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 ...

Author: Harry Suhl

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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 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2007 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd. www.biddles.co.uk ISBN 978–0–19–852802–9 1 3 5 7 9 10 8 6 4 2

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

PREFACE

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) idS −

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 = iδij 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 )− = iαβγ Siγ Siβ + Siβ Siγ = iαβγ 2Siγ Siβ + iβγδ Siδ = iαβγ 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 theenergy 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 < a2b2 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 B0varies 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 = ftr 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, ftr 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 ftr 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 ftr 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 )ftr (x , t )eνt w tr (x, t) = µ −∞ 1 ∞ dτ d3 ξ gtr (ξ, τ )ftr (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 )) ftr (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

ftr (x − ξ, t − |ξ|/ctr ) (3.19)

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

t

dΩ exp (−ντ ) ftr (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

ftr (x, t − τ )

⎞

⎠ dΩ ˆ (ξ · ∇)2 ftr (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 ∂ ftr (x, t) β2 ∂ 2 c2tr 2 + ∇ ftr (x, t) ftr (x, t) − + µ µ ∂t µ ∂t2 3

3 ∂ β3 2 ∂ 2 + ctr ∇ ftr (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)

− ···⎠

ftr (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 ftr 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 ftr (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 ftr , 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 ftr 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 fl 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 σe ∇ × 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) − ∇2h, so, from the third of eqns (3.32), we get ∇2h =

) 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 ⎠ = − d3ρ (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 << 1/τ , the deviation φ from feq will be small. eE

52

INTRINSIC DAMPING

So the equation for φ is ∂φ φ ∂feq dM + =− · ∂t τ dt ∂M

(3.59)

Actually, there are two such equations, with feq functions of plus and minus AM, respectively (up-spin and down-spin bands). If the impurities have no magnetic moment, there can be no spin-ﬂip scattering of electrons, so there cannot be scattering between these two bands. Also, in a quasi-thermodynamic treatment, is neglected. Therefore any spin ﬂip scattering of electrons by coupling to M there is one such equation for each band, and no cross terms linking the two. So from here on, we consider only one of the bands at a time, remembering to allow for the two separate relaxation processes at the end. For example, if it is possible to describe the processes by two relaxation times, the eﬀective reciprocal relaxation time for the two together will be the sum of reciprocals of the two separately. Note that, in the presence of spin-orbit coupling, either the total number of electrons (or, equivalently, the chemical potential) will become . So does the magnitude Ms , but presumably a function of the orientation of M to a much lesser extent. Thus we cannot expect d3 k φ to vanish, as it does, for example, in the case of electrical conductivity in the absence of s.o. coupling. The deviation φ, satisﬁes eqn (3.59), one such equation for each of the two bands. The solution, on the assumption that, in the remote past, the system was in equilibrium, is t (t ) ∂feq dM e−(t−t )/τ (3.60) φ=− · (t ) dt ∂M −∞ t (t ) ∂feq ∂s.o. (t ) dM e−(t−t )/τ s.o. =− · (t ) ∂ (t ) ∂ M dt −∞ (t )). The eﬀective ﬁeld is ∂s.o. /∂ M . Then, if where s.o. (t ) stands for s.o. (M Ω0 is the volume of the unit cell, the torque is Ω0 ∂s.o. 3 d Q = M (t) × k φ (3.61) (t) 8π 3 ∂M

t (t ) ∂feq ∂s.o. (t) ∂s.o. (t ) dM Ω0 3 −(t−t )/τ M (t) × d k = dt e · (t) (t ) 8π 3 ∂s.o. (t ) ∂ M dt ∂M −∞ with one such expression for each of the two bands, and the total torque is the sum of the two. Obviously there is a valid moment expansion. The zeroeth moment is a tensor version of Gilbert damping:

s.o. s.o. (t) ∂ (t) (t ) ∂f ∂ d M eq =τ M (t) × d3 k (3.62) Q · (t) (t ) ∂s.o. (t ) ∂ M dt ∂M

LOSS TORQUE IN MAGNETIC METALS

53

The odd moments correspond to non-dissipative shifts in the torque; only even moments contribute to the damping torque. Moments beyond the zeroeth involve higher time derivatives of M, in a manner similar to the case of damping by magnetostriction. For comparison, in the case of an applied electric ﬁeld E, ∂feq ∂φ φ + = −eE(t) · ∂t τ ∂k ∂(k) ∂feq = −eE(t) · ∂k ∂(k)

(3.63)

so that the current is

t 2 k) k) Ω ∂( ∂f ∂( e 0 eq ) j = dt e−(t−t )/τ d3 k · E(t m 8π 3 ∂(k) ∂k ∂k −∞ and the zeroeth moment is 2 j(t) = τ e Ω0 m 8π 3

∂feq ∂(k) d k ∂(k) ∂k 3

(3.64)

∂(k) · E(t) ∂k

(3.65)

So both the electrical conductivity and the damping torque are proportional to the relaxation time. It may seem strange that the damping torque is smaller the shorter the relaxation time. The reason is that Q is the torque at a given rate of . Similarly, in the case of the applied electric ﬁeld, in a given electric change of M ﬁeld, the dissipation rate is proportional to the conductivity. Note that, as usual, the derivative of feq in the k-space integral causes all the other k-dependent parts of the integrand to be conﬁned to one of the two Fermi surfaces (k) ± AM = µ. 3.3.2.2 Scattering by phonons At normal temperatures, electron-phonon scattering dominates, and presents a case in which a moment expansion needs examination. To discuss this case, we ﬁrst note that even though spin-orbit coupling does allow spin-ﬂip scattering with emission or absorption of a circularly polarized acoustic phonon, we can rule out this process in any material for which AM kB Θ, where Θ is the Debye temperature. Assuming this inequality to be satisﬁed, as it is in standard cases, the scatterings will be conﬁned to spin-conserving transitions within one or other of the two Fermi seas, just as for scattering by nonmagnetic impurities. To ﬁnd the response to a small applied force, one writes f = feq + φ to obtain a linear equation for φ, just as in the case of the simple relaxation equation discussed in Section 3.2.1, but there the resemblance ends. Whereas in that simple case, the linearized collision operator ∂/∂t + 1/τ was independent of the character of the perturbing driving term, this is no longer true when the relaxation is caused by phonons (or any other system with its own dynamics). In the relaxation time approximation above, it was clear that the deviation φ had exactly the same k-dependence as the driving

54

INTRINSIC DAMPING

term. This resulted “automatically”, simply because τ was totally structureless. In the case of relaxation by phonons, it must still be true that φ and the driving term must have the same k-dependence. However, whereas in the absence of a drive, it is possible to single out the direction of the k-vector as polar axis in doing the integration over the direction of the electron-phonon momentum transfer q, the driving term now involves its own preferred directions: the anisotropy axes in the magnetic case, and the direction of the E-ﬁeld in the conductivity problem. Therefore integrations over the q-direction becomes more involved. In our problem, the linearized version of the Boltzmann equation, including our magnetic driving term reads ⎡

(nq + 1)δ − ((1 − feq (˜k )) φk+q − feq (˜k+q )φk )

⎢ −(nq + 1) δ + ((1 − feq (˜k+q )) φk − φk+q feq (˜k )) ⎢ 2π |Vq |2 ⎢ ∂φk ⎢ +nq δ + ((1 − feq (˜k )) φk+q − feq (˜k+q )φk ) = ⎢ ∂t q N ⎢ −nq δ − ((1 − feq (˜k+q )) φk − φk+q feq (˜k )) ⎢ ⎣ ∂feq (˜k ) dM − · dt ∂M

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.66)

with δ ± = δ(˜k+q − ˜k ± ωq ) and Vq the electron-phonon matrix element. We are concerned with long wavelength phonons, so that ωq kB T, and that q < 2kFermi . Therefore it is appropriate to expand the collision expression in powers of q (or, equivalently, ωq ), and to classify the successive terms with respect to the physical eﬀects they cause. At large T , the occupation number nq becomes practically kB T /ωq , and nq + 1 becomes eωq /kB T kB T /ωq . At q = 0, i.e. ωq = 0, the terms multiplying these phonon occupation numbers in the collision sum cancel exactly. The expansion of these terms thus starts with q (or ωq ), and can be classiﬁed as follows: a. q set equal to zero in φk+q but not elsewhere. This makes the coeﬃcients of the occupation numbers proportional to φk . Assuming that its coeﬃcient (whose lowest power will be ωq ) is negative, this gives an ordinary relaxation term, with reciprocal relaxation term equal to that coeﬃcient. b. q not set equal to zero in φk+q , and not elsewhere. For small q this generates new terms of the form A(ωq , k)q·(∂φk /∂k) + B(ωq , k)(q · ∂/(∂k))2 φk + o(q 3 ). This is weighted by the appropriate phonon occupation nunmbers and summed over k) = A(ωq , k)q can be written q. If it turns out that this weighted sum A( 2 ∂/∂k B(ω q , k)q , we2 obtain a diﬀusion on the energy surface, with diﬀusion coeﬃcient B(ωq , k)q c. If (b) does not quite hold, then there will in addition be a kinematic “wave propagation” term on the energy surface, proportional to ∂φk /∂k, with a velocity depending on k. At high temperatures, the ordinary relaxation term described in (a) will dominate, and a moment expansion of the response to the driving term becomes

LOSS TORQUE IN MAGNETIC METALS

55

possible, even if (b) and (c) are taken into account. (b) and (c) cause only modest changes in the successive moments. The coeﬃcient of φk is # " nq (eωq /kB T − 1) (δ − − δ + )(1 − feq (˜k ) − feq (˜k+q ))

(3.67)

= (δ − − δ + ) (1 − feq (˜k ) − feq (˜k+q )) + o(ωq /kB T ) But δ − −δ + = δ(˜k+q −˜k −ωq )−δ(˜k+q −˜k +ωq ) = −2ωq ∂δ(˜k+q −˜k )/∂˜k+q , so that (δ − − δ + ) (1 − feq (˜k ) − feq (˜k+q )) = 2ωq =−

∂feq (˜k ) δ(˜k ) ∂˜k

(3.68)

e˜k /(kB T ) 2ωq δ(˜k ) kB T e˜k /(kB T ) + 1 2

Thus the reciprocal relaxation time of φk is 1 πδ(˜k ) |Vq |2 2ωq = τ (˜k ) 2 N kB T q =

Ω0 |V |2 δ(˜k ) 2πc3 kB T

kB Θ/

(3.69)

ωq3 dωq

0

Ω0 |V |2 (kB Θ)4 δ(˜k ) = 2πc3 kB T 4 where Vq , assumed to vary slowly with q, has been replaced by a constant V. In the presence of the driving term, the equation for φk has the form (t) ∂φk ∂feq ∂s.o. (t) dM φk = − s.o. · + ∂t τk ∂ dt ∂ M (t)

(3.70)

For small spin-orbit coupling, we may replace ∂feq /∂s.o. by ∂feq /∂. For simplicity, consider the uniaxial case, and assume to be a function of |k| only. Then the right hand side of eqn (3.70), using eqn (3.54), takes the form −2

dM3 ∂feq Λ(k)k32 M3 ∂ dt

and Λ(k) can be written as Λ() so that φk is necessarily of the form φk = k32 Φ(, t)

∂feq ∂

(3.71)

56

INTRINSIC DAMPING

where Φ varies relatively slowly with , and may be replaced by Φ(0, t) where it occurs, multiplied by the rate of change of feq . So φ(0, t) satisﬁes the equation ∂feq ∂Φ(0, t) Φ(0, t) + ∂ ∂t τk

∂feq ∂

= −2 =0

dM3 ∂feq Λ()M3 ∂ dt

(3.72)

Integrating over all , remembering the δ-function in 1/τk , and using (∂Feq /∂)=0 = −1/(4kB T ), we get 1 dM3 ∂Φ(0, t) + Φ(0, t) = 2Λ(0) M3 ∂t τ dt

(3.73)

Here, 1/τ = Ω0 |V |2 (kB Θ)4 /[8π c3 (kB T )2 4 ]. The solution is

t

Φ(0, t) = 2Λ(0) −∞

dt e−(t−t )/τ M3 (t )

dM3 (t ) dt

(3.74)

The expectation value of the ﬁeld has only a 3-component, equal to

∂s.o. = d kφk ∂M3 3

=

d3 kk32 Φ(0, t)

∂feq ∂s.o. ∂ ∂M3 (t)

(3.75)

1 (Λ(0))2 (kFermi )4 /(kB T )M3 (t) 4 t dM3 (t ) dt e−(t−t )/τ M3 (t ) dt × dt −∞

This allows a moment expansion, and the zeroeth moment is τ dM3 (t) 1 (Λ(0))2 (kFermi )4 (M3 (t))2 4 (kB T ) dt

(3.76)

This is proportional to kB T, just as in the case of the relaxation approximation. The ultimate reason is that, because the distinction between nq and nq + 1 is negligible at high temperatures, the scattering of electrons is practically elastic, just as in the case of scattering at ﬁxed impurities. For uniaxial anisotropy, the loss torque is evidently purely transverse, since the eﬀective ﬁeld is entirely in the 3-direction. Calculations under (b) and (c) become very involved, and may not be worthwhile to consider in detail. If (b) is included, a grossly oversimpliﬁed equation for φk is ∂φk ∂ 2 φk φk =D − Zk (t) + ∂t τk ∂2k

(3.77)

FLUCTUATIONS IN MEDIUM PROPERTIES

57

where D is a diﬀusion coeﬃcient, and Zk (t) the driving term. The solution is φk ≈ −

3

t

d k

−∞

e−(t−t )/τk

−(k −k )2 /(4D(t−t )

4πD(t − t )

Zk (t )

(3.78)

and evidently allows a moment expansion. Note that in the corresponding calculation of phonon eﬀects on electrical conductivity, similar diﬀusive modiﬁcations of straight exponential decay will occur, but are rarely, if ever, discussed, presumably because they are “small”, at least at high temperatures. The old literature, in particular Wilson (1953) in Chapter 9 of that book, explicitly discards terms that, in the time domain, would appear as diﬀusion and/or kinematic wave motion on the Fermi surface. This matter has, however, been considered by Lifshitz and Pitaevski (1978). At very low temperatures, relaxation equations for the nq due to scattering of phonons by electrons must be taken into account. One of the resultant eﬀects is phonon drag, tending to conserve total crystal momentum, leaving aside Umklapp processes. But even if the metal has “open” Fermi surfaces traversing many Brioullin zones, making it unnecessary to consider Umklapp, there will also be relaxation eﬀects. In principle, one can solve the transport equation for the nq in terms of the fk , and substitute the result in the equations for the fk . This will introduce retardation eﬀects, and the existence of moments is not guaranteed. There are two reasons why one might be hesitant to explore this regime: On the one hand, very pure material is needed to avoid the eﬀects being masked by ordinary impurity scattering. On the other hand, in many metals, the resistivity is linear in T, perhaps undeservedly, even a long way below the Debye temperature Θ. Judging by the similarities of the electrical conductivity and the loss torque calculations, the latter probably also works far below Θ.

3.4

Fluctuations in medium properties

Certain themodynamic ﬂuctuations in the magnetic attributes of the medium can also lead to a loss-torque. In this section we consider as an example a particular kind of valence ﬂuctuations of the ionic constituents of the medium. In the case of magnetic insulators the simplest ﬂuctuation comes about by thermally activated hopping of electrons from one type of ion to another, thereby changing the valence of the two types of sites, and thereby the magnetic anisotropy of the sites. This happens particularly in the case of uncompensated ions introduced substitutionally or interstitially into a host lattice. Some 50 years ago, a case of this type was investigated experimentally by Galt (1954) and Wijn and van der Heide (1950) and theoretically by Clogston (1955). The material considered was nickel ferrite, with small amounts of divalent iron replacing the nickel on the octahedral sites. The divalent nickel normally occcupying these sites has negligible anisotropy energy, since its spin is 1, which, in a ﬁeld of octahedral

58

INTRINSIC DAMPING

symmetry, cannot give anisotropy. Fe++ , on the other hand, has spin 2, and will have anisotropy. If Ni (t) is the number of electrons (or holes) occupying site i at time t, and i (M (t)) is the energy of an electron (or hole) at that site, partly composed of (t)). This assumes that magnetic anisotropy, the total energy is Ni (t) i (M i is not also a function of Ni . But at least, in the typical case of nickel ferrite, this does not matter, since the actual values of Ni involved are zero or one. An initial arbitrary distribution of the Ni will decay to their equilibrium distribution by thermal processes not particularly related to the magnetization ﬁeld. The equilibrium distribution may be the Boltzmann distribution Ni∞ ∝ )/kB T ). It is assumed that, even when M is not in its equilibrium exp(−i (M state and is a function of time, the occupation numbers will still tend to the (t))/kB T ), but the processes driving them there distribution Ni∞ ∝ exp(−i (M (t). Their time evolution is assumed to obey will not appreciably depend on M a set of master equations. For simplicity, Clogston replaces this set by a single equation for all the Ni : Ni∞ − Ni dNi = dt τ

(3.79)

with τ their common relaxation time. This appears to be a good approximation if there are only a few diﬀerent types of site. (When there are just two types, this form is exact.) At any stage of the process, the torque on the magnetic system is =M × Q

Ni

i

∂i ∂M

(3.80)

) = −kB T ( Ni∞ /∂ M )= and it is zero in equilibrium, because Ni∞ (∂i /∂ M 0. So the torque on the system away from equilibrium is =M × Q

i

ni

∂i ∂M

(3.81)

where ni = Ni − Ni∞ t dNi∞ (t ) −(t−t )/τ dt e =− dt −∞

(3.82)

is a slightly unconventional form of the solution of eqn (3.79). (Here, Ni∞ (t ) (t ))/kB T ).) stands for exp(−i (M

FLUCTUATIONS IN MEDIUM PROPERTIES

59

Substituting this in (3.81), and exhibiting all time dependences gives t 1 M (t) × dt [e−(t−t )/τ Q= kB T −∞ hi (t) hi (t ) · dM (t )/dt Ni∞ (t )]

(3.83)

i

where hi (t) = hi (M (t)) =

∂i (t) ∂M

(3.84)

are local anisotropy ﬁelds. Thanks to the exponential decay factor in the integral in eqn (3.83), a moment expansion is possible here. The zeroeth moment (t)/dt, but it is more compliresembles the Gilbert form in that it involves dM cated, since the anisotropy ﬁelds strongly depend on magnetization direction. For example, one of the model cases that Clogston considers in his work has three inequivalent sites with i = καi2 , with i = 1, 2, 3, where the αi are the , and κ is a constant. If it is assumed that Q is small three direction cosines of M compared with the torque M × H in the eﬀective magnetic ﬁeld H along the 3-direction, the Gilbert-like form (3.83) can be replaced by the Landau-Lifshitz × H. In terms of the direction (t )/dt in (3.83) by M like form, replacing dM cosines, the equations of motion then take on the somewhat more compact form α˙ 1 = α2 H − µα2 α3

t

−∞

α˙ 2 = −α1 H − µα3 α1 α˙ 3 = 2µα1 α2

t

dt e−(t−t )/τ α1 (t )α2 (t )

t

−∞

(3.85)

dt e−(t−t )/τ α1 (t )α2 (t )

−∞

dt e−(t−t )/τ α1 (t )α2 (t )

where µ=H

< Ni > κ2 kB T

with all occupation numbers replaced by a single common average for simplicity. Breaking oﬀ the moment expansion at the zeroeth moment gives α˙ 1 = α2 H − τ µα1 α22 α3 α˙ 2 = −α1 H − α˙ 3 = 2τ µα12 α22

τ µα12

α2 α3

(3.86)

60

INTRINSIC DAMPING

If one were to replace both α12 and α22 in these equations by the same constant average value, one would arrive at the small-motion result familiar from ferromagnetic resonance. For large motions, such as encountered in magnetization reversal, all direction cosines must be allowed to vary. Suppose that the reversal is induced by sudden application of a magnetic ﬁeld step in a direction opposite to the initial magnetization. The reversal’s progress according to the usual form of the Landau-Lifshitz equations is more gentle than predicted by eqns (3.86). (For analytic solutions of these equations, and the resulting pictures, see Suhl, 2001.) The LL equation results in a more gentle switch than equations (3.86), because the squares of the transverse direction cosines in the ﬁrst two equations become substantial only after the precession has built up to a substantial value. Once it has done so, the switch proceeds rapidly. In conclusion, we note that the moment expansion proceeds in powers of τ. Thus, for very slow valence ﬂuctuates even more ﬂuctuations the expansion converges poorly, unless M slowly. The valence ﬂuctuations discussed so far are of an incoherent kind in that no quantum-mechanical phase relations between the diﬀerent valencies is assumed to exist. Valence exchange is probably a better term for the physics considered in this section. In general, valence ﬂuctuations, particularly at low temperatures, involve a marked phase coherence. (Examples are some of the manganites, and certain rare earth compounds.) A serious evaluation of the impact of such coher alone is well beyond the scope ence on the form of the dynamic equations of M of this book, and must await considerably deeper understanding of ﬂuctuating valence in general.

3.5

Relaxation due to weakly coupled magnetic impurities

The classic case here is that of magnetic rare earth impurities in yttrium iron garnet. The magnetization of the ferric ions see the eﬀective spins of the impurities via a moderately weak exchange coupling. The latter are coupled to phonons by spin-orbit interaction and relax relatively rapidly, except at low temperatures. will relax via the impurities. This process, which So the magnetization ﬁeld M has a certain resemblance to the one discussed in the previous section, makes a large contribution to the ferromagnetic line width. A simple description of the process in purely phenomenological terms and in the limit of small motions, is given in Haas and Callen (1963), in the section entitled “Other magnon-phonon processes”, and may apply in some limiting cases. This description assumes the impurities to give rise to a saturated magnetization ﬁeld with its own (LandauLifshitz) damping. That magnetization is exchange coupled to the magnetization of the host. As the result of that coupling, the damping of the auxilﬁeld M . This straightforward iary ﬁeld makes its appearance in the damping term of M approach is rather unconvincing in the case of dilute impurities which cannot very well form their own fully magnetized sublattice. We here consider only the

RELAXATION DUE TO WEAKLY COUPLED MAGNETIC IMPURITIES 61

dilute case. For an early microscopic treatment of this problem, see de Gennes et al. (1959); among more recent less detailed papers is that of Safanov and Bertram (2000a), and references therein. The impurity spins will be treated quantummechanically, while the magnetization ﬁeld will still be treated classically, and there will be no restriction to small motions. Such a semi-classical treatment is possible for the same reason that a semi-classical treatment of the electromagnetic ﬁeld interacting with matter is possible. In most practical problems of optics, it is unnecessary to quantize the electromagnetic ﬁeld in order to determine how it is attenuated by the medium, even though the medium is treated quantum-mechanically. Similarly here, there is no good reason for quantizing the magnetization ﬁeld of the host material. The impurity ions, such as transition metal ions in a magnetic host metal, or magnetic rare earth ions in yttrium iron garnet and others, are subject to the crystal ﬁeld, to anisotropic exchange interaction with the magnetization ﬁeld, and to spin-orbit coupling. The latter coupling, in so far as it includes spin-other-orbit interaction, results in coupling to phonons, which we assume to provide an eﬀective equilibration mechanism. The exchange interaction with the host ions will split any Kramers degeneracy of a ground state doublet left behind by crystal ﬁeld splitting of an outer d-shell. (A typical case is the Ti+++ ion which, in isolation, is in a 2 D state.) But whatever the origin of the two lowest states, we shall represent them by a spin one-half, assuming that they are magnetically active. 3.5.1

Slow relaxation

In the literature, a distinction is made between so-called slow relaxing and fast relaxing scenarios. The “slow” process bears a resemblance to the Clogston model discussed in Section 3.3, and magnetic resonance data on systems with impurity ions are sometimes interpreted in terms of that model. Here we consider the impurity case on its own terms. The impurity spins have energy levels determined by the crystal ﬁeld, and by generally anisotropic exchange coupling to the host with magnetization. One speaks of slow relaxation if the rate of variation of M time is too slow to induce ordinary Fermi “golden rule” transitions between these energy levels. That is to say, d log Mi /dt, for the relevant components Mi , is assumed to be much less than the splitting of the magnetic impurity levels due to exchange coupling to the host magnetization. Then the most important eﬀect is to modulate the spacing of the split levels. The very rapid equilibration of M by phonons then senses that slow modulation as a change in the phonon-caused transition probabilities with time. Under these conditions, it is suﬃcient to treat the dynamics of the impurities using a simple kinetic master equation for the occupation numbers of their levels. We here follow in spirit the treatment of this problem by van Vleck and Orbach (1963). For simplicity, consider just one pair of levels, described by ﬁctitious spin one-half. Any particular impurity located “next to” position x is in generally anisotropic exchange interaction s A x, t) with the host magnetization. The master equations for the i ij Mj ( i,j

62

INTRINSIC DAMPING

“up” and “down” occupation numbers of the two spin levels are n˙ ↑ = w↓↑ n↓ − w↑↓ n↑

(3.87)

n˙ ↓ = w↑↓ n↑ − w↓↑ n↓ “up” and “down” denoting 3 directions parallel and antiparallel to the exchange ﬁeld with components j=1 Aij Mj (x, t) acting on the impurity spin. With the given anisotropic interaction, the two slowly modulated energies are $ ⎛ ⎞2 % % 3 3 1% ⎝ ± (x, t) = ± & Akj Mj (x, t)⎠ 2 j=1

(3.88)

k=1

as is seen by taking the direction of the vector j Aij Mj (x, t) as quantization axis. The phonons rapidly establish thermal equilibrium between these two levels, so that detailed balance requires w↑↓ = exp w↓↑

(+ − − ) kB T

,

(3.89)

is time independent. Assuming this relation to hold also for suﬃciently if M slow time dependence, the occupation numbers will change slowly according to equations (3.87). It will be necessary to normalize the w’s so that w↓↑ =

w0 exp(−β+ ) exp(−β+ ) + exp(−β− )

w↑↓ =

w0 exp(−β− ) exp(−β+ ) + exp(−β− )

(3.90)

where β = 1/kB T , and where w0 /2 is the rate of equilibration in the absence . The normalization is necessary to ensure that, even at of any coupling to M T = 0, both transition probabilities remain ﬁnite. Since n↑ + n↓ = 1, equation (3.80) may be replaced by a single equation n˙ ↑ − n˙ ↓ = (w↓↑ − w↑↓ ) − (w↑↓ + w↓↑ )(n↑ − n↓ ) = −w0 tanh(β(x, t)) − w0 (n↑ − n↓ ) where (t) is the magnitude of either ± (t), and β = 1/kB T. The solution is (n↑ − n↓ ) = −w0

t

−∞

dt e−w0 (t−t ) tanh(β(x, t ))

(3.91)

RELAXATION DUE TO WEAKLY COUPLED MAGNETIC IMPURITIES 63

The total magnetization ﬁeld (host+impurities) is tot (x, t) = M (x, t) + 1 u(x, t)(n↑ (x, t) − n↓ (x, t)) M 2

(3.92)

where u(x, t) is a unit vector in the direction of the exchange ﬁeld with com3 ponents j=1 Aij Mj (x, t). Equation (3.91) is the solution for the motion of the is (besides the usual terms impurity magnetization. The equation of motion of M due to applied and anisotropy ﬁeld) 1 dM = M × u(x, t) (n↑ (x, t) − n↓ (x, t)) dt 2

(3.93)

(We have assumed here that every site x has an impurity next to it, each coupled to the host with the same anisotropic exchange. More generally, Aij will itself be a function of position x, and, in particular, a random funtion if the system is a disordered alloy.) To derive the damping torque, note that (n↑ − n↓ ) given by equation (3.91), has a valid moment expansion of which the ﬁrst three terms are 1 ∂β (n↑ (x, t) − n↓ (x, t)) = − tanh(β(x, t)) + sech2 (β(x, t)) w0 ∂t ⎡ ⎤ 2 ∂ 2 β sech (β(x, t)) ∂t2 + 1 ⎣ 2 ⎦ + · · · − 2 w0 + tanh(β(x, t))sech2 (β(x, t)) ∂β ∂t (3.94) The ﬁrst term in the bracket gives only a non-dissipative torque, exerted on (x, t) by the exchange ﬁeld (along with an applied ﬁeld, if any). The second M term gives the low frequency damping torque 1 1 ∂β =M ( Q x, t) × u(x, t)sech2 (β(x, t)) 2 w0 ∂t

(3.95) ⎞ 3 3 β sech2 (β(x, t)) ∂M ( x , t) j (x, t) × ⎝ ⎠ = M u(x, t) Akj Akj Mj (x, t) 2( x, t)w0 ∂t k=1 ⎛

j,j =1

, and the Note that, in the case of purely isotropic exchange, u is along M damping torque vanishes, in accordance with the conclusion of van Vleck and Orbach (1963). The graph of torque versus temperature has the characteristic hump with a peak at kB T approximately equal to the magnitude of the prevailing exchange ﬁeld, but the precise location of the peak will depend on the direction of the exchange ﬁeld acting on the impurity “next to” x. That peak will of course be broadened if either the magnitude and/or the direction is diﬀerent for diﬀerent impurities. In as much as our treatment has eﬀectively been conﬁned

64

INTRINSIC DAMPING

to magnetic samples small enough so that spatially non-uniform excitations of are excluded, the total loss torque to be used in the equation of motion for M will simply be the sum of the loss torques over the Aij of all the that uniform M impurities. We now examine the limits of validity of the above result. It is valid only for a , henceforth denoted by ωM . Obviously ωM suﬃciently slow rate of variation of M must be be small enough so that no ordinary “golden rule” transitions between the two levels of the impurity spin can occur, requiring ωM + − − . At low temperatures, another limitation arises from a breakdown of the master equation. As shown by van Hove (1962), deriving a simple master equation from the full equation for the statistical operator, or density matrix, by perturbation theory is valid only in the limit V → 0 and t → ∞ in such a way that V 2 t/2 remains ﬁnite, and possibly small. In our problem, V is the spin-orbit coupling energy Vs.o. of the phonons to the impurity spin, but its square must be multiplied by the number of phonons available (at low temperatures) for inducing transitions between the two impurity levels, in the present case t ≈ 1/ωM . Therefore validity suﬃciently slow so of the master equations (3.91) is conﬁned to variations of M 2 −β(+ −−) 2 that (ωM ) e Vs.o. . When this condition fails, the full density matrix must be treated non-perturbatively.

3.5.2

Corrections to the adiabatic limit

Evidently, the notion of slow relaxation is based on the adiabatic approximation (t). At ordinary temperatures, to the impurity levels, which are modulated by M and slowly varying M (t), this adiabatic approximation agrees fairly well with observation. For the approximation to hold at a given rate of variation ω, the exchange splitting must be much greater than ω. Thus, at ω equal to 28 GHz, the exchange ﬁeld acting on the impurity spin must be much greater than 10,000 oersteds. If magnetic recording rates ever reach such large values, the adiabatic results must be corrected. (There will always be some correction, even at modest (t) will have a Fourier component values of ω, since any general variation of M + − , the problem that matches the level splitting − . For small motions of M can be treated in linear response theory, without concerns about relative rates of variation. The work of de Gennes et al. (1959) presents an essentially complete solution in this limit.) There is an extensive literature on corrections to the adiabatic limit, dating back to Zener (1932), Landau (1932), Dykhne (1962), Davis and Pechukas (1976), and continuing to more recent times in works by Berry (1989, and references therein), Marinov and Strahov (2001) and many others. The subject is of special interest to chemists, in the discussion of corrections to the Born-Oppenheimer approximation needed to describe chemical reactions. Typically, one seeks to determine how a slow variation of a parameter involved in the Hamiltonian induces transitions between two energy levels of a system that never cross as

RELAXATION DUE TO WEAKLY COUPLED MAGNETIC IMPURITIES 65

that parameter varies in time. Here we give only a brief description of the result, using a particular case of the exchange energy, eqn (3.88), namely ± (x, t) = ± a 2 M12 (t) + b2 M22 (t) + c2 M32 (t).

(3.96)

Evidently + − − is never zero, so the levels never cross, at least not for real values of t. So there is no way to get ﬁnite overlap of the adiabatic wavefunction t + i ψ+ = e− 0 (t )dt |+ of the impurity spin with the conjugate wavefunction t − i ∗ ψ− = e 0 (t )dt −| of the reversed spin for real values of t. So there is no transition. However, the form of the approximate solutions do not change if the time integrations in the exponent run along contours in the complex t− have no singularities there. In the complex plane, assuming the components of M t− plane, , and the original Hamiltonian, have branch points at the zeroes of a2 M12 (t) + b2 M22 (t) + c2 M32 (t), which occur in complex conjugate pairs. Consider the particular pair ξ ± iτ with imaginary parts closest to the real axis and ξ > 0. For simplicity, ignore all other pairs for now. Then all relevant functions will be analytic in the t-plane cut from ξ − iτ to ξ + iτ . Starting with the wavefunction |+ at t = 0 on the real axis, run the integration along the real axis up to t = ξ − 0 to the left of the cut. Then run it downwards to ξ − 0 − iτ along the left of the cut, then around the singularity and upwards to t = ξ + 0. to the right of the cut. Finally integrate along the real axis up to t = t. Evidently, (t ) has an imaginary jump across the cut precisely equal to i times the diﬀerence in up- and down-spin energies, whereas the real part of is continuous all the way from zero to t ; therefore the real part gives only the usual adiabatic exponential phase factor. But the integrals along the edges of the cut iτ do not cancel, and give the decay factor exp − 2 Im 0 (t)dt of the |+ state. Since it can decay only into the |− state, this factor must also be the transition amplitude from |+ to |−. The nonadiabatic part of the transition probability is therefore

iτ 4 exp − Im (t)dt 0

(3.97)

unrestrained by the condition of ﬁxed M 2 , this is still not complete. There For M may be an additional, so-called “geometric” contribution independent of and of 2 ﬁxed, the geometric contribution cannot all features of the dynamics. With M has only arise; three independent degrees of freedom would be needed, and M two. Both the uncorrected adiabatic and the nonadiabatic correction (3.97) are reminiscent of the semiclassical approximation to states of a system with slowly varying potentials (slowly varying with time in the present case). Deﬁne a long (Υ)) ≡ (Υ). time scale Υ = κt, where κ is a small number, and write = (M

66

INTRINSIC DAMPING

Then (3.97) becomes

4 Im exp − κ

iτ

(Υ)dΥ

(3.98)

0

where τ is now measured on the slow time scale. So plays a dual role: κ does not have to be particularly small if is small enough. This is quite general. If the Hamiltonian is a function of κt, and ψ(, t) is a solution of i∂ψ/∂t = H(κt)ψ(, t), then φ = ψ(κ, t) is a solution of i∂φ/∂t = H(t)φ. (For a more general statement, see Hwang and Pechukas (1977).) The transition probability (3.97) should be added to the transition probability caused by spinorbit coupling of the impurity. To see if it may be neglected, we estimate it for a precesses in the equatorial plane, with particularly simple case. Suppose that M M1 = M cos ωt, M2 = M sin ωt, M3 = 0. The zeroes of are at tan2 ωt = −(a/b)2

(3.99)

and, assuming b > a, the ones with the smallest imaginary parts are at t = ± iτ = ± i ω −1 arctanh(a/b)4 . Then

iτ

iτ

(t)dt = M 0

0

=

iM b ω

a2 − (a2 − b2 ) sin2 ωtdt

arctanh|a/b|

(3.100)

(a/b)2 − ((a/b)2 − 1) sinh2 xdx

0

Except when a is almost equal to b, the integral varies very slowly with a/b, around a value of about 0.3. Therefore, the transition rate is approximately

Mb exp −1.2 (3.101) ω M b is of order of the exchange coupling. As a approches b to within about 10%, the integral in eqn (3.100) becomes very large, so that the transition rate becomes extremely small. This shows that the more “isotropic” the motion of , the smaller the transition rate. Finally, we determine how large ω must be M in order to be comparable with the transition rate due to spin-orbit coupling. The transition probability due to spin-orbit coupling is approximately s.o /(M b), provided this ratio is less than 1. So the rate (3.94) becomes comparable with the latter for frequencies such that

Mb exp −1.2 (3.102) ≈ s.o /(M b) ω 4 Roots with larger imaginary parts only give small additive corrections to the rate discussed here.

APPENDIX 3A. INCLUSION OF DISPLACEMENT CURRENT

67

or ω ≈ 1.2(M b/) ln(M b/s.o. ). As one might expect, the nonadiabatic correction to the adiabatic result becomes dominant only well above the exchange frequency (M b/). None of this aﬀects the temperature dependence of the ﬁnal transition rate: the actual observable transition rate involves the densities of initial and ﬁnal continuum of states, here due entirely to the phonon bath. The only way the nonadiabatic rate enters the ﬁnal result is through the factor w0 in formula (3.91), which is now the sum of the spin-orbit induced and the nonadi via at least one abatic probabilities. So w0 will acquire a part depending on M complex zero of the exchange expression. This makes it essentially impossible to ; to know the position write down the damping torque as a simple function of M , whose form of that zero, one would have to solve the equations of motion of M involves that zero. The only hope is then to iterate: 1. make a guess at the posi , using that guess, and 3. from tions of that zero, 2. solve for the motion of M the solution ﬁnd the new (hopefully improved) position of the zero, and continue this process in the hope that a ﬁnal ﬁxed point will be reached. This would seem to be a case in which a numerical simulation of the system is the only feasible strategy.

3.6

Appendix 3A. Inclusion of displacement current in Section 3.3.1 is so slow as to result For many applications, the timescale of change of M in a skin depth large compared with the size of the magnetic specimen. Then there is no need for higher moments, since these are multiplied by higher (and . For high recording rates, or for very therefore smaller) time derivatives of M well-conducting material, this is not the case, and a complete Green function based on unabbreviated Maxwell equations is needed to ensure the existence of higher moments. This also allows a more precise criterion for the rates of change at which retention of higher moments is indicated. Maxwell’s equations ˙ ∇ × h = e/c ˙ + 4πσe/c, ∇ × e = −h/c, lead to an equation for the Green function G(x − x , t − t ): ∇2 G −

1 ∂ 2 G 4πσ ∂G = δ(x − x )δ(t − t ), t > t − 2 c2 ∂t2 c ∂t = 0, t < t

(3.103)

(For simplicity, dielectric constant and magnetic permeability have been taken equal to 1. The inequalities imposed on the times ensure that eﬀects are not observed before they are produced.) The equation for the spatial Fourier transform 1 ˜ k, t) = d3 x eik·x G(x, t) G( (2π)3/2

68

INTRINSIC DAMPING

˜ can be written of G

∂2 ∂ ˜ G(k, t) = δ(t ), t > 0 c2 k 2 + 2 + 4πσ ∂t ∂t

(3.104)

= 0, t < 0. The general solutions of the corresponding homogeneous equation are Aeλ−t and Beλ+t where λ± (k) = −2πσ ± 4π 2 σ 2 − c2 k 2 . Then the required particular solution of equation (3.104) is λ+ t − eλ− t ˜ k, t) = e , t>0 G( λ+ − λ−

= 0, t < 0. (see, for example, Zwillinger, 1989). Returning to conﬁguration space gives, for t > t (which will henceforth be implied), )

e−2πσ(t−t G( x− x ,t − t ) = (2π)3/2

−2πσ(t−t )

=

4πe (2π)3 R

√ 3

d ke

−i k·( x− x )

e

4π 2 σ 2 −c2 k2 (t−t )

√

− e−

√

4π 2 σ 2 −c2 k2 (t−t )

2 4π 2 σ 2 − c2 k2 √ √ 2 2 2 2 2 2 2 2 ∞ e 4π σ −c k (t−t ) − e− 4π σ −c k (t−t ) dkk sin(kR) , √ 2 4π 2 σ 2 − c2 k2 0 (3.105)

where R = |x − x |. Two requirements must now be imposed: (1) for very large σ, the result should reduce to the diﬀusion Green function, and (2) as σ tends to zero, the result should reduce to the retarded solution. Both requirements will be met if we keep only the ﬁrst term in the numerator of the integrand. For 2πσ >> ck, expand the square root to ﬁrst order in k 2 /σ. The integration over k then delivers the diﬀusion form. The integrand is even in k, so the limits of the integral may be extended to −∞. Writing sin kR in exponential form, it is seen that as σ tends to zero, only the part with negative exponent gives the retarded wave, the only one of interest here. The second term in the numerator of (3.105) does not meet requirement 1. So we ﬁnally keep only √ 2 2 2 2 2iπe−2πσ(t−t ) ∞ e−{ikR− 4π σ −c k (t−t )} √ G(x − x , t − t ) = − kdk (2π)3 R 4π 2 σ 2 − c2 k 2 0 (3.106) Well within and well outside the light cone this integral is conveniently evaluated by the saddle point method in the complex k–plane. The saddle point is at ' 1 , (3.107) ck = σR R2 − c2 τ 2

APPENDIX 3A. INCLUSION OF DISPLACEMENT CURRENT

69

where τ = t − t . Well within the light cone ck becomes −iσR/cτ + iσR3 / (2c2 τ 3 ) + . . . , and substitution of this in equation 3.106 gives a leading term of diﬀusion form. Well outside the light cone, ck is approximately equal to σ(1 + c2 τ 2 /R2 + . . .). Substitution in 3.106 then gives a steadily decaying oscillation. Thus a moment expansion becomes possible. For practical purposes, one may assume the diﬀusion form of G to be valid up to R = cτ , and to calculate moments using the truncated expression, with G = 0 outside the light cone.

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4 FLUCTUATIONS 4.1

Introduction

In equilibrium statistical mechanics the system of interest, call it A, is assumed to be in weak contact with a much larger system, B, the ‘bath’. The total phase space accessible to the degrees of freedom of the total system A+B is taken to be constant, equal to Γ, say, and equilibrium is, by def inition, the state in which Γ is a maximum. But A can exchange energy, momentum, etc., with B. As a result, the smaller phase space, γ, say, covered by the degrees of freedom of A can ﬂuctuate, but only in such a way as to leave Γ unchanged. If, for a given value of γ, the phase space accessible to B is Γγ , we must have Γ = γΓγ , since, for any one of the conﬁgurations within γ, we can have B located anywhere in its diminshed space Γγ . If absolutely nothing else is known about a system, a reasonable deﬁnition of the probability of the state of a system is the size of its accessible phase space. Since phase space volume must be a positive quantity, it is proper to write Γ = eS and γ = es . Then Γγ = Γ/γ = eS−s is the probability of the occurrence of γ. It is further assumed that the so-called entropy S is fully determined by a complete set of conserved quantities, such as total energy, momentum, particle number etc., and that the total system A+B is completely isolated. Under these conditions, the equilibrium probability distribution of energy, momentum, particle number etc. of the subsystem A is found without need for a knowledge of the detailed dynamics of either system A or B. Processes in which a parameter of the system is changed from one conserved value to another, but reversal of the change returns the system to its original state, can be fully described by equilibrium theory. In other words, such changes imply that the equilibrium entropy depends only on the value of the conserved quantities, and not on the past histories of these quantities. However, in many situations we have irreversibility. Forces are applied to the system that result in a violation of one or other of the above listed assumptions. To evaluate fully the response to such forces, detailed dynamical solutions of the equations of motion would be needed, and these are rarely obtainable. So one settles for much less ambitious projects: to establish relations between sets of diﬀerent quantities none of which can be calculated separately. The Onsager relations are classic examples of this: phenomenological responses such as electrical conductivity, thermoelectric coeﬃcients, thermal conductivity, etc. cannot generally be calculated separately, but they obey the said relations. The ﬂuctuation-dissipation theorem is another case in point; it establishes a relation between the time dependent linear response of a quantity to a small impulse and 71

72

FLUCTUATIONS

the time autocorrelation function of the same quantity in thermal equilibrium. From a purely theoretical point of view, such relations do little or nothing to solve the basic problem, but they can be of considerable practical value. For example, the measured resistance of a circuit element will, by the ﬂuctuation dissipation theorem, determine the noise power in the element without requiring measurement. Also, a high quality, though approximate, calculation of the resistance should enable the theorem to deliver correspondingly approximate ﬂuctuation characteristics of similar quality. Frequently, the quantity of most immediate interest is the response of a system to externally imposed forces, and the related ﬂuctuations may be regarded as a reﬁnement that may or may not be important depending on context. These are cases in which there is no impairment to the system’s motion other than friction. Once the friction has been calculated, the motion of the system is ‘deterministic’ and no explicit reference to ﬂuctuations is needed. On the other hand, if the motion is impeded by barriers of any kind, ﬂuctuations have to be taken into account immediately, unless the forces driving the system are so large as to render the barriers irrelevant. For smaller forces, thermal ﬂuctuations provide the only means of surmounting any barriers. In analyzing these cases it is necessary to clearly deﬁne the ‘system of interest’, a small subsystem, A, of the total system A+B. We would like to know as much as possible about the motion of A, while knowing as little as possible about the motion of B. Two distinct cases must be considered: (1) the constituents of the bath B have essentially the same physical features as A, and (2) the constituents of B belong to a physical species other than those of A. As an example of case 1, the system of interest A may be the spatial average of the magnetization vector, the uniform mode, while B is the set of all other possible modes, the spin waves, in linear approximation. In Chapter 2, the response of the uniform mode to a small rf ﬁeld, especially the damping part, was discussed. An example of case 2 is the (x) (system A) to the strain ﬁeld coupling of the entire magnetization ﬁeld M of the underlying lattice (system B), which was discussed in Chapter 3. Case 1 may be called ‘distributive damping’, because it merely redistributes the energy extracted from the degrees of freedom of mode A among modes of B sharing the same physical origin. Case 2, on the other hand, was called intrinsic damping. As already discussed in Chapters 2 and 3, one important (but frequently overlooked) diﬀerence between cases 1 and 2 is that, in case 1, the equation of motion of A, after elimination of the degrees of freedom of B, does not preserve vector. Case 2 on the other hand does the magnitude of the resulting quasi-M preserve the magnitude of M everywhere, since none of its modes have been eliminated in the reduction to A. Another important diﬀerence comes to the fore in a discussion of magnetization reversal in the presence of anisotropy barriers. In particular, consider the reversal process for an array of weakly interacting magnetic particles, whose magnetization vectors, initially all aligned, are to be switched to the opposite direction by a suddenly applied ﬁeld. In case 1, the reservoir B is composed of the particles themselves, isolated from any contact

INTRODUCTION

73

with outside agencies. Interaction with the others allows any one of the particles to extract the energy needed to overcome its anisotropy barrier from the others. In general, this process requires the collaboration of many members of the array, especially if the barrier is high. To allow one to speak of noticeable reversal, a ﬁnite fraction of particles must have their magnetizations reversed. With weak interactions, this takes a long time. In case 2, on the other hand, each of the particles can independently extract the necessary energy from the external bath B; therefore, to reverse a ﬁnite fraction of the individual magnetizations takes not much longer than to reverse a single one. It follows that, in eﬀecting reversal in a sparse array of particles, intrinsic processes play the dominant role. In other words, it may be argued that dynamic processes ending in rearrangement of a ﬁnite fraction of a system take much longer in a microcanonic ensemble than in the canonic ensemble. This diﬀers from equilibrium statistics in which there is little diﬀerence in averages of the same quantity calculated in either ensemble. Up to this point, the objective has been to derive purely deterministic equations of motion for the magnetization ﬁeld. In Sections 3.3 and 3.4, thermal ﬂuctuations did play a role, but only in the form of thermodynamic averages, such as the occupation numbers of phonon or electronic states of the medium, calculated as equilibrium quantities consistent with a slowly varying, determin . In Section 3.2, and in the ﬁrst part of 3.3, thermal ﬂuctuations istic ﬁeld M were not involved at all. As just noted, this picture cannot very well describe the irreversible process of magnetization reversal in the presence of anisotropy barriers. We must now examine how the magnetization senses random ﬂuctuations. In Chapter 3, we eliminated the unwanted nonmagnetic degrees of freedom of the lattice (lattice strains, eddy currents, etc., system B) by solving for them in (the favored system A) and substituting the solution in the equation terms of M . The motions of B, the unwanted degrees, were considered to of motion of M be deterministic. Their contact with a thermal reservoir, call it B*, entered only through the back door in the form of damping of these unwanted degrees. However, these unwanted degrees of B will still have a ﬂuctuating part, neglected so far. B ‘sees’ the random motions of the reservoir B*, albeit through the ‘veil’ of its response function, especially through the dissipative part of the response. To keep track of these ﬂuctuations, we must eliminate B, the unwanted degrees of only after including the random terms in B inherited from freedom, in favor of M reservoir B*. Those random terms will then appear in the resulting equation of , but in ‘doubly veiled’ form (veiled once by the response function motion of M of B, and further veiled by the response of the magnetization). If B is the strain ﬁeld of the underlying lattice, and the motion of these strains does not involve activation barriers, it is safe to restrict the motions of B to low frequencies and long wavelengths. B* is then composed of the high frequency, short wavelength phonons, and the dissipative aspect of B is then distributive in origin. One may ask why one should not take into account the direct stochastic eﬀect of these phonons on the magnetization via spin-orbit coupling. Doing this would risk

74

FLUCTUATIONS

B

NB

B*

NB*

A

NB

Fig. 4.1. Reservoir B produces random ﬂuctuations NB resulting in a dissipativee eﬀect on reservoir B∗ , whose response ‘ﬁlters’ NB to produce a noise NB∗ . The degrees of freedom of B∗ , together with NB∗ are eliminated to obtain an equation of motion for A, which then still shows a trace of NB∗ . Direct coupling of B to A (dashed line) is not considered here. In the case of magnetostrictive coupling including it risks overcounting. In the case of magnetic metals it would involve direct scattering of electrons in the magnetization ﬁeld, raising self-consistency questions noted in Chapter 1.

overcounting. Spin-orbit coupling is the source of coupling to the strain ﬁeld, and if this is eliminated together with its ﬂuctuations, the latter should not be counted again. A consistent way to view this process is shown in Fig. 4.1. Mathematical descriptions of stochastic processes of physical origin involve one or other of various incarnations of the so-called ﬂuctuation-dissipation theorem. In this connection, it is worthwhile to consider an unphysical but instructive case in which the magnetization is in direct contact with a reservoir, such as black body radiation modelled by a set of harmonic oscillators representing a random magnetic ﬁeld. This brings out the general form of the relation between dissipation and random force. Sections 4.2 and 4.3 will be devoted to standard forms of that theorem, with some emphasis on magnetization dynamics.

4.2

Fluctuation-dissipation theorem

We begin with a description of a relation between the linear response of a system to a small signal and thermal ﬂuctuations in the responding variable. This relationship is exact and was established by Callen and Welton (1951) on the basis of quantum mechanics. Since it is exact, it must hold true in the classical limit also. Nevertheless, a classical derivation does require further thought. In as much as the problems considered here relate to classical aspects of the ﬂuctuation-dissipation theorem (FDT), we examine this matter further. Consider a system of particles with Hamiltonian 1 p2i + V (q1 , q2 , ....) 2m i i

(4.1)

FLUCTUATION-DISSIPATION THEOREM

75

(The label i has two components: particle label and cartesian coordinate label of that particle.) Add to this Hamiltonian a frequently encountered form of energy in an applied ﬁeld: − i qi fi (t) Here, fi (t) is a small force whose origin is of no immediate concern. The equations of motion are q˙i = pi /mi

(4.2)

p˙i = fi (t) − ∂V /∂qi As a matter of common experience, the resulting response of the system is usually a linear function of the small fi ’s. For example, let fi (t) be a small impulse ri (0)δ(t) around t = 0, changing pi (0) by an amount ∆pi (0). This change is found by integrating equations (4.2) from time −0 to time +0: ∆pi (0) = ri (0)

(4.3)

and the changes in qi and V are negligible in that time interval. If, at a later time t, the equations of motion of the unperturbed system would have had the solution pj (t, {pi (0)}, {qi (0)}); qj (t, {pi (0)}, {qi (0)}) in terms of the sets {pi (0)}, {qi (0)} of initial coordinates and momenta, the eﬀect of the perturbing pulse might be expected to change these ﬁnal variables by ∆pj =

∂pj (t) ri (0) ∂pi (0) i

∆qj =

∂qj (t) ri (0) ∂pi (0) i

(4.4)

to ﬁrst order in r. So the momentum and position responses ∆pj , ∆qj are linear in r. Suppose now that the initial probability distribution is the Boltzmann distribution ≈ exp(−H/(kB T )). Integrating by parts, the thermal averages of these responses per unit ri (0) will be ) ( ∂pj (t) 1 1 = pj (t)pi (0) ∂pi (0) kB T i mi i ) ( ∂qj (t) 1 1 = qj (t)pi (0) ∂pi (0) kB T i mi i

(4.5)

So the response is closely related to the ﬂuctuations. Usually one is more interested in response to a small signal with general time dependence. A general signal ri (t) can be written as a sum of short pulses applied at closely spaced times. Invoking the linear superposition principle, the response should be a linear sum of the responses to these applied pulses. Thus one would hope the momentum

76

FLUCTUATIONS

response of particle j to be ) ( ∂pj (t) Rj (t) = ri (t ) dt ) ∂p (t i −∞ i t 1 1 = dt pj (t)pi (t ) ri (t ) kB T −∞ m i i

t

(4.6)

Equations (4.5) and (4.6) are the classical form of the quantum FDT. But even though they have proved their worth in countless practical situations, on the face of it, the above derivation has a serious shortcoming, as pointed out by van Kampen (1971). The reason is that, in a majority of complex classical systems, a small initial deviation is exponentially ampliﬁed in time, so that a Taylor expansion in the initial disturbance is meaningless. In technical language: most complex systems have at least one positive Lyapunov exponent. A seemingly more convincing demonstration is based on Liouville’s equation ρ˙ = −Lρ, for the distribution function ρ, where the operator

∂H0,1 ∂ ∂H0,1 ∂ − L= ∂qi ∂pi ∂pi ∂qi i

To keep it simple, let H = H0 + H1 , where H1 = δ(t) i gi qi is a short signal pulse of small coupling strength g. The objective is to calculate the average of the vector pj , to the disturbance H1 , If we are willing to believe that this changes the distribution function only slightly, to ρ0 + ρ1 , where ρ0 is the equilibrium distribution (independent of time). Then, to ﬁrst order in g, ρ˙ 1 = −L0 ρ1 − L1 ρ0

∂H0,1 ∂ ∂H0,1 ∂ where L0,1 (t) = i − . The solution is ∂∂qi ∂pi ∂pi ∂qi t dt e−L0 (t−t ) L1 (t )ρ0 ρ1 = − 0

= − e−L0 t L1 (0)ρ0 This does not resolve the contradiction, since, for a chaotic system, ρ1 cannot remain small for very long. But in view of the practical success of the FDT, several investigators have mounted rescue operations, in particular Bianucci and Mannella (1996). A very clear account is found in that reference, and is brieﬂy presented in Appendix 4A. A much less appealing form of FDT results, if: (1) the Hamiltonian is not the sum of kinetic and potential energies and (2) the applied force and the measured response are general functions of the canonic variables. Contingency (1) arises for the magnetic system when its Hamiltonian is formulated in terms of polar

LANGEVIN EQUATION, AND GENERALIZED LANGEVIN EQUATION

77

coordinates, and then (2) will generally apply also. It is not diﬃcult to derive the analog of eqn (4.6). For simplicity, consider the case of one degree of freedom, with momentum p and position q. (In the case of a small, single-domain magnetic particle in polar coordinates, p = cos θ, q = φ.) If the Hamiltonian of the initial impulse is I(p(0), q(0))δ(t), and the quantity to be measured has the form F (p(t), q(t)), then the response per small unit impulse is

+ * ∂F ∂F (q, I)− + (p, I)− (I, H)− (4.7) R(t) = ∂q(t) ∂p(t)

∂q(t) ∂I(0) ∂I(0) ∂q(t) where (q, I)− is the Poisson bracket − and similarly ∂q(0) ∂p(0) ∂q(0) ∂p(0) for (p, I)− . The average is over the Boltzmann distribution of the initial variables. The right hand side of (7) is still a correlation function, but without much intuitive appeal. Also, the integration by parts that led to formulae (4.5) is now much less beneﬁcial. The FDT as given by eqn (4.6) is simply an identity that gives no clue of the function of the reservoir. In the next section a somewhat diﬀerent approach is presented that comes a little closer to illuminating that function. 4.3

Langevin equation, and generalized Langevin equation

A direct calculation of the autocorrelation function involved in the FDT is usually diﬃcult. A less direct but more helpful method starts with an equation of motion for the quantity of interest, known as the Langevin equation (LE). In its simplest form it follows the motion of a particle in its response to a random force. For example, a free particle of mass m moving in one dimension with momentum p will satisfy the equation p/m ˙ + ηp = f (t),

(4.8)

the particle being subjected to a friction force −ηp. This equation is solved exactly. In this language, the FDT emerges from the requirement that after transients have died out, the distribution function of p must be the Maxwell distribution. After disappearance of the transients, the solution is t e−η(t−t ) f (t )dt (4.9) p(t) = 0

At this point it is tempting to make some assumptions about the random force. For example, if f (t) is taken to be a Gaussian random process then so is p(t). To ﬁnd the variance of p, it is assumed that the autocorrelation function of f (t) decays rapidly, relative to the ‘slow’ time scale set by 1/η, so that we may write f (t )f (t ) = σ 2 δ(t − t ). (In other words, σ 2 is the variance of the ∆t integral 0 f (t)dt, where 1/η ∆t autocorrelation time of f (t).) Then,

78

FLUCTUATIONS

using eqn (4.9) and some familiar facts about Gaussian distributions, it is found that the variance of p2 is , 2σ2 p = (1 − e−2ηt ) 2η →

(4.10)

σ2 2η

But we know that the distribution , - of p must tend to a Maxwellian, ∝ exp −p2 /(2mkB T . Evaluation p2 with this weight function gives a common form of the FDT: η=

1 σ2 4mkB T

(4.11)

If it is possible to measure σ 2 , relation (4.11) delivers η and is much more useful than the identity (4.6). Conversely, a measurement of η delivers σ 2 , at least if f is very nearly white noise. From the point of view of exact theory, however, eqn (4.11) is only an approximation, as will become evident later in this chapter. in a model Below, we give a detailed account of this situation for the motion of M heat bath, and, upon linearizing the resulting nonlinear equations of motion of , derive the analog of η and generalizations thereof. M If the particle is moving in a general potential ﬁeld, the equations of motion become nonlinear and usually cannot be solved even without the random applied force. Also, the random force may depend on the particle variables (a case known as parametric or multiplicative noise as distinct from the case just discussed, which is called additive noise). The resulting diﬃculties in these situations are alleviated to some extent by means of the so-called Fokker-Planck equation (FPE) discussed in the next section. This is a second order partial diﬀerential equation, but it is linear, and therefore seems more tractable. (In the same way, replacing an unsolvable nonlinear classical equation of motion in an arbitrary potential by the linear Schroedinger equation brieﬂy creates the illusion of easier solvability.) Actually, certain aspects of the motion are indeed easier to discuss approximately in terms of FPE rather than in terms of the LE. At this point, we consider the magnetic case, in particular the case of a small are nonlinear. single domain particle. In this case, the equations of motion of M If we adopt a particular form of damping, for example the Gilbert form, we can write down the analog of the Langevin equation (4.8): dM ×H eﬀ + αM =M × h(t) −M dt eﬀ . and linearize it around the 3-direction, taken to be the direction of a large H The random ﬁeld h being assumed small, the term on the right, restricted to lowest order, still represents additive rather than multiplicative noise. It then

LANGEVIN EQUATION, AND GENERALIZED LANGEVIN EQUATION

79

becomes a two-component version of equation (4.8), and gives a relation between α, h2 , and the mean square equilibrium ﬂuctuation of the small transverse . As already mentioned above, there is not much hope of going components of M further. However, for the simple model of the bath discussed next, we can derive the full nonlinear generalization of the Langevin equation, and furthermore determine the form of the damping term. We write down the nonlinear equations of variables, including their coupling to the reservoir, and then motion of the M solve the equations of motion of the degrees of freedom of the reservoir. These , and will have added terms representing the free, solutions will be driven by M uncoupled motions. They are then substituted into the equations for the ‘good’ . A dissipative term will then arise, that will stand in a deﬁnite variables, M relation to the ﬂuctuations in the reservoir variables. This is the same technique was that was employed in Chapter 3, but there the ‘reservoir’ coupled to M regarded as deterministic. The random forces exerted on that reservoir by an even bigger heat bath composed of external forces, or else of ‘uninteresting’, fast internal degrees of freedom, led to deterministic dissipative terms (lattice viscosity, electric resistivity, etc.). The magnetization sees the random force only via the interposed medium. That medium derived its characteristic dissipation by contact with yet another reservoir. In principle, one could start out by excluding dissipation terms from the equation of motion of the underlying medium as well, and then use coupling to the bigger reservoir to calculate dissipation terms for both the medium and the magnetization equations in one fell swoop. But such a calculation would be much more involved. Before discussing this in more detail, let us study the simplest model by far: that of coupling to a bath of three-dimensional isotropic harmonic oscillators {ak }, with of M frequencies Ωk . The coupling is assumed linear in the ak , and, for present pur . Purely for notational convenience, we describe the oscillators poses, linear in M quantum-mechanically. The energy of the harmonic oscillators is then written a†k · ak . To keep it simple, the coupling energy p.u.v. is taken linear in k Ωk (a restriction that can be avoided): M Ec = −

· (Λkak + Λ† a† ) M k k

(4.12)

k

where the Λk ’s are three-by-three hermitian matrices. The ak ’s are dimensionless column vectors, so the Λk ’s have the dimensions of a magnetic ﬁeld. The equation is of motion of M dM ×H eﬀ (M k ) + γM × A = γM dt

(4.13)

k

eﬀ (M ) = ∂E(M )/∂ M , with E(M ) the energy expression of earlier where H † † pages, and Ak = (Λkak + Λkak ). The Heisenberg equations of motion for the

80

FLUCTUATIONS

oscillators are found from the commutator relation (ai ,

j

Λjm a†j )− = Λim . Thus

dak / = iΩkak − iΛ†k M dt

(4.14)

Integrating these equations gives ak (t) = −i

t

−∞

(t )/)eiΩk (t−t ) + ak (0)eiΩk t dt (Λ†k M

(4.15)

The ﬁrst term on the right represents the recoil of the bath caused by coupling to , i.e. the degree to which the bath is thrown out of equilibrium. Substitution M in equation (4.13) gives dM (t) × H eﬀ (M ) = γM dt

(t ) × + γM

1 2 dt |Λk | M (t ) sin(Ωk (t − t )) −∞

k

t

(4.16)

× (Λkak (0)eiΩk t + comp.conj.) + γM or

t dM = γ M × Heﬀ (M ) + dt Γ(t − t )M (t ) dt −∞ × Λk (ak (0)eiΩk t + c.c.) + γM k

= γ M × Heﬀ (M ) + × + γM

∞

(4.17)

(t − τ ) dτ Γ(τ )M

0

Λk (ak (0)eiΩk t + c.c.)

k

where Γ(τ ) is the memory kernel, with dimensions (ﬁeld/time): Γ(τ ) = Γk (τ ) k

=2

(4.18)

2 |Λk | / sin Ωk τ

k

whose action is non-local in time. (Note the ‘left over’ torque exerted by the solutions of the harmonic oscillator equations that eﬀectively constitute the random noise.) In this situation one speaks of a generalized Langevin equation (GLE) (West and Lindenberg, 1990). Replacing the summation by an integration over a

LANGEVIN EQUATION, AND GENERALIZED LANGEVIN EQUATION

81

continuum of k-values, shows that Γ(t) goes to zero as t tends to inﬁnity, according to the Riemann-Lebesgue lemma. If it declines suﬃciently rapidly, which is not guaranteed, a moment expansion becomes possible, which usually converges only asymptotically. If Γ is just a scalar times the unit matrix, the lowest non-vanishing moment is the ﬁrst, and gives the Gilbert damping term × −αM

dM dt

(4.19)

with relaxation constant

∞

tΓ(t)dt

α=

(4.20)

0

If Γ is a matrix, the zeroeth moment does not vanish, and then that moment is simply a correction to the anisotropy part of Heﬀ . In that case, the ﬁrst moment, the damping coeﬃcient, becomes a matrix. Thus anisotropic coupling to the reservoir will change the anisotropic part of Heﬀ , but the opposite is not true: the anisotropic part of Heﬀ cannot change the damping. Anisotropies ) do not aﬀect the result. Note that, if the in the deterministic part of E(M moment expansion is carried one step further, the equation of motion changes, /dt2 , as already discussed in Chapter 3. × d2 M acquiring an additional torque M In the small-motion limit, this may simply be viewed as a detuning of the smallsignal resonance frequency. (Obviously, the even moments are non-dissipative, in contrast to all the odd moments.) From eqn (4.13) and (4.14), it is seen that Γ involves the correlation function of the a’s. In a typical case in which the correlation function decreases exponentially with time, µn will grow indeﬁnitely with n. (For example, the nth moment of e−γt is proportional to (n!/γ n ).) So the moment series strictly diverges, and one needs to ﬁnd the best place to terminate the series to minimize the error. This optimization determines the that will allow retention of all moments fastest permitted rate of change of M th up to the N , say. If a particular process is expected to involve a rate of change that is of order ω, the best place to terminate the series is found by of M minimizing µN ω N . For example, with exponential decay e−λt of the correlation function, the best value of N is the root of d ω N 0= N! (4.21) dN λ ≈

d exp N (log(ω/λ) + log N ) dN

which gives N ≈ (λ/ω)e−1 . So ω must certainly be less than λ. For example, if we want the Gilbert form of damping to be an adequate description, N = 1, so that must be no greater than λ. If ω is greater than λ, but the rate of change of M less than twice λ, it is necessary to add to the Gilbert term the non-dissipative

82

FLUCTUATIONS

torque mentioned above. If the correlation decays only as an inverse power law with time, a moment expansion is ruled out altogether. It is easily veriﬁed that a more general coupling to the bath k (Υk (M ) ak + c.c.), with the matrices Υk depending on a multinomial in the components , gives very similar results, but it gives a more elaborate moment expansion of M determining α, etc., even when such an expansion is permitted. (Coupling to the magnetoelastic medium discussed in Chapter 3 is a case in point.) Also, the procedure described above can be used for the position-dependent case with a coupling energy (x) · u(x) g d3 x M to a reservoir displacement ﬁeld u(x). If the free Lagrangian density of with that ﬁeld is i (∂ui /∂t)2 − (∇ui )2 , so that it propagates as simple waves, the results are almost the same as equation (17), except that the matrix Γ(t − t ) is t replaced by a Greens function matrix Γ(t−t , x −x ), and the integral 0 dt Γ(t− (t ) is replaced by t )M

t

dt

(t , x ) d3 x Γ(t − t , x − x )M

0

We have already encountered this situation in Chapter 3, and will not consider it further in the present context. Much more than equation (4.17) can be said in the case of small motions of . Most importantly, there is then no need to invoke the moment expansion of M Γ(t). For small motions, equation (4.17) may be linearized, and the linearized form may be Fourier transformed with respect to time. With Γ(t) deﬁned to be zero for negative times, the Laplace transform has essentially the structure ˜ p − G − Γ(p) m(p) = f˜(p) (4.22) × H, and Γ ˜ the Laplace where G is the matrix resulting from linearizing M transform of Γ. Finally, f˜(p) is now the Laplace transform of iΩk t f (t) = M0 ( Λk (a+ + c.c.).1 k (0)e k

˜ According1 to equation (4.18), Γ(p) is a scalar function. G is still a matrix (if the eﬀective ﬁeld is in the 3-direction, G is the transverse part of (....× Heﬀ )), 1 Taking Laplace transforms rather than the familiar Fourier transform is appropriate in (t − τ ) equation (4.17). The integration over τ runs only from 0 to t, so that neither Γ(τ ) nor M ever depend on negative times. Therefore they may as well be assumed to be zero for negative times.

LANGEVIN EQUATION, AND GENERALIZED LANGEVIN EQUATION

83

and f˜ is still a vector. Equation (4.18), in the limit of continuous k values, reads

1 V 1 † 3 ˜ Γ(p) = −i 3 − d kΛk Λk (4.23) 8π p + iΩk p − iΩk in volume V. If ρ(Ωk ) is the density of the oscillator states, this could be converted into simple principal value integrals, if the analytic properties of ρ(Ωk )Λk Λ†k were known as a function of Ωk in the upper half plane. If this is not the case, direct numerical integration will generally be needed. Setting p = iω + δ in equation (4.23) gives m(ω) =

g(ω) iω + δ − G − Θ(ω)

(4.24)

i(Ωk −ω)t Λk (a+ + c.c.) k (0)e

(4.25)

where g(ω) = M0

k

and the complex function Θ(ω) = dΩk ρ(Ωk )

1 1 − ω − iδ + iΩk ω − iδ − iΩk

.

(4.26)

The real part of Θ can be absorbed in the diagonal elements of G, and the imaginary part is the analog of η/m in the frequency space version of the simplest Langevin equation (4.8). So, once again, we obtain the ﬂuctuation dissipation theorem. One interesting feature now emerges from a comparison of eqn (4.22) with the solution for the small-signal response of the uniform mode m0 in the presence of scattering via triple or higher terms in the spin wave expansion of the energy function. Written in frequency space, m0 will again have essentially the form (4.22b), with Θ arising from the elimination of the spin waves by solving for them in terms of m0 etc. This means that the ﬂuctuation dissipation theorem holds for distributive damping also. In other words, it can safely be applied to the case in which the observations measure only the spatial average . But in that case it cannot yield an immediate relation between kB T and of M the damping coeﬃcient, because the exponent in the Boltzmann factor does not depend only on a spatial average of m. In contrast, (4.22) does deliver such a relation. This can be seen by ﬁrst using Parseval’s theorem relating 2 2 dω |m(ω)| to dt |m(t)| , invoking the equality of time averages and system averages, and ﬁnally expanding the exponent of the Boltzmann factor to second order in m. Perhaps the harmonic oscillator bath just discussed represents direct interac with black body radiation fairly realistically, but this may be a small tion of M

84

FLUCTUATIONS

eﬀect. In practice, as already pointed out, damping will occur via an intermediate reservoir, with its own loss characteristics determined by contact with yet another heat bath. Continuing in this manner would lead to an inﬁnite hierarchy of reservoirs; hopefully the results for the magnetic systems are not sensitive to forcible termination of the hierarchy at an early stage. For example, in the case of losses due to magnetoelastic coupling, reservoir I is the lattice, which is coupled to reservoir II, a ‘super reservoir’ furnishing both the loss characteristics of the lattice motion and a random force on it by the same elimination procedure that led to equation (4.17). In the case of eddy current damping, reservoir I is the electromagnetic ﬁeld, while electric current ﬂuctuations (Johnson noise) constitute reservoir II. In the magnetoelastic case, reservoir I, the lattice displacement ﬁeld, is agitated by reservoir II with very short autocorrelation times, and satisﬁes a set of LEs with viscous damping, plus random terms from contact with II. Except for these random terms, this latter set of LEs is identical with the usual equations of motion of linear viscoelasticity, which contain the coupling as in Section 4.2. Solving for the strains in terms of M gives the terms to M equations of motion of Section 3.2, except for a random torque originating in the ‘left over’ random terms from reservoir II that still occurred in the LE of reservoir I (see Fig. 4.1). A simpliﬁed two-stage calculation of this kind is shown in Appendix 4B. The ‘left overs’ are only coupled to reservoir I and are not . Therefore they cannot directly contribute to the loss kernel of sensitive to M in the manner of the above derivations. Nevertheless, they the equation for M over any barriers. Incidentally, the above results show that help in propelling M whenever it is possible to exactly solve for the degrees of freedom of the reser , the distinction between additive and multiplicative noise voir in terms of M becomes moot. Assuming the big reservoir II to consist of harmonic oscillators, one can solve for them as a sum of ‘polarizations’ induced by their coupling to the small reservoir I, plus noise terms. This decomposition depends critically on the linearity of the oscillator equations of motion. In the case of nonlinear systems, there is no such additivity, and it becomes much harder to extract the systematic damping part of the motion of the small reservoir and to characterize its ﬂuctuations. For a certain limiting case, this matter is discussed in detail in the book by Gardiner (2003). There is also a large literature on random noise as a driving term of unspeciﬁed or unknown physical provenance (such as might be encountered in the ﬁnance business), which has given rise to two distinct approaches to the Fokker-Planck equation: ˆIto’s (1944) stochastic calculus, with its own rules, and Statonovich’s, which is based on conventional calculus and comes closer to what one would get from a full knowledge of the dynamics of the heat bath. For discussions of thermally activated processes, a formulation in terms of is much more convenient than the Langevin forthe distribution function of M mulation, even though exact solutions are diﬃcult to achieve. We here give a brief derivation of the Fokker-Planck equation, untroubled by the caveats just mentioned. We ﬁrst derive it in cartesians, and then in the more natural polar

FOKKER-PLANCK EQUATION–CARTESIANS

85

coordinates. The treatment is specialized to magnetic systems; the reader interested in more general aspects might consult books by Gardiner (2003), Risken (1984) or by Coﬀey et al. (1986).

4.4

Fokker-Planck equation–cartesians 2 = M 2 makes a cartesian formulation rather awkward. The restriction M s However, since the most common derivations in the literature employ cartesian coordinates, we brieﬂy follow that practice here. Consider the distribution , t) of M . In the absence of dissipation, ρ must be conserved, so that function ρ(M ∂ρ dM ∂ρ dρ = + · =0 dt ∂t dt ∂ M

(4.27)

∂ρ = −Lρ ∂t

(4.28)

×H ef f (M )= M ) · ∂ L(M ∂M

(4.29)

or

where the operator

The total eﬀective ﬁeld is ) 0 (M )+H 1 (t, M eff = H H

(4.30)

0 = −∂E0 /∂ M is the systematic ﬁeld, and H 1 (t, M ) = −∂E1 (M , t)/∂ M where H is a small random ﬁeld resulting from a coupling energy to randomly moving degrees of freedom of a reservoir. The simplest assumption would be to set , t) = −M ·H 1 (t), where H 1 (t) is a randomly varying ﬁeld of unspeciﬁed E1 (M physical origin, with zero average. That would be incorrect, because it would , which is responsible for neglect the recoil of the reservoir to the motion of M the systematic damping of M (see Section 4.3 for the speciﬁc example of a 1 (t) = 0 would harmonic oscillator bath). In the present context, setting H be equivalent to writing down Langevin’s equation without damping term. To 1 (t) is proportional allow for the bath recoil, it is necessary to assume that H /dt, further assuming that long memory eﬀects are negligible. This comto dM plication can be avoided by augmenting L0 ρ to include a Gilbert damping term 1 (t) = 0. As explained in Section 4.3, and in L0 , which allows one to write H further elaborated in Appendix 4C, the reservoir makes itself felt in the motion , be it directly or via a surrogate medium. If this is done, the decomposition of M L = L0 + L1

(4.31)

86

FLUCTUATIONS

leads to a Fokker-Planck equation consistent with the ﬂuctuation-dissipation theorem or the Einstein relation, and it is permissible to treat L1 in the equation ∂ρ , t)ρ = −L0 (M )ρ − L1 (M ∂t

(4.32)

without worrying about its provenance, and assuming it to be zero on average. From here on, we follow the treatment of Zwanzig (2001). Equation (4.32) is converted to an integral equation t , t) = e−L0 t ρ(M , 0) − , t )ρ(M , t ). dt e−L0 (t−t ) L1 (M (4.33) ρ(M −∞

This is substituted in the second term on the right hand side of eqn (4.32), which then reads: ∂ρ , t)e−L0 t ρ(M , 0) = −L0 (M )ρ − L1 (M ∂t t , t)e−L0 (t−t ) L1 (M , t )ρ(M , t ) dt L1 (M +

(4.34)

−∞

). At this point 1 (t, M This result is now averaged over the random process H it is assumed that the correlation functions H1i (t)H1j (t ) of the random ﬁeld components are all proportional to δij δ(t−t ). With that assumption, the random , t ) inside the integral does not contribute to the average, because part of ρ(M , t ), where t < t . So, for example, iterating this equation would involve ρ(M , t ) and ρ(M , t ) will involve δ(t − t ). Then t would have the average of L1 (M to be t, which is inconsistent with having t > t > t . Thus, in averaging the integral, ρ may be considered to be slowly varying, and then the result is ∂ρ )ρ = −L0 (M )ρ + G(M ∂t where the average t / . G(M ) = dt L1 (M , t)L1 (M , t ) = −∞

∞

(4.35)

/ . , t)L1 (M , t − t ) dt L1 (M

0

(4.36) , t) = 0. If the random process is stationary, and it has been assumed that L1 (M G is independent of t. In evaluating G, it is convenient to write H1i (t, M ) = ). For the simple case of δ-function correlation of the components of H1i (t)h(M , we then H1 (t), with H1i (t)H1j (t ) = δij δ(t − t ), and h2 independent of M have ) = (M × h(M )∂/∂ M )·(M × h(M )∂/∂ M ) G(M

(4.37)

FOKKER-PLANCK EQUATION–CARTESIANS

87

The FP equation is then ∂ρ × h(M )∂/∂ M · M × h(M )∂/∂ M ρ = −L0 (M )ρ + M ∂t

(4.38)

. Then it can This result is simple only if h is independent of the direction of M be written ∂ρ × ∂/∂ M )D(M )(M × ∂/∂ M )ρ + L0 (M )ρ = (M ∂t

(4.39)

where D = h2 (M ) is a diﬀusion coeﬃcient. The left hand side gives the deterministic part of the motion, including a systematic loss term. For a ﬁnite decay time of the correlation of the random ﬁeld, the result is more complicated. For example, with autocorrelation decreasing like e−λt , a factor proportional to λe−L0 (t−t )−λ(t−t ) will intervene between the two L1 ’s in equation (4.34). Only if the rate measured by L0 is small compared with the decay rate λ can that intervening factor be replaced by unity. We assume this to be the case from now on. It is easily veriﬁed that the last equation may be written ∂ρ + L0 (M )ρ = D(∇2M ρ)surface of unit sphere ∂t

(4.40)

where D = h2 acts as diﬀusion coeﬃcient. Note that this argument strictly holds only in the case of δ- function time correlation of pairs of random ﬁeld components. When the correlations range over a time interval of the same order as the deterministic motion or longer, the stochastic part of ρ in the averaging of the integral in equation (4.34) cannot be ignored. One is then forced to fall back on the generalized Langevin equation. Alternatively, one could continue iterations of the integral equation (4.34) until the earliest time integration variable is so much earlier than t, that the correlations have essentially died down. This would result in a correspondingly higher than second order partial diﬀerential equation – not a very useful concept. is furnished by the interaction of An example of resevoir coupling linear in M M with electromagnetic ﬁelds and currents. But, in general, the coupling to the reservoir is not linear. For example, if, in the case of magnetostrictive coupling, the uniform shear strain components ﬂuctuate, they generate random ﬁelds with components H1i = B j Mj eij (t), and then ) = B 2 eil ejk Ml (M × ∂/∂ M )i Mk (M × ∂/∂ M )j (summation convention) G(M (4.41) If eil ejk = δij δlk e2 , there will be one term of the same form as before, with h2 replaced by B 2 e2 Ms2 , and one more term of the form 0 1 × ∂/∂ M )i Ml (M × ∂/∂ M )i B 2 e2 Ml (M (4.42)

88

FLUCTUATIONS

which is found to vanish. So the forms of equations (4.38) and (4.39) remain intact in this particular case. A more serious case of M-dependence of the random ﬁeld arises if a substantial crystalline anisotropy is modulated by lattice vibrations, even if the vibrations do not cause ﬂuctuations in symmetry, but merely modulate the anisotropy constant for the given symmetry. For example,with cubic symmetry, modulation δK(t) of the constant K in the form − 14 K Mi4 gives a random ﬁeld H1i = δK(t)Mi3 . In that case (summation convention), , × ∂/∂ M )i Mj3 (M × ∂/∂ M )j ) = δK(t)2 Mi3 (M G(M

(4.43)

and the FP equation then looks very diﬀerent. This kind of ﬂuctuation can result from random variations in hydrostatic pressure. Serious problems with the Fokker-Planck approach arise if the autocorrelation of the random ﬁeld decays more slowly than an exponential. As we have seen already, the GLE does not admit of a moment expansion in that case. Then the integral formulation, equation (4.34), must be retained as is, with correspondingly diminished returns vis-` a-vis the GLE approach. For a particularly clear exposition of this limitation, see Zaslavski (2005), which demonstrates the diﬃculty by comparing a Gaussian random process with a L´evy process. In the following section, as well as in the following chapter, it will be assumed that a moment expansion of the memory kernel is permitted. 4.4.1

Fokker-Planck equation in polar angles

In the next chapter, magnetization reversal in small particles is discussed. For is more convenient, since the conthis purpose, the polar representation of M coupling to 2 is then automatic. In the simplest case of linear-in-M stancy of M the reservoir, equation (4.39) becomes

∂ρ 1 ∂2ρ 1 ∂ ∂ρ + L0 (M )ρ = D sin θ + (4.44) ∂t sin θ ∂θ ∂θ sin2 θ ∂ 2 φ ∂ ∂ + φ˙ , with θ˙ and φ˙ shown in several earlier chapters. In more where L0 is θ˙ ∂θ ∂φ complicated cases, it is best to go to the spherical metric directly, and it is then easier to dispense with cartesians altogether. In spherical polar coordinates, the are (0, 0, Ms ), and those of Ms ∇M are ∂ , 1 ∂ , 0 in components of M ∂θ sin θ ∂φ the directions of θ, φ and M increasing. Therefore

× ∇M ) = − 1 ∂ , ∂ , 0 (M (4.45) sin θ ∂φ ∂θ in polar coordinates, it is also easy to write the in this axis system. With M dissipative component of L0 ρ and move it over to the right hand side of the

FOKKER-PLANCK EQUATION–CARTESIANS

89

equation so that it becomes the divergence of a total current (see Chapter 5). But an average like (4.43) looks complicated even in polar coordinates. This is inevitably the case whenever the symmetry of the random energy expression clashes with the basically spherical symmetry of the direction of the ﬁxed length magnetization vector. 4.4.2

Fokker-Planck equation in the absence of well-deﬁned canonical variables

In some cases, deﬁning conjugate variables is awkward or at least artiﬁcial. An example, which we shall encounter in Chapter 6, is the relaxation of only one variable: the magnetic energy E of a particle system being degraded by intrinsic damping. There, an equation for ρ(E, t) is needed. In principle, one could insist on the canonical formulation by making a contact transformation from the usual conjugate variables to new variables, one pair of which is the energy itself, with the time as conjugate variable. A much more direct derivation makes no reference at all to a Hamiltonian. It is used in innumerable texts but, for completeness, we brieﬂy derive it here: A single variable x is subject to random forces, possibly in addition to systematic forces. In a time interval ∆t it undergoes a change ∆x with normalized probability p(x; ∆x). Then, assuming the process to be markovian, ρ(x + ∆x, t )p(x + ∆x; ∆x)d∆x

ρ(x, t + ∆t) =

(4.46)

all ∆x

⎞ ∂ ρ(x , t ) all ∆x ∆xp(x ; ∆x) ρ(x, t )p(x; ∆x) + ⎟ ⎜ ∂x

=⎝ ⎠ ∆x2 ∂2 p(x ; ∆x)d∆x + · · · + 2 ρ(x , t ) all ∆x ∂x 2 ⎛

Because p(x; ∆x) is normalized, this can be written, to ﬁrst order in ∆t, ∂ρ ∂ρ + ∂t ∂x

dx dt

= syst

∂ ∂2 (ρ(x, t)µ1 (x)) + 2 (ρ(x, t)µ2 (x)) + · · · ∂x ∂x

(4.47)

where we have allowed for the systematic (relatively slow, non-dissipative) motion of x that contributes to (ρ(x, t + ∆t) − ρ(x, t ))/∆t, and where 1 µ1 (x) = ∆t µ2 (x) =

1 2∆t

∆xp(x; ∆x)d∆x all ∆x

∆x2 p(x; ∆x)d∆x all ∆x

(4.48)

90

FLUCTUATIONS

µ2 , usually by D(x), is the well-known measure of the ‘random walk’ , denoted motion ∆x2 /2∆t, which remains ﬁnite as ∆t tends to zero. So equation (4.47) reads:

∂ ∂ρ ∂ρ dx ∂ (µ2 (x)ρ) + µ1 (x)ρ = (4.49) + ∂t ∂x dt syst ∂x ∂x The right hand side is the divergence of a total dissipative current: diﬀusion and damping. The latter comes from the ﬁrst moment of p(x; ∆x), which must not vanish, as explained in section 4.4.1, because of recoil of the bath. Alternatively, µ1 (x) can be equated to zero, and the damping included as a systematic term on the left hand side of (4.49). A systematic loss term would add to the left hand side the divergence of a current (dx/dt)loss ρ (see Chapter 5 for an example). The result will be the same, because µ1 (x) and (dx/dt)loss arise from the same mechanism. Appendix 4A. Conditions for validity of linear response theory Consider the full response. The distribution function just after the pulse (obtained by integrating ¯(0) = ρ0 − ρ1 ≡

Liouville’s equation from −0 to +0) is ρ ∂H0 ρ0 − ρ0 i gi , (since ρ0 does not change appreciably as a function ∂pi t=0 of qi during the pulse). Then, at a later time, pj = Πi (dqi dpi )pj ρ ¯(0, t) ¯(0) = Πi (dqi dpi )pj e−L0 t ρ =−

Πi (dqi dpi )

i

−L0 t

gi pj e

∂H0 ∂pi

ρ0 t=0

Integration by parts allows us to shift the time to pj .2 Then, assuming evolution 2 pi /(2m), and remembering that the momentum dependent part of H0 to be ρ0 does not change with time, we get 1 Πi (dqi dpi ) pj = gi pj (−t)pi (0)ρ0 m i The burden of the FDT has thereby been placed on time evolution of the observables, and ρ1 is involved only at time t = 0. Its subsequent motion is irrelevant. In fact, chaotic motion of the observables is necessary. For example, if the momenta did not show ergodic behavior (also called mixing), and stayed conﬁned to a special corner of phase space, the last equation would still be correct, 2 Otherwise

said, L0 is usually antihermitian.

Appendix 4B. Coupling of magnetization via an intermediate medium

91

but it would be wrong to expect the response to be related to the correlation function on the right hand side evaluated over the usual thermal distribution function. An obvious corollary of this conclusion is that the rate of change of interest to the observed quantities must be slow compared with the time needed for adequate ‘mixing’. For a general time dependent signal, not just a pulse, one might be inclined to invoke the linear superposition principle, but a more careful step-by-step derivation is given in M. Bianucci and R. Mannella (1996). Appendix 4B. Coupling of magnetization via an intermediate medium We generalize the process leading to equations (4.12) through (4.17): Reservoir II is represented by a set of harmonic oscillators, {ak }, linearly coupled to reservoir I. For simplicity, reservoir I is represented by a single harmonic oscillator, b, in . The total coupling energy is turn linearly coupled to M bΞka† + ak Ξ†b† + g M · b + b† , − (4.50) k k with all constants in frequency units. The equations of motion (Heisenberg’s for ) are the oscillators, the usual one for M dak = iΩkak + iΞkb dt † db = iΩbb + i Ξkak + ig M dt

(4.51)

k

dM ×H ef f (M ) + g M × b + b† =M dt where the Ξk are three-by-three hermitian matrices, and g is a coupling constant. Solving the ﬁrst of these for the ak ’s and substituting them in the second gives

t db −iΩk (t−t ) T = iΩb b + ig M − dt e Ξk Ξk b(t ) dt −∞ k † + Ξkak (−∞)eiΩk t , (4.52) k

the GLE of reservoir I. If k e−iΩk t Ξk ΞTk is a rapidly decaying function of t, a moment expansion of the time integral becomes possible, and eqn (4.52) becomes an ordinary LE equation with a solution of the form t t † (t ) + b = ig e(iΩb −µ)(t−t ) M Ξkak (−∞) dt e(iΩb −µ)(t−t )+iΩk t −∞

k

−∞

(4.53)

92

FLUCTUATIONS

where µ is the zeroeth moment of the memory kernel in eqn (4.52). (The last term is the random term ‘inherited’ from reservoir II.) Substituted in the last equation of (4.51), it gives t dM ) + 2(g)2 M (t )dt × ×H ef f (M =M e−µ(t−t ) sin Ωb (t − t )M dt −∞

t iΩk t +(iΩb −µ)(t−t ) + 2g M × Ξk Re ak (−∞) dt e (4.54) k

−∞

which is the GLE equation for the magnetization, and it includes the ‘inherited’ noise. Note, however, that this inheritance is no longer totally white, even if the noise of reservoir II was white. If µ is large enough, the memory kernel may again be expanded. The zeroth moment merely gives zero torque, but the ﬁrst moment gives a Gilbert term. The following conclusion may be drawn from this model: Leaving aside the random terms in eqn (4.54), that equation is just the macroscopic, phenomenological equation one would write down for reservoir I. Therefore, when studying a dynamical system, M, in contact with a lossy system, B, whose equations of motion are known in terms of phenomenological parameters, the important loss terms in the equations for M can be found by eliminating the degrees of freedom of B, using only its phenomenological equations. Although purely schematic, these results resemble those for magnetoelastic loss calculated in Section 4.2 quite closely. The most important diﬀerence is that to reservoir I, the lattice, was not linear in M . Since only the coupling of M systematic terms were retained, the equations of the lattice were the same as the usual equations for the strains of a magnetoelastic medium; this simply meant -dependent, and therefore got involved in that the coupling energy g became M . If, however, the Ξk the integrand of the memory kernel of the equation for M coupling the two reservoirs were to depend on ak , the problem would be much , some harder. If it is assumed that the ak vary much faster than either b or M help may be provided by the cited reference in Gardiner’s book.

Appendix 4C. Generalization: FP equation in a reservoir of harmonic oscillators The original formulation of Langevin’s equation treats the friction as a systematic force and establishes the relation between that friction and the applied random force only after solving the equation and invoking eventual attainment of thermal equilibrium of the system. It is not meant to yield the friction coeﬃcient directly. Similarly, the corresponding Fokker-Planck (FP) equation must allow for a dissipative term to contribute to the operator L0 . The generalized Langevin equation directly delivers a memory term, part of which results in friction if a moment expansion of the memory kernel is permitted. This was demonstrated explicitly in Section 4.3 in the case of a thermal reservoir of harmonic oscillators. Correspondingly, one can write down a generalized

Appendix 4C. Generalization

93

Fokker-Planck equation (GFPE), which delivers such a memory term as part of the systematic force, and furthermore delivers a generalized diﬀusion term which likewise has memory. We return to equation (4.34), applied to the total in a harmonic oscillator reservoir. ρ is then a function of M and the system: M ak ’s, and, with h.c. denoting hermitian conjugate,

∂ ∂ × ∂E · ∂ − i Ωk ak − a†k ∗ (4.55) L0 = M ∂ak ∂ak ∂M ∂M k

∂ ∂ i M · Λk L1 = −M × (Λkak + h.c.) · − h.c. − ∂ak ∂M k ≡ L11 + L12 , say For easy reference, we restate equation (4.34): ∂ρ , t)e−L0 t ρ(M , 0)+ = −L0 (M )ρ − L1 (M ∂t t , t)e−L0 (t−t ) L1 (M , t )ρ(M , t ) dt L1 (M +

(4.56)

−∞

It will be seen that, among the terms under the integral sign, the cross-term in the L12 e−L0 (t−t ) L11 ρ turns into a ‘systematic’ friction term acting on M −L0 (t−t ) L12 is a corresponding systematic recoil term GFPE. The term L11 e aﬀecting the ak ’s whereas the terms L11 e−L0 (t−t ) L11 ρ and L12 e−L0 (t−t ) L12 ρ space and the diﬀusion of the harmonic respectively describe the diﬀusion in M . (To describe the diﬀusion and friction oscillators due to their recoil from M process of the harmonic oscillators in full, we would of course have to take account of their coupling to yet another reservoir.) First, consider

t

, t)e−L0 (t−t ) L11 (M , t )ρ(M , t ) dt L12 (M

0

which equals i

t

dt

0

(t) · Λk M

k

∂ ∂ − † ∂ak ∂ak

e−L0 (t−t ) M

(4.57)

, t ) ∂ρ(M (t ) ∂M

∂ i t ∂ (t) · =− dt Λk M − † (Λkak eiΩk (t−t ) + h.c.) 0 ∂ak ∂ak k k ×

(Λkak + h.c.) ·

(t − t ) · ∂ρ(M , t ) ×M (t ) ∂M

94

FLUCTUATIONS

=2 0

t

(t) × M (t ) · ∂ρ(M , t ) dt Γ(t − t )M (t ) ∂M

(An easy way to derive the third line from the second line is to write out the latter in component form.) If ρ is now considered as varying suﬃciently slowly, its , t)/∂ M (t). derivative may be pulled out of the integrand, and replaced by ∂ρ(M Then this term looks like a contribution to the systematic −L0 ρ, part of which will be dissipative. In fact, if a moment expansion is permitted, the ﬁrst moment of Γ(t − t ) gives Gilbert damping

× dM −α M dt

·

∂ρ ∂M

(t) × M (t ) is not if Γ is taken to be scalar. More generally, however, Γ(t − t )M (t) × Γ(t − t )M (t ) in equation quite the same as the corresponding term M (4.17) leading to the GLE. In the simplest case of totally isotropic coupling, with Λ’s proportional to unit matrices, this makes no diﬀerence. If Γ(t) declines suﬃciently rapidly, a moment expansion can again be made for times greater than the correlation time. Note that, in the isotropic case, dropping the assumption of slow variation of ρ will not cause it to contribute to the ﬁrst moment, only to higher moments. For example, the second moment will involve ρ. If ρ depends on time only through M, that second moment contributes to the diﬀusion term to be discussed next. If ρ also depends on time explicitly, part of the second moment results in a renormalization of the left hand side, ∂ρ/∂t, of the FP equation. Next,

t

, t)e−L0 (t−t ) L11 (M , t )ρ(M , t ) dt L11 (M

0

=

intt0 dt ×

k

(t) × M

∂ (Λkak + h.c) · (t) ∂M

(Λkak eiΩk (t−t ) + h.c.) ·

(t ) M

(4.58)

, t ) ∂ρ(M (t ) ∂M

At this point, we ignore the modiﬁcation of the equilibrium distribution of the , and factor ρ: oscillators due to their coupling to M , t)ρeq ({ak }) ρ → ρ(M

Appendix 4C. Generalization

95

, t) is a function of M alone. Then we average over the where from now on ρ(M reservoir. The expression (4.58), becomes

t , t ) ∂ρ( M ∂ ˜ ij (t − t ) M (t ) × (t) × Γ dt M (4.59) (t) (t ) ∂M ∂M 0 i j where ˜ ij (t ) = 2 Γ

3 k

2

l=1

3 k

l=1

(Λk )il (Λk )jl

(Λk )il (Λk )jl

cos Ωk t → exp(Ωk /kB T ) − 1

kB T cos Ωk t Ωk

, t) is assumed to vary slowly, so that at high temperatures. If ρ(M , t )/∂ M (t ) may be replaced by ∂ρ(M , t)/∂ M (t), one obtains the gen∂ρ(M eralized diﬀusion term

t ∂ ∂ ˜ , t) Γij (t − t ) M (t ) × dt M (t) × ρ(M (4.60) (t ) (t) ∂ M ∂M 0 i j ˜ is a scalar and a moment expansion is permitted, the zeroeth moment does If Γ not vanish and gives the diﬀusion constant D. The expression (4.60) is then of the same form as obtained in the traditional treatment, i.e eqn (4.42). To achieve equilibrium, it is necessary that the diﬀusion constant D, the zeroeth moment of ˜ ij (t), be related to α, the ﬁrst moment based on the last of eqns (4.57). This is Γ the Einstein relation, another form of the ubiquitous FD theorem. These general results can be transformed into spherical polar coordinates. For the case of scalar couplings, the resulting FP form is equation (4.44).

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5 MAGNETIZATION REVERSAL IN A VERY DILUTE ARRAY OF SMALL PARTICLES 5.1

Introduction

In this chapter, some of the details of magnetization reversal due to two kinds of forces are examined: a relatively large applied magnetic ﬁeld, and a comparatively small, but persistent, stochastic ﬁeld. From an experimentalist’s point of view, most of the interest is in a plot of the switching ﬁeld (aka the coercive ﬁeld) versus the time needed to achieve magnetization reversal, suitably deﬁned. Most of this chapter is devoted to this relationship in an array of small particles so dilute that their interactions may be neglected. Typically, it will be assumed that the magnetization starts out very close to an absolute energy minimum. A suﬃciently large magnetic ﬁeld applied in an appropriate direction can render that position unstable, but even in the absence of a systematic applied ﬁeld, the stochastic ﬁeld will gradually cause the magnetization direction to become statistically distributed. Even if there are substantial barriers separating the initial equilibrium position from other stable or metastable energy minima, the stochastic ﬁeld will eventually ﬁll these minima partially, according to a Boltzmann distribution. (This is responsible for the ﬁnite lifetime of magnetic recordings.) A distinction must be made between applied-ﬁeld-induced reversal in particles with linear dimensions small compared with a typical domain wall width, and larger particles. In the former, the magnetization will tend to reverse as a single domain; in the latter, reversal will occur by nucleation of one or more domain walls, followed by relatively rapid wall motion, leading to merging of domains oriented in the reversed direction, or annihilation of domain walls at the boundaries. In either case, except in large applied ﬁelds, a barrier formed by anisotropy energies of various origins must be overcome. This involves thermal activation. In small specimens, the barrier energy is proportional to the sample volume. In large specimens, the barrier energy is proportional to the volume of the region participating in the formation of the nucleus, which is much smaller than the sample volume. Certain theories for idealized large samples free of nucleation centers treat the reversal process as essentially single domain reversal, by processes such as curling or buckling (Frei et al., 1957). Since these involve the entire sample volume, the barrier energy will be proportional to the entire volume. Measurements of activation energy, on the other hand (see, for example, Li and Metzger, 1997), conclude that the eﬀective volume (called the ‘magnetic volume’) is quite small, and not directly related to the total volume.

97

98

MAGNETIZATION REVERSAL

Needless to say, a theory of the small specimen is the easier than that of large ones, especially for the simple models discussed in this chapter. 5.2

General observations

It is customary in experimental work on magnetization reversal to plot the results as a curve of the ﬁeld needed to switch (the coercive ﬁeld) versus the time needed to complete the switch. This plot covers not only the values of ﬁeld greatly exceeding the anisotropy barrier ﬁeld, but also ﬁelds less than the barrier. In fact, the smallest ﬁelds reported may be less than those needed to saturate the magnetization. In magnetic recording, the switching ﬁeld furnished by the signal is normally larger than the eﬀective ﬁeld posed by the barrier opposing the reversal. For such large ﬁelds, one may be tempted to treat the problem using a purely ballistic calculation, without taking into account thermally induced processes. This will not ‘work’ for the following reasons: The fully reversed position is an equilibrium position. There, by deﬁnition, all components of the torque exerted are ﬁrst order on the magnetization are zero. But the equations of motion for M diﬀerential equations in the time variable. So as the torques get smaller upon /dt. Therefore the time taken to approach to the new equilibrium, so does dM achieve total reversal is strictly inﬁnite (an ‘inertial’ term would be needed to to the desired position on a ﬁnite time scale). From the point of view carry M of applications, one may be content to deem the magnetization to have switched when it reaches some assigned angle close to equilibrium. In that case, a ballistic calculation may suﬃce. However, if the ﬁeld is equal to or slightly less than the barrier ﬁeld, the same dilemma arises: the barrier, though an unstable stationary /dt is zero point, is still a zero of all energy derivatives. Hence the ‘velocity’ dM there, and the magnetization would never make it across. To account for thermal agitation one could, in principle, use the Langevin approach, but in practice, the Fokker-Planck method is easier to apply. In either treatment, some violence must be done to the exact formulation. The main problem is that the Langevin approach involves a memory kernel, precluding a solution which depends only on initial conditions. In the Fokker-Planck approach, a prolonged memory would imply a high order diﬀerential equation. For these reasons, we must hope that a moment expansion of the memory term is possible, and furthermore, may be stopped at the lowest non-trivial term. This ensures a damping term local in time, and hence the validity of a Fokker-Planck equation of the usual form. The ﬁrst step is to determine in what sense a small enough particle will reverse its magnetization uniformly. Even when it is so small that ‘mature’ domain walls within it are ruled out, the magnetization cannot be truly uniform in a real sample. Usually, the magnetization at the surface will be subject to a diﬀerent crystalline anisotropy, one form of which is surface pinning. Even though the , a exchange interaction will tend to smooth out any resulting variation in M residual non-uniformity must remain. A way to deal with this is to replace the actual particle P by P∗ , an ‘eﬀective’ particle. P∗ is to have position independent

GENERAL OBSERVATIONS

99

anisotropy constants K ∗ , in contrast with P, whose constants K(x) depend on ∗ of P∗ is position. Also, for a given applied ﬁeld, the magnetization vector M to be independent of position, in contrast with M (x) of the actual particle P. In the static problem, this replacement is not diﬃcult to achieve to lowest nontrivial order in K(x)/J, where J is the large exchange energy. We brieﬂy sketch the procedure, with dipolar forces omitted. The total energy for P is written as usual

3 1 (x) + J ·M (x) + Ki (x)Pi M (∇Mn )2 E = d3 x −H 2 n=1 i . The Ki are Here, the Pi are various even polynomials of the components of M the actual anisotropy constants, and H is the applied ﬁeld. On the other hand, the energy of P∗ is

∗ ∗ ∗ ∗ K Pi M E = v −H · M + i

i

∗ satisﬁes where v is the volume. The equilibrium position of M ∂ ∗ =0 M −Hn + Ki∗ P i ∂Mn∗ i (x) will diﬀer Now consider E/J instead of E. Hopefully, for large J, the actual M ∗ ∗ only by a small vector δ M (x) from M , but the Ki must involve some averaging of the Ki . We have ⎞ ⎛ ⎛ ∗ + δM (x) − M K ( x )P i i E∗ E 1 1 · δM (x) + ⎝ ⎠ = + d 3 x ⎝− H ∗ J J J J −K ∗ Pi M i

⎞

+

i

3 1 (∇δMn )2 ⎠ 2 n=1

This expression is stationary (hopefully a minimum) with respect to δM, if, to lowest order, ⎞ ⎫ ⎧ ⎛ ∗ ⎬ ⎨ ∂Pi M 1 ⎝(Ki (x) − Ki∗ ) ⎠−H ∇2 δMn = ⎭ J⎩ ∂Mn∗ i

⎞ ⎛ ∗ 3 ∂2P i M 1 + Ki (x) ⎝ δMl ⎠ J i ∂Ml∗ ∂Mn∗ l=1

100

MAGNETIZATION REVERSAL

The solution, as expected, is small, of order 1/J, and so contributes only of order 1/J 2 to E/J. Neglecting this contribution leaves only E = E∗ +

d3 x

∗ (Ki (x) − Ki∗ ) Pi M

i

The most obvious, though not the most general, choice giving E = E∗ , is Ki∗ =

1 v

d3 xKi (x),

where v is the specimen volume. It is then plausible to assume that the eﬀective magnetization vector obeys the equation ∗ dM ∂E ∗ =− ∗ dt ∂M ∗ . This simple view holds only if the exchange energy everywhere is much for M larger than the anisotropy energy. It will fail if there is strong surface pinning, especially if the exchange energy at the surface is smaller than in the interior. From here on we assume that we are dealing with a single domain eﬀective particle, and drop the asterisks on the symbols. The Fokker-Planck equation will , equation (4.46). The drift term be written in terms of the polar angles of M ∂ρ ∂ρ + φ˙ , or, in Hamiltonian form, L0 ρ = θ˙ ∂θ ∂φ L0 ρ =

∂E ∂ρ ∂E ∂ρ − ∂ cos θ ∂φ ∂φ ∂ cos θ

has a further systematic contribution due to dissipation. To appreciate the need for this term, recall the two-reservoir model discussed in Chapter 4, , saw the thermal reservoir (II) Appendix 4B. There, the object of interest, M of harmonic oscillators a only indirectly through an intermediate reservoir (I) denoted by b. The equation of motion of b, through its coupling to a, acquired a systematic dissipative term, in accordance with the FDT or GFDT relation. This through its coupling to b, and thereby the drift term aﬀects the motion of M term acting on M must acquire a non-Hamiltonian dissipative contribution. (In our model, the b-system had only one degree of freedom, but the same argument holds if it has many degrees of freedom.) In order that the FP equation for the distribution function W (θ, t) remains of the standard parabolic form of a diﬀusion equation, the memory kernel must be local in time, and we will make this simpliﬁcation. As explained in Chapter 4,

GENERAL OBSERVATIONS

101

∂W ∂W the drift term θ˙ + φ˙ requires supplementation since now ∂θ ∂φ ∂E ∂E −α sin θ∂φ ∂θ ∂E ∂E sin θφ˙ = − −α . ∂θ sin θ∂φ θ˙ =

where α is the loss constant. In the presence of loss, the total time rate of change of ρ is no longer zero as in the absence of loss, but is negative, equal to minus ∂E ∂E W, α W. When this the divergence of the current with components α ∂θ sin θ∂φ is taken into account, the FP equation, written out in full, becomes ∂W ∂E ∂W ∂E ∂W − + ∂t ∂θ sin θ∂φ sin θ∂φ ∂ θ

1 ∂ ∂W 1 ∂2W =D sin θ + sin θ ∂θ ∂θ sin2 θ ∂ 2 φ

∂ ∂E ∂ ∂E W + +α W ∂θ ∂θ sin θ∂φ sin θ∂φ

(5.1)

The diﬀusion coeﬃcient D is calculated by evaluating the expectation value of the autocorrelation of the random term that will be ‘left over’ from interaction of reservoirs I and II when the Langevin solution for b in terms of a is substituted .1 In general coordinates, eqn (5.1) reads into the equation of motion of M ∂W = ∇ · (D∇W + αW ∇E) ∂t In equilibrium,

(5.2)

∂W = 0, and then a solution of eqns (5.1) or (5.2) will be ∂t W = const. × exp (−(α/D)E)

(5.3)

This is the Boltzmann distribution, if another version of the FDT, the Einstein relation, D = kB T α

(5.4)

holds. That this equation should hold, imposes certain restrictions on the statistics of the big reservoir a. The argument of Gibbs, on which the Boltzmann 1 In the case of magnetostriction, that left-over coupling term represents multiplicative noise, since it is of the form ijk mj ml e0lk (t), where e0lk (t) is the random ﬁeld in the Langevin equation for the strain components caused by coupling to a. But as we saw in the cartesian formulation of the FP equation in the previous chapter, this special structure of the multiplicative noise reduces it to additive noise. In more general forms of coupling, there will be a Stratonovich (1963) type drift term leading to a small extra systematic loss term in eqn 5.1.

102

MAGNETIZATION REVERSAL

distribution is based, assumes that the system of interest is extremely weakly coupled to its surroundings, yet coupled strongly enough to achieve equilibrium. These assumptions require that α and D both go to zero, but in such a way that D/α remains ﬁnite. In our two-reservoir model, the only way the ratio may be forced to remain ﬁnite is to impose a condition on the statistics of reservoir a. This recalls eqns (4.8) through (4.11), in which a knowledge of the damping coeﬃcient in a simple LE, together with an insistence on the Boltzmann distribution of the solution, imposed a restriction on the statistics of the random force. In the following, we assume that these conditions for the validity of eqns (5.1) through (5.4) are met. 5.3

Reversal in 2d

We begin by considering a relatively simple model. In that model, the rotation of the magnetization vector is very nearly conﬁned to a plane, as it will be if the sample is a thin oblate spheroid. If we impose the Einstein relation, the FP equation may be written

∂ ∂W v∂E ∂W = + W (5.5) D∂t ∂θ ∂θ kB T ∂θ where we have furthermore assumed that the diﬀusion constant is independent . Here vE is the total magnetic energy in the volume v of the direction of M of the sample. In the simplest non-trivial case, the energy per unit volume has the form 1 E = −HMs cos θ − HK Ms cos2 θ 2

(5.6)

where the anisotropy ﬁeld HK measures the strength of a uniaxial anisotropy, and it is assumed that the applied ﬁeld is along the anisotropy axis, assumed to lie in the plane of the thin oblate sample. When |H| is less than HK , there are two minima, at θ = 0 and θ = π. For positive (negative) H the deeper minimum is at θ = 0 (θ = π). The two minima are separated by a maximum at cosθ = H/HK . With increasing, positive H, that maximum moves towards the more shallow minimum at θ = π. Finally, when H reaches HK, the maximum reaches the former minimum at π, resulting in an extremely ﬂat maximum varying as −(δθ)4 for small deviations from π, whereas the deeper minimum at θ = 0 remains a deep quadratic minimum. For negative H, the roles of 0 and π are reversed. When H is negative, and |H| > HK , both extrema, at 0 and π, are quadratic: θ = 0 is a maximum and θ = π a minimum, and there is no extremum in between. Figures 5.1a&b show the position of the barrier when |H| < HK , and its height above the prevailing minimum, divided by HK . Suppose H was zero up to time t = 0, and the magnetization direction was forcibly held close to the minimum θ = 0. At t = 0, a small ﬁeld is applied in is released. If the negative direction, favoring the minimum at θ = π, and M

REVERSAL IN 2D

103

π

(a)

3π/4

π/2 π/4

0.2

0.4

0.6

1

0.8

H/HK (b)

Eb 0.5 0.4 0.3 0.2 0.1

0.2

0.4

0.6

0.8

1

|H|/HK

Fig. 5.1. (a) Position of energy maximum for negative H with magnitude less than HK , (b) barrier height (in units of HK /Ms ) above shallower minimum at θ = 0 versus H/HK .

will still have to diﬀuse over the barrier near π/2 to reach the |H|/HK << 1, M deeper minimum at θ = π, slightly assisted by the negative H. It turns out that more insight into the diﬀusion process can be gained by ﬁrst transforming eqn (5.5) into Schr¨ odinger-like form, as proposed, for example, by van Kampen (1977). (Excellent use of this transformation was made by Caroli et al. (1979) in their discussion of thermal activation.) For convenience, time will be measured in units of 1/D, and energy vE in units of kB T, so that eqn (5.5)

104

MAGNETIZATION REVERSAL

reads ∂W ∂ = ∂t ∂θ

∂E ∂W + W ∂θ ∂θ

(5.7)

The required transformation to a new dependent variable u is W (θ, t) = e−E(θ)/2 u(θ, t)

(5.8)

∂u ∂2u = + V (θ)u, ∂t ∂θ2

(5.9)

and the new equation is

a Schr¨ odinger-type equation for imaginary time, in a potential

2 1 ∂2E 1 ∂E V (θ) = − 2 ∂θ2 4 ∂θ

(5.10)

Written out in full, V (θ) looks complicated. In fact, using the energy, expression (5.6) gives V (θ) =

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

(5.11)

which, for some purposes, may be more conveniently expressed in terms of multiple angles: V (θ) =

1 " 2 −(HK + 4H 2 ) − 4H(HK − 4) cos θ + 4(4HK + H 2 ) cos 2θ 32 # 2 +4HHK cos 3θ + HK cos 4θ (5.12)

The ﬁelds have been written in terms of magnetic energy in units of kB T , i.e. vMs H/kB T → H

(5.13)

vMs H/kB T → HK .2 The formidable appearance of V (θ) is largely compensated for by the fact that this formulation can draw on the rich existing lore concerning an electron in one-dimensional motion through a periodic lattice. More importantly, it allows a seamless connection of switching time for values of H exceeding the anisotropy barrier and values of H less than the barrier. The usual treatments of that transition require additional semi-intuitive considerations. 2 In either of the two limits, H → 0 and/or H K → 0, the Hill eqn (5.9), using (5.12) for V, becomes somewhat simpler. In the limits of very large |H| at ﬁxed HK , or very large HK at ﬁxed |H|, eqn (5.9) reduces to Mathieu’s equation with well-tabulated solutions. Also, in either of the two limits, H → 0, ﬁnite HK and HK → 0 for ﬁnite H, the Hill equation becomes somewhat simpler, involving only two, rather than four, periods.

REVERSAL IN 2D

105

Before tackling eqn (5.9) for general values of the ﬁelds, consider the simple problem of unhindered diﬀusion on the circumference of a circle. Certain features of this trivial case also appear in the full problem. The equation for free diﬀusion on the circle is ∂2u ∂u = ∂θ2 ∂t

(5.14)

If θ were not an angle between 0 and 2π, but extended from −∞ to +∞, the source solution of this equation, with the particle located exactly at θ = 0 at time t = 0, would be, apart from a constant factor, u(θ, t) = (2πt)−1/2 exp(−θ2 /4t)

(5.15)

which becomes proportional to δ(θ) as t tends to zero. But this is not acceptable here, since we require u to be single-valued on the circle 0 < θ < 2π. To meet this requirement it is necessary to assume that u(θ, 0) is proportional to the so-called periodic delta function δP (θ) =

+∞

δ(θ − 2nπ)

(5.16)

−∞

Then, by the linear superposition principle applied to the solution of eqn (5.14), the correct solution is u(θ, t) ≈ (2πt)−1/2

+∞

exp(−(θ − 2nπ)2 /4t)

(5.17)

−∞

which is periodic in θ and single valued, as required. It becomes uniform as t → ∞. Note that, for very small values of t, only the term with n = 0 contributes appreciably to the sum. Note also that, at θ = π, du/dθ is zero, so no net diﬀusion current ﬂows through θ = π. A solution resulting in a totally absorbing barrier at π is produced, if the initial condition is the antiperiodic δ-function δAP (θ) =

+∞

(−)n δ(θ − 2nπ)

(5.18)

−∞

The subsequent solution is then u(θ, t) = (2πt)−1/2

+∞ 1 (−)n exp(−(θ − 2nπ)2 /4t) N −∞

(5.19)

and goes to zero as t → ∞. The two solutions (5.17) and (5.19), are known as theta functions ϑ3 (θ, t) and ϑ4 (θ, t) respectively (Erd´elyi et al., 1953). They are satisfactory representations for small t; for large t they converge very slowly. For large t, a better form of solution is based on cosine and sine eigenfunctions of

106

MAGNETIZATION REVERSAL

equation (5.14). In terms of the cosines, the above solution for the reﬂecting and absorbing barriers are proportional to ∞

2

e−n t cos nθ

(5.20)

n=0

and ∞

1 2

(−)n e−(n+ 2 ) t cos(n +

1 2)

θ

(5.21)

n=0

respectively. (Degenerate with the cosines, there are also sine solutions, but only the cosine solutions are needed here.) For small values of t, these series converge very poorly, but they obviously converge very rapidly for large t. (The solutions previously written in terms of ϑ functions can be expanded in these eigenfunctions (Erd´elyi et al., 1953)). More generally, eigenfunctions appropriate to a sink at position θs , are the cosines with arguments (n + 12 )θπ/θs , and corresponding decay constants (n+ 12 )2 π 2 /θs2 . Certain features, shared by the more complicated problem (5.9), should be noted here: Suppose the periodicity requirement had been ignored, so that θ is the coordinate on an inﬁnite straight line. Then the solution arising from the initial delta function is (5.15), and the eigenfunctions are eikθ , with eigenvalues k forming a continuous spectrum. Imposing the periodicity requirement collapses this continuum into a discrete set −n2 or −(n + 1/2)2 as if one is dealing with a set of discrete bound states. Of course, these states are ‘bound’ only as a result of the kinematic constraint of periodicity, and do not arise from any potential wells. If, at time zero, the angle had been at θ0 2 ∞ rather than at 0, the solution would have been n=0 e−n t cos n(θ − θ0 ). Using the addition formula for the cosine shows that, in this case, we should have to use the sin nθ eigenfunctions, as well as the cosines. This performance can be repeated in the more complicated case (5.9). We ﬁrst write down the solution, ignoring the requirement of single-valuedness. As shown in many texts on solid state physics, e.g. Ashcroft and Mermin (1976), the eigensolutions then have the form ψk,n (θ) = eikθ un (θ, k)

(5.22)

with energies λ(k, n), and with θ allowed to range from minus to plus inﬁnity. We shall be interested only in the real part of this function. Here, k is a continuous variable, likewise ranging over all positive and negative values. In the case of free diﬀusion, λ = −k 2 . In the present case, the curve of λ versus k breaks up into zones, with jumps at so-called zone boundaries every time k is of the form K/2, where K is any vector of the ‘reciprocal lattice’. Here, the ‘lattice spacing’ is 2π, and K is any integral multiple of 2π/(lattice spacing). So K is any integer, and the ‘zone boundaries’ are at all half-integers and integers. The curve of λ versus k has the form in Fig. 5.2. The so-called Bloch functions un (θ, k) are periodic

REVERSAL IN 2D

107

λ

K=1/2

K=1

k

Fig. 5.2. Band structure of a one-dimensional periodic lattice. The solution of the transformed diﬀusion problem has the same structure in k-space. functions of θ. Evidently at a zone boundary with integral K, the functions ψK,n are likewise periodic in θ; for half-integral K they are antiperiodic. Thus it appears that the role of periodic eigenfunctions cos nθ of the free diﬀusion problem are now played by the functions un (θ, K), with integral K. In fact, K might as well be replaced by the index n, on the understanding that K (or equivalently, n) is an integer, since we are not interested in an absorbing barrier. (We need the reﬂecting barrier at π, since the probabilities of ‘clockwise’ and ‘anticlockwise’ diﬀusion are equal, so that zero current ﬂows through π.) To ﬁnd how a particle placed into the lattice at a deﬁnite position θ = 0 will in time diﬀuse, Green’s function for this problem: (5.23) u∗n (0, k)un (θ, k) e−λ(k,n)t−ikθ dk ψ(θ, 0; t) = Re n

must be found (of course, in the quantum case, t is replaced by it). The case of an initial ﬁnite θ requires only a trivial generalization, stated at the end of this section. The k-integration is broken up into integrals from the bottoms, n, of the even bands to their tops, n + 1/2 :

∞

−∞

f (k)dk →

n

0

1 2

f (k, n)dk

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MAGNETIZATION REVERSAL

where f (k, n) is the value of f (k) at the bottom of the nth band. In particular, λ(k, n) = λn + n (k),

(5.24)

where n (k) starts out quadratically with positive curvature near k = 0, and ends up with negative curvature near k = 1/2. From expression (5.23), the response to the periodic δ-function δP (θ − θ1 ) follows by superposition: ψ(θ + 2πm, 0; t). (5.25) u(θ, 0; t) = m

Since the un (θ, k) are periodic in θ, the sum over m involves the exponential in (5.23) only, and is negligible unless k equals an integer or zero. This integer must 1 be 0, if it is not to lie outside the integrand 02 . The result is thus u(θ, 0; t) =

u∗n (0, n)un (θ, n)e−λn t

(5.26)

n

This is exactly the result that would have been obtained, had we restricted θ to the range 0 to 2π, and solved the diﬀusion equation with reﬂecting boundary conditions at π. The eigenfunctions found that way are exactly the Bloch functions un (θ), at the bottom of every other band, and they will henceforth be denoted by un (θ). They correspond to the cos nθ in the case of free diﬀusion on the circle and λn corresponds to n2 . We may call u(θ, 0; t) the Green function −n2 t of our problem. It is the generalization of the series e cos n0 cos nθ for free diﬀusion. Just as in free diﬀusion, u(θ, 0; t) converges slowly for small t, but extremely well for large t. Thus, if one is interested in pursuing the reversal process almost all the way to completion, it is best to solve the problem in the single cell, and keep only the ﬁrst few terms. From the viewpoint of magnetic recording practice, such experiments are not particularly useful. In practice, it is suﬃcient to consider reversal to be complete when the ﬁnal θ has a value considerably less than π. The reason is that the individual particle sees the switching ﬁeld only for a limited time. If within that time the ﬁeld has managed to get ‘most’ of the magnetization well over the anisotropy barrier that will remain after the applied ﬁeld has departed, the particle will take care of itself and more or less reach the reversed equilibrium state by unaided diﬀusion. (Of course, this means that one must tolerate a certain error due to the magnetization vector returning to θ = 0, but this error is not vastly greater than the error entailed by ignoring the ﬁnite lifetime of the fully reversed state.) Thus, for practical purposes, one needs to examine the complete series for an angle suﬃciently large to ensure eventual switched equilibrium after departure of the ﬁeld. One needs an expression that interpolates between the solution (5.26) useful only for large times, and the analog of ϑ3 , good only for short times. In common with the long-time limit, this expression likewise involves features of the eigenfunction expansion, so we begin with a discussion of that limit.

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5.3.1

109

Reversal in the long time limit

We now return to the pursuit of u all the way to the immediate neighborhood of π, for values of t well beyond the ‘singular’ regime, with the solution a series of decaying exponentials. The eigenfunctions describing reversal from the initial state δP (θ) are even solutions of equation (5.9), with a totally reﬂecting wall at π, corresponding to the cosnθ for free rotation. Figure 5.3 shows the potential −V (θ) for two values of H, with HK = 5. (Equation (5.9), rewritten as a ‘genuine’ Schr¨ odinger equation, will involve −V rather than V ). The lowest eigenvalue is λ = 0, with eigenfunction u0 = exp(−E(θ)/2), corresponding to the equilibrium solution ρ = exp(−E(θ)) – see eqn (5.8).) At ﬁrst sight, it would seem that the

–V(θ) H= 1, HK =5

3 2 1

π/4

π/2

3π/4

π

θ

–1 –2 –3

–V(θ) 20 15

H =8, HK =5

10 5 π/4

π/2

3π/4

π

θ

–5 –10

Fig. 5.3. The potential in the Schroedinger type equation for Hk = 5, and two values of H.

110

MAGNETIZATION REVERSAL

potential wells in these pictures are so deep that they might cause highly localized, genuine bound states with negative λ to appear. However, the deeper the wells, the greater their curvature at the minima. The result is that the zero point energy is so large that the only marginally ‘bound state’ has λ = 0, and there are no states with λ below zero, a result conﬁrmed by the computations that follow. As the ﬁelds become very large, the ratio of zero point energy to well depth approaches one (from above). This means that, if the range of θ were unlimited, there would be no bound state, but a resonant scattering state in which θ spends a long time near the θmin ’s. (Of course, the periodicity requirement once again causes the continuum states to collapse into a discrete set.) A few properties of the eigenfunctions un and eigenvalues λn of the characteristic equation −λn un =

∂ 2 un + V (θ)un ∂θ2

(5.27)

should be noted here: 1. Evidently V (θ, H) = V (π − θ, −H). A change of variable from θ to π − θ in eqn (5.27) then shows that the eigenvalues λn must be independent of the sign of H. This may seem surprising at ﬁrst sight (but only at ﬁrst sight) because one feels that the eigenvalues are a measure of switching speed for large negative H, and quite small for positive H. But this is not the case; the eigenvalue spectrum is a measure of attainment of equilibrium, whatever initial conditions are imposed, and whatever ﬁnal conditions are prescribed. 2π 2. According to eqn (5.9), 0 W (θ, t)dθ is obviously a constant independent 2π of t. As the result, the integrals 0 un (θ)e−E(θ)/2 dθ are zero for all n, with the exception of n = 0. To prove this, multiply the left hand side of the characteristic equation for un by e−E(θ)/2 , and keep on integrating by parts the term involving un . This exactly undoes the process by which the equation for the un was found 2π from the equation for W, and ﬁnally cancels 0 V (θ)un (θ)e−E(θ)/2 dθ. . The potential V (θ) is even in θ. Therefore the eigenfunctions have period 2π, and are either even or odd in θ. Unless the magnetization initially point away from θ = 0, the odd ones are of no interest here, since they go to zero at θ = 0. As already noted, the lowest eigenfunction is u0 (θ) = exp(−E(θ)/2), and is positive everywhere in (0,2π). Its eigenvalue λ0 is zero, for any and all values of H and HK . The next lowest eigenstate is odd about θ = π, going to zero at π, corresponding to an absorbing barrier there. The next one is even about π, and has two zeroes, symmetrically disposed about π, and so on. Here, we are interested only in the eigenfunctions even about π since we do not have absorption at π. The index n will be assumed to refer to these eigensolutions only. If W (θ, t) starts out proportional to δP (θ), then u(θ, 0; t) starts out as eE(0)/2 δP (θ). Subsequently, u becomes u(θ, 0; t) = eE(0)/2

∞ n=0

un (0)un (θ)e−λn t

(5.28)

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111

Then the unnormalized W (θ, t) is e−(E(θ)−E(0))/2

∞

un (0)un (θ)e−λn t

n=0

To normalize W to unity, this is dividedby the integral of this expression from 2π 0 to 2π. But (because of orthogonality) 0 e−E(θ)/2 un (θ)dθ = 0 for n > 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.

114

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

REVERSAL IN 2D

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.

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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.

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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

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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

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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

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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.

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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

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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

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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

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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 <2, there is no separatrix if r < 1, so that all curves are closed. For r = 1, the separatrix is µ = ±2. For r slightly less than one, it becomes almost rectangular as in Fig. 6.1b), crossing the ψ-axis just above −π/2 and below π/2. This entire picture is repeated, in a slightly diﬀerent order, centered on ψ equal to any integral multiple of π. As a result, under conditions of either chaos or complex discrete spectra, the orbit will frequently jump between curves around these various centers. So much for the phase portrait. To ﬁnd an analytic solution in the time domain, the solution (6.9) for µ is substituted in the right hand side of the

134

MAGNETIZATION REVERSAL

second of eqns (6.7), which becomes dψ = µ (r − cos ψ) dt = ± 2 (r + cos ψ) (r + cos ψ − c)

(6.10)

with time measured in units of 1/κ12 . This can be solved in terms of a combination of elliptic integrals, but is more easily evaluated numerically when needed. If r < c + 1, that is if, at ﬁxed r, |µ0 | < 8/(r + 1), the system oscillates with a period given by arccos(c−r) dψ T =4 (6.11) (r + cos ψ)(r + cos ψ − c) − arccos(c−r) otherwise ψ acquires a secularly growing part. On the separatrix, c = r − 1, the period T becomes inﬁnite, as is always the case on a separatrix. In the nonoscillatory region, i.e. outside the separatix, time t and phase ψ are related by ψ dψ t=2 (6.12) (r + cos ψ )(r + cos ψ − c) 0 Setting ψ = 2πn + φ and t = γn + τ, where n is any integer, this gives 2π dψ γ=2 , (r + cos ψ )(r + cos ψ − c) 0 φ dψ τ =2 (r + cos ψ )(r + cos ψ − c) 0

(6.13)

If s212 = 2/3, and the ‘single phase’ terms in eqn (6.5) are ignored, the roles of µ and m are reversed: µ is a constant of the motion, while m can oscillate. For µ = 0, the phase plane proﬁle is now given by 1 − 1/4m2 sin χ dm = −κ12 (6.14) dχ 2H + m ((K + 1)/κ12 − 1) cos χ which is easily integrated only for H= 0. When H =0, the results for s12 = 0 can be taken over directly, replacing 12 K + + κ12 by 12 K + − κ12 . If H = 0, a necessary and, for practical purposes, suﬃcient condition for closed trajectories in the phase plane is that there be a vertical tangent somewhere. According to eqn (6.11), this requires that m=−

−2H ((K + 1)/κ12 − 1) cos χ

(6.15)

somewhere. But the absolute value of m must be less than 2. Thus, even in the most favorable case, (χ = π), oscillations can occur only if |H| < |(K+1)/κ12 −1|.

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135

When single-phase terms are retained, and c is close to its separatrix value, the motion will become chaotic in all cases, even for s12 = 0, in the same manner as the driven pendulum. When s12 is neither zero nor 2/3, the system has two degrees of freedom, and stochastic motion becomes possible, whether or not the single-phase terms are retained.2 In the absence of single-phase terms, there will still be some islands of regular motion in some regions of parameter space. An extensive study of these and the associated KAM tori (see Lichtenberg and Lieberman, 1983) is beyond the objectives of this book. Figures 6.1c and 6.1d are calculated for s12 = 0, so that m is no longer a constant of the motion. The points of intersection of the orbit in the now four dimensional (ψ, µ, χ, m) space with the (ψ, µ) plane form an irregular set, the so-called Poincar´e map, even for quite small values of s12 . Still, this map does bear vague traces of the integrable case of Figs. 6.1a and 6.1b. In fact, the irregularity does not necessarily mean that the motion is chaotic in the accepted sense of the term; rather, it may be multiply periodic. To decide the issue, the Fourier power spectra of the separate variables must be considered. If these spectra are ‘clean’, the motion, though complex, is not deemed chaotic in the sense of extreme sensitivity to initial conditions. To be certiﬁably chaotic, the frequency spectrum must resemble that of broad band random noise. Figures 6.2a through 6.2d show the spectra of ψ, µ, χ and m, when the ‘single phase’ terms are retained in the equations of motion. However, for applied d.c. ﬁelds well in excess of anisotropy and dipolar ﬁelds, the spectra are ‘clean’, as shown in Figs. 6.2e through 6.2g, indicating multiple periodicity rather than chaos. It might be interesting to determine which ‘route to chaos’ (Landau, Rouelle, Feigenbaum or others) the system follows as H is decreased from large (a)

ln|c(v)| 6 4 2

200

400

600

800

1000

1200

1400

v

–2

Fig. 6.2. Continued 2 For

κ12 much less than the anisotropy or applied ﬁelds, the sytem is called ‘almost integrable’, and stochastic aspects of its motion have been studied rather extensively. In the present case, we regard pairs with small κ12 as isolated singles, with only negligible chaotic motion. The case of κ12 of the same order as the ﬁelds is obviously more relevant in practical situations.

136

MAGNETIZATION REVERSAL (b)

ln(|m(v)|) 3 2 1

200

400

600

800

1000

1200

1400

v

–1 –2 (c)

ln(|m(v)|) 2

1

200

400

600

800

1000

1200

1400

v

–1

–2

–3

–4 (d)

ln(|dx(v)|/2π) 4 3 2 1

100

200

300

–1 –2 –3 –4

Fig. 6.2. Continued

400

500

v

RELAXATION DUE TO MAGNETIC MOMENT INTERACTION (e)

137

ln|c(v)| 6 4 2

200

400

600

800

1000

1200

1400

200

400

600

800

1000

1200

1400

v

–2

(f)

ln|m(v)| 2

v

–2 –4 –6 (g)

ln(|(v)|)m 1

200

400

600

800

1000

1200

1400

v

–1 –2 –3 –4

Fig. 6.2. (a) through (d) Two dipoles in chaotic motion when the line joining their centers is slightly inclined to their anisotropy axes and when the single phase terms are retained. Applied ﬁeld H = .3, K1 = K2 = .1, s12 = .01. (a) logplot of absolute value of Fourier transformed φ versus frequency ω. (b), (c) and (d), corresponding plots for µ, m, and δχ . The last of these is the diﬀerence δχ left over when the secularly increasing part of the sum-phase χ is subtracted out. Figures (e), (f), and (g) show multiple periodicity rather than chaos when the magnetic ﬁeld is increased to H = 1.5. All other parameter values are the same as in (a) through (d).

138

MAGNETIZATION REVERSAL

values towards the stochastic regime, but we do not examine this question here. When conditions are such that the phases have a secularly increasing component, such as beyond the separatrix in the integrable case, their spectra diverge at zero frequency. In practice, only trigonometric functions of the phases are needed, so that only mod(ψ, 2πn), mod(χ, 2πn) are of interest, and the spectra of these are ﬁnite at the origin. This section is concluded with a consideration in support of the argument in the introduction, that, for weak enough coupling to the reservoir, stochasticity will be maintained even in the presence of intrinsic damping. Consider the simple undamped pendulum in the absence of any driving ﬁeld, so that its motion is not chaotic. But, especially if its motion in the phase plane is close to the separatrix, a very small driving ﬁeld renders it stochastic. If damping unrelated to the driving ﬁeld is added, the stochasticity is suppressed, and the orbit spirals into a point. But if the drive is large enough, it can maintain the orbit suﬃciently close to the separatrix to ensure continued chaos. As shown in Chapter 4, if the heat bath is composed of harmonic oscillators, the damping constant is equal to the square of the coupling constant g to the bath, times the ﬁrst moment, τ, say, of the autocorrelation function of the bath, whereas the root mean square value of the random driving ﬁeld is only linear in the coupling constant and is roughly equal to g kB T /ω, where ω is a typical frequency of the individual reservoir oscillators. Ignoring the nonlinearity of the pendulum, the damping term in its equation of motion near the separatrix is approximately equal to g 2 πΩτ /2, for an angular excursion π/2, and a natural pendulum frequency Ω. So the damping term is down from the driving term by one power of the coupling constant. In addition, taking into account the nonlinearity means that Ω goes to zero at the separatrix. This suggests that, for weak coupling, chaos persists in the presence of damping. 6.2.2.1 Diﬀusion in the absence of thermal agitation In previous chapters, derivations of the Fokker-Planck equations were based on coupling to an external reservoir. In an appropriate sense, a system in stochastic motion should be able to provide its own reservoir, without the beneﬁt of thermal agitation. In areas such as ﬂuid dynamics and plasma physics that draw heavily on the methods of nonlinear dynamics, this possibility is taken into account. Undoubtedly, well established chaos allows the phasepoint to visit much of the accessible phase space. So one might hope to establish a diﬀusion equations for the probability distribution of the ‘slow’ quantities, usually the action variables. Unfortunately, this is not generally possible (see Zaslavski, 2005). Particularly in a system with only two degrees of freedom, chaotic motion does not normally cover the accessible phase space uniformly. Chaotic regions are bordered by KAM tori, leaving some uncovered islands. Some regions, though chaotic, may turn out to be ‘sticky’, with the phase point favoring them excessively. Quite obviously, if the averaged autocorrelation functions of deviations from the mean path decay more

MORE DENSE ARRAYS OF MANY INTERACTING PARTICLES

139

slowly than exponentially, so that all their moments diverge, the Fokker-Planck equation cannot hold in its simplest form.3 For systems with more than two degrees of freedom, KAM tori no longer divide the phase space generically, chaos becomes more uniform, and an F-P equation becomes somewhat more plausible (Kaufman, 1971). Nevertheless, we shall not pursue this matter further, and in Section 6.3 adopt a less risky (but much harder to evaluate quantitatively) approach to the establishment of more or less uniform phase space coverage. We conclude this section with a numerical example that shows that interaction of two closely spaced particles with a third more remote one favors chaos, as one would expect. In the simplest case, the three interacting particles, with coupling constants κ12 , κ13 , and κ23 are assumed to have their anisotropy axes all lined up along the lines joining them, so that s12 = s13 = s23 = 0, and the interactions reduce to pure exchange. Therefore m1 + m2 + m3 is a constant of the motion, and therefore there are just two degrees of freedom. Any two particles, with coupling to the third neglected, gives an integrable motion, a (µ, ψ)-resonance of the pair, as described in Section 6.2.1. Suppose that particles 1 and 2 are close together, with particle 3 more distant from 1 and 2. Then κ12 is larger than κ13 and κ23 . If the distance is very large, particle 3 becomes irrelevant, and the motion of 1-2 is integrable. As the distance diminishes, i.e. as κ13 and κ23 grow, the possibility of 1-3 and 2-3 resonances cannot be ignored. The motion of all three then becomes chaotic, the more so, the closer the approach. Figure 6.3 was computed for the case κ13 and κ23 practically equal, and equal to 20% of κ12. . The Fourier spectra indicate chaos. ‘Single phase’ terms are absent, so the stochasticity cannot be due to time dependent drive. If any or all of the sij are not zero, the system has three degrees of freedom, and therefore will be ‘even more’ chaotic.

6.3

More dense arrays of many interacting particles

In as much as dipolar coupling has no characteristic length scale, deciding whether the picture of an array of pairs, uncoupled from one another and from an array of single particles, makes sense is a subjective matter. However, the extent to which such a separation might be reasonable may be judged as follows. If the total density of particles is ρ, the probability of ﬁnding ν particles within a volume υ ≈ rc3 is Pν =

e−υρ (υρ)ν ν!

3 Of course, this problem does not arise, if, as is the usual practice in establishing the FP equation, the randomness is imposed mathematically rather than produced physically by chaotic dynamics. For example, a Gaussian random process with rapidly declining correlations raises no such questions. This contrasts with a L´evy process, which may be closer to dynamics (see Zaslavski, 2005).

140

MAGNETIZATION REVERSAL

ln(|m2(ω)|)

ln(|m1(v)|) 4

4

2 50

100

150

200

250

300

–2

2

v

–2

50

100

150

200

250

300

v

–4

–4

–6

–6

–8

–8

ln(|m3(v)|)

ln(|m1(v)|)– ln(|m2(v)|)

2 –2 –4 –6 –8 –10 –12

50

100

150

200

250

300

v

1 50

100

150

200

250

300

v

–1 –2 –3

Fig. 6.3. Chaotic motion of three interacting dipoles, with κ13 and κ23 equal, each equal to 20% of κ12 . All three anisotropy axes aligned, and aligned with the line joining all three, so that s12 = s13 = s23 = 0 and single phase terms are absent. Because the sum of the 3-components of the m’s is conserved, the system has only two degrees of freedom. rc is some cutoﬀ radius. The ratio of pairs to singles is thus P2 r3 ρ ≈ c P1 2 So in an array in which the available volume 1/ρ per particle, is much larger than rc3 , the separation makes sense. For arrays with higher density, it is still possible to gain some insight by beginning with a weak coupling approach. For this purpose, we have to outline the procedure for ‘almost integrable’ nonlinear systems of N degrees of freedom in general. The Hamiltonian for the system is written in terms of the set of N action variables Ji and N angle variables φi . + κV (J , φ ) H = H0 (J) where κ is regarded as small, at least to begin with. For κ = 0, the equations of motion are J˙i = −∂H0 /φi = 0, φ˙ i = ∂H0 /Ji = ωi (J ), say.

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141

V is expanded in a multiple Fourier series: i(n·φ) V = Vn (J)e n

In lowest approximation, φ is replaced by ω t, where ω is the vector with components ωi . In that lowest approximation )t i(n·ω(J) Vn (J)e (6.16) V = n

Almost all the terms in this series ﬂuctuate wildly, and contribute almost nothing to the next approximation: t ni ωi Vn dt ei(n·ω(J))t (6.17) Ji = −iκ n

0

The only exceptions are the secular terms, for which = 0, n · ω (J)

(6.18)

known as resonances. Taken at face value, they would result in an indeﬁnite, unsustainable growth of the Ji , so they cannot be handled by simple perturbation theory. We have already encountered this situation in Section 6.2, where the resonance condition could be met by the phases of the two interacting particles. This made it necessary to treat their linear combinations (sums and diﬀerences) nonperturbatively, on the same basis as the action variables (cos θ1 , cos θ2 ). When there are more than two particles, their dipolar interactions are still simpler than the most general case of a nonlinear system, because sums and diﬀerences of the phases of only two particles at a time can resonate. On the other hand, each particle can meet the resonance condition with any other particle, so the problem is still not trivial. In the next section, we outline the systematics that provides at least a qualitative description of the stochastics of a general ‘almost integrable’ nonlinear system. 6.3.1

The Arnold web

First consider a system with three degrees of freedom. For κ = 0, the twodimensional surface of constant energy is E = H0 (J1, J2, J3 )

(6.19)

in the three-dimensional space of the three J components. Resonances will occur for any triplet of n values such that + n2 ω2 (J) + n3 ω3 (J) = 0, n1 ω1 (J)

(6.20)

142

MAGNETIZATION REVERSAL

provided the three components of J lie on the energy surface. So, for a given set must lie on the curve of intersection of the surfaces n1 , n2 , n3 , the admissible J’s (6.19) and (6.20). For diﬀerent points on that curve, the resonance will have diﬀerent amplitudes and (small) frequencies proportional to κ. For suﬃciently small κ, chaos, if it occurs at all, is conﬁned to a narrow strip around the intersection curve. The actual orbit, of course, lies on a surface of total energy, not just H0. This may be visualized using the Poincar´e map: every time the actual orbit crosses the surface H0 = E, the crossing point is noted on that surface. Thus the original curve spreads out into a set of dots, hugging the original intersection curve (Zaslavski, 2005; Lichtenberg and Lieberman, 1983). For diﬀerent sets n, diﬀerent curves of intersection with H0 = E are obtained. Generically, the curves arising from two diﬀerent sets n intersect one another, forming a so-called Arnold web covering the surface H0 = E. At the points of intersection, the system can change from one of the intersecting resonance curves to the other. Thus by such changes at successive intersections, the phasepoint of the system gradually diﬀuses over the entire energy surface. This process, known as Arnold diﬀusion, is usually quite slow. However, as κ increases, the separatrices of the diﬀerent resonances bulge out. Eventually, they should overlap. According to Chirikov (1979), when overlapping becomes widespread, chaotic motion becomes general. Similarly, for a system with N degrees of freedom, the N − 1-dimensional energy surface intersects the diﬀerent N − 1-dimensional resonant surfaces in a set of N − 2-dimensional manifolds. In turn, these intersect in N − 3-dimensional conﬁgurations. The latter facilitate the spreading of resonances over the entire energy surface. For the present case of dipolar interaction, the Fourier series is rather primitive: ni and mi can only equal plus or minus one for each particle i. As an example, consider the case of three particles, in zero applied ﬁeld, and with anisotropy constants K1,2,3 respectively. Then ωi = Ki Ji and the conditions for resonance of particles (1,2) and (1,3) are planes: K1 J1 ± K2 J2 = 0 and K1 J1 ± K3 J3 = 0. Their curves of intersection with the energy surface E = (K1 J12 + K2 J22 + K3 J32 )/2 in turn intersect in a point with ﬁrst coordinate 8 K2 K3 , (6.21) J1 = ± K1 (K2 K3 + K3 K1 + K1 K2 ) the two others following by cyclic interchanges. In this simple case, one of the two possible resonances of particles 2 and 3 occur automatically with the resonances (1,2) and (1,3); however, if (1,2) and (1,3) intersect at (J1 , J2 , J3 ) that extra (2,3)-resonance curve intersects the (1,2) resonance curve at (J2 , J1 , J3 ). When there is an applied ﬁeld H, then if the gyromagnetic ratios of the particles are all equal, the origin of the energy ellipsoid is displaced to a position (H/2K1 , H/2K2 , H/2K3 ), whereas the resonant surfaces are still planes through the origin. Hence, for large enough ﬁeld H, they do not cut the displaced ellipsoid at all, and the resonances, and any possible chaos, is suppressed, as expected. This conclusion applies also in the case of general values of N. On the other hand,

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143

for values of H less than the smallest anisotropy ﬁeld, and reasonable values of κ, widespread stochastic behavior should result. 6.3.2

Relevance to magnetic relaxation and reversal

From previous sections it appears that, for H well in excess of anisotropy, the resonances cannot result in self-generated stochastics. The dipolar interaction may be rather complicated, but, barring collective eﬀects, suﬃciently high order perturbation theory may then suﬃce. On the other hand, in the ‘tail end’ of the coercive ﬁeld versus switching pulse length (i.e. for small coercive ﬁelds) it becomes necessary to assess the eﬀect of stochastics for coupling to the bath less than the critical value discussed in Section 6.2.1. The same considerations are necessary in an estimate of the lifetime of a non-equilibrium conﬁguration upon completion of a reversal, once the switching ﬁeld is removed. Actually, an argument may be made that the essential complexity of chaos can lead to essential simplicity of the relaxation process. We present the argument here, but cannot deny that it is rather speculative. As already noted, intrinsic chaotic motion alone cannot switch the system, since that motion conserves energy, and a suddenly applied switching ﬁeld leaves the system in a relatively high energy state. Contact with a thermal reservoir is needed. As discussed in Section 6.2.1, if that reservoir is such that its coupling to the system is greater than a certain critical value, chaos will tend to be suppressed, and the calculations of Chapter 5 will not be greatly aﬀected. If global chaos does prevail, some fairly drastic assumptions are needed to allow further progress. Our main assumption will be that in the N -particle system, resonance overlap has ensured a uniform and rapid coverage of the energy surface at any given value of energy, whereas that energy itself changes slowly. In a suﬃciently dense array of particles, this is not necessarily an unlikely scenario. (The rate at which uniformity over an energy surface is attained must be proportional to κ, whereas the rate of decay of energy must be of order α, and cases of α << κ can certainly arise.) Then at each stage of the reversal process, the phasepoint should uniformly cover the energy surface H = E, and we have only a single reactant, the energy itself. The situation then resembles the weak friction limit discussed by Kramers for the case of one degree of freedom (Kramers, 1940), except that here we are faced with a many-body problem. Kramers assumed that a single reactant would move on a given orbit in its own phase space a long time before undergoing a collision moving it towards and over the barrier. He speciﬁed the orbit in terms of the exact action variable of that orbit. In the present case we should seek a diﬀusion equation for the entire set of action variables alone; the phase variables are of no interest, except in so far as they form resonances. Derivation of such an equation was presented, with certain caveats, by Kaufman (1971), (see also Suhl, 1988). Even if this procedure turns out to be rigorously correct, which is not certain, it would be very diﬃcult to evaluate in the present case. We therefore adopt a greatly simpliﬁed picture. Attainment of uniformity

144

MAGNETIZATION REVERSAL

over each energy surface is assumed to be so rapid that the probability distribution ρ is a function of energy E alone. This assumption ignores the possible ‘patchiness’ of coverage of the energy surface noted in Zaslavski (2005); however, the global chaos associated with the large number of degrees of freedom makes uniform coverage quite likely. We start with the equation of continuity in phase space, allowing for dissipative current ﬂow j = α(E)ρ∇E, where the gradient operator is still with respect to all degrees of freedom: ∂ρ = {E, ρ} + ∇ · j ∂t

(6.22)

(In simple cases, α may be independent of E, but not for very general coupling to the reservoir.) On an extremely short time scale, the phasepoint stays on a surface of given E, so j has no component within that surface, and the attainment of uniform coverage is governed by ∂ρ = {E, ρ} = 0 ∂t

(6.23)

alone. Elimination of the phases by the Kaufman (1971) perturbative procedure, for α = 0, leads from eqn (6.22) to an equation for the slow action variables. Carried to second order in the coupling constants κ, eqn (6.22) leads to a diﬀusion equation in the space of the action variables alone: ∂ρ ∂ ∂ρ = Dα,β (J) ∂t ∂Jα ∂Jβ α,β

We adopt the gross simpliﬁcation of choosing the energy E itself as the dominant slow variable, and replace the diﬀusion tensor by a scalar function D(E) alone. Furthermore, we allow ρ to depend on all variables only through E. Then, instead of (6.22), we get ∂ρ = ∇ · (D(E)∇ρ(E) + α(E)ρ(E)∇E) ∂t

(6.24)

∂ρ . Further, by hypothesis, and, by hypothesis, ∇ ρ(E) is to be replaced by ∇E ∂E the components of ∇E within the surface E do not change ρ; therefore we are interested only in the rate of change of E normal to the surface E. That rate of change is ∇E · (∇E/|∇E|) = |∇E| so eqn (6.24) becomes

∂ρ ∂ ∂ρ = + α(E)ρ(E) (6.25) |∇E| D(E) ∂t ∂E ∂E in close analogy with the ‘weak friction’ equation in the Kramers’ (1940) paper. This corresponds to Kramers’ (1940) treatment of the velocity dependent FokkerPlanck term, in which |∇E| simply appears as the velocity u, eventually averaged over the energy surface of his single degree of freedom. Analogously, we here

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145

replace |∇E| by |∇E|E , its average over the energy surface E, and identify that average with the reciprocal density of states 1/ν(E). The ﬁnal assumption is that, even for this emaciated form, the Einstein relation will hold. So, to achieve Boltzmann equilibrium, the ratio α(E)/D(E) must be independent of E, and equal to 1/(kB T ). We forego a full analysis of the time dependent solutions of (6.25), and restrict the remaining argument to the conventional view of reaction kinetics. This assumes an inexhaustible reservoir of reactants in the originating well at zero energy with density ρ0 , producing a steady current I over the energy barrier. The reaction rate is then deﬁned as R = I/ρ0 . So we need to solve

1 ∂ρ + α(E)ρ(E) = I D(E) ν(E) ∂E Assuming the Einstein relation α(E) = D(E)/(kB T ) to hold at each energy, the equation can be written ∂ E/kB T ν(E) I e−E/kB T ρe = ∂E D(E) and integrating it from the well at zero energy to EB , the barrier energy, gives a reaction rate −1

EB ν(E) dE (6.26) R= D(E) 0 −1

EB ν(E) = (kB T ) dE eE/kB T α(E) 0 if the density at the top of the barrier is negligibly small. Assuming the Einstein relation at each and every energy E implies that, for some reason, the distributive mechanism of chaos described by D is not totally independent of the intrinsic damping mechanism described by α. One can, of course, write down a formula for the rate without making this assumption if one has other information on the energy dependence of D/α. 6.3.3

Eﬀective single-variable relaxation from causes other than chaos

Consider a regular lattice of particles, and ignore the question of chaos altogether. Some kind of systematic collective motions will be most natural in this case. Randomly placed imperfections in the lattice will disorganise the collective motions, usually elastically, so that the total energy does not change. Once again we have distributive damping. If the character of the imperfections is ‘sufﬁciently’ random, the energy surface will be covered uniformly in a short time. If that time is short compared with the intrinsic damping rate, measured by α in Section 6.3.2, we have essentially the same situation as in Section 6.3.2, and the Kramers’ low friction regime applies once again. The question arises how

146

MAGNETIZATION REVERSAL

one might determine the validity of this picture in practice. Possibly the linear dependence of the rate on temperature can oﬀer a clue. 6.4

Magnetization reversal and the magnetization process in large, dense systems

In a very dense array of strongly coupled particles or in a continuous ferromagnetic medium, the starting point for a description of the reversal process should be a spatially ordered state of the magnetization. In the absence of imperfections, any non-uniformity in the order should be the result of domain formation needed to minimize total free energy. Normally the required number of domain walls needed to achieve this is not very large. To begin with, consider a very long cylindrical sample, magnetized along its axis. There is, then, no appreciable demagnetizing ﬁeld, and no need to introduce domain walls. Suppose that the magnetization direction is to be reversed completely coherently, simultaneously along its entire length, without any non-uniformities appearing. If the sample is of volume V, an energy barrier of height πM 2 V arises due to the transverse demagnetizing ﬁeld 2πM, reached when the magnetization is at right angles to the axis. At zero absolute temperature there is no thermal assistance of the reversal, and then a ﬁeld at least equal to 2πM is needed to annul the barrier and to ensure the switch. So the coercive ﬁeld is 2πM, and the energy required for reversal is πM 2 V. Switching the ﬁeld back and forth then gives an M versus H rectangular curve known as the Stoner-Wohlfarth hysteresis loop. When thermal agitation is present, and time is no object, there will still be switching even in zero ﬁeld, with probability given by the Boltzmann factor proportional to exp(−πM 2 V /kB T ). (For faster switching, a ﬁeld is needed and is given by the curves in Chapter 5. The necessary coercive ﬁeld increases rapidly with decreasing switching time.) From these considerations it should be clear that uniform reversal is energetically unfavorable. Even though spatial non-uniformity costs some exchange energy, a compromise may be reached if the non-uniformity reduces the demagnetizing energy, and the compromise may reduce the eﬀective barrier height. For example, in the case of the cylindrical rod, we may consider a so-called buckling process in which the transverse component of the magnetization ﬂuctuates between positive and negative values. This reduces the average surface charge on the cylinder walls and hence the demagnetizing energy. Similarly, if the tip of the magnetization vector traces out a spiral around the cylinder axis, the transverse demagnetization energy is diminished, and for the right choice of the pitch of the spiral, this reduction is not totally cancelled by the attendant exchange energy. (This process is called curl to small perturbations ing.) Calculations of these eﬀects begin by subjecting M having either buckling or curling symmetry, and periodic dependence on the z-direction (see, for example, Frei et al., 1957). If, in the presence of the switching ﬁeld, these perturbations show unstable growth for the appropriate choice of period, it is reasonable to suppose that, when extended beyond the linearized

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

147

approximation, they will result in reversal. This was indeed found to be the case in computer simulations (see, for example, Schabes and Bertram, 1988; Fredkin and Koehler, 1988). As expected, the resulting hysteresis loop is much thinner than the Stoner-Wohlfarth loop. However, because these non-uniform states still involve the entire sample, the barrier, though less than in the Stoner-Wohlfarth case, is still proportional to sample volume. Experiments, on the other hand, show that this is not the case (Li and Metzger, 1997). Fitting the data to the Boltzmann factor yields a so-called magnetic volume very much smaller than the physical volume. This suggests that reversal takes place by highly localized nucleation, followed by domain wall motion. If domain wall motion is considered barrier-less and rapid, the apparent volume is then reduced to the total magnetic volume of the nucleation centers. (However, in a real sample, imperfections may hang up the domain walls, and the magnetic volume will then be larger by an amount depending on the barriers impeding the wall motion.) Evidently, localized nucleation occurs most readily around some kind of imperfection, but it may also occur ‘spontaneously’ as the result of some large localized thermal ﬂuctuation. Here, we consider only the ‘imperfection’ mechanism. First, suppose that the sample whose magnetization is to be reversed is perfect, so there are no nucleation centers in its interior. However, the surface still remains fertile ground for localized nucleation. This is due to the fact that boundary conditions are, in practice, not homogeneous in the magnetization ﬁelds, contrary to the usual assumption. Making this assumption inevitably leads to sample-wide nucleation such as buckling or curling, causing the magnetic volume to be proportional to the physical volume, at odds with observations. It is easy to see why this happens on the basis of small-amplitude spin wave theory. Recall the formula for the frequency of a spin wave of wavenumber k, traveling at an angle θk to the applied ﬁeld H and to the anisotropy ﬁeld HK : ωk =

(H + HK + Jk 2 ) H + HK + Jk 2 + 4πM sin2 θk

and consider long waves with very small k almost equal to zero but with θk not equal to zero. As long as H is positive, this frequency is ﬁnite for all k, including k = 0. Then if the ﬁeld H is reversed, so that it reads −|H| in this formula, ωk will become imaginary when |H| is slightly larger than HK . This signals instability and marks the onset of uniform (i.e. Stoner-Wohlfarth) reversal of the magnetization, since k is essentially zero. For a cylindrical sample, the formula for the frequency looks rather diﬀerent. k is replaced by azimuthal, radial and axial quantum numbers, but once again, for a certain value of |H|, ωk goes to zero, and turns imaginary as |H| is increased further. The smallest quantum numbers for which this happens hopefully give the pattern of the magnetization (curling or buckling) as M reverses. These results are obtained by assuming homogeneous boundary conditions; the most popular one is vanishing along the normal to the surface. For a truly continuous of the derivative of M

148

MAGNETIZATION REVERSAL

medium, with nothing to distinguish the surface from the interior, this boundary condition is very reasonable. However, at the surface, microscopic aspects have to be taken into account. For example, in a model of magnetism with localized atomic spins, the absence of one or more nearest neighbors to a spin located at the surface weakens its tendency to maintain lineup with the interior spins, and makes its orientation more sensitive to both the applied ﬁeld and to surface pinning eﬀects. When this is taken into account, a zero frequency mode (i.e. a time independent state) localized at the surface becomes possible even for a range of positive values of H for the following reasons: The spatial dependence of all the small modes, localized or otherwise, is ultimately given by a certain second order diﬀerential equation with constant coeﬃcients. When homogeneous boundary conditions are imposed, the solutions are wavelike of the form aeikx + be−ikx , and their time dependence is oscillatory if the applied ﬁeld is less than the switching ﬁeld. Inhomogeneous boundary conditions on the other hand can be satisﬁed with an imaginary value of k = iκ. The solution is then a superposition of a decaying and a growing exponential and the latter exponential can be discarded on the grounds that the solution should go to zero as x tends to inﬁnity. Thus one ﬁnds a zero frequency mode, a bump in energy, localized close to the surface. For a large enough reversed ﬁeld it goes unstable in linearized approximation, but beyond the linearized regime, it turns into a propagating domain wall solution that sweeps out the old unswitched orientation of the magnetization. An exact mathematical treatment of localized small amplitude modes arising from inhomogeneous boundary conditions was given by Suhl and Bertram (1997) for the case of a semi-inﬁnite medium with a plane boundary. An appendix in the same reference extends that result to the full nonlinear case. (That non-perturbative treatment actually requires less mathematical equipment than the linearized case, but is harder to extend to more general geometries.) Here we brieﬂy sketch the full nonlinear planar case; the detailed treatment is found in the aforementioned appendix. The total magnetic energy per unit cross section of a semi-inﬁnite medium may be written 2 2 ∞ Hex Ms 2 dθ dφ 1 dx + sin2 θ − HMs cos θ − HK Ms cos2 θ E= 2 dx dx 2 0 where Hex Ms is the exchange energy, a length of order of a lattice spacing, Ms the saturation magnetization, and HK the total anisotropy ﬁeld, demagnetizing −4πMs2 , plus crystalline Hcrys Ms /2 (assumed uniaxial, with easy axis normal to the surface at x = 0). In extremal states, the functional derivatives of E with respect to θ and φ are zero. It is easily seen that one such √ possible state has φ independent of x. With distance measured in units of / 2, energy measured in units of Hex Ms , an extremum of E must then satisfy the equation d2 θ − h sin θ − hK sin θ cos θ = 0 dx2

(6.27)

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS

149

where h = H/Hex , and hK = −4πMs /Hex + Hcrys /Hex . To ﬁnd a localized state, we look for a solution such that both θ and dθ/dx go to zero as x goes to inﬁnity, while at x = 0 it is ‘pinned’ at some assigned value θ = θ0 . The ﬁrst integral of this equation, obtained to satisfy the ﬁrst two conditions is

dθ dx

2

θ = 4 sin 2 2

θ h + hK cos 2

2

(6.28)

Meeting the pinning condition for arbitrary values of θ0 and arbitrary positive h is possible only if the crystalline ﬁeld exceeds the demagnetizing ﬁeld so that hK cos2 (θ/2) is positive, for all θ0 , no matter how small. When the demagnetizing ﬁeld exceeds the anisotropy ﬁeld, cos2 (θ0 /2) must be smaller than h/|hK |, which can be satisﬁed for arbitrary values of θ0 only for large enough h. If the pinning value θ0 does not meet this condition, the integration constant in the ﬁrst integral cannot be chosen to give a localized state. Instead, one obtains a ‘buckling’ state in which θ oscillates as a function of x. Suppose that we start with some positive h, and a value of θ0 consistent with a localized state. If now a negative ﬁeld −|h | is applied, the initial localized state changes. In principle, a new and diﬀerent θ0 > |h |, but, more likely, localized state could arise eventually, provided hK cos2 2 the magnetization will attain a uniform reversed state by domain wall motion, as discussed below. Before considering reversal, we evaluate the height of the barrier for the case h + hK > 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
never

˜ 2 < hK (1 − h2 /h2 ) U K

˜ 2 − hK (1 − h2 /h2 ) < U ˜2 0
(π, 0) cos−1 hh , 0 K

Table 6.1.

hK and h will be assumed positive throughout. Then for h > hK , the orbit always ˜ 2 , goes leaves (0,0) along the unstable saddle direction, and for suﬃciently large U directly to the focus at (π, 0) along the stable direction of the latter. However, for ˜ 2 < h − hK the orbit instead spirals into (π, 0). If h < hK , then, for suﬃciently U ˜ , the orbit leaves (0,0) and enters the spiral ﬁxed point corresponding to small U |h|/hK < 1 (or rather one of them, see Footnote 4), and reversal is incomplete. ˜ , the orbit arrives at a ﬁxed point determined by the values Also, at a ﬁxed U of θ(0) and θ (0). When θ (0) is suﬃciently large, the simplest third ﬁxed points are avoided, and the proﬁle settles down at one of the many higher branches of the arccos function. For moderate initial values, the switch is incomplete; the wall can only switch to θ = arccos h/hK ± π, and thermal assistance would be required to reach θ = π. (To properly include thermal assistance would require establishing, and solving, a Fokker-Planck equation in the function space of θ(x). This will not be attempted here.) Figure 6.5a shows the phase plane picture in

THE MAGNETIZATION PROCESS IN LARGE DENSE SYSTEMS (a)

U=0

.8

155

du(j)/dj

.2

1

1.5 0.5

2 0.6

0.8

1.2

1.4

u(j)/π

–0.5 –1

θ(j)

(b)

3π/2

U=0 .4

π

.8 1.2 2

π/2

–20

–10

10

20

j

π/2

˜ from zero to 2 Fig. 6.5. (a) Curves in the phase plane for velocities U = αU ˜ = 0, the with h = 1.5, hK = 1, θ(0) = 4π/4, θ (0) = 0. Note that for U ˜ near 2, pattern is periodic, and the ﬁxed point is never reached, while for U it goes to the ﬁxed point with minimal delay. (b) Domain wall viewed in a ˜ . For U ˜ = 0, periodic buckling occurs on frame moving with velocity U = αU ˜ both sides of the wall, and for 0 < U < 2, buckling decays and becomes very ˜ = 2. small near U

156

MAGNETIZATION REVERSAL

the case of a complete switch (i.e. h/hK > 1) for various assumed values of the ˜ , and modest initial values of θ and θ . Figure 6.5b shows complete velocity U ˜ . Note that, for U ˜ = 0, proﬁles of moving domain walls for various values of U we ﬁnd a buckling state (i.e. persistent oscillation of θ as a function of ξ.) ˜ have been presented. So far, no means for determining the actual value of U Presumably its precise value depends on initial conditions. The class of initial conditions underlying the Kolmogorov et al. (1937) theory of the special Fisher case a = 1, b = 1 enabled these authors to ﬁnd the exact value. How˜ -values. As the switch ever, it is relatively easy to ﬁnd the permitted range of U approaches completion as the variable in the moving frame tends to inﬁnity, eqn (6.33) becomes linear in the deviation from the ﬁnal ﬁxed point π, and the above ﬁxed √ point analysis shows that non-oscillatory decay to π is possible only ˜ > 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 dx/dt = (dx/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 ∇

dx

,

(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

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