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

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2.2 Cerium and Ytterbium Systems

A large number of cerium compounds and alloys ex-

hibit an EC ranging between the 4f

and the 4f

state,

thus behaving as IV materials. The most striking

feature of IV, however, occurs in f.c.c. cerium as a

response to applied pressure. At a temperature-

dependent critical pressure (p

¼7 kbar at room tem-

perature, Franceschi and Olcese 1969) cerium crosses

over from the g state (lattice parameter a ¼5.15 A

)to

the a state (a ¼4.85 A

). This transition is accompa-

nied by a volume change of about 15% which does

not, however, distort the cubic crystal symmetry.

While the g state is primarily trivalent (4f

(5d6s)

the a state behaves almost tetravalent (4f

(5d6s)

with a valence n ¼3.67 (Koskenmaki 1978).

Intermediate valence in ytterbium systems is con-

strained by divalent ytterbium (EC 4f

) and trivalent

Yb (EC 4f

). While the former is a nonmagnetic

state, the latter is magnetic with a total angular mo-

mentum J ¼7/2. Pressure can be used to subtly tune

the valence of the ytterbium ion and eventually drive

it towards the trivalent magnetic ground state. For

proper ytterbium compounds and alloys, intermedi-

ate valence is then completely adjustable from the

2 þ

towards Yb

3 þ

state (Bauer et al. 1995).

In addition to pressure, temperature may also

cause a dramatic alteration of the ytterbium valence.

A temperature driven ﬁrst order type of transition, as

shown in Fig. 4, has been observed for YbCu

(Felner and Nowik 1986). The particular transition

temperature T

, however, depends sensitively on de-

tails of the stoichiometry (Lo

ffert et al. 1999). Qua-

litatively, the high-temperature state corresponds

to the magnetic Yb

3 þ

while the low-temperature

state is the nonmagnetic Yb

2 þ

. Quantitatively,

however, the valence at T

only changes by about 0.2,

and the principal effect observed is the nearly com-

plete Kondo compensation of the local moment of

ytterbium. This scenario is rendered also from the

change of the Kondo temperature, which is small for

T4T

B25 K), but substantially larger for ToT



B500 K) (Cornelius et al. 1997).

Concomitantly, there is a large change in the

carrier density which occurs at the phase transition,

from trivalent semimetallic behavior at high temper-

ature to IV metallic behavior at low temperatures.

Although the change of the volume at T ¼T

(about

0.5%) is reminiscent of the ga transition of cerium,

a ‘‘Kondo volume expansion’’ in YbCu

In would

require an extraordinary and unrealistic large

Gru

neisen parameter O ¼dlnT

/dlnVB4000 in

order to match the conditions of the Kondo volume

collapse of cerium.

3. Concluding Remarks

Several rare earth, actinide, and transition metal

compounds are characterized by two almost degen-

erate consecutive f or d levels. Their electronic

conﬁguration results from a quantum mechanical

mixture of both levels, causing the valence of such

systems to adopt noninteger numbers. This hybrid-

ization distinguishes IV materials from the classical

type of ‘‘mixed valent’’ behavior due to inequivalent

lattice sites with different oxidation states.

Intermediate valence can originate in a number of

outstanding low temperature features of physical

properties. Among them are significantly enhanced

Sommerfeld values of the specific heat and Pauli sus-

ceptibility, a temperature dependent cross-over from

a magnetic to a nonmagnetic state, pressure driven

semiconductor to metal transitions, or the formation

of an excitonic insulating state.

See also: Heavy-fermion Systems; Kondo Systems

and Heavy Fermions: Transport Phenomena

Bibliography

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Rev. 124, 41–52

Batlogg B, Kaldis E, Ott H R 1977 Magnetic mixed valent

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Batlogg B, Ott H R, Kaldis E, Thoni W, Wachter P 1979

Magnetic mixed valent TmSe. Phys. Rev. B 19, 247–59

Bauer E, Tuan L, Hauser R, Gratz E, Holubar T, Hilscher G,

Michor H, Perthold W, Godart C, Alleno E, Hiebl K 1995

Evolution of a magnetic state in YbCu

5x

. Phys. Rev. B

52, 4327–35

Bjerrum-Moller H, Shapiro S M, Birgenau R J 1977 Field-

dependent magnetic phase transitions in mixed-valent TmSe.

Phys. Rev. Lett. 39, 1021–5

Bucher E, Narayanamurti V, Jayaraman A 1971 Magnetism,

metal-insulator transition, and optical properties in Sm- and

Figure 4

Temperature-dependent resistivity r (left axis) and

lattice constant a (right axis) of YbCu

In.

320

Intermediate Valence Systems

some other divalent rare-earth monochalcogenides. J. Appl.

Phys. 42, 1741–5

Cornelius A L, Lawrence J M, Sarrao J L, Fisk Z, Hundley M

F, Kwei G H, Thompson J D, Booth C H, Bridges F 1997

Experimental studies of the phase transition in

YbIn

1x

. Phys. Rev. B 56, 7993–8000

Felner I, Nowik I 1986 First-order valence phase transition in

cubic Yb

1x

. Phys. Rev. B 33, 617–9

Franceschi E, Olcese G L 1969 A new allotropic form of cerium

due to its transition under pressure to the tetravalent state.

Phys. Rev. Lett. 22, 1299–300

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F 1984 M-edge X-ray absorption spectroscopy of 4f insta-

bilities in rare-earth systems. J. Appl. Phys. 55, 1910–5

Koskenmaki D C, Gschneidner K A Jr. 1978 Cerium. In:

Gschneidner K A Jr., Eyring L (eds.) 1978 Handbook on the

Physics and Chemistry of Rare Earths. Vol. 1. Elsevier, Am-

sterdam, pp. 337–7

Jayaraman A, Narayanamurti V, Bucher E, Maines R G 1970

Pressure-induced metal-semiconductor transition and 4f elec-

tron delocalization in SmTe. Phys. Rev. Lett. 25, 368–70

Lawrence J M, Riseborough P S, Parks R D 1981 Valence

ﬂuctuation phenomena. Rep. Progr. Phys. 4, 1–84

ffert A, Hautsch S, Ritter F, Assmus W 1999 The phase

diagram of YbInCu

. Physica B 259–61, 134–5

Martin R M 1982 Fermi-surface sum rule and its consequences

for periodic Kondo and mixed-valence systems. Phys. Rev.

Lett. 48, 362–5

Moser M, Wachter P, Hulliger F, Etourneau J R 1985 Point

contact spectroscopy of YbB

and CeAl

. Solid State

Commun. 54, 241–4

Neuenschwander J, Wachter P 1990 Pressure-driven semicon-

ductor-metal transition in intermediate-valence TmSe

1x

and the concept of an excitonic insulator. Phys. Rev. B 41,

12693–703

Wachter P 1994 Intermediate valence and heavy fermions. In:

Gschneidner K A Jr., Eyring L, Lander G H, Choppin G R

(eds.) 1994 Handbook on the Physics and Chemistry of Rare

Earths, Vol. 19. Elsevier, Amsterdam, pp. 383–478

E. Bauer

Technische Universita

t Wien, Austria

Intermetallic Compounds: Electrical

Resistivity

The aim of this article is to show what information

can be obtained about the properties of intermetallics

from resistivity measurements. The inﬂuence of scat-

tering of conduction electrons by impurities, lattice

vibrations (phonons), and magnetic moments will be

discussed in detail. It will also be shown how changes

in the crystallographic structure and in the spin ar-

rangements (magnetic phase transitions) are mani-

fested in the temperature variation of resistivity.

The temperature variation of the electrical resistiv-

ity (r(T)) of different binary yttrium–transition metal

compounds is shown in Fig. 1. All these compounds

crystallize in the cubic Laves phase structure where

the rare-earth (RE) atoms form a diamond lattice.

The remaining space inside the cell is occupied by

regular tetrahedrons consisting of the transition metal

atoms or aluminum (see below). Figure 1 shows the

remarkable variety of behavior within this family of

Laves phases when the measurements are performed

across an extended temperature range up to 1000 K.

With the exception of YMn

, the residual resistivity

) has been subtracted from the r vs. T curves.

The reasons for choosing mainly Laves phases to

demonstrate the temperature dependence of r are as

follows:

(i) These compounds have a cubic structure, and

therefore the anisotropy of the resistivity in the dif-

ferent crystallographic directions can be neglected.

(ii) The physical properties of these yttrium-based

intermetallics are known in great detail, and hence a

deeper understanding of the resistivity behavior can

be expected. As a consequence, general conclusions

can be drawn from a discussion of r(T ) data.

(iii) More than 730 binary and ternary compounds

have been shown to have a cubic Laves phase struc-

ture (e.g., Villars 1997), and all have been partially

investigated with respect to their physical properties.

Referring to Fig. 1, we note the following.

YAl

. This is a Pauli paramagnetic compound with

a weakly temperature-dependent susceptibility. At

room temperature w ¼0.106 10

3

emu mol

1

.Band-

structure calculations reveal a moderately enhanced

electronic density of states (DOS) at the Fermi level

owing to Y(4d ) states. Investigations of the lattice dy-

namics have revealed a phonon DOS (PDOS) ranging

up to about 30 meV; the center of gravity of the PDOS

is approximately 15 meV. YAl

is an ideal prototype

for demonstrating the electron–phonon scattering

mechanism, which in the simplest approximation can

be described by the Bloch–Gru

neisen relationship.

Figure 1

Temperature dependence of the electrical resistivity of

yttrium-based cubic Laves phase compounds with 3d

transition metals (manganese, iron, cobalt, and nickel)

and aluminum.

321

Intermetallic Compounds: Electrical Res istivity

YNi

. The resistivity curve of this nonmagnetic

compound has been selected in order to demonstrate

the inﬂuence of a temperature-induced structural

transition on the electrical resistivity (Gratz et al.

1996). A detailed discussion of the inﬂuence of phase

transitions on r vs. T by means of the pseudobinary

LaAg

1x

series is given below.

YCo

. From a magnetic point of view this is a

paramagnetic compound. However, the susceptibility

is strongly enhanced and has a maximum in the w vs.

T plot around room temperature. It has been shown

that YCo

belongs to the group of cobalt-based com-

pounds whose properties are strongly inﬂuenced by

spin ﬂuctuations (see Spin Fluctuations). A compari-

son of the r vs. T plot for YAl

and YCo

clearly

shows the strong inﬂuence of spin-ﬂuctuation scat-

tering processes on the resistivity, giving rise to the

pronounced negative curvature of r(T) around room

temperature in YCo

YFe

. This compound represents a ferromagnetic

Laves phase with a Curie temperature of 550 K.

Studies of the magnetic properties reveal the localized

character of the iron moments which are coupled via

a direct 3d–3d interaction. The breakdown of the

long-range magnetic order at T

can clearly be seen

in the kink of the r(T) curve at 550 K.

YMn

. Among all these compounds, YMn

exhib-

its the most complex magnetic properties. There is an

antiferromagnetic transition of a ﬁrst-order type

around 110 K (Ne

el temperature). This magnetic

transition is accompanied by a very large volume ex-

pansion (about 5%) and a distortion of the cubic unit

cell. The enormous volume expansion prevents resis-

tivity experiments in the magnetically ordered state

since the sample falls apart. The arrangement of the

manganese magnetic moments appears to be sinus-

oidally modulated along the [111] direction with a

very long periodicity. The manganese moments show

pronounced temperature dependence. They change

from approximately 2.2 m

f.u.

1

at 4 K, decreasing

to 1.7 m

f.u.

1

at the Ne

el temperature, and then

increasing to 2 m

f.u.

1

at room temperature. The

complex magnetic behavior of YMn

is still a matter

of discussion. The r(T) curve of YMn

has been in-

cluded in order to show that even in a compound

with such a relatively simple crystal structure (cubic

Laves phase) a temperature variation of the resistivity

can be observed for which it is difﬁcult to give a

qualitative explanation. This r(T) curve is character-

ized by values of more than 200 mO cm, and a de-

crease in the whole temperature range above its Ne

temperature up to 1000 K. The Boltzmann equation

and scattering mechanism conditions are provided in

Boltzmann Equation and Scattering Mechanisms, de-

tailing the circumstances under which the Boltzmann

formalism is no longer applicable, and YMn

is a

good example of this. A rigorous quantum mechan-

ical treatment would be necessary to give at least a

qualitative explanation for this r(T) dependence.

1. Analysis of Resistivity Data

A quantitative analysis of resistivity data is in prac-

tice only possible if the sum rule can be used (Matt-

hiessen’s rule) (see Boltzmann Equation and Scattering

Mechanisms). This is because a calculation of the

temperature dependence of the resistivity in reality is

only practicable for a single scattering mechanism

(e.g., for the electron–phonon scattering) otherwise

the mathematical complexity would be too great.

However, the measured temperature dependence of

the resistivity (total resistivity) is the result of all the

various scattering processes.

Under the assumption that Matthiessen’s rule is

valid, the total resistivity is simply the sum of all the

calculated terms

tot

¼ r

þ r

mag

ð1Þ

where the subscripts denote the residual resistivity

(0), the resistivity owing to phonon scattering (ph)

and the resistivity owing to spin-dependent scattering

(magnetic scattering processes, mag). If the calculated

total and experimental resistivities are in agreement,

all the underlying assumptions and the procedure of

the calculation can be taken as being conﬁrmed. If,

however, there is no agreement between theory and

experiment it is necessary to check whether the ex-

perimental conditions (and/or the assumptions) are

correct. One of the most serious experimental prob-

lems is the correct determination of the sample geo-

metry (sample cross-section), especially if the sample

material is brittle and the measured cross-section is

uncertain owing to cracks inside the sample. This can

be the case for many samples. When the true sample

geometry is not known it is often helpful to consider

the temperature variation of the normalized resistiv-

ity, i.e., r(T)/r(300 K) vs. T.

In cases where the conditions for Matthiessen’s rule

are not fulﬁlled, i.e., the individual scattering processes

are coupled, a quantitative analysis of the resistivity

data becomes intractable. Finally, it should be em-

phasized that for resistivity values of the order of

250 mO cm and higher, the whole concept of the Boltz-

mann formalism is not applicable. An estimation of

the mean free path, l, of the conduction electrons re-

veals that in these cases values of l are comparable to

the size of the interatomic distances. A rigorous quan-

tum mechanical treatment of the transport phenomena

may then lead to useful results (Kubo 1959).

1.1 Scattering Mechanism

(a) Impurity scattering

In a nonmagnetic, single-phase sample where scat-

tering contributions from vacancies and disloca-

tions are negligible, the residual resistivity (r

)is

solely because of a low concentration of foreign

impurity atoms. With the exception of the lowest

322

Intermetallic Compounds: Electrical Resistivity

temperatures, r

is much smaller than the tempera-

ture-dependent phonon term ( r

), and scattering

of electrons by impurity defects is invariably elastic.

As the temperature rises, r

remains constant (see

Boltzmann Equation and Scattering Mechanisms) and

increases rapidly. The effect of the variable

‘‘impurity’’ scattering can best be seen in pseudo-

binary systems. The concentration dependence of r

vs. the concentration of the partner element of three

different pseudobinary systems is shown in Fig. 2.

The simplest example represents the Lu

1x

Ag sys-

tem, which crystallizes in the cubic CsCl-type struc-

ture over the whole concentration range. The r

vs. x

curve is parabolic (see Fig. 2(a)), in good agreement

with the so-called Nordheim rule, which states that

the composition-dependent part of r

is proportional

to x(1x), where x is the composition. However,

these contributions to r

caused by dislocations,

stacking faults, etc., are superimposed in any case.

If one of the lattice sites is statistically occupied by

a magnetic ion, then a further contribution to r

ap-

pears owing to the disorder in the magnetic moment

arrangement. As an example, the pseudobinary

1x

Ni system is shown in Fig. 2(b). This sys-

tem crystallizes in the orthorhombic CBr-type struc-

ture. Gadolinium bears a magnetic moment, leading

to a long-range ferromagnetic order for gadolinium

concentrations higher than 30% (Gratz et al. 1986).

As can be seen from Fig. 2(b), there is a discontinuity

in r

vs. x at about the onset of ferromagnetism. The

increase of r

in the ordered state is obviously be-

cause of the domain wall scattering and the disorder

in the magnetic moment arrangement within the do-

mains if not all of the RE places are uniformly oc-

cupied by gadolinium atoms. Note that r

is very

small in the boundary compound GdNi. The third

pseudobinary system, (Gd

1x

)

crystallizes in

the hexagonal Ho

-type structure. In this system

a linear increase of the Curie temperature with the

gadolinium content has been found, with the highest

ordering temperature for Gd

¼220 K)

(Gratz et al. 1980). The asymmetric shape of the r

vs. x curve is referred to the increasing disorder in the

magnetic subsystem which exists even at the lowest

temperatures. Note that Gd

again shows a com-

paratively low r

value.

(b) Electron–phonon scattering

When an electron interacts with a quantized lattice

vibration of frequency o, an energy quantum, _o,is

exchanged and such events are inelastic. Assuming

that the conduction electrons are part of one and

the same s-conduction band and they are scattered

into vacant energy states within the same band, then

the r

(T) dependence can be approximated by the

Bloch–Gru

neisen law, derived using a Debye phonon

spectrum and a spherical Fermi surface.

This is expressed by (Blatt 1968)

¼ 4R



Y=T

ðe

 1Þð1  e

x

ð2Þ

where the constant R

includes, among other funda-

mental physical quantities, the electron–phonon

coupling constant, an average atomic mass of the

different types of atoms, and the Debye tempera-

ture Y (see Boltzmann Equation and Scattering

Mechanisms).

In order to demonstrate what can be concluded

from a ﬁt of Eqn. (2) to experimental data, the

temperature variation of the resistivity of three

cubic Laves phases (YAl

, LaAl

, LuAl

) have been

Figure 2

Concentration dependence of the residual resistivity (r

)

of three pseudobinary systems with different crystal

structures: (a) Lu

1x

Ag (cubic CsCl-type structure);

(b) Gd

1x

Ni (orthorhombic CBr-type structure) and

(Gd

1x

)

(hexagonal Ho

-type structure). A

ferromagnetic order exists in Gd

1x

Ni for 0pxo0.7

(Gratz et al. 1986), and in (Gd

1x

)

for 0pxp1.0

(Gratz et al. 1980). Solid lines are guides for the eyes.

323

Intermetallic Compounds: Electrical Res istivity

selected and their r(T) curves are depicted in Fig. 3

(Bauer et al. 1985). In a ﬁt of the function r(T) ¼r

þ r

to these experimental data (given by the solid

lines in Fig. 3), r

, R

, and Y are used as ﬁt param-

eters. These values are listed in Table 1.

One can ask if these ﬁt parameters have any sig-

nificance. (Note that R

gives the resistivity at the

corresponding Debye temperature of a material.) A

closer inspection of these parameters reveals an in-

teresting picture, with a much higher Debye temper-

ature obtained for YAl

(Y ¼310 K), which is related

to the higher frequency of the yttrium-dominated

optical phonon modes. By contrast, LuAl

(owing

to the larger atomic mass of lutetium) has a lower

Debye temperature (Y ¼275 K). The much smaller

Debye temperature in LaAl

(Y ¼205 K), the high

value (45 mO cm), plus the stronger curvature of

the experimental r(T) data at elevated temperatures

(see Fig. 3) are indicative of a strong electron–pho-

non coupling. This is a necessary condition for su-

perconductivity and we note that superconductivity is

observed in LaAl

below 3.3 K (Fig. 3 inset).

There are two mechanisms thought to be respon-

sible for the stronger electron–phonon interaction in

LaAl

, which might also occur in other lanthanum-

based compounds (see below). The ﬁrst of these

comes from band-structure studies. It follows from

the calculation that the electronic DOS at E

is en-

hanced owing to the proximity of the empty 4f band

at the Fermi level (Hasegawa and Yanase 1980). The

second mechanism comes from studies of the lattice

dynamics by inelastic neutron diffraction, which re-

veal soft phonon modes in LaAl

(Yeh et al. 1981). In

this investigation the lattice dynamics of YAl

and

LaAl

have been compared, and it is shown that the

phonon DOS has a pronounced gap at about 17 meV

in LaAl

. The trends outlined above have also been

observed for LaAg, which is isostructural with YAg

and LuAg (Bauer et al. 1986).

In conclusion, measurements of the resistivity

provide a rough estimate of the electron–phonon

interaction strength, especially for cases where iso-

structural compounds may be compared.

resistivity measurements

Changes of the crystal structure inﬂuence the resis-

tivity, since if there is a change in symmetry then the

lattice dynamics (described by the phonon dispersion

relationship) will change resulting from the different

interatomic distances in the emerging structure. In

addition, changes in structure will be accompanied by

a change in the electronic properties (described in the

electronic band structure). An example revealing the

inﬂuence of structural transitions on the resistivity is

provided by LaAg

1x

. These pseudobinary com-

pounds crystallize in the cubic CsCl-type structure

at room temperature. For xo0.95 the compounds

undergo a temperature-induced structural transition

(Ihrig et al. 1973). Figure 4 shows the resistivity for

some x values of this pseudobinary system. The hys-

teresis in the r(T) curves indicates a ﬁrst-order struc-

tural transition. Studies of the lattice dynamics reveal

a soft mode in the phonon spectrum (Knorr et al.

1980). The underlying mechanism of the anomalies in

r(T) are a result of a change in the electron–phonon

scattering mechanism for the two different structures

leading to a destabilization of the cubic structure.

These curves present convincing examples that such

a martensitic type of transition can be investigated

using resistivity measurements.

(d) Spin-dependent scattering

The third term on the right-hand side of Eqn. (1),

mag

, is introduced to describe spin-dependent

Figure 3

Temperature dependence of the electrical resistivity of

three nonmagnetic RAl

cubic Laves phases (r

values

are subtracted). The solid lines depict the temperature

dependence according to the Bloch–Gru

neisen

relationship, with the ﬁt parameters given in Table 1.

The inset shows the superconducting transition of LaAl

in r vs. T at 3.3 K (Bauer et al. 1985).

Table 1

Fit parameters obtained by a ﬁt of r(T) ¼r

þ r

(T)

to the experimental resistivity data, with r

(T) from the

Bloch–Gru

neisen law (see Eqn. (2)). r

, Y, and R

are

the residual resistivity, the Debye temperature, and the

prefactor in the Bloch–Gru

neisen law (which depends on

the electron–phonon interaction strength).

Compound r

(mO cm) Y (K) R

(mO cm)

LuAl

7.0 275 24

YAl

4.4 310 35

LaAl

3.1 205 45

324

Intermetallic Compounds: Electrical Resistivity

scattering phenomena. These interactions between

the conduction electron system and the matrix nat-

urally depend on the magnetic state of the lattice.

Assuming that the lattice contains magnetic ions

(e.g., RE ions), any disorder in the arrangement of

the associated moments will give rise to magnetic

scattering. RFe

(R being a heavy RE element) has

been selected as an example of the temperature vari-

ation of the resistivity in compounds with stable

magnetic moments in both sublattices of the Laves

phase (Gratz et al. 1989). The series is not complete

since single-phase samples of light RE elements (pra-

seodymium, neodymium, and samarium) are excep-

tionally difﬁcult to prepare, with the exception of

CeFe

which is included in the discussion. Among

this series there are two examples (YFe

and LuFe

)

where only the iron moments determine the magnet-

ism. In all the others the RE 4f moments and the iron

moments are involved in magnetism as well as in the

scattering of the conduction electrons.

The localization of the iron moments owing to

the itinerant character of the 3d electrons are less

pronounced than the 4f moments. However, the iron

moments are much more localized than the cobalt

moments in the RCo

compounds (see below). The

temperature variation of the resistivity of these ferri-

magnetic compounds (the RE and iron moments are

coupled antiparallel, see Alloys of 4f (R) and 3d (T)

Elements: Magnetism ) is shown in Fig. 5. The kinks

in the r vs. T curves indicate the Curie temperatures,

with the arrows showing the temperature at which

there is an inﬂection point in dr/dT vs. T. The high

magnetic ordering temperature of RFe

is mainly be-

cause of the strong interaction between nearest-neigh-

bor iron atoms (these have a separation almost

identical to that in elemental iron). There is an in-

crease in T

when proceeding from LuFe

to GdFe

owing to the additional 4f–4f interaction, which is

largest for the gadolinium compound. With the ex-

ception of CeFe

, the r(T) curves ﬁt to the theoretical

r(T) curves for a magnetic compound in Boltzmann

Equation and Scattering Mechanisms.

The r

spd

values (obtained by an extrapolation from

high temperatures to T ¼0) vs. the deGennes factor,

(g –1)

J(J þ1), for the different RFe

compounds are

given in Fig. 6. It is evident that there is no propor-

tionality r

spd

p (g  1)

J(J þ1), since the r

spd

values

in the RFe

series are dominated by the 3d moment

scattering, which is, however, constant within the

series. Many RE compounds with nonmagnetic part-

ner elements (e.g., RAl

) do show the proportionality

spd

p (g  1)

J(J þ1). Examples of these are RAl

and RPt, both of which are included in Fig. 6. At low

temperatures in magnetically ordered compounds

(T5T

ord

) spin waves are often important and spin-

wave scattering will dominate the low-temperature

increase in r vs. T. In the case of ferromagnetism, the

Figure 4

Temperature dependence of the electrical resistivity of

the pseudobinary LaAg

1x

series. The hysteresis in

the r(T) curves indicates structural transitions of

martensitic origin.

Figure 5

Temperature dependence of the electrical resistivity of

RFe

Laves phases in the range up to 1000 K (Gratz

et al. 1989). The arrows indicate the Curie temperatures

(note there is a shift of the zero points).

325

Intermetallic Compounds: Electrical Res istivity

spin-wave dispersion is given by _o ¼Dq

(where D is

the spin-wave stiffness and q the wave vector). For

RFe

this relationship is obviously fulﬁlled since

there is a T

dependence of r

mag

(Fig. 6 inset) (Gratz

et al. 1989). The curvature in r vs. T

for TmFe

and

ErFe

is because of low-lying crystal ﬁeld excitations.

It can be asked why CeFe

does not ﬁt into the

general behavior of these RFe

compounds. In early

investigations the unusual situation of CeFe

had

been noted (the much smaller Curie temperature

¼270 K), the strongly enhanced electronic spe-

cific heat coefﬁcient (g

¼56 mJ mol

1

2

), etc.).

Eriksson et al. (1988) suggested that the cerium 4f

electron moments may be itinerant in this compound.

They suggested that cerium might behave quite dif-

ferently from the localized 4f electrons associated

with the other RFe

compounds. It is found exper-

imentally that the net moment on the cerium site is

0.15 m

, consisting mostly of the spin component and

being polarized opposite to the iron moment of

1.2 m

. The latter value is much smaller than that in

the other RFe

compounds. The delocalization of

both the cerium 4f and the iron 3d moments are be-

cause of the hybridization of both states (Eriksson

et al. 1988). CeFe

is an example of an itinerant

ferromagnetic system, where spin ﬂuctuations are

important for the physical properties. In the temper-

ature variation of the resistivity of CeFe

there are

basically two features. These are the much stronger

dependence at low temperatures and the negative

curvature at elevated temperatures in the magneti-

cally ordered state (see below).

Figure 7 shows the resistivity variations of RCo

compounds (Gratz et al. 1995). The reason for having

selected these compounds is to show the effect of

spin-ﬂuctuation scattering on resistivity (see Spin

Fluctuations). Whereas r

mag

in YCo

is determined by

the spin ﬂuctuations in the cobalt sublattice only (see

Fig. 1 and earlier comments), in RCo

compounds

containing R atoms with an intrinsic moment, the

magnetization is a result of the strong interplay be-

tween the two magnetic sublattices, and thus r

mag

depends on both. It leads to magnetic ordering at

temperatures indicated by the arrows in Fig. 7. In the

paramagnetic state (above T

) there are spin ﬂuctu-

ations in the cobalt sublattice that cause the pro-

nounced negative curvature in r vs. T (as in YCo

see Fig. 1). However, there is a temperature-inde-

pendent spin-disorder scattering, r

spd

, superimposed

(owing to the disordered 4f moments). The onset of

long-range magnetic order in the R sublattice below

causes a suppression of the spin ﬂuctuations in the

cobalt 3d band, and an induced moment of the order

of 1 m

per cobalt atom on the cobalt sites. Interest-

ingly, this strong interaction of both sublattices caus-

es the magnetic transition at T

to be of ﬁrst order

(normally second order) in the compounds with

R ¼Dy, Ho, Er, and Tm. The sudden disappearance

of the spin-ﬂuctuation scattering below T

gives rise

to the decrease in resistivity, within a few tenths of a

degree, by approximately a factor six in the case of

ErCo

Below T

, r

mag

depends on the dynamics of the

localized moments only. This turns out to be much

smaller than the spin-ﬂuctuation scattering contribu-

tion to r

mag

. The magnitude of the discontinuity in

r vs. T depends on the temperature at which the

magnetic transition takes place. The higher is T

the smaller is the discontinuity in r vs. T. These

Figure 6

spd

as a function of the deGennes factor (g1)

J(J þ 1)

for RFe

, RAl

, and RPt. The inset shows r vs. T

in the

low-temperature region for RFe

Figure 7

Temperature dependence of the electrical resistivity of

RCo

Laves phases (Gratz et al. 1995). The arrows

indicate the magnetic transition at T

(ﬁrst-order type

transitions). The inset shows details for TmCo

at low

temperatures (the hysteresis is owing to a further

magnetic transition at 3.2 K).

326

Intermetallic Compounds: Electrical Resistivity

ﬁrst-order transitions have a strong dependence on

external parameters such as pressure or external

magnetic ﬁelds. These transitions can also be driven

into a second-order-type transition by a substitution

of the magnetic R atoms by yttrium, e.g., Er

1x

(Hauser et al. 1997). The resistivities of these

RCo

compounds underline convincingly the sensi-

tivity of the resistivity to spin-ﬂuctuation scattering.

The inﬂuence is more pronounced when compared to

other dynamic measuring methods such as NMR or

neutron scattering experiments. It has been argued

that a reason for the dominant inﬂuence of spin-ﬂuc-

tuation scattering on the resistivity derives from the

characteristic timescales of spin ﬂuctuations and the

transport phenomena (such as the electrical resistiv-

ity), which are of the same order of magnitude. A

successful attempt to prove this argument, where in-

elastic neutron experiments and resistivity data for

YCo

have been compared, seems to conﬁrm this

timescale argument (Gratz et al. 2000).

2. Conclusion

In this article we have selected compounds of rela-

tively simple crystal structures in order to present the

inﬂuences of the different scattering mechanisms on

the temperature dependence of the resistivity, and to

discuss what can be learned from resistivity meas-

urements with respect to properties of a given mate-

rial. The contribution of the phonon scattering to the

resistivity is discussed using the Bloch–Gru

neisen

law. Furthermore, it is shown that the temperature-

induced changes of the crystal structure are readily

detected using resistivity as a probe. The inﬂuence of

the spin-dependent scattering processes on the resis-

tivity is shown on selected cubic Laves phases (RFe

A pronounced kink at T

or T

in r vs. T is an

indication of well-localized magnetic moments. The

effect of spin-ﬂuctuation scattering has been demon-

strated in the RCo

series. Finally, the resistivity of

YMn

is shown as an example that, although this

compound crystallizes in the cubic Laves phase, its

resistivity behavior is difﬁcult to explain in the scope

of a classical (Boltzmann equation) concept.

See also: Electron Systems: Strong Correlations;

Electronic Conﬁguration of 3d,4f and 5f Elements:

Properties and Simulations; Kondo Systems and

Heavy-fermions: Transport Phenomena; Magneto-

resistance: Magnetic and Nonmagnetic Intermetallics

Bibliography

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Electrical resistivity, thermal conductivity and thermopower

of nonmagnetic REAl

compounds. J. Less Comm. Met. 111,

369–73

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electronic structure of nonmagnetic RAg compounds. Z.

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m L, Brooks M S, Johansson B 1988

4f-Band magnetism in CeFe

. Phys. Rev. Lett. 60, 2523–6

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1989 Temperature dependence of the electrical resistivity of

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compounds. Solid State Commun. 69, 1007–10

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timate the electrical resistivity of YCo

. J. Phys. Condens.

Matter 12, 5507–18

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1x

)Ni

(0pxp1). J. Magn. Magn. Mater. 54–7, 459–60

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1x

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E. Gratz

Vienna University of Technology, Austria

Intermetallics: Hall Effect

The basic theories and experiments on the Hall effect

are ﬁrst described, and then typical Hall effect results

in f-electron intermetallic compounds are presented.

327

Intermetallics: Hall Ef fect

1. What is the Hall Effect?

In isotropic metals and alloys, a resistive voltage de-

velops only along the current direction (x) in a zero

magnetic ﬁeld. Under a magnetic ﬁeld B applied nor-

mal (z) to the current direction (density ¼j) (Fig. 1),

there appears a transverse ﬁeld:

¼ R

B  j ð1Þ

which is known as the Hall effect, and R

is deﬁned

as the Hall coefﬁcient (Hurd 1972, 1980). Experi-

mentally, R

measured on a thin plate with a thick-

ness of d can be determined as:

¼ V

d=IB ð2Þ

where V

is the transverse voltage, I is the current,

and B is the magnetic ﬁeld strength.

Empirically, the general expression for the Hall

resistivity in magnetic materials is given as:

¼ R

B þ R

M ð3Þ

The origin of the ﬁrst term is a Lorenz force ev

B

experienced by the conduction electrons with a drift

velocity of v

, which is balanced by the force from the

Hall ﬁeld eE

in a steady state:

ev  B ¼ eE

ð4Þ

as shown in Fig. 1. R

is called the normal Hall

coefﬁcient. The second term in Eqn. (3) is the anom-

alous Hall resistivity, resulting from the asymmetric

scattering of conduction electrons at magnetic ions,

which often dominates in magnetic materials.

In the analysis of the Hall effect in magnetic inter-

metallics, special care should be paid to separate the

various contributions to the Hall voltages. The tem-

perature dependence of R

in magnetic materials has

often been ascribed wholly to the anomalous Hall

effect. However, several other possibilities should be

taken into account. One is the change in carrier

number resulting from the Fermi surface reconstruc-

tion, if the system exhibits phase transitions. More

importantly, the variation of the wave vector de-

pendence of the conduction electron relaxation time

t(k) with temperatures (e.g., for scattering by phon-

ons) changes from isotropic large angle scattering at

high temperatures to smaller angle scattering at lower

temperatures. This should not be ignored, because it

leads to the temperature dependence of R

. In fact,

even in normal metals, an apparent temperature

dependence of R

resulting from the tempera-

ture dependence of anisotropy in t(k) was reported

(Hurd 1972). Furthermore, the low-ﬁeld/high-ﬁeld

transition in the cyclotron motion should be taken

into account, especially in samples having a low

residual resistivity.

2. Brief Summary of the Theories

2.1 Normal Hall Effect

In the free electron model, the current density can be

written in terms of the carrier density n as:

j ¼ nev

ð5Þ

Substituting Eqns. (4) and (5) into Eqn. (1), we

obtain:

¼ 1=ne ð6Þ

Thus, the measurement of R

allows us to determine

the carrier density n. For simple metals at room tem-

perature R

, calculated from the free electron den-

sity, agrees with experimental results. For instance,

the experimental value of R

¼3.4(70.1) 10

11

1

in aluminum agrees well with the estimated

value of 3.45 10

11

1

from Eqn. (6) assum-

ing three conduction electrons per ion. However,

such an agreement is accidental except for the simple

(alkaline) metals. In general, the Hall coefﬁcient de-

pends on the magnetic ﬁeld direction. Even for metals

with the cubic crystal structure, the Hall coefﬁcient

is ﬁeld- and temperature-dependent, reﬂecting a

non-spherical Fermi surface. All these behaviors are

beyond the free electron model.

For cubic metals and low ﬁeld condition, a general

formula was given by Tsuji (1985) as:

12p



tðkÞ

vðkÞ

rðkÞ

1

tðkÞvðkÞdS



ð7Þ

where t(k), v(k) and r(k)

1

are the relaxation time,

velocity and mean local curvature of the Fermi sur-

face at k. This formula shows that, even for the cubic

metals, the Hall coefﬁcient can be temperature-

dependent through the temperature dependence of

t(k). The sign of local mean curvature r

1

(k) on the

Fermi surface in the numerator determines the sign of

the Hall coefﬁcient through the Fermi surface inte-

gral in the numerator with weight factor t(k).

The two-carrier model, the simplest possible ap-

proach to explain the temperature and ﬁeld depend-

ences, is derived from the Tsuji formula by dividing

the Fermi surface integral into integrals over two

Figure 1

Experimental geometry for measurement of the Hall

effect.

328

Intermetallics: Hall Effect

spherical Fermi spheres. In the model, we assume

that the two Fermi spheres have different charges, q

( ¼eo0) and q

( ¼7e7), different carrier numbers n

and n

, different effective masses m

and m

and dif-

ferent relaxation times t

and t

. The two Fermi

spheres carry the electrical current in parallel; i.e., the

total conductivity is written as the sum of the con-

tributions from the two spheres as:

¼ s

þ s

ð8aÞ

¼ n

ðx : electron or holeÞð8bÞ

In the low ﬁeld condition (o

t51), the Hall coefﬁ-

cient is written as:

¼½s

þ s

=s

ð9Þ

Even in this simplest model, R

possibly varies with

temperature, if t

and t

have different temperature

dependences.

2.2 Extraordinary Hall Effect

(a) Skew scattering and side jump mechanisms

In magnetic materials, the second term in Eqn. (3),

resulting from the magnetic scattering of conduction

electrons, often dominates (Fert and Hamzic 1980,

Berger and Bergmann 1980).

The origin of the anomalous Hall voltage can be

described as follows. There remains a net unbalance

between the scattering probabilities, from the same

incident wave vector of k, to the right (state k

) and to

the left (state k

) with the y-component of different

sign (k

¼k

ð10Þ

For simplicity, we assume the magnetic ﬁeld (mag-

netization) and the current directions as z- and

x-axes, respectively. More generally, the breakdown

of reversal symmetry in the transition probability

k,k

(i.e., W

k,k

) generates a current along the

y-direction which is compensated by the anomalous

Hall voltage along the y-direction.

More intuitively, an example to produce such an

asymmetry in the scattering probability is shown in

Fig. 2, where the asymmetry is caused by the coupling

of an orbital motion (represented by the angular mo-

mentum l) of conduction electrons with an ion’s spin

S. The pass along the left-hand half circle is preferred,

since the energy is lower for S//-l. If the conduction

electron spins (s) are polarized as in 3d-ferromagnetic

metals, even for the nonmagnetic scattering centers,

the intrinsic spin–orbit interaction (s l) leads to the

asymmetric scattering of conduction electrons (Hurd

1972).

The magnetic conductors are classiﬁed roughly in-

to two categories, depending on whether the electrons

responsible for magnetism are itinerant or localized.

Several theories based on the itinerant magnetic elec-

trons have been reported, and were successfully ap-

plied to 3d-magnetic systems (Hurd 1972). However,

such theories are not suitable for rare-earth inter-

metallics with highly localized 4f electrons. Here we

briefly summarize mechanisms that are more suitable

for systems containing localized magnetic electrons.

The origin of the left–right asymmetric scattering,

the orbital exchange interaction, is introduced

(Kondo 1962) via the relation:

O:Ex

Exð2  g

Þl  J; ð11Þ

where g

and J are the Lande

g-factor and the total

angular momentum of the localized electrons (ions),

respectively. However, the orbital exchange term is

apparently not enough to explain the experimental

results in 4f-electron systems, since gadolinium-con-

taining materials, where the orbital angular momen-

tum is absent (g

¼2), also show a large anomalous

Hall effect. It is understood as a combined effect of

the spin exchange scattering (Bs J) and the spin–

orbit scattering (due to the intrinsic spin–orbit cou-

pling Bs l) (Fert and Hamzic 1980). The combina-

tion of the two mechanisms (the orbital exchange and

the spin–orbit coupled spin exchange scattering) well

explains the experimental results, both for dilute rare-

earth impurities in noble metals and for rare-earth

intermetallic compounds, though several additional

mechanisms have been proposed.

In the ferromagnetic state with a large magnetic

scattering, the ﬁeld dependence of R

mimics that of

the magnetization, since the second term in Eqn. (3)

prevails in R

. Figure 3 shows how to decompose the

normal and anomalous Hall components in the ﬁeld-

dependent Hall resistivity for ferromagnetic metal

thin plates based on Eqn. (2). r

shows a tendency to

saturate around a demagnetization ﬁeld B

(the de-

magnetizing coefﬁcient is B1 for thin plates). The

slope above B

gives the normal part of the Hall co-

efﬁcient R

, while the intercept of the extrapolated

line of the slope with the vertical axis gives R

The anomalous Hall term can be further decom-

posed into so-called skew-scattering and side jump

mechanisms. Another possible mechanism (spin-

effect, Fert et al. 1981), originating from the differ-

ence in the relaxation time between the split up- and

Figure 2

An example of conduction electron orbital motion

leading to asymmetric scattering.

329

Intermetallics: Hall Ef fect