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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
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COF’TEMP. PHYS.,
1982, voi.. 23, NO. 5, 469493
Varieties of Magnetic Order in Solids
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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
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C. M. Hurd
Vcrrieties of mtrgnetic order in solids
47 1
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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.
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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
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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.
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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
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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
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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
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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.
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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
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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.
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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.
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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).
~~~
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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
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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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