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OCR for page 95
4
Magnetism
INTRODUCTION
With respect to their magnetic properties, solids can be divided into
two categories depending on the direction of the moment induced by an
applied magnetic field. If the moment is opposite in direction to the
field, the material is said to be diamagnetic; materials where the
moment is parallel to the field are paramagnetic. Apart from its role in
superconductivity, diamagnetism is a weak eject. In contrast, concen-
trated paramagnetic materials often display very large responses
corresponding to local fields of the order of several hundred tesla. At
present most of the research in magnetism involves systems that show
paramagnetic behavior.
In paramagnetic materials the magnetism is associated with partially
filled inner shells of atomic electrons. These are the ad shell (transition
metal compounds), the 4fshell (rare-earth compounds), and the Sfshell
(actinide compounds). The classification can be extended further
depending on whether the electrons in the partially filled shells are
localized, as happens in insulators and semiconductors, or itinerant, as
in many metals. In magnetic insulators each atom with an unfilled shell
possesses an intrinsic magnetic dipole moment. In addition to their
interaction with the applied field, the dipoles interact with one another
through long-range dipolar forces and short-range exchange interac-
tions, the latter arising from the interplay of electrostatics and quantum
95
OCR for page 96
96 A DECADE OF CONDENSED-MATTER PHYSICS
symmetry. Besides their mutual interactions, the moments also inter-
act with the charges on the surrounding ions, which create a crystalline
electric field. The moments in itinerant magnets are delocalized,
extending throughout the system. As in the case of localized moments,
there are both dipolar and exchange interactions.
Strictly speaking, in many materials the paramagnetic behavior
alluded to earlier is observed only at high temperatures. As the
temperature is lowered the system undergoes a phase transition to a
state characterized by long-range magnetic order. The most familiar
example of this behavior occurs in ferromagnets where the long-range
order appears as a spontaneous magnetization. The transitions are
caused by the exchange interactions between the moments and gener-
ally occur at temperatures comparable to the strength of the interaction
between neighboring moments.
Studies of the magnetic properties of materials with localized mo-
ments occupy an unusual position in solid-state physics. This happens
because these properties can be characterized using models involving
only the individual atomic moments, rather than the full array of atomic
electrons. These models, generally referred to as spin Hamiltonians,
have generic forms that depend on the symmetry of the system and
interaction parameters that reflect the microscopic environment of the
magnetic ions. The study of spin Hamiltonians is a central part of
many-body theory and statistical mechanics. The spin Hamiltonian
formalism makes possible a direct connection between real materials
and formal theory that has been a driving force for much of the
research in magnetism in recent years. Experimental studies have
provided crucial tests of theories of collective excitations and critical
phenomena. In turn, the availability of accurate data has stimulated the
development of increasingly precise theories.
The past decade has been one of remarkable progress in magnetism.
By 1970 it can be said that the behavior of isolated magnetic ions in
insulators was well understood. The low-temperature properties of
ideal, three-dimensional (3-D) magnetic insulators were successfully
interpreted in terms of elementary excitations. The high-temperature
behavior had also been investigated, and theoretical studies utilizing
high-temperature expansions had given useful insights. There was
growing interest in magnetic critical phenomena, but no unifying
microscopic theory was available. The magnetic properties of metallic
systems were not well understood. The behavior of isolated magnetic
ions in nonmagnetic metallic hosts had revealed unexpected complex-
ities at low temperatures. In addition, insight into the behavior of
common magnets like iron and nickel had not advanced much beyond
OCR for page 97
MAGNETISM 97
the level of molecular field theory. As will be discussed below, in the
intervening years there have been significant advances in our under-
standing of both insulating and metallic magnets. These occurred not
only in ideal systems but in various types of disordered materials as
well.
The period since 1970 has also been an era of rapid growth in
computer simulation studies of various static and dynamic properties
of magnets. Such a development would have been impossible without
large, high-speed digital computers.
In addition to its place in basic research, magnetism continues to
make important contributions to technology through improved and
novel materials for information storage and power generation, for
example.
MAGNETIC INSULATORS
Low-Dimensional Systems
Recently there has been a shift in emphasis away from studies of
ideal, 3-D magnetic insulators. Increasing attention has been directed
toward low-dimensional (low-D) systems, an area of research that has
grown rapidly in the past 15 years. Before this period, the most
important single research achievement in the field was undoubtedly the
famous Onsager solution of the two-dimensional (2-D) Ising model.
The Ising model is a simplified version of a physically realistic model of
magnetism. The importance of the Onsager solution is that it shows
that a phase transition is possible in a simple model with short-range
magnetic interactions, a matter previously open to doubt. Further-
more, the nature of the critical behavior is significantly different from
the older, molecular-field type of approximate theories of critical
behavior in magnets. In the 40 years since its appearance in 1944, the
Onsager solution has been exploited in a variety of ingenious ways,
which have provided much of the basis of the modern theory of critical
phenomena.
The one-dimensional (1-D) version of the Ising model was solved
back in 1925 but was not considered interesting because the solution
did not show a phase transition. Around the early 1960s, however, a
number of other 1-D models of magnetism were solved, either exactly
or numerically. At that time, the feeling was widespread that 1-D
models were merely amusing toys for mathematical physicists to play
with, with little or no relation to the real, 3-D, physical world.
Nevertheless, the appearance on the scene of a number of 1-D model
OCR for page 98
98 A DECADE OF CONDENSED-MATTER PHYSICS
solutions attracted the attention of experimental physicists, who
searched successfully for chemical systems with highly spatially
anisotropic magnetic properties. They were able to show that the
experimental behavior of the real systems agreed well with that of the
1-D theoretical models. Subsequently, the process of molecular engi-
neering was developed, whereby quasi-2-D and quasi-l-D magnetic
systems were prepared according to specifications by inserting large
nonmagnetic spacer molecular complexes (usually organic) into suit-
able systems to increase the physical separation between planes or
chains of magnetic ions, respectively. In this way, the magnetic
interactions were substantially reduced in one or more crystalline
directions.
As noted, low-D physics is continuing to grow in importance relative
to the traditional 3-D variety for the following reasons:
1. In general, the ease of solution of a particular model of coopera-
tive magnetism increases as the dimensionality decreases. Hence, a
variety of exact solutions of varied and nontrivial models is now
available in 1-D, whereas there are hardly any exact solutions in 3-D.
(The value of exact solutions can hardly be overestimated.) A useful
secondary feature of 1-D exact solutions is their ability to serve as
testing grounds to give insight into the degree of reliability of the
various approximate calculational techniques that must, of necessity,
be employed in 3-D.
2. A feature of great importance is the wealth of novel and interest-
ing physical phenomena that are peculiar to low-D. Examples include
prominent quantum effects in low-D, e.g., in the area of low-
temperature spin dynamics, and the enhancement, by virtue of topo-
logical considerations, of the effects of impurities and randomness in
low-D. Further, the current trend in physics is to move away from the
traditional approach of the linear (harmonic) approximation to consider
nonlinear effects. In the linear approximation to a model magnetic
system, the small-amplitude collective excitations are called magnons,
and their behavior has been studied for decades. Recently, the impor-
tance and interesting properties of nonlinear (large-amplitude) excita-
tions have been recognized. Various types of such phenomena exist,
called, for example, solitons, kinks, vortices, breathers, instantons,
and domain walls. These excitations are important in many areas of
physics, including plasma physics, turbulence, and field theory, but
they appear to be most easily investigated, theoretically and experi-
mentally, in magnetic systems, particularly in 1-D, but also in 2-D,
systems (Figure 4. 11.
OCR for page 99
''t, t t /'1 ~ t / t t t ,' t \ t
''t t '''/ t t t / t t t', t /,
'~''''/ ~ / ~ t 1 ~ ~ t~t t t J
/ / t / / ~ ''',/ I t / t', / t I
t I''t t //~// t t / t J t t
t t t t / t t ~ t t t t / / ,/ / t t t
t t I ~ t','-t t t t t t ,' t t t
~t t '''t t t t t t t'/ t't ~ t
t ~ t t / t t~t t i t''/~/ t f
t t t t ~ ~ t t t t ~ ~ ~ i ,'/ / t
t ~ t I t / t t ~ / t'////~'t /
\~ t \ ~ t / t ~ t /'/~'--~ ~ ~
~ ~ ~ t t t / t t t t /-------~-
\ ~ \ ~ t t / / t \ / ~------
-~ ~ ~ t t t \ \ / ~ ~----`'` I
---'~ ~ t / t / / /----`x ~ J
---' \ \ / / t ~ J / ~ ~-----
~ \ ~ ~ \ t t / / / /~/ /'---`x
`~\~\ ~ t \ t t t t /~/--- -
~ ~ ~ ~ ~ \ \ t \ \ / / / ~ ~--~-
`~\ t t t ~ t / ~
-`~` ~ t \ t / I
`~`--~ ~ t t I I / t
\~_~\ ~ \ \ \ / 1
\ \ t \ \ `~` \ -I I I t
\\ - \~`-\N 1 1 ~
\ ~ \ ~ \'N'\ ~ \ ~ \
t t ' ~ ~ ~ ~ t ~ ~ ~ ~ ~
\ \ t ~ 1 1 1 \~`
t t / t I t ~ \'\ \ ~\
t t t t t t \\~`
\ t \ \ \ \ ' ' ' ~
t ~ ~ t ~ I I' - '
t t I I t \ x\'-
/ \ ~ 1 ~ 1 \~\'
'/ t l t \'\
t t t ~ I I ~`'~
I t \\` \~\ \~
\ \ t t I \ t \
t t I t,/ t
MAGNETISM 99
t t t / t 1 t t / t
/ t ~ ~ ~ I t t I t
'/~t t / t I //
~t t \t\\ \ I t
\ t t t \\~/\\
/ t t \ t tN t I I
t I t I t / / I I I
//~} t t t t t t
/ \~\ t t \~\ \
/ ~-`~` ~ \ ~ ~
t ~-~ ~ ~ _ _ ~ ~
~ ~ ___~____
~ _ ~ _ _ _ _
__, ______,
_ _ _
1 1 ~ ~ ''~ ~ ~ ~
~ 1 J I J/~'
NN \ ~ 1 1 1 1~'
J /~' l
I I J I ~'
J~J 1~/
1 / 1 1 1 /--'
1 1 ~ 1 1 /'-~
1 ~ 1 1 1 /~'-
~1 / I I J/''
'1 /'/ /.~'
I J ~ J I I /'
I J Jil'J'
,,' J ~',
~ J J ~ ~ ''
, /,, ~ _,
/,__
~ J I ~-- ~
~ ~ 1 ~---
_,, ___ _
___ ~ _ %~ _
--~ ~ __ ~
FIGURE 4.1 Nonlinear vortex excitations in the two-dimensional XY model. The dark
and open circles denote the centers of spin vortices of opposite circulation. [S. Miya-
shita, H. Nishimori, A. Kuroda. and M. Suzuki. Prog. Theor. Phys. 6C, 1669 (1978).1
3. A fascinating new development of recent years is a phenomenon
that may be termed, for convenience, mapping. Mapping refers to the
discovery that apparently different physical phenomena are, in fact,
related to one another through an underlying mathematical description.
The same mathematics has been found to describe a variety of physical
systems, with appropriate definitions of the relevant mathematical
parameters. This result may be characterized as obtaining several
solutions for the price of one. Mappings have been discovered between
systems of the same or different dimensionalities. A well-known
example of the former case is the class of systems that are isomorphous
to the 2-D XY model of magnetism. This class includes 2-D superfluids.
2-D melting solids, smectic (layered) liquid crystals, and 2-D Coulomb
gases.
Possibly the most famous mapping between systems of different
dimensionality involves the model many-body system consisting of a
single magnetic impurity exchange-coupled to a sea of conduction
OCR for page 100
100 A DECADE OF CONDENSED-MATTER PHYSICS
electrons. Experimentally, dilute solutions of such impurities in other-
wise normal metals display noticeable anomalies in susceptibility, in
specific heat, and in their temperature-dependent resistivities. These
anomalies are referred to collectively as the Kondo effect. Their
explanation constitutes the Kondo problem. In a remarkable develop-
ment, the 3-D Kondo problem has been mapped onto a solvable 1-D
quantum model. The calculated susceptibility and specific heat agree
well with experiment. At the same time, it is a tribute to the power of
the renormalization group method (Chapter 3) that the solution of the
Kondo problem obtained through its use, though manifestly an approx-
imation, is nevertheless demonstrably accurate when compared with
the exact solution.
The Kondo mapping is presumably the first of many mappings from
3-D to a much more tractable lower dimensionality. This factor,
together with the fact that new mappings are turning up in rapid
succession. makes low-D physics applicable to more areas than one
might, at first sight, suppose.
Critical Phenomena
As noted in Chapter 3, the 1970s was a period of intense activity in
the field of phase transitions and critical phenomena. Studies of phase
transitions in magnetic materials, primarily insulators, confirmed many
of the predictions of high-temperature series and renormalization group
calculations. In addition, they provided important evidence in support
of the concepts of scaling and universality. In the first part of the
decade most of the research pertained to ideal, 3-D magnets. Cur-
rently, greater emphasis is being placed on studies of critical phenom-
ena in disordered and lower-D systems.
METALLIC MAGNETS
Transition-Metal Ferromagnets
Metallic magnets can be divided into two classes depending on
whether the magnetic atoms belong to the transition-metal series or to
the rare-earth and actinide series. Recent advances in the theory of
transition-metal ferromagnets have led to a better understanding of the
nature of their ground state and of their magnetic properties at finite
temperatures.
1. After 50 years of discussion, it is now widely accepted that the
ground state of iron, nickel, and most other transition-metal fer
OCR for page 101
MAGNETISM 10 1
romagnets is best described by an itinerant or band picture, as opposed
to a localized picture. de Haas-van Alphen measurements have gener-
ally agreed with the results of band calculations. The calculations
themselves have now been greatly improved by using better algorithms
and larger machines. The greatest advance, however, is the use of a
new theory that incorporates electron-electron effects into the single-
particle states. Such calculations now correctly predict which transi-
tion elements are ferromagnetic, and they give improved agreement
with Fermi surface data.
These band calculations are also used to determine cohesive energies
and bulk moduli of whole series of materials with great success. In
particular, the anomalously large lattice constants of the magnetic
transition metals, as compared with the trend of their nonmagnetic
neighbors, can be understood from the computed magnetic compo-
nents of their cohesive energies. Another extension is to the calculation
of the spin-wave scattering, which also agrees well with measurements.
A stringent test of the band picture is given by angle-resolved
photoelectron spectroscopy, which determines both the energy and
wave vector of the emitted electron. An improvement to include
spin-polarization analysis was recently announced (Figure 4.21. Large-
scale angle-resolved measurements became feasible with the advent of
high-luminosity synchrotron radiation sources (although some impor-
tant high-resolution work is done with conventional sources). By and
large, the measured dispersion curves are in agreement with band
calculations, giving direct evidence for band states in a wide class of
materials, magnets in particular. There is, however' still much to be
done to achieve an understanding of the various effects of the real holes
created in the photoemission process. A beginning has been made, but
the problems are formidable.
2. The natural model for the temperature dependence of band
ferromagnetism is that of Stoner, which is unsatisfactory, at least for
some materials, in several respects. It predicts too high a Curie
temperature; it does not incorporate directly the thermodynamically
dominant spin-wave excitations; and it does not admit the persistence
of magnetic correlation effects above the Curie temperature. That such
effects exist is shown by the Curie-Weiss susceptibility in the paramag-
netic phase and, more dramatically, by the continued existence of
spin-wave excitations far above T`. in iron and nickel that are seen in
inelastic neutron scattering.
These observations and others have led to a number of new
theoretical schemes for extending the ground-state band picture to
finite temperatures. The competition between these schemes has
revived the localized versus itinerant controversy in a new form.
OCR for page 102
102 A DECADE OF CONDENSED-MATTER PHYSICS
.
.
a
..
x2
.
.
.~.t t t..t
..
.
.
.
.
·1 _
.
-
l ~ . _
O=EF
- o. 9 - 0.6 - 0.3
Initial Energy (eV)
-
-
40 ~
C'
O
c,
-40t~
c
· _
CL
(a
FIGURE 4.2 Electron distribution curves and corresponding spin polarization for
photoemission from the (110) face of nickel. [R. Rune, H. Hopster, and R. Clauberg,
Phys. Rev. Lett. 50, 1623 ( 1983).]
Paramagnon, or weak itinerant, models consider magnetic fluctuations
about a basically nonmagnetic state. Though useful for enhanced
paramagnets, and quite possibly for weak ferromagnets, they provide
less correlation than is needed to describe iron and nickel. Phenome-
nological attempts to extend the paramagnon models to strongly
correlated cases have been made and have met with some success.
Two approaches more directly concerned with the underlying elec-
tronic structure are the local-band theory and the alloy analogy. Each
assumes a disordered magnetization configuration, approximately
solves for the electronic states in the mean exchange field produced by
the configuration, and demands that the configuration be reproduced
self-consistently. The local-band scheme assumes that the important
configurations are characterized by a short-range order, sufficient to
define the exchange split bands locally even above T`.. In the alloy
analogy the magnetizations at different sites are statistically indepen-
dent, and electronic states are computed in the coherent potential
approximation. Evidence in favor of short-range order at high temper-
ature has been reported, but the interpretation of the results has been
OCR for page 103
MAGNETISM 1 03
challenged. Currently, then, the contention is between the picture that
the magnetization (although not the electrons making it up) is localized
for a relatively long time at a single site, and the picture that it has
coherent structure on a larger, 10-15 A scale.
Rare-Earth and Actinide Magnets
There are three great conceptual distinctions between f (rare-earth
and actinide) magnetism and d (transition-metal) magnetism. One is
that, overall, orbital (as opposed to spin) magnetic effects are qualita-
tively more apparent infmagnetism. This shows up in many properties
where the magnetic behavior has peculiarities associated with the
coupling of the orbital moment to the crystalline lattice. Second is that
in metallic systems the f electrons tend to delocalize as one moves
toward the light end of the 4f or Sf row. Thus at cerium in the rare
earths, or at plutonium, neptunium, and uranium in the actinides, one
can study and hope to understand the effects involved in the transition
from localized (Gd-like) to itinerant (Ni-like) magnetism. The lattice
property most obviously correlated with this transition is the lattice
parameter (or more strictly thefatom "of-atom spacing); however, the
effects of the detailed electronic structure can significantly alter this
correlation. Third is that the proximity of a sharp 4f level to the Fermi
energy can lead to instabilities of the charge configurations (valence)
and the magnetic moment.
There have been striking conceptual advances in the past decade
associated with all three of these distinctive features of f-electron
behavior. A discussion of these advances and their interrelationships
follows.
1. The most characteristic consequence of the orbital contribution to
the moment is strongly anisotropic magnetization behavior, with
related peculiarities of magnetic structures and excitations. In the past
decade this has been strikingly evidenced in cerium metallic and
semimetallic compounds and, more recently, in the actinides, with
most recent work in plutonium compounds. The association of excep-
tionally strong anisotropy in magnetic properties with the region where
the local-to-nonlocal f transition occurs suggests a strong connection
between the two phenomena. The availability of single crystals of
actinide compounds, including those containing plutonium, has been a
key element in making possible these advances.
2. Much of the intellectual excitement inf-electron magnetism in the
past decade has been associated with a shift in experimental emphasis
OCR for page 104
104 A DECADE OF CONDENSED-MATTER PHYSICS
from heavier rare-earth systems to cerium systems and into the light
actinides. This excitement has arisen out of a variety of striking
experimental observations, which in one way or another have tended
to relate to the central question of the f-electron localized-to-
delocalized transition. A variety of experimental techniques has been
important in this conceptual opening. They have included high field
magnetism, elastic and inelastic neutron scattering, and electromag-
netic (e.g., de Haas-van Alphen, optical) and electron emission
spectroscopies. On the theoretical side there have been two major
advances. One of these has emerged from electronic (band) structure
studies of the actinide elements, showing a transition in f-electron
behavior from nonbonding (localized) at americium to bonding
(delocalized) at plutonium. The other major theoretical advance has
come out of Anderson-lattice model and band calculations pertinent to
cerium and light actinide metallic, semimetallic, or semiconducting
compounds. This theory shows that when the f electrons are moder-
ately delocalized (intermediate between localized and band behavior,
so as to be slightly bonding), f-electron-band electron hybridization
(mixing) can explain a variety of otherwise anomalous properties
including extreme anisotropy of magnetization and highly unusual
magnetic structures and transitions.
3. The past decade has seen the birth and coming to fruition of a
major area of research in valence instability. This interest was initiated
by high-pressure experiments on samarium compounds at the begin-
ning of the 1 970s. Resistivity, volume change, and susceptibility
measurements, done in rapid sequence, indicated the occurrence of a
valence change of the samarium, with consequent semiconductor-to-
metal and magnetic-to-nonmagnetic transitions. This, in turn, was
followed by a vigorous research effort that showed that the candidate
rare earths for mixed-valence behavior are Sm and Eu at the middle of
the 4f row, Tm and Yb at the end of the row, and Ce at the beginning.
The necessary condition for mixed-valence behavior is that two
bonding states (i.e., with different f-occupation numbers) of the rare
earth in the solid be nearly degenerate (Figure 4.31. It has become clear
that the cause of mixed-valence behavior in cerium materials is
different from that in the heavier rare earths. For the heavier rare
earths mixed valence involves a fluctuation between two degenerate
nearly localized states, whereas for cerium the mixed-valence behavior
may be associated with a 4f localization/delocalization transition.
Whether this is indeed so is a question of great interest. In the coming
period, we can expect to see a lively search for mixed-valence behavior
in actinide systems. It will be important to see whether light actinide
OCR for page 105
MAGNETISM 1 05
8
6
a)
4
llJ
2
o
1 /
-
J =0 / J = 7/2
/:
J = 5/2
0 1 2 3
nf
FIGURE 4.3 Schematic energy-level diagram for the intermediate valence Ce atom. nf
is the number off electrons and J denotes the total angular momentum. [D. M. Newns,
A. C. Hewson, J. W. Rasul, and N. Read, J. Appl. Phys. 53, 7877 (1982).]
systems characteristically show cerium-type or samarium-type mixed
valency, or a mixture of the two. On the theoretical side, we anticipate
an exceptionally active effort in trying to understand the ground-state
properties of lattices of mixed-valence ions on the basis of Hamilton-
ians that include narrow f states, broad conduction bands, hybridiza-
tion, and If correlation effects.
DISORDERED SYSTEMS
Introduction
As in other fields of condensed-matter physics, the study of disor-
dered materials has been an area of intense activity in magnetism in
recent years. In ideal magnets the atoms are arranged on a lattice. The
lattice structure is characterized by translational invariance. This is to
say, atoms that are separated by one or more fundamental repeat
distances have the same local environment. The existence of this
invariance often simplifies the theoretical analysis, especially at low
temperatures, where there is negligible thermal disorder.
In disordered magnets the translational invariance can be broken in
OCR for page 106
106 A DECADE OF CONDENSED-MATTER PHYSICS
different ways. The underlying lattice structure of substitutionally
disordered materials is preserved. However, the sites of the magnetic
atoms are occupied at random either by one of two (or more) different
species of magnetic atoms, as in FerMn~_xF2, or by either a magnetic
or nonmagnetic atom, as in Cd~_~.MnxTe. In amorphous magnets the
lattice is absent altogether. In this case the material is said to be
topologically disordered. Examples of materials that can be prepared in
the amorphous state include YFe2, (Fe-Ni)sOP~4B6, and Fe~_xBx. Usu-
ally the preparation involves rapid quenching from the melt so as to
avoid crystallization.
The fundamental problem in the study of disordered magnets is to
understand the effects of the disorder on the magnetic properties. If the
disorder is weak, i.e., if only a few nonmagnetic atoms are present in
an otherwise fully occupied magnetic lattice or a low concentration of
magnetic atoms is present in a nonmagnetic host, one can interpret the
behavior as a superposition of effects due to isolated impurities.
Although the study of impurities is an important topic in its own right,
the main emphasis currently is on highly disordered systems where an
analysis based on the single-impurity picture is not applicable.
At high temperatures thermal disorder generally dominates any
substitutional or topological disorder with the consequence that ideal
and disordered magnetic materials behave in a qualitatively similar
manner. However, as the temperature is lowered toward the regime
where the energy associated with the thermal fluctuations becomes
comparable to the strength of the interactions between the individual
moments, the absence of translational invariance becomes increasingly
important. The question then arises as to whether there is a transition
out of the high-temperature phase. Should this be the case, is it to a
state of conventional magnetic order or to a low-temperature disor-
dered phase not present in the ideal magnets?
Disordered Ferromagnets, Antiferromagnets, and Paramagnets
The title of this subsection refers to systems that undergo phase
transitions to states of conventional long-range order characteristic of
ideal magnets or else are sufficiently dilute that they remain in their
high-temperature or paramagnetic phase at all temperatures. An im-
portant question pertaining to those systems that do undergo transi-
tions is the influence of disorder on the critical temperature, critical
indices (Chapter 3), and other properties characteristic of the transi-
tion.
The behavior of the disordered systems at low temperatures is also
OCR for page 107
MAGNETISM 107
interesting. As with the ideal magnets one can interpret the static and
dynamic properties in terms of a nearly ideal gas of magnon excita-
tions. Even in the lowest-order, or linear, approximation the calcula-
tion of the spectrum of excitations is a formidable problem. Neverthe-
less considerable progress was made toward its solution in the past
decade. This progress came about in a number of ways. Experimental
studies involving inelastic neutron and light scattering have provided
detailed information about the magnons in various disordered magnets.
Paralleling the experimental work there have been two types of theo-
retical investigation. The first involved purely analytic work mostly
under the general heading of the coherent potential approximation, a
name that reflects its origin as an approximation introduced in the cal-
culation of the electronic properties of disordered alloys. The second en-
tailed the development of computer simulation techniques, which made
possible a direct calculation of the neutron-scattering cross section by
integrating the linearized equations of motion of the spins (Figure 4.4~.
Studies of magnons in substitutionally disordered magnetic insula-
tors have provided important general tests of our understanding of
NEUTRON SCATTERING FROM Rb2 MnO54 M9046 F4 AT 4.0K AS
COMPaRED WITH COMPUTER SIMULATIONS
I ' I I 1 '
2
1
·^
Q=(0.5,0, 2.9) ~
V)
'c O \ -lo'
<' 3 :'h Q=(07,0,2.9)
_
>_ 2 _
1 _
it
i_ 16 ~
Z 12 _;
8 _ ~
4- \~. .
'it.
Q =(0.9,0,2.9) _
2 4
10
20
10
l
Q=(0.6,0, 2.9)
N
3~
Q=(0.8,0,2.9)
l ~
Q = (1.0,0,2.9)
6 8 V 2 4 6 8
ENERGY TRANSFER (meV)
FIGURE 4.4 Distribution of scattered neutrons from the dilute antiferromagnet
Rb2MnO.s4Mgo.46F4. The solid lines are computer simulations. [R. A. Cowley, G. Shirane,
R. J. Birgeneau, and H. J. Guggenheim, Phys. Rev. B 15, 4292 (1977).]
OCR for page 108
108 A DECADE OF CONDENSED-MATTER PHYSICS
collective excitations in disordered systems. The reason for this is that
the interactions in insulators are of short range so that the model spin
Hamiltonian is characterized by only one or two parameters, which can
often be inferred from independent measurements. In such a situation
differences between experiment and theory cannot be explained away
by a suitable adjustment of the parameters. This situation contrasts
with studies of the electronic states and lattice vibrations in disordered
materials. These provide a much less stringent test of theories like the
coherent potential approximation since there are many more unknown,
and hence potentially adjustable, parameters in the models.
One of the most interesting topics in the area of dilute magnetic
insulators concerns their behavior near the critical percolation concen-
tration. The percolation concentration refers to the concentration
below which there is no longer an infinite cluster of mutually interact-
ing magnetic atoms. Near the percolation point there is a direct
competition between the thermal disorder due to the temperature and
the substitutional disorder coming from the dilution. In addition to the
critical behavior one is also interested in the nature of the magnon
excitations and the extent to which the ideal gas model, which works
well at higher concentrations, is useful. One aspect of the behavior near
the percolation point that has recently been recognized is the frac-
tional effective dimension, or fractal character, of the magnetic clusters.
Because of this connection, studies near the percolation point may
provide insight into magnetism in effectively nonintegral dimensions.
Spin Glasses
Certain disordered magnets undergo transitions to a state commonly
referred to as a spin-glass state, rather than to the more familiar
ferromagnetic or antiferromagnetic states. The spin-glass state, which
has also been found in arrays of electric dipoles and quadrupoles, is
characterized by the absence of long-range order, a property it shares
with the paramagnetic phase, and by the presence of hysteresis, which
is a characteristic of ferromagnetism. The appearance of the spin-glass
state is signaled on the microscopic scale by a rapid decrease in the rate
of fluctuations in the local field, a phenomenon sometimes referred to
as spin freezing. Although the unique properties of spin glasses have
been recognized for little more than a decade, they are a major topic of
basic research. Originally discovered in semidilute magnetic alloys
such as CuMn and AuFe, spin-glass behavior has also been observed
in magnetic insulators like EuxSr~_xS (Figure 4.5) and Cd~_xMnxTe.
The characteristic features of spin-glass behavior are believed to
OCR for page 109
MAGNETISM 109
I I I 1 1 ' ' ' ' I
I3J
10
J
5
Euxsr'-xs
PM ~ FM
O ~1 , 1 it, ,
0.0 0.5
CONCENTRATI ON x
1.0
FIGURE 4.5 Phase diagram of the insulating spin glass EuvSr,_`S. PM, paramagnetic;
FM, ferromagnetic; SO, spin glass. [H. Maletta, J. Appl. Phys. 53, 2185 (1982).]
arise from the presence of a large number of local minima in the free
energy of the system. As the temperature is reduced, the system
becomes trapped in one of these local minima. Transitions between the
minima give rise to the irreversible behavior reflected in the hysteresis.
The large number of minima is a consequence of a property known as
frustration. Frustration refers to the absence of a unique arrangement
of moments in the ground state. Unlike a ferromagnet, where the
moments in the ground state are parallel, or an antiferromagnet, where
there are interpenetrating lattices of oppositely directed moments, the
spin glass has a large number of nearly degenerate ground states with
widely differing noncollinear spin arrangements. The multiplicity of
ground states can arise in a number of different ways. In the archetypal
systems like CuMn and AuFe the frustration is induced by the random
distribution of magnetic atoms and the oscillatory nature of the
exchange interaction between them, which favors parallel or antiparal-
lel alignment depending on the separation between the sites. In spin
glasses like Cd~_xMnxTe the interactions are short ranged and favor
antiparallel alignment of the moments. In this case the frustration
arises from the dilution and the topology of the lattice.
Research on spin glasses focuses on understanding the onset of
OCR for page 110
110 A DECADE OF CONDENSED-MATTER PHYSICS
irreversible behavior and the properties of the ground states. Despite
intense theoretical activity there appears to be no clear consensus on
the nature of the spin-glass state as yet. Some analytic and simulation
studies indicate that it is not a true equilibrium phase; rather, it is a
metastable state analogous to that found in ordinary window glass. In
contrast many experimental studies indicate behavior indistinguishable
from that of a thermodynamically stable phase. It is likely that the
spin-glass state is characterized by a broad range of relaxation times
extending to values at least as long as the duration of the experiments.
If this is the case, in an operational sense it may not be important
whether the spin-glass state is truly stable.
Theoretical studies of the spin-glass transition have generally in-
volved the analysis of infinite-range models, with the expectation that
they retain some of the features of the real systems. Efforts are being
made to understand the appearance of the long relaxation times at the
onset of the transition. On the low-temperature side, the properties of
the ground states are being analyzed along with the corresponding
elementary excitations and their contribution to the specific heat and
inelastic neutron scattering, for example.
Spin-glass behavior has been established in a great many materials,
seemingly rivaling in number those showing conventional magnetic
order. Many of the spin glasses have been studied in detail both in
terms of their static behavior, as reflected in the magnetization and
specific heat, and their dynamics, the latter being probed by use of
magnetic resonance, inelastic neutron scattering, Mossbauer effect,
and ac susceptibility, muon spin rotation, and ultrasonic measure-
ments. Recently, two topics in the field have achieved considerable
prominence. The first pertains to the study of so-called re-entrant spin
glasses, which are systems that pass from the paramagnetic to fer-
romagnetic and then to the spin-glass phase with decreasing tempera-
ture (e.g., EuxSr~_xS for 0.52 < x < 0.651. The issue here is whether the
spin-glass state in a re-entrant spin glass is different from the spin-glass
state in a system that has no intervening ferromagnetic phase. The
second area is the nature of the macroscopic anisotropy in spin glasses,
which is being probed in torque and electron paramagnetic resonance
measurements. In this case the important question is the range of
validity of various novel three-axis or triad models for the anisotropy
and the low-frequency dynamics.
COMPUTER SIMULATIONS IN MAGNETISM
Few magnetic models are exactly soluble, and approximate methods
of solution turn out to be either inaccurate or complex. This situation
OCR for page 111
MAGNETISM l l l
poses an increasingly difficult problem since current models of purely
theoretical interest as well as those appropriate to real, physical
systems are themselves relatively complex. Computer simulation stud-
ies span the gulf between theory and experiment. Often one can change
the model to make it more like the physical system. One can in a
controlled manner examine the effects of finite system size and
surfaces, imperfections, and more complicated interactions between
magnetic moments, for example (Figure 4.61. Different simulation
methods have now been developed for addressing different problems in
magnetism. For example, the bulk, macroscopic behavior of magnetic
models and the dependence on variations with temperature and mag-
netic field can be determined by Monte Carlo methods. In principle, we
could calculate the properties of these models in terms of properly
weighted averages over all the possible microscopic states of the
system. In practice, however, there are too many states to enumerate,
and one is simply unable to carry out the calculation. Using various
Monte Carlo methods, we can estimate the behavior of the system
accurately by sampling only a small fraction of the possible configura-
tions. Different ways have now been developed for carrying out this
1 .0
M
0.5
0.0
0.5 1.0 1.5
art,
I ~_.
l
. . .
2.0 2.5 3.0 3.0
kT / K on
FIGURE 4.6 Monte Carlo calculations of the magnetization versus temperature for
finite-size, two-dimensional Ising models. The various curves display the results from
different size arrays ranging from 4 x 4 to 60 x 60. The broken curve is the exact result
for the infinite lattice. [D. P. Landau, Magnetism and Magnetic Materials, AIP
Conference Proc. No. 24 (Am. Inst. of Phys., New York, 1974), p. 304.]
OCR for page 112
112 A DECADE OF CONDENSED-MATTER PHYSICS
sampling. The Monte Carlo method has proven successful, particularly
in providing information about magnetic systems in the vicinity of their
phase transitions. Simulations have been used to locate transition
temperatures and to determine if the transition is first order (discon-
tinuous) or second order (continuous).
Solutions to other problems in magnetism demand knowledge of the
time dependence of the microscopic fluctuations in various magnetic
models. Such behavior may be probed with very high accuracy using
magnetic resonance or neutron scattering. Although we can predict the
behavior of each magnetic moment for a short period of time, we find
that the time development is determined by the environment (i.e.,
other nearby magnetic moments), which is itself changing. A computer
simulation method known as spin dynamics is used to update the
environment of each moment constantly, and hence determine the time
development of the system as a whole. This dependence in time and
space may then be analyzed using suitable Fourier transform tech-
niques so that elementary excitations such as magnons or solitons can
be detected. Most studies involve only classical models. Over the last
decade, however, progress has been made in understanding various
ways in which quantum lattice models may be simulated. Early work
has provided information about the properties of low-dimensional XY
and Heisenberg magnets.
FUTURE DEVELOPMENTS
Looking ahead to the next decade one can identify a number of areas
where significant progress is expected. In particular, the utilization of
various mappings to solve problems in lower-dimensional systems is
one field where major advances are likely to be made. In the area of
metallic systems, the behavior of intermediate valence compounds is
becoming better understood as are the low-temperature properties of
magnetic impurities in nonmagnetic hosts. One also looks forward with
guarded optimism to continued progress in solving what may be the
oldest and most difficult problem in this field: transition-metal fer-
romagnetism.
The study of disordered magnets is likely to become even more
important, particularly since spin-glass-like behavior is being found in
a rapidly growing number of materials. There is a need for a unifying
picture similar to that provided by the renormalization group approach
that will bring together the results obtained from a variety of measure-
ments in different systems. In such a development it is likely computer
simulations will play an important role in testing theories under
controlled conditions.
Representative terms from entire chapter:
spin glasses
