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

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to a quantum phase transition, or single-ion effects as

the multichannel Kondo effect or a distribution of

Kondo temperatures.

1. Classical and Quantum Phase Transitions

Magnetic phase transitions—like many other types of

phase transition (PT)—are usually observed by var-

ying the temperature (T) and monitoring changes in

thermodynamic quantities like the magnetic suscep-

tibility, the specific heat, or other properties (see

Magnetism in Solids: General Introduction). Second-

order transitions where the ﬁrst derivatives of the free

energy (i.e., the magnetization or entropy) vary con-

tinuously are strongly inﬂuenced by thermal ﬂuctu-

ations. These ﬂuctuations determine how, at the

critical temperature (T

), the system goes from a dis-

ordered state for T4T

, to a magnetically ordered

state at ToT

. The ﬂuctuations diverge spatially as

is approached, as characterized by the correlation

length (xB7TT

n

) with a critical exponent (n),

and have a typical lifetime given by tBx

, where z is

called the dynamical critical exponent.

Renormalization group theory predicts that the

static properties near a PT depend only on a few pa-

rameters, namely the spatial dimension of the system

(d), the symmetry of the order parameter (OP), and

the range of the interaction leading to magnetic order.

These parameters determine the universality class of

the system. For magnetic systems, the OP is usually

the spontaneous magnetization—for antiferromag-

nets and incommensurate magnets, the staggered

magnetization—which is zero above T

and acquires

a ﬁnite value below T

. In many cases, e.g., when the

magnetic moments are spatially localized, the sym-

metry of the OP is described completely by the

number (n) of its components, i.e., n ¼1 (Ising model),

n ¼2 (XY model), or n ¼3 (Heisenberg model).

While the critical behavior of static properties such

as the specific heat or magnetization for these exam-

ples is entirely determined by the universality classes,

the dynamic behavior within a given class may differ

depending on z. Fluctuations can actually suppress a

PT altogether; the easier it is to generate ﬂuctuations

(i.e., the larger n is), the more PT is suppressed. A PT

to long-range order is only possible above a lower

critical dimension (d

) where d

¼2 for n ¼3, d

¼2

for n ¼2, and d

¼1 for n ¼1. (For n ¼2 and d ¼2a

special type of PT, the Kosterlitz–Thouless transi-

tion, exists.) On the other hand, there is an upper

critical dimension (d

) above which ﬂuctuations in

local-moment systems are deemed unimportant be-

cause they effectively average out (with d

¼4 for a

wide range of models). In general, for d4d

, mean

ﬁeld theory (MFT) is sufﬁcient to describe the PT.

In certain cases, T

can be driven to zero by some

external parameter (d). The picture of a ‘‘classical’’

phase transition (CPT) has to be modiﬁed when T

vanishes at a critical value d

.AtT ¼0, only quantum

ﬂuctuations are possible—for instance at the transi-

tion from a paramagnet to an ordered magnet. For

such a quantum phase transition (QPT), the relevant

length scales and time scales of these ﬂuctuations are

coupled via the uncertainty principle, leading to the

time entering as additional z dimensions. Formally,

the inverse temperature corresponds to an imaginary

time, hence the system acquires its full effective di-

mensionality (d

eff

¼d þz) only at T ¼0 (Sondhi et al.

1997). The added time dimensions drive the system

towards mean ﬁeld behavior, where above d

eff

¼4

only Gaussian ﬂuctuations are observed and genuine

critical behavior is absent. However, the static prop-

erties in all cases of a QPT are inﬂuenced by z. This

leads to a behavior even at ﬁnite T close to d

, which

is distinctly different from that at a CPT. A partic-

ularly interesting QPT is that depicted in Fig. 1 where

an itinerant electron system is driven towards (or

away from) magnetic order by d. While far from d

the low-temperature properties of the system can be

described in the framework of a Fermi liquid (see

Heavy-Fermion Systems), unusual ‘‘non-Fermi liq-

uid’’ (NFL) behavior is observed at ﬁnite T around

. This QPT has been observed in weakly magnetic

transition metal compounds (Sect. 2.2) and in heavy-

fermion systems based on rare earths or actinides

(Sect. 2.3). On the other hand, NFL behavior in

metals does not necessarily arise from proximity to a

magnetic instability. Single-ion effects such as the

multichannel Kondo effect (Sect. 3.1) or a distribu-

tion of Kondo temperatures (Sect. 3.2) may also lead

to NFL-like properties.

An example of a QPT in an insulating solid

is provided by the dipolar-coupled Ising magnet,

e.g., LiHo

1–x

, in a transverse magnetic ﬁeld

(Rosenbaum 1996).

Figure 1

Schematic phase diagram of a metal with an instability

towards magnetic order driven by a control parameter

(d) which may be pressure or composition.

980

Non-Fermi Liquid Behavior: Quantum Phase Transitions

2. Quantum Phase Transitions and Non- Fermi

Liquid Behavior in Metallic Magnets

2.1 Fermi Liquids

In a metal, the conduction electrons interact with

each other due to their Coulomb repulsion. At the

same time, there is a strong correlation by virtue of

the Pauli priniciple, which states that two electrons

(or, more generally, two indistinguishable fermions,

i.e., particles with half-integer spin) cannot be in the

same quantum state. Much of the complexity of

the interacting system may be captured in terms of

the physics of a non-interacting electron gas. For

kT5E

, the specific heat of a non-interacting elec-

tron system is given by C ¼g

T with g

¼(p

/3)

N(E

) where N(E

) is the density of levels at the

Fermi energy E

and k the Boltzmann constant.

The Pauli spin susceptibility is independent of T,

¼m

N(E

) where m

¼4p 10

7

Vs/Am and m

is the Bohr magneton. The contribution to the elec-

trical resistivity due to electron–electron (e–e) scat-

tering is given by r

e–e

¼c (kT/E

)

where c is a

microscopic scattering cross section.

Assuming a one-to-one correspondence between

the excitations of an interacting electron system

(termed ‘‘quasiparticles’’) and of the non-interacting

electron gas, Landau postulated that the low-T prop-

erties for the interacting system obey the same laws as

for the Fermi gas, with a renormalized effective mass

m* (with respect to the free-electron mass m

) and a

few additional parameters taking account of the re-

sidual interactions among the quasiparticles. This

special case of an interacting electron system that can

be obtained from the non-interacting Fermi gas by

adiabatically turning on the interaction, is called a

(Landau) Fermi liquid. In such a Fermi liquid (FL),

the specific heat is given by C ¼gT ¼(m*/m

T, the

spin susceptibility by w ¼w

(m*/m

)(1 þF

) where F

is an additional Landau parameter, and the resistivity

ee

is again B(kT/E

)

where E

may be modiﬁed

due to the interactions.

2.2 Weak Itinerant Magnets

Besides the traditional itinerant magnets like Fe, Co,

Ni with rather large Curie temperatures T

and

ordered magnetic moments of the order of 1m

, there

exist a number of intermetallic transition metal com-

pounds and alloys with low T

and ordered moments

of only a small fraction of 1m

(see Itinerant Electron

Systems: Magnetism (Ferromagnetism)). ExaN

1mples of these weak itinerant magnets are MnSi,

ZrZn

, and YMn

. In some of these materials, the

magnetic order can be suppressed by hydrostatic

pressure (p) which serves as a control parameter

(cf. Fig. 1). Figure 2 shows, as an example, resistivity

data for MnSi, which at ambient pressure is an in-

commensurate magnet with T

¼30 K. Because of the

large 2p/Q ( ¼190 A

), MnSi is often described as an

effective ferromagnet (FM).

The onset of magnetic order is seen as a kink in the

electrical resistivity r(T) (Fig. 2) (Pﬂeiderer et al. 1997).

falls monotonically with p and tends to zero at

¼14.6 kbar, the critical pressure of the magnetic

QPT. For p4p

, r(T) ¼r

þbT

with m between 1.5

and 1.6 is observed over a large T range. This NFL-

like behavior (mo2) arises from a long-range interac-

tion between quasiparticles, mediated by overdamped

spin ﬂuctuations. Assuming a space- and time-depend-

ent magnetization m(r, t), which is treated as a molec-

ular ﬁeld, the imaginary part of the susceptibility is

given by:

ðq; oÞ¼ow

=ðo

þ G

Þð1Þ

where G

describes the temporal relaxation rate of the

Fourier component m

(t)ofm(r, t). For cubic, nearly

ferromagnetic materials, q ¼7q7 and:

¼ gq

ðw

1

þ cq

Þð2Þ

where g and c are constants, and w is the T dependent

uniform (q ¼0) susceptibility.

The case n ¼1 corresponds to damping by quasi-

particles in a homogeneous FL. As the QPT is ap-

proached, w

1

-0 and G

, i.e., z ¼3. Assuming

that _G

¼kT, the temperature dependence of r, var-

ying as T

, is given by m ¼(d þ2)/z. This is in fair

agreement with the behavior of MnSi. T

is predicted

by this model to vary as T

B7pp

3/4

, which is con-

sistent with experimental results not too close to p

(inset of Fig. 2). The stronger dependence of T

7pp

7 in the vicinity of p

might signal a ﬁrst-order

transition, in which case the ﬂuctuations are cut off.

In an antiferromagnet (AF), the critical ﬂuc-

tuations are not centered at qE0, but at the AF

Figure 2

Resistivity r(T) of MnSi at various pressures p. Inset

shows the magnetic transitions temperature T

(p)as

obtained from the kink in r(T) (adapted by permission

of American Physical Society from Phys. Rev. B, 1997,

55, 8330–8).

981

Non-Fermi Liquid Behavior: Quantum Phase Transitions

wavevector Q, which is of the order of the dimension

of the Brillouin zone. Hence:

¼ g

ðw

1

þ c

Þð3Þ

The exponent n is effectively zero, and hence for

1

-0 at the transition, G

Q þq

is obtained, i.e.,

z ¼2 leading to m ¼d/z. For the MFT described here,

sometimes termed quantum Ginzburg–Landau

(QGL) model, it is required that d þz44.

The above description of a QPT involving long-

range interactions between quasiparticles (Moriya

1985, Lonzarich and Taillefer 1985, Lonzarich 1997)

can also be derived in a more sophisticated renor-

malization-group treatment (Hertz 1976, Millis

1993). The results for the limiting behavior of r and

C in the QGL model are listed in Table 1.

There is an additional caveat for AF metals at the

QPT. The simple expectation m ¼1.5 ignores the fact

that the scattering of quasiparticles by ﬂuctuations

occurs only on those parts of the Fermi surface that

are connected by Q. However, this scattering may be

‘‘short-circuited’’ by quasiparticles on other parts of

the Fermi surface. As was shown by Rosch (1999), an

asymptotic FL-like T

dependence for T-0isonly

obtained for extremely pure samples without impurity

scattering. Impure samples yield m ¼1.5, while effec-

tive exponents 1tmt2 can be obtained in a wide T

region for samples of moderate and high purity. The

different T dependences are due to the existence of

two scattering processes with different q depend-

ences—strongly anisotropic scattering by magnetic

ﬂuctuations and isotropic impurity scattering.

2.3 Heavy-fermion Systems

Heavy-fermion systems (HFS), with an (often) regu-

lar sub-lattice of 4f or 5f atoms, notably Ce, Yb, or

U, are close to a magnetic instability giving way to

long-range magnetic order arising from the interac-

tion between the 4f or 5f magnetic moments via the

conduction electrons (see Heavy-fermion Systems). It

is instructive to consider the case of an isolated mag-

netic impurity in a nonmagnetic metallic host where a

strong hybridization between conduction electrons

and f electrons may lead to a complete screening of

the magnetic moment by the former, resulting in a

singlet ground state. This can be described by the

Kondo Hamiltonian H ¼JSs where S is the impu-

rity spin and s the conduction-electron spin, with an

effective antiferromagnetic exchange interaction (J)

(see Electron Systems: Strong Correlations).

At sufﬁciently low T well below the characteristic

Kondo temperature T

, a local FL is formed around

the magnetic impurity. In a simpliﬁed picture, HFS

can be thought of as a lattice-coherent superposition

of these local FLs. Hence at low T, most HFS can be

described within the framework of FL theory with,

however, a very large effective mass m* derived from

the huge linear specific heat coefﬁcient (g: ¼C/T) and

a correspondingly large Pauli susceptibility (w), both

being only weakly dependent on T. The contribution

elel

B(T/T

)

is observable in HFS because

B10 K is much smaller than E

in typical metals.

The competition between on-site Kondo interaction

quenching the 4f or 5f localized magnetic moments

and intersite Ruderman–Kittel–Kasuya–Yosida

(RKKY) interaction between these moments via the

conduction electrons allows for nonmagnetic or mag-

netically ordered ground states in HFS. This compe-

tition is governed by a single parameter, namely the

effective exchange constant (J), which enters the char-

acteristic energy scales kT

Bexp(1/N(E

)J) and

RKKY

N(E

) for Kondo and RKKY interac-

tions, respectively. The strength of J is usually tuned

by composition or pressure. Because of the expo-

nential dependence of kT

on J (which depends on

the hybridization), volume changes are often the

dominant effect in producing the magnetic–non-

magnetic transition. The hybridization can of course

be strongly affected when alloying with non-isoelec-

tronic ligands.

A large number of investigations have been carried

out on HFS close to the magnetic instability. NFL

behavior, i.e., deviations from the FL predictions in

the low-T thermodynamic and transport properties

has been reported for many systems. The main ob-

servations in many cases are a specific heat coefﬁcient

(g) exhibiting a logarithmic T dependence, saturating

towards low T in a number of systems, and a T

dependence of the electrical resistivity (r) with m o2.

There are relatively few magnetization studies. The

most decisive probe for magnetic ﬂuctuations near a

QPT, inelastic neutron scattering (INS), has only

been employed for a few systems so far.

Since many of the HFS are driven through the

QPT by changing the composition (hence introducing

disorder), its effect on the critical behavior is an

important issue. One has to distinguish between

disorder by dilution of the 4f or 5f site as, e.g., in

1–x

, or by altering the ligand conﬁgura-

tion as for example in CeCu

6–x

. In the ﬁrst case, a

‘‘Kondo hole’’ introduced by dilution might lead

to substantial quasiparticle scattering and ultimate

loss of coherence. In the second case, the Ce atoms

Table 1

Predictions of the QGL model for the temperature

dependence of the linear specific-heat coefﬁcient C/T

and the electrical resistivity r at the QPT.

d ¼3 d ¼2

z ¼3 (FM) z ¼2 (AF) z ¼3 (FM) z ¼2 (AF)

C/T Bln(T/T

) BgbOT BT

1/3

Bln(T/T

)

r BT

5/3

3/2

4/3

982

Non-Fermi Liquid Behavior: Quantum Phase Transitions

experience different local environments and this may

lead to different local Kondo temperatures. In

CeCu

6–x

, the QPT was tuned both by composi-

tion and pressure, with no difference in the functional

C(T) and r(T) dependencies. Disorder may, however,

play an important role in other systems (Sect. 3.2).

Practically, all HFS systems investigated so far are

close to an AF instability (as opposed to a FM in-

stability). A number of systems, notably Ce

1x

, have been compared in detail with the QGL

model (Table 1). Long-range AF is introduced into

CeRu

by moderate La doping and is due to the

concomitant lattice expansion. The behavior near the

QPT, occurring for x

E0.075, is broadly consistent

with expectation, although some differences between

INS-derived and specific heat-derived parameters re-

main (Raymond et al. 1997). In CeCu

6x

(the

most extensively studied system up to now), the QPT

can be tuned by the Au concentration (x), again by

virtue of a weakening of J upon expansion of the

CeCu

lattice. At the critical concentration x

E0.1,

NFL behavior occurs with g ¼a ln(T

/T) over nearly

two decades in T (Fig. 3), and rEr

þAT between

0.02 K and 0.5 K.

These T dependences can be understood in terms

of the spin-ﬂuctuation model when two-dimensional

critical ﬂuctuations coupled to quasiparticles with

three-dimensional dynamics are assumed. Indeed,

INS experiments for x ¼0.1 reveal quasi-one-dimen-

sional features in the dynamical structure factor

(S(q, E ¼const.)) which correspond to quasi-two-

dimensional ‘‘rods’’ in real space (Stockert et al.

1998). Here the neutron energy transfer was

E ¼0.15 meV. For strictly two-dimensional ﬂuctua-

tions, the system would be at the upper critical di-

mension since d

eff

¼d þz ¼4.

The energy dependence of the critical ﬂuctuations

measured at Q ¼(0.8, 0, 0) has revealed an anomalous

scaling with E and T of the imaginary part of the dy-

namical susceptibility w

(q, E, T) ¼S(q, E, T) (1exp

(E/kT)) (Schro

der et al. 1998), i.e., w

(E, T) ¼

a

g(E/kT)witha ¼0.74 (Fig. 4). The scaling func-

tion g is consistent with a simple form of the dynamical

susceptibility:

wðq; E; TÞ¼cðf ðqÞþðiE þ bTÞ

1

ð4Þ

where c and b are constants. The anomalous T de-

pendence of the uniform static magnetic susceptibility

has initially been described as w(0, 0, T) ¼w

(1BT

1/2

)

between 0.08 K and 3 K, but can be modeled up to 7 K

by w ¼c

1

þb

)

1

with a ¼0.8 (Schro

der et al.

1998). The fact that a as obtained from INS for

Q ¼(0.8,0,0) is equal to a (q ¼0, E ¼0) may imply

that (i) the non local physics is contained in f(q)which

vanishes as q-Q where Q is a critical vector, i.e., be-

longing to the quasi-one-dimensional features of

S(q, E ¼const), and (ii) the anomalous scaling expo-

nent is a local property since it appears to be inde-

pendent of q. This ‘‘local’’ behavior is, however,

different from a simple local Lorentzian ﬂuctuation

spectrum that would be described by a ¼1.

The QPT in CeCu

6–x

has not only been tuned

by x but also by pressure. Fig. 3 shows that samples

with x4x

can be driven to the QPT, i.e., T

¼0by

appropriate hydrostatic pressure (p). Near the QPT, C

Figure 3

Specific heat C of CeCu

6x

plotted as C/T vs. log T

for different hydrostatic pressures p. Note that when

¼0, i.e., for x ¼0.3 at 8.2 kbar and for x ¼0.2 at

4.1 kbar, the C/T curve falls on the same ‘‘universal

curve’’ as for x ¼0.1 where T

¼0 at ambient pressure.

Here a pressure of 6.0 kbar leads to Fermi-liquid

behavior C/TEconst. Dashed line indicates C/T for

CeCu

. Inset shows the Ne

el temperature T

vs. x at

p ¼0 (reproduced by permission of Elsevier Science

from J. Magn. Magn. Mater., 1999, 200, 532–51).

Figure 4

Scaling plot inelastic neutron-scattering data for

CeCu

5.9

0.1

at q ¼(0.8, 0, 0) vs. E/k

T. The inset shows

how the quality of the scaling collapse varies with a in

Eqn. (5). Solid line in the main part corresponds to a ﬁt

with a ¼0.74 (reproduced by permission of American

Physical Society from Phys. Rev. Lett., 1998, 80, 5623–6).

983

Non-Fermi Liquid Behavior: Quantum Phase Transitions

is identical for all samples. Likewise, the sample

with x ¼0.1 can be driven by p away from the QPT

towards FL behavior C/TEconst. approximately

followed by pure CeCu

. The linear T

(p)andT

(x)

dependencies are in agreement with the scenario

of two-dimensional ﬂuctuations. The detailed mecha-

nism of how local moments and long-range ﬂuc

tuations work together to produce the unique quan-

tum-critical behavior of CeCu

6x

has yet to be

disentangled.

Tuning of the QPT in the related system Ce-

6x

by a magnetic ﬁeld has been inferred from

C and r (Heuser et al. 1998) with T dependencies

compatible with the QGL scenario with d ¼3 and

z ¼2 (Table 1), although this scenario does not in-

clude a magnetic ﬁeld.

A number of stoichiometric HFS where the QPT

can be tuned by pressure have also been investi-

gated, e.g., CeCu

with its intriguing—but not yet

fully understood—interplay between superconducti-

vity (SC) and magnetism (see Heavy-fermion Systems).

A particularly interesting case is CePd

(Fig. 5),

which in a small p range where T

is suppressed to

zero, exhibits SC below a large T-range of NFL be-

havior (Mathur et al. 1998). Analogous behavior was

found for CeIn

. This is suggestive of spin-ﬂuctuation

mediated SC. Up to now, however, only resistivity

measurements have been reported, and the detailed

interplay between superconductivity and AF order is

not yet known. Even more intriguing is the recent ob-

servation of superconductivity in ferromagnetic UGe

(Saxena et al. 2000).

3. Non-Fermi Liquid Behavior due to Single-ion

Effects

3.1 Multichannel Kondo Effect

Perfect screening of the magnetic moment via the

Kondo effect leads to a local FL around a magnetic

impurity in a metal. In general, however, several

conduction-electron channels (N) may be present and

the Kondo Hamiltonian reads H ¼S

a ¼1

, where

a is the channel index (Schlottmann and Sacramento

1993). For No2S, the screening is not complete;

N ¼2 S leads to perfect screening for T-0. In the

case of degenerate channels, J

¼J for all a, the im-

purity spin will be overscreened if N42S.

For the special case S ¼1/2 and N ¼2, NFL be-

havior is predicted with C/TBwBln(T/T

) and

rB1(T/T

)

1/2

for T-0. Up to now, the experi-

mental evidence for NFL behavior by overscreening

of the magnetic moment of an impurity has been

scarce.

In a more general sense, the Kondo effect may also

pertain to orbital degrees of freedom of an impurity

acting as a pseudospin instead of the spin. This is

discussed for the interaction of conduction electrons

with tunneling states, ﬁrst in amorphous metals and

later in metallic nanoconstrictions (van Delft et al.

1998). In these cases, the two channels are provided

by the spectator spin of the conduction electrons. For

certain rare-earth and actinide metals, a quadrupolar

Kondo effect (QKE) has been considered (Cox and

Jarrell 1996), where the pseudospin of the impurity is

the quadrupolar moment arising from the aspherical

charge distribution of the 4f or 5f electrons.

Again, the two channels are provided by the spin

degeneracy of the conduction electrons. Indeed, the

NFL anomalies as ﬁrst reported for U

0.2

0.8

(Seaman et al. 1991) were suggested to arise from the

QKE. Here, NFL anomalies persist to rather dilute U

alloys (Maple et al. 1995), thus favoring the single-ion

QKE model. On the other hand, the NFL behavior

might be due to an AF instability (Andraka and

Tsvelik 1991), supported by long-range AF order in

0.4

0.6

. In addition, a divergence of the quad-

rupolar susceptibility for T-0 as predicted for the

QKE scenario was not found. Hence the situation for

1x

is not clear.

The clearest signature of a QKE is found in

1–x

(Aliev et al. 1996). In U

1x

single-ion scaling of NFL properties compatible with

the QKE is observed, while U

1x

exhibits

Figure 5

Temperature–pressure phase diagram of a high-purity

CePd

single crystal. Superconductivity appears

below T

in a narrow window where the Ne

temperature T

tends to absolute zero. For clarity, the

values of T

have been scaled by a factor of three, and

the origin of the inset has been set at 5 K below absolute

zero. The inset shows that the resistivity r exhibits a T

1.2

dependence on temperature over a wide T range

(reproduced by permission of Macmillan from Nature,

1998, 394, 39–43).

984

Non-Fermi Liquid Behavior: Quantum Phase Transitions

signatures of a magnetic instability around x

¼0.7

whence the NFL properties in this case might be due

to a QPT (Maple et al. 1999).

3.2 Distribution of Kondo Temperatures

Features of NFL behavior may arise from a quite

different single-ion scenario, i.e., a distribution of

Kondo temperatures (P(T

)) in disordered systems.

Fluctuations in J or the local N(E

) near a disorder-

induced metal–insulator transition may lead to a wide

distribution of P(T

). Indeed, the specific heat of

heavily doped Si:P on the metallic side of the tran-

sition shows a contribution DCBT

with aE0.2,

which is attributed to P(T

) (Lakner et al. 1994). Here

the local moments arise from the statistical distribu-

tion of P donor atoms in the single crystalline host,

with a distribution of hybridization strengths to the

conduction electrons and hence of J. In the case of

heavy-fermion solid solutions, the statistical distribu-

tion of atoms may lead to a distribution of T

, as ﬁrst

put forward for the case of UCu

5x

(x ¼1 and 1.5)

on the basis of NMR measurements which directly

showed a distribution of Knight shifts, viz., Kondo

temperatures (Bernal et al. 1996). Even for a rather

narrow distribution of J and/or N(E

), a broad P(T

)

with a ﬁnite value for T

-0 may occur because of

the exponential dependence of T

on J and N(E

On the other hand, an interesting single-ion NFL

scaling was found in the dynamic structure factor

S(q, o) determined from INS (Aronson et al . 1995).

This system is also at the verge of a magnetic insta-

bility as evident from spin-glass behavior at the low-

est temperatures (Vollmer et al. 1997). A model

combining the proximity to a magnetic instability and

disorder, the so-called Grifﬁths phase model, has

been tested for a number of U systems (de Andrade

et al. 1998).

See also: Magnetic Excitations in Solids

Bibliography

Aliev F G, Vieira S, Villar R, Moshchalkov V V 1996 The

quadrupolar Kondo ground state in U

1x

. J. Phys.:

Condens. Matter 8, 9807–23

Andraka B, Tsvelik A M 1991 Observation of non-Fermi-liquid

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0.2

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H. v. Lo

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Universita

t Karlsruhe, Germany

Nonlinear Magneto-optics

Just as linear magneto-optics describes the interac-

tion of polarized light with magnetic (or magnetized)

materials, nonlinear magneto-optics describes their

nonlinear optical response. Physically, magneto-op-

tical (MO) effects are related to the breaking of the

time-reversal operation (see Faraday and Kerr Rota-

tion: Phenomenological Theory). This leads to well-

known phenomena such as the Faraday effect and

magnetic circular dichroism in transmission and the

magneto-optical Kerr effect (MOKE) in reﬂection.

Similarly, nonlinear MO effects are related to a com-

bination of time-reversal symmetry breaking with the

breaking of space-inversion symmetry (at least within

the electric dipole approximation).

The Faraday and Kerr effects have been known for

over a century. However, the ﬁeld of nonlinear mag-

neto-optics started much more recently with the pre-

diction (Pan et al. 1989) of magnetization-induced

contributions to the nonlinear optical response of

magnetic surfaces and interfaces. This was followed

by experimental observation of such effects by Reif

et al. (1991) for an Fe surface and by Spierings et al.

(1993) for Au/Co interfaces. (Actually, ﬁrst predic-

tions of possible nonlinear MO effects were already

published as early as 1967 by Lajzerowicz and Vallade

(1967) for antiferromagnetic materials and by Akhe-

miev et al. (1985) for magnetic crystals. The ‘‘revival’’

and recent strong development of nonlinear magneto-

optics is clearly related to the enormous interest in the

study and applications of magnetic multilayers and

nanostructures and by the development of solid state

mode-locked femto second lasers that are particularly

suitable for these kinds of studies (Rasing 1998).

Apart from the nonlinear optical equivalents of the

Faraday and Kerr effects, the higher order optical

response also leads to new nonlinear MO phenomena

that have no equivalence in linear magneto-optics,

e.g., transverse Faraday and Kerr effects that are

linear in magnetization (Pavlov et al. 1997, Wierenga

et al. 1995), the breaking of the equivalence between

magnetization reversal, the change of light helicity

(Kirilyuk et al. 2000b), and the sensitivity for anti-

ferromagnetic ordering (Fiebig et al. 1995).

1. Theory

An incident electromagnetic wave can induce a po-

larization in a medium that serves as a source for the

transmitted and reﬂected light. This polarization P

can be written in the electric dipole approximation as

an expansion in powers of the electric ﬁeld E(o)of

the incident wave:

¼ w

ð1Þ

ðoÞþw

ð2Þ

ijk

ðoÞE

ðoÞþ? ð1Þ

The tensor w

(1)

is the linear optical susceptibility,

leading to the linear reﬂection and refraction of light.

The second term describes nonlinear optical effects

such as second harmonic generation. In the presence of

a magnetization M both w

(1)

and w

(2)

should be fur-

ther expanded in powers of M:

ð1Þ

¼ w

ð1Þ

M þ ? ð2aÞ

ð2Þ

¼ w

ð2Þ

M þ ? ð2bÞ

The ﬁrst term in both Eqns 2(a) and 2(b) could

be thought of as describing purely crystallographic

effects while the second one only exists in the pres-

ence of magnetization. In principle in both cases the

magnetic contribution is described by a higher rank

tensor; in practice these magnetic contributions can

be written as effective w

(1)

and w

(2)

components, re-

spectively. For example, a nonzero magnetization

along the zˆ -axis of an optically isotropic material

yield nonzero components w

(1)

¼w

(1)

that lead to

linear MO effects that are proportional to the ratio

(1)

The crystallographic nonlinear optical tensor w

(2)

only nonzero in media without inversion symmetry.

Consequently, for isotropic media, w

(2)

only gives a

contribution from surfaces and interfaces where the

inversion symmetry is broken. The presence of a

magnetization does not affect the inversion symmetry

(as M is an axial vector) but does lower the symmetry

of w

(2)

and consequently leads to new nonzero, non-

linear susceptibility components. These elements can

be derived in a straightforward way and are summa-

rized in Table 1 for the standard longitudinal, trans-

versal, and polar MO conﬁgurations (Rasing 1998).

From Table 1, the effects of magnetization-induced

second harmonic generation (MSHG) can directly

be understood. For example, in the longitudinal

conﬁguration, an s-polarized (parallel to y) input

986

Nonlinear Magneto-optics

beam will give rise to one even component (w

(2)

zyy

) and

one odd component (w

(2)

yyy

). The even term will yield a

p-polarized and the odd term an s-polarized ﬁeld,

respectively. Thus, the output polarization can be

varied by changing the direction of M, yielding a

nonlinear Kerr rotation y

(2)

Barctan (w

(2)

yyy

(2)

zyy

) in the

same way as in linear magneto-optics, but with values

for y

(2)

that are three orders of magnitude larger than

the linear equivalent (Koopmans et al. 1995). Simi-

larly, it follows from Table 1 that in the transverse

geometry both s- and p-polarized input beams will

always generate a p-polarized output. Consequently,

as in linear MOKE, there is no nonlinear Kerr

rotation. However, in contrast to the linear case,

there is a strong magnetization-induced intensity

change, that is linear in the magnetization.

Table 1 shows another important difference be-

tween the linear and the nonlinear MO effects.

Whereas in both cases the MO effects are propor-

tional to a ratio between odd and even tensor com-

ponents, in the linear case the odd components are

small off-diagonal elements. In the nonlinear case the

odd components are of a similar type as the even ones

(compare the components in Table 1). This fact,

which is also supported by calculations of w

(2)

based

on spin-dependent band structure (Hu

bner and Ben-

neman 1989) is responsible for the very strong non-

linear MO effects that have been observed in various

experiments and that are typically three orders of

magnitude larger than the linear equivalents.

The relation between the nonlinear MO suscepti-

bility and the spin-dependent band structure can be

written as:

ð2Þ

ijk

ð2oÞ

a;b;c

/a9i9cS/c 9j9bS/b9k9aS

ð2_o  E

þ i_G

Þð_o  E

þ i_G

ð3Þ

Equation (3) shows that spectroscopic MSHG exper-

iments in principle allow the probing of the spin-

dependent surface or interface density of electronic

states which is not only of fundamental interest but

also highly relevant for the understanding of spin-

dependent tunneling processes in magnetic tunnel

junctions (de Boer et al. 1998).

All the effects so far have been described within the

electric dipole approximation only. An extension to

include higher order contributions to the nonlinear

MO response can be found in Rasing (1998), whereas

the application of magnetization-induced sum frequen-

cy generation can be found in Kirilyuk et al. (2000a).

2. Experimental Considerations

For each polarization combination, the total MSHG

response from a magnetic material can be simpliﬁed

Pð2o; 7MÞ¼ðw

even

eff

ð7MÞ7w

odd

eff

ð7MÞÞE

ð4Þ

where w

even

eff

and w

odd

eff

are effective tensor components

that are even and odd in the magnetization and de-

scribe the crystallographic and magnetic contribu-

tions to the total response, respectively. As both these

contributions are complex quantities, the total

(MSHG) signal is thus given by

Ið2o Þ B7w

even

eff

þ 7w

odd

eff

727w

odd

eff

77w

odd

eff

7cosDF ð5Þ

where DF is the phase difference between the two

contributions. The importance of the latter is obvi-

ous: when DF ¼P/2, the interference term is zero and

changing the magnetization direction will have no

effect on the total MSHG signal.

Though generally phase information is lost in in-

tensity measurements, fortunately DF can be meas-

ured quite easily in nonlinear optics by using

interference techniques (Stolle et al. 1997). The latter

can also be exploited in the case where there is only a

purely odd response by adding a nonmagnetic refer-

ence signal, as it is the interference between even and

odd terms that gives rise to the nonlinear MO effects.

Though MSHG signals give large relative MO ef-

fects, being a nonlinear optical technique the absolute

intensities are rather small (10–10

photons sec

1

However, they are easily detectable with modern

photon counting or charge coupled devices (CCD),

though care should be taken to ﬁlter out the two

signal versus the much stronger fundamental signal at

o (see Fig. 1).

Because of the simplicity of the experimental con-

ﬁguration coupled with the large effects, the transverse

geometry is often used for experimental studies. One

can then deﬁne a magnetic contrast or asymmetry as:

A ¼

Ið2o; þMÞI ð2o; MÞ

Ið2o; þMÞþI ð2o; MÞ

ð6aÞ

27w

odd

eff

7=7w

even

eff

1 þ 7w

odd

eff

odd

eff

cosDF ð6bÞ

Table 1

The nonzero w

(2)

components for the longitudinal

(M:x), transversal (M:y), and polar (M:z)MO

conﬁgurations (xz is the plane of incidence). For

convenience, only the indices are indicated. The odd

components are indicated in bold.

M:x xxz, zxx, zzz zyy P

out

yxx, yzz yyy S

out

M:y xzx, zxx, zzz, xxx, xzz, zzx zyy, xyy P

out

M:z xxz, zxx, zzz zyy P

out

yzx S

out

987

Nonlinear Magneto-optics

Because A is normalized with respect to the total

SHG intensity, it does not depend on the intensity

or shape of the fundamental light pulses, nor on

the spectral properties of optical components such as

ﬁlters in the optical setup. Together with the already

mentioned simplicity, this makes A a useful para-

meter for quantitative investigations. One should,

however, realize that the appearance of large effects

that result from the large magnetic tensor compo-

nents also means that, in contrast to most linear

MO effects, the nonlinear effects are often not sim-

ply proportional to the magnetization as directly

follows from Eqns (5) and (6). This can for example

strongly affect the shape of MSHG loops (see

Veenstra 2000).

3. Symmetry Considerations

As a second-order nonlinear optical technique de-

scribed by a third-rank tensor, MSHG can be ex-

ploited to probe other changes in symmetry, e.g., the

appearance of steps on vicinal surfaces (Leermakers

in press), magnetic moments at step edges (Jin et al.

1998), and the changes (phase transitions) in these

properties. Because of the use of femtosecond laser

sources for these experiments the latter can be per-

formed with unprecedented time resolution. This has

already led to exciting results, such as the disappear-

ance of magnetic order within picoseconds, by using

strong, but ultrashort, excitation pulses (Hohlfeld

et al. 1999).

Since the MSHG response originates from the in-

terference of both crystallographic (even) and mag-

netic (odd) contributions to the nonlinear optical

response, a detailed analysis of the symmetry prop-

erties of the MSHG signal can provide deeper insight

into both the crystallographic and magnetic symme-

try of crystalline material. Experimentally this is done

by using various polarization combinations (both

circular and linear) in combination with azimuthal

rotation dependence of the MSHG signal. These ef-

fects have been beautifully demonstrated for mag-

netic garnets (Pavlov et al. 1997, Kirilyuk et al. 1999)

and for hexagonal manganites (Fiebig et al. 2000a),

where in the last case even very complicated non-

collinear spin structures could unambiguously be re-

solved. In both classes of material, the power of

MSHG to probe the local spin structure, i.e., to do

domain microscopy has been demonstrated as well

(Fiebig et al. 2000b, Kirilyuk et al. 1997).

Figure 1

Schematic conﬁguration for nonlinear MO experiments. Photomultipliers or CCD cameras (for domain imaging) can

be used as detectors.

988

Nonlinear Magneto-optics

4. Concluding Remarks

Nonlinear Magneto-optics, or more specifically,

MSHG has yielded an interesting and powerful new

technique to investigate the magnetic structure of ma-

terials. The higher order optical response gives, in par-

ticular, the possibility of probing magnetic ordering in

otherwise complicated or inaccessible structures (such

as buried magnetic interfaces or antiferromagnetic ma-

terials) and has been shown to lead to intrinsically new

effects. Therefore, this new area of magneto-optics is

not only interesting from a fundamental point of view

but it will also be quite important for applications such

as the investigations of magnetic nanostructures and

the spin dynamics in these systems.

5. Acknowledgments

Part of this work was supported by the Stichting voor

Fundamenteel Onderzoek der Materie (FOM), which

is ﬁnancially supported by the Nederlandse Organ-

isatie voor Wetenschappelijk Onderzoek (NWO), the

TMR Network NOMOKE (ERBFMRXCT960015)

and INTAS 97-0705.

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Nuclear Magnetic Resonance

Spectrometry

Nuclear magnetic resonance (NMR) spectroscopy is

an analytical technique used extensively in the mo-

lecular structure determination of organic, organo-

metallic, biological, and inorganic materials. As the

name implies, NMR is concerned with the magnetic

properties of atomic nuclei, which give rise to the

observed phenomenon. In an NMR experiment,

samples are placed in a strong, homogeneous mag-

netic ﬁeld and radio-frequency (RF) electromagnetic

989

Nuclear Magnetic Resonance Spectrometry