization and the change in magnetization.

Table 7.1 Hysteron Magnetization State

Region Previous State Preisach Models

II

III

IV

v

VI

+

+

+

+

+

+

+

+

+

same

same

+

+

same

same

same

same

We will assume that the mediumis a single-quadrantmedium,so that Ptu,v)is zero if either u is negative or v is positive. The state variable Q can take anyvalue from -1 to 1, to account for the dilution of the region due to the motion ofhysterons in the Preisach plane. If we define the component of the Preisachfunction due to regionj (j =I, II, III, etc.) by

Pj = f p(u,v)dudv , (7.9)j

then the normalization is

LP. =. }J

J p(u, v)dudv = 1.u>v

(7.10)

We will also assume that Q(u, v) is constant in anyregionof the Preisachplane anddefine Mj to be the remanencecontributiondue to region j in the operative plane;then

Thus,

Mj = Qj SMs fp(u,v)dudv = SMsQjPj"j

M; = SMs E QjPj.j

(7.11)

(7.12)

SECTION 7.5 PULSED BEHAVIOR 175

If a fieldHI is appliedto a medium that is negatively saturated, the Preisachplaneis dividedas shownin Fig. 7.5, wherehi is the operative field,HI + aM, and ex isthe moving parameter. Forpulsessuchthat tp >tR, thestatevariable QI' associatedwithregionI, willbe+1; however, if tN < tp < tit, it willbe dilutedto a smaller, butstillpositive, value. The subsequent application of fieldH2 willincreasethe valueof Q in regionII from -1 to a maximum of +1, if it is held for a sufficiently longtime.

We willdefineM(H2)to betheremanence aftera negatively saturatedmediumhas been subjected to a field pulse, H2, and we will define M(H., H2) to be theremanence after the samenegatively saturated medium has beenfirst subjectedtoa field pulseHI, followed bya fieldH2 , whereHI < H2• The experiment describedearlier [3] compares M(H2) withM(Ht,H2) ; these authors found that fewerof thehysterons switched in the second case, especially when H2 is the order of thecoercivity. It is notedthatM(H2) =M(O, H2) .

In the model presented here,twoeffectsaccountfor thisdifference. Thefirstis due to the difference in operative fields. For a singlepulse, the operativefieldis H2 + a [M,(H2) - S Ms], where M,(H2) is the reversible component ofmagnetization whenH2 isapplied. Whentwopulsesareapplied, theoperativefieldat the secondpulse is H2 + a[MI + M,(H2) ] , whereM1 is the magnetization due tothe first pulse. This field operative is morepositive than -S Ms. For positivea,M(H1, H2) is a monotonically increasing function of HI; for negative a it is amonotonically decreasing function. In particular, if thereis noreversible magneti­zationand the Preisach function is Gaussian, the remanence is proportional to theerror function of HI .

Thesecondeffectisduetoaccommodation [3].Thestatevariable inregionIII,after the application of HI , is givenby

QIIIl = QII10 + P <M> dM, (7.13)

Figure 7.S Division of theoperative planewhen fields b, andh2 are applied.

176 CHAPTER 7 PREISACH APPLICATIONS

where QUI 0 and QUIt are the initial and final state variables, respectively, p is anaccommodation constant that determines the fraction of hysterons at a point on thePreisach plane that come from other regions, <M> is the steady-state averageremanence, and 11M is the total change in magnetization. Since 11M is equal to 2PIand <M> is equal to SMs (PI- Pn - PIlI)' we see that

QIIIl = -1 + yPI' (7.14)

where y =4 pSMs . Then, from (7.12), after the application of the second pulse,the resulting remanence is

2M2(Hl'H2) = PI + PII - PilI + VPIII PI . (7.15)

In the case of a single pulse equal to the coercivity, we have Pn = PilI = 0.5, andM(O, H 2) = O. For a double pulse, we see that M(H]t H 2) = 1 - 2Pul + Vp

2III' andtherefore, M(H} , He) =Vp211I. For example, if HJ is chosen such that PI=0.25, thenfor H2 still at the coercivity, Pn =0.25 -Psu=0.5, and thus, M(H t , H2) =v/32 .

To reproduce the results in [3], one could assume that a is negative and thatthe two effects described above are roughly equal when HI =H2 • A calculation ofthe remanence difference, L1 =M(H., H2) - M(O, H2) , as a function of HI /H 2 , forhigh squareness media is shown in Fig. 7.6. This is similar to the result obtainedexperimentally in that paper [3]. A quantitative analysis of this effect wouldrequire the identification of a complete set of the model parameters for a givenmedium.

Modeling the overwrite process in very high frequency recording requires asimple model that calculates the variation of the remanence with pulse height andwidth of the applied field. The model we present here is the DOK model, a movingmodel with magnetization-dependent locally reversible magnetization [6], to whichwe have added accommodation effects [3]. This extension of our results [7],assumes that once the critical field for a hysteron has been reached, itsmagnetization will start to change only after a nucleation time, tN' whereupon, it

Figure 7.6 Ratioof remanences as a function of HI /H2•

SECTION 7.5 PULSED BEHAVIOR 177

(7.16)

will rotate at such a rate that its magnetization varies linearly from state to state ina time, tR, even if the applied field is then removed.

We will compute the variation of the remanence of a medium that has beeninitially saturated negatively (down) after the application of two positive pulses(up) of various heights and lengths and compare these results with themeasurements of Doyle et al. [3]. Before further refinement of the model isundertaken, one must identify the medium's parameters through careful analysisand compare the model results quantitatively with experiments.

The irreversible component is obtained by integrating the product of thePreisach function P(u, v) and the state function Q(u, v) where u and vare the "up-"and "down-" switching fields, respectively. Thus,

mi = f Q(u,v)p(u,v)dudv =~ QjPj'u>v }

The state function is either +1 or -1 for the classical Preisach model, but in theaccommodation model, because of dilution, it can take an intermediate value.There are three ranges of applied field to be considered: If the applied field islarger than the value of u in a region, Q is set to +1; if the applied field is smallerthan the value of v in a region, Q is set to -1; otherwise, Q is unaffected by thatfield.

Accommodation occurs when the magnetizationchanges and the interactionfield changes at all hysterons. Thus, the positions of hysterons in the operativeplane change. Therefore, the value of Q in an unaffected region is modified byhysterons coming into that region from other parts of the plane, carrying with themtheir original magnetization. As in Chapter 5, we will assume that the value of Qin such a region is given by

Q = (l-pam)Q'+pldml<m>, (7.17)

where Q' is the old valueof Q,p is the accommodationconstant, !1mis the changein normalized magnetization, and <m> is the average normalized magnetization.

In this model we will use the DOK characterization of the locally reversiblecomponent of the magnetization, so that

m.+l m.-lm, = -'2-.f{H) + -'Z-f( -H), (7.18)

wheref(H) is the variationof the reversible magnetizationwhen a hysteron is in the"up" state. In the following simulation, we will use the following function for.f(h):

.f(h) = 1 - exp( -~:). (7.19)

Although this is a monotonic increasing function, its slope is a monotonicdecreasing function. Sincej{H) is zero if H is zero, andj{H) is always greater thanj{-H), if H is held constant, then from (7.18), as m, increases, m, will decrease.

178 CHAPTER 7 PREISACH APPLICATIONS

Defining the region RJ to be the physical region of the operativeplane to theleft of the line h = hi is convenient. If we assume that the Preisach function isGaussian, then it has been shown [8] thatPI is given by

(7.20)

where erf is the error function. Defining the remainderof the physical region ofthe operative plane to be R2, when an "up" field of strength H is applied to amediumthat is in the "down" state, we have

m; = PI + Q2P2' (7.21)

whereQ2wouldbe -1 if therewerenoaccommodation, but nowis given by (7.17),where Q' is -1.

When an "up" pulse whose time duration is greater than tN is applied to ahysteron, we will assume that the variationof its momentwith time is given by

-1 if t < tN

t - (tN + tR/2) ifget) IN < I < tN + tR (7.22)

tR/2

Therefore, m, as a function of time is given by

P2m;(t) = PI get) + 2' [Q2 + 1 + g(t)(Q2 -1)],

where the state of region 2 varies from Q=-1 to Q= Q2'

7.5.2 Single-Pulse Simulation

(7.23)

We willnow assumethat the mediumis saturated"down" and that at t =0 an "up"pulse whosestrengthis HI and whosedurationis tOJ is applied. As longas t is lessthan tN'nothinghappens. After that, themagnetization willstart tochangelinearly;however, for positive a as the magnetization changes, the operative field willincrease, thereby increasingthe slope. If the duration of the pulse is long enoughto permitall the hysteronsthat are going to switch to completetheir switching,thesystem will be in equilibriumat the conclusionof the pulse.

Although the applied field is constant during the pulse, the operative field hvaries with the magnetization. The irreversiblemagnetization varies accordingto(7.23) and, althoughJ{H) and .f{-H) remainconstant, the reversiblemagnetizationvaries because the state changes according to (7.18). Thus, the operative field isgiven by

SECTION 7.5 PULSED BEHAVIOR 179

(7.24)

................. m,----- m,---m

I:I :

I :I ..

II

II

I/

I______1-0.4

~ .

0.6/B 0.'1 ..1

.+:; ~----.g 0.2

u~ 0«I

~ -0.2

For positive pulses, mi(t) will increase, which in turn causes m,(t) to decrease.Thus, these two magnetization changes are in opposite directions. To solve for themagnetization, one must substitute this operative field into (7.20) to compute m,using (7.21) and obtaining m, from (7.18). Since these equations are implicit in mi,

they have to be solved iteratively.Figure 7.7 illustrates the variation of the magnetization with time for a pulse

whose duration, 6 arbitrary units, is less than the reversal time of a hysteron. Whenthe pulse is applied, m, immediately responds. The change in m,after the nucleationtime of 3 units, causes m,to decrease, since it is state dependent. At the conclusionof the pulse, m, immediately decreases to zero; however, the model assumes thatm,and m both continue to change until tR• The total magnetization, the solid line,is simply the sum of these two components.

The calculated variation in the remanence is a step function of the pulse width,as shown in Fig. 7.8. There is no change in the remanence until the nucleation timeis reached. After that, the remanence changes whether the pulse is there or not.When the pulse is finished, the change in magnetization will cause the change inlocation of hysterons in the Preisach plane that is the cause of accommodation;however, the motion of hysterons in the plane cannot change the remanence unlessthey encounter an applied field, which is now zero, greater than their switchingfield. The pulse width dependence changes only the initial conditions for theapplication of a second pulse.

The step function behavior is due to the model's assumption that once itscritical field has been exceeded, a hysteron will continue to reverse, even if theapplied field is turned off. If one modifies this behavior to that of reversing onlya fraction of the hysterons depending on the fraction of the magnetization changethat has occurred, then one would get a ramp increase in the remanence with pulse

0.8

-0.6'-- -L- ---'

o 2 4 8 8Time (arbitrary units)

Figure 7.7 Variation of the total magnetization and its components whena singlepulseis applied.

180 CHAPTER 7 PREISACH APPLICATIONS

0.6

0.4

0.2

8

J 0

-0.2

-0.4

II

J

-0.6o 6

Pulsewidth12

Figure 7.8 Pulse width dependence of theremanence.

width after the nucleation time.The pulseheight dependence of the remanence for the samepulse shown in

Fig. 7.7 is illustrated in Fig. 7.9. It is seen that the remanence varies from-Sto +S,where in this case, Sis 0.5. For this choice of material parameters, this curve isessentially the sameas the major remanence loopfor this material.

542 3Pulse height

~~

IV

/-7~L'

-0.6o

-0.4

0.4

0.6

0.2ug~ 0

!-0.2

Figure 7.9 Pulseheightdependence of the remanence .

SECTION 7.6 NOISE

7.5.3 Double-Pulse Simulation

181

We now assume that the medium is saturated in the "down" direction. At t =0, apulse in the "up" direction whose strength is HI and whose duration is tD I isapplied. This is followed by a second "up" pulse, at t = t1, whose strength is H2 andwhose duration is tD2• The initial condition is different when the two pulses areapplied. For the first pulse the entire fourth quadrant of the Preisach plane had aQ of -1.

For the second pulse the fourth quadrant is divided vertically in two regions.To the left of the line v =hmax, Q=1 at the end of the pulse if tDI is greater than thesum of tN and tR• It will then accommodate to a value determined by the change inm; For shorter pulses the change in magnetization will be increased by thecompletion in the change in m; These two changes will have the opposite effect.To the right of the line v = hmax, the value of Q starts from -1 at the beginning ofthe pulse and will accommodate to a more positive value. In both cases the valueof Q is computed from (7.17).

When the second pulse is applied, m,will change immediately. The region tothe left of the operative field will then start reversing as indicated by (7.22). In thiscase, the value of Q will increase only slightly, since only the hysterons that haveaccommodated into that region need to be reversed. The analysis is morecomplicated if the height of the second pulse is different from that of the firstpulse.

For second pulse heights greater than the first pulse, the analysis is similar tothat given for the first pulse. For smaller pulse heights, the Preisach plane isdivided into three regions: the region to the left of the new operative field, thatbetween the new operative field and the old one, and that to the right of the oldoperative field. The first region will have Q= 1as long as the pulse is applied.; thesecond region will have a Q somewhat less than 1; and in the third region Q willhave a value somewhat more than -1. The difference between the magnitude ofthese Q's and 1 is due to accommodation.

7.6 NOISE

The theory of Barkhausen noise in recording media has been studied extensivelyfor recording media consisting of noninteracting hysterons. This noise occursbecause the magnetization changes in discrete steps, and as a result, themagnetization curve is a staircase instead of a smooth curve, as shown in Fig. 7.10.A smooth curve would have no noise. Interaction between hysterons increasesnoise by reducing the number of independent magnetic states available to thesystem by the cooperative magnetization of otherwise independent hysterons [9].The inclusion of interaction into this theory requires a physical model of themagnetizing process. In this section we will use an extended Preisach model thatincludes accommodation and noncongruency effects.

182

Smooth, noiselessmagnetization curve

CHAPTER 7 PREISACH APPLICATIONS

M

H

Realmagnetizing processwithBarkhausen noise

Figure 7.10 Staircase ascending majorloop as a resultof Barkhausen noisein the magnetizingprocess as contrasted to a smoothnoiseless magnetization curve.

In addition to the other sources of noise in a recording system, Mallinsonsummarizes the theory of Barkhausen noise in noninteracting particulate recordingmedia in his excellent summary article [10]. He shows that the noise power PN ofa fully saturated recorded bit, if all the hysterons in the medium are identical, isgiven by

(7.25)

where m is the dipole moment of each hysteron, N is the number of hysterons perunit volume, w is the track width, Vis the head-to-medium velocity, l) is the coatingthickness, and d is the head-to-medium spacing. This formula is derived with theseassumptions:

The head efficiency is 100%.The head is able to capture all the flux from the recorded bit.The recording medium is very thin.The head has one turn.There is no gap loss.The head is connected into a one-Ohm load.The hysterons do not interact.

The assumption that the recording mediumis thin implies that the magnetizationis uniform throughout the thickness of the coating. This assumption, like theothers, can easily be corrected.

The effect of hysteron interaction, on the other hand, requires some knowledgeof how interaction affects the recording process. We will now examine the effectof medium thickness.

SECTION 7.6 NOISE 183

(7.27)

Let us define K to be the number of hysterons in a half-wavelength, 'A/2. Forthin coatings, say less than one-third of a wavelength, we can assume that therecording is uniform throughout the coating. For thicker coatings, however, thepenetration depth of the recording into the coating is limited by the wavelength. Wecan adopt the following rule of thumb for K:

K = { Nw'Ao, if 0 s 'A/3 (7.26)Nw}..,2/3 , if 0 > 'A/3.

In the remaining equations in this section, we will assume that 0 ~ 'A/3, so that(7.25) may be written

P = 41tm 2KV2 d + 0/2N 'Ad 2(d +0)

The noise power spectral density in a wave interval number ~k is given by

e;(k) 41tm)..20KV2Ikl(l_e -21Id6)e -2l1ddak, (7.28)

where k is given by

k = 21tf.V

(7.29)

For sine wave recording, the maximum possible signal power spectrum is given by

2(k) [1tmKV(l -fkI6) -fkld] 2 (7.30)es =~ -e e .

Thus, the maximum signal-to-noise ratio SNR is given by

K(1- e -1k16)2

t>(l - e -21k16)ak(7.31)

It is seen that the SNR is independent of hysteron moment or the head-to-mediumvelocity; however, this is only the SNR due to Barkhausen noise. The contributionto the SNR due to the remainder of the noise does vary with the hysteron moment.We shall modify these formulas by including the effect of interaction calculated bythe eMH model with accommodation.

7.6.1 The Magnetization Model

A system of K noninteracting hysterons has 2K possible states. Of these states, amagnetizing process starting at negative saturation and going to positive saturationtraverses K of these states. As the field is increased, hysterons with the lowestcritical field will switch first. The distribution of critical fields affects only thelinearity of the magnetizing process. For example, if the distribution is Gaussian,

184 CHAPTER 7 PREISACH APPLICATIONS

the magnetization curve is an error function. If a system of interacting hysteronsis describable by the classical Preisach model, then when the field is increased justenough to switch one hysteron, only that hysteron will switch. Therefore, thehysterons may switch in a different order depending on the history of the process,but there will be no effect on the number of available magnetization states. Thus,the main effects of this type of interaction are to modify the magnetization curveand to redistribute the magnetization noise, not the total noise power associatedwith the process; but the system will still traverse K states in going from negativesaturation to positive saturation.

The effect of interaction normally increases the amount of noise by reducingthe number of independent hysterons that can switch. That is, interaction candecrease the number of independent states when pairs of hysterons switch as asingle unit. Hence, this additional noise will be referred to as excess Barkhausennoise. This effect may also modify the magnetization curve by redistributing theswitching fields.

Two modifications have been made to the classical Preisach model whichaffect the number of available states: the moving model modification [11] and theaccommodation model modification [3]. These two modifications have removedthe congruency property limitation and the deletion property limitation,respectively, of the classical Preisach model. Although the product model [12] alsoremoves the congruency property, it does not appear to be applicable to recordingmedia.

7.6.2 The Effect ofthe Moving Model

We will assume that K, the total number of hysterons in the system, is the sum ofthe number of hysterons that switch independently, Kind, and the number of groupsof hysterons that switch cooperatively, Kcoop; that is

K = K;nd + Kcoop . (7.32)

When the applied field is increased by aH, two regions are switched in theoperative plane, as shown in Fig. 7.11. In this process, ilK hysterons are switched;IlKind of them are switched independently, and ~Kcoop are switched cooperatively.Thus,

IiKind = Jp{u. v)dudv,J

and

IiKcoop = Jp{u,v)dudv,Jl

where P is the Preisach function. Then,

t:.K = 1 + llKcoopt:.K;nd dK;nd .

(7.33)

(7.34)

(7.35)

SECTION 7.6 NOISE

v

u

RegionI (switchedindependently)

RegionIl (switchedcooperatively)

~aAM

185

(7.38)

Figure 7.11 Regionsof the operativeplane that are switchedwhenthe appliedfieldin increasedfromHtoH+ sn.

For this change in the applied field, the number of hysterons that wereindependently switched is now given by

t!KbaK;nd = (7.36)

1 + 4Kcoop/4K;nd

For smallchanges in dB, the ratioof dKind to dKcoop is givenby a dM/4R. Thus,(7.36) can be rewritten

dK. = 11K 11Kmtl 1+aliMlliH - 1 +ax I (7.37)

where Xis the susceptibility. It is seen that the numberof independent states isnormally smaller than the number of hysterons, since X is positive and a isnormallypositive. If a wereever to be negative, the SNR in some cases could begreaterthan the casefor noninteracting hysterons; however, thismaybe permitted,since there are many more states than just those traversed when the hysterons donot interact. Furthermore,since Xis a function of both the applied field and themagnetization, the decrease in the numberof independent states depends on boththe magnetization and the appliedfield. The total numberof independentstates isthen obtained by integrating(7.37); that is,

K. = fa> dKldH dH.ind -00 1 + ax

It is seen that if ex is zero, then the numberof independentstates is the same as thetotal numberof hysterons.

186 CHAPTER 7 PREISACH APPLICATIONS

We will decompose the susceptibility Xinto a reversible component~ and anirreversible component 'Xi. Thus,

X = X, + Xi· (7.39)

This is a function of both the magnetic state and the applied field. In particular, ifthe reversible function, f{H), can be factored, in the same way as in the CMHmodel, the reversible susceptibility is given by

For the simplified case, the DOK model, we have

m;+ 1 _ mi-la+ - ---2--- and a - ---2---'

(7.40)

(7.41)

where m,is the normalized irreversible component of the magnetization. Otherwise,the a's are Preisach-like integrals.

The irreversible component of the susceptibility as computed by the Preisachmodel is

H+aM

X.;<H,M) = J p(H +aM,v) dv I (7.42)

where p is the Preisach function. Substituting these equations into (7.38) gives thenumber of independent states as computed from the eMH model alone.

7.6.3 The Effect ofthe Accommodation Model

The accommodation model introduces a second type of cooperative effect thatoften occurs in hysteretic many-body problems. Not only does the Preisachfunction move in the plane, but also hysterons will change their position within thefunction. Whenever hysterons change their position, they may cross the linedefined by the applied field H as shown in Fig. 7.12. If the change inmagnetization causes the hysteron in the illustration to move from position 1 toposition 2 in the hatched region, the hysteron will switch. This is an additionalexample of cooperative switching which further reduces the number of statesavailable.

The degree of saturation of a region, Q, can be computed by theaccommodation model as

dQ = P(Mi-SMsQ)ldM;,dH S2M; dH (7.43)

where pis the accommodation constant. We see that if pis zero, there will be nodilution of the magnetization in any region. In this case, the rate of change in thenumber of independent states is given by

SECTION 7.6 NOISE

u

187

Figure 7.12 When a field is applied, the hatched region is magnetized in the positive direction. Asa hysteronmoves from position 1 to position 2, the local field becomes sufficient tomagnetizeit positively.

dK

dH-K~~ f f p(u,v)dudv.

H<H+a.M(7.44)

It is noted that this quantity is negative, since Qincreases when the field increasesand the Preisach integral is always a positive fraction. We can write this as

aK = P(M;-SMsQ) aM. f P(u,v)dudv.K 2 2 I (7.45)

S Ms H<.H+a.M

It is seen that if pis zero, there will be no change in the number of available states,hence, no excess Barkhausen noise. The region of integration is the region wherethe hysterons have a positive switching field that is smaller than the appliedoperative field. It is noted that the quantity on the right-hand side of (7.45) is lessthan 1, so that the number of states is again smaller. We see that if pis zero (i.e.,there is no minor loop accommodation), there is no decrease in the number ofindependent hysterons due to this process. It is noted that although the major loopdoes not accommodate, it is still susceptible to this type of excess Barkhausennoise.

In a system with both motion and accommodation, the excess noise is the sumof the two effects. Furthermore, the two effects interact: Any accommodationproduces a change in magnetization, which moves the Preisach function and resultsin a loss of independent states due to motion; also any motion changes themagnetization, which in turn causes accommodation. For completeness, the effectof reversible magnetization must be included into the accommodation calculations.

It is noted that the cooperative effect is not the same for all magnetizations.In particular, the moving model produces less excess noise when the susceptibilityis small, such as the case of near saturation. The accommodation model also

188 CHAPTER 7 PREISACH APPLICATIONS

produces less excess noise near saturation, since the accommodation model isdriven by the change in magnetization.

The analysis above was carried out for an increasing applied field. For anapplied field decreasing from positive saturation to negative saturation, the signsof dM/dH must be changed. In this case, the overall effect is still the same: Boththe moving model and the accommodation model decrease the number ofindependent states.

7.7 MAGNETOSTRICTION

Highly magnetostrictive media, such as Terfenol-D, are useful for transducerapplications [13], but are also hysteretic. Their usefulness as linear actuators islimited to a small fraction of their capability unless they can be accuratelycontrolled [14]. The first step in controlling these materials is to develop anaccurate, efficient model. Modeling of this material has been extensively discussedin earlier work [15] and [16,17]. Here, we modify Preisach models with state­dependent reversible magnetization to model magnetostrictive behavior [18].

Both the magnetization and the strain of a magnetostrictive material arehysteretic when viewed as a function of applied field. Two moving Preisachmodels-the DOK model [19] with magnetization-dependent reversiblemagnetization, and the more accurate eMH model [20] with state-dependentreversible magnetization--ean accurately characterize the magnetization of somemedia. In this section, the DOK model is modified to also characterize the strainof magnetostricti ve material.

Figure 7.13 shows a typical plot of measured strain versus applied field for theparticular magnetostrictive material Terfenol-D [21]. For this material, strain is an

3000oOl.---"'----"""'---~--'"""---------'

-3000

1000

Applied field(oe)

Figure. 7.13 Measured strain vs applied field for Terfenol-D (courtesy of J. E. Ostensenand D. C. Jiles).

SECTION 7.7 MAGNETOSTRICTION 189

Compresive ~ Expansive

Applied field

Figure 7.14 Effectof an appliedfieldon an acicularparticle.

expansive, even function of the applied field; that is, it elongates in the presenceof a field.

To model this behavior, we will assume that the medium consists of hysterons,which are either particles or grains whose shape may be acicular or platelet.Because of the anisotropy of the hysterons, if their axes are not perfectly alignedwith the applied field, the medium will not have unity squareness, When a field isapplied to this medium, a torque is applied to each hysteron, which in turn appliesa stress to the medium, since the hysterons have shape anisotropy.

The torque, and consequently the stress, depends on the direction of themagnetization along the hysteron's easy axis, and thus is state dependent. Asillustrated in Fig. 7.14, if the applied field makes an obtuse angle with themagnetization, which we will call the "negative magnetization state," the stress iscompressive for acicular hysterons. If it makes an acute angle, which we will callthe "positive magnetization state," the stress is expansive. This set of definitionsimplies that if the medium is demagnetized, the stress field is zero. However, if thematerial is magnetized, it is not zero.

Change in magnetization is due to the rotation effected by the torque suppliedby the applied field. This rotation from the hysteron's easy axis is opposed by thevariation in the demagnetizing field for hysterons with shape anisotropy. When theapplied field is removed, the magnetization will return to the easy axis. Thus, therotational energy supplied by the applied field is returned when the field isremoved.

An applied field also produces a torque on the hysteron, which attempts torotate it in the same direction that the magnetization is rotated. In the case of themagnetization, there is a restoring torque due to the hysteron's shape or due to themagnetocrystalline anisotropy of the particle/grain. In the case ofmagnetostriction,the rotation is opposed by the binder that holds the material together ~ Assumingthat the magnetization of the hysteron is constrained to its long axis, then in bothcases, a certain amount of rotation produces the same fractional increase inmagnetization as the fractional increase in length (strain).

In both the DOK and the CMH models, the reversible component of themagnetization M, is given by

190 CHAPTER 7 PREISACH APPLICATIONS

(7.46)

(7.47)

whereS is the squareness,determined by the angulardistribution of the hysterons,Msis thesaturationmagnetization, and.f{H) is thenormalized reversiblecomponentof themagnetization whenthehysteronis in itspositivestate.Thesquarenessis theratioof themaximumremanence to thesaturationmagnetization. Thefunctionj{H)is essentiallydeterminedby the hysteron's anisotropyand is a monotonic functionthat approachesunityasymptotically as H becomes large. The differencebetweenthe DOK and CMH models is in the methodof calculatingthe a's. In the DOKmodel, the a's are given by

M; + Msa+ = and a

2Ms

where M i is the irreversible magnetization. That is, the DOK model ismagnetization dependent. TheeMH model, whichis statedependent, uses a morecomplexexpressionfor the a's.

Since the functional variation of the reversible magnetization and themagnetostriction are the same in a given state, we will use the same function forboth; however, for a magnetostrictive material, the stress and the reversiblemagnetization have the opposite effect when the magnetization is in the negativestate. Therefore, we will describe the stress T by

T(H,M;) = K [a+(M;) j{vH) + a_(M;) f{-vH)] , (7.48)

wherethea's are givenby(7.47). Sincef{H)approaches unityasHbecomes large,wehaveto introducethe factorK for lengthnormalization. Similarly, weintroducethefactorvas theratioofmagnetostrictive susceptibility tomagnetic susceptibility.The latter factor is determined by the relative effectiveness of the magneticanisotropy and the binding forces that try to keep the hysteron oriented in aparticularway. The constantK is positivefor acicularhysterons and negativeforplatelet hysterons. This produces the desiredpropertiesthat Tis zero if H is zeroor if a. =a, (zero magnetization in the OOKmodel). Since the DOKmodeldoesnot differentiate betweenthe demagnetized states and the CMH model does, thetwo models will behavedifferentlyfor demagnetized media.

In the present simulation we assumethat the magnetic material is representedby the DOK modeland that the movingconstant is zero. We will further assumethat thePreisachfunctionisGaussianandthatthereversiblefieldvariation is givenby

(7.49)

where~ is thenormalized zerofieldsusceptibility. Startingwithadc-demagnetizedspecimen,the irreversible componentof the magnetization is given by

SECTION 7.7 MAGNETOSTRICTION

( H-Hrem)Mj =SMs erf 0 '

191

(7.50)

where Hrem is the remanent coercivity and 0 is the standard deviation of theswitching field.

Figure 7.15 plots both T and TIH as the field is increased from thedemagnetized state (lower curve) and then reduced back to zero. The T-H plotcould becompared tothefirst quadrant ofFig. 7.13, if thestress-strain relationshipwere linear. A nonlinear relationship would further modify this curve. Note thatsincef{O) is zero, Tis zero when the applied field is zero. Furthermore, since in theDOKmodel a+(O) =-a.(O), we have

dT~~O)IH=O = K[aJO) + a_(O)] = 0, (7.51)

and the slope of T is zero. Additionally, since

a+(M;) = -a_( -M), (7.52)

starting from a demagnetized state, and since the error function is an odd function,T is aneven function ofH, which is consistent with what is observed inFig. 7.14.Since the curve has even symmetry, it is plotted for positive fields only.

As the field is increased, the ratio TIH increases at first. Sincej{H) is positiveandj{-H) is negative, the increase in T/H is due to increase in a, at the expense of

1

-,

0.8-,

rnrn

i 0.6

10.40Z

0.2

00

T

T/H

2Applied field

3

Figure 7.15 Calculated stress relationship using the OOK model.

192 CHAPTER 7 PREISACH APPLICATIONS

a: The ratio then decreases as a result of the saturationof the numeratorand thecontinued increase in the denominator. When the field decreases, the ratioincreases,since the denominator decreases faster than the numerator.

It is seen that this model generates hysteresis close to that seen in thesematerials, witha few exceptions. The stress in this modelis zero in the absenceofan appliedfield, contraryto the measurements. If the stress werea functionof theoperative field, h = H + a.M, instead of the applied field H, there would be aremanentstress in the material. Future versions of the modelwill be based on theoperativefield rather than the appliedfield. Furthermore, the slope of T at H = 0is not observed to be zero, as the model predicts. This reflects the model'sassumptionof startingwithacompletely demagnetized sample,whilethemeasureddata was takenon a samplethat was notdemagnetized. It mayalso be the result ofa nonlinearstress-strain relationship.

The actual straindependson the stress-strain relationship of the mediumandthe load placed on the transducer. For a linear stress-strain relationship, themedium strain is found by Young's modulus times the stress. For a hystereticrelationship, the strain could be calculated by a second Preisach model. In thelattercase there mightbe someresidualstressat the conclusionof this process. Iffound to be necessary, this relatively simple modification of the model requireslittle additional computing time and introduces only one additional arbitraryconstant.

Thismodelhas thecapabilityof predictingminorloopbehavioras seen in Fig.7.16. In this case, the material is assumedto be single-quadrant material, so thatthere is no changein state as the appliedfield is decreased. Thus, the a; and a. donot change,and the shapeof the curve is determined entirelyby the shapeoff{H).

Examination of Fig. 7.17 illustrates the effect of varying v, as defined in(7.48), for the valuesof 0.15,0.6, 1.35,and 2.4. It is seen that as it increases,boththe slope and heightof the hysteresis loop increasefor a givenfield. Thesecurvesare normalized in Fig. 7.18 so that the subtle changes in shape are more easily

Applied fieldFigure 7.16 Minorloopscalculated by the magnetostriction model.

SECTION 7.7 MAGNETOSTRICTION

v

0.15

0.6

1.35

2.4

Applied field

Figure 7.17 Theeffectof v on themodel' s magnetostriction behavior.

193

compared. As v is increased, the curvature of the stress increases as the field isdecreased. As the field is increased, the effectof increasing v is to emphasize thedecreasein slope at largerfields making the curve more"S" shaped.

A Preisach-type model has beenpresented for modeling magnetostriction ofparticulate or granularmedia. The stress-applied field relationship generatedbythismodeldisplays thehysteresis observed inmagnetostrictive media, eventhoughonly a very simplified DOK model was used. Again, to validate the model, onemust identify the parameters completely, to upgrade to a eMU model, and toaccurately model the stress-strainrelationship in thecontextof the overallmodel.

v

0.150.81.352.4

Applied field

Figure 7.18 Normalized magnetostriction behavior for different values of v.

194

7.8 THE INVERSE PROBLEM

CHAPTER 7 PREISACH APPLICATIONS

An important application of hysteresis modeling is the inverse problem, in whichan appropriate circuit is obtained to condition the input to the hysteretic transducerso that the overall circuit does not appear to have hysteresis. Visone et al. haveshown [22] that the stop model is the inverse of the play model (see Appendix A).We will now show how to obtain the inverse of the differential equation form ofthe Preisach model.

We visualize the Preisach model as consisting of three components, as shownin Fig. 7.19: a differentiation, a susceptibility computer, and an integrator. The timederivative of the operative field is obtained as the output of a differentiation. Forthe Gaussian Preisach density function, the susceptibility X is computed using(2.22), to obtain the irreversible component. In addition, we must obtain a suitablemodel for the reversible component - for example, by means of (3.12). The choiceof the a's is determined by whether one uses the magnetization-dependent or thestate-dependent model for the reversible magnetization.

The history of the process is maintained by a stack in the box labeled"Compute X." The time derivative of the operative field is then multiplied by thesusceptibility to obtain the time derivative of the magnetization, which is in turnintegrated to obtain the magnetization. Other features could be added to this model,such as accommodation and aftereffect; however, we will not do so, since theirpresence would complicate this picture.

Since the susceptibility is a scalar, the inverse of this transducer involvesreplacing the multiplication by X with division by X, as shown in Fig. 7.20. Thereciprocal susceptibility is computed the same way that the susceptibility iscomputed, and the same stack is used to maintain the magnetizing history. Since theoutput of this circuit is integrated, it will be less sensitive to noise. The errorsassociated with this inverse are associated with the approximation of the model tothe real system. In particular, the parameters of the system may change with time

r---------- ~ Compute ._.-----...,X

Ioooo-----------t (X~-------

Figure 7.19 Blockdiagramof the differential Preisachmodel.

REFERENCES

M

195

'--------_alUI-----------IFigure 7.20 Block diagram of the inverse differential Preisach model.

as the temperature of the transducer changes, and the model may not track themcorrectly. Other errors are associated with approximating the critical field Preisachdensity and with approximating the reversible variation.

This model has a self-correcting property. Whenever the applied field becomeslarge, the material and the inverse model go to a unique state, the saturation state.Furthermore, the errors associated with the improper registration of a corner of thehistory staircase are deleted whenever that corner is deleted by the applied field.This inverse is both a left inverse and a right inverse.

7.9 CONCLUSIONS

The classical Preisach model is able to describe hysteresis but is limited by thecongruency property and the deletion property. These properties are not found inmagnetic materials, and so the model must be modified accordingly. Furthermore,the model is a scalar one, and real magnetizing processes are vector ones. In earlierchapters we showed how physical arguments could be used to modify theseproperties. The results were accurate models that had relatively few parameters andgave some insight into the magnetizing process.

In this chapter we showed how to introduce dynamics into the rate­independent Preisach model. One can also obtain a robust model that is capable ofdescribing far more thanjust the magnetization characteristics of the material. Oneexample of such an extension of the model is the magnetostriction model. Inaddition, since the Preisach model possesses an inverse, it can be used if desiredto modify the input so that the resulting transducer appears to have no hysteresis.

REFERENCES

[1] G. Bertotti, "Dynamic generalization of the scalar Preisach model ofhysteresis," IEEE Trans. Magn., MAG·2S, September 1992, pp. 2599-2601.

196 CHAPTER 7 PREISACH APPLICATIONS

[2] E. Della Torre, "An analysis of the frequency response of the magneticrecording process," IEEE Trans. Audio Electroacoust., AE-13, May-June1965, pp. 61-65.

[3] W. D. Doyle, L. Varga, L. He, and P. J. Flanders, "Reptation and viscosityin particulate recording media in the time-limited switching regime," J. Appl.Phys., 75, May 1994, pp. 5547-5549.

[4] P. J. Flanders, W. D. Doyle, and L. Varga, "Magnetization reversal inmagnetic tapes with sequential field pulses," IEEETrans. Magn.,MAG-30,November 1994, pp. 4089-4091.

[5] Y. D. Yan and E. Della Torre, "Particle interaction in numerical micromagne­tic modeling," J. Appl. Phys., 67(9), May 1990, pp. 5370-5372.

[6] E. Della Torre, J. Oti, and G. Kadar, "Preisach modeling and reversiblemagnetization," IEEE Trans. Magn., MAG-26, November 1990, pp.3052-3058.

[7] E. Della Torre, "Dynamics in the Preisach accommodation model," IEEETrans. Magn., MAG·31, November 1995, pp. 3799-3801.

[8] E. Della Torre and F. Vajda, "Parameter identification of the complete­moving hysteresis model using major loop data," IEEETransMagn.,MAG­30,November 1994, pp. 4987-5000.

[9] E. Della Torre, "Effect ofparticle interaction on recording noise," Physica B,223, 1997,pp.337-341.

[10] J. C. Mallinson, in Magnetic Recording, Vol. I, C. D. Mee and E. D. Daniels,eds. McGraw-Hill: New York, 1987, pp. 337-375.

[11] E. Della Torre, "Effect of interaction on the magnetization of single domainparticles:' IEEETrans. AudioElectroacoust., AE·14,June 1966, pp. 86-93.

[12] G. Kadar, "On the Preisach function of ferromagnetic hysteresis," J. Appl.Phys.,61, April 1987, pp. 4013-4015.

[13] M. B. Moffet, A. E. Clarke, M. Wun-Fogle, J. Linberg, J. P. Teter, and E. A.McLaughlin, "Characterization of Terfenol-D for magnetostrictiontransducers," J. Acoust. Soc. Am., 89(3), 1991, pp. 1448-1455.

[14] F. T. Calkins, and A. B. Flatau, "Transducer based measurements ofTerfenol-D material properties," SPIE 1996 Proc.: Smart Structures andIntegrated Systems, 2717, 1996, pp. 709-719.

[15] J. B. Restorff, H. P. Savage, A. E. Clark, and M. Wun-Fogle, "Preisachmodeling of hysteresis in Terfenol," J. Appl. Phys., 67(9), May 1990, pp.5016-5018.

[16] I. D. Mayergoyz, Mathematical Models ofHysteresis, Springer-Verlag: NewYork, 1990, pp. 122-129.

[17] A. Adly and I. D. Mayergoyz, "Magnetostriction simulation by usinganisotropic vector Preisach models," IEEE Trans. Magn., MAG·32,November 1996, pp. 4147-4149.

[18] E. Della Torre and A. Reimers, "A Preisach-type magnetostriction model formagnetic media," IEEE Trans. Magn., MAG·33, Sepember 1997, pp.3967-3999.

REFERENCES 197

[19] E. Della Torre, J. Oti, and G. Kadar, "Preisach modeling and reversiblemagnetization," IEEE Trans. Magn., MAG-26, November 1990, pp.3052-3058.

[20] E. Della Torre and F. Vajda, "Parameter identification of the complete­moving-hysteresis model using major loop data," IEEETrans. Magn., MAG­30, November 1994, pp. 4987-5000.

[21] J. E. Ostenson and D. C. Jiles, Ames Laboratory, Iowa State University,private communication.

[22] C. Miano, C. Serpico, and C. Visone, "A new model of magnetic hysteresis,based on stop hysterons: an application to the magnetic field diffusion,"IEEE Trans. Magn., MAG-32, May 1996, pp. 1132-1135.

APPENDIXA

THE PLAY AND STOP MODELS

The play model* is another method of handling certain types of hysteresis.Hysteresis observedin mechanical systems, suchas geartrains,is calledbacklash.In particularin a gear train, there is a rangeof input,called the dead zone, whichproduces no changein output.When that rangehas beenexceeded, the changeinoutputis directlyproportional to thechangein inputuntilthedirection of the inputis changed. At that point, the gear train reenters the dead zone. The ratio of thechangein output to the changein input is called the gear ratio.

A graphical description of this behavior is shownin Fig. A.t. The slopinglinegoingthroughpointa is followed onlywhenever the inputis increasing. The slopeof this line is givenby the gearratio.Similarly, the linegoingthroughpointb alsohas a slope given by the gear ratio and is followed only whenever the input isdecreasing. Theseslopinglineswillbereferredtoasthebounding lines. Theregionbetweenthese lines is the dead zone.At pointc, if the input increases, the outputfollows the line going throughpoint a, but if the input decreases, it follows thehorizontallineintothedeadzone.This is anexample of a branching in thismodel.The horizontal line can be traversed in eitherdirection, and one followsthis lineas long as one is in the dead zone. A reversible component in this model can beintroduced by replacing the horizontal lines in the dead zone withcurvesof finite

*M.A. Krasnosel'skii andA. V. Pokrovskii, Systems withHysteresis, Springer-Verlag: Berlin,1989.

199

200 APPENDIX A THE PLAY AND STOP MODELS

Output

Figure A.I Dlustration of the play model.

slope.The outputof this modelcan beof anyvalueand does notchangeas longas the

input has a rangeof valuesdefined by the widthof the dead zone. Similarly for agiven input, the output can have a rangeof valuesdefined by the dead zone. Theparticular valueof output for a given inputdependson the history,so this systemexhibits hysteresis. It is noted that the output of this modeldoes not saturateas inthePreisachmodel. Thus, to use thismodeltocharacterize magnetic hysteresis, theoutput of the model must be fed into a saturating nonlinearity, as shown in Fig.A.2.T hiscumbersome additionto themodellimitsits usefulness, especiallywhen

1.5oInput

-I l-..-._----'--__..L--_----L.__--L...-_--'-__--'

-1.5

Ir-----r---.,.-----,.----r---~-___,

=~Oo

Figure A.2 Saturating nonlinearity.

APPENDIX A THE PLAY AND STOP MODELS 201

trying to relate the model parameters to physical quantities.The inverse of this model, the stop model, is a similar model where the slope

of each of the lines in the inverse model are the reciprocal of the slope in the playmodel. In the dead zone, the slope of the curves would be infinite if there were noreversible component leading to discontinuities in the behavior. This is the sameproblem that the inverse of the Preisach model would have when the material is insaturation.

APPENDIXB

THE LOG-NORMAL DISTRIBUTION

Thelog-normal distribution isa modification of thenormal (Gaussian) distributionfor random variables that are constrained to be positive. We will define a log­normal distribution function by:

j{x) :AeXp{-[In~b)II (B.l)

where band c are positive. A plot of this function is shown in Fig. B.I for A = I,b =1 and three values of c.

..- -'.: .: , / \ e

j(x) / \ 0.09

: / \ 0.3

I \ 1

I \0.5

I \I

\-,

I -,/ <,

<,

/ '-0

0 1.00 2.00 3.00x

Figure B.l Log-normal distribution.

203

204 APPENDIX B THE LOG-NORMAL DISTRIBUTION

The moment generatingfunctionMGF for this distribution is

MGF = £·x"f(x)dx = AL·X/lexp{-[ In~b)r}dx. (B.2)

If we make the substitution

then

_ In(xlb)u---

2c '(B.3)

Thus, since

(B.4)

r:exp[-(x- c>,"]dx = fi, (B. 5)

(B.2) becomes

MGF =r:Ucb/l+ Ie -112

+2cu(/I+I)du

=Ucb/l+ lec~/I+ 1)1r:exp{-[u-c(n + 1)]2}du (B.6)

=2fiAcb"+ Ie c2(" + 1)2.

We selectA by normalizingthe distribution, that is, setting theMGFto I for n = O.Then

e- C2

A=--.2bcfi

Therefore,

f(x) = -----------

and the MGF becomes

(B.?)

(B.8)

(B.9)

The expected value of this distribution is obtained letting n = I in (B.9),so that

<r> = be3c 2,

and the expected value of x2 is obtainedby setting n = 2 in (B.9), so that

(B.IO)

205

(B.II)

Then the varianceofj{x), that is, the squareof its standarddeviation, is given by

0 2 = <X2>_<X~ = b2{e8c2_e6c2) = 2b2e7C2sinh(c~. (B.12)

Wecan nowshowthatthelog-normaldistributionreducesto theGaussiandistribution if the standard deviation is small compared to the mean, that is, b islarge and c is small.We note that for small values of c we have

<x» s:: b and a s:: bc{i. (B.13)

Note that if x is smallcomparedto b, then In(xIb) is approximately (xlb -1). Underthese conditions, the distribution reduces to

f(x) s:: _l_~,L 1(X-<x>tj, (B.14)o{ii ~1 2 0 2

which is a standardGaussiandistribution.

APPENDIX C

DEFINITIONS

Term Symbol Comment

Accommodation p Describes motion of hysterons inconstant Preisach plane when M changes

Apparent reversible Irreversible magnetization that has aunique state in zero field

Applied field HA Applied magnetic field due to externalsources

Applied field energy WHA Energy due to interaction of themagnetization with an applied field

Coercivity He Field required to reduce the saturationmagnetization to zero

Critical field Hk Field required to switch a noninteractinghysteron

Critical field t; Average of hysteron switching operativeexpectation fields

207

208 DEFINITIONS

Cubic anisotropy Wcubic Magnetocrystalline energy of a cubicenergy crystal

Demagnetization D Ratio of demagnetizing field tofactor magnetization of a material

Demagnetizing field HD Magnetic field due to the material'smagnetization

Exchange constant A Exchange integral density for a simplecubic crystal equal to JSl/a

Exchange integral J Exchange energy between adjacentatoms

Exchange energy Wtx Total exchange energy in a crystal

Hysteron A minimum unit of magnetization withtwo stable states

Interaction field H; Field due the magnetization of otherhysterons

Irreversible M; Component of the magnetization thatmagnetization changes irreversibly

Lattice spacing a Distance between adjacent atoms

Locally reversible M, State-dependent reversiblemagnetization magnetization

Magnetic state Q(u, v) Magnetization state of a hysteron withcritical fields u and v

Magnetization M Total magnetization per unit volume(the sum of the irreversible and locallyreversible magnetization)

Moving constant a Ratio of expected value of interactionfield to M

Negative critical field v Field required to switch a hysteron intothe negative state

Positive critical field u Field required to switch a hysteron intothe positive state

DEFINITIONS 209

Preisach function Ptu, v) Densityfunction of hysteronswithcritical fields u and v

Remanence Mo Magnetization when H is zero

Remanentcoercivity HRC Field required to make the remanencezero

Saturation Ms Maximummagnetization for givenmagnetization material

Squareness S Ratio of maximum remanenceto Ms

Susceptibility X dM/dH, a subscript may be added toidentify type of magnetization

Uniaxial anisotropy Ku Differencein energydensity betweenconstant easy axis and hard plane

Uniaxial anisotropy w, Magnetocrystalline energy of a crystalenergy with uniaxial symmetry

Wall width lw Width of a domain wall, classicallyequal to nJAIKu

Virgin magnetization Mv Magnetization curve for an ac-curve demagnetized specimen

Zeemanenergy WH Energyof a hysteron in a magneticfield

INDEX

Aac demagnetization 33ac-magnetizing process 38accommodation 26, 125accommodation process 39, 131, 139activation volume 25aftereffect 26, 112anhysteretic magnetizing process 36anisotropy constant 9anisotropy energy 18anisotropy hysteresis 149antiferromagnetism 8apparent reversible behavior 40, 68,

88appliedfield accommodation 39, 139Arrheniuslaw 112ascendingmajorcurve 31

Bbacklash 199Barkhausenjump 29, 149Barkhausen noise 183Bloch wall 12blockingtemperature 7Bohr magneton 3Boltzmann's constant 2Brillouinfunction 4

Ccentral limit theorem 77classicalPreisachmodel 50CMHmodel 59coefficientof magnetic viscosity 117coercivity 16, 32coherentmagnetization model 17compensation temperature 8compound selectionrule 155congruency property 36, 49coupled-hysteron models 154crossovercondition 71Curie temperature 4, 6Curie-Weiss temperature 7curve fitting 47

ode magnetizing process 36dead zone 199deletionproperty 36, 49, 104, 125,

143demagnetizing factor 18, 28demagnetizing factor accommodation

139diamagnetism 2DOKmodel 55domain 12

211

212

domain wall 12down-switching field 33dynamic accommodation model 173

Eeddy currents 28, 167, 168ellipsoidally magnetizable 150energy barrier 112Everett integral 41excess Barkhausen noise 184exchange energy 5, 9exchange field 6exchange integral 9

FFermi-Dirac statistics 5ferrimagnetism 8ferromagnetism 5first order reversal curves 33fluctuation field 115frequency response 170

GGaussian Preisach function 41Gudermannian 14gyromagnetic ratio 3, 26gyromagnetic effects 26

HHenkel plots 93hysteretic many-body problem 29

Iinteraction field 34interpolation 46inverse problem 194irreversible magnetization 53, 54

JJacobian 42

LLangevin function 3locally reversible magnetization 56

INDEX

log-normal function 41, 203loss property 147, 164

Mmagnetization-dependent model 55magnetizing curve 33magnetocrystalline anisotropy energy

9magnetostriction 188major hysteresis loop 31Mayergoyz vector model 148, 151method of tails 84micromagnetism 8, 11minor loop 33, 36molecular field constant 6moving constant 77moving model 78

N~eel temperature 8Neel wall 12, 15noise 181nonlinear congruency 99normalized Preisach function 39nucleation volume 25

ooperative field accommodation 139operative plane 78

pparamagnetism 2parameter identification 66, 80physically realizable region 53, 54Preisach differential equation 40Preisach function 33Preisach measurement axis 153Preisach model 33Preisach state function 39product model 92pseudoparticle models 152pulse height-dependence 180pulsed behavior 172

INDEX

Rrate-independent phenomenon 26, 31reentrant 17remanence 32remanence loop 32remanent coercivity 32remanent susceptibility 32replacement factor 128reversible magnetization 54

Ssaturation magnetization 2, 32saturation property 147shape anisotropy 18simple selection rule 155simplified vector model 159single-domain particle 18single-quadrant Preisach functions 40Slonczewski asteroid 22squareness 32staircase 35state-dependent magnetization 59Stoner-Wohlfarth model 17

213

superexchange 6superparamagnetism 8, 18susceptibility 3, 32symmetry method 80

Tthree-quadrant Preisach functions 40turning points 33

Uup-switching field 33

Vvariable-variance model 86virgin magnetizing curve 33

WWalker velocity 28wall mobility 28

ZZeeman energy 10

ABOUT THE AUTHOR

Edward Della Torre received the B. E. E. degree from Brooklyn PolytechnicInstitutein 1954, theM. Sc. in electrical engineering fromPrinceton University in1956, the M. Sc. in physics from Rutgers University in 1961,and the D. E. Sc.fromColumbiaUniversity in 1964. He hastaughtatRutgers University, McMasterUniversity, and Wayne State University, and he chaired the Electrical andComputerEngineering Departments atthelattertwouniversities. Hewasamemberof the technical staff at the BellTelephone Laboratories in MurrayHill, NJ. He isProfessor of Engineering and Applied Science at The George WashingtonUniversity.

Dr. Della Torre has made fundamental contributions to the modeling ofmagnetic materials. A proponent of using physical principles to guide thedevelopment of models of magnetic materials, hedeveloped themoving model, thestate-dependentreversible magnetization model, theaccommodation model, andthesimplified vectormodel. Currently he isworking on thePreisach-Arrhenius modelfor magnetic aftereffect, whichdetermines the lifetime of magnetization.

AFellowof boththeIEEEand theAmerican Physical Society, Dr.DellaTorreis currently president of the Magnetics Society. He has served the MagneticsSocietyin manycapacities including chairingseveralINTERMAG Conferences.He is the coauthor of The Electromagnetic Field with Charles V. Longo andMagnetic BubbleswithAndrew H. Bobeck. A member of EtaKappaNu, TauBetaPi and Sigma Xi, he is the author of almost 200 technical papers in refereedjournals and has presented over 150papersat technical conferences. He holds 18patents.

215


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