Magnetism and the Kondo effect in cerium heavy-fermion compounds cerium-aluminum-3 and cerium-lead-3

MAGNETISM AND THE KONDO EFFECT IN CERIUM HEAVY-FERMION
COMPOUNDS CERIUM-ALUMINUM-3 AND CERIUM-LEAD-3

By

RICHARD PIETRI

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2001

....... .' ..

.:::. .... :
:: .. .: ::.;- :**** :. :,, i iii! : .

ACKNOWLEDGMENTS

I would like to dedicate this work to my parents Gilberto Pietri and Palmira

Santiago, who made it possible for me to complete my education. There is no

way to measure the amount of support and advice I have received from these two

wonderful human beings. I give thanks to an all-powerful, everlasting God for my

parents, and for the opportunity to pursue my goals and dreams. I also thank my

relatives for all their support during my years at UF.

The most influential person in this project was my research advisor, Dr. Bohdan

Andraka. He was the source behind many of the ideas on this dissertation. He

was also a great mentor in the lab, from whom I learned countless experimental

"tricks." He has my deepest appreciation. The second most influential person was

Prof. Greg Stewart, an endless source of information. I thank him very much for

letting me work in his lab. His written work inspired me throughout my gradu-

ate career. I would also like to thank my other committee members, Prof. Mark

Meisel, Prof. Pradeep Kumar, and Prof. Cammy Abernathy for their patience in

reading this work, for many discussions, and for their advice regarding this dis-

sertation. My appreciation also goes to people whom I worked with in the lab

over many years. I thank Dr. Jungsoo Kim and Dr. Steve Thomas for their train-

ing and technical advice, and Josh Alwood and Dr. Hiroyuki Tsujii for help in

the lab and with some of the experiments. Greg Labbe and the people at the

Cryogenics Lab were also very helpful, especially while using the magnet dewar.

Other people in this field I would like to acknowledge are Prof. Kevin Ingersent, for

many discussions about my research and for an excellent collaboration; Prof. Peter

Hirschfeld for introducing me to the theory of heavy-fermions and to the Kondo

effect; and Dr. Ray Osborn and Dr. Eugene Goremychkin, whose work motivated

part of this study, for very enlightening discussions over the last year and during

the 2000 APS March Meeting. I am indebted to Dr. Youli Kanev and my good

friend Dr. Mike Jones for developing the I#IX UF thesis template, which greatly

simplified all of the formatting work, and to my fellow graduate students, especially

Rich Haas, Dr. Tony Rubiera, and Brian Baker for interesting physics discussions

and advice. My thanks go also to Susan Rizzo and Darlene Latimer for all the

grad-school related paperwork and for taking care of my registration over the years.

Finally, my life would have been unbearable without the company and emo-

tional support of many people here in Gainesville, FL. They helped me stay

motivated and cope with the ups and downs of Physics Graduate School. I would

like to thank my dearest friends James Bailey, Ferdinand Rosa, Dr. Carlos ("Caco")

Ortiz, Ivan Guzmin, Clinton Kaiser, Dr. Fernando G6mez, Soraya Benitez, Cristine

Plaza, Diana Serrano, Jorge Carranza, Franco Ortiz, Lyvia Rodriguez, Anthony

Wells, Diana Hambrick, and Charles and Sarah Reagor. I apologize to the countless

others who are not on this list, including the people at the Southwest Recreation

Center, the Worldwide Church of God, Latin nights at the Soul House, Saoca,

La Sala, Rhythm, and all the "tailgators" over the years.

TABLE OF CONTENTS

page
ii

ACKNOWLEDGMENTS

ABSTRACT .. ..................

CHAPTERS

1 INTRODUCTION ................

2 THEORETICAL BACKGROUND ........
2.1 Landau Fermi-Liquid Theory ...........
2.1.1 Theoretical Basis for a Fermi-liquid ....
2.1.2 Thermodynamic and Transport Properties
2.2 Localized Magnetic Moments in Metals ......
2.2.1 Electronic States of Magnetic Ions .....

2.3
2.4
2.5
2.6

2.2.2 Anderson Model
Single-ion Kondo Model
Anisotropic Kondo Model
Kondo Lattice .......
Non-Fermi-Liquid Effects .

3 PROPERTIES OF CeAla AND CePbs .
3.1 Properties of CeAl . . .
3.1.1 Crystal Structure ........
3.1.2 Specific Heat .........
3.1.3 Magnetic Susceptibility ....
3.1.4 Transport Measurements .
3.1.5 Nuclear Magnetic Resonance .
3.1.6 Muon Spin Rotation .....
3.1.7 Neutron Scattering ......
3.1.8 Chemical Substitution Studies
3.2 Properties of CePb . . .
3.2.1 Crystal Structure .......
3.2.2 Specific Heat . . .
3.2.3 Sound Velocity Measurements
3.2.4 Transport Measurements .
3.2.5 Magnetic Susceptibility ..
3.2.6 Neutron Scattering ......

3.2.7 Chemical Substitution Studies ................

4 MOTIVATION . . . . . . .
4.1 Importance of CeA13 and CePbs ....................
4.2 Objectives ................... .............
4.2.1 Magnetism and Heavy-Fermion Behavior in Ce Kondo Lattices

4.2.2 Ground State of CeAl3

. . 71

5 EXPERIMENTAL METHODS .
5.1 Sample Preparation . . .
5.1.1 Synthesis . . .
5.1.2 Annealing . . .
5.2 Diffraction of X-Rays . .
5.3 Magnetic Measurements . .
5.4 Specific Heat Measurements .
5.4.1 Equipment . . .
5.4.2 Thermal Relaxation Method
5.5 Experimental Probes . .
5.5.1 Experiments on CeA13 .
5.5.2 Experiments on CePb3 .

6 STRUCTURAL AND THERMODYNAMIC PROPERTIES
LOY S . . . . . .
6.1 Lattice Parameter Study of CeA13 Alloys . .
6.1.1 Lanthanum Doping: Cel_-LaAl3 . .
6.1.2 Yttrium Doping: CeYAl3 . . .
6.1.3 Mixed Doping: Ceo.8(La-xY)o.2A3 . .
6.1.4 Summary ....................
6.2 Thermodynamic Measurements of Celx-LaAl3 Alloys
6.2.1 Magnetic Susceptibility .............
6.2.2 Specific Heat ..................

6.2.3 Discussion .......
6.3 Thermodynamic Measurements
6.3.1 Magnetic Susceptibility
6.3.2 Specific Heat .....
6.3.3 Discussion .......
6.4 Thermodynamic Measurements
6.4.1 Magnetic Susceptibility
6.4.2 Specific Heat .....
6.4.3 Discussion .......
6.5 Heat Capacity of Ceo.8Lao.2Al3
6.5.1 Results .........
6.5.2 Discussion .......

C

.

............
Cel-,YAla Alloys .

Ce......(Lao...
Ceo.8(Lal-xY=)0.2Al3

)F CeAla

.....

...o.
. .

* .

* .

Alloys
* .
* *

. . . . . . 147

and Ceo.sLao.,7A3 in Magnetic
.................
.................

Fields

. .

AL-
93
.. .93
.96
. 100
S. 108
..113
S. 116
S. 116
S.118
..127
. 134
S. 134
..137
..139
. 141
. 141
..144

153
154
157

7 MAGNETIC FIELD STUDY OF CePba ALLOYS . .. 165
7.1 Specific Heat of CePb3 in Magnetic Fields ............. 165
7.2 Single-Ion Kondo Behavior of Ceo.6Lao.4Pb3 in Magnetic Fields .. 175
7.2.1 Results ............................... 175
7.2.2 Discussion ............................ 179

8 CONCLUSION .. .. .. .. .. .. ... 184
8.1 Summary ................... ........ ... 184
8.1.1 Ideas for Future Work .......... ...... 187

REFERENCES ... .. . .. ... .. .. ... .189

BIOGRAPHICAL SKETCH .................... 198

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MAGNETISM AND THE KONDO EFFECT IN CERIUM HEAVY-FERMION
COMPOUNDS CERIUM-ALUMINUM-3 AND CERIUM-LEAD-3

By

Richard Pietri

August 2001

Chairman: Bohdan Andraka
Major Department: Physics

Measurements of the lattice parameters, magnetic susceptibility, and specific

heat between 0.4 and 10 K in magnetic fields up to 14 T have been conducted on

Cei-,M Al3 alloys, with M = La (0 < x 1) and Y (0
of CePb3 and Ce0.6La0.4Pb3 was also measured up to 14 T. The above experiments

were performed to study the anomalies in the specific heat of CeAl3 and CePb3,

and to better understand the interplay between magnetism and Kondo behavior

in the ground state of Ce heavy-fermion systems.

Data for x-ray diffraction of Ce-_.,MAl3 confirmed an anisotropic lattice

volume expansion for M = La (decreasing c/a ratio) and a contraction for M =

Y. The low-temperature magnetic susceptibility and specific heat of Cel_,LazAl3

are consistent with Doniach's Kondo necklace model. The electronic coefficient

7 decreases with Y concentration, and has a nonmonotonic dependence for M

= La with a minimum at x = 0.2. The temperature position of the anomaly

Tm has a maximum around x = 0.3 for La doping. The lack of a suppression

of T, for Y x < 0.2 suggests a dependence of this maximum on the absolute-

value change in c/a. Magnetic field measurements on La-doped CeA13 alloys

revealed that the field dependence of T, is inconsistent with the anisotropic Kondo

model, with Tm for Ceo.8Lao.2A13 decreasing only by 0.4K at 14T. Experiments

on Ceo.8(Lai-_Y,)o.2Al3 revealed that C/T oc x oc T-1+A for x = 0.4, with A

comparable to that of heavy-fermion alloys with scaling similar to that associated

with a quantum Griffiths phase.

Specific heat measurements up to 14 T on polycrystalline CePb3 indicated

a shift in TN to lower values, disappearing for H > 6 T. The ratio A/72 is field-

dependent below 6 T. Studies on Ceo.6Lao.4Pb3 revealed that the electronic specific

heat AC of this alloy can be described by the single-ion Kondo model in magnetic

fields, with TK 2.3 K. A previously undetected anomaly in C/T was found below

2 K, shifting toward higher temperatures with increasing field. This maximum

appears to be a feature of the Kondo model in magnetic fields.

viii

CHAPTER 1
INTRODUCTION

Over the last century, our current understanding of the metallic state developed

as a result of substantial experimental and theoretical work based on the discov-

ery of the electron by J. J. Thomson in 1897 and the advent of modern quantum

physics. The behavior of solids has long been described in terms of the dynam-

ics of its constituents, electrons and nuclei; with the former being responsible for

electrical conduction and dominating the thermodynamic properties at very low

temperatures. This single-electron picture of the solid state has been remarkably

successful in describing the properties of many body systems that, as a whole,

are much more than a simple array of atoms. The current picture of a lattice

of ions embedded in a gas of electrons obeying Fermi-Dirac statistics is justified

by the theoretical framework set by Landau on his Fermi-liquid theory, for which

he won the Nobel Prize in 1962. Based on the principle of adiabatic continuity,

the theory states that the metallic state at low temperatures can be described

quantum-mechanically in terms of a fluid of weakly-interacting particles (Fermi-

liquid, see Chapter 2). The properties of this quantum fluid are similar in form

to those of a gas of noninteracting electrons. Landau's Fermi-liquid theory has

been successfully applied to a variety of systems, including liquid 3He and normal

metals like Au and Ag. It is one of the foundations of modern condensed matter

physics, rivaled in its scope only by the standard model of particle physics.

Since the development of Fermi-liquid theory, the synthesis of new materials

displaying unusual properties presented challenges to this well-established descrip-

tion of condensed matter systems. A large number of these materials exhibit strong

electron correlations in their normal paramagneticc) state, stretching the limits of

applicability of Fermi-liquid theory. In some materials, the effect of these interac-

tions is reflected in the deviations of their thermodynamic and transport properties

from the predictions of this theory. This group includes the normal state of high-

temperature superconductors and non-Fermi-liquid systems [1, 2, 3]. In others,

their normal-state properties remarkably agree with Fermi-liquid theory, despite

the presence of strong interactions between electrons and even the coexistence with

a magnetic phase. It is in this group that we find most heavy-fermion compounds.

Heavy-fermion systems are alloys where one of their constituents is a member

of the lanthanide (Ce, Yb) or actinide (U, Np) family. They are so called because

the effective mass of the particles dominating the thermodynamics, which have

half-integer spin fermionss), is hundreds of times that of a free electron (heavy).

Extensive reviews on these systems have been written over the last two decades [4,

5, 6, 7, 8]. In these systems, the interactions between localized f electrons and the

conduction band reduce the f magnetic moment and give rise to a Fermi-liquid-

like state at low temperatures. The large effective mass m* is a consequence of the

large density of states at the Fermi energy N(O).

The most widely used experimental parameter to determine both the density

of states and the effective mass of these particles is the Sommerfeld coefficient

of the specific heat 7. In Fermi-liquid theory, 7 is proportional to both m* and

N(0). The specific heat of metals in their normal state at low temperatures is

approximated by the following formula [9, 10]:

C = T + T3, (1.1)

where 7 is the electronic contribution and f is the Debye contribution from lattice

vibrations. Values of 7 for heavy-fermion compounds typically range from several

hundred to several thousand mJ/K mol, compared to less than one for normal

metals like Cu and Au. The presence of additional contributions to the specific

heat makes the determination of y more difficult, and y is usually represented as

the extrapolated value of C/T at zero temperature.

The heavy-fermion character is also reflected in other properties, like mag-

netic susceptibility and electrical resistivity. The magnetic susceptibility at high

temperatures follows the Curie-Weiss form [9, 10],

C
S= (1.2)
T + Ocw'

where C is the Curie constant and Ocw is the Curie-Weiss temperature. At lower

temperatures, the susceptibility reaches a constant value (~.10 to 100 memu/mol),

proportional to the density of states N(0) according to Fermi-liquid theory. The

electrical resistivity of metals at very low temperatures is given by

p = Po + AT2. (1.3)

Here, po is the temperature-independent term due to scattering off impurities and

defects, and A is the Fermi-liquid term. Values for A in heavy fermions are in

the order of tens of pQ cm/K2, much larger than those corresponding to normal

metals.

An intriguing fact of heavy-fermion systems is that the observed Fermi-

liquid properties are not exclusive to the normal state of these materials. The

variety of ground states for these compounds [5, 6] ranges from nonmagnetic, as

in UPt4Au [11], to antiferromagnetic (UCus, U2Zn17, CeA12) to superconduct-

ing (UBei3, CeCuaSi2), to both magnetic and superconducting (UPt3, URu2Si2,

UPd2A13, UNi2Al3). The presence of magnetism and/or superconductivity in these

compounds indicates that the heavy Fermi-liquid ground state coexists with a dif-

ferent phase.

This unconventional ground state, when tuned as a function of pressure,

magnetic field, and/or chemical disorder, can completely move away from Fermi-

liquid behavior. These non-Fermi-liquid (NFL) alloys have been widely studied

during the last decade [3, 12]. Their thermodynamic and transport properties are

characterized by power laws in temperature. Theoretical models for the description

of these effects are currently under development. Examples of these systems [3, 12]

include UCus.,Pda, CeCu6-.Au,, UI-_Y.Pd3, Ce7Ni3 (pressure-induced NFL),

and CeNi2Ge2 [13], U2Pt2In, and U2Co2Sn [14] (NFL compounds).

Among the many unresolved issues in heavy-fermion materials is the coex-

istence of magnetic and Fermi-liquid degrees of freedom giving rise to the ground

state. In addition, a recent interpretation of the ground state in terms of an

anisotropic interaction between f electrons and the conduction band has been

proposed for these systems [15]. Both topics are confronted in this dissertation

by studying structural and thermodynamic properties of two well-studied canoni-

cal heavy-fermion compounds: CeA13 and CePb3. Cerium-based compounds were

chosen because of their simpler electronic configuration. There is only one 4f spin

per Ce ionic site, as opposed to two or three 5f spins per U ionic site. The ground

state properties of the above compounds are not well understood, despite more

than 20 years of study. The experiments presented here will help clarify these

issues in order to motivate further discussion of these topics on both theoretical

and experimental grounds.

The outline of the dissertation is as follows: The necessary theoretical back-

ground behind heavy-fermion physics is presented in Chapter 2. The chapter

begins with an overview of Landau's Fermi-liquid theory, followed by a discussion

of the energies involved in the determination of the ionic ground state and mag-

netic moments in metals. The Kondo effect, the mechanism responsible for the

Fermi-liquid state at low temperatures in heavy fermions, is then presented along

with its anisotropic version. The concept of a Kondo lattice is also introduced,

and the consequences of extending the Kondo model to a concentrated system

are discussed. Chapter 3 gives an experimental review of the essential physical

properties of both CeA13 and CePb3. It is then followed by a discussion of the

motivation behind this study (Chapter 4). Chapter 5 gives a general description of

the experimental apparatus and methods used in this dissertation. The results of

structural and thermodynamic measurements on CeA13 and CePb3 alloys are then

explained in Chapters 6 and 7, respectively. Finally, Chapter 8 summarizes the

main findings of the dissertation and elaborates on its contributions to the field.

The dissertation ends by pointing out unresolved issues and elaborating on ideas

for future studies.

CHAPTER 2
THEORETICAL BACKGROUND

This chapter discusses the current theoretical models describing the charac-

teristics and behavior of heavy-fermion systems, such as Fermi-liquid theory, ionic

configurations in solids, and the Kondo effect.

2.1 Landau Fermi-Liquid Theory

Landau's theory of interacting fermions at low temperatures [16] stands as

one of the most remarkable achievements of theoretical condensed matter physics.

It has often been compared to the standard model of elementary particle physics,

as far as its scope and prediction of physical properties is concerned. The basis

of its success is the adaptation of the Fermi gas model of noninteracting electrons

to a system of interacting fermions at low densities and energies. This mapping

allows for a single-particle description of thermodyamic and transport properties

of Fermi systems like liquid 3He and normal metals like copper, silver, and gold.

Although Landau's Fermi-liquid theory has been successfully applied in a large

number of condensed-matter systems, its validity relies on a series of assumptions

that apply mostly to weak interactions and isotropic scattering between fermions.

Heavy-fermion systems, often described as having a Fermi-liquid ground state,

exhibit strong many-particle correlations that lead to magnetic order in many cases.

The relation between magnetism and Fermi-liquid behavior in heavy fermions is

at present not fully understood. Nevertheless, the theory has been successful

in predicting the properties of these compounds. In this section, the differences

between Fermi-gas and Fermi-liquid models are outlined, followed by a description

of thermodynamic and transport properties of the Fermi liquid.

2.1.1 Theoretical Basis for a Fermi-liquid

For a system of noninteracting particles obeying Fermi-Dirac statistics, with

mass m, momentum p and spin a, the probability of finding a particle with energy

e is given by the Fermi distribution function n(e) [17],
1
n() = 1 + T' (2.1)
1 + e(E-p)/keT'

where kB is Boltzmann's constant and p = 6F, the Fermi energy. The spins are

assumed to be quantized along the z-axis.

In the absence of an external field, the energy of a particle becomes e = Ep =

p2/2m, and the ground state distribution np0 is given by

nPa (2.2)
0 p> PF

where PF is the Fermi momentum. The ground state energy of the system Eo is

equal to

Eo npo ep. (2.3)
pa

The total energy is the sum of the ground state energy and the excitation energies

of the system. The number of excitations is given by the difference between the

ground-state and excited-states distribution functions:

bnp = np, np, (2.4)

where 6np, > 0 corresponds to a particle excitation and np, < 0 to a hole excita-

tion. Since the excitation energies depend on the number of excitations, the total

energy of the system can be expressed as

E = Eo + e 6np. (2.5)
P1.

Despite the strong electrostatic forces between electrons in a solid, the Fermi

gas model for noninteracting electrons is capable of describing their behavior in

metals. At metallic electron densities, the kinetic and Coulomb energy terms are

comparable in magnitude to each other. The justification for the predictions of this

model come from their close resemblance to those of the interacting case. Through

adiabatic continuity [16], it is possible to label the states of an interacting Fermi

system in terms of the states of a Fermi gas. When the interaction potential is

treated as a perturbation, and is turned on slowly enough to prevent a change in

the eigenstates of the Hamiltonian, there is a one-to-one correspondence between

the initial and final states. The excitation energies of the final state are different

from those of the Fermi gas because of the additional interaction term in the

Hamiltonian. The final state has also the same entropy and can be described by the

same distribution function as the noninteracting Fermi gas. The system resulting

from the adiabatic perturbation is called a Fermi liquid. The excited states of a

Fermi liquid are no longer associated with independent electrons, but to negatively

charged, spin-1/2 fermions called quasiparticles, with an effective mass m* different

from that of a free electron. These quasiparticles have a sufficiently long lifetime

7 between collisions at low temperatures. The condition for the applicability of

Fermi-liquid theory is that the uncertainty in the energy of a particle, of order

h/7r c (ksT)2, is much smaller than the width of the excitation spectrum of the

Fermi distribution function, of order kT [18]:

h/7r < kBT. (2.6)

This condition applies to a system with excitation energies much smaller than kT.

Due to the mutual interaction between quasiparticles, the total energy of

the system is no longer represented by the sum of ground state and individual

excitation energies. As a consequence, each quasiparticle is under the influence

of a self-consistent field from other quasiparticles. This self-consistent field affects

both potential and kinetic energy terms of each individual quasiparticle. The

energy E then becomes a functional E{np } of the distribution function. The

excitation (quasiparticle) energy, which itself is a functional of the distribution

function, (e = e{npa}), has an additional term corresponding to the interaction

energy between two quasiparticles fpa,p'a', each with momentum and spin p a and

p'a', respectively. This energy term is also a functional f{npa} of the distribution
function, so that the quasiparticle energy becomes an expansion in terms of the

number of excitations 6npr [19]:

Ep = pa + fpa,p'a' np'a' + ..., (2.7)
p'at
where ega is the ground-state quasiparticle energy. As a result, the total energy of

the system is also an expansion in dnpa:

E =Eo + np + f pa, npa bp, + .... (2.8)
pa pa,p'a'

When considering an ensemble of quasiparticles with spins quantized along

different axes, the distribution function pa, should be treated as a 2 x 2 matrix in

spin space, that is, as a linear combination of the Pauli matrices. In the absence of

higher-order scattering processes, like spin-orbit coupling, the interaction energy

can be expressed as the sum of symmetric and antisymmetric (spin-dependent)

terms

fpp' = fpp + f pp, r ', (2.9)

where fp', and fp'p are the symmetric and antisymmetric terms, respectively,

and r, r' are Pauli matrices. Both fsp, and fap, are dependent on the angle

between p and p', and can be expressed as an expansion in Legendre polynomials,

with coefficients ft and ff, in the case of isotropic scattering (spherical Fermi

surface). In some metals, the presence of crystal-field and spin-orbit coupling

effects significantly distorts the Fermi surface, changing the angular dependence of

fp,, and fp,,. The Landau parameters Fl and FlP are defined with respect to the
coefficients ff and ff corresponding to isotropic scattering:
F8 N(0) f, (2.10)
(2.10)
F, N(O0) ff,

where N(0) is the density of states at the Fermi energy.

2.1.2 Thermodynamic and Transport Properties

Since the total energy of the system of quasiparticles is an expansion in terms

of the variation in the distribution function 6np0, it follows that the thermodynamic

properties are expansions in powers of the temperature. The first term of the

expansion corresponds to the result for the noninteracting Fermi gas. Subsequent

terms are finite temperature corrections due to coupling with spin fluctuations

within the interacting fermion fluid.

The specific heat of a Fermi liquid is given by:

C = T + aT3n T + ..., (2.11)

where the Sommerfeld coefficient 7 is
27r2k2 k2m*pF
S= 2 N(o) = 3 (2.12)

The first term is linear in temperature, and proportional to the effective quasipar-

ticle mass m*. The effective mass is related to the free-electron mass m by
m* 1
= 1 + -FI, (2.13)
m 3
where Ff is one of the Fermi-liquid parameters. The second term in the specific

heat is a smaller correction and originates from quasiparticle coupling to spin

fluctuations.

The magnetic susceptibility is independent of the temperature to first order:

2. 72 N(0)
x = 21N 2)N(0) +.. + ""( (2.14)
4 1 + FJ

where /.,f corresponds to the quasiparticle effective magnetic moment, -y is the

linear coefficient of the specific heat, and F' is a Fermi-liquid parameter. The

second term in the expansion is of order T2 In T.

The electrical resistivity due to quasiparticle scattering is inversely propor-

tional to the time between collisions r, and proportional to the square of the

temperature [20]:

T2 r2e2 m(76.06) T
p= AT'N = 1( (2.15)
16N(0)h3 Tp '

where e is the electronic charge, m is the mass of a free electron, h is Planck's

constant, and Tt is the effective Fermi temperature of the Fermi liquid.

2.2 Localized Magnetic Moments in Metals

Electrons in metals are not entirely free particles. They are constantly under

the influence of a periodic potential due to a charged lattice. In addition, the

distances between electrons are close enough for the Pauli exclusion principle to

play an important role in the formation of energy levels. In general, electrons with
energies in the vicinity of the Fermi energy tend to be delocalized and form part of

the conduction band. To a first approximation, the equation of motion of nearly-

free electrons is given in the Hartree-Fock form. Orbital states within a single ion

are formed by electrons with energies below EF, and are more localized. Their wave

functions retain some ionic character. For the most part, the thermodynamics of

a metallic system in its normal state can be described by taking into account

the individual contributions of quasiparticles (Fermi-liquid theory) and localized

free spins. However, in many systems, the lattice of localized electrons near or

below the Fermi level strongly interacts with conduction electrons. The resulting

potential can have a major effect on the thermodynamics not accounted for by

nearly-free electron models. In order to understand the behavior of 4f magnetic

moments in metals, it is important to have a knowledge of the interactions that

give rise to their formation.

2.2.1 Electronic States of Magnetic Ions

The localized states of electrons in metals are similar to those of free magnetic

ions [21]. For each energy level n, there are (2s+1)(21+1) degenerate states, where

n, 1, and s are the principal, orbital, and spin quantum numbers, respectively. The

degeneracy is partially lifted by the electron-electron Coulomb interaction, of order

10 eV. These energy levels, called multiplets, are filled up according to Hund's rules

and the Pauli exclusion principle. Once all 2(21 + 1) levels are fully occupied, the

sum total of spin and orbital angular moment equals zero, so that a filled shell

has no magnetic moment.

In an incompletely filled shell, one of two relevant interactions responsible

for lifting any additional degeneracies is spin-orbit coupling. The spin of each

orbiting electron couples with an effective magnetic field due to its motion about

the nucleus. The effective field is proportional to the orbital angular momentum

of the electron. The total spin-orbit interaction is then given by

ito = A(L.S) = gp Z, ) (L.S), (2.16)

where g is the electron g-value, p, is the Bohr magneton, Zf is the effective

atomic number, and L and S are the total orbital and spin angular moment,

respectively. The coefficient A is positive when the shell is less than half-filled, and

negative for more than half-filled. The coupling between L and S has an effect on

the eigenstates of the ionic Hamiltonian. Both operators are no longer constants

of the motion, and the states are now labeled by the total angular momentum

J = L + S. As a consequence, the degenerate states of each multiple split into

2S+1 levels for L > S or 2L+1 levels for L < S, each carrying a 2J+1 degeneracy.

The second interaction responsible for the splitting of degenerate energy lev-

els of a multiple is due to the surrounding ions. Crystal-field effects represent the

influence of Coulomb interactions from neighboring charges on localized states.

The crystal-field contribution is given by the net Coulomb energy due to point

charges located at the different crystallographic sites, and by the direct Coulomb

interaction between the outermost localized orbitals of surrounding ions. To a first

approximation,

WCEF = -e S VCEF(r) = -e Zer (2.17)

where Rj and Zej are the position vector and charge of the jth ion, respectively,

and r, and e indicate the position and charge of the electrons. The potential

VCEF can be expressed in polar coordinates and expanded in terms of the spherical
harmonics Yj,(0, 0). The result is an expansion in powers of (r) and of the angular

momentum operators L2 and L. (or J2, Jz). The crystal-field interaction partially

lifts the degeneracy of the ionic states. The number of states is determined by

the symmetry of the crystal structure, and typically increases for structures of low

point-group symmetry.

In solids with magnetic ions, the relative strength of spin-orbit and crystal-

field energies depends on the localized character of the wave function corresponding

to the incompletely-filled shell. The spin-orbit interaction increases as the distance

from the nucleus decreases ({iso oc (1/r3)). The crystal-field contribution 2CEF, on

the other hand, increases with the radial extent of the wave function. For electrons

in incomplete d orbitals, 7CEF > 7so due to their direct interaction with orbitals

from neighboring ions. In contrast, electrons in incomplete f orbitals are very

localized and reside close to the nucleus. Therefore, the spin-orbit interaction is

very large (> 0.1 eV), and the crystal-field contribution HCEF comparatively smaller

(> 0.01 eV). As a consequence, the lowest-lying multiple is first split by the spin-

orbit interaction, and each of these levels is split further by the crystal field. The

ground state of the system is the crystal-field ground state. For example, in Ce3+,

there is only one 4f electron (S = 1), and the lowest-lying multiple corresponds

to L = 3. 7,so splits the multiple into two 6-fold degenerate levels: IJ = ) and

IJ = ). The lowest-energy level (J = ) is then split by H.CEF into a double
and a quartet for cubic crystal symmetry and into three doublets in the case of

hexagonal symmetry. For a crystal-field doublet ground state, the effective total

angular momentum of Ce3+ is J = .

2.2.2 Anderson Model

The fundamental problem in magnetic alloys (including heavy-fermion sys-

tems) is the coexistence and interaction of the electron liquid with localized atomic

orbital states. From this point of view, the conduction band is formed primarily

of electrons in the outermost s and p shells, and the localized states consist of d

or f orbitals in iron-group and rare-earth ions, respectively. The following discus-

sion will focus on localized f states. Electrons in a partially-filled f shell have

a finite probability of mixing and are free to interact with the conduction band

if their energy is close to the Fermi level. The interaction with the conduction

electrons regulates the average occupancy and magnetic moment of the f level.

This problem was described by Anderson [22] in the following Hamiltonian for a

single impurity embedded in a free-electron environment:

NAndu = 710 + 7Of + 7/ff + cf,/ (2.18)

The first term is the unperturbed free-electron Hamiltonian:

710 = EknkE. (2.19)
ka
Here, Eknka is the energy of a free-electron state with wave number k and spin a,

and nko is the number operator

nkoa = akaka, (2.20)

with atk and ak0 the creation and anihilation operators, respectively, for a free

electron state with labels k and a. The second term is the unperturbed energy of

the localized f level:

Qof= Efnf1, (2.21)

where Ei corresponds to the energy of the f level and

nfa = ac4afa. (2.22)

The third term represents the on-site Coulomb repulsion between two f elec-

trons of opposite spin:

Wff = UnfTnfi, (2.23)

with U the Coulomb integral between the two f states, and n/t and nyI the number

operators for f states with up and down spin, respectively. The last term denotes

the mixing between conduction electrons and the f orbital:

7tf = Vkf(atkafo + aaka). (2.24)
ka
Here Vkf is the hybridization matrix element between localized and conduction

electronic states.

The effect of the Anderson Hamiltonian on the localized f states depends on

the relative magnitudes of the Coulomb and mixing terms. The Coulomb repulsion

U determines the separation of the up and down spin f levels with respect to each

other. The hybridization term Vkf is responsible for a broadening of the f levels,

which determines the overlap between the lowest f state and the Fermi energy.

These levels are represented by a Lorentzian of width 2r, where

r = jrIVkJ2N(0), (2.25)

and N(O) is the density of states at the Fermi energy. Figure 2.1 illustrates the

density of states of up and down-spin free-electrons and localized levels for different

relative strenghts of Coulomb repulsion and mixing width. For U I VkIl, the

localized up and down-spin levels (d or f) have a small width 2F and are well

separated by U. The down-spin level resides far above the Fermi energy and is

therefore unoccupied, favoring the formation of a strong local magnetic moment. If

the energy of the up-spin state is close to the Fermi energy in the limit U oo, the

localized moment couples strongly with the conduction band (Kondo effect). This

scenario corresponds to integer valence, and is conducive to the formation of the

heavy-fermion state when the impurity concentration is of the order of Avogadro's

number NA and the magnetic ions achieve the periodicity of the crystal lattice. For

U l Vkfl, both localized levels are significantly broad and might overlap with the

Fermi energy due to a reduction in U. An overlap with the conduction band results

in partial occupancy of both up and down-spin levels, leading to mixed valence and

the formation of a weak local magnetic moment. In the limit IVkf I > U 0, both

levels have the same energy and occupancy and the impurity loses its magnetic

moment.

By studying the limit in which F<
perform a canonical transformation on the Anderson Hamiltonian that eliminates

the hybridization term Vkf. Instead, the transformed Hamiltonian is expressed in

terms of 7Wo, 7o/0, li, and an exchange interaction between f-ion and conduction

electron spins

7t, Jkk'Sk Si, (2.26)
kk'
where Sk and Sf are the spin polarization of the conduction electrons and the spin

of the impurity, respectively, and Jkk' the exchange coupling constant. Close to

the Fermi level, k, k' kF, and Jkk, becomes

NI(E) Nt(E) NJ(E) Nt(E)

U U=0

2r
2r 2r 2r

N1 (E Nt (E) N (E) N (E)
Figure 2.1: Spin-up and spin-down electronic density of states distributions for a
localized d orbital embedded in a sea of conduction electrons. Upper left: U =
IVkf = 0; upper right: U> VkJI; lower left: U IVkf ; lower right: U < IVkf; (U =
0) (from Mydosh, 1993) [23].

.

U
JkFkF J = 2|Vk/ 12 < 0, (2.27)
Ef (Ef + U)
where J is the Kondo coupling constant. In this manner, the Anderson Hamilto-

nian effectively transforms into the Kondo Hamiltonian in the limit r << Ef (or

N(0)J<< 1).

2.3 Single-ion Kondo Model

The Kondo problem is that of a single localized magnetic impurity in a metal-

lic host. This scenario corresponds to the above-mentioned U -, oo limit of the

single-impurity Anderson model, with Ef close to the Fermi level. The following

discussion refers to the case of a spin- impurity in a sea of conduction electrons, as

in the crystal-field ground state of Ce3+. As the temperature decreases, the local-

ized f orbital hybridizes with the conduction band, spin-flip scattering increases,

and a scattering resonance appears near the Fermi level, known as the Kondo

or Abrikosov-Suhl resonance. The Hamiltonian describing these processes is the

Kondo (or s-d) Hamiltonian, of the form

'HKondo = -J (r)(si-S), (2.28)
i
where J is the effective coupling constant between f and conduction electrons (as

in Eq. 2.27), S is the localized spin, and si and r4 represent the ith conduction-

electron spin and position vector, respectively. In the case where both Ef and

U + Ef are symmetric with respect to the Fermi energy (U/2 = 6F Ef|),

J oc Vkfl (2.29)
IEF- EA/'

where EF is the Fermi energy. A perturbation treatment of tKondo beyond the

Born approximation leads to an expansion of the thermodynamic and transport

properties in powers of JN(O) ln(kT/D). Here, N(O) is the density of states at the

Fermi energy and D is the bandwidth of scattering states. The electrical resistivity
was calculated by Kondo [25] using third-order perturbation theory:

p = B (l + 2JN(0) In k (2.30)

The constant term
3 mr J2
PB S(S + 1), (2.31)
2ne2h 4EF

obtained from the Born approximation, is a residual resistivity term due to the
presence of the magnetic impurity. The third-order term diverges at low tempera-
tures. The specific heat and magnetic susceptibility due to the impurity are given
by

C = (-JN(0))47rS(S + 1)k 1 + 4JN(0) In k + ... (2.32)

and
g212S(S + 1) + )T
X = T 1 + JN(O) (1- JN(O) In (2.33)

respectively. The perturbation treatment for J < 0 breaks down at a temperature
k.TK = Dexp ( ). (2.34)
(JN(w) (2.34)
The temperature TK is called the Kondo temperature.
At low temperatures (T << TK), the impurity spin strongly couples with
the conduction electron spin polarization, forming a many-body singlet that com-
pletely suppresses the localized magnetic moment at T = 0. In this range, the
thermodynamic and transport properties can be described by Fermi-liquid theory
due to the absence of an impurity spin. The zero-temperature susceptibility of the
impurity is inversely proportional to the Kondo temperature [26],

= (1 )' 129
Xo= -9 k-" (2.35)
2 wk.TK'

and the linear coefficient of the specific heat 7 is given by

2 kB
y = 1.29- (2.36)
6TK

The ratio of the magnetic susceptibility to the electronic specific heat coefficient

y, called the Wilson ratio, is given by

Xo 3 g1t!2
= -3 I. (2.37)
7 2 7r kB

This value is twice that corresponding to the noninteracting electron gas.

The exact solution to the Kondo Hamiltonian and its thermodynamic prop-

erties in terms of T < TK and T > TK and a range of magnetic fields were

obtained using the Bethe ansatz [26, 27, 28, 29]. The above equations follow the

exact solution obtained with this method. Numerical solutions for the specific heat

and the magnetic susceptibility of a spin-! impurity in different magnetic fields are

illustrated in Figs. 2.2 and 2.3. The zero-field specific heat reaches a maximum

at a temperature just below TK. Both the magnitude and the temperature posi-

tion of the maximum increase with field, reaching a shape corresponding to the

Schottky anomaly of a free uncompensated spin-j at large fields gp.H > kTK,

where g is the g-factor of the magnetic impurity. The zero-field magnetic suscep-

tibility shows a Curie-like increase for T > TK, and then saturates until it reaches

a temperature-independent value well below TK. A maximum associated with the

Schottky anomaly of the specific heat appears around TK for gS.H/k.TK = 2 [30].

Its temperature position increases, while its magnitude decreases with increasing

field.

2.4 Anisotropic Kondo Model

The anisotropic Kondo model (AKM) [31, 32] refers to the problem of a

single magnetic impurity coupled to the conduction electrons via an anisotropic

exchange interaction J -J Jl J., where J1g > JL. The Hamiltonian is given by

0.35-
S=1/2
0.30-

S0.25 -
2.
0.20 1.0
*-
c 0.15 H11=0.0
0.10-
0.05 -

10- 10-" 10.1

Figure 2.2: Specific heat of a S = 1 K(
different magnetic fields (H gpiH/kB

1.8 i -'"" "
=0.0

1.4 0 0

0.6- 2.0

0.2- 4.1
.....m ......I .
10- 10- 10-1

Figure 2.3: Magnetic susceptibility of a
T/TK for different magnetic fields (H --

)ndo impurity as a function of T/TK for
TK) [30].

S=1/2

I-

).0.
100 10' 102 10s
T/TK

S = Kondo impurity as a function of
g9BH/kBTK) [30].

T/TK

IAKM E- k C ,Cc,a + (C tCkt'l + cJIckITS) +
k,7 kk'

Skk'

where cL and co are the conduction electron creation and anihilation operators,
S+ and S- are the impurity spin raising and lowering operator eigenvalues, and S,

is the impurity spin value in the z direction. The first term in 7AKM represents the

conduction-electron energies, the second and third terms represent the in-plane

(J) and easy-axis (J11) exchange interactions between a localized spin and the
conduction electrons, respectively, and the last term corresponds to the Zeeman

energy due to a local magnetic field h applied only to the impurity spin S. The

Kondo temperature for an anisotropic exchange interaction (JII < 0) is given in

terms of J11 and Ji as [21, 33]

-1 J J
kTK = Dexp x tanh-'1 (2.39)
N(0) J J2 -JI(

where N(O) is the density of states and D is the bandwidth. The exponential

dependence of the Kondo temperature in the parameter Jll is qualitatively similar

to the J dependence of TK in the isotropic case.

The Hamiltonian for an anisotropic Kondo interaction has been used suc-

cessfully to evaluate the properties of the spin-boson Hamiltonian [34, 35, 36, 37],

which describes the dissipation in the dynamics of a two-level system by an Ohmic

bosonic bath. A mapping of the spin-boson model [38] onto the AKM has been

exploited to calculate the thermodynamic properties of the former model. Further-

more, the parameters of the spin-boson model have recently been used to describe

the properties of the AKM applied to the heavy-fermion system Cei_,.La.Al3 [15].

The spin-boson Hamiltonian has the form

s= 1AaU, + I c Wa aa,, + +
2 2 1 +
1 CO (an ) (2.40)
2 a 2mo + a

Here oa and oz are Pauli matrices, A is the tunneling energy between the two

states and e is an external bias applied to the system. The third term corresponds

to the energy of the bosonic bath and the last term represents the coupling of the

two-level system to the bath, with coupling constants Co. In the case of Ohmic

dissipation, the spectral function of the system is J(w) = 27r a w for w < we, where

a is a measure of the strength of the dissipation and we is a cutoff frequency. For

a 0, the tunneling energy A (h = 1) is renormalized into
a
A, = A (- (2.41)

with A,/k,B equivalent to the Kondo temperature TK in the AKM.

The low temperature behavior of both spin-boson and AKM systems is that

of a Fermi liquid. The linear coefficient of the specific heat per total mole is given

by [35, 36]
72 k2. r2 R
7 = a NA = r (2.42)
3A, 3TK

where NA is Avogadro's number and R = kBNA is the gas constant, and the

magnetic susceptibility of the spin-boson model per total mole at T = 0 is
2 2 NA g2 2 NA
XSB = 2A 2kBTK (2.43)
2Ar 2kTK

where g is the g-factor of the impurity spin. The susceptibility of the AKM at

T = 0 differs from XSB by a factor of a: XAKM = aXSB. The Wilson ratios for both

models are related as follows:
4 7r 2c XAKM
RAKM 4 2k XAKM 2
3 (gpB)2 7

4 7r2 k2 XSB 2
RsB = --- = _- (2.44)
3(gp,)2 7 a

where RAKM = aRSB.

The thermodynamic properties of the AKM are given in terms of the exchange

interactions (Jll) and (Ji), and therefore can also be expressed in terms of the

parameters a and Ar of the spin-boson model [35, 36]:

Ar =p Ji,

a= 1 + -tan (2.45)
7r 4

Figure 2.4 illustrates the temperature dependence of the static susceptibility and

specific heat as C/T for different values of the dissipation a and E = 0. The

parameter a is a good measure of the Kondo anisotropy of the system, since

it decreases sharply with increasing J11. Both curves are universal functions of

(T/Ar) (T/TK). For e = 0, the electronic coefficient of the specific heat is given

by 7 = a/A,, and C/T reaches a maximum at a temperature corresponding to

Ar for a < 0.3. This maximum is reduced in magnitude with increasing a. The

susceptibility expressed as kTXs, has a finite value at T = 0, as in the isotropic

Kondo model, and reaches the free-spin value at high temperatures. The main

effect of a is to increase the temperature at which this latter value is attained. The

temperature Ar indicates the crossover between Kondo and free-spin behavior.

The behavior for a finite bias c > 0 is described in Fig. 2.5 for a = 0.2. The

quantity E is equivalent to a magnetic energy gp.h acting on the impurity spin in

the AKM. The temperature A, is renormalized by e, and becomes [37]

Ar = A+2. (2.46)

The effects of a field on the specific heat are a strong reduction of 7, an attenuation

of the maximum in C/T, and an increase of its temperature position given by

Ar. The low-temperature susceptibility strongly decreases as a function of the

parameter e. It also shows a maximum for fields of order A, and above, with a

temperature position that increases with A,.

2.5 Kondo Lattice

Certain types of metallic compounds, including heavy-fermion systems, can

be described as a lattice of Kondo impurities embedded in a metallic host [39,

40, 41]. This class of materials is commonly referred to as concentrated Kondo

systems. In these alloys, a giant Abrikosov-Suhl resonance of width TK appears

in the density of states near the Fermi level for T
I Kondo scatterers, the resonance lies right at the Fermi energy. This feature
2
indicates the crossover to a strong-coupling regime in the scattering between f

and conduction electrons, growing in size as the number of impurities approaches

Avogadro's number NA. Consequently, there is a substantial increase in the density

of states at EF. Figure 2.6 illustrates the evolution of the Abrikosov-Suhl resonance

for different temperatures. In heavy-fermion compounds, the 4f level is located

well below the Fermi energy. As a result, the localized orbital has integer valence.

The large resonance in the density of states has an effect on the effective mass

m*, as indicated by Fermi-liquid theory. At high temperatures (T > TK), the

Abrikosov-Suhl resonance disappears, and the system behaves as an ensemble of

classical free spins.

Two other characteristics of the Kondo lattice are the appearance of coherence

effects and interactions between magnetic impurities. Below a temperature Tb,

the electronic properties change from those described by scattering off independent

Kondo impurities to those reflecting the periodicity of the lattice via Bloch's theo-

rem. This crossover is usually described in terms of a maximum in the temperature

dependence of the specific heat as CIT and the electrical resistivity around T T,.

A consequence of coherence is an increase of indirect exchange interactions between

impurity spins. At distances larger than the 4f radius (r4f < 0.5A) but less than

1.00

0.80

So0.60

S0.40
0

0.20

0.00
V(

0.30

0.10
0.10

10'
kqT/A,

Figure 2.4: Thermodynamic properties of the anisotropic Kondo model for c = 0
and different values of a. a) Specific heat expressed as ArC/kTT vs T/Ar. b)
Universal static susceptibility curves expressed as kTx.b vs T/A, [37].

0.30

0.20

0 0.10

a= I/5

kaT/5
a= 1/5

0.20

0.15

E, 0.10

0.05

0.00
11

10,

kOT/A,

Figure 2.5: Thermodynamic properties of the AKM for a = 0.2 and different values
of e (in units of A,). a) Specific heat as ArC/kDT vs k.T/A,. b) Susceptibility
curves expressed as A,x.b vs kBT/Ar [37].

g(E)

T <

T4 TK

T>TK

E, EF E4+U E

Figure 2.6: Density of states of a nonmagnetic Kondo lattice at different temper-
atures, showing the evolution of the giant Abrikosov-Suhl resonance [39].

the size of the Kondo compensation cloud for a single impurity, the presence of

closely-spaced uncompensated spins leads to the Ruderman-Kittel-Kasuya-Yosida

(RKKY) interaction between localized f orbitals

7"RKKY = J(r)S. Sj, (2.47)

where
J cos(2kr) (2.48)
J(r) ~ (2(2.48)
(2kFT)3

is the RKKY coupling at large distances, J is the Kondo coupling, and kF is

the Fermi wavevector. In most heavy-fermions J(r) leads to antiferromagnetic

coupling between impurity spins.

The state of a concentrated Kondo system depends on the competition between

the two energies represented by the Kondo and RKKY temperatures TK and TRKKY,

This competition has been described in a simple form through the Kondo necklace

model, developed by Doniach [42, 43]. Both TK and TRKKY depend on the Kondo

coupling J and the concentration of magnetic impurities. The Doniach model relies

on the assumption that the ground state of the system depends on the relative

magnitude of the coupling J only. The phase diagram for this model is shown in

Fig. 2.7. The Kondo temperature depends exponentially on the parameter J, as

discussed previously, while TRKKY ~ J2 N(0). At low values of J, TRKKY > TK, the

material is a magnetic 4f metal, and the Kondo effect is absent. As J increases,

TK > TRKKY, the Kondo effect appears before magnetic order, and the material is a

magnetic Kondo lattice. At even larger values of J (TK TRKKY), magnetic order

disappears altogether and the material is a nonmagnetic Kondo lattice. Heavy-

fermion compounds exist in the region around the magnetic-nonmagetic phase

boundary, and those with a magnetic ground state exhibit mostly antiferromag-

netic order.

A modified form of the Doniach diagram has been recently proposed [44, 45]

to account for the effect of intersite magnetic correlations on the Kondo tempera-

ture in the nonmagnetic region. Instead of continuing to increase exponentially as

in the single-impurity case, TK reaches a saturation value, after which it decreases

slightly with increasing J. Thus, TK in nonmagnetic Kondo lattices may not nece-

sarily follow single-impurity behavior. On the other hand, a complete theoretical

explanation of the effect of magnetic interactions on the Kondo temperature has

yet to be developed.

At a value of the Kondo coupling J = Jc, the magnetic ordering temperature

TM approaches zero at a critical point. The ground state of some heavy fermions at

or near Jc is neither magnetically ordered nor Fermi-liquid-like. A large number of

intermetallics falling in this category are commonly referred to as non-Fermi-liquid

(NFL) systems. Their thermodynamic and transport properties can in some cases

be described by either logarithmic divergences or power-law behavior according to

different theoretical models [3, 12].

2.6 Non-Fermi-Liquid Effects

Current models of non-Fermi-liquid phenomena can be divided into two

groups: theories describing a possible single-ion origin to these effects and those

attributing them to intersite interactions. A member of the first group is the

two-channel quadrupolar Kondo effect [46], a particular scenario within the more

general multichannel Kondo problem [47]. The quadrupolar Kondo effect consists

of the quenching of a nonmagnetic quadrupolar level by two degenerate conduction-

electron bands, and has been used to explain the properties of heavy-fermion

systems like U1-.Th.Bel3 [48]. In this model, NFL behavior is associated with

fluctuations of the quadrupolar degrees of freedom, rather than spin fluctuations.

Another possible single-ion mechanism towards non-Fermi-liquid behavior

is Kondo disorder [49, 50, 51]. The material exhibits a random distribution of

the quantity pJ, where p is the density of states and J is the Kondo coupling

constant. Thus, variations in either the Kondo couplings or the local density of

states gives as a result a distribution of Kondo temperatures. The probability

distribution function P(TK) = P(pJ)d(pJ)/dTK acquires a log-normal form for

strong disorder:

P(TK) = (4iru)4 exp n2oJe'u1(/TK)] (2.49)
TK ln( F /TK ) 4iru

where Po is the average density of states, and u is a dimensionless parameter cor-

responding to the amount of disorder in the system. For weak disorder, P(TK)

takes the form of a Gaussian. At a given temperature T, there are regions where

the local Kondo temperature TK
probability of having uncompensated spins at T = 0, P(TK = 0) 0, the thermo-

dynamic properties are dominated by the contribution from free spins, leading to

non-Fermi-liquid behavior.

The first model involving collective behavior applied to NFL alloys was based

on a description of the physical properties in terms of their proximity to a quantum

critical point (QCP). The system exhibits critical fluctuations of the order param-

eter in the vicinity of a quantum phase transition at T 0 [52, 53, 54, 55, 56].

At finite temperatures, the characteristic frequency w* associated with the critical

fluctuations of the order parameter is much smaller than the transition temper-

ature Tc, so that the system behaves classically at hIw* < k,Tc [56]. A quantum

phase transition at T = 0 is not achieved by a change in temperature, but rather

by a change in a parameter of the Hamiltionian. Under this model, non-Fermi-

liquid effects in heavy-fermion systems arise as a consequence of a near-zero anti-

ferromagnetic transition temperature, so that a quantum-mechanical treatment is

necessary. The thermodynamic properties are dominated by the collective modes

due to critical fluctuations rather than by Fermi-liquid-like elementary excitations,

and are described by various scaling laws [53, 54] depending on the effective dimen-

sionality and the nature of the magnetic transition. As a result, the system is said

to have a 'generalized' (non-Landau) Fermi-liquid ground state, with an enhanced

quasiparticle mass m* due to the presence of long-range spin fluctuations [57].

A recent explanation for NFL behavior relies on the competition between

anisotropic Kondo and RKKY interactions in a disordered system [58, 59]. Around

the QCP corresponding to Jc, for TK > TRKKY, free spins arrange into clusters,

which increase in size as TK -- TRKKY The spin clusters form a granular magnetic

phase, coexisting with the metallic phase, and the system exhibits a Griffiths

singularity at zero temperature [60]. Non-Fermi-liquid effects are attributed to the

dynamics of large spin clusters in the Griffiths phase. A percolation limit for these

clusters is reached at the QCP, which for Tc # 0 leads to an antiferromagnetic,

spin-glass, or ferromagnetic transition [58]. The temperature dependence of the

thermodynamic properties obey power laws, with exponents determined by the

crystal symmetry and the values of the local exchange constants. The nonuniversal

nature of these exponents offers a common description of NFL effects in heavy-

fermion alloys within the Griffiths phase model.

//W
Magnetic 4f -metal Magnetic CKS Non-magnetic CKS
I
'T/ K
2 f
o-

E / i- "

Mognetic 4f- metal Magnetic CKS | Non-magnetic CKS

Figure 2.7: Phase diagram of the Kondo lattice [39], illustrating the different
dependence of TK and TRKK on the parameter J/W, where J represents the
Kondo coupling and W is the bandwidth. The dependence of the magnetic ordering
temperature TM on J/W dictates the regions corresponding to magnetic metal,
magnetic concentrated Kondo system (CKS), and nonmagnetic CKS.

CHAPTER 3
PROPERTIES OF CeAl3 AND CePb3

This chapter gives an overview of structural, thermodynamic, transport, and

magnetic properties of CeAl3 and CePb3 alloys that are relevant to the problems

addressed in this dissertation.

3.1 Properties of CeAls

3.1.1 Crystal Structure

The compound CeA3l crystallizes in the hexagonal Ni3Sn structure (DO19),

Pearson symbol hP8, space group P63/mmc, number 194. This structure con-

sists of two alternating hexagonal layers. The most recently published lattice

parameter measurements give a = 6.547 A and c = 4.608 A [61]. The above val-

ues correspond to a c/a ratio of 0.704, much smaller than the close packed ratio

(0.816), and a lattice volume V = 171.05 A3. A study of the structure of rare-

earth trialuminides[62] attributed the formation of a particular structure and its

c/a ratio to the rare-earth/aluminum ratio RRE/RA,. This ratio is largest for the

hexagonal LaAl3, PrAl3, and CeAl3, and smallest for Yb, Tm and Sc trialuminides,

which crystallize in the cubic Cu3Au structure. As RE/RA, decreases, the crystal

structure is modified from hexagonal to cubic, the layer stacking changes, and the

c/a ratio increases.

Figure 3.1 shows the idealized (Ni3Sn) unit cell of CeAl3. The cell contains

two formula units. The atom positions with respect to the origin are given in

Table 3.1 in terms of the lattice parameters a (x, y axes) and c (z axis). Figure 3.2

is an extended scheme showing the hexagonal stacking and the periodicity of the

34
~~ ,1 .......................... ... ;:............

Figure 3.1: Hexagonal Ni3Sn structure of CeAl3.

Al Ce
a. ,at

AI.

Figure 3.2: Hexagonal Ni3Sn structure of CeA13 (extended scheme).

N

Table 3.1: Cell Content of Ni3Sn structure of CeA13 [64].
Atom Multiplicity Coordinates
(Wyckoff notation) x y z
Ce 2c 1/3 2/3 1/4
2/3 1/3 3/4
Al 6h 0.833 0.666 1/4
0.833 0.167 1/4
0.334 0.167 1/4
0.167 0.334 3/4
0.666 0.833 3/4
0.167 0.833 3/4

unit cell. Each Ce atom has 6 Al nearest neighbors, at a distance dc,-Al = 3.27 A,

and 6 Ce nearest neighbors at a distance dcc, = 4.428 A [63]. The central Ce

atom is surrounded by six nearest neighbors (3 Al and 3 Ce atoms) above and six

below the basal plane. It is important to point out that all nearest neighbors are

located in the layers above and below the central Ce atoms, and their distances are

not along the c-axis direction, but rather at an angle. These off-axis neighboring

distances might have some implications regarding the hybridization between Ce

and Al atoms, as well as the effects of the RKKY interaction on the magnetic

properties of CeAl3 (see Chapter 7).

3.1.2 Specific Heat

Early measurements of the specific heat of CeAl3 below 10 K proved to be

unreliable [65, 66] due to anomalies caused by the presence of the secondary phases.

Later measurements by Brodale et al. [67] demonstrated a significant reduction of

these anomalies. In their study, the low temperature specific heat showed a maxi-

mum around 0.4 K when plotted as C/T vs T. The value of the electronic specific

heat coefficient y extrapolated from C/T vs T2 is -y = 1250 mJ/K2 mol. This max-

imum in C/T has been the subject of intense controversy about the ground state of

CeAla. It was initially proposed that its origin is due to the formation of a Kondo

lattice state in which the conduction electrons undergo coherent scattering [68].

Later experiments [69, 70, 71] suggested that the maximum was due to either

magnetic correlations or a possible antiferromagnetic order in this compound.

The anomaly in C/T has also been studied at different pressures and mag-

netic fields. Magnetic field measurements up to 4T [68] showed that both the

maximum and its temperature position decrease in field, while there is an increase

of C/T values below 0.2 K (see Fig. 3.3). Measurements above 1 K and at 23 T[72]

indicated a decrease in C/T values below 4-5 K (more than 15% at 1 K) and an

increase in values above the same temperature (around 20% near 10K). These

results seem to indicate an initial increase of the electronic coefficient 7 with field,

followed by a marked decrease at higher fields. The pressure dependence of the

specific heat as C/T vs T is shown in Fig. 3.3 [73]. The specific heat is very sen-

sitive to pressure. C/T values at 0.4 K were found to decrease with pressure as
P'1/. There is no sign of the specific heat anomaly at a pressure of 0.4 kbar. The

coefficient y is reduced from 1250mJ/K2 mol at atmospheric pressure to about

550 mJ/K2 mol at 8.2 kbar. Values of C/T are essentially constant below 1 K for

pressures around and above 2 kbar.

An attempt was also made to measure specific heat on very small single crys-

tals of CeAla [74]. The results proved to be sample-dependent. Some of the crystals

showed peaks in the specific heat resembling antiferromagnetic phase transitions.

It remains to be understood whether there is any relationship between these peaks

in the specific heat and the maximum observed in C/T for polycrystalline samples.

3.1.3 Magnetic Susceptibility

Avenel et al. [75] measured the magnetic susceptibility of polycrystalline

CeAl3 down to 0.8 mK. The results show a broad maximum around 0.5 K, resem-

bling the anomaly in C/T near 0.4 K (see Fig. 3.4). The susceptibility becomes

temperature independent below 40 mK (x(T = 0) i 29.5memu/mol), consistent

2.1

0

N

I-
.-J

1.8

I.e

1.8

1.5

1.2

0.5

O

1.4 -

IO0
0

0.6 F

0.2L
0.

2

ant-a

)kbor

1.4" I^"
.4
0 M0M= W 9 OOgl

T K)

LaMeIaaSIaMaIa a,
1 I _nfo Iekfl*C f t E ,1*,O* 1

SI I I jI..r

2
T(K)

5 10 20

Figure 3.3: Magnetic field and pressure dependence of the specific heat of CeAl3.
Upper part: C/T vs T of CeA13 in magnetic fields up to 4 T (0T: circles, 2T:
diamonds, and 4 T: triangles) [68]. Lower part: Pressure dependence of C/T vs T
for CeA13 up to 8.2 kbar [73].

l* Ii j i-i I I

-%

SIPw *

- S

l0l

I a I
7?-,

45.0

35.0

za3o

15.0

0.0

5.0
T (K)

5.0
T (K)

10.0

Figure 3.4: Magnetic susceptibility of CeAls below 10 K [75]. The inset shows the
inverse susceptibility.

a
a
a
a
a
a
a
*

~~ ______. _.____._.___

~ r I 1 1

rN,

with Fermi-liquid behavior. The inverse susceptibility follows Curie-Weiss law

above 150 K, with an effective magnetic moment close to that of a free Ce3+ ion,

APe = 2.54p8, and Ocw = -30 6 K. The susceptibility of single crystals above
4K was also measured with the field parallel (XII) and perpendicular (xi) to the

c-axis [76]. The susceptibility along the c-axis Xll is at least three times as large as

XI around 4 K, indicating a large anisotropic magnetic behavior.

3.1.4 Transport Measurements

Figure 3.5 shows the electrical resistivity of CeAl3 below 300 K. It can gener-

ally be described by a Kondo-like increase down to 50 K, a maximum around 35 K,

possibly signaling the crossover from single-impurity to Kondo-lattice behavior,

and a sharp decrease below 10 K. At temperatures below 100mK, the resistivity

has the form of a Fermi-liquid, with a coefficient A = 35 p cm/K2 (see Fig. 3.5).

No sign of a magnetic phase transition (i.e. kink in the resistivity curve) has been

detected in electrical resistivity measurements around 0.4-0.5 K. When pressure is

applied, there is an increase in both the temperature and magnitude of the maxi-

mum [77]. In addition, the A coefficient decreases, and resistivity values above the

temperature of the maximum are enhanced as pressure increases.

The low temperature magnetoresistance of polycrystalline samples was found

to change sign at a field of 2T, becoming positive at lower fields [79, 80]. The

results are shown in Fig. 3.6. The resistivity values are dependent on the field

direction with respect to the current. This anisotropic behavior increases with

applied field and at low temperatures. The magnetoresistance at 4.2 K and field

perpendicular to the current becomes less negative with increasing pressure for

fields larger than 2 T [77]. In single-crystal measurements, the electrical resistivity

in zero field along the basal plane is more than twice that along the c-axis [76, 81].

The field dependence of the A coefficient parallel to the c-axis shows a peak around

..................... ....... *. H 'i *

P 240

no0

100
HO

140

120

100

60

40

0to

0 4 6 8 10 10"3
T2 (K )

Figure 3.5: Transport measurements on CeAl3. Upper part: Electrical resistivity

below 300 K [78]. Lower part: p vs T2 below 100 mK [20].

-G.4

"%' m "..

W

-
r-

- 10 5

2 T. The authors found this result to be in qualitative agreement with theoretical

models describing weakly-antiferromagnetic metals.

3.1.5 Nuclear Magnetic Resonance

Measurements on 27Al nuclear magnetic resonance (NMR) on CeAl3 down

to 0.3 K by Nakamura et a/.[82] are part of a series of microscopic measurements

arguing against the coherence interpretation of the anomalies in C/T and the mag-

netic susceptibility. The temperature dependence of the spin-lattice relaxation rate

at 0.98 MHz increases by one order of magnitude at the lowest temperatures in a

nonlinear fashion. The relaxation rate reaches a maximum at 1.2 K. The authors

attributed this maximum to the onset of antiferromagnetic order at this tempera-

ture. Later measurements by Wong and Clark [83] and Gavilano et al. [70] revealed

not only the absence of a maximum in the relaxation rate at low temperature, but

a Korringa-like (T1 T = const.) behavior below 0.6 K as well. The reason for these

discrepancies might be related to a large sensitivity of the ground state to lattice

strains and sample preparation for NMR measurements. Powdered samples have

grains with typical linear dimensions around 50 pm. The nonuniform strains cre-

ated by preparing the powder can have a dramatic effect on the physical properties

of CeAla below 1 K. The presence of secondary phases can also have an effect on the

results, since it is more probable to find entire grains of either CeAl2 or Ce3Al11, as

proposed by Wong and Clark [83]. Gavilano et al. also measured the NMR spec-

tra of partially oriented powder (c-axis along the direction of the applied field) at

6.968 MHz, and observed two distinct components (Fig. 3.7). They concluded that

these components correspond to two different regions of the sample being stud-

ied: the spectral lines seen in Fig. 3.7 were attributed to a normal paramagnetic

phase, while the broad structure was ascribed to a phase where static magnetic

correlations take place. The Ce moments of this latter phase were estimated to be

Figure 3.6: Magnetoresistance of CeAla down to 100 mK [79].

c T = 1.4 K

c T = 0.98 K

5.5 6.0 6.5 7.0
Field (kGauss)

Figure 3.7: NMR spectra of partially oriented powder at 6.968 MHz for different
temperatures [70].

less than 0.05ps. The presence of magnetic correlations in CeA13 argues against a

simple interpretation of its ground state in terms of a non-magnetic Fermi-liquid.

3.1.6 Muon Spin Rotation

The only muon spin rotation (gSR) experiments on pure CeA13 available to

date are those of Barth et al. [69, 84]. The authors measured the time-dependent

muon polarization on two polycrystalline samples, as seen in Fig. 3.8. The muon

polarization signal was described as the sum of several time-dependent components,

two of which correspond to the response of muons from different magnetic envi-

ronments. The most significant finding was the detection of a spontaneous muon

spin precession frequency in zero field below 0.7 K from one of these components.

This Larmor frequency, proportional to the local magnetic field, has a very small

temperature dependence below 0.7 K. Its extrapolated value at T = 0 is just above

3 MHz, which corresponds to an average local field of 220 G. In agreement with this

estimate, the muon precession signal could not be observed at an external applied

field of 750 G. Both the oscillating component and the fast relaxation of the muon

polarization are commonly associated with spin-density-wave behavior [85]. The

presence of the local field at the muon sites was interpreted as the development

of short-range, quasistatic magnetic correlations in CeAl3 below 0.7K. As the

temperature decreases, these correlated moments, estimated to be around 0.5/,,

develop some coherence in a spatially inhomogeneous manner. The appearance

of this almost percolative effect was attributed to magnetic frustration. Electron

paramagnetic resonance (EPR) measurements by Coles et al. on GdA13 (Ni3Sn

structure) [86], also contributed to the development of this idea, arguing that the

magnetic behavior in CeAl3 might be mediated by frustrated antiferromagnetism

in the triangular sublattice of the hexagonal a-b planes.

46

-CLIO--

-020 C0.05 K .

S" 0t i I ..-- I ...
0

S-0.05 -

-0.10

S-0.05

S-0.2,0 0.5 K
I
0"- _O-A-------

-a'
-015

-0.20 I K

0 0.5 1.0 1.5 2.0
time (psec)

Figure 3.8: Muon polarization as a function of time in zero external field for
T = 0.05, 0.5, and 1 K [69].

120 0
T 40K

so80 \ o

I I
io

40

so
SL20 "
40 1-

OI

-15 -10 -5 0 5 10 15 20 25 30
Enery =t f [o V]

Figure 3.9: Magnetic contribution to the inelastic scattering function of CeA13 at
T = 20 and 40 K [87]. The solid line is a fit to a three-Lorentzian model. The
dotted lines represent the individual fit components.

3.1.7 Neutron Scattering

Inelastic neutron scattering is one of the most direct methods of determining

electronic energies and crystal fields in metallic compounds. In CeAl3, the cerium

ions occupy positions of low point symmetry. In hexagonal structures, the Ce3+

IJ = 5) multiple splits into three doublets under the influence of a crystalline
electric field (CEF): F7: | ), rF : | ), and r : | 3). In cerium heavy-
2 2 2 "
fermion compounds, the neutron scattering spectrum can be described in terms

of two components: a quasielastic peak around zero energy transfer and a width

of order TK at T w 0, and an inelastic peak at an energy that coincides with the

characteristic energy of crystal-field excitations.

In addition to the quasielastic peak, the most recent measurements [87] dis-

played a single inelastic peak at an energy e -6.4 meV for T = 20 K (Fig. 3.9). With

the help of previous single-crystal magnetic susceptibility data [76], the authors cal-

culated the crystal-field parameters for CeAI3 and determined the ground state to

be F : I 3), followed by s8 : | ) at 6.1meV (T = 71K), and 17 :1 )

at 6.4 meV (T = 74 K). By comparing the parameters to those of other rare-earth

trialuminides with Ni3Sn structure, they concluded that the hybridization of Ce

4f electrons with the conduction band is the dominant contribution to the CEF

potential, as proposed by some theories of the Kondo effect in crystal fields [88].

Thus, the hybridization is responsible for both Kondo and CEF energy scales.

3.1.8 Chemical Substitution Studies

By far the most interesting doping studies on CeAls to date are those of La

impurities on the Ce sites. Recent specific heat studies of Cel_,LazAls, performed

after evidence for magnetic correlations was found for the pure compound [69, 84],

added to the already existing controversy about the nature of the anomalies in

CeA13. An enhancement of the anomaly in C/T was found for 0 < x < 0.2 [71],

and a corresponding peak appears in the specific heat, as seen in Fig. 3.10. The

magnetic susceptibility also shows an enhancement in its corresponding maximum,

with a temperature around 2.5 K for x = 0.2. A T3 dependence of the specific

heat below this maximum for the La-doped alloys led to the conclusion that the

anomalies represented the development of an antiferromagnetic transition. Two

reasons for this development were proposed. The first one is the application of a

negative chemical pressure by the larger La atoms and a subsequent decrease in

hybridization between f ions and conduction electrons. This effect is in accordance

with the Kondo necklace model (see Chapter 2). The second possibility is the

reduction of magnetic frustration in the basal-plane triangular lattice of Ce ions [86,

89]. As the Ce ions are substituted by non-magnetic La atoms in the triangular

sites, a number of the Ce moments are relieved from the frustration constraint and

are free to interact with others. This explanation relies on the assumption that the

in-plane interactions are much stronger than the interactions between two adjacent

planes.

More recent neutron scattering and pSR studies on CelxLazAl3 [15, 90] have

shown that the temperature at which the maximum in the specific heat for x = 0.2

develops coincides with both the appearance of an inelastic peak in the neutron

scattering function and the divergence of the uSR relaxation rate. The divergence

of the muon relaxation rate was interpreted as evidence for either short-range mag-

netic correlations, as found for pure CeAl3 [69, 84], or long-range magnetic order

of small moments. Bragg scattering on powdered samples did not show evidence

of long range order within the resolution of the measurement. The magnitude of

the Ce moments was estimated as < 0.05B.. The position of the inelastic peak for

x = 0.2 is weakly temperature-dependent, with an estimated energy of 0.54 meV

at T = 0. It was argued that the magnetic correlations in this sample were too

small to be responsible for the behavior of both the inelastic peak and the thermo-

dynamics below 2 K. In the search for an alternate explanation, the specific heat

and the inelastic peak were described in terms of the anisotropic Kondo model

(discussed in Chapter 2), which shows a similar response function and a maximum

in C/T for specific parameter values. This interpretation was not able to account

for the magnetic behavior inferred from the pSR results. Instead, the AKM proved

to be useful in providing an explanation for the anomalies in terms of a single-ion

mechanism, rather than cooperative behavior. Numerical results for the specific

heat of the AKM will be compared to specific heat measurements in magnetic field

of La-doped CeAI3 alloys in Chapter 6.

Only one study reports doping of CeAl3-based alloys on the Al ligand sites [61].

Corsepius et al. found that the alloys were single-phased for doping levels less than

x = 0.1, and that substitution of Ga, Si, and Ge contracts the lattice, while Sn

expands it. All of the above elements have the same effect on the specific heat

and the magnetic susceptibility. The anomaly in C/T for the pure compound is

shifted to higher temperatures, as much as 4.2 K for Ce(Alo.9Sno.1)3. A maximum

at a slightly higher temperature is also seen in the susceptibility between 0.1 and

70 kG. The maxima were attributed to the development of an antiferromagnetic

phase transition. All samples except those with Ga impurities exhibit discrepan-

cies between zero-field-cooled and field-cooled susceptibilities, and only those above

x = 0.1 show a time-dependent maximum (spin-glass-like). The development of an

apparent phase transition in the thermodynamic properties does not seem to be

exclusively related to an isotropic volume change of the hexagonal lattice, since

these features were seen in alloys with both smaller and larger lattice parameters

than those of CeAl3. Instead, the authors argued that the change in the tem-

perature position of the anomaly in C/T is related to the absolute-value change

(increase or decrease) in the c/a ratio.

Table 3.2: Cell Content of Cu3Au structure of CePb3 [64].
Atom Multiplicity Coordinates
(Wyckoff notation) x y z
Ce la 0 0 0
Pb 3c 1/2 1/2 0
1/2 0 1/2
0 1/2 1/2

3.2 Properties of CePb3

3.2.1 Crystal Structure

The compound CePb3 crystallizes in the face-centered cubic CuaAu struc-

ture, Pearson symbol cP4, space group Pm3m, number 221. The Ce sites cor-

respond to the corners of the cube, while the Pb atoms occupy the face-centered

positions. The structure forms directly from the melt at 11700C on the Ce-Pb

phase diagram [91]. Unlike CeAl3, there are no secondary phases that might af-

fect the physical properties and the formation of single crystals of this compound.

The lattice constant is a = 4.8760.002A [92], corresponding to a lattice volume

V = 115.93 A3.

Figure 3.11 shows the CuaAu unit cell of CePb3. The cell contains one

formula unit. The atomic coordinates with respect to the origin are given in units

of a in Table 3.2. In an fcc structure, the Ce atoms have 6 Ce nearest neighbors at

a distance equal to the lattice constant, and 12 Pb nearest neighbors at a distance

dcepb =a/f2 = 3.448 A.

3.2.2 Specific Heat

The low-temperature specific heat, plotted as C/T vs T2, is shown in Fig. 3.12.

It has a peak around 1.1 K due to an antiferromagnetic transition. The magnitude

of the peak is close to 3.5 J/K2 mol, and the extrapolated electronic coefficient 7

reaches a value around 1000 mJ/K2 mol. The effect of high magnetic fields was

4000 1
Ce La I
O A
E
w 3000 A A a K-0.05
U
XY A ic-. I

E_ K A *

2000

1000 ar
0 1 2 3 4 5 S 7

T (K)

Figure 3.10: Specific heat of Cel_,LaAls alloys (x = 0.05, 0.1, and 0.2) [71].

Ce

Pb

a a ab

Figure 3.11: Cubic Cu3Au structure of CePbs.

I 1% .e **3e
ap a s 1 I a
0 2o

1.0 20 40 60 80 2

0 5 10 15 20
T2 (K')

Figure 3.12: Specific heat plotted as C/T vs T2 of CePb3 between 0.6 and 4K.
The inset shows C/T vs T2 from 1.5 to 10 K [93].

F1--

first studied by Fortune et al. [94]. Magnetic fields between 10 and 20 T were found

to suppress the antiferromagnetic state and reduce the electronic coefficient.

Specific heat studies under pressure [95] revealed the existence of a pressure-

induced magnetic phase above 0.7 GPa. Below the critical pressure, the antiferro-

magnetic temperature TN is suppressed down to 0.6 K; above 0.7 GPa, the temper-

ature of this pressure-induced type-II antiferromagnetic phase increases from 0.6 K

to 1 K at 1.3 GPa. Figure 3.13 illustrates the temperature-pressure phase diagram,

with TN decreasing up to 0.7 GPa and increasing at higher pressures. This behav-

ior is rather unusual since a continuous decrease of TN with pressure is expected for

Kondo lattices, especially when the Kondo temperature TK is about three times

as large as the transition temperature, as in CePbs [96]. In addition, contrary to

other Ce Kondo lattices like CeCu6-sAua (x > 0.1) [97], and CeRu2Ge2 [45], no

pressure-induced suppression of TN to zero was observed for this compound.

3.2.3 Sound Velocity Measurements

The temperature dependence of elastic constants was determined from mea-

surements on a CePb3 single crystal along the (100) and (110) directions [98]. Fig-

ure 3.14 illustrates the magnetic field dependence of the relative change in velocity

of an elastic mode in the (110) direction at 10 MHz. Two phase boundaries (indi-

cated by arrows) can be distinguished at 0.38 K. The lower one signals the antifer-

romagnetic phase transition. The high-field boundary corresponds to an unknown

phase, possibly a spin-flop state [98]. The exact nature of this field-induced phase

remains to be determined by neutron diffraction experiments. Nevertheless, the

discovery of this field-induced transition in the (110) direction motivated further

investigation of the properties of CePbs single crystals in magnetic fields.

12,

.8
z
OWN

.6

.8 10
p (GPa)

12 U1

Figure 3.13: Transition temperature-pressure phase diagram for CePb3 up to
1.4 GPa [95]; the graph shows specific heat measurements (crosses), neutron scat-
tering (circles), and transport measurements (triangles). The broken line indicates
a crossover between two distinct magnetic phases (see text).

S--I I I II- I I"

r C*Pb3
Io
4-
-..
+ &.

'te
0 ++

iA-l
I
I t I '! I I .

Av 2
V

tK
UIK

0 5 10 15 20 (T)

Figure 3.14: Magnetic field dependence of the relative change in sound velocity for
the (cn c12)/2 elastic mode at 10 MHz [98].

-. 40
r yr/ 10.

H = 03 T 1 92 33
cp20 sT(K)

06 0 0 2 0 0 M I D
T (K)

Figure 3.15: Magnetic contribution to the electrical resistivity of CePb3 at H = 1 T
below room temperature. The inset shows the resistivity between 0.2 and 4 K at
H = 0.93T [93].

3.2.4 Transport Measurements

In order to measure the electrical resistivity of CePbs, it is important to

measure in magnetic fields of order 1 T in order to suppress the superconducting

transition due to the presence of Pb on the surface of the sample [93]. The reaction

of CePb3 with oxygen from air causes the separation of the two elements, eventually

followed by oxidation of Ce and Pb. Figure 3.15 displays the magnetic resistivity

between 0.2 and 4 K. It shows a logarithmic, Kondo-like increase from room tem-

perature down to 40 K, followed by two maxima, and finally by a drop below 2 K.

The maximum around 20 K has been attributed to the decrease in Kondo scatter-

ing due to a depopulation of the excited crystal-field levels [99]. The maximum

at 3.3 K is thought to be due to a coherence effect of the Kondo lattice. There is

also a rapid change in slope around 1 K, indicative of the antiferromagnetic phase

transition, as shown in the inset to the figure.

The pressure dependence of the magnetic resistivity was measured on a single

crystal [99]. There is a shift of the maximum at 3.3 K toward higher temperatures.

Only one broad maximum was detected for pressures above 11.5 kbar. This result

is consistent with an increase of the Kondo temperature TK. The magnetore-

sistance was recently measured along the (110) crystallographic direction [100].

Two field-induced anomalies were found for the magnetoresistance curves below

400mK at 5 and 9.5T, respectively (see Fig. 3.16). The resistivity increases up

to 5 T, decreasing sharply above the first transition, and becoming almost field-

independent after the second. A magnetic field-temperature phase diagram was

constructed, in good agreement with previous sound velocity measurements. The

angle dependence near the (110) direction was also measured in order to verify the

orientational dependence of the field-induced phase above 5 T, detected by sound

velocity measurements. A large increase in the magnetoresistance was observed as

the field direction was rotated toward the (100) direction, at which point the sharp

0 4 8 12 16
Magnetic Field (T)

Figure 3.16: Magnetoresistance curves between 1 and 16 T for temperatures in the
range 20 mK to 8 K. The magnetic field is along the (110) direction [100].

drop at 5 T could not be detected. The low-temperature resistivity was found to be

proportional to T2 with a field-dependent A coefficient. At 5 T, A reaches a max-

imum, the range of T2 dependence becomes smaller, and the resistivity acquires

a linear term, all coinciding with the field-induced transition. This enhancement

of A with field points to a corresponding enhancement of the specific heat coeffi-

cient y, as the ratio A/72 is expected to remain constant for heavy fermions [101].

At 10T, there is a small bump in the A coefficient, indicating a transition to a

ferromagnetically-polarized paramagnetic state [100].

3.2.5 Magnetic Susceptibility

Measurements of the magnetic susceptibility on a CePb3 polycrystal below

4K [102] revealed a maximum at 1.25 K, similar to that found for the specific

heat at 1.1 K. Figure 3.17 shows the data measured at 2.6kG. This maximum

is reminiscent of an antiferromagnetic phase transition, and coincides with the

appearance of a maximum in the specific heat at 1.1 K. The estimated value of

x(T = 0) is somewhere between 32 and 33 memu/mol. The inverse susceptibility
follows a Curie-Weiss behavior, and gives a high temperature effective moment

pe, = 2.5 p, and a Curie-Weiss temperature Ecw = -25K. An investigation of
the pressure dependence of the inverse susceptibility [99] found an increase of ,cw

from 0 to 15 kbar, a trend consistent with an increase of TK.

Recently, the ac susceptibility of a CePb3 single crystal was measured as

a function of crystallographic direction to verify the phase diagram and the field-

induced (presumably spin-flop) phase transition [103]. Their phase diagrams along

the (100) and (110) directions indicated that the range of the field-induced phase

depends on the crystallographic direction. Between 20 and 600 mK, with H ||

(100), the range is about 1T, while for H 11 (110), it is close to 5T. The phase

diagram determined from ac susceptibility data along (110) is in agreement with

previous studies, as shown in Fig. 3.18.

3.6

3.4,

S3.2

- 3.0
X

2 .-
0 1 2 3
T (K)

Figure 3.17: Magnetic susceptibility of a CePb3 polycrystal below 4K at H =
2.6kG [102].

'.%

H = 2.6 kG '-e

_

62

-- -- -- -- -----S
12- RnaWetic Stat

10 ----.. *

.A

A o
Spim-'lop phase u

6n- a '\ -

2- '

00 02 0.4 0.6 0.8 10 1 2

TeapeaBs(E)

Figure 3.18: Phase diagram (H T) for CePb3, with the field along the (110)
direction (Solid circles: ac susceptibility [103], open circles: sound velocity [98],
and open triangles: magnetoresistance [100]).

3.2.6 Neutron Scattering

Neutron scattering studies are essential in the determination of the ordered
moment at low temperatures and the crystal-field parameters of heavy-fermion
systems. The Cu3Au cubic structure of CePb3 provides a high degree of crystal
symmetry. In the cubic environment of Ce3+ ions in CePb3, the crystal-field (CEF)
potential splits the IJ = ) multiple into a F7 doublet and a F8 quartet [104]:
|r7) = al + ) )

) (3.1)

I[s) =
b ) + al F 3)

where a = (1)1/2 and b = (Q)1/2
The magnetic scattering function of polycrystalline CePb3 is shown in Fig. 3.19,
which shows the inelastic, quasielastic, and elastic peaks. A fit to the scattering
function [105] determined that the ground state is the TF doublet. The CEF
splitting between the doublet and the first excited state is around 72 K [106].
Bragg scattering studies on a single crystal led to the conclusion that the magnetic
structure of CePb3 is antiferromagnetic, and that the moments are aligned along
the (100) direction [106]. The magnetism is incommensurate, with a modula-
tion amplitude of 0.55P, at 30 mK. A similar incommensurate structure has also
been detected for CeA12 [107], another cubic heavy-fermion compound. Vettier et
al. [106] concluded from a comparative study of Ce Kondo lattices that cubic com-
pounds are more magnetic than those with a large crystal anisotropy, like CeAl3,
CeCu6, and CeCu2Si2. This statement has important implications regarding a
possible role of crystalline anisotropy in regulating the competition between the
Kondo and RKKY energy scales.

64

4, H Ce Pb3 ..
&%4K
30 Q=.9A' */

U, / I *
ar .
E 0- alai

-2 0 2 4 6 8 1
E meV]

Figure 3.19: Magnetic neutron scattering function of a CePb3 polycrystal [105].
The solid line is a fit to the data. The dashed line represents the determined
quasielastic component, and the dash-dotted line corresponds to the inelastic com-
ponent.

3.2.7 Chemical Substitution Studies

Alloying studies on the Ce sites of CePb3 were first reported using La [96].

These studies are particularly important and have fundamental significance, because

they constitute evidence of single-impurity effects in a concentrated heavy-fermion

system. The specific heat, magnetic susceptibility, and electrical resistivity all

scale with Ce concentration. Electrical resistivity measurements revealed that the

crystal-field splitting is also unaffected by La doping. The electronic specific heat

data for alloys with La x = 0.4, 0.6, and 0.96 are shown in Fig. 3.20, along with the

theoretical prediction for S = 1. The Kondo temperature is constant throughout

the series, implying a constant value of J. The transition temperature TN goes to

zero near a La concentration x = 0.2. The suppression of magnetism as a result

of a lattice expansion upon La substitution seems to indicate that the decrease

in TRKKY with respect to TK is due to an increase in the average Ce-Ce distance,

rather than to an overall change in J. Indeed, Cel-.LaxPb3 is a unique system in

the sense that TK and the coupling J seem to remain unaffected by La doping.

While thermodynamic and transport properties of Ce-lLaPb3 seem to be

unaffected by the electronic environment surrounding the Ce3+ ions, experiments

on Cel_-MPb3 (M = Y, Th) [109] confirmed that the single-impurity scaling

observed by La doping on the Ce sites is the exception rather than the rule. Instead,

a rather unusual behavior is observed upon either Y or Th doping. The magnetic

susceptibility at 1.8 K increases with Y concentration. The Kondo susceptibility

is inversely proportional to TK, so this result implies an unusual decrease of the

Kondo temperature as the lattice contracts (increasing J). Substitution of Th on

the Ce sites also contracts the lattice, and at the same time leads to magnetic-like

anomalies in both specific heat and susceptibility for x = 0.3, 0.5. The differences

in the outer electronic structure between Ce, Y, and Th seem to play an important

role in the evolution of the ground state properties of Ce1-=MPb3.

66

2.0
A X=0.4
B X-O.6
X0.96

TK =3.3K

T/TK

Figure 3.20: Electronic specific heat vs T/TK for Cei_,LaaPb3 alloys, x = 0.4, 0.6,
and 0.96 [96]. The data are in good agreement with the prediction from the spin-!
Kondo specific heat [108]. The only adjustable parameter is TK = 3.3 K.

67

Chemical substitution studies were also performed on both f-ion and ligand

sites of the CePb3 structure. In Ce(Pbl-x_.hM)3 studies with M = TI, In, and

Sn [110, 111], the antiferromagnetic transition temperature decreased toward zero

for a Sn concentration x = 0.4, and increased for both TI and In. For the latter

two dopants, there is a maximum towards the center of the TN x phase diagram.

Substitution of Sn for Pb on the ligand sites suppresses TN and greatly increases

the Kondo temperature [112, 113].

CHAPTER 4
MOTIVATION

This chapter begins with a discussion on the importance of the study of CeAla

and CePbs, followed by a presentation of the objectives of the current study.

4.1 Importance of CeAla and CePb3

Both CeA13 and CePb3 are canonical, well-documented heavy-fermion sys-

tems, with values of the 7 coefficient surpassing 1 J/K2 mol, crystal-field doublet

ground states, and a low temperature resistivity characteristic of Kondo lattices.

Studies on these compounds over the last 25 years made a substantial contribution

to the standard interpretation of heavy-fermion systems, based on the Kondo effect

and Fermi-liquid theory. However, deviations from this standard model have

been observed in these and other compounds through the coexistence of mag-

netic order and heavy electrons, the presence of unaccountable anomalies in the

thermodynamic properties, and non-Fermi-liquid effects. These are all topics of

current interest, yet they are among the least understood aspects of heavy-fermion

physics. Any information obtained from the study of the above two compounds

might be utilized in the development of new interpretations for the heavy-fermion

state. The current work will concentrate on the coexistence of heavy fermions

and magnetic order in CePb3, the nature of the anomaly seen in the specific heat

(plotted as C/T) of CeAl3, and the heavy-fermion behavior of both compounds in

magnetic fields.

In 1975, specific heat and electrical resistivity measurements below 100 mK

by Andres, Graebner, and Ott led to the discovery of CeA13 as the first heavy-

fermion compound [20]. Despite its significance in the field of strongly-correlated

electron systems, CeA13 is probably one of the least understood among these com-

pounds. Ever since its discovery, it has been considered a canonical, nonmagnetic

heavy-fermion system. Yet later experimental results (see Chapter 3) challenged

its nonmagnetic status, and pointed to a possible magnetically-ordered ground

state for CeAla. Whether the ground state in this compound is magnetic or not

has been a long-standing debate, and remains an important topic in the study of

heavy-fermion systems.

The compound CePb3 ranks among the most extensively studied magnetic

Kondo lattices. The magnetic transition has little effect in reducing the large value

of the electronic specific heat coefficient, -y 1000mJ/K2 mol. The electrical re-

sistivity has a large T2 coefficient, A = 45 pf cm/K2, and the ratio A/y2 is around

4 x 10-5 'cm K2 mol2/J2. When taking into account the relatively large value of

-y for this compound, the above suggests that the ground state is some superposi-

tion of ordered local moments and heavy electrons. Very little is known about the

nature of the magnetic ground state of heavy-fermion materials. Measurements of

thermodynamic properties of paramagnetic and magnetic states in this compound

may be useful to understand the coexistence of magnetic order and heavy electrons.

Another important characteristic of CePb3 is the observation of single-ion

scaling of thermodynamic and transport properties in a concentrated 4f system.

The study of Cel_-LaPb3 by Lin et al. [96] revealed that the normal state of

alloys over the range (0 < x < 1) can be described in terms of a single-ion picture

(see Chapter 3). It is the only Ce heavy-fermion system to date exhibiting such
behavior. The reason why such a concentrated system can exist with apparently

noninteracting 4f sites remains unclear.

4.2 Objectives

4.2.1 Magnetism and Heavy-Fermion Behavior in Ce Kondo Lattices

The studies on CeA3l and CePb3 alloys presented in this dissertation are

motivated by a fundamentally important topic in heavy-fermion research: the

need for a full understanding of the interdependence between magnetic correlations

and/or magnetic order and the heavy-fermion state. The ground state of rare-earth

intermetallics is generally described in terms of the competition between two energy

scales, TK and TRKKY, discussed in Chapter 2. The former represents a single-ion

effect due to the local Kondo interaction between conduction electrons and the f

orbital. The latter portrays a collective effect due to indirect exchange interactions

between ionic spins. The schematics of this delicate balance were shown in Fig. 2.7.

For TRKKY > TK, magnetic order occurs and the moments are unquenched at zero

temperature. The size of the moments is close to that corresponding to the crystal-

field ground state. Concentrated Kondo systems falling into this category have

relatively low values of 7, of order 100mJ/K2mol (e.g., CeCu2 and CeAl2 [6]).

Whenever TK > TRKKY, the Kondo effect develops without magnetic order. This

regime corresponds to most nonmagnetic Kondo lattices, with Kondo temperatures

larger than 10 K. For TK TRKKY, the formation of heavy electrons occurs, with 7

values in excess of several hundred mJ/K2 mol. This is the least understood area

of the Doniach phase diagram. The applicability of this model to heavy-fermion

Kondo lattices, in particular to CeA3l alloys, will be discussed as part of a study

on the anomaly present in this system.

Two empirical correlations have been postulated in order to distinguish between

magnetic and nonmagnetic heavy-fermion ground states: the Wilson ratio R and

the Kadowaki-Woods ratio. The experimental Wilson ratio R [5] is defined as

7r2kBXo/p -7, where Xo is the zero-temperature susceptibility and p.A is the effective

moment at room temperature. Values of R are usually much larger for magnetically-

ordered than for nonmagnetic Kondo lattices [5]. Nevertheless, the experimental

ratios for CeAla and CePb3 are both around 0.7, a value within the range corre-

sponding to nonmagnetic heavy fermions. Thus, this ratio does not seem to account

for the magnetic order observed in CePb3, as well as for a possible magnetic order

in CeAl3.

In most heavy-fermion compounds, the empirical relation A/72 lies somewhat

close to the Kadowaki-Woods ratio A/72 = 1 x 10-5 2 cmK2 mo2/J2 [101]. This

ratio is about an order of magnitude larger than that corresponding to transition-

metal alloys. The magnetic field dependence of this relation has not been exten-

sively studied. The ratio A/72 has been observed to remain constant with field in

nonmagnetic CeCus.gAuo.1 [114], the only published study of the field dependence

of this ratio. In order to verify whether A/>2 remains the same for both param-

agnetic and ordered states, it would be of interest to explore the field dependence

of this ratio in a magnetically-ordered heavy-fermion system.

Previous thermodynamic and transport measurements on Ce0.6Lao.4Pb3 [96]

suggested a single-ion mechanism for the heavy-fermion behavior in this system. A

study of the specific heat in magnetic field of Ceo.6Lao.4Pb3, a nonmagnetic coun-

terpart of CePb3, was conducted in this dissertation to search for further evidence

of a single-ion Kondo origin for the heavy-fermion state in Ce-based systems.

4.2.2 Ground State of CeAl3

The experiments on CeAls alloys presented in this dissertation are motivated

by the existing controversy about the ground state of CeA13. The nature of the

anomalies in the thermodynamic properties of CeAl3 systems below 1 K is not well

understood. It is a major topic of interest in the field of strongly-correlated electron

systems. There are at least three competing interpretations for the origin of these

anomalies. One explanation is that the weak maxima seen in C/T and in the

magnetic susceptibility between 0.3 and 0.5 K is due to a reduction in the density

of states caused by the formation of coherent states in the Kondo lattice [68].

Another interpretation argues for an unconventional ground state in which heavy

electrons coexist with either magnetic correlations or magnetic order. There is

now enough evidence [61, 70, 69, 71] for the existence of magentic correlations

below 1 K in CeAl3 through NMR and pSR studies, casting serious doubt on the

so-called coherence interpretation [68]. However, it is not clear at the present time

whether the magnetic correlations are short-ranged, frustrated, or whether they

lead to long range order. The third and most recent interpretation suggests that the

anisotropic Kondo model provides an alternative explanation to the ground state

properties, as driven by single-ion dynamics, and dependent on the anisotropy of

the Kondo interaction [15, 90]. Under this point of view, the question remains of

how to reconcile the presence of magnetic correlations in CeAl3 with a single-ion

Kondo description of its thermodynamic features.

CHAPTER 5
EXPERIMENTAL METHODS

5.1 Sample Preparation

5.1.1 Synthesis

Alloys used in this dissertation were synthesized by melting its respective

constituents in an Edmund-Biihler arc furnace under a high-purity argon atmo-

sphere. The arc-melting apparatus consisted of a stainless-steel vacuum chamber

with a water-cooled copper crucible at the bottom and a hydraulic mechanism sup-

porting an electrode at the top. The tip of the electrode is made out of a tungsten

alloy, and it is capable of carrying well over 100 A of current.

Prior to melting, each of the consituent elements was carefully cleaned to

eliminate any oxide layer on the surface, and later weighed to an accuracy of

0.03 mg. Their molecular weights and stoichiometric ratios were used to calcu-

late the appropriate relative masses. The total mass of an average sample was

about 500 mg, and the diameter of a sample bead ranged between 0.5 and 1 cm.

The Cu hearth on the arc-melter was thoroughly cleaned to avoid the presence

of unwanted impurities during sample preparation. The element with the high-

est vapor pressure was placed on the Cu crucible below those with lower vapor

pressures. This procedure minimizes direct contact between the Ar arc and the

material with highest vapor pressure, therefore reducing its mass loss, and mini-

mizing the discrepancy between predicted and actual stoichiometries for the alloy

being synthesized. The chamber was then pumped and subsequently flushed with

high-purity Ar. After this procedure was repeated three to four times, the cham-

Weight Percent Cerium
so go

Al Atomic Percent Cerium Ce

Figure 5.1: Phase diagram of Ce-Al [91].

ber was filled to 0.5 atm of Ar gas. In order to avoid the unwanted presence of

oxygen and water vapor, two measures were taken. First, the high-purity Ar goes

through a purifier before entering the arc-furnace. Second, a zirconium bead is

placed inside the furnace and melted before sample synthesis. Zirconium is known

for its high absorbing capacity for oxygen.

At the start of the melting process, a relatively low current was sent through

the tungsten electrode. The arc was moved slowly towards the elements to avoid

any thermal stresses and motion or splashing of material due to the arc pressure.

During melting, enough time was allowed for the liquid components to mix via arc

pressure. To ensure homogeneity, the above process was repeated several times and

the sample bead was turned over after each melt. The mass loss during melting

was obtained as a percentage difference (typically < 0.1 0.3%) between the total

masses before and after sample synthesis.

Alloys of CeAl3

Alloys of Cel_-,MAl3 (M = La, Y) were synthesized using the purest avail-

able materials: cerium and lanthanum from Ames Laboratory, and Johnson Matthey

(AESAR) aluminum (99.999% purity). The weighing of constituents required spe-

cial attention due to the sensitivity of the crystal structure of CeAl3 to small

changes in the relative concentration of Ce and Al atoms. The synthesis of CeAl3

alloys is always accompanied by the formation of a large amount of the secondary

phases CeAl2 and Ce3Aln1. The presence of these unwanted phases is substantially

reduced by proper annealing conditions.

The cerium-aluminum phase diagram has been studied by several groups [91],

its latest addition being CeA13 [62]. It contains four other compounds: Ce3A11,

CeAl2, CeA1, and Ce3Al (see Fig. 5.1). Both CeA12 and Ce3Al form directly

from the liquid solution, CeAl and Ce3A111 form peritectically, and CeA13 forms

peritectoidally at 1135C. A peritectic reaction is one in which the compound

melts incongruently [115], that is, the composition of the liquid just above the

melting point has a different composition than the solid before melting. Only part

of the solid forms a liquid solution, with the remaining part forming crystallites

floating around in the liquid. As the temperature reaches the melting point, the

mixture solidifies into a single phase. The peritectoid reaction in CeAl3 is similar

to a peritectic reaction, except that the compound does not melt into a liquid-

crystallite mixture. Rather, it separates into a solid phase mixture of CeA12 and

P-Ce3Al11, which in turn melts into CeAl2 crystallites embedded in a liquid solution

matrix.

The transformation of a mixture of Ce-Al neighboring phases into the CeA3l

phase upon cooling has a marked effect on the way samples crystallize. The pres-

ence of secondary phases is the cause of many sample dependence of thermo-

dynamic and transport measurements. Polycrystals synthesized by arc melting

consist of a mixture of CeAl3 with large amounts of CeAl2 and Ce3A111. Anneal-

ing has been found to reduce the proportion of secondary phases to the point of

becoming undetectable by conventional x-ray diffraction methods. Magnetic sus-

ceptibility measurements on annealed samples are an efficient way of detecting the

above second phases, since CeA12 is antiferromagnetic below 3.8 K, and Ce3Al is

ferromagnetic with transitions at 3.2 and 6.2 K [116]. Specific heat data has also

been used successfully by some groups to detect irregularities at these tempera-

tures.

Alloys of CePb3

Lanthanum-doped CePb3 alloys were made using Ames Laboratory Ce and

La, and Johnson Matthey Pb with 99.9999% purity. Special care was also taken

in the making of both CePb3 and Ce0.6Lao.4Pb3 due to the large vapor pressure of

lead. Therefore, Ce should be melted first, then Pb. Unfortunately, this procedure

was not enough to significantly reduce Pb mass loss due to vapor pressure at

0.5 atm of Ar gas. In order to compensate for this mass loss, an additional 3%

of the calculated mass for Pb was added to the constituents before the first melt.

The mass loss for each bead after melting was mostly due to lead, usually around

3%. The sample was remelted in case the mass loss was less than the extra amount

of Pb. Correspondingly, more Pb was added in the event that the mass loss was

greater than expected. After melting the sample, the stoichiometry was verified

by recalculating the atomic percentages based on the final mass of the sample.

CePb3-based alloys are generally free of any secondary phases except pure Pb,

which can precipitate in the surface as the alloys react with air. As a result, the

samples were kept in a vacuum container along with Drierite acting as a moisture

absorber.

5.1.2 Annealing

Annealing helps relieve stresses inside the samples not removed during crys-

tallization. It also reduces the amount of unwanted secondary phases in the final

melt. Typical annealing temperatures range between 2/3 and 3/4 of the melting

point of the alloy.

The final beads were broken into smaller pieces using a ceramic mortar instead

of a metal crusher to avoid the presence of iron impurities in the samples. Part of

each original bead was wrapped in a clean tantalum foil and placed inside a quartz

tube. The tubes were pumped and flushed with Ar gas several times. Right before

sealing, the Ar pressure inside was reduced to 100 mtorr. The quartz tubes were

then placed inside a Lindberg furnace and annealed according to a previously

tested prescription. Alloys of Cel_-LazAl3 were annealed at 830C for two weeks,

while those of Ce-l,YAl3 were annealed at 800C for two weeks, then 850C for

five days. Both CePbs and Ceo.6Lao.4Pbs were annealed at 800C for one week. In

all cases, annealing started with the furnace already at annealing temperature. At

the end of the prescribed annealing period, the samples were immediately removed

from the furnace and left to cool down at ambient temperature.

5.2 Diffraction of X-Rays

Measurements of x-ray diffraction were used as a means to verify whether

the arc melting and annealing processes led to the formation of the desired crystal

structure. From the diffraction pattern, it was also possible to determine the lattice

parameters and the presence of secondary phases in the sample. The principle

behind the diffraction of x-rays in crystals is based on Bragg's Law:

A = 2dsin (5.1)

which for a first order (n = 1) spectrum relates the known Cu Ka wavelength to the

diffraction angle 0 and the distance between lattice planes d. The lattice constants

are then calculated from d and the intersection points of the lattice planes for the

desired space group number, given in terms of the Miller indices (h k 1).

The experimental setup consisted of a Phillips APD 3720 diffractometer, an

x-ray source with a water-cooled power supply, and a computer for data acquisition.

The APD 3720 consists primarily of x-ray beam slits, the sample holder, and

an electronic counter. Both the counter and the sample holder rotate about a

horizontal axis so that the angle of rotation of the counter is always twice that of

the holder. This latter angle corresponds to the angle of incidence/reflection from

the sample plane 0. The x-ray beam is of known wavelength: a Cu Ka line with

A = 1.540562 A.

Powder samples were ground out of annealed pieces from the original beads

using a ceramic mortar. About 1 cm2 of powder was then glued to a glass slide using

a 7:1 amyl acetate collodion mixture. With the slide in place, the diffractometer

power supply was set to 40 kV and 20 mA. The detector angular speed was set

to 6/min, and its range to 50 < 20 < 1200. The counting rate was set to 1000

counts/sec. All measurements were performed at room temperature.

The angular positions of the resulting intensities were compared to the the-

oretical positions and reflection indices obtained from a structure-generating soft-

ware. This procedure allows for identification of secondary-phase intensity lines

larger than the background intensity (~ 5% of maximum intensity line). For a

cubic system (i. e. CePbs alloys), the indices for primary-phase lines are obtained

from the following equation [117]:
A2
sin 8 = -(h2 + ki + 12). (5.2)
4a2
Similarly, for a hexagonal system (CeA13 alloys),

2 =h24 (h2 + k2 12) (5.3)2
sin2 0 = + ( 5 .3)
4 3 a2 C2(3
The indices (h k 1) and the angles 20 for the highest and narrowest intensity lines

were entered as data points into a least-squares fitting program, along with the

wavelength and structure type. The room-temperature lattice parameters and

their uncertainties were then obtained from a least-squares fit using one of the

above two equations, depending on the structure type of the sample.

5.3 Magnetic Measurements

All magnetization and magnetic susceptibility measurements were conducted

using a Quantum Design Magnetic Property Measurement System (MPMS) SQUID

magnetometer. The apparatus consisted of a liquid He dewar, the sample probe

assembly, the electronic console with temperature and gas controllers, the He gas

handling system, and a Hewlett Packard computer. The probe assembly is inserted

inside the dewar; it contains the sample space, thermometers, the sample heater,

an impedance controlling He flow, a superconducting magnet producing fields up

to 5.5 T, and the sample transport mechanism. The temperature is regulated by

the flow of He gas through the sample space and by the sample heater. Below

approximately 4.2 K, the liquid-helium vapor inside a pot is pumped in order to

reach temperatures down to 2 K.

The technique used for magnetization measurements on the MPMS detects

the change in flux induced by the sample under an applied field using a super-

conducting quantum interference device (SQUID) amplifier. The sample is first

enclosed in a 0.5 cm-long plastic straw segment, which is slid into a drinking straw

at the end of the support tube, serving as the sample holder. During each mea-

surement, the sample is moved upward along the axis of a series of pick-up coils

connected to the SQUID. The SQUID voltage is read at different position intervals

across the scan length. This voltage is proportional to the change in flux detected

by the coils, which in turn is proportional to the magnetization of the sample.

The accuracy of magnetization measurements is generally around 3%, while the

precision at a fixed temperature can be as low as 0.01%.

Magnetization curves as a function of magnetic field can also be obtained by

measuring at the lowest temperature (2 K) and measuring at each field, sweeping

the field from 0 to 5 T. The magnetization (in emu/mol) is obtained by multiplying

the signal by the molecular weight of the sample and dividing by its mass. The

magnetic susceptibility X = M/H (in memu/mol) is calculated from the signal

measured at a fixed field (typically 1 kG), multiplied by the molecular weight of the

alloy, and divided by its mass and the applied field. Each measurement sequence is

fully automated, and uses a version of the MPMS software from Quantum Design.

The convention used for units of magnetization and magnetic susceptibility in this

dissertation follows from the literature on heavy-fermion systems (e.g., Refs. [5]

and [6]).

brass can
pumping line

pins

platform

wim sample

block
thermometer

Cu block

S- brass can

Cu ring

Figure 5.2: View of the cryostat used for zero-field specific heat measurements
between 1 and 10 K.

5.4 Specific Heat Measurements

This section will discuss the necessary cryogenic and electrical equipment to

measure specific heat of small samples (< 100mg) with large heat capacity, and

the thermal relaxation method [118, 119, 120] used for this purpose.

5.4.1 Equipment

Electronic

The experimental setup for the measurement of specific heat in both zero

and magnetic fields by the thermal relaxation method consisted of three cryostats,

a liquid-He dewar, two Keithley 220 and a Keithley 224 programmable current

sources, a Keithley 195A, 196 digital multimeter for thermometer voltage measure-

ments, an EG&G Model 124A lock-in amplifier for platform thermometer current

detection, a variable decade resistor and a resistance box with three internal resis-

tances. The resistance box is connected to the decade resistor in a Wheatstone

bridge configuration. A more detailed explanation of the equipment is provided

elsewhere [118, 119, 120, 121]. A Dell PC was used for data acquisition and anal-

ysis. The computer was interfaced to the digital equipment using an AT-TNT

Plug and Play GPIB board from National Instruments. A 12-bit resolution Keith-

ley Metrabyte DAS-1402 A/D converter board interfaced the PC to the lock-in

amplifier. The data acquisition was monitored using two PC-based programs for

thermal conductance and specific heat measurements, respectively. The software

was designed by the author using LabVIEWT version 5.1 for Windows 95/98.

Cryogenic

The cryostats used for zero-field measurements are illustrated in Figs. 5.2

and 5.3. Figure 5.2 shows the probe used in the temperature range 1-10 K. The

electrical connections are enclosed by a brass can attached to a taper joint by

pumping on the enclosure. The cooldown procedure consisted of precooling in

liquid nitrogen for about 15 to 60 minutes, insertion into a dewar, and subsequent

transfer of liquid He into the dewar, which reduces the temperature to 4.2 K. A

temperature of 1 K was achieved by pumping the He vapor out of the dewar/probe

assembly for about an hour.

Measurements in the range 0.4-2 K were conducted using the cryostat described

in Fig. 5.3. After reaching a temperature of 4.2 K following the procedure above,

the 'He pot was filled with liquid He from the bath by opening the needle valve,

and 3He gas was transferred into the 3He pot. The needle valve was then closed,

and the He pot was pumped out to reach a temperature between 1 and 2 K. Al-

though this temperature can be sustained for many hours, the 4He pot can be

easily refilled if necessary. In order to reach a temperature of 0.4 K, the following

method was used. A Cu container full of activated charcoal resides at the lower

end of a rod inside the 3He-gas enclosure. At 1 K, the 3He gas condenses inside.

As the charcoal container is lowered towards the 3He pot, the condensed 3He is

attracted to the charcoal, which acts as an adsorption pump. Temperatures below

1 K could be achieved in 20 minutes and sustained up to several hours with this

technique. Once the charcoal saturates with 3He, it was warmed up to release the

gas and the above process was repeated.

Specific heat measurements in magnetic field were conducted in a specially-

designed dewar from Cryogenic Consultants Limited (CCL). The additional elec-

tronic equipment consisted of a GenRad 1689M RLC DigiBridge, used to measure

the capacitance of a thermometer used above 1 K, a CCL superconducting magnet

and a magnet power supply. The magnet is made of two inner coil sections of

niobium-tin wire and two outer coil sections of niobium-titanium wire. The cryo-

stat used below 1 K is the same as in Fig. 5.3, and the one used between 1-10 K

is illustrated in Fig. 5.4. The main difference between them is the lack of a 3He

enclosure for the higher-temperature probe.

He4 pot
pumping line

- brass can
pumping line

He3 pot
pumping line

He3 pot

platform
with sample

Cu ring

I_ --

I
I
I
I
I
I
I

I
I basa
I
SI
I
I
I

I

Figure 5.3: View of the 3He inner pot cryostat used in both zero and magnetic

field specific heat measurements between 0.4 and 2 K.

needle
valve

capillary

brass can
pumping line

He4 pot
neede pumping line
needle
valve
heat sink

capillary _, lI

brass can pins

He4 pot

in magnetic fields at temperatures between 2 and 10 K.
pins ***** ,
** ** -,-- Cu block
platform : i *****
with sample l_ J P .

Cu ring IA block
S thermometer

Figure 5.4: View of the 4He inner pot cryostat used for specific heat measurements
in magnetic fields at temperatures between 2 and 10 K.

All cryostats have a similar electronic design. They are equipped with radiation

shields from top to bottom, and the wires are coupled to the He bath by a heat

sink, as shown in Figs. 5.3, and 5.4. Additional wires are soldered from the heat

sink to the Cu block, and wrapped around the 4He pot to ensure thermal equi-

librium. The temperature of the block is regulated by a heater made of wrapped

manganin wire. It is monitored by a Lake Shore calibrated Ge thermometer in

the range 1-10 K, and by a Speer carbon resistor between 0.4 and 2K. In mag-

netic fields, a Lake Shore capacitance thermometer was used above 1 K due to its

negligible field dependence, and the Speer resistor was used from 0.4-2 K for its

known magnetoresistance [122]. All thermometers are linked to the block using

thermally-conductive Wakefield grease.

Sample Platform

The sample resides at the bottom of the cryostat, attached to a sapphire

platform by Wakefield grease. A flat surface at the bottom of the sample is impor-

tant in order to establish optimum thermal contact between platform and sample.

The platform is thermally linked to a copper ring, as shown in Fig. 5.5. Two types

of platforms were used in this study. Each platform has four wires soldered to

silver pads attached to the ring by thermally-conductive Stycast. The two pairs of

wires are connected to the platform heater and thermometer, respectively, using

EpoTek H31LV silver epoxy. The platform heater is an evaporated layer of 7%Ti-

Cr alloy. For measurements between 1-10 K, the platform thermometer used was

an elongated piece of doped Ge, and the platform wires were made of a Au-7%Cu

alloy. A thin piece of Speer carbon resistor and Pt-10%Rh platform wires were

used for measurements between 0.4 and 2 K.

87

Stycast

Silver pad
Sapphire disc Evaporated
7%Ti-Cr
heater

Wires:
H31LV < Au-7%Cu (T>1K)
Silver epoxy Pt-10%Rh (T<2K)

Thermometer:
Ge (T>l K)

Solder:
Cu-Sn-Cd (Au-7%Cu wires)
Pb-Sn (Pt-10%Rh wires)
Cu ring

Figure 5.5: Top view of the sample-platform/Cu-ring assembly at the bottom of
the cryostat.

5.4.2 Thermal Relaxation Method

A thermal relaxation technique consists of calculating the time constant of

the temperature decay of the sample linked to a heat bath by a small thermal

resistance [118, 119, 120]. The electrical analog of the system is that of an RC

circuit, where the time constant is proportional to the capacitance. When heat is

applied to the platform-sample system by means of a small current (in pA), the

temperature increases from a base value To by an amount AT. When the current

is turned off, the system temperature T(t) decays exponentially to To:

T(t) = To + A Te-'7t". (5.4)

The time constant '-1 is proportional to the total heat capacity (sample plus plat-

form) C,.ot,:

total
Ti = -, (5.5)

where n is the thermal conductance of the wires linking both platform and sample

at T = To + A T, and the Cu ring at T = To. The time constant was obtained

by measuring the time decay of the off-null voltage signal from a Wheatstone

bridge using a lock-in amplifier. Two arms of the Wheatstone bridge consisted

of a resistance box and the platform thermometer. By adjusting the resistance of

the box it is possible to balance the bridge and obtain the platform thermometer

resistance. The platform temperature is extracted from a previous calibration of

the platform thermometer. The accuracy of the time constant measurement in the

temperature range 0.4-10 K is 1-3%. The thermal conductance is given by

P = (5.6)
AT
Here, P = IV is the power applied to the platform heater. The above equations

are valid under the assumption of an ideal thermal contact (Kp, .-~ oo) between

sample and platform. In the event of a poor thermal contact between the sample

and the sapphire (K,.,,,pie~ ) the temperature decay can generally be described as

the sum of two exponentials

T(t) = To + Ae-'/'r + Be-'/n, (5.7)

where A and B are measurement parameters and r2 is the time constant between

sample and platform temperatures. The total heat capacity can be calculated

from T1, -2, and K. The thermal conductance is measured separately by applying a

current to the platform heater, calculating the power P = IV, and calculating A T

as a result of the power applied to the heater. The accuracy of this measurement

between 0.4-10K is 5%. The sample heat capacity is calculated by subtracting

the heat capacity of the addenda (sapphire platform, wires, silver epoxy, platform

thermometer, and thermal grease) from the total heat capacity. Finally, the specific

heat is obtained by multiplying by the molecular weight and dividing by the sample

mass.

5.5 Experimental Probes

In order to accomplish the objectives discussed in the previous chapter, two

mechanisms for the study of thermodynamic properties were used in this disser-

tation: alloying and magnetic fields. Alloying is a powerful tool that allows for

changes in the electronic structure, the lattice constants, and the properties of a

system. Magnetic fields allow to probe the energy scales relevant to heavy-fermion

systems at low temperatures and test their thermodynamic properties against the-

oretical predictions.

The two main types of doping on heavy-fermion compounds are Kondo-hole

and ligand-site doping. The first one consists of replacing the magnetic ion by a

nonmagnetic counterpart (e.g., La or Y instead of Ce). In this method, there is a

reduction of the number of magnetic moments in the sample and some disorder in

their electronic environment. In addition, the lattice structure changes significantly

due to an atomic size difference between the f ion and the dopant ion. Doping with

La usually leads to a lattice volume expansion, while Y substitution corresponds

to the application of a positive chemical pressure. Ligand-site doping consists of

substituting the ligand atoms of one species by another. The main effect here is

a dramatic change in the electronic environment of the magnetic ions, changing

the value of the local exchange constants. Maximum atomic disorder is introduced

using this method, which could complicate the analysis of properties. It is of

current interest to investigate the extent to which each method of doping affects

the electronic properties.

The measurement of thermodynamic properties as a function of applied mag-

netic field is an important, though not often implemented tool in the study of heavy

fermions. The relevant energy scales, both single-site and cooperative, are small

enough that magnetic fields easily accessible in a laboratory can help determine

their overall magnitude and their role in determining physical properties. The mag-

netic behavior of heavy-fermion compounds ranges from short-range correlations to

non-Fermi-liquid behavior to long-range antiferromagnetic order. Magnetic fields

are useful in understanding the different types of magnetic behavior through a

comparative study of changes in the density of states, the entropy, the specific

heat, and the magnetic characteristic temperature. Various theoretical models,

including the single-impurity Kondo description, have different predictions for the

magnetic field response of thermodynamic properties. Therefore, the use of mag-

netic fields as an external parameter is a convenient way of testing the applicability

of these models. Specific heat measurements in magnetic field on CePb3 and CeAls

alloys will be presented in this dissertation in order to study the trends followed

by parameters relevant to both Kondo and magnetic degrees of freedom in these

systems.

5.5.1 Experiments on CeAla

A doping study of the lattice parameters, specific heat, and magnetic sus-

ceptibility of Cel_-,MA13 alloys has been conducted, with M = La concentrations

0 < x < 1, and M = Y concentrations 0 < x < 0.2. The evolution of the lat-

tice parameters and their ratio c/a with La/Y concentration x was investigated

to determine how the relative variation of a with respect to c and changes in the

lattice volume are related to trends in the thermodynamic properties. In addition,

the specific heat, the anomaly in C/T, the magnetic susceptibility, and the Wilson

ratio expressed as /7y of Celx.LaAl3 were studied over the whole concentration

range to search for evidence for a magnetic origin of the anomaly in this system

by comparing the concentration dependence of TK and the temperature Tm of the

anomaly in C/T, with their dependence on the parameter J based on Doniach's

Kondo necklace model. The coupling J is proportional to the hybridization, which

is expected to decrease with La concentration (expansion of the lattice).

The specific heat of Ceo.sLao.2A13 and Ceo.3Lao.7Al3 was measured in magnetic

fields up to 14 T to compare to the predictions of the anisotropic Kondo model [15,

36, 37] and to search for clues regarding the magnetic character of the ground

state in these alloys. The measured field dependence will allow to determine a

connection between the maxima in C/T and those of the AKM. The specific heat

data of Y-doped samples will be compared to data as a function of pressure for

CeA13 to distinguish between the effects of chemical and hydrostatic pressure on

the anomaly in C/T.

Additional Ceo.8(Lal_,Y.2)0.2Al3 samples with x = 0.09,0.4 were also pre-

pared for specific heat and magnetic susceptibility studies. In this system, yttrium

doping of Ceo.sLao.2Ala was conducted to create a similar hybridization environ-

ment to that of CeA13 by reducing the lattice volume to that of the undoped

compound. Thermodynamic measurements will allow to test the magnetic inter-

pretation of the anomaly in C/T by assuming a constant coupling J, yet reducing

TRKKY by increasing the Ce-Ce distance with respect to CeAl3.

5.5.2 Experiments on CePb3

In CePb3, the increase in the A coefficient of the electrical resistivity along

(110) points to a possible enhancement of the heavy-fermion state in magnetic

fields based on the proportionality between A and 7. A study of the specific heat

of a CePb3 polycrystal in magnetic fields will be presented in order to describe

the changes of the Fermi-liquid parameters 7 and A/72 as a function of mag-

netic field. The phase diagram obtained from these measurements will be com-

pared to previous magnetoresistance results along (110) to search for evidence of

the field-induced transition detected by previous sound velocity and magnetoresis-

tance measurements, and for possible non-Fermi-liquid effects. The data should

be helpful in understanding the effects of a magnetic transition on the nature of

the heavy-fermion state.

Results from measurements of the heat capacity of Ce0.6Lao.4Pba in magnetic

fields up to 14 T will also be discussed in order to investigate further the single-

impurity nature of the paramagnetic heavy-fermion state of CePb3. The electronic

contribution to the specific heat below 10 K will be compared to predictions for the

S = single-impurity Kondo model in magnetic fields. The above measurements

on CePb3 and Ceo.6Lao.4Pb3 allow for an analysis of the electronic coefficient 7 and

the Kondo state in both nonmagnetic and magnetic heavy-fermion systems.

	Title Page
	Acknowledgement
	Table of Contents
	Abstract
	Introduction
	Theoretical background
	Properties of CeAl3 and CePb3
	Motivation
	Experimental methods
	Structural and thermodynamic properties...
	Magnetic field study of CePb3...
	Conclusion
	References
	Biographical sketch

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