Magnetism and Magnetic Materials PDF

This page intentionally left blank  Magnetism and Magnetic Materials Covering basic physical concepts, experimental methods, and applications, this book is an indispensable text on the fascinating science of magnetism, and an invaluable source of practical reference da...

This page intentionally left blank  Magnetism and Magnetic Materials Covering basic physical concepts, experimental methods, and applications, this book is an indispensable text on the fascinating science of magnetism, and an invaluable source of practical reference data. Accessible, authoritative, and assuming undergraduate familiarity with vectors, electromagnetism and quantum mechanics, this textbook is well suited to graduate courses. Emphasis is placed on practical calculations and numer- ical magnitudes – from nanoscale to astronomical scale – focussing on mod- ern applications, including permanent magnet structures and spin electronic devices. Each self-contained chapter begins with a summary, and ends with exercises and further reading. The book is thoroughly illustrated with over 600 figures to help convey concepts and clearly explain ideas. Easily digestible tables and data sheets provide a wealth of useful information on magnetic properties. The 38 principal magnetic materials, and many more related compounds, are treated in detail. J. M. D. Coey leads the Magnetism and Spin Electronics group at Trinity College, Dublin, where he is Erasmus Smith’s Professor of Natural and Exper- imental Philosophy. An authority on magnetism and its applications, he has been awarded the Gold Medal of the Royal Irish Academy and the Charles Chree Medal of the Institute of Physics for his work on magnetic materials.  Magnetism and Magnetic Materials J. M. D. COEY Trinity College, Dublin CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521816144 © J. Coey 2009 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010 ISBN-13 978-0-511-67743-4 eBook (NetLibrary) ISBN-13 978-0-521-81614-4 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.  Contents List of tables of numerical data ix Preface xi Acknowledgements xiii 1 Introduction 1 1.1 A brief history of magnetism 1 1.2 Magnetism and hysteresis 7 1.3 Magnet applications 13 1.4 Magnetism, the felicitous science 19 2 Magnetostatics 24 2.1 The magnetic dipole moment 24 2.2 Magnetic fields 28 2.3 Maxwell’s equations 41 2.4 Magnetic field calculations 43 2.5 Magnetostatic energy and forces 50 3 Magnetism of electrons 62 3.1 Orbital and spin moments 63 3.2 Magnetic field effects 74 3.3 Theory of electronic magnetism 87 3.4 Magnetism of electrons in solids 92 4 Magnetism of localized electrons on the atom 97 4.1 The hydrogenic atom and angular momentum 97 4.2 The many-electron atom 100 4.3 Paramagnetism 106 4.4 Ions in solids; crystal-field interactions 114 5 Ferromagnetism and exchange 128 5.1 Mean field theory 129 5.2 Exchange interactions 135 5.3 Band magnetism 144 5.4 Collective excitations 161 vi Contents 5.5 Anisotropy 168 5.6 Ferromagnetic phenomena 174 6 Antiferromagnetism and other magnetic order 195 6.1 Molecular field theory of antiferromagnetism 196 6.2 Ferrimagnets 200 6.3 Frustration 203 6.4 Amorphous magnets 209 6.5 Spin glasses 218 6.6 Magnetic models 221 7 Micromagnetism, domains and hysteresis 231 7.1 Micromagnetic energy 234 7.2 Domain theory 239 7.3 Reversal, pinning and nucleation 244 8 Nanoscale magnetism 264 8.1 Characteristic length scales 265 8.2 Thin films 267 8.3 Thin-film heterostructures 274 8.4 Wires and needles 293 8.5 Small particles 295 8.6 Bulk nanostructures 299 9 Magnetic resonance 305 9.1 Electron paramagnetic resonance 307 9.2 Ferromagnetic resonance 313 9.3 Nuclear magnetic resonance 318 9.4 Other methods 329 10 Experimental methods 333 10.1 Materials growth 333 10.2 Magnetic fields 340 10.3 Atomic-scale magnetism 343 10.4 Domain-scale measurements 353 10.5 Bulk magnetization measurements 360 10.6 Excitations 368 10.7 Numerical methods 370 11 Magnetic materials 374 11.1 Introduction 374 11.2 Iron group metals and alloys 384 vii Contents 11.3 Rare-earth metals and intermetallic compounds 398 11.4 Interstitial compounds 407 11.5 Oxides with ferromagnetic interactions 410 11.6 Oxides with antiferromagnetic interactions 417 11.7 Miscellaneous materials 432 12 Applications of soft magnets 439 12.1 Losses 441 12.2 Soft magnetic materials 448 12.3 Static applications 453 12.4 Low-frequency applications 454 12.5 High-frequency applications 457 13 Applications of hard magnets 464 13.1 Magnetic circuits 466 13.2 Permanent magnet materials 469 13.3 Static applications 473 13.4 Dynamic applications with mechanical recoil 481 13.5 Dynamic applications with active recoil 485 13.6 Magnetic microsystems 491 14 Spin electronics and magnetic recording 494 14.1 Spin-polarized currents 497 14.2 Materials for spin electronics 515 14.3 Magnetic sensors 516 14.4 Magnetic memory 522 14.5 Other topics 525 14.6 Magnetic recording 530 15 Special topics 542 15.1 Magnetic liquids 543 15.2 Magnetoelectrochemistry 547 15.3 Magnetic levitation 549 15.4 Magnetism in biology and medicine 555 15.5 Planetary and cosmic magnetism 565 Appendices 580 Appendix A Notation 580 Appendix B Units and dimensions 590 Appendix C Vector and trigonometric relations 595 Appendix D Demagnetizing factors for ellipsoids of revolution 596 viii Contents Appendix E Field, magnetization and susceptibility 597 Appendix F Quantum mechanical operators 598 Appendix G Reduced magnetization of ferromagnets 598 Appendix H Crystal field and anisotropy 599 Appendix I Magnetic point groups 600 Formula index 601 Index 604  List of tables of numerical data Unit conversions rear endpaper Physical constants rear endpaper The magnetic periodic table front endpaper Demagnetizing factors 596 Diamagnetic susceptibilities of ion cores 76 Properties of the free-electron gas 79 Susceptibilities of diamagnetic and paramagnetic materials 87 Spin-orbit coupling constants 105 Properties of 4f ions 114,125 Properties of 3d ions 115 Susceptibility of metals 134 Kondo temperatures 146 Intrinsic magnetic properties of Fe, Co, Ni 150 Energy contributions in a ferromagnet 179 Faraday and Kerr rotation 190,191 Reduced magnetization; Brillouin theory 598 Model critical exponents 224 Domain wall parameters for ferromagnets 242 Micromagnetic length scales for ferromagnets 266 Antiferromagnets for exchange bias 278 g-factors for ferromagnets 314 Magnetism of elementary particles 319 Nuclei for NMR 320 Nuclei for Mössbauer effect 330 Nuclear and magnetic scattering lengths for neutrons 347 Properties of selected magnetic materials 375 Magnetic parameters of useful magnetic materials 377 Metallic radii of elements 379 Ionic radii of ions 380 Soft materials for low-frequency applications 450 Soft materials for high-frequency applications 452 Properties of permanent magnets 471,473 Mean free paths and spin diffusion lengths 499 Properties of materials used for spin electronics 516 Properties of commercial ferrofluids and nanobeads 547  Preface This book offers a broad introduction to magnetism and its applications, designed for graduate students and advanced undergraduates as well as prac- tising scientists and engineers. The approach is descriptive and quantitative, treating concepts, phenomena, materials and devices in a way that emphasises numerical magnitudes, and provides a wealth of useful data. Magnetism is a venerable subject, which underwent four revolutionary changes in the course of the twentieth century – understanding of the physics, extension to high frequencies, the avalanche of consumer applications and, most recently, the emergence of spin electronics. The reader probably owns one or two hundred magnets, or some billions if you have a computer where each bit on the hard disc counts as an individually addressable magnet. Sixty years ago, the number would have been at best two or three. Magnetics, in part- nership with semiconductors, has created the information revolution, which in turn has given birth to new ways to research the subject – numerical simu- lation of physical theory, automatic data acquisition and web-based literature searches. The text is structured in five parts. First, there is a short overview of the field. Then come eight chapters devoted to concepts and principles. Two parts follow which treat experimental methods and materials, respectively. Finally there are four chapters on applications. An elementary knowledge of electromagnetism and quantum mechanics is needed for the second part. Each chapter ends with a short bibliography of secondary literature, and some exercises. SI units are used throughout, to avoid confusion and promote magnetic numeracy. A detailed conversion table for cgs units, which are still in widespread use, is provided inside the back cover. There is some attempt to place the study of magnetism in a global context; our activity is not only intellectual and practical, it is also social and economic. The text has grown out of courses given to undergraduates, postgraduates and engineers over the past 15 years in Dublin, San Diego, Tallahassee, Stras- bourg and Seagate, as well as from the activities of our own research group at Trinity College, Dublin. I am very grateful to many students, past and present, who contributed to the venture, as well as to numerous colleagues who took the trouble to read a chapter and let me have their criticism and advice, and correct at least some of the mistakes. I should mention particu- larly Sara McMurray, Plamen Stamenov and Munuswamy Venkatesan, as well as Grainne Costigan, Graham Green, Ma Qinli and Chen Junyang, who all xii Preface worked on the figures, and Emer Brady who helped me get the whole text into shape. Outlines of the solutions to the odd-numbered exercises are available at the Cambridge website www.cambridge.org/9780521816144. Comments, correc- tions and suggestions for improvements of the text are very welcome; please post them at www.tcd.physics/magnetism/coeybook. Finally, I am grateful to Wong May, thinking of everything we missed doing together when I was too busy with this. J. M. D. Coey Dublin, November 2009  Acknowledgements The following figures are reproduced with permission from the publishers: American Association for the Advancement of Science: 14.18, p.525 (margin), p.537 (margin),14.27; American Institute of Physics: 5.25, 5.31, 6.18, 8.5, 8.33, 10.12, 11.8; American Physical Society: 4.9, 5.35, 5.40, 6.27a, 6.27b, 8.3, 8.8, 8.9, 8.15, 8.17, 8.18, 8.21, 8.22, 8.26, 8.29, 9.5, p.360 (margin), 11.15, 14.16; American Geophysical Union p.572 (margin); United States Geological Survey Geomagnetism Program: 15.18, p.572 (margin); American Society for Metals: 5.35; Cambridge University Press: 4.15, 4.17, 7.8, 7.18, 9.12, 10.16, p.573 (margin); Elsevier: 6.23, 8.2, 8.4, 11.22, 14.22, 14.23, 14.26, 15.22; Institute of Electrical and Electronics Engineers: 5.32, 8.31, 8.34, 8.35, 9.6, 11.6, 11.7; MacMillan Publishers: 14.17, 15.4c; Oxford University Press: 5.26; National Academy of Sciences:15.1; Springer Verlag: 4.18, 14.13, 14.21, 15.8, 15.21; Taylor and Francis: 1.6, 2.8b, 10.2; Institution of Engineering and Technology: 11.20; University of Chicago Press: 1.1a; John Wiley: 5.21, 6.4, 6.15, 8.11a,b, 9.9, 12.10 Fermi surfaces are reproduced with kind permission of the University of Florida, Department of Physics, http://www.phys.ufl.edu/fermisurface. Thanks are due to Wiebke Drenckhan and Orphee Cugat for permission to reproduce the cartoons on pages 161 and 531. Figure 15.3 is reproduced by courtesy of Johannes Kluehspiess. Figure 15.5 is reproduced by courtesy of L. Nelemans, High Field Magnet Laboratory, Nijmegen. Figure 15.5 is reproduced by permission of Y. I.Wang, Figure 15.17 is repoduced by courtesy of N. Sadato; Figure 15.23 is reproduced by courtesy of P. Rochette.  1 Introduction After a short historical summary, the central concepts of magnetic order and hystere- sis are presented. Magnet applications are summarized, and magnetism is situated in relation to physics, materials science and industrial technology. 1.1 A brief history of magnetism The history of magnetism is coeval with the history of science. The mag- net’s ability to attract ferrous objects by remote control, acting at a distance, has captivated countless curious spirits over two millenia (not least the young Albert Einstein). To demonstrate a force field that can be manipulated at will, you need only two chunks of permanent magnet or one chunk of permanent magnet and a piece of temporary magnet such as iron. Feeble permanent mag- nets are quite widespread in nature in the form of lodestones – rocks rich in magnetite, the iron oxide Fe3 O4 – which were magnetized by huge elec- tric currents in lightning strikes. Priests and people in Sumer, ancient Greece, China and pre-Colomban America were familiar with the natural magic of these magnets. A lodestone carved in the shape of a Chinese spoon was the centrepiece of an early magnetic device, the ‘South pointer’. Used for geomancy in China at the beginning of our era (Fig. 1.1), the spoon turns on the base to align its handle with the Earth’s magnetic field. Evidence of the South pointer’s application can be seen in the grid-like street plans of certain Chinese towns, where the axes of quarters built at different times are misaligned because of the secular variation of the direction of the horizontal component of the Earth’s magnetic field. A propitious discovery, attributed to Zheng Gongliang in 1064, was that iron could acquire a thermoremanent magnetization when quenched from red heat. Steel needles thus magnetized in the Earth’s field were the first artificial perma- nent magnets. They aligned themselves with the field when floated or suitably suspended. A short step led to the invention of the navigational compass, which was described by Shen Kua around 1088. Reinvented in Europe a century later, the compass enabled the great voyages of discovery, including the European discovery of America by Christopher Columbus in 1492 and the earlier Chinese Sheng Kua, 1031–1095. discovery of Africa by the eunuch admiral Cheng Ho in 1433.  2 Introduction Figure 1.1 Some early magnetic devices: the ‘South pointer’ used for orientation in China around the beginning of the present era, and a Portuguese mariner’s compass from the fifteenth century. When we come to the middle ages, virtues and superstitions had accreted to the lodestone like iron filings. Some were associated with its name.1 People dreamt of perpetual motion and magnetic levitiation. The first European text on magnetism by Petrus Peregrinus describes a perpetuum mobile. Perpetual motion was not to be, except perhaps in the never-ending dance of electrons in atomic orbitals with quantized angular momentum, but purely passive magnetic levitation was eventually achieved at the end of the twentieth century. Much egregious fantasy was debunked by William Gilbert in his 1600 monograph De A perpetuum mobile, Magnete, which was arguably the first modern scientific text. Examination of the proposed by Petrus direction of the dipole field at the surface of a lodestone sphere or ‘terella’, and Peregrinus in 1269. relating it to the observation of dip which by then had been measured at many points on the Earth’s surface, led Gilbert to identify the source of the magnetic force which aligned the compass needle as the Earth itself, rather than the stars as previously assumed. He inferred that the Earth itself was a great magnet.2 The curious Greek notion that the magnet possessed a soul – it was animated because it moved – was to persist in Europe well into the seventeenth century, when it was finally laid to rest by Descartes. But other superstitions regarding the benign or malign influences of magnetic North and South poles remain alive and well, as a few minutes spent browsing the Internet will reveal. Magnetic research in the seventeenth and eighteenth centuries was mostly the domain of the military, particularly the British Navy. An important civilian advance, promoted by the Swiss polymath Daniel Bernoulli, was the inven- tion in 1743 of the horseshoe magnet. This was to become magnetism’s most enduring archetype. The horseshoe is an ingenious solution to the problem of making a reasonably compact magnet which will not destroy itself in its own William Gilbert, 1544–1603. demagnetizing field. It has remained the icon of magnetism up to the present 1 In English, the word ‘magnet’ is derived through Latin from the Greek for Magnesian stone (Ŏ µαγ νης λı̄θ oς), after sources of lodestones in Asia Minor. In Sanscrit ‘SÉÖ¨¤ÉE ’ and Romance languages – French ‘l’aimant’, Spanish ‘imán’, Portuguese ‘imã’ – the connotation is the attrac- tion of opposite poles, like that of man and woman. 2 ‘Magnus magnes ipse est globus terrestris’.  3 1.1 A brief history of magnetism day. Usually red, and marked with ‘North’ and ‘South’ poles, horseshoe mag- nets still feature in primary school science books all over the world, despite the fact that these horseshoes have been quite obsolete for the past 50 years. The obvious resemblances between magnetism and electricity, where like or unlike charges repel or attract, led to a search for a deeper connection between the two cousins. Luigi Galvani’s ‘animal electricity’, stemming from his cele- brated experiments on frogs and corpses, had a physical basis – nerves work by electricity. It inspired Anton Messmer to postulate a doctrine of ‘animal A lodestone ‘terella’ used magnetism’ which was enthusiastically embraced in Parisian salons for some by Gilbert to demonstrate years before Louis XVI was moved to appoint a Royal Commission to inves- how the magnetic field of tigate. Chaired by Benjamin Franklin, the Commission thoroughly discredited the Earth resembles that of the phenomenon, on the basis of a series of blind tests. Their report, published a magnet. in 1784, was a landmark of scientific rationality. It was in Denmark in 1820 that Hans-Christian Oersted eventually discov- ered the true connection between electricity and magnetism by accident. He demonstrated that a current-carrying wire produced a circumferential field capable of deflecting a compass needle. Within weeks, André-Marie Ampère and Dominique-François Arago in Paris wound wire into a coil and showed that the current-carrying coil was equivalent to a magnet. The electromagnetic revolution was launched. The remarkable sequence of events that ensued changed the world for ever. Michael Faraday’s intuition that the electric and magnetic forces could be con- ceived in terms of all-pervading fields was critical. He discovered electromag- Réné Descartes, netic induction (1821) and demonstrated the principle of the electric motor with 1596–1650. a steel magnet, a current-carrying wire and a dish of mercury. The discovery of a connection between magnetism and light followed with the magneto-optic Faraday effect (1845). All this experimental work inspired James Clerk Maxwell’s formulation3 of a unified theory of electricity, magnetism and light in 1864, which is summarized in the four famous equations that bear his name: ∇ · B = 0, (1.1a)  0 ∇ · E = ρ, (1.1b) (1/µ0 )∇ × B = j +  0 ∂ E/∂t, (1.1c) ∇ × E = −∂ B/∂t. (1.1d) These equations relate the electric and magnetic fields, E and B at a point in free space to the distributions of electric charge and current densities, ρ and j An eighteenth century in surrounding space. A spectacular consequence of Maxwell’s equations is the horseshoe magnet. existence of a solution representing coupled oscillatory electric and magnetic 3 ‘From a long view of the history of mankind there can be little doubt that the most significant event of the nineteenth century will be judged as Maxwell’s discovery of the laws of electrodynamics’ (R. Feynman The Feynman Lectures in Physics. Vol. II, Menlo Park: Addison-Wesley (1964)).  4 Introduction fields propagating at the speed of light. These electromagnetic waves extend over the entire spectrum, with wavelength and frequency f , related by c = f. The electric and magnetic constants  0 and µ0 depend on definitions and the system of units, but they are related by √ 1  0 µ0 = , (1.2) c where c is the speed of light in vacuum, 2.998 × 108 m s−1. This is also the ratio of the average values of E and B in the electromagnetic wave. Maxwell’s equa- tions are asymmetric in the fields E and B because no magnetic counterpart of electric charge has ever been identified in nature. Gilbert’s idea of North and André Marie Ampère, South magnetic poles, somehow analagous to Coulomb’s positive and negative 1775–1836. electric charges, has no physical reality, although poles remain a conceptual convenience and they simplify certain calculations. Ampère’s approach, regard- ing electric currents as the source of magnetic fields, has a sounder physical basis. Either approach can be used to describe ferromagnetic material such as magnetite or iron, whose magnetism is equally well represented by distributions of magnetic poles or electric currents. Nevertheless, the real building blocks of electricity and magnetism are electric charges and magnetic dipoles; the dipoles are equivalent to electric current loops. Dielectric and magnetic mate- rials are handled by introducing two auxiliary fields D and H, as discussed in Chapter 2. An additional equation, due to Lorentz, gives the force on a particle with charge q moving with velocity v, which is subject to electric and magnetic fields: Hans-Christian Oersted, 1777–1851. f = q(E + v × B). (1.3) Units of E are volts per metre (or newtons per coulomb), and the units of B are newtons per ampere per metre (or tesla). A technical landmark in the early nineteenth century was William Sturgeon’s invention of the iron-cored electromagnet in 1824. The horseshoe-shaped core was temporarily magnetized by the magnetic field produced by current flowing in the windings. Electromagnets proved more effective than the weak permanent magnets then available for excitation of electric motors and generators. By the time the electron was discovered in 1897,4 the electrification of the planet was already well advanced. Urban electrical distribution networks dispelled the tyranny of night with electric light and the stench of public streets was eliminated as horses were displaced by electric trams. Telegraph cables spanned the Earth, transmitting messages close to the speed of light for the equivalent of e20 a word. Michael Faraday, 4 The decisive step for the discovery of the electron was taken in England by Joseph John 1791–1867. ì Thompson, who measured the ratio of its charge to mass. The name, derived from ηλκτ ρoν the Greek word for amber, had been coined earlier (1891 in Dublin) by George Johnston Stoney.  5 1.1 A brief history of magnetism Despite the dazzling technical and intellectual triumphs of the electromag- netic revolution, the problem of explaining how a solid could possibly be ferro- magnetic was unsolved. The magnetization of iron, M = 1.76 × 106 amperes per metre, implies a perpetually circulating Ampèrian surface current density of the same magnitude. Currents of hundreds of thousands of amperes coursing around the surface of a magnetized iron bar appeared to be a wildly implausible proposition. Just as preposterous was Pierre Weiss’s molecular field theory, dat- ing from 1907, which successfully explained the phase transition at the Curie point where iron reversibly loses its ferromagnetism. The theory postulated an internal magnetic field parallel to, but some three orders of magnitude greater than, the magnetization. Although Maxwell’s equation (1.1a) proclaims that the magnetic field B should be continuous, no field remotely approaching that A nineteenth century magnitude has ever been detected outside a magnetized iron specimen. Fer- electromagnet. romagnetism therefore challenged the foundations of classical physics, and a satisfactory explanation only emerged after quantum mechanics and relativity, the twin pillars on which modern physics rests, were erected in the early years of the twentieth century. Strangely, the Ampèrian currents turned out to be associated with quantized angular momentum, and especially with the intrinsic spin of the electron, discov- ered by George Uhlenbeck and Samuel Goudsmit in 1925. The spin is quantized in such a way that it can have just two possible orientations in a magnetic field, ‘up’ and ‘down’. Spin is the source of the electron’s intrinsic magnetic moment, which is known as the Bohr magneton: µB = 9.274 × 10−24 A m2. The mag- netic properties of solids arise essentially from the magnetic moments of their atomic electrons. The interactions responsible for ferromagnetism represented James Clerk Maxwell, by the Weiss molecular field were shown by Werner Heisenberg in 1929 to be 1831–1879. electrostatic in nature, originating from the quantum mechanics of the Pauli principle. Heisenberg formulated a Hamiltonian to represent the interaction of two neighbouring atoms whose total electronic spins, in units of Planck’s constant
= 1.055 × 10−34 J s, are Si and Sj , namely H = −2J Si · Sj , (1.4) where J is the exchange constant; J /kB is typically in the range 1–100 K. Here kB is Boltzmann’s constant, 1.3807 × 10−23 J K−1. Atomic magnetic moments are associated with the electronic spins. The quantum revolution underpinning modern atomic and solid state physics and chemistry was essentially complete at the time of the sixth Solvay Congress in 1930 (Fig. 1.2). Filling in the details has proved to be astonishingly rich and endlessly useful.5 For instance, when the exchange interaction J is negative (antiferromagnetic) rather than 5 Already in 1930 there was the conviction that all the basic problems of the physics of solids had been solved in principle; Paul Dirac said ‘The underlying physical phenomena necessary for a mathematical explanation of a large part of physics and all of chemistry are now understood in principle, the only difficulty being that the exact application of these laws leads to equations much too complicated to be soluble’ (P. Dirac, Proc. Roy. Soc. A123, 714 (1929)).  6 Introduction Figure 1.2 Participants at the 1930 Solvay Congress, which was positive (ferromagnetic) there is a tendency for the spins at sites i and j to align devoted to magnetism. antiparallel rather than parallel. Louis Néel pointed out in 1936 and 1948 that this leads to antiferromagnetism or ferrimagnetism, depending on the topology of the crystal lattice. Magnetite, the archetypal natural magnetic material, is a ferrimagnet. One lesson from a study of the history of magnetism is that fundamen- tal understanding of the science may not be a prerequisite for technologi- cal progress. Yet fundamental understanding helps. The progression from the poorly differentiated set of hard and soft magnetic steels that existed at the start of the twentieth century to the wealth of different materials available today, with all sorts of useful properties described in this book, owes more to metallurgy and systematic crystal chemistry than it does to quantum physics. Only since the rare-earth elements began to be alloyed with cobalt and iron in new perma- nent magnets from the late 1960s onwards has quantum mechanics contributed significantly to magnetic materials development. Much progress in science is made empirically, with no recourse to basic theory. One area, however, where quantum mechanics has been of central importance for magnetism is in its interaction with electromagnetic radiation in the radiofrequency, microwave Louis Néel, 1904–2000. and optical ranges. The discovery of magnetic resonance methods in the 1940s  7 1.2 Magnetism and hysteresis Table 1.1. The seven ages of magnetism Period Dates Icon Drivers Materials Ancient period −2000–1500 Compass State, geomancers Iron, lodestone Early modern age 1500–1820 Horseshoe magnet Navy Iron, lodestone Electromagnetic age 1820–1900 Electromagnet Industry/infrastructure Electrical steel Age of understanding 1900–1935 Pauli matrices Academic (Alnico) High-frequency age 1935–1960 Magnetic resonance Military Ferrites Age of applications 1960–1995 Electric screwdriver Consumer market Sm-Co, Nd-Fe-B Age of spin electronics 1995– Read head Consumer market Multilayers and 1950s and the introduction of powerful spectroscopic and diffraction tech- niques led to new insights into the magnetic and electronic structure of solids. Technology for generating and manipulating microwaves had been developed in Great Britain for the Second World War. Recent decades have witnessed an immense expansion of magnetic appli- cations. The science developed over a century, mostly in Europe, was ripe for exploitation throughout the industrialized world. Advances in permanent mag- netism, magnetic recording and high-frequency materials underpin much of the progress that has been made with computers, telecommunications equipment and consumer goods that benefit most people on Earth. Permanent magnets Samuel Goudsmit, have come back to replace electromagnets in a billion tiny motors manufac- 1902–1978. tured every year. Magnetic recording sustains the information revolution and the Internet. There have been seminal advances in earth science, medical imag- ing and the theory of phase transitions that can be laid at the door of magnetism. This long and promising history of magnetism can be envisaged as seven ages, which are summarized in Table 1.1. The third millenium sees us at the thresh- old of the seventh age, that of spin electronics. Conventional electronics has ignored the spin on the electron. We are just now beginning to learn how to manipulate spin currents and to make good use of them. Georg Uhlenbeck, 1900–1988. 1.2 Magnetism and hysteresis The most striking manifestation of magnetism in solids is the spontaneous mag- netization of ferromagnetic materials such as iron or magnetite. Spontaneous magnetism is usually associated with hysteresis,6 a phenomenon studied by James Ewing, and named by him in 1881.7 6 ‘Hysteresis’ was coined from the greek ῠσ τ ριν, to lag behind. 7 Ewing, a Scot, was appointed as a foreign Professor of Engineering at the University of Tokyo by the Meiji government in 1878. He is regarded as the founder of magnetic research in Japan.  8 Introduction Figure 1.3 M The hysteresis loop of a Ms ferromagnet. Initially in an Mr unmagnetized, virgin state. Magnetization appears as an imposed magnetic field H , modifies and eventually −Hc Hc H eliminates the microstructure of ferromagnetic domains magnetized in different directions, to reveal the spontaneous magnetization M s. The remanence Mr which remains when the applied 1.2.1 The ferromagnetic hysteresis loop field is restored to zero, and the coercivity H c , The essential practical characteristic of any ferromagnetic material is the irre- which is the reverse field versible nonlinear response of magnetization M to an imposed magnetic field needed to reduce the magnetization to zero, are H. This response is epitomized by the hysteresis loop. The material responds to marked on the loop. H, rather than B, for reasons discussed in the next chapter where we distinguish the applied and internal fields. Magnetization, the magnetic dipole moment per unit volume of material, and the H -field are both measured in amperes per metre (A m−1 ). Since this is a rather small unit – the Earth’s magnetic field is about 50 A m−1 – the multiples kA m−1 and MA m−1 are often employed. The applied field must be comparable in magnitude to the magnetization in order to trace a hysteresis loop. The values of spontaneous magnetization Ms of the ferromagnetic elements Fe, Co and Ni at 296 K are 1720, 1370 and 485 kA m−1 , respectively. That of magnetite, Fe3 O4 , is 480 kA m−1. A large electromagnet may produce a field of 1000 kA m−1 (1 MA m−1 ). Hard magnetic materials8 have broad, square M(H ) loops. They are suitable for permanent magnets because, once magnetized by applying a field H ≥ Ms sufficient to saturate the magnetization, they remain in a magnetized state when the field is removed. Soft magnetic materials have very narrow loops. They are temporary magnets, readily losing their magnetization as soon as the field is removed. The applied field serves to unveil the spontaneous ferro- magnetic order that already exists on the scale of microscopic domains. These domain structures are illustrated schematically on the hysteresis loop of Fig. 1.3 for the unmagnetized state at the origin, the saturated state where M = Ms , the remanent state in zero field where M = Mr and the state at H = Hc , the coer- cive field where M changes sign. Mr and Hc are known as the remanence and the coercivity. Magnetic domains were proposed by James Ewing and the prin- ciples of domain theory were established by Lev Landau and Evgenii Lifschitz James Ewing, 1855–1935. in 1935. 8 The terms hard and soft for magnets originated from the mechanical properties of the corre- sponding magnetic steels.  9 1.2 Magnetism and hysteresis Figure 1.4 1.0 Magnetization, Ms(T )/Ms(0) Temperature dependence 0.8 of the spontaneous magnetization of nickel. 0.6 The Curie point at 628 K is marked. 0.4 0.2 Tc 0 0 628 Temperature (K) The hysteresis loop is central to technical magnetism; physicists endeav- our to explain it, materials scientists aim to improve it and engineers work to exploit it. The loop combines information on an intrinsic mag- netic property, the spontaneous magnetization Ms which exists within a domain of a ferromagnet, and two extrinsic properties, the remanence Mr and coercivity Hc , which depend on a host of extraneous factors including the sample shape, surface roughness, microscopic defects and thermal his- tory, as well as the rate at which the field is swept in order to trace the loop. 1.2.2 The Curie temperature The spontaneous magnetization due to alignment of the atomic magnetic moments depends on temperature, and it falls precipitously to zero at the Curie temperature TC. The magnetic ordering is a continuous thermodynamic phase transition with a λ-shaped anomaly in specific heat, associated with disordering of the atomic dipole moments. Above TC , Ms (T ) is zero; below TC , Ms (T ) is reversible. This behaviour is illustrated for nickel in Fig. 1.4. The Curie temperatures of the three ferromagnetic metals, iron, cobalt and nickel, are 1044 K, 1388 K and 628 K, respectively. No material is known to have a higher Curie temperature than cobalt. Magnetite has a Curie temperature of 856 K. 1.2.3 Coercivity The progress in the twentieth century which has spawned such a range of magnetic applications can be summarized in three words – mastery of coercivity. No new ferromagnetic material has been discovered with a magnetization greater than that of ‘permendur’, Fe65 Co35 , for which Ms = 1950 kA m−1 , but coercivity which barely spanned two orders of magnitude in 1900, from the Pierre Curie, 1859–1906. softest soft iron to the hardest magnet steel, now ranges over eight orders of  10 Introduction Figure 1.5 Hard 107 Sm–Co Nd–Fe Progress in expanding the 106 range of coercivity of Ba ferrite Co–Cr 105 magnetic materials during Alnico the twentieth century. Lodestone 104 Co steel W, Cr steel 103 Steel Steel Iron Iron 102 NiZn ferrite Ni–Fe 101 Ni–Fe–Mo 1 Soft aFe–Co–B 1000 1900 2000 Year magnitude, from less than 0.1 A m−1 to more than 10 MA m−1 , as shown in Fig. 1.5. 1.2.4 Anisotropy The natural direction of magnetization in a microscopic ferromagnetic domain H is usually constrained to lie along one or more easy axes. Since magnetism is q M associated with circulating electron currents, time reversal symmetry requires Eas that a state with a certain magnetization distribution M(r) should have the same ya xis energy as the state with reversed magnetization along the same axis, −M(r). Magnetization is not This tendency is represented by the anistropy energy Ea , of which the leading necessarily parallel to term is applied field, unless H is applied in an easy Ea = Ku sin2 θ , (1.5) direction. where θ is the angle between the direction of M and the easy axis. Here Ea and Ku , the anistropy constant, are measured in J m−3. Typical values range from less than 1 kJ m−3 to more than 10 MJ m−3. Anisotropy limits the coercivity available in hard magnets. We show in Chapter 7 that Hc < 2Ku /µ0 Ms , (1.6) where the magnetic constant µ0 is 4π × 10−7 J A−2 m−1. Anisotropy also leads to unwanted coercivity in soft magnets. It may be noted from the units that µ0 always multiplies H 2 or MH in expressions for magnetic energy per unit volume. Atomic densities in solids are around n = 1029 m−3 , so if anisotropy energy per atom is expressed in terms of an equivalent temperature using Ea /n = kB T , it is in the range 1 mK–10 K. The energy is usually small in relation to the Curie temperature, but it is nevertheless decisive in determining the hysteresis. 11 1.2 Magnetism and hysteresis 1.2.5 Susceptibility At temperatures above TC , where the ferromagnetic order collapses, and the material becomes paramagnetic, the atomic moments of a few Bohr magnetons experience random thermal fluctuations. Although Ms is zero, an applied field can induce some alignment of the atomic moments, leading to a small mag- netization M which varies linearly with H , except in very large fields or very close to the Curie point. The susceptibility, defined as χ = M/H, (1.7) is a dimensionless quantity, which diverges as T → TC from above. Above TC it often follows a Curie–Weiss law χ = C/(T − TC ), (1.8) where C is known as the Curie constant. Its value is of order 1 K. The magnetic response to an applied field of materials which do not order magnetically may be either paramagnetic or diamagnetic.9 In isotropic param- agnets, the induced magnetization M is in the same direction as H, whereas in diamagnets it is in the opposite direction. Superconductors exhibit diamagnetic hysteresis loops below their superconducting transition temperature Tsc , and their susceptibility can approach the limiting value of −1. The susceptibility of many paramagnets follows a Curie law, χ = C/T , (1.9) but for some metallic paramagnets and almost all diamagnets χ is independent of temperature. The sign of the room-temperature susceptibility is indicated on the magnetic periodic table (Table A, see endpapers) and the molar suscepti- bility χ mol is plotted for the elements in Fig. 3.5. There it is appropriate to look at the molar susceptibility because some of the elements are gasses at room temperature. A cubic metre of a solid contains roughly 105 moles, so χ mol is approximately five orders of magnitude less than χ. From Table A, it can be seen that the transition metals are paramagnetic, whereas main group elements are mostly diamagnetic. 1.2.6 Other types of magnetic order The spontaneous magnetization of a ferromagnet is the result of alignment of the magnetic moments of individual atoms. But parallel alignment is not the only – or even the most common – type of magnetic order. In an antifer- romagnet, the atomic moments form two equivalent but oppositely oriented 9 Faraday was the first to classify solids as diamagnetic, paramagnetic or ferromagnetic, according to their response to a magnetic field. 12 Introduction magnetic sublattices. Although Ms = 0, the material nonetheless exhibits a phase transition with a λ-shaped specific heat anomaly where the moments begin to order. The antiferromagnetic transition occurs at the Néel temperature TN. Occasionally it is possible to switch an antiferromagnet into a ferromagnet if a sufficiently large field is applied. This discontinuous change of magnetic order is known as a metamagnetic transition. If the sublattices are inequivalent, with sublattice magnetizations MA and MB where M A = −M B , there is a net spontaneous magnetization. The material is a ferrimagnet. Most of the useful magnetic oxides, including magnetite, are ferrimagnetic. The alignment of the atomic moments in the ordered state need not be collinear. Multiple noncollinear sublattices are found in manganese and some of its alloys. Other materials such as MnSi or Mn3 Au have helical or spiral magnetic structures that are incommensurate with the underlying crystal lattice. In some disordered and amorphous materials the atomic moments freeze in more or less random directions. Such random, noncollinear magnets are known collectively as spin glasses. The original spin glassses were magnetically dilute crystalline alloys, but several different varieties of random spin freezing are encountered in noncrystalline (amorphous) solids. Finally, we remark on the behaviour of ferromagnetic fine particles whose volume V is so small that the product Ku V is less than or comparable to the thermal energy kB T ; in that case the total sum moment, m, of all the coupled atoms fluctuates randomly like that of a large paramagnetic atom or macrospin. The susceptibility follows a Curie law with a huge value of C. The name superparamagnetism was coined by Néel for this phenomenon, which is important for ferrofluids (magnetic liquids which are really colloidal suspensions of ferrimagnetic fine particles) and in rock magnetism. Figure 1.6 portrays the magnetic family tree, summarizing the behaviour of the magnetization or susceptibility for the different types of magnetic order in crystalline and amorphous solids. 1.2.7 The magnetic periodic table Table A (endpapers) displays the magnetic properties of the elements, distin- guishing those that are paramagnetic, diamagnetic, ferromagnetic or antiferro- magnetic at room temperature, and those that order magnetically at some lower temperature. Only sixteen elements have a magnetically ordered ground state, and all but oxygen belong to the 3d or 4f transition series. Besides iron, cobalt and nickel, only gadolinium can be ferromagnetic at room temperature, but that depends on the weather! The Curie temperature of gadolinium is just 292 K. Many other elements become superconducting at low enough temperature. The remainder are neither magnetic nor superconducting. No element manages to be both at the same time.  13 1.3 Magnet applications Figure 1.6 The magnetic family tree CURIE + (after C. M. Hurd, Contemp. CURIE-WEISS 0 Phys. 23, L69 (1982)). C - σs σs M AF PARA 0 Tc T0 H 0 Tc T 0 TN T σ σ 77 K 200 K H H/T σs M M A M B T TORD 0 T 0 H 0 Tc T0 H 0 TORD T 0 H A B 1.3 Magnet applications 1.3.1 Overview of the world market Magnetic materials, recording media, heads and sensors constitute a market worth over $30 billion per year. Since the population of the Earth is approaching 7 billion, this means an average of about $5 per head. The world’s goods are unevenly distributed. The richest billion, living mainly in North America, Europe and East Asia, consume the lion’s share, but most people derive some Applied magnetism. A benefit from magnetic technology, whether in the form of a cassette recorder, Berlin postcard ca 1920. an electric pump in a tube well or a communal mobile phone.  14 Introduction Figure 1.7 Magnetic recording 40% Permanent magnets 20% 10 kA m−1 < Hc < 400 kA m−1 Hc > 400 kA m−1 Breakdown of the market for magnetic materials, based on material type and coercivity. The total pie represents about $30 billion per annum. Soft magnets 40% Hc < 10 kA m−1 A first view of the global market is given in Fig. 1.7. The breakdown is by material, distinguishing hard magnets with Hc > 400 kA m−1 , soft magnets with Hc < 10 kA m−1 and magnetic recording media with intermediate val- ues of coercivity. In this breakdown, it is easy to account for bulk permanent magnets and soft magnetic magnets which are commodities sold by the kilo- gram at a price depending on the grade and form. The disc and tape media used for magnetic recording incorporate a film of magnetic material on a rigid or flex- ible substrate. Sophisticated magnetic multilayer structures used in read/write heads for magnetic recording, magnetic sensors and magnetic random-access memory, are the first products of the spin electronic age. It is more difficult to assign a value to the magnetic constituent of a medium or a device which is composed of nonmagnetic as well as magnetic materials. The value added by the complex processing far exceeds the cost of the minuscule amounts of magnetic raw material involved. Further breakdowns are made in terms of materials and applications. In the hard magnet sector, the great bulk of production and over half the value is represented by the hard ferrites Ba2 Fe12 O19 and Sr2 Fe12 O19. These materials are used for colourful fridge magnets, as well as numerous motors, actuators, sensors and holding devices. Rare-earth compounds, especially Nd2 Fe14 B, are important in high-performance applications, and magnets based on Sm–Co alloys continue to be produced in smaller quantities. Hard discs generally use thin films of a Co–Pt alloy. Thin film heads for magnetic recording typically use films of Fe–Ni or Fe–Co alloys in the writer and thin film stacks comprising Fe–Co and Mn-based alloys in the reader. These are soft magnetic films with a good high-frequency response, except for the Mn alloy which is an antiferromagnet. For flexible magnetic recording media, tapes and floppy discs, acicular fine particles of Fe, or Co-doped γ -Fe2 O3 , are commonly used. Bulk soft magnetic materials are principally electrical steels. These Fe–Si alloys are produced in sheets about 300 µm thick for laminated temporary magnet cores in transformers and electrical machines. The better grades are grain-oriented with a specific crystalline texture. Soft ferrite is used for radiofre- quency and microwave applications. Ferromagnetic metallic glasses, thin rib- bons (≈ 50 µm thick) of rapidly quenched amorphous Fe- or Co-based alloys are used in an intermediate frequency range (kHz–MHz).  15 1.3 Magnet applications Figure 1.8 600 The distribution of magnetic ordering 500 temperatures of ferromagnetic and antiferromagnetic 400 materials (data are from T. Count F. Connolly and E. D. 300 Copenhover (editors), Bibliography of Magnetic 200 Materials, Oak Ridge National Laboratory, 1970). 100 αFe2O3 Co 0 0 200 400 600 800 1000 1200 1400 Magnetic ordering temperature ( K ) To put everything in context, imagine a shopping basket with the average person’s e5 worth of magnetic materials. It would include about 30 g of ferrite magnets, 1 g of rare-earth magnets, 1 m2 of flexible recording media, an eighth of a hard disc, a quarter of a thin film head, 0.25 m2 of electrical sheet steel, 30 g of soft ferrite and a few square centimetres of metallic glass. Perhaps 95% of the market for magnetic materials is accounted for by barely a dozen different ferromagnetic and ferrimagnetic materials. That this is only a tiny fraction of the thousands that are known to order magnetically is testimony to the difficulty of developing new materials with the right combination of properties to bring to the market. The Curie temperature, for example, must be well above the maximum operating temperature for any practical magnetic material. A typical operating temperature range is −50 to 120 ◦ C, so the Curie temperature needs to be at least 500 or 600 K. The distribution of magnetic ordering temperatures in Fig. 1.8 shows that only a small fraction of all known magnetic materials meet this requirement. Nevertheless, the magnetics indus- try has a far wider materials base than the semiconductor industry, with its overwhelming reliance on silicon. Figure 1.9 is an attempt to break down the market by materials and applica- tions. Magnetism is a pervasive and largely unnoticed component of the tech- nology underpinning modern life. Our electricity is generated by movement of conductors in a magnetic field. Key components of audiovisual equipment, telephones, kitchen machines and the microwave oven are magnetic. Electri- cal consumer goods, where something moves when you switch on, invariably involve temporary or permanent magnets. Powerful medical imaging depends on magnetic resonance. Magnetic sensors offer contactless monitoring of posi- tion or velocity. Unimaginable amounts of information are stored and retrieved from magnetic discs in computers and servers throughout the world. Some non- volatile memory is magnetic. In 2008, consumers bought 500 million hard disk  16 Introduction Figure 1.9 Hard magnets Soft Other Magnetic materials and Amorphous ferrite Hard ferrite their applications. Ni–Fe/Fe–Co Nd–Fe–B Fe–Si (oriented) Sm-Co Alnico Other Co–γFe2O3 Soft magnets CrO2 Iron Fe–Si Co–Cr Magnetic recording Iron Others Ni–Fe/Fe–Co drives and over a billion permanent-magnet motors. Magnetics is the partner of electronics in the global information revolution. Ask any one of the wealthy billion how many magnets they own. A correct answer could be a couple of hundred or some billions depending on whether or not they possess a computer. On a hard disc drive every bit counts as an individually addressable magnet. Fifty years ago the answer might have been two or three. Fifty years hence, who knows? 1.3.2 Economics Magnet applications depend critically on the cost and performance of ferro- magnetic materials. For bulk magnets, the raw material cost may be significant. To a rough approximation, this cost is related to the abundance of the element in the Earth’s crust. The composition of the crust is shown in Fig. 1.10. It can be expressed either as atom %, which emphasizes the light elements and relates to chemical formulae, or as weight %, which emphasizes the heavy elements. Note that abundances plotted in Fig. 1.10a) are in atomic % and those in Fig. 1.10b) are in weight %. Luckily, one ferromagnetic element, iron, features among the eight most abundant in the crust by either measure. Iron represents about 5% by weight of the crust, and it is the most abundant element overall when the composition of the entire globe is considered. In fact, it is 40 times as plentiful as all the other magnetic elements put together. We are truly fortunate that the cheapest metal is in many respects the best ferromag- net. Its rival, cobalt, is a thousand times scarcer, and about one hundred times more expensive. Some of the light rare-earth elements at the beginning of the 4f series have abundances comparable to cobalt (Fig. 1.11), but the heavy rare-earths live up to their name. Thulium, for example, sells for much more than gold or platinum. In thin-film devices, however, the cost of the element is  17 1.3 Magnet applications Figure 1.10 Si 20.6% Si 15% Elemental composition of: O 30% (a) the Earth’s crust in Al 6.1% Mg 13% atom %, and (b) the whole Earth in weight %. Na 2.6% Fe 2.1% H 2.1% Ni 2.4% Ca 1.9% S 1.9% Mg 1.8% K 1.5% Ca 1.1% Others 0.6% Al 1.1% Others 0.5% O 60.7% Fe 35% (a) (b) Figure 1.11 1 Crustal abundances of iron and other magnetic 0 elements, shown on a logarithmic scale. −1 log(A%) −2 −3 −4 −5 Fe Co Ni Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb 3d 4f largely irrelevant because the mass of material used per device is measured in micrograms or less. Ruthenium, for example, is a rare metal used in spin-value stacks in layers a nanometre thick. A single sputtering target suffices to coat hundreds of wafers, with millions of devices. The performance of magnetic materials improved by leaps and bounds in the twentieth century, although the records for magnetization and Curie temper- ature were not broken. Following from the progressive mastery of coercivity, and the relentless miniaturization in the feature size for magnetic recording, there has been an exponential improvement in properties in all three market segments. In soft materials, the 60 Hz core losses halved every 18 years throughout the twentieth century (Fig. 1.12); the maximum available susceptibility doubled every 6 years over the first half. Further improvements here seem pointless, although there is a pressing need to improve the properties of temporary mag- nets at frequencies above 1 MHz.  18 Introduction Figure 1.12 10 Improvements in energy Core loss (W kg−1) losses for soft magnetic materials. 1 0.1 1900 1920 1940 1960 1980 2000 Year Figure 1.13 Improvement in the maximum energy product 400 available for permanent Nd–Fe–B magnets.10 320 Sm2(Co–Fe–Cu–Zr)17 Sm2(Co–Fe–Cu)17 240 Sm–Pr–Co5 max Sm–Fe–N Sinte red SmCo5 160 Columnar alnico 80 SmCo5 Alnico 5 Ba–Sr–Ferrite MK-steel KS-steel Co-Ferrite YCo5 1920 1930 1940 1950 1960 1970 1980 1990 Year Hard magnets improved beyond recognition, breaking the shape barrier in 1950. This means that they could be made in any desired shape, without having to resort to horseshoes and bars to avoid self-demagnetization. The figure of merit here is the energy product, which is twice the energy stored in the magnetic field produced surrounding unit volume of an optimally shaped permanent magnet. Energy product has doubled every 14 years (Fig. 1.13). The best permanent magnets have square hysteresis loops with Hc > Mr /2. The energy An early eighteenth century lodestone, a ferrite product cannot exceed µ0 Mr2 /4. magnet (right) and a Nd–Fe–B magnet (front), which all store about a 10 In industry, units of MG Oe are used for energy product. 100 kJ m−3 = 12.57 MG Oe. Units joule of energy. are discussed in Appendix B.  19 1.4 Magnetism, the felicitous science Figure 1.14 1000 Moore’s law for transistor Areal density (bits µm−2) density in semiconductors 100 and for storage density in magnetic recording.11 10 1 Magnetic HDD DRAM 0.1 0.01 0.001 1970 1980 1990 2000 2010 Year Most remarkable of all has been the progress of magnetic recording, illus- trated in Fig. 1.14. For over a decade, the areal density doubled almost every year, the magnetic analogue of ‘Moore’s law’ for semiconductors. The extraordinary technological progress epitomized by the three magnetic exponentials has translated into an improvement in the capabilities of consumer goods and services that economics struggles to quantify. The consumer takes it all for granted, ignorant of the struggle of scientists and engineers to achieve their current mastery of nature, and of the science on which our civilization is based. One thing is certain: exponential improvement cannot continue indefinitely. Permanent magnets are approaching the limit of energy product, 1200 J m−3 , defined by the remanence of permendur, the alloy of iron and cobalt having greatest room-temperature magnetization. Magnetic recording densities may saturate as magnetic instability is encounterd at densities well in excess of 1000 bits µm−2 , where each bit is so tiny that the corresponding volume of magnetic medium is too small to withstand thermal fluctuations. Future improvements in the performance of bulk magnetic materials will probably focus on achieving desirable combinations of properties, e.g. a per- manent magnet stable at 500 ◦ C, a material combining low anisotropy and high magnetostriction, a multiferroic material that is both ferromagnetic and ferroelectric, all at the lowest possible cost. In devices, the trend is towards increasing integration of magnetic functionality with optics and electronics. As for the methods of investigation, intelligent experimentation will be supple- mented increasingly by computer simulation and combinational synthesis. 1.4 Magnetism, the felicitous science Magnetism is a wonderful example of how basic science, flowing from a mag- ical natural phenomenon can become ubiquitous in our lives, thanks to the 11 Industry prefers Gbits per square inch. 1 bit µm−2 = 0.645 Gbits per square inch.  20 Introduction Figure 1.15 An impression of activity Quantum mechanics over the twentieth century Wave mechanics Special relatvity in fundamental theory, Dirac theory of electron normal science, materials development and industrial transitions Phase production in relation to BCS permanent magnetism. Amorphous magnetism Spin glass simulation Spin waves AFM Spin Lorentz electronics Neutrons microscopy Mean field Slater–Pauling Superparamagnetism Synchrotron Paramagnetism Antiferromagnetism Micromagnetism Nanomagnetism Atomic spectra Microwaves EPR, FMR Mossbauer Nd–Fe–B Ferrite Sm–Fe–N Alincos SmCo Steels Powder metallurgy Thin films Casting Powder metallurgy Ceramics Melt spinning 1900 1950 2000 Year labours of successive generations of specialists. The twentieth century is a con- venient time frame in which to trace how leaps in theoretical understanding and advances in experimental practice can relate to the emergence of a technology that creates wealth and facilitates our life. Figure 1.15 portrays the work in basic theory, normal science, materials development and industrial production (the latter two in relation to permanent magnetism). The four are interlinked, but cause-and-effect relations are not always obvious. The context of magnetic research and development is schematized in Fig. 1.16. The activity employs roughly 30 000 people worldwide, at a cost  21 Further reading Figure 1.16 ment Govern Relations between academic and industrial Education research and development. Curiosity-driven research Public e dg Applications-oriented knowle research P r iva te dge knowle Industry Product development ess Busin of over e1 billion per year. It is instructive to contrast attitudes of academic and industrial scientists to new knowledge and technology. One seeks diffusion, the other ownership. Academic research is rewarded by peer recognition, indus- trial development by profit. Both share an understanding that by interrogating nature in a structured and rational way, trustworthy and objective knowledge of practical importance can be obtained. This is what unites the geomancer, the telegraph engineer, the PhD student struggling for data, the corporate scientist whose invention could trigger a billion dollar investment providing employ- ment for thousands and the professor attempting to draw the strands together in a book on this felicitous science. FURTHER READING J. D. Livingston, Driving Force, Cambridge, MA: Harvard University Press (1996). An enjoyable account of magnetism for the general reader. A. P. Guimares, From Lodestone to Supermagnets, Weinheim: Wiley-VCH (2005). Another popular account. A. Kloss, Geschichte des Magnetismus, Berlin: VDE (1994). Packed with facts, an excellent one-volume history of magnetism from its beginnings to the early twentieth century. D. C. Matthis, Theory of Magnetism Made Simple, Singapore: World Scientific (2006). An elegant introductory chapter outlines the history of ideas, especially in the nine- teenth and twentieth centuries. J. Needham, Science and Civilisation in China, Vol. IV:1, Cambridge: Cambridge Uni- versity Press (1962). The definitive scholarly account of Chinese contributions to science and technology. This volume treats magnetism. S. Blundell, Magnetism in Condensed Matter, Oxford: Oxford University Press (2001). A lively introduction for final-year undergraduates. 22 Introduction K. H. J. Buschow and F. R. de Boer, Physics of Magnetism and Magnetic Materials, Berlin: Springer (2003). A concise introduction to the principles and applications. N. Spaldin, Magnetic Materials, Fundaments and Device Applications, Cambridge: Cambridge University Press (2003). A short introduction for undergraduates. D. Jilles, Introduction to Magnetism and Magnetic Material, 2nd edition, London Chap- man and Hall (1998). An introduction for engineers, in question and answer format. R. D. Cullity and C. D. Graham, Introduction to Magnetic Materials 2nd edition, New York: Wiley (2008). A revision of the best-written book of its kind for materials scientists. R. M Bozorth, Ferromagnetism, Princeton: van Nostrand (1951). Although published over 50 years ago, Bozorth contains much information, especially on 3d metals and alloys which is still relevant. Reprinted in 1993 by IEEE Press. A. H. Morrish, The Physical Principles of Magnetism, New York: Wiley (1965). A classic text, reprinted by IEEE Press in 2001. S. Chiukazumi, Physics of Ferromagnetism, 2nd edition, Oxford: Oxford University Press (1997). Another classic general text. R. C. O’Handley, Modern Magnetic Materials; Principles and Applications, New York: Wiley (1999). A thorough treatment of modern materials and applications, including thin films. H. Kronmuller and S. S. P. Parkin (editors), Handbook of Magnetism and Advanced Magnetic Materials, 5 volumes, Chichester: Wiley (2007). A modern multi-author reference work, with a focus on contemporary topics. G. Rado and H. Suhl (editors), Magnetism, 5 volumes, New York: Academic Press (1960–73). A multiauthor treatise which details much of the modern understanding of the theory of magnetism. K. H. J. Buschow and E. P. Wohlfarth (editors), Handbook of Magnetic Materials, 16 volumes, Amsterdam: North Holland/Elsevier (1980–). A continuing multiauthor series which is a mine of information on magnetic materials. New volumes appear every year or two. EXERCISES 1.1 As a rule of thumb, a field 3 times greater than the spontaneous magnetization is needed to magnetize a permanent magnet. Given that the current in a lightning strike is 106 A, make an estimate of the time that will elapse before a particular rock outcrop of lodestone becomes magnetized. 1.2 Find one documented historical reference to magnetism, before 1200, from your own part of the world. 1.3 Estimate, and rank in decreasing order: (a) the magnetostatic energy stored in space around a 10 g permanent magnet; (b) the chemical energy stored in 10 g of cornflakes; (c) the gravitational potential energy in a 10 g pencil sitting on your desk; (d) the kinetic energy of a 10 g bullet moving at the speed of sound; (e) the mass energy released by fission of 10 g of 235 U. 1.4 When does the extrapolation of Fig. 1.14 ‘predict’ that a bit will have dimensions smaller than an atom? 23 Exercises 1.5 Given two apparently identical metal bars, one a temporary (soft) magnet, the other a permanent (hard) magnet and a piece of string, how could you distinguish between them? 1.6 Write two paragraphs on the area of magnetism which you think will have greatest commercial potential in 10 years time, explaining why. You may consult the last four chapters of the book. 2 Magnetostatics Back to basics The dipole moment m is the elementary magnetic quantity, and magnetization M(r) is its mesoscopic volume average. The primary magnetic field B is related to the auxiliary magnetic field H and the magnetization by B = µ0 (H + M). Sources of magnetic field are electric currents and magnetized material. The field produced by a given distribution of magnetization can be calculated by integrat- ing th