Buschow K.H.J. (Ed.) Concise Encyclopedia of Magnetic and Superconducting Materials

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with high values of Curie temperature (T

4500 K),

high saturation magnetization M

41 T, with

the permeability of free space (4p 10

7

1

)), and high magnetocrystalline anisotropy, H

In this article, these intrinsic properties which de-

pend on crystal structure and chemical composition

will be described. In addition, it will be elucidated

that a favorable combination of these values does

not automatically lead to a good hard magnet ma-

terial but can only be regarded as a prerequisite (for

more detailed descriptions see Cullity 1972, Coey

1996, Chikazumi 1997 and Buschow 1997). The ﬁnal

suitability can only be assessed when the extrinsic

properties such as coercive ﬁeld H

, remanent mag-

netization B

, and maximum energy product (BH )

max

derived from the intrinsic properties by the prepara-

tion of appropriate microstructures, fulﬁll certain

criteria.

Ferromagnetic materials are characterized by a

parallel orientation of the participating magnetic mo-

ments below the Curie temperature, i.e., the magnetic

ordering temperature. This is caused by an exchange

interaction H

that was described by Heisenberg in

1928 by:

¼2J

ð1Þ

where J

is the exchange integral, and S

and S

are

the localized magnetic spin vectors of the ions i and j.

For a ferromagnetic interaction, the exchange inte-

gral is positive, whereas it is negative for an anti-

ferromagnetic coupling between the two ions. The

coupling is sensitive to the interatomic spacing r

and

the ferromagnetic interaction is strongest for cobalt,

and significantly weaker for iron and nickel (Bethe–

Slater curve).

The above description requires localized electrons

and needs to be modiﬁed for transition metals (T–T

interactions) where the electrons are itinerant: that

means the 3d ions do not have an integral number of

electrons. Nonetheless, the atomic moment m of the

transition metal element can be deﬁned as

m ¼ gS

ð2Þ

where g is the Lande

factor (ratio of the magnetic

moment to the angular moment), S

an effective

spin and m

the Bohr magneton (1 m

¼9.274 10

24

1

). In a weak ferromagnet both spin-up and spin-

down bands are partially depleted, and in a strong

ferromagnet the majority spin-up band is completely

occupied (Fig. 1). Despite its relatively large moment

iron is a weak ferromagnet (2.22 m

at T ¼0 K) where-

as cobalt (1.72 m

) and nickel (0.61 m

) are strong

ferromagnets (Slater–Pauling curve, see Fig. 2). If iron

were a strong ferromagnet its moment could be

extrapolated to 2.7 m

In rare-earth–transition metal compounds three

types of exchange interactions may occur: T–T, R–T,

and R–R. In the case of SmCo

for example, where

the transition metal carries a well-established mag-

netic moment, the T–T interactions dominate. The

strongly ferromagnetic coupling is responsible for the

magnetic order up to the very high Curie tempera-

ture. The R–R exchange coupling proceeds indirectly

via 4f–5d–5d–4f and is generally very small compared

to the R–T interaction. The latter also proceeds via

an indirect mechanism (4f–5d–3d) in which the strong

interatomic direct 5d–3d exchange is transmitted to

the 4f electrons via the intra-atomic 4f–5d ferromag-

netic interaction (Brooks and Johansson 1993). The

5d–3d exchange is antiferromagnetic when the 5d

band is less than half full and the 3d band is more

than half full, as is the case for compounds of rare-

earth metals with ferromagnetic 3d metals.

Qualitatively it can be summarized that, as the

transition metal spin S

couples antiparallel to the

rare-earth spin S

, the magnetizations of R and

T sublattices couple parallel for the light rare-earth

Figure 2

Atomic magnetic moment m

of ferromagnetic metals

and binary alloys as a function of the number of outer

electrons (3d) per atom n

(Slater–Pauling curve).

Figure 1

Schematic band structures of (a) weak and (b) strong

ferromagnets.

270

Hard Magnet ic Materials, Basic Principles of

elements (J ¼LS) and antiparallel for the heavy

rare-earth elements (J ¼L þS). The orbital magnetic

moments in rare-earth metals remain unquenched by

crystalline ﬁelds because the magnetic moments are

produced by 4f electrons well inside the atoms and

protected from the surrounding atoms. If the total

magnetization rotates together with the orbits in the

crystal lattice, the electrostatic interaction between

the orbits and the lattice favors certain directions of

the magnetization. The result of this interaction is an

enormous magnetocrystalline anisotropy (see Mag-

netic Anisotropy) as illustrated in Fig. 3 for hexagonal

terbium. In that case the shape of the 4f orbit (pan-

cake-like electron cloud) and the crystal symmetry

(compressed along the c-axis) ﬁnally lead to easy and

hard directions of magnetization.

The intrinsic properties are determined by magnet-

ic moments and interactions on an atomic scale and

are independent of the microstructure of the specific

material. The disappearance of long-range magnetic

order at the Curie temperature is a second-order

thermodynamic phase transition, as can be seen from

specific heat capacity measurements, where a signif-

icant magnetic contribution arises. Above this tem-

perature the material becomes paramagnetic. The

transition metals cobalt, iron, and nickel exhibit

Curie temperatures of 1127 1C, 770 1C, and 358 1C

respectively, whereas of the rare-earth metals only

gadolinium (T

¼16 1C) is ferromagnetic close to

room temperature.

The magnetization of a ferromagnetic material has

its origin in the spin and orbital magnetic moments of

the electrons. In a continuum description called

micromagnetism, a local magnetization vector M

can be deﬁned by summing the elementary contribu-

tions per unit volume. In most materials the spatial

distribution of the magnetization is dominated by

magnetic domains (see Magnets, Soft and Hard:

Domains). The direction of magnetization varies from

domain to domain and normally lies along a cry-

stallographic axis. However, within one domain the

magnitude of M is uniform, and it is called the spon-

taneous magnetization M

. The magnetization over

the entire volume of a polycrystalline sample would

normally average out to zero, but a net magnetization

can be retained when the ferromagnetic material can

be prepared in a metastable state.

This remanent state can be achieved after satura-

tion in a magnetic ﬁeld which corresponds to a com-

plete alignment of all the moments (see also Fig. 4)

parallel to the ﬁeld and which eliminates the domain

walls in each crystallite. An easy-axis anisotropy is

indispensable for maintaining the metastable domain

conﬁguration. The crystalline electric ﬁeld mentioned

above, together with the spin-orbit coupling, gives

rise to the single-ion anisotropy on an atomic scale.

The atomic moment has local easy directions that

reﬂect the point symmetry of the site. The overall

crystal symmetry is described macroscopically by the

magnetocrystalline anisotropy that is obtained by

summing all single-ion contributions.

In a material with uniaxial symmetry (e.g., hexa-

gonal cobalt) the anisotropy energy of a crystal E

may be described by

¼K

sin

þ K

sin

y þ K

sin

y þ K

sin

y cos6 f ð3Þ

where K

, K

, etc. are the anisotropy constants

which vary with temperature and differ for different

Figure 4

Hysteresis loops for a permanent magnet: (left) M–H

loop and (right) B–H loop. The initial magnetization

curve after thermal demagnetization starts at the origin;

a minor loop arising after the application of ﬁelds

insufﬁcient for saturation is also shown. The energy

product (BH)

max

is the shaded area in the second

quadrant of the B–H loop.

Figure 3

Electron cloud of the 4f electron of a terbium atom in a

hexagonal lattice (Chikazumi 1997). The c-axis is the

hard direction of magnetization. The ideal c/a value for

a closed-packed hexagonal lattice is 1.63.

271

Hard Magnetic Materials, Basic Pr inciples of

materials; and where the direction of the spontaneous

magnetization relative to the single uniaxial direction

(c-axis) and the a-axis is given by the polar angles y

and f, respectively. With K

dominating and K

40,

the easy direction of magnetization will be along the

c-axis in hexagonal or tetragonal structures. (SI units

of E

and K

are J m

3

.) The anisotropy ﬁeld H

the intersection of the magnetization curves when

measured with the ﬁeld parallel and perpendicular to

the easy magnetization direction and, therefore, can

be used as an estimate for K

(for materials where K

is negligible, this is the case when the magnetization

measured with the ﬁeld applied perpendicular is

a straight line, e.g., tetragonal Nd

B at room

temperature).

¼ð2K

þ 4K

Þ=m

ð4Þ

The static magnetic properties are described by

hysteresis loops as shown in Fig. 4 where either mag-

netization M or magnetic ﬂux density B is plotted as a

function of ﬁeld H (M–H or B–H curves). Provided

that the applied ﬁeld, corrected for the demagnetiza-

tion effect (the demagnetization factor N is a pure

number between 0 and 1 and depends only on sample

shape, e.g., 1/3 for spherical particles and the sum in

the three orthogonal directions is unity), is sufﬁcient

to achieve saturation, one symmetric major loop is

traced (the intrinsic property M

is delineated in

the M–H presentation); otherwise minor loops are

obtained. In the SI system the relationship between B,

M, and H is given by:

B ¼ m

ðH þ MÞð5Þ

M–H or B–H loops have different contours, the

former being more relevant for fundamental aspects,

the latter of interest from an application point of

view and being usually presented in the magnet data

sheets.

For an ideal permanent magnet the second quad-

rant B– H loop is a straight line, with slope m

. The

remanence B

or m

is the ﬂux density or reman-

ent magnetization that persists after removing the

magnetizing ﬁeld H. Another extrinsic property is

given when the loop cuts the ﬁeld axis; with M ¼0

the coercivity H

is obtained (often referred to as

intrinsic coercivity , which is rather unfortunate be-

cause the coercivity is not an intrinsic property but

is very much microstructure-dependent). The max-

imum energy density ( BH)

max

, or also energy prod-

uct, of rare-earth permanent magnets is a measure

of the magnetostatic energy stored and, thus, de-

scribes the performance of the magnet (the shaded

area in Fig. 4). It is the most widely spread ﬁgure of

merit and is found where the product of B and H is

maximized, which is equal to twice the potential

energy of the magnetic ﬁeld outside the magnet

divided by its volume.

(BH)

max

is limited by the remanence, provided the

coercivity has at least half the value of the remanence:

ðBHÞ

max

p m

=4 ð6Þ

where, for example, the remanent magnetization M

is half of the saturation magnetization M

, as pos-

tulated by Stoner and Wohlfarth in 1948 for an as-

sembly of isotropic noninteracting single-domain

particles (or for an ideal permanent magnet:

¼M

). (BH)

max

doubled every 12 years during

the twentieth century (see Rare Earth Magnets:

Materials) and the spectacular development is illus-

trated in Fig. 5. The progress has been made due to

the improvement in coercivity, H

, and not in mag-

netization, M

. The coercivity H

is usually approx-

imated by (assuming K

¼ 2ðK

Þa  N

eff

ð7Þ

where a and N

eff

are microstructural parameters, the

former describing the reduction of the crystal ﬁeld by

defects as well as by misaligned grains and the latter

taking into account internal stray ﬁelds (Adler and

Hamann 1985, Durst and Kronmu

ller 1985).

The mechanism by which a hard magnet is locked

in the metastable state described above is controlled

by many factors (see Coercivity Mechanisms). Here it

is only stated that reverse domain nucleation and

domain wall propagation result in H

(the so-

called Brown’s paradox), which means that the

anisotropy ﬁeld represents an upper limit on the co-

ercivity H

, but typical coercivities in high-perform-

ance magnets do not exceed 20% of H

. This

reduction is principally attributed to microstructural

effects and it is remarked that the type of coercivity

Figure 5

Development in the energy density (BH)

max

of hard

magnetic materials in the twentieth century and

presentation of different types of materials with

comparable energy density.

272

Hard Magnet ic Materials, Basic Principles of

mechanism can be determined by analyzing the initial

magnetization curves after thermal and d.c.-ﬁeld de-

magnetization and the demagnetization curves in the

respective hysteresis loops.

Larger grains of a ferromagnetic phase are not

uniformly magnetized but consist of magnetic do-

mains in order to reduce the magnetostatic energy.

The domains are separated by Bloch walls and the

wall width d

in hard magnets is comparable in size

to the exchange length l

¼O(A/K

) with A being the

exchange stiffness in J m

1

(proportional to T

), the

latter describing the scale of the perturbed area when

a spin is unfavorably aligned. These micromagnetic

parameters are especially relevant for exchange-cou-

pled magnets, which are characterized by an unusual

coercivity mechanism when consisting of ﬁnely dis-

persed hard and soft magnetic phases (‘‘exchange

spring magnets,’’ Kneller and Hawig 1991; see Mag-

nets: Remanence-enhanced ). The critical single-do-

main particle size d

O(A/K

)/(m

)

describes

the size of the largest possible crystallite in which the

energy cost for the formation of a domain wall is

higher than the gain in magnetostatic energy.

Typical values for iron-based rare-earth–transition

metal permanent magnets are of the order of d

200–300 nm. There are two basic concepts (see Co-

ercivity Mechanisms) for the prevention of nucleation

and growth of reverse domains from the fully mag-

netized state (magnetization reversal): (i) for micro-

crystalline magnets (see Magnets: Sintered) and (ii)

for single-domain grains found in nanostructured

magnets (see Magnets: Mechanically Alloyed, Mag-

nets: HDDR Processed). The former concept is real-

ized ﬁrst in the nucleation type, as found in sintered

NdFeB-type magnets consisting of multidomain

grains in the thermally demagnetized, equilibrium

state. The wall motion within a grain is relatively easy

but the spontaneous nucleation of reverse domains is

hindered by smooth grain boundaries, which also

decouple the individual crystallites and thus impede

propagation. Second, in the pinning type as found in

sintered Sm

(Co, Fe, Cu, Zr)

magnets where the

growth of reverse domains is prevented by pinning

the existing domain walls at inclusions or defects. The

nucleation- and pinning-types can be easily recog-

nized from their respective initial magnetization

curve, the former showing a large low-ﬁeld suscep-

tibility whereas the latter is characterized by a weak

low-ﬁeld susceptibility.

Stable coercivities and magnetizations at tempera-

tures often exceeding 200 1C are required for appli-

cations as in motors, actuators, and electromagnetic

machines. Both quantities naturally decline as the

operating temperature approaches the Curie temper-

ature. Reversible temperature coefﬁcients of co-

ercivity (dH

/dT ) and remanence (dB

/dT ) are used

for magnet speciﬁcations in order to design a

machine. Ideally, irreversible magnetic losses are

avoided, i.e., they are only temporary on heating,

and the original values are recovered when returning

to room temperature.

See also: Alnicos and Hexaferrites; Ferrite Magnets:

Improved Performance; Magnetic Films: Hard;

Magnetic Materials: Hard

Bibliography

Adler E, Hamann P 1985 A contribution to the understanding

of coercivity and its temperature dependence in sintered

SmCo

and Nd

B magnets. In: Strnat K (ed.) Proc. 4th

Int. Symp. Magnetic Anisotropy and Coercivity in RE–TM

Alloys. Dayton, OH, pp. 747–60

Brooks M S S, Johansson B 1993 Density functional theory of

the ground state magnetic properties of rare earths and ac-

tinides. In: Buschow K H J (ed.) Handbook of Magnetic

Materials. Elsevier, Amsterdam, Vol. 7, Chap. 3

Buschow K H J 1997 Magnetism and processing of permanent

magnet materials. In: Buschow K H J (ed.) Handbook of

Magnetic Materials. Elsevier, Amsterdam, Vol. 10, Chap. 4

Coey J M D 1996 Rare Earth Iron Permanent Magnets. Claren-

don Press, Oxford

Chikazumi S 1997 Physics of Ferromagnetism., 2nd edn. Oxford

Science Publications, Oxford, pp. 276

Cullity B D 1972 Introduction to Magnetic Materials. Addison-

Wesley, Reading, MA

Durst K D, Kronmu

ller H 1985 Magnetic hardening mecha-

nisms in sintered Nd–Fe–B and Sm(Co,Cu,Fe,Zr)

7.6

perma-

nent magnets. In: Proc. 4th Int. Symp. Magnetic Anisotropy

and Coercivity in RE–TM Alloys. Dayton, OH, pp. 725–35

Kneller E F, Hawig R 1991 The exchange-spring magnet: a new

material principle for permanent magnets. IEEE Trans.

Magn. 27, 3588–600

Stoner E C, Wohlfarth E P 1948 A mechanism of magnetic

hysteresis in heterogeneous alloys. Philos. Trans. R. Soc.

London, Ser. A 240, 599

Strnat K, Hoffer G, Olson J, Ostertag W, Becker J J 1967

A family of new cobalt-base permanent magnetic materials.

J. Appl. Phys. 38, 1001–2

O. Gutﬂeisch

IFW, Dresden, Germany

Heavy-fermion Systems

Heavy-fermion metals represent a class of archetyp-

ical strongly correlated electron systems (Stewart

1984, Grewe and Steglich 1991). Mostly, these ma-

terials are based upon intermetallics containing the f

electron elements cerium, uranium, ytterbium, or

neptunium, and this review will focus entirely on

these ‘‘classic’’ heavy fermions. The heavy-fermion-

like behavior of a number of compounds not con-

taining these elements is beyond the scope of this

work; there is still some controversy over whether the

273

Heavy-fermion Systems

underlying physical mechanisms here are truly anal-

ogous to those of the classic heavy-fermion materials.

Historically, the term heavy fermion (HF) refers to

the observation of a largely enhanced electronic con-

tribution to the specific heat C

at low temperatures

T, compared to normal metals. The specific heat of a

nonmagnetic metal at T5Y

¼Debye temper-

ature) evolves as C

¼gT þbT

. The term gT repre-

sents the electronic specific heat, bT

the phonon

part. In free electron approximation the electronic

specific heat coefﬁcient is

g ¼

NðE



ð1Þ

where N(E

) is the Fermi level density of states, k

the Fermi wave vector, and m



is the effective elec-

tron mass. For simple metals such as sodium the

value of g is of the order of 1 mJK

2

mole

1

, which

corresponds to an effective mass m



almost equal to

the free electron mass. In contrast, in HF systems g

values of 100–1000 mJK

2

mole

1

are found, reﬂect-

ing an equally bigger effective electron mass—hence

the label heavy-fermion or heavy-electron materials .

HF behavior is intimately connected to the sup-

pression of magnetic order. In HF systems magnet-

ism is unstable due to the competition between two

processes involving the magnetic moment on the f ion

site and its hybridization G with the surrounding

conduction electrons: the indirect magnetic interac-

tion between f moments promotes long-range mag-

netic order, while a moment screening process, the

Kondo effect, tends to suppress magnetic ordering.

The strength of the hybridization G controls the

competition of these two processes, and by variation

of G a transition from a magnetic to a nonmagnetic

ground state can be achieved.

Experimentally, the hybridization of a material can

be tuned by applying chemical or hydrostatic pres-

sure. Ideally, both similarly change the material

properties, as is the case for CeRu

(Fig. 1).

Here, the crystallographic unit cell volume is reduced

by 5% by the isoelectronic substitution of silicon by

germanium (x ¼1 in CeRu

(Ge

1x

)

) or applied

pressure p ¼70 kbar, leading to an increase of G

(Su

llow et al. 1999). Physically, the ground state of

the system is tuned from a local moment ferromagnet

(FM) via two reduced moment antiferromagnetic

(AFM) phases, through the magnetic instability with

¼0 into a nonmagnetic Fermi liquid (FL) below a

characteristic temperature T

. The magnetic phase

diagram, derived from alloying (shaded symbols, top

scale) and pressure (open symbols, bottom scale)

studies, is plotted in Fig. 1, with the FM phase I and

the two AFM phases II and III. Included in the ﬁgure

are T

and the Kondo temperature T

(ﬁlled sym-

bols). The HF regime for CeRu

(Ge

1x

)

, with

large effective electron mass (i.e. g4100 mJK

2

mole

1

), consists of a narrow pressure range of less

than 10 kbar on both sides of the magnetic instability,

with the mass enhancement increasing on approach-

ing the instability. At the instability, with T

¼0, non-Fermi liquid (NFL) behavior is observed

(see Non-Fermi Liquid Behavior: Quantum Phase

Transitions).

Microscopically, the large mass enhancement in

the HF state reﬂects the appearance of a very large

resonance peak, the Kondo resonance, in the en-

hanced electron density of states at the Fermi energy

. Based on inelastic magnetic neutron scattering

and dilution studies on prototypical nonmagnetic HF

compounds such as CeCu

or CeRu

, this reso-

nance at least in part can be traced back to the single-

ion Kondo effect, which causes the screening of the

local magnetic moments by conduction electrons be-

low a characteristic temperature, the Kondo temper-

ature T

. On the other hand, the observation of a

quasielastic magnetic response at certain distinct

wave vectors in reciprocal space attests to the impor-

tance of intersite effects, which give rise to short-

range and short-lived antiferromagnetic ﬂuctuations.

As the intersite interactions compared to the onsite

Figure 1

The pressure/chemical pressure phase diagram of

CeRu

(Ge

1x

)

. Open symbols &, J, and B denote

the ferromagnetic FM I, antiferromagnetic AFM II and

AFM III transition temperatures determined in a

pressure experiment on CeRu

, the corresponding

shaded symbols those established from an alloying study

on CeRu

(Ge

1x

)

. Further, the single-ion Kondo

temperature T

, derived from a combination of inelastic

neutron scattering and pressure dependent resistivity

measurements, and the Fermi liquid temperature T

representing the temperature range of a resistivity

r ¼r

þAT

, are included in the ﬁgure (ﬁlled symbols).

Note that an increasing pressure corresponds to an

increasing hybridization G.

274

Heavy-fermion Systems

Kondo interactions gain importance, an antiferro-

magnetically ordered state will eventually be entered

which, sufﬁciently close to the magnetic instabil-

ity, is carried by even more strongly mass-enhanced

electrons.

While the fundamental properties of HF materials

derive from the single-ion Kondo model, they do not

fully correspond to it. In particular, as manifesta-

tion of single-ion Kondo behavior the resistivity

saturates far below T

with rp1 ðT=T

, while

in HF compounds below T

a metallic resistivity

r ¼r

þAT

occurs (r

being the residual resistivity).

This deviation from single-ion Kondo behavior is re-

ferred to as the transition into the coherent state.

Phenomenologically, it is viewed as the realization of

a ‘‘Bloch-wave state of Kondo scatterers,’’ i.e., the

coherent superposition of scattering amplitudes from

Kondo ions with the periodicity of the crystallo-

graphic lattice. Because of this transition into a co-

herent state, HF materials are also called Kondo

lattice systems (see Kondo Systems and Heavy Fermi-

ons: Transport Phenomena).

Closely linked to HF behavior, a rich variety of

ground state properties and exotic physical phenom-

ena is observed. There is the intricate relationship

between the HF state and FL as well as NFL be-

havior. For a number of HF compounds, supercon-

ductivity—possibly unconventional—and unusual

types of magnetic order are observed. Further, a

gap can open in the enhanced electron density of

states at E

, causing a semiconducting strongly cor-

related behavior. These, as the most relevant anom-

alies of the HF state, are briefly discussed below.

1. Fermi Liquid Behavior

The principal feature of nonmagnetic HF compounds

below T

, which is not zero, is that the materials

behave as if they were free electron materials, but

with largely enhanced electron mass m



. This high-

lights the presence of a novel kind of electronic

‘‘quasiparticles’’ (the ‘‘heavy fermions’’), whose con-

tribution to the electronic specific heat evolves like

gT, and whose susceptibility behaves Pauli paramag-

netic, saturating at a value w

. In addition, the resis-

tivity rr

obeys AT

, as expected for dominant

quasiparticle–quasiparticle scattering. The coefﬁ-

cients g, w

, and A reﬂect the mass enhanced state

with gpw

ﬃﬃﬃﬃ



. The heavy FL state, coher-

ently formed out of conduction electrons and f elec-

trons, weakly delocalized due to their hybridization

with the former, has been studied by de Haas–van

Alphen and Shubnikov–de Haas measurements (see

Magnetic Systems: De Haas–van Alphen Studies of

Fermi Surface). The experimentally determined re-

normalized band structures of HF systems are repro-

duced by calculations, if the mass renormalization

is taken into account as an empirically introduced

parameter. Altogether, the physical properties are

linked by a single parameter, the electron mass en-

hancement. The concept to describe such single-

parameter scaling is the Landau theory of the Fermi

liquid. Within this concept a one-to-one correspond-

ence between the quasiparticles of the HF state and

the electrons in a free electron gas is drawn via a

renormalization of the effective mass. In this way, the

Fermi liquid theory achieves a mapping of the HF

state onto the well-known non-interacting free elec-

tron gas.

At the magnetic instability, for T

¼T

¼0, the

description of the HF properties within the Landau

FL theory appears to fail. Here, temperature de-

pendencies of the physical properties are recorded

which do not adhere to the FL predictions, and which

are collectively referred to as NFL behavior.

2. Superconductivity and Magnetism

One of the most spectacular and elusive phenomena

of HF materials is the appearance of superconduc-

tivity in a few compounds (Zwicknagl 1992, Heffner

and Norman 1996). Presently, six materials are re-

garded as heavy-fermion superconductors (HFS):

CeCu

, UBe

, UPt

, URu

, UPd

, and

UNi

. Furthermore, CeCu

, CeRh

, CePd

, and CeIn

are probably HFS at high pressure.

The unifying aspect of HFS is that superconductivity

is carried by the heavy quasiparticles. Thus, super-

conductivity in HF systems is a property of the

strongly correlated state and involves those electrons

that also take part in the magnetic exchange, i.e., the

f electrons.

The electronic correlations manifest themselves in

highly unusual superconducting properties, such as

multiple superconducting phases, anomalous pinning

behavior, or impurity effects. In analogy to the su-

perﬂuidity of

He, it is assumed that in HFS the

symmetry of the superconducting order parameter is

highly anisotropic and that they behave as uncon-

ventional superconductors, i.e., show an order pa-

rameter with a lower symmetry than that of the

underlying crystal lattice. While for CeCu

, UBe

and UPd

the order parameter most probably is of

even parity, Knight-shift results suggest an odd-par-

ity superconducting state in UPt

(Tou et al. 1996).

Further candidates for this variant are URu

and

UNi

. An important question concerns the super-

conducting pairing mechanism. In contrast to the

classical, phonon-mediated superconductors, in HFS

the Cooper-pair formation may be, at least partly,

mediated by electronic exchange mechanisms. So far,

however, neither the unconventional superconducting

order parameter nor the specific pairing processes

have been established beyond doubt.

The main difﬁculty in determining the basic phys-

ical principles of HFS is the multitude of physical

275

Heavy-fermion Systems

phenomena observed in this class of compounds, in

particular with respect to the interdependence, com-

petition, and coexistence of superconductivity with

other ground states. In the cerium-based systems,

superconductivity is observed close to the magnetic

instability, implying an intimate connection of HFS

to the suppression of magnetic ordering. In contrast,

in UPt

, URu

, UPd

, and UNi

magnet-

ism and superconductivity coexist on a microscopic

scale. Yet, in these uranium systems the magnetically

ordered ground states are vastly different. In UPt

spin-density-wave (SDW) state of ultralow moments

ord

¼0.02 m

) forms. Apparently similar to UPt

,an

ultralow-moment SDW state exists in URu

. How-

ever, here magnetism probably does not constitute

the primary order parameter; instead, quadrupolar

ordering mechanisms are considered. UPd

and

UNi

both exhibit a more common type of antif-

erromagnetic order, but the ordered moment in the

palladium system (m

ord

¼0.85 m

) is huge compared

to the nickel compound (m

ord

¼0.24 m

) and to all

other HFS. Again in variance, no long-range mag-

netic order is observed in pure UBe

, but alloying

the system with a few percent thorium triggers ultra-

low-moment magnetism and leads to a complicated

superconducting phase diagram.

Possibly the most remarkable anomaly of HFS is

the multiple superconducting B–T phase diagram of

UPt

shown in Fig. 2. This phase diagram constitutes

the strongest evidence for the unconventional super-

conductivity in HFS. Theoretical proposals to ex-

plain the multiple phase diagram started from the

assumption of two different anisotropic supercon-

ducting states A and C, each with lower symmetry

than the crystalline lattice and energetically almost

degenerate. The corresponding phase boundaries

cross at a tetracritical point. The phase B below

, as a combined state of the phases A and C, shows

magnetic signatures that have been attributed to the

nonunitary property of this state (Tou et al. 1998).

However, a conclusive experimental veriﬁcation of

unconventional superconducting order parameters in

UPt

or other HFS is still lacking. An even greater

challenge remaining is the determination of the su-

perconducting pairing mechanisms. While there is an

abundance of theoretical studies on electronic pair-

ing, only a few experiments have yet been presented

in support of such pairing mechanisms in HFS. The

nonuniversality of the superconducting phenomena

has led to the proposal that more than one coupling

mechanism may have to be considered for all HFS.

Aside from the superconducting properties of HF

metals, their magnetic behavior is also remarkable

(Amato 1997). Most important, low- and ultralow-

moment magnetism is observed, i.e., the transition

into antiferromagnetically ordered states with stag-

gered magnetic moments as low as 0.001 m

. To some

extent, this behavior can be understood by proximity

of the magnetic instability in HF materials. On ap-

proaching the instability within the magnetically or-

dered phase, the f moment is increasingly screened by

the Kondo effect, which thus can attain any arbitrary

small value.

This mechanism for low-moment magnetism

relates the size of the moment to the magnetic

transition temperature, which both decrease in a

corresponding way. For a number of HF systems,

however, such a link does not exist; here additional

effects come into play. First, a magnetic order some-

times realized in a SDW indicates that nesting of the

renormalized Fermi surface plays a role. Second, in a

few HF materials the magnetic ordering appears to

be only a secondary, ‘‘spurious’’ effect, which masks

the underlying order/disorder transition of the ‘‘true’’

order parameter, frequently being of quadrupolar

nature. Finally, for some of the ‘‘ultralow-moment’’

systems, i.e., those with an ordered moment of

B0.01 m

, the distinction between ‘‘long-range mag-

netic ordered’’ and ‘‘slowly ﬂuctuating’’ apparently

gradually disappears. For all these materials the

magnetic behavior remains very much a mystery to

date.

3. Kondo Insulators

These materials are HF or intermediate valence (IV)

compounds with a small gap in their quasiparticle

density of states (Aeppli and Fisk 1992, Degiorgi

1999). Accordingly, at temperatures TbD (D ¼width

of the energy gap; typically D10–100 K) the magnetic

properties of Kondo insulators are those of para-

magnetic HF metals, with a Curie–Weiss local

f-moment susceptibility. But, as T decreases, thermal

depopulation controls the T dependence of the sus-

ceptibility. Hence, w goes through a maximum at

Figure 2

Schematic B–T phase diagram of the three

superconducting phases of UPt

276

Heavy-fermion Systems

TED and saturates at a value w

, which is much

smaller than in HF metals. Similarly, the transport

properties at TbD resemble a dirty HF metal, but as

T is lowered below D, semiconducting behavior de-

velops. The effective carrier mass in the semicon-

ducting state is as large as in HF or IV metals,

reﬂecting a strongly enhanced density of states at the

gap edges. And while the electronic specific heat co-

efﬁcient is small (B1 mJK

2

mole

1

), upon closing

the gap of a Kondo insulator due to doping, semi-

conductivity is wiped out, and a large g value

(4100 mJK

2

mole

1

) is observed in the specific

heat. In summary, Kondo insulators are nonmagnet-

ic, mass-enhanced semiconductors at T5D, while at

TbD they are reminiscent of HF/IV metals.

The essential properties of Kondo insulators are

captured in a simple model of a ﬂat f-band hybrid-

izing with a single broad conduction band, generating

the gap D. The total number of valence electrons is

even, including f electrons in the count. At T ¼0 the

electrons ﬁll the lower hybridized band. Then, for

ﬁelds H smaller than the energy gap D the suscepti-

bility is zero due to the equal number of spin up

and spin down electrons. As T increases, w increases

because of a thermally activated Pauli-like suscepti-

bility, until at TbD Curie–Weiss paramagnetism

develops. Similarly, the transport properties show a

transition from semiconducting to metallic behavior

as function of temperature. This model represents the

realization of an Anderson lattice Hamiltonian, with

two electronic states per f site applied to a system

containing one f- and one conduction band, and the

f onsite Coulomb repulsion U.

This simple picture qualitatively accounts for the

fundamental properties of Kondo insulators, but it

fails to fully explain the Kondo insulating state. For

instance, it has not conclusively been resolved whe-

ther the opening of a gap in Kondo insulators is

entirely a band structure effect. Surely, electronic

correlations do play an important role. Further, four

phenomena, which appear to be characteristic fea-

tures of Kondo insulators, elude simple models: (i) In

Kondo insulators the band structure is renormalized

in a fashion similar to HF metals. It causes the piling

up of states at the gap edges, and thus the mass en-

hancement. To understand this behavior, electronic

interactions have to be taken into account. (ii) The

energy gap is renormalized in temperature, i.e., its

signatures already disappear well below T ¼D/k

(iii) In contrast to magnetic semiconductors or HF

metals, in which magnetic interactions play a funda-

mental role and eventually cause the formation of

magnetically ordered ground states, for Kondo in-

sulators the intersite magnetic interaction J vanishes

in the limit o-0. Thus, in spite of a dense array of

magnetic ions in Kondo insulators the net magnetic

exchange is zero. (iv) There are a number of prob-

lems related to the gap structure of Kondo insula-

tors. Most important, it is not clear whether a gap or

a pseudogap intrinsically opens up in the density of

states.

See also: Non-Fermi Liquid Behavior: Quantum

Phase Transitions; Electron Systems: Strong Corre-

lations; Intermediate Valence Systems

Bibliography

Aeppli G, Fisk Z 1992 Kondo insulators. Comments Condens.

Matter Phys. 16, 155–70

Amato A 1997 Heavy-fermion systems studied by mSR tech-

nique. Rev. Mod. Phys. 69, 1119–79

Degiorgi L 1999 The electrodynamic response of heavy-electron

compounds. Rev. Mod. Phys. 71, 687–734

Grewe N, Steglich F 1991 Heavy fermions. In: Gschneidner K

A, Eyring L (eds.) Handbook on the Physics and Chemistry of

the Rare Earths. Elsevier, Amsterdam, Vol. 14, pp. 343–474

Heffner R H, Norman M R 1996 Heavy fermion superconduc-

tivity. Comments Condens. Matter Phys. 17, 361–408

Stewart G R 1984 Heavy-fermion systems. Rev. Mod. Phys. 58,

755–87

llow S, Aronson M C, Rainford B D, Haen P 1999 Doniach

phase diagram, revisited: from ferromagnet to fermi liquid in

pressurized CeRu

. Phys. Rev. Lett. 82, 2963–6

Tou H, Kitaoka Y, Asayama K, Kimura N, Onuki Y, Yama-

moto E, Maezawa K 1996 Odd-parity superconductivity with

parallel spin pairing in UPt

: evidence from

195

Pt Knight shift

study. Phys. Rev. Lett. 77, 1374–7

Tou H, Kitaoka Y, Ishida K, Asayama K, Kimura N, Onuki Y,

Yamamoto E, Haga Y, Maezawa K 1998 Nonunitary spin-

triplet superconductivity in UPt

: evidence from

195

Pt Knight

shift study. Phys. Rev. Lett. 80, 3129–32

Zwicknagl G 1992 Quasi-particles in heavy fermion systems.

Adv. Phys. 41, 203–302

F. Steglich

Max-Planck-Institut fu

r Chemische Physik fester

Stoffe, Dresden, Germany

S. Su

llow

Technische Universita

t Braunschweig, Germany

High-resolution Fourier Transform

Spectroscopy: Application to Magnetic

Insulators

Optical spectroscopy is used widely for studying

magnetic insulators. Various aspects of these studies

have been reviewed by Hu

fner (1982), Eremenko

et al. (1986), and Cone and Meltzer (1987). Typically,

the spectral resolution of grating spectrometers used

for these studies is 0.5–1 cm

1

in the visible range of

the spectrum. For magnetic compounds containing

rare earths (REs) the inhomogeneous width of some

277

High-resolution Fourier Transf orm Spectroscopy: Application to Magnetic Insulators

spectral lines is less than 0.1 cm

1

(Popova 1998). The

most intense optical transitions allowed for a free ion

lie in the infrared and occupy broad spectral regions

for the majority of RE

3 þ

ions. It is advantageous to

meter (FTS) rather than by a classical one.

Substantial gain in sensitivity of a FTS compared

with grating spectrometers permits the study of weak

spectra of RE ions introduced in a small amount as a

probe into a magnetic material, even if the latter is

available in a polycrystalline form only. Far-infrared

(FIR) study of magnetic compounds that undergo a

magnetoelastic phase transition is another ﬁeld where

FTS can be successfully applied. High-resolution

Fourier transform spectroscopy offers new possibil-

ities for studying magnetic phase transitions, short-

range order, and magnetic structures in magnetic

insulators.

1. Fourier Transform Spectroscopy

The principal parts of a FTS are the Michelson inter-

ferometer with a moving mirror (see Fig. 1) and a

computer that performs the Fourier transform of the

interferogram and thus calculates the spectrum of an

incident radiation E(t). A detector registers the aver-

aged intensity at the output of the Michelson inter-

ferometer

FðtÞB½EðtÞþEðt þ tÞ

¼ E

ðtÞþE

ðt þ tÞþ2EðtÞEðt þ tÞð1Þ

Here t ¼l/c is the time delay between the two inter-

fering beams, l being the path difference. Under certain

conditions, that usually are fulﬁlled, the ﬁrst two terms

in Eqn. (1) are equal to the time-independent mean

intensity while the third term represents the auto-

correlation function

cðtÞ¼EðtÞEðt þ tÞð2Þ

which depends on the time delay t. The Fourier trans-

formation of c(t) yields the spectrum, namely,

BðoÞ¼

cðtÞcos otdt ð3Þ

A real instrument delivers c(t) for the interval [0,

L/c], where L is the maximum path difference which

depends on the maximum displacement of the moving

mirror of the FTS.

The L-dependent instrumental function, that is the

response of an instrument to a monochromatic inci-

dent light E(t) ¼cos o

t, can be written as

sinðo  o

ÞL=c

ðo  o

ð4Þ

L-N

dðo  o

Þ, where d(oo

) is the delta-

function; for a ﬁnite L, f

exhibits secondary maxi-

ma, and the distance between the ﬁrst zeros of f

do ¼2pc/L, or, in wavenumbers (s ¼1/l ¼o/2pc),

ds ¼1/L, characterizes the spectral resolution. In

high-resolution instruments L may reach several me-

ters which corresponds to the better than 0.01 cm

1

resolution.

The free spectral range Ds for the FTS depends on

a number N of points registered in the interferogram:

Ds ¼ Nds ð5Þ

Typically, NB10

, and so even for high-resolution

(dsB10

3

1

) instruments DsB10

1

, and thus

broadband high-resolution spectroscopy is possible

with FTS.

FTS analyzes all the spectral elements simulta-

neously which results in the so-called Fellgett or

multiplex advantage of FTS over conventional

instruments in signal-to-noise ratio. For the intensi-

ty-independent noise of a detector (infrared spectral

region) the Fellgett advantage is maximum, namely,

ﬃﬃﬃﬃﬃ

, where M ¼Ds/ds is the number of registered

spectral elements. Additional gain comes from the

large throughput of a FTS in comparison with a slit

spectrometer (Jacquinot advantage). In Fourier

transform spectroscopy, the wave number scale is es-

tablished relative to the frequency of a stabilized laser

used for the measurement of the path difference and,

as a result, is precise in the whole spectral region

(Connes advantage). General advantages of the FTS

over conventional instruments make them a useful

tool for material research.

Figure 1

Scheme of the Michelson interferometer.

278

High-resolution Fourier Transform Spectroscopy: Application to Magnetic Insulators

2. Rare Earth Spectroscopic Probe in a Magnetic

Compound

A crystal ﬁeld of any symmetry lower than cubic

raises all but the Kramers degeneracy of the levels of

a free RE

3 þ

ion with an odd number of electrons.

Kramers doublets are further split by the magnetic

interactions in a magnetically ordered state. These

splittings are usually called exchange splittings. If the

crystal unit cell contains n equivalent RE ions,

additional so-called Davydov or factor-group split-

tings of the excited levels into n components occur,

due to the interactions between these ions. Usually,

Davydov splittings in the RE spectra are small

(B0.1 cm

1

) and not observable and a single-ion

crystal ﬁeld model shown schematically in Fig. 2 is

suitable even for concentrated materials. Each spec-

tral line of a paramagnetic compound splits into a

maximum of four components in a magnetically or-

dered state. Using the fact that the population of the

upper component of the split ground Kramers dou-

blet diminishes with decreasing temperature, it is

possible to assign all the components of a split spec-

tral line and to ﬁnd the ground and excited state

splittings from optical measurements. Figure 3 gives

an example of the spectra of a Kramers ion in a

magnetic compound and of the spectral line identi-

ﬁcation. It shows the absorption of Er

at dif-

ferent temperatures, both higher and lower than the

el temperature T

¼28 K. A single-ion crystal ﬁeld

model is true for this concentrated material.

For magnetic compounds containing both RE and

transition metal (TM) ions, the main contribution to

the splitting comes from the electronic exchange in-

teraction between the RE ion and neighboring TM

ions. Usually, the electron clouds of intervening an-

ions are involved, so the superexchange interaction is

to be considered. In purely RE compounds, the mag-

netic dipole–dipole interaction makes comparable or

even major contributions to magnetic splittings.

The magnetic dipole–dipole contributions to the

splittings of the Kramers levels of a particular RE ion

may be described by an action of an effective mag-

netic ﬁeld H

dip

created by all the magnetic ions of the

lattice at the RE site and can be calculated accurately.

As for the exchange interaction, a concept of an ef-

fective ﬁeld fails to describe the splittings in this case.

The effective exchange potential can be written as an

expansion in products of spherical tensor operators

for electrons at the two sites, with coefﬁcients that are

linear combinations of the two-electron exchange in-

tegrals. Only a small number of symmetry-dependent

coefﬁcients is required. These may be regarded as

empirical parameters and may be found from a set of

level splittings measured by optical spectroscopy,

which is analogous to the case of crystal ﬁeld

parameters describing the single-ion energy levels of

Figure 2

Levels of a Kramers ion in a magnetically ordered

crystal.

Figure 3

Absorption in the region of the

15/2

13/2

optical

transition in Er

at 88 K4T

(lower trace) and

6.2 KoT

(upper trace). Identiﬁcation is given

according to the scheme of Fig. 2. Inset shows the

temperature dependence of the line IB. The following

values of the level splittings (in cm

1

) have been found

from the spectra: D ¼20.6 (19.7); D

¼3.3 (5.7); D

¼9.0

(7.2). Values and notations in parentheses refer to the

second crystallographic position for Er

3 þ

in Er

(after Popova 1998).

279

High-resolution Fourier Transf orm Spectroscopy: Application to Magnetic Insulators