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4 Magnetism INTRODUCTION With respect to their magnetic properties, solids can be divided into two categories depending on the direction of the moment induced by an applied magnetic field. If the moment is opposite in direction to the field, the material is said to be diamagnetic; materials where the moment is parallel to the field are paramagnetic. Apart from its role in superconductivity, diamagnetism is a weak eject. In contrast, concen- trated paramagnetic materials often display very large responses corresponding to local fields of the order of several hundred tesla. At present most of the research in magnetism involves systems that show paramagnetic behavior. In paramagnetic materials the magnetism is associated with partially filled inner shells of atomic electrons. These are the ad shell (transition metal compounds), the 4fshell (rare-earth compounds), and the Sfshell (actinide compounds). The classification can be extended further depending on whether the electrons in the partially filled shells are localized, as happens in insulators and semiconductors, or itinerant, as in many metals. In magnetic insulators each atom with an unfilled shell possesses an intrinsic magnetic dipole moment. In addition to their interaction with the applied field, the dipoles interact with one another through long-range dipolar forces and short-range exchange interac- tions, the latter arising from the interplay of electrostatics and quantum 95

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96 A DECADE OF CONDENSED-MATTER PHYSICS symmetry. Besides their mutual interactions, the moments also inter- act with the charges on the surrounding ions, which create a crystalline electric field. The moments in itinerant magnets are delocalized, extending throughout the system. As in the case of localized moments, there are both dipolar and exchange interactions. Strictly speaking, in many materials the paramagnetic behavior alluded to earlier is observed only at high temperatures. As the temperature is lowered the system undergoes a phase transition to a state characterized by long-range magnetic order. The most familiar example of this behavior occurs in ferromagnets where the long-range order appears as a spontaneous magnetization. The transitions are caused by the exchange interactions between the moments and gener- ally occur at temperatures comparable to the strength of the interaction between neighboring moments. Studies of the magnetic properties of materials with localized mo- ments occupy an unusual position in solid-state physics. This happens because these properties can be characterized using models involving only the individual atomic moments, rather than the full array of atomic electrons. These models, generally referred to as spin Hamiltonians, have generic forms that depend on the symmetry of the system and interaction parameters that reflect the microscopic environment of the magnetic ions. The study of spin Hamiltonians is a central part of many-body theory and statistical mechanics. The spin Hamiltonian formalism makes possible a direct connection between real materials and formal theory that has been a driving force for much of the research in magnetism in recent years. Experimental studies have provided crucial tests of theories of collective excitations and critical phenomena. In turn, the availability of accurate data has stimulated the development of increasingly precise theories. The past decade has been one of remarkable progress in magnetism. By 1970 it can be said that the behavior of isolated magnetic ions in insulators was well understood. The low-temperature properties of ideal, three-dimensional (3-D) magnetic insulators were successfully interpreted in terms of elementary excitations. The high-temperature behavior had also been investigated, and theoretical studies utilizing high-temperature expansions had given useful insights. There was growing interest in magnetic critical phenomena, but no unifying microscopic theory was available. The magnetic properties of metallic systems were not well understood. The behavior of isolated magnetic ions in nonmagnetic metallic hosts had revealed unexpected complex- ities at low temperatures. In addition, insight into the behavior of common magnets like iron and nickel had not advanced much beyond

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MAGNETISM 97 the level of molecular field theory. As will be discussed below, in the intervening years there have been significant advances in our under- standing of both insulating and metallic magnets. These occurred not only in ideal systems but in various types of disordered materials as well. The period since 1970 has also been an era of rapid growth in computer simulation studies of various static and dynamic properties of magnets. Such a development would have been impossible without large, high-speed digital computers. In addition to its place in basic research, magnetism continues to make important contributions to technology through improved and novel materials for information storage and power generation, for example. MAGNETIC INSULATORS Low-Dimensional Systems Recently there has been a shift in emphasis away from studies of ideal, 3-D magnetic insulators. Increasing attention has been directed toward low-dimensional (low-D) systems, an area of research that has grown rapidly in the past 15 years. Before this period, the most important single research achievement in the field was undoubtedly the famous Onsager solution of the two-dimensional (2-D) Ising model. The Ising model is a simplified version of a physically realistic model of magnetism. The importance of the Onsager solution is that it shows that a phase transition is possible in a simple model with short-range magnetic interactions, a matter previously open to doubt. Further- more, the nature of the critical behavior is significantly different from the older, molecular-field type of approximate theories of critical behavior in magnets. In the 40 years since its appearance in 1944, the Onsager solution has been exploited in a variety of ingenious ways, which have provided much of the basis of the modern theory of critical phenomena. The one-dimensional (1-D) version of the Ising model was solved back in 1925 but was not considered interesting because the solution did not show a phase transition. Around the early 1960s, however, a number of other 1-D models of magnetism were solved, either exactly or numerically. At that time, the feeling was widespread that 1-D models were merely amusing toys for mathematical physicists to play with, with little or no relation to the real, 3-D, physical world. Nevertheless, the appearance on the scene of a number of 1-D model

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98 A DECADE OF CONDENSED-MATTER PHYSICS solutions attracted the attention of experimental physicists, who searched successfully for chemical systems with highly spatially anisotropic magnetic properties. They were able to show that the experimental behavior of the real systems agreed well with that of the 1-D theoretical models. Subsequently, the process of molecular engi- neering was developed, whereby quasi-2-D and quasi-l-D magnetic systems were prepared according to specifications by inserting large nonmagnetic spacer molecular complexes (usually organic) into suit- able systems to increase the physical separation between planes or chains of magnetic ions, respectively. In this way, the magnetic interactions were substantially reduced in one or more crystalline directions. As noted, low-D physics is continuing to grow in importance relative to the traditional 3-D variety for the following reasons: 1. In general, the ease of solution of a particular model of coopera- tive magnetism increases as the dimensionality decreases. Hence, a variety of exact solutions of varied and nontrivial models is now available in 1-D, whereas there are hardly any exact solutions in 3-D. (The value of exact solutions can hardly be overestimated.) A useful secondary feature of 1-D exact solutions is their ability to serve as testing grounds to give insight into the degree of reliability of the various approximate calculational techniques that must, of necessity, be employed in 3-D. 2. A feature of great importance is the wealth of novel and interest- ing physical phenomena that are peculiar to low-D. Examples include prominent quantum effects in low-D, e.g., in the area of low- temperature spin dynamics, and the enhancement, by virtue of topo- logical considerations, of the effects of impurities and randomness in low-D. Further, the current trend in physics is to move away from the traditional approach of the linear (harmonic) approximation to consider nonlinear effects. In the linear approximation to a model magnetic system, the small-amplitude collective excitations are called magnons, and their behavior has been studied for decades. Recently, the impor- tance and interesting properties of nonlinear (large-amplitude) excita- tions have been recognized. Various types of such phenomena exist, called, for example, solitons, kinks, vortices, breathers, instantons, and domain walls. These excitations are important in many areas of physics, including plasma physics, turbulence, and field theory, but they appear to be most easily investigated, theoretically and experi- mentally, in magnetic systems, particularly in 1-D, but also in 2-D, systems (Figure 4. 11.

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''t, t t /'1 ~ t / t t t ,' t \ t ''t t '''/ t t t / t t t', t /, '~''''/ ~ / ~ t 1 ~ ~ t~t t t J / / t / / ~ ''',/ I t / t', / t I t I''t t //~// t t / t J t t t t t t / t t ~ t t t t / / ,/ / t t t t t I ~ t','-t t t t t t ,' t t t ~t t '''t t t t t t t'/ t't ~ t t ~ t t / t t~t t i t''/~/ t f t t t t ~ ~ t t t t ~ ~ ~ i ,'/ / t t ~ t I t / t t ~ / t'////~'t / \~ t \ ~ t / t ~ t /'/~'--~ ~ ~ ~ ~ ~ t t t / t t t t /-------~- \ ~ \ ~ t t / / t \ / ~------ -~ ~ ~ t t t \ \ / ~ ~----`'` I ---'~ ~ t / t / / /----`x ~ J ---' \ \ / / t ~ J / ~ ~----- ~ \ ~ ~ \ t t / / / /~/ /'---`x `~\~\ ~ t \ t t t t /~/--- - ~ ~ ~ ~ ~ \ \ t \ \ / / / ~ ~--~- `~\ t t t ~ t / ~ -`~` ~ t \ t / I `~`--~ ~ t t I I / t \~_~\ ~ \ \ \ / 1 \ \ t \ \ `~` \ -I I I t \\ - \~`-\N 1 1 ~ \ ~ \ ~ \'N'\ ~ \ ~ \ t t ' ~ ~ ~ ~ t ~ ~ ~ ~ ~ \ \ t ~ 1 1 1 \~` t t / t I t ~ \'\ \ ~\ t t t t t t \\~` \ t \ \ \ \ ' ' ' ~ t ~ ~ t ~ I I' - ' t t I I t \ x\'- / \ ~ 1 ~ 1 \~\' '/ t l t \'\ t t t ~ I I ~`'~ I t \\` \~\ \~ \ \ t t I \ t \ t t I t,/ t MAGNETISM 99 t t t / t 1 t t / t / t ~ ~ ~ I t t I t '/~t t / t I // ~t t \t\\ \ I t \ t t t \\~/\\ / t t \ t tN t I I t I t I t / / I I I //~} t t t t t t / \~\ t t \~\ \ / ~-`~` ~ \ ~ ~ t ~-~ ~ ~ _ _ ~ ~ ~ ~ ___~____ ~ _ ~ _ _ _ _ __, ______, _ _ _ 1 1 ~ ~ ''~ ~ ~ ~ ~ 1 J I J/~' NN \ ~ 1 1 1 1~' J /~' l I I J I ~' J~J 1~/ 1 / 1 1 1 /--' 1 1 ~ 1 1 /'-~ 1 ~ 1 1 1 /~'- ~1 / I I J/'' '1 /'/ /.~' I J ~ J I I /' I J Jil'J' ,,' J ~', ~ J J ~ ~ '' , /,, ~ _, /,__ ~ J I ~-- ~ ~ ~ 1 ~--- _,, ___ _ ___ ~ _ %~ _ --~ ~ __ ~ FIGURE 4.1 Nonlinear vortex excitations in the two-dimensional XY model. The dark and open circles denote the centers of spin vortices of opposite circulation. [S. Miya- shita, H. Nishimori, A. Kuroda. and M. Suzuki. Prog. Theor. Phys. 6C, 1669 (1978).1 3. A fascinating new development of recent years is a phenomenon that may be termed, for convenience, mapping. Mapping refers to the discovery that apparently different physical phenomena are, in fact, related to one another through an underlying mathematical description. The same mathematics has been found to describe a variety of physical systems, with appropriate definitions of the relevant mathematical parameters. This result may be characterized as obtaining several solutions for the price of one. Mappings have been discovered between systems of the same or different dimensionalities. A well-known example of the former case is the class of systems that are isomorphous to the 2-D XY model of magnetism. This class includes 2-D superfluids. 2-D melting solids, smectic (layered) liquid crystals, and 2-D Coulomb gases. Possibly the most famous mapping between systems of different dimensionality involves the model many-body system consisting of a single magnetic impurity exchange-coupled to a sea of conduction

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100 A DECADE OF CONDENSED-MATTER PHYSICS electrons. Experimentally, dilute solutions of such impurities in other- wise normal metals display noticeable anomalies in susceptibility, in specific heat, and in their temperature-dependent resistivities. These anomalies are referred to collectively as the Kondo effect. Their explanation constitutes the Kondo problem. In a remarkable develop- ment, the 3-D Kondo problem has been mapped onto a solvable 1-D quantum model. The calculated susceptibility and specific heat agree well with experiment. At the same time, it is a tribute to the power of the renormalization group method (Chapter 3) that the solution of the Kondo problem obtained through its use, though manifestly an approx- imation, is nevertheless demonstrably accurate when compared with the exact solution. The Kondo mapping is presumably the first of many mappings from 3-D to a much more tractable lower dimensionality. This factor, together with the fact that new mappings are turning up in rapid succession. makes low-D physics applicable to more areas than one might, at first sight, suppose. Critical Phenomena As noted in Chapter 3, the 1970s was a period of intense activity in the field of phase transitions and critical phenomena. Studies of phase transitions in magnetic materials, primarily insulators, confirmed many of the predictions of high-temperature series and renormalization group calculations. In addition, they provided important evidence in support of the concepts of scaling and universality. In the first part of the decade most of the research pertained to ideal, 3-D magnets. Cur- rently, greater emphasis is being placed on studies of critical phenom- ena in disordered and lower-D systems. METALLIC MAGNETS Transition-Metal Ferromagnets Metallic magnets can be divided into two classes depending on whether the magnetic atoms belong to the transition-metal series or to the rare-earth and actinide series. Recent advances in the theory of transition-metal ferromagnets have led to a better understanding of the nature of their ground state and of their magnetic properties at finite temperatures. 1. After 50 years of discussion, it is now widely accepted that the ground state of iron, nickel, and most other transition-metal fer

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MAGNETISM 10 1 romagnets is best described by an itinerant or band picture, as opposed to a localized picture. de Haas-van Alphen measurements have gener- ally agreed with the results of band calculations. The calculations themselves have now been greatly improved by using better algorithms and larger machines. The greatest advance, however, is the use of a new theory that incorporates electron-electron effects into the single- particle states. Such calculations now correctly predict which transi- tion elements are ferromagnetic, and they give improved agreement with Fermi surface data. These band calculations are also used to determine cohesive energies and bulk moduli of whole series of materials with great success. In particular, the anomalously large lattice constants of the magnetic transition metals, as compared with the trend of their nonmagnetic neighbors, can be understood from the computed magnetic compo- nents of their cohesive energies. Another extension is to the calculation of the spin-wave scattering, which also agrees well with measurements. A stringent test of the band picture is given by angle-resolved photoelectron spectroscopy, which determines both the energy and wave vector of the emitted electron. An improvement to include spin-polarization analysis was recently announced (Figure 4.21. Large- scale angle-resolved measurements became feasible with the advent of high-luminosity synchrotron radiation sources (although some impor- tant high-resolution work is done with conventional sources). By and large, the measured dispersion curves are in agreement with band calculations, giving direct evidence for band states in a wide class of materials, magnets in particular. There is, however' still much to be done to achieve an understanding of the various effects of the real holes created in the photoemission process. A beginning has been made, but the problems are formidable. 2. The natural model for the temperature dependence of band ferromagnetism is that of Stoner, which is unsatisfactory, at least for some materials, in several respects. It predicts too high a Curie temperature; it does not incorporate directly the thermodynamically dominant spin-wave excitations; and it does not admit the persistence of magnetic correlation effects above the Curie temperature. That such effects exist is shown by the Curie-Weiss susceptibility in the paramag- netic phase and, more dramatically, by the continued existence of spin-wave excitations far above T`. in iron and nickel that are seen in inelastic neutron scattering. These observations and others have led to a number of new theoretical schemes for extending the ground-state band picture to finite temperatures. The competition between these schemes has revived the localized versus itinerant controversy in a new form.

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102 A DECADE OF CONDENSED-MATTER PHYSICS . . a .. x2 . . .~.t t t..t .. . . . . 1 _ . - l ~ . _ O=EF - o. 9 - 0.6 - 0.3 Initial Energy (eV) - - 40 ~ C' O c, -40t~ c _ CL (a FIGURE 4.2 Electron distribution curves and corresponding spin polarization for photoemission from the (110) face of nickel. [R. Rune, H. Hopster, and R. Clauberg, Phys. Rev. Lett. 50, 1623 ( 1983).] Paramagnon, or weak itinerant, models consider magnetic fluctuations about a basically nonmagnetic state. Though useful for enhanced paramagnets, and quite possibly for weak ferromagnets, they provide less correlation than is needed to describe iron and nickel. Phenome- nological attempts to extend the paramagnon models to strongly correlated cases have been made and have met with some success. Two approaches more directly concerned with the underlying elec- tronic structure are the local-band theory and the alloy analogy. Each assumes a disordered magnetization configuration, approximately solves for the electronic states in the mean exchange field produced by the configuration, and demands that the configuration be reproduced self-consistently. The local-band scheme assumes that the important configurations are characterized by a short-range order, sufficient to define the exchange split bands locally even above T`.. In the alloy analogy the magnetizations at different sites are statistically indepen- dent, and electronic states are computed in the coherent potential approximation. Evidence in favor of short-range order at high temper- ature has been reported, but the interpretation of the results has been

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MAGNETISM 1 03 challenged. Currently, then, the contention is between the picture that the magnetization (although not the electrons making it up) is localized for a relatively long time at a single site, and the picture that it has coherent structure on a larger, 10-15 A scale. Rare-Earth and Actinide Magnets There are three great conceptual distinctions between f (rare-earth and actinide) magnetism and d (transition-metal) magnetism. One is that, overall, orbital (as opposed to spin) magnetic effects are qualita- tively more apparent infmagnetism. This shows up in many properties where the magnetic behavior has peculiarities associated with the coupling of the orbital moment to the crystalline lattice. Second is that in metallic systems the f electrons tend to delocalize as one moves toward the light end of the 4f or Sf row. Thus at cerium in the rare earths, or at plutonium, neptunium, and uranium in the actinides, one can study and hope to understand the effects involved in the transition from localized (Gd-like) to itinerant (Ni-like) magnetism. The lattice property most obviously correlated with this transition is the lattice parameter (or more strictly thefatom "of-atom spacing); however, the effects of the detailed electronic structure can significantly alter this correlation. Third is that the proximity of a sharp 4f level to the Fermi energy can lead to instabilities of the charge configurations (valence) and the magnetic moment. There have been striking conceptual advances in the past decade associated with all three of these distinctive features of f-electron behavior. A discussion of these advances and their interrelationships follows. 1. The most characteristic consequence of the orbital contribution to the moment is strongly anisotropic magnetization behavior, with related peculiarities of magnetic structures and excitations. In the past decade this has been strikingly evidenced in cerium metallic and semimetallic compounds and, more recently, in the actinides, with most recent work in plutonium compounds. The association of excep- tionally strong anisotropy in magnetic properties with the region where the local-to-nonlocal f transition occurs suggests a strong connection between the two phenomena. The availability of single crystals of actinide compounds, including those containing plutonium, has been a key element in making possible these advances. 2. Much of the intellectual excitement inf-electron magnetism in the past decade has been associated with a shift in experimental emphasis

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104 A DECADE OF CONDENSED-MATTER PHYSICS from heavier rare-earth systems to cerium systems and into the light actinides. This excitement has arisen out of a variety of striking experimental observations, which in one way or another have tended to relate to the central question of the f-electron localized-to- delocalized transition. A variety of experimental techniques has been important in this conceptual opening. They have included high field magnetism, elastic and inelastic neutron scattering, and electromag- netic (e.g., de Haas-van Alphen, optical) and electron emission spectroscopies. On the theoretical side there have been two major advances. One of these has emerged from electronic (band) structure studies of the actinide elements, showing a transition in f-electron behavior from nonbonding (localized) at americium to bonding (delocalized) at plutonium. The other major theoretical advance has come out of Anderson-lattice model and band calculations pertinent to cerium and light actinide metallic, semimetallic, or semiconducting compounds. This theory shows that when the f electrons are moder- ately delocalized (intermediate between localized and band behavior, so as to be slightly bonding), f-electron-band electron hybridization (mixing) can explain a variety of otherwise anomalous properties including extreme anisotropy of magnetization and highly unusual magnetic structures and transitions. 3. The past decade has seen the birth and coming to fruition of a major area of research in valence instability. This interest was initiated by high-pressure experiments on samarium compounds at the begin- ning of the 1 970s. Resistivity, volume change, and susceptibility measurements, done in rapid sequence, indicated the occurrence of a valence change of the samarium, with consequent semiconductor-to- metal and magnetic-to-nonmagnetic transitions. This, in turn, was followed by a vigorous research effort that showed that the candidate rare earths for mixed-valence behavior are Sm and Eu at the middle of the 4f row, Tm and Yb at the end of the row, and Ce at the beginning. The necessary condition for mixed-valence behavior is that two bonding states (i.e., with different f-occupation numbers) of the rare earth in the solid be nearly degenerate (Figure 4.31. It has become clear that the cause of mixed-valence behavior in cerium materials is different from that in the heavier rare earths. For the heavier rare earths mixed valence involves a fluctuation between two degenerate nearly localized states, whereas for cerium the mixed-valence behavior may be associated with a 4f localization/delocalization transition. Whether this is indeed so is a question of great interest. In the coming period, we can expect to see a lively search for mixed-valence behavior in actinide systems. It will be important to see whether light actinide

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MAGNETISM 1 05 8 6 a) 4 llJ 2 o 1 / - J =0 / J = 7/2 /: J = 5/2 0 1 2 3 nf FIGURE 4.3 Schematic energy-level diagram for the intermediate valence Ce atom. nf is the number off electrons and J denotes the total angular momentum. [D. M. Newns, A. C. Hewson, J. W. Rasul, and N. Read, J. Appl. Phys. 53, 7877 (1982).] systems characteristically show cerium-type or samarium-type mixed valency, or a mixture of the two. On the theoretical side, we anticipate an exceptionally active effort in trying to understand the ground-state properties of lattices of mixed-valence ions on the basis of Hamilton- ians that include narrow f states, broad conduction bands, hybridiza- tion, and If correlation effects. DISORDERED SYSTEMS Introduction As in other fields of condensed-matter physics, the study of disor- dered materials has been an area of intense activity in magnetism in recent years. In ideal magnets the atoms are arranged on a lattice. The lattice structure is characterized by translational invariance. This is to say, atoms that are separated by one or more fundamental repeat distances have the same local environment. The existence of this invariance often simplifies the theoretical analysis, especially at low temperatures, where there is negligible thermal disorder. In disordered magnets the translational invariance can be broken in

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106 A DECADE OF CONDENSED-MATTER PHYSICS different ways. The underlying lattice structure of substitutionally disordered materials is preserved. However, the sites of the magnetic atoms are occupied at random either by one of two (or more) different species of magnetic atoms, as in FerMn~_xF2, or by either a magnetic or nonmagnetic atom, as in Cd~_~.MnxTe. In amorphous magnets the lattice is absent altogether. In this case the material is said to be topologically disordered. Examples of materials that can be prepared in the amorphous state include YFe2, (Fe-Ni)sOP~4B6, and Fe~_xBx. Usu- ally the preparation involves rapid quenching from the melt so as to avoid crystallization. The fundamental problem in the study of disordered magnets is to understand the effects of the disorder on the magnetic properties. If the disorder is weak, i.e., if only a few nonmagnetic atoms are present in an otherwise fully occupied magnetic lattice or a low concentration of magnetic atoms is present in a nonmagnetic host, one can interpret the behavior as a superposition of effects due to isolated impurities. Although the study of impurities is an important topic in its own right, the main emphasis currently is on highly disordered systems where an analysis based on the single-impurity picture is not applicable. At high temperatures thermal disorder generally dominates any substitutional or topological disorder with the consequence that ideal and disordered magnetic materials behave in a qualitatively similar manner. However, as the temperature is lowered toward the regime where the energy associated with the thermal fluctuations becomes comparable to the strength of the interactions between the individual moments, the absence of translational invariance becomes increasingly important. The question then arises as to whether there is a transition out of the high-temperature phase. Should this be the case, is it to a state of conventional magnetic order or to a low-temperature disor- dered phase not present in the ideal magnets? Disordered Ferromagnets, Antiferromagnets, and Paramagnets The title of this subsection refers to systems that undergo phase transitions to states of conventional long-range order characteristic of ideal magnets or else are sufficiently dilute that they remain in their high-temperature or paramagnetic phase at all temperatures. An im- portant question pertaining to those systems that do undergo transi- tions is the influence of disorder on the critical temperature, critical indices (Chapter 3), and other properties characteristic of the transi- tion. The behavior of the disordered systems at low temperatures is also

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MAGNETISM 107 interesting. As with the ideal magnets one can interpret the static and dynamic properties in terms of a nearly ideal gas of magnon excita- tions. Even in the lowest-order, or linear, approximation the calcula- tion of the spectrum of excitations is a formidable problem. Neverthe- less considerable progress was made toward its solution in the past decade. This progress came about in a number of ways. Experimental studies involving inelastic neutron and light scattering have provided detailed information about the magnons in various disordered magnets. Paralleling the experimental work there have been two types of theo- retical investigation. The first involved purely analytic work mostly under the general heading of the coherent potential approximation, a name that reflects its origin as an approximation introduced in the cal- culation of the electronic properties of disordered alloys. The second en- tailed the development of computer simulation techniques, which made possible a direct calculation of the neutron-scattering cross section by integrating the linearized equations of motion of the spins (Figure 4.4~. Studies of magnons in substitutionally disordered magnetic insula- tors have provided important general tests of our understanding of NEUTRON SCATTERING FROM Rb2 MnO54 M9046 F4 AT 4.0K AS COMPaRED WITH COMPUTER SIMULATIONS I ' I I 1 ' 2 1 ^ Q=(0.5,0, 2.9) ~ V) 'c O \ -lo' <' 3 :'h Q=(07,0,2.9) _ >_ 2 _ 1 _ it i_ 16 ~ Z 12 _; 8 _ ~ 4- \~. . 'it. Q =(0.9,0,2.9) _ 2 4 10 20 10 l Q=(0.6,0, 2.9) N 3~ Q=(0.8,0,2.9) l ~ Q = (1.0,0,2.9) 6 8 V 2 4 6 8 ENERGY TRANSFER (meV) FIGURE 4.4 Distribution of scattered neutrons from the dilute antiferromagnet Rb2MnO.s4Mgo.46F4. The solid lines are computer simulations. [R. A. Cowley, G. Shirane, R. J. Birgeneau, and H. J. Guggenheim, Phys. Rev. B 15, 4292 (1977).]

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108 A DECADE OF CONDENSED-MATTER PHYSICS collective excitations in disordered systems. The reason for this is that the interactions in insulators are of short range so that the model spin Hamiltonian is characterized by only one or two parameters, which can often be inferred from independent measurements. In such a situation differences between experiment and theory cannot be explained away by a suitable adjustment of the parameters. This situation contrasts with studies of the electronic states and lattice vibrations in disordered materials. These provide a much less stringent test of theories like the coherent potential approximation since there are many more unknown, and hence potentially adjustable, parameters in the models. One of the most interesting topics in the area of dilute magnetic insulators concerns their behavior near the critical percolation concen- tration. The percolation concentration refers to the concentration below which there is no longer an infinite cluster of mutually interact- ing magnetic atoms. Near the percolation point there is a direct competition between the thermal disorder due to the temperature and the substitutional disorder coming from the dilution. In addition to the critical behavior one is also interested in the nature of the magnon excitations and the extent to which the ideal gas model, which works well at higher concentrations, is useful. One aspect of the behavior near the percolation point that has recently been recognized is the frac- tional effective dimension, or fractal character, of the magnetic clusters. Because of this connection, studies near the percolation point may provide insight into magnetism in effectively nonintegral dimensions. Spin Glasses Certain disordered magnets undergo transitions to a state commonly referred to as a spin-glass state, rather than to the more familiar ferromagnetic or antiferromagnetic states. The spin-glass state, which has also been found in arrays of electric dipoles and quadrupoles, is characterized by the absence of long-range order, a property it shares with the paramagnetic phase, and by the presence of hysteresis, which is a characteristic of ferromagnetism. The appearance of the spin-glass state is signaled on the microscopic scale by a rapid decrease in the rate of fluctuations in the local field, a phenomenon sometimes referred to as spin freezing. Although the unique properties of spin glasses have been recognized for little more than a decade, they are a major topic of basic research. Originally discovered in semidilute magnetic alloys such as CuMn and AuFe, spin-glass behavior has also been observed in magnetic insulators like EuxSr~_xS (Figure 4.5) and Cd~_xMnxTe. The characteristic features of spin-glass behavior are believed to

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MAGNETISM 109 I I I 1 1 ' ' ' ' I I3J 10 J 5 Euxsr'-xs PM ~ FM O ~1 , 1 it, , 0.0 0.5 CONCENTRATI ON x 1.0 FIGURE 4.5 Phase diagram of the insulating spin glass EuvSr,_`S. PM, paramagnetic; FM, ferromagnetic; SO, spin glass. [H. Maletta, J. Appl. Phys. 53, 2185 (1982).] arise from the presence of a large number of local minima in the free energy of the system. As the temperature is reduced, the system becomes trapped in one of these local minima. Transitions between the minima give rise to the irreversible behavior reflected in the hysteresis. The large number of minima is a consequence of a property known as frustration. Frustration refers to the absence of a unique arrangement of moments in the ground state. Unlike a ferromagnet, where the moments in the ground state are parallel, or an antiferromagnet, where there are interpenetrating lattices of oppositely directed moments, the spin glass has a large number of nearly degenerate ground states with widely differing noncollinear spin arrangements. The multiplicity of ground states can arise in a number of different ways. In the archetypal systems like CuMn and AuFe the frustration is induced by the random distribution of magnetic atoms and the oscillatory nature of the exchange interaction between them, which favors parallel or antiparal- lel alignment depending on the separation between the sites. In spin glasses like Cd~_xMnxTe the interactions are short ranged and favor antiparallel alignment of the moments. In this case the frustration arises from the dilution and the topology of the lattice. Research on spin glasses focuses on understanding the onset of

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110 A DECADE OF CONDENSED-MATTER PHYSICS irreversible behavior and the properties of the ground states. Despite intense theoretical activity there appears to be no clear consensus on the nature of the spin-glass state as yet. Some analytic and simulation studies indicate that it is not a true equilibrium phase; rather, it is a metastable state analogous to that found in ordinary window glass. In contrast many experimental studies indicate behavior indistinguishable from that of a thermodynamically stable phase. It is likely that the spin-glass state is characterized by a broad range of relaxation times extending to values at least as long as the duration of the experiments. If this is the case, in an operational sense it may not be important whether the spin-glass state is truly stable. Theoretical studies of the spin-glass transition have generally in- volved the analysis of infinite-range models, with the expectation that they retain some of the features of the real systems. Efforts are being made to understand the appearance of the long relaxation times at the onset of the transition. On the low-temperature side, the properties of the ground states are being analyzed along with the corresponding elementary excitations and their contribution to the specific heat and inelastic neutron scattering, for example. Spin-glass behavior has been established in a great many materials, seemingly rivaling in number those showing conventional magnetic order. Many of the spin glasses have been studied in detail both in terms of their static behavior, as reflected in the magnetization and specific heat, and their dynamics, the latter being probed by use of magnetic resonance, inelastic neutron scattering, Mossbauer effect, and ac susceptibility, muon spin rotation, and ultrasonic measure- ments. Recently, two topics in the field have achieved considerable prominence. The first pertains to the study of so-called re-entrant spin glasses, which are systems that pass from the paramagnetic to fer- romagnetic and then to the spin-glass phase with decreasing tempera- ture (e.g., EuxSr~_xS for 0.52 < x < 0.651. The issue here is whether the spin-glass state in a re-entrant spin glass is different from the spin-glass state in a system that has no intervening ferromagnetic phase. The second area is the nature of the macroscopic anisotropy in spin glasses, which is being probed in torque and electron paramagnetic resonance measurements. In this case the important question is the range of validity of various novel three-axis or triad models for the anisotropy and the low-frequency dynamics. COMPUTER SIMULATIONS IN MAGNETISM Few magnetic models are exactly soluble, and approximate methods of solution turn out to be either inaccurate or complex. This situation

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MAGNETISM l l l poses an increasingly difficult problem since current models of purely theoretical interest as well as those appropriate to real, physical systems are themselves relatively complex. Computer simulation stud- ies span the gulf between theory and experiment. Often one can change the model to make it more like the physical system. One can in a controlled manner examine the effects of finite system size and surfaces, imperfections, and more complicated interactions between magnetic moments, for example (Figure 4.61. Different simulation methods have now been developed for addressing different problems in magnetism. For example, the bulk, macroscopic behavior of magnetic models and the dependence on variations with temperature and mag- netic field can be determined by Monte Carlo methods. In principle, we could calculate the properties of these models in terms of properly weighted averages over all the possible microscopic states of the system. In practice, however, there are too many states to enumerate, and one is simply unable to carry out the calculation. Using various Monte Carlo methods, we can estimate the behavior of the system accurately by sampling only a small fraction of the possible configura- tions. Different ways have now been developed for carrying out this 1 .0 M 0.5 0.0 0.5 1.0 1.5 art, I ~_. l . . . 2.0 2.5 3.0 3.0 kT / K on FIGURE 4.6 Monte Carlo calculations of the magnetization versus temperature for finite-size, two-dimensional Ising models. The various curves display the results from different size arrays ranging from 4 x 4 to 60 x 60. The broken curve is the exact result for the infinite lattice. [D. P. Landau, Magnetism and Magnetic Materials, AIP Conference Proc. No. 24 (Am. Inst. of Phys., New York, 1974), p. 304.]

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112 A DECADE OF CONDENSED-MATTER PHYSICS sampling. The Monte Carlo method has proven successful, particularly in providing information about magnetic systems in the vicinity of their phase transitions. Simulations have been used to locate transition temperatures and to determine if the transition is first order (discon- tinuous) or second order (continuous). Solutions to other problems in magnetism demand knowledge of the time dependence of the microscopic fluctuations in various magnetic models. Such behavior may be probed with very high accuracy using magnetic resonance or neutron scattering. Although we can predict the behavior of each magnetic moment for a short period of time, we find that the time development is determined by the environment (i.e., other nearby magnetic moments), which is itself changing. A computer simulation method known as spin dynamics is used to update the environment of each moment constantly, and hence determine the time development of the system as a whole. This dependence in time and space may then be analyzed using suitable Fourier transform tech- niques so that elementary excitations such as magnons or solitons can be detected. Most studies involve only classical models. Over the last decade, however, progress has been made in understanding various ways in which quantum lattice models may be simulated. Early work has provided information about the properties of low-dimensional XY and Heisenberg magnets. FUTURE DEVELOPMENTS Looking ahead to the next decade one can identify a number of areas where significant progress is expected. In particular, the utilization of various mappings to solve problems in lower-dimensional systems is one field where major advances are likely to be made. In the area of metallic systems, the behavior of intermediate valence compounds is becoming better understood as are the low-temperature properties of magnetic impurities in nonmagnetic hosts. One also looks forward with guarded optimism to continued progress in solving what may be the oldest and most difficult problem in this field: transition-metal fer- romagnetism. The study of disordered magnets is likely to become even more important, particularly since spin-glass-like behavior is being found in a rapidly growing number of materials. There is a need for a unifying picture similar to that provided by the renormalization group approach that will bring together the results obtained from a variety of measure- ments in different systems. In such a development it is likely computer simulations will play an important role in testing theories under controlled conditions.