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This article was downloaded by: [University of Cambridge] On: 06 March 2012, At: 02:05 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Contemporary Physics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tcph20 Varieties of magnetic order in solids C. M. Hurd a a Solid State Chemistry, National Research Council of Canada, Ottawa, K1A OR9, Canada Available online: 13 Sep 2006 To cite this article: C. M. Hurd (1982): Varieties of magnetic order in solids, Contemporary Physics, 23:5, 469-493 To link to this article: http://dx.doi.org/10.1080/00107518208237096 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. COF’TEMP. PHYS., 1982, voi.. 23, NO. 5, 469493 Varieties of Magnetic Order in Solids Downloaded by [University of Cambridge] at 02:05 06 March 2012 C. M.HURD Solid State Chemistry, National Research Council of Canada, Ottawa. K I A OR9. Canada ABSTRACT.Mictomagnetism, metarnagnetism, asperomagnetism, sperimagnetism, speromagnetisrn, spin glass or cluster glass-this string of jargon is part of a score or more ofnew terms recently added to the lexicon ofmapetism. Few ofthem appear in standard textbooks. This article gives an elementary and descriptive review of the new kinds ofmagnetism and their relationships.We have tried to cover all the new terminology. Starting from the familiar types ofmagneticorder in solids, we review the consequences of amorphousnessand disorder for atomic moments, exchange interactions,and single-ion anisotropies.In all, fourteen different types of magnetism are considered. 1. Introduction Magnetism in solids used to be a tidy subject. Although the microscopic origins of some of the different types of magnetism were disputed, their classification seemed straightforward. Five basic types of magnetic behaviour were distinguished and associated with diamagnetism, paramagnetism, ferromagnetism, antiferromagnetism and ferrimagnetism.It had been established that, apart from closed shell diamagnetism and the diamagnetism and paramagnetism of conduction electrons, the behaviour comes from permanent, microscopic magnetic moments possessed by some or all ofthe ions in the solid; the difference between the behaviours lies in the internal arrangement of these moments. Magnetism in solids was an orderly subject that considered orderly systemswith identical magnetic ions distributed throughout a regular crystallinelattice on equivalent atomic sites. The subject was upset about a decade ago by a burst oftheoretical and experimental activity involving two related types of systems: amorphous solids, in which no two atomic sites are equivalent, and disordered solids, in which different atoms occupy irregularly the sites of a regular crystal lattice. New types of magnetic order were recognized that appear only where there is no long-range order, while others appear only in a regular crystal lattice. The subject has since expanded from the original five types of magnetic behaviour to comprise nearly three times that number, and the terminology has multiplied proportionately. It has been jokingly suggested (Nature 1973) that magnetism needs a taxonomist to cope with names like mictomagnetism, metamagnetism, asperomagnetism, speromagnetism, sperimagnetism,spin glass, cluster glass and a dozen others in current use. Clearly, the field presents a bewildering picture to an outsider. My aim is to give an elementary and descriptive review of the origms and connections of the various types of magnetic order in solids. I start from the five basic types (Section 3) and consider their relationship to nine other classes of behaviour (fig. 1).(I shall concentrate on the ground states of the systems and ignore the various types of magnetic excitations that are possible.) In each case I will give a short description of the microscopic origins of the order, together with some examples of its 0010 7514/82,2305 0469 5005.00 I 1982 Taylor & Francis Ltd Downloaded by [University of Cambridge] at 02:05 06 March 2012 470 C. M. Hurd Vcrrieties of mtrgnetic order in solids 47 1 Downloaded by [University of Cambridge] at 02:05 06 March 2012 occurrence.I have tried to cover all the new terminology. It is convenient to begin with a brief review of the two essential ingredients for magnetic order-magnetic ions and a coupling between them-before turning to the main description in Sections 3 and 4. 2. Requirements for magnetic order The interpretation of magnetic effects in solids has developed from two concepts. The first is that a discrete magnetic moment can be associated with ions in a solid. (Induced moments are produced by an externally applied magnetic field; spontaneous ones are present even in its absence.) The second concept is that these microscopic moments interact mutually not only through the ordinary dipole-dipole forces analogous to that felt when two bar magnets are pushed together-for these forces are far too weak to be important-but through quantum mechanical forces. These socalled exchange forces depend on the separation of the magnetic ions as well as their geometrical arrangement, leading to the variety of magnetic order in solids. 2.1. Magnetic ions The electron is the carrier of magnetism. As well as charge, it has an intrinsic angular momentum (‘spin’) that leads to an intrinsic magnetic moment (the Bohr J T-l). The origin of a permanent magnetic moment on magneton, pB =9.27 x an isolated atom was dealt with in an earlier Conrernportrry Physics article (Allen 1976). Briefly, an atom has a net moment when an inner d- or f-electron shell is incompletely filled so that the spin and orbital momenta of the electrons in the shell do not cancel exactly. (The electrons in such shells are frequently called ‘magnetic electrons’.) The Periodic Table shows five groups in which this occurs: the iron group (incomplete 3d shell), the palladium group (4d shell), the lanthanide group (4f shell), the platinum group (5d shell) and the actinide group (5f shell). The idea that a moment can persist when the atom becomes an ion in a solid, even a metal, is basic to the interpretation of magnetic order in solids. There is a strong interaction between electrons in the outer or valence shells ofneighbouring ions, particularly in a metal, leading to an energy band of delocalized states, but the d- or f-shells of ions in such circumstances are af€ected to different extents. The f-shell, which is the more localized and tightly bound in the ion, is less affected by neighbouring ions and it generally retains its atomic characteristics. The origin of its moment can thus be envisaged as in the isolated atom, with an integral number of Bohr magnetons per ion expected.This picture is particularly appropriate to a magnetic ion in an insulator, and much is known about its behaviour (Bates and Wood 1975),but in metals the delocalized electrons complicate the situation. Not only are they an extra source of discrete and itinerant magnetic moments-leading to Pauli paramagnetism-but their interaction with the magnetic ions undermines the atomic concept of a local moment of fixed magnitude. The interaction between an isolated magnetic ion and the delocalized electrons in a metal is described by the s-d interaction models of various types. Where local d-levels and s-conduction states have overlapping energy, it is convenient to talk in terms ofs-d mixing rather like the hybridization that occurs in pure transition metals, but where the mixing is weak it may be treated as a simple s-d exchange. The mixing approach leads to a virtual bound state model, where the 3d moment is modified by the so-called s-d inreruction. In this picture, which was previously described in Contemponrry Physics by Bell and Caplin (1975),an itinerant (s) electron becomes a temporary resident in a (d) atomic-like state about the ion before tunnelling back into the delocalized states. Downloaded by [University of Cambridge] at 02:05 06 March 2012 412 C. M. Hurd During its stay, it experiences the intra-atomic exchange force that couples its spin to that of other resident electrons. Whence the ion’s moment. But it is important to recognize that an ‘isolated’magnetic ion in a metal is a system comprising the ion (‘delectrons’) plus the surrounding itinerant electrons (‘s-electrons’)involved in the s-d interaction. Furthermore, the s-electron population around the ion is spin polarized, opposite to the ion’s net moment, by the effects discussed below. This ‘cloud’ of antiferromagneticspin polarization resonating about the magnetic ion is called Kondo binding or Nagaoka’s compensation after two of its investigators. Below a characteristic temperature ( TK,the Kondo temperature; Bell and Caplin 1975)’ the antiparallel spin cloud neutralizes the ion’s moment and reduces its observable magnitude to zero in the theoretical limit. The Kondo binding is destroyed by increasing temperature and by overlap of clouds from neighbouring ions. The Kondo regime therefore relates ideally to an isolated magnetic ion at absolute zero. This ideal can be approached approximately in a very dilute magnetic alloy at very low temperatures. The opposite extreme occurs when the concentration of magnetic ions is so great that the unfilled shells of neighbouring ions interact sufficiently to form a narrow energy band. The electrons responsible for the ion’s magnetism are then also itinerant to an appreciable extent and the atomic concept of a permanent, localized moment is lost. This occurs in some 3d transition metals and their alloys. The combination ofionic moments and itinerancy ofthe magneticelectrons is called itinerant electron magnetism, and to picture its origins we must consider the constraints acting on itinerant electrons in a narrow band. Firstly, there are exchange effects due to the Pauli principle: electrons of parallel spin stay out of each other’s way to a greater extent than those of antiparallel spin. A pair of electrons of like spin localized on an ion are lower in energy than a pair with opposite spin by an amount called the intra-atomic exchange energy (V). (This is about 1.5eV per spin for 3d electrons.) Consequently, there is a statistical correlation for electrons of like spin, with each surrounded by a void due to the local depletion of parallel-spin electrons. This is called an exchange or Fermi hole. An electron therefore sees a more attractive potential in the presence of parallel spin electrons because the repulsive Coulomb effects are reduced by the exchange hole. (Hence the origin of Hund’s rule for atoms: the ground state of an incomplete shell in a free atom is that of maximum spin.) Secondly, there are dynamical correlation effects: Coulomb repulsion tends to keep electrons ofwhatever spin as far apart as possible, so there is an additional void surrounding any electron-the so-called correlation hole. The cost in energy of the exchange and correlation effects can be reduced only by an increase in an electron’s kinetic energy. Electrons can keep out of each other’s way only by increasing their confinement in space, but the greater an electron’s localization, the higher its kinetic energy. Itinerant electron magnetism is the result of the competition between exchange and kinetic energies. A magnetic ion in this picture is a consequence of the exchange hole. Although the magnetic electrons have some itinerancy, a localized exclusion effect operates for a given spin in the small region ofan ion’s incomplete shell. We can picture a constant interchange ofelectrons by quantum tunnelling between energy states of the shell and the more delocalized ones ofthe band. The narrower the energy bandwidth, the more localized the electrons in it. When the bandwidth is comparable with U,some electrons stay long enough on an ion to interact and to align their spins so that the ion appears to have a local moment. Because of the local exclusion effect, the delocalized electrons are drawn from a pool having a preferred spin orientation, so the ion’s Downloaded by [University of Cambridge] at 02:05 06 March 2012 Vtrrieties of mtrgnetic order in solids 473 moment tends to be self perpetuating. On the time scale of a magnetic experiment, it appears to be permanent and localized on the ion. Since d electrons have a relatively large exchange interaction compared to their kinetic energy, metals and alloys of the iron, palladium and platinum groups show particularly varied types of magnetic ordering, especially ferromagnetism, antiferromagnetism, and exchange-enhanced paramagnetism, which derive from a reduction in exchange energy at the expense of increased kinetic energy for the magnetic electrons. The Stoner band model of itinerant electron magnetism was one of the earliest formulations of this competition. In this model, the energy states of the itinerant magnetic electrons are split into up- and down-spin bands, separated by the exchange interaction. The Stoner model assumes that the splitting is proportional to the spontaneous magnetisation via the Stoner parameter 1.The criterion for ferromagnetic splitting, IN(&),> 1, expresses the fact that the gain in exchange energy due to splitting exceeds the cost in increased kinetic energy for electrons that have to be promoted to higher energy levels (fig. 2). (N(&),is the density of states at the Fermi energy and is inversely proportional to the change in kinetic energy.) t E Fig. 2. Showingthe density ofelectron statesin energy N(E)for itinerant spin-upand spin-down electrons in an itinerant ferromagnet (Section 3.3). The ferromagnet is known as ‘strong’ (S) or ’weak‘ (W), depending on the position of the Fermi level. In addition to the idea that a localized moment on an ion is the time-averaged result of a dynamical process, we must also note the existenceof unequal moments ofthe same species in a given solid. This originates from the asymmetric charge distribution of an unfilled d- or f-shell ion, which interacts differently with the charge on neighbouring ions (the ‘ligands’). The exchange forces felt‘ by magnetic electrons in an ion’s incomplete shell therefore depend on the separation and geometrical arrangement of the ligands, particularly the number and arrangement of nearest neighbours of the same species. This dependence of an ion’s moment of its environment is found in crystalline materials-neither Ni nor Co ions, for example, are magnetic in Au but both acquire magnetism when the number of their like nearest neighbours increases sufficiently-but it is most pronounced in amorphous or disordered solids where the atomic sites are inequivalent. There is typically a distribution in magnitude of the ionic moments in these materials that ranges from a narrow distribution for f-shell ions to a very broad one for d-shell ions. In fact, a d-shell ion can sometimes be present in both magnetic and nonmagnetic states in the same material. Downloaded by [University of Cambridge] at 02:05 06 March 2012 474 C. M. Hurd 2.2. Coupling between moments A magnetic state arising from spontaneous moments in a solid is the net effect of competing influences. The orientation of each moment tends on the one hand to be randomized by its thermal energy while on the other it tends to be aligned by an ordering influence that depends on the magnetism involved. We can distinguish two types of magnetism: the non-cooperative type, where the individual moments behave independently and are unaware ofeach others’ existence, and a cooperative type, where the mutual interactions between the moments are intrinsically important. In a noncooperative magnetic state, the external applied field does the ordering, but in a cooperative one the ordering results from the exchange couplings between the moments. The external field is then frequentlyjust a means of making the microscopic ordering evident on a macroscopic scale. The quantum mechanical coupling between moments in cooperative magnetism is described metaphorically in several different ways depending on the context, but they all derive (see the table) from the influence ofthe Pauli principle and its manifestation as the Fermi hole described in Section 2.1. We can distinguish two classes of exchange coupling. Direct (or contact) exchange operates between moments on ions that are close enough to have significant overlap of their wave functions; it gives a strong but shortrange coupling which decreases rapidly as the ions are separated. Intlirecr exchange, o n the other hand, couples moments over relatively large distances. It acts through an intermediary which in metals can be the itinerant electrons or in insulators can be Hierarchy of exchange coupling. P P The Pauli exclusion principle is the basis of all exchange forces. E: An exchange interaction is a metaphorical description of the effects of the Pauli principle on the Coulomb repulsion between fermions. I: Indirect exchange is a coupling between quantum systems so far apart that some intermediary must be involved. D Direct exchange is a coupling between quantum systems close enough to have overlapping wave functions. R RKKY is an indirect exchange where itinerant electrons are the intermediaries. S: Superexchange is an indirect exchange where the intermediary is a ligand. DM,: The Dzyaloshinsky-Moriya coupling occurs when the spin information between the indirectly coupled systems is upset asymmetrically by spin-orbit effects. In this version, itinerant electrons are the intermediaries. DM,: As in DM, except in this version the spin-orbit coupling occurs at an intermediate ligand. Vcirieties of magnetic order in solids 475 nonmagnetic ions in the lattice. The coupling is then known as the RKKY or superexchange effect in metals or in insulators, respectively. We have seen that the effective electrostatic interaction between two electrons depends on the relative orientations of their magnetic moments (or spins). This is conveniently expressed as a spin-dependent coupling that is assumed to be isotropic and to depend only on the distance between interacting ions. Thus if ions i and j , separated by a distance rij, have spins Si and Sj,respectively, the exchange energy HEis written: H~= - C J(rij)S i * S j Downloaded by [University of Cambridge] at 02:05 06 March 2012 ij where J , is called the exchange parameter. For direct intra-atomic exchange-as between two electrons belonging to the same atom-J is positive leading to Hund’s rule. But for direct inter-atomic exchange, J can be positive or negative depending on the balance between Coulomb and kinetic energies as described in Section 2.1. In indirect exchange, J can be positive or negative, as in superexchange between magnetic ions in insulators (discussed at the end of this Section), or oscillatory as in the so-called RKKY interaction. The RKKY interaction-named after its principal investigators, Ruderman and Kittel, Kasuya and Yosida-is unique because J oscillates from positive to negative as the separation between the ions changes. It is restricted to materials containing itinerant electrons, which are the intermediaries in the coupling. A magnetic ion induces an oscillatory spin polarization in the conduction electrons in its neighbourhood. The reason why this polarization is oscillatory is that the conduction electrons try to screen out the magnetic moment on the ion by the means of their spins (just as by means of their charge they try to screen out the charge on an ion) but their wavefunctionshave a limited range of wavelengths (or wave numbers). In the simplest case ofa degenerate gas offree electrons the highest wavenumber available is 2kFwhere kF is the wave number of an electron on the Fermi surface. The process is analogous to the representation of a non-oscillatory function by means of an incomplete set of Fourier components: there is finally a residual oscillatory behaviour in the representation. The strength of this screening polarization decreases with increasing distance from the ion, but its effect has a relatively long range. This modulated spin polarization in the itinerant electrons is felt by the moments of other magnetic ions within range, leading to an oscillatory, indirect coupling. The existence of RKKY coupling means that in a disordered metallic system, where the separation between magnetic ions is random, positive or negative coupling between moments can be found. This leads to possible conflicts in the system on a microscopic scale as moments try to respond to antagonistic constraints (fig. 3). This is called frusfrution,a concept that has wider implications in other branches of science where it is sometimes called ‘structural disequilibrium’. A frustrated system is one which, not being able to achieve’astate that satisfies entirely its microscopic constraints, possesses a multiplicity of equally unsatisfied states. A frustrated system therefore has no unique microscopic arrangement for its ground state; there is an essentially infinite number of equivalent states that can be adopted. As a result, a frustrated magnetic system shows metastability, with hysteresis effects, time-dependent relaxation towards an equilibrium state, or dependence on the sample’s thermal or magnetic histories. Frustration is also possible in some ordered insulating magnetic materials. Thus MnO has an f.c.c. lattice of Mn2+ions in which nearest neighbours of a given ion are nearest neighbours C. M. Hurd Downloaded by [University of Cambridge] at 02:05 06 March 2012 476 Fig. 3. A triangular lattice of magnetic ions that can b e ’ u p or down-spin illustrates ‘frustration’. When the exchange parameter J between all moments is positive (a),the constraintthat all moments be parallel to their neighburs can be satisfied,so the system is nonfrustrated. But if J is always negative, or positive and negative (b), there is no arrangement that satisfies all the microscopic constraints.The system is thus hstrated. .of one another, and since the n.n. coupling is antiferromagnetic, TNis lowered and the ordered structure complex. Equation (1) has shortcomings, particularly for the interpretation of the newer types of magnetic order, because it involves only the interatomic distances as structural information. It does not account for an exchange interaction that is anisotropic-that is, depends on the direction in the material. Such anisotropy arises bom a coupling between an ion’s moment and the lattice and can have two sources. One is the anisotropic geometrical arrangement of the ligands, leading to an anisotropic electric field at the ion (the ‘crystal field’). The other is spin-orbit coupling. Let us consider first the case when the ligand field effect is dominant. Imagine an amorphous material where the crystal field varies from point to point. Each magnetic ion will have a preferred alignment for its moment along a local ‘easy axis’ determined by the local crystal field. This is called single-site anisotropy, and to include its effect in equation (1)it is convenient to assume the case of uniaxial symmetry. The crystal field energy at the ith site is then of the form -DS;,where D is the axial crystal field strength and S, is the total spin of the ion along the local easy direction z (Kanamori 1963).This must be summed over all sites and added to equation (1) to give the energy for the case with both the local anisotropy and exchange effects: H = -ID&); i -zJ(rij)S;Sj ri In an amorphous solid, D is fixed and positive but the easy axes, zi, are randomly distributed in direction. The first term is usually insignificant compared with the second for d-shell ions in an amorphous metal but the reverse usually applies for f-shell ions. Spin-orbit coupling is the other source of anisotropy. This coupling, which is a relativistic effect,acts on an electron’s intrinsic moment due to the effective magnetic Downloaded by [University of Cambridge] at 02:05 06 March 2012 Vurieties of rntrgneric order in solitls 477 field of its own orbiting charge (Cullity 1972, Hurd 1975). The net spin and orbital motions of the unpaired electrons in a magnetic ion are therefore coupled. Generally, the orbital motion is also coupled to the crystal field via the Coulomb force acting between the charge distribution in the unfilled orbital and the electric field from neighbouring ligands. So there is effectively a coupling between an ion’s moment and the crystal field. Spin-orbit coupling is thus a component of the single-site anisotropy described above, but here we are concerned with its role leading to anisotropic exchange between pairs of magnetic ions, coupled through superexchange. Superexchange was introduced to describean interaction between moments on ions too far apart to be connected by direct exchange but coupled over a relatively long distance through an intervening, non-magnetic ligand (White and Geballe 1979). Different forms of superexchangehave been postulated for different circumstances, but we consider the prototypical case of coupling between the moments on a pair of metal cations separated by a diamagnetic anion, as in a magnetic insulator. This trio can be regarded as a trimeric molecule, where superexchange is a consequence of the hybridization of overlapping orbitals. Figure 4 is a sketch of two cases: an axiallysymmetric molecule with strong spin-orbit coupling on one of the cations, and an axially-asymmetric molecule with spin-orbit coupling on the anion. The former is illustrated by the rare earth-iron (R-Fe) interaction in a garnet. The femc ion has a half-filled 3d shell and so has a spherically symmetric charge distribution ( S state ion). The triply-ionized rare-earth ion, on the other hand, is not symmetric and has a strong spin-orbit coupling; its charge distribution is coupled to its moment. The ion’s moments are coupled via superexchange,so turning the Fe moment alters the overlap ofthe R cation in the molecule(fig.4). This changes the magnitude ofboth the Coulomb and exchange interactions between the cations, leading to a coupling which depends on the moment’s orientation. Thus J in equation (1) depends on the orbital state ofthe rate earth ion, and the exchange energy is anisotropic. ““‘+WR3+ $+ S2’ Fig. 4. Superexchange combined with spin-orbit coupling can have two consequences sketched here: anisotropicexchange, as in the ferric-rare earth interaction in garnets, and the antisymmetric exchange (or Dzyaloshinsky-Moriya coupling) of P-MnS. Downloaded by [University of Cambridge] at 02:05 06 March 2012 478 C.M.Hurd The second example (fig. 4)is illustrated by 8-MnS. This has a cubic zinc blende structure but the ionic arrangement in certain planes is asymmetric with respect to the line of cation centres. When there is strong spin-orbit coupling of the anion, this arrangement allows the so-called DzyaloshinskpMoriya (D-M) interaction to couple the moments on the cations (Moriya 1963). The interaction arises because the spin information carried between the cations by the delocalized electrons of superexchange-or by itinerant electrons in the RKKY version of D-M coupling (Smith 1976)- is upset by the spin-orbit coupling in the anion orbitals. The upset depends on the direction of electron transfer imagined between the cations, and its net effect is zero when averaged over all equivalent molecular configurations of ions in the crystalunless the anion breaks the inversion symmetry with respect to the mid-point of the cations’centres. Then the combination of superexchangeand spin-orbit coupling gives a D-M interaction between the cations of the form HDM=dij(Six Sj) in place of equation (1)(Keffer 1962). This mechanism is important in systems called weak ferromagnets (not to be conhsed with the ‘weakitinerant ferromagnets’of Section 3.3) where the Si and Sj form separate sublattices with antiferromagnetic alignment (Moriya 1963).The lattices are equivalent but not exactly antiparallel, leaving a net magnetization. (Examples are given in Section 4.6.)In some cases HDMis opposed by a strong tendency towards ferroor antiferromagnetic alignment from exchange of the type expressed in equation (1). The D-M energy can then be reduced by canting of the sublattices, but at a cost of increased exchange energy. The cant angles in these systems are small. But in systems where the D-M and exchange energies are not in conflict the cant angles can be up to n/2,as in P-MnS sketched in fig. 4 (Keffer 1962). 3. Five basic types of magnetic order In this Section we review briefly the basic magnetic states familiar from any standard textbook (Cullity 1972, for example) and shown in the top row of fig. 1. 3.1. Diamagnetism We can regard diamagnetism as the result of shielding currents induced in an ion’s filled electron shells by an applied field. The currents are equivalent to an induced moment on each ion in the substance. Ideal diamagnetism is a noncooperative magnetism (Section 2 -2) characterized by a negative, temperature-independent magnetic susceptibility x (fig.5 ) . Diamagnetism is part ofall magnetic states, but it is usually neghgiblecompared with the magnetism arising fiom any spontaneous moments in the system. Examples ofdiamagnetic solids are Cu and NaCl fiom the classes ofcrystalline metals and insulators, and SiO, fiom the class of amorphous solids. 3.2.Ideal paramagnetism This is another non-cooperative magnetism but it arises from spontaneous moments which, in the ideal case, are identical and located in isotropic surroundings ( D = O in equation 2), sufficiently separated to be independent of the others’ existence. The orientation of each moment is on the one hand randomized by its thermal energy k,Twhile on the other it is aligned by an externally applied magnetic field. The magnetic order is thus the net alignment achieved by the applied field in the face of the thermal disarray existing at the ambient temperature. The measured magnetization is the time-averaged alignment of the moments, for their orientations fluctuate constantly and are in no sense fixed (as suggested by the arcs in fig. 6 et seq.). Ideal paramagnetism Viirieties of magnetic order in solids 479 -I AMORPHOUS CRYSTALLINE 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Downloaded by [University of Cambridge] at 02:05 06 March 2012 0 O 00 oooo o o o 0 Cu ,Na Cl O o o 0 0 S iO2 Fig. 5 . Diamagnetism. 0-0 CRYSTALLINE I AMORPHOUS Fig. 6. Ideal paramagnetism. is characterized by a susceptibility x which varies inversely with temperature T ( x = C/T;the Curie law, fig. 6). C is called the Curie constant. Ideal paramagnetism is the exception rather than the rule because there is normally an appreciable exchange coupling between the ionic moments in a solid. Curie's law is only a special case of the more general CurieWeiss law, x = C/(T- e), where 8 is a constant which expresses the interionic coupling. Curie-Weiss behaviour (fig. 6) thus implies a cooperative magnetism that can dominate if the thermal energy of the moments in the paramagnet is reduced sufficiently. As temperature is reduced, this domination occurs at what is, for practical purposes, a discrete temperature (T= TORD). Below TORD, a different magnetic state exists. When it is ferro- or ferrimagnetism, then ToRo is called the Curie temperature (TJ; it is called the Nee1 temperature ( TN)for antiferromagnetism;the.spin-glass temperature (7&)for the spin-glass state; and the C. M. Hurd 480 freezing temperature ( TF)for mictomagnetism. Sketches of metallic paramagnets in crystalline and amorphous forms are given in fig. 6. In each case the group 1B metal is, of course, the nonmagnetic constituent. Downloaded by [University of Cambridge] at 02:05 06 March 2012 3.3. Ferromagnetism Ferromagnetism is a cooperative magnetism (Section 2.2) leading to long-range colinear alignment of all the moments in the system (fig. 7). There is thus a magnetization even with no external field (the spontaneous magnetization). In ideal ferromagnetism, every ion has an identical, spontaneous moment and occupies an identical crystallographic site. The constraints acting on a moment are dominated by the inter-ion exchange coupling (IJI >> 101)and the exchange parameter is everywhere positive (J >0). To maximize its magnetostatic energy, a crystalline sample usually divides into domains which are spontaneously magnetized, nearly or completely to saturation, along a direction of easy magnetization determined by D . An external field H can change the size of these domains, enlarging those of favourable orientation between D and H at the expense of others, but it can make little difference to the intrinsic magnitude of magnetization within a domain. A ferromagnet is an example ofmagnetism where an external field isjust an agent to make evident on a macroscopic scale the ordering that exists microscopically. The field dependence of the reduced magnetization M/M,is typically (fig. 7) sharply rising at lower fields, as the domains with more favourable alignments expand at the expense of the others, and saturating when the maximum domain alignment is achieved. (This is called technical saturation.) Thereis a slight field dependence above saturation (which is called the paraprocess range in the German and Russian literature) due to the applied field's effect on the alignment within a domain. With increasing temperature in any cooperative state, a point is reached at which the moments' thermal energies are comparable to, and eventually exceed, the exchange energies. This occurs at the Curie temperature T, in a ferromagnet. The spontaneous magnetization u, decreases with temperature (fig. 7) to disappear at T,. Above T,, an IJblDl J>O C K W A LLIN E AMORPHOUS Fe Feeob.Feh Fig. 7. Ideal ferromagnetism. Downloaded by [University of Cambridge] at 02:05 06 March 2012 Vm-ieties of riitrgiieric order in soliils 48 1 ideal ferromagnet becomes a paramagnet and obeys a Curie-Weiss law; 1/x is linear in T. Iron, Co and Ni are examples of elemental, crystalline ferromagnets, and various compositions of Fe, Ni, Co, B (known by the trade name ‘Metglas Alloys’) are typical amorphous ferromagnets (Cahn 1980). Iron, Co and Ni, to different degrees, are also examples of itinerant electron ferromagnets(Section2.1). In the Stoner picture (fig.2), the ferromagnetismcomes from the unequal populations of up- and down-spin itinerant magnetic electrons. When the majority-spin band is completely full in this picture-as for S in fig. 2-the system is referred to as a strong itinerantferrornagnet;‘strong’meaning that hrther band splitting cannot increase the magnetism. When itinerant magnetic electrons are contained in both bands (as for W in fig. 2), the system is called a weak itinerantfenornagnet. Then the spontaneous magnetism depends crucially on the shape of the N(E)curve close to the Fermi energy.This can be sensitive to both applied field and temperature changes, leading to metamagnetism (Section 4.1) which may be described as thermal spontaneous mognetizutioii as in Y2Ni, (Gignoux et (11. 1980). Note that this ‘weak itinerant ferromagnetism’should not be confused with the ’weak ferromagnetism’of Sections2.2 and 4.6. Ferromagnetism and superconductivity should be incompatible if the scattering of electrons by the ordered magnetic ions interferes with Cooper pairing. The onset of long-range ferromagnetism can indeed destroy superconductivity, but very recent work on some exotic ternary compounds of holmium molybdenum sulphide (HOMO,$,) and particularly erbium rhodium boride (ErRh,B,) has shown that ferromagnetism and superconductivity can coexist over limited temperature ranges. The result is a strange state that is still controversial (Mook et al. 1982)but appears to be a mixture of normal but very small ferromagnetic domains and superconducting regions with sinusoidally modulated magnetic moments. It was suggested about 25 years ago that magnetic moments in a superconductor would find long-range ferromagnetism less favourable than a ferromagnetism broken into extremely small domains-this is so-called c.r.~ptc!frrromuyrietisrlr(Anderson and Suhl 1959kbut these arguments assumed a strong exchange coupling between the spins of the itinerant electrons and those of the magnetic ions. Such is not the case in the above compounds where the 4felectrons, tightly bound in the ion, have only a small interaction with the superconducting electrons. It is currently believed that electromagnetic rather than exchange interactions govern the competition between ferromagnetism and superconductivity in these compounds. 3.4. Ant@woinugnetisrn Antiferromagnetism, like ferromagnetism, is a cooperative magnetism of longrange order among identical, spontaneous moments. Ideally, the magnetic ions occupy crystallographicallyidentical sites. The exchange coupling dominates the constraints on the moments (IJI >> 1 0 1 )but the exchange parameter is negative (J <0).Thus an ion’s moment points in a given direction while that on a neighbouringion is exactly opposed. There is overall no spontaneous magnetization. Alternatively, a simple antiferromagnet can be regarded as two identical and ferromagnetic lattices of moments which interpenetrate so that the spins on one sublattice are exactly opposed by those on the other. Simple antiferromagnetism can exist only in a crystalline system since it is impossible to subdivide consistently an amorphous system into two sublattices. 482 C. M. Hurd Downloaded by [University of Cambridge] at 02:05 06 March 2012 Most antiferromagnets are ionic compounds such as oxides, sulphides, and so forth (FeO, COO,MnS, etc.), but elemental examples also exist (Crt, a-Mnt and some light lanthanides like Sm) and ordered alloys too (Fe,Mn, CrPt, Pt,Fe, etc.). The temperature dependence of the susceptibility of an antiferromagnet is peculiar (fig. 8). Above the Nee1 temperature (TN),where the thermal energy of a moment exceeds the exchange energy, the substance is paramagnetic and follows a Curie-Weiss law (Section 3.2). Below TN in the antiferromagnetic range (AF in fig. 8), the susceptibility falls with falling temperature as long-range order is increased. At absolute zero in an ideal antiferromagnet, the antiparallel arrangement of moments is perfect (fig. 8). Jh, J<O CRYSTALLINE Cr, X-Mn, MnO Fig.8. Antiferromagnetism. 3.5. F err imag net ism Ferrimagnetism differs from the other four types of basic magnetism because it involves two (or more) magnetic species that are chemically different (represented by the open and closed circles offig. 9). These may bejust two different valence states of the same ion (Fez+and Fe”, for example) or they may be two different elements (Gd and Co, for example).The most important crystalline ferrimagnetic substances are double oxides of Fe, as in XO.Fe,O, where X is a divalent metal like Mn, Ni, Fe, etc. These are based on the spinel structure, and the prototypical example is magnetite (FeO.Fe,O,). The magnetic ions in a crystalline ferrimagnet occupy two kinds oflattice sites, say A and By that have different crystallographic environments. Each of the A and B sublattices is occupied by one of the magnetic species, with ferromagnetic alignment between the moments on the sublattice. There is antiferromagnetic alignment, however, between the moments on A and B. Since the numbers ofA and B sites per unit cell are generally different, and since the magnitudes.of the A and B moments are also different, there is a resultant spontaneous magnetization. t These are itinerant (3d electron) antiferrornagnets and do not follow a Curie-Weiss law above T,. Vorieties of inugneric order in solids Downloaded by [University of Cambridge] at 02:05 06 March 2012 CRYSTALLINE . Spinel:XOFezO, 483 AMORPHOUS REFez,Qd,,Co, Fig.9. Ferrimagnetism. The temperature dependence of ferrimagnetism is qualitatively similar to that of ferromagnetism (compare figs. 7 and 9). In ferrimagnets, however, the spontaneous magnetisation a, usually decreasesmore rapidly with increasing temperature up to the Curie temperature T,,and in the paramagnetic range there is appreciable curvature from the Curie-Weiss law (Section 3.2), particularly close to T,. Amorphous ferrimagnetism exists, by direct analogy with the crystalline case, except that the ions in the A and B sublatticeshave random positions (fig. 9). Examples occur in alloys of the form RE-Fe2, where RE is a heavy rare earth like Tb or Gd.In these alloys, the Fe-Fe and RE-RE couplings are believed to be ferromagnetic,but the F e R E interaction is antiferromagnetic. 4. Nine other types of magnetic order Here we review the magnetic states shown in the second and third rows of fig. 1. They fall into three groups: ferromagnetism and its derivatives (Sections 4.1 to 4.3), paramagnetism and its derivatives (Sections 4.4 to 4.8), and ferrimagnetism and its derivatives (Section 4.9). These follow the linkages suggested in fig. 1. 4.1. Metmagnetism In a normal antiferrornagnet (Section 3.4) the moments feel constraints due to exchange (J)and to anisotropic crystal fields (0)that are far more substantial than the effects of any practical applied field. A metamagnet is an exception. Below its NM temperature, a typical metamagnet is an antiferromagnet,but increasing the applied field strengthcan ultimately overcome the crystal anisotropy forces to change abruptly the internal magnetic structure.The resulting field-induced magnetic transition from a state of low magnetization to one of relatively high magnetization-but with a low susceptibilityin each case-is called metmagnetism (Stryjewskiand Giordano 1977). It belongs to the family of ferro- and antiferromagnetism, where long-range colinear order exists among spontaneous moments constrained to point either parallel or antiparallel to an easy axis. C. M . Hurd 484 Downloaded by [University of Cambridge] at 02:05 06 March 2012 Suppose an applied field has a value H i inside an antiferrornagnet and is parallel to a critical value is reached where the the anisotropy axis D. As H i increases with T < TN, force on an unfavourably aligned moment exceeds the constraint from the crystal-field anisotropy . The consequences can be divided into two classes, depending on whether anisotropy is strong or weak (large or small D in fig. 10).For large D, an unfavourably aligned moment spontaneously reuerses its direction at the critical field; it spinflips to become parallel to the direction of the applied field. For small D,however, with H i along the easy axis, a moment gains magnetic energy by orienting perpendicular to Hi, rather than parallel or antiparallel to it. Above the critical field, the unfavourably aligned moments therefore tend to rotate away from the D axis, to become closer to the perpendicular orientation. This is called spin .Pop. Hi D LARGE FeCL.240 FeC12 D SMALL MnF,, GdAlO, Fig. 10. Metarnagnetism. Field-induced transitions. At lower temperatures, these field-induced transitions produce a ferrimagnetic phase when D is large (fig. lo), and a spin-flop phase when D is small. (The latter phase does not exist, of course, if H i is applied perpendicular to D.) At higher temperatures, the transition is to a saturated paramagnetic state in both cases, although it is sometimes called a ‘pseudo-ferromagnetic’ state. Figure 10 shows typical metamagnetic behaviours for the examples cited, but there are many variations. In particular, in some systems with large D the ferrirnagnetic phase is absent and the transition at lower temperatures is directly from antiferrornagnetism to saturated paramagnetism. But whatever the variations on fig. 10, the moments in the antiferro-, ferri- and paramagnetic phases are constrained by the crystal field to lie along the easy axis D. Itinerant electron metamagnetism is also believed to exist but the effect generally Ingenious experiments have generated large H i internally requires large values of Hi. through suitable doping with rare-earth ions, but nowadays the direct approach is possible since applied fields in the range 10-50T are available. Itinerant electron metamagnets are typically 3d-transition metal compounds (YCo,, TiBe, or FePt,, for example) and transitions from paramagnetism to ferromagnetism, and antiferromagnetism to ferromagnetism have been found (Wohlfarth 1980). 4.2. Incipient ferromagnetism The progression from ideal ferromagnet through metamagnet to incipient ferromagnet (fig. 1) can be described as the decreasing influences of exchange and anisotropy effectson the spontaneous moments in a system-the progression away Downloaded by [University of Cambridge] at 02:05 06 March 2012 Vurieties of mugnetic order in solids 485 from cooperative magnetism (Section 2.2). An incipienrferromagnet (also known as an exchrrnye enhunced meful) is a metal in which the exchange coupling of itinerant electron magnetism (Section 2.1) is not quite sufficient to produce the long-range order ofideal ferromagnetism.It is strongenough, however, to produce at low temperatures a temporary, ferromagnetic alignment of the ions’ moments and surrounding itinerant electrons over restricted regions ofthe metal. (The regions comprise a few hundred or few thousand ions.) The alignment is opposed by thermal disarray, but with falling temperature we can imagine the regions of aligned moments becoming increasingly persistent in time and space. Such a region is called a paramagnon or localized spin fluctuation. Palladium and Pt are prototypical examples among the elements that display such effects, while Ti&, is an example of a compound that is an enhanced itinerant paramagnet and a metamagnet too. Exchange enhancement of the magnetism originates from the mechanism described in Section 2.1 in connection with itinerant electron magnetism (and sketched in fig. 11). If a small evanescent region of spin correlation among the delocalized electrons occurs during their random spin fluctuations, it tends to be self-sustaining because of the Fermi and correlation holes. The energy separation of the u p and down-spin delocalized states is then greater than for the uncorrelated case, leading to enhanced Pauli paramagnetism of the itinerant electrons. The localized moments of surrounding ions are thus sustained from a pool of spin-polarizeddelocalized states (Section 2.1)and so their spin alignment also tends to be self-perpetuating.This mechanism occurs not only in incipient ferromagnets, where the moment on every ion can be enhanced, but also for isolated transition metal ions in a diamagnetic host. The latter case is called a spinjuctuation around an impurity. An ion in a metal (Mn in Al, for example) can be ‘nearly magnetic’ by exchange enhancement just as an incipient ferromagnet is ‘nearly ferromagnetic’. / PARAMAGNON Pd, Pt Fig. 11. Incipient ferromagnetism. 4.3. Superparamagnetism The effectivemoment ofa ferromagneticparticle is determined by its size.A sample of uniaxial ferromagnet having a volume Vgreater than a critical value divides into multiple magnetic domains, each magnetized along the D axis (Section 3.3) but in different directions. The multiple domain structure is, however, no longer favourable below the critical volume, and the particle becomes a single domain with ferromagnetic alignment of all its moments along the same D direction. Thermal fluctuations of the moments exist on a microscopic scale, but to reverse the direction of the single domain’s Downloaded by [University of Cambridge] at 02:05 06 March 2012 486 C . M. Hurd magnetization requires an energy A,? to overcome the crystal field anistropy. (AE is proportional to V.) Making the particle small enough releases moments from their constraints, permitting the magnetization of the single-domain particle to fluctuate between the D directions, as in an ideal paramagnet (Section 3.2). The probability of such a reversal by thermal activation is proportional to exp( -AE/k7') (fig. 12). This differs from conventional paramagnetism because the effective moment of the particle is the sum of its ionic constituents, which can be several thousand moments in a ferromagnetic particle small enough to show paramagnetism. Hence the name superparamagnetism. This is a technically important branch of magnetism that is not new (Bean and Livingston 1959),but it has found recent academic application in the interpretation of some newer types of magnetism. Superparamagnetism, as described here, clearly has antecedents in ideal ferromagnetism and paramagnetism (fig. 1) although any magnetically ordered structure whether collinear or random can exhibit superparamagnetic fluctuations when the material is finely divided into sub-micron sized particles. Two experimental features characterise superparamagnetism (Cullity 1972):there is no hysteresis in the field dependence of the magnetization (a versus H is a singlevalued curve at a given temperature) and a is a universal function of H/T(fig 12). It is important to note that superparamagnetism can be destroyed by cooling. This follows because the characteristic fluctuation time for a particle's moment varies exponentially with temperature, so the magnetization appears to switch sharply to a stable state as the temperature is reduced. The temperature at which this occurs is called the blocking temperature, and it depends linearly on the sample's volume and on the anisotropy strength. Nee1 (1949) first studied blocking in connection with superparamagnetic inclusions in rock, but the concept has found a modern application in mictomagnetism (Section 4.8). Other examples of superparamagnetism are found in systems containing a distribution of fine ferromagnetic particles such as Co spheres in Hg (fig. 12), or Fe particles in amorphous gels, and in independent clusters of magnetic ions in metallic solid solutions. .-P;Ce -AE/kT / AMORPHOUS: Fe ,Co(-406) in Hy Fig. 12. Superparamagnetism. 4.4. Speromagnetism A magnetic state in which localized moments of a given species are locked into random orientations, with no net magnetization nor regular pattern of local ordering Downloaded by [University of Cambridge] at 02:05 06 March 2012 Vririeties of nirrynetic order in solirls 487 beyond say nearest neighbours, is called a speromagnet (from Greek, meaning ‘to scatter in all directions’; Coey 1978). In other words, if P(4)is the probability that any moment makes an angle 4 with a fixed direction, then P@)/ sin 4 is constant (sin 4 is proportional to the solid angle between 4 and 4 + A4.) It is important to distinguish this structure from paramagnetism, where the moments’directions ffuctuate constantly and randomly with time (compare figs. 6 and 13). In the speromagnet below some ordering temperature, TORD, which can appear to be a sharp transition (fig. 13), the moments are ‘frozen’intoorientations which do not vary with time. The arrangement is not unique but one of many nearly degenerate ground states possessed by the system and determined by the balance of exchange interactions at each moment’s site. (Not every moment is necessarily frozen, for the constraints at some sites may cancel exactly, leaving so-called ‘loose moments’.) Speromagnetism is believed to occur mainly in systems where the J i j (equation 2) are essentially random (e.g. from RKKY interactions), or where they are negative (e.g. superexchange)but strongly frustrated. Superexchange (Section 2.2) can give a distribution of J in an amorphous system which may even extend to interactions ofboth signs because the interaction depends on the geometrical arrangment ofneighbouring moments, which varies from site to site. It is believed (Coey 1978) to give speromagnetism in different systems consisting of magnetic nodules, atomic clusters, or tiny spheres distributed amorphously in a nonmagnetic matrix. Speromagnetism also exists in amorphous magnetic insulators such as FeF, where an entirely antiferromagnetic exchange is expected, with frustrated exchange paths. (The crystalline equivalent is found in the random canted structures of disordered oxides and ferrites like (CoJn, -JFe,O,.) In the RKKY interaction, the sign ofJ depends on the moments’ separation (Section2.2), so a distribution of positive and negative exchangescan arise in metals having moments in an amorphous matrix or distributed randomly on a crystalline lattice. A class of crystalline alloys called ‘spin glasses’(Section 4.7),consisting of solid solutions of a few per cent of magnetic ions in a non-magnetic metal, most likely has speromagnetic order below TORD. CuMn is the canonical example (fig. 13) (Ford 1982). Amorphous speromagnets include mixtures of the 3d and 4f groups, and special preparations of the pure rare-earth metals, notably Tb and Dy. The latter are exceptional, however, because D>>J and the moments then are strongly coupled to the local easy axes, which are randomly oriented. CRYSTALLINE AMORPHOUS Fig. 13. Sperornagnetism. Downloaded by [University of Cambridge] at 02:05 06 March 2012 488 C. M . Hurd 4.5. Asperonitrgrietism A magnetic state formed from randomly-placed localized moments of a given species, locked in various orientations below some ordering temperature, TOR,,,but with some orientation(s)more likely than others, is called anhsperomagnet(C0ey 1978). It therefore has a net spontaneous magnetization. Asperomagnetism differs from speromagnetism (Section 4.4) in that P(4)/sin4depends on 4. Since the direction of a moment does not vary completely randomly from site to site, the possibility exists for local ordering within limited regions or domains. Since J > 0 is implied in asperornagnetism, each domain possesses a significant net moment, but two types of asperomagnet can be distinguisheddepending on the relative magnitude of IJI and (DI (equation 2). These are sketched in fig. 14, where the dotted circle represents a domain. When exchange is the dominant interaction (A of fig. 14), each domain can be represented by a single moment of fixed orientation that is relatively unaffected even by very large applied fields-thus the field dependenceon the magnetization shows little saturation (fig. 14). YFe3 is an example which does not saturate even in fields of 15 T. When ID[> IJI, however, the magnetic state is less rigid and more susceptible to external fields. In this case, (B of fig. 14), a local, random D axis is defined at each magnetic site. (These are designated by headless arrows in fig. 14.) The small but positive J causes a moment to select the direction along D that is most nearly parallel to that of its neighbours in the domain. The directions ofmoments in the domain thus fall within a hemisphere such that P(4)lsin4 = 1, (4 < 7r/2); or 0,(4> 7r/2), where 4 is the angle between any D and the optimum direction in the domain. (This is represented in fig. 14 by the semicircular (for hemispherical) spread of directions.) The field dependence of the magnetization (B in fig. 14) is analogous to that of an ideal ferromagnet (Section 3.3). For increasing applied field strength from zero, the initial part of the magnetization curve, where Hi is less influential than D, corresponds to the alignment of the asperomagnetic domains; those favourably oriented grow at the expense of the rest. The higher-field section of the magnetization curve represents the domination of H iover the local anisotropy fields with the reduction ofthe hemisphere AMORPHOUS r - . IJI~DI GdAg,YFa,,GdAI2 101 > I J I DyNi, , TbAg Fig. 14. Asperomagnetism. Vurieries of niLiynetic order in solitls 489 Downloaded by [University of Cambridge] at 02:05 06 March 2012 of allowed directions to a smaller and smaller cone. DyNi3 is an example that shows these sections of the magnetization curve corresponding to the alignment of the domains and the so-called 'closing of the fan'. 4.6. Helimagnetism A magnetic state formed from ordered, localized moments on a crystalline lattice, locked in different orientations below some ordering temperature but with some orientation more likely than the rest, can becalled a helimagnet.It is thecrystalline form of asperomagnetism (Section 4.9, and has antecedents in spero- and asperomagnetism. MnAu, is a prototypical example (fig. 15) of a spiral or helical structure where the preferred orientation ofthe moments varies systematically from one crystal plane to the next. The Mn ions are magnetic and form a b.c.t. structure. Their moments are parallel in any plane normal to the c-axis but their direction rotates about SO" from plane to plane along the c-axis. More complicated forms of helimagnetism occur among the rare-earth metals, with rotation of the moments' direction along the surface of a cone rather than in a plane. Another class of crystalline asperomagnets comprises two (or more) antiferromagnetic lattices that are canted at an angle and thus not quite colinear, leaving a net magnetization. They used to be called parasiticferromagnets but are nowadays known as canted antiferrornagnets or weak ferromagnets. Weak ferromagnetism has 'two origins. One is a difference between the single-ion anisotropies on the sublattices (Section 2.2),the other is the Dzyaloshinsky-Moriya interaction (Section 2.2). In some materials both may operate but in others one mechanism is clearly dominant. NiF, is a case where the single-ion anisotropy is the cause of weak ferromagnetism (Moriya 1963).This material has the rutile structure, which has different kinds of cation sites. The crystallineelectric fields around these sites are the same except that their principal axes are interchanged in the ab-plane. The moments lie in these planes, but below TN they align as two almost antiparallel sublattices, leaving a weak spontaneous magnetism in the plane. Weak ferromagnetism from the D-M interaction can occur only in particular crystal symmetries that have the correct disposition of neighbouring ions (fig. 4). Apart from /?-MnSshown in the figre, a-Fe203,MnSi, CrF, and CoCO, are materials showing weak ferromagnetism which is believed to be determined by the D-M coupling (Moriya 1963). Since weak ferromagnets are a delicate balance of opposing forces, it is not surprising to find that many are also metamagnets. Mn Au,, Dy, Ho, Er Fig. 15. Helimagnetism. Crystalline asperornagnetism. C. M. Hurd 490 Downloaded by [University of Cambridge] at 02:05 06 March 2012 4.1. The ideal spin glasst A crystalline alloy comprising magnetic ions of a given species incorporated into a nonmagnetic host can order speromagnetically (Section 4.4) when cooled below a This is called ‘spin-glass freezing’,and its classic signatures critical temperature TORD. are the appearance of a cusp in the low-field a.c. susceptibility at ToRD ( E TsG)(fig. 13) with metastability below. Prototypical spin glasses are Fe or Mn dissolved in Cu or Au (Ford 1982). The randomness of the orientations of the moments is reminiscent of the randomness in the positions of the constituents of ordinary glass; hence the name spin glass. We recall iiom Section 4.4 that the orientations of the iiozen moments are fixed indefinitely and do not fluctuate with time, but above TsG,in the paramagnetic state, the moments are free to follow the random fluctuations (fig. 16). The spin-glass state in crystalline alloys exists only within a limited range of concentration of magnetic solute. The concentration must be high enough to give mutual interactions via RKKY coupling yet low enough to avoid clusters or linkages of directly-coupled moments that extend throughout the sample. These are called the ‘dilute’and ‘percolation’limits, respectively, and occur in a f.c.c. alloy at about 0.01 and 20 at.%. For concentrations below the dilute limit, the Kondo binding (Section 2.1) is dominant and screens each magnetic ion from interaction with others. (This is called the ‘single-impurity’ or ‘Kondo regime’.) Direct exchange dominates when the concentration is above the percolation limit, leading to ferro- or antiferromagnetism. The ideul spin-yluss sture envisages isolated moments coupled by RKKY interaction with negligible effect of directly-coupled pairs or larger groups. This may be approached experimentally in some systems just above the dilute limit, but in others there may be metallurgical reasons why the solute ions are not distributed randomly in the host so that clusters of ions are important whatever the solute concentrations. The spin-glass phenomenon results ffom a random distribution in the magnitude and sign of the interactions between neighbouring moments. This can arise from the random distribution ofmagnetic ions in either the crystalline and amorphous cases (fig. 16) with RKKY coupling ( J = k),or with entirely antiferromagnetic coupling (J c 0) in a frustrated matrix (Section 2.2). Mn dissolved substitutionally in dilute Cu alloys is an tB B=O o& 0 2 0 O 0 0 0 0 0 0 o r 0 0 0 0‘50 0 0 0 ) J o t s o o o o T= 0 CRYSTALLINE CuMn, (x-0) 0 0 0 0 0‘5 0 0 0 0 o ” E 0 OF0 0 0 T>TSG AMORPHOUS La.0.x Gd, Au, (~41) Fig. 16. Ideal spin glass. J = f or ~ t For a detailed account see Ford (1982). ~~~ Downloaded by [University of Cambridge] at 02:05 06 March 2012 Varieties of rnugnetic order in solids 49 1 example of a spin glass with RKKY coupling, while Mn in ternary semiconductorslike Cd, -,Mn,Te or Hg, -,Mn,Se is an example of one with antiferromagnetic coupling on a frustrated lattice. Spin glasses have been the source of much recent experimental and theoretical activity (Rivier and Taylor 1975, Mydosh 1978, Ford 1982). Their humble origin in the unspectacular field of magnetism in dilute alloys belies their importance to magnetism in solids, for they forced magneticians to abandon emphasis on orderliness and to accept that randomness can be just as fundamental. Despite great effort, several basic qustions remain unanswered: what is the natuh of the spin-glass state? Is it a new phase ofmatter? And is the freezing of the moments at GGthe result of a sharp thermodynamic transition? Two main views ofspin-glass keezing haveemerged. The first, which is based on the ideal spin glass picture, sees the transition as a new type in physics-a collectiveprocess in which the frozen ground state can have many equivalent configurations. The second doubts that the ideal picture is ever applicable in practice since the moments are groups into clusters that have significant influence in all experimental systems. The net moments of these clusters are the magnetic particles. Spin-glass Freezing is then associated with the thermal blocking (Section 4.3) of the clusters’ moments. Some authors prefer the terms ‘mictomagnet’or‘clusterglass’ to describe a spin glass in which clusters of moments are important (Section 4.8). 4.8 M ictomagnet ism Before the spin glass concept was established, earlier work had mistakingly attributed the behaviour to a mixture offerro- and antiferrornagnetism.Hence the term ’mictomagnetism’was coined-the prefix is fiom the Greek meaning ’mixed’-for what later was called ‘spin glass behaviour’ (Beck 1978). The two terms were used interchangeably for a time but lately ‘spin glass’ has been associated with systems in which clustering ofmagnetic ions and short-range order can be ignored; ‘mictomagnet’ or ‘cluster glass’ refers to the opposite case. A mictomagnet is thus similar to the ideal spin glass (Section 4.7) except that the local correlations of the magnetic ions (as pairs, triplets, and so forth) are dominant, generally because the concentration of magnetic solute has been increased sufficiently. Small groups of ions are coupled by direct exchange that is characteristic of the particular species, and coexist as magnetic entities embedded in a spin-glass matrix (fig. 17). The clusters’ moments are coupled indirectly by the RKKY interaction, and they freeze cooperatively below some characteristic temperature &, as in speromagnetism (Section 4.4). The freezing in the mictomagnet is due to the thermal blocking of the otherwise superparamagnetic moments of the clusters (Section 4.3). Since there is a spread of cluster sizes, we expect a corresponding spread of blocking temperatures. This makes the freezing transition less pronounced than in a system with uniform cluster size. (Wohlfarth (1977) has pointed out that exactly this behaviour is found for magnetic inclusions in rock, where the cusp in x( T )at TFhas been compared with the ‘Hopkinson peak’ found in the low field susceptibility of some ferromagnets below TC.)Another characteristic feature of mictomagnetism is the dependence of the magnetization cr on the sample’smagnetichistory (fig. 17). Cooling a sample through & in a relatively small (- 1 T) applied field builds a preferred orientation into the configuration of the clusters’ frozen moments. This asperomagnetic state (B in fig. 17) has typically a shifted hysteresis loop and an enhanced cr below TF,corresponding to a ‘frozen-in’ effective field from the aligned moments. C. M. Hurd 492 Downloaded by [University of Cambridge] at 02:05 06 March 2012 Itr Fig. 17. Mictomagnetism. Cluster glass. 4.9. Sperimagnetism A sperimagnetic structure comprises two (or more) magnetic species with the moments of at least one species frozen in random orientations (Coey 1978). Sperimagnetism is to ferrimagnetism what, speromagnetism is to ferromagnetismexcept that a sperimagnet usually has a net spontaneous magnetization due to predominant ferromagnetic order among one of its constituents (fig. 18).In this case it can be regarded as a canted ferrimagnet (compare figs. 9 and 18),and we can distinguish two classes. In the first, the single-site anisotropy (Section 2.2) of the canted species dominates the exchange coupling between them. ( D l > J in fig. 18 where J is the exchange coupling between 3d ions; Dl, D, are the local anisotropy constants for the 3d and rare earth species, respectively. Exchange coupling between rare earth ions is negligible.)The cant angle is then the same for all ions ofa given species on a crystalline lattice, as for Fe in the example shown. In the second case ( D , < J in fig. 18), the 3d species is now coupled krromagnetically but the rare earth ions are non-collinear. The net moments of the two species are antiparallel, but the rare-earth moments are 0, > J CRYSTALLINE FePd,, +Fe Pt,, 0 Pd,R Dz< J AMORPHOUS DyC4 ,Tb Fez +Dy,Tb + Co,Fe Fig. 18. Sperimagnetism. Canted ferrimagnetism. Vm-ieties of niaynetic order in solids 493 oriented asperomagnetically within a cone (Section 4.5). In fact, a sperimagnetic structure can be regarded alternatively as one which, if all its constituents were equivalent, would be an asperomagnet. Descriptions of many of the topics covered above are not available outside the research literature. So I have tried to be as helpful as I can to the newcomer or nonspecialist with a descriptive and elementary treatment. The various magneiic excitations that occur in the different magnetic states, which have to be ignored here for reasons of space, are covered in the recentbook by Mattis (1981). 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