Relaxation Processes in Micromagnetics (International Series of Monographs on Physics)

INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS SERIES EDITORS J. B I R M A N CITY UNIVERSITY OF NEW YORK S. F. E D WA R D S UNIVERSITY OF CAMBRIDGE R. F R I E N D UNIVERSITY OF CAMBRIDGE M. R E E S UNIVERSITY OF CAMBRIDGE D. S H E R R I N G T O N U N I V E R S I T Y O F O X F O R D G. V E N E Z I A N O C E R N , G E N E VA

International Series of Monographs on Physics 135. 134. 133. 132. 131. 130. 129. 128. 127. 126. 125. 124. 123. 122. 121. 120. 119. 118. 117. 116. 115. 114. 113. 112. 111. 110. 109. 108. 107. 106. 105. 104. 103. 102. 101. 100. 99. 98. 97. 96. 95. 94. 91. 90. 88. 87. 86. 83. 82. 73. 71. 70. 69. 51. 46. 32. 27. 23.

V. Fortov, I. Iakubov, A. Khrapak: Physics of strongly coupled plasma G. Fredrickson: The equilibrium theory of inhomogeneous polymers H. Suhl: Relaxation processes in micromagnetics J. Terning: Modern supersymmetry M. Mari˜ no: Chern-Simons theory, matrix models, and topological strings V. Gantmakher: Electrons and disorder in solids W. Barford: Electronic and optical properties of conjugated polymers R. E. Raab, O. L. de Lange: Multipole theory in electromagnetism A. Larkin, A. Varlamov: Theory of fluctuations in superconductors P. Goldbart, N. Goldenfeld, D. Sherrington: Stealing the gold S. Atzeni, J. Meyer-ter-Vehn: The physics of inertial fusion C. Kiefer: Quantum gravity T. Fujimoto: Plasma spectroscopy K. Fujikawa, H. Suzuki: Path integrals and quantum anomalies T. Giamarchi: Quantum physics in one dimension M. Warner, E. Terentjev: Liquid crystal elastomers L. Jacak, P. Sitko, K. Wieczorek, A. Wojs: Quantum Hall systems J. Wesson: Tokamaks, Third edition G. Volovik: The Universe in a helium droplet L. Pitaevskii, S. Stringari: Bose-Einstein condensation G. Dissertori, I. G. Knowles, M. Schmelling: Quantum chromodynamics B. DeWitt: The global approach to quantum field theory J. Zinn-Justin: Quantum field theory and critical phenomena, Fourth edition R. M. Mazo: Brownian motion – fluctuations, dynamics, and applications H. Nishimori: Statistical physics of spin glasses and information processing – an introduction N. B. Kopnin: Theory of nonequilibrium superconductivity A. Aharoni: Introduction to the theory of ferromagnetism, Second edition R. Dobbs: Helium three R. Wigmans: Calorimetry J. K¨ ubler: Theory of itinerant electron magnetism Y. Kuramoto, Y. Kitaoka: Dynamics of heavy electrons D. Bardin, G. Passarino: The Standard Model in the making G. C. Branco, L. Lavoura, J. P. Silva: CP Violation T. C. Choy: Effective medium theory H. Araki: Mathematical theory of quantum fields L. M. Pismen: Vortices in nonlinear fields L. Mestel: Stellar magnetism K. H. Bennemann: Nonlinear optics in metals D. Salzmann: Atomic physics in hot plasmas M. Brambilla: Kinetic theory of plasma waves M. Wakatani: Stellarator and heliotron devices S. Chikazumi: Physics of ferromagnetism R. A. Bertlmann: Anomalies in quantum field theory P. K. Gosh: Ion traps S. L. Adler: Quaternionic quantum mechanics and quantum fields P. S. Joshi: Global aspects in gravitation and cosmology E. R. Pike, S. Sarkar: The quantum theory of radiation P. G. de Gennes, J. Prost: The physics of liquid crystals B. H. Bransden, M. R. C. McDowell: Charge exchange and the theory of ion-atom collision M. Doi, S. F. Edwards: The theory of polymer dynamics E. L. Wolf: Principles of electron tunneling spectroscopy H. K. Henisch: Semiconductor contacts S. Chandrasekhar: The mathematical theory of black holes C. Møller: The theory of relativity H. E. Stanley: Introduction to phase transitions and critical phenomena A. Abragam: Principles of nuclear magnetism P. A. M. Dirac: Principles of quantum mechanics R. E. Peierls: Quantum theory of solids

Relaxation Processes in Micromagnetics HARRY SUHL University of California, San Diego

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Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c Oxford University Press 2007
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Ode to Joyce

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PREFACE This book addresses the issue of translating a number of microscopic relaxation mechanisms into a language adapted to the peculiarities of the dynamics of the magnetization vector. It brings to light some previously unexplored, and some unexpected, features of that dynamics. Only modest mathematical equipment is employed throughout, but some familiarity with the simpler properties of ferromagnets at the level of advanced undergraduates or first year graduate students is assumed. The book should also be of some interest to professionals in the magnetic recording industry. Magnetic phenomena have fascinated people for several millennia, but besides their early use for determining the magnetic north, really major applications date back no more than two or three centuries. A big advance in the harnessing of magnetic phenomena went hand in hand with the growing understanding of electromagnetism that culminated in the nineteenth century. More recent times have seen a development of comparable import: the utilization of magnetism in information technology. Magnetic recording and storing of immense amounts of information in a small space, and at (so far) steadily diminishing cost has been one of the two mainstays of information technology, with semiconductor progress the other. The steady progress in magnetic recording technology is based on material science on the one hand and on advances in engineering of very small structures and particle assemblies on the other. As in many other fields of applied science, empiricism holds sway, and one may ask what role, if any, theory plays in the advance of magnetic recording. The most helpful kind of theory, intended to provide direct support and guidance to experimentalists, is based on principles described by very plausible equations that have proved their worth in many contexts. Aided by computer power, equally plausible solutions have been obtained in many cases, and often provide a reasonable fit to observation. As a result, the provenance of these equations is rarely questioned. This is not at all unreasonable if the focus is on empirical advances in the recording industry. Nonetheless a critical examination of the origin of the equations is in order, if only as an insurance policy against unexpected aspects that may lurk in future developments. In particular, the manner in which various relaxation mechanisms enter the ultimate form of the equations deserves attention. In so far as this relaxation has to do with only transfer of energy of magnetic motion to nonmagnetic degrees of freedom of the substrate, it is obviously very inconvenient to carry along these degrees of freedom in a calculation designed to match theory to observation. Experiments almost always measure only the magnetization. Accordingly, part of this book will be concerned with elimination of the unwanted degrees of freedom. The result is an equation of motion for the magnetization vector alone. However, a price must be paid: the resulting equation in general vii

viii

PREFACE

involves terms that are non-local in space and that involve the pre-history of the magnetization up to the time of interest. Asymptotic expansions of this result then determine the range of space and time variations over which these effects are of importance. In Chapter 3, five different relaxation mechanisms are considered in this light. Two of these describe the loss mechanism in magnetic metals: one (an obvious one) involves the eddy currents induced by a moving magnetization; the other, a quasithermodynamic one, is the result of ‘breathing’ of the Fermi surface owing to that motion. But there is one mechanism that has been left out in this work: direct relaxation of the magnetization in metallic ferromagnets owing to scattering of conduction electrons. In the opinion of this author, a credible account of this mechanism in transition metals must await clarification of the origin of their ferromagnetism. The same physics involved in damping of the motion of the magnetization must be involved in establishing the magnetization in the first place. Admittedly, it is tempting to describe the magnetization in terms of a sophisticated mean field theory, and to attribute the loss mechanism to inelastic electron scattering in this mean field, but to achieve a credible self-consistency would not be easy. Magnetic recording by its nature involves large motions of the magnetization vector. Most purely analytic studies, on the other hand, deal with small motions that allow linearization of the equation of motion. In the absence of an adequate analytic theory far beyond the linearized limit and its lowest nonlinear corrections, recourse is had to extensive computer simulations. Undoubtedly the more thoughtful studies of this kind can yield important insights, but in this book, the emphasis is on analytic solutions of the nonlinear equations with only minimal computational assistance. To focus on the essential qualitative features of large motions, only the simplest non-trivial model will be treated in detail: a magnetic specimen with uniaxial anisotropy of crystalline and/or demagnetizing origin. Also, these solutions are restricted to only the lowest term in the above-mentioned asymptotic expansion. Only one chapter is devoted to small motions. Analyzed judiciously, these occasionally provide clues or correspondences with the nonlinear situation. But, more importantly, linear theory and the lowest nonlinear extensions thereof hold the promise of applications, such as magnetic delay lines. These studies, very briefly sketched in Chapter 2, are based on series expansions in powers of small motion amplitudes and have provided elegant demonstrations (both theoretical and in the laboratory) of soliton-like propagation of magnetic disturbances. In this book, the magnetization field as a function of time and position is treated as a classical quantity. This classical field is a gross manifestation of the electronic spins and orbits and their largely quantum mechanical interactions in a magnetic medium, but these nonclassical features enter a coarse-grained description of the field only in the form of coefficients in its equations of motion. Except in specialized investigations employing scanning tunnelling microscopy, or magnetic force microscopy, of magnetic surfaces, measurements are coarse grained because of the limited resolution of equipment. The grains should be

PREFACE

ix

considered large enough so that the spins in a grain centered on position
x say,
(
x, t) are coupled to yield a large, essentially classical magnetic moment vector M of
x-and t-independent magnitude Ms well below the Curie temperature. This limits the description to excitations of energies sufficiently low so that the coupling of the spins within a grain is not significantly disrupted. Otherwise stated, this description is applicable only to phenomena of sufficiently gentle spatial variations. Just below the transition temperature, the magnitude of the magnetic moment is also variable and may have to be taken into account in so-called thermally assisted recording. A short section at the end of Chapter 1 addresses this matter. In theory, one can imagine quantum effects showing up even at the level of a coarse description. For example, one may consider spontaneous depinning of domain walls. However, the jury is still out on the question of submergence of such observations in extraneous effects. Quite often, observations deal with a magnetization field averaged over distances orders of magnitude larger than the coarse-grained distances referred to
 , is sometimes assumed to satabove. Such a spatial average vector, call it M isfy dynamic equations originally designed for special cases in which there is no significant distinction to be made between local and average values, as, for example, in magnetic particles smaller than a typical domain wall width. But,
 will satisfy equations with relaxation terms of a structure very in general, M
field. The reason is that M
 is different from those appropriate for the local M in contact with many other modes of magnetic motion of essentially zero spatial
 . The process by which any average that will detract from the magnitude of M one mode of magnetic motion loses energy through coupling to other modes of magnetic motion will be called distributive damping in these pages. In contrast, damping that results from transfer of energy to degrees of freedom of the host medium will be called intrinsic damping, and is treated in Chapter 3. In samples too small to support domain walls, it is the only allowed form of relaxation, but, in general, distributive and intrinsic relaxation will occur side by side. Relaxation processes cannot be adequately discussed without reference to the closely related subject of fluctuations. This close relation is particularly evident in the famous fluctuation-dissipation theorem. It relates the dissipative part of the linear response of a system to the mean square fluctuations in the responding degrees of freedom. A rigorous proof has been given only for linearized equations of motion, classical or quantum mechanical. But for reasons that are not totally clear (at least not to this author), when a statistical ensemble of systems with infinitely many degrees of freedom is considered, a seemingly more general form of the fluctuation-dissipation theorem appears. In such a system, the role of the mean square fluctuations is played by the diffusion coefficients of the system. In this formulation, linear response of any particular one of these degrees of freedom does not appear to be a requirement. At least, this is the case if the motion of the ensemble distribution obeys the Fokker-Planck equation. Chapter 4 deals with these matters with special reference to ferromagnetic systems, but,

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PREFACE

for the benefit of readers not especially familiar with this subject, some general background material is included. Chapter 5 is devoted entirely to the question of magnetization reversal in small, effectively single-domain particles with given uniaxial anisotropy, and sufficiently sparse so that their interaction is neglected, even though this may not be quite justified in view of the long-range character of dipolar forces. The treatment is based on the Fokker-Planck equation, transformed to look like the
vector is constrained Schroedinger equation of quantum mechanics. When the M to move in a plane, that Schroedinger equation is the same as that for a particle in a one-dimensional periodic lattice, a thoroughly explored subject. The main interest from the point of view of magnetic recording is the graph of the field, the coercive field, necessary to reverse the magnetization in a given time. The main advantage of the method presented here is that it avoids the need for separate considerations for applied fields greater than or less than the anisotropy field. Most of the work assumes the easy magnetization direction to be parallel to the applied field, but one small section deals with the case of misalignment of
is discussed, these directions. The generalization to unconstrained rotation of M still for purely uniaxial anisotropy, using the fact that the azimuthal motion
about the field direction cannot seriously affect reversal times and may of M be averaged out. In another section, the relation of the work presented here to standard reaction rate theory and first passage time theories is outlined. Chapter 6 consists of two parts. The first part examines the motion of a more dense array of single-domain particles, interacting by dipolar forces. It is found that, in applied fields less than the (still assumed uniaxial) anisotropy fields, the motion is in general chaotic. Applied fields substantially exceeding anisotropy fields tend to extinguish chaos, replacing it by ‘clean’, though multiperiodic, motion. The chapter begins by presenting a simple integrable case: just two particles with the line joining their centers parallel to the field applied along the easy direction. The slightest deviation from this lineup results in chaos except in sufficiently large fields. As one might expect, the motion of three such particles is shown to be even more prone to chaos. From these examples it is concluded that, in a sufficiently dense (but not too dense) array, general chaotic motion should prevail at low applied fields, and that its characteristic time scale may well be shorter than the time scale of intrinsic damping. Then chaos appears as an extreme case of distributive damping and we speculate that intrinsic relaxation may then be regarded as the ‘small friction’ scenario of Kramers’ diffusion model of chemical reactions. In that scenario, only one variable is relaxing: the energy of the system, the detailed motion of its particles becoming irrelevant. If our speculation turns out to be valid, simplicity will have emerged from chaos. The second part of the chapter deals with reversal in continuous media. As is well known, chaos in a system of discrete particles does not necessarily carry over to the limit of a system of infinitely small, but infinitely dense particles forming a continuum. Usually new and different physics arises. The question of irregular motion in a continuous medium must be considered ab initio. (The same may

PREFACE

xi

apply even to discrete, but extremely dense particle arrays.) Examining conditions of integrability of the partial differential equations of the magnetization field is obviously very hard. One might suspect that the instabilities of small motions discussed in Chapter 2 portend stochastics of large motions, but no evidence to this effect, experimental or theoretical, has come to light so far. Therefore the rest of Chapter 6 is confined to purely deterministic methods. In an infinite, perfect sample, magnetization reversal can occur by uniform rotation, or by domain wall motion. In the case of rotation, the magnetization vector everywhere turns from one easy direction to the opposite easy direction in a magnetic field applied along the latter direction. On the other hand, reversal by domain wall motion depends on the existence of a domain wall separating regions of opposite magnetization direction prior to the switch. In that case, some small region of the sample must already have its magnetization in the final direction to be reached by the entire sample as the applied field drives the wall along. In the case of rotation, the graph of magnetization versus field as that field is cycled between positive and negative values exceeding the coercive field is a rectangle: the hysteresis loop. In the case of reversal by domain wall motion, that loop shrinks to a line: no hysteresis occurs. However, real samples are finite, and even if they are free of imperfections, these conclusions require major modifications, mainly owing to dipolar forces as manifested by demagnetizing fields. These tend to render perfectly uniform rotation unstable. A stable reversal process requires the magnetization field to assume a distinctive non-uniform pattern. This pattern minimizes the effect of the demagnetizing field, at the expense of introducing exchange torques set up by non-uniformity. But, on balance, stable magnetization reversal will result. Cycling the applied field still results in a hysteresis loop, but a more ‘skinny’ one than that corresponding to uniform rotation. In applied fields less than the field equivalent of the total anisotropy barrier energy (crystalline and/or demagnetizing), magnetization reversal can occur only with the assistance of thermal fluctuations. In this range of applied fields, the theory for the ideal sample with ideal boundary conditions parts company with a more realistic treatment. The activation energy needed to overcome the barrier in the finite, but idealized sample is proportional to the total sample volume, contrary to observation. Instead, the barrier is overcome only in the immediate vicinity of a nonuniformity in the underlying physical properties, particularly at a surface. The activation energy is proportional to the quite small volume of the imperfection. Once the magnetization on the far side of the imperfection is reversed, it provides the seed for further reversal by domain wall motion. Thus the switching speed of a sample large enough to allow domain wall formation should be related to the speed of propagation of a domain wall. The only available theories are based on the assumption of a uniform wall velocity. As the result of this assumption, the partial differential equations describing the motion are replaced by ordinary nonlinear differential equations, which then describe the shape of the wall as viewed by an observer moving along with it.

xii

PREFACE

Much of Chapter 6 is devoted to the motion of a N´eel domain wall in these terms, with the rotation of the magnetization confined to a plane, as it might be in a thin film. This problem has some formal similarities with propagation problems in chemistry and biology. Permitted ranges of stable velocities are given in terms of applied and anisotropy fields, but the actual final steady value of the velocity cannot be determined without solving for the transient following launch of the wall. That transient obviously does not allow replacement of a time rate of change by a velocity times a spatial rate of change. Kolmogorov and co-workers have succeeded in relating a final steady velocity to a certain class of initial conditions. In the present problem, a plausible argument is presented for the speed of the wall shortly after launch; quite possibly that speed will persist. One weak point in the notion of a steady wave traveling with uniform velocity for ever is that it does not account for its fate upon arrival at its destination. The author is not aware of any literature addressing this question analytically. Here, an argument is presented suggesting that, as the state of complete magnetization reversal is approached, the speed of the wall steadily grows to a (formally) infinite value. The moving Bloch wall is considerably harder to analyze, since it involves both the azimuthal and the polar angles of the magnetization vector. There is only one known analytic solution of this problem, that of L. R. Walker. It is a tour de force, both elegant and exact, and for these reasons it is presented in these pages. However, its singular character makes it hard to judge if it is in some sense an isolated, not widely applicable triumph. The motion of actual domain walls, even in ideal media, conforms with these types of theories only in certain geometries and/or ranges of applied field strength. For applied fields well above saturation value, they should be more or less valid. In field values below saturation, the very concept of a wall velocity becomes dubious, except in special geometries. The final section of Chapter 6 discusses the behavior of domain walls in applied fields below saturation. No attempt is made to explore this subject in depth, and treatment is restricted to domain walls in the form of sheets that render the problem effectively two dimensional. In samples of dimensions larger than a typical domain wall, static domain walls form owing to demagnetizing effects alone, without significant involvement of the exchange field inside the walls. In the absence of an applied field, the magnetization distribution in the sample arranges itself so as to result in zero internal field also (any crystalline anisotropy field of course remains), resulting in a particular domain wall configuration. In a finite subsaturating applied field, the magnetization still attempts to shield the interior from the external field, reducing the interior field to zero as far as possible (some field penetration must occur at sharp edges or corners). As the applied field is increased adiabatically, the wall configuration moves in such a way as to increase the size of domains with magnetizations tending towards lineup with the field, pervading the entire sample when the applied field reaches saturation value. This process can be analyzed rigorously for a certain class of sample shapes, and from that analysis some physical conclusions may be drawn that allow a simple geometric construction

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xiii

for the general situation. When the external field is increased dynamically, a purely analytic treatment of the wall motion becomes impossible, even for that class of sample shapes. However, it is proposed that, in the dynamic case, the development of the wall configuration, even if quite complex, the time dependence is nearly the same as that of a primitive case, for which an exact solution is presented. The reason why the usual concept of domain wall velocity is not applicable here is simple: in the subsaturated case there is no length scale; configurations and their movements depend only on sample shapes since exchange length is not significantly involved. Standard wall motion theory on the other hand depends on that length. This book is focussed on analytic formulations of magnetic relaxation processes on the one hand, and on ways of solving the resulting equations in their full nonlinear form without significant resort to computer simulation on the other, at least for simple cases. Unfortunately, this narrow focus did not allow inclusion of certain highly topical subjects, such as relaxation processes in multilayer structures, the physics of exchange bias, magnetization fields in constricted structures, proximity effects and spin polarized tunneling into nonmagnetic media, etc. Hopefully, the methods adopted in this work will prove relevant in at least some of these areas.

ACKNOWLEDGMENTS Many of the topics analyzed in this book were discussed, sometimes at some length, with my colleagues at the Center for Magnetic Recording Research (CMRR) and in the Physics Department at UCSD. I am especially indebted to Neal. H. Bertram of CMRR for bringing several of the problems discussed in this book to my attention; one of these problems led to a joint publication. I can only hope that some of the results obtained in this book will be of some modest help to him in his pursuit of advanced magnetic recording techniques. In the Physics Department, I have had numerous discussions on nonlinear particle dynamics with Tom O’Neil and on nonlinear propagation problems with Patrick Diamond. Also, I must give credit to Ivan Schuller for innumerable general discussions of dynamic behavior of magnetic domains, and in particular for bringing to my attention time dependent features of certain hysteresis loops that seem to be related to one of the chapters in this book. This computationally challenged author must especially acknowledge virtually constant help with Mathematica received from Daniel Dubin. Much credit is also due to the experts at the Computer Help Desk of the Physics Department for their extraordinarily patient technical support. I wish to acknowledge Oxford University Press for their help with this project. Special thanks are due to Sonke Adlung of OUP for countenancing my habitual missing of completion deadlines and for his prompt attention to my frequent enquiries relating to preparation of this manuscript. Finally, Judy St Austin, my secretary, deserves special mention for general support, and especially for tracing and promptly providing me with hard copy of needed publications. Likewise, Betty Manoulian of CMRR must be praised for help in tracing the work of former students and post doctoral fellows at CMRR. On a personal level, I must thank my wife, Joyce, for her ungrudging willingness to put up with my reclusive behavior during the preparation of this manuscript.

xiv

CONTENTS

Preface

vii

Notations and conventions

xvii

1 The Classical Magnetization Field 1.1 Introduction 1.2 Equations of motion 1.2.1 Damping 1.3 Approaching the Curie temperature

1 1 4 9 11

2 Small motions of the Magnetization 2.1 Introduction 2.2 Models of small motions 2.2.1 Distributive damping 2.2.2 Instabilities and spin wave condensates

17 17 17 19 21

3 Intrinsic Damping 3.1 Introduction 3.2 Magnetostrictive coupling 3.2.1 Small samples 3.2.2 Large, homogeneous samples 3.3 Loss torque in magnetic metals 3.3.1 Eddy current damping 3.3.2 Direct coupling of conduction electrons to the magnetization field 3.4 Fluctuations in medium properties 3.5 Relaxation due to weakly coupled magnetic impurities 3.5.1 Slow relaxation 3.5.2 Corrections to the adiabatic limit 3.6 Appendix 3A. Inclusion of displacement current in Section 3.3.1

31 31 31 32 35 43 44

4 Fluctuations 4.1 Introduction 4.2 Fluctuation-dissipation theorem 4.3 Langevin equation, and generalized Langevin equation

71 71 74 77

xv

49 57 60 61 64 67

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CONTENTS

4.4

Fokker-Planck equation–cartesians 4.4.1 Fokker-Planck equation in polar angles 4.4.2 Fokker-Planck equation in the absence of well-defined canonical variables

5 Magnetization Reversal in a Very Dilute Array of Small Particles 5.1 Introduction 5.2 General observations 5.3 Reversal in 2d 5.3.1 Reversal in the long time limit 5.3.2 Intermediate time scales 5.3.3 Applied field and anisotropy axis misaligned 5.3.4 Relation to first-passage type theories 5.4 Rotation in 3d 6 Magnetization Reversal in Arrays of Particles and Continuous Media 6.1 Introduction 6.2 Relaxation due to magnetic moment interaction in a sparse medium 6.2.1 Equations of motion for dipolar interaction 6.2.2 A single pair 6.3 More dense arrays of many interacting particles 6.3.1 The Arnold web 6.3.2 Relevance to magnetic relaxation and reversal 6.3.3 Effective single-variable relaxation from causes other than chaos 6.4 Magnetization reversal and the magnetization process in large, dense systems 6.4.1 Simple model of magnetization reversal by domain wall motion 6.4.2 Motion of a Bloch domain wall 6.4.3 Magnetostatics and the magnetization process. Pre-existing domain walls 6.5 Appendix 6A: Vortex solutions in cylinder and disc: stability considerations

85 88 89

97 97 98 102 109 114 118 119 122

125 125 127 128 130 139 141 143 145 146 149 160 163 176

References

185

Subject Index

189

NOTATIONS AND CONVENTIONS Vectors are indicated by arrows over the symbols. The cartesian components of position vector
x are denoted by x1 , x2 , x3 . The direction of the xi -axis is called the i-direction. In volume integrals, the volume element is denoted by d3 x. Where products of vector components, or products of vector and tensor components occur, the summation convention (summation over repeated indices) is frequently used, without always being explicitely indicated. (Example: the vec
×B
is written ijk Aj Bk on occasions.) If T denotes a tensor, its tor product A divergence ∇ · T is defined to be a vector with components ∂Tij /∂xj .
, generally a function of
x and The magnetization vector is denoted by M time t. Except at the end of Chapter 1, its magnitude is considered a constant denoted by Ms , the total saturated magnetic moment per unit volume. The
2 have
are the same as those of the field H,
and (M
)2 and (H) dimensions of M the dimensions of energy per unit volume. Where convenient, Ms is equated to
and H
are measured in units of Ms and the energy per unit 1, and then M
will be specified in volume (p.u.v.) is measured in units of Ms2 . Frequently, M polar coordinates θ, φ.
is not usually The gyromagnetic ratio γ in the equations of motion of M shown. Where it is not shown, the fields are measured in frequency units. This requires no comment if Ms is not equated to 1. If it is equated to 1, so that H is in units of Ms , then this reduced H is still measured in frequency units, which implies that (the silent) γ will have been replaced by (silent) γMs . Similarly,
measured in units of Ms , and H in units of frequency, the Landauwith M
×M
×H
has the form −α M
×M
× H,
where Lifshit damping torque −γαLL M LL  αLL = αLL Ms . (In the text, the prime on the α will be omitted.) The Gilbert
× dM
/dt, in reduced variables, is likewise unchanged in damping torque αG M  form, with αG = αG Ms . The exchange energy p.u.v. is denoted by Al2 Σ3i=1 (∇Mi )2 . A is a dimensionless constant proportional to the microscopic exchange energy per pair of spins, and l is a distance of order of the lattice spacing. Uniaxial anisotropy energy p.u.v., with the 3-direction as axis, is commonly written KM32 /2. In this book, this energy is frequently written in the form HK M3 /2 where HK = KM3 is called the anisotropy field. For the purposes of
this book, functional differentiation at position
x of the total energy E = d3 xE(
x) with position dependent energy E p.u.v. is defined as ˜ dE δE . = 3lim ˜ ˜
(x)d3 x dM (x) d x→0 δ M xvii

xviii

NOTATIONS AND CONVENTIONS

Boltzmann’s constant is denoted by κB , and temperature by T .
vector Diffusion constant is denoted by D. Where polar coordinates for the M are used in this book, and Ms = 1, the dimensions of D are 1/time. Then time will be measured in units of 1/D. Maxwell’s equations are written in absolute units, so that the velocity of light, c, appears explicitly. This is to facilitate comparison with sound propagation results for magnetoelastic coupling. In both the magnetoelastic and the electromagnetic case, Ms is to be chosen to equal 1 only after the equations of
have been derived. motion of M

1 THE CLASSICAL MAGNETIZATION FIELD 1.1

Introduction

At the time of writing, all applications of magnetically ordered materials can safely avoid explicit reference to the quantum mechanical aspects of the underlying spin and orbital degrees of freedom of the constituent ions. Also, it is not normally necessary to keep track of the atomistic character of the material. The phenomena of interest are usually describable as spatial averages over many lattice spacings. In such a description, the microscopic quantum aspects appear only in the form of constants in the equations governing the statics and dynamics of the magnetization. There are exceptions (for example magnetic force microscopes) that probe the material on a more or less atomic scale. Also, with the advent of nanoscience, intrinsically quantum mechanical phenomena that have no classical limit (such as tunneling of the magnetization as such) may sooner or later require attention. This book will largely be confined to a macroscopic description in terms of a magnetization field as a continuous function of position, governed by classical equations of motion. We begin with an account of the plausibility of this point of view, at least for magnetic insulators in which the ionic spins (or effective spins) may be considered localized. At first sight, the case of metals would seem to lead to the classical magnetization field more readily, since the itinerant character of the electrons makes a quantum mechanical spin density field operator a logical starting point. In fact, the transition to the classical field description of magnetic metals requires rather advanced theoretical methods outside the scope of this work. For early efforts in that direction, see Herring and Kittel (1951).1 It is conceptually easier
i } of N spins. to consider the case of magnetic insulators with a localized set {S Let H denote their Hamiltonian, i.e. their total energy expressed in terms of the Si . Their Heisenberg equations of motion are  
i , H
i /dt = S i = 1...N (1.1) i
dS −

with the right hand side denoting the commutator. The Hamiltonian H is always
i , (α = 1, 2, 3). The commutation a multinomial in the components Siα of S

1 For a nonmagnetic electron gas, reducing the quantum mechanics to a more phenomenological description, such as Fermi liquid theory, has more recently been found possible using renormalization theory (Shankar, 1994). For possible extensions to magnetism, see Chubukov (2005) and Rech et al. (2006).

1

2

THE CLASSICAL MAGNETIZATION FIELD


i are relations of these components Siα of S Siα Sjβ − Sjβ Siα = i
δij αβγ Sjγ

(1.2)

Consider, for example, the commutator (Siα , (Siβ )2 )− that might be encountered during the evaluation of the commutator in equation (1.1). It is equal to   (Siα , Siβ )− Siβ + Siβ (Siα , Siβ )− = i
αβγ Siγ Siβ + Siβ Siγ   = i
αβγ 2Siγ Siβ + i
βγδ Siδ   = i
αβγ Siγ d(Siβ )2 /dSiβ + o(
) . Similar results hold for commutators of the Siα with the higher powers and/or products of spin components that occur in H. From this it follows that
i dS
i × ∂H + o(
) = −S
i dt ∂S

(1.3)

As
→ 0, this equation, though still in terms of operators, has the same form as the classical equations of motion of magnetic moments associated with the spins. Thus, (looking forward to eqn (1.4)), this obeys the correspondence principle, and becomes a classical equation in the limit of large spins. (For large spins, the right hand side of eqn (1.2) may be neglected compared with the quantities on the left. The latter then commute, i.e. behave classically.) However, this is no great comfort; in many materials the individual spins are not large. Thus, if we desire a classical description of a coarse-grained magnetization field, we need to examine the sense in which the spin may be considered large, even though the constituent spins may be small. This is made plausible as follows: A full quantum mechanical treatment of the system would supply a correlation length of the spin system. At temperatures well below the Curie temperature, this correlation length will be large, comprising n lattice sites, say, around position
x. Then over a region n3 , the spin
x)>. This averorientations will hardly change from an average direction, <S(
3 age may thus be viewed as a single, rigid spin of magnitude n S, independent of
x, with only very small fluctuations around this value. Similarly, its magnitude should fluctuate in time with only small amplitude. A continuum field view is then justified if n is large enough to comprise many spins, yet small on the scale of distances over which the direction of the magnetization changes. The spin
(
x, t) of constant system may then be represented by a magnetization field M magnitude M (x, t) practically equal to the saturation magnetization Ms at the ambient temperature T  Tc . The mathematics needed to justify this qualitative argument is known as renormalization theory.

INTRODUCTION

3

This proceeds by eliminating the rapidly varying (and usually unobservable) components of the discrete spin system in favor of the slow components accessible to observation. In this process, one finds a new coarse-grained “Hamiltonian” depending only on the retained variables. The constants in that “Hamiltonian” are not equal to the ones in the microscopic expression, but have so-called renormalized values. In addition, the new Hamiltonian contains terms of a structure not found in its original form. In principle, one can imagine solving the equations of motion for the unwanted variables at given values of the wanted ones. The solutions are then substituted in the equations of motion for the wanted variables. Unfortunately, these equations then contain terms that are non-local in space, and have a memory of past history. If it is assumed that the unwanted variables fluctuate rapidly with both time and position, this non-locality effect can be ignored. The offending terms can be replaced by their current, local values, and the now purely local equations of motion can be derived from an effective Hamiltonian. If the system is in thermodynamic equilibrium, and interest centers on a close vicinity of Tc , this tedious process can be avoided by “integrating out” the high wavenumber components of the variables, and rescaling these, as well as the distance scale, so that the new Hamiltonian describes the same physics as the old one. In particular, the new correlation length should still become infinite at an appropriate Tc . The whole process is repeated until a so-called fixed point is reached, at which the Hamiltonian no longer changes under further operations. Except in Section 1.3, interest is limited to temperatures far below Tc , so
(
x, t) is a robust concept. However, it must be rememthere is no doubt that M bered that experimental equipment has limited resolution, and sees only the coarse-grained vector field. So the constants in the effective Hamiltonian for the
(
x, t) have values not quite equal to the constants in the microscopic observed M Hamiltonian, except possibly at extremely low temperatures. Furthermore, not
integrated out of the observations, but so are all only is the “fast” part of M external degrees of freedom to which the spins may be coupled but hold no interest to the observer. This situation will be encountered in some later chapters, and some practical consequences will be noted. Evidently a price must be paid: one cannot use this description in the case of phenomena with a spatial variation greater than the reciprocal coherence distance of the magnitude Ms and/or involving times shorter than its correlation time. Otherwise stated, a greater rate of variation would be inconsistent with maintainance of a well-defined, large spin, implying excitation energetic enough to break up the coherence within a correlated cluster. The successes of the classical field approach in the description of various dynamical processes in magnetic recording would lead one to judge that correlation distances and times are, in fact, adequate. However, certain experiments can induce highly irregular motion, quite possibly cascading towards chaos. If such chaos is sufficiently fine grained, the classical field approach may become questionable. This would be the analog

4

THE CLASSICAL MAGNETIZATION FIELD

of fluid turbulence so fine-grained that the Navier-Stokes equation would have to be replaced by molecular dynamics. Throughout most of this work, the temperature is assumed to be well below the Curie temperature, and spatial variations of the magnetization direction sufficiently gentle so that the field approach is justified, and we begin with a description of the necessary formalism. (See, however, Section 1.3.)

1.2

Equations of motion


per unit volume in a constant magnetic A position-independent magnetization M



× dE/dM, field H experiences a torque M × H, which can also be written − M
·M
. The torque equals the rate of change of angular momentum, with E = −H and bearing in mind the angular momentum associated with spin, and there
, the equation of motion is ∂ M
/∂t = −γ M
× dE/dM, where γ is the fore M gyromagnetic ratio. This equation of motion must be generalized to the case in
depends on position as well as time. Then the total energy E can be which M written as a volume integral of an energy E(
x) per unit volume. The derivative of the
energy must now be written as the functional derivative of the total energy
(
x, t). Here, E(
x, t) is the energy per unit E = d3 xE(
x, t) with respect to M volume, including, where needed, the energy in time-dependent fields. Eqn (1.3) is thus replaced by ˜
∂M
× dE = −γ M ˜
∂t dM

(1.4)

where the tilde over the d denotes functional differentiation, defined by ˜ dE dE = 3lim ˜

d x→0 dM (
x)d3 x dM

(1.5)


or certain other For contributions to E that involve no differentiation of M operations, the functional derivative reduces to the ordinary derivative of the part of the energy density E describing those contributions. E is written as a sum Eex + Edip + Eem + Ean , (exchange, dipolar, electromagnetic, and anisotropy), plus any coupling energy to other degrees of freedom external to the spin system. (The last contribution will be discussed in detail later.) Eex is the exchange energy. For the individual ionic spins (or effective spins when the total angular momentum is a good quantum number), prior to 

coarse graining, this is usually written −J Si · Sj summed over near, usually nearest, neighbors. For the nearest neighbor case, this may be written, apart 

i )2 , where
denotes the various − S from an unimportant constant, J i, (S  i+
i by nearest neighbors to i. Were one to proceed naively and simply replace S 3
M (
x)Ω/µB , where µB is the magnetic moment of the spin, and Ω = an atomic

EQUATIONS OF MOTION

5

volume, one would write this sum as  2

J(Ω/µB )

 3 2  2 2


(∇Mi )2 d x M (
x + ) − M (
x) = J(Ω/µB )

d3 x 3



i=1


in order to restrict the result to sufficiently neglecting higher derivatives of M slow spatial variation. This form is consistent with the coarse-graining process, except that the value of the constant factor needs renormalizing. Since this renormalization program is beyond the scope of this work, we shall simply write Eex =

1 2 A

2

 d3 x

3 

(∇Mi )2

(1.6)

i=1

where A is a dimensionless constant, evidently proportional to J. Next, the dipolar energy has the form   Edip =

 3
(
x) · M
 M  (
x )/|
x −
x |− − 
(
x) · (
x −
x ) M
(
x ) · (
x −
x ) /|
x −
x |5 3 M



 d3 x d3 x (1.7)

Frequently, it is more conveniently described with the help of an auxiliary field Φ:

 Edip =

d3 x

1
(∇Φ)2 − 4π∇Φ · M 2

(1.8)

which, however, is assumed to have no “equation of motion”. This is particularly ˜ ˜ dip /dΦ useful if electromagnetic propagation effects can be neglected. Setting dE equal to zero gives Poisson’s equation
∇2 Φ = −4π∇ · M

(1.9)

Integrating this, and substituting the result in eqn (1.8), then gives the expression (1.7). No heed has so far been paid to the boundary conditions; to allow for these, it is necessary to add a solution of Laplace’s equation ∇2 χ = 0 to the usual inhomogeneous solution of eqn (1.9). This gives the needed flexibility for matching internal and external fluxes and fields at the boundaries. Then from eqn (1.9), ∇Φ + 4πM = ∇χ. The left hand side will be recognized as the vector
with ∇Φ the internal H-field.
B, ∇χ will be zero inside the sample, and equal

in vaccuo. One of the familiar to the B vector outside, which is equal to H boundary conditions is ∂Φ/∂n + 4πMn = ∂χ/∂n, where
n denotes the normal, and the other is
n ×∇Φ =
n ×∇χ (continuity of the tangential field components).

6

THE CLASSICAL MAGNETIZATION FIELD

More generally, the interaction energy with the electromagnetic field is written 
·H
em d3 x Eem = − M (1.10) We do not trouble to add here an expression for the free electromagnetic energy since we know that this leads to Maxwell’s equations to be included in a later chapter on relaxation processes. Finally, anisotropy energy Ean is the result of the spins being able to sense the lattice and its symmetry via spin-orbit coupling. One does not wish to be
of the electrons in discussing burdened with the orbital angular momentum L

is to treat the spin orbit the motion of M . A favorite way of eliminating L

coupling energy, ∝ L · S, as a perturbation of the Hamiltonian, and then apply standard Schroedinger perturbation theory, using only the orbital states as basis
as ordinary c-numbers. The states, effectively treating the components of S
only. Of course, resulting energy shift is then a function of the components of S this procedure is not quite correct, but it is acceptable if the true perturbed state contains only a small admixture of states with higher spin multiplicity, because of large energy denominators. Both the resulting spin Hamiltonian, and its coarse-grained limit will reflect the crystal symmetry. For a crystal with uniaxial symmetry around the z-axis,  1 (1.11) Ean = K M32 d3 x 2 whereas for cubic symmetry Ean =

1  K 4

 (M14 + M24 + M34 )d3 x

(1.12)


is large enough, there will also be terms in higher powers of M
, but these are If S usually much smaller. In most of this book we shall assume uniaxial anisotropy. (K, K  are constants, and the factors 1/2 and 1/4 are simply convenience factors.) The form of this so-called crystalline anisotropy is in some cases similar to dipolar energy. For example, if the formalism of eqn (1.9) and the discussion following it is applied to the case of an ellipsoid in a uniform magnetic field exceeding the saturation field, the dipolar energy can be written in the form
 1 Mx21 + Nx2 Mx22 + Nx3 Mx23 d3 x where the N are demagnetizing factors Nx1  2 obeying Nxi = 4π, and the principle axes of the ellipsoid are chosen as coordinate axes. This simple result works only if the M components are uniform, as they always will be above the saturation field, or for any size field in sufficiently small (somewhat idealized) samples, as discussed in later chapters. ˜ d˜M
, which we define as H
eff in the torque The next task is to evaluate −dE/ eqn (1.4). The only contribution to this quantity that perhaps requires slightly

EQUATIONS OF MOTION

7

˜ ex /d˜M
. We obtain this by writing the derivative in detailed consideration is dE the form ˜ ex
+ δM
}) − Eex ({M
}) dE Eex ({M = lim
d3 x
δM d˜M with both magnetization element and volume element going to zero. (Curly brackets are used to denote functional dependence on the entire field.) Using
to vanish on the the form of eqn (1.6), integrating by parts, and assuming δ M boundary, results in
ex = A 2 ∇2 M
H

(1.13)


, derived from the All other contributions to Heff have simply the form −∂E/∂ M other contributions to E. One awkward feature of the equation of motion (1.4)
∂M
×H
eff = γM ∂t

(1.14)


2 = constant (equal to the square of the saturation magneis that it gives M tization Ms ) for all
x and t, so we have three differential equations for only two unknown functions. Probably the simplest way to avoid this difficulty is to
in polar coordinates (θ, φ, Ms ): express M M1 = Ms sin θ cos φ, M2 = Ms sin θ sin φ, M3 = Ms cos θ

(1.15)

The directions of increasing θ, increasing φ and increasing M form a right handed system of axes (see Fig. 1.1). To transform eqn (1.14) into this axis
has components (Ms dθ, Ms sin θdφ, 0), system, note that the increment dM
while the vector M has components (0,0,Ms ). The components of Heff are then ˜ ˜ ˜ ˜ −dE/M s dθ, −dE/Ms sin θ dφ, 0, since the length of M is not allowed to vary. The equations of motion are therefore ˜ ˜ dE dE ∂θ ∂φ =− = , sin θ ˜ ˜ ∂t ∂t sin θdφ dθ

(1.16)

where, for convenience, Ms has been equated to unity, and γt has been replaced by t. These can also be written in Hamiltonian form as ˜ ˜ dE dE ∂ cos θ ∂φ = =− , ˜ ˜ ∂t dφ ∂t d cos θ

(1.17)


2 is conserved, with canonical variables cos θ, φ. Aside from the guarantee that M the formulation in terms of polar coordinates has a further advantage in the

8

THE CLASSICAL MAGNETIZATION FIELD

x3

θ

x2

φ

Μ

x1

Fig. 1.1. Magnetization vector in Polar coordinates. frequent case in which there is a strongly preferred direction, such as an applied steady field and/or a uniaxial anisotropy axis. This direction will be chosen as the zero of the θ− angle. The range of θ is 0 to π. The φ variable, on the other hand, is allowed to increase indefinitely with time as the magnetization vector precesses, even if haltingly, about the preferred direction. Writing φ = ωt + ψ, where ω is an average precession frequency, we have, in rotating axes, M2 /M1 = tan ψ. So if ψ has a random spatial and temporal fluctuation, the motion of the transverse components is being dephased. The θ− variable, on the other hand, can only nutate between zero and π and, in many situations, will do so slowly on the time scale of the fluctuations in ψ. In constructing E in polar coordinates, only the exchange part needs some attention. Substituting (1.15) in eqn (1.6) gives (with Ms = 1),   1 2 (1.18) Eex = A

d3 x (∇θ)2 + sin2 θ (∇φ)2 2 and functional differentiation then gives ˜ ex  dE = A 2 −∇2 θ + sin θ cos θ (∇φ)2 ˜ dθ ˜ dEex = −A 2 ∇ · (sin2 θ∇φ) ˜ sin θdφ

(1.19)

EQUATIONS OF MOTION

9

The other contributions to E are simply found by substituting the components from eqns (1.15) into the Cartesian expressions for E. The functional derivatives then become ordinary derivatives of the corresponding energy densities E.

1.2.1

Damping

Unless E has some explicit time dependence, the total magnetic energy E is a constant of the motion, as is seen by differentiating it with respect to time and ˜ d˜M
of the effective using eqn (1.14), together with the definition Heff = − dE/ field. A further term or terms are needed in eqn (1.4) to describe reversion of the magnetization field to equilibrium from some excited state. Evidently, for total equilibrium to be achieved, coupling to some sufficiently large external reservoir must be considered. This relaxation process will be called “intrinsic damping”. From a practical point of view, this is not the only kind of relaxation observed in a typical experiment. Normally, an experiment samples only
some particular
, in most cases the spatial average M  = Ω−1 M
d3 x over the function of M entire sample. In general, this is coupled to various spatially varying features
, which are excited by this coupling, without changing the total magnetic of M
, which will energy E. The result is a degradation of the observed feature of M be called “distributed damping”. One of the first phenomenological expressions for a damping term was that proposed by Landau and Lifshitz (1935) (LL). They proposed a form purely local in space and time, with a structure that forces the magnetization to seek complete alignment with the local effective field. The term
× (M
×H
eff ), −γαLL M

(1.20)

with αLL a positive constant, has the necessary form. It is readily shown that such a term describes intrinsic damping. Adding eqn (1.20) to the right hand side of eqn (1.14), one finds 

˜ dE ˜
dM (
x, t)  ˜ dE = αLL ˜
dM

dE = dt



˜ dE
d˜M 
= αLL M

= αLL


(
x, t) ∂M d3 x ∂t

 ˜ dE
× M
× · M d3 x ˜
dM

  ˜ ˜ dE dE 2
M
· · M − Ms d3 x ˜ ˜

dM dM

2 2 ˜ ˜ dE dE · − Ms2 d3 x ˜ ˜

dM dM

·

(1.21)

10

THE CLASSICAL MAGNETIZATION FIELD

In the final, fully relaxed, state, no more precession must take place, so that ˜ d˜M ˜ d˜M
must everywhere be along −dE/
, or dE/
= −λM
. Thus dE/dt =0, M as required in the relaxed state. Prior to that, the right hand side of the last equation is always negative, according to the inequality (
a ·
b)2 <
a2
b2 that holds at each point in the integrand. Thus E steadily decreases towards equilibrium. Hence LL damping involves intrinsic damping. A different form for the damping torque was proposed by Gilbert (1955):
× dM
/dt −γαG M

(1.22)

For weak damping, the two forms are equivalent. Substituting the undamped
/dt gives, to lowest order in the damping constants, form (1.13) for dM αLL = γαG Thus, in the case of weak damping, there is no difference to say which of the two forms should be used in the presentation of experimental data. Nevertheless, it is of interest to find out which of the two forms is nearer to the “truth” when the damping torque is derived from the physics of the coupling to the reservoir. As will be demonstrated in later chapters, neither form is quite correct; in general, the damping torque is neither local in time nor local in space. At best, an asymptotic series may be established, of which the leading term is a purely local torque. That leading term has the form advocated by Gilbert. However, there is at least one known example, discussed later, in which that torque is a more
than either the LL or the G form, complicated function of the components of M and this can become functionally (as opposed to just numerically) important for
. large motions of M
, such as its spaAs regards distributive damping of some linear function of M tial average, it must be noted that its magnitude is not a constant of the motion, except in the trivial case of only a uniform external field and its bulk demagnetizing field representing Heff . When Heff has significant spatial dependence,
 may still be imagined as precessing in the uniform contribution the average M to Heff , but its precession will be marred by scattering to other modes of magnetic motion (rewriting eqn (1.14) in Fourier components immediately makes this clear). The precessional motion will progressively be dephased, so that the magnetization components transverse to the uniform field eventually decay to zero. This process resembles the 1/T2 relaxation in nuclear resonance, whereas intrinsic damping resembles the 1/T1 relaxation rate. One may be tempted to regard distributive damping as a special case of intrinsic damping by considering the non-observed motions of the magnetization as the reservoir, but then it is necessary to solve the equations for these motions in terms of given values of the observed motion if a closed equation for the observed motion alone is to be obtained. This is not only difficult, but there is also a problem of principle, which is most easily noted in the linearized version of this reduction process. As noted in the next chapter, instabilities in the closing procedure

APPROACHING THE CURIE TEMPERATURE

11

arise that can be overcome only by a reliance on sufficient damping, most likely intrinsic damping. If, therefore one does not insist on closure, one is left with the dephasing picture discussed above, but it is generally applicable only for small
. When arbitrarily large motion of M
is allowed, the instabilities motions of M of course disappear (exponential growth cannot persist for ever), but at that point a nonlinear theory is called for. In this respect, a reservoir composed of the spin system itself is very different from reservoirs composed of non-magnetic degrees of freedom. The latter, if large enough, remain stable in the face of the
. motion of M
. Intrinsic damping takes a very simple form in polar representation of M
× dM
/dt has compoChoosing the Gilbert form, it is seen from Fig. 1.1 that M ˙ θ˙ along θ and φ increasing respectively. Hence from eqn (1.15), nents − sin θ φ, the Gilbert form, to lowest order in the damping, results in ˜ ˜ ˜ ˜ dE dE dE dE ∂θ ∂φ =− = − αG , sin θ − αG ˜ ˜ ˜ ˜ ∂t ∂t sin θdφ sin θdφ dθ dθ

(1.23)

The same form will be found (with appropriately changed damping constant) if the LL form of the damping torque is used. (As before, γt has been replaced by t, and Ms is chosen to be 1.) It is clear that no comparably simple polar form exists for the motion of M  in the case of distributive damping.

1.3

Approaching the Curie temperature


(
x, t) has been So far, it has been assumed that the magnitude of the vector M independent of
x and t; only its direction has been subject to change. This is appropriate at temperatures well below the Curie point Tc (see discussion in Section 1.1). But in recent years the trend to ever higher magnetic recording density, requiring very small magnetic particles, has led to work just below Tc . The direction of the magnetization vector of particles that are so small that the magnetic energy barrier to reversal is no bigger than the thermal energy kB T , is subject to large fluctuations. These seriously reduce the lifetime of magnetic recording (superparamagnetism). To avoid this problem, materials with large anisotropy (i.e. large energy barriers) can be employed. The problem there is that the correspondingly large coercive fields require large magnetic signals to imprint the recording. In principle, this difficulty can be overcome by utilizing the fact that crystalline anisotropy generally shows a marked temperature dependence, becoming very small just below Tc . Hence, if the sample is heated to a temperature just below Tc , a conveniently small magnetic field is needed to magnetize the particle. If it can then be cooled sufficiently rapidly to meet recording rate requirements, the revived anisotropy barrier should be large enough to ensure an acceptable lifetime of the recording. This process is called thermally assisted recording (Alex et al., 2001).

12

THE CLASSICAL MAGNETIZATION FIELD

To describe it mathematically, the dynamics of the magnitude of the magnetization, as well as its direction, must be studied. In as much as the proximity of the transition temperature is involved, it is necessary to take note of the theory of magnetic phase transitions. In its usual form, this theory concerns itself only
, not its direction. Before the advent of modwith M (T ), the magnitude of M ern renormalization theory, the Ginsburg-Landau theory of second order phase transitions was used, with M as order parameter (Landau and Lifshitz, 1958). The appropriate free energy in the vicinity of Tc is written in the form  F({M }) =



1 1 d x (∇M ) + a(T − Tc )M 2 + bM 4 + · · · 2 4 3



2

(1.24)

with higher powers of M neglected, and a and b positive constants. At equilibrium, this expression must be a minimum, requiring a spatially uniform M, and a(T − Tc )M + bM 3 = 0

(1.25)

For T > Tc , the only real solution is M = 0. For T < Tc , there is another solution  M = ± a(Tc − T )/b

(1.26)

with finite magnetization. Unfortunately this simple “mean field” type of theory breaks down very close to Tc , since it takes no account of fluctuations. Renormalization theory takes better account of these, with the result that M varies, not as (Tc − T )1/2 very close to the transition, but rather as (Tc − T )β , where β is not a rational fraction. The criterion of how far T must be below Tc for the simple mean field result to hold is known as the Ginsburg criterion (for an excellent account, see Goldenfeld 1992). For the problem at hand, we assume that the Ginsburg-Landau theory (GLT) is adequate, in the hope that, in thermally assisted recording, the Ginsburg criterion is always satisfied. But now the order parameter has three components. Presumably, the appropriate generalization of the free energy (1.23) is now 
}) = F({M



1 1 d3 x E(
x) + a(T − Tc )M 2 + bM 4 + · · · , 2 4

(1.27)

where E(
x) is the sum of the energy densities, exchange, dipolar, anisotropy etc., that composed the total magnetic energy E described above. But now the magnitude Ms is no longer a constant, and will henceforth be denoted by M, which in general depends on
x, and also on t in dynamic problems. At temperatures below, but close to, Tc we are allowed to ignore the terms in powers of M in the integrand of eqn (1.26), at least within the constraints of

APPROACHING THE CURIE TEMPERATURE

13

mean field theory. The generalization of eqn (1.25) to arbitrary temperatures is the Weiss molecular field equation M = Bs (λM/(kB T ))

(1.28)

where Bs is the Brillouin function for spin S, and λM a measure of the exchange field. Well below Tc (the temperature just below which this equation has a nonzero solution), the solution of eqn (1.28) approaches a constant. The energy expression (of which the Weiss equation is the derivative) then also becomes a constant, and as such can be ignored leaving only the contribution of E(
x) to
) Nearer Tc this is not allowed. The use of the polar coordinate the total F(M
is no longer especially helpful. For example representation of M

2  ∂M
(1.29) Eex = ∂xi i will now have cross-terms involving gardients of M multiplied by gradients of angular orientations, in addition to the more convenient terms (∇M )2 and M 2 (∇θ)2 , etc. As T approaches Tc it is no longer correct to assume a simple correspondence
(x, t) that was of the equations of motion of individual spins and those of M assumed in Section 1.2. We are interested only in spatial and temporal varia
that are within the limited range of available observational equipment. tions of M But this does not mean that motions outside that range do not affect observations within the range. In principle (but only in principle), we should solve for the equations of motion of the parts of the magnetization field outside the range in terms of the portions within range, and substitute that solution into the equa
. This leaves us with equations for the interesting tions for the in-range part of M quantities alone. Carrying out this process in practice is out of the question. But we can make reliable qualitative guesses of the ensuing equations, and maybe even get some quantitative results by sufficiently sophisticated perturbation theory. Perhaps the most plausible way to discuss this matter is by using Fourier
(
k, ω) of M (
x, t), with wavenumber
k and frequency ω. Obsertransforms M
and (0, Ω) in wavenumbers and vations are restricted to the ranges (0, K) frequencies. To begin with, ignore all energy contributions except the exchange
 (
k, ω) within the restricted range. First, energy, and consider the damping of M consider the entire spin system. In the presence 
of isotropic exchange coupling alone, the time rate of change of the sum Si is zero, and that sum is the Fourier component with
k = 0. If done correctly, the process of eliminating the unwanted part of the spectrum should leave this conservation law intact. This means that the wanted part should also be undamped at
k = 0. This implies that any damping constant must go to zero with
k, and, in an isotropic case with spatial inversion symmetry, it

14

THE CLASSICAL MAGNETIZATION FIELD

must vary as k 2 . With a small dose of realism injected into the problem (such as 
dipolar coupling on a lattice, for which Si is not conserved), the damping of
 need not go to zero, but it is reasonable to assume that it can be expanded in M powers of
k so as to read Γ0 + Γ1 k 2 for the range of k values of interest. Furthermore, the simple Landau-Lifshitz or Gilbert form for the damping torque cannot possibly hold for M  . The reason is that even though such a torque form allows   2
i to remain constant, the same cannot possibly S the squared magnitude be true for the more general linear combination of spins that compose only the
 . Or, in the language of a configuration space description, the wanted part M damping of a limited block of spins cannot have the magnitude-conserving damp
 ing of the totality of spins. Ignoring the non-dissipative part of the motion of M altogether, the form Γ0 + Γ1 k 2 for the damping suggests that in configuration space, M  decays partly diffusively. For example, an initial M  (
x, 0) will, in time t, turn into a form proportional to exp(−Γ0 t) √ t



 M  (
x , 0) exp −(
x −
x )2 /(4t Γ1 )

(1.30)

This argument seems to have proved too much. At no point has temperature been mentioned, so one may question whether the form of LL or G damping can ever be used, even well below Tc . The explanation becomes most evident in the limit of small vibrations of the magnetization field, known as spin waves, discussed in Chapter 2. It turns out that the low wavenumber components of small vibrations are coupled to higher wavenumber vibrations, and the degree of excitation of these increases with temperature. This coupling represents a drain on the observable components with wavenumbers k < K, say. At low temperatures this effect is small, and the magnitude of a block spin of general dimensions 1/K is then preserved almost as well as the magnitude of the total spin, though its direction may vary. So, well below Tc , the diffusive part of the damping due to draining to unwanted modes is small compared with Γ0 which is due to coupling to an external reservoir and does allow the magnetic energy to decay, as required by LL/G damping. As noted before, it should be borne in mind that, in the process of eliminating unwanted variables in favor of the wanted ones, some terms in the resulting equations for the wanted ones will depend on earlier times, not just the present. Fortunately, as Tc is approached from below, the “memory” of earlier times becomes much shorter than the times of interest in the evolution of the wanted variables, so the offending terms may simply be taken at their current value. However, in the case of thermally assisted recording there is one normally “uninteresting” variable whose memory cannot be treated in this cavalier manner: the temperature. Because of heat generated by the damping of M, the temperature T will rise at the same rate at which M is damped. So we require coupled equations of motion for M and T. For this purpose, we extend the form (1.24) by

APPROACHING THE CURIE TEMPERATURE

adding to it an energy term of the form

 1 1 2 1 3 2 2 d x w(∇Q) + uQ + gQM , 2 2 2

15

(1.31)

where Q is the quantity of heat evolved. Evidently the equation of motion for Q is the diffusion equation driven by the magnetization as a heat source, but we write it directly in terms of temperature, using the appropriate specific heat constant c. Within mean field theory, we can describe the entire process (heating, switching and then cooling) by adding to the equation of motion for T a source Q1 (t)/c to drive the system close to Tc , then applying the switching field up to time t1 , and finally subtracting Q2 (t−t1 )/c for t > t1 in order to restore the system to the low temperature regime. To cover the entire temperature range in mean field theory, it is necessary only to replace the contribution a(T − Tc )m + bm3 to the equation of motion for m by a slight generalization of the Weiss field expression (1.28) to allow for anisotropy fields. Leaving aside these obvious extensions, the coupled
and T are (complete with precession perpendicular equations of motion for M
and diffusion along M
): to M 





· ∇2 M
∂M M ∂E 1 2 2 2
× A ∇ M
−
Γ0 (a(T − Tc ) + bM ) − Γ1 =M −M
γ∂t M2 ∂M (1.32) ∂T = κ∇2 T + gcM 2 ∂t Here E1 is the energy density due to all the fields other than exchange, and κ = wc is the thermal conductivity. Readers interested in pursuing this matter all the way into the critical region T ∼ Tc should consult the chapter on the dynamic renormalization group in the book by Shang-keng Ma (1976).

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2 SMALL MOTIONS OF THE MAGNETIZATION 2.1

Introduction

In both their Cartesian and their polar forms, the equations of motion are nonlinear partial differential equations. In some situations, such as in ferromagnetic resonance at low signal levels, the magnetization vector precesses around a pre
, transverse ferred direction with only a small angle of opening. The part m
of M to the preferred direction, is then small, and the equations of motion may be linearized. In most applications, such as magnetic recording, the motion of the magnetization vector is large and the linearized theory is not immediately relevant. However, it frequently offers valuable insight into what to expect in more realistic situations. Therefore we here present a brief sketch of the linear results and the attempts to go beyond them. 2.2

Models of small motions

To establish the basic modes of small motion, the expression for the energy is expanded up to bilinear terms in the small transverse components which, for convenience, will be written in complex forms a(
x) = m+ = m1 (
x) + im2 (
x),

(2.1)

a∗ (
x) = m− = m1 (
x) − im2 (
x). Taking the preferred direction to be the 3-axis, we have, to second order  M3 = Ms2 − m+ m− = Ms −

1 a∗ a 2 Ms

Usually, a Fourier decomposition of a and a∗ is convenient:  1  d3 x a(
x)e−ik·x ak = Ω   a(
x) = ak eik·x

(2.2)

(2.3)

k

where Ω is the (large) sample volume, and the
k are the discrete points of the reciprocal lattice. (Choosing the
k to be discrete results in the energy E to 17

18

SMALL MOTIONS OF THE MAGNETIZATION

become a discrete sum for easy diagonalization.) To second order in the a’s one then finds, apart from constant terms,  1 k 2 a∗k ak , Eex = − A 2 Ω 2

Edip = πΩ

k

Eem

 (k + a∗ + k − ak )2 k , k2 k

HΩ  ∗ = ak ak 2Ms

(2.4)

k

and, if there is uniaxial anisotropy only, Ean =

 1 KΩ a∗k ak 2

(2.5)

k

Here, k ± stands for k1 ± ik2 . In deriving Edip , eqns (1.8) and (1.9) were used, rather than the more elaborate expression in eqn (1.7). Next, linear combinations of the a, a∗ are found that diagonalize the sum E of these energies. These combinations are known as spin waves or magnons. The equations of motion of the new variables (bk , say) have solutions of the form e±iωk t , where (Fig. 2.1) ωk2 = (ωH − ωM N3 + ωex 2 k 2 )(ωH − ωM Nz + ωex 2 k 2 + ωM sin2 ξk )

(2.6)

Here, ξk is the angle between
k and the z-axis, ωH = γH (plus 12 K if there is uniaxial anisotropy), and ωM = 4πγMs . A small slight of hand has been committed here: N3 is the demagnetizing factor responsible for reducing the external constant field to its value in the interior. But, strictly, the above analysis applies only to the infinite medium in which demagnetizing factors are not defined. In the finite medium, the characteristic modes will not be plane waves; more elaborate modes must be found that take into account the correct boundary conditions. On the other hand, modes with many nodes inside the sample ω(k) ξk = π/2

ω0

//////// //////// //////// //////// //////// //////// ////////

ξk = 0

---------- Q T

|k|

Fig. 2.1. Spin wave frequency versus wavenumber for short wave-length.

MODELS OF SMALL MOTIONS

19

will have frequencies that have about the same density in mode number space as the plane waves have in wave number space. The justification for using the demagnetized field in (2.6) corresponds to the fact that it is the field actually seen by small deviations of the magnetization rapidly varying in position within the sample dimensions. For variations with very few nodes, the exchange energy will be negligible, and then we have a purely magnetostatic problem, strongly involving the boundary conditions. This regime has been studied extensively by Walker (1963); the resulting eigenfunctions are known as Walker modes. The most widely known of these is the uniform precession set up in ferromagnetic resonance experiments on ellipsoidal samples. 2.2.1

Distributive damping

One way to study weakly nonlinear effects is to carry the expansion of E to higher powers in the a’s. These higher terms have been used extensively in the discussion of distributive damping in ferromagnetic resonance. Certain of the terms have the effect of degrading the uniform mode by excitation of spin waves synchronous with it. It turns out that the range of validity of this approach is restricted to situations with adequate intrinsic damping. Without it, the entire spin wave picture is rendered unstable by triple and higher terms in the expansion, at least in the ideal sample. In this brief sketch, to discuss distributive damping, and instabilities, we only carry the expansion to triple terms. Also, we do not exhibit the structure of the coefficients of the triple terms, referring the reader to the literature on the subject (for example, Suhl, 1957). For reasons which will become clear, we write the total energy E = E1 + E2 , where, in suitable units,   ωk b∗k bk + (pkk
bk
b∗k bk−k
+ comp. conj.) (2.7) E1 = k=0

E2 = ω0 b∗0 b0 +



k k
=k

({pk bk b−k b∗0 + c. c.} + {pk0 b0 + p∗k0 b∗0 }b∗k bk )

k=0

where ω0 is the resonant frequency of the mode b0 whose damping we wish to discuss, and pk are the aforementioned coefficients. Typically, b0 will be a magnetostatic, or Walker, mode, which we shall call the uniform mode. In addition, we shall consider damping by imperfections that break momentum conservation, and whose distributive damping effect is relatively easy to discuss, not only in the present context, but also in terms of the full nonlinear theory. The simplest such energy can be written  Eimp = {qkk
b∗k bk
+ (rk,k
bk bk
+ comp. conj.)} (2.8) k,k


To discuss distributive damping, C. W. Haas and H. B. Callen (1963) take the quantum mechanical view, treating the b and b∗ as dimensionless annihilation and creation operators, so that the p’s and q’s must have dimensions of energy.

20

SMALL MOTIONS OF THE MAGNETIZATION

Starting with a discussion of damping due to imperfections, Haas and Callen evaluate the net rate at which one quantum in the uniform mode is destroyed and  ∗ b one spin wave is created. They apply Fermi’s golden rule to the term q k0 k b0 ,  2 δ(
ω −
ω )|q | (n +1)n which describes that process. This gives a rate 2π k 0 k0  k 0,
δ(
ω − from which must be subtracted the rate for thereverse process, 2π k
δ(
ωk −
ω0 )|qk0 |2 (n0 − nk ).
ω0 )|qk0 |2 (n0 + 1)nk , leaving a net rate of 2π
In terms of some representative q, and an energy density of plane wave states ρ 2 synchronous with ω0 , the result may be written 2π
ρ|q| (n0 − nk ). In this result, the thermally excited nk may be neglected compared with n0 , the number of quanta excited in a resonance experiment. The same result can, of course, be derived classically by solving equations of motion for the b as classical variables. We briefly describe that method here, mainly in order to set the stage for a comparison with the corresponding results for large motions discussed in later chapters. Starting with the linearized equations of motion for the a (i.e. m+ ), we make the same linear combinations b that diagonalized the bilinear part of the E (without Eimp ). Then it is found that the motions of the b, under the action of Eimp , satisfy ib˙ 0 = ω0 b0 +



qk0 bk

(2.9)

k

ib˙ k = ωk bk + q0k b0 Setting b0 = B0 (t)e−iω0 t and bk = Bk (t)e−iωk t , where B0 and Bk vary slowly with time if q is small, we get  iB˙ 0 = qk0 Bk ei(ω0 −ωk )t (2.10) k

iB˙ k = q0k B0 ei(ωk −ω0 )t Solving the second equation for Bk (t), subject to Bk (−∞) = 0, and substituting in the first, gives B˙ 0 (t) = −



 |q0k |2

k

=−

 k

t




−∞

 |q0k |2

∞

B0 (t )ei(ω0 −ωk )(t−t ) dt

(2.11)




B0 (t − t )ei(ω0 −ωk )t dt

0

If B0
varies slowly, it may be taken outside the integration sign. Using the fact v that 0 eixt dt = iP (1/x) +πδ(x), one then finds B˙ 0 (t) = −iB0

 k

|q0k | P 2

1 ωres − ωk

− B0 π

 k

|q0k |2 δ(ωres − ωk )

(2.12)

MODELS OF SMALL MOTIONS

21

This is essentially the same result in frequency variables as that of Haas and Callen (1963) in energy variables, except that here we get a small bonus: the detuning of the resonance due to the principal part term. It is easy to extend this solution to large
values of the q’s, by writing the equations in terms of Laplace ∞ transforms ˆb0 = 0 e−pt b0 (t) and similarly for ˆbk. . The result is found to be b (0) 0 (2.13) 2 k |q0k | /(p − iωk )
3  Ω In the continuum limit for k, with k... → 8π d k . . . in total volume Ω, this 3 expression has a branch cut in the p-plane, extending along the positive imaginary axis. The usual approximation for small q’s amounts to shrinking the branch cut to a pole around p = iω0 , resulting in exponential decay. For unrestricted values of the q’s, the decay of b0 is not exponential, and the resonance line not quite Lorentzian. (Using S-matrix methods, a similar result can be derived quantum mechanically). Not surprisingly, since Eimp adds only bilinear terms to the diagonal part of the remaining E, the problem is solved completely. Reverting once more to the language of quantum mechanics, we see that the scattering due to Eimp  conserves both the total energy and the total number of excited quanta n0 + nk . The latter sum is also proportional to the increment in M3 . ˆb0 =

2.2.2

p − iωres −

Instabilities and spin wave condensates

Relaxation produced by the part E2 of E involves trilinear mode-mode coupling, but can still be treated by Fermi’s golden rule in very lowest order. The equation of motion method, however, bares a weakness of the entire expansion procedure. (Most likely, a corresponding weakness would show up in an S-matrix approach.) The contribution to E2 from the second curly bracket inside the summation amounts to a secular shift in the origin of the b0 variable, and may be ignored or forcibly removed by a contact transformation. The rest of it, from the quantum viewpoint, destroys one quantum in the uniform mode and creates two spin wave quanta with oppositely directed momenta. Energy (or frequency) and momentum must be conserved in the process, so that ω0 = ωk + ω−k

(2.14)

or ω0 = 2ωk in the present model. Figure 2.1, point T shows that this synchronism (called degeneracy in quantum mechanics) can in general be satisfied. The rate for this process, according to the rule, is proportional to n0 (nk +1)(n−k +1), and the rate for the inverse process (confluence of two magnons into one uniform quantum) is (n0 + 1)nk n−k . The difference is proportional to n0 (1 + nk + n−k ). In our case, we may assume that nk = n−k on the average, and that nk 1 (at any reasonable temperature). If also n0 nk , then the net rate of loss of the uniform mode is proportional to n0 nk . The trouble is that this is also the rate of gain of the nk , for any
k that satisfies ω0 = 2ωk . So, with n0 given, nk will

22

SMALL MOTIONS OF THE MAGNETIZATION

increase exponentially in time, unless either nk is sufficiently damped by some other process, or else the very growth of nk inhibits n0 . We can write dn0 = −r2 n0 n, dt

2

dnk = r 2 n0 nk dt

(2.15)

where r2 is proportional  to some average absolute square of the
matrix element nk , with the sum extending only over k’s that satisfy p0k , and where n = ω0 = 2ωk . (The factor two in the second equation arises from the assumption nk = n−k ). Evidently n0 + 2n is a constant of the motion equal to c, say. Then from (2.15), r2 dn0 = − n0 (c − n0 ) dt 2 whose integral is 2

cAe−r ct/2 n0 = 1 + Ae−r2 ct/2

(2.16)

c 1 + Ae−r2 ct/2

(2.17)

Hence 2n =

The constants A and c are then found from the values of these at time zero: n0 (0) =

(1/2)c cA ; n(0) = 1+A 1+A

Eqns (2.16) and (2.17) indicate that n0 steadily declines to zero, while 2nk grows until it reaches saturation at value c. A very small initial thermal value of n corresponds to a very large value of A. So c is essentially the initial value of n0 , all of which ends up in the spin wave manifold defined by ω0 = 2ωk . We have tacitly assumed that n0 (1 + 2nk ) − n2k > 0, which requires n0 > nk /(1 + 1/2nk ), as is always the case in ferromagnetic resonance. When damping, particularly intrinsic damping of both the spin waves and the uniform mode, is taken into account, all modes eventually decay to zero, unless one or the other (normally n0 ) is sustained by a driving field. Then growth of the other waves will occur only above a certain threshold value of the “big one”. However, this simple lowest order quantum picture misses all phase information, which has important ramifications in the present case. That information is easily captured by using classical equations of motion. Under the action of the relevant part of E2 alone, the equations of motion are  pk bk b−k − iλ0 b0 (2.18) ib˙ 0 = ω0 b0 + k

ib˙ k = ωk bk + pk b0 b∗−k − iλk bk

MODELS OF SMALL MOTIONS

23

where λ0 , λk are unspecified damping parameters. In terms of amplitudes, these equations read  B˙ 0 = −i pk ei(ω0 −2ωk )t Bk B−k − λ0 B0 (2.19) ∗ B˙ k = −ipk B0 B−k e−i(ω0 −2ωk )t − λk Bk

On the degenerate manifold, the second of these, together with its conjugate ∗ ∗ B˙ −k = ip∗−k B0∗ Bk e+i(ω0 −2ωk )t − λk B−k

will, at fixed B0 , have a solution varying as eΓt , where Γ = −λk ± |pk B0 |

(2.20)

Thus, at fixed B0 , the Bk can grow exponentially, if |B0 | exceeds the threshold value λk /|pk |. Of course, |B0 | does not remain fixed; in fact, in the absence of a drive, it declines to a value below threshold, and eventually to zero, along with Bk and B−k . To find out more, first note that, according to the second of equations (2.19), B0 has established a phase relation between Bk and B−k , whose phases are normally uncorrelated and random, with the average of Bk B−k equal to zero. Here, however, the average Bk B−k has acquired a life of its own, reminiscent of fermion pair formation in superconductivity, or boson formation in superfluidity. This aspect has been stressed by the Russian school, particularly by Zakharov et al. (1970), whose formulation of the problem is known as S-theory. Accordingly, we examine the equation of motion of ∆k = Bk B−k

(2.21)

on the degenerate manifold. Making repeated use of (2.19), we find ˙ k = −ipk (nk + n−k )B0 − 2λk ∆k ∆ n˙ k = −i(pk ∆∗k B0 − p∗k ∆k B0∗ ) − 2λk nk  B˙ 0 = −i pk ∆k − λ0 B0 + h

(2.22)

Here, nk = Bk∗ Bk , and we have added a driving term he−iω0 t in the form of a circularly polarized signal at frequency ω0 . First, consider these equations of motion in the absence of any drive. Then the orbits spiral inwards towards a “fixed point”, (0, 0, 0). Even though one may specify an initial value of B0 above the instability threshold, the resulting initial growth of ∆k feeds back into the third of equations (2.22), in such a way as to decrease B0 to values below threshold. When the driving term is added to that third equation, the fixed point (0, 0, 0) goes to (B0 = h/λ0 , 0, 0). As already shown, this point goes unstable if h/λ0 exceeds the smallest of the values of λk /|pk |. A new, stable fixed point arises at which the system comes to rest. For simplicity, let each of the k-dependent quantities be replaced by a single representative value on the

24

SMALL MOTIONS OF THE MAGNETIZATION

degenerate manifold. So the k- subscripts can be omitted. Furthermore, take all decay constants λk to be equal to a single λ. Without much loss of generality, we choose B0 and pk = p to be real, and ∆k to be purely imaginary, = −i∆ . Then equations (2.22) become `  = 2pnB0 − 2λ∆ ∆

(2.23)



n˙ = 2p∆ B0 − 2λn B˙ 0 = −ρp∆ − λB0 + h where ρ is a measure of the manifold to allow (albeit crudely) for the sum in the third of equation (2.22). The fixed points, stable or not, are obtained by equating the time derivatives on the left of these equations to zero. There are two solutions for the set (B0 , n, ∆ ), namely (h/λ, 0, 0), and (λ/p, n0 , ∆0 ), with n0 = ∆0 = (h − λ2 /p)/(ρp). The first of these is stable if h/λ < λ/p, and the second is unstable. When h exceeds λ2 /p, the first fixed point becomes unstable (as we have already seen), and the second takes over and becomes stable. (Note that, at the threshold of the first fixed point, B0 = h/λ = λ/p, and B0 is still λ/p at the second fixed point even though h/λ > λ/p there. So B0 is “stuck”, which accounts for the so-called saturation of the main resonance above the first critical signal level.) One suspects that the second fixed point, too, goes unstable as h is increased beyond a third threshold, but the crude model we are considering here does not seem to have a sufficiently high dimensionality to show a further sharp transition. Nonetheless, we can anticipate what might happen then, by considering equations (2.23) in the absence of both h and λ. Clearly all fixed points will then be telescoped down to h = 0. The first two of (2.23) then give n2 − ∆2 = c2 , a constant

(2.24)

so that, from the first and third, √ 2 c2 + ∆2 B0 d∆ =− dB0 ρ∆ or 

ρ

c2 + ∆2 + B02 = d2 , another constant

(2.25)

Write n = c cosh ξ, ∆ = c sinh ξ. Then the first of (2.23), together with (2.25), becomes ξ˙ = 2p



d2 − ρc cosh ξ

(2.26)

MODELS OF SMALL MOTIONS

This can be integrated in terms of elliptic functions. The final result is   ρ n(t) = n(0) − r− sin2 am r+ t 4p   ∆ (t) = n(t)2 − n(0)2 + ∆ (0)2

25

(2.27)

B02 (t) = ρ (n(0) − n(t))  where r± =

n(0) ±



n(0)2 − ∆ (0)2 , and ∞

amu =

 qn πu nπu +2 sin 2n 2K(κ) n(1 + q ) K(κ) 1

√ with K(κ) the complete elliptic integral of the first kind, K  (κ) = K( 1 − κ2 ), q = exp(−πK  /K) and finally κ = r− /r+ . All three quantities in (2.27) oscillate = 16K(κ)p/r+ , and harmonics. All their amplitudes with basic frequency ωlc  are about equal to n(0) − n(0)2 − ∆ (0)2 , but they are all centered differently. So we suspect that even with finite losses somehow compensated by a drive, a limit cycle may arise, with the second fixed point becoming unstable. This does not appear to be the case in this toy model, at least not when the h field is exactly in resonance, which the foregoing equations assume. However, it is true that, for large fields, the motion – even with B0 , ∆ , n started very close to the values appropriate to the second threshold – will not promptly attain these values. They will first spiral a long way out, as though searching for a limit cycle, then spiral in to attain their appropriate steady values only after many revolutions. (Fig 2.2). The complete treatment in Zhang and Suhl (1988), shows that a limit cycle does indeed arise for large enough drive. Undoubtedly, this is the origin of the relatively low frequency auto-oscillations observed in almost all experiments at high signal powers. Incidentally, from the first and second of eqns (2.23) it follows that, for any value of h beyond the first threshold, d 2 (n − ∆2 ) = −4λ(n2 − ∆2 ) dt so that, with n = c cosh ξ, and ∆ = c sinh ξ, the “amplitude” c now satisfies c˙ = −4λc, i.e. it decays exponentially. So, eventually, ∆ = ±n. This confirms that, for large enough h, the “anomalous” quantity ∆ attains the status of a c-number. These considerations indicate the vital role played by dissipation in ensuring even the limited validity of the spin wave picture of magnetic excitations. Evidently, the three-mode coupling energy will, in the absence of losses, cause instability of any pair of waves with wavenumbers
k ±
κ whose frequencies obey the conservation relation ω(
k) = ω(
k +
κ) + ω(
k −
κ). When dipolar terms are

26

SMALL MOTIONS OF THE MAGNETIZATION (a)

B0 2 1.5 1 0.5 –0.5

0.5

1

1.5

2

–0.5 –1 –1.5

B0 0.01 0.0075 0.005 0.0025 –0.001 –0.0005 –0.0025

0.0005

0.001

0.0015

∆''

0.002

–0.005 –0.0075 –0.01

(b)

B0 0.4 0.375 0.35 0.325 0.3 0.275 0.25 0.225

0.08

0.12

0.14

0.16

Fig. 2.2. Continued

0.18

∆''

∆''

MODELS OF SMALL MOTIONS (c)

27

Bo 0.6 0.4 0.2

9.5

10

10.5

∆''

–0.2 –0.4

Fig. 2.2. (a) Transient decay of driven precession amplitude B0 and the spin wave pair ∆ = iBk B−k versus time as parameter, towards the first fixed point with h below threshold. Insert: Detail of final approach. (b) Transient decay beyond the first towards the second fixed points. h = 0.5, λ = 0.3, p = 1, ρ = 3. Second fixed point has ∆ = h − λ2 /(pρ). (c) Third fixed “point” is a limit cycle (auto-oscillations), but is not quite reached in this figure (see text).

neglected, i.e. for a purely quadratic, angle-independent dispersion relation this condition cannot hold, but with dipolar forces included, it can − even though the “amount of phase space” will be less than in the case in which
k = 0, i.e. the uniform mode. Similar results apply to higher order coupling terms. The question remains whether distributive damping is sufficient to ensure stability, since it just spreads the excitation around to other spin waves, thereby enhancing their amplitudes slightly. Truly dissipative damping is obviously a safer bet. The purpose of this brief survey was to demonstrate that spin wave theory, supplemented by quite low-order mode-mode coupling, can deliver results that have been verified in experiments. The instabilities discussed here, at least their initiation, may be regarded as a parametric excitation of spin wave pairs. In the above discussion, it was the uniform precession that provided the modulated parameter. An even simpler mechanism was proposed by Schl¨ omann (1959), who showed that modulation of the magnetizing field H could induce instability of appropriate spin wave pairs of opposite momenta. This so-called “parallel pumping” technique is widely used as a probe of small magnetic modes (Schl¨ omann et al., 1960). 2.2.2.1 Use of spin wave clusters for signal propagation Another category of spin wave theories that has successfully stretched the limits of applicability of

28

SMALL MOTIONS OF THE MAGNETIZATION

the small motion picture is the work on propagation of spin wave packets or pulses (see Slavin et al., 1994, also earlier work by Kalinikos et al., 1983 and 1991.) Without mode-mode coupling, the initial pulse would soon lose its shape as it advances through the medium, because the spin wave spectrum is rendered dispersive by the exchange interaction responsible for the quadratic dependence of the spectrum on wavenumber. (Dipolar terms contribute to dispersion only to the extent that the spectrum depends on the angle of propagation of the wavevector, but not on its magnitude.) However, mode-mode coupling can yield an effective attraction among the constituents of the packet that will tend to keep the pulse together, counteracting the effect of dispersion. A highly condensed summary of Slavin et al. (1994) on the subject is presented here: Confining themselves to interaction energy involving the product of four spin waves, these authors start with the equations of motion within the manifold 2ωk = ωk+κ +ωk−κ to conserve energy and momentum (point Q, Fig. 2.1)  −ib˙ k = ωk bk − qb∗k bk+κ bk−κ (2.28) Combine these modes to form a narrow wavepacket Ψ = on
k0 . Write bk0 +κ = φ(κ)Bk0 +κ ei((κ·r)+ωk0 +κ t)



φ(κ)bk0 +κ , centered

(2.29)

and in equation (2.28) keep only terms with κ centered around 0. The frequency in the exponent is expanded: ωk0 +κ = ωk0 +
κ ·
vg + 12 (
κ · ∇k0 )2 ωk0 , where
vg is the group velocity ∇k0 ωko . Next, it is assumed that κ may be neglected in the subscripts of the B’s on the right hand side of equation (2.28). Substitute (2.29) in (2.28). If the triple terms in (2.28) were absent, the φ(
κ) would be constants. Assuming the interaction q to be sufficiently weak, φ(k) should change only slowly with time, satisfying

 1 ∂φ 2 +
κ ·
vg + (
κ · ∇k0 ) ωk0 φ = − −i (2.30) qb∗k bk+κ bk−κ ∂t 2  2 =− q |φ| φ where, on the right hand side, only terms near the center of the packet have been
retained. This is transformed into configuration space: φ(
x) = d3 κei(κ·x) φ(
κ), which gives



 ∂ 1 2 2 i +
vg · ∇x + (∇x · ∇k0 ) ωk0 φ − (2.31) q |φ| φ = 0 ∂t 2 For sufficiently simple geometries and/or dispersion relations, the “diffusion” term on the left hand side is diagonal, becoming proportional to ∇2x φ. Seeking solutions of the form φ(
x −
vs t) reduces the number of independent variables from

MODELS OF SMALL MOTIONS

29

four to three. Further specializing to solutions depending on only one spatial variequation able along the velocity
vs , turns it into an ordinary nonlinear  q differential (
x −
v t) , sometimes with a well-known solution, proportional to 1/cosh s D called a soliton. Here, D = ∂ 2 ωk0 /∂k02 , where the k0 -derivative is in the chosen direction. Of course, a solution assumed to depend only on the particular combination (
x −
vs t) is not necessarily stable. Slavin et al. (1994) describe stability conditions in some detail. Not surprisingly, one condition is that the diffusion coefficients be not too large, all else being equal. Losses, not mentioned so far, can be included approximately. In their simplest form, they may be represented by a term −α(∂φ/∂t) on the right hand equation (2.31). The approxi side  of q (
x −
vs t) . Actual observations mate solution then becomes e−αt /cosh e−αt D of these magnetic solitons were made in thin films of yttrium iron garnet. The modes with magnetization variation normal to the film are then treated as discrete, and for each of these discrete modes there will be a different soliton solution propagating in the plane of the film. Finally, a large body of experimental and theoretical work has been devoted to the question of chaos in ferromagnetic resonance. The reader further interested in this subject is referred to the following accounts: Gibson and Jeffries (1984), (Experimental observation of period doubling sequence, followed by apparent chaos and auto-oscillation of various periods); Bryant, et al. (1994), (A thorough experimental investigation, resulting in a “phase diagram” in parameter space separating regions of different nonlinear behavior); McMichael and Wigen (1991) (Parametrically coupled Walker modes in thin films); Caroll, et al. (1991), (Classification of experimental results in the low threshold coincidence regime of ferromagnetic resonance); Rezende and Azevedo (1994), (Effects of imperfections and boundaries and classification of instabilities). The main objective of these works has been to characterize the observations in terms of known universality principles of nonlinear dynamics, and there has been some success in this endeavor. To the extent that the experiments confirm bifurcation sequences or other, more direct, routes ending in chaos, and to the extent that these are explained by mode coupling theories, it must be concluded that the system of partial differential equations unrestricted to small motion should be non-integrable. This would preclude an exact analytic theory of large motions, but it does not rule out model theories based on approximations other than the small motion method, and more appropriate to large motions.

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3 INTRINSIC DAMPING 3.1

Introduction

In chapters 2, the discussion centered on the manner in which some particular mode of interest, usually the spatially uniform mode, dissipated into other modes of magnetic motion, solely as the result of coupling between the magnetic degrees of freedom. For this reason, we used the term distributive damping to describe the process. For small motions, a linearized analysis, augmented by inclusion of coupling terms between the linearized modes, is usually sufficient. For large motions, instability tendencies due to the coupling terms make it desirable to treat the magnetic system without such approximations. Where the linearized regime is appropriate, it is found that distributive damping of any particular mode is best described by the Bloch-Purcell type of equations familiar from nuclear resonance. Landau-Lifshitz-Gilbert damping terms are clearly inappro
vector occurring therein is interpreted as describing a single priate when the M
(
x, t), which is not mode of interest. (LLG terms preserve the magnitude of M the case for a single mode.) In this chapter, we consider damping due to transfer
alone. of magnetic motion to degrees of freedom that cannot be described by M Four different mechanisms will be considered: magnetostrictive coupling to a lossy lattice, magnetic losses in a metal, coupling to fluctuating valence sites, and inelastic coupling to localized magnetic impurities. It should be stressed that the aim here is not to discover new physics. Rather, the idea is to eliminate the parasitic degrees of freedom in favor of the magnetization field, ending up with an equation of motion for the magnetization field alone, which usually is the only object of interest. An alternative approach is to carry along all degrees of freedom in a computer program, but it is probably harder to gain qualitative insight in this way.

3.2

Magnetostrictive coupling


· S,
the spins, and therefore the magVia spin-orbit coupling, proportional to L netization field, can see the lattice. As already mentioned in Chapter 1, it is
in favor of convenient to eliminate the orbital angular momentum variables L the magnetization components. A quick way of doing this is to use Schr¨ odinger perturbation theory based on the atomic levels of the ions forming the static,
· S.
The components of S are perfect host lattice, the perturbation being L treated as ordinary c-numbers. The energy shift thus calculated is a polynomial in the spin components, and forms part of the spin Hamiltonian. Finally, the 31

32

INTRINSIC DAMPING

quantum character of S is reinstated (to allow for non-commutativity of the spin components, the expressions are symmetrized, if necessary). If the coarse-grained
are view is appropriate, and quantum effects are negligible, the components of S
replaced by those of M , and the resulting expression is called anisotropy energy, whose strength is renormalized relative to its microscopic value. It obviously has the same symmetry as the lattice. In one respect, this process is incomplete: if a more elaborate (and more suspect) form of perturbation theory is used, the energy denominators in the perturbation series are no longer constants, but are themselves functions of the energy shift to be calculated. The anisotropy constants would thus be functions of energy. This is not acceptable, and it sug
gests (since energy and time are conjugate variables) that the elimination of L should be carried out in the framework of time-dependent perturbation theory. This will give
the anisotropy energy as a non-local function of time, such as, for example, Mi (t)Kij (t − t )Mj (t )dtdt . For the case of the static lattice, with well-separated ionic levels, the effect is negligible; K is almost Kδ(t − t ). However, the lattice can vibrate, and excitation energies are small; consequently, spin-orbit coupling to lattice distortions will result in significant non-locality of
, once the lattice distortions are elimthe resulting equation of motion for M
to the lattice vibrations is inated. To begin with, the coupling energy of M usually stated in terms of the components Mi and the strain tensor components eij . When there is substantial magnetostriction, the viscous damping of the lattice manifests itself as a loss torque on the magnetization. (Suhl, 1998). In sufficiently small samples, this is related to the relaxation of the uniform strain towards its equilibrium value. For samples that are large in an appropriate sense, there is an additional “radiation reaction” loss due to magnetic energy radiated away into sound waves. In accordance with our ultimate aim, this torque is to be found by solving the equations of motion for the strain components in terms of the Mi , and substituting the solution in the equations of motion for the Mi . Strictly speaking, the strain tensor used here is really the deviation of that tensor from its equilibrium value. That equilibrium value is found by minimizing
. This shift gives a the total strain energy with respect to the eij at fixed M small correction to the magnetic anisotropy energy and, hence, a correction to
(Slonczewski, 1963). the equilibrium value of M

3.2.1

Small samples

We begin by considering the case of magnetic samples that are small compared with both the wavelength of sound and small compared with a domain wall width. In that case the resulting loss torque is independent of position. With the sound velocity effectively infinite in this limit, the magnetoelastic energy (Landau and

)), where, using summation convention Lifshitz, 1986) is E = dv(Fe +Fme +E(M

MAGNETOSTRICTIVE COUPLING

33

to sum over repeated indices, Fe = µ e2ij + 12 λe2ii δij

(3.1)

Fme = B eij Mi Mj (1 − δij ) + B  eij Mi Mj δij Fe is the elastic energy due to strain components eij = 1/2(∂ui /∂xj + ∂uj /∂xi ), where the ui (
x) is the ith component of the small lattice displacement from position
x. The first and second terms in Fme are the couplings to the shear and compressional strains respectively. The Lam´e coefficients µ and λ are related to K, the bulk modulus, by K = λ + 23 µ. For simplicity, the elastic medium is treated as isotropic, specified by just two elastic constants, even though we allow cubic symmetry for the magnetoelastic coupling. Since the sound velocity is taken to be infinite, the eij may be considered uniform throughout the sample. B, B  are the magneto elastic coupling coefficients. The elastic stress tensor is ∂(Fe + Fme ) = 2µeij (1 − δij ) + λeij δij + BMi Mj (1 − δij ) + B  Mi2 δij ∂eij When the eij are functions of time, this stress tensor must be balanced by a viscous stress tensor 2η e˙ ij . So the equations of motion for eij become η e˙ ij + µeij + 12 BMi Mj = 0,

i = j

η e˙ ij + (µ + 12 λ)eij + 12 B  Mi Mj = 0,

i=j

(3.2)

Note that, in the time independent case, the strain components given by these
. Substituted in the total equations minimize the total elastic energy at fixed M energy expression, they evidently yield some extra anisotropy energy of the form Mi4 or, equivalently, Mi2 Mj2 with appropriate coefficients B 2 /2(µ + 12 λ) and
. B 2 /2µ. This extra anisotropy must show up in the equation of motion for M
The equation of motion for M is (summation convention)

∂E ∂Fme M˙ i = γ ijk Mj + γijk Mj (3.3) ∂Mk ∂Mk

∂E + 2Bijk Mj Ml elk (1 − δlk ) + 2B  ijk Mj Mk ekk = γ ijk Mj ∂Mk

34

INTRINSIC DAMPING

Equation (3.2) may be formally solved for eij and the result substituted in eqn (3.3). This gives ⎛

⎞

∂E − ∂Mk

⎜ ⎜ B2
∞ ⎜ ˙ Mk (t − τ )Ml (t)Ml (t − τ )(1 − δlk )e−ντ dτ − Mi = γijk Mj ⎜ − ⎜ 2η 0 ⎝ B 2
∞
Mk (t)Mk (t − τ )Mk (t − τ )e−ν τ − 0 2η

⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (3.4)

where, expressed in more familiar material constants, ν = µ/η = Y /2η(1 + σ), and ν  = (µ + λ/2)/2η = Y (1 − σ)/(2η(1 + σ)), (Y = Young’s modulus, σ =
. As Poisson’s ratio). The integrated terms depend on the prior history of M η goes to infinity, ν goes to zero, and the memory becomes extremely long. Nevertheless, a moment expansion in powers of η is possible, even though it is only asymptotic. This is done by expanding M (t − τ ) in powers of τ and integrating. Making frequent use of the facts that Ml (t)Ml (t)|l=k = Ms2 − Mk2 and Ml (t)dMl (t)/dt|l=k = −Mk dMk /dt, etc., the result for the term arising from the shear is then (summation convention) B2 γijk Mj 2η



∞

Mk (t − τ ) (Ml (t)Ml (t − τ )) (1 − δlk )e−ντ dτ

(3.5)

0

which, when expanded, gives ⎛ γ

⎜ B2 ⎜ ijk Mj ⎜ 2η ⎝



 dMk 2 2 dMk3 α0 Mk Ms2 − Mk3 − α1 Ms − dt 3  dt



2 2 2 3 dMk d Mk 2 1 d Mk Ms + M k − − ··· +α2 dt2 dt 3 dt2

⎞ ⎟ ⎟ ⎟ ⎠

(3.6)

where n+1 η αn = µ

(3.7)

The zeroth moment gives only the anisotropy torque already mentioned. The
× (dM
/dt), corrected by a first moment results in damping of Gilbert’s form M term arising from the rate of change of the strain-induced anisotropy. The second moment involves only second time derivatives or squares of first time derivatives; as behooves such terms, it gives no damping at all, but only renormalizations. In the simple case ∂E/∂Mk = H, this is demonstrated by the fact that to zero

MAGNETOSTRICTIVE COUPLING

35

order, 2


× ((M
× H)
× H)

= γ2M
× d M = γM
× dM × H M 2 dt dt
× H)(
M
· H)
= γ 2 (M

(3.8)


to become
×d2 M
/dt2 causes only a renormalization of H so that the term γα2 M
+ γ 2 α2 (M
· H)).
Only terms with odd time derivatives of M − components H(1 affect damping. However, note that, according to expression (3.4), the value of
less than µ/ν. For the asymptotic expansion is limited to rates of variation of M example, the first moment Gilbert-like damping torque goes to infinity with η, whereas the exact expression (3.4) gives a finite torque with infinite memory in that limit! Expansion of the term arising from compressional strain has a slightly different structure. Here, we simply state the result: ⎛ γ

⎜ B2 ⎜ ijk Mj ⎜ 2η ⎝



2 dMk3 − 3 dt 



2 2 3 dM d M 1 1 k k  +α2 − Mk − ··· 3 dt2 3 dt +α0



Mk3



α1

⎞ ⎟ ⎟ ⎟ ⎠

(3.9)

with α0 = η/(µ + λ/2). It makes no contribution to the Gilbert term, which appears to arise entirely from shear. Only the terms due to the changing straininduced anisotropy are left. (The reason why they have signs opposite to those in the shear torque is simply the difference in sign between the two equivalent expressions for the cubic anisotropy.) The total torque in the equation of motion is the sum of the two expressions. Finally, we confirm the (perhaps intuitively obvious) fact that the change in saturation magnetization due to the volume change induced by the compressive strains does not contribute to the result. The volume V is changed by that strain to V (1+eii ); so the saturation magnetization is changed to Ms (1−eii ). Assuming that the anisotropy constant of the undistorted lattice is K(Ms ), then that of the distorted lattice will be K(Ms ) − K  Ms eii . This adds a constant term to the left hand side of the second of eqns (3.2), and the added term in the solution of that
, and gives zero torque. As for equation does not depend on the direction of M  a similar dependence of B and B on eii , this will only contribute higher order terms to the coupling energy, terms that go beyond the usual approximation for magnetoelastic media. 3.2.2

Large, homogeneous samples

For samples of a size comparable with, or larger than, the sound wavelength and
(
x) alone shows a domain wall thickness, the contracted equation of motion for M

36

INTRINSIC DAMPING

a torque on M (
x) non-local in space, as well as in time.1 The strain field then has a position dependent part, and may have a uniform part as well. Initially, the
. two must be analysed separately, although they are coupled indirectly via M An asymptotic expansion in position as well as time then exhibits damping that mixes time and space derivatives of the magnetization. The wavelength of sound in typical materials of interest is of order of one micrometer at 1GHz, which is also of the order of typical domain wall widths. Therefore in samples larger than about one micrometer and at frequencies in the GHz range and beyond, it is necessary to consider position dependence of the strain and magnetization fields. The equation of motion for the displacement vector
u(
x, t) is ρ

∂ 2
u = −∇ · Fˆ tot , ∂t2

where ∇ · Fˆ tot stands for the vector with components ∂Fijtot /∂xj (summation ˜ ij , and convention). Fˆ tot is the total stress tensor with components Fij + eij σ with σ ˜ ij = BMi Mj (1 − δij ) + B  Mi2 δij = Mi Mj (B + (B  − B)δij ), (summation convention). After some processing, the equation for
u, including the viscous stress, becomes (see Landau and Lifshitz, 1986) ∂ 2
u ρ 2 − ∂t



∂ µ+η ∂t





1 ∇(∇ ·
u) ∇
u + 1 − 2σ 2

= −∇ · σ ˜

(3.10)

where ∇ · σ ˜ is a vector with components ∂ σ ˜ ij /∂xj . The solution of this equation with the time derivatives set equal to zero gives the equilibrium strain field
, and substituting that solution in the expression for the magnetic at given M
-dependent shift in the original exchange constant and in the energy gives an M coefficients of any other explicitly position-sensitive terms of the magnetic energy. To the solution of eqn (3.10) should be added any solution of ∇· Fˆ tot = 0, in particular the solution for the eij of the equation Fˆ tot = 0. This gives the uniform strains as a function of the spatially uniform part of σ ˜ ij , and, in the time independent case, leads to the shift in the anisotropy constant discussed in Section 3.2. In the general case, the solutions of eqn (3.10) and of M˙ = M × ∂E/∂M are coupled nonlinearly via the M —components. A consistency condition should be imposed on them, but will be ignored in the following. A direct solution of eqn (3.10) is too involved; it is simplified by writing
u as the sum of a vector 1 It


only, contracting should be stressed that if one is content to consider small motions of M
the description to an equation of M alone is not worthwhile. Then in ordinary magnetic media, the strains set up are almost always within the limits of linearized behavior, so that the coupled
and

. The resulting linearized equations for M u need to be linearized only with respect to M system yields the so-called magnetoelastic waves that have been studied in detail (see, for example, Cracknell (1974) and references therein).

MAGNETOSTRICTIVE COUPLING

37


utr whose divergence vanishes, and a vector
u whose curl vanishes, yielding, respectively, transverse and longitudinal waves. The decomposition gives

∂ 2
utr ∂ 1 − 1+ ∇2
utr = f
tr 2 2 ctr ∂t ν∂t µ

2 ∂
u ∂ 1 − 2σ
− 1+ ∇2
u = f c2 ∂t2 ν∂t µ(1 − σ)

(3.11)

  where ctr = µ/ρ, c = 2µ(1 − σ)/(ρ(1 − 2σ)) are the transverse and longitudinal wave velocities, and where, as before, ν = µ/η. Also, f
tr is the transverse part of the vector f
= ∇ · σ ˜ , and f
 its longitudinal part. Unfortunately, this decomposition of ∇ · σ ˜ results in an inherent nonlocality. The only way to avoid this would be to solve eqn (3.10) directly, but the required inversion of the differential operator on the left poses other problems. Explicit expressions of f
tr and f
 in terms of the M -components will be given below. We begin by considering the infinite medium. It is useful to remember the analogy with electromagnetism, with
utr or
u analogous to the vector potential, and f
tr or f
 the radiating current density. In electromagnetism, the resulting radiation field will carry away energy, which must be supplied by the current. The equation of motion for the current must therefore include a loss-term (the so-called radiation reaction). In our present problem, this means that, even if the infinite lattice is totally loss
in the infinite medium must include a loss free, the equation of motion for M torque. However, as we shall see, in the loss-free medium, that loss torque is irremedially non-local even when the lattice has finite viscosity; a moment expansion would be possible, were it not for non-locality introduced by the decomposition of ∇ · σ ˜ . (The latter non-locality disappears if the Lam´e coefficient λ = 0 see below.) In solving the first of equations (3.11), we shift the burden of the 1+

∂ ν∂t



operator from the space derivatives onto the time derivatives by setting
utr = w
tr e−νt with the result that 1 c2tr



∂ −ν ∂t

2 w
tr − ∇2 w
tr =

1
νt ftr e µ

(3.12)

38

INTRINSIC DAMPING

The solution takes the form  
1 t  dt d3
x gtr (
x −
x , t − t )f
tr (
x , t )eνt w
tr (
x, t) = µ −∞   1 ∞ dτ d3 ξ gtr (
ξ, τ )f
tr (
x −
ξ, t − τ )e−ντ = µ 0 Here, g(
x −
x , t − t ) is the source solution of the equation

2 ∂ 1 2 − ν gtr = δ 3 (
x −
x )δ(t − t ) ∇ gtr − 2 ctr ∂t

(3.13)

(3.14)

1
∞ Setting g˜tr = √ gtr eiωt dt and similarly for δ(t − t )δ 3 (
x −
x ), we get 2π −∞  
1 2 ∇2 + 2 (ω + iν ) g˜tr = eiωt δ 3 (
x −
x ) (3.15) ctr Here we require the retarded of the two possible solutions for gtr . (Each solution is a function of |
x| only, since no preferred direction has been stipulated.) For a simple derivation, see Jackson (1999). The only change from the derivation in that reference is that, here, the propagation constant k is (ω + iν ) /c rather than ω/c. The retarded wave has the form − exp(ik|
x|)/(4π|
x|), so that  ∞ 1 g˜tr e−iωt dω (3.16) gtr (
x −
x , t) = √ 2π −∞  exp −ν|
x −
x |/ctr ∞ =− exp iω ((|
x −
x |/ctr − (t − t )) dω 8π 2 |
x −
x | −∞ =−

exp −ν|
x −
x |/ctr δ ((|
x −
x |/ctr − (t − t )) 4π |
x −
x |


tr e−νt , This result is substituted in eqn (3.11), and, since
utr = w  t  exp −ν (|
x −
x |/ctr − (t − t )) 1
utr (
x, t) = − dt d3 x 4πµ −∞ |
x −
x |


× δ ((|
x −
x |/ctr − (t − t )) f
tr (
x , t )e−ν(t−t )

(3.17)

This is evaluated to give  exp −ν|
x −
x |/ctr
 1 d3 x
utr (
x, t) = − ftr (
x , t − |
x −
x |/ctr )t>|x−x
|/ctr 4πµ |
x −
x | (3.18) and zero if t < |
x −
x |/ctr . It is seen that, thanks to finite viscosity, an (asymptotic) moment expansion, and hence a local description of the damping torque,

MAGNETOSTRICTIVE COUPLING

39

becomes possible. When ν is zero, all moments are obviously infinite, just as in the case of uniform strain considered in the previous section. Then we have purely radiative loss, which is incurably non-local, so the moment expansion is only asymptotic. Equation (3.18) offers two options: it can be written 1
utr (
x, t) = − 4πµ





ctr t

dΩ 0

ν|
ξ| |
ξ|d|
ξ| exp − ctr

 f
tr (
x −
ξ, t − |
ξ|/ctr ) (3.19)

or, equally well, as c2
utr (
x, t) = − tr 4πµ





t

dΩ exp (−ντ ) f
tr (
x − ˆξctr τ, t − τ )

τ dτ

(3.20)

0

where ˆξ is a unit vector in the direction of
ξ, and dΩ is an element of solid angle subtended by it. The latter form is the more convenient for the purpose of a moment expansion. First, by expanding with respect to the position variable at fixed τ, it is seen immediately that only terms in even derivatives of f
with respect to
x can survive the integration over the solid angle. To be consistent with the approximation that we have used for the exchange energy, we should not go beyond the second space derivatives, at most, in this expansion.2 So c2
utr = − t µ



t

0



2τ τ dτ exp − ν



⎛ ⎝

+

τ 2 c2t 2



f
tr (
x, t − τ )

⎞

⎠ dΩ ˆ (ξ · ∇)2 f
tr (
x, t − τ ) + · · · 4π (3.21)

The integrand is further expanded in powers of τ, and, using 

dΩ ˆ 1 (ξ · ∇)2 = ∇2 , 4π 3

(3.22)

the final result is −
utr =

β0
β1 ∂ f
tr (
x, t) β2 ∂ 2 c2tr 2
+ ∇ ftr (
x, t) ftr (
x, t) − + µ µ ∂t µ ∂t2 3

3 ∂ β3 2 ∂ 2 + ctr ∇ f
tr (
x, t) + · · · − µ ∂t3 ∂t

(3.23)

2 In fact, strictly speaking, we should not even go this far since f
already involves one position derivative.

40

INTRINSIC DAMPING

where βn =

c2tr 4πn!



t

τ n+1 dτ exp(−ντ ) = 0

c2tr 4π



1 2ν

n+2 

  1 − o e−νt (νt)n+1) (3.24)

The result (3.23) is analogous to that of Abraham and Lorentz for the reaction field problem in electrodynamics (see Jackson, 1998), Chapter 16). The fact that the moments still have a residual exponentially declining transient time dependence is due to retardation of the radiation field. The solution (3.18) is still not complete: it does not encompass the result of Section 3.2 for the uniform strain. To complete the solution, we first note that uniform fields in finite samples are not the norm. Close to the boundaries of large but finite samples, the fields are generally non-uniform as the result of boundary conditions, but they may become uniform deep in the interior. So a solution is needed that will tend to uniformity far from regions of significant spatial variation of the fields. It seems that this can be found only in the limit for samples that are small compared with c/ω, where ω is a typical time rate of change of the fields. In the limit of zero ω, the equation for
u becomes essentially Poisson’s equation, and the solution is still (3.18) in the time independent case.
, because of the derivative form of f
, but this does It gives zero for uniform M not mean that the magnetic stress is zero there. Magnetic surface charges, for example, can cause uniform stress. In the time independent case, we may add to (3.18) any solution of the homogeneous equation. The latter has a solution (in component form) ui = eij xj

(3.25)

where eij are precisely the strain components found in Section 3.2. So, at large distances from the boundary, the sum (3.18) + (3.25) meets the necessary conditions. However, when the fields are allowed to be time dependent, an error arises, ∂ 2 eij because the “solution” (3.25) will introduce a term 2 2 xj into the complete ctr ∂t wave equation; that error term can be tolerated only if the sample dimensions are small compared with a wavelength. When this is not the case, a solely time dependent strain field is inconsistent with spatial uniformity, except in the limit of infinite sound velocity. When ν is zero, so that there are no lattice losses, the moments will be finite only if the magnetization vector changes over small regions separated by
. In that case it is best to use the form (3.17) of the solution. essentially uniform M
Assume that f varies rapidly only within a neighborhood ∆ of
x. Then, with ν

MAGNETOSTRICTIVE COUPLING

41

infinite, Θ(
x, ctr t)
utr = 4πµ

Θ(
x, ctr t) = 4πµ





⎛

1 |
ξ|∂ dΩ + |
ξ|d|
ξ| ⎝1 − ctr ∂t 2 ∆



 dΩ

⎛

1 |
ξ|∂ + |
ξ|d|
ξ| ⎝1 − c ∂t 2 tr ∆

1 × 1 − (
ξ · ∇) + (
ξ · ∇)2 − · · · 2



|
ξ|∂ ctr ∂t

|
ξ|∂ c tr ∂t

2

⎞ − · · · ⎠ f
tr (
x −
ξ, t)

2

⎞

(3.26)

− ···⎠

f
tr (
x, t)

where Θ(
x, ct t) = 1 if at least one component of
x is greater than ctr t and zero otherwise. All moments are now finite. If there is a whole network of such regions, as, for example a set of domain walls, then the result will be a sum over all positions “up to”
x that are within the “range” ctr t. This means that the reaction field is again non-local, but perhaps easier to handle as a sum over discrete positions. The results for the longitudinal displacements u (
x, t) can be obtained from the above by replacing the subscript tr by , and replacing µ by µl = µ(1 − σ)/(1 − 2σ). To determine f
 and f
tr in terms of the total f
, we need only note from Chapter 2 that, in terms of Fourier transforms, the dipole
has the simple form
k(
k · M
(
k))/k 2 , which is strictly field generated by M longitudinal. Similarly, here, the longitudinal field generated by f
(
k) is
k(
k · f
(
k)) f
 (
k) = k2 So, in real space, it reads

  f
(
x , t) (f
(
x , t) · (
x −
x ))(
x −
x ) 3 
−3 f (
x, t) = d x |
x −
x |3 |
x −
x |5

(3.27)

and then f
tr (
x, t) = f
(
x, t) − f
 (
x, t)

(3.28)


, we recall that To finally evaluate the damping torque in terms of M ∂Fme = ekj (BMj (1 − δkj ) + B  Mj δkj ) = ekj Mj (B + (B  − B)δkj ) , ∂Mk

42

INTRINSIC DAMPING

and that ekj = 1/2(∂uk /∂xj + ∂uj /∂xk ). These strains enter the equations of
as before, and they now involve spatial derivatives of the compomotion for M

tr due nents of f . For example, the ith component Qtr,i of the reaction torque Q  to the transverse wave becomes, with (B − B) = ∆B,

Qtr,i =

  ∂ftr,k 1 ∂ftr,l ijk Mj (
x, t)Ml (
x, t) (B + ∆Bδkl ) Otr + ∂xl ∂xk 2µ

(3.29)

where the operator β3 β1 ∂ Otr = + µ ∂t µ



∂3 ∂ + c2tr ∇2 ∂t3 ∂t

+ ···

(3.30)

and we have kept only odd time derivatives (the others do not cause dissipation). In evaluating eqn (3.29), the f
part of f
tr , in eqn (3.29), is relatively easy to handle. We have ∂fl ∂fk + = (B + ∆Bδkl ) ∂xl ∂xk



∂2 ∂2 Mk M m + Ml Mm ∂xl ∂xm ∂xk ∂xm

(3.31)

and it contributes an amount ⎛

(1)

Qtr,i

⎞ ∂2 M M + k m ⎜ ∂x ∂x ⎟ B2 l m ⎜ ⎟ ijk Mj (
x, t)Ml (
x, t)Otr ⎜ = ⎟ 2 2µ ⎝ ⎠ ∂ + Ml M m ∂xk ∂xm +

(3.32)

2B∆B + ∆B 2 ∂2 ijk Mj (
x, t)Mk (
x, t)Otr Mk Mm µ ∂xk ∂xm

to Qtr,i , a result which does not look any more elegant in vector notation. The dipolar contribution −f
 to f
tr in equation (3.29) cannot be expanded in moments because of its long-range character. However, if only the lowest term, (β1 /µ)(∂/∂t), is retained in the expansion of O, there is one special case in which
 . Since µ = µ + λ/2, evithat contribution is cancelled exactly by the torque Q dently if Poisson’s ratio is exactly zero, then µ = µ , and the longitudinal terms
= Qtr + Q . This case, cancel exactly in the calculation of the total loss torque Q though very special, is at least not inconsistent with the well-known inequality −1 < σ < 1/2. To summarize: In the exceptional case σ = 0, with only the first

LOSS TORQUE IN MAGNETIC METALS

43


× (dM
/dt) must be order time derivative retained, the Gilbert loss term γαM


replaced by γ(β1 /µ)M × (∂ N /∂t), where the components of N are B2 Nk = Ml (
x, t) 2µ +



∂2 ∂2 Mk Mm + M l Mm ∂xl ∂xm ∂xk ∂xm

(3.33)

∂2 2B∆B + ∆B 2 Mk (
x, t) Mk Mm µ ∂xk ∂xm

If σ = 0, the torque due to f
l must be evaluated, and the result cannot be
and its space derivatives. expressed in terms of purely local values of the M

3.3

Loss torque in magnetic metals

The task is to solve for electromagnetic fields and currents in terms of the magnetization vector as a driving term. The magnetic field expressed in this manner
, and the loss mechanism is then substituted into the equation of motion of M associated with the conduction electrons then surfaces as a loss torque in that equation. There are two cases to be considered (a) the loss characteristics of the metal are caused almost entirely by electron interaction with lattice vibrations
in the conductivity mechanism, and and impurities, without involvement of M (b) losses that result from direct coupling of the conduction electrons to the magnetization field. Case (a) is rather straightforward: the loss torque comes from eddy currents induced by the changing magnetization. Case (b) is more
-dependent. complicated in as much as the loss mechanism itself may become M Case (a) is not very interesting from a basic physics point of view, but it is probably more important for practical purposes. The opposite is true for case (b) which may be further subdivided into uncompromising and compromising approaches. An uncompromising treatment must take into account the fact that
are also the interthe same electron interactions that yield a loss torque on M
actions that must cooperate to produce M in the first place. (This problem is particularly challenging in the case of a totally itinerant model of a magnetic metal – see footnote 1, Chapter 1.) A fully satisfactory understanding does not seem to be available at this time. A compromising approach assumes that some future theory will justify that it is permissible to treat these two functions of the electron interaction totally separately, so that the conduction electrons scatter from the magnetization field viewed as given (Berger, 1978). For rare earth metals, such as gadolinium with its well-localized magnetic f-shell, this approach is plausible; for the transition metal series and their d-shell, it is more questionable. In the next section we consider case (a), and in Section 3.3.2 we present case (b), but only in the compromising mode.

44

3.3.1

INTRINSIC DAMPING

Eddy current damping

In many practical situations, displacement current can be neglected compared with conduction current, and we begin by assuming that this is also the case
alone, in our program, which aims at establishing an equation of motion for M preferably a history-independent equation. The treatment will be restricted to the case of metals with normal skin effect (electron mean free path much smaller
, in component form, may be than the skin depth). The equation of motion of M written M˙ i = ijk Mj (Hk + hk + A∇2 Mk )

(3.34)


the steady applied magnetic field,
h the magnetic field satisfying where H Maxwell’s equations. Crystalline anisotropy fields have been neglected, since they do not seriously affect the results derived here. We ultimately wish to eliminate
h and the electric field
e between equations (3.34) and Maxwell’s equations σ
e ∇ ×
h = 4π c 


) ∂(
h + 4π M ∇ ×
e = − c∂t

(3.35)


=0 ∇ ·
h + 4π∇ · M where σ is the conductivity, and c the speed of light. Eliminating
e between the first two equations gives 


h + 4π M
∂ 4πσ =0 (3.36) ∇ × ∇ ×
h + 2 c ∂t But ∇ × ∇ ×
h = ∇(∇ ·
h) − ∇2
h, so, from the third of eqns (3.32), we get ∇2
h =


) 4πσ ∂(
h + 4π M
) + 4π∇(∇ · M 2 c ∂t

(3.37)


, we need a Green function G(
x −
x , To solve this equation for
h in terms of M  t − t ) that satisfies the equation ∇2 G −

4πσ ∂G = 0, c2 ∂t

(3.38)

but reduces to δ(
x −
x ) at time t = t . In infinite space, G is the source-solution: G=

σ (t − t )c2

3/2

(
x −
x )2 πσ exp − 2 c (t − t )

(3.39)

LOSS TORQUE IN MAGNETIC METALS

45


, both in position Use of this Green function makes
h a non-local function of M and time. The solution of equation (3.34) takes the form ⎞ ⎛
(
x , t )  t  16π 2 σ ∂ M
h(
x, t) = d3
x ⎠ dt G (|
x −
x |, (t − t )) ⎝ c2 ∂t −∞
(
x , t )) +4π∇(∇ · M ⎞ ⎛
(
x −
ρ, t − τ )   ∞ 16π 2 σ ∂ M ⎠ = − d3
ρ (3.40) dτ G (ρ, τ ) ⎝ c2 ∂τ 0
+4π∇(∇ · M (
x −
ρ, t − τ ))
×
h. A quasi-local and the loss torque, if any, should be at least part of M approximation to this solution depends on the existence of all moments of the Green function. G, as given by equation (3.36), obviously does not satisfy that criterion. The reason is that displacement current has been neglected in the derivation. When this is included, the retarded, damped Green function, taking the place of (3.36), does allow a moment expansion. This is very similar to the case of losses to sound waves discussed in Section 3.2.2. In this section, we shall use only the lowest moments, for which (3.36) is adequate for specimens smaller than a skin depth, leaving a brief discussion of the general case to Appendix 3A. As discussed previously, higher moments become important only in situations with sufficiently rapidly varying fields. First, consider some special cases:
independent of position. Then in the first term on the right of the first 1. M of equations (3.37), the volume integral only affects G, and gives unity for an
at time t, which infinite material, and the time integral then gives the value of M contributes nothing to the torque. This may seem strange, in as much as the time
in (3.37) seems to be the leading candidate for Gilbert damping. derivative of M
does not depend on position, it cannot produce The reason is quite simple: if M a current, and, without it, there can be no extra dissipation. This is seen easily by eliminating
h in favor of
e, resulting in a diffusion equation for
e that is driven
)/dt. For a finite sample, however, by an extra current proportional to d(∇ × M the integral of G over position is finite and time dependent, so the Gilbert-like damping torque remains finite, but history dependent. (From the point of view
in the interior must of the diffusion equation for
e rather than for
h, a uniform M still have a surface curl across the boundary.) Returning to equation (3.40), if
is independent of position, the second term on the right of (3.40) would seem M to be zero. However, this conclusion ignores boundary conditions that involve
. The correct procedure is to put the burden of the a surface divergence of M differentiations on G, using integration by parts. The result becomes familiar when . . .
is independent of time. Then only the second term on the right of (3.40) 2. M survives. The time integration involves only G, and, integrating equation (3.39) over time, it gives essentially G = 1/|
r −
r  |. Then, two integrations by parts make h the gradient of a volume integral of GdivM +a surface integral of G times

46

INTRINSIC DAMPING

the normal component of M over the bounding surface. This is just the well
(see, for example, known expression for the magnetostatic dipolar field due to M
Jackson (1999), in the section on magnetostatics). When M is not independent of time, then the second term describes what happens to the dipolar field under the diffusion-like propagation in this model. We consider this term no further in this chapter, even though it can contribute to an additional damping term
, if the appropriate moment expansion involving time and space derivatives of M exists.
, the non-local Hopefully, in the general case of a both
x and t dependent M expression can be expanded in an at least asymptotically convergent series of purely local terms. This procedure is quite successful in the case of the inter
with elastic lattice displacements via magnetostriction. Using the action of M expression (3.39) for an infinite magnetic medium, it is seen to fail completely in the present case. Write t = t − τ and
r  =
r −
ρ. Then, for example,



(
x , t )
(
x, t) ∂M ∂ ∂M (
x, t) ∂M − (
ρ · ∇) + τ = ∂t ∂t ∂t ∂t

2
∂ M (
x, t) ∂ 1 + ··· + (
ρ · ∇) + τ 2! ∂t ∂t

(3.41)

So, to evaluate (3.34), we need successive moments involving products of components of the vector
ρ Im,n1 ,n2 ,... =

 

 dρi ρni i

∞

dτ τ m G(ρ, τ )

(3.42)

0

i


As already noted above, the case m = 0 gives dτ G proportional to 1/ρ; therefore the case of all ni = m = 0 requires the evaluation of  I00 = 4π 0

∞

ρ2 dρ , ρ

(3.43)

which diverges. Higher moments diverge even more strongly. So, in the infinite medium, a purely diffusive Green function cannot give a quasi-local equation of
in the sense of the above expansion. On the other hand, for a finite motion of M
sample, at least I00 exists and is equal to d3 ρ/ρ, the integral extending only
is zero outside. Thus there is a leading loss over the sample volume, since M term of Gilbert form with α = 16π 2 σI00 /c2 .

(3.44)

Presumably higher moments also exist,  but they will not be examined here. The skin depth in the material is δ (ω) = c2 /(4πσω) for an a.c. field of frequency ω.

LOSS TORQUE IN MAGNETIC METALS

47

Hence, in the case of a sphere of radius R, the formula for α would be 4π R2

α=

(3.45)

2

δ (ω)

This is clearly not acceptable: it claims that the rate of loss is not an intrinsic quantity, i.e. it is size dependent. In fact, this may be acceptable only if the sample dimensions are smaller than the skin depth, ω being interpreted as the
. For large samples, displacement currents must dominant rate of variation of M be included, and α then attains a size-independent value. The complete Green function, including electromagnetic propagation is derived in Appendix 3A. Its retarded form is


iπe−2πσ(t−t ) G(
x −
x , t − t ) = − (2π)3 R 





∞

0

√

2

2

2 2




e−{ikR− 4π σ −c k (t−t )} √ kdk (3.46) 4π 2 σ 2 − c2 k 2

Evalution of this integral is complicated. For σ = 0, the integral gives just the usual δ-function on the light cone. For moderate values of σ, the δ-function is not significantly broadened, and then the main effect resides in the prefactor e−2πσt . This factor, together with the δ-function, assures that all time and space moments exist, and the result is very similar to the one discussed for lightly damped elastic waves in Section 3.2.2. (The spatial moments on the light cone turn into time moments, and these are finite because of the e−2πσt -factor.) However, in the case of a metal, the conduction current is not normally smaller than the displacement current, so this simple result is not quite appropriate. The effect of the conductivity is to considerably blur the surface of the light cone, but the spatial moments will converge nevertheless. For the purposes of evaluating moments, formula (3.46) can be scaled conveniently. Writing k=

3 ˆ G(2πσ) 2πσκ τ ξc , R= , t − t = , G= c 2πσ 2πσ c

(3.47)

it takes the universal form −τ ˆ τ) = e G(ξ, 2π 2 ξ



∞ √ 1−κ2 τ −iκξ

e

κdκ

(3.48)

0

independent of σ and c. This scaling makes it possible to state any one of the moments, such as the pth spatial and the q th temporal moment in the expansion of the internal field in terms of σ and c, multiplied by a numerical constant depending on p and q only: tq  xp11 xp22 xp33  = 4π 2 (2πσ)3−(q+p ) cp−1 ap1 p2 p3 gpq

(3.49)

48

INTRINSIC DAMPING


∞
∞ ˆ τ ), p = p1 + p2 + p3 , and ap p p is an angular where gpq = 0 τ q dτ 0 ξ p dξ G(ξ, 1 2 3 integral depending on the choice of the integers pi . 3.3.1.1 Boundary conditions Up to this point, boundary conditions (tangential H and normal B continuous across boundaries) have been ignored. Two cases are to be distinguished: externally applied fields large enough to produce saturation and subsaturating applied fields. As an extreme example of the latter
as a given. The boundary case, consider zero applied field, but treat the field M conditions then require that magnetic fields are set up outside the sample. These decline as various inverse powers of the distance from the sample, but they will nevertheless contain energy proportional to some bilinear form in the compo
. This is obviously not the lowest energy state of the system. In the nents of M absence of any crystalline anisotropy, the lowest energy state is reached if the
-field is free to rearrange itself in such a way as to give no normal component M at all at the sample surface, and hence no external field. If the sample is large enough, this should still be true, even if exchange is taken into account. If the sample has crystalline anisotropy, removing the external field will not result in complete demagnetization, if the total magnetic energy outside the sample is no greater than the energy lowering due to line-up (or at least partial line-up) of the magnetization in an easy direction. In a field large enough to saturate the sample, it is necessary to add to the expression for hint a solution of the homogeneous
, and therefore the magnitude Maxwell equations, because the magnitude of M of hint , is limited and cannot quite match the normal component of the external
-lines are held field. Of course, for applied fields well above saturation, the M quite rigid, and the linearized theory should then be adequate. A very complicated case can arise if the sample has sharp corners, unless the applied field is exactly zero. When it is finite, it develops nominally infinite values at such corners, and some field penetration into the sample will create localized saturated regions. These considerations are quasi-magnetostatic because they effectively have
) by using only its zeroeth time moment, neglected the time variation of ∇(∇ · M permissible only if the applied field varies sufficiently slowly. The problem becomes much more difficult for varying fields, and some highly simplified cases are discussed in Chapter 6. If the time dependent part of the applied field is very small, essentially exact results can be obtained by abandoning the attempt
, and simply to eliminate the fields in favor of the equations of motion of M linearizing the entire system. One feature that sets eddy current damping of the magnetization apart from other mechanisms is that the effective field entering the damping torque is a
(essentially because Maxwell’s equations linear function (or functional) of M are linear in all fields). This is not generally the case, as we have already seen in Section 3.2, and, as will be seen in the rest of this chapter. As the result, even where a moment expansion is acceptable, the damping torque, in general, is a
-component. Gilbert tensor function of more than one M

LOSS TORQUE IN MAGNETIC METALS

3.3.2

49

Direct coupling of conduction electrons to the magnetization field

A truly convincing account of magnetic relaxation in d-band metal would require
, not a thorough understanding of how mobile electrons can generate a viable M withstanding their itinerant aspects. Here we simply assume that the magnetization is given, and how its motion changes the equilibrium distribution of the electrons. The work necessary to produce this change appears as a loss
. This assumes that the electrons relax term in the equation of motion of M
. (A more ambitious treatextremely rapidly on the time scale of motion of M ment would take account of the rate of electronic relaxation, going beyond such a quasi-equilibrium treatment.) We here consider only the most primitive itinerant electron model that still reflects the essence of this type of relaxation: two interpenetrating Fermi liquids with a common chemical potential µ are composed of different numbers of quasi-free electrons with opposite spin orientations. Their exchange interactions are modeled by an average uniform exchange field, AM = A(n↑ − n↓ ), proportional to the net magnetization, that is to say proportional to the difference in the concentrations n↑ and n↓ of “up-spin” and “down-spin” electrons. The two liquids have a common Fermi level. Choose the zero of energy in such a way that  − AM and  + AM are the energies of upand down-spin electrons, where  is the kinetic energy of an electron. At a given electron concentration N , the saturation magnetization Ms and the chemical potential µ are found by solving the two equations  µ  µ ρ()f ( + AMs )d + ρ()f ( − AMs )d (3.50) n↑ + n ↓ = N = 0

0

Ms ∝ gµB (n↑ − n↓ )  µ  = gµB ρ()f ( + AMs )d − 0



µ

ρ()f ( − AMs )d

0


. Here, ρ() is the density of electron states for µ and for the magnitude of M per unit volume with kinetic energy , and f () the Fermi-Dirac distribution. At a given chemical potential, these equations determine N and M . This is the Stoner model of ferromagnetism of itinerant electrons. If spin orbit coupling is neglected, these equations also hold if the periodic lattice potential is taken into account, at least in semiclassical approximation. The anisotropy of the lattice structure is taken into account by writing ρ as a surface integral:  dS ρ() = . S |∇k |k = Then  = k depends on both magnitude and direction of
k, and the shapes of the Fermi surfaces k ± AMs = µ of both majority and minority spins are no longer spherical. In equilibrium, and in the absence of an external field, they have crystal symmetry. However, in the presence of spin orbit coupling, the electron energies

50

INTRINSIC DAMPING


. Then the shapes of the two Fermi surfaces become functions of both
k and M
, an effect sometimes called likewise become functions of the components of M “breathing of the Fermi surfaces”. This implies that a change in the direction
must be accompanied by variation in the scattering of electrons, which of M
. Attention to this mechanism was appears as a damping of the motion of M drawn by Kambersk´ y (1970) in his study of Landau-Lifshitz damping in magnetic metals. His procedure is extended here to allow for inherent time-delay aspects of the process, if any. The general expression for spin-orbit (s.o.) energy of strength λ is λ
s · (∇V (
x) × p
)

(3.51)

where V is the periodic lattice potential and p
the electron momentum. The effect on the single electron energies is found by perturbation theory3 . In a manner explained in Chapter 1, blocks of
s operators are carried along as c-numbers and
. The result can be guessed from finally made to correspond to components of M symmetry considerations. It must be ˜k = k + s.o. k

(3.52)

where, for a lattice of simple cubic symmetry, s.o. = Λ(k)(M12 k12 + M22 k22 + M32 k32 ) k

(3.53)

plus a constant of no importance. Similarly, in the case of uniaxial symmetry, the result must be = Λ(k)k32 M32 s.o. k

(3.54)

where Λ(
k) is a slowly varying function of
k. In real ferromagnetic metals, particularly in the 3d series, this calculation is quite complicated, mainly because the three energy bands spawned by the three-fold degeneracy of the levels at
k = 0 must be taken into account. For professional treatments of this problem, the reader is referred to the old literature discussing spin-orbit-coupling induced magnetic anisotropy in these metals (for example, Kanamori, 1963). Here, we only write down the generalization of the Stoner model needed to
, the electron include s.o. coupling. If m
is a unit vector in the direction of M s.o. energy ˜k , eqn (3.49) has the form ˜k (m,
Ms ) = k + k (m,
Ms ) because of the 3 Evidently, “raw” perturbation theory involving one spin at a time doesn’t work for conduction electrons, since all even powers of the spin components for spin 1/2 are constants. The block formation method of renormalization theory is required, and at least two spins per block are needed.

LOSS TORQUE IN MAGNETIC METALS

51

s.o. contribution. Then equations (3.50) take the form  n↑ + n↓ = N =

µ





ρ ()f ( + AMs )d + 0



Ms ∝ gµB (n↑ − n↓ ) = gµB  −

µ

ρ ()f ( − AMs )d

µ

ρ ()f ( + AMs )d−

0 µ

(3.55)

0

ρ ()f ( − AMs )d



0

where now ρ () =

 S

dS |∇k |=k +so (m,M  s)

(3.56)

k

and the two fermi surfaces in k− space are now given by k + so
Ms ) ± AMs = µ. k (m,

(3.57)


/Ms . Their dependence on Ms is They depend on the direction m
= M presumably not changed much from the one in the absence of s.o. coupling.
set up at extremely low temperaWe now consider the damping torque on M tures via a mechanism dominated by elastic electron scattering at non-magnetic impurities acting in combination with the orientation dependence of the Fermi surfaces. In this simple case, the damping torque set up by the time rate of
can be related to the current set up by an external electric field. change of M This is not surprising since, in both cases, we can use the Boltzmann equation
(or obeyed by the electron distribution function for small rates of change of M for a small electric field). 3.3.2.1 Scattering by fixed, nonmagnetic impurities When there is only impurity scattering of the electrons, the popular practice of replacing the collision integral in the Boltzmann equation by a simple relaxation term is actually exact (Wilson, 1953). Then, in the absence of any spatial variation, the Boltzmann equation reads f − feq Df =− Dt τ

(3.58)

where feq is the equilibrium Fermi distribution. Df /Dt, the total time rate of change consists of an explicit rate of change ∂f /∂t, plus a rate of change
, namely ∂f /∂ M
· dM
/dt. (In the case due to the time dependence of M
of an applied E-field, that extra rate of change is due to the “drift” term,
· ∂f /∂

k.) For dM
/dt)/Ms 0, and
2π −E(θ)
2π −E(θ)/2 e u0 (θ)dθ = 0 e dθ, the normalized W is 0 −E(θ)/2

W (θ, t) = e

∞

un (0)un (θ)e−λn t
2π u0 (0) 0 e−E(θ) dθ n=0

(5.29)


2π So, as t → ∞, this approaches exp (−E(θ)) / 0 e−E(θ) dθ, as it should. Consider now the switching process, with magnetization vector at θ0 = 0 at t = 0, and a field −|H| applied. We explicitly show this field as one of the arguments of the various functions. Then ∞ un (0, −|H|)un (θ, −|H|)e−λn t W (θ, t; −|H|) = e−E(θ,−|H|)/2 n=0 (5.30)
2π u0 (0, −|H|) 0 e−E(θ,−|H|) dθ After a sufficiently long observation time, only the terms with n = 0 and n = 1 will be important. In order to check if the time has been ‘sufficiently long’, it is necessary to compare the observed mean value < M3 (t) > obs , with the value < M3 >calc calculated from the Boltzmann distribution in field −|H|. Evidently in the long-time limit ln (< M3 (t) >calc − < M3 >obs ) 


2π u1 (0, −|H|) 0 dθe−E(θ,−|H|)/2 cos θ u1 (θ, −|H|) (5.31) = −λ1 t + ln −
2π u0 (0, −|H|) 0 e−E(θ,−|H|) dθ So, if measurements of the right hand side as a function of time give a very nearly straight line with negative slope, the observation time has been ‘long enough’. Also, the intercept of the straight line on the time axis gives a small amount of information on the first non-trivial eigenfunction. An uncomfortable feature of this result is that it defines switching time only in terms of observation. Even though the obviously crucial λ1 can be calculated, its reciprocal is not the switching time per se. It is necessary for the experimenters to decide at what value of the deviation from final equilibrium they consider the reversal to be complete. The need for such a subjective approach is simply due to the fact that exponential decay towards equilibrium is never strictly complete in any finite time. In search of a more objective criterion for completion of the reversal, some investigators choose mean first passage time, briefly discussed below.

112

MAGNETIZATION REVERSAL

This basically assumes the existence of an absorbing barrier, equating our loss of interest in a particle once it passes the barrier with an actual disappearance of the particle. An attempt to correct this unphysical viewpoint has been made by Safonov and Bertram (2006). There, the ‘return trip’ of the particle is allowed for by adding to the mean forward first passage time, a kind of mean reverse passage time. Possibly the sum of the two might provide a reasonably good objective criterion. Finally, it should be noted that the experimental determination of λ1 just described is not particularly tied to the switching process. Any deviation from final equilibrium, regardless of the sign of H would be equally acceptable, since, as shown above, the eigenvalues are independent of the sign of H. In other words, the method only describes the final approach to equilibrium. This may be of basic interest, but is probably not very relevant in practice, as discussed further below. Equation (5.27) is a form of Hill’s equation (see Whittacker and Watson, 1927). The formal procedure for solving it is well known, but applying it requires evaluation of an infinite determinant, with λ replaced by zero. If that determinant can be evaluated reliably, the eigenvalues λn can be found from a very simple transcendental equation, and the corresponding eigenfunctions are trigonometric series. The coefficients in the series are complicated functions of the parameters. This procedure works well in the case of Mathieu’s equation, where it ultimately reduces to a solution of a three-term linear recursion relation. In the present problem, the procedure would require solving a five-term recursion relation, whose solution requires an analytic solution of a quintic, known to be unobtainable. In a few restricted ranges of H and HK various methods requiring only simple quadratures can be employed, but involve an inordinate facility with certain advanced arithmetic techniques. Therefore we examine the diffusion equation numerically, which is not difficult as far as the eigenvalues are concerned. As for the wavefunctions un , the most inconvenient aspect of the problem is that the equation must be solved as a two-point boundary value problem. That is to say, it must be solved subject to the conditions un (π)=0, or, equivalently un (2π) = un (0), rather than subject to the usual specification of un (0) and un (0). The only general method of solution is called the ‘shooting method’. Typically, it is applied by specifying the initial slope for a fixed initial value, and then running the computation to the end point to see if it comes close to the required final value. If it does not, the initial slope is changed, and the computation is re-run to see if the final value comes closer to the one required. Shooting continues in this manner until the final value comes close enough to the one intended. The uniqueness of this way of solving two-point boundary problems is not guaranteed under all circumstances, but our particular problem seems to raise no such questions. As for the eigenvalues, we present here a method based on Mathematica software that requires only minimal familiarity with its program structure. It very rapidly finds the entire eigenvalue spectrum for any given pair of values (H, HK ) to within better than 10%. Finding the eigenfunctions takes a bit longer; fortunately they are much less needed. To find the eigenvalues,

REVERSAL IN 2D

113

proceed as follows: 1. For a given pair (H, HK ), make a table of closely spaced values of λ, ranging from λ = 0, to λ = 5, say. 2. Setting u(0) = 1 and u (0) = 0, run the numerical integration from 0 to 2π for each λ , starting with λ = 0, and listplot u (π) versus these λ values. From the earlier analysis, we know that u (π) should be zero for each eigenvalue relevant to our problem. This means that the correct eigenvalues are deep minima in the curves of u (π) versus λ, and, hopefully, ordinates at these minima are very nearly zero. If they are not, then more closely spaced values of λ are necessary. Eventually, a spacing will be found for which u (π) is as close to zero as desired. 3. Tabulate the values of λ at the minima by having the program pick out the values at which the signs of the slope of the curves reverses. These form the set of eigenvalues that lie between 0 and 5 for the given H and HK . 4. Repeat this procedure for various H and HK . Figure 5.4 shows the result for the first seven eigenvalues, for various H at given HK . Figure 5.5 shows the computed curves of switching field versus 1/λ1 for various values of HK . (As explained above, 1/λ1 is the ‘subjective’ switching time obtained by experiments measuring the deviation from switched equilibrium.) Figure 5.6 shows the contours of given 1/λ1 in the |H|, HK plane. As for the corresponding eigenfunctions, we have to contend with the well known syndrome that good approximate eigenvalues generally give poor eigenstates. Only by extreme luck will a solution of the eigenvalue problem using one of the eigenvalues just found – though very nearly correct – give an acceptable u versus θ curve. In particulars, the value u(2π) will usually be nowhere near that of u(0). In his book, Dubin (2003) describes a way of causing Mathematica to

80

λ

HK = 3

60

H=9

40 H=3 20

1

2

3

4 Eigenvalue number

5

6

Fig. 5.4. The first five or six eigenvalues λ at given HK and a few values of H.

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MAGNETIZATION REVERSAL

25 20 H 15 HK = 6

10 5 HK = 0 0.5

1

1.5

2

2.5

1/λ1

Fig. 5.5. ‘Coercive field’ (switching field) versus switching time (taken to be 1/λ1 .

interpolate between two close shooting results in order to achieve rapid improvement. In the problem at hand, almost as good results can be obtained as follows: Concentrating on one particular eigenvalue found by the previous procedure, blow up the vicinity of the corresponding minimum in the u (π) versus λ curve by preparing a table with close spacing of values between λmin ± . Then run the previously used program. The resulting curve should give a new and much better value of λmin . The improvement should show up in the u vs. θ curve. Figure 5.7 shows the ground state wavefunction exp(−E/2) and the next lowest relevant eigenfunction for a particular H, HK pair. The latter two was found by means of two rounds of blow-up of the lambda scale around λ0 and λ1 respectively.

5.3.2

Intermediate time scales

As noted above, in magnetic recording practice one needs to know how much |H| is needed to more or less ensure eventual attainment of reversal for a given duration of the reversal pulse. So it is not sufficient to consider only the immediate neighborhood of θ = π. This means that the eigenvalues λ are needed, not just for lattice vectors n = K, but also at general k-values in the successive bands, and the eigenfunctions in the various bands now have the form eikθ un (k, θ). Consider first the case of free diffusion. Suppose that one desires information about the distribution function at a particular time t, not necessarily long, and possibly quite short. Suppose that time of interest is t < (2N π)2 . Referring to the solution of form (5.17), with θ confined to the range (0, 2π), terms beyond

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115

12 0.15

0.1 11 10

0.12

0.2

9 8 H

0.3

7

0.4

6 5

0.5

4

0.6

1.2

3 0.7 2 1

1.

1.4 1.6 1.8 2. 3.

0.8

0 0

1

2

3

4

3.5 4. 5. 5

6

HK

Fig. 5.6. Shows contours of given 1/λ in the |H|, HK -plane.

the N th are small, so it is sufficient to use the series (5.17), terminated at the expansion (5.28), and N th term. Alternatively, one could use the eigenfunction √ truncate the sum at (eigenvalue) m of order 1/ t. In the case of free diffusion, we are at liberty to use either alternative. For small t, it is advantageous to use termination of the sum (5.17); for large t it is better to use the sum (5.28), greatly truncated. In either case, these truncations violate the single-valuedness requirement but, within 0 < θ < 2π, almost all of the true, single-valued distribution to Green’s function comes from these partial sums. We now show that in the presence of the potential V, we have a hybrid situation, in which the same sum describes both large and small t. To obtain the Green function necessary to describe this situation, we briefly revert to the case of a θ allowed to vary over the range (−∞, +∞), i.e. a Green function appropriate to a particle in a periodic potential. This involves not only summation over n, but also integration over the continuum of k values within the bands. Finally, ensuring periodicity of the solution that we really need will be postponed until after we have done the

116

MAGNETIZATION REVERSAL 0.8 0.6 0.4 0.2 π

π/2

3π/2

2π

Fig. 5.7. The (normalized) eigenfunctions u0 (ground state, eigenvalue 0, dashedline), and u1 (first excited state, eigenvalue λ1 , solid line), for H = −3 and HK = 2.5.

integration over k in some reasonable approximation. Near the bottom of the nth band, λ(n, k) has the form λ(n, k) = λn + µn k 2 + o(k 4 )

(5.32)

where k is measured from the bottom of the band. (In the quantum case, µn has the form
2 /2m∗n , with m∗n known as the effective band mass of the nth band.) This form does not hold throughout the entire band, but we adopt this approximation for the entire band, with k ranging from zero to the zone boundary k = 1/2. We have 

1 2

Re

u∗n (0, k)un (θ, k)e−µn k

2

t−ikθ

dk

(5.33)

0

1 ≈ 2

'

π un (0)un (θ) Re erf µn t

√

iθ µn t +√ 2 4µn t

exp −

θ2 4µn t

2
z where erf(z) is the error function √ 0 exp(−y 2 )dy. (This result follows from π Re erf(iθ/4) = 0), and from the assumption that, compared with the exponential factors, un (θ, k) varies only slowly with k, so that it can been replaced by un (θ), its value at the bottom of the band.) Also, un (0) has been chosen real. The expression (5.33) is the response to a single delta function located at θ = zero.

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117

Hence, the needed Green function in response to the periodic δ-function is now √

' π µn t (θ − 2πm)i u(θ, 0; t) = un (0)un (θ) Re erf + √ 4µn t 2 4µn t m n

−(θ − 2πm)2 × exp − λn t , (5.34) 4µn t At a given time t, and angle θ in the range (0, 2π), the first term in the exponent completely dominates the second if  (5.35) t  2|θ − 2πm| µn λn , and the second dominates the first if the direction of this ‘strong’ inequality is reversed. This means that, for given t, θ, the ‘curve’ in the ‘n, m-plane’,  (5.36) t = 2|θ − 2πm| µn λn = tcrit (θ, n, m) divides that plane into two regions, within one of which, (t < tcrit ), the straight exponential decay term has little effect, while in the other region, (t > tcrit ), the ‘singular’ term has little effect. In fact, along the critical curve (5.36), the contribution to (5.34) is maximal. To make use of these conclusions in practice, one may proceed as follows: Given that the available switching pulse length t is limited, one must decide on the angle θ beyond which near completion of the switch can reasonably be expected after the end of the pulse (obviously, in the present model that angle should well exceed the location of the barrier, so that π can be reached by pure diffusion in preference to a return to θ = 0). The pulse length t, and that angle, then determine the region in the n, m plane over which the series should be summed. As a crude approximation, one could keep only the two largest terms, solving (5.36) for two possible m-values

t 1 m= θ± √ 2π 2 µn λn at given n, θ, and t, leaving only a single sum over n. An approximate expression for µn may be found by means of so-called p·k perturbation theory. This gives an expression for µn in terms of quantities depending only on the zone boundaries already discussed. For a particularly clear account, the reader may wish to consult Harrison (1970). If one wishes to go beyond such approximations, one must remember that the criterion un (π) = 0, or equally well un (2π) = un (0), which was previously used, must now be replaced by the criterion u(2π) = cos(2πk)u(0), corresponding to λn + µn k 2 . The latter eigenvalue can be found by the same method as before. After an initial check for at least three cases (k = 0 and two different finite small k-values to ensure that the k-dependence

118

MAGNETIZATION REVERSAL

is indeed quadratic), it is necessary to use only k = 0 and a single k-value to determine both λn and µn . 5.3.3

Applied field and anisotropy axis misaligned

In practice, the anisotropy axes orientations of particles in recording media will not coincide with the direction π of the switching field. Figure 5.8 shows the final equilibrium angle in the reversed field as a function of the misalignment angle, for various values of H/HK . When that ratio is less than approximately 2, the final angle seeks some accommodation with the misalignment angle. For H/HK > 2, instability arises, and the angle of repose snaps to π. We restrict further analysis to this case only. A convenient form for the effective potential is now V (θ) =

1 (H cos θ + HK cos(2(θ − θm )) 2

H2 1 H 2 sin2 θ + K sin2 2(θ − θm ) + HHK sin θ sin 2(θ − θm ) , − 4 4 (5.37)

π 3π 4 θm

H/HK=0 H/HK=2

π 2 π 4

π 4

π 2 θeq

3π 4

π

Fig. 5.8. Anisotropy axis and (negative) applied field misaligned. Ordinate is
. Curves are the misalignment angle, abscissa is the equilibrium angle of M for various values of |H|/HK , from 0 to 2, in steps of .125. Instability develops when |H|/HK exceeds about .75, and θeq goes to π.

REVERSAL IN 2D

119

0.8 0.7

1/λ1

0.6 0.5

1/λ

0.4 0.3 0.2

1/λ5

0.1 0.5

1

1.5

2

θm Fig. 5.9. The longest five relaxation times as a function of misalignment angle. Not surprisingly, the relaxation times are the shorter, the closest θm gets to π.

which reduces to the form (5.11) when the misalignment angle θm is equated to zero. Before application of the reversal pulse, the magnetization points in the direction θm , rather than zero. Therefore, in the eigenfunction/eigenvalue analysis, we should also admit both of the degenerate periodic eigenfunctions that vanish at θ = π. However, in most situations, θm is quite small, so the weight of these eigenfunctions in the Green function expansion is also small, and has been neglected in the construction of Figure 5.9, which shows the first few lambdas for various misalignment angles for HK = 1, H = 3, when the zero field misalignment is small. 5.3.4

Relation to first-passage type theories

Probably the most simple-minded approach to the question of reversal rate when there is a potential barrier, that is if H < HK , is to imagine a perfectly absorbing sink placed at the barrier θB . Even though the systematic current wdE/dθ is zero there, the diffusion current −Ddw/dθ will be finite. Of course, if there is a sink, there has to be a source of current, here taken to be located at θ = 0. It is then plausible to seek a steady state solution as follows: A current of unit amplitude is slowly turned on, say −∞ to the present. This results
t at θ = 0, from time
∞ in a density W (θ) = −∞ w(θ, t − t )dt = 0 w(θ, t)dt at θ, and produces a
∞ diffusion current J(θ) = −D 0 dt (∂w(θ, t)/∂θ)θ=θB at θB . The total current is, of course, independent of θ. A reasonable definition of switching rate is then R→ = J(θB )/W (0)

(5.38)

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MAGNETIZATION REVERSAL

However, in the solution of the diffusion equation for w, to allow for the sink at π, we must keep only those eigenfunctions u ˜n (θ) that are finite at θ = 0 and zero at θB , corresponding to perfect absorption there. (Recall that in the case of free diffusion, these eigenfunctions were cos[(n + 1/2)πθ/θB ].) We now have  ˜ un (θ)e−λnt (5.39) u ˜n (0)˜ w(θ) = e−[E(θ)+E(0)]/2 where tilded symbols indicate that only the absorptive eigenfunctions and eigenvalues are used. (Note that there is no zero eigenvalue in this case!). Then W (θ) = e−[E(θ)+E(0)]/2 −E(0)/2

J(θ) = e

u ˜n (0)˜ un (θ) ˜n λ

,

(5.40)

 u ˜n (θ) /dθ ˜n (0)d e−E(θ)/2 u ˜n λ

and since dE (θ) /dθ is zero at the barrier,  1 ˜n (0) (d˜ un (θB )/dθB ) ˜n u λ  1  R→ = u ˜n (0) u ˜n (0) ˜ λ

(5.41)

n

All this assumes that there is an infinite supply at θ = 0, and that there is no backflow. Thus this result holds only at the very beginning of the switch, before w(0) is seriously depleted. So this formula is not a good way of tracing the switching of a single particle over at least a large part of its history. However, for magnetic recording purposes, it could make some sense. Then there is in fact a constant supply of unswitched particles as the medium passes under the recording head. The question of backflow can be addressed to some extent by using essentially the same calculation, but replacing the starting state with the switched state θ = π, and then solving the two master equations dw(0) = −R→ w(0) + R← w(π) dt dw(π) = −R← w(π) + R→ w(0) dt

(5.42)

subject to the conservation condition w(0) + w(π) = c, a constant, which should hold on some intermediate time scale (the true conservation condition
is w(θ)dθ = constant). Taking into account the approximate conservation conditions, these equations are easily solved for w(π), starting at zero:   R→ 1 − e−(R→ +R← )t (5.43) w(π) = R→ + R← These equations do not mention any constant current supply, which appears only implicitly in the calculation of the R’s, and so they are expected to hold only if the R’s are small so that the w’s change slowly.

REVERSAL IN 2D

121

Going beyond these steady state or quasi-steady state considerations requires a more detailed treatment. Here, we briefly summarize the generally accepted procedure. The probability that, starting at θ , one will arrive at θ in time t is given by Green’s function 1




˜(θ, θ ; t), P (θ|θ ; t) = e− 2 (E(θ)+E(θ )) u

(5.44)

and the chance of ‘still’ finding the system at θ after a time t is 

θB

dθ P (θ|θ ; t).

(5.45)

0

This must also be the probability P (θ, τ ) that the phase point near θ has left the range (0, θB ) in a time τ greater than t. The differential probability that it has left the vicinity of θ in times between t and t + dt is then 

θB

dθ (P (θ|θ ; t + dt) − P (θ|θ ; t))

p(θ, t)dt =

(5.46)

0



θB

dθ (∂P (θ|θ ; t)/∂t) dt

= 0

The average time of escape from the vicinity of θ is then  ∞ T (θ) = tp(θ, t)dt

(5.47)

0

This is usually called the mean f irst passage time over the barrier. The use of the adjective ‘first’ is due to the fact that, in this model of a perfectly absorbing barrier without backflow, the first passage time will be the only passage time. In the present problem, we want to know the average time in which the initial density at θ = 0 gets depleted by the reversal process. So we require 



∞

< T (0) > =

θB

tdt 

dθ (∂P (0|θ ; t)/∂t)

(5.48)

0

0



∞

=

θB

tdt 0



=− 0

=+




0



∞

dt 

1

˜(0, θ ; t)/∂t dθ e− 2 (E(0)+E(θ )) ∂ u

θB

1




˜(0, θ ; t) dθ e− 2 (E(0)+E(θ )) u

0 θB


dθ e− (E(0)+E(θ ))/2



˜n un (θ )/λ u ˜n (0)˜

0

The plus sign in the last expression derives from the fact that P (0|θ ; t) gives the probability of arriving near zero, after being definitely at θ > 0. This implies

122

MAGNETIZATION REVERSAL

backward propagation, and so the sign of t must be reversed in all the exponentials in the series for u ˜. (In other words, one uses the solution of the so-called backward FP equation.) At this point, how to account for backflow is not clear, apart from the plausible suggestion of Safanov and Bertram (2000b), to add to this time a similarly calculated mean first passage time from π to 0 in order to estimate the total time for the transition. 5.4

Rotation in 3d

The model considered in Section 5.2 applies to oblate spheroidal particles whose magnetization is unable to precess out of the plane because of strong demagnetizing. In this section, the slightly more general case of spherical particles is analyzed, in which the magnetization can precess freely during the reversal process. The completely general case, in which either crystalline or shape
without restriction anisotropy or both depend on both polar coordinates of M requires nontrivial computational efforts, and is not considered here. The FP equation in more than one dimension is: ∂w = ∇ (∇w + w∇E) ∂t

(5.49)

This is transformed into a Schr¨ odinger type equation by setting w = e−E/2 u. Then u satisfies

1 2 ∂u 1 2 2 = u+  E − (E) u (5.50) ∂t 2 4 with the spatial derivatives confined to the surface of a unit sphere. In terms of
vector the eigenvalue equation reads polar coordinates (θ, φ) of the M ∂u 1 ∂2u 1 ∂ sin θ + + (V (θ, φ) + λ)u = 0 sin θ ∂θ ∂θ sin2 θ ∂φ2 where



1 ∂ ∂EK 1 ∂ 2 EK sin θ + sin θ ∂θ ∂θ sin2 θ ∂φ2



2

2  ∂EK ∂EK 1 1 H sin θ + − + 4 ∂θ ∂φ sin2 θ

V (θ, φ) =

1 2

H cos θ +

(5.51)

(5.52)

and E = −H cos θ + EK (θ, φ). In the general case, anisotropy energy EK (θ, φ) does not have separable form, so that not much can be done. However, some progress is possible for uniaxial anisotropy EK = − 12 HK cos2 θ, and no dynamical demagnetizing energy as in a spherical particle. Then in the switching process we are not interested in the precessional motion, so we are entitled to average the equation over φ. Since u must be single valued in φ, its second derivative

ROTATION IN 3D

123

∂ 2 u/∂φ2 drops out in the averaging process, and the averaged u is a function of θ only. The equation for u is then du 1 d sin θ + (V (θ) + λ)u sin θ dθ dθ d2 u du + (V (θ) + λ)u = 0 = 2 + cot θ dθ dθ

0=

(5.53)

where V =

1 1 2 H cos θ + 3HK cos2 θ − HK − (H + Hk cos θ) sin2 θ 2 4

(5.54)

As in the 2-d case, we require that no net current flows at θ = 0 and θ = π, so that ∂u/∂θ = 0 at these two places, so we again have a two-point boundary problem. The way this was handled in Section 5.2.1.3 was by proceeding as for ordinary problems, with u(0) = 1 and u (0) = 0. For an arbitrarily assigned λ, the solution fails to meet u (π) = 0. Then we varied λ and found that, for a particular discrete set of values representing the spectrum of decay constants, the condition u (θ) = 0 was met. This method ‘worked’ very well for all values of H and HK , indicating that the conditions for the existence of a solution of this two-point problem were actually met for all values of H and HK . The 3-d problem can be attacked in the same way, but with somewhat different results. First, consider the problem of free diffusion on a sphere. If, at time t = 0, the magnetization vector is assumed to point in the direction θ1 , φ1 . Its probability distribution is then the delta function in spherical polar coordinates: δ(θ − θ1 , φ − φ1 ) =

+l ∞ 1   Yl,m (θ1 , φ1 )Yl,m (θ, φ), sin θ

(5.55)

l=0 m=−l

where the Y are the spherical harmonics, assumed normalized. At a later time, this distribution will evolve to Green’s function u(θ|θ1 , φ|φ1 ; t) =

+l ∞ 1   Yl,m (θ1 , φ1 )Yl,m (θ, φ)e−l(l+1)t sin θ

(5.56)

l=0 m=−l

but since we are interested only in averages over initial and final φ, we only need u(θ|θ1 ; t) =

∞ 1  Pl (cos θ1 )Pl (cos θ)e−l(l+1)t sin θ

(5.57)

l=0

with the Legendre polynomials P considered normalized. This assumes that θ = π is a reflecting barrier. If one wishes instead to have a sink at θ = θS , say, then the l’s are no longer integers, and the P ’s are no longer polynomials. The

124

MAGNETIZATION REVERSAL

required values of l are now the roots of the equation Pl (θS ) = 0.

(5.58)

So much for the case of free rotation. When V is introduced, the eigenvalue spectrum can again be searched by the same methods as in the 2-d case, but with not quite as unambiguous results. In the 2-d case, the eigenvalues for finite H and/or Hk were continuously connected to the eigenvalues of free diffusion on the circle for V = 0. Similarly, in the 3-d case, as V increases from zero, the great majority of eigenvalues (in particular λ1 ) evolve continuously from the form l(l + 1) appropriate to free diffusion on the sphere. However, it appears that at least one new small eigenvalue rises from zero, with no connection to the others, and could dominate long term relaxation. Likewise, large extra eigenvalues appear at large anisotropies. Conceivably, the reason may be the delicate behavior of equation (5.53) near the pole θ = 0. This matter calls for further study, probably requiring a more sophisticated shooting method. Finally, the case of anisotropy energy, given as a general function of both θ and φ, obviously calls for much more elaborate computer simulations. However,
itself, if one is satisfied with much less intimate details of the motion of M one might settle for the behavior of some ‘surrogate’ quantity, for example the purely magnetic part EM of the total energy. This part of the energy itself is a stochastic quantity. A purely formal way of treating it, already mentioned in Chapter 4, is the Mori-Zwanzig projection technique (Zwanzig, 2001). The non
to
does not change EM . The coupling of M dissipative part of the motion of M the reservoir (direct or indirect) must be projected onto EM . (This is easy to say, but certainly not trivial to do.) The result is a generalized Langevin equation, with a deterministic memory kernel, and a ‘left over’ random fluctuating driving field.

6 MAGNETIZATION REVERSAL IN ARRAYS OF PARTICLES AND CONTINUOUS MEDIA 6.1

Introduction

Most recording media involve arrays of magnetic particles confined to a thin layer. Their density has to be sufficiently large so that an adequate number of them pass under the read/recording head at the same time. This is to ensure adequate signal strength and low recording noise (Bertram, 1994). The interparticle distance is then so small that interaction between magnetic moments must be taken into account. The most important interaction is dipolar, although, for close packing of the particles, exchange coupling across their interfaces may have to be considered. To provide a transition from the isolated particle picture of Chapter 5, we begin by considering a sparse system of randomly positioned particles. A sparse arrangement is obviously not of much interest in applications like magnetic recording, but it does draw attention to certain complexities that may have a bearing on practical situations. First, we note a few rather obvious generalities. The array of particles is usually quenched as far as positions of the particles and their anisotropy axes are concerned. Therefore it would be wrong to average over particle positions or axis orientations early in the calculation. The evolution equation, and its solution must carry along explicitly all the particle positions and orientations. Averaging over positions etc. is permitted only for so called self-averaging quantities, such as the total magnetization components or their autocorrelation functions. Thus the average of M3 for the entire system must be calculated first as an expectation value using a distribution function calculated with particle positions, anisotropy axes, etc. f rozen in. Thereafter, the average over all particle positions etc. may be taken.1 This contrasts with the case in which the particle positions and orientations are allowed to move. In that situation a joint diffusion equation of their magnetizations and their positions etc. would have to be solved. Only if the positions and orientations diffuse extremely rapidly would it be permissible to speak of average particle properties from the outset. A novel feature of the dilute array with dipolar interaction is its tendency to chaotic motion of the individual magnetization vectors, particularly in weak applied magnetic fields. The author is not aware of any experimental verification 1 Practically all macroscopic observations of many particle properties involve symmetric sums over the positions of one particle or of pairs of particles, . . etc. with equal weight for each term in the sum. This looks very much like a spatial average times the volume–hence the term ‘self averaging’. The components of the total magnetization are of this kind.

125

126

MAGNETIZATION REVERSAL

of this prediction in connection with micromagnetics. However, in nuclear magnetic resonance spectroscopy both numerical simulation (Enss et al., 1999; and Khomeriki, 1998) and experimental work (Lin et al., 2000) appear to implicate dipolar interctions in such chaotic behavior. In this chapter, we examine this effect in some detail for the case of two interacting magnetic dipoles, for general values of the distance between them, and for general values of the relative orientation of their anisotropy axes in the absence of any coupling to a thermal reservoir. Except in sufficiently large applied magnetic fields, their magnetization vectors will in general move chaotically. Then we show that an entire dilute array will exhibit stochasticity due to overlap of pair resonances. When the sys
will decay, as discussed tem is coupled to a thermal reservoir, the motion of M in the previous chapter for the simplest kinds of coupling. This damping is due
; it is this recoil that can take to the recoil of the reservoir to the motion of M energy away from that motion. At first sight, one may be inclined to believe that this damping will suppress chaos. However, the same reservoir that causes the damping also tends to resuscitate the chaos. It succeeds in keeping it alive indefinitely, unless the coupling constant to the bath exceeds a critical threshold proportional to the square root of the temperature. Only if that coupling significantly exceeds this threshold will there be no persistent stochasticity. This situation is the large motions analogue of the spin wave instabilities discussed in Chapter 2. The argument concerning the critical coupling will be made at the end of Section 6.2. In the absence of an imposed reservoir, the stochastic motion provides the system with its own ‘internal’ reservoir, unrelated to temperature. Accordingly, one might hope that a diffusion equation could be established for it. However, magnetization reversal cannot be achieved in this way, even if such an equation can be established. The reason is that, without coupling to an imposed reservoir with which to exchange energy, the system is Hamiltonian, and is confined to a surface of constant energy. When the system starts out more or less aligned in the positive z direction, and a reversal pulse is applied in the negative z direction, the system is left behind in a higher energy state and an external heat bath is needed to get it off that high energy surface. For a sufficiently large reversal field, chaos is suppressed, and then a conventional treatment suffices. For example, if the system is sufficiently dilute, the reversal rate in large fields might be calculated by starting with the rate calculated for independent particles in Chapter 5, and using perturbation theory. As already mentioned, damping will suppress chaos only if the parameter, g, measuring the coupling strength to the reservoir, exceeds a certain critical value, gcrit . When g  gcrit , general chaos will be established at a rate greater than the rate governing attainment of the final energy state and the stochasticity must be taken into account, as described at the end of Section 6.2. When g gcrit , conventional theory again suffices: the energy equilibrates too fast to be affected by tendency to chaos. The problem becomes much harder when g is close to gcrit , and we cannot make any predictions in this case.

RELAXATION DUE TO MAGNETIC MOMENT INTERACTION

127

In the first few sections of this chapter, we presuppose spatial uniformity of the statistics (not, of course, coherent rotation of all the magnetic moments). In the case of magnetization reversal, this excludes various possibilities, such as reversal nucleating in a limited region, domain wall motion, spatio-temporal chaos, etc. Certainly these questions become more and more critical as the density of the array increases towards the limiting case of a continuous medium. It is well known that, as the number of degrees of freedom becomes very large, collective effects, such as soliton motion may appear, that have no analog in systems with a finite number of degrees of freedom. So, in the later sections, we begin with a conventional approach to the case of a dense medium and its reversal properties, ignoring the question of spatio-temporal chaos. In particular, we consider magnetization reversal in extended media large enough to support domain walls. We show that, except in special cases, reversal takes place by highly localized nucleation, followed by relatively rapid domain wall propagation. To a certain extent, nucleation of the reversal may have a time dependence similar to that of single particle reversal discussed in Chapter 5. Completion of reversal by domain wall motion will occur on a different time scale. 6.2

Relaxation due to magnetic moment interaction in a sparse medium

In Chapter 5, interaction of the magnetizations of different particles was completely neglected. The distribution function of the system as a whole is then the product of the single particle distributions. For less extreme dilution it becomes necessary to allow for interaction of particles within a ‘reasonable’ distance of one another. In a random and very dilute array, this interaction will mainly involve particles one pair at a time. The array as a whole will then be a mixture of such ‘dimers’ and isolated single particles. (In an ordered array, even if of low density, this picture is obviously inappropriate.) To highlight the relaxation process within the pairs, we begin by imagining coupling to the heat bath, and hence the mechanism needed to produce intrinsic damping constants like α, turned off completely. The ensemble of particles is then microcanonical, and total equilibrium (if it is indeed attained) corresponds to uniform distribution over a surface of given total energy. This process with α = 0 was, in previous chapters, called distributive damping; the total magnetic energy is not degraded thereby. The following notation will be used: The required probability distribu
i is written w({M
i }, t).
i } of all the magnetization vectors M tion of the set {M
We shall again find it profitable to specify Mi in polar coordinates φi , θi . So w is written as w({φi , θi }, t). To keep it simple, we restrict the discussion to the case with all particles having uniaxial anisotropy, with axis along θi = 0, which is also the direction of an applied field, if any. We are usually, but not always, interested only in the distribution function of the ‘slow’ variables of the system, which would be left over if all the ‘fast’ variables were somehow disposed of. In the present context, the angles θi vary slowly compared with the precession

128

MAGNETIZATION REVERSAL

angles φi , with the result that the latter, ‘fast’ angles rarely enter a measurement directly, and are of little interest to the observer. Thus, for practical purposes, one is more interested in a reduced probability distribution w({θi }, t). This is true, not only in our problem, but also in many problems in nonlinear dynamics, which are commonly framed in terms of action and angle variables. In the present case the action and angle variables are, respectively, θi (or convenient functions thereof, like cos θi ), and φi . However, certain linear combinations of φi ’s can conspire to produce a secular effect on the θi and their distribution. These combinations must of course be retained, and certain of these combinations do, in fact, play a major role in the present problem. 6.2.1

Equations of motion for dipolar interaction

The total energy is the sum E = E1 + E2 of single particle and dipolar energies. For definiteness, we adopt the same primitive model for the single particle energy E1 as in the previous chapter:

 1 2 E1 = − H cos θi + Ki cos θi (6.1) 2 i allowing different particles to have different anisotropy fields Ki . Other variabilities between different particles, for example a spread in the directions of their anisotropy axes, could equally well be accommodated in the following, but will not be considered here. The dipolar energy, in polar coordinates, has the form E2 =

  M 2 a3 s

i

j=i

x3ij

(cos ϑij − 3 cos Θi,ij cos Θj,ij )

(6.2)

where cos ϑij = cos θi cos θj + sin θi sin θj cos(φi − φj )

(6.3)

cos Θi,ij = cos θi cos θij + sin θi sin θij cos(φi − φij ) cos Θj,ij = cos θj cos θij + sin θj sin θij cos(φj − φij ) with θij and φij denoting the polar angles of
xij , the vector distance between particles i and j. All particles are assumed to have the same volume, a3 . From now on, we set Ms = 1, and define a coupling constant κij = a3 /x3ij . In this abbreviated notation ⎤ ⎡ ci cj (1 − 3c2ij ) + si sj (1 − 32 s2ij ) cos(φi − φj )  κij ⎣ −3sij cij (si cj cos(φi − φij ) + sj ci cos(φj − φij )) ⎦ (6.4) E2 = i j=i − 32 si sj s2ij cos(φi + φj − 2φij ) where si = sin θi , ci = cos θi , and similarly for the j-subscript, and where sij = sin θij , etc. In that formula, the product of the two cosines was written as a sum of cosines of added and subtracted arguments.

RELAXATION DUE TO MAGNETIC MOMENT INTERACTION

129

The equations of motion are, after some minor rearrangement, and with the additional notation cti = cot θi , ⎞ ⎛  ∂E (6.5) = H + ⎝ Ki ci − κij (1 − 3c2ij )cj ⎠ φ˙ i = − ∂ cos θi

+

 j=i

⎛ κij ⎝

j=i

cti sj (1 − 32 s2ij ) cos(φi − φj ) − 32 cti sj s2ij cos(φi + φj − 2φij )

⎞ ⎠

−3sij cij (cti sj cos(φj − φij ) + ctj si cos(φi − φij ))

φ˙ j = same, with i and j interchanged ⎞ ⎛ sj (1 − 32 s2ij ) sin(φi − φj )  ∂E ⎠ −3sij cij cj sin(φi − φij ) =− κij ⎝ θ˙ i = sin θi ∂φi 3 2 − 2 sj sij sin(φi + φj − 2φij ) j=i ⎞ ⎛ −si (1 − 32 s2ij ) sin(φi − φj )  ∂E θ˙ j = κij ⎝ −3sij cij ci sin(φj − φij ) ⎠ =− sin θj ∂φj − 32 si s2ij sin(φi + φj − 2φij ) j=i In the dilute limit, we may concentrate on the motion of a particular i, j pair by omitting the summation symbol on the right hand sides of the above equations. Then we have a system with just two degrees of freedom. Systems with more than one degree of freedom can have chaotic aspects to their motion and, except for large applied fields, this is the case here. (In one special case, sij = 0, the system reduces to only one degree of freedom, and then chaos is excluded.) In seeking ‘adequate’ solutions of eqns (6.5), we use a feature that has already been used in earlier chapters: disregarding for the moment the dipolar coupling, in a model with only uniaxial anisotropy, and with the external field parallel to the anisotropy axis, the time rate of change of the θ-angles is slow compared with the rate of change of the φ-angles. This fails only for θ values such that K cos θ + H is close to zero, which can be the case only for sufficiently small values of H. In that vicinity, θ and φ change at comparable rates because there is then no longer a total effective field imposing a preferred direction. In applying this general observation to eqns (6.5), we note that there are three types of φdependent terms in these equations: terms that depend only on phase differences φi − φj , terms that depend on sums φi + φj , and terms depending on single φ’s. In examining these for their potential for slow variation on the same time scale as the θ’s, note that the difference φi − φj is capable of slow variation for all values of H, no matter how large, since its rate of change does not involve H at all. Therefore φi − φj is a prime candidate for serious consideration. Next, the sum φi + φj is capable of slow variation only if H is small enough to allow 2H +Ki cos θi +Kj cos θj to be close to zero. However, it must still be considered seriously, since it involves behavior characteristic of pairs. Finally, terms involving only a single φi affect the behavior of pairs only in the manner of an applied

130

MAGNETIZATION REVERSAL

signal field. As such, they will contribute only to the stochastic aspects of the motion. For lack of a better term, we shall refer to the ranges in which the phase differences or phase sums can vary slowly as resonances. The difference and sum resonance will actually coincide in the very special case cos θi = −H/Ki and also cos θj = −H/Kj , which can occur if K1 and K2 are separately greater than H. When conditions are such that only one resonance can occur, the term failing to resonate simply adds to the stochastic behavior of the one in resonance. 6.2.2

A single pair

To begin, concentrate on a single pair i = 1, j = 2. Our ‘action variables’ will be m1 = cos θ1 and m2 = cos θ2 , the components of the magnetization along the common anisotropy axis. Before examining chaotic characteristics, we note that the chaos actually rides astride some regular features, which should be distinguishable experimentally. To see this, we rewrite eqns (6.5), leaving out the ‘single phase’ terms that are responsible for most of the stochastics. Introducing the variables m = m1 + m2 and µ = m1 − m2 , respectively conjugate to the phases χ = φ1 + φ2 and ψ = φ1 − φ2 , these equations are

3 2 (6.6) µ˙ = −2κ12 s1 s2 1 − s12 sin ψ 2 1 − K m + (K + + κ12 )µ ψ˙ = 2 



1 + 14 (m2 − µ2 ) 3 2 3 2 1 − s12 cos ψ − s12 cos χ − κ12 µ s1 s2 2 2 3 m ˙ = − κ12 s212 s1 s2 sin χ 2 
1 + χ˙ = 2H + (K − κ12 )m + K − µ 2 



1 − 14 (m2 − µ2 ) 3 2 3 2 + κ12 m 1 − s12 cos ψ − s12 cos χ s1 s2 2 2 where K ± = K1 ± K2 and κ12 = κ12 (1 − 3c212 ). For general values of s12 , this is a system with two degrees of freedom and, as such, still has chaotic solutions. But the chaos is much weaker than that due to the single phase terms. In the particular case s12 = 0, in which the anisotropy axes make zero angle with the line joining the particles, the single phase terms are altogether absent, and the motion is exactly integrable. For s12 = 0, only the exchange-like part of the dipolar interaction survives, and eqns (6.6) reduce to µ˙ = −2κ12 s1 s2 sin ψ



(6.7)

1 + 14 (m2 − µ2 ) 1 − K m + (K + + κ12 )µ + κ12 µ ψ˙ = s1 s2 2

cos ψ

RELAXATION DUE TO MAGNETIC MOMENT INTERACTION

131

  where s1 = 1 − 14 (m + µ)2 and s2 = 1 − 14 (m − µ)2 . Evidently, m is a constant of the motion in this pure exchange case, as behooves any component of the total magnetization, in particular the 3-component m. (The sum phase, χ, will also move, but only ‘comes along for the ride’, and cannot react back on the solution of eqns (6.6). In this simple case, therefore, there is only one degree of freedom: thus, for s12 = 0, there can be no chaos. Nevertheless some features of the motion should, by continuity, persist into the chaotic regime in ‘skeletal’ form. (Another rigorously integrable case of eqns (6.6) occurs when s212 = 2/3, i.e. at an angle of about 55o between anisotropy axis and the line joining the particles. But that solution becomes stochastic when the now inevitable single phase terms are included.) When s12 = 0, the situation is essentially the same as that of a planar pendulum: it is integrable in the absence of drive, and performs either clean oscillations or else clean rotations (but in the presence of a driving field of an arbitrary frequency, stochasticity results). The orbit in the phase plane that separates libration from 360o rotation of the pendulum is called the separatrix, and corresponds to a certain critical total energy of the undriven pendulum. As shown by Chiricov (1979), in appropriate neighborhoods of the phase plane, any problem of nonlinear oscillations can be reduced to that of a pendulum by a series of canonical transformations, but the problem considered here is sufficiently simple, so that we can discuss it without recourse to transformation theory. Consider the particularly simple case m = 0, and K1 = K2 = K, as well as s12 = 0. Then we have s1 s2 = 1 − µ2 /4 and the solution curves in the ψ, µ phase plane are found by integrating (1 − 14 µ2 ) sin ψ dµ = −2 dψ µ(r + cos ψ) where r = (K − 1)/(κ12 ). The solution may be written  µ = ±2 1 − c(r + cos ψ)−1

(6.8)

(6.9)

where c is a constant of integration. If µ = µ0 when ψ = 0, then c = (r + 1)(1 − µ20 /4). So µ can go to zero at a value of ψ given by cos ψ = 1 − (r + 1)µ20 /4, provided this exceeds −1, that is provided r < 8/µ20 − 1. For values of µ0 that satisfy this inequality, the µ vs. ψ curves are closed, otherwise they are open (see Fig. 6.1a). Thus, for given r, the value of µ0 corresponding

132

MAGNETIZATION REVERSAL µ

(a) 1.5 1 0.5

–10

–7.5

–5

–2.5

2.5

5

7.5

10

ψ

–0.5 –1 –1.5

µ

(b)

0.4

0.2

–4

–2

2

4

ψ

–0.2

–0.4

(c) 2

1.5

1

0.5




0

–0.5

–1

–1.5

–2 –2

–1.5

–1

–0.5

0

0.5



Fig. 6.1. Continued

1

1.5

2

RELAXATION DUE TO MAGNETIC MOMENT INTERACTION

133

(d) 2

1.5

1

0.5




0

–0.5

–1

–1.5

–2 –15

–10

–5

0

5



Fig. 6.1. (a) Integrable motion of one pair with anisotropy axes aligned along the line joining the two particles. Parameters: s12 = 0, κ = .3, K1 + .4K2 = 1, m = 0. (b) motion around the separatrix |µ| = ±2 when r is nearly equal to one. Parameters: s12 = 0, κ1 , K1 = 1, K2 = 1.2, m1 .5. Small changes easily shift the orbit to centers around φ equal to multiples of π. Exactly on the separatrix, φ runs away. (c) Poincar´e map of the non-integrable motion (intersections of the 4-d orbit with the (φ, µ)-plane) when the line joining the particles makes a small angle (s12 = .01) with their common anisotropy axis directions. The finely dotted curve is the integrable case, with the same parameter values, except s12 = 0. (d) Similarly non-integrable case of (b).  to the separatrix is µ0 = ± 8/(r + 1). But since |µ0 | is always 0. Equation (6.27) may be integrated further, giving a rather lengthy result in terms of the original variables. In its most compact form the result reads tanh

√ ψ ψ0 = e− h+hK x tanh 2 2

(6.29)

√ where sinhψ = κ tan(θ/2), and κ = h/(h + h K ). (For the explicit form, see Suhl and Bertram, 1997, equation A7.) Of particular interest is the height of the resulting energy barrier, which is calculated in the Suhl-Bertram reference. It is shown in Figure 6.4 as a function of h/hK for a range of initial values θ0 . Also shown in the figure is the boundary of possible localized barrier states; to the left of that boundary, only buckling states can be found. The contour labels are the values of the barrier heights, in units of J. 6.4.1

Simple model of magnetization reversal by domain wall motion

A domain wall is defined as the thin region separating relatively large regions of
vector. In the very simplest models the two oriendifferent orientations of the M tations differ by 180 degrees, but this is not generally the case, particularly when demagnetizing fields are taken into consideration. A domain wall carries with it positive exchange energy, but this is not necessarily energetically unfavorable, if it has the effect of sufficiently reducing some other energy, such as demagnetizing energy. Furthermore, a totally demagnetized equilibrium state at a finite

150

MAGNETIZATION REVERSAL 3

1.6

1.4

1.2 1.0

2.5

Initial Angle

0.8 2

0.6 0.45

1.5

0.3 0.2

1 0.5 0

0.1 0.05 0.02

0.5

0

1

0.5

1.5

2

off

H/Hk

Fig. 6.4. Contours of equal barrier heights in units of exchange energy due to pinning at the surface of a semi-infinite slab, as a function of applied field and pinning angle. (Figure by courtesy of Journal of Applied Physics).

temperature will carry a whole network of domain walls. Even if demagnetizing energy is small for topological or other reasons, the associated entropy tends to compensate for the increased energy. The total free energy is then minimized. This is true even in the ideal sample; any non-ideal effects complicate the matter further. A full analytic description of even the static state is probably out of the question, except possibly close to the Curie point. Describing the motion of the whole network during the magnetization process from zero to finite applied field is even further out of reach, except in special models. It is not out of the question that some extreme idealization allows the process to be described in terms of some crude paradigm sharing some features with the real situation. Here we begin by restricting consideration to the motion of a single domain wall, in a situation in which the dipolar field and its demagnetizing effect plays no major role. A further crucial simplification is to consider the domain wall to be uniform in a plane normal to the direction of motion. Ideally, one considers an infinite medium with magnetization directions opposite on the two sides of the wall, subjected to a magnetic field turning the magnetization direction towards it, thereby moving the wall. Even in this simple model, one is dealing with a nonlinear partial differential equation with two independent variables: time and position, with first order time derivative, and second order position derivative. This leads the investigator into an irresistible temptation: to view the motion in a coordinate system moving with (hopefully constant) wall velocity, U , and to

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

151

assume that, in that system, no explicit time dependence is found. The solution to the problem should then depend on space x and time t only through the single variable ξ = x − U t, and the partial derivative ∂/∂t becomes −U d/dξ. The equation then becomes an ordinary differential equation. The main problem raised by this procedure is to reconcile it with boundary conditions in a f inite sample. Thus one must show that some plausible class of initial spatial variation
at time t = 0 will lead to such a traveling wave. Similarly, one faces the of M problem of ultimate disposition of the wave at the far end of the specimen. In the infinite medium, the notion that an applied field results in a wave is reasonable, because the energy of a wall in the absence of the applied field is independent of the position of its center. Breaking that symmetry by localizing the wall should be accompanied by an excitation spectrum starting at zero energy. By contrast, in a finite sample, any wall heeding the boundary conditions will be localized, so there is no symmetry breaking and a finite excitation gap arises. In a large sample it is nevertheless likely that, on a limited time scale, and in central regions only, the displacement of the wall due to the applied field will have the appearance of a wave travelling with more or less constant speed. Two particular cases will be discussed here: the N´eel wall and the Bloch wall, both in idealized geometries. Motion of the N´eel wall is a relatively easy problem, especially since it can draw on the considerable literature on formally similar questions in reaction kinetics, particularly biology. Motion of the Bloch wall is a much harder problem. In particular, its solution is very ‘Ansatz-dependent’, as typified by the inspired guess of L. R. Walker as described in Dillon (1963). See also Schreyer and Walker (1974). For a simple discussion of the Walker solution, see O’Dell (1981).
vector varies 6.4.1.1 Simple model of a N´eel wall In this configuration the M with x1 = x, and is independent of x2 , x3 . In the directions of the 2- and 3-axes, the medium is considered infinite. There is crystalline anisotropy energy −0.5 HK cos2 θ + 0.5H∞ sin2 θ sin2 φ p.u.v. with easy axis x3 , and an enormously hard axis with large K2 along 0x2 (Ms is equated to 1 in the following). The exchange energy is 0.5A(dθ/dx)2 p.u.v., with A = J 2 Ms2 , in an earlier notation. With the applied field H equal to zero, the wall is stationary. The total energy is stationary, if θ satisfies Ad2 θ/dx2 = HK cos θ sin θ. For a sample, with dθ/dx = 0 as x → ±∞, the solution is tan(θ/2) =  tan(θ(0)/2) exp( HK /Ax). Choosing θ(0) to equal π/2, the explicit solution is    θ = arcsin 1/ cosh( HK /Ax)

(6.30)

152

MAGNETIZATION REVERSAL

but it must be noted that the ‘usual’ branch of arcsin used for x < 0 must be changed to the branch π − arcsin for x > 0. Then θ varies continuously from zero at minus infinity to π at plus infinity, with maximum slope at x = 0. When a magnetic field is applied along the easy direction, this pattern, apart from possibly moving along 0x, will try to precess about 0x3 . However, if K2 is very large (or, in the case of a thin film, demagnetizing normal to its plane is strong) the precession angle φ will be suppressed at value zero throughout. Then the remaining equation of motion becomes θ˙ = −α∂E/∂θ, or 2

∂ θ ∂θ =α − (h + h cos θ) sin θ (6.31) K ∂t ∂x2 and is known as a reaction-diffusion equation. Here, Gilbert damping has been assumed, time is measured in units of a reciprocal exchange frequency, h = H/J, hK = HK /J, and distance measured in units of lattice spacing. In suppressing precession, we forego the possibility of including the D¨ oring (1948) virtual ‘domain wall mass’ due to precession forcing the magnetization vector out of its plane and causing a local magnetic charge accumulation (Chikazumi and Charap, 1964). Writing the equation in the form

∂ ∂θ ∂θ + −α = −α(h + hK cos θ) sin θ ∂t ∂x ∂x shows that the ‘reaction’ term acts as a source, spoiling the conservation law that would otherwise connect charge θ with current α∂θ/∂x. This source term can result in propagation; without it, one would only have a simple diffusion process. Equation (6.30) bears a certain resemblance to Fisher’s equation used to ∂ model propagation effects in biology (Fisher, 1937; Murrey, 1989). To replace ∂t ∂ is not totally unreasonable. That some kind of propagation might by −U ∂x result is made plausible by the following argument: Without the source term in eqn (6.30), the solution subject to an initial condition of finite θ at time zero 2 and x = 0, but zero everywhere else has the form ∼ t−1/2 e−x /2t . When there is a source of strength f (x, t), that solution becomes  t  ∞
2 dτ dx (t − τ )−1/2 e−(x−x ) /2(t−τ ) f (x , τ ). 0

−∞

Without source, the peak around the origin would broaden and eventually die. But with the source in place, the peak gets replenished at the various places x , especially at time τ = t. So it is reasonable to assume that the solution will persist for large x. This, of course, does not guarantee a traveling wave for large x − U t when f depends on x and t, through θ itself. Even if there is such a solution, it risks running into trouble with the initial condition, usually specified as a shape θ0 (x) at time t = 0. However, in the case of Fisher’s equation at

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

153

least, Kolmogorov et al. (1937) have shown that a broad class of initial shapes can evolve into traveling wave solutions. In particular, their proof covers Fisher’s equation, whose source term in the present notation has the general form θa (1 − θb ), with a and b positive integers. Up to cubic terms, our source term reads

1 −αθ h + hK − hK θ2 2 and a minor change of variables brings it into that standard form, with a = 1 and b = 2. So, at least during the ‘launch’ phase of the wave, a solution of the form θ(x − U t) is on firm ground. A proof for our source term over the entire range of θ is beyond the scope of this work, and validity of the wave-like solution will simply be assumed. As it stands, it applies only to an infinite medium carrying one domain wall. We shall presently attempt to achieve greater realism, but, to begin with, we examine this highly idealized picture. The first step, then, is to determine if it is possible to find a value of U such that the equation 2

∂ θ ∂θ −U =α − (h + hK cos θ) sin θ , (6.32) ∂x ∂x2 has a stable solution with the desired properties at infinity. We write it as two equations: ∂θ =p ∂x ∂p α = −U p + α (h + hK cos θ) sin θ, ∂x

(6.33)

and examine these in the (θ, p) phase plane: α

−U p + α (h + hK cos θ) sin θ dp = . dθ p

(6.34)

9 9 9 h 9 If 99 99  1, there are three critical points within the domain of θ. They are hK 9 9 9 h 9 (θ = 0, p = 0), (θ = arccos(−h/hK ), p = 0), and (θ = π, p = 0). If 99 99  1, hK there are just two; the middle one disappears4 . To determine the character of any one of the critical points (θf , pf ), with pf = 0 at all of them, write θ = θf +δθeλx , p = δpeλx . Then: λδθ = δpλ

 αδp = −U δp + α (h + hK cos θf ) cos θf − hK sin2 θf δθ.

(6.35)

4 For positive h, the middle fixed point splits in two, with θ = arccos(h/h ) ± π. This K f feature actually shows up in profiles of the wall motion.

154

MAGNETIZATION REVERSAL

So λ satisfies the quadratic: λ(λα + U ) = α ((h cos θf + hK cos 2θf )) .

(6.36)

First, 9 9 consider the two critical points θf = 0 and θf = π common to both 9 h 9 9 9 ≶ 1. The roots are: 9 hK 9  ˜ 2 + (h + hK ), at θf = 0 U  ˜ ± ˜ 2 − (h − hK ), at θf = π U λ = −U 9 9 9 h 9 ˜ where U = U/(2α). The roots near the critical point for 99 99  1 are hK  ˜ ± U ˜ 2 − hK, (1 − h2 /h2 ) , at θf = arccos(−h/hK ) λ = −U K ˜± λ = −U

(6.37) (6.38)

(6.39)

The behavior of orbits 9 9 near the fixed points is given by Table 6.1. The last row 9 h 9 is present only if 99 99 < 1. hK

Fxt Pt.

Saddle

Spiral

Focus

(0, 0)

˜ 2 > −(hK + h) U ˜ 2 > (h − hK ) U

˜ 2 < − (hK + h) U ˜ 2 < (h − hK ) U

˜ 2 + (hK + h) < U ˜2 0 2 h − hK . In fact, Kolmogorov et al. show that, in the case of the Fisher if U equation, U is exactly equal to 2, if the initial form θ(x, t = 0) of θ is zero for all large enough x, whereas the fixed point analysis only requires U > 2. (The linearized version of the present equation is of the same form as the linearized Fisher equation). For cases more general than the Fisher equation only the inequality can be inferred. However, we can make a very crude argument (not depending on the assumption ∂/∂t = −U ∂/∂ξ) that the velocity of arrival at θ = π is given by the equality. Just before arrival, θ = π − ψ, where ψ is small and satisfies ∂ψ =α ∂t



∂2ψ − (h − hK )ψ ∂x2

(6.40)

The solution evolving from a given initial small value ψ0 is √

ψ0 exp 4παt



−x2 − (h − hK )αt 4αt

(6.41)

This solution decays in all directions in the (x, t) plane. However, it decays √ most slowly along a ridge given by the line x = 2α h − hK t, and shown in Fig. 6.6a except for a negligible correction due to the prefactor, so along that √ ridge the velocity is evidently U = 2α h − hK . The hope is that the nonlinear terms neglected here somehow sharpen the ridge and prevent it from declining. A similar argument can be made for the velocity of departure from θ = 0. For small θ, the equation is ∂θ =α ∂t



∂2θ − (h + hK )θ ∂x2



√ Starting from a small initial θ0 , the same argument now gives U = 2α h + hK for the departure velocity. If there is an initial gaussian distribution θ0 (x) ∝

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

157

(a) 20

15

t 10

5

0 0 (b)

5

10

20

15

X

25

t 0.5

1.

0.8

0.6

0.4

1.2

0.4

1.4 0.3

1.6 1.8

0.2

0.2

2.

0.1 0.1

3.

0 0.03

0 0

0.2

0.4

0.6

0.8

1

X

Fig. 6.6. (a) Terminal wall velocity in the unbounded medium. When θ approches π, the propagation equation may be linearized, and the maximum of the solution in the (x, t) plane follows the dashed line, whose slope is the reciprocal wall velocity. (b) In the bounded medium, the same considerations that lead to Figure 6.6a) show that the wave dissipates upon approach of the wall, and its velocity goes to infinity.

158

MAGNETIZATION REVERSAL

exp(−x2 /(4∆)), rather than a sharp initial value, the subsequent distribution is 1 √ 4παt







dx θ0 (x ) exp

 2 − (x − x ) − (h − hK )αt 4αt

√ The equation of the ridge is then x = 2α h + hK (t + ∆/α) and is shifted somwhat, but the departure velocity is the same as for a sharp initial value. 6.4.1.2 Finite sample length In Section 6.4.1.1 it was supposed that the specimen stretches all the way to infinity. Therefore, even if one is reconciled to the notion that a wide range of initial shapes θ(x, 0) at time zero can launch a wave with constant velocity, one has to come to terms with the fact that θ, or θ at the far end of the sample, may have assigned values independent of time. The author has not been able to find analytical work on this question in the literature; work has been essentially numerical. Some crude indication of what to expect can be gleaned from the following consideration of the linearized problem. To keep it simple, suppose that at the far end x = L, pinning forces cause θ to be zero for all time. On the other hand, let θ be proportional to δ(x) at time t = 0. First, assume that both h and hK are zero. Then we have a pure diffusion equation with a brief pulse injected at x = 0, and a sink at x = L. This equation is solved by one of the theta functions, ϑ2 (x, t). The diffusion equation with a linear term −gθ subtracted from θ has the solution ˜ Θ(x, t) = e−gt ϑ2 (x, t)

(6.42)

where, in our problem, g = h + hK . Let us assume (pending later verification) ˜ that, at a fixed value of x, the function ln Θ(x, t) has a maximum with respect to t, given by g=

∂ ln ϑ2 (x, t) ∂t

(6.43)

From the solution t = t1 (x) of this equation, a position dependent velocity U (x) = 1/(dt1 /dx) may be determined. From equation (6.42), dt1 =− dx



∂ 2 ln ϑ2 (x, t)(x, t) ∂x∂t



t=t1 (x)

∂ 2 ln ϑ2 (x, t)(x, t) ∂t2

−1 (6.44) t=t1 (x)

where, on the right, t is replaced by t1 (x) only after the differentiation with respect to x has been carried out. But we can proceed less formally by writing down two different representations of the theta function for short times and for long times as follows: The boundary condition θ = 0 at L can be assured by

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

159

constructing the antiperiodic δ-function ∞ 

(−)n δ(x − nL)

n=−∞

which, in the course of time, becomes proportional to ϑ2 ≈ √

∞  1 −(x − nL)2 , (−)n exp 4αt 4παt n=−∞

(6.45)

a form convenient for short times. At short times, just a single term dominates √ the sum in (6.44), and any one of these once again gives the velocity 2 g., just as for the infinite sample. For long times, ϑ2 is written as an expansion in the eigenfunctions ψn = cos(λn x) of the equation −λ2n ψn = d2 ψn /dx2 , with In terms of these eigenfunctions, the initial antiperiodic λn = (2n + 1)π/(2L). ∞ δ-function is n=−∞ cos(λn x), and later on becomes ϑ2 ≈

∞ 

2

e−λn αt cos(λn x)

(6.46)

n=−∞

˜ Figures 6.6b is a contour plot of Θ(x, t) = e−gt ϑ2 (x, t) for a typical value of g. As in the unbounded case shown in Fig. 6.6a, the contours bulge out, and the ˜ declines corresponding three dimensional plot will show a ‘ridge’, along which Θ most slowly. Assuming that the nonlinear term will some how make that ridge more sharp at the expense of the rest of the territory, the velocity along that crest gives some indication of the velocity in a more complete solution. In Fig. 6.7, as in Fig. 6.6, the ridge is marked by a dotted line. Along this line, dt1 /dx evidently tends to zero at the end of the sample as contour θ = 0 is approached. Thus the wall velocity starts out for small x just as in the infinite sample, but increases steadily to infinity as the boundary is approached. That rate of increase is the larger, the greater g. The following analysis shows that U goes to infinity like 1/(x − L)2 . For long times, the series (6.44) converges poorly, and (6.45) must be used. The ‘ridge’ along which the solution is maximum for given x is determined by solving ∂ ln ϑ2 ∂t ∞ 2 −λ2n αt cos(λn x) n=−∞ λn e  =− ∞ −λ2n αt cos(λ x) e n n=−∞

g=

(6.47)

Because of rapid convergence, only a few terms in the sums need to be retained at sufficiently large times. But to obtain any meaningful result, at least n = 0, 1, −1, −2 must be kept. (If only the term n = 0 is kept, t becomes independent

160

MAGNETIZATION REVERSAL

of x altogether, so the velocity would be infinite everywhere.) Then the last equation becomes 2

g=−

2

λ20 e−λ0 αt cos(λ0 x) + λ21 e−λ1 αt cos(λ1 x) 2 2 e−λ0 αt cos(λ0 x) + e−λ1 αt cos(λ1 x) 2

=−

(6.48)

2

λ20 cos(λ0 x) + λ21 e−(λ1 −λ0 )αt cos(λ1 x) 2 2 cos(λ0 x) + e−(λ1 −λ0 )αt cos(λ1 x)

so that 2

2

e−(λ1 −λ0 )αt = −

(g + λ20 ) cos(λ0 x) (g + λ21 ) cos(λ1 x)

(6.49)

This is meaningful only if (λ1 x) is in the third quadrant (but(λ0 x) is always in the first quadrant, since x < L). Hence early termination of the series holds good only for L > x > L/3. Near the end of the sample, write x = L − y, where y is small. Then cos(λ0 x) → sin(λ0 y) and cos(λ1 x) → − sin(λ1 y). Hence eqn (6.49) becomes 2

2

(g + λ20 ) sin(λ0 y) (g + λ21 ) sin(λ1 y)  (g + λ20 ) λ0 1 − (yλ0 )2 /3 = (g + λ21 )λ1 (1 − (yλ1 )2 /3)

e−(λ1 −λ0 )αt =

=

(6.50)

(g + λ20 ) λ0  1 + (L − x)2 (λ21 − λ20 )/3 (g + λ21 )λ1

From this follows α

2 dt1 (x) = (L − x) dx 3

(6.51)

whence ˜= U

3 2(L − x)

(6.52)

In this approximation (and in this approximation only) the final approach velocity is independent of g. The same result holds for the constraint θ = θL = 0 at L. The constraint ∂θ/∂x = 0 at x = L requires ∞ use of a different theta function. The appropriate initial δ-function is again n=−∞ cos(λn x), but now with λn = nπ/L. No major changes in the results are to be expected. 6.4.2

Motion of a Bloch domain wall

The classic stationary Bloch wall involves both angles θ and φ, usually chosen as indicated in Figure 1.1. The magnetization vector twists around the x-direction,

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

161

with angle θ(x) varying from zero to π, independent of y and z. The angle φ measures the angle between the x-axis and the projection of the magnetization vector on the plane z = 0. In the stationary wall, φ = π/2 and is independent of position. The specimen is assumed infinite in all directions. The crystalline anisotropy field is assumed to be along 0z as easy axis. When a field is applied along 0z, the wall will move. The most important difference between Bloch and N´eel wall motion previously discussed is that the former shows precessional effects among the α-independent terms in the equations of motion. Since we ultimately will need to make numerical comparisons, we restate the equations of motion with explicit definitions of the constants. Define the frequencies ωH = γH, ωK = γHK , ωD = πγMs , ωex = γHex ,

(6.53)

where Hex = JMs , and ωD is the frequency associated with the demagnetizing field resulting from the deviation of φ from π/2. Position x is measured in units of the lattice spacing . The equations of motion may be written θ˙ = 2ωD sin θ sin 2φ − αφ˙ sin θ

(6.54)

1 φ˙ sin θ = ωH sin θ + ωK cos θ sin θ + 2ωD sin 2θ cos2 φ 2 2 2 ∂φ ∂ θ 1 − ωex sin 2θ + αθ˙ − ωex 2 ∂x 2 ∂x if we use φ˙ sin θ = ∂E/∂θ, and θ˙ = −∂E/(sin θ∂φ) in place of our usual expression for the damping terms. We now show that for sufficiently large α, sufficiently small fields, or sufficiently short samples, the motion becomes the same as that of the N´eel wall. Assume for a moment that in the limit of large α, the wall velocity is large. Then, if it is large enough, the precession of φ away from its starting angle π/2 will not have advanced appreciably by the time the wall reaches the far end. Then all φ-dependent terms in the forgoing equations may be dropped, and the remaining equation is the√same as that for the N´eel wall. Its speed is then more ¯ , where ω ¯ is some average of ωH, ωK , and ωD . A or less equal to U ≈ 2α ωex ω sample of length L is traversed in a time L/U. The time needed for an appreciable precession of φ away from its value π/2 is of order 1/¯ ω . If that time is long compared with L/U, precession may be neglected. Thusthe condition for the ¯ /ωex  2α /L. For N´eel wall result to be applicable here is L/U  1/¯ ω , or ω ¯ about 109 Hz, and a high-loss specimen with α ≈ 1, ωex about 1013 Hz, and ω this inequality is satisfied for sample length less than 100 lattice spacings. For longer samples and lower loss materials, the variation of φ must be taken seriously. One can still look for a solution with ∂/∂t replaced by −U ∂/∂x, so that eqns (6.54) become a pair of ordinary differential equations for θ and φ. But now there is no possibility of a simple phase plane analysis, which provided crucial clues in the case of the N´eel wall. Qualitative understanding of the solution

162

MAGNETIZATION REVERSAL

would require a computer exploration of the five-dimensional parameter space subtended by the four ω’s and U/α. This need not deter one from seeking special analytic solutions, even if they do not necessarily have immediate practical relevance. An ingenious special solution was proposed by Walker as described by J. F. Dillon (1963) (see also Walker and Schreyer, 1974). This solution assumes that the wall, viewed in a frame moving with velocity U, retains the shape of the stationary wall in zero applied field, except that φ differs from π/2 by a fixed amount independent of position, and the constants in the expression for the stationary wall are renormalized. Walker seeks a solution with φ = φ0 = constant and makes the replacement ∂/∂t = U ∂/∂ξ, where ξ = x + U t. Then the foregoing equations become U

∂θ = 2ωD sin θ sin 2φ0 ∂ξ

(6.55)

0 = ωH sin θ + ωK cos θ sin θ + 2ωD sin 2θ cos2 φ0 − ωex

∂2θ ∂θ + αU ∂ξ 2 ∂ξ

Note that the first of these equations is identical  with the equation for the stationary wall, whose solution is tan θ/2 = exp ωK /ωex x. It is necessary only to replace ωK /ωex by 2ωD sin 2φ0 /U. Using the first of eqns (6.55) in the second and dividing it by sin θ gives  2 0 = ωH + ωK + 4ωD cos2 φ0 − 4(ωex ωD /U 2 ) sin2 2φ0 cos θ(ξ) + 2αωD sin 2φ0 (6.56) This equation can be satisfied for all ξ only if the coefficient of cos θ(ξ) vanishes, which determines φ0 . Then the remaining term must also vanish, so that ωH + 2αωD sin 2φ0 = 0

(6.57)

Using this in the vanishing of the coefficient of cos θ(ξ), we get ωK + 4ωD cos φ0 − 2

2 4(ωex ωD /U 2 )

ωH 2αωD

2 (6.58)

or 2

U2 =

2 ωH /(2αωD ) 4ωex ωD ωK + 4ωD cos2 φ0

(6.59)

Also ⎛ ⎞ 8

2 1 ω H ⎠ 1− cos2 φ0 = ⎝1 ± 2 2αωD

(6.60)

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

163

The negative square root must be chosen to ensure that φ0 tends to π/2 as ωH tends to zero. Note that the solution is real only if ωH < 2αωD . If that is satisfied ' 1 ωH ωex 8 (6.61) U=



 α ωK ωD 2 1+2 1 − ωH /(2αωD ) 1− ωK The largest allowed velocity, attained when ωH = 2αωD, is evidently 2ωD ωex /ωK . It may be too much to expect this extremely elegant, but highly singular solution to be of wide applicability in practice. The most important difference between the velocity given by eqn (60) and the velocity discussed in the previous section for N´eel walls is that, according to eqn (6.60), the velocity increases with decreasing α for sufficiently small ωH , while the latter increases indefinitely with increasing α, as is more likely from general considerations. 6.4.3

Magnetostatics and the magnetization process. Pre-existing domain walls

The theory in Section 6.4.2 has paid no heed to the effect of lateral boundaries and the demagnetizing fields they can set up. In the case of the N´eel wall, lateral effects were suppressed simply by assuming very large hard anisotropy along an axis in an undesired direction. In the case of the Bloch wall, however, the dimension of the medium normal to the plane formed by driving field and direction of wall motion must be considered infinite, or else toroidally closed on itself, if lateral demagnetizing effects are to be excluded. Allowing for boundary effects poses much more complicated problems. In fact, we shall argue in this section that the simple theory of wall motion outlined above survives in its pristine form only in special circumstances that involve dipolar forces only trivially or not at all. A classic example is a sample configuration in the shape of a picture frame or lozenge in an applied field furnished by a current in a coil wound around its sides and threading the frame. If the easy direction of magnetization runs parallel to the sides, and a N´eel domain wall is formed at the outer edge, an applied field can drive the wall to the inner edge essentially in accord with the above theory. Generally, one may expect this theory to work in regions in which both the applied field and the domain wall to be moved run parallel to the sample boundaries. In these cases one may expect that, at least for large driving fields, ‘lateral’ dipolar fields will not be important. However, in a process of magnetization reversal with the field switched from a large value in one direction to a similarly large value in the opposite direction, the value of the field must pass through a range of values below those needed to saturate the sample. Then, unless the sample is small enough to exclude formation of domain walls, the magnetization may collapse into a more complicated domain structure. In this

164

MAGNETIZATION REVERSAL

section we mainly formulate the general mathematical challenge presented by these processes even in reduced dimensions, particularly in subsaturating fields. However, the conclusions we draw do seem to provide a rough indication of what to expect in practical cases of specimens with linear boundaries. We begin by discussing the simple case of an infinite magnetized cylinder of sufficiently large radius in the absence of any externally imposed magnetic field. If composed of material with negligible crystalline anisotropy, it will be uniformly magnetized parallel to the cylinder axis in its state of lowest energy. Its exchange energy is then zero, and so will be all other forms of its magnetic energy. If its magnetization vector had a component perpendicular to the boundary, the required continuity of magnetic flux normal to the boundary would result in an external magnetic field and some associated magnetic energy (even in zero applied field). The external field thus generated would be just sufficient to ensure zero internal field. If it were less than that, an internal field would arise opposing the magnetization direction, an unstable condition. Next suppose that crystalline anisotropy in the plane normal to the cylinder axis is zero, but that the cylinder axis is a hard direction, with anisotropy energy (1/2) K cos2 θ, where θ is the angle between magnetization direction and cylinder axis, and where K is a positive constant. Then the same magnetostatic energy
vector at the boundary to be directed considerations as before require the M purely circumferentially. But this sets up some exchange interaction because of
direction, a rate of change that will be greater the resulting rate of change of M the smaller the cylinder radius. So for sufficiently small cylinder radii, the cost of exchange energy would exceed the cost in magnetostatic energy entailed in
with some component normal to the wall. But allowing a uniform inplane M for a cylinder with radius exceeding a critical value, the lowest state is a quasi
lines around the center. The internal and external vortex formed by the M magnetic fields and their associated energies are then zero, and there remains only the exchange energy, which is best calculated in polar coordinates. Let r
and and ψ specify position in the plane, and let θ be the angle between M the cylinder axis. In this vortex configuration, we can evidently choose φ, the
, to be equal to ψ. Remembering that ∇ in this case is other polar angle of M ∂φ = 1) is (∂/∂r, ∂/(r∂ψ)), the energy per unit cylinder length (since ∂ψ  2π 0

R

:

1 2 A

rdr 2



∂θ ∂r

2

sin2 θ + r2

;

 2

+ K cos θ

where R is the cylinder radius and the equation to be solved is A 2

∂2θ A 2 sin θ cos θ = 0. + K − ∂r2 r2

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

165

subject to a boundary condition that depends on what one believes the surface ∂θ boundary condition to be (as mentioned before, a popular choice is = 0). For ∂R
must practically line up with the cylinder axis, even though it is very small r, M the ‘hard’ direction (‘escape into the third dimension’ to avoid a singularity), so θ must go to zero with √ r. Forsmall θ, the solution in terms of a Bessel function is proportional to rJ√5/2 ( K/A 2 r). In this section, we are concerned with
large samples. For  large r, M must almost be confined to a plane, so θ = π/2− 2 if the cylinder radius R is much larger const × exp(− K/A r). Therefore,  than a domain wall thickness A 2 /K, the ‘inner’ region of the vortex may as well be replaced by a point, andthe exchange energy of the entire vortex is essentially the familiar one: ∼ K K/A 2 per unit length of cylinder. If the cylinder is of finite length, the calculation is more complicated (see Guslienko and Metlov, 2001). Evidently, the end faces of such a cylinder have magnetic charges, causing an external field whose energy must be taken into account. A more detailed analysis of the vortex state, including conditions for stability, are presented in an appendix to this chapter. 6.4.3.1 Small Specimen When an external field H is applied perpendicular to the axis, with a view to determining how the sample reaches saturation as a function of H by clearing out the vortex, replacing it by uniform magnetization, the problem becomes rather intractable analytically for small samples. Using reasonable approximations, Guslienko and Metlov (2001) found that, to begin with, the vortex distorts, and its center moves towards the sample boundary. Eventually, when H reaches saturation value, the center is swallowed up by the boundary, and the magnetization is then uniform. Returning for a moment to the cylinder, it is evident that if there is strong in-plane anisotropy energy Kip p.u. length of cylinder the vortex will be modified and will be prevented  altogether if Kip K K/( 2 A), because then the negative energy due to align
along the easy in-plane axis is not overcome by the positive vortex ment of M exchange energy, plus the positive magnetic energy that now arises outside the sample. This situation was tacitly assumed in the calculations of Chapter 5.  Actually, for R not much greater than A/K, the energy balance is even more favorable for uniform magnetization. For sufficiently soft magnetic materials, in nanomagnetic structures the Guslienko and Metlov (2001) calculations should be important. 6.4.3.2 Large Specimen However, in this section, the focus is on samples with linear dimensions much larger than a typical domain wall thickness. We consider only very soft materials with zero crystalline anisotropy (except in the ‘hard’ direction parallel to the cylinder axis), this being in order to concentrate on the dipolar, demagnetizing, source  of anisotropy. In a cylinder with transverse dimensions much larger than A/K, vortex formation still plays a role in zero

166

MAGNETIZATION REVERSAL

applied field, but in finite fields normal to the cylinder axis, the quasi-singularity at the center spreads into a line (or, more precisely, into a sheet everywhere parallel to the cylinder surface), and moves sideways. When H reaches saturation, that wall merges into the boundary, and the magnetization is then uniform. We note that, in large samples of general shape, exchange interaction plays virtually no role in determining the shapes and positions of the walls. Instead, the walls form as the result of dipolar forces in their incarnation as demagnetizing fields.5 At most, exchange energy will perturb these wall configurations slightly, for example by reducing their curvatures a little. A large, perfectly homogeneous sample in a very small applied field will harbor as many domain walls as needed to reduce the internal magnetic field to the lowest possible value consistent with its shape. To see how pure magnetostatics, with exchange energy neglected, determines the morphology of a single domain wall in a large sample, we briefly sketch the theory for an infinite cylinder with zero in-plane anisotropy and ‘hard’ anisotropy energy along the cylinder axis (van den Berg, 1987), and Bryant and Suhl, 1989a). Ignoring exchange coupling, the equations minimizing the energy are ∇2 φ = 0 outside sample,

(6.62)


inside sample, ∇ φ = −4π∇ · M 2


vector is confined to the plane everywhere, except possibly inside a and the M domain wall, where it may (but need not) rise up as it did in the vortex case (Green and Krantz, 1997). (At the center of the vortex, confining the magnetiz
indeterminate there.) For small tion to the plane would make the direction of M enough applied H normal to the axis, the internal field must be zero. Therefore
= 0. This can be satisfied for in-plane components of the form ∇·M ∂A ∂x2 ∂A M2 = ∂x1 M1 = −

(6.63)


2 = 1, the function A(x1 , x2 ) must satisfy and since M

∂A ∂x1



2 +

∂A ∂x2

2 = 1.

(6.64)

5 A thoroughly familiar example is the rectangular block in zero applied field. Usually the disposition of domain walls is as shown in Fig. 6.8e, but this cannot be quite right in even a small finite field because, at the sharp corners, the field is singular and will penetrate into the sample. For a block with rounded corners, the walls will not quite reach the edges; at each of the rounded corners, the walls will be as shown in Fig. 6.8b).

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

167

This equation, reminiscent of geometrical optics, may be solved in great generality (Courant and Hilbert, 1962). With an applied field H perpendicular to a general cylinder with cross-sectional shape that can be conformally mapped onto the complex plane, analytic solutions may be found with negligible computation. (The case of a thin disc with a field applied in its plane is somewhat more complicated, since a field applied in its plane pervades the whole (extremely thin)
is no longer zero. More serious computation is then required.) The disc, so ∇ · M equations are solved subject to boundary condition that the magnetic flux nor
·
n mal to the surface ∇φint ·
n + 4πMn be equal to the normal flux ∇φext ·
n = H arriving from outside. (Some cases are shown in Fig. 6.7, by courtesy of the Journal of Applied Physics.) The result is a domain wall that moves out from the quasi-singular center, spreading out into a shell independent of the z-direction. As H increases, the wall moves towards the boundary. Its surface increases, but
lines decreases, and the curvature of the lines the discontinuity in angle of the M away from the walls also decreases. This means that the exchange energy cost,  so far neglected, cannot be much different from its value K K/( 2 A) owing to the quasi-singularity of the vortex state at H = 0. When H reaches its saturation value, the wall merges with the boundary. Just before this value is reached, the internal field is still zero, so the magnetostatic energy of the interior is still zero. If there is non-zero in-plane anisotropy, the analysis becomes harder, but it is easy to see what happens qualitatively: to take advantage of easy anisotropy
lines will attempt to be as straight as possible along easy direcenergy, the M tions, consistent however with shielding out the field. An assessment of the effect of exchange energy on these walls of dipolar origin may be made by singular per
are not the same as turbation theory. Inside the thin wall, the equations for M the equations well outside it. This is analogous to the boundary layer problem in fluid dynamics, with the wall representing the thin boundary layer. Singular perturbation theory provides a generally successful approximation to the problem of joining the normal and boundary regions (Binder and Orszag, 1978). If H is changed to some new value, a new magnetostatic situation arises after a certain relaxation time. If the rate of change of H is slow compared with the magnetic relaxation rate, no further analysis is needed: we can then ignore details of the time dependent process leading to the changed configuration. If one still insists on speaking of a wall velocity, one may define it as the time rate of change of position of some special point on the (generally curved) domain wall. (An end point of a wall is particularly suitable, because an analytic expression for any end point can be written down rather easily (Bryant and Suhl, 1989b.) The position
x of such a point is a function of H. Therefore the wall speed defined in this way is d
x/dt = (d
x/dH) · dH/dt
/dt  dH/dt. If, however, H changes at a rate comparable provided that αdM with the magnetic relaxation time, the problem becomes much harder. Neglecting

168

MAGNETIZATION REVERSAL

h=0

h=.6

h=.4

h=.8

h=1

Fig. 6.7. Exact solutions of the magnetostatic equations in the subsaturation regime in the transversely magnetized circular cylinder as a function of applied field h, in units of 4πMs . exchange, the equations to be solved are now  
dM
×
h ,
×
h − αM
× M =M dt

(6.65)

together with Maxwell’s equations needed to give a relation between the internal
. (Because, in the static subsaturated case,
h is expelled, the field
h and M
= 0, plus boundary conditions, are sufficient to determine M
.) equation ∇ · M The Landau-Lifshitz form of the damping term is more convenient than the Gilbert form in the present context. Before formulating (without, however, solving) the general time dependent problem as an extension of the static case, we consider the utterly trivial case of a semi-infinite demagnetized sample with plane boundary being magnetized by a uniform magnetic field normal to the boundary. This will give some indication, discussed in Section 6.4.3.3, of what is involved in the general case of piecewise rectilinear domain walls and boundaries, like those of Fig. 6.8e. First, we note
× ∇φ has very little effect on the magnetization that the precessional term M process; it of course results in oscillations, but the envelope of these is kept under control by the damping term. We begin, at time t = 0, with an applied field well below saturation, and the internal field shielded out by the magnetization direction making the appropriate angle θ with the surface, and pointing towards it. Measuring that angle from the 2-direction normal to the surface, that angle is θ(t = 0) = θ0 = arccos(h0 /m2 ) where m2 = 4πM2 /Ms , and all fields are likewise

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

(a)

(c)

169

(b)

(d)

(e)

Fig. 6.8. Exact solutions for (a) the domain wall near a rounded ‘corner’ in zero field, and (b), same in an applied field about half-way to saturation. Rounded geometries are chosen to avoid field penetration at sharp corners in finite fields. Figures (c) and (d) corresponding to (a) and (b) for a rectangular corner, with field penetration and fringing fields neglected. (e) a rectangular block. Solution by conformal mapping is much harder in this case.

170

MAGNETIZATION REVERSAL

measured in units of 4πMs . At that angle, the internal field in the 2-direction, h = H0 −m2 , equals zero. Now suppose that the field is increased, to a new value h, still below saturation value 1, so that the new value of h is no longer zero, and θ will begin to decrease until it reaches the new value, arccos h, needed to shield. In that process, the internal field is equal to h − cos(θ(t)). Since precession is ignored, only m2 is involved; the other two components, m1 and m3 are absent. Also, the damping term can be written m(
m
· ∇φ) − ∇φ, so that equation of (6.64) becomes m ˙ 2 = −α(h − m2 )(m22 − 1)

(6.66)

θ˙ = −α(h − cos θ) sin θ

(6.67)

or, in terms of θ,

This can be integrated from θ0 to θ:  

(h − cos θ) sin θ0 1 tan(θ/2) αt = − ln h ln 1 − h2 tan(θ0 /2) (h − cos θ0 ) sin θ

(6.68)

In particular, starting from total demagnetization (m
parallel to the surface), the time taken to reach θ(t) is given by 1 αt = 1 − h2





(h − cos θ(t)) h ln tan(θ(t)/2) − ln h sin θ(t)



(6.69)

plotted in Fig. 6.9a for various values of the field exceeding the saturation value 1. The same formula (6.67) also holds when h > 1. Then there is no more shielding: θ ends up at zero. But the time to get there is logarithmically infinite. (This is because, as θ = 0 is approached, the speed θ˙ goes to zero, so the magnetization never quite gets there.) When h is exactly equal to one, the infinity is more violent, like 1/θ2 . (The exact formula at h equal to one is αt = (cos θ)/(4 sin2 (θ/2)) + (1/2) ln tan(θ/2).). Just as in the case of switching discussed in Chapter 5, the decision as to when saturation is deemed complete is subjective. Even for h less than 1, the time needed to shield out h goes to infinity logarithmically. Solution of the opposite problem, demagnetization upon sudden removal of the saturating field, is even simpler. With h equated to zero in equation (6.66), and θ0 close to zero, the solution is θ = arctan(tan θ0 eαt ), shown in Fig. 6.9b6 . Similarly, in the case of magnetization reversal, in which the initial angle θ0 is close to π, and with a large field applied along θ = 0, formula (6.67) can be used. Finally, Fig. 6.9c gives the ‘coercive field’ (the field needed to reverse magnetization in a given time T ). At this point, the reader may have noticed that the problem here is almost the same as the one considered in Chapter 5 dealing with switching of the 6 As usual, a small initial deviation is needed to move θ from its stationary but unstable maximun at θ = 0.

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

171

(a) t 10 8 6

ο=.11

4 2

ο=.01 0.25

0.5

0.75

1

1.25

1.5



t

(b) 200

h=1.05

150 100 70 50

h=1.15

30

h=1.25

20 0

0.5

1

1.5

2

2.5

3



h=2.05 (c) H/(4Ms)

3

2.5 2

1.5

1.5

2

2.5

T

Fig. 6.9. Magnetization and demagnetization of a semi-infinite magnetic medium, with field applied normal to its plane surface. (a) demagnetization process upon sudden removal of saturating field, (b) magnetization reversal, with θ starting close to π and rotating to θ ≈ 0 as the result of a suddenly applied field exceeding saturation value. (c) the coercive field as a function of α times the reversal pulse duration T .

172

MAGNETIZATION REVERSAL

magnetization of small particles. The only difference is that, there, we allowed for thermal agitation, which avoided both the difficulty of dislodging the magnetization from stationary, though unstable points on the energy surface, and the difficulty of arriving at the final state in a finite time. Also, here the demagnetizing field takes the place of the crystalline anisotropy. Had we omitted diffusion in Chapter 5, the result would have been identical to the one just derived. We could, of course, use the diffusion equation for the single θ variable here also, but in an extended sample the diffusion process must be described by a functional differential equation for the entire field θ(
x). However, for h > 1, the ‘ballistic’ regime ignoring diffusion should be adequate. The reason for discussing this simple case at such length is that, for lack of better, it can be used to construct the course of magnetization and reversal processes in large samples, at least if these have linear boundaries. Figure 6.10 demonstrates this construction for the square cross section. One may raise the question why, in the present context, we have abandoned the earlier considerations that attributed the reversal process to initial

(a)

(b)

H

(c)

Fig. 6.10. Magnetization process for a cylinder with square cross-section, analogous to the circular cross section shown in Figure 6.7. (a) zero field, (b) an intermediate field, (c) field slightly below saturation.

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

173

overcoming of a localized barrier, followed by rapid domain wall motion. This had explained the fact that the observed ‘magnetic volume’ was so much smaller than one would have expected from a process like curling involving the entire sample volume. The reason why the initial stage of the process (overcoming a barrier) is not needed here, is that there are pre-existing domain walls. In contrast with the curling or buckling process, there is here no critical field that must be exceeded to initiate reversal. The approximate treatment of linear boundaries and linear domain walls is simple, because the construction does not explicitely involve position. More generally, we have to solve eqn (6.65), taking into account spatial variation to the extent required by dipolar forces, but excluding exchange forces for the reasons stated earlier. As in the discussion in Section 3.3, we use only the zeroeth time moment of the dipolar part of the internal field hint , so that the relation between it and the magnetization is the usual magnetostatic one at every instant in time and does not involve any materials parameters, such as conductivity etc. (The next higher time-moment results in position dependent damping, which we ignore
can be written as the sum of the curl of a vector plus here.) Like any vector, M the gradient of a scalar:
=∇×A
+ ∇χ M

(6.70)

Since the right hand side involves two vectors, and the left hand side only one,
χ, conveniently chosen in the form: a constraint must be imposed on A,
· ∇χ = 0 ∇×A

(6.71)

Also, we can identify ∇χ with −
hint (from hereon called −
h), because we have
= ∇ · (
hint + 4π M
) = ∇ · (4π M
− ∇χ) 0=∇·B

(6.72)


=∇·∇×A
2 = 1, another constraint is Finally, since M
2 + (∇χ)2 = 1 (∇ × A)

(6.73)

because of condition (6.70). Discarding the precession term, the equation of motion (6.64) may be written  = < 
dM
(∇ × A
+ ∇χ) · ∇χ − ∇χ =α M dt  
(∇χ)2 − ∇χ =α M

(6.74)

174

MAGNETIZATION REVERSAL


(see in view of (6.65) and (6.70). The field ∇χ is expressed in terms of M Section 3.3.2.1.):

 ∇χ = ∇


(
x ) 
·M
(
x ) ·M dS +  |
x −
x | x −
x | surface |


3 ∇

d
x

 ,

(6.75)

using the zeroeth time moment only. Equations (6.73) and (6.74) determine the motion of a magnetization field with long-range dipolar interaction and we are unable to offer a simple solution procedure. However, based on the simple example of eqns (6.65) to (6.68), it is possible to draw some very rough conclusions concerning the motion of magnetostatics-dominated domain walls for simple geometries. 6.4.3.3 Time dependence of rectilinear wall arrays The work described in Bryant and Suhl (1989b, 1989c) characterized the walls as the loci of self-crossing
-lines. The portions beyond the self-crossing points must be points of the M
-lines, resulting in discarded, since along these portions they intersect other M
vector. This consideration must apply in all indeterminate direction of the M cases. In particular, it must apply near a corner of a cylinder with square cross section. In the absence of any field, the domain walls cut a cross-section of the cylinder in a familiar pattern shown in Fig. 6.10a. Any finite applied field must penetrate the sample near the sharp edges of the cylinder, because there the field becomes very large. For this reason, Bryant and Suhl (1989b) considered the case of a rounded corner, where field penetration need not occur (Figs 6.8a and 6.8b). The resultant wall for a few values of the field are shown in Fig. 6.7b. Consider now a crude approximation: assume that the simple picture of eqns (6.65) to (6.68) developed for an infinite plane boundary can be applied to the sides bounding the rectangular cylinder around one of its corners. In zero field this picture gives the right answer if we simply apply the principle that the
must be unique everywhere, and the angle at which M
-lines enter direction of M the wall must equal the angle at which they leave it (otherwise poles accumulate on the wall, causing a non-zero internal field). This means that the wall must
-lines running be a 45 degree line emerging from the corner (Fig. 6.8c). If the M parallel to the two sides were to cross that line, they would enter a region with
-direction. We apply the same principle to the case of a finite subnon-unique M saturating field applied normal to one of the two sides, ignoring the fact that there must be some field penetration near the corner. We use equation (6.78)
-lines turn away from the direction to determine the angle through which the M parallel to that side. (The lines parallel to the other side would, as a first approximation, remain parallel to that side, since they are not directly affected by the field.) The new domain wall near a corner is now turned as shown in Fig. 6.8c.
is As before, the construction avoids poles on the wall, and non-uniqueness of M avoided. Comparison with the purely static results of Fig. 6.7b strongly suggests

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

175

that the crude picture cannot be too far off the mark. This simple construction can be extended, step by step, to describe the static configuration of an array of linear domain walls in a sample with linear boundaries. Figure 6.10 illustrates the construction for the case of a square cross section, analogous to the rigorous solution for the circular cross section shown in Fig. 6.7. Field penetration near the corners, as well as fringing fields, are neglected. Turning of the magnetization vector is confined to the triangular domains at top and bottom, and, upon sudden application of a field exceeding saturation value, the magnetization direction in these domains must more or less follow the simple calculation for the semiinfinite specimen, as shown in Fig. 6.9b. It is reasonable to assume that the same construction may be applied at each instant in the time dependent case. If that is the case, Fig. 6.9c, which shows the field needed to reverse magnetization (the coercive field) as a function of reversal pulse length, applies generally. Crystalline anisotropy makes this description more difficult in detail, but not in principle. The reason is that in the infinite medium, domain wall energy is independent of position. In moving the wall, the energy increase involved in turning spins on one side of the wall against the anisotropy field is balanced by the energy decrease of spins on the opposite side of the wall aligning more nearly with the favored anisotropy direction. Only a small ‘kinetic energy’ due to the effective D¨ oring mass arises, and this is small compared with the demagnetizing energies involved here. However, for curved boundaries, and also for curved domain walls, it is necessary to solve equations (6.72) to (6.74). Finally, we examine the extent to which a wall propagation velocity might be defined in the present context. Evidently the notion of a constant velocity is inappropriate here. With demagnetizing being the dominant force, there is no length scale. So, to the extent that one insists on speaking of a velocity at all, one must define it in terms of rates of change of shape rather than position. For example, in the case of the corner just discussed, the idea of a linear velocity must be replaced by the angular velocity of the wall. But this angular velocity never attains a constant value in subsaturating fields. It starts at zero velocity, briefly attains a stationary maximum value, and then declines to zero as the shielding angle is approached. The same applies to the reversal process above the saturating field. There, θ starts at π and ends at 0, with almost vanishing velocities at both points. A stationary maximum speed is reached, the more briefly, the larger the saturating field. Clearly the notion of a constant, stable domain wall velocity is useful only in configurations or topologies in which dipolar forces have only minor effects. In a perfect specimen, starting from a demagnetized state, the only barrier to reaching a finite magnetic moment is the demagnetizing energy. In principle, if thermal diffusion is taken into account, the motion could attain that barrier, but for large samples, the time needed to reach it from the demagnetized state unassisted by an applied field would be enormously long. For practical purposes, therefore, diffusion may be neglected, and a purely ‘ballistic’ calculation of the kind leading to equation (6.67) is appropriate. In samples with imperfections

176

MAGNETIZATION REVERSAL

that can present barriers to domain wall motion, thermal activation will play a role, and some effective diffusion equation must be used, except of course for large fields. 6.5

Appendix 6A: Vortex solutions in cylinder and disc: stability considerations

Vortices in samples of general shape (even if spheroidal) seem to require serious computational effort in establishing vortex solutions. Therefore we consider only two extreme cases, long circular cylinders and very thin circular discs. Only minimal computational effort is needed in these cases. 1. The long circular cylinder For the infinite cylinder, the total energy of the vortex may be written    sin2 θ cos2 θ A 2 2 (dθ/dr) + + rdψdr (6.76) 2 r2 ∆2  where ∆ = A/K is a typical domain wall thickness. K is the axial anisotropy energy, which in this case must be of crystalline origin, since there is no appreciable axial magnetostatic energy. (However, if it is assumed that the results for a long, thin prolate spheroid will be more or less the same as for the infinite cylinder, anisotropy will be furnished by the demagnetizing energy Nz cos2 θ, where Nz is the small longitudinal demagnetizing factor.) The total energy expression comes about as follows: Anticipating a circular vortex configuration,
with base a distance r from the center will have the magnetization vector M components Mr = 0 radially outwards and Mφ = sinθ(r) along the circle passing through the point with polar coordinates (r, ψ) fixed in the body. Also, Mz = cos θ(r). Evidently φ = ψ + π/2, so that dφ/dψ = 1, and the usual contribution sin2 θ(dφ/r2 dψ)2 to the exchange energy expressed in fixed coordinates reduces to sin2 θ/r2 . The energy is an extremal along a curve given by

1 θ + 1 − 2 cos θ sin θ = 0 (6.77) r where r has been measured in units of ∆, which no longer appears in the equation. The closest that a solution of eqn (6.77) can come to the standard appearance of a mathematical vortex is to have θ = 0 at r = 0 (to ensure escape into 3d), but there are no considerations dictating the value of θ at r = 0. Instead, the value of θ at the boundary must be specified. For the long circular cylinder, that
would have a component normal to the value must be π/2. If it were not, M boundary pointing in the radial direction all around the cylinder. This would set up a radial magnetic field extending to infinity, which is not allowed since its divergence would not be zero. Equation (6.77) must be solved as a boundary, rather than a pure initial value problem, so that the ‘shooting method’ must be used to solve it. The curves in Fig. 6.11a shows the set of resulting solutions

VORTEX SOLUTIONS

177



(a) 3 5 2 2 3 2   2 –

(b)

0  2

5

10

15

20

r 3 2

  2

0 −

5

10

15

20

 2

(c) 

 2

0 5

−

10

15

20

 2

Fig. 6.11. (a) Orientation θ(r) of the magnetization vector versus distance r from center of cross section of a circular cylinder for a succession of initial slopes θ (0) at the center. In a narrow range of slopes, all curves initially rise to a very close vicinity of π/2, but eventually divide into two groups, one group ending close to −π/2, the other rising to plateaus at successively larger and larger odd multiples of π/2. As a certain critical slope approached from below, a typical curve begins its descent from π/2 to the lower group at larger and larger values of r. At the critical slope it remains at π/2 to infinite r. Beyond that critical initial slope, it rises to join the upper group. The critical slope is probably an irrational number that cannot be reached by the shooting method. The curve marked by an arrow is the closest to critical obtained with the set of slopes tried here (see text). (b) Same as 1a, but with a more refined set of initial slopes allowing a closer approach to the critical curve (see text). (c) Solid curve is the one closest to critical in Figure 6.11b. Along the particularly heavy portion (on the π/2-plateau), high order derivatives of θ(r) vanish (at the center of the plateau d5 θ/dr5 equals zero).

178

MAGNETIZATION REVERSAL

for various initial slopes around 11. For relatively large values of initial slopes, θ passes through a succession of increasing stationary points with θ’s close to odd multiples of π/2. Some of these stationary points are ‘plateaus’ (i.e. horizontal inflection points). For relatively small initial slopes, the curves oscillate around zero, with absolute values of the maxima and minima less than π/2. These curves are extremely sensitive to the value of the initial slope. As that slope is increased in tiny steps from the oscillatory towards the ‘plateau’ regime, a critical slope is reached at which the period of the oscillation goes to infinity, and the maximum amplitude remains stuck at a plateau with θ exactly equal to π/2. Most likely, that critical slope is an irrational number, unknown in the absence of an analytic theory. The shooting method must therefore be content with finding an initial slope that makes that infinitely long plateau finite, but as long as possible. Figure 6.11b shows θ as a function of r for a narrow range of initial slopes from 11.0012765 to 11.0012815 in equal steps of 0.000005 from 0.000065 to 0.000115. The plateau extending to about r = 20 was found for initial slope equal to 11.001278. This plateau, even though of only finite length, is a very high order inflection; we have found that the fifth derivative of θ vanishes around r = 10. We now show that only this lowest of the sequence of plateaus can result in a stable vortex. For stability, it is necessary (and sufficient) that the second variation of the total energy be positive definite. In the present model, the second order change in energy due to independent increments δθ(r) and δθ (r) in θ and its derivative is  1 2 2 (δθ (r)) − (1 − 1/r2 ) cos (2θ(r)) (δθ(r)) (6.78) δE = 2π 2 which, for stability, must be positive, even in the ‘worst case scenario’. That worst scenario has δθ (r) = 0, and δθ(r) finite in the range in which (1 − 1/r2 ) cos (2θ(r)) > 0,

(6.79)

and zero elsewhere. Any solution that satisfies this inequality for any finite range of r is unstable. The faint curve in Fig. 6.11c shows the plot of (1 − 1/r2 ) cos (2θ(r)) for our favored plateau at θ = π/2 which extends only up to r ≈ 20. The stability criterion is evidently satisfied up to r = 20, and will be satisfied for all r if the initial slope were to allow it to extend to infinity. (In a very small range of r-values near the origin, the inequality is very slightly violated even on that plateau, but this is most likely an artifact attributable to the singularity in the equation at r = 0. (For example, to obtain numerically stable results, it was necessary to start with θ = 0, not at r = 0, but at r = 10−6 . To avoid these difficulties, the continuum model must be replaced by a physically more realistic discrete model close to the origin.) As regards higher plateaus, these can also not be stable, just as in our favored case in which the most persistent curve eventually turned down to reach for a lower plateau or minimum at θ = −π/2.

VORTEX SOLUTIONS

179

The curve has to pass through a high-slope region in which the stability criterion fails. The same is true for a very slightly bigger initial slope, for which the favored curve turns upwards to reach a higher plateau. If an analytic solution to the problem could be found, it would, of course, be able to describe the lowest plateau and its stable vortex exactly for all values of r all the way to infinity. To the extent that our numerical solution can represent the exact result, it appears that the favored plateau is fully established at about r = 7, that is to say about seven typical domain wall widths, setting the lower limit of radial sample size that can support the vortex7 . Smaller radii would result in boundary values of θ unequal to π/2 and impermissible radial fields outside. 2. The very thin disc. For the long cylinder, the anisotropy constant K had to be of purely crystalline origin, since demagnetizing along the cylinder axis is very small. For more general, finite shapes, magnetostatic energy can furnish the equivalent of crystalline anisotropy, since significant energy will be stored in the external field, which must be taken into account. In the case of a very thin disc, a very crude approximation would assume that K is simply the constant 4π, with a demagnetizing energy 2π cos2 θ. However, the textbook form ∼ cos2 θ for the
, which demagnetizing energy is normally derived only for the case of uniform M is not the case here. To replace the textbook expression, it is necessary to calculate in some detail the energy stored in the external field in terms of the normal component cos θ(r), allowing for variation of θ with r. As usual, expressing the undesired part of a problem (the external field in this case), in terms of the desired part (the magnetization field), results in non-local terms in the equation governing the wanted part. In other words, the contribution to the energy replacto uniform magnetization must ing the textbook, purely local, cos2 θ appropriate
now be replaced by a term of the form K(r, r ) cos θ(r) cos θ(r ) rr dr dr. The resulting integro-differential equation analogous to (6.77) is no longer accessible to trivial computation. A rough calculation shows that K(r, r ) fairly rapidly goes to zero with distance between r and r (see Fig. 6.12), so that a moment expansion of cos θ(r ) around r − r seems reasonable. Truncating the moment series at an early stage would then give a ‘local’ differential equation. Unfortunately no such expansion is possible: already very low order moments diverge. This is not surprising: Laplace’s equation, and the fields derived therefrom do not have a characteristic length scale, a sine qua non condition for a successful moment expansion. However, if we are willing to settle for approximate results, we can go to the other extreme and treat K(r, r ) as a constant. This is plausible for the same reason that a moment expansion is implausible: the rate of variation of K(r, r ) is much slower than that of cosθ(r ), which is set by domain 7 This lower limit should not be confused with the well-known limit setting the smallest sample size for which the uniformly magnetized state will have higher energy than the vortex state.

180

MAGNETIZATION REVERSAL

r' − 40.

6

− 20. − 40.

5

50. 95. 6.

4

80.

50.

180.

3

120.

80.

−20.

150.

2

220.

−40. −40.

6.

1 1

2

3

4

5

6

r

Fig. 6.12. A rough contour map of a section of the kernel K(r, r ) illustrating its falloff along the ridge r = r , as well as perpendicularly to the ridge. The falloff in the latter direction is not rapid enough to allow a moment expansion of cos θ(r) around cos θ(r ). The values labelling the contours are on an arbitrary scale. But r and r are on the ordinary scale of electrostatics. If they were measured in units of domain wall width, the coordinate labels would be multiplied by a reciprocal domain wall width, indicating very little decline of K within such a width. This is the basis of the approximate model used in the text. wall thickness. Therefore K(r, r ) may be pulled out from under the integral for the magnetostatic part of the energy, and may reasonably be replaced by some constant mean value, which we shall denote by C/w. Then we have to solve the ‘simpler’ equation

 1 cos θ(r) cos θ(r )dr − sin θ(r) = 0 (6.80) A 2 θ + C w r2 It is not hard to guess what the orders of magnitude of C and w must be: C must be of order 4πMs2 , which it would be if a disc of very large radius w were uniformly magnetized. In the present problem, C will be somewhat smaller than, but still of order, 4πMs2 , and w will be the distance in which θ

VORTEX SOLUTIONS

181

changes significantly. Equation (6.80), while still non-local, now has the form of a relatively simple self-consistency problem as compared with the exact equation 

cos θ(r)     θ + K(r, r ) cos θ(r )dr − sin θ(r) = 0 (6.81) r2 Distances (including w and the integration variable r ) are now measured in units  of A/C. Let  1 cos θ(r )dr , I= (6.82) w The constant I will enter the solution of equation (6.80), which, when entered into the definition (6.82) will hopefully reproduce I, at least after a reasonable number of iterations. A major difference between results for the thin disc and the long cylinder now arises: if it is anticipated that a self-consistent solution leads to a self-consistent I, then at large r the equation is the same as that for a simple pendulum, with the time variable replaced by r. For small initial slopes (i.e. small velocities of the pendulum at time zero), θ(r) oscillates about θ = 0. For large enough initial slope and very large r, the pendulum reaches θ = π and stays there (albeit unstably), corresponding to complete reversal of the magnetization. It is not possible to achieve self-consistency in that case. The reason is that the new value of I resulting from integrating the solution with the old value of I will be negative, because on its way to the top, θ moves more slowly in the range θ > π/2 than in the range θ < π/2, so cos θ(r) is negative more of the ‘time’ than positive. One suspects (and numerical solution confirms) that the same difficulty arises even for finite r. However, for a particular initial slope, it seems to be possible to obtain a self-consistent solution for θ that oscillates with a fixed period between maxima and minima at plus and minus π/2. If the sample radius precisely equals any one of the corresponding r-values, the solution is acceptable, since it does not allow θ to stick out of the rim of the disc and therefore does not give a radial field. We found that the smallest such r-value is 3.1 and has θ = π/2, and the higher r-values, for which θ is again π/2 are 3.1 + 10.04 n, with n = 1, 2... . Similarly θ = −π/2 at radii 3.1 + 5.02 n. A caveat must be added here: only a finite number of iterations were used to achieve self-consistency. This, plus numerical errors, resulted in only 2% accuracy of the higher supposedly ±π/2 values, when the iterations produced an accuracy of 10−5 of the nominal π/2 value of θ at r = 3.1. Also, the question of stability of these states is not totally resolved, as will be described presently. In any case, we found, using only about six iterations, that a very nearly self-consistent state with θ = π/2 at the rim has Initial Slope = 0.00024805; disc radius = 3.1; I = 0.546142 with this final value of I differing from the initially assumed value, 0.556349, by only 0.000207. Also, the angle at the rim was found to be 1.57079, within 10−5 of

182

MAGNETIZATION REVERSAL

π/2. If these conclusions survive detailed stability considerations, discs of radii 3.1 plus several times 5.02 should be able to support ‘domains’ with in-plane components of the magnetization vector pointing in successively clockwise and anticlockwise directions. We now examine the question of stability within the above model. This question is more difficult to discuss than the cylindrical case, since the bilinear form in the δθ’s describing the second variation of the energy now has terms 2  with mixed derivatives (∂θ,θ
E)δθ(r)δθ(r ). Off hand, one might be inclined to assume that the final value θ = π/2 cannot be stable (After all, in the pendulum problem, θ = π/2 is not even stationary, let alone stable.) However, the presence of the (1/r2 ) sin θ cos θ will greatly affect the issue. The second order variation in the exchange energy, proportional to δ  (θ)2 , is always positive. The second order variation of the rest of the exact energy in our crude model is, with dimensionless r, 

 cos 2θ(r) 2   δ E = − rr drdr Iw cos θ(r) − δθ(r)2 r2 −2Kw sin θ(r) sin θ(r )δθ(r)δθ(r )]

(6.83)

1
cos θ(r )dr ). For stability, this bilinear form in the δθ’s must (recall that I = w be positive definite for arbitrary deviations from the solution path determined above. This will be the case if all the eigenvalues λ of the kernel Q(r, r ) = a(r)δ(r − r ) − 2w sin θ(r) sin θ(r )

(6.84)

are negative along the solution path. Here a(r) = Iw cos θ(r) − (cos 2θ(r))/r2 . The eigenvector v(r) satisfies the equation  Q(r, r )v(r )r dr = λv(r) (6.85) Write s =




v(r) sin θ(r)dr. Then this equation becomes (a(r) − λ)v(r) − ws sin θ(r) = 0

(6.86)

so that λ is given by  1=w

sin2 θ(r)dr (a(r) − λ)

(6.87)

with the integrand evaluated along the solution path. Leaving aside the possibility of a solitary ‘collective’ solution, there is an infinity of eigenvalues very close to a(r), with corresponding eigenvectors proportional to 1/(a(r) − λ). A graph of a(r) shows that there are positive eigenvalues for a range of values around π/2, and therefore no guaranteed stability. However, at this point, the second order increment in exchange energy, always positive and proportional to

VORTEX SOLUTIONS

183

(dδ(θ)/dr)2 , may well come to the rescue, since for any particular eigenvector this term is proportional to (a (r))2 /(a(r) − λ)2 , and this quantity is very large. Also, the eigenvalue of the collective mode, λc say, must be negative, since it must be outside the range in which the integrand of (6.87) has singularities. Hence, regardless of the sign of a along the solution path, λc must be more negative than the maximum value of |a|. However, a truly reliable decision on stability will require a serious treatment of K(r, r ). For completeness, we here derive its form, although we are in no position to solve the exact integro-differential equation for θ by primitive methods. The magnetic potential outside the very thin disc, allowing for its top and bottom faces, is V (r, z) = − ∆

∂ ∂z



cos θ(r )r dr dφrr
 r2 + r2 − 2rr cos φrr
+ z 2

(6.88)

where φrr
= φr − φr
. This is expanded in Bessel Functions:  ∞  ∂  dk cos θ(r )einφrr
Jn (kr)Jn (kr )e−k|z| r dr dφrr
∂z −∞  = 2π∆ ke−k|z| g(k)J0 (kr)dk (6.89)

V (r, z) = −∆


where g(k) = cos θ(r ) J0 (kr )r dr . The field components normal to the disc and along r are  Hz = 2π∆sgnz k 2 e−k|z| g(k)J0 (kr)dk  Hr = 2π∆sgnz k 2 e−k|z| g(k)J1 (kr)dk or  Hz (r, z} =  Hr (r, z} =

r dr Gz (r, r ; z) cos θ(r )

(6.90)

r dr Gr (r, r ; z) cos θ(r )

where 



Gz (r, r ; z) = 2π∆sgnz k 2 e−k|z| J0 (kr)J0 (kr )dk   k 2 e−k|z| J1 (kr)J0 (kr )dk Gr (r, r ; z) = −2π∆

(6.91)

184

MAGNETIZATION REVERSAL


2 Then (Hz + Hr2 )rdrdz/4π in the space outside the disc, becomes
  the energy    r r dr dr K(r , r ) cos θ(r ) cos θ(r ),with    2 rdrdzGz (r, r ; z)Gz (r, r ; z) + a similar term from Hr2 K(r , r ) = π∆ (6.92) We know that, at z = 0, the G’s must be essentially δ-functions in order to meet the boundary conditions on the faces of the disc. Similarly, prior to integration over z, K resembles a delta function of r − r , so one might have hoped for a moment expansion. But integration over z spoils this conclusion. (A sim
ple case
is furnished by the example δ(x)(1/2π) eikx dk = δ(x). The integral (1/2π) eikx−k|z| dkdz, on the other hand, is a constant independent of x.) A rough sketch of K(r, r ), with the necessary integrals evaluated by Simpson’s rule, is shown in Fig. 6.12.

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SUBJECT INDEX

Equation of motion of magnetization, just below TCurie 15

Action-angle variables 140, 141 Anisotropy barrier 6, 33, 57, 97, 164 Arnold diffusion 142 Arnold webb 141 Asymptotic expansion 34, 38, 46, 47 Auto oscillations 25

First passage time 119, 120, 121 Fixed points 24, 26, 27 Fluctuation-Dissipation theorem 75, 78 Fokker-Planck equation 85 Free diffusion (on circle) 105 Free diffusion (on sphere) 123

Ballistic motion 98 Barrier (see anisotropy barrier) Boundary layer (domain wall) 167 Breathing of Fermi surfaces 50

Generalized Fokker-Planck equation 95 Generalized Langevin equation 80 Gilbert damping term 10 Green’s function 38, 44, 47, 68, 82

Canonical variables 7 Chaos (pair of magnetic dipoles) 135 Chaos (widespread) 142 Chiricov criterion 142 Coarsegraining 2 Coercive field (pulse length dependent) 114 Condensate (spin waves) 21 Conformal mapping for domain structures in 2d 167 Correlation function 75

Hamiltonian equations of motion 7 Harmonic oscillator reservoir 79 Homogeneous boundary conditions 147 Instability thresholds 21 Intrinsic damping (definition) 9 Kramers’ weak friction r´egime 143

Damping (see distributive, and intrinsic) Decay constants 106, 107 Degeneracy 21, 28 Demagnetizing field 6 Derivative (functional) 4 Diffusion coefficient 87 Diffusion coefficient (with memory) 95 Diffusion equation 87, 88 Dipolar interaction 5, 128 Distribution function 87 Distributive damping (definition) 9 Domain wall motion (Bloch) 160 Domain wall motion (N´eel) 151 Domain walls 166, 168, 170, 172

Landau-Lifshitz damping term 9 Langevin equation 77 Lifetime of magnetic recording 97 Limit cycle 25 Localized barrier at surface 148, 149 Magnetic impurities induced losses 61 Magneto-elastic coupling 33, 34, 35 Misalignment of anisotropy axes and line joining dipoles 135 Misalignment of field and anisotropy axis in switching 118 Multiperiodicity distinct from chaos 137

Eddy current damping 44 Effective field 4 Eigenvalue spectrum 106, 107 Einstein relation 101 Elliptic theta function 105

Non-adiabatic transitions 65 Non-exponential decay 21

189

190

SUBJECT INDEX

Parallelc pumping Possible patchy coverage of energy surface 138 Relaxation equations in nuclear magnetic resonance 10, 126 Reversal time, arbitrariness of definition 111 Reversal time, extended time, small particles 111 intermediate time, small particles 116, 117 large specimen 170, 171 Sample shape dependent domain wall velocity 175 Schr¨ odinger equation equivalent of diffusion equation 104 Separatrix 133

Slow relaxation 61 Surface pinning 149, 150 Temperature diffusion coupled to magnetization 15 Tensor form of Gilbert loss term 81 Thermally assisted recording 15 Valence fluctuation induced losses 60 Velocity of wall, approximate at arrival at infinity 156 at launch 156 at sample end wall 158–160 Walker magnetostatic modes of small motion 19 Zakharov S-theory 23